LMFDB - Embedded newform 8009.2.a.b.1.11

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			1.11
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)

$f(q)$	$=$	$q-2.70301 q^{2} +1.53742 q^{3} +5.30627 q^{4} -0.0405993 q^{5} -4.15566 q^{6} +1.54013 q^{7} -8.93687 q^{8} -0.636347 q^{9} +O(q^{10})$	\(q-2.70301 q^{2} +1.53742 q^{3} +5.30627 q^{4} -0.0405993 q^{5} -4.15566 q^{6} +1.54013 q^{7} -8.93687 q^{8} -0.636347 q^{9} +0.109740 q^{10} +5.45156 q^{11} +8.15795 q^{12} -4.70483 q^{13} -4.16299 q^{14} -0.0624181 q^{15} +13.5439 q^{16} +0.965682 q^{17} +1.72005 q^{18} +1.62462 q^{19} -0.215431 q^{20} +2.36783 q^{21} -14.7356 q^{22} -5.15150 q^{23} -13.7397 q^{24} -4.99835 q^{25} +12.7172 q^{26} -5.59058 q^{27} +8.17235 q^{28} -0.0908200 q^{29} +0.168717 q^{30} +0.756515 q^{31} -18.7356 q^{32} +8.38133 q^{33} -2.61025 q^{34} -0.0625284 q^{35} -3.37663 q^{36} +8.46302 q^{37} -4.39136 q^{38} -7.23329 q^{39} +0.362831 q^{40} -1.40062 q^{41} -6.40026 q^{42} -2.93988 q^{43} +28.9275 q^{44} +0.0258353 q^{45} +13.9245 q^{46} -7.04753 q^{47} +20.8227 q^{48} -4.62799 q^{49} +13.5106 q^{50} +1.48466 q^{51} -24.9651 q^{52} -11.6264 q^{53} +15.1114 q^{54} -0.221330 q^{55} -13.7640 q^{56} +2.49771 q^{57} +0.245488 q^{58} +11.9252 q^{59} -0.331207 q^{60} +6.02580 q^{61} -2.04487 q^{62} -0.980059 q^{63} +23.5548 q^{64} +0.191013 q^{65} -22.6548 q^{66} +8.05061 q^{67} +5.12416 q^{68} -7.92000 q^{69} +0.169015 q^{70} +13.7416 q^{71} +5.68696 q^{72} +0.324012 q^{73} -22.8756 q^{74} -7.68455 q^{75} +8.62065 q^{76} +8.39613 q^{77} +19.5516 q^{78} +14.6951 q^{79} -0.549875 q^{80} -6.68602 q^{81} +3.78590 q^{82} +15.1979 q^{83} +12.5643 q^{84} -0.0392060 q^{85} +7.94652 q^{86} -0.139628 q^{87} -48.7199 q^{88} +4.86275 q^{89} -0.0698331 q^{90} -7.24606 q^{91} -27.3352 q^{92} +1.16308 q^{93} +19.0495 q^{94} -0.0659584 q^{95} -28.8045 q^{96} -4.43381 q^{97} +12.5095 q^{98} -3.46909 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.70301	−1.91132	−0.955659	−	0.294477i	$-0.904855\pi$
$2$	−2.70301	−1.91132	−0.955659	+	0.294477i	$0.904855\pi$
$3$	1.53742	0.887628	0.443814	−	0.896119i	$-0.353625\pi$
$3$	1.53742	0.887628	0.443814	+	0.896119i	$0.353625\pi$
$4$	5.30627	2.65313
$5$	−0.0405993	−0.0181566	−0.00907829	−	0.999959i	$-0.502890\pi$
$5$	−0.0405993	−0.0181566	−0.00907829	+	0.999959i	$0.502890\pi$
$6$	−4.15566	−1.69654
$7$	1.54013	0.582115	0.291058	−	0.956706i	$-0.405993\pi$
$7$	1.54013	0.582115	0.291058	+	0.956706i	$0.405993\pi$
$8$	−8.93687	−3.15966
$9$	−0.636347	−0.212116
$10$	0.109740	0.0347030
$11$	5.45156	1.64371	0.821854	−	0.569698i	$-0.192940\pi$
$11$	5.45156	1.64371	0.821854	+	0.569698i	$0.192940\pi$
$12$	8.15795	2.35500
$13$	−4.70483	−1.30488	−0.652442	−	0.757838i	$-0.726256\pi$
$13$	−4.70483	−1.30488	−0.652442	+	0.757838i	$0.726256\pi$
$14$	−4.16299	−1.11261
$15$	−0.0624181	−0.0161163
$16$	13.5439	3.38598
$17$	0.965682	0.234212	0.117106	−	0.993119i	$-0.462638\pi$
$17$	0.965682	0.234212	0.117106	+	0.993119i	$0.462638\pi$
$18$	1.72005	0.405421
$19$	1.62462	0.372713	0.186356	−	0.982482i	$-0.440332\pi$
$19$	1.62462	0.372713	0.186356	+	0.982482i	$0.440332\pi$
$20$	−0.215431	−0.0481718
$21$	2.36783	0.516702
$22$	−14.7356	−3.14165
$23$	−5.15150	−1.07416	−0.537081	−	0.843531i	$-0.680473\pi$
$23$	−5.15150	−1.07416	−0.537081	+	0.843531i	$0.680473\pi$
$24$	−13.7397	−2.80461
$25$	−4.99835	−0.999670
$26$	12.7172	2.49405
$27$	−5.59058	−1.07591
$28$	8.17235	1.54443
$29$	−0.0908200	−0.0168649	−0.00843243	−	0.999964i	$-0.502684\pi$
$29$	−0.0908200	−0.0168649	−0.00843243	+	0.999964i	$0.502684\pi$
$30$	0.168717	0.0308034
$31$	0.756515	0.135874	0.0679371	−	0.997690i	$-0.478358\pi$
$31$	0.756515	0.135874	0.0679371	+	0.997690i	$0.478358\pi$
$32$	−18.7356	−3.31203
$33$	8.38133	1.45900
$34$	−2.61025	−0.447654
$35$	−0.0625284	−0.0105692
$36$	−3.37663	−0.562772
$37$	8.46302	1.39131	0.695656	−	0.718375i	$-0.255114\pi$
$37$	8.46302	1.39131	0.695656	+	0.718375i	$0.255114\pi$
$38$	−4.39136	−0.712372
$39$	−7.23329	−1.15825
$40$	0.362831	0.0573687
$41$	−1.40062	−0.218741	−0.109370	−	0.994001i	$-0.534883\pi$
$41$	−1.40062	−0.218741	−0.109370	+	0.994001i	$0.534883\pi$
$42$	−6.40026	−0.987582
$43$	−2.93988	−0.448327	−0.224164	−	0.974552i	$-0.571965\pi$
$43$	−2.93988	−0.448327	−0.224164	+	0.974552i	$0.571965\pi$
$44$	28.9275	4.36098
$45$	0.0258353	0.00385130
$46$	13.9245	2.05306
$47$	−7.04753	−1.02799	−0.513994	−	0.857794i	$-0.671835\pi$
$47$	−7.04753	−1.02799	−0.513994	+	0.857794i	$0.671835\pi$
$48$	20.8227	3.00549
$49$	−4.62799	−0.661142
$50$	13.5106	1.91069
$51$	1.48466	0.207893
$52$	−24.9651	−3.46203
$53$	−11.6264	−1.59701	−0.798504	−	0.601989i	$-0.794375\pi$
$53$	−11.6264	−1.59701	−0.798504	+	0.601989i	$0.794375\pi$
$54$	15.1114	2.05640
$55$	−0.221330	−0.0298441
$56$	−13.7640	−1.83929
$57$	2.49771	0.330830
$58$	0.245488	0.0322341
$59$	11.9252	1.55253	0.776266	−	0.630406i	$-0.217112\pi$
$59$	11.9252	1.55253	0.776266	+	0.630406i	$0.217112\pi$
$60$	−0.331207	−0.0427587
$61$	6.02580	0.771525	0.385763	−	0.922598i	$-0.373938\pi$
$61$	6.02580	0.771525	0.385763	+	0.922598i	$0.373938\pi$
$62$	−2.04487	−0.259699
$63$	−0.980059	−0.123476
$64$	23.5548	2.94435
$65$	0.191013	0.0236922
$66$	−22.6548	−2.78862
$67$	8.05061	0.983539	0.491769	−	0.870726i	$-0.336350\pi$
$67$	8.05061	0.983539	0.491769	+	0.870726i	$0.336350\pi$
$68$	5.12416	0.621396
$69$	−7.92000	−0.953456
$70$	0.169015	0.0202011
$71$	13.7416	1.63083	0.815413	−	0.578879i	$-0.196510\pi$
$71$	13.7416	1.63083	0.815413	+	0.578879i	$0.196510\pi$
$72$	5.68696	0.670214
