Properties

Label 6012.2.a.f
Level 6012
Weight 2
Character orbit 6012.a
Self dual Yes
Analytic conductor 48.006
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{5} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} \) \( + ( 1 + 2 \beta_{1} ) q^{17} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{19} \) \( + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} \) \( + ( -2 - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{25} \) \( + ( 3 - \beta_{3} - 2 \beta_{4} ) q^{29} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{31} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{35} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} \) \( + ( 2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{41} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{43} \) \( + ( 4 - 3 \beta_{1} + 4 \beta_{3} + \beta_{4} ) q^{47} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{49} \) \( + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{53} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{61} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} ) q^{65} \) \( + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{67} \) \( + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{73} \) \( + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{77} \) \( + ( -4 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{79} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{83} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} \) \( + ( 5 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{89} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{91} \) \( + ( -\beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{95} \) \( + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 12q^{35} \) \(\mathstrut -\mathstrut 7q^{37} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 19q^{47} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut +\mathstrut 7q^{59} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut +\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut 35q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 11q^{83} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 32q^{89} \) \(\mathstrut +\mathstrut 5q^{91} \) \(\mathstrut +\mathstrut 19q^{95} \) \(\mathstrut +\mathstrut 11q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(6\) \(x^{3}\mathstrut +\mathstrut \) \(3\) \(x^{2}\mathstrut +\mathstrut \) \(5\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 3 \nu^{3} + 11 \nu^{2} - 11 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)

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
1.31991
−2.07823
−0.261082
2.56399
−0.544588
0 0 0 −1.28459 0 −1.96468 0 0 0
1.2 0 0 0 0.614948 0 −3.46328 0 0 0
1.3 0 0 0 1.38924 0 −0.871845 0 0 0
1.4 0 0 0 1.85181 0 2.41580 0 0 0
1.5 0 0 0 4.42860 0 1.88401 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{5} \) \(\mathstrut -\mathstrut 7 T_{5}^{4} \) \(\mathstrut +\mathstrut 11 T_{5}^{3} \) \(\mathstrut +\mathstrut 6 T_{5}^{2} \) \(\mathstrut -\mathstrut 21 T_{5} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).