LMFDB - Newform orbit 3276.2.bi.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,2,Mod(1945,3276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3276, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("3276.1945");

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.bi (of order $4$ , 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 $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 36x^{14} + 472x^{12} + 2912x^{10} + 8914x^{8} + 13164x^{6} + 8828x^{4} + 2648x^{2} + 289$ x^16 + 36x^14 + 472x^12 + 2912x^10 + 8914x^8 + 13164x^6 + 8828x^4 + 2648*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q + (\beta_{14} + \beta_{4}) q^{5} + ( - \beta_{8} + \beta_{5} + \cdots - \beta_{3}) q^{7} + ( - \beta_{12} - \beta_{10} - 1) q^{11} + ( - \beta_{14} - \beta_{7} + \cdots - \beta_1) q^{13} - \beta_1 q^{17}+ \cdots + (3 \beta_{14} + \beta_{9} + \cdots + \beta_1) q^{97}+O(q^{100})$ q + (b14 + b4) * q^5 + (-b8 + b5 + b4 - b3) * q^7 + (-b12 - b10 - 1) * q^11 + (-b14 - b7 + b5 - b1) * q^13 - b1 * q^17 + (-b14 - b9 + b8 - b6 + b3 - b1) * q^19 + (-b15 + b12 + b11 + 2b10 + b8 - b4) q^23 + (-b15 + b12 + b11 + b10 + b8 - b4) * q^25 + (-b3 - 1) * q^29 + (-b14 - b9 - b4 - b1) * q^31 + (-b14 + b13 + b12 - b11 - b9 + b8 - b4 + b3 + 3) * q^35 + (-b15 + b12 + b10 - b8 + b4 - b3 - b2) * q^37 + (-2b14 - b9 + b8 - 2b6 - b4 + b3 - b1) * q^41 + (2b15 - b12 - b11 - b10 + b8 - b4) q^43 + (-b13 + b9 - b8 - 3b7 + b5 + b4 - b3 - b1) q^47 + (b14 + b13 - b12 - b11 - b10 - b8 - b7 + b6 + b4) * q^49 + (b12 - b11 + b3 - 3b2 - 3) q^53 + (3b9 - 4b8 + 4b5 - 4b3) * q^55 + (-3b9 - b7 + 3b1) * q^59 + (-2b14 + 2b13 - 2b9 + 2b8 + b7 + b6 - 2b4 + 2b3) * q^61 + (2b12 - b11 - 2b10 - b8 + b4 - 2b2 + 1) q^65 + (-2b11 - b10 + 1) q^67 + (3b15 - 3b11 - 2b10 - 2b8 + 2b4 + 2b3 - 3b2 - 1) q^71 + (b13 + b8 + b7 - b5 - b4 + b3) * q^73 + (b15 - 2b14 - 2b13 - b10 - b8 - 3b5 - 2b4 + b1) * q^77 + (-b12 + b11 - b3 + 2b2 + 1) q^79 + (-5b14 - b9 + 2b8 - 3b4 + 2b3 - b1) * q^83 + (-b15 + b12 + b10 - b2) * q^85 + (-b13 - 3b8 - 3b7 + 5b5 + 3b4 - 3b3) q^89 + (2b15 + 2b14 - b13 - b12 - b11 - 3b10 + b9 - b8 - b7 + b6 + b5 + 2b4 - b3 - b2 + 2b1) q^91 + (2b12 + 2b11 + 3b10 - b8 + b4) q^95 + (3b14 + b9 + b8 + b6 + 4b4 + b3 + b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{7} - 12 q^{11} - 20 q^{29} + 40 q^{35} - 28 q^{53} + 20 q^{65} + 8 q^{67} + 4 q^{71} + 4 q^{79} + 4 q^{85} + 10 q^{91}+O(q^{100})$ 16 * q - 2 * q^7 - 12 * q^11 - 20 * q^29 + 40 * q^35 - 28 * q^53 + 20 * q^65 + 8 * q^67 + 4 * q^71 + 4 * q^79 + 4 * q^85 + 10 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 36x^{14} + 472x^{12} + 2912x^{10} + 8914x^{8} + 13164x^{6} + 8828x^{4} + 2648x^{2} + 289$ x^16 + 36*x^14 + 472*x^12 + 2912*x^10 + 8914*x^8 + 13164*x^6 + 8828*x^4 + 2648*x^2 + 289 :

