Properties

Label 22.8.a.c
Level 22
Weight 8
Character orbit 22.a
Self dual yes
Analytic conductor 6.872
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.87247056065\)
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

\(f(q)\) \(=\) \( q + 8q^{2} - 21q^{3} + 64q^{4} - 551q^{5} - 168q^{6} + 62q^{7} + 512q^{8} - 1746q^{9} + O(q^{10}) \) \( q + 8q^{2} - 21q^{3} + 64q^{4} - 551q^{5} - 168q^{6} + 62q^{7} + 512q^{8} - 1746q^{9} - 4408q^{10} - 1331q^{11} - 1344q^{12} + 1500q^{13} + 496q^{14} + 11571q^{15} + 4096q^{16} - 29930q^{17} - 13968q^{18} + 29512q^{19} - 35264q^{20} - 1302q^{21} - 10648q^{22} + 31499q^{23} - 10752q^{24} + 225476q^{25} + 12000q^{26} + 82593q^{27} + 3968q^{28} - 75168q^{29} + 92568q^{30} - 235845q^{31} + 32768q^{32} + 27951q^{33} - 239440q^{34} - 34162q^{35} - 111744q^{36} + 75507q^{37} + 236096q^{38} - 31500q^{39} - 282112q^{40} - 270288q^{41} - 10416q^{42} - 1028030q^{43} - 85184q^{44} + 962046q^{45} + 251992q^{46} - 771840q^{47} - 86016q^{48} - 819699q^{49} + 1803808q^{50} + 628530q^{51} + 96000q^{52} + 765778q^{53} + 660744q^{54} + 733381q^{55} + 31744q^{56} - 619752q^{57} - 601344q^{58} - 392007q^{59} + 740544q^{60} + 1248460q^{61} - 1886760q^{62} - 108252q^{63} + 262144q^{64} - 826500q^{65} + 223608q^{66} + 3498133q^{67} - 1915520q^{68} - 661479q^{69} - 273296q^{70} + 1101753q^{71} - 893952q^{72} - 1122996q^{73} + 604056q^{74} - 4734996q^{75} + 1888768q^{76} - 82522q^{77} - 252000q^{78} - 4362946q^{79} - 2256896q^{80} + 2084049q^{81} - 2162304q^{82} - 4437790q^{83} - 83328q^{84} + 16491430q^{85} - 8224240q^{86} + 1578528q^{87} - 681472q^{88} - 521233q^{89} + 7696368q^{90} + 93000q^{91} + 2015936q^{92} + 4952745q^{93} - 6174720q^{94} - 16261112q^{95} - 688128q^{96} - 2129831q^{97} - 6557592q^{98} + 2323926q^{99} + O(q^{100}) \)

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 −21.0000 64.0000 −551.000 −168.000 62.0000 512.000 −1746.00 −4408.00
\(n\): 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 22.8.a.c 1
3.b odd 2 1 198.8.a.b 1
4.b odd 2 1 176.8.a.c 1
5.b even 2 1 550.8.a.a 1
11.b odd 2 1 242.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.8.a.c 1 1.a even 1 1 trivial
176.8.a.c 1 4.b odd 2 1
198.8.a.b 1 3.b odd 2 1
242.8.a.b 1 11.b odd 2 1
550.8.a.a 1 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 21 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T \)
$3$ \( 1 + 21 T + 2187 T^{2} \)
$5$ \( 1 + 551 T + 78125 T^{2} \)
$7$ \( 1 - 62 T + 823543 T^{2} \)
$11$ \( 1 + 1331 T \)
$13$ \( 1 - 1500 T + 62748517 T^{2} \)
$17$ \( 1 + 29930 T + 410338673 T^{2} \)
$19$ \( 1 - 29512 T + 893871739 T^{2} \)
$23$ \( 1 - 31499 T + 3404825447 T^{2} \)
$29$ \( 1 + 75168 T + 17249876309 T^{2} \)
$31$ \( 1 + 235845 T + 27512614111 T^{2} \)
$37$ \( 1 - 75507 T + 94931877133 T^{2} \)
$41$ \( 1 + 270288 T + 194754273881 T^{2} \)
$43$ \( 1 + 1028030 T + 271818611107 T^{2} \)
$47$ \( 1 + 771840 T + 506623120463 T^{2} \)
$53$ \( 1 - 765778 T + 1174711139837 T^{2} \)
$59$ \( 1 + 392007 T + 2488651484819 T^{2} \)
$61$ \( 1 - 1248460 T + 3142742836021 T^{2} \)
$67$ \( 1 - 3498133 T + 6060711605323 T^{2} \)
$71$ \( 1 - 1101753 T + 9095120158391 T^{2} \)
$73$ \( 1 + 1122996 T + 11047398519097 T^{2} \)
$79$ \( 1 + 4362946 T + 19203908986159 T^{2} \)
$83$ \( 1 + 4437790 T + 27136050989627 T^{2} \)
$89$ \( 1 + 521233 T + 44231334895529 T^{2} \)
$97$ \( 1 + 2129831 T + 80798284478113 T^{2} \)
show more
show less