LMFDB - Embedded newform 2166.4.a.bf.1.1

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:	$6$
Coefficient field:	6.6.11196169353.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 87x^{4} + 179x^{3} + 2574x^{2} - 2664x - 25992 $ x^6 - 3x^5 - 87x^4 + 179x^3 + 2574x^2 - 2664*x - 25992
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.57904$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -11.4115 q^{5} -6.00000 q^{6} -8.90590 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -11.4115 q^{5} -6.00000 q^{6} -8.90590 q^{7} +8.00000 q^{8} +9.00000 q^{9} -22.8230 q^{10} +22.5981 q^{11} -12.0000 q^{12} -11.4947 q^{13} -17.8118 q^{14} +34.2345 q^{15} +16.0000 q^{16} +31.6273 q^{17} +18.0000 q^{18} -45.6460 q^{20} +26.7177 q^{21} +45.1962 q^{22} -49.6096 q^{23} -24.0000 q^{24} +5.22231 q^{25} -22.9895 q^{26} -27.0000 q^{27} -35.6236 q^{28} -23.8498 q^{29} +68.4690 q^{30} +155.008 q^{31} +32.0000 q^{32} -67.7942 q^{33} +63.2547 q^{34} +101.630 q^{35} +36.0000 q^{36} -88.9835 q^{37} +34.4842 q^{39} -91.2920 q^{40} -1.75763 q^{41} +53.4354 q^{42} +502.441 q^{43} +90.3923 q^{44} -102.703 q^{45} -99.2193 q^{46} +178.728 q^{47} -48.0000 q^{48} -263.685 q^{49} +10.4446 q^{50} -94.8820 q^{51} -45.9790 q^{52} -22.4292 q^{53} -54.0000 q^{54} -257.878 q^{55} -71.2472 q^{56} -47.6997 q^{58} +604.207 q^{59} +136.938 q^{60} -688.976 q^{61} +310.016 q^{62} -80.1531 q^{63} +64.0000 q^{64} +131.172 q^{65} -135.588 q^{66} +370.614 q^{67} +126.509 q^{68} +148.829 q^{69} +203.259 q^{70} -569.852 q^{71} +72.0000 q^{72} +686.896 q^{73} -177.967 q^{74} -15.6669 q^{75} -201.256 q^{77} +68.9684 q^{78} +610.842 q^{79} -182.584 q^{80} +81.0000 q^{81} -3.51525 q^{82} -625.214 q^{83} +106.871 q^{84} -360.915 q^{85} +1004.88 q^{86} +71.5495 q^{87} +180.785 q^{88} -2.01030 q^{89} -205.407 q^{90} +102.371 q^{91} -198.439 q^{92} -465.024 q^{93} +357.455 q^{94} -96.0000 q^{96} +178.624 q^{97} -527.370 q^{98} +203.383 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} + 27 q^{5} - 36 q^{6} - 42 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 + 27 * q^5 - 36 * q^6 - 42 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} + 27 q^{5} - 36 q^{6} - 42 q^{7} + 48 q^{8} + 54 q^{9} + 54 q^{10} - 57 q^{11} - 72 q^{12} + 21 q^{13} - 84 q^{14} - 81 q^{15} + 96 q^{16} - 45 q^{17} + 108 q^{18} + 108 q^{20} + 126 q^{21} - 114 q^{22} + 138 q^{23} - 144 q^{24} - 135 q^{25} + 42 q^{26} - 162 q^{27} - 168 q^{28} - 177 q^{29} - 162 q^{30} - 99 q^{31} + 192 q^{32} + 171 q^{33} - 90 q^{34} - 594 q^{35} + 216 q^{36} - 615 q^{37} - 63 q^{39} + 216 q^{40} - 171 q^{41} + 252 q^{42} + 456 q^{43} - 228 q^{44} + 243 q^{45} + 276 q^{46} + 228 q^{47} - 288 q^{48} + 168 q^{49} - 270 q^{50} + 135 q^{51} + 84 q^{52} - 357 q^{53} - 324 q^{54} - 1356 q^{55} - 336 q^{56} - 354 q^{58} - 741 q^{59} - 324 q^{60} - 66 q^{61} - 198 q^{62} - 378 q^{63} + 384 q^{64} + 843 q^{65} + 342 q^{66} + 966 q^{67} - 180 q^{68} - 414 q^{69} - 1188 q^{70} - 1338 q^{71} + 432 q^{72} + 1293 q^{73} - 1230 q^{74} + 405 q^{75} + 1299 q^{77} - 126 q^{78} - 1404 q^{79} + 432 q^{80} + 486 q^{81} - 342 q^{82} - 1440 q^{83} + 504 q^{84} - 1209 q^{85} + 912 q^{86} + 531 q^{87} - 456 q^{88} - 2991 q^{89} + 486 q^{90} - 4632 q^{91} + 552 q^{92} + 297 q^{93} + 456 q^{94} - 576 q^{96} - 2802 q^{97} + 336 q^{98} - 513 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 + 27 * q^5 - 36 * q^6 - 42 * q^7 + 48 * q^8 + 54 * q^9 + 54 * q^10 - 57 * q^11 - 72 * q^12 + 21 * q^13 - 84 * q^14 - 81 * q^15 + 96 * q^16 - 45 * q^17 + 108 * q^18 + 108 * q^20 + 126 * q^21 - 114 * q^22 + 138 * q^23 - 144 * q^24 - 135 * q^25 + 42 * q^26 - 162 * q^27 - 168 * q^28 - 177 * q^29 - 162 * q^30 - 99 * q^31 + 192 * q^32 + 171 * q^33 - 90 * q^34 - 594 * q^35 + 216 * q^36 - 615 * q^37 - 63 * q^39 + 216 * q^40 - 171 * q^41 + 252 * q^42 + 456 * q^43 - 228 * q^44 + 243 * q^45 + 276 * q^46 + 228 * q^47 - 288 * q^48 + 168 * q^49 - 270 * q^50 + 135 * q^51 + 84 * q^52 - 357 * q^53 - 324 * q^54 - 1356 * q^55 - 336 * q^56 - 354 * q^58 - 741 * q^59 - 324 * q^60 - 66 * q^61 - 198 * q^62 - 378 * q^63 + 384 * q^64 + 843 * q^65 + 342 * q^66 + 966 * q^67 - 180 * q^68 - 414 * q^69 - 1188 * q^70 - 1338 * q^71 + 432 * q^72 + 1293 * q^73 - 1230 * q^74 + 405 * q^75 + 1299 * q^77 - 126 * q^78 - 1404 * q^79 + 432 * q^80 + 486 * q^81 - 342 * q^82 - 1440 * q^83 + 504 * q^84 - 1209 * q^85 + 912 * q^86 + 531 * q^87 - 456 * q^88 - 2991 * q^89 + 486 * q^90 - 4632 * q^91 + 552 * q^92 + 297 * q^93 + 456 * q^94 - 576 * q^96 - 2802 * q^97 + 336 * q^98 - 513 * 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
$5$	−11.4115	−1.02068	−0.510338	−	0.859974i	$-0.670480\pi$
$5$	−11.4115	−1.02068	−0.510338	+	0.859974i	$0.670480\pi$
$6$	−6.00000	−0.408248
$7$	−8.90590	−0.480873	−0.240437	−	0.970665i	$-0.577291\pi$
$7$	−8.90590	−0.480873	−0.240437	+	0.970665i	$0.577291\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−22.8230	−0.721727
$11$	22.5981	0.619416	0.309708	−	0.950832i	$-0.399769\pi$
$11$	22.5981	0.619416	0.309708	+	0.950832i	$0.399769\pi$
$12$	−12.0000	−0.288675
$13$	−11.4947	−0.245236	−0.122618	−	0.992454i	$-0.539129\pi$
$13$	−11.4947	−0.245236	−0.122618	+	0.992454i	$0.539129\pi$
$14$	−17.8118	−0.340029
$15$	34.2345	0.589287
$16$	16.0000	0.250000
$17$	31.6273	0.451221	0.225610	−	0.974218i	$-0.427562\pi$
$17$	31.6273	0.451221	0.225610	+	0.974218i	$0.427562\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−45.6460	−0.510338
$21$	26.7177	0.277632
$22$	45.1962	0.437993
$23$	−49.6096	−0.449753	−0.224877	−	0.974387i	$-0.572198\pi$
$23$	−49.6096	−0.449753	−0.224877	+	0.974387i	$0.572198\pi$
$24$	−24.0000	−0.204124
$25$	5.22231	0.0417784
$26$	−22.9895	−0.173408
$27$	−27.0000	−0.192450
$28$	−35.6236	−0.240437
$29$	−23.8498	−0.152717	−0.0763586	−	0.997080i	$-0.524329\pi$
$29$	−23.8498	−0.152717	−0.0763586	+	0.997080i	$0.524329\pi$
$30$	68.4690	0.416689
$31$	155.008	0.898073	0.449037	−	0.893513i	$-0.351767\pi$
$31$	155.008	0.898073	0.449037	+	0.893513i	$0.351767\pi$
$32$	32.0000	0.176777
$33$	−67.7942	−0.357620
$34$	63.2547	0.319061
$35$	101.630	0.490816
