LMFDB - Newform orbit 483.2.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,2,Mod(160,483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("483.160");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	483.h (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:	$3.85677441763$
Analytic rank:	$0$
Dimension:	$12$
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} + 17x^{10} + 92x^{8} + 180x^{6} + 92x^{4} + 17x^{2} + 1 $ x^12 + 17x^10 + 92x^8 + 180x^6 + 92x^4 + 17*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{2} + \beta_{9} q^{3} + (\beta_{7} + 3) q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + (\beta_{3} + \beta_1) q^{7} + (\beta_{7} - 3 \beta_{2} - 3) q^{8} - q^{9} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{10}+ \cdots + (\beta_{10} - \beta_{3}) q^{99}+O(q^{100}) $ q - b2 * q^2 + b9 * q^3 + (b7 + 3) * q^4 - b5 * q^5 + b4 * q^6 + (b3 + b1) * q^7 + (b7 - 3b2 - 3) q^8 - q^9 + (-b6 + b5 - b1) * q^10 + (-b10 + b3) * q^11 + (3b9 + b8) q^12 + (2b9 - b4) q^13 + (b11 + b5 - b3 + 2b1) q^14 + b3 * q^15 + (2b7 + b2 + 6) q^16 + (-b6 + b5 - b1) * q^17 + b2 * q^18 + (-b5 + b1) * q^19 + (b6 - 4b5 - b1) q^20 + (-b10 + b5) * q^21 + (b11 - 2b10 - 2b3) * q^22 + (-b10 - b7 - 2b3 + 1) q^23 + (-3b9 + b8 + 3b4) * q^24 + (b2 - 1) * q^25 + (-5b9 - b8 + 2b4) * q^26 - b9 * q^27 + (-b11 + 2b10 + b6 + b5 + 4b3 + 3b1) q^28 + (b7 + b2 - 1) * q^29 + (b11 - b3) * q^30 + (-3b9 - b8 - b4) q^31 + (-b7 - 4b2 - 5) q^32 + (b5 - b1) * q^33 + (b6 - 6b5 - b1) q^34 + (4b9 - b7 - b4 + b2 + 1) q^35 + (-b7 - 3) * q^36 + (b11 + 2b10 + b3) q^37 + (-b6 + 2b5 + b1) q^38 + (-b2 - 2) * q^39 + (-2b6 + 5b5 - 4b1) q^40 + 5b9 q^41 + (-2b10 + b6 - b5 - b3 + b1) q^42 + (-b11 - 2b3) q^43 + (-2b11 + 3b3) * q^44 + b5 * q^45 + (-2b11 - 2b10 - b7 + b3 + b2 + 3) * q^46 + (2b9 + b8 - 3b4) * q^47 + (6b9 + 2b8 - b4) * q^48 + (2b9 - 2b8 - 2b4 - 2b2 - 1) * q^49 + (-b7 + b2 - 5) * q^50 + (b11 - b3) * q^51 + (9b9 + b8 - 5b4) * q^52 + (b11 + b10 - b3) * q^53 - b4 * q^54 + (5b9 - b8 - 2b4) * q^55 + (2b11 + 2b10 + b6 + 4b5 - 5b3 + 3b1) q^56 + (-b10 + b3) * q^57 + (-b2 - 8) * q^58 + (-5b9 + 2b8 + b4) * q^59 + (-b11 + 2b10 + 4b3) * q^60 + (-2b6 - b1) q^61 + (-2b9 - 2b8 - 5b4) q^62 + (-b3 - b1) * q^63 + (-b7 + 5b2 + 11) q^64 + (-b11 + 3b3) q^65 + (b6 - 2b5 - b1) q^66 - 3b3 q^67 + (-4b6 + 7b5 - 6b1) q^68 + (b9 - b8 - 2b5 - b1) q^69 + (-5b9 - b8 - 2b7 + 4b4 + b2 - 2) q^70 + (2b7 + b2 + 2) q^71 + (-b7 + 3b2 + 3) q^72 + (b9 + 5b8 + b4) q^73 + (b11 + 6b10 + 6b3) * q^74 + (-b9 - b4) * q^75 + (2b6 - 3b5 + 2b1) q^76 + (4b9 - b8 + b7 - 2b2 - 5) * q^77 + (b7 + 2b2 + 5) q^78 + (-b11 + 6b10 - b3) q^79 + (3b6 - 9b5 - b1) * q^80 + q^81 + 5b4 q^82 + (3b6 - b5 + 3b1) * q^83 + (-b11 - 2b10 - b6 + 4b5 - b3 + b1) * q^84 + (-b7 - 4b2 - 5) q^85 + (-2b11 - 2b10 - 3b3) q^86 + (-b9 + b8 - b4) * q^87 + (b11 - 9b3) q^88 + (b6 - 3b1) q^89 + (b6 - b5 + b1) * q^90 + (-b6 + 3b5 + b3 - b1) q^91 + (b11 - 6b10 - 9b3 - b2 - 4) * q^92 + (b7 - b2 + 3) * q^93 + (-18b9 - 2b8 + 4b4) q^94 + (-b7 + 2b2 + 5) q^95 + (-5b9 - b8 + 4b4) * q^96 + (-b6 + 3b5 + 7b1) * q^97 + (-4b9 - 4b8 + 2b7 - 2b4 + b2 + 10) * q^98 + (b10 - b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 36 q^{4} - 24 q^{8} - 12 q^{9} + 68 q^{16} - 4 q^{18} + 12 q^{23} - 16 q^{25} - 16 q^{29} - 44 q^{32} + 8 q^{35} - 36 q^{36} - 20 q^{39} + 32 q^{46} - 4 q^{49} - 64 q^{50} - 92 q^{58} + 112 q^{64}+ \cdots + 116 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 36 * q^4 - 24 * q^8 - 12 * q^9 + 68 * q^16 - 4 * q^18 + 12 * q^23 - 16 * q^25 - 16 * q^29 - 44 * q^32 + 8 * q^35 - 36 * q^36 - 20 * q^39 + 32 * q^46 - 4 * q^49 - 64 * q^50 - 92 * q^58 + 112 * q^64 - 28 * q^70 + 20 * q^71 + 24 * q^72 - 52 * q^77 + 52 * q^78 + 12 * q^81 - 44 * q^85 - 44 * q^92 + 40 * q^93 + 52 * q^95 + 116 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 17x^{10} + 92x^{8} + 180x^{6} + 92x^{4} + 17x^{2} + 1 $ x^12 + 17*x^10 + 92*x^8 + 180*x^6 + 92*x^4 + 17*x^2 + 1 :

