LMFDB - Embedded newform 2166.4.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$127.798137072$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.184225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 45x^{2} + 46x + 449 $ x^4 - 2x^3 - 45x^2 + 46*x + 449
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.17400$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.0134493 q^{5} +6.00000 q^{6} -2.50416 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.0134493 q^{5} +6.00000 q^{6} -2.50416 q^{7} -8.00000 q^{8} +9.00000 q^{9} -0.0268985 q^{10} +13.8918 q^{11} -12.0000 q^{12} +91.4873 q^{13} +5.00831 q^{14} -0.0403478 q^{15} +16.0000 q^{16} +30.9633 q^{17} -18.0000 q^{18} +0.0537970 q^{20} +7.51247 q^{21} -27.7835 q^{22} -108.563 q^{23} +24.0000 q^{24} -125.000 q^{25} -182.975 q^{26} -27.0000 q^{27} -10.0166 q^{28} -86.5794 q^{29} +0.0806956 q^{30} -187.854 q^{31} -32.0000 q^{32} -41.6753 q^{33} -61.9266 q^{34} -0.0336790 q^{35} +36.0000 q^{36} -144.685 q^{37} -274.462 q^{39} -0.107594 q^{40} +288.226 q^{41} -15.0249 q^{42} -155.380 q^{43} +55.5670 q^{44} +0.121043 q^{45} +217.127 q^{46} +11.9790 q^{47} -48.0000 q^{48} -336.729 q^{49} +250.000 q^{50} -92.8900 q^{51} +365.949 q^{52} +89.6081 q^{53} +54.0000 q^{54} +0.186834 q^{55} +20.0332 q^{56} +173.159 q^{58} +569.078 q^{59} -0.161391 q^{60} -121.164 q^{61} +375.709 q^{62} -22.5374 q^{63} +64.0000 q^{64} +1.23044 q^{65} +83.3506 q^{66} +637.772 q^{67} +123.853 q^{68} +325.690 q^{69} +0.0673581 q^{70} -100.727 q^{71} -72.0000 q^{72} -1115.33 q^{73} +289.371 q^{74} +374.999 q^{75} -34.7871 q^{77} +548.924 q^{78} +472.478 q^{79} +0.215188 q^{80} +81.0000 q^{81} -576.453 q^{82} +904.871 q^{83} +30.0499 q^{84} +0.416434 q^{85} +310.761 q^{86} +259.738 q^{87} -111.134 q^{88} -676.614 q^{89} -0.242087 q^{90} -229.099 q^{91} -434.253 q^{92} +563.563 q^{93} -23.9579 q^{94} +96.0000 q^{96} -496.960 q^{97} +673.458 q^{98} +125.026 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} - 32 q^{8} + 36 q^{9} + 56 q^{10} - 72 q^{11} - 48 q^{12} + 8 q^{13} + 34 q^{14} + 84 q^{15} + 64 q^{16} + 33 q^{17} - 72 q^{18} - 112 q^{20} + 51 q^{21} + 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} - 108 q^{27} - 68 q^{28} + 75 q^{29} - 168 q^{30} - 419 q^{31} - 128 q^{32} + 216 q^{33} - 66 q^{34} + 331 q^{35} + 144 q^{36} - 131 q^{37} - 24 q^{39} + 224 q^{40} + 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} + 176 q^{46} + 395 q^{47} - 192 q^{48} - 1033 q^{49} - 188 q^{50} - 99 q^{51} + 32 q^{52} + 625 q^{53} + 216 q^{54} + 413 q^{55} + 136 q^{56} - 150 q^{58} + 1328 q^{59} + 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} + 1817 q^{65} - 432 q^{66} + 99 q^{67} + 132 q^{68} + 264 q^{69} - 662 q^{70} + 1307 q^{71} - 288 q^{72} + 183 q^{73} + 262 q^{74} - 282 q^{75} - 513 q^{77} + 48 q^{78} + 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} + 204 q^{84} - 867 q^{85} - 152 q^{86} - 225 q^{87} + 576 q^{88} - 207 q^{89} + 504 q^{90} + 183 q^{91} - 352 q^{92} + 1257 q^{93} - 790 q^{94} + 384 q^{96} + 1331 q^{97} + 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 - 32 * q^8 + 36 * q^9 + 56 * q^10 - 72 * q^11 - 48 * q^12 + 8 * q^13 + 34 * q^14 + 84 * q^15 + 64 * q^16 + 33 * q^17 - 72 * q^18 - 112 * q^20 + 51 * q^21 + 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 - 108 * q^27 - 68 * q^28 + 75 * q^29 - 168 * q^30 - 419 * q^31 - 128 * q^32 + 216 * q^33 - 66 * q^34 + 331 * q^35 + 144 * q^36 - 131 * q^37 - 24 * q^39 + 224 * q^40 + 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 + 176 * q^46 + 395 * q^47 - 192 * q^48 - 1033 * q^49 - 188 * q^50 - 99 * q^51 + 32 * q^52 + 625 * q^53 + 216 * q^54 + 413 * q^55 + 136 * q^56 - 150 * q^58 + 1328 * q^59 + 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 + 1817 * q^65 - 432 * q^66 + 99 * q^67 + 132 * q^68 + 264 * q^69 - 662 * q^70 + 1307 * q^71 - 288 * q^72 + 183 * q^73 + 262 * q^74 - 282 * q^75 - 513 * q^77 + 48 * q^78 + 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 + 204 * q^84 - 867 * q^85 - 152 * q^86 - 225 * q^87 + 576 * q^88 - 207 * q^89 + 504 * q^90 + 183 * q^91 - 352 * q^92 + 1257 * q^93 - 790 * q^94 + 384 * q^96 + 1331 * q^97 + 2066 * q^98 - 648 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0.0134493	0.00120294	0.000601469	−	1.00000i	$-0.499809\pi$
$5$	0.0134493	0.00120294	0.000601469		1.00000i	$0.499809\pi$
$6$	6.00000	0.408248
$7$	−2.50416	−0.135212	−0.0676059	−	0.997712i	$-0.521536\pi$
$7$	−2.50416	−0.135212	−0.0676059	+	0.997712i	$0.521536\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−0.0268985	−0.000850606 0
$11$	13.8918	0.380775	0.190387	−	0.981709i	$-0.439026\pi$
$11$	13.8918	0.380775	0.190387	+	0.981709i	$0.439026\pi$
$12$	−12.0000	−0.288675
$13$	91.4873	1.95185	0.975924	−	0.218111i	$-0.0699894\pi$
$13$	91.4873	1.95185	0.975924	+	0.218111i	$0.0699894\pi$
$14$	5.00831	0.0956091
$15$	−0.0403478	−0.000694517 0
$16$	16.0000	0.250000
$17$	30.9633	0.441748	0.220874	−	0.975302i	$-0.429109\pi$
$17$	30.9633	0.441748	0.220874	+	0.975302i	$0.429109\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	0.0537970	0.000601469 0
$21$	7.51247	0.0780645
$22$	−27.7835	−0.269248
$23$	−108.563	−0.984218	−0.492109	−	0.870533i	$-0.663774\pi$
$23$	−108.563	−0.984218	−0.492109	+	0.870533i	$0.663774\pi$
$24$	24.0000	0.204124
$25$	−125.000	−0.999999
$26$	−182.975	−1.38016
$27$	−27.0000	−0.192450
$28$	−10.0166	−0.0676059
$29$	−86.5794	−0.554393	−0.277196	−	0.960813i	$-0.589405\pi$
$29$	−86.5794	−0.554393	−0.277196	+	0.960813i	$0.589405\pi$
$30$	0.0806956	0.000491097 0
$31$	−187.854	−1.08838	−0.544188	−	0.838963i	$-0.683162\pi$
$31$	−187.854	−1.08838	−0.544188	+	0.838963i	$0.683162\pi$
$32$	−32.0000	−0.176777
$33$	−41.6753	−0.219840
$34$	−61.9266	−0.312363
$35$	−0.0336790	−0.000162651 0
$36$	36.0000	0.166667
$37$	−144.685	−0.642869	−0.321434	−	0.946932i	$-0.604165\pi$
$37$	−144.685	−0.642869	−0.321434	+	0.946932i	$0.604165\pi$
$38$	0	0
$39$	−274.462	−1.12690
$40$	−0.107594	−0.000425303 0
$41$	288.226	1.09789	0.548944	−	0.835859i	$-0.315030\pi$
