# Properties

 Label 18.6.a.b Level $18$ Weight $6$ Character orbit 18.a Self dual yes Analytic conductor $2.887$ 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] = [18,6,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, 6, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$2.88690875663$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 16 q^{4} + 66 q^{5} + 176 q^{7} - 64 q^{8}+O(q^{10})$$ q - 4 * q^2 + 16 * q^4 + 66 * q^5 + 176 * q^7 - 64 * q^8 $$q - 4 q^{2} + 16 q^{4} + 66 q^{5} + 176 q^{7} - 64 q^{8} - 264 q^{10} + 60 q^{11} - 658 q^{13} - 704 q^{14} + 256 q^{16} + 414 q^{17} + 956 q^{19} + 1056 q^{20} - 240 q^{22} - 600 q^{23} + 1231 q^{25} + 2632 q^{26} + 2816 q^{28} - 5574 q^{29} - 3592 q^{31} - 1024 q^{32} - 1656 q^{34} + 11616 q^{35} - 8458 q^{37} - 3824 q^{38} - 4224 q^{40} - 19194 q^{41} + 13316 q^{43} + 960 q^{44} + 2400 q^{46} + 19680 q^{47} + 14169 q^{49} - 4924 q^{50} - 10528 q^{52} + 31266 q^{53} + 3960 q^{55} - 11264 q^{56} + 22296 q^{58} - 26340 q^{59} - 31090 q^{61} + 14368 q^{62} + 4096 q^{64} - 43428 q^{65} - 16804 q^{67} + 6624 q^{68} - 46464 q^{70} - 6120 q^{71} - 25558 q^{73} + 33832 q^{74} + 15296 q^{76} + 10560 q^{77} + 74408 q^{79} + 16896 q^{80} + 76776 q^{82} + 6468 q^{83} + 27324 q^{85} - 53264 q^{86} - 3840 q^{88} + 32742 q^{89} - 115808 q^{91} - 9600 q^{92} - 78720 q^{94} + 63096 q^{95} + 166082 q^{97} - 56676 q^{98}+O(q^{100})$$ q - 4 * q^2 + 16 * q^4 + 66 * q^5 + 176 * q^7 - 64 * q^8 - 264 * q^10 + 60 * q^11 - 658 * q^13 - 704 * q^14 + 256 * q^16 + 414 * q^17 + 956 * q^19 + 1056 * q^20 - 240 * q^22 - 600 * q^23 + 1231 * q^25 + 2632 * q^26 + 2816 * q^28 - 5574 * q^29 - 3592 * q^31 - 1024 * q^32 - 1656 * q^34 + 11616 * q^35 - 8458 * q^37 - 3824 * q^38 - 4224 * q^40 - 19194 * q^41 + 13316 * q^43 + 960 * q^44 + 2400 * q^46 + 19680 * q^47 + 14169 * q^49 - 4924 * q^50 - 10528 * q^52 + 31266 * q^53 + 3960 * q^55 - 11264 * q^56 + 22296 * q^58 - 26340 * q^59 - 31090 * q^61 + 14368 * q^62 + 4096 * q^64 - 43428 * q^65 - 16804 * q^67 + 6624 * q^68 - 46464 * q^70 - 6120 * q^71 - 25558 * q^73 + 33832 * q^74 + 15296 * q^76 + 10560 * q^77 + 74408 * q^79 + 16896 * q^80 + 76776 * q^82 + 6468 * q^83 + 27324 * q^85 - 53264 * q^86 - 3840 * q^88 + 32742 * q^89 - 115808 * q^91 - 9600 * q^92 - 78720 * q^94 + 63096 * q^95 + 166082 * q^97 - 56676 * 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
−4.00000 0 16.0000 66.0000 0 176.000 −64.0000 0 −264.000
 $$n$$: 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 18.6.a.b 1
3.b odd 2 1 6.6.a.a 1
4.b odd 2 1 144.6.a.j 1
5.b even 2 1 450.6.a.m 1
5.c odd 4 2 450.6.c.j 2
7.b odd 2 1 882.6.a.a 1
8.b even 2 1 576.6.a.j 1
8.d odd 2 1 576.6.a.i 1
9.c even 3 2 162.6.c.h 2
9.d odd 6 2 162.6.c.e 2
12.b even 2 1 48.6.a.c 1
15.d odd 2 1 150.6.a.d 1
15.e even 4 2 150.6.c.b 2
21.c even 2 1 294.6.a.m 1
21.g even 6 2 294.6.e.a 2
21.h odd 6 2 294.6.e.g 2
24.f even 2 1 192.6.a.g 1
24.h odd 2 1 192.6.a.o 1
33.d even 2 1 726.6.a.a 1
39.d odd 2 1 1014.6.a.c 1
48.i odd 4 2 768.6.d.c 2
48.k even 4 2 768.6.d.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 3.b odd 2 1
18.6.a.b 1 1.a even 1 1 trivial
48.6.a.c 1 12.b even 2 1
144.6.a.j 1 4.b odd 2 1
150.6.a.d 1 15.d odd 2 1
150.6.c.b 2 15.e even 4 2
162.6.c.e 2 9.d odd 6 2
162.6.c.h 2 9.c even 3 2
192.6.a.g 1 24.f even 2 1
192.6.a.o 1 24.h odd 2 1
294.6.a.m 1 21.c even 2 1
294.6.e.a 2 21.g even 6 2
294.6.e.g 2 21.h odd 6 2
450.6.a.m 1 5.b even 2 1
450.6.c.j 2 5.c odd 4 2
576.6.a.i 1 8.d odd 2 1
576.6.a.j 1 8.b even 2 1
726.6.a.a 1 33.d even 2 1
768.6.d.c 2 48.i odd 4 2
768.6.d.p 2 48.k even 4 2
882.6.a.a 1 7.b odd 2 1
1014.6.a.c 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 66$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T - 66$$
$7$ $$T - 176$$
$11$ $$T - 60$$
$13$ $$T + 658$$
$17$ $$T - 414$$
$19$ $$T - 956$$
$23$ $$T + 600$$
$29$ $$T + 5574$$
$31$ $$T + 3592$$
$37$ $$T + 8458$$
$41$ $$T + 19194$$
$43$ $$T - 13316$$
$47$ $$T - 19680$$
$53$ $$T - 31266$$
$59$ $$T + 26340$$
$61$ $$T + 31090$$
$67$ $$T + 16804$$
$71$ $$T + 6120$$
$73$ $$T + 25558$$
$79$ $$T - 74408$$
$83$ $$T - 6468$$
$89$ $$T - 32742$$
$97$ $$T - 166082$$