LMFDB - Embedded newform 4009.2.a.f.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30870 q^{2} -1.28180 q^{3} +3.33008 q^{4} +1.74419 q^{5} +2.95929 q^{6} -4.60684 q^{7} -3.07074 q^{8} -1.35699 q^{9} +O(q^{10})$	\(q-2.30870 q^{2} -1.28180 q^{3} +3.33008 q^{4} +1.74419 q^{5} +2.95929 q^{6} -4.60684 q^{7} -3.07074 q^{8} -1.35699 q^{9} -4.02679 q^{10} -0.0160486 q^{11} -4.26850 q^{12} -2.94863 q^{13} +10.6358 q^{14} -2.23570 q^{15} +0.429252 q^{16} -2.09416 q^{17} +3.13287 q^{18} -1.00000 q^{19} +5.80827 q^{20} +5.90506 q^{21} +0.0370514 q^{22} -7.23331 q^{23} +3.93608 q^{24} -1.95782 q^{25} +6.80750 q^{26} +5.58479 q^{27} -15.3411 q^{28} -8.17492 q^{29} +5.16155 q^{30} +0.941538 q^{31} +5.15047 q^{32} +0.0205712 q^{33} +4.83477 q^{34} -8.03519 q^{35} -4.51886 q^{36} +2.42152 q^{37} +2.30870 q^{38} +3.77956 q^{39} -5.35594 q^{40} -9.38574 q^{41} -13.6330 q^{42} -6.97664 q^{43} -0.0534432 q^{44} -2.36683 q^{45} +16.6995 q^{46} -6.54631 q^{47} -0.550216 q^{48} +14.2230 q^{49} +4.52001 q^{50} +2.68429 q^{51} -9.81917 q^{52} -13.2751 q^{53} -12.8936 q^{54} -0.0279918 q^{55} +14.1464 q^{56} +1.28180 q^{57} +18.8734 q^{58} +3.56720 q^{59} -7.44505 q^{60} +5.44523 q^{61} -2.17373 q^{62} +6.25142 q^{63} -12.7494 q^{64} -5.14296 q^{65} -0.0474926 q^{66} -15.2023 q^{67} -6.97370 q^{68} +9.27167 q^{69} +18.5508 q^{70} +6.00387 q^{71} +4.16695 q^{72} -8.14847 q^{73} -5.59056 q^{74} +2.50953 q^{75} -3.33008 q^{76} +0.0739336 q^{77} -8.72586 q^{78} +8.12239 q^{79} +0.748695 q^{80} -3.08763 q^{81} +21.6688 q^{82} +14.2503 q^{83} +19.6643 q^{84} -3.65260 q^{85} +16.1069 q^{86} +10.4786 q^{87} +0.0492812 q^{88} -14.4051 q^{89} +5.46430 q^{90} +13.5839 q^{91} -24.0875 q^{92} -1.20686 q^{93} +15.1134 q^{94} -1.74419 q^{95} -6.60187 q^{96} +4.23922 q^{97} -32.8366 q^{98} +0.0217778 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−2.30870	−1.63249	−0.816247	−	0.577703i	$-0.803949\pi$
$2$	−2.30870	−1.63249	−0.816247	+	0.577703i	$0.803949\pi$
$3$	−1.28180	−0.740048	−0.370024	−	0.929022i	$-0.620651\pi$
$3$	−1.28180	−0.740048	−0.370024	+	0.929022i	$0.620651\pi$
$4$	3.33008	1.66504
$5$	1.74419	0.780023	0.390012	−	0.920810i	$-0.372471\pi$
$5$	1.74419	0.780023	0.390012	+	0.920810i	$0.372471\pi$
$6$	2.95929	1.20812
$7$	−4.60684	−1.74122	−0.870611	−	0.491971i	$-0.836277\pi$
$7$	−4.60684	−1.74122	−0.870611	+	0.491971i	$0.836277\pi$
$8$	−3.07074	−1.08567
$9$	−1.35699	−0.452329
$10$	−4.02679	−1.27338
$11$	−0.0160486	−0.00483885	−0.00241942	−	0.999997i	$-0.500770\pi$
$11$	−0.0160486	−0.00483885	−0.00241942	+	0.999997i	$0.500770\pi$
$12$	−4.26850	−1.23221
$13$	−2.94863	−0.817804	−0.408902	−	0.912578i	$-0.634088\pi$
$13$	−2.94863	−0.817804	−0.408902	+	0.912578i	$0.634088\pi$
$14$	10.6358	2.84254
$15$	−2.23570	−0.577255
$16$	0.429252	0.107313
$17$	−2.09416	−0.507907	−0.253954	−	0.967216i	$-0.581731\pi$
$17$	−2.09416	−0.507907	−0.253954	+	0.967216i	$0.581731\pi$
$18$	3.13287	0.738424
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	5.80827	1.29877
$21$	5.90506	1.28859
$22$	0.0370514	0.00789939
$23$	−7.23331	−1.50825	−0.754125	−	0.656731i	$-0.771939\pi$
$23$	−7.23331	−1.50825	−0.754125	+	0.656731i	$0.771939\pi$
$24$	3.93608	0.803449
$25$	−1.95782	−0.391564
$26$	6.80750	1.33506
$27$	5.58479	1.07479
$28$	−15.3411	−2.89920
$29$	−8.17492	−1.51804	−0.759022	−	0.651065i	$-0.774323\pi$
$29$	−8.17492	−1.51804	−0.759022	+	0.651065i	$0.774323\pi$
$30$	5.16155	0.942365
$31$	0.941538	0.169105	0.0845526	−	0.996419i	$-0.473054\pi$
$31$	0.941538	0.169105	0.0845526	+	0.996419i	$0.473054\pi$
$32$	5.15047	0.910483
$33$	0.0205712	0.00358098
$34$	4.83477	0.829156
$35$	−8.03519	−1.35819
$36$	−4.51886	−0.753144
$37$	2.42152	0.398096	0.199048	−	0.979990i	$-0.436215\pi$
$37$	2.42152	0.398096	0.199048	+	0.979990i	$0.436215\pi$
$38$	2.30870	0.374520
$39$	3.77956	0.605214
$40$	−5.35594	−0.846848
$41$	−9.38574	−1.46581	−0.732904	−	0.680333i	$-0.761835\pi$
$41$	−9.38574	−1.46581	−0.732904	+	0.680333i	$0.761835\pi$
$42$	−13.6330	−2.10361
$43$	−6.97664	−1.06393	−0.531963	−	0.846767i	$-0.678546\pi$
$43$	−6.97664	−1.06393	−0.531963	+	0.846767i	$0.678546\pi$
$44$	−0.0534432	−0.00805686
$45$	−2.36683	−0.352827
$46$	16.6995	2.46221
$47$	−6.54631	−0.954877	−0.477438	−	0.878665i	$-0.658435\pi$
$47$	−6.54631	−0.954877	−0.477438	+	0.878665i	$0.658435\pi$
$48$	−0.550216	−0.0794168
$49$	14.2230	2.03186
$50$	4.52001	0.639225
$51$	2.68429	0.375876
$52$	−9.81917	−1.36167
$53$	−13.2751	−1.82347	−0.911735	−	0.410779i	$-0.865257\pi$
$53$	−13.2751	−1.82347	−0.911735	+	0.410779i	$0.865257\pi$
$54$	−12.8936	−1.75459
$55$	−0.0279918	−0.00377441
$56$	14.1464	1.89039
$57$	1.28180	0.169779
$58$	18.8734	2.47820
$59$	3.56720	0.464409	0.232205	−	0.972667i	$-0.425406\pi$
$59$	3.56720	0.464409	0.232205	+	0.972667i	$0.425406\pi$
$60$	−7.44505	−0.961151
$61$	5.44523	0.697190	0.348595	−	0.937274i	$-0.386659\pi$
$61$	5.44523	0.697190	0.348595	+	0.937274i	$0.386659\pi$
$62$	−2.17373	−0.276063
$63$	6.25142	0.787605
$64$	−12.7494	−1.59367
$65$	−5.14296	−0.637906
$66$	−0.0474926	−0.00584593
$67$	−15.2023	−1.85725	−0.928627	−	0.371014i	$-0.879010\pi$
$67$	−15.2023	−1.85725	−0.928627	+	0.371014i	$0.879010\pi$
$68$	−6.97370	−0.845685
$69$	9.27167	1.11618
$70$	18.5508	2.21724
$71$	6.00387	0.712528	0.356264	−	0.934385i	$-0.384050\pi$
$71$	6.00387	0.712528	0.356264	+	0.934385i	$0.384050\pi$
$72$	4.16695	0.491080
$73$	−8.14847	−0.953706	−0.476853	−	0.878983i	$-0.658223\pi$
$73$	−8.14847	−0.953706	−0.476853	+	0.878983i	$0.658223\pi$
$74$	−5.59056	−0.649889