$41$	288.226	1.09789	0.548944	+	0.835859i	$0.315030\pi$
$42$	−15.0249	−0.0552000
$43$	−155.380	−0.551054	−0.275527	−	0.961293i	$-0.588852\pi$
$43$	−155.380	−0.551054	−0.275527	+	0.961293i	$0.588852\pi$
$44$	55.5670	0.190387
$45$	0.121043	0.000400979 0
$46$	217.127	0.695947
$47$	11.9790	0.0371768	0.0185884	−	0.999827i	$-0.494083\pi$
$47$	11.9790	0.0371768	0.0185884	+	0.999827i	$0.494083\pi$
$48$	−48.0000	−0.144338
$49$	−336.729	−0.981718
$50$	250.000	0.707106
$51$	−92.8900	−0.255043
$52$	365.949	0.975924
$53$	89.6081	0.232238	0.116119	−	0.993235i	$-0.462955\pi$
$53$	89.6081	0.232238	0.116119	+	0.993235i	$0.462955\pi$
$54$	54.0000	0.136083
$55$	0.186834	0.000458049 0
$56$	20.0332	0.0478046
$57$	0	0
$58$	173.159	0.392015
$59$	569.078	1.25572	0.627861	−	0.778325i	$-0.283931\pi$
$59$	569.078	1.25572	0.627861	+	0.778325i	$0.283931\pi$
$60$	−0.161391	−0.000347258 0
$61$	−121.164	−0.254319	−0.127159	−	0.991882i	$-0.540586\pi$
$61$	−121.164	−0.254319	−0.127159	+	0.991882i	$0.540586\pi$
$62$	375.709	0.769598
$63$	−22.5374	−0.0450706
$64$	64.0000	0.125000
$65$	1.23044	0.00234795
$66$	83.3506	0.155451
$67$	637.772	1.16293	0.581464	−	0.813572i	$-0.302480\pi$
$67$	637.772	1.16293	0.581464	+	0.813572i	$0.302480\pi$
$68$	123.853	0.220874
$69$	325.690	0.568239
$70$	0.0673581	0.000115012 0
$71$	−100.727	−0.168367	−0.0841837	−	0.996450i	$-0.526828\pi$
$71$	−100.727	−0.168367	−0.0841837	+	0.996450i	$0.526828\pi$
$72$	−72.0000	−0.117851
$73$	−1115.33	−1.78822	−0.894110	−	0.447847i	$-0.852191\pi$
$73$	−1115.33	−1.78822	−0.894110	+	0.447847i	$0.852191\pi$
$74$	289.371	0.454577
$75$	374.999	0.577349
$76$	0	0
$77$	−34.7871	−0.0514852
$78$	548.924	0.796839
$79$	472.478	0.672885	0.336442	−	0.941704i	$-0.390776\pi$
$79$	472.478	0.672885	0.336442	+	0.941704i	$0.390776\pi$
$80$	0.215188	0.000300735 0
$81$	81.0000	0.111111
$82$	−576.453	−0.776324
$83$	904.871	1.19666	0.598329	−	0.801251i	$-0.295832\pi$
$83$	904.871	1.19666	0.598329	+	0.801251i	$0.295832\pi$
$84$	30.0499	0.0390323
$85$	0.416434	0.000531395 0
$86$	310.761	0.389654
$87$	259.738	0.320079
$88$	−111.134	−0.134624
$89$	−676.614	−0.805854	−0.402927	−	0.915232i	$-0.632007\pi$
$89$	−676.614	−0.805854	−0.402927	+	0.915232i	$0.632007\pi$
$90$	−0.242087	−0.000283535 0
$91$	−229.099	−0.263913
$92$	−434.253	−0.492109
$93$	563.563	0.628374
$94$	−23.9579	−0.0262880
$95$	0	0
$96$	96.0000	0.102062
$97$	−496.960	−0.520192	−0.260096	−	0.965583i	$-0.583754\pi$
$97$	−496.960	−0.520192	−0.260096	+	0.965583i	$0.583754\pi$
$98$	673.458	0.694179
$99$	125.026	0.126925
$100$	−499.999	−0.499999
$101$	−705.777	−0.695321	−0.347660	−	0.937621i	$-0.613024\pi$
$101$	−705.777	−0.695321	−0.347660	+	0.937621i	$0.613024\pi$
$102$	185.780	0.180343
$103$	530.519	0.507510	0.253755	−	0.967268i	$-0.418334\pi$
$103$	530.519	0.507510	0.253755	+	0.967268i	$0.418334\pi$
$104$	−731.899	−0.690082
$105$	0.101037	9.39068e−5 0
$106$	−179.216	−0.164217
$107$	−279.154	−0.252213	−0.126107	−	0.992017i	$-0.540248\pi$
$107$	−279.154	−0.252213	−0.126107	+	0.992017i	$0.540248\pi$
$108$	−108.000	−0.0962250
$109$	1000.90	0.879530	0.439765	−	0.898113i	$-0.355062\pi$
$109$	1000.90	0.879530	0.439765	+	0.898113i	$0.355062\pi$
$110$	−0.373668	−0.000323889 0
$111$	434.056	0.371160
$112$	−40.0665	−0.0338029
$113$	−957.025	−0.796720	−0.398360	−	0.917229i	$-0.630420\pi$
$113$	−957.025	−0.796720	−0.398360	+	0.917229i	$0.630420\pi$
$114$	0	0
$115$	−1.46010	−0.00118395
$116$	−346.318	−0.277196
$117$	823.386	0.650616
$118$	−1138.16	−0.887930
$119$	−77.5370	−0.0597294
$120$	0.322782	0.000245549 0
$121$	−1138.02	−0.855011
$122$	242.328	0.179831
$123$	−864.679	−0.633866
$124$	−751.418	−0.544188
$125$	−3.36231	−0.00240587
$126$	45.0748	0.0318697
$127$	525.464	0.367145	0.183573	−	0.983006i	$-0.441234\pi$
$127$	525.464	0.367145	0.183573	+	0.983006i	$0.441234\pi$
$128$	−128.000	−0.0883883
$129$	466.141	0.318151
$130$	−2.46087	−0.00166025
$131$	2860.40	1.90774	0.953870	−	0.300220i	$-0.0970603\pi$
$131$	2860.40	1.90774	0.953870	+	0.300220i	$0.0970603\pi$
$132$	−166.701	−0.109920
$133$	0	0
$134$	−1275.54	−0.822315
$135$	−0.363130	−0.000231506 0
$136$	−247.707	−0.156181
$137$	−2263.37	−1.41148	−0.705739	−	0.708472i	$-0.749385\pi$
$137$	−2263.37	−1.41148	−0.705739	+	0.708472i	$0.749385\pi$
$138$	−651.380	−0.401805
$139$	−1891.21	−1.15403	−0.577017	−	0.816732i	$-0.695783\pi$
$139$	−1891.21	−1.15403	−0.577017	+	0.816732i	$0.695783\pi$
$140$	−0.134716	−8.13257e−5 0
$141$	−35.9369	−0.0214641
$142$	201.454	0.119054
$143$	1270.92	0.743215
$144$	144.000	0.0833333
$145$	−1.16443	−0.000666900 0
$146$	2230.67	1.26446
$147$	1010.19	0.566795
$148$	−578.742	−0.321434
$149$	−934.852	−0.514000	−0.257000	−	0.966411i	$-0.582734\pi$
$149$	−934.852	−0.514000	−0.257000	+	0.966411i	$0.582734\pi$
$150$	−749.999	−0.408248
$151$	753.923	0.406314	0.203157	−	0.979146i	$-0.434880\pi$
$151$	753.923	0.406314	0.203157	+	0.979146i	$0.434880\pi$
$152$	0	0
$153$	278.670	0.147249
$154$	69.5743	0.0364055
$155$	−2.52650	−0.00130925
$156$	−1097.85	−0.563450
$157$	−734.825	−0.373538	−0.186769	−	0.982404i	$-0.559802\pi$
$157$	−734.825	−0.373538	−0.186769	+	0.982404i	$0.559802\pi$
$158$	−944.956	−0.475802
$159$	−268.824	−0.134083
$160$	−0.430376	−0.000212651 0
$161$	271.860	0.133078
$162$	−162.000	−0.0785674
$163$	1649.60	0.792679	0.396339	−	0.918104i	$-0.370280\pi$
$163$	1649.60	0.792679	0.396339	+	0.918104i	$0.370280\pi$
$164$	1152.91	0.548944
$165$	−0.560502	−0.000264454 0
$166$	−1809.74	−0.846164
$167$	−2753.81	−1.27602	−0.638012	−	0.770026i	$-0.720243\pi$
$167$	−2753.81	−1.27602	−0.638012	+	0.770026i	$0.720243\pi$
$168$	−60.0997	−0.0276000
$169$	6172.93	2.80971
$170$	−0.832867	−0.000375753 0
$171$	0	0
$172$	−621.522	−0.275527
$173$	1551.77	0.681959	0.340980	−	0.940071i	$-0.389241\pi$
$173$	1551.77	0.681959	0.340980	+	0.940071i	$0.389241\pi$
$174$	−519.477	−0.226330
$175$	313.019	0.135212
$176$	222.268	0.0951937
$177$	−1707.23	−0.724992
$178$	1353.23	0.569825
$179$	−809.190	−0.337887	−0.168943	−	0.985626i	$-0.554035\pi$
$179$	−809.190	−0.337887	−0.168943	+	0.985626i	$0.554035\pi$
