LMFDB - Embedded newform 8002.2.a.d.1.36

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:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.36
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.494918 q^{3} +1.00000 q^{4} -0.262672 q^{5} -0.494918 q^{6} +3.16413 q^{7} +1.00000 q^{8} -2.75506 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.494918 q^{3} +1.00000 q^{4} -0.262672 q^{5} -0.494918 q^{6} +3.16413 q^{7} +1.00000 q^{8} -2.75506 q^{9} -0.262672 q^{10} +4.49659 q^{11} -0.494918 q^{12} -2.40186 q^{13} +3.16413 q^{14} +0.130001 q^{15} +1.00000 q^{16} -2.69263 q^{17} -2.75506 q^{18} -7.13887 q^{19} -0.262672 q^{20} -1.56598 q^{21} +4.49659 q^{22} +0.817693 q^{23} -0.494918 q^{24} -4.93100 q^{25} -2.40186 q^{26} +2.84828 q^{27} +3.16413 q^{28} +3.95860 q^{29} +0.130001 q^{30} -2.73215 q^{31} +1.00000 q^{32} -2.22544 q^{33} -2.69263 q^{34} -0.831126 q^{35} -2.75506 q^{36} +2.76701 q^{37} -7.13887 q^{38} +1.18873 q^{39} -0.262672 q^{40} -9.60456 q^{41} -1.56598 q^{42} -4.55462 q^{43} +4.49659 q^{44} +0.723675 q^{45} +0.817693 q^{46} -6.90513 q^{47} -0.494918 q^{48} +3.01170 q^{49} -4.93100 q^{50} +1.33263 q^{51} -2.40186 q^{52} -8.97649 q^{53} +2.84828 q^{54} -1.18113 q^{55} +3.16413 q^{56} +3.53315 q^{57} +3.95860 q^{58} +10.8107 q^{59} +0.130001 q^{60} -11.9738 q^{61} -2.73215 q^{62} -8.71735 q^{63} +1.00000 q^{64} +0.630902 q^{65} -2.22544 q^{66} -0.309722 q^{67} -2.69263 q^{68} -0.404691 q^{69} -0.831126 q^{70} -5.80078 q^{71} -2.75506 q^{72} +8.54558 q^{73} +2.76701 q^{74} +2.44044 q^{75} -7.13887 q^{76} +14.2278 q^{77} +1.18873 q^{78} -16.1099 q^{79} -0.262672 q^{80} +6.85550 q^{81} -9.60456 q^{82} +7.64022 q^{83} -1.56598 q^{84} +0.707278 q^{85} -4.55462 q^{86} -1.95918 q^{87} +4.49659 q^{88} -9.56710 q^{89} +0.723675 q^{90} -7.59980 q^{91} +0.817693 q^{92} +1.35219 q^{93} -6.90513 q^{94} +1.87518 q^{95} -0.494918 q^{96} +2.87298 q^{97} +3.01170 q^{98} -12.3884 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	−0.494918	−0.285741	−0.142870	−	0.989741i	$-0.545633\pi$
$3$	−0.494918	−0.285741	−0.142870	+	0.989741i	$0.545633\pi$
$4$	1.00000	0.500000
$5$	−0.262672	−0.117470	−0.0587352	−	0.998274i	$-0.518707\pi$
$5$	−0.262672	−0.117470	−0.0587352	+	0.998274i	$0.518707\pi$
$6$	−0.494918	−0.202049
$7$	3.16413	1.19593	0.597964	−	0.801523i	$-0.295977\pi$
$7$	3.16413	1.19593	0.597964	+	0.801523i	$0.295977\pi$
$8$	1.00000	0.353553
$9$	−2.75506	−0.918352
$10$	−0.262672	−0.0830641
$11$	4.49659	1.35577	0.677887	−	0.735166i	$-0.262896\pi$
$11$	4.49659	1.35577	0.677887	+	0.735166i	$0.262896\pi$
$12$	−0.494918	−0.142870
$13$	−2.40186	−0.666157	−0.333079	−	0.942899i	$-0.608087\pi$
$13$	−2.40186	−0.666157	−0.333079	+	0.942899i	$0.608087\pi$
$14$	3.16413	0.845648
$15$	0.130001	0.0335661
$16$	1.00000	0.250000
$17$	−2.69263	−0.653059	−0.326530	−	0.945187i	$-0.605879\pi$
$17$	−2.69263	−0.653059	−0.326530	+	0.945187i	$0.605879\pi$
$18$	−2.75506	−0.649373
$19$	−7.13887	−1.63777	−0.818884	−	0.573959i	$-0.805407\pi$
$19$	−7.13887	−1.63777	−0.818884	+	0.573959i	$0.805407\pi$
$20$	−0.262672	−0.0587352
$21$	−1.56598	−0.341725
$22$	4.49659	0.958677
$23$	0.817693	0.170501	0.0852504	−	0.996360i	$-0.472831\pi$
$23$	0.817693	0.170501	0.0852504	+	0.996360i	$0.472831\pi$
$24$	−0.494918	−0.101025
$25$	−4.93100	−0.986201
$26$	−2.40186	−0.471044
$27$	2.84828	0.548152
$28$	3.16413	0.597964
$29$	3.95860	0.735094	0.367547	−	0.930005i	$-0.380198\pi$
$29$	3.95860	0.735094	0.367547	+	0.930005i	$0.380198\pi$
$30$	0.130001	0.0237348
$31$	−2.73215	−0.490708	−0.245354	−	0.969434i	$-0.578904\pi$
$31$	−2.73215	−0.490708	−0.245354	+	0.969434i	$0.578904\pi$
$32$	1.00000	0.176777
$33$	−2.22544	−0.387400
$34$	−2.69263	−0.461783
$35$	−0.831126	−0.140486
$36$	−2.75506	−0.459176
$37$	2.76701	0.454894	0.227447	−	0.973790i	$-0.426962\pi$
$37$	2.76701	0.454894	0.227447	+	0.973790i	$0.426962\pi$
$38$	−7.13887	−1.15808
$39$	1.18873	0.190348
$40$	−0.262672	−0.0415320
$41$	−9.60456	−1.49998	−0.749990	−	0.661449i	$-0.769942\pi$
$41$	−9.60456	−1.49998	−0.749990	+	0.661449i	$0.769942\pi$
$42$	−1.56598	−0.241636
$43$	−4.55462	−0.694573	−0.347287	−	0.937759i	$-0.612897\pi$
$43$	−4.55462	−0.694573	−0.347287	+	0.937759i	$0.612897\pi$
$44$	4.49659	0.677887
$45$	0.723675	0.107879
$46$	0.817693	0.120562
$47$	−6.90513	−1.00722	−0.503608	−	0.863932i	$-0.667994\pi$
$47$	−6.90513	−1.00722	−0.503608	+	0.863932i	$0.667994\pi$
$48$	−0.494918	−0.0714352
$49$	3.01170	0.430242
$50$	−4.93100	−0.697349
$51$	1.33263	0.186606
$52$	−2.40186	−0.333079
$53$	−8.97649	−1.23302	−0.616508	−	0.787349i	$-0.711453\pi$
$53$	−8.97649	−1.23302	−0.616508	+	0.787349i	$0.711453\pi$
$54$	2.84828	0.387602
$55$	−1.18113	−0.159263
$56$	3.16413	0.422824
$57$	3.53315	0.467978
$58$	3.95860	0.519790
$59$	10.8107	1.40743	0.703716	−	0.710482i	$-0.251523\pi$
$59$	10.8107	1.40743	0.703716	+	0.710482i	$0.251523\pi$
$60$	0.130001	0.0167830
$61$	−11.9738	−1.53309	−0.766547	−	0.642188i	$-0.778027\pi$
$61$	−11.9738	−1.53309	−0.766547	+	0.642188i	$0.778027\pi$
$62$	−2.73215	−0.346983
$63$	−8.71735	−1.09828
$64$	1.00000	0.125000
$65$	0.630902	0.0782537
$66$	−2.22544	−0.273933
$67$	−0.309722	−0.0378385	−0.0189193	−	0.999821i	$-0.506023\pi$
$67$	−0.309722	−0.0378385	−0.0189193	+	0.999821i	$0.506023\pi$
$68$	−2.69263	−0.326530
$69$	−0.404691	−0.0487190
$70$	−0.831126	−0.0993386
$71$	−5.80078	−0.688426	−0.344213	−	0.938892i	$-0.611854\pi$
$71$	−5.80078	−0.688426	−0.344213	+	0.938892i	$0.611854\pi$
$72$	−2.75506	−0.324686
$73$	8.54558	1.00018	0.500092	−	0.865972i	$-0.333299\pi$
