Properties

Label 6027.2.a.bc
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: 8.8.7457527933.1
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 6 x^{6} + 23 x^{5} - 4 x^{4} - 27 x^{3} + 8 x^{2} + 8 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 7 \nu^{5} + 14 \nu^{4} + 3 \nu^{3} - 10 \nu^{2} + \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 2 \nu^{5} + 8 \nu^{4} - 15 \nu^{3} - 10 \nu^{2} + 16 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 16 \nu - 6 \)
\(\beta_{5}\)\(=\)\( -\nu^{7} + 2 \nu^{6} + 8 \nu^{5} - 15 \nu^{4} - 11 \nu^{3} + 16 \nu^{2} + 7 \nu - 1 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - 4 \nu^{5} - 15 \nu^{4} + 29 \nu^{3} + 14 \nu^{2} - 26 \nu - 6 \)
\(\beta_{7}\)\(=\)\( -3 \nu^{7} + 6 \nu^{6} + 23 \nu^{5} - 44 \nu^{4} - 24 \nu^{3} + 42 \nu^{2} + 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{5} + \beta_{2} + 5 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 8 \beta_{3} + \beta_{2} - \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(9 \beta_{7} + \beta_{6} - 17 \beta_{5} + 2 \beta_{3} + 10 \beta_{2} + 31 \beta_{1} - 17\)
\(\nu^{6}\)\(=\)\(11 \beta_{7} + 10 \beta_{6} - 20 \beta_{5} + 38 \beta_{4} + 57 \beta_{3} + 13 \beta_{2} - 5 \beta_{1} + 97\)
\(\nu^{7}\)\(=\)\(68 \beta_{7} + 13 \beta_{6} - 125 \beta_{5} + 2 \beta_{4} + 26 \beta_{3} + 80 \beta_{2} + 205 \beta_{1} - 113\)

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
2.70360
1.35554
1.21768
−0.978012
−0.169079
1.70821
−2.52358
−0.314356
−2.23115 1.00000 2.97801 −0.893036 −2.23115 0 −2.18209 1.00000 1.99249
1.2 −2.04183 1.00000 2.16908 3.68950 −2.04183 0 −0.345232 1.00000 −7.53333
1.3 −1.51387 1.00000 0.291794 −3.80505 −1.51387 0 2.58600 1.00000 5.76034
1.4 −1.13860 1.00000 −0.703600 2.27123 −1.13860 0 3.07831 1.00000 −2.58601
1.5 1.62618 1.00000 0.644462 1.67363 1.62618 0 −2.20435 1.00000 2.72162
1.6 1.66803 1.00000 0.782321 4.32429 1.66803 0 −2.03112 1.00000 7.21304
1.7 2.07710 1.00000 2.31436 −2.30039 2.07710 0 0.652949 1.00000 −4.77814
1.8 2.55413 1.00000 4.52358 2.03983 2.55413 0 6.44554 1.00000 5.20999
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{8} - \cdots\)
\(T_{5}^{8} - \cdots\)
\( T_{13}^{4} - 5 T_{13}^{3} - 10 T_{13}^{2} + 44 T_{13} + 49 \)