$73$	0.324012	0.0379228	0.0189614	−	0.999820i	$-0.493964\pi$
$73$	0.324012	0.0379228	0.0189614	+	0.999820i	$0.493964\pi$
$74$	−22.8756	−2.65924
$75$	−7.68455	−0.887336
$76$	8.62065	0.988856
$77$	8.39613	0.956828
$78$	19.5516	2.21379
$79$	14.6951	1.65333	0.826664	−	0.562696i	$-0.190236\pi$
$79$	14.6951	1.65333	0.826664	+	0.562696i	$0.190236\pi$
$80$	−0.549875	−0.0614779
$81$	−6.68602	−0.742891
$82$	3.78590	0.418083
$83$	15.1979	1.66818	0.834091	−	0.551626i	$-0.185993\pi$
$83$	15.1979	1.66818	0.834091	+	0.551626i	$0.185993\pi$
$84$	12.5643	1.37088
$85$	−0.0392060	−0.00425249
$86$	7.94652	0.856896
$87$	−0.139628	−0.0149697
$88$	−48.7199	−5.19356
$89$	4.86275	0.515450	0.257725	−	0.966218i	$-0.417027\pi$
$89$	4.86275	0.515450	0.257725	+	0.966218i	$0.417027\pi$
$90$	−0.0698331	−0.00736105
$91$	−7.24606	−0.759593
$92$	−27.3352	−2.84989
$93$	1.16308	0.120606
$94$	19.0495	1.96481
$95$	−0.0659584	−0.00676718
$96$	−28.8045	−2.93985
$97$	−4.43381	−0.450185	−0.225093	−	0.974337i	$-0.572268\pi$
$97$	−4.43381	−0.450185	−0.225093	+	0.974337i	$0.572268\pi$
$98$	12.5095	1.26365
$99$	−3.46909	−0.348657
$100$	−26.5226	−2.65226
$101$	−8.84680	−0.880289	−0.440145	−	0.897927i	$-0.645073\pi$
$101$	−8.84680	−0.880289	−0.440145	+	0.897927i	$0.645073\pi$
$102$	−4.01304	−0.397350
$103$	8.30648	0.818462	0.409231	−	0.912431i	$-0.365797\pi$
$103$	8.30648	0.818462	0.409231	+	0.912431i	$0.365797\pi$
$104$	42.0465	4.12299
$105$	−0.0961322	−0.00938154
$106$	31.4263	3.05239
$107$	−17.5658	−1.69815	−0.849075	−	0.528273i	$-0.822840\pi$
$107$	−17.5658	−1.69815	−0.849075	+	0.528273i	$0.822840\pi$
$108$	−29.6651	−2.85453
$109$	10.7013	1.02500	0.512498	−	0.858688i	$-0.328720\pi$
$109$	10.7013	1.02500	0.512498	+	0.858688i	$0.328720\pi$
$110$	0.598257	0.0570416
$111$	13.0112	1.23497
$112$	20.8594	1.97103
$113$	10.1111	0.951169	0.475584	−	0.879670i	$-0.342237\pi$
$113$	10.1111	0.951169	0.475584	+	0.879670i	$0.342237\pi$
$114$	−6.75135	−0.632322
$115$	0.209147	0.0195031
$116$	−0.481915	−0.0447447
$117$	2.99391	0.276787
$118$	−32.2340	−2.96738
$119$	1.48728	0.136338
$120$	0.557823	0.0509220
$121$	18.7196	1.70178
$122$	−16.2878	−1.47463
$123$	−2.15334	−0.194160
$124$	4.01427	0.360492
$125$	0.405927	0.0363072
$126$	2.64911	0.236002
$127$	6.77966	0.601597	0.300799	−	0.953688i	$-0.402747\pi$
$127$	6.77966	0.601597	0.300799	+	0.953688i	$0.402747\pi$
$128$	−26.1975	−2.31556
$129$	−4.51982	−0.397948
$130$	−0.516310	−0.0452834
$131$	−12.0686	−1.05444	−0.527219	−	0.849730i	$-0.676765\pi$
$131$	−12.0686	−1.05444	−0.527219	+	0.849730i	$0.676765\pi$
$132$	44.4736	3.87093
$133$	2.50212	0.216962
$134$	−21.7609	−1.87985
$135$	0.226974	0.0195348
$136$	−8.63017	−0.740031
$137$	4.76806	0.407363	0.203682	−	0.979037i	$-0.434709\pi$
$137$	4.76806	0.407363	0.203682	+	0.979037i	$0.434709\pi$
$138$	21.4078	1.82236
$139$	17.3063	1.46790	0.733952	−	0.679201i	$-0.237674\pi$
$139$	17.3063	1.46790	0.733952	+	0.679201i	$0.237674\pi$
$140$	−0.331792	−0.0280416
$141$	−10.8350	−0.912471
$142$	−37.1437	−3.11703
$143$	−25.6487	−2.14485
$144$	−8.61865	−0.718221
$145$	0.00368723	0.000306208 0
$146$	−0.875809	−0.0724824
$147$	−7.11516	−0.586848
$148$	44.9070	3.69133
$149$	7.18746	0.588819	0.294410	−	0.955679i	$-0.404877\pi$
$149$	7.18746	0.588819	0.294410	+	0.955679i	$0.404877\pi$
$150$	20.7714	1.69598
$151$	−9.10769	−0.741173	−0.370586	−	0.928798i	$-0.620843\pi$
$151$	−9.10769	−0.741173	−0.370586	+	0.928798i	$0.620843\pi$
$152$	−14.5190	−1.17765
$153$	−0.614509	−0.0496801
$154$	−22.6948	−1.82880
$155$	−0.0307140	−0.00246701
$156$	−38.3817	−3.07300
$157$	16.3864	1.30778	0.653889	−	0.756591i	$-0.273136\pi$
$157$	16.3864	1.30778	0.653889	+	0.756591i	$0.273136\pi$
$158$	−39.7210	−3.16003
$159$	−17.8746	−1.41755
$160$	0.760655	0.0601350
$161$	−7.93399	−0.625286
$162$	18.0724	1.41990
$163$	6.52872	0.511369	0.255684	−	0.966760i	$-0.417699\pi$
$163$	6.52872	0.511369	0.255684	+	0.966760i	$0.417699\pi$
$164$	−7.43208	−0.580348
$165$	−0.340277	−0.0264905
$166$	−41.0800	−3.18843
$167$	−0.656606	−0.0508097	−0.0254048	−	0.999677i	$-0.508087\pi$
$167$	−0.656606	−0.0508097	−0.0254048	+	0.999677i	$0.508087\pi$
$168$	−21.1610	−1.63260
$169$	9.13541	0.702724
$170$	0.105974	0.00812786
$171$	−1.03382	−0.0790582
$172$	−15.5998	−1.18947
$173$	−9.12575	−0.693818	−0.346909	−	0.937899i	$-0.612769\pi$
$173$	−9.12575	−0.693818	−0.346909	+	0.937899i	$0.612769\pi$
$174$	0.377417	0.0286119
$175$	−7.69812	−0.581923
$176$	73.8356	5.56557
$177$	18.3340	1.37807
$178$	−13.1441	−0.985189
$179$	0.451074	0.0337149	0.0168574	−	0.999858i	$-0.494634\pi$
$179$	0.451074	0.0337149	0.0168574	+	0.999858i	$0.494634\pi$
$180$	0.137089	0.0102180
$181$	17.1011	1.27111	0.635556	−	0.772055i	$-0.280771\pi$
$181$	17.1011	1.27111	0.635556	+	0.772055i	$0.280771\pi$
$182$	19.5862	1.45182
$183$	9.26418	0.684828
$184$	46.0383	3.39399
$185$	−0.343593	−0.0252615
$186$	−3.14382	−0.230516
$187$	5.26448	0.384977
$188$	−37.3961	−2.72739
$189$	−8.61024	−0.626303
$190$	0.178286	0.0129342
$191$	−4.81760	−0.348589	−0.174295	−	0.984694i	$-0.555765\pi$
$191$	−4.81760	−0.348589	−0.174295	+	0.984694i	$0.555765\pi$
$192$	36.2135	2.61349
$193$	22.5646	1.62424	0.812119	−	0.583492i	$-0.198314\pi$
$193$	22.5646	1.62424	0.812119	+	0.583492i	$0.198314\pi$
$194$	11.9846	0.860446
$195$	0.293667	0.0210299
$196$	−24.5574	−1.75410
$197$	−5.08416	−0.362232	−0.181116	−	0.983462i	$-0.557971\pi$
$197$	−5.08416	−0.362232	−0.181116	+	0.983462i	$0.557971\pi$
$198$	9.37699	0.666393
$199$	19.3710	1.37317	0.686586	−	0.727049i	$-0.259109\pi$
$199$	19.3710	1.37317	0.686586	+	0.727049i	$0.259109\pi$
