Properties

Label 6015.2.a.a
Level 6015
Weight 2
Character orbit 6015.a
Self dual Yes
Analytic conductor 48.030
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.41421
−1.41421
1.00000 1.00000 −1.00000 1.00000 1.00000 −0.828427 −3.00000 1.00000 1.00000
1.2 1.00000 1.00000 −1.00000 1.00000 1.00000 4.82843 −3.00000 1.00000 1.00000
\(n\): 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
\(3\) \(-1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).