$180$	0.484173	0.000200490 0
$181$	−45.0007	−0.0184800	−0.00923999	−	0.999957i	$-0.502941\pi$
$181$	−45.0007	−0.0184800	−0.00923999	+	0.999957i	$0.502941\pi$
$182$	458.197	0.186614
$183$	363.492	0.146831
$184$	868.507	0.347974
$185$	−1.94591	−0.000773332 0
$186$	−1127.13	−0.444328
$187$	430.135	0.168206
$188$	47.9158	0.0185884
$189$	67.6122	0.0260215
$190$	0	0
$191$	4840.84	1.83388	0.916940	−	0.399025i	$-0.130651\pi$
$191$	4840.84	1.83388	0.916940	+	0.399025i	$0.130651\pi$
$192$	−192.000	−0.0721688
$193$	618.207	0.230567	0.115284	−	0.993333i	$-0.463222\pi$
$193$	618.207	0.230567	0.115284	+	0.993333i	$0.463222\pi$
$194$	993.919	0.367831
$195$	−3.69131	−0.00135559
$196$	−1346.92	−0.490859
$197$	−447.508	−0.161846	−0.0809229	−	0.996720i	$-0.525787\pi$
$197$	−447.508	−0.161846	−0.0809229	+	0.996720i	$0.525787\pi$
$198$	−250.052	−0.0897495
$199$	−4114.64	−1.46572	−0.732862	−	0.680377i	$-0.761816\pi$
$199$	−4114.64	−1.46572	−0.732862	+	0.680377i	$0.761816\pi$
$200$	999.999	0.353553
$201$	−1913.31	−0.671417
$202$	1411.55	0.491666
$203$	216.808	0.0749604
$204$	−371.560	−0.127522
$205$	3.87643	0.00132069
$206$	−1061.04	−0.358864
$207$	−977.070	−0.328073
$208$	1463.80	0.487962
$209$	0	0
$210$	−0.202074	−6.64021e−5 0
$211$	1861.18	0.607246	0.303623	−	0.952792i	$-0.401804\pi$
$211$	1861.18	0.607246	0.303623	+	0.952792i	$0.401804\pi$
$212$	358.432	0.116119
$213$	302.181	0.0972070
$214$	558.308	0.178342
$215$	−2.08975	−0.000662883 0
$216$	216.000	0.0680414
$217$	470.417	0.147161
$218$	−2001.80	−0.621922
$219$	3346.00	1.03243
$220$	0.747335	0.000229024 0
$221$	2832.75	0.862224
$222$	−868.113	−0.262450
$223$	−6029.78	−1.81069	−0.905345	−	0.424678i	$-0.860387\pi$
$223$	−6029.78	−1.81069	−0.905345	+	0.424678i	$0.860387\pi$
$224$	80.1330	0.0239023
$225$	−1125.00	−0.333333
$226$	1914.05	0.563366
$227$	4905.60	1.43434	0.717172	−	0.696896i	$-0.245436\pi$
$227$	4905.60	1.43434	0.717172	+	0.696896i	$0.245436\pi$
$228$	0	0
$229$	−3215.36	−0.927848	−0.463924	−	0.885875i	$-0.653559\pi$
$229$	−3215.36	−0.927848	−0.463924	+	0.885875i	$0.653559\pi$
$230$	2.92019	0.000837182 0
$231$	104.361	0.0297250
$232$	692.635	0.196007
$233$	−2808.37	−0.789625	−0.394812	−	0.918762i	$-0.629190\pi$
$233$	−2808.37	−0.789625	−0.394812	+	0.918762i	$0.629190\pi$
$234$	−1646.77	−0.460055
$235$	0.161108	4.47214e−5 0
$236$	2276.31	0.627861
$237$	−1417.43	−0.388490
$238$	155.074	0.0422351
$239$	−2821.57	−0.763648	−0.381824	−	0.924235i	$-0.624704\pi$
$239$	−2821.57	−0.763648	−0.381824	+	0.924235i	$0.624704\pi$
$240$	−0.645564	−0.000173629 0
$241$	−3872.65	−1.03510	−0.517551	−	0.855653i	$-0.673156\pi$
$241$	−3872.65	−1.03510	−0.517551	+	0.855653i	$0.673156\pi$
$242$	2276.04	0.604584
$243$	−243.000	−0.0641500
$244$	−484.656	−0.127159
$245$	−4.52876	−0.00118095
$246$	1729.36	0.448211
$247$	0	0
$248$	1502.84	0.384799
$249$	−2714.61	−0.690890
$250$	6.72462	0.00170121
$251$	−1115.55	−0.280530	−0.140265	−	0.990114i	$-0.544795\pi$
$251$	−1115.55	−0.280530	−0.140265	+	0.990114i	$0.544795\pi$
$252$	−90.1496	−0.0225353
$253$	−1508.14	−0.374766
$254$	−1050.93	−0.259611
$255$	−1.24930	−0.000306801 0
$256$	256.000	0.0625000
$257$	7438.46	1.80544	0.902720	−	0.430228i	$-0.141567\pi$
$257$	7438.46	1.80544	0.902720	+	0.430228i	$0.141567\pi$
$258$	−932.283	−0.224967
$259$	362.315	0.0869234
$260$	4.92175	0.00117398
$261$	−779.215	−0.184798
$262$	−5720.79	−1.34898
$263$	−4333.47	−1.01602	−0.508009	−	0.861351i	$-0.669619\pi$
$263$	−4333.47	−1.01602	−0.508009	+	0.861351i	$0.669619\pi$
$264$	333.402	0.0777253
$265$	1.20516	0.000279368 0
$266$	0	0
$267$	2029.84	0.465260
$268$	2551.09	0.581464
$269$	6239.20	1.41417	0.707083	−	0.707130i	$-0.250011\pi$
$269$	6239.20	1.41417	0.707083	+	0.707130i	$0.250011\pi$
$270$	0.726260	0.000163699 0
$271$	−4220.09	−0.945948	−0.472974	−	0.881076i	$-0.656820\pi$
$271$	−4220.09	−0.945948	−0.472974	+	0.881076i	$0.656820\pi$
$272$	495.413	0.110437
$273$	687.296	0.152370
$274$	4526.73	0.998065
$275$	−1736.47	−0.380774
$276$	1302.76	0.284119
$277$	−1998.89	−0.433580	−0.216790	−	0.976218i	$-0.569559\pi$
$277$	−1998.89	−0.433580	−0.216790	+	0.976218i	$0.569559\pi$
$278$	3782.43	0.816025
$279$	−1690.69	−0.362792
$280$	0.269432	5.75059e−5 0
$281$	−6799.68	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$281$	−6799.68	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$282$	71.8738	0.0151774
$283$	7248.10	1.52246	0.761228	−	0.648485i	$-0.224597\pi$
$283$	7248.10	1.52246	0.761228	+	0.648485i	$0.224597\pi$
$284$	−402.908	−0.0841837
$285$	0	0
$286$	−2541.84	−0.525532
$287$	−721.764	−0.148447
$288$	−288.000	−0.0589256
$289$	−3954.27	−0.804859
$290$	2.32886	0.000471570 0
$291$	1490.88	0.300333
$292$	−4461.34	−0.894110
$293$	−8932.77	−1.78109	−0.890543	−	0.454899i	$-0.849675\pi$
$293$	−8932.77	−1.78109	−0.890543	+	0.454899i	$0.849675\pi$
$294$	−2020.38	−0.400785
$295$	7.65368	0.00151056
$296$	1157.48	0.227288
$297$	−375.078	−0.0732801
$298$	1869.70	0.363453
$299$	−9932.17	−1.92104
$300$	1500.00	0.288675
$301$	389.097	0.0745089
$302$	−1507.85	−0.287307
$303$	2117.33	0.401444
$304$	0	0
$305$	−1.62956	−0.000305930 0
$306$	−557.340	−0.104121
$307$	−2247.54	−0.417830	−0.208915	−	0.977934i	$-0.566993\pi$
$307$	−2247.54	−0.417830	−0.208915	+	0.977934i	$0.566993\pi$
$308$	−139.149	−0.0257426
$309$	−1591.56	−0.293011
$310$	5.05301	0.000925779 0
$311$	−1166.15	−0.212624	−0.106312	−	0.994333i	$-0.533904\pi$
$311$	−1166.15	−0.212624	−0.106312	+	0.994333i	$0.533904\pi$
$312$	2195.70	0.398419
$313$	1137.84	0.205478	0.102739	−	0.994708i	$-0.467239\pi$
$313$	1137.84	0.205478	0.102739	+	0.994708i	$0.467239\pi$
$314$	1469.65	0.264131
$315$	−0.303111	−5.42171e−5 0
$316$	1889.91	0.336442
$317$	5135.99	0.909988	0.454994	−	0.890494i	$-0.349641\pi$
$317$	5135.99	0.909988	0.454994	+	0.890494i	$0.349641\pi$
$318$	537.648	0.0948108
$319$	−1202.74	−0.211099
$320$	0.860753	0.000150367 0
$321$	837.462	0.145615
$322$	−543.719	−0.0941003
$323$	0	0
$324$	324.000	0.0555556
$325$	−11435.9	−1.95185
$326$	−3299.20	−0.560509
$327$	−3002.70	−0.507797
$328$	−2305.81	−0.388162
