LMFDB - Embedded newform 8039.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, 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("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$0$
Dimension:	$391$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72603 q^{2} -3.12098 q^{3} +5.43123 q^{4} -3.26463 q^{5} +8.50788 q^{6} -3.62664 q^{7} -9.35363 q^{8} +6.74052 q^{9} +O(q^{10})$	\(q-2.72603 q^{2} -3.12098 q^{3} +5.43123 q^{4} -3.26463 q^{5} +8.50788 q^{6} -3.62664 q^{7} -9.35363 q^{8} +6.74052 q^{9} +8.89948 q^{10} -4.91027 q^{11} -16.9508 q^{12} -0.0363150 q^{13} +9.88632 q^{14} +10.1889 q^{15} +14.6358 q^{16} +0.0949410 q^{17} -18.3748 q^{18} +4.77740 q^{19} -17.7310 q^{20} +11.3187 q^{21} +13.3855 q^{22} -9.10526 q^{23} +29.1925 q^{24} +5.65782 q^{25} +0.0989958 q^{26} -11.6741 q^{27} -19.6971 q^{28} +4.29893 q^{29} -27.7751 q^{30} -7.15070 q^{31} -21.1903 q^{32} +15.3249 q^{33} -0.258812 q^{34} +11.8396 q^{35} +36.6093 q^{36} -5.16307 q^{37} -13.0233 q^{38} +0.113339 q^{39} +30.5362 q^{40} +12.3069 q^{41} -30.8550 q^{42} +8.69287 q^{43} -26.6688 q^{44} -22.0053 q^{45} +24.8212 q^{46} +11.0116 q^{47} -45.6780 q^{48} +6.15252 q^{49} -15.4234 q^{50} -0.296309 q^{51} -0.197235 q^{52} -10.0686 q^{53} +31.8239 q^{54} +16.0302 q^{55} +33.9222 q^{56} -14.9102 q^{57} -11.7190 q^{58} -5.09661 q^{59} +55.3380 q^{60} -2.10100 q^{61} +19.4930 q^{62} -24.4454 q^{63} +28.4939 q^{64} +0.118555 q^{65} -41.7760 q^{66} +14.0141 q^{67} +0.515646 q^{68} +28.4173 q^{69} -32.2752 q^{70} +7.79073 q^{71} -63.0483 q^{72} +0.900325 q^{73} +14.0747 q^{74} -17.6579 q^{75} +25.9471 q^{76} +17.8078 q^{77} -0.308964 q^{78} -11.8379 q^{79} -47.7805 q^{80} +16.2130 q^{81} -33.5489 q^{82} -13.8416 q^{83} +61.4743 q^{84} -0.309947 q^{85} -23.6970 q^{86} -13.4169 q^{87} +45.9289 q^{88} +3.99530 q^{89} +59.9871 q^{90} +0.131702 q^{91} -49.4527 q^{92} +22.3172 q^{93} -30.0180 q^{94} -15.5964 q^{95} +66.1346 q^{96} +11.8453 q^{97} -16.7719 q^{98} -33.0978 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) $ 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * 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 $

$2$	−2.72603	−1.92759	−0.963797	−	0.266639i	$-0.914087\pi$
$2$	−2.72603	−1.92759	−0.963797	+	0.266639i	$0.914087\pi$
$3$	−3.12098	−1.80190	−0.900949	−	0.433924i	$-0.857129\pi$
$3$	−3.12098	−1.80190	−0.900949	+	0.433924i	$0.857129\pi$
$4$	5.43123	2.71561
$5$	−3.26463	−1.45999	−0.729994	−	0.683454i	$-0.760477\pi$
$5$	−3.26463	−1.45999	−0.729994	+	0.683454i	$0.760477\pi$
$6$	8.50788	3.47333
$7$	−3.62664	−1.37074	−0.685371	−	0.728194i	$-0.740360\pi$
$7$	−3.62664	−1.37074	−0.685371	+	0.728194i	$0.740360\pi$
$8$	−9.35363	−3.30701
$9$	6.74052	2.24684
$10$	8.89948	2.81426
$11$	−4.91027	−1.48050	−0.740251	−	0.672330i	$-0.765293\pi$
$11$	−4.91027	−1.48050	−0.740251	+	0.672330i	$0.765293\pi$
$12$	−16.9508	−4.89326
$13$	−0.0363150	−0.0100720	−0.00503599	−	0.999987i	$-0.501603\pi$
$13$	−0.0363150	−0.0100720	−0.00503599	+	0.999987i	$0.501603\pi$
$14$	9.88632	2.64223
$15$	10.1889	2.63075
$16$	14.6358	3.65895
$17$	0.0949410	0.0230266	0.0115133	−	0.999934i	$-0.496335\pi$
$17$	0.0949410	0.0230266	0.0115133	+	0.999934i	$0.496335\pi$
$18$	−18.3748	−4.33099
$19$	4.77740	1.09601	0.548005	−	0.836475i	$-0.315387\pi$
$19$	4.77740	1.09601	0.548005	+	0.836475i	$0.315387\pi$
$20$	−17.7310	−3.96476
$21$	11.3187	2.46994
$22$	13.3855	2.85381
$23$	−9.10526	−1.89858	−0.949289	−	0.314406i	$-0.898195\pi$
$23$	−9.10526	−1.89858	−0.949289	+	0.314406i	$0.898195\pi$
$24$	29.1925	5.95889
$25$	5.65782	1.13156
$26$	0.0989958	0.0194147
$27$	−11.6741	−2.24668
$28$	−19.6971	−3.72240
$29$	4.29893	0.798290	0.399145	−	0.916888i	$-0.369307\pi$
$29$	4.29893	0.798290	0.399145	+	0.916888i	$0.369307\pi$
$30$	−27.7751	−5.07102
$31$	−7.15070	−1.28430	−0.642152	−	0.766577i	$-0.721958\pi$
$31$	−7.15070	−1.28430	−0.642152	+	0.766577i	$0.721958\pi$
$32$	−21.1903	−3.74596
$33$	15.3249	2.66772
$34$	−0.258812	−0.0443859
$35$	11.8396	2.00127
$36$	36.6093	6.10155
$37$	−5.16307	−0.848804	−0.424402	−	0.905474i	$-0.639516\pi$
$37$	−5.16307	−0.848804	−0.424402	+	0.905474i	$0.639516\pi$
$38$	−13.0233	−2.11266
$39$	0.113339	0.0181487
$40$	30.5362	4.82819
$41$	12.3069	1.92201	0.961005	−	0.276530i	$-0.0891846\pi$
$41$	12.3069	1.92201	0.961005	+	0.276530i	$0.0891846\pi$
$42$	−30.8550	−4.76103
$43$	8.69287	1.32565	0.662825	−	0.748774i	$-0.269357\pi$
$43$	8.69287	1.32565	0.662825	+	0.748774i	$0.269357\pi$
$44$	−26.6688	−4.02047
$45$	−22.0053	−3.28036
$46$	24.8212	3.65968
$47$	11.0116	1.60621	0.803105	−	0.595838i	$-0.203180\pi$
$47$	11.0116	1.60621	0.803105	+	0.595838i	$0.203180\pi$
$48$	−45.6780	−6.59306
$49$	6.15252	0.878931
$50$	−15.4234	−2.18120
$51$	−0.296309	−0.0414916
$52$	−0.197235	−0.0273516
$53$	−10.0686	−1.38303	−0.691517	−	0.722360i	$-0.743057\pi$
$53$	−10.0686	−1.38303	−0.691517	+	0.722360i	$0.743057\pi$
$54$	31.8239	4.33068
$55$	16.0302	2.16152
$56$	33.9222	4.53305
$57$	−14.9102	−1.97490
$58$	−11.7190	−1.53878
$59$	−5.09661	−0.663522	−0.331761	−	0.943364i	$-0.607643\pi$
$59$	−5.09661	−0.663522	−0.331761	+	0.943364i	$0.607643\pi$
$60$	55.3380	7.14410
$61$	−2.10100	−0.269006	−0.134503	−	0.990913i	$-0.542944\pi$
$61$	−2.10100	−0.269006	−0.134503	+	0.990913i	$0.542944\pi$
$62$	19.4930	2.47562
$63$	−24.4454	−3.07983
$64$	28.4939	3.56173
$65$	0.118555	0.0147050
$66$	−41.7760	−5.14227
$67$	14.0141	1.71209	0.856047	−	0.516898i	$-0.172914\pi$
$67$	14.0141	1.71209	0.856047	+	0.516898i	$0.172914\pi$
$68$	0.515646	0.0625313
$69$	28.4173	3.42104
