# Properties

 Label 162.3.d.b Level $162$ Weight $3$ Character orbit 162.d Analytic conductor $4.414$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,3,Mod(53,162)]

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([5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, 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: $$4.41418028264$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 3 \beta_1) q^{5} + ( - 4 \beta_{2} + 4) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 2*b2 * q^4 + (-3*b3 + 3*b1) * q^5 + (-4*b2 + 4) * q^7 + 2*b3 * q^8 $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 3 \beta_1) q^{5} + ( - 4 \beta_{2} + 4) q^{7} + 2 \beta_{3} q^{8} + 6 q^{10} + 12 \beta_1 q^{11} - 8 \beta_{2} q^{13} + ( - 4 \beta_{3} + 4 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + 9 \beta_{3} q^{17} - 16 q^{19} + 6 \beta_1 q^{20} + 24 \beta_{2} q^{22} + ( - 12 \beta_{3} + 12 \beta_1) q^{23} + (7 \beta_{2} - 7) q^{25} - 8 \beta_{3} q^{26} + 8 q^{28} + 3 \beta_1 q^{29} - 44 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (18 \beta_{2} - 18) q^{34} - 12 \beta_{3} q^{35} - 34 q^{37} - 16 \beta_1 q^{38} + 12 \beta_{2} q^{40} + (33 \beta_{3} - 33 \beta_1) q^{41} + ( - 40 \beta_{2} + 40) q^{43} + 24 \beta_{3} q^{44} + 24 q^{46} - 60 \beta_1 q^{47} + 33 \beta_{2} q^{49} + (7 \beta_{3} - 7 \beta_1) q^{50} + ( - 16 \beta_{2} + 16) q^{52} - 27 \beta_{3} q^{53} + 72 q^{55} + 8 \beta_1 q^{56} + 6 \beta_{2} q^{58} + (24 \beta_{3} - 24 \beta_1) q^{59} + (50 \beta_{2} - 50) q^{61} - 44 \beta_{3} q^{62} - 8 q^{64} - 24 \beta_1 q^{65} - 8 \beta_{2} q^{67} + (18 \beta_{3} - 18 \beta_1) q^{68} + ( - 24 \beta_{2} + 24) q^{70} + 36 \beta_{3} q^{71} - 16 q^{73} - 34 \beta_1 q^{74} - 32 \beta_{2} q^{76} + ( - 48 \beta_{3} + 48 \beta_1) q^{77} + ( - 76 \beta_{2} + 76) q^{79} + 12 \beta_{3} q^{80} - 66 q^{82} + 84 \beta_1 q^{83} + 54 \beta_{2} q^{85} + ( - 40 \beta_{3} + 40 \beta_1) q^{86} + (48 \beta_{2} - 48) q^{88} - 9 \beta_{3} q^{89} - 32 q^{91} + 24 \beta_1 q^{92} - 120 \beta_{2} q^{94} + (48 \beta_{3} - 48 \beta_1) q^{95} + (176 \beta_{2} - 176) q^{97} + 33 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 2*b2 * q^4 + (-3*b3 + 3*b1) * q^5 + (-4*b2 + 4) * q^7 + 2*b3 * q^8 + 6 * q^10 + 12*b1 * q^11 - 8*b2 * q^13 + (-4*b3 + 4*b1) * q^14 + (4*b2 - 4) * q^16 + 9*b3 * q^17 - 16 * q^19 + 6*b1 * q^20 + 24*b2 * q^22 + (-12*b3 + 12*b1) * q^23 + (7*b2 - 7) * q^25 - 8*b3 * q^26 + 8 * q^28 + 3*b1 * q^29 - 44*b2 * q^31 + (4*b3 - 4*b1) * q^32 + (18*b2 - 18) * q^34 - 12*b3 * q^35 - 34 * q^37 - 16*b1 * q^38 + 12*b2 * q^40 + (33*b3 - 33*b1) * q^41 + (-40*b2 + 40) * q^43 + 24*b3 * q^44 + 24 * q^46 - 60*b1 * q^47 + 33*b2 * q^49 + (7*b3 - 7*b1) * q^50 + (-16*b2 + 16) * q^52 - 27*b3 * q^53 + 72 * q^55 + 8*b1 * q^56 + 6*b2 * q^58 + (24*b3 - 24*b1) * q^59 + (50*b2 - 50) * q^61 - 44*b3 * q^62 - 8 * q^64 - 24*b1 * q^65 - 8*b2 * q^67 + (18*b3 - 18*b1) * q^68 + (-24*b2 + 24) * q^70 + 36*b3 * q^71 - 16 * q^73 - 34*b1 * q^74 - 32*b2 * q^76 + (-48*b3 + 48*b1) * q^77 + (-76*b2 + 76) * q^79 + 12*b3 * q^80 - 66 * q^82 + 84*b1 * q^83 + 54*b2 * q^85 + (-40*b3 + 40*b1) * q^86 + (48*b2 - 48) * q^88 - 9*b3 * q^89 - 32 * q^91 + 24*b1 * q^92 - 120*b2 * q^94 + (48*b3 - 48*b1) * q^95 + (176*b2 - 176) * q^97 + 33*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q + 4 * q^4 + 8 * q^7 $$4 q + 4 q^{4} + 8 q^{7} + 24 q^{10} - 16 q^{13} - 8 q^{16} - 64 q^{19} + 48 q^{22} - 14 q^{25} + 32 q^{28} - 88 q^{31} - 36 q^{34} - 136 q^{37} + 24 q^{40} + 80 q^{43} + 96 q^{46} + 66 q^{49} + 32 q^{52} + 288 q^{55} + 12 q^{58} - 100 q^{61} - 32 q^{64} - 16 q^{67} + 48 q^{70} - 64 q^{73} - 64 q^{76} + 152 q^{79} - 264 q^{82} + 108 q^{85} - 96 q^{88} - 128 q^{91} - 240 q^{94} - 352 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 + 8 * q^7 + 24 * q^10 - 16 * q^13 - 8 * q^16 - 64 * q^19 + 48 * q^22 - 14 * q^25 + 32 * q^28 - 88 * q^31 - 36 * q^34 - 136 * q^37 + 24 * q^40 + 80 * q^43 + 96 * q^46 + 66 * q^49 + 32 * q^52 + 288 * q^55 + 12 * q^58 - 100 * q^61 - 32 * q^64 - 16 * q^67 + 48 * q^70 - 64 * q^73 - 64 * q^76 + 152 * q^79 - 264 * q^82 + 108 * q^85 - 96 * q^88 - 128 * q^91 - 240 * q^94 - 352 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$83$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −3.67423 2.12132i 0 2.00000 + 3.46410i 2.82843i 0 6.00000
53.2 1.22474 0.707107i 0 1.00000 1.73205i 3.67423 + 2.12132i 0 2.00000 + 3.46410i 2.82843i 0 6.00000
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.67423 + 2.12132i 0 2.00000 3.46410i 2.82843i 0 6.00000
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.67423 2.12132i 0 2.00000 3.46410i 2.82843i 0 6.00000
 $$n$$: 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.d.b 4
3.b odd 2 1 inner 162.3.d.b 4
4.b odd 2 1 1296.3.q.f 4
9.c even 3 1 18.3.b.a 2
9.c even 3 1 inner 162.3.d.b 4
9.d odd 6 1 18.3.b.a 2
9.d odd 6 1 inner 162.3.d.b 4
12.b even 2 1 1296.3.q.f 4
36.f odd 6 1 144.3.e.b 2
36.f odd 6 1 1296.3.q.f 4
36.h even 6 1 144.3.e.b 2
36.h even 6 1 1296.3.q.f 4
45.h odd 6 1 450.3.d.f 2
45.j even 6 1 450.3.d.f 2
45.k odd 12 2 450.3.b.b 4
45.l even 12 2 450.3.b.b 4
63.g even 3 1 882.3.s.b 4
63.h even 3 1 882.3.s.b 4
63.i even 6 1 882.3.s.d 4
63.j odd 6 1 882.3.s.b 4
63.k odd 6 1 882.3.s.d 4
63.l odd 6 1 882.3.b.a 2
63.n odd 6 1 882.3.s.b 4
63.o even 6 1 882.3.b.a 2
