Newform orbit 312.2.h.b

• Label: 312.2.h.b
• Level: $312$
• Weight: $2$
• Character orbit: 312.h
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $12$
• CM discriminant: -104
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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.

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} + \beta_{7} q^{3} - 2 q^{4} - \beta_{9} q^{5} - \beta_{4} q^{6} + (\beta_{8} + \beta_{4}) q^{7} - 2 \beta_{3} q^{8} - \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) :

\(\beta_{1}\)	\(=\)	\( ( 2\nu^{9} - 21\nu^{6} + 16\nu^{3} + 372 ) / 105 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{10} + 18\nu^{7} - 113\nu^{4} + 144\nu ) / 30 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{9} + 49\nu^{6} - 269\nu^{3} + 107 ) / 70 \)
\(\beta_{4}\)	\(=\)	\( ( 6\nu^{11} - 98\nu^{8} + 573\nu^{5} - 529\nu^{2} ) / 105 \)
\(\beta_{5}\)	\(=\)	\( ( 40\nu^{11} - 9\nu^{10} - 595\nu^{8} + 147\nu^{7} + 3050\nu^{5} - 1017\nu^{4} + 475\nu^{2} + 2211\nu ) / 630 \)
\(\beta_{6}\)	\(=\)	\( ( 4\nu^{11} - 6\nu^{10} - 67\nu^{8} + 93\nu^{7} + 422\nu^{5} - 498\nu^{4} - 521\nu^{2} + 159\nu ) / 90 \)
\(\beta_{7}\)	\(=\)	\( ( 2\nu^{11} - 31\nu^{8} + 166\nu^{5} - 23\nu^{2} ) / 30 \)
\(\beta_{8}\)	\(=\)	\( ( 40\nu^{11} - 42\nu^{10} - 595\nu^{8} + 651\nu^{7} + 3050\nu^{5} - 3486\nu^{4} + 475\nu^{2} - 147\nu ) / 630 \)
\(\beta_{9}\)	\(=\)	\( ( 40\nu^{11} + 51\nu^{10} - 595\nu^{8} - 798\nu^{7} + 3050\nu^{5} + 4503\nu^{4} + 475\nu^{2} - 2064\nu ) / 630 \)
\(\beta_{10}\)	\(=\)	\( ( -4\nu^{11} - 12\nu^{10} + 67\nu^{8} + 186\nu^{7} - 422\nu^{5} - 996\nu^{4} + 521\nu^{2} + 318\nu ) / 90 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{9} - 15\nu^{6} + 83\nu^{3} - 33 ) / 6 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\)	\(79\)	\(145\)	\(157\)	\(209\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

155.1

−0.205073	−	0.809425i
−1.34767	−	1.57689i
−0.691788	+	1.95556i
0.803519	+	0.227114i
−0.598446	+	0.582311i
2.03946	−	0.378672i
−1.34767	+	1.57689i
−0.205073	+	0.809425i
0.803519	−	0.227114i
−0.691788	−	1.95556i
2.03946	+	0.378672i
−0.598446	−	0.582311i

