# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1,22,Mod(1,1)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1, base_ring=CyclotomicField(1))

chi = DirichletCharacter(H, H._module([]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 288 q^{2} - 128844 q^{3} - 2014208 q^{4} + 21640950 q^{5} + 37107072 q^{6} - 768078808 q^{7} + 1184071680 q^{8} + 6140423133 q^{9}+O(q^{10})$$ q - 288 * q^2 - 128844 * q^3 - 2014208 * q^4 + 21640950 * q^5 + 37107072 * q^6 - 768078808 * q^7 + 1184071680 * q^8 + 6140423133 * q^9 $$q - 288 q^{2} - 128844 q^{3} - 2014208 q^{4} + 21640950 q^{5} + 37107072 q^{6} - 768078808 q^{7} + 1184071680 q^{8} + 6140423133 q^{9} - 6232593600 q^{10} - 94724929188 q^{11} + 259518615552 q^{12} - 80621789794 q^{13} + 221206696704 q^{14} - 2788306561800 q^{15} + 3883087691776 q^{16} + 3052282930002 q^{17} - 1768441862304 q^{18} - 7920788351740 q^{19} - 43589374617600 q^{20} + 98962345937952 q^{21} + 27280779606144 q^{22} - 73845437470344 q^{23} - 152560531537920 q^{24} - 8506441300625 q^{25} + 23219075460672 q^{26} + 556597069939080 q^{27} + 15\!\cdots\!64 q^{28}+ \cdots - 58\!\cdots\!04 q^{99}+O(q^{100})$$ q - 288 * q^2 - 128844 * q^3 - 2014208 * q^4 + 21640950 * q^5 + 37107072 * q^6 - 768078808 * q^7 + 1184071680 * q^8 + 6140423133 * q^9 - 6232593600 * q^10 - 94724929188 * q^11 + 259518615552 * q^12 - 80621789794 * q^13 + 221206696704 * q^14 - 2788306561800 * q^15 + 3883087691776 * q^16 + 3052282930002 * q^17 - 1768441862304 * q^18 - 7920788351740 * q^19 - 43589374617600 * q^20 + 98962345937952 * q^21 + 27280779606144 * q^22 - 73845437470344 * q^23 - 152560531537920 * q^24 - 8506441300625 * q^25 + 23219075460672 * q^26 + 556597069939080 * q^27 + 1547070479704064 * q^28 - 4253031736469010 * q^29 + 803032289798400 * q^30 + 1900541176310432 * q^31 - 3601507547086848 * q^32 + 12204738776298672 * q^33 - 879057483840576 * q^34 - 16621955079987600 * q^35 - 12368089397873664 * q^36 + 22191429912035222 * q^37 + 2281187045301120 * q^38 + 10387633884218136 * q^39 + 25624436023296000 * q^40 - 20622803144546358 * q^41 - 28501155630130176 * q^42 - 193605854685795844 * q^43 + 190795710169903104 * q^44 + 132884590000096350 * q^45 + 21267485991459072 * q^46 + 146960504315611632 * q^47 - 500312550559186944 * q^48 + 31399191215416857 * q^49 + 2449855094580000 * q^50 - 393268341833177688 * q^51 + 162389053977393152 * q^52 + 2038267110310687206 * q^53 - 160299956142455040 * q^54 - 2049937456311048600 * q^55 - 909460364560957440 * q^56 + 1020546054391588560 * q^57 + 1224873140103074880 * q^58 - 5975882742742352820 * q^59 + 5616229383230054400 * q^60 + 6190617154478149262 * q^61 - 547355858777404416 * q^62 - 4716328880610265464 * q^63 - 7106190945422409728 * q^64 - 1744732121842464300 * q^65 - 3514964767574017536 * q^66 + 16961315295446680052 * q^67 - 6147932695873468416 * q^68 + 9514541545429002336 * q^69 + 4787123063036428800 * q^70 - 5632758963952293528 * q^71 + 7270701135002173440 * q^72 - 43284759511102937494 * q^73 - 6391131814666143936 * q^74 + 1096003922937727500 * q^75 + 15954115264381521920 * q^76 + 72756210698603447904 * q^77 - 2991638558654823168 * q^78 - 51264938664949064560 * q^79 + 84033706583339827200 * q^80 - 135945187666282668519 * q^81 + 5939367305629351104 * q^82 + 48911854702961049156 * q^83 - 199330748886990422016 * q^84 + 66054302274026781900 * q^85 + 55758486149509203072 * q^86 + 547977621053613124440 * q^87 - 112161106041516195840 * q^88 - 504303489899844009030 * q^89 - 38270761920027748800 * q^90 + 61923888203802085552 * q^91 + 148740070916266647552 * q^92 - 244873327320541300608 * q^93 - 42324625242896150016 * q^94 - 171413384680587753000 * q^95 + 464032638396857843712 * q^96 + 808275058155029184482 * q^97 - 9042967070040054816 * q^98 - 581651146457782106004 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

$p$ $F_p(T)$
$2$ $$T + 288$$
$3$ $$T + 128844$$
$5$ $$T - 21640950$$
$7$ $$T + 768078808$$
$11$ $$T + 94724929188$$
$13$ $$T + 80621789794$$
$17$ $$T - 3052282930002$$
$19$ $$T + 7920788351740$$
$23$ $$T + 73845437470344$$
$29$ $$T + 4253031736469010$$
$31$ $$T - 1900541176310432$$
$37$ $$T - 22\!\cdots\!22$$
$41$ $$T + 20\!\cdots\!58$$
$43$ $$T + 19\!\cdots\!44$$
$47$ $$T - 14\!\cdots\!32$$
$53$ $$T - 20\!\cdots\!06$$
$59$ $$T + 59\!\cdots\!20$$
$61$ $$T - 61\!\cdots\!62$$
$67$ $$T - 16\!\cdots\!52$$
$71$ $$T + 56\!\cdots\!28$$
$73$ $$T + 43\!\cdots\!94$$
$79$ $$T + 51\!\cdots\!60$$
$83$ $$T - 48\!\cdots\!56$$
$89$ $$T + 50\!\cdots\!30$$
$97$ $$T - 80\!\cdots\!82$$