LMFDB - Embedded newform 2013.4.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.449187 q^{2} -3.00000 q^{3} -7.79823 q^{4} +5.67323 q^{5} +1.34756 q^{6} +0.979067 q^{7} +7.09636 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.449187 q^{2} -3.00000 q^{3} -7.79823 q^{4} +5.67323 q^{5} +1.34756 q^{6} +0.979067 q^{7} +7.09636 q^{8} +9.00000 q^{9} -2.54834 q^{10} -11.0000 q^{11} +23.3947 q^{12} -33.6897 q^{13} -0.439784 q^{14} -17.0197 q^{15} +59.1983 q^{16} -99.2995 q^{17} -4.04268 q^{18} +14.5342 q^{19} -44.2411 q^{20} -2.93720 q^{21} +4.94106 q^{22} +12.6819 q^{23} -21.2891 q^{24} -92.8145 q^{25} +15.1330 q^{26} -27.0000 q^{27} -7.63499 q^{28} -48.9373 q^{29} +7.64502 q^{30} -61.5945 q^{31} -83.3620 q^{32} +33.0000 q^{33} +44.6040 q^{34} +5.55447 q^{35} -70.1841 q^{36} +391.538 q^{37} -6.52858 q^{38} +101.069 q^{39} +40.2593 q^{40} +297.916 q^{41} +1.31935 q^{42} +284.314 q^{43} +85.7805 q^{44} +51.0590 q^{45} -5.69652 q^{46} -37.9658 q^{47} -177.595 q^{48} -342.041 q^{49} +41.6911 q^{50} +297.898 q^{51} +262.720 q^{52} -265.705 q^{53} +12.1280 q^{54} -62.4055 q^{55} +6.94781 q^{56} -43.6026 q^{57} +21.9820 q^{58} -438.213 q^{59} +132.723 q^{60} -61.0000 q^{61} +27.6675 q^{62} +8.81160 q^{63} -436.141 q^{64} -191.129 q^{65} -14.8232 q^{66} -312.761 q^{67} +774.360 q^{68} -38.0456 q^{69} -2.49499 q^{70} -926.140 q^{71} +63.8672 q^{72} -1164.51 q^{73} -175.874 q^{74} +278.443 q^{75} -113.341 q^{76} -10.7697 q^{77} -45.3989 q^{78} +526.836 q^{79} +335.845 q^{80} +81.0000 q^{81} -133.820 q^{82} -528.876 q^{83} +22.9050 q^{84} -563.348 q^{85} -127.710 q^{86} +146.812 q^{87} -78.0599 q^{88} +1170.87 q^{89} -22.9351 q^{90} -32.9844 q^{91} -98.8961 q^{92} +184.784 q^{93} +17.0537 q^{94} +82.4559 q^{95} +250.086 q^{96} -1491.87 q^{97} +153.641 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9}+O(q^{10}) $ 38 * q - 2 * q^2 - 114 * q^3 + 142 * q^4 + 15 * q^5 + 6 * q^6 + 63 * q^7 - 45 * q^8 + 342 * q^9	\( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9} + 95 q^{10} - 418 q^{11} - 426 q^{12} + 13 q^{13} + 26 q^{14} - 45 q^{15} + 486 q^{16} - 224 q^{17} - 18 q^{18} + 367 q^{19} + 18 q^{20} - 189 q^{21} + 22 q^{22} + 51 q^{23} + 135 q^{24} + 773 q^{25} - 439 q^{26} - 1026 q^{27} + 22 q^{28} - 462 q^{29} - 285 q^{30} + 234 q^{31} - 597 q^{32} + 1254 q^{33} + 956 q^{34} - 522 q^{35} + 1278 q^{36} + 954 q^{37} + 705 q^{38} - 39 q^{39} + 1495 q^{40} - 740 q^{41} - 78 q^{42} + 1441 q^{43} - 1562 q^{44} + 135 q^{45} + 581 q^{46} + 1003 q^{47} - 1458 q^{48} + 2707 q^{49} + 388 q^{50} + 672 q^{51} + 788 q^{52} + 735 q^{53} + 54 q^{54} - 165 q^{55} + 1059 q^{56} - 1101 q^{57} + 177 q^{58} + 261 q^{59} - 54 q^{60} - 2318 q^{61} + 1251 q^{62} + 567 q^{63} + 5571 q^{64} - 1354 q^{65} - 66 q^{66} + 3495 q^{67} - 1856 q^{68} - 153 q^{69} + 542 q^{70} - 873 q^{71} - 405 q^{72} + 989 q^{73} - 3406 q^{74} - 2319 q^{75} + 1712 q^{76} - 693 q^{77} + 1317 q^{78} + 2313 q^{79} + 1593 q^{80} + 3078 q^{81} + 5170 q^{82} + 569 q^{83} - 66 q^{84} - 1271 q^{85} + 