LMFDB - Newform orbit 1502.2.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1502,2,Mod(1,1502)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1502, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1502.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1502 = 2 \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1502.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.9935303836$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 $ x^19 - 6x^18 - 23x^17 + 190x^16 + 128x^15 - 2394x^14 + 749x^13 + 15539x^12 - 11338x^11 - 56744x^10 + 50183x^9 + 120237x^8 - 102992x^7 - 148539x^6 + 99519x^5 + 104388x^4 - 41081x^3 - 35733x^2 + 5542x + 4222
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{18}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{13} + 1) q^{7} - q^{8} + (\beta_{18} - \beta_{17} + \beta_{14} + \cdots + 1) q^{9}+O(q^{10}) $ q - q^2 + b1 * q^3 + q^4 - b3 * q^5 - b1 * q^6 + (-b13 + 1) * q^7 - q^8 + (b18 - b17 + b14 + b7 + b1 + 1) * q^9	$ q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{13} + 1) q^{7} - q^{8} + (\beta_{18} - \beta_{17} + \beta_{14} + \cdots + 1) q^{9}+ \cdots + (\beta_{17} + 2 \beta_{16} - \beta_{15} + \cdots - 1) q^{99}+O(q^{100}) $ q - q^2 + b1 * q^3 + q^4 - b3 * q^5 - b1 * q^6 + (-b13 + 1) * q^7 - q^8 + (b18 - b17 + b14 + b7 + b1 + 1) * q^9 + b3 * q^10 + b11 * q^11 + b1 * q^12 + (b14 - b9 + b7 + 1) * q^13 + (b13 - 1) * q^14 + (b14 + b12 + b6 - b5 - b4) * q^15 + q^16 + (-b16 - b12 - b10 - b8 + 1) * q^17 + (-b18 + b17 - b14 - b7 - b1 - 1) * q^18 + (-b15 + b10 - b6 + b4 - b3) * q^19 - b3 * q^20 + (b15 - b14 - b13 + b5 + b2 + 2b1 + 1) q^21 - b11 * q^22 + (b16 - b14 + b10 + b9 + b8 - b2) * q^23 - b1 * q^24 + (b16 + b15 - b11 - b10 - b7 - b2 + 1) * q^25 + (-b14 + b9 - b7 - 1) * q^26 + (b18 - b17 + b16 - b15 + b8 + b2 + b1 + 1) * q^27 + (-b13 + 1) * q^28 + (b17 - b16 - b14 + b8 - b7 - b6 + b5) * q^29 + (-b14 - b12 - b6 + b5 + b4) * q^30 + (-b18 - b14 + b9 + b6 - b2 + 1) * q^31 - q^32 + (-b16 + b14 + b9 - b8 + b4 - b3 + b2 + 1) * q^33 + (b16 + b12 + b10 + b8 - 1) * q^34 + (-b18 - b16 - b14 - b8 - b7 - 2b3 + b2) q^35 + (b18 - b17 + b14 + b7 + b1 + 1) * q^36 + (-b18 + b15 - b14 + b13 - b2 + 2) * q^37 + (b15 - b10 + b6 - b4 + b3) * q^38 + (2b17 - b15 - b11 - b10 - b9 - b8 - b6 + 2b1 - 2) * q^39 + b3 * q^40 + (b16 - b12 - b11 - b10 - b9 - b5 - b4 + b3 + 1) * q^41 + (-b15 + b14 + b13 - b5 - b2 - 2b1 - 1) q^42 + (-b15 - b14 + b13 - b9 - b7 - b6 + 1) * q^43 + b11 * q^44 + (b16 + b15 + b14 + b13 + b12 + 3b10 + b8 + 2b6 - b3 + 1) * q^45 + (-b16 + b14 - b10 - b9 - b8 + b2) * q^46 + (-b18 + 2b17 - b16 + b15 - b14 + b12 + b11 + 2b10 + b9 - b7 + b6) * q^47 + b1 * q^48 + (-b15 + b14 - b13 - b12 + b7 - b6 - b5 - b4 + b3 + 3) * q^49 + (-b16 - b15 + b11 + b10 + b7 + b2 - 1) * q^50 + (-b18 + b16 + b14 - b12 - b10 - b8 + b6 - b5 + b3) * q^51 + (b14 - b9 + b7 + 1) * q^52 + (-b15 + b13 - b10 - b6 + b4 - b3 - 1) * q^53 + (-b18 + b17 - b16 + b15 - b8 - b2 - b1 - 1) * q^54 + (2b17 - b16 + b15 + b12 + b8 - b7 + 2b6 + b5 - b4 + 2b3 + 1) q^55 + (b13 - 1) * q^56 + (b18 - b15 + b13 - 2b10 - b9 + b7 - 2b6 + b5 + b3 + 2b1) q^57 + (-b17 + b16 + b14 - b8 + b7 + b6 - b5) * q^58 + (-2b17 + b16 + b14 + b13 + b10 + b8 + b7 - b5 + b4 - b2 - 1) q^59 + (b14 + b12 + b6 - b5 - b4) * q^60 + (-b18 - b16 - b14 - b10 - b8 - b7 + b4 - b3 + b2 + 1) * q^61 + (b18 + b14 - b9 - b6 + b2 - 1) * q^62 + (2b18 - 2b17 - b16 + b15 - b14 - b13 + b11 + b7 - b6 - b4 + 2b3 + 2b2 + 2b1 + 5) q^63 + q^64 + (b14 + b13 - 2b8 + b6 + b5 - b3 - b2) q^65 + (b16 - b14 - b9 + b8 - b4 + b3 - b2 - 1) * q^66 + (-b17 + b11 + 2b10 + 2b9 + b8 - b6 + b5 + b4 + 3) * q^67 + (-b16 - b12 - b10 - b8 + 1) * q^68 + (-b17 - 2b16 - b14 + b13 + b12 + b10 + b9 + b8 + 2b6 - b4 + b3 - b2 + 1) * q^69 + (b18 + b16 + b14 + b8 + b7 + 2b3 - b2) q^70 + (b16 + b15 - b11 - 2b10 - b9 + b3 - b2 - 1) q^71 + (-b18 + b17 - b14 - b7 - b1 - 1) * q^72 + (2b18 - b15 - b13 - b12 - b10 - b6 + b5 + b3 + b2 + 4) q^73 + (b18 - b15 + b14 - b13 + b2 - 2) * q^74 + (-2b18 + b16 + b15 + b14 - b11 - 4b10 - 2b7 - b4 + b3 - 2b2 + b1 - 2) * q^75 + (-b15 + b10 - b6 + b4 - b3) * q^76 + (-2b17 + 3b16 + b14 + b13 + b12 + b11 - b9 + 3b7 - b5 - 1) q^77 + (-2b17 + b15 + b11 + b10 + b9 + b8 + b6 - 2b1 + 2) * q^78 + (-b16 + b15 - b13 - b12 - b8 + b7 - b5 + b4 + b3 + b2 + 3) * q^79 - b3 * q^80 + (b18 - b17 - 2b16 - 2b14 + b12 + b11 + 4b10 + 2b9 + b8 + b6 + b4 - b3 + 2b2 + b1 + 3) q^81 + (-b16 + b12 + b11 + b10 + b9 + b5 + b4 - b3 - 1) * q^82 + (b18 - b16 - b15 + b14 - b12 - 2b10 - 2b6 + 2b5 + 2b3 - b2) * q^83 + (b15 - b14 - b13 + b5 + b2 + 2b1 + 1) q^84 + (b18 + 2b17 + b16 - b15 - b13 + 2b10 - b9 + b8 + b6 + 2b2 - 2b1 + 2) * q^85 + (b15 + b14 - b13 + b9 + b7 + b6 - 1) * q^86 + (b18 + b17 + 2b16 + b14 - b13 + b12 + 2b10 + 2b8 - 2b6 + b3 - b1) * q^87 - b11 * q^88 + (b17 + 2b16 - b15 + b13 - b11 + b8 - b7 + b4 - 2b2 - b1 - 2) * q^89 + (-b16 - b15 - b14 - b13 - b12 - 3b10 - b8 - 2b6 + b3 - 1) * q^90 + (2b17 + b15 - 3b13 - b12 + b11 - b10 - b9 - b8 - b6 - b4 + 2b3 + b2 + 4) q^91 + (b16 - b14 + b10 + b9 + b8 - b2) * q^92 + (-b18 - 2b17 + b15 - b14 + b13 + 2b10 + b9 + b7 + 3b6 - b5 - 2b2 + 2) * q^93 + (b18 - 2b17 + b16 - b15 + b14 - b12 - b11 - 2b10 - b9 + b7 - b6) * q^94 + (-b18 - b16 - b15 + b14 - b11 + 2b10 + b9 + b4 - 2b3 - b2 - 2b1 - 1) q^95 - b1 * q^96 + (-b16 - b15 + b13 - b12 - b8 - b5 - 2b1 + 5) q^97 + (b15 - b14 + b13 + b12 - b7 + b6 + b5 + b4 - b3 - 3) * q^98 + (b17 + 2b16 - b15 + 2b14 - b13 - b12 - b11 - 2b10 - b9 + b8 - 2b7 + b3 + b2 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) $ 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 $ x^19 - 6*x^18 - 23*x^17 + 190*x^16 + 128*x^15 - 2394*x^14 + 749*x^13 + 15539*x^12 - 11338*x^11 - 56744*x^10 + 50183*x^9 + 120237*x^8 - 102992*x^7 - 148539*x^6 + 99519*x^5 + 104388*x^4 - 41081*x^3 - 35733*x^2 + 5542*x + 4222 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 37\!\cdots\!86 \nu^{18} + \cdots - 31\!\cdots\!88 ) / 17\!\cdots\!02 $ (-3760520192170430386v^18 + 30305082839094078765v^17 + 41200800710110050060v^16 - 886378677694456159036v^15 + 907949322179800655710v^14 + 9829864376744292242242v^13 - 19458254457701180205140v^12 - 51174904871571611337619v^11 + 141922316336666980879649v^10 + 121951098427378259989987v^9 - 501893339143834545238647v^8 - 80600003893061145349438v^7 + 896205871816710953186330v^6 - 122113529794623773103574v^5 - 760681740768789785859421v^4 + 150786020791875092571053v^3 + 280020833321376854132087v^2 - 39882937575483697112922v - 31872343548678071734688) / 1764112429837355700902
$\beta_{3}$	$=$	$ ( - 29\!\cdots\!73 \nu^{18} + \cdots - 12\!\cdots\!10 ) / 35\!\cdots\!04 $ (-29076110460680765573v^18 + 206990547035840834085v^17 + 449984253396810000572v^16 - 6101941593449854293310v^15 + 2797270294997483076910v^14 + 68736174355783190880152v^13 - 96465481661643000176109v^12 - 370669531209545576582376v^11 + 744314723041094580645146v^10 + 974192543055183898521206v^9 - 2613862736587341024233861v^8 - 1051452713367576331560234v^7 + 4457592187429563078996866v^6 + 66062436696734714273185v^5 - 3452999863129750339575306v^4 + 327922948269301043803302v^3 + 1137688178086206599168283v^2 - 85125640017045839740932v - 122836999681487683614610) / 3528224859674711401804
$\beta_{4}$	$=$	$ ( - 30\!\cdots\!89 \nu^{18} + \cdots - 15\!\cdots\!54 ) / 35\!\cdots\!04 $ (-30945253987065060989v^18 + 219306674910305950519v^17 + 521676309588807770152v^16 - 6637331055069694956182v^15 + 1812199387042442266854v^14 + 78055232816059923303620v^13 - 91332498446351700215337v^12 - 454285839667464592518246v^11 + 748467525203016066124524v^10 + 1388661333356872083903068v^9 - 2762612032560116272703091v^8 - 2178547418940168131940534v^7 + 5028250121350414105405834v^6 + 1615298067719303956571269v^5 - 4277654130079105760072216v^4 - 538411709022266322520764v^3 + 1492995420294699602344401v^2 + 41402129559144245656416v - 151051038335736692981254) / 3528224859674711401804
$\beta_{5}$	$=$	$ ( 35\!\cdots\!67 \nu^{18} + \cdots + 26\!\cdots\!14 ) / 35\!\cdots\!04 $ (35514340071205248367v^18 - 242632159693174792561v^17 - 643305185654029693144v^16 + 7400901982469154931142v^15 - 817581316649995040554v^14 - 87979131560055003267392v^13 + 91618867761805274083903v^12 + 519709571191677577219486v^11 - 798377461348665252899432v^10 - 1619993422592754555171372v^9 + 3075165837805834008163477v^8 + 2598218311229720705507906v^7 - 5902151591410770828067210v^6 - 1931036361335959872817407v^5 + 5457929300039474936559668v^4 + 551106253558324847470664v^3 - 2161482440858026625606455v^2 - 15471705894405999436640v + 261764691393999547176914) / 3528224859674711401804
