Properties

Label 6024.2.a.k
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + ( -\beta_{1} - \beta_{2} ) q^{5} \) \( + ( -\beta_{3} - \beta_{7} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + ( -\beta_{1} - \beta_{2} ) q^{5} \) \( + ( -\beta_{3} - \beta_{7} ) q^{7} \) \(+ q^{9}\) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} \) \( + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{13} \) \( + ( -\beta_{1} - \beta_{2} ) q^{15} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{17} \) \( + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} \) \( + ( -\beta_{3} - \beta_{7} ) q^{21} \) \( + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{23} \) \( + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} \) \(+ q^{27}\) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{29} \) \( + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{33} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{35} \) \( + ( -4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{37} \) \( + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{39} \) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{41} \) \( + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} \) \( + ( -\beta_{1} - \beta_{2} ) q^{45} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} \) \( + ( -3 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{49} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{51} \) \( + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{53} \) \( + ( -4 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{55} \) \( + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} \) \( + ( 3 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{59} \) \( + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{61} \) \( + ( -\beta_{3} - \beta_{7} ) q^{63} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{65} \) \( + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{67} \) \( + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{69} \) \( + ( 2 - \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{71} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{6} ) q^{73} \) \( + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{77} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{79} \) \(+ q^{81}\) \( + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} \) \( + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{85} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{87} \) \( + ( \beta_{1} + \beta_{2} + 4 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{89} \) \( + ( -2 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{91} \) \( + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{93} \) \( + ( 2 - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{95} \) \( + ( -5 - \beta_{1} + \beta_{2} - \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{97} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut -\mathstrut 21q^{49} \) \(\mathstrut -\mathstrut 12q^{51} \) \(\mathstrut +\mathstrut 13q^{53} \) \(\mathstrut -\mathstrut 33q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 38q^{61} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut +\mathstrut 16q^{65} \) \(\mathstrut -\mathstrut 31q^{67} \) \(\mathstrut -\mathstrut 6q^{69} \) \(\mathstrut +\mathstrut 5q^{71} \) \(\mathstrut -\mathstrut 26q^{73} \) \(\mathstrut -\mathstrut 7q^{75} \) \(\mathstrut -\mathstrut 24q^{77} \) \(\mathstrut -\mathstrut 25q^{79} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 7q^{83} \) \(\mathstrut -\mathstrut 18q^{85} \) \(\mathstrut -\mathstrut 2q^{87} \) \(\mathstrut -\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut -\mathstrut 51q^{97} \) \(\mathstrut -\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(10\) \(x^{6}\mathstrut +\mathstrut \) \(25\) \(x^{5}\mathstrut +\mathstrut \) \(5\) \(x^{4}\mathstrut -\mathstrut \) \(36\) \(x^{3}\mathstrut +\mathstrut \) \(11\) \(x^{2}\mathstrut +\mathstrut \) \(3\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{7} + 9 \nu^{6} + 73 \nu^{5} - 124 \nu^{4} - 87 \nu^{3} + 183 \nu^{2} - 24 \nu - 13 \)\()/8\)
\(\beta_{3}\)\(=\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 25 \nu^{4} - 5 \nu^{3} + 36 \nu^{2} - 11 \nu - 3 \)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} + 15 \nu^{6} + 95 \nu^{5} - 196 \nu^{4} - 113 \nu^{3} + 305 \nu^{2} + 8 \nu - 35 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{7} + 19 \nu^{6} + 139 \nu^{5} - 252 \nu^{4} - 189 \nu^{3} + 381 \nu^{2} + 40 \nu - 47 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{7} + 15 \nu^{6} + 95 \nu^{5} - 192 \nu^{4} - 109 \nu^{3} + 277 \nu^{2} - 23 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} - 9 \nu^{6} - 53 \nu^{5} + 114 \nu^{4} + 59 \nu^{3} - 171 \nu^{2} + 6 \nu + 17 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(21\)
\(\nu^{5}\)\(=\)\(-\)\(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(69\) \(\beta_{1}\mathstrut -\mathstrut \) \(41\)
\(\nu^{6}\)\(=\)\(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(45\) \(\beta_{6}\mathstrut +\mathstrut \) \(139\) \(\beta_{5}\mathstrut -\mathstrut \) \(75\) \(\beta_{4}\mathstrut +\mathstrut \) \(70\) \(\beta_{3}\mathstrut -\mathstrut \) \(86\) \(\beta_{2}\mathstrut -\mathstrut \) \(150\) \(\beta_{1}\mathstrut +\mathstrut \) \(181\)
\(\nu^{7}\)\(=\)\(-\)\(102\) \(\beta_{7}\mathstrut +\mathstrut \) \(64\) \(\beta_{6}\mathstrut -\mathstrut \) \(330\) \(\beta_{5}\mathstrut +\mathstrut \) \(89\) \(\beta_{4}\mathstrut -\mathstrut \) \(180\) \(\beta_{3}\mathstrut +\mathstrut \) \(247\) \(\beta_{2}\mathstrut +\mathstrut \) \(628\) \(\beta_{1}\mathstrut -\mathstrut \) \(453\)

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
2.05340
−1.22404
2.38731
−0.297227
0.397795
0.291651
1.53651
−3.14539
0 1.00000 0 −3.76309 0 −2.52327 0 1.00000 0
1.2 0 1.00000 0 −2.55797 0 1.21480 0 1.00000 0
1.3 0 1.00000 0 −1.69055 0 2.31718 0 1.00000 0
1.4 0 1.00000 0 −1.13469 0 −3.14746 0 1.00000 0
1.5 0 1.00000 0 −0.220466 0 3.34369 0 1.00000 0
1.6 0 1.00000 0 0.624693 0 0.240445 0 1.00000 0
1.7 0 1.00000 0 1.27524 0 0.298981 0 1.00000 0
1.8 0 1.00000 0 2.46685 0 −0.744363 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} - \cdots\)