Properties

Label 6.12.a.b
Level 6
Weight 12
Character orbit 6.a
Self dual Yes
Analytic conductor 4.610
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut -\mathstrut 11730q^{5} \) \(\mathstrut -\mathstrut 7776q^{6} \) \(\mathstrut -\mathstrut 50008q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut -\mathstrut 11730q^{5} \) \(\mathstrut -\mathstrut 7776q^{6} \) \(\mathstrut -\mathstrut 50008q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 59049q^{9} \) \(\mathstrut +\mathstrut 375360q^{10} \) \(\mathstrut -\mathstrut 531420q^{11} \) \(\mathstrut +\mathstrut 248832q^{12} \) \(\mathstrut +\mathstrut 1332566q^{13} \) \(\mathstrut +\mathstrut 1600256q^{14} \) \(\mathstrut -\mathstrut 2850390q^{15} \) \(\mathstrut +\mathstrut 1048576q^{16} \) \(\mathstrut -\mathstrut 5109678q^{17} \) \(\mathstrut -\mathstrut 1889568q^{18} \) \(\mathstrut +\mathstrut 2901404q^{19} \) \(\mathstrut -\mathstrut 12011520q^{20} \) \(\mathstrut -\mathstrut 12151944q^{21} \) \(\mathstrut +\mathstrut 17005440q^{22} \) \(\mathstrut +\mathstrut 30597000q^{23} \) \(\mathstrut -\mathstrut 7962624q^{24} \) \(\mathstrut +\mathstrut 88764775q^{25} \) \(\mathstrut -\mathstrut 42642112q^{26} \) \(\mathstrut +\mathstrut 14348907q^{27} \) \(\mathstrut -\mathstrut 51208192q^{28} \) \(\mathstrut -\mathstrut 77006634q^{29} \) \(\mathstrut +\mathstrut 91212480q^{30} \) \(\mathstrut -\mathstrut 239418352q^{31} \) \(\mathstrut -\mathstrut 33554432q^{32} \) \(\mathstrut -\mathstrut 129135060q^{33} \) \(\mathstrut +\mathstrut 163509696q^{34} \) \(\mathstrut +\mathstrut 586593840q^{35} \) \(\mathstrut +\mathstrut 60466176q^{36} \) \(\mathstrut -\mathstrut 785041666q^{37} \) \(\mathstrut -\mathstrut 92844928q^{38} \) \(\mathstrut +\mathstrut 323813538q^{39} \) \(\mathstrut +\mathstrut 384368640q^{40} \) \(\mathstrut +\mathstrut 411252954q^{41} \) \(\mathstrut +\mathstrut 388862208q^{42} \) \(\mathstrut +\mathstrut 351233348q^{43} \) \(\mathstrut -\mathstrut 544174080q^{44} \) \(\mathstrut -\mathstrut 692644770q^{45} \) \(\mathstrut -\mathstrut 979104000q^{46} \) \(\mathstrut +\mathstrut 95821680q^{47} \) \(\mathstrut +\mathstrut 254803968q^{48} \) \(\mathstrut +\mathstrut 523473321q^{49} \) \(\mathstrut -\mathstrut 2840472800q^{50} \) \(\mathstrut -\mathstrut 1241651754q^{51} \) \(\mathstrut +\mathstrut 1364547584q^{52} \) \(\mathstrut -\mathstrut 1465857378q^{53} \) \(\mathstrut -\mathstrut 459165024q^{54} \) \(\mathstrut +\mathstrut 6233556600q^{55} \) \(\mathstrut +\mathstrut 1638662144q^{56} \) \(\mathstrut +\mathstrut 705041172q^{57} \) \(\mathstrut +\mathstrut 2464212288q^{58} \) \(\mathstrut +\mathstrut 5621152020q^{59} \) \(\mathstrut -\mathstrut 2918799360q^{60} \) \(\mathstrut -\mathstrut 10473587770q^{61} \) \(\mathstrut +\mathstrut 7661387264q^{62} \) \(\mathstrut -\mathstrut 2952922392q^{63} \) \(\mathstrut +\mathstrut 1073741824q^{64} \) \(\mathstrut -\mathstrut 15630999180q^{65} \) \(\mathstrut +\mathstrut 4132321920q^{66} \) \(\mathstrut +\mathstrut 4515307532q^{67} \) \(\mathstrut -\mathstrut 5232310272q^{68} \) \(\mathstrut +\mathstrut 7435071000q^{69} \) \(\mathstrut -\mathstrut 18771002880q^{70} \) \(\mathstrut -\mathstrut 8509579560q^{71} \) \(\mathstrut -\mathstrut 1934917632q^{72} \) \(\mathstrut +\mathstrut 2012496986q^{73} \) \(\mathstrut +\mathstrut 25121333312q^{74} \) \(\mathstrut +\mathstrut 21569840325q^{75} \) \(\mathstrut +\mathstrut 2971037696q^{76} \) \(\mathstrut +\mathstrut 26575251360q^{77} \) \(\mathstrut -\mathstrut 10362033216q^{78} \) \(\mathstrut -\mathstrut 22238409568q^{79} \) \(\mathstrut -\mathstrut 12299796480q^{80} \) \(\mathstrut +\mathstrut 3486784401q^{81} \) \(\mathstrut -\mathstrut 13160094528q^{82} \) \(\mathstrut +\mathstrut 6328647516q^{83} \) \(\mathstrut -\mathstrut 12443590656q^{84} \) \(\mathstrut +\mathstrut 59936522940q^{85} \) \(\mathstrut -\mathstrut 11239467136q^{86} \) \(\mathstrut -\mathstrut 18712612062q^{87} \) \(\mathstrut +\mathstrut 17413570560q^{88} \) \(\mathstrut -\mathstrut 50123706678q^{89} \) \(\mathstrut +\mathstrut 22164632640q^{90} \) \(\mathstrut -\mathstrut 66638960528q^{91} \) \(\mathstrut +\mathstrut 31331328000q^{92} \) \(\mathstrut -\mathstrut 58178659536q^{93} \) \(\mathstrut -\mathstrut 3066293760q^{94} \) \(\mathstrut -\mathstrut 34033468920q^{95} \) \(\mathstrut -\mathstrut 8153726976q^{96} \) \(\mathstrut +\mathstrut 94805961314q^{97} \) \(\mathstrut -\mathstrut 16751146272q^{98} \) \(\mathstrut -\mathstrut 31379819580q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −11730.0 −7776.00 −50008.0 −32768.0 59049.0 375360.
\(n\): 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 \(T_{5} \) \(\mathstrut +\mathstrut 11730 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).