Newform orbit 98.4.c.g

• Label: 98.4.c.g
• Level: $98$
• Weight: $4$
• Character orbit: 98.c
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{22})\)
Defining polynomial:	\( x^{4} + 22x^{2} + 484 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} + 8 q^{8} + ( - 61 \beta_{2} - 61) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 122 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 22x^{2} + 484 \) :

\(\beta_{1}\)	\(=\)	\( 2\nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 22 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 11 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(-1 - \beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

2.34521	+	4.06202i
−2.34521	−	4.06202i
2.34521	−	4.06202i
−2.34521	+	4.06202i

−1.00000

−

1.73205i

−4.69042

+

8.12404i

−2.00000

+

3.46410i

4.69042

+

8.12404i

18.7617

0

8.00000

−30.5000

−

52.8275i

9.38083

−

16.2481i

67.2

−1.00000

−

1.73205i

4.69042

−

8.12404i

−2.00000

+

3.46410i

−4.69042

−

8.12404i

−18.7617

0

8.00000

−30.5000

−

52.8275i

−9.38083

+

16.2481i

79.1

−1.00000

+

1.73205i

−4.69042

−

8.12404i

−2.00000

−

3.46410i

4.69042

−

8.12404i

18.7617

0

8.00000

−30.5000

+

52.8275i

9.38083

+

16.2481i

79.2

−1.00000

+

1.73205i

4.69042

+

8.12404i

−2.00000

−

3.46410i

−4.69042

+

8.12404i

−18.7617

0

8.00000

−30.5000

+

52.8275i

−9.38083

−

16.2481i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	98.4.c.g		4
3.b	odd	2	1		882.4.g.bi		4
7.b	odd	2	1	inner	98.4.c.g		4
7.c	even	3	1		98.4.a.h	✓	2
7.c	even	3	1	inner	98.4.c.g		4
7.d	odd	6	1		98.4.a.h	✓	2
7.d	odd	6	1	inner	98.4.c.g		4
21.c	even	2	1		882.4.g.bi		4
21.g	even	6	1		882.4.a.w		2
21.g	even	6	1		882.4.g.bi		4
21.h	odd	6	1		882.4.a.w		2
21.h	odd	6	1		882.4.g.bi		4
28.f	even	6	1		784.4.a.z		2
28.g	odd	6	1		784.4.a.z		2
35.i	odd	6	1		2450.4.a.bs		2
35.j	even	6	1		2450.4.a.bs		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.4.a.h	✓	2	7.c	even	3	1
98.4.a.h	✓	2	7.d	odd	6	1
98.4.c.g		4	1.a	even	1	1	trivial
98.4.c.g		4	7.b	odd	2	1	inner
98.4.c.g		4	7.c	even	3	1	inner
98.4.c.g		4	7.d	odd	6	1	inner
784.4.a.z		2	28.f	even	6	1
784.4.a.z		2	28.g	odd	6	1
882.4.a.w		2	21.g	even	6	1
882.4.a.w		2	21.h	odd	6	1
882.4.g.bi		4	3.b	odd	2	1
882.4.g.bi		4	21.c	even	2	1
882.4.g.bi		4	21.g	even	6	1
882.4.g.bi		4	21.h	odd	6	1
2450.4.a.bs		2	35.i	odd	6	1
2450.4.a.bs		2	35.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 88T_{3}^{2} + 7744 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 2 T + 4)^{2} \)
$3$	\( T^{4} + 88T^{2} + 7744 \)
$5$	\( T^{4} + 88T^{2} + 7744 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} + 20 T + 400)^{2} \)