Newform 403.2.a.e

• Label: 403.2.a.e
• Level: 403
• Weight: 2
• Character orbit: 403.a
• Self dual: Yes
• Analytic conductor: 3.218
• Analytic rank: 0
• Dimension: 8
• CM: No
• Inner twists: 1

Newspace parameters

Level:	\( N \)	=	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	403.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q -\beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} + \beta_{7} ) q^{5} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{8} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - q^{2} + 7q^{3} + 7q^{4} + 11q^{5} - 2q^{7} + 9q^{9} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 37 x^{4} - 33 x^{3} - 36 x^{2} + 33 x - 4\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - \nu^{2} - 4 \nu + 3 \)
\(\beta_{4}\)	\(=\)	\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 5 \nu^{2} + 10 \nu - 5 \)
\(\beta_{5}\)	\(=\)	\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 6 \nu^{3} - 15 \nu^{2} + 9 \nu + 2 \)
\(\beta_{6}\)	\(=\)	\( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 12 \nu^{3} + 5 \nu^{2} - 15 \nu + 5 \)
\(\beta_{7}\)	\(=\)	\( \nu^{7} - 10 \nu^{5} - \nu^{4} + 28 \nu^{3} + \nu^{2} - 21 \nu + 4 \)

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

2.53815
1.71590
1.53950
0.656875
0.147670
−1.38904
−2.06827
−2.14079

−2.53815

2.96968

4.44218

2.45818

−7.53747

−1.29216

−6.19862

5.81898

−6.23922

1.2

−1.71590

−1.27835

0.944315

−2.00831

2.19352

−3.76625

1.81145

−1.36583

3.44606

1.3

−1.53950

0.734778

0.370072

3.72666

−1.13119

3.41650

2.50928

−2.46010

−5.73720

1.4

−0.656875

2.67827

−1.56852

−1.55873

−1.75929

2.17607

2.34407

4.17313

1.02389

1.5

−0.147670

−2.43721

−1.97819

4.40045

0.359904

−2.90659

0.587461

2.94000

−0.649816

1.6

1.38904

1.62343

−0.0705747

3.01447

2.25501

2.19250

−2.87611

−0.364464

4.18722

1.7

2.06827

2.49221

2.27774

1.40222

5.15456

−5.08294

0.574435

3.21112

2.90017

1.8

2.14079

0.217190

2.58298

−0.434938

0.464957

3.26287

1.24803

−2.95283

−0.931109

Inner twists

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

Atkin-Lehner signs

\( p \)	Sign
\(13\)	\(-1\)
\(31\)	\(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).