LMFDB - Embedded newform 8009.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78850 q^{2} +0.327361 q^{3} +5.77575 q^{4} -3.12369 q^{5} -0.912847 q^{6} +1.40614 q^{7} -10.5287 q^{8} -2.89283 q^{9} +O(q^{10})$	\(q-2.78850 q^{2} +0.327361 q^{3} +5.77575 q^{4} -3.12369 q^{5} -0.912847 q^{6} +1.40614 q^{7} -10.5287 q^{8} -2.89283 q^{9} +8.71042 q^{10} -1.60671 q^{11} +1.89076 q^{12} +6.86097 q^{13} -3.92103 q^{14} -1.02257 q^{15} +17.8078 q^{16} +3.40196 q^{17} +8.06668 q^{18} -7.28137 q^{19} -18.0417 q^{20} +0.460315 q^{21} +4.48033 q^{22} +8.24707 q^{23} -3.44669 q^{24} +4.75744 q^{25} -19.1318 q^{26} -1.92908 q^{27} +8.12152 q^{28} +2.48883 q^{29} +2.85145 q^{30} -1.95607 q^{31} -28.5997 q^{32} -0.525975 q^{33} -9.48638 q^{34} -4.39234 q^{35} -16.7083 q^{36} +10.7638 q^{37} +20.3041 q^{38} +2.24601 q^{39} +32.8884 q^{40} -1.71143 q^{41} -1.28359 q^{42} +6.47839 q^{43} -9.27998 q^{44} +9.03632 q^{45} -22.9970 q^{46} -6.67303 q^{47} +5.82958 q^{48} -5.02277 q^{49} -13.2661 q^{50} +1.11367 q^{51} +39.6272 q^{52} +5.50964 q^{53} +5.37926 q^{54} +5.01887 q^{55} -14.8048 q^{56} -2.38364 q^{57} -6.94010 q^{58} -12.0792 q^{59} -5.90614 q^{60} +2.21258 q^{61} +5.45452 q^{62} -4.06773 q^{63} +44.1349 q^{64} -21.4315 q^{65} +1.46668 q^{66} +5.97319 q^{67} +19.6489 q^{68} +2.69977 q^{69} +12.2481 q^{70} +0.926605 q^{71} +30.4578 q^{72} +8.77594 q^{73} -30.0148 q^{74} +1.55740 q^{75} -42.0554 q^{76} -2.25926 q^{77} -6.26301 q^{78} +3.06460 q^{79} -55.6261 q^{80} +8.04700 q^{81} +4.77232 q^{82} +16.4511 q^{83} +2.65867 q^{84} -10.6267 q^{85} -18.0650 q^{86} +0.814745 q^{87} +16.9166 q^{88} +0.974709 q^{89} -25.1978 q^{90} +9.64748 q^{91} +47.6330 q^{92} -0.640342 q^{93} +18.6078 q^{94} +22.7447 q^{95} -9.36244 q^{96} -10.0869 q^{97} +14.0060 q^{98} +4.64796 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.78850	−1.97177	−0.985885	−	0.167425i	$-0.946455\pi$
$2$	−2.78850	−1.97177	−0.985885	+	0.167425i	$0.946455\pi$
$3$	0.327361	0.189002	0.0945010	−	0.995525i	$-0.469874\pi$
$3$	0.327361	0.189002	0.0945010	+	0.995525i	$0.469874\pi$
$4$	5.77575	2.88788
$5$	−3.12369	−1.39696	−0.698478	−	0.715631i	$-0.746139\pi$
$5$	−3.12369	−1.39696	−0.698478	+	0.715631i	$0.746139\pi$
$6$	−0.912847	−0.372668
$7$	1.40614	0.531471	0.265735	−	0.964046i	$-0.414385\pi$
$7$	1.40614	0.531471	0.265735	+	0.964046i	$0.414385\pi$
$8$	−10.5287	−3.72246
$9$	−2.89283	−0.964278
$10$	8.71042	2.75448
$11$	−1.60671	−0.484442	−0.242221	−	0.970221i	$-0.577876\pi$
$11$	−1.60671	−0.484442	−0.242221	+	0.970221i	$0.577876\pi$
$12$	1.89076	0.545814
$13$	6.86097	1.90289	0.951445	−	0.307820i	$-0.0995994\pi$
$13$	6.86097	1.90289	0.951445	+	0.307820i	$0.0995994\pi$
$14$	−3.92103	−1.04794
$15$	−1.02257	−0.264028
$16$	17.8078	4.45195
$17$	3.40196	0.825097	0.412548	−	0.910936i	$-0.364639\pi$
$17$	3.40196	0.825097	0.412548	+	0.910936i	$0.364639\pi$
$18$	8.06668	1.90133
$19$	−7.28137	−1.67046	−0.835231	−	0.549900i	$-0.814666\pi$
$19$	−7.28137	−1.67046	−0.835231	+	0.549900i	$0.814666\pi$
$20$	−18.0417	−4.03424
$21$	0.460315	0.100449
$22$	4.48033	0.955209
$23$	8.24707	1.71963	0.859816	−	0.510604i	$-0.170578\pi$
$23$	8.24707	1.71963	0.859816	+	0.510604i	$0.170578\pi$
$24$	−3.44669	−0.703552
$25$	4.75744	0.951487
$26$	−19.1318	−3.75206
$27$	−1.92908	−0.371252
$28$	8.12152	1.53482
$29$	2.48883	0.462164	0.231082	−	0.972934i	$-0.425773\pi$
$29$	2.48883	0.462164	0.231082	+	0.972934i	$0.425773\pi$
$30$	2.85145	0.520601
$31$	−1.95607	−0.351321	−0.175661	−	0.984451i	$-0.556206\pi$
$31$	−1.95607	−0.351321	−0.175661	+	0.984451i	$0.556206\pi$
$32$	−28.5997	−5.05577
$33$	−0.525975	−0.0915605
$34$	−9.48638	−1.62690
$35$	−4.39234	−0.742442
$36$	−16.7083	−2.78472
$37$	10.7638	1.76955	0.884777	−	0.466014i	$-0.154311\pi$
$37$	10.7638	1.76955	0.884777	+	0.466014i	$0.154311\pi$
$38$	20.3041	3.29376
$39$	2.24601	0.359650
$40$	32.8884	5.20011
$41$	−1.71143	−0.267280	−0.133640	−	0.991030i	$-0.542667\pi$
$41$	−1.71143	−0.267280	−0.133640	+	0.991030i	$0.542667\pi$
$42$	−1.28359	−0.198062
$43$	6.47839	0.987945	0.493973	−	0.869477i	$-0.335544\pi$
$43$	6.47839	0.987945	0.493973	+	0.869477i	$0.335544\pi$
$44$	−9.27998	−1.39901
$45$	9.03632	1.34705
$46$	−22.9970	−3.39072
$47$	−6.67303	−0.973361	−0.486681	−	0.873580i	$-0.661792\pi$
$47$	−6.67303	−0.973361	−0.486681	+	0.873580i	$0.661792\pi$
$48$	5.82958	0.841428
$49$	−5.02277	−0.717539
$50$	−13.2661	−1.87611
$51$	1.11367	0.155945
$52$	39.6272	5.49531
$53$	5.50964	0.756808	0.378404	−	0.925641i	$-0.376473\pi$
$53$	5.50964	0.756808	0.378404	+	0.925641i	$0.376473\pi$
$54$	5.37926	0.732024
$55$	5.01887	0.676745
$56$	−14.8048	−1.97838
$57$	−2.38364	−0.315720
$58$	−6.94010	−0.911280
$59$	−12.0792	−1.57258	−0.786291	−	0.617857i	$-0.788001\pi$
$59$	−12.0792	−1.57258	−0.786291	+	0.617857i	$0.788001\pi$
$60$	−5.90614	−0.762479
$61$	2.21258	0.283292	0.141646	−	0.989917i	$-0.454761\pi$
$61$	2.21258	0.283292	0.141646	+	0.989917i	$0.454761\pi$
$62$	5.45452	0.692724
$63$	−4.06773	−0.512486
$64$	44.1349	5.51686
$65$	−21.4315	−2.65825
$66$	1.46668	0.180536
$67$	5.97319	0.729741	0.364870	−	0.931058i	$-0.381113\pi$
$67$	5.97319	0.729741	0.364870	+	0.931058i	$0.381113\pi$
$68$	19.6489	2.38278
$69$	2.69977	0.325014
$70$	12.2481	1.46392
$71$	0.926605	0.109968	0.0549839	−	0.998487i	$-0.482489\pi$
$71$	0.926605	0.109968	0.0549839	+	0.998487i	$0.482489\pi$
$72$	30.4578	3.58948
$73$	8.77594	1.02715	0.513573	−	0.858046i	$-0.328322\pi$
$73$	8.77594	1.02715	0.513573	+	0.858046i	$0.328322\pi$
