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

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

$p$ $F_p(T)$
$2$ \( 288 + T \)
$3$ \( 128844 + T \)
$5$ \( -21640950 + T \)
$7$ \( 768078808 + T \)
$11$ \( 94724929188 + T \)
$13$ \( 80621789794 + T \)
$17$ \( -3052282930002 + T \)
$19$ \( 7920788351740 + T \)
$23$ \( 73845437470344 + T \)
$29$ \( 4253031736469010 + T \)
$31$ \( -1900541176310432 + T \)
$37$ \( -22191429912035222 + T \)
$41$ \( 20622803144546358 + T \)
$43$ \( 193605854685795844 + T \)
$47$ \( -146960504315611632 + T \)
$53$ \( -2038267110310687206 + T \)
$59$ \( 5975882742742352820 + T \)
$61$ \( -6190617154478149262 + T \)
$67$ \( -16961315295446680052 + T \)
$71$ \( 5632758963952293528 + T \)
$73$ \( 43284759511102937494 + T \)
$79$ \( 51264938664949064560 + T \)
$83$ \( -48911854702961049156 + T \)
$89$ \( \)\(50\!\cdots\!30\)\( + T \)
$97$ \( -\)\(80\!\cdots\!82\)\( + T \)
show more
show less