LMFDB - Newform orbit 3276.2.cf.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,2,Mod(1765,3276)]

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, 0, 5]))

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

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

chi := DirichletCharacter("3276.1765");

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.cf (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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704$ x^16 + 38x^14 + 587x^12 + 4762x^10 + 21849x^8 + 56552x^6 + 76456x^4 + 42624*x^2 + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 364)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{13} + \beta_{12} + \beta_{2}) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + (\beta_{10} - \beta_{8} + \cdots - 2 \beta_{3}) q^{11} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \cdots + 2) q^{13}+ \cdots + ( - 2 \beta_{15} + \beta_{13} + \cdots - 2) q^{97}+O(q^{100})$ q + (-b13 + b12 + b2) * q^5 + (-b3 - b2) * q^7 + (b10 - b8 - b7 - b5 - 2b3) q^11 + (-b15 + b13 - b12 - b10 + b9 + b8 + b6 + b4 - b2 + 2) * q^13 + (b13 - b11 + b9 + b8 + b7 + 2b5 + b4) q^17 + (b13 + b11 - b8 + b7) * q^19 + (-b15 + b14 + b10 + b8 - b6 - b2) * q^23 + (-b13 + b11 - b10 + b8 - b6 - b5 + b4 - 4b3 - 2b2 - b1 - 2) * q^25 + (-b15 - b14 - b10 + 2b9 - b8 + 2b7 - b3 - b2 + 2) * q^29 + (-b15 - 2b14 + b12 - b11 + b10 + 2b9 + 2b8 + b7 - b6 - b5 - b4 - b2 - b1 + 1) q^31 + (b13 - b11 - b9) * q^35 + (-2b13 + 2b11 + b10 - b8 - b7 - b6 + b5 + b3) * q^37 + (-3b15 + b13 + b12 - b11 - 3b10 + 3b8 - 2b5 + b3) * q^41 + (-2b15 + b13 - b11 - 2b10 - 3b9 + 2b8 + 2b7 - 4b5 - 2b4 + b3 - b2) q^43 + (-b13 + b12 - 2b9 - b5 - b4 - 5b2 - 1) * q^47 + (b9 + 1) * q^49 + (2b15 + 2b12 + 2b11 - 2b8 + 2b7 - 2b6 - 1) * q^53 + (-2b15 + b14 + b13 - b12 + b11 - 2b10 - b9 - 2b8 + 4b7 - 3b6 + b5 + 2b4 + b3 + b2 - 1) * q^55 + (-b14 + b13 + b9 - b8 + b7 + b6 + b4 - b3 + b1 + 2) * q^59 + (2b15 + 2b14 + b13 - 2b12 + 2b10 - 3b8 - 3b7 + b6 + b3 - 3b2 + 2b1) * q^61 + (-b15 - 3b14 + b13 - b11 - 2b10 + 2b8 + b6 - b5 - 3b4 - 3b3 + 3b2 - b1 + 2) * q^65 + (b15 + b14 - 2b13 + 2b12 + 2b11 + 2b10 - b9 - 2b8 - b7 - 2b6 - b5 - 5b3 - b2 + 2b1 + 1) * q^67 + (2b15 + 2b13 + 2b11 - 2b10 - 2b9 - 2b8 + 2b7 - 2b4 - 4b3 - 4b2 - 4) * q^71 + (-b15 + b12 - b11 + b10 + 2b8 + b7 - b6 + 2b5 + 2b4 + 3b2) * q^73 + (-b15 + b8 - b7 + b1 + 2) * q^77 + (3b15 + b13 - b11 - b10 - 2b8 + 3b7 + b6 + 2b5 - 2b4 + b1 - 2) q^79 + (-3b15 - 2b14 + b12 - b11 - b10 + 2b9 + 2b8 + 3b7 - b6 + b5 + b4 + 5b2 - b1 + 1) * q^83 + (-b14 - 2b11 + 2b9 + 3b8 - 3b7 + 2b6 - 2b4 + 5b3 + 6b2 + b1 + 4) * q^85 + (-b15 + 2b13 - 2b11 - 3b10 - 2b9 + 3b8 + 2b7 + b6 + 4b3 + 2) q^89 + (-b14 - b12 + b11 + b10 + b9 - b8 - b3 - b2) * q^91 + (-2b15 + b14 - 2b10 - 3b9 + b8 + b7 + 4b5 + 2b4 - 3b3 + 2b2 + b1) q^95 + (-2b15 + b13 - 2b11 + 2b10 - b9 + b8 - b7 + 3b6 + b4 - b3 - b2 - 2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{11} + 10 q^{13} - 2 q^{17} - 44 q^{25} + 22 q^{29} + 6 q^{35} + 12 q^{37} - 36 q^{41} + 6 q^{43} + 8 q^{49} - 8 q^{53} + 2 q^{55} + 18 q^{59} + 4 q^{61} + 30 q^{65} + 24 q^{67} - 36 q^{71} + 24 q^{77}+ \cdots - 42 q^{97}+O(q^{100})$ 16 * q - 6 * q^11 + 10 * q^13 - 2 * q^17 - 44 * q^25 + 22 * q^29 + 6 * q^35 + 12 * q^37 - 36 * q^41 + 6 * q^43 + 8 * q^49 - 8 * q^53 + 2 * q^55 + 18 * q^59 + 4 * q^61 + 30 * q^65 + 24 * q^67 - 36 * q^71 + 24 * q^77 + 8 * q^79 + 42 * q^85 + 36 * q^89 - 2 * q^91 + 30 * q^95 - 42 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704$ x^16 + 38*x^14 + 587*x^12 + 4762*x^10 + 21849*x^8 + 56552*x^6 + 76456*x^4 + 42624*x^2 + 2704 :