$75$	2.50953	0.289776
$76$	−3.33008	−0.381986
$77$	0.0739336	0.00842551
$78$	−8.72586	−0.988009
$79$	8.12239	0.913840	0.456920	−	0.889508i	$-0.348952\pi$
$79$	8.12239	0.913840	0.456920	+	0.889508i	$0.348952\pi$
$80$	0.748695	0.0837067
$81$	−3.08763	−0.343070
$82$	21.6688	2.39292
$83$	14.2503	1.56418	0.782090	−	0.623166i	$-0.214154\pi$
$83$	14.2503	1.56418	0.782090	+	0.623166i	$0.214154\pi$
$84$	19.6643	2.14555
$85$	−3.65260	−0.396180
$86$	16.1069	1.73685
$87$	10.4786	1.12343
$88$	0.0492812	0.00525339
$89$	−14.4051	−1.52693	−0.763466	−	0.645848i	$-0.776504\pi$
$89$	−14.4051	−1.52693	−0.763466	+	0.645848i	$0.776504\pi$
$90$	5.46430	0.575988
$91$	13.5839	1.42398
$92$	−24.0875	−2.51129
$93$	−1.20686	−0.125146
$94$	15.1134	1.55883
$95$	−1.74419	−0.178950
$96$	−6.60187	−0.673801
$97$	4.23922	0.430428	0.215214	−	0.976567i	$-0.430955\pi$
$97$	4.23922	0.430428	0.215214	+	0.976567i	$0.430955\pi$
$98$	−32.8366	−3.31700
$99$	0.0217778	0.00218875
$100$	−6.51968	−0.651968
$101$	9.50818	0.946099	0.473049	−	0.881036i	$-0.343153\pi$
$101$	9.50818	0.946099	0.473049	+	0.881036i	$0.343153\pi$
$102$	−6.19721	−0.613616
$103$	−5.86197	−0.577597	−0.288799	−	0.957390i	$-0.593256\pi$
$103$	−5.86197	−0.577597	−0.288799	+	0.957390i	$0.593256\pi$
$104$	9.05448	0.887865
$105$	10.2995	1.00513
$106$	30.6481	2.97680
$107$	14.0773	1.36091	0.680453	−	0.732791i	$-0.261783\pi$
$107$	14.0773	1.36091	0.680453	+	0.732791i	$0.261783\pi$
$108$	18.5978	1.78957
$109$	3.75659	0.359816	0.179908	−	0.983683i	$-0.442420\pi$
$109$	3.75659	0.359816	0.179908	+	0.983683i	$0.442420\pi$
$110$	0.0646245	0.00616171
$111$	−3.10391	−0.294610
$112$	−1.97750	−0.186856
$113$	−12.7246	−1.19703	−0.598514	−	0.801113i	$-0.704242\pi$
$113$	−12.7246	−1.19703	−0.598514	+	0.801113i	$0.704242\pi$
$114$	−2.95929	−0.277163
$115$	−12.6162	−1.17647
$116$	−27.2231	−2.52760
$117$	4.00125	0.369916
$118$	−8.23557	−0.758146
$119$	9.64745	0.884380
$120$	6.86525	0.626709
$121$	−10.9997	−0.999977
$122$	−12.5714	−1.13816
$123$	12.0307	1.08477
$124$	3.13539	0.281567
$125$	−12.1357	−1.08545
$126$	−14.4326	−1.28576
$127$	−8.82754	−0.783318	−0.391659	−	0.920111i	$-0.628099\pi$
$127$	−8.82754	−0.783318	−0.391659	+	0.920111i	$0.628099\pi$
$128$	19.1335	1.69118
$129$	8.94266	0.787357
$130$	11.8735	1.04138
$131$	4.45578	0.389303	0.194652	−	0.980872i	$-0.437642\pi$
$131$	4.45578	0.389303	0.194652	+	0.980872i	$0.437642\pi$
$132$	0.0685035	0.00596247
$133$	4.60684	0.399464
$134$	35.0975	3.03196
$135$	9.74091	0.838364
$136$	6.43061	0.551420
$137$	5.70460	0.487377	0.243689	−	0.969854i	$-0.421643\pi$
$137$	5.70460	0.487377	0.243689	+	0.969854i	$0.421643\pi$
$138$	−21.4055	−1.82215
$139$	12.5105	1.06113	0.530564	−	0.847645i	$-0.321980\pi$
$139$	12.5105	1.06113	0.530564	+	0.847645i	$0.321980\pi$
$140$	−26.7578	−2.26145
$141$	8.39106	0.706655
$142$	−13.8611	−1.16320
$143$	0.0473216	0.00395723
$144$	−0.582489	−0.0485407
$145$	−14.2586	−1.18411
$146$	18.8123	1.55692
$147$	−18.2311	−1.50367
$148$	8.06385	0.662844
$149$	−4.32361	−0.354204	−0.177102	−	0.984192i	$-0.556672\pi$
$149$	−4.32361	−0.354204	−0.177102	+	0.984192i	$0.556672\pi$
$150$	−5.79375	−0.473058
$151$	−21.8577	−1.77876	−0.889379	−	0.457170i	$-0.848863\pi$
$151$	−21.8577	−1.77876	−0.889379	+	0.457170i	$0.848863\pi$
$152$	3.07074	0.249070
$153$	2.84174	0.229741
$154$	−0.170690	−0.0137546
$155$	1.64222	0.131906
$156$	12.5862	1.00770
$157$	4.63302	0.369755	0.184878	−	0.982762i	$-0.440811\pi$
$157$	4.63302	0.369755	0.184878	+	0.982762i	$0.440811\pi$
$158$	−18.7521	−1.49184
$159$	17.0160	1.34946
$160$	8.98337	0.710198
$161$	33.3227	2.62620
$162$	7.12841	0.560060
$163$	−10.4335	−0.817218	−0.408609	−	0.912710i	$-0.633986\pi$
$163$	−10.4335	−0.817218	−0.408609	+	0.912710i	$0.633986\pi$
$164$	−31.2552	−2.44062
$165$	0.0358799	0.00279325
$166$	−32.8997	−2.55351
$167$	−13.4688	−1.04224	−0.521122	−	0.853482i	$-0.674487\pi$
$167$	−13.4688	−1.04224	−0.521122	+	0.853482i	$0.674487\pi$
$168$	−18.1329	−1.39898
$169$	−4.30556	−0.331197
$170$	8.43273	0.646761
$171$	1.35699	0.103771
$172$	−23.2327	−1.77148
$173$	3.45814	0.262917	0.131459	−	0.991322i	$-0.458034\pi$
$173$	3.45814	0.262917	0.131459	+	0.991322i	$0.458034\pi$
$174$	−24.1919	−1.83399
$175$	9.01936	0.681800
$176$	−0.00688891	−0.000519271 0
$177$	−4.57244	−0.343685
$178$	33.2569	2.49271
$179$	−23.7515	−1.77527	−0.887636	−	0.460546i	$-0.847653\pi$
$179$	−23.7515	−1.77527	−0.887636	+	0.460546i	$0.847653\pi$
$180$	−7.88174	−0.587470
$181$	−22.0240	−1.63703	−0.818514	−	0.574487i	$-0.805202\pi$
$181$	−22.0240	−1.63703	−0.818514	+	0.574487i	$0.805202\pi$
$182$	−31.3611	−2.32464
$183$	−6.97970	−0.515954
$184$	22.2116	1.63746
$185$	4.22358	0.310524
$186$	2.78628	0.204300
$187$	0.0336084	0.00245769
$188$	−21.7997	−1.58991
$189$	−25.7282	−1.87145
$190$	4.02679	0.292134
$191$	2.41072	0.174434	0.0872168	−	0.996189i	$-0.472203\pi$
$191$	2.41072	0.174434	0.0872168	+	0.996189i	$0.472203\pi$
$192$	16.3422	1.17939
$193$	−8.24272	−0.593324	−0.296662	−	0.954983i	$-0.595873\pi$
$193$	−8.24272	−0.593324	−0.296662	+	0.954983i	$0.595873\pi$
$194$	−9.78707	−0.702671
$195$	6.59226	0.472081
$196$	47.3637	3.38312
$197$	−21.6756	−1.54432	−0.772161	−	0.635427i	$-0.780824\pi$
$197$	−21.6756	−1.54432	−0.772161	+	0.635427i	$0.780824\pi$
$198$	−0.0502783	−0.00357312
$199$	15.0206	1.06478	0.532391	−	0.846499i	$-0.321294\pi$
$199$	15.0206	1.06478	0.532391	+	0.846499i	$0.321294\pi$
$200$	6.01195	0.425109
$201$	19.4863	1.37446
$202$	−21.9515	−1.54450
$203$	37.6606	2.64325
$204$	8.93890	0.625848
