LMFDB - Embedded newform 4019.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, 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("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.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:	$32.0918765724$
Analytic rank:	$0$
Dimension:	$186$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62160 q^{2} -1.74677 q^{3} +4.87277 q^{4} -3.59409 q^{5} +4.57932 q^{6} +5.00470 q^{7} -7.53124 q^{8} +0.0512031 q^{9} +O(q^{10})$	\(q-2.62160 q^{2} -1.74677 q^{3} +4.87277 q^{4} -3.59409 q^{5} +4.57932 q^{6} +5.00470 q^{7} -7.53124 q^{8} +0.0512031 q^{9} +9.42226 q^{10} +4.83308 q^{11} -8.51160 q^{12} +2.69395 q^{13} -13.1203 q^{14} +6.27805 q^{15} +9.99834 q^{16} +4.49415 q^{17} -0.134234 q^{18} -4.49596 q^{19} -17.5132 q^{20} -8.74206 q^{21} -12.6704 q^{22} -1.03340 q^{23} +13.1553 q^{24} +7.91749 q^{25} -7.06245 q^{26} +5.15087 q^{27} +24.3868 q^{28} +5.52907 q^{29} -16.4585 q^{30} +7.30673 q^{31} -11.1491 q^{32} -8.44228 q^{33} -11.7819 q^{34} -17.9874 q^{35} +0.249501 q^{36} +7.95655 q^{37} +11.7866 q^{38} -4.70571 q^{39} +27.0680 q^{40} +10.5430 q^{41} +22.9182 q^{42} +11.3587 q^{43} +23.5505 q^{44} -0.184029 q^{45} +2.70917 q^{46} -1.42485 q^{47} -17.4648 q^{48} +18.0470 q^{49} -20.7565 q^{50} -7.85025 q^{51} +13.1270 q^{52} -2.96560 q^{53} -13.5035 q^{54} -17.3705 q^{55} -37.6916 q^{56} +7.85340 q^{57} -14.4950 q^{58} +5.22309 q^{59} +30.5915 q^{60} +10.0176 q^{61} -19.1553 q^{62} +0.256256 q^{63} +9.23185 q^{64} -9.68230 q^{65} +22.1323 q^{66} +1.46016 q^{67} +21.8990 q^{68} +1.80512 q^{69} +47.1556 q^{70} +4.84033 q^{71} -0.385623 q^{72} -0.166246 q^{73} -20.8589 q^{74} -13.8300 q^{75} -21.9078 q^{76} +24.1881 q^{77} +12.3365 q^{78} -6.67132 q^{79} -35.9349 q^{80} -9.15099 q^{81} -27.6396 q^{82} +3.94157 q^{83} -42.5980 q^{84} -16.1524 q^{85} -29.7779 q^{86} -9.65801 q^{87} -36.3991 q^{88} -12.1444 q^{89} +0.482449 q^{90} +13.4824 q^{91} -5.03553 q^{92} -12.7632 q^{93} +3.73537 q^{94} +16.1589 q^{95} +19.4750 q^{96} -5.94555 q^{97} -47.3121 q^{98} +0.247469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) $ 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.62160	−1.85375	−0.926874	−	0.375372i	$-0.877515\pi$
$2$	−2.62160	−1.85375	−0.926874	+	0.375372i	$0.877515\pi$
$3$	−1.74677	−1.00850	−0.504249	−	0.863558i	$-0.668231\pi$
$3$	−1.74677	−1.00850	−0.504249	+	0.863558i	$0.668231\pi$
$4$	4.87277	2.43638
$5$	−3.59409	−1.60733	−0.803663	−	0.595084i	$-0.797119\pi$
$5$	−3.59409	−1.60733	−0.803663	+	0.595084i	$0.797119\pi$
$6$	4.57932	1.86950
$7$	5.00470	1.89160	0.945800	−	0.324750i	$-0.105280\pi$
$7$	5.00470	1.89160	0.945800	+	0.324750i	$0.105280\pi$
$8$	−7.53124	−2.66270
$9$	0.0512031	0.0170677
$10$	9.42226	2.97958
$11$	4.83308	1.45723	0.728615	−	0.684924i	$-0.240164\pi$
$11$	4.83308	1.45723	0.728615	+	0.684924i	$0.240164\pi$
$12$	−8.51160	−2.45709
$13$	2.69395	0.747167	0.373584	−	0.927597i	$-0.378129\pi$
$13$	2.69395	0.747167	0.373584	+	0.927597i	$0.378129\pi$
$14$	−13.1203	−3.50655
$15$	6.27805	1.62099
$16$	9.99834	2.49958
$17$	4.49415	1.08999	0.544996	−	0.838438i	$-0.316531\pi$
$17$	4.49415	1.08999	0.544996	+	0.838438i	$0.316531\pi$
$18$	−0.134234	−0.0316392
$19$	−4.49596	−1.03144	−0.515722	−	0.856756i	$-0.672476\pi$
$19$	−4.49596	−1.03144	−0.515722	+	0.856756i	$0.672476\pi$
$20$	−17.5132	−3.91607
$21$	−8.74206	−1.90767
$22$	−12.6704	−2.70134
$23$	−1.03340	−0.215479	−0.107740	−	0.994179i	$-0.534361\pi$
$23$	−1.03340	−0.215479	−0.107740	+	0.994179i	$0.534361\pi$
$24$	13.1553	2.68532
$25$	7.91749	1.58350
$26$	−7.06245	−1.38506
$27$	5.15087	0.991285
$28$	24.3868	4.60866
$29$	5.52907	1.02672	0.513361	−	0.858173i	$-0.328400\pi$
$29$	5.52907	1.02672	0.513361	+	0.858173i	$0.328400\pi$
$30$	−16.4585	−3.00490
$31$	7.30673	1.31233	0.656164	−	0.754619i	$-0.272178\pi$
$31$	7.30673	1.31233	0.656164	+	0.754619i	$0.272178\pi$
$32$	−11.1491	−1.97091
$33$	−8.44228	−1.46961
$34$	−11.7819	−2.02057
$35$	−17.9874	−3.04042
$36$	0.249501	0.0415835
$37$	7.95655	1.30805	0.654024	−	0.756474i	$-0.273079\pi$
$37$	7.95655	1.30805	0.654024	+	0.756474i	$0.273079\pi$
$38$	11.7866	1.91204
$39$	−4.70571	−0.753516
$40$	27.0680	4.27982
$41$	10.5430	1.64655	0.823273	−	0.567646i	$-0.192146\pi$
$41$	10.5430	1.64655	0.823273	+	0.567646i	$0.192146\pi$
$42$	22.9182	3.53635
$43$	11.3587	1.73218	0.866091	−	0.499887i	$-0.166625\pi$
$43$	11.3587	1.73218	0.866091	+	0.499887i	$0.166625\pi$
$44$	23.5505	3.55037
$45$	−0.184029	−0.0274334
$46$	2.70917	0.399445
$47$	−1.42485	−0.207835	−0.103918	−	0.994586i	$-0.533138\pi$
$47$	−1.42485	−0.207835	−0.103918	+	0.994586i	$0.533138\pi$
$48$	−17.4648	−2.52083
$49$	18.0470	2.57815
$50$	−20.7565	−2.93541
$51$	−7.85025	−1.09926
$52$	13.1270	1.82039
$53$	−2.96560	−0.407357	−0.203678	−	0.979038i	$-0.565290\pi$
$53$	−2.96560	−0.407357	−0.203678	+	0.979038i	$0.565290\pi$
$54$	−13.5035	−1.83759
$55$	−17.3705	−2.34224
$56$	−37.6916	−5.03676
$57$	7.85340	1.04021
$58$	−14.4950	−1.90329
$59$	5.22309	0.679988	0.339994	−	0.940428i	$-0.389575\pi$
$59$	5.22309	0.679988	0.339994	+	0.940428i	$0.389575\pi$
$60$	30.5915	3.94934
$61$	10.0176	1.28262	0.641311	−	0.767281i	$-0.278391\pi$
$61$	10.0176	1.28262	0.641311	+	0.767281i	$0.278391\pi$
$62$	−19.1553	−2.43273
$63$	0.256256	0.0322853
$64$	9.23185	1.15398
$65$	−9.68230	−1.20094
$66$	22.1323	2.72429
$67$	1.46016	0.178387	0.0891936	−	0.996014i	$-0.471571\pi$
$67$	1.46016	0.178387	0.0891936	+	0.996014i	$0.471571\pi$
$68$	21.8990	2.65564
$69$	1.80512	0.217311
$70$	47.1556	5.63617
