LMFDB - Newform orbit 735.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(734,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("735.734");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.g (of order $2$, degree $1$, 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:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.721389578983833600000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} + 2) q^{4} + \beta_{11} q^{5} + \beta_1 q^{6} + (\beta_{9} - 2 \beta_{3}) q^{8} - 3 q^{9} + \beta_{6} q^{10} + (\beta_{14} + 2 \beta_{5}) q^{12} + \beta_{10} q^{15}+ \cdots + ( - 3 \beta_{6} + 4 \beta_1) q^{96}+O(q^{100}) $ q - b3 * q^2 + b5 * q^3 + (-b8 + 2) * q^4 + b11 * q^5 + b1 * q^6 + (b9 - 2b3) q^8 - 3 * q^9 + b6 * q^10 + (b14 + 2b5) q^12 + b10 * q^15 + (b10 - 2b8 + 4) q^16 - b4 * q^17 + 3b3 q^18 + b13 * q^19 + (-b14 + 2b11 + b4) q^20 + b7 * q^23 + (b13 - b6 + 2b1) q^24 - 5 * q^25 - 3b5 q^27 + (b9 - b7) * q^30 - b12 * q^31 + (b9 - b7 - 4b3) q^32 + (-b13 - b12 - b6 - b1) * q^34 + (3b8 - 6) q^36 + (b14 + b11 + b4) * q^38 + (b12 + 2b6 - b1) q^40 - 3b11 q^45 + (-3b10 - b2) q^46 + (-2b14 + b4) q^47 + (2b14 - 3b11 + 4b5) q^48 + 5b3 q^50 + (-b8 - b2) * q^51 + (-2b9 + b7) q^53 - 3b1 q^54 + (-2b9 - b7) q^57 + (2b10 - 2b8 + b2) * q^60 + (-b12 + 2b1) q^61 + (b14 + b5 - 2b4) q^62 + (2b10 - 2b8 + b2 + 8) * q^64 + (-5b11 - 3b5 - 2b4) q^68 + (b13 + 2b6) q^69 + (-3b9 + 6b3) * q^72 - 5b5 q^75 + (b12 + b6 + 3b1) q^76 + (-3b8 + b2) q^79 + (-2b14 + 4b11 - 5b5 + 2b4) * q^80 + 9 * q^81 + (-2b14 - b4) q^83 + (-3b8 - b2) q^85 - 3b6 q^90 + (-b15 - 3b9 + 3b7) * q^92 + (-b15 + b3) * q^93 + (-b13 + b12 + 3b6 - 3b1) * q^94 + (b15 + b3) * q^95 + (-3b6 + 4b1) * q^96
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 64 * q^16 - 80 * q^25 - 96 * q^36 + 128 * q^64 + 144 * q^81

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8*x^14 + 44*x^12 + 128*x^10 + 223*x^8 - 464*x^6 - 724*x^4 + 784*x^2 + 2401 :

