LMFDB - Embedded newform 69.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,4,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	69.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:	$4.07113179040$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} -7.31371 q^{5} -5.48528 q^{6} -11.1716 q^{7} -23.1421 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} -7.31371 q^{5} -5.48528 q^{6} -11.1716 q^{7} -23.1421 q^{8} +9.00000 q^{9} -13.3726 q^{10} -16.2010 q^{11} +13.9706 q^{12} -13.0294 q^{13} -20.4264 q^{14} +21.9411 q^{15} -5.05887 q^{16} +33.8579 q^{17} +16.4558 q^{18} -6.11775 q^{19} +34.0589 q^{20} +33.5147 q^{21} -29.6224 q^{22} +23.0000 q^{23} +69.4264 q^{24} -71.5097 q^{25} -23.8234 q^{26} -27.0000 q^{27} +52.0244 q^{28} +84.6762 q^{29} +40.1177 q^{30} -236.735 q^{31} +175.887 q^{32} +48.6030 q^{33} +61.9066 q^{34} +81.7056 q^{35} -41.9117 q^{36} -63.6224 q^{37} -11.1859 q^{38} +39.0883 q^{39} +169.255 q^{40} +75.5391 q^{41} +61.2792 q^{42} -260.912 q^{43} +75.4457 q^{44} -65.8234 q^{45} +42.0538 q^{46} +224.902 q^{47} +15.1766 q^{48} -218.196 q^{49} -130.750 q^{50} -101.574 q^{51} +60.6762 q^{52} +44.2843 q^{53} -49.3675 q^{54} +118.489 q^{55} +258.534 q^{56} +18.3532 q^{57} +154.824 q^{58} -466.891 q^{59} -102.177 q^{60} -520.593 q^{61} -432.853 q^{62} -100.544 q^{63} +362.068 q^{64} +95.2935 q^{65} +88.8671 q^{66} -906.950 q^{67} -157.671 q^{68} -69.0000 q^{69} +149.393 q^{70} -920.930 q^{71} -208.279 q^{72} +251.608 q^{73} -116.329 q^{74} +214.529 q^{75} +28.4895 q^{76} +180.991 q^{77} +71.4701 q^{78} +1052.88 q^{79} +36.9991 q^{80} +81.0000 q^{81} +138.118 q^{82} +143.202 q^{83} -156.073 q^{84} -247.627 q^{85} -477.058 q^{86} -254.029 q^{87} +374.926 q^{88} +1081.94 q^{89} -120.353 q^{90} +145.559 q^{91} -107.108 q^{92} +710.205 q^{93} +411.216 q^{94} +44.7434 q^{95} -527.662 q^{96} +1159.76 q^{97} -398.955 q^{98} -145.809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 8 q^{5} + 6 q^{6} - 28 q^{7} - 18 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 8 * q^5 + 6 * q^6 - 28 * q^7 - 18 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 8 q^{5} + 6 q^{6} - 28 q^{7} - 18 q^{8} + 18 q^{9} - 72 q^{10} - 72 q^{11} - 6 q^{12} - 60 q^{13} + 44 q^{14} - 24 q^{15} - 78 q^{16} + 96 q^{17} - 18 q^{18} - 148 q^{19} + 136 q^{20} + 84 q^{21} + 184 q^{22} + 46 q^{23} + 54 q^{24} + 38 q^{25} + 156 q^{26} - 54 q^{27} - 60 q^{28} - 204 q^{29} + 216 q^{30} - 168 q^{31} + 414 q^{32} + 216 q^{33} - 176 q^{34} - 176 q^{35} + 18 q^{36} + 116 q^{37} + 532 q^{38} + 180 q^{39} + 248 q^{40} + 4 q^{41} - 132 q^{42} - 420 q^{43} - 296 q^{44} + 72 q^{45} - 46 q^{46} - 48 q^{47} + 234 q^{48} - 278 q^{49} - 550 q^{50} - 288 q^{51} - 252 q^{52} + 32 q^{53} + 54 q^{54} - 736 q^{55} + 172 q^{56} + 444 q^{57} + 1260 q^{58} - 40 q^{59} - 408 q^{60} - 764 q^{61} - 696 q^{62} - 252 q^{63} + 34 q^{64} - 624 q^{65} - 552 q^{66} - 988 q^{67} + 256 q^{68} - 138 q^{69} + 1136 q^{70} - 224 q^{71} - 162 q^{72} + 820 q^{73} - 804 q^{74} - 114 q^{75} - 916 q^{76} + 1120 q^{77} - 468 q^{78} + 1772 q^{79} - 1080 q^{80} + 162 q^{81} + 412 q^{82} + 1480 q^{83} + 180 q^{84} + 704 q^{85} + 132 q^{86} + 612 q^{87} + 88 q^{88} + 744 q^{89} - 648 q^{90} + 936 q^{91} + 46 q^{92} + 504 q^{93} + 1456 q^{94} - 2128 q^{95} - 1242 q^{96} - 260 q^{97} - 170 q^{98} - 648 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 8 * q^5 + 6 * q^6 - 28 * q^7 - 18 * q^8 + 18 * q^9 - 72 * q^10 - 72 * q^11 - 6 * q^12 - 60 * q^13 + 44 * q^14 - 24 * q^15 - 78 * q^16 + 96 * q^17 - 18 * q^18 - 148 * q^19 + 136 * q^20 + 84 * q^21 + 184 * q^22 + 46 * q^23 + 54 * q^24 + 38 * q^25 + 156 * q^26 - 54 * q^27 - 60 * q^28 - 204 * q^29 + 216 * q^30 - 168 * q^31 + 414 * q^32 + 216 * q^33 - 176 * q^34 - 176 * q^35 + 18 * q^36 + 116 * q^37 + 532 * q^38 + 180 * q^39 + 248 * q^40 + 4 * q^41 - 132 * q^42 - 420 * q^43 - 296 * q^44 + 72 * q^45 - 46 * q^46 - 48 * q^47 + 234 * q^48 - 278 * q^49 - 550 * q^50 - 288 * q^51 - 252 * q^52 + 32 * q^53 + 54 * q^54 - 736 * q^55 + 172 * q^56 + 444 * q^57 + 1260 * q^58 - 40 * q^59 - 408 * q^60 - 764 * q^61 - 696 * q^62 - 252 * q^63 + 34 * q^64 - 624 * q^65 - 552 * q^66 - 988 * q^67 + 256 * q^68 - 138 * q^69 + 1136 * q^70 - 224 * q^71 - 162 * q^72 + 820 * q^73 - 804 * q^74 - 114 * q^75 - 916 * q^76 + 1120 * q^77 - 468 * q^78 + 1772 * q^79 - 1080 * q^80 + 162 * q^81 + 412 * q^82 + 1480 * q^83 + 180 * q^84 + 704 * q^85 + 132 * q^86 + 612 * q^87 + 88 * q^88 + 744 * q^89 - 648 * q^90 + 936 * q^91 + 46 * q^92 + 504 * q^93 + 1456 * q^94 - 2128 * q^95 - 1242 * q^96 - 260 * q^97 - 170 * q^98 - 648 * 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 $

$2$	1.82843	0.646447	0.323223	−	0.946323i	$-0.395234\pi$
$2$	1.82843	0.646447	0.323223	+	0.946323i	$0.395234\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.65685	−0.582107
$5$	−7.31371	−0.654158	−0.327079	−	0.944997i	$-0.606064\pi$
$5$	−7.31371	−0.654158	−0.327079	+	0.944997i	$0.606064\pi$
$6$	−5.48528	−0.373226
$7$	−11.1716	−0.603208	−0.301604	−	0.953433i	$-0.597522\pi$
$7$	−11.1716	−0.603208	−0.301604	+	0.953433i	$0.597522\pi$
$8$	−23.1421	−1.02275
$9$	9.00000	0.333333
$10$	−13.3726	−0.422878
$11$	−16.2010	−0.444072	−0.222036	−	0.975039i	$-0.571270\pi$
$11$	−16.2010	−0.444072	−0.222036	+	0.975039i	$0.571270\pi$
$12$	13.9706	0.336080
$13$	−13.0294	−0.277978	−0.138989	−	0.990294i	$-0.544385\pi$
$13$	−13.0294	−0.277978	−0.138989	+	0.990294i	$0.544385\pi$
$14$	−20.4264	−0.389942
$15$	21.9411	0.377678
$16$	−5.05887	−0.0790449
$17$	33.8579	0.483043	0.241522	−	0.970395i	$-0.422353\pi$
$17$	33.8579	0.483043	0.241522	+	0.970395i	$0.422353\pi$
$18$	16.4558	0.215482
$19$	−6.11775	−0.0738688	−0.0369344	−	0.999318i	$-0.511759\pi$
$19$	−6.11775	−0.0738688	−0.0369344	+	0.999318i	$0.511759\pi$
$20$	34.0589	0.380790
$21$	33.5147	0.348262
$22$	−29.6224	−0.287069
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	69.4264	0.590484
$25$	−71.5097	−0.572077
$26$	−23.8234	−0.179698
$27$	−27.0000	−0.192450
$28$	52.0244	0.351132
$29$	84.6762	0.542206	0.271103	−	0.962550i	$-0.412612\pi$
$29$	84.6762	0.542206	0.271103	+	0.962550i	$0.412612\pi$
$30$	40.1177	0.244149
$31$	−236.735	−1.37158	−0.685788	−	0.727801i	$-0.740542\pi$