$71$	4.84033	0.574441	0.287220	−	0.957864i	$-0.407269\pi$
$71$	4.84033	0.574441	0.287220	+	0.957864i	$0.407269\pi$
$72$	−0.385623	−0.0454461
$73$	−0.166246	−0.0194577	−0.00972884	−	0.999953i	$-0.503097\pi$
$73$	−0.166246	−0.0194577	−0.00972884	+	0.999953i	$0.503097\pi$
$74$	−20.8589	−2.42479
$75$	−13.8300	−1.59696
$76$	−21.9078	−2.51299
$77$	24.1881	2.75650
$78$	12.3365	1.39683
$79$	−6.67132	−0.750582	−0.375291	−	0.926907i	$-0.622457\pi$
$79$	−6.67132	−0.750582	−0.375291	+	0.926907i	$0.622457\pi$
$80$	−35.9349	−4.01765
$81$	−9.15099	−1.01678
$82$	−27.6396	−3.05228
$83$	3.94157	0.432643	0.216322	−	0.976322i	$-0.430594\pi$
$83$	3.94157	0.432643	0.216322	+	0.976322i	$0.430594\pi$
$84$	−42.5980	−4.64783
$85$	−16.1524	−1.75197
$86$	−29.7779	−3.21103
$87$	−9.65801	−1.03545
$88$	−36.3991	−3.88016
$89$	−12.1444	−1.28731	−0.643653	−	0.765318i	$-0.722582\pi$
$89$	−12.1444	−1.28731	−0.643653	+	0.765318i	$0.722582\pi$
$90$	0.482449	0.0508546
$91$	13.4824	1.41334
$92$	−5.03553	−0.524991
$93$	−12.7632	−1.32348
$94$	3.73537	0.385274
$95$	16.1589	1.65787
$96$	19.4750	1.98765
$97$	−5.94555	−0.603679	−0.301840	−	0.953359i	$-0.597601\pi$
$97$	−5.94555	−0.603679	−0.301840	+	0.953359i	$0.597601\pi$
$98$	−47.3121	−4.77924
$99$	0.247469	0.0248716
$100$	38.5801	3.85801
$101$	−6.31979	−0.628842	−0.314421	−	0.949284i	$-0.601810\pi$
$101$	−6.31979	−0.628842	−0.314421	+	0.949284i	$0.601810\pi$
$102$	20.5802	2.03774
$103$	11.4102	1.12428	0.562141	−	0.827041i	$-0.309978\pi$
$103$	11.4102	1.12428	0.562141	+	0.827041i	$0.309978\pi$
$104$	−20.2888	−1.98948
$105$	31.4198	3.06626
$106$	7.77461	0.755137
$107$	−3.19930	−0.309288	−0.154644	−	0.987970i	$-0.549423\pi$
$107$	−3.19930	−0.309288	−0.154644	+	0.987970i	$0.549423\pi$
$108$	25.0990	2.41515
$109$	16.7146	1.60097	0.800486	−	0.599352i	$-0.204575\pi$
$109$	16.7146	1.60097	0.800486	+	0.599352i	$0.204575\pi$
$110$	45.5386	4.34193
$111$	−13.8983	−1.31916
$112$	50.0387	4.72821
$113$	−20.1898	−1.89930	−0.949649	−	0.313317i	$-0.898560\pi$
$113$	−20.1898	−1.89930	−0.949649	+	0.313317i	$0.898560\pi$
$114$	−20.5884	−1.92828
$115$	3.71414	0.346346
$116$	26.9419	2.50149
$117$	0.137939	0.0127524
$118$	−13.6928	−1.26053
$119$	22.4919	2.06183
$120$	−47.2815	−4.31619
$121$	12.3587	1.12352
$122$	−26.2621	−2.37766
$123$	−18.4162	−1.66054
$124$	35.6040	3.19733
$125$	−10.4857	−0.937873
$126$	−0.671801	−0.0598488
$127$	−1.67283	−0.148439	−0.0742196	−	0.997242i	$-0.523647\pi$
$127$	−1.67283	−0.148439	−0.0742196	+	0.997242i	$0.523647\pi$
$128$	−1.90391	−0.168284
$129$	−19.8410	−1.74690
$130$	25.3831	2.22624
$131$	10.4782	0.915483	0.457741	−	0.889085i	$-0.348659\pi$
$131$	10.4782	0.915483	0.457741	+	0.889085i	$0.348659\pi$
$132$	−41.1373	−3.58054
$133$	−22.5009	−1.95108
$134$	−3.82796	−0.330685
$135$	−18.5127	−1.59332
$136$	−33.8466	−2.90232
$137$	−2.12717	−0.181736	−0.0908681	−	0.995863i	$-0.528964\pi$
$137$	−2.12717	−0.181736	−0.0908681	+	0.995863i	$0.528964\pi$
$138$	−4.73229	−0.402839
$139$	11.8479	1.00493	0.502463	−	0.864599i	$-0.332427\pi$
$139$	11.8479	1.00493	0.502463	+	0.864599i	$0.332427\pi$
$140$	−87.6483	−7.40763
$141$	2.48888	0.209601
$142$	−12.6894	−1.06487
$143$	13.0201	1.08879
$144$	0.511946	0.0426622
$145$	−19.8720	−1.65028
$146$	0.435831	0.0360696
$147$	−31.5240	−2.60006
$148$	38.7704	3.18691
$149$	6.58977	0.539855	0.269927	−	0.962881i	$-0.413000\pi$
$149$	6.58977	0.539855	0.269927	+	0.962881i	$0.413000\pi$
$150$	36.2568	2.96035
$151$	−20.3780	−1.65834	−0.829169	−	0.558998i	$-0.811186\pi$
$151$	−20.3780	−1.65834	−0.829169	+	0.558998i	$0.811186\pi$
$152$	33.8601	2.74642
$153$	0.230115	0.0186037
$154$	−63.4116	−5.10985
$155$	−26.2611	−2.10934
$156$	−22.9298	−1.83586
$157$	−16.0314	−1.27945	−0.639724	−	0.768605i	$-0.720951\pi$
$157$	−16.0314	−1.27945	−0.639724	+	0.768605i	$0.720951\pi$
$158$	17.4895	1.39139
$159$	5.18022	0.410818
$160$	40.0710	3.16789
$161$	−5.17187	−0.407601
$162$	23.9902	1.88485
$163$	−10.1731	−0.796818	−0.398409	−	0.917208i	$-0.630438\pi$
$163$	−10.1731	−0.796818	−0.398409	+	0.917208i	$0.630438\pi$
$164$	51.3738	4.01162
$165$	30.3423	2.36215
$166$	−10.3332	−0.802012
$167$	−14.4360	−1.11709	−0.558546	−	0.829474i	$-0.688641\pi$
$167$	−14.4360	−1.11709	−0.558546	+	0.829474i	$0.688641\pi$
$168$	65.8386	5.07956
$169$	−5.74264	−0.441741
$170$	42.3451	3.24772
$171$	−0.230207	−0.0176044
$172$	55.3482	4.22026
$173$	14.1050	1.07239	0.536193	−	0.844095i	$-0.319862\pi$
$173$	14.1050	1.07239	0.536193	+	0.844095i	$0.319862\pi$
$174$	25.3194	1.91946
$175$	39.6247	2.99535
$176$	48.3228	3.64247
$177$	−9.12353	−0.685766
$178$	31.8378	2.38634
$179$	22.6885	1.69582	0.847908	−	0.530144i	$-0.177862\pi$
$179$	22.6885	1.69582	0.847908	+	0.530144i	$0.177862\pi$
$180$	−0.896730	−0.0668383
$181$	−13.4078	−0.996592	−0.498296	−	0.867007i	$-0.666041\pi$
$181$	−13.4078	−0.996592	−0.498296	+	0.867007i	$0.666041\pi$
$182$	−35.3455	−2.61998
$183$	−17.4984	−1.29352
$184$	7.78281	0.573756
$185$	−28.5966	−2.10246
$186$	33.4599	2.45340
$187$	21.7206	1.58837
$188$	−6.94295	−0.506367
$189$	25.7786	1.87511
$190$	−42.3621	−3.07327
$191$	−9.75439	−0.705803	−0.352901	−	0.935661i	$-0.614805\pi$
$191$	−9.75439	−0.705803	−0.352901	+	0.935661i	$0.614805\pi$
$192$	−16.1259	−1.16379
$193$	−23.0795	−1.66130	−0.830651	−	0.556793i	$-0.812032\pi$
$193$	−23.0795	−1.66130	−0.830651	+	0.556793i	$0.812032\pi$
$194$	15.5868	1.11907
$195$	16.9127	1.21115
$196$	87.9391	6.28136
$197$	−15.4919	−1.10375	−0.551875	−	0.833927i	$-0.686087\pi$
$197$	−15.4919	−1.10375	−0.551875	+	0.833927i	$0.686087\pi$