$\beta_{1}$	$=$	$ ( 65 \nu^{15} - 4078 \nu^{13} - 51682 \nu^{11} - 316357 \nu^{9} - 1225031 \nu^{7} + \cdots + 4240068 \nu ) / 4215813 $ (65v^15 - 4078v^13 - 51682v^11 - 316357v^9 - 1225031v^7 - 2770459v^5 - 1907888v^3 + 4240068v) / 4215813
$\beta_{2}$	$=$	$ ( 333 \nu^{14} + 1814 \nu^{12} + 20169 \nu^{10} + 61722 \nu^{8} + 226708 \nu^{6} - 249174 \nu^{4} + \cdots - 6475154 ) / 743967 $ (333v^14 + 1814v^12 + 20169v^10 + 61722v^8 + 226708v^6 - 249174v^4 - 1378692*v^2 - 6475154) / 743967
$\beta_{3}$	$=$	$ ( 467 \nu^{15} + 3796 \nu^{13} + 21270 \nu^{11} + 62179 \nu^{9} + 110885 \nu^{7} - 225705 \nu^{5} + \cdots - 366226 \nu ) / 743967 $ (467v^15 + 3796v^13 + 21270v^11 + 62179v^9 + 110885v^7 - 225705v^5 - 357836v^3 - 366226v) / 743967
$\beta_{4}$	$=$	$ ( - 416 \nu^{14} - 5038 \nu^{12} - 35507 \nu^{10} - 164752 \nu^{8} - 485725 \nu^{6} - 905496 \nu^{4} + \cdots + 1049286 ) / 602259 $ (-416v^14 - 5038v^12 - 35507v^10 - 164752v^8 - 485725v^6 - 905496v^4 + 203815*v^2 + 1049286) / 602259
$\beta_{5}$	$=$	$ ( 3152 \nu^{14} + 14536 \nu^{12} + 57408 \nu^{10} - 24278 \nu^{8} - 497536 \nu^{6} - 3376584 \nu^{4} + \cdots + 4619867 ) / 4215813 $ (3152v^14 + 14536v^12 + 57408v^10 - 24278v^8 - 497536v^6 - 3376584v^4 + 1229536*v^2 + 4619867) / 4215813
$\beta_{6}$	$=$	$ ( - 12788 \nu^{15} - 118391 \nu^{13} - 703899 \nu^{11} - 2491939 \nu^{9} - 6346228 \nu^{7} + \cdots - 11242315 \nu ) / 12647439 $ (-12788v^15 - 118391v^13 - 703899v^11 - 2491939v^9 - 6346228v^7 - 4121229v^5 + 1057316v^3 - 11242315v) / 12647439
$\beta_{7}$	$=$	$ ( 784 \nu^{15} + 1690 \nu^{13} + 2190 \nu^{11} - 72529 \nu^{9} - 255160 \nu^{7} - 896622 \nu^{5} + \cdots + 3139430 \nu ) / 743967 $ (784v^15 + 1690v^13 + 2190v^11 - 72529v^9 - 255160v^7 - 896622v^5 + 2788235v^3 + 3139430v) / 743967
$\beta_{8}$	$=$	$ ( 753 \nu^{14} + 6868 \nu^{12} + 36990 \nu^{10} + 108930 \nu^{8} + 163589 \nu^{6} - 387270 \nu^{4} + \cdots + 1299284 ) / 743967 $ (753v^14 + 6868v^12 + 36990v^10 + 108930v^8 + 163589v^6 - 387270v^4 - 1297401*v^2 + 1299284) / 743967
$\beta_{9}$	$=$	$ ( 1036 \nu^{15} + 6313 \nu^{13} + 31659 \nu^{11} + 62366 \nu^{9} + 79892 \nu^{7} - 710379 \nu^{5} + \cdots + 669095 \nu ) / 743967 $ (1036v^15 + 6313v^13 + 31659v^11 + 62366v^9 + 79892v^7 - 710379v^5 + 572429v^3 + 669095v) / 743967
$\beta_{10}$	$=$	$ ( 48\nu^{14} + 433\nu^{12} + 2308\nu^{10} + 7516\nu^{8} + 11488\nu^{6} - 26290\nu^{4} - 88652\nu^{2} + 52430 ) / 35427 $ (48v^14 + 433v^12 + 2308v^10 + 7516v^8 + 11488v^6 - 26290v^4 - 88652*v^2 + 52430) / 35427
$\beta_{11}$	$=$	$ ( -80\nu^{14} - 587\nu^{12} - 3204\nu^{10} - 8992\nu^{8} - 18880\nu^{6} + 26304\nu^{4} - 15484\nu^{2} - 44590 ) / 52479 $ (-80v^14 - 587v^12 - 3204v^10 - 8992v^8 - 18880v^6 + 26304v^4 - 15484*v^2 - 44590) / 52479
$\beta_{12}$	$=$	$ ( 28079 \nu^{15} + 207511 \nu^{13} + 1032996 \nu^{11} + 2276044 \nu^{9} + 1085018 \nu^{7} + \cdots + 85821932 \nu ) / 12647439 $ (28079v^15 + 207511v^13 + 1032996v^11 + 2276044v^9 + 1085018v^7 - 25896513v^5 - 29876738v^3 + 85821932v) / 12647439
$\beta_{13}$	$=$	$ ( - 37616 \nu^{15} - 315308 \nu^{13} - 1717926 \nu^{11} - 5390767 \nu^{9} - 10004680 \nu^{7} + \cdots - 70216510 \nu ) / 12647439 $ (-37616v^15 - 315308v^13 - 1717926v^11 - 5390767v^9 - 10004680v^7 + 9998436v^5 + 19314689v^3 - 70216510v) / 12647439
$\beta_{14}$	$=$	$ ( - 2311 \nu^{14} - 15992 \nu^{12} - 87844 \nu^{10} - 230258 \nu^{8} - 395405 \nu^{6} + \cdots - 1649928 ) / 602259 $ (-2311v^14 - 15992v^12 - 87844v^10 - 230258v^8 - 395405v^6 + 1020426v^4 - 518377*v^2 - 1649928) / 602259
$\beta_{15}$	$=$	$ ( 6422 \nu^{15} + 41353 \nu^{13} + 219822 \nu^{11} + 507391 \nu^{9} + 751901 \nu^{7} + \cdots + 1073786 \nu ) / 743967 $ (6422v^15 + 41353v^13 + 219822v^11 + 507391v^9 + 751901v^7 - 3701286v^5 + 729142v^3 + 1073786v) / 743967