$73$	8.54558	1.00018	0.500092	+	0.865972i	$0.333299\pi$
$74$	2.76701	0.321659
$75$	2.44044	0.281798
$76$	−7.13887	−0.818884
$77$	14.2278	1.62141
$78$	1.18873	0.134597
$79$	−16.1099	−1.81251	−0.906254	−	0.422733i	$-0.861071\pi$
$79$	−16.1099	−1.81251	−0.906254	+	0.422733i	$0.861071\pi$
$80$	−0.262672	−0.0293676
$81$	6.85550	0.761723
$82$	−9.60456	−1.06065
$83$	7.64022	0.838623	0.419311	−	0.907843i	$-0.362272\pi$
$83$	7.64022	0.838623	0.419311	+	0.907843i	$0.362272\pi$
$84$	−1.56598	−0.170863
$85$	0.707278	0.0767151
$86$	−4.55462	−0.491138
$87$	−1.95918	−0.210046
$88$	4.49659	0.479338
$89$	−9.56710	−1.01411	−0.507055	−	0.861914i	$-0.669266\pi$
$89$	−9.56710	−1.01411	−0.507055	+	0.861914i	$0.669266\pi$
$90$	0.723675	0.0762821
$91$	−7.59980	−0.796676
$92$	0.817693	0.0852504
$93$	1.35219	0.140215
$94$	−6.90513	−0.712209
$95$	1.87518	0.192389
$96$	−0.494918	−0.0505123
$97$	2.87298	0.291707	0.145854	−	0.989306i	$-0.453407\pi$
$97$	2.87298	0.291707	0.145854	+	0.989306i	$0.453407\pi$
$98$	3.01170	0.304227
$99$	−12.3884	−1.24508
$100$	−4.93100	−0.493100
$101$	14.6478	1.45751	0.728757	−	0.684773i	$-0.240099\pi$
$101$	14.6478	1.45751	0.728757	+	0.684773i	$0.240099\pi$
$102$	1.33263	0.131950
$103$	−5.88419	−0.579786	−0.289893	−	0.957059i	$-0.593620\pi$
$103$	−5.88419	−0.579786	−0.289893	+	0.957059i	$0.593620\pi$
$104$	−2.40186	−0.235522
$105$	0.411339	0.0401426
$106$	−8.97649	−0.871874
$107$	−12.8312	−1.24044	−0.620221	−	0.784427i	$-0.712957\pi$
$107$	−12.8312	−1.24044	−0.620221	+	0.784427i	$0.712957\pi$
$108$	2.84828	0.274076
$109$	8.19075	0.784532	0.392266	−	0.919852i	$-0.371691\pi$
$109$	8.19075	0.784532	0.392266	+	0.919852i	$0.371691\pi$
$110$	−1.18113	−0.112616
$111$	−1.36944	−0.129982
$112$	3.16413	0.298982
$113$	−1.49015	−0.140181	−0.0700907	−	0.997541i	$-0.522329\pi$
$113$	−1.49015	−0.140181	−0.0700907	+	0.997541i	$0.522329\pi$
$114$	3.53315	0.330910
$115$	−0.214785	−0.0200288
$116$	3.95860	0.367547
$117$	6.61727	0.611767
$118$	10.8107	0.995204
$119$	−8.51983	−0.781011
$120$	0.130001	0.0118674
$121$	9.21934	0.838122
$122$	−11.9738	−1.08406
$123$	4.75347	0.428606
$124$	−2.73215	−0.245354
$125$	2.60859	0.233320
$126$	−8.71735	−0.776603
$127$	16.3116	1.44742	0.723712	−	0.690102i	$-0.242434\pi$
$127$	16.3116	1.44742	0.723712	+	0.690102i	$0.242434\pi$
$128$	1.00000	0.0883883
$129$	2.25416	0.198468
$130$	0.630902	0.0553337
$131$	4.25260	0.371551	0.185776	−	0.982592i	$-0.440520\pi$
$131$	4.25260	0.371551	0.185776	+	0.982592i	$0.440520\pi$
$132$	−2.22544	−0.193700
$133$	−22.5883	−1.95865
$134$	−0.309722	−0.0267559
$135$	−0.748163	−0.0643916
$136$	−2.69263	−0.230891
$137$	−11.8221	−1.01003	−0.505016	−	0.863110i	$-0.668514\pi$
$137$	−11.8221	−1.01003	−0.505016	+	0.863110i	$0.668514\pi$
$138$	−0.404691	−0.0344496
$139$	−11.7104	−0.993261	−0.496631	−	0.867962i	$-0.665430\pi$
$139$	−11.7104	−0.993261	−0.496631	+	0.867962i	$0.665430\pi$
$140$	−0.831126	−0.0702430
$141$	3.41747	0.287803
$142$	−5.80078	−0.486791
$143$	−10.8002	−0.903158
$144$	−2.75506	−0.229588
$145$	−1.03981	−0.0863517
$146$	8.54558	0.707237
$147$	−1.49054	−0.122938
$148$	2.76701	0.227447
$149$	14.4380	1.18281	0.591404	−	0.806375i	$-0.298574\pi$
$149$	14.4380	1.18281	0.591404	+	0.806375i	$0.298574\pi$
$150$	2.44044	0.199261
$151$	6.75408	0.549640	0.274820	−	0.961496i	$-0.411382\pi$
$151$	6.75408	0.549640	0.274820	+	0.961496i	$0.411382\pi$
$152$	−7.13887	−0.579039
$153$	7.41835	0.599738
$154$	14.2278	1.14651
$155$	0.717658	0.0576436
$156$	1.18873	0.0951742
$157$	2.59998	0.207501	0.103751	−	0.994603i	$-0.466916\pi$
$157$	2.59998	0.207501	0.103751	+	0.994603i	$0.466916\pi$
$158$	−16.1099	−1.28164
$159$	4.44263	0.352323
$160$	−0.262672	−0.0207660
$161$	2.58728	0.203906
$162$	6.85550	0.538619
$163$	−15.8615	−1.24237	−0.621183	−	0.783665i	$-0.713348\pi$
$163$	−15.8615	−1.24237	−0.621183	+	0.783665i	$0.713348\pi$
$164$	−9.60456	−0.749990
$165$	0.584561	0.0455080
$166$	7.64022	0.592996
$167$	−14.2174	−1.10018	−0.550088	−	0.835107i	$-0.685406\pi$
$167$	−14.2174	−1.10018	−0.550088	+	0.835107i	$0.685406\pi$
$168$	−1.56598	−0.120818
$169$	−7.23105	−0.556235
$170$	0.707278	0.0542458
$171$	19.6680	1.50405
$172$	−4.55462	−0.347287
$173$	6.97459	0.530268	0.265134	−	0.964212i	$-0.414584\pi$
$173$	6.97459	0.530268	0.265134	+	0.964212i	$0.414584\pi$
$174$	−1.95918	−0.148525
$175$	−15.6023	−1.17942
$176$	4.49659	0.338943
$177$	−5.35040	−0.402161
$178$	−9.56710	−0.717084
$179$	−19.4848	−1.45636	−0.728181	−	0.685385i	$-0.759634\pi$
$179$	−19.4848	−1.45636	−0.728181	+	0.685385i	$0.759634\pi$
$180$	0.723675	0.0539396
$181$	11.3020	0.840068	0.420034	−	0.907508i	$-0.362018\pi$
$181$	11.3020	0.840068	0.420034	+	0.907508i	$0.362018\pi$
$182$	−7.59980	−0.563335
$183$	5.92607	0.438068
$184$	0.817693	0.0602811
$185$	−0.726816	−0.0534366
$186$	1.35219	0.0991473
$187$	−12.1077	−0.885400
$188$	−6.90513	−0.503608
$189$	9.01232	0.655550
$190$	1.87518	0.136040
$191$	−3.19008	−0.230826	−0.115413	−	0.993318i	$-0.536819\pi$
$191$	−3.19008	−0.230826	−0.115413	+	0.993318i	$0.536819\pi$
$192$	−0.494918	−0.0357176
$193$	2.70929	0.195019	0.0975096	−	0.995235i	$-0.468912\pi$
$193$	2.70929	0.195019	0.0975096	+	0.995235i	$0.468912\pi$
$194$	2.87298	0.206268
$195$	−0.312245	−0.0223603
$196$	3.01170	0.215121
$197$	17.2966	1.23233	0.616164	−	0.787618i	$-0.288686\pi$
$197$	17.2966	1.23233	0.616164	+	0.787618i	$0.288686\pi$
$198$	−12.3884	−0.880403
$199$	2.09299	0.148368	0.0741840	−	0.997245i	$-0.476365\pi$
$199$	2.09299	0.148368	0.0741840	+	0.997245i	$0.476365\pi$
$200$	−4.93100	−0.348675
$201$	0.153287	0.0108120