$205$	−16.3705	−1.14336
$206$	13.5335	0.942924
$207$	9.81550	0.682224
$208$	−1.26571	−0.0877610
$209$	0.0160486	0.00111011
$210$	−23.7784	−1.64087
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−44.2070	−3.03615
$213$	−7.69577	−0.527305
$214$	−32.5003	−2.22167
$215$	−12.1685	−0.829888
$216$	−17.1494	−1.16687
$217$	−4.33752	−0.294450
$218$	−8.67282	−0.587398
$219$	10.4447	0.705789
$220$	−0.0932148	−0.00628454
$221$	6.17490	0.415369
$222$	7.16598	0.480949
$223$	10.3453	0.692770	0.346385	−	0.938092i	$-0.387409\pi$
$223$	10.3453	0.692770	0.346385	+	0.938092i	$0.387409\pi$
$224$	−23.7274	−1.58535
$225$	2.65673	0.177115
$226$	29.3772	1.95414
$227$	15.3525	1.01898	0.509489	−	0.860477i	$-0.329834\pi$
$227$	15.3525	1.01898	0.509489	+	0.860477i	$0.329834\pi$
$228$	4.26850	0.282688
$229$	−16.3217	−1.07857	−0.539284	−	0.842124i	$-0.681305\pi$
$229$	−16.3217	−1.07857	−0.539284	+	0.842124i	$0.681305\pi$
$230$	29.1270	1.92058
$231$	−0.0947681	−0.00623529
$232$	25.1030	1.64810
$233$	−2.68852	−0.176131	−0.0880655	−	0.996115i	$-0.528068\pi$
$233$	−2.68852	−0.176131	−0.0880655	+	0.996115i	$0.528068\pi$
$234$	−9.23767	−0.603886
$235$	−11.4180	−0.744826
$236$	11.8790	0.773259
$237$	−10.4113	−0.676286
$238$	−22.2730	−1.44375
$239$	22.8458	1.47777	0.738887	−	0.673829i	$-0.235352\pi$
$239$	22.8458	1.47777	0.738887	+	0.673829i	$0.235352\pi$
$240$	−0.959678	−0.0619470
$241$	−19.9847	−1.28733	−0.643665	−	0.765307i	$-0.722587\pi$
$241$	−19.9847	−1.28733	−0.643665	+	0.765307i	$0.722587\pi$
$242$	25.3951	1.63246
$243$	−12.7966	−0.820905
$244$	18.1330	1.16085
$245$	24.8075	1.58490
$246$	−27.7751	−1.77088
$247$	2.94863	0.187617
$248$	−2.89122	−0.183593
$249$	−18.2661	−1.15757
$250$	28.0177	1.77199
$251$	19.9629	1.26005	0.630023	−	0.776577i	$-0.283045\pi$
$251$	19.9629	1.26005	0.630023	+	0.776577i	$0.283045\pi$
$252$	20.8177	1.31139
$253$	0.116085	0.00729819
$254$	20.3801	1.27876
$255$	4.68190	0.293192
$256$	−18.6746	−1.16716
$257$	−9.95149	−0.620757	−0.310378	−	0.950613i	$-0.600456\pi$
$257$	−9.95149	−0.620757	−0.310378	+	0.950613i	$0.600456\pi$
$258$	−20.6459	−1.28536
$259$	−11.1556	−0.693173
$260$	−17.1265	−1.06214
$261$	11.0932	0.686655
$262$	−10.2870	−0.635535
$263$	0.950162	0.0585895	0.0292947	−	0.999571i	$-0.490674\pi$
$263$	0.950162	0.0585895	0.0292947	+	0.999571i	$0.490674\pi$
$264$	−0.0631687	−0.00388776
$265$	−23.1542	−1.42235
$266$	−10.6358	−0.652123
$267$	18.4644	1.13000
$268$	−50.6248	−3.09240
$269$	14.3051	0.872200	0.436100	−	0.899898i	$-0.356359\pi$
$269$	14.3051	0.872200	0.436100	+	0.899898i	$0.356359\pi$
$270$	−22.4888	−1.36862
$271$	−16.6340	−1.01044	−0.505221	−	0.862990i	$-0.668589\pi$
$271$	−16.6340	−1.01044	−0.505221	+	0.862990i	$0.668589\pi$
$272$	−0.898921	−0.0545051
$273$	−17.4118	−1.05381
$274$	−13.1702	−0.795641
$275$	0.0314203	0.00189472
$276$	30.8754	1.85848
$277$	16.3445	0.982047	0.491024	−	0.871146i	$-0.336623\pi$
$277$	16.3445	0.982047	0.491024	+	0.871146i	$0.336623\pi$
$278$	−28.8829	−1.73228
$279$	−1.27765	−0.0764911
$280$	24.6740	1.47455
$281$	4.53183	0.270347	0.135173	−	0.990822i	$-0.456841\pi$
$281$	4.53183	0.270347	0.135173	+	0.990822i	$0.456841\pi$
$282$	−19.3724	−1.15361
$283$	−7.25080	−0.431015	−0.215508	−	0.976502i	$-0.569141\pi$
$283$	−7.25080	−0.431015	−0.215508	+	0.976502i	$0.569141\pi$
$284$	19.9933	1.18639
$285$	2.23570	0.132431
$286$	−0.109251	−0.00646015
$287$	43.2387	2.55230
$288$	−6.98911	−0.411837
$289$	−12.6145	−0.742030
$290$	32.9187	1.93305
$291$	−5.43384	−0.318537
$292$	−27.1350	−1.58796
$293$	7.15200	0.417824	0.208912	−	0.977934i	$-0.433008\pi$
$293$	7.15200	0.417824	0.208912	+	0.977934i	$0.433008\pi$
$294$	42.0900	2.45474
$295$	6.22185	0.362250
$296$	−7.43586	−0.432201
$297$	−0.0896283	−0.00520076
$298$	9.98191	0.578236
$299$	21.3284	1.23345
$300$	8.35694	0.482488
$301$	32.1403	1.85253
$302$	50.4629	2.90381
$303$	−12.1876	−0.700159
$304$	−0.429252	−0.0246193
$305$	9.49748	0.543824
$306$	−6.56071	−0.375051
$307$	7.73787	0.441624	0.220812	−	0.975316i	$-0.429129\pi$
$307$	7.73787	0.441624	0.220812	+	0.975316i	$0.429129\pi$
$308$	0.246204	0.0140288
$309$	7.51388	0.427450
$310$	−3.79138	−0.215336
$311$	−3.69560	−0.209558	−0.104779	−	0.994496i	$-0.533414\pi$
$311$	−3.69560	−0.209558	−0.104779	+	0.994496i	$0.533414\pi$
$312$	−11.6061	−0.657063
$313$	5.31533	0.300440	0.150220	−	0.988653i	$-0.452002\pi$
$313$	5.31533	0.300440	0.150220	+	0.988653i	$0.452002\pi$
$314$	−10.6962	−0.603623
$315$	10.9036	0.614350
$316$	27.0482	1.52158
$317$	−31.6938	−1.78010	−0.890052	−	0.455859i	$-0.849332\pi$
$317$	−31.6938	−1.78010	−0.890052	+	0.455859i	$0.849332\pi$
$318$	−39.2847	−2.20298
$319$	0.131196	0.00734559
$320$	−22.2373	−1.24310
$321$	−18.0443	−1.00714
$322$	−76.9321	−4.28726
$323$	2.09416	0.116522
$324$	−10.2821	−0.571225
$325$	5.77289	0.320222
$326$	24.0879	1.33410
$327$	−4.81520	−0.266281
$328$	28.8212	1.59138
$329$	30.1578	1.66265
$330$	−0.0828358	−0.00455996
$331$	−9.07313	−0.498704	−0.249352	−	0.968413i	$-0.580218\pi$
$331$	−9.07313	−0.498704	−0.249352	+	0.968413i	$0.580218\pi$
$332$	47.4547	2.60442
$333$	−3.28597	−0.180070
$334$	31.0953	1.70146
$335$	−26.5156	−1.44870
$336$	2.53476	0.138282
$337$	12.8459	0.699763	0.349882	−	0.936794i	$-0.386222\pi$
$337$	12.8459	0.699763	0.349882	+	0.936794i	$0.386222\pi$
$338$	9.94023	0.540677
$339$	16.3104	0.885858
$340$	−12.1634	−0.659654
$341$	−0.0151104	−0.000818275 0
$342$	−3.13287	−0.169406
$343$	−33.2752	−1.79669
$344$	21.4234	1.15507
$345$	16.1715	0.870645
$346$	−7.98378	−0.429211