$70$	−32.2752	−3.85762
$71$	7.79073	0.924590	0.462295	−	0.886726i	$-0.347026\pi$
$71$	7.79073	0.924590	0.462295	+	0.886726i	$0.347026\pi$
$72$	−63.0483	−7.43031
$73$	0.900325	0.105375	0.0526876	−	0.998611i	$-0.483221\pi$
$73$	0.900325	0.105375	0.0526876	+	0.998611i	$0.483221\pi$
$74$	14.0747	1.63615
$75$	−17.6579	−2.03896
$76$	25.9471	2.97634
$77$	17.8078	2.02939
$78$	−0.308964	−0.0349833
$79$	−11.8379	−1.33187	−0.665934	−	0.746011i	$-0.731967\pi$
$79$	−11.8379	−1.33187	−0.665934	+	0.746011i	$0.731967\pi$
$80$	−47.7805	−5.34202
$81$	16.2130	1.80145
$82$	−33.5489	−3.70485
$83$	−13.8416	−1.51932	−0.759658	−	0.650323i	$-0.774634\pi$
$83$	−13.8416	−1.51932	−0.759658	+	0.650323i	$0.774634\pi$
$84$	61.4743	6.70740
$85$	−0.309947	−0.0336185
$86$	−23.6970	−2.55531
$87$	−13.4169	−1.43844
$88$	45.9289	4.89603
$89$	3.99530	0.423501	0.211750	−	0.977324i	$-0.432084\pi$
$89$	3.99530	0.423501	0.211750	+	0.977324i	$0.432084\pi$
$90$	59.9871	6.32319
$91$	0.131702	0.0138061
$92$	−49.4527	−5.15581
$93$	22.3172	2.31419
$94$	−30.0180	−3.09612
$95$	−15.5964	−1.60016
$96$	66.1346	6.74984
$97$	11.8453	1.20271	0.601355	−	0.798982i	$-0.294628\pi$
$97$	11.8453	1.20271	0.601355	+	0.798982i	$0.294628\pi$
$98$	−16.7719	−1.69422
$99$	−33.0978	−3.32645
$100$	30.7289	3.07289
$101$	−12.6475	−1.25847	−0.629237	−	0.777214i	$-0.716632\pi$
$101$	−12.6475	−1.25847	−0.629237	+	0.777214i	$0.716632\pi$
$102$	0.807747	0.0799788
$103$	−0.657388	−0.0647744	−0.0323872	−	0.999475i	$-0.510311\pi$
$103$	−0.657388	−0.0647744	−0.0323872	+	0.999475i	$0.510311\pi$
$104$	0.339677	0.0333081
$105$	−36.9513	−3.60608
$106$	27.4474	2.66593
$107$	−12.9089	−1.24795	−0.623975	−	0.781445i	$-0.714483\pi$
$107$	−12.9089	−1.24795	−0.623975	+	0.781445i	$0.714483\pi$
$108$	−63.4046	−6.10111
$109$	−3.17993	−0.304582	−0.152291	−	0.988336i	$-0.548665\pi$
$109$	−3.17993	−0.304582	−0.152291	+	0.988336i	$0.548665\pi$
$110$	−43.6989	−4.16652
$111$	16.1138	1.52946
$112$	−53.0788	−5.01547
$113$	−2.35577	−0.221613	−0.110806	−	0.993842i	$-0.535343\pi$
$113$	−2.35577	−0.221613	−0.110806	+	0.993842i	$0.535343\pi$
$114$	40.6455	3.80680
$115$	29.7253	2.77190
$116$	23.3485	2.16785
$117$	−0.244782	−0.0226301
$118$	13.8935	1.27900
$119$	−0.344317	−0.0315635
$120$	−95.3027	−8.69991
$121$	13.1108	1.19189
$122$	5.72739	0.518534
$123$	−38.4095	−3.46327
$124$	−38.8371	−3.48767
$125$	−2.14754	−0.192082
$126$	66.6389	5.93667
$127$	12.9626	1.15025	0.575124	−	0.818067i	$-0.304954\pi$
$127$	12.9626	1.15025	0.575124	+	0.818067i	$0.304954\pi$
$128$	−35.2944	−3.11961
$129$	−27.1303	−2.38869
$130$	−0.323185	−0.0283452
$131$	−7.27346	−0.635485	−0.317742	−	0.948177i	$-0.602925\pi$
$131$	−7.27346	−0.635485	−0.317742	+	0.948177i	$0.602925\pi$
$132$	83.2328	7.24449
$133$	−17.3259	−1.50235
$134$	−38.2028	−3.30022
$135$	38.1116	3.28012
$136$	−0.888043	−0.0761490
$137$	7.64749	0.653369	0.326685	−	0.945133i	$-0.394069\pi$
$137$	7.64749	0.653369	0.326685	+	0.945133i	$0.394069\pi$
$138$	−77.4664	−6.59438
$139$	2.46871	0.209393	0.104697	−	0.994504i	$-0.466613\pi$
$139$	2.46871	0.209393	0.104697	+	0.994504i	$0.466613\pi$
$140$	64.3038	5.43467
$141$	−34.3670	−2.89423
$142$	−21.2378	−1.78223
$143$	0.178317	0.0149116
$144$	98.6528	8.22107
$145$	−14.0344	−1.16549
$146$	−2.45431	−0.203120
$147$	−19.2019	−1.58374
$148$	−28.0418	−2.30502
$149$	10.8643	0.890037	0.445018	−	0.895521i	$-0.353197\pi$
$149$	10.8643	0.890037	0.445018	+	0.895521i	$0.353197\pi$
$150$	48.1361	3.93029
$151$	9.70734	0.789972	0.394986	−	0.918687i	$-0.370749\pi$
$151$	9.70734	0.789972	0.394986	+	0.918687i	$0.370749\pi$
$152$	−44.6860	−3.62451
$153$	0.639951	0.0517370
$154$	−48.5445	−3.91183
$155$	23.3444	1.87507
$156$	0.615568	0.0492848
$157$	4.36115	0.348058	0.174029	−	0.984741i	$-0.444321\pi$
$157$	4.36115	0.348058	0.174029	+	0.984741i	$0.444321\pi$
$158$	32.2705	2.56730
$159$	31.4240	2.49209
$160$	69.1787	5.46905
$161$	33.0215	2.60246
$162$	−44.1972	−3.47246
$163$	2.89619	0.226847	0.113424	−	0.993547i	$-0.463818\pi$
$163$	2.89619	0.226847	0.113424	+	0.993547i	$0.463818\pi$
$164$	66.8414	5.21944
$165$	−50.0300	−3.89483
$166$	37.7327	2.92862
$167$	−3.40140	−0.263209	−0.131604	−	0.991302i	$-0.542013\pi$
$167$	−3.40140	−0.263209	−0.131604	+	0.991302i	$0.542013\pi$
$168$	−105.871	−8.16810
$169$	−12.9987	−0.999899
$170$	0.844925	0.0648028
$171$	32.2021	2.46256
$172$	47.2130	3.59995
$173$	−12.2570	−0.931881	−0.465941	−	0.884816i	$-0.654284\pi$
$173$	−12.2570	−0.931881	−0.465941	+	0.884816i	$0.654284\pi$
$174$	36.5747	2.77272
$175$	−20.5189	−1.55108
$176$	−71.8657	−5.41708
$177$	15.9064	1.19560
$178$	−10.8913	−0.816338
$179$	−8.73266	−0.652709	−0.326355	−	0.945247i	$-0.605820\pi$
$179$	−8.73266	−0.652709	−0.326355	+	0.945247i	$0.605820\pi$
$180$	−119.516	−8.90819
$181$	−5.22149	−0.388110	−0.194055	−	0.980991i	$-0.562164\pi$
$181$	−5.22149	−0.388110	−0.194055	+	0.980991i	$0.562164\pi$
$182$	−0.359022	−0.0266125
$183$	6.55719	0.484721
$184$	85.1672	6.27861
$185$	16.8555	1.23924
$186$	−60.8373	−4.46081
$187$	−0.466186	−0.0340909
$188$	59.8066	4.36185
$189$	42.3377	3.07961
$190$	42.5163	3.08446
$191$	−21.2060	−1.53442	−0.767208	−	0.641399i	$-0.778355\pi$
$191$	−21.2060	−1.53442	−0.767208	+	0.641399i	$0.778355\pi$
$192$	−88.9288	−6.41788
$193$	1.29110	0.0929353	0.0464677	−	0.998920i	$-0.485204\pi$
$193$	1.29110	0.0929353	0.0464677	+	0.998920i	$0.485204\pi$
$194$	−32.2907	−2.31833
$195$	−0.370009	−0.0264969
$196$	33.4157	2.38684
$197$	17.5401	1.24968	0.624841	−	0.780752i	$-0.285164\pi$
$197$	17.5401	1.24968	0.624841	+	0.780752i	$0.285164\pi$