$74$	−30.0148	−3.48915
$75$	1.55740	0.179833
$76$	−42.0554	−4.82408
$77$	−2.25926	−0.257467
$78$	−6.26301	−0.709147
$79$	3.06460	0.344795	0.172397	−	0.985027i	$-0.444849\pi$
$79$	3.06460	0.344795	0.172397	+	0.985027i	$0.444849\pi$
$80$	−55.6261	−6.21918
$81$	8.04700	0.894111
$82$	4.77232	0.527014
$83$	16.4511	1.80574	0.902869	−	0.429915i	$-0.141456\pi$
$83$	16.4511	1.80574	0.902869	+	0.429915i	$0.141456\pi$
$84$	2.65867	0.290084
$85$	−10.6267	−1.15262
$86$	−18.0650	−1.94800
$87$	0.814745	0.0873498
$88$	16.9166	1.80332
$89$	0.974709	0.103319	0.0516595	−	0.998665i	$-0.483549\pi$
$89$	0.974709	0.103319	0.0516595	+	0.998665i	$0.483549\pi$
$90$	−25.1978	−2.65608
$91$	9.64748	1.01133
$92$	47.6330	4.96608
$93$	−0.640342	−0.0664004
$94$	18.6078	1.91924
$95$	22.7447	2.33356
$96$	−9.36244	−0.955550
$97$	−10.0869	−1.02417	−0.512083	−	0.858936i	$-0.671126\pi$
$97$	−10.0869	−1.02417	−0.512083	+	0.858936i	$0.671126\pi$
$98$	14.0060	1.41482
$99$	4.64796	0.467137
$100$	27.4778	2.74778
$101$	−1.28937	−0.128297	−0.0641487	−	0.997940i	$-0.520433\pi$
$101$	−1.28937	−0.128297	−0.0641487	+	0.997940i	$0.520433\pi$
$102$	−3.10547	−0.307487
$103$	−2.31191	−0.227799	−0.113899	−	0.993492i	$-0.536334\pi$
$103$	−2.31191	−0.227799	−0.113899	+	0.993492i	$0.536334\pi$
$104$	−72.2370	−7.08342
$105$	−1.43788	−0.140323
$106$	−15.3637	−1.49225
$107$	−17.2115	−1.66390	−0.831949	−	0.554853i	$-0.812775\pi$
$107$	−17.2115	−1.66390	−0.831949	+	0.554853i	$0.812775\pi$
$108$	−11.1419	−1.07213
$109$	−16.0453	−1.53686	−0.768431	−	0.639933i	$-0.778962\pi$
$109$	−16.0453	−1.53686	−0.768431	+	0.639933i	$0.778962\pi$
$110$	−13.9951	−1.33438
$111$	3.52364	0.334449
$112$	25.0403	2.36608
$113$	5.23344	0.492321	0.246160	−	0.969229i	$-0.420831\pi$
$113$	5.23344	0.492321	0.246160	+	0.969229i	$0.420831\pi$
$114$	6.64678	0.622528
$115$	−25.7613	−2.40225
$116$	14.3749	1.33467
$117$	−19.8476	−1.83491
$118$	33.6830	3.10077
$119$	4.78363	0.438515
$120$	10.7664	0.982831
$121$	−8.41847	−0.765316
$122$	−6.16979	−0.558587
$123$	−0.560254	−0.0505164
$124$	−11.2978	−1.01457
$125$	0.757694	0.0677702
$126$	11.3429	1.01050
$127$	−7.84611	−0.696230	−0.348115	−	0.937452i	$-0.613178\pi$
$127$	−7.84611	−0.696230	−0.348115	+	0.937452i	$0.613178\pi$
$128$	−65.8707	−5.82221
$129$	2.12077	0.186724
$130$	59.7619	5.24146
$131$	−6.41291	−0.560299	−0.280149	−	0.959956i	$-0.590384\pi$
$131$	−6.41291	−0.560299	−0.280149	+	0.959956i	$0.590384\pi$
$132$	−3.03790	−0.264416
$133$	−10.2386	−0.887801
$134$	−16.6563	−1.43888
$135$	6.02586	0.518624
$136$	−35.8182	−3.07139
$137$	−2.27467	−0.194339	−0.0971693	−	0.995268i	$-0.530979\pi$
$137$	−2.27467	−0.194339	−0.0971693	+	0.995268i	$0.530979\pi$
$138$	−7.52831	−0.640853
$139$	16.4575	1.39591	0.697955	−	0.716142i	$-0.254094\pi$
$139$	16.4575	1.39591	0.697955	+	0.716142i	$0.254094\pi$
$140$	−25.3691	−2.14408
$141$	−2.18449	−0.183967
$142$	−2.58384	−0.216831
$143$	−11.0236	−0.921840
$144$	−51.5151	−4.29292
$145$	−7.77432	−0.645622
$146$	−24.4717	−2.02529
$147$	−1.64426	−0.135616
$148$	62.1689	5.11025
$149$	−6.33752	−0.519190	−0.259595	−	0.965718i	$-0.583589\pi$
$149$	−6.33752	−0.519190	−0.259595	+	0.965718i	$0.583589\pi$
$150$	−4.34281	−0.354589
$151$	−7.98795	−0.650050	−0.325025	−	0.945705i	$-0.605373\pi$
$151$	−7.98795	−0.650050	−0.325025	+	0.945705i	$0.605373\pi$
$152$	76.6634	6.21822
$153$	−9.84131	−0.795623
$154$	6.29996	0.507666
$155$	6.11017	0.490780
$156$	12.9724	1.03862
$157$	−8.14132	−0.649748	−0.324874	−	0.945757i	$-0.605322\pi$
$157$	−8.14132	−0.649748	−0.324874	+	0.945757i	$0.605322\pi$
$158$	−8.54565	−0.679856
$159$	1.80364	0.143038
$160$	89.3367	7.06269
$161$	11.5965	0.913934
$162$	−22.4391	−1.76298
$163$	4.47513	0.350519	0.175259	−	0.984522i	$-0.443924\pi$
$163$	4.47513	0.350519	0.175259	+	0.984522i	$0.443924\pi$
$164$	−9.88477	−0.771871
$165$	1.64298	0.127906
$166$	−45.8738	−3.56050
$167$	−19.5377	−1.51187	−0.755936	−	0.654645i	$-0.772818\pi$
$167$	−19.5377	−1.51187	−0.755936	+	0.654645i	$0.772818\pi$
$168$	−4.84652	−0.373917
$169$	34.0728	2.62099
$170$	29.6325	2.27271
$171$	21.0638	1.61079
$172$	37.4176	2.85306
$173$	−1.78815	−0.135950	−0.0679752	−	0.997687i	$-0.521654\pi$
$173$	−1.78815	−0.135950	−0.0679752	+	0.997687i	$0.521654\pi$
$174$	−2.27192	−0.172234
$175$	6.68962	0.505688
$176$	−28.6120	−2.15671
$177$	−3.95427	−0.297221
$178$	−2.71798	−0.203721
$179$	1.71094	0.127882	0.0639408	−	0.997954i	$-0.479633\pi$
$179$	1.71094	0.127882	0.0639408	+	0.997954i	$0.479633\pi$
$180$	52.1915	3.89013
$181$	−23.2324	−1.72685	−0.863424	−	0.504480i	$-0.831684\pi$
$181$	−23.2324	−1.72685	−0.863424	+	0.504480i	$0.831684\pi$
$182$	−26.9020	−1.99411
$183$	0.724313	0.0535428
$184$	−86.8309	−6.40126
$185$	−33.6227	−2.47199
$186$	1.78560	0.130926
$187$	−5.46598	−0.399712
$188$	−38.5418	−2.81095
$189$	−2.71256	−0.197310
$190$	−63.4238	−4.60125
$191$	2.23404	0.161650	0.0808248	−	0.996728i	$-0.474245\pi$
$191$	2.23404	0.161650	0.0808248	+	0.996728i	$0.474245\pi$
$192$	14.4480	1.04270
$193$	18.0550	1.29963	0.649815	−	0.760092i	$-0.274846\pi$
$193$	18.0550	1.29963	0.649815	+	0.760092i	$0.274846\pi$
$194$	28.1272	2.01942
$195$	−7.01585	−0.502415
$196$	−29.0103	−2.07216
$197$	15.5863	1.11048	0.555240	−	0.831690i	$-0.312626\pi$
$197$	15.5863	1.11048	0.555240	+	0.831690i	$0.312626\pi$
$198$	−12.9608	−0.921087
$199$	0.965114	0.0684151	0.0342076	−	0.999415i	$-0.489109\pi$
$199$	0.965114	0.0684151	0.0342076	+	0.999415i	$0.489109\pi$
$200$	−50.0896	−3.54187