$347$	−7.68441	−0.412521	−0.206260	−	0.978497i	$-0.566129\pi$
$347$	−7.68441	−0.412521	−0.206260	+	0.978497i	$0.566129\pi$
$348$	34.8946	1.87055
$349$	28.8817	1.54600	0.773001	−	0.634405i	$-0.218755\pi$
$349$	28.8817	1.54600	0.773001	+	0.634405i	$0.218755\pi$
$350$	−20.8230	−1.11303
$351$	−16.4675	−0.878970
$352$	−0.0826580	−0.00440569
$353$	28.7459	1.52999	0.764994	−	0.644037i	$-0.222742\pi$
$353$	28.7459	1.52999	0.764994	+	0.644037i	$0.222742\pi$
$354$	10.5564	0.561065
$355$	10.4719	0.555789
$356$	−47.9699	−2.54240
$357$	−12.3661	−0.654484
$358$	54.8350	2.89812
$359$	12.3300	0.650750	0.325375	−	0.945585i	$-0.394509\pi$
$359$	12.3300	0.650750	0.325375	+	0.945585i	$0.394509\pi$
$360$	7.26793	0.383054
$361$	1.00000	0.0526316
$362$	50.8466	2.67244
$363$	14.0995	0.740031
$364$	45.2354	2.37098
$365$	−14.2124	−0.743913
$366$	16.1140	0.842292
$367$	7.58539	0.395954	0.197977	−	0.980207i	$-0.436563\pi$
$367$	7.58539	0.395954	0.197977	+	0.980207i	$0.436563\pi$
$368$	−3.10491	−0.161855
$369$	12.7363	0.663026
$370$	−9.75097	−0.506929
$371$	61.1561	3.17507
$372$	−4.01895	−0.208373
$373$	11.6349	0.602433	0.301217	−	0.953556i	$-0.402607\pi$
$373$	11.6349	0.602433	0.301217	+	0.953556i	$0.402607\pi$
$374$	−0.0775915	−0.00401216
$375$	15.5556	0.803287
$376$	20.1020	1.03668
$377$	24.1048	1.24146
$378$	59.3987	3.05514
$379$	4.68979	0.240898	0.120449	−	0.992719i	$-0.461567\pi$
$379$	4.68979	0.240898	0.120449	+	0.992719i	$0.461567\pi$
$380$	−5.80827	−0.297958
$381$	11.3152	0.579693
$382$	−5.56562	−0.284762
$383$	−2.66287	−0.136066	−0.0680330	−	0.997683i	$-0.521672\pi$
$383$	−2.66287	−0.136066	−0.0680330	+	0.997683i	$0.521672\pi$
$384$	−24.5253	−1.25155
$385$	0.128954	0.00657210
$386$	19.0299	0.968598
$387$	9.46719	0.481244
$388$	14.1169	0.716678
$389$	22.8884	1.16049	0.580245	−	0.814442i	$-0.302957\pi$
$389$	22.8884	1.16049	0.580245	+	0.814442i	$0.302957\pi$
$390$	−15.2195	−0.770670
$391$	15.1477	0.766051
$392$	−43.6751	−2.20593
$393$	−5.71142	−0.288103
$394$	50.0424	2.52110
$395$	14.1670	0.712817
$396$	0.0725216	0.00364435
$397$	−0.802080	−0.0402552	−0.0201276	−	0.999797i	$-0.506407\pi$
$397$	−0.802080	−0.0402552	−0.0201276	+	0.999797i	$0.506407\pi$
$398$	−34.6780	−1.73825
$399$	−5.90506	−0.295623
$400$	−0.840398	−0.0420199
$401$	−25.9352	−1.29514	−0.647570	−	0.762006i	$-0.724215\pi$
$401$	−25.9352	−1.29514	−0.647570	+	0.762006i	$0.724215\pi$
$402$	−44.9880	−2.24380
$403$	−2.77625	−0.138295
$404$	31.6629	1.57529
$405$	−5.38540	−0.267603
$406$	−86.9468	−4.31510
$407$	−0.0388621	−0.00192632
$408$	−8.24276	−0.408078
$409$	28.9175	1.42988	0.714940	−	0.699185i	$-0.246454\pi$
$409$	28.9175	1.42988	0.714940	+	0.699185i	$0.246454\pi$
$410$	37.7944	1.86653
$411$	−7.31217	−0.360683
$412$	−19.5208	−0.961721
$413$	−16.4335	−0.808640
$414$	−22.6610	−1.11373
$415$	24.8552	1.22010
$416$	−15.1868	−0.744596
$417$	−16.0360	−0.785285
$418$	−0.0370514	−0.00181224
$419$	−9.40337	−0.459385	−0.229692	−	0.973263i	$-0.573772\pi$
$419$	−9.40337	−0.459385	−0.229692	+	0.973263i	$0.573772\pi$
$420$	34.2982	1.67358
$421$	−2.94363	−0.143464	−0.0717318	−	0.997424i	$-0.522853\pi$
$421$	−2.94363	−0.143464	−0.0717318	+	0.997424i	$0.522853\pi$
$422$	−2.30870	−0.112386
$423$	8.88324	0.431918
$424$	40.7643	1.97969
$425$	4.09998	0.198878
$426$	17.7672	0.860823
$427$	−25.0853	−1.21396
$428$	46.8786	2.26596
$429$	−0.0606568	−0.00292854
$430$	28.0935	1.35479
$431$	37.8513	1.82323	0.911616	−	0.411043i	$-0.134835\pi$
$431$	37.8513	1.82323	0.911616	+	0.411043i	$0.134835\pi$
$432$	2.39728	0.115339
$433$	9.22601	0.443374	0.221687	−	0.975118i	$-0.428844\pi$
$433$	9.22601	0.443374	0.221687	+	0.975118i	$0.428844\pi$
$434$	10.0140	0.480688
$435$	18.2767	0.876299
$436$	12.5097	0.599107
$437$	7.23331	0.346016
$438$	−24.1137	−1.15220
$439$	−18.5698	−0.886288	−0.443144	−	0.896450i	$-0.646137\pi$
$439$	−18.5698	−0.886288	−0.443144	+	0.896450i	$0.646137\pi$
$440$	0.0859555	0.00409777
$441$	−19.3004	−0.919067
$442$	−14.2560	−0.678087
$443$	28.4653	1.35243	0.676215	−	0.736705i	$-0.263619\pi$
$443$	28.4653	1.35243	0.676215	+	0.736705i	$0.263619\pi$
$444$	−10.3363	−0.490537
$445$	−25.1251	−1.19104
$446$	−23.8841	−1.13094
$447$	5.54201	0.262128
$448$	58.7343	2.77494
$449$	17.9201	0.845700	0.422850	−	0.906200i	$-0.361030\pi$
$449$	17.9201	0.845700	0.422850	+	0.906200i	$0.361030\pi$
$450$	−6.13358	−0.289140
$451$	0.150628	0.00709282
$452$	−42.3738	−1.99310
$453$	28.0173	1.31637
$454$	−35.4441	−1.66348
$455$	23.6928	1.11074
$456$	−3.93608	−0.184324
$457$	−36.9253	−1.72729	−0.863645	−	0.504100i	$-0.831824\pi$
$457$	−36.9253	−1.72729	−0.863645	+	0.504100i	$0.831824\pi$
$458$	37.6818	1.76076
$459$	−11.6954	−0.545896
$460$	−42.0130	−1.95887
$461$	−0.920475	−0.0428708	−0.0214354	−	0.999770i	$-0.506824\pi$
$461$	−0.920475	−0.0428708	−0.0214354	+	0.999770i	$0.506824\pi$
$462$	0.218791	0.0101791
$463$	−37.1664	−1.72727	−0.863634	−	0.504119i	$-0.831817\pi$
$463$	−37.1664	−1.72727	−0.863634	+	0.504119i	$0.831817\pi$
$464$	−3.50910	−0.162906
$465$	−2.10500	−0.0976168
$466$	6.20698	0.287533
$467$	−4.69352	−0.217190	−0.108595	−	0.994086i	$-0.534635\pi$
$467$	−4.69352	−0.217190	−0.108595	+	0.994086i	$0.534635\pi$
$468$	13.3245	0.615924
$469$	70.0345	3.23389
$470$	26.3606	1.21592
$471$	−5.93861	−0.273637
$472$	−10.9539	−0.504196
$473$	0.111966	0.00514818
$474$	24.0365	1.10403
$475$	1.95782	0.0898309
$476$	32.1267	1.47253
$477$	18.0141	0.824808
$478$	−52.7441	−2.41246
$479$	29.2971	1.33862	0.669309	−	0.742984i	$-0.266590\pi$
$479$	29.2971	1.33862	0.669309	+	0.742984i	$0.266590\pi$
