Properties

Label 6027.2.a.s
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

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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: \(3\)
Coefficient field: 3.3.316.1
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)

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
−1.81361
0.470683
2.34292
−1.81361 1.00000 1.28917 1.10278 −1.81361 0 1.28917 1.00000 −2.00000
1.2 0.470683 1.00000 −1.77846 −4.24914 0.470683 0 −1.77846 1.00000 −2.00000
1.3 2.34292 1.00000 3.48929 −0.853635 2.34292 0 3.48929 1.00000 −2.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{3} \) \(\mathstrut -\mathstrut T_{2}^{2} \) \(\mathstrut -\mathstrut 4 T_{2} \) \(\mathstrut +\mathstrut 2 \)
\(T_{5}^{3} \) \(\mathstrut +\mathstrut 4 T_{5}^{2} \) \(\mathstrut -\mathstrut 2 T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{13}^{3} \) \(\mathstrut +\mathstrut 8 T_{13}^{2} \) \(\mathstrut +\mathstrut 14 T_{13} \) \(\mathstrut -\mathstrut 4 \)