Properties

Label 22.6.a.a
Level 22
Weight 6
Character orbit 22.a
Self dual yes
Analytic conductor 3.528
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52844403589\)
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

\(f(q)\) \(=\) \( q - 4q^{2} - 21q^{3} + 16q^{4} + 81q^{5} + 84q^{6} + 98q^{7} - 64q^{8} + 198q^{9} + O(q^{10}) \) \( q - 4q^{2} - 21q^{3} + 16q^{4} + 81q^{5} + 84q^{6} + 98q^{7} - 64q^{8} + 198q^{9} - 324q^{10} + 121q^{11} - 336q^{12} + 824q^{13} - 392q^{14} - 1701q^{15} + 256q^{16} + 978q^{17} - 792q^{18} - 2140q^{19} + 1296q^{20} - 2058q^{21} - 484q^{22} + 3699q^{23} + 1344q^{24} + 3436q^{25} - 3296q^{26} + 945q^{27} + 1568q^{28} + 3480q^{29} + 6804q^{30} - 7813q^{31} - 1024q^{32} - 2541q^{33} - 3912q^{34} + 7938q^{35} + 3168q^{36} - 13597q^{37} + 8560q^{38} - 17304q^{39} - 5184q^{40} + 6492q^{41} + 8232q^{42} + 14234q^{43} + 1936q^{44} + 16038q^{45} - 14796q^{46} - 20352q^{47} - 5376q^{48} - 7203q^{49} - 13744q^{50} - 20538q^{51} + 13184q^{52} - 366q^{53} - 3780q^{54} + 9801q^{55} - 6272q^{56} + 44940q^{57} - 13920q^{58} + 9825q^{59} - 27216q^{60} + 26132q^{61} + 31252q^{62} + 19404q^{63} + 4096q^{64} + 66744q^{65} + 10164q^{66} + 17093q^{67} + 15648q^{68} - 77679q^{69} - 31752q^{70} - 23583q^{71} - 12672q^{72} - 35176q^{73} + 54388q^{74} - 72156q^{75} - 34240q^{76} + 11858q^{77} + 69216q^{78} - 42490q^{79} + 20736q^{80} - 67959q^{81} - 25968q^{82} + 22674q^{83} - 32928q^{84} + 79218q^{85} - 56936q^{86} - 73080q^{87} - 7744q^{88} - 17145q^{89} - 64152q^{90} + 80752q^{91} + 59184q^{92} + 164073q^{93} + 81408q^{94} - 173340q^{95} + 21504q^{96} - 30727q^{97} + 28812q^{98} + 23958q^{99} + O(q^{100}) \)

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
−4.00000 −21.0000 16.0000 81.0000 84.0000 98.0000 −64.0000 198.000 −324.000
\(n\): 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 22.6.a.a 1
3.b odd 2 1 198.6.a.d 1
4.b odd 2 1 176.6.a.d 1
5.b even 2 1 550.6.a.g 1
5.c odd 4 2 550.6.b.g 2
7.b odd 2 1 1078.6.a.b 1
8.b even 2 1 704.6.a.i 1
8.d odd 2 1 704.6.a.b 1
11.b odd 2 1 242.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.a 1 1.a even 1 1 trivial
176.6.a.d 1 4.b odd 2 1
198.6.a.d 1 3.b odd 2 1
242.6.a.c 1 11.b odd 2 1
550.6.a.g 1 5.b even 2 1
550.6.b.g 2 5.c odd 4 2
704.6.a.b 1 8.d odd 2 1
704.6.a.i 1 8.b even 2 1
1078.6.a.b 1 7.b odd 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 + 21 T + 243 T^{2} \)
$5$ \( 1 - 81 T + 3125 T^{2} \)
$7$ \( 1 - 98 T + 16807 T^{2} \)
$11$ \( 1 - 121 T \)
$13$ \( 1 - 824 T + 371293 T^{2} \)
$17$ \( 1 - 978 T + 1419857 T^{2} \)
$19$ \( 1 + 2140 T + 2476099 T^{2} \)
$23$ \( 1 - 3699 T + 6436343 T^{2} \)
$29$ \( 1 - 3480 T + 20511149 T^{2} \)
$31$ \( 1 + 7813 T + 28629151 T^{2} \)
$37$ \( 1 + 13597 T + 69343957 T^{2} \)
$41$ \( 1 - 6492 T + 115856201 T^{2} \)
$43$ \( 1 - 14234 T + 147008443 T^{2} \)
$47$ \( 1 + 20352 T + 229345007 T^{2} \)
$53$ \( 1 + 366 T + 418195493 T^{2} \)
$59$ \( 1 - 9825 T + 714924299 T^{2} \)
$61$ \( 1 - 26132 T + 844596301 T^{2} \)
$67$ \( 1 - 17093 T + 1350125107 T^{2} \)
$71$ \( 1 + 23583 T + 1804229351 T^{2} \)
$73$ \( 1 + 35176 T + 2073071593 T^{2} \)
$79$ \( 1 + 42490 T + 3077056399 T^{2} \)
$83$ \( 1 - 22674 T + 3939040643 T^{2} \)
$89$ \( 1 + 17145 T + 5584059449 T^{2} \)
$97$ \( 1 + 30727 T + 8587340257 T^{2} \)
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