LMFDB - Newform orbit 1224.2.bq.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(145,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 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("1224.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.bq (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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4$ x^12 + 28x^10 + 258x^8 + 880x^6 + 1033x^4 + 132*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 136)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{9} q^{5} + (\beta_{10} - \beta_{9} - \beta_{8} + \cdots + 1) q^{7} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 1) q^{11} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{13} + ( - \beta_{8} - 2 \beta_{7} + \cdots - \beta_{3}) q^{17}+ \cdots + (2 \beta_{9} - 3 \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{97}+O(q^{100})$ q - b9 * q^5 + (b10 - b9 - b8 - b7 - b5 + 1) * q^7 + (-b10 - b9 + b7 - b6 + b2 + b1 + 1) * q^11 + (b7 - b6 - b5 + b3 + 2b1) q^13 + (-b8 - 2b7 - b6 - b5 - b3) q^17 + (-b10 + b8 + b5 - b4 + b1 - 1) * q^19 + (b11 + b7 + 2b6 - b4 - b3 - b2 - 2b1 + 1) * q^23 + (-b11 + b9 - b6 + b5 + b4 - b3 + b2 - 2b1 - 2) q^25 + (b9 + b7 + b6 - b5 + b2 - b1 - 1) * q^29 + (-b10 - 3b6 - b5 - b4 - b2 - 3) q^31 + (-b11 - b10 - 3b7 - 3b6 - b5 - b3 + 2) * q^35 + (b11 - b10 + b8 + 2b7 + 2b1) * q^37 + (b10 - b9 - 2b8 - 3b6 + 2b4 - 2b3 - 3b1) q^41 + (-b11 + b10 + b9 + b8 + 2b6 - b5 - b4) q^43 + (-b11 + b10 - b9 + b8 + 2b7 - 2b6 - b5 + b4 + b3 - b2 - 2b1) q^47 + (-b11 + 2b10 - b9 - 2b8 - 5b7 - 2b5 + 2b4 - b3 - b2 - 5b1 + 5) * q^49 + (-b10 + b8 - b5 + b4 + b3 + b2) * q^53 + (b11 - b10 + 5b7 - 5b6 + b5 - b3 - 4b1) q^55 + (-b11 + b10 + b9 + b8 - 2b6 - b5 - b4 - 2b1 - 2) * q^59 + (-b8 - 4b7 - 2b6 + 2b4 - 2b3 - 2b1 + 4) q^61 + (-2b10 + b7 + b6 - b5 - 2b4 - b2 + b1 + 1) * q^65 + (b9 + b8 + 3b7 + 3b6 + b5 + b4 + b3 + b2 - 4) * q^67 + (-b10 + 2b7 - 3b6 + b5 - b4 + b2 + 2b1 - 3) q^71 + (b11 - 2b9 - b8 - b7 - 2b6 - 2b5 + 2b3 + 2b2 + b1 + 2) q^73 + (3b11 - 2b10 - 3b9 - 2b8 - 10b6 - 2b5 - 2b4 - b3 + b2 + 3b1 + 3) * q^77 + (-b11 - 2b10 - 2b9 - b7 + 2b6 - b4 - b3 + b2 - 2b1 - 1) * q^79 + (-b11 - b10 - b9 + b8 - 2b7 + b3 + b2 + 4b1 - 4) * q^83 + (2b11 - b9 - 2b8 + 3b7 + 3b6 - b4 - b3 - 2b2 - b1 + 5) q^85 + (-3b11 + 3b10 - 4b7 + 4b6 + b4 - b2 + 2b1) q^89 + (4b11 - 2b10 - 2b9 - 2b7 - 4b6 - 2b4 - 2b3 + 2b2 + 4b1 - 2) q^91 + (-b10 + b9 + 3b7 - 3b6 + 2b5 + b4 - b3 - 3b1 - 3) * q^95 + (2b9 - 3b7 + 2b6 + b5 - 2b3 - b2 + 3b1 - 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{5} + 12 q^{11} - 4 q^{17} - 4 q^{19} + 8 q^{23} - 16 q^{25} - 8 q^{29} - 32 q^{31} + 32 q^{35} + 4 q^{37} - 16 q^{41} + 8 q^{43} + 44 q^{49} + 8 q^{53} - 16 q^{59} + 44 q^{61} + 20 q^{65} - 40 q^{67}+ \cdots - 16 q^{97}+O(q^{100})$ 12 * q - 4 * q^5 + 12 * q^11 - 4 * q^17 - 4 * q^19 + 8 * q^23 - 16 * q^25 - 8 * q^29 - 32 * q^31 + 32 * q^35 + 4 * q^37 - 16 * q^41 + 8 * q^43 + 44 * q^49 + 8 * q^53 - 16 * q^59 + 44 * q^61 + 20 * q^65 - 40 * q^67 - 32 * q^71 + 8 * q^73 + 12 * q^77 - 8 * q^79 - 40 * q^83 + 40 * q^85 - 40 * q^91 - 28 * q^95 - 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4$ x^12 + 28*x^10 + 258*x^8 + 880*x^6 + 1033*x^4 + 132*x^2 + 4 :