$198$	90.2255	6.41204
$199$	−10.6033	−0.751647	−0.375824	−	0.926691i	$-0.622640\pi$
$199$	−10.6033	−0.751647	−0.375824	+	0.926691i	$0.622640\pi$
$200$	−52.9212	−3.74209
$201$	−43.7377	−3.08502
$202$	34.4774	2.42582
$203$	−15.5907	−1.09425
$204$	−1.60932	−0.112675
$205$	−40.1774	−2.80611
$206$	1.79206	0.124859
$207$	−61.3741	−4.26580
$208$	−0.531500	−0.0368529
$209$	−23.4583	−1.62265
$210$	100.730	6.95105
$211$	−12.0459	−0.829276	−0.414638	−	0.909986i	$-0.636092\pi$
$211$	−12.0459	−0.829276	−0.414638	+	0.909986i	$0.636092\pi$
$212$	−54.6851	−3.75579
$213$	−24.3147	−1.66602
$214$	35.1900	2.40554
$215$	−28.3790	−1.93543
$216$	109.195	7.42978
$217$	25.9330	1.76045
$218$	8.66857	0.587110
$219$	−2.80990	−0.189875
$220$	87.0638	5.86984
$221$	−0.00344779	−0.000231923 0
$222$	−43.9268	−2.94817
$223$	−16.3100	−1.09220	−0.546100	−	0.837720i	$-0.683888\pi$
$223$	−16.3100	−1.09220	−0.546100	+	0.837720i	$0.683888\pi$
$224$	76.8497	5.13474
$225$	38.1366	2.54244
$226$	6.42191	0.427179
$227$	−3.15235	−0.209229	−0.104614	−	0.994513i	$-0.533361\pi$
$227$	−3.15235	−0.209229	−0.104614	+	0.994513i	$0.533361\pi$
$228$	−80.9805	−5.36307
$229$	12.0217	0.794418	0.397209	−	0.917728i	$-0.369979\pi$
$229$	12.0217	0.794418	0.397209	+	0.917728i	$0.369979\pi$
$230$	−81.0320	−5.34309
$231$	−55.5777	−3.65675
$232$	−40.2106	−2.63995
$233$	−18.5071	−1.21244	−0.606220	−	0.795297i	$-0.707315\pi$
$233$	−18.5071	−1.21244	−0.606220	+	0.795297i	$0.707315\pi$
$234$	0.667283	0.0436217
$235$	−35.9489	−2.34505
$236$	−27.6808	−1.80187
$237$	36.9459	2.39989
$238$	0.938617	0.0608415
$239$	−21.6926	−1.40318	−0.701588	−	0.712583i	$-0.747525\pi$
$239$	−21.6926	−1.40318	−0.701588	+	0.712583i	$0.747525\pi$
$240$	149.122	9.62578
$241$	5.42310	0.349332	0.174666	−	0.984628i	$-0.444115\pi$
$241$	5.42310	0.349332	0.174666	+	0.984628i	$0.444115\pi$
$242$	−35.7403	−2.29747
$243$	−15.5783	−0.999347
$244$	−11.4110	−0.730516
$245$	−20.0857	−1.28323
$246$	104.705	6.67577
$247$	−0.173491	−0.0110390
$248$	66.8850	4.24720
$249$	43.1994	2.73765
$250$	5.85426	0.370256
$251$	4.64203	0.293002	0.146501	−	0.989211i	$-0.453199\pi$
$251$	4.64203	0.293002	0.146501	+	0.989211i	$0.453199\pi$
$252$	−132.769	−8.36364
$253$	44.7093	2.81085
$254$	−35.3365	−2.21721
$255$	0.967340	0.0605772
$256$	39.2258	2.45161
$257$	−6.76906	−0.422242	−0.211121	−	0.977460i	$-0.567711\pi$
$257$	−6.76906	−0.422242	−0.211121	+	0.977460i	$0.567711\pi$
$258$	73.9579	4.60442
$259$	18.7246	1.16349
$260$	0.643901	0.0399330
$261$	28.9770	1.79363
$262$	19.8276	1.22496
$263$	25.9069	1.59749	0.798743	−	0.601672i	$-0.205499\pi$
$263$	25.9069	1.59749	0.798743	+	0.601672i	$0.205499\pi$
$264$	−143.343	−8.82216
$265$	32.8704	2.01921
$266$	47.2309	2.89591
$267$	−12.4693	−0.763106
$268$	76.1137	4.64939
$269$	18.0809	1.10241	0.551205	−	0.834370i	$-0.314168\pi$
$269$	18.0809	1.10241	0.551205	+	0.834370i	$0.314168\pi$
$270$	−103.893	−6.32274
$271$	2.16312	0.131400	0.0656999	−	0.997839i	$-0.479072\pi$
$271$	2.16312	0.131400	0.0656999	+	0.997839i	$0.479072\pi$
$272$	1.38954	0.0842531
$273$	−0.411038	−0.0248772
$274$	−20.8473	−1.25943
$275$	−27.7814	−1.67528
$276$	154.341	9.29024
$277$	−7.78138	−0.467538	−0.233769	−	0.972292i	$-0.575106\pi$
$277$	−7.78138	−0.467538	−0.233769	+	0.972292i	$0.575106\pi$
$278$	−6.72977	−0.403625
$279$	−48.1994	−2.88562
$280$	−110.744	−6.61820
$281$	−28.9639	−1.72784	−0.863920	−	0.503629i	$-0.831998\pi$
$281$	−28.9639	−1.72784	−0.863920	+	0.503629i	$0.831998\pi$
$282$	93.6855	5.57889
$283$	−22.2893	−1.32496	−0.662480	−	0.749080i	$-0.730496\pi$
$283$	−22.2893	−1.32496	−0.662480	+	0.749080i	$0.730496\pi$
$284$	42.3133	2.51083
$285$	48.6762	2.88333
$286$	−0.486096	−0.0287435
$287$	−44.6326	−2.63458
$288$	−142.834	−8.41657
$289$	−16.9910	−0.999470
$290$	38.2582	2.24660
$291$	−36.9690	−2.16716
$292$	4.88987	0.286158
$293$	−1.25446	−0.0732866	−0.0366433	−	0.999328i	$-0.511667\pi$
$293$	−1.25446	−0.0732866	−0.0366433	+	0.999328i	$0.511667\pi$
$294$	52.3449	3.05282
$295$	16.6385	0.968734
$296$	48.2935	2.80700
$297$	57.3229	3.32621
$298$	−29.6163	−1.71563
$299$	0.330658	0.0191224
$300$	−95.9044	−5.53704
$301$	−31.5259	−1.81712
$302$	−26.4625	−1.52275
$303$	39.4726	2.26764
$304$	69.9210	4.01025
$305$	6.85900	0.392745
$306$	−1.74453	−0.0997279
$307$	−16.9471	−0.967223	−0.483611	−	0.875283i	$-0.660675\pi$
$307$	−16.9471	−0.967223	−0.483611	+	0.875283i	$0.660675\pi$
$308$	96.7182	5.51103
$309$	2.05170	0.116717
$310$	−63.6375	−3.61437
$311$	−12.2526	−0.694782	−0.347391	−	0.937720i	$-0.612932\pi$
$311$	−12.2526	−0.694782	−0.347391	+	0.937720i	$0.612932\pi$
$312$	−1.06013	−0.0600178
$313$	3.00262	0.169718	0.0848591	−	0.996393i	$-0.472956\pi$
$313$	3.00262	0.169718	0.0848591	+	0.996393i	$0.472956\pi$
$314$	−11.8886	−0.670914
$315$	79.8053	4.49652
$316$	−64.2944	−3.61684
$317$	13.0166	0.731088	0.365544	−	0.930794i	$-0.380883\pi$
$317$	13.0166	0.731088	0.365544	+	0.930794i	$0.380883\pi$
$318$	−85.6628	−4.80373
$319$	−21.1089	−1.18187
$320$	−93.0220	−5.20009
$321$	40.2884	2.24868
$322$	−90.0175	−5.01648
$323$	0.453571	0.0252374
$324$	88.0566	4.89204
$325$	−0.205464	−0.0113971
$326$	−7.89511	−0.437270
$327$	9.92449	0.548826
$328$	−115.114	−6.35610
$329$	−39.9352	−2.20170
$330$	136.383	7.50765
$331$	−27.3694	−1.50436	−0.752178	−	0.658960i	$-0.770997\pi$
$331$	−27.3694	−1.50436	−0.752178	+	0.658960i	$0.770997\pi$
$332$	−75.1771	−4.12588
$333$	−34.8018	−1.90713
$334$	9.27233	0.507359
$335$	−45.7508	−2.49964
$336$	165.658	9.03737
$337$	−12.6992	−0.691767	−0.345884	−	0.938277i	$-0.612421\pi$