$\beta_{1}$	$=$	$( 590 \nu^{14} + 20965 \nu^{12} + 268794 \nu^{10} + 1595550 \nu^{8} + 4545302 \nu^{6} + 5780895 \nu^{4} + \cdots + 433464 ) / 14924$ (590v^14 + 20965v^12 + 268794v^10 + 1595550v^8 + 4545302v^6 + 5780895v^4 + 2750276*v^2 + 433464) / 14924
$\beta_{2}$	$=$	$( 590 \nu^{14} + 20965 \nu^{12} + 268794 \nu^{10} + 1595550 \nu^{8} + 4545302 \nu^{6} + 5780895 \nu^{4} + \cdots + 396154 ) / 7462$ (590v^14 + 20965v^12 + 268794v^10 + 1595550v^8 + 4545302v^6 + 5780895v^4 + 2742814*v^2 + 396154) / 7462
$\beta_{3}$	$=$	$( - 2102 \nu^{14} - 74891 \nu^{12} - 964252 \nu^{10} - 5760993 \nu^{8} - 16583714 \nu^{6} + \cdots - 1685303 ) / 14924$ (-2102v^14 - 74891v^12 - 964252v^10 - 5760993v^8 - 16583714v^6 - 21486785v^4 - 10614108*v^2 - 1685303) / 14924
$\beta_{4}$	$=$	$( - 33003 \nu^{15} + 23477 \nu^{14} - 1151473 \nu^{13} + 844135 \nu^{12} - 14284923 \nu^{11} + \cdots + 40958865 ) / 2029664$ (-33003v^15 + 23477v^14 - 1151473v^13 + 844135v^12 - 14284923v^11 + 11049133v^10 - 79742797v^9 + 68039219v^8 - 199279357v^7 + 207895091v^6 - 176254047v^5 + 306043129v^4 + 4632487v^3 + 198119343v^2 + 20626753*v + 40958865) / 2029664
$\beta_{5}$	$=$	$( 33003 \nu^{15} - 262395 \nu^{14} + 1151473 \nu^{13} - 9341041 \nu^{12} + 14284923 \nu^{11} + \cdots - 188242343 ) / 2029664$ (33003v^15 - 262395v^14 + 1151473v^13 - 9341041v^12 + 14284923v^11 - 120089139v^10 + 79742797v^9 - 715455829v^8 + 199279357v^7 - 2047490013v^6 + 176254047v^5 - 2616159631v^4 - 4632487v^3 - 1245399345v^2 - 20626753*v - 188242343) / 2029664
$\beta_{6}$	$=$	$( 39901 \nu^{15} + 53805 \nu^{14} + 1427545 \nu^{13} + 1919657 \nu^{12} + 18511581 \nu^{11} + \cdots + 48177439 ) / 1014832$ (39901v^15 + 53805v^14 + 1427545v^13 + 1919657v^12 + 18511581v^11 + 24778333v^10 + 111956893v^9 + 148703505v^8 + 329628295v^7 + 431561847v^6 + 447287131v^5 + 567705259v^4 + 245051627v^3 + 288834675v^2 + 45755883*v + 48177439) / 1014832
$\beta_{7}$	$=$	$( 39901 \nu^{15} - 53805 \nu^{14} + 1427545 \nu^{13} - 1919657 \nu^{12} + 18511581 \nu^{11} + \cdots - 48177439 ) / 1014832$ (39901v^15 - 53805v^14 + 1427545v^13 - 1919657v^12 + 18511581v^11 - 24778333v^10 + 111956893v^9 - 148703505v^8 + 329628295v^7 - 431561847v^6 + 447287131v^5 - 567705259v^4 + 245051627v^3 - 288834675v^2 + 45755883*v - 48177439) / 1014832
$\beta_{8}$	$=$	$( 93265 \nu^{15} + 23477 \nu^{14} + 3323523 \nu^{13} + 844135 \nu^{12} + 42807025 \nu^{11} + \cdots + 40958865 ) / 2029664$ (93265v^15 + 23477v^14 + 3323523v^13 + 844135v^12 + 42807025v^11 + 11049133v^10 + 255924679v^9 + 68039219v^8 + 737592567v^7 + 207895091v^6 + 956800189v^5 + 306043129v^4 + 467944667v^3 + 198119343v^2 + 66804429*v + 40958865) / 2029664
