Properties

Label 22.10.a.a
Level 22
Weight 10
Character orbit 22.a
Self dual yes
Analytic conductor 11.331
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3307883956\)
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

\(f(q)\) \(=\) \( q + 16q^{2} - 41q^{3} + 256q^{4} - 1039q^{5} - 656q^{6} - 3482q^{7} + 4096q^{8} - 18002q^{9} + O(q^{10}) \) \( q + 16q^{2} - 41q^{3} + 256q^{4} - 1039q^{5} - 656q^{6} - 3482q^{7} + 4096q^{8} - 18002q^{9} - 16624q^{10} + 14641q^{11} - 10496q^{12} - 199796q^{13} - 55712q^{14} + 42599q^{15} + 65536q^{16} + 164038q^{17} - 288032q^{18} - 277560q^{19} - 265984q^{20} + 142762q^{21} + 234256q^{22} - 1211721q^{23} - 167936q^{24} - 873604q^{25} - 3196736q^{26} + 1545085q^{27} - 891392q^{28} + 4248880q^{29} + 681584q^{30} + 9112927q^{31} + 1048576q^{32} - 600281q^{33} + 2624608q^{34} + 3617798q^{35} - 4608512q^{36} + 10500403q^{37} - 4440960q^{38} + 8191636q^{39} - 4255744q^{40} - 844768q^{41} + 2284192q^{42} + 1083514q^{43} + 3748096q^{44} + 18704078q^{45} - 19387536q^{46} - 45843752q^{47} - 2686976q^{48} - 28229283q^{49} - 13977664q^{50} - 6725558q^{51} - 51147776q^{52} + 5568394q^{53} + 24721360q^{54} - 15211999q^{55} - 14262272q^{56} + 11379960q^{57} + 67982080q^{58} - 106773315q^{59} + 10905344q^{60} - 98810468q^{61} + 145806832q^{62} + 62682964q^{63} + 16777216q^{64} + 207588044q^{65} - 9604496q^{66} - 168277647q^{67} + 41993728q^{68} + 49680561q^{69} + 57884768q^{70} + 67984277q^{71} - 73736192q^{72} - 65392116q^{73} + 168006448q^{74} + 35817764q^{75} - 71055360q^{76} - 50979962q^{77} + 131066176q^{78} + 85785910q^{79} - 68091904q^{80} + 290984881q^{81} - 13516288q^{82} - 103589846q^{83} + 36547072q^{84} - 170435482q^{85} + 17336224q^{86} - 174204080q^{87} + 59969536q^{88} - 809499425q^{89} + 299265248q^{90} + 695689672q^{91} - 310200576q^{92} - 373630007q^{93} - 733500032q^{94} + 288384840q^{95} - 42991616q^{96} + 859612633q^{97} - 451668528q^{98} - 263567282q^{99} + O(q^{100}) \)

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
16.0000 −41.0000 256.000 −1039.00 −656.000 −3482.00 4096.00 −18002.0 −16624.0
\(n\): 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 22.10.a.a 1
3.b odd 2 1 198.10.a.d 1
4.b odd 2 1 176.10.a.c 1
11.b odd 2 1 242.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.a 1 1.a even 1 1 trivial
176.10.a.c 1 4.b odd 2 1
198.10.a.d 1 3.b odd 2 1
242.10.a.b 1 11.b odd 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 \( T_{3} + 41 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 + 41 T + 19683 T^{2} \)
$5$ \( 1 + 1039 T + 1953125 T^{2} \)
$7$ \( 1 + 3482 T + 40353607 T^{2} \)
$11$ \( 1 - 14641 T \)
$13$ \( 1 + 199796 T + 10604499373 T^{2} \)
$17$ \( 1 - 164038 T + 118587876497 T^{2} \)
$19$ \( 1 + 277560 T + 322687697779 T^{2} \)
$23$ \( 1 + 1211721 T + 1801152661463 T^{2} \)
$29$ \( 1 - 4248880 T + 14507145975869 T^{2} \)
$31$ \( 1 - 9112927 T + 26439622160671 T^{2} \)
$37$ \( 1 - 10500403 T + 129961739795077 T^{2} \)
$41$ \( 1 + 844768 T + 327381934393961 T^{2} \)
$43$ \( 1 - 1083514 T + 502592611936843 T^{2} \)
$47$ \( 1 + 45843752 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 5568394 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 106773315 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 98810468 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 168277647 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 67984277 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 65392116 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 85785910 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 103589846 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 809499425 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 859612633 T + 760231058654565217 T^{2} \)
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