Newform orbit 760.2.p.g

• Label: 760.2.p.g
• Level: $760$
• Weight: $2$
• Character orbit: 760.p
• Analytic conductor: $6.069$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.06863055362\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.1499238400.2
Defining polynomial:	\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\)	\(=\)	\( q + (\beta_{5} + 1) q^{2} + q^{3} + 2 \beta_{5} q^{4} + (\beta_{5} - \beta_{4}) q^{5} + (\beta_{5} + 1) q^{6} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{7} + (2 \beta_{5} - 2) q^{8} - 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{6} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) :

\(\beta_{1}\)	\(=\)	\( ( -3\nu^{7} + 8\nu^{6} - 34\nu^{5} + 50\nu^{4} - 82\nu^{3} + 67\nu^{2} - 56\nu + 30 ) / 5 \)
\(\beta_{2}\)	\(=\)	\( \nu^{6} - 3\nu^{5} + 12\nu^{4} - 19\nu^{3} + 28\nu^{2} - 19\nu + 8 \)
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{7} - 13\nu^{6} + 49\nu^{5} - 105\nu^{4} + 167\nu^{3} - 162\nu^{2} + 111\nu - 20 ) / 5 \)
\(\beta_{4}\)	\(=\)	\( -\nu^{7} + 3\nu^{6} - 13\nu^{5} + 22\nu^{4} - 39\nu^{3} + 35\nu^{2} - 27\nu + 8 \)
\(\beta_{5}\)	\(=\)	\( ( 6\nu^{7} - 21\nu^{6} + 83\nu^{5} - 155\nu^{4} + 249\nu^{3} - 229\nu^{2} + 157\nu - 45 ) / 5 \)
\(\beta_{6}\)	\(=\)	\( -\nu^{7} + 4\nu^{6} - 16\nu^{5} + 33\nu^{4} - 56\nu^{3} + 56\nu^{2} - 40\nu + 12 \)
\(\beta_{7}\)	\(=\)	\( ( 12\nu^{7} - 42\nu^{6} + 176\nu^{5} - 335\nu^{4} + 598\nu^{3} - 583\nu^{2} + 474\nu - 150 ) / 5 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(381\)	\(401\)	\(457\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

379.1

0.500000	+	2.41267i
0.500000	+	1.08454i
0.500000	−	0.0845405i
0.500000	−	1.41267i
0.500000	−	2.41267i
0.500000	−	1.08454i
0.500000	+	0.0845405i
0.500000	+	1.41267i

1.00000

−

1.00000i

1.00000

−

2.00000i

−1.91267

+

1.15831i

1.00000

−

1.00000i

−3.21974

−2.00000

−

2.00000i

−2.00000

−0.754359

+

3.07098i

379.2

1.00000

−

1.00000i

1.00000

−

2.00000i

−0.584541

−

2.15831i

1.00000

−

1.00000i

−4.86140

−2.00000

−

2.00000i

−2.00000

−2.74285

−

1.57377i

379.3

1.00000

−

1.00000i

1.00000

−

2.00000i

0.584541

−

2.15831i

1.00000

−

1.00000i

4.86140

−2.00000

−

2.00000i

−2.00000

−1.57377

−

2.74285i

379.4

1.00000

−

1.00000i

1.00000

−

2.00000i

1.91267

+

1.15831i

1.00000

−

1.00000i

3.21974

−2.00000

−

2.00000i

−2.00000

3.07098

−

0.754359i

379.5

1.00000

+

1.00000i

1.00000

2.00000i

−1.91267

−

1.15831i

1.00000

+

1.00000i

−3.21974

−2.00000

+

2.00000i

−2.00000

−0.754359

−

3.07098i

379.6

1.00000

+

1.00000i

1.00000

2.00000i

−0.584541

+

2.15831i

1.00000

+

1.00000i

−4.86140

−2.00000

+

2.00000i

−2.00000

−2.74285

+

1.57377i

379.7

1.00000

+

1.00000i

1.00000

2.00000i

0.584541

+

2.15831i

1.00000

+

1.00000i

4.86140

−2.00000

+

2.00000i

−2.00000

−1.57377

+

2.74285i

379.8

1.00000

+

1.00000i

1.00000

2.00000i

1.91267

−

1.15831i

1.00000

+

1.00000i

3.21974

−2.00000

+

2.00000i

−2.00000

3.07098

+

0.754359i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
95.d	odd	2	1	inner
760.p	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	760.2.p.g	yes	8
5.b	even	2	1		760.2.p.d	✓	8
8.d	odd	2	1	inner	760.2.p.g	yes	8
19.b	odd	2	1		760.2.p.d	✓	8
40.e	odd	2	1		760.2.p.d	✓	8
95.d	odd	2	1	inner	760.2.p.g	yes	8
152.b	even	2	1		760.2.p.d	✓	8
760.p	even	2	1	inner	760.2.p.g	yes	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.p.d	✓	8	5.b	even	2	1
760.2.p.d	✓	8	19.b	odd	2	1
760.2.p.d	✓	8	40.e	odd	2	1
760.2.p.d	✓	8	152.b	even	2	1
760.2.p.g	yes	8	1.a	even	1	1	trivial
760.2.p.g	yes	8	8.d	odd	2	1	inner
760.2.p.g	yes	8	95.d	odd	2	1	inner
760.2.p.g	yes	8	760.p	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3} - 1 \)

\( T_{7}^{4} - 34T_{7}^{2} + 245 \)

\( T_{29}^{4} - 130T_{29}^{2} + 3125 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2 T + 2)^{4} \)
$3$	\( (T - 1)^{8} \)
$5$	T^8 + 4*T^6 + 10*T^4 + 100*T^2 + 625
$7$	\( (T^{4} - 34 T^{2} + 245)^{2} \)
$11$	\( (T^{2} - 2 T - 10)^{4} \)