LMFDB - Newform orbit 3276.2.hi.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,2,Mod(1297,3276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3276, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 5]))

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

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

chi := DirichletCharacter("3276.1297");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3276.hi (of order $6$ , degree $2$ , 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:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{14} - 3 x^{13} + 10 x^{12} + x^{11} - 171 x^{10} + 105 x^{9} + 1781 x^{8} + 6966 x^{7} + 16542 x^{6} + \cdots + 1251$ x^14 - 3x^13 + 10x^12 + x^11 - 171x^10 + 105x^9 + 1781x^8 + 6966x^7 + 16542x^6 + 23794x^5 + 26827x^4 + 22359x^3 + 12768x^2 + 5346x + 1251
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1092)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{13} q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{4} + \cdots + 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{3}) q^{11} + (\beta_{13} - \beta_{8} + \cdots - \beta_{2}) q^{13} + (\beta_{11} + \beta_{10}) q^{17}+ \cdots + (3 \beta_{13} + 4 \beta_{12} + \cdots - 1) q^{97}+O(q^{100})$ q - b13 * q^5 + (-b10 + b8 + b4 + b1 + 1) * q^7 + (-b11 - b10 - b7 + b4 + b3) * q^11 + (b13 - b8 - b6 + b5 + b3 - b2) * q^13 + (b11 + b10) * q^17 + (-b10 + 2b9 - b8 + b5 - 2) q^19 + (b13 + b12 - b11 - b10 + b5 + b4 + 2b3 - b2 - 1) q^23 + (-b12 - b11 + 2b10 - 2b8 + b6 + 3b5 + b4 + b3 - 2b2 - b1) * q^25 + (-2b11 - b8 - 2b7 + b3) * q^29 + (-b10 + 2b9 + b5) q^31 + (3b13 + 2b12 - 3b9 + b8 - b7 - 2b6 - 2b5 + b3 + b1) q^35 + (-2b11 - 2b10 - 4b8 - 4b7 + 2b4 + 2b3 - b2 + 3b1 - 2) q^37 + (-b12 + b11 + b10 + b9 + b8 - b6 - b4 - b1 + 2) * q^41 + (-2b11 - b10 - 2b7 + 2b4 + b3 - b2 + 3b1) * q^43 + (-b11 + b8 + b7 + 2b5 + b4 + 2b3 - 3b2 - b1 - 1) q^47 + (-b13 - 2b12 + b10 + b8 - b7 + b6 + 2b5 + 2b4 - 3b2 + b1) * q^49 + (-2b13 + 2b11 - 3b10 + b9 - b8 - 2b7 - 3b5 - 2b3 + 4b2 + 4b1) * q^53 + (b12 + 2b11 - b10 - b8 - b7 - b6 - b5 - b3 + 3b2 + 3b1) q^55 + (-2b13 + 3b11 - b10 + 2b9 + 2b7 - 3b5 - 4b4 - 3b3 + 5b2) * q^59 + (-2b13 - 2b12 + b11 + b9 + b8 + b7 + 2b6 + b5 + b4 - b3 - b2) q^61 + (-3b13 - b12 + 3b11 + b10 + 2b9 + b8 + b7 + b6 + 3b5 - 2b4 - 2b3 - b2 - 2b1 + 1) q^65 + (-b13 + 2b11 + 3b10 + 2b7 + b5 - 3b4 - 2b3 - b1) q^67 + (-2b11 - b8 - 2b7 + b5 - b4 + b3 + b2 - 2b1 + 1) q^71 + (2b11 + b10 + b9 + b7 + b5 - 2b4 - 2b3 + b2 - 3b1) * q^73 + (3b11 + 2b10 + b7 - b4 - 3b3 + 2b2 - 2b1 + 2) q^77 + (2b11 + 2b9 + 4b8 - 2b6 - 2b4 - 2b3 - 2b1 + 4) q^79 + (-2b13 - b12 - b11 + b10 + b9 + 2b6 + b5 + b4 - b3 - 2b2 - b1) q^83 + (b13 - 3b11 - b10 - b7 + 4b5 + 3b4 + 5b3 - 4b2 - 2b1) * q^85 + (b13 - b11 + b10 - b9 + 4b5 + 4b4 + b3 - 2b2 + 2b1) * q^89 + (-2b11 - 3b10 + 3b9 + b8 - 2b7 - b6 - b5 - b3 - b2 + 2b1 + 1) q^91 + (-b13 + 2b11 + 2b10 - b9 - 4b5 - 3b3 + b2 + 3b1 - 7) q^95 + (3b13 + 4b12 + b11 - b10 - 2b9 + b8 - 2b7 - 2b6 - 3b5 + 4b2 + 2b1 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$14 q + 4 q^{7} - 6 q^{11} + 10 q^{13} + 12 q^{17} - 18 q^{19} - 8 q^{23} + 17 q^{25} + q^{29} + 3 q^{31} - 4 q^{35} + 27 q^{41} - 15 q^{47} - 4 q^{49} + 4 q^{53} + 13 q^{55} - 7 q^{61} + 16 q^{65} + 12 q^{67}+ \cdots - 12 q^{97}+O(q^{100})$ 14 * q + 4 * q^7 - 6 * q^11 + 10 * q^13 + 12 * q^17 - 18 * q^19 - 8 * q^23 + 17 * q^25 + q^29 + 3 * q^31 - 4 * q^35 + 27 * q^41 - 15 * q^47 - 4 * q^49 + 4 * q^53 + 13 * q^55 - 7 * q^61 + 16 * q^65 + 12 * q^67 + 3 * q^71 - 3 * q^73 + 28 * q^77 + 28 * q^79 - 2 * q^91 - 80 * q^95 - 12 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{14} - 3 x^{13} + 10 x^{12} + x^{11} - 171 x^{10} + 105 x^{9} + 1781 x^{8} + 6966 x^{7} + 16542 x^{6} + \cdots + 1251$ x^14 - 3*x^13 + 10*x^12 + x^11 - 171*x^10 + 105*x^9 + 1781*x^8 + 6966*x^7 + 16542*x^6 + 23794*x^5 + 26827*x^4 + 22359*x^3 + 12768*x^2 + 5346*x + 1251 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 33\!