$\beta_{6}$	$=$	$ ( - 39\!\cdots\!95 \nu^{18} + \cdots - 22\!\cdots\!10 ) / 35\!\cdots\!04 $ (-39225674615895018295v^18 + 281334979929612838243v^17 + 630357296227855912720v^16 - 8430966253267278684034v^15 + 3188656397738161913338v^14 + 97614079406512640434288v^13 - 125532268617349703729955v^12 - 553483373863916062832792v^11 + 999509744710647745384134v^10 + 1614712864453446395048338v^9 - 3630156788064959123947035v^8 - 2309127089727213676127182v^7 + 6513098057013774939701338v^6 + 1393894842156114864238095v^5 - 5474664469504159485153834v^4 - 329634434562021923542522v^3 + 1952055797874724118604737v^2 + 7043498701436966126708v - 222304348635242813768410) / 3528224859674711401804
$\beta_{7}$	$=$	$ ( - 11\!\cdots\!11 \nu^{18} + \cdots - 61\!\cdots\!57 ) / 88\!\cdots\!51 $ (-11167946556581994711v^18 + 72037020882250373706v^17 + 218163849798413458391v^16 - 2198104257192141994412v^15 - 249022931837379544406v^14 + 26155264541052107927708v^13 - 22385598655266651602489v^12 - 154856996848546561721898v^11 + 209171645728351253723877v^10 + 485395555103885711253293v^9 - 815202509303176415989788v^8 - 790394723234322452109492v^7 + 1547205525833040825470336v^6 + 618091829102644345876033v^5 - 1382690686240782796797340v^4 - 216675894834980889747933v^3 + 521040532835462169581003v^2 + 23377396899492401072662v - 61007845242638016139457) / 882056214918677850451
$\beta_{8}$	$=$	$ ( 22\!\cdots\!63 \nu^{18} + \cdots + 11\!\cdots\!02 ) / 17\!\cdots\!02 $ (22935642479960504263v^18 - 163807917546078032844v^17 - 347968100546541242034v^16 + 4811256116260884393776v^15 - 2415755773575146538442v^14 - 53879781520772923569932v^13 + 78553475030489475053325v^12 + 287609418859943312771113v^11 - 601867975227054779042041v^10 - 740938814962541543108443v^9 + 2111650992028433615016010v^8 + 758295797594366651969202v^7 - 3615592773829894301652558v^6 + 12016842140245612195017v^5 + 2840490829818975545731353v^4 - 242616546224541959763249v^3 - 970625595458513263192090v^2 + 49046681426265715429604v + 113592698179520862754802) / 1764112429837355700902
$\beta_{9}$	$=$	$ ( - 53\!\cdots\!13 \nu^{18} + \cdots - 33\!\cdots\!30 ) / 35\!\cdots\!04 $ (-53078987331776809413v^18 + 367515426498576955637v^17 + 920244526033840804776v^16 - 11088118004222127647018v^15 + 2323875647558612679514v^14 + 129631647355211696363740v^13 - 147215378581633743827297v^12 - 745654129012549222047316v^11 + 1227483135034000590929390v^10 + 2223128080408016070228306v^9 - 4564940454465900858809637v^8 - 3293148157720498569728226v^7 + 8377236903027542588927042v^6 + 2119801390785974910893953v^5 - 7282681049875190648528770v^4 - 555446990794750228761258v^3 + 2758013642041956597019251v^2 + 43217652384396165704284v - 338893975678974074603630) / 3528224859674711401804
$\beta_{10}$	$=$	$ ( 65\!\cdots\!89 \nu^{18} + \cdots + 38\!\cdots\!54 ) / 35\!\cdots\!04 $ (65269994698618887489v^18 - 440295935608668625411v^17 - 1191848115016691865296v^16 + 13359502166014633995958v^15 - 1037892829669082123282v^14 - 157625826309534025609356v^13 + 159778105580821061796529v^12 + 921037838225169820617946v^11 - 1388161951758038181773760v^10 - 2825851115465353418466920v^9 + 5260830789395044175743819v^8 + 4431949649138633371027590v^7 - 9808515862489234476993834v^6 - 3227048649923522292410445v^5 + 8655297992211802941703232v^4 + 1006944353978834637406320v^3 - 3252303057734070582886901v^2 - 76711896678285078567952v + 382673437272331542837754) / 3528224859674711401804
$\beta_{11}$	$=$	$ ( - 67\!\cdots\!05 \nu^{18} + \cdots - 40\!\cdots\!90 ) / 35\!\cdots\!04 $ (-67545888761056757405v^18 + 474152691703187535957v^17 + 1126972007891038411660v^16 - 14225479148011647792746v^15 + 4245207934669709738222v^14 + 164933151038642750448804v^13 - 201754926509586295161889v^12 - 936527712809011504410176v^11 + 1640288763668577806266250v^10 + 2733320468157187722611582v^9 - 6025875980509308282125301v^8 - 3890467207784257373413434v^7 + 10950927147978590859975302v^6 + 2281174130293399234730025v^5 - 9407015588513958382958478v^4 - 481304642842418863805234v^3 + 3472206780176237103939911v^2 + 23882619695694000437704v - 407486381451920829795090) / 3528224859674711401804
$\beta_{12}$	$=$	$ ( - 69\!\cdots\!03 \nu^{18} + \cdots - 22\!\cdots\!42 ) / 35\!\cdots\!04 $ (-69985589363716372003v^18 + 452117294523362098281v^17 + 1325535324928609159564v^16 - 13563574351565447656442v^15 - 639254816371122214178v^14 + 157208079346273319443112v^13 - 145053125102334518860899v^12 - 891743569213297416919646v^11 + 1282552727139659807366264v^10 + 2595492162936333398605104v^9 - 4739692707772156654829369v^8 - 3671011968791771277480298v^7 + 8342749374284723742911106v^6 + 2133527757290567026500795v^5 - 6635766416826814603288452v^4 - 482217178587693989306880v^3 + 2185026163458131991656443v^2 + 28482245194557351640248v - 221905016240633020395442) / 3528224859674711401804
$\beta_{13}$	$=$	$ ( - 77\!\cdots\!15 \nu^{18} + \cdots - 44\!\cdots\!50 ) / 35\!\cdots\!04 $ (-77065122442043030315v^18 + 532774562852707940065v^17 + 1309795665184004200420v^16 - 15947559575155028954070v^15 + 4066746067386985452170v^14 + 184141182031040943735476v^13 - 220131935427482750177679v^12 - 1037180746879497347929890v^11 + 1804220288341667306366968v^10 + 2973324222468615128719668v^9 - 6624784111405967801651453v^8 - 4032909788018291328760214v^7 + 11979292489662292335093490v^6 + 1963412732253456474997411v^5 - 10207686105122915636409520v^4 - 72364707768952431818544v^3 + 3775019299987169463270923v^2 - 93743526820595676419864v - 448047703088435065929750) / 3528224859674711401804
$\beta_{14}$	$=$	$ ( 81\!\cdots\!29 \nu^{18} + \cdots + 43\!\cdots\!06 ) / 35\!\cdots\!04 $ (81246465791995337029v^18 - 538011472036996170959v^17 - 1500040891301178160836v^16 + 16239099098267565245182v^15 - 682749423888030571250v^14 - 190058582570187933079240v^13 + 189727613796164956802949v^12 + 1096000891963530403875206v^11 - 1656270139070271276372200v^10 - 3286800831286664099437744v^9 + 6237831720989536635596223v^8 + 4936824531945407856072402v^7 - 11476874966336529882099158v^6 - 3283786466870076576469109v^5 + 9927586089252498464492112v^4 + 881333673434794369135432v^3 - 3681466686787631497562293v^2 - 47898124722861190738136v + 432163679569144496110306) / 3528224859674711401804
$\beta_{15}$	$=$	$ ( - 47\!\cdots\!28 \nu^{18} + \cdots - 27\!\cdots\!58 ) / 17\!\cdots\!02 $ (-47097892268680919528v^18 + 319743617906841950481v^17 + 837922334641744413414v^16 - 9633843421968099827224v^15 + 1369200205499709724344v^14 + 112406080492905974955444v^13 - 121720435612565003591624v^12 - 644497773229142363627215v^11 + 1031071031664700952304797v^10 + 1910019844408435737913565v^9 - 3845750444490116295482927v^8 - 2787652479313939222193926v^7 + 7047960430634142289825628v^6 + 1696565935560911310021946v^5 - 6094487564041673976730955v^4 - 326408167210916415098145v^3 + 2271252467228465080858335v^2 - 14489750421562796486650v - 271400599812957179459358) / 1764112429837355700902
$\beta_{16}$	$=$	$ ( - 95\!\cdots\!25 \nu^{18} + \cdots - 50\!\cdots\!54 ) / 35\!\cdots\!04 $ (-95971349547274628625v^18 + 656629516719657132985v^17 + 1661993776848826883504v^16 - 19670759651570058856394v^15 + 4075172162552562465278v^14 + 227474470867889711197860v^13 - 261446093195608246486625v^12 - 1285291669865683973133804v^11 + 2162449824953311239457686v^10 + 3712796511016076787639470v^9 - 7933994510972599758883509v^8 - 5155450907022371409992946v^7 + 14246641802290284696365898v^6 + 2801906095971079748896029v^5 - 11951769961894352195903022v^4 - 450996964601953574266394v^3 + 4322881678425745449196591v^2 - 22864874827629484464976v - 507524935709983375201054) / 3528224859674711401804
$\beta_{17}$	$=$	$ ( - 11\!\cdots\!69 \nu^{18} + \cdots - 63\!\cdots\!18 ) / 35\!\cdots\!04 $ (-111180934604728577169v^18 + 754491441872043631203v^17 + 1983118773266591338776v^16 - 22754311006284932937834v^15 + 3094221402718334038530v^14 + 265927825342822788548944v^13 - 285979842814695639634801v^12 - 1529488751053174975282250v^11 + 2428243393839957414429780v^10 + 4563172264386688163453912v^9 - 9069817744293933606095027v^8 - 6775592737231410424291170v^7 + 16641663522586577497468190v^6 + 4371124193456046186265857v^5 - 14402873239663841582626060v^4 - 1084873524382554347794788v^3 + 5368607597817917780318525v^2 + 46591581792452054756420v - 636544201841169330766818) / 3528224859674711401804
$\beta_{18}$	$=$	$ ( - 73\!\cdots\!77 \nu^{18} + \cdots - 41\!\cdots\!56 ) / 17\!\cdots\!02 $ (-73877807085197967677v^18 + 502177415190019153669v^17 + 1305252132687057833024v^16 - 15100496537891965102684v^15 + 2386531276977941393702v^14 + 175682674874401144958676v^13 - 193082530994896995013897v^12 - 1003030827811259566134932v^11 + 1623913474998411837953236v^10 + 2954195437628904708939242v^9 - 6023419714035382288866049v^8 - 4275419188119764235962802v^7 + 10964858192795472038843002v^6 + 2591271671957772689615417v^5 - 9399848291976604429964406v^4 - 549751809238712578969244v^3 + 3484720189061687655479305v^2 - 1274052971165535098948v - 419394699939230303963256) / 1764112429837355700902