$31$	−236.735	−1.37158	−0.685788	+	0.727801i	$0.740542\pi$
$32$	175.887	0.971649
$33$	48.6030	0.256385
$34$	61.9066	0.312262
$35$	81.7056	0.394593
$36$	−41.9117	−0.194036
$37$	−63.6224	−0.282688	−0.141344	−	0.989961i	$-0.545142\pi$
$37$	−63.6224	−0.282688	−0.141344	+	0.989961i	$0.545142\pi$
$38$	−11.1859	−0.0477523
$39$	39.0883	0.160491
$40$	169.255	0.669038
$41$	75.5391	0.287737	0.143869	−	0.989597i	$-0.454046\pi$
$41$	75.5391	0.287737	0.143869	+	0.989597i	$0.454046\pi$
$42$	61.2792	0.225133
$43$	−260.912	−0.925318	−0.462659	−	0.886536i	$-0.653105\pi$
$43$	−260.912	−0.925318	−0.462659	+	0.886536i	$0.653105\pi$
$44$	75.4457	0.258497
$45$	−65.8234	−0.218053
$46$	42.0538	0.134793
$47$	224.902	0.697984	0.348992	−	0.937126i	$-0.386524\pi$
$47$	224.902	0.697984	0.348992	+	0.937126i	$0.386524\pi$
$48$	15.1766	0.0456366
$49$	−218.196	−0.636140
$50$	−130.750	−0.369817
$51$	−101.574	−0.278885
$52$	60.6762	0.161813
$53$	44.2843	0.114772	0.0573860	−	0.998352i	$-0.481723\pi$
$53$	44.2843	0.114772	0.0573860	+	0.998352i	$0.481723\pi$
$54$	−49.3675	−0.124409
$55$	118.489	0.290493
$56$	258.534	0.616930
$57$	18.3532	0.0426482
$58$	154.824	0.350507
$59$	−466.891	−1.03024	−0.515119	−	0.857118i	$-0.672252\pi$
$59$	−466.891	−1.03024	−0.515119	+	0.857118i	$0.672252\pi$
$60$	−102.177	−0.219849
$61$	−520.593	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$61$	−520.593	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$62$	−432.853	−0.886651
$63$	−100.544	−0.201069
$64$	362.068	0.707164
$65$	95.2935	0.181842
$66$	88.8671	0.165739
$67$	−906.950	−1.65376	−0.826878	−	0.562382i	$-0.809885\pi$
$67$	−906.950	−1.65376	−0.826878	+	0.562382i	$0.809885\pi$
$68$	−157.671	−0.281183
$69$	−69.0000	−0.120386
$70$	149.393	0.255084
$71$	−920.930	−1.53936	−0.769678	−	0.638432i	$-0.779583\pi$
$71$	−920.930	−1.53936	−0.769678	+	0.638432i	$0.779583\pi$
$72$	−208.279	−0.340916
$73$	251.608	0.403404	0.201702	−	0.979447i	$-0.435353\pi$
$73$	251.608	0.403404	0.201702	+	0.979447i	$0.435353\pi$
$74$	−116.329	−0.182743
$75$	214.529	0.330289
$76$	28.4895	0.0429996
$77$	180.991	0.267868
$78$	71.4701	0.103749
$79$	1052.88	1.49947	0.749734	−	0.661740i	$-0.230182\pi$
$79$	1052.88	1.49947	0.749734	+	0.661740i	$0.230182\pi$
$80$	36.9991	0.0517079
$81$	81.0000	0.111111
$82$	138.118	0.186007
$83$	143.202	0.189379	0.0946894	−	0.995507i	$-0.469814\pi$
$83$	143.202	0.189379	0.0946894	+	0.995507i	$0.469814\pi$
$84$	−156.073	−0.202726
$85$	−247.627	−0.315987
$86$	−477.058	−0.598168
$87$	−254.029	−0.313043
$88$	374.926	0.454173
$89$	1081.94	1.28859	0.644297	−	0.764775i	$-0.277150\pi$
$89$	1081.94	1.28859	0.644297	+	0.764775i	$0.277150\pi$
$90$	−120.353	−0.140959
$91$	145.559	0.167679
$92$	−107.108	−0.121378
$93$	710.205	0.791880
$94$	411.216	0.451210
$95$	44.7434	0.0483219
$96$	−527.662	−0.560982
$97$	1159.76	1.21398	0.606990	−	0.794709i	$-0.292377\pi$
$97$	1159.76	1.21398	0.606990	+	0.794709i	$0.292377\pi$
$98$	−398.955	−0.411230
$99$	−145.809	−0.148024
$100$	333.010	0.333010
$101$	1172.72	1.15535	0.577676	−	0.816266i	$-0.303960\pi$
$101$	1172.72	1.15535	0.577676	+	0.816266i	$0.303960\pi$
$102$	−185.720	−0.180284
$103$	981.003	0.938458	0.469229	−	0.883077i	$-0.344532\pi$
$103$	981.003	0.938458	0.469229	+	0.883077i	$0.344532\pi$
$104$	301.529	0.284301
$105$	−245.117	−0.227819
$106$	80.9706	0.0741939
$107$	−899.543	−0.812730	−0.406365	−	0.913711i	$-0.633204\pi$
$107$	−899.543	−0.812730	−0.406365	+	0.913711i	$0.633204\pi$
$108$	125.735	0.112027
$109$	1710.34	1.50294	0.751470	−	0.659767i	$-0.229345\pi$
$109$	1710.34	1.50294	0.751470	+	0.659767i	$0.229345\pi$
$110$	216.649	0.187788
$111$	190.867	0.163210
$112$	56.5156	0.0476805
$113$	−1079.79	−0.898919	−0.449459	−	0.893301i	$-0.648383\pi$
$113$	−1079.79	−0.898919	−0.449459	+	0.893301i	$0.648383\pi$
$114$	33.5576	0.0275698
$115$	−168.215	−0.136401
$116$	−394.325	−0.315622
$117$	−117.265	−0.0926594
$118$	−853.677	−0.665994
$119$	−378.246	−0.291376
$120$	−507.765	−0.386270
$121$	−1068.53	−0.802800
$122$	−951.866	−0.706376
$123$	−226.617	−0.166125
$124$	1102.44	0.798404
$125$	1437.21	1.02839
$126$	−183.838	−0.129981
$127$	−361.324	−0.252459	−0.126230	−	0.992001i	$-0.540288\pi$
$127$	−361.324	−0.252459	−0.126230	+	0.992001i	$0.540288\pi$
$128$	−745.083	−0.514505
$129$	782.735	0.534232
$130$	174.237	0.117551
$131$	−1056.56	−0.704672	−0.352336	−	0.935873i	$-0.614613\pi$
$131$	−1056.56	−0.704672	−0.352336	+	0.935873i	$0.614613\pi$
$132$	−226.337	−0.149243
$133$	68.3449	0.0445583
$134$	−1658.29	−1.06906
$135$	197.470	0.125893
$136$	−783.543	−0.494031
$137$	2273.44	1.41776	0.708879	−	0.705331i	$-0.249201\pi$
$137$	2273.44	1.41776	0.708879	+	0.705331i	$0.249201\pi$
$138$	−126.161	−0.0778230
$139$	−1544.41	−0.942411	−0.471206	−	0.882023i	$-0.656181\pi$
$139$	−1544.41	−0.942411	−0.471206	+	0.882023i	$0.656181\pi$
$140$	−380.491	−0.229696
$141$	−674.705	−0.402981
$142$	−1683.85	−0.995112
$143$	211.090	0.123442
$144$	−45.5299	−0.0263483
$145$	−619.297	−0.354688
$146$	460.047	0.260779
$147$	654.588	0.367275
$148$	296.280	0.164555
$149$	665.076	0.365672	0.182836	−	0.983143i	$-0.441472\pi$
$149$	665.076	0.365672	0.182836	+	0.983143i	$0.441472\pi$
$150$	392.251	0.213514
$151$	−3559.98	−1.91859	−0.959294	−	0.282410i	$-0.908866\pi$
$151$	−3559.98	−1.91859	−0.959294	+	0.282410i	$0.908866\pi$
$152$	141.578	0.0755492
$153$	304.721	0.161014
$154$	330.928	0.173162
$155$	1731.41	0.897228
$156$	−182.029	−0.0934228
$157$	−1391.25	−0.707221	−0.353611	−	0.935393i	$-0.615046\pi$
$157$	−1391.25	−0.707221	−0.353611	+	0.935393i	$0.615046\pi$
$158$	1925.11	0.969326
$159$	−132.853	−0.0662636
$160$	−1286.39	−0.635612
$161$	−256.946	−0.125778
$162$	148.103	0.0718274
$163$	975.960	0.468976	0.234488	−	0.972119i	$-0.424659\pi$
$163$	975.960	0.468976	0.234488	+	0.972119i	$0.424659\pi$
$164$	−351.775	−0.167494
$165$	−355.468	−0.167716
$166$	261.834	0.122423
$167$	2053.06	0.951323	0.475661	−	0.879628i	$-0.342209\pi$
$167$	2053.06	0.951323	0.475661	+	0.879628i	$0.342209\pi$
$168$	−775.602	−0.356185
$169$	−2027.23	−0.922728
$170$	−452.767	−0.204269
$171$	−55.0597	−0.0246229
$172$	1215.03	0.538634
$173$	1308.25	0.574937	0.287468	−	0.957790i	$-0.407186\pi$
$173$	1308.25	0.574937	0.287468	+	0.957790i	$0.407186\pi$