$202$	14.6478	1.03062
$203$	12.5255	0.879119
$204$	1.33263	0.0933029
$205$	2.52285	0.176203
$206$	−5.88419	−0.409971
$207$	−2.25279	−0.156580
$208$	−2.40186	−0.166539
$209$	−32.1006	−2.22044
$210$	0.411339	0.0283851
$211$	15.0641	1.03706	0.518528	−	0.855061i	$-0.326480\pi$
$211$	15.0641	1.03706	0.518528	+	0.855061i	$0.326480\pi$
$212$	−8.97649	−0.616508
$213$	2.87091	0.196712
$214$	−12.8312	−0.877125
$215$	1.19637	0.0815918
$216$	2.84828	0.193801
$217$	−8.64486	−0.586851
$218$	8.19075	0.554748
$219$	−4.22936	−0.285794
$220$	−1.18113	−0.0796316
$221$	6.46733	0.435040
$222$	−1.36944	−0.0919111
$223$	−23.8998	−1.60045	−0.800225	−	0.599699i	$-0.795287\pi$
$223$	−23.8998	−1.60045	−0.800225	+	0.599699i	$0.795287\pi$
$224$	3.16413	0.211412
$225$	13.5852	0.905679
$226$	−1.49015	−0.0991232
$227$	12.7845	0.848535	0.424267	−	0.905537i	$-0.360532\pi$
$227$	12.7845	0.848535	0.424267	+	0.905537i	$0.360532\pi$
$228$	3.53315	0.233989
$229$	−18.5823	−1.22795	−0.613977	−	0.789324i	$-0.710431\pi$
$229$	−18.5823	−1.22795	−0.613977	+	0.789324i	$0.710431\pi$
$230$	−0.214785	−0.0141625
$231$	−7.04159	−0.463302
$232$	3.95860	0.259895
$233$	27.0111	1.76956	0.884778	−	0.466013i	$-0.154310\pi$
$233$	27.0111	1.76956	0.884778	+	0.466013i	$0.154310\pi$
$234$	6.61727	0.432585
$235$	1.81378	0.118318
$236$	10.8107	0.703716
$237$	7.97309	0.517908
$238$	−8.51983	−0.552258
$239$	−16.5131	−1.06815	−0.534073	−	0.845439i	$-0.679339\pi$
$239$	−16.5131	−1.06815	−0.534073	+	0.845439i	$0.679339\pi$
$240$	0.130001	0.00839152
$241$	−0.426785	−0.0274916	−0.0137458	−	0.999906i	$-0.504376\pi$
$241$	−0.426785	−0.0274916	−0.0137458	+	0.999906i	$0.504376\pi$
$242$	9.21934	0.592642
$243$	−11.9378	−0.765807
$244$	−11.9738	−0.766547
$245$	−0.791087	−0.0505407
$246$	4.75347	0.303070
$247$	17.1466	1.09101
$248$	−2.73215	−0.173492
$249$	−3.78128	−0.239629
$250$	2.60859	0.164982
$251$	6.77066	0.427361	0.213680	−	0.976904i	$-0.431455\pi$
$251$	6.77066	0.427361	0.213680	+	0.976904i	$0.431455\pi$
$252$	−8.71735	−0.549141
$253$	3.67683	0.231160
$254$	16.3116	1.02348
$255$	−0.350045	−0.0219206
$256$	1.00000	0.0625000
$257$	−3.61656	−0.225595	−0.112797	−	0.993618i	$-0.535981\pi$
$257$	−3.61656	−0.225595	−0.112797	+	0.993618i	$0.535981\pi$
$258$	2.25416	0.140338
$259$	8.75518	0.544021
$260$	0.630902	0.0391269
$261$	−10.9062	−0.675075
$262$	4.25260	0.262726
$263$	29.6136	1.82606	0.913028	−	0.407898i	$-0.133738\pi$
$263$	29.6136	1.82606	0.913028	+	0.407898i	$0.133738\pi$
$264$	−2.22544	−0.136967
$265$	2.35787	0.144843
$266$	−22.5883	−1.38498
$267$	4.73493	0.289773
$268$	−0.309722	−0.0189193
$269$	−16.7286	−1.01996	−0.509979	−	0.860187i	$-0.670347\pi$
$269$	−16.7286	−1.01996	−0.509979	+	0.860187i	$0.670347\pi$
$270$	−0.748163	−0.0455317
$271$	15.6633	0.951477	0.475738	−	0.879587i	$-0.342181\pi$
$271$	15.6633	0.951477	0.475738	+	0.879587i	$0.342181\pi$
$272$	−2.69263	−0.163265
$273$	3.76128	0.227643
$274$	−11.8221	−0.714201
$275$	−22.1727	−1.33706
$276$	−0.404691	−0.0243595
$277$	9.91687	0.595847	0.297924	−	0.954590i	$-0.403706\pi$
$277$	9.91687	0.595847	0.297924	+	0.954590i	$0.403706\pi$
$278$	−11.7104	−0.702342
$279$	7.52722	0.450643
$280$	−0.831126	−0.0496693
$281$	19.2755	1.14988	0.574940	−	0.818195i	$-0.305025\pi$
$281$	19.2755	1.14988	0.574940	+	0.818195i	$0.305025\pi$
$282$	3.41747	0.203507
$283$	4.05471	0.241027	0.120514	−	0.992712i	$-0.461546\pi$
$283$	4.05471	0.241027	0.120514	+	0.992712i	$0.461546\pi$
$284$	−5.80078	−0.344213
$285$	−0.928059	−0.0549735
$286$	−10.8002	−0.638629
$287$	−30.3901	−1.79387
$288$	−2.75506	−0.162343
$289$	−9.74974	−0.573514
$290$	−1.03981	−0.0610599
$291$	−1.42189	−0.0833527
$292$	8.54558	0.500092
$293$	6.34473	0.370663	0.185331	−	0.982676i	$-0.440664\pi$
$293$	6.34473	0.370663	0.185331	+	0.982676i	$0.440664\pi$
$294$	−1.49054	−0.0869302
$295$	−2.83966	−0.165331
$296$	2.76701	0.160829
$297$	12.8076	0.743170
$298$	14.4380	0.836372
$299$	−1.96399	−0.113580
$300$	2.44044	0.140899
$301$	−14.4114	−0.830659
$302$	6.75408	0.388654
$303$	−7.24947	−0.416471
$304$	−7.13887	−0.409442
$305$	3.14519	0.180093
$306$	7.41835	0.424079
$307$	31.2612	1.78417	0.892086	−	0.451867i	$-0.149242\pi$
$307$	31.2612	1.78417	0.892086	+	0.451867i	$0.149242\pi$
$308$	14.2278	0.810703
$309$	2.91219	0.165669
$310$	0.717658	0.0407602
$311$	−8.81789	−0.500016	−0.250008	−	0.968244i	$-0.580433\pi$
$311$	−8.81789	−0.500016	−0.250008	+	0.968244i	$0.580433\pi$
$312$	1.18873	0.0672983
$313$	−22.1111	−1.24979	−0.624896	−	0.780708i	$-0.714859\pi$
$313$	−22.1111	−1.24979	−0.624896	+	0.780708i	$0.714859\pi$
$314$	2.59998	0.146726
$315$	2.28980	0.129016
$316$	−16.1099	−0.906254
$317$	2.45616	0.137952	0.0689758	−	0.997618i	$-0.478027\pi$
$317$	2.45616	0.137952	0.0689758	+	0.997618i	$0.478027\pi$
$318$	4.44263	0.249130
$319$	17.8002	0.996620
$320$	−0.262672	−0.0146838
$321$	6.35041	0.354445
$322$	2.58728	0.144184
$323$	19.2223	1.06956
$324$	6.85550	0.380861
$325$	11.8436	0.656965
$326$	−15.8615	−0.878486
$327$	−4.05375	−0.224173
$328$	−9.60456	−0.530323
$329$	−21.8487	−1.20456
$330$	0.584561	0.0321790
$331$	−14.9052	−0.819265	−0.409632	−	0.912251i	$-0.634343\pi$
$331$	−14.9052	−0.819265	−0.409632	+	0.912251i	$0.634343\pi$
$332$	7.64022	0.419311
$333$	−7.62328	−0.417753
$334$	−14.2174	−0.777942
$335$	0.0813551	0.00444490
$336$	−1.56598	−0.0854314
$337$	5.66795	0.308753	0.154376	−	0.988012i	$-0.450663\pi$
$337$	5.66795	0.308753	0.154376	+	0.988012i	$0.450663\pi$
$338$	−7.23105	−0.393317
$339$	0.737501	0.0400556
$340$	0.707278	0.0383575
$341$	−12.2854	−0.665289
$342$	19.6680	1.06352
