LMFDB - Newform orbit 4641.2.a.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4641 = 3 \cdot 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4641.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:	$37.0585715781$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 29 x^{15} + 26 x^{14} + 339 x^{13} - 266 x^{12} - 2047 x^{11} + 1356 x^{10} + \cdots + 18 $ x^17 - x^16 - 29x^15 + 26x^14 + 339x^13 - 266x^12 - 2047x^11 + 1356x^10 + 6792x^9 - 3573x^8 - 12125x^7 + 4456x^6 + 10521x^5 - 1858x^4 - 3551x^3 + 34x^2 + 323*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
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 $1,\beta_1,\ldots,\beta_{16}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{17} - x^{16} - 29 x^{15} + 26 x^{14} + 339 x^{13} - 266 x^{12} - 2047 x^{11} + 1356 x^{10} + \cdots + 18 $ x^17 - x^16 - 29*x^15 + 26*x^14 + 339*x^13 - 266*x^12 - 2047*x^11 + 1356*x^10 + 6792*x^9 - 3573*x^8 - 12125*x^7 + 4456*x^6 + 10521*x^5 - 1858*x^4 - 3551*x^3 + 34*x^2 + 323*x + 18 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 6\nu $ v^3 - 6*v
$\beta_{4}$	$=$	$ ( - 182363 \nu^{16} - 635232 \nu^{15} + 6314037 \nu^{14} + 15925689 \nu^{13} - 85417176 \nu^{12} + \cdots + 20926898 ) / 34598792 $ (-182363v^16 - 635232v^15 + 6314037v^14 + 15925689v^13 - 85417176v^12 - 153410128v^11 + 574078911v^10 + 713828025v^9 - 1998606291v^8 - 1677211886v^7 + 3369029929v^6 + 2010475901v^5 - 2190624084v^4 - 1399314554v^3 + 309928699v^2 + 419594857v + 20926898) / 34598792
$\beta_{5}$	$=$	$ ( 692887 \nu^{16} - 1390708 \nu^{15} - 20297337 \nu^{14} + 39425547 \nu^{13} + 237741260 \nu^{12} + \cdots - 155767906 ) / 34598792 $ (692887v^16 - 1390708v^15 - 20297337v^14 + 39425547v^13 + 237741260v^12 - 447357408v^11 - 1411430687v^10 + 2582725711v^9 + 4407146167v^8 - 7922139050v^7 - 6641859893v^6 + 12079502691v^5 + 3464959712v^4 - 7297203162v^3 + 124836205v^2 + 1130273211v - 155767906) / 34598792
$\beta_{6}$	$=$	$ ( - 423045 \nu^{16} + 737824 \nu^{15} + 10477388 \nu^{14} - 18208849 \nu^{13} - 97946412 \nu^{12} + \cdots + 90963824 ) / 17299396 $ (-423045v^16 + 737824v^15 + 10477388v^14 - 18208849v^13 - 97946412v^12 + 173680677v^11 + 420007317v^10 - 811902374v^9 - 754359126v^8 + 1977394680v^7 + 137109609v^6 - 2552296817v^5 + 855700275v^4 + 1729606451v^3 - 525645306v^2 - 349293924v + 90963824) / 17299396
$\beta_{7}$	$=$	$ ( 635478 \nu^{16} - 967674 \nu^{15} - 17958577 \nu^{14} + 26527272 \nu^{13} + 202125058 \nu^{12} + \cdots + 46631182 ) / 17299396 $ (635478v^16 - 967674v^15 - 17958577v^14 + 26527272v^13 + 202125058v^12 - 290920709v^11 - 1150434514v^10 + 1627972321v^9 + 3456707697v^8 - 4884174294v^7 - 5142556628v^6 + 7427053020v^5 + 3062418029v^4 - 4637970759v^3 - 506408405v^2 + 697731055v + 46631182) / 17299396
$\beta_{8}$	$=$	$ ( 1303891 \nu^{16} - 424866 \nu^{15} - 37778005 \nu^{14} + 11085539 \nu^{13} + 436467742 \nu^{12} + \cdots + 217121182 ) / 34598792 $ (1303891v^16 - 424866v^15 - 37778005v^14 + 11085539v^13 + 436467742v^12 - 118724556v^11 - 2553309369v^10 + 675621221v^9 + 7899508999v^8 - 2163990686v^7 - 12160109833v^6 + 3660922501v^5 + 7629647306v^4 - 2629218780v^3 - 1408008313v^2 + 700431403v + 217121182) / 34598792
$\beta_{9}$	$=$	$ ( 677433 \nu^{16} - 1435409 \nu^{15} - 17950299 \nu^{14} + 35723525 \nu^{13} + 189342297 \nu^{12} + \cdots + 59864298 ) / 17299396 $ (677433v^16 - 1435409v^15 - 17950299v^14 + 35723525v^13 + 189342297v^12 - 340633412v^11 - 1024683338v^10 + 1548350632v^9 + 3082176817v^8 - 3359468672v^7 - 5238155771v^6 + 2872134606v^5 + 4710451227v^4 - 212861191v^3 - 1546297220v^2 - 248624959v + 59864298) / 17299396
$\beta_{10}$	$=$	$ ( - 1487999 \nu^{16} + 2116602 \nu^{15} + 41661515 \nu^{14} - 52876099 \nu^{13} - 469276218 \nu^{12} + \cdots - 333493610 ) / 34598792 $ (-1487999v^16 + 2116602v^15 + 41661515v^14 - 52876099v^13 - 469276218v^12 + 510740530v^11 + 2736901125v^10 - 2393005083v^9 - 8869471105v^8 + 5554500714v^7 + 15844668969v^6 - 5665324489v^5 - 14173139620v^4 + 1580500642v^3 + 4669413407v^2 + 58645071v - 333493610) / 34598792
$\beta_{11}$	$=$	$ ( 890113 \nu^{16} - 1848408 \nu^{15} - 25201461 \nu^{14} + 50298183 \nu^{13} + 287053302 \nu^{12} + \cdots + 164926066 ) / 17299396 $ (890113v^16 - 1848408v^15 - 25201461v^14 + 50298183v^13 + 287053302v^12 - 542497424v^11 - 1684211681v^10 + 2944594579v^9 + 5401094085v^8 - 8402810002v^7 - 9206740559v^6 + 11840081529v^5 + 7423820830v^4 - 6627129518v^3 - 2257987903v^2 + 876370625v + 164926066) / 17299396
$\beta_{12}$	$=$	$ ( 1138463 \nu^{16} - 1238074 \nu^{15} - 31941509 \nu^{14} + 31700493 \nu^{13} + 357588194 \nu^{12} + \cdots + 174176538 ) / 17299396 $ (1138463v^16 - 1238074v^15 - 31941509v^14 + 31700493v^13 + 357588194v^12 - 319196958v^11 - 2039660379v^10 + 1608101971v^9 + 6277134995v^8 - 4260204586v^7 - 10154043025v^6 + 5631031305v^5 + 7787047246v^4 - 2978389830v^3 - 2319954655v^2 + 282683483v + 174176538) / 17299396
$\beta_{13}$	$=$	$ ( 1276182 \nu^{16} - 1030299 \nu^{15} - 37196105 \nu^{14} + 26997258 \nu^{13} + 435167065 \nu^{12} + \cdots + 193871046 ) / 17299396 $ (1276182v^16 - 1030299v^15 - 37196105v^14 + 26997258v^13 + 435167065v^12 - 281583623v^11 - 2609268291v^10 + 1490924220v^9 + 8473227223v^8 - 4203411192v^7 - 14412473368v^6 + 5914433689v^5 + 11344959736v^4 - 3263050046v^3 - 3238945204v^2 + 493402689v + 193871046) / 17299396
$\beta_{14}$	$=$	$ ( 640187 \nu^{16} - 1167581 \nu^{15} - 17092575 \nu^{14} + 29572871 \nu^{13} + 180025891 \nu^{12} + \cdots + 1124404 ) / 8649698 $ (640187v^16 - 1167581v^15 - 17092575v^14 + 29572871v^13 + 180025891v^12 - 290564082v^11 - 951449720v^10 + 1394158334v^9 + 2657744209v^8 - 3370511750v^7 - 3784613073v^6 + 3770627330v^5 + 2395122365v^4 - 1440908151v^3 - 426600076v^2 - 23685433v + 1124404) / 8649698
$\beta_{15}$	$=$	$ ( 2693881 \nu^{16} - 3916108 \nu^{15} - 74131217 \nu^{14} + 99720533 \nu^{13} + 810695188 \nu^{12} + \cdots + 218262630 ) / 34598792 $ (2693881v^16 - 3916108v^15 - 74131217v^14 + 99720533v^13 + 810695188v^12 - 991730034v^11 - 4493333329v^10 + 4872937155v^9 + 13336094307v^8 - 12317610370v^7 - 20506809719v^6 + 15038163389v^5 + 14332726626v^4 - 7264398784v^3 - 3283219271v^2 + 1035838955v + 218262630) / 34598792
$\beta_{16}$	$=$	$ ( - 2766512 \nu^{16} + 4280000 \nu^{15} + 77737369 \nu^{14} - 113386126 \nu^{13} - 873487896 \nu^{12} + \cdots - 94972362 ) / 17299396 $ (-2766512v^16 + 4280000v^15 + 77737369v^14 - 113386126v^13 - 873487896v^12 + 1187347343v^11 + 5010320418v^10 - 6245563495v^9 - 15490381733v^8 + 17277415138v^7 + 24828243438v^6 - 23707951512v^5 - 17681762345v^4 + 13103528687v^3 + 3672725155v^2 - 1858004099v - 94972362) / 17299396

