Properties

Label 6.20.a.b
Level $6$
Weight $20$
Character orbit 6.a
Self dual yes
Analytic conductor $13.729$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 512q^{2} + 19683q^{3} + 262144q^{4} - 5849490q^{5} - 10077696q^{6} + 173530952q^{7} - 134217728q^{8} + 387420489q^{9} + O(q^{10}) \) \( q - 512q^{2} + 19683q^{3} + 262144q^{4} - 5849490q^{5} - 10077696q^{6} + 173530952q^{7} - 134217728q^{8} + 387420489q^{9} + 2994938880q^{10} - 7312828380q^{11} + 5159780352q^{12} - 41845065034q^{13} - 88847847424q^{14} - 115135511670q^{15} + 68719476736q^{16} - 95834399598q^{17} - 198359290368q^{18} - 2419072521316q^{19} - 1533408706560q^{20} + 3415609728216q^{21} + 3744168130560q^{22} - 13218544831800q^{23} - 2641807540224q^{24} + 15143046931975q^{25} + 21424673297408q^{26} + 7625597484987q^{27} + 45490097881088q^{28} + 22096708325526q^{29} + 58949381975040q^{30} + 54205000762928q^{31} - 35184372088832q^{32} - 143938401003540q^{33} + 49067212594176q^{34} - 1015067568414480q^{35} + 101559956668416q^{36} - 754675410892066q^{37} + 1238565130913792q^{38} - 823636415064222q^{39} + 785105257758720q^{40} + 1015505924861274q^{41} - 1748792180846592q^{42} + 2307401507879108q^{43} - 1917014082846720q^{44} - 2266212276200610q^{45} + 6767894953881600q^{46} + 73656034083120q^{47} + 1352605460594688q^{48} + 18714096116653161q^{49} - 7753240029171200q^{50} - 1886308487287434q^{51} - 10969432728272896q^{52} - 5772296141217378q^{53} - 3904305912313344q^{54} + 42776316480526200q^{55} - 23290930115117056q^{56} - 47614604437062828q^{57} - 11313514662669312q^{58} - 129569039139755820q^{59} - 30182083571220480q^{60} + 114049208167000550q^{61} - 27752960390619136q^{62} + 67229446280475528q^{63} + 18014398509481984q^{64} + 244772289465732660q^{65} + 73696461313812480q^{66} - 131076909617853748q^{67} - 25122412848218112q^{68} - 260180617924319400q^{69} + 519714595028213760q^{70} + 532256691369812760q^{71} - 51998697814228992q^{72} - 801680088264316774q^{73} + 386393810376737792q^{74} + 298060592762063925q^{75} - 634145347027861504q^{76} - 1269002070594017760q^{77} + 421701844512881664q^{78} - 799386550683767488q^{79} - 401973891972464640q^{80} + 150094635296999121q^{81} - 519939033528972288q^{82} + 1021049179204582236q^{83} + 895381596593455104q^{84} + 560582362104505020q^{85} - 1181389572034103296q^{86} + 434929509971328258q^{87} + 981511210417520640q^{88} + 4064785168167821322q^{89} + 1160300685414712320q^{90} - 7261413971851932368q^{91} - 3465162216387379200q^{92} + 1066917030016711824q^{93} - 37711889450557440q^{94} + 14150340522712728840q^{95} - 692533995824480256q^{96} + 11014962791774968034q^{97} - 9581617211726418432q^{98} - 2833139546952677820q^{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. −5.84949e6 −1.00777e7 1.73531e8 −1.34218e8 3.87420e8 2.99494e9
\(n\): 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.b 1
3.b odd 2 1 18.20.a.g 1
4.b odd 2 1 48.20.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.20.a.b 1 1.a even 1 1 trivial
18.20.a.g 1 3.b odd 2 1
48.20.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 5849490 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 512 + T \)
$3$ \( -19683 + T \)
$5$ \( 5849490 + T \)
$7$ \( -173530952 + T \)
$11$ \( 7312828380 + T \)
$13$ \( 41845065034 + T \)
$17$ \( 95834399598 + T \)
$19$ \( 2419072521316 + T \)
$23$ \( 13218544831800 + T \)
$29$ \( -22096708325526 + T \)
$31$ \( -54205000762928 + T \)
$37$ \( 754675410892066 + T \)
$41$ \( -1015505924861274 + T \)
$43$ \( -2307401507879108 + T \)
$47$ \( -73656034083120 + T \)
$53$ \( 5772296141217378 + T \)
$59$ \( 129569039139755820 + T \)
$61$ \( -114049208167000550 + T \)
$67$ \( 131076909617853748 + T \)
$71$ \( -532256691369812760 + T \)
$73$ \( 801680088264316774 + T \)
$79$ \( 799386550683767488 + T \)
$83$ \( -1021049179204582236 + T \)
$89$ \( -4064785168167821322 + T \)
$97$ \( -11014962791774968034 + T \)
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