Properties

Label 6.18.a.a
Level 6
Weight 18
Character orbit 6.a
Self dual Yes
Analytic conductor 10.993
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q - 256q^{2} - 6561q^{3} + 65536q^{4} + 645150q^{5} + 1679616q^{6} + 3974432q^{7} - 16777216q^{8} + 43046721q^{9} + O(q^{10}) \) \( q - 256q^{2} - 6561q^{3} + 65536q^{4} + 645150q^{5} + 1679616q^{6} + 3974432q^{7} - 16777216q^{8} + 43046721q^{9} - 165158400q^{10} - 500068668q^{11} - 429981696q^{12} - 5425661314q^{13} - 1017454592q^{14} - 4232829150q^{15} + 4294967296q^{16} - 5466992958q^{17} - 11019960576q^{18} - 53889877060q^{19} + 42280550400q^{20} - 26076248352q^{21} + 128017579008q^{22} + 578906836536q^{23} + 110075314176q^{24} - 346720930625q^{25} + 1388969296384q^{26} - 282429536481q^{27} + 260468375552q^{28} - 4619583681690q^{29} + 1083604262400q^{30} - 6802815567448q^{31} - 1099511627776q^{32} + 3280950530748q^{33} + 1399550197248q^{34} + 2564104804800q^{35} + 2821109907456q^{36} - 19571909422138q^{37} + 13795808527360q^{38} + 35597763881154q^{39} - 10823820902400q^{40} + 57213620756922q^{41} + 6675519578112q^{42} - 24501250225084q^{43} - 32772500226048q^{44} + 27771592053150q^{45} - 148200150153216q^{46} + 184283998832832q^{47} - 28179280429056q^{48} - 216834404264583q^{49} + 88760558240000q^{50} + 35868940797438q^{51} - 355576139874304q^{52} - 206542562280354q^{53} + 72301961339136q^{54} - 322619301160200q^{55} - 66679904141312q^{56} + 353571483390660q^{57} + 1182613422512640q^{58} - 418648048246140q^{59} - 277402691174400q^{60} + 2501287878088382q^{61} + 1741520785266688q^{62} + 171086265437472q^{63} + 281474976710656q^{64} - 3500365396727100q^{65} - 839923335871488q^{66} - 145692866050948q^{67} - 358284850495488q^{68} - 3798207754512696q^{69} - 656410830028800q^{70} - 5364313152664248q^{71} - 722204136308736q^{72} + 3302058927938186q^{73} + 5010408812067328q^{74} + 2274836025830625q^{75} - 3531726983004160q^{76} - 1987488916296576q^{77} - 9113027553575424q^{78} + 22067463278260760q^{79} + 2770898151014400q^{80} + 1853020188851841q^{81} - 14646686913772032q^{82} + 20438378406354876q^{83} - 1708933011996672q^{84} - 3527030506853700q^{85} + 6272320057621504q^{86} + 30309088535568090q^{87} + 8389760057868288q^{88} - 56063805950152710q^{89} - 7109527565606400q^{90} - 21563921947523648q^{91} + 37939238439223296q^{92} + 44633272938026328q^{93} - 47176703701204992q^{94} - 34767054185259000q^{95} + 7213895789838336q^{96} - 118254406396110718q^{97} + 55509607491733248q^{98} - 21526316432237628q^{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 645150. 1.67962e6 3.97443e6 −1.67772e7 4.30467e7 −1.65158e8
\(n\): 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 \( T_{5} - 645150 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\).