$36$	36.0000	0.166667
$37$	−88.9835	−0.395373	−0.197686	−	0.980265i	$-0.563343\pi$
$37$	−88.9835	−0.395373	−0.197686	+	0.980265i	$0.563343\pi$
$38$	0	0
$39$	34.4842	0.141587
$40$	−91.2920	−0.360863
$41$	−1.75763	−0.00669500	−0.00334750	−	0.999994i	$-0.501066\pi$
$41$	−1.75763	−0.00669500	−0.00334750	+	0.999994i	$0.501066\pi$
$42$	53.4354	0.196316
$43$	502.441	1.78190	0.890949	−	0.454104i	$-0.150040\pi$
$43$	502.441	1.78190	0.890949	+	0.454104i	$0.150040\pi$
$44$	90.3923	0.309708
$45$	−102.703	−0.340225
$46$	−99.2193	−0.318024
$47$	178.728	0.554683	0.277341	−	0.960771i	$-0.410547\pi$
$47$	178.728	0.554683	0.277341	+	0.960771i	$0.410547\pi$
$48$	−48.0000	−0.144338
$49$	−263.685	−0.768761
$50$	10.4446	0.0295418
$51$	−94.8820	−0.260512
$52$	−45.9790	−0.122618
$53$	−22.4292	−0.0581299	−0.0290650	−	0.999578i	$-0.509253\pi$
$53$	−22.4292	−0.0581299	−0.0290650	+	0.999578i	$0.509253\pi$
$54$	−54.0000	−0.136083
$55$	−257.878	−0.632223
$56$	−71.2472	−0.170014
$57$	0	0
$58$	−47.6997	−0.107987
$59$	604.207	1.33324	0.666619	−	0.745399i	$-0.267741\pi$
$59$	604.207	1.33324	0.666619	+	0.745399i	$0.267741\pi$
$60$	136.938	0.294644
$61$	−688.976	−1.44614	−0.723069	−	0.690776i	$-0.757269\pi$
$61$	−688.976	−1.44614	−0.723069	+	0.690776i	$0.757269\pi$
$62$	310.016	0.635034
$63$	−80.1531	−0.160291
$64$	64.0000	0.125000
$65$	131.172	0.250306
$66$	−135.588	−0.252876
$67$	370.614	0.675786	0.337893	−	0.941184i	$-0.390286\pi$
$67$	370.614	0.675786	0.337893	+	0.941184i	$0.390286\pi$
$68$	126.509	0.225610
$69$	148.829	0.259665
$70$	203.259	0.347059
$71$	−569.852	−0.952522	−0.476261	−	0.879304i	$-0.658008\pi$
$71$	−569.852	−0.952522	−0.476261	+	0.879304i	$0.658008\pi$
$72$	72.0000	0.117851
$73$	686.896	1.10130	0.550652	−	0.834735i	$-0.314379\pi$
$73$	686.896	1.10130	0.550652	+	0.834735i	$0.314379\pi$
$74$	−177.967	−0.279571
$75$	−15.6669	−0.0241208
$76$	0	0
$77$	−201.256	−0.297861
$78$	68.9684	0.100117
$79$	610.842	0.869937	0.434969	−	0.900446i	$-0.356759\pi$
$79$	610.842	0.869937	0.434969	+	0.900446i	$0.356759\pi$
$80$	−182.584	−0.255169
$81$	81.0000	0.111111
$82$	−3.51525	−0.00473408
$83$	−625.214	−0.826821	−0.413410	−	0.910545i	$-0.635663\pi$
$83$	−625.214	−0.826821	−0.413410	+	0.910545i	$0.635663\pi$
$84$	106.871	0.138816
$85$	−360.915	−0.460550
$86$	1004.88	1.25999
$87$	71.5495	0.0881714
$88$	180.785	0.218997
$89$	−2.01030	−0.00239428	−0.00119714	−	0.999999i	$-0.500381\pi$
$89$	−2.01030	−0.00239428	−0.00119714	+	0.999999i	$0.500381\pi$
$90$	−205.407	−0.240576
$91$	102.371	0.117927
$92$	−198.439	−0.224877
$93$	−465.024	−0.518503
$94$	357.455	0.392220
$95$	0	0
$96$	−96.0000	−0.102062
$97$	178.624	0.186974	0.0934870	−	0.995621i	$-0.470199\pi$
$97$	178.624	0.186974	0.0934870	+	0.995621i	$0.470199\pi$
$98$	−527.370	−0.543596
$99$	203.383	0.206472
$100$	20.8892	0.0208892
$101$	209.428	0.206325	0.103163	−	0.994665i	$-0.467104\pi$
$101$	209.428	0.206325	0.103163	+	0.994665i	$0.467104\pi$
$102$	−189.764	−0.184210
$103$	335.696	0.321137	0.160569	−	0.987025i	$-0.448667\pi$
$103$	335.696	0.321137	0.160569	+	0.987025i	$0.448667\pi$
$104$	−91.9579	−0.0867040
$105$	−304.889	−0.283373
$106$	−44.8584	−0.0411041
$107$	−1600.65	−1.44617	−0.723086	−	0.690758i	$-0.757277\pi$
$107$	−1600.65	−1.44617	−0.723086	+	0.690758i	$0.757277\pi$
$108$	−108.000	−0.0962250
$109$	−796.409	−0.699836	−0.349918	−	0.936780i	$-0.613791\pi$
$109$	−796.409	−0.699836	−0.349918	+	0.936780i	$0.613791\pi$
$110$	−515.756	−0.447049
$111$	266.950	0.228269
$112$	−142.494	−0.120218
$113$	−1222.16	−1.01744	−0.508722	−	0.860931i	$-0.669882\pi$
$113$	−1222.16	−1.01744	−0.508722	+	0.860931i	$0.669882\pi$
$114$	0	0
$115$	566.120	0.459052
$116$	−95.3993	−0.0763586
$117$	−103.453	−0.0817453
$118$	1208.41	0.942741
$119$	−281.670	−0.216980
$120$	273.876	0.208345
$121$	−820.327	−0.616324
$122$	−1377.95	−1.02257
$123$	5.27288	0.00386536
$124$	620.032	0.449037
$125$	1366.84	0.978033
$126$	−160.306	−0.113343
$127$	−1460.41	−1.02040	−0.510199	−	0.860056i	$-0.670428\pi$
$127$	−1460.41	−1.02040	−0.510199	+	0.860056i	$0.670428\pi$
$128$	128.000	0.0883883
$129$	−1507.32	−1.02878
$130$	262.344	0.176993
$131$	−1838.66	−1.22629	−0.613146	−	0.789970i	$-0.710096\pi$
$131$	−1838.66	−1.22629	−0.613146	+	0.789970i	$0.710096\pi$
$132$	−271.177	−0.178810
$133$	0	0
$134$	741.228	0.477853
$135$	308.110	0.196429
$136$	253.019	0.159531
$137$	−2147.39	−1.33915	−0.669576	−	0.742743i	$-0.733524\pi$
$137$	−2147.39	−1.33915	−0.669576	+	0.742743i	$0.733524\pi$
$138$	297.658	0.183611
$139$	−2942.22	−1.79537	−0.897684	−	0.440640i	$-0.854752\pi$
$139$	−2942.22	−1.79537	−0.897684	+	0.440640i	$0.854752\pi$
$140$	406.519	0.245408
$141$	−536.183	−0.320246
$142$	−1139.70	−0.673534
$143$	−259.759	−0.151903
$144$	144.000	0.0833333
$145$	272.162	0.155875
$146$	1373.79	0.778739
$147$	791.055	0.443844
$148$	−355.934	−0.197686
$149$	2973.88	1.63510	0.817550	−	0.575857i	$-0.195332\pi$
$149$	2973.88	1.63510	0.817550	+	0.575857i	$0.195332\pi$
$150$	−31.3338	−0.0170560
$151$	−718.878	−0.387427	−0.193713	−	0.981058i	$-0.562053\pi$
$151$	−718.878	−0.387427	−0.193713	+	0.981058i	$0.562053\pi$
$152$	0	0
$153$	284.646	0.150407
$154$	−402.512	−0.210619
$155$	−1768.87	−0.916641
$156$	137.937	0.0707935
$157$	1658.74	0.843199	0.421599	−	0.906782i	$-0.361469\pi$
$157$	1658.74	0.843199	0.421599	+	0.906782i	$0.361469\pi$
$158$	1221.68	0.615139
$159$	67.2875	0.0335613
$160$	−365.168	−0.180432
$161$	441.818	0.216274
$162$	162.000	0.0785674
$163$	−942.731	−0.453008	−0.226504	−	0.974010i	$-0.572730\pi$
$163$	−942.731	−0.453008	−0.226504	+	0.974010i	$0.572730\pi$
$164$	−7.03050	−0.00334750
$165$	773.634	0.365014
$166$	−1250.43	−0.584651
$167$	−1960.33	−0.908353	−0.454176	−	0.890912i	$-0.650066\pi$
$167$	−1960.33	−0.908353	−0.454176	+	0.890912i	$0.650066\pi$
$168$	213.742	0.0981579
$169$	−2064.87	−0.939859
$170$	−721.830	−0.325658
$171$	0	0
$172$	2009.77	0.890949
$173$	−3342.82	−1.46907	−0.734537	−	0.678568i	$-0.762601\pi$
$173$	−3342.82	−1.46907	−0.734537	+	0.678568i	$0.762601\pi$
$174$	143.099	0.0623466
$175$	−46.5093	−0.0200901
$176$	361.569	0.154854