\cdots\!39 \nu^{13} + \cdots + 11\!\cdots\!46 ) / 57\!\cdots\!14$ (33984914455022598139v^13 - 114004465530902807448v^12 + 377811647115313393234v^11 - 93501583151974826126v^10 - 5780602322912319362661v^9 + 5501826760913225431068v^8 + 59404441574538147258077v^7 + 214878310102892852633898v^6 + 479073577604758923418302v^5 + 629373151377022458288583v^4 + 658824343175338712888950v^3 + 522529308764075823780249v^2 + 281077214873710985547510*v + 116909338563438723957546) / 57153602893591836195714
$\beta_{3}$	$=$	$( 28\!\cdots\!56 \nu^{13} + \cdots + 19\!\cdots\!62 ) / 13\!\cdots\!66$ (282209250797246700856v^13 - 1329761974342862698625v^12 + 4634106541420376152330v^11 - 5780054927283428235538v^10 - 44549449221965265449666v^9 + 110400009169142542289733v^8 + 392753501883757673183078v^7 + 1162367985933136599635531v^6 + 1917300576138362227083966v^5 + 1061511078266443603708600v^4 + 1220478612417797398776509v^3 - 79063296017243939380934v^2 - 208078047315104649845409*v + 195711609368501484305262) / 133358406751714284456666
$\beta_{4}$	$=$	$( - 13\!\cdots\!34 \nu^{13} + \cdots - 18\!\cdots\!01 ) / 40\!\cdots\!98$ (-1358214452702038832434v^13 + 5696716041572236976940v^12 - 19555625424038383243543v^11 + 18439602830238846360830v^10 + 222061468620157930534884v^9 - 418046622005242646754528v^8 - 2062337781714045542245649v^7 - 6717652574600110948011246v^6 - 13150350181805096155625679v^5 - 12782039607998939340892372v^4 - 14056057245962981882155546v^3 - 8424031511557427230148271v^2 - 3097697209298392429560972*v - 1866237872559961570608501) / 400075220255142853369998
$\beta_{5}$	$=$	$( - 17\!\cdots\!89 \nu^{13} + \cdots - 10\!\cdots\!76 ) / 40\!\cdots\!98$ (-1741292467418172399289v^13 + 6317201610982194061584v^12 - 21253847536072596497854v^11 + 11601054017046333774434v^10 + 290137874857667910221511v^9 - 359261890697473885096230v^8 - 2903650025891833502788223v^7 - 10349828058195395272436454v^6 - 21975284447791517344608660v^5 - 26196564519450069843283159v^4 - 26314437304273678433852548v^3 - 15769606886509737179598597v^2 - 5769148880351587203934464*v - 1058208380765827835415876) / 400075220255142853369998
$\beta_{6}$	$=$	$( - 65\!\cdots\!66 \nu^{13} + \cdots - 80\!\cdots\!59 ) / 13\!\cdots\!66$ (-655816572655256173666v^13 + 2621795709923698581799v^12 - 8959073740805095866529v^11 + 7590742954590928537014v^10 + 106741305652195573627534v^9 - 175445456788942424323467v^8 - 1030000812023891235975941v^7 - 3513171046375026813448063v^6 - 6941699530669444172908485v^5 - 7240222730099762622434032v^4 - 7457309088618210464192289v^3 - 4073569757384823466780871v^2 - 2051047740757306036667883*v - 807628119295547266180359) / 133358406751714284456666
$\beta_{7}$	$=$	$( - 21533854618238 \nu^{13} + 82305414388808 \nu^{12} - 282696080660293 \nu^{11} + \cdots - 39\!\cdots\!67 ) / 41\!\cdots\!38$ (-21533854618238v^13 + 82305414388808v^12 - 282696080660293v^11 + 208529521379666v^10 + 3518001233130454v^9 - 5162749382962420v^8 - 34185558542409345v^7 - 121531212012501894v^6 - 256075826473851519v^5 - 302525716260970376v^4 - 332822288669934246v^3 - 222055115782592859v^2 - 103523915401840572*v - 39009446378285067) / 4188226826337738
