Properties

Label 15.4.a.b
Level 15
Weight 4
Character orbit 15.a
Self dual yes
Analytic conductor 0.885
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.885028650086\)
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

\(f(q)\) \(=\) \( q + 3q^{2} - 3q^{3} + q^{4} - 5q^{5} - 9q^{6} + 20q^{7} - 21q^{8} + 9q^{9} + O(q^{10}) \) \( q + 3q^{2} - 3q^{3} + q^{4} - 5q^{5} - 9q^{6} + 20q^{7} - 21q^{8} + 9q^{9} - 15q^{10} - 24q^{11} - 3q^{12} + 74q^{13} + 60q^{14} + 15q^{15} - 71q^{16} + 54q^{17} + 27q^{18} - 124q^{19} - 5q^{20} - 60q^{21} - 72q^{22} - 120q^{23} + 63q^{24} + 25q^{25} + 222q^{26} - 27q^{27} + 20q^{28} - 78q^{29} + 45q^{30} + 200q^{31} - 45q^{32} + 72q^{33} + 162q^{34} - 100q^{35} + 9q^{36} - 70q^{37} - 372q^{38} - 222q^{39} + 105q^{40} + 330q^{41} - 180q^{42} + 92q^{43} - 24q^{44} - 45q^{45} - 360q^{46} - 24q^{47} + 213q^{48} + 57q^{49} + 75q^{50} - 162q^{51} + 74q^{52} + 450q^{53} - 81q^{54} + 120q^{55} - 420q^{56} + 372q^{57} - 234q^{58} + 24q^{59} + 15q^{60} - 322q^{61} + 600q^{62} + 180q^{63} + 433q^{64} - 370q^{65} + 216q^{66} - 196q^{67} + 54q^{68} + 360q^{69} - 300q^{70} - 288q^{71} - 189q^{72} - 430q^{73} - 210q^{74} - 75q^{75} - 124q^{76} - 480q^{77} - 666q^{78} - 520q^{79} + 355q^{80} + 81q^{81} + 990q^{82} + 156q^{83} - 60q^{84} - 270q^{85} + 276q^{86} + 234q^{87} + 504q^{88} + 1026q^{89} - 135q^{90} + 1480q^{91} - 120q^{92} - 600q^{93} - 72q^{94} + 620q^{95} + 135q^{96} - 286q^{97} + 171q^{98} - 216q^{99} + O(q^{100}) \)

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
3.00000 −3.00000 1.00000 −5.00000 −9.00000 20.0000 −21.0000 9.00000 −15.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 15.4.a.b 1
3.b odd 2 1 45.4.a.b 1
4.b odd 2 1 240.4.a.f 1
5.b even 2 1 75.4.a.a 1
5.c odd 4 2 75.4.b.a 2
7.b odd 2 1 735.4.a.i 1
8.b even 2 1 960.4.a.bi 1
8.d odd 2 1 960.4.a.l 1
9.c even 3 2 405.4.e.d 2
9.d odd 6 2 405.4.e.k 2
11.b odd 2 1 1815.4.a.a 1
12.b even 2 1 720.4.a.r 1
15.d odd 2 1 225.4.a.g 1
15.e even 4 2 225.4.b.d 2
20.d odd 2 1 1200.4.a.o 1
20.e even 4 2 1200.4.f.m 2
21.c even 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 1.a even 1 1 trivial
45.4.a.b 1 3.b odd 2 1
75.4.a.a 1 5.b even 2 1
75.4.b.a 2 5.c odd 4 2
225.4.a.g 1 15.d odd 2 1
225.4.b.d 2 15.e even 4 2
240.4.a.f 1 4.b odd 2 1
405.4.e.d 2 9.c even 3 2
405.4.e.k 2 9.d odd 6 2
720.4.a.r 1 12.b even 2 1
735.4.a.i 1 7.b odd 2 1
960.4.a.l 1 8.d odd 2 1
960.4.a.bi 1 8.b even 2 1
1200.4.a.o 1 20.d odd 2 1
1200.4.f.m 2 20.e even 4 2
1815.4.a.a 1 11.b odd 2 1
2205.4.a.c 1 21.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(15))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 8 T^{2} \)
$3$ \( 1 + 3 T \)
$5$ \( 1 + 5 T \)
$7$ \( 1 - 20 T + 343 T^{2} \)
$11$ \( 1 + 24 T + 1331 T^{2} \)
$13$ \( 1 - 74 T + 2197 T^{2} \)
$17$ \( 1 - 54 T + 4913 T^{2} \)
$19$ \( 1 + 124 T + 6859 T^{2} \)
$23$ \( 1 + 120 T + 12167 T^{2} \)
$29$ \( 1 + 78 T + 24389 T^{2} \)
$31$ \( 1 - 200 T + 29791 T^{2} \)
$37$ \( 1 + 70 T + 50653 T^{2} \)
$41$ \( 1 - 330 T + 68921 T^{2} \)
$43$ \( 1 - 92 T + 79507 T^{2} \)
$47$ \( 1 + 24 T + 103823 T^{2} \)
$53$ \( 1 - 450 T + 148877 T^{2} \)
$59$ \( 1 - 24 T + 205379 T^{2} \)
$61$ \( 1 + 322 T + 226981 T^{2} \)
$67$ \( 1 + 196 T + 300763 T^{2} \)
$71$ \( 1 + 288 T + 357911 T^{2} \)
$73$ \( 1 + 430 T + 389017 T^{2} \)
$79$ \( 1 + 520 T + 493039 T^{2} \)
$83$ \( 1 - 156 T + 571787 T^{2} \)
$89$ \( 1 - 1026 T + 704969 T^{2} \)
$97$ \( 1 + 286 T + 912673 T^{2} \)
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