LMFDB - Newform orbit 612.2.w.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [612,2,Mod(145,612)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(612, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("612.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$612 = 2^{2} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	612.w (of order $8$ , degree $4$ , minimal)

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:	no
Analytic conductor:	$4.88684460370$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 60x^{14} + 1434x^{12} + 17508x^{10} + 116445x^{8} + 414440x^{6} + 715372x^{4} + 462752x^{2} + 4$ x^16 + 60x^14 + 1434x^12 + 17508x^10 + 116445x^8 + 414440x^6 + 715372x^4 + 462752*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{5}$
Twist minimal:	no (minimal twist has level 204)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{5} + ( - \beta_{15} - \beta_{14} + \beta_{12} + \cdots + 1) q^{7} + (\beta_{13} + \beta_{8} + \cdots + \beta_{2}) q^{11} + ( - \beta_{15} + \beta_{12} - \beta_{10} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{14} - \beta_{13} - 2 \beta_{11} + \cdots - 1) q^{97}+O(q^{100})$ q + (b15 + b14 - b12 + b11 + b10 + b9 + b5 - b4 - 2b2 - 1) q^5 + (-b15 - b14 + b12 - b11 - b10 + b6 - b5 - b3 + b2 + 1) * q^7 + (b13 + b8 + b7 + b6 - b3 + b2) * q^11 + (-b15 + b12 - b10 + 2b7 + b6 - b3 + b2 + 1) q^13 + (-b14 + b12 - b8 - 1) * q^17 + (b13 - b12 + b11 + b9 + b8 + b6 + b5 - 3b4 - b2 + b1) q^19 + (-b15 - b11 - b10 - b8 + b7 - b5 + b4 - b3 + b1) * q^23 + (b15 + b14 + b13 - b12 + b10 + b9 + b8 + b7 + 2b5 - b4 - 3b3 - b2 - b1 - 1) * q^25 + (b11 - 2b7 + b5 - 2b4 - 2b3 + b1 + 2) q^29 + (-b14 + b13 + b12 - b9 - b8 + b6 - 2b5 - b3 + b2 + b1) q^31 + (-b15 - b14 - 2b11 - 2b10 - b9 + b8 + b6 - b5 - 2b4 + 2b3 + 2b2 - 2b1 + 2) * q^35 + (-b14 + b12 - b10 - 2b8 - b7 + b6 + b4 - b3) q^37 + (-2b15 - b14 + b13 + 2b12 + b9 + b7 + b6 - b5 + 2b4 - 2b3 + b2 + b1 + 4) * q^41 + (b15 + b14 - b13 - b12 + b10 + 3b9 - b8 + 2b7 - b4 - 3b2 + b1) q^43 + (-2b15 + 2b12 - 2b11 - 2b5 + 2b4 + 2b2 + 2) * q^47 + (b15 + 2b14 + 2b13 - b12 + 3b11 + 4b10 + 2b9 - 2b8 - 2b7 + b5 + 3b4 + b3 - 4b2 + 2b1) * q^49 + (b15 - b13 - b12 - b9 + b8 + b7 + b5 + b3 - b2 - b1 - 3) * q^53 + (b14 - b12 - b11 + 2b10 + 2b9 + 2b8 + 2b7 + b5 - 3b4 - 2b3 - b2 - 1) * q^55 + (b15 - b13 + 2b11 + b9 - b8 + 2b7 - b3 - b2 + b1 + 2) * q^59 + (-2b12 - 2b10 - 2b9 + 4b8 - b7 + 2b5 - 2b4 - b3 + 2b2 - 2b1 - 2) * q^61 + (-b14 - 3b13 + b12 - 3b11 - 4b10 - b9 - 2b7 + b6 - b5 + 2b4 + 2b3 + b2 + 3) * q^65 + (b15 + b14 + b11 + b10 - 2b9 + 2b8 - b6 + b5 - 4b4 + 4b3 - 2b1 - 2) q^67 + (-b14 + 2b13 + 2b11 + b10 + b9 - b8 + 4b7 + b6 + b5 - 4b4 - b3) * q^71 + (2b15 + b14 + b13 + 2b11 + b9 - b8 + 2b7 - b6 + 2b4 + b3 - 2b2 - b1 - 2) q^73 + (2b15 + b14 + 3b13 - 2b12 + 3b11 + b10 - b9 + 3b8 - 2b7 + 2b5 - b4 + b2 - 3b1 - 4) * q^77 + (-b15 - b14 - 2b13 + b12 - b11 - 3b10 - 2b8 + b6 - b5 - b3 + b2 + 2b1 + 3) * q^79 + (2b15 + 4b14 - 2b13 - 2b12 + 2b10 - 2b9 + 2b8 - 2b7 - 2b6 + 2b5 + 2b4 + 2b3 - 2b2 - 2b1 - 2) * q^83 + (-2b15 - 3b14 - b13 + 2b12 - b11 - 5b10 - 3b9 + b8 - b5 + 3b4 + 2b3 + 5b2 - b1 + 3) * q^85 + (2b14 + 2b13 - 2b12 + 2b11 + 2b10 + b9 + b8 - 2b7 - 2b6 - b4 + b3 - 2b2 - 2) * q^89 + (-3b15 - b14 - 3b13 + b12 - 3b11 - 3b9 - b8 - 2b7 - b6 - 3b5 + 6b4 + b3 + 4b2 - b1 - 1) * q^91 + (-b14 - b13 - 2b12 - 2b11 + b10 + b9 - 3b7 + b5 - 6b4 - 2b3 - b2 + 2b1 - 6) * q^95 + (-b14 - b13 - 2b11 - 3b9 + b8 - 3b7 - b6 - b4 + 2b3 + 2b2 - b1 - 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{5} - 8 q^{11} - 24 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{25} + 24 q^{29} - 24 q^{31} + 32 q^{35} - 8 q^{37} + 32 q^{41} + 24 q^{43} + 24 q^{49} - 24 q^{53} + 32 q^{59} - 16 q^{61} + 32 q^{65}+ \cdots - 24 q^{97}+O(q^{100})$ 16 * q + 8 * q^5 - 8 * q^11 - 24 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^25 + 24 * q^29 - 24 * q^31 + 32 * q^35 - 8 * q^37 + 32 * q^41 + 24 * q^43 + 24 * q^49 - 24 * q^53 + 32 * q^59 - 16 * q^61 + 32 * q^65 - 16 * q^67 - 8 * q^73 - 32 * q^77 + 16 * q^79 + 16 * q^83 - 48 * q^91 - 88 * q^95 - 24 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 60x^{14} + 1434x^{12} + 17508x^{10} + 116445x^{8} + 414440x^{6} + 715372x^{4} + 462752x^{2} + 4$ x^16 + 60*x^14 + 1434*x^12 + 17508*x^10 + 116445*x^8 + 414440*x^6 + 715372*x^4 + 462752*x^2 + 4 :

