LMFDB - Embedded newform 471.4.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40822 q^{2} -3.00000 q^{3} +3.61596 q^{4} -8.48893 q^{5} +10.2247 q^{6} +10.4594 q^{7} +14.9418 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.40822 q^{2} -3.00000 q^{3} +3.61596 q^{4} -8.48893 q^{5} +10.2247 q^{6} +10.4594 q^{7} +14.9418 q^{8} +9.00000 q^{9} +28.9321 q^{10} -10.0003 q^{11} -10.8479 q^{12} -5.54647 q^{13} -35.6479 q^{14} +25.4668 q^{15} -79.8525 q^{16} +128.098 q^{17} -30.6740 q^{18} -120.613 q^{19} -30.6956 q^{20} -31.3782 q^{21} +34.0831 q^{22} -109.315 q^{23} -44.8253 q^{24} -52.9380 q^{25} +18.9036 q^{26} -27.0000 q^{27} +37.8207 q^{28} +11.8562 q^{29} -86.7964 q^{30} -93.8626 q^{31} +152.621 q^{32} +30.0008 q^{33} -436.584 q^{34} -88.7892 q^{35} +32.5436 q^{36} -88.0091 q^{37} +411.074 q^{38} +16.6394 q^{39} -126.840 q^{40} +156.426 q^{41} +106.944 q^{42} +226.089 q^{43} -36.1605 q^{44} -76.4004 q^{45} +372.568 q^{46} -396.304 q^{47} +239.558 q^{48} -233.601 q^{49} +180.424 q^{50} -384.293 q^{51} -20.0558 q^{52} +541.351 q^{53} +92.0219 q^{54} +84.8916 q^{55} +156.282 q^{56} +361.838 q^{57} -40.4086 q^{58} +603.502 q^{59} +92.0868 q^{60} -721.681 q^{61} +319.904 q^{62} +94.1346 q^{63} +118.656 q^{64} +47.0836 q^{65} -102.249 q^{66} -188.860 q^{67} +463.195 q^{68} +327.944 q^{69} +302.613 q^{70} -658.994 q^{71} +134.476 q^{72} +1001.51 q^{73} +299.954 q^{74} +158.814 q^{75} -436.130 q^{76} -104.597 q^{77} -56.7108 q^{78} +975.300 q^{79} +677.863 q^{80} +81.0000 q^{81} -533.134 q^{82} -1127.02 q^{83} -113.462 q^{84} -1087.41 q^{85} -770.560 q^{86} -35.5687 q^{87} -149.422 q^{88} +555.492 q^{89} +260.389 q^{90} -58.0128 q^{91} -395.277 q^{92} +281.588 q^{93} +1350.69 q^{94} +1023.87 q^{95} -457.862 q^{96} +1674.62 q^{97} +796.163 q^{98} -90.0024 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * 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 $

$2$	−3.40822	−1.20499	−0.602494	−	0.798124i	$-0.705826\pi$
$2$	−3.40822	−1.20499	−0.602494	+	0.798124i	$0.705826\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	3.61596	0.451994
$5$	−8.48893	−0.759273	−0.379637	−	0.925136i	$-0.623951\pi$
$5$	−8.48893	−0.759273	−0.379637	+	0.925136i	$0.623951\pi$
$6$	10.2247	0.695700
$7$	10.4594	0.564755	0.282377	−	0.959303i	$-0.408877\pi$
$7$	10.4594	0.564755	0.282377	+	0.959303i	$0.408877\pi$
$8$	14.9418	0.660340
$9$	9.00000	0.333333
$10$	28.9321	0.914915
$11$	−10.0003	−0.274108	−0.137054	−	0.990564i	$-0.543763\pi$
$11$	−10.0003	−0.274108	−0.137054	+	0.990564i	$0.543763\pi$
$12$	−10.8479	−0.260959
$13$	−5.54647	−0.118332	−0.0591659	−	0.998248i	$-0.518844\pi$
$13$	−5.54647	−0.118332	−0.0591659	+	0.998248i	$0.518844\pi$
$14$	−35.6479	−0.680522
$15$	25.4668	0.438367
$16$	−79.8525	−1.24770
$17$	128.098	1.82754	0.913771	−	0.406230i	$-0.133157\pi$
$17$	128.098	1.82754	0.913771	+	0.406230i	$0.133157\pi$
$18$	−30.6740	−0.401662
$19$	−120.613	−1.45634	−0.728170	−	0.685397i	$-0.759629\pi$
$19$	−120.613	−1.45634	−0.728170	+	0.685397i	$0.759629\pi$
$20$	−30.6956	−0.343187
$21$	−31.3782	−0.326061
$22$	34.0831	0.330297
$23$	−109.315	−0.991030	−0.495515	−	0.868599i	$-0.665021\pi$
$23$	−109.315	−0.991030	−0.495515	+	0.868599i	$0.665021\pi$
$24$	−44.8253	−0.381247
$25$	−52.9380	−0.423504
$26$	18.9036	0.142588
$27$	−27.0000	−0.192450
$28$	37.8207	0.255266
$29$	11.8562	0.0759188	0.0379594	−	0.999279i	$-0.487914\pi$
$29$	11.8562	0.0759188	0.0379594	+	0.999279i	$0.487914\pi$
$30$	−86.7964	−0.528226
$31$	−93.8626	−0.543813	−0.271907	−	0.962324i	$-0.587654\pi$
$31$	−93.8626	−0.543813	−0.271907	+	0.962324i	$0.587654\pi$
$32$	152.621	0.843118
$33$	30.0008	0.158257
$34$	−436.584	−2.20217
$35$	−88.7892	−0.428803
$36$	32.5436	0.150665
$37$	−88.0091	−0.391043	−0.195522	−	0.980699i	$-0.562640\pi$
$37$	−88.0091	−0.391043	−0.195522	+	0.980699i	$0.562640\pi$
$38$	411.074	1.75487
$39$	16.6394	0.0683189
$40$	−126.840	−0.501378
$41$	156.426	0.595845	0.297923	−	0.954590i	$-0.403706\pi$
$41$	156.426	0.595845	0.297923	+	0.954590i	$0.403706\pi$
$42$	106.944	0.392900
$43$	226.089	0.801819	0.400909	−	0.916118i	$-0.368694\pi$
$43$	226.089	0.801819	0.400909	+	0.916118i	$0.368694\pi$
$44$	−36.1605	−0.123896
$45$	−76.4004	−0.253091
$46$	372.568	1.19418
$47$	−396.304	−1.22993	−0.614967	−	0.788553i	$-0.710831\pi$
$47$	−396.304	−1.22993	−0.614967	+	0.788553i	$0.710831\pi$
$48$	239.558	0.720357
$49$	−233.601	−0.681052
$50$	180.424	0.510317
$51$	−384.293	−1.05513
$52$	−20.0558	−0.0534854
$53$	541.351	1.40302	0.701512	−	0.712658i	$-0.252509\pi$
$53$	541.351	1.40302	0.701512	+	0.712658i	$0.252509\pi$
$54$	92.0219	0.231900
$55$	84.8916	0.208123
$56$	156.282	0.372930
$57$	361.838	0.840818
$58$	−40.4086	−0.0914812
$59$	603.502	1.33168	0.665842	−	0.746093i	$-0.268073\pi$
$59$	603.502	1.33168	0.665842	+	0.746093i	$0.268073\pi$
$60$	92.0868	0.198139
$61$	−721.681	−1.51478	−0.757391	−	0.652961i	$-0.773526\pi$
$61$	−721.681	−1.51478	−0.757391	+	0.652961i	$0.773526\pi$
$62$	319.904	0.655288
$63$	94.1346	0.188252
$64$	118.656	0.231749
$65$	47.0836	0.0898462
$66$	−102.249	−0.190697
$67$	−188.860	−0.344371	−0.172186	−	0.985065i	$-0.555083\pi$
$67$	−188.860	−0.344371	−0.172186	+	0.985065i	$0.555083\pi$
$68$	463.195	0.826039
$69$	327.944	0.572171
$70$	302.613	0.516702
$71$	−658.994	−1.10152	−0.550762	−	0.834662i	$-0.685663\pi$
$71$	−658.994	−1.10152	−0.550762	+	0.834662i	$0.685663\pi$
$72$	134.476	0.220113
$73$	1001.51	1.60572	0.802861	−	0.596166i	$-0.203310\pi$
$73$	1001.51	1.60572	0.802861	+	0.596166i	$0.203310\pi$
$74$	299.954	0.471202
