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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 41 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).