$\beta_{1}$	$=$	$( 125 \nu^{14} + 4204 \nu^{12} + 55047 \nu^{10} + 355436 \nu^{8} + 1184157 \nu^{6} + 1944230 \nu^{4} + \cdots + 104520 ) / 70144$ (125v^14 + 4204v^12 + 55047v^10 + 355436v^8 + 1184157v^6 + 1944230v^4 + 1318700*v^2 + 104520) / 70144
$\beta_{2}$	$=$	$( - 1005 \nu^{15} - 34940 \nu^{13} - 480631 \nu^{11} - 3354588 \nu^{9} - 12716909 \nu^{7} + \cdots - 8550920 \nu ) / 1823744$ (-1005v^15 - 34940v^13 - 480631v^11 - 3354588v^9 - 12716909v^7 - 26046678v^5 - 26288300v^3 - 8550920v) / 1823744
$\beta_{3}$	$=$	$( 1005 \nu^{15} - 56472 \nu^{14} + 34940 \nu^{13} - 1972672 \nu^{12} + 480631 \nu^{11} + \cdots - 53684800 ) / 3647488$ (1005v^15 - 56472v^14 + 34940v^13 - 1972672v^12 + 480631v^11 - 27102088v^10 + 3354588v^9 - 185927040v^8 + 12716909v^7 - 665206360v^6 + 26046678v^5 - 1162607472v^4 + 26288300v^3 - 779180064v^2 + 8550920*v - 53684800) / 3647488
$\beta_{4}$	$=$	$( - 29 \nu^{14} - 1012 \nu^{12} - 13879 \nu^{10} - 94948 \nu^{8} - 338349 \nu^{6} - 588174 \nu^{4} + \cdots - 25064 ) / 1024$ (-29v^14 - 1012v^12 - 13879v^10 - 94948v^8 - 338349v^6 - 588174v^4 - 390604v^2 + 512v - 25064) / 1024
$\beta_{5}$	$=$	$( 29 \nu^{14} + 1012 \nu^{12} + 13879 \nu^{10} + 94948 \nu^{8} + 338349 \nu^{6} + 588174 \nu^{4} + \cdots + 25064 ) / 1024$ (29v^14 + 1012v^12 + 13879v^10 + 94948v^8 + 338349v^6 + 588174v^4 + 390604v^2 + 512v + 25064) / 1024
$\beta_{6}$	$=$	$( - 6205 \nu^{15} - 37973 \nu^{14} - 205552 \nu^{13} - 1333644 \nu^{12} - 2636655 \nu^{11} + \cdots - 33629960 ) / 1823744$ (-6205v^15 - 37973v^14 - 205552v^13 - 1333644v^12 - 2636655v^11 - 18423951v^10 - 16549224v^9 - 126995596v^8 - 52753685v^7 - 455425269v^6 - 78865210v^5 - 793690742v^4 - 41280316v^3 - 524477772v^2 - 503192*v - 33629960) / 1823744
$\beta_{7}$	$=$	$( - 14471 \nu^{15} - 26962 \nu^{14} - 505724 \nu^{13} - 935168 \nu^{12} - 6952053 \nu^{11} + \cdots - 16802032 ) / 1823744$ (-14471v^15 - 26962v^14 - 505724v^13 - 935168v^12 - 6952053v^11 - 12726246v^10 - 47752076v^9 - 86194160v^8 - 171512567v^7 - 303079426v^6 - 304113450v^5 - 516819524v^4 - 217566628v^3 - 332049432v^2 - 31407032*v - 16802032) / 1823744
$\beta_{8}$	$=$	$( 15621 \nu^{15} + 56823 \nu^{14} + 542976 \nu^{13} + 1976156 \nu^{12} + 7402151 \nu^{11} + \cdots + 49562552 ) / 1823744$ (15621v^15 + 56823v^14 + 542976v^13 + 1976156v^12 + 7402151v^11 + 26992901v^10 + 50129176v^9 + 183778348v^8 + 175193597v^7 + 651076647v^6 + 291147418v^5 + 1123398250v^4 + 166058332v^3 + 739694436v^2 - 12937832*v + 49562552) / 1823744
