LMFDB - Embedded newform 8002.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.34684 q^{3} +1.00000 q^{4} -3.62971 q^{5} +3.34684 q^{6} -3.29144 q^{7} -1.00000 q^{8} +8.20136 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.34684 q^{3} +1.00000 q^{4} -3.62971 q^{5} +3.34684 q^{6} -3.29144 q^{7} -1.00000 q^{8} +8.20136 q^{9} +3.62971 q^{10} +4.76800 q^{11} -3.34684 q^{12} +6.14900 q^{13} +3.29144 q^{14} +12.1481 q^{15} +1.00000 q^{16} -6.09136 q^{17} -8.20136 q^{18} -2.51517 q^{19} -3.62971 q^{20} +11.0159 q^{21} -4.76800 q^{22} +0.550827 q^{23} +3.34684 q^{24} +8.17480 q^{25} -6.14900 q^{26} -17.4081 q^{27} -3.29144 q^{28} -5.87746 q^{29} -12.1481 q^{30} -3.36269 q^{31} -1.00000 q^{32} -15.9578 q^{33} +6.09136 q^{34} +11.9470 q^{35} +8.20136 q^{36} -9.17804 q^{37} +2.51517 q^{38} -20.5797 q^{39} +3.62971 q^{40} +9.88542 q^{41} -11.0159 q^{42} +2.24034 q^{43} +4.76800 q^{44} -29.7686 q^{45} -0.550827 q^{46} -7.54847 q^{47} -3.34684 q^{48} +3.83357 q^{49} -8.17480 q^{50} +20.3868 q^{51} +6.14900 q^{52} +2.27248 q^{53} +17.4081 q^{54} -17.3065 q^{55} +3.29144 q^{56} +8.41787 q^{57} +5.87746 q^{58} +3.13834 q^{59} +12.1481 q^{60} -7.33251 q^{61} +3.36269 q^{62} -26.9943 q^{63} +1.00000 q^{64} -22.3191 q^{65} +15.9578 q^{66} +8.96318 q^{67} -6.09136 q^{68} -1.84353 q^{69} -11.9470 q^{70} -8.20801 q^{71} -8.20136 q^{72} -3.01659 q^{73} +9.17804 q^{74} -27.3598 q^{75} -2.51517 q^{76} -15.6936 q^{77} +20.5797 q^{78} +9.80445 q^{79} -3.62971 q^{80} +33.6582 q^{81} -9.88542 q^{82} +2.97943 q^{83} +11.0159 q^{84} +22.1099 q^{85} -2.24034 q^{86} +19.6709 q^{87} -4.76800 q^{88} +18.7005 q^{89} +29.7686 q^{90} -20.2390 q^{91} +0.550827 q^{92} +11.2544 q^{93} +7.54847 q^{94} +9.12933 q^{95} +3.34684 q^{96} +11.3129 q^{97} -3.83357 q^{98} +39.1041 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.34684	−1.93230	−0.966150	−	0.257979i	$-0.916943\pi$
$3$	−3.34684	−1.93230	−0.966150	+	0.257979i	$0.916943\pi$
$4$	1.00000	0.500000
$5$	−3.62971	−1.62326	−0.811628	−	0.584174i	$-0.801418\pi$
$5$	−3.62971	−1.62326	−0.811628	+	0.584174i	$0.801418\pi$
$6$	3.34684	1.36634
$7$	−3.29144	−1.24405	−0.622023	−	0.782999i	$-0.713689\pi$
$7$	−3.29144	−1.24405	−0.622023	+	0.782999i	$0.713689\pi$
$8$	−1.00000	−0.353553
$9$	8.20136	2.73379
$10$	3.62971	1.14782
$11$	4.76800	1.43761	0.718804	−	0.695213i	$-0.244690\pi$
$11$	4.76800	1.43761	0.718804	+	0.695213i	$0.244690\pi$
$12$	−3.34684	−0.966150
$13$	6.14900	1.70542	0.852712	−	0.522381i	$-0.174956\pi$
$13$	6.14900	1.70542	0.852712	+	0.522381i	$0.174956\pi$
$14$	3.29144	0.879674
$15$	12.1481	3.13662
$16$	1.00000	0.250000
$17$	−6.09136	−1.47737	−0.738686	−	0.674050i	$-0.764553\pi$
$17$	−6.09136	−1.47737	−0.738686	+	0.674050i	$0.764553\pi$
$18$	−8.20136	−1.93308
$19$	−2.51517	−0.577019	−0.288509	−	0.957477i	$-0.593160\pi$
$19$	−2.51517	−0.577019	−0.288509	+	0.957477i	$0.593160\pi$
$20$	−3.62971	−0.811628
$21$	11.0159	2.40387
$22$	−4.76800	−1.01654
$23$	0.550827	0.114855	0.0574276	−	0.998350i	$-0.481710\pi$
$23$	0.550827	0.114855	0.0574276	+	0.998350i	$0.481710\pi$
$24$	3.34684	0.683172
$25$	8.17480	1.63496
$26$	−6.14900	−1.20592
$27$	−17.4081	−3.35020
$28$	−3.29144	−0.622023
$29$	−5.87746	−1.09142	−0.545709	−	0.837975i	$-0.683740\pi$
$29$	−5.87746	−1.09142	−0.545709	+	0.837975i	$0.683740\pi$
$30$	−12.1481	−2.21792
$31$	−3.36269	−0.603957	−0.301978	−	0.953315i	$-0.597647\pi$
$31$	−3.36269	−0.603957	−0.301978	+	0.953315i	$0.597647\pi$
$32$	−1.00000	−0.176777
$33$	−15.9578	−2.77789
$34$	6.09136	1.04466
$35$	11.9470	2.01941
$36$	8.20136	1.36689
$37$	−9.17804	−1.50886	−0.754431	−	0.656380i	$-0.772087\pi$
$37$	−9.17804	−1.50886	−0.754431	+	0.656380i	$0.772087\pi$
$38$	2.51517	0.408014
$39$	−20.5797	−3.29539
$40$	3.62971	0.573908
$41$	9.88542	1.54384	0.771922	−	0.635718i	$-0.219296\pi$
$41$	9.88542	1.54384	0.771922	+	0.635718i	$0.219296\pi$
$42$	−11.0159	−1.69979
$43$	2.24034	0.341649	0.170825	−	0.985301i	$-0.445357\pi$
$43$	2.24034	0.341649	0.170825	+	0.985301i	$0.445357\pi$
$44$	4.76800	0.718804
$45$	−29.7686	−4.43764
$46$	−0.550827	−0.0812149
$47$	−7.54847	−1.10106	−0.550529	−	0.834816i	$-0.685574\pi$
$47$	−7.54847	−1.10106	−0.550529	+	0.834816i	$0.685574\pi$
$48$	−3.34684	−0.483075
$49$	3.83357	0.547653
$50$	−8.17480	−1.15609
$51$	20.3868	2.85473
$52$	6.14900	0.852712
$53$	2.27248	0.312149	0.156074	−	0.987745i	$-0.450116\pi$
$53$	2.27248	0.312149	0.156074	+	0.987745i	$0.450116\pi$
$54$	17.4081	2.36895
$55$	−17.3065	−2.33361
$56$	3.29144	0.439837
$57$	8.41787	1.11497
$58$	5.87746	0.771749
$59$	3.13834	0.408577	0.204289	−	0.978911i	$-0.434512\pi$
$59$	3.13834	0.408577	0.204289	+	0.978911i	$0.434512\pi$
$60$	12.1481	1.56831
$61$	−7.33251	−0.938832	−0.469416	−	0.882977i	$-0.655535\pi$
$61$	−7.33251	−0.938832	−0.469416	+	0.882977i	$0.655535\pi$
$62$	3.36269	0.427062
$63$	−26.9943	−3.40096
$64$	1.00000	0.125000
$65$	−22.3191	−2.76834
$66$	15.9578	1.96427
$67$	8.96318	1.09503	0.547513	−	0.836797i	$-0.315575\pi$
$67$	8.96318	1.09503	0.547513	+	0.836797i	$0.315575\pi$
$68$	−6.09136	−0.738686
$69$	−1.84353	−0.221935
$70$	−11.9470	−1.42794
$71$	−8.20801	−0.974111	−0.487056	−	0.873371i	$-0.661929\pi$
$71$	−8.20801	−0.974111	−0.487056	+	0.873371i	$0.661929\pi$
$72$	−8.20136	−0.966540
$73$	−3.01659	−0.353065	−0.176532	−	0.984295i	$-0.556488\pi$
$73$	−3.01659	−0.353065	−0.176532	+	0.984295i	$0.556488\pi$
$74$	9.17804	1.06693
$75$	−27.3598	−3.15924
$76$	−2.51517	−0.288509
$77$	−15.6936	−1.78845
