Properties

Label 22.10.a.b
Level 22
Weight 10
Character orbit 22.a
Self dual Yes
Analytic conductor 11.331
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 137q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 595q^{5} \) \(\mathstrut +\mathstrut 2192q^{6} \) \(\mathstrut +\mathstrut 11354q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut -\mathstrut 914q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 137q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 595q^{5} \) \(\mathstrut +\mathstrut 2192q^{6} \) \(\mathstrut +\mathstrut 11354q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut -\mathstrut 914q^{9} \) \(\mathstrut -\mathstrut 9520q^{10} \) \(\mathstrut -\mathstrut 14641q^{11} \) \(\mathstrut +\mathstrut 35072q^{12} \) \(\mathstrut +\mathstrut 55620q^{13} \) \(\mathstrut +\mathstrut 181664q^{14} \) \(\mathstrut -\mathstrut 81515q^{15} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut +\mathstrut 421550q^{17} \) \(\mathstrut -\mathstrut 14624q^{18} \) \(\mathstrut -\mathstrut 435872q^{19} \) \(\mathstrut -\mathstrut 152320q^{20} \) \(\mathstrut +\mathstrut 1555498q^{21} \) \(\mathstrut -\mathstrut 234256q^{22} \) \(\mathstrut +\mathstrut 779077q^{23} \) \(\mathstrut +\mathstrut 561152q^{24} \) \(\mathstrut -\mathstrut 1599100q^{25} \) \(\mathstrut +\mathstrut 889920q^{26} \) \(\mathstrut -\mathstrut 2821789q^{27} \) \(\mathstrut +\mathstrut 2906624q^{28} \) \(\mathstrut +\mathstrut 1206768q^{29} \) \(\mathstrut -\mathstrut 1304240q^{30} \) \(\mathstrut -\mathstrut 7626195q^{31} \) \(\mathstrut +\mathstrut 1048576q^{32} \) \(\mathstrut -\mathstrut 2005817q^{33} \) \(\mathstrut +\mathstrut 6744800q^{34} \) \(\mathstrut -\mathstrut 6755630q^{35} \) \(\mathstrut -\mathstrut 233984q^{36} \) \(\mathstrut -\mathstrut 19473681q^{37} \) \(\mathstrut -\mathstrut 6973952q^{38} \) \(\mathstrut +\mathstrut 7619940q^{39} \) \(\mathstrut -\mathstrut 2437120q^{40} \) \(\mathstrut -\mathstrut 9906168q^{41} \) \(\mathstrut +\mathstrut 24887968q^{42} \) \(\mathstrut -\mathstrut 20662730q^{43} \) \(\mathstrut -\mathstrut 3748096q^{44} \) \(\mathstrut +\mathstrut 543830q^{45} \) \(\mathstrut +\mathstrut 12465232q^{46} \) \(\mathstrut +\mathstrut 2751600q^{47} \) \(\mathstrut +\mathstrut 8978432q^{48} \) \(\mathstrut +\mathstrut 88559709q^{49} \) \(\mathstrut -\mathstrut 25585600q^{50} \) \(\mathstrut +\mathstrut 57752350q^{51} \) \(\mathstrut +\mathstrut 14238720q^{52} \) \(\mathstrut +\mathstrut 78527114q^{53} \) \(\mathstrut -\mathstrut 45148624q^{54} \) \(\mathstrut +\mathstrut 8711395q^{55} \) \(\mathstrut +\mathstrut 46505984q^{56} \) \(\mathstrut -\mathstrut 59714464q^{57} \) \(\mathstrut +\mathstrut 19308288q^{58} \) \(\mathstrut -\mathstrut 105727893q^{59} \) \(\mathstrut -\mathstrut 20867840q^{60} \) \(\mathstrut -\mathstrut 24639860q^{61} \) \(\mathstrut -\mathstrut 122019120q^{62} \) \(\mathstrut -\mathstrut 10377556q^{63} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut -\mathstrut 33093900q^{65} \) \(\mathstrut -\mathstrut 32093072q^{66} \) \(\mathstrut -\mathstrut 94817369q^{67} \) \(\mathstrut +\mathstrut 107916800q^{68} \) \(\mathstrut +\mathstrut 106733549q^{69} \) \(\mathstrut -\mathstrut 108090080q^{70} \) \(\mathstrut +\mathstrut 338924343q^{71} \) \(\mathstrut -\mathstrut 3743744q^{72} \) \(\mathstrut +\mathstrut 10287972q^{73} \) \(\mathstrut -\mathstrut 311578896q^{74} \) \(\mathstrut -\mathstrut 219076700q^{75} \) \(\mathstrut -\mathstrut 111583232q^{76} \) \(\mathstrut -\mathstrut 166233914q^{77} \) \(\mathstrut +\mathstrut 121919040q^{78} \) \(\mathstrut +\mathstrut 556386626q^{79} \) \(\mathstrut -\mathstrut 38993920q^{80} \) \(\mathstrut -\mathstrut 368594831q^{81} \) \(\mathstrut -\mathstrut 158498688q^{82} \) \(\mathstrut +\mathstrut 22479190q^{83} \) \(\mathstrut +\mathstrut 398207488q^{84} \) \(\mathstrut -\mathstrut 250822250q^{85} \) \(\mathstrut -\mathstrut 330603680q^{86} \) \(\mathstrut +\mathstrut 165327216q^{87} \) \(\mathstrut -\mathstrut 59969536q^{88} \) \(\mathstrut -\mathstrut 213504377q^{89} \) \(\mathstrut +\mathstrut 8701280q^{90} \) \(\mathstrut +\mathstrut 631509480q^{91} \) \(\mathstrut +\mathstrut 199443712q^{92} \) \(\mathstrut -\mathstrut 1044788715q^{93} \) \(\mathstrut +\mathstrut 44025600q^{94} \) \(\mathstrut +\mathstrut 259343840q^{95} \) \(\mathstrut +\mathstrut 143654912q^{96} \) \(\mathstrut +\mathstrut 1512644953q^{97} \) \(\mathstrut +\mathstrut 1416955344q^{98} \) \(\mathstrut +\mathstrut 13381874q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 137.000 256.000 −595.000 2192.00 11354.0 4096.00 −914.000 −9520.00
\(n\): 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 \(T_{3} \) \(\mathstrut -\mathstrut 137 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).