$177$	−1812.62	−0.769745
$178$	−4.02060	−0.00169302
$179$	−2848.38	−1.18937	−0.594686	−	0.803958i	$-0.702724\pi$
$179$	−2848.38	−1.18937	−0.594686	+	0.803958i	$0.702724\pi$
$180$	−410.814	−0.170113
$181$	−2373.50	−0.974699	−0.487350	−	0.873207i	$-0.662036\pi$
$181$	−2373.50	−0.974699	−0.487350	+	0.873207i	$0.662036\pi$
$182$	204.742	0.0833873
$183$	2066.93	0.834928
$184$	−396.877	−0.159012
$185$	1015.43	0.403547
$186$	−930.048	−0.366637
$187$	714.717	0.279493
$188$	714.910	0.277341
$189$	240.459	0.0925441
$190$	0	0
$191$	4815.13	1.82414	0.912070	−	0.410034i	$-0.134483\pi$
$191$	4815.13	1.82414	0.912070	+	0.410034i	$0.134483\pi$
$192$	−192.000	−0.0721688
$193$	−1984.38	−0.740097	−0.370049	−	0.929012i	$-0.620659\pi$
$193$	−1984.38	−0.740097	−0.370049	+	0.929012i	$0.620659\pi$
$194$	357.247	0.132211
$195$	−393.517	−0.144514
$196$	−1054.74	−0.384380
$197$	421.647	0.152493	0.0762464	−	0.997089i	$-0.475706\pi$
$197$	421.647	0.152493	0.0762464	+	0.997089i	$0.475706\pi$
$198$	406.765	0.145998
$199$	−1722.81	−0.613703	−0.306852	−	0.951757i	$-0.599276\pi$
$199$	−1722.81	−0.613703	−0.306852	+	0.951757i	$0.599276\pi$
$200$	41.7784	0.0147709
$201$	−1111.84	−0.390165
$202$	418.856	0.145894
$203$	212.404	0.0734377
$204$	−379.528	−0.130256
$205$	20.0571	0.00683342
$206$	671.393	0.227078
$207$	−446.487	−0.149918
$208$	−183.916	−0.0613090
$209$	0	0
$210$	−609.778	−0.200375
$211$	−906.017	−0.295606	−0.147803	−	0.989017i	$-0.547220\pi$
$211$	−906.017	−0.295606	−0.147803	+	0.989017i	$0.547220\pi$
$212$	−89.7167	−0.0290650
$213$	1709.56	0.549939
$214$	−3201.30	−1.02260
$215$	−5733.61	−1.81874
$216$	−216.000	−0.0680414
$217$	−1380.49	−0.431859
$218$	−1592.82	−0.494859
$219$	−2060.69	−0.635838
$220$	−1031.51	−0.316111
$221$	−363.548	−0.110656
$222$	533.901	0.161410
$223$	−5229.06	−1.57024	−0.785120	−	0.619344i	$-0.787399\pi$
$223$	−5229.06	−1.57024	−0.785120	+	0.619344i	$0.787399\pi$
$224$	−284.989	−0.0850072
$225$	47.0007	0.0139261
$226$	−2444.32	−0.719442
$227$	−44.6697	−0.0130609	−0.00653047	−	0.999979i	$-0.502079\pi$
$227$	−44.6697	−0.0130609	−0.00653047	+	0.999979i	$0.502079\pi$
$228$	0	0
$229$	3847.75	1.11033	0.555167	−	0.831739i	$-0.312654\pi$
$229$	3847.75	1.11033	0.555167	+	0.831739i	$0.312654\pi$
$230$	1132.24	0.324599
$231$	603.769	0.171970
$232$	−190.799	−0.0539937
$233$	−1134.64	−0.319024	−0.159512	−	0.987196i	$-0.550992\pi$
$233$	−1134.64	−0.319024	−0.159512	+	0.987196i	$0.550992\pi$
$234$	−206.905	−0.0578027
$235$	−2039.55	−0.566151
$236$	2416.83	0.666619
$237$	−1832.53	−0.502259
$238$	−563.340	−0.153428
$239$	4302.04	1.16433	0.582167	−	0.813070i	$-0.302205\pi$
$239$	4302.04	1.16433	0.582167	+	0.813070i	$0.302205\pi$
$240$	547.752	0.147322
$241$	3633.24	0.971110	0.485555	−	0.874206i	$-0.338618\pi$
$241$	3633.24	0.971110	0.485555	+	0.874206i	$0.338618\pi$
$242$	−1640.65	−0.435807
$243$	−243.000	−0.0641500
$244$	−2755.91	−0.723069
$245$	3009.04	0.784655
$246$	10.5458	0.00273322
$247$	0	0
$248$	1240.06	0.317517
$249$	1875.64	0.477365
$250$	2733.69	0.691574
$251$	−2787.48	−0.700973	−0.350487	−	0.936568i	$-0.613984\pi$
$251$	−2787.48	−0.700973	−0.350487	+	0.936568i	$0.613984\pi$
$252$	−320.612	−0.0801456
$253$	−1121.08	−0.278584
$254$	−2920.82	−0.721531
$255$	1082.75	0.265899
$256$	256.000	0.0625000
$257$	−3709.83	−0.900440	−0.450220	−	0.892918i	$-0.648654\pi$
$257$	−3709.83	−0.900440	−0.450220	+	0.892918i	$0.648654\pi$
$258$	−3014.65	−0.727457
$259$	792.478	0.190124
$260$	524.689	0.125153
$261$	−214.648	−0.0509058
$262$	−3677.31	−0.867119
$263$	1472.97	0.345350	0.172675	−	0.984979i	$-0.444759\pi$
$263$	1472.97	0.345350	0.172675	+	0.984979i	$0.444759\pi$
$264$	−542.354	−0.126438
$265$	255.951	0.0593318
$266$	0	0
$267$	6.03090	0.00138234
$268$	1482.46	0.337893
$269$	−2136.66	−0.484292	−0.242146	−	0.970240i	$-0.577851\pi$
$269$	−2136.66	−0.484292	−0.242146	+	0.970240i	$0.577851\pi$
$270$	616.221	0.138896
$271$	4135.19	0.926919	0.463460	−	0.886118i	$-0.346608\pi$
$271$	4135.19	0.926919	0.463460	+	0.886118i	$0.346608\pi$
$272$	506.037	0.112805
$273$	−307.113	−0.0680854
$274$	−4294.78	−0.946924
$275$	118.014	0.0258782
$276$	595.316	0.129833
$277$	−3980.29	−0.863366	−0.431683	−	0.902025i	$-0.642080\pi$
$277$	−3980.29	−0.863366	−0.431683	+	0.902025i	$0.642080\pi$
$278$	−5884.45	−1.26952
$279$	1395.07	0.299358
$280$	813.037	0.173530
$281$	−5639.72	−1.19729	−0.598643	−	0.801016i	$-0.704293\pi$
$281$	−5639.72	−1.19729	−0.598643	+	0.801016i	$0.704293\pi$
$282$	−1072.37	−0.226448
$283$	−6784.86	−1.42515	−0.712576	−	0.701594i	$-0.752472\pi$
$283$	−6784.86	−1.42515	−0.712576	+	0.701594i	$0.752472\pi$
$284$	−2279.41	−0.476261
$285$	0	0
$286$	−519.518	−0.107412
$287$	15.6532	0.00321945
$288$	288.000	0.0589256
$289$	−3912.71	−0.796400
$290$	544.325	0.110220
$291$	−535.871	−0.107949
$292$	2747.59	0.550652
$293$	2129.26	0.424549	0.212275	−	0.977210i	$-0.431913\pi$
$293$	2129.26	0.424549	0.212275	+	0.977210i	$0.431913\pi$
$294$	1582.11	0.313845
$295$	−6894.90	−1.36080
$296$	−711.868	−0.139785
$297$	−610.148	−0.119207
$298$	5947.77	1.15619
$299$	570.250	0.110296
$300$	−62.6677	−0.0120604
$301$	−4474.69	−0.856867
$302$	−1437.76	−0.273952
$303$	−628.283	−0.119122
$304$	0	0
$305$	7862.25	1.47604
$306$	569.292	0.106354
$307$	5402.40	1.00434	0.502168	−	0.864770i	$-0.332536\pi$
$307$	5402.40	1.00434	0.502168	+	0.864770i	$0.332536\pi$
$308$	−805.025	−0.148930
$309$	−1007.09	−0.185409
$310$	−3537.75	−0.648163
$311$	5553.87	1.01264	0.506320	−	0.862346i	$-0.331006\pi$
$311$	5553.87	1.01264	0.506320	+	0.862346i	$0.331006\pi$
$312$	275.874	0.0500586
$313$	−7915.75	−1.42947	−0.714735	−	0.699395i	$-0.753453\pi$
$313$	−7915.75	−1.42947	−0.714735	+	0.699395i	$0.753453\pi$
$314$	3317.49	0.596232
$315$	914.667	0.163605
$316$	2443.37	0.434969
$317$	9137.51	1.61897	0.809486	−	0.587140i	$-0.199746\pi$
$317$	9137.51	1.61897	0.809486	+	0.587140i	$0.199746\pi$
$318$	134.575	0.0237314
$319$	−538.960	−0.0945956
$320$	−730.336	−0.127584
$321$	4801.94	0.834948
$322$	883.637	0.152929
$323$	0	0
$324$	324.000	0.0555556
$325$	−60.0290	−0.0102456
$326$	−1885.46	−0.320325