$174$	−464.473	−0.202365
$175$	798.875	0.345082
$176$	81.9589	0.0351016
$177$	1400.67	0.594809
$178$	1978.24	0.833007
$179$	−829.584	−0.346403	−0.173201	−	0.984886i	$-0.555411\pi$
$179$	−829.584	−0.346403	−0.173201	+	0.984886i	$0.555411\pi$
$180$	306.530	0.126930
$181$	−2431.29	−0.998431	−0.499216	−	0.866478i	$-0.666378\pi$
$181$	−2431.29	−0.998431	−0.499216	+	0.866478i	$0.666378\pi$
$182$	266.145	0.108395
$183$	1561.78	0.630874
$184$	−532.269	−0.213258
$185$	465.315	0.184923
$186$	1298.56	0.511908
$187$	−548.532	−0.214506
$188$	−1047.33	−0.406301
$189$	301.632	0.116087
$190$	81.8101	0.0312375
$191$	−2878.90	−1.09063	−0.545314	−	0.838232i	$-0.683590\pi$
$191$	−2878.90	−1.09063	−0.545314	+	0.838232i	$0.683590\pi$
$192$	−1086.20	−0.408281
$193$	1423.90	0.531060	0.265530	−	0.964103i	$-0.414453\pi$
$193$	1423.90	0.531060	0.265530	+	0.964103i	$0.414453\pi$
$194$	2120.54	0.784773
$195$	−285.881	−0.104986
$196$	1016.11	0.370301
$197$	−3517.78	−1.27224	−0.636121	−	0.771590i	$-0.719462\pi$
$197$	−3517.78	−1.27224	−0.636121	+	0.771590i	$0.719462\pi$
$198$	−266.601	−0.0956895
$199$	4069.01	1.44947	0.724735	−	0.689028i	$-0.241962\pi$
$199$	4069.01	1.44947	0.724735	+	0.689028i	$0.241962\pi$
$200$	1654.89	0.585091
$201$	2720.85	0.954796
$202$	2144.24	0.746873
$203$	−945.966	−0.327063
$204$	473.013	0.162341
$205$	−552.471	−0.188226
$206$	1793.69	0.606663
$207$	207.000	0.0695048
$208$	65.9143	0.0219728
$209$	99.1137	0.0328031
$210$	−448.178	−0.147273
$211$	−2631.29	−0.858510	−0.429255	−	0.903183i	$-0.641224\pi$
$211$	−2631.29	−0.858510	−0.429255	+	0.903183i	$0.641224\pi$
$212$	−206.225	−0.0668095
$213$	2762.79	0.888748
$214$	−1644.75	−0.525387
$215$	1908.23	0.605304
$216$	624.838	0.196828
$217$	2644.70	0.827346
$218$	3127.22	0.971570
$219$	−754.824	−0.232905
$220$	−551.788	−0.169098
$221$	−441.149	−0.134276
$222$	348.987	0.105507
$223$	−2886.47	−0.866780	−0.433390	−	0.901206i	$-0.642683\pi$
$223$	−2886.47	−0.866780	−0.433390	+	0.901206i	$0.642683\pi$
$224$	−1964.94	−0.586107
$225$	−643.587	−0.190692
$226$	−1974.31	−0.581103
$227$	1819.49	0.532000	0.266000	−	0.963973i	$-0.414298\pi$
$227$	1819.49	0.532000	0.266000	+	0.963973i	$0.414298\pi$
$228$	−85.4684	−0.0248258
$229$	1038.46	0.299664	0.149832	−	0.988711i	$-0.452127\pi$
$229$	1038.46	0.299664	0.149832	+	0.988711i	$0.452127\pi$
$230$	−307.569	−0.0881762
$231$	−542.972	−0.154653
$232$	−1959.59	−0.554540
$233$	−1432.80	−0.402858	−0.201429	−	0.979503i	$-0.564559\pi$
$233$	−1432.80	−0.402858	−0.201429	+	0.979503i	$0.564559\pi$
$234$	−214.410	−0.0598993
$235$	−1644.86	−0.456592
$236$	2174.25	0.599709
$237$	−3158.63	−0.865718
$238$	−691.595	−0.188359
$239$	−6548.04	−1.77221	−0.886103	−	0.463488i	$-0.846598\pi$
$239$	−6548.04	−1.77221	−0.886103	+	0.463488i	$0.846598\pi$
$240$	−110.997	−0.0298535
$241$	1815.36	0.485217	0.242609	−	0.970124i	$-0.421997\pi$
$241$	1815.36	0.485217	0.242609	+	0.970124i	$0.421997\pi$
$242$	−1953.72	−0.518968
$243$	−243.000	−0.0641500
$244$	2424.33	0.636072
$245$	1595.82	0.416136
$246$	−414.353	−0.107391
$247$	79.7108	0.0205339
$248$	5478.55	1.40278
$249$	−429.606	−0.109338
$250$	2627.84	0.664797
$251$	−7405.10	−1.86218	−0.931088	−	0.364796i	$-0.881139\pi$
$251$	−7405.10	−1.86218	−0.931088	+	0.364796i	$0.881139\pi$
$252$	468.219	0.117044
$253$	−372.623	−0.0925953
$254$	−660.654	−0.163201
$255$	742.880	0.182435
$256$	−4258.88	−1.03976
$257$	−203.697	−0.0494408	−0.0247204	−	0.999694i	$-0.507870\pi$
$257$	−203.697	−0.0494408	−0.0247204	+	0.999694i	$0.507870\pi$
$258$	1431.17	0.345353
$259$	710.762	0.170520
$260$	−443.768	−0.105851
$261$	762.086	0.180735
$262$	−1931.84	−0.455533
$263$	4807.59	1.12718	0.563591	−	0.826054i	$-0.309419\pi$
$263$	4807.59	1.12718	0.563591	+	0.826054i	$0.309419\pi$
$264$	−1124.78	−0.262217
$265$	−323.882	−0.0750790
$266$	124.964	0.0288046
$267$	−3245.81	−0.743970
$268$	4223.54	0.962662
$269$	−7183.57	−1.62822	−0.814108	−	0.580714i	$-0.802773\pi$
$269$	−7183.57	−1.62822	−0.814108	+	0.580714i	$0.802773\pi$
$270$	361.060	0.0813829
$271$	7099.96	1.59148	0.795742	−	0.605636i	$-0.207081\pi$
$271$	7099.96	1.59148	0.795742	+	0.605636i	$0.207081\pi$
$272$	−171.283	−0.0381821
$273$	−436.678	−0.0968093
$274$	4156.81	0.916504
$275$	1158.53	0.254043
$276$	321.323	0.0700774
$277$	−2195.58	−0.476244	−0.238122	−	0.971235i	$-0.576532\pi$
$277$	−2195.58	−0.476244	−0.238122	+	0.971235i	$0.576532\pi$
$278$	−2823.84	−0.609219
$279$	−2130.62	−0.457192
$280$	−1890.84	−0.403570
$281$	−646.472	−0.137243	−0.0686215	−	0.997643i	$-0.521860\pi$
$281$	−646.472	−0.137243	−0.0686215	+	0.997643i	$0.521860\pi$
$282$	−1233.65	−0.260506
$283$	−3031.71	−0.636806	−0.318403	−	0.947955i	$-0.603147\pi$
$283$	−3031.71	−0.636806	−0.318403	+	0.947955i	$0.603147\pi$
$284$	4288.64	0.896070
$285$	−134.230	−0.0278987
$286$	385.963	0.0797988
$287$	−843.891	−0.173565
$288$	1582.99	0.323883
$289$	−3766.65	−0.766669
$290$	−1132.34	−0.229287
$291$	−3479.29	−0.700892
$292$	−1171.70	−0.234824
$293$	3944.40	0.786466	0.393233	−	0.919439i	$-0.371357\pi$
$293$	3944.40	0.786466	0.393233	+	0.919439i	$0.371357\pi$
$294$	1196.87	0.237424
$295$	3414.71	0.673939
$296$	1472.36	0.289118
$297$	437.427	0.0854616
$298$	1216.04	0.236388
$299$	−299.677	−0.0579624
$300$	−999.030	−0.192263
$301$	2914.79	0.558159
$302$	−6509.16	−1.24026
$303$	−3518.17	−0.667042
$304$	30.9489	0.00583896
$305$	3807.46	0.714803
$306$	557.160	0.104087
$307$	−3403.21	−0.632676	−0.316338	−	0.948647i	$-0.602453\pi$
$307$	−3403.21	−0.632676	−0.316338	+	0.948647i	$0.602453\pi$
$308$	−842.848	−0.155928
$309$	−2943.01	−0.541819
$310$	3165.76	0.580010
$311$	−5024.59	−0.916136	−0.458068	−	0.888917i	$-0.651458\pi$
$311$	−5024.59	−0.916136	−0.458068	+	0.888917i	$0.651458\pi$
$312$	−904.587	−0.164142
$313$	6242.51	1.12731	0.563654	−	0.826011i	$-0.309395\pi$
$313$	6242.51	1.12731	0.563654	+	0.826011i	$0.309395\pi$
$314$	−2543.80	−0.457181
$315$	735.351	0.131531
$316$	−4903.10	−0.872850
$317$	5029.02	0.891034	0.445517	−	0.895273i	$-0.353020\pi$
$317$	5029.02	0.891034	0.445517	+	0.895273i	$0.353020\pi$
$318$	−242.912	−0.0428359
$319$	−1371.84	−0.240778
$320$	−2648.06	−0.462597
$321$	2698.63	0.469230
$322$	−469.807	−0.0813085
$323$	−207.134	−0.0356819
$324$	−377.205	−0.0646785