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{18} - \beta_{17} + \beta_{14} + \beta_{7} + \beta _1 + 4 $ b18 - b17 + b14 + b7 + b1 + 4
$\nu^{3}$	$=$	$ \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{8} + \beta_{2} + 7\beta _1 + 1 $ b18 - b17 + b16 - b15 + b8 + b2 + 7*b1 + 1
$\nu^{4}$	$=$	$ 10 \beta_{18} - 10 \beta_{17} - 2 \beta_{16} + 7 \beta_{14} + \beta_{12} + \beta_{11} + 4 \beta_{10} + \cdots + 30 $ 10b18 - 10b17 - 2b16 + 7b14 + b12 + b11 + 4b10 + 2b9 + b8 + 9b7 + b6 + b4 - b3 + 2b2 + 10*b1 + 30
$\nu^{5}$	$=$	$ 13 \beta_{18} - 15 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} + \cdots + 18 $ 13b18 - 15b17 + 10b16 - 10b15 - 2b14 - 2b13 + b12 + 4b11 + 6b10 + 3b9 + 15b8 + 2b7 + 3b6 - 2b4 + 2b3 + 13b2 + 58b1 + 18
$\nu^{6}$	$=$	$ 96 \beta_{18} - 101 \beta_{17} - 30 \beta_{16} + 4 \beta_{15} + 46 \beta_{14} - 5 \beta_{13} + \cdots + 270 $ 96b18 - 101b17 - 30b16 + 4b15 + 46b14 - 5b13 + 15b12 + 16b11 + 62b10 + 34b9 + 18b8 + 80b7 + 22b6 + 9b4 - 12b3 + 32b2 + 99*b1 + 270
$\nu^{7}$	$=$	$ 145 \beta_{18} - 193 \beta_{17} + 85 \beta_{16} - 79 \beta_{15} - 46 \beta_{14} - 41 \beta_{13} + \cdots + 257 $ 145b18 - 193b17 + 85b16 - 79b15 - 46b14 - 41b13 + 21b12 + 66b11 + 115b10 + 62b9 + 177b8 + 33b7 + 69b6 - 5b5 - 38b4 + 29b3 + 149b2 + 525b1 + 257
$\nu^{8}$	$=$	$ 930 \beta_{18} - 1072 \beta_{17} - 351 \beta_{16} + 94 \beta_{15} + 278 \beta_{14} - 106 \beta_{13} + \cdots + 2634 $ 930b18 - 1072b17 - 351b16 + 94b15 + 278b14 - 106b13 + 186b12 + 211b11 + 768b10 + 451b9 + 254b8 + 736b7 + 349b6 - 11b5 + 42b4 - 115b3 + 401b2 + 1012b1 + 2634
$\nu^{9}$	$=$	$ 1573 \beta_{18} - 2397 \beta_{17} + 666 \beta_{16} - 530 \beta_{15} - 738 \beta_{14} - 590 \beta_{13} + \cdots + 3380 $ 1573b18 - 2397b17 + 666b16 - 530b15 - 738b14 - 590b13 + 327b12 + 852b11 + 1664b10 + 964b9 + 1961b8 + 444b7 + 1115b6 - 119b5 - 538b4 + 310b3 + 1648b2 + 5036b1 + 3380
$\nu^{10}$	$=$	$ 9175 \beta_{18} - 11769 \beta_{17} - 3807 \beta_{16} + 1508 \beta_{15} + 1251 \beta_{14} - 1589 \beta_{13} + \cdots + 26870 $ 9175b18 - 11769b17 - 3807b16 + 1508b15 + 1251b14 - 1589b13 + 2189b12 + 2612b11 + 8964b10 + 5554b9 + 3307b8 + 6985b7 + 4848b6 - 307b5 - 183b4 - 1029b3 + 4663b2 + 10683b1 + 26870
$\nu^{11}$	$=$	$ 17145 \beta_{18} - 29325 \beta_{17} + 4745 \beta_{16} - 2619 \beta_{15} - 10236 \beta_{14} + \cdots + 42768 $ 17145b18 - 29325b17 + 4745b16 - 2619b15 - 10236b14 - 7453b13 + 4519b12 + 10217b11 + 21795b10 + 13362b9 + 21420b8 + 5710b7 + 15698b6 - 1975b5 - 6883b4 + 2992b3 + 18039b2 + 50367b1 + 42768
$\nu^{12}$	$=$	$ 92445 \beta_{18} - 131860 \beta_{17} - 40241 \beta_{16} + 20806 \beta_{15} - 1416 \beta_{14} + \cdots + 282144 $ 92445b18 - 131860b17 - 40241b16 + 20806b15 - 1416b14 - 20898b13 + 25311b12 + 31401b11 + 102868b10 + 66431b9 + 41484b8 + 68177b7 + 63006b6 - 5642b5 - 8331b4 - 8937b3 + 52854b2 + 115912b1 + 282144
$\nu^{13}$	$=$	$ 189444 \beta_{18} - 355071 \beta_{17} + 28095 \beta_{16} + 467 \beta_{15} - 131894 \beta_{14} + \cdots + 529320 $ 189444b18 - 355071b17 + 28095b16 + 467b15 - 131894b14 - 88916b13 + 58752b12 + 119285b11 + 272586b10 + 174263b9 + 234511b8 + 72291b7 + 206215b6 - 28423b5 - 84230b4 + 27678b3 + 197598b2 + 520148b1 + 529320
$\nu^{14}$	$=$	$ 951790 \beta_{18} - 1495641 \beta_{17} - 422941 \beta_{16} + 266482 \beta_{15} - 148463 \beta_{14} + \cdots + 3025986 $ 951790b18 - 1495641b17 - 422941b16 + 266482b15 - 148463b14 - 258558b13 + 291063b12 + 371861b11 + 1176880b10 + 784039b9 + 509590b8 + 683060b7 + 788675b6 - 86676b5 - 151568b4 - 76389b3 + 595243b2 + 1285093b1 + 3025986
$\nu^{15}$	$=$	$ 2123444 \beta_{18} - 4263971 \beta_{17} + 84246 \beta_{16} + 270917 \beta_{15} - 1630800 \beta_{14} + \cdots + 6456756 $ 2123444b18 - 4263971b17 + 84246b16 + 270917b15 - 1630800b14 - 1033607b13 + 737547b12 + 1379282b11 + 3325956b10 + 2191644b9 + 2587562b8 + 905228b7 + 2603486b6 - 380335b5 - 1008395b4 + 251131b3 + 2177452b2 + 5510799b1 + 6456756
$\nu^{16}$	$=$	$ 10004896 \beta_{18} - 17096882 \beta_{17} - 4461039 \beta_{16} + 3277301 \beta_{15} - 2835698 \beta_{14} + \cdots + 32999887 $ 10004896b18 - 17096882b17 - 4461039b16 + 3277301b15 - 2835698b14 - 3101641b13 + 3344826b12 + 4369096b11 + 13486035b10 + 9191526b9 + 6176669b8 + 7011607b7 + 9648683b6 - 1209261b5 - 2237308b4 - 646233b3 + 6713785b2 + 14477716b1 + 32999887
$\nu^{17}$	$=$	$ 24090869 \beta_{18} - 50866953 \beta_{17} - 1189598 \beta_{16} + 5419328 \beta_{15} - 19678325 \beta_{14} + \cdots + 77950265 $ 24090869b18 - 50866953b17 - 1189598b16 + 5419328b15 - 19678325b14 - 11883466b13 + 9060210b12 + 15905311b11 + 39994481b10 + 26934897b9 + 28809860b8 + 11215969b7 + 32068711b6 - 4876496b5 - 11932955b4 + 2258019b3 + 24189418b2 + 59605928b1 + 77950265
$\nu^{18}$	$=$	$ 107185878 \beta_{18} - 196451573 \beta_{17} - 47431680 \beta_{16} + 39376536 \beta_{15} + \cdots + 364797939 $ 107185878b18 - 196451573b17 - 47431680b16 + 39376536b15 - 42400156b14 - 36615716b13 + 38484796b12 + 51124315b11 + 154989621b10 + 107374656b9 + 74180123b8 + 73579411b7 + 116313191b6 - 15923475b5 - 30123224b4 - 5421373b3 + 76069064b2 + 164981093b1 + 364797939

