Properties

Label 6012.2.a.g
Level 6012
Weight 2
Character orbit 6012.a
Self dual Yes
Analytic conductor 48.006
Analytic rank 1
Dimension 7
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{11} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} \) \( + ( 2 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{17} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{19} \) \( + ( 4 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{23} \) \( + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{25} \) \( + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{29} \) \( + ( -4 + 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{31} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{35} \) \( + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} ) q^{37} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{41} \) \( + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{47} \) \( + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{49} \) \( + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{55} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{59} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{61} \) \( + ( 3 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{65} \) \( + ( -6 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} \) \( + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{71} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{73} \) \( + ( 5 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{77} \) \( + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{79} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{83} \) \( + ( -4 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} \) \( + ( -3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{89} \) \( + ( \beta_{1} - 7 \beta_{2} + \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{91} \) \( + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} \) \( + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut +\mathstrut q^{17} \) \(\mathstrut -\mathstrut 11q^{19} \) \(\mathstrut +\mathstrut 19q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut -\mathstrut 13q^{31} \) \(\mathstrut +\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 26q^{37} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut 11q^{47} \) \(\mathstrut +\mathstrut 19q^{49} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 5q^{61} \) \(\mathstrut -\mathstrut 13q^{65} \) \(\mathstrut -\mathstrut 42q^{67} \) \(\mathstrut -\mathstrut 9q^{71} \) \(\mathstrut +\mathstrut 27q^{73} \) \(\mathstrut -\mathstrut 12q^{77} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 16q^{83} \) \(\mathstrut -\mathstrut 27q^{85} \) \(\mathstrut -\mathstrut 9q^{89} \) \(\mathstrut -\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut -\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(11\) \(x^{5}\mathstrut -\mathstrut \) \(7\) \(x^{4}\mathstrut +\mathstrut \) \(21\) \(x^{3}\mathstrut +\mathstrut \) \(17\) \(x^{2}\mathstrut -\mathstrut \) \(4\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} - 6 \nu^{5} - 73 \nu^{4} + 13 \nu^{3} + 149 \nu^{2} + 5 \nu - 50 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{6} - 6 \nu^{5} - 95 \nu^{4} - \nu^{3} + 191 \nu^{2} + 35 \nu - 58 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{6} + 10 \nu^{5} + 135 \nu^{4} - 11 \nu^{3} - 263 \nu^{2} - 31 \nu + 70 \)\()/4\)
\(\beta_{5}\)\(=\)\( -6 \nu^{6} + 4 \nu^{5} + 63 \nu^{4} - 123 \nu^{2} - 18 \nu + 33 \)
\(\beta_{6}\)\(=\)\((\)\( 27 \nu^{6} - 18 \nu^{5} - 285 \nu^{4} + \nu^{3} + 565 \nu^{2} + 81 \nu - 154 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(29\)
\(\nu^{5}\)\(=\)\(-\)\(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(48\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{6}\)\(=\)\(-\)\(107\) \(\beta_{6}\mathstrut -\mathstrut \) \(125\) \(\beta_{5}\mathstrut +\mathstrut \) \(92\) \(\beta_{4}\mathstrut +\mathstrut \) \(51\) \(\beta_{3}\mathstrut +\mathstrut \) \(90\) \(\beta_{2}\mathstrut +\mathstrut \) \(71\) \(\beta_{1}\mathstrut +\mathstrut \) \(260\)

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
0.495342
1.47217
3.27771
−2.38961
−1.47685
−0.721798
−0.656969
0 0 0 −4.03761 0 −3.68648 0 0 0
1.2 0 0 0 −1.35854 0 −3.73540 0 0 0
1.3 0 0 0 −0.610181 0 −0.184306 0 0 0
1.4 0 0 0 0.836956 0 1.84506 0 0 0
1.5 0 0 0 1.35423 0 −2.86706 0 0 0
1.6 0 0 0 2.77086 0 1.71364 0 0 0
1.7 0 0 0 3.04428 0 −5.08545 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(T_{5}^{7} \) \(\mathstrut -\mathstrut 2 T_{5}^{6} \) \(\mathstrut -\mathstrut 17 T_{5}^{5} \) \(\mathstrut +\mathstrut 42 T_{5}^{4} \) \(\mathstrut +\mathstrut 28 T_{5}^{3} \) \(\mathstrut -\mathstrut 88 T_{5}^{2} \) \(\mathstrut +\mathstrut 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).