Properties

Label 6027.2.a.k
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
−2.56155 1.00000 4.56155 −2.00000 −2.56155 0 −6.56155 1.00000 5.12311
1.2 1.56155 1.00000 0.438447 −2.00000 1.56155 0 −2.43845 1.00000 −3.12311
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5} \) \(\mathstrut +\mathstrut 2 \)
\(T_{13}^{2} \) \(\mathstrut +\mathstrut 2 T_{13} \) \(\mathstrut -\mathstrut 16 \)