$\beta_{9}$	$=$	$( 24620 \nu^{15} + 876290 \nu^{13} + 11264235 \nu^{11} + 67123942 \nu^{9} + 192338330 \nu^{7} + \cdots + 18692776 \nu ) / 253708$ (24620v^15 + 876290v^13 + 11264235v^11 + 67123942v^9 + 192338330v^7 + 246827546v^5 + 119070145v^3 + 18692776v) / 253708
$\beta_{10}$	$=$	$( 24620 \nu^{15} + 876290 \nu^{13} + 11264235 \nu^{11} + 67123942 \nu^{9} + 192338330 \nu^{7} + \cdots + 18439068 \nu ) / 253708$ (24620v^15 + 876290v^13 + 11264235v^11 + 67123942v^9 + 192338330v^7 + 246827546v^5 + 119070145v^3 + 18439068v) / 253708
$\beta_{11}$	$=$	$( - 56088 \nu^{15} - 70312 \nu^{14} - 2001743 \nu^{13} - 2497538 \nu^{12} - 25853206 \nu^{11} + \cdots - 43679358 ) / 507416$ (-56088v^15 - 70312v^14 - 2001743v^13 - 2497538v^12 - 25853206v^11 - 31995360v^10 - 155342727v^9 - 189566694v^8 - 452152838v^7 - 537437048v^6 - 599563459v^5 - 674438450v^4 - 310933340v^3 - 308403528v^2 - 51029543*v - 43679358) / 507416
$\beta_{12}$	$=$	$( - 56088 \nu^{15} + 70312 \nu^{14} - 2001743 \nu^{13} + 2497538 \nu^{12} - 25853206 \nu^{11} + \cdots + 43679358 ) / 507416$ (-56088v^15 + 70312v^14 - 2001743v^13 + 2497538v^12 - 25853206v^11 + 31995360v^10 - 155342727v^9 + 189566694v^8 - 452152838v^7 + 537437048v^6 - 599563459v^5 + 674438450v^4 - 310933340v^3 + 308403528v^2 - 51029543*v + 43679358) / 507416
$\beta_{13}$	$=$	$( 259235 \nu^{15} + 268345 \nu^{14} + 9242377 \nu^{13} + 9562143 \nu^{12} + 119142639 \nu^{11} + \cdots + 205735581 ) / 2029664$ (259235v^15 + 268345v^14 + 9242377v^13 + 9562143v^12 + 119142639v^11 + 123143869v^10 + 713330057v^9 + 735919783v^8 + 2060989045v^7 + 2118410919v^6 + 2686420031v^5 + 2739428161v^4 + 1335890661v^3 + 1336744935v^2 + 215558683*v + 205735581) / 2029664
$\beta_{14}$	$=$	$( - 259235 \nu^{15} - 17527 \nu^{14} - 9242377 \nu^{13} - 623033 \nu^{12} - 119142639 \nu^{11} + \cdots - 23465627 ) / 2029664$ (-259235v^15 - 17527v^14 - 9242377v^13 - 623033v^12 - 119142639v^11 - 7994403v^10 - 713330057v^9 - 47575265v^8 - 2060989045v^7 - 136974185v^6 - 2686420031v^5 - 182774599v^4 - 1335890661v^3 - 106773753v^2 - 215558683*v - 23465627) / 2029664
$\beta_{15}$	$=$	$( - 39210 \nu^{15} - 1396175 \nu^{13} - 17958972 \nu^{11} - 107123534 \nu^{9} - 307406526 \nu^{7} + \cdots - 29255540 \nu ) / 253708$ (-39210v^15 - 1396175v^13 - 17958972v^11 - 107123534v^9 - 307406526v^7 - 395379877v^5 - 191385598v^3 - 29255540v) / 253708

