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

$q$-expansion

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

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 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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.10.a.b 1
3.b odd 2 1 198.10.a.b 1
4.b odd 2 1 176.10.a.b 1
11.b odd 2 1 242.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.b 1 1.a even 1 1 trivial
176.10.a.b 1 4.b odd 2 1
198.10.a.b 1 3.b odd 2 1
242.10.a.c 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 \( T_{3} - 137 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 - 137 T + 19683 T^{2} \)
$5$ \( 1 + 595 T + 1953125 T^{2} \)
$7$ \( 1 - 11354 T + 40353607 T^{2} \)
$11$ \( 1 + 14641 T \)
$13$ \( 1 - 55620 T + 10604499373 T^{2} \)
$17$ \( 1 - 421550 T + 118587876497 T^{2} \)
$19$ \( 1 + 435872 T + 322687697779 T^{2} \)
$23$ \( 1 - 779077 T + 1801152661463 T^{2} \)
$29$ \( 1 - 1206768 T + 14507145975869 T^{2} \)
$31$ \( 1 + 7626195 T + 26439622160671 T^{2} \)
$37$ \( 1 + 19473681 T + 129961739795077 T^{2} \)
$41$ \( 1 + 9906168 T + 327381934393961 T^{2} \)
$43$ \( 1 + 20662730 T + 502592611936843 T^{2} \)
$47$ \( 1 - 2751600 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 78527114 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 105727893 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 24639860 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 94817369 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 338924343 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 10287972 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 556386626 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 22479190 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 213504377 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 1512644953 T + 760231058654565217 T^{2} \)
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