$329$	−29.9972	−0.00502674
$330$	1.12100	0.000186998 0
$331$	−5292.01	−0.878777	−0.439388	−	0.898297i	$-0.644805\pi$
$331$	−5292.01	−0.878777	−0.439388	+	0.898297i	$0.644805\pi$
$332$	3619.49	0.598329
$333$	−1302.17	−0.214290
$334$	5507.61	0.902285
$335$	8.57756	0.00139893
$336$	120.199	0.0195161
$337$	−6271.57	−1.01375	−0.506875	−	0.862019i	$-0.669200\pi$
$337$	−6271.57	−1.01375	−0.506875	+	0.862019i	$0.669200\pi$
$338$	−12345.9	−1.98677
$339$	2871.07	0.459986
$340$	1.66573	0.000265697 0
$341$	−2609.63	−0.414426
$342$	0	0
$343$	1702.15	0.267951
$344$	1243.04	0.194827
$345$	4.38029	0.000683556 0
$346$	−3103.54	−0.482218
$347$	4036.89	0.624530	0.312265	−	0.949995i	$-0.398912\pi$
$347$	4036.89	0.624530	0.312265	+	0.949995i	$0.398912\pi$
$348$	1038.95	0.160039
$349$	−10921.0	−1.67504	−0.837520	−	0.546407i	$-0.815995\pi$
$349$	−10921.0	−1.67504	−0.837520	+	0.546407i	$0.815995\pi$
$350$	−626.038	−0.0956090
$351$	−2470.16	−0.375633
$352$	−444.536	−0.0673121
$353$	1402.83	0.211516	0.105758	−	0.994392i	$-0.466273\pi$
$353$	1402.83	0.211516	0.105758	+	0.994392i	$0.466273\pi$
$354$	3414.47	0.512647
$355$	−1.35470	−0.000202536 0
$356$	−2706.46	−0.402927
$357$	232.611	0.0344848
$358$	1618.38	0.238922
$359$	11094.1	1.63099	0.815493	−	0.578767i	$-0.196466\pi$
$359$	11094.1	1.63099	0.815493	+	0.578767i	$0.196466\pi$
$360$	−0.968347	−0.000141768 0
$361$	0	0
$362$	90.0014	0.0130673
$363$	3414.06	0.493641
$364$	−916.394	−0.131956
$365$	−15.0004	−0.00215112
$366$	−726.983	−0.103825
$367$	−3426.46	−0.487356	−0.243678	−	0.969856i	$-0.578354\pi$
$367$	−3426.46	−0.487356	−0.243678	+	0.969856i	$0.578354\pi$
$368$	−1737.01	−0.246055
$369$	2594.04	0.365963
$370$	3.89182	0.000546828 0
$371$	−224.393	−0.0314013
$372$	2254.25	0.314187
$373$	−8397.46	−1.16569	−0.582847	−	0.812582i	$-0.698061\pi$
$373$	−8397.46	−1.16569	−0.582847	+	0.812582i	$0.698061\pi$
$374$	−860.270	−0.118940
$375$	10.0869	0.00138903
$376$	−95.8317	−0.0131440
$377$	−7920.92	−1.08209
$378$	−135.224	−0.0184000
$379$	3637.68	0.493022	0.246511	−	0.969140i	$-0.420716\pi$
$379$	3637.68	0.493022	0.246511	+	0.969140i	$0.420716\pi$
$380$	0	0
$381$	−1576.39	−0.211971
$382$	−9681.69	−1.29675
$383$	269.458	0.0359494	0.0179747	−	0.999838i	$-0.494278\pi$
$383$	269.458	0.0359494	0.0179747	+	0.999838i	$0.494278\pi$
$384$	384.000	0.0510310
$385$	−0.467861	−6.19335e−5 0
$386$	−1236.41	−0.163036
$387$	−1398.42	−0.183685
$388$	−1987.84	−0.260096
$389$	−9678.09	−1.26144	−0.630718	−	0.776012i	$-0.717239\pi$
$389$	−9678.09	−1.26144	−0.630718	+	0.776012i	$0.717239\pi$
$390$	7.38262	0.000958548 0
$391$	−3361.48	−0.434776
$392$	2693.83	0.347090
$393$	−8581.19	−1.10143
$394$	895.016	0.114442
$395$	6.35448	0.000809439 0
$396$	500.103	0.0634625
$397$	1538.10	0.194446	0.0972229	−	0.995263i	$-0.469004\pi$
$397$	1538.10	0.194446	0.0972229	+	0.995263i	$0.469004\pi$
$398$	8229.28	1.03642
$399$	0	0
$400$	−2000.00	−0.250000
$401$	194.468	0.0242176	0.0121088	−	0.999927i	$-0.496146\pi$
$401$	194.468	0.0242176	0.0121088	+	0.999927i	$0.496146\pi$
$402$	3826.63	0.474764
$403$	−17186.3	−2.12435
$404$	−2823.11	−0.347660
$405$	1.08939	0.000133660 0
$406$	−433.617	−0.0530050
$407$	−2009.94	−0.244788
$408$	743.120	0.0901713
$409$	4397.51	0.531645	0.265823	−	0.964022i	$-0.414356\pi$
$409$	4397.51	0.531645	0.265823	+	0.964022i	$0.414356\pi$
$410$	−7.75286	−0.000933870 0
$411$	6790.10	0.814917
$412$	2122.08	0.253755
$413$	−1425.06	−0.169788
$414$	1954.14	0.231982
$415$	12.1698	0.00143950
$416$	−2927.60	−0.345041
$417$	5673.64	0.666282
$418$	0	0
$419$	−11672.3	−1.36093	−0.680466	−	0.732780i	$-0.738222\pi$
$419$	−11672.3	−1.36093	−0.680466	+	0.732780i	$0.738222\pi$
$420$	0.404149	4.69534e−5 0
$421$	−8397.50	−0.972136	−0.486068	−	0.873921i	$-0.661569\pi$
$421$	−8397.50	−0.972136	−0.486068	+	0.873921i	$0.661569\pi$
$422$	−3722.36	−0.429388
$423$	107.811	0.0123923
$424$	−716.865	−0.0821085
$425$	−3870.41	−0.441747
$426$	−604.362	−0.0687357
$427$	303.413	0.0343869
$428$	−1116.62	−0.126107
$429$	−3812.76	−0.429095
$430$	4.17950	0.000468729 0
$431$	−11591.9	−1.29550	−0.647751	−	0.761852i	$-0.724290\pi$
$431$	−11591.9	−1.29550	−0.647751	+	0.761852i	$0.724290\pi$
$432$	−432.000	−0.0481125
$433$	−14187.7	−1.57463	−0.787316	−	0.616550i	$-0.788530\pi$
$433$	−14187.7	−1.57463	−0.787316	+	0.616550i	$0.788530\pi$
$434$	−940.834	−0.104059
$435$	3.49329	0.000385035 0
$436$	4003.60	0.439765
$437$	0	0
$438$	−6692.01	−0.730038
$439$	−12553.9	−1.36484	−0.682419	−	0.730961i	$-0.739072\pi$
$439$	−12553.9	−1.36484	−0.682419	+	0.730961i	$0.739072\pi$
$440$	−1.49467	−0.000161945 0
$441$	−3030.56	−0.327239
$442$	−5665.50	−0.609684
$443$	10847.0	1.16333	0.581667	−	0.813427i	$-0.302401\pi$
$443$	10847.0	1.16333	0.581667	+	0.813427i	$0.302401\pi$
$444$	1736.23	0.185580
$445$	−9.09996	−0.000969392 0
$446$	12059.6	1.28035
$447$	2804.55	0.296758
$448$	−160.266	−0.0169015
$449$	16076.9	1.68979	0.844897	−	0.534929i	$-0.179662\pi$
$449$	16076.9	1.68979	0.844897	+	0.534929i	$0.179662\pi$
$450$	2250.00	0.235702
$451$	4003.97	0.418048
$452$	−3828.10	−0.398360
$453$	−2261.77	−0.234586
$454$	−9811.20	−1.01423
$455$	−3.08121	−0.000317471 0
$456$	0	0
$457$	11333.0	1.16003	0.580016	−	0.814605i	$-0.303046\pi$
$457$	11333.0	1.16003	0.580016	+	0.814605i	$0.303046\pi$
$458$	6430.73	0.656088
$459$	−836.010	−0.0850143
$460$	−5.84039	−0.000591977 0
$461$	13366.2	1.35038	0.675191	−	0.737643i	$-0.264061\pi$
$461$	13366.2	1.35038	0.675191	+	0.737643i	$0.264061\pi$
$462$	−208.723	−0.0210188
$463$	−9941.35	−0.997870	−0.498935	−	0.866639i	$-0.666275\pi$
$463$	−9941.35	−0.997870	−0.498935	+	0.866639i	$0.666275\pi$
$464$	−1385.27	−0.138598
$465$	7.57951	0.000755896 0
$466$	5616.75	0.558349
$467$	−17101.5	−1.69457	−0.847284	−	0.531141i	$-0.821764\pi$
$467$	−17101.5	−1.69457	−0.847284	+	0.531141i	$0.821764\pi$
$468$	3293.54	0.325308
$469$	−1597.08	−0.157242
$470$	−0.322216	−3.16228e−5 0
$471$	2204.47	0.215662
$472$	−4552.62	−0.443965
$473$	−2158.51	−0.209827
$474$	2834.87	0.274704
$475$	0	0
$476$	−310.148	−0.0298647
$477$	806.473	0.0774127
$478$	5643.13	0.539981