$75$	158.814	0.244510
$76$	−436.130	−0.658257
$77$	−104.597	−0.154804
$78$	−56.7108	−0.0823235
$79$	975.300	1.38898	0.694492	−	0.719500i	$-0.255629\pi$
$79$	975.300	1.38898	0.694492	+	0.719500i	$0.255629\pi$
$80$	677.863	0.947342
$81$	81.0000	0.111111
$82$	−533.134	−0.717986
$83$	−1127.02	−1.49044	−0.745219	−	0.666819i	$-0.767655\pi$
$83$	−1127.02	−1.49044	−0.745219	+	0.666819i	$0.767655\pi$
$84$	−113.462	−0.147378
$85$	−1087.41	−1.38760
$86$	−770.560	−0.966182
$87$	−35.5687	−0.0438317
$88$	−149.422	−0.181005
$89$	555.492	0.661596	0.330798	−	0.943702i	$-0.392682\pi$
$89$	555.492	0.661596	0.330798	+	0.943702i	$0.392682\pi$
$90$	260.389	0.304972
$91$	−58.0128	−0.0668285
$92$	−395.277	−0.447940
$93$	281.588	0.313971
$94$	1350.69	1.48205
$95$	1023.87	1.10576
$96$	−457.862	−0.486774
$97$	1674.62	1.75290	0.876450	−	0.481492i	$-0.159905\pi$
$97$	1674.62	1.75290	0.876450	+	0.481492i	$0.159905\pi$
$98$	796.163	0.820659
$99$	−90.0024	−0.0913695
$100$	−191.422	−0.191422
$101$	1098.27	1.08200	0.541001	−	0.841022i	$-0.318046\pi$
$101$	1098.27	1.08200	0.541001	+	0.841022i	$0.318046\pi$
$102$	1309.75	1.27142
$103$	317.565	0.303792	0.151896	−	0.988396i	$-0.451462\pi$
$103$	317.565	0.303792	0.151896	+	0.988396i	$0.451462\pi$
$104$	−82.8742	−0.0781392
$105$	266.368	0.247570
$106$	−1845.04	−1.69063
$107$	2163.39	1.95461	0.977305	−	0.211838i	$-0.0679449\pi$
$107$	2163.39	1.95461	0.977305	+	0.211838i	$0.0679449\pi$
$108$	−97.6308	−0.0869864
$109$	−712.563	−0.626158	−0.313079	−	0.949727i	$-0.601360\pi$
$109$	−712.563	−0.626158	−0.313079	+	0.949727i	$0.601360\pi$
$110$	−289.329	−0.250786
$111$	264.027	0.225769
$112$	−835.210	−0.704642
$113$	2042.49	1.70037	0.850183	−	0.526488i	$-0.176492\pi$
$113$	2042.49	1.70037	0.850183	+	0.526488i	$0.176492\pi$
$114$	−1233.22	−1.01317
$115$	927.965	0.752463
$116$	42.8716	0.0343149
$117$	−49.9182	−0.0394440
$118$	−2056.87	−1.60466
$119$	1339.82	1.03211
$120$	380.519	0.289471
$121$	−1230.99	−0.924865
$122$	2459.65	1.82529
$123$	−469.278	−0.344011
$124$	−339.403	−0.245801
$125$	1510.50	1.08083
$126$	−320.831	−0.226841
$127$	−95.3274	−0.0666058	−0.0333029	−	0.999445i	$-0.510603\pi$
$127$	−95.3274	−0.0666058	−0.0333029	+	0.999445i	$0.510603\pi$
$128$	−1625.37	−1.12237
$129$	−678.266	−0.462930
$130$	−160.471	−0.108264
$131$	2048.48	1.36623	0.683116	−	0.730310i	$-0.260624\pi$
$131$	2048.48	1.36623	0.683116	+	0.730310i	$0.260624\pi$
$132$	108.482	0.0715311
$133$	−1261.54	−0.822474
$134$	643.675	0.414963
$135$	229.201	0.146122
$136$	1914.01	1.20680
$137$	−2146.85	−1.33881	−0.669407	−	0.742896i	$-0.733452\pi$
$137$	−2146.85	−1.33881	−0.669407	+	0.742896i	$0.733452\pi$
$138$	−1117.71	−0.689459
$139$	−1098.93	−0.670578	−0.335289	−	0.942115i	$-0.608834\pi$
$139$	−1098.93	−0.670578	−0.335289	+	0.942115i	$0.608834\pi$
$140$	−321.058	−0.193817
$141$	1188.91	0.710103
$142$	2246.00	1.32732
$143$	55.4662	0.0324358
$144$	−718.673	−0.415898
$145$	−100.647	−0.0576431
$146$	−3413.36	−1.93488
$147$	700.803	0.393206
$148$	−318.237	−0.176749
$149$	1660.90	0.913199	0.456599	−	0.889672i	$-0.349067\pi$
$149$	1660.90	0.913199	0.456599	+	0.889672i	$0.349067\pi$
$150$	−541.273	−0.294632
$151$	568.625	0.306451	0.153225	−	0.988191i	$-0.451034\pi$
$151$	568.625	0.306451	0.153225	+	0.988191i	$0.451034\pi$
$152$	−1802.17	−0.961679
$153$	1152.88	0.609181
$154$	356.489	0.186537
$155$	796.793	0.412903
$156$	60.1674	0.0308798
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−3324.04	−1.67371
$159$	−1624.05	−0.810036
$160$	−1295.59	−0.640157
$161$	−1143.37	−0.559689
$162$	−276.066	−0.133887
$163$	−694.135	−0.333551	−0.166776	−	0.985995i	$-0.553336\pi$
$163$	−694.135	−0.333551	−0.166776	+	0.985995i	$0.553336\pi$
$164$	565.630	0.269319
$165$	−254.675	−0.120160
$166$	3841.13	1.79596
$167$	−1983.44	−0.919063	−0.459531	−	0.888161i	$-0.651983\pi$
$167$	−1983.44	−0.919063	−0.459531	+	0.888161i	$0.651983\pi$
$168$	−468.846	−0.215311
$169$	−2166.24	−0.985998
$170$	3706.14	1.67204
$171$	−1085.51	−0.485446
$172$	817.527	0.362418
$173$	−3530.51	−1.55156	−0.775780	−	0.631004i	$-0.782643\pi$
$173$	−3530.51	−1.55156	−0.775780	+	0.631004i	$0.782643\pi$
$174$	121.226	0.0528167
$175$	−553.700	−0.239176
$176$	798.546	0.342004
$177$	−1810.51	−0.768848
$178$	−1893.24	−0.797215
$179$	3665.01	1.53037	0.765184	−	0.643811i	$-0.222648\pi$
$179$	3665.01	1.53037	0.765184	+	0.643811i	$0.222648\pi$
$180$	−276.260	−0.114396
$181$	−590.599	−0.242535	−0.121268	−	0.992620i	$-0.538696\pi$
$181$	−590.599	−0.242535	−0.121268	+	0.992620i	$0.538696\pi$
$182$	197.720	0.0805275
$183$	2165.04	0.874560
$184$	−1633.36	−0.654417
$185$	747.103	0.296909
$186$	−959.712	−0.378331
$187$	−1281.01	−0.500945
$188$	−1433.02	−0.555923
$189$	−282.404	−0.108687
$190$	−3489.58	−1.33243
$191$	21.0434	0.00797199	0.00398599	−	0.999992i	$-0.498731\pi$
$191$	21.0434	0.00797199	0.00398599	+	0.999992i	$0.498731\pi$
$192$	−355.967	−0.133801
$193$	2254.63	0.840890	0.420445	−	0.907318i	$-0.361874\pi$
$193$	2254.63	0.840890	0.420445	+	0.907318i	$0.361874\pi$
$194$	−5707.45	−2.11222
$195$	−141.251	−0.0518727
$196$	−844.690	−0.307832
$197$	249.117	0.0900958	0.0450479	−	0.998985i	$-0.485656\pi$
$197$	249.117	0.0900958	0.0450479	+	0.998985i	$0.485656\pi$
$198$	306.748	0.110099
$199$	2866.06	1.02095	0.510477	−	0.859892i	$-0.329469\pi$
$199$	2866.06	1.02095	0.510477	+	0.859892i	$0.329469\pi$
$200$	−790.989	−0.279657
$201$	566.579	0.198823
$202$	−3743.15	−1.30380
$203$	124.009	0.0428755
$204$	−1389.59	−0.476914
$205$	−1327.89	−0.452409
$206$	−1082.33	−0.366066
