Properties

Label 6.18.a.b
Level 6
Weight 18
Character orbit 6.a
Self dual Yes
Analytic conductor 10.993
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 6.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.9933252407\)
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

\(f(q)\) \(=\) \( q - 256q^{2} + 6561q^{3} + 65536q^{4} - 72186q^{5} - 1679616q^{6} - 8640184q^{7} - 16777216q^{8} + 43046721q^{9} + O(q^{10}) \) \( q - 256q^{2} + 6561q^{3} + 65536q^{4} - 72186q^{5} - 1679616q^{6} - 8640184q^{7} - 16777216q^{8} + 43046721q^{9} + 18479616q^{10} + 1159304460q^{11} + 429981696q^{12} + 2801062862q^{13} + 2211887104q^{14} - 473612346q^{15} + 4294967296q^{16} + 32979662226q^{17} - 11019960576q^{18} + 5778498836q^{19} - 4730781696q^{20} - 56688247224q^{21} - 296781941760q^{22} + 169116994200q^{23} - 110075314176q^{24} - 757728634529q^{25} - 717072092672q^{26} + 282429536481q^{27} - 566243098624q^{28} + 3631735478814q^{29} + 121244760576q^{30} + 6880978560608q^{31} - 1099511627776q^{32} + 7606196562060q^{33} - 8442793529856q^{34} + 623700322224q^{35} + 2821109907456q^{36} - 35464500749338q^{37} - 1479295702016q^{38} + 18377773437582q^{39} + 1211080114176q^{40} - 8923766734806q^{41} + 14512191289344q^{42} - 129966457018324q^{43} + 75976177090560q^{44} - 3107370602106q^{45} - 43293950515200q^{46} + 129499777218480q^{47} + 28179280429056q^{48} - 157977734433351q^{49} + 193978530439424q^{50} + 216379563864786q^{51} + 183570455724032q^{52} + 218262107088054q^{53} - 72301961339136q^{54} - 83685551749560q^{55} + 144958233247744q^{56} + 37912730862996q^{57} - 929724282576384q^{58} - 1783401246652740q^{59} - 31038658707456q^{60} + 1469145893932670q^{61} - 1761530511515648q^{62} - 371931590036664q^{63} + 281474976710656q^{64} - 202197523756332q^{65} - 1947186319887360q^{66} + 5051560974054596q^{67} + 2161355143643136q^{68} + 1109576598946200q^{69} - 159667282489344q^{70} - 793480696785720q^{71} - 722204136308736q^{72} + 6343500933237962q^{73} + 9078912191830528q^{74} - 4971457571144769q^{75} + 378699699716096q^{76} - 10016603846420640q^{77} - 4704710000020992q^{78} - 8292883305185392q^{79} - 310036509229056q^{80} + 1853020188851841q^{81} + 2284484284110336q^{82} - 24031501915598508q^{83} - 3715120970072064q^{84} - 2380669897446036q^{85} + 33271412996690944q^{86} + 23827816476498654q^{87} - 19449901335183360q^{88} - 15466463339248422q^{89} + 795486874139136q^{90} - 24201698523246608q^{91} + 11083251331891200q^{92} + 45146100336149088q^{93} - 33151942967930880q^{94} - 417126716975496q^{95} - 7213895789838336q^{96} + 79745962551777122q^{97} + 40442300014937856q^{98} + 49904255643675660q^{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
−256.000 6561.00 65536.0 −72186.0 −1.67962e6 −8.64018e6 −1.67772e7 4.30467e7 1.84796e7
\(n\): 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 \( T_{5} + 72186 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\).