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 $1$

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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

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

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:

Atkin-Lehner signs

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

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 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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T + 21 \) Copy content Toggle raw display
$5$ \( T + 551 \) Copy content Toggle raw display
$7$ \( T - 62 \) Copy content Toggle raw display
$11$ \( T + 1331 \) Copy content Toggle raw display
$13$ \( T - 1500 \) Copy content Toggle raw display
$17$ \( T + 29930 \) Copy content Toggle raw display
$19$ \( T - 29512 \) Copy content Toggle raw display
$23$ \( T - 31499 \) Copy content Toggle raw display
$29$ \( T + 75168 \) Copy content Toggle raw display
$31$ \( T + 235845 \) Copy content Toggle raw display
$37$ \( T - 75507 \) Copy content Toggle raw display
$41$ \( T + 270288 \) Copy content Toggle raw display
$43$ \( T + 1028030 \) Copy content Toggle raw display
$47$ \( T + 771840 \) Copy content Toggle raw display
$53$ \( T - 765778 \) Copy content Toggle raw display
$59$ \( T + 392007 \) Copy content Toggle raw display
$61$ \( T - 1248460 \) Copy content Toggle raw display
$67$ \( T - 3498133 \) Copy content Toggle raw display
$71$ \( T - 1101753 \) Copy content Toggle raw display
$73$ \( T + 1122996 \) Copy content Toggle raw display
$79$ \( T + 4362946 \) Copy content Toggle raw display
$83$ \( T + 4437790 \) Copy content Toggle raw display
$89$ \( T + 521233 \) Copy content Toggle raw display
$97$ \( T + 2129831 \) Copy content Toggle raw display
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