Newform orbit 363.2.f.h

• Label: 363.2.f.h
• Level: $363$
• Weight: $2$
• Character orbit: 363.f
• Analytic conductor: $2.899$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.f (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89856959337\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	16.0.26873856000000000000.1
Defining polynomial:	\( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b13 + b12 - b8 - 2*b6 + b2) * q^2 + (-b15 + 2*b5) * q^3 + (4*b15 + 4*b9 + 4*b7 - 4*b3 - 4*b1 - 4) * q^4 + (b15 + b11 + b9 - b7 - b5 - b3 - b1) * q^5 - 3*b14 * q^6 - b8 * q^7 + (2*b13 + 4*b10) * q^8 + (3*b11 - 3*b1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} - 16 q^{4} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{4} ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{6} ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{8} ) / 16 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{9} ) / 16 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{10} ) / 32 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{11} ) / 32 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{12} ) / 64 \)
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} ) / 128 \)
\(\beta_{13}\)	\(=\)	\( ( -\nu^{11} + 32\nu ) / 32 \)
\(\beta_{14}\)	\(=\)	\( ( \nu^{13} - 32\nu^{3} ) / 64 \)
\(\beta_{15}\)	\(=\)	\( ( \nu^{14} - 4\nu^{10} - 8\nu^{8} + 64\nu^{2} + 128 ) / 128 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(1 + \beta_{1} + \beta_{3} - \beta_{7} - \beta_{9} - \beta_{15}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

161.1

−0.294032	−	1.38331i
1.05097	−	0.946294i
0.294032	+	1.38331i
−1.05097	+	0.946294i
1.40647	+	0.147826i
0.575212	+	1.29195i
−1.40647	−	0.147826i
−0.575212	−	1.29195i
1.40647	−	0.147826i
0.575212	−	1.29195i
−1.40647	+	0.147826i
−0.575212	+	1.29195i
−0.294032	+	1.38331i
1.05097	+	0.946294i
0.294032	−	1.38331i
−1.05097	−	0.946294i

