# Properties

 Label 162.5.d.e Level $162$ Weight $5$ Character orbit 162.d Analytic conductor $16.746$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,5,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, 5, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ 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: $$16.7459340196$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{10}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{5} + 2 \beta_{3}) q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - \beta_{6} + 7 \beta_{5}) q^{5} + ( - \beta_{2} + 13 \beta_1) q^{7} + 16 \beta_{3} q^{8}+O(q^{10})$$ q + (2*b5 + 2*b3) * q^2 + (-8*b1 + 8) * q^4 + (-b6 + 7*b5) * q^5 + (-b2 + 13*b1) * q^7 + 16*b3 * q^8 $$q + (2 \beta_{5} + 2 \beta_{3}) q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - \beta_{6} + 7 \beta_{5}) q^{5} + ( - \beta_{2} + 13 \beta_1) q^{7} + 16 \beta_{3} q^{8} + ( - 2 \beta_{4} + 30) q^{10} + (2 \beta_{7} - 2 \beta_{6} + 53 \beta_{5} + 55 \beta_{3}) q^{11} + (10 \beta_{4} - 10 \beta_{2} - 73 \beta_1 + 73) q^{13} + ( - 4 \beta_{6} + 24 \beta_{5}) q^{14} - 64 \beta_1 q^{16} + ( - 37 \beta_{7} - 38 \beta_{3}) q^{17} + ( - 31 \beta_{4} + 185) q^{19} + (8 \beta_{7} - 8 \beta_{6} + 56 \beta_{5} + 64 \beta_{3}) q^{20} + ( - 4 \beta_{4} + 4 \beta_{2} - 216 \beta_1 + 216) q^{22} + (40 \beta_{6} + 155 \beta_{5}) q^{23} + ( - 15 \beta_{2} - 391 \beta_1) q^{25} + ( - 40 \beta_{7} + 126 \beta_{3}) q^{26} + ( - 8 \beta_{4} + 104) q^{28} + (59 \beta_{7} - 59 \beta_{6} - 34 \beta_{5} + 25 \beta_{3}) q^{29} + (18 \beta_{4} - 18 \beta_{2} - 1264 \beta_1 + 1264) q^{31} - 128 \beta_{5} q^{32} + ( - 74 \beta_{2} + 78 \beta_1) q^{34} + ( - 28 \beta_{7} - 233 \beta_{3}) q^{35} + ( - 83 \beta_{4} + 1106) q^{37} + (124 \beta_{7} - 124 \beta_{6} + 308 \beta_{5} + 432 \beta_{3}) q^{38} + ( - 16 \beta_{4} + 16 \beta_{2} - 240 \beta_1 + 240) q^{40} + (206 \beta_{6} + 7 \beta_{5}) q^{41} + ( - 141 \beta_{2} - 107 \beta_1) q^{43} + (16 \beta_{7} + 440 \beta_{3}) q^{44} + (80 \beta_{4} + 540) q^{46} + ( - 54 \beta_{7} + 54 \beta_{6} - 1812 \beta_{5} - 1866 \beta_{3}) q^{47} + (26 \beta_{4} - 26 \beta_{2} - 1989 \beta_1 + 1989) q^{49} + ( - 60 \beta_{6} - 812 \beta_{5}) q^{50} + ( - 80 \beta_{2} - 584 \beta_1) q^{52} + (34 \beta_{7} - 2047 \beta_{3}) q^{53} + ( - 69 \beta_{4} + 1053) q^{55} + (32 \beta_{7} - 32 \beta_{6} + 192 \beta_{5} + 224 \beta_{3}) q^{56} + ( - 118 \beta_{4} + 118 \beta_{2} + 18 \beta_1 - 18) q^{58} + (274 \beta_{6} + 272 \beta_{5}) q^{59} + (189 \beta_{2} - 1124 \beta_1) q^{61} + ( - 72 \beta_{7} + 2492 \beta_{3}) q^{62} - 512 q^{64} + ( - 77 \beta_{7} + 77 \beta_{6} - 629 \beta_{5} - 706 \beta_{3}) q^{65} + (129 \beta_{4} - 129 \beta_{2} + 4109 \beta_1 - 4109) q^{67} + ( - 296 \beta_{6} + 8 \beta_{5}) q^{68} + ( - 56 \beta_{2} + 876 \beta_1) q^{70} + (32 \beta_{7} - 1277 \beta_{3}) q^{71} + ( - 21 \beta_{4} - 5200) q^{73} + (332 \beta_{7} - 332 \beta_{6} + 2046 \beta_{5} + 2378 \beta_{3}) q^{74} + ( - 248 \beta_{4} + 248 \beta_{2} - 1480 \beta_1 + 1480) q^{76} + ( - 134 \beta_{6} + 878 \beta_{5}) q^{77} + (439 \beta_{2} + 4183 \beta_1) q^{79} + (64 \beta_{7} + 512 \beta_{3}) q^{80} + (412 \beta_{4} - 384) q^{82} + ( - 696 \beta_{7} + 696 \beta_{6} + 3450 \beta_{5} + 2754 \beta_{3}) q^{83} + (297 \beta_{4} - 297 \beta_{2} + 4788 \beta_1 - 4788) q^{85} + ( - 564 \beta_{6} - 496 \beta_{5}) q^{86} + (32 \beta_{2} - 1728 \beta_1) q^{88} + (365 \beta_{7} + 973 \beta_{3}) q^{89} + (57 \beta_{4} - 1481) q^{91} + ( - 320 \beta_{7} + 320 \beta_{6} + 1240 \beta_{5} + 920 \beta_{3}) q^{92} + (108 \beta_{4} - 108 \beta_{2} + 7356 \beta_1 - 7356) q^{94} + ( - 650 \beta_{6} + 4829 \beta_{5}) q^{95} + (766 \beta_{2} + 952 \beta_1) q^{97} + ( - 104 \beta_{7} + 3926 \beta_{3}) q^{98}+O(q^{100})$$ q + (2*b5 + 2*b3) * q^2 + (-8*b1 + 8) * q^4 + (-b6 + 7*b5) * q^5 + (-b2 + 13*b1) * q^7 + 16*b3 * q^8 + (-2*b4 + 30) * q^10 + (2*b7 - 2*b6 + 53*b5 + 55*b3) * q^11 + (10*b4 - 10*b2 - 73*b1 + 73) * q^13 + (-4*b6 + 24*b5) * q^14 - 64*b1 * q^16 + (-37*b7 - 38*b3) * q^17 + (-31*b4 + 185) * q^19 + (8*b7 - 8*b6 + 56*b5 + 64*b3) * q^20 + (-4*b4 + 4*b2 - 216*b1 + 216) * q^22 + (40*b6 + 155*b5) * q^23 + (-15*b2 - 391*b1) * q^25 + (-40*b7 + 126*b3) * q^26 + (-8*b4 + 104) * q^28 + (59*b7 - 59*b6 - 34*b5 + 25*b3) * q^29 + (18*b4 - 18*b2 - 1264*b1 + 1264) * q^31 - 128*b5 * q^32 + (-74*b2 + 78*b1) * q^34 + (-28*b7 - 233*b3) * q^35 + (-83*b4 + 1106) * q^37 + (124*b7 - 124*b6 + 308*b5 + 432*b3) * q^38 + (-16*b4 + 16*b2 - 240*b1 + 240) * q^40 + (206*b6 + 7*b5) * q^41 + (-141*b2 - 107*b1) * q^43 + (16*b7 + 440*b3) * q^44 + (80*b4 + 540) * q^46 + (-54*b7 + 54*b6 - 1812*b5 - 1866*b3) * q^47 + (26*b4 - 26*b2 - 1989*b1 + 1989) * q^49 + (-60*b6 - 812*b5) * q^50 + (-80*b2 - 584*b1) * q^52 + (34*b7 - 2047*b3) * q^53 + (-69*b4 + 1053) * q^55 + (32*b7 - 32*b6 + 192*b5 + 224*b3) * q^56 + (-118*b4 + 118*b2 + 18*b1 - 18) * q^58 + (274*b6 + 272*b5) * q^59 + (189*b2 - 1124*b1) * q^61 + (-72*b7 + 2492*b3) * q^62 - 512 * q^64 + (-77*b7 + 77*b6 - 629*b5 - 706*b3) * q^65 + (129*b4 - 129*b2 + 4109*b1 - 4109) * q^67 + (-296*b6 + 8*b5) * q^68 + (-56*b2 + 876*b1) * q^70 + (32*b7 - 1277*b3) * q^71 + (-21*b4 - 5200) * q^73 + (332*b7 - 332*b6 + 2046*b5 + 2378*b3) * q^74 + (-248*b4 + 248*b2 - 1480*b1 + 1480) * q^76 + (-134*b6 + 878*b5) * q^77 + (439*b2 + 4183*b1) * q^79 + (64*b7 + 512*b3) * q^80 + (412*b4 - 384) * q^82 + (-696*b7 + 696*b6 + 3450*b5 + 2754*b3) * q^83 + (297*b4 - 297*b2 + 4788*b1 - 4788) * q^85 + (-564*b6 - 496*b5) * q^86 + (32*b2 - 1728*b1) * q^88 + (365*b7 + 973*b3) * q^89 + (57*b4 - 1481) * q^91 + (-320*b7 + 320*b6 + 1240*b5 + 920*b3) * q^92 + (108*b4 - 108*b2 + 7356*b1 - 7356) * q^94 + (-650*b6 + 4829*b5) * q^95 + (766*b2 + 952*b1) * q^97 + (-104*b7 + 3926*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 32 q^{4} + 52 q^{7}+O(q^{10})$$ 8 * q + 32 * q^4 + 52 * q^7 $$8 q + 32 q^{4} + 52 q^{7} + 240 q^{10} + 292 q^{13} - 256 q^{16} + 1480 q^{19} + 864 q^{22} - 1564 q^{25} + 832 q^{28} + 5056 q^{31} + 312 q^{34} + 8848 q^{37} + 960 q^{40} - 428 q^{43} + 4320 q^{46} + 7956 q^{49} - 2336 q^{52} + 8424 q^{55} - 72 q^{58} - 4496 q^{61} - 4096 q^{64} - 16436 q^{67} + 3504 q^{70} - 41600 q^{73} + 5920 q^{76} + 16732 q^{79} - 3072 q^{82} - 19152 q^{85} - 6912 q^{88} - 11848 q^{91} - 29424 q^{94} + 3808 q^{97}+O(q^{100})$$ 8 * q + 32 * q^4 + 52 * q^7 + 240 * q^10 + 292 * q^13 - 256 * q^16 + 1480 * q^19 + 864 * q^22 - 1564 * q^25 + 832 * q^28 + 5056 * q^31 + 312 * q^34 + 8848 * q^37 + 960 * q^40 - 428 * q^43 + 4320 * q^46 + 7956 * q^49 - 2336 * q^52 + 8424 * q^55 - 72 * q^58 - 4496 * q^61 - 4096 * q^64 - 16436 * q^67 + 3504 * q^70 - 41600 * q^73 + 5920 * q^76 + 16732 * q^79 - 3072 * q^82 - 19152 * q^85 - 6912 * q^88 - 11848 * q^91 - 29424 * q^94 + 3808 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$9\zeta_{24}^{6} + 9\zeta_{24}^{2}$$ 9*v^6 + 9*v^2 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ -v^5 - v^3 + v $$\beta_{4}$$ $$=$$ $$-9\zeta_{24}^{6} + 18\zeta_{24}^{2}$$ -9*v^6 + 18*v^2 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{6}$$ $$=$$ $$5\zeta_{24}^{7} - 5\zeta_{24}^{5} + 4\zeta_{24}^{3} + 9\zeta_{24}$$ 5*v^7 - 5*v^5 + 4*v^3 + 9*v $$\beta_{7}$$ $$=$$ $$9\zeta_{24}^{7} + 5\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24}$$ 9*v^7 + 5*v^5 - 4*v^3 + 4*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 14\beta_{5} + 14\beta_{3} ) / 27$$ (b7 + b6 + 14*b5 + 14*b3) / 27 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{2} ) / 27$$ (b4 + b2) / 27 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + \beta_{5} - 14\beta_{3} ) / 27$$ (-b7 + 2*b6 + b5 - 14*b3) / 27 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} - \beta_{6} + 13\beta_{5} + \beta_{3} ) / 27$$ (2*b7 - b6 + 13*b5 + b3) / 27 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{2} ) / 27$$ (-b4 + 2*b2) / 27 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - 13\beta_{5} - 13\beta_{3} ) / 27$$ (b7 + b6 - 13*b5 - 13*b3) / 27

## Character values

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

 $$n$$ $$83$$ $$\chi(n)$$ $$1 - \beta_{1}$$

## 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
 −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i
−2.44949 + 1.41421i 0 4.00000 6.92820i −18.7315 10.8147i 0 14.2942 + 24.7583i 22.6274i 0 61.1769