$\beta_{1}$	$=$	$( - 10183231 \nu^{14} - 730091944 \nu^{12} - 20013985161 \nu^{10} - 264194956239 \nu^{8} + \cdots + 1177839845590 ) / 450248976368$ (-10183231v^14 - 730091944v^12 - 20013985161v^10 - 264194956239v^8 - 1728005430420v^6 - 5118882005523v^4 - 4765884821288*v^2 + 1177839845590) / 450248976368
$\beta_{2}$	$=$	$( - 86362753 \nu^{14} - 4947624218 \nu^{12} - 110704440654 \nu^{10} - 1225172767250 \nu^{8} + \cdots + 834139859368 ) / 450248976368$ (-86362753v^14 - 4947624218v^12 - 110704440654v^10 - 1225172767250v^8 - 6973443020659v^6 - 18805602643828v^4 - 17202998438278*v^2 + 834139859368) / 450248976368
$\beta_{3}$	$=$	$( - 28803044 \nu^{15} - 766785 \nu^{14} - 1725343305 \nu^{13} - 41712081 \nu^{12} + \cdots - 37457113132 ) / 52970467808$ (-28803044v^15 - 766785v^14 - 1725343305v^13 - 41712081v^12 - 41140786928v^11 - 872350943v^10 - 500634663986v^9 - 8866998107v^8 - 3313465288556v^7 - 46130941900v^6 - 11706031006803v^5 - 121753921784v^4 - 19981915899780v^3 - 154574621844v^2 - 12739898076102*v - 37457113132) / 52970467808
$\beta_{4}$	$=$	$( - 28803044 \nu^{15} + 766785 \nu^{14} - 1725343305 \nu^{13} + 41712081 \nu^{12} + \cdots + 37457113132 ) / 52970467808$ (-28803044v^15 + 766785v^14 - 1725343305v^13 + 41712081v^12 - 41140786928v^11 + 872350943v^10 - 500634663986v^9 + 8866998107v^8 - 3313465288556v^7 + 46130941900v^6 - 11706031006803v^5 + 121753921784v^4 - 19981915899780v^3 + 154574621844v^2 - 12739898076102*v + 37457113132) / 52970467808
$\beta_{5}$	$=$	$( 470397100 \nu^{15} + 94265158 \nu^{14} + 28257173258 \nu^{13} + 5215141977 \nu^{12} + \cdots + 691188026456 ) / 900497952736$ (470397100v^15 + 94265158v^14 + 28257173258v^13 + 5215141977v^12 + 676673874128v^11 + 111434060969v^10 + 8290208940598v^9 + 1159917430273v^8 + 55492341655432v^7 + 6081895983335v^6 + 199956544935140v^5 + 14685750256246v^4 + 353089832963568v^3 + 12105874045474v^2 + 235059249358272*v + 691188026456) / 900497952736
$\beta_{6}$	$=$	$( 470397100 \nu^{15} - 94265158 \nu^{14} + 28257173258 \nu^{13} - 5215141977 \nu^{12} + \cdots - 691188026456 ) / 900497952736$ (470397100v^15 - 94265158v^14 + 28257173258v^13 - 5215141977v^12 + 676673874128v^11 - 111434060969v^10 + 8290208940598v^9 - 1159917430273v^8 + 55492341655432v^7 - 6081895983335v^6 + 199956544935140v^5 - 14685750256246v^4 + 353089832963568v^3 - 12105874045474v^2 + 235059249358272*v - 691188026456) / 900497952736
$\beta_{7}$	$=$	$( 1789 \nu^{15} + 107628 \nu^{13} + 2581674 \nu^{11} + 31676582 \nu^{9} + 212104547 \nu^{7} + \cdots + 874905524 \nu ) / 2572256$ (1789v^15 + 107628v^13 + 2581674v^11 + 31676582v^9 + 212104547v^7 + 761904098v^5 + 1331873950v^3 + 874905524v) / 2572256
$\beta_{8}$	$=$	$( 630241980 \nu^{15} + 30183059 \nu^{14} + 37723759683 \nu^{13} + 1769619153 \nu^{12} + \cdots - 345529433242 ) / 900497952736$ (630241980v^15 + 30183059v^14 + 37723759683v^13 + 1769619153v^12 + 899051581736v^11 + 41289910746v^10 + 10941952531944v^9 + 489648271146v^8 + 72530523488424v^7 + 3092284932373v^6 + 257208904350375v^5 + 9591550356055v^4 + 441876241326876v^3 + 10340297968278v^2 + 283805077784822*v - 345529433242) / 900497952736