$201$	1.95539	0.137922
$202$	3.59542	0.252973
$203$	3.49964	0.245627
$204$	6.43228	0.450350
$205$	5.34596	0.373378
$206$	6.44676	0.449167
$207$	−23.8574	−1.65820
$208$	122.179	8.47157
$209$	11.6991	0.809242
$210$	4.00954	0.276685
$211$	12.4818	0.859281	0.429640	−	0.903000i	$-0.358640\pi$
$211$	12.4818	0.859281	0.429640	+	0.903000i	$0.358640\pi$
$212$	31.8223	2.18557
$213$	0.303334	0.0207841
$214$	47.9943	3.28082
$215$	−20.2365	−1.38012
$216$	20.3107	1.38197
$217$	−2.75051	−0.186717
$218$	44.7424	3.03034
$219$	2.87290	0.194133
$220$	28.9878	1.95436
$221$	23.3407	1.57007
$222$	−9.82569	−0.659457
$223$	25.2296	1.68950	0.844749	−	0.535162i	$-0.179750\pi$
$223$	25.2296	1.68950	0.844749	+	0.535162i	$0.179750\pi$
$224$	−40.2152	−2.68699
$225$	−13.7625	−0.917498
$226$	−14.5935	−0.970743
$227$	7.82148	0.519130	0.259565	−	0.965726i	$-0.416421\pi$
$227$	7.82148	0.519130	0.259565	+	0.965726i	$0.416421\pi$
$228$	−13.7673	−0.911761
$229$	−11.0808	−0.732238	−0.366119	−	0.930568i	$-0.619314\pi$
$229$	−11.0808	−0.732238	−0.366119	+	0.930568i	$0.619314\pi$
$230$	71.8354	4.73669
$231$	−0.739595	−0.0486618
$232$	−26.2041	−1.72038
$233$	−10.7685	−0.705470	−0.352735	−	0.935723i	$-0.614748\pi$
$233$	−10.7685	−0.705470	−0.352735	+	0.935723i	$0.614748\pi$
$234$	55.3452	3.61803
$235$	20.8445	1.35974
$236$	−69.7666	−4.54142
$237$	1.00323	0.0651669
$238$	−13.3392	−0.864650
$239$	16.0726	1.03965	0.519827	−	0.854272i	$-0.325997\pi$
$239$	16.0726	1.03965	0.519827	+	0.854272i	$0.325997\pi$
$240$	−18.2098	−1.17544
$241$	3.35170	0.215902	0.107951	−	0.994156i	$-0.465571\pi$
$241$	3.35170	0.215902	0.107951	+	0.994156i	$0.465571\pi$
$242$	23.4749	1.50903
$243$	8.42153	0.540241
$244$	12.7793	0.818112
$245$	15.6896	1.00237
$246$	1.56227	0.0996068
$247$	−49.9572	−3.17870
$248$	20.5949	1.30778
$249$	5.38544	0.341288
$250$	−2.11283	−0.133627
$251$	3.65518	0.230713	0.115357	−	0.993324i	$-0.463199\pi$
$251$	3.65518	0.230713	0.115357	+	0.993324i	$0.463199\pi$
$252$	−23.4942	−1.48000
$253$	−13.2507	−0.833063
$254$	21.8789	1.37281
$255$	−3.47876	−0.217848
$256$	95.4111	5.96319
$257$	4.74264	0.295838	0.147919	−	0.989000i	$-0.452743\pi$
$257$	4.74264	0.295838	0.147919	+	0.989000i	$0.452743\pi$
$258$	−5.91378	−0.368176
$259$	15.1354	0.940467
$260$	−123.783	−7.67671
$261$	−7.19977	−0.445654
$262$	17.8824	1.10478
$263$	14.1976	0.875459	0.437729	−	0.899107i	$-0.355783\pi$
$263$	14.1976	0.875459	0.437729	+	0.899107i	$0.355783\pi$
$264$	5.53784	0.340830
$265$	−17.2104	−1.05723
$266$	28.5504	1.75054
$267$	0.319082	0.0195275
$268$	34.4996	2.10740
$269$	30.1149	1.83614	0.918069	−	0.396420i	$-0.129748\pi$
$269$	30.1149	1.83614	0.918069	+	0.396420i	$0.129748\pi$
$270$	−16.8031	−1.02261
$271$	15.9892	0.971275	0.485637	−	0.874160i	$-0.338588\pi$
$271$	15.9892	0.971275	0.485637	+	0.874160i	$0.338588\pi$
$272$	60.5815	3.67329
$273$	3.15821	0.191143
$274$	6.34294	0.383191
$275$	−7.64384	−0.460941
$276$	15.5932	0.938600
$277$	−17.6619	−1.06120	−0.530600	−	0.847622i	$-0.678034\pi$
$277$	−17.6619	−1.06120	−0.530600	+	0.847622i	$0.678034\pi$
$278$	−45.8919	−2.75241
$279$	5.65860	0.338771
$280$	46.2457	2.76371
$281$	0.339813	0.0202715	0.0101358	−	0.999949i	$-0.496774\pi$
$281$	0.339813	0.0202715	0.0101358	+	0.999949i	$0.496774\pi$
$282$	6.09146	0.362741
$283$	−1.42129	−0.0844870	−0.0422435	−	0.999107i	$-0.513451\pi$
$283$	−1.42129	−0.0844870	−0.0422435	+	0.999107i	$0.513451\pi$
$284$	5.35184	0.317573
$285$	7.44574	0.441048
$286$	30.7394	1.81766
$287$	−2.40650	−0.142051
$288$	82.7343	4.87517
$289$	−5.42666	−0.319215
$290$	21.6787	1.27302
$291$	−3.30204	−0.193569
$292$	50.6876	2.96627
$293$	−31.8116	−1.85845	−0.929226	−	0.369512i	$-0.879525\pi$
$293$	−31.8116	−1.85845	−0.929226	+	0.369512i	$0.879525\pi$
$294$	4.58502	0.267404
$295$	37.7318	2.19683
$296$	−113.329	−6.58709
$297$	3.09949	0.179850
$298$	17.6722	1.02372
$299$	56.5828	3.27227
$300$	8.99515	0.519335
$301$	9.10952	0.525064
$302$	22.2744	1.28175
$303$	−0.422091	−0.0242485
$304$	−129.665	−7.43681
$305$	−6.91142	−0.395747
$306$	27.4425	1.56879
$307$	12.1221	0.691842	0.345921	−	0.938264i	$-0.387567\pi$
$307$	12.1221	0.691842	0.345921	+	0.938264i	$0.387567\pi$
$308$	−13.0489	−0.743533
$309$	−0.756828	−0.0430544
$310$	−17.0382	−0.967706
$311$	19.0622	1.08092	0.540458	−	0.841371i	$-0.318251\pi$
$311$	19.0622	1.08092	0.540458	+	0.841371i	$0.318251\pi$
$312$	−23.6476	−1.33878
$313$	−2.78565	−0.157454	−0.0787272	−	0.996896i	$-0.525086\pi$
$313$	−2.78565	−0.157454	−0.0787272	+	0.996896i	$0.525086\pi$
$314$	22.7021	1.28115
$315$	12.7063	0.715920
$316$	17.7004	0.995724
$317$	−29.2765	−1.64433	−0.822166	−	0.569247i	$-0.807235\pi$
$317$	−29.2765	−1.64433	−0.822166	+	0.569247i	$0.807235\pi$
$318$	−5.02946	−0.282038
$319$	−3.99883	−0.223892
$320$	−137.864	−7.70681
$321$	−5.63437	−0.314480
$322$	−32.3370	−1.80207
$323$	−24.7709	−1.37829
$324$	46.4775	2.58208
$325$	32.6406	1.81057
$326$	−12.4789	−0.691143
$327$	−5.25261	−0.290470
$328$	18.0191	0.994938
$329$	−9.38321	−0.517313
$330$	−4.58147	−0.252201
$331$	6.02428	0.331124	0.165562	−	0.986199i	$-0.447056\pi$
$331$	6.02428	0.331124	0.165562	+	0.986199i	$0.447056\pi$
$332$	95.0173	5.21475
$333$	−31.1378	−1.70634
$334$	54.4809	2.98106
$335$	−18.6584	−1.01942
$336$	8.19721	0.447194
$337$	−35.7745	−1.94876	−0.974381	−	0.224905i	$-0.927793\pi$
$337$	−35.7745	−1.94876	−0.974381	+	0.224905i	$0.927793\pi$
$338$	−95.0122	−5.16798
$339$	1.71322	0.0930496
$340$	−61.3770	−3.32864
$341$	3.14285	0.170195