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

$\nu$	$=$	$ ( 2\beta_{12} - 7\beta_{3} - 3\beta_1 ) / 14 $ (2b12 - 7b3 - 3*b1) / 14
$\nu^{2}$	$=$	$ ( -5\beta_{14} - 5\beta_{8} - 14\beta_{5} + 4\beta_{4} - 2\beta_{2} - 14 ) / 14 $ (-5b14 - 5b8 - 14b5 + 4b4 - 2*b2 - 14) / 14
$\nu^{3}$	$=$	$ ( -3\beta_{15} - 4\beta_{13} - 3\beta_{12} + 4\beta_{9} + 6\beta_{7} + 8\beta_{6} + 27\beta_{3} - 6\beta_1 ) / 14 $ (-3b15 - 4b13 - 3b12 + 4b9 + 6b7 + 8b6 + 27b3 - 6b1) / 14
$\nu^{4}$	$=$	$ ( 4\beta_{14} + 21\beta_{11} - 21\beta_{10} + 32\beta_{8} + 42\beta_{5} - 20\beta_{4} - 4\beta_{2} - 42 ) / 14 $ (4b14 + 21b11 - 21b10 + 32b8 + 42b5 - 20b4 - 4*b2 - 42) / 14
$\nu^{5}$	$=$	$ ( 10\beta_{15} + 3\beta_{13} - 10\beta_{12} - 67\beta_{9} + 8\beta_{7} - 41\beta_{6} - 20\beta_{3} + 50\beta_1 ) / 14 $ (10b15 + 3b13 - 10b12 - 67b9 + 8b7 - 41b6 - 20b3 + 50b1) / 14
$\nu^{6}$	$=$	$ ( 92\beta_{14} - 252\beta_{11} - 27\beta_{8} + 23\beta_{4} + 27\beta_{2} + 280 ) / 14 $ (92b14 - 252b11 - 27b8 + 23b4 + 27*b2 + 280) / 14
$\nu^{7}$	$=$	$ ( - 21 \beta_{15} + 98 \beta_{13} + 65 \beta_{12} + 315 \beta_{9} - 91 \beta_{7} - 147 \beta_{6} + \cdots - 24 \beta_1 ) / 14 $ (-21b15 + 98b13 + 65b12 + 315b9 - 91b7 - 147b6 - 259b3 - 24b1) / 14
$\nu^{8}$	$=$	$ ( -264\beta_{14} + 588\beta_{11} + 588\beta_{10} - 768\beta_{8} - 161\beta_{5} - 24\beta_{4} - 72\beta_{2} - 161 ) / 14 $ (-264b14 + 588b11 + 588b10 - 768b8 - 161b5 - 24b4 - 72*b2 - 161) / 14
$\nu^{9}$	$=$	$ ( 25 \beta_{15} - 732 \beta_{13} - 311 \beta_{12} - 24 \beta_{9} - 36 \beta_{7} + 1464 \beta_{6} + \cdots - 622 \beta_1 ) / 14 $ (25b15 - 732b13 - 311b12 - 24b9 - 36b7 + 1464b6 - 225b3 - 622b1) / 14
$\nu^{10}$	$=$	$ ( - 1333 \beta_{14} + 2520 \beta_{11} - 2520 \beta_{10} + 3196 \beta_{8} - 1526 \beta_{5} + 197 \beta_{4} + \cdots + 1526 ) / 14 $ (-1333b14 + 2520b11 - 2520b10 + 3196b8 - 1526b5 + 197b4 + 409*b2 + 1526) / 14
$\nu^{11}$	$=$	$ ( 297 \beta_{15} + 2103 \beta_{13} + 1551 \beta_{12} - 5618 \beta_{9} + 1849 \beta_{7} + \cdots - 1287 \beta_1 ) / 14 $ (297b15 + 2103b13 + 1551b12 - 5618b9 + 1849b7 - 2428b6 + 5874b3 - 1287b1) / 14
$\nu^{12}$	$=$	$ ( 3784\beta_{14} - 9387\beta_{11} + 1530\beta_{8} + 946\beta_{4} - 1530\beta_{2} - 15246 ) / 7 $ (3784b14 - 9387b11 + 1530b8 + 946b4 - 1530*b2 - 15246) / 7
$\nu^{13}$	$=$	$ ( - 4368 \beta_{15} + 140 \beta_{13} - 4940 \beta_{12} + 21735 \beta_{9} - 2065 \beta_{7} + \cdots + 22698 \beta_1 ) / 14 $ (-4368b15 + 140b13 - 4940b12 + 21735b9 - 2065b7 - 13755b6 + 10738b3 + 22698b1) / 14
$\nu^{14}$	$=$	$ ( 8101 \beta_{14} + 27930 \beta_{11} + 27930 \beta_{10} - 46268 \beta_{8} + 95732 \beta_{5} + \cdots + 95732 ) / 14 $ (8101b14 + 27930b11 + 27930b10 - 46268b8 + 95732b5 - 31853b4 + 6865*b2 + 95732) / 14
$\nu^{15}$	$=$	$ ( 26170 \beta_{15} - 24343 \beta_{13} - 10076 \beta_{12} - 21710 \beta_{9} - 32565 \beta_{7} + \cdots - 20152 \beta_1 ) / 14 $ (26170b15 - 24343b13 - 10076b12 - 21710b9 - 32565b7 + 48686b6 - 235530b3 - 20152b1) / 14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