$\beta_{8}$	$=$	$( - 31182611013817 \nu^{13} + 115081687659689 \nu^{12} - 394131524526978 \nu^{11} + \cdots - 63\!\cdots\!10 ) / 41\!\cdots\!38$ (-31182611013817v^13 + 115081687659689v^12 - 394131524526978v^11 + 251513469646476v^10 + 5123696961983041v^9 - 6792175389581239v^8 - 50373480832645657v^7 - 183032509779839877v^6 - 394291539378058920v^5 - 485883219988910179v^4 - 534010189406698283v^3 - 364389710988000057v^2 - 176084461641822597*v - 63178323078025110) / 4188226826337738
$\beta_{9}$	$=$	$( - 12\!\cdots\!37 \nu^{13} + \cdots - 20\!\cdots\!31 ) / 13\!\cdots\!66$ (-1292875188440463886637v^13 + 5044299055571848865740v^12 - 17393694875400052764897v^11 + 13908638727123798182268v^10 + 210295047540703667689313v^9 - 328441375655822029345616v^8 - 2017907081052098649531926v^7 - 7141017572562646996675416v^6 - 14859021514859107777780917v^5 - 17158672527478457739997325v^4 - 19071806304939377823065260v^3 - 12270536239128476169663780v^2 - 5812440102989495912289354*v - 2050217426295375688782831) / 133358406751714284456666
$\beta_{10}$	$=$	$( 22\!\cdots\!49 \nu^{13} + \cdots + 43\!\cdots\!70 ) / 20\!\cdots\!99$ (2228111118284671151849v^13 - 8331672429664159739892v^12 + 28502691818078486949647v^11 - 19045752532055709100828v^10 - 366194740305550434957897v^9 + 504331660457959500076569v^8 + 3585594964071891227617501v^7 + 12879381460339741066797627v^6 + 27429105748070247948155679v^5 + 33159701828622258307616783v^4 + 36302725728110063174296010v^3 + 24418708661391023845119303v^2 + 12274875290797601297328177*v + 4338365839913048067304470) / 200037610127571426684999
$\beta_{11}$	$=$	$( - 27\!\cdots\!42 \nu^{13} + \cdots - 54\!\cdots\!68 ) / 20\!\cdots\!99$ (-2720816824926069833942v^13 + 9837799193658608156019v^12 - 33389905755542084509478v^11 + 18396740983875651179968v^10 + 451973933455075756206096v^9 - 561687874190454746017671v^8 - 4480591555104479486226799v^7 - 16237222877140510156785681v^6 - 35186913965419806867371970v^5 - 43635482616791033915049077v^4 - 47123707619006479343031236v^3 - 32788984548177963215069835v^2 - 15851021131564099448385876*v - 5408742487458917731524768) / 200037610127571426684999
$\beta_{12}$	$=$	$( - 19\!\cdots\!28 \nu^{13} + \cdots - 29\!\cdots\!81 ) / 13\!\cdots\!66$ (-1985469993210884210128v^13 + 7402484512750162263472v^12 - 25257644171895100176815v^11 + 16589864745362213483746v^10 + 326944468041603850155380v^9 - 445641070132937019723620v^8 - 3207940218450241326028059v^7 - 11522912172768970146713952v^6 - 24467905469425222094811297v^5 - 29186778495010457239544152v^4 - 31091379852759478117619784v^3 - 19812928931425264054094217v^2 - 8838741407682062426046726*v - 2973709795774406767277781) / 133358406751714284456666
$\beta_{13}$	$=$	$( 26\!\cdots\!31 \nu^{13} + \cdots + 43\!\cdots\!82 ) / 13\!\cdots\!66$ (2611149776540102525231v^13 - 9325428253760128953270v^12 + 31406316705782406177288v^11 - 15378744014265076814552v^10 - 437580538146345244146571v^9 + 522754107461387257149428v^8 + 4357170112760915263086125v^7 + 15722988152476020659180762v^6 + 34126491721982963558127306v^5 + 42117799105089209656717649v^4 + 44500955341146095883163886v^3 + 30300453993869606024890885v^2 + 13315364507826515377396998*v + 4311459504239174449239582) / 133358406751714284456666