$\beta_{1}$	$=$	$ ( -19\nu^{10} - 322\nu^{8} - 1730\nu^{6} - 3310\nu^{4} - 1478\nu^{2} - 141 ) / 20 $ (-19v^10 - 322v^8 - 1730v^6 - 3310v^4 - 1478*v^2 - 141) / 20
$\beta_{2}$	$=$	$ \nu^{10} + 17\nu^{8} + 92\nu^{6} + 180\nu^{4} + 91\nu^{2} + 11 $ v^10 + 17v^8 + 92v^6 + 180v^4 + 91v^2 + 11
$\beta_{3}$	$=$	$ -\nu^{11} - 17\nu^{9} - 92\nu^{7} - 180\nu^{5} - 92\nu^{3} - 16\nu $ -v^11 - 17v^9 - 92v^7 - 180v^5 - 92v^3 - 16*v
$\beta_{4}$	$=$	$ ( 6\nu^{11} + 98\nu^{9} + 485\nu^{7} + 730\nu^{5} - 63\nu^{3} - 56\nu ) / 5 $ (6v^11 + 98v^9 + 485v^7 + 730v^5 - 63v^3 - 56v) / 5
$\beta_{5}$	$=$	$ ( 33\nu^{10} + 554\nu^{8} + 2920\nu^{6} + 5340\nu^{4} + 1966\nu^{2} + 187 ) / 10 $ (33v^10 + 554v^8 + 2920v^6 + 5340v^4 + 1966*v^2 + 187) / 10
$\beta_{6}$	$=$	$ ( 93\nu^{10} + 1554\nu^{8} + 8110\nu^{6} + 14450\nu^{4} + 4526\nu^{2} + 267 ) / 20 $ (93v^10 + 1554v^8 + 8110v^6 + 14450v^4 + 4526*v^2 + 267) / 20
$\beta_{7}$	$=$	$ 5\nu^{10} + 84\nu^{8} + 443\nu^{6} + 809\nu^{4} + 292\nu^{2} + 26 $ 5v^10 + 84v^8 + 443v^6 + 809v^4 + 292*v^2 + 26
$\beta_{8}$	$=$	$ ( -63\nu^{11} - 1054\nu^{9} - 5510\nu^{7} - 9830\nu^{5} - 3046\nu^{3} - 97\nu ) / 20 $ (-63v^11 - 1054v^9 - 5510v^7 - 9830v^5 - 3046v^3 - 97v) / 20
$\beta_{9}$	$=$	$ ( 83\nu^{11} + 1394\nu^{9} + 7350\nu^{7} + 13430\nu^{5} + 4886\nu^{3} + 457\nu ) / 20 $ (83v^11 + 1394v^9 + 7350v^7 + 13430v^5 + 4886v^3 + 457v) / 20
$\beta_{10}$	$=$	$ ( -11\nu^{11} - 184\nu^{9} - 962\nu^{7} - 1720\nu^{5} - 550\nu^{3} - 39\nu ) / 2 $ (-11v^11 - 184v^9 - 962v^7 - 1720v^5 - 550v^3 - 39v) / 2
$\beta_{11}$	$=$	$ 9\nu^{11} + 152\nu^{9} + 811\nu^{7} + 1528\nu^{5} + 649\nu^{3} + 69\nu $ 9v^11 + 152v^9 + 811v^7 + 1528v^5 + 649v^3 + 69v