734.1

−1.36434	−	1.59774i
−1.36434	+	1.59774i
−1.22796	−	0.279124i
−1.22796	+	0.279124i
−0.701515	−	1.98043i
−0.701515	+	1.98043i
−0.372250	+	1.20300i
−0.372250	−	1.20300i
0.372250	−	1.20300i
0.372250	+	1.20300i
0.701515	+	1.98043i
0.701515	−	1.98043i
1.22796	+	0.279124i
1.22796	−	0.279124i
1.36434	+	1.59774i
1.36434	−	1.59774i

−2.72868

−

1.73205i

5.44572

2.23607i

4.72622i

−9.40228

−3.00000

−

6.10152i

734.2

−2.72868

1.73205i

5.44572

−

2.23607i

−

4.72622i

−9.40228

−3.00000

6.10152i

734.3

−2.45591

−

1.73205i

4.03151

−

2.23607i

4.25377i

−4.98920

−3.00000

5.49159i

734.4

−2.45591

1.73205i

4.03151

2.23607i

−

4.25377i

−4.98920

−3.00000

−

5.49159i

734.5

−1.40303

−

1.73205i

−0.0315060

−

2.23607i

2.43012i

2.85026

−3.00000

3.13727i

734.6

−1.40303

1.73205i

−0.0315060

2.23607i

−

2.43012i

2.85026

−3.00000

−

3.13727i

734.7

−0.744500

−

1.73205i

−1.44572

2.23607i

1.28951i

2.56534

−3.00000

−

1.66475i

734.8

−0.744500

1.73205i

−1.44572

−

2.23607i

−

1.28951i

2.56534

−3.00000

1.66475i

734.9

0.744500

−

1.73205i

−1.44572

2.23607i

−

1.28951i

−2.56534

−3.00000

1.66475i

734.10

0.744500

1.73205i

−1.44572

−

2.23607i

1.28951i

−2.56534

−3.00000

−

1.66475i

734.11

1.40303

−

1.73205i

−0.0315060

−

2.23607i

−

2.43012i

−2.85026

−3.00000

−

3.13727i

734.12

1.40303

1.73205i

−0.0315060

2.23607i

2.43012i

−2.85026

−3.00000

3.13727i

734.13

2.45591

−

1.73205i

4.03151

−

2.23607i

−

4.25377i

4.98920

−3.00000

−

5.49159i

734.14

2.45591

1.73205i

4.03151

2.23607i

4.25377i

4.98920

−3.00000

5.49159i

734.15

2.72868

−

1.73205i

5.44572

2.23607i

−

4.72622i

9.40228

−3.00000

6.10152i

734.16

2.72868

1.73205i

5.44572

−

2.23607i

4.72622i

9.40228

−3.00000

−

6.10152i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.g.a	✓	16
3.b	odd	2	1	inner	735.2.g.a	✓	16
5.b	even	2	1	inner	735.2.g.a	✓	16
7.b	odd	2	1	inner	735.2.g.a	✓	16
7.c	even	3	1		735.2.p.d		16
7.c	even	3	1		735.2.p.e		16
7.d	odd	6	1		735.2.p.d		16
7.d	odd	6	1		735.2.p.e		16
15.d	odd	2	1	CM	735.2.g.a	✓	16
21.c	even	2	1	inner	735.2.g.a	✓	16
21.g	even	6	1		735.2.p.d		16
21.g	even	6	1		735.2.p.e		16
21.h	odd	6	1		735.2.p.d		16
21.h	odd	6	1		735.2.p.e		16
35.c	odd	2	1	inner	735.2.g.a	✓	16
35.i	odd	6	1		735.2.p.d		16
35.i	odd	6	1		735.2.p.e		16
35.j	even	6	1		735.2.p.d		16
35.j	even	6	1		735.2.p.e		16
105.g	even	2	1	inner	735.2.g.a	✓	16
105.o	odd	6	1		735.2.p.d		16
105.o	odd	6	1		735.2.p.e		16
105.p	even	6	1		735.2.p.d		16
105.p	even	6	1		735.2.p.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.g.a	✓	16	1.a	even	1	1	trivial
735.2.g.a	✓	16	3.b	odd	2	1	inner
735.2.g.a	✓	16	5.b	even	2	1	inner
735.2.g.a	✓	16	7.b	odd	2	1	inner
735.2.g.a	✓	16	15.d	odd	2	1	CM
735.2.g.a	✓	16	21.c	even	2	1	inner
735.2.g.a	✓	16	35.c	odd	2	1	inner
735.2.g.a	✓	16	105.g	even	2	1	inner
735.2.p.d		16	7.c	even	3	1
735.2.p.d		16	7.d	odd	6	1
735.2.p.d		16	21.g	even	6	1
735.2.p.d		16	21.h	odd	6	1
735.2.p.d		16	35.i	odd	6	1
735.2.p.d		16	35.j	even	6	1