$\beta_{9}$	$=$	$( 630241980 \nu^{15} - 30183059 \nu^{14} + 37723759683 \nu^{13} - 1769619153 \nu^{12} + \cdots + 345529433242 ) / 900497952736$ (630241980v^15 - 30183059v^14 + 37723759683v^13 - 1769619153v^12 + 899051581736v^11 - 41289910746v^10 + 10941952531944v^9 - 489648271146v^8 + 72530523488424v^7 - 3092284932373v^6 + 257208904350375v^5 - 9591550356055v^4 + 441876241326876v^3 - 10340297968278v^2 + 283805077784822*v + 345529433242) / 900497952736
$\beta_{10}$	$=$	$( - 384330901 \nu^{15} - 5768030 \nu^{14} - 23059716203 \nu^{13} - 333947021 \nu^{12} + \cdots + 487357836308 ) / 450248976368$ (-384330901v^15 - 5768030v^14 - 23059716203v^13 - 333947021v^12 - 550744767037v^11 - 7315411723v^10 - 6709646651825v^9 - 73093876203v^8 - 44397804203222v^7 - 297914865297v^6 - 156317046040216v^5 - 99090963372v^4 - 264225212267472v^3 + 1371589038938v^2 - 165761866021268*v + 487357836308) / 450248976368
$\beta_{11}$	$=$	$( 384330901 \nu^{15} - 5768030 \nu^{14} + 23059716203 \nu^{13} - 333947021 \nu^{12} + \cdots + 487357836308 ) / 450248976368$ (384330901v^15 - 5768030v^14 + 23059716203v^13 - 333947021v^12 + 550744767037v^11 - 7315411723v^10 + 6709646651825v^9 - 73093876203v^8 + 44397804203222v^7 - 297914865297v^6 + 156317046040216v^5 - 99090963372v^4 + 264225212267472v^3 + 1371589038938v^2 + 165761866021268*v + 487357836308) / 450248976368
$\beta_{12}$	$=$	$( 848959971 \nu^{15} + 88634985 \nu^{14} + 50982942440 \nu^{13} + 5266939953 \nu^{12} + \cdots + 1180083186368 ) / 900497952736$ (848959971v^15 + 88634985v^14 + 50982942440v^13 + 5266939953v^12 + 1220103229442v^11 + 123337412129v^10 + 14926761716220v^9 + 1442431117793v^8 + 99592230993357v^7 + 8749033688262v^6 + 356174500011984v^5 + 25301012335574v^4 + 618686634269978v^3 + 25565490162696v^2 + 401308473215848*v + 1180083186368) / 900497952736
$\beta_{13}$	$=$	$( - 841692656 \nu^{15} + 5768030 \nu^{14} - 50607784084 \nu^{13} + 333947021 \nu^{12} + \cdots - 487357836308 ) / 450248976368$ (-841692656v^15 + 5768030v^14 - 50607784084v^13 + 333947021v^12 - 1212588675134v^11 + 7315411723v^10 - 14849116624604v^9 + 73093876203v^8 - 99105919846354v^7 + 297914865297v^6 - 354203774305028v^5 + 99090963372v^4 - 614687276659692v^3 - 1371589038938v^2 - 400184344456296*v - 487357836308) / 450248976368
$\beta_{14}$	$=$	$( - 1997867746 \nu^{15} - 153784274 \nu^{14} - 119731104237 \nu^{13} - 8466451041 \nu^{12} + \cdots + 2719728082012 ) / 900497952736$ (-1997867746v^15 - 153784274v^14 - 119731104237v^13 - 8466451041v^12 - 2857862201528v^11 - 180044740671v^10 - 34847511079170v^9 - 1870905126755v^8 - 231543839462682v^7 - 9899692593497v^6 - 823930529271495v^5 - 24727944611256v^4 - 1424112015933552v^3 - 21061790265862v^2 - 925127497705982*v + 2719728082012) / 900497952736
$\beta_{15}$	$=$	$( 1997867746 \nu^{15} - 153784274 \nu^{14} + 119731104237 \nu^{13} - 8466451041 \nu^{12} + \cdots + 2719728082012 ) / 900497952736$ (1997867746v^15 - 153784274v^14 + 119731104237v^13 - 8466451041v^12 + 2857862201528v^11 - 180044740671v^10 + 34847511079170v^9 - 1870905126755v^8 + 231543839462682v^7 - 9899692593497v^6 + 823930529271495v^5 - 24727944611256v^4 + 1424112015933552v^3 - 21061790265862v^2 + 925127497705982*v + 2719728082012) / 900497952736

