Properties

Label 6.16.a.b
Level $6$
Weight $16$
Character orbit 6.a
Self dual yes
Analytic conductor $8.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.56161030600\)
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

\(f(q)\) \(=\) \( q + 128q^{2} - 2187q^{3} + 16384q^{4} - 114810q^{5} - 279936q^{6} - 3034528q^{7} + 2097152q^{8} + 4782969q^{9} + O(q^{10}) \) \( q + 128q^{2} - 2187q^{3} + 16384q^{4} - 114810q^{5} - 279936q^{6} - 3034528q^{7} + 2097152q^{8} + 4782969q^{9} - 14695680q^{10} - 103451700q^{11} - 35831808q^{12} - 104365834q^{13} - 388419584q^{14} + 251089470q^{15} + 268435456q^{16} + 997689762q^{17} + 612220032q^{18} + 4934015444q^{19} - 1881047040q^{20} + 6636512736q^{21} - 13241817600q^{22} + 8324920200q^{23} - 4586471424q^{24} - 17336242025q^{25} - 13358826752q^{26} - 10460353203q^{27} - 49717706752q^{28} + 104128242846q^{29} + 32139452160q^{30} - 296696681512q^{31} + 34359738368q^{32} + 226248867900q^{33} + 127704289536q^{34} + 348394159680q^{35} + 78364164096q^{36} - 178337455666q^{37} + 631553976832q^{38} + 228248078958q^{39} - 240774021120q^{40} - 1790882416086q^{41} + 849473630208q^{42} - 2863459422772q^{43} - 1694952652800q^{44} - 549132670890q^{45} + 1065589785600q^{46} + 4332907521600q^{47} - 587068342272q^{48} + 4460798672841q^{49} - 2219038979200q^{50} - 2181947509494q^{51} - 1709929824256q^{52} + 9732317104422q^{53} - 1338925209984q^{54} + 11877289677000q^{55} - 6363866464256q^{56} - 10790691776028q^{57} + 13328415084288q^{58} - 13514837176500q^{59} + 4113849876480q^{60} + 5352663511190q^{61} - 37977175233536q^{62} - 14514053353632q^{63} + 4398046511104q^{64} + 11982241401540q^{65} + 28959855091200q^{66} - 53233909720108q^{67} + 16346149060608q^{68} - 18206600477400q^{69} + 44594452439040q^{70} - 20229661643400q^{71} + 10030613004288q^{72} + 26264166466106q^{73} - 22827194325248q^{74} + 37914361308675q^{75} + 80838909034496q^{76} + 313927080297600q^{77} + 29215754106624q^{78} - 339031361615128q^{79} - 30819074703360q^{80} + 22876792454961q^{81} - 229232949259008q^{82} + 131684771045076q^{83} + 108732624666624q^{84} - 114544761575220q^{85} - 366522806114816q^{86} - 227728467104202q^{87} - 216953939558400q^{88} - 39352148322678q^{89} - 70288981873920q^{90} + 316701045516352q^{91} + 136395492556800q^{92} + 648875642466744q^{93} + 554612162764800q^{94} - 566474313125640q^{95} - 75144747810816q^{96} + 1128750908801474q^{97} + 570982230123648q^{98} - 494806274097300q^{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
128.000 −2187.00 16384.0 −114810. −279936. −3.03453e6 2.09715e6 4.78297e6 −1.46957e7
\(n\): 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 6.16.a.b 1
3.b odd 2 1 18.16.a.b 1
4.b odd 2 1 48.16.a.d 1
5.b even 2 1 150.16.a.f 1
5.c odd 4 2 150.16.c.a 2
12.b even 2 1 144.16.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 1.a even 1 1 trivial
18.16.a.b 1 3.b odd 2 1
48.16.a.d 1 4.b odd 2 1
144.16.a.j 1 12.b even 2 1
150.16.a.f 1 5.b even 2 1
150.16.c.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 114810 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -128 + T \)
$3$ \( 2187 + T \)
$5$ \( 114810 + T \)
$7$ \( 3034528 + T \)
$11$ \( 103451700 + T \)
$13$ \( 104365834 + T \)
$17$ \( -997689762 + T \)
$19$ \( -4934015444 + T \)
$23$ \( -8324920200 + T \)
$29$ \( -104128242846 + T \)
$31$ \( 296696681512 + T \)
$37$ \( 178337455666 + T \)
$41$ \( 1790882416086 + T \)
$43$ \( 2863459422772 + T \)
$47$ \( -4332907521600 + T \)
$53$ \( -9732317104422 + T \)
$59$ \( 13514837176500 + T \)
$61$ \( -5352663511190 + T \)
$67$ \( 53233909720108 + T \)
$71$ \( 20229661643400 + T \)
$73$ \( -26264166466106 + T \)
$79$ \( 339031361615128 + T \)
$83$ \( -131684771045076 + T \)
$89$ \( 39352148322678 + T \)
$97$ \( -1128750908801474 + T \)
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