735.2.p.d		16	105.o	odd	6	1
735.2.p.d		16	105.p	even	6	1
735.2.p.e		16	7.c	even	3	1
735.2.p.e		16	7.d	odd	6	1
735.2.p.e		16	21.g	even	6	1
735.2.p.e		16	21.h	odd	6	1
735.2.p.e		16	35.i	odd	6	1
735.2.p.e		16	35.j	even	6	1
735.2.p.e		16	105.o	odd	6	1
735.2.p.e		16	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 16T_{2}^{6} + 80T_{2}^{4} - 128T_{2}^{2} + 49 $ T2^8 - 16*T2^6 + 80*T2^4 - 128*T2^2 + 49 acting on $S_{2}^{\mathrm{new}}(735, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 16 T^{6} + \cdots + 49)^{2} $ (T^8 - 16T^6 + 80T^4 - 128*T^2 + 49)^2
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{2} + 5)^{8} $ (T^2 + 5)^8
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{4} + 68 T^{2} + 196)^{4} $ (T^4 + 68*T^2 + 196)^4
$19$	$ (T^{8} + 152 T^{6} + \cdots + 56644)^{2} $ (T^8 + 152T^6 + 7220T^4 + 109744*T^2 + 56644)^2
$23$	$ (T^{8} - 184 T^{6} + \cdots + 9604)^{2} $ (T^8 - 184T^6 + 10580T^4 - 194672*T^2 + 9604)^2
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} + 248 T^{6} + \cdots + 3678724)^{2} $ (T^8 + 248T^6 + 19220T^4 + 476656*T^2 + 3678724)^2
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{4} + 188 T^{2} + 196)^{4} $ (T^4 + 188*T^2 + 196)^4
$53$	$ (T^{8} - 424 T^{6} + \cdots + 3161284)^{2} $ (T^8 - 424T^6 + 56180T^4 - 2382032*T^2 + 3161284)^2
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} + 488 T^{6} + \cdots + 42016324)^{2} $ (T^8 + 488T^6 + 74420T^4 + 3631696*T^2 + 42016324)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ (T^{4} - 316 T^{2} + 9604)^{4} $ (T^4 - 316*T^2 + 9604)^4
$83$	$ (T^{4} + 332 T^{2} + 23716)^{4} $ (T^4 + 332*T^2 + 23716)^4
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} + 2) q^{4} + \beta_{11} q^{5} + \beta_1 q^{6} + (\beta_{9} - 2 \beta_{3}) q^{8} - 3 q^{9} + \beta_{6} q^{10} + (\beta_{14} + 2 \beta_{5}) q^{12} + \beta_{10} q^{15}+ \cdots + ( - 3 \beta_{6} + 4 \beta_1) q^{96}+O(q^{100}) \) q - b3 * q^2 + b5 * q^3 + (-b8 + 2) * q^4 + b11 * q^5 + b1 * q^6 + (b9 - 2b3) q^8 - 3 * q^9 + b6 * q^10 + (b14 + 2b5) q^12 + b10 * q^15 + (b10 - 2b8 + 4) q^16 - b4 * q^17 + 3b3 q^18 + b13 * q^19 + (-b14 + 2b11 + b4) q^20 + b7 * q^23 + (b13 - b6 + 2b1) q^24 - 5 * q^25 - 3b5 q^27 + (b9 - b7) * q^30 - b12 * q^31 + (b9 - b7 - 4b3) q^32 + (-b13 - b12 - b6 - b1) * q^34 + (3b8 - 6) q^36 + (b14 + b11 + b4) * q^38 + (b12 + 2b6 - b1) q^40 - 3b11 q^45 + (-3b10 - b2) q^46 + (-2b14 + b4) q^47 + (2b14 - 3b11 + 4b5) q^48 + 5b3 q^50 + (-b8 - b2) * q^51 + (-2b9 + b7) q^53 - 3b1 q^54 + (-2b9 - b7) q^57 + (2b10 - 2b8 + b2) * q^60 + (-b12 + 2b1) q^61 + (b14 + b5 - 2b4) q^62 + (2b10 - 2b8 + b2 + 8) * q^64 + (-5b11 - 3b5 - 2b4) q^68 + (b13 + 2b6) q^69 + (-3b9 + 6b3) * q^72 - 5b5 q^75 + (b12 + b6 + 3b1) q^76 + (-3b8 + b2) q^79 + (-2b14 + 4b11 - 5b5 + 2b4) * q^80 + 9 * q^81 + (-2b14 - b4) q^83 + (-3b8 - b2) q^85 - 3b6 q^90 + (-b15 - 3b9 + 3b7) * q^92 + (-b15 + b3) * q^93 + (-b13 + b12 + 3b6 - 3b1) * q^94 + (b15 + b3) * q^95 + (-3b6 + 4b1) * q^96
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 64 * q^16 - 80 * q^25 - 96 * q^36 + 128 * q^64 + 144 * q^81