$480$	−11.5149	−0.525581
$481$	−7.14018	−0.325564
$482$	46.1387	2.10156
$483$	−42.7131	−1.94351
$484$	−36.6300	−1.66500
$485$	7.39398	0.335744
$486$	29.5435	1.34012
$487$	−24.0057	−1.08780	−0.543901	−	0.839150i	$-0.683053\pi$
$487$	−24.0057	−1.08780	−0.543901	+	0.839150i	$0.683053\pi$
$488$	−16.7209	−0.756918
$489$	13.3737	0.604781
$490$	−57.2731	−2.58733
$491$	39.1170	1.76532	0.882662	−	0.470008i	$-0.155749\pi$
$491$	39.1170	1.76532	0.882662	+	0.470008i	$0.155749\pi$
$492$	40.0630	1.80618
$493$	17.1196	0.771026
$494$	−6.80750	−0.306284
$495$	0.0379845	0.00170727
$496$	0.404157	0.0181472
$497$	−27.6589	−1.24067
$498$	42.1709	1.88972
$499$	−15.2707	−0.683609	−0.341805	−	0.939771i	$-0.611038\pi$
$499$	−15.2707	−0.683609	−0.341805	+	0.939771i	$0.611038\pi$
$500$	−40.4129	−1.80732
$501$	17.2643	0.771312
$502$	−46.0882	−2.05702
$503$	−31.0449	−1.38422	−0.692111	−	0.721791i	$-0.743319\pi$
$503$	−31.0449	−1.38422	−0.692111	+	0.721791i	$0.743319\pi$
$504$	−19.1965	−0.855079
$505$	16.5840	0.737979
$506$	−0.268005	−0.0119143
$507$	5.51887	0.245102
$508$	−29.3964	−1.30425
$509$	34.6755	1.53697	0.768483	−	0.639870i	$-0.221012\pi$
$509$	34.6755	1.53697	0.768483	+	0.639870i	$0.221012\pi$
$510$	−10.8091	−0.478634
$511$	37.5387	1.66062
$512$	4.84709	0.214213
$513$	−5.58479	−0.246574
$514$	22.9750	1.01338
$515$	−10.2244	−0.450539
$516$	29.7797	1.31098
$517$	0.105059	0.00462050
$518$	25.7548	1.13160
$519$	−4.43264	−0.194571
$520$	15.7927	0.692556
$521$	−13.4245	−0.588139	−0.294069	−	0.955784i	$-0.595010\pi$
$521$	−13.4245	−0.588139	−0.294069	+	0.955784i	$0.595010\pi$
$522$	−25.6109	−1.12096
$523$	−25.7939	−1.12789	−0.563944	−	0.825813i	$-0.690717\pi$
$523$	−25.7939	−1.12789	−0.563944	+	0.825813i	$0.690717\pi$
$524$	14.8381	0.648204
$525$	−11.5610	−0.504565
$526$	−2.19363	−0.0956470
$527$	−1.97173	−0.0858898
$528$	0.00883022	0.000384286 0
$529$	29.3208	1.27482
$530$	53.4559	2.32198
$531$	−4.84063	−0.210066
$532$	15.3411	0.665123
$533$	27.6751	1.19874
$534$	−42.6287	−1.84472
$535$	24.5535	1.06154
$536$	46.6823	2.01637
$537$	30.4447	1.31379
$538$	−33.0262	−1.42386
$539$	−0.228260	−0.00983185
$540$	32.4380	1.39591
$541$	−4.74421	−0.203969	−0.101985	−	0.994786i	$-0.532519\pi$
$541$	−4.74421	−0.203969	−0.101985	+	0.994786i	$0.532519\pi$
$542$	38.4028	1.64954
$543$	28.2303	1.21148
$544$	−10.7859	−0.462441
$545$	6.55219	0.280665
$546$	40.1987	1.72034
$547$	−16.8048	−0.718520	−0.359260	−	0.933238i	$-0.616971\pi$
$547$	−16.8048	−0.718520	−0.359260	+	0.933238i	$0.616971\pi$
$548$	18.9968	0.811502
$549$	−7.38909	−0.315359
$550$	−0.0725400	−0.00309311
$551$	8.17492	0.348263
$552$	−28.4709	−1.21180
$553$	−37.4186	−1.59120
$554$	−37.7345	−1.60319
$555$	−5.41379	−0.229803
$556$	41.6609	1.76682
$557$	11.5526	0.489497	0.244749	−	0.969587i	$-0.421295\pi$
$557$	11.5526	0.489497	0.244749	+	0.969587i	$0.421295\pi$
$558$	2.94971	0.124871
$559$	20.5715	0.870083
$560$	−3.44912	−0.145752
$561$	−0.0430792	−0.00181881
$562$	−10.4626	−0.441339
$563$	24.1997	1.01990	0.509949	−	0.860205i	$-0.329664\pi$
$563$	24.1997	1.01990	0.509949	+	0.860205i	$0.329664\pi$
$564$	27.9429	1.17661
$565$	−22.1940	−0.933709
$566$	16.7399	0.703630
$567$	14.2242	0.597362
$568$	−18.4363	−0.773571
$569$	15.7393	0.659826	0.329913	−	0.944011i	$-0.392981\pi$
$569$	15.7393	0.659826	0.329913	+	0.944011i	$0.392981\pi$
$570$	−5.16155	−0.216193
$571$	6.09103	0.254902	0.127451	−	0.991845i	$-0.459321\pi$
$571$	6.09103	0.254902	0.127451	+	0.991845i	$0.459321\pi$
$572$	0.157584	0.00658893
$573$	−3.09006	−0.129089
$574$	−99.8249	−4.16661
$575$	14.1615	0.590576
$576$	17.3007	0.720863
$577$	−34.0920	−1.41927	−0.709633	−	0.704571i	$-0.751139\pi$
$577$	−34.0920	−1.41927	−0.709633	+	0.704571i	$0.751139\pi$
$578$	29.1231	1.21136
$579$	10.5655	0.439088
$580$	−47.4821	−1.97159
$581$	−65.6491	−2.72358
$582$	12.5451	0.520010
$583$	0.213047	0.00882349
$584$	25.0218	1.03541
$585$	6.97893	0.288543
$586$	−16.5118	−0.682096
$587$	17.1526	0.707962	0.353981	−	0.935253i	$-0.384828\pi$
$587$	17.1526	0.707962	0.353981	+	0.935253i	$0.384828\pi$
$588$	−60.7108	−2.50367
$589$	−0.941538	−0.0387954
$590$	−14.3644	−0.591371
$591$	27.7838	1.14287
$592$	1.03944	0.0427209
$593$	−11.2529	−0.462103	−0.231052	−	0.972941i	$-0.574217\pi$
$593$	−11.2529	−0.462103	−0.231052	+	0.972941i	$0.574217\pi$
$594$	0.206924	0.00849021
$595$	16.8269	0.689837
$596$	−14.3980	−0.589763
$597$	−19.2534	−0.787990
$598$	−49.2408	−2.01360
$599$	−22.4065	−0.915504	−0.457752	−	0.889080i	$-0.651345\pi$
$599$	−22.4065	−0.915504	−0.457752	+	0.889080i	$0.651345\pi$
$600$	−7.70613	−0.314601
$601$	−27.4586	−1.12006	−0.560030	−	0.828473i	$-0.689210\pi$
$601$	−27.4586	−1.12006	−0.560030	+	0.828473i	$0.689210\pi$
$602$	−74.2021	−3.02425
$603$	20.6293	0.840089
$604$	−72.7880	−2.96170
$605$	−19.1856	−0.780005
$606$	28.1374	1.14301
$607$	40.0914	1.62726	0.813631	−	0.581382i	$-0.197488\pi$
$607$	40.0914	1.62726	0.813631	+	0.581382i	$0.197488\pi$
$608$	−5.15047	−0.208879
$609$	−48.2734	−1.95614
$610$	−21.9268	−0.887790
$611$	19.3027	0.780902
$612$	9.46321	0.382527
$613$	11.8630	0.479143	0.239571	−	0.970879i	$-0.422993\pi$
$613$	11.8630	0.479143	0.239571	+	0.970879i	$0.422993\pi$
$614$	−17.8644	−0.720948
$615$	20.9837	0.846144
$616$	−0.227031	−0.00914733
$617$	−44.1869	−1.77890	−0.889449	−	0.457035i	$-0.848912\pi$
$617$	−44.1869	−1.77890	−0.889449	+	0.457035i	$0.848912\pi$
$618$	−17.3473	−0.697809
$619$	20.6764	0.831055	0.415528	−	0.909581i	$-0.363597\pi$
$619$	20.6764	0.831055	0.415528	+	0.909581i	$0.363597\pi$
