LMFDB - Embedded newform 8038.2.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.12834 q^{3} +1.00000 q^{4} -2.40910 q^{5} +2.12834 q^{6} +0.412842 q^{7} -1.00000 q^{8} +1.52982 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.12834 q^{3} +1.00000 q^{4} -2.40910 q^{5} +2.12834 q^{6} +0.412842 q^{7} -1.00000 q^{8} +1.52982 q^{9} +2.40910 q^{10} -0.688867 q^{11} -2.12834 q^{12} -4.35737 q^{13} -0.412842 q^{14} +5.12738 q^{15} +1.00000 q^{16} +3.49510 q^{17} -1.52982 q^{18} +6.05088 q^{19} -2.40910 q^{20} -0.878668 q^{21} +0.688867 q^{22} +4.90455 q^{23} +2.12834 q^{24} +0.803779 q^{25} +4.35737 q^{26} +3.12904 q^{27} +0.412842 q^{28} +0.0642078 q^{29} -5.12738 q^{30} +8.89632 q^{31} -1.00000 q^{32} +1.46614 q^{33} -3.49510 q^{34} -0.994580 q^{35} +1.52982 q^{36} -5.56717 q^{37} -6.05088 q^{38} +9.27396 q^{39} +2.40910 q^{40} -6.37795 q^{41} +0.878668 q^{42} +6.78695 q^{43} -0.688867 q^{44} -3.68550 q^{45} -4.90455 q^{46} -2.25867 q^{47} -2.12834 q^{48} -6.82956 q^{49} -0.803779 q^{50} -7.43876 q^{51} -4.35737 q^{52} +7.26546 q^{53} -3.12904 q^{54} +1.65955 q^{55} -0.412842 q^{56} -12.8783 q^{57} -0.0642078 q^{58} +1.34514 q^{59} +5.12738 q^{60} -1.59456 q^{61} -8.89632 q^{62} +0.631574 q^{63} +1.00000 q^{64} +10.4974 q^{65} -1.46614 q^{66} -6.72173 q^{67} +3.49510 q^{68} -10.4385 q^{69} +0.994580 q^{70} +1.05719 q^{71} -1.52982 q^{72} -4.88351 q^{73} +5.56717 q^{74} -1.71071 q^{75} +6.05088 q^{76} -0.284394 q^{77} -9.27396 q^{78} -3.83905 q^{79} -2.40910 q^{80} -11.2491 q^{81} +6.37795 q^{82} +17.4609 q^{83} -0.878668 q^{84} -8.42006 q^{85} -6.78695 q^{86} -0.136656 q^{87} +0.688867 q^{88} -8.36608 q^{89} +3.68550 q^{90} -1.79891 q^{91} +4.90455 q^{92} -18.9344 q^{93} +2.25867 q^{94} -14.5772 q^{95} +2.12834 q^{96} -2.31602 q^{97} +6.82956 q^{98} -1.05384 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−2.12834	−1.22880	−0.614398	−	0.788996i	$-0.710601\pi$
$3$	−2.12834	−1.22880	−0.614398	+	0.788996i	$0.710601\pi$
$4$	1.00000	0.500000
$5$	−2.40910	−1.07738	−0.538692	−	0.842503i	$-0.681081\pi$
$5$	−2.40910	−1.07738	−0.538692	+	0.842503i	$0.681081\pi$
$6$	2.12834	0.868890
$7$	0.412842	0.156040	0.0780198	−	0.996952i	$-0.475140\pi$
$7$	0.412842	0.156040	0.0780198	+	0.996952i	$0.475140\pi$
$8$	−1.00000	−0.353553
$9$	1.52982	0.509940
$10$	2.40910	0.761825
$11$	−0.688867	−0.207701	−0.103851	−	0.994593i	$-0.533116\pi$
$11$	−0.688867	−0.207701	−0.103851	+	0.994593i	$0.533116\pi$
$12$	−2.12834	−0.614398
$13$	−4.35737	−1.20852	−0.604259	−	0.796788i	$-0.706531\pi$
$13$	−4.35737	−1.20852	−0.604259	+	0.796788i	$0.706531\pi$
$14$	−0.412842	−0.110337
$15$	5.12738	1.32389
$16$	1.00000	0.250000
$17$	3.49510	0.847687	0.423843	−	0.905735i	$-0.360681\pi$
$17$	3.49510	0.847687	0.423843	+	0.905735i	$0.360681\pi$
$18$	−1.52982	−0.360582
$19$	6.05088	1.38817	0.694084	−	0.719894i	$-0.255810\pi$
$19$	6.05088	1.38817	0.694084	+	0.719894i	$0.255810\pi$
$20$	−2.40910	−0.538692
$21$	−0.878668	−0.191741
$22$	0.688867	0.146867
$23$	4.90455	1.02267	0.511335	−	0.859381i	$-0.329151\pi$
$23$	4.90455	1.02267	0.511335	+	0.859381i	$0.329151\pi$
$24$	2.12834	0.434445
$25$	0.803779	0.160756
$26$	4.35737	0.854552
$27$	3.12904	0.602184
$28$	0.412842	0.0780198
$29$	0.0642078	0.0119231	0.00596155	−	0.999982i	$-0.498102\pi$
$29$	0.0642078	0.0119231	0.00596155	+	0.999982i	$0.498102\pi$
$30$	−5.12738	−0.936128
$31$	8.89632	1.59783	0.798914	−	0.601446i	$-0.205408\pi$
$31$	8.89632	1.59783	0.798914	+	0.601446i	$0.205408\pi$
$32$	−1.00000	−0.176777
$33$	1.46614	0.255223
$34$	−3.49510	−0.599405
$35$	−0.994580	−0.168115
$36$	1.52982	0.254970
$37$	−5.56717	−0.915237	−0.457619	−	0.889149i	$-0.651297\pi$
$37$	−5.56717	−0.915237	−0.457619	+	0.889149i	$0.651297\pi$
$38$	−6.05088	−0.981582
$39$	9.27396	1.48502
$40$	2.40910	0.380913
$41$	−6.37795	−0.996068	−0.498034	−	0.867157i	$-0.665945\pi$
$41$	−6.37795	−0.996068	−0.498034	+	0.867157i	$0.665945\pi$
$42$	0.878668	0.135581
$43$	6.78695	1.03500	0.517500	−	0.855683i	$-0.326863\pi$
$43$	6.78695	1.03500	0.517500	+	0.855683i	$0.326863\pi$
$44$	−0.688867	−0.103851
$45$	−3.68550	−0.549401
$46$	−4.90455	−0.723137
$47$	−2.25867	−0.329461	−0.164730	−	0.986339i	$-0.552675\pi$
$47$	−2.25867	−0.329461	−0.164730	+	0.986339i	$0.552675\pi$
$48$	−2.12834	−0.307199
$49$	−6.82956	−0.975652
$50$	−0.803779	−0.113671
$51$	−7.43876	−1.04163
$52$	−4.35737	−0.604259
$53$	7.26546	0.997987	0.498994	−	0.866606i	$-0.333703\pi$
$53$	7.26546	0.997987	0.498994	+	0.866606i	$0.333703\pi$
$54$	−3.12904	−0.425808
$55$	1.65955	0.223774
$56$	−0.412842	−0.0551684
$57$	−12.8783	−1.70577
$58$	−0.0642078	−0.00843090
$59$	1.34514	0.175122	0.0875610	−	0.996159i	$-0.472093\pi$
$59$	1.34514	0.175122	0.0875610	+	0.996159i	$0.472093\pi$
$60$	5.12738	0.661943
$61$	−1.59456	−0.204163	−0.102081	−	0.994776i	$-0.532550\pi$
$61$	−1.59456	−0.204163	−0.102081	+	0.994776i	$0.532550\pi$
$62$	−8.89632	−1.12983
$63$	0.631574	0.0795709
$64$	1.00000	0.125000
$65$	10.4974	1.30204
$66$	−1.46614	−0.180470
$67$	−6.72173	−0.821189	−0.410595	−	0.911818i	$-0.634679\pi$
$67$	−6.72173	−0.821189	−0.410595	+	0.911818i	$0.634679\pi$
$68$	3.49510	0.423843
$69$	−10.4385	−1.25665
$70$	0.994580	0.118875
$71$	1.05719	0.125465	0.0627326	−	0.998030i	$-0.480018\pi$
$71$	1.05719	0.125465	0.0627326	+	0.998030i	$0.480018\pi$
$72$	−1.52982	−0.180291
$73$	−4.88351	−0.571571	−0.285786	−	0.958294i	$-0.592255\pi$
$73$	−4.88351	−0.571571	−0.285786	+	0.958294i	$0.592255\pi$
$74$	5.56717	0.647170
$75$	−1.71071	−0.197536
$76$	6.05088	0.694084
$77$	−0.284394	−0.0324097