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + 6\beta_1 $ b3 + 6*b1
$\nu^{4}$	$=$	$ -\beta_{13} + \beta_{12} + \beta_{8} + \beta_{6} + \beta_{4} + 9\beta_{2} + 16 $ -b13 + b12 + b8 + b6 + b4 + 9*b2 + 16
$\nu^{5}$	$=$	$ \beta_{15} - \beta_{14} + \beta_{10} + \beta_{9} + 10\beta_{3} + 40\beta_1 $ b15 - b14 + b10 + b9 + 10b3 + 40b1
$\nu^{6}$	$=$	$ - \beta_{16} - 11 \beta_{13} + 11 \beta_{12} - \beta_{11} + \beta_{10} + 10 \beta_{8} + \beta_{7} + \cdots + 98 $ -b16 - 11b13 + 11b12 - b11 + b10 + 10b8 + b7 + 12b6 - 2b5 + 11b4 - b3 + 72*b2 + b1 + 98
$\nu^{7}$	$=$	$ 12 \beta_{15} - 13 \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} + 13 \beta_{10} + 14 \beta_{9} + \cdots + 3 $ 12b15 - 13b14 - b13 + 2b12 - b11 + 13b10 + 14b9 + b7 + b6 + b5 + 84b3 + b2 + 282*b1 + 3
$\nu^{8}$	$=$	$ - 14 \beta_{16} + \beta_{15} - 94 \beta_{13} + 95 \beta_{12} - 14 \beta_{11} + 17 \beta_{10} + \cdots + 645 $ -14b16 + b15 - 94b13 + 95b12 - 14b11 + 17b10 + b9 + 80b8 + 17b7 + 111b6 - 32b5 + 96b4 - 15b3 + 557b2 + 17*b1 + 645
$\nu^{9}$	$=$	$ - \beta_{16} + 107 \beta_{15} - 126 \beta_{14} - 19 \beta_{13} + 39 \beta_{12} - 16 \beta_{11} + \cdots + 58 $ -b16 + 107b15 - 126b14 - 19b13 + 39b12 - 16b11 + 128b10 + 142b9 + 5b8 + 9b7 + 18b6 + 20b5 + 8b4 + 670b3 + 24b2 + 2052*b1 + 58
$\nu^{10}$	$=$	$ - 142 \beta_{16} + 16 \beta_{15} + \beta_{14} - 739 \beta_{13} + 753 \beta_{12} - 143 \beta_{11} + \cdots + 4436 $ -142b16 + 16b15 + b14 - 739b13 + 753b12 - 143b11 + 195b10 + 20b9 + 602b8 + 200b7 + 936b6 - 357b5 + 778b4 - 152b3 + 4253b2 + 207*b1 + 4436
$\nu^{11}$	$=$	$ - 20 \beta_{16} + 862 \beta_{15} - 1096 \beta_{14} - 244 \beta_{13} + 504 \beta_{12} - 173 \beta_{11} + \cdots + 756 $ -20b16 + 862b15 - 1096b14 - 244b13 + 504b12 - 173b11 + 1144b10 + 1275b9 + 107b8 + 24b7 + 224b6 + 261b5 + 162b4 + 5232b3 + 358b2 + 15206b1 + 756
$\nu^{12}$	$=$	$ - 1275 \beta_{16} + 170 \beta_{15} + 17 \beta_{14} - 5614 \beta_{13} + 5735 \beta_{12} - 1295 \beta_{11} + \cdots + 31380 $ -1275b16 + 170b15 + 17b14 - 5614b13 + 5735b12 - 1295b11 + 1894b10 + 265b9 + 4449b8 + 2024b7 + 7555b6 - 3447b5 + 6106b4 - 1303b3 + 32303b2 + 2207b1 + 31380
$\nu^{13}$	$=$	$ - 265 \beta_{16} + 6657 \beta_{15} - 9061 \beta_{14} - 2652 \beta_{13} + 5450 \beta_{12} - 1586 \beta_{11} + \cdots + 8335 $ -265b16 + 6657b15 - 9061b14 - 2652b13 + 5450b12 - 1586b11 + 9765b10 + 10785b9 + 1474b8 - 451b7 + 2376b6 + 2820b5 + 2126b4 + 40449b3 + 4318b2 + 113913b1 + 8335
$\nu^{14}$	$=$	$ - 10785 \beta_{16} + 1534 \beta_{15} + 168 \beta_{14} - 42030 \beta_{13} + 42777 \beta_{12} + \cdots + 226156 $ -10785b16 + 1534b15 + 168b14 - 42030b13 + 42777b12 - 11041b11 + 16853b10 + 2940b9 + 32771b8 + 18930b7 + 59552b6 - 30915b5 + 47121b4 - 10172b3 + 244901b2 + 21894b1 + 226156
$\nu^{15}$	$=$	$ - 2940 \beta_{16} + 50464 \beta_{15} - 72945 \beta_{14} - 26309 \beta_{13} + 53438 \beta_{12} + \cdots + 83783 $ -2940b16 + 50464b15 - 72945b14 - 26309b13 + 53438b12 - 13322b11 + 81182b10 + 88264b9 + 16717b8 - 9535b7 + 23046b6 + 27436b5 + 23043b4 + 311136b3 + 46236b2 + 859178b1 + 83783
$\nu^{16}$	$=$	$ - 88264 \beta_{16} + 12812 \beta_{15} + 1127 \beta_{14} - 312931 \beta_{13} + 315449 \beta_{12} + \cdots + 1650863 $ -88264b16 + 12812b15 + 1127b14 - 312931b13 + 315449b12 - 90915b11 + 142512b10 + 29591b9 + 241894b8 + 168832b7 + 462949b6 - 265694b5 + 359976b4 - 74579b3 + 1856252b2 + 207313b1 + 1650863