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

$\nu$	$=$	$-\beta_{10} + \beta_{9}$ -b10 + b9
$\nu^{2}$	$=$	$-\beta_{2} + 2\beta _1 - 5$ -b2 + 2*b1 - 5
$\nu^{3}$	$=$	$3\beta_{15} - \beta_{14} + \beta_{13} + 10\beta_{10} - 9\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}$ 3b15 - b14 + b13 + 10b10 - 9*b9 + b7 + b6 + b5 - b4
$\nu^{4}$	$=$	$- 4 \beta_{14} - 4 \beta_{13} - \beta_{12} + \beta_{11} - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 51$ -4b14 - 4b13 - b12 + b11 - 4b7 + 4b6 - 4b5 - 4b4 + 2b3 + 16b2 - 28*b1 + 51
$\nu^{5}$	$=$	$- 60 \beta_{15} + 20 \beta_{14} - 20 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} - 111 \beta_{10} + \cdots + 2 \beta_{3}$ -60b15 + 20b14 - 20b13 + 5b12 + 5b11 - 111b10 + 109b9 - 8b8 - 21b7 - 21b6 - 22b5 + 30b4 + 2*b3
$\nu^{6}$	$=$	$80 \beta_{14} + 80 \beta_{13} + 27 \beta_{12} - 27 \beta_{11} + 86 \beta_{7} - 86 \beta_{6} + 92 \beta_{5} + \cdots - 653$ 80b14 + 80b13 + 27b12 - 27b11 + 86b7 - 86b6 + 92b5 + 92b4 - 52b3 - 255b2 + 390*b1 - 653
$\nu^{7}$	$=$	$1001 \beta_{15} - 335 \beta_{14} + 335 \beta_{13} - 119 \beta_{12} - 119 \beta_{11} + \cdots - 66 \beta_{3}$ 1001b15 - 335b14 + 335b13 - 119b12 - 119b11 + 1408b10 - 1499b9 + 158b8 + 366b7 + 366b6 + 401b5 - 559b4 - 66*b3
$\nu^{8}$	$=$	$- 1336 \beta_{14} - 1336 \beta_{13} - 520 \beta_{12} + 520 \beta_{11} - 1472 \beta_{7} + 1472 \beta_{6} + \cdots + 9269$ -1336b14 - 1336b13 - 520b12 + 520b11 - 1472b7 + 1472b6 - 1640b5 - 1640b4 + 956b3 + 4034b2 - 5672*b1 + 9269
$\nu^{9}$	$=$	$- 15930 \beta_{15} + 5370 \beta_{14} - 5370 \beta_{13} + 2160 \beta_{12} + 2160 \beta_{11} + \cdots + 1344 \beta_{3}$ -15930b15 + 5370b14 - 5370b13 + 2160b12 + 2160b11 - 19579b10 + 21969b9 - 2556b8 - 5942b7 - 5942b6 - 6714b5 + 9270b4 + 1344*b3
$\nu^{10}$	$=$	$21300 \beta_{14} + 21300 \beta_{13} + 8874 \beta_{12} - 8874 \beta_{11} + 23612 \beta_{7} + \cdots - 137957$ 21300b14 + 21300b13 + 8874b12 - 8874b11 + 23612b7 - 23612b6 + 26964b5 + 26964b4 - 15784b3 - 63211b2 + 85018*b1 - 137957
$\nu^{11}$	$=$	$249381 \beta_{15} - 84511 \beta_{14} + 84511 \beta_{13} - 35838 \beta_{12} - 35838 \beta_{11} + \cdots - 23412 \beta_{3}$ 249381b15 - 84511b14 + 84511b13 - 35838b12 - 35838b11 + 287200b10 - 331649b9 + 39596b8 + 93733b7 + 93733b6 + 107923b5 - 147519b4 - 23412*b3
$\nu^{12}$	$=$	$- 333892 \beta_{14} - 333892 \beta_{13} - 143761 \beta_{12} + 143761 \beta_{11} - 370284 \beta_{7} + \cdots + 2097467$ -333892b14 - 333892b13 - 143761b12 + 143761b11 - 370284b7 + 370284b6 - 428980b5 - 428980b4 + 250474b3 + 984342b2 - 1295316*b1 + 2097467
$\nu^{13}$	$=$	$- 3878186 \beta_{15} + 1318234 \beta_{14} - 1318234 \beta_{13} + 572741 \beta_{12} + \cdots + 382610 \beta_{3}$ -3878186b15 + 1318234b14 - 1318234b13 + 572741b12 + 572741b11 - 4331289b10 + 5073945b9 - 608432b8 - 1461339b7 - 1461339b6 - 1700844b5 + 2309276b4 + 382610*b3
$\nu^{14}$	$=$	$5196420 \beta_{14} + 5196420 \beta_{13} + 2273585 \beta_{12} - 2273585 \beta_{11} + 5757826 \beta_{7} + \cdots - 32193153$ 5196420b14 + 5196420b13 + 2273585b12 - 2273585b11 + 5757826b7 - 5757826b6 + 6724512b5 + 6724512b4 - 3913720b3 - 15276497b2 + 19894310*b1 - 32193153
$\nu^{15}$	$=$	$60135483 \beta_{15} - 20472917 \beta_{14} + 20472917 \beta_{13} - 8998097 \beta_{12} + \cdots - 6075262 \beta_{3}$ 60135483b15 - 20472917b14 + 20472917b13 - 8998097b12 - 8998097b11 + 66206746b10 - 78093287b9 + 9354110b8 + 22674458b7 + 22674458b6 + 26548179b5 - 35902289b4 - 6075262*b3