$78$	−9.27396	−1.05007
$79$	−3.83905	−0.431927	−0.215964	−	0.976401i	$-0.569289\pi$
$79$	−3.83905	−0.431927	−0.215964	+	0.976401i	$0.569289\pi$
$80$	−2.40910	−0.269346
$81$	−11.2491	−1.24990
$82$	6.37795	0.704327
$83$	17.4609	1.91658	0.958291	−	0.285794i	$-0.0922574\pi$
$83$	17.4609	1.91658	0.958291	+	0.285794i	$0.0922574\pi$
$84$	−0.878668	−0.0958705
$85$	−8.42006	−0.913284
$86$	−6.78695	−0.731855
$87$	−0.136656	−0.0146511
$88$	0.688867	0.0734335
$89$	−8.36608	−0.886803	−0.443402	−	0.896323i	$-0.646228\pi$
$89$	−8.36608	−0.886803	−0.443402	+	0.896323i	$0.646228\pi$
$90$	3.68550	0.388485
$91$	−1.79891	−0.188577
$92$	4.90455	0.511335
$93$	−18.9344	−1.96340
$94$	2.25867	0.232964
$95$	−14.5772	−1.49559
$96$	2.12834	0.217223
$97$	−2.31602	−0.235156	−0.117578	−	0.993064i	$-0.537513\pi$
$97$	−2.31602	−0.235156	−0.117578	+	0.993064i	$0.537513\pi$
$98$	6.82956	0.689890
$99$	−1.05384	−0.105915
$100$	0.803779	0.0803779
$101$	−4.22348	−0.420252	−0.210126	−	0.977674i	$-0.567387\pi$
$101$	−4.22348	−0.420252	−0.210126	+	0.977674i	$0.567387\pi$
$102$	7.43876	0.736547
$103$	−5.94656	−0.585932	−0.292966	−	0.956123i	$-0.594642\pi$
$103$	−5.94656	−0.585932	−0.292966	+	0.956123i	$0.594642\pi$
$104$	4.35737	0.427276
$105$	2.11680	0.206579
$106$	−7.26546	−0.705684
$107$	10.1285	0.979160	0.489580	−	0.871958i	$-0.337150\pi$
$107$	10.1285	0.979160	0.489580	+	0.871958i	$0.337150\pi$
$108$	3.12904	0.301092
$109$	−3.65849	−0.350419	−0.175210	−	0.984531i	$-0.556060\pi$
$109$	−3.65849	−0.350419	−0.175210	+	0.984531i	$0.556060\pi$
$110$	−1.65955	−0.158232
$111$	11.8488	1.12464
$112$	0.412842	0.0390099
$113$	−14.6305	−1.37632	−0.688160	−	0.725559i	$-0.741581\pi$
$113$	−14.6305	−1.37632	−0.688160	+	0.725559i	$0.741581\pi$
$114$	12.8783	1.20616
$115$	−11.8156	−1.10181
$116$	0.0642078	0.00596155
$117$	−6.66600	−0.616272
$118$	−1.34514	−0.123830
$119$	1.44293	0.132273
$120$	−5.12738	−0.468064
$121$	−10.5255	−0.956860
$122$	1.59456	0.144365
$123$	13.5744	1.22397
$124$	8.89632	0.798914
$125$	10.1091	0.904188
$126$	−0.631574	−0.0562651
$127$	1.37153	0.121704	0.0608518	−	0.998147i	$-0.480618\pi$
$127$	1.37153	0.121704	0.0608518	+	0.998147i	$0.480618\pi$
$128$	−1.00000	−0.0883883
$129$	−14.4449	−1.27180
$130$	−10.4974	−0.920680
$131$	3.96822	0.346705	0.173352	−	0.984860i	$-0.444540\pi$
$131$	3.96822	0.346705	0.173352	+	0.984860i	$0.444540\pi$
$132$	1.46614	0.127611
$133$	2.49806	0.216609
$134$	6.72173	0.580669
$135$	−7.53818	−0.648783
$136$	−3.49510	−0.299703
$137$	18.9507	1.61907	0.809535	−	0.587071i	$-0.199719\pi$
$137$	18.9507	1.61907	0.809535	+	0.587071i	$0.199719\pi$
$138$	10.4385	0.888588
$139$	−5.69872	−0.483359	−0.241680	−	0.970356i	$-0.577698\pi$
$139$	−5.69872	−0.483359	−0.241680	+	0.970356i	$0.577698\pi$
$140$	−0.994580	−0.0840573
$141$	4.80721	0.404840
$142$	−1.05719	−0.0887173
$143$	3.00165	0.251011
$144$	1.52982	0.127485
$145$	−0.154683	−0.0128458
$146$	4.88351	0.404162
$147$	14.5356	1.19888
$148$	−5.56717	−0.457619
$149$	1.70730	0.139867	0.0699337	−	0.997552i	$-0.477721\pi$
$149$	1.70730	0.139867	0.0699337	+	0.997552i	$0.477721\pi$
$150$	1.71071	0.139679
$151$	2.41476	0.196510	0.0982551	−	0.995161i	$-0.468674\pi$
$151$	2.41476	0.196510	0.0982551	+	0.995161i	$0.468674\pi$
$152$	−6.05088	−0.490791
$153$	5.34688	0.432269
$154$	0.284394	0.0229171
$155$	−21.4322	−1.72147
$156$	9.27396	0.742511
$157$	−2.66871	−0.212986	−0.106493	−	0.994313i	$-0.533962\pi$
$157$	−2.66871	−0.212986	−0.106493	+	0.994313i	$0.533962\pi$
$158$	3.83905	0.305419
$159$	−15.4633	−1.22632
$160$	2.40910	0.190456
$161$	2.02481	0.159577
$162$	11.2491	0.883814
$163$	−1.65547	−0.129666	−0.0648332	−	0.997896i	$-0.520652\pi$
$163$	−1.65547	−0.129666	−0.0648332	+	0.997896i	$0.520652\pi$
$164$	−6.37795	−0.498034
$165$	−3.53209	−0.274973
$166$	−17.4609	−1.35523
$167$	−18.7835	−1.45351	−0.726756	−	0.686896i	$-0.758973\pi$
$167$	−18.7835	−1.45351	−0.726756	+	0.686896i	$0.758973\pi$
$168$	0.878668	0.0677907
$169$	5.98672	0.460517
$170$	8.42006	0.645789
$171$	9.25676	0.707882
$172$	6.78695	0.517500
$173$	24.3918	1.85448	0.927239	−	0.374470i	$-0.122175\pi$
$173$	24.3918	1.85448	0.927239	+	0.374470i	$0.122175\pi$
$174$	0.136656	0.0103599
$175$	0.331834	0.0250843
$176$	−0.688867	−0.0519253
$177$	−2.86291	−0.215189
$178$	8.36608	0.627065
$179$	8.24363	0.616158	0.308079	−	0.951361i	$-0.400314\pi$
$179$	8.24363	0.616158	0.308079	+	0.951361i	$0.400314\pi$
$180$	−3.68550	−0.274701
$181$	14.3309	1.06521	0.532603	−	0.846365i	$-0.321214\pi$
$181$	14.3309	1.06521	0.532603	+	0.846365i	$0.321214\pi$
$182$	1.79891	0.133344
$183$	3.39376	0.250874
$184$	−4.90455	−0.361568
$185$	13.4119	0.986062
$186$	18.9344	1.38834
$187$	−2.40766	−0.176066
$188$	−2.25867	−0.164730
$189$	1.29180	0.0939646
$190$	14.5772	1.05754
$191$	−15.8423	−1.14631	−0.573154	−	0.819447i	$-0.694280\pi$
$191$	−15.8423	−1.14631	−0.573154	+	0.819447i	$0.694280\pi$
$192$	−2.12834	−0.153600
$193$	−15.8537	−1.14118	−0.570589	−	0.821236i	$-0.693285\pi$
$193$	−15.8537	−1.14118	−0.570589	+	0.821236i	$0.693285\pi$
$194$	2.31602	0.166280
$195$	−22.3419	−1.59994
$196$	−6.82956	−0.487826
$197$	17.6910	1.26043	0.630216	−	0.776420i	$-0.282966\pi$
$197$	17.6910	1.26043	0.630216	+	0.776420i	$0.282966\pi$
$198$	1.05384	0.0748934
$199$	−6.26136	−0.443856	−0.221928	−	0.975063i	$-0.571235\pi$
$199$	−6.26136	−0.443856	−0.221928	+	0.975063i	$0.571235\pi$
$200$	−0.803779	−0.0568357
$201$	14.3061	1.00907
$202$	4.22348	0.297163
$203$	0.0265077	0.00186048
$204$	−7.43876	−0.520817
$205$	15.3651	1.07315
$206$	5.94656	0.414317