$78$	20.5797	2.33019
$79$	9.80445	1.10309	0.551544	−	0.834146i	$-0.314039\pi$
$79$	9.80445	1.10309	0.551544	+	0.834146i	$0.314039\pi$
$80$	−3.62971	−0.405814
$81$	33.6582	3.73980
$82$	−9.88542	−1.09166
$83$	2.97943	0.327035	0.163517	−	0.986540i	$-0.447716\pi$
$83$	2.97943	0.327035	0.163517	+	0.986540i	$0.447716\pi$
$84$	11.0159	1.20194
$85$	22.1099	2.39815
$86$	−2.24034	−0.241582
$87$	19.6709	2.10895
$88$	−4.76800	−0.508271
$89$	18.7005	1.98225	0.991125	−	0.132932i	$-0.0424393\pi$
$89$	18.7005	1.98225	0.991125	+	0.132932i	$0.0424393\pi$
$90$	29.7686	3.13788
$91$	−20.2390	−2.12163
$92$	0.550827	0.0574276
$93$	11.2544	1.16703
$94$	7.54847	0.778565
$95$	9.12933	0.936649
$96$	3.34684	0.341586
$97$	11.3129	1.14865	0.574324	−	0.818628i	$-0.305265\pi$
$97$	11.3129	1.14865	0.574324	+	0.818628i	$0.305265\pi$
$98$	−3.83357	−0.387249
$99$	39.1041	3.93011
$100$	8.17480	0.817480
$101$	−11.0743	−1.10194	−0.550968	−	0.834527i	$-0.685741\pi$
$101$	−11.0743	−1.10194	−0.550968	+	0.834527i	$0.685741\pi$
$102$	−20.3868	−2.01860
$103$	−10.9463	−1.07857	−0.539287	−	0.842122i	$-0.681306\pi$
$103$	−10.9463	−1.07857	−0.539287	+	0.842122i	$0.681306\pi$
$104$	−6.14900	−0.602959
$105$	−39.9846	−3.90210
$106$	−2.27248	−0.220722
$107$	−7.54294	−0.729204	−0.364602	−	0.931164i	$-0.618795\pi$
$107$	−7.54294	−0.729204	−0.364602	+	0.931164i	$0.618795\pi$
$108$	−17.4081	−1.67510
$109$	11.3256	1.08479	0.542397	−	0.840122i	$-0.317517\pi$
$109$	11.3256	1.08479	0.542397	+	0.840122i	$0.317517\pi$
$110$	17.3065	1.65011
$111$	30.7175	2.91557
$112$	−3.29144	−0.311012
$113$	−18.0569	−1.69865	−0.849324	−	0.527872i	$-0.822990\pi$
$113$	−18.0569	−1.69865	−0.849324	+	0.527872i	$0.822990\pi$
$114$	−8.41787	−0.788406
$115$	−1.99934	−0.186440
$116$	−5.87746	−0.545709
$117$	50.4301	4.66227
$118$	−3.13834	−0.288908
$119$	20.0493	1.83792
$120$	−12.1481	−1.10896
$121$	11.7339	1.06672
$122$	7.33251	0.663854
$123$	−33.0850	−2.98317
$124$	−3.36269	−0.301978
$125$	−11.5236	−1.03070
$126$	26.9943	2.40484
$127$	3.49389	0.310033	0.155016	−	0.987912i	$-0.450457\pi$
$127$	3.49389	0.310033	0.155016	+	0.987912i	$0.450457\pi$
$128$	−1.00000	−0.0883883
$129$	−7.49808	−0.660169
$130$	22.3191	1.95751
$131$	−4.02485	−0.351653	−0.175826	−	0.984421i	$-0.556260\pi$
$131$	−4.02485	−0.351653	−0.175826	+	0.984421i	$0.556260\pi$
$132$	−15.9578	−1.38895
$133$	8.27851	0.717838
$134$	−8.96318	−0.774301
$135$	63.1865	5.43823
$136$	6.09136	0.522330
$137$	15.0686	1.28739	0.643697	−	0.765281i	$-0.277400\pi$
$137$	15.0686	1.28739	0.643697	+	0.765281i	$0.277400\pi$
$138$	1.84353	0.156932
$139$	−8.21108	−0.696455	−0.348227	−	0.937410i	$-0.613216\pi$
$139$	−8.21108	−0.696455	−0.348227	+	0.937410i	$0.613216\pi$
$140$	11.9470	1.00970
$141$	25.2635	2.12757
$142$	8.20801	0.688801
$143$	29.3184	2.45173
$144$	8.20136	0.683447
$145$	21.3335	1.77165
$146$	3.01659	0.249655
$147$	−12.8304	−1.05823
$148$	−9.17804	−0.754431
$149$	7.30208	0.598210	0.299105	−	0.954220i	$-0.403312\pi$
$149$	7.30208	0.598210	0.299105	+	0.954220i	$0.403312\pi$
$150$	27.3598	2.23392
$151$	8.64273	0.703335	0.351668	−	0.936125i	$-0.385615\pi$
$151$	8.64273	0.703335	0.351668	+	0.936125i	$0.385615\pi$
$152$	2.51517	0.204007
$153$	−49.9575	−4.03882
$154$	15.6936	1.26463
$155$	12.2056	0.980377
$156$	−20.5797	−1.64770
$157$	18.7737	1.49830	0.749151	−	0.662399i	$-0.230462\pi$
$157$	18.7737	1.49830	0.749151	+	0.662399i	$0.230462\pi$
$158$	−9.80445	−0.780000
$159$	−7.60562	−0.603165
$160$	3.62971	0.286954
$161$	−1.81301	−0.142885
$162$	−33.6582	−2.64444
$163$	11.1170	0.870748	0.435374	−	0.900250i	$-0.356616\pi$
$163$	11.1170	0.870748	0.435374	+	0.900250i	$0.356616\pi$
$164$	9.88542	0.771922
$165$	57.9221	4.50923
$166$	−2.97943	−0.231248
$167$	−11.7078	−0.905975	−0.452988	−	0.891517i	$-0.649642\pi$
$167$	−11.7078	−0.905975	−0.452988	+	0.891517i	$0.649642\pi$
$168$	−11.0159	−0.849897
$169$	24.8101	1.90847
$170$	−22.1099	−1.69575
$171$	−20.6278	−1.57745
$172$	2.24034	0.170825
$173$	−3.26499	−0.248233	−0.124116	−	0.992268i	$-0.539610\pi$
$173$	−3.26499	−0.248233	−0.124116	+	0.992268i	$0.539610\pi$
$174$	−19.6709	−1.49125
$175$	−26.9069	−2.03397
$176$	4.76800	0.359402
$177$	−10.5035	−0.789494
$178$	−18.7005	−1.40166
$179$	9.26498	0.692497	0.346249	−	0.938143i	$-0.387455\pi$
$179$	9.26498	0.692497	0.346249	+	0.938143i	$0.387455\pi$
$180$	−29.7686	−2.21882
$181$	13.7649	1.02314	0.511570	−	0.859242i	$-0.329064\pi$
$181$	13.7649	1.02314	0.511570	+	0.859242i	$0.329064\pi$
$182$	20.2390	1.50022
$183$	24.5408	1.81411
$184$	−0.550827	−0.0406075
$185$	33.3137	2.44927
$186$	−11.2544	−0.825212
$187$	−29.0436	−2.12388
$188$	−7.54847	−0.550529
$189$	57.2978	4.16780
$190$	−9.12933	−0.662311
$191$	−4.45215	−0.322146	−0.161073	−	0.986942i	$-0.551495\pi$
$191$	−4.45215	−0.322146	−0.161073	+	0.986942i	$0.551495\pi$
$192$	−3.34684	−0.241538
$193$	6.89957	0.496642	0.248321	−	0.968678i	$-0.420121\pi$
$193$	6.89957	0.496642	0.248321	+	0.968678i	$0.420121\pi$
$194$	−11.3129	−0.812216
$195$	74.6985	5.34927
$196$	3.83357	0.273826
$197$	−10.5798	−0.753781	−0.376890	−	0.926258i	$-0.623007\pi$
$197$	−10.5798	−0.753781	−0.376890	+	0.926258i	$0.623007\pi$
$198$	−39.1041	−2.77901
$199$	−20.4684	−1.45096	−0.725482	−	0.688241i	$-0.758383\pi$
$199$	−20.4684	−1.45096	−0.725482	+	0.688241i	$0.758383\pi$
$200$	−8.17480	−0.578046
$201$	−29.9984	−2.11592
$202$	11.0743	0.779186
$203$	19.3453	1.35777
$204$	20.3868	1.42736
$205$	−35.8812	−2.50605
$206$	10.9463	0.762668
