# Properties

 Label 162.3.b.b Level $162$ Weight $3$ Character orbit 162.b Analytic conductor $4.414$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 162.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 2 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2) q^{7} + 2 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 2 * q^4 + (-b2 + b1) * q^5 + (2*b3 + 2) * q^7 + 2*b1 * q^8 $$q - \beta_1 q^{2} - 2 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2) q^{7} + 2 \beta_1 q^{8} + ( - \beta_{3} + 3) q^{10} + ( - 4 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{3} - 16) q^{13} + ( - 4 \beta_{2} - 4 \beta_1) q^{14} + 4 q^{16} + ( - 5 \beta_{2} - 10 \beta_1) q^{17} + ( - 2 \beta_{3} + 14) q^{19} + (2 \beta_{2} - 2 \beta_1) q^{20} - 4 \beta_{3} q^{22} + (4 \beta_{2} + 2 \beta_1) q^{23} + (3 \beta_{3} + 7) q^{25} + ( - 2 \beta_{2} + 15 \beta_1) q^{26} + ( - 4 \beta_{3} - 4) q^{28} + ( - \beta_{2} + 22 \beta_1) q^{29} + 8 q^{31} - 4 \beta_1 q^{32} + ( - 5 \beta_{3} - 15) q^{34} + (4 \beta_{2} - 22 \beta_1) q^{35} + (8 \beta_{3} - 19) q^{37} + (4 \beta_{2} - 12 \beta_1) q^{38} + (2 \beta_{3} - 6) q^{40} + (8 \beta_{2} + \beta_1) q^{41} + ( - 6 \beta_{3} - 22) q^{43} + (8 \beta_{2} + 4 \beta_1) q^{44} + 4 \beta_{3} q^{46} - 12 \beta_1 q^{47} + (8 \beta_{3} + 63) q^{49} + ( - 6 \beta_{2} - 10 \beta_1) q^{50} + ( - 2 \beta_{3} + 32) q^{52} + (8 \beta_{2} + 43 \beta_1) q^{53} + (6 \beta_{3} - 54) q^{55} + (8 \beta_{2} + 8 \beta_1) q^{56} + ( - \beta_{3} + 45) q^{58} + (4 \beta_{2} + 56 \beta_1) q^{59} - 13 q^{61} - 8 \beta_1 q^{62} - 8 q^{64} + (19 \beta_{2} - 28 \beta_1) q^{65} + ( - 6 \beta_{3} - 10) q^{67} + (10 \beta_{2} + 20 \beta_1) q^{68} + (4 \beta_{3} - 48) q^{70} + (16 \beta_{2} - 22 \beta_1) q^{71} + (3 \beta_{3} + 56) q^{73} + ( - 16 \beta_{2} + 11 \beta_1) q^{74} + (4 \beta_{3} - 28) q^{76} + ( - 8 \beta_{2} - 112 \beta_1) q^{77} + ( - 14 \beta_{3} + 26) q^{79} + ( - 4 \beta_{2} + 4 \beta_1) q^{80} + (8 \beta_{3} - 6) q^{82} + ( - 12 \beta_{2} + 36 \beta_1) q^{83} - 45 q^{85} + (12 \beta_{2} + 28 \beta_1) q^{86} + 8 \beta_{3} q^{88} + ( - 29 \beta_{2} + 5 \beta_1) q^{89} + ( - 30 \beta_{3} + 22) q^{91} + ( - 8 \beta_{2} - 4 \beta_1) q^{92} - 24 q^{94} + ( - 20 \beta_{2} + 38 \beta_1) q^{95} + (16 \beta_{3} + 8) q^{97} + ( - 16 \beta_{2} - 71 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - 2 * q^4 + (-b2 + b1) * q^5 + (2*b3 + 2) * q^7 + 2*b1 * q^8 + (-b3 + 3) * q^10 + (-4*b2 - 2*b1) * q^11 + (b3 - 16) * q^13 + (-4*b2 - 4*b1) * q^14 + 4 * q^16 + (-5*b2 - 10*b1) * q^17 + (-2*b3 + 14) * q^19 + (2*b2 - 2*b1) * q^20 - 4*b3 * q^22 + (4*b2 + 2*b1) * q^23 + (3*b3 + 7) * q^25 + (-2*b2 + 15*b1) * q^26 + (-4*b3 - 4) * q^28 + (-b2 + 22*b1) * q^29 + 8 * q^31 - 4*b1 * q^32 + (-5*b3 - 15) * q^34 + (4*b2 - 22*b1) * q^35 + (8*b3 - 19) * q^37 + (4*b2 - 12*b1) * q^38 + (2*b3 - 6) * q^40 + (8*b2 + b1) * q^41 + (-6*b3 - 22) * q^43 + (8*b2 + 4*b1) * q^44 + 4*b3 * q^46 - 12*b1 * q^47 + (8*b3 + 63) * q^49 + (-6*b2 - 10*b1) * q^50 + (-2*b3 + 32) * q^52 + (8*b2 + 43*b1) * q^53 + (6*b3 - 54) * q^55 + (8*b2 + 8*b1) * q^56 + (-b3 + 45) * q^58 + (4*b2 + 56*b1) * q^59 - 13 * q^61 - 8*b1 * q^62 - 8 * q^64 + (19*b2 - 28*b1) * q^65 + (-6*b3 - 10) * q^67 + (10*b2 + 20*b1) * q^68 + (4*b3 - 48) * q^70 + (16*b2 - 22*b1) * q^71 + (3*b3 + 56) * q^73 + (-16*b2 + 11*b1) * q^74 + (4*b3 - 28) * q^76 + (-8*b2 - 112*b1) * q^77 + (-14*b3 + 26) * q^79 + (-4*b2 + 4*b1) * q^80 + (8*b3 - 6) * q^82 + (-12*b2 + 36*b1) * q^83 - 45 * q^85 + (12*b2 + 28*b1) * q^86 + 8*b3 * q^88 + (-29*b2 + 5*b1) * q^89 + (-30*b3 + 22) * q^91 + (-8*b2 - 4*b1) * q^92 - 24 * q^94 + (-20*b2 + 38*b1) * q^95 + (16*b3 + 8) * q^97 + (-16*b2 - 71*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^4 + 8 * q^7 $$4 q - 8 q^{4} + 8 q^{7} + 12 q^{10} - 64 q^{13} + 16 q^{16} + 56 q^{19} + 28 q^{25} - 16 q^{28} + 32 q^{31} - 60 q^{34} - 76 q^{37} - 24 q^{40} - 88 q^{43} + 252 q^{49} + 128 q^{52} - 216 q^{55} + 180 q^{58} - 52 q^{61} - 32 q^{64} - 40 q^{67} - 192 q^{70} + 224 q^{73} - 112 q^{76} + 104 q^{79} - 24 q^{82} - 180 q^{85} + 88 q^{91} - 96 q^{94} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^4 + 8 * q^7 + 12 * q^10 - 64 * q^13 + 16 * q^16 + 56 * q^19 + 28 * q^25 - 16 * q^28 + 32 * q^31 - 60 * q^34 - 76 * q^37 - 24 * q^40 - 88 * q^43 + 252 * q^49 + 128 * q^52 - 216 * q^55 + 180 * q^58 - 52 * q^61 - 32 * q^64 - 40 * q^67 - 192 * q^70 + 224 * q^73 - 112 * q^76 + 104 * q^79 - 24 * q^82 - 180 * q^85 + 88 * q^91 - 96 * q^94 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 6$$ 3*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 3$$ (b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 3$$ (b3 - 6) / 3 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

