LMFDB - Newform orbit 3645.2.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,2,Mod(1,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.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:	$29.1054715368$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 9 x^{16} + 112 x^{15} - 51 x^{14} - 810 x^{13} + 936 x^{12} + 2844 x^{11} - 4536 x^{10} + \cdots + 9 $ x^18 - 6x^17 - 9x^16 + 112x^15 - 51x^14 - 810x^13 + 936x^12 + 2844x^11 - 4536x^10 - 4934x^9 + 10173x^8 + 3672x^7 - 11144x^6 - 456x^5 + 5490x^4 - 354x^3 - 882x^2 - 54*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{5} $
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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 6 x^{17} - 9 x^{16} + 112 x^{15} - 51 x^{14} - 810 x^{13} + 936 x^{12} + 2844 x^{11} - 4536 x^{10} + \cdots + 9 $ x^18 - 6*x^17 - 9*x^16 + 112*x^15 - 51*x^14 - 810*x^13 + 936*x^12 + 2844*x^11 - 4536*x^10 - 4934*x^9 + 10173*x^8 + 3672*x^7 - 11144*x^6 - 456*x^5 + 5490*x^4 - 354*x^3 - 882*x^2 - 54*x + 9 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - \nu^{2} - 4\nu + 2 $ v^3 - v^2 - 4*v + 2
$\beta_{4}$	$=$	$ ( 260 \nu^{17} - 1019 \nu^{16} - 4489 \nu^{15} + 20229 \nu^{14} + 28191 \nu^{13} - 159909 \nu^{12} + \cdots + 34029 ) / 7461 $ (260v^17 - 1019v^16 - 4489v^15 + 20229v^14 + 28191v^13 - 159909v^12 - 72663v^11 + 644193v^10 + 38247v^9 - 1407583v^8 + 114322v^7 + 1630061v^6 - 79611v^5 - 856863v^4 - 107322v^3 + 80712v^2 + 70281*v + 34029) / 7461
$\beta_{5}$	$=$	$ ( - 314 \nu^{17} + 1186 \nu^{16} + 5568 \nu^{15} - 23493 \nu^{14} - 35940 \nu^{13} + 183192 \nu^{12} + \cdots - 6489 ) / 7461 $ (-314v^17 + 1186v^16 + 5568v^15 - 23493v^14 - 35940v^13 + 183192v^12 + 88845v^11 - 707337v^10 + 43284v^9 + 1377325v^8 - 662657v^7 - 1148712v^6 + 1260999v^5 + 70770v^4 - 865035v^3 + 248256v^2 + 153702*v - 6489) / 7461
$\beta_{6}$	$=$	$ ( 1273 \nu^{17} - 6255 \nu^{16} - 18175 \nu^{15} + 121896 \nu^{14} + 68688 \nu^{13} - 940854 \nu^{12} + \cdots - 10071 ) / 7461 $ (1273v^17 - 6255v^16 - 18175v^15 + 121896v^14 + 68688v^13 - 940854v^12 + 132228v^11 + 3669384v^10 - 1495221v^9 - 7674794v^8 + 3598935v^7 + 8491703v^6 - 3356310v^5 - 4631352v^4 + 980490v^3 + 993321v^2 + 56772*v - 10071) / 7461
$\beta_{7}$	$=$	$ ( - 1368 \nu^{17} + 6994 \nu^{16} + 19321 \nu^{15} - 138231 \nu^{14} - 69471 \nu^{13} + 1089723 \nu^{12} + \cdots + 33651 ) / 7461 $ (-1368v^17 + 6994v^16 + 19321v^15 - 138231v^14 - 69471v^13 + 1089723v^12 - 176988v^11 - 4390062v^10 + 1744629v^9 + 9656481v^8 - 4145153v^7 - 11540648v^6 + 3868881v^5 + 7021512v^4 - 1126749v^3 - 1731033v^2 - 83169*v + 33651) / 7461
$\beta_{8}$	$=$	$ ( - 1439 \nu^{17} + 7014 \nu^{16} + 22106 \nu^{15} - 142062 \nu^{14} - 104115 \nu^{13} + 1156950 \nu^{12} + \cdots + 37530 ) / 7461 $ (-1439v^17 + 7014v^16 + 22106v^15 - 142062v^14 - 104115v^13 + 1156950v^12 + 15615v^11 - 4867965v^10 + 1238229v^9 + 11336596v^8 - 3579243v^7 - 14527561v^6 + 3643614v^5 + 9479463v^4 - 867930v^3 - 2463246v^2 - 359163*v + 37530) / 7461
$\beta_{9}$	$=$	$ ( 1534 \nu^{17} - 6924 \nu^{16} - 24910 \nu^{15} + 138501 \nu^{14} + 139716 \nu^{13} - 1106859 \nu^{12} + \cdots - 16344 ) / 7461 $ (1534v^17 - 6924v^16 - 24910v^15 + 138501v^14 + 139716v^13 - 1106859v^12 - 261834v^11 + 4524207v^10 - 261546v^9 - 10080209v^8 + 1420434v^7 + 12115883v^6 - 1273752v^5 - 7305978v^4 - 62682v^3 + 1753524v^2 + 310950*v - 16344) / 7461
$\beta_{10}$	$=$	$ ( 1554 \nu^{17} - 7385 \nu^{16} - 22577 \nu^{15} + 143118 \nu^{14} + 91188 \nu^{13} - 1096968 \nu^{12} + \cdots - 9135 ) / 7461 $ (1554v^17 - 7385v^16 - 22577v^15 + 143118v^14 + 91188v^13 - 1096968v^12 + 109644v^11 + 4242798v^10 - 1658211v^9 - 8797869v^8 + 4159954v^7 + 9679471v^6 - 4015002v^5 - 5301750v^4 + 1253103v^3 + 1167444v^2 + 56943*v - 9135) / 7461
$\beta_{11}$	$=$	$ ( - 1508 \nu^{17} + 7734 \nu^{16} + 20399 \nu^{15} - 150654 \nu^{14} - 58059 \nu^{13} + 1162245 \nu^{12} + \cdots - 9351 ) / 7461 $ (-1508v^17 + 7734v^16 + 20399v^15 - 150654v^14 - 58059v^13 + 1162245v^12 - 350022v^11 - 4529175v^10 + 2597928v^9 + 9451750v^8 - 6188850v^7 - 10367746v^6 + 6315147v^5 + 5461734v^4 - 2383818v^3 - 1023228v^2 + 60921*v - 9351) / 7461
$\beta_{12}$	$=$	$ ( 1643 \nu^{17} - 8566 \nu^{16} - 21024 \nu^{15} + 163788 \nu^{14} + 41370 \nu^{13} - 1226670 \nu^{12} + \cdots + 7650 ) / 7461 $ (1643v^17 - 8566v^16 - 21024v^15 + 163788v^14 + 41370v^13 - 1226670v^12 + 535884v^11 + 4542789v^10 - 3370035v^9 - 8610109v^8 + 7760720v^7 + 7676733v^6 - 8030091v^5 - 2276628v^4 + 3458052v^3 - 246852v^2 - 385857*v + 7650) / 7461
$\beta_{13}$	$=$	$ ( - 1784 \nu^{17} + 8956 \nu^{16} + 24348 \nu^{15} - 172587 \nu^{14} - 75282 \nu^{13} + 1310262 \nu^{12} + \cdots + 26955 ) / 7461 $ (-1784v^17 + 8956v^16 + 24348v^15 - 172587v^14 - 75282v^13 + 1310262v^12 - 337779v^11 - 4982064v^10 + 2642421v^9 + 10002577v^8 - 6071621v^7 - 10320921v^6 + 5730195v^5 + 4979334v^4 - 1936404v^3 - 873828v^2 + 69993*v + 26955) / 7461
$\beta_{14}$	$=$	$ ( - 1844 \nu^{17} + 9510 \nu^{16} + 23981 \nu^{15} - 183951 \nu^{14} - 49074 \nu^{13} + 1402452 \nu^{12} + \cdots + 35172 ) / 7461 $ (-1844v^17 + 9510v^16 + 23981v^15 - 183951v^14 - 49074v^13 + 1402452v^12 - 629016v^11 - 5353980v^10 + 4121586v^9 + 10756507v^8 - 9961143v^7 - 10931122v^6 + 10870065v^5 + 4855866v^4 - 4821810v^3 - 600327v^2 + 458964*v + 35172) / 7461
$\beta_{15}$	$=$	$ ( - 3050 \nu^{17} + 13516 \nu^{16} + 50013 \nu^{15} - 270111 \nu^{14} - 286797 \nu^{13} + 2156217 \nu^{12} + \cdots + 16047 ) / 7461 $ (-3050v^17 + 13516v^16 + 50013v^15 - 270111v^14 - 286797v^13 + 2156217v^12 + 585423v^11 - 8803536v^10 + 291981v^9 + 19600981v^8 - 2418785v^7 - 23552715v^6 + 2209251v^5 + 14146125v^4 + 131976v^3 - 3303342v^2 - 519984*v + 16047) / 7461
$\beta_{16}$	$=$	$ ( - 3128 \nu^{17} + 15231 \nu^{16} + 45308 \nu^{15} - 298314 \nu^{14} - 178614 \nu^{13} + 2318343 \nu^{12} + \cdots + 11061 ) / 7461 $ (-3128v^17 + 15231v^16 + 45308v^15 - 298314v^14 - 178614v^13 + 2318343v^12 - 277404v^11 - 9129351v^10 + 3631242v^9 + 19357486v^8 - 9104646v^7 - 21799537v^6 + 8861238v^5 + 12061176v^4 - 2650614v^3 - 2554596v^2 - 285156*v + 11061) / 7461
$\beta_{17}$	$=$	$ ( - 4376 \nu^{17} + 20288 \nu^{16} + 67021 \nu^{15} - 398895 \nu^{14} - 320397 \nu^{13} + 3114258 \nu^{12} + \cdots + 20817 ) / 7461 $ (-4376v^17 + 20288v^16 + 67021v^15 - 398895v^14 - 320397v^13 + 3114258v^12 + 147978v^11 - 12325434v^10 + 3064161v^9 + 26247685v^8 - 8851417v^7 - 29576411v^6 + 8827047v^5 + 16225848v^4 - 2488125v^3 - 3362814v^2 - 400167*v + 20817) / 7461

