Properties

Label 15.4.a.a
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut +\mathstrut 52q^{11} \) \(\mathstrut -\mathstrut 21q^{12} \) \(\mathstrut +\mathstrut 22q^{13} \) \(\mathstrut -\mathstrut 24q^{14} \) \(\mathstrut +\mathstrut 15q^{15} \) \(\mathstrut +\mathstrut 41q^{16} \) \(\mathstrut -\mathstrut 14q^{17} \) \(\mathstrut +\mathstrut 9q^{18} \) \(\mathstrut -\mathstrut 20q^{19} \) \(\mathstrut -\mathstrut 35q^{20} \) \(\mathstrut -\mathstrut 72q^{21} \) \(\mathstrut +\mathstrut 52q^{22} \) \(\mathstrut -\mathstrut 168q^{23} \) \(\mathstrut -\mathstrut 45q^{24} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 22q^{26} \) \(\mathstrut +\mathstrut 27q^{27} \) \(\mathstrut +\mathstrut 168q^{28} \) \(\mathstrut +\mathstrut 230q^{29} \) \(\mathstrut +\mathstrut 15q^{30} \) \(\mathstrut -\mathstrut 288q^{31} \) \(\mathstrut +\mathstrut 161q^{32} \) \(\mathstrut +\mathstrut 156q^{33} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut -\mathstrut 120q^{35} \) \(\mathstrut -\mathstrut 63q^{36} \) \(\mathstrut -\mathstrut 34q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut +\mathstrut 66q^{39} \) \(\mathstrut -\mathstrut 75q^{40} \) \(\mathstrut +\mathstrut 122q^{41} \) \(\mathstrut -\mathstrut 72q^{42} \) \(\mathstrut -\mathstrut 188q^{43} \) \(\mathstrut -\mathstrut 364q^{44} \) \(\mathstrut +\mathstrut 45q^{45} \) \(\mathstrut -\mathstrut 168q^{46} \) \(\mathstrut +\mathstrut 256q^{47} \) \(\mathstrut +\mathstrut 123q^{48} \) \(\mathstrut +\mathstrut 233q^{49} \) \(\mathstrut +\mathstrut 25q^{50} \) \(\mathstrut -\mathstrut 42q^{51} \) \(\mathstrut -\mathstrut 154q^{52} \) \(\mathstrut -\mathstrut 338q^{53} \) \(\mathstrut +\mathstrut 27q^{54} \) \(\mathstrut +\mathstrut 260q^{55} \) \(\mathstrut +\mathstrut 360q^{56} \) \(\mathstrut -\mathstrut 60q^{57} \) \(\mathstrut +\mathstrut 230q^{58} \) \(\mathstrut +\mathstrut 100q^{59} \) \(\mathstrut -\mathstrut 105q^{60} \) \(\mathstrut +\mathstrut 742q^{61} \) \(\mathstrut -\mathstrut 288q^{62} \) \(\mathstrut -\mathstrut 216q^{63} \) \(\mathstrut -\mathstrut 167q^{64} \) \(\mathstrut +\mathstrut 110q^{65} \) \(\mathstrut +\mathstrut 156q^{66} \) \(\mathstrut -\mathstrut 84q^{67} \) \(\mathstrut +\mathstrut 98q^{68} \) \(\mathstrut -\mathstrut 504q^{69} \) \(\mathstrut -\mathstrut 120q^{70} \) \(\mathstrut -\mathstrut 328q^{71} \) \(\mathstrut -\mathstrut 135q^{72} \) \(\mathstrut -\mathstrut 38q^{73} \) \(\mathstrut -\mathstrut 34q^{74} \) \(\mathstrut +\mathstrut 75q^{75} \) \(\mathstrut +\mathstrut 140q^{76} \) \(\mathstrut -\mathstrut 1248q^{77} \) \(\mathstrut +\mathstrut 66q^{78} \) \(\mathstrut -\mathstrut 240q^{79} \) \(\mathstrut +\mathstrut 205q^{80} \) \(\mathstrut +\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 122q^{82} \) \(\mathstrut +\mathstrut 1212q^{83} \) \(\mathstrut +\mathstrut 504q^{84} \) \(\mathstrut -\mathstrut 70q^{85} \) \(\mathstrut -\mathstrut 188q^{86} \) \(\mathstrut +\mathstrut 690q^{87} \) \(\mathstrut -\mathstrut 780q^{88} \) \(\mathstrut +\mathstrut 330q^{89} \) \(\mathstrut +\mathstrut 45q^{90} \) \(\mathstrut -\mathstrut 528q^{91} \) \(\mathstrut +\mathstrut 1176q^{92} \) \(\mathstrut -\mathstrut 864q^{93} \) \(\mathstrut +\mathstrut 256q^{94} \) \(\mathstrut -\mathstrut 100q^{95} \) \(\mathstrut +\mathstrut 483q^{96} \) \(\mathstrut +\mathstrut 866q^{97} \) \(\mathstrut +\mathstrut 233q^{98} \) \(\mathstrut +\mathstrut 468q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000 3.00000 −7.00000 5.00000 3.00000 −24.0000 −15.0000 9.00000 5.00000
\(n\): 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 \(T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(15))\).