$198$	−0.648764	−0.0461057
$199$	0.714014	0.0506151	0.0253076	−	0.999680i	$-0.491943\pi$
$199$	0.714014	0.0506151	0.0253076	+	0.999680i	$0.491943\pi$
$200$	−59.6286	−4.21638
$201$	−2.55057	−0.179903
$202$	16.5679	1.16572
$203$	27.6713	1.94215
$204$	−38.2525	−2.67821
$205$	−37.8926	−2.64654
$206$	−29.9130	−2.08414
$207$	−0.0529135	−0.00367774
$208$	26.9350	1.86761
$209$	−21.7293	−1.50305
$210$	−82.3700	−5.68407
$211$	−2.03028	−0.139770	−0.0698852	−	0.997555i	$-0.522263\pi$
$211$	−2.03028	−0.139770	−0.0698852	+	0.997555i	$0.522263\pi$
$212$	−14.4507	−0.992478
$213$	−8.45493	−0.579322
$214$	8.38726	0.573342
$215$	−40.8241	−2.78418
$216$	−38.7924	−2.63949
$217$	36.5680	2.48240
$218$	−43.8190	−2.96780
$219$	0.290394	0.0196230
$220$	−84.6427	−5.70661
$221$	12.1070	0.814407
$222$	36.4356	2.44540
$223$	−25.6355	−1.71668	−0.858340	−	0.513082i	$-0.828504\pi$
$223$	−25.6355	−1.71668	−0.858340	+	0.513082i	$0.828504\pi$
$224$	−55.7981	−3.72817
$225$	0.405401	0.0270267
$226$	52.9295	3.52082
$227$	11.9297	0.791805	0.395903	−	0.918293i	$-0.370432\pi$
$227$	11.9297	0.791805	0.395903	+	0.918293i	$0.370432\pi$
$228$	38.2678	2.53435
$229$	24.6375	1.62809	0.814047	−	0.580798i	$-0.197260\pi$
$229$	24.6375	1.62809	0.814047	+	0.580798i	$0.197260\pi$
$230$	−9.73699	−0.642038
$231$	−42.2511	−2.77992
$232$	−41.6408	−2.73385
$233$	−1.05780	−0.0692990	−0.0346495	−	0.999400i	$-0.511031\pi$
$233$	−1.05780	−0.0692990	−0.0346495	+	0.999400i	$0.511031\pi$
$234$	−0.361620	−0.0236398
$235$	5.12103	0.334059
$236$	25.4509	1.65671
$237$	11.6533	0.756961
$238$	−58.9647	−3.82211
$239$	20.6100	1.33315	0.666576	−	0.745437i	$-0.267759\pi$
$239$	20.6100	1.33315	0.666576	+	0.745437i	$0.267759\pi$
$240$	62.7701	4.05179
$241$	13.9640	0.899499	0.449750	−	0.893155i	$-0.351513\pi$
$241$	13.9640	0.899499	0.449750	+	0.893155i	$0.351513\pi$
$242$	−32.3995	−2.08272
$243$	0.532061	0.0341317
$244$	48.8135	3.12496
$245$	−64.8627	−4.14393
$246$	48.2800	3.07822
$247$	−12.1119	−0.770661
$248$	−55.0287	−3.49433
$249$	−6.88501	−0.436320
$250$	27.4894	1.73858
$251$	16.4612	1.03902	0.519509	−	0.854465i	$-0.326115\pi$
$251$	16.4612	1.03902	0.519509	+	0.854465i	$0.326115\pi$
$252$	1.24868	0.0786594
$253$	−4.99452	−0.314003
$254$	4.38547	0.275169
$255$	28.2145	1.76686
$256$	−13.4724	−0.842025
$257$	6.16096	0.384310	0.192155	−	0.981365i	$-0.438452\pi$
$257$	6.16096	0.384310	0.192155	+	0.981365i	$0.438452\pi$
$258$	52.0151	3.23832
$259$	39.8202	2.47430
$260$	−47.1796	−2.92596
$261$	0.283106	0.0175238
$262$	−27.4696	−1.69707
$263$	−21.7710	−1.34246	−0.671229	−	0.741250i	$-0.734233\pi$
$263$	−21.7710	−1.34246	−0.671229	+	0.741250i	$0.734233\pi$
$264$	63.5809	3.91313
$265$	10.6586	0.654755
$266$	58.9884	3.61681
$267$	21.2135	1.29824
$268$	7.11503	0.434620
$269$	−27.1690	−1.65652	−0.828262	−	0.560341i	$-0.810670\pi$
$269$	−27.1690	−1.65652	−0.828262	+	0.560341i	$0.810670\pi$
$270$	48.5328	2.95361
$271$	10.7469	0.652830	0.326415	−	0.945227i	$-0.394159\pi$
$271$	10.7469	0.652830	0.326415	+	0.945227i	$0.394159\pi$
$272$	44.9341	2.72453
$273$	−23.5507	−1.42535
$274$	5.57657	0.336893
$275$	38.2659	2.30752
$276$	8.79592	0.529452
$277$	12.9230	0.776469	0.388234	−	0.921561i	$-0.373085\pi$
$277$	12.9230	0.776469	0.388234	+	0.921561i	$0.373085\pi$
$278$	−31.0604	−1.86288
$279$	0.374127	0.0223984
$280$	135.467	8.09571
$281$	−21.7945	−1.30015	−0.650075	−	0.759870i	$-0.725263\pi$
$281$	−21.7945	−1.30015	−0.650075	+	0.759870i	$0.725263\pi$
$282$	−6.52483	−0.388548
$283$	−13.5466	−0.805259	−0.402630	−	0.915363i	$-0.631904\pi$
$283$	−13.5466	−0.805259	−0.402630	+	0.915363i	$0.631904\pi$
$284$	23.5858	1.39956
$285$	−28.2258	−1.67195
$286$	−34.1334	−2.01835
$287$	52.7647	3.11460
$288$	−0.570870	−0.0336389
$289$	3.19743	0.188084
$290$	52.0963	3.05920
$291$	10.3855	0.608809
$292$	−0.810080	−0.0474064
$293$	−18.3408	−1.07148	−0.535742	−	0.844382i	$-0.679968\pi$
$293$	−18.3408	−1.07148	−0.535742	+	0.844382i	$0.679968\pi$
$294$	82.6433	4.81985
$295$	−18.7722	−1.09296
$296$	−59.9227	−3.48294
$297$	24.8946	1.44453
$298$	−17.2757	−1.00075
$299$	−2.78394	−0.160999
$300$	−67.3906	−3.89080
$301$	56.8468	3.27659
$302$	53.4229	3.07414
$303$	11.0392	0.634186
$304$	−44.9521	−2.57818
$305$	−36.0042	−2.06159
$306$	−0.603268	−0.0344865
$307$	−11.7553	−0.670913	−0.335457	−	0.942056i	$-0.608891\pi$
$307$	−11.7553	−0.670913	−0.335457	+	0.942056i	$0.608891\pi$
$308$	117.863	6.71588
$309$	−19.9310	−1.13384
$310$	68.8459	3.91018
$311$	5.02533	0.284960	0.142480	−	0.989798i	$-0.454492\pi$
$311$	5.02533	0.284960	0.142480	+	0.989798i	$0.454492\pi$
$312$	35.4398	2.00639
$313$	−28.9391	−1.63574	−0.817868	−	0.575406i	$-0.804844\pi$
$313$	−28.9391	−1.63574	−0.817868	+	0.575406i	$0.804844\pi$
$314$	42.0279	2.37177
$315$	−0.921009	−0.0518930
$316$	−32.5078	−1.82871
$317$	8.02755	0.450872	0.225436	−	0.974258i	$-0.427619\pi$
$317$	8.02755	0.450872	0.225436	+	0.974258i	$0.427619\pi$
$318$	−13.5805	−0.761554
$319$	26.7225	1.49617
$320$	−33.1801	−1.85482
$321$	5.58843	0.311916
$322$	13.5586	0.755590
$323$	−20.2055	−1.12427
$324$	−44.5906	−2.47726
$325$	21.3293	1.18314
$326$	26.6698	1.47710
$327$	−29.1966	−1.61458
$328$	−79.4021	−4.38425
$329$	−7.13093	−0.393141
$330$	−79.5454	−4.37883
$331$	−3.64933	−0.200585	−0.100293	−	0.994958i	$-0.531978\pi$
$331$	−3.64933	−0.200585	−0.100293	+	0.994958i	$0.531978\pi$
$332$	19.2064	1.05409
$333$	0.407400	0.0223254
$334$	37.8454	2.07081
$335$	−5.24796	−0.286727
$336$	−87.4061	−4.76839
$337$	−25.3134	−1.37891	−0.689453	−	0.724330i	$-0.742149\pi$
