LMFDB - Embedded newform 8009.2.a.a.1.10

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:	$1$
Dimension:	$306$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-2.63380 q^{2} -2.43471 q^{3} +4.93693 q^{4} -0.366932 q^{5} +6.41255 q^{6} +4.18826 q^{7} -7.73529 q^{8} +2.92781 q^{9} +O(q^{10})$	\(q-2.63380 q^{2} -2.43471 q^{3} +4.93693 q^{4} -0.366932 q^{5} +6.41255 q^{6} +4.18826 q^{7} -7.73529 q^{8} +2.92781 q^{9} +0.966426 q^{10} +1.51562 q^{11} -12.0200 q^{12} -2.14727 q^{13} -11.0311 q^{14} +0.893372 q^{15} +10.4994 q^{16} -0.693877 q^{17} -7.71127 q^{18} -7.76088 q^{19} -1.81151 q^{20} -10.1972 q^{21} -3.99184 q^{22} -1.22620 q^{23} +18.8332 q^{24} -4.86536 q^{25} +5.65550 q^{26} +0.175772 q^{27} +20.6772 q^{28} +5.26445 q^{29} -2.35297 q^{30} -6.72409 q^{31} -12.1828 q^{32} -3.69008 q^{33} +1.82754 q^{34} -1.53681 q^{35} +14.4544 q^{36} +6.34590 q^{37} +20.4406 q^{38} +5.22798 q^{39} +2.83832 q^{40} +11.5105 q^{41} +26.8574 q^{42} -4.07457 q^{43} +7.48248 q^{44} -1.07430 q^{45} +3.22957 q^{46} +5.38403 q^{47} -25.5630 q^{48} +10.5416 q^{49} +12.8144 q^{50} +1.68939 q^{51} -10.6009 q^{52} +4.92769 q^{53} -0.462949 q^{54} -0.556127 q^{55} -32.3974 q^{56} +18.8955 q^{57} -13.8655 q^{58} -3.88189 q^{59} +4.41051 q^{60} -9.07259 q^{61} +17.7099 q^{62} +12.2624 q^{63} +11.0882 q^{64} +0.787902 q^{65} +9.71896 q^{66} +2.06269 q^{67} -3.42562 q^{68} +2.98544 q^{69} +4.04765 q^{70} -11.7190 q^{71} -22.6474 q^{72} +2.81340 q^{73} -16.7139 q^{74} +11.8457 q^{75} -38.3149 q^{76} +6.34780 q^{77} -13.7695 q^{78} +9.56514 q^{79} -3.85256 q^{80} -9.21137 q^{81} -30.3165 q^{82} +7.25190 q^{83} -50.3428 q^{84} +0.254605 q^{85} +10.7316 q^{86} -12.8174 q^{87} -11.7237 q^{88} +2.49236 q^{89} +2.82951 q^{90} -8.99335 q^{91} -6.05366 q^{92} +16.3712 q^{93} -14.1805 q^{94} +2.84771 q^{95} +29.6615 q^{96} -8.10881 q^{97} -27.7644 q^{98} +4.43743 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) $ 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * 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 $

$2$	−2.63380	−1.86238	−0.931191	−	0.364533i	$-0.881229\pi$
$2$	−2.63380	−1.86238	−0.931191	+	0.364533i	$0.881229\pi$
$3$	−2.43471	−1.40568	−0.702840	−	0.711348i	$-0.748085\pi$
$3$	−2.43471	−1.40568	−0.702840	+	0.711348i	$0.748085\pi$
$4$	4.93693	2.46846
$5$	−0.366932	−0.164097	−0.0820484	−	0.996628i	$-0.526146\pi$
$5$	−0.366932	−0.164097	−0.0820484	+	0.996628i	$0.526146\pi$
$6$	6.41255	2.61791
$7$	4.18826	1.58302	0.791508	−	0.611159i	$-0.209297\pi$
$7$	4.18826	1.58302	0.791508	+	0.611159i	$0.209297\pi$
$8$	−7.73529	−2.73484
$9$	2.92781	0.975935
$10$	0.966426	0.305611
$11$	1.51562	0.456975	0.228488	−	0.973547i	$-0.426622\pi$
$11$	1.51562	0.456975	0.228488	+	0.973547i	$0.426622\pi$
$12$	−12.0200	−3.46987
$13$	−2.14727	−0.595546	−0.297773	−	0.954637i	$-0.596244\pi$
$13$	−2.14727	−0.595546	−0.297773	+	0.954637i	$0.596244\pi$
$14$	−11.0311	−2.94818
$15$	0.893372	0.230668
$16$	10.4994	2.62485
$17$	−0.693877	−0.168290	−0.0841449	−	0.996454i	$-0.526816\pi$
$17$	−0.693877	−0.168290	−0.0841449	+	0.996454i	$0.526816\pi$
$18$	−7.71127	−1.81756
$19$	−7.76088	−1.78047	−0.890234	−	0.455503i	$-0.849459\pi$
$19$	−7.76088	−1.78047	−0.890234	+	0.455503i	$0.849459\pi$
$20$	−1.81151	−0.405067
$21$	−10.1972	−2.22521
$22$	−3.99184	−0.851062
$23$	−1.22620	−0.255680	−0.127840	−	0.991795i	$-0.540804\pi$
$23$	−1.22620	−0.255680	−0.127840	+	0.991795i	$0.540804\pi$
$24$	18.8332	3.84431
$25$	−4.86536	−0.973072
$26$	5.65550	1.10913
$27$	0.175772	0.0338274
$28$	20.6772	3.90762
$29$	5.26445	0.977584	0.488792	−	0.872400i	$-0.337438\pi$
$29$	5.26445	0.977584	0.488792	+	0.872400i	$0.337438\pi$
$30$	−2.35297	−0.429591
$31$	−6.72409	−1.20768	−0.603841	−	0.797105i	$-0.706364\pi$
$31$	−6.72409	−1.20768	−0.603841	+	0.797105i	$0.706364\pi$
$32$	−12.1828	−2.15363
$33$	−3.69008	−0.642361
$34$	1.82754	0.313420
$35$	−1.53681	−0.259768
$36$	14.4544	2.40906
$37$	6.34590	1.04326	0.521630	−	0.853172i	$-0.325324\pi$
$37$	6.34590	1.04326	0.521630	+	0.853172i	$0.325324\pi$
$38$	20.4406	3.31591
$39$	5.22798	0.837147
$40$	2.83832	0.448778
$41$	11.5105	1.79764	0.898821	−	0.438316i	$-0.144425\pi$
$41$	11.5105	1.79764	0.898821	+	0.438316i	$0.144425\pi$
$42$	26.8574	4.14419
$43$	−4.07457	−0.621367	−0.310683	−	0.950513i	$-0.600558\pi$
$43$	−4.07457	−0.621367	−0.310683	+	0.950513i	$0.600558\pi$
$44$	7.48248	1.12803
$45$	−1.07430	−0.160148
$46$	3.22957	0.476174
$47$	5.38403	0.785341	0.392671	−	0.919679i	$-0.371551\pi$
$47$	5.38403	0.785341	0.392671	+	0.919679i	$0.371551\pi$
$48$	−25.5630	−3.68970
$49$	10.5416	1.50594
$50$	12.8144	1.81223
$51$	1.68939	0.236562
$52$	−10.6009	−1.47008
$53$	4.92769	0.676871	0.338435	−	0.940990i	$-0.390102\pi$
$53$	4.92769	0.676871	0.338435	+	0.940990i	$0.390102\pi$
$54$	−0.462949	−0.0629994
$55$	−0.556127	−0.0749882
$56$	−32.3974	−4.32929
$57$	18.8955	2.50277
$58$	−13.8655	−1.82063
$59$	−3.88189	−0.505380	−0.252690	−	0.967547i	$-0.581315\pi$
$59$	−3.88189	−0.505380	−0.252690	+	0.967547i	$0.581315\pi$
$60$	4.41051	0.569394
$61$	−9.07259	−1.16163	−0.580813	−	0.814037i	$-0.697265\pi$
$61$	−9.07259	−1.16163	−0.580813	+	0.814037i	$0.697265\pi$
$62$	17.7099	2.24916
$63$	12.2624	1.54492
$64$	11.0882	1.38603
$65$	0.787902	0.0977273
$66$	9.71896	1.19632
$67$	2.06269	0.251997	0.125999	−	0.992030i	$-0.459787\pi$
$67$	2.06269	0.251997	0.125999	+	0.992030i	$0.459787\pi$
$68$	−3.42562	−0.415417
$69$	2.98544	0.359405
$70$	4.04765	0.483787
$71$	−11.7190	−1.39078	−0.695392	−	0.718630i	$-0.744769\pi$