$\beta_{1}$	$=$	$( -9\nu^{11} - 250\nu^{9} - 2274\nu^{7} - 7596\nu^{5} - 8901\nu^{3} - 1658\nu ) / 272$ (-9v^11 - 250v^9 - 2274v^7 - 7596v^5 - 8901v^3 - 1658v) / 272
$\beta_{2}$	$=$	$( - 39 \nu^{11} + 17 \nu^{10} - 1089 \nu^{9} + 459 \nu^{8} - 9973 \nu^{7} + 3927 \nu^{6} - 33443 \nu^{5} + \cdots + 612 ) / 1088$ (-39v^11 + 17v^10 - 1089v^9 + 459v^8 - 9973v^7 + 3927v^6 - 33443v^5 + 11305v^4 - 37228v^3 + 10064v^2 - 3524*v + 612) / 1088
$\beta_{3}$	$=$	$( 39 \nu^{11} - 25 \nu^{10} + 1089 \nu^{9} - 651 \nu^{8} + 9973 \nu^{7} - 5223 \nu^{6} + 33443 \nu^{5} + \cdots - 756 ) / 1088$ (39v^11 - 25v^10 + 1089v^9 - 651v^8 + 9973v^7 - 5223v^6 + 33443v^5 - 12889v^4 + 37228v^3 - 8184v^2 + 2436*v - 756) / 1088
$\beta_{4}$	$=$	$( 39 \nu^{11} + 17 \nu^{10} + 1089 \nu^{9} + 459 \nu^{8} + 9973 \nu^{7} + 3927 \nu^{6} + 33443 \nu^{5} + \cdots + 612 ) / 1088$ (39v^11 + 17v^10 + 1089v^9 + 459v^8 + 9973v^7 + 3927v^6 + 33443v^5 + 11305v^4 + 37228v^3 + 10064v^2 + 3524*v + 612) / 1088
$\beta_{5}$	$=$	$( - 39 \nu^{11} - 25 \nu^{10} - 1089 \nu^{9} - 651 \nu^{8} - 9973 \nu^{7} - 5223 \nu^{6} - 33443 \nu^{5} + \cdots - 756 ) / 1088$ (-39v^11 - 25v^10 - 1089v^9 - 651v^8 - 9973v^7 - 5223v^6 - 33443v^5 - 12889v^4 - 37228v^3 - 8184v^2 - 2436*v - 756) / 1088
$\beta_{6}$	$=$	$( 153 \nu^{11} - 39 \nu^{10} + 4267 \nu^{9} - 1089 \nu^{8} + 39015 \nu^{7} - 9973 \nu^{6} + 130713 \nu^{5} + \cdots - 2436 ) / 1088$ (153v^11 - 39v^10 + 4267v^9 - 1089v^8 + 39015v^7 - 9973v^6 + 130713v^5 - 33443v^4 + 146744v^3 - 37228v^2 + 10132*v - 2436) / 1088
