Newform orbit 7623.2.a.bc

• Label: 7623.2.a.bc
• Level: $7623$
• Weight: $2$
• Character orbit: 7623.a
• Self dual: yes
• Analytic conductor: $60.870$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(60.8699614608\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} - q^{7} - 3 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \)

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

2.30278
−1.30278

−2.30278

0

3.30278

−3.60555

0

−1.00000

−3.00000

0

8.30278

1.2

1.30278

0

−0.302776

3.60555

0

−1.00000

−3.00000

0

4.69722

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(7\)	\(1\)
\(11\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7623.2.a.bc		2
3.b	odd	2	1		847.2.a.g	yes	2
11.b	odd	2	1		7623.2.a.bs		2
21.c	even	2	1		5929.2.a.p		2
33.d	even	2	1		847.2.a.e	✓	2
33.f	even	10	4		847.2.f.r		8
33.h	odd	10	4		847.2.f.o		8
231.h	odd	2	1		5929.2.a.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.a.e	✓	2	33.d	even	2	1
847.2.a.g	yes	2	3.b	odd	2	1
847.2.f.o		8	33.h	odd	10	4
847.2.f.r		8	33.f	even	10	4
5929.2.a.k		2	231.h	odd	2	1
5929.2.a.p		2	21.c	even	2	1
7623.2.a.bc		2	1.a	even	1	1	trivial
7623.2.a.bs		2	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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

\( T_{5}^{2} - 13 \)

\( T_{13}^{2} - 6T_{13} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 3 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 13 \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} \)