Properties

Label 6.22.a.c
Level $6$
Weight $22$
Character orbit 6.a
Self dual yes
Analytic conductor $16.769$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.7686406572\)
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

\(f(q)\) \(=\) \( q + 1024q^{2} + 59049q^{3} + 1048576q^{4} - 23245050q^{5} + 60466176q^{6} - 1322977768q^{7} + 1073741824q^{8} + 3486784401q^{9} + O(q^{10}) \) \( q + 1024q^{2} + 59049q^{3} + 1048576q^{4} - 23245050q^{5} + 60466176q^{6} - 1322977768q^{7} + 1073741824q^{8} + 3486784401q^{9} - 23802931200q^{10} - 109174443828q^{11} + 61917364224q^{12} + 468325115966q^{13} - 1354729234432q^{14} - 1372596957450q^{15} + 1099511627776q^{16} + 2654798072562q^{17} + 3570467226624q^{18} - 43712786306860q^{19} - 24374201548800q^{20} - 78120514222632q^{21} - 111794630479872q^{22} - 216861233964744q^{23} + 63403380965376q^{24} + 63495191299375q^{25} + 479564918749184q^{26} + 205891132094649q^{27} - 1387242736058368q^{28} + 2535247265345310q^{29} - 1405539284428800q^{30} + 5132915444930672q^{31} + 1125899906842624q^{32} - 6446641733599572q^{33} + 2718513226303488q^{34} + 30752684366048400q^{35} + 3656158440062976q^{36} - 8126962096433578q^{37} - 44761893178224640q^{38} + 27654129772676334q^{39} - 24959182385971200q^{40} - 28546174551317718q^{41} - 79995406563975168q^{42} + 60426656396902316q^{43} - 114477701611388928q^{44} - 81050477740465050q^{45} - 222065903579897856q^{46} - 316578527337771888q^{47} + 64925062108545024q^{48} + 1191724310538977817q^{49} + 65019075890560000q^{50} + 156763171386713538q^{51} + 491074476799164416q^{52} + 237962956198086486q^{53} + 210832519264920576q^{54} + 2537765405504051400q^{55} - 1420536561723768832q^{56} - 2581196318633776140q^{57} + 2596093199713597440q^{58} - 1932323017293179940q^{59} - 1439272227255091200q^{60} - 5540333075504971378q^{61} + 5256105415609008128q^{62} - 4612938244332196968q^{63} + 1152921504606846976q^{64} - 10886240736885468300q^{65} - 6601361135205961728q^{66} + 19977690098489923172q^{67} + 2783757543734771712q^{68} - 12805439004384168456q^{69} + 31490748790833561600q^{70} - 43257338948428460568q^{71} + 3743906242624487424q^{72} + 1683404149334166506q^{73} - 8322009186747983872q^{74} + 3749327551036794375q^{75} - 45836178614502031360q^{76} + 144435362018208815904q^{77} + 28317828887220566016q^{78} + 159081535530354452960q^{79} - 25558202763234508800q^{80} + 12157665459056928801q^{81} - 29231282740549343232q^{82} - 265853420659068579564q^{83} - 81915296321510572032q^{84} - 61710913936607318100q^{85} + 61876896150427971584q^{86} + 149703815771375210190q^{87} - 117225166450062262272q^{88} - 17496064778396968710q^{89} - 82995689206236211200q^{90} - 619583716619039843888q^{91} - 227395485265815404544q^{92} + 303093524107711250928q^{93} - 324176411993878413312q^{94} + 1016105903342276043000q^{95} + 66483263599150104576q^{96} - 361656712539028124638q^{97} + 1220325693991913284608q^{98} - 380667747727321127028q^{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
1024.00 59049.0 1.04858e6 −2.32450e7 6.04662e7 −1.32298e9 1.07374e9 3.48678e9 −2.38029e10
\(n\): 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 6.22.a.c 1
3.b odd 2 1 18.22.a.c 1
4.b odd 2 1 48.22.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.c 1 1.a even 1 1 trivial
18.22.a.c 1 3.b odd 2 1
48.22.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 23245050 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1024 + T \)
$3$ \( -59049 + T \)
$5$ \( 23245050 + T \)
$7$ \( 1322977768 + T \)
$11$ \( 109174443828 + T \)
$13$ \( -468325115966 + T \)
$17$ \( -2654798072562 + T \)
$19$ \( 43712786306860 + T \)
$23$ \( 216861233964744 + T \)
$29$ \( -2535247265345310 + T \)
$31$ \( -5132915444930672 + T \)
$37$ \( 8126962096433578 + T \)
$41$ \( 28546174551317718 + T \)
$43$ \( -60426656396902316 + T \)
$47$ \( 316578527337771888 + T \)
$53$ \( -237962956198086486 + T \)
$59$ \( 1932323017293179940 + T \)
$61$ \( 5540333075504971378 + T \)
$67$ \( -19977690098489923172 + T \)
$71$ \( 43257338948428460568 + T \)
$73$ \( -1683404149334166506 + T \)
$79$ \( -\)\(15\!\cdots\!60\)\( + T \)
$83$ \( \)\(26\!\cdots\!64\)\( + T \)
$89$ \( 17496064778396968710 + T \)
$97$ \( \)\(36\!\cdots\!38\)\( + T \)
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