$337$	−12.6992	−0.691767	−0.345884	+	0.938277i	$0.612421\pi$
$338$	35.4348	1.92740
$339$	7.35232	0.399323
$340$	−1.68340	−0.0912949
$341$	35.1119	1.90142
$342$	−87.7839	−4.74681
$343$	3.07351	0.165954
$344$	−81.3099	−4.38393
$345$	−92.7721	−4.99468
$346$	33.4129	1.79629
$347$	−23.3782	−1.25501	−0.627503	−	0.778614i	$-0.715923\pi$
$347$	−23.3782	−1.25501	−0.627503	+	0.778614i	$0.715923\pi$
$348$	−72.8701	−3.90625
$349$	−24.8516	−1.33028	−0.665138	−	0.746720i	$-0.731627\pi$
$349$	−24.8516	−1.33028	−0.665138	+	0.746720i	$0.731627\pi$
$350$	55.9350	2.98985
$351$	0.423945	0.0226285
$352$	104.050	5.54590
$353$	−19.8287	−1.05537	−0.527687	−	0.849439i	$-0.676941\pi$
$353$	−19.8287	−1.05537	−0.527687	+	0.849439i	$0.676941\pi$
$354$	−43.3613	−2.30463
$355$	−25.4339	−1.34989
$356$	21.6994	1.15007
$357$	1.07461	0.0568742
$358$	23.8055	1.25816
$359$	−11.7116	−0.618115	−0.309057	−	0.951043i	$-0.600014\pi$
$359$	−11.7116	−0.618115	−0.309057	+	0.951043i	$0.600014\pi$
$360$	205.829	10.8482
$361$	3.82353	0.201238
$362$	14.2339	0.748118
$363$	−40.9184	−2.14766
$364$	0.715302	0.0374920
$365$	−2.93923	−0.153846
$366$	−17.8751	−0.934345
$367$	4.14761	0.216503	0.108252	−	0.994124i	$-0.465475\pi$
$367$	4.14761	0.216503	0.108252	+	0.994124i	$0.465475\pi$
$368$	−133.263	−6.94680
$369$	82.9547	4.31845
$370$	−45.9486	−2.38876
$371$	36.5153	1.89578
$372$	121.210	6.28444
$373$	3.11786	0.161437	0.0807183	−	0.996737i	$-0.474279\pi$
$373$	3.11786	0.161437	0.0807183	+	0.996737i	$0.474279\pi$
$374$	1.27084	0.0657134
$375$	6.70243	0.346112
$376$	−102.999	−5.31175
$377$	−0.156116	−0.00804037
$378$	−115.414	−5.93624
$379$	−24.2984	−1.24813	−0.624063	−	0.781374i	$-0.714519\pi$
$379$	−24.2984	−1.24813	−0.624063	+	0.781374i	$0.714519\pi$
$380$	−84.7079	−4.34542
$381$	−40.4561	−2.07263
$382$	57.8083	2.95773
$383$	−6.03714	−0.308484	−0.154242	−	0.988033i	$-0.549293\pi$
$383$	−6.03714	−0.308484	−0.154242	+	0.988033i	$0.549293\pi$
$384$	110.153	5.62123
$385$	−58.1359	−2.96288
$386$	−3.51957	−0.179141
$387$	58.5944	2.97852
$388$	64.3346	3.26609
$389$	−19.5669	−0.992083	−0.496041	−	0.868299i	$-0.665214\pi$
$389$	−19.5669	−0.992083	−0.496041	+	0.868299i	$0.665214\pi$
$390$	1.00865	0.0510752
$391$	−0.864462	−0.0437177
$392$	−57.5484	−2.90663
$393$	22.7003	1.14508
$394$	−47.8149	−2.40888
$395$	38.6464	1.94451
$396$	−179.762	−9.03336
$397$	−23.1657	−1.16265	−0.581326	−	0.813671i	$-0.697466\pi$
$397$	−23.1657	−1.16265	−0.581326	+	0.813671i	$0.697466\pi$
$398$	28.9049	1.44887
$399$	54.0738	2.70708
$400$	82.8067	4.14034
$401$	31.3826	1.56717	0.783585	−	0.621284i	$-0.213389\pi$
$401$	31.3826	1.56717	0.783585	+	0.621284i	$0.213389\pi$
$402$	119.230	5.94666
$403$	0.259678	0.0129355
$404$	−68.6915	−3.41753
$405$	−52.9295	−2.63009
$406$	42.5006	2.10927
$407$	25.3521	1.25666
$408$	2.77156	0.137213
$409$	−27.1354	−1.34176	−0.670881	−	0.741565i	$-0.734084\pi$
$409$	−27.1354	−1.34176	−0.670881	+	0.741565i	$0.734084\pi$
$410$	109.525	5.40904
$411$	−23.8677	−1.17731
$412$	−3.57043	−0.175902
$413$	18.4836	0.909517
$414$	167.308	8.22272
$415$	45.1878	2.21818
$416$	0.769528	0.0377292
$417$	−7.70479	−0.377305
$418$	63.9480	3.12780
$419$	16.2925	0.795942	0.397971	−	0.917398i	$-0.369714\pi$
$419$	16.2925	0.795942	0.397971	+	0.917398i	$0.369714\pi$
$420$	−200.691	−9.79272
$421$	−0.349854	−0.0170508	−0.00852542	−	0.999964i	$-0.502714\pi$
$421$	−0.349854	−0.0170508	−0.00852542	+	0.999964i	$0.502714\pi$
$422$	32.8376	1.59851
$423$	74.2240	3.60890
$424$	94.1783	4.57370
$425$	0.537159	0.0260560
$426$	66.2826	3.21140
$427$	7.61958	0.368737
$428$	−70.1111	−3.38895
$429$	−0.556523	−0.0268692
$430$	77.3620	3.73073
$431$	3.33404	0.160595	0.0802975	−	0.996771i	$-0.474413\pi$
$431$	3.33404	0.160595	0.0802975	+	0.996771i	$0.474413\pi$
$432$	−170.859	−8.22048
$433$	24.8858	1.19593	0.597966	−	0.801521i	$-0.295976\pi$
$433$	24.8858	1.19593	0.597966	+	0.801521i	$0.295976\pi$
$434$	−70.6941	−3.39343
$435$	43.8011	2.10010
$436$	−17.2709	−0.827127
$437$	−43.4994	−2.08086
$438$	7.65986	0.366002
$439$	−31.8492	−1.52008	−0.760040	−	0.649876i	$-0.774821\pi$
$439$	−31.8492	−1.52008	−0.760040	+	0.649876i	$0.774821\pi$
$440$	−149.941	−7.14815
$441$	41.4712	1.97482
$442$	0.00939876	0.000447053 0
$443$	−26.1925	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$443$	−26.1925	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$444$	87.5180	4.15342
$445$	−13.0432	−0.618306
$446$	44.4616	2.10532
$447$	−33.9072	−1.60376
$448$	−103.337	−4.88221
$449$	−6.05576	−0.285789	−0.142894	−	0.989738i	$-0.545641\pi$
$449$	−6.05576	−0.285789	−0.142894	+	0.989738i	$0.545641\pi$
$450$	−103.962	−4.90079
$451$	−60.4301	−2.84554
$452$	−12.7948	−0.601814
$453$	−30.2964	−1.42345
$454$	8.59338	0.403307
$455$	−0.429957	−0.0201567
$456$	139.464	6.53101
$457$	0.635528	0.0297288	0.0148644	−	0.999890i	$-0.495268\pi$
$457$	0.635528	0.0297288	0.0148644	+	0.999890i	$0.495268\pi$
$458$	−32.7716	−1.53132
$459$	−1.10835	−0.0517333
$460$	161.445	7.52741
$461$	11.4480	0.533189	0.266594	−	0.963809i	$-0.414102\pi$
$461$	11.4480	0.533189	0.266594	+	0.963809i	$0.414102\pi$
$462$	151.507	7.04872
$463$	−1.70682	−0.0793229	−0.0396614	−	0.999213i	$-0.512628\pi$
$463$	−1.70682	−0.0793229	−0.0396614	+	0.999213i	$0.512628\pi$
$464$	62.9182	2.92090
$465$	−72.8574	−3.37868
$466$	50.4509	2.33709
$467$	6.97335	0.322688	0.161344	−	0.986898i	$-0.448417\pi$
$467$	6.97335	0.322688	0.161344	+	0.986898i	$0.448417\pi$
$468$	−1.32947	−0.0614547
$469$	−50.8241	−2.34684
$470$	97.9977	4.52030