3065 q^{86} + 1386 q^{87} + 495 q^{88} - 2917 q^{89} + 855 q^{90} + 2740 q^{91} + 1083 q^{92} - 702 q^{93} + 3272 q^{94} + 2696 q^{95} + 1791 q^{96} + 4250 q^{97} + 5952 q^{98} - 3762 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 - 114 * q^3 + 142 * q^4 + 15 * q^5 + 6 * q^6 + 63 * q^7 - 45 * q^8 + 342 * q^9 + 95 * q^10 - 418 * q^11 - 426 * q^12 + 13 * q^13 + 26 * q^14 - 45 * q^15 + 486 * q^16 - 224 * q^17 - 18 * q^18 + 367 * q^19 + 18 * q^20 - 189 * q^21 + 22 * q^22 + 51 * q^23 + 135 * q^24 + 773 * q^25 - 439 * q^26 - 1026 * q^27 + 22 * q^28 - 462 * q^29 - 285 * q^30 + 234 * q^31 - 597 * q^32 + 1254 * q^33 + 956 * q^34 - 522 * q^35 + 1278 * q^36 + 954 * q^37 + 705 * q^38 - 39 * q^39 + 1495 * q^40 - 740 * q^41 - 78 * q^42 + 1441 * q^43 - 1562 * q^44 + 135 * q^45 + 581 * q^46 + 1003 * q^47 - 1458 * q^48 + 2707 * q^49 + 388 * q^50 + 672 * q^51 + 788 * q^52 + 735 * q^53 + 54 * q^54 - 165 * q^55 + 1059 * q^56 - 1101 * q^57 + 177 * q^58 + 261 * q^59 - 54 * q^60 - 2318 * q^61 + 1251 * q^62 + 567 * q^63 + 5571 * q^64 - 1354 * q^65 - 66 * q^66 + 3495 * q^67 - 1856 * q^68 - 153 * q^69 + 542 * q^70 - 873 * q^71 - 405 * q^72 + 989 * q^73 - 3406 * q^74 - 2319 * q^75 + 1712 * q^76 - 693 * q^77 + 1317 * q^78 + 2313 * q^79 + 1593 * q^80 + 3078 * q^81 + 5170 * q^82 + 569 * q^83 - 66 * q^84 - 1271 * q^85 + 3065 * q^86 + 1386 * q^87 + 495 * q^88 - 2917 * q^89 + 855 * q^90 + 2740 * q^91 + 1083 * q^92 - 702 * q^93 + 3272 * q^94 + 2696 * q^95 + 1791 * q^96 + 4250 * q^97 + 5952 * q^98 - 3762 * 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 $

$2$	−0.449187	−0.158812	−0.0794058	−	0.996842i	$-0.525302\pi$
$2$	−0.449187	−0.158812	−0.0794058	+	0.996842i	$0.525302\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.79823	−0.974779
$5$	5.67323	0.507429	0.253714	−	0.967279i	$-0.418348\pi$
$5$	5.67323	0.507429	0.253714	+	0.967279i	$0.418348\pi$
$6$	1.34756	0.0916899
$7$	0.979067	0.0528646	0.0264323	−	0.999651i	$-0.491585\pi$
$7$	0.979067	0.0528646	0.0264323	+	0.999651i	$0.491585\pi$
$8$	7.09636	0.313618
$9$	9.00000	0.333333
$10$	−2.54834	−0.0805856
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	23.3947	0.562789
$13$	−33.6897	−0.718757	−0.359378	−	0.933192i	$-0.617011\pi$
$13$	−33.6897	−0.718757	−0.359378	+	0.933192i	$0.617011\pi$
$14$	−0.439784	−0.00839551
$15$	−17.0197	−0.292964
$16$	59.1983	0.924973
$17$	−99.2995	−1.41669	−0.708343	−	0.705868i	$-0.750557\pi$
$17$	−99.2995	−1.41669	−0.708343	+	0.705868i	$0.750557\pi$
$18$	−4.04268	−0.0529372
$19$	14.5342	0.175494	0.0877468	−	0.996143i	$-0.472033\pi$
$19$	14.5342	0.175494	0.0877468	+	0.996143i	$0.472033\pi$
$20$	−44.2411	−0.494631
$21$	−2.93720	−0.0305214
$22$	4.94106	0.0478835
$23$	12.6819	0.114972	0.0574859	−	0.998346i	$-0.481692\pi$
$23$	12.6819	0.114972	0.0574859	+	0.998346i	$0.481692\pi$
$24$	−21.2891	−0.181067
$25$	−92.8145	−0.742516
$26$	15.1330	0.114147
$27$	−27.0000	−0.192450
$28$	−7.63499	−0.0515313
$29$	−48.9373	−0.313359	−0.156680	−	0.987649i	$-0.550079\pi$
$29$	−48.9373	−0.313359	−0.156680	+	0.987649i	$0.550079\pi$