$479$	12777.2	1.21880	0.609398	−	0.792864i	$-0.291411\pi$
$479$	12777.2	1.21880	0.609398	+	0.792864i	$0.291411\pi$
$480$	1.29113	0.000122774 0
$481$	−13236.9	−1.25478
$482$	7745.30	0.731927
$483$	−815.579	−0.0768325
$484$	−4552.08	−0.427505
$485$	−6.68374	−0.000625759 0
$486$	486.000	0.0453609
$487$	1127.27	0.104890	0.0524452	−	0.998624i	$-0.483299\pi$
$487$	1127.27	0.104890	0.0524452	+	0.998624i	$0.483299\pi$
$488$	969.311	0.0899153
$489$	−4948.80	−0.457653
$490$	9.05752	0.000835055 0
$491$	−19693.7	−1.81011	−0.905055	−	0.425294i	$-0.860171\pi$
$491$	−19693.7	−1.81011	−0.905055	+	0.425294i	$0.860171\pi$
$492$	−3458.72	−0.316933
$493$	−2680.79	−0.244902
$494$	0	0
$495$	1.68150	0.000152683 0
$496$	−3005.67	−0.272094
$497$	252.236	0.0227653
$498$	5429.23	0.488533
$499$	6078.97	0.545355	0.272677	−	0.962106i	$-0.412091\pi$
$499$	6078.97	0.545355	0.272677	+	0.962106i	$0.412091\pi$
$500$	−13.4492	−0.00120294
$501$	8261.42	0.736713
$502$	2231.10	0.198365
$503$	16809.2	1.49003	0.745016	−	0.667046i	$-0.232442\pi$
$503$	16809.2	1.49003	0.745016	+	0.667046i	$0.232442\pi$
$504$	180.299	0.0159349
$505$	−9.49217	−0.000836428 0
$506$	3016.27	0.264999
$507$	−18518.8	−1.62219
$508$	2101.86	0.183573
$509$	13927.0	1.21278	0.606388	−	0.795169i	$-0.292618\pi$
$509$	13927.0	1.21278	0.606388	+	0.795169i	$0.292618\pi$
$510$	2.49860	0.000216941 0
$511$	2792.97	0.241788
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−14876.9	−1.27664
$515$	7.13509	0.000610504 0
$516$	1864.57	0.159075
$517$	166.409	0.0141560
$518$	−724.630	−0.0614641
$519$	−4655.31	−0.393729
$520$	−9.84350	−0.000830127 0
$521$	−18922.4	−1.59118	−0.795592	−	0.605833i	$-0.792840\pi$
$521$	−18922.4	−1.59118	−0.795592	+	0.605833i	$0.792840\pi$
$522$	1558.43	0.130672
$523$	−3288.11	−0.274912	−0.137456	−	0.990508i	$-0.543893\pi$
$523$	−3288.11	−0.274912	−0.137456	+	0.990508i	$0.543893\pi$
$524$	11441.6	0.953870
$525$	−939.057	−0.0780644
$526$	8666.93	0.718434
$527$	−5816.60	−0.480788
$528$	−666.804	−0.0549601
$529$	−381.000	−0.0313142
$530$	−2.41032	−0.000197543 0
$531$	5121.70	0.418574
$532$	0	0
$533$	26369.1	2.14291
$534$	−4059.69	−0.328988
$535$	−3.75441	−0.000303397 0
$536$	−5102.17	−0.411157
$537$	2427.57	0.195079
$538$	−12478.4	−0.999967
$539$	−4677.76	−0.373813
$540$	−1.45252	−0.000115753 0
$541$	−750.018	−0.0596041	−0.0298020	−	0.999556i	$-0.509488\pi$
$541$	−750.018	−0.0596041	−0.0298020	+	0.999556i	$0.509488\pi$
$542$	8440.17	0.668887
$543$	135.002	0.0106694
$544$	−990.826	−0.0780907
$545$	13.4614	0.00105802
$546$	−1374.59	−0.107742
$547$	8202.66	0.641171	0.320585	−	0.947220i	$-0.396120\pi$
$547$	8202.66	0.641171	0.320585	+	0.947220i	$0.396120\pi$
$548$	−9053.47	−0.705739
$549$	−1090.48	−0.0847729
$550$	3472.93	0.269248
$551$	0	0
$552$	−2605.52	−0.200903
$553$	−1183.16	−0.0909819
$554$	3997.78	0.306588
$555$	5.83774	0.000446483 0
$556$	−7564.86	−0.577017
$557$	7541.68	0.573701	0.286850	−	0.957975i	$-0.407392\pi$
$557$	7541.68	0.573701	0.286850	+	0.957975i	$0.407392\pi$
$558$	3381.38	0.256533
$559$	−14215.3	−1.07557
$560$	−0.538865	−4.06628e−5 0
$561$	−1290.40	−0.0971140
$562$	13599.4	1.02074
$563$	−2970.58	−0.222371	−0.111185	−	0.993800i	$-0.535465\pi$
$563$	−2970.58	−0.222371	−0.111185	+	0.993800i	$0.535465\pi$
$564$	−143.748	−0.0107320
$565$	−12.8713	−0.000958405 0
$566$	−14496.2	−1.07654
$567$	−202.837	−0.0150235
$568$	805.816	0.0595269
$569$	25168.3	1.85432	0.927162	−	0.374662i	$-0.122241\pi$
$569$	25168.3	1.85432	0.927162	+	0.374662i	$0.122241\pi$
$570$	0	0
$571$	−9302.04	−0.681748	−0.340874	−	0.940109i	$-0.610723\pi$
$571$	−9302.04	−0.681748	−0.340874	+	0.940109i	$0.610723\pi$
$572$	5083.68	0.371607
$573$	−14522.5	−1.05879
$574$	1443.53	0.104968
$575$	13570.4	0.984217
$576$	576.000	0.0416667
$577$	−13665.0	−0.985927	−0.492964	−	0.870050i	$-0.664086\pi$
$577$	−13665.0	−0.985927	−0.492964	+	0.870050i	$0.664086\pi$
$578$	7908.55	0.569121
$579$	−1854.62	−0.133118
$580$	−4.65772	−0.000333450 0
$581$	−2265.94	−0.161802
$582$	−2981.76	−0.212367
$583$	1244.81	0.0884304
$584$	8922.68	0.632231
$585$	11.0739	0.000782651 0
$586$	17865.5	1.25942
$587$	1121.54	0.0788601	0.0394301	−	0.999222i	$-0.487446\pi$
$587$	1121.54	0.0788601	0.0394301	+	0.999222i	$0.487446\pi$
$588$	4040.75	0.283398
$589$	0	0
$590$	−15.3074	−0.00106812
$591$	1342.52	0.0934418
$592$	−2314.97	−0.160717
$593$	−21606.7	−1.49625	−0.748127	−	0.663555i	$-0.769047\pi$
$593$	−21606.7	−1.49625	−0.748127	+	0.663555i	$0.769047\pi$
$594$	750.155	0.0518169
$595$	−1.04281	−7.18508e−5 0
$596$	−3739.41	−0.257000
$597$	12343.9	0.846237
$598$	19864.3	1.35838
$599$	73.1647	0.00499070	0.00249535	−	0.999997i	$-0.499206\pi$
$599$	73.1647	0.00499070	0.00249535	+	0.999997i	$0.499206\pi$
$600$	−3000.00	−0.204124
$601$	−26049.2	−1.76800	−0.884000	−	0.467486i	$-0.845160\pi$
$601$	−26049.2	−1.76800	−0.884000	+	0.467486i	$0.845160\pi$
$602$	−778.194	−0.0526857
$603$	5739.94	0.387643
$604$	3015.69	0.203157
$605$	−15.3055	−0.00102852
$606$	−4234.66	−0.283863
$607$	−12814.5	−0.856875	−0.428437	−	0.903571i	$-0.640936\pi$
$607$	−12814.5	−0.856875	−0.428437	+	0.903571i	$0.640936\pi$
$608$	0	0
$609$	−650.425	−0.0432784
$610$	3.25913	0.000216325 0
$611$	1095.92	0.0725635
$612$	1114.68	0.0736246
$613$	6912.95	0.455484	0.227742	−	0.973722i	$-0.426866\pi$
$613$	6912.95	0.455484	0.227742	+	0.973722i	$0.426866\pi$
$614$	4495.08	0.295451
$615$	−11.6293	−0.000762502 0
$616$	278.297	0.0182028
$617$	−11626.4	−0.758611	−0.379305	−	0.925272i	$-0.623837\pi$
$617$	−11626.4	−0.758611	−0.379305	+	0.925272i	$0.623837\pi$
$618$	3183.11	0.207190
$619$	6130.51	0.398071	0.199036	−	0.979992i	$-0.436219\pi$
$619$	6130.51	0.398071	0.199036	+	0.979992i	$0.436219\pi$
$620$	−10.1060	−0.000654625 0
$621$	2931.21	0.189413
$622$	2332.29	0.150348
$623$	1694.35	0.108961
$624$	−4391.39	−0.281725
$625$	15624.9	0.999996
$626$	−2275.68	−0.145295
$627$	0	0
$628$	−2939.30	−0.186769
$629$	−4479.94	−0.283986
$630$	0.606223	3.83373e−5 0
$631$	−27656.8	−1.74485	−0.872425	−	0.488749i	$-0.837453\pi$
$631$	−27656.8	−1.74485	−0.872425	+	0.488749i	$0.837453\pi$
