Properties

Label 6032.2.a.u
Level 6032
Weight 2
Character orbit 6032.a
Self dual Yes
Analytic conductor 48.166
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6032.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1657624992\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{3} \) \( + ( -1 + \beta_{2} - \beta_{6} ) q^{5} \) \( + ( -1 + \beta_{5} ) q^{7} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{3} \) \( + ( -1 + \beta_{2} - \beta_{6} ) q^{5} \) \( + ( -1 + \beta_{5} ) q^{7} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{9} \) \( + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{11} \) \(- q^{13}\) \( + ( -3 + \beta_{3} - \beta_{5} - \beta_{6} ) q^{15} \) \( + ( 2 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{17} \) \( + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{19} \) \( + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{21} \) \( + ( -1 + \beta_{1} - \beta_{5} ) q^{23} \) \( + ( 3 - 2 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{25} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{27} \) \(+ q^{29}\) \( + ( -\beta_{1} + \beta_{2} + \beta_{4} ) q^{31} \) \( + ( 6 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{33} \) \( + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{35} \) \( + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{37} \) \( + \beta_{2} q^{39} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} \) \( + ( -5 + 4 \beta_{1} + \beta_{4} + \beta_{5} ) q^{43} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{45} \) \( + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{47} \) \( + ( -1 - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{49} \) \( + ( -3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{51} \) \( + ( -1 + 5 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{53} \) \( + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{5} - 2 \beta_{6} ) q^{55} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{57} \) \( + ( -2 + 3 \beta_{1} - 2 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{59} \) \( + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{61} \) \( + ( 1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{63} \) \( + ( 1 - \beta_{2} + \beta_{6} ) q^{65} \) \( + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{6} ) q^{69} \) \( + ( -3 - 6 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{71} \) \( + ( 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{73} \) \( + ( 5 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} ) q^{75} \) \( + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{77} \) \( + ( -7 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{79} \) \( + ( 3 - 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{81} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{83} \) \( + ( -4 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{85} \) \( -\beta_{2} q^{87} \) \( + ( -3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} ) q^{89} \) \( + ( 1 - \beta_{5} ) q^{91} \) \( + ( -3 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{93} \) \( + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 5 \beta_{5} - \beta_{6} ) q^{95} \) \( + ( 1 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{97} \) \( + ( 4 - 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut -\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut +\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut +\mathstrut 7q^{29} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut +\mathstrut 6q^{35} \) \(\mathstrut +\mathstrut 9q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut -\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut +\mathstrut 9q^{47} \) \(\mathstrut -\mathstrut 10q^{49} \) \(\mathstrut -\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 13q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut -\mathstrut 13q^{59} \) \(\mathstrut +\mathstrut 3q^{61} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 7q^{67} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 22q^{71} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 29q^{75} \) \(\mathstrut -\mathstrut 30q^{77} \) \(\mathstrut -\mathstrut 48q^{79} \) \(\mathstrut +\mathstrut 15q^{81} \) \(\mathstrut +\mathstrut 30q^{83} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 2q^{87} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut +\mathstrut 7q^{91} \) \(\mathstrut -\mathstrut 27q^{93} \) \(\mathstrut -\mathstrut 23q^{95} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 25q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{6}\mathstrut -\mathstrut \) \(7\) \(x^{5}\mathstrut +\mathstrut \) \(11\) \(x^{4}\mathstrut +\mathstrut \) \(15\) \(x^{3}\mathstrut -\mathstrut \) \(15\) \(x^{2}\mathstrut -\mathstrut \) \(8\) \(x\mathstrut +\mathstrut \) \(6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} + 5 \nu - 5 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 9 \nu^{3} + 10 \nu^{2} - 7 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{6}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(35\) \(\beta_{2}\mathstrut +\mathstrut \) \(56\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)

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
−1.86207
2.56721
−1.33494
2.05973
−0.893322
0.887894
0.575509
0 −3.32938 0 1.40530 0 −1.72070 0 8.08475 0
1.2 0 −2.02334 0 1.17887 0 −0.399284 0 1.09391 0
1.3 0 −1.11702 0 2.27865 0 2.88565 0 −1.75226 0
1.4 0 −0.182751 0 1.30841 0 −4.37827 0 −2.96660 0
1.5 0 0.308654 0 −3.95109 0 −2.36054 0 −2.90473 0
1.6 0 2.09954 0 −3.72490 0 0.555254 0 1.40806 0
1.7 0 2.24430 0 −0.495240 0 −1.58211 0 2.03688 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
\(2\) \(-1\)
\(13\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\(T_{3}^{7} \) \(\mathstrut +\mathstrut 2 T_{3}^{6} \) \(\mathstrut -\mathstrut 11 T_{3}^{5} \) \(\mathstrut -\mathstrut 16 T_{3}^{4} \) \(\mathstrut +\mathstrut 30 T_{3}^{3} \) \(\mathstrut +\mathstrut 33 T_{3}^{2} \) \(\mathstrut -\mathstrut 6 T_{3} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{7} \) \(\mathstrut +\mathstrut 2 T_{5}^{6} \) \(\mathstrut -\mathstrut 18 T_{5}^{5} \) \(\mathstrut -\mathstrut 7 T_{5}^{4} \) \(\mathstrut +\mathstrut 106 T_{5}^{3} \) \(\mathstrut -\mathstrut 111 T_{5}^{2} \) \(\mathstrut -\mathstrut 8 T_{5} \) \(\mathstrut +\mathstrut 36 \)