$207$	4.51753	0.313990
$208$	6.14900	0.426356
$209$	−11.9923	−0.829526
$210$	39.9846	2.75920
$211$	−2.76482	−0.190338	−0.0951692	−	0.995461i	$-0.530339\pi$
$211$	−2.76482	−0.190338	−0.0951692	+	0.995461i	$0.530339\pi$
$212$	2.27248	0.156074
$213$	27.4709	1.88228
$214$	7.54294	0.515625
$215$	−8.13180	−0.554584
$216$	17.4081	1.18447
$217$	11.0681	0.751351
$218$	−11.3256	−0.767066
$219$	10.0960	0.682228
$220$	−17.3065	−1.16680
$221$	−37.4558	−2.51955
$222$	−30.7175	−2.06162
$223$	−2.68844	−0.180031	−0.0900157	−	0.995940i	$-0.528692\pi$
$223$	−2.68844	−0.180031	−0.0900157	+	0.995940i	$0.528692\pi$
$224$	3.29144	0.219918
$225$	67.0445	4.46963
$226$	18.0569	1.20113
$227$	5.47044	0.363086	0.181543	−	0.983383i	$-0.441891\pi$
$227$	5.47044	0.363086	0.181543	+	0.983383i	$0.441891\pi$
$228$	8.41787	0.557487
$229$	5.70290	0.376858	0.188429	−	0.982087i	$-0.439660\pi$
$229$	5.70290	0.376858	0.188429	+	0.982087i	$0.439660\pi$
$230$	1.99934	0.131833
$231$	52.5240	3.45583
$232$	5.87746	0.385874
$233$	5.54122	0.363017	0.181509	−	0.983389i	$-0.441902\pi$
$233$	5.54122	0.363017	0.181509	+	0.983389i	$0.441902\pi$
$234$	−50.4301	−3.29672
$235$	27.3988	1.78730
$236$	3.13834	0.204289
$237$	−32.8140	−2.13150
$238$	−20.0493	−1.29961
$239$	−22.5418	−1.45811	−0.729055	−	0.684455i	$-0.760040\pi$
$239$	−22.5418	−1.45811	−0.729055	+	0.684455i	$0.760040\pi$
$240$	12.1481	0.784155
$241$	3.66446	0.236049	0.118024	−	0.993011i	$-0.462344\pi$
$241$	3.66446	0.236049	0.118024	+	0.993011i	$0.462344\pi$
$242$	−11.7339	−0.754282
$243$	−60.4244	−3.87623
$244$	−7.33251	−0.469416
$245$	−13.9147	−0.888980
$246$	33.0850	2.10942
$247$	−15.4657	−0.984062
$248$	3.36269	0.213531
$249$	−9.97168	−0.631929
$250$	11.5236	0.728818
$251$	−3.34129	−0.210900	−0.105450	−	0.994425i	$-0.533628\pi$
$251$	−3.34129	−0.210900	−0.105450	+	0.994425i	$0.533628\pi$
$252$	−26.9943	−1.70048
$253$	2.62634	0.165117
$254$	−3.49389	−0.219226
$255$	−73.9983	−4.63395
$256$	1.00000	0.0625000
$257$	−31.1142	−1.94085	−0.970426	−	0.241400i	$-0.922393\pi$
$257$	−31.1142	−1.94085	−0.970426	+	0.241400i	$0.922393\pi$
$258$	7.49808	0.466810
$259$	30.2090	1.87709
$260$	−22.3191	−1.38417
$261$	−48.2032	−2.98370
$262$	4.02485	0.248656
$263$	28.4092	1.75179	0.875893	−	0.482505i	$-0.160273\pi$
$263$	28.4092	1.75179	0.875893	+	0.482505i	$0.160273\pi$
$264$	15.9578	0.982133
$265$	−8.24843	−0.506697
$266$	−8.27851	−0.507588
$267$	−62.5877	−3.83030
$268$	8.96318	0.547513
$269$	−8.20760	−0.500426	−0.250213	−	0.968191i	$-0.580501\pi$
$269$	−8.20760	−0.500426	−0.250213	+	0.968191i	$0.580501\pi$
$270$	−63.1865	−3.84541
$271$	−6.34263	−0.385288	−0.192644	−	0.981269i	$-0.561706\pi$
$271$	−6.34263	−0.385288	−0.192644	+	0.981269i	$0.561706\pi$
$272$	−6.09136	−0.369343
$273$	67.7369	4.09962
$274$	−15.0686	−0.910324
$275$	38.9775	2.35043
$276$	−1.84353	−0.110967
$277$	30.2391	1.81689	0.908447	−	0.418001i	$-0.137269\pi$
$277$	30.2391	1.81689	0.908447	+	0.418001i	$0.137269\pi$
$278$	8.21108	0.492468
$279$	−27.5786	−1.65109
$280$	−11.9470	−0.713968
$281$	−4.77744	−0.284998	−0.142499	−	0.989795i	$-0.545514\pi$
$281$	−4.77744	−0.284998	−0.142499	+	0.989795i	$0.545514\pi$
$282$	−25.2635	−1.50442
$283$	−23.0923	−1.37269	−0.686347	−	0.727274i	$-0.740787\pi$
$283$	−23.0923	−1.37269	−0.686347	+	0.727274i	$0.740787\pi$
$284$	−8.20801	−0.487056
$285$	−30.5544	−1.80989
$286$	−29.3184	−1.73364
$287$	−32.5373	−1.92061
$288$	−8.20136	−0.483270
$289$	20.1047	1.18263
$290$	−21.3335	−1.25275
$291$	−37.8624	−2.21953
$292$	−3.01659	−0.176532
$293$	−30.4633	−1.77968	−0.889842	−	0.456269i	$-0.849185\pi$
$293$	−30.4633	−1.77968	−0.889842	+	0.456269i	$0.849185\pi$
$294$	12.8304	0.748281
$295$	−11.3913	−0.663226
$296$	9.17804	0.533463
$297$	−83.0021	−4.81627
$298$	−7.30208	−0.422998
$299$	3.38703	0.195877
$300$	−27.3598	−1.57962
$301$	−7.37395	−0.425027
$302$	−8.64273	−0.497333
$303$	37.0640	2.12927
$304$	−2.51517	−0.144255
$305$	26.6149	1.52396
$306$	49.9575	2.85588
$307$	−15.3350	−0.875215	−0.437608	−	0.899166i	$-0.644174\pi$
$307$	−15.3350	−0.875215	−0.437608	+	0.899166i	$0.644174\pi$
$308$	−15.6936	−0.894225
$309$	36.6357	2.08413
$310$	−12.2056	−0.693231
$311$	12.8484	0.728564	0.364282	−	0.931289i	$-0.381314\pi$
$311$	12.8484	0.728564	0.364282	+	0.931289i	$0.381314\pi$
$312$	20.5797	1.16510
$313$	33.3394	1.88445	0.942227	−	0.334975i	$-0.108728\pi$
$313$	33.3394	1.88445	0.942227	+	0.334975i	$0.108728\pi$
$314$	−18.7737	−1.05946
$315$	97.9814	5.52063
$316$	9.80445	0.551544
$317$	10.6938	0.600624	0.300312	−	0.953841i	$-0.402909\pi$
$317$	10.6938	0.600624	0.300312	+	0.953841i	$0.402909\pi$
$318$	7.60562	0.426502
$319$	−28.0238	−1.56903
$320$	−3.62971	−0.202907
$321$	25.2450	1.40904
$322$	1.81301	0.101035
$323$	15.3208	0.852471
$324$	33.6582	1.86990
$325$	50.2668	2.78830
$326$	−11.1170	−0.615712
$327$	−37.9050	−2.09615
$328$	−9.88542	−0.545831
$329$	24.8453	1.36977
$330$	−57.9221	−3.18851
$331$	26.8605	1.47639	0.738193	−	0.674590i	$-0.235680\pi$
$331$	26.8605	1.47639	0.738193	+	0.674590i	$0.235680\pi$
$332$	2.97943	0.163517
$333$	−75.2725	−4.12491
$334$	11.7078	0.640621
$335$	−32.5338	−1.77751
$336$	11.0159	0.600968
$337$	−13.6528	−0.743713	−0.371856	−	0.928290i	$-0.621279\pi$
$337$	−13.6528	−0.743713	−0.371856	+	0.928290i	$0.621279\pi$
$338$	−24.8101	−1.34949
$339$	60.4335	3.28230
$340$	22.1099	1.19908
$341$	−16.0333	−0.868253
$342$	20.6278	1.11542
$343$	10.4221	0.562741
$344$	−2.24034	−0.120791
$345$	6.69148	0.360257
$346$	3.26499	0.175527
