Newform orbit 196.2.d.b

• Label: 196.2.d.b
• Level: $196$
• Weight: $2$
• Character orbit: 196.d
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_1 + 1) q^{2} + \beta_{3} q^{3} + 2 \beta_1 q^{4} - \beta_{2} q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (2 \beta_1 - 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{3} \)
\(\beta_{2}\)	\(=\)	\( 2\zeta_{12}^{2} - 1 \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

195.1

−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i

1.00000

−

1.00000i

−1.73205

−

2.00000i

−

1.73205i

−1.73205

+

1.73205i

0

−2.00000

−

2.00000i

0

−1.73205

−

1.73205i

195.2

1.00000

−

1.00000i

1.73205

−

2.00000i

1.73205i

1.73205

−

1.73205i

0

−2.00000

−

2.00000i

0

1.73205

+

1.73205i

195.3

1.00000

+

1.00000i

−1.73205

2.00000i

1.73205i

−1.73205

−

1.73205i

0

−2.00000

+

2.00000i

0

−1.73205

+

1.73205i

195.4

1.00000

+

1.00000i

1.73205

2.00000i

−

1.73205i

1.73205

+

1.73205i

0

−2.00000

+

2.00000i

0

1.73205

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
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	196.2.d.b		4
3.b	odd	2	1		1764.2.b.a		4
4.b	odd	2	1	inner	196.2.d.b		4
7.b	odd	2	1	inner	196.2.d.b		4
7.c	even	3	1		28.2.f.a	✓	4
7.c	even	3	1		196.2.f.a		4
7.d	odd	6	1		28.2.f.a	✓	4
7.d	odd	6	1		196.2.f.a		4
8.b	even	2	1		3136.2.f.e		4
8.d	odd	2	1		3136.2.f.e		4
12.b	even	2	1		1764.2.b.a		4
21.c	even	2	1		1764.2.b.a		4
21.g	even	6	1		252.2.bf.e		4
21.h	odd	6	1		252.2.bf.e		4
28.d	even	2	1	inner	196.2.d.b		4
28.f	even	6	1		28.2.f.a	✓	4
28.f	even	6	1		196.2.f.a		4
28.g	odd	6	1		28.2.f.a	✓	4
28.g	odd	6	1		196.2.f.a		4
35.i	odd	6	1		700.2.p.a		4
35.j	even	6	1		700.2.p.a		4
35.k	even	12	1		700.2.t.a		4
35.k	even	12	1		700.2.t.b		4
35.l	odd	12	1		700.2.t.a		4
35.l	odd	12	1		700.2.t.b		4
56.e	even	2	1		3136.2.f.e		4
56.h	odd	2	1		3136.2.f.e		4
56.j	odd	6	1		448.2.p.d		4
56.k	odd	6	1		448.2.p.d		4
56.m	even	6	1		448.2.p.d		4
56.p	even	6	1		448.2.p.d		4
84.h	odd	2	1		1764.2.b.a		4
84.j	odd	6	1		252.2.bf.e		4
84.n	even	6	1		252.2.bf.e		4
140.p	odd	6	1		700.2.p.a		4
140.s	even	6	1		700.2.p.a		4
140.w	even	12	1		700.2.t.a		4
140.w	even	12	1		700.2.t.b		4
140.x	odd	12	1		700.2.t.a		4
140.x	odd	12	1		700.2.t.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.f.a	✓	4	7.c	even	3	1
28.2.f.a	✓	4	7.d	odd	6	1
28.2.f.a	✓	4	28.f	even	6	1
28.2.f.a	✓	4	28.g	odd	6	1
196.2.d.b		4	1.a	even	1	1	trivial
196.2.d.b		4	4.b	odd	2	1	inner
196.2.d.b		4	7.b	odd	2	1	inner
196.2.d.b		4	28.d	even	2	1	inner
196.2.f.a		4	7.c	even	3	1
196.2.f.a		4	7.d	odd	6	1
196.2.f.a		4	28.f	even	6	1
196.2.f.a		4	28.g	odd	6	1
252.2.bf.e		4	21.g	even	6	1
252.2.bf.e		4	21.h	odd	6	1
252.2.bf.e		4	84.j	odd	6	1
252.2.bf.e		4	84.n	even	6	1
448.2.p.d		4	56.j	odd	6	1
448.2.p.d		4	56.k	odd	6	1
448.2.p.d		4	56.m	even	6	1
448.2.p.d		4	56.p	even	6	1
700.2.p.a		4	35.i	odd	6	1
700.2.p.a		4	35.j	even	6	1
700.2.p.a		4	140.p	odd	6	1
700.2.p.a		4	140.s	even	6	1
700.2.t.a		4	35.k	even	12	1
700.2.t.a		4	35.l	odd	12	1
700.2.t.a		4	140.w	even	12	1
700.2.t.a		4	140.x	odd	12	1
700.2.t.b		4	35.k	even	12	1
700.2.t.b		4	35.l	odd	12	1
700.2.t.b		4	140.w	even	12	1
700.2.t.b		4	140.x	odd	12	1
1764.2.b.a		4	3.b	odd	2	1
1764.2.b.a		4	12.b	even	2	1
1764.2.b.a		4	21.c	even	2	1
1764.2.b.a		4	84.h	odd	2	1
3136.2.f.e		4	8.b	even	2	1
3136.2.f.e		4	8.d	odd	2	1
3136.2.f.e		4	56.e	even	2	1
3136.2.f.e		4	56.h	odd	2	1

Hecke kernels

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

Hecke characteristic polynomials

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