# Properties

 Label 18.2.c.a Level $18$ Weight $2$ Character orbit 18.c Analytic conductor $0.144$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,2,Mod(7,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("18.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 18.c (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.143730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} + (\zeta_{6} + 1) q^{6} - 2 \zeta_{6} q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q - z * q^2 + (z - 2) * q^3 + (z - 1) * q^4 + (z + 1) * q^6 - 2*z * q^7 + q^8 + (-3*z + 3) * q^9 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} + (\zeta_{6} + 1) q^{6} - 2 \zeta_{6} q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + 3 \zeta_{6} q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + (2 \zeta_{6} - 2) q^{13} + (2 \zeta_{6} - 2) q^{14} - \zeta_{6} q^{16} - 3 q^{17} - 3 q^{18} - q^{19} + (2 \zeta_{6} + 2) q^{21} + ( - 3 \zeta_{6} + 3) q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + (\zeta_{6} - 2) q^{24} + 5 \zeta_{6} q^{25} + 2 q^{26} + (6 \zeta_{6} - 3) q^{27} + 2 q^{28} - 6 \zeta_{6} q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} + ( - 3 \zeta_{6} - 3) q^{33} + 3 \zeta_{6} q^{34} + 3 \zeta_{6} q^{36} - 4 q^{37} + \zeta_{6} q^{38} + ( - 4 \zeta_{6} + 2) q^{39} + (9 \zeta_{6} - 9) q^{41} + ( - 4 \zeta_{6} + 2) q^{42} + \zeta_{6} q^{43} - 3 q^{44} - 6 q^{46} + 6 \zeta_{6} q^{47} + (\zeta_{6} + 1) q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + ( - 5 \zeta_{6} + 5) q^{50} + ( - 3 \zeta_{6} + 6) q^{51} - 2 \zeta_{6} q^{52} + 12 q^{53} + ( - 3 \zeta_{6} + 6) q^{54} - 2 \zeta_{6} q^{56} + ( - \zeta_{6} + 2) q^{57} + (6 \zeta_{6} - 6) q^{58} + (3 \zeta_{6} - 3) q^{59} - 8 \zeta_{6} q^{61} - 4 q^{62} - 6 q^{63} + q^{64} + (6 \zeta_{6} - 3) q^{66} + (5 \zeta_{6} - 5) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + (12 \zeta_{6} - 6) q^{69} - 12 q^{71} + ( - 3 \zeta_{6} + 3) q^{72} + 11 q^{73} + 4 \zeta_{6} q^{74} + ( - 5 \zeta_{6} - 5) q^{75} + ( - \zeta_{6} + 1) q^{76} + ( - 6 \zeta_{6} + 6) q^{77} + (2 \zeta_{6} - 4) q^{78} + 4 \zeta_{6} q^{79} - 9 \zeta_{6} q^{81} + 9 q^{82} - 12 \zeta_{6} q^{83} + (2 \zeta_{6} - 4) q^{84} + ( - \zeta_{6} + 1) q^{86} + (6 \zeta_{6} + 6) q^{87} + 3 \zeta_{6} q^{88} + 6 q^{89} + 4 q^{91} + 6 \zeta_{6} q^{92} + (8 \zeta_{6} - 4) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} + ( - 2 \zeta_{6} + 1) q^{96} - 5 \zeta_{6} q^{97} - 3 q^{98} + 9 q^{99} +O(q^{100})$$ q - z * q^2 + (z - 2) * q^3 + (z - 1) * q^4 + (z + 1) * q^6 - 2*z * q^7 + q^8 + (-3*z + 3) * q^9 + 3*z * q^11 + (-2*z + 1) * q^12 + (2*z - 2) * q^13 + (2*z - 2) * q^14 - z * q^16 - 3 * q^17 - 3 * q^18 - q^19 + (2*z + 2) * q^21 + (-3*z + 3) * q^22 + (-6*z + 6) * q^23 + (z - 2) * q^24 + 5*z * q^25 + 2 * q^26 + (6*z - 3) * q^27 + 2 * q^28 - 6*z * q^29 + (-4*z + 4) * q^31 + (z - 1) * q^32 + (-3*z - 3) * q^33 + 3*z * q^34 + 3*z * q^36 - 4 * q^37 + z * q^38 + (-4*z + 2) * q^39 + (9*z - 9) * q^41 + (-4*z + 2) * q^42 + z * q^43 - 3 * q^44 - 6 * q^46 + 6*z * q^47 + (z + 1) * q^48 + (-3*z + 3) * q^49 + (-5*z + 5) * q^50 + (-3*z + 6) * q^51 - 2*z * q^52 + 12 * q^53 + (-3*z + 6) * q^54 - 2*z * q^56 + (-z + 2) * q^57 + (6*z - 6) * q^58 + (3*z - 3) * q^59 - 8*z * q^61 - 4 * q^62 - 6 * q^63 + q^64 + (6*z - 3) * q^66 + (5*z - 5) * q^67 + (-3*z + 3) * q^68 + (12*z - 6) * q^69 - 12 * q^71 + (-3*z + 3) * q^72 + 11 * q^73 + 4*z * q^74 + (-5*z - 5) * q^75 + (-z + 1) * q^76 + (-6*z + 6) * q^77 + (2*z - 4) * q^78 + 4*z * q^79 - 9*z * q^81 + 9 * q^82 - 12*z * q^83 + (2*z - 4) * q^84 + (-z + 1) * q^86 + (6*z + 6) * q^87 + 3*z * q^88 + 6 * q^89 + 4 * q^91 + 6*z * q^92 + (8*z - 4) * q^93 + (-6*z + 6) * q^94 + (-2*z + 1) * q^96 - 5*z * q^97 - 3 * q^98 + 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} + 3 q^{6} - 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 3 * q^6 - 2 * q^7 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} + 3 q^{6} - 2 q^{7} + 2 q^{8} + 3 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} - 6 q^{17} - 6 q^{18} - 2 q^{19} + 6 q^{21} + 3 q^{22} + 6 q^{23} - 3 q^{24} + 5 q^{25} + 4 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 9 q^{33} + 3 q^{34} + 3 q^{36} - 8 q^{37} + q^{38} - 9 q^{41} + q^{43} - 6 q^{44} - 12 q^{46} + 6 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} + 9 q^{51} - 2 q^{52} + 24 q^{53} + 9 q^{54} - 2 q^{56} + 3 q^{57} - 6 q^{58} - 3 q^{59} - 8 q^{61} - 8 q^{62} - 12 q^{63} + 2 q^{64} - 5 q^{67} + 3 q^{68} - 24 q^{71} + 3 q^{72} + 22 q^{73} + 4 q^{74} - 15 q^{75} + q^{76} + 6 q^{77} - 6 q^{78} + 4 q^{79} - 9 q^{81} + 18 q^{82} - 12 q^{83} - 6 q^{84} + q^{86} + 18 q^{87} + 3 q^{88} + 12 q^{89} + 8 q^{91} + 6 q^{92} + 6 q^{94} - 5 q^{97} - 6 q^{98} + 18 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 3 * q^6 - 2 * q^7 + 2 * q^8 + 3 * q^9 + 3 * q^11 - 2 * q^13 - 2 * q^14 - q^16 - 6 * q^17 - 6 * q^18 - 2 * q^19 + 6 * q^21 + 3 * q^22 + 6 * q^23 - 3 * q^24 + 5 * q^25 + 4 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 - q^32 - 9 * q^33 + 3 * q^34 + 3 * q^36 - 8 * q^37 + q^38 - 9 * q^41 + q^43 - 6 * q^44 - 12 * q^46 + 6 * q^47 + 3 * q^48 + 3 * q^49 + 5 * q^50 + 9 * q^51 - 2 * q^52 + 24 * q^53 + 9 * q^54 - 2 * q^56 + 3 * q^57 - 6 * q^58 - 3 * q^59 - 8 * q^61 - 8 * q^62 - 12 * q^63 + 2 * q^64 - 5 * q^67 + 3 * q^68 - 24 * q^71 + 3 * q^72 + 22 * q^73 + 4 * q^74 - 15 * q^75 + q^76 + 6 * q^77 - 6 * q^78 + 4 * q^79 - 9 * q^81 + 18 * q^82 - 12 * q^83 - 6 * q^84 + q^86 + 18 * q^87 + 3 * q^88 + 12 * q^89 + 8 * q^91 + 6 * q^92 + 6 * q^94 - 5 * q^97 - 6 * q^98 + 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 0 1.50000 0.866025i −1.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 0
13.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 0 1.50000 + 0.866025i −1.00000 1.73205i 1.00000 1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.2.c.a 2
3.b odd 2 1 54.2.c.a 2
4.b odd 2 1 144.2.i.c 2
5.b even 2 1 450.2.e.i 2
5.c odd 4 2 450.2.j.e 4
