# Properties

 Label 162.5.d.d 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{3} q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - 5 \beta_{6} + 22 \beta_{5}) q^{5} + ( - 13 \beta_{2} + 13 \beta_1) q^{7} + (16 \beta_{5} - 16 \beta_{3}) q^{8}+O(q^{10})$$ q - 2*b3 * q^2 + (-8*b1 + 8) * q^4 + (-5*b6 + 22*b5) * q^5 + (-13*b2 + 13*b1) * q^7 + (16*b5 - 16*b3) * q^8 $$q - 2 \beta_{3} q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - 5 \beta_{6} + 22 \beta_{5}) q^{5} + ( - 13 \beta_{2} + 13 \beta_1) q^{7} + (16 \beta_{5} - 16 \beta_{3}) q^{8} + (10 \beta_{4} - 78) q^{10} + (10 \beta_{7} - 10 \beta_{6} + 59 \beta_{3}) q^{11} + ( - 14 \beta_{4} + 14 \beta_{2} + 143 \beta_1 - 143) q^{13} + (52 \beta_{6} - 52 \beta_{5}) q^{14} - 64 \beta_1 q^{16} + (31 \beta_{7} - 116 \beta_{5} + 116 \beta_{3}) q^{17} + ( - 43 \beta_{4} - 31) q^{19} + (40 \beta_{7} - 40 \beta_{6} + 176 \beta_{3}) q^{20} + (20 \beta_{4} - 20 \beta_{2} + 216 \beta_1 - 216) q^{22} + (200 \beta_{6} + 35 \beta_{5}) q^{23} + ( - 195 \beta_{2} + 473 \beta_1) q^{25} + ( - 56 \beta_{7} - 258 \beta_{5} + 258 \beta_{3}) q^{26} + ( - 104 \beta_{4} + 104) q^{28} + (295 \beta_{7} - 295 \beta_{6} - 37 \beta_{3}) q^{29} + ( - 54 \beta_{4} + 54 \beta_{2} + 896 \beta_1 - 896) q^{31} + 128 \beta_{5} q^{32} + ( - 62 \beta_{2} + 402 \beta_1) q^{34} + ( - 572 \beta_{7} + 1417 \beta_{5} - 1417 \beta_{3}) q^{35} + (\beta_{4} - 2350) q^{37} + ( - 172 \beta_{7} + 172 \beta_{6} - 24 \beta_{3}) q^{38} + (80 \beta_{4} - 80 \beta_{2} + 624 \beta_1 - 624) q^{40} + ( - 266 \beta_{6} - 203 \beta_{5}) q^{41} + ( - 177 \beta_{2} - 755 \beta_1) q^{43} + (80 \beta_{7} - 472 \beta_{5} + 472 \beta_{3}) q^{44} + ( - 400 \beta_{4} - 540) q^{46} + (594 \beta_{7} - 594 \beta_{6} - 510 \beta_{3}) q^{47} + (338 \beta_{4} - 338 \beta_{2} + 2331 \beta_1 - 2331) q^{49} + (780 \beta_{6} - 1336 \beta_{5}) q^{50} + (112 \beta_{2} + 1144 \beta_1) q^{52} + ( - 262 \beta_{7} - 421 \beta_{5} + 421 \beta_{3}) q^{53} + ( - 465 \beta_{4} + 2781) q^{55} + ( - 416 \beta_{7} + 416 \beta_{6} - 416 \beta_{3}) q^{56} + (590 \beta_{4} - 590 \beta_{2} - 738 \beta_1 + 738) q^{58} + ( - 358 \beta_{6} + 530 \beta_{5}) q^{59} + ( - 351 \beta_{2} + 1036 \beta_1) q^{61} + ( - 216 \beta_{7} - 1684 \beta_{5} + 1684 \beta_{3}) q^{62} - 512 q^{64} + ( - 169 \beta_{7} + 169 \beta_{6} - 1928 \beta_{3}) q^{65} + ( - 1131 \beta_{4} + 1131 \beta_{2} - 3019 \beta_1 + 3019) q^{67} + (248 \beta_{6} - 928 \beta_{5}) q^{68} + (1144 \beta_{2} - 4524 \beta_1) q^{70} + (592 \beta_{7} + 3757 \beta_{5} - 3757 \beta_{3}) q^{71} + (1455 \beta_{4} - 448) q^{73} + (4 \beta_{7} - 4 \beta_{6} + 4702 \beta_{3}) q^{74} + ( - 344 \beta_{4} + 344 \beta_{2} + 248 \beta_1 - 248) q^{76} + ( - 1534 \beta_{6} + 3224 \beta_{5}) q^{77} + ( - 53 \beta_{2} + 3751 \beta_1) q^{79} + (320 \beta_{7} - 1408 \beta_{5} + 1408 \beta_{3}) q^{80} + (532 \beta_{4} + 1344) q^{82} + (408 \beta_{7} - 408 \beta_{6} + 450 \beta_{3}) q^{83} + ( - 1107 \beta_{4} + 1107 \beta_{2} - 6012 \beta_1 + 6012) q^{85} + (708 \beta_{6} + 1156 \beta_{5}) q^{86} + ( - 160 \beta_{2} + 