LMFDB - Newform orbit 1323.2.h.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(226,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1323.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1323.h (of order $3$ , degree $2$ , not 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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.991381711347.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$ x^10 - 2x^9 + 9x^8 - 8x^7 + 40x^6 - 36x^5 + 90x^4 - 3x^3 + 36x^2 - 9*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{9} - \beta_{6}) q^{5} + ( - \beta_{8} - \beta_{4} + 1) q^{8} + (\beta_{9} - \beta_{8} - 2 \beta_{6}) q^{10} + (\beta_{7} - \beta_{6} + \beta_{4} - 1) q^{11}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + \cdots + 4 \beta_{3}) q^{97}+O(q^{100})$ q + (-b5 + b1) * q^2 + (b3 + 1) * q^4 + (b9 - b6) * q^5 + (-b8 - b4 + 1) * q^8 + (b9 - b8 - 2b6) q^10 + (b7 - b6 + b4 - 1) * q^11 + (b6 + b2 + b1 + 1) * q^13 + (b9 - b8 - b4 - b3 - b2) * q^16 + (-b7 - 3b6 + b5 + b3) q^17 + (-b7 - b6 - b4 + b2 + 2b1 - 1) q^19 + (-2b8 - b7 - 2b6 + b3) * q^20 + (-b7 - b6 + b2 + b1 - 1) * q^22 + (b9 - 2b8 + 2b7 - 2b3) q^23 + (b7 + 2b2 - b1) q^25 + (b7 + b6 + b4 + b2 + 2b1 + 1) q^26 + (b8 - 2b7 + b6 + b5 + 2b3) * q^29 + (-b9 + b8 + b4 + b3 + b2 - 1) * q^31 + (2b9 + b8 + 3b5 + b4 + b3 - 2b2 - 3b1 + 1) * q^32 + (-b8 + b7 + 2b6 - 4b5 - b3) * q^34 + (-2b4 + 2b1) * q^37 + (3b7 + 5b6 + b4 - 2b1 + 5) q^38 + (-b8 - 2b7 - b6 - b5 + 2b3) * q^40 + (b7 + b4 + 2b2 + b1) q^41 + (-3b9 + b8 - b7 + 3b6 - 2b5 + b3) q^43 + (-b7 + 2b6 + b2 - b1 + 2) q^44 + (3b9 - b8 - 2b7 - 4b6 + 5b5 + 2b3) q^46 + (b9 - b8 + b5 - b4 + 2b3 - b2 - b1 - 4) q^47 + (-b7 - 6b6 + b4 + 2b2 + 3b1 - 6) q^50 + (b7 + b6 + b4 + 2b1 + 1) q^52 + (2b9 - 5b6 + b5) * q^53 + (-b9 - b8 + b5 - b4 - b3 + b2 - b1) * q^55 + (-b9 - b8 + 2b7 + 3b6 - 2b5 - 2b3) * q^58 + (b9 + b8 + b5 + b4 - b3 - b2 - b1 - 6) * q^59 + (-b9 - b8 + b5 - b4 + b3 + b2 - b1 - 2) * q^61 + (-2b9 + b8 - 2b5 + b4 - b3 + 2b2 + 2b1 - 3) * q^62 + (-b9 - b5 - 2b3 + b2 + b1 - 6) q^64 + (b9 - b8 + b5 - b4 + b3 - b2 - b1 - 1) * q^65 + (b9 + b8 - 4b5 + b4 + 2b3 - b2 + 4b1 - 1) q^67 + (b9 - 3b7 - 7b6 + 2b5 + 3b3) * q^68 + (-2b9 + 3b8 - 2b5 + 3b4 - b3 + 2b2 + 2b1 - 2) * q^71 + (3b8 - b7 + 4b6 + b3) * q^73 + (4b7 + 10b6 - 2b4 - 2b2 - 2b1 + 10) q^74 + (-b7 - 3b6 - b2 + 5b1 - 3) * q^76 + (b9 + 4b5 + 2b3 - b2 - 4b1 + 3) q^79 + (b9 + b8 - 3b6 - 2b5) * q^80 + (-2b6 + 2b4 + 3b2 + 4b1 - 2) * q^82 + (2b9 + 4b8 - 2b7 - b6 + 2b3) * q^83 + (-b7 - 2b6 + b4 + b2 - 2) q^85 + (-4b9 + 3b8 - b7 + b6 - 2b5 + b3) q^86 + (b7 - 4b6 + 2b4 - b2 - 4) * q^88 + (7b6 - 2b4 + b2 - 2b1 + 7) q^89 + (2b9 - 2b8 + 5b6 - 2b5) * q^92 + (2b9 - 4b8 + 4b5 - 4b4 - 2b2 - 4b1 + 3) * q^94 + (-b8 - b4 + 2b3 - 2) q^95 + (-2b9 + b8 - 4b7 - 2b6 - b5 + 4b3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 4 q^{2} + 8 q^{4} + 4 q^{5} + 6 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 4 q^{16} + 12 q^{17} - q^{19} + 5 q^{20} - q^{22} - 3 q^{23} - q^{25} + 11 q^{26} - 7 q^{29} - 6 q^{31} - 4 q^{32} - 3 q^{34}+ \cdots + 12 q^{97}+O(q^{100})$ 10 * q + 4 * q^2 + 8 * q^4 + 4 * q^5 + 6 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 4 * q^16 + 12 * q^17 - q^19 + 5 * q^20 - q^22 - 3 * q^23 - q^25 + 11 * q^26 - 7 * q^29 - 6 * q^31 - 4 * q^32 - 3 * q^34 + 20 * q^38 + 3 * q^40 + 5 * q^41 - 7 * q^43 + 10 * q^44 + 3 * q^46 - 54 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 - 4 * q^55 - 10 * q^58 - 60 * q^59 - 28 * q^61 - 12 * q^62 - 50 * q^64 - 22 * q^65 + 4 * q^67 + 27 * q^68 + 6 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 8 * q^79 + 20 * q^80 + 5 * q^82 + 9 * q^83 - 6 * q^85 + 8 * q^86 - 18 * q^88 + 28 * q^89 - 27 * q^92 - 6 * q^94 - 28 * q^95 + 12 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$ x^10 - 2*x^9 + 9*x^8 - 8*x^7 + 40*x^6 - 36*x^5 + 90*x^4 - 3*x^3 + 36*x^2 - 9*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{9} - 9\nu^{8} - 3\nu^{7} - 61\nu^{6} - 72\nu^{5} - 282\nu^{4} - 204\nu^{3} - 387\nu^{2} - 873\nu - 117 ) / 189$ (-v^9 - 9v^8 - 3v^7 - 61v^6 - 72v^5 - 282v^4 - 204v^3 - 387v^2 - 873v - 117) / 189
$\beta_{3}$	$=$	$( 7\nu^{9} - 12\nu^{8} + 48\nu^{7} - 23\nu^{6} + 204\nu^{5} - 240\nu^{4} + 303\nu^{3} - 108\nu^{2} + 36\nu - 1557 ) / 567$ (7v^9 - 12v^8 + 48v^7 - 23v^6 + 204v^5 - 240v^4 + 303v^3 - 108v^2 + 36*v - 1557) / 567
$\beta_{4}$	$=$	$( 2\nu^{9} - \nu^{8} + 12\nu^{7} + 8\nu^{6} + 68\nu^{5} + 30\nu^{4} + 123\nu^{3} + 204\nu^{2} + 270\nu + 63 ) / 63$ (2v^9 - v^8 + 12v^7 + 8v^6 + 68v^5 + 30v^4 + 123v^3 + 204v^2 + 270v + 63) / 63
$\beta_{5}$	$=$	$( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} + \cdots - 180 ) / 567$ (16v^9 - 39v^8 + 156v^7 - 176v^6 + 663v^5 - 780v^4 + 1680v^3 - 351v^2 + 684*v - 180) / 567
$\beta_{6}$	$=$	$( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + \cdots - 63 ) / 567$ (20v^9 - 24v^8 + 141v^7 - 4v^6 + 624v^5 - 57v^4 + 1020v^3 + 1620v^2 + 369*v - 63) / 567
$\beta_{7}$	$=$	$( - 53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} + \cdots - 1368 ) / 567$ (-53v^9 + 60v^8 - 375v^7 - 11v^6 - 1668v^5 - 69v^4 - 2757v^3 - 4401v^2 - 1071*v - 1368) / 567
$\beta_{8}$	$=$	$( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} + \cdots + 720 ) / 567$ (-82v^9 + 165v^8 - 732v^7 + 632v^6 - 3264v^5 + 2850v^4 - 7260v^3 - 432v^2 - 2898*v + 720) / 567
$\beta_{9}$	$=$	$( - 91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + \cdots + 801 ) / 567$ (-91v^9 + 174v^8 - 813v^7 + 704v^6 - 3633v^5 + 3174v^4 - 8070v^3 + 648v^2 - 3222*v + 801) / 567