$325$	931.731	0.159025
$326$	1784.47	0.303168
$327$	−5131.01	−0.867723
$328$	−1748.14	−0.294283
$329$	−2512.50	−0.421030
$330$	−649.948	−0.108420
$331$	6574.03	1.09167	0.545833	−	0.837894i	$-0.316213\pi$
$331$	6574.03	1.09167	0.545833	+	0.837894i	$0.316213\pi$
$332$	−666.870	−0.110239
$333$	−572.601	−0.0942293
$334$	3753.88	0.614979
$335$	6633.17	1.08182
$336$	−169.547	−0.0275284
$337$	6000.96	0.970010	0.485005	−	0.874511i	$-0.338818\pi$
$337$	6000.96	0.970010	0.485005	+	0.874511i	$0.338818\pi$
$338$	−3706.65	−0.596494
$339$	3239.36	0.518991
$340$	1153.16	0.183938
$341$	3835.35	0.609078
$342$	−100.673	−0.0159174
$343$	6269.44	0.986933
$344$	6038.05	0.946366
$345$	504.646	0.0787514
$346$	2392.03	0.371666
$347$	3854.32	0.596285	0.298143	−	0.954521i	$-0.403633\pi$
$347$	3854.32	0.596285	0.298143	+	0.954521i	$0.403633\pi$
$348$	1182.97	0.182224
$349$	4251.34	0.652061	0.326030	−	0.945359i	$-0.394289\pi$
$349$	4251.34	0.652061	0.326030	+	0.945359i	$0.394289\pi$
$350$	1460.69	0.223077
$351$	351.795	0.0534969
$352$	−2849.55	−0.431482
$353$	11058.1	1.66731	0.833656	−	0.552284i	$-0.186244\pi$
$353$	11058.1	1.66731	0.833656	+	0.552284i	$0.186244\pi$
$354$	2561.03	0.384512
$355$	6735.41	1.00698
$356$	−5038.41	−0.750100
$357$	1134.74	0.168226
$358$	−1516.83	−0.223931
$359$	445.389	0.0654784	0.0327392	−	0.999464i	$-0.489577\pi$
$359$	445.389	0.0654784	0.0327392	+	0.999464i	$0.489577\pi$
$360$	1523.29	0.223013
$361$	−6821.57	−0.994543
$362$	−4445.43	−0.645432
$363$	3205.58	0.463497
$364$	−677.848	−0.0976069
$365$	−1840.19	−0.263890
$366$	2855.60	0.407827
$367$	3992.27	0.567833	0.283917	−	0.958849i	$-0.408366\pi$
$367$	3992.27	0.567833	0.283917	+	0.958849i	$0.408366\pi$
$368$	−116.354	−0.0164820
$369$	679.852	0.0959124
$370$	850.795	0.119543
$371$	−494.725	−0.0692314
$372$	−3307.32	−0.460959
$373$	−6176.60	−0.857405	−0.428703	−	0.903446i	$-0.641029\pi$
$373$	−6176.60	−0.857405	−0.428703	+	0.903446i	$0.641029\pi$
$374$	−1002.95	−0.138667
$375$	−4311.64	−0.593739
$376$	−5204.70	−0.713862
$377$	−1103.28	−0.150721
$378$	551.513	0.0750444
$379$	−3069.15	−0.415968	−0.207984	−	0.978132i	$-0.566690\pi$
$379$	−3069.15	−0.415968	−0.207984	+	0.978132i	$0.566690\pi$
$380$	−208.364	−0.0281285
$381$	1083.97	0.145757
$382$	−5263.86	−0.705033
$383$	1017.81	0.135790	0.0678952	−	0.997692i	$-0.478372\pi$
$383$	1017.81	0.135790	0.0678952	+	0.997692i	$0.478372\pi$
$384$	2235.25	0.297050
$385$	−1323.71	−0.175228
$386$	2603.50	0.343302
$387$	−2348.21	−0.308439
$388$	−5400.85	−0.706666
$389$	−10231.0	−1.33350	−0.666748	−	0.745283i	$-0.732314\pi$
$389$	−10231.0	−1.33350	−0.666748	+	0.745283i	$0.732314\pi$
$390$	−522.712	−0.0678680
$391$	778.731	0.100722
$392$	5049.52	0.650610
$393$	3169.68	0.406843
$394$	−6432.01	−0.822436
$395$	−7700.44	−0.980889
$396$	679.012	0.0861657
$397$	−7312.11	−0.924394	−0.462197	−	0.886777i	$-0.652939\pi$
$397$	−7312.11	−0.924394	−0.462197	+	0.886777i	$0.652939\pi$
$398$	7439.88	0.937004
$399$	−205.035	−0.0257257
$400$	361.758	0.0452198
$401$	−2859.00	−0.356039	−0.178020	−	0.984027i	$-0.556969\pi$
$401$	−2859.00	−0.356039	−0.178020	+	0.984027i	$0.556969\pi$
$402$	4974.88	0.617225
$403$	3084.52	0.381268
$404$	−5461.21	−0.672538
$405$	−592.410	−0.0726842
$406$	−1729.63	−0.211429
$407$	1030.75	0.125534
$408$	2350.63	0.285229
$409$	−2100.40	−0.253932	−0.126966	−	0.991907i	$-0.540524\pi$
$409$	−2100.40	−0.253932	−0.126966	+	0.991907i	$0.540524\pi$
$410$	−1010.15	−0.121678
$411$	−6820.31	−0.818542
$412$	−4568.39	−0.546283
$413$	5215.91	0.621449
$414$	378.484	0.0449311
$415$	−1047.34	−0.123884
$416$	−2291.71	−0.270097
$417$	4633.23	0.544101
$418$	181.222	0.0212054
$419$	−8941.83	−1.04257	−0.521285	−	0.853382i	$-0.674547\pi$
$419$	−8941.83	−1.04257	−0.521285	+	0.853382i	$0.674547\pi$
$420$	1141.47	0.132615
$421$	−11876.0	−1.37483	−0.687413	−	0.726266i	$-0.741254\pi$
$421$	−11876.0	−1.37483	−0.687413	+	0.726266i	$0.741254\pi$
$422$	−4811.13	−0.554981
$423$	2024.11	0.232661
$424$	−1024.83	−0.117383
$425$	−2421.16	−0.276338
$426$	5051.56	0.574528
$427$	5815.84	0.659130
$428$	4189.04	0.473096
$429$	−633.270	−0.0712694
$430$	3489.06	0.391297
$431$	2474.23	0.276518	0.138259	−	0.990396i	$-0.455849\pi$
$431$	2474.23	0.276518	0.138259	+	0.990396i	$0.455849\pi$
$432$	136.590	0.0152122
$433$	−8210.49	−0.911249	−0.455624	−	0.890172i	$-0.650584\pi$
$433$	−8210.49	−0.911249	−0.455624	+	0.890172i	$0.650584\pi$
$434$	4835.65	0.534835
$435$	1857.89	0.204779
$436$	−7964.78	−0.874871
$437$	−140.708	−0.0154027
$438$	−1380.14	−0.150561
$439$	15542.0	1.68970	0.844850	−	0.535003i	$-0.179689\pi$
$439$	15542.0	1.68970	0.844850	+	0.535003i	$0.179689\pi$
$440$	−2742.10	−0.297101
$441$	−1963.76	−0.212047
$442$	−806.609	−0.0868019
$443$	−6576.52	−0.705328	−0.352664	−	0.935750i	$-0.614724\pi$
$443$	−6576.52	−0.705328	−0.352664	+	0.935750i	$0.614724\pi$
$444$	−888.840	−0.0950056
$445$	−7912.96	−0.842944
$446$	−5277.69	−0.560327
$447$	−1995.23	−0.211121
$448$	−4044.87	−0.426567
$449$	−6648.47	−0.698799	−0.349400	−	0.936974i	$-0.613614\pi$
$449$	−6648.47	−0.698799	−0.349400	+	0.936974i	$0.613614\pi$
$450$	−1176.75	−0.123272
$451$	−1223.81	−0.127776
$452$	5028.41	0.523267
$453$	10679.9	1.10770
$454$	3326.81	0.343910
$455$	−1064.58	−0.109688
$456$	−424.733	−0.0436183
$457$	−14901.1	−1.52526	−0.762631	−	0.646834i	$-0.776093\pi$
$457$	−14901.1	−1.52526	−0.762631	+	0.646834i	$0.776093\pi$
$458$	1898.74	0.193717
$459$	−914.162	−0.0929618
$460$	783.354	0.0794002
$461$	3180.92	0.321367	0.160684	−	0.987006i	$-0.448630\pi$
$461$	3180.92	0.321367	0.160684	+	0.987006i	$0.448630\pi$
$462$	−992.785	−0.0999752
$463$	10841.3	1.08821	0.544104	−	0.839018i	$-0.316870\pi$
$463$	10841.3	1.08821	0.544104	+	0.839018i	$0.316870\pi$
$464$	−428.366	−0.0428586
$465$	−5194.23	−0.518015
$466$	−2619.77	−0.260426
$467$	16873.2	1.67195	0.835973	−	0.548770i	$-0.184904\pi$
$467$	16873.2	1.67195	0.835973	+	0.548770i	$0.184904\pi$
$468$	546.086	0.0539377
$469$	10132.1	0.997559
$470$	−3007.52	−0.295162
$471$	4173.75	0.408314
$472$	10804.9	1.05367
$473$	4227.03	0.410907
$474$	−5775.33	−0.559640
$475$	437.478	0.0422587
$476$	1761.43	0.169612
$477$	398.558	0.0382573
$478$	−11972.6	−1.14564
$479$	5673.54	0.541192	0.270596	−	0.962693i	$-0.412779\pi$