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$- 3 \beta_{11} - 4 \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots + 1$ -3b11 - 4b10 - b9 + b8 - 2b7 + b5 + 3b4 + 3b3 - b2 + 4b1 + 1
$\nu^{3}$	$=$	$- \beta_{13} - 2 \beta_{12} - 18 \beta_{11} - 13 \beta_{10} - \beta_{9} + 18 \beta_{8} - 17 \beta_{7} + \cdots + 6$ -b13 - 2b12 - 18b11 - 13b10 - b9 + 18b8 - 17b7 + 3b6 + 6b5 + 17b4 + 17b3 - 18b2 + 11*b1 + 6
$\nu^{4}$	$=$	$- 2 \beta_{12} - 26 \beta_{11} + 2 \beta_{10} - 9 \beta_{9} + 90 \beta_{8} - 74 \beta_{7} + 16 \beta_{6} + \cdots + 76$ -2b12 - 26b11 + 2b10 - 9b9 + 90b8 - 74b7 + 16b6 + 2b5 + 35b4 + 14b3 - 45b2 + 36b1 + 76
$\nu^{5}$	$=$	$60 \beta_{13} + 15 \beta_{12} + 131 \beta_{11} + 55 \beta_{10} - 35 \beta_{9} + 105 \beta_{8} + \cdots + 145$ 60b13 + 15b12 + 131b11 + 55b10 - 35b9 + 105b8 - 203b7 + 12b6 + 24b5 + 55b4 - 45b3 + 104b2 + 255*b1 + 145
$\nu^{6}$	$=$	$248 \beta_{13} - 71 \beta_{12} + 161 \beta_{11} - 578 \beta_{10} - 76 \beta_{9} - 286 \beta_{8} + \cdots - 846$ 248b13 - 71b12 + 161b11 - 578b10 - 76b9 - 286b8 - 403b7 - 90b6 + 524b5 + 614b4 + 558b3 + 872b2 + 968*b1 - 846
$\nu^{7}$	$=$	$- 155 \beta_{13} - 1340 \beta_{12} - 3804 \beta_{11} - 3873 \beta_{10} + 665 \beta_{9} + 1563 \beta_{8} + \cdots - 6077$ -155b13 - 1340b12 - 3804b11 - 3873b10 + 665b9 + 1563b8 - 1915b7 + 239b6 + 3011b5 + 4041b4 + 5214b3 - 526b2 - 640*b1 - 6077
$\nu^{8}$	$=$	$- 3299 \beta_{13} - 4927 \beta_{12} - 11076 \beta_{11} + 3716 \beta_{10} + 4830 \beta_{9} + 20418 \beta_{8} + \cdots - 7639$ -3299b13 - 4927b12 - 11076b11 + 3716b10 + 4830b9 + 20418b8 - 8780b7 + 4029b6 + 3672b5 + 4782b4 + 6143b3 - 13571b2 - 22657*b1 - 7639
$\nu^{9}$	$=$	$2637 \beta_{13} + 3399 \beta_{12} + 81374 \beta_{11} + 112330 \beta_{10} + 13965 \beta_{9} + \cdots + 24275$ 2637b13 + 3399b12 + 81374b11 + 112330b10 + 13965b9 + 17449b8 + 10745b7 + 4515b6 - 30956b5 - 72530b4 - 94128b3 + 26193b2 - 89165*b1 + 24275
$\nu^{10}$	$=$	$83383 \beta_{13} + 63420 \beta_{12} + 592171 \beta_{11} + 354097 \beta_{10} + 25464 \beta_{9} + \cdots - 215822$ 83383b13 + 63420b12 + 592171b11 + 354097b10 + 25464b9 - 513614b8 + 358027b7 - 93366b6 - 104688b5 - 401389b4 - 441496b3 + 575801b2 - 135075*b1 - 215822
$\nu^{11}$	$=$	$83469 \beta_{13} - 40939 \beta_{12} + 622166 \beta_{11} - 466458 \beta_{10} + 220603 \beta_{9} + \cdots - 2760396$ 83469b13 - 40939b12 + 622166b11 - 466458b10 + 220603b9 - 2573157b8 + 1896408b7 - 461459b6 + 274146b5 - 552001b4 + 73804b3 + 1747483b2 - 712969*b1 - 2760396
$\nu^{12}$	$=$	$- 1970212 \beta_{13} - 1359828 \beta_{12} - 5892349 \beta_{11} - 2979929 \beta_{10} + 1728787 \beta_{9} + \cdots - 7756273$ -1970212b13 - 1359828b12 - 5892349b11 - 2979929b10 + 1728787b9 - 1439308b8 + 4612206b7 - 50604b6 + 909906b5 + 283071b4 + 3836420b3 - 3885125b2 - 8994311*b1 - 7756273
$\nu^{13}$	$=$	$- 8448626 \beta_{13} + 99309 \beta_{12} - 4758205 \beta_{11} + 24594061 \beta_{10} + 5023298 \beta_{9} + \cdots + 21825281$ -8448626b13 + 99309b12 - 4758205b11 + 24594061b10 + 5023298b9 + 17592077b8 + 9111332b7 + 4446804b6 - 15226542b5 - 19874021b4 - 18813265b3 - 29083981b2 - 43661981*b1 + 21825281