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{7} + 3\beta_{6} - \beta_{3}$ b7 + 3*b6 - b3
$\nu^{3}$	$=$	$\beta_{8} + 4\beta_{5} + \beta_{4} - 4\beta _1 - 1$ b8 + 4b5 + b4 - 4b1 - 1
$\nu^{4}$	$=$	$-5\beta_{7} - 14\beta_{6} + \beta_{4} + \beta_{2} - 14$ -5b7 - 14b6 + b4 + b2 - 14
$\nu^{5}$	$=$	$2\beta_{9} - 7\beta_{8} - \beta_{7} - 9\beta_{6} - 17\beta_{5} + \beta_{3}$ 2b9 - 7b8 - b7 - 9b6 - 17b5 + b3
$\nu^{6}$	$=$	$9\beta_{9} - 10\beta_{8} - \beta_{5} - 10\beta_{4} + 24\beta_{3} - 9\beta_{2} + \beta _1 + 70$ 9b9 - 10b8 - b5 - 10b4 + 24b3 - 9*b2 + b1 + 70
$\nu^{7}$	$=$	$11\beta_{7} + 65\beta_{6} - 43\beta_{4} - 19\beta_{2} + 75\beta _1 + 65$ 11b7 + 65b6 - 43b4 - 19b2 + 75*b1 + 65
$\nu^{8}$	$=$	$-62\beta_{9} + 73\beta_{8} + 118\beta_{7} + 360\beta_{6} + 14\beta_{5} - 118\beta_{3}$ -62b9 + 73b8 + 118b7 + 360b6 + 14b5 - 118b3
$\nu^{9}$	$=$	$-135\beta_{9} + 253\beta_{8} + 343\beta_{5} + 253\beta_{4} - 87\beta_{3} + 135\beta_{2} - 343\beta _1 - 430$ -135b9 + 253b8 + 343b5 + 253b4 - 87b3 + 135b2 - 343*b1 - 430