$\beta_{9}$	$=$	$( - 241 \nu^{15} - 8404 \nu^{13} - 115155 \nu^{11} - 786788 \nu^{9} - 2796961 \nu^{7} - 4831958 \nu^{5} + \cdots - 13312 ) / 26624$ (-241v^15 - 8404v^13 - 115155v^11 - 786788v^9 - 2796961v^7 - 4831958v^5 - 3133372v^3 - 116680v - 13312) / 26624
$\beta_{10}$	$=$	$( 15621 \nu^{15} - 56823 \nu^{14} + 542976 \nu^{13} - 1976156 \nu^{12} + 7402151 \nu^{11} + \cdots - 49562552 ) / 1823744$ (15621v^15 - 56823v^14 + 542976v^13 - 1976156v^12 + 7402151v^11 - 26992901v^10 + 50129176v^9 - 183778348v^8 + 175193597v^7 - 651076647v^6 + 291147418v^5 - 1123398250v^4 + 166058332v^3 - 739694436v^2 - 12937832*v - 49562552) / 1823744
$\beta_{11}$	$=$	$( 19521 \nu^{15} - 39260 \nu^{14} + 686964 \nu^{13} - 1365416 \nu^{12} + 9521411 \nu^{11} + \cdots - 33180992 ) / 1823744$ (19521v^15 - 39260v^14 + 686964v^13 - 1365416v^12 + 9521411v^11 - 18669508v^10 + 65993284v^9 - 127456888v^8 + 238899889v^7 - 454035244v^6 + 423225494v^5 - 790598848v^4 + 288500860v^3 - 525824624v^2 + 22039240*v - 33180992) / 1823744
$\beta_{12}$	$=$	$( - 25726 \nu^{15} - 77233 \nu^{14} - 892516 \nu^{13} - 2699060 \nu^{12} - 12158066 \nu^{11} + \cdots - 66810952 ) / 1823744$ (-25726v^15 - 77233v^14 - 892516v^13 - 2699060v^12 - 12158066v^11 - 37093459v^10 - 82542508v^9 - 254452484v^8 - 291653574v^7 - 909460513v^6 - 502090704v^5 - 1584289590v^4 - 329781176v^3 - 1050302396v^2 - 22542432*v - 66810952) / 1823744
$\beta_{13}$	$=$	$( 25726 \nu^{15} - 77233 \nu^{14} + 892516 \nu^{13} - 2699060 \nu^{12} + 12158066 \nu^{11} + \cdots - 66810952 ) / 1823744$ (25726v^15 - 77233v^14 + 892516v^13 - 2699060v^12 + 12158066v^11 - 37093459v^10 + 82542508v^9 - 254452484v^8 + 291653574v^7 - 909460513v^6 + 502090704v^5 - 1584289590v^4 + 329781176v^3 - 1050302396v^2 + 22542432*v - 66810952) / 1823744
$\beta_{14}$	$=$	$( - 29241 \nu^{15} - 1625 \nu^{14} - 1021276 \nu^{13} - 54652 \nu^{12} - 14031675 \nu^{11} + \cdots - 1358760 ) / 1823744$ (-29241v^15 - 1625v^14 - 1021276v^13 - 54652v^12 - 14031675v^11 - 715611v^10 - 96318108v^9 - 4620668v^8 - 345320089v^7 - 15394041v^6 - 607350414v^5 - 25274990v^4 - 415878332v^3 - 17143100v^2 - 35393320*v - 1358760) / 1823744
$\beta_{15}$	$=$	$( 30092 \nu^{15} + 29861 \nu^{14} + 1048700 \nu^{13} + 1040988 \nu^{12} + 14354204 \nu^{11} + \cdots + 32760520 ) / 1823744$ (30092v^15 + 29861v^14 + 1048700v^13 + 1040988v^12 + 14354204v^11 + 14266655v^10 + 97881252v^9 + 97584188v^8 + 346706164v^7 + 347997221v^6 + 595260868v^5 + 606578726v^4 + 383624960v^3 + 407645004v^2 + 18469200*v + 32760520) / 1823744