$207$	7.50308	0.521500
$208$	−4.35737	−0.302130
$209$	−4.16825	−0.288324
$210$	−2.11680	−0.146073
$211$	−5.55288	−0.382276	−0.191138	−	0.981563i	$-0.561218\pi$
$211$	−5.55288	−0.382276	−0.191138	+	0.981563i	$0.561218\pi$
$212$	7.26546	0.498994
$213$	−2.25005	−0.154171
$214$	−10.1285	−0.692371
$215$	−16.3505	−1.11509
$216$	−3.12904	−0.212904
$217$	3.67278	0.249324
$218$	3.65849	0.247784
$219$	10.3938	0.702345
$220$	1.65955	0.111887
$221$	−15.2295	−1.02444
$222$	−11.8488	−0.795241
$223$	7.87765	0.527526	0.263763	−	0.964587i	$-0.415036\pi$
$223$	7.87765	0.527526	0.263763	+	0.964587i	$0.415036\pi$
$224$	−0.412842	−0.0275842
$225$	1.22964	0.0819758
$226$	14.6305	0.973205
$227$	−13.8829	−0.921441	−0.460720	−	0.887545i	$-0.652409\pi$
$227$	−13.8829	−0.921441	−0.460720	+	0.887545i	$0.652409\pi$
$228$	−12.8783	−0.852887
$229$	−17.4618	−1.15391	−0.576954	−	0.816777i	$-0.695759\pi$
$229$	−17.4618	−1.15391	−0.576954	+	0.816777i	$0.695759\pi$
$230$	11.8156	0.779096
$231$	0.605285	0.0398249
$232$	−0.0642078	−0.00421545
$233$	20.7897	1.36198	0.680989	−	0.732293i	$-0.261550\pi$
$233$	20.7897	1.36198	0.680989	+	0.732293i	$0.261550\pi$
$234$	6.66600	0.435770
$235$	5.44137	0.354956
$236$	1.34514	0.0875610
$237$	8.17080	0.530751
$238$	−1.44293	−0.0935310
$239$	−4.80190	−0.310609	−0.155305	−	0.987867i	$-0.549636\pi$
$239$	−4.80190	−0.310609	−0.155305	+	0.987867i	$0.549636\pi$
$240$	5.12738	0.330971
$241$	−0.393550	−0.0253508	−0.0126754	−	0.999920i	$-0.504035\pi$
$241$	−0.393550	−0.0253508	−0.0126754	+	0.999920i	$0.504035\pi$
$242$	10.5255	0.676602
$243$	14.5548	0.933690
$244$	−1.59456	−0.102081
$245$	16.4531	1.05115
$246$	−13.5744	−0.865474
$247$	−26.3660	−1.67763
$248$	−8.89632	−0.564917
$249$	−37.1627	−2.35509
$250$	−10.1091	−0.639358
$251$	−15.8733	−1.00191	−0.500957	−	0.865472i	$-0.667019\pi$
$251$	−15.8733	−1.00191	−0.500957	+	0.865472i	$0.667019\pi$
$252$	0.631574	0.0397854
$253$	−3.37859	−0.212410
$254$	−1.37153	−0.0860574
$255$	17.9207	1.12224
$256$	1.00000	0.0625000
$257$	−2.55223	−0.159204	−0.0796018	−	0.996827i	$-0.525365\pi$
$257$	−2.55223	−0.159204	−0.0796018	+	0.996827i	$0.525365\pi$
$258$	14.4449	0.899301
$259$	−2.29836	−0.142813
$260$	10.4974	0.651019
$261$	0.0982265	0.00608007
$262$	−3.96822	−0.245157
$263$	21.7922	1.34377	0.671884	−	0.740657i	$-0.265485\pi$
$263$	21.7922	1.34377	0.671884	+	0.740657i	$0.265485\pi$
$264$	−1.46614	−0.0902348
$265$	−17.5032	−1.07522
$266$	−2.49806	−0.153166
$267$	17.8059	1.08970
$268$	−6.72173	−0.410595
$269$	−22.1652	−1.35143	−0.675717	−	0.737161i	$-0.736166\pi$
$269$	−22.1652	−1.35143	−0.675717	+	0.737161i	$0.736166\pi$
$270$	7.53818	0.458759
$271$	−8.29267	−0.503744	−0.251872	−	0.967761i	$-0.581046\pi$
$271$	−8.29267	−0.503744	−0.251872	+	0.967761i	$0.581046\pi$
$272$	3.49510	0.211922
$273$	3.82868	0.231722
$274$	−18.9507	−1.14486
$275$	−0.553697	−0.0333892
$276$	−10.4385	−0.628326
$277$	−17.4954	−1.05119	−0.525597	−	0.850734i	$-0.676158\pi$
$277$	−17.4954	−1.05119	−0.525597	+	0.850734i	$0.676158\pi$
$278$	5.69872	0.341787
$279$	13.6098	0.814796
$280$	0.994580	0.0594375
$281$	−7.67985	−0.458141	−0.229071	−	0.973410i	$-0.573569\pi$
$281$	−7.67985	−0.458141	−0.229071	+	0.973410i	$0.573569\pi$
$282$	−4.80721	−0.286265
$283$	3.07719	0.182920	0.0914600	−	0.995809i	$-0.470847\pi$
$283$	3.07719	0.182920	0.0914600	+	0.995809i	$0.470847\pi$
$284$	1.05719	0.0627326
$285$	31.0252	1.83777
$286$	−3.00165	−0.177491
$287$	−2.63309	−0.155426
$288$	−1.52982	−0.0901455
$289$	−4.78426	−0.281427
$290$	0.154683	0.00908332
$291$	4.92926	0.288958
$292$	−4.88351	−0.285786
$293$	25.0365	1.46265	0.731324	−	0.682031i	$-0.238903\pi$
$293$	25.0365	1.46265	0.731324	+	0.682031i	$0.238903\pi$
$294$	−14.5356	−0.847734
$295$	−3.24057	−0.188674
$296$	5.56717	0.323585
$297$	−2.15549	−0.125074
$298$	−1.70730	−0.0989012
$299$	−21.3710	−1.23592
$300$	−1.71071	−0.0987680
$301$	2.80194	0.161501
$302$	−2.41476	−0.138954
$303$	8.98899	0.516404
$304$	6.05088	0.347042
$305$	3.84146	0.219961
$306$	−5.34688	−0.305661
$307$	16.7591	0.956491	0.478246	−	0.878226i	$-0.341273\pi$
$307$	16.7591	0.956491	0.478246	+	0.878226i	$0.341273\pi$
$308$	−0.284394	−0.0162048
$309$	12.6563	0.719991
$310$	21.4322	1.21727
$311$	−17.5898	−0.997426	−0.498713	−	0.866767i	$-0.666194\pi$
$311$	−17.5898	−0.997426	−0.498713	+	0.866767i	$0.666194\pi$
$312$	−9.27396	−0.525035
$313$	−29.3505	−1.65899	−0.829495	−	0.558514i	$-0.811372\pi$
$313$	−29.3505	−1.65899	−0.829495	+	0.558514i	$0.811372\pi$
$314$	2.66871	0.150604
$315$	−1.52153	−0.0857284
$316$	−3.83905	−0.215964
$317$	4.63705	0.260442	0.130221	−	0.991485i	$-0.458431\pi$
$317$	4.63705	0.260442	0.130221	+	0.991485i	$0.458431\pi$
$318$	15.4633	0.867141
$319$	−0.0442307	−0.00247644
$320$	−2.40910	−0.134673
$321$	−21.5569	−1.20319
$322$	−2.02481	−0.112838
$323$	21.1484	1.17673
$324$	−11.2491	−0.624951
$325$	−3.50237	−0.194276
$326$	1.65547	0.0916880
$327$	7.78649	0.430594
$328$	6.37795	0.352163
$329$	−0.932474	−0.0514090
$330$	3.53209	0.194435
$331$	21.6619	1.19064	0.595322	−	0.803487i	$-0.297024\pi$
$331$	21.6619	1.19064	0.595322	+	0.803487i	$0.297024\pi$
$332$	17.4609	0.958291
$333$	−8.51677	−0.466716
$334$	18.7835	1.02779
$335$	16.1933	0.884736
$336$	−0.878668	−0.0479352
$337$	1.24228	0.0676712	0.0338356	−	0.999427i	$-0.489228\pi$
$337$	1.24228	0.0676712	0.0338356	+	0.999427i	$0.489228\pi$
$338$	−5.98672	−0.325634
$339$	31.1386	1.69122
$340$	−8.42006	−0.456642
$341$	−6.12839	−0.331871
$342$	−9.25676	−0.500548
$343$	−5.70943	−0.308280
$344$	−6.78695	−0.365928
