Properties

Label 8015.2.a.g
Level 8015
Weight 2
Character orbit 8015.a
Self dual Yes
Analytic conductor 64.000
Analytic rank 1
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0000972201\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( \beta_{1} + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( -3 - \beta_{2} ) q^{6} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{8} \) \( + ( 4 - 2 \beta_{1} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( \beta_{1} + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( -3 - \beta_{2} ) q^{6} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{8} \) \( + ( 4 - 2 \beta_{1} ) q^{9} \) \( -\beta_{1} q^{10} \) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{11} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{12} \) \( + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{13} \) \( -\beta_{1} q^{14} \) \( + ( \beta_{1} - \beta_{2} ) q^{15} \) \( + ( -1 - 2 \beta_{2} ) q^{16} \) \( + ( -2 + 2 \beta_{1} ) q^{17} \) \( + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{18} \) \( + ( -\beta_{1} - \beta_{2} ) q^{20} \) \( + ( \beta_{1} - \beta_{2} ) q^{21} \) \( + ( -3 - \beta_{1} - \beta_{2} ) q^{22} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{23} \) \( + ( 4 - 3 \beta_{1} ) q^{24} \) \(+ q^{25}\) \( + ( 4 + 3 \beta_{1} + 3 \beta_{2} ) q^{26} \) \( + ( 6 - \beta_{1} + 3 \beta_{2} ) q^{27} \) \( + ( -\beta_{1} - \beta_{2} ) q^{28} \) \( + ( 3 + \beta_{1} - \beta_{2} ) q^{29} \) \( + ( 3 + \beta_{2} ) q^{30} \) \( + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{31} \) \( + ( -3 \beta_{1} - 2 \beta_{2} ) q^{32} \) \( + ( 7 - \beta_{1} - \beta_{2} ) q^{33} \) \( + ( 4 + 2 \beta_{2} ) q^{34} \) \(+ q^{35}\) \( + ( -2 + 2 \beta_{2} ) q^{36} \) \( + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{37} \) \( + ( -1 - 2 \beta_{1} - 7 \beta_{2} ) q^{39} \) \( + ( -1 - \beta_{2} ) q^{40} \) \( + ( -9 + \beta_{1} - \beta_{2} ) q^{41} \) \( + ( 3 + \beta_{2} ) q^{42} \) \( + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} \) \( + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{44} \) \( + ( -4 + 2 \beta_{1} ) q^{45} \) \( + ( -5 + \beta_{1} - \beta_{2} ) q^{46} \) \( + ( -3 + \beta_{1} + \beta_{2} ) q^{47} \) \( + ( -8 + 5 \beta_{1} + \beta_{2} ) q^{48} \) \(+ q^{49}\) \( + \beta_{1} q^{50} \) \( + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{51} \) \( + ( 7 + 4 \beta_{1} - \beta_{2} ) q^{52} \) \( + ( 1 - \beta_{1} - \beta_{2} ) q^{53} \) \( + ( -5 + 8 \beta_{1} - \beta_{2} ) q^{54} \) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{55} \) \( + ( -1 - \beta_{2} ) q^{56} \) \( + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{58} \) \( + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{59} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{60} \) \( + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{61} \) \( + ( 9 + \beta_{1} + 3 \beta_{2} ) q^{62} \) \( + ( -4 + 2 \beta_{1} ) q^{63} \) \( + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} \) \( + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{65} \) \( + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{66} \) \( + ( -5 + 3 \beta_{1} + 3 \beta_{2} ) q^{67} \) \( + ( 2 + 2 \beta_{1} ) q^{68} \) \( + ( 15 - 5 \beta_{1} - 3 \beta_{2} ) q^{69} \) \( + \beta_{1} q^{70} \) \( + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{71} \) \( + ( 6 - 4 \beta_{1} + 4 \beta_{2} ) q^{72} \) \( + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} \) \( + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{74} \) \( + ( -\beta_{1} + \beta_{2} ) q^{75} \) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{77} \) \( + ( 3 - 10 \beta_{1} - 2 \beta_{2} ) q^{78} \) \( + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{79} \) \( + ( 1 + 2 \beta_{2} ) q^{80} \) \( + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} \) \( + ( 3 - 9 \beta_{1} + \beta_{2} ) q^{82} \) \( + ( -2 - \beta_{1} - \beta_{2} ) q^{83} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{84} \) \( + ( 2 - 2 \beta_{1} ) q^{85} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{86} \) \( + ( -7 - \beta_{1} + 3 \beta_{2} ) q^{87} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{88} \) \( + ( -6 \beta_{1} - 8 \beta_{2} ) q^{89} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{90} \) \( + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{91} \) \( + ( 5 - 3 \beta_{1} - 5 \beta_{2} ) q^{92} \) \( + ( -21 + 5 \beta_{1} + \beta_{2} ) q^{93} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{94} \) \( + ( 1 + 4 \beta_{1} + 5 \beta_{2} ) q^{96} \) \( + ( -8 - 4 \beta_{2} ) q^{97} \) \( + \beta_{1} q^{98} \) \( + ( 2 - 2 \beta_{1} + 6 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut q^{12} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut -\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 10q^{18} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 15q^{26} \) \(\mathstrut +\mathstrut 17q^{27} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 9q^{30} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 3q^{32} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut +\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 6q^{36} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 5q^{39} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 26q^{41} \) \(\mathstrut +\mathstrut 9q^{42} \) \(\mathstrut +\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 14q^{46} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 19q^{48} \) \(\mathstrut +\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 25q^{52} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 7q^{54} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 15q^{59} \) \(\mathstrut -\mathstrut q^{60} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 28q^{62} \) \(\mathstrut -\mathstrut 10q^{63} \) \(\mathstrut -\mathstrut 11q^{64} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut +\mathstrut 2q^{66} \) \(\mathstrut -\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut +\mathstrut 40q^{69} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut +\mathstrut 14q^{72} \) \(\mathstrut -\mathstrut 15q^{73} \) \(\mathstrut +\mathstrut 12q^{74} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut q^{78} \) \(\mathstrut +\mathstrut 5q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 7q^{83} \) \(\mathstrut -\mathstrut q^{84} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 22q^{87} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 10q^{90} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut -\mathstrut 58q^{93} \) \(\mathstrut +\mathstrut 2q^{94} \) \(\mathstrut +\mathstrut 7q^{96} \) \(\mathstrut -\mathstrut 24q^{97} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(3\) \(x\mathstrut +\mathstrut \) \(1\):

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

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
−1.48119
0.311108
2.17009
−1.48119 3.15633 0.193937 −1.00000 −4.67513 −1.00000 2.67513 6.96239 1.48119
1.2 0.311108 −2.52543 −1.90321 −1.00000 −0.785680 −1.00000 −1.21432 3.37778 −0.311108
1.3 2.17009 −1.63090 2.70928 −1.00000 −3.53919 −1.00000 1.53919 −0.340173 −2.17009
\(n\): 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
\(5\) \(1\)
\(7\) \(1\)
\(229\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\(T_{2}^{3} \) \(\mathstrut -\mathstrut T_{2}^{2} \) \(\mathstrut -\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3}^{3} \) \(\mathstrut +\mathstrut T_{3}^{2} \) \(\mathstrut -\mathstrut 9 T_{3} \) \(\mathstrut -\mathstrut 13 \)