# Properties

 Label 162.5.d.c 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: $$2^{6}\cdot 3^{12}$$ Twist minimal: no (minimal twist has level 54) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + 8 \beta_{2} q^{4} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 17 \beta_{2} - 17) q^{7} + 8 \beta_{4} q^{8}+O(q^{10})$$ q - b5 * q^2 + 8*b2 * q^4 + (-3*b5 - 3*b4 - b3 + b1) * q^5 + (b7 - b6 + 17*b2 - 17) * q^7 + 8*b4 * q^8 $$q - \beta_{5} q^{2} + 8 \beta_{2} q^{4} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 17 \beta_{2} - 17) q^{7} + 8 \beta_{4} q^{8} + ( - \beta_{7} + 24) q^{10} + ( - 30 \beta_{5} - \beta_1) q^{11} + (2 \beta_{6} + 130 \beta_{2}) q^{13} + (17 \beta_{5} + 17 \beta_{4} + 8 \beta_{3} - 8 \beta_1) q^{14} + (64 \beta_{2} - 64) q^{16} + (108 \beta_{4} + 2 \beta_{3}) q^{17} + ( - 2 \beta_{7} + 272) q^{19} + ( - 24 \beta_{5} + 8 \beta_1) q^{20} + (\beta_{6} + 240 \beta_{2}) q^{22} + (114 \beta_{5} + 114 \beta_{4} - 20 \beta_{3} + 20 \beta_1) q^{23} + ( - 6 \beta_{7} + 6 \beta_{6} - 176 \beta_{2} + 176) q^{25} + (130 \beta_{4} - 16 \beta_{3}) q^{26} + (8 \beta_{7} - 136) q^{28} + ( - 264 \beta_{5} - 22 \beta_1) q^{29} + ( - 15 \beta_{6} - 203 \beta_{2}) q^{31} + (64 \beta_{5} + 64 \beta_{4}) q^{32} + (2 \beta_{7} - 2 \beta_{6} + 864 \beta_{2} - 864) q^{34} + (780 \beta_{4} + 41 \beta_{3}) q^{35} + (14 \beta_{7} - 268) q^{37} + ( - 272 \beta_{5} + 16 \beta_1) q^{38} + ( - 8 \beta_{6} + 192 \beta_{2}) q^{40} + (330 \beta_{5} + 330 \beta_{4} + 68 \beta_{3} - 68 \beta_1) q^{41} + (6 \beta_{7} - 6 \beta_{6} - 1774 \beta_{2} + 1774) q^{43} + (240 \beta_{4} - 8 \beta_{3}) q^{44} + ( - 20 \beta_{7} - 912) q^{46} + ( - 804 \beta_{5} + 30 \beta_1) q^{47} + (34 \beta_{6} - 3720 \beta_{2}) q^{49} + ( - 176 \beta_{5} - 176 \beta_{4} - 48 \beta_{3} + 48 \beta_1) q^{50} + ( - 16 \beta_{7} + 16 \beta_{6} + 1040 \beta_{2} - 1040) q^{52} + (549 \beta_{4} - 119 \beta_{3}) q^{53} + ( - 27 \beta_{7} - 9) q^{55} + (136 \beta_{5} - 64 \beta_1) q^{56} + (22 \beta_{6} + 2112 \beta_{2}) q^{58} + ( - 228 \beta_{5} - 228 \beta_{4} - 98 \beta_{3} + 98 \beta_1) q^{59} + (24 \beta_{7} - 24 \beta_{6} - 3484 \beta_{2} + 3484) q^{61} + ( - 203 \beta_{4} + 120 \beta_{3}) q^{62} - 512 q^{64} + (1068 \beta_{5} + 82 \beta_1) q^{65} + (6 \beta_{6} - 806 \beta_{2}) q^{67} + (864 \beta_{5} + 864 \beta_{4} + 16 \beta_{3} - 16 \beta_1) q^{68} + (41 \beta_{7} - 41 \beta_{6} + 6240 \beta_{2} - 6240) q^{70} + ( - 1998 \beta_{4} + 62 \beta_{3}) q^{71} + ( - 18 \beta_{7} + 2153) q^{73} + (268 \beta_{5} - 112 \beta_1) q^{74} + ( - 16 \beta_{6} + 2176 \beta_{2}) q^{76} + ( - 219 \beta_{5} - 219 \beta_{4} + 223 \beta_{3} - 223 \beta_1) q^{77} + ( - 28 \beta_{7} + 28 \beta_{6} - 2890 \beta_{2} + 2890) q^{79} + (192 \beta_{4} + 64 \beta_{3}) q^{80} + (68 \beta_{7} - 2640) q^{82} + (1698 \beta_{5} - 27 \beta_1) q^{83} + ( - 114 \beta_{6} + 4050 \beta_{2}) q^{85} + ( - 1774 \beta_{5} - 1774 \beta_{4} + 48 \beta_{3} - 48 \beta_1) q^{86} + ( - 8 \beta_{7} + 8 \beta_{6} + 1920 \beta_{2} - 1920) q^{88} + ( - 2826 \beta_{4} - 262 \beta_{3}) q^{89} + (96 \beta_{7} + 9454) q^{91} + (912 \beta_{5} + 160 \beta_1) q^{92} + ( - 30 \beta_{6} + 6432 \beta_{2}) q^{94} + ( - 2274 \beta_{5} - 2274 \beta_{4} - 320 \beta_{3} + 320 \beta_1) q^{95} + ( - 88 \beta_{7} + 88 \beta_{6} + 395 \beta_{2} - 395) q^{97} + ( - 3720 \beta_{4} - 272 \beta_{3}) q^{98}+O(q^{100})$$ q - b5 * q^2 + 8*b2 * q^4 + (-3*b5 - 3*b4 - b3 + b1) * q^5 + (b7 - b6 + 17*b2 - 17) * q^7 + 8*b4 * q^8 + (-b7 + 24) * q^10 + (-30*b5 - b1) * q^11 + (2*b6 + 130*b2) * q^13 + (17*b5 + 17*b4 + 8*b3 - 8*b1) * q^14 + (64*b2 - 64) * q^16 + (108*b4 + 2*b3) * q^17 + (-2*b7 + 272) * q^19 + (-24*b5 + 8*b1) * q^20 + (b6 + 240*b2) * q^22 + (114*b5 + 114*b4 - 20*b3 + 20*b1) * q^23 + (-6*b7 + 6*b6 - 176*b2 + 176) * q^25 + (130*b4 - 16*b3) * q^26 + (8*b7 - 136) * q^28 + (-264*b5 - 22*b1) * q^29 + (-15*b6 - 203*b2) * q^31 + (64*b5 + 64*b4) * q^32 + (2*b7 - 2*b6 + 864*b2 - 864) * q^34 + (780*b4 + 41*b3) * q^35 + (14*b7 - 268) * q^37 + (-272*b5 + 16*b1) * q^38 + (-8*b6 + 192*b2) * q^40 + (330*b5 + 330*b4 + 68*b3 - 68*b1) * q^41 + (6*b7 - 6*b6 - 1774*b2 + 1774) * q^43 + (240*b4 - 8*b3) * q^44 + (-20*b7 - 912) * q^46 + (-804*b5 + 30*b1) * q^47 + (34*b6 - 3720*b2) * q^49 + (-176*b5 - 176*b4 - 48*b3 + 48*b1) * q^50 + (-16*b7 + 16*b6 + 1040*b2 - 1040) * q^52 + (549*b4 - 119*b3) * q^53 + (-27*b7 - 9) * q^55 + (136*b5 - 64*b1) * q^56 + (22*b6 + 2112*b2) * q^58 + (-228*b5 - 228*b4 - 98*b3 + 98*b1) * q^59 + (24*b7 - 24*b6 - 3484*b2 + 3484) * q^61 + (-203*b4 + 120*b3) * q^62 - 512 * q^64 + (1068*b5 + 82*b1) * q^65 + (6*b6 - 806*b2) * q^67 + (864*b5 + 864*b4 + 16*b3 - 16*b1) * q^68 + (41*b7 - 41*b6 + 6240*b2 - 6240) * q^70 + (-1998*b4 + 62*b3) * q^71 + (-18*b7 + 2153) * q^73 + (268*b5 - 112*b1) * q^74 + (-16*b6 + 2176*b2) * q^76 + (-219*b5 - 219*b4 + 223*b3 - 223*b1) * q^77 + (-28*b7 + 28*b6 - 2890*b2 + 2890) * q^79 + (192*b4 + 64*b3) * q^80 + (68*b7 - 2640) * q^82 + (1698*b5 - 27*b1) * q^83 + (-114*b6 + 4050*b2) * q^85 + (-1774*b5 - 1774*b4 + 48*b3 - 48*b1) * q^86 + (-8*b7 + 8*b6 + 1920*b2 - 1920) * q^88 + (-2826*b4 - 262*b3) * q^89 + (96*b7 + 9454) * q^91 + (912*b5 + 160*b1) * q^92 + (-30*b6 + 6432*b2) * q^94 + (-2274*b5 - 2274*b4 - 320*b3 + 320*b1) * q^95 + (-88*b7 + 88*b6 + 395*b2 - 395) * q^97 + (-3720*b4 - 272*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 32 