$620$	5.46871	0.219629
$621$	−40.3965	−1.62106
$622$	8.53201	0.342103
$623$	66.3618	2.65873
$624$	1.62238	0.0649474
$625$	−11.3779	−0.455114
$626$	−12.2715	−0.490467
$627$	−0.0205712	−0.000821533 0
$628$	15.4283	0.615656
$629$	−5.07104	−0.202196
$630$	−25.1732	−1.00292
$631$	0.519973	0.0206998	0.0103499	−	0.999946i	$-0.496705\pi$
$631$	0.519973	0.0206998	0.0103499	+	0.999946i	$0.496705\pi$
$632$	−24.9417	−0.992130
$633$	−1.28180	−0.0509470
$634$	73.1715	2.90601
$635$	−15.3969	−0.611006
$636$	56.6645	2.24690
$637$	−41.9384	−1.66166
$638$	−0.302892	−0.0119916
$639$	−8.14717	−0.322297
$640$	33.3723	1.31916
$641$	−22.0303	−0.870143	−0.435072	−	0.900396i	$-0.643277\pi$
$641$	−22.0303	−0.870143	−0.435072	+	0.900396i	$0.643277\pi$
$642$	41.6589	1.64415
$643$	−9.79822	−0.386404	−0.193202	−	0.981159i	$-0.561887\pi$
$643$	−9.79822	−0.386404	−0.193202	+	0.981159i	$0.561887\pi$
$644$	110.967	4.37272
$645$	15.5977	0.614157
$646$	−4.83477	−0.190221
$647$	−15.0973	−0.593536	−0.296768	−	0.954950i	$-0.595909\pi$
$647$	−15.0973	−0.593536	−0.296768	+	0.954950i	$0.595909\pi$
$648$	9.48132	0.372461
$649$	−0.0572487	−0.00224721
$650$	−13.3278	−0.522761
$651$	5.55984	0.217907
$652$	−34.7445	−1.36070
$653$	−2.14715	−0.0840243	−0.0420121	−	0.999117i	$-0.513377\pi$
$653$	−2.14715	−0.0840243	−0.0420121	+	0.999117i	$0.513377\pi$
$654$	11.1168	0.434703
$655$	7.77170	0.303666
$656$	−4.02885	−0.157300
$657$	11.0574	0.431389
$658$	−69.6252	−2.71427
$659$	25.7270	1.00218	0.501090	−	0.865395i	$-0.332933\pi$
$659$	25.7270	1.00218	0.501090	+	0.865395i	$0.332933\pi$
$660$	0.119483	0.00465086
$661$	−42.1891	−1.64097	−0.820484	−	0.571670i	$-0.806296\pi$
$661$	−42.1891	−1.64097	−0.820484	+	0.571670i	$0.806296\pi$
$662$	20.9471	0.814132
$663$	−7.91499	−0.307393
$664$	−43.7591	−1.69818
$665$	8.03519	0.311591
$666$	7.58630	0.293963
$667$	59.1317	2.28959
$668$	−44.8520	−1.73538
$669$	−13.2606	−0.512683
$670$	61.2165	2.36500
$671$	−0.0873885	−0.00337359
$672$	30.4138	1.17324
$673$	−16.7243	−0.644673	−0.322337	−	0.946625i	$-0.604468\pi$
$673$	−16.7243	−0.644673	−0.322337	+	0.946625i	$0.604468\pi$
$674$	−29.6574	−1.14236
$675$	−10.9340	−0.420850
$676$	−14.3378	−0.551456
$677$	−7.75161	−0.297918	−0.148959	−	0.988843i	$-0.547592\pi$
$677$	−7.75161	−0.297918	−0.148959	+	0.988843i	$0.547592\pi$
$678$	−37.6557	−1.44616
$679$	−19.5294	−0.749470
$680$	11.2162	0.430121
$681$	−19.6788	−0.754093
$682$	0.0348853	0.00133583
$683$	−35.6955	−1.36585	−0.682926	−	0.730488i	$-0.739293\pi$
$683$	−35.6955	−1.36585	−0.682926	+	0.730488i	$0.739293\pi$
$684$	4.51886	0.172783
$685$	9.94989	0.380166
$686$	76.8224	2.93309
$687$	20.9212	0.798193
$688$	−2.99474	−0.114173
$689$	39.1433	1.49124
$690$	−37.3351	−1.42132
$691$	13.7706	0.523859	0.261929	−	0.965087i	$-0.415641\pi$
$691$	13.7706	0.523859	0.261929	+	0.965087i	$0.415641\pi$
$692$	11.5159	0.437767
$693$	−0.100327	−0.00381110
$694$	17.7410	0.673438
$695$	21.8206	0.827704
$696$	−32.1771	−1.21967
$697$	19.6552	0.744494
$698$	−66.6791	−2.52384
$699$	3.44615	0.130345
$700$	30.0352	1.13522
$701$	−39.8618	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$701$	−39.8618	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$702$	38.0184	1.43491
$703$	−2.42152	−0.0913294
$704$	0.204610	0.00771153
$705$	14.6356	0.551207
$706$	−66.3655	−2.49770
$707$	−43.8027	−1.64737
$708$	−15.2266	−0.572249
$709$	50.4669	1.89532	0.947662	−	0.319276i	$-0.103440\pi$
$709$	50.4669	1.89532	0.947662	+	0.319276i	$0.103440\pi$
$710$	−24.1763	−0.907322
$711$	−11.0220	−0.413356
$712$	44.2342	1.65775
$713$	−6.81044	−0.255053
$714$	28.5496	1.06844
$715$	0.0825376	0.00308673
$716$	−79.0943	−2.95589
$717$	−29.2838	−1.09362
$718$	−28.4661	−1.06235
$719$	23.3381	0.870364	0.435182	−	0.900343i	$-0.356684\pi$
$719$	23.3381	0.870364	0.435182	+	0.900343i	$0.356684\pi$
$720$	−1.01597	−0.0378629
$721$	27.0052	1.00573
$722$	−2.30870	−0.0859208
$723$	25.6165	0.952686
$724$	−73.3414	−2.72571
$725$	16.0050	0.594411
$726$	−32.5514	−1.20810
$727$	1.34730	0.0499687	0.0249843	−	0.999688i	$-0.492046\pi$
$727$	1.34730	0.0499687	0.0249843	+	0.999688i	$0.492046\pi$
$728$	−41.7126	−1.54597
$729$	25.6656	0.950579
$730$	32.8122	1.21443
$731$	14.6102	0.540376
$732$	−23.2429	−0.859083
$733$	21.4987	0.794071	0.397035	−	0.917803i	$-0.370039\pi$
$733$	21.4987	0.794071	0.397035	+	0.917803i	$0.370039\pi$
$734$	−17.5124	−0.646393
$735$	−31.7983	−1.17290
$736$	−37.2549	−1.37324
$737$	0.243976	0.00898697
$738$	−29.4043	−1.08239
$739$	−19.8191	−0.729058	−0.364529	−	0.931192i	$-0.618770\pi$
$739$	−19.8191	−0.729058	−0.364529	+	0.931192i	$0.618770\pi$
$740$	14.0648	0.517034
$741$	−3.77956	−0.138846
$742$	−141.191	−5.18328
$743$	−15.0797	−0.553222	−0.276611	−	0.960982i	$-0.589211\pi$
$743$	−15.0797	−0.553222	−0.276611	+	0.960982i	$0.589211\pi$
$744$	3.70597	0.135867
$745$	−7.54118	−0.276288
$746$	−26.8615	−0.983469
$747$	−19.3375	−0.707523
$748$	0.111918	0.00409214
$749$	−64.8521	−2.36964
$750$	−35.9131	−1.31136
$751$	−22.5746	−0.823760	−0.411880	−	0.911238i	$-0.635128\pi$
$751$	−22.5746	−0.823760	−0.411880	+	0.911238i	$0.635128\pi$
$752$	−2.81002	−0.102471
$753$	−25.5884	−0.932495
$754$	−55.6507	−2.02668
$755$	−38.1240	−1.38747
$756$	−85.6770	−3.11604
$757$	−6.12109	−0.222475	−0.111237	−	0.993794i	$-0.535481\pi$
$757$	−6.12109	−0.222475	−0.111237	+	0.993794i	$0.535481\pi$
$758$	−10.8273	−0.393265
$759$	−0.148798	−0.00540101
$760$	5.35594	0.194280
$761$	−47.0977	−1.70729	−0.853645	−	0.520856i	$-0.825613\pi$