$207$	−983.832	−0.330343
$208$	442.900	0.147642
$209$	1206.16	0.399195
$210$	−907.839	−0.298318
$211$	−493.774	−0.161103	−0.0805517	−	0.996750i	$-0.525668\pi$
$211$	−493.774	−0.161103	−0.0805517	+	0.996750i	$0.525668\pi$
$212$	1957.50	0.634159
$213$	1976.98	0.635965
$214$	−7373.32	−2.35528
$215$	−1919.25	−0.608800
$216$	−403.428	−0.127082
$217$	−981.746	−0.307121
$218$	2428.57	0.754512
$219$	−3004.53	−0.927065
$220$	306.964	0.0940705
$221$	−710.489	−0.216256
$222$	−899.863	−0.272049
$223$	3238.03	0.972353	0.486176	−	0.873861i	$-0.338391\pi$
$223$	3238.03	0.972353	0.486176	+	0.873861i	$0.338391\pi$
$224$	1596.32	0.476155
$225$	−476.442	−0.141168
$226$	−6961.25	−2.04892
$227$	−2936.91	−0.858720	−0.429360	−	0.903133i	$-0.641261\pi$
$227$	−2936.91	−0.858720	−0.429360	+	0.903133i	$0.641261\pi$
$228$	1308.39	0.380045
$229$	−332.930	−0.0960725	−0.0480363	−	0.998846i	$-0.515296\pi$
$229$	−332.930	−0.0960725	−0.0480363	+	0.998846i	$0.515296\pi$
$230$	−3162.71	−0.906708
$231$	313.790	0.0893762
$232$	177.153	0.0501322
$233$	−871.811	−0.245125	−0.122563	−	0.992461i	$-0.539111\pi$
$233$	−871.811	−0.245125	−0.122563	+	0.992461i	$0.539111\pi$
$234$	170.132	0.0475295
$235$	3364.20	0.933856
$236$	2182.24	0.601914
$237$	−2925.90	−0.801931
$238$	−4566.41	−1.24368
$239$	4121.00	1.11534	0.557668	−	0.830064i	$-0.311696\pi$
$239$	4121.00	1.11534	0.557668	+	0.830064i	$0.311696\pi$
$240$	−2033.59	−0.546948
$241$	2403.84	0.642511	0.321256	−	0.946993i	$-0.395895\pi$
$241$	2403.84	0.642511	0.321256	+	0.946993i	$0.395895\pi$
$242$	4195.50	1.11445
$243$	−243.000	−0.0641500
$244$	−2609.57	−0.684673
$245$	1983.02	0.517105
$246$	1599.40	0.414529
$247$	668.975	0.172331
$248$	−1402.47	−0.359102
$249$	3381.06	0.860505
$250$	−5148.13	−1.30238
$251$	4062.77	1.02167	0.510836	−	0.859678i	$-0.329336\pi$
$251$	4062.77	1.02167	0.510836	+	0.859678i	$0.329336\pi$
$252$	340.387	0.0850887
$253$	1093.18	0.271650
$254$	324.897	0.0802592
$255$	3262.23	0.801133
$256$	4590.37	1.12070
$257$	1761.65	0.427583	0.213792	−	0.976879i	$-0.431419\pi$
$257$	1761.65	0.427583	0.213792	+	0.976879i	$0.431419\pi$
$258$	2311.68	0.557825
$259$	−920.522	−0.220844
$260$	170.252	0.0406100
$261$	106.706	0.0253063
$262$	−6981.66	−1.64629
$263$	2961.28	0.694297	0.347148	−	0.937810i	$-0.387150\pi$
$263$	2961.28	0.694297	0.347148	+	0.937810i	$0.387150\pi$
$264$	448.265	0.104503
$265$	−4595.49	−1.06528
$266$	4299.59	0.991071
$267$	−1666.48	−0.381973
$268$	−682.908	−0.155654
$269$	−6822.06	−1.54628	−0.773139	−	0.634237i	$-0.781314\pi$
$269$	−6822.06	−1.54628	−0.773139	+	0.634237i	$0.781314\pi$
$270$	−781.168	−0.176075
$271$	6248.42	1.40061	0.700304	−	0.713845i	$-0.253048\pi$
$271$	6248.42	1.40061	0.700304	+	0.713845i	$0.253048\pi$
$272$	−10228.9	−2.28022
$273$	174.038	0.0385834
$274$	7316.92	1.61325
$275$	529.394	0.116086
$276$	1185.83	0.258618
$277$	526.662	0.114238	0.0571192	−	0.998367i	$-0.481808\pi$
$277$	526.662	0.114238	0.0571192	+	0.998367i	$0.481808\pi$
$278$	3745.41	0.808038
$279$	−844.763	−0.181271
$280$	−1326.67	−0.283156
$281$	7163.79	1.52084	0.760420	−	0.649432i	$-0.224993\pi$
$281$	7163.79	1.52084	0.760420	+	0.649432i	$0.224993\pi$
$282$	−4052.07	−0.855665
$283$	−4513.53	−0.948062	−0.474031	−	0.880508i	$-0.657201\pi$
$283$	−4513.53	−0.948062	−0.474031	+	0.880508i	$0.657201\pi$
$284$	−2382.89	−0.497883
$285$	−3071.62	−0.638410
$286$	−189.041	−0.0390847
$287$	1636.12	0.336506
$288$	1373.59	0.281039
$289$	11496.0	2.33991
$290$	343.026	0.0694592
$291$	−5023.85	−1.01204
$292$	3621.41	0.725778
$293$	6217.62	1.23972	0.619859	−	0.784713i	$-0.287190\pi$
$293$	6217.62	1.23972	0.619859	+	0.784713i	$0.287190\pi$
$294$	−2388.49	−0.473808
$295$	−5123.09	−1.01111
$296$	−1315.01	−0.258221
$297$	270.007	0.0527522
$298$	−5660.73	−1.10039
$299$	606.311	0.117270
$300$	574.265	0.110517
$301$	2364.75	0.452831
$302$	−1938.00	−0.369269
$303$	−3294.82	−0.624694
$304$	9631.22	1.81707
$305$	6126.30	1.15013
$306$	−3929.26	−0.734055
$307$	−518.161	−0.0963291	−0.0481645	−	0.998839i	$-0.515337\pi$
$307$	−518.161	−0.0963291	−0.0481645	+	0.998839i	$0.515337\pi$
$308$	−378.217	−0.0699706
$309$	−952.696	−0.175395
$310$	−2715.64	−0.497543
$311$	6164.47	1.12397	0.561986	−	0.827147i	$-0.310038\pi$
$311$	6164.47	1.12397	0.561986	+	0.827147i	$0.310038\pi$
$312$	248.622	0.0451137
$313$	6490.48	1.17209	0.586044	−	0.810279i	$-0.300684\pi$
$313$	6490.48	1.17209	0.586044	+	0.810279i	$0.300684\pi$
$314$	535.090	0.0961685
$315$	−799.103	−0.142934
$316$	3526.64	0.627814
$317$	−3938.76	−0.697863	−0.348932	−	0.937148i	$-0.613455\pi$
$317$	−3938.76	−0.697863	−0.348932	+	0.937148i	$0.613455\pi$
$318$	5535.13	0.976083
$319$	−118.565	−0.0208100
$320$	−1007.26	−0.175961
$321$	−6490.18	−1.12849
$322$	3896.84	0.674418
$323$	−15450.2	−2.66152
$324$	292.892	0.0502216
$325$	293.619	0.0501140
$326$	2365.76	0.401925
$327$	2137.69	0.361512
$328$	2337.28	0.393460
$329$	−4145.10	−0.694611
$330$	867.987	0.144791
$331$	6785.80	1.12683	0.563416	−	0.826174i	$-0.309487\pi$
$331$	6785.80	1.12683	0.563416	+	0.826174i	$0.309487\pi$
$332$	−4075.25	−0.673670
$333$	−792.082	−0.130348
$334$	6760.01	1.10746
$335$	1603.22	0.261472
$336$	2505.63	0.406825
$337$	8362.11	1.35167	0.675835	−	0.737053i	$-0.263783\pi$
$337$	8362.11	1.35167	0.675835	+	0.737053i	$0.263783\pi$
$338$	7383.01	1.18811
$339$	−6127.47	−0.981706
$340$	−3932.03	−0.627189
$341$	938.650	0.149064
$342$	3699.67	0.584957
$343$	−6030.90	−0.949382
$344$	3378.17	0.529473
$345$	−2783.90	−0.434434
$346$	12032.8	1.86961
$347$	−12379.1	−1.91512	−0.957558	−	0.288241i	$-0.906930\pi$
