Properties

Label 6042.2.a.y
Level 6042
Weight 2
Character orbit 6042.a
Self dual Yes
Analytic conductor 48.246
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6042.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2456129013\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( 1 - \beta_{1} ) q^{5} \) \(- q^{6}\) \( + ( -1 - \beta_{3} ) q^{7} \) \(- q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( 1 - \beta_{1} ) q^{5} \) \(- q^{6}\) \( + ( -1 - \beta_{3} ) q^{7} \) \(- q^{8}\) \(+ q^{9}\) \( + ( -1 + \beta_{1} ) q^{10} \) \( + ( -1 + \beta_{3} + \beta_{6} ) q^{11} \) \(+ q^{12}\) \( + ( -1 + \beta_{1} - \beta_{6} ) q^{13} \) \( + ( 1 + \beta_{3} ) q^{14} \) \( + ( 1 - \beta_{1} ) q^{15} \) \(+ q^{16}\) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{17} \) \(- q^{18}\) \(- q^{19}\) \( + ( 1 - \beta_{1} ) q^{20} \) \( + ( -1 - \beta_{3} ) q^{21} \) \( + ( 1 - \beta_{3} - \beta_{6} ) q^{22} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{23} \) \(- q^{24}\) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{25} \) \( + ( 1 - \beta_{1} + \beta_{6} ) q^{26} \) \(+ q^{27}\) \( + ( -1 - \beta_{3} ) q^{28} \) \( + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{29} \) \( + ( -1 + \beta_{1} ) q^{30} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{31} \) \(- q^{32}\) \( + ( -1 + \beta_{3} + \beta_{6} ) q^{33} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{34} \) \( + ( -1 + 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} \) \(+ q^{36}\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{37} \) \(+ q^{38}\) \( + ( -1 + \beta_{1} - \beta_{6} ) q^{39} \) \( + ( -1 + \beta_{1} ) q^{40} \) \( + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} \) \( + ( 1 + \beta_{3} ) q^{42} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{43} \) \( + ( -1 + \beta_{3} + \beta_{6} ) q^{44} \) \( + ( 1 - \beta_{1} ) q^{45} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{46} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{47} \) \(+ q^{48}\) \( + ( -1 - \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{49} \) \( + ( \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{50} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{51} \) \( + ( -1 + \beta_{1} - \beta_{6} ) q^{52} \) \(+ q^{53}\) \(- q^{54}\) \( + ( -3 - 2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{55} \) \( + ( 1 + \beta_{3} ) q^{56} \) \(- q^{57}\) \( + ( 1 - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{59} \) \( + ( 1 - \beta_{1} ) q^{60} \) \( + ( -3 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{61} \) \( + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{62} \) \( + ( -1 - \beta_{3} ) q^{63} \) \(+ q^{64}\) \( + ( -3 + 2 \beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{65} \) \( + ( 1 - \beta_{3} - \beta_{6} ) q^{66} \) \( + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{67} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{68} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{69} \) \( + ( 1 - 3 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{70} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{71} \) \(- q^{72}\) \( + ( 1 - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{73} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{74} \) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{75} \) \(- q^{76}\) \( + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{77} \) \( + ( 1 - \beta_{1} + \beta_{6} ) q^{78} \) \( + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{79} \) \( + ( 1 - \beta_{1} ) q^{80} \) \(+ q^{81}\) \( + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{82} \) \( + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{6} ) q^{83} \) \( + ( -1 - \beta_{3} ) q^{84} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{85} \) \( + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{86} \) \( + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{87} \) \( + ( 1 - \beta_{3} - \beta_{6} ) q^{88} \) \( + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{89} \) \( + ( -1 + \beta_{1} ) q^{90} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{91} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{92} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{93} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{94} \) \( + ( -1 + \beta_{1} ) q^{95} \) \(- q^{96}\) \( + ( -2 - \beta_{1} + 3 \beta_{4} - 2 \beta_{6} ) q^{97} \) \( + ( 1 + \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{98} \) \( + ( -1 + \beta_{3} + \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 7q^{12} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut -\mathstrut 7q^{18} \) \(\mathstrut -\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut 9q^{21} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 7q^{24} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 7q^{26} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 9q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 4q^{30} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 7q^{32} \) \(\mathstrut -\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 10q^{34} \) \(\mathstrut +\mathstrut 7q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 7q^{38} \) \(\mathstrut -\mathstrut 7q^{39} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 9q^{42} \) \(\mathstrut -\mathstrut 26q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 17q^{47} \) \(\mathstrut +\mathstrut 7q^{48} \) \(\mathstrut -\mathstrut 3q^{50} \) \(\mathstrut -\mathstrut 10q^{51} \) \(\mathstrut -\mathstrut 7q^{52} \) \(\mathstrut +\mathstrut 7q^{53} \) \(\mathstrut -\mathstrut 7q^{54} \) \(\mathstrut -\mathstrut 23q^{55} \) \(\mathstrut +\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 7q^{57} \) \(\mathstrut +\mathstrut 6q^{58} \) \(\mathstrut +\mathstrut 7q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 18q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 7q^{64} \) \(\mathstrut -\mathstrut 23q^{65} \) \(\mathstrut +\mathstrut 2q^{66} \) \(\mathstrut -\mathstrut 3q^{67} \) \(\mathstrut -\mathstrut 10q^{68} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 2q^{71} \) \(\mathstrut -\mathstrut 7q^{72} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut +\mathstrut 3q^{75} \) \(\mathstrut -\mathstrut 7q^{76} \) \(\mathstrut -\mathstrut 23q^{77} \) \(\mathstrut +\mathstrut 7q^{78} \) \(\mathstrut -\mathstrut 18q^{79} \) \(\mathstrut +\mathstrut 4q^{80} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut +\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 17q^{83} \) \(\mathstrut -\mathstrut 9q^{84} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 26q^{86} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 2q^{88} \) \(\mathstrut -\mathstrut 13q^{89} \) \(\mathstrut -\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 17q^{94} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 7q^{96} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(14\) \(x^{5}\mathstrut +\mathstrut \) \(29\) \(x^{4}\mathstrut +\mathstrut \) \(48\) \(x^{3}\mathstrut -\mathstrut \) \(14\) \(x^{2}\mathstrut -\mathstrut \) \(35\) \(x\mathstrut -\mathstrut \) \(10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 16 \nu^{6} - 51 \nu^{5} - 216 \nu^{4} + 517 \nu^{3} + 682 \nu^{2} - 480 \nu - 445 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( 24 \nu^{6} - 89 \nu^{5} - 274 \nu^{4} + 888 \nu^{3} + 548 \nu^{2} - 695 \nu - 405 \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{6} + 108 \nu^{5} + 303 \nu^{4} - 1086 \nu^{3} - 456 \nu^{2} + 915 \nu + 335 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{6} + 34 \nu^{5} + 99 \nu^{4} - 338 \nu^{3} - 163 \nu^{2} + 260 \nu + 115 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -93 \nu^{6} + 348 \nu^{5} + 1043 \nu^{4} - 3466 \nu^{3} - 1886 \nu^{2} + 2665 \nu + 1285 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(20\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(43\)
\(\nu^{5}\)\(=\)\(44\) \(\beta_{6}\mathstrut -\mathstrut \) \(60\) \(\beta_{5}\mathstrut -\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(20\) \(\beta_{2}\mathstrut +\mathstrut \) \(147\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{6}\)\(=\)\(249\) \(\beta_{6}\mathstrut -\mathstrut \) \(354\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(240\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(384\) \(\beta_{1}\mathstrut +\mathstrut \) \(554\)

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
4.05485
3.00316
0.982384
−0.456607
−0.724990
−0.835135
−3.02367
−1.00000 1.00000 1.00000 −3.05485 −1.00000 −2.36098 −1.00000 1.00000 3.05485
1.2 −1.00000 1.00000 1.00000 −2.00316 −1.00000 −4.20447 −1.00000 1.00000 2.00316
1.3 −1.00000 1.00000 1.00000 0.0176159 −1.00000 0.282315 −1.00000 1.00000 −0.0176159
1.4 −1.00000 1.00000 1.00000 1.45661 −1.00000 1.71471 −1.00000 1.00000 −1.45661
1.5 −1.00000 1.00000 1.00000 1.72499 −1.00000 −0.765322 −1.00000 1.00000 −1.72499
1.6 −1.00000 1.00000 1.00000 1.83513 −1.00000 0.943730 −1.00000 1.00000 −1.83513
1.7 −1.00000 1.00000 1.00000 4.02367 −1.00000 −4.60998 −1.00000 1.00000 −4.02367
\(n\): 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
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)
\(53\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{7} \) \(\mathstrut -\mathstrut 4 T_{5}^{6} \) \(\mathstrut -\mathstrut 11 T_{5}^{5} \) \(\mathstrut +\mathstrut 51 T_{5}^{4} \) \(\mathstrut -\mathstrut T_{5}^{3} \) \(\mathstrut -\mathstrut 140 T_{5}^{2} \) \(\mathstrut +\mathstrut 116 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut 9 T_{7}^{6} \) \(\mathstrut +\mathstrut 16 T_{7}^{5} \) \(\mathstrut -\mathstrut 39 T_{7}^{4} \) \(\mathstrut -\mathstrut 80 T_{7}^{3} \) \(\mathstrut +\mathstrut 56 T_{7}^{2} \) \(\mathstrut +\mathstrut 48 T_{7} \) \(\mathstrut -\mathstrut 16 \)
\(T_{11}^{7} \) \(\mathstrut +\mathstrut 2 T_{11}^{6} \) \(\mathstrut -\mathstrut 29 T_{11}^{5} \) \(\mathstrut -\mathstrut 11 T_{11}^{4} \) \(\mathstrut +\mathstrut 193 T_{11}^{3} \) \(\mathstrut -\mathstrut 14 T_{11}^{2} \) \(\mathstrut -\mathstrut 330 T_{11} \) \(\mathstrut +\mathstrut 190 \)