Introduction

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Modular forms

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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	735.2.g.a

Level	$735$
Weight	$2$
Character orbit	735.g
Analytic conductor	$5.869$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.721389578983833600000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 \) x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8}\cdot 7^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 65 \nu^{15} - 4078 \nu^{13} - 51682 \nu^{11} - 316357 \nu^{9} - 1225031 \nu^{7} + \cdots + 4240068 \nu ) / 4215813 \) (65v^15 - 4078v^13 - 51682v^11 - 316357v^9 - 1225031v^7 - 2770459v^5 - 1907888v^3 + 4240068v) / 4215813
\(\beta_{2}\)	\(=\)	\( ( 333 \nu^{14} + 1814 \nu^{12} + 20169 \nu^{10} + 61722 \nu^{8} + 226708 \nu^{6} - 249174 \nu^{4} + \cdots - 6475154 ) / 743967 \) (333v^14 + 1814v^12 + 20169v^10 + 61722v^8 + 226708v^6 - 249174v^4 - 1378692*v^2 - 6475154) / 743967
\(\beta_{3}\)	\(=\)	\( ( 467 \nu^{15} + 3796 \nu^{13} + 21270 \nu^{11} + 62179 \nu^{9} + 110885 \nu^{7} - 225705 \nu^{5} + \cdots - 366226 \nu ) / 743967 \) (467v^15 + 3796v^13 + 21270v^11 + 62179v^9 + 110885v^7 - 225705v^5 - 357836v^3 - 366226v) / 743967
\(\beta_{4}\)	\(=\)	\( ( - 416 \nu^{14} - 5038 \nu^{12} - 35507 \nu^{10} - 164752 \nu^{8} - 485725 \nu^{6} - 905496 \nu^{4} + \cdots + 1049286 ) / 602259 \) (-416v^14 - 5038v^12 - 35507v^10 - 164752v^8 - 485725v^6 - 905496v^4 + 203815*v^2 + 1049286) / 602259
\(\beta_{5}\)	\(=\)	\( ( 3152 \nu^{14} + 14536 \nu^{12} + 57408 \nu^{10} - 24278 \nu^{8} - 497536 \nu^{6} - 3376584 \nu^{4} + \cdots + 4619867 ) / 4215813 \) (3152v^14 + 14536v^12 + 57408v^10 - 24278v^8 - 497536v^6 - 3376584v^4 + 1229536*v^2 + 4619867) / 4215813
\(\beta_{6}\)	\(=\)	\( ( - 12788 \nu^{15} - 118391 \nu^{13} - 703899 \nu^{11} - 2491939 \nu^{9} - 6346228 \nu^{7} + \cdots - 11242315 \nu ) / 12647439 \) (-12788v^15 - 118391v^13 - 703899v^11 - 2491939v^9 - 6346228v^7 - 4121229v^5 + 1057316v^3 - 11242315v) / 12647439
\(\beta_{7}\)	\(=\)	\( ( 784 \nu^{15} + 1690 \nu^{13} + 2190 \nu^{11} - 72529 \nu^{9} - 255160 \nu^{7} - 896622 \nu^{5} + \cdots + 3139430 \nu ) / 743967 \) (784v^15 + 1690v^13 + 2190v^11 - 72529v^9 - 255160v^7 - 896622v^5 + 2788235v^3 + 3139430v) / 743967
\(\beta_{8}\)	\(=\)	\( ( 753 \nu^{14} + 6868 \nu^{12} + 36990 \nu^{10} + 108930 \nu^{8} + 163589 \nu^{6} - 387270 \nu^{4} + \cdots + 1299284 ) / 743967 \) (753v^14 + 6868v^12 + 36990v^10 + 108930v^8 + 163589v^6 - 387270v^4 - 1297401*v^2 + 1299284) / 743967
\(\beta_{9}\)	\(=\)	\( ( 1036 \nu^{15} + 6313 \nu^{13} + 31659 \nu^{11} + 62366 \nu^{9} + 79892 \nu^{7} - 710379 \nu^{5} + \cdots + 669095 \nu ) / 743967 \) (1036v^15 + 6313v^13 + 31659v^11 + 62366v^9 + 79892v^7 - 710379v^5 + 572429v^3 + 669095v) / 743967
\(\beta_{10}\)	\(=\)	\( ( 48\nu^{14} + 433\nu^{12} + 2308\nu^{10} + 7516\nu^{8} + 11488\nu^{6} - 26290\nu^{4} - 88652\nu^{2} + 52430 ) / 35427 \) (48v^14 + 433v^12 + 2308v^10 + 7516v^8 + 11488v^6 - 26290v^4 - 88652*v^2 + 52430) / 35427
\(\beta_{11}\)	\(=\)	\( ( -80\nu^{14} - 587\nu^{12} - 3204\nu^{10} - 8992\nu^{8} - 18880\nu^{6} + 26304\nu^{4} - 15484\nu^{2} - 44590 ) / 52479 \) (-80v^14 - 587v^12 - 3204v^10 - 8992v^8 - 18880v^6 + 26304v^4 - 15484*v^2 - 44590) / 52479
\(\beta_{12}\)	\(=\)	\( ( 28079 \nu^{15} + 207511 \nu^{13} + 1032996 \nu^{11} + 2276044 \nu^{9} + 1085018 \nu^{7} + \cdots + 85821932 \nu ) / 12647439 \) (28079v^15 + 207511v^13 + 1032996v^11 + 2276044v^9 + 1085018v^7 - 25896513v^5 - 29876738v^3 + 85821932v) / 12647439
\(\beta_{13}\)	\(=\)	\( ( - 37616 \nu^{15} - 315308 \nu^{13} - 1717926 \nu^{11} - 5390767 \nu^{9} - 10004680 \nu^{7} + \cdots - 70216510 \nu ) / 12647439 \) (-37616v^15 - 315308v^13 - 1717926v^11 - 5390767v^9 - 10004680v^7 + 9998436v^5 + 19314689v^3 - 70216510v) / 12647439
\(\beta_{14}\)	\(=\)	\( ( - 2311 \nu^{14} - 15992 \nu^{12} - 87844 \nu^{10} - 230258 \nu^{8} - 395405 \nu^{6} + \cdots - 1649928 ) / 602259 \) (-2311v^14 - 15992v^12 - 87844v^10 - 230258v^8 - 395405v^6 + 1020426v^4 - 518377*v^2 - 1649928) / 602259
\(\beta_{15}\)	\(=\)	\( ( 6422 \nu^{15} + 41353 \nu^{13} + 219822 \nu^{11} + 507391 \nu^{9} + 751901 \nu^{7} + \cdots + 1073786 \nu ) / 743967 \) (6422v^15 + 41353v^13 + 219822v^11 + 507391v^9 + 751901v^7 - 3701286v^5 + 729142v^3 + 1073786v) / 743967