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

$\nu$	$=$	$( - 2 \beta_{14} + 2 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots + 2 ) / 2$ (-2b14 + 2b12 - b11 - b10 - 2b9 - 2b8 - 2b7 - 2b5 + 2b4 + 2b2 + 2) / 2
$\nu^{2}$	$=$	$\beta_{15} + \beta_{14} + \beta_{11} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{4} + \beta_{3} + \cdots - 8$ b15 + b14 + b11 + b10 + 2b9 - 2b8 - b4 + b3 - 3b2 + 2b1 - 8
$\nu^{3}$	$=$	$( 6 \beta_{15} + 24 \beta_{14} - 30 \beta_{12} + 19 \beta_{11} + 11 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} + \cdots - 30 ) / 2$ (6b15 + 24b14 - 30b12 + 19b11 + 11b10 + 24b9 + 24b8 + 22b7 - 2b6 + 28b5 - 14b4 + 16b3 - 30*b2 - 30) / 2
$\nu^{4}$	$=$	$- 14 \beta_{15} - 14 \beta_{14} - 13 \beta_{11} - 13 \beta_{10} - 34 \beta_{9} + 34 \beta_{8} + \cdots + 100$ -14b15 - 14b14 - 13b11 - 13b10 - 34b9 + 34b8 - 7b6 + 7b5 + 17b4 - 17b3 + 53b2 - 36b1 + 100
$\nu^{5}$	$=$	$( - 112 \beta_{15} - 334 \beta_{14} - 24 \beta_{13} + 446 \beta_{12} - 339 \beta_{11} - 131 \beta_{10} + \cdots + 446 ) / 2$ (-112b15 - 334b14 - 24b13 + 446b12 - 339b11 - 131b10 - 344b9 - 344b8 - 302b7 + 10b6 - 436b5 + 100b4 - 346b3 + 446b2 + 446) / 2
$\nu^{6}$	$=$	$208 \beta_{15} + 208 \beta_{14} + 172 \beta_{11} + 172 \beta_{10} + 546 \beta_{9} - 546 \beta_{8} + \cdots - 1438$ 208b15 + 208b14 + 172b11 + 172b10 + 546b9 - 546b8 + 173b6 - 173b5 - 207b4 + 207b3 - 875b2 + 600b1 - 1438
$\nu^{7}$	$=$	$( 1916 \beta_{15} + 5074 \beta_{14} + 724 \beta_{13} - 6990 \beta_{12} + 5949 \beta_{11} + 1765 \beta_{10} + \cdots - 6990 ) / 2$ (1916b15 + 5074b14 + 724b13 - 6990b12 + 5949b11 + 1765b10 + 5276b9 + 5276b8 + 4638b7 + 174b6 + 7164b5 - 676b4 + 6314b3 - 6990b2 - 6990) / 2
$\nu^{8}$	$=$	$- 3222 \beta_{15} - 3222 \beta_{14} - 2217 \beta_{11} - 2217 \beta_{10} - 8606 \beta_{9} + 8606 \beta_{8} + \cdots + 22070$ -3222b15 - 3222b14 - 2217b11 - 2217b10 - 8606b9 + 8606b8 - 3345b6 + 3345b5 + 2247b4 - 2247b3 + 14195b2 - 9868b1 + 22070
$\nu^{9}$	$=$	$( - 32568 \beta_{15} - 80942 \beta_{14} - 16624 \beta_{13} + 113510 \beta_{12} - 103867 \beta_{11} + \cdots + 113510 ) / 2$ (-32568b15 - 80942b14 - 16624b13 + 113510b12 - 103867b11 - 26267b10 - 83612b9 - 83612b8 - 76406b7 - 6930b6 - 120440b5 + 3848b4 - 109662b3 + 113510b2 + 113510) / 2
$\nu^{10}$	$=$	$51210 \beta_{15} + 51210 \beta_{14} + 27368 \beta_{11} + 27368 \beta_{10} + 134274 \beta_{9} + \cdots - 350586$ 51210b15 + 51210b14 + 27368b11 + 27368b10 + 134274b9 - 134274b8 + 59631b6 - 59631b5 - 20811b4 + 20811b3 - 229131b2 + 162244b1 - 350586
$\nu^{11}$	$=$	$( 554016 \beta_{15} + 1326254 \beta_{14} + 347652 \beta_{13} - 1880270 \beta_{12} + 1810189 \beta_{11} + \cdots - 1880270 ) / 2$ (554016b15 + 1326254b14 + 347652b13 - 1880270b12 + 1810189b11 + 417733b10 + 1350148b9 + 1350148b8 + 1312718b7 + 164934b6 + 2045204b5 - 11288b4 + 1868982b3 - 1880270b2 - 1880270) / 2
$\nu^{12}$	$=$	$- 826054 \beta_{15} - 826054 \beta_{14} - 312587 \beta_{11} - 312587 \beta_{10} - 2082902 \beta_{9} + \cdots + 5678658$ -826054b15 - 826054b14 - 312587b11 - 312587b10 - 2082902b9 + 2082902b8 - 1028905b6 + 1028905b5 + 113979b4 - 113979b3 + 3696543b2 - 2676900b1 + 5678658
$\nu^{13}$	$=$	$( - 9438728 \beta_{15} - 22060974 \beta_{14} - 6942264 \beta_{13} + 31499702 \beta_{12} - 31526063 \beta_{11} + \cdots + 31499702 ) / 2$ (-9438728b15 - 22060974b14 - 6942264b13 + 31499702b12 - 31526063b11 - 6915903b10 - 22064916b9 - 22064916b8 - 23054438b7 - 3405026b6 - 34904728b5 - 121600b4 - 31621302b3 + 31499702b2 + 31499702) / 2
$\nu^{14}$	$=$	$13442558 \beta_{15} + 13442558 \beta_{14} + 3020394 \beta_{11} + 3020394 \beta_{10} + 32213074 \beta_{9} + \cdots - 93075922$ 13442558b15 + 13442558b14 + 3020394b11 + 3020394b10 + 32213074b9 - 32213074b8 + 17515135b6 - 17515135b5 + 1482029b4 - 1482029b3 - 59729331b2 + 44367492b1 - 93075922
$\nu^{15}$	$=$	$( 161081800 \beta_{15} + 370280198 \beta_{14} + 134696124 \beta_{13} - 531361998 \beta_{12} + \cdots - 531361998 ) / 2$ (161081800b15 + 370280198b14 + 134696124b13 - 531361998b12 + 548890785b11 + 117167337b10 + 363621812b9 + 363621812b8 + 408821646b7 + 66141726b6 + 597503724b5 + 2474400b4 + 533836398b3 - 531361998b2 - 531361998) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/612\mathbb{Z}\right)^\times$ .