$343$	−12.6195	−0.681389
$344$	−4.55462	−0.245569
$345$	0.106301	0.00572304
$346$	6.97459	0.374956
$347$	−30.6079	−1.64312	−0.821558	−	0.570125i	$-0.806895\pi$
$347$	−30.6079	−1.64312	−0.821558	+	0.570125i	$0.806895\pi$
$348$	−1.95918	−0.105023
$349$	−31.4542	−1.68371	−0.841853	−	0.539707i	$-0.818535\pi$
$349$	−31.4542	−1.68371	−0.841853	+	0.539707i	$0.818535\pi$
$350$	−15.6023	−0.833979
$351$	−6.84118	−0.365155
$352$	4.49659	0.239669
$353$	4.19569	0.223314	0.111657	−	0.993747i	$-0.464384\pi$
$353$	4.19569	0.223314	0.111657	+	0.993747i	$0.464384\pi$
$354$	−5.35040	−0.284371
$355$	1.52370	0.0808697
$356$	−9.56710	−0.507055
$357$	4.21661	0.223167
$358$	−19.4848	−1.02980
$359$	8.49132	0.448155	0.224077	−	0.974571i	$-0.428063\pi$
$359$	8.49132	0.448155	0.224077	+	0.974571i	$0.428063\pi$
$360$	0.723675	0.0381410
$361$	31.9634	1.68229
$362$	11.3020	0.594018
$363$	−4.56282	−0.239486
$364$	−7.59980	−0.398338
$365$	−2.24468	−0.117492
$366$	5.92607	0.309761
$367$	−22.5646	−1.17786	−0.588930	−	0.808184i	$-0.700451\pi$
$367$	−22.5646	−1.17786	−0.588930	+	0.808184i	$0.700451\pi$
$368$	0.817693	0.0426252
$369$	26.4611	1.37751
$370$	−0.726816	−0.0377854
$371$	−28.4028	−1.47460
$372$	1.35219	0.0701077
$373$	−22.8897	−1.18518	−0.592591	−	0.805503i	$-0.701895\pi$
$373$	−22.8897	−1.18518	−0.592591	+	0.805503i	$0.701895\pi$
$374$	−12.1077	−0.626073
$375$	−1.29104	−0.0666690
$376$	−6.90513	−0.356105
$377$	−9.50802	−0.489688
$378$	9.01232	0.463544
$379$	−14.2387	−0.731393	−0.365696	−	0.930734i	$-0.619169\pi$
$379$	−14.2387	−0.731393	−0.365696	+	0.930734i	$0.619169\pi$
$380$	1.87518	0.0961946
$381$	−8.07292	−0.413588
$382$	−3.19008	−0.163219
$383$	−5.43567	−0.277750	−0.138875	−	0.990310i	$-0.544349\pi$
$383$	−5.43567	−0.277750	−0.138875	+	0.990310i	$0.544349\pi$
$384$	−0.494918	−0.0252562
$385$	−3.73724	−0.190467
$386$	2.70929	0.137899
$387$	12.5482	0.637863
$388$	2.87298	0.145854
$389$	−7.63892	−0.387309	−0.193654	−	0.981070i	$-0.562034\pi$
$389$	−7.63892	−0.387309	−0.193654	+	0.981070i	$0.562034\pi$
$390$	−0.312245	−0.0158111
$391$	−2.20175	−0.111347
$392$	3.01170	0.152114
$393$	−2.10469	−0.106167
$394$	17.2966	0.871388
$395$	4.23162	0.212916
$396$	−12.3884	−0.622539
$397$	−26.7157	−1.34082	−0.670412	−	0.741989i	$-0.733883\pi$
$397$	−26.7157	−1.34082	−0.670412	+	0.741989i	$0.733883\pi$
$398$	2.09299	0.104912
$399$	11.1793	0.559667
$400$	−4.93100	−0.246550
$401$	25.2039	1.25862	0.629310	−	0.777154i	$-0.283337\pi$
$401$	25.2039	1.25862	0.629310	+	0.777154i	$0.283337\pi$
$402$	0.153287	0.00764525
$403$	6.56225	0.326889
$404$	14.6478	0.728757
$405$	−1.80075	−0.0894798
$406$	12.5255	0.621631
$407$	12.4421	0.616734
$408$	1.33263	0.0659751
$409$	−17.8527	−0.882761	−0.441381	−	0.897320i	$-0.645511\pi$
$409$	−17.8527	−0.882761	−0.441381	+	0.897320i	$0.645511\pi$
$410$	2.52285	0.124595
$411$	5.85098	0.288608
$412$	−5.88419	−0.289893
$413$	34.2064	1.68319
$414$	−2.25279	−0.110719
$415$	−2.00687	−0.0985133
$416$	−2.40186	−0.117761
$417$	5.79567	0.283815
$418$	−32.1006	−1.57009
$419$	−38.8900	−1.89990	−0.949951	−	0.312400i	$-0.898867\pi$
$419$	−38.8900	−1.89990	−0.949951	+	0.312400i	$0.898867\pi$
$420$	0.411339	0.0200713
$421$	−8.40529	−0.409649	−0.204824	−	0.978799i	$-0.565662\pi$
$421$	−8.40529	−0.409649	−0.204824	+	0.978799i	$0.565662\pi$
$422$	15.0641	0.733309
$423$	19.0240	0.924979
$424$	−8.97649	−0.435937
$425$	13.2774	0.644047
$426$	2.87091	0.139096
$427$	−37.8868	−1.83347
$428$	−12.8312	−0.620221
$429$	5.34521	0.258069
$430$	1.19637	0.0576941
$431$	8.86712	0.427114	0.213557	−	0.976931i	$-0.431495\pi$
$431$	8.86712	0.427114	0.213557	+	0.976931i	$0.431495\pi$
$432$	2.84828	0.137038
$433$	−17.9839	−0.864253	−0.432126	−	0.901813i	$-0.642237\pi$
$433$	−17.9839	−0.864253	−0.432126	+	0.901813i	$0.642237\pi$
$434$	−8.64486	−0.414966
$435$	0.514622	0.0246742
$436$	8.19075	0.392266
$437$	−5.83740	−0.279241
$438$	−4.22936	−0.202087
$439$	4.07642	0.194557	0.0972785	−	0.995257i	$-0.468986\pi$
$439$	4.07642	0.194557	0.0972785	+	0.995257i	$0.468986\pi$
$440$	−1.18113	−0.0563080
$441$	−8.29739	−0.395114
$442$	6.46733	0.307620
$443$	−13.0641	−0.620692	−0.310346	−	0.950624i	$-0.600445\pi$
$443$	−13.0641	−0.620692	−0.310346	+	0.950624i	$0.600445\pi$
$444$	−1.36944	−0.0649910
$445$	2.51301	0.119128
$446$	−23.8998	−1.13169
$447$	−7.14563	−0.337977
$448$	3.16413	0.149491
$449$	15.2717	0.720715	0.360358	−	0.932814i	$-0.382655\pi$
$449$	15.2717	0.720715	0.360358	+	0.932814i	$0.382655\pi$
$450$	13.5852	0.640412
$451$	−43.1878	−2.03363
$452$	−1.49015	−0.0700907
$453$	−3.34272	−0.157055
$454$	12.7845	0.600005
$455$	1.99625	0.0935858
$456$	3.53315	0.165455
$457$	−42.6358	−1.99442	−0.997210	−	0.0746534i	$-0.976215\pi$
$457$	−42.6358	−1.99442	−0.997210	+	0.0746534i	$0.976215\pi$
$458$	−18.5823	−0.868294
$459$	−7.66937	−0.357976
$460$	−0.214785	−0.0100144
$461$	−29.0500	−1.35299	−0.676496	−	0.736447i	$-0.736502\pi$
$461$	−29.0500	−1.35299	−0.676496	+	0.736447i	$0.736502\pi$
$462$	−7.04159	−0.327604
$463$	−22.1464	−1.02923	−0.514615	−	0.857421i	$-0.672065\pi$
$463$	−22.1464	−1.02923	−0.514615	+	0.857421i	$0.672065\pi$
$464$	3.95860	0.183773
$465$	−0.355182	−0.0164712
$466$	27.0111	1.25126
$467$	12.4454	0.575905	0.287953	−	0.957645i	$-0.407025\pi$
$467$	12.4454	0.575905	0.287953	+	0.957645i	$0.407025\pi$
$468$	6.61727	0.305883
$469$	−0.979998	−0.0452521
$470$	1.81378	0.0836635
$471$	−1.28678	−0.0592916
$472$	10.8107	0.497602
$473$	−20.4803	−0.941684
$474$	7.97309	0.366216
$475$	35.2018	1.61517
$476$	−8.51983	−0.390506
$477$	24.7307	1.13234