$632$	−3779.82	−0.237901
$633$	−5583.54	−0.350594
$634$	−10272.0	−0.643459
$635$	7.06711	0.000441653 0
$636$	−1075.30	−0.0670413
$637$	−30806.5	−1.91616
$638$	2405.48	0.149269
$639$	−906.543	−0.0561225
$640$	−1.72151	−0.000106326 0
$641$	−15534.7	−0.957226	−0.478613	−	0.878026i	$-0.658860\pi$
$641$	−15534.7	−0.957226	−0.478613	+	0.878026i	$0.658860\pi$
$642$	−1674.92	−0.102966
$643$	30077.1	1.84467	0.922337	−	0.386386i	$-0.126277\pi$
$643$	30077.1	1.84467	0.922337	+	0.386386i	$0.126277\pi$
$644$	1087.44	0.0665389
$645$	6.26926	0.000382716 0
$646$	0	0
$647$	−17338.3	−1.05354	−0.526769	−	0.850009i	$-0.676597\pi$
$647$	−17338.3	−1.05354	−0.526769	+	0.850009i	$0.676597\pi$
$648$	−648.000	−0.0392837
$649$	7905.49	0.478148
$650$	22871.8	1.38016
$651$	−1411.25	−0.0849636
$652$	6598.40	0.396339
$653$	16872.7	1.01115	0.505573	−	0.862784i	$-0.331281\pi$
$653$	16872.7	1.01115	0.505573	+	0.862784i	$0.331281\pi$
$654$	6005.40	0.359067
$655$	38.4702	0.00229489
$656$	4611.62	0.274472
$657$	−10038.0	−0.596073
$658$	59.9944	0.00355444
$659$	20828.4	1.23120	0.615598	−	0.788060i	$-0.288914\pi$
$659$	20828.4	1.23120	0.615598	+	0.788060i	$0.288914\pi$
$660$	−2.24201	−0.000132227 0
$661$	19437.7	1.14378	0.571889	−	0.820331i	$-0.306211\pi$
$661$	19437.7	1.14378	0.571889	+	0.820331i	$0.306211\pi$
$662$	10584.0	0.621389
$663$	−8498.26	−0.497805
$664$	−7238.97	−0.423082
$665$	0	0
$666$	2604.34	0.151526
$667$	9399.35	0.545644
$668$	−11015.2	−0.638012
$669$	18089.3	1.04540
$670$	−17.1551	−0.000989194 0
$671$	−1683.18	−0.0968382
$672$	−240.399	−0.0138000
$673$	−5879.59	−0.336763	−0.168381	−	0.985722i	$-0.553854\pi$
$673$	−5879.59	−0.336763	−0.168381	+	0.985722i	$0.553854\pi$
$674$	12543.1	0.716830
$675$	3375.00	0.192450
$676$	24691.7	1.40486
$677$	−26176.1	−1.48601	−0.743005	−	0.669286i	$-0.766600\pi$
$677$	−26176.1	−1.48601	−0.743005	+	0.669286i	$0.766600\pi$
$678$	−5742.15	−0.325259
$679$	1244.46	0.0703360
$680$	−3.33147	−0.000187876 0
$681$	−14716.8	−0.828119
$682$	5219.26	0.293044
$683$	−27091.0	−1.51773	−0.758865	−	0.651248i	$-0.774246\pi$
$683$	−27091.0	−1.51773	−0.758865	+	0.651248i	$0.774246\pi$
$684$	0	0
$685$	−30.4406	−0.00169792
$686$	−3404.30	−0.189470
$687$	9646.09	0.535693
$688$	−2486.09	−0.137763
$689$	8198.01	0.453293
$690$	−8.76058	−0.000483347 0
$691$	−15508.6	−0.853801	−0.426901	−	0.904299i	$-0.640395\pi$
$691$	−15508.6	−0.853801	−0.426901	+	0.904299i	$0.640395\pi$
$692$	6207.08	0.340980
$693$	−313.084	−0.0171617
$694$	−8073.78	−0.441609
$695$	−25.4354	−0.00138823
$696$	−2077.91	−0.113165
$697$	8924.45	0.484989
$698$	21842.0	1.18443
$699$	8425.12	0.455890
$700$	1252.08	0.0676058
$701$	−26731.7	−1.44029	−0.720145	−	0.693824i	$-0.755925\pi$
$701$	−26731.7	−1.44029	−0.720145	+	0.693824i	$0.755925\pi$
$702$	4940.32	0.265613
$703$	0	0
$704$	889.073	0.0475969
$705$	−0.483324	−2.58199e−5 0
$706$	−2805.66	−0.149564
$707$	1767.37	0.0940155
$708$	−6828.93	−0.362496
$709$	13748.0	0.728235	0.364118	−	0.931353i	$-0.381371\pi$
$709$	13748.0	0.728235	0.364118	+	0.931353i	$0.381371\pi$
$710$	2.70941	0.000143214 0
$711$	4252.30	0.224295
$712$	5412.91	0.284912
$713$	20394.1	1.07120
$714$	−465.222	−0.0243844
$715$	17.0929	0.000894041 0
$716$	−3236.76	−0.168943
$717$	8464.70	0.440893
$718$	−22188.2	−1.15328
$719$	−32100.7	−1.66503	−0.832514	−	0.554004i	$-0.813099\pi$
$719$	−32100.7	−1.66503	−0.832514	+	0.554004i	$0.813099\pi$
$720$	1.93669	0.000100245 0
$721$	−1328.50	−0.0686214
$722$	0	0
$723$	11618.0	0.597616
$724$	−180.003	−0.00923999
$725$	10822.4	0.554392
$726$	−6828.11	−0.349057
$727$	21009.6	1.07181	0.535903	−	0.844280i	$-0.319971\pi$
$727$	21009.6	1.07181	0.535903	+	0.844280i	$0.319971\pi$
$728$	1832.79	0.0933072
$729$	729.000	0.0370370
$730$	30.0009	0.00152107
$731$	−4811.10	−0.243427
$732$	1453.97	0.0734155
$733$	18667.1	0.940636	0.470318	−	0.882497i	$-0.344139\pi$
$733$	18667.1	0.940636	0.470318	+	0.882497i	$0.344139\pi$
$734$	6852.92	0.344613
$735$	13.5863	0.000681819 0
$736$	3474.03	0.173987
$737$	8859.77	0.442814
$738$	−5188.08	−0.258775
$739$	12002.5	0.597454	0.298727	−	0.954339i	$-0.403438\pi$
$739$	12002.5	0.597454	0.298727	+	0.954339i	$0.403438\pi$
$740$	−7.78365	−0.000386666 0
$741$	0	0
$742$	448.785	0.0222041
$743$	19695.6	0.972492	0.486246	−	0.873822i	$-0.338366\pi$
$743$	19695.6	0.972492	0.486246	+	0.873822i	$0.338366\pi$
$744$	−4508.51	−0.222164
$745$	−12.5731	−0.000618311 0
$746$	16794.9	0.824271
$747$	8143.84	0.398886
$748$	1720.54	0.0841032
$749$	699.045	0.0341022
$750$	−20.1739	−0.000982194 0
$751$	−29729.4	−1.44453	−0.722266	−	0.691616i	$-0.756899\pi$
$751$	−29729.4	−1.44453	−0.722266	+	0.691616i	$0.756899\pi$
$752$	191.663	0.00929421
$753$	3346.66	0.161964
$754$	15841.8	0.765154
$755$	10.1397	0.000488771 0
$756$	270.449	0.0130108
$757$	23217.9	1.11476	0.557378	−	0.830259i	$-0.311807\pi$
$757$	23217.9	1.11476	0.557378	+	0.830259i	$0.311807\pi$
$758$	−7275.36	−0.348619
$759$	4524.41	0.216371
$760$	0	0
$761$	30950.8	1.47433	0.737164	−	0.675714i	$-0.236164\pi$
$761$	30950.8	1.47433	0.737164	+	0.675714i	$0.236164\pi$
$762$	3152.79	0.149886
$763$	−2506.41	−0.118923
$764$	19363.4	0.916940
$765$	3.74790	0.000177132 0
$766$	−538.915	−0.0254201
$767$	52063.4	2.45098
$768$	−768.000	−0.0360844
$769$	−36399.4	−1.70689	−0.853445	−	0.521184i	$-0.825491\pi$
$769$	−36399.4	−1.70689	−0.853445	+	0.521184i	$0.825491\pi$
$770$	0.935722	4.37936e−5 0
$771$	−22315.4	−1.04237
$772$	2472.83	0.115284
$773$	−11609.8	−0.540202	−0.270101	−	0.962832i	$-0.587057\pi$
$773$	−11609.8	−0.540202	−0.270101	+	0.962832i	$0.587057\pi$
$774$	2796.85	0.129885
$775$	23481.8	1.08837
$776$	3975.68	0.183916
$777$	−1086.95	−0.0501852
$778$	19356.2	0.891970
$779$	0	0
$780$	−14.7652	−0.000677796 0
$781$	−1399.27	−0.0641101
$782$	6722.96	0.307433
$783$	2337.64	0.106693
$784$	−5387.67	−0.245429
$785$	−9.88285	−0.000449343 0
$786$	17162.4	0.778831
$787$	−11434.0	−0.517887	−0.258944	−	0.965892i	$-0.583374\pi$
$787$	−11434.0	−0.517887	−0.258944	+	0.965892i	$0.583374\pi$
$788$	−1790.03	−0.0809229
$789$	13000.4	0.586599