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + \beta_{2} + 4\beta _1 + 1 $ b3 + b2 + 4*b1 + 1
$\nu^{4}$	$=$	$ -\beta_{16} + \beta_{13} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 $ -b16 + b13 + b8 - b7 - b6 + b3 + 7*b2 + b1 + 15
$\nu^{5}$	$=$	$ \beta_{17} - \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} + \cdots + 11 $ b17 - b16 - b15 + b13 + b12 + b10 + b7 - b6 + b5 - b4 + 8b3 + 10b2 + 20*b1 + 11
$\nu^{6}$	$=$	$ \beta_{17} - 11 \beta_{16} - \beta_{15} + 10 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{10} + \cdots + 87 $ b17 - 11b16 - b15 + 10b13 + b12 + 2b11 + b10 - b9 + 9b8 - 9b7 - 10b6 - 2b4 + 11b3 + 46b2 + 11b1 + 87
$\nu^{7}$	$=$	$ 13 \beta_{17} - 15 \beta_{16} - 13 \beta_{15} + 14 \beta_{13} + 12 \beta_{12} + 2 \beta_{11} + 12 \beta_{10} + \cdots + 96 $ 13b17 - 15b16 - 13b15 + 14b13 + 12b12 + 2b11 + 12b10 - 2b9 + 2b8 + 9b7 - 12b6 + 10b5 - 13b4 + 56b3 + 81b2 + 111b1 + 96
$\nu^{8}$	$=$	$ 17 \beta_{17} - 95 \beta_{16} - 18 \beta_{15} + \beta_{14} + 81 \beta_{13} + 16 \beta_{12} + 25 \beta_{11} + \cdots + 541 $ 17b17 - 95b16 - 18b15 + b14 + 81b13 + 16b12 + 25b11 + 15b10 - 17b9 + 65b8 - 62b7 - 79b6 + b5 - 29b4 + 95b3 + 307b2 + 94*b1 + 541
$\nu^{9}$	$=$	$ 120 \beta_{17} - 161 \beta_{16} - 121 \beta_{15} + 2 \beta_{14} + 142 \beta_{13} + 106 \beta_{12} + \cdots + 771 $ 120b17 - 161b16 - 121b15 + 2b14 + 142b13 + 106b12 + 35b11 + 104b10 - 37b9 + 34b8 + 55b7 - 108b6 + 72b5 - 125b4 + 387b3 + 618b2 + 653*b1 + 771
$\nu^{10}$	$=$	$ 191 \beta_{17} - 760 \beta_{16} - 208 \beta_{15} + 16 \beta_{14} + 621 \beta_{13} + 171 \beta_{12} + \cdots + 3509 $ 191b17 - 760b16 - 208b15 + 16b14 + 621b13 + 171b12 + 229b11 + 155b10 - 192b9 + 447b8 - 394b7 - 576b6 + 15b5 - 301b4 + 755b3 + 2093b2 + 739*b1 + 3509
$\nu^{11}$	$=$	$ 981 \beta_{17} - 1490 \beta_{16} - 1004 \beta_{15} + 36 \beta_{14} + 1263 \beta_{13} + 843 \beta_{12} + \cdots + 5958 $ 981b17 - 1490b16 - 1004b15 + 36b14 + 1263b13 + 843b12 + 398b11 + 807b10 - 445b9 + 384b8 + 269b7 - 881b6 + 460b5 - 1078b4 + 2693b3 + 4605b2 + 4005*b1 + 5958
$\nu^{12}$	$=$	$ 1800 \beta_{17} - 5879 \beta_{16} - 1995 \beta_{15} + 174 \beta_{14} + 4676 \beta_{13} + 1557 \beta_{12} + \cdots + 23444 $ 1800b17 - 5879b16 - 1995b15 + 174b14 + 4676b13 + 1557b12 + 1886b11 + 1377b10 - 1831b9 + 3057b8 - 2446b7 - 4062b6 + 152b5 - 2724b4 + 5777b3 + 14519b2 + 5603*b1 + 23444
$\nu^{13}$	$=$	$ 7596 \beta_{17} - 12704 \beta_{16} - 7920 \beta_{15} + 423 \beta_{14} + 10484 \beta_{13} + 6417 \beta_{12} + \cdots + 45099 $ 7596b17 - 12704b16 - 7920b15 + 423b14 + 10484b13 + 6417b12 + 3783b11 + 5985b10 - 4440b9 + 3641b8 + 988b7 - 6866b6 + 2796b5 - 8805b4 + 18917b3 + 33938b2 + 25403*b1 + 45099
$\nu^{14}$	$=$	$ 15449 \beta_{17} - 44645 \beta_{16} - 17354 \beta_{15} + 1611 \beta_{14} + 34958 \beta_{13} + \cdots + 160117 $ 15449b17 - 44645b16 - 17354b15 + 1611b14 + 34958b13 + 13085b12 + 14832b11 + 11336b10 - 15978b9 + 21054b8 - 15199b7 - 28277b6 + 1334b5 - 22910b4 + 43342b3 + 102035b2 + 41719*b1 + 160117
$\nu^{15}$	$=$	$ 57224 \beta_{17} - 103030 \beta_{16} - 60878 \beta_{15} + 4113 \beta_{14} + 83564 \beta_{13} + \cdots + 337218 $ 57224b17 - 103030b16 - 60878b15 + 4113b14 + 83564b13 + 47900b12 + 32761b11 + 43544b10 - 40013b9 + 31437b8 + 1059b7 - 52175b6 + 16746b5 - 69619b4 + 134017b3 + 248720b2 + 165706*b1 + 337218
$\nu^{16}$	$=$	$ 125414 \beta_{17} - 335073 \beta_{16} - 142550 \beta_{15} + 13680 \beta_{14} + 260395 \beta_{13} + \cdots + 1111863 $ 125414b17 - 335073b16 - 142550b15 + 13680b14 + 260395b13 + 105099b12 + 113983b11 + 89439b10 - 132280b9 + 146437b8 - 95643b7 - 196290b6 + 11075b5 - 184337b4 + 321631b3 + 723972b2 + 307593*b1 + 1111863
$\nu^{17}$	$=$	$ 424612 \beta_{17} - 808918 \beta_{16} - 461019 \beta_{15} + 35975 \beta_{14} + 649323 \beta_{13} + \cdots + 2501963 $ 424612b17 - 808918b16 - 461019b15 + 35975b14 + 649323b13 + 354392b12 + 268769b11 + 314484b10 - 338935b9 + 256675b8 - 28411b7 - 390392b6 + 101189b5 - 538756b4 + 956104b3 + 1817402b2 + 1106438*b1 + 2501963

