Newform orbit 2304.2.a.t

• Label: 2304.2.a.t
• Level: $2304$
• Weight: $2$
• Character orbit: 2304.a
• Self dual: yes
• Analytic conductor: $18.398$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -8
• 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:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 128)
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.

\(f(q)\)	\(=\)	\( q+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

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

0

0

0

0

0

0

1.2

0

0

0

0

0

0

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
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
4.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.t		2
3.b	odd	2	1		256.2.a.e		2
4.b	odd	2	1	inner	2304.2.a.t		2
8.b	even	2	1	inner	2304.2.a.t		2
8.d	odd	2	1	CM	2304.2.a.t		2
12.b	even	2	1		256.2.a.e		2
15.d	odd	2	1		6400.2.a.by		2
16.e	even	4	2		1152.2.d.c		2
16.f	odd	4	2		1152.2.d.c		2
24.f	even	2	1		256.2.a.e		2
24.h	odd	2	1		256.2.a.e		2
48.i	odd	4	2		128.2.b.a	✓	2
48.k	even	4	2		128.2.b.a	✓	2
60.h	even	2	1		6400.2.a.by		2
96.o	even	8	2		1024.2.e.a		2
96.o	even	8	2		1024.2.e.f		2
96.p	odd	8	2		1024.2.e.a		2
96.p	odd	8	2		1024.2.e.f		2
120.i	odd	2	1		6400.2.a.by		2
120.m	even	2	1		6400.2.a.by		2
240.t	even	4	2		3200.2.d.c		2
240.z	odd	4	2		3200.2.f.o		4
240.bb	even	4	2		3200.2.f.o		4
240.bd	odd	4	2		3200.2.f.o		4
240.bf	even	4	2		3200.2.f.o		4
240.bm	odd	4	2		3200.2.d.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.2.b.a	✓	2	48.i	odd	4	2
128.2.b.a	✓	2	48.k	even	4	2
256.2.a.e		2	3.b	odd	2	1
256.2.a.e		2	12.b	even	2	1
256.2.a.e		2	24.f	even	2	1
256.2.a.e		2	24.h	odd	2	1
1024.2.e.a		2	96.o	even	8	2
1024.2.e.a		2	96.p	odd	8	2
1024.2.e.f		2	96.o	even	8	2
1024.2.e.f		2	96.p	odd	8	2
1152.2.d.c		2	16.e	even	4	2
1152.2.d.c		2	16.f	odd	4	2
2304.2.a.t		2	1.a	even	1	1	trivial
2304.2.a.t		2	4.b	odd	2	1	inner
2304.2.a.t		2	8.b	even	2	1	inner
2304.2.a.t		2	8.d	odd	2	1	CM
3200.2.d.c		2	240.t	even	4	2
3200.2.d.c		2	240.bm	odd	4	2
3200.2.f.o		4	240.z	odd	4	2
3200.2.f.o		4	240.bb	even	4	2
3200.2.f.o		4	240.bd	odd	4	2
3200.2.f.o		4	240.bf	even	4	2
6400.2.a.by		2	15.d	odd	2	1
6400.2.a.by		2	60.h	even	2	1
6400.2.a.by		2	120.i	odd	2	1
6400.2.a.by		2	120.m	even	2	1

Hecke kernels

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

\( T_{5} \)

\( T_{7} \)

\( T_{11}^{2} - 8 \)

\( T_{13} \)

\( T_{19}^{2} - 72 \)

Hecke characteristic polynomials

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