$790$	−12.7090	−0.000572360 0
$791$	2396.54	0.107726
$792$	−1000.21	−0.0448747
$793$	−11085.0	−0.496392
$794$	−3076.20	−0.137494
$795$	−3.61549	−0.000161293 0
$796$	−16458.6	−0.732862
$797$	−25067.2	−1.11408	−0.557042	−	0.830485i	$-0.688064\pi$
$797$	−25067.2	−1.11408	−0.557042	+	0.830485i	$0.688064\pi$
$798$	0	0
$799$	370.908	0.0164228
$800$	3999.99	0.176776
$801$	−6089.53	−0.268618
$802$	−388.936	−0.0171244
$803$	−15494.0	−0.680909
$804$	−7653.26	−0.335709
$805$	3.65631	0.000160084 0
$806$	34372.6	1.50214
$807$	−18717.6	−0.816470
$808$	5646.21	0.245833
$809$	−37770.8	−1.64147	−0.820736	−	0.571308i	$-0.806436\pi$
$809$	−37770.8	−1.64147	−0.820736	+	0.571308i	$0.806436\pi$
$810$	−2.17878	−9.45118e−5 0
$811$	13157.2	0.569682	0.284841	−	0.958575i	$-0.408059\pi$
$811$	13157.2	0.569682	0.284841	+	0.958575i	$0.408059\pi$
$812$	867.234	0.0374802
$813$	12660.3	0.546144
$814$	4019.87	0.173091
$815$	22.1859	0.000953544 0
$816$	−1486.24	−0.0637608
$817$	0	0
$818$	−8795.02	−0.375930
$819$	−2061.89	−0.0879709
$820$	15.5057	0.000660346 0
$821$	34699.7	1.47506	0.737532	−	0.675312i	$-0.235991\pi$
$821$	34699.7	1.47506	0.737532	+	0.675312i	$0.235991\pi$
$822$	−13580.2	−0.576233
$823$	18181.1	0.770052	0.385026	−	0.922906i	$-0.374192\pi$
$823$	18181.1	0.770052	0.385026	+	0.922906i	$0.374192\pi$
$824$	−4244.15	−0.179432
$825$	5209.40	0.219840
$826$	2850.12	0.120059
$827$	−20195.3	−0.849165	−0.424583	−	0.905389i	$-0.639579\pi$
$827$	−20195.3	−0.849165	−0.424583	+	0.905389i	$0.639579\pi$
$828$	−3908.28	−0.164036
$829$	−30894.3	−1.29433	−0.647167	−	0.762349i	$-0.724046\pi$
$829$	−30894.3	−1.29433	−0.647167	+	0.762349i	$0.724046\pi$
$830$	−24.3397	−0.00101788
$831$	5996.67	0.250328
$832$	5855.19	0.243981
$833$	−10426.3	−0.433671
$834$	−11347.3	−0.471132
$835$	−37.0367	−0.00153498
$836$	0	0
$837$	5072.07	0.209458
$838$	23344.6	0.962324
$839$	−22423.8	−0.922714	−0.461357	−	0.887215i	$-0.652637\pi$
$839$	−22423.8	−0.922714	−0.461357	+	0.887215i	$0.652637\pi$
$840$	−0.808297	−3.32011e−5 0
$841$	−16893.0	−0.692648
$842$	16795.0	0.687404
$843$	20399.0	0.833428
$844$	7444.72	0.303623
$845$	83.0214	0.00337991
$846$	−215.621	−0.00876266
$847$	2849.78	0.115607
$848$	1433.73	0.0580595
$849$	−21744.3	−0.878990
$850$	7740.82	0.312362
$851$	15707.5	0.632723
$852$	1208.72	0.0486035
$853$	−17422.5	−0.699337	−0.349668	−	0.936874i	$-0.613706\pi$
$853$	−17422.5	−0.699337	−0.349668	+	0.936874i	$0.613706\pi$
$854$	−606.827	−0.0243152
$855$	0	0
$856$	2233.23	0.0891709
$857$	21675.3	0.863959	0.431979	−	0.901883i	$-0.357815\pi$
$857$	21675.3	0.863959	0.431979	+	0.901883i	$0.357815\pi$
$858$	7625.52	0.303416
$859$	12937.6	0.513881	0.256941	−	0.966427i	$-0.417286\pi$
$859$	12937.6	0.513881	0.256941	+	0.966427i	$0.417286\pi$
$860$	−8.35901	−0.000331442 0
$861$	2165.29	0.0857061
$862$	23183.8	0.916059
$863$	18115.6	0.714556	0.357278	−	0.933998i	$-0.383705\pi$
$863$	18115.6	0.714556	0.357278	+	0.933998i	$0.383705\pi$
$864$	864.000	0.0340207
$865$	20.8702	0.000820355 0
$866$	28375.3	1.11343
$867$	11862.8	0.464686
$868$	1881.67	0.0735806
$869$	6563.55	0.256218
$870$	−6.98657	−0.000272261 0
$871$	58348.0	2.26986
$872$	−8007.20	−0.310961
$873$	−4472.64	−0.173397
$874$	0	0
$875$	8.41975	0.000325302 0
$876$	13384.0	0.516215
$877$	−4644.60	−0.178834	−0.0894168	−	0.995994i	$-0.528500\pi$
$877$	−4644.60	−0.178834	−0.0894168	+	0.995994i	$0.528500\pi$
$878$	25107.8	0.965087
$879$	26798.3	1.02831
$880$	2.98934	0.000114512 0
$881$	20869.2	0.798072	0.399036	−	0.916935i	$-0.369345\pi$
$881$	20869.2	0.798072	0.399036	+	0.916935i	$0.369345\pi$
$882$	6061.13	0.231393
$883$	−1104.49	−0.0420941	−0.0210471	−	0.999778i	$-0.506700\pi$
$883$	−1104.49	−0.0420941	−0.0210471	+	0.999778i	$0.506700\pi$
$884$	11331.0	0.431112
$885$	−22.9610	−0.000872120 0
$886$	−21694.0	−0.822602
$887$	−16985.1	−0.642959	−0.321479	−	0.946917i	$-0.604180\pi$
$887$	−16985.1	−0.642959	−0.321479	+	0.946917i	$0.604180\pi$
$888$	−3472.45	−0.131225
$889$	−1315.84	−0.0496423
$890$	18.1999	0.000685464 0
$891$	1125.23	0.0423083
$892$	−24119.1	−0.905345
$893$	0	0
$894$	−5609.11	−0.209840
$895$	−10.8830	−0.000406457 0
$896$	320.532	0.0119511
$897$	29796.5	1.10912
$898$	−32153.9	−1.19486
$899$	16264.3	0.603388
$900$	−4499.99	−0.166666
$901$	2774.56	0.102591
$902$	−8007.94	−0.295605
$903$	−1167.29	−0.0430177
$904$	7656.20	0.281683
$905$	−0.605226	−2.22303e−5 0
$906$	4523.54	0.165877
$907$	30402.2	1.11300	0.556498	−	0.830849i	$-0.312145\pi$
$907$	30402.2	1.11300	0.556498	+	0.830849i	$0.312145\pi$
$908$	19622.4	0.717172
$909$	−6351.99	−0.231774
$910$	6.16241	0.000224486 0
$911$	15030.4	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$911$	15030.4	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$912$	0	0
$913$	12570.3	0.455657
$914$	−22666.0	−0.820267
$915$	4.88869	0.000176629 0
$916$	−12861.5	−0.463924
$917$	−7162.88	−0.257949
$918$	1672.02	0.0601142
$919$	−44831.3	−1.60919	−0.804596	−	0.593822i	$-0.797618\pi$
$919$	−44831.3	−1.60919	−0.804596	+	0.593822i	$0.797618\pi$
$920$	11.6808	0.000418591 0
$921$	6742.62	0.241234
$922$	−26732.4	−0.954864
$923$	−9215.24	−0.328628
$924$	417.446	0.0148625
$925$	18085.7	0.642868
$926$	19882.7	0.705601
$927$	4774.67	0.169170
$928$	2770.54	0.0980037
$929$	−11629.6	−0.410717	−0.205359	−	0.978687i	$-0.565836\pi$
$929$	−11629.6	−0.410717	−0.205359	+	0.978687i	$0.565836\pi$
$930$	−15.1590	−0.000534499 0
$931$	0	0
$932$	−11233.5	−0.394812
$933$	3498.44	0.122758
$934$	34203.0	1.19824
$935$	5.78500	0.000202342 0
$936$	−6587.09	−0.230027
$937$	−40181.9	−1.40095	−0.700473	−	0.713679i	$-0.747028\pi$
$937$	−40181.9	−1.40095	−0.700473	+	0.713679i	$0.747028\pi$
$938$	3194.16	0.111187
$939$	−3413.52	−0.118633
$940$	0.644433	2.23607e−5 0
$941$	−39832.6	−1.37992	−0.689960	−	0.723848i	$-0.742372\pi$
$941$	−39832.6	−1.37992	−0.689960	+	0.723848i	$0.742372\pi$
$942$	−4408.95	−0.152496
$943$	−31290.8	−1.08056
$944$	9105.25	0.313931
$945$	0.909334	3.13023e−5 0
$946$	4317.02	0.148370
$947$	41431.9	1.42171	0.710853	−	0.703341i	$-0.248309\pi$
$947$	41431.9	1.42171	0.710853	+	0.703341i	$0.248309\pi$
