Newform orbit 2304.2.a.q

• Label: 2304.2.a.q
• Level: $2304$
• Weight: $2$
• Character orbit: 2304.a
• Self dual: yes
• Analytic conductor: $18.398$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -24
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.3975326257\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	\(1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

0

0

0

−2.82843

0

−2.00000

0

0

0

1.2

0

0

0

2.82843

0

−2.00000

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by \(\Q(\sqrt{-6}) \)
3.b	odd	2	1	inner
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2304.2.a.q		2
3.b	odd	2	1	inner	2304.2.a.q		2
4.b	odd	2	1		2304.2.a.y		2
8.b	even	2	1	inner	2304.2.a.q		2
8.d	odd	2	1		2304.2.a.y		2
12.b	even	2	1		2304.2.a.y		2
16.e	even	4	2		72.2.d.a	✓	2
16.f	odd	4	2		288.2.d.a		2
24.f	even	2	1		2304.2.a.y		2
24.h	odd	2	1	CM	2304.2.a.q		2
48.i	odd	4	2		72.2.d.a	✓	2
48.k	even	4	2		288.2.d.a		2
80.i	odd	4	2		1800.2.d.n		4
80.j	even	4	2		7200.2.d.p		4
80.k	odd	4	2		7200.2.k.h		2
80.q	even	4	2		1800.2.k.e		2
80.s	even	4	2		7200.2.d.p		4
80.t	odd	4	2		1800.2.d.n		4
144.u	even	12	4		2592.2.r.i		4
144.v	odd	12	4		2592.2.r.i		4
144.w	odd	12	4		648.2.n.h		4
144.x	even	12	4		648.2.n.h		4
240.t	even	4	2		7200.2.k.h		2
240.z	odd	4	2		7200.2.d.p		4
240.bb	even	4	2		1800.2.d.n		4
240.bd	odd	4	2		7200.2.d.p		4
240.bf	even	4	2		1800.2.d.n		4
240.bm	odd	4	2		1800.2.k.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.d.a	✓	2	16.e	even	4	2
72.2.d.a	✓	2	48.i	odd	4	2
288.2.d.a		2	16.f	odd	4	2
288.2.d.a		2	48.k	even	4	2
648.2.n.h		4	144.w	odd	12	4
648.2.n.h		4	144.x	even	12	4
1800.2.d.n		4	80.i	odd	4	2
1800.2.d.n		4	80.t	odd	4	2
1800.2.d.n		4	240.bb	even	4	2
1800.2.d.n		4	240.bf	even	4	2
1800.2.k.e		2	80.q	even	4	2
1800.2.k.e		2	240.bm	odd	4	2
2304.2.a.q		2	1.a	even	1	1	trivial
2304.2.a.q		2	3.b	odd	2	1	inner
2304.2.a.q		2	8.b	even	2	1	inner
2304.2.a.q		2	24.h	odd	2	1	CM
2304.2.a.y		2	4.b	odd	2	1
2304.2.a.y		2	8.d	odd	2	1
2304.2.a.y		2	12.b	even	2	1
2304.2.a.y		2	24.f	even	2	1
2592.2.r.i		4	144.u	even	12	4
2592.2.r.i		4	144.v	odd	12	4
7200.2.d.p		4	80.j	even	4	2
7200.2.d.p		4	80.s	even	4	2
7200.2.d.p		4	240.z	odd	4	2
7200.2.d.p		4	240.bd	odd	4	2
7200.2.k.h		2	80.k	odd	4	2
7200.2.k.h		2	240.t	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5}^{2} - 8 \)

\( T_{7} + 2 \)

\( T_{11}^{2} - 32 \)

\( T_{13} \)

\( T_{19} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 8 \)
$7$	\( (T + 2)^{2} \)
$11$	\( T^{2} - 32 \)