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

−2.33155
−2.26753
−1.93622
−1.55859
−1.12960
−0.956682
−0.352457
−0.159695
0.0750551
0.891553
1.04746
1.23218
1.31041
2.03424
2.33441
2.36771
2.69411
2.70519

−2.33155

3.43613

1.00000

−1.42557

−3.34841

−2.33155

1.2

−2.26753

3.14168

1.00000

0.777155

−2.58880

−2.26753

1.3

−1.93622

1.74896

1.00000

−0.704113

0.486063

−1.93622

1.4

−1.55859

0.429203

1.00000

4.77211

2.44823

−1.55859

1.5

−1.12960

−0.724011

1.00000

1.66257

3.07703

−1.12960

1.6

−0.956682

−1.08476

1.00000

1.77032

2.95113

−0.956682

1.7

−0.352457

−1.87577

1.00000

−0.865944

1.36604

−0.352457

1.8

−0.159695

−1.97450

1.00000

−2.32101

0.634707

−0.159695

1.9

0.0750551

−1.99437

1.00000

−4.31915

−0.299798

0.0750551

1.10

0.891553

−1.20513

1.00000

4.60717

−2.85755

0.891553

1.11

1.04746

−0.902824

1.00000

4.44267

−3.04060

1.04746

1.12

1.23218

−0.481722

1.00000

0.309652

−3.05794

1.23218

1.13

1.31041

−0.282837

1.00000

−2.01323

−2.99144

1.31041

1.14

2.03424

2.13815

1.00000

2.56652

0.281035

2.03424

1.15

2.33441

3.44947

1.00000

1.85122

3.38365

2.33441

1.16

2.36771

3.60605

1.00000

−3.86768

3.80266

2.36771

1.17

2.69411

5.25825

1.00000

2.33612

8.77809

2.69411

1.18

2.70519

5.31803

1.00000

2.42117

8.97588

2.70519

Atkin-Lehner signs

$ p $	Sign
$3$	$1$
$5$	$-1$

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	3645.2.a.j	yes	18
3.b	odd	2	1		3645.2.a.i	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3645.2.a.i	✓	18	3.b	odd	2	1
3645.2.a.j	yes	18	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{18} - 6 T_{2}^{17} - 9 T_{2}^{16} + 112 T_{2}^{15} - 51 T_{2}^{14} - 810 T_{2}^{13} + 936 T_{2}^{12} + \cdots + 9 $ T2^18 - 6*T2^17 - 9*T2^16 + 112*T2^15 - 51*T2^14 - 810*T2^13 + 936*T2^12 + 2844*T2^11 - 4536*T2^10 - 4934*T2^9 + 10173*T2^8 + 3672*T2^7 - 11144*T2^6 - 456*T2^5 + 5490*T2^4 - 354*T2^3 - 882*T2^2 - 54*T2 + 9 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3645))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} - 6 T^{17} + \cdots + 9 $ T^18 - 6T^17 - 9T^16 + 112T^15 - 51T^14 - 810T^13 + 936T^12 + 2844T^11 - 4536T^10 - 4934T^9 + 10173T^8 + 3672T^7 - 11144T^6 - 456T^5 + 5490T^4 - 354T^3 - 882T^2 - 54*T + 9
$3$	$ T^{18} $ T^18
$5$	$ (T - 1)^{18} $ (T - 1)^18
$7$	$ T^{18} - 12 T^{17} + \cdots - 126144 $ T^18 - 12T^17 + 3T^16 + 476T^15 - 1485T^14 - 5430T^13 + 30322T^12 + 4818T^11 - 224781T^10 + 220698T^9 + 701316T^8 - 1197270T^7 - 787107T^6 + 2304900T^5 - 46980T^4 - 1730016T^3 + 407808T^2 + 430272*T - 126144
$11$	$ T^{18} - 24 T^{17} + \cdots - 366687 $ T^18 - 24T^17 + 195T^16 - 272T^15 - 4761T^14 + 24336T^13 + 2973T^12 - 285084T^11 + 480888T^10 + 1093840T^9 - 3265593T^8 - 1136280T^7 + 8247133T^6 - 1267776T^5 - 8602767T^4 + 2395116T^3 + 3348783T^2 - 796068*T - 366687
$13$	$ T^{18} - 12 T^{17} + \cdots + 541 $ T^18 - 12T^17 - 30T^16 + 760T^15 - 513T^14 - 18000T^13 + 27863T^12 + 196248T^11 - 343257T^10 - 992024T^9 + 1515288T^8 + 2218800T^7 - 2305648T^6 - 1466496T^5 + 1314117T^4 - 7052T^3 - 92829T^2 + 8592*T + 541
$17$	$ T^{18} - 12 T^{17} + \cdots + 2436813 $ T^18 - 12T^17 - 57T^16 + 1066T^15 + 336T^14 - 36234T^13 + 33264T^12 + 607104T^11 - 745047T^10 - 5576042T^9 + 5759061T^8 + 29788116T^7 - 15161204T^6 - 88549890T^5 - 14415327T^4 + 107484702T^3 + 91844073T^2 + 26803170*T + 2436813
$19$	$ T^{18} + \cdots + 143557201 $ T^18 - 150T^16 + 58T^15 + 8811T^14 - 4122T^13 - 268183T^12 + 101268T^11 + 4614783T^10 - 806282T^9 - 45246204T^8 - 4866276T^7 + 238546007T^6 + 104429376T^5 - 579845043T^4 - 446782334T^3 + 457764498T^2 + 566507604T + 143557201
$23$	$ T^{18} + \cdots - 128982699 $ T^18 - 24T^17 + 120T^16 + 1554T^15 - 20043T^14 + 51534T^13 + 328560T^12 - 2098170T^11 + 688770T^10 + 21686778T^9 - 43431147T^8 - 73792296T^7 + 286329816T^6 - 14845356T^5 - 665963370T^4 + 351499014T^3 + 588265821T^2 - 336933594*T - 128982699
$29$	$ T^{18} + \cdots - 3062298303 $ T^18 - 24T^17 + 15T^16 + 4088T^15 - 31359T^14 - 147348T^13 + 2644893T^12 - 5825340T^11 - 59687037T^10 + 344136992T^9 - 92810640T^8 - 3620548860T^7 + 8596977241T^6 + 2357981076T^5 - 27416198601T^4 + 19645455048T^3 + 12404376783T^2 - 8142561396*T - 3062298303
$31$	$ T^{18} + 6 T^{17} + \cdots + 3405888 $ T^18 + 6T^17 - 261T^16 - 1666T^15 + 22725T^14 + 159246T^13 - 707576T^12 - 6140940T^11 + 2529954T^10 + 82928976T^9 + 122171553T^8 - 273415554T^7 - 970678503T^6 - 1039261104T^5 - 347543136T^4 + 164179872T^3 + 162981072T^2 + 43343424*T + 3405888