−

1.41421i

−1.55274

−

0.767460i

−2.00000

0.380805i

−1.08535

+

2.19591i

−2.28519

2.82843i

1.82201

+

2.38333i

0.538540

155.2

−

1.41421i

−1.55274

+

0.767460i

−2.00000

0.380805i

1.08535

+

2.19591i

2.28519

2.82843i

1.82201

−

2.38333i

0.538540

155.3

−

1.41421i

0.111731

−

1.72844i

−2.00000

−

4.04932i

−2.44439

−

0.158012i

2.99062

2.82843i

−2.97503

−

0.386242i

−5.72660

155.4

−

1.41421i

0.111731

+

1.72844i

−2.00000

−

4.04932i

2.44439

−

0.158012i

−2.99062

2.82843i

−2.97503

+

0.386242i

−5.72660

155.5

−

1.41421i

1.44101

−

0.960984i

−2.00000

3.66851i

−1.35904

−

2.03790i

5.27581

2.82843i

1.15302

−

2.76957i

5.18806

155.6

−

1.41421i

1.44101

+

0.960984i

−2.00000

3.66851i

1.35904

−

2.03790i

−5.27581

2.82843i

1.15302

+

2.76957i

5.18806

155.7

1.41421i

−1.55274

−

0.767460i

−2.00000

−

0.380805i

1.08535

−

2.19591i

2.28519

−

2.82843i

1.82201

+

2.38333i

0.538540

155.8

1.41421i

−1.55274

+

0.767460i

−2.00000

−

0.380805i

−1.08535

−

2.19591i

−2.28519

−

2.82843i

1.82201

−

2.38333i

0.538540

155.9

1.41421i

0.111731

−

1.72844i

−2.00000

4.04932i

2.44439

+

0.158012i

−2.99062

−

2.82843i

−2.97503

−

0.386242i

−5.72660

155.10

1.41421i

0.111731

+

1.72844i

−2.00000

4.04932i

−2.44439

+

0.158012i

2.99062

−

2.82843i

−2.97503

+

0.386242i

−5.72660

155.11

1.41421i

1.44101

−

0.960984i

−2.00000

−

3.66851i

1.35904

+

2.03790i

−5.27581

−

2.82843i

1.15302

−

2.76957i

5.18806

155.12

1.41421i

1.44101

+

0.960984i

−2.00000

−

3.66851i

−1.35904

+

2.03790i

5.27581

−

2.82843i

1.15302

+

2.76957i

5.18806

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by \(\Q(\sqrt{-26}) \)
3.b	odd	2	1	inner
8.d	odd	2	1	inner
13.b	even	2	1	inner
24.f	even	2	1	inner
39.d	odd	2	1	inner
312.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.2.h.b	✓	12
3.b	odd	2	1	inner	312.2.h.b	✓	12
4.b	odd	2	1		1248.2.h.b		12
8.b	even	2	1		1248.2.h.b		12
8.d	odd	2	1	inner	312.2.h.b	✓	12
12.b	even	2	1		1248.2.h.b		12
13.b	even	2	1	inner	312.2.h.b	✓	12
24.f	even	2	1	inner	312.2.h.b	✓	12
24.h	odd	2	1		1248.2.h.b		12
39.d	odd	2	1	inner	312.2.h.b	✓	12
52.b	odd	2	1		1248.2.h.b		12
104.e	even	2	1		1248.2.h.b		12
104.h	odd	2	1	CM	312.2.h.b	✓	12
156.h	even	2	1		1248.2.h.b		12
312.b	odd	2	1		1248.2.h.b		12
312.h	even	2	1	inner	312.2.h.b	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.h.b	✓	12	1.a	even	1	1	trivial
312.2.h.b	✓	12	3.b	odd	2	1	inner
312.2.h.b	✓	12	8.d	odd	2	1	inner
312.2.h.b	✓	12	13.b	even	2	1	inner
312.2.h.b	✓	12	24.f	even	2	1	inner
312.2.h.b	✓	12	39.d	odd	2	1	inner
312.2.h.b	✓	12	104.h	odd	2	1	CM
312.2.h.b	✓	12	312.h	even	2	1	inner
1248.2.h.b		12	4.b	odd	2	1
1248.2.h.b		12	8.b	even	2	1
1248.2.h.b		12	12.b	even	2	1
1248.2.h.b		12	24.h	odd	2	1
1248.2.h.b		12	52.b	odd	2	1
1248.2.h.b		12	104.e	even	2	1
1248.2.h.b		12	156.h	even	2	1
1248.2.h.b		12	312.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{6} \)
$3$	\( (T^{6} + 2 T^{3} + 27)^{2} \)
$5$	(T^6 + 30*T^4 + 225*T^2 + 32)^2
$7$	(T^6 - 42*T^4 + 441*T^2 - 1300)^2
$11$	\( T^{12} \)