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−3.01539
−2.98144
−2.13636
−1.94953
−1.64365
−0.862574
−0.803709
−0.725860
−0.443463
0.493679
0.748702
1.28683
1.74260
2.07470
2.33529
2.66959
2.88063
2.91512
3.41483

−1.00000

−3.01539

1.00000

−2.64906

3.01539

2.56889

−1.00000

6.09259

2.64906

1.2

−1.00000

−2.98144

1.00000

3.72248

2.98144

−0.567696

−1.00000

5.88899

−3.72248

1.3

−1.00000

−2.13636

1.00000

−2.42355

2.13636

4.87888

−1.00000

1.56402

2.42355

1.4

−1.00000

−1.94953

1.00000

2.00911

1.94953

0.0633867

−1.00000

0.800679

−2.00911

1.5

−1.00000

−1.64365

1.00000

−0.210142

1.64365

−4.29873

−1.00000

−0.298402

0.210142

1.6

−1.00000

−0.862574

1.00000

−3.34357

0.862574

−0.0695492

−1.00000

−2.25597

3.34357

1.7

−1.00000

−0.803709

1.00000

2.36229

0.803709

4.01698

−1.00000

−2.35405

−2.36229

1.8

−1.00000

−0.725860

1.00000

3.71965

0.725860

−1.04918

−1.00000

−2.47313

−3.71965

1.9

−1.00000

−0.443463

1.00000

−2.41711

0.443463

−1.05600

−1.00000

−2.80334

2.41711

1.10

−1.00000

0.493679

1.00000

0.378923

−0.493679

2.20566

−1.00000

−2.75628

−0.378923

1.11

−1.00000

0.748702

1.00000

−1.18128

−0.748702

−4.67174

−1.00000

−2.43945

1.18128

1.12

−1.00000

1.28683

1.00000

−3.54885

−1.28683

3.93701

−1.00000

−1.34406

3.54885

1.13

−1.00000

1.74260

1.00000

1.47569

−1.74260

4.24779

−1.00000

0.0366584

−1.47569

1.14

−1.00000

2.07470

1.00000

4.37927

−2.07470

2.25344

−1.00000

1.30439

−4.37927

1.15

−1.00000

2.33529

1.00000

−4.05193

−2.33529

0.675791

−1.00000

2.45359

4.05193

1.16

−1.00000

2.66959

1.00000

1.32740

−2.66959

−3.70251

−1.00000

4.12668

−1.32740

1.17

−1.00000

2.88063

1.00000

2.69966

−2.88063

−2.78599

−1.00000

5.29806

−2.69966

1.18

−1.00000

2.91512

1.00000

−1.49286

−2.91512

3.25503

−1.00000

5.49794

1.49286

1.19

−1.00000

3.41483

1.00000

1.24390

−3.41483

3.09855

−1.00000

8.66107

−1.24390

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$751$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1502.2.a.h	✓	19

