Newform orbit 208.2.w.c

• Label: 208.2.w.c
• Level: $208$
• Weight: $2$
• Character orbit: 208.w
• Analytic conductor: $1.661$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.195105024.2
Defining polynomial:	\( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 + (b5 - b4 + b2) * q^3 + (b7 + b6 + b5 + b3 + b1) * q^5 + (-b5 + b3) * q^7 + (-2*b7 - 2*b5 - b2 - b1 - 2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} + 42\nu^{2} - 45\nu - 27 ) / 108 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 10\nu^{4} - 10\nu^{3} + 66\nu^{2} + 45\nu - 81 ) / 108 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} - 66\nu^{2} + 171\nu - 27 ) / 108 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 2\nu^{6} - 10\nu^{5} - 2\nu^{4} + 22\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 54 \)
\(\beta_{5}\)	\(=\)	\( ( -5\nu^{7} + 8\nu^{6} - 4\nu^{5} - 26\nu^{4} + 52\nu^{3} + 18\nu^{2} - 153\nu + 162 ) / 108 \)
\(\beta_{6}\)	\(=\)	\( ( 16\nu^{7} - 37\nu^{6} + 26\nu^{5} + 64\nu^{4} - 158\nu^{3} - 24\nu^{2} + 450\nu - 729 ) / 108 \)
\(\beta_{7}\)	\(=\)	\( ( 19\nu^{7} - 49\nu^{6} + 14\nu^{5} + 130\nu^{4} - 218\nu^{3} - 150\nu^{2} + 801\nu - 891 ) / 108 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(79\)	\(145\)
\(\chi(n)\)	\(1\)	\(1\)	\(1 + \beta_{5}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

17.1

1.30512	+	1.13871i
1.72124	−	0.193255i
−1.58726	+	0.693255i
0.560908	−	1.63871i
1.30512	−	1.13871i
1.72124	+	0.193255i
−1.58726	−	0.693255i
0.560908	+	1.63871i

0

−1.63871

+

2.83834i

0

2.61023i

0

0.971521

−

0.560908i

0

−3.87076

−

6.70436i

0

17.2

0

−0.693255

+

1.20075i

0

−

3.44247i

0

2.74922

−

1.58726i

0

0.538796

+

0.933222i

0

17.3

0

0.193255

−

0.334727i

0

3.17452i

0

−2.98127

+

1.72124i

0

1.42531

+

2.46870i

0

17.4

0

1.13871

−

1.97231i

0

1.12182i

0

2.26053

−

1.30512i

0

−1.09334

−

1.89372i

0

49.1

0

−1.63871

−

2.83834i

0

−

2.61023i

0

0.971521

+

0.560908i

0

−3.87076

+

6.70436i

0

49.2

0

−0.693255

−

1.20075i

0

3.44247i

0

2.74922

+

1.58726i

0

0.538796

−

0.933222i

0

49.3

0

0.193255

+

0.334727i

0

−

3.17452i

0

−2.98127

−

1.72124i

0

1.42531

−

2.46870i

0

49.4

0

1.13871

+

1.97231i

0

−

1.12182i

0

2.26053

+

1.30512i

0

−1.09334

+

1.89372i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.2.w.c		8
3.b	odd	2	1		1872.2.by.n		8
4.b	odd	2	1		104.2.o.a	✓	8
8.b	even	2	1		832.2.w.i		8
8.d	odd	2	1		832.2.w.g		8
12.b	even	2	1		936.2.bi.b		8
13.c	even	3	1		2704.2.f.q		8
13.e	even	6	1	inner	208.2.w.c		8
13.e	even	6	1		2704.2.f.q		8
13.f	odd	12	1		2704.2.a.bd		4
13.f	odd	12	1		2704.2.a.be		4
39.h	odd	6	1		1872.2.by.n		8
52.b	odd	2	1		1352.2.o.f		8
52.f	even	4	1		1352.2.i.k		8
52.f	even	4	1		1352.2.i.l		8
52.i	odd	6	1		104.2.o.a	✓	8
52.i	odd	6	1		1352.2.f.f		8
52.j	odd	6	1		1352.2.f.f		8
52.j	odd	6	1		1352.2.o.f		8
52.l	even	12	1		1352.2.a.k		4
52.l	even	12	1		1352.2.a.l		4
52.l	even	12	1		1352.2.i.k		8
52.l	even	12	1		1352.2.i.l		8
104.p	odd	6	1		832.2.w.g		8
104.s	even	6	1		832.2.w.i		8
156.r	even	6	1		936.2.bi.b		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.o.a	✓	8	4.b	odd	2	1
104.2.o.a	✓	8	52.i	odd	6	1
208.2.w.c		8	1.a	even	1	1	trivial
208.2.w.c		8	13.e	even	6	1	inner
832.2.w.g		8	8.d	odd	2	1
832.2.w.g		8	104.p	odd	6	1
832.2.w.i		8	8.b	even	2	1
832.2.w.i		8	104.s	even	6	1
936.2.bi.b		8	12.b	even	2	1
936.2.bi.b		8	156.r	even	6	1
1352.2.a.k		4	52.l	even	12	1
1352.2.a.l		4	52.l	even	12	1
1352.2.f.f		8	52.i	odd	6	1
1352.2.f.f		8	52.j	odd	6	1
1352.2.i.k		8	52.f	even	4	1
1352.2.i.k		8	52.l	even	12	1
1352.2.i.l		8	52.f	even	4	1
1352.2.i.l		8	52.l	even	12	1
1352.2.o.f		8	52.b	odd	2	1
1352.2.o.f		8	52.j	odd	6	1
1872.2.by.n		8	3.b	odd	2	1
1872.2.by.n		8	39.h	odd	6	1
2704.2.a.bd		4	13.f	odd	12	1
2704.2.a.be		4	13.f	odd	12	1
2704.2.f.q		8	13.c	even	3	1
2704.2.f.q		8	13.e	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 11T_{3}^{6} + 2T_{3}^{5} + 61T_{3}^{4} + 40T_{3}^{3} + 92T_{3}^{2} - 32T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(208, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	T^8 + 2*T^7 + 11*T^6 + 2*T^5 + 61*T^4 + 40*T^3 + 92*T^2 - 32*T + 16
$5$	T^8 + 30*T^6 + 305*T^4 + 1152*T^2 + 1024
$7$	T^8 - 6*T^7 + 3*T^6 + 54*T^5 - 31*T^4 - 648*T^3 + 2016*T^2 - 2304*T + 1024
$11$	T^8 + 6*T^7 + 3*T^6 - 54*T^5 - 31*T^4 + 648*T^3 + 2016*T^2 + 2304*T + 1024