Properties

Label 287.2.a.a
Level 287
Weight 2
Character orbit 287.a
Self dual Yes
Analytic conductor 2.292
Analytic rank 1
Dimension 2
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.61803 −2.61803 0.618034 0.618034 4.23607 1.00000 2.23607 3.85410 −1.00000
1.2 0.618034 −0.381966 −1.61803 −1.61803 −0.236068 1.00000 −2.23607 −2.85410 −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
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

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

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