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1502.2.a.h	✓	19	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{19} - 6 T_{3}^{18} - 23 T_{3}^{17} + 190 T_{3}^{16} + 128 T_{3}^{15} - 2394 T_{3}^{14} + \cdots + 4222 $ T3^19 - 6*T3^18 - 23*T3^17 + 190*T3^16 + 128*T3^15 - 2394*T3^14 + 749*T3^13 + 15539*T3^12 - 11338*T3^11 - 56744*T3^10 + 50183*T3^9 + 120237*T3^8 - 102992*T3^7 - 148539*T3^6 + 99519*T3^5 + 104388*T3^4 - 41081*T3^3 - 35733*T3^2 + 5542*T3 + 4222 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1502))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 1)^{19} $ (T + 1)^19
$3$	$ T^{19} - 6 T^{18} + \cdots + 4222 $ T^19 - 6T^18 - 23T^17 + 190T^16 + 128T^15 - 2394T^14 + 749T^13 + 15539T^12 - 11338T^11 - 56744T^10 + 50183T^9 + 120237T^8 - 102992T^7 - 148539T^6 + 99519T^5 + 104388T^4 - 41081T^3 - 35733T^2 + 5542T + 4222
$5$	$ T^{19} - 2 T^{18} + \cdots + 198336 $ T^19 - 2T^18 - 64T^17 + 119T^16 + 1685T^15 - 2936T^14 - 23650T^13 + 39164T^12 + 191703T^11 - 307853T^10 - 907658T^9 + 1448759T^8 + 2403291T^7 - 3932551T^6 - 3128876T^5 + 5542868T^4 + 1342688T^3 - 3176464T^2 + 272768T + 198336
$7$	$ T^{19} - 13 T^{18} + \cdots - 16384 $ T^19 - 13T^18 - 3T^17 + 716T^16 - 2346T^15 - 12178T^14 + 73421T^13 + 28542T^12 - 840580T^11 + 1029185T^10 + 3654176T^9 - 8333888T^8 - 3506704T^7 + 18123744T^6 - 1476096T^5 - 15360000T^4 + 1391616T^3 + 3792896T^2 + 16384T - 16384
$11$	$ T^{19} + 7 T^{18} + \cdots - 22147038 $ T^19 + 7T^18 - 102T^17 - 775T^16 + 3881T^15 + 34139T^14 - 64090T^13 - 762553T^12 + 293745T^11 + 9126950T^10 + 3782096T^9 - 57187113T^8 - 46697525T^7 + 173756417T^6 + 161674992T^5 - 220699057T^4 - 160178222T^3 + 118255823T^2 + 46099456T - 22147038
$13$	$ T^{19} - 19 T^{18} + \cdots + 28157056 $ T^19 - 19T^18 + 23T^17 + 1691T^16 - 10005T^15 - 36781T^14 + 458421T^13 - 429256T^12 - 7246108T^11 + 21524435T^10 + 29197792T^9 - 190401899T^8 + 99863768T^7 + 525181345T^6 - 681420452T^5 - 227787612T^4 + 726595952T^3 - 277076832T^2 - 45545536T + 28157056
$17$	$ T^{19} + \cdots - 2458536000 $ T^19 - 11T^18 - 94T^17 + 1176T^16 + 3987T^15 - 53333T^14 - 104994T^13 + 1335177T^12 + 1947696T^11 - 20064357T^10 - 25996433T^9 + 183631742T^8 + 238268165T^7 - 982595445T^6 - 1381766668T^5 + 2686785892T^4 + 4470472912T^3 - 2196435120T^2 - 6080804800T - 2458536000
$19$	$ T^{19} - 7 T^{18} + \cdots + 2000512 $ T^19 - 7T^18 - 150T^17 + 931T^16 + 9018T^15 - 46780T^14 - 282602T^13 + 1141658T^12 + 4935904T^11 - 14557383T^10 - 46921968T^9 + 99571346T^8 + 219483173T^7 - 387108185T^6 - 435180192T^5 + 799012968T^4 + 163955648T^3 - 606093936T^2 + 201811840T + 2000512
$23$	$ T^{19} + \cdots + 2499323904 $ T^19 - 12T^18 - 174T^17 + 2289T^16 + 11597T^15 - 172030T^14 - 381054T^13 + 6604311T^12 + 6600251T^11 - 140761469T^10 - 59521244T^9 + 1695988392T^8 + 264164715T^7 - 11212811777T^6 - 656370188T^5 + 36908081016T^4 + 966991232T^3 - 45089997120T^2 + 3860619264T + 2499323904
$29$	$ T^{19} + \cdots - 6319814328 $ T^19 + 8T^18 - 271T^17 - 2030T^16 + 29067T^15 + 191480T^14 - 1682804T^13 - 8934051T^12 + 58092374T^11 + 223263107T^10 - 1209651211T^9 - 2945845537T^8 + 14571815494T^7 + 17646919521T^6 - 91271784658T^5 - 14377526710T^4 + 219553291740T^3 - 186761444868T^2 + 58392042224T - 6319814328
$31$	$ T^{19} + \cdots - 12085879168 $ T^19 - 32T^18 + 225T^17 + 2895T^16 - 43610T^15 + 3291T^14 + 2377347T^13 - 6934699T^12 - 60585294T^11 + 275543920T^10 + 791276176T^9 - 4908438432T^8 - 4790687009T^7 + 44497288429T^6 + 4174315500T^5 - 189236768568T^4 + 73637972896T^3 + 262786776432T^2 - 158976812032T - 12085879168
$37$	$ T^{19} + \cdots + 918964640768 $ T^19 - 41T^18 + 557T^17 - 697T^16 - 52666T^15 + 415947T^14 + 783344T^13 - 21663637T^12 + 47934404T^11 + 448927849T^10 - 2030479017T^9 - 3795421157T^8 + 31997245606T^7 + 2189768456T^6 - 248686317800T^5 + 144981550544T^4 + 964471825856T^3 - 709561191168T^2 - 1496215054848T + 918964640768
$41$	$ T^{19} + \cdots + 491698922496 $ T^19 - 15T^18 - 241T^17 + 4837T^16 + 8919T^15 - 510640T^14 + 1251454T^13 + 21018112T^12 - 90563152T^11 - 397394545T^10 + 2280477675T^9 + 3709493583T^8 - 27226533788T^7 - 19528514988T^6 + 164860565808T^5 + 86062380688T^4 - 498919571392T^3 - 312449767232T^2 + 596072093696T + 491698922496
$43$	$ T^{19} + \cdots + 62830554180608 $ T^19 - 15T^18 - 360T^17 + 6195T^16 + 46133T^15 - 1016212T^14 - 2272470T^13 + 86680239T^12 - 16433963T^11 - 4191729558T^10 + 6365562426T^9 + 116825081946T^8 - 269084946147T^7 - 1809700862715T^6 + 4989342100972T^5 + 14085627775160T^4 - 40930212458544T^3 - 46043293625824T^2 + 110400685730304T + 62830554180608
$47$	$ T^{19} + \cdots + 31509244224 $ T^19 + 6T^18 - 383T^17 - 2601T^16 + 56547T^15 + 424851T^14 - 4075408T^13 - 33609815T^12 + 151822995T^11 + 1371100055T^10 - 3025619103T^9 - 28732282899T^8 + 37649053709T^7 + 289783085785T^6 - 381512864324T^5 - 1109372646788T^4 + 2149226791200T^3 - 627352970832T^2 - 380101158784T + 31509244224
$53$	$ T^{19} + \cdots - 869610298176 $ T^19 - 6T^18 - 331T^17 + 2171T^16 + 41236T^15 - 303316T^14 - 2381375T^13 + 20705528T^12 + 61604233T^11 - 735598986T^10 - 417682538T^9 + 13959810749T^8 - 11097660632T^7 - 135617283565T^6 + 231618577912T^5 + 552873958980T^4 - 1436211489520T^3 - 224776932080T^2 + 2165529532096T - 869610298176
$59$	$ T^{19} + \cdots + 2924927705088 $ T^19 + 4T^18 - 571T^17 - 2566T^16 + 126015T^15 + 637937T^14 - 13684928T^13 - 80600906T^12 + 752060389T^11 + 5513687477T^10 - 17264353908T^9 - 195125226608T^8 - 67976307968T^7 + 2866534233552T^6 + 7497271871936T^5 - 2401853026048T^4 - 32421175495424T^3 - 41457775769088T^2 - 13472912867328T + 2924927705088
$61$	$ T^{19} + \cdots + 20341939328 $ T^19 - 13T^18 - 302T^17 + 3823T^16 + 33939T^15 - 425696T^14 - 1692369T^13 + 22847114T^12 + 33567381T^11 - 611835565T^10 - 54459416T^9 + 7761904334T^8 - 3787589153T^7 - 46190770571T^6 + 27427416672T^5 + 122937171588T^4 - 40765126000T^3 - 135444605600T^2 - 14719374784T + 20341939328
$67$	$ T^{19} + \cdots - 94534189842024 $ T^19 - 47T^18 + 293T^17 + 19122T^16 - 352870T^15 - 729111T^14 + 68062076T^13 - 468052157T^12 - 2924510227T^11 + 51266479187T^10 - 164726788003T^9 - 779268556575T^8 + 6188722470644T^7 - 6094617852989T^6 - 50238385182902T^5 + 137375159094586T^4 + 29272887775068T^3 - 445982882460100T^2 + 434010257584320T - 94534189842024
$71$	$ T^{19} + \cdots - 12303059328 $ T^19 - 444T^17 + 489T^16 + 74394T^15 - 140913T^14 - 5986885T^13 + 14508819T^12 + 249178149T^11 - 699406127T^10 - 5291866118T^9 + 16450338911T^8 + 50869220176T^7 - 173412821023T^6 - 147473704265T^5 + 610722540741T^4 - 28997030720T^3 - 341378288865T^2 + 126278034560*T - 12303059328
$73$	$ T^{19} + \cdots - 2739304789504 $ T^19 - 64T^18 + 1502T^17 - 11208T^16 - 145642T^15 + 3538861T^14 - 20831634T^13 - 103849256T^12 + 2016311009T^11 - 7913470382T^10 - 28433738653T^9 + 364511718339T^8 - 1064705643498T^7 - 1463397890608T^6 + 19013652779848T^5 - 58859795080304T^4 + 92740845684992T^3 - 77146618346240T^2 + 29259860813312T - 2739304789504
$79$	$ T^{19} + \cdots - 769315246221952 $ T^19 - 34T^18 - 321T^17 + 23072T^16 - 98990T^15 - 5525343T^14 + 57461830T^13 + 501159237T^12 - 9291423625T^11 + 251945698T^10 + 611715355772T^9 - 2407671711095T^8 - 12916374059579T^7 + 98162487571845T^6 - 62776351310208T^5 - 696114795502320T^4 + 892201431154704T^3 + 1665421264650576T^2 - 1578225340043904T - 769315246221952
$83$	$ T^{19} + \cdots - 266808806442714 $ T^19 - 4T^18 - 918T^17 + 3126T^16 + 335563T^15 - 885345T^14 - 62336879T^13 + 103200158T^12 + 6228227891T^11 - 2824062987T^10 - 326017609082T^9 - 270146869430T^8 + 8095408195067T^7 + 13953619906597T^6 - 83247485713101T^5 - 187777090904796T^4 + 224505936944838T^3 + 496602920528139T^2 - 253901504066740T - 266808806442714
$89$	$ T^{19} + \cdots - 199659293904768 $ T^19 - 18T^18 - 615T^17 + 12006T^16 + 141397T^15 - 3216296T^14 - 13900817T^13 + 440701235T^12 + 282779369T^11 - 32515908126T^10 + 48946392907T^9 + 1243632209876T^8 - 3617257407005T^7 - 21392816120940T^6 + 88108099181189T^5 + 93705989146379T^4 - 694539760920858T^3 + 581180905438764T^2 + 210190376822336T - 199659293904768
$97$	$ T^{19} + \cdots - 57640403406224 $ T^19 - 82T^18 + 2745T^17 - 44794T^16 + 242774T^15 + 3750857T^14 - 79651417T^13 + 538472281T^12 + 363562367T^11 - 30199032996T^10 + 193323251095T^9 - 301395475868T^8 - 2272274995840T^7 + 12966424008709T^6 - 20347781175175T^5 - 21869592574339T^4 + 82283527566818T^3 + 12961014379128T^2 - 113387483142824T - 57640403406224
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1502 = 2 \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1502.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{13} + 1) q^{7} - q^{8} + (\beta_{18} - \beta_{17} + \beta_{14} + \cdots + 1) q^{9}+O(q^{10}) \) q - q^2 + b1 * q^3 + q^4 - b3 * q^5 - b1 * q^6 + (-b13 + 1) * q^7 - q^8 + (b18 - b17 + b14 + b7 + b1 + 1) * q^9	\( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{13} + 1) q^{7} - q^{8} + (\beta_{18} - \beta_{17} + \beta_{14} + \cdots + 1) q^{9}+ \cdots + (\beta_{17} + 2 \beta_{16} - \beta_{15} + \cdots - 1) q^{99}+O(q^{100}) \) q - q^2 + b1 * q^3 + q^4 - b3 * q^5 - b1 * q^6 + (-b13 + 1) * q^7 - q^8 + (b18 - b17 + b14 + b7 + b1 + 1) * q^9 + b3 * q^10 + b11 * q^11 + b1 * q^12 + (b14 - b9 + b7 + 1) * q^13 + (b13 - 1) * q^14 + (b14 + b12 + b6 - b5 - b4) * q^15 + q^16 + (-b16 - b12 - b10 - b8 + 1) * q^17 + (-b18 + b17 - b14 - b7 - b1 - 1) * q^18 + (-b15 + b10 - b6 + b4 - b3) * q^19 - b3 * q^20 + (b15 - b14 - b13 + b5 + b2 + 2b1 + 1) q^21 - b11 * q^22 + (b16 - b14 + b10 + b9 + b8 - b2) * q^23 - b1 * q^24 + (b16 + b15 - b11 - b10 - b7 - b2 + 1) * q^25 + (-b14 + b9 - b7 - 1) * q^26 + (b18 - b17 + b16 - b15 + b8 + b2 + b1 + 1) * q^27 + (-b13 + 1) * q^28 + (b17 - b16 - b14 + b8 - b7 - b6 + b5) * q^29 + (-b14 - b12 - b6 + b5 + b4) * q^30 + (-b18 - b14 + b9 + b6 - b2 + 1) * q^31 - q^32 + (-b16 + b14 + b9 - b8 + b4 - b3 + b2 + 1) * q^33 + (b16 + b12 + b10 + b8 - 1) * q^34 + (-b18 - b16 - b14 - b8 - b7 - 2b3 + b2) q^35 + (b18 - b17 + b14 + b7 + b1 + 1) * q^36 + (-b18 + b15 - b14 + b13 - b2 + 2) * q^37 + (b15 - b10 + b6 - b4 + b3) * q^38 + (2b17 - b15 - b11 - b10 - b9 - b8 - b6 + 2b1 - 2) * q^39 + b3 * q^40 + (b16 - b12 - b11 - b10 - b9 - b5 - b4 + b3 + 1) * q^41 + (-b15 + b14 + b13 - b5 - b2 - 2b1 - 1) q^42 + (-b15 - b14 + b13 - b9 - b7 - b6 + 1) * q^43 + b11 * q^44 + (b16 + b15 + b14 + b13 + b12 + 3b10 + b8 + 2b6 - b3 + 1) * q^45 + (-b16 + b14 - b10 - b9 - b8 + b2) * q^46 + (-b18 + 2b17 - b16 + b15 - b14 + b12 + b11 + 2b10 + b9 - b7 + b6) * q^47 + b1 * q^48 + (-b15 + b14 - b13 - b12 + b7 - b6 - b5 - b4 + b3 + 3) * q^49 + (-b16 - b15 + b11 + b10 + b7 + b2 - 1) * q^50 + (-b18 + b16 + b14 - b12 - b10 - b8 + b6 - b5 + b3) * q^51 + (b14 - b9 + b7 + 1) * q^52 + (-b15 + b13 - b10 - b6 + b4 - b3 - 1) * q^53 + (-b18 + b17 - b16 + b15 - b8 - b2 - b1 - 1) * q^54 + (2b17 - b16 + b15 + b12 + b8 - b7 + 2b6 + b5 - b4 + 2b3 + 1) q^55 + (b13 - 1) * q^56 + (b18 - b15 + b13 - 2b10 - b9 + b7 - 2b6 + b5 + b3 + 2b1) q^57 + (-b17 + b16 + b14 - b8 + b7 + b6 - b5) * q^58 + (-2b17 + b16 + b14 + b13 + b10 + b8 + b7 - b5 + b4 - b2 - 1) q^59 + (b14 + b12 + b6 - b5 - b4) * q^60 + (-b18 - b16 - b14 - b10 - b8 - b7 + b4 - b3 + b2 + 1) * q^61 + (b18 + b14 - b9 - b6 + b2 - 1) * q^62 + (2b18 - 2b17 - b16 + b15 - b14 - b13 + b11 + b7 - b6 - b4 + 2b3 + 2b2 + 2b1 + 5) q^63 + q^64 + (b14 + b13 - 2b8 + b6 + b5 - b3 - b2) q^65 + (b16 - b14 - b9 + b8 - b4 + b3 - b2 - 1) * q^66 + (-b17 + b11 + 2b10 + 2b9 + b8 - b6 + b5 + b4 + 3) * q^67 + (-b16 - b12 - b10 - b8 + 1) * q^68 + (-b17 - 2b16 - b14 + b13 + b12 + b10 + b9 + b8 + 2b6 - b4 + b3 - b2 + 1) * q^69 + (b18 + b16 + b14 + b8 + b7 + 2b3 - b2) q^70 + (b16 + b15 - b11 - 2b10 - b9 + b3 - b2 - 1) q^71 + (-b18 + b17 - b14 - b7 - b1 - 1) * q^72 + (2b18 - b15 - b13 - b12 - b10 - b6 + b5 + b3 + b2 + 4) q^73 + (b18 - b15 + b14 - b13 + b2 - 2) * q^74 + (-2b18 + b16 + b15 + b14 - b11 - 4b10 - 2b7 - b4 + b3 - 2b2 + b1 - 2) * q^75 + (-b15 + b10 - b6 + b4 - b3) * q^76 + (-2b17 + 3b16 + b14 + b13 + b12 + b11 - b9 + 3b7 - b5 - 1) q^77 + (-2b17 + b15 + b11 + b10 + b9 + b8 + b6 - 2b1 + 2) * q^78 + (-b16 + b15 - b13 - b12 - b8 + b7 - b5 + b4 + b3 + b2 + 3) * q^79 - b3 * q^80 + (b18 - b17 - 2b16 - 2b14 + b12 + b11 + 4b10 + 2b9 + b8 + b6 + b4 - b3 + 2b2 + b1 + 3) q^81 + (-b16 + b12 + b11 + b10 + b9 + b5 + b4 - b3 - 1) * q^82 + (b18 - b16 - b15 + b14 - b12 - 2b10 - 2b6 + 2b5 + 2b3 - b2) * q^83 + (b15 - b14 - b13 + b5 + b2 + 2b1 + 1) q^84 + (b18 + 2b17 + b16 - b15 - b13 + 2b10 - b9 + b8 + b6 + 2b2 - 2b1 + 2) * q^85 + (b15 + b14 - b13 + b9 + b7 + b6 - 1) * q^86 + (b18 + b17 + 2b16 + b14 - b13 + b12 + 2b10 + 2b8 - 2b6 + b3 - b1) * q^87 - b11 * q^88 + (b17 + 2b16 - b15 + b13 - b11 + b8 - b7 + b4 - 2b2 - b1 - 2) * q^89 + (-b16 - b15 - b14 - b13 - b12 - 3b10 - b8 - 2b6 + b3 - 1) * q^90 + (2b17 + b15 - 3b13 - b12 + b11 - b10 - b9 - b8 - b6 - b4 + 2b3 + b2 + 4) q^91 + (b16 - b14 + b10 + b9 + b8 - b2) * q^92 + (-b18 - 2b17 + b15 - b14 + b13 + 2b10 + b9 + b7 + 3b6 - b5 - 2b2 + 2) * q^93 + (b18 - 2b17 + b16 - b15 + b14 - b12 - b11 - 2b10 - b9 + b7 - b6) * q^94 + (-b18 - b16 - b15 + b14 - b11 + 2b10 + b9 + b4 - 2b3 - b2 - 2b1 - 1) q^95 - b1 * q^96 + (-b16 - b15 + b13 - b12 - b8 - b5 - 2b1 + 5) q^97 + (b15 - b14 + b13 + b12 - b7 + b6 + b5 + b4 - b3 - 3) * q^98 + (b17 + 2b16 - b15 + 2b14 - b13 - b12 - b11 - 2b10 - b9 + b8 - 2b7 + b3 + b2 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1502.2.a.h