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$ .

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$1 + \beta_{8}$	$-1 - \beta_{8}$	$1$

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}

1297.1

−0.715832	+	0.0321480i
1.01424	+	4.42940i
−0.0617823	−	1.05131i
3.90955	−	1.59252i
−2.04761	+	0.151491i
−0.455169	−	1.74670i
−0.143399	+	0.643509i
−0.715832	−	0.0321480i
1.01424	−	4.42940i
−0.0617823	+	1.05131i
3.90955	+	1.59252i
−2.04761	−	0.151491i
−0.455169	+	1.74670i
−0.143399	−	0.643509i

−2.94714

−

1.70153i

−0.0930505

−

2.64411i

1297.2

−2.54554

−

1.46967i

−2.28666

1.33086i

1297.3

−2.09200

−

1.20782i

2.52468

0.791199i

1297.4

0.983604

0.567884i

1.92982

−

1.80991i

1297.5

1.05101

0.606802i

1.21552

−

2.35000i

1297.6

1.77894

1.02707i

1.06574

2.42161i

1297.7

3.77112

2.17726i

−2.35606

−

1.20374i

1369.1

−2.94714

1.70153i

−0.0930505

2.64411i

1369.2

−2.54554

1.46967i

−2.28666

−

1.33086i

1369.3

−2.09200

1.20782i

2.52468

−

0.791199i

1369.4

0.983604

−

0.567884i

1.92982

1.80991i

1369.5

1.05101

−

0.606802i

1.21552

2.35000i

1369.6

1.77894

−

1.02707i

1.06574

−

2.42161i

1369.7

3.77112

−

2.17726i

−2.35606

1.20374i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.k	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3276.2.hi.g		14
3.b	odd	2	1		1092.2.cx.d	yes	14
7.c	even	3	1		3276.2.fe.g		14
13.e	even	6	1		3276.2.fe.g		14
21.h	odd	6	1		1092.2.cf.d	✓	14
39.h	odd	6	1		1092.2.cf.d	✓	14
91.k	even	6	1	inner	3276.2.hi.g		14
273.bp	odd	6	1		1092.2.cx.d	yes	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1092.2.cf.d	✓	14	21.h	odd	6	1
1092.2.cf.d	✓	14	39.h	odd	6	1
1092.2.cx.d	yes	14	3.b	odd	2	1
1092.2.cx.d	yes	14	273.bp	odd	6	1
3276.2.fe.g		14	7.c	even	3	1
3276.2.fe.g		14	13.e	even	6	1
3276.2.hi.g		14	1.a	even	1	1	trivial
3276.2.hi.g		14	91.k	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3276, [\chi])$ :

T_{5}^{14} - 26 T_{5}^{12} + 494 T_{5}^{10} + 294 T_{5}^{9} - 3887 T_{5}^{8} - 2154 T_{5}^{7} + \cdots + 88752

T_{19}^{14} + 18 T_{19}^{13} + 95 T_{19}^{12} - 234 T_{19}^{11} - 2853 T_{19}^{10} + 14358 T_{19}^{9} + \cdots + 143883

