Properties

Label 22.4.a.c
Level 22
Weight 4
Character orbit 22.a
Self dual yes
Analytic conductor 1.298
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.29804202013\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 10q^{7} + 8q^{8} - 26q^{9} + O(q^{10}) \) \( q + 2q^{2} + q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 10q^{7} + 8q^{8} - 26q^{9} - 6q^{10} + 11q^{11} + 4q^{12} - 16q^{13} - 20q^{14} - 3q^{15} + 16q^{16} + 42q^{17} - 52q^{18} + 116q^{19} - 12q^{20} - 10q^{21} + 22q^{22} + 189q^{23} + 8q^{24} - 116q^{25} - 32q^{26} - 53q^{27} - 40q^{28} - 120q^{29} - 6q^{30} - 163q^{31} + 32q^{32} + 11q^{33} + 84q^{34} + 30q^{35} - 104q^{36} - 409q^{37} + 232q^{38} - 16q^{39} - 24q^{40} + 468q^{41} - 20q^{42} + 110q^{43} + 44q^{44} + 78q^{45} + 378q^{46} + 144q^{47} + 16q^{48} - 243q^{49} - 232q^{50} + 42q^{51} - 64q^{52} + 90q^{53} - 106q^{54} - 33q^{55} - 80q^{56} + 116q^{57} - 240q^{58} - 453q^{59} - 12q^{60} + 20q^{61} - 326q^{62} + 260q^{63} + 64q^{64} + 48q^{65} + 22q^{66} - 97q^{67} + 168q^{68} + 189q^{69} + 60q^{70} - 465q^{71} - 208q^{72} + 848q^{73} - 818q^{74} - 116q^{75} + 464q^{76} - 110q^{77} - 32q^{78} - 742q^{79} - 48q^{80} + 649q^{81} + 936q^{82} + 438q^{83} - 40q^{84} - 126q^{85} + 220q^{86} - 120q^{87} + 88q^{88} - 273q^{89} + 156q^{90} + 160q^{91} + 756q^{92} - 163q^{93} + 288q^{94} - 348q^{95} + 32q^{96} + 761q^{97} - 486q^{98} - 286q^{99} + O(q^{100}) \)

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
2.00000 1.00000 4.00000 −3.00000 2.00000 −10.0000 8.00000 −26.0000 −6.00000
\(n\): 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 22.4.a.c 1
3.b odd 2 1 198.4.a.b 1
4.b odd 2 1 176.4.a.c 1
5.b even 2 1 550.4.a.e 1
5.c odd 4 2 550.4.b.g 2
7.b odd 2 1 1078.4.a.f 1
8.b even 2 1 704.4.a.e 1
8.d odd 2 1 704.4.a.g 1
11.b odd 2 1 242.4.a.a 1
11.c even 5 4 242.4.c.d 4
11.d odd 10 4 242.4.c.k 4
12.b even 2 1 1584.4.a.k 1
33.d even 2 1 2178.4.a.r 1
44.c even 2 1 1936.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.c 1 1.a even 1 1 trivial
176.4.a.c 1 4.b odd 2 1
198.4.a.b 1 3.b odd 2 1
242.4.a.a 1 11.b odd 2 1
242.4.c.d 4 11.c even 5 4
242.4.c.k 4 11.d odd 10 4
550.4.a.e 1 5.b even 2 1
550.4.b.g 2 5.c odd 4 2
704.4.a.e 1 8.b even 2 1
704.4.a.g 1 8.d odd 2 1
1078.4.a.f 1 7.b odd 2 1
1584.4.a.k 1 12.b even 2 1
1936.4.a.h 1 44.c even 2 1
2178.4.a.r 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 - T + 27 T^{2} \)
$5$ \( 1 + 3 T + 125 T^{2} \)
$7$ \( 1 + 10 T + 343 T^{2} \)
$11$ \( 1 - 11 T \)
$13$ \( 1 + 16 T + 2197 T^{2} \)
$17$ \( 1 - 42 T + 4913 T^{2} \)
$19$ \( 1 - 116 T + 6859 T^{2} \)
$23$ \( 1 - 189 T + 12167 T^{2} \)
$29$ \( 1 + 120 T + 24389 T^{2} \)
$31$ \( 1 + 163 T + 29791 T^{2} \)
$37$ \( 1 + 409 T + 50653 T^{2} \)
$41$ \( 1 - 468 T + 68921 T^{2} \)
$43$ \( 1 - 110 T + 79507 T^{2} \)
$47$ \( 1 - 144 T + 103823 T^{2} \)
$53$ \( 1 - 90 T + 148877 T^{2} \)
$59$ \( 1 + 453 T + 205379 T^{2} \)
$61$ \( 1 - 20 T + 226981 T^{2} \)
$67$ \( 1 + 97 T + 300763 T^{2} \)
$71$ \( 1 + 465 T + 357911 T^{2} \)
$73$ \( 1 - 848 T + 389017 T^{2} \)
$79$ \( 1 + 742 T + 493039 T^{2} \)
$83$ \( 1 - 438 T + 571787 T^{2} \)
$89$ \( 1 + 273 T + 704969 T^{2} \)
$97$ \( 1 - 761 T + 912673 T^{2} \)
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