Newform orbit 196.3.b.a

• Label: 196.3.b.a
• Level: $196$
• Weight: $3$
• Character orbit: 196.b
• Analytic conductor: $5.341$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.34061318146\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^3 + b * q^5 + 6 * q^9 + 15 * q^11 - 8*b * q^13 + 3 * q^15 - 17*b * q^17 + 9*b * q^19 - 9 * q^23 + 22 * q^25 - 15*b * q^27 - 6 * q^29 + 7*b * q^31 - 15*b * q^33 + 31 * q^37 - 24 * q^39 + 32*b * q^41 + 10 * q^43 + 6*b * q^45 + 25*b * q^47 - 51 * q^51 - 57 * q^53 + 15*b * q^55 + 27 * q^57 + 47*b * q^59 - 47*b * q^61 + 24 * q^65 - 49 * q^67 + 9*b * q^69 - 126 * q^71 + 15*b * q^73 - 22*b * q^75 - 73 * q^79 + 9 * q^81 + 8*b * q^83 + 51 * q^85 + 6*b * q^87 + 33*b * q^89 + 21 * q^93 - 27 * q^95 - 16*b * q^97 + 90 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 12 * q^9 + 30 * q^11 + 6 * q^15 - 18 * q^23 + 44 * q^25 - 12 * q^29 + 62 * q^37 - 48 * q^39 + 20 * q^43 - 102 * q^51 - 114 * q^53 + 54 * q^57 + 48 * q^65 - 98 * q^67 - 252 * q^71 - 146 * q^79 + 18 * q^81 + 102 * q^85 + 42 * q^93 - 54 * q^95 + 180 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

97.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−

1.73205i

0

1.73205i

0

0

0

6.00000

0

97.2

0

1.73205i

0

−

1.73205i

0

0

0

6.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.3.b.a		2
3.b	odd	2	1		1764.3.d.a		2
4.b	odd	2	1		784.3.c.a		2
7.b	odd	2	1	inner	196.3.b.a		2
7.c	even	3	1		28.3.h.a	✓	2
7.c	even	3	1		196.3.h.a		2
7.d	odd	6	1		28.3.h.a	✓	2
7.d	odd	6	1		196.3.h.a		2
21.c	even	2	1		1764.3.d.a		2
21.g	even	6	1		252.3.z.a		2
21.g	even	6	1		1764.3.z.f		2
21.h	odd	6	1		252.3.z.a		2
21.h	odd	6	1		1764.3.z.f		2
28.d	even	2	1		784.3.c.a		2
28.f	even	6	1		112.3.s.a		2
28.f	even	6	1		784.3.s.b		2
28.g	odd	6	1		112.3.s.a		2
28.g	odd	6	1		784.3.s.b		2
35.i	odd	6	1		700.3.s.a		2
35.j	even	6	1		700.3.s.a		2
35.k	even	12	2		700.3.o.a		4
35.l	odd	12	2		700.3.o.a		4
56.j	odd	6	1		448.3.s.a		2
56.k	odd	6	1		448.3.s.b		2
56.m	even	6	1		448.3.s.b		2
56.p	even	6	1		448.3.s.a		2
84.j	odd	6	1		1008.3.cg.c		2
84.n	even	6	1		1008.3.cg.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.3.h.a	✓	2	7.c	even	3	1
28.3.h.a	✓	2	7.d	odd	6	1
112.3.s.a		2	28.f	even	6	1
112.3.s.a		2	28.g	odd	6	1
196.3.b.a		2	1.a	even	1	1	trivial
196.3.b.a		2	7.b	odd	2	1	inner
196.3.h.a		2	7.c	even	3	1
196.3.h.a		2	7.d	odd	6	1
252.3.z.a		2	21.g	even	6	1
252.3.z.a		2	21.h	odd	6	1
448.3.s.a		2	56.j	odd	6	1
448.3.s.a		2	56.p	even	6	1
448.3.s.b		2	56.k	odd	6	1
448.3.s.b		2	56.m	even	6	1
700.3.o.a		4	35.k	even	12	2
700.3.o.a		4	35.l	odd	12	2
700.3.s.a		2	35.i	odd	6	1
700.3.s.a		2	35.j	even	6	1
784.3.c.a		2	4.b	odd	2	1
784.3.c.a		2	28.d	even	2	1
784.3.s.b		2	28.f	even	6	1
784.3.s.b		2	28.g	odd	6	1
1008.3.cg.c		2	84.j	odd	6	1
1008.3.cg.c		2	84.n	even	6	1
1764.3.d.a		2	3.b	odd	2	1
1764.3.d.a		2	21.c	even	2	1
1764.3.z.f		2	21.g	even	6	1
1764.3.z.f		2	21.h	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} + 3 \)
$7$	\( T^{2} \)
$11$	\( (T - 15)^{2} \)