$\beta_{7}$	$=$	$( - 153 \nu^{11} - 39 \nu^{10} - 4267 \nu^{9} - 1089 \nu^{8} - 39015 \nu^{7} - 9973 \nu^{6} + \cdots - 2436 ) / 1088$ (-153v^11 - 39v^10 - 4267v^9 - 1089v^8 - 39015v^7 - 9973v^6 - 130713v^5 - 33443v^4 - 146744v^3 - 37228v^2 - 10132*v - 2436) / 1088
$\beta_{8}$	$=$	$( 119 \nu^{11} - 27 \nu^{10} + 3315 \nu^{9} - 767 \nu^{8} + 30243 \nu^{7} - 7247 \nu^{6} + 100725 \nu^{5} + \cdots - 2016 ) / 544$ (119v^11 - 27v^10 + 3315v^9 - 767v^8 + 30243v^7 - 7247v^6 + 100725v^5 - 25865v^4 + 110670v^3 - 31582v^2 + 3808*v - 2016) / 544
$\beta_{9}$	$=$	$( - 119 \nu^{11} - 27 \nu^{10} - 3315 \nu^{9} - 767 \nu^{8} - 30243 \nu^{7} - 7247 \nu^{6} + \cdots - 2016 ) / 544$ (-119v^11 - 27v^10 - 3315v^9 - 767v^8 - 30243v^7 - 7247v^6 - 100725v^5 - 25865v^4 - 110670v^3 - 31582v^2 - 3808*v - 2016) / 544
$\beta_{10}$	$=$	$( 123 \nu^{11} - 41 \nu^{10} + 3445 \nu^{9} - 1137 \nu^{8} + 31741 \nu^{7} - 10297 \nu^{6} + 107943 \nu^{5} + \cdots - 1384 ) / 544$ (123v^11 - 41v^10 + 3445v^9 - 1137v^8 + 31741v^7 - 10297v^6 + 107943v^5 - 33839v^4 + 123568v^3 - 36486v^2 + 8300*v - 1384) / 544
$\beta_{11}$	$=$	$( - 123 \nu^{11} - 41 \nu^{10} - 3445 \nu^{9} - 1137 \nu^{8} - 31741 \nu^{7} - 10297 \nu^{6} + \cdots - 1384 ) / 544$ (-123v^11 - 41v^10 - 3445v^9 - 1137v^8 - 31741v^7 - 10297v^6 - 107943v^5 - 33839v^4 - 123568v^3 - 36486v^2 - 8300*v - 1384) / 544