$345$	25.1475	1.35390
$346$	−24.3918	−1.31131
$347$	3.21435	0.172556	0.0862778	−	0.996271i	$-0.472503\pi$
$347$	3.21435	0.172556	0.0862778	+	0.996271i	$0.472503\pi$
$348$	−0.136656	−0.00732553
$349$	15.8697	0.849485	0.424742	−	0.905314i	$-0.360365\pi$
$349$	15.8697	0.849485	0.424742	+	0.905314i	$0.360365\pi$
$350$	−0.331834	−0.0177373
$351$	−13.6344	−0.727750
$352$	0.688867	0.0367168
$353$	−15.1288	−0.805223	−0.402612	−	0.915371i	$-0.631897\pi$
$353$	−15.1288	−0.805223	−0.402612	+	0.915371i	$0.631897\pi$
$354$	2.86291	0.152162
$355$	−2.54688	−0.135174
$356$	−8.36608	−0.443402
$357$	−3.07103	−0.162536
$358$	−8.24363	−0.435689
$359$	−11.1770	−0.589899	−0.294950	−	0.955513i	$-0.595303\pi$
$359$	−11.1770	−0.589899	−0.294950	+	0.955513i	$0.595303\pi$
$360$	3.68550	0.194243
$361$	17.6132	0.927008
$362$	−14.3309	−0.753214
$363$	22.4017	1.17579
$364$	−1.79891	−0.0942884
$365$	11.7649	0.615802
$366$	−3.39376	−0.177395
$367$	33.4706	1.74715	0.873577	−	0.486687i	$-0.161795\pi$
$367$	33.4706	1.74715	0.873577	+	0.486687i	$0.161795\pi$
$368$	4.90455	0.255667
$369$	−9.75712	−0.507935
$370$	−13.4119	−0.697251
$371$	2.99949	0.155726
$372$	−18.9344	−0.981702
$373$	11.7721	0.609538	0.304769	−	0.952426i	$-0.401421\pi$
$373$	11.7721	0.609538	0.304769	+	0.952426i	$0.401421\pi$
$374$	2.40766	0.124497
$375$	−21.5156	−1.11106
$376$	2.25867	0.116482
$377$	−0.279778	−0.0144093
$378$	−1.29180	−0.0664430
$379$	24.9220	1.28016	0.640080	−	0.768308i	$-0.278901\pi$
$379$	24.9220	1.28016	0.640080	+	0.768308i	$0.278901\pi$
$380$	−14.5772	−0.747794
$381$	−2.91908	−0.149549
$382$	15.8423	0.810563
$383$	−25.7303	−1.31476	−0.657379	−	0.753560i	$-0.728335\pi$
$383$	−25.7303	−1.31476	−0.657379	+	0.753560i	$0.728335\pi$
$384$	2.12834	0.108611
$385$	0.685133	0.0349176
$386$	15.8537	0.806934
$387$	10.3828	0.527788
$388$	−2.31602	−0.117578
$389$	8.67673	0.439928	0.219964	−	0.975508i	$-0.429406\pi$
$389$	8.67673	0.439928	0.219964	+	0.975508i	$0.429406\pi$
$390$	22.3419	1.13133
$391$	17.1419	0.866904
$392$	6.82956	0.344945
$393$	−8.44571	−0.426030
$394$	−17.6910	−0.891260
$395$	9.24868	0.465352
$396$	−1.05384	−0.0529576
$397$	−5.37179	−0.269602	−0.134801	−	0.990873i	$-0.543040\pi$
$397$	−5.37179	−0.269602	−0.134801	+	0.990873i	$0.543040\pi$
$398$	6.26136	0.313854
$399$	−5.31671	−0.266169
$400$	0.803779	0.0401889
$401$	20.5641	1.02692	0.513462	−	0.858112i	$-0.328363\pi$
$401$	20.5641	1.02692	0.513462	+	0.858112i	$0.328363\pi$
$402$	−14.3061	−0.713523
$403$	−38.7646	−1.93100
$404$	−4.22348	−0.210126
$405$	27.1003	1.34662
$406$	−0.0265077	−0.00131556
$407$	3.83504	0.190096
$408$	7.43876	0.368273
$409$	27.3530	1.35252	0.676261	−	0.736663i	$-0.263599\pi$
$409$	27.3530	1.35252	0.676261	+	0.736663i	$0.263599\pi$
$410$	−15.3651	−0.758830
$411$	−40.3336	−1.98951
$412$	−5.94656	−0.292966
$413$	0.555329	0.0273260
$414$	−7.50308	−0.368756
$415$	−42.0651	−2.06489
$416$	4.35737	0.213638
$417$	12.1288	0.593950
$418$	4.16825	0.203876
$419$	6.07380	0.296724	0.148362	−	0.988933i	$-0.452600\pi$
$419$	6.07380	0.296724	0.148362	+	0.988933i	$0.452600\pi$
$420$	2.11680	0.103289
$421$	−1.63530	−0.0796995	−0.0398498	−	0.999206i	$-0.512688\pi$
$421$	−1.63530	−0.0796995	−0.0398498	+	0.999206i	$0.512688\pi$
$422$	5.55288	0.270310
$423$	−3.45536	−0.168005
$424$	−7.26546	−0.352842
$425$	2.80929	0.136271
$426$	2.25005	0.109015
$427$	−0.658302	−0.0318575
$428$	10.1285	0.489580
$429$	−6.38853	−0.308441
$430$	16.3505	0.788489
$431$	35.1462	1.69293	0.846466	−	0.532443i	$-0.178726\pi$
$431$	35.1462	1.69293	0.846466	+	0.532443i	$0.178726\pi$
$432$	3.12904	0.150546
$433$	1.48649	0.0714361	0.0357180	−	0.999362i	$-0.488628\pi$
$433$	1.48649	0.0714361	0.0357180	+	0.999362i	$0.488628\pi$
$434$	−3.67278	−0.176299
$435$	0.329218	0.0157848
$436$	−3.65849	−0.175210
$437$	29.6769	1.41964
$438$	−10.3938	−0.496633
$439$	20.7289	0.989335	0.494667	−	0.869082i	$-0.335290\pi$
$439$	20.7289	0.989335	0.494667	+	0.869082i	$0.335290\pi$
$440$	−1.65955	−0.0791161
$441$	−10.4480	−0.497524
$442$	15.2295	0.724392
$443$	15.5590	0.739229	0.369615	−	0.929185i	$-0.379490\pi$
$443$	15.5590	0.739229	0.369615	+	0.929185i	$0.379490\pi$
$444$	11.8488	0.562320
$445$	20.1548	0.955427
$446$	−7.87765	−0.373017
$447$	−3.63371	−0.171868
$448$	0.412842	0.0195050
$449$	−4.08339	−0.192707	−0.0963535	−	0.995347i	$-0.530718\pi$
$449$	−4.08339	−0.192707	−0.0963535	+	0.995347i	$0.530718\pi$
$450$	−1.22964	−0.0579657
$451$	4.39356	0.206885
$452$	−14.6305	−0.688160
$453$	−5.13942	−0.241471
$454$	13.8829	0.651557
$455$	4.33376	0.203170
$456$	12.8783	0.603082
$457$	−0.725445	−0.0339349	−0.0169675	−	0.999856i	$-0.505401\pi$
$457$	−0.725445	−0.0339349	−0.0169675	+	0.999856i	$0.505401\pi$
$458$	17.4618	0.815936
$459$	10.9363	0.510463
$460$	−11.8156	−0.550904
$461$	−29.6625	−1.38152	−0.690760	−	0.723084i	$-0.742724\pi$
$461$	−29.6625	−1.38152	−0.690760	+	0.723084i	$0.742724\pi$
$462$	−0.605285	−0.0281604
$463$	−13.7467	−0.638862	−0.319431	−	0.947610i	$-0.603492\pi$
$463$	−13.7467	−0.638862	−0.319431	+	0.947610i	$0.603492\pi$
$464$	0.0642078	0.00298077
$465$	45.6149	2.11534
$466$	−20.7897	−0.963064
$467$	−12.1513	−0.562296	−0.281148	−	0.959664i	$-0.590715\pi$
$467$	−12.1513	−0.562296	−0.281148	+	0.959664i	$0.590715\pi$
$468$	−6.66600	−0.308136
$469$	−2.77501	−0.128138
$470$	−5.44137	−0.250992
$471$	5.67992	0.261717
$472$	−1.34514	−0.0619149
$473$	−4.67531	−0.214971
$474$	−8.17080	−0.375297
$475$	4.86357	0.223156
$476$	1.44293	0.0661364
$477$	11.1148	0.508914
$478$	4.80190	0.219634
