Properties

Label 6.10.a.a
Level 6
Weight 10
Character orbit 6.a
Self dual yes
Analytic conductor 3.090
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.09021501698\)
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

\(f(q)\) \(=\) \( q - 16q^{2} + 81q^{3} + 256q^{4} + 2694q^{5} - 1296q^{6} - 3544q^{7} - 4096q^{8} + 6561q^{9} + O(q^{10}) \) \( q - 16q^{2} + 81q^{3} + 256q^{4} + 2694q^{5} - 1296q^{6} - 3544q^{7} - 4096q^{8} + 6561q^{9} - 43104q^{10} + 29580q^{11} + 20736q^{12} - 44818q^{13} + 56704q^{14} + 218214q^{15} + 65536q^{16} - 101934q^{17} - 104976q^{18} - 895084q^{19} + 689664q^{20} - 287064q^{21} - 473280q^{22} - 1113000q^{23} - 331776q^{24} + 5304511q^{25} + 717088q^{26} + 531441q^{27} - 907264q^{28} - 2357346q^{29} - 3491424q^{30} + 175808q^{31} - 1048576q^{32} + 2395980q^{33} + 1630944q^{34} - 9547536q^{35} + 1679616q^{36} - 2919418q^{37} + 14321344q^{38} - 3630258q^{39} - 11034624q^{40} + 26218794q^{41} + 4593024q^{42} - 18762964q^{43} + 7572480q^{44} + 17675334q^{45} + 17808000q^{46} - 20966160q^{47} + 5308416q^{48} - 27793671q^{49} - 84872176q^{50} - 8256654q^{51} - 11473408q^{52} + 57251574q^{53} - 8503056q^{54} + 79688520q^{55} + 14516224q^{56} - 72501804q^{57} + 37717536q^{58} + 33587580q^{59} + 55862784q^{60} + 82260830q^{61} - 2812928q^{62} - 23252184q^{63} + 16777216q^{64} - 120739692q^{65} - 38335680q^{66} - 188455804q^{67} - 26095104q^{68} - 90153000q^{69} + 152760576q^{70} + 80924040q^{71} - 26873856q^{72} - 236140918q^{73} + 46710688q^{74} + 429665391q^{75} - 229141504q^{76} - 104831520q^{77} + 58084128q^{78} + 526909808q^{79} + 176553984q^{80} + 43046721q^{81} - 419500704q^{82} + 18346452q^{83} - 73488384q^{84} - 274610196q^{85} + 300207424q^{86} - 190945026q^{87} - 121159680q^{88} + 690643098q^{89} - 282805344q^{90} + 158834992q^{91} - 284928000q^{92} + 14240448q^{93} + 335458560q^{94} - 2411356296q^{95} - 84934656q^{96} - 438251038q^{97} + 444698736q^{98} + 194074380q^{99} + O(q^{100}) \)

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
−16.0000 81.0000 256.000 2694.00 −1296.00 −3544.00 −4096.00 6561.00 −43104.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 6.10.a.a 1
3.b odd 2 1 18.10.a.c 1
4.b odd 2 1 48.10.a.d 1
5.b even 2 1 150.10.a.h 1
5.c odd 4 2 150.10.c.d 2
7.b odd 2 1 294.10.a.a 1
8.b even 2 1 192.10.a.a 1
8.d odd 2 1 192.10.a.h 1
9.c even 3 2 162.10.c.f 2
9.d odd 6 2 162.10.c.e 2
12.b even 2 1 144.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.10.a.a 1 1.a even 1 1 trivial
18.10.a.c 1 3.b odd 2 1
48.10.a.d 1 4.b odd 2 1
144.10.a.a 1 12.b even 2 1
150.10.a.h 1 5.b even 2 1
150.10.c.d 2 5.c odd 4 2
162.10.c.e 2 9.d odd 6 2
162.10.c.f 2 9.c even 3 2
192.10.a.a 1 8.b even 2 1
192.10.a.h 1 8.d odd 2 1
294.10.a.a 1 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T \)
$3$ \( 1 - 81 T \)
$5$ \( 1 - 2694 T + 1953125 T^{2} \)
$7$ \( 1 + 3544 T + 40353607 T^{2} \)
$11$ \( 1 - 29580 T + 2357947691 T^{2} \)
$13$ \( 1 + 44818 T + 10604499373 T^{2} \)
$17$ \( 1 + 101934 T + 118587876497 T^{2} \)
$19$ \( 1 + 895084 T + 322687697779 T^{2} \)
$23$ \( 1 + 1113000 T + 1801152661463 T^{2} \)
$29$ \( 1 + 2357346 T + 14507145975869 T^{2} \)
$31$ \( 1 - 175808 T + 26439622160671 T^{2} \)
$37$ \( 1 + 2919418 T + 129961739795077 T^{2} \)
$41$ \( 1 - 26218794 T + 327381934393961 T^{2} \)
$43$ \( 1 + 18762964 T + 502592611936843 T^{2} \)
$47$ \( 1 + 20966160 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 57251574 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 33587580 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 82260830 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 188455804 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 80924040 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 236140918 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 526909808 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 18346452 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 690643098 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 438251038 T + 760231058654565217 T^{2} \)
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