Properties

Label 140.1.h.b
Level 140
Weight 1
Character orbit 140.h
Self dual Yes
Analytic conductor 0.070
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -35
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 140.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0698691017686\)
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 3.1.140.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.2.98000.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - q^{7} + O(q^{10}) \) \( q + q^{3} - q^{5} - q^{7} - q^{11} + q^{13} - q^{15} + q^{17} - q^{21} + q^{25} - q^{27} - q^{29} - q^{33} + q^{35} + q^{39} + q^{47} + q^{49} + q^{51} + q^{55} - q^{65} + 2q^{71} - 2q^{73} + q^{75} + q^{77} - q^{79} - q^{81} - 2q^{83} - q^{85} - q^{87} - q^{91} + q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0
0 1.00000 0 −1.00000 0 −1.00000 0 0 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
35.c Odd 1 CM by \(\Q(\sqrt{-35}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(140, [\chi])\).