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

$\nu$	$=$	$ ( \beta_{9} + \beta_{8} + \beta_{3} ) / 2 $ (b9 + b8 + b3) / 2
$\nu^{2}$	$=$	$ ( -\beta_{6} + 2\beta_{5} - \beta_{2} + \beta _1 - 6 ) / 2 $ (-b6 + 2*b5 - b2 + b1 - 6) / 2
$\nu^{3}$	$=$	$ ( \beta_{11} - 8\beta_{9} - 4\beta_{8} + 3\beta_{4} - 8\beta_{3} ) / 2 $ (b11 - 8b9 - 4b8 + 3b4 - 8b3) / 2
$\nu^{4}$	$=$	$ ( \beta_{7} + 8\beta_{6} - 17\beta_{5} + 12\beta_{2} - 2\beta _1 + 39 ) / 2 $ (b7 + 8b6 - 17b5 + 12b2 - 2b1 + 39) / 2
$\nu^{5}$	$=$	$ -7\beta_{11} + \beta_{10} + 37\beta_{9} + 11\beta_{8} - 17\beta_{4} + 30\beta_{3} $ -7b11 + b10 + 37b9 + 11b8 - 17b4 + 30*b3
$\nu^{6}$	$=$	$ ( -17\beta_{7} - 64\beta_{6} + 147\beta_{5} - 112\beta_{2} - 10\beta _1 - 291 ) / 2 $ (-17b7 - 64b6 + 147b5 - 112b2 - 10*b1 - 291) / 2
$\nu^{7}$	$=$	$ ( 143\beta_{11} - 36\beta_{10} - 672\beta_{9} - 140\beta_{8} + 319\beta_{4} - 480\beta_{3} ) / 2 $ (143b11 - 36b10 - 672b9 - 140b8 + 319b4 - 480b3) / 2
$\nu^{8}$	$=$	$ ( 196\beta_{7} + 523\beta_{6} - 1278\beta_{5} + 985\beta_{2} + 189\beta _1 + 2318 ) / 2 $ (196b7 + 523b6 - 1278b5 + 985b2 + 189*b1 + 2318) / 2
$\nu^{9}$	$=$	$ ( -1324\beta_{11} + 428\beta_{10} + 5973\beta_{9} + 997\beta_{8} - 2832\beta_{4} + 3979\beta_{3} ) / 2 $ (-1324b11 + 428b10 + 5973b9 + 997b8 - 2832b4 + 3979b3) / 2
$\nu^{10}$	$=$	$ -974\beta_{7} - 2176\beta_{6} + 5540\beta_{5} - 4254\beta_{2} - 1012\beta _1 - 9565 $ -974b7 - 2176b6 + 5540b5 - 4254b2 - 1012*b1 - 9565
$\nu^{11}$	$=$	$ ( 11780\beta_{11} - 4324\beta_{10} - 52317\beta_{9} - 7677\beta_{8} + 24640\beta_{4} - 33565\beta_{3} ) / 2 $ (11780b11 - 4324b10 - 52317b9 - 7677b8 + 24640b4 - 33565b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$.

$n$	$323$	$346$	$442$
$\chi(n)$	$1$	$-1$	$-1$

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} $

160.1

	−	0.341398i
	−	2.92914i
		0.341398i
		2.92914i
		2.14584i
		0.466018i
	−	2.14584i
	−	0.466018i
	−	0.562016i
	−	1.77931i
		0.562016i
		1.77931i

