Properties

Label 87.1.d.a
Level 87
Weight 1
Character orbit 87.d
Self dual Yes
Analytic conductor 0.043
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -87
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 87 = 3 \cdot 29 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 87.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0434186560991\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.87.1
Artin image size \(6\)
Artin image $S_3$
Artin field Galois closure of 3.1.87.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} - q^{11} - q^{13} + q^{14} - q^{16} - q^{17} - q^{18} - q^{21} + q^{22} + q^{24} + q^{25} + q^{26} + q^{27} + q^{29} - q^{33} + q^{34} - q^{39} + 2q^{41} + q^{42} - q^{47} - q^{48} - q^{50} - q^{51} - q^{54} - q^{56} - q^{58} - q^{63} + q^{64} + q^{66} - q^{67} + q^{72} + q^{75} + q^{77} + q^{78} + q^{81} - 2q^{82} + q^{87} - q^{88} - q^{89} + q^{91} + q^{94} - q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
0
−1.00000 1.00000 0 0 −1.00000 −1.00000 1.00000 1.00000 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
87.d Odd 1 CM by \(\Q(\sqrt{-87}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(87, [\chi])\).