LMFDB - Newform orbit 460.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,2,Mod(459,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, 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("460.459");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	460.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:	$3.67311849298$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.207360000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 $ x^8 + 6x^6 + 32x^4 + 24*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
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 $1,\beta_1,\ldots,\beta_{7}$ 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_{4} + \beta_{2}) q^{3} + ( - \beta_{3} - 1) q^{4} + \beta_1 q^{5} + (2 \beta_{3} - 2) q^{6} + \beta_{6} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8} + 5 q^{9}+O(q^{10}) $ q - b2 * q^2 + (-b4 + b2) * q^3 + (-b3 - 1) * q^4 + b1 * q^5 + (2b3 - 2) q^6 + b6 * q^7 + (-b4 + b2) * q^8 + 5 * q^9	\( q - \beta_{2} q^{2} + ( - \beta_{4} + \beta_{2}) q^{3} + ( - \beta_{3} - 1) q^{4} + \beta_1 q^{5} + (2 \beta_{3} - 2) q^{6} + \beta_{6} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8} + 5 q^{9} + \beta_{5} q^{10} + (2 \beta_{4} + 2 \beta_{2}) q^{12} + (\beta_{4} + 3 \beta_{2}) q^{13} - \beta_{7} q^{14} + (\beta_{7} - \beta_{5}) q^{15} + (2 \beta_{3} - 2) q^{16} - \beta_1 q^{17} - 5 \beta_{2} q^{18} + ( - \beta_{7} + \beta_{5}) q^{19} + ( - \beta_{6} - \beta_1) q^{20} + (\beta_{7} + 3 \beta_{5}) q^{21} + ( - \beta_{6} + \beta_{4} - \beta_{2}) q^{23} + 8 q^{24} + 5 q^{25} + (2 \beta_{3} + 6) q^{26} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{27} + ( - \beta_{6} + 3 \beta_1) q^{28} + 3 q^{29} + (2 \beta_{6} - 2 \beta_1) q^{30} + \beta_{3} q^{31} + (2 \beta_{4} + 2 \beta_{2}) q^{32} - \beta_{5} q^{34} + 5 \beta_{3} q^{35} + ( - 5 \beta_{3} - 5) q^{36} + 3 \beta_1 q^{37} + ( - 2 \beta_{6} + 2 \beta_1) q^{38} - 8 \beta_{3} q^{39} + (\beta_{7} - \beta_{5}) q^{40} - 9 q^{41} + ( - 2 \beta_{6} - 6 \beta_1) q^{42} + 2 \beta_{6} q^{43} + 5 \beta_1 q^{45} + (\beta_{7} - 2 \beta_{3} + 2) q^{46} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{47} - 8 \beta_{2} q^{48} - 8 q^{49} - 5 \beta_{2} q^{50} + ( - \beta_{7} + \beta_{5}) q^{51} + (2 \beta_{4} - 6 \beta_{2}) q^{52} - \beta_1 q^{53} + (4 \beta_{3} - 4) q^{54} + (\beta_{7} + 3 \beta_{5}) q^{56} - 8 \beta_1 q^{57} - 3 \beta_{2} q^{58} + 5 \beta_{3} q^{59} + ( - 2 \beta_{7} - 2 \beta_{5}) q^{60} + ( - \beta_{7} - 3 \beta_{5}) q^{61} + \beta_{4} q^{62} + 5 \beta_{6} q^{63} + 8 q^{64} + ( - \beta_{7} - 3 \beta_{5}) q^{65} - 3 \beta_{6} q^{67} + (\beta_{6} + \beta_1) q^{68} + ( - \beta_{7} - 3 \beta_{5} - 8) q^{69} + 5 \beta_{4} q^{70} + \beta_{3} q^{71} + ( - 5 \beta_{4} + 