−2.69639

−

1.00000i

5.27053

−2.58774

2.69639i

−0.551006

2.58774i

−8.81864

−1.00000

6.97756

160.2

−2.69639

−

1.00000i

5.27053

2.58774

2.69639i

0.551006

−

2.58774i

−8.81864

−1.00000

−6.97756

160.3

−2.69639

1.00000i

5.27053

−2.58774

−

2.69639i

−0.551006

−

2.58774i

−8.81864

−1.00000

6.97756

160.4

−2.69639

1.00000i

5.27053

2.58774

−

2.69639i

0.551006

2.58774i

−8.81864

−1.00000

−6.97756

160.5

1.17819

−

1.00000i

−0.611859

−1.67982

−

1.17819i

−2.04406

1.67982i

−3.07728

−1.00000

−1.97916

160.6

1.17819

−

1.00000i

−0.611859

1.67982

−

1.17819i

2.04406

−

1.67982i

−3.07728

−1.00000

1.97916

160.7

1.17819

1.00000i

−0.611859

−1.67982

1.17819i

−2.04406

−

1.67982i

−3.07728

−1.00000

−1.97916

160.8

1.17819

1.00000i

−0.611859

1.67982

1.17819i

2.04406

1.67982i

−3.07728

−1.00000

1.97916

160.9

2.51820

−

1.00000i

4.34132

−1.21729

−

2.51820i

2.34908

1.21729i

5.89592

−1.00000

−3.06538

160.10

2.51820

−

1.00000i

4.34132

1.21729

−

2.51820i

−2.34908

−

1.21729i

5.89592

−1.00000

3.06538

160.11

2.51820

1.00000i

4.34132

−1.21729

2.51820i

2.34908

−

1.21729i

5.89592

−1.00000

−3.06538

160.12

2.51820

1.00000i

4.34132

1.21729

2.51820i

−2.34908

1.21729i

5.89592

−1.00000

3.06538

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
23.b	odd	2	1	inner
161.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	483.2.h.c	✓	12
3.b	odd	2	1		1449.2.h.e		12
7.b	odd	2	1	inner	483.2.h.c	✓	12
21.c	even	2	1		1449.2.h.e		12
23.b	odd	2	1	inner	483.2.h.c	✓	12
69.c	even	2	1		1449.2.h.e		12
161.c	even	2	1	inner	483.2.h.c	✓	12
483.c	odd	2	1		1449.2.h.e		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
483.2.h.c	✓	12	1.a	even	1	1	trivial
483.2.h.c	✓	12	7.b	odd	2	1	inner
483.2.h.c	✓	12	23.b	odd	2	1	inner
483.2.h.c	✓	12	161.c	even	2	1	inner
1449.2.h.e		12	3.b	odd	2	1
1449.2.h.e		12	21.c	even	2	1
1449.2.h.e		12	69.c	even	2	1
1449.2.h.e		12	483.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{3} - T_{2}^{2} - 7T_{2} + 8 $ T2^3 - T2^2 - 7*T2 + 8 acting on $S_{2}^{\mathrm{new}}(483, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{3} - T^{2} - 7 T + 8)^{4} $ (T^3 - T^2 - 7*T + 8)^4
$3$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$5$	$ (T^{6} - 11 T^{4} + \cdots - 28)^{2} $ (T^6 - 11T^4 + 33T^2 - 28)^2
$7$	$ T^{12} + 2 T^{10} + \cdots + 117649 $ T^12 + 2T^10 + 31T^8 + 476T^6 + 1519T^4 + 4802*T^2 + 117649