$347$	23.0119	1.23534	0.617672	−	0.786436i	$-0.288076\pi$
$347$	23.0119	1.23534	0.617672	+	0.786436i	$0.288076\pi$
$348$	19.6709	1.05447
$349$	11.6617	0.624235	0.312118	−	0.950043i	$-0.398962\pi$
$349$	11.6617	0.624235	0.312118	+	0.950043i	$0.398962\pi$
$350$	26.9069	1.43823
$351$	−107.043	−5.71351
$352$	−4.76800	−0.254135
$353$	34.3079	1.82603	0.913013	−	0.407930i	$-0.133749\pi$
$353$	34.3079	1.82603	0.913013	+	0.407930i	$0.133749\pi$
$354$	10.5035	0.558257
$355$	29.7927	1.58123
$356$	18.7005	0.991125
$357$	−67.1020	−3.55142
$358$	−9.26498	−0.489670
$359$	−19.8821	−1.04934	−0.524668	−	0.851307i	$-0.675810\pi$
$359$	−19.8821	−1.04934	−0.524668	+	0.851307i	$0.675810\pi$
$360$	29.7686	1.56894
$361$	−12.6739	−0.667049
$362$	−13.7649	−0.723469
$363$	−39.2714	−2.06121
$364$	−20.2390	−1.06081
$365$	10.9493	0.573115
$366$	−24.5408	−1.28277
$367$	1.28908	0.0672893	0.0336446	−	0.999434i	$-0.489289\pi$
$367$	1.28908	0.0672893	0.0336446	+	0.999434i	$0.489289\pi$
$368$	0.550827	0.0287138
$369$	81.0739	4.22054
$370$	−33.3137	−1.73189
$371$	−7.47971	−0.388327
$372$	11.2544	0.583513
$373$	−13.9778	−0.723745	−0.361872	−	0.932228i	$-0.617862\pi$
$373$	−13.9778	−0.723745	−0.361872	+	0.932228i	$0.617862\pi$
$374$	29.0436	1.50181
$375$	38.5677	1.99163
$376$	7.54847	0.389283
$377$	−36.1405	−1.86133
$378$	−57.2978	−2.94708
$379$	22.7640	1.16931	0.584654	−	0.811283i	$-0.301230\pi$
$379$	22.7640	1.16931	0.584654	+	0.811283i	$0.301230\pi$
$380$	9.12933	0.468325
$381$	−11.6935	−0.599076
$382$	4.45215	0.227792
$383$	−1.92783	−0.0985075	−0.0492538	−	0.998786i	$-0.515684\pi$
$383$	−1.92783	−0.0985075	−0.0492538	+	0.998786i	$0.515684\pi$
$384$	3.34684	0.170793
$385$	56.9632	2.90311
$386$	−6.89957	−0.351179
$387$	18.3739	0.933996
$388$	11.3129	0.574324
$389$	36.9430	1.87308	0.936542	−	0.350556i	$-0.114007\pi$
$389$	36.9430	1.87308	0.936542	+	0.350556i	$0.114007\pi$
$390$	−74.6985	−3.78250
$391$	−3.35528	−0.169684
$392$	−3.83357	−0.193624
$393$	13.4705	0.679499
$394$	10.5798	0.533004
$395$	−35.5873	−1.79059
$396$	39.1041	1.96506
$397$	27.8196	1.39622	0.698112	−	0.715989i	$-0.254024\pi$
$397$	27.8196	1.39622	0.698112	+	0.715989i	$0.254024\pi$
$398$	20.4684	1.02599
$399$	−27.7069	−1.38708
$400$	8.17480	0.408740
$401$	−17.2982	−0.863833	−0.431917	−	0.901914i	$-0.642163\pi$
$401$	−17.2982	−0.863833	−0.431917	+	0.901914i	$0.642163\pi$
$402$	29.9984	1.49618
$403$	−20.6772	−1.03000
$404$	−11.0743	−0.550968
$405$	−122.170	−6.07066
$406$	−19.3453	−0.960092
$407$	−43.7610	−2.16915
$408$	−20.3868	−1.00930
$409$	31.6120	1.56311	0.781556	−	0.623835i	$-0.214426\pi$
$409$	31.6120	1.56311	0.781556	+	0.623835i	$0.214426\pi$
$410$	35.8812	1.77205
$411$	−50.4321	−2.48763
$412$	−10.9463	−0.539287
$413$	−10.3297	−0.508289
$414$	−4.51753	−0.222024
$415$	−10.8145	−0.530861
$416$	−6.14900	−0.301479
$417$	27.4812	1.34576
$418$	11.9923	0.586564
$419$	35.6701	1.74260	0.871298	−	0.490754i	$-0.163279\pi$
$419$	35.6701	1.74260	0.871298	+	0.490754i	$0.163279\pi$
$420$	−39.9846	−1.95105
$421$	0.872239	0.0425103	0.0212552	−	0.999774i	$-0.493234\pi$
$421$	0.872239	0.0425103	0.0212552	+	0.999774i	$0.493234\pi$
$422$	2.76482	0.134590
$423$	−61.9077	−3.01006
$424$	−2.27248	−0.110361
$425$	−49.7957	−2.41545
$426$	−27.4709	−1.33097
$427$	24.1345	1.16795
$428$	−7.54294	−0.364602
$429$	−98.1242	−4.73748
$430$	8.13180	0.392150
$431$	−20.5771	−0.991165	−0.495583	−	0.868561i	$-0.665045\pi$
$431$	−20.5771	−0.991165	−0.495583	+	0.868561i	$0.665045\pi$
$432$	−17.4081	−0.837550
$433$	36.4438	1.75137	0.875687	−	0.482878i	$-0.160409\pi$
$433$	36.4438	1.75137	0.875687	+	0.482878i	$0.160409\pi$
$434$	−11.0681	−0.531285
$435$	−71.3999	−3.42336
$436$	11.3256	0.542397
$437$	−1.38542	−0.0662736
$438$	−10.0960	−0.482408
$439$	−25.1091	−1.19839	−0.599197	−	0.800602i	$-0.704513\pi$
$439$	−25.1091	−1.19839	−0.599197	+	0.800602i	$0.704513\pi$
$440$	17.3065	0.825054
$441$	31.4405	1.49717
$442$	37.4558	1.78159
$443$	−14.7204	−0.699388	−0.349694	−	0.936864i	$-0.613714\pi$
$443$	−14.7204	−0.699388	−0.349694	+	0.936864i	$0.613714\pi$
$444$	30.7175	1.45779
$445$	−67.8775	−3.21770
$446$	2.68844	0.127301
$447$	−24.4389	−1.15592
$448$	−3.29144	−0.155506
$449$	−0.194038	−0.00915720	−0.00457860	−	0.999990i	$-0.501457\pi$
$449$	−0.194038	−0.00915720	−0.00457860	+	0.999990i	$0.501457\pi$
$450$	−67.0445	−3.16051
$451$	47.1337	2.21944
$452$	−18.0569	−0.849324
$453$	−28.9259	−1.35906
$454$	−5.47044	−0.256740
$455$	73.4619	3.44395
$456$	−8.41787	−0.394203
$457$	−7.03428	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$457$	−7.03428	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$458$	−5.70290	−0.266479
$459$	106.039	4.94949
$460$	−1.99934	−0.0932198
$461$	−20.5502	−0.957118	−0.478559	−	0.878055i	$-0.658841\pi$
$461$	−20.5502	−0.957118	−0.478559	+	0.878055i	$0.658841\pi$
$462$	−52.5240	−2.44364
$463$	−1.78224	−0.0828279	−0.0414140	−	0.999142i	$-0.513186\pi$
$463$	−1.78224	−0.0828279	−0.0414140	+	0.999142i	$0.513186\pi$
$464$	−5.87746	−0.272854
$465$	−40.8502	−1.89438
$466$	−5.54122	−0.256692
$467$	−26.1573	−1.21042	−0.605209	−	0.796067i	$-0.706910\pi$
$467$	−26.1573	−1.21042	−0.605209	+	0.796067i	$0.706910\pi$
$468$	50.4301	2.33113
$469$	−29.5018	−1.36226
$470$	−27.3988	−1.26381
$471$	−62.8326	−2.89517
$472$	−3.13834	−0.144454
$473$	10.6820	0.491157
$474$	32.8140	1.50720
$475$	−20.5610	−0.943403
$476$	20.0493	0.918960
$477$	18.6374	0.853348
$478$	22.5418	1.03104
