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

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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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Atkin-Lehner signs

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

Hecke kernels

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