$11$	$ (T^{6} + 25 T^{4} + \cdots + 175)^{2} $ (T^6 + 25T^4 + 167T^2 + 175)^2
$13$	$ (T^{6} + 23 T^{4} + 21 T^{2} + 4)^{2} $ (T^6 + 23T^4 + 21T^2 + 4)^2
$17$	$ (T^{6} - 62 T^{4} + \cdots - 1792)^{2} $ (T^6 - 62T^4 + 685T^2 - 1792)^2
$19$	$ (T^{6} - 25 T^{4} + \cdots - 175)^{2} $ (T^6 - 25T^4 + 167T^2 - 175)^2
$23$	$ (T^{6} - 6 T^{5} + \cdots + 12167)^{2} $ (T^6 - 6T^5 + 41T^4 - 292T^3 + 943T^2 - 3174*T + 12167)^2
$29$	$ (T^{3} + 4 T^{2} - 19 T - 50)^{4} $ (T^3 + 4T^2 - 19T - 50)^4
$31$	$ (T^{6} + 54 T^{4} + \cdots + 100)^{2} $ (T^6 + 54T^4 + 329T^2 + 100)^2
$37$	$ (T^{6} + 114 T^{4} + \cdots + 1372)^{2} $ (T^6 + 114T^4 + 1053T^2 + 1372)^2
$41$	$ (T^{2} + 25)^{6} $ (T^2 + 25)^6
$43$	$ (T^{6} + 95 T^{4} + \cdots + 1372)^{2} $ (T^6 + 95T^4 + 2281T^2 + 1372)^2
$47$	$ (T^{6} + 197 T^{4} + \cdots + 71824)^{2} $ (T^6 + 197T^4 + 7228T^2 + 71824)^2
$53$	$ (T^{6} + 86 T^{4} + 1662 T^{2} + 7)^{2} $ (T^6 + 86T^4 + 1662T^2 + 7)^2
$59$	$ (T^{6} + 132 T^{4} + \cdots + 49)^{2} $ (T^6 + 132T^4 + 1220T^2 + 49)^2
$61$	$ (T^{6} - 194 T^{4} + \cdots - 229327)^{2} $ (T^6 - 194T^4 + 11938T^2 - 229327)^2
$67$	$ (T^{6} + 99 T^{4} + \cdots + 20412)^{2} $ (T^6 + 99T^4 + 2673T^2 + 20412)^2
$71$	$ (T^{3} - 5 T^{2} + \cdots + 128)^{4} $ (T^3 - 5T^2 - 53T + 128)^4
$73$	$ (T^{6} + 450 T^{4} + \cdots + 2458624)^{2} $ (T^6 + 450T^4 + 59633T^2 + 2458624)^2
$79$	$ (T^{6} + 386 T^{4} + \cdots + 51772)^{2} $ (T^6 + 386T^4 + 23757T^2 + 51772)^2
$83$	$ (T^{6} - 470 T^{4} + \cdots - 2903152)^{2} $ (T^6 - 470T^4 + 66741T^2 - 2903152)^2
$89$	$ (T^{6} - 171 T^{4} + \cdots - 17500)^{2} $ (T^6 - 171T^4 + 4457T^2 - 17500)^2
$97$	$ (T^{6} - 614 T^{4} + \cdots - 4435312)^{2} $ (T^6 - 614T^4 + 107173T^2 - 4435312)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + \beta_{9} q^{3} + (\beta_{7} + 3) q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + (\beta_{3} + \beta_1) q^{7} + (\beta_{7} - 3 \beta_{2} - 3) q^{8} - q^{9} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{10}+ \cdots + (\beta_{10} - \beta_{3}) q^{99}+O(q^{100}) \) q - b2 * q^2 + b9 * q^3 + (b7 + 3) * q^4 - b5 * q^5 + b4 * q^6 + (b3 + b1) * q^7 + (b7 - 3b2 - 3) q^8 - q^9 + (-b6 + b5 - b1) * q^10 + (-b10 + b3) * q^11 + (3b9 + b8) q^12 + (2b9 - b4) q^13 + (b11 + b5 - b3 + 2b1) q^14 + b3 * q^15 + (2b7 + b2 + 6) q^16 + (-b6 + b5 - b1) * q^17 + b2 * q^18 + (-b5 + b1) * q^19 + (b6 - 4b5 - b1) q^20 + (-b10 + b5) * q^21 + (b11 - 2b10 - 2b3) * q^22 + (-b10 - b7 - 2b3 + 1) q^23 + (-3b9 + b8 + 3b4) * q^24 + (b2 - 1) * q^25 + (-5b9 - b8 + 2b4) * q^26 - b9 * q^27 + (-b11 + 2b10 + b6 + b5 + 4b3 + 3b1) q^28 + (b7 + b2 - 1) * q^29 + (b11 - b3) * q^30 + (-3b9 - b8 - b4) q^31 + (-b7 - 4b2 - 5) q^32 + (b5 - b1) * q^33 + (b6 - 6b5 - b1) q^34 + (4b9 - b7 - b4 + b2 + 1) q^35 + (-b7 - 3) * q^36 + (b11 + 2b10 + b3) q^37 + (-b6 + 2b5 + b1) q^38 + (-b2 - 2) * q^39 + (-2b6 + 5b5 - 4b1) q^40 + 5b9 q^41 + (-2b10 + b6 - b5 - b3 + b1) q^42 + (-b11 - 2b3) q^43 + (-2b11 + 3b3) * q^44 + b5 * q^45 + (-2b11 - 2b10 - b7 + b3 + b2 + 3) * q^46 + (2b9 + b8 - 3b4) * q^47 + (6b9 + 2b8 - b4) * q^48 + (2b9 - 2b8 - 2b4 - 2b2 - 1) * q^49 + (-b7 + b2 - 5) * q^50 + (b11 - b3) * q^51 + (9b9 + b8 - 5b4) * q^52 + (b11 + b10 - b3) * q^53 - b4 * q^54 + (5b9 - b8 - 2b4) * q^55 + (2b11 + 2b10 + b6 + 4b5 - 5b3 + 3b1) q^56 + (-b10 + b3) * q^57 + (-b2 - 8) * q^58 + (-5b9 + 2b8 + b4) * q^59 + (-b11 + 2b10 + 4b3) * q^60 + (-2b6 - b1) q^61 + (-2b9 - 2b8 - 5b4) q^62 + (-b3 - b1) * q^63 + (-b7 + 5b2 + 11) q^64 + (-b11 + 3b3) q^65 + (b6 - 2b5 - b1) q^66 - 3b3 q^67 + (-4b6 + 7b5 - 6b1) q^68 + (b9 - b8 - 2b5 - b1) q^69 + (-5b9 - b8 - 2b7 + 4b4 + b2 - 2) q^70 + (2b7 + b2 + 2) q^71 + (-b7 + 3b2 + 3) q^72 + (b9 + 5b8 + b4) q^73 + (b11 + 6b10 + 6b3) * q^74 + (-b9 - b4) * q^75 + (2b6 - 3b5 + 2b1) q^76 + (4b9 - b8 + b7 - 2b2 - 5) * q^77 + (b7 + 2b2 + 5) q^78 + (-b11 + 6b10 - b3) q^79 + (3b6 - 9b5 - b1) * q^80 + q^81 + 5b4 q^82 + (3b6 - b5 + 3b1) * q^83 + (-b11 - 2b10 - b6 + 4b5 - b3 + b1) * q^84 + (-b7 - 4b2 - 5) q^85 + (-2b11 - 2b10 - 3b3) q^86 + (-b9 + b8 - b4) * q^87 + (b11 - 9b3) q^88 + (b6 - 3b1) q^89 + (b6 - b5 + b1) * q^90 + (-b6 + 3b5 + b3 - b1) q^91 + (b11 - 6b10 - 9b3 - b2 - 4) * q^92 + (b7 - b2 + 3) * q^93 + (-18b9 - 2b8 + 4b4) q^94 + (-b7 + 2b2 + 5) q^95 + (-5b9 - b8 + 4b4) * q^96 + (-b6 + 3b5 + 7b1) * q^97 + (-4b9 - 4b8 + 2b7 - 2b4 + b2 + 10) * q^98 + (b10 - b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 36 q^{4} - 24 q^{8} - 12 q^{9} + 68 q^{16} - 4 q^{18} + 12 q^{23} - 16 q^{25} - 16 q^{29} - 44 q^{32} + 8 q^{35} - 36 q^{36} - 20 q^{39} + 32 q^{46} - 4 q^{49} - 64 q^{50} - 92 q^{58} + 112 q^{64}+ \cdots + 116 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 36 * q^4 - 24 * q^8 - 12 * q^9 + 68 * q^16 - 4 * q^18 + 12 * q^23 - 16 * q^25 - 16 * q^29 - 44 * q^32 + 8 * q^35 - 36 * q^36 - 20 * q^39 + 32 * q^46 - 4 * q^49 - 64 * q^50 - 92 * q^58 + 112 * q^64 - 28 * q^70 + 20 * q^71 + 24 * q^72 - 52 * q^77 + 52 * q^78 + 12 * q^81 - 44 * q^85 - 44 * q^92 + 40 * q^93 + 52 * q^95 + 116 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	483.2.h.c

