# 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: $$x^{2} - x - 26$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + ( - \beta - 5) q^{5} + ( - \beta + 9) q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + (-b - 5) * q^5 + (-b + 9) * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + ( - \beta - 5) q^{5} + ( - \beta + 9) q^{7} - 8 q^{8} + (2 \beta + 10) q^{10} + (2 \beta - 11) q^{11} + (\beta + 31) q^{13} + (2 \beta - 18) q^{14} + 16 q^{16} + (\beta + 2) q^{17} + ( - 5 \beta + 64) q^{19} + ( - 4 \beta - 20) q^{20} + ( - 4 \beta + 22) q^{22} + (\beta + 35) q^{23} + (9 \beta + 136) q^{25} + ( - 2 \beta - 62) q^{26} + ( - 4 \beta + 36) q^{28} + ( - 7 \beta + 115) q^{29} + (3 \beta + 107) q^{31} - 32 q^{32} + ( - 2 \beta - 4) q^{34} + ( - 5 \beta + 191) q^{35} + (14 \beta + 138) q^{37} + (10 \beta - 128) q^{38} + (8 \beta + 40) q^{40} + (2 \beta + 235) q^{41} + ( - 24 \beta - 55) q^{43} + (8 \beta - 44) q^{44} + ( - 2 \beta - 70) q^{46} + (15 \beta + 249) q^{47} + ( - 19 \beta - 26) q^{49} + ( - 18 \beta - 272) q^{50} + (4 \beta + 124) q^{52} + (14 \beta + 82) q^{53} + (3 \beta - 417) q^{55} + (8 \beta - 72) q^{56} + (14 \beta - 230) q^{58} + ( - 2 \beta + 83) q^{59} + (15 \beta - 517) q^{61} + ( - 6 \beta - 214) q^{62} + 64 q^{64} + ( - 35 \beta - 391) q^{65} + (12 \beta - 577) q^{67} + (4 \beta + 8) q^{68} + (10 \beta - 382) q^{70} + ( - 32 \beta - 172) q^{71} + ( - 21 \beta - 166) q^{73} + ( - 28 \beta - 276) q^{74} + ( - 20 \beta + 256) q^{76} + (31 \beta - 571) q^{77} + (13 \beta + 181) q^{79} + ( - 16 \beta - 80) q^{80} + ( - 4 \beta - 470) q^{82} + (21 \beta + 621) q^{83} + ( - 6 \beta - 246) q^{85} + (48 \beta + 110) q^{86} + ( - 16 \beta + 88) q^{88} + (40 \beta - 226) q^{89} + ( - 21 \beta + 43) q^{91} + (4 \beta + 140) q^{92} + ( - 30 \beta - 498) q^{94} + ( - 44 \beta + 860) q^{95} + (22 \beta - 53) q^{97} + (38 \beta + 52) q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + (-b - 5) * q^5 + (-b + 9) * q^7 - 8 * q^8 + (2*b + 10) * q^10 + (2*b - 11) * q^11 + (b + 31) * q^13 + (2*b - 18) * q^14 + 16 * q^16 + (b + 2) * q^17 + (-5*b + 64) * q^19 + (-4*b - 20) * q^20 + (-4*b + 22) * q^22 + (b + 35) * q^23 + (9*b + 136) * q^25 + (-2*b - 62) * q^26 + (-4*b + 36) * q^28 + (-7*b + 115) * q^29 + (3*b + 107) * q^31 - 32 * q^32 + (-2*b - 4) * q^34 + (-5*b + 191) * q^35 + (14*b + 138) * q^37 + (10*b - 128) * q^38 + (8*b + 40) * q^40 + (2*b + 235) * q^41 + (-24*b - 55) * q^43 + (8*b - 44) * q^44 + (-2*b - 70) * q^46 + (15*b + 249) * q^47 + (-19*b - 26) * q^49 + (-18*b - 272) * q^50 + (4*b + 124) * q^52 + (14*b + 82) * q^53 + (3*b - 417) * q^55 + (8*b - 72) * q^56 + (14*b - 230) * q^58 + (-2*b + 83) * q^59 + (15*b - 517) * q^61 + (-6*b - 214) * q^62 + 64 * q^64 + (-35*b - 391) * q^65 + (12*b - 577) * q^67 + (4*b + 8) * q^68 + (10*b - 382) * q^70 + (-32*b - 172) * q^71 + (-21*b - 166) * q^73 + (-28*b - 276) * q^74 + (-20*b + 256) * q^76 + (31*b - 571) * q^77 + (13*b + 181) * q^79 + (-16*b - 80) * q^80 + (-4*b - 470) * q^82 + (21*b + 621) * q^83 + (-6*b - 246) * q^85 + (48*b + 110) * q^86 + (-16*b + 88) * q^88 + (40*b - 226) * q^89 + (-21*b + 43) * q^91 + (4*b + 140) * q^92 + (-30*b - 498) * q^94 + (-44*b + 860) * q^95 + (22*b - 53) * q^97 + (38*b + 52) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q - 4 * q^2 + 8 * q^4 - 9 * q^5 + 19 * q^7 - 16 * q^8 $$2 q - 4 q^{2} + 8 q^{4} - 9 q^{5} + 19 q^{7} - 16 q^{8} + 18 q^{10} - 24 q^{11} + 61 q^{13} - 38 q^{14} + 32 q^{16} + 3 q^{17} + 133 q^{19} - 36 q^{20} + 48 q^{22} + 69 q^{23} + 263 q^{25} - 122 q^{26} + 76 q^{28} + 237 q^{29} + 211 q^{31} - 64 q^{32} - 6 q^{34} + 387 q^{35} + 262 q^{37} - 266 q^{38} + 72 q^{40} + 468 q^{41} - 86 q^{43} - 96 q^{44} - 138 q^{46} + 483 q^{47} - 33 q^{49} - 526 q^{50} + 244 q^{52} + 150 q^{53} - 837 q^{55} - 152 q^{56} - 474 q^{58} + 168 q^{59} - 1049 q^{61} - 422 q^{62} + 128 q^{64} - 747 q^{65} - 1166 q^{67} + 12 q^{68} - 774 q^{70} - 312 q^{71} - 311 q^{73} - 524 q^{74} + 532 q^{76} - 1173 q^{77} + 349 q^{79} - 144 q^{80} - 936 q^{82} + 1221 q^{83} - 486 q^{85} + 172 q^{86} + 192 q^{88} - 492 q^{89} + 107 q^{91} + 276 q^{92} - 966 q^{94} + 1764 q^{95} - 128 q^{97} + 66 q^{98}+O(q^{100})$$ 2 * q - 4 * q^2 + 8 * q^4 - 9 * q^5 + 19 * q^7 - 16 * q^8 + 18 * q^10 - 24 * q^11 + 61 * q^13 - 38 * q^14 + 32 * q^16 + 3 * q^17 + 133 * q^19 - 36 * q^20 + 48 * q^22 + 69 * q^23 + 263 * q^25 - 122 * q^26 + 76 * q^28 + 237 * q^29 + 211 * q^31 - 64 * q^32 - 6 * q^34 + 387 * q^35 + 262 * q^37 - 266 * q^38 + 72 * q^40 + 468 * q^41 - 86 * q^43 - 96 * q^44 - 138 * q^46 + 483 * q^47 - 33 * q^49 - 526 * q^50 + 244 * q^52 + 150 * q^53 - 837 * q^55 - 152 * q^56 - 474 * q^58 + 168 * q^59 - 1049 * q^61 - 422 * q^62 + 128 * q^64 - 747 * q^65 - 1166 * q^67 + 12 * q^68 - 774 * q^70 - 312 * q^71 - 311 * q^73 - 524 * q^74 + 532 * q^76 - 1173 * q^77 + 349 * q^79 - 144 * q^80 - 936 * q^82 + 1221 * q^83 - 486 * q^85 + 172 * q^86 + 192 * q^88 - 492 * q^89 + 107 * q^91 + 276 * q^92 - 966 * q^94 + 1764 * q^95 - 128 * q^97 + 66 * 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
 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 $$T_{5}^{2} + 9T_{5} - 216$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(162))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9T - 216$$
$7$ $$T^{2} - 19T - 146$$
$11$ $$T^{2} + 24T - 801$$
$13$ $$T^{2} - 61T + 694$$
$17$ $$T^{2} - 3T - 234$$
$19$ $$T^{2} - 133T - 1484$$
$23$ $$T^{2} - 69T + 954$$
$29$ $$T^{2} - 237T + 2466$$
$31$ $$T^{2} - 211T + 9004$$
$37$ $$T^{2} - 262T - 29144$$
$41$ $$T^{2} - 468T + 53811$$
$43$ $$T^{2} + 86T - 134231$$
$47$ $$T^{2} - 483T + 5166$$
$53$ $$T^{2} - 150T - 40680$$
$59$ $$T^{2} - 168T + 6111$$
$61$ $$T^{2} + 1049 T + 221944$$
$67$ $$T^{2} + 1166 T + 305869$$
$71$ $$T^{2} + 312T - 217584$$
$73$ $$T^{2} + 311T - 80006$$
$79$ $$T^{2} - 349T - 9476$$
$83$ $$T^{2} - 1221 T + 268524$$
$89$ $$T^{2} + 492T - 317484$$
$97$ $$T^{2} + 128T - 110249$$