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

$\nu$	$=$	$\beta_{5} + \beta_{4}$ b5 + b4
$\nu^{2}$	$=$	$\beta_{15} - \beta_{10} + \beta_{7} + 2\beta_{3} + \beta_{2} - \beta _1 - 5$ b15 - b10 + b7 + 2*b3 + b2 - b1 - 5
$\nu^{3}$	$=$	$2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} + 2 \beta_{9} + \beta_{6} - 6 \beta_{5} - 6 \beta_{4} + \cdots + 1$ 2b14 + b13 - 2b12 + b11 + 2b9 + b6 - 6b5 - 6*b4 - b2 + b1 + 1
$\nu^{4}$	$=$	$- 10 \beta_{15} + \beta_{13} + 2 \beta_{12} + \beta_{11} + 9 \beta_{10} + \beta_{8} - 10 \beta_{7} + \cdots + 38$ -10b15 + b13 + 2b12 + b11 + 9b10 + b8 - 10b7 - b6 - b5 + b4 - 28b3 - 14b2 + 9*b1 + 38
$\nu^{5}$	$=$	$2 \beta_{15} - 26 \beta_{14} - 12 \beta_{13} + 25 \beta_{12} - 13 \beta_{11} - \beta_{10} - 22 \beta_{9} + \cdots - 11$ 2b15 - 26b14 - 12b13 + 25b12 - 13b11 - b10 - 22b9 - 3b8 - 2b7 - 13b6 + 44b5 + 44b4 + 25b2 - 13*b1 - 11
$\nu^{6}$	$=$	$99 \beta_{15} - 15 \beta_{13} - 28 \beta_{12} - 13 \beta_{11} - 82 \beta_{10} - 17 \beta_{8} + \cdots - 331$ 99b15 - 15b13 - 28b12 - 13b11 - 82b10 - 17b8 + 99b7 + 13b6 + 15b5 - 15b4 + 306b3 + 153b2 - 70*b1 - 331
$\nu^{7}$	$=$	$- 28 \beta_{15} + 280 \beta_{14} + 123 \beta_{13} - 265 \beta_{12} + 142 \beta_{11} + 27 \beta_{10} + \cdots + 124$ -28b15 + 280b14 + 123b13 - 265b12 + 142b11 + 27b10 + 248b9 + 55b8 + 28b7 + 142b6 - 358b5 - 358b4 - 374b2 + 140b1 + 124
$\nu^{8}$	$=$	$- 977 \beta_{15} + 188 \beta_{13} + 320 \beta_{12} + 132 \beta_{11} + 771 \beta_{10} + 206 \beta_{8} + \cdots + 3065$ -977b15 + 188b13 + 320b12 + 132b11 + 771b10 + 206b8 - 977b7 - 132b6 - 158b5 + 158b4 - 3150b3 - 1575b2 + 529*b1 + 3065
$\nu^{9}$	$=$	$312 \beta_{15} - 2826 \beta_{14} - 1223 \beta_{13} + 2708 \beta_{12} - 1485 \beta_{11} - 460 \beta_{10} + \cdots - 1463$ 312b15 - 2826b14 - 1223b13 + 2708b12 - 1485b11 - 460b10 - 2926b9 - 772b8 - 312b7 - 1485b6 + 3086b5 + 3086b4 + 4671b2 - 1413b1 - 1463
