Properties

Label 287.1.d.b
Level 287
Weight 1
Character orbit 287.d
Self dual yes
Analytic conductor 0.143
Analytic rank 0
Dimension 3
Projective image \(D_{7}\)
CM discriminant -287
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 287 = 7 \cdot 41 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 287.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.143231658626\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.23639903.1
Artin image $D_{14}$
Artin field Galois closure of 14.2.22912645567825769.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( 1 - \beta ) q^{6} - q^{7} + ( -1 + \beta ) q^{8} + ( -1 + \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( 1 - \beta ) q^{6} - q^{7} + ( -1 + \beta ) q^{8} + ( -1 + \beta^{2} ) q^{9} + ( \beta - \beta^{2} ) q^{12} + ( -1 - \beta + \beta^{2} ) q^{13} + ( -1 - \beta + \beta^{2} ) q^{14} + ( -1 - \beta + \beta^{2} ) q^{16} + ( 2 - \beta^{2} ) q^{17} - q^{18} + ( 2 - \beta^{2} ) q^{19} -\beta q^{21} -\beta q^{23} + ( -\beta + \beta^{2} ) q^{24} + q^{25} + ( -2 + \beta ) q^{26} + ( -1 + \beta^{2} ) q^{27} + ( -1 + \beta ) q^{28} - q^{32} + ( 2 + \beta - \beta^{2} ) q^{34} -\beta q^{36} + ( 1 + \beta - \beta^{2} ) q^{37} + ( 2 + \beta - \beta^{2} ) q^{38} + ( -1 + \beta ) q^{39} - q^{41} + ( -1 + \beta ) q^{42} + ( -2 + \beta^{2} ) q^{43} + ( -1 + \beta ) q^{46} + ( -1 - \beta + \beta^{2} ) q^{47} + ( -1 + \beta ) q^{48} + q^{49} + ( 1 + \beta - \beta^{2} ) q^{50} + ( 1 - \beta^{2} ) q^{51} + ( -2 \beta + \beta^{2} ) q^{52} - q^{54} + ( 1 - \beta ) q^{56} + ( 1 - \beta^{2} ) q^{57} + ( 1 - \beta^{2} ) q^{63} + q^{68} -\beta^{2} q^{69} + \beta q^{72} + ( 2 - \beta ) q^{74} + \beta q^{75} + q^{76} + ( -2 \beta + \beta^{2} ) q^{78} + \beta q^{81} + ( -1 - \beta + \beta^{2} ) q^{82} + ( -\beta + \beta^{2} ) q^{84} + ( -2 - \beta + \beta^{2} ) q^{86} + ( -1 - \beta + \beta^{2} ) q^{89} + ( 1 + \beta - \beta^{2} ) q^{91} + ( -\beta + \beta^{2} ) q^{92} + ( -2 + \beta ) q^{94} -\beta q^{96} + \beta q^{97} + ( 1 + \beta - \beta^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + q^{3} + 2q^{4} + 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 3q - q^{2} + q^{3} + 2q^{4} + 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} - 4q^{12} + q^{13} + q^{14} + q^{16} + q^{17} - 3q^{18} + q^{19} - q^{21} - q^{23} + 4q^{24} + 3q^{25} - 5q^{26} + 2q^{27} - 2q^{28} - 3q^{32} + 2q^{34} - q^{36} - q^{37} + 2q^{38} - 2q^{39} - 3q^{41} - 2q^{42} - q^{43} - 2q^{46} + q^{47} - 2q^{48} + 3q^{49} - q^{50} - 2q^{51} + 3q^{52} - 3q^{54} + 2q^{56} - 2q^{57} - 2q^{63} + 3q^{68} - 5q^{69} + q^{72} + 5q^{74} + q^{75} + 3q^{76} + 3q^{78} + q^{81} + q^{82} + 4q^{84} - 2q^{86} + q^{89} - q^{91} + 4q^{92} - 5q^{94} - q^{96} + q^{97} - q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(206\) \(211\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
286.1
−1.24698
1.80194
0.445042
−1.80194 −1.24698 2.24698 0 2.24698 −1.00000 −2.24698 0.554958 0
286.2 −0.445042 1.80194 −0.801938 0 −0.801938 −1.00000 0.801938 2.24698 0
286.3 1.24698 0.445042 0.554958 0 0.554958 −1.00000 −0.554958 −0.801938 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by \(\Q(\sqrt{-287}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.1.d.b yes 3
3.b odd 2 1 2583.1.f.a 3
7.b odd 2 1 287.1.d.a 3
7.c even 3 2 2009.1.i.a 6
7.d odd 6 2 2009.1.i.b 6
21.c even 2 1 2583.1.f.b 3
41.b even 2 1 287.1.d.a 3
123.b odd 2 1 2583.1.f.b 3
287.d odd 2 1 CM 287.1.d.b yes 3
287.i odd 6 2 2009.1.i.a 6
287.j even 6 2 2009.1.i.b 6
861.e even 2 1 2583.1.f.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.1.d.a 3 7.b odd 2 1
287.1.d.a 3 41.b even 2 1
287.1.d.b yes 3 1.a even 1 1 trivial
287.1.d.b yes 3 287.d odd 2 1 CM
2009.1.i.a 6 7.c even 3 2
2009.1.i.a 6 287.i odd 6 2
2009.1.i.b 6 7.d odd 6 2
2009.1.i.b 6 287.j even 6 2
2583.1.f.a 3 3.b odd 2 1
2583.1.f.a 3 861.e even 2 1
2583.1.f.b 3 21.c even 2 1
2583.1.f.b 3 123.b odd 2 1

Hecke kernels

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