Properties

Label 3.18.a.a
Level 3
Weight 18
Character orbit 3.a
Self dual Yes
Analytic conductor 5.497
Analytic rank 1
Dimension 1
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(5.49666262034\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 204q^{2} \) \(\mathstrut -\mathstrut 6561q^{3} \) \(\mathstrut -\mathstrut 89456q^{4} \) \(\mathstrut -\mathstrut 163554q^{5} \) \(\mathstrut -\mathstrut 1338444q^{6} \) \(\mathstrut -\mathstrut 20846560q^{7} \) \(\mathstrut -\mathstrut 44987712q^{8} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 204q^{2} \) \(\mathstrut -\mathstrut 6561q^{3} \) \(\mathstrut -\mathstrut 89456q^{4} \) \(\mathstrut -\mathstrut 163554q^{5} \) \(\mathstrut -\mathstrut 1338444q^{6} \) \(\mathstrut -\mathstrut 20846560q^{7} \) \(\mathstrut -\mathstrut 44987712q^{8} \) \(\mathstrut +\mathstrut 43046721q^{9} \) \(\mathstrut -\mathstrut 33365016q^{10} \) \(\mathstrut +\mathstrut 817372356q^{11} \) \(\mathstrut +\mathstrut 586920816q^{12} \) \(\mathstrut +\mathstrut 299589758q^{13} \) \(\mathstrut -\mathstrut 4252698240q^{14} \) \(\mathstrut +\mathstrut 1073077794q^{15} \) \(\mathstrut +\mathstrut 2547683584q^{16} \) \(\mathstrut -\mathstrut 44775606078q^{17} \) \(\mathstrut +\mathstrut 8781531084q^{18} \) \(\mathstrut +\mathstrut 78748651964q^{19} \) \(\mathstrut +\mathstrut 14630886624q^{20} \) \(\mathstrut +\mathstrut 136774280160q^{21} \) \(\mathstrut +\mathstrut 166743960624q^{22} \) \(\mathstrut -\mathstrut 704672009160q^{23} \) \(\mathstrut +\mathstrut 295164378432q^{24} \) \(\mathstrut -\mathstrut 736189542209q^{25} \) \(\mathstrut +\mathstrut 61116310632q^{26} \) \(\mathstrut -\mathstrut 282429536481q^{27} \) \(\mathstrut +\mathstrut 1864849871360q^{28} \) \(\mathstrut -\mathstrut 163793785242q^{29} \) \(\mathstrut +\mathstrut 218907869976q^{30} \) \(\mathstrut +\mathstrut 1049860831400q^{31} \) \(\mathstrut +\mathstrut 6416356838400q^{32} \) \(\mathstrut -\mathstrut 5362780027716q^{33} \) \(\mathstrut -\mathstrut 9134223639912q^{34} \) \(\mathstrut +\mathstrut 3409538274240q^{35} \) \(\mathstrut -\mathstrut 3850787473776q^{36} \) \(\mathstrut -\mathstrut 19805735857210q^{37} \) \(\mathstrut +\mathstrut 16064725000656q^{38} \) \(\mathstrut -\mathstrut 1965608402238q^{39} \) \(\mathstrut +\mathstrut 7357920248448q^{40} \) \(\mathstrut +\mathstrut 14660035932090q^{41} \) \(\mathstrut +\mathstrut 27901953152640q^{42} \) \(\mathstrut +\mathstrut 116038864682564q^{43} \) \(\mathstrut -\mathstrut 73118861478336q^{44} \) \(\mathstrut -\mathstrut 7040463406434q^{45} \) \(\mathstrut -\mathstrut 143753089868640q^{46} \) \(\mathstrut -\mathstrut 176606594594112q^{47} \) \(\mathstrut -\mathstrut 16715351994624q^{48} \) \(\mathstrut +\mathstrut 201948549846393q^{49} \) \(\mathstrut -\mathstrut 150182666610636q^{50} \) \(\mathstrut +\mathstrut 293772751477758q^{51} \) \(\mathstrut -\mathstrut 26800101391648q^{52} \) \(\mathstrut +\mathstrut 152863496635230q^{53} \) \(\mathstrut -\mathstrut 57615625442124q^{54} \) \(\mathstrut -\mathstrut 133684518313224q^{55} \) \(\mathstrut +\mathstrut 937839037470720q^{56} \) \(\mathstrut -\mathstrut 516669905535804q^{57} \) \(\mathstrut -\mathstrut 33413932189368q^{58} \) \(\mathstrut -\mathstrut 262797291296124q^{59} \) \(\mathstrut -\mathstrut 95993247140064q^{60} \) \(\mathstrut -\mathstrut 1358552281482562q^{61} \) \(\mathstrut +\mathstrut 214171609605600q^{62} \) \(\mathstrut -\mathstrut 897376052129760q^{63} \) \(\mathstrut +\mathstrut 975006812311552q^{64} \) \(\mathstrut -\mathstrut 48999103279932q^{65} \) \(\mathstrut -\mathstrut 1094007125654064q^{66} \) \(\mathstrut +\mathstrut 444863620615292q^{67} \) \(\mathstrut +\mathstrut 4005446617313568q^{68} \) \(\mathstrut +\mathstrut 4623353052098760q^{69} \) \(\mathstrut +\mathstrut 695545807944960q^{70} \) \(\mathstrut -\mathstrut 4003270764790968q^{71} \) \(\mathstrut -\mathstrut 1936573486892352q^{72} \) \(\mathstrut +\mathstrut 924832535317130q^{73} \) \(\mathstrut -\mathstrut 4040370114870840q^{74} \) \(\mathstrut +\mathstrut 4830139586433249q^{75} \) \(\mathstrut -\mathstrut 7044539410091584q^{76} \) \(\mathstrut -\mathstrut 17039401861695360q^{77} \) \(\mathstrut -\mathstrut 400984114056552q^{78} \) \(\mathstrut +\mathstrut 14747307742797080q^{79} \) \(\mathstrut -\mathstrut 416683840897536q^{80} \) \(\mathstrut +\mathstrut 1853020188851841q^{81} \) \(\mathstrut +\mathstrut 2990647330146360q^{82} \) \(\mathstrut +\mathstrut 26422963268810172q^{83} \) \(\mathstrut -\mathstrut 12235280005992960q^{84} \) \(\mathstrut +\mathstrut 7323229476481212q^{85} \) \(\mathstrut +\mathstrut 23671928395243056q^{86} \) \(\mathstrut +\mathstrut 1074651024972762q^{87} \) \(\mathstrut -\mathstrut 36771712148489472q^{88} \) \(\mathstrut -\mathstrut 38883748080645126q^{89} \) \(\mathstrut -\mathstrut 1436254534912536q^{90} \) \(\mathstrut -\mathstrut 6245415865532480q^{91} \) \(\mathstrut +\mathstrut 63037139251416960q^{92} \) \(\mathstrut -\mathstrut 6888136914815400q^{93} \) \(\mathstrut -\mathstrut 36027745297198848q^{94} \) \(\mathstrut -\mathstrut 12879657023320056q^{95} \) \(\mathstrut -\mathstrut 42097717216742400q^{96} \) \(\mathstrut -\mathstrut 25374394856250238q^{97} \) \(\mathstrut +\mathstrut 41197504168664172q^{98} \) \(\mathstrut +\mathstrut 35185199761844676q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
204.000 −6561.00 −89456.0 −163554. −1.33844e6 −2.08466e7 −4.49877e7 4.30467e7 −3.33650e7
\(n\): 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
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 204 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(3))\).