# Properties

 Label 162.3.d.c Level $162$ Weight $3$ Character orbit 162.d Analytic conductor $4.414$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.41418028264$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + 2 \beta_{3}) q^{5} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8}+O(q^{10})$$ q + b5 * q^2 + 2*b1 * q^4 + (b7 - b6 + 2*b3) * q^5 + (-2*b4 + 2*b2 + 2*b1 - 2) * q^7 + (2*b5 - 2*b3) * q^8 $$q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + 2 \beta_{3}) q^{5} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + ( - \beta_{4} + 3) q^{10} + (4 \beta_{6} - 2 \beta_{5}) q^{11} + ( - \beta_{2} + 16 \beta_1) q^{13} + (4 \beta_{7} - 4 \beta_{6}) q^{14} + (4 \beta_1 - 4) q^{16} + ( - 5 \beta_{7} - 5 \beta_{5} + 5 \beta_{3}) q^{17} + ( - 2 \beta_{4} + 14) q^{19} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{20} + 4 \beta_{2} q^{22} + ( - 4 \beta_{7} + 4 \beta_{6} - 2 \beta_{3}) q^{23} + ( - 3 \beta_{4} + 3 \beta_{2} + 7 \beta_1 - 7) q^{25} + ( - 2 \beta_{7} + 17 \beta_{5} - 17 \beta_{3}) q^{26} + ( - 4 \beta_{4} - 4) q^{28} + (\beta_{6} - 23 \beta_{5}) q^{29} - 8 \beta_1 q^{31} - 4 \beta_{3} q^{32} + (5 \beta_{4} - 5 \beta_{2} - 15 \beta_1 + 15) q^{34} + (4 \beta_{7} - 26 \beta_{5} + 26 \beta_{3}) q^{35} + (8 \beta_{4} - 19) q^{37} + ( - 4 \beta_{6} + 16 \beta_{5}) q^{38} + ( - 2 \beta_{2} + 6 \beta_1) q^{40} + ( - 8 \beta_{7} + 8 \beta_{6} - 7 \beta_{3}) q^{41} + (6 \beta_{4} - 6 \beta_{2} - 22 \beta_1 + 22) q^{43} + (8 \beta_{7} - 4 \beta_{5} + 4 \beta_{3}) q^{44} + 4 \beta_{4} q^{46} + 12 \beta_{5} q^{47} + ( - 8 \beta_{2} - 63 \beta_1) q^{49} + (6 \beta_{7} - 6 \beta_{6} - 4 \beta_{3}) q^{50} + (2 \beta_{4} - 2 \beta_{2} + 32 \beta_1 - 32) q^{52} + (8 \beta_{7} + 35 \beta_{5} - 35 \beta_{3}) q^{53} + (6 \beta_{4} - 54) q^{55} - 8 \beta_{6} q^{56} + (\beta_{2} - 45 \beta_1) q^{58} + ( - 4 \beta_{7} + 4 \beta_{6} + 52 \beta_{3}) q^{59} + ( - 13 \beta_1 + 13) q^{61} + ( - 8 \beta_{5} + 8 \beta_{3}) q^{62} - 8 q^{64} + ( - 19 \beta_{6} + 47 \beta_{5}) q^{65} + (6 \beta_{2} + 10 \beta_1) q^{67} + ( - 10 \beta_{7} + 10 \beta_{6} + 10 \beta_{3}) q^{68} + ( - 4 \beta_{4} + 4 \beta_{2} - 48 \beta_1 + 48) q^{70} + (16 \beta_{7} - 38 \beta_{5} + 38 \beta_{3}) q^{71} + (3 \beta_{4} + 56) q^{73} + (16 \beta_{6} - 27 \beta_{5}) q^{74} + ( - 4 \beta_{2} + 28 \beta_1) q^{76} + (8 \beta_{7} - 8 \beta_{6} - 104 \beta_{3}) q^{77} + (14 \beta_{4} - 14 \beta_{2} + 26 \beta_1 - 26) q^{79} + ( - 4 \beta_{7} + 8 \beta_{5} - 8 \beta_{3}) q^{80} + (8 \beta_{4} - 6) q^{82} + (12 \beta_{6} - 48 \beta_{5}) q^{83} + 45 \beta_1 q^{85} + ( - 12 \beta_{7} + 12 \beta_{6} + 16 \beta_{3}) q^{86} + ( - 8 \beta_{4} + 8 \beta_{2}) q^{88} + ( - 29 \beta_{7} + 34 \beta_{5} - 34 \beta_{3}) q^{89} + ( - 30 \beta_{4} + 22) q^{91} + (8 \beta_{6} - 4 \beta_{5}) q^{92} + 24 \beta_1 