q^{4} - 68 q^{7}+O(q^{10})$$ 8 * q + 32 * q^4 - 68 * q^7 $$8 q + 32 q^{4} - 68 q^{7} + 192 q^{10} + 520 q^{13} - 256 q^{16} + 2176 q^{19} + 960 q^{22} + 704 q^{25} - 1088 q^{28} - 812 q^{31} - 3456 q^{34} - 2144 q^{37} + 768 q^{40} + 7096 q^{43} - 7296 q^{46} - 14880 q^{49} - 4160 q^{52} - 72 q^{55} + 8448 q^{58} + 13936 q^{61} - 4096 q^{64} - 3224 q^{67} - 24960 q^{70} + 17224 q^{73} + 8704 q^{76} + 11560 q^{79} - 21120 q^{82} + 16200 q^{85} - 7680 q^{88} + 75632 q^{91} + 25728 q^{94} - 1580 q^{97}+O(q^{100})$$ 8 * q + 32 * q^4 - 68 * q^7 + 192 * q^10 + 520 * q^13 - 256 * q^16 + 2176 * q^19 + 960 * q^22 + 704 * q^25 - 1088 * q^28 - 812 * q^31 - 3456 * q^34 - 2144 * q^37 + 768 * q^40 + 7096 * q^43 - 7296 * q^46 - 14880 * q^49 - 4160 * q^52 - 72 * q^55 + 8448 * q^58 + 13936 * q^61 - 4096 * q^64 - 3224 * q^67 - 24960 * q^70 + 17224 * q^73 + 8704 * q^76 + 11560 * q^79 - 21120 * q^82 + 16200 * q^85 - 7680 * q^88 + 75632 * q^91 + 25728 * q^94 - 1580 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$27\zeta_{24}^{2}$$ 27*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$27\zeta_{24}^{6}$$ 27*v^6 $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}$$ -2*v^5 - 2*v^3 + 2*v $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3}$$ -2*v^7 + 2*v^5 + 2*v^3 $$\beta_{6}$$ $$=$$ $$54\zeta_{24}^{7} + 54\zeta_{24}$$ 54*v^7 + 54*v $$\beta_{7}$$ $$=$$ $$-54\zeta_{24}^{5} + 54\zeta_{24}^{3} + 54\zeta_{24}$$ -54*v^5 + 54*v^3 + 54*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{6} + 27\beta_{5} + 27\beta_{4} ) / 108$$ (b6 + 27*b5 + 27*b4) / 108 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_1 ) / 27$$ (b1) / 27 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} - 27\beta_{4} ) / 108$$ (b7 - 27*b4) / 108 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + 27\beta_{5} ) / 108$$ (-b7 + b6 + 27*b5) / 108 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_{3} ) / 27$$ (b3) / 27 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{6} - 27\beta_{5} - 27\beta_{4} ) / 108$$ (b6 - 27*b5 - 27*b4) / 108

## 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
 0.258819 + 0.965926i 0.965926 − 0.258819i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i −0.965926 − 0.258819i
−2.44949 + 1.41421i 0 4.00000 6.92820i −30.7312 17.7426i 0 −46.6838 80.8587i 22.6274i 0 100.368
53.2 −2.44949 + 1.41421i 0 4.00000 6.92820i 16.0342 + 9.25736i 0 29.6838 + 51.4138i 22.6274i 0 −52.3675
53.3 2.44949 1.41421i 0 4.00000 6.92820i −16.0342 9.25736i 0 29.6838 + 51.4138i 22.6274i 0 −52.3675
53.4 2.44949 1.41421i 0 4.00000 6.92820i 30.7312 + 17.7426i 0 −46.6838 80.8587i 22.6274i 0 100.368
107.1 −2.44949 1.41421i 0 4.00000 + 6.92820i −30.7312 + 17.7426i 0 −46.6838 + 80.8587i 22.6274i 0 100.368
