Newform orbit 456.2.p.a

• Label: 456.2.p.a
• Level: $456$
• Weight: $2$
• Character orbit: 456.p
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $12$
• CM discriminant: -152
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 22x^{6} + 729 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} + \beta_{8} q^{3} - 2 q^{4} - \beta_{2} q^{6} + ( - \beta_{11} + \beta_{10}) q^{7} + 2 \beta_{3} q^{8} - \beta_{11} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 22x^{6} + 729 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 5\nu ) / 12 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{8} + 5\nu^{2} ) / 36 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} + 5\nu^{3} ) / 108 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} + 31\nu ) / 12 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{9} + 49\nu^{3} ) / 54 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} - 11 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( 2\nu^{10} + 9\nu^{8} + 118\nu^{4} - 279\nu^{2} ) / 324 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} + 22\nu^{5} ) / 243 \)
\(\beta_{9}\)	\(=\)	\( ( 5\nu^{11} + 133\nu^{5} + 972\nu ) / 972 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} - 3\nu^{8} + 5\nu^{4} + 93\nu^{2} ) / 108 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{10} - 22\nu^{4} ) / 81 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

341.1

−1.70027	+	0.330283i
−1.13617	+	1.30733i
−0.564101	−	1.63762i
0.564101	−	1.63762i
1.13617	+	1.30733i
1.70027	+	0.330283i
−1.70027	−	0.330283i
−1.13617	−	1.30733i
−0.564101	+	1.63762i
0.564101	+	1.63762i
1.13617	−	1.30733i
1.70027	−	0.330283i

−

1.41421i

−1.70027

−

0.330283i

−2.00000

0

−0.467090

+

2.40454i

4.62947

2.82843i

2.78183

+

1.12314i

0

341.2

−

1.41421i

−1.13617

−

1.30733i

−2.00000

0

−1.84885

+

1.60678i

−4.53419

2.82843i

−0.418247

+

2.97070i

0

341.3

−

1.41421i

−0.564101

+

1.63762i

−2.00000

0

2.31594

+

0.797759i

−0.0952793

2.82843i

−2.36358

−

1.84756i

0

341.4

−

1.41421i

0.564101

+

1.63762i

−2.00000

0

2.31594

−

0.797759i

−0.0952793

2.82843i

−2.36358

+

1.84756i

0

341.5

−

1.41421i

1.13617

−

1.30733i

−2.00000

0

−1.84885

−

1.60678i

−4.53419

2.82843i

−0.418247

−

2.97070i

0

341.6

−

1.41421i

1.70027

−

0.330283i

−2.00000

0

−0.467090

−

2.40454i

4.62947

2.82843i

2.78183

−

1.12314i

0

341.7

1.41421i

−1.70027

+

0.330283i

−2.00000

0

−0.467090

−

2.40454i

4.62947

−

2.82843i

2.78183

−

1.12314i

0

341.8

1.41421i

−1.13617

+

1.30733i

−2.00000

0

−1.84885

−

1.60678i

−4.53419

−

2.82843i

−0.418247

−

2.97070i

0

341.9

1.41421i

−0.564101

−

1.63762i

−2.00000

0

2.31594

−

0.797759i

−0.0952793

−

2.82843i

−2.36358

+

1.84756i

0

341.10

1.41421i

0.564101

−

1.63762i

−2.00000

0

2.31594

+

0.797759i

−0.0952793

−

2.82843i

−2.36358

−

1.84756i

0

341.11

1.41421i

1.13617

+

1.30733i

−2.00000

0

−1.84885

+

1.60678i

−4.53419

−

2.82843i

−0.418247

+

2.97070i

0

341.12

1.41421i

1.70027

+

0.330283i

−2.00000

0

−0.467090

+

2.40454i

4.62947

−

2.82843i

2.78183

+

1.12314i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
152.g	odd	2	1	CM by \(\Q(\sqrt{-38}) \)
3.b	odd	2	1	inner
8.b	even	2	1	inner
19.b	odd	2	1	inner
24.h	odd	2	1	inner
57.d	even	2	1	inner
456.p	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	456.2.p.a	✓	12
3.b	odd	2	1	inner	456.2.p.a	✓	12
4.b	odd	2	1		1824.2.p.a		12
8.b	even	2	1	inner	456.2.p.a	✓	12
8.d	odd	2	1		1824.2.p.a		12
12.b	even	2	1		1824.2.p.a		12
19.b	odd	2	1	inner	456.2.p.a	✓	12
24.f	even	2	1		1824.2.p.a		12
24.h	odd	2	1	inner	456.2.p.a	✓	12
57.d	even	2	1	inner	456.2.p.a	✓	12
76.d	even	2	1		1824.2.p.a		12
152.b	even	2	1		1824.2.p.a		12
152.g	odd	2	1	CM	456.2.p.a	✓	12
228.b	odd	2	1		1824.2.p.a		12
456.l	odd	2	1		1824.2.p.a		12
456.p	even	2	1	inner	456.2.p.a	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
456.2.p.a	✓	12	1.a	even	1	1	trivial
456.2.p.a	✓	12	3.b	odd	2	1	inner
456.2.p.a	✓	12	8.b	even	2	1	inner
456.2.p.a	✓	12	19.b	odd	2	1	inner
456.2.p.a	✓	12	24.h	odd	2	1	inner
456.2.p.a	✓	12	57.d	even	2	1	inner
456.2.p.a	✓	12	152.g	odd	2	1	CM
456.2.p.a	✓	12	456.p	even	2	1	inner
1824.2.p.a		12	4.b	odd	2	1
1824.2.p.a		12	8.d	odd	2	1
1824.2.p.a		12	12.b	even	2	1
1824.2.p.a		12	24.f	even	2	1
1824.2.p.a		12	76.d	even	2	1
1824.2.p.a		12	152.b	even	2	1
1824.2.p.a		12	228.b	odd	2	1
1824.2.p.a		12	456.l	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{6} \)
$3$	\( T^{12} - 22T^{6} + 729 \)
$5$	\( T^{12} \)
$7$	\( (T^{3} - 21 T - 2)^{4} \)
$11$	\( T^{12} \)