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

−2.74789
−2.65153
−2.20635
−1.87087
−1.83369
−1.01973
−0.524651
−0.396729
−0.0581595
0.370016
0.744396
1.52099
1.63714
2.10320
2.35965
2.77865
2.79555

−2.74789

−1.00000

5.55089

3.19340

2.74789

−1.00000

−9.75745

1.00000

−8.77510

1.2

−2.65153

−1.00000

5.03063

−3.48911

2.65153

−1.00000

−8.03583

1.00000

9.25150

1.3

−2.20635

−1.00000

2.86797

0.538411

2.20635

−1.00000

−1.91505

1.00000

−1.18792

1.4

−1.87087

−1.00000

1.50015

−3.86075

1.87087

−1.00000

0.935160

1.00000

7.22294

1.5

−1.83369

−1.00000

1.36241

1.21290

1.83369

−1.00000

1.16914

1.00000

−2.22409

1.6

−1.01973

−1.00000

−0.960160

3.04739

1.01973

−1.00000

3.01855

1.00000

−3.10751

1.7

−0.524651

−1.00000

−1.72474

−2.36230

0.524651

−1.00000

1.95419

1.00000

1.23938

1.8

−0.396729

−1.00000

−1.84261

−1.99480

0.396729

−1.00000

1.52447

1.00000

0.791394

1.9

−0.0581595

−1.00000

−1.99662

−0.0629841

0.0581595

−1.00000

0.232441

1.00000

0.00366313

1.10

0.370016

−1.00000

−1.86309

3.67290

−0.370016

−1.00000

−1.42940

1.00000

1.35903

1.11

0.744396

−1.00000

−1.44587

−1.18048

−0.744396

−1.00000

−2.56510

1.00000

−0.878746

1.12

1.52099

−1.00000

0.313402

0.556831

−1.52099

−1.00000

−2.56529

1.00000

0.846933

1.13

1.63714

−1.00000

0.680235

−3.04998

−1.63714

−1.00000

−2.16064

1.00000

−4.99325

1.14

2.10320

−1.00000

2.42343

2.55731

−2.10320

−1.00000

0.890557

1.00000

5.37851

1.15

2.35965

−1.00000

3.56795

−4.35328

−2.35965

−1.00000

3.69982

1.00000

−10.2722

1.16

2.77865

−1.00000

5.72089

3.32548

−2.77865

−1.00000

10.3390

1.00000

9.24035

1.17

2.79555

−1.00000

5.81513

−1.75095

−2.79555

−1.00000

10.6654

1.00000

−4.89488

Atkin-Lehner signs

$ p $	Sign
$3$	$1$
$7$	$1$
$13$	$-1$
$17$	$1$

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	4641.2.a.ba	✓	17

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4641.2.a.ba	✓	17	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4641))$:

$ T_{2}^{17} - T_{2}^{16} - 29 T_{2}^{15} + 26 T_{2}^{14} + 339 T_{2}^{13} - 266 T_{2}^{12} - 2047 T_{2}^{11} + \cdots + 18 $

