Properties

Label 22.4.a.a
Level $22$
Weight $4$
Character orbit 22.a
Self dual yes
Analytic conductor $1.298$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 7 q^{3} + 4 q^{4} - 19 q^{5} + 14 q^{6} + 14 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 7 q^{3} + 4 q^{4} - 19 q^{5} + 14 q^{6} + 14 q^{7} - 8 q^{8} + 22 q^{9} + 38 q^{10} + 11 q^{11} - 28 q^{12} - 72 q^{13} - 28 q^{14} + 133 q^{15} + 16 q^{16} - 46 q^{17} - 44 q^{18} - 20 q^{19} - 76 q^{20} - 98 q^{21} - 22 q^{22} - 107 q^{23} + 56 q^{24} + 236 q^{25} + 144 q^{26} + 35 q^{27} + 56 q^{28} + 120 q^{29} - 266 q^{30} + 117 q^{31} - 32 q^{32} - 77 q^{33} + 92 q^{34} - 266 q^{35} + 88 q^{36} - 201 q^{37} + 40 q^{38} + 504 q^{39} + 152 q^{40} - 228 q^{41} + 196 q^{42} - 242 q^{43} + 44 q^{44} - 418 q^{45} + 214 q^{46} - 96 q^{47} - 112 q^{48} - 147 q^{49} - 472 q^{50} + 322 q^{51} - 288 q^{52} + 458 q^{53} - 70 q^{54} - 209 q^{55} - 112 q^{56} + 140 q^{57} - 240 q^{58} + 435 q^{59} + 532 q^{60} - 668 q^{61} - 234 q^{62} + 308 q^{63} + 64 q^{64} + 1368 q^{65} + 154 q^{66} + 439 q^{67} - 184 q^{68} + 749 q^{69} + 532 q^{70} - 1113 q^{71} - 176 q^{72} - 72 q^{73} + 402 q^{74} - 1652 q^{75} - 80 q^{76} + 154 q^{77} - 1008 q^{78} - 70 q^{79} - 304 q^{80} - 839 q^{81} + 456 q^{82} + 358 q^{83} - 392 q^{84} + 874 q^{85} + 484 q^{86} - 840 q^{87} - 88 q^{88} + 895 q^{89} + 836 q^{90} - 1008 q^{91} - 428 q^{92} - 819 q^{93} + 192 q^{94} + 380 q^{95} + 224 q^{96} + 409 q^{97} + 294 q^{98} + 242 q^{99}+O(q^{100}) \) 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 −7.00000 4.00000 −19.0000 14.0000 14.0000 −8.00000 22.0000 38.0000
\(n\): 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.a 1
3.b odd 2 1 198.4.a.g 1
4.b odd 2 1 176.4.a.f 1
5.b even 2 1 550.4.a.n 1
5.c odd 4 2 550.4.b.k 2
7.b odd 2 1 1078.4.a.d 1
8.b even 2 1 704.4.a.l 1
8.d odd 2 1 704.4.a.b 1
11.b odd 2 1 242.4.a.d 1
11.c even 5 4 242.4.c.l 4
11.d odd 10 4 242.4.c.e 4
12.b even 2 1 1584.4.a.v 1
33.d even 2 1 2178.4.a.l 1
44.c even 2 1 1936.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.a 1 1.a even 1 1 trivial
176.4.a.f 1 4.b odd 2 1
198.4.a.g 1 3.b odd 2 1
242.4.a.d 1 11.b odd 2 1
242.4.c.e 4 11.d odd 10 4
242.4.c.l 4 11.c even 5 4
550.4.a.n 1 5.b even 2 1
550.4.b.k 2 5.c odd 4 2
704.4.a.b 1 8.d odd 2 1
704.4.a.l 1 8.b even 2 1
1078.4.a.d 1 7.b odd 2 1
1584.4.a.v 1 12.b even 2 1
1936.4.a.n 1 44.c even 2 1
2178.4.a.l 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 7 \) 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
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T + 19 \) Copy content Toggle raw display
$7$ \( T - 14 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 72 \) Copy content Toggle raw display
$17$ \( T + 46 \) Copy content Toggle raw display
$19$ \( T + 20 \) Copy content Toggle raw display
$23$ \( T + 107 \) Copy content Toggle raw display
$29$ \( T - 120 \) Copy content Toggle raw display
$31$ \( T - 117 \) Copy content Toggle raw display
$37$ \( T + 201 \) Copy content Toggle raw display
$41$ \( T + 228 \) Copy content Toggle raw display
$43$ \( T + 242 \) Copy content Toggle raw display
$47$ \( T + 96 \) Copy content Toggle raw display
$53$ \( T - 458 \) Copy content Toggle raw display
$59$ \( T - 435 \) Copy content Toggle raw display
$61$ \( T + 668 \) Copy content Toggle raw display
$67$ \( T - 439 \) Copy content Toggle raw display
$71$ \( T + 1113 \) Copy content Toggle raw display
$73$ \( T + 72 \) Copy content Toggle raw display
$79$ \( T + 70 \) Copy content Toggle raw display
$83$ \( T - 358 \) Copy content Toggle raw display
$89$ \( T - 895 \) Copy content Toggle raw display
$97$ \( T - 409 \) Copy content Toggle raw display
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