# Properties

 Label 18.22.a.a Level $18$ Weight $22$ Character orbit 18.a Self dual yes Analytic conductor $50.306$ 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] = [18,22,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, 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("18.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 18.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: $$50.3059219717$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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)

 $$f(q)$$ $$=$$ $$q - 1024 q^{2} + 1048576 q^{4} - 12954174 q^{5} - 479513104 q^{7} - 1073741824 q^{8}+O(q^{10})$$ q - 1024 * q^2 + 1048576 * q^4 - 12954174 * q^5 - 479513104 * q^7 - 1073741824 * q^8 $$q - 1024 q^{2} + 1048576 q^{4} - 12954174 q^{5} - 479513104 q^{7} - 1073741824 q^{8} + 13265074176 q^{10} - 115657781700 q^{11} + 295658246702 q^{13} + 491021418496 q^{14} + 1099511627776 q^{16} - 6626983431906 q^{17} + 28576184164796 q^{19} - 13583435956224 q^{20} + 118433568460800 q^{22} - 335385196791000 q^{23} - 309026534180849 q^{25} - 302754044622848 q^{26} - 502805932539904 q^{28} + 699224214482106 q^{29} - 34\!\cdots\!92 q^{31}+ \cdots + 33\!\cdots\!84 q^{98}+O(q^{100})$$ q - 1024 * q^2 + 1048576 * q^4 - 12954174 * q^5 - 479513104 * q^7 - 1073741824 * q^8 + 13265074176 * q^10 - 115657781700 * q^11 + 295658246702 * q^13 + 491021418496 * q^14 + 1099511627776 * q^16 - 6626983431906 * q^17 + 28576184164796 * q^19 - 13583435956224 * q^20 + 118433568460800 * q^22 - 335385196791000 * q^23 - 309026534180849 * q^25 - 302754044622848 * q^26 - 502805932539904 * q^28 + 699224214482106 * q^29 - 3484957262657992 * q^31 - 1125899906842624 * q^32 + 6786031034271744 * q^34 + 6211696184496096 * q^35 - 35181531093012298 * q^37 - 29262012584751104 * q^38 + 13909438419173376 * q^40 - 6132056240639994 * q^41 + 233260850096910596 * q^43 - 121275974103859200 * q^44 + 343434441513984000 * q^46 + 580205712121346400 * q^47 - 328613047175569191 * q^49 + 316443171001189376 * q^50 + 310020141693796352 * q^52 + 1394471665941750306 * q^53 + 1498251028595815800 * q^55 + 514873274920861696 * q^56 - 716005595629676544 * q^58 - 2352476807159705700 * q^59 + 9920628300330384590 * q^61 + 3568596236961783808 * q^62 + 1152921504606846976 * q^64 - 3830008372312634148 * q^65 + 26068981808996843996 * q^67 - 6948895779094265856 * q^68 - 6360776892924002304 * q^70 + 13336955952504341400 * q^71 + 9037529597968684202 * q^73 + 36025887839244593152 * q^74 + 29964300886785130496 * q^76 + 55459421904721396800 * q^77 - 77283864571811027992 * q^79 - 14243264941233537024 * q^80 + 6279225590415353856 * q^82 + 155698418876868248388 * q^83 + 85847096472027475644 * q^85 - 238859110499236450304 * q^86 + 124186597482351820800 * q^88 - 253837312813381912218 * q^89 - 141772003599273783008 * q^91 - 351676868110319616000 * q^92 - 594130649212258713600 * q^94 - 370180861926812058504 * q^95 - 1030722092535365121598 * q^97 + 336499760307782851584 * 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
−1024.00 0 1.04858e6 −1.29542e7 0 −4.79513e8 −1.07374e9 0 1.32651e10
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## 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 18.22.a.a 1
3.b odd 2 1 6.22.a.b 1
12.b even 2 1 48.22.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.b 1 3.b odd 2 1
18.22.a.a 1 1.a even 1 1 trivial
48.22.a.f 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 12954174$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1024$$
$3$ $$T$$
$5$ $$T + 12954174$$
$7$ $$T + 479513104$$
$11$ $$T + 115657781700$$
$13$ $$T - 295658246702$$
$17$ $$T + 6626983431906$$
$19$ $$T - 28576184164796$$
$23$ $$T + 335385196791000$$
$29$ $$T - 699224214482106$$
$31$ $$T + 3484957262657992$$
$37$ $$T + 35\!\cdots\!98$$
$41$ $$T + 6132056240639994$$
$43$ $$T - 23\!\cdots\!96$$
$47$ $$T - 58\!\cdots\!00$$
$53$ $$T - 13\!\cdots\!06$$
$59$ $$T + 23\!\cdots\!00$$
$61$ $$T - 99\!\cdots\!90$$
$67$ $$T - 26\!\cdots\!96$$
$71$ $$T - 13\!\cdots\!00$$
$73$ $$T - 90\!\cdots\!02$$
$79$ $$T + 77\!\cdots\!92$$
$83$ $$T - 15\!\cdots\!88$$
$89$ $$T + 25\!\cdots\!18$$
$97$ $$T + 10\!\cdots\!98$$