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{14}$ T^14
$3$	$T^{14}$ T^14
$5$	$T^{14} - 26 T^{12} + \cdots + 88752$ T^14 - 26T^12 + 494T^10 + 294T^9 - 3887T^8 - 2154T^7 + 22670T^6 + 2496T^5 - 71690T^4 + 24540T^3 + 156961T^2 - 211044*T + 88752
$7$	$T^{14} - 4 T^{13} + \cdots + 823543$ T^14 - 4T^13 + 10T^12 + 6T^11 - 48T^10 + 6T^9 + 752T^8 - 2843T^7 + 5264T^6 + 294T^5 - 16464T^4 + 14406T^3 + 168070T^2 - 470596*T + 823543
$11$	$T^{14} + 6 T^{13} + \cdots + 23232$ T^14 + 6T^13 - 25T^12 - 222T^11 + 795T^10 + 3372T^9 - 13325T^8 - 28950T^7 + 224358T^6 - 456600T^5 + 401267T^4 - 80157T^3 - 85679T^2 + 20328*T + 23232
$13$	$T^{14} - 10 T^{13} + \cdots + 62748517$ T^14 - 10T^13 + 60T^12 - 221T^11 + 380T^10 + 1503T^9 - 15938T^8 + 74537T^7 - 207194T^6 + 254007T^5 + 834860T^4 - 6311981T^3 + 22277580T^2 - 48268090*T + 62748517
$17$	$(T^{7} - 6 T^{6} + \cdots + 1626)^{2}$ (T^7 - 6T^6 - 33T^5 + 152T^4 + 432T^3 - 924T^2 - 1523T + 1626)^2
$19$	$T^{14} + 18 T^{13} + \cdots + 143883$ T^14 + 18T^13 + 95T^12 - 234T^11 - 2853T^10 + 14358T^9 + 260080T^8 + 1417161T^7 + 4164441T^6 + 7044285T^5 + 6304364T^4 + 1892400T^3 - 646871T^2 - 312075*T + 143883
$23$	$(T^{7} + 4 T^{6} + \cdots - 16344)^{2}$ (T^7 + 4T^6 - 63T^5 - 248T^4 + 1123T^3 + 4317T^2 - 4629T - 16344)^2
$29$	$T^{14} + \cdots + 1984702500$ T^14 - T^13 + 95T^12 + 62T^11 + 6034T^10 + 6316T^9 + 212486T^8 + 438553T^7 + 5417518T^6 + 11219748T^5 + 82760670T^4 + 191191968T^3 + 849318129T^2 + 1221694650T + 1984702500
$31$	$T^{14} - 3 T^{13} + \cdots + 53868$ T^14 - 3T^13 - 55T^12 + 174T^11 + 3201T^10 + 5685T^9 - 10865T^8 - 29445T^7 + 40599T^6 + 133593T^5 + 44972T^4 - 111747T^3 - 40337T^2 + 77586*T + 53868
$37$	$T^{14} + 303 T^{12} + \cdots + 5250987$ T^14 + 303T^12 + 33372T^10 + 1672245T^8 + 41188743T^6 + 483618600T^4 + 2158441425T^2 + 5250987
$41$	$T^{14} + \cdots + 797070000$ T^14 - 27T^13 + 234T^12 + 243T^11 - 11982T^10 + 6702T^9 + 494957T^8 - 966027T^7 - 7684530T^6 + 16969644T^5 + 95675073T^4 - 211864440T^3 - 202862375T^2 + 528364500*T + 797070000
$43$	$T^{14} + 96 T^{12} + \cdots + 1890625$ T^14 + 96T^12 + 654T^11 + 8306T^10 + 36541T^9 + 204365T^8 + 689663T^7 + 2995471T^6 + 8112442T^5 + 21477996T^4 + 28443162T^3 + 32461319T^2 - 6927250T + 1890625
$47$	$T^{14} + \cdots + 4380012300$ T^14 + 15T^13 - 61T^12 - 2040T^11 + 6294T^10 + 157134T^9 - 185669T^8 - 6458184T^7 + 15570687T^6 + 107860746T^5 - 198162487T^4 - 1208460330T^3 + 2183570845T^2 + 6302930550*T + 4380012300