$327$	2389.23	0.404050
$328$	−14.0610	−0.00236704
$329$	−1591.73	−0.266732
$330$	1547.27	0.258104
$331$	−6052.53	−1.00507	−0.502534	−	0.864558i	$-0.667599\pi$
$331$	−6052.53	−1.00507	−0.502534	+	0.864558i	$0.667599\pi$
$332$	−2500.86	−0.413410
$333$	−800.851	−0.131791
$334$	−3920.66	−0.642302
$335$	−4229.26	−0.689759
$336$	427.483	0.0694081
$337$	−4682.35	−0.756866	−0.378433	−	0.925629i	$-0.623537\pi$
$337$	−4682.35	−0.756866	−0.378433	+	0.925629i	$0.623537\pi$
$338$	−4129.74	−0.664581
$339$	3666.48	0.587422
$340$	−1443.66	−0.230275
$341$	3502.88	0.556281
$342$	0	0
$343$	5403.08	0.850550
$344$	4019.53	0.629996
$345$	−1698.36	−0.265034
$346$	−6685.64	−1.03879
$347$	10835.8	1.67635	0.838177	−	0.545398i	$-0.183621\pi$
$347$	10835.8	1.67635	0.838177	+	0.545398i	$0.183621\pi$
$348$	286.198	0.0440857
$349$	−2515.07	−0.385755	−0.192878	−	0.981223i	$-0.561782\pi$
$349$	−2515.07	−0.385755	−0.192878	+	0.981223i	$0.561782\pi$
$350$	−93.0186	−0.0142059
$351$	310.358	0.0471957
$352$	723.139	0.109498
$353$	6382.88	0.962398	0.481199	−	0.876611i	$-0.340201\pi$
$353$	6382.88	0.962398	0.481199	+	0.876611i	$0.340201\pi$
$354$	−3625.24	−0.544292
$355$	6502.87	0.972215
$356$	−8.04120	−0.00119714
$357$	845.009	0.125273
$358$	−5696.76	−0.841014
$359$	9122.32	1.34111	0.670554	−	0.741861i	$-0.266056\pi$
$359$	9122.32	1.34111	0.670554	+	0.741861i	$0.266056\pi$
$360$	−821.628	−0.120288
$361$	0	0
$362$	−4746.99	−0.689216
$363$	2460.98	0.355835
$364$	409.484	0.0589637
$365$	−7838.52	−1.12407
$366$	4133.86	0.590383
$367$	5891.16	0.837918	0.418959	−	0.908005i	$-0.362395\pi$
$367$	5891.16	0.837918	0.418959	+	0.908005i	$0.362395\pi$
$368$	−793.754	−0.112438
$369$	−15.8186	−0.00223167
$370$	2030.87	0.285351
$371$	199.752	0.0279531
$372$	−1860.10	−0.259251
$373$	−8439.77	−1.17157	−0.585784	−	0.810468i	$-0.699213\pi$
$373$	−8439.77	−1.17157	−0.585784	+	0.810468i	$0.699213\pi$
$374$	1429.43	0.197632
$375$	−4100.53	−0.564668
$376$	1429.82	0.196110
$377$	274.147	0.0374518
$378$	480.919	0.0654386
$379$	−3600.45	−0.487976	−0.243988	−	0.969778i	$-0.578456\pi$
$379$	−3600.45	−0.487976	−0.243988	+	0.969778i	$0.578456\pi$
$380$	0	0
$381$	4381.24	0.589127
$382$	9630.27	1.28986
$383$	12097.7	1.61400	0.807001	−	0.590550i	$-0.201089\pi$
$383$	12097.7	1.61400	0.807001	+	0.590550i	$0.201089\pi$
$384$	−384.000	−0.0510310
$385$	2296.64	0.304019
$386$	−3968.76	−0.523328
$387$	4521.97	0.593966
$388$	714.494	0.0934870
$389$	−8898.22	−1.15979	−0.579894	−	0.814692i	$-0.696906\pi$
$389$	−8898.22	−1.15979	−0.579894	+	0.814692i	$0.696906\pi$
$390$	−787.033	−0.102187
$391$	−1569.02	−0.202938
$392$	−2109.48	−0.271798
$393$	5515.97	0.707999
$394$	843.293	0.107829
$395$	−6970.62	−0.887924
$396$	813.531	0.103236
$397$	6367.09	0.804924	0.402462	−	0.915437i	$-0.368155\pi$
$397$	6367.09	0.804924	0.402462	+	0.915437i	$0.368155\pi$
$398$	−3445.63	−0.433954
$399$	0	0
$400$	83.5569	0.0104446
$401$	−9439.78	−1.17556	−0.587781	−	0.809020i	$-0.699998\pi$
$401$	−9439.78	−1.17556	−0.587781	+	0.809020i	$0.699998\pi$
$402$	−2223.68	−0.275889
$403$	−1781.78	−0.220240
$404$	837.711	0.103163
$405$	−924.331	−0.113408
$406$	424.808	0.0519283
$407$	−2010.86	−0.244900
$408$	−759.056	−0.0921051
$409$	−6532.56	−0.789766	−0.394883	−	0.918731i	$-0.629215\pi$
$409$	−6532.56	−0.789766	−0.394883	+	0.918731i	$0.629215\pi$
$410$	40.1143	0.00483196
$411$	6442.17	0.773160
$412$	1342.79	0.160569
$413$	−5381.00	−0.641118
$414$	−892.973	−0.106008
$415$	7134.63	0.843916
$416$	−367.832	−0.0433520
$417$	8826.67	1.03656
$418$	0	0
$419$	1449.51	0.169005	0.0845026	−	0.996423i	$-0.473070\pi$
$419$	1449.51	0.169005	0.0845026	+	0.996423i	$0.473070\pi$
$420$	−1219.56	−0.141686
$421$	−3286.42	−0.380452	−0.190226	−	0.981740i	$-0.560922\pi$
$421$	−3286.42	−0.380452	−0.190226	+	0.981740i	$0.560922\pi$
$422$	−1812.03	−0.209025
$423$	1608.55	0.184894
$424$	−179.433	−0.0205520
$425$	165.168	0.0188513
$426$	3419.11	0.388865
$427$	6135.95	0.695409
$428$	−6402.59	−0.723086
$429$	779.277	0.0877013
$430$	−11467.2	−1.28604
$431$	−12362.6	−1.38164	−0.690820	−	0.723027i	$-0.742750\pi$
$431$	−12362.6	−1.38164	−0.690820	+	0.723027i	$0.742750\pi$
$432$	−432.000	−0.0481125
$433$	15249.7	1.69250	0.846252	−	0.532784i	$-0.178854\pi$
$433$	15249.7	1.69250	0.846252	+	0.532784i	$0.178854\pi$
$434$	−2760.97	−0.305371
$435$	−816.487	−0.0899944
$436$	−3185.63	−0.349918
$437$	0	0
$438$	−4121.38	−0.449605
$439$	−10069.7	−1.09476	−0.547381	−	0.836883i	$-0.684375\pi$
$439$	−10069.7	−1.09476	−0.547381	+	0.836883i	$0.684375\pi$
$440$	−2063.02	−0.223525
$441$	−2373.16	−0.256254
$442$	−727.096	−0.0782453
$443$	−3914.56	−0.419834	−0.209917	−	0.977719i	$-0.567319\pi$
$443$	−3914.56	−0.419834	−0.209917	+	0.977719i	$0.567319\pi$
$444$	1067.80	0.114134
$445$	22.9405	0.00244379
$446$	−10458.1	−1.11033
$447$	−8921.65	−0.944026
$448$	−569.978	−0.0601092
$449$	−7976.98	−0.838434	−0.419217	−	0.907886i	$-0.637695\pi$
$449$	−7976.98	−0.838434	−0.419217	+	0.907886i	$0.637695\pi$
$450$	94.0015	0.00984727
$451$	−39.7190	−0.00414699
$452$	−4888.64	−0.508722
$453$	2156.63	0.223681
$454$	−89.3394	−0.00923548
$455$	−1168.21	−0.120366
$456$	0	0
$457$	14557.8	1.49012	0.745061	−	0.666996i	$-0.232420\pi$
$457$	14557.8	1.49012	0.745061	+	0.666996i	$0.232420\pi$
$458$	7695.50	0.785125
$459$	−853.938	−0.0868375
$460$	2264.48	0.229526
$461$	14712.2	1.48637	0.743185	−	0.669086i	$-0.233314\pi$
$461$	14712.2	1.48637	0.743185	+	0.669086i	$0.233314\pi$
$462$	1207.54	0.121601
$463$	−18580.1	−1.86499	−0.932494	−	0.361186i	$-0.882372\pi$
$463$	−18580.1	−1.86499	−0.932494	+	0.361186i	$0.882372\pi$
$464$	−381.597	−0.0381793
$465$	5306.62	0.529223
$466$	−2269.28	−0.225584
$467$	2580.33	0.255681	0.127841	−	0.991795i	$-0.459195\pi$
$467$	2580.33	0.255681	0.127841	+	0.991795i	$0.459195\pi$
$468$	−413.811	−0.0408727
$469$	−3300.65	−0.324968
$470$	−4079.10	−0.400329
$471$	−4976.23	−0.486821
$472$	4833.65	0.471371
$473$	11354.2	1.10374
$474$	−3665.05	−0.355150
$475$	0	0
$476$	−1126.68	−0.108490
$477$	−201.863	−0.0193766
$478$	8604.07	0.823308
$479$	6293.95	0.600371	0.300186	−	0.953881i	$-0.402951\pi$
