# Properties

 Label 9.22.a.a Level $9$ Weight $22$ Character orbit 9.a Self dual yes Analytic conductor $25.153$ Analytic rank $0$ 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] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 9.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: $$25.1529609858$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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)

 $$f(q)$$ $$=$$ $$q - 1728 q^{2} + 888832 q^{4} + 41512770 q^{5} + 538429808 q^{7} + 2087976960 q^{8}+O(q^{10})$$ q - 1728 * q^2 + 888832 * q^4 + 41512770 * q^5 + 538429808 * q^7 + 2087976960 * q^8 $$q - 1728 q^{2} + 888832 q^{4} + 41512770 q^{5} + 538429808 q^{7} + 2087976960 q^{8} - 71734066560 q^{10} + 64113040188 q^{11} - 130980107986 q^{13} - 930406708224 q^{14} - 5472039993344 q^{16} - 8242029723618 q^{17} + 13492101753020 q^{19} + 36897878384640 q^{20} - 110787333444864 q^{22} + 233184825844776 q^{23} + 12\!\cdots\!75 q^{25}+ \cdots + 46\!\cdots\!04 q^{98}+O(q^{100})$$ q - 1728 * q^2 + 888832 * q^4 + 41512770 * q^5 + 538429808 * q^7 + 2087976960 * q^8 - 71734066560 * q^10 + 64113040188 * q^11 - 130980107986 * q^13 - 930406708224 * q^14 - 5472039993344 * q^16 - 8242029723618 * q^17 + 13492101753020 * q^19 + 36897878384640 * q^20 - 110787333444864 * q^22 + 233184825844776 * q^23 + 1246472914869775 * q^25 + 226333626599808 * q^26 + 478573643104256 * q^28 + 2024562031123770 * q^29 - 6869194988701768 * q^31 + 5076880050880512 * q^32 + 14242227362411904 * q^34 + 22351712780648160 * q^35 + 3443998107027638 * q^37 - 23314351829218560 * q^38 + 86677707305779200 * q^40 + 21842403084625158 * q^41 - 71792816814133756 * q^43 + 56985721736380416 * q^44 - 402943379059772928 * q^46 - 283544719418655648 * q^47 - 268639205940367143 * q^49 - 2153905196894971200 * q^50 - 116419311341412352 * q^52 + 2172285419049898146 * q^53 + 2661509891325200760 * q^55 + 1124229033681223680 * q^56 - 3498443189781874560 * q^58 - 1534831476719068260 * q^59 + 4311589520797626062 * q^61 + 11869968940476655104 * q^62 + 2702850888199831552 * q^64 - 5437347097397981220 * q^65 + 9243910904037307868 * q^67 - 7325779763302834176 * q^68 - 38623759684960020480 * q^70 + 20387361256404760728 * q^71 + 16617754439328636074 * q^73 - 5951228728943758464 * q^74 + 11992211785340272640 * q^76 + 34520371918721123904 * q^77 + 67940304745507627880 * q^79 - 227159537674491002880 * q^80 - 37743672530232273024 * q^82 - 39503732340682314684 * q^83 - 342149484249717601860 * q^85 + 124057987454823130368 * q^86 + 133866550748098068480 * q^88 - 41611676186839694490 * q^89 - 70523594394721246688 * q^91 + 207262135125263941632 * q^92 + 489965275155436959744 * q^94 + 560094516889716065400 * q^95 + 57181473208903260098 * q^97 + 464208547864954423104 * q^98

## 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
−1728.00 0 888832. 4.15128e7 0 5.38430e8 2.08798e9 0 −7.17341e10
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## 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 9.22.a.a 1
3.b odd 2 1 3.22.a.b 1
12.b even 2 1 48.22.a.d 1
15.d odd 2 1 75.22.a.a 1
15.e even 4 2 75.22.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.b 1 3.b odd 2 1
9.22.a.a 1 1.a even 1 1 trivial
48.22.a.d 1 12.b even 2 1
75.22.a.a 1 15.d odd 2 1
75.22.b.b 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1728$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1728$$
$3$ $$T$$
$5$ $$T - 41512770$$
$7$ $$T - 538429808$$
$11$ $$T - 64113040188$$
$13$ $$T + 130980107986$$
$17$ $$T + 8242029723618$$
$19$ $$T - 13492101753020$$
$23$ $$T - 233184825844776$$
$29$ $$T - 2024562031123770$$
$31$ $$T + 6869194988701768$$
$37$ $$T - 3443998107027638$$
$41$ $$T - 21\!\cdots\!58$$
$43$ $$T + 71\!\cdots\!56$$
$47$ $$T + 28\!\cdots\!48$$
$53$ $$T - 21\!\cdots\!46$$
$59$ $$T + 15\!\cdots\!60$$
$61$ $$T - 43\!\cdots\!62$$
$67$ $$T - 92\!\cdots\!68$$
$71$ $$T - 20\!\cdots\!28$$
$73$ $$T - 16\!\cdots\!74$$
$79$ $$T - 67\!\cdots\!80$$
$83$ $$T + 39\!\cdots\!84$$
$89$ $$T + 41\!\cdots\!90$$
$97$ $$T - 57\!\cdots\!98$$