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

Character values

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

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$-\beta_{10}$	$-1$	$1$

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}

1945.1

	−	0.703701i
		0.631533i
		3.93284i
	−	0.536261i
		2.53626i
	−	1.93284i
		1.36847i
		2.70370i
		0.703701i
	−	0.631533i
	−	3.93284i
		0.536261i
	−	2.53626i
		1.93284i
	−	1.36847i
	−	2.70370i

−2.45467

2.45467i

−1.75980

−

1.97563i

1945.2

−1.64651

1.64651i

−0.468972

2.60386i

1945.3

−1.32637

1.32637i

−2.51586

−

0.818817i

1945.4

−0.0659529

0.0659529i

2.36200

−

1.19204i

1945.5

0.0659529

−

0.0659529i

1.19204

−

2.36200i

1945.6

1.32637

−

1.32637i

0.818817

2.51586i

1945.7

1.64651

−

1.64651i

−2.60386

0.468972i

1945.8

2.45467

−

2.45467i

1.97563

1.75980i

2449.1

−2.45467

−

2.45467i

−1.75980

1.97563i

2449.2

−1.64651

−

1.64651i

−0.468972

−

2.60386i

2449.3

−1.32637

−

1.32637i

−2.51586

0.818817i

2449.4

−0.0659529

−

0.0659529i

2.36200

1.19204i

2449.5

0.0659529

0.0659529i

1.19204

2.36200i

2449.6

1.32637

1.32637i

0.818817

−

2.51586i

2449.7

1.64651

1.64651i

−2.60386

−

0.468972i

2449.8

2.45467

2.45467i

1.97563

−

1.75980i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
13.d	odd	4	1	inner
91.i	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3276.2.bi.c		16
3.b	odd	2	1		364.2.n.a	✓	16
7.b	odd	2	1	inner	3276.2.bi.c		16
13.d	odd	4	1	inner	3276.2.bi.c		16
21.c	even	2	1		364.2.n.a	✓	16
39.f	even	4	1		364.2.n.a	✓	16
91.i	even	4	1	inner	3276.2.bi.c		16
273.o	odd	4	1		364.2.n.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.n.a	✓	16	3.b	odd	2	1
364.2.n.a	✓	16	21.c	even	2	1
364.2.n.a	✓	16	39.f	even	4	1
364.2.n.a	✓	16	273.o	odd	4	1
3276.2.bi.c		16	1.a	even	1	1	trivial
3276.2.bi.c		16	7.b	odd	2	1	inner
3276.2.bi.c		16	13.d	odd	4	1	inner
3276.2.bi.c		16	91.i	even	4	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}^{16} + 187T_{5}^{12} + 6431T_{5}^{8} + 52853T_{5}^{4} + 4

