# Properties

 Label 18.6.a.c 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: 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + 96 q^{5} - 148 q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + 96 * q^5 - 148 * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} + 96 q^{5} - 148 q^{7} + 64 q^{8} + 384 q^{10} - 384 q^{11} - 334 q^{13} - 592 q^{14} + 256 q^{16} - 576 q^{17} - 664 q^{19} + 1536 q^{20} - 1536 q^{22} + 3840 q^{23} + 6091 q^{25} - 1336 q^{26} - 2368 q^{28} - 96 q^{29} - 4564 q^{31} + 1024 q^{32} - 2304 q^{34} - 14208 q^{35} + 5798 q^{37} - 2656 q^{38} + 6144 q^{40} + 6720 q^{41} - 14872 q^{43} - 6144 q^{44} + 15360 q^{46} + 19200 q^{47} + 5097 q^{49} + 24364 q^{50} - 5344 q^{52} - 7776 q^{53} - 36864 q^{55} - 9472 q^{56} - 384 q^{58} + 13056 q^{59} + 42782 q^{61} - 18256 q^{62} + 4096 q^{64} - 32064 q^{65} + 36656 q^{67} - 9216 q^{68} - 56832 q^{70} - 64512 q^{71} - 16810 q^{73} + 23192 q^{74} - 10624 q^{76} + 56832 q^{77} + 28076 q^{79} + 24576 q^{80} + 26880 q^{82} + 66432 q^{83} - 55296 q^{85} - 59488 q^{86} - 24576 q^{88} + 81792 q^{89} + 49432 q^{91} + 61440 q^{92} + 76800 q^{94} - 63744 q^{95} - 29938 q^{97} + 20388 q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + 96 * q^5 - 148 * q^7 + 64 * q^8 + 384 * q^10 - 384 * q^11 - 334 * q^13 - 592 * q^14 + 256 * q^16 - 576 * q^17 - 664 * q^19 + 1536 * q^20 - 1536 * q^22 + 3840 * q^23 + 6091 * q^25 - 1336 * q^26 - 2368 * q^28 - 96 * q^29 - 4564 * q^31 + 1024 * q^32 - 2304 * q^34 - 14208 * q^35 + 5798 * q^37 - 2656 * q^38 + 6144 * q^40 + 6720 * q^41 - 14872 * q^43 - 6144 * q^44 + 15360 * q^46 + 19200 * q^47 + 5097 * q^49 + 24364 * q^50 - 5344 * q^52 - 7776 * q^53 - 36864 * q^55 - 9472 * q^56 - 384 * q^58 + 13056 * q^59 + 42782 * q^61 - 18256 * q^62 + 4096 * q^64 - 32064 * q^65 + 36656 * q^67 - 9216 * q^68 - 56832 * q^70 - 64512 * q^71 - 16810 * q^73 + 23192 * q^74 - 10624 * q^76 + 56832 * q^77 + 28076 * q^79 + 24576 * q^80 + 26880 * q^82 + 66432 * q^83 - 55296 * q^85 - 59488 * q^86 - 24576 * q^88 + 81792 * q^89 + 49432 * q^91 + 61440 * q^92 + 76800 * q^94 - 63744 * q^95 - 29938 * q^97 + 20388 * 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 96.0000 0 −148.000 64.0000 0 384.000
 $$n$$: 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.c yes 1
3.b odd 2 1 18.6.a.a 1
4.b odd 2 1 144.6.a.l 1
5.b even 2 1 450.6.a.k 1
5.c odd 4 2 450.6.c.c 2
7.b odd 2 1 882.6.a.l 1
8.b even 2 1 576.6.a.a 1
8.d odd 2 1 576.6.a.b 1
9.c even 3 2 162.6.c.a 2
9.d odd 6 2 162.6.c.l 2
12.b even 2 1 144.6.a.a 1
15.d odd 2 1 450.6.a.v 1
15.e even 4 2 450.6.c.m 2
21.c even 2 1 882.6.a.k 1
24.f even 2 1 576.6.a.bi 1
24.h odd 2 1 576.6.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 3.b odd 2 1
18.6.a.c yes 1 1.a even 1 1 trivial
144.6.a.a 1 12.b even 2 1
144.6.a.l 1 4.b odd 2 1
162.6.c.a 2 9.c even 3 2
162.6.c.l 2 9.d odd 6 2
450.6.a.k 1 5.b even 2 1
450.6.a.v 1 15.d odd 2 1
450.6.c.c 2 5.c odd 4 2
450.6.c.m 2 15.e even 4 2
576.6.a.a 1 8.b even 2 1
576.6.a.b 1 8.d odd 2 1
576.6.a.bh 1 24.h odd 2 1
576.6.a.bi 1 24.f even 2 1
882.6.a.k 1 21.c even 2 1
882.6.a.l 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 96$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T - 96$$
$7$ $$T + 148$$
$11$ $$T + 384$$
$13$ $$T + 334$$
$17$ $$T + 576$$
$19$ $$T + 664$$
$23$ $$T - 3840$$
$29$ $$T + 96$$
$31$ $$T + 4564$$
$37$ $$T - 5798$$
$41$ $$T - 6720$$
$43$ $$T + 14872$$
$47$ $$T - 19200$$
$53$ $$T + 7776$$
$59$ $$T - 13056$$
$61$ $$T - 42782$$
$67$ $$T - 36656$$
$71$ $$T + 64512$$
$73$ $$T + 16810$$
$79$ $$T - 28076$$
$83$ $$T - 66432$$
$89$ $$T - 81792$$
$97$ $$T + 29938$$