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

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T + 10 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 16 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T - 116 \) Copy content Toggle raw display
$23$ \( T - 189 \) Copy content Toggle raw display
$29$ \( T + 120 \) Copy content Toggle raw display
$31$ \( T + 163 \) Copy content Toggle raw display
$37$ \( T + 409 \) Copy content Toggle raw display
$41$ \( T - 468 \) Copy content Toggle raw display
$43$ \( T - 110 \) Copy content Toggle raw display
$47$ \( T - 144 \) Copy content Toggle raw display
$53$ \( T - 90 \) Copy content Toggle raw display
$59$ \( T + 453 \) Copy content Toggle raw display
$61$ \( T - 20 \) Copy content Toggle raw display
$67$ \( T + 97 \) Copy content Toggle raw display
$71$ \( T + 465 \) Copy content Toggle raw display
$73$ \( T - 848 \) Copy content Toggle raw display
$79$ \( T + 742 \) Copy content Toggle raw display
$83$ \( T - 438 \) Copy content Toggle raw display
$89$ \( T + 273 \) Copy content Toggle raw display
$97$ \( T - 761 \) Copy content Toggle raw display
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