Newform orbit 2312.2.a.g

• Label: 2312.2.a.g
• Level: $2312$
• Weight: $2$
• Character orbit: 2312.a
• Self dual: yes
• Analytic conductor: $18.461$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.4614129473\)
Analytic rank:	\(0\)
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 136)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 - b * q^5 + b * q^7 - 3 * q^9 - 2*b * q^11 + 2 * q^13 + 4 * q^19 - b * q^23 + 3 * q^25 + 3*b * q^29 - b * q^31 - 8 * q^35 + b * q^37 - 2*b * q^41 - 4 * q^43 + 3*b * q^45 + 8 * q^47 + q^49 + 10 * q^53 + 16 * q^55 + 12 * q^59 + b * q^61 - 3*b * q^63 - 2*b * q^65 + 12 * q^67 - 3*b * q^71 + 4*b * q^73 - 16 * q^77 + 3*b * q^79 + 9 * q^81 + 4 * q^83 - 2 * q^89 + 2*b * q^91 - 4*b * q^95 - 6*b * q^97 + 6*b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 4 q^{13} + 8 q^{19} + 6 q^{25} - 16 q^{35} - 8 q^{43} + 16 q^{47} + 2 q^{49} + 20 q^{53} + 32 q^{55} + 24 q^{59} + 24 q^{67} - 32 q^{77} + 18 q^{81} + 8 q^{83} - 4 q^{89}+O(q^{100}) \)

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

2.82843

0

−3.00000

0

1.2

0

0

0

2.82843

0

−2.82843

0

−3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(17\)	\( +1 \)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.2.a.g		2
4.b	odd	2	1		4624.2.a.m		2
17.b	even	2	1	inner	2312.2.a.g		2
17.c	even	4	2		136.2.b.b	✓	2
51.f	odd	4	2		1224.2.c.e		2
68.d	odd	2	1		4624.2.a.m		2
68.f	odd	4	2		272.2.b.d		2
85.f	odd	4	2		3400.2.o.e		4
85.i	odd	4	2		3400.2.o.e		4
85.j	even	4	2		3400.2.c.e		2
136.i	even	4	2		1088.2.b.h		2
136.j	odd	4	2		1088.2.b.g		2
204.l	even	4	2		2448.2.c.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.2.b.b	✓	2	17.c	even	4	2
272.2.b.d		2	68.f	odd	4	2
1088.2.b.g		2	136.j	odd	4	2
1088.2.b.h		2	136.i	even	4	2
1224.2.c.e		2	51.f	odd	4	2
2312.2.a.g		2	1.a	even	1	1	trivial
2312.2.a.g		2	17.b	even	2	1	inner
2448.2.c.m		2	204.l	even	4	2
3400.2.c.e		2	85.j	even	4	2
3400.2.o.e		4	85.f	odd	4	2
3400.2.o.e		4	85.i	odd	4	2
4624.2.a.m		2	4.b	odd	2	1
4624.2.a.m		2	68.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\):

\( T_{3} \)

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

\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

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