$37$	$ T^{18} + \cdots - 462226412267 $ T^18 - 6T^17 - 351T^16 + 2196T^15 + 46770T^14 - 295938T^13 - 3118437T^12 + 19421832T^11 + 114009030T^10 - 673875862T^9 - 2337619749T^8 + 12330097884T^7 + 26355542508T^6 - 111325536156T^5 - 152311605867T^4 + 437784456042T^3 + 395952784383T^2 - 547687749672*T - 462226412267
$41$	$ T^{18} + \cdots - 32501442396627 $ T^18 - 24T^17 - 180T^16 + 8528T^15 - 20898T^14 - 1098180T^13 + 7229940T^12 + 59383188T^11 - 650570481T^10 - 800407660T^9 + 25794867768T^8 - 41174376444T^7 - 444329970254T^6 + 1485612047304T^5 + 2249407302237T^4 - 13841061481152T^3 + 5376358767519T^2 + 32960498368464*T - 32501442396627
$43$	$ T^{18} + \cdots + 6246133919317 $ T^18 - 24T^17 - 132T^16 + 7540T^15 - 30837T^14 - 763632T^13 + 6857963T^12 + 21064746T^11 - 450269562T^10 + 875309242T^9 + 10921626348T^8 - 57297752526T^7 - 18708198364T^6 + 831223359948T^5 - 2138000138064T^4 + 74305162618T^3 + 7941593294439T^2 - 12665521137984*T + 6246133919317
$47$	$ T^{18} + \cdots - 147452956947 $ T^18 - 24T^17 - 186T^16 + 8830T^15 - 27066T^14 - 1062846T^13 + 8057535T^12 + 44101044T^11 - 610167321T^10 + 220782850T^9 + 17232598956T^8 - 45877462890T^7 - 142103581367T^6 + 528987996744T^5 + 462598265178T^4 - 1365722204928T^3 - 2028339312903T^2 - 945702082188*T - 147452956947
$53$	$ T^{18} + \cdots - 5114078939583 $ T^18 - 24T^17 - 159T^16 + 8056T^15 - 24168T^14 - 955962T^13 + 7004454T^12 + 42390738T^11 - 561264156T^10 + 111267988T^9 + 17777649018T^8 - 53603072250T^7 - 161632261805T^6 + 964729341582T^5 - 506668281219T^4 - 3744634590006T^3 + 4749690118014T^2 + 3269443357818*T - 5114078939583
$59$	$ T^{18} + \cdots + 18\!\cdots\!41 $ T^18 - 48T^17 + 327T^16 + 20320T^15 - 402183T^14 - 966204T^13 + 93529182T^12 - 675449388T^11 - 6383599308T^10 + 114483279652T^9 - 286432758633T^8 - 5126958376944T^7 + 43571850843772T^6 - 60786313192836T^5 - 749014200404190T^4 + 3991210679093580T^3 - 5796330956077860T^2 - 6117540551087940*T + 18190228704633441
$61$	$ T^{18} + \cdots - 25\!\cdots\!59 $ T^18 + 24T^17 - 438T^16 - 15320T^15 + 17451T^14 + 3579870T^13 + 17804510T^12 - 367246338T^11 - 3447842283T^10 + 13620765100T^9 + 253946863317T^8 + 257304335586T^7 - 7787615049028T^6 - 27810035988930T^5 + 69392129811846T^4 + 456538043713660T^3 + 123076676913972T^2 - 2150518084083144*T - 2585260898197559
$67$	$ T^{18} + \cdots - 72094667182983 $ T^18 + 6T^17 - 657T^16 - 3760T^15 + 174039T^14 + 936660T^13 - 23904767T^12 - 118436292T^11 + 1823354679T^10 + 8020808598T^9 - 77096488410T^8 - 281408999904T^7 + 1727654186130T^6 + 4582913196708T^5 - 18918233353449T^4 - 29435224799160T^3 + 76581124908768T^2 + 70028192339154*T - 72094667182983
$71$	$ T^{18} + \cdots - 529074984674403 $ T^18 - 48T^17 + 432T^16 + 13676T^15 - 276468T^14 - 421092T^13 + 44794566T^12 - 199658592T^11 - 2994881760T^10 + 23990158748T^9 + 84361126476T^8 - 1127569436208T^7 - 479561394323T^6 + 24990640652580T^5 - 17749285908219T^4 - 237914065179924T^3 + 226007913633930T^2 + 508615175096604*T - 529074984674403
$73$	$ T^{18} + \cdots - 998613683 $ T^18 - 6T^17 - 420T^16 + 1048T^15 + 57594T^14 + 27102T^13 - 2554084T^12 - 3130974T^11 + 51813432T^10 + 72872290T^9 - 538131720T^8 - 722003772T^7 + 2878687439T^6 + 3410451690T^5 - 7106467089T^4 - 7629955466T^3 + 6246565722T^2 + 5605629624*T - 998613683
$79$	$ T^{18} + \cdots - 790954750503 $ T^18 + 6T^17 - 753T^16 - 3700T^15 + 218061T^14 + 854232T^13 - 31082327T^12 - 96287910T^11 + 2304006411T^10 + 5689699272T^9 - 85380383556T^8 - 171476003070T^7 + 1345302883113T^6 + 2235378516732T^5 - 4933571048253T^4 - 8992240965900T^3 + 556442484867T^2 + 3356287234146*T - 790954750503
$83$	$ T^{18} + \cdots + 33896022129 $ T^18 - 60T^17 + 1107T^16 + 1302T^15 - 262305T^14 + 1939464T^13 + 18625518T^12 - 244894752T^11 - 423726795T^10 + 12211146306T^9 + 227203218T^8 - 296112371976T^7 - 93109860855T^6 + 3463111006632T^5 + 4900747372866T^4 - 8077817492928T^3 - 16706450023263T^2 - 4246986528288*T + 33896022129
$89$	$ T^{18} + \cdots + 15261639728409 $ T^18 - 24T^17 - 657T^16 + 17556T^15 + 151569T^14 - 4862052T^13 - 12679614T^12 + 626984388T^11 - 162292896T^10 - 37138958244T^9 + 66782084823T^8 + 881802873144T^7 - 1789590732912T^6 - 9485842741812T^5 + 14571763462026T^4 + 46431714727620T^3 - 29864246250960T^2 - 67025099907144*T + 15261639728409
$97$	$ T^{18} + \cdots - 333200929985747 $ T^18 - 6T^17 - 927T^16 + 2538T^15 + 353895T^14 + 73638T^13 - 70203177T^12 - 189161754T^11 + 7501328622T^10 + 35994287144T^9 - 401613908811T^8 - 2779100907678T^7 + 7995667199823T^6 + 89330680587480T^5 + 40436084022939T^4 - 846948695344254T^3 - 1064991339956103T^2 + 1383652590264270*T - 333200929985747
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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:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3645.2.a.j