Level	$1502$
Weight	$2$
Character orbit	1502.a
Self dual	yes
Analytic conductor	$11.994$
Analytic rank	$0$
Dimension	$19$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(11.9935303836\)
Analytic rank:	\(0\)
Dimension:	\(19\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 \) x^19 - 6x^18 - 23x^17 + 190x^16 + 128x^15 - 2394x^14 + 749x^13 + 15539x^12 - 11338x^11 - 56744x^10 + 50183x^9 + 120237x^8 - 102992x^7 - 148539x^6 + 99519x^5 + 104388x^4 - 41081x^3 - 35733x^2 + 5542x + 4222
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 37\!\cdots\!86 \nu^{18} + \cdots - 31\!\cdots\!88 ) / 17\!\cdots\!02 \) (-3760520192170430386v^18 + 30305082839094078765v^17 + 41200800710110050060v^16 - 886378677694456159036v^15 + 907949322179800655710v^14 + 9829864376744292242242v^13 - 19458254457701180205140v^12 - 51174904871571611337619v^11 + 141922316336666980879649v^10 + 121951098427378259989987v^9 - 501893339143834545238647v^8 - 80600003893061145349438v^7 + 896205871816710953186330v^6 - 122113529794623773103574v^5 - 760681740768789785859421v^4 + 150786020791875092571053v^3 + 280020833321376854132087v^2 - 39882937575483697112922v - 31872343548678071734688) / 1764112429837355700902
\(\beta_{3}\)	\(=\)	\( ( - 29\!\cdots\!73 \nu^{18} + \cdots - 12\!\cdots\!10 ) / 35\!\cdots\!04 \) (-29076110460680765573v^18 + 206990547035840834085v^17 + 449984253396810000572v^16 - 6101941593449854293310v^15 + 2797270294997483076910v^14 + 68736174355783190880152v^13 - 96465481661643000176109v^12 - 370669531209545576582376v^11 + 744314723041094580645146v^10 + 974192543055183898521206v^9 - 2613862736587341024233861v^8 - 1051452713367576331560234v^7 + 4457592187429563078996866v^6 + 66062436696734714273185v^5 - 3452999863129750339575306v^4 + 327922948269301043803302v^3 + 1137688178086206599168283v^2 - 85125640017045839740932v - 122836999681487683614610) / 3528224859674711401804
\(\beta_{4}\)	\(=\)	\( ( - 30\!\cdots\!89 \nu^{18} + \cdots - 15\!\cdots\!54 ) / 35\!\cdots\!04 \) (-30945253987065060989v^18 + 219306674910305950519v^17 + 521676309588807770152v^16 - 6637331055069694956182v^15 + 1812199387042442266854v^14 + 78055232816059923303620v^13 - 91332498446351700215337v^12 - 454285839667464592518246v^11 + 748467525203016066124524v^10 + 1388661333356872083903068v^9 - 2762612032560116272703091v^8 - 2178547418940168131940534v^7 + 5028250121350414105405834v^6 + 1615298067719303956571269v^5 - 4277654130079105760072216v^4 - 538411709022266322520764v^3 + 1492995420294699602344401v^2 + 41402129559144245656416v - 151051038335736692981254) / 3528224859674711401804
\(\beta_{5}\)	\(=\)	\( ( 35\!\cdots\!67 \nu^{18} + \cdots + 26\!\cdots\!14 ) / 35\!\cdots\!04 \) (35514340071205248367v^18 - 242632159693174792561v^17 - 643305185654029693144v^16 + 7400901982469154931142v^15 - 817581316649995040554v^14 - 87979131560055003267392v^13 + 91618867761805274083903v^12 + 519709571191677577219486v^11 - 798377461348665252899432v^10 - 1619993422592754555171372v^9 + 3075165837805834008163477v^8 + 2598218311229720705507906v^7 - 5902151591410770828067210v^6 - 1931036361335959872817407v^5 + 5457929300039474936559668v^4 + 551106253558324847470664v^3 - 2161482440858026625606455v^2 - 15471705894405999436640v + 261764691393999547176914) / 3528224859674711401804
\(\beta_{6}\)	\(=\)	\( ( - 39\!\cdots\!95 \nu^{18} + \cdots - 22\!\cdots\!10 ) / 35\!\cdots\!04 \) (-39225674615895018295v^18 + 281334979929612838243v^17 + 630357296227855912720v^16 - 8430966253267278684034v^15 + 3188656397738161913338v^14 + 97614079406512640434288v^13 - 125532268617349703729955v^12 - 553483373863916062832792v^11 + 999509744710647745384134v^10 + 1614712864453446395048338v^9 - 3630156788064959123947035v^8 - 2309127089727213676127182v^7 + 6513098057013774939701338v^6 + 1393894842156114864238095v^5 - 5474664469504159485153834v^4 - 329634434562021923542522v^3 + 1952055797874724118604737v^2 + 7043498701436966126708v - 222304348635242813768410) / 3528224859674711401804
\(\beta_{7}\)	\(=\)	\( ( - 11\!\cdots\!11 \nu^{18} + \cdots - 61\!\cdots\!57 ) / 88\!\cdots\!51 \) (-11167946556581994711v^18 + 72037020882250373706v^17 + 218163849798413458391v^16 - 2198104257192141994412v^15 - 249022931837379544406v^14 + 26155264541052107927708v^13 - 22385598655266651602489v^12 - 154856996848546561721898v^11 + 209171645728351253723877v^10 + 485395555103885711253293v^9 - 815202509303176415989788v^8 - 790394723234322452109492v^7 + 1547205525833040825470336v^6 + 618091829102644345876033v^5 - 1382690686240782796797340v^4 - 216675894834980889747933v^3 + 521040532835462169581003v^2 + 23377396899492401072662v - 61007845242638016139457) / 882056214918677850451
\(\beta_{8}\)	\(=\)	\( ( 22\!\cdots\!63 \nu^{18} + \cdots + 11\!\cdots\!02 ) / 17\!\cdots\!02 \) (22935642479960504263v^18 - 163807917546078032844v^17 - 347968100546541242034v^16 + 4811256116260884393776v^15 - 2415755773575146538442v^14 - 53879781520772923569932v^13 + 78553475030489475053325v^12 + 287609418859943312771113v^11 - 601867975227054779042041v^10 - 740938814962541543108443v^9 + 2111650992028433615016010v^8 + 758295797594366651969202v^7 - 3615592773829894301652558v^6 + 12016842140245612195017v^5 + 2840490829818975545731353v^4 - 242616546224541959763249v^3 - 970625595458513263192090v^2 + 49046681426265715429604v + 113592698179520862754802) / 1764112429837355700902
\(\beta_{9}\)	\(=\)	\( ( - 53\!\cdots\!13 \nu^{18} + \cdots - 33\!\cdots\!30 ) / 35\!\cdots\!04 \) (-53078987331776809413v^18 + 367515426498576955637v^17 + 920244526033840804776v^16 - 11088118004222127647018v^15 + 2323875647558612679514v^14 + 129631647355211696363740v^13 - 147215378581633743827297v^12 - 745654129012549222047316v^11 + 1227483135034000590929390v^10 + 2223128080408016070228306v^9 - 4564940454465900858809637v^8 - 3293148157720498569728226v^7 + 8377236903027542588927042v^6 + 2119801390785974910893953v^5 - 7282681049875190648528770v^4 - 555446990794750228761258v^3 + 2758013642041956597019251v^2 + 43217652384396165704284v - 338893975678974074603630) / 3528224859674711401804
\(\beta_{10}\)	\(=\)	\( ( 65\!\cdots\!89 \nu^{18} + \cdots + 38\!\cdots\!54 ) / 35\!\cdots\!04 \) (65269994698618887489v^18 - 440295935608668625411v^17 - 1191848115016691865296v^16 + 13359502166014633995958v^15 - 1037892829669082123282v^14 - 157625826309534025609356v^13 + 159778105580821061796529v^12 + 921037838225169820617946v^11 - 1388161951758038181773760v^10 - 2825851115465353418466920v^9 + 5260830789395044175743819v^8 + 4431949649138633371027590v^7 - 9808515862489234476993834v^6 - 3227048649923522292410445v^5 + 8655297992211802941703232v^4 + 1006944353978834637406320v^3 - 3252303057734070582886901v^2 - 76711896678285078567952v + 382673437272331542837754) / 3528224859674711401804
\(\beta_{11}\)	\(=\)	\( ( - 67\!\cdots\!05 \nu^{18} + \cdots - 40\!\cdots\!90 ) / 35\!\cdots\!04 \) (-67545888761056757405v^18 + 474152691703187535957v^17 + 1126972007891038411660v^16 - 14225479148011647792746v^15 + 4245207934669709738222v^14 + 164933151038642750448804v^13 - 201754926509586295161889v^12 - 936527712809011504410176v^11 + 1640288763668577806266250v^10 + 2733320468157187722611582v^9 - 6025875980509308282125301v^8 - 3890467207784257373413434v^7 + 10950927147978590859975302v^6 + 2281174130293399234730025v^5 - 9407015588513958382958478v^4 - 481304642842418863805234v^3 + 3472206780176237103939911v^2 + 23882619695694000437704v - 407486381451920829795090) / 3528224859674711401804
\(\beta_{12}\)	\(=\)	\( ( - 69\!\cdots\!03 \nu^{18} + \cdots - 22\!\cdots\!42 ) / 35\!\cdots\!04 \) (-69985589363716372003v^18 + 452117294523362098281v^17 + 1325535324928609159564v^16 - 13563574351565447656442v^15 - 639254816371122214178v^14 + 157208079346273319443112v^13 - 145053125102334518860899v^12 - 891743569213297416919646v^11 + 1282552727139659807366264v^10 + 2595492162936333398605104v^9 - 4739692707772156654829369v^8 - 3671011968791771277480298v^7 + 8342749374284723742911106v^6 + 2133527757290567026500795v^5 - 6635766416826814603288452v^4 - 482217178587693989306880v^3 + 2185026163458131991656443v^2 + 28482245194557351640248v - 221905016240633020395442) / 3528224859674711401804
\(\beta_{13}\)	\(=\)	\( ( - 77\!\cdots\!15 \nu^{18} + \cdots - 44\!\cdots\!50 ) / 35\!\cdots\!04 \) (-77065122442043030315v^18 + 532774562852707940065v^17 + 1309795665184004200420v^16 - 15947559575155028954070v^15 + 4066746067386985452170v^14 + 184141182031040943735476v^13 - 220131935427482750177679v^12 - 1037180746879497347929890v^11 + 1804220288341667306366968v^10 + 2973324222468615128719668v^9 - 6624784111405967801651453v^8 - 4032909788018291328760214v^7 + 11979292489662292335093490v^6 + 1963412732253456474997411v^5 - 10207686105122915636409520v^4 - 72364707768952431818544v^3 + 3775019299987169463270923v^2 - 93743526820595676419864v - 448047703088435065929750) / 3528224859674711401804
\(\beta_{14}\)	\(=\)	\( ( 81\!\cdots\!29 \nu^{18} + \cdots + 43\!\cdots\!06 ) / 35\!\cdots\!04 \) (81246465791995337029v^18 - 538011472036996170959v^17 - 1500040891301178160836v^16 + 16239099098267565245182v^15 - 682749423888030571250v^14 - 190058582570187933079240v^13 + 189727613796164956802949v^12 + 1096000891963530403875206v^11 - 1656270139070271276372200v^10 - 3286800831286664099437744v^9 + 6237831720989536635596223v^8 + 4936824531945407856072402v^7 - 11476874966336529882099158v^6 - 3283786466870076576469109v^5 + 9927586089252498464492112v^4 + 881333673434794369135432v^3 - 3681466686787631497562293v^2 - 47898124722861190738136v + 432163679569144496110306) / 3528224859674711401804
\(\beta_{15}\)	\(=\)	\( ( - 47\!\cdots\!28 \nu^{18} + \cdots - 27\!\cdots\!58 ) / 17\!\cdots\!02 \) (-47097892268680919528v^18 + 319743617906841950481v^17 + 837922334641744413414v^16 - 9633843421968099827224v^15 + 1369200205499709724344v^14 + 112406080492905974955444v^13 - 121720435612565003591624v^12 - 644497773229142363627215v^11 + 1031071031664700952304797v^10 + 1910019844408435737913565v^9 - 3845750444490116295482927v^8 - 2787652479313939222193926v^7 + 7047960430634142289825628v^6 + 1696565935560911310021946v^5 - 6094487564041673976730955v^4 - 326408167210916415098145v^3 + 2271252467228465080858335v^2 - 14489750421562796486650v - 271400599812957179459358) / 1764112429837355700902
\(\beta_{16}\)	\(=\)	\( ( - 95\!\cdots\!25 \nu^{18} + \cdots - 50\!\cdots\!54 ) / 35\!\cdots\!04 \) (-95971349547274628625v^18 + 656629516719657132985v^17 + 1661993776848826883504v^16 - 19670759651570058856394v^15 + 4075172162552562465278v^14 + 227474470867889711197860v^13 - 261446093195608246486625v^12 - 1285291669865683973133804v^11 + 2162449824953311239457686v^10 + 3712796511016076787639470v^9 - 7933994510972599758883509v^8 - 5155450907022371409992946v^7 + 14246641802290284696365898v^6 + 2801906095971079748896029v^5 - 11951769961894352195903022v^4 - 450996964601953574266394v^3 + 4322881678425745449196591v^2 - 22864874827629484464976v - 507524935709983375201054) / 3528224859674711401804
\(\beta_{17}\)	\(=\)	\( ( - 11\!\cdots\!69 \nu^{18} + \cdots - 63\!\cdots\!18 ) / 35\!\cdots\!04 \) (-111180934604728577169v^18 + 754491441872043631203v^17 + 1983118773266591338776v^16 - 22754311006284932937834v^15 + 3094221402718334038530v^14 + 265927825342822788548944v^13 - 285979842814695639634801v^12 - 1529488751053174975282250v^11 + 2428243393839957414429780v^10 + 4563172264386688163453912v^9 - 9069817744293933606095027v^8 - 6775592737231410424291170v^7 + 16641663522586577497468190v^6 + 4371124193456046186265857v^5 - 14402873239663841582626060v^4 - 1084873524382554347794788v^3 + 5368607597817917780318525v^2 + 46591581792452054756420v - 636544201841169330766818) / 3528224859674711401804
\(\beta_{18}\)	\(=\)	\( ( - 73\!\cdots\!77 \nu^{18} + \cdots - 41\!\cdots\!56 ) / 17\!\cdots\!02 \) (-73877807085197967677v^18 + 502177415190019153669v^17 + 1305252132687057833024v^16 - 15100496537891965102684v^15 + 2386531276977941393702v^14 + 175682674874401144958676v^13 - 193082530994896995013897v^12 - 1003030827811259566134932v^11 + 1623913474998411837953236v^10 + 2954195437628904708939242v^9 - 6023419714035382288866049v^8 - 4275419188119764235962802v^7 + 10964858192795472038843002v^6 + 2591271671957772689615417v^5 - 9399848291976604429964406v^4 - 549751809238712578969244v^3 + 3484720189061687655479305v^2 - 1274052971165535098948v - 419394699939230303963256) / 1764112429837355700902