$479$	6293.95	0.600371	0.300186	+	0.953881i	$0.402951\pi$
$480$	1095.50	0.104172
$481$	1022.84	0.0969596
$482$	7266.48	0.686678
$483$	−1325.46	−0.124866
$484$	−3281.31	−0.308162
$485$	−2038.36	−0.190840
$486$	−486.000	−0.0453609
$487$	−13084.2	−1.21746	−0.608730	−	0.793378i	$-0.708321\pi$
$487$	−13084.2	−1.21746	−0.608730	+	0.793378i	$0.708321\pi$
$488$	−5511.81	−0.511287
$489$	2828.19	0.261544
$490$	6018.08	0.554835
$491$	−9209.00	−0.846429	−0.423214	−	0.906030i	$-0.639098\pi$
$491$	−9209.00	−0.846429	−0.423214	+	0.906030i	$0.639098\pi$
$492$	21.0915	0.00193268
$493$	−754.306	−0.0689092
$494$	0	0
$495$	−2320.90	−0.210741
$496$	2480.13	0.224518
$497$	5075.05	0.458042
$498$	3751.28	0.337548
$499$	16314.8	1.46363	0.731814	−	0.681504i	$-0.238674\pi$
$499$	16314.8	1.46363	0.731814	+	0.681504i	$0.238674\pi$
$500$	5467.37	0.489017
$501$	5880.99	0.524438
$502$	−5574.96	−0.495663
$503$	1752.64	0.155361	0.0776804	−	0.996978i	$-0.475249\pi$
$503$	1752.64	0.155361	0.0776804	+	0.996978i	$0.475249\pi$
$504$	−641.225	−0.0566715
$505$	−2389.88	−0.210591
$506$	−2242.17	−0.196989
$507$	6194.61	0.542628
$508$	−5841.65	−0.510199
$509$	−13310.5	−1.15909	−0.579546	−	0.814939i	$-0.696770\pi$
$509$	−13310.5	−1.15909	−0.579546	+	0.814939i	$0.696770\pi$
$510$	2165.49	0.188019
$511$	−6117.43	−0.529587
$512$	512.000	0.0441942
$513$	0	0
$514$	−7419.67	−0.636707
$515$	−3830.80	−0.327777
$516$	−6029.30	−0.514390
$517$	4038.90	0.343579
$518$	1584.96	0.134438
$519$	10028.5	0.848171
$520$	1049.38	0.0884966
$521$	−8874.93	−0.746291	−0.373146	−	0.927773i	$-0.621721\pi$
$521$	−8874.93	−0.746291	−0.373146	+	0.927773i	$0.621721\pi$
$522$	−429.297	−0.0359958
$523$	−11948.7	−0.999009	−0.499504	−	0.866311i	$-0.666485\pi$
$523$	−11948.7	−0.999009	−0.499504	+	0.866311i	$0.666485\pi$
$524$	−7354.62	−0.613146
$525$	139.528	0.0115990
$526$	2945.93	0.244199
$527$	4902.49	0.405229
$528$	−1084.71	−0.0894050
$529$	−9705.88	−0.797722
$530$	511.901	0.0419539
$531$	5437.86	0.444412
$532$	0	0
$533$	20.2034	0.00164185
$534$	12.0618	0.000977463 0
$535$	18265.8	1.47607
$536$	2964.91	0.238927
$537$	8545.13	0.686685
$538$	−4273.32	−0.342446
$539$	−5958.77	−0.476183
$540$	1232.44	0.0982145
$541$	13510.3	1.07367	0.536834	−	0.843688i	$-0.319620\pi$
$541$	13510.3	1.07367	0.536834	+	0.843688i	$0.319620\pi$
$542$	8270.39	0.655431
$543$	7120.49	0.562743
$544$	1012.07	0.0797653
$545$	9088.22	0.714305
$546$	−614.226	−0.0481437
$547$	−4364.02	−0.341119	−0.170559	−	0.985347i	$-0.554557\pi$
$547$	−4364.02	−0.341119	−0.170559	+	0.985347i	$0.554557\pi$
$548$	−8589.56	−0.669576
$549$	−6200.79	−0.482046
$550$	236.028	0.0182987
$551$	0	0
$552$	1190.63	0.0918055
$553$	−5440.09	−0.418330
$554$	−7960.58	−0.610492
$555$	−3046.30	−0.232988
$556$	−11768.9	−0.897684
$557$	−7108.39	−0.540740	−0.270370	−	0.962756i	$-0.587146\pi$
$557$	−7108.39	−0.540740	−0.270370	+	0.962756i	$0.587146\pi$
$558$	2790.14	0.211678
$559$	−5775.43	−0.436985
$560$	1626.07	0.122704
$561$	−2144.15	−0.161366
$562$	−11279.4	−0.846609
$563$	16491.3	1.23450	0.617250	−	0.786767i	$-0.288247\pi$
$563$	16491.3	1.23450	0.617250	+	0.786767i	$0.288247\pi$
$564$	−2144.73	−0.160123
$565$	13946.7	1.03848
$566$	−13569.7	−1.00774
$567$	−721.378	−0.0534304
$568$	−4558.82	−0.336767
$569$	−9189.59	−0.677061	−0.338531	−	0.940955i	$-0.609930\pi$
$569$	−9189.59	−0.677061	−0.338531	+	0.940955i	$0.609930\pi$
$570$	0	0
$571$	13551.8	0.993213	0.496606	−	0.867976i	$-0.334579\pi$
$571$	13551.8	0.993213	0.496606	+	0.867976i	$0.334579\pi$
$572$	−1039.04	−0.0759515
$573$	−14445.4	−1.05317
$574$	31.3065	0.00227649
$575$	−259.077	−0.0187900
$576$	576.000	0.0416667
$577$	−15075.1	−1.08767	−0.543836	−	0.839192i	$-0.683029\pi$
$577$	−15075.1	−1.08767	−0.543836	+	0.839192i	$0.683029\pi$
$578$	−7825.42	−0.563140
$579$	5953.14	0.427295
$580$	1088.65	0.0779374
$581$	5568.09	0.397596
$582$	−1071.74	−0.0763318
$583$	−506.856	−0.0360066
$584$	5495.17	0.389369
$585$	1180.55	0.0834354
$586$	4258.53	0.300202
$587$	−830.183	−0.0583736	−0.0291868	−	0.999574i	$-0.509292\pi$
$587$	−830.183	−0.0583736	−0.0291868	+	0.999574i	$0.509292\pi$
$588$	3164.22	0.221922
$589$	0	0
$590$	−13789.8	−0.962233
$591$	−1264.94	−0.0880418
$592$	−1423.74	−0.0988432
$593$	27716.6	1.91936	0.959682	−	0.281087i	$-0.0906951\pi$
$593$	27716.6	1.91936	0.959682	+	0.281087i	$0.0906951\pi$
$594$	−1220.30	−0.0842919
$595$	3214.27	0.221466
$596$	11895.5	0.817550
$597$	5168.44	0.354322
$598$	1140.50	0.0779908
$599$	−8672.10	−0.591540	−0.295770	−	0.955259i	$-0.595576\pi$
$599$	−8672.10	−0.591540	−0.295770	+	0.955259i	$0.595576\pi$
$600$	−125.335	−0.00852799
$601$	6303.27	0.427813	0.213907	−	0.976854i	$-0.431381\pi$
$601$	6303.27	0.427813	0.213907	+	0.976854i	$0.431381\pi$
$602$	−8949.39	−0.605897
$603$	3335.52	0.225262
$604$	−2875.51	−0.193713
$605$	9361.16	0.629066
$606$	−1256.57	−0.0842319
$607$	−12646.6	−0.845650	−0.422825	−	0.906211i	$-0.638962\pi$
$607$	−12646.6	−0.845650	−0.422825	+	0.906211i	$0.638962\pi$
$608$	0	0
$609$	−637.212	−0.0423993
$610$	15724.5	1.04372
$611$	−2054.43	−0.136028
$612$	1138.58	0.0752035
$613$	−10791.9	−0.711062	−0.355531	−	0.934665i	$-0.615700\pi$
$613$	−10791.9	−0.711062	−0.355531	+	0.934665i	$0.615700\pi$
$614$	10804.8	0.710173
$615$	−60.1714	−0.00394528
$616$	−1610.05	−0.105310
$617$	−21538.9	−1.40539	−0.702694	−	0.711492i	$-0.748020\pi$
$617$	−21538.9	−1.40539	−0.702694	+	0.711492i	$0.748020\pi$
$618$	−2014.18	−0.131104
$619$	−21147.7	−1.37318	−0.686588	−	0.727046i	$-0.740893\pi$
$619$	−21147.7	−1.37318	−0.686588	+	0.727046i	$0.740893\pi$
$620$	−7075.50	−0.458321
$621$	1339.46	0.0865550
$622$	11107.7	0.716044
$623$	17.9035	0.00115135
$624$	551.747	0.0353968
$625$	−16250.5	−1.04003
$626$	−15831.5	−1.01079
$627$	0	0
$628$	6634.98	0.421599
$629$	−2814.31	−0.178400
$630$	1829.33	0.115686
$631$	8694.83	0.548551	0.274276	−	0.961651i	$-0.411562\pi$
$631$	8694.83	0.548551	0.274276	+	0.961651i	$0.411562\pi$
$632$	4886.73	0.307569
$633$	2718.05	0.170668
$634$	18275.0	1.14479
$635$	16665.5	1.04150
$636$	269.150	0.0167807
$637$	3030.99	0.188528
$638$	−1077.92	−0.0668892
