Newform orbit 98.4.c.f

• Label: 98.4.c.f
• Level: $98$
• Weight: $4$
• Character orbit: 98.c
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 8 q^{3} - 4 q^{4} - 14 q^{5} + 32 q^{6} - 16 q^{8} - 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(-\zeta_{6}\)

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

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

+

1.73205i

4.00000

−

6.92820i

−2.00000

+

3.46410i

−7.00000

−

12.1244i

16.0000

0

−8.00000

−18.5000

−

32.0429i

14.0000

−

24.2487i

79.1

1.00000

−

1.73205i

4.00000

+

6.92820i

−2.00000

−

3.46410i

−7.00000

+

12.1244i

16.0000

0

−8.00000

−18.5000

+

32.0429i

14.0000

+

24.2487i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	98.4.c.f		2
3.b	odd	2	1		882.4.g.k		2
7.b	odd	2	1		98.4.c.d		2
7.c	even	3	1		98.4.a.a		1
7.c	even	3	1	inner	98.4.c.f		2
7.d	odd	6	1		14.4.a.a	✓	1
7.d	odd	6	1		98.4.c.d		2
21.c	even	2	1		882.4.g.b		2
21.g	even	6	1		126.4.a.h		1
21.g	even	6	1		882.4.g.b		2
21.h	odd	6	1		882.4.a.i		1
21.h	odd	6	1		882.4.g.k		2
28.f	even	6	1		112.4.a.a		1
28.g	odd	6	1		784.4.a.s		1
35.i	odd	6	1		350.4.a.l		1
35.j	even	6	1		2450.4.a.bo		1
35.k	even	12	2		350.4.c.b		2
56.j	odd	6	1		448.4.a.b		1
56.m	even	6	1		448.4.a.o		1
77.i	even	6	1		1694.4.a.g		1
84.j	odd	6	1		1008.4.a.s		1
91.s	odd	6	1		2366.4.a.h		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.a.a	✓	1	7.d	odd	6	1
98.4.a.a		1	7.c	even	3	1
98.4.c.d		2	7.b	odd	2	1
98.4.c.d		2	7.d	odd	6	1
98.4.c.f		2	1.a	even	1	1	trivial
98.4.c.f		2	7.c	even	3	1	inner
112.4.a.a		1	28.f	even	6	1
126.4.a.h		1	21.g	even	6	1
350.4.a.l		1	35.i	odd	6	1
350.4.c.b		2	35.k	even	12	2
448.4.a.b		1	56.j	odd	6	1
448.4.a.o		1	56.m	even	6	1
784.4.a.s		1	28.g	odd	6	1
882.4.a.i		1	21.h	odd	6	1
882.4.g.b		2	21.c	even	2	1
882.4.g.b		2	21.g	even	6	1
882.4.g.k		2	3.b	odd	2	1
882.4.g.k		2	21.h	odd	6	1
1008.4.a.s		1	84.j	odd	6	1
1694.4.a.g		1	77.i	even	6	1
2366.4.a.h		1	91.s	odd	6	1
2450.4.a.bo		1	35.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8T_{3} + 64 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} - 8T + 64 \)
$5$	\( T^{2} + 14T + 196 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 28T + 784 \)