Properties

Label 6.20.a.c
Level $6$
Weight $20$
Character orbit 6.a
Self dual yes
Analytic conductor $13.729$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7290017934\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 512q^{2} + 19683q^{3} + 262144q^{4} + 1953390q^{5} + 10077696q^{6} + 40488776q^{7} + 134217728q^{8} + 387420489q^{9} + O(q^{10}) \) \( q + 512q^{2} + 19683q^{3} + 262144q^{4} + 1953390q^{5} + 10077696q^{6} + 40488776q^{7} + 134217728q^{8} + 387420489q^{9} + 1000135680q^{10} + 1916860452q^{11} + 5159780352q^{12} + 3132480182q^{13} + 20730253312q^{14} + 38448575370q^{15} + 68719476736q^{16} + 607659965586q^{17} + 198359290368q^{18} + 2507511106460q^{19} + 512069468160q^{20} + 796940578008q^{21} + 981432551424q^{22} - 13588841327928q^{23} + 2641807540224q^{24} - 15257753836025q^{25} + 1603829853184q^{26} + 7625597484987q^{27} + 10613889695744q^{28} - 95129927516010q^{29} + 19685670589440q^{30} - 128131172993488q^{31} + 35184372088832q^{32} + 37729564276716q^{33} + 311121902380032q^{34} + 79090370150640q^{35} + 101559956668416q^{36} - 306931332936994q^{37} + 1283845686507520q^{38} + 61656607422306q^{39} + 262179567697920q^{40} + 1928507793837402q^{41} + 408033575940096q^{42} - 6036474006185788q^{43} + 502493466329088q^{44} + 756783309007710q^{45} - 6957486759899136q^{46} - 2205402168106704q^{47} + 1352605460594688q^{48} - 9759554203394967q^{49} - 7811969964044800q^{50} + 11960571102629238q^{51} + 821160884830208q^{52} + 30561502311272862q^{53} + 3904305912313344q^{54} + 3744376038332280q^{55} + 5434311524220928q^{56} + 49355341108452180q^{57} - 48706522888197120q^{58} + 51122948758448340q^{59} + 10079063341793280q^{60} - 62140463356892698q^{61} - 65603160572665856q^{62} + 15686181396931464q^{63} + 18014398509481984q^{64} + 6118955462716980q^{65} + 19317536909678592q^{66} - 137732367675340084q^{67} + 159294414018576384q^{68} - 267469163857606824q^{69} + 40494269517127680q^{70} - 138268877451263208q^{71} + 51998697814228992q^{72} - 331152017279915878q^{73} - 157148842463740928q^{74} - 300318368754480075q^{75} + 657328991491850240q^{76} + 77611333464286752q^{77} + 31568183000220672q^{78} + 1695082134983531840q^{79} + 134235938661335040q^{80} + 150094635296999121q^{81} + 987395990444749824q^{82} + 336651125543852892q^{83} + 208913190881329152q^{84} + 1186996900176036540q^{85} - 3090674691167123456q^{86} - 1872442363297624830q^{87} + 257276654760493056q^{88} - 2059907328938598390q^{89} + 387473054211947520q^{90} + 126830288413437232q^{91} - 3562233221068357632q^{92} - 2522005878030824304q^{93} - 1129165910070632448q^{94} + 4898147120247899400q^{95} + 692533995824480256q^{96} - 2808994919565485854q^{97} - 4996891752138223104q^{98} + 742631013658601028q^{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
512.000 19683.0 262144. 1.95339e6 1.00777e7 4.04888e7 1.34218e8 3.87420e8 1.00014e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.20.a.c 1
3.b odd 2 1 18.20.a.a 1
4.b odd 2 1 48.20.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.20.a.c 1 1.a even 1 1 trivial
18.20.a.a 1 3.b odd 2 1
48.20.a.b 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1953390 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -512 + T \)
$3$ \( -19683 + T \)
$5$ \( -1953390 + T \)
$7$ \( -40488776 + T \)
$11$ \( -1916860452 + T \)
$13$ \( -3132480182 + T \)
$17$ \( -607659965586 + T \)
$19$ \( -2507511106460 + T \)
$23$ \( 13588841327928 + T \)
$29$ \( 95129927516010 + T \)
$31$ \( 128131172993488 + T \)
$37$ \( 306931332936994 + T \)
$41$ \( -1928507793837402 + T \)
$43$ \( 6036474006185788 + T \)
$47$ \( 2205402168106704 + T \)
$53$ \( -30561502311272862 + T \)
$59$ \( -51122948758448340 + T \)
$61$ \( 62140463356892698 + T \)
$67$ \( 137732367675340084 + T \)
$71$ \( 138268877451263208 + T \)
$73$ \( 331152017279915878 + T \)
$79$ \( -1695082134983531840 + T \)
$83$ \( -336651125543852892 + T \)
$89$ \( 2059907328938598390 + T \)
$97$ \( 2808994919565485854 + T \)
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