$200$	44.6696	3.15862
$201$	12.3772	0.873017
$202$	23.9130	1.68251
$203$	−0.139875	−0.00981729
$204$	7.87798	0.551569
$205$	0.0568644	0.00397158
$206$	−22.4525	−1.56434
$207$	3.27814	0.227847
$208$	−63.7219	−4.41832
$209$	8.85670	0.612631
$210$	0.259846	0.0179311
$211$	−5.90141	−0.406270	−0.203135	−	0.979151i	$-0.565113\pi$
$211$	−5.90141	−0.406270	−0.203135	+	0.979151i	$0.565113\pi$
$212$	−61.6928	−4.23708
$213$	21.1266	1.44757
$214$	47.4805	3.24570
$215$	0.119357	0.00814009
$216$	49.9623	3.39951
$217$	1.16513	0.0790944
$218$	−28.9257	−1.95909
$219$	0.498142	0.0336613
$220$	−1.17444	−0.0791804
$221$	−4.54337	−0.305620
$222$	−35.1694	−2.36041
$223$	19.9361	1.33502	0.667510	−	0.744601i	$-0.267360\pi$
$223$	19.9361	1.33502	0.667510	+	0.744601i	$0.267360\pi$
$224$	−28.8554	−1.92798
$225$	3.18069	0.212046
$226$	−27.3303	−1.81799
$227$	2.20704	0.146486	0.0732430	−	0.997314i	$-0.476665\pi$
$227$	2.20704	0.146486	0.0732430	+	0.997314i	$0.476665\pi$
$228$	13.2535	0.877737
$229$	0.123873	0.00818577	0.00409288	−	0.999992i	$-0.498697\pi$
$229$	0.123873	0.00818577	0.00409288	+	0.999992i	$0.498697\pi$
$230$	−0.565328	−0.0372766
$231$	12.9084	0.849308
$232$	0.811647	0.0532873
$233$	3.14175	0.205823	0.102912	−	0.994691i	$-0.467184\pi$
$233$	3.14175	0.205823	0.102912	+	0.994691i	$0.467184\pi$
$234$	−8.09256	−0.529027
$235$	0.286125	0.0186647
$236$	63.2784	4.11907
$237$	22.5925	1.46754
$238$	−4.02013	−0.260586
$239$	27.0156	1.74749	0.873746	−	0.486382i	$-0.161684\pi$
$239$	27.0156	1.74749	0.873746	+	0.486382i	$0.161684\pi$
$240$	−0.845387	−0.0545695
$241$	3.70411	0.238603	0.119301	−	0.992858i	$-0.461935\pi$
$241$	3.70411	0.238603	0.119301	+	0.992858i	$0.461935\pi$
$242$	−50.5992	−3.25264
$243$	6.49255	0.416497
$244$	31.9745	2.04696
$245$	0.187893	0.0120041
$246$	5.82051	0.371102
$247$	−7.64354	−0.486347
$248$	−6.76088	−0.429316
$249$	23.3655	1.48073
$250$	−1.09722	−0.0693945
$251$	−7.89464	−0.498305	−0.249153	−	0.968464i	$-0.580152\pi$
$251$	−7.89464	−0.498305	−0.249153	+	0.968464i	$0.580152\pi$
$252$	−5.20046	−0.327598
$253$	−28.0837	−1.76561
$254$	−18.3255	−1.14984
$255$	−0.0602760	−0.00377463
$256$	23.7027	1.48142
$257$	−16.2250	−1.01209	−0.506044	−	0.862508i	$-0.668893\pi$
$257$	−16.2250	−1.01209	−0.506044	+	0.862508i	$0.668893\pi$
$258$	12.2171	0.760605
$259$	13.0342	0.809904
$260$	1.01357	0.0628587
$261$	0.0577931	0.00357730
$262$	32.6215	2.01536
$263$	14.4881	0.893377	0.446688	−	0.894690i	$-0.352603\pi$
$263$	14.4881	0.893377	0.446688	+	0.894690i	$0.352603\pi$
$264$	−74.9029	−4.60995
$265$	0.472024	0.0289962
$266$	−6.76327	−0.414683
$267$	7.47608	0.457528
$268$	42.7187	2.60946
$269$	25.8125	1.57381	0.786907	−	0.617072i	$-0.211681\pi$
$269$	25.8125	1.57381	0.786907	+	0.617072i	$0.211681\pi$
$270$	−0.613513	−0.0373372
$271$	8.92510	0.542161	0.271081	−	0.962557i	$-0.412619\pi$
$271$	8.92510	0.542161	0.271081	+	0.962557i	$0.412619\pi$
$272$	13.0791	0.793038
$273$	−11.1402	−0.674237
$274$	−12.8881	−0.778600
$275$	−27.2488	−1.64317
$276$	−42.0256	−2.52965
$277$	11.9666	0.719001	0.359500	−	0.933145i	$-0.382947\pi$
$277$	11.9666	0.719001	0.359500	+	0.933145i	$0.382947\pi$
$278$	−46.7792	−2.80563
$279$	−0.481407	−0.0288211
$280$	0.558808	0.0333952
$281$	21.7267	1.29611	0.648054	−	0.761594i	$-0.275583\pi$
$281$	21.7267	1.29611	0.648054	+	0.761594i	$0.275583\pi$
$282$	29.2871	1.74402
$283$	4.05949	0.241312	0.120656	−	0.992694i	$-0.461500\pi$
$283$	4.05949	0.241312	0.120656	+	0.992694i	$0.461500\pi$
$284$	72.9165	4.32680
$285$	−0.101406	−0.00600675
$286$	69.3286	4.09949
$287$	−2.15715	−0.127332
$288$	11.9224	0.702533
$289$	−16.0675	−0.945145
$290$	−0.00996663	−0.000585261 0
$291$	−6.81661	−0.399597
$292$	1.71930	0.100614
$293$	−14.7539	−0.861931	−0.430965	−	0.902368i	$-0.641827\pi$
$293$	−14.7539	−0.861931	−0.430965	+	0.902368i	$0.641827\pi$
$294$	19.2323	1.12165
$295$	−0.484156	−0.0281887
$296$	−75.6329	−4.39607
$297$	−30.4774	−1.76848
$298$	−19.4278	−1.12542
$299$	24.2369	1.40166
$300$	−40.7763	−2.35422
$301$	−4.52780	−0.260978
$302$	24.6182	1.41662
$303$	−13.6012	−0.781370
$304$	22.0037	1.26200
$305$	−0.244644	−0.0140083
$306$	1.66102	0.0949544
$307$	−2.34783	−0.133998	−0.0669990	−	0.997753i	$-0.521342\pi$
$307$	−2.34783	−0.133998	−0.0669990	+	0.997753i	$0.521342\pi$
$308$	44.5521	2.53859
$309$	12.7705	0.726490
$310$	0.0830203	0.00471524
$311$	−7.94702	−0.450634	−0.225317	−	0.974285i	$-0.572342\pi$
$311$	−7.94702	−0.450634	−0.225317	+	0.974285i	$0.572342\pi$
$312$	64.6430	3.65969
$313$	−20.9404	−1.18362	−0.591812	−	0.806076i	$-0.701587\pi$
$313$	−20.9404	−1.18362	−0.591812	+	0.806076i	$0.701587\pi$
$314$	−44.2926	−2.49958
$315$	0.0397898	0.00224190
$316$	77.9761	4.38650
$317$	22.0931	1.24087	0.620437	−	0.784257i	$-0.286955\pi$
$317$	22.0931	1.24087	0.620437	+	0.784257i	$0.286955\pi$
$318$	48.3153	2.70939
$319$	−0.495111	−0.0277209
$320$	−0.956309	−0.0534593
$321$	−27.0060	−1.50733
$322$	21.4456	1.19512
$323$	1.56886	0.0872938
$324$	−35.4778	−1.97099
$325$	23.5164	1.30445
$326$	−17.6472	−0.977388
$327$	16.4523	0.909816
$328$	12.5172	0.691146
$329$	−10.8541	−0.598407
$330$	0.919771	0.0506317
$331$	12.8661	0.707184	0.353592	−	0.935400i	$-0.384960\pi$
$331$	12.8661	0.707184	0.353592	+	0.935400i	$0.384960\pi$
$332$	80.6439	4.42591
$333$	−5.38542	−0.295119
$334$	1.77481	0.0971134
$335$	−0.326850	−0.0178577
$336$	32.0697	1.74954
$337$	16.2748	0.886545	0.443273	−	0.896387i	$-0.353817\pi$
$337$	16.2748	0.886545	0.443273	+	0.896387i	$0.353817\pi$
$338$	−24.6931	−1.34313
$339$	15.5449	0.844285
$340$	−0.208038	−0.0112824
$341$	4.12419	0.223338
