# Properties

 Label 162.4.c.h 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} - 8 q^{8} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 + 12*z * q^5 + (-7*z + 7) * q^7 - 8 * q^8 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} - 8 q^{8} + 24 q^{10} + ( - 60 \zeta_{6} + 60) q^{11} + 79 \zeta_{6} q^{13} - 14 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 108 q^{17} + 11 q^{19} + ( - 48 \zeta_{6} + 48) q^{20} - 120 \zeta_{6} q^{22} - 132 \zeta_{6} q^{23} + (19 \zeta_{6} - 19) q^{25} + 158 q^{26} - 28 q^{28} + ( - 96 \zeta_{6} + 96) q^{29} - 20 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( - 216 \zeta_{6} + 216) q^{34} + 84 q^{35} - 169 q^{37} + ( - 22 \zeta_{6} + 22) q^{38} - 96 \zeta_{6} q^{40} + 192 \zeta_{6} q^{41} + (488 \zeta_{6} - 488) q^{43} - 240 q^{44} - 264 q^{46} + ( - 204 \zeta_{6} + 204) q^{47} + 294 \zeta_{6} q^{49} + 38 \zeta_{6} q^{50} + ( - 316 \zeta_{6} + 316) q^{52} - 360 q^{53} + 720 q^{55} + (56 \zeta_{6} - 56) q^{56} - 192 \zeta_{6} q^{58} + 156 \zeta_{6} q^{59} + (83 \zeta_{6} - 83) q^{61} - 40 q^{62} + 64 q^{64} + (948 \zeta_{6} - 948) q^{65} - 47 \zeta_{6} q^{67} - 432 \zeta_{6} q^{68} + ( - 168 \zeta_{6} + 168) q^{70} - 216 q^{71} - 511 q^{73} + (338 \zeta_{6} - 338) q^{74} - 44 \zeta_{6} q^{76} - 420 \zeta_{6} q^{77} + ( - 529 \zeta_{6} + 529) q^{79} - 192 q^{80} + 384 q^{82} + (1128 \zeta_{6} - 1128) q^{83} + 1296 \zeta_{6} q^{85} + 976 \zeta_{6} q^{86} + (480 \zeta_{6} - 480) q^{88} - 36 q^{89} + 553 q^{91} + (528 \zeta_{6} - 528) q^{92} - 408 \zeta_{6} q^{94} + 132 \zeta_{6} q^{95} + (605 \zeta_{6} - 605) q^{97} + 588 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 + 12*z * q^5 + (-7*z + 7) * q^7 - 8 * q^8 + 24 * q^10 + (-60*z + 60) * q^11 + 79*z * q^13 - 14*z * q^14 + (16*z - 16) * q^16 + 108 * q^17 + 11 * q^19 + (-48*z + 48) * q^20 - 120*z * q^22 - 132*z * q^23 + (19*z - 19) * q^25 + 158 * q^26 - 28 * q^28 + (-96*z + 96) * q^29 - 20*z * q^31 + 32*z * q^32 + (-216*z + 216) * q^34 + 84 * q^35 - 169 * q^37 + (-22*z + 22) * q^38 - 96*z * q^40 + 192*z * q^41 + (488*z - 488) * q^43 - 240 * q^44 - 264 * q^46 + (-204*z + 204) * q^47 + 294*z * q^49 + 38*z * q^50 + (-316*z + 316) * q^52 - 360 * q^53 + 720 * q^55 + (56*z - 56) * q^56 - 192*z * q^58 + 156*z * q^59 + (83*z - 83) * q^61 - 40 * q^62 + 64 * q^64 + (948*z - 948) * q^65 - 47*z * q^67 - 432*z * q^68 + (-168*z + 168) * q^70 - 216 * q^71 - 511 * q^73 + (338*z - 338) * q^74 - 44*z * q^76 - 420*z * q^77 + (-529*z + 529) * q^79 - 192 * q^80 + 384 * q^82 + (1128*z - 1128) * q^83 + 1296*z * q^85 + 976*z * q^86 + (480*z - 480) * q^88 - 36 * q^89 + 553 * q^91 + (528*z - 528) * q^92 - 408*z * q^94 + 132*z * q^95 + (605*z - 605) * q^97 + 588 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} + 7 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 12 * q^5 + 7 * q^7 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} + 7 q^{7} - 16 q^{8} + 48 q^{10} + 60 q^{11} + 79 q^{13} - 14 q^{14} - 16 