$479$	−29.9727	−1.36949	−0.684743	−	0.728785i	$-0.740085\pi$
$479$	−29.9727	−1.36949	−0.684743	+	0.728785i	$0.740085\pi$
$480$	−12.1481	−0.554481
$481$	−56.4358	−2.57325
$482$	−3.66446	−0.166912
$483$	6.06787	0.276097
$484$	11.7339	0.533358
$485$	−41.0624	−1.86455
$486$	60.4244	2.74091
$487$	10.3610	0.469503	0.234751	−	0.972055i	$-0.424572\pi$
$487$	10.3610	0.469503	0.234751	+	0.972055i	$0.424572\pi$
$488$	7.33251	0.331927
$489$	−37.2068	−1.68255
$490$	13.9147	0.628604
$491$	−5.15526	−0.232654	−0.116327	−	0.993211i	$-0.537112\pi$
$491$	−5.15526	−0.232654	−0.116327	+	0.993211i	$0.537112\pi$
$492$	−33.0850	−1.49158
$493$	35.8018	1.61243
$494$	15.4657	0.695837
$495$	−141.937	−6.37958
$496$	−3.36269	−0.150989
$497$	27.0161	1.21184
$498$	9.97168	0.446842
$499$	8.31889	0.372405	0.186202	−	0.982511i	$-0.440382\pi$
$499$	8.31889	0.372405	0.186202	+	0.982511i	$0.440382\pi$
$500$	−11.5236	−0.515352
$501$	39.1841	1.75062
$502$	3.34129	0.149129
$503$	27.4474	1.22382	0.611909	−	0.790928i	$-0.290402\pi$
$503$	27.4474	1.22382	0.611909	+	0.790928i	$0.290402\pi$
$504$	26.9943	1.20242
$505$	40.1966	1.78872
$506$	−2.62634	−0.116755
$507$	−83.0357	−3.68774
$508$	3.49389	0.155016
$509$	37.7452	1.67303	0.836513	−	0.547947i	$-0.184590\pi$
$509$	37.7452	1.67303	0.836513	+	0.547947i	$0.184590\pi$
$510$	73.9983	3.27670
$511$	9.92891	0.439229
$512$	−1.00000	−0.0441942
$513$	43.7844	1.93313
$514$	31.1142	1.37239
$515$	39.7321	1.75080
$516$	−7.49808	−0.330084
$517$	−35.9911	−1.58289
$518$	−30.2090	−1.32731
$519$	10.9274	0.479660
$520$	22.3191	0.978756
$521$	−14.3082	−0.626852	−0.313426	−	0.949613i	$-0.601477\pi$
$521$	−14.3082	−0.626852	−0.313426	+	0.949613i	$0.601477\pi$
$522$	48.2032	2.10980
$523$	3.51214	0.153575	0.0767877	−	0.997047i	$-0.475534\pi$
$523$	3.51214	0.153575	0.0767877	+	0.997047i	$0.475534\pi$
$524$	−4.02485	−0.175826
$525$	90.0531	3.93024
$526$	−28.4092	−1.23870
$527$	20.4834	0.892269
$528$	−15.9578	−0.694473
$529$	−22.6966	−0.986808
$530$	8.24843	0.358289
$531$	25.7387	1.11696
$532$	8.27851	0.358919
$533$	60.7854	2.63291
$534$	62.5877	2.70843
$535$	27.3787	1.18368
$536$	−8.96318	−0.387150
$537$	−31.0085	−1.33811
$538$	8.20760	0.353855
$539$	18.2785	0.787309
$540$	63.1865	2.71911
$541$	30.4109	1.30747	0.653733	−	0.756725i	$-0.273202\pi$
$541$	30.4109	1.30747	0.653733	+	0.756725i	$0.273202\pi$
$542$	6.34263	0.272439
$543$	−46.0691	−1.97701
$544$	6.09136	0.261165
$545$	−41.1086	−1.76090
$546$	−67.7369	−2.89887
$547$	−32.2540	−1.37908	−0.689542	−	0.724246i	$-0.742188\pi$
$547$	−32.2540	−1.37908	−0.689542	+	0.724246i	$0.742188\pi$
$548$	15.0686	0.643697
$549$	−60.1366	−2.56657
$550$	−38.9775	−1.66201
$551$	14.7828	0.629768
$552$	1.84353	0.0784659
$553$	−32.2708	−1.37229
$554$	−30.2391	−1.28474
$555$	−111.496	−4.73272
$556$	−8.21108	−0.348227
$557$	6.44558	0.273108	0.136554	−	0.990633i	$-0.456397\pi$
$557$	6.44558	0.273108	0.136554	+	0.990633i	$0.456397\pi$
$558$	27.5786	1.16750
$559$	13.7759	0.582657
$560$	11.9470	0.504852
$561$	97.2045	4.10398
$562$	4.77744	0.201524
$563$	21.3290	0.898910	0.449455	−	0.893303i	$-0.351618\pi$
$563$	21.3290	0.898910	0.449455	+	0.893303i	$0.351618\pi$
$564$	25.2635	1.06379
$565$	65.5413	2.75734
$566$	23.0923	0.970642
$567$	−110.784	−4.65249
$568$	8.20801	0.344400
$569$	8.88692	0.372559	0.186280	−	0.982497i	$-0.440357\pi$
$569$	8.88692	0.372559	0.186280	+	0.982497i	$0.440357\pi$
$570$	30.5544	1.27978
$571$	−27.5937	−1.15476	−0.577381	−	0.816475i	$-0.695925\pi$
$571$	−27.5937	−1.15476	−0.577381	+	0.816475i	$0.695925\pi$
$572$	29.3184	1.22587
$573$	14.9006	0.622483
$574$	32.5373	1.35808
$575$	4.50290	0.187784
$576$	8.20136	0.341723
$577$	7.39296	0.307773	0.153886	−	0.988089i	$-0.450821\pi$
$577$	7.39296	0.307773	0.153886	+	0.988089i	$0.450821\pi$
$578$	−20.1047	−0.836245
$579$	−23.0918	−0.959662
$580$	21.3335	0.885825
$581$	−9.80660	−0.406846
$582$	37.8624	1.56945
$583$	10.8352	0.448747
$584$	3.01659	0.124827
$585$	−183.047	−7.56805
$586$	30.4633	1.25843
$587$	27.3088	1.12715	0.563577	−	0.826063i	$-0.309425\pi$
$587$	27.3088	1.12715	0.563577	+	0.826063i	$0.309425\pi$
$588$	−12.8304	−0.529115
$589$	8.45772	0.348494
$590$	11.3913	0.468971
$591$	35.4090	1.45653
$592$	−9.17804	−0.377215
$593$	−14.6972	−0.603541	−0.301770	−	0.953381i	$-0.597578\pi$
$593$	−14.6972	−0.603541	−0.301770	+	0.953381i	$0.597578\pi$
$594$	83.0021	3.40562
$595$	−72.7733	−2.98342
$596$	7.30208	0.299105
$597$	68.5045	2.80370
$598$	−3.38703	−0.138506
$599$	−27.6024	−1.12780	−0.563901	−	0.825842i	$-0.690700\pi$
$599$	−27.6024	−1.12780	−0.563901	+	0.825842i	$0.690700\pi$
$600$	27.3598	1.11696
$601$	20.0105	0.816244	0.408122	−	0.912927i	$-0.366184\pi$
$601$	20.0105	0.816244	0.408122	+	0.912927i	$0.366184\pi$
$602$	7.37395	0.300540
$603$	73.5103	2.99357
$604$	8.64273	0.351668
$605$	−42.5905	−1.73155
$606$	−37.0640	−1.50562
$607$	45.2401	1.83624	0.918120	−	0.396301i	$-0.129706\pi$
$607$	45.2401	1.83624	0.918120	+	0.396301i	$0.129706\pi$
$608$	2.51517	0.102003
$609$	−64.7457	−2.62363
$610$	−26.6149	−1.07761
$611$	−46.4155	−1.87777
$612$	−49.9575	−2.01941
$613$	3.56611	0.144034	0.0720170	−	0.997403i	$-0.477056\pi$
$613$	3.56611	0.144034	0.0720170	+	0.997403i	$0.477056\pi$
$614$	15.3350	0.618871
$615$	120.089	4.84245
$616$	15.6936	0.632313
$617$	11.3000	0.454922	0.227461	−	0.973787i	$-0.426958\pi$
$617$	11.3000	0.454922	0.227461	+	0.973787i	$0.426958\pi$
$618$	−36.6357	−1.47370
$619$	−8.67836	−0.348813	−0.174406	−	0.984674i	$-0.555801\pi$