$478$	−16.5131	−0.755293
$479$	8.31828	0.380072	0.190036	−	0.981777i	$-0.439140\pi$
$479$	8.31828	0.380072	0.190036	+	0.981777i	$0.439140\pi$
$480$	0.130001	0.00593370
$481$	−6.64599	−0.303031
$482$	−0.426785	−0.0194395
$483$	−1.28049	−0.0582644
$484$	9.21934	0.419061
$485$	−0.754651	−0.0342669
$486$	−11.9378	−0.541507
$487$	30.0468	1.36155	0.680776	−	0.732492i	$-0.261643\pi$
$487$	30.0468	1.36155	0.680776	+	0.732492i	$0.261643\pi$
$488$	−11.9738	−0.542031
$489$	7.85013	0.354995
$490$	−0.791087	−0.0357377
$491$	25.7412	1.16168	0.580842	−	0.814016i	$-0.302723\pi$
$491$	25.7412	1.16168	0.580842	+	0.814016i	$0.302723\pi$
$492$	4.75347	0.214303
$493$	−10.6591	−0.480060
$494$	17.1466	0.771462
$495$	3.25407	0.146260
$496$	−2.73215	−0.122677
$497$	−18.3544	−0.823308
$498$	−3.78128	−0.169443
$499$	−17.7380	−0.794061	−0.397030	−	0.917805i	$-0.629959\pi$
$499$	−17.7380	−0.794061	−0.397030	+	0.917805i	$0.629959\pi$
$500$	2.60859	0.116660
$501$	7.03645	0.314365
$502$	6.77066	0.302190
$503$	−13.3404	−0.594820	−0.297410	−	0.954750i	$-0.596123\pi$
$503$	−13.3404	−0.594820	−0.297410	+	0.954750i	$0.596123\pi$
$504$	−8.71735	−0.388301
$505$	−3.84757	−0.171215
$506$	3.67683	0.163455
$507$	3.57878	0.158939
$508$	16.3116	0.723712
$509$	−2.74833	−0.121818	−0.0609088	−	0.998143i	$-0.519400\pi$
$509$	−2.74833	−0.121818	−0.0609088	+	0.998143i	$0.519400\pi$
$510$	−0.350045	−0.0155002
$511$	27.0393	1.19615
$512$	1.00000	0.0441942
$513$	−20.3335	−0.897746
$514$	−3.61656	−0.159520
$515$	1.54561	0.0681077
$516$	2.25416	0.0992340
$517$	−31.0495	−1.36556
$518$	8.75518	0.384681
$519$	−3.45185	−0.151519
$520$	0.630902	0.0276669
$521$	−24.9913	−1.09489	−0.547444	−	0.836842i	$-0.684399\pi$
$521$	−24.9913	−1.09489	−0.547444	+	0.836842i	$0.684399\pi$
$522$	−10.9062	−0.477350
$523$	−27.9947	−1.22412	−0.612060	−	0.790811i	$-0.709659\pi$
$523$	−27.9947	−1.22412	−0.612060	+	0.790811i	$0.709659\pi$
$524$	4.25260	0.185776
$525$	7.72187	0.337010
$526$	29.6136	1.29122
$527$	7.35667	0.320461
$528$	−2.22544	−0.0968500
$529$	−22.3314	−0.970930
$530$	2.35787	0.102419
$531$	−29.7841	−1.29252
$532$	−22.5883	−0.979326
$533$	23.0689	0.999223
$534$	4.73493	0.204900
$535$	3.37040	0.145715
$536$	−0.309722	−0.0133779
$537$	9.64337	0.416142
$538$	−16.7286	−0.721219
$539$	13.5424	0.583311
$540$	−0.748163	−0.0321958
$541$	22.9117	0.985052	0.492526	−	0.870298i	$-0.336074\pi$
$541$	22.9117	0.985052	0.492526	+	0.870298i	$0.336074\pi$
$542$	15.6633	0.672796
$543$	−5.59354	−0.240042
$544$	−2.69263	−0.115446
$545$	−2.15148	−0.0921592
$546$	3.76128	0.160968
$547$	40.4198	1.72823	0.864113	−	0.503298i	$-0.167880\pi$
$547$	40.4198	1.72823	0.864113	+	0.503298i	$0.167880\pi$
$548$	−11.8221	−0.505016
$549$	32.9886	1.40792
$550$	−22.1727	−0.945448
$551$	−28.2599	−1.20391
$552$	−0.404691	−0.0172248
$553$	−50.9738	−2.16763
$554$	9.91687	0.421327
$555$	0.359714	0.0152690
$556$	−11.7104	−0.496631
$557$	−8.74405	−0.370497	−0.185249	−	0.982692i	$-0.559309\pi$
$557$	−8.74405	−0.370497	−0.185249	+	0.982692i	$0.559309\pi$
$558$	7.52722	0.318653
$559$	10.9396	0.462695
$560$	−0.831126	−0.0351215
$561$	5.99230	0.252995
$562$	19.2755	0.813089
$563$	0.748691	0.0315536	0.0157768	−	0.999876i	$-0.494978\pi$
$563$	0.748691	0.0315536	0.0157768	+	0.999876i	$0.494978\pi$
$564$	3.41747	0.143901
$565$	0.391420	0.0164672
$566$	4.05471	0.170432
$567$	21.6917	0.910965
$568$	−5.80078	−0.243395
$569$	29.4940	1.23645	0.618226	−	0.786000i	$-0.287851\pi$
$569$	29.4940	1.23645	0.618226	+	0.786000i	$0.287851\pi$
$570$	−0.928059	−0.0388721
$571$	−12.4795	−0.522249	−0.261125	−	0.965305i	$-0.584093\pi$
$571$	−12.4795	−0.522249	−0.261125	+	0.965305i	$0.584093\pi$
$572$	−10.8002	−0.451579
$573$	1.57883	0.0659564
$574$	−30.3901	−1.26846
$575$	−4.03205	−0.168148
$576$	−2.75506	−0.114794
$577$	10.6774	0.444506	0.222253	−	0.974989i	$-0.428659\pi$
$577$	10.6774	0.444506	0.222253	+	0.974989i	$0.428659\pi$
$578$	−9.74974	−0.405536
$579$	−1.34088	−0.0557250
$580$	−1.03981	−0.0431758
$581$	24.1746	1.00293
$582$	−1.42189	−0.0589393
$583$	−40.3636	−1.67169
$584$	8.54558	0.353619
$585$	−1.73817	−0.0718645
$586$	6.34473	0.262098
$587$	39.1214	1.61471	0.807357	−	0.590063i	$-0.200897\pi$
$587$	39.1214	1.61471	0.807357	+	0.590063i	$0.200897\pi$
$588$	−1.49054	−0.0614689
$589$	19.5044	0.803666
$590$	−2.83966	−0.116907
$591$	−8.56037	−0.352127
$592$	2.76701	0.113724
$593$	24.6337	1.01158	0.505792	−	0.862656i	$-0.331200\pi$
$593$	24.6337	1.01158	0.505792	+	0.862656i	$0.331200\pi$
$594$	12.8076	0.525500
$595$	2.23792	0.0917457
$596$	14.4380	0.591404
$597$	−1.03586	−0.0423948
$598$	−1.96399	−0.0803134
$599$	−34.4286	−1.40672	−0.703358	−	0.710836i	$-0.748317\pi$
$599$	−34.4286	−1.40672	−0.703358	+	0.710836i	$0.748317\pi$
$600$	2.44044	0.0996306
$601$	6.41087	0.261505	0.130752	−	0.991415i	$-0.458261\pi$
$601$	6.41087	0.261505	0.130752	+	0.991415i	$0.458261\pi$
$602$	−14.4114	−0.587365
$603$	0.853301	0.0347491
$604$	6.75408	0.274820
$605$	−2.42166	−0.0984545
$606$	−7.24947	−0.294490
$607$	29.2061	1.18544	0.592720	−	0.805409i	$-0.298054\pi$
$607$	29.2061	1.18544	0.592720	+	0.805409i	$0.298054\pi$
$608$	−7.13887	−0.289519
$609$	−6.19910	−0.251200
$610$	3.14519	0.127345
$611$	16.5852	0.670964
$612$	7.41835	0.299869
$613$	45.7579	1.84814	0.924071	−	0.382220i	$-0.124840\pi$
$613$	45.7579	1.84814	0.924071	+	0.382220i	$0.124840\pi$
$614$	31.2612	1.26160
$615$	−1.24860	−0.0503485
$616$	14.2278	0.573254
$617$	19.3896	0.780594	0.390297	−	0.920689i	$-0.372372\pi$
$617$	19.3896	0.780594	0.390297	+	0.920689i	$0.372372\pi$
$618$	2.91219	0.117145