\(\nu\)	\(=\)	\( ( 2\beta_{12} - 7\beta_{3} - 3\beta_1 ) / 14 \) (2b12 - 7b3 - 3*b1) / 14
\(\nu^{2}\)	\(=\)	\( ( -5\beta_{14} - 5\beta_{8} - 14\beta_{5} + 4\beta_{4} - 2\beta_{2} - 14 ) / 14 \) (-5b14 - 5b8 - 14b5 + 4b4 - 2*b2 - 14) / 14
\(\nu^{3}\)	\(=\)	\( ( -3\beta_{15} - 4\beta_{13} - 3\beta_{12} + 4\beta_{9} + 6\beta_{7} + 8\beta_{6} + 27\beta_{3} - 6\beta_1 ) / 14 \) (-3b15 - 4b13 - 3b12 + 4b9 + 6b7 + 8b6 + 27b3 - 6b1) / 14
\(\nu^{4}\)	\(=\)	\( ( 4\beta_{14} + 21\beta_{11} - 21\beta_{10} + 32\beta_{8} + 42\beta_{5} - 20\beta_{4} - 4\beta_{2} - 42 ) / 14 \) (4b14 + 21b11 - 21b10 + 32b8 + 42b5 - 20b4 - 4*b2 - 42) / 14
\(\nu^{5}\)	\(=\)	\( ( 10\beta_{15} + 3\beta_{13} - 10\beta_{12} - 67\beta_{9} + 8\beta_{7} - 41\beta_{6} - 20\beta_{3} + 50\beta_1 ) / 14 \) (10b15 + 3b13 - 10b12 - 67b9 + 8b7 - 41b6 - 20b3 + 50b1) / 14
\(\nu^{6}\)	\(=\)	\( ( 92\beta_{14} - 252\beta_{11} - 27\beta_{8} + 23\beta_{4} + 27\beta_{2} + 280 ) / 14 \) (92b14 - 252b11 - 27b8 + 23b4 + 27*b2 + 280) / 14
\(\nu^{7}\)	\(=\)	\( ( - 21 \beta_{15} + 98 \beta_{13} + 65 \beta_{12} + 315 \beta_{9} - 91 \beta_{7} - 147 \beta_{6} + \cdots - 24 \beta_1 ) / 14 \) (-21b15 + 98b13 + 65b12 + 315b9 - 91b7 - 147b6 - 259b3 - 24b1) / 14
\(\nu^{8}\)	\(=\)	\( ( -264\beta_{14} + 588\beta_{11} + 588\beta_{10} - 768\beta_{8} - 161\beta_{5} - 24\beta_{4} - 72\beta_{2} - 161 ) / 14 \) (-264b14 + 588b11 + 588b10 - 768b8 - 161b5 - 24b4 - 72*b2 - 161) / 14
\(\nu^{9}\)	\(=\)	\( ( 25 \beta_{15} - 732 \beta_{13} - 311 \beta_{12} - 24 \beta_{9} - 36 \beta_{7} + 1464 \beta_{6} + \cdots - 622 \beta_1 ) / 14 \) (25b15 - 732b13 - 311b12 - 24b9 - 36b7 + 1464b6 - 225b3 - 622b1) / 14
\(\nu^{10}\)	\(=\)	\( ( - 1333 \beta_{14} + 2520 \beta_{11} - 2520 \beta_{10} + 3196 \beta_{8} - 1526 \beta_{5} + 197 \beta_{4} + \cdots + 1526 ) / 14 \) (-1333b14 + 2520b11 - 2520b10 + 3196b8 - 1526b5 + 197b4 + 409*b2 + 1526) / 14
\(\nu^{11}\)	\(=\)	\( ( 297 \beta_{15} + 2103 \beta_{13} + 1551 \beta_{12} - 5618 \beta_{9} + 1849 \beta_{7} + \cdots - 1287 \beta_1 ) / 14 \) (297b15 + 2103b13 + 1551b12 - 5618b9 + 1849b7 - 2428b6 + 5874b3 - 1287b1) / 14
\(\nu^{12}\)	\(=\)	\( ( 3784\beta_{14} - 9387\beta_{11} + 1530\beta_{8} + 946\beta_{4} - 1530\beta_{2} - 15246 ) / 7 \) (3784b14 - 9387b11 + 1530b8 + 946b4 - 1530*b2 - 15246) / 7
\(\nu^{13}\)	\(=\)	\( ( - 4368 \beta_{15} + 140 \beta_{13} - 4940 \beta_{12} + 21735 \beta_{9} - 2065 \beta_{7} + \cdots + 22698 \beta_1 ) / 14 \) (-4368b15 + 140b13 - 4940b12 + 21735b9 - 2065b7 - 13755b6 + 10738b3 + 22698b1) / 14
\(\nu^{14}\)	\(=\)	\( ( 8101 \beta_{14} + 27930 \beta_{11} + 27930 \beta_{10} - 46268 \beta_{8} + 95732 \beta_{5} + \cdots + 95732 ) / 14 \) (8101b14 + 27930b11 + 27930b10 - 46268b8 + 95732b5 - 31853b4 + 6865*b2 + 95732) / 14
\(\nu^{15}\)	\(=\)	\( ( 26170 \beta_{15} - 24343 \beta_{13} - 10076 \beta_{12} - 21710 \beta_{9} - 32565 \beta_{7} + \cdots - 20152 \beta_1 ) / 14 \) (26170b15 - 24343b13 - 10076b12 - 21710b9 - 32565b7 + 48686b6 - 235530b3 - 20152b1) / 14