$479$	5673.54	0.541192	0.270596	+	0.962693i	$0.412779\pi$
$480$	3859.17	0.366971
$481$	828.964	0.0785811
$482$	3319.25	0.313667
$483$	770.839	0.0726177
$484$	4975.98	0.467316
$485$	−8482.17	−0.794135
$486$	−444.308	−0.0414696
$487$	16387.7	1.52484	0.762421	−	0.647082i	$-0.224011\pi$
$487$	16387.7	1.52484	0.762421	+	0.647082i	$0.224011\pi$
$488$	12047.6	1.11756
$489$	−2927.88	−0.270763
$490$	2917.84	0.269010
$491$	16060.1	1.47614	0.738068	−	0.674726i	$-0.235738\pi$
$491$	16060.1	1.47614	0.738068	+	0.674726i	$0.235738\pi$
$492$	1055.32	0.0967026
$493$	2866.95	0.261909
$494$	145.745	0.0132741
$495$	1066.41	0.0968310
$496$	1197.61	0.108416
$497$	10288.2	0.928552
$498$	−785.503	−0.0706811
$499$	−6254.22	−0.561077	−0.280539	−	0.959843i	$-0.590513\pi$
$499$	−6254.22	−0.561077	−0.280539	+	0.959843i	$0.590513\pi$
$500$	−6692.90	−0.598631
$501$	−6159.19	−0.549246
$502$	−13539.7	−1.20380
$503$	14085.9	1.24862	0.624312	−	0.781175i	$-0.285379\pi$
$503$	14085.9	1.24862	0.624312	+	0.781175i	$0.285379\pi$
$504$	2326.81	0.205643
$505$	−8576.97	−0.755782
$506$	−681.314	−0.0598579
$507$	6081.70	0.532737
$508$	1682.63	0.146958
$509$	6902.54	0.601080	0.300540	−	0.953769i	$-0.402833\pi$
$509$	6902.54	0.601080	0.300540	+	0.953769i	$0.402833\pi$
$510$	1358.30	0.117934
$511$	−2810.86	−0.243337
$512$	−1826.38	−0.157647
$513$	165.179	0.0142161
$514$	−372.446	−0.0319608
$515$	−7174.77	−0.613899
$516$	−3645.08	−0.310980
$517$	−3643.63	−0.309955
$518$	1299.58	0.110232
$519$	−3924.74	−0.331940
$520$	−2205.30	−0.185978
$521$	−13831.4	−1.16308	−0.581538	−	0.813519i	$-0.697549\pi$
$521$	−13831.4	−1.16308	−0.581538	+	0.813519i	$0.697549\pi$
$522$	1393.42	0.116836
$523$	4141.92	0.346297	0.173149	−	0.984896i	$-0.444606\pi$
$523$	4141.92	0.346297	0.173149	+	0.984896i	$0.444606\pi$
$524$	4920.25	0.410195
$525$	−2396.63	−0.199233
$526$	8790.34	0.728663
$527$	−8015.34	−0.662531
$528$	−245.877	−0.0202659
$529$	529.000	0.0434783
$530$	−592.195	−0.0485345
$531$	−4202.02	−0.343413
$532$	−318.272	−0.0259377
$533$	−984.232	−0.0799847
$534$	−5934.72	−0.480937
$535$	6579.00	0.531654
$536$	20988.8	1.69137
$537$	2488.75	0.199996
$538$	−13134.6	−1.05255
$539$	3534.99	0.282492
$540$	−919.590	−0.0732830
$541$	−4441.37	−0.352956	−0.176478	−	0.984305i	$-0.556470\pi$
$541$	−4441.37	−0.352956	−0.176478	+	0.984305i	$0.556470\pi$
$542$	12981.8	1.02881
$543$	7293.86	0.576445
$544$	5955.17	0.469349
$545$	−12508.9	−0.983160
$546$	−798.434	−0.0625821
$547$	−2858.01	−0.223399	−0.111700	−	0.993742i	$-0.535629\pi$
$547$	−2858.01	−0.223399	−0.111700	+	0.993742i	$0.535629\pi$
$548$	−10587.1	−0.825286
$549$	−4685.34	−0.364235
$550$	2118.29	0.164225
$551$	−518.028	−0.0400521
$552$	1596.81	0.123124
$553$	−11762.3	−0.904491
$554$	−4014.46	−0.307866
$555$	−1395.95	−0.106765
$556$	7192.09	0.548584
$557$	24614.7	1.87246	0.936230	−	0.351388i	$-0.114290\pi$
$557$	24614.7	1.87246	0.936230	+	0.351388i	$0.114290\pi$
$558$	−3895.68	−0.295550
$559$	3399.53	0.257218
$560$	−413.339	−0.0311906
$561$	1645.59	0.123845
$562$	−1182.03	−0.0887202
$563$	−21799.4	−1.63186	−0.815928	−	0.578154i	$-0.803773\pi$
$563$	−21799.4	−1.63186	−0.815928	+	0.578154i	$0.803773\pi$
$564$	3142.00	0.234578
$565$	7897.25	0.588035
$566$	−5543.25	−0.411661
$567$	−904.897	−0.0670231
$568$	21312.3	1.57437
$569$	−18323.3	−1.35000	−0.675002	−	0.737816i	$-0.735857\pi$
$569$	−18323.3	−1.35000	−0.675002	+	0.737816i	$0.735857\pi$
$570$	−245.430	−0.0180350
$571$	8424.79	0.617454	0.308727	−	0.951151i	$-0.400097\pi$
$571$	8424.79	0.617454	0.308727	+	0.951151i	$0.400097\pi$
$572$	−983.016	−0.0718565
$573$	8636.70	0.629674
$574$	−1542.99	−0.112201
$575$	−1644.72	−0.119286
$576$	3258.61	0.235721
$577$	−9565.96	−0.690184	−0.345092	−	0.938569i	$-0.612152\pi$
$577$	−9565.96	−0.690184	−0.345092	+	0.938569i	$0.612152\pi$
$578$	−6887.04	−0.495611
$579$	−4271.70	−0.306608
$580$	2883.98	0.206467
$581$	−1599.79	−0.114235
$582$	−6361.63	−0.453089
$583$	−717.450	−0.0509670
$584$	−5822.75	−0.412581
$585$	857.642	0.0606139
$586$	7212.05	0.508408
$587$	−27041.1	−1.90137	−0.950686	−	0.310156i	$-0.899619\pi$
$587$	−27041.1	−1.90137	−0.950686	+	0.310156i	$0.899619\pi$
$588$	−3048.32	−0.213794
$589$	1448.29	0.101317
$590$	6243.55	0.435666
$591$	10553.3	0.734529
$592$	321.858	0.0223450
$593$	−25057.0	−1.73519	−0.867595	−	0.497272i	$-0.834335\pi$
$593$	−25057.0	−1.73519	−0.867595	+	0.497272i	$0.834335\pi$
$594$	799.804	0.0552464
$595$	2766.38	0.190606
$596$	−3097.16	−0.212860
$597$	−12207.0	−0.836851
$598$	−547.938	−0.0374696
$599$	−4596.80	−0.313556	−0.156778	−	0.987634i	$-0.550111\pi$
$599$	−4596.80	−0.313556	−0.156778	+	0.987634i	$0.550111\pi$
$600$	−4964.66	−0.337802
$601$	7811.09	0.530152	0.265076	−	0.964228i	$-0.414603\pi$
$601$	7811.09	0.530152	0.265076	+	0.964228i	$0.414603\pi$
$602$	5329.49	0.360820
$603$	−8162.55	−0.551252
$604$	16578.3	1.11682
$605$	7814.90	0.525158
$606$	−6432.73	−0.431207
$607$	15143.2	1.01259	0.506296	−	0.862360i	$-0.331014\pi$
$607$	15143.2	1.01259	0.506296	+	0.862360i	$0.331014\pi$
$608$	−1076.03	−0.0717746
$609$	2837.90	0.188830
$610$	6961.67	0.462082
$611$	−2930.34	−0.194024
$612$	−1419.04	−0.0937276
$613$	−20487.4	−1.34988	−0.674941	−	0.737872i	$-0.735831\pi$
$613$	−20487.4	−1.34988	−0.674941	+	0.737872i	$0.735831\pi$
$614$	−6222.52	−0.408991
$615$	1657.41	0.108672
$616$	−4188.51	−0.273961
$617$	25547.8	1.66696	0.833480	−	0.552549i	$-0.186345\pi$
$617$	25547.8	1.66696	0.833480	+	0.552549i	$0.186345\pi$
$618$	−5381.08	−0.350257
$619$	15251.1	0.990294	0.495147	−	0.868809i	$-0.335114\pi$
$619$	15251.1	0.990294	0.495147	+	0.868809i	$0.335114\pi$
$620$	−8062.93	−0.522282
$621$	−621.000	−0.0401286
$622$	−9187.10	−0.592233
$623$	−12086.9	−0.777291
$624$	−197.743	−0.0126860
$625$	−1572.66	−0.100650
$626$	11414.0	0.728745
$627$	−297.341	−0.0189389
$628$	6478.84	0.411678
$629$	−2154.12	−0.136551
$630$	1344.54	0.0850279
$631$	−11888.3	−0.750024	−0.375012	−	0.927020i	$-0.622361\pi$
$631$	−11888.3	−0.750024	−0.375012	+	0.927020i	$0.622361\pi$
$632$	−24365.8	−1.53358
$633$	7893.88	0.495661
$634$	9195.19	0.576006
$635$	2642.62	0.165148
$636$	618.676	0.0385725
$637$	2842.97	0.176833
$638$	−2508.31	−0.155650
$639$	−8288.37	−0.513119
$640$	5449.32	0.336568
