Properties

Label 6.22.a.a
Level $6$
Weight $22$
Character orbit 6.a
Self dual yes
Analytic conductor $16.769$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - 1024q^{2} + 59049q^{3} + 1048576q^{4} + 26444550q^{5} - 60466176q^{6} + 166115864q^{7} - 1073741824q^{8} + 3486784401q^{9} + O(q^{10}) \) \( q - 1024q^{2} + 59049q^{3} + 1048576q^{4} + 26444550q^{5} - 60466176q^{6} + 166115864q^{7} - 1073741824q^{8} + 3486784401q^{9} - 27079219200q^{10} - 104878761780q^{11} + 61917364224q^{12} + 335591325758q^{13} - 170102644736q^{14} + 1561524232950q^{15} + 1099511627776q^{16} + 14596144763634q^{17} - 3570467226624q^{18} + 3569529974996q^{19} + 27729120460800q^{20} + 9808975653336q^{21} + 107395852062720q^{22} + 222369240588600q^{23} - 63403380965376q^{24} + 222477066499375q^{25} - 343645517576192q^{26} + 205891132094649q^{27} + 174185108209664q^{28} + 2194109701319454q^{29} - 1599000814540800q^{30} - 8723627187590032q^{31} - 1125899906842624q^{32} - 6192986004347220q^{33} - 14946452237961216q^{34} + 4392859271341200q^{35} + 3656158440062976q^{36} + 37470891663324758q^{37} - 3655198694395904q^{38} + 19816332194684142q^{39} - 28394619351859200q^{40} + 86616741616565034q^{41} - 10044391069016064q^{42} + 131416928813078444q^{43} - 109973352512225280q^{44} + 92206444431464550q^{45} - 227706102362726400q^{46} + 339041180377015440q^{47} + 64925062108545024q^{48} - 530951383810817511q^{49} - 227816516095360000q^{50} + 861887752147824066q^{51} + 351893009998020608q^{52} - 1571494796445297834q^{53} - 210832519264920576q^{54} - 2773471659829299000q^{55} - 178365550806695936q^{56} + 210777175493538804q^{57} - 2246768334151120896q^{58} + 5232984701774509020q^{59} + 1637376834089779200q^{60} - 4788384962739867250q^{61} + 8932994240092192768q^{62} + 579210203353837464q^{63} + 1152921504606846976q^{64} + 8874561593573718900q^{65} + 6341617668451553280q^{66} - 15480328743911983516q^{67} + 15305167091672285184q^{68} + 13130681287516241400q^{69} - 4498287893853388800q^{70} - 12930906477499746840q^{71} - 3743906242624487424q^{72} - 44257184658687636502q^{73} - 38370193063244552192q^{74} + 13137048299721594375q^{75} + 3742923463061405696q^{76} - 17422026128334877920q^{77} - 20291924167356561408q^{78} - 14888578935758942752q^{79} + 29076090216303820800q^{80} + 12157665459056928801q^{81} - 88695543415362594816q^{82} + 37085068910999181588q^{83} + 10285456454672449536q^{84} + 385988480009157494700q^{85} - 134570935104592326656q^{86} + 129559983753212439246q^{87} + 112612712972518686720q^{88} - 105572017962561697542q^{89} - 94419399097819699200q^{90} + 55747043029195624912q^{91} + 233171048819431833600q^{92} - 515121461800003799568q^{93} - 347178168706063810560q^{94} + 94394613900280471800q^{95} - 66483263599150104576q^{96} + 1381092294370554379298q^{97} + 543694217022277131264q^{98} - 365689630570698993780q^{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 2.64446e7 −6.04662e7 1.66116e8 −1.07374e9 3.48678e9 −2.70792e10
\(n\): 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.a 1
3.b odd 2 1 18.22.a.d 1
4.b odd 2 1 48.22.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.a 1 1.a even 1 1 trivial
18.22.a.d 1 3.b odd 2 1
48.22.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 26444550 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 + T \)
$3$ \( -59049 + T \)
$5$ \( -26444550 + T \)
$7$ \( -166115864 + T \)
$11$ \( 104878761780 + T \)
$13$ \( -335591325758 + T \)
$17$ \( -14596144763634 + T \)
$19$ \( -3569529974996 + T \)
$23$ \( -222369240588600 + T \)
$29$ \( -2194109701319454 + T \)
$31$ \( 8723627187590032 + T \)
$37$ \( -37470891663324758 + T \)
$41$ \( -86616741616565034 + T \)
$43$ \( -131416928813078444 + T \)
$47$ \( -339041180377015440 + T \)
$53$ \( 1571494796445297834 + T \)
$59$ \( -5232984701774509020 + T \)
$61$ \( 4788384962739867250 + T \)
$67$ \( 15480328743911983516 + T \)
$71$ \( 12930906477499746840 + T \)
$73$ \( 44257184658687636502 + T \)
$79$ \( 14888578935758942752 + T \)
$83$ \( -37085068910999181588 + T \)
$89$ \( \)\(10\!\cdots\!42\)\( + T \)
$97$ \( -\)\(13\!\cdots\!98\)\( + T \)
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