# Properties

 Label 18.4.a.a Level $18$ Weight $4$ Character orbit 18.a Self dual yes Analytic conductor $1.062$ 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] = [18,4,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, 4, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$1.06203438010$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 16 q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 16 * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 16 q^{7} + 8 q^{8} - 12 q^{10} - 12 q^{11} + 38 q^{13} - 32 q^{14} + 16 q^{16} + 126 q^{17} + 20 q^{19} - 24 q^{20} - 24 q^{22} - 168 q^{23} - 89 q^{25} + 76 q^{26} - 64 q^{28} - 30 q^{29} - 88 q^{31} + 32 q^{32} + 252 q^{34} + 96 q^{35} + 254 q^{37} + 40 q^{38} - 48 q^{40} - 42 q^{41} - 52 q^{43} - 48 q^{44} - 336 q^{46} + 96 q^{47} - 87 q^{49} - 178 q^{50} + 152 q^{52} - 198 q^{53} + 72 q^{55} - 128 q^{56} - 60 q^{58} + 660 q^{59} - 538 q^{61} - 176 q^{62} + 64 q^{64} - 228 q^{65} + 884 q^{67} + 504 q^{68} + 192 q^{70} - 792 q^{71} + 218 q^{73} + 508 q^{74} + 80 q^{76} + 192 q^{77} - 520 q^{79} - 96 q^{80} - 84 q^{82} + 492 q^{83} - 756 q^{85} - 104 q^{86} - 96 q^{88} - 810 q^{89} - 608 q^{91} - 672 q^{92} + 192 q^{94} - 120 q^{95} + 1154 q^{97} - 174 q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 16 * q^7 + 8 * q^8 - 12 * q^10 - 12 * q^11 + 38 * q^13 - 32 * q^14 + 16 * q^16 + 126 * q^17 + 20 * q^19 - 24 * q^20 - 24 * q^22 - 168 * q^23 - 89 * q^25 + 76 * q^26 - 64 * q^28 - 30 * q^29 - 88 * q^31 + 32 * q^32 + 252 * q^34 + 96 * q^35 + 254 * q^37 + 40 * q^38 - 48 * q^40 - 42 * q^41 - 52 * q^43 - 48 * q^44 - 336 * q^46 + 96 * q^47 - 87 * q^49 - 178 * q^50 + 152 * q^52 - 198 * q^53 + 72 * q^55 - 128 * q^56 - 60 * q^58 + 660 * q^59 - 538 * q^61 - 176 * q^62 + 64 * q^64 - 228 * q^65 + 884 * q^67 + 504 * q^68 + 192 * q^70 - 792 * q^71 + 218 * q^73 + 508 * q^74 + 80 * q^76 + 192 * q^77 - 520 * q^79 - 96 * q^80 - 84 * q^82 + 492 * q^83 - 756 * q^85 - 104 * q^86 - 96 * q^88 - 810 * q^89 - 608 * q^91 - 672 * q^92 + 192 * q^94 - 120 * q^95 + 1154 * q^97 - 174 * 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
2.00000 0 4.00000 −6.00000 0 −16.0000 8.00000 0 −12.0000
 $$n$$: 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 18.4.a.a 1
3.b odd 2 1 6.4.a.a 1
4.b odd 2 1 144.4.a.c 1
5.b even 2 1 450.4.a.h 1
5.c odd 4 2 450.4.c.e 2
7.b odd 2 1 882.4.a.n 1
7.c even 3 2 882.4.g.i 2
7.d odd 6 2 882.4.g.f 2
8.b even 2 1 576.4.a.q 1
8.d odd 2 1 576.4.a.r 1
9.c even 3 2 162.4.c.c 2
9.d odd 6 2 162.4.c.f 2
11.b odd 2 1 2178.4.a.e 1
12.b even 2 1 48.4.a.c 1
15.d odd 2 1 150.4.a.i 1
15.e even 4 2 150.4.c.d 2
21.c even 2 1 294.4.a.e 1
21.g even 6 2 294.4.e.g 2
21.h odd 6 2 294.4.e.h 2
24.f even 2 1 192.4.a.c 1
24.h odd 2 1 192.4.a.i 1
33.d even 2 1 726.4.a.f 1
39.d odd 2 1 1014.4.a.g 1
39.f even 4 2 1014.4.b.d 2
48.i odd 4 2 768.4.d.n 2
48.k even 4 2 768.4.d.c 2
51.c odd 2 1 1734.4.a.d 1
57.d even 2 1 2166.4.a.i 1
60.h even 2 1 1200.4.a.b 1
60.l odd 4 2 1200.4.f.j 2
84.h odd 2 1 2352.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 3.b odd 2 1
18.4.a.a 1 1.a even 1 1 trivial
48.4.a.c 1 12.b even 2 1
144.4.a.c 1 4.b odd 2 1
150.4.a.i 1 15.d odd 2 1
150.4.c.d 2 15.e even 4 2
162.4.c.c 2 9.c even 3 2
162.4.c.f 2 9.d odd 6 2
192.4.a.c 1 24.f even 2 1
192.4.a.i 1 24.h odd 2 1
294.4.a.e 1 21.c even 2 1
294.4.e.g 2 21.g even 6 2
294.4.e.h 2 21.h odd 6 2
450.4.a.h 1 5.b even 2 1
450.4.c.e 2 5.c odd 4 2
576.4.a.q 1 8.b even 2 1
576.4.a.r 1 8.d odd 2 1
726.4.a.f 1 33.d even 2 1
768.4.d.c 2 48.k even 4 2
768.4.d.n 2 48.i odd 4 2
882.4.a.n 1 7.b odd 2 1
882.4.g.f 2 7.d odd 6 2
882.4.g.i 2 7.c even 3 2
1014.4.a.g 1 39.d odd 2 1
1014.4.b.d 2 39.f even 4 2
1200.4.a.b 1 60.h even 2 1
1200.4.f.j 2 60.l odd 4 2
1734.4.a.d 1 51.c odd 2 1
2166.4.a.i 1 57.d even 2 1
2178.4.a.e 1 11.b odd 2 1
2352.4.a.e 1 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T + 16$$
$11$ $$T + 12$$
$13$ $$T - 38$$
$17$ $$T - 126$$
$19$ $$T - 20$$
$23$ $$T + 168$$
$29$ $$T + 30$$
$31$ $$T + 88$$
$37$ $$T - 254$$
$41$ $$T + 42$$
$43$ $$T + 52$$
$47$ $$T - 96$$
$53$ $$T + 198$$
$59$ $$T - 660$$
$61$ $$T + 538$$
$67$ $$T - 884$$
$71$ $$T + 792$$
$73$ $$T - 218$$
$79$ $$T + 520$$
$83$ $$T - 492$$
$89$ $$T + 810$$
$97$ $$T - 1154$$