$342$	2.79443	0.151105
$343$	−17.9086	−0.966976
$344$	26.2733	1.41656
$345$	0.321547	0.0173115
$346$	24.6670	1.32611
$347$	−0.671446	−0.0360451	−0.0180226	−	0.999838i	$-0.505737\pi$
$347$	−0.671446	−0.0360451	−0.0180226	+	0.999838i	$0.505737\pi$
$348$	−0.740905	−0.0397167
$349$	28.6640	1.53435	0.767174	−	0.641439i	$-0.221662\pi$
$349$	28.6640	1.53435	0.767174	+	0.641439i	$0.221662\pi$
$350$	20.8081	1.11224
$351$	26.3027	1.40394
$352$	−102.139	−5.44400
$353$	3.24595	0.172764	0.0863822	−	0.996262i	$-0.472469\pi$
$353$	3.24595	0.172764	0.0863822	+	0.996262i	$0.472469\pi$
$354$	−49.5571	−2.63393
$355$	−0.557900	−0.0296102
$356$	25.8030	1.36756
$357$	2.28657	0.121018
$358$	−1.21926	−0.0644398
$359$	−0.431289	−0.0227626	−0.0113813	−	0.999935i	$-0.503623\pi$
$359$	−0.431289	−0.0227626	−0.0113813	+	0.999935i	$0.503623\pi$
$360$	−0.230887	−0.0121688
$361$	−16.3606	−0.861085
$362$	−46.2243	−2.42950
$363$	28.7798	1.51055
$364$	−38.4495	−2.01530
$365$	−0.0131547	−0.000688548 0
$366$	−25.0412	−1.30892
$367$	12.3724	0.645832	0.322916	−	0.946428i	$-0.395337\pi$
$367$	12.3724	0.645832	0.322916	+	0.946428i	$0.395337\pi$
$368$	−69.7715	−3.63709
$369$	0.891283	0.0463983
$370$	0.928735	0.0482826
$371$	−17.9062	−0.929643
$372$	6.17161	0.319983
$373$	−26.9504	−1.39544	−0.697720	−	0.716370i	$-0.745802\pi$
$373$	−26.9504	−1.39544	−0.697720	+	0.716370i	$0.745802\pi$
$374$	−14.2299	−0.735812
$375$	0.624079	0.0322273
$376$	62.9829	3.24809
$377$	0.427293	0.0220067
$378$	23.2736	1.19706
$379$	−38.3224	−1.96849	−0.984245	−	0.176812i	$-0.943422\pi$
$379$	−38.3224	−1.96849	−0.984245	+	0.176812i	$0.943422\pi$
$380$	−0.349993	−0.0179542
$381$	10.4232	0.533995
$382$	13.0220	0.666265
$383$	−34.3491	−1.75516	−0.877579	−	0.479431i	$-0.840843\pi$
$383$	−34.3491	−1.75516	−0.877579	+	0.479431i	$0.840843\pi$
$384$	−40.2766	−2.05535
$385$	−0.340877	−0.0173727
$386$	−60.9924	−3.10443
$387$	1.87078	0.0950973
$388$	−23.5270	−1.19440
$389$	−11.7364	−0.595058	−0.297529	−	0.954713i	$-0.596163\pi$
$389$	−11.7364	−0.595058	−0.297529	+	0.954713i	$0.596163\pi$
$390$	−0.793784	−0.0401948
$391$	−4.97470	−0.251582
$392$	41.3598	2.08898
$393$	−18.5545	−0.935949
$394$	13.7425	0.692339
$395$	−0.596611	−0.0300188
$396$	−18.4079	−0.925032
$397$	−2.57006	−0.128988	−0.0644939	−	0.997918i	$-0.520543\pi$
$397$	−2.57006	−0.128988	−0.0644939	+	0.997918i	$0.520543\pi$
$398$	−52.3599	−2.62457
$399$	3.84681	0.192581
$400$	−67.6973	−3.38487
$401$	−35.7428	−1.78491	−0.892456	−	0.451134i	$-0.851020\pi$
$401$	−35.7428	−1.78491	−0.892456	+	0.451134i	$0.851020\pi$
$402$	−33.4556	−1.66861
$403$	−3.55927	−0.177300
$404$	−46.9435	−2.33553
$405$	0.271448	0.0134884
$406$	0.378083	0.0187640
$407$	46.1367	2.28691
$408$	−13.2682	−0.656873
$409$	−7.66224	−0.378874	−0.189437	−	0.981893i	$-0.560666\pi$
$409$	−7.66224	−0.378874	−0.189437	+	0.981893i	$0.560666\pi$
$410$	−0.153705	−0.00759095
$411$	7.33050	0.361587
$412$	44.0764	2.17149
$413$	18.3664	0.903752
$414$	−8.86085	−0.435487
$415$	−0.617024	−0.0302885
$416$	88.1480	4.32181
$417$	26.6071	1.30295
$418$	−23.9398	−1.17093
$419$	6.74363	0.329448	0.164724	−	0.986340i	$-0.447327\pi$
$419$	6.74363	0.329448	0.164724	+	0.986340i	$0.447327\pi$
$420$	−0.510103	−0.0248905
$421$	2.26626	0.110451	0.0552253	−	0.998474i	$-0.482412\pi$
$421$	2.26626	0.110451	0.0552253	+	0.998474i	$0.482412\pi$
$422$	15.9516	0.776510
$423$	4.48468	0.218052
$424$	103.904	5.04601
$425$	−4.82682	−0.234135
$426$	−57.1053	−2.76676
$427$	9.28054	0.449117
$428$	−93.2088	−4.50542
$429$	−39.4327	−1.90383
$430$	−0.322624	−0.0155583
$431$	−8.49626	−0.409251	−0.204625	−	0.978840i	$-0.565598\pi$
$431$	−8.49626	−0.409251	−0.204625	+	0.978840i	$0.565598\pi$
$432$	−75.7185	−3.64301
$433$	25.2401	1.21296	0.606481	−	0.795098i	$-0.292581\pi$
$433$	25.2401	1.21296	0.606481	+	0.795098i	$0.292581\pi$
$434$	−3.14937	−0.151175
$435$	0.00566882	0.000271799 0
$436$	56.7838	2.71945
$437$	−8.36920	−0.400353
$438$	−1.34648	−0.0643375
$439$	−24.6619	−1.17705	−0.588524	−	0.808480i	$-0.700291\pi$
$439$	−24.6619	−1.17705	−0.588524	+	0.808480i	$0.700291\pi$
$440$	1.97800	0.0942973
$441$	2.94501	0.140239
$442$	12.2808	0.584136
$443$	−5.15955	−0.245138	−0.122569	−	0.992460i	$-0.539113\pi$
$443$	−5.15955	−0.245138	−0.122569	+	0.992460i	$0.539113\pi$
$444$	69.0408	3.27653
$445$	−0.197424	−0.00935882
$446$	−53.8875	−2.55165
$447$	11.0501	0.522653
$448$	36.2775	1.71395
$449$	5.89576	0.278238	0.139119	−	0.990276i	$-0.455573\pi$
$449$	5.89576	0.278238	0.139119	+	0.990276i	$0.455573\pi$
$450$	−8.59743	−0.405287
$451$	−7.63559	−0.359546
$452$	53.6520	2.52358
$453$	−14.0023	−0.657886
$454$	−5.96564	−0.279981
$455$	0.294185	0.0137916
$456$	−22.3218	−1.04531
$457$	17.3035	0.809422	0.404711	−	0.914445i	$-0.367372\pi$
$457$	17.3035	0.809422	0.404711	+	0.914445i	$0.367372\pi$
$458$	−0.334830	−0.0156456
$459$	−5.39872	−0.251991
$460$	1.10979	0.0517443
$461$	−5.62700	−0.262075	−0.131038	−	0.991377i	$-0.541831\pi$
$461$	−5.62700	−0.262075	−0.131038	+	0.991377i	$0.541831\pi$
$462$	−34.8914	−1.62330
$463$	17.6231	0.819014	0.409507	−	0.912307i	$-0.365701\pi$
$463$	17.6231	0.819014	0.409507	+	0.912307i	$0.365701\pi$
$464$	−1.23006	−0.0571041
$465$	−0.0472203	−0.00218979
$466$	−8.49219	−0.393393
$467$	−15.6202	−0.722816	−0.361408	−	0.932408i	$-0.617704\pi$
$467$	−15.6202	−0.722816	−0.361408	+	0.932408i	$0.617704\pi$
$468$	15.8865	0.734352
$469$	12.3990	0.572533
$470$	−0.773399	−0.0356742
$471$	25.1928	1.16082
$472$	−106.574	−4.90547
$473$	−16.0269	−0.736919
$474$	−61.0678	−2.80493
$475$	−8.12040	−0.372590