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

$\nu$	$=$	$( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 2$ (b5 + b4 - b3 - b2) / 2
$\nu^{2}$	$=$	$( 2\beta_{11} + 2\beta_{10} - 4\beta_{7} - 4\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 8 ) / 2$ (2b11 + 2b10 - 4b7 - 4b6 - b5 - b4 - b3 - b2 - 8) / 2
$\nu^{3}$	$=$	$( 2\beta_{7} - 2\beta_{6} - 13\beta_{5} - 7\beta_{4} + 13\beta_{3} + 7\beta_{2} - 4\beta_1 ) / 2$ (2b7 - 2b6 - 13b5 - 7b4 + 13b3 + 7b2 - 4*b1) / 2
$\nu^{4}$	$=$	$( - 20 \beta_{11} - 20 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + 52 \beta_{7} + 52 \beta_{6} + 9 \beta_{5} + \cdots + 80 ) / 2$ (-20b11 - 20b10 - 6b9 - 6b8 + 52b7 + 52b6 + 9b5 + 17b4 + 9b3 + 17b2 + 80) / 2
$\nu^{5}$	$=$	$( 2 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 6 \beta_{8} - 42 \beta_{7} + 42 \beta_{6} + 155 \beta_{5} + \cdots + 64 \beta_1 ) / 2$ (2b11 - 2b10 + 6b9 - 6b8 - 42b7 + 42b6 + 155b5 + 69b4 - 155b3 - 69b2 + 64*b1) / 2
$\nu^{6}$	$=$	$( 218 \beta_{11} + 218 \beta_{10} + 84 \beta_{9} + 84 \beta_{8} - 612 \beta_{7} - 612 \beta_{6} + \cdots - 884 ) / 2$ (218b11 + 218b10 + 84b9 + 84b8 - 612b7 - 612b6 - 107b5 - 243b4 - 107b3 - 243b2 - 884) / 2
$\nu^{7}$	$=$	$( - 52 \beta_{11} + 52 \beta_{10} - 132 \beta_{9} + 132 \beta_{8} + 670 \beta_{7} - 670 \beta_{6} + \cdots - 964 \beta_1 ) / 2$ (-52b11 + 52b10 - 132b9 + 132b8 + 670b7 - 670b6 - 1817b5 - 767b4 + 1817b3 + 767b2 - 964*b1) / 2
$\nu^{8}$	$=$	$( - 2452 \beta_{11} - 2452 \beta_{10} - 998 \beta_{9} - 998 \beta_{8} + 7084 \beta_{7} + 7084 \beta_{6} + \cdots + 10072 ) / 2$ (-2452b11 - 2452b10 - 998b9 - 998b8 + 7084b7 + 7084b6 + 1381b5 + 3285b4 + 1381b3 + 3285b2 + 10072) / 2
$\nu^{9}$	$=$	$( 906 \beta_{11} - 906 \beta_{10} + 2214 \beta_{9} - 2214 \beta_{8} - 9690 \beta_{7} + 9690 \beta_{6} + \cdots + 13760 \beta_1 ) / 2$ (906b11 - 906b10 + 2214b9 - 2214b8 - 9690b7 + 9690b6 + 21307b5 + 8869b4 - 21307b3 - 8869b2 + 13760*b1) / 2
$\nu^{10}$	$=$	$( 27962 \beta_{11} + 27962 \beta_{10} + 11532 \beta_{9} + 11532 \beta_{8} - 82108 \beta_{7} + \cdots - 116276 ) / 2$ (27962b11 + 27962b10 + 11532b9 + 11532b8 - 82108b7 - 82108b6 - 17963b5 - 43211b4 - 17963b3 - 43211b2 - 116276) / 2
$\nu^{11}$	$=$	$( - 13716 \beta_{11} + 13716 \beta_{10} - 33212 \beta_{9} + 33212 \beta_{8} + 133350 \beta_{7} + \cdots - 188772 \beta_1 ) / 2$ (-13716b11 + 13716b10 - 33212b9 + 33212b8 + 133350b7 - 133350b6 - 250913b5 - 104063b4 + 250913b3 + 104063b2 - 188772*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$ .

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$\beta_{6}$	$1$

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

	−	0.306239i
		3.49562i
	−	3.18938i
	−	1.54190i
		1.75800i
	−	0.216105i
		1.54190i
	−	1.75800i
		0.216105i
		0.306239i
	−	3.49562i
		3.18938i