$347$	−12379.1	−1.91512	−0.957558	+	0.288241i	$0.906930\pi$
$348$	−128.615	−0.0198117
$349$	−12084.0	−1.85341	−0.926705	−	0.375790i	$-0.877372\pi$
$349$	−12084.0	−1.85341	−0.926705	+	0.375790i	$0.877372\pi$
$350$	1887.13	0.288204
$351$	149.755	0.0227730
$352$	−1526.25	−0.231106
$353$	3447.19	0.519761	0.259881	−	0.965641i	$-0.416317\pi$
$353$	3447.19	0.519761	0.259881	+	0.965641i	$0.416317\pi$
$354$	6170.61	0.926452
$355$	5594.15	0.836357
$356$	2008.64	0.299038
$357$	−4019.47	−0.595891
$358$	−12491.2	−1.84407
$359$	−403.741	−0.0593556	−0.0296778	−	0.999560i	$-0.509448\pi$
$359$	−403.741	−0.0593556	−0.0296778	+	0.999560i	$0.509448\pi$
$360$	−1141.56	−0.167126
$361$	7688.42	1.12092
$362$	2012.89	0.292252
$363$	3692.98	0.533971
$364$	−209.772	−0.0302061
$365$	−8501.74	−1.21918
$366$	−7378.94	−1.05383
$367$	10425.0	1.48278	0.741391	−	0.671073i	$-0.234166\pi$
$367$	10425.0	1.48278	0.741391	+	0.671073i	$0.234166\pi$
$368$	8729.05	1.23650
$369$	1407.83	0.198615
$370$	−2546.29	−0.357771
$371$	5662.21	0.792364
$372$	1018.21	0.141913
$373$	3378.53	0.468992	0.234496	−	0.972117i	$-0.424656\pi$
$373$	3378.53	0.468992	0.234496	+	0.972117i	$0.424656\pi$
$374$	4365.96	0.603632
$375$	−4531.51	−0.624017
$376$	−5921.49	−0.812174
$377$	−65.7602	−0.00898361
$378$	962.494	0.130967
$379$	−1844.86	−0.250037	−0.125019	−	0.992154i	$-0.539899\pi$
$379$	−1844.86	−0.250037	−0.125019	+	0.992154i	$0.539899\pi$
$380$	3702.28	0.499797
$381$	285.982	0.0384549
$382$	−71.7206	−0.00960615
$383$	−6360.82	−0.848623	−0.424311	−	0.905516i	$-0.639484\pi$
$383$	−6360.82	−0.848623	−0.424311	+	0.905516i	$0.639484\pi$
$384$	4876.11	0.648002
$385$	887.915	0.117539
$386$	−7684.27	−1.01326
$387$	2034.80	0.267273
$388$	6055.33	0.792302
$389$	13614.0	1.77445	0.887223	−	0.461341i	$-0.152632\pi$
$389$	13614.0	1.77445	0.887223	+	0.461341i	$0.152632\pi$
$390$	481.414	0.0625060
$391$	−14002.9	−1.81115
$392$	−3490.41	−0.449726
$393$	−6145.44	−0.788795
$394$	−849.047	−0.108564
$395$	−8279.25	−1.05462
$396$	−325.445	−0.0412985
$397$	−3739.45	−0.472739	−0.236370	−	0.971663i	$-0.575958\pi$
$397$	−3739.45	−0.472739	−0.236370	+	0.971663i	$0.575958\pi$
$398$	−9768.16	−1.23024
$399$	3784.61	0.474856
$400$	4227.23	0.528404
$401$	9952.18	1.23937	0.619686	−	0.784850i	$-0.287260\pi$
$401$	9952.18	1.23937	0.619686	+	0.784850i	$0.287260\pi$
$402$	−1931.02	−0.239579
$403$	520.606	0.0643504
$404$	3971.30	0.489059
$405$	−687.604	−0.0843637
$406$	−422.650	−0.0516644
$407$	880.114	0.107188
$408$	−5742.02	−0.696746
$409$	1746.10	0.211098	0.105549	−	0.994414i	$-0.466340\pi$
$409$	1746.10	0.211098	0.105549	+	0.994414i	$0.466340\pi$
$410$	4525.74	0.545147
$411$	6440.54	0.772965
$412$	1148.30	0.137313
$413$	6312.28	0.752075
$414$	3353.12	0.398060
$415$	9567.19	1.13165
$416$	−846.506	−0.0997677
$417$	3296.80	0.387158
$418$	−4110.85	−0.481025
$419$	−7741.06	−0.902566	−0.451283	−	0.892381i	$-0.649034\pi$
$419$	−7741.06	−0.902566	−0.451283	+	0.892381i	$0.649034\pi$
$420$	963.173	0.111900
$421$	12818.3	1.48390	0.741952	−	0.670453i	$-0.233900\pi$
$421$	12818.3	1.48390	0.741952	+	0.670453i	$0.233900\pi$
$422$	1682.89	0.194128
$423$	−3566.74	−0.409978
$424$	8088.74	0.926472
$425$	−6781.23	−0.773972
$426$	−6737.99	−0.766330
$427$	−7548.35	−0.855481
$428$	7822.74	0.883473
$429$	−166.399	−0.0187268
$430$	6541.23	0.733596
$431$	11574.0	1.29351	0.646754	−	0.762698i	$-0.276126\pi$
$431$	11574.0	1.29351	0.646754	+	0.762698i	$0.276126\pi$
$432$	2156.02	0.240119
$433$	14731.3	1.63497	0.817485	−	0.575950i	$-0.195368\pi$
$433$	14731.3	1.63497	0.817485	+	0.575950i	$0.195368\pi$
$434$	3346.01	0.370077
$435$	301.940	0.0332803
$436$	−2576.60	−0.283020
$437$	13184.7	1.44328
$438$	10240.1	1.11710
$439$	16543.4	1.79857	0.899286	−	0.437361i	$-0.144087\pi$
$439$	16543.4	1.79857	0.899286	+	0.437361i	$0.144087\pi$
$440$	1268.43	0.137432
$441$	−2102.41	−0.227017
$442$	2421.50	0.260586
$443$	−6002.43	−0.643756	−0.321878	−	0.946781i	$-0.604314\pi$
$443$	−6002.43	−0.643756	−0.321878	+	0.946781i	$0.604314\pi$
$444$	954.711	0.102046
$445$	−4715.54	−0.502332
$446$	−11035.9	−1.17167
$447$	−4982.71	−0.527236
$448$	1241.07	0.130882
$449$	−6183.01	−0.649876	−0.324938	−	0.945735i	$-0.605343\pi$
$449$	−6183.01	−0.649876	−0.324938	+	0.945735i	$0.605343\pi$
$450$	1623.82	0.170106
$451$	−1564.30	−0.163326
$452$	7385.55	0.768556
$453$	−1705.88	−0.176929
$454$	10009.6	1.03475
$455$	492.467	0.0507411
$456$	5406.50	0.555225
$457$	3451.03	0.353244	0.176622	−	0.984279i	$-0.443483\pi$
$457$	3451.03	0.353244	0.176622	+	0.984279i	$0.443483\pi$
$458$	1134.70	0.115766
$459$	−3458.63	−0.351711
$460$	3355.48	0.340109
$461$	−222.397	−0.0224687	−0.0112344	−	0.999937i	$-0.503576\pi$
$461$	−222.397	−0.0224687	−0.0112344	+	0.999937i	$0.503576\pi$
$462$	−1069.47	−0.107697
$463$	−14991.9	−1.50482	−0.752409	−	0.658696i	$-0.771108\pi$
$463$	−14991.9	−1.50482	−0.752409	+	0.658696i	$0.771108\pi$
$464$	−946.749	−0.0947235
$465$	−2390.38	−0.238390
$466$	2971.32	0.295373
$467$	9977.15	0.988624	0.494312	−	0.869285i	$-0.335420\pi$
$467$	9977.15	0.988624	0.494312	+	0.869285i	$0.335420\pi$
$468$	−180.502	−0.0178285
$469$	−1975.36	−0.194485
$470$	−11465.9	−1.12528
$471$	471.000	0.0460776
$472$	9017.40	0.879363
$473$	−2260.95	−0.219785
$474$	9972.11	0.966316
$475$	6385.00	0.616766
$476$	4844.74	0.466509
$477$	4872.16	0.467674
$478$	−14045.3	−1.34397
$479$	−11229.4	−1.07115	−0.535576	−	0.844487i	$-0.679906\pi$
$479$	−11229.4	−1.07115	−0.535576	+	0.844487i	$0.679906\pi$
$480$	3886.76	0.369595
$481$	488.140	0.0462729
