Newform orbit 6422.2.a.bj

• Label: 6422.2.a.bj
• Level: $6422$
• Weight: $2$
• Character orbit: 6422.a
• Self dual: yes
• Analytic conductor: $51.280$
• Analytic rank: $0$
• Dimension: $9$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(51.2799281781\)
Analytic rank:	\(0\)
Dimension:	\(9\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial:	\( x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - q^2 + b6 * q^3 + q^4 + (b7 + b4 + b1 - 1) * q^5 - b6 * q^6 + (b8 + b7 + b6 + 2*b3 + 2) * q^7 - q^8 + (b5 - b3 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( -2\nu^{7} - \nu^{6} + 21\nu^{5} + 3\nu^{4} - 50\nu^{3} - 5\nu^{2} + 14\nu \)
\(\beta_{3}\)	\(=\)	\( \nu^{8} + \nu^{7} - 11\nu^{6} - 7\nu^{5} + 32\nu^{4} + 15\nu^{3} - 23\nu^{2} - 4\nu + 3 \)
\(\beta_{4}\)	\(=\)	\( 2\nu^{7} + 2\nu^{6} - 20\nu^{5} - 13\nu^{4} + 44\nu^{3} + 28\nu^{2} - 3\nu - 4 \)
\(\beta_{5}\)	\(=\)	\( \nu^{8} + 2\nu^{7} - 10\nu^{6} - 17\nu^{5} + 26\nu^{4} + 38\nu^{3} - 12\nu^{2} - 9\nu + 3 \)
\(\beta_{6}\)	\(=\)	\( -\nu^{8} + \nu^{7} + 10\nu^{6} - 15\nu^{5} - 14\nu^{4} + 38\nu^{3} - 21\nu^{2} - 14\nu + 5 \)
\(\beta_{7}\)	\(=\)	\( 3\nu^{7} + 2\nu^{6} - 31\nu^{5} - 9\nu^{4} + 72\nu^{3} + 15\nu^{2} - 14\nu + 1 \)
\(\beta_{8}\)	\(=\)	\( 2\nu^{8} - \nu^{7} - 22\nu^{6} + 18\nu^{5} + 53\nu^{4} - 44\nu^{3} - 18\nu^{2} + 8\nu - 1 \)

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

−0.588380
0.380781
−1.66806
1.89407
1.91149
−2.97314
0.219034
−0.490336
2.31454

−1.00000

−2.01699

1.00000

−2.32439

2.01699

−1.49240

−1.00000

1.06824

2.32439

1.2

−1.00000

−1.66099

1.00000

1.69394

1.66099

−0.961642

−1.00000

−0.241101

−1.69394

1.3

−1.00000

−1.57968

1.00000

−1.25818

1.57968

0.329623

−1.00000

−0.504623

1.25818

1.4

−1.00000

−0.953982

1.00000

3.75463

0.953982

3.81192

−1.00000

−2.08992

−3.75463

1.5

−1.00000

0.0255071

1.00000

−1.15077

−0.0255071

−2.71462

−1.00000

−2.99935

1.15077

1.6

−1.00000

1.04618

1.00000

−0.929245

−1.04618

2.89421

−1.00000

−1.90551

0.929245

1.7

−1.00000

1.28668

1.00000

−4.29839

−1.28668

4.70931

−1.00000

−1.34446

4.29839

1.8

−1.00000

2.08053

1.00000

0.703813

−2.08053

4.73726

−1.00000

1.32860

−0.703813

1.9

−1.00000

2.77275

1.00000

2.80860

−2.77275

1.68633

−1.00000

4.68813

−2.80860

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(13\)	\(1\)
\(19\)	\(-1\)

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	6422.2.a.bj	✓	9
13.b	even	2	1		6422.2.a.bl	yes	9

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6422.2.a.bj	✓	9	1.a	even	1	1	trivial
6422.2.a.bl	yes	9	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{9} - T_{3}^{8} - 12T_{3}^{7} + 7T_{3}^{6} + 49T_{3}^{5} - 14T_{3}^{4} - 77T_{3}^{3} + 10T_{3}^{2} + 39T_{3} - 1 \)

\( T_{5}^{9} + T_{5}^{8} - 26T_{5}^{7} - 21T_{5}^{6} + 182T_{5}^{5} + 168T_{5}^{4} - 371T_{5}^{3} - 402T_{5}^{2} + 130T_{5} + 169 \)

\( T_{7}^{9} - 13T_{7}^{8} + 44T_{7}^{7} + 70T_{7}^{6} - 574T_{7}^{5} + 357T_{7}^{4} + 1631T_{7}^{3} - 1291T_{7}^{2} - 1375T_{7} + 533 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{9} \)
$3$	T^9 - T^8 - 12*T^7 + 7*T^6 + 49*T^5 - 14*T^4 - 77*T^3 + 10*T^2 + 39*T - 1
$5$	T^9 + T^8 - 26*T^7 - 21*T^6 + 182*T^5 + 168*T^4 - 371*T^3 - 402*T^2 + 130*T + 169
$7$	T^9 - 13*T^8 + 44*T^7 + 70*T^6 - 574*T^5 + 357*T^4 + 1631*T^3 - 1291*T^2 - 1375*T + 533
$11$	T^9 - 3*T^8 - 59*T^7 + 189*T^6 + 1141*T^5 - 3780*T^4 - 8603*T^3 + 29024*T^2 + 21554*T - 73177