$479$	0.759888	0.0347202	0.0173601	−	0.999849i	$-0.494474\pi$
$479$	0.759888	0.0347202	0.0173601	+	0.999849i	$0.494474\pi$
$480$	−5.12738	−0.234032
$481$	24.2583	1.10608
$482$	0.393550	0.0179257
$483$	−4.30947	−0.196088
$484$	−10.5255	−0.478430
$485$	5.57952	0.253353
$486$	−14.5548	−0.660219
$487$	−6.02677	−0.273099	−0.136550	−	0.990633i	$-0.543601\pi$
$487$	−6.02677	−0.273099	−0.136550	+	0.990633i	$0.543601\pi$
$488$	1.59456	0.0721824
$489$	3.52340	0.159334
$490$	−16.4531	−0.743276
$491$	−5.77206	−0.260489	−0.130245	−	0.991482i	$-0.541576\pi$
$491$	−5.77206	−0.260489	−0.130245	+	0.991482i	$0.541576\pi$
$492$	13.5744	0.611983
$493$	0.224413	0.0101071
$494$	26.3660	1.18626
$495$	2.53882	0.114111
$496$	8.89632	0.399457
$497$	0.436452	0.0195776
$498$	37.1627	1.66530
$499$	3.09211	0.138422	0.0692109	−	0.997602i	$-0.477952\pi$
$499$	3.09211	0.138422	0.0692109	+	0.997602i	$0.477952\pi$
$500$	10.1091	0.452094
$501$	39.9776	1.78607
$502$	15.8733	0.708461
$503$	−32.2109	−1.43621	−0.718106	−	0.695933i	$-0.754991\pi$
$503$	−32.2109	−1.43621	−0.718106	+	0.695933i	$0.754991\pi$
$504$	−0.631574	−0.0281326
$505$	10.1748	0.452773
$506$	3.37859	0.150196
$507$	−12.7418	−0.565881
$508$	1.37153	0.0608518
$509$	−8.26649	−0.366406	−0.183203	−	0.983075i	$-0.558647\pi$
$509$	−8.26649	−0.366406	−0.183203	+	0.983075i	$0.558647\pi$
$510$	−17.9207	−0.793543
$511$	−2.01612	−0.0891878
$512$	−1.00000	−0.0441942
$513$	18.9334	0.835932
$514$	2.55223	0.112574
$515$	14.3259	0.631274
$516$	−14.4449	−0.635902
$517$	1.55592	0.0684295
$518$	2.29836	0.100984
$519$	−51.9141	−2.27878
$520$	−10.4974	−0.460340
$521$	18.5253	0.811608	0.405804	−	0.913960i	$-0.366992\pi$
$521$	18.5253	0.811608	0.405804	+	0.913960i	$0.366992\pi$
$522$	−0.0982265	−0.00429926
$523$	14.5046	0.634243	0.317122	−	0.948385i	$-0.397284\pi$
$523$	14.5046	0.634243	0.317122	+	0.948385i	$0.397284\pi$
$524$	3.96822	0.173352
$525$	−0.706254	−0.0308235
$526$	−21.7922	−0.950187
$527$	31.0936	1.35446
$528$	1.46614	0.0638057
$529$	1.05464	0.0458538
$530$	17.5032	0.760292
$531$	2.05782	0.0893017
$532$	2.49806	0.108305
$533$	27.7911	1.20377
$534$	−17.8059	−0.770535
$535$	−24.4006	−1.05493
$536$	6.72173	0.290334
$537$	−17.5452	−0.757132
$538$	22.1652	0.955608
$539$	4.70466	0.202644
$540$	−7.53818	−0.324391
$541$	19.9758	0.858828	0.429414	−	0.903108i	$-0.358720\pi$
$541$	19.9758	0.858828	0.429414	+	0.903108i	$0.358720\pi$
$542$	8.29267	0.356201
$543$	−30.5009	−1.30892
$544$	−3.49510	−0.149851
$545$	8.81367	0.377536
$546$	−3.82868	−0.163853
$547$	19.6453	0.839971	0.419985	−	0.907531i	$-0.362035\pi$
$547$	19.6453	0.839971	0.419985	+	0.907531i	$0.362035\pi$
$548$	18.9507	0.809535
$549$	−2.43939	−0.104111
$550$	0.553697	0.0236097
$551$	0.388514	0.0165513
$552$	10.4385	0.444294
$553$	−1.58492	−0.0673978
$554$	17.4954	0.743306
$555$	−28.5450	−1.21167
$556$	−5.69872	−0.241680
$557$	10.2192	0.433001	0.216501	−	0.976282i	$-0.430536\pi$
$557$	10.2192	0.433001	0.216501	+	0.976282i	$0.430536\pi$
$558$	−13.6098	−0.576148
$559$	−29.5733	−1.25082
$560$	−0.994580	−0.0420287
$561$	5.12432	0.216349
$562$	7.67985	0.323955
$563$	6.08787	0.256573	0.128287	−	0.991737i	$-0.459052\pi$
$563$	6.08787	0.256573	0.128287	+	0.991737i	$0.459052\pi$
$564$	4.80721	0.202420
$565$	35.2463	1.48282
$566$	−3.07719	−0.129344
$567$	−4.64411	−0.195034
$568$	−1.05719	−0.0443586
$569$	−5.60775	−0.235089	−0.117545	−	0.993068i	$-0.537502\pi$
$569$	−5.60775	−0.235089	−0.117545	+	0.993068i	$0.537502\pi$
$570$	−31.0252	−1.29950
$571$	28.6870	1.20051	0.600257	−	0.799807i	$-0.295065\pi$
$571$	28.6870	1.20051	0.600257	+	0.799807i	$0.295065\pi$
$572$	3.00165	0.125505
$573$	33.7178	1.40858
$574$	2.63309	0.109903
$575$	3.94218	0.164400
$576$	1.52982	0.0637425
$577$	−14.3509	−0.597434	−0.298717	−	0.954342i	$-0.596559\pi$
$577$	−14.3509	−0.597434	−0.298717	+	0.954342i	$0.596559\pi$
$578$	4.78426	0.198999
$579$	33.7421	1.40227
$580$	−0.154683	−0.00642288
$581$	7.20859	0.299063
$582$	−4.92926	−0.204325
$583$	−5.00494	−0.207283
$584$	4.88351	0.202081
$585$	16.0591	0.663961
$586$	−25.0365	−1.03425
$587$	24.0355	0.992050	0.496025	−	0.868308i	$-0.334792\pi$
$587$	24.0355	0.992050	0.496025	+	0.868308i	$0.334792\pi$
$588$	14.5356	0.599439
$589$	53.8306	2.21805
$590$	3.24057	0.133412
$591$	−37.6524	−1.54881
$592$	−5.56717	−0.228809
$593$	29.7003	1.21964	0.609822	−	0.792538i	$-0.291241\pi$
$593$	29.7003	1.21964	0.609822	+	0.792538i	$0.291241\pi$
$594$	2.15549	0.0884409
$595$	−3.47616	−0.142509
$596$	1.70730	0.0699337
$597$	13.3263	0.545409
$598$	21.3710	0.873924
$599$	42.7999	1.74876	0.874378	−	0.485246i	$-0.161270\pi$
$599$	42.7999	1.74876	0.874378	+	0.485246i	$0.161270\pi$
$600$	1.71071	0.0698395
$601$	1.26918	0.0517710	0.0258855	−	0.999665i	$-0.491759\pi$
$601$	1.26918	0.0517710	0.0258855	+	0.999665i	$0.491759\pi$
$602$	−2.80194	−0.114198
$603$	−10.2830	−0.418757
$604$	2.41476	0.0982551
$605$	25.3569	1.03091
$606$	−8.98899	−0.365153
$607$	33.3911	1.35530	0.677652	−	0.735383i	$-0.262998\pi$
$607$	33.3911	1.35530	0.677652	+	0.735383i	$0.262998\pi$
$608$	−6.05088	−0.245396
$609$	−0.0564173	−0.00228615
$610$	−3.84146	−0.155536
$611$	9.84187	0.398160
$612$	5.34688	0.216135
$613$	−29.9651	−1.21028	−0.605139	−	0.796120i	$-0.706882\pi$
$613$	−29.9651	−1.21028	−0.605139	+	0.796120i	$0.706882\pi$
$614$	−16.7591	−0.676341
$615$	−32.7022	−1.31868
$616$	0.284394	0.0114585
$617$	45.3426	1.82542	0.912711	−	0.408605i	$-0.133985\pi$
$617$	45.3426	1.82542	0.912711	+	0.408605i	$0.133985\pi$
$618$	−12.6563	−0.509111
$619$	3.00079	0.120612	0.0603060	−	0.998180i	$-0.480792\pi$