$476$	7.89189	0.361724
$477$	7.39843	0.338751
$478$	−73.0234	−3.34001
$479$	−30.1082	−1.37568	−0.687840	−	0.725862i	$-0.741441\pi$
$479$	−30.1082	−1.37568	−0.687840	+	0.725862i	$0.741441\pi$
$480$	1.16944	0.0533776
$481$	−39.8170	−1.81550
$482$	−10.0123	−0.456046
$483$	−12.1978	−0.555021
$484$	99.3310	4.51504
$485$	0.180010	0.00817382
$486$	−17.5494	−0.796058
$487$	−25.3707	−1.14966	−0.574828	−	0.818274i	$-0.694931\pi$
$487$	−25.3707	−1.14966	−0.574828	+	0.818274i	$0.694931\pi$
$488$	−53.8518	−2.43776
$489$	10.0374	0.453905
$490$	−0.507878	−0.0229436
$491$	15.7361	0.710160	0.355080	−	0.934836i	$-0.384454\pi$
$491$	15.7361	0.710160	0.355080	+	0.934836i	$0.384454\pi$
$492$	−11.4262	−0.515133
$493$	−0.0877032	−0.00394996
$494$	20.6606	0.929563
$495$	0.140843	0.00633041
$496$	10.2462	0.460068
$497$	21.1639	0.949329
$498$	−63.1571	−2.83014
$499$	9.53523	0.426856	0.213428	−	0.976959i	$-0.431537\pi$
$499$	9.53523	0.426856	0.213428	+	0.976959i	$0.431537\pi$
$500$	2.15395	0.0963278
$501$	−1.00948	−0.0451001
$502$	21.3393	0.952419
$503$	24.4722	1.09116	0.545581	−	0.838058i	$-0.316309\pi$
$503$	24.4722	1.09116	0.545581	+	0.838058i	$0.316309\pi$
$504$	8.75867	0.390142
$505$	0.359174	0.0159830
$506$	75.9106	3.37464
$507$	14.0449	0.623757
$508$	35.9747	1.59612
$509$	24.0659	1.06670	0.533352	−	0.845893i	$-0.320932\pi$
$509$	24.0659	1.06670	0.533352	+	0.845893i	$0.320932\pi$
$510$	0.162927	0.00721452
$511$	0.499022	0.0220754
$512$	−11.6735	−0.515900
$513$	−9.08255	−0.401005
$514$	43.8564	1.93442
$515$	−0.337238	−0.0148605
$516$	−23.9834	−1.05581
$517$	−38.4201	−1.68971
$518$	−35.2315	−1.54798
$519$	−14.0301	−0.615852
$520$	−1.70706	−0.0748595
$521$	−18.0312	−0.789961	−0.394980	−	0.918690i	$-0.629249\pi$
$521$	−18.0312	−0.789961	−0.394980	+	0.918690i	$0.629249\pi$
$522$	−0.156215	−0.00683736
$523$	−40.3559	−1.76464	−0.882320	−	0.470650i	$-0.844019\pi$
$523$	−40.3559	−1.76464	−0.882320	+	0.470650i	$0.844019\pi$
$524$	−64.0392	−2.79756
$525$	−11.8352	−0.516532
$526$	−39.1616	−1.70753
$527$	0.730553	0.0318234
$528$	113.516	4.94016
$529$	3.53791	0.153822
$530$	−1.27589	−0.0554210
$531$	−7.58858	−0.329316
$532$	13.2769	0.575628
$533$	6.58969	0.285431
$534$	−20.2079	−0.874482
$535$	0.713160	0.0308326
$536$	−71.9473	−3.10765
$537$	0.693489	0.0299263
$538$	−69.7714	−3.00806
$539$	−25.2298	−1.08672
$540$	1.20438	0.0518285
$541$	41.5188	1.78503	0.892516	−	0.451016i	$-0.148938\pi$
$541$	41.5188	1.78503	0.892516	+	0.451016i	$0.148938\pi$
$542$	−24.1247	−1.03624
$543$	26.2915	1.12827
$544$	−18.0927	−0.775717
$545$	−0.434465	−0.0186104
$546$	30.1121	1.28868
$547$	12.5231	0.535448	0.267724	−	0.963496i	$-0.413728\pi$
$547$	12.5231	0.535448	0.267724	+	0.963496i	$0.413728\pi$
$548$	25.3006	1.08079
$549$	−3.83450	−0.163653
$550$	73.6539	3.14061
$551$	−0.147548	−0.00628574
$552$	70.7800	3.01260
$553$	22.6324	0.962427
$554$	−32.3457	−1.37424
$555$	−0.528246	−0.0224228
$556$	91.8320	3.89454
$557$	0.279615	0.0118477	0.00592383	−	0.999982i	$-0.498114\pi$
$557$	0.279615	0.0118477	0.00592383	+	0.999982i	$0.498114\pi$
$558$	1.30125	0.0550862
$559$	13.8316	0.585015
$560$	−0.846880	−0.0357872
$561$	8.09370	0.341716
$562$	−58.7276	−2.47727
$563$	10.4560	0.440668	0.220334	−	0.975425i	$-0.429285\pi$
$563$	10.4560	0.440668	0.220334	+	0.975425i	$0.429285\pi$
$564$	−57.4934	−2.42091
$565$	−0.410503	−0.0172700
$566$	−10.9728	−0.461223
$567$	−10.2974	−0.432448
$568$	−122.807	−5.15286
$569$	11.4522	0.480099	0.240050	−	0.970761i	$-0.422836\pi$
$569$	11.4522	0.480099	0.240050	+	0.970761i	$0.422836\pi$
$570$	0.274100	0.0114808
$571$	−31.3275	−1.31102	−0.655508	−	0.755188i	$-0.727545\pi$
$571$	−31.3275	−1.31102	−0.655508	+	0.755188i	$0.727545\pi$
$572$	−136.099	−5.69057
$573$	−7.40666	−0.309418
$574$	5.83079	0.243372
$575$	25.7490	1.07381
$576$	−14.9890	−0.624543
$577$	−24.7121	−1.02878	−0.514390	−	0.857557i	$-0.671982\pi$
$577$	−24.7121	−1.02878	−0.514390	+	0.857557i	$0.671982\pi$
$578$	43.4305	1.80647
$579$	34.6913	1.44172
$580$	0.0195654	0.000812411 0
$581$	23.4067	0.971075
$582$	18.4254	0.763757
$583$	−63.3821	−2.62502
$584$	−2.89566	−0.119823
$585$	−0.121551	−0.00502550
$586$	39.8799	1.64742
$587$	−19.6585	−0.811392	−0.405696	−	0.914008i	$-0.632971\pi$
$587$	−19.6585	−0.811392	−0.405696	+	0.914008i	$0.632971\pi$
$588$	−37.7549	−1.55699
$589$	1.22905	0.0506420
$590$	1.30868	0.0538775
$591$	−7.81648	−0.321527
$592$	114.623	4.71096
$593$	38.9573	1.59978	0.799892	−	0.600144i	$-0.204890\pi$
$593$	38.9573	1.59978	0.799892	+	0.600144i	$0.204890\pi$
$594$	82.3808	3.38013
$595$	−0.0603825	−0.00247544
$596$	38.1386	1.56222
$597$	29.7813	1.21887
$598$	−65.5126	−2.67901
$599$	−34.6999	−1.41780	−0.708899	−	0.705310i	$-0.750808\pi$
$599$	−34.6999	−1.41780	−0.708899	+	0.705310i	$0.750808\pi$
$600$	68.6759	2.80368
$601$	22.3604	0.912101	0.456050	−	0.889954i	$-0.349264\pi$
$601$	22.3604	0.912101	0.456050	+	0.889954i	$0.349264\pi$
$602$	12.2387	0.498812
$603$	−5.12299	−0.208624
$604$	−48.3278	−1.96643
$605$	−0.760002	−0.0308985
$606$	36.7643	1.49345
$607$	−23.4154	−0.950400	−0.475200	−	0.879878i	$-0.657624\pi$
$607$	−23.4154	−0.950400	−0.475200	+	0.879878i	$0.657624\pi$
$608$	−30.4382	−1.23443
$609$	−0.215046	−0.00871411
$610$	0.661274	0.0267742
$611$	33.1574	1.34141
$612$	−3.26075	−0.131808
$613$	12.3502	0.498821	0.249410	−	0.968398i	$-0.419763\pi$
$613$	12.3502	0.498821	0.249410	+	0.968398i	$0.419763\pi$
$614$	6.34622	0.256113
$615$	0.0874243	0.00352529
$616$	−75.0352	−3.02325
$617$	−46.3509	−1.86602	−0.933009	−	0.359853i	$-0.882827\pi$
$617$	−46.3509	−1.86602	−0.933009	+	0.359853i	$0.882827\pi$