$948$	−5669.73	−0.194245
$949$	−102039.	−3.49033
$950$	0	0
$951$	−15408.0	−0.525382
$952$	620.296	0.0211175
$953$	20902.0	0.710473	0.355237	−	0.934776i	$-0.384400\pi$
$953$	20902.0	0.710473	0.355237	+	0.934776i	$0.384400\pi$
$954$	−1612.95	−0.0547390
$955$	65.1058	0.00220604
$956$	−11286.3	−0.381824
$957$	3608.22	0.121878
$958$	−25554.3	−0.861819
$959$	5667.82	0.190848
$960$	−2.58226	−8.68146e−5 0
$961$	5498.31	0.184563
$962$	26473.8	0.887265
$963$	−2512.39	−0.0840711
$964$	−15490.6	−0.517551
$965$	8.31442	0.000277358 0
$966$	1631.16	0.0543288
$967$	45497.3	1.51302	0.756511	−	0.653980i	$-0.226902\pi$
$967$	45497.3	1.51302	0.756511	+	0.653980i	$0.226902\pi$
$968$	9104.15	0.302292
$969$	0	0
$970$	13.3675	0.000442478 0
$971$	18145.3	0.599703	0.299851	−	0.953986i	$-0.403063\pi$
$971$	18145.3	0.599703	0.299851	+	0.953986i	$0.403063\pi$
$972$	−972.000	−0.0320750
$973$	4735.90	0.156039
$974$	−2254.54	−0.0741687
$975$	34307.7	1.12690
$976$	−1938.62	−0.0635797
$977$	3852.83	0.126165	0.0630824	−	0.998008i	$-0.479907\pi$
$977$	3852.83	0.126165	0.0630824	+	0.998008i	$0.479907\pi$
$978$	9897.60	0.323610
$979$	−9399.36	−0.306849
$980$	−18.1150	−0.000590473 0
$981$	9008.10	0.293177
$982$	39387.4	1.27994
$983$	−14016.1	−0.454775	−0.227388	−	0.973804i	$-0.573018\pi$
$983$	−14016.1	−0.454775	−0.227388	+	0.973804i	$0.573018\pi$
$984$	6917.43	0.224105
$985$	−6.01865	−0.000194691 0
$986$	5361.57	0.173172
$987$	89.9916	0.00290219
$988$	0	0
$989$	16868.6	0.542357
$990$	−3.36301	−0.000107963 0
$991$	−22760.9	−0.729591	−0.364795	−	0.931088i	$-0.618861\pi$
$991$	−22760.9	−0.729591	−0.364795	+	0.931088i	$0.618861\pi$
$992$	6011.34	0.192400
$993$	15876.0	0.507362
$994$	−504.472	−0.0160975
$995$	−55.3389	−0.00176318
$996$	−10858.5	−0.345445
$997$	44875.1	1.42548	0.712742	−	0.701426i	$-0.247453\pi$
$997$	44875.1	1.42548	0.712742	+	0.701426i	$0.247453\pi$
$998$	−12157.9	−0.385624
$999$	3906.51	0.123720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.x.1.3	✓	4
19.18	odd	2		2166.4.a.ba.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.x.1.3	✓	4	1.1	even	1	trivial
2166.4.a.ba.1.3	yes	4	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.0134493 q^{5} +6.00000 q^{6} -2.50416 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.0134493 q^{5} +6.00000 q^{6} -2.50416 q^{7} -8.00000 q^{8} +9.00000 q^{9} -0.0268985 q^{10} +13.8918 q^{11} -12.0000 q^{12} +91.4873 q^{13} +5.00831 q^{14} -0.0403478 q^{15} +16.0000 q^{16} +30.9633 q^{17} -18.0000 q^{18} +0.0537970 q^{20} +7.51247 q^{21} -27.7835 q^{22} -108.563 q^{23} +24.0000 q^{24} -125.000 q^{25} -182.975 q^{26} -27.0000 q^{27} -10.0166 q^{28} -86.5794 q^{29} +0.0806956 q^{30} -187.854 q^{31} -32.0000 q^{32} -41.6753 q^{33} -61.9266 q^{34} -0.0336790 q^{35} +36.0000 q^{36} -144.685 q^{37} -274.462 q^{39} -0.107594 q^{40} +288.226 q^{41} -15.0249 q^{42} -155.380 q^{43} +55.5670 q^{44} +0.121043 q^{45} +217.127 q^{46} +11.9790 q^{47} -48.0000 q^{48} -336.729 q^{49} +250.000 q^{50} -92.8900 q^{51} +365.949 q^{52} +89.6081 q^{53} +54.0000 q^{54} +0.186834 q^{55} +20.0332 q^{56} +173.159 q^{58} +569.078 q^{59} -0.161391 q^{60} -121.164 q^{61} +375.709 q^{62} -22.5374 q^{63} +64.0000 q^{64} +1.23044 q^{65} +83.3506 q^{66} +637.772 q^{67} +123.853 q^{68} +325.690 q^{69} +0.0673581 q^{70} -100.727 q^{71} -72.0000 q^{72} -1115.33 q^{73} +289.371 q^{74} +374.999 q^{75} -34.7871 q^{77} +548.924 q^{78} +472.478 q^{79} +0.215188 q^{80} +81.0000 q^{81} -576.453 q^{82} +904.871 q^{83} +30.0499 q^{84} +0.416434 q^{85} +310.761 q^{86} +259.738 q^{87} -111.134 q^{88} -676.614 q^{89} -0.242087 q^{90} -229.099 q^{91} -434.253 q^{92} +563.563 q^{93} -23.9579 q^{94} +96.0000 q^{96} -496.960 q^{97} +673.458 q^{98} +125.026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} - 32 q^{8} + 36 q^{9} + 56 q^{10} - 72 q^{11} - 48 q^{12} + 8 q^{13} + 34 q^{14} + 84 q^{15} + 64 q^{16} + 33 q^{17} - 72 q^{18} - 112 q^{20} + 51 q^{21} + 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} - 108 q^{27} - 68 q^{28} + 75 q^{29} - 168 q^{30} - 419 q^{31} - 128 q^{32} + 216 q^{33} - 66 q^{34} + 331 q^{35} + 144 q^{36} - 131 q^{37} - 24 q^{39} + 224 q^{40} + 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} + 176 q^{46} + 395 q^{47} - 192 q^{48} - 1033 q^{49} - 188 q^{50} - 99 q^{51} + 32 q^{52} + 625 q^{53} + 216 q^{54} + 413 q^{55} + 136 q^{56} - 150 q^{58} + 1328 q^{59} + 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} + 1817 q^{65} - 432 q^{66} + 99 q^{67} + 132 q^{68} + 264 q^{69} - 662 q^{70} + 1307 q^{71} - 288 q^{72} + 183 q^{73} + 262 q^{74} - 282 q^{75} - 513 q^{77} + 48 q^{78} + 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} + 204 q^{84} - 867 q^{85} - 152 q^{86} - 225 q^{87} + 576 q^{88} - 207 q^{89} + 504 q^{90} + 183 q^{91} - 352 q^{92} + 1257 q^{93} - 790 q^{94} + 384 q^{96} + 1331 q^{97} + 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 - 32 * q^8 + 36 * q^9 + 56 * q^10 - 72 * q^11 - 48 * q^12 + 8 * q^13 + 34 * q^14 + 84 * q^15 + 64 * q^16 + 33 * q^17 - 72 * q^18 - 112 * q^20 + 51 * q^21 + 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 - 108 * q^27 - 68 * q^28 + 75 * q^29 - 168 * q^30 - 419 * q^31 - 128 * q^32 + 216 * q^33 - 66 * q^34 + 331 * q^35 + 144 * q^36 - 131 * q^37 - 24 * q^39 + 224 * q^40 + 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 + 176 * q^46 + 395 * q^47 - 192 * q^48 - 1033 * q^49 - 188 * q^50 - 99 * q^51 + 32 * q^52 + 625 * q^53 + 216 * q^54 + 413 * q^55 + 136 * q^56 - 150 * q^58 + 1328 * q^59 + 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 + 1817 * q^65 - 432 * q^66 + 99 * q^67 + 132 * q^68 + 264 * q^69 - 662 * q^70 + 1307 * q^71 - 288 * q^72 + 183 * q^73 + 262 * q^74 - 282 * q^75 - 513 * q^77 + 48 * q^78 + 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 + 204 * q^84 - 867 * q^85 - 152 * q^86 - 225 * q^87 + 576 * q^88 - 207 * q^89 + 504 * q^90 + 183 * q^91 - 352 * q^92 + 1257 * q^93 - 790 * q^94 + 384 * q^96 + 1331 * q^97 + 2066 * q^98 - 648 * q^99

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