$ T_{5}^{17} + 4 T_{5}^{16} - 53 T_{5}^{15} - 209 T_{5}^{14} + 1127 T_{5}^{13} + 4405 T_{5}^{12} + \cdots + 12128 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{17} - T^{16} + \cdots + 18 $ T^17 - T^16 - 29T^15 + 26T^14 + 339T^13 - 266T^12 - 2047T^11 + 1356T^10 + 6792T^9 - 3573T^8 - 12125T^7 + 4456T^6 + 10521T^5 - 1858T^4 - 3551T^3 + 34T^2 + 323*T + 18
$3$	$ (T + 1)^{17} $ (T + 1)^17
$5$	$ T^{17} + 4 T^{16} + \cdots + 12128 $ T^17 + 4T^16 - 53T^15 - 209T^14 + 1127T^13 + 4405T^12 - 12160T^11 - 47740T^10 + 68943T^9 + 279570T^8 - 190646T^7 - 843504T^6 + 219424T^5 + 1121030T^4 - 191216T^3 - 524176T^2 + 160576T + 12128
$7$	$ (T + 1)^{17} $ (T + 1)^17
$11$	$ T^{17} - 4 T^{16} + \cdots - 309248 $ T^17 - 4T^16 - 115T^15 + 426T^14 + 5098T^13 - 16925T^12 - 110158T^11 + 313772T^10 + 1196216T^9 - 2770808T^8 - 6066140T^7 + 10935760T^6 + 13372352T^5 - 18410464T^4 - 8545792T^3 + 12156928T^2 - 1517568T - 309248
$13$	$ (T - 1)^{17} $ (T - 1)^17
$17$	$ (T + 1)^{17} $ (T + 1)^17
$19$	$ T^{17} + \cdots + 394938368 $ T^17 - 13T^16 - 155T^15 + 2686T^14 + 4781T^13 - 204702T^12 + 349074T^11 + 6879593T^10 - 25899165T^9 - 89085876T^8 + 565734512T^7 - 18157584T^6 - 4204791696T^5 + 6298708048T^4 + 3068231040T^3 - 10618463488T^2 + 4517445632T + 394938368
$23$	$ T^{17} + \cdots + 1701568512 $ T^17 - 8T^16 - 280T^15 + 2183T^14 + 31073T^13 - 231555T^12 - 1768177T^11 + 12154394T^10 + 56059435T^9 - 333067378T^8 - 997590512T^7 + 4527580784T^6 + 9198103664T^5 - 23755072544T^4 - 33342155264T^3 - 3203046400T^2 + 6690811904T + 1701568512
$29$	$ T^{17} + \cdots + 52921496864 $ T^17 + 3T^16 - 314T^15 - 987T^14 + 39043T^13 + 120396T^12 - 2472375T^11 - 6924934T^10 + 85836567T^9 + 197546152T^8 - 1626354394T^7 - 2664288088T^6 + 15394879840T^5 + 15077850066T^4 - 56966295536T^3 - 43861724240T^2 + 65198418112T + 52921496864
$31$	$ T^{17} + \cdots + 127975866368 $ T^17 - 10T^16 - 280T^15 + 3038T^14 + 28091T^13 - 353211T^12 - 1129643T^11 + 19692496T^10 + 7781492T^9 - 546305828T^8 + 578425020T^7 + 7345061308T^6 - 14222186808T^5 - 40496138960T^4 + 110049881600T^3 + 31112383488T^2 - 223185100800T + 127975866368
$37$	$ T^{17} + \cdots + 1209986145536 $ T^17 - 354T^15 - 167T^14 + 51556T^13 + 55487T^12 - 4011036T^11 - 6972476T^10 + 178902060T^9 + 432731568T^8 - 4499492614T^7 - 14083881604T^6 + 56694518312T^5 + 229062480240T^4 - 224674914080T^3 - 1511077955776T^2 - 803192696960*T + 1209986145536
$41$	$ T^{17} + \cdots - 215269310016 $ T^17 + 12T^16 - 317T^15 - 3759T^14 + 40586T^13 + 473792T^12 - 2662020T^11 - 30872569T^10 + 91332002T^9 + 1108929688T^8 - 1354845526T^7 - 21407241544T^6 - 1721679282T^5 + 189709056908T^4 + 204663874800T^3 - 383208918368T^2 - 616308884384T - 215269310016
$43$	$ T^{17} + \cdots + 242460229632 $ T^17 - 29T^16 + 42T^15 + 6646T^14 - 66881T^13 - 240830T^12 + 7014987T^11 - 27008971T^10 - 154141145T^9 + 1450715812T^8 - 1754650648T^7 - 18499354912T^6 + 71914259264T^5 - 13073701504T^4 - 416067123200T^3 + 926312345600T^2 - 792369971200T + 242460229632
$47$	$ T^{17} + \cdots - 21342578176 $ T^17 + 4T^16 - 368T^15 - 1302T^14 + 50117T^13 + 147937T^12 - 3341535T^11 - 8097004T^10 + 117601306T^9 + 245201784T^8 - 2162939430T^7 - 4458240866T^6 + 18942567020T^5 + 45922257352T^4 - 49193369152T^3 - 200172027136T^2 - 152858832128T - 21342578176