$n$	$37$	$137$	$307$
$\chi(n)$	$\beta_{3}$	$1$	$1$

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}

145.1

	−	4.16775i
		1.29733i
	−	1.53196i
		2.40238i
	−	3.91262i
		2.85070i
	−	0.00294008i
		3.06486i
		4.16775i
	−	1.29733i
		1.53196i
	−	2.40238i
		3.91262i
	−	2.85070i
		0.00294008i
	−	3.06486i

−0.919353

2.21951i

−1.55193

−

3.74669i

145.2

−0.586256

1.41535i

−1.15081

−

2.77830i

145.3

0.496466

−

1.19857i

1.20907

2.91895i

145.4

1.59493

−

3.85050i

0.0794551

0.191822i

253.1

−2.83156

1.17287i

−0.119959

−

0.0496888i

253.2

−0.00271628

0.00112512i

3.72964

1.54487i

253.3

2.63370

−

1.09092i

−4.87029

−

2.01734i

253.4

3.61479

−

1.49729i

2.67483

1.10795i

325.1

−0.919353

−

2.21951i

−1.55193

3.74669i

325.2

−0.586256

−

1.41535i

−1.15081

2.77830i

325.3

0.496466

1.19857i

1.20907

−

2.91895i

325.4

1.59493

3.85050i

0.0794551

−

0.191822i

433.1

−2.83156

−

1.17287i

−0.119959

0.0496888i

433.2

−0.00271628

−

0.00112512i

3.72964

−

1.54487i

433.3

2.63370

1.09092i

−4.87029

2.01734i

433.4

3.61479

1.49729i

2.67483

−

1.10795i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.d	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	612.2.w.c		16
3.b	odd	2	1		204.2.o.a	✓	16
12.b	even	2	1		816.2.bq.d		16
17.d	even	8	1	inner	612.2.w.c		16
51.g	odd	8	1		204.2.o.a	✓	16
51.i	even	16	1		3468.2.a.o		8
51.i	even	16	1		3468.2.a.p		8
51.i	even	16	2		3468.2.b.h		16
51.i	even	16	2		3468.2.j.h		16
51.i	even	16	2		3468.2.j.i		16
204.p	even	8	1		816.2.bq.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
204.2.o.a	✓	16	3.b	odd	2	1
204.2.o.a	✓	16	51.g	odd	8	1
612.2.w.c		16	1.a	even	1	1	trivial
612.2.w.c		16	17.d	even	8	1	inner
816.2.bq.d		16	12.b	even	2	1
816.2.bq.d		16	204.p	even	8	1
3468.2.a.o		8	51.i	even	16	1
3468.2.a.p		8	51.i	even	16	1
3468.2.b.h		16	51.i	even	16	2
3468.2.j.h		16	51.i	even	16	2
3468.2.j.i		16	51.i	even	16	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{16} - 8 T_{5}^{15} + 28 T_{5}^{14} - 32 T_{5}^{13} - 120 T_{5}^{12} + 296 T_{5}^{11} + 1344 T_{5}^{10} + \cdots + 4$ T5^16 - 8*T5^15 + 28*T5^14 - 32*T5^13 - 120*T5^12 + 296*T5^11 + 1344*T5^10 - 2504*T5^9 + 6565*T5^8 - 54720*T5^7 + 126468*T5^6 - 123784*T5^5 + 363976*T5^4 - 216880*T5^3 + 461560*T5^2 + 2512*T5 + 4 acting on $S_{2}^{\mathrm{new}}(612, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16}$ T^16
$5$	$T^{16} - 8 T^{15} + \cdots + 4$ T^16 - 8T^15 + 28T^14 - 32T^13 - 120T^12 + 296T^11 + 1344T^10 - 2504T^9 + 6565T^8 - 54720T^7 + 126468T^6 - 123784T^5 + 363976T^4 - 216880T^3 + 461560T^2 + 2512*T + 4