1728 \beta_1) q^{88} + (2041 \beta_{7} + 1309 \beta_{5} - 1309 \beta_{3}) q^{89} + (1677 \beta_{4} + 3055) q^{91} + ( - 1600 \beta_{7} + 1600 \beta_{6} + 280 \beta_{3}) q^{92} + (1188 \beta_{4} - 1188 \beta_{2} - 3228 \beta_1 + 3228) q^{94} + ( - 1522 \beta_{6} + 3059 \beta_{5}) q^{95} + (310 \beta_{2} - 11576 \beta_1) q^{97} + (1352 \beta_{7} - 5338 \beta_{5} + 5338 \beta_{3}) q^{98}+O(q^{100})$$ q - 2*b3 * q^2 + (-8*b1 + 8) * q^4 + (-5*b6 + 22*b5) * q^5 + (-13*b2 + 13*b1) * q^7 + (16*b5 - 16*b3) * q^8 + (10*b4 - 78) * q^10 + (10*b7 - 10*b6 + 59*b3) * q^11 + (-14*b4 + 14*b2 + 143*b1 - 143) * q^13 + (52*b6 - 52*b5) * q^14 - 64*b1 * q^16 + (31*b7 - 116*b5 + 116*b3) * q^17 + (-43*b4 - 31) * q^19 + (40*b7 - 40*b6 + 176*b3) * q^20 + (20*b4 - 20*b2 + 216*b1 - 216) * q^22 + (200*b6 + 35*b5) * q^23 + (-195*b2 + 473*b1) * q^25 + (-56*b7 - 258*b5 + 258*b3) * q^26 + (-104*b4 + 104) * q^28 + (295*b7 - 295*b6 - 37*b3) * q^29 + (-54*b4 + 54*b2 + 896*b1 - 896) * q^31 + 128*b5 * q^32 + (-62*b2 + 402*b1) * q^34 + (-572*b7 + 1417*b5 - 1417*b3) * q^35 + (b4 - 2350) * q^37 + (-172*b7 + 172*b6 - 24*b3) * q^38 + (80*b4 - 80*b2 + 624*b1 - 624) * q^40 + (-266*b6 - 203*b5) * q^41 + (-177*b2 - 755*b1) * q^43 + (80*b7 - 472*b5 + 472*b3) * q^44 + (-400*b4 - 540) * q^46 + (594*b7 - 594*b6 - 510*b3) * q^47 + (338*b4 - 338*b2 + 2331*b1 - 2331) * q^49 + (780*b6 - 1336*b5) * q^50 + (112*b2 + 1144*b1) * q^52 + (-262*b7 - 421*b5 + 421*b3) * q^53 + (-465*b4 + 2781) * q^55 + (-416*b7 + 416*b6 - 416*b3) * q^56 + (590*b4 - 590*b2 - 738*b1 + 738) * q^58 + (-358*b6 + 530*b5) * q^59 + (-351*b2 + 1036*b1) * q^61 + (-216*b7 - 1684*b5 + 1684*b3) * q^62 - 512 * q^64 + (-169*b7 + 169*b6 - 1928*b3) * q^65 + (-1131*b4 + 1131*b2 - 3019*b1 + 3019) * q^67 + (248*b6 - 928*b5) * q^68 + (1144*b2 - 4524*b1) * q^70 + (592*b7 + 3757*b5 - 3757*b3) * q^71 + (1455*b4 - 448) * q^73 + (4*b7 - 4*b6 + 4702*b3) * q^74 + (-344*b4 + 344*b2 + 248*b1 - 248) * q^76 + (-1534*b6 + 3224*b5) * q^77 + (-53*b2 + 3751*b1) * q^79 + (320*b7 - 1408*b5 + 1408*b3) * q^80 + (532*b4 + 1344) * q^82 + (408*b7 - 408*b6 + 450*b3) * q^83 + (-1107*b4 + 1107*b2 - 6012*b1 + 6012) * q^85 + (708*b6 + 1156*b5) * q^86 + (-160*b2 + 1728*b1) * q^88 + (2041*b7 + 1309*b5 - 1309*b3) * q^89 + (1677*b4 + 3055) * q^91 + (-1600*b7 + 1600*b6 + 280*b3) * q^92 + (1188*b4 - 1188*b2 - 3228*b1 + 3228) * q^94 + (-1522*b6 + 3059*b5) * q^95 + (310*b2 - 11576*b1) * q^97 + (1352*b7 - 5338*b5 + 5338*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 $$8 q + 32 q^{4} + 52 q^{7} - 624 q^{10} - 572 q^{13} - 256 q^{16} - 248 q^{19} - 864 q^{22} + 1892 q^{25} + 832 q^{28} - 3584 q^{31} + 1608 q^{34} - 18800 q^{37} - 2496 q^{40} - 3020 q^{43} - 4320 q^{46} - 9324 q^{49} + 4576 q^{52} + 22248 q^{55} + 2952 q^{58} + 4144 q^{61} - 4096 q^{64} + 12076 q^{67} - 18096 q^{70} - 3584 q^{73} - 992 q^{76} + 15004 q^{79} + 10752 q^{82} + 24048 q^{85} + 6912 q^{88} + 24440 q^{91} + 12912 q^{94} - 46304 q^{97}+O(q^{100})$$ 8 * q + 32 * q^4 + 52 * q^7 - 624 * q^10 - 572 * q^13 - 256 * q^16 - 248 * q^19 - 864 * q^22 + 1892 * q^25 + 832 * q^28 - 3584 * q^31 + 1608 * q^34 - 18800 * q^37 - 2496 * q^40 - 3020 * q^43 - 4320 * q^46 - 9324 * q^49 + 4576 * q^52 + 22248 * q^55 + 2952 * q^58 + 4144 * q^61 - 4096 * q^64 + 12076 * q^67 - 18096 * q^70 - 3584 * q^73 - 992 * q^76 + 15004 * q^79 + 10752 * q^82 + 24048 * q^85 + 6912 * q^88 + 24440 * q^91 + 12912 * q^94 - 46304 * q^97