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 734.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 16 T^{6} + \cdots + 49)^{2} \) (T^8 - 16T^6 + 80T^4 - 128*T^2 + 49)^2
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{2} + 5)^{8} \) (T^2 + 5)^8
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{4} + 68 T^{2} + 196)^{4} \) (T^4 + 68*T^2 + 196)^4
$19$	\( (T^{8} + 152 T^{6} + \cdots + 56644)^{2} \) (T^8 + 152T^6 + 7220T^4 + 109744*T^2 + 56644)^2
$23$	\( (T^{8} - 184 T^{6} + \cdots + 9604)^{2} \) (T^8 - 184T^6 + 10580T^4 - 194672*T^2 + 9604)^2
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} + 248 T^{6} + \cdots + 3678724)^{2} \) (T^8 + 248T^6 + 19220T^4 + 476656*T^2 + 3678724)^2
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{4} + 188 T^{2} + 196)^{4} \) (T^4 + 188*T^2 + 196)^4
$53$	\( (T^{8} - 424 T^{6} + \cdots + 3161284)^{2} \) (T^8 - 424T^6 + 56180T^4 - 2382032*T^2 + 3161284)^2
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} + 488 T^{6} + \cdots + 42016324)^{2} \) (T^8 + 488T^6 + 74420T^4 + 3631696*T^2 + 42016324)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( (T^{4} - 316 T^{2} + 9604)^{4} \) (T^4 - 316*T^2 + 9604)^4
$83$	\( (T^{4} + 332 T^{2} + 23716)^{4} \) (T^4 + 332*T^2 + 23716)^4
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)