$471$	−13.6111	−0.627165
$472$	47.6718	2.19427
$473$	−42.6843	−1.96263
$474$	−100.715	−4.62601
$475$	27.0297	1.24021
$476$	−1.87006	−0.0857142
$477$	−67.8679	−3.10746
$478$	59.1346	2.70475
$479$	−34.3439	−1.56921	−0.784607	−	0.619994i	$-0.787135\pi$
$479$	−34.3439	−1.56921	−0.784607	+	0.619994i	$0.787135\pi$
$480$	−215.905	−9.85468
$481$	0.187497	0.00854913
$482$	−14.7835	−0.673371
$483$	−103.059	−4.68937
$484$	71.2076	3.23671
$485$	−38.6706	−1.75594
$486$	42.4668	1.92633
$487$	40.8443	1.85083	0.925416	−	0.378952i	$-0.123715\pi$
$487$	40.8443	1.85083	0.925416	+	0.378952i	$0.123715\pi$
$488$	19.6520	0.889604
$489$	−9.03897	−0.408756
$490$	54.7542	2.47354
$491$	19.0811	0.861116	0.430558	−	0.902563i	$-0.358317\pi$
$491$	19.0811	0.861116	0.430558	+	0.902563i	$0.358317\pi$
$492$	−208.611	−9.40490
$493$	0.408144	0.0183819
$494$	0.472942	0.0212787
$495$	108.052	4.85658
$496$	−104.656	−4.69920
$497$	−28.2542	−1.26737
$498$	−117.763	−5.27708
$499$	−12.2446	−0.548145	−0.274072	−	0.961709i	$-0.588371\pi$
$499$	−12.2446	−0.548145	−0.274072	+	0.961709i	$0.588371\pi$
$500$	−11.6638	−0.521621
$501$	10.6157	0.474275
$502$	−12.6543	−0.564789
$503$	18.4226	0.821422	0.410711	−	0.911766i	$-0.365281\pi$
$503$	18.4226	0.821422	0.410711	+	0.911766i	$0.365281\pi$
$504$	228.653	10.1850
$505$	41.2894	1.83736
$506$	−121.879	−5.41817
$507$	40.5686	1.80172
$508$	70.4030	3.12363
$509$	31.0504	1.37629	0.688143	−	0.725575i	$-0.258426\pi$
$509$	31.0504	1.37629	0.688143	+	0.725575i	$0.258426\pi$
$510$	−2.63700	−0.116768
$511$	−3.26516	−0.144442
$512$	−36.3419	−1.60610
$513$	−55.7717	−2.46238
$514$	18.4526	0.813911
$515$	2.14613	0.0945698
$516$	−147.351	−6.48675
$517$	−54.0700	−2.37800
$518$	−51.0438	−2.24274
$519$	38.2538	1.67916
$520$	−1.10892	−0.0486294
$521$	29.0700	1.27358	0.636790	−	0.771037i	$-0.280262\pi$
$521$	29.0700	1.27358	0.636790	+	0.771037i	$0.280262\pi$
$522$	−78.9921	−3.45739
$523$	21.7353	0.950420	0.475210	−	0.879872i	$-0.342372\pi$
$523$	21.7353	0.950420	0.475210	+	0.879872i	$0.342372\pi$
$524$	−39.5038	−1.72573
$525$	64.0390	2.79489
$526$	−70.6229	−3.07930
$527$	−0.678895	−0.0295731
$528$	224.292	9.76104
$529$	59.9057	2.60460
$530$	−89.6057	−3.89222
$531$	−34.3538	−1.49083
$532$	−94.1010	−4.07979
$533$	−0.446925	−0.0193585
$534$	33.9915	1.47096
$535$	42.1428	1.82199
$536$	−131.083	−5.66191
$537$	27.2544	1.17612
$538$	−49.2890	−2.12500
$539$	−30.2105	−1.30126
$540$	206.993	8.90755
$541$	−1.38439	−0.0595198	−0.0297599	−	0.999557i	$-0.509474\pi$
$541$	−1.38439	−0.0595198	−0.0297599	+	0.999557i	$0.509474\pi$
$542$	−5.89671	−0.253285
$543$	16.2962	0.699335
$544$	−2.01183	−0.0862566
$545$	10.3813	0.444686
$546$	1.12050	0.0479530
$547$	20.8758	0.892587	0.446293	−	0.894887i	$-0.352744\pi$
$547$	20.8758	0.892587	0.446293	+	0.894887i	$0.352744\pi$
$548$	41.5353	1.77430
$549$	−14.1618	−0.604413
$550$	75.7330	3.22926
$551$	20.5377	0.874934
$552$	−265.805	−11.3134
$553$	42.9318	1.82565
$554$	21.2123	0.901223
$555$	−52.6058	−2.23299
$556$	13.4081	0.568631
$557$	−32.5770	−1.38033	−0.690167	−	0.723650i	$-0.742463\pi$
$557$	−32.5770	−1.38033	−0.690167	+	0.723650i	$0.742463\pi$
$558$	131.393	5.56231
$559$	−0.315682	−0.0133519
$560$	173.283	7.32253
$561$	1.45496	0.0614283
$562$	78.9564	3.33057
$563$	−40.5752	−1.71004	−0.855020	−	0.518595i	$-0.826455\pi$
$563$	−40.5752	−1.71004	−0.855020	+	0.518595i	$0.826455\pi$
$564$	−186.655	−7.85961
$565$	7.69073	0.323552
$566$	60.7612	2.55398
$567$	−58.7988	−2.46932
$568$	−72.8716	−3.05763
$569$	21.3977	0.897040	0.448520	−	0.893773i	$-0.351951\pi$
$569$	21.3977	0.897040	0.448520	+	0.893773i	$0.351951\pi$
$570$	−132.693	−5.55788
$571$	−24.8201	−1.03869	−0.519345	−	0.854565i	$-0.673824\pi$
$571$	−24.8201	−1.03869	−0.519345	+	0.854565i	$0.673824\pi$
$572$	0.968479	0.0404941
$573$	66.1836	2.76486
$574$	121.670	5.07840
$575$	−51.5159	−2.14836
$576$	192.063	8.00264
$577$	4.02792	0.167684	0.0838422	−	0.996479i	$-0.473281\pi$
$577$	4.02792	0.167684	0.0838422	+	0.996479i	$0.473281\pi$
$578$	46.3179	1.92657
$579$	−4.02949	−0.167460
$580$	−76.2241	−3.16503
$581$	50.1986	2.08259
$582$	100.778	4.17740
$583$	49.4398	2.04759
$584$	−8.42131	−0.348476
$585$	0.799124	0.0330397
$586$	3.41970	0.141267
$587$	−7.40965	−0.305829	−0.152914	−	0.988239i	$-0.548866\pi$
$587$	−7.40965	−0.305829	−0.152914	+	0.988239i	$0.548866\pi$
$588$	−104.290	−4.30084
$589$	−34.1617	−1.40761
$590$	−45.3571	−1.86732
$591$	−54.7424	−2.25180
$592$	−75.5657	−3.10573
$593$	14.8326	0.609103	0.304551	−	0.952496i	$-0.401493\pi$
$593$	14.8326	0.609103	0.304551	+	0.952496i	$0.401493\pi$
$594$	−156.264	−6.41158
$595$	1.12407	0.0460823
$596$	59.0064	2.41700
$597$	33.0927	1.35439
$598$	−0.901382	−0.0368603
$599$	1.92363	0.0785973	0.0392987	−	0.999228i	$-0.487488\pi$
$599$	1.92363	0.0785973	0.0392987	+	0.999228i	$0.487488\pi$
$600$	165.166	6.74287
$601$	26.7929	1.09290	0.546452	−	0.837491i	$-0.315978\pi$
$601$	26.7929	1.09290	0.546452	+	0.837491i	$0.315978\pi$
$602$	85.9405	3.50267
$603$	94.4622	3.84680
$604$	52.7228	2.14526
$605$	−42.8018	−1.74014
$606$	−107.603	−4.37109
$607$	−43.6684	−1.77244	−0.886222	−	0.463260i	$-0.846680\pi$
$607$	−43.6684	−1.77244	−0.886222	+	0.463260i	$0.846680\pi$
$608$	−101.235	−4.10561
$609$	48.6581	1.97173
$610$	−18.6978	−0.757053
$611$	−0.399887	−0.0161777
$612$	3.47572	0.140498
$613$	31.0723	1.25500	0.627498	−	0.778618i	$-0.284079\pi$
$613$	31.0723	1.25500	0.627498	+	0.778618i	$0.284079\pi$
$614$	46.1983	1.86441
$615$	125.393	5.05633
$616$	−166.567	−6.71119
