Properties

Label 6027.2.a.j
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.

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

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 −0.561553 2.56155 0 −6.56155 1.00000 1.43845
1.2 1.56155 −1.00000 0.438447 3.56155 −1.56155 0 −2.43845 1.00000 5.56155
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

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