63.s even 6 1 882.3.s.d 4
63.t odd 6 1 882.3.s.d 4
72.j odd 6 1 576.3.e.c 2
72.l even 6 1 576.3.e.f 2
72.n even 6 1 576.3.e.c 2
72.p odd 6 1 576.3.e.f 2
99.g even 6 1 2178.3.c.d 2
99.h odd 6 1 2178.3.c.d 2
117.n odd 6 1 3042.3.c.e 2
117.t even 6 1 3042.3.c.e 2
117.y odd 12 2 3042.3.d.a 4
117.z even 12 2 3042.3.d.a 4
144.u even 12 2 2304.3.h.c 4
144.v odd 12 2 2304.3.h.c 4
144.w odd 12 2 2304.3.h.f 4
144.x even 12 2 2304.3.h.f 4
180.n even 6 1 3600.3.l.d 2
180.p odd 6 1 3600.3.l.d 2
180.v odd 12 2 3600.3.c.b 4
180.x even 12 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 9.c even 3 1
18.3.b.a 2 9.d odd 6 1
144.3.e.b 2 36.f odd 6 1
144.3.e.b 2 36.h even 6 1
162.3.d.b 4 1.a even 1 1 trivial
162.3.d.b 4 3.b odd 2 1 inner
162.3.d.b 4 9.c even 3 1 inner
162.3.d.b 4 9.d odd 6 1 inner
450.3.b.b 4 45.k odd 12 2
450.3.b.b 4 45.l even 12 2
450.3.d.f 2 45.h odd 6 1
450.3.d.f 2 45.j even 6 1
576.3.e.c 2 72.j odd 6 1
576.3.e.c 2 72.n even 6 1
576.3.e.f 2 72.l even 6 1
576.3.e.f 2 72.p odd 6 1
882.3.b.a 2 63.l odd 6 1
882.3.b.a 2 63.o even 6 1
882.3.s.b 4 63.g even 3 1
882.3.s.b 4 63.h even 3 1
882.3.s.b 4 63.j odd 6 1
882.3.s.b 4 63.n odd 6 1
882.3.s.d 4 63.i even 6 1
882.3.s.d 4 63.k odd 6 1
882.3.s.d 4 63.s even 6 1
882.3.s.d 4 63.t odd 6 1
1296.3.q.f 4 4.b odd 2 1
1296.3.q.f 4 12.b even 2 1
1296.3.q.f 4 36.f odd 6 1
1296.3.q.f 4 36.h even 6 1
2178.3.c.d 2 99.g even 6 1
2178.3.c.d 2 99.h odd 6 1
2304.3.h.c 4 144.u even 12 2
2304.3.h.c 4 144.v odd 12 2
2304.3.h.f 4 144.w odd 12 2
2304.3.h.f 4 144.x even 12 2
3042.3.c.e 2 117.n odd 6 1
3042.3.c.e 2 117.t even 6 1
3042.3.d.a 4 117.y odd 12 2
3042.3.d.a 4 117.z even 12 2
3600.3.c.b 4 180.v odd 12 2
3600.3.c.b 4 180.x even 12 2
3600.3.l.d 2 180.n even 6 1
3600.3.l.d 2 180.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 18T_{5}^{2} + 324$$ acting on $$S_{3}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 18T^{2} + 324$$
$7$ $$(T^{2} - 4 T + 16)^{2}$$
$11$ $$T^{4} - 288 T^{2} + 82944$$
$13$ $$(T^{2} + 8 T + 64)^{2}$$
$17$ $$(T^{2} + 162)^{2}$$
$19$ $$(T + 16)^{4}$$
$23$ $$T^{4} - 288 T^{2} + 82944$$
$29$ $$T^{4} - 18T^{2} + 324$$
$31$ $$(T^{2} + 44 T + 1936)^{2}$$
$37$ $$(T + 34)^{4}$$
$41$ $$T^{4} - 2178 T^{2} + \cdots + 4743684$$
$43$ $$(T^{2} - 40 T + 1600)^{2}$$
$47$ $$T^{4} - 7200 T^{2} + \cdots + 51840000$$
$53$ $$(T^{2} + 1458)^{2}$$
$59$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$61$ $$(T^{2} + 50 T + 2500)^{2}$$
$67$ $$(T^{2} + 8 T + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T + 16)^{4}$$
$79$ $$(T^{2} - 76 T + 5776)^{2}$$
$83$ $$T^{4} - 14112 T^{2} + \cdots + 199148544$$
$89$ $$(T^{2} + 162)^{2}$$
$97$ $$(T^{2} + 176 T + 30976)^{2}$$