Properties

Label 22.4.a.b
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

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{3} + 4q^{4} + 14q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 11q^{9} + O(q^{10}) \) \( q - 2q^{2} + 4q^{3} + 4q^{4} + 14q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 11q^{9} - 28q^{10} - 11q^{11} + 16q^{12} - 50q^{13} + 16q^{14} + 56q^{15} + 16q^{16} + 130q^{17} + 22q^{18} - 108q^{19} + 56q^{20} - 32q^{21} + 22q^{22} - 96q^{23} - 32q^{24} + 71q^{25} + 100q^{26} - 152q^{27} - 32q^{28} + 142q^{29} - 112q^{30} + 40q^{31} - 32q^{32} - 44q^{33} - 260q^{34} - 112q^{35} - 44q^{36} + 382q^{37} + 216q^{38} - 200q^{39} - 112q^{40} - 118q^{41} + 64q^{42} + 220q^{43} - 44q^{44} - 154q^{45} + 192q^{46} + 520q^{47} + 64q^{48} - 279q^{49} - 142q^{50} + 520q^{51} - 200q^{52} + 238q^{53} + 304q^{54} - 154q^{55} + 64q^{56} - 432q^{57} - 284q^{58} - 852q^{59} + 224q^{60} + 190q^{61} - 80q^{62} + 88q^{63} + 64q^{64} - 700q^{65} + 88q^{66} - 12q^{67} + 520q^{68} - 384q^{69} + 224q^{70} - 112q^{71} + 88q^{72} - 6q^{73} - 764q^{74} + 284q^{75} - 432q^{76} + 88q^{77} + 400q^{78} + 304q^{79} + 224q^{80} - 311q^{81} + 236q^{82} + 820q^{83} - 128q^{84} + 1820q^{85} - 440q^{86} + 568q^{87} + 88q^{88} + 202q^{89} + 308q^{90} + 400q^{91} - 384q^{92} + 160q^{93} - 1040q^{94} - 1512q^{95} - 128q^{96} - 1406q^{97} + 558q^{98} + 121q^{99} + O(q^{100}) \)

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 4.00000 4.00000 14.0000 −8.00000 −8.00000 −8.00000 −11.0000 −28.0000
\(n\): 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.b 1
3.b odd 2 1 198.4.a.d 1
4.b odd 2 1 176.4.a.b 1
5.b even 2 1 550.4.a.k 1
5.c odd 4 2 550.4.b.b 2
7.b odd 2 1 1078.4.a.a 1
8.b even 2 1 704.4.a.d 1
8.d odd 2 1 704.4.a.i 1
11.b odd 2 1 242.4.a.f 1
11.c even 5 4 242.4.c.h 4
11.d odd 10 4 242.4.c.b 4
12.b even 2 1 1584.4.a.b 1
33.d even 2 1 2178.4.a.a 1
44.c even 2 1 1936.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.b 1 1.a even 1 1 trivial
176.4.a.b 1 4.b odd 2 1
198.4.a.d 1 3.b odd 2 1
242.4.a.f 1 11.b odd 2 1
242.4.c.b 4 11.d odd 10 4
242.4.c.h 4 11.c even 5 4
550.4.a.k 1 5.b even 2 1
550.4.b.b 2 5.c odd 4 2
704.4.a.d 1 8.b even 2 1
704.4.a.i 1 8.d odd 2 1
1078.4.a.a 1 7.b odd 2 1
1584.4.a.b 1 12.b even 2 1
1936.4.a.g 1 44.c even 2 1
2178.4.a.a 1 33.d even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 - 4 T + 27 T^{2} \)
$5$ \( 1 - 14 T + 125 T^{2} \)
$7$ \( 1 + 8 T + 343 T^{2} \)
$11$ \( 1 + 11 T \)
$13$ \( 1 + 50 T + 2197 T^{2} \)
$17$ \( 1 - 130 T + 4913 T^{2} \)
$19$ \( 1 + 108 T + 6859 T^{2} \)
$23$ \( 1 + 96 T + 12167 T^{2} \)
$29$ \( 1 - 142 T + 24389 T^{2} \)
$31$ \( 1 - 40 T + 29791 T^{2} \)
$37$ \( 1 - 382 T + 50653 T^{2} \)
$41$ \( 1 + 118 T + 68921 T^{2} \)
$43$ \( 1 - 220 T + 79507 T^{2} \)
$47$ \( 1 - 520 T + 103823 T^{2} \)
$53$ \( 1 - 238 T + 148877 T^{2} \)
$59$ \( 1 + 852 T + 205379 T^{2} \)
$61$ \( 1 - 190 T + 226981 T^{2} \)
$67$ \( 1 + 12 T + 300763 T^{2} \)
$71$ \( 1 + 112 T + 357911 T^{2} \)
$73$ \( 1 + 6 T + 389017 T^{2} \)
$79$ \( 1 - 304 T + 493039 T^{2} \)
$83$ \( 1 - 820 T + 571787 T^{2} \)
$89$ \( 1 - 202 T + 704969 T^{2} \)
$97$ \( 1 + 1406 T + 912673 T^{2} \)
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