$342$	−58.7365	−3.17611
$343$	−16.9057	−0.912822
$344$	−68.2090	−3.67758
$345$	−8.43324	−0.454030
$346$	4.98626	0.268063
$347$	13.8101	0.741366	0.370683	−	0.928759i	$-0.379124\pi$
$347$	13.8101	0.741366	0.370683	+	0.928759i	$0.379124\pi$
$348$	4.70577	0.252256
$349$	−1.96800	−0.105345	−0.0526723	−	0.998612i	$-0.516774\pi$
$349$	−1.96800	−0.105345	−0.0526723	+	0.998612i	$0.516774\pi$
$350$	−18.6540	−0.997100
$351$	−13.2354	−0.706452
$352$	45.9516	2.44923
$353$	−20.2209	−1.07625	−0.538125	−	0.842865i	$-0.680867\pi$
$353$	−20.2209	−1.07625	−0.538125	+	0.842865i	$0.680867\pi$
$354$	11.0265	0.586051
$355$	−2.89443	−0.153620
$356$	5.62968	0.298372
$357$	1.56597	0.0828802
$358$	−4.77096	−0.252153
$359$	−17.9403	−0.946855	−0.473427	−	0.880833i	$-0.656983\pi$
$359$	−17.9403	−0.946855	−0.473427	+	0.880833i	$0.656983\pi$
$360$	−95.1407	−5.01435
$361$	34.0184	1.79044
$362$	64.7835	3.40495
$363$	−2.75588	−0.144646
$364$	55.7214	2.92060
$365$	−27.4133	−1.43488
$366$	−2.01975	−0.105574
$367$	15.1916	0.792993	0.396497	−	0.918036i	$-0.370226\pi$
$367$	15.1916	0.792993	0.396497	+	0.918036i	$0.370226\pi$
$368$	146.862	7.65572
$369$	4.95087	0.257732
$370$	93.7570	4.87420
$371$	7.74733	0.402221
$372$	−3.69846	−0.191756
$373$	−0.620600	−0.0321334	−0.0160667	−	0.999871i	$-0.505114\pi$
$373$	−0.620600	−0.0321334	−0.0160667	+	0.999871i	$0.505114\pi$
$374$	15.2419	0.788140
$375$	0.248040	0.0128087
$376$	70.2583	3.62330
$377$	17.0758	0.879446
$378$	7.56399	0.389050
$379$	1.96635	0.101005	0.0505023	−	0.998724i	$-0.483918\pi$
$379$	1.96635	0.101005	0.0505023	+	0.998724i	$0.483918\pi$
$380$	131.368	6.73904
$381$	−2.56851	−0.131589
$382$	−6.22963	−0.318736
$383$	20.0307	1.02352	0.511761	−	0.859128i	$-0.328993\pi$
$383$	20.0307	1.02352	0.511761	+	0.859128i	$0.328993\pi$
$384$	−21.5635	−1.10041
$385$	7.05724	0.359670
$386$	−50.3465	−2.56257
$387$	−18.7409	−0.952654
$388$	−58.2592	−2.95766
$389$	10.7719	0.546158	0.273079	−	0.961992i	$-0.411958\pi$
$389$	10.7719	0.546158	0.273079	+	0.961992i	$0.411958\pi$
$390$	19.5637	0.990647
$391$	28.0562	1.41886
$392$	52.8832	2.67101
$393$	−2.09934	−0.105898
$394$	−43.4625	−2.18961
$395$	−9.57286	−0.481663
$396$	26.8454	1.34903
$397$	21.6331	1.08573	0.542867	−	0.839819i	$-0.317339\pi$
$397$	21.6331	1.08573	0.542867	+	0.839819i	$0.317339\pi$
$398$	−2.69122	−0.134899
$399$	−3.35173	−0.167796
$400$	84.7195	4.23598
$401$	−11.9611	−0.597309	−0.298655	−	0.954361i	$-0.596538\pi$
$401$	−11.9611	−0.597309	−0.298655	+	0.954361i	$0.596538\pi$
$402$	−5.45261	−0.271951
$403$	−13.4205	−0.668525
$404$	−7.44710	−0.370507
$405$	−25.1363	−1.24903
$406$	−9.75876	−0.484319
$407$	−17.2943	−0.857247
$408$	−11.7255	−0.580498
$409$	12.9981	0.642716	0.321358	−	0.946958i	$-0.395861\pi$
$409$	12.9981	0.642716	0.321358	+	0.946958i	$0.395861\pi$
$410$	−14.9072	−0.736216
$411$	−0.744640	−0.0367304
$412$	−13.3530	−0.657855
$413$	−16.9851	−0.835781
$414$	66.5264	3.26960
$415$	−51.3880	−2.52254
$416$	−196.222	−9.62057
$417$	5.38755	0.263830
$418$	−32.6229	−1.59564
$419$	26.3185	1.28574	0.642872	−	0.765973i	$-0.277743\pi$
$419$	26.3185	1.28574	0.642872	+	0.765973i	$0.277743\pi$
$420$	−8.30485	−0.405235
$421$	−38.9054	−1.89613	−0.948066	−	0.318074i	$-0.896964\pi$
$421$	−38.9054	−1.89613	−0.948066	+	0.318074i	$0.896964\pi$
$422$	−34.8055	−1.69430
$423$	19.3040	0.938591
$424$	−58.0094	−2.81718
$425$	16.1846	0.785069
$426$	−0.845849	−0.0409815
$427$	3.11120	0.150561
$428$	−99.4093	−4.80513
$429$	−3.60870	−0.174230
$430$	56.4295	2.72127
$431$	12.3316	0.593991	0.296995	−	0.954879i	$-0.404015\pi$
$431$	12.3316	0.593991	0.296995	+	0.954879i	$0.404015\pi$
$432$	−34.3528	−1.65280
$433$	−11.4162	−0.548628	−0.274314	−	0.961640i	$-0.588451\pi$
$433$	−11.4162	−0.548628	−0.274314	+	0.961640i	$0.588451\pi$
$434$	7.66981	0.368163
$435$	−2.54501	−0.122024
$436$	−92.6737	−4.43827
$437$	−60.0499	−2.87258
$438$	−8.01109	−0.382785
$439$	1.46905	0.0701138	0.0350569	−	0.999385i	$-0.488839\pi$
$439$	1.46905	0.0701138	0.0350569	+	0.999385i	$0.488839\pi$
$440$	−52.8422	−2.51915
$441$	14.5300	0.691907
$442$	−65.0857	−3.09581
$443$	14.6808	0.697506	0.348753	−	0.937215i	$-0.386605\pi$
$443$	14.6808	0.697506	0.348753	+	0.937215i	$0.386605\pi$
$444$	20.3517	0.965848
$445$	−3.04469	−0.144332
$446$	−70.3528	−3.33130
$447$	−2.07466	−0.0981280
$448$	62.0598	2.93205
$449$	−11.1775	−0.527498	−0.263749	−	0.964591i	$-0.584959\pi$
$449$	−11.1775	−0.527498	−0.263749	+	0.964591i	$0.584959\pi$
$450$	38.3767	1.80910
$451$	2.74977	0.129482
$452$	30.2271	1.42176
$453$	−2.61494	−0.122861
$454$	−21.8102	−1.02360
$455$	−30.1357	−1.41278
$456$	25.0966	1.17526
$457$	10.0018	0.467864	0.233932	−	0.972253i	$-0.424841\pi$
$457$	10.0018	0.467864	0.233932	+	0.972253i	$0.424841\pi$
$458$	30.8988	1.44380
$459$	−6.56267	−0.306319
$460$	−148.791	−6.93740
$461$	−16.7080	−0.778169	−0.389085	−	0.921202i	$-0.627209\pi$
$461$	−16.7080	−0.778169	−0.389085	+	0.921202i	$0.627209\pi$
$462$	2.06236	0.0959498
$463$	29.9540	1.39208	0.696040	−	0.718003i	$-0.254943\pi$
$463$	29.9540	1.39208	0.696040	+	0.718003i	$0.254943\pi$
$464$	44.3206	2.05753
$465$	2.00023	0.0927584
$466$	30.0281	1.39102
$467$	32.4150	1.49999	0.749993	−	0.661446i	$-0.230057\pi$
$467$	32.4150	1.49999	0.749993	+	0.661446i	$0.230057\pi$
$468$	−114.635	−5.29901
$469$	8.39914	0.387836
$470$	−58.1249	−2.68110
$471$	−2.66515	−0.122804
$472$	127.179	5.85387
$473$	−10.4089	−0.478602
$474$	−2.79751	−0.128494
$475$	−34.6407	−1.58942
$476$	27.6291	1.26638