$482$	−8192.83	−0.774218
$483$	3430.10	0.323137
$484$	−4451.22	−0.418034
$485$	−14215.7	−1.33093
$486$	828.197	0.0773000
$487$	−18719.6	−1.74182	−0.870910	−	0.491442i	$-0.836470\pi$
$487$	−18719.6	−1.74182	−0.870910	+	0.491442i	$0.836470\pi$
$488$	−10783.2	−1.00027
$489$	2082.41	0.192576
$490$	−6758.57	−0.623104
$491$	−9573.91	−0.879969	−0.439984	−	0.898005i	$-0.645016\pi$
$491$	−9573.91	−0.879969	−0.439984	+	0.898005i	$0.645016\pi$
$492$	−1696.89	−0.155491
$493$	1518.75	0.138745
$494$	−2280.01	−0.207657
$495$	764.024	0.0693744
$496$	7495.16	0.678513
$497$	−6892.68	−0.622091
$498$	−11523.4	−1.03690
$499$	3049.00	0.273531	0.136766	−	0.990603i	$-0.456329\pi$
$499$	3049.00	0.273531	0.136766	+	0.990603i	$0.456329\pi$
$500$	5461.92	0.488529
$501$	5950.33	0.530621
$502$	−13846.8	−1.23110
$503$	11400.0	1.01053	0.505267	−	0.862963i	$-0.331394\pi$
$503$	11400.0	1.01053	0.505267	+	0.862963i	$0.331394\pi$
$504$	1406.54	0.124310
$505$	−9323.16	−0.821535
$506$	−3725.78	−0.327334
$507$	6498.71	0.569266
$508$	−344.700	−0.0301055
$509$	6800.74	0.592215	0.296107	−	0.955155i	$-0.404311\pi$
$509$	6800.74	0.592215	0.296107	+	0.955155i	$0.404311\pi$
$510$	−11118.4	−0.965356
$511$	10475.2	0.906840
$512$	−2642.03	−0.228051
$513$	3256.54	0.280273
$514$	−6004.10	−0.515232
$515$	−2695.79	−0.230661
$516$	−2452.58	−0.209242
$517$	3963.15	0.337135
$518$	3137.34	0.266114
$519$	10591.5	0.895794
$520$	703.513	0.0593290
$521$	6062.85	0.509824	0.254912	−	0.966964i	$-0.417954\pi$
$521$	6062.85	0.509824	0.254912	+	0.966964i	$0.417954\pi$
$522$	−363.677	−0.0304937
$523$	6429.76	0.537579	0.268789	−	0.963199i	$-0.413376\pi$
$523$	6429.76	0.537579	0.268789	+	0.963199i	$0.413376\pi$
$524$	7407.21	0.617529
$525$	1661.10	0.138088
$526$	−10092.7	−0.836619
$527$	−12023.6	−0.993842
$528$	−2395.64	−0.197456
$529$	−217.295	−0.0178593
$530$	15662.4	1.28365
$531$	5431.52	0.443894
$532$	−4561.66	−0.371754
$533$	−867.613	−0.0705075
$534$	5679.72	0.460272
$535$	−18364.9	−1.48408
$536$	−2821.90	−0.227402
$537$	−10995.0	−0.883559
$538$	23251.1	1.86324
$539$	2336.07	0.186682
$540$	828.781	0.0660464
$541$	9287.07	0.738045	0.369022	−	0.929420i	$-0.379693\pi$
$541$	9287.07	0.738045	0.369022	+	0.929420i	$0.379693\pi$
$542$	−21296.0	−1.68771
$543$	1771.80	0.140028
$544$	19550.3	1.54083
$545$	6048.90	0.475425
$546$	−593.161	−0.0464926
$547$	−13825.5	−1.08069	−0.540345	−	0.841444i	$-0.681706\pi$
$547$	−13825.5	−1.08069	−0.540345	+	0.841444i	$0.681706\pi$
$548$	−7762.90	−0.605137
$549$	−6495.13	−0.504928
$550$	−1804.29	−0.139882
$551$	−1430.01	−0.110564
$552$	4900.07	0.377828
$553$	10201.1	0.784436
$554$	−1794.98	−0.137656
$555$	−2241.31	−0.171420
$556$	−3973.70	−0.303098
$557$	−6858.40	−0.521723	−0.260861	−	0.965376i	$-0.584007\pi$
$557$	−6858.40	−0.521723	−0.260861	+	0.965376i	$0.584007\pi$
$558$	2879.14	0.218429
$559$	−1253.99	−0.0948807
$560$	7090.04	0.535016
$561$	3843.03	0.289221
$562$	−24415.8	−1.83259
$563$	−22889.3	−1.71345	−0.856724	−	0.515776i	$-0.827504\pi$
$563$	−22889.3	−1.71345	−0.856724	+	0.515776i	$0.827504\pi$
$564$	4299.05	0.320963
$565$	−17338.6	−1.29104
$566$	15383.1	1.14240
$567$	847.212	0.0627505
$568$	−9846.54	−0.727380
$569$	−9362.84	−0.689826	−0.344913	−	0.938635i	$-0.612092\pi$
$569$	−9362.84	−0.689826	−0.344913	+	0.938635i	$0.612092\pi$
$570$	10468.7	0.769276
$571$	−5253.08	−0.384999	−0.192500	−	0.981297i	$-0.561659\pi$
$571$	−5253.08	−0.384999	−0.192500	+	0.981297i	$0.561659\pi$
$572$	200.563	0.0146608
$573$	−63.1303	−0.00460263
$574$	−5576.27	−0.405486
$575$	5786.91	0.419705
$576$	1067.90	0.0772498
$577$	−5286.69	−0.381434	−0.190717	−	0.981645i	$-0.561081\pi$
$577$	−5286.69	−0.381434	−0.190717	+	0.981645i	$0.561081\pi$
$578$	−39180.8	−2.81956
$579$	−6763.89	−0.485488
$580$	−363.934	−0.0260544
$581$	−11787.9	−0.841732
$582$	17122.4	1.21949
$583$	−5413.65	−0.384581
$584$	14964.3	1.06032
$585$	423.753	0.0299487
$586$	−21191.0	−1.49384
$587$	−4391.35	−0.308774	−0.154387	−	0.988010i	$-0.549340\pi$
$587$	−4391.35	−0.308774	−0.154387	+	0.988010i	$0.549340\pi$
$588$	2534.07	0.177727
$589$	11321.0	0.791977
$590$	17460.6	1.21838
$591$	−747.352	−0.0520169
$592$	7027.75	0.487903
$593$	9887.06	0.684676	0.342338	−	0.939577i	$-0.388781\pi$
$593$	9887.06	0.684676	0.342338	+	0.939577i	$0.388781\pi$
$594$	−920.243	−0.0635657
$595$	−11373.7	−0.783656
$596$	6005.76	0.412761
$597$	−8598.18	−0.589448
$598$	−2066.44	−0.141309
$599$	−10266.1	−0.700268	−0.350134	−	0.936700i	$-0.613864\pi$
$599$	−10266.1	−0.700268	−0.350134	+	0.936700i	$0.613864\pi$
$600$	2372.97	0.161460
$601$	−5774.34	−0.391914	−0.195957	−	0.980613i	$-0.562781\pi$
$601$	−5774.34	−0.391914	−0.195957	+	0.980613i	$0.562781\pi$
$602$	−8059.60	−0.545656
$603$	−1699.74	−0.114790
$604$	2056.12	0.138514
$605$	10449.8	0.702225
$606$	11229.5	0.752748
$607$	−19362.3	−1.29471	−0.647356	−	0.762188i	$-0.724125\pi$
$607$	−19362.3	−1.29471	−0.647356	+	0.762188i	$0.724125\pi$
$608$	−18408.0	−1.22787
$609$	−372.027	−0.0247542
$610$	−20879.8	−1.38590
$611$	2198.09	0.145540
$612$	4168.76	0.275346
$613$	−1618.67	−0.106652	−0.0533258	−	0.998577i	$-0.516982\pi$
$613$	−1618.67	−0.106652	−0.0533258	+	0.998577i	$0.516982\pi$
$614$	1766.01	0.116075
$615$	3983.67	0.261199
$616$	−1562.86	−0.102223
$617$	23617.8	1.54103	0.770515	−	0.637421i	$-0.219999\pi$
$617$	23617.8	1.54103	0.770515	+	0.637421i	$0.219999\pi$
$618$	3246.99	0.211348
$619$	−11748.0	−0.762831	−0.381415	−	0.924404i	$-0.624563\pi$
$619$	−11748.0	−0.762831	−0.381415	+	0.924404i	$0.624563\pi$
$620$	2881.17	0.186630