$617$	15.9925	0.643835	0.321918	−	0.946768i	$-0.395673\pi$
$617$	15.9925	0.643835	0.321918	+	0.946768i	$0.395673\pi$
$618$	−5.59298	−0.224983
$619$	10.5593	0.424414	0.212207	−	0.977225i	$-0.431935\pi$
$619$	10.5593	0.424414	0.212207	+	0.977225i	$0.431935\pi$
$620$	126.789	5.09196
$621$	106.296	4.26549
$622$	33.4010	1.33926
$623$	−14.4895	−0.580510
$624$	1.65880	0.0664051
$625$	−21.2782	−0.851127
$626$	−8.18523	−0.327148
$627$	73.2129	2.92384
$628$	23.6864	0.945191
$629$	−0.490187	−0.0195450
$630$	−217.552	−8.66746
$631$	33.6193	1.33836	0.669181	−	0.743099i	$-0.266645\pi$
$631$	33.6193	1.33836	0.669181	+	0.743099i	$0.266645\pi$
$632$	110.727	4.40450
$633$	37.5951	1.49427
$634$	−35.4837	−1.40924
$635$	−42.3182	−1.67935
$636$	170.671	6.76755
$637$	−0.223429	−0.00885258
$638$	57.5434	2.27817
$639$	52.5136	2.07740
$640$	115.223	4.55460
$641$	27.9149	1.10257	0.551285	−	0.834317i	$-0.314138\pi$
$641$	27.9149	1.10257	0.551285	+	0.834317i	$0.314138\pi$
$642$	−109.827	−4.33454
$643$	−3.35399	−0.132268	−0.0661341	−	0.997811i	$-0.521067\pi$
$643$	−3.35399	−0.132268	−0.0661341	+	0.997811i	$0.521067\pi$
$644$	179.347	7.06727
$645$	88.5703	3.48745
$646$	−1.23645	−0.0486474
$647$	4.85255	0.190774	0.0953868	−	0.995440i	$-0.469591\pi$
$647$	4.85255	0.190774	0.0953868	+	0.995440i	$0.469591\pi$
$648$	−151.651	−5.95740
$649$	25.0257	0.982346
$650$	0.560101	0.0219690
$651$	−80.9364	−3.17215
$652$	15.7299	0.616030
$653$	−23.0578	−0.902322	−0.451161	−	0.892443i	$-0.648990\pi$
$653$	−23.0578	−0.902322	−0.451161	+	0.892443i	$0.648990\pi$
$654$	−27.0544	−1.05791
$655$	23.7452	0.927800
$656$	180.121	7.03254
$657$	6.06866	0.236761
$658$	108.864	4.24398
$659$	5.31113	0.206892	0.103446	−	0.994635i	$-0.467013\pi$
$659$	5.31113	0.206892	0.103446	+	0.994635i	$0.467013\pi$
$660$	−271.725	−10.5769
$661$	−45.6518	−1.77565	−0.887825	−	0.460182i	$-0.847784\pi$
$661$	−45.6518	−1.77565	−0.887825	+	0.460182i	$0.847784\pi$
$662$	74.6097	2.89979
$663$	0.0107605	0.000417902 0
$664$	129.469	5.02439
$665$	56.5627	2.19341
$666$	94.8706	3.67616
$667$	−39.1428	−1.51562
$668$	−18.4738	−0.714773
$669$	50.9033	1.96803
$670$	124.718	4.81828
$671$	10.3165	0.398264
$672$	−239.846	−9.25228
$673$	12.3801	0.477218	0.238609	−	0.971116i	$-0.423309\pi$
$673$	12.3801	0.477218	0.238609	+	0.971116i	$0.423309\pi$
$674$	34.6183	1.33345
$675$	−66.0498	−2.54226
$676$	−70.5988	−2.71534
$677$	19.8133	0.761489	0.380744	−	0.924680i	$-0.375668\pi$
$677$	19.8133	0.761489	0.380744	+	0.924680i	$0.375668\pi$
$678$	−20.0426	−0.769733
$679$	−42.9587	−1.64860
$680$	2.89913	0.111177
$681$	9.83841	0.377009
$682$	−95.7160	−3.66515
$683$	11.4020	0.436285	0.218143	−	0.975917i	$-0.430000\pi$
$683$	11.4020	0.436285	0.218143	+	0.975917i	$0.430000\pi$
$684$	174.897	6.68736
$685$	−24.9662	−0.953911
$686$	−8.37848	−0.319892
$687$	−37.5196	−1.43146
$688$	127.227	4.85049
$689$	0.365643	0.0139299
$690$	252.899	9.62772
$691$	27.4703	1.04502	0.522510	−	0.852633i	$-0.324996\pi$
$691$	27.4703	1.04502	0.522510	+	0.852633i	$0.324996\pi$
$692$	−66.5705	−2.53063
$693$	120.034	4.55970
$694$	63.7296	2.41914
$695$	−8.05942	−0.305711
$696$	125.496	4.75693
$697$	1.16843	0.0442573
$698$	67.7462	2.56423
$699$	57.7603	2.18470
$700$	−111.443	−4.21214
$701$	28.4261	1.07364	0.536820	−	0.843697i	$-0.319625\pi$
$701$	28.4261	1.07364	0.536820	+	0.843697i	$0.319625\pi$
$702$	−1.15569	−0.0436185
$703$	−24.6660	−0.930297
$704$	−139.913	−5.27316
$705$	112.196	4.22554
$706$	54.0536	2.03433
$707$	45.8679	1.72504
$708$	86.3914	3.24679
$709$	−6.72424	−0.252534	−0.126267	−	0.991996i	$-0.540300\pi$
$709$	−6.72424	−0.252534	−0.126267	+	0.991996i	$0.540300\pi$
$710$	69.3334	2.60204
$711$	−79.7936	−2.99249
$712$	−37.3706	−1.40052
$713$	65.1090	2.43835
$714$	−2.92941	−0.109630
$715$	−0.582138	−0.0217707
$716$	−47.4291	−1.77251
$717$	67.7021	2.52838
$718$	31.9262	1.19147
$719$	−47.6800	−1.77816	−0.889081	−	0.457749i	$-0.848656\pi$
$719$	−47.6800	−1.77816	−0.889081	+	0.457749i	$0.848656\pi$
$720$	−322.065	−12.0027
$721$	2.38411	0.0887889
$722$	−10.4230	−0.387906
$723$	−16.9254	−0.629462
$724$	−28.3591	−1.05396
$725$	24.3225	0.903317
$726$	111.545	4.13982
$727$	−48.7066	−1.80643	−0.903213	−	0.429192i	$-0.858798\pi$
$727$	−48.7066	−1.80643	−0.903213	+	0.429192i	$0.858798\pi$
$728$	−1.23189	−0.0456568
$729$	−0.0195691	−0.000724781 0
$730$	8.01243	0.296553
$731$	0.825310	0.0305252
$732$	35.6136	1.31632
$733$	5.09424	0.188160	0.0940801	−	0.995565i	$-0.470009\pi$
$733$	5.09424	0.188160	0.0940801	+	0.995565i	$0.470009\pi$
$734$	−11.3065	−0.417330
$735$	62.6871	2.31225
$736$	192.943	7.11199
$737$	−68.8130	−2.53476
$738$	−226.137	−8.32421
$739$	−39.8796	−1.46700	−0.733498	−	0.679691i	$-0.762114\pi$
$739$	−39.8796	−1.46700	−0.733498	+	0.679691i	$0.762114\pi$
$740$	91.5462	3.36531
$741$	0.541463	0.0198911
$742$	−99.5418	−3.65430
$743$	−8.56746	−0.314310	−0.157155	−	0.987574i	$-0.550232\pi$
$743$	−8.56746	−0.314310	−0.157155	+	0.987574i	$0.550232\pi$
$744$	−208.747	−7.65303
$745$	−35.4679	−1.29944
$746$	−8.49937	−0.311184
$747$	−93.2997	−3.41366
$748$	−2.53196	−0.0925778
$749$	46.8159	1.71062
$750$	−18.2710	−0.667163
$751$	38.0069	1.38689	0.693446	−	0.720509i	$-0.256092\pi$
$751$	38.0069	1.38689	0.693446	+	0.720509i	$0.256092\pi$
$752$	161.164	5.87704
$753$	−14.4877	−0.527960
$754$	0.425576	0.0154986
$755$	−31.6909	−1.15335
$756$	229.946	8.36304
$757$	−40.0322	−1.45500	−0.727498	−	0.686110i	$-0.759317\pi$
$757$	−40.0322	−1.45500	−0.727498	+	0.686110i	$0.759317\pi$
$758$	66.2382	2.40588
$759$	−139.537	−5.06486
$760$	145.883	5.29175