$477$	−15.9385	−0.729773
$478$	−44.8186	−2.04996
$479$	4.09579	0.187142	0.0935708	−	0.995613i	$-0.470172\pi$
$479$	4.09579	0.187142	0.0935708	+	0.995613i	$0.470172\pi$
$480$	29.2454	1.33486
$481$	73.8499	3.36727
$482$	−9.34624	−0.425709
$483$	3.79625	0.172735
$484$	−48.6230	−2.21014
$485$	31.5082	1.43071
$486$	−23.4835	−1.06523
$487$	−13.2890	−0.602181	−0.301090	−	0.953596i	$-0.597351\pi$
$487$	−13.2890	−0.602181	−0.301090	+	0.953596i	$0.597351\pi$
$488$	−23.2956	−1.05454
$489$	1.46498	0.0662488
$490$	−43.7504	−1.97644
$491$	−6.16553	−0.278247	−0.139123	−	0.990275i	$-0.544428\pi$
$491$	−6.16553	−0.278247	−0.139123	+	0.990275i	$0.544428\pi$
$492$	−3.23589	−0.145885
$493$	8.46689	0.381330
$494$	139.306	6.26767
$495$	−14.5188	−0.652570
$496$	−34.8334	−1.56406
$497$	1.30294	0.0584447
$498$	−15.0173	−0.672942
$499$	−41.5898	−1.86181	−0.930907	−	0.365257i	$-0.880981\pi$
$499$	−41.5898	−1.86181	−0.930907	+	0.365257i	$0.880981\pi$
$500$	4.37625	0.195712
$501$	−6.39588	−0.285747
$502$	−10.1925	−0.454913
$503$	−2.92881	−0.130589	−0.0652946	−	0.997866i	$-0.520799\pi$
$503$	−2.92881	−0.130589	−0.0652946	+	0.997866i	$0.520799\pi$
$504$	42.8279	1.90771
$505$	4.02760	0.179226
$506$	36.9495	1.64261
$507$	11.1541	0.495372
$508$	−45.3172	−2.01063
$509$	19.8000	0.877620	0.438810	−	0.898580i	$-0.355400\pi$
$509$	19.8000	0.877620	0.438810	+	0.898580i	$0.355400\pi$
$510$	9.70053	0.429547
$511$	12.3402	0.545898
$512$	−134.313	−5.93584
$513$	14.0464	0.620163
$514$	−13.2249	−0.583324
$515$	7.22168	0.318225
$516$	12.2491	0.539235
$517$	10.7216	0.471537
$518$	−42.2051	−1.85438
$519$	−0.585370	−0.0256949
$520$	225.646	9.89523
$521$	27.4605	1.20306	0.601532	−	0.798849i	$-0.294557\pi$
$521$	27.4605	1.20306	0.601532	+	0.798849i	$0.294557\pi$
$522$	20.0766	0.878728
$523$	−15.0777	−0.659302	−0.329651	−	0.944103i	$-0.606931\pi$
$523$	−15.0777	−0.659302	−0.329651	+	0.944103i	$0.606931\pi$
$524$	−37.0394	−1.61807
$525$	2.18992	0.0955760
$526$	−39.5899	−1.72620
$527$	−6.65448	−0.289874
$528$	−9.36647	−0.407623
$529$	45.0141	1.95713
$530$	47.9913	2.08461
$531$	34.9432	1.51641
$532$	−59.1358	−2.56386
$533$	−11.7420	−0.508604
$534$	−0.889761	−0.0385037
$535$	53.7633	2.32439
$536$	−62.8899	−2.71643
$537$	0.560095	0.0241699
$538$	−83.9755	−3.62044
$539$	8.07015	0.347606
$540$	34.8039	1.49772
$541$	−23.3768	−1.00505	−0.502525	−	0.864563i	$-0.667595\pi$
$541$	−23.3768	−1.00505	−0.502525	+	0.864563i	$0.667595\pi$
$542$	−44.5860	−1.91513
$543$	−7.60537	−0.326378
$544$	−97.2952	−4.17150
$545$	50.1205	2.14693
$546$	−8.80667	−0.376891
$547$	−19.2789	−0.824304	−0.412152	−	0.911115i	$-0.635223\pi$
$547$	−19.2789	−0.824304	−0.412152	+	0.911115i	$0.635223\pi$
$548$	−13.1380	−0.561226
$549$	−6.40063	−0.273172
$550$	21.3149	0.908869
$551$	−18.1221	−0.772026
$552$	−28.4250	−1.20985
$553$	4.30926	0.183248
$554$	49.2503	2.09244
$555$	−11.0068	−0.467211
$556$	95.0546	4.03121
$557$	38.1914	1.61822	0.809110	−	0.587657i	$-0.199949\pi$
$557$	38.1914	1.61822	0.809110	+	0.587657i	$0.199949\pi$
$558$	−15.7790	−0.667979
$559$	44.4480	1.87995
$560$	−78.2180	−3.30531
$561$	−1.78935	−0.0755463
$562$	−0.947569	−0.0399708
$563$	14.2015	0.598522	0.299261	−	0.954171i	$-0.403260\pi$
$563$	14.2015	0.598522	0.299261	+	0.954171i	$0.403260\pi$
$564$	−12.6171	−0.531274
$565$	−16.3476	−0.687751
$566$	3.96327	0.166589
$567$	11.3152	0.475194
$568$	−9.75594	−0.409350
$569$	−10.2004	−0.427625	−0.213812	−	0.976875i	$-0.568588\pi$
$569$	−10.2004	−0.427625	−0.213812	+	0.976875i	$0.568588\pi$
$570$	−20.7625	−0.869644
$571$	27.9671	1.17039	0.585193	−	0.810894i	$-0.301019\pi$
$571$	27.9671	1.17039	0.585193	+	0.810894i	$0.301019\pi$
$572$	−63.6696	−2.66216
$573$	0.731338	0.0305521
$574$	6.71055	0.280093
$575$	39.2349	1.63621
$576$	−127.675	−5.31979
$577$	23.2424	0.967595	0.483797	−	0.875180i	$-0.339257\pi$
$577$	23.2424	0.967595	0.483797	+	0.875180i	$0.339257\pi$
$578$	15.1323	0.629419
$579$	5.91052	0.245633
$580$	−44.9026	−1.86448
$581$	23.1325	0.959698
$582$	9.20776	0.381674
$583$	−8.85242	−0.366630
$584$	−92.3992	−3.82351
$585$	61.9979	2.56330
$586$	88.7067	3.66444
$587$	34.2656	1.41429	0.707147	−	0.707067i	$-0.249982\pi$
$587$	34.2656	1.41429	0.707147	+	0.707067i	$0.249982\pi$
$588$	−9.49683	−0.391643
$589$	14.2429	0.586868
$590$	−105.215	−4.33164
$591$	5.10236	0.209883
$592$	191.679	7.87797
$593$	1.74315	0.0715825	0.0357913	−	0.999359i	$-0.488605\pi$
$593$	1.74315	0.0715825	0.0357913	+	0.999359i	$0.488605\pi$
$594$	−8.64293	−0.354624
$595$	−14.9426	−0.612586
$596$	−36.6040	−1.49936
$597$	0.315941	0.0129306
$598$	−157.781	−6.45216
$599$	−24.6525	−1.00727	−0.503637	−	0.863915i	$-0.668005\pi$
$599$	−24.6525	−1.00727	−0.503637	+	0.863915i	$0.668005\pi$
$600$	−16.3974	−0.669420
$601$	20.7171	0.845070	0.422535	−	0.906347i	$-0.361140\pi$
$601$	20.7171	0.845070	0.422535	+	0.906347i	$0.361140\pi$
$602$	−25.4019	−1.03531
$603$	−17.2794	−0.703673
$604$	−46.1364	−1.87726
$605$	26.2967	1.06911
$606$	1.17700	0.0478124
$607$	27.3106	1.10850	0.554251	−	0.832350i	$-0.313005\pi$
$607$	27.3106	1.10850	0.554251	+	0.832350i	$0.313005\pi$
$608$	208.245	8.44546
$609$	1.14565	0.0464239
$610$	19.2725	0.780321
$611$	−45.7834	−1.85220
$612$	−56.8410	−2.29766
$613$	33.4116	1.34948	0.674740	−	0.738055i	$-0.264256\pi$
$613$	33.4116	1.34948	0.674740	+	0.738055i	$0.264256\pi$
$614$	−33.8024	−1.36415
$615$	1.75006	0.0705692
$616$	23.7871	0.958410
$617$	19.9478	0.803068	0.401534	−	0.915844i	$-0.368477\pi$
$617$	19.9478	0.803068	0.401534	+	0.915844i	$0.368477\pi$