$\nu^{10}$	$=$	$9654 \beta_{15} - 2207 \beta_{13} - 3480 \beta_{12} - 1273 \beta_{11} - 7405 \beta_{10} - 2249 \beta_{8} + \cdots - 29296$ 9654b15 - 2207b13 - 3480b12 - 1273b11 - 7405b10 - 2249b8 + 9654b7 + 1273b6 + 1435b5 - 1435b4 + 31776b3 + 15888b2 - 3949*b1 - 29296
$\nu^{11}$	$=$	$- 3360 \beta_{15} + 27686 \beta_{14} + 12158 \beta_{13} - 27499 \beta_{12} + 15341 \beta_{11} + \cdots + 17351$ -3360b15 + 27686b14 + 12158b13 - 27499b12 + 15341b11 + 6435b10 + 34702b9 + 9795b8 + 3360b7 + 15341b6 - 27558b5 - 27558b4 - 53881b2 + 13843b1 + 17351
$\nu^{12}$	$=$	$- 95721 \beta_{15} + 24959 \beta_{13} + 37294 \beta_{12} + 12335 \beta_{11} + 72152 \beta_{10} + \cdots + 285301$ -95721b15 + 24959b13 + 37294b12 + 12335b11 + 72152b10 + 23569b8 - 95721b7 - 12335b6 - 11833b5 + 11833b4 - 318406b3 - 159203b2 + 28862*b1 + 285301
$\nu^{13}$	$=$	$36406 \beta_{15} - 267484 \beta_{14} - 121781 \beta_{13} + 279755 \beta_{12} - 157974 \beta_{11} + \cdots - 202962$ 36406b15 - 267484b14 - 121781b13 + 279755b12 - 157974b11 - 81375b10 - 405924b9 - 117781b8 - 36406b7 - 157974b6 + 251910b5 + 251910b4 + 597556b2 - 133742b1 - 202962
$\nu^{14}$	$=$	$953123 \beta_{15} - 275542 \beta_{13} - 397536 \beta_{12} - 121994 \beta_{11} - 710589 \beta_{10} + \cdots - 2812765$ 953123b15 - 275542b13 - 397536b12 - 121994b11 - 710589b10 - 242534b8 + 953123b7 + 121994b6 + 88754b5 - 88754b4 + 3187822b3 + 1593911b2 - 201593*b1 - 2812765
$\nu^{15}$	$=$	$- 397768 \beta_{15} + 2570258 \beta_{14} + 1230119 \beta_{13} - 2856320 \beta_{12} + 1626201 \beta_{11} + \cdots + 2331497$ -397768b15 + 2570258b14 + 1230119b13 - 2856320b12 + 1626201b11 + 971326b10 + 4662994b9 + 1369094b8 + 397768b7 + 1626201b6 - 2341280b5 - 2341280b4 - 6489141b2 + 1285129b1 + 2331497