Level	$3645$
Weight	$2$
Character orbit	3645.a
Self dual	yes
Analytic conductor	$29.105$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(29.1054715368\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - 6 x^{17} - 9 x^{16} + 112 x^{15} - 51 x^{14} - 810 x^{13} + 936 x^{12} + 2844 x^{11} - 4536 x^{10} + \cdots + 9 \) x^18 - 6x^17 - 9x^16 + 112x^15 - 51x^14 - 810x^13 + 936x^12 + 2844x^11 - 4536x^10 - 4934x^9 + 10173x^8 + 3672x^7 - 11144x^6 - 456x^5 + 5490x^4 - 354x^3 - 882x^2 - 54*x + 9
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{5} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( \nu^{3} - \nu^{2} - 4\nu + 2 \) v^3 - v^2 - 4*v + 2
\(\beta_{4}\)	\(=\)	\( ( 260 \nu^{17} - 1019 \nu^{16} - 4489 \nu^{15} + 20229 \nu^{14} + 28191 \nu^{13} - 159909 \nu^{12} + \cdots + 34029 ) / 7461 \) (260v^17 - 1019v^16 - 4489v^15 + 20229v^14 + 28191v^13 - 159909v^12 - 72663v^11 + 644193v^10 + 38247v^9 - 1407583v^8 + 114322v^7 + 1630061v^6 - 79611v^5 - 856863v^4 - 107322v^3 + 80712v^2 + 70281*v + 34029) / 7461
\(\beta_{5}\)	\(=\)	\( ( - 314 \nu^{17} + 1186 \nu^{16} + 5568 \nu^{15} - 23493 \nu^{14} - 35940 \nu^{13} + 183192 \nu^{12} + \cdots - 6489 ) / 7461 \) (-314v^17 + 1186v^16 + 5568v^15 - 23493v^14 - 35940v^13 + 183192v^12 + 88845v^11 - 707337v^10 + 43284v^9 + 1377325v^8 - 662657v^7 - 1148712v^6 + 1260999v^5 + 70770v^4 - 865035v^3 + 248256v^2 + 153702*v - 6489) / 7461
\(\beta_{6}\)	\(=\)	\( ( 1273 \nu^{17} - 6255 \nu^{16} - 18175 \nu^{15} + 121896 \nu^{14} + 68688 \nu^{13} - 940854 \nu^{12} + \cdots - 10071 ) / 7461 \) (1273v^17 - 6255v^16 - 18175v^15 + 121896v^14 + 68688v^13 - 940854v^12 + 132228v^11 + 3669384v^10 - 1495221v^9 - 7674794v^8 + 3598935v^7 + 8491703v^6 - 3356310v^5 - 4631352v^4 + 980490v^3 + 993321v^2 + 56772*v - 10071) / 7461
\(\beta_{7}\)	\(=\)	\( ( - 1368 \nu^{17} + 6994 \nu^{16} + 19321 \nu^{15} - 138231 \nu^{14} - 69471 \nu^{13} + 1089723 \nu^{12} + \cdots + 33651 ) / 7461 \) (-1368v^17 + 6994v^16 + 19321v^15 - 138231v^14 - 69471v^13 + 1089723v^12 - 176988v^11 - 4390062v^10 + 1744629v^9 + 9656481v^8 - 4145153v^7 - 11540648v^6 + 3868881v^5 + 7021512v^4 - 1126749v^3 - 1731033v^2 - 83169*v + 33651) / 7461
\(\beta_{8}\)	\(=\)	\( ( - 1439 \nu^{17} + 7014 \nu^{16} + 22106 \nu^{15} - 142062 \nu^{14} - 104115 \nu^{13} + 1156950 \nu^{12} + \cdots + 37530 ) / 7461 \) (-1439v^17 + 7014v^16 + 22106v^15 - 142062v^14 - 104115v^13 + 1156950v^12 + 15615v^11 - 4867965v^10 + 1238229v^9 + 11336596v^8 - 3579243v^7 - 14527561v^6 + 3643614v^5 + 9479463v^4 - 867930v^3 - 2463246v^2 - 359163*v + 37530) / 7461
\(\beta_{9}\)	\(=\)	\( ( 1534 \nu^{17} - 6924 \nu^{16} - 24910 \nu^{15} + 138501 \nu^{14} + 139716 \nu^{13} - 1106859 \nu^{12} + \cdots - 16344 ) / 7461 \) (1534v^17 - 6924v^16 - 24910v^15 + 138501v^14 + 139716v^13 - 1106859v^12 - 261834v^11 + 4524207v^10 - 261546v^9 - 10080209v^8 + 1420434v^7 + 12115883v^6 - 1273752v^5 - 7305978v^4 - 62682v^3 + 1753524v^2 + 310950*v - 16344) / 7461
\(\beta_{10}\)	\(=\)	\( ( 1554 \nu^{17} - 7385 \nu^{16} - 22577 \nu^{15} + 143118 \nu^{14} + 91188 \nu^{13} - 1096968 \nu^{12} + \cdots - 9135 ) / 7461 \) (1554v^17 - 7385v^16 - 22577v^15 + 143118v^14 + 91188v^13 - 1096968v^12 + 109644v^11 + 4242798v^10 - 1658211v^9 - 8797869v^8 + 4159954v^7 + 9679471v^6 - 4015002v^5 - 5301750v^4 + 1253103v^3 + 1167444v^2 + 56943*v - 9135) / 7461
\(\beta_{11}\)	\(=\)	\( ( - 1508 \nu^{17} + 7734 \nu^{16} + 20399 \nu^{15} - 150654 \nu^{14} - 58059 \nu^{13} + 1162245 \nu^{12} + \cdots - 9351 ) / 7461 \) (-1508v^17 + 7734v^16 + 20399v^15 - 150654v^14 - 58059v^13 + 1162245v^12 - 350022v^11 - 4529175v^10 + 2597928v^9 + 9451750v^8 - 6188850v^7 - 10367746v^6 + 6315147v^5 + 5461734v^4 - 2383818v^3 - 1023228v^2 + 60921*v - 9351) / 7461
\(\beta_{12}\)	\(=\)	\( ( 1643 \nu^{17} - 8566 \nu^{16} - 21024 \nu^{15} + 163788 \nu^{14} + 41370 \nu^{13} - 1226670 \nu^{12} + \cdots + 7650 ) / 7461 \) (1643v^17 - 8566v^16 - 21024v^15 + 163788v^14 + 41370v^13 - 1226670v^12 + 535884v^11 + 4542789v^10 - 3370035v^9 - 8610109v^8 + 7760720v^7 + 7676733v^6 - 8030091v^5 - 2276628v^4 + 3458052v^3 - 246852v^2 - 385857*v + 7650) / 7461
\(\beta_{13}\)	\(=\)	\( ( - 1784 \nu^{17} + 8956 \nu^{16} + 24348 \nu^{15} - 172587 \nu^{14} - 75282 \nu^{13} + 1310262 \nu^{12} + \cdots + 26955 ) / 7461 \) (-1784v^17 + 8956v^16 + 24348v^15 - 172587v^14 - 75282v^13 + 1310262v^12 - 337779v^11 - 4982064v^10 + 2642421v^9 + 10002577v^8 - 6071621v^7 - 10320921v^6 + 5730195v^5 + 4979334v^4 - 1936404v^3 - 873828v^2 + 69993*v + 26955) / 7461
\(\beta_{14}\)	\(=\)	\( ( - 1844 \nu^{17} + 9510 \nu^{16} + 23981 \nu^{15} - 183951 \nu^{14} - 49074 \nu^{13} + 1402452 \nu^{12} + \cdots + 35172 ) / 7461 \) (-1844v^17 + 9510v^16 + 23981v^15 - 183951v^14 - 49074v^13 + 1402452v^12 - 629016v^11 - 5353980v^10 + 4121586v^9 + 10756507v^8 - 9961143v^7 - 10931122v^6 + 10870065v^5 + 4855866v^4 - 4821810v^3 - 600327v^2 + 458964*v + 35172) / 7461
\(\beta_{15}\)	\(=\)	\( ( - 3050 \nu^{17} + 13516 \nu^{16} + 50013 \nu^{15} - 270111 \nu^{14} - 286797 \nu^{13} + 2156217 \nu^{12} + \cdots + 16047 ) / 7461 \) (-3050v^17 + 13516v^16 + 50013v^15 - 270111v^14 - 286797v^13 + 2156217v^12 + 585423v^11 - 8803536v^10 + 291981v^9 + 19600981v^8 - 2418785v^7 - 23552715v^6 + 2209251v^5 + 14146125v^4 + 131976v^3 - 3303342v^2 - 519984*v + 16047) / 7461
\(\beta_{16}\)	\(=\)	\( ( - 3128 \nu^{17} + 15231 \nu^{16} + 45308 \nu^{15} - 298314 \nu^{14} - 178614 \nu^{13} + 2318343 \nu^{12} + \cdots + 11061 ) / 7461 \) (-3128v^17 + 15231v^16 + 45308v^15 - 298314v^14 - 178614v^13 + 2318343v^12 - 277404v^11 - 9129351v^10 + 3631242v^9 + 19357486v^8 - 9104646v^7 - 21799537v^6 + 8861238v^5 + 12061176v^4 - 2650614v^3 - 2554596v^2 - 285156*v + 11061) / 7461
\(\beta_{17}\)	\(=\)	\( ( - 4376 \nu^{17} + 20288 \nu^{16} + 67021 \nu^{15} - 398895 \nu^{14} - 320397 \nu^{13} + 3114258 \nu^{12} + \cdots + 20817 ) / 7461 \) (-4376v^17 + 20288v^16 + 67021v^15 - 398895v^14 - 320397v^13 + 3114258v^12 + 147978v^11 - 12325434v^10 + 3064161v^9 + 26247685v^8 - 8851417v^7 - 29576411v^6 + 8827047v^5 + 16225848v^4 - 2488125v^3 - 3362814v^2 - 400167*v + 20817) / 7461

