# Properties

 Label 162.4.c.d 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: yes 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + 21*z * q^5 + (8*z - 8) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} - 42 q^{10} + ( - 36 \zeta_{6} + 36) q^{11} + 49 \zeta_{6} q^{13} - 16 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 21 q^{17} - 112 q^{19} + ( - 84 \zeta_{6} + 84) q^{20} + 72 \zeta_{6} q^{22} + 180 \zeta_{6} q^{23} + (316 \zeta_{6} - 316) q^{25} - 98 q^{26} + 32 q^{28} + (135 \zeta_{6} - 135) q^{29} - 308 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 42 \zeta_{6} + 42) q^{34} - 168 q^{35} - q^{37} + ( - 224 \zeta_{6} + 224) q^{38} + 168 \zeta_{6} q^{40} - 42 \zeta_{6} q^{41} + (20 \zeta_{6} - 20) q^{43} - 144 q^{44} - 360 q^{46} + ( - 84 \zeta_{6} + 84) q^{47} + 279 \zeta_{6} q^{49} - 632 \zeta_{6} q^{50} + ( - 196 \zeta_{6} + 196) q^{52} + 174 q^{53} + 756 q^{55} + (64 \zeta_{6} - 64) q^{56} - 270 \zeta_{6} q^{58} + 504 \zeta_{6} q^{59} + ( - 385 \zeta_{6} + 385) q^{61} + 616 q^{62} + 64 q^{64} + (1029 \zeta_{6} - 1029) q^{65} - 272 \zeta_{6} q^{67} + 84 \zeta_{6} q^{68} + ( - 336 \zeta_{6} + 336) q^{70} + 888 q^{71} + 371 q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 448 \zeta_{6} q^{76} + 288 \zeta_{6} q^{77} + ( - 652 \zeta_{6} + 652) q^{79} - 336 q^{80} + 84 q^{82} + ( - 84 \zeta_{6} + 84) q^{83} - 441 \zeta_{6} q^{85} - 40 \zeta_{6} q^{86} + ( - 288 \zeta_{6} + 288) q^{88} - 21 q^{89} - 392 q^{91} + ( - 720 \zeta_{6} + 720) q^{92} + 168 \zeta_{6} q^{94} - 2352 \zeta_{6} q^{95} + ( - 1246 \zeta_{6} + 1246) q^{97} - 558 q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + 21*z * q^5 + (8*z - 8) * q^7 + 8 * q^8 - 42 * q^10 + (-36*z + 36) * q^11 + 49*z * q^13 - 16*z * q^14 + (16*z - 16) * q^16 - 21 * q^17 - 112 * q^19 + (-84*z + 84) * q^20 + 72*z * q^22 + 180*z * q^23 + (316*z - 316) * q^25 - 98 * q^26 + 32 * q^28 + (135*z - 135) * q^29 - 308*z * q^31 - 32*z * q^32 + (-42*z + 42) * q^34 - 168 * q^35 - q^37 + (-224*z + 224) * q^38 + 168*z * q^40 - 42*z * q^41 + (20*z - 20) * q^43 - 144 * q^44 - 360 * q^46 + (-84*z + 84) * q^47 + 279*z * q^49 - 632*z * q^50 + (-196*z + 196) * q^52 + 174 * q^53 + 756 * q^55 + (64*z - 64) * q^56 - 270*z * q^58 + 504*z * q^59 + (-385*z + 385) * q^61 + 616 * q^62 + 64 * q^64 + (1029*z - 1029) * q^65 - 272*z * q^67 + 84*z * q^68 + (-336*z + 336) * q^70 + 888 * q^71 + 371 * q^73 + (-2*z + 2) * q^74 + 448*z * q^76 + 288*z * q^77 + (-652*z + 652) * q^79 - 336 * q^80 + 84 * q^82 + (-84*z + 84) * q^83 - 441*z * q^85 - 40*z * q^86 + (-288*z + 288) * q^88 - 21 * q^89 - 392 * q^91 + (-720*z + 720) * q^92 + 168*z * q^94 - 2352*z * q^95 + (-1246*z + 1246) * q^97 - 558 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 21 * q^5 - 8 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26} + 64 q^{28} - 135 q^{29} - 308 q^{31} - 32 q^{32} + 