Properties

Label 287.2.a.c
Level 287
Weight 2
Character orbit 287.a
Self dual Yes
Analytic conductor 2.292
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 287 = 7 \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 287.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + 2 q^{5} + ( -3 + \beta_{1} ) q^{6} + q^{7} + \beta_{2} q^{8} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + 2 q^{5} + ( -3 + \beta_{1} ) q^{6} + q^{7} + \beta_{2} q^{8} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{9} + 2 \beta_{1} q^{10} -2 q^{11} + ( 3 - \beta_{1} - \beta_{2} ) q^{12} + ( 3 - \beta_{2} ) q^{13} + \beta_{1} q^{14} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{15} + ( -2 + \beta_{1} - \beta_{2} ) q^{16} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{18} + ( 4 + \beta_{1} ) q^{19} + ( 2 + 2 \beta_{2} ) q^{20} + ( -\beta_{1} + \beta_{2} ) q^{21} -2 \beta_{1} q^{22} + ( -4 + \beta_{1} - \beta_{2} ) q^{23} + ( 3 - 2 \beta_{2} ) q^{24} - q^{25} + ( 2 \beta_{1} - \beta_{2} ) q^{26} + ( -3 - \beta_{1} + 4 \beta_{2} ) q^{27} + ( 1 + \beta_{2} ) q^{28} + ( 4 - 2 \beta_{1} ) q^{29} + ( -6 + 2 \beta_{1} ) q^{30} -2 \beta_{1} q^{31} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{34} + 2 q^{35} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 1 + 5 \beta_{2} ) q^{37} + ( 3 + 4 \beta_{1} + \beta_{2} ) q^{38} + ( -3 - 3 \beta_{1} + 5 \beta_{2} ) q^{39} + 2 \beta_{2} q^{40} - q^{41} + ( -3 + \beta_{1} ) q^{42} + ( 4 - 3 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -2 - 2 \beta_{2} ) q^{44} + ( 6 - 2 \beta_{1} - 4 \beta_{2} ) q^{45} + ( 3 - 5 \beta_{1} ) q^{46} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -6 + 3 \beta_{1} ) q^{48} + q^{49} -\beta_{1} q^{50} + ( -9 + \beta_{1} + 4 \beta_{2} ) q^{51} + ( -\beta_{1} + 3 \beta_{2} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{54} -4 q^{55} + \beta_{2} q^{56} + ( -3 - 3 \beta_{1} + 4 \beta_{2} ) q^{57} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{58} -4 q^{59} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{60} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -6 - 2 \beta_{2} ) q^{62} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{63} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{64} + ( 6 - 2 \beta_{2} ) q^{65} + ( 6 - 2 \beta_{1} ) q^{66} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -6 + \beta_{2} ) q^{68} + ( -6 + 5 \beta_{1} - 2 \beta_{2} ) q^{69} + 2 \beta_{1} q^{70} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -6 - 3 \beta_{1} + 4 \beta_{2} ) q^{72} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 6 \beta_{1} + 5 \beta_{2} ) q^{74} + ( \beta_{1} - \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} + 5 \beta_{2} ) q^{76} -2 q^{77} + ( -9 + 2 \beta_{1} + 2 \beta_{2} ) q^{78} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{80} + ( 6 + 5 \beta_{1} - 5 \beta_{2} ) q^{81} -\beta_{1} q^{82} + ( -2 + 4 \beta_{1} ) q^{83} + ( 3 - \beta_{1} - \beta_{2} ) q^{84} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{85} + ( -9 - 7 \beta_{2} ) q^{86} + ( 6 - 6 \beta_{1} + 4 \beta_{2} ) q^{87} -2 \beta_{2} q^{88} + ( -1 + 8 \beta_{1} - \beta_{2} ) q^{89} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{90} + ( 3 - \beta_{2} ) q^{91} + ( -7 + \beta_{1} - 3 \beta_{2} ) q^{92} + ( 6 - 2 \beta_{1} ) q^{93} + ( 6 - 6 \beta_{1} - \beta_{2} ) q^{94} + ( 8 + 2 \beta_{1} ) q^{95} + ( 3 - 6 \beta_{1} + 7 \beta_{2} ) q^{96} + ( 10 - 3 \beta_{1} + \beta_{2} ) q^{97} + \beta_{1} q^{98} + ( -6 + 2 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - q^{3} + 3q^{4} + 6q^{5} - 8q^{6} + 3q^{7} + 8q^{9} + O(q^{10}) \) \( 3q + q^{2} - q^{3} + 3q^{4} + 6q^{5} - 8q^{6} + 3q^{7} + 8q^{9} + 2q^{10} - 6q^{11} + 8q^{12} + 9q^{13} + q^{14} - 2q^{15} - 5q^{16} + q^{17} - 8q^{18} + 13q^{19} + 6q^{20} - q^{21} - 2q^{22} - 11q^{23} + 9q^{24} - 3q^{25} + 2q^{26} - 10q^{27} + 3q^{28} + 10q^{29} - 16q^{30} - 2q^{31} + 6q^{32} + 2q^{33} + 7q^{34} + 6q^{35} - 13q^{36} + 3q^{37} + 13q^{38} - 12q^{39} - 3q^{41} - 8q^{42} + 9q^{43} - 6q^{44} + 16q^{45} + 4q^{46} - 7q^{47} - 15q^{48} + 3q^{49} - q^{50} - 26q^{51} - q^{52} + 2q^{53} - 8q^{54} - 12q^{55} - 12q^{57} - 14q^{58} - 12q^{59} + 16q^{60} - 4q^{61} - 18q^{62} + 8q^{63} - 16q^{64} + 18q^{65} + 16q^{66} + 10q^{67} - 18q^{68} - 13q^{69} + 2q^{70} - 14q^{71} - 21q^{72} - 4q^{73} + 6q^{74} + q^{75} + 14q^{76} - 6q^{77} - 25q^{78} + 14q^{79} - 10q^{80} + 23q^{81} - q^{82} - 2q^{83} + 8q^{84} + 2q^{85} - 27q^{86} + 12q^{87} + 5q^{89} - 16q^{90} + 9q^{91} - 20q^{92} + 16q^{93} + 12q^{94} + 26q^{95} + 3q^{96} + 27q^{97} + q^{98} - 16q^{99} + O(q^{100}) \)

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

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

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
−1.91223
0.713538
2.19869
−1.91223 2.56885 1.65662 2.00000 −4.91223 1.00000 0.656620 3.59899 −3.82446
1.2 0.713538 −3.20440 −1.49086 2.00000 −2.28646 1.00000 −2.49086 7.26819 1.42708
1.3 2.19869 −0.364448 2.83424 2.00000 −0.801309 1.00000 1.83424 −2.86718 4.39738
\(n\): 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
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(287))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 3 \)
\( T_{3}^{3} + T_{3}^{2} - 8 T_{3} - 3 \)