# Properties

 Label 162.4.c.b Level $162$ Weight $4$ Character orbit 162.c Analytic conductor $9.558$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.55830942093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 3*z * q^5 + (29*z - 29) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 8 q^{8} + 6 q^{10} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{13} - 58 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 72 q^{17} - 106 q^{19} + (12 \zeta_{6} - 12) q^{20} + 114 \zeta_{6} q^{22} - 174 \zeta_{6} q^{23} + ( - 116 \zeta_{6} + 116) q^{25} + 40 q^{26} + 116 q^{28} + ( - 210 \zeta_{6} + 210) q^{29} - 47 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 144 \zeta_{6} + 144) q^{34} + 87 q^{35} + 2 q^{37} + ( - 212 \zeta_{6} + 212) q^{38} - 24 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + (218 \zeta_{6} - 218) q^{43} - 228 q^{44} + 348 q^{46} + (474 \zeta_{6} - 474) q^{47} - 498 \zeta_{6} q^{49} + 232 \zeta_{6} q^{50} + (80 \zeta_{6} - 80) q^{52} + 81 q^{53} - 171 q^{55} + (232 \zeta_{6} - 232) q^{56} + 420 \zeta_{6} q^{58} - 84 \zeta_{6} q^{59} + (56 \zeta_{6} - 56) q^{61} + 94 q^{62} + 64 q^{64} + (60 \zeta_{6} - 60) q^{65} + 142 \zeta_{6} q^{67} + 288 \zeta_{6} q^{68} + (174 \zeta_{6} - 174) q^{70} + 360 q^{71} - 1159 q^{73} + (4 \zeta_{6} - 4) q^{74} + 424 \zeta_{6} q^{76} + 1653 \zeta_{6} q^{77} + ( - 160 \zeta_{6} + 160) q^{79} + 48 q^{80} - 12 q^{82} + (735 \zeta_{6} - 735) q^{83} + 216 \zeta_{6} q^{85} - 436 \zeta_{6} q^{86} + ( - 456 \zeta_{6} + 456) q^{88} - 954 q^{89} + 580 q^{91} + (696 \zeta_{6} - 696) q^{92} - 948 \zeta_{6} q^{94} + 318 \zeta_{6} q^{95} + (191 \zeta_{6} - 191) q^{97} + 996 q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 3*z * q^5 + (29*z - 29) * q^7 + 8 * q^8 + 6 * q^10 + (-57*z + 57) * q^11 - 20*z * q^13 - 58*z * q^14 + (16*z - 16) * q^16 - 72 * q^17 - 106 * q^19 + (12*z - 12) * q^20 + 114*z * q^22 - 174*z * q^23 + (-116*z + 116) * q^25 + 40 * q^26 + 116 * q^28 + (-210*z + 210) * q^29 - 47*z * q^31 - 32*z * q^32 + (-144*z + 144) * q^34 + 87 * q^35 + 2 * q^37 + (-212*z + 212) * q^38 - 24*z * q^40 + 6*z * q^41 + (218*z - 218) * q^43 - 228 * q^44 + 348 * q^46 + (474*z - 474) * q^47 - 498*z * q^49 + 232*z * q^50 + (80*z - 80) * q^52 + 81 * q^53 - 171 * q^55 + (232*z - 232) * q^56 + 420*z * q^58 - 84*z * q^59 + (56*z - 56) * q^61 + 94 * q^62 + 64 * q^64 + (60*z - 60) * q^65 + 142*z * q^67 + 288*z * q^68 + (174*z - 174) * q^70 + 360 * q^71 - 1159 * q^73 + (4*z - 4) * q^74 + 424*z * q^76 + 1653*z * q^77 + (-160*z + 160) * q^79 + 48 * q^80 - 12 * q^82 + (735*z - 735) * q^83 + 216*z * q^85 - 436*z * q^86 + (-456*z + 456) * q^88 - 954 * q^89 + 580 * q^91 + (696*z - 696) * q^92 - 948*z * q^94 + 318*z * q^95 + (191*z - 191) * q^97 + 996 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 3 q^{5} - 29 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 3 * q^5 - 29 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 3 q^{5} - 29 q^{7} + 16 q^{8} + 12 q^{10} + 57 q^{11} - 20 q^{13} - 58 q^{14} - 16 q^{16} - 144 q^{17} - 212 q^{19} - 12 q^{20} + 114 q^{22} - 174 q^{23} + 116 q^{25} + 80 q^{26} + 232 q^{28} + 210 q^{29} - 47 q^{31} - 32 q^{32} + 144 q^{34} + 174 q^{35} + 4 q^{37} + 212 q^{38} - 24 q^{40} + 6 q^{41} - 218 q^{43} - 456 q^{44} + 696 q^{46} - 474 q^{47} - 498 q^{49} + 232 q^{50} - 80 q^{52} + 162 q^{53} - 342 q^{55} - 232 q^{56} + 420 q^{58} - 84 q^{59} - 56 q^{61} + 188 q^{62} + 128 q^{64} - 60 q^{65} + 142 q^{67} + 288 q^{68} - 174 q^{70} + 720 q^{71} - 2318 q^{73} - 4 q^{74} + 424 q^{76} + 1653 q^{77} + 160 q^{79} + 96 q^{80} - 24 q^{82} - 735 q^{83} + 216 q^{85} - 436 q^{86} + 456 q^{88} - 1908 q^{89} + 1160 q^{91} - 696 q^{92} - 948 q^{94} + 318 q^{95} - 191 q^{97} + 1992 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 3 * q^5 - 29 * q^7 + 16 * q^8 + 12 * q^10 + 57 * q^11 - 20 * q^13 - 58 * q^14 - 16 * q^16 - 144 * q^17 - 212 * q^19 - 12 * q^20 + 114 * q^22 - 174 * q^23 + 116 * q^25 + 80 * q^26 + 232 * q^28 + 210 * q^29 - 47 * q^31 - 32 * q^32 + 144 * q^34 + 174 * q^35 + 4 * q^37 + 212 * q^38 - 24 * q^40 + 6 * q^41 - 218 * q^43 - 456 * q^44 + 696 * q^46 - 474 * q^47 - 498 * q^49 + 232 * q^50 - 80 * q^52 + 162 * q^53 - 342 * q^55 - 232 * q^56 + 420 * q^58 - 84 * q^59 - 56 * q^61 + 188 * q^62 + 128 * q^64 - 60 * q^65 + 142 * q^67 + 288 * q^68 - 174 * q^70 + 720 * q^71 - 2318 * q^73 - 4 * q^74 + 424 * q^76 + 1653 * q^77 + 160 * q^79 + 96 * q^80 - 24 * q^82 - 735 * q^83 + 216 * q^85 - 436 * q^86 + 456 * q^88 - 1908 * q^89 + 1160 * q^91 - 696 * q^92 - 948 * q^94 + 318 * q^95 - 191 * q^97 + 1992 * q^98

