Properties

Label 6.12.a.a
Level 6
Weight 12
Character orbit 6.a
Self dual Yes
Analytic conductor 4.610
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 6.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.61005908336\)
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

\(f(q)\) \(=\) \( q - 32q^{2} - 243q^{3} + 1024q^{4} + 5766q^{5} + 7776q^{6} + 72464q^{7} - 32768q^{8} + 59049q^{9} + O(q^{10}) \) \( q - 32q^{2} - 243q^{3} + 1024q^{4} + 5766q^{5} + 7776q^{6} + 72464q^{7} - 32768q^{8} + 59049q^{9} - 184512q^{10} - 408948q^{11} - 248832q^{12} + 1367558q^{13} - 2318848q^{14} - 1401138q^{15} + 1048576q^{16} + 5422914q^{17} - 1889568q^{18} + 15166100q^{19} + 5904384q^{20} - 17608752q^{21} + 13086336q^{22} - 52194072q^{23} + 7962624q^{24} - 15581369q^{25} - 43761856q^{26} - 14348907q^{27} + 74203136q^{28} + 118581150q^{29} + 44836416q^{30} - 57652408q^{31} - 33554432q^{32} + 99374364q^{33} - 173533248q^{34} + 417827424q^{35} + 60466176q^{36} - 375985186q^{37} - 485315200q^{38} - 332316594q^{39} - 188940288q^{40} + 856316202q^{41} + 563480064q^{42} - 1245189172q^{43} - 418762752q^{44} + 340476534q^{45} + 1670210304q^{46} - 1306762656q^{47} - 254803968q^{48} + 3273704553q^{49} + 498603808q^{50} - 1317768102q^{51} + 1400379392q^{52} + 409556358q^{53} + 459165024q^{54} - 2357994168q^{55} - 2374500352q^{56} - 3685362300q^{57} - 3794596800q^{58} - 2882866260q^{59} - 1434765312q^{60} + 5731767302q^{61} + 1844877056q^{62} + 4278926736q^{63} + 1073741824q^{64} + 7885339428q^{65} - 3179979648q^{66} + 3893272244q^{67} + 5553063936q^{68} + 12683159496q^{69} - 13370477568q^{70} - 9075890088q^{71} - 1934917632q^{72} - 15571822822q^{73} + 12031525952q^{74} + 3786272667q^{75} + 15530086400q^{76} - 29634007872q^{77} + 10634131008q^{78} - 30196762600q^{79} + 6046089216q^{80} + 3486784401q^{81} - 27402118464q^{82} + 23135252628q^{83} - 18031362048q^{84} + 31268522124q^{85} + 39846053504q^{86} - 28815219450q^{87} + 13400408064q^{88} - 25614819990q^{89} - 10895249088q^{90} + 99098722912q^{91} - 53446729728q^{92} + 14009535144q^{93} + 41816404992q^{94} + 87447732600q^{95} + 8153726976q^{96} - 61937553406q^{97} - 104758545696q^{98} - 24147970452q^{99} + O(q^{100}) \)

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
−32.0000 −243.000 1024.00 5766.00 7776.00 72464.0 −32768.0 59049.0 −184512.
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5} - 5766 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).