$53$	$ T^{17} + \cdots - 391066812416 $ T^17 + 8T^16 - 328T^15 - 2422T^14 + 42823T^13 + 281915T^12 - 2887967T^11 - 16174578T^10 + 108539580T^9 + 486144544T^8 - 2298296272T^7 - 7471951520T^6 + 27515785792T^5 + 50279734784T^4 - 186592526336T^3 - 75220008960T^2 + 556734152704T - 391066812416
$59$	$ T^{17} + \cdots - 728570689536 $ T^17 - 25T^16 - 177T^15 + 8256T^14 - 8477T^13 - 1050155T^12 + 3757202T^11 + 65850558T^10 - 312081056T^9 - 2120393402T^8 + 11234692030T^7 + 32433566572T^6 - 174977906872T^5 - 194482074272T^4 + 830128152576T^3 + 883054034176T^2 - 669073720832T - 728570689536
$61$	$ T^{17} + \cdots + 78610370593728 $ T^17 + 5T^16 - 458T^15 - 2486T^14 + 84143T^13 + 483394T^12 - 8134606T^11 - 48962043T^10 + 451537220T^9 + 2843442204T^8 - 14630773752T^7 - 96958457160T^6 + 267328750630T^5 + 1891441741132T^4 - 2482571123920T^3 - 19275841780896T^2 + 8761411196128T + 78610370593728
$67$	$ T^{17} + \cdots - 2573929852928 $ T^17 - 15T^16 - 448T^15 + 6317T^14 + 82191T^13 - 1025339T^12 - 7934294T^11 + 81830298T^10 + 430604056T^9 - 3478830722T^8 - 13547068542T^7 + 79607917912T^6 + 244067350824T^5 - 918641707040T^4 - 2317639362432T^3 + 4280403953152T^2 + 8895949698048T - 2573929852928
$71$	$ T^{17} + \cdots - 920071883563008 $ T^17 - 32T^16 - 332T^15 + 21171T^14 - 83250T^13 - 4865239T^12 + 50447460T^11 + 399171280T^10 - 7928025688T^9 + 6774425712T^8 + 488240187824T^7 - 2523454802752T^6 - 7258710867968T^5 + 94423886799232T^4 - 222765148786688T^3 - 192190187290624T^2 + 1235135848325120T - 920071883563008
$73$	$ T^{17} + \cdots - 202589086278656 $ T^17 + 15T^16 - 582T^15 - 8407T^14 + 131883T^13 + 1790501T^12 - 15548519T^11 - 186496416T^10 + 1106873292T^9 + 10374588536T^8 - 50900323296T^7 - 307592148944T^6 + 1486511596064T^5 + 4267157852352T^4 - 24665789872128T^3 - 11198280789760T^2 + 173593748448768T - 202589086278656
$79$	$ T^{17} + \cdots - 62725724831744 $ T^17 - 27T^16 - 629T^15 + 21601T^14 + 97628T^13 - 6227871T^12 + 6781461T^11 + 804764308T^10 - 2490343104T^9 - 49754163944T^8 + 175480946000T^7 + 1436365591120T^6 - 4343478126912T^5 - 19184369757952T^4 + 39225614880768T^3 + 94726572605440T^2 - 109688933122048T - 62725724831744
$83$	$ T^{17} + \cdots + 16457777289216 $ T^17 + 24T^16 - 565T^15 - 18571T^14 + 42904T^13 + 4942570T^12 + 25232167T^11 - 491618330T^10 - 5075564626T^9 + 8251311084T^8 + 269868498424T^7 + 609771443102T^6 - 4665603171440T^5 - 20021043003712T^4 + 13899703165952T^3 + 152006907563392T^2 + 151723028409344T + 16457777289216
$89$	$ T^{17} + \cdots - 2616109361152 $ T^17 + 8T^16 - 930T^15 - 8562T^14 + 327315T^13 + 3476075T^12 - 52347195T^11 - 671344404T^10 + 3195855744T^9 + 61358764104T^8 + 52639139104T^7 - 2013507693952T^6 - 9644163324928T^5 - 13799568790272T^4 + 3461033867520T^3 + 15634932220928T^2 - 463257153536T - 2616109361152
$97$	$ T^{17} + \cdots - 2639165403136 $ T^17 + 22T^16 - 798T^15 - 18138T^14 + 227947T^13 + 5250263T^12 - 32249479T^11 - 669569522T^10 + 2818173084T^9 + 37946710576T^8 - 150532510896T^7 - 782692642960T^6 + 3117148215488T^5 + 4741064948096T^4 - 20756327777792T^3 + 8145946161408T^2 + 5621056407552T - 2639165403136
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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:	\( N \)	\(=\)	\( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4641.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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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	4641.2.a.ba

