Properties

Label 6027.2.a.v
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.981328.1
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 6 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 10 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)

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
−1.74401
−0.844325
−0.243417
1.90925
2.92250
−2.74401 1.00000 5.52961 0.811635 −2.74401 0 −9.68531 1.00000 −2.22714
1.2 −1.84433 1.00000 1.40154 1.91836 −1.84433 0 1.10376 1.00000 −3.53808
1.3 −1.24342 1.00000 −0.453913 −3.27559 −1.24342 0 3.05124 1.00000 4.07293
1.4 0.909252 1.00000 −1.17326 2.40229 0.909252 0 −2.88529 1.00000 2.18428
1.5 1.92250 1.00000 1.69602 −2.85669 1.92250 0 −0.584397 1.00000 −5.49199
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{5} \) \(\mathstrut +\mathstrut 3 T_{2}^{4} \) \(\mathstrut -\mathstrut 4 T_{2}^{3} \) \(\mathstrut -\mathstrut 14 T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut +\mathstrut 11 \)
\(T_{5}^{5} \) \(\mathstrut +\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 14 T_{5}^{3} \) \(\mathstrut -\mathstrut 2 T_{5}^{2} \) \(\mathstrut +\mathstrut 53 T_{5} \) \(\mathstrut -\mathstrut 35 \)
\(T_{13}^{5} \) \(\mathstrut +\mathstrut 3 T_{13}^{4} \) \(\mathstrut -\mathstrut 18 T_{13}^{3} \) \(\mathstrut -\mathstrut 26 T_{13}^{2} \) \(\mathstrut +\mathstrut 91 T_{13} \) \(\mathstrut -\mathstrut 1 \)