42 q^{34} - 336 q^{35} - 2 q^{37} + 224 q^{38} + 168 q^{40} - 42 q^{41} - 20 q^{43} - 288 q^{44} - 720 q^{46} + 84 q^{47} + 279 q^{49} - 632 q^{50} + 196 q^{52} + 348 q^{53} + 1512 q^{55} - 64 q^{56} - 270 q^{58} + 504 q^{59} + 385 q^{61} + 1232 q^{62} + 128 q^{64} - 1029 q^{65} - 272 q^{67} + 84 q^{68} + 336 q^{70} + 1776 q^{71} + 742 q^{73} + 2 q^{74} + 448 q^{76} + 288 q^{77} + 652 q^{79} - 672 q^{80} + 168 q^{82} + 84 q^{83} - 441 q^{85} - 40 q^{86} + 288 q^{88} - 42 q^{89} - 784 q^{91} + 720 q^{92} + 168 q^{94} - 2352 q^{95} + 1246 q^{97} - 1116 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 21 * q^5 - 8 * q^7 + 16 * q^8 - 84 * q^10 + 36 * q^11 + 49 * q^13 - 16 * q^14 - 16 * q^16 - 42 * q^17 - 224 * q^19 + 84 * q^20 + 72 * q^22 + 180 * q^23 - 316 * q^25 - 196 * q^26 + 64 * q^28 - 135 * q^29 - 308 * q^31 - 32 * q^32 + 42 * q^34 - 336 * q^35 - 2 * q^37 + 224 * q^38 + 168 * q^40 - 42 * q^41 - 20 * q^43 - 288 * q^44 - 720 * q^46 + 84 * q^47 + 279 * q^49 - 632 * q^50 + 196 * q^52 + 348 * q^53 + 1512 * q^55 - 64 * q^56 - 270 * q^58 + 504 * q^59 + 385 * q^61 + 1232 * q^62 + 128 * q^64 - 1029 * q^65 - 272 * q^67 + 84 * q^68 + 336 * q^70 + 1776 * q^71 + 742 * q^73 + 2 * q^74 + 448 * q^76 + 288 * q^77 + 652 * q^79 - 672 * q^80 + 168 * q^82 + 84 * q^83 - 441 * q^85 - 40 * q^86 + 288 * q^88 - 42 * q^89 - 784 * q^91 + 720 * q^92 + 168 * q^94 - 2352 * q^95 + 1246 * q^97 - 1116 * 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 10.5000 + 18.1865i 0 −4.00000 + 6.92820i 8.00000 0 −42.0000
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 10.5000 18.1865i 0 −4.00000 6.92820i 8.00000 0 −42.0000
 $$n$$: 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.d 2
3.b odd 2 1 162.4.c.e 2
9.c even 3 1 162.4.a.c yes 1
9.c even 3 1 inner 162.4.c.d 2
9.d odd 6 1 162.4.a.b 1
9.d odd 6 1 162.4.c.e 2
36.f odd 6 1 1296.4.a.a 1
36.h even 6 1 1296.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 9.d odd 6 1
162.4.a.c yes 1 9.c even 3 1
162.4.c.d 2 1.a even 1 1 trivial
162.4.c.d 2 9.c even 3 1 inner
162.4.c.e 2 3.b odd 2 1
162.4.c.e 2 9.d odd 6 1
1296.4.a.a 1 36.f odd 6 1
1296.4.a.h 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 21T_{5} + 441$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 21T + 441$$
$7$ $$T^{2} + 8T + 64$$
$11$ $$T^{2} - 36T + 1296$$
$13$ $$T^{2} - 49T + 2401$$
$17$ $$(T + 21)^{2}$$
$19$ $$(T + 112)^{2}$$
$23$ $$T^{2} - 180T + 32400$$
$29$ $$T^{2} + 135T + 18225$$
$31$ $$T^{2} + 308T + 94864$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 42T + 1764$$
$43$ $$T^{2} + 20T + 400$$
$47$ $$T^{2} - 84T + 7056$$
$53$ $$(T - 174)^{2}$$
$59$ $$T^{2} - 504T + 254016$$
$61$ $$T^{2} - 385T + 148225$$
$67$ $$T^{2} + 272T + 73984$$
$71$ $$(T - 888)^{2}$$
$73$ $$(T - 371)^{2}$$
$79$ $$T^{2} - 652T + 425104$$
$83$ $$T^{2} - 84T + 7056$$
$89$ $$(T + 21)^{2}$$
$97$ $$T^{2} - 1246 T + 1552516$$