Basis of coefficient ring

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

## 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
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i
−2.44949 + 1.41421i 0 4.00000 6.92820i 7.97262 + 4.60300i 0 −27.2750 47.2417i 22.6274i 0 −26.0385
53.2 −2.44949 + 1.41421i 0 4.00000 6.92820i 39.7924 + 22.9742i 0 40.2750 + 69.7583i 22.6274i 0 −129.962
53.3 2.44949 1.41421i 0 4.00000 6.92820i −39.7924 22.9742i 0 40.2750 + 69.7583i 22.6274i 0 −129.962
53.4 2.44949 1.41421i 0 4.00000 6.92820i −7.97262 4.60300i 0 −27.2750 47.2417i 22.6274i 0 −26.0385
107.1 −2.44949 1.41421i 0 4.00000 + 6.92820i 7.97262 4.60300i 0 −27.2750 + 47.2417i 22.6274i 0 −26.0385
107.2 −2.44949 1.41421i 0 4.00000 + 6.92820i 39.7924 22.9742i 0 40.2750 69.7583i 22.6274i 0 −129.962
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −39.7924 + 22.9742i 0 40.2750 69.7583i 22.6274i 0 −129.962
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i −7.97262 + 4.60300i 0 −27.2750 + 47.2417i 22.6274i 0 −26.0385
 $$n$$: 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.d 8
3.b odd 2 1 inner 162.5.d.d 8
9.c even 3 1 162.5.b.a 4
9.c even 3 1 inner 162.5.d.d 8
9.d odd 6 1 162.5.b.a 4
9.d odd 6 1 inner 162.5.d.d 8
36.f odd 6 1 1296.5.e.b 4
36.h even 6 1 1296.5.e.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 2196T_{5}^{6} + 4643487T_{5}^{4} - 392928084T_{5}^{2} + 32015587041$$ acting on $$S_{5}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2196 T^{6} + \cdots + 32015587041$$
$7$ $$(T^{4} - 26 T^{3} + 5070 T^{2} + \cdots + 19307236)^{2}$$
$11$ $$T^{8} + \cdots + 403540761128976$$
$13$ $$(T^{4} + 286 T^{3} + 66639 T^{2} + \cdots + 229734649)^{2}$$
$17$ $$(T^{4} + 66348 T^{2} + 52229529)^{2}$$
$19$ $$(T^{2} + 62 T - 48962)^{4}$$
$23$ $$T^{8} - 1152900 T^{6} + \cdots + 64\!\cdots\!00$$
$29$ $$T^{8} - 2485836 T^{6} + \cdots + 15\!\cdots\!01$$
$31$ $$(T^{4} + 1792 T^{3} + \cdots + 524297639056)^{2}$$
$37$ $$(T^{2} + 4700 T + 5522473)^{4}$$
$41$ $$T^{8} - 2361996 T^{6} + \cdots + 28\!\cdots\!16$$
$43$ $$(T^{4} + 1510 T^{3} + \cdots + 76097636164)^{2}$$
$47$ $$T^{8} - 12131568 T^{6} + \cdots + 14\!\cdots\!36$$
$53$ $$(T^{4} + 3072204 T^{2} + \cdots + 100670405796)^{2}$$
$59$ $$T^{8} - 3953232 T^{6} + \cdots + 48\!\cdots\!36$$
$61$ $$(T^{4} - 2072 T^{3} + \cdots + 5076599303161)^{2}$$
$67$ $$(T^{4} - 6038 T^{3} + \cdots + 646328217156196)^{2}$$
$71$ $$(T^{4} + 75169764 T^{2} + \cdots + 790866794501316)^{2}$$
$73$ $$(T^{2} + 896 T - 56958971)^{4}$$
$79$ $$(T^{4} - 7502 T^{3} + \cdots + 195836458128964)^{2}$$
$83$ $$T^{8} - 4736592 T^{6} + \cdots + 20\!\cdots\!76$$
$89$ $$(T^{4} + 134179668 T^{2} + \cdots + 20\!\cdots\!09)^{2}$$
$97$ $$(T^{4} + 23152 T^{3} + \cdots + 17\!\cdots\!76)^{2}$$