T_{19}^{16} + 1767T_{19}^{12} + 920035T_{19}^{8} + 157342549T_{19}^{4} + 3262922884

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16}$ T^16
$5$	$T^{16} + 187 T^{12} + \cdots + 4$ T^16 + 187T^12 + 6431T^8 + 52853*T^4 + 4
$7$	$T^{16} + 2 T^{15} + \cdots + 5764801$ T^16 + 2T^15 + 2T^14 + 30T^13 + 68T^12 + 118T^11 + 550T^10 + 1482T^9 + 3910T^8 + 10374T^7 + 26950T^6 + 40474T^5 + 163268T^4 + 504210T^3 + 235298T^2 + 1647086*T + 5764801
$11$	$(T^{8} + 6 T^{7} + \cdots + 1156)^{2}$ (T^8 + 6T^7 + 18T^6 - 46T^5 + 188T^4 + 596T^3 + 1250T^2 - 1700*T + 1156)^2
$13$	$T^{16} + \cdots + 815730721$ T^16 - 30T^14 + 876T^12 - 15258T^10 + 237270T^8 - 2578602T^6 + 25019436T^4 - 144804270*T^2 + 815730721
$17$	$(T^{8} - 14 T^{6} + 54 T^{4} + \cdots + 8)^{2}$ (T^8 - 14T^6 + 54T^4 - 66*T^2 + 8)^2
$19$	$T^{16} + \cdots + 3262922884$ T^16 + 1767T^12 + 920035T^8 + 157342549*T^4 + 3262922884
$23$	$(T^{8} + 83 T^{6} + \cdots + 484)^{2}$ (T^8 + 83T^6 + 1215T^4 + 2365*T^2 + 484)^2
$29$	$(T^{4} + 5 T^{3} - 3 T^{2} + \cdots - 22)^{4}$ (T^4 + 5T^3 - 3T^2 - 33*T - 22)^4
$31$	$T^{16} + 1347 T^{12} + \cdots + 334084$ T^16 + 1347T^12 + 40567T^8 + 312037*T^4 + 334084
$37$	$(T^{8} - 74 T^{5} + \cdots + 784996)^{2}$ (T^8 - 74T^5 + 3128T^4 - 5180T^3 + 2738T^2 + 65564*T + 784996)^2
$41$	$T^{16} + \cdots + 14178141184$ T^16 + 10308T^12 + 26979136T^8 + 5533018112*T^4 + 14178141184
$43$	$(T^{8} + 183 T^{6} + \cdots + 391876)^{2}$ (T^8 + 183T^6 + 9007T^4 + 109885*T^2 + 391876)^2
$47$	$T^{16} + \cdots + 1488209686084$ T^16 + 35495T^12 + 313380355T^8 + 53841772341*T^4 + 1488209686084
$53$	$(T^{4} + 7 T^{3} + \cdots + 2074)^{4}$ (T^4 + 7T^3 - 113T^2 - 143*T + 2074)^4
$59$	$T^{16} + \cdots + 78261181504$ T^16 + 21088T^12 + 129996400T^8 + 231367138704*T^4 + 78261181504
$61$	$(T^{8} + 196 T^{6} + \cdots + 86528)^{2}$ (T^8 + 196T^6 + 11600T^4 + 186272*T^2 + 86528)^2
$67$	$(T^{8} - 4 T^{7} + \cdots + 633616)^{2}$ (T^8 - 4T^7 + 8T^6 + 264T^5 + 6152T^4 - 4560T^3 + 3872T^2 + 70048*T + 633616)^2
$71$	$(T^{8} - 2 T^{7} + \cdots + 502566724)^{2}$ (T^8 - 2T^7 + 2T^6 + 582T^5 + 56288T^4 + 27664T^3 + 1458T^2 + 1210572*T + 502566724)^2
$73$	$T^{16} + 1423 T^{12} + \cdots + 521284$ T^16 + 1423T^12 + 431515T^8 + 2541285*T^4 + 521284
$79$	$(T^{4} - T^{3} - 65 T^{2} + \cdots + 724)^{4}$ (T^4 - T^3 - 65T^2 + 53T + 724)^4
$83$	$T^{16} + \cdots + 59341934404$ T^16 + 87363T^12 + 57033943T^8 + 9362370373*T^4 + 59341934404
$89$	$T^{16} + \cdots + 77940681873604$ T^16 + 101263T^12 + 2004422315T^8 + 9049264966901*T^4 + 77940681873604
$97$	$T^{16} + \cdots + 408528258774724$ T^16 + 46631T^12 + 562817267T^8 + 1435075611973*T^4 + 408528258774724
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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 1945.8
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.bi.c

Level	$3276$
Weight	$2$
Character orbit	3276.bi
Analytic conductor	$26.159$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$