Properties

Label 31.1.b.a
Level 31
Weight 1
Character orbit 31.b
Self dual Yes
Analytic conductor 0.015
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -31
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 31.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0154710153916\)
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 3.1.31.1
Artin image size \(6\)
Artin image $S_3$
Artin field Galois closure of 3.1.31.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9} + q^{10} + q^{14} - q^{16} - q^{18} - q^{19} + q^{31} + q^{35} + q^{38} - q^{40} - q^{41} - q^{45} + 2q^{47} - q^{56} - q^{59} - q^{62} - q^{63} + q^{64} + 2q^{67} - q^{70} - q^{71} + q^{72} + q^{80} + q^{81} + q^{82} + q^{90} - 2q^{94} + q^{95} - q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
0
−1.00000 0 0 −1.00000 0 −1.00000 1.00000 1.00000 1.00000
\(n\): 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
31.b Odd 1 CM by \(\Q(\sqrt{-31}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(31, [\chi])\).