Properties

Label 15.8.a.b
Level 15
Weight 8
Character orbit 15.a
Self dual Yes
Analytic conductor 4.686
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.68577538226\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 41q^{4} \) \(\mathstrut -\mathstrut 125q^{5} \) \(\mathstrut +\mathstrut 351q^{6} \) \(\mathstrut +\mathstrut 1380q^{7} \) \(\mathstrut +\mathstrut 1131q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 41q^{4} \) \(\mathstrut -\mathstrut 125q^{5} \) \(\mathstrut +\mathstrut 351q^{6} \) \(\mathstrut +\mathstrut 1380q^{7} \) \(\mathstrut +\mathstrut 1131q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut 1625q^{10} \) \(\mathstrut -\mathstrut 3304q^{11} \) \(\mathstrut -\mathstrut 1107q^{12} \) \(\mathstrut +\mathstrut 8506q^{13} \) \(\mathstrut -\mathstrut 17940q^{14} \) \(\mathstrut +\mathstrut 3375q^{15} \) \(\mathstrut -\mathstrut 19951q^{16} \) \(\mathstrut -\mathstrut 9994q^{17} \) \(\mathstrut -\mathstrut 9477q^{18} \) \(\mathstrut +\mathstrut 41236q^{19} \) \(\mathstrut -\mathstrut 5125q^{20} \) \(\mathstrut -\mathstrut 37260q^{21} \) \(\mathstrut +\mathstrut 42952q^{22} \) \(\mathstrut +\mathstrut 84120q^{23} \) \(\mathstrut -\mathstrut 30537q^{24} \) \(\mathstrut +\mathstrut 15625q^{25} \) \(\mathstrut -\mathstrut 110578q^{26} \) \(\mathstrut -\mathstrut 19683q^{27} \) \(\mathstrut +\mathstrut 56580q^{28} \) \(\mathstrut +\mathstrut 132802q^{29} \) \(\mathstrut -\mathstrut 43875q^{30} \) \(\mathstrut -\mathstrut 55800q^{31} \) \(\mathstrut +\mathstrut 114595q^{32} \) \(\mathstrut +\mathstrut 89208q^{33} \) \(\mathstrut +\mathstrut 129922q^{34} \) \(\mathstrut -\mathstrut 172500q^{35} \) \(\mathstrut +\mathstrut 29889q^{36} \) \(\mathstrut +\mathstrut 228170q^{37} \) \(\mathstrut -\mathstrut 536068q^{38} \) \(\mathstrut -\mathstrut 229662q^{39} \) \(\mathstrut -\mathstrut 141375q^{40} \) \(\mathstrut -\mathstrut 139670q^{41} \) \(\mathstrut +\mathstrut 484380q^{42} \) \(\mathstrut -\mathstrut 755492q^{43} \) \(\mathstrut -\mathstrut 135464q^{44} \) \(\mathstrut -\mathstrut 91125q^{45} \) \(\mathstrut -\mathstrut 1093560q^{46} \) \(\mathstrut +\mathstrut 836984q^{47} \) \(\mathstrut +\mathstrut 538677q^{48} \) \(\mathstrut +\mathstrut 1080857q^{49} \) \(\mathstrut -\mathstrut 203125q^{50} \) \(\mathstrut +\mathstrut 269838q^{51} \) \(\mathstrut +\mathstrut 348746q^{52} \) \(\mathstrut +\mathstrut 1641650q^{53} \) \(\mathstrut +\mathstrut 255879q^{54} \) \(\mathstrut +\mathstrut 413000q^{55} \) \(\mathstrut +\mathstrut 1560780q^{56} \) \(\mathstrut -\mathstrut 1113372q^{57} \) \(\mathstrut -\mathstrut 1726426q^{58} \) \(\mathstrut -\mathstrut 989656q^{59} \) \(\mathstrut +\mathstrut 138375q^{60} \) \(\mathstrut -\mathstrut 1658162q^{61} \) \(\mathstrut +\mathstrut 725400q^{62} \) \(\mathstrut +\mathstrut 1006020q^{63} \) \(\mathstrut +\mathstrut 1063993q^{64} \) \(\mathstrut -\mathstrut 1063250q^{65} \) \(\mathstrut -\mathstrut 1159704q^{66} \) \(\mathstrut -\mathstrut 4523844q^{67} \) \(\mathstrut -\mathstrut 409754q^{68} \) \(\mathstrut -\mathstrut 2271240q^{69} \) \(\mathstrut +\mathstrut 2242500q^{70} \) \(\mathstrut -\mathstrut 389408q^{71} \) \(\mathstrut +\mathstrut 824499q^{72} \) \(\mathstrut +\mathstrut 5617330q^{73} \) \(\mathstrut -\mathstrut 2966210q^{74} \) \(\mathstrut -\mathstrut 421875q^{75} \) \(\mathstrut +\mathstrut 1690676q^{76} \) \(\mathstrut -\mathstrut 4559520q^{77} \) \(\mathstrut +\mathstrut 2985606q^{78} \) \(\mathstrut +\mathstrut 3901080q^{79} \) \(\mathstrut +\mathstrut 2493875q^{80} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut +\mathstrut 1815710q^{82} \) \(\mathstrut -\mathstrut 9394116q^{83} \) \(\mathstrut -\mathstrut 1527660q^{84} \) \(\mathstrut +\mathstrut 1249250q^{85} \) \(\mathstrut +\mathstrut 9821396q^{86} \) \(\mathstrut -\mathstrut 3585654q^{87} \) \(\mathstrut -\mathstrut 3736824q^{88} \) \(\mathstrut +\mathstrut 2803746q^{89} \) \(\mathstrut +\mathstrut 1184625q^{90} \) \(\mathstrut +\mathstrut 11738280q^{91} \) \(\mathstrut +\mathstrut 3448920q^{92} \) \(\mathstrut +\mathstrut 1506600q^{93} \) \(\mathstrut -\mathstrut 10880792q^{94} \) \(\mathstrut -\mathstrut 5154500q^{95} \) \(\mathstrut -\mathstrut 3094065q^{96} \) \(\mathstrut +\mathstrut 5099426q^{97} \) \(\mathstrut -\mathstrut 14051141q^{98} \) \(\mathstrut -\mathstrut 2408616q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−13.0000 −27.0000 41.0000 −125.000 351.000 1380.00 1131.00 729.000 1625.00
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 13 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).