$761$	24.9521	0.904515	0.452257	−	0.891888i	$-0.350619\pi$
$761$	24.9521	0.904515	0.452257	+	0.891888i	$0.350619\pi$
$762$	110.284	3.99519
$763$	11.5325	0.417503
$764$	−115.175	−4.16688
$765$	−2.08921	−0.0755354
$766$	16.4574	0.594631
$767$	0.185083	0.00668298
$768$	−122.423	−4.41756
$769$	−31.6464	−1.14120	−0.570600	−	0.821228i	$-0.693289\pi$
$769$	−31.6464	−1.14120	−0.570600	+	0.821228i	$0.693289\pi$
$770$	158.480	5.71122
$771$	21.1261	0.760838
$772$	7.01225	0.252376
$773$	−43.0732	−1.54923	−0.774617	−	0.632431i	$-0.782057\pi$
$773$	−43.0732	−1.54923	−0.774617	+	0.632431i	$0.782057\pi$
$774$	−159.730	−5.74138
$775$	−40.4574	−1.45327
$776$	−110.797	−3.97737
$777$	−58.4391	−2.09649
$778$	53.3400	1.91233
$779$	58.7948	2.10654
$780$	−2.00960	−0.0719553
$781$	−38.2546	−1.36886
$782$	2.35655	0.0842700
$783$	−50.1860	−1.79350
$784$	90.0470	3.21596
$785$	−14.2376	−0.508160
$786$	−61.8817	−2.20725
$787$	−28.7085	−1.02335	−0.511674	−	0.859180i	$-0.670974\pi$
$787$	−28.7085	−1.02335	−0.511674	+	0.859180i	$0.670974\pi$
$788$	95.2644	3.39365
$789$	−80.8549	−2.87851
$790$	−105.351	−3.74823
$791$	8.54354	0.303773
$792$	309.584	11.0006
$793$	0.0762980	0.00270942
$794$	63.1503	2.24112
$795$	−102.588	−3.63842
$796$	−57.5889	−2.04118
$797$	−36.1792	−1.28153	−0.640767	−	0.767736i	$-0.721383\pi$
$797$	−36.1792	−1.28153	−0.640767	+	0.767736i	$0.721383\pi$
$798$	−147.407	−5.21814
$799$	1.04545	0.0369855
$800$	−119.891	−4.23879
$801$	26.9304	0.951539
$802$	−85.5498	−3.02087
$803$	−4.42084	−0.156008
$804$	−237.549	−8.37772
$805$	−107.803	−3.79956
$806$	−0.707890	−0.0249343
$807$	−56.4301	−1.98643
$808$	118.300	4.16178
$809$	0.566738	0.0199255	0.00996273	−	0.999950i	$-0.496829\pi$
$809$	0.566738	0.0199255	0.00996273	+	0.999950i	$0.496829\pi$
$810$	144.287	5.06974
$811$	52.8651	1.85635	0.928173	−	0.372150i	$-0.121379\pi$
$811$	52.8651	1.85635	0.928173	+	0.372150i	$0.121379\pi$
$812$	−84.6764	−2.97156
$813$	−6.75104	−0.236769
$814$	−69.1105	−2.42232
$815$	−9.45501	−0.331195
$816$	−4.33672	−0.151815
$817$	41.5293	1.45293
$818$	73.9720	2.58637
$819$	0.887737	0.0310200
$820$	−218.213	−7.62032
$821$	22.1958	0.774638	0.387319	−	0.921946i	$-0.373401\pi$
$821$	22.1958	0.774638	0.387319	+	0.921946i	$0.373401\pi$
$822$	65.0639	2.26937
$823$	−15.2631	−0.532037	−0.266019	−	0.963968i	$-0.585708\pi$
$823$	−15.2631	−0.532037	−0.266019	+	0.963968i	$0.585708\pi$
$824$	6.14897	0.214209
$825$	86.7053	3.01869
$826$	−50.3867	−1.75318
$827$	−11.6600	−0.405456	−0.202728	−	0.979235i	$-0.564981\pi$
$827$	−11.6600	−0.405456	−0.202728	+	0.979235i	$0.564981\pi$
$828$	−333.337	−11.5843
$829$	−10.2139	−0.354742	−0.177371	−	0.984144i	$-0.556759\pi$
$829$	−10.2139	−0.354742	−0.177371	+	0.984144i	$0.556759\pi$
$830$	−123.183	−4.27575
$831$	24.2855	0.842456
$832$	−1.03476	−0.0358737
$833$	0.584126	0.0202388
$834$	21.0035	0.727291
$835$	11.1043	0.384281
$836$	−127.408	−4.40648
$837$	83.4779	2.88542
$838$	−44.4139	−1.53425
$839$	−8.57866	−0.296168	−0.148084	−	0.988975i	$-0.547311\pi$
$839$	−8.57866	−0.296168	−0.148084	+	0.988975i	$0.547311\pi$
$840$	345.629	11.9253
$841$	−10.5192	−0.362732
$842$	0.953712	0.0328671
$843$	90.3957	3.11339
$844$	−65.4242	−2.25199
$845$	42.4359	1.45984
$846$	−202.337	−6.95648
$847$	−47.5480	−1.63377
$848$	−147.363	−5.06045
$849$	69.5644	2.38744
$850$	−1.46431	−0.0502254
$851$	47.0111	1.61152
$852$	−132.059	−4.52426
$853$	−35.3976	−1.21199	−0.605996	−	0.795468i	$-0.707225\pi$
$853$	−35.3976	−1.21199	−0.605996	+	0.795468i	$0.707225\pi$
$854$	−20.7712	−0.710775
$855$	−105.128	−3.59531
$856$	120.745	4.12698
$857$	2.82188	0.0963937	0.0481969	−	0.998838i	$-0.484653\pi$
$857$	2.82188	0.0963937	0.0481969	+	0.998838i	$0.484653\pi$
$858$	1.51710	0.0517928
$859$	35.0326	1.19530	0.597648	−	0.801759i	$-0.296102\pi$
$859$	35.0326	1.19530	0.597648	+	0.801759i	$0.296102\pi$
$860$	−154.133	−5.25589
$861$	139.297	4.74724
$862$	−9.08869	−0.309562
$863$	25.8618	0.880347	0.440173	−	0.897913i	$-0.354917\pi$
$863$	25.8618	0.880347	0.440173	+	0.897913i	$0.354917\pi$
$864$	247.378	8.41596
$865$	40.0145	1.36054
$866$	−67.8393	−2.30527
$867$	53.0285	1.80094
$868$	140.848	4.78070
$869$	58.1273	1.97183
$870$	−119.403	−4.04814
$871$	−0.508922	−0.0172442
$872$	29.7439	1.00725
$873$	79.8435	2.70229
$874$	118.581	4.01105
$875$	7.78836	0.263295
$876$	−15.2612	−0.515628
$877$	−37.1950	−1.25598	−0.627992	−	0.778219i	$-0.716123\pi$
$877$	−37.1950	−1.25598	−0.627992	+	0.778219i	$0.716123\pi$
$878$	86.8218	2.93010
$879$	3.91516	0.132055
$880$	234.615	7.90888
$881$	36.5207	1.23041	0.615207	−	0.788366i	$-0.289073\pi$
$881$	36.5207	1.23041	0.615207	+	0.788366i	$0.289073\pi$
$882$	−113.052	−3.80664
$883$	2.79159	0.0939444	0.0469722	−	0.998896i	$-0.485043\pi$
$883$	2.79159	0.0939444	0.0469722	+	0.998896i	$0.485043\pi$
$884$	−0.0187257	−0.000629814 0
$885$	−51.9286	−1.74556
$886$	71.4015	2.39878
$887$	14.8106	0.497293	0.248646	−	0.968594i	$-0.420014\pi$
$887$	14.8106	0.497293	0.248646	+	0.968594i	$0.420014\pi$
$888$	−150.723	−5.05793
$889$	−47.0108	−1.57669
$890$	35.5561	1.19184
$891$	−79.6103	−2.66705
$892$	−88.5835	−2.96600
$893$	52.6069	1.76042
$894$	92.4320	3.09139
$895$	28.5089	0.952948
$896$	128.000	4.27618
$897$	−1.03198	−0.0344567
$898$	16.5082	0.550885
$899$	−30.7403	−1.02525
$900$	207.129	6.90429
$901$	−0.955927	−0.0318465
$902$	164.734	5.48505
$903$	98.3917	3.27427
$904$	22.0350	0.732874
$905$	17.0462	0.566636
$906$	82.5889	2.74383
$907$	7.26287	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$907$	7.26287	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$908$	−17.1211	−0.568184