$621$	2951.50	0.190724
$622$	−21009.9	−1.35437
$623$	5810.12	0.373640
$624$	−1328.70	−0.0852412
$625$	−6205.31	−0.397140
$626$	−22121.0	−1.41235
$627$	−3618.48	−0.230475
$628$	−567.705	−0.0360731
$629$	−11273.7	−0.714648
$630$	2723.52	0.172234
$631$	−19641.3	−1.23915	−0.619577	−	0.784936i	$-0.712696\pi$
$631$	−19641.3	−1.23915	−0.619577	+	0.784936i	$0.712696\pi$
$632$	14572.7	0.917202
$633$	1481.32	0.0930131
$634$	13424.1	0.840916
$635$	809.228	0.0505720
$636$	−5872.50	−0.366132
$637$	1295.66	0.0805902
$638$	404.097	0.0250758
$639$	−5930.94	−0.367175
$640$	13797.7	0.852188
$641$	−16920.7	−1.04263	−0.521315	−	0.853364i	$-0.674558\pi$
$641$	−16920.7	−1.04263	−0.521315	+	0.853364i	$0.674558\pi$
$642$	22120.0	1.35982
$643$	18387.9	1.12776	0.563879	−	0.825857i	$-0.309308\pi$
$643$	18387.9	1.12776	0.563879	+	0.825857i	$0.309308\pi$
$644$	−4134.36	−0.252976
$645$	5757.76	0.351491
$646$	52657.6	3.20710
$647$	23051.3	1.40068	0.700341	−	0.713808i	$-0.253031\pi$
$647$	23051.3	1.40068	0.700341	+	0.713808i	$0.253031\pi$
$648$	1210.28	0.0733711
$649$	−6035.18	−0.365026
$650$	−1000.72	−0.0603868
$651$	2945.24	0.177316
$652$	−2509.96	−0.150763
$653$	12253.2	0.734313	0.367157	−	0.930159i	$-0.380331\pi$
$653$	12253.2	0.734313	0.367157	+	0.930159i	$0.380331\pi$
$654$	−7285.72	−0.435618
$655$	−17389.4	−1.03734
$656$	−12491.0	−0.743433
$657$	9013.58	0.535241
$658$	14127.4	0.836997
$659$	−26381.8	−1.55947	−0.779735	−	0.626110i	$-0.784646\pi$
$659$	−26381.8	−1.55947	−0.779735	+	0.626110i	$0.784646\pi$
$660$	−920.892	−0.0543117
$661$	−10600.7	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$661$	−10600.7	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$662$	−23127.5	−1.35782
$663$	2131.47	0.124856
$664$	−16839.7	−0.984196
$665$	10709.1	0.624483
$666$	2699.59	0.157067
$667$	−1296.06	−0.0752378
$668$	−7172.05	−0.415411
$669$	−9714.09	−0.561388
$670$	−5464.11	−0.315070
$671$	7217.00	0.415215
$672$	−4788.96	−0.274908
$673$	26804.5	1.53527	0.767634	−	0.640888i	$-0.221434\pi$
$673$	26804.5	1.53527	0.767634	+	0.640888i	$0.221434\pi$
$674$	−28499.9	−1.62875
$675$	1429.33	0.0815034
$676$	−7833.02	−0.445665
$677$	−609.777	−0.0346169	−0.0173084	−	0.999850i	$-0.505510\pi$
$677$	−609.777	−0.0346169	−0.0173084	+	0.999850i	$0.505510\pi$
$678$	20883.8	1.18294
$679$	17515.5	0.989959
$680$	−16247.9	−0.916290
$681$	8810.73	0.495782
$682$	−3199.13	−0.179620
$683$	1573.14	0.0881327	0.0440664	−	0.999029i	$-0.485969\pi$
$683$	1573.14	0.0881327	0.0440664	+	0.999029i	$0.485969\pi$
$684$	−3925.17	−0.219419
$685$	18224.4	1.01653
$686$	20554.6	1.14399
$687$	998.789	0.0554675
$688$	−18053.8	−1.00043
$689$	−3002.59	−0.166022
$690$	9488.12	0.523488
$691$	7568.75	0.416684	0.208342	−	0.978056i	$-0.433193\pi$
$691$	7568.75	0.416684	0.208342	+	0.978056i	$0.433193\pi$
$692$	−12766.2	−0.701297
$693$	−941.371	−0.0516013
$694$	42190.7	2.30769
$695$	9328.77	0.509152
$696$	−531.459	−0.0289438
$697$	20037.8	1.08893
$698$	41184.8	2.23334
$699$	2615.43	0.141523
$700$	−2002.16	−0.108106
$701$	−8562.34	−0.461334	−0.230667	−	0.973033i	$-0.574091\pi$
$701$	−8562.34	−0.461334	−0.230667	+	0.973033i	$0.574091\pi$
$702$	−510.397	−0.0274412
$703$	10615.0	0.569492
$704$	−1186.59	−0.0635245
$705$	−10092.6	−0.539162
$706$	−11748.8	−0.626306
$707$	11487.3	0.611066
$708$	−6546.71	−0.347515
$709$	15813.7	0.837653	0.418826	−	0.908066i	$-0.362442\pi$
$709$	15813.7	0.837653	0.418826	+	0.908066i	$0.362442\pi$
$710$	−19066.1	−1.00780
$711$	8777.70	0.462995
$712$	8300.05	0.436878
$713$	10260.6	0.538935
$714$	13699.2	0.718041
$715$	−470.849	−0.0246276
$716$	13252.5	0.691718
$717$	−12363.0	−0.643940
$718$	1376.04	0.0715227
$719$	4560.92	0.236570	0.118285	−	0.992980i	$-0.462260\pi$
$719$	4560.92	0.236570	0.118285	+	0.992980i	$0.462260\pi$
$720$	6100.76	0.315781
$721$	3321.54	0.171568
$722$	−26203.8	−1.35070
$723$	−7211.53	−0.370954
$724$	−2135.58	−0.109625
$725$	−627.645	−0.0321519
$726$	−12586.5	−0.643428
$727$	925.436	0.0472112	0.0236056	−	0.999721i	$-0.492485\pi$
$727$	925.436	0.0472112	0.0236056	+	0.999721i	$0.492485\pi$
$728$	−866.814	−0.0441295
$729$	729.000	0.0370370
$730$	28975.8	1.46910
$731$	28961.4	1.46536
$732$	7828.70	0.395296
$733$	−5598.38	−0.282102	−0.141051	−	0.990002i	$-0.545048\pi$
$733$	−5598.38	−0.282102	−0.141051	+	0.990002i	$0.545048\pi$
$734$	−35530.7	−1.78673
$735$	−5949.07	−0.298550
$736$	−16683.7	−0.835555
$737$	1888.65	0.0943951
$738$	−4798.21	−0.239329
$739$	662.240	0.0329647	0.0164823	−	0.999864i	$-0.494753\pi$
$739$	662.240	0.0329647	0.0164823	+	0.999864i	$0.494753\pi$
$740$	2701.49	0.134201
$741$	−2006.92	−0.0994955
$742$	−19298.0	−0.954789
$743$	−19340.7	−0.954968	−0.477484	−	0.878641i	$-0.658451\pi$
$743$	−19340.7	−0.954968	−0.477484	+	0.878641i	$0.658451\pi$
$744$	4207.42	0.207327
$745$	−14099.3	−0.693367
$746$	−11514.8	−0.565129
$747$	−10143.2	−0.496813
$748$	−4632.07	−0.226424
$749$	22627.8	1.10387
$750$	15444.4	0.751932
$751$	−39737.0	−1.93079	−0.965396	−	0.260789i	$-0.916017\pi$
$751$	−39737.0	−1.93079	−0.965396	+	0.260789i	$0.916017\pi$
$752$	31645.9	1.53458
$753$	−12188.3	−0.589863
$754$	224.125	0.0108251
$755$	−4827.02	−0.232680
$756$	−1021.16	−0.0491260
$757$	14076.5	0.675850	0.337925	−	0.941173i	$-0.390275\pi$
$757$	14076.5	0.675850	0.337925	+	0.941173i	$0.390275\pi$
$758$	6287.68	0.301291
$759$	−3279.53	−0.156837
$760$	15298.5	0.730177
$761$	−31721.5	−1.51104	−0.755522	−	0.655123i	$-0.772617\pi$
$761$	−31721.5	−1.51104	−0.755522	+	0.655123i	$0.772617\pi$
$762$	−974.690	−0.0463377
$763$	−7452.99	−0.353626