$53$	$T^{14} + \cdots + 1208794700304$ T^14 - 4T^13 + 241T^12 - 752T^11 + 38092T^10 - 111955T^9 + 3328955T^8 - 8168752T^7 + 206266354T^6 - 500891045T^5 + 7465840726T^4 - 16200021127T^3 + 182464268893T^2 - 387799808892*T + 1208794700304
$59$	$T^{14} + \cdots + 1159898258412$ T^14 + 561T^12 + 129677T^10 + 15817380T^8 + 1076086498T^6 + 39014873235T^4 + 611884860669T^2 + 1159898258412
$61$	$T^{14} + 7 T^{13} + \cdots + 5593225$ T^14 + 7T^13 + 164T^12 + 1635T^11 + 24348T^10 + 179789T^9 + 1181266T^8 + 3965035T^7 + 11522939T^6 + 7238384T^5 + 31002029T^4 + 34175790T^3 + 35691645T^2 + 15609000*T + 5593225
$67$	$T^{14} + \cdots + 2106078410700$ T^14 - 12T^13 - 199T^12 + 2964T^11 + 32109T^10 - 409482T^9 - 3027667T^8 + 33282462T^7 + 257155108T^6 - 1398571662T^5 - 13124119851T^4 + 24905904309T^3 + 512446271931T^2 + 1745830339110*T + 2106078410700
$71$	$T^{14} + \cdots + 4340544367500$ T^14 - 3T^13 - 279T^12 + 846T^11 + 54018T^10 - 99576T^9 - 5605362T^8 + 5039415T^7 + 430045686T^6 + 477935316T^5 - 17446271994T^4 - 43661145528T^3 + 477293598441T^2 + 2671846199550*T + 4340544367500
$73$	$T^{14} + \cdots + 8265915243$ T^14 + 3T^13 - 157T^12 - 480T^11 + 18723T^10 + 15021T^9 - 962765T^8 - 220596T^7 + 36827307T^6 - 38158821T^5 - 387171508T^4 + 565393311T^3 + 3241105450T^2 - 9613884123*T + 8265915243
$79$	$T^{14} + \cdots + 60397977600$ T^14 - 28T^13 + 720T^12 - 9216T^11 + 130464T^10 - 1136352T^9 + 13607424T^8 - 91601664T^7 + 872383488T^6 - 3613766656T^5 + 31893867520T^4 - 88408233984T^3 + 742295506944T^2 + 207570862080*T + 60397977600
$83$	$T^{14} + 340 T^{12} + \cdots + 1080000$ T^14 + 340T^12 + 33756T^10 + 954706T^8 + 8462056T^6 + 23746455T^4 + 15462225T^2 + 1080000
$89$	$T^{14} + \cdots + 36318923597232$ T^14 + 934T^12 + 341589T^10 + 62313685T^8 + 5986956088T^6 + 291192589755T^4 + 6107596944129T^2 + 36318923597232
$97$	$T^{14} + \cdots + 42203904094323$ T^14 + 12T^13 - 429T^12 - 5724T^11 + 142356T^10 + 1908312T^9 - 19454038T^8 - 272587740T^7 + 2447291154T^6 + 21273812244T^5 - 192851546964T^4 - 804882885312T^3 + 12445400293309T^2 - 38253114960192*T + 42203904094323
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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 1297.7
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