$641$	11163.5	0.687881	0.343941	−	0.938991i	$-0.388238\pi$
$641$	11163.5	0.687881	0.343941	+	0.938991i	$0.388238\pi$
$642$	4934.25	0.303332
$643$	−22456.6	−1.37730	−0.688648	−	0.725096i	$-0.741795\pi$
$643$	−22456.6	−1.37730	−0.688648	+	0.725096i	$0.741795\pi$
$644$	1196.56	0.0732160
$645$	−5724.70	−0.349472
$646$	−378.729	−0.0230664
$647$	27307.0	1.65927	0.829635	−	0.558306i	$-0.188549\pi$
$647$	27307.0	1.65927	0.829635	+	0.558306i	$0.188549\pi$
$648$	−1874.51	−0.113639
$649$	7564.11	0.457500
$650$	1703.60	0.102801
$651$	−7934.11	−0.477669
$652$	−4544.90	−0.272994
$653$	−21965.4	−1.31634	−0.658171	−	0.752868i	$-0.728670\pi$
$653$	−21965.4	−1.31634	−0.658171	+	0.752868i	$0.728670\pi$
$654$	−9381.67	−0.560936
$655$	7727.37	0.460967
$656$	−382.143	−0.0227442
$657$	2264.47	0.134468
$658$	−4593.93	−0.272173
$659$	−27620.9	−1.63271	−0.816356	−	0.577549i	$-0.804009\pi$
$659$	−27620.9	−1.63271	−0.816356	+	0.577549i	$0.804009\pi$
$660$	1655.36	0.0976288
$661$	4999.00	0.294158	0.147079	−	0.989125i	$-0.453013\pi$
$661$	4999.00	0.294158	0.147079	+	0.989125i	$0.453013\pi$
$662$	12020.1	0.705703
$663$	1323.45	0.0775240
$664$	−3314.00	−0.193687
$665$	−499.855	−0.0291482
$666$	−1046.96	−0.0609142
$667$	1947.55	0.113058
$668$	−9560.82	−0.553771
$669$	8659.40	0.500436
$670$	12128.3	0.699337
$671$	8434.13	0.485240
$672$	5894.81	0.338389
$673$	−18499.9	−1.05961	−0.529807	−	0.848118i	$-0.677736\pi$
$673$	−18499.9	−1.05961	−0.529807	+	0.848118i	$0.677736\pi$
$674$	10972.3	0.627059
$675$	1930.76	0.110096
$676$	9440.53	0.537126
$677$	−1611.99	−0.0915121	−0.0457560	−	0.998953i	$-0.514570\pi$
$677$	−1611.99	−0.0915121	−0.0457560	+	0.998953i	$0.514570\pi$
$678$	5922.94	0.335500
$679$	−12956.4	−0.732283
$680$	5730.61	0.323175
$681$	−5458.48	−0.307151
$682$	7012.65	0.393737
$683$	2745.18	0.153794	0.0768972	−	0.997039i	$-0.475499\pi$
$683$	2745.18	0.153794	0.0768972	+	0.997039i	$0.475499\pi$
$684$	256.405	0.0143332
$685$	−16627.2	−0.927437
$686$	11463.2	0.638000
$687$	−3115.37	−0.173011
$688$	1319.92	0.0731417
$689$	−576.999	−0.0319041
$690$	922.708	0.0509086
$691$	6774.46	0.372956	0.186478	−	0.982459i	$-0.440293\pi$
$691$	6774.46	0.372956	0.186478	+	0.982459i	$0.440293\pi$
$692$	−6092.31	−0.334675
$693$	1628.92	0.0892892
$694$	7047.35	0.385467
$695$	11295.4	0.616486
$696$	5878.76	0.320164
$697$	2557.59	0.138990
$698$	7773.27	0.421522
$699$	4298.41	0.232590
$700$	−3720.25	−0.200874
$701$	19717.2	1.06235	0.531176	−	0.847261i	$-0.321750\pi$
$701$	19717.2	1.06235	0.531176	+	0.847261i	$0.321750\pi$
$702$	643.231	0.0345829
$703$	389.226	0.0208818
$704$	−5865.87	−0.314032
$705$	4934.59	0.263614
$706$	20218.9	1.07783
$707$	−13101.2	−0.696918
$708$	−6522.74	−0.346242
$709$	25897.2	1.37178	0.685889	−	0.727706i	$-0.259414\pi$
$709$	25897.2	1.37178	0.685889	+	0.727706i	$0.259414\pi$
$710$	12315.2	0.650960
$711$	9475.89	0.499823
$712$	−25038.3	−1.31791
$713$	−5444.91	−0.285993
$714$	2074.78	0.108749
$715$	−1543.85	−0.0807507
$716$	3863.25	0.201643
$717$	19644.1	1.02318
$718$	814.362	0.0423283
$719$	14013.5	0.726865	0.363433	−	0.931620i	$-0.381605\pi$
$719$	14013.5	0.726865	0.363433	+	0.931620i	$0.381605\pi$
$720$	332.992	0.0172360
$721$	−10959.4	−0.566085
$722$	−12472.7	−0.642919
$723$	−5446.07	−0.280140
$724$	11322.1	0.581194
$725$	−6055.17	−0.310184
$726$	5861.17	0.299626
$727$	−6116.58	−0.312038	−0.156019	−	0.987754i	$-0.549866\pi$
$727$	−6116.58	−0.312038	−0.156019	+	0.987754i	$0.549866\pi$
$728$	−3368.55	−0.171493
$729$	729.000	0.0370370
$730$	−3364.65	−0.170591
$731$	−8833.91	−0.446969
$732$	−7272.98	−0.367236
$733$	−25067.8	−1.26317	−0.631583	−	0.775308i	$-0.717594\pi$
$733$	−25067.8	−1.26317	−0.631583	+	0.775308i	$0.717594\pi$
$734$	7299.58	0.367074
$735$	−4787.46	−0.240256
$736$	4045.41	0.202603
$737$	14693.5	0.734386
$738$	1243.06	0.0620023
$739$	14908.2	0.742095	0.371047	−	0.928614i	$-0.378999\pi$
$739$	14908.2	0.742095	0.371047	+	0.928614i	$0.378999\pi$
$740$	−2166.91	−0.107645
$741$	−239.132	−0.0118553
$742$	−904.569	−0.0447544
$743$	17125.0	0.845566	0.422783	−	0.906231i	$-0.361053\pi$
$743$	17125.0	0.845566	0.422783	+	0.906231i	$0.361053\pi$
$744$	−16435.7	−0.809893
$745$	−4864.18	−0.239208
$746$	−11293.5	−0.554267
$747$	1288.82	0.0631263
$748$	2554.43	0.124865
$749$	10049.3	0.490246
$750$	−7883.53	−0.383821
$751$	−16547.1	−0.804009	−0.402004	−	0.915638i	$-0.631686\pi$
$751$	−16547.1	−0.804009	−0.402004	+	0.915638i	$0.631686\pi$
$752$	−1137.75	−0.0551721
$753$	22215.3	1.07513
$754$	−2017.27	−0.0974333
$755$	26036.6	1.25506
$756$	−1404.66	−0.0675753
$757$	33624.0	1.61438	0.807189	−	0.590293i	$-0.200988\pi$
$757$	33624.0	1.61438	0.807189	+	0.590293i	$0.200988\pi$
$758$	−5611.72	−0.268901
$759$	1117.87	0.0534599
$760$	−1035.46	−0.0494211
$761$	1430.59	0.0681459	0.0340729	−	0.999419i	$-0.489152\pi$
$761$	1430.59	0.0681459	0.0340729	+	0.999419i	$0.489152\pi$
$762$	1981.96	0.0942243
$763$	−19107.1	−0.906586
$764$	13406.6	0.634862
$765$	−2228.64	−0.105329
$766$	1860.99	0.0877813
$767$	6083.33	0.286384
$768$	12776.6	0.600308
$769$	−32902.7	−1.54291	−0.771457	−	0.636282i	$-0.780471\pi$
$769$	−32902.7	−1.54291	−0.771457	+	0.636282i	$0.780471\pi$
$770$	−2420.31	−0.113275
$771$	611.092	0.0285447
$772$	−6630.90	−0.309134
$773$	15196.9	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$773$	15196.9	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$774$	−4293.52	−0.199389
$775$	16928.8	0.784648
$776$	−26839.4	−1.24160
$777$	−2132.29	−0.0984496
$778$	−18706.6	−0.862034
$779$	−462.129	−0.0212548
$780$	1331.30	0.0611132
$781$	14920.0	0.683585
$782$	1423.85	0.0651111
$783$	−2286.26	−0.104348
$784$	1103.83	0.0502836
$785$	10175.2	0.462634
$786$	5795.53	0.263002
$787$	−33540.5	−1.51917	−0.759586	−	0.650406i	$-0.774599\pi$
$787$	−33540.5	−1.51917	−0.759586	+	0.650406i	$0.774599\pi$
$788$	16381.8	0.740580
$789$	−14422.8	−0.650779
$790$	−14079.7	−0.634092
$791$	12062.9	0.542235
$792$	3374.33	0.151391
$793$	6783.03	0.303749
$794$	−13369.7	−0.597571
$795$	971.647	0.0433469
$796$	−18948.8	−0.843746
$797$	10451.6	0.464508	0.232254	−	0.972655i	$-0.425390\pi$
$797$	10451.6	0.464508	0.232254	+	0.972655i	$0.425390\pi$
$798$	−374.891	−0.0166303
$799$	7614.69	0.337157
$800$	−12577.6	−0.555859
$801$	9737.42	0.429531
