Newform orbit 196.4.e.e

• Label: 196.4.e.e
• Level: $196$
• Weight: $4$
• Character orbit: 196.e
• Analytic conductor: $11.564$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5643743611\)
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:	yes
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.

\(f(q)\)	\(=\)	\( q + ( - 4 \zeta_{6} + 4) q^{3} + 20 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 20 q^{5} + 11 q^{9}+O(q^{10}) \)

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

2.00000

−

3.46410i

0

10.0000

+

17.3205i

0

0

0

5.50000

+

9.52628i

0

177.1

0

2.00000

+

3.46410i

0

10.0000

−

17.3205i

0

0

0

5.50000

−

9.52628i

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	196.4.e.e		2
3.b	odd	2	1		1764.4.k.a		2
7.b	odd	2	1		196.4.e.b		2
7.c	even	3	1		196.4.a.a	✓	1
7.c	even	3	1	inner	196.4.e.e		2
7.d	odd	6	1		196.4.a.c	yes	1
7.d	odd	6	1		196.4.e.b		2
21.c	even	2	1		1764.4.k.p		2
21.g	even	6	1		1764.4.a.a		1
21.g	even	6	1		1764.4.k.p		2
21.h	odd	6	1		1764.4.a.m		1
21.h	odd	6	1		1764.4.k.a		2
28.f	even	6	1		784.4.a.f		1
28.g	odd	6	1		784.4.a.m		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.4.a.a	✓	1	7.c	even	3	1
196.4.a.c	yes	1	7.d	odd	6	1
196.4.e.b		2	7.b	odd	2	1
196.4.e.b		2	7.d	odd	6	1
196.4.e.e		2	1.a	even	1	1	trivial
196.4.e.e		2	7.c	even	3	1	inner
784.4.a.f		1	28.f	even	6	1
784.4.a.m		1	28.g	odd	6	1
1764.4.a.a		1	21.g	even	6	1
1764.4.a.m		1	21.h	odd	6	1
1764.4.k.a		2	3.b	odd	2	1
1764.4.k.a		2	21.h	odd	6	1
1764.4.k.p		2	21.c	even	2	1
1764.4.k.p		2	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{2} - 4T_{3} + 16 \)

\( T_{5}^{2} - 20T_{5} + 400 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 4T + 16 \)
$5$	\( T^{2} - 20T + 400 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 44T + 1936 \)