Properties

Label 6027.2.a.w
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

Related objects

Downloads

Learn more about

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: 5.5.626512.1
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 3 \nu + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 2 \nu + 2 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + 3 \nu^{3} + 3 \nu^{2} - 6 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(11 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 15 \beta_{1} + 23\)

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
3.30065
1.11295
0.302390
−1.14904
−1.56695
−2.30065 −1.00000 3.29299 −1.89893 2.30065 0 −2.97472 1.00000 4.36878
1.2 −0.112947 −1.00000 −1.98724 2.19021 0.112947 0 0.450348 1.00000 −0.247378
1.3 0.697610 −1.00000 −1.51334 −3.10064 −0.697610 0 −2.45094 1.00000 −2.16304
1.4 2.14904 −1.00000 2.61836 1.12223 −2.14904 0 1.32887 1.00000 2.41172
1.5 2.56695 −1.00000 4.58924 −1.31287 −2.56695 0 6.64645 1.00000 −3.37008
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{5} - 3 T_{2}^{4} - 4 T_{2}^{3} + 16 T_{2}^{2} - 7 T_{2} - 1 \)
\( T_{5}^{5} + 3 T_{5}^{4} - 6 T_{5}^{3} - 18 T_{5}^{2} + 5 T_{5} + 19 \)
\( T_{13}^{5} + 5 T_{13}^{4} - 10 T_{13}^{3} - 48 T_{13}^{2} + 23 T_{13} + 103 \)