$7$	$T^{16} - 12 T^{14} + \cdots + 4096$ T^16 - 12T^14 - 24T^13 + 72T^12 + 240T^11 + 4760T^10 - 27824T^9 + 131844T^8 - 401344T^7 + 1023424T^6 - 2990848T^5 + 5407232T^4 + 397312T^3 + 100352T^2 + 40960T + 4096
$11$	$T^{16} + 8 T^{15} + \cdots + 4734976$ T^16 + 8T^15 + 16T^14 + 8T^13 + 192T^12 - 376T^11 + 2288T^10 - 16712T^9 + 6977T^8 + 184400T^7 - 301344T^6 - 984576T^5 + 5693952T^4 - 12879872T^3 + 15839232T^2 - 10584064*T + 4734976
$13$	$T^{16} + \cdots + 121264144$ T^16 + 148T^14 + 8926T^12 + 286204T^10 + 5292993T^8 + 56647464T^6 + 326167800T^4 + 792919776*T^2 + 121264144
$17$	$T^{16} + \cdots + 6975757441$ T^16 + 24T^15 + 284T^14 + 2184T^13 + 12182T^12 + 52056T^11 + 176236T^10 + 511752T^9 + 1704418T^8 + 8699784T^7 + 50932204T^6 + 255751128T^5 + 1017452822T^4 + 3100967688T^3 + 6855069596T^2 + 9848128152*T + 6975757441
$19$	$T^{16} + \cdots + 4421718016$ T^16 - 8T^15 + 32T^14 - 56T^13 + 2210T^12 - 17816T^11 + 73376T^10 - 108920T^9 + 1239233T^8 - 9715792T^7 + 40714368T^6 - 55693184T^5 + 188608640T^4 - 1296589824T^3 + 5575680000T^2 - 7021977600*T + 4421718016
$23$	$T^{16} + \cdots + 2264237056$ T^16 + 8T^15 + 8T^14 + 440T^13 + 4064T^12 - 16792T^11 - 41720T^10 + 452040T^9 - 653695T^8 - 1893616T^7 + 59177840T^6 - 77330752T^5 + 1487778944T^4 - 5344024576T^3 + 7465552384T^2 - 4814739456*T + 2264237056
$29$	$T^{16} + \cdots + 14168617024$ T^16 - 24T^15 + 268T^14 - 1352T^13 - 3064T^12 + 87232T^11 - 301336T^10 - 4449008T^9 + 71895572T^8 - 554546688T^7 + 2820210304T^6 - 10096975232T^5 + 25412739200T^4 - 43336409600T^3 + 49226733440T^2 - 35976231680*T + 14168617024
$31$	$T^{16} + 24 T^{15} + \cdots + 18939904$ T^16 + 24T^15 + 216T^14 + 224T^13 - 12768T^12 - 91584T^11 + 400960T^10 + 7649792T^9 + 34175040T^8 + 95450112T^7 + 502135808T^6 + 1004474368T^5 + 2135564288T^4 + 1344667648T^3 + 253198336T^2 + 6684672*T + 18939904
$37$	$T^{16} + 8 T^{15} + \cdots + 2347024$ T^16 + 8T^15 + 128T^14 + 1248T^13 + 11008T^12 + 67728T^11 + 219456T^10 + 68064T^9 + 3953032T^8 - 10102240T^7 + 29640192T^6 - 157831040T^5 + 282099200T^4 + 436950080T^3 + 198954240T^2 + 35542400*T + 2347024
$41$	$T^{16} + \cdots + 48146014084$ T^16 - 32T^15 + 548T^14 - 6784T^13 + 68232T^12 - 578072T^11 + 4316256T^10 - 24266200T^9 + 103181765T^8 - 334334632T^7 + 854834524T^6 - 1958084472T^5 + 3432095368T^4 + 1811938464T^3 - 17764100536T^2 + 2666416144*T + 48146014084
$43$	$T^{16} + \cdots + 18339659776$ T^16 - 24T^15 + 288T^14 - 2008T^13 + 19362T^12 - 281912T^11 + 3205664T^10 - 22066136T^9 + 92443841T^8 - 203537568T^7 + 238998016T^6 - 580019200T^5 + 4212147712T^4 - 7807016960T^3 + 4437573632T^2 + 12758024192*T + 18339659776
$47$	$T^{16} + \cdots + 303038464$ T^16 + 304T^14 + 33760T^12 + 1724672T^10 + 41951488T^8 + 476307456T^6 + 2320990208T^4 + 3462922240*T^2 + 303038464