$619$	42.8651	1.72289	0.861447	−	0.507847i	$-0.169558\pi$
$619$	42.8651	1.72289	0.861447	+	0.507847i	$0.169558\pi$
$620$	0.717658	0.0288218
$621$	2.32902	0.0934603
$622$	−8.81789	−0.353565
$623$	−30.2715	−1.21280
$624$	1.18873	0.0475871
$625$	23.9698	0.958793
$626$	−22.1111	−0.883736
$627$	15.8872	0.634472
$628$	2.59998	0.103751
$629$	−7.45055	−0.297073
$630$	2.28980	0.0912278
$631$	2.14179	0.0852633	0.0426317	−	0.999091i	$-0.486426\pi$
$631$	2.14179	0.0852633	0.0426317	+	0.999091i	$0.486426\pi$
$632$	−16.1099	−0.640818
$633$	−7.45549	−0.296329
$634$	2.45616	0.0975465
$635$	−4.28461	−0.170029
$636$	4.44263	0.176162
$637$	−7.23368	−0.286609
$638$	17.8002	0.704717
$639$	15.9815	0.632218
$640$	−0.262672	−0.0103830
$641$	13.3901	0.528875	0.264438	−	0.964403i	$-0.414814\pi$
$641$	13.3901	0.528875	0.264438	+	0.964403i	$0.414814\pi$
$642$	6.35041	0.250631
$643$	31.5260	1.24326	0.621632	−	0.783310i	$-0.286470\pi$
$643$	31.5260	1.24326	0.621632	+	0.783310i	$0.286470\pi$
$644$	2.58728	0.101953
$645$	−0.592105	−0.0233141
$646$	19.2223	0.756293
$647$	−23.2876	−0.915530	−0.457765	−	0.889073i	$-0.651350\pi$
$647$	−23.2876	−0.915530	−0.457765	+	0.889073i	$0.651350\pi$
$648$	6.85550	0.269310
$649$	48.6112	1.90816
$650$	11.8436	0.464544
$651$	4.27850	0.167687
$652$	−15.8615	−0.621183
$653$	21.6995	0.849166	0.424583	−	0.905389i	$-0.360421\pi$
$653$	21.6995	0.849166	0.424583	+	0.905389i	$0.360421\pi$
$654$	−4.05375	−0.158514
$655$	−1.11704	−0.0436463
$656$	−9.60456	−0.374995
$657$	−23.5435	−0.918521
$658$	−21.8487	−0.851751
$659$	−17.0759	−0.665184	−0.332592	−	0.943071i	$-0.607923\pi$
$659$	−17.0759	−0.665184	−0.332592	+	0.943071i	$0.607923\pi$
$660$	0.584561	0.0227540
$661$	27.5118	1.07008	0.535042	−	0.844826i	$-0.320296\pi$
$661$	27.5118	1.07008	0.535042	+	0.844826i	$0.320296\pi$
$662$	−14.9052	−0.579308
$663$	−3.20080	−0.124309
$664$	7.64022	0.296498
$665$	5.93330	0.230084
$666$	−7.62328	−0.295396
$667$	3.23692	0.125334
$668$	−14.2174	−0.550088
$669$	11.8285	0.457314
$670$	0.0813551	0.00314302
$671$	−53.8415	−2.07853
$672$	−1.56598	−0.0604091
$673$	49.6830	1.91514	0.957569	−	0.288205i	$-0.0930586\pi$
$673$	49.6830	1.91514	0.957569	+	0.288205i	$0.0930586\pi$
$674$	5.66795	0.218321
$675$	−14.0449	−0.540588
$676$	−7.23105	−0.278117
$677$	−33.0800	−1.27137	−0.635684	−	0.771950i	$-0.719282\pi$
$677$	−33.0800	−1.27137	−0.635684	+	0.771950i	$0.719282\pi$
$678$	0.737501	0.0283236
$679$	9.09048	0.348861
$680$	0.707278	0.0271229
$681$	−6.32726	−0.242461
$682$	−12.2854	−0.470430
$683$	−21.5470	−0.824473	−0.412236	−	0.911077i	$-0.635252\pi$
$683$	−21.5470	−0.824473	−0.412236	+	0.911077i	$0.635252\pi$
$684$	19.6680	0.752024
$685$	3.10534	0.118649
$686$	−12.6195	−0.481815
$687$	9.19672	0.350877
$688$	−4.55462	−0.173643
$689$	21.5603	0.821383
$690$	0.106301	0.00404680
$691$	−6.37198	−0.242401	−0.121201	−	0.992628i	$-0.538674\pi$
$691$	−6.37198	−0.242401	−0.121201	+	0.992628i	$0.538674\pi$
$692$	6.97459	0.265134
$693$	−39.1984	−1.48902
$694$	−30.6079	−1.16186
$695$	3.07598	0.116679
$696$	−1.95918	−0.0742626
$697$	25.8616	0.979576
$698$	−31.4542	−1.19056
$699$	−13.3683	−0.505635
$700$	−15.6023	−0.589712
$701$	−5.89360	−0.222598	−0.111299	−	0.993787i	$-0.535501\pi$
$701$	−5.89360	−0.222598	−0.111299	+	0.993787i	$0.535501\pi$
$702$	−6.84118	−0.258204
$703$	−19.7533	−0.745012
$704$	4.49659	0.169472
$705$	−0.897673	−0.0338083
$706$	4.19569	0.157907
$707$	46.3476	1.74308
$708$	−5.35040	−0.201080
$709$	42.5508	1.59803	0.799015	−	0.601312i	$-0.205355\pi$
$709$	42.5508	1.59803	0.799015	+	0.601312i	$0.205355\pi$
$710$	1.52370	0.0571835
$711$	44.3838	1.66452
$712$	−9.56710	−0.358542
$713$	−2.23406	−0.0836661
$714$	4.21661	0.157803
$715$	2.83691	0.106094
$716$	−19.4848	−0.728181
$717$	8.17264	0.305213
$718$	8.49132	0.316893
$719$	−37.0310	−1.38102	−0.690511	−	0.723322i	$-0.742614\pi$
$719$	−37.0310	−1.38102	−0.690511	+	0.723322i	$0.742614\pi$
$720$	0.723675	0.0269698
$721$	−18.6183	−0.693382
$722$	31.9634	1.18956
$723$	0.211223	0.00785548
$724$	11.3020	0.420034
$725$	−19.5199	−0.724950
$726$	−4.56282	−0.169342
$727$	16.9223	0.627612	0.313806	−	0.949487i	$-0.398396\pi$
$727$	16.9223	0.627612	0.313806	+	0.949487i	$0.398396\pi$
$728$	−7.59980	−0.281667
$729$	−14.6583	−0.542900
$730$	−2.24468	−0.0830794
$731$	12.2639	0.453597
$732$	5.92607	0.219034
$733$	−42.7513	−1.57906	−0.789528	−	0.613715i	$-0.789674\pi$
$733$	−42.7513	−1.57906	−0.789528	+	0.613715i	$0.789674\pi$
$734$	−22.5646	−0.832873
$735$	0.391523	0.0144416
$736$	0.817693	0.0301406
$737$	−1.39269	−0.0513005
$738$	26.4611	0.974047
$739$	53.8621	1.98135	0.990674	−	0.136252i	$-0.0435058\pi$
$739$	53.8621	1.98135	0.990674	+	0.136252i	$0.0435058\pi$
$740$	−0.726816	−0.0267183
$741$	−8.48615	−0.311747
$742$	−28.4028	−1.04270
$743$	−22.0910	−0.810440	−0.405220	−	0.914219i	$-0.632805\pi$
$743$	−22.0910	−0.810440	−0.405220	+	0.914219i	$0.632805\pi$
$744$	1.35219	0.0495736
$745$	−3.79246	−0.138945
$746$	−22.8897	−0.838051
$747$	−21.0492	−0.770151
$748$	−12.1077	−0.442700
$749$	−40.5996	−1.48348
$750$	−1.29104	−0.0471421
$751$	−49.4636	−1.80495	−0.902476	−	0.430741i	$-0.858252\pi$
$751$	−49.4636	−1.80495	−0.902476	+	0.430741i	$0.858252\pi$
$752$	−6.90513	−0.251804
$753$	−3.35092	−0.122114
$754$	−9.50802	−0.346262
$755$	−1.77411	−0.0645663
$756$	9.01232	0.327775
$757$	−18.4313	−0.669899	−0.334949	−	0.942236i	$-0.608719\pi$
$757$	−18.4313	−0.669899	−0.334949	+	0.942236i	$0.608719\pi$
$758$	−14.2387	−0.517173
$759$	−1.81973	−0.0660520
$760$	1.87518	0.0680199
$761$	4.25981	0.154418	0.0772090	−	0.997015i	$-0.475399\pi$
