Properties

Label 15.12.a.a
Level 15
Weight 12
Character orbit 15.a
Self dual Yes
Analytic conductor 11.525
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.5251477084\)
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

\(f(q)\) \(=\) \( q - 56q^{2} - 243q^{3} + 1088q^{4} + 3125q^{5} + 13608q^{6} + 27984q^{7} + 53760q^{8} + 59049q^{9} + O(q^{10}) \) \( q - 56q^{2} - 243q^{3} + 1088q^{4} + 3125q^{5} + 13608q^{6} + 27984q^{7} + 53760q^{8} + 59049q^{9} - 175000q^{10} - 112028q^{11} - 264384q^{12} - 1096922q^{13} - 1567104q^{14} - 759375q^{15} - 5238784q^{16} - 249566q^{17} - 3306744q^{18} - 13712420q^{19} + 3400000q^{20} - 6800112q^{21} + 6273568q^{22} + 41395728q^{23} - 13063680q^{24} + 9765625q^{25} + 61427632q^{26} - 14348907q^{27} + 30446592q^{28} - 4533850q^{29} + 42525000q^{30} - 265339008q^{31} + 183271424q^{32} + 27222804q^{33} + 13975696q^{34} + 87450000q^{35} + 64245312q^{36} - 212136946q^{37} + 767895520q^{38} + 266552046q^{39} + 168000000q^{40} - 1266969958q^{41} + 380806272q^{42} + 14129548q^{43} - 121886464q^{44} + 184528125q^{45} - 2318160768q^{46} - 2657273336q^{47} + 1273024512q^{48} - 1194222487q^{49} - 546875000q^{50} + 60644538q^{51} - 1193451136q^{52} + 2402699278q^{53} + 803538792q^{54} - 350087500q^{55} + 1504419840q^{56} + 3332118060q^{57} + 253895600q^{58} + 7498737220q^{59} - 826200000q^{60} - 4064828858q^{61} + 14858984448q^{62} + 1652427216q^{63} + 465829888q^{64} - 3427881250q^{65} - 1524477024q^{66} + 6871514244q^{67} - 271527808q^{68} - 10059161904q^{69} - 4897200000q^{70} - 13283734648q^{71} + 3174474240q^{72} - 28875844262q^{73} + 11879668976q^{74} - 2373046875q^{75} - 14919112960q^{76} - 3134991552q^{77} - 14926914576q^{78} + 27100302240q^{79} - 16371200000q^{80} + 3486784401q^{81} + 70950317648q^{82} - 34365255132q^{83} - 7398521856q^{84} - 779893750q^{85} - 791254688q^{86} + 1101725550q^{87} - 6022625280q^{88} - 63500412630q^{89} - 10333575000q^{90} - 30696265248q^{91} + 45038552064q^{92} + 64477378944q^{93} + 148807306816q^{94} - 42851312500q^{95} - 44534956032q^{96} + 19634495234q^{97} + 66876459272q^{98} - 6615141372q^{99} + O(q^{100}) \)

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
−56.0000 −243.000 1088.00 3125.00 13608.0 27984.0 53760.0 59049.0 −175000.
\(n\): 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 \( T_{2} + 56 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).