$619$	3.00079	0.120612	0.0603060	+	0.998180i	$0.480792\pi$
$620$	−21.4322	−0.860736
$621$	15.3465	0.615835
$622$	17.5898	0.705287
$623$	−3.45387	−0.138376
$624$	9.27396	0.371256
$625$	−28.3728	−1.13491
$626$	29.3505	1.17308
$627$	8.87145	0.354292
$628$	−2.66871	−0.106493
$629$	−19.4578	−0.775835
$630$	1.52153	0.0606191
$631$	−21.0483	−0.837920	−0.418960	−	0.908005i	$-0.637605\pi$
$631$	−21.0483	−0.837920	−0.418960	+	0.908005i	$0.637605\pi$
$632$	3.83905	0.152709
$633$	11.8184	0.469739
$634$	−4.63705	−0.184161
$635$	−3.30416	−0.131121
$636$	−15.4633	−0.613161
$637$	29.7590	1.17909
$638$	0.0442307	0.00175111
$639$	1.61731	0.0639797
$640$	2.40910	0.0952282
$641$	−24.4778	−0.966817	−0.483408	−	0.875395i	$-0.660601\pi$
$641$	−24.4778	−0.966817	−0.483408	+	0.875395i	$0.660601\pi$
$642$	21.5569	0.850783
$643$	20.0893	0.792243	0.396122	−	0.918198i	$-0.370356\pi$
$643$	20.0893	0.792243	0.396122	+	0.918198i	$0.370356\pi$
$644$	2.02481	0.0797886
$645$	34.7993	1.37022
$646$	−21.1484	−0.832074
$647$	41.6170	1.63613	0.818066	−	0.575124i	$-0.195046\pi$
$647$	41.6170	1.63613	0.818066	+	0.575124i	$0.195046\pi$
$648$	11.2491	0.441907
$649$	−0.926621	−0.0363731
$650$	3.50237	0.137374
$651$	−7.81691	−0.306369
$652$	−1.65547	−0.0648332
$653$	−19.9007	−0.778775	−0.389388	−	0.921074i	$-0.627313\pi$
$653$	−19.9007	−0.778775	−0.389388	+	0.921074i	$0.627313\pi$
$654$	−7.78649	−0.304476
$655$	−9.55985	−0.373534
$656$	−6.37795	−0.249017
$657$	−7.47089	−0.291467
$658$	0.932474	0.0363516
$659$	33.3653	1.29973	0.649864	−	0.760051i	$-0.274826\pi$
$659$	33.3653	1.29973	0.649864	+	0.760051i	$0.274826\pi$
$660$	−3.53209	−0.137486
$661$	30.4861	1.18577	0.592886	−	0.805286i	$-0.297988\pi$
$661$	30.4861	1.18577	0.592886	+	0.805286i	$0.297988\pi$
$662$	−21.6619	−0.841913
$663$	32.4135	1.25883
$664$	−17.4609	−0.677614
$665$	−6.01808	−0.233371
$666$	8.51677	0.330018
$667$	0.314911	0.0121934
$668$	−18.7835	−0.726756
$669$	−16.7663	−0.648222
$670$	−16.1933	−0.625603
$671$	1.09844	0.0424048
$672$	0.878668	0.0338953
$673$	−0.548251	−0.0211335	−0.0105668	−	0.999944i	$-0.503364\pi$
$673$	−0.548251	−0.0211335	−0.0105668	+	0.999944i	$0.503364\pi$
$674$	−1.24228	−0.0478507
$675$	2.51505	0.0968045
$676$	5.98672	0.230258
$677$	41.2567	1.58562	0.792812	−	0.609467i	$-0.208616\pi$
$677$	41.2567	1.58562	0.792812	+	0.609467i	$0.208616\pi$
$678$	−31.1386	−1.19587
$679$	−0.956149	−0.0366936
$680$	8.42006	0.322895
$681$	29.5475	1.13226
$682$	6.12839	0.234668
$683$	−29.4895	−1.12838	−0.564192	−	0.825644i	$-0.690812\pi$
$683$	−29.4895	−1.12838	−0.564192	+	0.825644i	$0.690812\pi$
$684$	9.25676	0.353941
$685$	−45.6543	−1.74436
$686$	5.70943	0.217987
$687$	37.1646	1.41792
$688$	6.78695	0.258750
$689$	−31.6583	−1.20609
$690$	−25.1475	−0.957350
$691$	−37.5848	−1.42979	−0.714897	−	0.699230i	$-0.753526\pi$
$691$	−37.5848	−1.42979	−0.714897	+	0.699230i	$0.753526\pi$
$692$	24.3918	0.927239
$693$	−0.435071	−0.0165270
$694$	−3.21435	−0.122015
$695$	13.7288	0.520763
$696$	0.136656	0.00517993
$697$	−22.2916	−0.844354
$698$	−15.8697	−0.600676
$699$	−44.2475	−1.67359
$700$	0.331834	0.0125421
$701$	−15.4951	−0.585241	−0.292621	−	0.956229i	$-0.594527\pi$
$701$	−15.4951	−0.585241	−0.292621	+	0.956229i	$0.594527\pi$
$702$	13.6344	0.514597
$703$	−33.6863	−1.27050
$704$	−0.688867	−0.0259627
$705$	−11.5811	−0.436168
$706$	15.1288	0.569379
$707$	−1.74363	−0.0655760
$708$	−2.86291	−0.107595
$709$	−13.8912	−0.521694	−0.260847	−	0.965380i	$-0.584002\pi$
$709$	−13.8912	−0.521694	−0.260847	+	0.965380i	$0.584002\pi$
$710$	2.54688	0.0955826
$711$	−5.87306	−0.220257
$712$	8.36608	0.313532
$713$	43.6325	1.63405
$714$	3.07103	0.114931
$715$	−7.23129	−0.270435
$716$	8.24363	0.308079
$717$	10.2201	0.381675
$718$	11.1770	0.417122
$719$	22.9205	0.854792	0.427396	−	0.904064i	$-0.359431\pi$
$719$	22.9205	0.854792	0.427396	+	0.904064i	$0.359431\pi$
$720$	−3.68550	−0.137350
$721$	−2.45499	−0.0914287
$722$	−17.6132	−0.655494
$723$	0.837608	0.0311510
$724$	14.3309	0.532603
$725$	0.0516089	0.00191671
$726$	−22.4017	−0.831406
$727$	36.1329	1.34010	0.670048	−	0.742318i	$-0.266274\pi$
$727$	36.1329	1.34010	0.670048	+	0.742318i	$0.266274\pi$
$728$	1.79891	0.0666720
$729$	2.76983	0.102586
$730$	−11.7649	−0.435437
$731$	23.7211	0.877356
$732$	3.39376	0.125437
$733$	14.2513	0.526384	0.263192	−	0.964743i	$-0.415225\pi$
$733$	14.2513	0.526384	0.263192	+	0.964743i	$0.415225\pi$
$734$	−33.4706	−1.23542
$735$	−35.0178	−1.29165
$736$	−4.90455	−0.180784
$737$	4.63038	0.170562
$738$	9.75712	0.359164
$739$	−2.36621	−0.0870425	−0.0435212	−	0.999053i	$-0.513858\pi$
$739$	−2.36621	−0.0870425	−0.0435212	+	0.999053i	$0.513858\pi$
$740$	13.4119	0.493031
$741$	56.1156	2.06146
$742$	−2.99949	−0.110115
$743$	−1.73805	−0.0637628	−0.0318814	−	0.999492i	$-0.510150\pi$
$743$	−1.73805	−0.0637628	−0.0318814	+	0.999492i	$0.510150\pi$
$744$	18.9344	0.694168
$745$	−4.11306	−0.150691
$746$	−11.7721	−0.431009
$747$	26.7120	0.977342
$748$	−2.40766	−0.0880328
$749$	4.18148	0.152788
$750$	21.5156	0.785640
$751$	−21.8218	−0.796290	−0.398145	−	0.917322i	$-0.630346\pi$
$751$	−21.8218	−0.796290	−0.398145	+	0.917322i	$0.630346\pi$
$752$	−2.25867	−0.0823652
$753$	33.7838	1.23115
$754$	0.279778	0.0101889
$755$	−5.81740	−0.211717
$756$	1.29180	0.0469823
$757$	−15.7984	−0.574202	−0.287101	−	0.957900i	$-0.592692\pi$
$757$	−15.7984	−0.574202	−0.287101	+	0.957900i	$0.592692\pi$
$758$	−24.9220	−0.905210
$759$	7.19077	0.261008
$760$	14.5772	0.528770
$761$	−40.3954	−1.46433	−0.732166	−	0.681126i	$-0.761491\pi$