5 \beta_{2}) q^{72} + (\beta_{4} + 3 \beta_{2}) q^{73} + 3 \beta_{5} q^{74} + ( - 5 \beta_{4} + 5 \beta_{2}) q^{75} + (2 \beta_{7} + 2 \beta_{5}) q^{76} - 8 \beta_{4} q^{78} + ( - 2 \beta_{7} + 2 \beta_{5}) q^{79} + (2 \beta_{6} - 2 \beta_1) q^{80} + q^{81} + 9 \beta_{2} q^{82} + \beta_{6} q^{83} + (2 \beta_{7} - 6 \beta_{5}) q^{84} - 5 q^{85} - 2 \beta_{7} q^{86} + ( - 3 \beta_{4} + 3 \beta_{2}) q^{87} + (\beta_{7} + 3 \beta_{5}) q^{89} + 5 \beta_{5} q^{90} + (3 \beta_{7} - 3 \beta_{5}) q^{91} + (\beta_{6} - 2 \beta_{4} + \cdots - 3 \beta_1) q^{92}+ \cdots + 8 \beta_{2} q^{98}+O(q^{100}) \) q - b2 * q^2 + (-b4 + b2) * q^3 + (-b3 - 1) * q^4 + b1 * q^5 + (2b3 - 2) q^6 + b6 * q^7 + (-b4 + b2) * q^8 + 5 * q^9 + b5 * q^10 + (2b4 + 2b2) * q^12 + (b4 + 3b2) q^13 - b7 * q^14 + (b7 - b5) * q^15 + (2b3 - 2) q^16 - b1 * q^17 - 5b2 q^18 + (-b7 + b5) * q^19 + (-b6 - b1) * q^20 + (b7 + 3b5) q^21 + (-b6 + b4 - b2) * q^23 + 8 * q^24 + 5 * q^25 + (2b3 + 6) q^26 + (-2b4 + 2b2) * q^27 + (-b6 + 3b1) q^28 + 3 * q^29 + (2b6 - 2b1) * q^30 + b3 * q^31 + (2b4 + 2b2) * q^32 - b5 * q^34 + 5b3 q^35 + (-5b3 - 5) q^36 + 3b1 q^37 + (-2b6 + 2b1) * q^38 - 8b3 q^39 + (b7 - b5) * q^40 - 9 * q^41 + (-2b6 - 6b1) * q^42 + 2b6 q^43 + 5b1 q^45 + (b7 - 2b3 + 2) q^46 + (-2b4 + 2b2) * q^47 - 8b2 q^48 - 8 * q^49 - 5b2 q^50 + (-b7 + b5) * q^51 + (2b4 - 6b2) * q^52 - b1 * q^53 + (4b3 - 4) q^54 + (b7 + 3b5) q^56 - 8b1 q^57 - 3b2 q^58 + 5b3 q^59 + (-2b7 - 2b5) * q^60 + (-b7 - 3b5) q^61 + b4 * q^62 + 5b6 q^63 + 8 * q^64 + (-b7 - 3b5) q^65 - 3b6 q^67 + (b6 + b1) * q^68 + (-b7 - 3b5 - 8) q^69 + 5b4 q^70 + b3 * q^71 + (-5b4 + 5b2) * q^72 + (b4 + 3b2) q^73 + 3b5 q^74 + (-5b4 + 5b2) * q^75 + (2b7 + 2b5) * q^76 - 8b4 q^78 + (-2b7 + 2b5) * q^79 + (2b6 - 2b1) * q^80 + q^81 + 9b2 q^82 + b6 * q^83 + (2b7 - 6b5) * q^84 - 5 * q^85 - 2b7 q^86 + (-3b4 + 3b2) * q^87 + (b7 + 3b5) q^89 + 5b5 q^90 + (3b7 - 3b5) * q^91 + (b6 - 2b4 - 2b2 - 3b1) q^92 + (-b4 - 3b2) q^93 + (4b3 - 4) q^94 + (5b4 - 5b2) * q^95 + (-8b3 - 8) q^96 + 6b1 q^97 + 8b2 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 16 q^{6} + 40 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 - 16 * q^6 + 40 * q^9	$ 8 q - 8 q^{4} - 16 q^{6} + 40 q^{9} - 16 q^{16} + 64 q^{24} + 40 q^{25} + 48 q^{26} + 24 q^{29} - 40 q^{36} - 72 q^{41} + 16 q^{46} - 64 q^{49} - 32 q^{54} + 64 q^{64} - 64 q^{69} + 8 q^{81} - 40 q^{85} - 32 q^{94} - 64 q^{96}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^6 + 40 * q^9 - 16 * q^16 + 64 * q^24 + 40 * q^25 + 48 * q^26 + 24 * q^29 - 40 * q^36 - 72 * q^41 + 16 * q^46 - 64 * q^49 - 32 * q^54 + 64 * q^64 - 64 * q^69 + 8 * q^81 - 40 * q^85 - 32 * q^94 - 64 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 $ x^8 + 6*x^6 + 32*x^4 + 24*x^2 + 16 :