$618$	−34.5189	−1.38855
$619$	−13.0570	−0.524804	−0.262402	−	0.964959i	$-0.584515\pi$
$619$	−13.0570	−0.524804	−0.262402	+	0.964959i	$0.584515\pi$
$620$	−0.162977	−0.00654531
$621$	28.7999	1.15570
$622$	21.4809	0.861305
$623$	7.48928	0.300052
$624$	−97.9671	−3.92182
$625$	24.9753	0.999011
$626$	56.6022	2.26228
$627$	13.6164	0.543789
$628$	86.9507	3.46971
$629$	8.17258	0.325862
$630$	−0.107552	−0.00428498
$631$	−28.3939	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$631$	−28.3939	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$632$	−131.328	−5.22396
$633$	−9.07293	−0.360617
$634$	−59.7179	−2.37170
$635$	−0.275250	−0.0109230
$636$	−94.8475	−3.76095
$637$	21.7739	0.862714
$638$	1.33829	0.0529835
$639$	−8.74443	−0.345924
$640$	1.06360	0.0420426
$641$	−40.4730	−1.59859	−0.799294	−	0.600940i	$-0.794793\pi$
$641$	−40.4730	−1.59859	−0.799294	+	0.600940i	$0.794793\pi$
$642$	72.9974	2.88098
$643$	45.1359	1.77999	0.889993	−	0.455975i	$-0.150709\pi$
$643$	45.1359	1.77999	0.889993	+	0.455975i	$0.150709\pi$
$644$	−42.0998	−1.65897
$645$	0.183502	0.00722538
$646$	−4.24065	−0.166846
$647$	17.1970	0.676083	0.338041	−	0.941131i	$-0.390236\pi$
$647$	17.1970	0.676083	0.338041	+	0.941131i	$0.390236\pi$
$648$	59.7521	2.34728
$649$	65.0111	2.55191
$650$	−63.5650	−2.49323
$651$	1.79130	0.0702065
$652$	34.6431	1.35673
$653$	6.67973	0.261398	0.130699	−	0.991422i	$-0.458278\pi$
$653$	6.67973	0.261398	0.130699	+	0.991422i	$0.458278\pi$
$654$	−44.4708	−1.73895
$655$	0.489977	0.0191450
$656$	−18.9699	−0.740652
$657$	−0.206184	−0.00804402
$658$	29.3388	1.14375
$659$	−19.0695	−0.742843	−0.371422	−	0.928464i	$-0.621130\pi$
$659$	−19.0695	−0.742843	−0.371422	+	0.928464i	$0.621130\pi$
$660$	−1.80560	−0.0702828
$661$	5.12699	0.199417	0.0997084	−	0.995017i	$-0.468209\pi$
$661$	5.12699	0.199417	0.0997084	+	0.995017i	$0.468209\pi$
$662$	−34.7771	−1.35165
$663$	−6.98505	−0.271277
$664$	−135.821	−5.27089
$665$	−0.101585	−0.00393928
$666$	14.5568	0.564066
$667$	0.467859	0.0181156
$668$	−3.48413	−0.134805
$669$	30.6501	1.18500
$670$	0.883478	0.0341317
$671$	32.8501	1.26816
$672$	−44.3628	−1.71133
$673$	−33.8304	−1.30407	−0.652034	−	0.758190i	$-0.726084\pi$
$673$	−33.8304	−1.30407	−0.652034	+	0.758190i	$0.726084\pi$
$674$	−43.9910	−1.69447
$675$	27.9437	1.07555
$676$	48.4749	1.86442
$677$	−34.3422	−1.31988	−0.659939	−	0.751319i	$-0.729418\pi$
$677$	−34.3422	−1.31988	−0.659939	+	0.751319i	$0.729418\pi$
$678$	−42.0181	−1.61370
$679$	−6.82865	−0.262060
$680$	0.350379	0.0134364
$681$	3.39314	0.130025
$682$	−11.1477	−0.426869
$683$	2.49247	0.0953719	0.0476859	−	0.998862i	$-0.484815\pi$
$683$	2.49247	0.0953719	0.0476859	+	0.998862i	$0.484815\pi$
$684$	−5.48573	−0.209752
$685$	−0.193580	−0.00739632
$686$	48.4073	1.84820
$687$	0.190445	0.00726592
$688$	−39.8175	−1.51803
$689$	54.7002	2.08391
$690$	−0.869144	−0.0330878
$691$	26.1755	0.995765	0.497882	−	0.867245i	$-0.334111\pi$
$691$	26.1755	0.995765	0.497882	+	0.867245i	$0.334111\pi$
$692$	−48.4236	−1.84079
$693$	−5.34286	−0.202958
$694$	1.81493	0.0688936
$695$	−0.702626	−0.0266521
$696$	1.24784	0.0472993
$697$	−1.35256	−0.0512317
$698$	−77.4790	−2.93263
$699$	4.83018	0.182694
$700$	−40.8483	−1.54392
$701$	19.1802	0.724428	0.362214	−	0.932095i	$-0.382021\pi$
$701$	19.1802	0.724428	0.362214	+	0.932095i	$0.382021\pi$
$702$	−71.0966	−2.68337
$703$	13.7492	0.518559
$704$	128.410	4.83965
$705$	0.439894	0.0165674
$706$	−8.77383	−0.330207
$707$	−13.6252	−0.512430
$708$	97.2853	3.65621
$709$	9.57738	0.359686	0.179843	−	0.983695i	$-0.442441\pi$
$709$	9.57738	0.359686	0.179843	+	0.983695i	$0.442441\pi$
$710$	1.50801	0.0565945
$711$	−9.35119	−0.350697
$712$	−43.4578	−1.62865
$713$	−3.89719	−0.145951
$714$	−6.18061	−0.231304
$715$	1.04132	0.0389431
$716$	2.39352	0.0894500
$717$	41.5342	1.55112
$718$	1.16578	0.0435065
$719$	−45.9801	−1.71477	−0.857383	−	0.514678i	$-0.827911\pi$
$719$	−45.9801	−1.71477	−0.857383	+	0.514678i	$0.827911\pi$
$720$	0.349911	0.0130404
$721$	12.7931	0.476439
$722$	44.2229	1.64581
$723$	5.69477	0.211791
$724$	90.7427	3.37243
$725$	0.453951	0.0168593
$726$	−77.7920	−2.88713
$727$	36.5797	1.35667	0.678333	−	0.734755i	$-0.262703\pi$
$727$	36.5797	1.35667	0.678333	+	0.734755i	$0.262703\pi$
$728$	64.7571	2.40006
$729$	30.0398	1.11259
$730$	0.0355573	0.00131603
$731$	−2.83899	−0.105004
$732$	49.1582	1.81694
$733$	−3.76791	−0.139171	−0.0695856	−	0.997576i	$-0.522168\pi$
$733$	−3.76791	−0.139171	−0.0695856	+	0.997576i	$0.522168\pi$
$734$	−33.4426	−1.23439
$735$	0.288871	0.0106552
$736$	96.5166	3.55765
$737$	43.8884	1.61665
$738$	−2.40915	−0.0886820
$739$	−43.9409	−1.61639	−0.808196	−	0.588913i	$-0.799556\pi$
$739$	−43.9409	−1.61639	−0.808196	+	0.588913i	$0.799556\pi$
$740$	−1.82320	−0.0670220
$741$	−11.7513	−0.431695
$742$	48.4006	1.77684
$743$	−26.0035	−0.953975	−0.476987	−	0.878910i	$-0.658271\pi$
$743$	−26.0035	−0.953975	−0.476987	+	0.878910i	$0.658271\pi$
$744$	−10.3943	−0.381073
$745$	−0.291806	−0.0106909
$746$	72.8473	2.66713
$747$	−9.67113	−0.353848
$748$	27.9347	1.02139
$749$	−27.0537	−0.988519
$750$	−1.68689	−0.0615965
$751$	32.3972	1.18219	0.591096	−	0.806601i	$-0.298696\pi$
$751$	32.3972	1.18219	0.591096	+	0.806601i	$0.298696\pi$
$752$	−95.4512	−3.48075
$753$	−12.1374	−0.442310
$754$	−1.15498	−0.0420618
$755$	0.369766	0.0134572
$756$	−45.6882	−1.66166
$757$	31.2567	1.13604	0.568021	−	0.823014i	$-0.307709\pi$
$757$	31.2567	1.13604	0.568021	+	0.823014i	$0.307709\pi$
$758$	103.586	3.76241
$759$	−43.1764	−1.56720
$760$	0.589462	0.0213820