$619$	−8.67836	−0.348813	−0.174406	+	0.984674i	$0.555801\pi$
$620$	12.2056	0.490188
$621$	−9.58887	−0.384788
$622$	−12.8484	−0.515173
$623$	−61.5516	−2.46601
$624$	−20.5797	−0.823848
$625$	0.953390	0.0381356
$626$	−33.3394	−1.33251
$627$	40.1364	1.60289
$628$	18.7737	0.749151
$629$	55.9068	2.22915
$630$	−97.9814	−3.90367
$631$	16.8880	0.672302	0.336151	−	0.941808i	$-0.390875\pi$
$631$	16.8880	0.672302	0.336151	+	0.941808i	$0.390875\pi$
$632$	−9.80445	−0.390000
$633$	9.25343	0.367791
$634$	−10.6938	−0.424705
$635$	−12.6818	−0.503262
$636$	−7.60562	−0.301582
$637$	23.5726	0.933980
$638$	28.0238	1.10947
$639$	−67.3168	−2.66301
$640$	3.62971	0.143477
$641$	40.4668	1.59834	0.799172	−	0.601102i	$-0.205272\pi$
$641$	40.4668	1.59834	0.799172	+	0.601102i	$0.205272\pi$
$642$	−25.2450	−0.996342
$643$	−5.82030	−0.229530	−0.114765	−	0.993393i	$-0.536612\pi$
$643$	−5.82030	−0.229530	−0.114765	+	0.993393i	$0.536612\pi$
$644$	−1.81301	−0.0714427
$645$	27.2158	1.07162
$646$	−15.3208	−0.602788
$647$	−6.29622	−0.247530	−0.123765	−	0.992312i	$-0.539497\pi$
$647$	−6.29622	−0.247530	−0.123765	+	0.992312i	$0.539497\pi$
$648$	−33.6582	−1.32222
$649$	14.9636	0.587374
$650$	−50.2668	−1.97163
$651$	−37.0431	−1.45184
$652$	11.1170	0.435374
$653$	27.6763	1.08306	0.541529	−	0.840682i	$-0.317846\pi$
$653$	27.6763	1.08306	0.541529	+	0.840682i	$0.317846\pi$
$654$	37.9050	1.48220
$655$	14.6091	0.570823
$656$	9.88542	0.385961
$657$	−24.7401	−0.965204
$658$	−24.8453	−0.968572
$659$	21.2725	0.828658	0.414329	−	0.910127i	$-0.364016\pi$
$659$	21.2725	0.828658	0.414329	+	0.910127i	$0.364016\pi$
$660$	57.9221	2.25461
$661$	−35.2463	−1.37092	−0.685461	−	0.728109i	$-0.740400\pi$
$661$	−35.2463	−1.37092	−0.685461	+	0.728109i	$0.740400\pi$
$662$	−26.8605	−1.04396
$663$	125.359	4.86852
$664$	−2.97943	−0.115624
$665$	−30.0486	−1.16524
$666$	75.2725	2.91675
$667$	−3.23746	−0.125355
$668$	−11.7078	−0.452988
$669$	8.99779	0.347875
$670$	32.5338	1.25689
$671$	−34.9614	−1.34967
$672$	−11.0159	−0.424949
$673$	−25.4814	−0.982234	−0.491117	−	0.871094i	$-0.663411\pi$
$673$	−25.4814	−0.982234	−0.491117	+	0.871094i	$0.663411\pi$
$674$	13.6528	0.525884
$675$	−142.308	−5.47744
$676$	24.8101	0.954236
$677$	−32.3921	−1.24493	−0.622464	−	0.782648i	$-0.713868\pi$
$677$	−32.3921	−1.24493	−0.622464	+	0.782648i	$0.713868\pi$
$678$	−60.4335	−2.32094
$679$	−37.2356	−1.42897
$680$	−22.1099	−0.847875
$681$	−18.3087	−0.701591
$682$	16.0333	0.613947
$683$	1.55460	0.0594851	0.0297425	−	0.999558i	$-0.490531\pi$
$683$	1.55460	0.0594851	0.0297425	+	0.999558i	$0.490531\pi$
$684$	−20.6278	−0.788723
$685$	−54.6945	−2.08977
$686$	−10.4221	−0.397918
$687$	−19.0867	−0.728204
$688$	2.24034	0.0854123
$689$	13.9734	0.532346
$690$	−6.69148	−0.254740
$691$	18.6960	0.711231	0.355615	−	0.934632i	$-0.384271\pi$
$691$	18.6960	0.711231	0.355615	+	0.934632i	$0.384271\pi$
$692$	−3.26499	−0.124116
$693$	−128.709	−4.88924
$694$	−23.0119	−0.873520
$695$	29.8039	1.13052
$696$	−19.6709	−0.745625
$697$	−60.2157	−2.28083
$698$	−11.6617	−0.441401
$699$	−18.5456	−0.701459
$700$	−26.9069	−1.01698
$701$	8.29029	0.313120	0.156560	−	0.987668i	$-0.449960\pi$
$701$	8.29029	0.313120	0.156560	+	0.987668i	$0.449960\pi$
$702$	107.043	4.04006
$703$	23.0843	0.870641
$704$	4.76800	0.179701
$705$	−91.6994	−3.45360
$706$	−34.3079	−1.29120
$707$	36.4504	1.37086
$708$	−10.5035	−0.394747
$709$	−15.1051	−0.567286	−0.283643	−	0.958930i	$-0.591543\pi$
$709$	−15.1051	−0.567286	−0.283643	+	0.958930i	$0.591543\pi$
$710$	−29.7927	−1.11810
$711$	80.4098	3.01560
$712$	−18.7005	−0.700831
$713$	−1.85226	−0.0693676
$714$	67.1020	2.51123
$715$	−106.417	−3.97979
$716$	9.26498	0.346249
$717$	75.4440	2.81751
$718$	19.8821	0.741992
$719$	5.72666	0.213569	0.106784	−	0.994282i	$-0.465945\pi$
$719$	5.72666	0.213569	0.106784	+	0.994282i	$0.465945\pi$
$720$	−29.7686	−1.10941
$721$	36.0292	1.34180
$722$	12.6739	0.471675
$723$	−12.2644	−0.456117
$724$	13.7649	0.511570
$725$	−48.0471	−1.78442
$726$	39.2714	1.45750
$727$	−13.1053	−0.486047	−0.243024	−	0.970020i	$-0.578139\pi$
$727$	−13.1053	−0.486047	−0.243024	+	0.970020i	$0.578139\pi$
$728$	20.2390	0.750109
$729$	101.256	3.75024
$730$	−10.9493	−0.405253
$731$	−13.6467	−0.504743
$732$	24.5408	0.907053
$733$	−14.2789	−0.527403	−0.263701	−	0.964604i	$-0.584943\pi$
$733$	−14.2789	−0.527403	−0.263701	+	0.964604i	$0.584943\pi$
$734$	−1.28908	−0.0475807
$735$	46.5705	1.71778
$736$	−0.550827	−0.0203037
$737$	42.7365	1.57422
$738$	−81.0739	−2.98437
$739$	27.1887	1.00015	0.500077	−	0.865981i	$-0.333305\pi$
$739$	27.1887	1.00015	0.500077	+	0.865981i	$0.333305\pi$
$740$	33.3137	1.22463
$741$	51.7614	1.90150
$742$	7.47971	0.274589
$743$	−9.06543	−0.332578	−0.166289	−	0.986077i	$-0.553179\pi$
$743$	−9.06543	−0.332578	−0.166289	+	0.986077i	$0.553179\pi$
$744$	−11.2544	−0.412606
$745$	−26.5044	−0.971047
$746$	13.9778	0.511765
$747$	24.4354	0.894043
$748$	−29.0436	−1.06194
$749$	24.8271	0.907164
$750$	−38.5677	−1.40829
$751$	14.9202	0.544444	0.272222	−	0.962234i	$-0.412241\pi$
$751$	14.9202	0.544444	0.272222	+	0.962234i	$0.412241\pi$
$752$	−7.54847	−0.275264
$753$	11.1828	0.407523
$754$	36.1405	1.31616
$755$	−31.3706	−1.14169
$756$	57.2978	2.08390
$757$	19.2585	0.699962	0.349981	−	0.936757i	$-0.386188\pi$
$757$	19.2585	0.699962	0.349981	+	0.936757i	$0.386188\pi$
$758$	−22.7640	−0.826826
$759$	−8.78996	−0.319055
$760$	−9.12933	−0.331155
$761$	−9.18792	−0.333062	−0.166531	−	0.986036i	$-0.553257\pi$
