# Properties

 Label 162.4.a.h Level $162$ Weight $4$ Character orbit 162.a Self dual yes Analytic conductor $9.558$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + (\beta + 6) q^{5} + (2 \beta + 8) q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + (b + 6) * q^5 + (2*b + 8) * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + (\beta + 6) q^{5} + (2 \beta + 8) q^{7} + 8 q^{8} + (2 \beta + 12) q^{10} + ( - 8 \beta + 18) q^{11} + ( - 14 \beta + 5) q^{13} + (4 \beta + 16) q^{14} + 16 q^{16} + (11 \beta + 60) q^{17} + (22 \beta - 4) q^{19} + (4 \beta + 24) q^{20} + ( - 16 \beta + 36) q^{22} + ( - 4 \beta + 90) q^{23} + (12 \beta - 62) q^{25} + ( - 28 \beta + 10) q^{26} + (8 \beta + 32) q^{28} + (7 \beta + 162) q^{29} + ( - 36 \beta - 124) q^{31} + 32 q^{32} + (22 \beta + 120) q^{34} + (20 \beta + 102) q^{35} + (2 \beta - 217) q^{37} + (44 \beta - 8) q^{38} + (8 \beta + 48) q^{40} + ( - 44 \beta + 96) q^{41} + ( - 6 \beta - 304) q^{43} + ( - 32 \beta + 72) q^{44} + ( - 8 \beta + 180) q^{46} + ( - 36 \beta - 192) q^{47} + (32 \beta - 171) q^{49} + (24 \beta - 124) q^{50} + ( - 56 \beta + 20) q^{52} + (76 \beta - 204) q^{53} + ( - 30 \beta - 108) q^{55} + (16 \beta + 64) q^{56} + (14 \beta + 324) q^{58} + (32 \beta - 504) q^{59} + ( - 18 \beta + 371) q^{61} + ( - 72 \beta - 248) q^{62} + 64 q^{64} + ( - 79 \beta - 348) q^{65} + (138 \beta - 52) q^{67} + (44 \beta + 240) q^{68} + (40 \beta + 204) q^{70} + (8 \beta - 570) q^{71} + ( - 96 \beta + 425) q^{73} + (4 \beta - 434) q^{74} + (88 \beta - 16) q^{76} + ( - 28 \beta - 288) q^{77} + ( - 50 \beta - 220) q^{79} + (16 \beta + 96) q^{80} + ( - 88 \beta + 192) q^{82} + ( - 60 \beta + 132) q^{83} + (126 \beta + 657) q^{85} + ( - 12 \beta - 608) q^{86} + ( - 64 \beta + 144) q^{88} + (89 \beta + 384) q^{89} + ( - 102 \beta - 716) q^{91} + ( - 16 \beta + 360) q^{92} + ( - 72 \beta - 384) q^{94} + (128 \beta + 570) q^{95} + ( - 56 \beta - 382) q^{97} + (64 \beta - 342) q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + (b + 6) * q^5 + (2*b + 8) * q^7 + 8 * q^8 + (2*b + 12) * q^10 + (-8*b + 18) * q^11 + (-14*b + 5) * q^13 + (4*b + 16) * q^14 + 16 * q^16 + (11*b + 60) * q^17 + (22*b - 4) * q^19 + (4*b + 24) * q^20 + (-16*b + 36) * q^22 + (-4*b + 90) * q^23 + (12*b - 62) * q^25 + (-28*b + 10) * q^26 + (8*b + 32) * q^28 + (7*b + 162) * q^29 + (-36*b - 124) * q^31 + 32 * q^32 + (22*b + 120) * q^34 + (20*b + 102) * q^35 + (2*b - 217) * q^37 + (44*b - 8) * q^38 + (8*b + 48) * q^40 + (-44*b + 96) * q^41 + (-6*b - 304) * q^43 + (-32*b + 72) * q^44 + (-8*b + 180) * q^46 + (-36*b - 192) * q^47 + (32*b - 171) * q^49 + (24*b - 124) * q^50 + (-56*b + 20) * q^52 + (76*b - 204) * q^53 + (-30*b - 108) * q^55 + (16*b + 64) * q^56 + (14*b + 324) * q^58 + (32*b - 504) * q^59 + (-18*b + 371) * q^61 + (-72*b - 248) * q^62 + 64 * q^64 + (-79*b - 348) * q^65 + (138*b - 52) * q^67 + (44*b + 240) * q^68 + (40*b + 204) * q^70 + (8*b - 570) * q^71 + (-96*b + 425) * q^73 + (4*b - 434) * q^74 + (88*b - 16) * q^76 + (-28*b - 288) * q^77 + (-50*b - 220) * q^79 + (16*b + 96) * q^80 + (-88*b + 192) * q^82 + (-60*b + 132) * q^83 + (126*b + 657) * q^85 + (-12*b - 608) * q^86 + (-64*b + 144) * q^88 + (89*b + 384) * q^89 + (-102*b - 716) * q^91 + (-16*b + 360) * q^92 + (-72*b - 384) * q^94 + (128*b + 570) * q^95 + (-56*b - 382) * q^97 + (64*b - 342) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 $$2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 + 24 * q^10 + 36 * q^11 + 10 * q^13 + 32 * q^14 + 32 * q^16 + 120 * q^17 - 8 * q^19 + 48 * q^20 + 72 * q^22 + 180 * q^23 - 124 * q^25 + 20 * q^26 + 64 * q^28 + 324 * q^29 - 248 * q^31 + 64 * q^32 + 240 * q^34 + 204 * q^35 - 434 * q^37 - 16 * q^38 + 96 * q^40 + 192 * q^41 - 608 * q^43 + 144 * q^44 + 360 * q^46 - 384 * q^47 - 342 * q^49 - 248 * q^50 + 40 * q^52 - 408 * q^53 - 216 * q^55 + 128 * q^56 + 648 * q^58 - 1008 * q^59 + 742 * q^61 - 496 * q^62 + 128 * q^64 - 696 * q^65 - 104 * q^67 + 480 * q^68 + 408 * q^70 - 1140 * q^71 + 850 * q^73 - 868 * q^74 - 32 * q^76 - 576 * q^77 - 440 * q^79 + 192 * q^80 + 384 * q^82 + 264 * q^83 + 1314 * q^85 - 1216 * q^86 + 288 * q^88 + 768 * q^89 - 1432 * q^91 + 720 * q^92 - 768 * q^94 + 1140 * q^95 - 764 * q^97 - 684 * 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
 −1.73205 1.73205
2.00000 0 4.00000 0.803848 0 −2.39230 8.00000 0 1.60770
1.2 2.00000 0 4.00000 11.1962 0 18.3923 8.00000 0 22.3923
 $$n$$: 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.h yes 2
3.b odd 2 1 162.4.a.e 2
4.b odd 2 1 1296.4.a.s 2
9.c even 3 2 162.4.c.i 4
9.d odd 6 2 162.4.c.j 4
12.b even 2 1 1296.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.e 2 3.b odd 2 1
162.4.a.h yes 2 1.a even 1 1 trivial
162.4.c.i 4 9.c even 3 2
162.4.c.j 4 9.d odd 6 2
1296.4.a.j 2 12.b even 2 1
1296.4.a.s 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12T_{5} + 9$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(162))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12T + 9$$
$7$ $$T^{2} - 16T - 44$$
$11$ $$T^{2} - 36T - 1404$$
$13$ $$T^{2} - 10T - 5267$$
$17$ $$T^{2} - 120T + 333$$
$19$ $$T^{2} + 8T - 13052$$
$23$ $$T^{2} - 180T + 7668$$
$29$ $$T^{2} - 324T + 24921$$
$31$ $$T^{2} + 248T - 19616$$
$37$ $$T^{2} + 434T + 46981$$
$41$ $$T^{2} - 192T - 43056$$
$43$ $$T^{2} + 608T + 91444$$
$47$ $$T^{2} + 384T + 1872$$
$53$ $$T^{2} + 408T - 114336$$
$59$ $$T^{2} + 1008 T + 226368$$
$61$ $$T^{2} - 742T + 128893$$
$67$ $$T^{2} + 104T - 511484$$
$71$ $$T^{2} + 1140 T + 323172$$
$73$ $$T^{2} - 850T - 68207$$
$79$ $$T^{2} + 440T - 19100$$
$83$ $$T^{2} - 264T - 79776$$
$89$ $$T^{2} - 768T - 66411$$
$97$ $$T^{2} + 764T + 61252$$