$\beta_{1}$	$=$	$ ( \nu^{6} - 72 ) / 32 $ (v^6 - 72) / 32
$\beta_{2}$	$=$	$ ( 5\nu^{7} + 32\nu^{5} + 160\nu^{3} + 120\nu ) / 64 $ (5v^7 + 32v^5 + 160v^3 + 120v) / 64
$\beta_{3}$	$=$	$ ( 3\nu^{6} + 16\nu^{4} + 96\nu^{2} + 40 ) / 32 $ (3v^6 + 16v^4 + 96*v^2 + 40) / 32
$\beta_{4}$	$=$	$ ( -7\nu^{7} - 32\nu^{5} - 160\nu^{3} + 88\nu ) / 64 $ (-7v^7 - 32v^5 - 160v^3 + 88v) / 64
$\beta_{5}$	$=$	$ ( 11\nu^{7} + 64\nu^{5} + 352\nu^{3} + 264\nu ) / 64 $ (11v^7 + 64v^5 + 352v^3 + 264v) / 64
$\beta_{6}$	$=$	$ ( -\nu^{6} - 6\nu^{4} - 28\nu^{2} - 12 ) / 4 $ (-v^6 - 6v^4 - 28v^2 - 12) / 4
$\beta_{7}$	$=$	$ ( -13\nu^{7} - 64\nu^{5} - 352\nu^{3} + 200\nu ) / 64 $ (-13v^7 - 64v^5 - 352v^3 + 200v) / 64