$30$	7.64502	0.0465261
$31$	−61.5945	−0.356861	−0.178431	−	0.983952i	$-0.557102\pi$
$31$	−61.5945	−0.356861	−0.178431	+	0.983952i	$0.557102\pi$
$32$	−83.3620	−0.460514
$33$	33.0000	0.174078
$34$	44.6040	0.224986
$35$	5.55447	0.0268250
$36$	−70.1841	−0.324926
$37$	391.538	1.73969	0.869843	−	0.493328i	$-0.164220\pi$
$37$	391.538	1.73969	0.869843	+	0.493328i	$0.164220\pi$
$38$	−6.52858	−0.0278704
$39$	101.069	0.414974
$40$	40.2593	0.159139
$41$	297.916	1.13480	0.567398	−	0.823444i	$-0.307950\pi$
$41$	297.916	1.13480	0.567398	+	0.823444i	$0.307950\pi$
$42$	1.31935	0.00484715
$43$	284.314	1.00831	0.504157	−	0.863612i	$-0.331803\pi$
$43$	284.314	1.00831	0.504157	+	0.863612i	$0.331803\pi$
$44$	85.7805	0.293907
$45$	51.0590	0.169143
$46$	−5.69652	−0.0182588
$47$	−37.9658	−0.117827	−0.0589136	−	0.998263i	$-0.518764\pi$
$47$	−37.9658	−0.117827	−0.0589136	+	0.998263i	$0.518764\pi$
$48$	−177.595	−0.534033
$49$	−342.041	−0.997205
$50$	41.6911	0.117920
$51$	297.898	0.817924
$52$	262.720	0.700629
$53$	−265.705	−0.688631	−0.344315	−	0.938854i	$-0.611889\pi$
$53$	−265.705	−0.688631	−0.344315	+	0.938854i	$0.611889\pi$
$54$	12.1280	0.0305633
$55$	−62.4055	−0.152996
$56$	6.94781	0.0165793
$57$	−43.6026	−0.101321
$58$	21.9820	0.0497651
$59$	−438.213	−0.966957	−0.483478	−	0.875356i	$-0.660627\pi$
$59$	−438.213	−0.966957	−0.483478	+	0.875356i	$0.660627\pi$
$60$	132.723	0.285575
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	27.6675	0.0566737
$63$	8.81160	0.0176215
$64$	−436.141	−0.851838
$65$	−191.129	−0.364718
$66$	−14.8232	−0.0276455
$67$	−312.761	−0.570296	−0.285148	−	0.958483i	$-0.592043\pi$
$67$	−312.761	−0.570296	−0.285148	+	0.958483i	$0.592043\pi$
$68$	774.360	1.38096
$69$	−38.0456	−0.0663790
$70$	−2.49499	−0.00426013
$71$	−926.140	−1.54807	−0.774033	−	0.633146i	$-0.781763\pi$
$71$	−926.140	−1.54807	−0.774033	+	0.633146i	$0.781763\pi$
$72$	63.8672	0.104539
$73$	−1164.51	−1.86706	−0.933532	−	0.358494i	$-0.883290\pi$
$73$	−1164.51	−1.86706	−0.933532	+	0.358494i	$0.883290\pi$
$74$	−175.874	−0.276282
$75$	278.443	0.428692
$76$	−113.341	−0.171067
$77$	−10.7697	−0.0159393
$78$	−45.3989	−0.0659027
$79$	526.836	0.750299	0.375150	−	0.926964i	$-0.377591\pi$
$79$	526.836	0.750299	0.375150	+	0.926964i	$0.377591\pi$
$80$	335.845	0.469358
$81$	81.0000	0.111111
$82$	−133.820	−0.180219
$83$	−528.876	−0.699418	−0.349709	−	0.936858i	$-0.613719\pi$
$83$	−528.876	−0.699418	−0.349709	+	0.936858i	$0.613719\pi$
$84$	22.9050	0.0297516
$85$	−563.348	−0.718867
$86$	−127.710	−0.160132
$87$	146.812	0.180918
$88$	−78.0599	−0.0945593
$89$	1170.87	1.39452	0.697259	−	0.716819i	$-0.254403\pi$
$89$	1170.87	1.39452	0.697259	+	0.716819i	$0.254403\pi$
$90$	−22.9351	−0.0268619
$91$	−32.9844	−0.0379968
$92$	−98.8961	−0.112072
$93$	184.784	0.206034
$94$	17.0537	0.0187123
$95$	82.4559	0.0890505
$96$	250.086	0.265878
$97$	−1491.87	−1.56162	−0.780808	−	0.624771i	$-0.785192\pi$
$97$	−1491.87	−1.56162	−0.780808	+	0.624771i	$0.785192\pi$
$98$	153.641	0.158368
$99$	−99.0000	−0.100504