−1.33137

3.21420i

0.934059

2.25502i

145.2

0.392531

−

0.947653i

−0.893542

−

2.15720i

145.3

1.35305

−

3.26655i

0.666590

1.60929i

433.1

−3.31685

−

1.37389i

−3.66830

1.51946i

433.2

−0.194339

−

0.0804980i

−1.76317

0.730328i

433.3

1.09698

0.454383i

4.72436

−

1.95689i

865.1

−3.31685

1.37389i

−3.66830

−

1.51946i

865.2

−0.194339

0.0804980i

−1.76317

−

0.730328i

865.3

1.09698

−

0.454383i

4.72436

1.95689i

937.1

−1.33137

−

3.21420i

0.934059

−

2.25502i

937.2

0.392531

0.947653i

−0.893542

2.15720i

937.3

1.35305

3.26655i

0.666590

−

1.60929i

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	1224.2.bq.c		12
3.b	odd	2	1		136.2.n.c	✓	12
12.b	even	2	1		272.2.v.f		12
17.d	even	8	1	inner	1224.2.bq.c		12
51.g	odd	8	1		136.2.n.c	✓	12
51.i	even	16	2		2312.2.a.w		12
51.i	even	16	2		2312.2.b.n		12
204.p	even	8	1		272.2.v.f		12
204.t	odd	16	2		4624.2.a.bt		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.2.n.c	✓	12	3.b	odd	2	1
136.2.n.c	✓	12	51.g	odd	8	1
272.2.v.f		12	12.b	even	2	1
272.2.v.f		12	204.p	even	8	1
1224.2.bq.c		12	1.a	even	1	1	trivial
1224.2.bq.c		12	17.d	even	8	1	inner
2312.2.a.w		12	51.i	even	16	2
2312.2.b.n		12	51.i	even	16	2
4624.2.a.bt		12	204.t	odd	16	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{12} + 4 T_{5}^{11} + 16 T_{5}^{10} + 56 T_{5}^{9} + 160 T_{5}^{8} + 328 T_{5}^{7} + 288 T_{5}^{6} + \cdots + 128$ T5^12 + 4*T5^11 + 16*T5^10 + 56*T5^9 + 160*T5^8 + 328*T5^7 + 288*T5^6 - 2624*T5^5 + 4228*T5^4 - 3232*T5^3 + 1120*T5^2 + 896*T5 + 128 acting on $S_{2}^{\mathrm{new}}(1224, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12}$ T^12
$5$	$T^{12} + 4 T^{11} + \cdots + 128$ T^12 + 4T^11 + 16T^10 + 56T^9 + 160T^8 + 328T^7 + 288T^6 - 2624T^5 + 4228T^4 - 3232T^3 + 1120T^2 + 896*T + 128
$7$	$T^{12} - 22 T^{10} + \cdots + 147968$ T^12 - 22T^10 - 12T^9 + 242T^8 + 632T^7 + 3352T^6 + 9024T^5 + 25488T^4 + 49408T^3 + 62848T^2 + 95744T + 147968
$11$	$T^{12} - 12 T^{11} + \cdots + 16928$ T^12 - 12T^11 + 48T^10 + 32T^9 - 960T^8 + 2592T^7 + 10088T^6 - 68416T^5 + 107044T^4 - 6640T^3 + 118608T^2 - 29440*T + 16928
$13$	$T^{12} + 92 T^{10} + \cdots + 262144$ T^12 + 92T^10 + 3060T^8 + 44448T^6 + 265280T^4 + 483328*T^2 + 262144
$17$	$T^{12} + 4 T^{11} + \cdots + 24137569$ T^12 + 4T^11 - 40T^10 - 180T^9 + 559T^8 + 1936T^7 - 5136T^6 + 32912T^5 + 161551T^4 - 884340T^3 - 3340840T^2 + 5679428*T + 24137569
$19$	$T^{12} + 4 T^{11} + \cdots + 1024$ T^12 + 4T^11 + 8T^10 + 8T^9 + 676T^8 + 2912T^7 + 6272T^6 + 128T^5 + 59776T^4 + 250368T^3 + 524288T^2 + 32768*T + 1024
$23$	$T^{12} - 8 T^{11} + \cdots + 43655168$ T^12 - 8T^11 - 6T^10 - 68T^9 + 1970T^8 + 5416T^7 - 24200T^6 - 227040T^5 - 277488T^4 + 2947072T^3 + 21729792T^2 + 29601792*T + 43655168
