Properties

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

Related objects

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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: \(1\)
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

\(f(q)\) \(=\) \( q - 4q^{2} + q^{3} + 16q^{4} - 51q^{5} - 4q^{6} - 166q^{7} - 64q^{8} - 242q^{9} + O(q^{10}) \) \( q - 4q^{2} + q^{3} + 16q^{4} - 51q^{5} - 4q^{6} - 166q^{7} - 64q^{8} - 242q^{9} + 204q^{10} - 121q^{11} + 16q^{12} + 692q^{13} + 664q^{14} - 51q^{15} + 256q^{16} - 738q^{17} + 968q^{18} + 1424q^{19} - 816q^{20} - 166q^{21} + 484q^{22} - 1779q^{23} - 64q^{24} - 524q^{25} - 2768q^{26} - 485q^{27} - 2656q^{28} - 2064q^{29} + 204q^{30} + 6245q^{31} - 1024q^{32} - 121q^{33} + 2952q^{34} + 8466q^{35} - 3872q^{36} - 14785q^{37} - 5696q^{38} + 692q^{39} + 3264q^{40} + 5304q^{41} + 664q^{42} + 17798q^{43} - 1936q^{44} + 12342q^{45} + 7116q^{46} - 17184q^{47} + 256q^{48} + 10749q^{49} + 2096q^{50} - 738q^{51} + 11072q^{52} - 30726q^{53} + 1940q^{54} + 6171q^{55} + 10624q^{56} + 1424q^{57} + 8256q^{58} - 34989q^{59} - 816q^{60} - 45940q^{61} - 24980q^{62} + 40172q^{63} + 4096q^{64} - 35292q^{65} + 484q^{66} + 25343q^{67} - 11808q^{68} - 1779q^{69} - 33864q^{70} + 13311q^{71} + 15488q^{72} - 53260q^{73} + 59140q^{74} - 524q^{75} + 22784q^{76} + 20086q^{77} - 2768q^{78} + 77234q^{79} - 13056q^{80} + 58321q^{81} - 21216q^{82} + 55014q^{83} - 2656q^{84} + 37638q^{85} - 71192q^{86} - 2064q^{87} + 7744q^{88} + 125415q^{89} - 49368q^{90} - 114872q^{91} - 28464q^{92} + 6245q^{93} + 68736q^{94} - 72624q^{95} - 1024q^{96} - 88807q^{97} - 42996q^{98} + 29282q^{99} + O(q^{100}) \)

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 1.00000 16.0000 −51.0000 −4.00000 −166.000 −64.0000 −242.000 204.000
\(n\): 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.b 1
3.b odd 2 1 198.6.a.i 1
4.b odd 2 1 176.6.a.b 1
5.b even 2 1 550.6.a.f 1
5.c odd 4 2 550.6.b.f 2
7.b odd 2 1 1078.6.a.a 1
8.b even 2 1 704.6.a.e 1
8.d odd 2 1 704.6.a.f 1
11.b odd 2 1 242.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 1.a even 1 1 trivial
176.6.a.b 1 4.b odd 2 1
198.6.a.i 1 3.b odd 2 1
242.6.a.d 1 11.b odd 2 1
550.6.a.f 1 5.b even 2 1
550.6.b.f 2 5.c odd 4 2
704.6.a.e 1 8.b even 2 1
704.6.a.f 1 8.d odd 2 1
1078.6.a.a 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 \( T_{3} - 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 - T + 243 T^{2} \)
$5$ \( 1 + 51 T + 3125 T^{2} \)
$7$ \( 1 + 166 T + 16807 T^{2} \)
$11$ \( 1 + 121 T \)
$13$ \( 1 - 692 T + 371293 T^{2} \)
$17$ \( 1 + 738 T + 1419857 T^{2} \)
$19$ \( 1 - 1424 T + 2476099 T^{2} \)
$23$ \( 1 + 1779 T + 6436343 T^{2} \)
$29$ \( 1 + 2064 T + 20511149 T^{2} \)
$31$ \( 1 - 6245 T + 28629151 T^{2} \)
$37$ \( 1 + 14785 T + 69343957 T^{2} \)
$41$ \( 1 - 5304 T + 115856201 T^{2} \)
$43$ \( 1 - 17798 T + 147008443 T^{2} \)
$47$ \( 1 + 17184 T + 229345007 T^{2} \)
$53$ \( 1 + 30726 T + 418195493 T^{2} \)
$59$ \( 1 + 34989 T + 714924299 T^{2} \)
$61$ \( 1 + 45940 T + 844596301 T^{2} \)
$67$ \( 1 - 25343 T + 1350125107 T^{2} \)
$71$ \( 1 - 13311 T + 1804229351 T^{2} \)
$73$ \( 1 + 53260 T + 2073071593 T^{2} \)
$79$ \( 1 - 77234 T + 3077056399 T^{2} \)
$83$ \( 1 - 55014 T + 3939040643 T^{2} \)
$89$ \( 1 - 125415 T + 5584059449 T^{2} \)
$97$ \( 1 + 88807 T + 8587340257 T^{2} \)
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