$100$	723.789	0.723789
$101$	263.430	0.259528	0.129764	−	0.991545i	$-0.458578\pi$
$101$	263.430	0.259528	0.129764	+	0.991545i	$0.458578\pi$
$102$	−133.812	−0.129896
$103$	542.940	0.519393	0.259696	−	0.965690i	$-0.416377\pi$
$103$	542.940	0.519393	0.259696	+	0.965690i	$0.416377\pi$
$104$	−239.074	−0.225415
$105$	−16.6634	−0.0154874
$106$	119.351	0.109363
$107$	−1831.11	−1.65439	−0.827197	−	0.561912i	$-0.810066\pi$
$107$	−1831.11	−1.65439	−0.827197	+	0.561912i	$0.810066\pi$
$108$	210.552	0.187596
$109$	813.247	0.714632	0.357316	−	0.933983i	$-0.383692\pi$
$109$	813.247	0.714632	0.357316	+	0.933983i	$0.383692\pi$
$110$	28.0317	0.0242975
$111$	−1174.61	−1.00441
$112$	57.9590	0.0488983
$113$	1555.48	1.29493	0.647465	−	0.762096i	$-0.275829\pi$
$113$	1555.48	1.29493	0.647465	+	0.762096i	$0.275829\pi$
$114$	19.5857	0.0160910
$115$	71.9470	0.0583400
$116$	381.624	0.305456
$117$	−303.207	−0.239586
$118$	196.839	0.153564
$119$	−97.2208	−0.0748926
$120$	−120.778	−0.0918788
$121$	121.000	0.0909091
$122$	27.4004	0.0203337
$123$	−893.748	−0.655175
$124$	480.328	0.347861
$125$	−1235.71	−0.884203
$126$	−3.95806	−0.00279850
$127$	−257.102	−0.179639	−0.0898193	−	0.995958i	$-0.528629\pi$
$127$	−257.102	−0.179639	−0.0898193	+	0.995958i	$0.528629\pi$
$128$	862.804	0.595796
$129$	−852.943	−0.582150
$130$	85.8528	0.0579214
$131$	−879.531	−0.586603	−0.293301	−	0.956020i	$-0.594754\pi$
$131$	−879.531	−0.586603	−0.293301	+	0.956020i	$0.594754\pi$
$132$	−257.342	−0.169687
$133$	14.2300	0.00927740
$134$	140.488	0.0905696
$135$	−153.177	−0.0976547
$136$	−704.665	−0.444298
$137$	2354.67	1.46842	0.734208	−	0.678925i	$-0.237554\pi$
$137$	2354.67	1.46842	0.734208	+	0.678925i	$0.237554\pi$
$138$	17.0896	0.0105417
$139$	−1428.63	−0.871763	−0.435882	−	0.900004i	$-0.643563\pi$
$139$	−1428.63	−0.871763	−0.435882	+	0.900004i	$0.643563\pi$
$140$	−43.3150	−0.0261485
$141$	113.897	0.0680276
$142$	416.010	0.245851
$143$	370.587	0.216713
$144$	532.784	0.308324
$145$	−277.632	−0.159008
$146$	523.083	0.296511
$147$	1026.12	0.575737
$148$	−3053.30	−1.69581
$149$	960.163	0.527917	0.263958	−	0.964534i	$-0.414972\pi$
$149$	960.163	0.527917	0.263958	+	0.964534i	$0.414972\pi$
$150$	−125.073	−0.0680812
$151$	−834.577	−0.449781	−0.224891	−	0.974384i	$-0.572202\pi$
$151$	−834.577	−0.449781	−0.224891	+	0.974384i	$0.572202\pi$
$152$	103.140	0.0550379
$153$	−893.695	−0.472229
$154$	4.83762	0.00253134
$155$	−349.440	−0.181082
$156$	−788.160	−0.404508
$157$	1364.31	0.693527	0.346763	−	0.937953i	$-0.387281\pi$
$157$	1364.31	0.693527	0.346763	+	0.937953i	$0.387281\pi$
$158$	−236.648	−0.119156
$159$	797.116	0.397581
$160$	−472.931	−0.233678
$161$	12.4164	0.00607794
$162$	−36.3841	−0.0176457
$163$	313.623	0.150705	0.0753523	−	0.997157i	$-0.475992\pi$
$163$	313.623	0.150705	0.0753523	+	0.997157i	$0.475992\pi$
$164$	−2323.22	−1.10618
$165$	187.216	0.0883320
$166$	237.564	0.111076
$167$	3718.15	1.72287	0.861434	−	0.507869i	$-0.169567\pi$
$167$	3718.15	1.72287	0.861434	+	0.507869i	$0.169567\pi$
$168$	−20.8434	−0.00957205
