Newform orbit 304.2.i.b

• Label: 304.2.i.b
• Level: $304$
• Weight: $2$
• Character orbit: 304.i
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
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:	\( 1 \)
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0.500000

−

0.866025i

0

−1.50000

+

2.59808i

0

0

0

1.00000

+

1.73205i

0

273.1

0

0.500000

+

0.866025i

0

−1.50000

−

2.59808i

0

0

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.2.i.b		2
3.b	odd	2	1		2736.2.s.q		2
4.b	odd	2	1		152.2.i.a	✓	2
8.b	even	2	1		1216.2.i.e		2
8.d	odd	2	1		1216.2.i.i		2
12.b	even	2	1		1368.2.s.g		2
19.c	even	3	1	inner	304.2.i.b		2
19.c	even	3	1		5776.2.a.h		1
19.d	odd	6	1		5776.2.a.o		1
57.h	odd	6	1		2736.2.s.q		2
76.f	even	6	1		2888.2.a.c		1
76.g	odd	6	1		152.2.i.a	✓	2
76.g	odd	6	1		2888.2.a.e		1
152.k	odd	6	1		1216.2.i.i		2
152.p	even	6	1		1216.2.i.e		2
228.m	even	6	1		1368.2.s.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.i.a	✓	2	4.b	odd	2	1
152.2.i.a	✓	2	76.g	odd	6	1
304.2.i.b		2	1.a	even	1	1	trivial
304.2.i.b		2	19.c	even	3	1	inner
1216.2.i.e		2	8.b	even	2	1
1216.2.i.e		2	152.p	even	6	1
1216.2.i.i		2	8.d	odd	2	1
1216.2.i.i		2	152.k	odd	6	1
1368.2.s.g		2	12.b	even	2	1
1368.2.s.g		2	228.m	even	6	1
2736.2.s.q		2	3.b	odd	2	1
2736.2.s.q		2	57.h	odd	6	1
2888.2.a.c		1	76.f	even	6	1
2888.2.a.e		1	76.g	odd	6	1
5776.2.a.h		1	19.c	even	3	1
5776.2.a.o		1	19.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)

\( T_{5}^{2} + 3T_{5} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( (T - 4)^{2} \)