Newform orbit 98.10.c.b

• Label: 98.10.c.b
• Level: $98$
• Weight: $10$
• Character orbit: 98.c
• Analytic conductor: $50.474$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(50.4735119441\)
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 2)
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.

\(f(q)\)	\(=\)	\( q - 16 \zeta_{6} q^{2} + (156 \zeta_{6} - 156) q^{3} + (256 \zeta_{6} - 256) q^{4} + 870 \zeta_{6} q^{5} + 2496 q^{6} + 4096 q^{8} - 4653 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 156 q^{3} - 256 q^{4} + 870 q^{5} + 4992 q^{6} + 8192 q^{8} - 4653 q^{9}+O(q^{10}) \)

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

−8.00000

−

13.8564i

−78.0000

+

135.100i

−128.000

+

221.703i

435.000

+

753.442i

2496.00

0

4096.00

−2326.50

−

4029.62i

6960.00

−

12055.1i

79.1

−8.00000

+

13.8564i

−78.0000

−

135.100i

−128.000

−

221.703i

435.000

−

753.442i

2496.00

0

4096.00

−2326.50

+

4029.62i

6960.00

+

12055.1i

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	98.10.c.b		2
7.b	odd	2	1		98.10.c.c		2
7.c	even	3	1		98.10.a.c		1
7.c	even	3	1	inner	98.10.c.b		2
7.d	odd	6	1		2.10.a.a	✓	1
7.d	odd	6	1		98.10.c.c		2
21.g	even	6	1		18.10.a.a		1
28.f	even	6	1		16.10.a.d		1
35.i	odd	6	1		50.10.a.c		1
35.k	even	12	2		50.10.b.a		2
56.j	odd	6	1		64.10.a.h		1
56.m	even	6	1		64.10.a.b		1
63.i	even	6	1		162.10.c.i		2
63.k	odd	6	1		162.10.c.b		2
63.s	even	6	1		162.10.c.i		2
63.t	odd	6	1		162.10.c.b		2
77.i	even	6	1		242.10.a.a		1
84.j	odd	6	1		144.10.a.d		1
91.s	odd	6	1		338.10.a.a		1
112.v	even	12	2		256.10.b.e		2
112.x	odd	12	2		256.10.b.g		2
140.s	even	6	1		400.10.a.b		1
140.x	odd	12	2		400.10.c.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.10.a.a	✓	1	7.d	odd	6	1
16.10.a.d		1	28.f	even	6	1
18.10.a.a		1	21.g	even	6	1
50.10.a.c		1	35.i	odd	6	1
50.10.b.a		2	35.k	even	12	2
64.10.a.b		1	56.m	even	6	1
64.10.a.h		1	56.j	odd	6	1
98.10.a.c		1	7.c	even	3	1
98.10.c.b		2	1.a	even	1	1	trivial
98.10.c.b		2	7.c	even	3	1	inner
98.10.c.c		2	7.b	odd	2	1
98.10.c.c		2	7.d	odd	6	1
144.10.a.d		1	84.j	odd	6	1
162.10.c.b		2	63.k	odd	6	1
162.10.c.b		2	63.t	odd	6	1
162.10.c.i		2	63.i	even	6	1
162.10.c.i		2	63.s	even	6	1
242.10.a.a		1	77.i	even	6	1
256.10.b.e		2	112.v	even	12	2
256.10.b.g		2	112.x	odd	12	2
338.10.a.a		1	91.s	odd	6	1
400.10.a.b		1	140.s	even	6	1
400.10.c.d		2	140.x	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 156T_{3} + 24336 \) acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 16T + 256 \)
$3$	\( T^{2} + 156T + 24336 \)
$5$	\( T^{2} - 870T + 756900 \)
$7$	\( T^{2} \)
$11$	T^2 - 56148*T + 3152597904