$802$	−5227.47	−0.230160
$803$	−4076.31	−0.179140
$804$	−12670.6	−0.555793
$805$	1879.23	0.0822784
$806$	5639.83	0.246470
$807$	21550.7	0.940050
$808$	−27139.4	−1.18163
$809$	44402.8	1.92969	0.964845	−	0.262820i	$-0.0846525\pi$
$809$	44402.8	1.92969	0.964845	+	0.262820i	$0.0846525\pi$
$810$	−1083.18	−0.0469865
$811$	−17243.5	−0.746611	−0.373305	−	0.927708i	$-0.621776\pi$
$811$	−17243.5	−0.746611	−0.373305	+	0.927708i	$0.621776\pi$
$812$	4405.23	0.190386
$813$	−21299.9	−0.918844
$814$	1884.65	0.0811508
$815$	−7137.88	−0.306784
$816$	513.848	0.0220445
$817$	1596.19	0.0683521
$818$	−3840.43	−0.164154
$819$	1310.03	0.0558929
$820$	2572.78	0.109567
$821$	−22572.4	−0.959541	−0.479771	−	0.877394i	$-0.659280\pi$
$821$	−22572.4	−0.959541	−0.479771	+	0.877394i	$0.659280\pi$
$822$	−12470.4	−0.529144
$823$	−25704.3	−1.08870	−0.544348	−	0.838859i	$-0.683223\pi$
$823$	−25704.3	−1.08870	−0.544348	+	0.838859i	$0.683223\pi$
$824$	−22702.5	−0.959805
$825$	−3475.59	−0.146672
$826$	9536.92	0.401733
$827$	−6681.58	−0.280945	−0.140472	−	0.990085i	$-0.544862\pi$
$827$	−6681.58	−0.280945	−0.140472	+	0.990085i	$0.544862\pi$
$828$	−963.969	−0.0404592
$829$	20180.8	0.845487	0.422743	−	0.906249i	$-0.361067\pi$
$829$	20180.8	0.845487	0.422743	+	0.906249i	$0.361067\pi$
$830$	−1914.98	−0.0800842
$831$	6586.74	0.274960
$832$	−4717.54	−0.196576
$833$	−7387.65	−0.307283
$834$	8471.53	0.351733
$835$	−15015.5	−0.622315
$836$	−461.558	−0.0190949
$837$	6391.85	0.263960
$838$	−16349.5	−0.673966
$839$	15142.7	0.623104	0.311552	−	0.950229i	$-0.399151\pi$
$839$	15142.7	0.623104	0.311552	+	0.950229i	$0.399151\pi$
$840$	5672.53	0.233001
$841$	−17218.9	−0.706013
$842$	−21714.5	−0.888752
$843$	1939.42	0.0792373
$844$	12253.6	0.499745
$845$	14826.6	0.603610
$846$	3700.95	0.150403
$847$	11937.1	0.484256
$848$	−224.029	−0.00907214
$849$	9095.12	0.367660
$850$	−4426.92	−0.178638
$851$	−1463.31	−0.0589445
$852$	−12865.9	−0.517346
$853$	−29073.6	−1.16701	−0.583507	−	0.812108i	$-0.698320\pi$
$853$	−29073.6	−1.16701	−0.583507	+	0.812108i	$0.698320\pi$
$854$	10633.8	0.426092
$855$	402.691	0.0161073
$856$	20817.4	0.831218
$857$	−24579.3	−0.979713	−0.489857	−	0.871803i	$-0.662951\pi$
$857$	−24579.3	−0.979713	−0.489857	+	0.871803i	$0.662951\pi$
$858$	−1157.89	−0.0460719
$859$	20359.2	0.808670	0.404335	−	0.914611i	$-0.367503\pi$
$859$	20359.2	0.808670	0.404335	+	0.914611i	$0.367503\pi$
$860$	−8886.36	−0.352352
$861$	2531.67	0.100208
$862$	4523.95	0.178754
$863$	−31440.0	−1.24013	−0.620064	−	0.784551i	$-0.712893\pi$
$863$	−31440.0	−1.24013	−0.620064	+	0.784551i	$0.712893\pi$
$864$	−4748.96	−0.186994
$865$	−9568.13	−0.376100
$866$	−15012.3	−0.589074
$867$	11299.9	0.442637
$868$	−12316.0	−0.481604
$869$	−17057.7	−0.665871
$870$	3397.02	0.132379
$871$	11817.1	0.459708
$872$	−39580.8	−1.53713
$873$	10437.9	0.404660
$874$	−257.275	−0.00995703
$875$	−16055.9	−0.620331
$876$	3515.11	0.135576
$877$	33232.1	1.27955	0.639777	−	0.768561i	$-0.279027\pi$
$877$	33232.1	1.27955	0.639777	+	0.768561i	$0.279027\pi$
$878$	28417.4	1.09230
$879$	−11833.2	−0.454066
$880$	−599.423	−0.0229620
$881$	46530.5	1.77940	0.889700	−	0.456545i	$-0.150913\pi$
$881$	46530.5	1.77940	0.889700	+	0.456545i	$0.150913\pi$
$882$	−3590.60	−0.137077
$883$	−9010.26	−0.343397	−0.171698	−	0.985150i	$-0.554925\pi$
$883$	−9010.26	−0.343397	−0.171698	+	0.985150i	$0.554925\pi$
$884$	2054.37	0.0781627
$885$	−10244.1	−0.389099
$886$	−12024.7	−0.455957
$887$	16773.0	0.634930	0.317465	−	0.948270i	$-0.397168\pi$
$887$	16773.0	0.634930	0.317465	+	0.948270i	$0.397168\pi$
$888$	−4417.07	−0.166923
$889$	4036.56	0.152285
$890$	−14468.3	−0.544918
$891$	−1312.28	−0.0493413
$892$	13441.9	0.504559
$893$	−1375.89	−0.0515593
$894$	−3648.13	−0.136478
$895$	6067.34	0.226602
$896$	8323.75	0.310354
$897$	899.031	0.0334646
$898$	−12156.3	−0.451736
$899$	−20045.8	−0.743677
$900$	2997.09	0.111003
$901$	1499.37	0.0554398
$902$	−2237.65	−0.0826003
$903$	−8744.38	−0.322253
$904$	24988.6	0.919367
$905$	17781.7	0.653132
$906$	19527.5	0.716067
$907$	−42971.0	−1.57313	−0.786565	−	0.617507i	$-0.788143\pi$
$907$	−42971.0	−1.57313	−0.786565	+	0.617507i	$0.788143\pi$
$908$	−8473.12	−0.309681
$909$	10554.5	0.385117
$910$	−1946.50	−0.0709077
$911$	−4801.23	−0.174612	−0.0873061	−	0.996182i	$-0.527826\pi$
$911$	−4801.23	−0.174612	−0.0873061	+	0.996182i	$0.527826\pi$
$912$	−92.8468	−0.00337112
$913$	−2320.02	−0.0840978
$914$	−27245.6	−0.986001
$915$	−11422.4	−0.412692
$916$	−4835.94	−0.174437
$917$	11803.4	0.425064
$918$	−1671.48	−0.0600948
$919$	31232.7	1.12108	0.560539	−	0.828128i	$-0.310594\pi$
$919$	31232.7	1.12108	0.560539	+	0.828128i	$0.310594\pi$
$920$	3892.86	0.139504
$921$	10209.6	0.365275
$922$	5816.09	0.207747
$923$	11999.2	0.427907
$924$	2528.54	0.0900248
$925$	4549.61	0.161719
$926$	19822.6	0.703468
$927$	8829.03	0.312819
$928$	14893.5	0.526834
$929$	−23023.4	−0.813104	−0.406552	−	0.913628i	$-0.633269\pi$
$929$	−23023.4	−0.813104	−0.406552	+	0.913628i	$0.633269\pi$
$930$	−9497.28	−0.334869
$931$	1334.87	0.0469909
$932$	6672.35	0.234507
$933$	15073.8	0.528932
$934$	30851.4	1.08082
$935$	4011.80	0.140321
$936$	2713.76	0.0947672
$937$	−11599.2	−0.404405	−0.202203	−	0.979344i	$-0.564810\pi$
$937$	−11599.2	−0.404405	−0.202203	+	0.979344i	$0.564810\pi$
$938$	18525.7	0.644869
$939$	−18727.5	−0.650852
$940$	7659.89	0.265785
$941$	−20423.2	−0.707522	−0.353761	−	0.935336i	$-0.615097\pi$
$941$	−20423.2	−0.707522	−0.353761	+	0.935336i	$0.615097\pi$
$942$	7631.39	0.263953
$943$	1737.40	0.0599974
$944$	2361.95	0.0814351
$945$	−2206.05	−0.0759396
$946$	7728.82	0.265630
$947$	22021.1	0.755638	0.377819	−	0.925880i	$-0.376674\pi$
$947$	22021.1	0.755638	0.377819	+	0.925880i	$0.376674\pi$
$948$	14709.3	0.503940
$949$	−3278.31	−0.112138
$950$	799.897	0.0273180
$951$	−15087.1	−0.514439
$952$	8753.41	0.298004
$953$	46234.5	1.57154	0.785772	−	0.618517i	$-0.212266\pi$
$953$	46234.5	1.57154	0.785772	+	0.618517i	$0.212266\pi$
$954$	728.735	0.0247313
$955$	21055.4	0.713443
$956$	30493.3	1.03161
$957$	4115.52	0.139013
$958$	10373.7	0.349852
$959$	−25397.9	−0.855203
$960$	7944.18	0.267081
$961$	26252.5	0.881222
$962$	1515.70	0.0507985
$963$	−8095.89	−0.270910
$964$	−8453.85	−0.282448
$965$	−10414.0	−0.347397