$169$	−1062.00	−0.483389
$170$	253.049	0.114164
$171$	130.808	0.0584978
$172$	−2217.15	−0.982883
$173$	1798.36	0.790328	0.395164	−	0.918611i	$-0.370688\pi$
$173$	1798.36	0.790328	0.395164	+	0.918611i	$0.370688\pi$
$174$	−65.9459	−0.0287319
$175$	−90.8716	−0.0392528
$176$	−651.181	−0.278890
$177$	1314.64	0.558273
$178$	−525.940	−0.221466
$179$	3457.64	1.44378	0.721888	−	0.692010i	$-0.243274\pi$
$179$	3457.64	1.44378	0.721888	+	0.692010i	$0.243274\pi$
$180$	−398.170	−0.164877
$181$	2210.69	0.907842	0.453921	−	0.891042i	$-0.350025\pi$
$181$	2210.69	0.907842	0.453921	+	0.891042i	$0.350025\pi$
$182$	14.8162	0.00603433
$183$	183.000	0.0739221
$184$	89.9950	0.0360572
$185$	2221.28	0.882767
$186$	−83.0024	−0.0327206
$187$	1092.29	0.427147
$188$	296.066	0.114856
$189$	−26.4348	−0.0101738
$190$	−37.0381	−0.0141422
$191$	−2002.34	−0.758556	−0.379278	−	0.925283i	$-0.623828\pi$
$191$	−2002.34	−0.758556	−0.379278	+	0.925283i	$0.623828\pi$
$192$	1308.42	0.491809
$193$	−878.621	−0.327692	−0.163846	−	0.986486i	$-0.552390\pi$
$193$	−878.621	−0.327692	−0.163846	+	0.986486i	$0.552390\pi$
$194$	670.130	0.248003
$195$	573.388	0.210570
$196$	2667.32	0.972055
$197$	442.287	0.159958	0.0799788	−	0.996797i	$-0.474515\pi$
$197$	442.287	0.159958	0.0799788	+	0.996797i	$0.474515\pi$
$198$	44.4695	0.0159612
$199$	4251.39	1.51444	0.757218	−	0.653162i	$-0.226558\pi$
$199$	4251.39	1.51444	0.757218	+	0.653162i	$0.226558\pi$
$200$	−658.645	−0.232866
$201$	938.283	0.329261
$202$	−118.329	−0.0412160
$203$	−47.9128	−0.0165656
$204$	−2323.08	−0.797295
$205$	1690.14	0.575828
$206$	−243.882	−0.0824856
$207$	114.137	0.0383239
$208$	−1994.37	−0.664830
$209$	−159.876	−0.0529133
$210$	7.48498	0.00245958
$211$	−478.923	−0.156258	−0.0781290	−	0.996943i	$-0.524895\pi$
$211$	−478.923	−0.156258	−0.0781290	+	0.996943i	$0.524895\pi$
$212$	2072.03	0.671263
$213$	2778.42	0.893776
$214$	822.511	0.262737
$215$	1612.98	0.511648
$216$	−191.602	−0.0603558
$217$	−60.3052	−0.0188653
$218$	−365.300	−0.113492
$219$	3493.53	1.07795
$220$	486.652	0.149137
$221$	3345.37	1.01825
$222$	527.621	0.159512
$223$	3759.87	1.12906	0.564528	−	0.825414i	$-0.309058\pi$
$223$	3759.87	1.12906	0.564528	+	0.825414i	$0.309058\pi$
$224$	−81.6169	−0.0243449
$225$	−835.330	−0.247505
$226$	−698.700	−0.205650
$227$	4051.29	1.18455	0.592277	−	0.805735i	$-0.298229\pi$
$227$	4051.29	1.18455	0.592277	+	0.805735i	$0.298229\pi$
$228$	340.023	0.0987658
$229$	−4621.89	−1.33372	−0.666862	−	0.745181i	$-0.732363\pi$
$229$	−4621.89	−1.33372	−0.666862	+	0.745181i	$0.732363\pi$
$230$	−32.3177	−0.00926506
$231$	32.3092	0.00920255
$232$	−347.276	−0.0982751
$233$	−3605.31	−1.01370	−0.506850	−	0.862035i	$-0.669190\pi$
$233$	−3605.31	−1.01370	−0.506850	+	0.862035i	$0.669190\pi$
$234$	136.197	0.0380490
$235$	−215.389	−0.0597890
$236$	3417.28	0.942569
$237$	−1580.51	−0.433185
$238$	43.6703	0.0118938
$239$	1432.89	0.387806	0.193903	−	0.981021i	$-0.437885\pi$
$239$	1432.89	0.387806	0.193903	+	0.981021i	$0.437885\pi$
$240$	−1007.54	−0.270984