$71$	−11.7190	−1.39078	−0.695392	+	0.718630i	$0.744769\pi$
$72$	−22.6474	−2.66903
$73$	2.81340	0.329284	0.164642	−	0.986353i	$-0.447353\pi$
$73$	2.81340	0.329284	0.164642	+	0.986353i	$0.447353\pi$
$74$	−16.7139	−1.94295
$75$	11.8457	1.36783
$76$	−38.3149	−4.39502
$77$	6.34780	0.723399
$78$	−13.7695	−1.55909
$79$	9.56514	1.07616	0.538081	−	0.842893i	$-0.319149\pi$
$79$	9.56514	1.07616	0.538081	+	0.842893i	$0.319149\pi$
$80$	−3.85256	−0.430729
$81$	−9.21137	−1.02349
$82$	−30.3165	−3.34789
$83$	7.25190	0.796000	0.398000	−	0.917385i	$-0.369704\pi$
$83$	7.25190	0.796000	0.398000	+	0.917385i	$0.369704\pi$
$84$	−50.3428	−5.49286
$85$	0.254605	0.0276158
$86$	10.7316	1.15722
$87$	−12.8174	−1.37417
$88$	−11.7237	−1.24975
$89$	2.49236	0.264189	0.132095	−	0.991237i	$-0.457830\pi$
$89$	2.49236	0.264189	0.132095	+	0.991237i	$0.457830\pi$
$90$	2.82951	0.298256
$91$	−8.99335	−0.942759
$92$	−6.05366	−0.631138
$93$	16.3712	1.69761
$94$	−14.1805	−1.46260
$95$	2.84771	0.292169
$96$	29.6615	3.02731
$97$	−8.10881	−0.823325	−0.411663	−	0.911336i	$-0.635052\pi$
$97$	−8.10881	−0.823325	−0.411663	+	0.911336i	$0.635052\pi$
$98$	−27.7644	−2.80463
$99$	4.43743	0.445978
$100$	−24.0199	−2.40199
$101$	6.89261	0.685841	0.342920	−	0.939364i	$-0.388584\pi$
$101$	6.89261	0.685841	0.342920	+	0.939364i	$0.388584\pi$
$102$	−4.44952	−0.440568
$103$	−13.1515	−1.29586	−0.647928	−	0.761701i	$-0.724364\pi$
$103$	−13.1515	−1.29586	−0.647928	+	0.761701i	$0.724364\pi$
$104$	16.6098	1.62872
$105$	3.74168	0.365150
$106$	−12.9786	−1.26059
$107$	0.437159	0.0422618	0.0211309	−	0.999777i	$-0.493273\pi$
$107$	0.437159	0.0422618	0.0211309	+	0.999777i	$0.493273\pi$
$108$	0.867774	0.0835016
$109$	19.1762	1.83675	0.918373	−	0.395716i	$-0.129503\pi$
$109$	19.1762	1.83675	0.918373	+	0.395716i	$0.129503\pi$
$110$	1.46473	0.139657
$111$	−15.4504	−1.46649
$112$	43.9742	4.15517
$113$	−8.45554	−0.795431	−0.397715	−	0.917509i	$-0.630197\pi$
$113$	−8.45554	−0.795431	−0.397715	+	0.917509i	$0.630197\pi$
$114$	−49.7670	−4.66111
$115$	0.449932	0.0419563
$116$	25.9902	2.41313
$117$	−6.28680	−0.581215
$118$	10.2241	0.941209
$119$	−2.90614	−0.266405
$120$	−6.91049	−0.630838
$121$	−8.70291	−0.791174
$122$	23.8954	2.16339
$123$	−28.0248	−2.52691
$124$	−33.1963	−2.98112
$125$	3.61991	0.323775
$126$	−32.2968	−2.87723
$127$	19.8373	1.76028	0.880140	−	0.474715i	$-0.157449\pi$
$127$	19.8373	1.76028	0.880140	+	0.474715i	$0.157449\pi$
$128$	−4.83873	−0.427688
$129$	9.92040	0.873442
$130$	−2.07518	−0.182005
$131$	−18.5509	−1.62080	−0.810400	−	0.585877i	$-0.800750\pi$
$131$	−18.5509	−1.62080	−0.810400	+	0.585877i	$0.800750\pi$
$132$	−18.2177	−1.58564
$133$	−32.5046	−2.81851
$134$	−5.43271	−0.469315
$135$	−0.0644963	−0.00555096
$136$	5.36734	0.460245
$137$	−7.77216	−0.664020	−0.332010	−	0.943276i	$-0.607727\pi$
$137$	−7.77216	−0.664020	−0.332010	+	0.943276i	$0.607727\pi$
$138$	−7.86307	−0.669349
$139$	16.1255	1.36775	0.683875	−	0.729600i	$-0.260294\pi$
$139$	16.1255	1.36775	0.683875	+	0.729600i	$0.260294\pi$
$140$	−7.58710	−0.641227
$141$	−13.1085	−1.10394
$142$	30.8654	2.59017
$143$	−3.25444	−0.272150
$144$	30.7402	2.56168
$145$	−1.93169	−0.160418
$146$	−7.40995	−0.613252
$147$	−25.6656	−2.11687
$148$	31.3293	2.57525
$149$	8.14557	0.667311	0.333656	−	0.942695i	$-0.391718\pi$
$149$	8.14557	0.667311	0.333656	+	0.942695i	$0.391718\pi$
$150$	−31.1994	−2.54742
$151$	−6.64156	−0.540483	−0.270241	−	0.962793i	$-0.587103\pi$
$151$	−6.64156	−0.540483	−0.270241	+	0.962793i	$0.587103\pi$
$152$	60.0327	4.86929
$153$	−2.03154	−0.164240
$154$	−16.7189	−1.34724
$155$	2.46728	0.198177
$156$	25.8102	2.06647
$157$	−15.0611	−1.20200	−0.601002	−	0.799247i	$-0.705232\pi$
$157$	−15.0611	−1.20200	−0.601002	+	0.799247i	$0.705232\pi$
$158$	−25.1927	−2.00422
$159$	−11.9975	−0.951463
$160$	4.47024	0.353404
$161$	−5.13565	−0.404746
$162$	24.2610	1.90612
$163$	−1.13384	−0.0888093	−0.0444046	−	0.999014i	$-0.514139\pi$
$163$	−1.13384	−0.0888093	−0.0444046	+	0.999014i	$0.514139\pi$
$164$	56.8266	4.43741
$165$	1.35401	0.105409
$166$	−19.1001	−1.48245
$167$	5.24352	0.405756	0.202878	−	0.979204i	$-0.434971\pi$
$167$	5.24352	0.405756	0.202878	+	0.979204i	$0.434971\pi$
$168$	78.8783	6.08560
$169$	−8.38922	−0.645324
$170$	−0.670581	−0.0514312
$171$	−22.7223	−1.73762
$172$	−20.1159	−1.53382
$173$	7.08127	0.538379	0.269190	−	0.963087i	$-0.413244\pi$
$173$	7.08127	0.538379	0.269190	+	0.963087i	$0.413244\pi$
$174$	33.7585	2.55923
$175$	−20.3774	−1.54039
$176$	15.9130	1.19949
$177$	9.45128	0.710402
$178$	−6.56438	−0.492021
$179$	12.2448	0.915217	0.457609	−	0.889154i	$-0.348706\pi$
$179$	12.2448	0.915217	0.457609	+	0.889154i	$0.348706\pi$
$180$	−5.30376	−0.395319
$181$	−2.27832	−0.169346	−0.0846731	−	0.996409i	$-0.526985\pi$
$181$	−2.27832	−0.169346	−0.0846731	+	0.996409i	$0.526985\pi$
$182$	23.6867	1.75578
$183$	22.0891	1.63287
$184$	9.48502	0.699245
$185$	−2.32851	−0.171196
$186$	−43.1185	−3.16160
$187$	−1.05165	−0.0769043
$188$	26.5805	1.93859
$189$	0.736180	0.0535492
$190$	−7.50032	−0.544130
$191$	15.9167	1.15169	0.575846	−	0.817558i	$-0.304673\pi$
$191$	15.9167	1.15169	0.575846	+	0.817558i	$0.304673\pi$
$192$	−26.9966	−1.94831
$193$	14.2626	1.02665	0.513323	−	0.858196i	$-0.328414\pi$
$193$	14.2626	1.02665	0.513323	+	0.858196i	$0.328414\pi$
$194$	21.3570	1.53335
$195$	−1.91831	−0.137373
$196$	52.0429	3.71735
$197$	−8.16829	−0.581966	−0.290983	−	0.956728i	$-0.593982\pi$
$197$	−8.16829	−0.581966	−0.290983	+	0.956728i	$0.593982\pi$
$198$	−11.6873	−0.830582
$199$	−22.1107	−1.56738	−0.783692	−	0.621150i	$-0.786666\pi$