## Character values

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

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −1.50000 2.59808i 0 −14.5000 + 25.1147i 8.00000 0 6.00000
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.50000 + 2.59808i 0 −14.5000 25.1147i 8.00000 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.c.b 2
3.b odd 2 1 162.4.c.g 2
9.c even 3 1 54.4.a.c yes 1
9.c even 3 1 inner 162.4.c.b 2
9.d odd 6 1 54.4.a.b 1
9.d odd 6 1 162.4.c.g 2
36.f odd 6 1 432.4.a.j 1
36.h even 6 1 432.4.a.e 1
45.h odd 6 1 1350.4.a.o 1
45.j even 6 1 1350.4.a.a 1
45.k odd 12 2 1350.4.c.b 2
45.l even 12 2 1350.4.c.s 2
72.j odd 6 1 1728.4.a.v 1
72.l even 6 1 1728.4.a.u 1
72.n even 6 1 1728.4.a.l 1
72.p odd 6 1 1728.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 9.d odd 6 1
54.4.a.c yes 1 9.c even 3 1
162.4.c.b 2 1.a even 1 1 trivial
162.4.c.b 2 9.c even 3 1 inner
162.4.c.g 2 3.b odd 2 1
162.4.c.g 2 9.d odd 6 1
432.4.a.e 1 36.h even 6 1
432.4.a.j 1 36.f odd 6 1
1350.4.a.a 1 45.j even 6 1
1350.4.a.o 1 45.h odd 6 1
1350.4.c.b 2 45.k odd 12 2
1350.4.c.s 2 45.l even 12 2
1728.4.a.k 1 72.p odd 6 1
1728.4.a.l 1 72.n even 6 1
1728.4.a.u 1 72.l even 6 1
1728.4.a.v 1 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3T_{5} + 9$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 29T + 841$$
$11$ $$T^{2} - 57T + 3249$$
$13$ $$T^{2} + 20T + 400$$
$17$ $$(T + 72)^{2}$$
$19$ $$(T + 106)^{2}$$
$23$ $$T^{2} + 174T + 30276$$
$29$ $$T^{2} - 210T + 44100$$
$31$ $$T^{2} + 47T + 2209$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 218T + 47524$$
$47$ $$T^{2} + 474T + 224676$$
$53$ $$(T - 81)^{2}$$
$59$ $$T^{2} + 84T + 7056$$
$61$ $$T^{2} + 56T + 3136$$
$67$ $$T^{2} - 142T + 20164$$
$71$ $$(T - 360)^{2}$$
$73$ $$(T + 1159)^{2}$$
$79$ $$T^{2} - 160T + 25600$$
$83$ $$T^{2} + 735T + 540225$$
$89$ $$(T + 954)^{2}$$
$97$ $$T^{2} + 191T + 36481$$