# Properties

 Label 162.4.c.a 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -12 \zeta_{6} q^{5} + ( 7 - 7 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -12 \zeta_{6} q^{5} + ( 7 - 7 \zeta_{6} ) q^{7} + 8 q^{8} + 24 q^{10} + ( -60 + 60 \zeta_{6} ) q^{11} + 79 \zeta_{6} q^{13} + 14 \zeta_{6} q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} -108 q^{17} + 11 q^{19} + ( -48 + 48 \zeta_{6} ) q^{20} -120 \zeta_{6} q^{22} + 132 \zeta_{6} q^{23} + ( -19 + 19 \zeta_{6} ) q^{25} -158 q^{26} -28 q^{28} + ( -96 + 96 \zeta_{6} ) q^{29} -20 \zeta_{6} q^{31} -32 \zeta_{6} q^{32} + ( 216 - 216 \zeta_{6} ) q^{34} -84 q^{35} -169 q^{37} + ( -22 + 22 \zeta_{6} ) q^{38} -96 \zeta_{6} q^{40} -192 \zeta_{6} q^{41} + ( -488 + 488 \zeta_{6} ) q^{43} + 240 q^{44} -264 q^{46} + ( -204 + 204 \zeta_{6} ) q^{47} + 294 \zeta_{6} q^{49} -38 \zeta_{6} q^{50} + ( 316 - 316 \zeta_{6} ) q^{52} + 360 q^{53} + 720 q^{55} + ( 56 - 56 \zeta_{6} ) q^{56} -192 \zeta_{6} q^{58} -156 \zeta_{6} q^{59} + ( -83 + 83 \zeta_{6} ) q^{61} + 40 q^{62} + 64 q^{64} + ( 948 - 948 \zeta_{6} ) q^{65} -47 \zeta_{6} q^{67} + 432 \zeta_{6} q^{68} + ( 168 - 168 \zeta_{6} ) q^{70} + 216 q^{71} -511 q^{73} + ( 338 - 338 \zeta_{6} ) q^{74} -44 \zeta_{6} q^{76} + 420 \zeta_{6} q^{77} + ( 529 - 529 \zeta_{6} ) q^{79} + 192 q^{80} + 384 q^{82} + ( 1128 - 1128 \zeta_{6} ) q^{83} + 1296 \zeta_{6} q^{85} -976 \zeta_{6} q^{86} + ( -480 + 480 \zeta_{6} ) q^{88} + 36 q^{89} + 553 q^{91} + ( 528 - 528 \zeta_{6} ) q^{92} -408 \zeta_{6} q^{94} -132 \zeta_{6} q^{95} + ( -605 + 605 \zeta_{6} ) q^{97} -588 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 12q^{5} + 7q^{7} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 12q^{5} + 7q^{7} + 16q^{8} + 48q^{10} - 60q^{11} + 79q^{13} + 14q^{14} - 16q^{16} - 216q^{17} + 22q^{19} - 48q^{20} - 120q^{22} + 132q^{23} - 19q^{25} - 316q^{26} - 56q^{28} - 96q^{29} - 20q^{31} - 32q^{32} + 216q^{34} - 168q^{35} - 338q^{37} - 22q^{38} - 96q^{40} - 192q^{41} - 488q^{43} + 480q^{44} - 528q^{46} - 204q^{47} + 294q^{49} - 38q^{50} + 316q^{52} + 720q^{53} + 1440q^{55} + 56q^{56} - 192q^{58} - 156q^{59} - 83q^{61} + 80q^{62} + 128q^{64} + 948q^{65} - 47q^{67} + 432q^{68} + 168q^{70} + 432q^{71} - 1022q^{73} + 338q^{74} - 44q^{76} + 420q^{77} + 529q^{79} + 384q^{80} + 768q^{82} + 1128q^{83} + 1296q^{85} - 976q^{86} - 480q^{88} + 72q^{89} + 1106q^{91} + 528q^{92} - 408q^{94} - 132q^{95} - 605q^{97} - 1176q^{98} + O(q^{100})$$

## 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.a 2
3.b odd 2 1 162.4.c.h 2
9.c even 3 1 54.4.a.d yes 1
9.c even 3 1 inner 162.4.c.a 2
9.d odd 6 1 54.4.a.a 1
9.d odd 6 1 162.4.c.h 2
36.f odd 6 1 432.4.a.m 1
36.h even 6 1 432.4.a.b 1
45.h odd 6 1 1350.4.a.v 1
45.j even 6 1 1350.4.a.h 1
45.k odd 12 2 1350.4.c.t 2
45.l even 12 2 1350.4.c.a 2
72.j odd 6 1 1728.4.a.ba 1
72.l even 6 1 1728.4.a.bb 1
72.n even 6 1 1728.4.a.e 1
72.p odd 6 1 1728.4.a.f 1

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

## Hecke kernels

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