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

Character values

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

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$-\beta_{9}$	$1$	$1$

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}

1765.1

	−	2.42977i
	−	3.23100i
		1.17707i
		0.268953i
		2.98100i
	−	1.75101i
		1.77673i
	−	2.25607i
		2.25607i
	−	1.77673i
		1.75101i
	−	2.98100i
	−	0.268953i
	−	1.17707i
		3.23100i
		2.42977i

−

3.63781i

−0.866025

−

0.500000i

1765.2

−

3.14769i

0.866025

0.500000i

1765.3

−

1.46614i

−0.866025

−

0.500000i

1765.4

−

1.35585i

0.866025

0.500000i

1765.5

0.118179i

0.866025

0.500000i

1765.6

1.38536i

0.866025

0.500000i

1765.7

3.82804i

−0.866025

−

0.500000i

1765.8

4.27591i

−0.866025

−

0.500000i

2773.1

−

4.27591i

−0.866025

0.500000i

2773.2

−

3.82804i

−0.866025

0.500000i

2773.3

−

1.38536i

0.866025

−

0.500000i

2773.4

−

0.118179i

0.866025

−

0.500000i

2773.5

1.35585i

0.866025

−

0.500000i

2773.6

1.46614i

−0.866025

0.500000i

2773.7

3.14769i

0.866025

−

0.500000i

2773.8

3.63781i

−0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	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.cf.c		16
3.b	odd	2	1		364.2.u.a	✓	16
12.b	even	2	1		1456.2.cc.f		16
13.e	even	6	1	inner	3276.2.cf.c		16
21.c	even	2	1		2548.2.u.c		16
21.g	even	6	1		2548.2.bb.c		16
21.g	even	6	1		2548.2.bq.c		16
21.h	odd	6	1		2548.2.bb.d		16
21.h	odd	6	1		2548.2.bq.e		16
39.h	odd	6	1		364.2.u.a	✓	16
39.h	odd	6	1		4732.2.g.k		16
39.i	odd	6	1		4732.2.g.k		16
39.k	even	12	1		4732.2.a.s		8
39.k	even	12	1		4732.2.a.t		8
156.r	even	6	1		1456.2.cc.f		16
273.u	even	6	1		2548.2.u.c		16
273.x	odd	6	1		2548.2.bb.d		16
273.y	even	6	1		2548.2.bb.c		16
273.bp	odd	6	1		2548.2.bq.e		16
273.br	even	6	1		2548.2.bq.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.u.a	✓	16	3.b	odd	2	1
364.2.u.a	✓	16	39.h	odd	6	1
1456.2.cc.f		16	12.b	even	2	1
1456.2.cc.f		16	156.r	even	6	1
2548.2.u.c		16	21.c	even	2	1
2548.2.u.c		16	273.u	even	6	1
2548.2.bb.c		16	21.g	even	6	1
2548.2.bb.c		16	273.y	even	6	1
2548.2.bb.d		16	21.h	odd	6	1
2548.2.bb.d		16	273.x	odd	6	1
2548.2.bq.c		16	21.g	even	6	1
2548.2.bq.c		16	273.br	even	6	1
2548.2.bq.e		16	21.h	odd	6	1
2548.2.bq.e		16	273.bp	odd	6	1
3276.2.cf.c		16	1.a	even	1	1	trivial
3276.2.cf.c		16	13.e	even	6	1	inner
4732.2.a.s		8	39.k	even	12	1
4732.2.a.t		8	39.k	even	12	1
4732.2.g.k		16	39.h	odd	6	1
4732.2.g.k		16	39.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{16} + 62 T_{5}^{14} + 1505 T_{5}^{12} + 18058 T_{5}^{10} + 111420 T_{5}^{8} + 339950 T_{5}^{6} + \cdots + 3721$ T5^16 + 62*T5^14 + 1505*T5^12 + 18058*T5^10 + 111420*T5^8 + 339950*T5^6 + 492193*T5^4 + 273234*T5^2 + 3721 acting on $S_{2}^{\mathrm{new}}(3276, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16}$ T^16
