Newform orbit 208.4.i.e

• Label: 208.4.i.e
• Level: $208$
• Weight: $4$
• Character orbit: 208.i
• Analytic conductor: $12.272$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2723972812\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial:	\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b2 + 3*b1) * q^3 + (-5*b3 - 10) * q^5 + (-b3 - 7*b2 - b1 + 7) * q^7 + (15*b3 + 10*b2 + 15*b1 - 10) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + 4 ) / 5 \)

Character values

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

\(n\)	\(53\)	\(79\)	\(145\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 + \beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

81.1

−0.780776	−	1.35234i
1.28078	+	2.21837i
−0.780776	+	1.35234i
1.28078	−	2.21837i

0

−1.84233

−

3.19101i

0

−17.8078

0

2.71922

−

4.70983i

0

6.71165

−

11.6249i

0

81.2

0

4.34233

+

7.52113i

0

2.80776

0

4.78078

−

8.28055i

0

−24.2116

+

41.9358i

0

113.1

0

−1.84233

+

3.19101i

0

−17.8078

0

2.71922

+

4.70983i

0

6.71165

+

11.6249i

0

113.2

0

4.34233

−

7.52113i

0

2.80776

0

4.78078

+

8.28055i

0

−24.2116

−

41.9358i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.4.i.e		4
4.b	odd	2	1		13.4.c.b	✓	4
12.b	even	2	1		117.4.g.d		4
13.c	even	3	1	inner	208.4.i.e		4
52.b	odd	2	1		169.4.c.f		4
52.f	even	4	2		169.4.e.g		8
52.i	odd	6	1		169.4.a.j		2
52.i	odd	6	1		169.4.c.f		4
52.j	odd	6	1		13.4.c.b	✓	4
52.j	odd	6	1		169.4.a.f		2
52.l	even	12	2		169.4.b.e		4
52.l	even	12	2		169.4.e.g		8
156.p	even	6	1		117.4.g.d		4
156.p	even	6	1		1521.4.a.t		2
156.r	even	6	1		1521.4.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.c.b	✓	4	4.b	odd	2	1
13.4.c.b	✓	4	52.j	odd	6	1
117.4.g.d		4	12.b	even	2	1
117.4.g.d		4	156.p	even	6	1
169.4.a.f		2	52.j	odd	6	1
169.4.a.j		2	52.i	odd	6	1
169.4.b.e		4	52.l	even	12	2
169.4.c.f		4	52.b	odd	2	1
169.4.c.f		4	52.i	odd	6	1
169.4.e.g		8	52.f	even	4	2
169.4.e.g		8	52.l	even	12	2
208.4.i.e		4	1.a	even	1	1	trivial
208.4.i.e		4	13.c	even	3	1	inner
1521.4.a.l		2	156.r	even	6	1
1521.4.a.t		2	156.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 5T_{3}^{3} + 57T_{3}^{2} + 160T_{3} + 1024 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	T^4 - 5*T^3 + 57*T^2 + 160*T + 1024
$5$	\( (T^{2} + 15 T - 50)^{2} \)
$7$	T^4 - 15*T^3 + 173*T^2 - 780*T + 2704
$11$	T^4 - 17*T^3 + 1173*T^2 + 15028*T + 781456