Properties

Label 6018.2.a.q
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 6
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5173625.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 8 x^{4} + 8 x^{3} + 10 x^{2} - 7 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + \nu^{4} + 8 \nu^{3} - 7 \nu^{2} - 10 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 8 \nu^{2} + 10 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{5} - 2 \nu^{4} - 24 \nu^{3} + 16 \nu^{2} + 32 \nu - 10 \)
\(\beta_{5}\)\(=\)\( 3 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 16 \nu^{2} + 37 \nu - 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 9 \beta_{4} - 2 \beta_{3} + 28 \beta_{1} - 10\)

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
−2.56798
−1.16459
0.225865
0.353486
1.72409
2.42914
1.00000 −1.00000 1.00000 −3.56798 −1.00000 −0.530708 1.00000 1.00000 −3.56798
1.2 1.00000 −1.00000 1.00000 −2.16459 −1.00000 −4.86606 1.00000 1.00000 −2.16459
1.3 1.00000 −1.00000 1.00000 −0.774135 −1.00000 1.30916 1.00000 1.00000 −0.774135
1.4 1.00000 −1.00000 1.00000 −0.646514 −1.00000 4.78830 1.00000 1.00000 −0.646514
1.5 1.00000 −1.00000 1.00000 0.724086 −1.00000 −0.160419 1.00000 1.00000 0.724086
1.6 1.00000 −1.00000 1.00000 1.42914 −1.00000 −1.54027 1.00000 1.00000 1.42914
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(17\) \(1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{6} + 5 T_{5}^{5} + 2 T_{5}^{4} - 14 T_{5}^{3} - 9 T_{5}^{2} + 6 T_{5} + 4 \)
\( T_{7}^{6} + T_{7}^{5} - 25 T_{7}^{4} - 23 T_{7}^{3} + 41 T_{7}^{2} + 32 T_{7} + 4 \)