$761$	4.25981	0.154418	0.0772090	+	0.997015i	$0.475399\pi$
$762$	−8.07292	−0.292451
$763$	25.9166	0.938243
$764$	−3.19008	−0.115413
$765$	−1.94859	−0.0704515
$766$	−5.43567	−0.196399
$767$	−25.9658	−0.937571
$768$	−0.494918	−0.0178588
$769$	43.8882	1.58265	0.791325	−	0.611395i	$-0.209392\pi$
$769$	43.8882	1.58265	0.791325	+	0.611395i	$0.209392\pi$
$770$	−3.73724	−0.134681
$771$	1.78990	0.0644617
$772$	2.70929	0.0975096
$773$	−26.3732	−0.948577	−0.474289	−	0.880369i	$-0.657295\pi$
$773$	−26.3732	−0.948577	−0.474289	+	0.880369i	$0.657295\pi$
$774$	12.5482	0.451037
$775$	13.4722	0.483937
$776$	2.87298	0.103134
$777$	−4.33310	−0.155449
$778$	−7.63892	−0.273869
$779$	68.5657	2.45662
$780$	−0.312245	−0.0111801
$781$	−26.0838	−0.933350
$782$	−2.20175	−0.0787342
$783$	11.2752	0.402943
$784$	3.01170	0.107561
$785$	−0.682942	−0.0243753
$786$	−2.10469	−0.0750717
$787$	−6.63659	−0.236569	−0.118284	−	0.992980i	$-0.537739\pi$
$787$	−6.63659	−0.236569	−0.118284	+	0.992980i	$0.537739\pi$
$788$	17.2966	0.616164
$789$	−14.6563	−0.521779
$790$	4.23162	0.150554
$791$	−4.71502	−0.167647
$792$	−12.3884	−0.440201
$793$	28.7595	1.02128
$794$	−26.7157	−0.948106
$795$	−1.16695	−0.0413875
$796$	2.09299	0.0741840
$797$	5.93718	0.210306	0.105153	−	0.994456i	$-0.466467\pi$
$797$	5.93718	0.210306	0.105153	+	0.994456i	$0.466467\pi$
$798$	11.1793	0.395744
$799$	18.5930	0.657772
$800$	−4.93100	−0.174337
$801$	26.3579	0.931310
$802$	25.2039	0.889979
$803$	38.4260	1.35602
$804$	0.153287	0.00540601
$805$	−0.679606	−0.0239530
$806$	6.56225	0.231145
$807$	8.27926	0.291444
$808$	14.6478	0.515309
$809$	40.8540	1.43635	0.718175	−	0.695862i	$-0.244978\pi$
$809$	40.8540	1.43635	0.718175	+	0.695862i	$0.244978\pi$
$810$	−1.80075	−0.0632718
$811$	8.59804	0.301918	0.150959	−	0.988540i	$-0.451764\pi$
$811$	8.59804	0.301918	0.150959	+	0.988540i	$0.451764\pi$
$812$	12.5255	0.439559
$813$	−7.75204	−0.271876
$814$	12.4421	0.436097
$815$	4.16636	0.145941
$816$	1.33263	0.0466514
$817$	32.5148	1.13755
$818$	−17.8527	−0.624207
$819$	20.9379	0.731629
$820$	2.52285	0.0881016
$821$	−22.6483	−0.790432	−0.395216	−	0.918588i	$-0.629330\pi$
$821$	−22.6483	−0.790432	−0.395216	+	0.918588i	$0.629330\pi$
$822$	5.85098	0.204077
$823$	−0.519419	−0.0181058	−0.00905290	−	0.999959i	$-0.502882\pi$
$823$	−0.519419	−0.0181058	−0.00905290	+	0.999959i	$0.502882\pi$
$824$	−5.88419	−0.204985
$825$	10.9737	0.382054
$826$	34.2064	1.19019
$827$	−4.86293	−0.169101	−0.0845503	−	0.996419i	$-0.526945\pi$
$827$	−4.86293	−0.169101	−0.0845503	+	0.996419i	$0.526945\pi$
$828$	−2.25279	−0.0782898
$829$	−0.466594	−0.0162055	−0.00810275	−	0.999967i	$-0.502579\pi$
$829$	−0.466594	−0.0162055	−0.00810275	+	0.999967i	$0.502579\pi$
$830$	−2.00687	−0.0696594
$831$	−4.90804	−0.170258
$832$	−2.40186	−0.0832697
$833$	−8.10939	−0.280974
$834$	5.79567	0.200688
$835$	3.73451	0.129238
$836$	−32.1006	−1.11022
$837$	−7.78192	−0.268983
$838$	−38.8900	−1.34343
$839$	−33.3443	−1.15117	−0.575587	−	0.817741i	$-0.695226\pi$
$839$	−33.3443	−1.15117	−0.575587	+	0.817741i	$0.695226\pi$
$840$	0.411339	0.0141926
$841$	−13.3295	−0.459637
$842$	−8.40529	−0.289665
$843$	−9.53980	−0.328568
$844$	15.0641	0.518528
$845$	1.89939	0.0653411
$846$	19.0240	0.654059
$847$	29.1712	1.00233
$848$	−8.97649	−0.308254
$849$	−2.00675	−0.0688714
$850$	13.2774	0.455410
$851$	2.26257	0.0775598
$852$	2.87091	0.0983558
$853$	−28.9131	−0.989965	−0.494982	−	0.868903i	$-0.664825\pi$
$853$	−28.9131	−0.989965	−0.494982	+	0.868903i	$0.664825\pi$
$854$	−37.8868	−1.29646
$855$	−5.16622	−0.176681
$856$	−12.8312	−0.438562
$857$	−11.7001	−0.399667	−0.199834	−	0.979830i	$-0.564040\pi$
$857$	−11.7001	−0.399667	−0.199834	+	0.979830i	$0.564040\pi$
$858$	5.34521	0.182483
$859$	0.390732	0.0133316	0.00666580	−	0.999978i	$-0.497878\pi$
$859$	0.390732	0.0133316	0.00666580	+	0.999978i	$0.497878\pi$
$860$	1.19637	0.0407959
$861$	15.0406	0.512582
$862$	8.86712	0.302015
$863$	−39.9881	−1.36121	−0.680606	−	0.732649i	$-0.738284\pi$
$863$	−39.9881	−1.36121	−0.680606	+	0.732649i	$0.738284\pi$
$864$	2.84828	0.0969005
$865$	−1.83203	−0.0622908
$866$	−17.9839	−0.611119
$867$	4.82532	0.163876
$868$	−8.64486	−0.293426
$869$	−72.4398	−2.45735
$870$	0.514622	0.0174473
$871$	0.743909	0.0252064
$872$	8.19075	0.277374
$873$	−7.91523	−0.267890
$874$	−5.83740	−0.197453
$875$	8.25392	0.279033
$876$	−4.22936	−0.142897
$877$	23.3548	0.788635	0.394317	−	0.918974i	$-0.370981\pi$
$877$	23.3548	0.788635	0.394317	+	0.918974i	$0.370981\pi$
$878$	4.07642	0.137573
$879$	−3.14012	−0.105914
$880$	−1.18113	−0.0398158
$881$	15.2883	0.515074	0.257537	−	0.966268i	$-0.417089\pi$
$881$	15.2883	0.515074	0.257537	+	0.966268i	$0.417089\pi$
$882$	−8.29739	−0.279388
$883$	21.3212	0.717514	0.358757	−	0.933431i	$-0.383201\pi$
$883$	21.3212	0.717514	0.358757	+	0.933431i	$0.383201\pi$
$884$	6.46733	0.217520
$885$	1.40540	0.0472420
$886$	−13.0641	−0.438895
$887$	5.94686	0.199676	0.0998380	−	0.995004i	$-0.468168\pi$
$887$	5.94686	0.199676	0.0998380	+	0.995004i	$0.468168\pi$
$888$	−1.36944	−0.0459556
$889$	51.6121	1.73101
$890$	2.51301	0.0842361
$891$	30.8264	1.03272
$892$	−23.8998	−0.800225
$893$	49.2948	1.64959
$894$	−7.14563	−0.238986
$895$	5.11810	0.171079
$896$	3.16413	0.105706
$897$	0.972012	0.0324545
$898$	15.2717	0.509623
$899$	−10.8155	−0.360716
$900$	13.5852	0.452840
$901$	24.1704	0.805232
$902$	−43.1878	−1.43800
$903$	7.13246	0.237353
$904$	−1.49015	−0.0495616
$905$	−2.96870	−0.0986831
$906$	−3.34272	−0.111054
$907$	−58.3745	−1.93829	−0.969147	−	0.246483i	$-0.920725\pi$