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{18} - \beta_{17} + \beta_{14} + \beta_{7} + \beta _1 + 4 \) b18 - b17 + b14 + b7 + b1 + 4
\(\nu^{3}\)	\(=\)	\( \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{8} + \beta_{2} + 7\beta _1 + 1 \) b18 - b17 + b16 - b15 + b8 + b2 + 7*b1 + 1
\(\nu^{4}\)	\(=\)	\( 10 \beta_{18} - 10 \beta_{17} - 2 \beta_{16} + 7 \beta_{14} + \beta_{12} + \beta_{11} + 4 \beta_{10} + \cdots + 30 \) 10b18 - 10b17 - 2b16 + 7b14 + b12 + b11 + 4b10 + 2b9 + b8 + 9b7 + b6 + b4 - b3 + 2b2 + 10*b1 + 30
\(\nu^{5}\)	\(=\)	\( 13 \beta_{18} - 15 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} + \cdots + 18 \) 13b18 - 15b17 + 10b16 - 10b15 - 2b14 - 2b13 + b12 + 4b11 + 6b10 + 3b9 + 15b8 + 2b7 + 3b6 - 2b4 + 2b3 + 13b2 + 58b1 + 18
\(\nu^{6}\)	\(=\)	\( 96 \beta_{18} - 101 \beta_{17} - 30 \beta_{16} + 4 \beta_{15} + 46 \beta_{14} - 5 \beta_{13} + \cdots + 270 \) 96b18 - 101b17 - 30b16 + 4b15 + 46b14 - 5b13 + 15b12 + 16b11 + 62b10 + 34b9 + 18b8 + 80b7 + 22b6 + 9b4 - 12b3 + 32b2 + 99*b1 + 270
\(\nu^{7}\)	\(=\)	\( 145 \beta_{18} - 193 \beta_{17} + 85 \beta_{16} - 79 \beta_{15} - 46 \beta_{14} - 41 \beta_{13} + \cdots + 257 \) 145b18 - 193b17 + 85b16 - 79b15 - 46b14 - 41b13 + 21b12 + 66b11 + 115b10 + 62b9 + 177b8 + 33b7 + 69b6 - 5b5 - 38b4 + 29b3 + 149b2 + 525b1 + 257
\(\nu^{8}\)	\(=\)	\( 930 \beta_{18} - 1072 \beta_{17} - 351 \beta_{16} + 94 \beta_{15} + 278 \beta_{14} - 106 \beta_{13} + \cdots + 2634 \) 930b18 - 1072b17 - 351b16 + 94b15 + 278b14 - 106b13 + 186b12 + 211b11 + 768b10 + 451b9 + 254b8 + 736b7 + 349b6 - 11b5 + 42b4 - 115b3 + 401b2 + 1012b1 + 2634
\(\nu^{9}\)	\(=\)	\( 1573 \beta_{18} - 2397 \beta_{17} + 666 \beta_{16} - 530 \beta_{15} - 738 \beta_{14} - 590 \beta_{13} + \cdots + 3380 \) 1573b18 - 2397b17 + 666b16 - 530b15 - 738b14 - 590b13 + 327b12 + 852b11 + 1664b10 + 964b9 + 1961b8 + 444b7 + 1115b6 - 119b5 - 538b4 + 310b3 + 1648b2 + 5036b1 + 3380
\(\nu^{10}\)	\(=\)	\( 9175 \beta_{18} - 11769 \beta_{17} - 3807 \beta_{16} + 1508 \beta_{15} + 1251 \beta_{14} - 1589 \beta_{13} + \cdots + 26870 \) 9175b18 - 11769b17 - 3807b16 + 1508b15 + 1251b14 - 1589b13 + 2189b12 + 2612b11 + 8964b10 + 5554b9 + 3307b8 + 6985b7 + 4848b6 - 307b5 - 183b4 - 1029b3 + 4663b2 + 10683b1 + 26870
\(\nu^{11}\)	\(=\)	\( 17145 \beta_{18} - 29325 \beta_{17} + 4745 \beta_{16} - 2619 \beta_{15} - 10236 \beta_{14} + \cdots + 42768 \) 17145b18 - 29325b17 + 4745b16 - 2619b15 - 10236b14 - 7453b13 + 4519b12 + 10217b11 + 21795b10 + 13362b9 + 21420b8 + 5710b7 + 15698b6 - 1975b5 - 6883b4 + 2992b3 + 18039b2 + 50367b1 + 42768
\(\nu^{12}\)	\(=\)	\( 92445 \beta_{18} - 131860 \beta_{17} - 40241 \beta_{16} + 20806 \beta_{15} - 1416 \beta_{14} + \cdots + 282144 \) 92445b18 - 131860b17 - 40241b16 + 20806b15 - 1416b14 - 20898b13 + 25311b12 + 31401b11 + 102868b10 + 66431b9 + 41484b8 + 68177b7 + 63006b6 - 5642b5 - 8331b4 - 8937b3 + 52854b2 + 115912b1 + 282144
\(\nu^{13}\)	\(=\)	\( 189444 \beta_{18} - 355071 \beta_{17} + 28095 \beta_{16} + 467 \beta_{15} - 131894 \beta_{14} + \cdots + 529320 \) 189444b18 - 355071b17 + 28095b16 + 467b15 - 131894b14 - 88916b13 + 58752b12 + 119285b11 + 272586b10 + 174263b9 + 234511b8 + 72291b7 + 206215b6 - 28423b5 - 84230b4 + 27678b3 + 197598b2 + 520148b1 + 529320
\(\nu^{14}\)	\(=\)	\( 951790 \beta_{18} - 1495641 \beta_{17} - 422941 \beta_{16} + 266482 \beta_{15} - 148463 \beta_{14} + \cdots + 3025986 \) 951790b18 - 1495641b17 - 422941b16 + 266482b15 - 148463b14 - 258558b13 + 291063b12 + 371861b11 + 1176880b10 + 784039b9 + 509590b8 + 683060b7 + 788675b6 - 86676b5 - 151568b4 - 76389b3 + 595243b2 + 1285093b1 + 3025986
\(\nu^{15}\)	\(=\)	\( 2123444 \beta_{18} - 4263971 \beta_{17} + 84246 \beta_{16} + 270917 \beta_{15} - 1630800 \beta_{14} + \cdots + 6456756 \) 2123444b18 - 4263971b17 + 84246b16 + 270917b15 - 1630800b14 - 1033607b13 + 737547b12 + 1379282b11 + 3325956b10 + 2191644b9 + 2587562b8 + 905228b7 + 2603486b6 - 380335b5 - 1008395b4 + 251131b3 + 2177452b2 + 5510799b1 + 6456756
\(\nu^{16}\)	\(=\)	\( 10004896 \beta_{18} - 17096882 \beta_{17} - 4461039 \beta_{16} + 3277301 \beta_{15} - 2835698 \beta_{14} + \cdots + 32999887 \) 10004896b18 - 17096882b17 - 4461039b16 + 3277301b15 - 2835698b14 - 3101641b13 + 3344826b12 + 4369096b11 + 13486035b10 + 9191526b9 + 6176669b8 + 7011607b7 + 9648683b6 - 1209261b5 - 2237308b4 - 646233b3 + 6713785b2 + 14477716b1 + 32999887
\(\nu^{17}\)	\(=\)	\( 24090869 \beta_{18} - 50866953 \beta_{17} - 1189598 \beta_{16} + 5419328 \beta_{15} - 19678325 \beta_{14} + \cdots + 77950265 \) 24090869b18 - 50866953b17 - 1189598b16 + 5419328b15 - 19678325b14 - 11883466b13 + 9060210b12 + 15905311b11 + 39994481b10 + 26934897b9 + 28809860b8 + 11215969b7 + 32068711b6 - 4876496b5 - 11932955b4 + 2258019b3 + 24189418b2 + 59605928b1 + 77950265
\(\nu^{18}\)	\(=\)	\( 107185878 \beta_{18} - 196451573 \beta_{17} - 47431680 \beta_{16} + 39376536 \beta_{15} + \cdots + 364797939 \) 107185878b18 - 196451573b17 - 47431680b16 + 39376536b15 - 42400156b14 - 36615716b13 + 38484796b12 + 51124315b11 + 154989621b10 + 107374656b9 + 74180123b8 + 73579411b7 + 116313191b6 - 15923475b5 - 30123224b4 - 5421373b3 + 76069064b2 + 164981093b1 + 364797939