$764$	76.0921	0.00360329
$765$	−9786.70	−0.462535
$766$	21679.1	1.02258
$767$	−3347.31	−0.157581
$768$	−13771.1	−0.647034
$769$	31852.7	1.49368	0.746838	−	0.665005i	$-0.231571\pi$
$769$	31852.7	1.49368	0.746838	+	0.665005i	$0.231571\pi$
$770$	−3026.21	−0.141632
$771$	−5284.96	−0.246865
$772$	8152.64	0.380078
$773$	−18635.8	−0.867117	−0.433559	−	0.901125i	$-0.642742\pi$
$773$	−18635.8	−0.867117	−0.433559	+	0.901125i	$0.642742\pi$
$774$	−6935.04	−0.322061
$775$	4968.90	0.230307
$776$	25021.7	1.15751
$777$	2761.57	0.127504
$778$	−46399.6	−2.13818
$779$	−18867.0	−0.867753
$780$	−510.757	−0.0234462
$781$	6590.11	0.301937
$782$	47725.1	2.18241
$783$	−320.118	−0.0146106
$784$	18653.6	0.849746
$785$	1332.76	0.0605966
$786$	20945.0	0.950487
$787$	−33929.5	−1.53679	−0.768395	−	0.639975i	$-0.778945\pi$
$787$	−33929.5	−1.53679	−0.768395	+	0.639975i	$0.778945\pi$
$788$	900.798	0.0407228
$789$	−8883.83	−0.400853
$790$	28217.5	1.27080
$791$	21363.2	0.960289
$792$	−1344.80	−0.0603349
$793$	4002.78	0.179247
$794$	12744.9	0.569645
$795$	13786.5	0.615039
$796$	10363.6	0.461465
$797$	9361.88	0.416079	0.208039	−	0.978120i	$-0.433292\pi$
$797$	9361.88	0.416079	0.208039	+	0.978120i	$0.433292\pi$
$798$	−12898.8	−0.572195
$799$	−50765.6	−2.24776
$800$	−8079.43	−0.357064
$801$	4999.43	0.220532
$802$	−33919.2	−1.49343
$803$	−10015.4	−0.440142
$804$	2048.72	0.0898668
$805$	9705.96	0.424957
$806$	−1774.34	−0.0775415
$807$	20466.2	0.892744
$808$	16410.1	0.714489
$809$	29671.3	1.28948	0.644739	−	0.764403i	$-0.276966\pi$
$809$	29671.3	1.28948	0.644739	+	0.764403i	$0.276966\pi$
$810$	2343.50	0.101657
$811$	−3375.51	−0.146153	−0.0730765	−	0.997326i	$-0.523282\pi$
$811$	−3375.51	−0.146153	−0.0730765	+	0.997326i	$0.523282\pi$
$812$	448.411	0.0193795
$813$	−18745.3	−0.808641
$814$	−2999.62	−0.129161
$815$	5892.47	0.253257
$816$	30686.7	1.31648
$817$	−27269.2	−1.16772
$818$	−5951.10	−0.254371
$819$	−522.115	−0.0222762
$820$	−4801.59	−0.204487
$821$	8874.78	0.377262	0.188631	−	0.982048i	$-0.439595\pi$
$821$	8874.78	0.377262	0.188631	+	0.982048i	$0.439595\pi$
$822$	−21950.8	−0.931413
$823$	39558.2	1.67547	0.837736	−	0.546076i	$-0.183879\pi$
$823$	39558.2	1.67547	0.837736	+	0.546076i	$0.183879\pi$
$824$	4744.99	0.200606
$825$	−1588.18	−0.0670223
$826$	−21513.6	−0.906240
$827$	−22212.5	−0.933983	−0.466992	−	0.884262i	$-0.654662\pi$
$827$	−22212.5	−0.933983	−0.466992	+	0.884262i	$0.654662\pi$
$828$	−3557.49	−0.149313
$829$	14964.8	0.626957	0.313479	−	0.949595i	$-0.398506\pi$
$829$	14964.8	0.626957	0.313479	+	0.949595i	$0.398506\pi$
$830$	−32607.1	−1.36362
$831$	−1579.99	−0.0659556
$832$	−658.121	−0.0274233
$833$	−29923.7	−1.24465
$834$	−11236.2	−0.466521
$835$	16837.3	0.697820
$836$	4361.42	0.180434
$837$	2534.29	0.104657
$838$	26383.2	1.08758
$839$	10074.6	0.414557	0.207279	−	0.978282i	$-0.433539\pi$
$839$	10074.6	0.414557	0.207279	+	0.978282i	$0.433539\pi$
$840$	3980.01	0.163480
$841$	−24248.4	−0.994236
$842$	−43687.4	−1.78809
$843$	−21491.4	−0.878057
$844$	−1785.47	−0.0728179
$845$	18389.0	0.748642
$846$	12156.2	0.494018
$847$	−12875.5	−0.522322
$848$	−43228.2	−1.75055
$849$	13540.6	0.547364
$850$	23111.9	0.932626
$851$	9620.69	0.387536
$852$	7148.68	0.287453
$853$	−21992.7	−0.882785	−0.441392	−	0.897314i	$-0.645515\pi$
$853$	−21992.7	−0.882785	−0.441392	+	0.897314i	$0.645515\pi$
$854$	25726.4	1.03084
$855$	9214.86	0.368586
$856$	32325.0	1.29071
$857$	−24754.1	−0.986678	−0.493339	−	0.869837i	$-0.664224\pi$
$857$	−24754.1	−0.986678	−0.493339	+	0.869837i	$0.664224\pi$
$858$	567.123	0.0225656
$859$	−34616.4	−1.37497	−0.687484	−	0.726199i	$-0.741285\pi$
$859$	−34616.4	−1.37497	−0.687484	+	0.726199i	$0.741285\pi$
$860$	−6939.93	−0.275174
$861$	−4908.37	−0.194282
$862$	−39446.9	−1.55866
$863$	−3248.74	−0.128144	−0.0640721	−	0.997945i	$-0.520409\pi$
$863$	−3248.74	−0.128144	−0.0640721	+	0.997945i	$0.520409\pi$
$864$	−4120.76	−0.162258
$865$	29970.3	1.17806
$866$	−50207.5	−1.97012
$867$	−34487.9	−1.35095
$868$	−3549.95	−0.138817
$869$	−9753.26	−0.380732
$870$	−1029.08	−0.0401023
$871$	1047.50	0.0407501
$872$	−10647.0	−0.413477
$873$	15071.5	0.584300
$874$	−44936.5	−1.73913
$875$	15799.0	0.610403
$876$	−10864.2	−0.419028
$877$	−50458.7	−1.94284	−0.971419	−	0.237373i	$-0.923714\pi$
$877$	−50458.7	−1.94284	−0.971419	+	0.237373i	$0.923714\pi$
$878$	−56383.5	−2.16726
$879$	−18652.9	−0.715751
$880$	−6778.80	−0.259674
$881$	23222.9	0.888080	0.444040	−	0.896007i	$-0.353545\pi$
$881$	23222.9	0.888080	0.444040	+	0.896007i	$0.353545\pi$
$882$	7165.47	0.273553
$883$	−6096.02	−0.232330	−0.116165	−	0.993230i	$-0.537060\pi$
$883$	−6096.02	−0.232330	−0.116165	+	0.993230i	$0.537060\pi$
$884$	−2569.10	−0.0977467
$885$	15369.3	0.583766
$886$	20457.6	0.775718
$887$	16271.3	0.615936	0.307968	−	0.951397i	$-0.400351\pi$
$887$	16271.3	0.615936	0.307968	+	0.951397i	$0.400351\pi$
$888$	3945.04	0.149084
$889$	−997.068	−0.0376160
$890$	16071.6	0.605304
$891$	−810.021	−0.0304565
$892$	11708.6	0.439498
$893$	47799.3	1.79120
$894$	16982.2	0.635312
$895$	−31112.0	−1.16197
$896$	−17000.4	−0.633865
$897$	−1818.93	−0.0677061
$898$	21073.0	0.783092
$899$	−1112.86	−0.0412857
$900$	−1722.79	−0.0638072
$901$	69345.7	2.56408
$902$	5331.48	0.196806
$903$	−7094.26	−0.261442
$904$	30518.4	1.12282
$905$	5013.55	0.184150
$906$	5814.00	0.213198
$907$	−16228.8	−0.594123	−0.297062	−	0.954858i	$-0.596007\pi$
$907$	−16228.8	−0.594123	−0.297062	+	0.954858i	$0.596007\pi$
$908$	−10619.7	−0.388137
$909$	9884.45	0.360667
$910$	−1678.43	−0.0611423