$53$	$T^{16} + 24 T^{15} + \cdots + 1183744$ T^16 + 24T^15 + 288T^14 + 1264T^13 + 1896T^12 - 3744T^11 + 162944T^10 + 390592T^9 + 435728T^8 - 2091520T^7 + 16089088T^6 + 15520768T^5 + 7613952T^4 - 1236992T^3 + 3276800T^2 + 2785280*T + 1183744
$59$	$T^{16} - 32 T^{15} + \cdots + 4734976$ T^16 - 32T^15 + 512T^14 - 3600T^13 + 13196T^12 - 37088T^11 + 910464T^10 - 6221664T^9 + 20239876T^8 + 15668928T^7 + 207045120T^6 - 1446039040T^5 + 4929552896T^4 + 8977870848T^3 + 8093990912T^2 + 276856832*T + 4734976
$61$	$T^{16} + \cdots + 2539777694224$ T^16 + 16T^15 + 72T^14 - 736T^13 - 11232T^12 + 7456T^11 + 1989552T^10 + 30133696T^9 + 400397960T^8 + 4116003136T^7 + 31620630688T^6 + 179720312960T^5 + 749158527104T^4 + 2228340739712T^3 + 4546407032256T^2 + 5208566000384*T + 2539777694224
$67$	$(T^{8} + 8 T^{7} + \cdots - 1482496)^{2}$ (T^8 + 8T^7 - 204T^6 - 1936T^5 + 5124T^4 + 78208T^3 + 55488T^2 - 808960*T - 1482496)^2
$71$	$T^{16} + 144 T^{14} + \cdots + 39337984$ T^16 + 144T^14 - 3904T^13 + 10368T^12 - 57088T^11 + 5479552T^10 + 26050688T^9 + 236543504T^8 + 264410880T^7 + 646878976T^6 + 4091692032T^5 + 1358383104T^4 - 16306798592T^3 + 14003118080T^2 - 1337491456T + 39337984
$73$	$T^{16} + \cdots + 502805791744$ T^16 + 8T^15 + 108T^14 - 2200T^13 - 17656T^12 + 147776T^11 + 2457448T^10 + 17164848T^9 + 113235972T^8 + 643098048T^7 + 4714447296T^6 - 8009505536T^5 - 224737553920T^4 + 39228608512T^3 + 5294989423616T^2 - 1252260753408*T + 502805791744
$79$	$T^{16} + \cdots + 58866405376$ T^16 - 16T^15 + 276T^14 - 3832T^13 + 45128T^12 - 375376T^11 + 1770904T^10 + 4606032T^9 - 40605052T^8 + 376206336T^7 + 2529658432T^6 - 3096340736T^5 + 413282816T^4 + 14187919360T^3 - 17071785984T^2 - 28478234624*T + 58866405376
$83$	$T^{16} + \cdots + 34\!\cdots\!04$ T^16 - 16T^15 + 128T^14 + 960T^13 + 50400T^12 - 761856T^11 + 6199296T^10 + 47563776T^9 + 533995776T^8 - 6129963008T^7 + 53156937728T^6 + 433811783680T^5 + 2132804009984T^4 - 8073184018432T^3 + 84429294796800T^2 + 768140252282880*T + 3494281508552704
$89$	$T^{16} + \cdots + 641585799247936$ T^16 + 848T^14 + 287084T^12 + 49899408T^10 + 4805998196T^8 + 261457580448T^6 + 7835068023200T^4 + 117037148351872*T^2 + 641585799247936
$97$	$T^{16} + \cdots + 1671373209856$ T^16 + 24T^15 + 300T^14 + 3480T^13 + 38664T^12 + 311152T^11 + 1789080T^10 + 6872368T^9 + 19703940T^8 + 74201952T^7 - 540887136T^6 - 6488098560T^5 + 53362150528T^4 + 646611200256T^3 + 1535467750656T^2 - 810854195200*T + 1671373209856
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 145.4
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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