$761$	−47.0977	−1.70729	−0.853645	+	0.520856i	$0.825613\pi$
$762$	−26.1232	−0.946345
$763$	−17.3060	−0.626520
$764$	8.02788	0.290439
$765$	4.95652	0.179203
$766$	6.14775	0.222127
$767$	−10.5184	−0.379796
$768$	23.9372	0.863758
$769$	3.19192	0.115104	0.0575518	−	0.998343i	$-0.481671\pi$
$769$	3.19192	0.115104	0.0575518	+	0.998343i	$0.481671\pi$
$770$	−0.297715	−0.0107289
$771$	12.7558	0.459390
$772$	−27.4489	−0.987907
$773$	4.11657	0.148063	0.0740314	−	0.997256i	$-0.476414\pi$
$773$	4.11657	0.148063	0.0740314	+	0.997256i	$0.476414\pi$
$774$	−21.8569	−0.785629
$775$	−1.84336	−0.0662155
$776$	−13.0175	−0.467302
$777$	14.2992	0.512982
$778$	−52.8425	−1.89449
$779$	9.38574	0.336279
$780$	21.9527	0.786033
$781$	−0.0963540	−0.00344782
$782$	−34.9714	−1.25057
$783$	−45.6552	−1.63158
$784$	6.10525	0.218045
$785$	8.08084	0.288418
$786$	13.1859	0.470327
$787$	36.7434	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$787$	36.7434	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$788$	−72.1814	−2.57136
$789$	−1.21792	−0.0433590
$790$	−32.7072	−1.16367
$791$	58.6201	2.08429
$792$	−0.0668739	−0.00237626
$793$	−16.0560	−0.570164
$794$	1.85176	0.0657164
$795$	29.6790	1.05261
$796$	50.0197	1.77290
$797$	40.7945	1.44502	0.722508	−	0.691362i	$-0.242989\pi$
$797$	40.7945	1.44502	0.722508	+	0.691362i	$0.242989\pi$
$798$	13.6330	0.482602
$799$	13.7090	0.484989
$800$	−10.0837	−0.356512
$801$	19.5474	0.690675
$802$	59.8764	2.11431
$803$	0.130772	0.00461484
$804$	64.8909	2.28853
$805$	58.1210	2.04850
$806$	6.40952	0.225766
$807$	−18.3363	−0.645470
$808$	−29.1971	−1.02715
$809$	52.4368	1.84358	0.921790	−	0.387689i	$-0.126727\pi$
$809$	52.4368	1.84358	0.921790	+	0.387689i	$0.126727\pi$
$810$	12.4333	0.436860
$811$	−41.4379	−1.45508	−0.727542	−	0.686064i	$-0.759337\pi$
$811$	−41.4379	−1.45508	−0.727542	+	0.686064i	$0.759337\pi$
$812$	125.413	4.40112
$813$	21.3214	0.747775
$814$	0.0897208	0.00314471
$815$	−18.1980	−0.637449
$816$	1.15224	0.0403364
$817$	6.97664	0.244082
$818$	−66.7618	−2.33427
$819$	−18.4331	−0.644106
$820$	−54.5149	−1.90374
$821$	−8.71912	−0.304300	−0.152150	−	0.988357i	$-0.548620\pi$
$821$	−8.71912	−0.304300	−0.152150	+	0.988357i	$0.548620\pi$
$822$	16.8816	0.588812
$823$	−13.9001	−0.484527	−0.242264	−	0.970210i	$-0.577890\pi$
$823$	−13.9001	−0.484527	−0.242264	+	0.970210i	$0.577890\pi$
$824$	18.0006	0.627080
$825$	−0.0402746	−0.00140218
$826$	37.9400	1.32010
$827$	−1.24207	−0.0431912	−0.0215956	−	0.999767i	$-0.506875\pi$
$827$	−1.24207	−0.0431912	−0.0215956	+	0.999767i	$0.506875\pi$
$828$	32.6864	1.13593
$829$	−13.7383	−0.477152	−0.238576	−	0.971124i	$-0.576681\pi$
$829$	−13.7383	−0.477152	−0.238576	+	0.971124i	$0.576681\pi$
$830$	−57.3832	−1.99180
$831$	−20.9504	−0.726762
$832$	37.5932	1.30331
$833$	−29.7852	−1.03200
$834$	37.0222	1.28197
$835$	−23.4920	−0.812975
$836$	0.0534432	0.00184837
$837$	5.25829	0.181753
$838$	21.7095	0.749943
$839$	12.6904	0.438122	0.219061	−	0.975711i	$-0.429701\pi$
$839$	12.6904	0.438122	0.219061	+	0.975711i	$0.429701\pi$
$840$	−31.6271	−1.09124
$841$	37.8293	1.30446
$842$	6.79594	0.234203
$843$	−5.80891	−0.200070
$844$	3.33008	0.114626
$845$	−7.50970	−0.258341
$846$	−20.5087	−0.705104
$847$	50.6741	1.74118
$848$	−5.69835	−0.195682
$849$	9.29408	0.318972
$850$	−9.46560	−0.324667
$851$	−17.5156	−0.600428
$852$	−25.6275	−0.877983
$853$	46.2562	1.58378	0.791891	−	0.610663i	$-0.209097\pi$
$853$	46.2562	1.58378	0.791891	+	0.610663i	$0.209097\pi$
$854$	57.9143	1.98179
$855$	2.36683	0.0809440
$856$	−43.2278	−1.47750
$857$	−38.5382	−1.31644	−0.658220	−	0.752826i	$-0.728690\pi$
$857$	−38.5382	−1.31644	−0.658220	+	0.752826i	$0.728690\pi$
$858$	0.140038	0.00478082
$859$	41.5290	1.41695	0.708476	−	0.705735i	$-0.249383\pi$
$859$	41.5290	1.41695	0.708476	+	0.705735i	$0.249383\pi$
$860$	−40.5222	−1.38179
$861$	−55.4234	−1.88882
$862$	−87.3871	−2.97642
$863$	−6.11137	−0.208034	−0.104017	−	0.994576i	$-0.533170\pi$
$863$	−6.11137	−0.208034	−0.104017	+	0.994576i	$0.533170\pi$
$864$	28.7643	0.978580
$865$	6.03163	0.205082
$866$	−21.3001	−0.723805
$867$	16.1693	0.549138
$868$	−14.4443	−0.490270
$869$	−0.130353	−0.00442193
$870$	−42.1952	−1.43055
$871$	44.8260	1.51887
$872$	−11.5355	−0.390642
$873$	−5.75256	−0.194695
$874$	−16.6995	−0.564870
$875$	55.9074	1.89001
$876$	34.7817	1.17517
$877$	24.8000	0.837435	0.418718	−	0.908116i	$-0.362480\pi$
$877$	24.8000	0.837435	0.418718	+	0.908116i	$0.362480\pi$
$878$	42.8720	1.44686
$879$	−9.16744	−0.309210
$880$	−0.0120155	−0.000405044 0
$881$	−19.7502	−0.665400	−0.332700	−	0.943033i	$-0.607960\pi$
$881$	−19.7502	−0.665400	−0.332700	+	0.943033i	$0.607960\pi$
$882$	44.5588	1.50037
$883$	13.7727	0.463487	0.231744	−	0.972777i	$-0.425557\pi$
$883$	13.7727	0.463487	0.231744	+	0.972777i	$0.425557\pi$
$884$	20.5629	0.691604
$885$	−7.97518	−0.268083
$886$	−65.7178	−2.20783
$887$	33.8134	1.13534	0.567671	−	0.823256i	$-0.307845\pi$
$887$	33.8134	1.13534	0.567671	+	0.823256i	$0.307845\pi$
$888$	9.53130	0.319849
$889$	40.6671	1.36393
$890$	58.0062	1.94437
$891$	0.0495523	0.00166007
$892$	34.4505	1.15349
$893$	6.54631	0.219064
$894$	−12.7948	−0.427923
$895$	−41.4270	−1.38475
$896$	−88.1449	−2.94471
$897$	−27.3388	−0.912814
$898$	−41.3719	−1.38060
$899$	−7.69700	−0.256709
$900$	8.84712	0.294904
$901$	27.8001	0.926154
$902$	−0.347755	−0.0115790
$903$	−41.1974	−1.37096
$904$	39.0738	1.29958
$905$	−38.4139	−1.27692
$906$	−64.6834	−2.14896
$907$	11.7483	0.390095	0.195048	−	0.980794i	$-0.437514\pi$
$907$	11.7483	0.390095	0.195048	+	0.980794i	$0.437514\pi$