$618$	2.11042	0.0848934
$619$	−20.5054	−0.824182	−0.412091	−	0.911143i	$-0.635201\pi$
$619$	−20.5054	−0.824182	−0.412091	+	0.911143i	$0.635201\pi$
$620$	35.2908	1.41731
$621$	−15.9093	−0.638418
$622$	−53.1549	−2.13132
$623$	1.37058	0.0549110
$624$	39.9966	1.60114
$625$	−26.1540	−1.04616
$626$	7.76780	0.310464
$627$	3.82982	0.152948
$628$	−47.0222	−1.87639
$629$	36.6180	1.46005
$630$	−35.4316	−1.41163
$631$	−39.7605	−1.58284	−0.791420	−	0.611273i	$-0.790658\pi$
$631$	−39.7605	−1.58284	−0.791420	+	0.611273i	$0.790658\pi$
$632$	−32.2663	−1.28348
$633$	4.08605	0.162406
$634$	81.6377	3.24225
$635$	24.5088	0.972603
$636$	10.4174	0.413077
$637$	−34.4611	−1.36540
$638$	11.1508	0.441463
$639$	−2.68052	−0.106040
$640$	205.760	8.13337
$641$	−47.0612	−1.85881	−0.929403	−	0.369068i	$-0.879677\pi$
$641$	−47.0612	−1.85881	−0.929403	+	0.369068i	$0.879677\pi$
$642$	15.7115	0.620082
$643$	−3.21174	−0.126659	−0.0633294	−	0.997993i	$-0.520172\pi$
$643$	−3.21174	−0.126659	−0.0633294	+	0.997993i	$0.520172\pi$
$644$	66.9787	2.63933
$645$	−6.62463	−0.260845
$646$	69.0739	2.71767
$647$	7.34160	0.288628	0.144314	−	0.989532i	$-0.453902\pi$
$647$	7.34160	0.288628	0.144314	+	0.989532i	$0.453902\pi$
$648$	−84.7244	−3.32829
$649$	19.4079	0.761825
$650$	−91.0184	−3.57004
$651$	−0.900410	−0.0352899
$652$	25.8472	1.01226
$653$	−0.0502430	−0.00196616	−0.000983081	−	1.00000i	$-0.500313\pi$
$653$	−0.0502430	−0.00196616	−0.000983081		1.00000i	$0.500313\pi$
$654$	14.6469	0.572740
$655$	20.0320	0.782713
$656$	−30.4768	−1.18992
$657$	−25.3873	−0.990454
$658$	26.1651	1.02002
$659$	8.19501	0.319232	0.159616	−	0.987179i	$-0.448974\pi$
$659$	8.19501	0.319232	0.159616	+	0.987179i	$0.448974\pi$
$660$	9.48947	0.369377
$661$	−31.0716	−1.20854	−0.604272	−	0.796778i	$-0.706536\pi$
$661$	−31.0716	−1.20854	−0.604272	+	0.796778i	$0.706536\pi$
$662$	−16.7987	−0.652901
$663$	7.64085	0.296746
$664$	−173.208	−6.72179
$665$	31.9823	1.24022
$666$	86.8280	3.36452
$667$	20.5255	0.794751
$668$	−112.845	−4.36610
$669$	8.25918	0.319319
$670$	52.0290	2.01005
$671$	−3.55498	−0.137239
$672$	−13.1649	−0.507847
$673$	39.8256	1.53517	0.767583	−	0.640950i	$-0.221459\pi$
$673$	39.8256	1.53517	0.767583	+	0.640950i	$0.221459\pi$
$674$	99.7573	3.84251
$675$	−9.17750	−0.353242
$676$	196.796	7.56909
$677$	25.9325	0.996667	0.498334	−	0.866985i	$-0.333945\pi$
$677$	25.9325	0.996667	0.498334	+	0.866985i	$0.333945\pi$
$678$	−4.77733	−0.183472
$679$	−14.1835	−0.544314
$680$	111.885	4.29059
$681$	2.56045	0.0981166
$682$	−8.76384	−0.335585
$683$	7.98486	0.305532	0.152766	−	0.988262i	$-0.451182\pi$
$683$	7.98486	0.305532	0.152766	+	0.988262i	$0.451182\pi$
$684$	121.659	4.65176
$685$	7.10538	0.271483
$686$	47.1416	1.79987
$687$	−3.62741	−0.138394
$688$	115.366	4.39829
$689$	37.8015	1.44012
$690$	23.5161	0.895243
$691$	−32.3105	−1.22915	−0.614574	−	0.788859i	$-0.710672\pi$
$691$	−32.3105	−1.22915	−0.614574	+	0.788859i	$0.710672\pi$
$692$	−10.3279	−0.392608
$693$	6.53568	0.248270
$694$	−38.5096	−1.46180
$695$	−51.4082	−1.95002
$696$	−8.57821	−0.325156
$697$	−5.82221	−0.220532
$698$	5.48778	0.207715
$699$	−3.52520	−0.133335
$700$	38.6376	1.46036
$701$	−38.7808	−1.46473	−0.732365	−	0.680913i	$-0.761583\pi$
$701$	−38.7808	−1.46473	−0.732365	+	0.680913i	$0.761583\pi$
$702$	36.9069	1.39296
$703$	−78.3751	−2.95597
$704$	−70.9121	−2.67260
$705$	6.82367	0.256994
$706$	56.3860	2.12212
$707$	−1.81304	−0.0681864
$708$	−22.8389	−0.858337
$709$	10.3602	0.389085	0.194542	−	0.980894i	$-0.437678\pi$
$709$	10.3602	0.389085	0.194542	+	0.980894i	$0.437678\pi$
$710$	8.07112	0.302904
$711$	−8.86539	−0.332478
$712$	−10.2624	−0.384600
$713$	−16.1319	−0.604143
$714$	−4.36673	−0.163421
$715$	34.4343	1.28777
$716$	9.88197	0.369306
$717$	5.26156	0.196497
$718$	50.0267	1.86698
$719$	−45.8356	−1.70938	−0.854689	−	0.519140i	$-0.826252\pi$
$719$	−45.8356	−1.70938	−0.854689	+	0.519140i	$0.826252\pi$
$720$	160.917	5.99702
$721$	−3.25086	−0.121068
$722$	−94.8603	−3.53034
$723$	1.09722	0.0408059
$724$	−134.184	−4.98692
$725$	11.8404	0.439743
$726$	7.68478	0.285209
$727$	9.17882	0.340424	0.170212	−	0.985407i	$-0.445555\pi$
$727$	9.17882	0.340424	0.170212	+	0.985407i	$0.445555\pi$
$728$	−101.575	−3.76463
$729$	−21.3841	−0.792004
$730$	76.4421	2.82925
$731$	22.0392	0.815150
$732$	4.18345	0.154625
$733$	12.2497	0.452453	0.226227	−	0.974075i	$-0.427361\pi$
$733$	12.2497	0.452453	0.226227	+	0.974075i	$0.427361\pi$
$734$	−42.3617	−1.56360
$735$	5.13616	0.189450
$736$	−235.864	−8.69406
$737$	−9.59720	−0.353517
$738$	−13.8055	−0.508188
$739$	−8.58673	−0.315868	−0.157934	−	0.987450i	$-0.550483\pi$
$739$	−8.58673	−0.315868	−0.157934	+	0.987450i	$0.550483\pi$
$740$	−194.196	−7.13880
$741$	−16.3541	−0.600781
$742$	−21.6035	−0.793088
$743$	47.7922	1.75332	0.876662	−	0.481106i	$-0.159765\pi$
$743$	47.7922	1.75332	0.876662	+	0.481106i	$0.159765\pi$
$744$	6.74197	0.247173
$745$	19.7965	0.725286
$746$	1.73054	0.0633597
$747$	−47.5902	−1.74123
$748$	−31.5701	−1.15432
$749$	−24.2018	−0.884313
$750$	−0.691659	−0.0252558
$751$	28.0761	1.02451	0.512256	−	0.858833i	$-0.328810\pi$
$751$	28.0761	1.02451	0.512256	+	0.858833i	$0.328810\pi$
$752$	−118.832	−4.33336
$753$	1.19656	0.0436052
$754$	−47.6158	−1.73407
$755$	24.9519	0.908091
$756$	−15.6671	−0.569806
$757$	29.9230	1.08757	0.543785	−	0.839225i	$-0.316991\pi$
$757$	29.9230	1.08757	0.543785	+	0.839225i	$0.316991\pi$
$758$	−5.48317	−0.199158
$759$	−4.33775	−0.157450
$760$	−239.473	−8.68658