$5$	$T^{16} + 62 T^{14} + \cdots + 3721$ T^16 + 62T^14 + 1505T^12 + 18058T^10 + 111420T^8 + 339950T^6 + 492193T^4 + 273234*T^2 + 3721
$7$	$(T^{4} - T^{2} + 1)^{4}$ (T^4 - T^2 + 1)^4
$11$	$T^{16} + \cdots + 185722384$ T^16 + 6T^15 - 39T^14 - 306T^13 + 1192T^12 + 10302T^11 - 14155T^10 - 195282T^9 + 81429T^8 + 2731320T^7 + 2485576T^6 - 19785144T^5 - 35166196T^4 + 91075584T^3 + 359132864T^2 + 421595808*T + 185722384
$13$	$T^{16} + \cdots + 815730721$ T^16 - 10T^15 + 56T^14 - 232T^13 + 630T^12 - 338T^11 - 7816T^10 + 56710T^9 - 245457T^8 + 737230T^7 - 1320904T^6 - 742586T^5 + 17993430T^4 - 86139976T^3 + 270301304T^2 - 627485170*T + 815730721
$17$	$T^{16} + 2 T^{15} + \cdots + 2982529$ T^16 + 2T^15 + 80T^14 + 212T^13 + 4662T^12 + 12190T^11 + 132152T^10 + 365786T^9 + 2744479T^8 + 6163134T^7 + 20284408T^6 + 18227802T^5 + 48773366T^4 + 37353820T^3 + 86101744T^2 + 16403046*T + 2982529
$19$	$T^{16} + \cdots + 1235663104$ T^16 - 89T^14 + 5884T^12 - 9906T^11 - 143793T^10 + 273618T^9 + 2618657T^8 - 4173480T^7 - 23933912T^6 + 32157600T^5 + 165076656T^4 - 116633088T^3 - 528837504T^2 + 263218176*T + 1235663104
$23$	$T^{16} + 92 T^{14} + \cdots + 88510464$ T^16 + 92T^14 - 64T^13 + 6444T^12 - 6144T^11 + 169136T^10 - 502656T^9 + 3376720T^8 - 7875072T^7 + 29188224T^6 - 58812416T^5 + 166387456T^4 - 250822656T^3 + 379029504T^2 - 202309632T + 88510464
$29$	$T^{16} + \cdots + 530426961$ T^16 - 22T^15 + 367T^14 - 3570T^13 + 30289T^12 - 182956T^11 + 1140486T^10 - 5378824T^9 + 27059074T^8 - 85677832T^7 + 294268214T^6 - 550124156T^5 + 1808842705T^4 - 2377835082T^3 + 5990273703T^2 + 1647868050*T + 530426961
$31$	$T^{16} + \cdots + 1643965037584$ T^16 + 384T^14 + 61390T^12 + 5275544T^10 + 262193913T^8 + 7529012968T^6 + 117129257624T^4 + 833486342048*T^2 + 1643965037584
$37$	$T^{16} + \cdots + 592240896$ T^16 - 12T^15 - 50T^14 + 1176T^13 + 5475T^12 - 46056T^11 - 97002T^10 + 881628T^9 + 1527209T^8 - 10838256T^7 - 9715444T^6 + 76831872T^5 + 68221312T^4 - 318996288T^3 - 121582656T^2 + 592240896*T + 592240896
$41$	$T^{16} + \cdots + 1637621852416$ T^16 + 36T^15 + 378T^14 - 1944T^13 - 52725T^12 + 87960T^11 + 6146002T^10 + 19462308T^9 - 229659991T^8 - 849692880T^7 + 7193009532T^6 + 14626121088T^5 - 120877931840T^4 - 98475622080T^3 + 1579494482368T^2 - 2733123528960*T + 1637621852416
$43$	$T^{16} + \cdots + 5967799496464$ T^16 - 6T^15 + 305T^14 - 1462T^13 + 60232T^12 - 275526T^11 + 6572285T^10 - 25613022T^9 + 497606725T^8 - 1925953752T^7 + 20147105240T^6 - 65069335784T^5 + 529193165612T^4 - 1648161979456T^3 + 5654690954240T^2 - 6308233383712*T + 5967799496464