$639$	−5128.67	−0.317507
$640$	−1460.67	−0.0902158
$641$	7230.41	0.445529	0.222764	−	0.974872i	$-0.428492\pi$
$641$	7230.41	0.445529	0.222764	+	0.974872i	$0.428492\pi$
$642$	9603.89	0.590397
$643$	3209.68	0.196854	0.0984271	−	0.995144i	$-0.468619\pi$
$643$	3209.68	0.196854	0.0984271	+	0.995144i	$0.468619\pi$
$644$	1767.27	0.108137
$645$	17200.8	1.05005
$646$	0	0
$647$	−17589.5	−1.06880	−0.534401	−	0.845231i	$-0.679463\pi$
$647$	−17589.5	−1.06880	−0.534401	+	0.845231i	$0.679463\pi$
$648$	648.000	0.0392837
$649$	13653.9	0.825829
$650$	−120.058	−0.00724471
$651$	4141.46	0.249334
$652$	−3770.92	−0.226504
$653$	−4309.56	−0.258264	−0.129132	−	0.991627i	$-0.541219\pi$
$653$	−4309.56	−0.258264	−0.129132	+	0.991627i	$0.541219\pi$
$654$	4778.45	0.285707
$655$	20981.8	1.25165
$656$	−28.1220	−0.00167375
$657$	6182.07	0.367101
$658$	−3183.46	−0.188608
$659$	−32200.1	−1.90339	−0.951697	−	0.307039i	$-0.900662\pi$
$659$	−32200.1	−1.90339	−0.951697	+	0.307039i	$0.900662\pi$
$660$	3094.54	0.182507
$661$	23276.6	1.36967	0.684836	−	0.728697i	$-0.259874\pi$
$661$	23276.6	1.36967	0.684836	+	0.728697i	$0.259874\pi$
$662$	−12105.1	−0.710690
$663$	1090.64	0.0638870
$664$	−5001.71	−0.292325
$665$	0	0
$666$	−1601.70	−0.0931903
$667$	1183.18	0.0686851
$668$	−7841.32	−0.454176
$669$	15687.2	0.906578
$670$	−8458.52	−0.487733
$671$	−15569.5	−0.895761
$672$	854.966	0.0490789
$673$	9207.85	0.527394	0.263697	−	0.964605i	$-0.415058\pi$
$673$	9207.85	0.527394	0.263697	+	0.964605i	$0.415058\pi$
$674$	−9364.70	−0.535185
$675$	−141.002	−0.00804026
$676$	−8259.48	−0.469930
$677$	17786.3	1.00972	0.504862	−	0.863200i	$-0.331543\pi$
$677$	17786.3	1.00972	0.504862	+	0.863200i	$0.331543\pi$
$678$	7332.96	0.415370
$679$	−1590.80	−0.0899108
$680$	−2887.32	−0.162829
$681$	134.009	0.00754074
$682$	7005.77	0.393350
$683$	27734.5	1.55378	0.776888	−	0.629638i	$-0.216797\pi$
$683$	27734.5	1.55378	0.776888	+	0.629638i	$0.216797\pi$
$684$	0	0
$685$	24504.9	1.36684
$686$	10806.2	0.601430
$687$	−11543.3	−0.641052
$688$	8039.06	0.445474
$689$	257.818	0.0142555
$690$	−3396.72	−0.187407
$691$	−4636.28	−0.255242	−0.127621	−	0.991823i	$-0.540734\pi$
$691$	−4636.28	−0.255242	−0.127621	+	0.991823i	$0.540734\pi$
$692$	−13371.3	−0.734537
$693$	−1811.31	−0.0992869
$694$	21671.6	1.18536
$695$	33575.2	1.83249
$696$	572.396	0.0311733
$697$	−55.5890	−0.00302092
$698$	−5030.14	−0.272770
$699$	3403.92	0.184189
$700$	−186.037	−0.0100451
$701$	19023.6	1.02498	0.512490	−	0.858693i	$-0.328723\pi$
$701$	19023.6	1.02498	0.512490	+	0.858693i	$0.328723\pi$
$702$	620.716	0.0333724
$703$	0	0
$704$	1446.28	0.0774270
$705$	6118.65	0.326867
$706$	12765.8	0.680518
$707$	−1865.14	−0.0992163
$708$	−7250.48	−0.384872
$709$	29121.4	1.54256	0.771281	−	0.636495i	$-0.219616\pi$
$709$	29121.4	1.54256	0.771281	+	0.636495i	$0.219616\pi$
$710$	13005.7	0.687460
$711$	5497.58	0.289979
$712$	−16.0824	−0.000846508 0
$713$	−7689.89	−0.403911
$714$	1690.02	0.0885817
$715$	2964.24	0.155044
$716$	−11393.5	−0.594686
$717$	−12906.1	−0.672228
$718$	18244.6	0.948307
$719$	10801.5	0.560259	0.280130	−	0.959962i	$-0.409623\pi$
$719$	10801.5	0.560259	0.280130	+	0.959962i	$0.409623\pi$
$720$	−1643.26	−0.0850563
$721$	−2989.68	−0.154426
$722$	0	0
$723$	−10899.7	−0.560670
$724$	−9493.98	−0.487350
$725$	−124.551	−0.00638029
$726$	4921.96	0.251613
$727$	−18235.3	−0.930276	−0.465138	−	0.885238i	$-0.653995\pi$
$727$	−18235.3	−0.930276	−0.465138	+	0.885238i	$0.653995\pi$
$728$	818.968	0.0416936
$729$	729.000	0.0370370
$730$	−15677.0	−0.794840
$731$	15890.9	0.804029
$732$	8267.72	0.417464
$733$	5867.74	0.295675	0.147838	−	0.989012i	$-0.452769\pi$
$733$	5867.74	0.295675	0.147838	+	0.989012i	$0.452769\pi$
$734$	11782.3	0.592498
$735$	−9027.12	−0.453021
$736$	−1587.51	−0.0795059
$737$	8375.16	0.418593
$738$	−31.6373	−0.00157803
$739$	−28368.8	−1.41213	−0.706065	−	0.708147i	$-0.749531\pi$
$739$	−28368.8	−1.41213	−0.706065	+	0.708147i	$0.749531\pi$
$740$	4061.74	0.201774
$741$	0	0
$742$	399.504	0.0197658
$743$	−2691.78	−0.132909	−0.0664547	−	0.997789i	$-0.521169\pi$
$743$	−2691.78	−0.132909	−0.0664547	+	0.997789i	$0.521169\pi$
$744$	−3720.19	−0.183318
$745$	−33936.5	−1.66891
$746$	−16879.5	−0.828423
$747$	−5626.92	−0.275607
$748$	2858.87	0.139747
$749$	14255.2	0.695426
$750$	−8201.06	−0.399280
$751$	−2589.67	−0.125830	−0.0629151	−	0.998019i	$-0.520040\pi$
$751$	−2589.67	−0.125830	−0.0629151	+	0.998019i	$0.520040\pi$
$752$	2859.64	0.138671
$753$	8362.44	0.404707
$754$	548.295	0.0264824
$755$	8203.47	0.395437
$756$	961.837	0.0462721
$757$	12079.4	0.579965	0.289983	−	0.957032i	$-0.406350\pi$
$757$	12079.4	0.579965	0.289983	+	0.957032i	$0.406350\pi$
$758$	−7200.90	−0.345051
$759$	3363.25	0.160841
$760$	0	0
$761$	−15595.8	−0.742900	−0.371450	−	0.928453i	$-0.621139\pi$
$761$	−15595.8	−0.742900	−0.371450	+	0.928453i	$0.621139\pi$
$762$	8762.47	0.416576
$763$	7092.73	0.336532
$764$	19260.5	0.912070
$765$	−3248.24	−0.153517
$766$	24195.4	1.14127
$767$	−6945.20	−0.326958
$768$	−768.000	−0.0360844
$769$	16923.3	0.793587	0.396794	−	0.917908i	$-0.370123\pi$
$769$	16923.3	0.793587	0.396794	+	0.917908i	$0.370123\pi$
$770$	4593.27	0.214974
$771$	11129.5	0.519869
$772$	−7937.52	−0.370049
$773$	9481.75	0.441184	0.220592	−	0.975366i	$-0.429201\pi$
$773$	9481.75	0.441184	0.220592	+	0.975366i	$0.429201\pi$
$774$	9043.95	0.419997
$775$	809.499	0.0375201
$776$	1428.99	0.0661053
$777$	−2377.43	−0.109768
$778$	−17796.4	−0.820094
$779$	0	0
$780$	−1574.07	−0.0722572
$781$	−12877.6	−0.590007
$782$	−3138.04	−0.143499
$783$	643.945	0.0293905
$784$	−4218.96	−0.192190
$785$	−18928.8	−0.860632
$786$	11031.9	0.500631
$787$	−1689.69	−0.0765323	−0.0382662	−	0.999268i	$-0.512183\pi$
$787$	−1689.69	−0.0765323	−0.0382662	+	0.999268i	$0.512183\pi$
$788$	1686.59	0.0762464
$789$	−4418.90	−0.199388
$790$	−13941.2	−0.627857
$791$	10884.4	0.489262
$792$	1627.06	0.0729989
$793$	7919.60	0.354645
$794$	12734.2	0.569167
$795$	−767.852	−0.0342552
$796$	−6891.25	−0.306852
$797$	4265.09	0.189557	0.0947787	−	0.995498i	$-0.469786\pi$
$797$	4265.09	0.189557	0.0947787	+	0.995498i	$0.469786\pi$