107.2 −2.44949 1.41421i 0 4.00000 + 6.92820i 16.0342 9.25736i 0 29.6838 51.4138i 22.6274i 0 −52.3675
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −16.0342 + 9.25736i 0 29.6838 51.4138i 22.6274i 0 −52.3675
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i 30.7312 17.7426i 0 −46.6838 + 80.8587i 22.6274i 0 100.368
 $$n$$: 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.c 8
3.b odd 2 1 inner 162.5.d.c 8
9.c even 3 1 54.5.b.b 4
9.c even 3 1 inner 162.5.d.c 8
9.d odd 6 1 54.5.b.b 4
9.d odd 6 1 inner 162.5.d.c 8
36.f odd 6 1 432.5.e.h 4
36.h even 6 1 432.5.e.h 4
45.h odd 6 1 1350.5.d.c 4
45.j even 6 1 1350.5.d.c 4
45.k odd 12 1 1350.5.b.a 4
45.k odd 12 1 1350.5.b.c 4
45.l even 12 1 1350.5.b.a 4
45.l even 12 1 1350.5.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.5.b.b 4 9.c even 3 1
54.5.b.b 4 9.d odd 6 1
162.5.d.c 8 1.a even 1 1 trivial
162.5.d.c 8 3.b odd 2 1 inner
162.5.d.c 8 9.c even 3 1 inner
162.5.d.c 8 9.d odd 6 1 inner
432.5.e.h 4 36.f odd 6 1
432.5.e.h 4 36.h even 6 1
1350.5.b.a 4 45.k odd 12 1
1350.5.b.a 4 45.l even 12 1
1350.5.b.c 4 45.k odd 12 1
1350.5.b.c 4 45.l even 12 1
1350.5.d.c 4 45.h odd 6 1
1350.5.d.c 4 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 1602T_{5}^{6} + 2134755T_{5}^{4} - 691501698T_{5}^{2} + 186320859201$$ acting on $$S_{5}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 1602 T^{6} + \cdots + 186320859201$$
$7$ $$(T^{4} + 34 T^{3} + 6699 T^{2} + \cdots + 30724849)^{2}$$
$11$ $$T^{8} - 15858 T^{6} + \cdots + 17\!\cdots\!81$$
$13$ $$(T^{4} - 260 T^{3} + 74028 T^{2} + \cdots + 41319184)^{2}$$
$17$ $$(T^{4} + 192456 T^{2} + \cdots + 8171436816)^{2}$$
$19$ $$(T^{2} - 544 T + 50656)^{4}$$
$23$ $$T^{8} - 791136 T^{6} + \cdots + 12\!\cdots\!76$$
$29$ $$T^{8} - 1820808 T^{6} + \cdots + 17\!\cdots\!76$$
$31$ $$(T^{4} + 406 T^{3} + \cdots + 1615418122081)^{2}$$
$37$ $$(T^{2} + 536 T - 1071248)^{4}$$
$41$ $$T^{8} - 8484192 T^{6} + \cdots + 39\!\cdots\!56$$
$43$ $$(T^{4} - 3548 T^{3} + \cdots + 8626697391376)^{2}$$
$47$ $$T^{8} - 11654856 T^{6} + \cdots + 41\!\cdots\!56$$
$53$ $$(T^{4} + 25469154 T^{2} + \cdots + 62602291689921)^{2}$$
$59$ $$T^{8} - 14834376 T^{6} + \cdots + 18\!\cdots\!96$$
$61$ $$(T^{4} - 6968 T^{3} + \cdots + 77071262392576)^{2}$$
$67$ $$(T^{4} + 1612 T^{3} + \cdots + 193322019856)^{2}$$
$71$ $$(T^{4} + 69476616 T^{2} + \cdots + 848775738667536)^{2}$$
$73$ $$(T^{2} - 4306 T + 2745841)^{4}$$
$79$ $$(T^{4} - 5780 T^{3} + \cdots + 14286978755344)^{2}$$
$83$ $$T^{8} - 47194146 T^{6} + \cdots + 25\!\cdots\!61$$
$89$ $$(T^{4} + 227863368 T^{2} + \cdots + 191787378007824)^{2}$$
$97$ $$(T^{4} + 790 T^{3} + \cdots + 20\!\cdots\!89)^{2}$$