$337$	−25.3134	−1.37891	−0.689453	+	0.724330i	$0.742149\pi$
$338$	15.0549	0.818877
$339$	35.2669	1.91544
$340$	−78.7069	−4.26848
$341$	35.3140	1.91236
$342$	0.603510	0.0326341
$343$	55.2872	2.98523
$344$	−85.5449	−4.61227
$345$	−6.48775	−0.349289
$346$	−36.9777	−1.98793
$347$	22.6162	1.21410	0.607051	−	0.794663i	$-0.292352\pi$
$347$	22.6162	1.21410	0.607051	+	0.794663i	$0.292352\pi$
$348$	−47.0612	−2.52275
$349$	18.6963	1.00079	0.500396	−	0.865797i	$-0.333188\pi$
$349$	18.6963	1.00079	0.500396	+	0.865797i	$0.333188\pi$
$350$	−103.880	−5.55262
$351$	13.8762	0.740656
$352$	−53.8847	−2.87206
$353$	13.9921	0.744723	0.372362	−	0.928088i	$-0.378548\pi$
$353$	13.9921	0.744723	0.372362	+	0.928088i	$0.378548\pi$
$354$	23.9182	1.27124
$355$	−17.3966	−0.923314
$356$	−59.1769	−3.13637
$357$	−39.2882	−2.07935
$358$	−59.4800	−3.14362
$359$	−15.7751	−0.832578	−0.416289	−	0.909232i	$-0.636670\pi$
$359$	−15.7751	−0.832578	−0.416289	+	0.909232i	$0.636670\pi$
$360$	1.38596	0.0730468
$361$	1.21363	0.0638755
$362$	35.1498	1.84743
$363$	−21.5878	−1.13307
$364$	65.6967	3.44344
$365$	0.597505	0.0312748
$366$	45.8738	2.39786
$367$	25.0061	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$367$	25.0061	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$368$	−10.3323	−0.538609
$369$	0.539836	0.0281028
$370$	74.9686	3.89743
$371$	−14.8420	−0.770556
$372$	−62.1920	−3.22450
$373$	−18.2775	−0.946373	−0.473187	−	0.880962i	$-0.656896\pi$
$373$	−18.2775	−0.946373	−0.473187	+	0.880962i	$0.656896\pi$
$374$	−56.9427	−2.94444
$375$	18.3162	0.945843
$376$	10.7309	0.553402
$377$	14.8950	0.767133
$378$	−67.5810	−3.47599
$379$	1.91196	0.0982107	0.0491054	−	0.998794i	$-0.484363\pi$
$379$	1.91196	0.0982107	0.0491054	+	0.998794i	$0.484363\pi$
$380$	78.7385	4.03920
$381$	2.92204	0.149701
$382$	25.5721	1.30838
$383$	−8.87430	−0.453456	−0.226728	−	0.973958i	$-0.572803\pi$
$383$	−8.87430	−0.453456	−0.226728	+	0.973958i	$0.572803\pi$
$384$	3.32570	0.169714
$385$	−86.9344	−4.43059
$386$	60.5053	3.07964
$387$	0.581600	0.0295644
$388$	−28.9713	−1.47080
$389$	21.5022	1.09021	0.545103	−	0.838369i	$-0.316491\pi$
$389$	21.5022	1.09021	0.545103	+	0.838369i	$0.316491\pi$
$390$	−44.3384	−2.24516
$391$	−4.64427	−0.234871
$392$	−135.917	−6.86483
$393$	−18.3030	−0.923262
$394$	40.6134	2.04607
$395$	23.9773	1.20643
$396$	1.20586	0.0605967
$397$	−0.646998	−0.0324719	−0.0162359	−	0.999868i	$-0.505168\pi$
$397$	−0.646998	−0.0324719	−0.0162359	+	0.999868i	$0.505168\pi$
$398$	−1.87186	−0.0938277
$399$	39.3039	1.96766
$400$	79.1618	3.95809
$401$	2.92636	0.146135	0.0730677	−	0.997327i	$-0.476721\pi$
$401$	2.92636	0.146135	0.0730677	+	0.997327i	$0.476721\pi$
$402$	6.68656	0.333495
$403$	19.6840	0.980528
$404$	−30.7949	−1.53210
$405$	32.8895	1.63429
$406$	−72.5431	−3.60025
$407$	38.4547	1.90613
$408$	59.1221	2.92698
$409$	5.97515	0.295452	0.147726	−	0.989028i	$-0.452805\pi$
$409$	5.97515	0.295452	0.147726	+	0.989028i	$0.452805\pi$
$410$	99.3392	4.90601
$411$	3.71567	0.183280
$412$	55.5994	2.73918
$413$	26.1400	1.28626
$414$	0.138718	0.00681761
$415$	−14.1664	−0.695399
$416$	−30.0352	−1.47260
$417$	−20.6956	−1.01347
$418$	56.9656	2.78628
$419$	−1.92065	−0.0938298	−0.0469149	−	0.998899i	$-0.514939\pi$
$419$	−1.92065	−0.0938298	−0.0469149	+	0.998899i	$0.514939\pi$
$420$	153.101	7.47058
$421$	−26.6011	−1.29646	−0.648228	−	0.761446i	$-0.724490\pi$
$421$	−26.6011	−1.29646	−0.648228	+	0.761446i	$0.724490\pi$
$422$	5.32258	0.259099
$423$	−0.0729566	−0.00354727
$424$	22.3347	1.08467
$425$	35.5824	1.72600
$426$	22.1654	1.07392
$427$	50.1351	2.42621
$428$	−15.5894	−0.753544
$429$	−22.7431	−1.09805
$430$	107.024	5.16117
$431$	21.1280	1.01770	0.508850	−	0.860855i	$-0.330071\pi$
$431$	21.1280	1.01770	0.508850	+	0.860855i	$0.330071\pi$
$432$	51.5001	2.47780
$433$	27.8264	1.33725	0.668625	−	0.743600i	$-0.266883\pi$
$433$	27.8264	1.33725	0.668625	+	0.743600i	$0.266883\pi$
$434$	−95.8666	−4.60174
$435$	34.7118	1.66430
$436$	81.4465	3.90058
$437$	4.64614	0.222255
$438$	−0.761296	−0.0363761
$439$	−12.1390	−0.579362	−0.289681	−	0.957123i	$-0.593549\pi$
$439$	−12.1390	−0.579362	−0.289681	+	0.957123i	$0.593549\pi$
$440$	130.822	6.23668
$441$	0.924065	0.0440031
$442$	−31.7397	−1.50971
$443$	−21.0719	−1.00116	−0.500579	−	0.865691i	$-0.666879\pi$
$443$	−21.0719	−1.00116	−0.500579	+	0.865691i	$0.666879\pi$
$444$	−67.7230	−3.21399
$445$	43.6481	2.06912
$446$	67.2059	3.18229
$447$	−11.5108	−0.544442
$448$	46.2026	2.18287
$449$	8.85250	0.417775	0.208888	−	0.977940i	$-0.433016\pi$
$449$	8.85250	0.417775	0.208888	+	0.977940i	$0.433016\pi$
$450$	−1.06280	−0.0501007
$451$	50.9554	2.39939
$452$	−98.3803	−4.62742
$453$	35.5957	1.67243
$454$	−31.2750	−1.46781
$455$	−48.4570	−2.27170
$456$	−59.1459	−2.76976
$457$	−9.49040	−0.443942	−0.221971	−	0.975053i	$-0.571249\pi$
$457$	−9.49040	−0.443942	−0.221971	+	0.975053i	$0.571249\pi$
$458$	−64.5897	−3.01808
$459$	23.1488	1.08049
$460$	18.0982	0.843832
$461$	5.91355	0.275422	0.137711	−	0.990472i	$-0.456026\pi$
$461$	5.91355	0.275422	0.137711	+	0.990472i	$0.456026\pi$
$462$	110.765	5.15327
$463$	−1.20527	−0.0560137	−0.0280068	−	0.999608i	$-0.508916\pi$
$463$	−1.20527	−0.0560137	−0.0280068	+	0.999608i	$0.508916\pi$
$464$	55.2815	2.56638
$465$	45.8720	2.12726
$466$	2.77313	0.128463
$467$	16.7156	0.773506	0.386753	−	0.922183i	$-0.373597\pi$
$467$	16.7156	0.773506	0.386753	+	0.922183i	$0.373597\pi$
$468$	0.672143	0.0310698
$469$	7.30768	0.337437
$470$	−13.4253	−0.619262
$471$	28.0032	1.29032
$472$	−39.3363	−1.81060
$473$	54.8974	2.52419