$761$	−9.18792	−0.333062	−0.166531	+	0.986036i	$0.553257\pi$
$762$	11.6935	0.423611
$763$	−37.2775	−1.34954
$764$	−4.45215	−0.161073
$765$	181.331	6.55604
$766$	1.92783	0.0696553
$767$	19.2976	0.696798
$768$	−3.34684	−0.120769
$769$	−14.1159	−0.509033	−0.254516	−	0.967068i	$-0.581916\pi$
$769$	−14.1159	−0.509033	−0.254516	+	0.967068i	$0.581916\pi$
$770$	−56.9632	−2.05281
$771$	104.134	3.75031
$772$	6.89957	0.248321
$773$	−31.6456	−1.13821	−0.569106	−	0.822264i	$-0.692711\pi$
$773$	−31.6456	−1.13821	−0.569106	+	0.822264i	$0.692711\pi$
$774$	−18.3739	−0.660435
$775$	−27.4893	−0.987446
$776$	−11.3129	−0.406108
$777$	−101.105	−3.62711
$778$	−36.9430	−1.32447
$779$	−24.8635	−0.890826
$780$	74.6985	2.67463
$781$	−39.1358	−1.40039
$782$	3.35528	0.119985
$783$	102.316	3.65647
$784$	3.83357	0.136913
$785$	−68.1430	−2.43213
$786$	−13.4705	−0.480479
$787$	13.3739	0.476728	0.238364	−	0.971176i	$-0.423389\pi$
$787$	13.3739	0.476728	0.238364	+	0.971176i	$0.423389\pi$
$788$	−10.5798	−0.376890
$789$	−95.0811	−3.38498
$790$	35.5873	1.26614
$791$	59.4331	2.11320
$792$	−39.1041	−1.38950
$793$	−45.0876	−1.60111
$794$	−27.8196	−0.987279
$795$	27.6062	0.979091
$796$	−20.4684	−0.725482
$797$	−45.9898	−1.62904	−0.814520	−	0.580135i	$-0.803000\pi$
$797$	−45.9898	−1.62904	−0.814520	+	0.580135i	$0.803000\pi$
$798$	27.7069	0.980813
$799$	45.9805	1.62667
$800$	−8.17480	−0.289023
$801$	153.370	5.41905
$802$	17.2982	0.610822
$803$	−14.3831	−0.507569
$804$	−29.9984	−1.05796
$805$	6.58071	0.231940
$806$	20.6772	0.728322
$807$	27.4695	0.966974
$808$	11.0743	0.389593
$809$	21.5864	0.758938	0.379469	−	0.925205i	$-0.376107\pi$
$809$	21.5864	0.758938	0.379469	+	0.925205i	$0.376107\pi$
$810$	122.170	4.29260
$811$	−11.6918	−0.410556	−0.205278	−	0.978704i	$-0.565810\pi$
$811$	−11.6918	−0.410556	−0.205278	+	0.978704i	$0.565810\pi$
$812$	19.3453	0.678887
$813$	21.2278	0.744491
$814$	43.7610	1.53382
$815$	−40.3514	−1.41345
$816$	20.3868	0.713682
$817$	−5.63483	−0.197138
$818$	−31.6120	−1.10529
$819$	−165.988	−5.80008
$820$	−35.8812	−1.25303
$821$	−9.26900	−0.323490	−0.161745	−	0.986833i	$-0.551712\pi$
$821$	−9.26900	−0.323490	−0.161745	+	0.986833i	$0.551712\pi$
$822$	50.4321	1.75902
$823$	6.35052	0.221365	0.110683	−	0.993856i	$-0.464696\pi$
$823$	6.35052	0.221365	0.110683	+	0.993856i	$0.464696\pi$
$824$	10.9463	0.381334
$825$	−130.452	−4.54174
$826$	10.3297	0.359415
$827$	43.4746	1.51176	0.755879	−	0.654711i	$-0.227210\pi$
$827$	43.4746	1.51176	0.755879	+	0.654711i	$0.227210\pi$
$828$	4.51753	0.156995
$829$	37.5953	1.30574	0.652870	−	0.757470i	$-0.273565\pi$
$829$	37.5953	1.30574	0.652870	+	0.757470i	$0.273565\pi$
$830$	10.8145	0.375375
$831$	−101.206	−3.51078
$832$	6.14900	0.213178
$833$	−23.3516	−0.809087
$834$	−27.4812	−0.951596
$835$	42.4959	1.47063
$836$	−11.9923	−0.414763
$837$	58.5382	2.02337
$838$	−35.6701	−1.23220
$839$	−27.3303	−0.943548	−0.471774	−	0.881719i	$-0.656386\pi$
$839$	−27.3303	−0.943548	−0.471774	+	0.881719i	$0.656386\pi$
$840$	39.9846	1.37960
$841$	5.54458	0.191192
$842$	−0.872239	−0.0300593
$843$	15.9894	0.550703
$844$	−2.76482	−0.0951692
$845$	−90.0536	−3.09794
$846$	61.9077	2.12843
$847$	−38.6213	−1.32704
$848$	2.27248	0.0780371
$849$	77.2863	2.65246
$850$	49.7957	1.70798
$851$	−5.05551	−0.173301
$852$	27.4709	0.941138
$853$	−30.9521	−1.05978	−0.529889	−	0.848067i	$-0.677766\pi$
$853$	−30.9521	−1.05978	−0.529889	+	0.848067i	$0.677766\pi$
$854$	−24.1345	−0.825866
$855$	74.8729	2.56060
$856$	7.54294	0.257812
$857$	−27.3494	−0.934239	−0.467120	−	0.884194i	$-0.654708\pi$
$857$	−27.3494	−0.934239	−0.467120	+	0.884194i	$0.654708\pi$
$858$	98.1242	3.34991
$859$	−9.11864	−0.311124	−0.155562	−	0.987826i	$-0.549719\pi$
$859$	−9.11864	−0.311124	−0.155562	+	0.987826i	$0.549719\pi$
$860$	−8.13180	−0.277292
$861$	108.897	3.71120
$862$	20.5771	0.700860
$863$	−21.3823	−0.727861	−0.363931	−	0.931426i	$-0.618565\pi$
$863$	−21.3823	−0.727861	−0.363931	+	0.931426i	$0.618565\pi$
$864$	17.4081	0.592237
$865$	11.8510	0.402945
$866$	−36.4438	−1.23841
$867$	−67.2873	−2.28520
$868$	11.0681	0.375675
$869$	46.7477	1.58581
$870$	71.3999	2.42068
$871$	55.1145	1.86749
$872$	−11.3256	−0.383533
$873$	92.7809	3.14016
$874$	1.38542	0.0468625
$875$	37.9293	1.28224
$876$	10.0960	0.341114
$877$	54.2271	1.83112	0.915559	−	0.402184i	$-0.131749\pi$
$877$	54.2271	1.83112	0.915559	+	0.402184i	$0.131749\pi$
$878$	25.1091	0.847392
$879$	101.956	3.43888
$880$	−17.3065	−0.583401
$881$	−51.1891	−1.72461	−0.862303	−	0.506393i	$-0.830979\pi$
$881$	−51.1891	−1.72461	−0.862303	+	0.506393i	$0.830979\pi$
$882$	−31.4405	−1.05866
$883$	19.8501	0.668008	0.334004	−	0.942572i	$-0.391600\pi$
$883$	19.8501	0.668008	0.334004	+	0.942572i	$0.391600\pi$
$884$	−37.4558	−1.25977
$885$	38.1248	1.28155
$886$	14.7204	0.494542
$887$	13.0556	0.438365	0.219183	−	0.975684i	$-0.429661\pi$
$887$	13.0556	0.438365	0.219183	+	0.975684i	$0.429661\pi$
$888$	−30.7175	−1.03081
$889$	−11.4999	−0.385695
$890$	67.8775	2.27526
$891$	160.483	5.37637
$892$	−2.68844	−0.0900157
$893$	18.9857	0.635331
$894$	24.4389	0.817360
$895$	−33.6292	−1.12410
$896$	3.29144	0.109959
$897$	−11.3359	−0.378493
$898$	0.194038	0.00647512
$899$	19.7641	0.659169
$900$	67.0445	2.23482
$901$	−13.8425	−0.461160
$902$	−47.1337	−1.56938
$903$	24.6795	0.821281
$904$	18.0569	0.600563
$905$	−49.9627	−1.66082
$906$	28.9259	0.960997
$907$	−45.1445	−1.49900	−0.749499	−	0.662005i	$-0.769706\pi$
$907$	−45.1445	−1.49900	−0.749499	+	0.662005i	$0.769706\pi$