$29$	$T^{12} + \cdots + 118210688$ T^12 + 8T^11 + 20T^10 + 8T^9 + 8T^8 - 96T^7 + 17480T^6 - 70160T^5 + 571140T^4 + 2766336T^3 + 2452512T^2 + 25462656*T + 118210688
$31$	$T^{12} + \cdots + 103219712$ T^12 + 32T^11 + 386T^10 + 1868T^9 + 1154T^8 - 15752T^7 - 39864T^6 - 39200T^5 + 1676816T^4 + 1519616T^3 + 18451328T^2 + 5747200*T + 103219712
$37$	$T^{12} - 4 T^{11} + \cdots + 512$ T^12 - 4T^11 + 56T^10 - 328T^9 + 2048T^8 - 8616T^7 + 24832T^6 - 72992T^5 + 200260T^4 - 267872T^3 + 147520T^2 - 15360*T + 512
$41$	$T^{12} + \cdots + 591542408$ T^12 + 16T^11 + 174T^10 + 404T^9 - 6558T^8 - 61328T^7 + 25104T^6 + 3730464T^5 + 28852132T^4 + 49805920T^3 + 150106584T^2 + 497916496*T + 591542408
$43$	$T^{12} + \cdots + 148254976$ T^12 - 8T^11 + 32T^10 + 336T^9 + 2264T^8 - 6560T^7 + 36480T^6 + 433984T^5 + 2328464T^4 + 1186816T^3 + 739328T^2 + 14806016*T + 148254976
$47$	$T^{12} + \cdots + 440664064$ T^12 + 360T^10 + 46032T^8 + 2438912T^6 + 46576640T^4 + 288784384*T^2 + 440664064
$53$	$T^{12} - 8 T^{11} + \cdots + 25080064$ T^12 - 8T^11 + 32T^10 + 512T^9 + 1880T^8 - 896T^7 + 78080T^6 + 1028224T^5 + 5908880T^4 + 19102336T^3 + 38158848T^2 + 43749888*T + 25080064
$59$	$T^{12} + 16 T^{11} + \cdots + 9339136$ T^12 + 16T^11 + 128T^10 + 592T^9 + 15192T^8 + 216288T^7 + 1691264T^6 + 8008640T^5 + 24599952T^4 + 47992960T^3 + 56775168T^2 + 32564736*T + 9339136
$61$	$T^{12} + \cdots + 118949888$ T^12 - 44T^11 + 824T^10 - 8632T^9 + 61056T^8 - 338280T^7 + 1608928T^6 - 7793088T^5 + 32975108T^4 + 9096640T^3 + 511760768T^2 + 455810048*T + 118949888
$67$	$(T^{6} + 20 T^{5} + \cdots + 28928)^{2}$ (T^6 + 20T^5 + 58T^4 - 880T^3 - 4784T^2 + 1664*T + 28928)^2
$71$	$T^{12} + 32 T^{11} + \cdots + 10913792$ T^12 + 32T^11 + 578T^10 + 6868T^9 + 57090T^8 + 336872T^7 + 1384040T^6 + 3716800T^5 + 7623056T^4 + 14803584T^3 + 24433920T^2 + 24817664*T + 10913792
$73$	$T^{12} + \cdots + 545424392$ T^12 - 8T^11 + 158T^10 - 460T^9 + 7074T^8 - 77360T^7 - 429104T^6 + 5009888T^5 + 43060708T^4 + 86347968T^3 + 196591256T^2 - 757133872*T + 545424392
$79$	$T^{12} + \cdots + 1130596352$ T^12 + 8T^11 - 118T^10 - 2524T^9 - 4654T^8 + 385432T^7 + 5894168T^6 + 1103008T^5 + 32598160T^4 - 293761152T^3 + 374777600T^2 + 101951488*T + 1130596352
$83$	$T^{12} + 40 T^{11} + \cdots + 256$ T^12 + 40T^11 + 800T^10 + 8192T^9 + 51672T^8 + 208640T^7 + 562432T^6 + 999296T^5 + 1150352T^4 + 774528T^3 + 270848T^2 + 11776*T + 256
$89$	$T^{12} + \cdots + 12558340096$ T^12 + 552T^10 + 107672T^8 + 9151008T^6 + 325147152T^4 + 3825710592*T^2 + 12558340096
$97$	$T^{12} + 16 T^{11} + \cdots + 30451208$ T^12 + 16T^11 + 334T^10 + 20T^9 - 13022T^8 - 77296T^7 + 235472T^6 + 5368608T^5 + 27214564T^4 + 57668896T^3 + 42001240T^2 - 905264*T + 30451208
show more
show less

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 145.3
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more