Newform orbit 98.2.c.b

• Label: 98.2.c.b
• Level: $98$
• Weight: $2$
• Character orbit: 98.c
• Analytic conductor: $0.783$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.782533939809\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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.

\(f(q)\)	\(=\)	\( q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} + 2 q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \)

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

0.500000

+

0.866025i

1.00000

−

1.73205i

−0.500000

+

0.866025i

0

2.00000

0

−1.00000

−0.500000

−

0.866025i

0

79.1

0.500000

−

0.866025i

1.00000

+

1.73205i

−0.500000

−

0.866025i

0

2.00000

0

−1.00000

−0.500000

+

0.866025i

0

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	98.2.c.b		2
3.b	odd	2	1		882.2.g.c		2
4.b	odd	2	1		784.2.i.c		2
7.b	odd	2	1		98.2.c.a		2
7.c	even	3	1		14.2.a.a	✓	1
7.c	even	3	1	inner	98.2.c.b		2
7.d	odd	6	1		98.2.a.a		1
7.d	odd	6	1		98.2.c.a		2
21.c	even	2	1		882.2.g.d		2
21.g	even	6	1		882.2.a.i		1
21.g	even	6	1		882.2.g.d		2
21.h	odd	6	1		126.2.a.b		1
21.h	odd	6	1		882.2.g.c		2
28.d	even	2	1		784.2.i.i		2
28.f	even	6	1		784.2.a.b		1
28.f	even	6	1		784.2.i.i		2
28.g	odd	6	1		112.2.a.c		1
28.g	odd	6	1		784.2.i.c		2
35.i	odd	6	1		2450.2.a.t		1
35.j	even	6	1		350.2.a.f		1
35.k	even	12	2		2450.2.c.c		2
35.l	odd	12	2		350.2.c.d		2
56.j	odd	6	1		3136.2.a.e		1
56.k	odd	6	1		448.2.a.a		1
56.m	even	6	1		3136.2.a.z		1
56.p	even	6	1		448.2.a.g		1
63.g	even	3	1		1134.2.f.l		2
63.h	even	3	1		1134.2.f.l		2
63.j	odd	6	1		1134.2.f.f		2
63.n	odd	6	1		1134.2.f.f		2
77.h	odd	6	1		1694.2.a.e		1
84.j	odd	6	1		7056.2.a.bd		1
84.n	even	6	1		1008.2.a.h		1
91.r	even	6	1		2366.2.a.j		1
91.z	odd	12	2		2366.2.d.b		2
105.o	odd	6	1		3150.2.a.i		1
105.x	even	12	2		3150.2.g.j		2
112.u	odd	12	2		1792.2.b.g		2
112.w	even	12	2		1792.2.b.c		2
119.j	even	6	1		4046.2.a.f		1
133.r	odd	6	1		5054.2.a.c		1
140.p	odd	6	1		2800.2.a.g		1
140.w	even	12	2		2800.2.g.h		2
161.f	odd	6	1		7406.2.a.a		1
168.s	odd	6	1		4032.2.a.w		1
168.v	even	6	1		4032.2.a.r		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.2.a.a	✓	1	7.c	even	3	1
98.2.a.a		1	7.d	odd	6	1
98.2.c.a		2	7.b	odd	2	1
98.2.c.a		2	7.d	odd	6	1
98.2.c.b		2	1.a	even	1	1	trivial
98.2.c.b		2	7.c	even	3	1	inner
112.2.a.c		1	28.g	odd	6	1
126.2.a.b		1	21.h	odd	6	1
350.2.a.f		1	35.j	even	6	1
350.2.c.d		2	35.l	odd	12	2
448.2.a.a		1	56.k	odd	6	1
448.2.a.g		1	56.p	even	6	1
784.2.a.b		1	28.f	even	6	1
784.2.i.c		2	4.b	odd	2	1
784.2.i.c		2	28.g	odd	6	1
784.2.i.i		2	28.d	even	2	1
784.2.i.i		2	28.f	even	6	1
882.2.a.i		1	21.g	even	6	1
882.2.g.c		2	3.b	odd	2	1
882.2.g.c		2	21.h	odd	6	1
882.2.g.d		2	21.c	even	2	1
882.2.g.d		2	21.g	even	6	1
1008.2.a.h		1	84.n	even	6	1
1134.2.f.f		2	63.j	odd	6	1
1134.2.f.f		2	63.n	odd	6	1
1134.2.f.l		2	63.g	even	3	1
1134.2.f.l		2	63.h	even	3	1
1694.2.a.e		1	77.h	odd	6	1
1792.2.b.c		2	112.w	even	12	2
1792.2.b.g		2	112.u	odd	12	2
2366.2.a.j		1	91.r	even	6	1
2366.2.d.b		2	91.z	odd	12	2
2450.2.a.t		1	35.i	odd	6	1
2450.2.c.c		2	35.k	even	12	2
2800.2.a.g		1	140.p	odd	6	1
2800.2.g.h		2	140.w	even	12	2
3136.2.a.e		1	56.j	odd	6	1
3136.2.a.z		1	56.m	even	6	1
3150.2.a.i		1	105.o	odd	6	1
3150.2.g.j		2	105.x	even	12	2
4032.2.a.r		1	168.v	even	6	1
4032.2.a.w		1	168.s	odd	6	1
4046.2.a.f		1	119.j	even	6	1
5054.2.a.c		1	133.r	odd	6	1
7056.2.a.bd		1	84.j	odd	6	1
7406.2.a.a		1	161.f	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)