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

 $$n$$ $$83$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.517638i − 1.93185i 1.93185i − 0.517638i
1.41421i 0 −2.00000 1.55291i 0 12.3923 2.82843i 0 −2.19615
161.2 1.41421i 0 −2.00000 5.79555i 0 −8.39230 2.82843i 0 8.19615
161.3 1.41421i 0 −2.00000 5.79555i 0 −8.39230 2.82843i 0 8.19615
161.4 1.41421i 0 −2.00000 1.55291i 0 12.3923 2.82843i 0 −2.19615
 $$n$$: 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
3.b odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 36T_{5}^{2} + 81$$ acting on $$S_{3}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 36T^{2} + 81$$
$7$ $$(T^{2} - 4 T - 104)^{2}$$
$11$ $$(T^{2} + 216)^{2}$$
$13$ $$(T^{2} + 32 T + 229)^{2}$$
$17$ $$T^{4} + 900 T^{2} + 50625$$
$19$ $$(T^{2} - 28 T + 88)^{2}$$
$23$ $$(T^{2} + 216)^{2}$$
$29$ $$T^{4} + 2052 T^{2} + 998001$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} + 38 T - 1367)^{2}$$
$41$ $$T^{4} + 1764 T^{2} + 715716$$
$43$ $$(T^{2} + 44 T - 488)^{2}$$
$47$ $$(T^{2} + 288)^{2}$$
$53$ $$T^{4} + 7812 T^{2} + \cdots + 4743684$$
$59$ $$T^{4} + 12096 T^{2} + \cdots + 31539456$$
$61$ $$(T + 13)^{4}$$
$67$ $$(T^{2} + 20 T - 872)^{2}$$
$71$ $$T^{4} + 10512 T^{2} + \cdots + 2742336$$
$73$ $$(T^{2} - 112 T + 2893)^{2}$$
$79$ $$(T^{2} - 52 T - 4616)^{2}$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 2509056$$
$89$ $$T^{4} + 24228 T^{2} + \cdots + 112211649$$
$97$ $$(T^{2} - 16 T - 6848)^{2}$$