$761$	38.9208	1.41088	0.705440	−	0.708770i	$-0.250750\pi$
$761$	38.9208	1.41088	0.705440	+	0.708770i	$0.250750\pi$
$762$	7.16230	0.259463
$763$	−22.5619	−0.816797
$764$	12.9033	0.466824
$765$	30.7412	1.11145
$766$	−55.8557	−2.01815
$767$	−82.8752	−2.99245
$768$	31.2339	1.12706
$769$	38.1329	1.37511	0.687555	−	0.726133i	$-0.258684\pi$
$769$	38.1329	1.37511	0.687555	+	0.726133i	$0.258684\pi$
$770$	−19.6791	−0.709187
$771$	1.55256	0.0559139
$772$	104.281	3.75317
$773$	32.4816	1.16828	0.584140	−	0.811653i	$-0.301432\pi$
$773$	32.4816	1.16828	0.584140	+	0.811653i	$0.301432\pi$
$774$	52.2591	1.87841
$775$	−9.30589	−0.334278
$776$	106.201	3.81241
$777$	4.95473	0.177750
$778$	−30.0376	−1.07690
$779$	12.4615	0.446481
$780$	−40.5218	−1.45091
$781$	−1.48879	−0.0532730
$782$	−78.2348	−2.79767
$783$	−4.80116	−0.171579
$784$	−89.4445	−3.19445
$785$	25.4309	0.907669
$786$	5.85401	0.208806
$787$	34.0627	1.21420	0.607102	−	0.794624i	$-0.292332\pi$
$787$	34.0627	1.21420	0.607102	+	0.794624i	$0.292332\pi$
$788$	90.0228	3.20693
$789$	4.64773	0.165463
$790$	26.6940	0.949729
$791$	7.35895	0.261654
$792$	−48.9369	−1.73890
$793$	15.1804	0.539073
$794$	−60.3240	−2.14082
$795$	−5.63402	−0.199818
$796$	5.57426	0.197574
$797$	45.4659	1.61049	0.805243	−	0.592945i	$-0.202035\pi$
$797$	45.4659	1.61049	0.805243	+	0.592945i	$0.202035\pi$
$798$	9.34630	0.330856
$799$	−22.7014	−0.803117
$800$	−136.061	−4.81050
$801$	−2.81967	−0.0996282
$802$	33.3536	1.17776
$803$	−14.1004	−0.497593
$804$	11.2938	0.398303
$805$	−36.2240	−1.27673
$806$	37.4233	1.31818
$807$	9.85845	0.347034
$808$	13.5754	0.477582
$809$	−42.7388	−1.50262	−0.751309	−	0.659951i	$-0.770577\pi$
$809$	−42.7388	−1.50262	−0.751309	+	0.659951i	$0.770577\pi$
$810$	70.0927	2.46281
$811$	18.7760	0.659315	0.329658	−	0.944101i	$-0.393067\pi$
$811$	18.7760	0.659315	0.329658	+	0.944101i	$0.393067\pi$
$812$	20.2130	0.709339
$813$	5.23424	0.183573
$814$	48.2252	1.69029
$815$	−13.9789	−0.489660
$816$	19.8320	0.694259
$817$	−47.1716	−1.65032
$818$	−36.2453	−1.26729
$819$	−27.9086	−0.975204
$820$	30.8770	1.07827
$821$	−16.1533	−0.563755	−0.281878	−	0.959450i	$-0.590957\pi$
$821$	−16.1533	−0.563755	−0.281878	+	0.959450i	$0.590957\pi$
$822$	2.07643	0.0724238
$823$	−6.42356	−0.223911	−0.111956	−	0.993713i	$-0.535711\pi$
$823$	−6.42356	−0.223911	−0.111956	+	0.993713i	$0.535711\pi$
$824$	24.3414	0.847972
$825$	−2.50229	−0.0871187
$826$	47.3630	1.64797
$827$	2.93670	0.102119	0.0510596	−	0.998696i	$-0.483740\pi$
$827$	2.93670	0.102119	0.0510596	+	0.998696i	$0.483740\pi$
$828$	−137.794	−4.78869
$829$	32.5523	1.13059	0.565295	−	0.824889i	$-0.308762\pi$
$829$	32.5523	1.13059	0.565295	+	0.824889i	$0.308762\pi$
$830$	143.296	4.97386
$831$	−5.78182	−0.200569
$832$	302.808	10.4980
$833$	−17.0873	−0.592039
$834$	−15.0232	−0.520211
$835$	61.0297	2.11202
$836$	67.5710	2.33699
$837$	3.77343	0.130429
$838$	−73.3893	−2.53519
$839$	−36.4115	−1.25706	−0.628532	−	0.777784i	$-0.716344\pi$
$839$	−36.4115	−1.25706	−0.628532	+	0.777784i	$0.716344\pi$
$840$	15.1390	0.522346
$841$	−22.8057	−0.786405
$842$	108.488	3.73874
$843$	0.111241	0.00383136
$844$	72.0916	2.48150
$845$	−106.433	−3.66141
$846$	−53.8292	−1.85069
$847$	−11.8375	−0.406743
$848$	98.1147	3.36927
$849$	−0.465275	−0.0159682
$850$	−45.1308	−1.54798
$851$	88.7696	3.04298
$852$	1.75198	0.0600220
$853$	−32.6608	−1.11829	−0.559143	−	0.829071i	$-0.688870\pi$
$853$	−32.6608	−1.11829	−0.559143	+	0.829071i	$0.688870\pi$
$854$	−8.67559	−0.296873
$855$	−65.7968	−2.25020
$856$	181.215	6.19379
$857$	−15.6974	−0.536213	−0.268107	−	0.963389i	$-0.586398\pi$
$857$	−15.6974	−0.536213	−0.268107	+	0.963389i	$0.586398\pi$
$858$	10.0629	0.343541
$859$	19.2065	0.655317	0.327658	−	0.944796i	$-0.393741\pi$
$859$	19.2065	0.655317	0.327658	+	0.944796i	$0.393741\pi$
$860$	−116.881	−3.98561
$861$	−0.787796	−0.0268480
$862$	−34.3866	−1.17121
$863$	1.16703	0.0397262	0.0198631	−	0.999803i	$-0.493677\pi$
$863$	1.16703	0.0397262	0.0198631	+	0.999803i	$0.493677\pi$
$864$	55.1713	1.87697
$865$	5.58562	0.189917
$866$	31.8341	1.08177
$867$	−1.77648	−0.0603323
$868$	−15.8863	−0.539215
$869$	−4.92394	−0.167033
$870$	7.09677	0.240603
$871$	40.9818	1.38862
$872$	168.936	5.72090
$873$	29.1796	0.987580
$874$	167.449	5.66406
$875$	1.06542	0.0360179
$876$	16.5932	0.560631
$877$	−6.12825	−0.206936	−0.103468	−	0.994633i	$-0.532994\pi$
$877$	−6.12825	−0.206936	−0.103468	+	0.994633i	$0.532994\pi$
$878$	−4.09644	−0.138248
$879$	−10.4139	−0.351251
$880$	89.3751	3.01284
$881$	37.2657	1.25551	0.627756	−	0.778410i	$-0.283973\pi$
$881$	37.2657	1.25551	0.627756	+	0.778410i	$0.283973\pi$
$882$	−40.5171	−1.36428
$883$	42.0658	1.41563	0.707813	−	0.706399i	$-0.249682\pi$
$883$	42.0658	1.41563	0.707813	+	0.706399i	$0.249682\pi$
$884$	134.810	4.53416
$885$	12.3519	0.415205
$886$	−40.9374	−1.37532
$887$	−2.95435	−0.0991972	−0.0495986	−	0.998769i	$-0.515794\pi$
$887$	−2.95435	−0.0991972	−0.0495986	+	0.998769i	$0.515794\pi$
$888$	−37.0994	−1.24497
$889$	−11.0327	−0.370026
$890$	8.49013	0.284590
$891$	−12.9292	−0.433145
$892$	145.720	4.87906
$893$	48.5888	1.62596
$894$	5.78519	0.193486
$895$	−5.34445	−0.178645
$896$	−92.6235	−3.09433
$897$	18.5230	0.618465
$898$	31.1684	1.04010
$899$	−4.86833	−0.162368
$900$	−79.4887	−2.64962
$901$	18.7436	0.624440
$902$	−7.66775	−0.255308
$903$	2.98210	0.0992382
$904$	−55.1013	−1.83264
$905$	72.5707	2.41233
$906$	7.29178	0.242253
$907$	27.0470	0.898081	0.449041	−	0.893511i	$-0.351766\pi$
