Newform orbit 196.10.e.b

• Label: 196.10.e.b
• Level: $196$
• Weight: $10$
• Character orbit: 196.e
• Analytic conductor: $100.947$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(100.947023888\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 4)
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.

\(f(q)\)	\(=\)	\( q + ( - 228 \zeta_{6} + 228) q^{3} - 666 \zeta_{6} q^{5} - 32301 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 228 q^{3} - 666 q^{5} - 32301 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

165.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

114.000

−

197.454i

0

−333.000

−

576.773i

0

0

0

−16150.5

−

27973.5i

0

177.1

0

114.000

+

197.454i

0

−333.000

+

576.773i

0

0

0

−16150.5

+

27973.5i

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	196.10.e.b		2
7.b	odd	2	1		196.10.e.a		2
7.c	even	3	1		196.10.a.a		1
7.c	even	3	1	inner	196.10.e.b		2
7.d	odd	6	1		4.10.a.a	✓	1
7.d	odd	6	1		196.10.e.a		2
21.g	even	6	1		36.10.a.b		1
28.f	even	6	1		16.10.a.a		1
35.i	odd	6	1		100.10.a.a		1
35.k	even	12	2		100.10.c.a		2
56.j	odd	6	1		64.10.a.a		1
56.m	even	6	1		64.10.a.i		1
63.i	even	6	1		324.10.e.b		2
63.k	odd	6	1		324.10.e.e		2
63.s	even	6	1		324.10.e.b		2
63.t	odd	6	1		324.10.e.e		2
84.j	odd	6	1		144.10.a.j		1
112.v	even	12	2		256.10.b.b		2
112.x	odd	12	2		256.10.b.j		2
140.s	even	6	1		400.10.a.k		1
140.x	odd	12	2		400.10.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.10.a.a	✓	1	7.d	odd	6	1
16.10.a.a		1	28.f	even	6	1
36.10.a.b		1	21.g	even	6	1
64.10.a.a		1	56.j	odd	6	1
64.10.a.i		1	56.m	even	6	1
100.10.a.a		1	35.i	odd	6	1
100.10.c.a		2	35.k	even	12	2
144.10.a.j		1	84.j	odd	6	1
196.10.a.a		1	7.c	even	3	1
196.10.e.a		2	7.b	odd	2	1
196.10.e.a		2	7.d	odd	6	1
196.10.e.b		2	1.a	even	1	1	trivial
196.10.e.b		2	7.c	even	3	1	inner
256.10.b.b		2	112.v	even	12	2
256.10.b.j		2	112.x	odd	12	2
324.10.e.b		2	63.i	even	6	1
324.10.e.b		2	63.s	even	6	1
324.10.e.e		2	63.k	odd	6	1
324.10.e.e		2	63.t	odd	6	1
400.10.a.k		1	140.s	even	6	1
400.10.c.a		2	140.x	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 228T_{3} + 51984 \) acting on \(S_{10}^{\mathrm{new}}(196, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 228T + 51984 \)
$5$	\( T^{2} + 666T + 443556 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 30420 T + 925376400 \)