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} ) / 4 $ (b7 + b5 - b4 - b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{6} + 3\beta_{3} - \beta _1 - 3 ) / 2 $ (b6 + 3*b3 - b1 - 3) / 2
$\nu^{3}$	$=$	$ -\beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{2} $ -b7 + b5 + 2b4 - 2b2
$\nu^{4}$	$=$	$ -3\beta_{6} - 7\beta_{3} - 3\beta _1 - 7 $ -3b6 - 7b3 - 3*b1 - 7
$\nu^{5}$	$=$	$ -10\beta_{5} + 22\beta_{2} $ -10b5 + 22b2
$\nu^{6}$	$=$	$ 32\beta _1 + 72 $ 32*b1 + 72
$\nu^{7}$	$=$	$ 26\beta_{7} + 26\beta_{5} - 58\beta_{4} - 58\beta_{2} $ 26b7 + 26b5 - 58b4 - 58b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$.

$n$	$231$	$277$	$281$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

459.1

−0.437016	+	0.756934i
1.14412	−	1.98168i
−0.437016	−	0.756934i
1.14412	+	1.98168i
0.437016	+	0.756934i
−1.14412	−	1.98168i
0.437016	−	0.756934i
−1.14412	+	1.98168i

−0.707107

−

1.22474i

2.82843

−1.00000

1.73205i

−2.23607

−2.00000

−

3.46410i

3.87298i

2.82843

5.00000

1.58114

2.73861i

459.2

−0.707107

−

1.22474i

2.82843

−1.00000

1.73205i

2.23607

−2.00000

−

3.46410i

−

3.87298i

2.82843

5.00000

−1.58114

−

2.73861i

459.3

−0.707107

1.22474i

2.82843

−1.00000

−

1.73205i

−2.23607

−2.00000

3.46410i

−

3.87298i

2.82843

5.00000

1.58114

−

2.73861i

459.4

−0.707107

1.22474i

2.82843

−1.00000

−

1.73205i

2.23607

−2.00000

3.46410i

3.87298i

2.82843

5.00000

−1.58114

2.73861i

459.5

0.707107

−

1.22474i

−2.82843

−1.00000

−

1.73205i

−2.23607

−2.00000

3.46410i

−

3.87298i

−2.82843

5.00000

−1.58114

2.73861i

459.6

0.707107

−

1.22474i

−2.82843

−1.00000

−

1.73205i

2.23607

−2.00000

3.46410i

3.87298i

−2.82843

5.00000

1.58114

−

2.73861i

459.7

0.707107

1.22474i

−2.82843

−1.00000

1.73205i

−2.23607

−2.00000

−

3.46410i

3.87298i

−2.82843

5.00000

−1.58114

−

2.73861i

459.8

0.707107

1.22474i

−2.82843

−1.00000

1.73205i

2.23607

−2.00000

−

3.46410i

−

3.87298i

−2.82843

5.00000

1.58114

2.73861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner
23.b	odd	2	1	inner
92.b	even	2	1	inner
115.c	odd	2	1	inner
460.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	460.2.g.b	✓	8
4.b	odd	2	1	inner	460.2.g.b	✓	8
5.b	even	2	1	inner	460.2.g.b	✓	8
20.d	odd	2	1	inner	460.2.g.b	✓	8
23.b	odd	2	1	inner	460.2.g.b	✓	8
92.b	even	2	1	inner	460.2.g.b	✓	8
115.c	odd	2	1	inner	460.2.g.b	✓	8
460.g	even	2	1	inner	460.2.g.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
460.2.g.b	✓	8	1.a	even	1	1	trivial
460.2.g.b	✓	8	4.b	odd	2	1	inner
460.2.g.b	✓	8	5.b	even	2	1	inner
460.2.g.b	✓	8	20.d	odd	2	1	inner
460.2.g.b	✓	8	23.b	odd	2	1	inner
460.2.g.b	✓	8	92.b	even	2	1	inner
460.2.g.b	✓	8	115.c	odd	2	1	inner
460.2.g.b	✓	8	460.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} - 8 $ T3^2 - 8 acting on $S_{2}^{\mathrm{new}}(460, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 2 T^{2} + 4)^{2} $ (T^4 + 2*T^2 + 4)^2
$3$	$ (T^{2} - 8)^{4} $ (T^2 - 8)^4
$5$	$ (T^{2} - 5)^{4} $ (T^2 - 5)^4
$7$	$ (T^{2} + 15)^{4} $ (T^2 + 15)^4
$11$	$ T^{8} $ T^8
$13$	$ (T^{2} + 24)^{4} $ (T^2 + 24)^4
$17$	$ (T^{2} - 5)^{4} $ (T^2 - 5)^4
$19$	$ (T^{2} - 40)^{4} $ (T^2 - 40)^4
$23$	$ (T^{4} + 14 T^{2} + 529)^{2} $ (T^4 + 14*T^2 + 529)^2
$29$	$ (T - 3)^{8} $ (T - 3)^8
$31$	$ (T^{2} + 3)^{4} $ (T^2 + 3)^4
$37$	$ (T^{2} - 45)^{4} $ (T^2 - 45)^4
$41$	$ (T + 9)^{8} $ (T + 9)^8
$43$	$ (T^{2} + 60)^{4} $ (T^2 + 60)^4
$47$	$ (T^{2} - 32)^{4} $ (T^2 - 32)^4
$53$	$ (T^{2} - 5)^{4} $ (T^2 - 5)^4
$59$	$ (T^{2} + 75)^{4} $ (T^2 + 75)^4
$61$	$ (T^{2} + 120)^{4} $ (T^2 + 120)^4
$67$	$ (T^{2} + 135)^{4} $ (T^2 + 135)^4
$71$	$ (T^{2} + 3)^{4} $ (T^2 + 3)^4
$73$	$ (T^{2} + 24)^{4} $ (T^2 + 24)^4
$79$	$ (T^{2} - 160)^{4} $ (T^2 - 160)^4
$83$	$ (T^{2} + 15)^{4} $ (T^2 + 15)^4
$89$	$ (T^{2} + 120)^{4} $ (T^2 + 120)^4
$97$	$ (T^{2} - 180)^{4} $ (T^2 - 180)^4
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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:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.g (of order \(2\), degree \(1\), minimal)

