Properties

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

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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: \(1\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 356q^{4} \) \(\mathstrut -\mathstrut 125q^{5} \) \(\mathstrut -\mathstrut 594q^{6} \) \(\mathstrut -\mathstrut 420q^{7} \) \(\mathstrut -\mathstrut 5016q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 356q^{4} \) \(\mathstrut -\mathstrut 125q^{5} \) \(\mathstrut -\mathstrut 594q^{6} \) \(\mathstrut -\mathstrut 420q^{7} \) \(\mathstrut -\mathstrut 5016q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut 2750q^{10} \) \(\mathstrut -\mathstrut 2944q^{11} \) \(\mathstrut +\mathstrut 9612q^{12} \) \(\mathstrut -\mathstrut 11006q^{13} \) \(\mathstrut +\mathstrut 9240q^{14} \) \(\mathstrut -\mathstrut 3375q^{15} \) \(\mathstrut +\mathstrut 64784q^{16} \) \(\mathstrut -\mathstrut 16546q^{17} \) \(\mathstrut -\mathstrut 16038q^{18} \) \(\mathstrut -\mathstrut 25364q^{19} \) \(\mathstrut -\mathstrut 44500q^{20} \) \(\mathstrut -\mathstrut 11340q^{21} \) \(\mathstrut +\mathstrut 64768q^{22} \) \(\mathstrut -\mathstrut 5880q^{23} \) \(\mathstrut -\mathstrut 135432q^{24} \) \(\mathstrut +\mathstrut 15625q^{25} \) \(\mathstrut +\mathstrut 242132q^{26} \) \(\mathstrut +\mathstrut 19683q^{27} \) \(\mathstrut -\mathstrut 149520q^{28} \) \(\mathstrut +\mathstrut 163042q^{29} \) \(\mathstrut +\mathstrut 74250q^{30} \) \(\mathstrut -\mathstrut 201600q^{31} \) \(\mathstrut -\mathstrut 783200q^{32} \) \(\mathstrut -\mathstrut 79488q^{33} \) \(\mathstrut +\mathstrut 364012q^{34} \) \(\mathstrut +\mathstrut 52500q^{35} \) \(\mathstrut +\mathstrut 259524q^{36} \) \(\mathstrut +\mathstrut 120530q^{37} \) \(\mathstrut +\mathstrut 558008q^{38} \) \(\mathstrut -\mathstrut 297162q^{39} \) \(\mathstrut +\mathstrut 627000q^{40} \) \(\mathstrut -\mathstrut 115910q^{41} \) \(\mathstrut +\mathstrut 249480q^{42} \) \(\mathstrut -\mathstrut 19148q^{43} \) \(\mathstrut -\mathstrut 1048064q^{44} \) \(\mathstrut -\mathstrut 91125q^{45} \) \(\mathstrut +\mathstrut 129360q^{46} \) \(\mathstrut +\mathstrut 841016q^{47} \) \(\mathstrut +\mathstrut 1749168q^{48} \) \(\mathstrut -\mathstrut 647143q^{49} \) \(\mathstrut -\mathstrut 343750q^{50} \) \(\mathstrut -\mathstrut 446742q^{51} \) \(\mathstrut -\mathstrut 3918136q^{52} \) \(\mathstrut +\mathstrut 501890q^{53} \) \(\mathstrut -\mathstrut 433026q^{54} \) \(\mathstrut +\mathstrut 368000q^{55} \) \(\mathstrut +\mathstrut 2106720q^{56} \) \(\mathstrut -\mathstrut 684828q^{57} \) \(\mathstrut -\mathstrut 3586924q^{58} \) \(\mathstrut -\mathstrut 1586176q^{59} \) \(\mathstrut -\mathstrut 1201500q^{60} \) \(\mathstrut -\mathstrut 372962q^{61} \) \(\mathstrut +\mathstrut 4435200q^{62} \) \(\mathstrut -\mathstrut 306180q^{63} \) \(\mathstrut +\mathstrut 8938048q^{64} \) \(\mathstrut +\mathstrut 1375750q^{65} \) \(\mathstrut +\mathstrut 1748736q^{66} \) \(\mathstrut +\mathstrut 4561044q^{67} \) \(\mathstrut -\mathstrut 5890376q^{68} \) \(\mathstrut -\mathstrut 158760q^{69} \) \(\mathstrut -\mathstrut 1155000q^{70} \) \(\mathstrut +\mathstrut 1512832q^{71} \) \(\mathstrut -\mathstrut 3656664q^{72} \) \(\mathstrut -\mathstrut 1522910q^{73} \) \(\mathstrut -\mathstrut 2651660q^{74} \) \(\mathstrut +\mathstrut 421875q^{75} \) \(\mathstrut -\mathstrut 9029584q^{76} \) \(\mathstrut +\mathstrut 1236480q^{77} \) \(\mathstrut +\mathstrut 6537564q^{78} \) \(\mathstrut +\mathstrut 4231920q^{79} \) \(\mathstrut -\mathstrut 8098000q^{80} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut +\mathstrut 2550020q^{82} \) \(\mathstrut -\mathstrut 1854204q^{83} \) \(\mathstrut -\mathstrut 4037040q^{84} \) \(\mathstrut +\mathstrut 2068250q^{85} \) \(\mathstrut +\mathstrut 421256q^{86} \) \(\mathstrut +\mathstrut 4402134q^{87} \) \(\mathstrut +\mathstrut 14767104q^{88} \) \(\mathstrut -\mathstrut 6888174q^{89} \) \(\mathstrut +\mathstrut 2004750q^{90} \) \(\mathstrut +\mathstrut 4622520q^{91} \) \(\mathstrut -\mathstrut 2093280q^{92} \) \(\mathstrut -\mathstrut 5443200q^{93} \) \(\mathstrut -\mathstrut 18502352q^{94} \) \(\mathstrut +\mathstrut 3170500q^{95} \) \(\mathstrut -\mathstrut 21146400q^{96} \) \(\mathstrut +\mathstrut 3700034q^{97} \) \(\mathstrut +\mathstrut 14237146q^{98} \) \(\mathstrut -\mathstrut 2146176q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−22.0000 27.0000 356.000 −125.000 −594.000 −420.000 −5016.00 729.000 2750.00
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 22 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).