Properties

Label 6.14.a.a
Level $6$
Weight $14$
Character orbit 6.a
Self dual yes
Analytic conductor $6.434$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.43385573711\)
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

\(f(q)\) \(=\) \( q + 64q^{2} - 729q^{3} + 4096q^{4} + 54654q^{5} - 46656q^{6} + 176336q^{7} + 262144q^{8} + 531441q^{9} + O(q^{10}) \) \( q + 64q^{2} - 729q^{3} + 4096q^{4} + 54654q^{5} - 46656q^{6} + 176336q^{7} + 262144q^{8} + 531441q^{9} + 3497856q^{10} + 6612420q^{11} - 2985984q^{12} - 24028978q^{13} + 11285504q^{14} - 39842766q^{15} + 16777216q^{16} - 154665054q^{17} + 34012224q^{18} + 190034876q^{19} + 223862784q^{20} - 128548944q^{21} + 423194880q^{22} - 352957800q^{23} - 191102976q^{24} + 1766356591q^{25} - 1537854592q^{26} - 387420489q^{27} + 722272256q^{28} - 2804086266q^{29} - 2549937024q^{30} + 2763661208q^{31} + 1073741824q^{32} - 4820454180q^{33} - 9898563456q^{34} + 9637467744q^{35} + 2176782336q^{36} + 20030257622q^{37} + 12162232064q^{38} + 17517124962q^{39} + 14327218176q^{40} - 39624547206q^{41} - 8227132416q^{42} - 81486174844q^{43} + 27084472320q^{44} + 29045376414q^{45} - 22589299200q^{46} - 34136017440q^{47} - 12230590464q^{48} - 65794625511q^{49} + 113046821824q^{50} + 112750824366q^{51} - 98422693888q^{52} - 21810829986q^{53} - 24794911296q^{54} + 361395202680q^{55} + 46225424384q^{56} - 138535424604q^{57} - 179461521024q^{58} + 229219661220q^{59} - 163195969536q^{60} + 9799736750q^{61} + 176874317312q^{62} + 93712180176q^{63} + 68719476736q^{64} - 1313279763612q^{65} - 308509067520q^{66} + 789042707996q^{67} - 633508061184q^{68} + 257306236200q^{69} + 616797935616q^{70} - 369504705240q^{71} + 139314069504q^{72} - 693077725078q^{73} + 1281936487808q^{74} - 1287673954839q^{75} + 778382852096q^{76} + 1166007693120q^{77} + 1121095997568q^{78} + 2231309995208q^{79} + 916941963264q^{80} + 282429536481q^{81} - 2535971021184q^{82} + 2084328707772q^{83} - 526536474624q^{84} - 8453063861316q^{85} - 5215115190016q^{86} + 2044178887914q^{87} + 1733406228480q^{88} + 2221961096538q^{89} + 1858904090496q^{90} - 4237173864608q^{91} - 1445715148800q^{92} - 2014709020632q^{93} - 2184705116160q^{94} + 10386166112904q^{95} - 782757789696q^{96} + 10268379896642q^{97} - 4210856032704q^{98} + 3514111097220q^{99} + O(q^{100}) \)

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
64.0000 −729.000 4096.00 54654.0 −46656.0 176336. 262144. 531441. 3.49786e6
\(n\): 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 6.14.a.a 1
3.b odd 2 1 18.14.a.a 1
4.b odd 2 1 48.14.a.e 1
5.b even 2 1 150.14.a.b 1
5.c odd 4 2 150.14.c.d 2
8.b even 2 1 192.14.a.f 1
8.d odd 2 1 192.14.a.a 1
12.b even 2 1 144.14.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.14.a.a 1 1.a even 1 1 trivial
18.14.a.a 1 3.b odd 2 1
48.14.a.e 1 4.b odd 2 1
144.14.a.b 1 12.b even 2 1
150.14.a.b 1 5.b even 2 1
150.14.c.d 2 5.c odd 4 2
192.14.a.a 1 8.d odd 2 1
192.14.a.f 1 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -64 + T \)
$3$ \( 729 + T \)
$5$ \( -54654 + T \)
$7$ \( -176336 + T \)
$11$ \( -6612420 + T \)
$13$ \( 24028978 + T \)
$17$ \( 154665054 + T \)
$19$ \( -190034876 + T \)
$23$ \( 352957800 + T \)
$29$ \( 2804086266 + T \)
$31$ \( -2763661208 + T \)
$37$ \( -20030257622 + T \)
$41$ \( 39624547206 + T \)
$43$ \( 81486174844 + T \)
$47$ \( 34136017440 + T \)
$53$ \( 21810829986 + T \)
$59$ \( -229219661220 + T \)
$61$ \( -9799736750 + T \)
$67$ \( -789042707996 + T \)
$71$ \( 369504705240 + T \)
$73$ \( 693077725078 + T \)
$79$ \( -2231309995208 + T \)
$83$ \( -2084328707772 + T \)
$89$ \( -2221961096538 + T \)
$97$ \( -10268379896642 + T \)
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