# Properties

 Label 162.4.c.f Level $162$ Weight $4$ Character orbit 162.c Analytic conductor $9.558$ 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] = [162,4,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, 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("162.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 162.c (of order $$3$$, degree $$2$$, not 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: $$9.55830942093$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 6 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} - 8 q^{8} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 - 6*z * q^5 + (-16*z + 16) * q^7 - 8 * q^8 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 6 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} - 8 q^{8} - 12 q^{10} + (12 \zeta_{6} - 12) q^{11} - 38 \zeta_{6} q^{13} - 32 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 126 q^{17} + 20 q^{19} + (24 \zeta_{6} - 24) q^{20} + 24 \zeta_{6} q^{22} - 168 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 76 q^{26} - 64 q^{28} + (30 \zeta_{6} - 30) q^{29} + 88 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + (252 \zeta_{6} - 252) q^{34} - 96 q^{35} + 254 q^{37} + ( - 40 \zeta_{6} + 40) q^{38} + 48 \zeta_{6} q^{40} - 42 \zeta_{6} q^{41} + ( - 52 \zeta_{6} + 52) q^{43} + 48 q^{44} - 336 q^{46} + ( - 96 \zeta_{6} + 96) q^{47} + 87 \zeta_{6} q^{49} - 178 \zeta_{6} q^{50} + (152 \zeta_{6} - 152) q^{52} + 198 q^{53} + 72 q^{55} + (128 \zeta_{6} - 128) q^{56} + 60 \zeta_{6} q^{58} + 660 \zeta_{6} q^{59} + ( - 538 \zeta_{6} + 538) q^{61} + 176 q^{62} + 64 q^{64} + (228 \zeta_{6} - 228) q^{65} - 884 \zeta_{6} q^{67} + 504 \zeta_{6} q^{68} + (192 \zeta_{6} - 192) q^{70} + 792 q^{71} + 218 q^{73} + ( - 508 \zeta_{6} + 508) q^{74} - 80 \zeta_{6} q^{76} + 192 \zeta_{6} q^{77} + ( - 520 \zeta_{6} + 520) q^{79} + 96 q^{80} - 84 q^{82} + ( - 492 \zeta_{6} + 492) q^{83} + 756 \zeta_{6} q^{85} - 104 \zeta_{6} q^{86} + ( - 96 \zeta_{6} + 96) q^{88} + 810 q^{89} - 608 q^{91} + (672 \zeta_{6} - 672) q^{92} - 192 \zeta_{6} q^{94} - 120 \zeta_{6} q^{95} + (1154 \zeta_{6} - 1154) q^{97} + 174 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 - 6*z * q^5 + (-16*z + 16) * q^7 - 8 * q^8 - 12 * q^10 + (12*z - 12) * q^11 - 38*z * q^13 - 32*z * q^14 + (16*z - 16) * q^16 - 126 * q^17 + 20 * q^19 + (24*z - 24) * q^20 + 24*z * q^22 - 168*z * q^23 + (-89*z + 89) * q^25 - 76 * q^26 - 64 * q^28 + (30*z - 30) * q^29 + 88*z * q^31 + 32*z * q^32 + (252*z - 252) * q^34 - 96 * q^35 + 254 * q^37 + (-40*z + 40) * q^38 + 48*z * q^40 - 42*z * q^41 + (-52*z + 52) * q^43 + 48 * q^44 - 336 * q^46 + (-96*z + 96) * q^47 + 87*z * q^49 - 178*z * q^50 + (152*z - 152) * q^52 + 198 * q^53 + 72 * q^55 + (128*z - 128) * q^56 + 60*z * q^58 + 660*z * q^59 + (-538*z + 538) * q^61 + 176 * q^62 + 64 * q^64 + (228*z - 228) * q^65 - 884*z * q^67 + 504*z * q^68 + (192*z - 192) * q^70 + 792 * q^71 + 218 * q^73 + (-508*z + 508) * q^74 - 80*z * q^76 + 192*z * q^77 + (-520*z + 520) * q^79 + 96 * q^80 - 84 * q^82 + (-492*z + 492) * q^83 + 756*z * q^85 - 104*z * q^86 + (-96*z + 96) * q^88 + 810 * q^89 - 608 * q^91 + (672*z - 672) * q^92 - 192*z * q^94 - 120*z * q^95 + (1154*z - 1154) * q^97 + 174 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 - 6 * q^5 + 16 * q^7 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{7} - 16 q^{8} - 24 q^{10} - 12 q^{11} - 38 q^{13} - 32 q^{14} - 16 q^{16} - 252 q^{17} + 40 q^{19} - 24 q^{20} + 24 q^{22} - 168 q^{23} + 89 q^{25} - 152 q^{26} - 128 q^{28} - 30 q^{29} + 88 q^{31} + 32 q^{32} - 252 q^{34} - 192 q^{35} + 508 q^{37} + 40 q^{38} + 48 q^{40} - 42 q^{41} + 52 q^{43} + 96 q^{44} - 672 q^{46} + 96 q^{47} + 87 q^{49} - 178 q^{50} - 152 q^{52} + 396 q^{53} + 144 q^{55} - 128 q^{56} + 60 q^{58} + 660 q^{59} + 538 q^{61} + 352 q^{62} + 128 q^{64} - 228 q^{65} - 884 q^{67} + 504 q^{68} - 192 q^{70} + 1584 q^{71} + 436 q^{73} + 508 q^{74} - 80 q^{76} + 192 q^{77} + 520 q^{79} + 192 q^{80} - 168 q^{82} + 492 q^{83} + 756 q^{85} - 104 q^{86} + 96 q^{88} + 1620 q^{89} - 1216 q^{91} - 672 q^{92} - 192 q^{94} - 120 q^{95} - 1154 q^{97} + 348 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 - 6 * q^5 + 16 * q^7 - 16 * q^8 - 24 * q^10 - 12 * q^11 - 38 * q^13 - 32 * q^14 - 16 * q^16 - 252 * q^17 + 40 * q^19 - 24 * q^20 + 24 * q^22 - 168 * q^23 + 89 * q^25 - 152 * q^26 - 128 * q^28 - 30 * q^29 + 88 * q^31 + 32 * q^32 - 252 * q^34 - 192 * q^35 + 508 * q^37 + 40 * q^38 + 48 * q^40 - 42 * q^41 + 52 * q^43 + 96 * q^44 - 672 * q^46 + 96 * q^47 + 87 * q^49 - 178 * q^50 - 152 * q^52 + 396 * q^53 + 144 * q^55 - 128 * q^56 + 60 * q^58 + 660 * q^59 + 538 * q^61 + 352 * q^62 + 128 * q^64 - 228 * q^65 - 884 * q^67 + 504 * q^68 - 192 * q^70 + 1584 * q^71 + 436 * q^73 + 508 * q^74 - 80 * q^76 + 192 * q^77 + 520 * q^79 + 192 * q^80 - 168 * q^82 + 492 * q^83 + 756 * q^85 - 104 * q^86 + 96 * q^88 + 1620 * q^89 - 1216 * q^91 - 672 * q^92 - 192 * q^94 - 120 * q^95 - 1154 * q^97 + 348 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i −3.00000 5.19615i 0 8.00000 13.8564i −8.00000 0 −12.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.00000 + 5.19615i 0 8.00000 + 13.8564i −8.00000 0 −12.0000
 $$n$$: 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 162.4.c.f 2
