Properties

Label 287.2.r.a
Level 287
Weight 2
Character orbit 287.r
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.r (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(206\) \(211\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.500000 + 1.86603i −0.500000 0.866025i −1.50000 0.866025i 1.36603 1.36603i 2.50000 0.866025i 3.00000i −0.633975 0.366025i 0.866025 + 1.50000i
32.1 −0.866025 + 0.500000i −0.500000 1.86603i −0.500000 + 0.866025i −1.50000 + 0.866025i 1.36603 + 1.36603i 2.50000 + 0.866025i 3.00000i −0.633975 + 0.366025i 0.866025 1.50000i
114.1 0.866025 + 0.500000i −0.500000 0.133975i −0.500000 0.866025i −1.50000 0.866025i −0.366025 0.366025i 2.50000 0.866025i 3.00000i −2.36603 1.36603i −0.866025 1.50000i
214.1 0.866025 0.500000i −0.500000 + 0.133975i −0.500000 + 0.866025i −1.50000 + 0.866025i −0.366025 + 0.366025i 2.50000 + 0.866025i 3.00000i −2.36603 + 1.36603i −0.866025 + 1.50000i
\(n\): 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
287.r Even 1 yes

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \)
\( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} + 4 T_{3} + 1 \)