$761$	−40.3954	−1.46433	−0.732166	+	0.681126i	$0.761491\pi$
$762$	2.91908	0.105747
$763$	−1.51038	−0.0546793
$764$	−15.8423	−0.573154
$765$	−12.8812	−0.465720
$766$	25.7303	0.929674
$767$	−5.86127	−0.211638
$768$	−2.12834	−0.0767998
$769$	−1.24962	−0.0450626	−0.0225313	−	0.999746i	$-0.507173\pi$
$769$	−1.24962	−0.0450626	−0.0225313	+	0.999746i	$0.507173\pi$
$770$	−0.685133	−0.0246905
$771$	5.43200	0.195629
$772$	−15.8537	−0.570589
$773$	24.1091	0.867142	0.433571	−	0.901119i	$-0.357253\pi$
$773$	24.1091	0.867142	0.433571	+	0.901119i	$0.357253\pi$
$774$	−10.3828	−0.373202
$775$	7.15068	0.256860
$776$	2.31602	0.0831401
$777$	4.89169	0.175488
$778$	−8.67673	−0.311076
$779$	−38.5922	−1.38271
$780$	−22.3419	−0.799970
$781$	−0.728263	−0.0260593
$782$	−17.1419	−0.612994
$783$	0.200909	0.00717989
$784$	−6.82956	−0.243913
$785$	6.42920	0.229468
$786$	8.44571	0.301249
$787$	43.3893	1.54666	0.773330	−	0.634003i	$-0.218589\pi$
$787$	43.3893	1.54666	0.773330	+	0.634003i	$0.218589\pi$
$788$	17.6910	0.630216
$789$	−46.3812	−1.65122
$790$	−9.24868	−0.329053
$791$	−6.04008	−0.214760
$792$	1.05384	0.0374467
$793$	6.94810	0.246734
$794$	5.37179	0.190638
$795$	37.2528	1.32122
$796$	−6.26136	−0.221928
$797$	50.3203	1.78244	0.891218	−	0.453575i	$-0.149852\pi$
$797$	50.3203	1.78244	0.891218	+	0.453575i	$0.149852\pi$
$798$	5.31671	0.188210
$799$	−7.89428	−0.279280
$800$	−0.803779	−0.0284179
$801$	−12.7986	−0.452217
$802$	−20.5641	−0.726145
$803$	3.36409	0.118716
$804$	14.3061	0.504537
$805$	−4.87797	−0.171926
$806$	38.7646	1.36543
$807$	47.1749	1.66064
$808$	4.22348	0.148582
$809$	−29.8868	−1.05077	−0.525383	−	0.850866i	$-0.676078\pi$
$809$	−29.8868	−1.05077	−0.525383	+	0.850866i	$0.676078\pi$
$810$	−27.1003	−0.952206
$811$	31.9425	1.12165	0.560826	−	0.827934i	$-0.310484\pi$
$811$	31.9425	1.12165	0.560826	+	0.827934i	$0.310484\pi$
$812$	0.0265077	0.000930238 0
$813$	17.6496	0.618999
$814$	−3.83504	−0.134418
$815$	3.98820	0.139700
$816$	−7.43876	−0.260409
$817$	41.0670	1.43675
$818$	−27.3530	−0.956377
$819$	−2.75201	−0.0961629
$820$	15.3651	0.536574
$821$	22.4361	0.783026	0.391513	−	0.920173i	$-0.371952\pi$
$821$	22.4361	0.783026	0.391513	+	0.920173i	$0.371952\pi$
$822$	40.3336	1.40679
$823$	−22.6093	−0.788111	−0.394055	−	0.919087i	$-0.628928\pi$
$823$	−22.6093	−0.788111	−0.394055	+	0.919087i	$0.628928\pi$
$824$	5.94656	0.207158
$825$	1.17845	0.0410285
$826$	−0.555329	−0.0193224
$827$	−42.6328	−1.48249	−0.741243	−	0.671236i	$-0.765764\pi$
$827$	−42.6328	−1.48249	−0.741243	+	0.671236i	$0.765764\pi$
$828$	7.50308	0.260750
$829$	−3.01694	−0.104783	−0.0523913	−	0.998627i	$-0.516684\pi$
$829$	−3.01694	−0.104783	−0.0523913	+	0.998627i	$0.516684\pi$
$830$	42.0651	1.46010
$831$	37.2360	1.29170
$832$	−4.35737	−0.151065
$833$	−23.8700	−0.827047
$834$	−12.1288	−0.419986
$835$	45.2514	1.56599
$836$	−4.16825	−0.144162
$837$	27.8369	0.962185
$838$	−6.07380	−0.209816
$839$	7.82247	0.270062	0.135031	−	0.990841i	$-0.456887\pi$
$839$	7.82247	0.270062	0.135031	+	0.990841i	$0.456887\pi$
$840$	−2.11680	−0.0730366
$841$	−28.9959	−0.999858
$842$	1.63530	0.0563561
$843$	16.3453	0.562962
$844$	−5.55288	−0.191138
$845$	−14.4226	−0.496153
$846$	3.45536	0.118798
$847$	−4.34535	−0.149308
$848$	7.26546	0.249497
$849$	−6.54930	−0.224771
$850$	−2.80929	−0.0963578
$851$	−27.3045	−0.935986
$852$	−2.25005	−0.0770856
$853$	−49.1360	−1.68239	−0.841193	−	0.540735i	$-0.818146\pi$
$853$	−49.1360	−1.68239	−0.841193	+	0.540735i	$0.818146\pi$
$854$	0.658302	0.0225266
$855$	−22.3005	−0.762661
$856$	−10.1285	−0.346185
$857$	14.9883	0.511990	0.255995	−	0.966678i	$-0.417597\pi$
$857$	14.9883	0.511990	0.255995	+	0.966678i	$0.417597\pi$
$858$	6.38853	0.218101
$859$	−12.1340	−0.414006	−0.207003	−	0.978340i	$-0.566371\pi$
$859$	−12.1340	−0.414006	−0.207003	+	0.978340i	$0.566371\pi$
$860$	−16.3505	−0.557546
$861$	5.60410	0.190987
$862$	−35.1462	−1.19708
$863$	−10.9845	−0.373918	−0.186959	−	0.982368i	$-0.559863\pi$
$863$	−10.9845	−0.373918	−0.186959	+	0.982368i	$0.559863\pi$
$864$	−3.12904	−0.106452
$865$	−58.7625	−1.99798
$866$	−1.48649	−0.0505129
$867$	10.1825	0.345817
$868$	3.67278	0.124662
$869$	2.64460	0.0897119
$870$	−0.329218	−0.0111615
$871$	29.2891	0.992422
$872$	3.65849	0.123892
$873$	−3.54309	−0.119915
$874$	−29.6769	−1.00383
$875$	4.17348	0.141089
$876$	10.3938	0.351172
$877$	−3.37586	−0.113995	−0.0569974	−	0.998374i	$-0.518153\pi$
$877$	−3.37586	−0.113995	−0.0569974	+	0.998374i	$0.518153\pi$
$878$	−20.7289	−0.699565
$879$	−53.2861	−1.79730
$880$	1.65955	0.0559435
$881$	38.2330	1.28810	0.644050	−	0.764983i	$-0.277253\pi$
$881$	38.2330	1.28810	0.644050	+	0.764983i	$0.277253\pi$
$882$	10.4480	0.351803
$883$	−40.3575	−1.35814	−0.679069	−	0.734074i	$-0.737616\pi$
$883$	−40.3575	−1.35814	−0.679069	+	0.734074i	$0.737616\pi$
$884$	−15.2295	−0.512222
$885$	6.89703	0.231841
$886$	−15.5590	−0.522714
$887$	−22.7126	−0.762614	−0.381307	−	0.924448i	$-0.624526\pi$
$887$	−22.7126	−0.762614	−0.381307	+	0.924448i	$0.624526\pi$
$888$	−11.8488	−0.397620
$889$	0.566225	0.0189906
$890$	−20.1548	−0.675589
$891$	7.74915	0.259606
$892$	7.87765	0.263763
$893$	−13.6669	−0.457347
$894$	3.63371	0.121529
$895$	−19.8598	−0.663838
$896$	−0.412842	−0.0137921
$897$	45.4846	1.51869
$898$	4.08339	0.136264
$899$	0.571214	0.0190510
$900$	1.22964	0.0409879
$901$	25.3935	0.845981
$902$	−4.39356	−0.146290
$903$	−5.96347	−0.198452
$904$	14.6305	0.486602
$905$	−34.5246	−1.14764
$906$	5.13942	0.170746
$907$	19.4788	0.646784	0.323392	−	0.946265i	$-0.395177\pi$
$907$	19.4788	0.646784	0.323392	+	0.946265i	$0.395177\pi$
