Newform orbit 7623.2.a.cp

• Label: 7623.2.a.cp
• Level: $7623$
• Weight: $2$
• Character orbit: 7623.a
• Self dual: yes
• Analytic conductor: $60.870$
• Analytic rank: $1$
• Dimension: $6$
• 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:	\(6\)
Coefficient field:	6.6.7674048.1
Defining polynomial:	\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{4} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{5} + q^{7} + (\beta_{3} - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
\(\beta_{4}\)	\(=\)	\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
\(\beta_{5}\)	\(=\)	\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2 \)

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

−1.70320
−1.10939
−0.276564
0.879640
1.82356
2.38595

−2.70320

0

5.30727

0.445072

0

1.00000

−8.94020

0

−1.20312

1.2

−2.10939

0

2.44952

−0.492391

0

1.00000

−0.948212

0

1.03864

1.3

−1.27656

0

−0.370384

4.09144

0

1.00000

3.02595

0

−5.22298

1.4

−0.120360

0

−1.98551

2.80853

0

1.00000

0.479696

0

−0.338034

1.5

0.823556

0

−1.32176

−2.98565

0

1.00000

−2.73565

0

−2.45885

1.6

1.38595

0

−0.0791355

0.133004

0

1.00000

−2.88158

0

0.184338

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.cp		6
3.b	odd	2	1		847.2.a.n	yes	6
11.b	odd	2	1		7623.2.a.cs		6
21.c	even	2	1		5929.2.a.bm		6
33.d	even	2	1		847.2.a.m	✓	6
33.f	even	10	4		847.2.f.z		24
33.h	odd	10	4		847.2.f.y		24
231.h	odd	2	1		5929.2.a.bj		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.a.m	✓	6	33.d	even	2	1
847.2.a.n	yes	6	3.b	odd	2	1
847.2.f.y		24	33.h	odd	10	4
847.2.f.z		24	33.f	even	10	4
5929.2.a.bj		6	231.h	odd	2	1
5929.2.a.bm		6	21.c	even	2	1
7623.2.a.cp		6	1.a	even	1	1	trivial
7623.2.a.cs		6	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{6} + 4T_{2}^{5} - 12T_{2}^{3} - 4T_{2}^{2} + 8T_{2} + 1 \)

\( T_{5}^{6} - 4T_{5}^{5} - 9T_{5}^{4} + 36T_{5}^{3} - T_{5}^{2} - 8T_{5} + 1 \)

\( T_{13}^{6} - 4T_{13}^{5} - 19T_{13}^{4} + 72T_{13}^{3} - 36T_{13}^{2} - 32T_{13} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 + 4*T^5 - 12*T^3 - 4*T^2 + 8*T + 1
$3$	\( T^{6} \)
$5$	T^6 - 4*T^5 - 9*T^4 + 36*T^3 - T^2 - 8*T + 1
$7$	\( (T - 1)^{6} \)
$11$	\( T^{6} \)