Properties

Label 15.2.a.a
Level $15$
Weight $2$
Character orbit 15.a
Self dual yes
Analytic conductor $0.120$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.119775603032\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} - q^{20} + 4 q^{22} - 3 q^{24} + q^{25} + 2 q^{26} - q^{27} - 2 q^{29} + q^{30} - 5 q^{32} + 4 q^{33} - 2 q^{34} - q^{36} - 10 q^{37} - 4 q^{38} + 2 q^{39} + 3 q^{40} + 10 q^{41} + 4 q^{43} + 4 q^{44} + q^{45} + 8 q^{47} + q^{48} - 7 q^{49} - q^{50} - 2 q^{51} + 2 q^{52} - 10 q^{53} + q^{54} - 4 q^{55} - 4 q^{57} + 2 q^{58} - 4 q^{59} + q^{60} - 2 q^{61} + 7 q^{64} - 2 q^{65} - 4 q^{66} + 12 q^{67} - 2 q^{68} - 8 q^{71} + 3 q^{72} + 10 q^{73} + 10 q^{74} - q^{75} - 4 q^{76} - 2 q^{78} - q^{80} + q^{81} - 10 q^{82} + 12 q^{83} + 2 q^{85} - 4 q^{86} + 2 q^{87} - 12 q^{88} - 6 q^{89} - q^{90} - 8 q^{94} + 4 q^{95} + 5 q^{96} + 2 q^{97} + 7 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)\eta(3z)\eta(5z)\eta(15z)=q\prod_{n=1}^\infty(1 - q^{n})^{}(1 - q^{3n})^{}(1 - q^{5n})^{}(1 - q^{15n})^{}\)

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
−1.00000 −1.00000 −1.00000 1.00000 1.00000 0 3.00000 1.00000 −1.00000
\(n\): 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 15.2.a.a 1
3.b odd 2 1 45.2.a.a 1
4.b odd 2 1 240.2.a.d 1
5.b even 2 1 75.2.a.b 1
5.c odd 4 2 75.2.b.b 2
7.b odd 2 1 735.2.a.c 1
7.c even 3 2 735.2.i.e 2
7.d odd 6 2 735.2.i.d 2
8.b even 2 1 960.2.a.l 1
8.d odd 2 1 960.2.a.a 1
9.c even 3 2 405.2.e.f 2
9.d odd 6 2 405.2.e.c 2
11.b odd 2 1 1815.2.a.d 1
12.b even 2 1 720.2.a.c 1
13.b even 2 1 2535.2.a.j 1
15.d odd 2 1 225.2.a.b 1
15.e even 4 2 225.2.b.b 2
16.e even 4 2 3840.2.k.m 2
16.f odd 4 2 3840.2.k.r 2
17.b even 2 1 4335.2.a.c 1
19.b odd 2 1 5415.2.a.j 1
20.d odd 2 1 1200.2.a.e 1
20.e even 4 2 1200.2.f.h 2
21.c even 2 1 2205.2.a.i 1
23.b odd 2 1 7935.2.a.d 1
24.f even 2 1 2880.2.a.bc 1
24.h odd 2 1 2880.2.a.y 1
33.d even 2 1 5445.2.a.c 1
35.c odd 2 1 3675.2.a.j 1
39.d odd 2 1 7605.2.a.g 1
40.e odd 2 1 4800.2.a.bz 1
40.f even 2 1 4800.2.a.t 1
40.i odd 4 2 4800.2.f.bf 2
40.k even 4 2 4800.2.f.c 2
55.d odd 2 1 9075.2.a.g 1
60.h even 2 1 3600.2.a.u 1
60.l odd 4 2 3600.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 1.a even 1 1 trivial
45.2.a.a 1 3.b odd 2 1
75.2.a.b 1 5.b even 2 1
75.2.b.b 2 5.c odd 4 2
225.2.a.b 1 15.d odd 2 1
225.2.b.b 2 15.e even 4 2
240.2.a.d 1 4.b odd 2 1
405.2.e.c 2 9.d odd 6 2
405.2.e.f 2 9.c even 3 2
720.2.a.c 1 12.b even 2 1
735.2.a.c 1 7.b odd 2 1
735.2.i.d 2 7.d odd 6 2
735.2.i.e 2 7.c even 3 2
960.2.a.a 1 8.d odd 2 1
960.2.a.l 1 8.b even 2 1
1200.2.a.e 1 20.d odd 2 1
1200.2.f.h 2 20.e even 4 2
1815.2.a.d 1 11.b odd 2 1
2205.2.a.i 1 21.c even 2 1
2535.2.a.j 1 13.b even 2 1
2880.2.a.y 1 24.h odd 2 1
2880.2.a.bc 1 24.f even 2 1
3600.2.a.u 1 60.h even 2 1
3600.2.f.e 2 60.l odd 4 2
3675.2.a.j 1 35.c odd 2 1
3840.2.k.m 2 16.e even 4 2
3840.2.k.r 2 16.f odd 4 2
4335.2.a.c 1 17.b even 2 1
4800.2.a.t 1 40.f even 2 1
4800.2.a.bz 1 40.e odd 2 1
4800.2.f.c 2 40.k even 4 2
4800.2.f.bf 2 40.i odd 4 2
5415.2.a.j 1 19.b odd 2 1
5445.2.a.c 1 33.d even 2 1
7605.2.a.g 1 39.d odd 2 1
7935.2.a.d 1 23.b odd 2 1
9075.2.a.g 1 55.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\).

Hecke characteristic polynomials

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