Level	$483$
Weight	$2$
Character orbit	483.h
Analytic conductor	$3.857$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 17x^{10} + 92x^{8} + 180x^{6} + 92x^{4} + 17x^{2} + 1 \) x^12 + 17x^10 + 92x^8 + 180x^6 + 92x^4 + 17*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -19\nu^{10} - 322\nu^{8} - 1730\nu^{6} - 3310\nu^{4} - 1478\nu^{2} - 141 ) / 20 \) (-19v^10 - 322v^8 - 1730v^6 - 3310v^4 - 1478*v^2 - 141) / 20
\(\beta_{2}\)	\(=\)	\( \nu^{10} + 17\nu^{8} + 92\nu^{6} + 180\nu^{4} + 91\nu^{2} + 11 \) v^10 + 17v^8 + 92v^6 + 180v^4 + 91v^2 + 11
\(\beta_{3}\)	\(=\)	\( -\nu^{11} - 17\nu^{9} - 92\nu^{7} - 180\nu^{5} - 92\nu^{3} - 16\nu \) -v^11 - 17v^9 - 92v^7 - 180v^5 - 92v^3 - 16*v
\(\beta_{4}\)	\(=\)	\( ( 6\nu^{11} + 98\nu^{9} + 485\nu^{7} + 730\nu^{5} - 63\nu^{3} - 56\nu ) / 5 \) (6v^11 + 98v^9 + 485v^7 + 730v^5 - 63v^3 - 56v) / 5
\(\beta_{5}\)	\(=\)	\( ( 33\nu^{10} + 554\nu^{8} + 2920\nu^{6} + 5340\nu^{4} + 1966\nu^{2} + 187 ) / 10 \) (33v^10 + 554v^8 + 2920v^6 + 5340v^4 + 1966*v^2 + 187) / 10
\(\beta_{6}\)	\(=\)	\( ( 93\nu^{10} + 1554\nu^{8} + 8110\nu^{6} + 14450\nu^{4} + 4526\nu^{2} + 267 ) / 20 \) (93v^10 + 1554v^8 + 8110v^6 + 14450v^4 + 4526*v^2 + 267) / 20
\(\beta_{7}\)	\(=\)	\( 5\nu^{10} + 84\nu^{8} + 443\nu^{6} + 809\nu^{4} + 292\nu^{2} + 26 \) 5v^10 + 84v^8 + 443v^6 + 809v^4 + 292*v^2 + 26
\(\beta_{8}\)	\(=\)	\( ( -63\nu^{11} - 1054\nu^{9} - 5510\nu^{7} - 9830\nu^{5} - 3046\nu^{3} - 97\nu ) / 20 \) (-63v^11 - 1054v^9 - 5510v^7 - 9830v^5 - 3046v^3 - 97v) / 20
\(\beta_{9}\)	\(=\)	\( ( 83\nu^{11} + 1394\nu^{9} + 7350\nu^{7} + 13430\nu^{5} + 4886\nu^{3} + 457\nu ) / 20 \) (83v^11 + 1394v^9 + 7350v^7 + 13430v^5 + 4886v^3 + 457v) / 20
\(\beta_{10}\)	\(=\)	\( ( -11\nu^{11} - 184\nu^{9} - 962\nu^{7} - 1720\nu^{5} - 550\nu^{3} - 39\nu ) / 2 \) (-11v^11 - 184v^9 - 962v^7 - 1720v^5 - 550v^3 - 39v) / 2
\(\beta_{11}\)	\(=\)	\( 9\nu^{11} + 152\nu^{9} + 811\nu^{7} + 1528\nu^{5} + 649\nu^{3} + 69\nu \) 9v^11 + 152v^9 + 811v^7 + 1528v^5 + 649v^3 + 69v

