# Properties

 Label 162.4.a.f 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: $$\Q(\sqrt{105})$$ Defining polynomial: $$x^{2} - x - 26$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(-1 + 3\sqrt{105})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -5 - \beta ) q^{5} + ( 9 - \beta ) q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -5 - \beta ) q^{5} + ( 9 - \beta ) q^{7} -8 q^{8} + ( 10 + 2 \beta ) q^{10} + ( -11 + 2 \beta ) q^{11} + ( 31 + \beta ) q^{13} + ( -18 + 2 \beta ) q^{14} + 16 q^{16} + ( 2 + \beta ) q^{17} + ( 64 - 5 \beta ) q^{19} + ( -20 - 4 \beta ) q^{20} + ( 22 - 4 \beta ) q^{22} + ( 35 + \beta ) q^{23} + ( 136 + 9 \beta ) q^{25} + ( -62 - 2 \beta ) q^{26} + ( 36 - 4 \beta ) q^{28} + ( 115 - 7 \beta ) q^{29} + ( 107 + 3 \beta ) q^{31} -32 q^{32} + ( -4 - 2 \beta ) q^{34} + ( 191 - 5 \beta ) q^{35} + ( 138 + 14 \beta ) q^{37} + ( -128 + 10 \beta ) q^{38} + ( 40 + 8 \beta ) q^{40} + ( 235 + 2 \beta ) q^{41} + ( -55 - 24 \beta ) q^{43} + ( -44 + 8 \beta ) q^{44} + ( -70 - 2 \beta ) q^{46} + ( 249 + 15 \beta ) q^{47} + ( -26 - 19 \beta ) q^{49} + ( -272 - 18 \beta ) q^{50} + ( 124 + 4 \beta ) q^{52} + ( 82 + 14 \beta ) q^{53} + ( -417 + 3 \beta ) q^{55} + ( -72 + 8 \beta ) q^{56} + ( -230 + 14 \beta ) q^{58} + ( 83 - 2 \beta ) q^{59} + ( -517 + 15 \beta ) q^{61} + ( -214 - 6 \beta ) q^{62} + 64 q^{64} + ( -391 - 35 \beta ) q^{65} + ( -577 + 12 \beta ) q^{67} + ( 8 + 4 \beta ) q^{68} + ( -382 + 10 \beta ) q^{70} + ( -172 - 32 \beta ) q^{71} + ( -166 - 21 \beta ) q^{73} + ( -276 - 28 \beta ) q^{74} + ( 256 - 20 \beta ) q^{76} + ( -571 + 31 \beta ) q^{77} + ( 181 + 13 \beta ) q^{79} + ( -80 - 16 \beta ) q^{80} + ( -470 - 4 \beta ) q^{82} + ( 621 + 21 \beta ) q^{83} + ( -246 - 6 \beta ) q^{85} + ( 110 + 48 \beta ) q^{86} + ( 88 - 16 \beta ) q^{88} + ( -226 + 40 \beta ) q^{89} + ( 43 - 21 \beta ) q^{91} + ( 140 + 4 \beta ) q^{92} + ( -498 - 30 \beta ) q^{94} + ( 860 - 44 \beta ) q^{95} + ( -53 + 22 \beta ) q^{97} + ( 52 + 38 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 9q^{5} + 19q^{7} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 9q^{5} + 19q^{7} - 16q^{8} + 18q^{10} - 24q^{11} + 61q^{13} - 38q^{14} + 32q^{16} + 3q^{17} + 133q^{19} - 36q^{20} + 48q^{22} + 69q^{23} + 263q^{25} - 122q^{26} + 76q^{28} + 237q^{29} + 211q^{31} - 64q^{32} - 6q^{34} + 387q^{35} + 262q^{37} - 266q^{38} + 72q^{40} + 468q^{41} - 86q^{43} - 96q^{44} - 138q^{46} + 483q^{47} - 33q^{49} - 526q^{50} + 244q^{52} + 150q^{53} - 837q^{55} - 152q^{56} - 474q^{58} + 168q^{59} - 1049q^{61} - 422q^{62} + 128q^{64} - 747q^{65} - 1166q^{67} + 12q^{68} - 774q^{70} - 312q^{71} - 311q^{73} - 524q^{74} + 532q^{76} - 1173q^{77} + 349q^{79} - 144q^{80} - 936q^{82} + 1221q^{83} - 486q^{85} + 172q^{86} + 192q^{88} - 492q^{89} + 107q^{91} + 276q^{92} - 966q^{94} + 1764q^{95} - 128q^{97} + 66q^{98} + O(q^{100})$$

## 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
 5.62348 −4.62348
−2.00000 0 4.00000 −19.8704 0 −5.87043 −8.00000 0 39.7409
1.2 −2.00000 0 4.00000 10.8704 0 24.8704 −8.00000 0 −21.7409
 $$n$$: 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.f 2
3.b odd 2 1 162.4.a.g 2
4.b odd 2 1 1296.4.a.l 2
9.c even 3 2 18.4.c.b 4
9.d odd 6 2 54.4.c.b 4
12.b even 2 1 1296.4.a.r 2
36.f odd 6 2 144.4.i.b 4
36.h even 6 2 432.4.i.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 9 T_{5} - 216$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(162))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ 1
$5$ $$1 + 9 T + 34 T^{2} + 1125 T^{3} + 15625 T^{4}$$
$7$ $$1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4}$$
$11$ $$1 + 24 T + 1861 T^{2} + 31944 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 69 T + 25288 T^{2} - 839523 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 237 T + 51244 T^{2} - 5780193 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 468 T + 191653 T^{2} - 32255028 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 483 T + 212812 T^{2} - 50146509 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 168 T + 416869 T^{2} - 34503672 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 1221 T + 1412098 T^{2} - 698151927 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4}$$