$474$	−30.5501	−1.40321
$475$	−35.5967	−1.63329
$476$	109.598	5.02341
$477$	−0.151848	−0.00695265
$478$	−54.0312	−2.47133
$479$	−17.9870	−0.821848	−0.410924	−	0.911670i	$-0.634794\pi$
$479$	−17.9870	−0.821848	−0.410924	+	0.911670i	$0.634794\pi$
$480$	−69.9948	−3.19481
$481$	21.4345	0.977331
$482$	−36.6079	−1.66745
$483$	9.03407	0.411065
$484$	60.2211	2.73732
$485$	21.3689	0.970310
$486$	−1.39485	−0.0632716
$487$	17.1059	0.775144	0.387572	−	0.921839i	$-0.373314\pi$
$487$	17.1059	0.775144	0.387572	+	0.921839i	$0.373314\pi$
$488$	−75.4450	−3.41523
$489$	17.7701	0.803589
$490$	170.044	7.68180
$491$	−24.2680	−1.09520	−0.547599	−	0.836741i	$-0.684458\pi$
$491$	−24.2680	−1.09520	−0.547599	+	0.836741i	$0.684458\pi$
$492$	−89.7381	−4.04571
$493$	24.8485	1.11912
$494$	31.7525	1.42861
$495$	−0.889426	−0.0399767
$496$	73.0552	3.28027
$497$	24.2244	1.08661
$498$	18.0497	0.808827
$499$	−16.9863	−0.760413	−0.380206	−	0.924902i	$-0.624147\pi$
$499$	−16.9863	−0.760413	−0.380206	+	0.924902i	$0.624147\pi$
$500$	−51.0946	−2.28502
$501$	25.2164	1.12658
$502$	−43.1545	−1.92608
$503$	−15.2632	−0.680552	−0.340276	−	0.940326i	$-0.610520\pi$
$503$	−15.2632	−0.680552	−0.340276	+	0.940326i	$0.610520\pi$
$504$	−1.92993	−0.0859659
$505$	22.7139	1.01075
$506$	13.0936	0.582083
$507$	10.0311	0.445495
$508$	−8.15129	−0.361655
$509$	−0.311340	−0.0137999	−0.00689996	−	0.999976i	$-0.502196\pi$
$509$	−0.311340	−0.0137999	−0.00689996	+	0.999976i	$0.502196\pi$
$510$	−73.9671	−3.27532
$511$	−0.832014	−0.0368061
$512$	39.1270	1.72919
$513$	−23.1581	−1.02245
$514$	−16.1515	−0.712414
$515$	−41.0094	−1.80709
$516$	−96.6805	−4.25612
$517$	−6.88640	−0.302864
$518$	−104.392	−4.58674
$519$	−24.6383	−1.08150
$520$	72.9198	3.19774
$521$	8.09571	0.354680	0.177340	−	0.984150i	$-0.443251\pi$
$521$	8.09571	0.354680	0.177340	+	0.984150i	$0.443251\pi$
$522$	−0.742189	−0.0324847
$523$	−1.50381	−0.0657570	−0.0328785	−	0.999459i	$-0.510467\pi$
$523$	−1.50381	−0.0657570	−0.0328785	+	0.999459i	$0.510467\pi$
$524$	51.0577	2.23047
$525$	−69.2152	−3.02080
$526$	57.0748	2.48858
$527$	32.8376	1.43043
$528$	−84.4088	−3.67342
$529$	−21.9321	−0.953569
$530$	−27.9427	−1.21375
$531$	0.267438	0.0116058
$532$	−109.642	−4.75358
$533$	28.4024	1.23024
$534$	−55.6132	−2.40662
$535$	11.4986	0.497126
$536$	−10.9968	−0.474991
$537$	−39.6315	−1.71023
$538$	71.2262	3.07078
$539$	87.2229	3.75696
$540$	−90.2081	−3.88194
$541$	27.3025	1.17382	0.586912	−	0.809651i	$-0.300343\pi$
$541$	27.3025	1.17382	0.586912	+	0.809651i	$0.300343\pi$
$542$	−28.1741	−1.21018
$543$	23.4203	1.00506
$544$	−50.1059	−2.14827
$545$	−60.0739	−2.57328
$546$	61.7404	2.64224
$547$	7.56196	0.323326	0.161663	−	0.986846i	$-0.448314\pi$
$547$	7.56196	0.323326	0.161663	+	0.986846i	$0.448314\pi$
$548$	−10.3652	−0.442779
$549$	0.512933	0.0218914
$550$	−100.318	−4.27757
$551$	−24.8585	−1.05901
$552$	−13.5948	−0.578632
$553$	−33.3880	−1.41980
$554$	−33.8789	−1.43938
$555$	49.9516	2.12033
$556$	57.7321	2.44839
$557$	−5.45719	−0.231229	−0.115614	−	0.993294i	$-0.536884\pi$
$557$	−5.45719	−0.231229	−0.115614	+	0.993294i	$0.536884\pi$
$558$	−0.980811	−0.0415210
$559$	30.5997	1.29423
$560$	−179.844	−7.59979
$561$	−37.9409	−1.60187
$562$	57.1364	2.41015
$563$	12.2065	0.514444	0.257222	−	0.966352i	$-0.417193\pi$
$563$	12.2065	0.514444	0.257222	+	0.966352i	$0.417193\pi$
$564$	12.1277	0.510670
$565$	72.5640	3.05279
$566$	35.5136	1.49275
$567$	−45.7980	−1.92333
$568$	−36.4537	−1.52956
$569$	−2.95666	−0.123950	−0.0619748	−	0.998078i	$-0.519740\pi$
$569$	−2.95666	−0.123950	−0.0619748	+	0.998078i	$0.519740\pi$
$570$	73.9968	3.09938
$571$	−25.8570	−1.08208	−0.541041	−	0.840996i	$-0.681970\pi$
$571$	−25.8570	−1.08208	−0.541041	+	0.840996i	$0.681970\pi$
$572$	63.4439	2.65272
$573$	17.0387	0.711800
$574$	−138.328	−5.77369
$575$	−8.18196	−0.341211
$576$	0.472699	0.0196958
$577$	−8.12175	−0.338113	−0.169056	−	0.985606i	$-0.554072\pi$
$577$	−8.12175	−0.338113	−0.169056	+	0.985606i	$0.554072\pi$
$578$	−8.38237	−0.348661
$579$	40.3146	1.67542
$580$	−96.8316	−4.02071
$581$	19.7264	0.818388
$582$	−27.2266	−1.12858
$583$	−14.3330	−0.593612
$584$	1.25204	0.0518099
$585$	−0.495764	−0.0204973
$586$	48.0823	1.98626
$587$	−22.7370	−0.938458	−0.469229	−	0.883077i	$-0.655468\pi$
$587$	−22.7370	−0.938458	−0.469229	+	0.883077i	$0.655468\pi$
$588$	−153.609	−6.33474
$589$	−32.8507	−1.35359
$590$	49.2133	2.02608
$591$	27.0607	1.11313
$592$	79.5523	3.26958
$593$	21.4016	0.878860	0.439430	−	0.898277i	$-0.355180\pi$
$593$	21.4016	0.878860	0.439430	+	0.898277i	$0.355180\pi$
$594$	−65.2635	−2.67780
$595$	−80.8380	−3.31403
$596$	32.1104	1.31529
$597$	−1.24722	−0.0510452
$598$	7.29836	0.298452
$599$	5.52344	0.225682	0.112841	−	0.993613i	$-0.464005\pi$
$599$	5.52344	0.225682	0.112841	+	0.993613i	$0.464005\pi$
$600$	104.157	4.25221
$601$	−13.6050	−0.554959	−0.277479	−	0.960732i	$-0.589499\pi$
$601$	−13.6050	−0.554959	−0.277479	+	0.960732i	$0.589499\pi$
$602$	−149.029	−6.07398
$603$	0.0747649	0.00304466
$604$	−99.2973	−4.04035
$605$	−44.4183	−1.80586
$606$	−28.9404	−1.17562
$607$	−28.2370	−1.14610	−0.573052	−	0.819519i	$-0.694241\pi$
$607$	−28.2370	−1.14610	−0.573052	+	0.819519i	$0.694241\pi$
$608$	50.1260	2.03288
$609$	−48.3355	−1.95865
$610$	94.3884	3.82168
$611$	−3.83846	−0.155288
$612$	1.12130	0.0453257
$613$	−20.4912	−0.827634	−0.413817	−	0.910360i	$-0.635805\pi$
$613$	−20.4912	−0.827634	−0.413817	+	0.910360i	$0.635805\pi$
$614$	30.8178	1.24370
$615$	66.1897	2.66903
$616$	−182.167	−7.33971
$617$	22.5927	0.909548	0.454774	−	0.890607i	$-0.349720\pi$
