# Properties

 Label 162.4.a.a Level $162$ Weight $4$ Character orbit 162.a Self dual yes Analytic conductor $9.558$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 162.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.55830942093$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + 9 * q^5 - 31 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8} - 18 q^{10} + 15 q^{11} - 37 q^{13} + 62 q^{14} + 16 q^{16} + 42 q^{17} - 28 q^{19} + 36 q^{20} - 30 q^{22} - 195 q^{23} - 44 q^{25} + 74 q^{26} - 124 q^{28} - 111 q^{29} - 205 q^{31} - 32 q^{32} - 84 q^{34} - 279 q^{35} - 166 q^{37} + 56 q^{38} - 72 q^{40} + 261 q^{41} - 43 q^{43} + 60 q^{44} + 390 q^{46} - 177 q^{47} + 618 q^{49} + 88 q^{50} - 148 q^{52} - 114 q^{53} + 135 q^{55} + 248 q^{56} + 222 q^{58} - 159 q^{59} + 191 q^{61} + 410 q^{62} + 64 q^{64} - 333 q^{65} - 421 q^{67} + 168 q^{68} + 558 q^{70} - 156 q^{71} + 182 q^{73} + 332 q^{74} - 112 q^{76} - 465 q^{77} + 1133 q^{79} + 144 q^{80} - 522 q^{82} + 1083 q^{83} + 378 q^{85} + 86 q^{86} - 120 q^{88} + 1050 q^{89} + 1147 q^{91} - 780 q^{92} + 354 q^{94} - 252 q^{95} - 901 q^{97} - 1236 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + 9 * q^5 - 31 * q^7 - 8 * q^8 - 18 * q^10 + 15 * q^11 - 37 * q^13 + 62 * q^14 + 16 * q^16 + 42 * q^17 - 28 * q^19 + 36 * q^20 - 30 * q^22 - 195 * q^23 - 44 * q^25 + 74 * q^26 - 124 * q^28 - 111 * q^29 - 205 * q^31 - 32 * q^32 - 84 * q^34 - 279 * q^35 - 166 * q^37 + 56 * q^38 - 72 * q^40 + 261 * q^41 - 43 * q^43 + 60 * q^44 + 390 * q^46 - 177 * q^47 + 618 * q^49 + 88 * q^50 - 148 * q^52 - 114 * q^53 + 135 * q^55 + 248 * q^56 + 222 * q^58 - 159 * q^59 + 191 * q^61 + 410 * q^62 + 64 * q^64 - 333 * q^65 - 421 * q^67 + 168 * q^68 + 558 * q^70 - 156 * q^71 + 182 * q^73 + 332 * q^74 - 112 * q^76 - 465 * q^77 + 1133 * q^79 + 144 * q^80 - 522 * q^82 + 1083 * q^83 + 378 * q^85 + 86 * q^86 - 120 * q^88 + 1050 * q^89 + 1147 * q^91 - 780 * q^92 + 354 * q^94 - 252 * q^95 - 901 * q^97 - 1236 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 0 4.00000 9.00000 0 −31.0000 −8.00000 0 −18.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.a.a 1
3.b odd 2 1 162.4.a.d 1
4.b odd 2 1 1296.4.a.g 1
9.c even 3 2 54.4.c.a 2
9.d odd 6 2 18.4.c.a 2
12.b even 2 1 1296.4.a.b 1
36.f odd 6 2 432.4.i.a 2
36.h even 6 2 144.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 9.d odd 6 2
54.4.c.a 2 9.c even 3 2
144.4.i.a 2 36.h even 6 2
162.4.a.a 1 1.a even 1 1 trivial
162.4.a.d 1 3.b odd 2 1
432.4.i.a 2 36.f odd 6 2
1296.4.a.b 1 12.b even 2 1
1296.4.a.g 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 9$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(162))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T + 31$$
$11$ $$T - 15$$
$13$ $$T + 37$$
$17$ $$T - 42$$
$19$ $$T + 28$$
$23$ $$T + 195$$
$29$ $$T + 111$$
$31$ $$T + 205$$
$37$ $$T + 166$$
$41$ $$T - 261$$
$43$ $$T + 43$$
$47$ $$T + 177$$
$53$ $$T + 114$$
$59$ $$T + 159$$
$61$ $$T - 191$$
$67$ $$T + 421$$
$71$ $$T + 156$$
$73$ $$T - 182$$
$79$ $$T - 1133$$
$83$ $$T - 1083$$
$89$ $$T - 1050$$
$97$ $$T + 901$$