Properties

Label 6.8.a.a
Level 6
Weight 8
Character orbit 6.a
Self dual yes
Analytic conductor 1.874
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: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.87431015290\)
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

\(f(q)\) \(=\) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 114q^{5} + 216q^{6} - 1576q^{7} + 512q^{8} + 729q^{9} + O(q^{10}) \) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 114q^{5} + 216q^{6} - 1576q^{7} + 512q^{8} + 729q^{9} - 912q^{10} + 7332q^{11} + 1728q^{12} - 3802q^{13} - 12608q^{14} - 3078q^{15} + 4096q^{16} - 6606q^{17} + 5832q^{18} + 24860q^{19} - 7296q^{20} - 42552q^{21} + 58656q^{22} + 41448q^{23} + 13824q^{24} - 65129q^{25} - 30416q^{26} + 19683q^{27} - 100864q^{28} - 41610q^{29} - 24624q^{30} + 33152q^{31} + 32768q^{32} + 197964q^{33} - 52848q^{34} + 179664q^{35} + 46656q^{36} - 36466q^{37} + 198880q^{38} - 102654q^{39} - 58368q^{40} - 639078q^{41} - 340416q^{42} - 156412q^{43} + 469248q^{44} - 83106q^{45} + 331584q^{46} - 433776q^{47} + 110592q^{48} + 1660233q^{49} - 521032q^{50} - 178362q^{51} - 243328q^{52} + 786078q^{53} + 157464q^{54} - 835848q^{55} - 806912q^{56} + 671220q^{57} - 332880q^{58} + 745140q^{59} - 196992q^{60} - 1660618q^{61} + 265216q^{62} - 1148904q^{63} + 262144q^{64} + 433428q^{65} + 1583712q^{66} - 3290836q^{67} - 422784q^{68} + 1119096q^{69} + 1437312q^{70} + 5716152q^{71} + 373248q^{72} + 2659898q^{73} - 291728q^{74} - 1758483q^{75} + 1591040q^{76} - 11555232q^{77} - 821232q^{78} + 3807440q^{79} - 466944q^{80} + 531441q^{81} - 5112624q^{82} + 2229468q^{83} - 2723328q^{84} + 753084q^{85} - 1251296q^{86} - 1123470q^{87} + 3753984q^{88} + 5991210q^{89} - 664848q^{90} + 5991952q^{91} + 2652672q^{92} + 895104q^{93} - 3470208q^{94} - 2834040q^{95} + 884736q^{96} - 4060126q^{97} + 13281864q^{98} + 5345028q^{99} + O(q^{100}) \)

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
8.00000 27.0000 64.0000 −114.000 216.000 −1576.00 512.000 729.000 −912.000
\(n\): 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 6.8.a.a 1
3.b odd 2 1 18.8.a.a 1
4.b odd 2 1 48.8.a.b 1
5.b even 2 1 150.8.a.e 1
5.c odd 4 2 150.8.c.k 2
7.b odd 2 1 294.8.a.l 1
7.c even 3 2 294.8.e.c 2
7.d odd 6 2 294.8.e.d 2
8.b even 2 1 192.8.a.f 1
8.d odd 2 1 192.8.a.n 1
9.c even 3 2 162.8.c.d 2
9.d odd 6 2 162.8.c.i 2
12.b even 2 1 144.8.a.h 1
15.d odd 2 1 450.8.a.ba 1
15.e even 4 2 450.8.c.a 2
24.f even 2 1 576.8.a.i 1
24.h odd 2 1 576.8.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.8.a.a 1 1.a even 1 1 trivial
18.8.a.a 1 3.b odd 2 1
48.8.a.b 1 4.b odd 2 1
144.8.a.h 1 12.b even 2 1
150.8.a.e 1 5.b even 2 1
150.8.c.k 2 5.c odd 4 2
162.8.c.d 2 9.c even 3 2
162.8.c.i 2 9.d odd 6 2
192.8.a.f 1 8.b even 2 1
192.8.a.n 1 8.d odd 2 1
294.8.a.l 1 7.b odd 2 1
294.8.e.c 2 7.c even 3 2
294.8.e.d 2 7.d odd 6 2
450.8.a.ba 1 15.d odd 2 1
450.8.c.a 2 15.e even 4 2
576.8.a.h 1 24.h odd 2 1
576.8.a.i 1 24.f even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T \)
$3$ \( 1 - 27 T \)
$5$ \( 1 + 114 T + 78125 T^{2} \)
$7$ \( 1 + 1576 T + 823543 T^{2} \)
$11$ \( 1 - 7332 T + 19487171 T^{2} \)
$13$ \( 1 + 3802 T + 62748517 T^{2} \)
$17$ \( 1 + 6606 T + 410338673 T^{2} \)
$19$ \( 1 - 24860 T + 893871739 T^{2} \)
$23$ \( 1 - 41448 T + 3404825447 T^{2} \)
$29$ \( 1 + 41610 T + 17249876309 T^{2} \)
$31$ \( 1 - 33152 T + 27512614111 T^{2} \)
$37$ \( 1 + 36466 T + 94931877133 T^{2} \)
$41$ \( 1 + 639078 T + 194754273881 T^{2} \)
$43$ \( 1 + 156412 T + 271818611107 T^{2} \)
$47$ \( 1 + 433776 T + 506623120463 T^{2} \)
$53$ \( 1 - 786078 T + 1174711139837 T^{2} \)
$59$ \( 1 - 745140 T + 2488651484819 T^{2} \)
$61$ \( 1 + 1660618 T + 3142742836021 T^{2} \)
$67$ \( 1 + 3290836 T + 6060711605323 T^{2} \)
$71$ \( 1 - 5716152 T + 9095120158391 T^{2} \)
$73$ \( 1 - 2659898 T + 11047398519097 T^{2} \)
$79$ \( 1 - 3807440 T + 19203908986159 T^{2} \)
$83$ \( 1 - 2229468 T + 27136050989627 T^{2} \)
$89$ \( 1 - 5991210 T + 44231334895529 T^{2} \)
$97$ \( 1 + 4060126 T + 80798284478113 T^{2} \)
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