$761$	−11.2106	−0.406384	−0.203192	−	0.979139i	$-0.565132\pi$
$761$	−11.2106	−0.406384	−0.203192	+	0.979139i	$0.565132\pi$
$762$	−28.1739	−1.02063
$763$	16.4814	0.596666
$764$	−25.5635	−0.924854
$765$	0.0249487	0.000902021 0
$766$	92.8461	3.35466
$767$	−56.1061	−2.02587
$768$	36.4409	1.31495
$769$	34.7912	1.25460	0.627301	−	0.778777i	$-0.284160\pi$
$769$	34.7912	1.25460	0.627301	+	0.778777i	$0.284160\pi$
$770$	0.921395	0.0332048
$771$	−24.9446	−0.898358
$772$	119.734	4.30932
$773$	24.3730	0.876635	0.438317	−	0.898820i	$-0.355575\pi$
$773$	24.3730	0.876635	0.438317	+	0.898820i	$0.355575\pi$
$774$	−5.05675	−0.181761
$775$	−3.78133	−0.135829
$776$	39.6244	1.42243
$777$	20.0390	0.718893
$778$	31.7236	1.13734
$779$	−2.27548	−0.0815274
$780$	1.55827	0.0557951
$781$	74.9132	2.68060
$782$	13.4467	0.480852
$783$	0.507737	0.0181450
$784$	−62.6812	−2.23861
$785$	−0.665277	−0.0237448
$786$	50.1529	1.78890
$787$	24.0098	0.855857	0.427928	−	0.903813i	$-0.359244\pi$
$787$	24.0098	0.855857	0.427928	+	0.903813i	$0.359244\pi$
$788$	−26.9779	−0.961049
$789$	22.2743	0.792987
$790$	1.61265	0.0573754
$791$	15.5724	0.553690
$792$	31.0028	1.10164
$793$	−28.3504	−1.00675
$794$	6.94691	0.246537
$795$	0.725698	0.0257379
$796$	102.787	3.64321
$797$	0.903988	0.0320209	0.0160105	−	0.999872i	$-0.494903\pi$
$797$	0.903988	0.0320209	0.0160105	+	0.999872i	$0.494903\pi$
$798$	−10.3980	−0.368084
$799$	−6.80567	−0.240767
$800$	93.6473	3.31093
$801$	−3.09440	−0.109335
$802$	96.6133	3.41153
$803$	1.76637	0.0623340
$804$	65.6765	2.31623
$805$	0.322115	0.0113530
$806$	9.62076	0.338877
$807$	39.6845	1.39696
$808$	79.0627	2.78142
$809$	32.2311	1.13318	0.566592	−	0.823998i	$-0.308261\pi$
$809$	32.2311	1.13318	0.566592	+	0.823998i	$0.308261\pi$
$810$	−0.733727	−0.0257805
$811$	−1.10231	−0.0387073	−0.0193536	−	0.999813i	$-0.506161\pi$
$811$	−1.10231	−0.0387073	−0.0193536	+	0.999813i	$0.506161\pi$
$812$	−0.742213	−0.0260466
$813$	13.7216	0.481238
$814$	−124.708	−4.37101
$815$	−0.265062	−0.00928471
$816$	20.1081	0.703923
$817$	−4.77618	−0.167097
$818$	20.7111	0.724147
$819$	4.61101	0.161122
$820$	0.301738	0.0105371
$821$	46.2631	1.61459	0.807297	−	0.590145i	$-0.200929\pi$
$821$	46.2631	1.61459	0.807297	+	0.590145i	$0.200929\pi$
$822$	−19.8144	−0.691108
$823$	−8.76719	−0.305605	−0.152803	−	0.988257i	$-0.548830\pi$
$823$	−8.76719	−0.305605	−0.152803	+	0.988257i	$0.548830\pi$
$824$	−74.2339	−2.58606
$825$	−41.8928	−1.45852
$826$	−49.6446	−1.72736
$827$	32.5174	1.13074	0.565370	−	0.824837i	$-0.308733\pi$
$827$	32.5174	1.13074	0.565370	+	0.824837i	$0.308733\pi$
$828$	17.3947	0.604507
$829$	−18.3693	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$829$	−18.3693	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$830$	1.66782	0.0578909
$831$	18.3976	0.638205
$832$	−110.821	−3.84203
$833$	−4.46917	−0.154847
$834$	−71.9192	−2.49036
$835$	0.0266578	0.000922530 0
$836$	46.9960	1.62539
$837$	−4.22936	−0.146188
$838$	−18.2281	−0.629680
$839$	−30.8867	−1.06633	−0.533164	−	0.846012i	$-0.678997\pi$
$839$	−30.8867	−1.06633	−0.533164	+	0.846012i	$0.678997\pi$
$840$	0.859121	0.0296425
$841$	−28.9918	−0.999716
$842$	−6.12572	−0.211106
$843$	33.4031	1.15046
$844$	−31.3144	−1.07789
$845$	−0.370892	−0.0127591
$846$	−12.1221	−0.416767
$847$	28.8306	0.990631
$848$	−157.467	−5.40744
$849$	6.24113	0.214195
$850$	13.0469	0.447506
$851$	−43.5972	−1.49449
$852$	112.103	3.84059
$853$	41.5195	1.42160	0.710799	−	0.703395i	$-0.248333\pi$
$853$	41.5195	1.42160	0.710799	+	0.703395i	$0.248333\pi$
$854$	−25.0854	−0.858404
$855$	0.0419724	0.00143543
$856$	156.983	5.36558
$857$	8.78797	0.300191	0.150096	−	0.988671i	$-0.452042\pi$
$857$	8.78797	0.300191	0.150096	+	0.988671i	$0.452042\pi$
$858$	106.587	3.63882
$859$	37.8338	1.29087	0.645436	−	0.763815i	$-0.276676\pi$
$859$	37.8338	1.29087	0.645436	+	0.763815i	$0.276676\pi$
$860$	0.633341	0.0215967
$861$	−3.31643	−0.113024
$862$	22.9655	0.782208
$863$	−45.3614	−1.54412	−0.772060	−	0.635550i	$-0.780773\pi$
$863$	−45.3614	−1.54412	−0.772060	+	0.635550i	$0.780773\pi$
$864$	104.743	3.56344
$865$	0.370499	0.0125974
$866$	−68.2243	−2.31836
$867$	−24.7024	−0.838937
$868$	6.18251	0.209848
$869$	80.1113	2.71759
$870$	−0.0153229	−0.000519494 0
$871$	−37.8767	−1.28340
$872$	−95.6360	−3.23864
$873$	2.82144	0.0954914
$874$	22.6220	0.765202
$875$	0.625181	0.0211350
$876$	2.64328	0.0893080
$877$	17.4359	0.588769	0.294384	−	0.955687i	$-0.404885\pi$
$877$	17.4359	0.588769	0.294384	+	0.955687i	$0.404885\pi$
$878$	66.6614	2.24971
$879$	−22.6829	−0.765074
$880$	−2.99768	−0.101052
$881$	6.55548	0.220860	0.110430	−	0.993884i	$-0.464777\pi$
$881$	6.55548	0.220860	0.110430	+	0.993884i	$0.464777\pi$
$882$	−7.96040	−0.268041
$883$	−7.12949	−0.239927	−0.119963	−	0.992778i	$-0.538278\pi$
$883$	−7.12949	−0.239927	−0.119963	+	0.992778i	$0.538278\pi$
$884$	−24.1083	−0.810850
$885$	−0.744350	−0.0250211
$886$	13.9463	0.468536
$887$	−1.59823	−0.0536633	−0.0268317	−	0.999640i	$-0.508542\pi$
$887$	−1.59823	−0.0536633	−0.0268317	+	0.999640i	$0.508542\pi$
$888$	−116.279	−3.90208
$889$	10.4416	0.350199
$890$	0.533640	0.0178877
$891$	−36.4493	−1.22110
$892$	105.786	3.54198
$893$	−11.4495	−0.383144
$894$	−29.8686	−0.998955
$895$	−0.0183133	−0.000612146 0
$896$	−40.3477	−1.34792
$897$	37.2622	1.24415
$898$	−15.9363	−0.531801
$899$	−0.0687068	−0.00229150
$900$	16.8776	0.562586
$901$	−11.2274	−0.374039
$902$	20.6391	0.687206
$903$	−6.96112	−0.231652
$904$	−90.3613	−3.00537
$905$	−0.694291	−0.0230790
$906$	37.8484	1.25743
$907$	−12.8341	−0.426148	−0.213074	−	0.977036i	$-0.568347\pi$