$907$	−58.3745	−1.93829	−0.969147	+	0.246483i	$0.920725\pi$
$908$	12.7845	0.424267
$909$	−40.3556	−1.33851
$910$	1.99625	0.0661751
$911$	9.46492	0.313587	0.156793	−	0.987631i	$-0.449884\pi$
$911$	9.46492	0.313587	0.156793	+	0.987631i	$0.449884\pi$
$912$	3.53315	0.116994
$913$	34.3549	1.13698
$914$	−42.6358	−1.41027
$915$	−1.55661	−0.0514600
$916$	−18.5823	−0.613977
$917$	13.4558	0.444348
$918$	−7.66937	−0.253127
$919$	49.6148	1.63664	0.818320	−	0.574763i	$-0.194906\pi$
$919$	49.6148	1.63664	0.818320	+	0.574763i	$0.194906\pi$
$920$	−0.214785	−0.00708124
$921$	−15.4717	−0.509811
$922$	−29.0500	−0.956709
$923$	13.9327	0.458600
$924$	−7.04159	−0.231651
$925$	−13.6442	−0.448617
$926$	−22.1464	−0.727776
$927$	16.2113	0.532448
$928$	3.95860	0.129947
$929$	−22.5594	−0.740149	−0.370074	−	0.929002i	$-0.620668\pi$
$929$	−22.5594	−0.740149	−0.370074	+	0.929002i	$0.620668\pi$
$930$	−0.355182	−0.0116469
$931$	−21.5001	−0.704637
$932$	27.0111	0.884778
$933$	4.36413	0.142875
$934$	12.4454	0.407227
$935$	3.18034	0.104008
$936$	6.61727	0.216292
$937$	−33.0415	−1.07942	−0.539709	−	0.841852i	$-0.681466\pi$
$937$	−33.0415	−1.07942	−0.539709	+	0.841852i	$0.681466\pi$
$938$	−0.979998	−0.0319981
$939$	10.9432	0.357117
$940$	1.81378	0.0591590
$941$	35.0442	1.14241	0.571204	−	0.820808i	$-0.306477\pi$
$941$	35.0442	1.14241	0.571204	+	0.820808i	$0.306477\pi$
$942$	−1.28678	−0.0419255
$943$	−7.85358	−0.255748
$944$	10.8107	0.351858
$945$	−2.36728	−0.0770077
$946$	−20.4803	−0.665871
$947$	26.5043	0.861276	0.430638	−	0.902525i	$-0.358289\pi$
$947$	26.5043	0.861276	0.430638	+	0.902525i	$0.358289\pi$
$948$	7.97309	0.258954
$949$	−20.5253	−0.666280
$950$	35.2018	1.14210
$951$	−1.21560	−0.0394184
$952$	−8.51983	−0.276129
$953$	−43.7995	−1.41880	−0.709402	−	0.704804i	$-0.751035\pi$
$953$	−43.7995	−1.41880	−0.709402	+	0.704804i	$0.751035\pi$
$954$	24.7307	0.800687
$955$	0.837943	0.0271152
$956$	−16.5131	−0.534073
$957$	−8.80964	−0.284775
$958$	8.31828	0.268752
$959$	−37.4067	−1.20793
$960$	0.130001	0.00419576
$961$	−23.5354	−0.759206
$962$	−6.64599	−0.214275
$963$	35.3508	1.13916
$964$	−0.426785	−0.0137458
$965$	−0.711655	−0.0229090
$966$	−1.28049	−0.0411992
$967$	54.9279	1.76636	0.883181	−	0.469032i	$-0.155397\pi$
$967$	54.9279	1.76636	0.883181	+	0.469032i	$0.155397\pi$
$968$	9.21934	0.296321
$969$	−9.51348	−0.305617
$970$	−0.754651	−0.0242304
$971$	−33.5446	−1.07650	−0.538249	−	0.842786i	$-0.680914\pi$
$971$	−33.5446	−1.07650	−0.538249	+	0.842786i	$0.680914\pi$
$972$	−11.9378	−0.382904
$973$	−37.0531	−1.18787
$974$	30.0468	0.962763
$975$	−5.86161	−0.187722
$976$	−11.9738	−0.383273
$977$	31.2469	0.999676	0.499838	−	0.866119i	$-0.333393\pi$
$977$	31.2469	0.999676	0.499838	+	0.866119i	$0.333393\pi$
$978$	7.85013	0.251019
$979$	−43.0193	−1.37490
$980$	−0.791087	−0.0252704
$981$	−22.5660	−0.720476
$982$	25.7412	0.821435
$983$	39.7001	1.26624	0.633118	−	0.774056i	$-0.281775\pi$
$983$	39.7001	1.26624	0.633118	+	0.774056i	$0.281775\pi$
$984$	4.75347	0.151535
$985$	−4.54331	−0.144762
$986$	−10.6591	−0.339453
$987$	10.8133	0.344191
$988$	17.1466	0.545506
$989$	−3.72428	−0.118425
$990$	3.25407	0.103421
$991$	53.9645	1.71424	0.857119	−	0.515118i	$-0.172252\pi$
$991$	53.9645	1.71424	0.857119	+	0.515118i	$0.172252\pi$
$992$	−2.73215	−0.0867458
$993$	7.37686	0.234097
$994$	−18.3544	−0.582166
$995$	−0.549769	−0.0174288
$996$	−3.78128	−0.119814
$997$	−17.4783	−0.553545	−0.276772	−	0.960936i	$-0.589265\pi$
$997$	−17.4783	−0.553545	−0.276772	+	0.960936i	$0.589265\pi$
$998$	−17.7380	−0.561486
$999$	7.88123	0.249351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.36	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.36	✓	69	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.494918 q^{3} +1.00000 q^{4} -0.262672 q^{5} -0.494918 q^{6} +3.16413 q^{7} +1.00000 q^{8} -2.75506 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.494918 q^{3} +1.00000 q^{4} -0.262672 q^{5} -0.494918 q^{6} +3.16413 q^{7} +1.00000 q^{8} -2.75506 q^{9} -0.262672 q^{10} +4.49659 q^{11} -0.494918 q^{12} -2.40186 q^{13} +3.16413 q^{14} +0.130001 q^{15} +1.00000 q^{16} -2.69263 q^{17} -2.75506 q^{18} -7.13887 q^{19} -0.262672 q^{20} -1.56598 q^{21} +4.49659 q^{22} +0.817693 q^{23} -0.494918 q^{24} -4.93100 q^{25} -2.40186 q^{26} +2.84828 q^{27} +3.16413 q^{28} +3.95860 q^{29} +0.130001 q^{30} -2.73215 q^{31} +1.00000 q^{32} -2.22544 q^{33} -2.69263 q^{34} -0.831126 q^{35} -2.75506 q^{36} +2.76701 q^{37} -7.13887 q^{38} +1.18873 q^{39} -0.262672 q^{40} -9.60456 q^{41} -1.56598 q^{42} -4.55462 q^{43} +4.49659 q^{44} +0.723675 q^{45} +0.817693 q^{46} -6.90513 q^{47} -0.494918 q^{48} +3.01170 q^{49} -4.93100 q^{50} +1.33263 q^{51} -2.40186 q^{52} -8.97649 q^{53} +2.84828 q^{54} -1.18113 q^{55} +3.16413 q^{56} +3.53315 q^{57} +3.95860 q^{58} +10.8107 q^{59} +0.130001 q^{60} -11.9738 q^{61} -2.73215 q^{62} -8.71735 q^{63} +1.00000 q^{64} +0.630902 q^{65} -2.22544 q^{66} -0.309722 q^{67} -2.69263 q^{68} -0.404691 q^{69} -0.831126 q^{70} -5.80078 q^{71} -2.75506 q^{72} +8.54558 q^{73} +2.76701 q^{74} +2.44044 q^{75} -7.13887 q^{76} +14.2278 q^{77} +1.18873 q^{78} -16.1099 q^{79} -0.262672 q^{80} +6.85550 q^{81} -9.60456 q^{82} +7.64022 q^{83} -1.56598 q^{84} +0.707278 q^{85} -4.55462 q^{86} -1.95918 q^{87} +4.49659 q^{88} -9.56710 q^{89} +0.723675 q^{90} -7.59980 q^{91} +0.817693 q^{92} +1.35219 q^{93} -6.90513 q^{94} +1.87518 q^{95} -0.494918 q^{96} +2.87298 q^{97} +3.01170 q^{98} -12.3884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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