$199$	−22.1107	−1.56738	−0.783692	+	0.621150i	$0.786666\pi$
$200$	37.6350	2.66120
$201$	−5.02204	−0.354227
$202$	−18.1538	−1.27730
$203$	22.0489	1.54753
$204$	8.34038	0.583944
$205$	−4.22357	−0.294987
$206$	34.6385	2.41338
$207$	−3.59008	−0.249528
$208$	−22.5451	−1.56322
$209$	−11.7625	−0.813630
$210$	−9.85484	−0.680049
$211$	8.15523	0.561429	0.280715	−	0.959791i	$-0.409429\pi$
$211$	8.15523	0.561429	0.280715	+	0.959791i	$0.409429\pi$
$212$	24.3277	1.67083
$213$	28.5322	1.95500
$214$	−1.15139	−0.0787075
$215$	1.49509	0.101964
$216$	−1.35965	−0.0925124
$217$	−28.1623	−1.91178
$218$	−50.5063	−3.42072
$219$	−6.84981	−0.462867
$220$	−2.74556	−0.185106
$221$	1.48994	0.100224
$222$	40.6934	2.73116
$223$	−9.89708	−0.662757	−0.331379	−	0.943498i	$-0.607514\pi$
$223$	−9.89708	−0.662757	−0.331379	+	0.943498i	$0.607514\pi$
$224$	−51.0247	−3.40923
$225$	−14.2448	−0.949655
$226$	22.2702	1.48139
$227$	−14.4547	−0.959390	−0.479695	−	0.877435i	$-0.659253\pi$
$227$	−14.4547	−0.959390	−0.479695	+	0.877435i	$0.659253\pi$
$228$	93.2856	6.17799
$229$	−19.6315	−1.29729	−0.648645	−	0.761091i	$-0.724664\pi$
$229$	−19.6315	−1.29729	−0.648645	+	0.761091i	$0.724664\pi$
$230$	−1.18503	−0.0781387
$231$	−15.4550	−1.01687
$232$	−40.7221	−2.67354
$233$	15.2231	0.997300	0.498650	−	0.866803i	$-0.333829\pi$
$233$	15.2231	0.997300	0.498650	+	0.866803i	$0.333829\pi$
$234$	16.5582	1.08244
$235$	−1.97557	−0.128872
$236$	−19.1646	−1.24751
$237$	−23.2883	−1.51274
$238$	7.65420	0.496148
$239$	−16.7393	−1.08277	−0.541387	−	0.840774i	$-0.682100\pi$
$239$	−16.7393	−1.08277	−0.541387	+	0.840774i	$0.682100\pi$
$240$	9.37986	0.605467
$241$	−27.3520	−1.76190	−0.880949	−	0.473211i	$-0.843095\pi$
$241$	−27.3520	−1.76190	−0.880949	+	0.473211i	$0.843095\pi$
$242$	22.9218	1.47347
$243$	21.8997	1.40487
$244$	−44.7907	−2.86743
$245$	−3.86803	−0.247119
$246$	73.8118	4.70607
$247$	16.6647	1.06035
$248$	52.0128	3.30281
$249$	−17.6563	−1.11892
$250$	−9.53414	−0.602992
$251$	−23.8160	−1.50325	−0.751626	−	0.659589i	$-0.770730\pi$
$251$	−23.8160	−1.50325	−0.751626	+	0.659589i	$0.770730\pi$
$252$	60.5387	3.81358
$253$	−1.85845	−0.116840
$254$	−52.2477	−3.27831
$255$	−0.619890	−0.0388190
$256$	−9.43221	−0.589513
$257$	−15.6029	−0.973282	−0.486641	−	0.873602i	$-0.661778\pi$
$257$	−15.6029	−0.973282	−0.486641	+	0.873602i	$0.661778\pi$
$258$	−26.1284	−1.62668
$259$	26.5783	1.65150
$260$	3.88982	0.241236
$261$	15.4133	0.954059
$262$	48.8594	3.01855
$263$	20.4281	1.25965	0.629825	−	0.776737i	$-0.283127\pi$
$263$	20.4281	1.25965	0.629825	+	0.776737i	$0.283127\pi$
$264$	28.5439	1.75675
$265$	−1.80813	−0.111072
$266$	85.6108	5.24914
$267$	−6.06816	−0.371365
$268$	10.1833	0.622046
$269$	9.99835	0.609610	0.304805	−	0.952415i	$-0.401409\pi$
$269$	9.99835	0.609610	0.304805	+	0.952415i	$0.401409\pi$
$270$	0.169871	0.0103380
$271$	−13.4124	−0.814743	−0.407372	−	0.913262i	$-0.633555\pi$
$271$	−13.4124	−0.814743	−0.407372	+	0.913262i	$0.633555\pi$
$272$	−7.28528	−0.441735
$273$	21.8962	1.32522
$274$	20.4703	1.23666
$275$	−7.37402	−0.444670
$276$	14.7389	0.887178
$277$	32.1576	1.93216	0.966081	−	0.258239i	$-0.0831423\pi$
$277$	32.1576	1.93216	0.966081	+	0.258239i	$0.0831423\pi$
$278$	−42.4715	−2.54727
$279$	−19.6868	−1.17862
$280$	11.8876	0.710423
$281$	−22.2302	−1.32615	−0.663073	−	0.748555i	$-0.730748\pi$
$281$	−22.2302	−1.32615	−0.663073	+	0.748555i	$0.730748\pi$
$282$	34.5253	2.05595
$283$	28.5823	1.69904	0.849520	−	0.527556i	$-0.176891\pi$
$283$	28.5823	1.69904	0.849520	+	0.527556i	$0.176891\pi$
$284$	−57.8556	−3.43310
$285$	−6.93335	−0.410696
$286$	8.57156	0.506847
$287$	48.2091	2.84569
$288$	−35.6688	−2.10180
$289$	−16.5185	−0.971679
$290$	5.08770	0.298760
$291$	19.7426	1.15733
$292$	13.8896	0.812825
$293$	22.1539	1.29424	0.647121	−	0.762387i	$-0.275973\pi$
$293$	22.1539	1.29424	0.647121	+	0.762387i	$0.275973\pi$
$294$	67.5982	3.94241
$295$	1.42439	0.0829312
$296$	−49.0874	−2.85315
$297$	0.266403	0.0154583
$298$	−21.4538	−1.24279
$299$	2.63299	0.152270
$300$	58.4815	3.37643
$301$	−17.0654	−0.983633
$302$	17.4926	1.00658
$303$	−16.7815	−0.964072
$304$	−81.4845	−4.67346
$305$	3.32902	0.190619
$306$	5.35067	0.305877
$307$	15.1694	0.865761	0.432880	−	0.901451i	$-0.357497\pi$
$307$	15.1694	0.865761	0.432880	+	0.901451i	$0.357497\pi$
$308$	31.3386	1.78568
$309$	32.0201	1.82156
$310$	−6.49833	−0.369080
$311$	2.90782	0.164888	0.0824438	−	0.996596i	$-0.473728\pi$
$311$	2.90782	0.164888	0.0824438	+	0.996596i	$0.473728\pi$
$312$	−40.4400	−2.28946
$313$	−16.3847	−0.926115	−0.463058	−	0.886328i	$-0.653248\pi$
$313$	−16.3847	−0.926115	−0.463058	+	0.886328i	$0.653248\pi$
$314$	39.6679	2.23859
$315$	−4.49947	−0.253516
$316$	47.2224	2.65647
$317$	24.5509	1.37891	0.689457	−	0.724327i	$-0.257849\pi$
$317$	24.5509	1.37891	0.689457	+	0.724327i	$0.257849\pi$
$318$	31.5991	1.77199
$319$	7.97889	0.446732
$320$	−4.06863	−0.227443
$321$	−1.06435	−0.0594065
$322$	13.5263	0.753791
$323$	5.38509	0.299635
$324$	−45.4759	−2.52644
$325$	10.4473	0.579510
$326$	2.98632	0.165397
$327$	−46.6884	−2.58188
$328$	−89.0373	−4.91626
$329$	22.5497	1.24321
$330$	−3.56619	−0.196312
$331$	−2.13417	−0.117305	−0.0586523	−	0.998278i	$-0.518680\pi$
$331$	−2.13417	−0.117305	−0.0586523	+	0.998278i	$0.518680\pi$
$332$	35.8021	1.96490
$333$	18.5796	1.01815
$334$	−13.8104	−0.755672
$335$	−0.756865	−0.0413519
$336$	−107.064	−5.84084
$337$	4.79965	0.261454	0.130727	−	0.991418i	$-0.458269\pi$
$337$	4.79965	0.261454	0.130727	+	0.991418i	$0.458269\pi$
$338$	22.0956	1.20184