$966$	1409.42	0.0469435
$967$	32443.4	1.07891	0.539456	−	0.842014i	$-0.318630\pi$
$967$	32443.4	1.07891	0.539456	+	0.842014i	$0.318630\pi$
$968$	24728.0	0.821062
$969$	621.402	0.0206009
$970$	−15509.0	−0.513366
$971$	10183.9	0.336579	0.168290	−	0.985738i	$-0.446176\pi$
$971$	10183.9	0.336579	0.168290	+	0.985738i	$0.446176\pi$
$972$	1131.62	0.0373422
$973$	17253.5	0.568470
$974$	29963.7	0.985729
$975$	−2795.19	−0.0918131
$976$	2633.61	0.0863729
$977$	−26704.7	−0.874472	−0.437236	−	0.899347i	$-0.644043\pi$
$977$	−26704.7	−0.874472	−0.437236	+	0.899347i	$0.644043\pi$
$978$	−5353.41	−0.175034
$979$	−17528.4	−0.572228
$980$	−7431.51	−0.242236
$981$	15393.0	0.500980
$982$	29364.8	0.954243
$983$	−12350.6	−0.400735	−0.200368	−	0.979721i	$-0.564214\pi$
$983$	−12350.6	−0.400735	−0.200368	+	0.979721i	$0.564214\pi$
$984$	5244.41	0.169904
$985$	25728.0	0.832247
$986$	5242.02	0.169310
$987$	7537.51	0.243082
$988$	−371.202	−0.0119529
$989$	−6000.97	−0.192942
$990$	1949.84	0.0625961
$991$	18050.6	0.578602	0.289301	−	0.957238i	$-0.406577\pi$
$991$	18050.6	0.578602	0.289301	+	0.957238i	$0.406577\pi$
$992$	−41638.7	−1.33269
$993$	−19722.1	−0.630273
$994$	18811.3	0.600260
$995$	−29759.5	−0.948182
$996$	2000.61	0.0636464
$997$	14290.5	0.453946	0.226973	−	0.973901i	$-0.427117\pi$
$997$	14290.5	0.453946	0.226973	+	0.973901i	$0.427117\pi$
$998$	−11435.4	−0.362706
$999$	1717.80	0.0544033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.a.b.1.2	✓	2
3.2	odd	2		207.4.a.b.1.1		2
4.3	odd	2		1104.4.a.q.1.1		2
5.4	even	2		1725.4.a.m.1.1		2
23.22	odd	2		1587.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.b.1.2	✓	2	1.1	even	1	trivial
207.4.a.b.1.1		2	3.2	odd	2
1104.4.a.q.1.1		2	4.3	odd	2
1587.4.a.c.1.2		2	23.22	odd	2
1725.4.a.m.1.1		2	5.4	even	2

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} -7.31371 q^{5} -5.48528 q^{6} -11.1716 q^{7} -23.1421 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} -7.31371 q^{5} -5.48528 q^{6} -11.1716 q^{7} -23.1421 q^{8} +9.00000 q^{9} -13.3726 q^{10} -16.2010 q^{11} +13.9706 q^{12} -13.0294 q^{13} -20.4264 q^{14} +21.9411 q^{15} -5.05887 q^{16} +33.8579 q^{17} +16.4558 q^{18} -6.11775 q^{19} +34.0589 q^{20} +33.5147 q^{21} -29.6224 q^{22} +23.0000 q^{23} +69.4264 q^{24} -71.5097 q^{25} -23.8234 q^{26} -27.0000 q^{27} +52.0244 q^{28} +84.6762 q^{29} +40.1177 q^{30} -236.735 q^{31} +175.887 q^{32} +48.6030 q^{33} +61.9066 q^{34} +81.7056 q^{35} -41.9117 q^{36} -63.6224 q^{37} -11.1859 q^{38} +39.0883 q^{39} +169.255 q^{40} +75.5391 q^{41} +61.2792 q^{42} -260.912 q^{43} +75.4457 q^{44} -65.8234 q^{45} +42.0538 q^{46} +224.902 q^{47} +15.1766 q^{48} -218.196 q^{49} -130.750 q^{50} -101.574 q^{51} +60.6762 q^{52} +44.2843 q^{53} -49.3675 q^{54} +118.489 q^{55} +258.534 q^{56} +18.3532 q^{57} +154.824 q^{58} -466.891 q^{59} -102.177 q^{60} -520.593 q^{61} -432.853 q^{62} -100.544 q^{63} +362.068 q^{64} +95.2935 q^{65} +88.8671 q^{66} -906.950 q^{67} -157.671 q^{68} -69.0000 q^{69} +149.393 q^{70} -920.930 q^{71} -208.279 q^{72} +251.608 q^{73} -116.329 q^{74} +214.529 q^{75} +28.4895 q^{76} +180.991 q^{77} +71.4701 q^{78} +1052.88 q^{79} +36.9991 q^{80} +81.0000 q^{81} +138.118 q^{82} +143.202 q^{83} -156.073 q^{84} -247.627 q^{85} -477.058 q^{86} -254.029 q^{87} +374.926 q^{88} +1081.94 q^{89} -120.353 q^{90} +145.559 q^{91} -107.108 q^{92} +710.205 q^{93} +411.216 q^{94} +44.7434 q^{95} -527.662 q^{96} +1159.76 q^{97} -398.955 q^{98} -145.809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 8 q^{5} + 6 q^{6} - 28 q^{7} - 18 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 8 * q^5 + 6 * q^6 - 28 * q^7 - 18 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 8 q^{5} + 6 q^{6} - 28 q^{7} - 18 q^{8} + 18 q^{9} - 72 q^{10} - 72 q^{11} - 6 q^{12} - 60 q^{13} + 44 q^{14} - 24 q^{15} - 78 q^{16} + 96 q^{17} - 18 q^{18} - 148 q^{19} + 136 q^{20} + 84 q^{21} + 184 q^{22} + 46 q^{23} + 54 q^{24} + 38 q^{25} + 156 q^{26} - 54 q^{27} - 60 q^{28} - 204 q^{29} + 216 q^{30} - 168 q^{31} + 414 q^{32} + 216 q^{33} - 176 q^{34} - 176 q^{35} + 18 q^{36} + 116 q^{37} + 532 q^{38} + 180 q^{39} + 248 q^{40} + 4 q^{41} - 132 q^{42} - 420 q^{43} - 296 q^{44} + 72 q^{45} - 46 q^{46} - 48 q^{47} + 234 q^{48} - 278 q^{49} - 550 q^{50} - 288 q^{51} - 252 q^{52} + 32 q^{53} + 54 q^{54} - 736 q^{55} + 172 q^{56} + 444 q^{57} + 1260 q^{58} - 40 q^{59} - 408 q^{60} - 764 q^{61} - 696 q^{62} - 252 q^{63} + 34 q^{64} - 624 q^{65} - 552 q^{66} - 988 q^{67} + 256 q^{68} - 138 q^{69} + 1136 q^{70} - 224 q^{71} - 162 q^{72} + 820 q^{73} - 804 q^{74} - 114 q^{75} - 916 q^{76} + 1120 q^{77} - 468 q^{78} + 1772 q^{79} - 1080 q^{80} + 162 q^{81} + 412 q^{82} + 1480 q^{83} + 180 q^{84} + 704 q^{85} + 132 q^{86} + 612 q^{87} + 88 q^{88} + 744 q^{89} - 648 q^{90} + 936 q^{91} + 46 q^{92} + 504 q^{93} + 1456 q^{94} - 2128 q^{95} - 1242 q^{96} - 260 q^{97} - 170 q^{98} - 648 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 8 * q^5 + 6 * q^6 - 28 * q^7 - 18 * q^8 + 18 * q^9 - 72 * q^10 - 72 * q^11 - 6 * q^12 - 60 * q^13 + 44 * q^14 - 24 * q^15 - 78 * q^16 + 96 * q^17 - 18 * q^18 - 148 * q^19 + 136 * q^20 + 84 * q^21 + 184 * q^22 + 46 * q^23 + 54 * q^24 + 38 * q^25 + 156 * q^26 - 54 * q^27 - 60 * q^28 - 204 * q^29 + 216 * q^30 - 168 * q^31 + 414 * q^32 + 216 * q^33 - 176 * q^34 - 176 * q^35 + 18 * q^36 + 116 * q^37 + 532 * q^38 + 180 * q^39 + 248 * q^40 + 4 * q^41 - 132 * q^42 - 420 * q^43 - 296 * q^44 + 72 * q^45 - 46 * q^46 - 48 * q^47 + 234 * q^48 - 278 * q^49 - 550 * q^50 - 288 * q^51 - 252 * q^52 + 32 * q^53 + 54 * q^54 - 736 * q^55 + 172 * q^56 + 444 * q^57 + 1260 * q^58 - 40 * q^59 - 408 * q^60 - 764 * q^61 - 696 * q^62 - 252 * q^63 + 34 * q^64 - 624 * q^65 - 552 * q^66 - 988 * q^67 + 256 * q^68 - 138 * q^69 + 1136 * q^70 - 224 * q^71 - 162 * q^72 + 820 * q^73 - 804 * q^74 - 114 * q^75 - 916 * q^76 + 1120 * q^77 - 468 * q^78 + 1772 * q^79 - 1080 * q^80 + 162 * q^81 + 412 * q^82 + 1480 * q^83 + 180 * q^84 + 704 * q^85 + 132 * q^86 + 612 * q^87 + 88 * q^88 + 744 * q^89 - 648 * q^90 + 936 * q^91 + 46 * q^92 + 504 * q^93 + 1456 * q^94 - 2128 * q^95 - 1242 * q^96 - 260 * q^97 - 170 * q^98 - 648 * q^99

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