Newform orbit 392.2.e.e

• Label: 392.2.e.e
• Level: $392$
• Weight: $2$
• Character orbit: 392.e
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.144054149089536.2
Defining polynomial:	\( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} - \beta_{10} q^{3} + \beta_1 q^{4} - \beta_{9} q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{3}) q^{6} + (\beta_{11} - 1) q^{8} + (\beta_{8} + \beta_{5} + \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) :

\(\nu\)	\(=\)	\( ( -\beta_{11} - 2\beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 4 \)
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} + 4\beta_{10} - \beta_{9} - 2\beta_{8} + 5\beta_{7} - 3\beta_{6} - 2\beta_{5} + \beta_{4} + 6\beta _1 + 3 ) / 4 \)
\(\nu^{3}\)	\(=\)	(-4*b11 - 5*b10 + 3*b8 - 9*b7 - b6 + 6*b5 - 3*b4 - b3 - 6*b2 + 4) / 2
\(\nu^{4}\)	\(=\)	(5*b11 + 20*b10 - 11*b9 - 36*b8 + 19*b7 + 13*b6 + 37*b4 + 8*b3 + 28*b2 + 4*b1 - 41) / 4
\(\nu^{5}\)	\(=\)	(-9*b11 + 24*b10 + 53*b9 + 24*b8 + 23*b7 - 103*b6 - 84*b5 - 71*b4 - 32*b3 - 128*b2 + 64*b1 + 141) / 4
\(\nu^{6}\)	\(=\)	(-67*b11 - 149*b10 - 27*b9 - 6*b8 - 184*b7 + 131*b6 + 144*b5 + 82*b4 + 60*b3 + 66*b2 - 111*b1 - 201) / 2
\(\nu^{7}\)	\(=\)	(483*b11 + 1036*b10 + 187*b9 - 526*b8 + 1281*b7 - 451*b6 - 1052*b5 + 203*b4 + 30*b3 - 36*b2 + 650*b1 + 113) / 4
\(\nu^{8}\)	\(=\)	(-1347*b11 - 2728*b10 + 731*b9 + 2024*b8 - 3487*b7 - 61*b6 + 1368*b5 - 1781*b4 - 224*b3 - 2176*b2 - 1088*b1 + 1375) / 4
\(\nu^{9}\)	\(=\)	(1108*b11 + 1440*b10 - 1185*b9 - 2853*b8 + 2079*b7 + 2568*b6 + 90*b5 + 3717*b4 + 1455*b3 + 4302*b2 - 657*b1 - 5540) / 2
\(\nu^{10}\)	\(=\)	(2529*b11 + 8128*b10 + 10329*b9 + 9188*b8 + 9739*b7 - 20859*b6 - 18376*b5 - 19343*b4 - 7820*b3 - 23872*b2 + 12664*b1 + 33643) / 4
\(\nu^{11}\)	\(=\)	(-26995*b11 - 68422*b10 - 19207*b9 + 10204*b8 - 82791*b7 + 61673*b6 + 76280*b5 + 23791*b4 + 18276*b3 + 38004*b2 - 57006*b1 - 73457) / 4

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

195.1

2.00233	+	0.854000i
1.09935	+	0.468876i
1.09935	−	0.468876i
2.00233	−	0.854000i
0.186445	−	1.54034i
−0.0263223	+	0.217464i
−0.0263223	−	0.217464i
0.186445	+	1.54034i
0.609850	−	0.457915i
−2.37165	+	1.78079i
−2.37165	−	1.78079i
0.609850	+	0.457915i