$339$	20.5868	1.11812
$340$	1.25697	0.0681686
$341$	−10.1911	−0.551881
$342$	59.8462	3.23611
$343$	14.8330	0.800906
$344$	31.5180	1.69934
$345$	−1.09545	−0.0589772
$346$	−18.6507	−1.00267
$347$	−6.46649	−0.347139	−0.173570	−	0.984822i	$-0.555530\pi$
$347$	−6.46649	−0.347139	−0.173570	+	0.984822i	$0.555530\pi$
$348$	−63.2786	−3.39209
$349$	−2.15643	−0.115431	−0.0577156	−	0.998333i	$-0.518382\pi$
$349$	−2.15643	−0.115431	−0.0577156	+	0.998333i	$0.518382\pi$
$350$	53.6701	2.86879
$351$	−0.377431	−0.0201458
$352$	−18.4644	−0.984156
$353$	4.66487	0.248286	0.124143	−	0.992264i	$-0.460382\pi$
$353$	4.66487	0.248286	0.124143	+	0.992264i	$0.460382\pi$
$354$	−24.8928	−1.32304
$355$	4.30006	0.228223
$356$	12.3046	0.652141
$357$	7.07560	0.374481
$358$	−32.2503	−1.70448
$359$	−18.5255	−0.977739	−0.488870	−	0.872357i	$-0.662591\pi$
$359$	−18.5255	−0.977739	−0.488870	+	0.872357i	$0.662591\pi$
$360$	8.31006	0.437979
$361$	41.2313	2.17007
$362$	6.00065	0.315387
$363$	21.1890	1.11214
$364$	−44.3995	−2.32717
$365$	−1.03233	−0.0540344
$366$	−58.1784	−3.04104
$367$	11.7119	0.611358	0.305679	−	0.952135i	$-0.401117\pi$
$367$	11.7119	0.611358	0.305679	+	0.952135i	$0.401117\pi$
$368$	−12.8744	−0.671122
$369$	33.7006	1.75438
$370$	6.13285	0.318831
$371$	20.6385	1.07150
$372$	80.8234	4.19050
$373$	−16.5175	−0.855245	−0.427622	−	0.903957i	$-0.640649\pi$
$373$	−16.5175	−0.855245	−0.427622	+	0.903957i	$0.640649\pi$
$374$	2.76984	0.143225
$375$	−8.81343	−0.455124
$376$	−41.6470	−2.14778
$377$	−11.3042	−0.582197
$378$	−1.93895	−0.0997291
$379$	0.477602	0.0245328	0.0122664	−	0.999925i	$-0.496095\pi$
$379$	0.477602	0.0245328	0.0122664	+	0.999925i	$0.496095\pi$
$380$	14.0589	0.721209
$381$	−48.2981	−2.47439
$382$	−41.9215	−2.14489
$383$	24.5066	1.25223	0.626114	−	0.779731i	$-0.284644\pi$
$383$	24.5066	1.25223	0.626114	+	0.779731i	$0.284644\pi$
$384$	11.7809	0.601192
$385$	−2.32921	−0.118707
$386$	−37.5649	−1.91200
$387$	−11.9296	−0.606414
$388$	−40.0326	−2.03235
$389$	−31.9521	−1.62004	−0.810018	−	0.586405i	$-0.800543\pi$
$389$	−31.9521	−1.62004	−0.810018	+	0.586405i	$0.800543\pi$
$390$	5.05246	0.255841
$391$	0.850832	0.0430284
$392$	−81.5420	−4.11849
$393$	45.1660	2.27833
$394$	21.5137	1.08384
$395$	−3.50975	−0.176595
$396$	21.9073	1.10088
$397$	−13.0095	−0.652930	−0.326465	−	0.945209i	$-0.605857\pi$
$397$	−13.0095	−0.652930	−0.326465	+	0.945209i	$0.605857\pi$
$398$	58.2352	2.91907
$399$	79.1393	3.96192
$400$	−51.0833	−2.55417
$401$	22.9877	1.14795	0.573975	−	0.818873i	$-0.305401\pi$
$401$	22.9877	1.14795	0.573975	+	0.818873i	$0.305401\pi$
$402$	13.2271	0.659706
$403$	14.4384	0.719230
$404$	34.0283	1.69297
$405$	3.37994	0.167951
$406$	−58.0725	−2.88209
$407$	9.61795	0.476744
$408$	−13.0679	−0.646958
$409$	−32.4766	−1.60587	−0.802933	−	0.596069i	$-0.796729\pi$
$409$	−32.4766	−1.60587	−0.802933	+	0.596069i	$0.796729\pi$
$410$	11.1241	0.549379
$411$	18.9229	0.933400
$412$	−64.9280	−3.19877
$413$	−16.2584	−0.800023
$414$	9.45556	0.464715
$415$	−2.66095	−0.130621
$416$	26.1597	1.28259
$417$	−39.2610	−1.92262
$418$	30.9802	1.51529
$419$	10.3959	0.507872	0.253936	−	0.967221i	$-0.418275\pi$
$419$	10.3959	0.507872	0.253936	+	0.967221i	$0.418275\pi$
$420$	18.4724	0.901360
$421$	10.7208	0.522498	0.261249	−	0.965272i	$-0.415866\pi$
$421$	10.7208	0.522498	0.261249	+	0.965272i	$0.415866\pi$
$422$	−21.4793	−1.04559
$423$	15.7634	0.766442
$424$	−38.1171	−1.85113
$425$	3.37596	0.163758
$426$	−75.1484	−3.64095
$427$	−37.9984	−1.83887
$428$	2.15822	0.104322
$429$	7.92362	0.382556
$430$	−3.93777	−0.189896
$431$	−14.5319	−0.699975	−0.349988	−	0.936754i	$-0.613814\pi$
$431$	−14.5319	−0.699975	−0.349988	+	0.936754i	$0.613814\pi$
$432$	1.84550	0.0887917
$433$	33.9776	1.63286	0.816429	−	0.577446i	$-0.195950\pi$
$433$	33.9776	1.63286	0.816429	+	0.577446i	$0.195950\pi$
$434$	74.1739	3.56046
$435$	4.70311	0.225497
$436$	94.6715	4.53394
$437$	9.51639	0.455231
$438$	18.0411	0.862036
$439$	−30.1177	−1.43744	−0.718720	−	0.695299i	$-0.755272\pi$
$439$	−30.1177	−1.43744	−0.718720	+	0.695299i	$0.755272\pi$
$440$	4.30181	0.205081
$441$	30.8636	1.46970
$442$	−3.92422	−0.186656
$443$	22.4503	1.06664	0.533322	−	0.845912i	$-0.320943\pi$
$443$	22.4503	1.06664	0.533322	+	0.845912i	$0.320943\pi$
$444$	−76.2776	−3.61998
$445$	−0.914524	−0.0433526
$446$	26.0670	1.23431
$447$	−19.8321	−0.938026
$448$	46.4405	2.19411
$449$	8.51961	0.402065	0.201033	−	0.979585i	$-0.435570\pi$
$449$	8.51961	0.402065	0.201033	+	0.979585i	$0.435570\pi$
$450$	37.5181	1.76862
$451$	17.4455	0.821478
$452$	−41.7444	−1.96349
$453$	16.1703	0.759745
$454$	38.0708	1.78675
$455$	3.29994	0.154704
$456$	−146.162	−6.84467
$457$	−20.5433	−0.960974	−0.480487	−	0.877002i	$-0.659540\pi$
$457$	−20.5433	−0.960974	−0.480487	+	0.877002i	$0.659540\pi$
$458$	51.7057	2.41605
$459$	−0.121964	−0.00569280
$460$	2.22128	0.103568
$461$	38.9101	1.81222	0.906111	−	0.423040i	$-0.139037\pi$
$461$	38.9101	1.81222	0.906111	+	0.423040i	$0.139037\pi$
$462$	40.7056	1.89379
$463$	−30.2047	−1.40373	−0.701865	−	0.712310i	$-0.747649\pi$
$463$	−30.2047	−1.40373	−0.701865	+	0.712310i	$0.747649\pi$
$464$	55.2736	2.56601
$465$	−6.00711	−0.278573
$466$	−40.0947	−1.85735
$467$	−37.2684	−1.72458	−0.862288	−	0.506418i	$-0.830969\pi$
$467$	−37.2684	−1.72458	−0.862288	+	0.506418i	$0.830969\pi$
$468$	−31.0375	−1.43471
$469$	8.63907	0.398915
$470$	5.20326	0.240009
$471$	36.6693	1.68963
$472$	30.0276	1.38213
$473$	−6.17549	−0.283949
$474$	61.3369	2.81730
$475$	37.7595	1.73252
$476$	−14.3474	−0.657612
