Newform 5.20.a.b

• Label: 5.20.a.b
• Level: 5
• Weight: 20
• Character orbit: 5.a
• Self dual: Yes
• Analytic conductor: 11.441
• Analytic rank: 0
• Dimension: 4
• CM: No
• Inner twists: 1

Newspace parameters

Level:	\( N \)	=	\( 5 \)
Weight:	\( k \)	=	\( 20 \)
Character orbit:	\([\chi]\)	=	5.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	\(11.4408348278\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q\) \( + ( -105 - \beta_{1} ) q^{2} \) \( + ( 770 + 14 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 521248 - 18 \beta_{1} - 26 \beta_{2} - \beta_{3} ) q^{4} \) \( -1953125 q^{5} \) \( + ( -14517938 - 24482 \beta_{1} + 256 \beta_{2} + 96 \beta_{3} ) q^{6} \) \( + ( 53505350 - 49970 \beta_{1} - 81 \beta_{2} - 720 \beta_{3} ) q^{7} \) \( + ( 20434440 - 698512 \beta_{1} - 968 \beta_{2} + 2660 \beta_{3} ) q^{8} \) \( + ( 38516137 + 508024 \beta_{1} + 10108 \beta_{2} - 5472 \beta_{3} ) q^{9} \) \(+O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\(4q \) \(\mathstrut -\mathstrut 420q^{2} \) \(\mathstrut +\mathstrut 3080q^{3} \) \(\mathstrut +\mathstrut 2084992q^{4} \) \(\mathstrut -\mathstrut 7812500q^{5} \) \(\mathstrut -\mathstrut 58071752q^{6} \) \(\mathstrut +\mathstrut 214021400q^{7} \) \(\mathstrut +\mathstrut 81737760q^{8} \) \(\mathstrut +\mathstrut 154064548q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(323561\) \(x^{2}\mathstrut -\mathstrut \) \(26738538\) \(x\mathstrut +\mathstrut \) \(10870990650\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\((\)\( -50 \nu^{3} + 12712 \nu^{2} + 10576678 \nu - 1034892417 \)\()/1360737\)
\(\beta_{2}\)	\(=\)	\((\)\( -100 \nu^{3} + 25424 \nu^{2} + 57439676 \nu - 2087927994 \)\()/453579\)
\(\beta_{3}\)	\(=\)	\((\)\( -340 \nu^{3} - 224584 \nu^{2} + 120441404 \nu + 43256659674 \)\()/194391\)

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

−462.664
150.720
581.867
−267.923

−1387.10

48079.9

1.39976e6

−1.95313e6

−6.66917e7

9.13351e7

−1.21437e9

1.14942e9

2.70918e9

1.2

−602.379

−7268.55

−161427.

−1.95313e6

4.37842e6

−1.76809e8

4.13061e8

−1.10943e9

1.17652e9

1.3

208.721

−48249.1

−480724.

−1.95313e6

−1.00706e7

1.75500e8

−2.09767e8

1.16572e9

−4.07657e8

1.4

1360.76

10517.7

1.32738e6

−1.95313e6

1.43121e7

1.23996e8

1.09282e9

−1.05164e9

−2.65773e9

Inner twists

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

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 420 T_{2}^{3} \) \(\mathstrut -\mathstrut 2002872 T_{2}^{2} \) \(\mathstrut -\mathstrut 746347520 T_{2} \) \(\mathstrut +\mathstrut 237314973696 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\).