Properties

Label 1.22.a.a
Level 1
Weight 22
Character orbit 1.a
Self dual yes
Analytic conductor 2.795
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79477344287\)
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

\(f(q)\) \(=\) \( q - 288q^{2} - 128844q^{3} - 2014208q^{4} + 21640950q^{5} + 37107072q^{6} - 768078808q^{7} + 1184071680q^{8} + 6140423133q^{9} + O(q^{10}) \) \( q - 288q^{2} - 128844q^{3} - 2014208q^{4} + 21640950q^{5} + 37107072q^{6} - 768078808q^{7} + 1184071680q^{8} + 6140423133q^{9} - 6232593600q^{10} - 94724929188q^{11} + 259518615552q^{12} - 80621789794q^{13} + 221206696704q^{14} - 2788306561800q^{15} + 3883087691776q^{16} + 3052282930002q^{17} - 1768441862304q^{18} - 7920788351740q^{19} - 43589374617600q^{20} + 98962345937952q^{21} + 27280779606144q^{22} - 73845437470344q^{23} - 152560531537920q^{24} - 8506441300625q^{25} + 23219075460672q^{26} + 556597069939080q^{27} + 1547070479704064q^{28} - 4253031736469010q^{29} + 803032289798400q^{30} + 1900541176310432q^{31} - 3601507547086848q^{32} + 12204738776298672q^{33} - 879057483840576q^{34} - 16621955079987600q^{35} - 12368089397873664q^{36} + 22191429912035222q^{37} + 2281187045301120q^{38} + 10387633884218136q^{39} + 25624436023296000q^{40} - 20622803144546358q^{41} - 28501155630130176q^{42} - 193605854685795844q^{43} + 190795710169903104q^{44} + 132884590000096350q^{45} + 21267485991459072q^{46} + 146960504315611632q^{47} - 500312550559186944q^{48} + 31399191215416857q^{49} + 2449855094580000q^{50} - 393268341833177688q^{51} + 162389053977393152q^{52} + 2038267110310687206q^{53} - 160299956142455040q^{54} - 2049937456311048600q^{55} - 909460364560957440q^{56} + 1020546054391588560q^{57} + 1224873140103074880q^{58} - 5975882742742352820q^{59} + 5616229383230054400q^{60} + 6190617154478149262q^{61} - 547355858777404416q^{62} - 4716328880610265464q^{63} - 7106190945422409728q^{64} - 1744732121842464300q^{65} - 3514964767574017536q^{66} + 16961315295446680052q^{67} - 6147932695873468416q^{68} + 9514541545429002336q^{69} + 4787123063036428800q^{70} - 5632758963952293528q^{71} + 7270701135002173440q^{72} - 43284759511102937494q^{73} - 6391131814666143936q^{74} + 1096003922937727500q^{75} + 15954115264381521920q^{76} + 72756210698603447904q^{77} - 2991638558654823168q^{78} - 51264938664949064560q^{79} + 84033706583339827200q^{80} - 135945187666282668519q^{81} + 5939367305629351104q^{82} + 48911854702961049156q^{83} - 199330748886990422016q^{84} + 66054302274026781900q^{85} + 55758486149509203072q^{86} + 547977621053613124440q^{87} - 112161106041516195840q^{88} - 504303489899844009030q^{89} - 38270761920027748800q^{90} + 61923888203802085552q^{91} + 148740070916266647552q^{92} - 244873327320541300608q^{93} - 42324625242896150016q^{94} - 171413384680587753000q^{95} + 464032638396857843712q^{96} + 808275058155029184482q^{97} - 9042967070040054816q^{98} - 581651146457782106004q^{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
−288.000 −128844. −2.01421e6 2.16410e7 3.71071e7 −7.68079e8 1.18407e9 6.14042e9 −6.23259e9
\(n\): 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 1.22.a.a 1
3.b odd 2 1 9.22.a.c 1
4.b odd 2 1 16.22.a.c 1
5.b even 2 1 25.22.a.a 1
5.c odd 4 2 25.22.b.a 2
7.b odd 2 1 49.22.a.a 1
8.b even 2 1 64.22.a.g 1
8.d odd 2 1 64.22.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.22.a.a 1 1.a even 1 1 trivial
9.22.a.c 1 3.b odd 2 1
16.22.a.c 1 4.b odd 2 1
25.22.a.a 1 5.b even 2 1
25.22.b.a 2 5.c odd 4 2
49.22.a.a 1 7.b odd 2 1
64.22.a.a 1 8.d odd 2 1
64.22.a.g 1 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 288 T + 2097152 T^{2} \)
$3$ \( 1 + 128844 T + 10460353203 T^{2} \)
$5$ \( 1 - 21640950 T + 476837158203125 T^{2} \)
$7$ \( 1 + 768078808 T + 558545864083284007 T^{2} \)
$11$ \( 1 + 94724929188 T + \)\(74\!\cdots\!11\)\( T^{2} \)
$13$ \( 1 + 80621789794 T + \)\(24\!\cdots\!13\)\( T^{2} \)
$17$ \( 1 - 3052282930002 T + \)\(69\!\cdots\!17\)\( T^{2} \)
$19$ \( 1 + 7920788351740 T + \)\(71\!\cdots\!19\)\( T^{2} \)
$23$ \( 1 + 73845437470344 T + \)\(39\!\cdots\!23\)\( T^{2} \)
$29$ \( 1 + 4253031736469010 T + \)\(51\!\cdots\!29\)\( T^{2} \)
$31$ \( 1 - 1900541176310432 T + \)\(20\!\cdots\!31\)\( T^{2} \)
$37$ \( 1 - 22191429912035222 T + \)\(85\!\cdots\!37\)\( T^{2} \)
$41$ \( 1 + 20622803144546358 T + \)\(73\!\cdots\!41\)\( T^{2} \)
$43$ \( 1 + 193605854685795844 T + \)\(20\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - 146960504315611632 T + \)\(13\!\cdots\!47\)\( T^{2} \)
$53$ \( 1 - 2038267110310687206 T + \)\(16\!\cdots\!53\)\( T^{2} \)
$59$ \( 1 + 5975882742742352820 T + \)\(15\!\cdots\!59\)\( T^{2} \)
$61$ \( 1 - 6190617154478149262 T + \)\(31\!\cdots\!61\)\( T^{2} \)
$67$ \( 1 - 16961315295446680052 T + \)\(22\!\cdots\!67\)\( T^{2} \)
$71$ \( 1 + 5632758963952293528 T + \)\(75\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 43284759511102937494 T + \)\(13\!\cdots\!73\)\( T^{2} \)
$79$ \( 1 + 51264938664949064560 T + \)\(70\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - 48911854702961049156 T + \)\(19\!\cdots\!83\)\( T^{2} \)
$89$ \( 1 + \)\(50\!\cdots\!30\)\( T + \)\(86\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - \)\(80\!\cdots\!82\)\( T + \)\(52\!\cdots\!97\)\( T^{2} \)
show more
show less