$908$	5.47044	0.181543
$909$	−90.8244	−3.01246
$910$	−73.4619	−2.43524
$911$	−26.7988	−0.887882	−0.443941	−	0.896056i	$-0.646420\pi$
$911$	−26.7988	−0.887882	−0.443941	+	0.896056i	$0.646420\pi$
$912$	8.41787	0.278743
$913$	14.2059	0.470147
$914$	7.03428	0.232673
$915$	−89.0759	−2.94476
$916$	5.70290	0.188429
$917$	13.2476	0.437473
$918$	−106.039	−3.49982
$919$	−46.0766	−1.51993	−0.759963	−	0.649966i	$-0.774783\pi$
$919$	−46.0766	−1.51993	−0.759963	+	0.649966i	$0.774783\pi$
$920$	1.99934	0.0659163
$921$	51.3239	1.69118
$922$	20.5502	0.676785
$923$	−50.4710	−1.66127
$924$	52.5240	1.72791
$925$	−75.0287	−2.46693
$926$	1.78224	0.0585682
$927$	−89.7749	−2.94859
$928$	5.87746	0.192937
$929$	−20.8926	−0.685463	−0.342731	−	0.939433i	$-0.611352\pi$
$929$	−20.8926	−0.685463	−0.342731	+	0.939433i	$0.611352\pi$
$930$	40.8502	1.33953
$931$	−9.64206	−0.316006
$932$	5.54122	0.181509
$933$	−43.0015	−1.40781
$934$	26.1573	0.855894
$935$	105.420	3.44760
$936$	−50.4301	−1.64836
$937$	−18.9974	−0.620617	−0.310308	−	0.950636i	$-0.600432\pi$
$937$	−18.9974	−0.620617	−0.310308	+	0.950636i	$0.600432\pi$
$938$	29.5018	0.963266
$939$	−111.582	−3.64133
$940$	27.3988	0.893649
$941$	26.6083	0.867405	0.433702	−	0.901056i	$-0.357207\pi$
$941$	26.6083	0.867405	0.433702	+	0.901056i	$0.357207\pi$
$942$	62.8326	2.04720
$943$	5.44515	0.177319
$944$	3.13834	0.102144
$945$	−207.975	−6.76541
$946$	−10.6820	−0.347301
$947$	−45.4362	−1.47648	−0.738239	−	0.674539i	$-0.764343\pi$
$947$	−45.4362	−1.47648	−0.738239	+	0.674539i	$0.764343\pi$
$948$	−32.8140	−1.06575
$949$	−18.5490	−0.602126
$950$	20.5610	0.667087
$951$	−35.7905	−1.16059
$952$	−20.0493	−0.649803
$953$	−34.5800	−1.12016	−0.560078	−	0.828440i	$-0.689229\pi$
$953$	−34.5800	−1.12016	−0.560078	+	0.828440i	$0.689229\pi$
$954$	−18.6374	−0.603408
$955$	16.1600	0.522926
$956$	−22.5418	−0.729055
$957$	93.7912	3.03184
$958$	29.9727	0.968373
$959$	−49.5972	−1.60158
$960$	12.1481	0.392077
$961$	−19.6923	−0.635236
$962$	56.4358	1.81956
$963$	−61.8624	−1.99349
$964$	3.66446	0.118024
$965$	−25.0435	−0.806177
$966$	−6.06787	−0.195230
$967$	−48.3039	−1.55335	−0.776675	−	0.629901i	$-0.783095\pi$
$967$	−48.3039	−1.55335	−0.776675	+	0.629901i	$0.783095\pi$
$968$	−11.7339	−0.377141
$969$	−51.2763	−1.64723
$970$	41.0624	1.31844
$971$	6.15130	0.197405	0.0987024	−	0.995117i	$-0.468531\pi$
$971$	6.15130	0.197405	0.0987024	+	0.995117i	$0.468531\pi$
$972$	−60.4244	−1.93811
$973$	27.0263	0.866422
$974$	−10.3610	−0.331988
$975$	−168.235	−5.38784
$976$	−7.33251	−0.234708
$977$	−46.4165	−1.48499	−0.742497	−	0.669849i	$-0.766359\pi$
$977$	−46.4165	−1.48499	−0.742497	+	0.669849i	$0.766359\pi$
$978$	37.2068	1.18974
$979$	89.1641	2.84970
$980$	−13.9147	−0.444490
$981$	92.8853	2.96560
$982$	5.15526	0.164511
$983$	−27.5247	−0.877900	−0.438950	−	0.898511i	$-0.644649\pi$
$983$	−27.5247	−0.877900	−0.438950	+	0.898511i	$0.644649\pi$
$984$	33.0850	1.05471
$985$	38.4017	1.22358
$986$	−35.8018	−1.14016
$987$	−83.1534	−2.64680
$988$	−15.4657	−0.492031
$989$	1.23404	0.0392402
$990$	141.937	4.51104
$991$	−57.6706	−1.83197	−0.915984	−	0.401215i	$-0.868588\pi$
$991$	−57.6706	−1.83197	−0.915984	+	0.401215i	$0.868588\pi$
$992$	3.36269	0.106765
$993$	−89.8978	−2.85282
$994$	−27.0161	−0.856900
$995$	74.2943	2.35529
$996$	−9.97168	−0.315965
$997$	−29.6901	−0.940295	−0.470147	−	0.882588i	$-0.655799\pi$
$997$	−29.6901	−0.940295	−0.470147	+	0.882588i	$0.655799\pi$
$998$	−8.31889	−0.263330
$999$	159.773	5.05498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.1	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.1	✓	89	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.34684 q^{3} +1.00000 q^{4} -3.62971 q^{5} +3.34684 q^{6} -3.29144 q^{7} -1.00000 q^{8} +8.20136 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.34684 q^{3} +1.00000 q^{4} -3.62971 q^{5} +3.34684 q^{6} -3.29144 q^{7} -1.00000 q^{8} +8.20136 q^{9} +3.62971 q^{10} +4.76800 q^{11} -3.34684 q^{12} +6.14900 q^{13} +3.29144 q^{14} +12.1481 q^{15} +1.00000 q^{16} -6.09136 q^{17} -8.20136 q^{18} -2.51517 q^{19} -3.62971 q^{20} +11.0159 q^{21} -4.76800 q^{22} +0.550827 q^{23} +3.34684 q^{24} +8.17480 q^{25} -6.14900 q^{26} -17.4081 q^{27} -3.29144 q^{28} -5.87746 q^{29} -12.1481 q^{30} -3.36269 q^{31} -1.00000 q^{32} -15.9578 q^{33} +6.09136 q^{34} +11.9470 q^{35} +8.20136 q^{36} -9.17804 q^{37} +2.51517 q^{38} -20.5797 q^{39} +3.62971 q^{40} +9.88542 q^{41} -11.0159 q^{42} +2.24034 q^{43} +4.76800 q^{44} -29.7686 q^{45} -0.550827 q^{46} -7.54847 q^{47} -3.34684 q^{48} +3.83357 q^{49} -8.17480 q^{50} +20.3868 q^{51} +6.14900 q^{52} +2.27248 q^{53} +17.4081 q^{54} -17.3065 q^{55} +3.29144 q^{56} +8.41787 q^{57} +5.87746 q^{58} +3.13834 q^{59} +12.1481 q^{60} -7.33251 q^{61} +3.36269 q^{62} -26.9943 q^{63} +1.00000 q^{64} -22.3191 q^{65} +15.9578 q^{66} +8.96318 q^{67} -6.09136 q^{68} -1.84353 q^{69} -11.9470 q^{70} -8.20801 q^{71} -8.20136 q^{72} -3.01659 q^{73} +9.17804 q^{74} -27.3598 q^{75} -2.51517 q^{76} -15.6936 q^{77} +20.5797 q^{78} +9.80445 q^{79} -3.62971 q^{80} +33.6582 q^{81} -9.88542 q^{82} +2.97943 q^{83} +11.0159 q^{84} +22.1099 q^{85} -2.24034 q^{86} +19.6709 q^{87} -4.76800 q^{88} +18.7005 q^{89} +29.7686 q^{90} -20.2390 q^{91} +0.550827 q^{92} +11.2544 q^{93} +7.54847 q^{94} +9.12933 q^{95} +3.34684 q^{96} +11.3129 q^{97} -3.83357 q^{98} +39.1041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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