Properties

Label 6.18.a.c
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

\(f(q)\) \(=\) \( q + 256q^{2} - 6561q^{3} + 65536q^{4} - 199650q^{5} - 1679616q^{6} + 24959264q^{7} + 16777216q^{8} + 43046721q^{9} + O(q^{10}) \) \( q + 256q^{2} - 6561q^{3} + 65536q^{4} - 199650q^{5} - 1679616q^{6} + 24959264q^{7} + 16777216q^{8} + 43046721q^{9} - 51110400q^{10} + 125556420q^{11} - 429981696q^{12} + 4227195518q^{13} + 6389571584q^{14} + 1309903650q^{15} + 4294967296q^{16} + 35551782594q^{17} + 11019960576q^{18} - 64354589764q^{19} - 13084262400q^{20} - 163757731104q^{21} + 32142443520q^{22} - 245819296200q^{23} - 110075314176q^{24} - 723079330625q^{25} + 1082162052608q^{26} - 282429536481q^{27} + 1635730325504q^{28} - 2280393162906q^{29} + 335335334400q^{30} + 4349964811688q^{31} + 1099511627776q^{32} - 823775671620q^{33} + 9101256344064q^{34} - 4983117057600q^{35} + 2821109907456q^{36} + 20770411877318q^{37} - 16474774979584q^{38} - 27734629793598q^{39} - 3349571174400q^{40} - 97624823830086q^{41} - 41921979162624q^{42} + 76137596568644q^{43} + 8228465541120q^{44} - 8594277847650q^{45} - 62929739827200q^{46} + 296069387010240q^{47} - 28179280429056q^{48} + 390334345434489q^{49} - 185108308640000q^{50} - 233255245599234q^{51} + 277033485467648q^{52} - 213113313107874q^{53} - 72301961339136q^{54} - 25067339253000q^{55} + 418746963329024q^{56} + 422230463441604q^{57} - 583780649703936q^{58} - 1776690045107580q^{59} + 85845845606400q^{60} - 1424434275760450q^{61} + 1113590991792128q^{62} + 1074414473773344q^{63} + 281474976710656q^{64} - 843959585168700q^{65} - 210886571934720q^{66} - 1599652965063556q^{67} + 2329921624080384q^{68} + 1612820402368200q^{69} - 1275677966745600q^{70} + 5439386569413960q^{71} + 722204136308736q^{72} - 3725056002188662q^{73} + 5317225440593408q^{74} + 4744123488230625q^{75} - 4217542394773504q^{76} + 3133795833674880q^{77} - 7100065227161088q^{78} + 10282676957218328q^{79} - 857490220646400q^{80} + 1853020188851841q^{81} - 24991954900502016q^{82} - 29457780904474692q^{83} - 10732026665631744q^{84} - 7097913394892100q^{85} + 19491224721572864q^{86} + 14961659541826266q^{87} + 2106487178526720q^{88} - 43414503538999302q^{89} - 2200135128998400q^{90} + 105507688913378752q^{91} - 16110013395763200q^{92} - 28540119129484968q^{93} + 75793763074621440q^{94} + 12848393846382600q^{95} - 7213895789838336q^{96} + 34754667389544578q^{97} + 99925592431229184q^{98} + 5404792181498820q^{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 −199650. −1.67962e6 2.49593e7 1.67772e7 4.30467e7 −5.11104e7
\(n\): 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 \( T_{5} + 199650 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\).