Newform orbit 192.2.c.a

• Label: 192.2.c.a
• Level: $192$
• Weight: $2$
• Character orbit: 192.c
• Analytic conductor: $1.533$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.53312771881\)
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 48)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 2 \beta q^{7} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

191.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−

1.73205i

0

0

0

−

3.46410i

0

−3.00000

0

191.2

0

1.73205i

0

0

0

3.46410i

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.2.c.a		2
3.b	odd	2	1	CM	192.2.c.a		2
4.b	odd	2	1	inner	192.2.c.a		2
8.b	even	2	1		48.2.c.a	✓	2
8.d	odd	2	1		48.2.c.a	✓	2
12.b	even	2	1	inner	192.2.c.a		2
16.e	even	4	2		768.2.f.d		4
16.f	odd	4	2		768.2.f.d		4
24.f	even	2	1		48.2.c.a	✓	2
24.h	odd	2	1		48.2.c.a	✓	2
40.e	odd	2	1		1200.2.h.e		2
40.f	even	2	1		1200.2.h.e		2
40.i	odd	4	2		1200.2.o.i		4
40.k	even	4	2		1200.2.o.i		4
48.i	odd	4	2		768.2.f.d		4
48.k	even	4	2		768.2.f.d		4
56.e	even	2	1		2352.2.h.c		2
56.h	odd	2	1		2352.2.h.c		2
72.j	odd	6	1		1296.2.s.b		2
72.j	odd	6	1		1296.2.s.e		2
72.l	even	6	1		1296.2.s.b		2
72.l	even	6	1		1296.2.s.e		2
72.n	even	6	1		1296.2.s.b		2
72.n	even	6	1		1296.2.s.e		2
72.p	odd	6	1		1296.2.s.b		2
72.p	odd	6	1		1296.2.s.e		2
120.i	odd	2	1		1200.2.h.e		2
120.m	even	2	1		1200.2.h.e		2
120.q	odd	4	2		1200.2.o.i		4
120.w	even	4	2		1200.2.o.i		4
168.e	odd	2	1		2352.2.h.c		2
168.i	even	2	1		2352.2.h.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.2.c.a	✓	2	8.b	even	2	1
48.2.c.a	✓	2	8.d	odd	2	1
48.2.c.a	✓	2	24.f	even	2	1
48.2.c.a	✓	2	24.h	odd	2	1
192.2.c.a		2	1.a	even	1	1	trivial
192.2.c.a		2	3.b	odd	2	1	CM
192.2.c.a		2	4.b	odd	2	1	inner
192.2.c.a		2	12.b	even	2	1	inner
768.2.f.d		4	16.e	even	4	2
768.2.f.d		4	16.f	odd	4	2
768.2.f.d		4	48.i	odd	4	2
768.2.f.d		4	48.k	even	4	2
1200.2.h.e		2	40.e	odd	2	1
1200.2.h.e		2	40.f	even	2	1
1200.2.h.e		2	120.i	odd	2	1
1200.2.h.e		2	120.m	even	2	1
1200.2.o.i		4	40.i	odd	4	2
1200.2.o.i		4	40.k	even	4	2
1200.2.o.i		4	120.q	odd	4	2
1200.2.o.i		4	120.w	even	4	2
1296.2.s.b		2	72.j	odd	6	1
1296.2.s.b		2	72.l	even	6	1
1296.2.s.b		2	72.n	even	6	1
1296.2.s.b		2	72.p	odd	6	1
1296.2.s.e		2	72.j	odd	6	1
1296.2.s.e		2	72.l	even	6	1
1296.2.s.e		2	72.n	even	6	1
1296.2.s.e		2	72.p	odd	6	1
2352.2.h.c		2	56.e	even	2	1
2352.2.h.c		2	56.h	odd	2	1
2352.2.h.c		2	168.e	odd	2	1
2352.2.h.c		2	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 12 \)
$11$	\( T^{2} \)