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

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

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
−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 \(T_{2} \) \(\mathstrut +\mathstrut 56 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).