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) b3 + b2 + 4*b1 + 1
\(\nu^{4}\)	\(=\)	\( -\beta_{16} + \beta_{13} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) -b16 + b13 + b8 - b7 - b6 + b3 + 7*b2 + b1 + 15
\(\nu^{5}\)	\(=\)	\( \beta_{17} - \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} + \cdots + 11 \) b17 - b16 - b15 + b13 + b12 + b10 + b7 - b6 + b5 - b4 + 8b3 + 10b2 + 20*b1 + 11
\(\nu^{6}\)	\(=\)	\( \beta_{17} - 11 \beta_{16} - \beta_{15} + 10 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{10} + \cdots + 87 \) b17 - 11b16 - b15 + 10b13 + b12 + 2b11 + b10 - b9 + 9b8 - 9b7 - 10b6 - 2b4 + 11b3 + 46b2 + 11b1 + 87
\(\nu^{7}\)	\(=\)	\( 13 \beta_{17} - 15 \beta_{16} - 13 \beta_{15} + 14 \beta_{13} + 12 \beta_{12} + 2 \beta_{11} + 12 \beta_{10} + \cdots + 96 \) 13b17 - 15b16 - 13b15 + 14b13 + 12b12 + 2b11 + 12b10 - 2b9 + 2b8 + 9b7 - 12b6 + 10b5 - 13b4 + 56b3 + 81b2 + 111b1 + 96
\(\nu^{8}\)	\(=\)	\( 17 \beta_{17} - 95 \beta_{16} - 18 \beta_{15} + \beta_{14} + 81 \beta_{13} + 16 \beta_{12} + 25 \beta_{11} + \cdots + 541 \) 17b17 - 95b16 - 18b15 + b14 + 81b13 + 16b12 + 25b11 + 15b10 - 17b9 + 65b8 - 62b7 - 79b6 + b5 - 29b4 + 95b3 + 307b2 + 94*b1 + 541
\(\nu^{9}\)	\(=\)	\( 120 \beta_{17} - 161 \beta_{16} - 121 \beta_{15} + 2 \beta_{14} + 142 \beta_{13} + 106 \beta_{12} + \cdots + 771 \) 120b17 - 161b16 - 121b15 + 2b14 + 142b13 + 106b12 + 35b11 + 104b10 - 37b9 + 34b8 + 55b7 - 108b6 + 72b5 - 125b4 + 387b3 + 618b2 + 653*b1 + 771
\(\nu^{10}\)	\(=\)	\( 191 \beta_{17} - 760 \beta_{16} - 208 \beta_{15} + 16 \beta_{14} + 621 \beta_{13} + 171 \beta_{12} + \cdots + 3509 \) 191b17 - 760b16 - 208b15 + 16b14 + 621b13 + 171b12 + 229b11 + 155b10 - 192b9 + 447b8 - 394b7 - 576b6 + 15b5 - 301b4 + 755b3 + 2093b2 + 739*b1 + 3509
\(\nu^{11}\)	\(=\)	\( 981 \beta_{17} - 1490 \beta_{16} - 1004 \beta_{15} + 36 \beta_{14} + 1263 \beta_{13} + 843 \beta_{12} + \cdots + 5958 \) 981b17 - 1490b16 - 1004b15 + 36b14 + 1263b13 + 843b12 + 398b11 + 807b10 - 445b9 + 384b8 + 269b7 - 881b6 + 460b5 - 1078b4 + 2693b3 + 4605b2 + 4005*b1 + 5958
\(\nu^{12}\)	\(=\)	\( 1800 \beta_{17} - 5879 \beta_{16} - 1995 \beta_{15} + 174 \beta_{14} + 4676 \beta_{13} + 1557 \beta_{12} + \cdots + 23444 \) 1800b17 - 5879b16 - 1995b15 + 174b14 + 4676b13 + 1557b12 + 1886b11 + 1377b10 - 1831b9 + 3057b8 - 2446b7 - 4062b6 + 152b5 - 2724b4 + 5777b3 + 14519b2 + 5603*b1 + 23444
\(\nu^{13}\)	\(=\)	\( 7596 \beta_{17} - 12704 \beta_{16} - 7920 \beta_{15} + 423 \beta_{14} + 10484 \beta_{13} + 6417 \beta_{12} + \cdots + 45099 \) 7596b17 - 12704b16 - 7920b15 + 423b14 + 10484b13 + 6417b12 + 3783b11 + 5985b10 - 4440b9 + 3641b8 + 988b7 - 6866b6 + 2796b5 - 8805b4 + 18917b3 + 33938b2 + 25403*b1 + 45099
\(\nu^{14}\)	\(=\)	\( 15449 \beta_{17} - 44645 \beta_{16} - 17354 \beta_{15} + 1611 \beta_{14} + 34958 \beta_{13} + \cdots + 160117 \) 15449b17 - 44645b16 - 17354b15 + 1611b14 + 34958b13 + 13085b12 + 14832b11 + 11336b10 - 15978b9 + 21054b8 - 15199b7 - 28277b6 + 1334b5 - 22910b4 + 43342b3 + 102035b2 + 41719*b1 + 160117
\(\nu^{15}\)	\(=\)	\( 57224 \beta_{17} - 103030 \beta_{16} - 60878 \beta_{15} + 4113 \beta_{14} + 83564 \beta_{13} + \cdots + 337218 \) 57224b17 - 103030b16 - 60878b15 + 4113b14 + 83564b13 + 47900b12 + 32761b11 + 43544b10 - 40013b9 + 31437b8 + 1059b7 - 52175b6 + 16746b5 - 69619b4 + 134017b3 + 248720b2 + 165706*b1 + 337218
\(\nu^{16}\)	\(=\)	\( 125414 \beta_{17} - 335073 \beta_{16} - 142550 \beta_{15} + 13680 \beta_{14} + 260395 \beta_{13} + \cdots + 1111863 \) 125414b17 - 335073b16 - 142550b15 + 13680b14 + 260395b13 + 105099b12 + 113983b11 + 89439b10 - 132280b9 + 146437b8 - 95643b7 - 196290b6 + 11075b5 - 184337b4 + 321631b3 + 723972b2 + 307593*b1 + 1111863
\(\nu^{17}\)	\(=\)	\( 424612 \beta_{17} - 808918 \beta_{16} - 461019 \beta_{15} + 35975 \beta_{14} + 649323 \beta_{13} + \cdots + 2501963 \) 424612b17 - 808918b16 - 461019b15 + 35975b14 + 649323b13 + 354392b12 + 268769b11 + 314484b10 - 338935b9 + 256675b8 - 28411b7 - 390392b6 + 101189b5 - 538756b4 + 956104b3 + 1817402b2 + 1106438*b1 + 2501963