q^{16} + 216 q^{17} + 22 q^{19} + 48 q^{20} - 120 q^{22} - 132 q^{23} - 19 q^{25} + 316 q^{26} - 56 q^{28} + 96 q^{29} - 20 q^{31} + 32 q^{32} + 216 q^{34} + 168 q^{35} - 338 q^{37} + 22 q^{38} - 96 q^{40} + 192 q^{41} - 488 q^{43} - 480 q^{44} - 528 q^{46} + 204 q^{47} + 294 q^{49} + 38 q^{50} + 316 q^{52} - 720 q^{53} + 1440 q^{55} - 56 q^{56} - 192 q^{58} + 156 q^{59} - 83 q^{61} - 80 q^{62} + 128 q^{64} - 948 q^{65} - 47 q^{67} - 432 q^{68} + 168 q^{70} - 432 q^{71} - 1022 q^{73} - 338 q^{74} - 44 q^{76} - 420 q^{77} + 529 q^{79} - 384 q^{80} + 768 q^{82} - 1128 q^{83} + 1296 q^{85} + 976 q^{86} - 480 q^{88} - 72 q^{89} + 1106 q^{91} - 528 q^{92} - 408 q^{94} + 132 q^{95} - 605 q^{97} + 1176 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 12 * q^5 + 7 * q^7 - 16 * q^8 + 48 * q^10 + 60 * q^11 + 79 * q^13 - 14 * q^14 - 16 * q^16 + 216 * q^17 + 22 * q^19 + 48 * q^20 - 120 * q^22 - 132 * q^23 - 19 * q^25 + 316 * q^26 - 56 * q^28 + 96 * q^29 - 20 * q^31 + 32 * q^32 + 216 * q^34 + 168 * q^35 - 338 * q^37 + 22 * q^38 - 96 * q^40 + 192 * q^41 - 488 * q^43 - 480 * q^44 - 528 * q^46 + 204 * q^47 + 294 * q^49 + 38 * q^50 + 316 * q^52 - 720 * q^53 + 1440 * q^55 - 56 * q^56 - 192 * q^58 + 156 * q^59 - 83 * q^61 - 80 * q^62 + 128 * q^64 - 948 * q^65 - 47 * q^67 - 432 * q^68 + 168 * q^70 - 432 * q^71 - 1022 * q^73 - 338 * q^74 - 44 * q^76 - 420 * q^77 + 529 * q^79 - 384 * q^80 + 768 * q^82 - 1128 * q^83 + 1296 * q^85 + 976 * q^86 - 480 * q^88 - 72 * q^89 + 1106 * q^91 - 528 * q^92 - 408 * q^94 + 132 * q^95 - 605 * q^97 + 1176 * 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 6.00000 + 10.3923i 0 3.50000 6.06218i −8.00000 0 24.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 6.00000 10.3923i 0 3.50000 + 6.06218i −8.00000 0 24.0000
 $$n$$: 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.h 2
3.b odd 2 1 162.4.c.a 2
9.c even 3 1 54.4.a.a 1
9.c even 3 1 inner 162.4.c.h 2
9.d odd 6 1 54.4.a.d yes 1
9.d odd 6 1 162.4.c.a 2
36.f odd 6 1 432.4.a.b 1
36.h even 6 1 432.4.a.m 1
45.h odd 6 1 1350.4.a.h 1
45.j even 6 1 1350.4.a.v 1
45.k odd 12 2 1350.4.c.a 2
45.l even 12 2 1350.4.c.t 2
72.j odd 6 1 1728.4.a.e 1
72.l even 6 1 1728.4.a.f 1
72.n even 6 1 1728.4.a.ba 1
72.p odd 6 1 1728.4.a.bb 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12T_{5} + 144$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12T + 144$$
$7$ $$T^{2} - 7T + 49$$
$11$ $$T^{2} - 60T + 3600$$
$13$ $$T^{2} - 79T + 6241$$
$17$ $$(T - 108)^{2}$$
$19$ $$(T - 11)^{2}$$
$23$ $$T^{2} + 132T + 17424$$
$29$ $$T^{2} - 96T + 9216$$
$31$ $$T^{2} + 20T + 400$$
$37$ $$(T + 169)^{2}$$
$41$ $$T^{2} - 192T + 36864$$
$43$ $$T^{2} + 488T + 238144$$
$47$ $$T^{2} - 204T + 41616$$
$53$ $$(T + 360)^{2}$$
$59$ $$T^{2} - 156T + 24336$$
$61$ $$T^{2} + 83T + 6889$$
$67$ $$T^{2} + 47T + 2209$$
$71$ $$(T + 216)^{2}$$
$73$ $$(T + 511)^{2}$$
$79$ $$T^{2} - 529T + 279841$$
$83$ $$T^{2} + 1128 T + 1272384$$
$89$ $$(T + 36)^{2}$$
$97$ $$T^{2} + 605T + 366025$$