q^{94} + (20 \beta_{7} - 20 \beta_{6} + 58 \beta_{3}) q^{95} + ( - 16 \beta_{4} + 16 \beta_{2} + 8 \beta_1 - 8) q^{97} + ( - 16 \beta_{7} - 55 \beta_{5} + 55 \beta_{3}) q^{98}+O(q^{100})$$ q + b5 * q^2 + 2*b1 * q^4 + (b7 - b6 + 2*b3) * q^5 + (-2*b4 + 2*b2 + 2*b1 - 2) * q^7 + (2*b5 - 2*b3) * q^8 + (-b4 + 3) * q^10 + (4*b6 - 2*b5) * q^11 + (-b2 + 16*b1) * q^13 + (4*b7 - 4*b6) * q^14 + (4*b1 - 4) * q^16 + (-5*b7 - 5*b5 + 5*b3) * q^17 + (-2*b4 + 14) * q^19 + (-2*b6 + 4*b5) * q^20 + 4*b2 * q^22 + (-4*b7 + 4*b6 - 2*b3) * q^23 + (-3*b4 + 3*b2 + 7*b1 - 7) * q^25 + (-2*b7 + 17*b5 - 17*b3) * q^26 + (-4*b4 - 4) * q^28 + (b6 - 23*b5) * q^29 - 8*b1 * q^31 - 4*b3 * q^32 + (5*b4 - 5*b2 - 15*b1 + 15) * q^34 + (4*b7 - 26*b5 + 26*b3) * q^35 + (8*b4 - 19) * q^37 + (-4*b6 + 16*b5) * q^38 + (-2*b2 + 6*b1) * q^40 + (-8*b7 + 8*b6 - 7*b3) * q^41 + (6*b4 - 6*b2 - 22*b1 + 22) * q^43 + (8*b7 - 4*b5 + 4*b3) * q^44 + 4*b4 * q^46 + 12*b5 * q^47 + (-8*b2 - 63*b1) * q^49 + (6*b7 - 6*b6 - 4*b3) * q^50 + (2*b4 - 2*b2 + 32*b1 - 32) * q^52 + (8*b7 + 35*b5 - 35*b3) * q^53 + (6*b4 - 54) * q^55 - 8*b6 * q^56 + (b2 - 45*b1) * q^58 + (-4*b7 + 4*b6 + 52*b3) * q^59 + (-13*b1 + 13) * q^61 + (-8*b5 + 8*b3) * q^62 - 8 * q^64 + (-19*b6 + 47*b5) * q^65 + (6*b2 + 10*b1) * q^67 + (-10*b7 + 10*b6 + 10*b3) * q^68 + (-4*b4 + 4*b2 - 48*b1 + 48) * q^70 + (16*b7 - 38*b5 + 38*b3) * q^71 + (3*b4 + 56) * q^73 + (16*b6 - 27*b5) * q^74 + (-4*b2 + 28*b1) * q^76 + (8*b7 - 8*b6 - 104*b3) * q^77 + (14*b4 - 14*b2 + 26*b1 - 26) * q^79 + (-4*b7 + 8*b5 - 8*b3) * q^80 + (8*b4 - 6) * q^82 + (12*b6 - 48*b5) * q^83 + 45*b1 * q^85 + (-12*b7 + 12*b6 + 16*b3) * q^86 + (-8*b4 + 8*b2) * q^88 + (-29*b7 + 34*b5 - 34*b3) * q^89 + (-30*b4 + 22) * q^91 + (8*b6 - 4*b5) * q^92 + 24*b1 * q^94 + (20*b7 - 20*b6 + 58*b3) * q^95 + (-16*b4 + 16*b2 + 8*b1 - 8) * q^97 + (-16*b7 - 55*b5 + 55*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} - 8 q^{7}+O(q^{10})$$ 8 * q + 8 * q^4 - 8 * q^7 $$8 q + 8 q^{4} - 8 q^{7} + 24 q^{10} + 64 q^{13} - 16 q^{16} + 112 q^{19} - 28 q^{25} - 32 q^{28} - 32 q^{31} + 60 q^{34} - 152 q^{37} + 24 q^{40} + 88 q^{43} - 252 q^{49} - 128 q^{52} - 432 q^{55} - 180 q^{58} + 52 q^{61} - 64 q^{64} + 40 q^{67} + 192 q^{70} + 448 q^{73} + 112 q^{76} - 104 q^{79} - 48 q^{82} + 180 q^{85} + 176 q^{91} + 96 q^{94} - 32 q^{97}+O(q^{100})$$ 8 * q + 8 * q^4 - 8 * q^7 + 24 * q^10 + 64 * q^13 - 16 * q^16 + 112 * q^19 - 28 * q^25 - 32 * q^28 - 32 * q^31 + 60 * q^34 - 152 * q^37 + 24 * q^40 + 88 * q^43 - 252 * q^49 - 128 * q^52 - 432 * q^55 - 180 * q^58 + 52 * q^61 - 64 * q^64 + 40 * q^67 + 192 * q^70 + 448 * q^73 + 112 * q^76 - 104 * q^79 - 48 * q^82 + 180 * q^85 + 176 * q^91 + 96 * q^94 - 32 * 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 $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

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