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$
$\chi(n)$	$-1 - \beta_{6}$	$-1 - \beta_{6}$

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}

226.1

−1.02682	+	1.77851i
−0.335166	+	0.580525i
0.247934	−	0.429435i
0.920620	−	1.59456i
1.19343	−	2.06709i
−1.02682	−	1.77851i
−0.335166	−	0.580525i
0.247934	+	0.429435i
0.920620	+	1.59456i
1.19343	+	2.06709i

−2.05365

2.21746

0.0731228

0.126652i

−0.446582

−0.150168

−

0.260099i

226.2

−0.670333

−1.55065

−0.712469

−

1.23403i

2.38012

0.477591

0.827212i

226.3

0.495868

−1.75411

1.84629

3.19787i

−1.86155

0.915516

1.58572i

226.4

1.84124

1.39017

−0.667377

−

1.15593i

−1.12285

−1.22880

−

2.12835i

226.5

2.38687

3.69714

1.46043

2.52954i

4.05086

3.48586

6.03769i

802.1

−2.05365

2.21746

0.0731228

−

0.126652i

−0.446582

−0.150168

0.260099i

802.2

−0.670333

−1.55065

−0.712469

1.23403i

2.38012

0.477591

−

0.827212i

802.3

0.495868

−1.75411

1.84629

−

3.19787i

−1.86155

0.915516

−

1.58572i

802.4

1.84124

1.39017

−0.667377

1.15593i

−1.12285

−1.22880

2.12835i

802.5

2.38687

3.69714

1.46043

−

2.52954i

4.05086

3.48586

−

6.03769i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.h.f		10
3.b	odd	2	1		441.2.h.f		10
7.b	odd	2	1		189.2.h.b		10
7.c	even	3	1		1323.2.f.f		10
7.c	even	3	1		1323.2.g.f		10
7.d	odd	6	1		189.2.g.b		10
7.d	odd	6	1		1323.2.f.e		10
9.c	even	3	1		1323.2.g.f		10
9.d	odd	6	1		441.2.g.f		10
21.c	even	2	1		63.2.h.b	yes	10
21.g	even	6	1		63.2.g.b	✓	10
21.g	even	6	1		441.2.f.e		10
21.h	odd	6	1		441.2.f.f		10
21.h	odd	6	1		441.2.g.f		10
28.d	even	2	1		3024.2.q.i		10
28.f	even	6	1		3024.2.t.i		10
63.g	even	3	1		1323.2.f.f		10
63.h	even	3	1	inner	1323.2.h.f		10
63.h	even	3	1		3969.2.a.bb		5
63.i	even	6	1		63.2.h.b	yes	10
63.i	even	6	1		3969.2.a.z		5
63.j	odd	6	1		441.2.h.f		10
63.j	odd	6	1		3969.2.a.ba		5
63.k	odd	6	1		567.2.e.e		10
63.k	odd	6	1		1323.2.f.e		10
63.l	odd	6	1		189.2.g.b		10
63.l	odd	6	1		567.2.e.e		10
63.n	odd	6	1		441.2.f.f		10
63.o	even	6	1		63.2.g.b	✓	10
63.o	even	6	1		567.2.e.f		10
63.s	even	6	1		441.2.f.e		10
63.s	even	6	1		567.2.e.f		10
63.t	odd	6	1		189.2.h.b		10
63.t	odd	6	1		3969.2.a.bc		5
84.h	odd	2	1		1008.2.q.i		10
84.j	odd	6	1		1008.2.t.i		10
252.r	odd	6	1		1008.2.q.i		10
252.s	odd	6	1		1008.2.t.i		10
252.bi	even	6	1		3024.2.t.i		10
252.bj	even	6	1		3024.2.q.i		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.g.b	✓	10	21.g	even	6	1
63.2.g.b	✓	10	63.o	even	6	1
63.2.h.b	yes	10	21.c	even	2	1
63.2.h.b	yes	10	63.i	even	6	1
189.2.g.b		10	7.d	odd	6	1
189.2.g.b		10	63.l	odd	6	1
189.2.h.b		10	7.b	odd	2	1
189.2.h.b		10	63.t	odd	6	1
441.2.f.e		10	21.g	even	6	1
441.2.f.e		10	63.s	even	6	1
441.2.f.f		10	21.h	odd	6	1
441.2.f.f		10	63.n	odd	6	1
441.2.g.f		10	9.d	odd	6	1
441.2.g.f		10	21.h	odd	6	1
441.2.h.f		10	3.b	odd	2	1
441.2.h.f		10	63.j	odd	6	1
567.2.e.e		10	63.k	odd	6	1
567.2.e.e		10	63.l	odd	6	1
567.2.e.f		10	63.o	even	6	1
567.2.e.f		10	63.s	even	6	1
1008.2.q.i		10	84.h	odd	2	1
1008.2.q.i		10	252.r	odd	6	1
1008.2.t.i		10	84.j	odd	6	1
1008.2.t.i		10	252.s	odd	6	1
1323.2.f.e		10	7.d	odd	6	1
1323.2.f.e		10	63.k	odd	6	1
1323.2.f.f		10	7.c	even	3	1
1323.2.f.f		10	63.g	even	3	1
1323.2.g.f		10	7.c	even	3	1
1323.2.g.f		10	9.c	even	3	1
1323.2.h.f		10	1.a	even	1	1	trivial
1323.2.h.f		10	63.h	even	3	1	inner
3024.2.q.i		10	28.d	even	2	1
3024.2.q.i		10	252.bj	even	6	1
3024.2.t.i		10	28.f	even	6	1
3024.2.t.i		10	252.bi	even	6	1
3969.2.a.z		5	63.i	even	6	1
3969.2.a.ba		5	63.j	odd	6	1
3969.2.a.bb		5	63.h	even	3	1
3969.2.a.bc		5	63.t	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1323, [\chi])$ :