$907$	−12.8341	−0.426148	−0.213074	+	0.977036i	$0.568347\pi$
$908$	11.7111	0.388647
$909$	5.62964	0.186723
$910$	−0.795186	−0.0263601
$911$	−24.7741	−0.820804	−0.410402	−	0.911905i	$-0.634612\pi$
$911$	−24.7741	−0.820804	−0.410402	+	0.911905i	$0.634612\pi$
$912$	33.8289	1.12019
$913$	82.8522	2.74201
$914$	−46.7715	−1.54706
$915$	−0.376119	−0.0124341
$916$	0.657304	0.0217179
$917$	−18.5872	−0.613804
$918$	14.5928	0.481634
$919$	21.0705	0.695052	0.347526	−	0.937670i	$-0.387022\pi$
$919$	21.0705	0.695052	0.347526	+	0.937670i	$0.387022\pi$
$920$	−1.86912	−0.0616232
$921$	−3.60960	−0.118940
$922$	15.2098	0.500909
$923$	−64.6518	−2.12804
$924$	68.4952	2.25333
$925$	−42.3011	−1.39085
$926$	−47.6354	−1.56540
$927$	−5.28581	−0.173609
$928$	1.70157	0.0558568
$929$	−19.8483	−0.651201	−0.325600	−	0.945508i	$-0.605566\pi$
$929$	−19.8483	−0.651201	−0.325600	+	0.945508i	$0.605566\pi$
$930$	0.127637	0.00418538
$931$	−7.51871	−0.246416
$932$	16.6710	0.546076
$933$	−12.2179	−0.399996
$934$	42.2216	1.38153
$935$	−0.213734	−0.00698986
$936$	−26.7562	−0.874552
$937$	16.9074	0.552341	0.276170	−	0.961109i	$-0.410935\pi$
$937$	16.9074	0.552341	0.276170	+	0.961109i	$0.410935\pi$
$938$	−33.5146	−1.09429
$939$	−32.1942	−1.05062
$940$	1.51826	0.0495200
$941$	9.54384	0.311120	0.155560	−	0.987826i	$-0.450282\pi$
$941$	9.54384	0.311120	0.155560	+	0.987826i	$0.450282\pi$
$942$	−68.0963	−2.21870
$943$	7.21531	0.234963
$944$	161.514	5.25684
$945$	0.349570	0.0113715
$946$	43.3210	1.40849
$947$	−7.42824	−0.241385	−0.120693	−	0.992690i	$-0.538512\pi$
$947$	−7.42824	−0.241385	−0.120693	+	0.992690i	$0.538512\pi$
$948$	119.882	3.89358
$949$	−1.52442	−0.0494848
$950$	21.9495	0.712137
$951$	33.9663	1.10143
$952$	−13.2916	−0.430784
$953$	58.4301	1.89274	0.946369	−	0.323088i	$-0.104721\pi$
$953$	58.4301	1.89274	0.946369	+	0.323088i	$0.104721\pi$
$954$	−19.9980	−0.647460
$955$	0.195591	0.00632919
$956$	143.352	4.63633
$957$	−0.761193	−0.0246059
$958$	81.3828	2.62936
$959$	7.34345	0.237132
$960$	−1.47025	−0.0474520
$961$	−30.4277	−0.981538
$962$	107.626	3.47000
$963$	11.1780	0.360204
$964$	19.6550	0.633045
$965$	−0.916109	−0.0294906
$966$	32.9709	1.06082
$967$	−19.6536	−0.632017	−0.316009	−	0.948756i	$-0.602343\pi$
$967$	−19.6536	−0.632017	−0.316009	+	0.948756i	$0.602343\pi$
$968$	−167.294	−5.37704
$969$	2.41200	0.0774845
$970$	−0.486568	−0.0156228
$971$	3.77853	0.121259	0.0606294	−	0.998160i	$-0.480689\pi$
$971$	3.77853	0.121259	0.0606294	+	0.998160i	$0.480689\pi$
$972$	34.4512	1.10502
$973$	26.6540	0.854489
$974$	68.5772	2.19736
$975$	36.1545	1.15787
$976$	81.6131	2.61237
$977$	−36.5570	−1.16956	−0.584780	−	0.811192i	$-0.698819\pi$
$977$	−36.5570	−1.16956	−0.584780	+	0.811192i	$0.698819\pi$
$978$	−27.1311	−0.867557
$979$	26.5096	0.847250
$980$	0.997013	0.0318484
$981$	−6.80973	−0.217418
$982$	−42.5348	−1.35734
$983$	49.1302	1.56701	0.783505	−	0.621385i	$-0.213430\pi$
$983$	49.1302	1.56701	0.783505	+	0.621385i	$0.213430\pi$
$984$	19.2442	0.613481
$985$	0.206414	0.00657689
$986$	0.237063	0.00754962
$987$	−16.6873	−0.531163
$988$	−40.5587	−1.29034
$989$	15.1448	0.481576
$990$	−0.380699	−0.0120994
$991$	−47.6543	−1.51379	−0.756894	−	0.653538i	$-0.773284\pi$
$991$	−47.6543	−1.51379	−0.756894	+	0.653538i	$0.773284\pi$
$992$	−14.1738	−0.450019
$993$	19.7805	0.627716
$994$	−57.2062	−1.81447
$995$	−0.786448	−0.0249321
$996$	123.983	3.92856
$997$	41.5840	1.31698	0.658489	−	0.752591i	$-0.271196\pi$
$997$	41.5840	1.31698	0.658489	+	0.752591i	$0.271196\pi$
$998$	−25.7738	−0.815857
$999$	−47.3132	−1.49692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.11	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-2.70301 q^{2} +1.53742 q^{3} +5.30627 q^{4} -0.0405993 q^{5} -4.15566 q^{6} +1.54013 q^{7} -8.93687 q^{8} -0.636347 q^{9} +O(q^{10})\)	\(q-2.70301 q^{2} +1.53742 q^{3} +5.30627 q^{4} -0.0405993 q^{5} -4.15566 q^{6} +1.54013 q^{7} -8.93687 q^{8} -0.636347 q^{9} +0.109740 q^{10} +5.45156 q^{11} +8.15795 q^{12} -4.70483 q^{13} -4.16299 q^{14} -0.0624181 q^{15} +13.5439 q^{16} +0.965682 q^{17} +1.72005 q^{18} +1.62462 q^{19} -0.215431 q^{20} +2.36783 q^{21} -14.7356 q^{22} -5.15150 q^{23} -13.7397 q^{24} -4.99835 q^{25} +12.7172 q^{26} -5.59058 q^{27} +8.17235 q^{28} -0.0908200 q^{29} +0.168717 q^{30} +0.756515 q^{31} -18.7356 q^{32} +8.38133 q^{33} -2.61025 q^{34} -0.0625284 q^{35} -3.37663 q^{36} +8.46302 q^{37} -4.39136 q^{38} -7.23329 q^{39} +0.362831 q^{40} -1.40062 q^{41} -6.40026 q^{42} -2.93988 q^{43} +28.9275 q^{44} +0.0258353 q^{45} +13.9245 q^{46} -7.04753 q^{47} +20.8227 q^{48} -4.62799 q^{49} +13.5106 q^{50} +1.48466 q^{51} -24.9651 q^{52} -11.6264 q^{53} +15.1114 q^{54} -0.221330 q^{55} -13.7640 q^{56} +2.49771 q^{57} +0.245488 q^{58} +11.9252 q^{59} -0.331207 q^{60} +6.02580 q^{61} -2.04487 q^{62} -0.980059 q^{63} +23.5548 q^{64} +0.191013 q^{65} -22.6548 q^{66} +8.05061 q^{67} +5.12416 q^{68} -7.92000 q^{69} +0.169015 q^{70} +13.7416 q^{71} +5.68696 q^{72} +0.324012 q^{73} -22.8756 q^{74} -7.68455 q^{75} +8.62065 q^{76} +8.39613 q^{77} +19.5516 q^{78} +14.6951 q^{79} -0.549875 q^{80} -6.68602 q^{81} +3.78590 q^{82} +15.1979 q^{83} +12.5643 q^{84} -0.0392060 q^{85} +7.94652 q^{86} -0.139628 q^{87} -48.7199 q^{88} +4.86275 q^{89} -0.0698331 q^{90} -7.24606 q^{91} -27.3352 q^{92} +1.16308 q^{93} +19.0495 q^{94} -0.0659584 q^{95} -28.8045 q^{96} -4.43381 q^{97} +12.5095 q^{98} -3.46909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 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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