$477$	14.4273	0.660582
$478$	44.0880	2.01654
$479$	−8.90852	−0.407041	−0.203520	−	0.979071i	$-0.565238\pi$
$479$	−8.90852	−0.407041	−0.203520	+	0.979071i	$0.565238\pi$
$480$	−10.8837	−0.496772
$481$	−13.6264	−0.621310
$482$	72.0399	3.28133
$483$	12.5038	0.568943
$484$	−42.9656	−1.95298
$485$	2.97538	0.135105
$486$	−57.6795	−2.61640
$487$	−19.6079	−0.888518	−0.444259	−	0.895898i	$-0.646533\pi$
$487$	−19.6079	−0.888518	−0.444259	+	0.895898i	$0.646533\pi$
$488$	70.1792	3.17686
$489$	2.76057	0.124837
$490$	10.1876	0.460231
$491$	10.2614	0.463092	0.231546	−	0.972824i	$-0.425622\pi$
$491$	10.2614	0.463092	0.231546	+	0.972824i	$0.425622\pi$
$492$	−138.356	−6.23758
$493$	−3.65288	−0.164517
$494$	−43.8916	−1.97478
$495$	−1.62823	−0.0731836
$496$	−70.5988	−3.16998
$497$	−49.0821	−2.20163
$498$	46.5032	2.08386
$499$	33.7815	1.51227	0.756134	−	0.654417i	$-0.227086\pi$
$499$	33.7815	1.51227	0.756134	+	0.654417i	$0.227086\pi$
$500$	17.8712	0.799226
$501$	−12.7664	−0.570363
$502$	62.7267	2.79963
$503$	23.2034	1.03459	0.517293	−	0.855808i	$-0.326940\pi$
$503$	23.2034	1.03459	0.517293	+	0.855808i	$0.326940\pi$
$504$	−94.8534	−4.22511
$505$	−2.52912	−0.112544
$506$	4.89479	0.217600
$507$	20.4253	0.907119
$508$	97.9355	4.34518
$509$	22.9290	1.01631	0.508154	−	0.861266i	$-0.330328\pi$
$509$	22.9290	1.01631	0.508154	+	0.861266i	$0.330328\pi$
$510$	1.63267	0.0722958
$511$	11.7833	0.521261
$512$	34.5201	1.52559
$513$	−1.36415	−0.0602285
$514$	41.0950	1.81262
$515$	4.82570	0.212646
$516$	48.9763	2.15606
$517$	8.16012	0.358882
$518$	−70.0021	−3.07572
$519$	−17.2408	−0.756789
$520$	−6.09465	−0.267268
$521$	23.3280	1.02202	0.511010	−	0.859575i	$-0.329272\pi$
$521$	23.3280	1.02202	0.511010	+	0.859575i	$0.329272\pi$
$522$	−40.5956	−1.77682
$523$	−3.87352	−0.169377	−0.0846886	−	0.996407i	$-0.526990\pi$
$523$	−3.87352	−0.169377	−0.0846886	+	0.996407i	$0.526990\pi$
$524$	−91.5844	−4.00088
$525$	49.6131	2.16529
$526$	−53.8036	−2.34595
$527$	4.66569	0.203240
$528$	−38.7436	−1.68610
$529$	−21.4964	−0.934628
$530$	4.76225	0.206859
$531$	−11.3654	−0.493218
$532$	−160.473	−6.95738
$533$	−24.7162	−1.07058
$534$	15.9823	0.691624
$535$	−0.160407	−0.00693502
$536$	−15.9555	−0.689172
$537$	−29.8125	−1.28650
$538$	−26.3337	−1.13533
$539$	15.9770	0.688176
$540$	−0.318414	−0.0137023
$541$	−17.1038	−0.735351	−0.367676	−	0.929954i	$-0.619846\pi$
$541$	−17.1038	−0.735351	−0.367676	+	0.929954i	$0.619846\pi$
$542$	35.3256	1.51736
$543$	5.54705	0.238047
$544$	8.45334	0.362434
$545$	−7.03635	−0.301404
$546$	−57.6703	−2.46806
$547$	−8.54350	−0.365294	−0.182647	−	0.983179i	$-0.558466\pi$
$547$	−8.54350	−0.365294	−0.182647	+	0.983179i	$0.558466\pi$
$548$	−38.3706	−1.63911
$549$	−26.5628	−1.13367
$550$	19.4217	0.828145
$551$	−40.8568	−1.74056
$552$	−23.0933	−0.982914
$553$	40.0613	1.70358
$554$	−84.6968	−3.59842
$555$	5.66925	0.240646
$556$	79.6105	3.37624
$557$	−17.1112	−0.725024	−0.362512	−	0.931979i	$-0.618081\pi$
$557$	−17.1112	−0.725024	−0.362512	+	0.931979i	$0.618081\pi$
$558$	51.8512	2.19504
$559$	8.74922	0.370053
$560$	−16.1355	−0.681851
$561$	2.56046	0.108103
$562$	58.5501	2.46979
$563$	−29.4521	−1.24126	−0.620628	−	0.784105i	$-0.713122\pi$
$563$	−29.4521	−1.24126	−0.620628	+	0.784105i	$0.713122\pi$
$564$	−64.7159	−2.72503
$565$	3.10261	0.130528
$566$	−75.2802	−3.16426
$567$	−38.5797	−1.62019
$568$	90.6496	3.80357
$569$	−36.9515	−1.54909	−0.774543	−	0.632521i	$-0.782020\pi$
$569$	−36.9515	−1.54909	−0.774543	+	0.632521i	$0.782020\pi$
$570$	18.2611	0.764873
$571$	−28.5034	−1.19283	−0.596414	−	0.802677i	$-0.703409\pi$
$571$	−28.5034	−1.19283	−0.596414	+	0.802677i	$0.703409\pi$
$572$	−16.0669	−0.671792
$573$	−38.7525	−1.61891
$574$	−126.973	−5.29977
$575$	5.96591	0.248796
$576$	32.4642	1.35268
$577$	12.8318	0.534195	0.267097	−	0.963670i	$-0.413935\pi$
$577$	12.8318	0.534195	0.267097	+	0.963670i	$0.413935\pi$
$578$	43.5066	1.80964
$579$	−34.7253	−1.44313
$580$	−9.53663	−0.395987
$581$	30.3729	1.26008
$582$	−51.9981	−2.15539
$583$	7.46849	0.309313
$584$	−21.7625	−0.900538
$585$	2.30683	0.0953755
$586$	−58.3490	−2.41037
$587$	−34.7573	−1.43459	−0.717294	−	0.696770i	$-0.754620\pi$
$587$	−34.7573	−1.43459	−0.717294	+	0.696770i	$0.754620\pi$
$588$	−126.709	−5.22540
$589$	52.1848	2.15024
$590$	−3.75156	−0.154449
$591$	19.8874	0.818058
$592$	66.6281	2.73840
$593$	−42.2923	−1.73674	−0.868368	−	0.495921i	$-0.834831\pi$
$593$	−42.2923	−1.73674	−0.868368	+	0.495921i	$0.834831\pi$
$594$	−0.701653	−0.0287892
$595$	1.06635	0.0437163
$596$	40.2141	1.64723
$597$	53.8330	2.20324
$598$	−6.93477	−0.283584
$599$	−20.2942	−0.829200	−0.414600	−	0.910004i	$-0.636078\pi$
$599$	−20.2942	−0.829200	−0.414600	+	0.910004i	$0.636078\pi$
$600$	−91.6302	−3.74079
$601$	−24.4362	−0.996771	−0.498386	−	0.866955i	$-0.666074\pi$
$601$	−24.4362	−0.996771	−0.498386	+	0.866955i	$0.666074\pi$
$602$	44.9469	1.83190
$603$	6.03914	0.245933
$604$	−32.7889	−1.33416
$605$	3.19337	0.129829
$606$	44.1992	1.79547
$607$	34.4256	1.39729	0.698646	−	0.715467i	$-0.253786\pi$
$607$	34.4256	1.39729	0.698646	+	0.715467i	$0.253786\pi$
$608$	94.5490	3.83447
$609$	−53.6827	−2.17533
$610$	−8.76799	−0.355006
$611$	−11.5610	−0.467707
$612$	−10.0295	−0.405420
$613$	5.23964	0.211627	0.105814	−	0.994386i	$-0.466255\pi$
$613$	5.23964	0.211627	0.105814	+	0.994386i	$0.466255\pi$
$614$	−39.9531	−1.61238
$615$	10.2832	0.414658
$616$	−49.1021	−1.97838
$617$	20.9135	0.841944	0.420972	−	0.907074i	$-0.361689\pi$
$617$	20.9135	0.841944	0.420972	+	0.907074i	$0.361689\pi$
$618$	−84.3346	−3.39244
