Properties

Label 6.28.a.a
Level $6$
Weight $28$
Character orbit 6.a
Self dual yes
Analytic conductor $27.711$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.7113344903\)
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

\(f(q)\) \(=\) \( q - 8192q^{2} + 1594323q^{3} + 67108864q^{4} + 1992850350q^{5} - 13060694016q^{6} - 321751224088q^{7} - 549755813888q^{8} + 2541865828329q^{9} + O(q^{10}) \) \( q - 8192q^{2} + 1594323q^{3} + 67108864q^{4} + 1992850350q^{5} - 13060694016q^{6} - 321751224088q^{7} - 549755813888q^{8} + 2541865828329q^{9} - 16325430067200q^{10} - 25382713790940q^{11} + 106993205379072q^{12} - 618784763833834q^{13} + 2635786027728896q^{14} + 3177247148563050q^{15} + 4503599627370496q^{16} + 42768440409795282q^{17} - 20822964865671168q^{18} + 42070253342628764q^{19} + 133737923110502400q^{20} - 512975376841652424q^{21} + 207935191375380480q^{22} - 3028321281867281400q^{23} - 876488338465357824q^{24} - 3479128079428705625q^{25} + 5069084785326768128q^{26} + 4052555153018976267q^{27} - 21592359139155116032q^{28} - 100366681142476985514q^{29} - 26028008641028505600q^{30} + 55016279561029110608q^{31} - 36893488147419103232q^{32} - 40468244399312833620q^{33} - 350359063837042950144q^{34} - 641202039536699230800q^{35} + 170581728179578208256q^{36} - 2821694106703256608066q^{37} - 344639515382814834688q^{38} - 986542781029849724382q^{39} - 1095581066121235660800q^{40} - 3427515768510486304806q^{41} + 4202294287086816657408q^{42} + 20535927846768765196868q^{43} - 1703405087747116892160q^{44} + 5065558205638487565150q^{45} + 24808007941056769228800q^{46} - 8141280020337107297040q^{47} + 7180192468708211294208q^{48} + 37811487838592111292201q^{49} + 28501017226679956480000q^{50} + 68186708219466043384086q^{51} - 41525942561396884504576q^{52} - 116044242102163951943778q^{53} - 33198531813531453579264q^{54} - 50583950062224605829000q^{55} + 176884606067958710534144q^{56} + 67073572519979918906772q^{57} + 822203851919171465330688q^{58} - 1158685835874911959210860q^{59} + 213221446787305517875200q^{60} - 1947671858682488283713530q^{61} - 450693362163950474100736q^{62} - 817848441732313817588952q^{63} + 302231454903657293676544q^{64} - 1233145433180923428741900q^{65} + 331515858119170733015040q^{66} - 1975718741948908545255028q^{67} + 2870141450953055847579648q^{68} - 4828122271070489683492200q^{69} + 5252727107884640098713600q^{70} + 2230407467653040108533080q^{71} - 1397405517247104682033152q^{72} + 14212344156312963793333466q^{73} + 23115318122113078133276672q^{74} - 5546853916979012238166875q^{75} + 2823286910016019125764096q^{76} + 8166919232910303924162720q^{77} + 8081758462196528942137344q^{78} - 4298873468785234776056608q^{79} + 8975000093665162533273600q^{80} + 6461081889226673298932241q^{81} + 28078209175637903808970752q^{82} - 97472194121103881757652644q^{83} - 34425194799815202057486336q^{84} + 85231101439614671162048700q^{85} - 168230320920729724492742656q^{86} - 160016908179117334975637022q^{87} + 13954294478824381580574720q^{88} - 92903297497219352080554678q^{89} - 41497052820590490133708800q^{90} + 199094755210540081330193392q^{91} - 203227201053137053522329600q^{92} + 87713719878578614711878384q^{93} + 66693365926601582977351680q^{94} + 83839719098446402257467400q^{95} - 58820136703657666922151936q^{96} + 1265152186081280981741120354q^{97} - 309751708373746575705710592q^{98} - 64519452815445634835539260q^{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
−8192.00 1.59432e6 6.71089e7 1.99285e9 −1.30607e10 −3.21751e11 −5.49756e11 2.54187e12 −1.63254e13
\(n\): 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 6.28.a.a 1
3.b odd 2 1 18.28.a.d 1
4.b odd 2 1 48.28.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.28.a.a 1 1.a even 1 1 trivial
18.28.a.d 1 3.b odd 2 1
48.28.a.b 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1992850350 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8192 + T \)
$3$ \( -1594323 + T \)
$5$ \( -1992850350 + T \)
$7$ \( 321751224088 + T \)
$11$ \( 25382713790940 + T \)
$13$ \( 618784763833834 + T \)
$17$ \( -42768440409795282 + T \)
$19$ \( -42070253342628764 + T \)
$23$ \( 3028321281867281400 + T \)
$29$ \( \)\(10\!\cdots\!14\)\( + T \)
$31$ \( -55016279561029110608 + T \)
$37$ \( \)\(28\!\cdots\!66\)\( + T \)
$41$ \( \)\(34\!\cdots\!06\)\( + T \)
$43$ \( -\)\(20\!\cdots\!68\)\( + T \)
$47$ \( \)\(81\!\cdots\!40\)\( + T \)
$53$ \( \)\(11\!\cdots\!78\)\( + T \)
$59$ \( \)\(11\!\cdots\!60\)\( + T \)
$61$ \( \)\(19\!\cdots\!30\)\( + T \)
$67$ \( \)\(19\!\cdots\!28\)\( + T \)
$71$ \( -\)\(22\!\cdots\!80\)\( + T \)
$73$ \( -\)\(14\!\cdots\!66\)\( + T \)
$79$ \( \)\(42\!\cdots\!08\)\( + T \)
$83$ \( \)\(97\!\cdots\!44\)\( + T \)
$89$ \( \)\(92\!\cdots\!78\)\( + T \)
$97$ \( -\)\(12\!\cdots\!54\)\( + T \)
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