\(\nu\)	\(=\)	\( ( \beta_{9} + \beta_{8} + \beta_{3} ) / 2 \) (b9 + b8 + b3) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{6} + 2\beta_{5} - \beta_{2} + \beta _1 - 6 ) / 2 \) (-b6 + 2*b5 - b2 + b1 - 6) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} - 8\beta_{9} - 4\beta_{8} + 3\beta_{4} - 8\beta_{3} ) / 2 \) (b11 - 8b9 - 4b8 + 3b4 - 8b3) / 2
\(\nu^{4}\)	\(=\)	\( ( \beta_{7} + 8\beta_{6} - 17\beta_{5} + 12\beta_{2} - 2\beta _1 + 39 ) / 2 \) (b7 + 8b6 - 17b5 + 12b2 - 2b1 + 39) / 2
\(\nu^{5}\)	\(=\)	\( -7\beta_{11} + \beta_{10} + 37\beta_{9} + 11\beta_{8} - 17\beta_{4} + 30\beta_{3} \) -7b11 + b10 + 37b9 + 11b8 - 17b4 + 30*b3
\(\nu^{6}\)	\(=\)	\( ( -17\beta_{7} - 64\beta_{6} + 147\beta_{5} - 112\beta_{2} - 10\beta _1 - 291 ) / 2 \) (-17b7 - 64b6 + 147b5 - 112b2 - 10*b1 - 291) / 2
\(\nu^{7}\)	\(=\)	\( ( 143\beta_{11} - 36\beta_{10} - 672\beta_{9} - 140\beta_{8} + 319\beta_{4} - 480\beta_{3} ) / 2 \) (143b11 - 36b10 - 672b9 - 140b8 + 319b4 - 480b3) / 2
\(\nu^{8}\)	\(=\)	\( ( 196\beta_{7} + 523\beta_{6} - 1278\beta_{5} + 985\beta_{2} + 189\beta _1 + 2318 ) / 2 \) (196b7 + 523b6 - 1278b5 + 985b2 + 189*b1 + 2318) / 2
\(\nu^{9}\)	\(=\)	\( ( -1324\beta_{11} + 428\beta_{10} + 5973\beta_{9} + 997\beta_{8} - 2832\beta_{4} + 3979\beta_{3} ) / 2 \) (-1324b11 + 428b10 + 5973b9 + 997b8 - 2832b4 + 3979b3) / 2
\(\nu^{10}\)	\(=\)	\( -974\beta_{7} - 2176\beta_{6} + 5540\beta_{5} - 4254\beta_{2} - 1012\beta _1 - 9565 \) -974b7 - 2176b6 + 5540b5 - 4254b2 - 1012*b1 - 9565
\(\nu^{11}\)	\(=\)	\( ( 11780\beta_{11} - 4324\beta_{10} - 52317\beta_{9} - 7677\beta_{8} + 24640\beta_{4} - 33565\beta_{3} ) / 2 \) (11780b11 - 4324b10 - 52317b9 - 7677b8 + 24640b4 - 33565b3) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 160.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{3} - T^{2} - 7 T + 8)^{4} \) (T^3 - T^2 - 7*T + 8)^4
$3$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$5$	\( (T^{6} - 11 T^{4} + \cdots - 28)^{2} \) (T^6 - 11T^4 + 33T^2 - 28)^2
$7$	\( T^{12} + 2 T^{10} + \cdots + 117649 \) T^12 + 2T^10 + 31T^8 + 476T^6 + 1519T^4 + 4802*T^2 + 117649
$11$	\( (T^{6} + 25 T^{4} + \cdots + 175)^{2} \) (T^6 + 25T^4 + 167T^2 + 175)^2
$13$	\( (T^{6} + 23 T^{4} + 21 T^{2} + 4)^{2} \) (T^6 + 23T^4 + 21T^2 + 4)^2
$17$	\( (T^{6} - 62 T^{4} + \cdots - 1792)^{2} \) (T^6 - 62T^4 + 685T^2 - 1792)^2
$19$	\( (T^{6} - 25 T^{4} + \cdots - 175)^{2} \) (T^6 - 25T^4 + 167T^2 - 175)^2
$23$	\( (T^{6} - 6 T^{5} + \cdots + 12167)^{2} \) (T^6 - 6T^5 + 41T^4 - 292T^3 + 943T^2 - 3174*T + 12167)^2
$29$	\( (T^{3} + 4 T^{2} - 19 T - 50)^{4} \) (T^3 + 4T^2 - 19T - 50)^4
$31$	\( (T^{6} + 54 T^{4} + \cdots + 100)^{2} \) (T^6 + 54T^4 + 329T^2 + 100)^2
$37$	\( (T^{6} + 114 T^{4} + \cdots + 1372)^{2} \) (T^6 + 114T^4 + 1053T^2 + 1372)^2
$41$	\( (T^{2} + 25)^{6} \) (T^2 + 25)^6
$43$	\( (T^{6} + 95 T^{4} + \cdots + 1372)^{2} \) (T^6 + 95T^4 + 2281T^2 + 1372)^2
$47$	\( (T^{6} + 197 T^{4} + \cdots + 71824)^{2} \) (T^6 + 197T^4 + 7228T^2 + 71824)^2
$53$	\( (T^{6} + 86 T^{4} + 1662 T^{2} + 7)^{2} \) (T^6 + 86T^4 + 1662T^2 + 7)^2
$59$	\( (T^{6} + 132 T^{4} + \cdots + 49)^{2} \) (T^6 + 132T^4 + 1220T^2 + 49)^2
$61$	\( (T^{6} - 194 T^{4} + \cdots - 229327)^{2} \) (T^6 - 194T^4 + 11938T^2 - 229327)^2
$67$	\( (T^{6} + 99 T^{4} + \cdots + 20412)^{2} \) (T^6 + 99T^4 + 2673T^2 + 20412)^2
$71$	\( (T^{3} - 5 T^{2} + \cdots + 128)^{4} \) (T^3 - 5T^2 - 53T + 128)^4
$73$	\( (T^{6} + 450 T^{4} + \cdots + 2458624)^{2} \) (T^6 + 450T^4 + 59633T^2 + 2458624)^2
$79$	\( (T^{6} + 386 T^{4} + \cdots + 51772)^{2} \) (T^6 + 386T^4 + 23757T^2 + 51772)^2
$83$	\( (T^{6} - 470 T^{4} + \cdots - 2903152)^{2} \) (T^6 - 470T^4 + 66741T^2 - 2903152)^2
$89$	\( (T^{6} - 171 T^{4} + \cdots - 17500)^{2} \) (T^6 - 171T^4 + 4457T^2 - 17500)^2
$97$	\( (T^{6} - 614 T^{4} + \cdots - 4435312)^{2} \) (T^6 - 614T^4 + 107173T^2 - 4435312)^2
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\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)