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.19
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T + 1)^{19} \) (T + 1)^19
$3$	\( T^{19} - 6 T^{18} + \cdots + 4222 \) T^19 - 6T^18 - 23T^17 + 190T^16 + 128T^15 - 2394T^14 + 749T^13 + 15539T^12 - 11338T^11 - 56744T^10 + 50183T^9 + 120237T^8 - 102992T^7 - 148539T^6 + 99519T^5 + 104388T^4 - 41081T^3 - 35733T^2 + 5542T + 4222
$5$	\( T^{19} - 2 T^{18} + \cdots + 198336 \) T^19 - 2T^18 - 64T^17 + 119T^16 + 1685T^15 - 2936T^14 - 23650T^13 + 39164T^12 + 191703T^11 - 307853T^10 - 907658T^9 + 1448759T^8 + 2403291T^7 - 3932551T^6 - 3128876T^5 + 5542868T^4 + 1342688T^3 - 3176464T^2 + 272768T + 198336
$7$	\( T^{19} - 13 T^{18} + \cdots - 16384 \) T^19 - 13T^18 - 3T^17 + 716T^16 - 2346T^15 - 12178T^14 + 73421T^13 + 28542T^12 - 840580T^11 + 1029185T^10 + 3654176T^9 - 8333888T^8 - 3506704T^7 + 18123744T^6 - 1476096T^5 - 15360000T^4 + 1391616T^3 + 3792896T^2 + 16384T - 16384
$11$	\( T^{19} + 7 T^{18} + \cdots - 22147038 \) T^19 + 7T^18 - 102T^17 - 775T^16 + 3881T^15 + 34139T^14 - 64090T^13 - 762553T^12 + 293745T^11 + 9126950T^10 + 3782096T^9 - 57187113T^8 - 46697525T^7 + 173756417T^6 + 161674992T^5 - 220699057T^4 - 160178222T^3 + 118255823T^2 + 46099456T - 22147038
$13$	\( T^{19} - 19 T^{18} + \cdots + 28157056 \) T^19 - 19T^18 + 23T^17 + 1691T^16 - 10005T^15 - 36781T^14 + 458421T^13 - 429256T^12 - 7246108T^11 + 21524435T^10 + 29197792T^9 - 190401899T^8 + 99863768T^7 + 525181345T^6 - 681420452T^5 - 227787612T^4 + 726595952T^3 - 277076832T^2 - 45545536T + 28157056
$17$	\( T^{19} + \cdots - 2458536000 \) T^19 - 11T^18 - 94T^17 + 1176T^16 + 3987T^15 - 53333T^14 - 104994T^13 + 1335177T^12 + 1947696T^11 - 20064357T^10 - 25996433T^9 + 183631742T^8 + 238268165T^7 - 982595445T^6 - 1381766668T^5 + 2686785892T^4 + 4470472912T^3 - 2196435120T^2 - 6080804800T - 2458536000
$19$	\( T^{19} - 7 T^{18} + \cdots + 2000512 \) T^19 - 7T^18 - 150T^17 + 931T^16 + 9018T^15 - 46780T^14 - 282602T^13 + 1141658T^12 + 4935904T^11 - 14557383T^10 - 46921968T^9 + 99571346T^8 + 219483173T^7 - 387108185T^6 - 435180192T^5 + 799012968T^4 + 163955648T^3 - 606093936T^2 + 201811840T + 2000512
$23$	\( T^{19} + \cdots + 2499323904 \) T^19 - 12T^18 - 174T^17 + 2289T^16 + 11597T^15 - 172030T^14 - 381054T^13 + 6604311T^12 + 6600251T^11 - 140761469T^10 - 59521244T^9 + 1695988392T^8 + 264164715T^7 - 11212811777T^6 - 656370188T^5 + 36908081016T^4 + 966991232T^3 - 45089997120T^2 + 3860619264T + 2499323904
$29$	\( T^{19} + \cdots - 6319814328 \) T^19 + 8T^18 - 271T^17 - 2030T^16 + 29067T^15 + 191480T^14 - 1682804T^13 - 8934051T^12 + 58092374T^11 + 223263107T^10 - 1209651211T^9 - 2945845537T^8 + 14571815494T^7 + 17646919521T^6 - 91271784658T^5 - 14377526710T^4 + 219553291740T^3 - 186761444868T^2 + 58392042224T - 6319814328
$31$	\( T^{19} + \cdots - 12085879168 \) T^19 - 32T^18 + 225T^17 + 2895T^16 - 43610T^15 + 3291T^14 + 2377347T^13 - 6934699T^12 - 60585294T^11 + 275543920T^10 + 791276176T^9 - 4908438432T^8 - 4790687009T^7 + 44497288429T^6 + 4174315500T^5 - 189236768568T^4 + 73637972896T^3 + 262786776432T^2 - 158976812032T - 12085879168
$37$	\( T^{19} + \cdots + 918964640768 \) T^19 - 41T^18 + 557T^17 - 697T^16 - 52666T^15 + 415947T^14 + 783344T^13 - 21663637T^12 + 47934404T^11 + 448927849T^10 - 2030479017T^9 - 3795421157T^8 + 31997245606T^7 + 2189768456T^6 - 248686317800T^5 + 144981550544T^4 + 964471825856T^3 - 709561191168T^2 - 1496215054848T + 918964640768
$41$	\( T^{19} + \cdots + 491698922496 \) T^19 - 15T^18 - 241T^17 + 4837T^16 + 8919T^15 - 510640T^14 + 1251454T^13 + 21018112T^12 - 90563152T^11 - 397394545T^10 + 2280477675T^9 + 3709493583T^8 - 27226533788T^7 - 19528514988T^6 + 164860565808T^5 + 86062380688T^4 - 498919571392T^3 - 312449767232T^2 + 596072093696T + 491698922496
$43$	\( T^{19} + \cdots + 62830554180608 \) T^19 - 15T^18 - 360T^17 + 6195T^16 + 46133T^15 - 1016212T^14 - 2272470T^13 + 86680239T^12 - 16433963T^11 - 4191729558T^10 + 6365562426T^9 + 116825081946T^8 - 269084946147T^7 - 1809700862715T^6 + 4989342100972T^5 + 14085627775160T^4 - 40930212458544T^3 - 46043293625824T^2 + 110400685730304T + 62830554180608
$47$	\( T^{19} + \cdots + 31509244224 \) T^19 + 6T^18 - 383T^17 - 2601T^16 + 56547T^15 + 424851T^14 - 4075408T^13 - 33609815T^12 + 151822995T^11 + 1371100055T^10 - 3025619103T^9 - 28732282899T^8 + 37649053709T^7 + 289783085785T^6 - 381512864324T^5 - 1109372646788T^4 + 2149226791200T^3 - 627352970832T^2 - 380101158784T + 31509244224
$53$	\( T^{19} + \cdots - 869610298176 \) T^19 - 6T^18 - 331T^17 + 2171T^16 + 41236T^15 - 303316T^14 - 2381375T^13 + 20705528T^12 + 61604233T^11 - 735598986T^10 - 417682538T^9 + 13959810749T^8 - 11097660632T^7 - 135617283565T^6 + 231618577912T^5 + 552873958980T^4 - 1436211489520T^3 - 224776932080T^2 + 2165529532096T - 869610298176
$59$	\( T^{19} + \cdots + 2924927705088 \) T^19 + 4T^18 - 571T^17 - 2566T^16 + 126015T^15 + 637937T^14 - 13684928T^13 - 80600906T^12 + 752060389T^11 + 5513687477T^10 - 17264353908T^9 - 195125226608T^8 - 67976307968T^7 + 2866534233552T^6 + 7497271871936T^5 - 2401853026048T^4 - 32421175495424T^3 - 41457775769088T^2 - 13472912867328T + 2924927705088
$61$	\( T^{19} + \cdots + 20341939328 \) T^19 - 13T^18 - 302T^17 + 3823T^16 + 33939T^15 - 425696T^14 - 1692369T^13 + 22847114T^12 + 33567381T^11 - 611835565T^10 - 54459416T^9 + 7761904334T^8 - 3787589153T^7 - 46190770571T^6 + 27427416672T^5 + 122937171588T^4 - 40765126000T^3 - 135444605600T^2 - 14719374784T + 20341939328
$67$	\( T^{19} + \cdots - 94534189842024 \) T^19 - 47T^18 + 293T^17 + 19122T^16 - 352870T^15 - 729111T^14 + 68062076T^13 - 468052157T^12 - 2924510227T^11 + 51266479187T^10 - 164726788003T^9 - 779268556575T^8 + 6188722470644T^7 - 6094617852989T^6 - 50238385182902T^5 + 137375159094586T^4 + 29272887775068T^3 - 445982882460100T^2 + 434010257584320T - 94534189842024
$71$	\( T^{19} + \cdots - 12303059328 \) T^19 - 444T^17 + 489T^16 + 74394T^15 - 140913T^14 - 5986885T^13 + 14508819T^12 + 249178149T^11 - 699406127T^10 - 5291866118T^9 + 16450338911T^8 + 50869220176T^7 - 173412821023T^6 - 147473704265T^5 + 610722540741T^4 - 28997030720T^3 - 341378288865T^2 + 126278034560*T - 12303059328
$73$	\( T^{19} + \cdots - 2739304789504 \) T^19 - 64T^18 + 1502T^17 - 11208T^16 - 145642T^15 + 3538861T^14 - 20831634T^13 - 103849256T^12 + 2016311009T^11 - 7913470382T^10 - 28433738653T^9 + 364511718339T^8 - 1064705643498T^7 - 1463397890608T^6 + 19013652779848T^5 - 58859795080304T^4 + 92740845684992T^3 - 77146618346240T^2 + 29259860813312T - 2739304789504
$79$	\( T^{19} + \cdots - 769315246221952 \) T^19 - 34T^18 - 321T^17 + 23072T^16 - 98990T^15 - 5525343T^14 + 57461830T^13 + 501159237T^12 - 9291423625T^11 + 251945698T^10 + 611715355772T^9 - 2407671711095T^8 - 12916374059579T^7 + 98162487571845T^6 - 62776351310208T^5 - 696114795502320T^4 + 892201431154704T^3 + 1665421264650576T^2 - 1578225340043904T - 769315246221952
$83$	\( T^{19} + \cdots - 266808806442714 \) T^19 - 4T^18 - 918T^17 + 3126T^16 + 335563T^15 - 885345T^14 - 62336879T^13 + 103200158T^12 + 6228227891T^11 - 2824062987T^10 - 326017609082T^9 - 270146869430T^8 + 8095408195067T^7 + 13953619906597T^6 - 83247485713101T^5 - 187777090904796T^4 + 224505936944838T^3 + 496602920528139T^2 - 253901504066740T - 266808806442714
$89$	\( T^{19} + \cdots - 199659293904768 \) T^19 - 18T^18 - 615T^17 + 12006T^16 + 141397T^15 - 3216296T^14 - 13900817T^13 + 440701235T^12 + 282779369T^11 - 32515908126T^10 + 48946392907T^9 + 1243632209876T^8 - 3617257407005T^7 - 21392816120940T^6 + 88108099181189T^5 + 93705989146379T^4 - 694539760920858T^3 + 581180905438764T^2 + 210190376822336T - 199659293904768
$97$	\( T^{19} + \cdots - 57640403406224 \) T^19 - 82T^18 + 2745T^17 - 44794T^16 + 242774T^15 + 3750857T^14 - 79651417T^13 + 538472281T^12 + 363562367T^11 - 30199032996T^10 + 193323251095T^9 - 301395475868T^8 - 2272274995840T^7 + 12966424008709T^6 - 20347781175175T^5 - 21869592574339T^4 + 82283527566818T^3 + 12961014379128T^2 - 113387483142824T - 57640403406224
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\(2\)	\(1\)
\(751\)	\(-1\)