$908$	−13.8829	−0.460720
$909$	−6.46117	−0.214303
$910$	−4.33376	−0.143663
$911$	16.9166	0.560473	0.280237	−	0.959931i	$-0.409587\pi$
$911$	16.9166	0.560473	0.280237	+	0.959931i	$0.409587\pi$
$912$	−12.8783	−0.426444
$913$	−12.0282	−0.398077
$914$	0.725445	0.0239956
$915$	−8.17593	−0.270288
$916$	−17.4618	−0.576954
$917$	1.63825	0.0540997
$918$	−10.9363	−0.360952
$919$	12.0568	0.397717	0.198859	−	0.980028i	$-0.436277\pi$
$919$	12.0568	0.397717	0.198859	+	0.980028i	$0.436277\pi$
$920$	11.8156	0.389548
$921$	−35.6690	−1.17533
$922$	29.6625	0.976882
$923$	−4.60657	−0.151627
$924$	0.605285	0.0199124
$925$	−4.47477	−0.147130
$926$	13.7467	0.451743
$927$	−9.09717	−0.298790
$928$	−0.0642078	−0.00210773
$929$	19.2557	0.631758	0.315879	−	0.948799i	$-0.397701\pi$
$929$	19.2557	0.631758	0.315879	+	0.948799i	$0.397701\pi$
$930$	−45.6149	−1.49577
$931$	−41.3249	−1.35437
$932$	20.7897	0.680989
$933$	37.4370	1.22563
$934$	12.1513	0.397603
$935$	5.80031	0.189690
$936$	6.66600	0.217885
$937$	−27.6450	−0.903122	−0.451561	−	0.892240i	$-0.649133\pi$
$937$	−27.6450	−0.903122	−0.451561	+	0.892240i	$0.649133\pi$
$938$	2.77501	0.0906074
$939$	62.4679	2.03856
$940$	5.44137	0.177478
$941$	50.9707	1.66160	0.830799	−	0.556573i	$-0.187884\pi$
$941$	50.9707	1.66160	0.830799	+	0.556573i	$0.187884\pi$
$942$	−5.67992	−0.185062
$943$	−31.2810	−1.01865
$944$	1.34514	0.0437805
$945$	−3.11208	−0.101236
$946$	4.67531	0.152007
$947$	−31.5019	−1.02367	−0.511837	−	0.859083i	$-0.671035\pi$
$947$	−31.5019	−1.02367	−0.511837	+	0.859083i	$0.671035\pi$
$948$	8.17080	0.265375
$949$	21.2793	0.690754
$950$	−4.86357	−0.157795
$951$	−9.86920	−0.320031
$952$	−1.44293	−0.0467655
$953$	15.0196	0.486532	0.243266	−	0.969960i	$-0.421781\pi$
$953$	15.0196	0.486532	0.243266	+	0.969960i	$0.421781\pi$
$954$	−11.1148	−0.359856
$955$	38.1657	1.23501
$956$	−4.80190	−0.155305
$957$	0.0941378	0.00304304
$958$	−0.759888	−0.0245509
$959$	7.82366	0.252639
$960$	5.12738	0.165486
$961$	48.1446	1.55305
$962$	−24.2583	−0.782117
$963$	15.4948	0.499313
$964$	−0.393550	−0.0126754
$965$	38.1933	1.22949
$966$	4.30947	0.138655
$967$	2.90508	0.0934212	0.0467106	−	0.998908i	$-0.485126\pi$
$967$	2.90508	0.0934212	0.0467106	+	0.998908i	$0.485126\pi$
$968$	10.5255	0.338301
$969$	−45.0110	−1.44596
$970$	−5.57952	−0.179148
$971$	−37.4050	−1.20038	−0.600192	−	0.799856i	$-0.704909\pi$
$971$	−37.4050	−1.20038	−0.600192	+	0.799856i	$0.704909\pi$
$972$	14.5548	0.466845
$973$	−2.35267	−0.0754232
$974$	6.02677	0.193110
$975$	7.45422	0.238726
$976$	−1.59456	−0.0510406
$977$	11.0458	0.353387	0.176693	−	0.984266i	$-0.443460\pi$
$977$	11.0458	0.353387	0.176693	+	0.984266i	$0.443460\pi$
$978$	−3.52340	−0.112666
$979$	5.76312	0.184190
$980$	16.4531	0.525576
$981$	−5.59683	−0.178693
$982$	5.77206	0.184194
$983$	55.7836	1.77922	0.889610	−	0.456720i	$-0.150976\pi$
$983$	55.7836	1.77922	0.889610	+	0.456720i	$0.150976\pi$
$984$	−13.5744	−0.432737
$985$	−42.6195	−1.35797
$986$	−0.224413	−0.00714676
$987$	1.98462	0.0631712
$988$	−26.3660	−0.838813
$989$	33.2869	1.05846
$990$	−2.53882	−0.0806889
$991$	−20.6402	−0.655656	−0.327828	−	0.944737i	$-0.606317\pi$
$991$	−20.6402	−0.655656	−0.327828	+	0.944737i	$0.606317\pi$
$992$	−8.89632	−0.282459
$993$	−46.1038	−1.46306
$994$	−0.436452	−0.0138434
$995$	15.0843	0.478204
$996$	−37.1627	−1.17754
$997$	−6.37980	−0.202050	−0.101025	−	0.994884i	$-0.532212\pi$
$997$	−6.37980	−0.202050	−0.101025	+	0.994884i	$0.532212\pi$
$998$	−3.09211	−0.0978790
$999$	−17.4199	−0.551141

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.14	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.14	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.12834 q^{3} +1.00000 q^{4} -2.40910 q^{5} +2.12834 q^{6} +0.412842 q^{7} -1.00000 q^{8} +1.52982 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.12834 q^{3} +1.00000 q^{4} -2.40910 q^{5} +2.12834 q^{6} +0.412842 q^{7} -1.00000 q^{8} +1.52982 q^{9} +2.40910 q^{10} -0.688867 q^{11} -2.12834 q^{12} -4.35737 q^{13} -0.412842 q^{14} +5.12738 q^{15} +1.00000 q^{16} +3.49510 q^{17} -1.52982 q^{18} +6.05088 q^{19} -2.40910 q^{20} -0.878668 q^{21} +0.688867 q^{22} +4.90455 q^{23} +2.12834 q^{24} +0.803779 q^{25} +4.35737 q^{26} +3.12904 q^{27} +0.412842 q^{28} +0.0642078 q^{29} -5.12738 q^{30} +8.89632 q^{31} -1.00000 q^{32} +1.46614 q^{33} -3.49510 q^{34} -0.994580 q^{35} +1.52982 q^{36} -5.56717 q^{37} -6.05088 q^{38} +9.27396 q^{39} +2.40910 q^{40} -6.37795 q^{41} +0.878668 q^{42} +6.78695 q^{43} -0.688867 q^{44} -3.68550 q^{45} -4.90455 q^{46} -2.25867 q^{47} -2.12834 q^{48} -6.82956 q^{49} -0.803779 q^{50} -7.43876 q^{51} -4.35737 q^{52} +7.26546 q^{53} -3.12904 q^{54} +1.65955 q^{55} -0.412842 q^{56} -12.8783 q^{57} -0.0642078 q^{58} +1.34514 q^{59} +5.12738 q^{60} -1.59456 q^{61} -8.89632 q^{62} +0.631574 q^{63} +1.00000 q^{64} +10.4974 q^{65} -1.46614 q^{66} -6.72173 q^{67} +3.49510 q^{68} -10.4385 q^{69} +0.994580 q^{70} +1.05719 q^{71} -1.52982 q^{72} -4.88351 q^{73} +5.56717 q^{74} -1.71071 q^{75} +6.05088 q^{76} -0.284394 q^{77} -9.27396 q^{78} -3.83905 q^{79} -2.40910 q^{80} -11.2491 q^{81} +6.37795 q^{82} +17.4609 q^{83} -0.878668 q^{84} -8.42006 q^{85} -6.78695 q^{86} -0.136656 q^{87} +0.688867 q^{88} -8.36608 q^{89} +3.68550 q^{90} -1.79891 q^{91} +4.90455 q^{92} -18.9344 q^{93} +2.25867 q^{94} -14.5772 q^{95} +2.12834 q^{96} -2.31602 q^{97} +6.82956 q^{98} -1.05384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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