T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 9T_{2}^{2} + 3T_{2} - 3

T_{5}^{10} - 4T_{5}^{9} + 21T_{5}^{8} - 16T_{5}^{7} + 79T_{5}^{6} + 51T_{5}^{5} + 402T_{5}^{4} + 294T_{5}^{3} + 378T_{5}^{2} - 54T_{5} + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{5} - 2 T^{4} - 5 T^{3} + \cdots - 3)^{2}$ (T^5 - 2T^4 - 5T^3 + 9T^2 + 3T - 3)^2
$3$	$T^{10}$ T^10
$5$	$T^{10} - 4 T^{9} + \cdots + 9$ T^10 - 4T^9 + 21T^8 - 16T^7 + 79T^6 + 51T^5 + 402T^4 + 294T^3 + 378T^2 - 54*T + 9
$7$	$T^{10}$ T^10
$11$	$T^{10} + 4 T^{9} + \cdots + 225$ T^10 + 4T^9 + 24T^8 - 2T^7 + 112T^6 + 39T^5 + 261T^4 + 60T^3 + 369T^2 + 180*T + 225
$13$	$T^{10} - 8 T^{9} + \cdots + 25$ T^10 - 8T^9 + 51T^8 - 130T^7 + 296T^6 - 204T^5 + 428T^4 - 169T^3 + 594T^2 + 115*T + 25
$17$	$T^{10} - 12 T^{9} + \cdots + 81$ T^10 - 12T^9 + 99T^8 - 420T^7 + 1287T^6 - 2259T^5 + 2898T^4 - 1890T^3 + 864T^2 + 162*T + 81
$19$	$T^{10} + T^{9} + \cdots + 185761$ T^10 + T^9 + 42T^8 - 133T^7 + 1376T^6 - 2835T^5 + 13166T^4 - 23428T^3 + 86907T^2 - 111629T + 185761
$23$	$T^{10} + 3 T^{9} + \cdots + 2595321$ T^10 + 3T^9 + 72T^8 - 87T^7 + 3042T^6 - 4878T^5 + 75474T^4 - 258066T^3 + 1084239T^2 - 1739880*T + 2595321
$29$	$T^{10} + 7 T^{9} + \cdots + 81$ T^10 + 7T^9 + 69T^8 + 190T^7 + 1690T^6 + 5199T^5 + 24462T^4 + 22635T^3 + 19710T^2 - 1215*T + 81
$31$	$(T^{5} + 3 T^{4} + \cdots + 285)^{2}$ (T^5 + 3T^4 - 21T^3 - 64T^2 + 93T + 285)^2
$37$	$T^{10} + 96 T^{8} + \cdots + 82944$ T^10 + 96T^8 + 560T^7 + 8832T^6 + 27168T^5 + 115264T^4 - 52224T^3 + 228096T^2 + 110592T + 82944
$41$	$T^{10} - 5 T^{9} + \cdots + 2025$ T^10 - 5T^9 + 69T^8 + 118T^7 + 2020T^6 - 579T^5 + 9900T^4 + 4761T^3 + 31536T^2 - 7695*T + 2025
$43$	$T^{10} + 7 T^{9} + \cdots + 687241$ T^10 + 7T^9 + 138T^8 + 837T^7 + 14496T^6 + 84651T^5 + 408318T^4 + 921888T^3 + 1541055T^2 + 1214485*T + 687241
$47$	$(T^{5} + 27 T^{4} + \cdots - 6615)^{2}$ (T^5 + 27T^4 + 213T^3 + 93T^2 - 4032T - 6615)^2
$53$	$T^{10} - 21 T^{9} + \cdots + 178929$ T^10 - 21T^9 + 306T^8 - 2415T^7 + 14238T^6 - 45693T^5 + 110088T^4 - 25380T^3 + 267759T^2 - 178929*T + 178929
$59$	$(T^{5} + 30 T^{4} + \cdots - 5625)^{2}$ (T^5 + 30T^4 + 306T^3 + 1113T^2 - 108T - 5625)^2
$61$	$(T^{5} + 14 T^{4} + 34 T^{3} + \cdots - 1)^{2}$ (T^5 + 14T^4 + 34T^3 - 7T^2 - 14T - 1)^2
$67$	$(T^{5} - 2 T^{4} + \cdots - 7121)^{2}$ (T^5 - 2T^4 - 203T^3 + 340T^2 + 6784T - 7121)^2
$71$	$(T^{5} - 3 T^{4} - 168 T^{3} + \cdots + 81)^{2}$ (T^5 - 3T^4 - 168T^3 + 567T^2 + 1053T + 81)^2
$73$	$T^{10} + 15 T^{9} + \cdots + 772641$ T^10 + 15T^9 + 231T^8 + 784T^7 + 5394T^6 - 34167T^5 + 211336T^4 - 533637T^3 + 1048686T^2 - 1052163*T + 772641
$79$	$(T^{5} - 4 T^{4} + \cdots + 193)^{2}$ (T^5 - 4T^4 - 95T^3 - 224T^2 - 22T + 193)^2
$83$	$T^{10} + \cdots + 218123361$ T^10 - 9T^9 + 267T^8 - 2538T^7 + 56277T^6 - 455571T^5 + 3795093T^4 - 11237130T^3 + 38540043T^2 + 40275063*T + 218123361
$89$	$T^{10} - 28 T^{9} + \cdots + 7080921$ T^10 - 28T^9 + 549T^8 - 5740T^7 + 45157T^6 - 190791T^5 + 648528T^4 - 540030T^3 + 3980484T^2 - 4502412*T + 7080921
$97$	$T^{10} + \cdots + 2307745521$ T^10 - 12T^9 + 288T^8 - 1958T^7 + 40326T^6 - 252807T^5 + 3183925T^4 - 9179814T^3 + 94916553T^2 - 121346514*T + 2307745521
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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 226.5
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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