Properties

Label 6.22.a.b
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

\(f(q)\) \(=\) \( q + 1024q^{2} - 59049q^{3} + 1048576q^{4} + 12954174q^{5} - 60466176q^{6} - 479513104q^{7} + 1073741824q^{8} + 3486784401q^{9} + O(q^{10}) \) \( q + 1024q^{2} - 59049q^{3} + 1048576q^{4} + 12954174q^{5} - 60466176q^{6} - 479513104q^{7} + 1073741824q^{8} + 3486784401q^{9} + 13265074176q^{10} + 115657781700q^{11} - 61917364224q^{12} + 295658246702q^{13} - 491021418496q^{14} - 764931020526q^{15} + 1099511627776q^{16} + 6626983431906q^{17} + 3570467226624q^{18} + 28576184164796q^{19} + 13583435956224q^{20} + 28314769278096q^{21} + 118433568460800q^{22} + 335385196791000q^{23} - 63403380965376q^{24} - 309026534180849q^{25} + 302754044622848q^{26} - 205891132094649q^{27} - 502805932539904q^{28} - 699224214482106q^{29} - 783289365018624q^{30} - 3484957262657992q^{31} + 1125899906842624q^{32} - 6829476351603300q^{33} + 6786031034271744q^{34} - 6211696184496096q^{35} + 3656158440062976q^{36} - 35181531093012298q^{37} + 29262012584751104q^{38} - 17458323809506398q^{39} + 13909438419173376q^{40} + 6132056240639994q^{41} + 28994323740770304q^{42} + 233260850096910596q^{43} + 121275974103859200q^{44} + 45168411831039774q^{45} + 343434441513984000q^{46} - 580205712121346400q^{47} - 64925062108545024q^{48} - 328613047175569191q^{49} - 316443171001189376q^{50} - 391316744670617394q^{51} + 310020141693796352q^{52} - 1394471665941750306q^{53} - 210832519264920576q^{54} + 1498251028595815800q^{55} - 514873274920861696q^{56} - 1687395098747039004q^{57} - 716005595629676544q^{58} + 2352476807159705700q^{59} - 802088309779070976q^{60} + 9920628300330384590q^{61} - 3568596236961783808q^{62} - 1671958811102290704q^{63} + 1152921504606846976q^{64} + 3830008372312634148q^{65} - 6993383784041779200q^{66} + 26068981808996843996q^{67} + 6948895779094265856q^{68} - 19804160485311759000q^{69} - 6360776892924002304q^{70} - 13336955952504341400q^{71} + 3743906242624487424q^{72} + 9037529597968684202q^{73} - 36025887839244593152q^{74} + 18247707816844952601q^{75} + 29964300886785130496q^{76} - 55459421904721396800q^{77} - 17877323580934551552q^{78} - 77283864571811027992q^{79} + 14243264941233537024q^{80} + 12157665459056928801q^{81} + 6279225590415353856q^{82} - 155698418876868248388q^{83} + 29690187510548791296q^{84} + 85847096472027475644q^{85} + 238859110499236450304q^{86} + 41288490640953877194q^{87} + 124186597482351820800q^{88} + 253837312813381912218q^{89} + 46252453714984728576q^{90} - 141772003599273783008q^{91} + 351676868110319616000q^{92} + 205783241402691769608q^{93} - 594130649212258713600q^{94} + 370180861926812058504q^{95} - 66483263599150104576q^{96} - 1030722092535365121598q^{97} - 336499760307782851584q^{98} + 403273749085823261700q^{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 1.29542e7 −6.04662e7 −4.79513e8 1.07374e9 3.48678e9 1.32651e10
\(n\): 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.b 1
3.b odd 2 1 18.22.a.a 1
4.b odd 2 1 48.22.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.b 1 1.a even 1 1 trivial
18.22.a.a 1 3.b odd 2 1
48.22.a.f 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 12954174 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1024 + T \)
$3$ \( 59049 + T \)
$5$ \( -12954174 + T \)
$7$ \( 479513104 + T \)
$11$ \( -115657781700 + T \)
$13$ \( -295658246702 + T \)
$17$ \( -6626983431906 + T \)
$19$ \( -28576184164796 + T \)
$23$ \( -335385196791000 + T \)
$29$ \( 699224214482106 + T \)
$31$ \( 3484957262657992 + T \)
$37$ \( 35181531093012298 + T \)
$41$ \( -6132056240639994 + T \)
$43$ \( -233260850096910596 + T \)
$47$ \( 580205712121346400 + T \)
$53$ \( 1394471665941750306 + T \)
$59$ \( -2352476807159705700 + T \)
$61$ \( -9920628300330384590 + T \)
$67$ \( -26068981808996843996 + T \)
$71$ \( 13336955952504341400 + T \)
$73$ \( -9037529597968684202 + T \)
$79$ \( 77283864571811027992 + T \)
$83$ \( \)\(15\!\cdots\!88\)\( + T \)
$89$ \( -\)\(25\!\cdots\!18\)\( + T \)
$97$ \( \)\(10\!\cdots\!98\)\( + T \)
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