$619$	−2.43159	−0.0977337	−0.0488669	−	0.998805i	$-0.515561\pi$
$619$	−2.43159	−0.0977337	−0.0488669	+	0.998805i	$0.515561\pi$
$620$	12.1808	0.489192
$621$	−0.215532	−0.00864899
$622$	−7.65864	−0.307084
$623$	10.4386	0.418215
$624$	54.8907	2.19739
$625$	22.9985	0.919942
$626$	43.1540	1.72478
$627$	28.6383	1.14370
$628$	−74.3554	−2.96710
$629$	−4.40327	−0.175570
$630$	11.8507	0.472144
$631$	−5.24948	−0.208978	−0.104489	−	0.994526i	$-0.533321\pi$
$631$	−5.24948	−0.208978	−0.104489	+	0.994526i	$0.533321\pi$
$632$	−73.9891	−2.94313
$633$	−19.8556	−0.789189
$634$	−64.6622	−2.56806
$635$	−7.27895	−0.288856
$636$	−59.2308	−2.34865
$637$	−22.6356	−0.896855
$638$	−21.0148	−0.831985
$639$	−34.3108	−1.35732
$640$	1.77548	0.0701822
$641$	−19.4943	−0.769978	−0.384989	−	0.922921i	$-0.625795\pi$
$641$	−19.4943	−0.769978	−0.384989	+	0.922921i	$0.625795\pi$
$642$	2.80330	0.110638
$643$	−17.5692	−0.692862	−0.346431	−	0.938076i	$-0.612606\pi$
$643$	−17.5692	−0.692862	−0.346431	+	0.938076i	$0.612606\pi$
$644$	−25.3543	−0.999101
$645$	−3.64011	−0.143329
$646$	−14.1833	−0.558034
$647$	−32.2286	−1.26704	−0.633519	−	0.773727i	$-0.718390\pi$
$647$	−32.2286	−1.26704	−0.633519	+	0.773727i	$0.718390\pi$
$648$	71.2526	2.79907
$649$	−5.88346	−0.230946
$650$	−27.5160	−1.07927
$651$	68.5669	2.68735
$652$	−5.59769	−0.219222
$653$	30.5058	1.19379	0.596893	−	0.802321i	$-0.296402\pi$
$653$	30.5058	1.19379	0.596893	+	0.802321i	$0.296402\pi$
$654$	122.968	4.80844
$655$	6.80691	0.265968
$656$	120.854	4.71854
$657$	8.23709	0.321360
$658$	−59.3916	−2.31532
$659$	−5.36742	−0.209085	−0.104542	−	0.994520i	$-0.533338\pi$
$659$	−5.36742	−0.209085	−0.104542	+	0.994520i	$0.533338\pi$
$660$	6.68464	0.260199
$661$	39.2253	1.52569	0.762843	−	0.646584i	$-0.223803\pi$
$661$	39.2253	1.52569	0.762843	+	0.646584i	$0.223803\pi$
$662$	5.62099	0.218466
$663$	−3.62758	−0.140883
$664$	−56.0956	−2.17693
$665$	11.9270	0.462508
$666$	−48.9350	−1.89619
$667$	−6.45527	−0.249949
$668$	25.8869	1.00159
$669$	24.0965	0.931625
$670$	1.99343	0.0770131
$671$	−13.7506	−0.530835
$672$	124.230	4.79228
$673$	43.4843	1.67620	0.838098	−	0.545519i	$-0.183668\pi$
$673$	43.4843	1.67620	0.838098	+	0.545519i	$0.183668\pi$
$674$	−12.6413	−0.486926
$675$	−0.855195	−0.0329165
$676$	−41.4170	−1.59296
$677$	6.36165	0.244498	0.122249	−	0.992499i	$-0.460989\pi$
$677$	6.36165	0.244498	0.122249	+	0.992499i	$0.460989\pi$
$678$	−54.2216	−2.08237
$679$	−33.9619	−1.30334
$680$	−1.96945	−0.0755248
$681$	35.1929	1.34859
$682$	26.8414	1.02781
$683$	−15.0312	−0.575153	−0.287577	−	0.957758i	$-0.592850\pi$
$683$	−15.0312	−0.575153	−0.287577	+	0.957758i	$0.592850\pi$
$684$	−112.179	−4.28926
$685$	2.85185	0.108964
$686$	−39.0672	−1.49159
$687$	47.7971	1.82357
$688$	−42.7806	−1.63099
$689$	−10.5811	−0.403108
$690$	2.88521	0.109838
$691$	−28.4294	−1.08150	−0.540752	−	0.841182i	$-0.681860\pi$
$691$	−28.4294	−1.08150	−0.540752	+	0.841182i	$0.681860\pi$
$692$	34.9597	1.32897
$693$	18.5851	0.705991
$694$	17.0315	0.646506
$695$	−5.91696	−0.224443
$696$	99.1464	3.75813
$697$	−7.98688	−0.302525
$698$	5.67962	0.214977
$699$	−37.0639	−1.40188
$700$	−100.602	−3.80239
$701$	1.50485	0.0568374	0.0284187	−	0.999596i	$-0.490953\pi$
$701$	1.50485	0.0568374	0.0284187	+	0.999596i	$0.490953\pi$
$702$	0.994079	0.0375191
$703$	−49.2498	−1.85749
$704$	16.8055	0.633382
$705$	4.80994	0.181153
$706$	−12.2864	−0.462403
$707$	28.8681	1.08570
$708$	46.6603	1.75360
$709$	−50.1581	−1.88373	−0.941864	−	0.335993i	$-0.890928\pi$
$709$	−50.1581	−1.88373	−0.941864	+	0.335993i	$0.890928\pi$
$710$	−11.3255	−0.425039
$711$	28.0049	1.05026
$712$	−19.2791	−0.722515
$713$	8.24508	0.308781
$714$	−18.6358	−0.697425
$715$	1.19416	0.0446590
$716$	60.4515	2.25918
$717$	40.7552	1.52203
$718$	48.7926	1.82092
$719$	−12.4235	−0.463319	−0.231660	−	0.972797i	$-0.574416\pi$
$719$	−12.4235	−0.463319	−0.231660	+	0.972797i	$0.574416\pi$
$720$	−11.2795	−0.420364
$721$	−55.0820	−2.05136
$722$	−108.595	−4.04149
$723$	66.5942	2.47666
$724$	−11.2479	−0.418025
$725$	−25.6135	−0.951260
$726$	−55.8078	−2.07122
$727$	22.9740	0.852058	0.426029	−	0.904709i	$-0.359912\pi$
$727$	22.9740	0.852058	0.426029	+	0.904709i	$0.359912\pi$
$728$	69.5662	2.57829
$729$	−25.6852	−0.951305
$730$	2.71895	0.100633
$731$	2.82725	0.104570
$732$	109.052	4.03069
$733$	−24.0478	−0.888227	−0.444113	−	0.895971i	$-0.646481\pi$
$733$	−24.0478	−0.888227	−0.444113	+	0.895971i	$0.646481\pi$
$734$	−30.8469	−1.13858
$735$	9.41753	0.347371
$736$	14.9385	0.550641
$737$	3.12624	0.115157
$738$	−88.7607	−3.26733
$739$	39.0113	1.43506	0.717528	−	0.696530i	$-0.245274\pi$
$739$	39.0113	1.43506	0.717528	+	0.696530i	$0.245274\pi$
$740$	−11.4957	−0.422590
$741$	−40.5738	−1.49051
$742$	−54.3577	−1.99553
$743$	−14.3236	−0.525482	−0.262741	−	0.964866i	$-0.584626\pi$
$743$	−14.3236	−0.525482	−0.262741	+	0.964866i	$0.584626\pi$
$744$	−126.636	−4.64270
$745$	−2.98887	−0.109504
$746$	43.5039	1.59279
$747$	21.2322	0.776844
$748$	−5.19192	−0.189835
$749$	1.83094	0.0669010
$750$	23.2129	0.847614
$751$	−0.704240	−0.0256981	−0.0128490	−	0.999917i	$-0.504090\pi$
$751$	−0.704240	−0.0256981	−0.0128490	+	0.999917i	$0.504090\pi$
$752$	56.5290	2.06140
$753$	57.9850	2.11309
$754$	29.7731	1.08427
$755$	2.43700	0.0886915
$756$	3.63447	0.132184
$757$	16.6201	0.604067	0.302033	−	0.953297i	$-0.402335\pi$
$757$	16.6201	0.604067	0.302033	+	0.953297i	$0.402335\pi$
$758$	−1.25791	−0.0456894
$759$	4.52478	0.164239
$760$	−22.0279	−0.799035
$761$	0.596087	0.0216082	0.0108041	−	0.999942i	$-0.496561\pi$