$908$	51.1248	1.69664
$909$	−12.9025	−0.427948
$910$	−54.6995	−1.81327
$911$	−6.10592	−0.202298	−0.101149	−	0.994871i	$-0.532252\pi$
$911$	−6.10592	−0.202298	−0.101149	+	0.994871i	$0.532252\pi$
$912$	0.550216	0.0182195
$913$	−0.228699	−0.00756882
$914$	85.2492	2.81979
$915$	−12.1739	−0.402456
$916$	−54.3525	−1.79586
$917$	−20.5271	−0.677864
$918$	27.0012	0.891171
$919$	−53.6603	−1.77009	−0.885045	−	0.465506i	$-0.845872\pi$
$919$	−53.6603	−1.77009	−0.885045	+	0.465506i	$0.845872\pi$
$920$	38.7412	1.27726
$921$	−9.91841	−0.326823
$922$	2.12510	0.0699863
$923$	−17.7032	−0.582708
$924$	−0.315585	−0.0103820
$925$	−4.74090	−0.155880
$926$	85.8059	2.81976
$927$	7.95461	0.261264
$928$	−42.1047	−1.38215
$929$	−53.7393	−1.76313	−0.881564	−	0.472064i	$-0.843509\pi$
$929$	−53.7393	−1.76313	−0.881564	+	0.472064i	$0.843509\pi$
$930$	4.85979	0.159359
$931$	−14.2230	−0.466140
$932$	−8.95298	−0.293265
$933$	4.73702	0.155083
$934$	10.8359	0.354562
$935$	0.0586192	0.00191705
$936$	−12.2868	−0.401607
$937$	−16.6122	−0.542698	−0.271349	−	0.962481i	$-0.587470\pi$
$937$	−16.6122	−0.542698	−0.271349	+	0.962481i	$0.587470\pi$
$938$	−161.688	−5.27932
$939$	−6.81320	−0.222340
$940$	−38.0227	−1.24016
$941$	47.1160	1.53594	0.767969	−	0.640487i	$-0.221267\pi$
$941$	47.1160	1.53594	0.767969	+	0.640487i	$0.221267\pi$
$942$	13.7104	0.446710
$943$	67.8900	2.21080
$944$	1.53123	0.0498372
$945$	−44.8748	−1.45978
$946$	−0.258494	−0.00840437
$947$	17.8707	0.580719	0.290359	−	0.956918i	$-0.406225\pi$
$947$	17.8707	0.580719	0.290359	+	0.956918i	$0.406225\pi$
$948$	−34.6704	−1.12604
$949$	24.0269	0.779945
$950$	−4.52001	−0.146648
$951$	40.6252	1.31736
$952$	−29.6248	−0.960145
$953$	−31.2158	−1.01118	−0.505589	−	0.862775i	$-0.668725\pi$
$953$	−31.2158	−1.01118	−0.505589	+	0.862775i	$0.668725\pi$
$954$	−41.5890	−1.34649
$955$	4.20474	0.136062
$956$	76.0784	2.46055
$957$	−0.168168	−0.00543609
$958$	−67.6381	−2.18529
$959$	−26.2802	−0.848632
$960$	28.5037	0.919954
$961$	−30.1135	−0.971403
$962$	16.4845	0.531482
$963$	−19.1027	−0.615577
$964$	−66.5507	−2.14345
$965$	−14.3768	−0.462807
$966$	98.6116	3.17278
$967$	13.1981	0.424423	0.212212	−	0.977224i	$-0.431933\pi$
$967$	13.1981	0.424423	0.212212	+	0.977224i	$0.431933\pi$
$968$	33.7773	1.08564
$969$	−2.68429	−0.0862319
$970$	−17.0705	−0.548099
$971$	4.17779	0.134072	0.0670359	−	0.997751i	$-0.478646\pi$
$971$	4.17779	0.134072	0.0670359	+	0.997751i	$0.478646\pi$
$972$	−42.6138	−1.36684
$973$	−57.6339	−1.84766
$974$	55.4218	1.77583
$975$	−7.39970	−0.236980
$976$	2.33737	0.0748175
$977$	−37.9001	−1.21253	−0.606266	−	0.795262i	$-0.707333\pi$
$977$	−37.9001	−1.21253	−0.606266	+	0.795262i	$0.707333\pi$
$978$	−30.8759	−0.987301
$979$	0.231181	0.00738859
$980$	82.6110	2.63891
$981$	−5.09764	−0.162755
$982$	−90.3092	−2.88188
$983$	21.8434	0.696698	0.348349	−	0.937365i	$-0.386742\pi$
$983$	21.8434	0.696698	0.348349	+	0.937365i	$0.386742\pi$
$984$	−36.9430	−1.17770
$985$	−37.8063	−1.20461
$986$	−39.5238	−1.25870
$987$	−38.6563	−1.23044
$988$	9.81917	0.312389
$989$	50.4642	1.60467
$990$	−0.0876946	−0.00278712
$991$	−41.8275	−1.32869	−0.664347	−	0.747424i	$-0.731290\pi$
$991$	−41.8275	−1.32869	−0.664347	+	0.747424i	$0.731290\pi$
$992$	4.84936	0.153967
$993$	11.6300	0.369065
$994$	63.8560	2.02539
$995$	26.1987	0.830554
$996$	−60.8275	−1.92739
$997$	59.2130	1.87530	0.937648	−	0.347586i	$-0.112999\pi$
$997$	59.2130	1.87530	0.937648	+	0.347586i	$0.112999\pi$
$998$	35.2553	1.11599
$999$	13.5237	0.427871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.10	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.10	✓	83	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30870 q^{2} -1.28180 q^{3} +3.33008 q^{4} +1.74419 q^{5} +2.95929 q^{6} -4.60684 q^{7} -3.07074 q^{8} -1.35699 q^{9} +O(q^{10})\)	\(q-2.30870 q^{2} -1.28180 q^{3} +3.33008 q^{4} +1.74419 q^{5} +2.95929 q^{6} -4.60684 q^{7} -3.07074 q^{8} -1.35699 q^{9} -4.02679 q^{10} -0.0160486 q^{11} -4.26850 q^{12} -2.94863 q^{13} +10.6358 q^{14} -2.23570 q^{15} +0.429252 q^{16} -2.09416 q^{17} +3.13287 q^{18} -1.00000 q^{19} +5.80827 q^{20} +5.90506 q^{21} +0.0370514 q^{22} -7.23331 q^{23} +3.93608 q^{24} -1.95782 q^{25} +6.80750 q^{26} +5.58479 q^{27} -15.3411 q^{28} -8.17492 q^{29} +5.16155 q^{30} +0.941538 q^{31} +5.15047 q^{32} +0.0205712 q^{33} +4.83477 q^{34} -8.03519 q^{35} -4.51886 q^{36} +2.42152 q^{37} +2.30870 q^{38} +3.77956 q^{39} -5.35594 q^{40} -9.38574 q^{41} -13.6330 q^{42} -6.97664 q^{43} -0.0534432 q^{44} -2.36683 q^{45} +16.6995 q^{46} -6.54631 q^{47} -0.550216 q^{48} +14.2230 q^{49} +4.52001 q^{50} +2.68429 q^{51} -9.81917 q^{52} -13.2751 q^{53} -12.8936 q^{54} -0.0279918 q^{55} +14.1464 q^{56} +1.28180 q^{57} +18.8734 q^{58} +3.56720 q^{59} -7.44505 q^{60} +5.44523 q^{61} -2.17373 q^{62} +6.25142 q^{63} -12.7494 q^{64} -5.14296 q^{65} -0.0474926 q^{66} -15.2023 q^{67} -6.97370 q^{68} +9.27167 q^{69} +18.5508 q^{70} +6.00387 q^{71} +4.16695 q^{72} -8.14847 q^{73} -5.59056 q^{74} +2.50953 q^{75} -3.33008 q^{76} +0.0739336 q^{77} -8.72586 q^{78} +8.12239 q^{79} +0.748695 q^{80} -3.08763 q^{81} +21.6688 q^{82} +14.2503 q^{83} +19.6643 q^{84} -3.65260 q^{85} +16.1069 q^{86} +10.4786 q^{87} +0.0492812 q^{88} -14.4051 q^{89} +5.46430 q^{90} +13.5839 q^{91} -24.0875 q^{92} -1.20686 q^{93} +15.1134 q^{94} -1.74419 q^{95} -6.60187 q^{96} +4.23922 q^{97} -32.8366 q^{98} +0.0217778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

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