Properties

Label 6.26.a.b
Level $6$
Weight $26$
Character orbit 6.a
Self dual yes
Analytic conductor $23.760$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7598067971\)
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

\(f(q)\) \(=\) \( q - 4096q^{2} + 531441q^{3} + 16777216q^{4} + 590425734q^{5} - 2176782336q^{6} + 57857417576q^{7} - 68719476736q^{8} + 282429536481q^{9} + O(q^{10}) \) \( q - 4096q^{2} + 531441q^{3} + 16777216q^{4} + 590425734q^{5} - 2176782336q^{6} + 57857417576q^{7} - 68719476736q^{8} + 282429536481q^{9} - 2418383806464q^{10} + 9494266240140q^{11} + 8916100448256q^{12} - 134968021061458q^{13} - 236983982391296q^{14} + 313776442502694q^{15} + 281474976710656q^{16} - 2526114016804014q^{17} - 1156831381426176q^{18} + 11468758872260756q^{19} + 9905700071276544q^{20} + 30747803854007016q^{21} - 38888514519613440q^{22} + 113342630802000600q^{23} - 36520347436056576q^{24} + 50579323492485631q^{25} + 552829014267731968q^{26} + 150094635296999121q^{27} + 970686391874748416q^{28} + 1081348899350530974q^{29} - 1285228308491034624q^{30} + 4649090467326833408q^{31} - 1152921504606846976q^{32} + 5045642344926241740q^{33} + 10346963012829241344q^{34} + 34160508239654300784q^{35} + 4738381338321616896q^{36} - 46093370056702003258q^{37} - 46976036340780056576q^{38} - 71727540080922300978q^{39} - 40573747491948724224q^{40} + 51449233931826001194q^{41} - 125943004586012737536q^{42} - 369342639690619984084q^{43} + 159287355472336650240q^{44} + 166753666380074202054q^{45} - 464251415764994457600q^{46} - 49106637730499080080q^{47} + 149587343098087735296q^{48} + 2006412148899668814969q^{49} - 207172909025221144576q^{50} - 1342480559204342004174q^{51} - 2264387642440630140928q^{52} + 4440077625909370178934q^{53} - 614787626176508399616q^{54} + 5605659113626079762760q^{55} - 3975931461118969511936q^{56} + 6094968683833128429396q^{57} - 4429205091739774869504q^{58} + 22549343358865698156540q^{59} + 5264295151579277819904q^{60} - 12310641025418994171490q^{61} - 19042674554170709639168q^{62} + 16340643627977342590056q^{63} + 4722366482869645213696q^{64} - 79688592901738798760172q^{65} - 20666951044817886167040q^{66} + 6385919851217871016196q^{67} - 42381160500548572545024q^{68} + 60234921056046000864600q^{69} - 139921441749624016011264q^{70} + 60276767662482517683720q^{71} - 19408409961765342806016q^{72} - 268812100727467655130358q^{73} + 188798443752251405344768q^{74} + 26879926256170056224271q^{75} + 192413844851835111735296q^{76} + 549313726433499472700640q^{77} + 293796004171457744805888q^{78} - 262861199604551219808592q^{79} + 166190069727021974421504q^{80} + 79766443076872509863361q^{81} - 210736062184759300890624q^{82} + 688505757189913419282132q^{83} + 515862546784308172947456q^{84} - 1491482722539198300096276q^{85} + 1512827452172779454808064q^{86} + 574673140419745531353534q^{87} - 652441008014690919383040q^{88} - 2558749855718408389837542q^{89} - 683023017492783931613184q^{90} - 7808901153959138265385808q^{91} + 1901573798973417298329600q^{92} + 2470717287046639673180928q^{93} + 201140788144124232007680q^{94} + 6771450375223569100694904q^{95} - 612709757329767363772416q^{96} + 2595207926228515475808482q^{97} - 8218264161893043466113024q^{98} + 2681461213429946836547340q^{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
−4096.00 531441. 1.67772e7 5.90426e8 −2.17678e9 5.78574e10 −6.87195e10 2.82430e11 −2.41838e12
\(n\): 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 6.26.a.b 1
3.b odd 2 1 18.26.a.b 1
4.b odd 2 1 48.26.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.26.a.b 1 1.a even 1 1 trivial
18.26.a.b 1 3.b odd 2 1
48.26.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 590425734 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4096 + T \)
$3$ \( -531441 + T \)
$5$ \( -590425734 + T \)
$7$ \( -57857417576 + T \)
$11$ \( -9494266240140 + T \)
$13$ \( 134968021061458 + T \)
$17$ \( 2526114016804014 + T \)
$19$ \( -11468758872260756 + T \)
$23$ \( -113342630802000600 + T \)
$29$ \( -1081348899350530974 + T \)
$31$ \( -4649090467326833408 + T \)
$37$ \( 46093370056702003258 + T \)
$41$ \( -51449233931826001194 + T \)
$43$ \( \)\(36\!\cdots\!84\)\( + T \)
$47$ \( 49106637730499080080 + T \)
$53$ \( -\)\(44\!\cdots\!34\)\( + T \)
$59$ \( -\)\(22\!\cdots\!40\)\( + T \)
$61$ \( \)\(12\!\cdots\!90\)\( + T \)
$67$ \( -\)\(63\!\cdots\!96\)\( + T \)
$71$ \( -\)\(60\!\cdots\!20\)\( + T \)
$73$ \( \)\(26\!\cdots\!58\)\( + T \)
$79$ \( \)\(26\!\cdots\!92\)\( + T \)
$83$ \( -\)\(68\!\cdots\!32\)\( + T \)
$89$ \( \)\(25\!\cdots\!42\)\( + T \)
$97$ \( -\)\(25\!\cdots\!82\)\( + T \)
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