$761$	0.596087	0.0216082	0.0108041	+	0.999942i	$0.496561\pi$
$762$	127.208	4.60825
$763$	80.3150	2.90760
$764$	78.5796	2.84291
$765$	0.745435	0.0269513
$766$	−64.5456	−2.33213
$767$	8.33549	0.300977
$768$	22.9647	0.828667
$769$	12.4077	0.447433	0.223716	−	0.974654i	$-0.428181\pi$
$769$	12.4077	0.447433	0.223716	+	0.974654i	$0.428181\pi$
$770$	6.13468	0.221079
$771$	37.9885	1.36812
$772$	70.4135	2.53424
$773$	7.24574	0.260611	0.130306	−	0.991474i	$-0.458404\pi$
$773$	7.24574	0.260611	0.130306	+	0.991474i	$0.458404\pi$
$774$	31.4201	1.12937
$775$	32.7151	1.17516
$776$	62.7240	2.25166
$777$	−64.7105	−2.32147
$778$	84.1556	3.01712
$779$	−89.3318	−3.20064
$780$	−9.47057	−0.339101
$781$	−17.7614	−0.635554
$782$	−2.24092	−0.0801353
$783$	0.925344	0.0330691
$784$	110.680	3.95286
$785$	5.52638	0.197245
$786$	−118.959	−4.24311
$787$	0.866570	0.0308899	0.0154449	−	0.999881i	$-0.495084\pi$
$787$	0.866570	0.0308899	0.0154449	+	0.999881i	$0.495084\pi$
$788$	−40.3262	−1.43656
$789$	−49.7364	−1.77066
$790$	9.24400	0.328887
$791$	−35.4141	−1.25918
$792$	−34.3248	−1.21968
$793$	19.4813	0.691803
$794$	34.2646	1.21600
$795$	4.40226	0.156132
$796$	−109.159	−3.86903
$797$	13.6472	0.483408	0.241704	−	0.970350i	$-0.422294\pi$
$797$	13.6472	0.483408	0.241704	+	0.970350i	$0.422294\pi$
$798$	−208.437	−7.37860
$799$	−3.73585	−0.132165
$800$	59.2736	2.09564
$801$	7.29713	0.257832
$802$	−60.5451	−2.13792
$803$	4.26404	0.150475
$804$	−24.7934	−0.874397
$805$	1.88443	0.0664175
$806$	−38.0281	−1.33948
$807$	−24.3431	−0.856917
$808$	−53.3164	−1.87566
$809$	−27.0801	−0.952086	−0.476043	−	0.879422i	$-0.657929\pi$
$809$	−27.0801	−0.952086	−0.476043	+	0.879422i	$0.657929\pi$
$810$	−8.90211	−0.312788
$811$	22.2409	0.780983	0.390492	−	0.920606i	$-0.372305\pi$
$811$	22.2409	0.780983	0.390492	+	0.920606i	$0.372305\pi$
$812$	108.854	3.82002
$813$	32.6552	1.14527
$814$	−25.3318	−0.887879
$815$	0.416042	0.0145733
$816$	17.7375	0.620938
$817$	31.6223	1.10632
$818$	85.5371	2.99074
$819$	−26.3308	−0.920072
$820$	−20.8515	−0.728165
$821$	−5.86933	−0.204841	−0.102420	−	0.994741i	$-0.532659\pi$
$821$	−5.86933	−0.204841	−0.102420	+	0.994741i	$0.532659\pi$
$822$	−49.8393	−1.73835
$823$	−36.5845	−1.27525	−0.637627	−	0.770345i	$-0.720084\pi$
$823$	−36.5845	−1.27525	−0.637627	+	0.770345i	$0.720084\pi$
$824$	101.731	3.54396
$825$	17.9536	0.625064
$826$	42.8214	1.48995
$827$	−19.7208	−0.685761	−0.342880	−	0.939379i	$-0.611403\pi$
$827$	−19.7208	−0.685761	−0.342880	+	0.939379i	$0.611403\pi$
$828$	−17.7239	−0.615950
$829$	−40.0209	−1.38998	−0.694991	−	0.719018i	$-0.744592\pi$
$829$	−40.0209	−1.38998	−0.694991	+	0.719018i	$0.744592\pi$
$830$	7.00843	0.243266
$831$	−78.2943	−2.71600
$832$	−23.8095	−0.825446
$833$	−7.31454	−0.253434
$834$	103.406	3.58065
$835$	−1.92401	−0.0665832
$836$	−58.0707	−2.00842
$837$	−1.18191	−0.0408527
$838$	−27.3807	−0.945851
$839$	−7.55150	−0.260707	−0.130353	−	0.991468i	$-0.541611\pi$
$839$	−7.55150	−0.260707	−0.130353	+	0.991468i	$0.541611\pi$
$840$	−28.9430	−0.998627
$841$	−1.28554	−0.0443290
$842$	−28.2364	−0.973089
$843$	54.1242	1.86414
$844$	40.2618	1.38587
$845$	3.07827	0.105896
$846$	−41.5177	−1.42741
$847$	−36.4501	−1.25244
$848$	51.7378	1.77668
$849$	−69.5896	−2.38831
$850$	−8.89162	−0.304980
$851$	−7.78135	−0.266741
$852$	140.862	4.82584
$853$	5.91294	0.202455	0.101228	−	0.994863i	$-0.467723\pi$
$853$	5.91294	0.202455	0.101228	+	0.994863i	$0.467723\pi$
$854$	100.080	3.42468
$855$	8.33755	0.285138
$856$	−3.38155	−0.115579
$857$	−30.0474	−1.02640	−0.513199	−	0.858269i	$-0.671540\pi$
$857$	−30.0474	−1.02640	−0.513199	+	0.858269i	$0.671540\pi$
$858$	−20.8693	−0.712465
$859$	−27.6662	−0.943960	−0.471980	−	0.881609i	$-0.656461\pi$
$859$	−27.6662	−0.943960	−0.471980	+	0.881609i	$0.656461\pi$
$860$	7.38115	0.251695
$861$	−117.375	−4.00013
$862$	38.2741	1.30362
$863$	5.30282	0.180510	0.0902550	−	0.995919i	$-0.471232\pi$
$863$	5.30282	0.180510	0.0902550	+	0.995919i	$0.471232\pi$
$864$	−2.14139	−0.0728516
$865$	−2.59834	−0.0883463
$866$	−89.4903	−3.04100
$867$	40.2178	1.36587
$868$	−139.035	−4.71915
$869$	14.4971	0.491779
$870$	−12.3871	−0.419961
$871$	−4.42915	−0.150076
$872$	−148.333	−5.02320
$873$	−23.7410	−0.803512
$874$	−25.0643	−0.847813
$875$	15.1612	0.512541
$876$	−33.8170	−1.14257
$877$	−41.9103	−1.41521	−0.707604	−	0.706609i	$-0.750224\pi$
$877$	−41.9103	−1.41521	−0.707604	+	0.706609i	$0.750224\pi$
$878$	79.3242	2.67706
$879$	−53.9382	−1.81929
$880$	−5.83900	−0.196833
$881$	−45.2864	−1.52574	−0.762868	−	0.646554i	$-0.776209\pi$
$881$	−45.2864	−1.52574	−0.762868	+	0.646554i	$0.776209\pi$
$882$	−81.2888	−2.73714
$883$	23.0636	0.776152	0.388076	−	0.921627i	$-0.373140\pi$
$883$	23.0636	0.776152	0.388076	+	0.921627i	$0.373140\pi$
$884$	7.35574	0.247400
$885$	−3.46797	−0.116575
$886$	−59.1296	−1.98650
$887$	46.4688	1.56027	0.780135	−	0.625611i	$-0.215150\pi$
$887$	46.4688	1.56027	0.780135	+	0.625611i	$0.215150\pi$
$888$	119.514	4.01061
$889$	83.0840	2.78655
$890$	2.40868	0.0807391
$891$	−13.9609	−0.467708
$892$	−48.8612	−1.63599
$893$	−41.7848	−1.39827
$894$	52.2339	1.74696
$895$	−4.49299	−0.150184
$896$	−20.2659	−0.677036
$897$	−6.41056	−0.214042
$898$	−22.4390	−0.748799
$899$	−35.3986	−1.18061
$900$	−70.3257	−2.34419
$901$	−3.41921	−0.113910
$902$	−45.9481	−1.52991
$903$	41.5493	1.38267
$904$	65.4061	2.17537
$905$	0.835988	0.0277892
$906$	−42.5893	−1.41494
$907$	−13.1034	−0.435091	−0.217546	−	0.976050i	$-0.569805\pi$
$907$	−13.1034	−0.435091	−0.217546	+	0.976050i	$0.569805\pi$
