Properties

Label 287.2.e.a
Level 287
Weight 2
Character orbit 287.e
Analytic conductor 2.292
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 287 = 7 \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 287.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
−0.809017 1.40126i −0.500000 + 0.866025i −0.309017 + 0.535233i 0.500000 + 0.866025i 1.61803 −2.00000 + 1.73205i −2.23607 1.00000 + 1.73205i 0.809017 1.40126i
165.2 0.309017 + 0.535233i −0.500000 + 0.866025i 0.809017 1.40126i 0.500000 + 0.866025i −0.618034 −2.00000 + 1.73205i 2.23607 1.00000 + 1.73205i −0.309017 + 0.535233i
247.1 −0.809017 + 1.40126i −0.500000 0.866025i −0.309017 0.535233i 0.500000 0.866025i 1.61803 −2.00000 1.73205i −2.23607 1.00000 1.73205i 0.809017 + 1.40126i
247.2 0.309017 0.535233i −0.500000 0.866025i 0.809017 + 1.40126i 0.500000 0.866025i −0.618034 −2.00000 1.73205i 2.23607 1.00000 1.73205i −0.309017 0.535233i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 2 T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\).