$617$	22.5927	0.909548	0.454774	+	0.890607i	$0.349720\pi$
$618$	52.2511	2.10185
$619$	−13.9119	−0.559167	−0.279583	−	0.960121i	$-0.590196\pi$
$619$	−13.9119	−0.559167	−0.279583	+	0.960121i	$0.590196\pi$
$620$	−127.964	−5.13916
$621$	−5.32292	−0.213602
$622$	−13.1744	−0.528245
$623$	−60.7792	−2.43507
$624$	−47.0493	−1.88348
$625$	−1.90075	−0.0760301
$626$	75.8667	3.03224
$627$	37.9561	1.51582
$628$	−78.1175	−3.11723
$629$	35.7580	1.42576
$630$	2.41451	0.0961966
$631$	23.0393	0.917181	0.458590	−	0.888648i	$-0.348355\pi$
$631$	23.0393	0.917181	0.458590	+	0.888648i	$0.348355\pi$
$632$	50.2433	1.99857
$633$	3.54643	0.140958
$634$	−21.0450	−0.835803
$635$	6.01229	0.238590
$636$	25.2420	1.00091
$637$	48.6178	1.92631
$638$	−70.0555	−2.77352
$639$	0.247840	0.00980439
$640$	6.84284	0.270487
$641$	44.0942	1.74162	0.870808	−	0.491623i	$-0.163596\pi$
$641$	44.0942	1.74162	0.870808	+	0.491623i	$0.163596\pi$
$642$	−14.6506	−0.578214
$643$	−10.7601	−0.424336	−0.212168	−	0.977233i	$-0.568052\pi$
$643$	−10.7601	−0.424336	−0.212168	+	0.977233i	$0.568052\pi$
$644$	−25.2014	−0.993072
$645$	71.3103	2.80784
$646$	52.9707	2.08411
$647$	36.2437	1.42489	0.712443	−	0.701730i	$-0.247589\pi$
$647$	36.2437	1.42489	0.712443	+	0.701730i	$0.247589\pi$
$648$	68.9183	2.70737
$649$	25.2436	0.990898
$650$	−55.9169	−2.19324
$651$	−63.8759	−2.50349
$652$	−49.5711	−1.94136
$653$	34.5179	1.35079	0.675396	−	0.737456i	$-0.263973\pi$
$653$	34.5179	1.35079	0.675396	+	0.737456i	$0.263973\pi$
$654$	76.5417	2.99302
$655$	−37.6595	−1.47148
$656$	105.413	4.11568
$657$	−0.00851234	−0.000332098 0
$658$	18.6944	0.728785
$659$	−12.2573	−0.477478	−0.238739	−	0.971084i	$-0.576734\pi$
$659$	−12.2573	−0.477478	−0.238739	+	0.971084i	$0.576734\pi$
$660$	147.851	5.75510
$661$	6.76439	0.263104	0.131552	−	0.991309i	$-0.458004\pi$
$661$	6.76439	0.263104	0.131552	+	0.991309i	$0.458004\pi$
$662$	9.56706	0.371834
$663$	−21.1482	−0.821327
$664$	−29.6849	−1.15200
$665$	80.8704	3.13602
$666$	−1.06804	−0.0413857
$667$	−5.71376	−0.221238
$668$	−70.3433	−2.72167
$669$	44.7793	1.73127
$670$	13.7580	0.531519
$671$	48.4159	1.86908
$672$	97.4664	3.75985
$673$	−16.3569	−0.630514	−0.315257	−	0.949006i	$-0.602091\pi$
$673$	−16.3569	−0.630514	−0.315257	+	0.949006i	$0.602091\pi$
$674$	66.3614	2.55615
$675$	40.7820	1.56970
$676$	−27.9825	−1.07625
$677$	−47.1334	−1.81148	−0.905742	−	0.423829i	$-0.860686\pi$
$677$	−47.1334	−1.81148	−0.905742	+	0.423829i	$0.860686\pi$
$678$	−92.4557	−3.55074
$679$	−29.7557	−1.14192
$680$	121.648	4.66497
$681$	−20.8385	−0.798534
$682$	−92.5792	−3.54504
$683$	−38.1299	−1.45900	−0.729499	−	0.683982i	$-0.760247\pi$
$683$	−38.1299	−1.45900	−0.729499	+	0.683982i	$0.760247\pi$
$684$	−1.12175	−0.0428910
$685$	7.64523	0.292109
$686$	−144.941	−5.53386
$687$	−43.0361	−1.64193
$688$	113.568	4.32974
$689$	−7.98918	−0.304364
$690$	17.0083	0.647494
$691$	−27.6199	−1.05071	−0.525356	−	0.850883i	$-0.676068\pi$
$691$	−27.6199	−1.05071	−0.525356	+	0.850883i	$0.676068\pi$
$692$	68.7306	2.61275
$693$	1.23851	0.0470471
$694$	−59.2906	−2.25064
$695$	−42.5825	−1.61525
$696$	72.7368	2.75708
$697$	47.3820	1.79472
$698$	−49.0143	−1.85522
$699$	1.84774	0.0698879
$700$	193.082	7.29782
$701$	−51.1254	−1.93098	−0.965490	−	0.260441i	$-0.916132\pi$
$701$	−51.1254	−1.93098	−0.965490	+	0.260441i	$0.916132\pi$
$702$	−36.3777	−1.37299
$703$	−35.7723	−1.34918
$704$	44.6183	1.68161
$705$	−8.94526	−0.336898
$706$	−36.6816	−1.38053
$707$	−31.6287	−1.18952
$708$	−44.4568	−1.67079
$709$	6.24782	0.234642	0.117321	−	0.993094i	$-0.462569\pi$
$709$	6.24782	0.234642	0.117321	+	0.993094i	$0.462569\pi$
$710$	45.6068	1.71159
$711$	−0.341593	−0.0128107
$712$	91.4625	3.42770
$713$	−7.55080	−0.282780
$714$	102.998	3.85459
$715$	−46.7954	−1.75005
$716$	110.556	4.13166
$717$	−36.0010	−1.34448
$718$	41.3559	1.54339
$719$	28.1094	1.04830	0.524152	−	0.851625i	$-0.324382\pi$
$719$	28.1094	1.04830	0.524152	+	0.851625i	$0.324382\pi$
$720$	−1.83998	−0.0685721
$721$	57.1048	2.12669
$722$	−3.18166	−0.118409
$723$	−24.3919	−0.907143
$724$	−65.3330	−2.42808
$725$	43.7764	1.62581
$726$	56.5945	2.10042
$727$	−34.0068	−1.26124	−0.630621	−	0.776091i	$-0.717200\pi$
$727$	−34.0068	−1.26124	−0.630621	+	0.776091i	$0.717200\pi$
$728$	−101.539	−3.76330
$729$	26.5236	0.982355
$730$	−1.56642	−0.0579757
$731$	51.0476	1.88807
$732$	−85.2659	−3.15152
$733$	−1.37261	−0.0506984	−0.0253492	−	0.999679i	$-0.508070\pi$
$733$	−1.37261	−0.0506984	−0.0253492	+	0.999679i	$0.508070\pi$
$734$	−65.5559	−2.41971
$735$	113.300	4.17914
$736$	11.5215	0.424690
$737$	7.05709	0.259951
$738$	−1.41523	−0.0520954
$739$	−51.5090	−1.89479	−0.947394	−	0.320070i	$-0.896294\pi$
$739$	−51.5090	−1.89479	−0.947394	+	0.320070i	$0.896294\pi$
$740$	−139.344	−5.12240
$741$	21.1567	0.777210
$742$	38.9096	1.42842
$743$	8.77792	0.322031	0.161015	−	0.986952i	$-0.448523\pi$
$743$	8.77792	0.322031	0.161015	+	0.986952i	$0.448523\pi$
$744$	96.1225	3.52402
$745$	−23.6842	−0.867723
$746$	47.9162	1.75434
$747$	0.201821	0.00738423
$748$	105.840	3.86988
$749$	−16.0115	−0.585048
$750$	−48.0176	−1.75336
$751$	−47.4783	−1.73251	−0.866255	−	0.499603i	$-0.833479\pi$
$751$	−47.4783	−1.73251	−0.866255	+	0.499603i	$0.833479\pi$
$752$	−14.2461	−0.519502
$753$	−28.7538	−1.04785
$754$	−39.0488	−1.42207
$755$	73.2404	2.66549
$756$	125.613	4.56850
$757$	21.9311	0.797100	0.398550	−	0.917147i	$-0.369514\pi$
$757$	21.9311	0.797100	0.398550	+	0.917147i	$0.369514\pi$
$758$	−5.01238	−0.182058
$759$	8.72428	0.316671
$760$	−121.696	−4.41439