Level	$4641$
Weight	$2$
Character orbit	4641.a
Self dual	yes
Analytic conductor	$37.059$
Analytic rank	$0$
Dimension	$17$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(37.0585715781\)
Analytic rank:	\(0\)
Dimension:	\(17\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{17} - x^{16} - 29 x^{15} + 26 x^{14} + 339 x^{13} - 266 x^{12} - 2047 x^{11} + 1356 x^{10} + \cdots + 18 \) x^17 - x^16 - 29x^15 + 26x^14 + 339x^13 - 266x^12 - 2047x^11 + 1356x^10 + 6792x^9 - 3573x^8 - 12125x^7 + 4456x^6 + 10521x^5 - 1858x^4 - 3551x^3 + 34x^2 + 323*x + 18
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 6\nu \) v^3 - 6*v
\(\beta_{4}\)	\(=\)	\( ( - 182363 \nu^{16} - 635232 \nu^{15} + 6314037 \nu^{14} + 15925689 \nu^{13} - 85417176 \nu^{12} + \cdots + 20926898 ) / 34598792 \) (-182363v^16 - 635232v^15 + 6314037v^14 + 15925689v^13 - 85417176v^12 - 153410128v^11 + 574078911v^10 + 713828025v^9 - 1998606291v^8 - 1677211886v^7 + 3369029929v^6 + 2010475901v^5 - 2190624084v^4 - 1399314554v^3 + 309928699v^2 + 419594857v + 20926898) / 34598792
\(\beta_{5}\)	\(=\)	\( ( 692887 \nu^{16} - 1390708 \nu^{15} - 20297337 \nu^{14} + 39425547 \nu^{13} + 237741260 \nu^{12} + \cdots - 155767906 ) / 34598792 \) (692887v^16 - 1390708v^15 - 20297337v^14 + 39425547v^13 + 237741260v^12 - 447357408v^11 - 1411430687v^10 + 2582725711v^9 + 4407146167v^8 - 7922139050v^7 - 6641859893v^6 + 12079502691v^5 + 3464959712v^4 - 7297203162v^3 + 124836205v^2 + 1130273211v - 155767906) / 34598792
\(\beta_{6}\)	\(=\)	\( ( - 423045 \nu^{16} + 737824 \nu^{15} + 10477388 \nu^{14} - 18208849 \nu^{13} - 97946412 \nu^{12} + \cdots + 90963824 ) / 17299396 \) (-423045v^16 + 737824v^15 + 10477388v^14 - 18208849v^13 - 97946412v^12 + 173680677v^11 + 420007317v^10 - 811902374v^9 - 754359126v^8 + 1977394680v^7 + 137109609v^6 - 2552296817v^5 + 855700275v^4 + 1729606451v^3 - 525645306v^2 - 349293924v + 90963824) / 17299396
\(\beta_{7}\)	\(=\)	\( ( 635478 \nu^{16} - 967674 \nu^{15} - 17958577 \nu^{14} + 26527272 \nu^{13} + 202125058 \nu^{12} + \cdots + 46631182 ) / 17299396 \) (635478v^16 - 967674v^15 - 17958577v^14 + 26527272v^13 + 202125058v^12 - 290920709v^11 - 1150434514v^10 + 1627972321v^9 + 3456707697v^8 - 4884174294v^7 - 5142556628v^6 + 7427053020v^5 + 3062418029v^4 - 4637970759v^3 - 506408405v^2 + 697731055v + 46631182) / 17299396
\(\beta_{8}\)	\(=\)	\( ( 1303891 \nu^{16} - 424866 \nu^{15} - 37778005 \nu^{14} + 11085539 \nu^{13} + 436467742 \nu^{12} + \cdots + 217121182 ) / 34598792 \) (1303891v^16 - 424866v^15 - 37778005v^14 + 11085539v^13 + 436467742v^12 - 118724556v^11 - 2553309369v^10 + 675621221v^9 + 7899508999v^8 - 2163990686v^7 - 12160109833v^6 + 3660922501v^5 + 7629647306v^4 - 2629218780v^3 - 1408008313v^2 + 700431403v + 217121182) / 34598792
\(\beta_{9}\)	\(=\)	\( ( 677433 \nu^{16} - 1435409 \nu^{15} - 17950299 \nu^{14} + 35723525 \nu^{13} + 189342297 \nu^{12} + \cdots + 59864298 ) / 17299396 \) (677433v^16 - 1435409v^15 - 17950299v^14 + 35723525v^13 + 189342297v^12 - 340633412v^11 - 1024683338v^10 + 1548350632v^9 + 3082176817v^8 - 3359468672v^7 - 5238155771v^6 + 2872134606v^5 + 4710451227v^4 - 212861191v^3 - 1546297220v^2 - 248624959v + 59864298) / 17299396
\(\beta_{10}\)	\(=\)	\( ( - 1487999 \nu^{16} + 2116602 \nu^{15} + 41661515 \nu^{14} - 52876099 \nu^{13} - 469276218 \nu^{12} + \cdots - 333493610 ) / 34598792 \) (-1487999v^16 + 2116602v^15 + 41661515v^14 - 52876099v^13 - 469276218v^12 + 510740530v^11 + 2736901125v^10 - 2393005083v^9 - 8869471105v^8 + 5554500714v^7 + 15844668969v^6 - 5665324489v^5 - 14173139620v^4 + 1580500642v^3 + 4669413407v^2 + 58645071v - 333493610) / 34598792
\(\beta_{11}\)	\(=\)	\( ( 890113 \nu^{16} - 1848408 \nu^{15} - 25201461 \nu^{14} + 50298183 \nu^{13} + 287053302 \nu^{12} + \cdots + 164926066 ) / 17299396 \) (890113v^16 - 1848408v^15 - 25201461v^14 + 50298183v^13 + 287053302v^12 - 542497424v^11 - 1684211681v^10 + 2944594579v^9 + 5401094085v^8 - 8402810002v^7 - 9206740559v^6 + 11840081529v^5 + 7423820830v^4 - 6627129518v^3 - 2257987903v^2 + 876370625v + 164926066) / 17299396
\(\beta_{12}\)	\(=\)	\( ( 1138463 \nu^{16} - 1238074 \nu^{15} - 31941509 \nu^{14} + 31700493 \nu^{13} + 357588194 \nu^{12} + \cdots + 174176538 ) / 17299396 \) (1138463v^16 - 1238074v^15 - 31941509v^14 + 31700493v^13 + 357588194v^12 - 319196958v^11 - 2039660379v^10 + 1608101971v^9 + 6277134995v^8 - 4260204586v^7 - 10154043025v^6 + 5631031305v^5 + 7787047246v^4 - 2978389830v^3 - 2319954655v^2 + 282683483v + 174176538) / 17299396
\(\beta_{13}\)	\(=\)	\( ( 1276182 \nu^{16} - 1030299 \nu^{15} - 37196105 \nu^{14} + 26997258 \nu^{13} + 435167065 \nu^{12} + \cdots + 193871046 ) / 17299396 \) (1276182v^16 - 1030299v^15 - 37196105v^14 + 26997258v^13 + 435167065v^12 - 281583623v^11 - 2609268291v^10 + 1490924220v^9 + 8473227223v^8 - 4203411192v^7 - 14412473368v^6 + 5914433689v^5 + 11344959736v^4 - 3263050046v^3 - 3238945204v^2 + 493402689v + 193871046) / 17299396
\(\beta_{14}\)	\(=\)	\( ( 640187 \nu^{16} - 1167581 \nu^{15} - 17092575 \nu^{14} + 29572871 \nu^{13} + 180025891 \nu^{12} + \cdots + 1124404 ) / 8649698 \) (640187v^16 - 1167581v^15 - 17092575v^14 + 29572871v^13 + 180025891v^12 - 290564082v^11 - 951449720v^10 + 1394158334v^9 + 2657744209v^8 - 3370511750v^7 - 3784613073v^6 + 3770627330v^5 + 2395122365v^4 - 1440908151v^3 - 426600076v^2 - 23685433v + 1124404) / 8649698
\(\beta_{15}\)	\(=\)	\( ( 2693881 \nu^{16} - 3916108 \nu^{15} - 74131217 \nu^{14} + 99720533 \nu^{13} + 810695188 \nu^{12} + \cdots + 218262630 ) / 34598792 \) (2693881v^16 - 3916108v^15 - 74131217v^14 + 99720533v^13 + 810695188v^12 - 991730034v^11 - 4493333329v^10 + 4872937155v^9 + 13336094307v^8 - 12317610370v^7 - 20506809719v^6 + 15038163389v^5 + 14332726626v^4 - 7264398784v^3 - 3283219271v^2 + 1035838955v + 218262630) / 34598792
\(\beta_{16}\)	\(=\)	\( ( - 2766512 \nu^{16} + 4280000 \nu^{15} + 77737369 \nu^{14} - 113386126 \nu^{13} - 873487896 \nu^{12} + \cdots - 94972362 ) / 17299396 \) (-2766512v^16 + 4280000v^15 + 77737369v^14 - 113386126v^13 - 873487896v^12 + 1187347343v^11 + 5010320418v^10 - 6245563495v^9 - 15490381733v^8 + 17277415138v^7 + 24828243438v^6 - 23707951512v^5 - 17681762345v^4 + 13103528687v^3 + 3672725155v^2 - 1858004099v - 94972362) / 17299396

