Properties

Label 287.2.e.b
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: \(\Q(\sqrt{-3}, \sqrt{13})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 4 x^{2} + 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} - 4 \nu - 3 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 7 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{3} - 7\)

Character Values

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

\(n\) \(206\) \(211\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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
1.15139 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 1.12824i
0 −1.15139 + 1.99426i 1.00000 1.73205i −0.651388 1.12824i 0 −0.500000 2.59808i 0 −1.15139 1.99426i 0
165.2 0 0.651388 1.12824i 1.00000 1.73205i 1.15139 + 1.99426i 0 −0.500000 2.59808i 0 0.651388 + 1.12824i 0
247.1 0 −1.15139 1.99426i 1.00000 + 1.73205i −0.651388 + 1.12824i 0 −0.500000 + 2.59808i 0 −1.15139 + 1.99426i 0
247.2 0 0.651388 + 1.12824i 1.00000 + 1.73205i 1.15139 1.99426i 0 −0.500000 + 2.59808i 0 0.651388 1.12824i 0
\(n\): 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 \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\).