$761$	23.2955	0.844461	0.422231	−	0.906488i	$-0.361247\pi$
$761$	23.2955	0.844461	0.422231	+	0.906488i	$0.361247\pi$
$762$	−7.66041	−0.277507
$763$	83.6518	3.02840
$764$	−47.5309	−1.71961
$765$	−0.827054	−0.0299022
$766$	23.2648	0.840593
$767$	14.0707	0.508065
$768$	23.5332	0.849180
$769$	21.3481	0.769832	0.384916	−	0.922952i	$-0.374230\pi$
$769$	21.3481	0.769832	0.384916	+	0.922952i	$0.374230\pi$
$770$	227.907	8.21320
$771$	−10.7618	−0.387576
$772$	−112.461	−4.04757
$773$	32.1762	1.15730	0.578649	−	0.815576i	$-0.303580\pi$
$773$	32.1762	1.15730	0.578649	+	0.815576i	$0.303580\pi$
$774$	−1.52472	−0.0548049
$775$	57.8510	2.07807
$776$	44.7774	1.60742
$777$	−69.5566	−2.49533
$778$	−56.3701	−2.02097
$779$	−47.4010	−1.69832
$780$	82.4119	2.95082
$781$	23.3937	0.837092
$782$	12.1754	0.435392
$783$	28.4795	1.01777
$784$	180.441	6.44430
$785$	57.6184	2.05649
$786$	47.9830	1.71150
$787$	17.9477	0.639765	0.319883	−	0.947457i	$-0.396356\pi$
$787$	17.9477	0.639765	0.319883	+	0.947457i	$0.396356\pi$
$788$	−75.4883	−2.68916
$789$	38.0289	1.35387
$790$	−62.8589	−2.23642
$791$	−101.044	−3.59271
$792$	−1.86375	−0.0662254
$793$	26.9869	0.958333
$794$	1.69617	0.0601947
$795$	−18.6182	−0.660319
$796$	3.47923	0.123318
$797$	−1.71188	−0.0606378	−0.0303189	−	0.999540i	$-0.509652\pi$
$797$	−1.71188	−0.0606378	−0.0303189	+	0.999540i	$0.509652\pi$
$798$	−103.039	−3.64754
$799$	−6.40348	−0.226539
$800$	−88.2732	−3.12093
$801$	−0.621832	−0.0219714
$802$	−7.67173	−0.270898
$803$	−0.803483	−0.0283543
$804$	−12.4283	−0.438313
$805$	18.5882	0.655148
$806$	−51.6034	−1.81765
$807$	47.4580	1.67060
$808$	47.5958	1.67442
$809$	−31.7245	−1.11537	−0.557686	−	0.830052i	$-0.688311\pi$
$809$	−31.7245	−1.11537	−0.557686	+	0.830052i	$0.688311\pi$
$810$	−86.2230	−3.02957
$811$	−49.9569	−1.75422	−0.877112	−	0.480285i	$-0.840533\pi$
$811$	−49.9569	−1.75422	−0.877112	+	0.480285i	$0.840533\pi$
$812$	134.836	4.73182
$813$	−18.7724	−0.658377
$814$	−100.813	−3.53348
$815$	36.5630	1.28075
$816$	−78.4895	−2.74768
$817$	−51.0681	−1.78665
$818$	−15.6644	−0.547694
$819$	0.690342	0.0241225
$820$	−184.642	−6.44798
$821$	9.39202	0.327784	0.163892	−	0.986478i	$-0.447595\pi$
$821$	9.39202	0.327784	0.163892	+	0.986478i	$0.447595\pi$
$822$	−9.74099	−0.339756
$823$	46.5051	1.62107	0.810533	−	0.585692i	$-0.199177\pi$
$823$	46.5051	1.62107	0.810533	+	0.585692i	$0.199177\pi$
$824$	−85.9331	−2.99362
$825$	−66.8417	−2.32713
$826$	−68.5285	−2.38441
$827$	44.7603	1.55647	0.778234	−	0.627975i	$-0.216116\pi$
$827$	44.7603	1.55647	0.778234	+	0.627975i	$0.216116\pi$
$828$	−0.257835	−0.00896039
$829$	−18.6308	−0.647074	−0.323537	−	0.946216i	$-0.604872\pi$
$829$	−18.6308	−0.647074	−0.323537	+	0.946216i	$0.604872\pi$
$830$	37.1385	1.28910
$831$	−22.5735	−0.783067
$832$	24.8701	0.862216
$833$	81.1062	2.81016
$834$	54.2554	1.87871
$835$	51.8843	1.79553
$836$	−105.882	−3.66201
$837$	37.6360	1.30089
$838$	5.03517	0.173937
$839$	18.0186	0.622072	0.311036	−	0.950398i	$-0.399324\pi$
$839$	18.0186	0.622072	0.311036	+	0.950398i	$0.399324\pi$
$840$	−236.630	−8.16451
$841$	1.57060	0.0541587
$842$	69.7372	2.40330
$843$	38.0700	1.31120
$844$	−9.89309	−0.340534
$845$	20.6396	0.710022
$846$	0.191263	0.00657575
$847$	61.8516	2.12525
$848$	−29.6511	−1.01822
$849$	23.6627	0.812102
$850$	−93.2828	−3.19957
$851$	−8.22232	−0.281857
$852$	−41.1989	−1.41145
$853$	−38.5727	−1.32070	−0.660352	−	0.750956i	$-0.729593\pi$
$853$	−38.5727	−1.32070	−0.660352	+	0.750956i	$0.729593\pi$
$854$	−131.434	−4.49758
$855$	0.827385	0.0282960
$856$	24.0947	0.823539
$857$	20.0781	0.685855	0.342927	−	0.939362i	$-0.388582\pi$
$857$	20.0781	0.685855	0.342927	+	0.939362i	$0.388582\pi$
$858$	59.6232	2.03550
$859$	54.9282	1.87413	0.937063	−	0.349160i	$-0.113533\pi$
$859$	54.9282	1.87413	0.937063	+	0.349160i	$0.113533\pi$
$860$	−198.926	−6.78334
$861$	−92.1678	−3.14107
$862$	−55.3891	−1.88656
$863$	−33.1526	−1.12853	−0.564264	−	0.825594i	$-0.690840\pi$
$863$	−33.1526	−1.12853	−0.564264	+	0.825594i	$0.690840\pi$
$864$	−57.4277	−1.95373
$865$	−50.6948	−1.72368
$866$	−72.9495	−2.47892
$867$	−5.58517	−0.189682
$868$	178.187	6.04808
$869$	−32.2431	−1.09377
$870$	−91.0002	−3.08520
$871$	3.93360	0.133285
$872$	−125.882	−4.26290
$873$	−0.304431	−0.0103034
$874$	−12.1803	−0.412005
$875$	−52.4780	−1.77408
$876$	1.41502	0.0478092
$877$	−12.3301	−0.416357	−0.208179	−	0.978091i	$-0.566754\pi$
$877$	−12.3301	−0.416357	−0.208179	+	0.978091i	$0.566754\pi$
$878$	31.8235	1.07399
$879$	32.0372	1.08059
$880$	−173.677	−5.85464
$881$	35.5259	1.19690	0.598450	−	0.801160i	$-0.295784\pi$
$881$	35.5259	1.19690	0.598450	+	0.801160i	$0.295784\pi$
$882$	−2.42253	−0.0815707
$883$	−0.878130	−0.0295514	−0.0147757	−	0.999891i	$-0.504703\pi$
$883$	−0.878130	−0.0295514	−0.0147757	+	0.999891i	$0.504703\pi$
$884$	58.9947	1.98421
$885$	32.7908	1.10225
$886$	55.2421	1.85589
$887$	−16.6357	−0.558572	−0.279286	−	0.960208i	$-0.590098\pi$
$887$	−16.6357	−0.558572	−0.279286	+	0.960208i	$0.590098\pi$
$888$	104.671	3.51253
$889$	−8.37199	−0.280788
$890$	−114.428	−3.83563
$891$	−44.2275	−1.48168
$892$	−124.916	−4.18249
$893$	6.40605	0.214370
$894$	30.1767	1.00926
$895$	−81.5444	−2.72573
$896$	−9.52852	−0.318325
$897$	4.86289	0.162367
$898$	−23.2077	−0.774450
$899$	40.3994	1.34740
$900$	1.97542	0.0658474
$901$	−13.3279	−0.444016
$902$	−133.584	−4.44787
$903$	−99.2982	−3.30444
$904$	152.054	5.05725
$905$	48.1888	1.60185
$906$	−93.3175	−3.10027
$907$	34.1227	1.13303	0.566513	−	0.824053i	$-0.308292\pi$
$907$	34.1227	1.13303	0.566513	+	0.824053i	$0.308292\pi$