## 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.

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
−1.22474 + 0.707107i 0 1.00000 1.73205i −5.01910 2.89778i 0 4.19615 + 7.26795i 2.82843i 0 8.19615
53.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.34486 + 0.776457i 0 −6.19615 10.7321i 2.82843i 0 −2.19615
53.3 1.22474 0.707107i 0 1.00000 1.73205i −1.34486 0.776457i 0 −6.19615 10.7321i 2.82843i 0 −2.19615
53.4 1.22474 0.707107i 0 1.00000 1.73205i 5.01910 + 2.89778i 0 4.19615 + 7.26795i 2.82843i 0 8.19615
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −5.01910 + 2.89778i 0 4.19615 7.26795i 2.82843i 0 8.19615
107.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.34486 0.776457i 0 −6.19615 + 10.7321i 2.82843i 0 −2.19615
107.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.34486 + 0.776457i 0 −6.19615 + 10.7321i 2.82843i 0 −2.19615
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 5.01910 2.89778i 0 4.19615 7.26795i 2.82843i 0 8.19615
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 107.4 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 162.3.d.c 8
3.b odd 2 1 inner 162.3.d.c 8
4.b odd 2 1 1296.3.q.o 8
9.c even 3 1 162.3.b.b 4
9.c even 3 1 inner 162.3.d.c 8
9.d odd 6 1 162.3.b.b 4
9.d odd 6 1 inner 162.3.d.c 8
12.b even 2 1 1296.3.q.o 8
36.f odd 6 1 1296.3.e.d 4
36.f odd 6 1 1296.3.q.o 8
36.h even 6 1 1296.3.e.d 4
36.h even 6 1 1296.3.q.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 9.c even 3 1
162.3.b.b 4 9.d odd 6 1
162.3.d.c 8 1.a even 1 1 trivial
162.3.d.c 8 3.b odd 2 1 inner
162.3.d.c 8 9.c even 3 1 inner
162.3.d.c 8 9.d odd 6 1 inner
1296.3.e.d 4 36.f odd 6 1
1296.3.e.d 4 36.h even 6 1
1296.3.q.o 8 4.b odd 2 1
1296.3.q.o 8 12.b even 2 1
1296.3.q.o 8 36.f odd 6 1
1296.3.q.o 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 36T_{5}^{6} + 1215T_{5}^{4} - 2916T_{5}^{2} + 6561$$ acting on $$S_{3}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 36 T^{6} + 1215 T^{4} + \cdots + 6561$$
$7$ $$(T^{4} + 4 T^{3} + 120 T^{2} - 416 T + 10816)^{2}$$
$11$ $$(T^{4} - 216 T^{2} + 46656)^{2}$$
$13$ $$(T^{4} - 32 T^{3} + 795 T^{2} + \cdots + 52441)^{2}$$
$17$ $$(T^{4} + 900 T^{2} + 50625)^{2}$$
$19$ $$(T^{2} - 28 T + 88)^{4}$$
$23$ $$(T^{4} - 216 T^{2} + 46656)^{2}$$
$29$ $$T^{8} - 2052 T^{6} + \cdots + 996005996001$$
$31$ $$(T^{2} + 8 T + 64)^{4}$$
$37$ $$(T^{2} + 38 T - 1367)^{4}$$
$41$ $$T^{8} - 1764 T^{6} + \cdots + 512249392656$$
$43$ $$(T^{4} - 44 T^{3} + 2424 T^{2} + \cdots + 238144)^{2}$$
$47$ $$(T^{4} - 288 T^{2} + 82944)^{2}$$
$53$ $$(T^{4} + 7812 T^{2} + 4743684)^{2}$$
$59$ $$T^{8} + \cdots + 994737284775936$$
$61$ $$(T^{2} - 13 T + 169)^{4}$$
$67$ $$(T^{4} - 20 T^{3} + 1272 T^{2} + \cdots + 760384)^{2}$$
$71$ $$(T^{4} + 10512 T^{2} + 2742336)^{2}$$
$73$ $$(T^{2} - 112 T + 2893)^{4}$$
$79$ $$(T^{4} + 52 T^{3} + 7320 T^{2} + \cdots + 21307456)^{2}$$
$83$ $$T^{8} - 10944 T^{6} + \cdots + 6295362011136$$
$89$ $$(T^{4} + 24228 T^{2} + \cdots + 112211649)^{2}$$
$97$ $$(T^{4} + 16 T^{3} + 7104 T^{2} + \cdots + 46895104)^{2}$$