Properties

Label 6027.2.a.x
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1197392.1
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(8\) \(x^{3}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut +\mathstrut \) \(14\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 7 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 4 \nu^{3} - \nu^{2} + 11 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(19\)

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
3.48439
1.61299
0.136381
−1.54585
−1.68791
−2.48439 1.00000 4.17219 4.10940 −2.48439 0 −5.39657 1.00000 −10.2093
1.2 −0.612990 1.00000 −1.62424 1.48910 −0.612990 0 2.22162 1.00000 −0.912805
1.3 0.863619 1.00000 −1.25416 3.66459 0.863619 0 −2.81036 1.00000 3.16481
1.4 2.54585 1.00000 4.48134 2.36161 2.54585 0 6.31711 1.00000 6.01230
1.5 2.68791 1.00000 5.22488 −2.62470 2.68791 0 8.66818 1.00000 −7.05495
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{5} \) \(\mathstrut -\mathstrut 3 T_{2}^{4} \) \(\mathstrut -\mathstrut 6 T_{2}^{3} \) \(\mathstrut +\mathstrut 20 T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut -\mathstrut 9 \)
\(T_{5}^{5} \) \(\mathstrut -\mathstrut 9 T_{5}^{4} \) \(\mathstrut +\mathstrut 18 T_{5}^{3} \) \(\mathstrut +\mathstrut 42 T_{5}^{2} \) \(\mathstrut -\mathstrut 171 T_{5} \) \(\mathstrut +\mathstrut 139 \)
\(T_{13}^{5} \) \(\mathstrut -\mathstrut 3 T_{13}^{4} \) \(\mathstrut -\mathstrut 56 T_{13}^{3} \) \(\mathstrut +\mathstrut 192 T_{13}^{2} \) \(\mathstrut +\mathstrut 473 T_{13} \) \(\mathstrut -\mathstrut 1681 \)