$908$	−71.3616	−2.36822
$909$	20.1802	0.669336
$910$	−8.69141	−0.288117
$911$	−17.5401	−0.581129	−0.290565	−	0.956855i	$-0.593843\pi$
$911$	−17.5401	−0.581129	−0.290565	+	0.956855i	$0.593843\pi$
$912$	198.391	6.56939
$913$	10.9911	0.363752
$914$	54.1070	1.78970
$915$	−8.10520	−0.267950
$916$	−96.9195	−3.20231
$917$	−77.6961	−2.56575
$918$	0.321230	0.0106022
$919$	−2.92987	−0.0966474	−0.0483237	−	0.998832i	$-0.515388\pi$
$919$	−2.92987	−0.0966474	−0.0483237	+	0.998832i	$0.515388\pi$
$920$	−3.48035	−0.114744
$921$	−36.9329	−1.21698
$922$	−102.481	−3.37505
$923$	25.1638	0.828277
$924$	−76.3004	−2.51010
$925$	−30.8751	−1.01517
$926$	79.5532	2.61428
$927$	−38.5051	−1.26467
$928$	−64.1356	−2.10535
$929$	41.5737	1.36399	0.681994	−	0.731358i	$-0.261113\pi$
$929$	41.5737	1.36399	0.681994	+	0.731358i	$0.261113\pi$
$930$	15.8215	0.518809
$931$	−81.8118	−2.68127
$932$	75.1554	2.46180
$933$	−7.07971	−0.231779
$934$	98.1577	3.21182
$935$	0.385884	0.0126197
$936$	48.6302	1.58953
$937$	29.3946	0.960281	0.480140	−	0.877192i	$-0.340586\pi$
$937$	29.3946	0.960281	0.480140	+	0.877192i	$0.340586\pi$
$938$	−22.7536	−0.742933
$939$	39.8919	1.30182
$940$	−9.75324	−0.318116
$941$	45.5784	1.48581	0.742906	−	0.669396i	$-0.233447\pi$
$941$	45.5784	1.48581	0.742906	+	0.669396i	$0.233447\pi$
$942$	−96.5798	−3.14674
$943$	−14.1142	−0.459622
$944$	−40.7575	−1.32654
$945$	−0.270128	−0.00878726
$946$	16.2650	0.528822
$947$	−8.32490	−0.270523	−0.135261	−	0.990810i	$-0.543187\pi$
$947$	−8.32490	−0.270523	−0.135261	+	0.990810i	$0.543187\pi$
$948$	−114.973	−3.73414
$949$	−6.04114	−0.196104
$950$	−99.4511	−3.22662
$951$	−59.7742	−1.93831
$952$	22.4798	0.728576
$953$	−48.0409	−1.55620	−0.778099	−	0.628142i	$-0.783816\pi$
$953$	−48.0409	−1.55620	−0.778099	+	0.628142i	$0.783816\pi$
$954$	−37.9988	−1.23026
$955$	−5.84034	−0.188989
$956$	−82.6405	−2.67279
$957$	−19.4263	−0.627962
$958$	23.4633	0.758065
$959$	−32.5519	−1.05115
$960$	9.90592	0.319712
$961$	14.2133	0.458495
$962$	35.8892	1.15712
$963$	1.27992	0.0412447
$964$	−135.035	−4.34918
$965$	−5.23340	−0.168469
$966$	−32.9326	−1.05959
$967$	−26.3507	−0.847381	−0.423690	−	0.905807i	$-0.639266\pi$
$967$	−26.3507	−0.847381	−0.423690	+	0.905807i	$0.639266\pi$
$968$	67.3195	2.16373
$969$	−13.1111	−0.421190
$970$	−7.83657	−0.251617
$971$	60.4289	1.93925	0.969627	−	0.244587i	$-0.0786524\pi$
$971$	60.4289	1.93925	0.969627	+	0.244587i	$0.0786524\pi$
$972$	108.117	3.46786
$973$	67.5380	2.16517
$974$	51.6433	1.65476
$975$	−25.4360	−0.814605
$976$	−95.2567	−3.04909
$977$	−15.0679	−0.482064	−0.241032	−	0.970517i	$-0.577486\pi$
$977$	−15.0679	−0.482064	−0.241032	+	0.970517i	$0.577486\pi$
$978$	−7.27081	−0.232495
$979$	3.77745	0.120728
$980$	−19.0962	−0.610005
$981$	56.1442	1.79255
$982$	−27.0266	−0.862454
$983$	−1.29141	−0.0411895	−0.0205948	−	0.999788i	$-0.506556\pi$
$983$	−1.29141	−0.0411895	−0.0205948	+	0.999788i	$0.506556\pi$
$984$	216.780	6.91069
$985$	2.99720	0.0954988
$986$	9.62097	0.306394
$987$	−54.9020	−1.74755
$988$	82.2726	2.61744
$989$	4.99624	0.158871
$990$	4.28845	0.136296
$991$	−28.4348	−0.903260	−0.451630	−	0.892205i	$-0.649157\pi$
$991$	−28.4348	−0.903260	−0.451630	+	0.892205i	$0.649157\pi$
$992$	81.9180	2.60090
$993$	5.19608	0.164893
$994$	129.273	4.10028
$995$	8.11311	0.257203
$996$	−87.1677	−2.76201
$997$	−4.01030	−0.127007	−0.0635037	−	0.997982i	$-0.520227\pi$
$997$	−4.01030	−0.127007	−0.0635037	+	0.997982i	$0.520227\pi$
$998$	−88.9739	−2.81642
$999$	1.11543	0.0352907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.10	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.10	✓	306	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)

\(f(q)\)	\(=\)	\(q-2.63380 q^{2} -2.43471 q^{3} +4.93693 q^{4} -0.366932 q^{5} +6.41255 q^{6} +4.18826 q^{7} -7.73529 q^{8} +2.92781 q^{9} +O(q^{10})\)	\(q-2.63380 q^{2} -2.43471 q^{3} +4.93693 q^{4} -0.366932 q^{5} +6.41255 q^{6} +4.18826 q^{7} -7.73529 q^{8} +2.92781 q^{9} +0.966426 q^{10} +1.51562 q^{11} -12.0200 q^{12} -2.14727 q^{13} -11.0311 q^{14} +0.893372 q^{15} +10.4994 q^{16} -0.693877 q^{17} -7.71127 q^{18} -7.76088 q^{19} -1.81151 q^{20} -10.1972 q^{21} -3.99184 q^{22} -1.22620 q^{23} +18.8332 q^{24} -4.86536 q^{25} +5.65550 q^{26} +0.175772 q^{27} +20.6772 q^{28} +5.26445 q^{29} -2.35297 q^{30} -6.72409 q^{31} -12.1828 q^{32} -3.69008 q^{33} +1.82754 q^{34} -1.53681 q^{35} +14.4544 q^{36} +6.34590 q^{37} +20.4406 q^{38} +5.22798 q^{39} +2.83832 q^{40} +11.5105 q^{41} +26.8574 q^{42} -4.07457 q^{43} +7.48248 q^{44} -1.07430 q^{45} +3.22957 q^{46} +5.38403 q^{47} -25.5630 q^{48} +10.5416 q^{49} +12.8144 q^{50} +1.68939 q^{51} -10.6009 q^{52} +4.92769 q^{53} -0.462949 q^{54} -0.556127 q^{55} -32.3974 q^{56} +18.8955 q^{57} -13.8655 q^{58} -3.88189 q^{59} +4.41051 q^{60} -9.07259 q^{61} +17.7099 q^{62} +12.2624 q^{63} +11.0882 q^{64} +0.787902 q^{65} +9.71896 q^{66} +2.06269 q^{67} -3.42562 q^{68} +2.98544 q^{69} +4.04765 q^{70} -11.7190 q^{71} -22.6474 q^{72} +2.81340 q^{73} -16.7139 q^{74} +11.8457 q^{75} -38.3149 q^{76} +6.34780 q^{77} -13.7695 q^{78} +9.56514 q^{79} -3.85256 q^{80} -9.21137 q^{81} -30.3165 q^{82} +7.25190 q^{83} -50.3428 q^{84} +0.254605 q^{85} +10.7316 q^{86} -12.8174 q^{87} -11.7237 q^{88} +2.49236 q^{89} +2.82951 q^{90} -8.99335 q^{91} -6.05366 q^{92} +16.3712 q^{93} -14.1805 q^{94} +2.84771 q^{95} +29.6615 q^{96} -8.10881 q^{97} -27.7644 q^{98} +4.43743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * q^99

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