Newform orbit 196.5.h.a

• Label: 196.5.h.a
• Level: $196$
• Weight: $5$
• Character orbit: 196.h
• Analytic conductor: $20.261$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	196.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.2605127644\)
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 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - 4 \zeta_{6} - 4) q^{3} + ( - 12 \zeta_{6} + 24) q^{5} - 33 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{3} + 36 q^{5} - 33 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

117.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−6.00000

+

3.46410i

0

18.0000

+

10.3923i

0

0

0

−16.5000

+

28.5788i

0

129.1

0

−6.00000

−

3.46410i

0

18.0000

−

10.3923i

0

0

0

−16.5000

−

28.5788i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.5.h.a		2
7.b	odd	2	1		196.5.h.b		2
7.c	even	3	1		28.5.b.a	✓	2
7.c	even	3	1		196.5.h.b		2
7.d	odd	6	1		28.5.b.a	✓	2
7.d	odd	6	1	inner	196.5.h.a		2
21.g	even	6	1		252.5.d.a		2
21.h	odd	6	1		252.5.d.a		2
28.f	even	6	1		112.5.c.b		2
28.g	odd	6	1		112.5.c.b		2
35.i	odd	6	1		700.5.d.a		2
35.j	even	6	1		700.5.d.a		2
35.k	even	12	2		700.5.h.a		4
35.l	odd	12	2		700.5.h.a		4
56.j	odd	6	1		448.5.c.c		2
56.k	odd	6	1		448.5.c.d		2
56.m	even	6	1		448.5.c.d		2
56.p	even	6	1		448.5.c.c		2
84.j	odd	6	1		1008.5.f.c		2
84.n	even	6	1		1008.5.f.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.5.b.a	✓	2	7.c	even	3	1
28.5.b.a	✓	2	7.d	odd	6	1
112.5.c.b		2	28.f	even	6	1
112.5.c.b		2	28.g	odd	6	1
196.5.h.a		2	1.a	even	1	1	trivial
196.5.h.a		2	7.d	odd	6	1	inner
196.5.h.b		2	7.b	odd	2	1
196.5.h.b		2	7.c	even	3	1
252.5.d.a		2	21.g	even	6	1
252.5.d.a		2	21.h	odd	6	1
448.5.c.c		2	56.j	odd	6	1
448.5.c.c		2	56.p	even	6	1
448.5.c.d		2	56.k	odd	6	1
448.5.c.d		2	56.m	even	6	1
700.5.d.a		2	35.i	odd	6	1
700.5.d.a		2	35.j	even	6	1
700.5.h.a		4	35.k	even	12	2
700.5.h.a		4	35.l	odd	12	2
1008.5.f.c		2	84.j	odd	6	1
1008.5.f.c		2	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 12T_{3} + 48 \) acting on \(S_{5}^{\mathrm{new}}(196, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 12T + 48 \)
$5$	\( T^{2} - 36T + 432 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 18T + 324 \)