Properties

Label 6027.2.a.o
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: 3.3.148.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
2.17009
0.311108
−1.48119
−2.17009 1.00000 2.70928 −2.70928 −2.17009 0 −1.53919 1.00000 5.87936
1.2 −0.311108 1.00000 −1.90321 1.90321 −0.311108 0 1.21432 1.00000 −0.592104
1.3 1.48119 1.00000 0.193937 −0.193937 1.48119 0 −2.67513 1.00000 −0.287258
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{3} + T_{2}^{2} - 3 T_{2} - 1 \)
\( T_{5}^{3} + T_{5}^{2} - 5 T_{5} - 1 \)
\( T_{13}^{3} - 7 T_{13}^{2} + T_{13} + 43 \)