$241$	−1238.93	−0.331146	−0.165573	−	0.986198i	$-0.552947\pi$
$241$	−1238.93	−0.331146	−0.165573	+	0.986198i	$0.552947\pi$
$242$	−54.3516	−0.0144374
$243$	−243.000	−0.0641500
$244$	475.692	0.124808
$245$	−1940.48	−0.506011
$246$	401.460	0.104049
$247$	−489.653	−0.126137
$248$	−437.097	−0.111918
$249$	1586.63	0.403809
$250$	555.065	0.140422
$251$	−6918.49	−1.73980	−0.869902	−	0.493224i	$-0.835818\pi$
$251$	−6918.49	−1.73980	−0.869902	+	0.493224i	$0.835818\pi$
$252$	−68.7149	−0.0171771
$253$	−139.500	−0.0346653
$254$	115.487	0.0285287
$255$	1690.05	0.415038
$256$	3101.57	0.757219
$257$	5830.66	1.41520	0.707601	−	0.706613i	$-0.249778\pi$
$257$	5830.66	1.41520	0.707601	+	0.706613i	$0.249778\pi$
$258$	383.131	0.0924522
$259$	383.342	0.0919679
$260$	1490.47	0.355519
$261$	−440.435	−0.104453
$262$	395.074	0.0931593
$263$	−3033.48	−0.711225	−0.355613	−	0.934633i	$-0.615728\pi$
$263$	−3033.48	−0.711225	−0.355613	+	0.934633i	$0.615728\pi$
$264$	234.180	0.0545938
$265$	−1507.41	−0.349431
$266$	−6.39191	−0.00147336
$267$	−3512.61	−0.805125
$268$	2438.98	0.555913
$269$	8109.46	1.83808	0.919038	−	0.394169i	$-0.128968\pi$
$269$	8109.46	1.83808	0.919038	+	0.394169i	$0.128968\pi$
$270$	68.8052	0.0155087
$271$	6639.24	1.48821	0.744106	−	0.668062i	$-0.232876\pi$
$271$	6639.24	1.48821	0.744106	+	0.668062i	$0.232876\pi$
$272$	−5878.36	−1.31040
$273$	98.9533	0.0219375
$274$	−1057.69	−0.233201
$275$	1020.96	0.223877
$276$	296.688	0.0647048
$277$	−1873.16	−0.406308	−0.203154	−	0.979147i	$-0.565119\pi$
$277$	−1873.16	−0.406308	−0.203154	+	0.979147i	$0.565119\pi$
$278$	641.723	0.138446
$279$	−554.351	−0.118954
$280$	39.4165	0.00841281
$281$	254.127	0.0539499	0.0269750	−	0.999636i	$-0.491413\pi$
$281$	254.127	0.0539499	0.0269750	+	0.999636i	$0.491413\pi$
$282$	−51.1612	−0.0108036
$283$	2353.79	0.494410	0.247205	−	0.968963i	$-0.420488\pi$
$283$	2353.79	0.494410	0.247205	+	0.968963i	$0.420488\pi$
$284$	7222.26	1.50902
$285$	−247.368	−0.0514133
$286$	−166.463	−0.0344166
$287$	291.680	0.0599906
$288$	−750.258	−0.153505
$289$	4947.39	1.00700
$290$	124.709	0.0252522
$291$	4475.62	0.901599
$292$	9081.12	1.81997
$293$	−1476.57	−0.294409	−0.147205	−	0.989106i	$-0.547028\pi$
$293$	−1476.57	−0.294409	−0.147205	+	0.989106i	$0.547028\pi$
$294$	−460.922	−0.0914337
$295$	−2486.08	−0.490662
$296$	2778.49	0.545597
$297$	297.000	0.0580259
$298$	−431.292	−0.0838393
$299$	−427.248	−0.0826367
$300$	−2171.37	−0.417880
$301$	278.363	0.0533042
$302$	374.881	0.0714304
$303$	−790.291	−0.149838
$304$	860.400	0.162327
$305$	−346.067	−0.0649696
$306$	401.436	0.0749954
$307$	4371.52	0.812691	0.406345	−	0.913720i	$-0.366803\pi$
$307$	4371.52	0.812691	0.406345	+	0.913720i	$0.366803\pi$
$308$	83.9849	0.0155373
$309$	−1628.82	−0.299872
$310$	156.964	0.0287579
$311$	−1744.54	−0.318083	−0.159042	−	0.987272i	$-0.550840\pi$
$311$	−1744.54	−0.318083	−0.159042	+	0.987272i	$0.550840\pi$
$312$	717.222	0.130143
$313$	3572.30	0.645107	0.322554	−	0.946551i	$-0.395459\pi$
$313$	3572.30	0.645107	0.322554	+	0.946551i	$0.395459\pi$
