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

Related objects

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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:

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4 \) 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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