$47$	$T^{16} + \cdots + 5725040896$ T^16 + 304T^14 + 34990T^12 + 1926624T^10 + 53054657T^8 + 702423856T^6 + 3981918048T^4 + 8403401472*T^2 + 5725040896
$53$	$(T^{8} + 4 T^{7} + \cdots + 117909)^{2}$ (T^8 + 4T^7 - 200T^6 - 684T^5 + 10722T^4 + 25164T^3 - 160560T^2 - 122148*T + 117909)^2
$59$	$T^{16} + \cdots + 119508624$ T^16 - 18T^15 + 30T^14 + 1404T^13 - 4781T^12 - 95340T^11 + 659246T^10 + 1212462T^9 - 17282291T^8 - 13940688T^7 + 349237900T^6 - 250919256T^5 - 1968840020T^4 + 1067407968T^3 + 9261802608T^2 + 1815324192*T + 119508624
$61$	$T^{16} + \cdots + 1654989623296$ T^16 - 4T^15 + 278T^14 - 640T^13 + 56043T^12 - 147224T^11 + 4196582T^10 - 13208884T^9 + 231129265T^8 - 609321432T^7 + 5622334624T^6 - 4480768128T^5 + 70013015744T^4 - 3285883904T^3 + 654017284096T^2 + 751624310784*T + 1654989623296
$67$	$T^{16} + \cdots + 62066753424$ T^16 - 24T^15 + 70T^14 + 2928T^13 - 13737T^12 - 344784T^11 + 3538326T^10 - 848280T^9 - 98332435T^8 + 124744992T^7 + 2141381852T^6 - 4255209504T^5 - 22447734452T^4 + 41091558144T^3 + 187259539632T^2 - 193198876416*T + 62066753424
$71$	$T^{16} + \cdots + 7345574313984$ T^16 + 36T^15 + 240T^14 - 6912T^13 - 78560T^12 + 1766976T^11 + 43321856T^10 + 275275008T^9 - 1070992640T^8 - 18118917120T^7 + 63460089856T^6 + 2210173440000T^5 + 15928809791488T^4 + 51153786765312T^3 + 56775948435456T^2 - 38305640153088*T + 7345574313984
$73$	$T^{16} + \cdots + 89262317824$ T^16 + 396T^14 + 47350T^12 + 2633732T^10 + 78468801T^8 + 1307926552T^6 + 11951716784T^4 + 54149796224*T^2 + 89262317824
$79$	$(T^{8} - 4 T^{7} + \cdots - 1070784)^{2}$ (T^8 - 4T^7 - 328T^6 + 1056T^5 + 30996T^4 - 101168T^3 - 862448T^2 + 3524352*T - 1070784)^2
$83$	$T^{16} + 656 T^{14} + \cdots + 1971216$ T^16 + 656T^14 + 157262T^12 + 17668920T^10 + 953546361T^8 + 21016522744T^6 + 71565626392T^4 + 38267407584*T^2 + 1971216
$89$	$T^{16} + \cdots + 33199755264$ T^16 - 36T^15 + 395T^14 + 1332T^13 - 36796T^12 - 41094T^11 + 2193483T^10 + 4611030T^9 - 67386279T^8 - 274634016T^7 + 1184061808T^6 + 12901908096T^5 + 49150842304T^4 + 105719691264T^3 + 136875125760T^2 + 100823703552*T + 33199755264
$97$	$T^{16} + 42 T^{15} + \cdots + 44302336$ T^16 + 42T^15 + 483T^14 - 4410T^13 - 106596T^12 + 384276T^11 + 17660299T^10 + 32284188T^9 - 1268856919T^8 - 4004939904T^7 + 70788698592T^6 + 352170240000T^5 - 1006255221248T^4 - 5142892093440T^3 + 20857866108928T^2 + 52656734208*T + 44302336
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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 1765.8
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	3276.2.cf.c

Level	$3276$
Weight	$2$
Character orbit	3276.cf
Analytic conductor	$26.159$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$