$798$	0	0
$799$	5652.67	0.250284
$800$	167.114	0.00738545
$801$	−18.0927	−0.000798095 0
$802$	−18879.6	−0.831247
$803$	15522.5	0.682165
$804$	−4447.37	−0.195083
$805$	−5041.81	−0.220746
$806$	−3563.55	−0.155733
$807$	6409.99	0.279606
$808$	1675.42	0.0729470
$809$	1622.16	0.0704969	0.0352484	−	0.999379i	$-0.488778\pi$
$809$	1622.16	0.0704969	0.0352484	+	0.999379i	$0.488778\pi$
$810$	−1848.66	−0.0801918
$811$	−39797.7	−1.72316	−0.861582	−	0.507619i	$-0.830526\pi$
$811$	−39797.7	−1.72316	−0.861582	+	0.507619i	$0.830526\pi$
$812$	849.617	0.0367188
$813$	−12405.6	−0.535157
$814$	−4021.71	−0.173171
$815$	10758.0	0.462375
$816$	−1518.11	−0.0651281
$817$	0	0
$818$	−13065.1	−0.558449
$819$	921.339	0.0393091
$820$	80.2286	0.00341671
$821$	−29942.3	−1.27283	−0.636414	−	0.771347i	$-0.719583\pi$
$821$	−29942.3	−1.27283	−0.636414	+	0.771347i	$0.719583\pi$
$822$	12884.3	0.546707
$823$	661.147	0.0280026	0.0140013	−	0.999902i	$-0.495543\pi$
$823$	661.147	0.0280026	0.0140013	+	0.999902i	$0.495543\pi$
$824$	2685.57	0.113539
$825$	−354.042	−0.0149408
$826$	−10762.0	−0.453339
$827$	22338.6	0.939287	0.469643	−	0.882856i	$-0.344382\pi$
$827$	22338.6	0.939287	0.469643	+	0.882856i	$0.344382\pi$
$828$	−1785.95	−0.0749589
$829$	40601.8	1.70104	0.850519	−	0.525945i	$-0.176288\pi$
$829$	40601.8	1.70104	0.850519	+	0.525945i	$0.176288\pi$
$830$	14269.3	0.596739
$831$	11940.9	0.498465
$832$	−735.663	−0.0306545
$833$	−8339.65	−0.346881
$834$	17653.3	0.732956
$835$	22370.3	0.927133
$836$	0	0
$837$	−4185.22	−0.172834
$838$	2899.02	0.119505
$839$	−16173.5	−0.665519	−0.332760	−	0.943012i	$-0.607980\pi$
$839$	−16173.5	−0.665519	−0.332760	+	0.943012i	$0.607980\pi$
$840$	−2439.11	−0.100187
$841$	−23820.2	−0.976677
$842$	−6572.84	−0.269020
$843$	16919.2	0.691254
$844$	−3624.07	−0.147803
$845$	23563.3	0.959291
$846$	3217.10	0.130740
$847$	7305.75	0.296374
$848$	−358.867	−0.0145325
$849$	20354.6	0.822812
$850$	330.335	0.0133299
$851$	4414.44	0.177820
$852$	6838.23	0.274969
$853$	10860.7	0.435947	0.217973	−	0.975955i	$-0.430055\pi$
$853$	10860.7	0.435947	0.217973	+	0.975955i	$0.430055\pi$
$854$	12271.9	0.491728
$855$	0	0
$856$	−12805.2	−0.511299
$857$	−33673.4	−1.34220	−0.671098	−	0.741369i	$-0.734177\pi$
$857$	−33673.4	−1.34220	−0.671098	+	0.741369i	$0.734177\pi$
$858$	1558.55	0.0620142
$859$	−25309.8	−1.00531	−0.502654	−	0.864488i	$-0.667643\pi$
$859$	−25309.8	−1.00531	−0.502654	+	0.864488i	$0.667643\pi$
$860$	−22934.4	−0.909370
$861$	−46.9597	−0.00185875
$862$	−24725.2	−0.976967
$863$	26842.5	1.05878	0.529392	−	0.848378i	$-0.322420\pi$
$863$	26842.5	1.05878	0.529392	+	0.848378i	$0.322420\pi$
$864$	−864.000	−0.0340207
$865$	38146.6	1.49945
$866$	30499.4	1.19678
$867$	11738.1	0.459802
$868$	−5521.94	−0.215930
$869$	13803.9	0.538853
$870$	−1632.97	−0.0636356
$871$	−4260.11	−0.165727
$872$	−6371.27	−0.247429
$873$	1607.61	0.0623247
$874$	0	0
$875$	−12173.0	−0.470310
$876$	−8242.76	−0.317919
$877$	27145.7	1.04521	0.522603	−	0.852576i	$-0.324961\pi$
$877$	27145.7	1.04521	0.522603	+	0.852576i	$0.324961\pi$
$878$	−20139.4	−0.774114
$879$	−6387.79	−0.245114
$880$	−4126.05	−0.158056
$881$	−11515.1	−0.440356	−0.220178	−	0.975460i	$-0.570664\pi$
$881$	−11515.1	−0.440356	−0.220178	+	0.975460i	$0.570664\pi$
$882$	−4746.33	−0.181199
$883$	28678.2	1.09298	0.546488	−	0.837467i	$-0.315964\pi$
$883$	28678.2	1.09298	0.546488	+	0.837467i	$0.315964\pi$
$884$	−1454.19	−0.0553278
$885$	20684.7	0.785660
$886$	−7829.13	−0.296868
$887$	−966.906	−0.0366015	−0.0183008	−	0.999833i	$-0.505826\pi$
$887$	−966.906	−0.0366015	−0.0183008	+	0.999833i	$0.505826\pi$
$888$	2135.60	0.0807051
$889$	13006.3	0.490683
$890$	45.8811	0.00172802
$891$	1830.44	0.0688240
$892$	−20916.2	−0.785120
$893$	0	0
$894$	−17843.3	−0.667527
$895$	32504.3	1.21396
$896$	−1139.96	−0.0425036
$897$	−1710.75	−0.0636792
$898$	−15954.0	−0.592863
$899$	−3696.91	−0.137151
$900$	188.003	0.00696307
$901$	−709.375	−0.0262294
$902$	−79.4379	−0.00293237
$903$	13424.1	0.494712
$904$	−9777.28	−0.359721
$905$	27085.1	0.994852
$906$	4313.27	0.158166
$907$	−38400.2	−1.40580	−0.702899	−	0.711290i	$-0.748111\pi$
$907$	−38400.2	−1.40580	−0.702899	+	0.711290i	$0.748111\pi$
$908$	−178.679	−0.00653047
$909$	1884.85	0.0687751
$910$	−2336.41	−0.0851113
$911$	16835.2	0.612265	0.306133	−	0.951989i	$-0.400965\pi$
$911$	16835.2	0.612265	0.306133	+	0.951989i	$0.400965\pi$
$912$	0	0
$913$	−14128.6	−0.512146
$914$	29115.6	1.05368
$915$	−23586.8	−0.852190
$916$	15391.0	0.555167
$917$	16374.9	0.589691
$918$	−1707.88	−0.0614034
$919$	−37958.3	−1.36249	−0.681246	−	0.732055i	$-0.738561\pi$
$919$	−37958.3	−1.36249	−0.681246	+	0.732055i	$0.738561\pi$
$920$	4528.96	0.162299
$921$	−16207.2	−0.579854
$922$	29424.5	1.05102
$923$	6550.30	0.233592
$924$	2415.07	0.0859850
$925$	−464.699	−0.0165181
$926$	−37160.1	−1.31875
$927$	3021.27	0.107046
$928$	−763.194	−0.0269969
$929$	29436.6	1.03959	0.519797	−	0.854290i	$-0.326007\pi$
$929$	29436.6	1.03959	0.519797	+	0.854290i	$0.326007\pi$
$930$	10613.2	0.374217
$931$	0	0
$932$	−4538.56	−0.159512
$933$	−16661.6	−0.584648
$934$	5160.65	0.180794
$935$	−8155.99	−0.285272
$936$	−827.621	−0.0289013
$937$	9758.20	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$937$	9758.20	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$938$	−6601.30	−0.229787
$939$	23747.2	0.825305
$940$	−8158.19	−0.283076
$941$	47858.3	1.65795	0.828977	−	0.559282i	$-0.188923\pi$
$941$	47858.3	1.65795	0.828977	+	0.559282i	$0.188923\pi$
$942$	−9952.47	−0.344235
$943$	87.1952	0.00301110
$944$	9667.30	0.333309
$945$	−2744.00	−0.0944575
$946$	22708.4	0.780459
$947$	31063.9	1.06593	0.532967	−	0.846136i	$-0.321077\pi$
$947$	31063.9	1.06593	0.532967	+	0.846136i	$0.321077\pi$
$948$	−7330.10	−0.251129
$949$	−7895.69	−0.270079
$950$	0	0
$951$	−27412.5	−0.934714
$952$	−2253.36	−0.0767140
$953$	−32651.6	−1.10985	−0.554927	−	0.831899i	$-0.687254\pi$
$953$	−32651.6	−1.10985	−0.554927	+	0.831899i	$0.687254\pi$
$954$	−403.725	−0.0137014
$955$	−54947.9	−1.86186
$956$	17208.1	0.582167
$957$	1616.88	0.0546148
$958$	12587.9	0.424527
$959$	19124.4	0.643963