7.b odd 2 1 882.2.f.d 2
7.c even 3 1 882.2.e.i 2
7.c even 3 1 882.2.h.c 2
7.d odd 6 1 882.2.e.g 2
7.d odd 6 1 882.2.h.b 2
8.b even 2 1 576.2.i.g 2
8.d odd 2 1 576.2.i.a 2
9.c even 3 1 inner 18.2.c.a 2
9.c even 3 1 162.2.a.c 1
9.d odd 6 1 54.2.c.a 2
9.d odd 6 1 162.2.a.b 1
12.b even 2 1 432.2.i.b 2
15.d odd 2 1 1350.2.e.c 2
15.e even 4 2 1350.2.j.a 4
21.c even 2 1 2646.2.f.g 2
21.g even 6 1 2646.2.e.c 2
21.g even 6 1 2646.2.h.i 2
21.h odd 6 1 2646.2.e.b 2
21.h odd 6 1 2646.2.h.h 2
24.f even 2 1 1728.2.i.f 2
24.h odd 2 1 1728.2.i.e 2
36.f odd 6 1 144.2.i.c 2
36.f odd 6 1 1296.2.a.g 1
36.h even 6 1 432.2.i.b 2
36.h even 6 1 1296.2.a.f 1
45.h odd 6 1 1350.2.e.c 2
45.h odd 6 1 4050.2.a.v 1
45.j even 6 1 450.2.e.i 2
45.j even 6 1 4050.2.a.c 1
45.k odd 12 2 450.2.j.e 4
45.k odd 12 2 4050.2.c.c 2
45.l even 12 2 1350.2.j.a 4
45.l even 12 2 4050.2.c.r 2
63.g even 3 1 882.2.e.i 2
63.h even 3 1 882.2.h.c 2
63.i even 6 1 2646.2.h.i 2
63.j odd 6 1 2646.2.h.h 2
63.k odd 6 1 882.2.e.g 2
63.l odd 6 1 882.2.f.d 2
63.l odd 6 1 7938.2.a.x 1
63.n odd 6 1 2646.2.e.b 2
63.o even 6 1 2646.2.f.g 2
63.o even 6 1 7938.2.a.i 1
63.s even 6 1 2646.2.e.c 2
63.t odd 6 1 882.2.h.b 2
72.j odd 6 1 1728.2.i.e 2
72.j odd 6 1 5184.2.a.q 1
72.l even 6 1 1728.2.i.f 2
72.l even 6 1 5184.2.a.p 1
72.n even 6 1 576.2.i.g 2
72.n even 6 1 5184.2.a.r 1
72.p odd 6 1 576.2.i.a 2
72.p odd 6 1 5184.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 1.a even 1 1 trivial
18.2.c.a 2 9.c even 3 1 inner
54.2.c.a 2 3.b odd 2 1
54.2.c.a 2 9.d odd 6 1
144.2.i.c 2 4.b odd 2 1
144.2.i.c 2 36.f odd 6 1
162.2.a.b 1 9.d odd 6 1
162.2.a.c 1 9.c even 3 1
432.2.i.b 2 12.b even 2 1
432.2.i.b 2 36.h even 6 1
450.2.e.i 2 5.b even 2 1
450.2.e.i 2 45.j even 6 1
450.2.j.e 4 5.c odd 4 2
450.2.j.e 4 45.k odd 12 2
576.2.i.a 2 8.d odd 2 1
576.2.i.a 2 72.p odd 6 1
576.2.i.g 2 8.b even 2 1
576.2.i.g 2 72.n even 6 1
882.2.e.g 2 7.d odd 6 1
882.2.e.g 2 63.k odd 6 1
882.2.e.i 2 7.c even 3 1
882.2.e.i 2 63.g even 3 1
882.2.f.d 2 7.b odd 2 1
882.2.f.d 2 63.l odd 6 1
882.2.h.b 2 7.d odd 6 1
882.2.h.b 2 63.t odd 6 1
882.2.h.c 2 7.c even 3 1
882.2.h.c 2 63.h even 3 1
1296.2.a.f 1 36.h even 6 1
1296.2.a.g 1 36.f odd 6 1
1350.2.e.c 2 15.d odd 2 1
1350.2.e.c 2 45.h odd 6 1
1350.2.j.a 4 15.e even 4 2
1350.2.j.a 4 45.l even 12 2
1728.2.i.e 2 24.h odd 2 1
1728.2.i.e 2 72.j odd 6 1
1728.2.i.f 2 24.f even 2 1
1728.2.i.f 2 72.l even 6 1
2646.2.e.b 2 21.h odd 6 1
2646.2.e.b 2 63.n odd 6 1
2646.2.e.c 2 21.g even 6 1
2646.2.e.c 2 63.s even 6 1
2646.2.f.g 2 21.c even 2 1
2646.2.f.g 2 63.o even 6 1
2646.2.h.h 2 21.h odd 6 1
2646.2.h.h 2 63.j odd 6 1
2646.2.h.i 2 21.g even 6 1
2646.2.h.i 2 63.i even 6 1
4050.2.a.c 1 45.j even 6 1
4050.2.a.v 1 45.h odd 6 1
4050.2.c.c 2 45.k odd 12 2
4050.2.c.r 2 45.l even 12 2
5184.2.a.o 1 72.p odd 6 1
5184.2.a.p 1 72.l even 6 1
5184.2.a.q 1 72.j odd 6 1
5184.2.a.r 1 72.n even 6 1
7938.2.a.i 1 63.o even 6 1
7938.2.a.x 1 63.l odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 5T + 25$$