$908$	58.1309	1.92914
$909$	−0.323593	−0.0107329
$910$	127.035	4.21116
$911$	−50.5156	−1.67366	−0.836829	−	0.547465i	$-0.815593\pi$
$911$	−50.5156	−1.67366	−0.836829	+	0.547465i	$0.815593\pi$
$912$	78.5210	2.60009
$913$	19.0499	0.630461
$914$	24.8800	0.822958
$915$	62.8910	2.07911
$916$	120.053	3.96667
$917$	52.4402	1.73173
$918$	−60.6868	−2.00296
$919$	23.3013	0.768640	0.384320	−	0.923200i	$-0.374436\pi$
$919$	23.3013	0.768640	0.384320	+	0.923200i	$0.374436\pi$
$920$	−27.9721	−0.922214
$921$	20.5339	0.676614
$922$	−15.5029	−0.510562
$923$	13.0396	0.429203
$924$	−205.880	−6.77295
$925$	62.9959	2.07129
$926$	3.15973	0.103835
$927$	0.584239	0.0191889
$928$	−61.6443	−2.02357
$929$	−32.9585	−1.08133	−0.540666	−	0.841237i	$-0.681828\pi$
$929$	−32.9585	−1.08133	−0.540666	+	0.841237i	$0.681828\pi$
$930$	−120.258	−3.94341
$931$	−81.1388	−2.65922
$932$	−5.15443	−0.168839
$933$	−8.77809	−0.287382
$934$	−43.8216	−1.43388
$935$	−78.0659	−2.55303
$936$	−1.03885	−0.0339559
$937$	−46.0785	−1.50532	−0.752659	−	0.658411i	$-0.771229\pi$
$937$	−46.0785	−1.50532	−0.752659	+	0.658411i	$0.771229\pi$
$938$	−19.1578	−0.625524
$939$	50.5500	1.64964
$940$	24.9536	0.813896
$941$	−44.5795	−1.45325	−0.726624	−	0.687035i	$-0.758912\pi$
$941$	−44.5795	−1.45325	−0.726624	+	0.687035i	$0.758912\pi$
$942$	−73.4131	−2.39193
$943$	−10.8952	−0.354797
$944$	52.2222	1.69969
$945$	−92.6505	−3.01392
$946$	−143.919	−4.67921
$947$	−0.707003	−0.0229745	−0.0114873	−	0.999934i	$-0.503657\pi$
$947$	−0.707003	−0.0229745	−0.0114873	+	0.999934i	$0.503657\pi$
$948$	56.7836	1.84425
$949$	−0.447859	−0.0145381
$950$	93.3202	3.02771
$951$	−14.0223	−0.454703
$952$	−169.392	−5.49003
$953$	−49.5961	−1.60657	−0.803287	−	0.595592i	$-0.796917\pi$
$953$	−49.5961	−1.60657	−0.803287	+	0.595592i	$0.796917\pi$
$954$	0.398084	0.0128885
$955$	35.0582	1.13446
$956$	100.428	3.24807
$957$	−46.6780	−1.50888
$958$	47.1547	1.52350
$959$	−10.6458	−0.343772
$960$	57.9580	1.87059
$961$	22.3883	0.722203
$962$	−56.1927	−1.81173
$963$	−0.163814	−0.00527883
$964$	68.0433	2.19153
$965$	82.9500	2.67026
$966$	−23.6837	−0.762010
$967$	−1.42966	−0.0459747	−0.0229874	−	0.999736i	$-0.507318\pi$
$967$	−1.42966	−0.0459747	−0.0229874	+	0.999736i	$0.507318\pi$
$968$	−93.0764	−2.99159
$969$	35.2944	1.13382
$970$	−56.0205	−1.79871
$971$	35.8930	1.15186	0.575931	−	0.817498i	$-0.304640\pi$
$971$	35.8930	1.15186	0.575931	+	0.817498i	$0.304640\pi$
$972$	2.59261	0.0831580
$973$	59.2953	1.90092
$974$	−44.8449	−1.43692
$975$	−37.2574	−1.19319
$976$	100.159	3.20602
$977$	1.54033	0.0492796	0.0246398	−	0.999696i	$-0.492156\pi$
$977$	1.54033	0.0492796	0.0246398	+	0.999696i	$0.492156\pi$
$978$	−46.5859	−1.48965
$979$	−58.6950	−1.87590
$980$	−316.061	−10.0962
$981$	0.855842	0.0273249
$982$	63.6208	2.03022
$983$	−33.4724	−1.06760	−0.533801	−	0.845610i	$-0.679237\pi$
$983$	−33.4724	−1.06760	−0.533801	+	0.845610i	$0.679237\pi$
$984$	138.697	4.42151
$985$	55.6792	1.77409
$986$	−65.1427	−2.07457
$987$	12.4561	0.396482
$988$	−59.0184	−1.87763
$989$	−11.7381	−0.373249
$990$	2.33172	0.0741068
$991$	−16.6246	−0.528097	−0.264049	−	0.964509i	$-0.585058\pi$
$991$	−16.6246	−0.528097	−0.264049	+	0.964509i	$0.585058\pi$
$992$	−81.4637	−2.58647
$993$	6.37453	0.202290
$994$	−63.5066	−2.01431
$995$	−2.56623	−0.0813550
$996$	−33.5491	−1.06304
$997$	−2.00053	−0.0633574	−0.0316787	−	0.999498i	$-0.510085\pi$
$997$	−2.00053	−0.0633574	−0.0316787	+	0.999498i	$0.510085\pi$
$998$	44.5313	1.40961
$999$	40.9831	1.29665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.b.1.9	✓	186

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.b.1.9	✓	186	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62160 q^{2} -1.74677 q^{3} +4.87277 q^{4} -3.59409 q^{5} +4.57932 q^{6} +5.00470 q^{7} -7.53124 q^{8} +0.0512031 q^{9} +O(q^{10})\)	\(q-2.62160 q^{2} -1.74677 q^{3} +4.87277 q^{4} -3.59409 q^{5} +4.57932 q^{6} +5.00470 q^{7} -7.53124 q^{8} +0.0512031 q^{9} +9.42226 q^{10} +4.83308 q^{11} -8.51160 q^{12} +2.69395 q^{13} -13.1203 q^{14} +6.27805 q^{15} +9.99834 q^{16} +4.49415 q^{17} -0.134234 q^{18} -4.49596 q^{19} -17.5132 q^{20} -8.74206 q^{21} -12.6704 q^{22} -1.03340 q^{23} +13.1553 q^{24} +7.91749 q^{25} -7.06245 q^{26} +5.15087 q^{27} +24.3868 q^{28} +5.52907 q^{29} -16.4585 q^{30} +7.30673 q^{31} -11.1491 q^{32} -8.44228 q^{33} -11.7819 q^{34} -17.9874 q^{35} +0.249501 q^{36} +7.95655 q^{37} +11.7866 q^{38} -4.70571 q^{39} +27.0680 q^{40} +10.5430 q^{41} +22.9182 q^{42} +11.3587 q^{43} +23.5505 q^{44} -0.184029 q^{45} +2.70917 q^{46} -1.42485 q^{47} -17.4648 q^{48} +18.0470 q^{49} -20.7565 q^{50} -7.85025 q^{51} +13.1270 q^{52} -2.96560 q^{53} -13.5035 q^{54} -17.3705 q^{55} -37.6916 q^{56} +7.85340 q^{57} -14.4950 q^{58} +5.22309 q^{59} +30.5915 q^{60} +10.0176 q^{61} -19.1553 q^{62} +0.256256 q^{63} +9.23185 q^{64} -9.68230 q^{65} +22.1323 q^{66} +1.46016 q^{67} +21.8990 q^{68} +1.80512 q^{69} +47.1556 q^{70} +4.84033 q^{71} -0.385623 q^{72} -0.166246 q^{73} -20.8589 q^{74} -13.8300 q^{75} -21.9078 q^{76} +24.1881 q^{77} +12.3365 q^{78} -6.67132 q^{79} -35.9349 q^{80} -9.15099 q^{81} -27.6396 q^{82} +3.94157 q^{83} -42.5980 q^{84} -16.1524 q^{85} -29.7779 q^{86} -9.65801 q^{87} -36.3991 q^{88} -12.1444 q^{89} +0.482449 q^{90} +13.4824 q^{91} -5.03553 q^{92} -12.7632 q^{93} +3.73537 q^{94} +16.1589 q^{95} +19.4750 q^{96} -5.94555 q^{97} -47.3121 q^{98} +0.247469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

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