53.2 −2.44949 + 1.41421i 0 4.00000 6.92820i 0.360355 + 0.208051i 0 −1.29423 2.24167i 22.6274i 0 −1.17691
53.3 2.44949 1.41421i 0 4.00000 6.92820i −0.360355 0.208051i 0 −1.29423 2.24167i 22.6274i 0 −1.17691
53.4 2.44949 1.41421i 0 4.00000 6.92820i 18.7315 + 10.8147i 0 14.2942 + 24.7583i 22.6274i 0 61.1769
107.1 −2.44949 1.41421i 0 4.00000 + 6.92820i −18.7315 + 10.8147i 0 14.2942 24.7583i 22.6274i 0 61.1769
107.2 −2.44949 1.41421i 0 4.00000 + 6.92820i 0.360355 0.208051i 0 −1.29423 + 2.24167i 22.6274i 0 −1.17691
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −0.360355 + 0.208051i 0 −1.29423 + 2.24167i 22.6274i 0 −1.17691
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i 18.7315 10.8147i 0 14.2942 24.7583i 22.6274i 0 61.1769
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.4 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 162.5.d.e 8
3.b odd 2 1 inner 162.5.d.e 8
9.c even 3 1 162.5.b.b 4
9.c even 3 1 inner 162.5.d.e 8
9.d odd 6 1 162.5.b.b 4
9.d odd 6 1 inner 162.5.d.e 8
36.f odd 6 1 1296.5.e.a 4
36.h even 6 1 1296.5.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.5.b.b 4 9.c even 3 1
162.5.b.b 4 9.d odd 6 1
162.5.d.e 8 1.a even 1 1 trivial
162.5.d.e 8 3.b odd 2 1 inner
162.5.d.e 8 9.c even 3 1 inner
162.5.d.e 8 9.d odd 6 1 inner
1296.5.e.a 4 36.f odd 6 1
1296.5.e.a 4 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 468T_{5}^{6} + 218943T_{5}^{4} - 37908T_{5}^{2} + 6561$$ acting on $$S_{5}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 468 T^{6} + 218943 T^{4} + \cdots + 6561$$
$7$ $$(T^{4} - 26 T^{3} + 750 T^{2} + 1924 T + 5476)^{2}$$
$11$ $$T^{8} + \cdots + 816800166640656$$
$13$ $$(T^{4} - 146 T^{3} + 40287 T^{2} + \cdots + 359898841)^{2}$$
$17$ $$(T^{4} + 334188 T^{2} + \cdots + 27414418329)^{2}$$
$19$ $$(T^{2} - 370 T - 199298)^{4}$$
$23$ $$T^{8} - 461700 T^{6} + \cdots + 62\!\cdots\!00$$
$29$ $$T^{8} - 845964 T^{6} + \cdots + 31\!\cdots\!01$$
$31$ $$(T^{4} - 2528 T^{3} + \cdots + 2307251633296)^{2}$$
$37$ $$(T^{2} - 2212 T - 450791)^{4}$$
$41$ $$T^{8} - 10348812 T^{6} + \cdots + 69\!\cdots\!96$$
$43$ $$(T^{4} + 214 T^{3} + \cdots + 23228871893956)^{2}$$
$47$ $$T^{8} - 14236272 T^{6} + \cdots + 16\!\cdots\!16$$
$53$ $$(T^{4} + 17321292 T^{2} + \cdots + 70220008948644)^{2}$$
$59$ $$T^{8} - 18316368 T^{6} + \cdots + 68\!\cdots\!36$$
$61$ $$(T^{4} + 2248 T^{3} + \cdots + 55009322747929)^{2}$$
$67$ $$(T^{4} + 8218 T^{3} + \cdots + 164868630253924)^{2}$$
$71$ $$(T^{4} + 6936228 T^{2} + \cdots + 10363776595524)^{2}$$
$73$ $$(T^{2} + 10400 T + 26932837)^{4}$$
$79$ $$(T^{4} - 8366 T^{3} + \cdots + 860466777033796)^{2}$$
$83$ $$T^{8} - 156202704 T^{6} + \cdots + 24\!\cdots\!16$$
$89$ $$(T^{4} + 34873236 T^{2} + \cdots + 223115671821249)^{2}$$
$97$ $$(T^{4} - 1904 T^{3} + \cdots + 20\!\cdots\!16)^{2}$$