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.18
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} - 6 T^{17} + \cdots + 9 \) T^18 - 6T^17 - 9T^16 + 112T^15 - 51T^14 - 810T^13 + 936T^12 + 2844T^11 - 4536T^10 - 4934T^9 + 10173T^8 + 3672T^7 - 11144T^6 - 456T^5 + 5490T^4 - 354T^3 - 882T^2 - 54*T + 9
$3$	\( T^{18} \) T^18
$5$	\( (T - 1)^{18} \) (T - 1)^18
$7$	\( T^{18} - 12 T^{17} + \cdots - 126144 \) T^18 - 12T^17 + 3T^16 + 476T^15 - 1485T^14 - 5430T^13 + 30322T^12 + 4818T^11 - 224781T^10 + 220698T^9 + 701316T^8 - 1197270T^7 - 787107T^6 + 2304900T^5 - 46980T^4 - 1730016T^3 + 407808T^2 + 430272*T - 126144
$11$	\( T^{18} - 24 T^{17} + \cdots - 366687 \) T^18 - 24T^17 + 195T^16 - 272T^15 - 4761T^14 + 24336T^13 + 2973T^12 - 285084T^11 + 480888T^10 + 1093840T^9 - 3265593T^8 - 1136280T^7 + 8247133T^6 - 1267776T^5 - 8602767T^4 + 2395116T^3 + 3348783T^2 - 796068*T - 366687
$13$	\( T^{18} - 12 T^{17} + \cdots + 541 \) T^18 - 12T^17 - 30T^16 + 760T^15 - 513T^14 - 18000T^13 + 27863T^12 + 196248T^11 - 343257T^10 - 992024T^9 + 1515288T^8 + 2218800T^7 - 2305648T^6 - 1466496T^5 + 1314117T^4 - 7052T^3 - 92829T^2 + 8592*T + 541
$17$	\( T^{18} - 12 T^{17} + \cdots + 2436813 \) T^18 - 12T^17 - 57T^16 + 1066T^15 + 336T^14 - 36234T^13 + 33264T^12 + 607104T^11 - 745047T^10 - 5576042T^9 + 5759061T^8 + 29788116T^7 - 15161204T^6 - 88549890T^5 - 14415327T^4 + 107484702T^3 + 91844073T^2 + 26803170*T + 2436813
$19$	\( T^{18} + \cdots + 143557201 \) T^18 - 150T^16 + 58T^15 + 8811T^14 - 4122T^13 - 268183T^12 + 101268T^11 + 4614783T^10 - 806282T^9 - 45246204T^8 - 4866276T^7 + 238546007T^6 + 104429376T^5 - 579845043T^4 - 446782334T^3 + 457764498T^2 + 566507604T + 143557201
$23$	\( T^{18} + \cdots - 128982699 \) T^18 - 24T^17 + 120T^16 + 1554T^15 - 20043T^14 + 51534T^13 + 328560T^12 - 2098170T^11 + 688770T^10 + 21686778T^9 - 43431147T^8 - 73792296T^7 + 286329816T^6 - 14845356T^5 - 665963370T^4 + 351499014T^3 + 588265821T^2 - 336933594*T - 128982699
$29$	\( T^{18} + \cdots - 3062298303 \) T^18 - 24T^17 + 15T^16 + 4088T^15 - 31359T^14 - 147348T^13 + 2644893T^12 - 5825340T^11 - 59687037T^10 + 344136992T^9 - 92810640T^8 - 3620548860T^7 + 8596977241T^6 + 2357981076T^5 - 27416198601T^4 + 19645455048T^3 + 12404376783T^2 - 8142561396*T - 3062298303
$31$	\( T^{18} + 6 T^{17} + \cdots + 3405888 \) T^18 + 6T^17 - 261T^16 - 1666T^15 + 22725T^14 + 159246T^13 - 707576T^12 - 6140940T^11 + 2529954T^10 + 82928976T^9 + 122171553T^8 - 273415554T^7 - 970678503T^6 - 1039261104T^5 - 347543136T^4 + 164179872T^3 + 162981072T^2 + 43343424*T + 3405888
$37$	\( T^{18} + \cdots - 462226412267 \) T^18 - 6T^17 - 351T^16 + 2196T^15 + 46770T^14 - 295938T^13 - 3118437T^12 + 19421832T^11 + 114009030T^10 - 673875862T^9 - 2337619749T^8 + 12330097884T^7 + 26355542508T^6 - 111325536156T^5 - 152311605867T^4 + 437784456042T^3 + 395952784383T^2 - 547687749672*T - 462226412267
$41$	\( T^{18} + \cdots - 32501442396627 \) T^18 - 24T^17 - 180T^16 + 8528T^15 - 20898T^14 - 1098180T^13 + 7229940T^12 + 59383188T^11 - 650570481T^10 - 800407660T^9 + 25794867768T^8 - 41174376444T^7 - 444329970254T^6 + 1485612047304T^5 + 2249407302237T^4 - 13841061481152T^3 + 5376358767519T^2 + 32960498368464*T - 32501442396627
$43$	\( T^{18} + \cdots + 6246133919317 \) T^18 - 24T^17 - 132T^16 + 7540T^15 - 30837T^14 - 763632T^13 + 6857963T^12 + 21064746T^11 - 450269562T^10 + 875309242T^9 + 10921626348T^8 - 57297752526T^7 - 18708198364T^6 + 831223359948T^5 - 2138000138064T^4 + 74305162618T^3 + 7941593294439T^2 - 12665521137984*T + 6246133919317
