Newform orbit 196.3.c.a

• Label: 196.3.c.a
• Level: $196$
• Weight: $3$
• Character orbit: 196.c
• Self dual: yes
• Analytic conductor: $5.341$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	196.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.34061318146\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 7\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 q^{2} + 4 q^{4} + \beta q^{5} - 8 q^{8} + 9 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

99.1

−1.41421
1.41421

−2.00000

0

4.00000

−9.89949

0

0

−8.00000

9.00000

19.7990

99.2

−2.00000

0

4.00000

9.89949

0

0

−8.00000

9.00000

−19.7990

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.3.c.a	✓	2
4.b	odd	2	1	CM	196.3.c.a	✓	2
7.b	odd	2	1	inner	196.3.c.a	✓	2
7.c	even	3	2		196.3.g.e		4
7.d	odd	6	2		196.3.g.e		4
28.d	even	2	1	inner	196.3.c.a	✓	2
28.f	even	6	2		196.3.g.e		4
28.g	odd	6	2		196.3.g.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.3.c.a	✓	2	1.a	even	1	1	trivial
196.3.c.a	✓	2	4.b	odd	2	1	CM
196.3.c.a	✓	2	7.b	odd	2	1	inner
196.3.c.a	✓	2	28.d	even	2	1	inner
196.3.g.e		4	7.c	even	3	2
196.3.g.e		4	7.d	odd	6	2
196.3.g.e		4	28.f	even	6	2
196.3.g.e		4	28.g	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(196, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} - 98 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 98 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)