Properties

Label 135.1.d.b
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 discriminant -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 \(2\), degree \(1\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.135.1
Artin image $D_6$
Artin field Galois closure of 6.0.54675.1

$q$-expansion

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.1.d.b yes 1
3.b odd 2 1 135.1.d.a 1
4.b odd 2 1 2160.1.c.a 1
5.b even 2 1 135.1.d.a 1
5.c odd 4 2 675.1.c.c 2
9.c even 3 2 405.1.h.a 2
9.d odd 6 2 405.1.h.b 2
12.b even 2 1 2160.1.c.b 1
15.d odd 2 1 CM 135.1.d.b yes 1
15.e even 4 2 675.1.c.c 2
20.d odd 2 1 2160.1.c.b 1
27.e even 9 6 3645.1.n.e 6
27.f odd 18 6 3645.1.n.d 6
45.h odd 6 2 405.1.h.a 2
45.j even 6 2 405.1.h.b 2
45.k odd 12 4 2025.1.j.c 4
45.l even 12 4 2025.1.j.c 4
60.h even 2 1 2160.1.c.a 1
135.n odd 18 6 3645.1.n.e 6
135.p even 18 6 3645.1.n.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 3.b odd 2 1
135.1.d.a 1 5.b even 2 1
135.1.d.b yes 1 1.a even 1 1 trivial
135.1.d.b yes 1 15.d odd 2 1 CM
405.1.h.a 2 9.c even 3 2
405.1.h.a 2 45.h odd 6 2
405.1.h.b 2 9.d odd 6 2
405.1.h.b 2 45.j even 6 2
675.1.c.c 2 5.c odd 4 2
675.1.c.c 2 15.e even 4 2
2025.1.j.c 4 45.k odd 12 4
2025.1.j.c 4 45.l even 12 4
2160.1.c.a 1 4.b odd 2 1
2160.1.c.a 1 60.h even 2 1
2160.1.c.b 1 12.b even 2 1
2160.1.c.b 1 20.d odd 2 1
3645.1.n.d 6 27.f odd 18 6
3645.1.n.d 6 135.p even 18 6
3645.1.n.e 6 27.e even 9 6
3645.1.n.e 6 135.n odd 18 6

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ 1
$5$ \( 1 + T \)
$7$ \( ( 1 - T )( 1 + T ) \)
$11$ \( ( 1 - T )( 1 + T ) \)
$13$ \( ( 1 - T )( 1 + T ) \)
$17$ \( 1 - T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( 1 + T + T^{2} \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( ( 1 + T )^{2} \)
$53$ \( 1 - T + T^{2} \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( 1 + T + T^{2} \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( ( 1 - T )( 1 + T ) \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( 1 + T + T^{2} \)
$83$ \( 1 - T + T^{2} \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( ( 1 - T )( 1 + T ) \)
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