Newform orbit 192.3.g.a

• Label: 192.3.g.a
• Level: $192$
• Weight: $3$
• Character orbit: 192.g
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23162107572\)
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:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 6 q^{5} + 4 \beta q^{7} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−

1.73205i

0

−6.00000

0

6.92820i

0

−3.00000

0

127.2

0

1.73205i

0

−6.00000

0

−

6.92820i

0

−3.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.3.g.a		2
3.b	odd	2	1		576.3.g.i		2
4.b	odd	2	1	inner	192.3.g.a		2
8.b	even	2	1		48.3.g.a	✓	2
8.d	odd	2	1		48.3.g.a	✓	2
12.b	even	2	1		576.3.g.i		2
16.e	even	4	2		768.3.b.b		4
16.f	odd	4	2		768.3.b.b		4
24.f	even	2	1		144.3.g.b		2
24.h	odd	2	1		144.3.g.b		2
40.e	odd	2	1		1200.3.e.h		2
40.f	even	2	1		1200.3.e.h		2
40.i	odd	4	2		1200.3.j.a		4
40.k	even	4	2		1200.3.j.a		4
48.i	odd	4	2		2304.3.b.n		4
48.k	even	4	2		2304.3.b.n		4
56.e	even	2	1		2352.3.m.a		2
56.h	odd	2	1		2352.3.m.a		2
72.j	odd	6	1		1296.3.o.n		2
72.j	odd	6	1		1296.3.o.p		2
72.l	even	6	1		1296.3.o.n		2
72.l	even	6	1		1296.3.o.p		2
72.n	even	6	1		1296.3.o.a		2
72.n	even	6	1		1296.3.o.c		2
72.p	odd	6	1		1296.3.o.a		2
72.p	odd	6	1		1296.3.o.c		2
120.i	odd	2	1		3600.3.e.t		2
120.m	even	2	1		3600.3.e.t		2
120.q	odd	4	2		3600.3.j.i		4
120.w	even	4	2		3600.3.j.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.3.g.a	✓	2	8.b	even	2	1
48.3.g.a	✓	2	8.d	odd	2	1
144.3.g.b		2	24.f	even	2	1
144.3.g.b		2	24.h	odd	2	1
192.3.g.a		2	1.a	even	1	1	trivial
192.3.g.a		2	4.b	odd	2	1	inner
576.3.g.i		2	3.b	odd	2	1
576.3.g.i		2	12.b	even	2	1
768.3.b.b		4	16.e	even	4	2
768.3.b.b		4	16.f	odd	4	2
1200.3.e.h		2	40.e	odd	2	1
1200.3.e.h		2	40.f	even	2	1
1200.3.j.a		4	40.i	odd	4	2
1200.3.j.a		4	40.k	even	4	2
1296.3.o.a		2	72.n	even	6	1
1296.3.o.a		2	72.p	odd	6	1
1296.3.o.c		2	72.n	even	6	1
1296.3.o.c		2	72.p	odd	6	1
1296.3.o.n		2	72.j	odd	6	1
1296.3.o.n		2	72.l	even	6	1
1296.3.o.p		2	72.j	odd	6	1
1296.3.o.p		2	72.l	even	6	1
2304.3.b.n		4	48.i	odd	4	2
2304.3.b.n		4	48.k	even	4	2
2352.3.m.a		2	56.e	even	2	1
2352.3.m.a		2	56.h	odd	2	1
3600.3.e.t		2	120.i	odd	2	1
3600.3.e.t		2	120.m	even	2	1
3600.3.j.i		4	120.q	odd	4	2
3600.3.j.i		4	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 6 \) acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( (T + 6)^{2} \)
$7$	\( T^{2} + 48 \)
$11$	\( T^{2} + 432 \)