$907$	27.0470	0.898081	0.449041	+	0.893511i	$0.351766\pi$
$908$	45.1749	1.49918
$909$	3.72994	0.123714
$910$	84.0336	2.78569
$911$	57.0088	1.88878	0.944392	−	0.328821i	$-0.106651\pi$
$911$	57.0088	1.88878	0.944392	+	0.328821i	$0.106651\pi$
$912$	−42.4474	−1.40557
$913$	−26.4321	−0.874776
$914$	−27.8900	−0.922521
$915$	−2.26253	−0.0747969
$916$	−63.9998	−2.11461
$917$	−9.01745	−0.297783
$918$	18.3000	0.603991
$919$	−14.6520	−0.483325	−0.241662	−	0.970360i	$-0.577693\pi$
$919$	−14.6520	−0.483325	−0.241662	+	0.970360i	$0.577693\pi$
$920$	271.233	8.94228
$921$	3.96829	0.130760
$922$	46.5903	1.53437
$923$	6.35740	0.209257
$924$	−4.27172	−0.140529
$925$	51.2080	1.68371
$926$	−83.5269	−2.74486
$927$	6.68796	0.219662
$928$	−71.1798	−2.33659
$929$	23.4014	0.767774	0.383887	−	0.923380i	$-0.374585\pi$
$929$	23.4014	0.767774	0.383887	+	0.923380i	$0.374585\pi$
$930$	−5.57765	−0.182898
$931$	36.5727	1.19862
$932$	−62.1964	−2.03731
$933$	6.24021	0.204295
$934$	−90.3892	−2.95763
$935$	17.0740	0.558380
$936$	208.970	6.83039
$937$	41.3841	1.35196	0.675979	−	0.736921i	$-0.263721\pi$
$937$	41.3841	1.35196	0.675979	+	0.736921i	$0.263721\pi$
$938$	−23.4210	−0.764723
$939$	−0.911914	−0.0297592
$940$	120.392	3.92677
$941$	50.3674	1.64193	0.820965	−	0.570979i	$-0.193436\pi$
$941$	50.3674	1.64193	0.820965	+	0.570979i	$0.193436\pi$
$942$	7.43178	0.242140
$943$	−14.1142	−0.459623
$944$	−215.105	−7.00106
$945$	8.47320	0.275633
$946$	29.0253	0.943694
$947$	5.39828	0.175420	0.0877102	−	0.996146i	$-0.472045\pi$
$947$	5.39828	0.175420	0.0877102	+	0.996146i	$0.472045\pi$
$948$	5.79441	0.188194
$949$	60.2114	1.95454
$950$	96.5956	3.13397
$951$	−9.58399	−0.310782
$952$	−50.3654	−1.63235
$953$	57.2524	1.85459	0.927293	−	0.374336i	$-0.122129\pi$
$953$	57.2524	1.85459	0.927293	+	0.374336i	$0.122129\pi$
$954$	44.4445	1.43895
$955$	−6.97845	−0.225817
$956$	92.8316	3.00239
$957$	−1.30906	−0.0423160
$958$	−11.4211	−0.369000
$959$	−3.19851	−0.103285
$960$	−45.1312	−1.45660
$961$	−27.1738	−0.876573
$962$	−205.931	−6.63947
$963$	49.7900	1.60446
$964$	19.3586	0.623499
$965$	−56.3983	−1.81553
$966$	−10.5859	−0.340594
$967$	16.2749	0.523364	0.261682	−	0.965154i	$-0.415723\pi$
$967$	16.2749	0.523364	0.261682	+	0.965154i	$0.415723\pi$
$968$	88.6356	2.84885
$969$	−8.10904	−0.260500
$970$	−87.8607	−2.82104
$971$	41.5664	1.33393	0.666965	−	0.745089i	$-0.267593\pi$
$971$	41.5664	1.33393	0.666965	+	0.745089i	$0.267593\pi$
$972$	48.6406	1.56015
$973$	23.1416	0.741885
$974$	37.0563	1.18736
$975$	10.6853	0.342202
$976$	39.4012	1.26120
$977$	−28.4039	−0.908722	−0.454361	−	0.890818i	$-0.650132\pi$
$977$	−28.4039	−0.908722	−0.454361	+	0.890818i	$0.650132\pi$
$978$	−4.08511	−0.130627
$979$	−1.56608	−0.0500521
$980$	90.6191	2.89472
$981$	46.4164	1.48196
$982$	17.1926	0.548638
$983$	−32.8526	−1.04783	−0.523917	−	0.851769i	$-0.675530\pi$
$983$	−32.8526	−1.04783	−0.523917	+	0.851769i	$0.675530\pi$
$984$	5.89875	0.188045
$985$	−48.6869	−1.55129
$986$	−23.6100	−0.751894
$987$	−3.07170	−0.0977732
$988$	−288.541	−9.17970
$989$	53.4277	1.69890
$990$	40.4856	1.28672
$991$	42.0064	1.33438	0.667188	−	0.744889i	$-0.267498\pi$
$991$	42.0064	1.33438	0.667188	+	0.744889i	$0.267498\pi$
$992$	55.9432	1.77620
$993$	1.97211	0.0625831
$994$	−3.63324	−0.115239
$995$	−3.01472	−0.0955729
$996$	31.1049	0.985598
$997$	31.1357	0.986078	0.493039	−	0.870007i	$-0.335886\pi$
$997$	31.1357	0.986078	0.493039	+	0.870007i	$0.335886\pi$
$998$	115.973	3.67107
$999$	−20.7642	−0.656951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.5	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.5	✓	361	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78850 q^{2} +0.327361 q^{3} +5.77575 q^{4} -3.12369 q^{5} -0.912847 q^{6} +1.40614 q^{7} -10.5287 q^{8} -2.89283 q^{9} +O(q^{10})\)	\(q-2.78850 q^{2} +0.327361 q^{3} +5.77575 q^{4} -3.12369 q^{5} -0.912847 q^{6} +1.40614 q^{7} -10.5287 q^{8} -2.89283 q^{9} +8.71042 q^{10} -1.60671 q^{11} +1.89076 q^{12} +6.86097 q^{13} -3.92103 q^{14} -1.02257 q^{15} +17.8078 q^{16} +3.40196 q^{17} +8.06668 q^{18} -7.28137 q^{19} -18.0417 q^{20} +0.460315 q^{21} +4.48033 q^{22} +8.24707 q^{23} -3.44669 q^{24} +4.75744 q^{25} -19.1318 q^{26} -1.92908 q^{27} +8.12152 q^{28} +2.48883 q^{29} +2.85145 q^{30} -1.95607 q^{31} -28.5997 q^{32} -0.525975 q^{33} -9.48638 q^{34} -4.39234 q^{35} -16.7083 q^{36} +10.7638 q^{37} +20.3041 q^{38} +2.24601 q^{39} +32.8884 q^{40} -1.71143 q^{41} -1.28359 q^{42} +6.47839 q^{43} -9.27998 q^{44} +9.03632 q^{45} -22.9970 q^{46} -6.67303 q^{47} +5.82958 q^{48} -5.02277 q^{49} -13.2661 q^{50} +1.11367 q^{51} +39.6272 q^{52} +5.50964 q^{53} +5.37926 q^{54} +5.01887 q^{55} -14.8048 q^{56} -2.38364 q^{57} -6.94010 q^{58} -12.0792 q^{59} -5.90614 q^{60} +2.21258 q^{61} +5.45452 q^{62} -4.06773 q^{63} +44.1349 q^{64} -21.4315 q^{65} +1.46668 q^{66} +5.97319 q^{67} +19.6489 q^{68} +2.69977 q^{69} +12.2481 q^{70} +0.926605 q^{71} +30.4578 q^{72} +8.77594 q^{73} -30.0148 q^{74} +1.55740 q^{75} -42.0554 q^{76} -2.25926 q^{77} -6.26301 q^{78} +3.06460 q^{79} -55.6261 q^{80} +8.04700 q^{81} +4.77232 q^{82} +16.4511 q^{83} +2.65867 q^{84} -10.6267 q^{85} -18.0650 q^{86} +0.814745 q^{87} +16.9166 q^{88} +0.974709 q^{89} -25.1978 q^{90} +9.64748 q^{91} +47.6330 q^{92} -0.640342 q^{93} +18.6078 q^{94} +22.7447 q^{95} -9.36244 q^{96} -10.0869 q^{97} +14.0060 q^{98} +4.64796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

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