# Properties

 Label 162.4.a.e Level 162 Weight 4 Character orbit 162.a Self dual yes Analytic conductor 9.558 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + 3 \beta ) q^{5} + ( 8 - 6 \beta ) q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + 3 \beta ) q^{5} + ( 8 - 6 \beta ) q^{7} -8 q^{8} + ( 12 - 6 \beta ) q^{10} + ( -18 - 24 \beta ) q^{11} + ( 5 + 42 \beta ) q^{13} + ( -16 + 12 \beta ) q^{14} + 16 q^{16} + ( -60 + 33 \beta ) q^{17} + ( -4 - 66 \beta ) q^{19} + ( -24 + 12 \beta ) q^{20} + ( 36 + 48 \beta ) q^{22} + ( -90 - 12 \beta ) q^{23} + ( -62 - 36 \beta ) q^{25} + ( -10 - 84 \beta ) q^{26} + ( 32 - 24 \beta ) q^{28} + ( -162 + 21 \beta ) q^{29} + ( -124 + 108 \beta ) q^{31} -32 q^{32} + ( 120 - 66 \beta ) q^{34} + ( -102 + 60 \beta ) q^{35} + ( -217 - 6 \beta ) q^{37} + ( 8 + 132 \beta ) q^{38} + ( 48 - 24 \beta ) q^{40} + ( -96 - 132 \beta ) q^{41} + ( -304 + 18 \beta ) q^{43} + ( -72 - 96 \beta ) q^{44} + ( 180 + 24 \beta ) q^{46} + ( 192 - 108 \beta ) q^{47} + ( -171 - 96 \beta ) q^{49} + ( 124 + 72 \beta ) q^{50} + ( 20 + 168 \beta ) q^{52} + ( 204 + 228 \beta ) q^{53} + ( -108 + 90 \beta ) q^{55} + ( -64 + 48 \beta ) q^{56} + ( 324 - 42 \beta ) q^{58} + ( 504 + 96 \beta ) q^{59} + ( 371 + 54 \beta ) q^{61} + ( 248 - 216 \beta ) q^{62} + 64 q^{64} + ( 348 - 237 \beta ) q^{65} + ( -52 - 414 \beta ) q^{67} + ( -240 + 132 \beta ) q^{68} + ( 204 - 120 \beta ) q^{70} + ( 570 + 24 \beta ) q^{71} + ( 425 + 288 \beta ) q^{73} + ( 434 + 12 \beta ) q^{74} + ( -16 - 264 \beta ) q^{76} + ( 288 - 84 \beta ) q^{77} + ( -220 + 150 \beta ) q^{79} + ( -96 + 48 \beta ) q^{80} + ( 192 + 264 \beta ) q^{82} + ( -132 - 180 \beta ) q^{83} + ( 657 - 378 \beta ) q^{85} + ( 608 - 36 \beta ) q^{86} + ( 144 + 192 \beta ) q^{88} + ( -384 + 267 \beta ) q^{89} + ( -716 + 306 \beta ) q^{91} + ( -360 - 48 \beta ) q^{92} + ( -384 + 216 \beta ) q^{94} + ( -570 + 384 \beta ) q^{95} + ( -382 + 168 \beta ) q^{97} + ( 342 + 192 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 12q^{5} + 16q^{7} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 12q^{5} + 16q^{7} - 16q^{8} + 24q^{10} - 36q^{11} + 10q^{13} - 32q^{14} + 32q^{16} - 120q^{17} - 8q^{19} - 48q^{20} + 72q^{22} - 180q^{23} - 124q^{25} - 20q^{26} + 64q^{28} - 324q^{29} - 248q^{31} - 64q^{32} + 240q^{34} - 204q^{35} - 434q^{37} + 16q^{38} + 96q^{40} - 192q^{41} - 608q^{43} - 144q^{44} + 360q^{46} + 384q^{47} - 342q^{49} + 248q^{50} + 40q^{52} + 408q^{53} - 216q^{55} - 128q^{56} + 648q^{58} + 1008q^{59} + 742q^{61} + 496q^{62} + 128q^{64} + 696q^{65} - 104q^{67} - 480q^{68} + 408q^{70} + 1140q^{71} + 850q^{73} + 868q^{74} - 32q^{76} + 576q^{77} - 440q^{79} - 192q^{80} + 384q^{82} - 264q^{83} + 1314q^{85} + 1216q^{86} + 288q^{88} - 768q^{89} - 1432q^{91} - 720q^{92} - 768q^{94} - 1140q^{95} - 764q^{97} + 684q^{98} + O(q^{100})$$

## 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 −11.1962 0 18.3923 −8.00000 0 22.3923
1.2 −2.00000 0 4.00000 −0.803848 0 −2.39230 −8.00000 0 1.60770
 $$n$$: 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.e 2
3.b odd 2 1 162.4.a.h yes 2
4.b odd 2 1 1296.4.a.j 2
9.c even 3 2 162.4.c.j 4
9.d odd 6 2 162.4.c.i 4
12.b even 2 1 1296.4.a.s 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ 1
$5$ $$1 + 12 T + 259 T^{2} + 1500 T^{3} + 15625 T^{4}$$
$7$ $$1 - 16 T + 642 T^{2} - 5488 T^{3} + 117649 T^{4}$$
$11$ $$1 + 36 T + 1258 T^{2} + 47916 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 10 T - 873 T^{2} - 21970 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 120 T + 10159 T^{2} + 589560 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 180 T + 32002 T^{2} + 2190060 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 324 T + 73699 T^{2} + 7902036 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 248 T + 39966 T^{2} + 7388168 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 192 T + 94786 T^{2} + 13232832 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 608 T + 250458 T^{2} + 48340256 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 384 T + 209518 T^{2} - 39868032 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 408 T + 183418 T^{2} - 60741816 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 1008 T + 637126 T^{2} - 207022032 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 742 T + 582855 T^{2} - 168419902 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 104 T + 90042 T^{2} + 31279352 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 1140 T + 1038994 T^{2} - 408018540 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 440 T + 966978 T^{2} + 216937160 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 264 T + 1063798 T^{2} + 150951768 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 768 T + 1343527 T^{2} + 541416192 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 764 T + 1886598 T^{2} + 697282172 T^{3} + 832972004929 T^{4}$$