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 6\beta_1 \) b3 + 6*b1
\(\nu^{4}\)	\(=\)	\( -\beta_{13} + \beta_{12} + \beta_{8} + \beta_{6} + \beta_{4} + 9\beta_{2} + 16 \) -b13 + b12 + b8 + b6 + b4 + 9*b2 + 16
\(\nu^{5}\)	\(=\)	\( \beta_{15} - \beta_{14} + \beta_{10} + \beta_{9} + 10\beta_{3} + 40\beta_1 \) b15 - b14 + b10 + b9 + 10b3 + 40b1
\(\nu^{6}\)	\(=\)	\( - \beta_{16} - 11 \beta_{13} + 11 \beta_{12} - \beta_{11} + \beta_{10} + 10 \beta_{8} + \beta_{7} + \cdots + 98 \) -b16 - 11b13 + 11b12 - b11 + b10 + 10b8 + b7 + 12b6 - 2b5 + 11b4 - b3 + 72*b2 + b1 + 98
\(\nu^{7}\)	\(=\)	\( 12 \beta_{15} - 13 \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} + 13 \beta_{10} + 14 \beta_{9} + \cdots + 3 \) 12b15 - 13b14 - b13 + 2b12 - b11 + 13b10 + 14b9 + b7 + b6 + b5 + 84b3 + b2 + 282*b1 + 3
\(\nu^{8}\)	\(=\)	\( - 14 \beta_{16} + \beta_{15} - 94 \beta_{13} + 95 \beta_{12} - 14 \beta_{11} + 17 \beta_{10} + \cdots + 645 \) -14b16 + b15 - 94b13 + 95b12 - 14b11 + 17b10 + b9 + 80b8 + 17b7 + 111b6 - 32b5 + 96b4 - 15b3 + 557b2 + 17*b1 + 645
\(\nu^{9}\)	\(=\)	\( - \beta_{16} + 107 \beta_{15} - 126 \beta_{14} - 19 \beta_{13} + 39 \beta_{12} - 16 \beta_{11} + \cdots + 58 \) -b16 + 107b15 - 126b14 - 19b13 + 39b12 - 16b11 + 128b10 + 142b9 + 5b8 + 9b7 + 18b6 + 20b5 + 8b4 + 670b3 + 24b2 + 2052*b1 + 58
\(\nu^{10}\)	\(=\)	\( - 142 \beta_{16} + 16 \beta_{15} + \beta_{14} - 739 \beta_{13} + 753 \beta_{12} - 143 \beta_{11} + \cdots + 4436 \) -142b16 + 16b15 + b14 - 739b13 + 753b12 - 143b11 + 195b10 + 20b9 + 602b8 + 200b7 + 936b6 - 357b5 + 778b4 - 152b3 + 4253b2 + 207*b1 + 4436
\(\nu^{11}\)	\(=\)	\( - 20 \beta_{16} + 862 \beta_{15} - 1096 \beta_{14} - 244 \beta_{13} + 504 \beta_{12} - 173 \beta_{11} + \cdots + 756 \) -20b16 + 862b15 - 1096b14 - 244b13 + 504b12 - 173b11 + 1144b10 + 1275b9 + 107b8 + 24b7 + 224b6 + 261b5 + 162b4 + 5232b3 + 358b2 + 15206b1 + 756
\(\nu^{12}\)	\(=\)	\( - 1275 \beta_{16} + 170 \beta_{15} + 17 \beta_{14} - 5614 \beta_{13} + 5735 \beta_{12} - 1295 \beta_{11} + \cdots + 31380 \) -1275b16 + 170b15 + 17b14 - 5614b13 + 5735b12 - 1295b11 + 1894b10 + 265b9 + 4449b8 + 2024b7 + 7555b6 - 3447b5 + 6106b4 - 1303b3 + 32303b2 + 2207b1 + 31380
\(\nu^{13}\)	\(=\)	\( - 265 \beta_{16} + 6657 \beta_{15} - 9061 \beta_{14} - 2652 \beta_{13} + 5450 \beta_{12} - 1586 \beta_{11} + \cdots + 8335 \) -265b16 + 6657b15 - 9061b14 - 2652b13 + 5450b12 - 1586b11 + 9765b10 + 10785b9 + 1474b8 - 451b7 + 2376b6 + 2820b5 + 2126b4 + 40449b3 + 4318b2 + 113913b1 + 8335
\(\nu^{14}\)	\(=\)	\( - 10785 \beta_{16} + 1534 \beta_{15} + 168 \beta_{14} - 42030 \beta_{13} + 42777 \beta_{12} + \cdots + 226156 \) -10785b16 + 1534b15 + 168b14 - 42030b13 + 42777b12 - 11041b11 + 16853b10 + 2940b9 + 32771b8 + 18930b7 + 59552b6 - 30915b5 + 47121b4 - 10172b3 + 244901b2 + 21894b1 + 226156
\(\nu^{15}\)	\(=\)	\( - 2940 \beta_{16} + 50464 \beta_{15} - 72945 \beta_{14} - 26309 \beta_{13} + 53438 \beta_{12} + \cdots + 83783 \) -2940b16 + 50464b15 - 72945b14 - 26309b13 + 53438b12 - 13322b11 + 81182b10 + 88264b9 + 16717b8 - 9535b7 + 23046b6 + 27436b5 + 23043b4 + 311136b3 + 46236b2 + 859178b1 + 83783
\(\nu^{16}\)	\(=\)	\( - 88264 \beta_{16} + 12812 \beta_{15} + 1127 \beta_{14} - 312931 \beta_{13} + 315449 \beta_{12} + \cdots + 1650863 \) -88264b16 + 12812b15 + 1127b14 - 312931b13 + 315449b12 - 90915b11 + 142512b10 + 29591b9 + 241894b8 + 168832b7 + 462949b6 - 265694b5 + 359976b4 - 74579b3 + 1856252b2 + 207313b1 + 1650863