3.b odd 2 1 162.4.c.c 2
9.c even 3 1 6.4.a.a 1
9.c even 3 1 inner 162.4.c.f 2
9.d odd 6 1 18.4.a.a 1
9.d odd 6 1 162.4.c.c 2
36.f odd 6 1 48.4.a.c 1
36.h even 6 1 144.4.a.c 1
45.h odd 6 1 450.4.a.h 1
45.j even 6 1 150.4.a.i 1
45.k odd 12 2 150.4.c.d 2
45.l even 12 2 450.4.c.e 2
63.g even 3 1 294.4.e.h 2
63.h even 3 1 294.4.e.h 2
63.i even 6 1 882.4.g.f 2
63.j odd 6 1 882.4.g.i 2
63.k odd 6 1 294.4.e.g 2
63.l odd 6 1 294.4.a.e 1
63.n odd 6 1 882.4.g.i 2
63.o even 6 1 882.4.a.n 1
63.s even 6 1 882.4.g.f 2
63.t odd 6 1 294.4.e.g 2
72.j odd 6 1 576.4.a.q 1
72.l even 6 1 576.4.a.r 1
72.n even 6 1 192.4.a.i 1
72.p odd 6 1 192.4.a.c 1
99.g even 6 1 2178.4.a.e 1
99.h odd 6 1 726.4.a.f 1
117.t even 6 1 1014.4.a.g 1
117.y odd 12 2 1014.4.b.d 2
144.v odd 12 2 768.4.d.c 2
144.x even 12 2 768.4.d.n 2
153.h even 6 1 1734.4.a.d 1
171.o odd 6 1 2166.4.a.i 1
180.p odd 6 1 1200.4.a.b 1
180.x even 12 2 1200.4.f.j 2
252.bi even 6 1 2352.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 9.c even 3 1
18.4.a.a 1 9.d odd 6 1
48.4.a.c 1 36.f odd 6 1
144.4.a.c 1 36.h even 6 1
150.4.a.i 1 45.j even 6 1
150.4.c.d 2 45.k odd 12 2
162.4.c.c 2 3.b odd 2 1
162.4.c.c 2 9.d odd 6 1
162.4.c.f 2 1.a even 1 1 trivial
162.4.c.f 2 9.c even 3 1 inner
192.4.a.c 1 72.p odd 6 1
192.4.a.i 1 72.n even 6 1
294.4.a.e 1 63.l odd 6 1
294.4.e.g 2 63.k odd 6 1
294.4.e.g 2 63.t odd 6 1
294.4.e.h 2 63.g even 3 1
294.4.e.h 2 63.h even 3 1
450.4.a.h 1 45.h odd 6 1
450.4.c.e 2 45.l even 12 2
576.4.a.q 1 72.j odd 6 1
576.4.a.r 1 72.l even 6 1
726.4.a.f 1 99.h odd 6 1
768.4.d.c 2 144.v odd 12 2
768.4.d.n 2 144.x even 12 2
882.4.a.n 1 63.o even 6 1
882.4.g.f 2 63.i even 6 1
882.4.g.f 2 63.s even 6 1
882.4.g.i 2 63.j odd 6 1
882.4.g.i 2 63.n odd 6 1
1014.4.a.g 1 117.t even 6 1
1014.4.b.d 2 117.y odd 12 2
1200.4.a.b 1 180.p odd 6 1
1200.4.f.j 2 180.x even 12 2
1734.4.a.d 1 153.h even 6 1
2166.4.a.i 1 171.o odd 6 1
2178.4.a.e 1 99.g even 6 1
2352.4.a.e 1 252.bi even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$T^{2} - 16T + 256$$
$11$ $$T^{2} + 12T + 144$$
$13$ $$T^{2} + 38T + 1444$$
$17$ $$(T + 126)^{2}$$
$19$ $$(T - 20)^{2}$$
$23$ $$T^{2} + 168T + 28224$$
$29$ $$T^{2} + 30T + 900$$
$31$ $$T^{2} - 88T + 7744$$
$37$ $$(T - 254)^{2}$$
$41$ $$T^{2} + 42T + 1764$$
$43$ $$T^{2} - 52T + 2704$$
$47$ $$T^{2} - 96T + 9216$$
$53$ $$(T - 198)^{2}$$
$59$ $$T^{2} - 660T + 435600$$
$61$ $$T^{2} - 538T + 289444$$
$67$ $$T^{2} + 884T + 781456$$
$71$ $$(T - 792)^{2}$$
$73$ $$(T - 218)^{2}$$
$79$ $$T^{2} - 520T + 270400$$
$83$ $$T^{2} - 492T + 242064$$
$89$ $$(T - 810)^{2}$$
$97$ $$T^{2} + 1154 T + 1331716$$