Properties

Label 135.4.a.a
Level $135$
Weight $4$
Character orbit 135.a
Self dual yes
Analytic conductor $7.965$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 24 q^{8} + O(q^{10}) \) \( q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 24 q^{8} - 10 q^{10} + 10 q^{11} - 80 q^{13} - 16 q^{16} + 7 q^{17} - 113 q^{19} - 20 q^{20} - 20 q^{22} - 81 q^{23} + 25 q^{25} + 160 q^{26} - 220 q^{29} - 189 q^{31} - 160 q^{32} - 14 q^{34} + 170 q^{37} + 226 q^{38} + 120 q^{40} - 130 q^{41} + 10 q^{43} - 40 q^{44} + 162 q^{46} + 160 q^{47} - 343 q^{49} - 50 q^{50} + 320 q^{52} + 631 q^{53} + 50 q^{55} + 440 q^{58} - 560 q^{59} + 229 q^{61} + 378 q^{62} + 448 q^{64} - 400 q^{65} + 750 q^{67} - 28 q^{68} + 890 q^{71} - 890 q^{73} - 340 q^{74} + 452 q^{76} - 27 q^{79} - 80 q^{80} + 260 q^{82} + 429 q^{83} + 35 q^{85} - 20 q^{86} + 240 q^{88} - 750 q^{89} + 324 q^{92} - 320 q^{94} - 565 q^{95} - 1480 q^{97} + 686 q^{98} + O(q^{100}) \)

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 −4.00000 5.00000 0 0 24.0000 0 −10.0000
\(n\): 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 135.4.a.a 1
3.b odd 2 1 135.4.a.d yes 1
4.b odd 2 1 2160.4.a.n 1
5.b even 2 1 675.4.a.i 1
5.c odd 4 2 675.4.b.d 2
9.c even 3 2 405.4.e.j 2
9.d odd 6 2 405.4.e.e 2
12.b even 2 1 2160.4.a.d 1
15.d odd 2 1 675.4.a.b 1
15.e even 4 2 675.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 1.a even 1 1 trivial
135.4.a.d yes 1 3.b odd 2 1
405.4.e.e 2 9.d odd 6 2
405.4.e.j 2 9.c even 3 2
675.4.a.b 1 15.d odd 2 1
675.4.a.i 1 5.b even 2 1
675.4.b.c 2 15.e even 4 2
675.4.b.d 2 5.c odd 4 2
2160.4.a.d 1 12.b even 2 1
2160.4.a.n 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -5 + T \)
$7$ \( T \)
$11$ \( -10 + T \)
$13$ \( 80 + T \)
$17$ \( -7 + T \)
$19$ \( 113 + T \)
$23$ \( 81 + T \)
$29$ \( 220 + T \)
$31$ \( 189 + T \)
$37$ \( -170 + T \)
$41$ \( 130 + T \)
$43$ \( -10 + T \)
$47$ \( -160 + T \)
$53$ \( -631 + T \)
$59$ \( 560 + T \)
$61$ \( -229 + T \)
$67$ \( -750 + T \)
$71$ \( -890 + T \)
$73$ \( 890 + T \)
$79$ \( 27 + T \)
$83$ \( -429 + T \)
$89$ \( 750 + T \)
$97$ \( 1480 + T \)
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