# Properties

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

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

chi := DirichletCharacter("18.7");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$1.06203438010$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + ( - 9 \zeta_{6} + 9) q^{5} + (6 \zeta_{6} - 12) q^{6} + 31 \zeta_{6} q^{7} + 8 q^{8} - 27 q^{9} +O(q^{10})$$ q - 2*z * q^2 + (-6*z + 3) * q^3 + (4*z - 4) * q^4 + (-9*z + 9) * q^5 + (6*z - 12) * q^6 + 31*z * q^7 + 8 * q^8 - 27 * q^9 $$q - 2 \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + ( - 9 \zeta_{6} + 9) q^{5} + (6 \zeta_{6} - 12) q^{6} + 31 \zeta_{6} q^{7} + 8 q^{8} - 27 q^{9} - 18 q^{10} + 15 \zeta_{6} q^{11} + (12 \zeta_{6} + 12) q^{12} + ( - 37 \zeta_{6} + 37) q^{13} + ( - 62 \zeta_{6} + 62) q^{14} + ( - 27 \zeta_{6} - 27) q^{15} - 16 \zeta_{6} q^{16} - 42 q^{17} + 54 \zeta_{6} q^{18} - 28 q^{19} + 36 \zeta_{6} q^{20} + ( - 93 \zeta_{6} + 186) q^{21} + ( - 30 \zeta_{6} + 30) q^{22} + (195 \zeta_{6} - 195) q^{23} + ( - 48 \zeta_{6} + 24) q^{24} + 44 \zeta_{6} q^{25} - 74 q^{26} + (162 \zeta_{6} - 81) q^{27} - 124 q^{28} - 111 \zeta_{6} q^{29} + (108 \zeta_{6} - 54) q^{30} + ( - 205 \zeta_{6} + 205) q^{31} + (32 \zeta_{6} - 32) q^{32} + ( - 45 \zeta_{6} + 90) q^{33} + 84 \zeta_{6} q^{34} + 279 q^{35} + ( - 108 \zeta_{6} + 108) q^{36} - 166 q^{37} + 56 \zeta_{6} q^{38} + ( - 111 \zeta_{6} - 111) q^{39} + ( - 72 \zeta_{6} + 72) q^{40} + ( - 261 \zeta_{6} + 261) q^{41} + ( - 186 \zeta_{6} - 186) q^{42} + 43 \zeta_{6} q^{43} - 60 q^{44} + (243 \zeta_{6} - 243) q^{45} + 390 q^{46} - 177 \zeta_{6} q^{47} + (48 \zeta_{6} - 96) q^{48} + (618 \zeta_{6} - 618) q^{49} + ( - 88 \zeta_{6} + 88) q^{50} + (252 \zeta_{6} - 126) q^{51} + 148 \zeta_{6} q^{52} + 114 q^{53} + ( - 162 \zeta_{6} + 324) q^{54} + 135 q^{55} + 248 \zeta_{6} q^{56} + (168 \zeta_{6} - 84) q^{57} + (222 \zeta_{6} - 222) q^{58} + (159 \zeta_{6} - 159) q^{59} + ( - 108 \zeta_{6} + 216) q^{60} - 191 \zeta_{6} q^{61} - 410 q^{62} - 837 \zeta_{6} q^{63} + 64 q^{64} - 333 \zeta_{6} q^{65} + ( - 90 \zeta_{6} - 90) q^{66} + ( - 421 \zeta_{6} + 421) q^{67} + ( - 168 \zeta_{6} + 168) q^{68} + (585 \zeta_{6} + 585) q^{69} - 558 \zeta_{6} q^{70} + 156 q^{71} - 216 q^{72} + 182 q^{73} + 332 \zeta_{6} q^{74} + ( - 132 \zeta_{6} + 264) q^{75} + ( - 112 \zeta_{6} + 112) q^{76} + (465 \zeta_{6} - 465) q^{77} + (444 \zeta_{6} - 222) q^{78} - 1133 \zeta_{6} q^{79} - 144 q^{80} + 729 q^{81} - 522 q^{82} + 1083 \zeta_{6} q^{83} + (744 \zeta_{6} - 372) q^{84} + (378 \zeta_{6} - 378) q^{85} + ( - 86 \zeta_{6} + 86) q^{86} + (333 \zeta_{6} - 666) q^{87} + 120 \zeta_{6} q^{88} - 1050 q^{89} + 486 q^{90} + 1147 q^{91} - 780 \zeta_{6} q^{92} + ( - 615 \zeta_{6} - 615) q^{93} + (354 \zeta_{6} - 354) q^{94} + (252 \zeta_{6} - 252) q^{95} + (96 \zeta_{6} + 96) q^{96} + 901 \zeta_{6} q^{97} + 1236 q^{98} - 405 \zeta_{6} q^{99} +O(q^{100})$$ q - 2*z * q^2 + (-6*z + 3) * q^3 + (4*z - 4) * q^4 + (-9*z + 9) * q^5 + (6*z - 12) * q^6 + 31*z * q^7 + 8 * q^8 - 27 * q^9 - 18 * q^10 + 15*z * q^11 + (12*z + 12) * q^12 + (-37*z + 37) * q^13 + (-62*z + 62) * q^14 + (-27*z - 27) * q^15 - 16*z * q^16 - 42 * q^17 + 54*z * q^18 - 28 * q^19 + 36*z * q^20 + (-93*z + 186) * q^21 + (-30*z + 30) * q^22 + (195*z - 195) * q^23 + (-48*z + 24) * q^24 + 44*z * q^25 - 74 * q^26 + (162*z - 81) * q^27 - 124 * q^28 - 111*z * q^29 + (108*z - 54) * q^30 + (-205*z + 205) * q^31 + (32*z - 32) * q^32 + (-45*z + 90) * q^33 + 84*z * q^34 + 279 * q^35 + (-108*z + 108) * q^36 - 166 * q^37 + 56*z * q^38 + (-111*z - 111) * q^39 + (-72*z + 72) * q^40 + (-261*z + 261) * q^41 + (-186*z - 186) * q^42 + 43*z * q^43 - 60 * q^44 + (243*z - 243) * q^45 + 390 * q^46 - 177*z * q^47 + (48*z - 96) * q^48 + (618*z - 618) * q^49 + (-88*z + 88) * q^50 + (252*z - 126) * q^51 + 148*z * q^52 + 114 * q^53 + (-162*z + 324) * q^54 + 135 * q^55 + 248*z * q^56 + (168*z - 84) * q^57 + (222*z - 222) * q^58 + (159*z - 159) * q^59 + (-108*z + 216) * q^60 - 191*z * q^61 - 410 * q^62 - 837*z * q^63 + 64 * q^64 - 333*z * q^65 + (-90*z - 90) * q^66 + (-421*z + 421) * q^67 + (-168*z + 168) * q^68 + (585*z + 585) * q^69 - 558*z * q^70 + 156 * q^71 - 216 * q^72 + 182 * q^73 + 332*z * q^74 + (-132*z + 264) * q^75 + (-112*z + 112) * q^76 + (465*z - 465) * q^77 + (444*z - 222) * q^78 - 1133*z * q^79 - 144 * q^80 + 729 * q^81 - 522 * q^82 + 1083*z * q^83 + (744*z - 372) * q^84 + (378*z - 378) * q^85 + (-86*z + 86) * q^86 + (333*z - 666) * q^87 + 120*z * q^88 - 1050 * q^89 + 486 * q^90 + 1147 * q^91 - 780*z * q^92 + (-615*z - 615) * q^93 + (354*z - 354) * q^94 + (252*z - 252) * q^95 + (96*z + 96) * q^96 + 901*z * q^97 + 1236 * q^98 - 405*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 9 q^{5} - 18 q^{6} + 31 q^{7} + 16 q^{8} - 54 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 9 * q^5 - 18 * q^6 + 31 * q^7 + 16 * q^8 - 54 * q^9 $$2 q - 2 q^{2} - 4 q^{4} + 9 q^{5} - 18 q^{6} + 31 q^{7} + 16 q^{8} - 54 q^{9} - 36 q^{10} + 15 q^{11} + 36 q^{12} + 37 q^{13} + 62 q^{14} - 81 q^{15} - 16 q^{16} - 84 q^{17} + 54 q^{18} - 56 q^{19} + 36 q^{20} + 279 q^{21} + 30 q^{22} - 195 q^{23} + 44 q^{25} - 148 q^{26} - 248 q^{28} - 111 q^{29} + 205 q^{31} - 32 q^{32} + 135 q^{33} + 84 q^{34} + 558 q^{35} + 108 q^{36} - 332 q^{37} + 56 q^{38} - 333 q^{39} + 72 q^{40} + 261 q^{41} - 558 q^{42} + 43 q^{43} - 120 q^{44} - 243 q^{45} + 780 q^{46} - 177 q^{47} - 144 q^{48} - 618 q^{49} + 88 q^{50} + 148 q^{52} + 228 q^{53} + 486 q^{54} + 270 q^{55} + 248 q^{56} - 222 q^{58} - 159 q^{59} + 324 q^{60} - 191 q^{61} - 820 q^{62} - 837 q^{63} + 128 q^{64} - 333 q^{65} - 270 q^{66} + 421 q^{67} + 168 q^{68} + 1755 q^{69} - 558 q^{70} + 312 q^{71} - 432 q^{72} + 364 q^{73} + 332 q^{74} + 396 q^{75} + 112 q^{76} - 465 q^{77} - 1133 q^{79} - 288 q^{80} + 1458 q^{81} - 1044 q^{82} + 1083 q^{83} - 378 q^{85} + 86 q^{86} - 999 q^{87} + 120 q^{88} - 2100 q^{89} + 972 q^{90} + 2294 q^{91} - 780 q^{92} - 1845 q^{93} - 354 q^{94} - 252 q^{95} + 288 