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

Related objects

Downloads

Learn more about

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

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

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
−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 \(T_{5} \) \(\mathstrut -\mathstrut 5766 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).