Properties

Label 135.1.d.a
Level 135
Weight 1
Character orbit 135.d
Self dual Yes
Analytic conductor 0.067
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -15
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 135.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0673737767055\)
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 3.1.135.1
Artin image size \(6\)
Artin image $S_3$
Artin field Galois closure of 3.1.135.1

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut q^{34} \) \(\mathstrut +\mathstrut q^{38} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut q^{62} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
0
−1.00000 0 0 1.00000 0 0 1.00000 0 −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
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(135, [\chi])\).