q^{96} + 901 q^{97} + 2472 q^{98} - 405 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 9 * q^5 - 18 * q^6 + 31 * q^7 + 16 * q^8 - 54 * q^9 - 36 * q^10 + 15 * q^11 + 36 * q^12 + 37 * q^13 + 62 * q^14 - 81 * q^15 - 16 * q^16 - 84 * q^17 + 54 * q^18 - 56 * q^19 + 36 * q^20 + 279 * q^21 + 30 * q^22 - 195 * q^23 + 44 * q^25 - 148 * q^26 - 248 * q^28 - 111 * q^29 + 205 * q^31 - 32 * q^32 + 135 * q^33 + 84 * q^34 + 558 * q^35 + 108 * q^36 - 332 * q^37 + 56 * q^38 - 333 * q^39 + 72 * q^40 + 261 * q^41 - 558 * q^42 + 43 * q^43 - 120 * q^44 - 243 * q^45 + 780 * q^46 - 177 * q^47 - 144 * q^48 - 618 * q^49 + 88 * q^50 + 148 * q^52 + 228 * q^53 + 486 * q^54 + 270 * q^55 + 248 * q^56 - 222 * q^58 - 159 * q^59 + 324 * q^60 - 191 * q^61 - 820 * q^62 - 837 * q^63 + 128 * q^64 - 333 * q^65 - 270 * q^66 + 421 * q^67 + 168 * q^68 + 1755 * q^69 - 558 * q^70 + 312 * q^71 - 432 * q^72 + 364 * q^73 + 332 * q^74 + 396 * q^75 + 112 * q^76 - 465 * q^77 - 1133 * q^79 - 288 * q^80 + 1458 * q^81 - 1044 * q^82 + 1083 * q^83 - 378 * q^85 + 86 * q^86 - 999 * q^87 + 120 * q^88 - 2100 * q^89 + 972 * q^90 + 2294 * q^91 - 780 * q^92 - 1845 * q^93 - 354 * q^94 - 252 * q^95 + 288 * q^96 + 901 * q^97 + 2472 * q^98 - 405 * 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
−1.00000 + 1.73205i 5.19615i −2.00000 3.46410i 4.50000 + 7.79423i −9.00000 5.19615i 15.5000 26.8468i 8.00000 −27.0000 −18.0000
13.1 −1.00000 1.73205i 5.19615i −2.00000 + 3.46410i 4.50000 7.79423i −9.00000 + 5.19615i 15.5000 + 26.8468i 8.00000 −27.0000 −18.0000
 $$n$$: 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 18.4.c.a 2
3.b odd 2 1 54.4.c.a 2
4.b odd 2 1 144.4.i.a 2
9.c even 3 1 inner 18.4.c.a 2
9.c even 3 1 162.4.a.d 1
9.d odd 6 1 54.4.c.a 2
9.d odd 6 1 162.4.a.a 1
12.b even 2 1 432.4.i.a 2
36.f odd 6 1 144.4.i.a 2
36.f odd 6 1 1296.4.a.b 1
36.h even 6 1 432.4.i.a 2
36.h even 6 1 1296.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 1.a even 1 1 trivial
18.4.c.a 2 9.c even 3 1 inner
54.4.c.a 2 3.b odd 2 1
54.4.c.a 2 9.d odd 6 1
144.4.i.a 2 4.b odd 2 1
144.4.i.a 2 36.f odd 6 1
162.4.a.a 1 9.d odd 6 1
162.4.a.d 1 9.c even 3 1
432.4.i.a 2 12.b even 2 1
432.4.i.a 2 36.h even 6 1
1296.4.a.b 1 36.f odd 6 1
1296.4.a.g 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 9T_{5} + 81$$ acting on $$S_{4}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 27$$
$5$ $$T^{2} - 9T + 81$$
$7$ $$T^{2} - 31T + 961$$
$11$ $$T^{2} - 15T + 225$$
$13$ $$T^{2} - 37T + 1369$$
$17$ $$(T + 42)^{2}$$
$19$ $$(T + 28)^{2}$$
$23$ $$T^{2} + 195T + 38025$$
$29$ $$T^{2} + 111T + 12321$$
$31$ $$T^{2} - 205T + 42025$$
$37$ $$(T + 166)^{2}$$
$41$ $$T^{2} - 261T + 68121$$
$43$ $$T^{2} - 43T + 1849$$
$47$ $$T^{2} + 177T + 31329$$
$53$ $$(T - 114)^{2}$$
$59$ $$T^{2} + 159T + 25281$$
$61$ $$T^{2} + 191T + 36481$$
$67$ $$T^{2} - 421T + 177241$$
$71$ $$(T - 156)^{2}$$
$73$ $$(T - 182)^{2}$$
$79$ $$T^{2} + 1133 T + 1283689$$
$83$ $$T^{2} - 1083 T + 1172889$$
$89$ $$(T + 1050)^{2}$$
$97$ $$T^{2} - 901T + 811801$$