$314$	−612.830	−0.110140
$315$	49.9902	0.00894168
$316$	−4108.39	−0.731376
$317$	1385.62	0.245503	0.122751	−	0.992437i	$-0.460828\pi$
$317$	1385.62	0.245503	0.122751	+	0.992437i	$0.460828\pi$
$318$	−358.054	−0.0631405
$319$	538.310	0.0944814
$320$	−2474.33	−0.432247
$321$	5493.33	0.955164
$322$	−5.57728	−0.000965247 0
$323$	−1443.24	−0.248619
$324$	−631.657	−0.108309
$325$	3126.89	0.533688
$326$	−140.875	−0.0239336
$327$	−2439.74	−0.412593
$328$	2114.12	0.355892
$329$	−37.1710	−0.00622889
$330$	−84.0952	−0.0140281
$331$	6145.41	1.02049	0.510246	−	0.860029i	$-0.329555\pi$
$331$	6145.41	1.02049	0.510246	+	0.860029i	$0.329555\pi$
$332$	4124.30	0.681777
$333$	3523.84	0.579896
$334$	−1670.14	−0.273611
$335$	−1774.36	−0.289385
$336$	−173.877	−0.0282315
$337$	9580.35	1.54859	0.774295	−	0.632824i	$-0.218105\pi$
$337$	9580.35	1.54859	0.774295	+	0.632824i	$0.218105\pi$
$338$	477.039	0.0767677
$339$	−4666.43	−0.747628
$340$	4393.12	0.700737
$341$	677.540	0.107598
$342$	−58.7572	−0.00929013
$343$	−670.701	−0.105582
$344$	2017.60	0.316225
$345$	−215.841	−0.0336826
$346$	−807.800	−0.125513
$347$	1298.01	0.200809	0.100405	−	0.994947i	$-0.467986\pi$
$347$	1298.01	0.200809	0.100405	+	0.994947i	$0.467986\pi$
$348$	−1144.87	−0.176355
$349$	3120.01	0.478540	0.239270	−	0.970953i	$-0.423092\pi$
$349$	3120.01	0.478540	0.239270	+	0.970953i	$0.423092\pi$
$350$	40.8183	0.00623380
$351$	909.622	0.138325
$352$	916.981	0.138850
$353$	244.835	0.0369157	0.0184579	−	0.999830i	$-0.494124\pi$
$353$	244.835	0.0369157	0.0184579	+	0.999830i	$0.494124\pi$
$354$	−590.518	−0.0886602
$355$	−5254.20	−0.785533
$356$	−9130.72	−1.35935
$357$	291.662	0.0432392
$358$	−1553.13	−0.229288
$359$	2228.58	0.327632	0.163816	−	0.986491i	$-0.447620\pi$
$359$	2228.58	0.327632	0.163816	+	0.986491i	$0.447620\pi$
$360$	362.333	0.0530462
$361$	−6647.76	−0.969202
$362$	−993.014	−0.144176
$363$	−363.000	−0.0524864
$364$	257.220	0.0370385
$365$	−6606.53	−0.947402
$366$	−82.2012	−0.0117397
$367$	−234.839	−0.0334019	−0.0167010	−	0.999861i	$-0.505316\pi$
$367$	−234.839	−0.0334019	−0.0167010	+	0.999861i	$0.505316\pi$
$368$	750.744	0.106346
$369$	2681.24	0.378265
$370$	−997.771	−0.140194
$371$	−260.143	−0.0364042
$372$	−1440.99	−0.200838
$373$	12840.4	1.78244	0.891220	−	0.453572i	$-0.149851\pi$
$373$	12840.4	1.78244	0.891220	+	0.453572i	$0.149851\pi$
$374$	−490.644	−0.0678359
$375$	3707.13	0.510495
$376$	−269.419	−0.0369527
$377$	1648.68	0.225229
$378$	11.8742	0.00161572
$379$	3747.91	0.507960	0.253980	−	0.967209i	$-0.418260\pi$
$379$	3747.91	0.507960	0.253980	+	0.967209i	$0.418260\pi$
$380$	−643.010	−0.0868045
$381$	771.306	0.103714
$382$	899.425	0.120468
$383$	3659.24	0.488195	0.244097	−	0.969751i	$-0.421508\pi$
$383$	3659.24	0.488195	0.244097	+	0.969751i	$0.421508\pi$
$384$	−2588.41	−0.343983
$385$	−61.0991	−0.00808805
$386$	394.665	0.0520413
$387$	2558.83	0.336105
$388$	11634.0	1.52223
$389$	−11391.3	−1.48474	−0.742368	−	0.669993i	$-0.766297\pi$
$389$	−11391.3	−1.48474	−0.742368	+	0.669993i	$0.766297\pi$
