Newform orbit 760.2.p.f

• Label: 760.2.p.f
• 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:	8.0.5473632256.3
Defining polynomial:	\( x^{8} - 4x^{7} + 14x^{6} - 12x^{5} + 34x^{4} + 24x^{3} + 56x^{2} + 32x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
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.

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} - q^{3} + ( - \beta_{6} - \beta_{4} - \beta_{3} - 1) q^{4} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{5} + \beta_{6} q^{6} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{6} - 2 \beta_{3}) q^{8} - 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 8 q^{3} - 4 q^{4} - 4 q^{6} + 4 q^{8} - 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 12x^{5} + 34x^{4} + 24x^{3} + 56x^{2} + 32x + 16 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} + 8\nu^{4} + 50\nu^{3} + 124\nu^{2} + 304\nu + 208 ) / 216 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} + 8\nu^{5} - 56\nu^{4} + 10\nu^{3} - 82\nu^{2} - 196\nu - 148 ) / 108 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} + 7\nu^{6} - 28\nu^{5} + 60\nu^{4} - 90\nu^{3} + 82\nu^{2} - 28\nu + 64 ) / 36 \)
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{7} - 46\nu^{6} + 146\nu^{5} - 80\nu^{4} + 166\nu^{3} + 404\nu^{2} + 320\nu + 176 ) / 216 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{7} - 9\nu^{6} + 32\nu^{5} - 37\nu^{4} + 74\nu^{3} + 30\nu^{2} + 68\nu + 12 ) / 18 \)
\(\beta_{7}\)	\(=\)	\( ( 23\nu^{7} - 124\nu^{6} + 446\nu^{5} - 704\nu^{4} + 1126\nu^{3} - 448\nu^{2} + 440\nu - 496 ) / 216 \)

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

−0.404265	+	0.468519i
2.11137	−	2.44696i
−0.404265	−	0.468519i
2.11137	+	2.44696i
0.556194	+	1.97874i
−0.263301	−	0.936733i
0.556194	−	1.97874i
−0.263301	+	0.936733i

−0.207107

−

1.39897i

−1.00000

−1.91421

+

0.579471i

−1.04201

+

1.97844i

0.207107

+

1.39897i

−1.47363

1.20711

+

2.55791i

−2.00000

2.98357

+

1.04799i

379.2

−0.207107

−

1.39897i

−1.00000

−1.91421

+

0.579471i

1.04201

+

1.97844i

0.207107

+

1.39897i

1.47363

1.20711

+

2.55791i

−2.00000

2.55196

−

1.86749i

379.3

−0.207107

+

1.39897i

−1.00000

−1.91421

−

0.579471i

−1.04201

−

1.97844i

0.207107

−

1.39897i

−1.47363

1.20711

−

2.55791i

−2.00000

2.98357

−

1.04799i

379.4

−0.207107

+

1.39897i

−1.00000

−1.91421

−

0.579471i

1.04201

−

1.97844i

0.207107

−

1.39897i

1.47363

1.20711

−

2.55791i

−2.00000

2.55196

+

1.86749i

379.5

1.20711

−

0.736813i

−1.00000

0.914214

−

1.77882i

−1.97844

−

1.04201i

−1.20711

+

0.736813i

2.79793

−0.207107

−

2.82083i

−2.00000

−3.15595

+

0.199920i

379.6

1.20711

−

0.736813i

−1.00000

0.914214

−

1.77882i

1.97844

−

1.04201i

−1.20711

+

0.736813i

−2.79793

−0.207107

−

2.82083i

−2.00000

1.62042

−

2.71556i

379.7

1.20711

+

0.736813i

−1.00000

0.914214

+

1.77882i

−1.97844

+

1.04201i

−1.20711

−

0.736813i

2.79793

−0.207107

+

2.82083i

−2.00000

−3.15595

−

0.199920i

379.8

1.20711

+

0.736813i

−1.00000

0.914214

+

1.77882i

1.97844

+

1.04201i

−1.20711

−

0.736813i

−2.79793

−0.207107

+

2.82083i

−2.00000

1.62042

+

2.71556i

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.f	yes	8
5.b	even	2	1		760.2.p.e	✓	8
8.d	odd	2	1	inner	760.2.p.f	yes	8
19.b	odd	2	1		760.2.p.e	✓	8
40.e	odd	2	1		760.2.p.e	✓	8
95.d	odd	2	1	inner	760.2.p.f	yes	8
152.b	even	2	1		760.2.p.e	✓	8
760.p	even	2	1	inner	760.2.p.f	yes	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.p.e	✓	8	5.b	even	2	1
760.2.p.e	✓	8	19.b	odd	2	1
760.2.p.e	✓	8	40.e	odd	2	1
760.2.p.e	✓	8	152.b	even	2	1
760.2.p.f	yes	8	1.a	even	1	1	trivial
760.2.p.f	yes	8	8.d	odd	2	1	inner
760.2.p.f	yes	8	95.d	odd	2	1	inner
760.2.p.f	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 \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3} + 1 \)

\( T_{7}^{4} - 10T_{7}^{2} + 17 \)

\( T_{29}^{4} - 74T_{29}^{2} + 17 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^4 - 2*T^3 + 3*T^2 - 4*T + 4)^2
$3$	\( (T + 1)^{8} \)
$5$	\( T^{8} + 18T^{4} + 625 \)
$7$	\( (T^{4} - 10 T^{2} + 17)^{2} \)
$11$	\( (T^{2} + 8 T + 14)^{4} \)