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

\(f(q)\) \(=\) \( q - 2q^{2} - 7q^{3} + 4q^{4} - 19q^{5} + 14q^{6} + 14q^{7} - 8q^{8} + 22q^{9} + O(q^{10}) \) \( q - 2q^{2} - 7q^{3} + 4q^{4} - 19q^{5} + 14q^{6} + 14q^{7} - 8q^{8} + 22q^{9} + 38q^{10} + 11q^{11} - 28q^{12} - 72q^{13} - 28q^{14} + 133q^{15} + 16q^{16} - 46q^{17} - 44q^{18} - 20q^{19} - 76q^{20} - 98q^{21} - 22q^{22} - 107q^{23} + 56q^{24} + 236q^{25} + 144q^{26} + 35q^{27} + 56q^{28} + 120q^{29} - 266q^{30} + 117q^{31} - 32q^{32} - 77q^{33} + 92q^{34} - 266q^{35} + 88q^{36} - 201q^{37} + 40q^{38} + 504q^{39} + 152q^{40} - 228q^{41} + 196q^{42} - 242q^{43} + 44q^{44} - 418q^{45} + 214q^{46} - 96q^{47} - 112q^{48} - 147q^{49} - 472q^{50} + 322q^{51} - 288q^{52} + 458q^{53} - 70q^{54} - 209q^{55} - 112q^{56} + 140q^{57} - 240q^{58} + 435q^{59} + 532q^{60} - 668q^{61} - 234q^{62} + 308q^{63} + 64q^{64} + 1368q^{65} + 154q^{66} + 439q^{67} - 184q^{68} + 749q^{69} + 532q^{70} - 1113q^{71} - 176q^{72} - 72q^{73} + 402q^{74} - 1652q^{75} - 80q^{76} + 154q^{77} - 1008q^{78} - 70q^{79} - 304q^{80} - 839q^{81} + 456q^{82} + 358q^{83} - 392q^{84} + 874q^{85} + 484q^{86} - 840q^{87} - 88q^{88} + 895q^{89} + 836q^{90} - 1008q^{91} - 428q^{92} - 819q^{93} + 192q^{94} + 380q^{95} + 224q^{96} + 409q^{97} + 294q^{98} + 242q^{99} + O(q^{100}) \)

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 −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 \(S_{4}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 + 7 T + 27 T^{2} \)
$5$ \( 1 + 19 T + 125 T^{2} \)
$7$ \( 1 - 14 T + 343 T^{2} \)
$11$ \( 1 - 11 T \)
$13$ \( 1 + 72 T + 2197 T^{2} \)
$17$ \( 1 + 46 T + 4913 T^{2} \)
$19$ \( 1 + 20 T + 6859 T^{2} \)
$23$ \( 1 + 107 T + 12167 T^{2} \)
$29$ \( 1 - 120 T + 24389 T^{2} \)
$31$ \( 1 - 117 T + 29791 T^{2} \)
$37$ \( 1 + 201 T + 50653 T^{2} \)
$41$ \( 1 + 228 T + 68921 T^{2} \)
$43$ \( 1 + 242 T + 79507 T^{2} \)
$47$ \( 1 + 96 T + 103823 T^{2} \)
$53$ \( 1 - 458 T + 148877 T^{2} \)
$59$ \( 1 - 435 T + 205379 T^{2} \)
$61$ \( 1 + 668 T + 226981 T^{2} \)
$67$ \( 1 - 439 T + 300763 T^{2} \)
$71$ \( 1 + 1113 T + 357911 T^{2} \)
$73$ \( 1 + 72 T + 389017 T^{2} \)
$79$ \( 1 + 70 T + 493039 T^{2} \)
$83$ \( 1 - 358 T + 571787 T^{2} \)
$89$ \( 1 - 895 T + 704969 T^{2} \)
$97$ \( 1 - 409 T + 912673 T^{2} \)
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