$960$	2191.01	0.0736609
$961$	−5763.51	−0.193465
$962$	2045.68	0.0685608
$963$	−14405.8	−0.482058
$964$	14533.0	0.485555
$965$	22644.7	0.755399
$966$	−2650.91	−0.0882936
$967$	20568.7	0.684016	0.342008	−	0.939697i	$-0.388893\pi$
$967$	20568.7	0.684016	0.342008	+	0.939697i	$0.388893\pi$
$968$	−6562.61	−0.217903
$969$	0	0
$970$	−4076.73	−0.134944
$971$	−42239.2	−1.39601	−0.698003	−	0.716095i	$-0.745928\pi$
$971$	−42239.2	−1.39601	−0.698003	+	0.716095i	$0.745928\pi$
$972$	−972.000	−0.0320750
$973$	26203.1	0.863345
$974$	−26168.5	−0.860874
$975$	180.087	0.00591528
$976$	−11023.6	−0.361534
$977$	16357.1	0.535630	0.267815	−	0.963470i	$-0.413698\pi$
$977$	16357.1	0.535630	0.267815	+	0.963470i	$0.413698\pi$
$978$	5656.38	0.184940
$979$	−45.4289	−0.00148306
$980$	12036.2	0.392328
$981$	−7167.68	−0.233279
$982$	−18418.0	−0.598515
$983$	34375.0	1.11535	0.557676	−	0.830059i	$-0.311693\pi$
$983$	34375.0	1.11535	0.557676	+	0.830059i	$0.311693\pi$
$984$	42.1830	0.00136661
$985$	−4811.62	−0.155646
$986$	−1508.61	−0.0487262
$987$	4775.19	0.153998
$988$	0	0
$989$	−24925.9	−0.801414
$990$	−4641.80	−0.149016
$991$	16405.4	0.525867	0.262934	−	0.964814i	$-0.415310\pi$
$991$	16405.4	0.525867	0.262934	+	0.964814i	$0.415310\pi$
$992$	4960.26	0.158758
$993$	18157.6	0.580276
$994$	10150.1	0.323885
$995$	19659.9	0.626392
$996$	7502.57	0.238683
$997$	−37217.7	−1.18224	−0.591121	−	0.806583i	$-0.701314\pi$
$997$	−37217.7	−1.18224	−0.591121	+	0.806583i	$0.701314\pi$
$998$	32629.6	1.03494
$999$	2402.55	0.0760895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bf.1.1		6
19.4	even	9		114.4.i.a.73.1	yes	12
19.5	even	9		114.4.i.a.25.1	✓	12
19.18	odd	2		2166.4.a.bd.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.a.25.1	✓	12	19.5	even	9
114.4.i.a.73.1	yes	12	19.4	even	9
2166.4.a.bd.1.1		6	19.18	odd	2
2166.4.a.bf.1.1		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -11.4115 q^{5} -6.00000 q^{6} -8.90590 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -11.4115 q^{5} -6.00000 q^{6} -8.90590 q^{7} +8.00000 q^{8} +9.00000 q^{9} -22.8230 q^{10} +22.5981 q^{11} -12.0000 q^{12} -11.4947 q^{13} -17.8118 q^{14} +34.2345 q^{15} +16.0000 q^{16} +31.6273 q^{17} +18.0000 q^{18} -45.6460 q^{20} +26.7177 q^{21} +45.1962 q^{22} -49.6096 q^{23} -24.0000 q^{24} +5.22231 q^{25} -22.9895 q^{26} -27.0000 q^{27} -35.6236 q^{28} -23.8498 q^{29} +68.4690 q^{30} +155.008 q^{31} +32.0000 q^{32} -67.7942 q^{33} +63.2547 q^{34} +101.630 q^{35} +36.0000 q^{36} -88.9835 q^{37} +34.4842 q^{39} -91.2920 q^{40} -1.75763 q^{41} +53.4354 q^{42} +502.441 q^{43} +90.3923 q^{44} -102.703 q^{45} -99.2193 q^{46} +178.728 q^{47} -48.0000 q^{48} -263.685 q^{49} +10.4446 q^{50} -94.8820 q^{51} -45.9790 q^{52} -22.4292 q^{53} -54.0000 q^{54} -257.878 q^{55} -71.2472 q^{56} -47.6997 q^{58} +604.207 q^{59} +136.938 q^{60} -688.976 q^{61} +310.016 q^{62} -80.1531 q^{63} +64.0000 q^{64} +131.172 q^{65} -135.588 q^{66} +370.614 q^{67} +126.509 q^{68} +148.829 q^{69} +203.259 q^{70} -569.852 q^{71} +72.0000 q^{72} +686.896 q^{73} -177.967 q^{74} -15.6669 q^{75} -201.256 q^{77} +68.9684 q^{78} +610.842 q^{79} -182.584 q^{80} +81.0000 q^{81} -3.51525 q^{82} -625.214 q^{83} +106.871 q^{84} -360.915 q^{85} +1004.88 q^{86} +71.5495 q^{87} +180.785 q^{88} -2.01030 q^{89} -205.407 q^{90} +102.371 q^{91} -198.439 q^{92} -465.024 q^{93} +357.455 q^{94} -96.0000 q^{96} +178.624 q^{97} -527.370 q^{98} +203.383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} + 27 q^{5} - 36 q^{6} - 42 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 + 27 * q^5 - 36 * q^6 - 42 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} + 27 q^{5} - 36 q^{6} - 42 q^{7} + 48 q^{8} + 54 q^{9} + 54 q^{10} - 57 q^{11} - 72 q^{12} + 21 q^{13} - 84 q^{14} - 81 q^{15} + 96 q^{16} - 45 q^{17} + 108 q^{18} + 108 q^{20} + 126 q^{21} - 114 q^{22} + 138 q^{23} - 144 q^{24} - 135 q^{25} + 42 q^{26} - 162 q^{27} - 168 q^{28} - 177 q^{29} - 162 q^{30} - 99 q^{31} + 192 q^{32} + 171 q^{33} - 90 q^{34} - 594 q^{35} + 216 q^{36} - 615 q^{37} - 63 q^{39} + 216 q^{40} - 171 q^{41} + 252 q^{42} + 456 q^{43} - 228 q^{44} + 243 q^{45} + 276 q^{46} + 228 q^{47} - 288 q^{48} + 168 q^{49} - 270 q^{50} + 135 q^{51} + 84 q^{52} - 357 q^{53} - 324 q^{54} - 1356 q^{55} - 336 q^{56} - 354 q^{58} - 741 q^{59} - 324 q^{60} - 66 q^{61} - 198 q^{62} - 378 q^{63} + 384 q^{64} + 843 q^{65} + 342 q^{66} + 966 q^{67} - 180 q^{68} - 414 q^{69} - 1188 q^{70} - 1338 q^{71} + 432 q^{72} + 1293 q^{73} - 1230 q^{74} + 405 q^{75} + 1299 q^{77} - 126 q^{78} - 1404 q^{79} + 432 q^{80} + 486 q^{81} - 342 q^{82} - 1440 q^{83} + 504 q^{84} - 1209 q^{85} + 912 q^{86} + 531 q^{87} - 456 q^{88} - 2991 q^{89} + 486 q^{90} - 4632 q^{91} + 552 q^{92} + 297 q^{93} + 456 q^{94} - 576 q^{96} - 2802 q^{97} + 336 q^{98} - 513 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 + 27 * q^5 - 36 * q^6 - 42 * q^7 + 48 * q^8 + 54 * q^9 + 54 * q^10 - 57 * q^11 - 72 * q^12 + 21 * q^13 - 84 * q^14 - 81 * q^15 + 96 * q^16 - 45 * q^17 + 108 * q^18 + 108 * q^20 + 126 * q^21 - 114 * q^22 + 138 * q^23 - 144 * q^24 - 135 * q^25 + 42 * q^26 - 162 * q^27 - 168 * q^28 - 177 * q^29 - 162 * q^30 - 99 * q^31 + 192 * q^32 + 171 * q^33 - 90 * q^34 - 594 * q^35 + 216 * q^36 - 615 * q^37 - 63 * q^39 + 216 * q^40 - 171 * q^41 + 252 * q^42 + 456 * q^43 - 228 * q^44 + 243 * q^45 + 276 * q^46 + 228 * q^47 - 288 * q^48 + 168 * q^49 - 270 * q^50 + 135 * q^51 + 84 * q^52 - 357 * q^53 - 324 * q^54 - 1356 * q^55 - 336 * q^56 - 354 * q^58 - 741 * q^59 - 324 * q^60 - 66 * q^61 - 198 * q^62 - 378 * q^63 + 384 * q^64 + 843 * q^65 + 342 * q^66 + 966 * q^67 - 180 * q^68 - 414 * q^69 - 1188 * q^70 - 1338 * q^71 + 432 * q^72 + 1293 * q^73 - 1230 * q^74 + 405 * q^75 + 1299 * q^77 - 126 * q^78 - 1404 * q^79 + 432 * q^80 + 486 * q^81 - 342 * q^82 - 1440 * q^83 + 504 * q^84 - 1209 * q^85 + 912 * q^86 + 531 * q^87 - 456 * q^88 - 2991 * q^89 + 486 * q^90 - 4632 * q^91 + 552 * q^92 + 297 * q^93 + 456 * q^94 - 576 * q^96 - 2802 * q^97 + 336 * q^98 - 513 * q^99

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