Properties

Label 135.2.a.b
Level $135$
Weight $2$
Character orbit 135.a
Self dual yes
Analytic conductor $1.078$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7} + 2 q^{10} + 2 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 4 q^{22} - 6 q^{23} + q^{25} - 10 q^{26} - 6 q^{28} - 2 q^{29} - 8 q^{32} + 16 q^{34} - 3 q^{35} + 5 q^{37} + 2 q^{38} + 10 q^{41} + 4 q^{43} + 4 q^{44} - 12 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{50} - 10 q^{52} + 2 q^{53} + 2 q^{55} - 4 q^{58} + 8 q^{59} + 7 q^{61} - 8 q^{64} - 5 q^{65} - 9 q^{67} + 16 q^{68} - 6 q^{70} - 2 q^{71} - 5 q^{73} + 10 q^{74} + 2 q^{76} - 6 q^{77} - 3 q^{79} - 4 q^{80} + 20 q^{82} - 6 q^{83} + 8 q^{85} + 8 q^{86} + 12 q^{89} + 15 q^{91} - 12 q^{92} - 8 q^{94} + q^{95} - 13 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 0 2.00000 1.00000 0 −3.00000 0 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.a.b yes 1
3.b odd 2 1 135.2.a.a 1
4.b odd 2 1 2160.2.a.v 1
5.b even 2 1 675.2.a.a 1
5.c odd 4 2 675.2.b.b 2
7.b odd 2 1 6615.2.a.j 1
8.b even 2 1 8640.2.a.c 1
8.d odd 2 1 8640.2.a.bb 1
9.c even 3 2 405.2.e.b 2
9.d odd 6 2 405.2.e.h 2
12.b even 2 1 2160.2.a.j 1
15.d odd 2 1 675.2.a.i 1
15.e even 4 2 675.2.b.a 2
21.c even 2 1 6615.2.a.a 1
24.f even 2 1 8640.2.a.ce 1
24.h odd 2 1 8640.2.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.a 1 3.b odd 2 1
135.2.a.b yes 1 1.a even 1 1 trivial
405.2.e.b 2 9.c even 3 2
405.2.e.h 2 9.d odd 6 2
675.2.a.a 1 5.b even 2 1
675.2.a.i 1 15.d odd 2 1
675.2.b.a 2 15.e even 4 2
675.2.b.b 2 5.c odd 4 2
2160.2.a.j 1 12.b even 2 1
2160.2.a.v 1 4.b odd 2 1
6615.2.a.a 1 21.c even 2 1
6615.2.a.j 1 7.b odd 2 1
8640.2.a.c 1 8.b even 2 1
8640.2.a.bb 1 8.d odd 2 1
8640.2.a.bh 1 24.h odd 2 1
8640.2.a.ce 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(135))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 8 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 5 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 4 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T - 7 \) Copy content Toggle raw display
$67$ \( T + 9 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T + 5 \) Copy content Toggle raw display
$79$ \( T + 3 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 13 \) Copy content Toggle raw display
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