$47$	\( T^{18} + \cdots - 147452956947 \) T^18 - 24T^17 - 186T^16 + 8830T^15 - 27066T^14 - 1062846T^13 + 8057535T^12 + 44101044T^11 - 610167321T^10 + 220782850T^9 + 17232598956T^8 - 45877462890T^7 - 142103581367T^6 + 528987996744T^5 + 462598265178T^4 - 1365722204928T^3 - 2028339312903T^2 - 945702082188*T - 147452956947
$53$	\( T^{18} + \cdots - 5114078939583 \) T^18 - 24T^17 - 159T^16 + 8056T^15 - 24168T^14 - 955962T^13 + 7004454T^12 + 42390738T^11 - 561264156T^10 + 111267988T^9 + 17777649018T^8 - 53603072250T^7 - 161632261805T^6 + 964729341582T^5 - 506668281219T^4 - 3744634590006T^3 + 4749690118014T^2 + 3269443357818*T - 5114078939583
$59$	\( T^{18} + \cdots + 18\!\cdots\!41 \) T^18 - 48T^17 + 327T^16 + 20320T^15 - 402183T^14 - 966204T^13 + 93529182T^12 - 675449388T^11 - 6383599308T^10 + 114483279652T^9 - 286432758633T^8 - 5126958376944T^7 + 43571850843772T^6 - 60786313192836T^5 - 749014200404190T^4 + 3991210679093580T^3 - 5796330956077860T^2 - 6117540551087940*T + 18190228704633441
$61$	\( T^{18} + \cdots - 25\!\cdots\!59 \) T^18 + 24T^17 - 438T^16 - 15320T^15 + 17451T^14 + 3579870T^13 + 17804510T^12 - 367246338T^11 - 3447842283T^10 + 13620765100T^9 + 253946863317T^8 + 257304335586T^7 - 7787615049028T^6 - 27810035988930T^5 + 69392129811846T^4 + 456538043713660T^3 + 123076676913972T^2 - 2150518084083144*T - 2585260898197559
$67$	\( T^{18} + \cdots - 72094667182983 \) T^18 + 6T^17 - 657T^16 - 3760T^15 + 174039T^14 + 936660T^13 - 23904767T^12 - 118436292T^11 + 1823354679T^10 + 8020808598T^9 - 77096488410T^8 - 281408999904T^7 + 1727654186130T^6 + 4582913196708T^5 - 18918233353449T^4 - 29435224799160T^3 + 76581124908768T^2 + 70028192339154*T - 72094667182983
$71$	\( T^{18} + \cdots - 529074984674403 \) T^18 - 48T^17 + 432T^16 + 13676T^15 - 276468T^14 - 421092T^13 + 44794566T^12 - 199658592T^11 - 2994881760T^10 + 23990158748T^9 + 84361126476T^8 - 1127569436208T^7 - 479561394323T^6 + 24990640652580T^5 - 17749285908219T^4 - 237914065179924T^3 + 226007913633930T^2 + 508615175096604*T - 529074984674403
$73$	\( T^{18} + \cdots - 998613683 \) T^18 - 6T^17 - 420T^16 + 1048T^15 + 57594T^14 + 27102T^13 - 2554084T^12 - 3130974T^11 + 51813432T^10 + 72872290T^9 - 538131720T^8 - 722003772T^7 + 2878687439T^6 + 3410451690T^5 - 7106467089T^4 - 7629955466T^3 + 6246565722T^2 + 5605629624*T - 998613683
$79$	\( T^{18} + \cdots - 790954750503 \) T^18 + 6T^17 - 753T^16 - 3700T^15 + 218061T^14 + 854232T^13 - 31082327T^12 - 96287910T^11 + 2304006411T^10 + 5689699272T^9 - 85380383556T^8 - 171476003070T^7 + 1345302883113T^6 + 2235378516732T^5 - 4933571048253T^4 - 8992240965900T^3 + 556442484867T^2 + 3356287234146*T - 790954750503
$83$	\( T^{18} + \cdots + 33896022129 \) T^18 - 60T^17 + 1107T^16 + 1302T^15 - 262305T^14 + 1939464T^13 + 18625518T^12 - 244894752T^11 - 423726795T^10 + 12211146306T^9 + 227203218T^8 - 296112371976T^7 - 93109860855T^6 + 3463111006632T^5 + 4900747372866T^4 - 8077817492928T^3 - 16706450023263T^2 - 4246986528288*T + 33896022129
$89$	\( T^{18} + \cdots + 15261639728409 \) T^18 - 24T^17 - 657T^16 + 17556T^15 + 151569T^14 - 4862052T^13 - 12679614T^12 + 626984388T^11 - 162292896T^10 - 37138958244T^9 + 66782084823T^8 + 881802873144T^7 - 1789590732912T^6 - 9485842741812T^5 + 14571763462026T^4 + 46431714727620T^3 - 29864246250960T^2 - 67025099907144*T + 15261639728409
$97$	\( T^{18} + \cdots - 333200929985747 \) T^18 - 6T^17 - 927T^16 + 2538T^15 + 353895T^14 + 73638T^13 - 70203177T^12 - 189161754T^11 + 7501328622T^10 + 35994287144T^9 - 401613908811T^8 - 2779100907678T^7 + 7995667199823T^6 + 89330680587480T^5 + 40436084022939T^4 - 846948695344254T^3 - 1064991339956103T^2 + 1383652590264270*T - 333200929985747
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\( p \)	Sign
\(3\)	\(1\)
\(5\)	\(-1\)