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

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	460.2.g.b

Level	$460$
Weight	$2$
Character orbit	460.g
Analytic conductor	$3.673$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.207360000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) x^8 + 6x^6 + 32x^4 + 24*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} - 72 ) / 32 \) (v^6 - 72) / 32
\(\beta_{2}\)	\(=\)	\( ( 5\nu^{7} + 32\nu^{5} + 160\nu^{3} + 120\nu ) / 64 \) (5v^7 + 32v^5 + 160v^3 + 120v) / 64
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{6} + 16\nu^{4} + 96\nu^{2} + 40 ) / 32 \) (3v^6 + 16v^4 + 96*v^2 + 40) / 32
\(\beta_{4}\)	\(=\)	\( ( -7\nu^{7} - 32\nu^{5} - 160\nu^{3} + 88\nu ) / 64 \) (-7v^7 - 32v^5 - 160v^3 + 88v) / 64
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{7} + 64\nu^{5} + 352\nu^{3} + 264\nu ) / 64 \) (11v^7 + 64v^5 + 352v^3 + 264v) / 64
\(\beta_{6}\)	\(=\)	\( ( -\nu^{6} - 6\nu^{4} - 28\nu^{2} - 12 ) / 4 \) (-v^6 - 6v^4 - 28v^2 - 12) / 4
\(\beta_{7}\)	\(=\)	\( ( -13\nu^{7} - 64\nu^{5} - 352\nu^{3} + 200\nu ) / 64 \) (-13v^7 - 64v^5 - 352v^3 + 200v) / 64

\(\nu\)	\(=\)	\( ( \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} ) / 4 \) (b7 + b5 - b4 - b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{6} + 3\beta_{3} - \beta _1 - 3 ) / 2 \) (b6 + 3*b3 - b1 - 3) / 2
\(\nu^{3}\)	\(=\)	\( -\beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{2} \) -b7 + b5 + 2b4 - 2b2
\(\nu^{4}\)	\(=\)	\( -3\beta_{6} - 7\beta_{3} - 3\beta _1 - 7 \) -3b6 - 7b3 - 3*b1 - 7
\(\nu^{5}\)	\(=\)	\( -10\beta_{5} + 22\beta_{2} \) -10b5 + 22b2
\(\nu^{6}\)	\(=\)	\( 32\beta _1 + 72 \) 32*b1 + 72
\(\nu^{7}\)	\(=\)	\( 26\beta_{7} + 26\beta_{5} - 58\beta_{4} - 58\beta_{2} \) 26b7 + 26b5 - 58b4 - 58b2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 459.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 2 T^{2} + 4)^{2} \) (T^4 + 2*T^2 + 4)^2
$3$	\( (T^{2} - 8)^{4} \) (T^2 - 8)^4
$5$	\( (T^{2} - 5)^{4} \) (T^2 - 5)^4
$7$	\( (T^{2} + 15)^{4} \) (T^2 + 15)^4
$11$	\( T^{8} \) T^8
$13$	\( (T^{2} + 24)^{4} \) (T^2 + 24)^4
$17$	\( (T^{2} - 5)^{4} \) (T^2 - 5)^4
$19$	\( (T^{2} - 40)^{4} \) (T^2 - 40)^4
$23$	\( (T^{4} + 14 T^{2} + 529)^{2} \) (T^4 + 14*T^2 + 529)^2
$29$	\( (T - 3)^{8} \) (T - 3)^8
$31$	\( (T^{2} + 3)^{4} \) (T^2 + 3)^4
$37$	\( (T^{2} - 45)^{4} \) (T^2 - 45)^4
$41$	\( (T + 9)^{8} \) (T + 9)^8
$43$	\( (T^{2} + 60)^{4} \) (T^2 + 60)^4
$47$	\( (T^{2} - 32)^{4} \) (T^2 - 32)^4
$53$	\( (T^{2} - 5)^{4} \) (T^2 - 5)^4
$59$	\( (T^{2} + 75)^{4} \) (T^2 + 75)^4
$61$	\( (T^{2} + 120)^{4} \) (T^2 + 120)^4
$67$	\( (T^{2} + 135)^{4} \) (T^2 + 135)^4
$71$	\( (T^{2} + 3)^{4} \) (T^2 + 3)^4
$73$	\( (T^{2} + 24)^{4} \) (T^2 + 24)^4
$79$	\( (T^{2} - 160)^{4} \) (T^2 - 160)^4
$83$	\( (T^{2} + 15)^{4} \) (T^2 + 15)^4
$89$	\( (T^{2} + 120)^{4} \) (T^2 + 120)^4
$97$	\( (T^{2} - 180)^{4} \) (T^2 - 180)^4
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\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)