−1.30084

−

0.554812i

−

0.480901i

1.38437

+

1.44344i

3.19427

−0.266810

+

0.625575i

0

−1.00000

−

2.64575i

2.76873

−4.15523

−

1.77222i

195.2

−1.30084

−

0.554812i

0.480901i

1.38437

+

1.44344i

−3.19427

0.266810

−

0.625575i

0

−1.00000

−

2.64575i

2.76873

4.15523

+

1.77222i

195.3

−1.30084

+

0.554812i

−

0.480901i

1.38437

−

1.44344i

−3.19427

0.266810

+

0.625575i

0

−1.00000

+

2.64575i

2.76873

4.15523

−

1.77222i

195.4

−1.30084

+

0.554812i

0.480901i

1.38437

−

1.44344i

3.19427

−0.266810

−

0.625575i

0

−1.00000

+

2.64575i

2.76873

−4.15523

+

1.77222i

195.5

0.169938

−

1.40397i

−

2.62383i

−1.94224

−

0.477176i

−2.07852

−3.68377

−

0.445890i

0

−1.00000

+

2.64575i

−3.88448

−0.353220

+

2.91817i

195.6

0.169938

−

1.40397i

2.62383i

−1.94224

−

0.477176i

2.07852

3.68377

+

0.445890i

0

−1.00000

+

2.64575i

−3.88448

0.353220

−

2.91817i

195.7

0.169938

+

1.40397i

−

2.62383i

−1.94224

+

0.477176i

2.07852

3.68377

−

0.445890i

0

−1.00000

−

2.64575i

−3.88448

0.353220

+

2.91817i

195.8

0.169938

+

1.40397i

2.62383i

−1.94224

+

0.477176i

−2.07852

−3.68377

+

0.445890i

0

−1.00000

−

2.64575i

−3.88448

−0.353220

−

2.91817i

195.9

1.13090

−

0.849154i

−

1.37268i

0.557875

−

1.92062i

0.690214

−1.16562

−

1.55237i

0

−1.00000

−

2.64575i

1.11575

0.780564

−

0.586098i

195.10

1.13090

−

0.849154i

1.37268i

0.557875

−

1.92062i

−0.690214

1.16562

+

1.55237i

0

−1.00000

−

2.64575i

1.11575

−0.780564

+

0.586098i

195.11

1.13090

+

0.849154i

−

1.37268i

0.557875

+

1.92062i

−0.690214

1.16562

−

1.55237i

0

−1.00000

+

2.64575i

1.11575

−0.780564

−

0.586098i

195.12

1.13090

+

0.849154i

1.37268i

0.557875

+

1.92062i

0.690214

−1.16562

+

1.55237i

0

−1.00000

+

2.64575i

1.11575

0.780564

+

0.586098i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.d	odd	2	1	inner
56.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.2.e.e		12
4.b	odd	2	1		1568.2.e.e		12
7.b	odd	2	1	inner	392.2.e.e		12
7.c	even	3	1		56.2.m.a	✓	12
7.c	even	3	1		392.2.m.g		12
7.d	odd	6	1		56.2.m.a	✓	12
7.d	odd	6	1		392.2.m.g		12
8.b	even	2	1		1568.2.e.e		12
8.d	odd	2	1	inner	392.2.e.e		12
21.g	even	6	1		504.2.bk.a		12
21.h	odd	6	1		504.2.bk.a		12
28.d	even	2	1		1568.2.e.e		12
28.f	even	6	1		224.2.q.a		12
28.f	even	6	1		1568.2.q.g		12
28.g	odd	6	1		224.2.q.a		12
28.g	odd	6	1		1568.2.q.g		12
56.e	even	2	1	inner	392.2.e.e		12
56.h	odd	2	1		1568.2.e.e		12
56.j	odd	6	1		224.2.q.a		12
56.j	odd	6	1		1568.2.q.g		12
56.k	odd	6	1		56.2.m.a	✓	12
56.k	odd	6	1		392.2.m.g		12
56.m	even	6	1		56.2.m.a	✓	12
56.m	even	6	1		392.2.m.g		12
56.p	even	6	1		224.2.q.a		12
56.p	even	6	1		1568.2.q.g		12
84.j	odd	6	1		2016.2.bs.a		12
84.n	even	6	1		2016.2.bs.a		12
168.s	odd	6	1		2016.2.bs.a		12
168.v	even	6	1		504.2.bk.a		12
168.ba	even	6	1		2016.2.bs.a		12
168.be	odd	6	1		504.2.bk.a		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.m.a	✓	12	7.c	even	3	1
56.2.m.a	✓	12	7.d	odd	6	1
56.2.m.a	✓	12	56.k	odd	6	1
56.2.m.a	✓	12	56.m	even	6	1
224.2.q.a		12	28.f	even	6	1
224.2.q.a		12	28.g	odd	6	1
224.2.q.a		12	56.j	odd	6	1
224.2.q.a		12	56.p	even	6	1
392.2.e.e		12	1.a	even	1	1	trivial
392.2.e.e		12	7.b	odd	2	1	inner
392.2.e.e		12	8.d	odd	2	1	inner
392.2.e.e		12	56.e	even	2	1	inner
392.2.m.g		12	7.c	even	3	1
392.2.m.g		12	7.d	odd	6	1
392.2.m.g		12	56.k	odd	6	1
392.2.m.g		12	56.m	even	6	1
504.2.bk.a		12	21.g	even	6	1
504.2.bk.a		12	21.h	odd	6	1
504.2.bk.a		12	168.v	even	6	1
504.2.bk.a		12	168.be	odd	6	1
1568.2.e.e		12	4.b	odd	2	1
1568.2.e.e		12	8.b	even	2	1
1568.2.e.e		12	28.d	even	2	1
1568.2.e.e		12	56.h	odd	2	1
1568.2.q.g		12	28.f	even	6	1
1568.2.q.g		12	28.g	odd	6	1
1568.2.q.g		12	56.j	odd	6	1
1568.2.q.g		12	56.p	even	6	1
2016.2.bs.a		12	84.j	odd	6	1
2016.2.bs.a		12	84.n	even	6	1
2016.2.bs.a		12	168.s	odd	6	1
2016.2.bs.a		12	168.ba	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{6} + 9T_{3}^{4} + 15T_{3}^{2} + 3 \)

\( T_{5}^{6} - 15T_{5}^{4} + 51T_{5}^{2} - 21 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{6} + 2 T^{3} + 8)^{2} \)
$3$	\( (T^{6} + 9 T^{4} + 15 T^{2} + 3)^{2} \)
$5$	(T^6 - 15*T^4 + 51*T^2 - 21)^2
$7$	\( T^{12} \)
$11$	\( (T^{3} - 3 T^{2} - 3 T + 7)^{4} \)