−1.98168

−

1.43977i

−1.28716

+

1.15897i

1.23607

+

3.80423i

1.01807

+

1.40126i

4.21940

−

0.443477i

1.34500

−

0.437016i

1.51387

−

4.65921i

0.313585

−

2.98357i

−

4.24264i

161.2

−1.98168

−

1.43977i

0.360114

+

1.69420i

1.23607

+

3.80423i

−1.01807

−

1.40126i

1.72564

−

3.87585i

−1.34500

+

0.437016i

1.51387

−

4.65921i

−2.74064

+

1.22021i

4.24264i

161.3

1.98168

+

1.43977i

−1.28716

+

1.15897i

1.23607

+

3.80423i

1.01807

+

1.40126i

−4.21940

+

0.443477i

−1.34500

+

0.437016i

−1.51387

+

4.65921i

0.313585

−

2.98357i

4.24264i

161.4

1.98168

+

1.43977i

0.360114

+

1.69420i

1.23607

+

3.80423i

−1.01807

−

1.40126i

−1.72564

+

3.87585i

1.34500

−

0.437016i

−1.51387

+

4.65921i

−2.74064

+

1.22021i

−

4.24264i

215.1

−0.756934

−

2.32960i

0.704489

+

1.58231i

−3.23607

+

2.35114i

−1.64728

−

0.535233i

3.15290

−

2.83888i

−0.831254

−

1.14412i

3.96336

+

2.87955i

−2.00739

+

2.22943i

4.24264i

215.2

−0.756934

−

2.32960i

1.72256

+

0.181049i

−3.23607

+

2.35114i

1.64728

+

0.535233i

−0.882095

−

4.14993i

0.831254

+

1.14412i

3.96336

+

2.87955i

2.93444

+

0.623735i

−

4.24264i

215.3

0.756934

+

2.32960i

0.704489

+

1.58231i

−3.23607

+

2.35114i

−1.64728

−

0.535233i

−3.15290

+

2.83888i

0.831254

+

1.14412i

−3.96336

−

2.87955i

−2.00739

+

2.22943i

−

4.24264i

215.4

0.756934

+

2.32960i

1.72256

+

0.181049i

−3.23607

+

2.35114i

1.64728

+

0.535233i

0.882095

+

4.14993i

−0.831254

−

1.14412i

−3.96336

−

2.87955i

2.93444

+

0.623735i

4.24264i

233.1

−0.756934

+

2.32960i

0.704489

−

1.58231i

−3.23607

−

2.35114i

−1.64728

+

0.535233i

3.15290

+

2.83888i

−0.831254

+

1.14412i

3.96336

−

2.87955i

−2.00739

−

2.22943i

−

4.24264i

233.2

−0.756934

+

2.32960i

1.72256

−

0.181049i

−3.23607

−

2.35114i

1.64728

−

0.535233i

−0.882095

+

4.14993i

0.831254

−

1.14412i

3.96336

−

2.87955i

2.93444

−

0.623735i

4.24264i

233.3

0.756934

−

2.32960i

0.704489

−

1.58231i

−3.23607

−

2.35114i

−1.64728

+

0.535233i

−3.15290

−

2.83888i

0.831254

−

1.14412i

−3.96336

+

2.87955i

−2.00739

−

2.22943i

4.24264i

233.4

0.756934

−

2.32960i

1.72256

−

0.181049i

−3.23607

−

2.35114i

1.64728

−

0.535233i

0.882095

−

4.14993i

−0.831254

+

1.14412i

−3.96336

+

2.87955i

2.93444

−

0.623735i

−

4.24264i

239.1

−1.98168

+

1.43977i

−1.28716

−

1.15897i

1.23607

−

3.80423i

1.01807

−

1.40126i

4.21940

+

0.443477i

1.34500

+

0.437016i

1.51387

+

4.65921i

0.313585

+

2.98357i

4.24264i

239.2

−1.98168

+

1.43977i

0.360114

−

1.69420i

1.23607

−

3.80423i

−1.01807

+

1.40126i

1.72564

+

3.87585i

−1.34500

−

0.437016i

1.51387

+

4.65921i

−2.74064

−

1.22021i

−

4.24264i

239.3

1.98168

−

1.43977i

−1.28716

−

1.15897i

1.23607

−

3.80423i

1.01807

−

1.40126i

−4.21940

−

0.443477i

−1.34500

−

0.437016i

−1.51387

−

4.65921i

0.313585

+

2.98357i

−

4.24264i

239.4

1.98168

−

1.43977i

0.360114

−

1.69420i

1.23607

−

3.80423i

−1.01807

+

1.40126i

−1.72564

−

3.87585i

1.34500

+

0.437016i

−1.51387

−

4.65921i

−2.74064

−

1.22021i

4.24264i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.2.f.h		16
3.b	odd	2	1	inner	363.2.f.h		16
11.b	odd	2	1	inner	363.2.f.h		16
11.c	even	5	1		363.2.d.c	✓	4
11.c	even	5	3	inner	363.2.f.h		16
11.d	odd	10	1		363.2.d.c	✓	4
11.d	odd	10	3	inner	363.2.f.h		16
33.d	even	2	1	inner	363.2.f.h		16
33.f	even	10	1		363.2.d.c	✓	4
33.f	even	10	3	inner	363.2.f.h		16
33.h	odd	10	1		363.2.d.c	✓	4
33.h	odd	10	3	inner	363.2.f.h		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.2.d.c	✓	4	11.c	even	5	1
363.2.d.c	✓	4	11.d	odd	10	1
363.2.d.c	✓	4	33.f	even	10	1
363.2.d.c	✓	4	33.h	odd	10	1
363.2.f.h		16	1.a	even	1	1	trivial
363.2.f.h		16	3.b	odd	2	1	inner
363.2.f.h		16	11.b	odd	2	1	inner
363.2.f.h		16	11.c	even	5	3	inner
363.2.f.h		16	11.d	odd	10	3	inner
363.2.f.h		16	33.d	even	2	1	inner
363.2.f.h		16	33.f	even	10	3	inner
363.2.f.h		16	33.h	odd	10	3	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{8} + 6T_{2}^{6} + 36T_{2}^{4} + 216T_{2}^{2} + 1296 \)

\( T_{5}^{8} - 3T_{5}^{6} + 9T_{5}^{4} - 27T_{5}^{2} + 81 \)

\( T_{7}^{8} - 2T_{7}^{6} + 4T_{7}^{4} - 8T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^8 + 6*T^6 + 36*T^4 + 216*T^2 + 1296)^2
$3$	(T^8 - 3*T^7 + 6*T^6 - 9*T^5 + 9*T^4 - 27*T^3 + 54*T^2 - 81*T + 81)^2
$5$	(T^8 - 3*T^6 + 9*T^4 - 27*T^2 + 81)^2
$7$	(T^8 - 2*T^6 + 4*T^4 - 8*T^2 + 16)^2
$11$	\( T^{16} \)