Properties

Label 1.16.a.a
Level 1
Weight 16
Character orbit 1.a
Self dual Yes
Analytic conductor 1.427
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.426935051\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 216q^{2} \) \(\mathstrut -\mathstrut 3348q^{3} \) \(\mathstrut +\mathstrut 13888q^{4} \) \(\mathstrut +\mathstrut 52110q^{5} \) \(\mathstrut -\mathstrut 723168q^{6} \) \(\mathstrut +\mathstrut 2822456q^{7} \) \(\mathstrut -\mathstrut 4078080q^{8} \) \(\mathstrut -\mathstrut 3139803q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 216q^{2} \) \(\mathstrut -\mathstrut 3348q^{3} \) \(\mathstrut +\mathstrut 13888q^{4} \) \(\mathstrut +\mathstrut 52110q^{5} \) \(\mathstrut -\mathstrut 723168q^{6} \) \(\mathstrut +\mathstrut 2822456q^{7} \) \(\mathstrut -\mathstrut 4078080q^{8} \) \(\mathstrut -\mathstrut 3139803q^{9} \) \(\mathstrut +\mathstrut 11255760q^{10} \) \(\mathstrut +\mathstrut 20586852q^{11} \) \(\mathstrut -\mathstrut 46497024q^{12} \) \(\mathstrut -\mathstrut 190073338q^{13} \) \(\mathstrut +\mathstrut 609650496q^{14} \) \(\mathstrut -\mathstrut 174464280q^{15} \) \(\mathstrut -\mathstrut 1335947264q^{16} \) \(\mathstrut +\mathstrut 1646527986q^{17} \) \(\mathstrut -\mathstrut 678197448q^{18} \) \(\mathstrut +\mathstrut 1563257180q^{19} \) \(\mathstrut +\mathstrut 723703680q^{20} \) \(\mathstrut -\mathstrut 9449582688q^{21} \) \(\mathstrut +\mathstrut 4446760032q^{22} \) \(\mathstrut +\mathstrut 9451116072q^{23} \) \(\mathstrut +\mathstrut 13653411840q^{24} \) \(\mathstrut -\mathstrut 27802126025q^{25} \) \(\mathstrut -\mathstrut 41055841008q^{26} \) \(\mathstrut +\mathstrut 58552201080q^{27} \) \(\mathstrut +\mathstrut 39198268928q^{28} \) \(\mathstrut -\mathstrut 36902568330q^{29} \) \(\mathstrut -\mathstrut 37684284480q^{30} \) \(\mathstrut +\mathstrut 71588483552q^{31} \) \(\mathstrut -\mathstrut 154934083584q^{32} \) \(\mathstrut -\mathstrut 68924780496q^{33} \) \(\mathstrut +\mathstrut 355650044976q^{34} \) \(\mathstrut +\mathstrut 147078182160q^{35} \) \(\mathstrut -\mathstrut 43605584064q^{36} \) \(\mathstrut -\mathstrut 1033652081554q^{37} \) \(\mathstrut +\mathstrut 337663550880q^{38} \) \(\mathstrut +\mathstrut 636365535624q^{39} \) \(\mathstrut -\mathstrut 212508748800q^{40} \) \(\mathstrut +\mathstrut 1641974018202q^{41} \) \(\mathstrut -\mathstrut 2041109860608q^{42} \) \(\mathstrut -\mathstrut 492403109308q^{43} \) \(\mathstrut +\mathstrut 285910200576q^{44} \) \(\mathstrut -\mathstrut 163615134330q^{45} \) \(\mathstrut +\mathstrut 2041441071552q^{46} \) \(\mathstrut -\mathstrut 3410684952624q^{47} \) \(\mathstrut +\mathstrut 4472751439872q^{48} \) \(\mathstrut +\mathstrut 3218696361993q^{49} \) \(\mathstrut -\mathstrut 6005259221400q^{50} \) \(\mathstrut -\mathstrut 5512575697128q^{51} \) \(\mathstrut -\mathstrut 2639738518144q^{52} \) \(\mathstrut +\mathstrut 6797151655902q^{53} \) \(\mathstrut +\mathstrut 12647275433280q^{54} \) \(\mathstrut +\mathstrut 1072780857720q^{55} \) \(\mathstrut -\mathstrut 11510201364480q^{56} \) \(\mathstrut -\mathstrut 5233785038640q^{57} \) \(\mathstrut -\mathstrut 7970954759280q^{58} \) \(\mathstrut +\mathstrut 9858856815540q^{59} \) \(\mathstrut -\mathstrut 2422959920640q^{60} \) \(\mathstrut +\mathstrut 4931842626902q^{61} \) \(\mathstrut +\mathstrut 15463112447232q^{62} \) \(\mathstrut -\mathstrut 8861955816168q^{63} \) \(\mathstrut +\mathstrut 10310557892608q^{64} \) \(\mathstrut -\mathstrut 9904721643180q^{65} \) \(\mathstrut -\mathstrut 14887752587136q^{66} \) \(\mathstrut -\mathstrut 28837826625364q^{67} \) \(\mathstrut +\mathstrut 22866980669568q^{68} \) \(\mathstrut -\mathstrut 31642336609056q^{69} \) \(\mathstrut +\mathstrut 31768887346560q^{70} \) \(\mathstrut +\mathstrut 125050114914552q^{71} \) \(\mathstrut +\mathstrut 12804367818240q^{72} \) \(\mathstrut -\mathstrut 82171455513478q^{73} \) \(\mathstrut -\mathstrut 223268849615664q^{74} \) \(\mathstrut +\mathstrut 93081517931700q^{75} \) \(\mathstrut +\mathstrut 21710515715840q^{76} \) \(\mathstrut +\mathstrut 58105483948512q^{77} \) \(\mathstrut +\mathstrut 137454955694784q^{78} \) \(\mathstrut -\mathstrut 25413078694480q^{79} \) \(\mathstrut -\mathstrut 69616211927040q^{80} \) \(\mathstrut -\mathstrut 150980027970519q^{81} \) \(\mathstrut +\mathstrut 354666387931632q^{82} \) \(\mathstrut -\mathstrut 281736730890468q^{83} \) \(\mathstrut -\mathstrut 131235804370944q^{84} \) \(\mathstrut +\mathstrut 85800573350460q^{85} \) \(\mathstrut -\mathstrut 106359071610528q^{86} \) \(\mathstrut +\mathstrut 123549798768840q^{87} \) \(\mathstrut -\mathstrut 83954829404160q^{88} \) \(\mathstrut +\mathstrut 715618564776810q^{89} \) \(\mathstrut -\mathstrut 35340869015280q^{90} \) \(\mathstrut -\mathstrut 536473633278128q^{91} \) \(\mathstrut +\mathstrut 131257100007936q^{92} \) \(\mathstrut -\mathstrut 239678242932096q^{93} \) \(\mathstrut -\mathstrut 736707949766784q^{94} \) \(\mathstrut +\mathstrut 81461331649800q^{95} \) \(\mathstrut +\mathstrut 518719311839232q^{96} \) \(\mathstrut +\mathstrut 612786136081826q^{97} \) \(\mathstrut +\mathstrut 695238414190488q^{98} \) \(\mathstrut -\mathstrut 64638659670156q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
216.000 −3348.00 13888.0 52110.0 −723168. 2.82246e6 −4.07808e6 −3.13980e6 1.12558e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\).