$911$	−37215.2	−1.35345	−0.676725	−	0.736236i	$-0.736602\pi$
$911$	−37215.2	−1.35345	−0.676725	+	0.736236i	$0.736602\pi$
$912$	−28893.7	−1.04908
$913$	11270.5	0.408542
$914$	−11761.9	−0.425655
$915$	−18378.9	−0.664030
$916$	−1203.86	−0.0434243
$917$	21425.9	0.771586
$918$	11787.8	0.423807
$919$	−4216.57	−0.151351	−0.0756756	−	0.997132i	$-0.524111\pi$
$919$	−4216.57	−0.151351	−0.0756756	+	0.997132i	$0.524111\pi$
$920$	13865.5	0.496881
$921$	1554.48	0.0556156
$922$	757.979	0.0270745
$923$	3655.09	0.130345
$924$	1134.65	0.0403975
$925$	4659.03	0.165608
$926$	51095.6	1.81329
$927$	2858.09	0.101264
$928$	1809.50	0.0640085
$929$	15674.1	0.553552	0.276776	−	0.960934i	$-0.410734\pi$
$929$	15674.1	0.553552	0.276776	+	0.960934i	$0.410734\pi$
$930$	8146.93	0.287256
$931$	28175.2	0.991843
$932$	−3152.43	−0.110795
$933$	−18493.4	−0.648925
$934$	−34004.3	−1.19128
$935$	10874.4	0.380354
$936$	−745.867	−0.0260464
$937$	28335.2	0.987909	0.493954	−	0.869488i	$-0.335551\pi$
$937$	28335.2	0.987909	0.493954	+	0.869488i	$0.335551\pi$
$938$	6732.46	0.234352
$939$	−19471.5	−0.676706
$940$	12164.8	0.422098
$941$	35690.7	1.23643	0.618216	−	0.786008i	$-0.287856\pi$
$941$	35690.7	1.23643	0.618216	+	0.786008i	$0.287856\pi$
$942$	−1605.27	−0.0555229
$943$	−17099.7	−0.590501
$944$	−48191.2	−1.66154
$945$	2397.31	0.0825232
$946$	7705.80	0.264839
$947$	−5225.75	−0.179318	−0.0896589	−	0.995973i	$-0.528578\pi$
$947$	−5225.75	−0.179318	−0.0896589	+	0.995973i	$0.528578\pi$
$948$	−10579.9	−0.362468
$949$	−5554.84	−0.190008
$950$	−21761.5	−0.743195
$951$	11816.3	0.402912
$952$	20019.4	0.681545
$953$	22484.7	0.764270	0.382135	−	0.924106i	$-0.375189\pi$
$953$	22484.7	0.764270	0.382135	+	0.924106i	$0.375189\pi$
$954$	−16605.4	−0.563542
$955$	−178.636	−0.00605292
$956$	14901.4	0.504126
$957$	355.696	0.0120147
$958$	38272.1	1.29073
$959$	−22454.7	−0.756101
$960$	3021.78	0.101591
$961$	−20980.8	−0.704267
$962$	−1663.69	−0.0557582
$963$	19470.5	0.651537
$964$	8692.20	0.290412
$965$	−19139.4	−0.638465
$966$	−11690.5	−0.389375
$967$	−38828.5	−1.29125	−0.645626	−	0.763654i	$-0.723404\pi$
$967$	−38828.5	−1.29125	−0.645626	+	0.763654i	$0.723404\pi$
$968$	−18393.3	−0.610725
$969$	46350.6	1.53663
$970$	48450.2	1.60375
$971$	43762.3	1.44634	0.723171	−	0.690669i	$-0.242684\pi$
$971$	43762.3	1.44634	0.723171	+	0.690669i	$0.242684\pi$
$972$	−878.677	−0.0289955
$973$	−11494.2	−0.378712
$974$	63800.6	2.09887
$975$	−880.858	−0.0289334
$976$	57628.0	1.88999
$977$	17347.9	0.568075	0.284038	−	0.958813i	$-0.408326\pi$
$977$	17347.9	0.568075	0.284038	+	0.958813i	$0.408326\pi$
$978$	−7097.29	−0.232052
$979$	−5555.07	−0.181349
$980$	7170.52	0.233728
$981$	−6413.07	−0.208719
$982$	32630.0	1.06035
$983$	10377.5	0.336714	0.168357	−	0.985726i	$-0.446154\pi$
$983$	10377.5	0.336714	0.168357	+	0.985726i	$0.446154\pi$
$984$	−7011.85	−0.227164
$985$	−2114.74	−0.0684073
$986$	−5176.24	−0.167186
$987$	12435.3	0.401034
$988$	2418.98	0.0778928
$989$	−24714.8	−0.794627
$990$	−2603.96	−0.0835953
$991$	−2684.26	−0.0860426	−0.0430213	−	0.999074i	$-0.513698\pi$
$991$	−2684.26	−0.0860426	−0.0430213	+	0.999074i	$0.513698\pi$
$992$	−14325.4	−0.458499
$993$	−20357.4	−0.650576
$994$	23491.8	0.749611
$995$	−24329.8	−0.775182
$996$	12225.8	0.388944
$997$	−18156.4	−0.576749	−0.288374	−	0.957518i	$-0.593115\pi$
$997$	−18156.4	−0.576749	−0.288374	+	0.957518i	$0.593115\pi$
$998$	−10391.7	−0.329602
$999$	2376.24	0.0752563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.6	✓	22
3.2	odd	2		1413.4.a.e.1.17		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.6	✓	22	1.1	even	1	trivial
1413.4.a.e.1.17		22	3.2	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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-3.40822 q^{2} -3.00000 q^{3} +3.61596 q^{4} -8.48893 q^{5} +10.2247 q^{6} +10.4594 q^{7} +14.9418 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.40822 q^{2} -3.00000 q^{3} +3.61596 q^{4} -8.48893 q^{5} +10.2247 q^{6} +10.4594 q^{7} +14.9418 q^{8} +9.00000 q^{9} +28.9321 q^{10} -10.0003 q^{11} -10.8479 q^{12} -5.54647 q^{13} -35.6479 q^{14} +25.4668 q^{15} -79.8525 q^{16} +128.098 q^{17} -30.6740 q^{18} -120.613 q^{19} -30.6956 q^{20} -31.3782 q^{21} +34.0831 q^{22} -109.315 q^{23} -44.8253 q^{24} -52.9380 q^{25} +18.9036 q^{26} -27.0000 q^{27} +37.8207 q^{28} +11.8562 q^{29} -86.7964 q^{30} -93.8626 q^{31} +152.621 q^{32} +30.0008 q^{33} -436.584 q^{34} -88.7892 q^{35} +32.5436 q^{36} -88.0091 q^{37} +411.074 q^{38} +16.6394 q^{39} -126.840 q^{40} +156.426 q^{41} +106.944 q^{42} +226.089 q^{43} -36.1605 q^{44} -76.4004 q^{45} +372.568 q^{46} -396.304 q^{47} +239.558 q^{48} -233.601 q^{49} +180.424 q^{50} -384.293 q^{51} -20.0558 q^{52} +541.351 q^{53} +92.0219 q^{54} +84.8916 q^{55} +156.282 q^{56} +361.838 q^{57} -40.4086 q^{58} +603.502 q^{59} +92.0868 q^{60} -721.681 q^{61} +319.904 q^{62} +94.1346 q^{63} +118.656 q^{64} +47.0836 q^{65} -102.249 q^{66} -188.860 q^{67} +463.195 q^{68} +327.944 q^{69} +302.613 q^{70} -658.994 q^{71} +134.476 q^{72} +1001.51 q^{73} +299.954 q^{74} +158.814 q^{75} -436.130 q^{76} -104.597 q^{77} -56.7108 q^{78} +975.300 q^{79} +677.863 q^{80} +81.0000 q^{81} -533.134 q^{82} -1127.02 q^{83} -113.462 q^{84} -1087.41 q^{85} -770.560 q^{86} -35.5687 q^{87} -149.422 q^{88} +555.492 q^{89} +260.389 q^{90} -58.0128 q^{91} -395.277 q^{92} +281.588 q^{93} +1350.69 q^{94} +1023.87 q^{95} -457.862 q^{96} +1674.62 q^{97} +796.163 q^{98} -90.0024 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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