$909$	−85.2507	−2.82759
$910$	1.17208	0.0388539
$911$	−25.4222	−0.842274	−0.421137	−	0.906997i	$-0.638369\pi$
$911$	−25.4222	−0.842274	−0.421137	+	0.906997i	$0.638369\pi$
$912$	−218.222	−7.22606
$913$	67.9661	2.24935
$914$	−1.73247	−0.0573049
$915$	−21.4068	−0.707687
$916$	65.2928	2.15733
$917$	26.3782	0.871085
$918$	3.02139	0.0997207
$919$	−4.68266	−0.154467	−0.0772333	−	0.997013i	$-0.524609\pi$
$919$	−4.68266	−0.154467	−0.0772333	+	0.997013i	$0.524609\pi$
$920$	−278.040	−9.16669
$921$	52.8916	1.74284
$922$	−31.2077	−1.02777
$923$	−0.282921	−0.00931245
$924$	−301.856	−9.93032
$925$	−29.2117	−0.960476
$926$	4.65285	0.152902
$927$	−4.43114	−0.145538
$928$	−91.0957	−2.99036
$929$	−43.3166	−1.42117	−0.710586	−	0.703610i	$-0.751570\pi$
$929$	−43.3166	−1.42117	−0.710586	+	0.703610i	$0.751570\pi$
$930$	198.611	6.51272
$931$	29.3930	0.963317
$932$	−100.516	−3.29252
$933$	38.2402	1.25193
$934$	−19.0095	−0.622011
$935$	1.52193	0.0497723
$936$	2.28960	0.0748380
$937$	13.0795	0.427287	0.213644	−	0.976912i	$-0.431467\pi$
$937$	13.0795	0.427287	0.213644	+	0.976912i	$0.431467\pi$
$938$	138.548	4.52375
$939$	−9.37112	−0.305815
$940$	−195.247	−6.36824
$941$	−47.8102	−1.55857	−0.779285	−	0.626670i	$-0.784417\pi$
$941$	−47.8102	−1.55857	−0.779285	+	0.626670i	$0.784417\pi$
$942$	37.1042	1.20892
$943$	−112.057	−3.64909
$944$	−74.5929	−2.42779
$945$	−138.217	−4.49620
$946$	116.359	3.78315
$947$	52.1522	1.69472	0.847359	−	0.531020i	$-0.178191\pi$
$947$	52.1522	1.69472	0.847359	+	0.531020i	$0.178191\pi$
$948$	200.661	6.51718
$949$	−0.0326954	−0.00106134
$950$	−73.6836	−2.39061
$951$	−40.6247	−1.31735
$952$	3.22061	0.104381
$953$	19.9420	0.645985	0.322993	−	0.946402i	$-0.395311\pi$
$953$	19.9420	0.645985	0.322993	+	0.946402i	$0.395311\pi$
$954$	185.010	5.98991
$955$	69.2299	2.24023
$956$	−117.817	−3.81049
$957$	65.8804	2.12961
$958$	93.6225	3.02481
$959$	−27.7347	−0.895600
$960$	290.320	9.37003
$961$	20.1325	0.649436
$962$	−0.511122	−0.0164792
$963$	−87.0126	−2.80394
$964$	29.4541	0.948652
$965$	−4.21496	−0.135684
$966$	280.943	9.03919
$967$	13.7679	0.442746	0.221373	−	0.975189i	$-0.428946\pi$
$967$	13.7679	0.442746	0.221373	+	0.975189i	$0.428946\pi$
$968$	−122.633	−3.94158
$969$	−1.41559	−0.0454752
$970$	105.417	3.38474
$971$	22.5688	0.724266	0.362133	−	0.932126i	$-0.382049\pi$
$971$	22.5688	0.724266	0.362133	+	0.932126i	$0.382049\pi$
$972$	−84.6092	−2.71384
$973$	−8.95311	−0.287024
$974$	−111.343	−3.56765
$975$	0.641249	0.0205364
$976$	−30.7498	−0.984279
$977$	−10.0341	−0.321019	−0.160510	−	0.987034i	$-0.551314\pi$
$977$	−10.0341	−0.321019	−0.160510	+	0.987034i	$0.551314\pi$
$978$	24.6405	0.787916
$979$	−19.6180	−0.626994
$980$	−109.090	−3.48475
$981$	−21.4344	−0.684346
$982$	−52.0155	−1.65988
$983$	−2.66826	−0.0851042	−0.0425521	−	0.999094i	$-0.513549\pi$
$983$	−2.66826	−0.0851042	−0.0425521	+	0.999094i	$0.513549\pi$
$984$	359.268	11.4531
$985$	−57.2620	−1.82452
$986$	−1.11261	−0.0354328
$987$	124.637	3.96724
$988$	−0.942272	−0.0299777
$989$	−79.1508	−2.51685
$990$	−294.553	−9.36150
$991$	39.6219	1.25863	0.629315	−	0.777150i	$-0.283335\pi$
$991$	39.6219	1.25863	0.629315	+	0.777150i	$0.283335\pi$
$992$	151.526	4.81095
$993$	85.4192	2.71070
$994$	77.0217	2.44298
$995$	34.6158	1.09740
$996$	234.626	7.43441
$997$	−15.4529	−0.489398	−0.244699	−	0.969599i	$-0.578689\pi$
$997$	−15.4529	−0.489398	−0.244699	+	0.969599i	$0.578689\pi$
$998$	33.3792	1.05660
$999$	60.2741	1.90699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.b.1.11	✓	391

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.b.1.11	✓	391	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72603 q^{2} -3.12098 q^{3} +5.43123 q^{4} -3.26463 q^{5} +8.50788 q^{6} -3.62664 q^{7} -9.35363 q^{8} +6.74052 q^{9} +O(q^{10})\)	\(q-2.72603 q^{2} -3.12098 q^{3} +5.43123 q^{4} -3.26463 q^{5} +8.50788 q^{6} -3.62664 q^{7} -9.35363 q^{8} +6.74052 q^{9} +8.89948 q^{10} -4.91027 q^{11} -16.9508 q^{12} -0.0363150 q^{13} +9.88632 q^{14} +10.1889 q^{15} +14.6358 q^{16} +0.0949410 q^{17} -18.3748 q^{18} +4.77740 q^{19} -17.7310 q^{20} +11.3187 q^{21} +13.3855 q^{22} -9.10526 q^{23} +29.1925 q^{24} +5.65782 q^{25} +0.0989958 q^{26} -11.6741 q^{27} -19.6971 q^{28} +4.29893 q^{29} -27.7751 q^{30} -7.15070 q^{31} -21.1903 q^{32} +15.3249 q^{33} -0.258812 q^{34} +11.8396 q^{35} +36.6093 q^{36} -5.16307 q^{37} -13.0233 q^{38} +0.113339 q^{39} +30.5362 q^{40} +12.3069 q^{41} -30.8550 q^{42} +8.69287 q^{43} -26.6688 q^{44} -22.0053 q^{45} +24.8212 q^{46} +11.0116 q^{47} -45.6780 q^{48} +6.15252 q^{49} -15.4234 q^{50} -0.296309 q^{51} -0.197235 q^{52} -10.0686 q^{53} +31.8239 q^{54} +16.0302 q^{55} +33.9222 q^{56} -14.9102 q^{57} -11.7190 q^{58} -5.09661 q^{59} +55.3380 q^{60} -2.10100 q^{61} +19.4930 q^{62} -24.4454 q^{63} +28.4939 q^{64} +0.118555 q^{65} -41.7760 q^{66} +14.0141 q^{67} +0.515646 q^{68} +28.4173 q^{69} -32.2752 q^{70} +7.79073 q^{71} -63.0483 q^{72} +0.900325 q^{73} +14.0747 q^{74} -17.6579 q^{75} +25.9471 q^{76} +17.8078 q^{77} -0.308964 q^{78} -11.8379 q^{79} -47.7805 q^{80} +16.2130 q^{81} -33.5489 q^{82} -13.8416 q^{83} +61.4743 q^{84} -0.309947 q^{85} -23.6970 q^{86} -13.4169 q^{87} +45.9289 q^{88} +3.99530 q^{89} +59.9871 q^{90} +0.131702 q^{91} -49.4527 q^{92} +22.3172 q^{93} -30.0180 q^{94} -15.5964 q^{95} +66.1346 q^{96} +11.8453 q^{97} -16.7719 q^{98} -33.0978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

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