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.17
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{17} - T^{16} + \cdots + 18 \) T^17 - T^16 - 29T^15 + 26T^14 + 339T^13 - 266T^12 - 2047T^11 + 1356T^10 + 6792T^9 - 3573T^8 - 12125T^7 + 4456T^6 + 10521T^5 - 1858T^4 - 3551T^3 + 34T^2 + 323*T + 18
$3$	\( (T + 1)^{17} \) (T + 1)^17
$5$	\( T^{17} + 4 T^{16} + \cdots + 12128 \) T^17 + 4T^16 - 53T^15 - 209T^14 + 1127T^13 + 4405T^12 - 12160T^11 - 47740T^10 + 68943T^9 + 279570T^8 - 190646T^7 - 843504T^6 + 219424T^5 + 1121030T^4 - 191216T^3 - 524176T^2 + 160576T + 12128
$7$	\( (T + 1)^{17} \) (T + 1)^17
$11$	\( T^{17} - 4 T^{16} + \cdots - 309248 \) T^17 - 4T^16 - 115T^15 + 426T^14 + 5098T^13 - 16925T^12 - 110158T^11 + 313772T^10 + 1196216T^9 - 2770808T^8 - 6066140T^7 + 10935760T^6 + 13372352T^5 - 18410464T^4 - 8545792T^3 + 12156928T^2 - 1517568T - 309248
$13$	\( (T - 1)^{17} \) (T - 1)^17
$17$	\( (T + 1)^{17} \) (T + 1)^17
$19$	\( T^{17} + \cdots + 394938368 \) T^17 - 13T^16 - 155T^15 + 2686T^14 + 4781T^13 - 204702T^12 + 349074T^11 + 6879593T^10 - 25899165T^9 - 89085876T^8 + 565734512T^7 - 18157584T^6 - 4204791696T^5 + 6298708048T^4 + 3068231040T^3 - 10618463488T^2 + 4517445632T + 394938368
$23$	\( T^{17} + \cdots + 1701568512 \) T^17 - 8T^16 - 280T^15 + 2183T^14 + 31073T^13 - 231555T^12 - 1768177T^11 + 12154394T^10 + 56059435T^9 - 333067378T^8 - 997590512T^7 + 4527580784T^6 + 9198103664T^5 - 23755072544T^4 - 33342155264T^3 - 3203046400T^2 + 6690811904T + 1701568512
$29$	\( T^{17} + \cdots + 52921496864 \) T^17 + 3T^16 - 314T^15 - 987T^14 + 39043T^13 + 120396T^12 - 2472375T^11 - 6924934T^10 + 85836567T^9 + 197546152T^8 - 1626354394T^7 - 2664288088T^6 + 15394879840T^5 + 15077850066T^4 - 56966295536T^3 - 43861724240T^2 + 65198418112T + 52921496864
$31$	\( T^{17} + \cdots + 127975866368 \) T^17 - 10T^16 - 280T^15 + 3038T^14 + 28091T^13 - 353211T^12 - 1129643T^11 + 19692496T^10 + 7781492T^9 - 546305828T^8 + 578425020T^7 + 7345061308T^6 - 14222186808T^5 - 40496138960T^4 + 110049881600T^3 + 31112383488T^2 - 223185100800T + 127975866368
$37$	\( T^{17} + \cdots + 1209986145536 \) T^17 - 354T^15 - 167T^14 + 51556T^13 + 55487T^12 - 4011036T^11 - 6972476T^10 + 178902060T^9 + 432731568T^8 - 4499492614T^7 - 14083881604T^6 + 56694518312T^5 + 229062480240T^4 - 224674914080T^3 - 1511077955776T^2 - 803192696960*T + 1209986145536
$41$	\( T^{17} + \cdots - 215269310016 \) T^17 + 12T^16 - 317T^15 - 3759T^14 + 40586T^13 + 473792T^12 - 2662020T^11 - 30872569T^10 + 91332002T^9 + 1108929688T^8 - 1354845526T^7 - 21407241544T^6 - 1721679282T^5 + 189709056908T^4 + 204663874800T^3 - 383208918368T^2 - 616308884384T - 215269310016
$43$	\( T^{17} + \cdots + 242460229632 \) T^17 - 29T^16 + 42T^15 + 6646T^14 - 66881T^13 - 240830T^12 + 7014987T^11 - 27008971T^10 - 154141145T^9 + 1450715812T^8 - 1754650648T^7 - 18499354912T^6 + 71914259264T^5 - 13073701504T^4 - 416067123200T^3 + 926312345600T^2 - 792369971200T + 242460229632
$47$	\( T^{17} + \cdots - 21342578176 \) T^17 + 4T^16 - 368T^15 - 1302T^14 + 50117T^13 + 147937T^12 - 3341535T^11 - 8097004T^10 + 117601306T^9 + 245201784T^8 - 2162939430T^7 - 4458240866T^6 + 18942567020T^5 + 45922257352T^4 - 49193369152T^3 - 200172027136T^2 - 152858832128T - 21342578176
$53$	\( T^{17} + \cdots - 391066812416 \) T^17 + 8T^16 - 328T^15 - 2422T^14 + 42823T^13 + 281915T^12 - 2887967T^11 - 16174578T^10 + 108539580T^9 + 486144544T^8 - 2298296272T^7 - 7471951520T^6 + 27515785792T^5 + 50279734784T^4 - 186592526336T^3 - 75220008960T^2 + 556734152704T - 391066812416
$59$	\( T^{17} + \cdots - 728570689536 \) T^17 - 25T^16 - 177T^15 + 8256T^14 - 8477T^13 - 1050155T^12 + 3757202T^11 + 65850558T^10 - 312081056T^9 - 2120393402T^8 + 11234692030T^7 + 32433566572T^6 - 174977906872T^5 - 194482074272T^4 + 830128152576T^3 + 883054034176T^2 - 669073720832T - 728570689536
$61$	\( T^{17} + \cdots + 78610370593728 \) T^17 + 5T^16 - 458T^15 - 2486T^14 + 84143T^13 + 483394T^12 - 8134606T^11 - 48962043T^10 + 451537220T^9 + 2843442204T^8 - 14630773752T^7 - 96958457160T^6 + 267328750630T^5 + 1891441741132T^4 - 2482571123920T^3 - 19275841780896T^2 + 8761411196128T + 78610370593728
$67$	\( T^{17} + \cdots - 2573929852928 \) T^17 - 15T^16 - 448T^15 + 6317T^14 + 82191T^13 - 1025339T^12 - 7934294T^11 + 81830298T^10 + 430604056T^9 - 3478830722T^8 - 13547068542T^7 + 79607917912T^6 + 244067350824T^5 - 918641707040T^4 - 2317639362432T^3 + 4280403953152T^2 + 8895949698048T - 2573929852928
$71$	\( T^{17} + \cdots - 920071883563008 \) T^17 - 32T^16 - 332T^15 + 21171T^14 - 83250T^13 - 4865239T^12 + 50447460T^11 + 399171280T^10 - 7928025688T^9 + 6774425712T^8 + 488240187824T^7 - 2523454802752T^6 - 7258710867968T^5 + 94423886799232T^4 - 222765148786688T^3 - 192190187290624T^2 + 1235135848325120T - 920071883563008
$73$	\( T^{17} + \cdots - 202589086278656 \) T^17 + 15T^16 - 582T^15 - 8407T^14 + 131883T^13 + 1790501T^12 - 15548519T^11 - 186496416T^10 + 1106873292T^9 + 10374588536T^8 - 50900323296T^7 - 307592148944T^6 + 1486511596064T^5 + 4267157852352T^4 - 24665789872128T^3 - 11198280789760T^2 + 173593748448768T - 202589086278656
$79$	\( T^{17} + \cdots - 62725724831744 \) T^17 - 27T^16 - 629T^15 + 21601T^14 + 97628T^13 - 6227871T^12 + 6781461T^11 + 804764308T^10 - 2490343104T^9 - 49754163944T^8 + 175480946000T^7 + 1436365591120T^6 - 4343478126912T^5 - 19184369757952T^4 + 39225614880768T^3 + 94726572605440T^2 - 109688933122048T - 62725724831744
$83$	\( T^{17} + \cdots + 16457777289216 \) T^17 + 24T^16 - 565T^15 - 18571T^14 + 42904T^13 + 4942570T^12 + 25232167T^11 - 491618330T^10 - 5075564626T^9 + 8251311084T^8 + 269868498424T^7 + 609771443102T^6 - 4665603171440T^5 - 20021043003712T^4 + 13899703165952T^3 + 152006907563392T^2 + 151723028409344T + 16457777289216
$89$	\( T^{17} + \cdots - 2616109361152 \) T^17 + 8T^16 - 930T^15 - 8562T^14 + 327315T^13 + 3476075T^12 - 52347195T^11 - 671344404T^10 + 3195855744T^9 + 61358764104T^8 + 52639139104T^7 - 2013507693952T^6 - 9644163324928T^5 - 13799568790272T^4 + 3461033867520T^3 + 15634932220928T^2 - 463257153536T - 2616109361152
$97$	\( T^{17} + \cdots - 2639165403136 \) T^17 + 22T^16 - 798T^15 - 18138T^14 + 227947T^13 + 5250263T^12 - 32249479T^11 - 669569522T^10 + 2818173084T^9 + 37946710576T^8 - 150532510896T^7 - 782692642960T^6 + 3117148215488T^5 + 4741064948096T^4 - 20756327777792T^3 + 8145946161408T^2 + 5621056407552T - 2639165403136
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\( p \)	Sign
\(3\)	\(1\)
\(7\)	\(1\)
\(13\)	\(-1\)
\(17\)	\(1\)