$390$	−257.558	−0.0334410
$391$	−1259.30	−0.162879
$392$	−2427.25	−0.312741
$393$	2638.59	0.338675
$394$	−198.669	−0.0254031
$395$	2988.86	0.380723
$396$	772.025	0.0979690
$397$	−12196.5	−1.54188	−0.770938	−	0.636910i	$-0.780212\pi$
$397$	−12196.5	−1.54188	−0.770938	+	0.636910i	$0.780212\pi$
$398$	−1909.67	−0.240510
$399$	−42.6899	−0.00535631
$400$	−5494.46	−0.686807
$401$	1333.23	0.166031	0.0830154	−	0.996548i	$-0.473545\pi$
$401$	1333.23	0.166031	0.0830154	+	0.996548i	$0.473545\pi$
$402$	−421.465	−0.0522904
$403$	2075.10	0.256497
$404$	−2054.29	−0.252982
$405$	459.531	0.0563810
$406$	21.5218	0.00263081
$407$	−4306.92	−0.524535
$408$	2113.99	0.256515
$409$	−6157.30	−0.744398	−0.372199	−	0.928153i	$-0.621396\pi$
$409$	−6157.30	−0.744398	−0.372199	+	0.928153i	$0.621396\pi$
$410$	−759.191	−0.0914482
$411$	−7064.01	−0.847790
$412$	−4233.97	−0.506293
$413$	−429.039	−0.0511178
$414$	−51.2687	−0.00608628
$415$	−3000.43	−0.354905
$416$	2808.44	0.330998
$417$	4285.90	0.503313
$418$	71.8144	0.00840324
$419$	−1863.59	−0.217285	−0.108642	−	0.994081i	$-0.534650\pi$
$419$	−1863.59	−0.217285	−0.108642	+	0.994081i	$0.534650\pi$
$420$	129.945	0.0150968
$421$	14357.7	1.66211	0.831057	−	0.556187i	$-0.187736\pi$
$421$	14357.7	1.66211	0.831057	+	0.556187i	$0.187736\pi$
$422$	215.126	0.0248156
$423$	−341.692	−0.0392758
$424$	−1885.54	−0.215967
$425$	9216.43	1.05191
$426$	−1248.03	−0.141942
$427$	−59.7231	−0.00676862
$428$	14279.4	1.61267
$429$	−1111.76	−0.125120
$430$	−724.529	−0.0812556
$431$	8040.26	0.898574	0.449287	−	0.893387i	$-0.351678\pi$
$431$	8040.26	0.898574	0.449287	+	0.893387i	$0.351678\pi$
$432$	−1598.35	−0.178011
$433$	10096.7	1.12060	0.560299	−	0.828291i	$-0.310686\pi$
$433$	10096.7	1.12060	0.560299	+	0.828291i	$0.310686\pi$
$434$	27.0883	0.00299604
$435$	832.897	0.0918031
$436$	−6341.89	−0.696609
$437$	184.321	0.0201768
$438$	−1569.25	−0.171191
$439$	−4136.25	−0.449687	−0.224843	−	0.974395i	$-0.572187\pi$
$439$	−4136.25	−0.449687	−0.224843	+	0.974395i	$0.572187\pi$
$440$	−442.852	−0.0479821
$441$	−3078.37	−0.332402
$442$	−1502.70	−0.161710
$443$	−4691.25	−0.503133	−0.251566	−	0.967840i	$-0.580946\pi$
$443$	−4691.25	−0.503133	−0.251566	+	0.967840i	$0.580946\pi$
$444$	9159.91	0.979076
$445$	6642.62	0.707619
$446$	−1688.88	−0.179307
$447$	−2880.49	−0.304793
$448$	−427.011	−0.0450321
$449$	−1523.66	−0.160147	−0.0800734	−	0.996789i	$-0.525515\pi$
$449$	−1523.66	−0.160147	−0.0800734	+	0.996789i	$0.525515\pi$
$450$	375.220	0.0393067
$451$	−3277.08	−0.342154
$452$	−12130.0	−1.26227
$453$	2503.73	0.259681
$454$	−1819.79	−0.188121
$455$	−187.128	−0.0192807
$456$	−309.420	−0.0317761
$457$	6488.94	0.664201	0.332100	−	0.943244i	$-0.392243\pi$
$457$	6488.94	0.664201	0.332100	+	0.943244i	$0.392243\pi$
$458$	2076.09	0.211811
$459$	2681.09	0.272641
$460$	−561.060	−0.0568686
$461$	−14619.4	−1.47699	−0.738497	−	0.674257i	$-0.764464\pi$
$461$	−14619.4	−1.47699	−0.738497	+	0.674257i	$0.764464\pi$
$462$	−14.5129	−0.00146147
$463$	15204.4	1.52616	0.763078	−	0.646306i	$-0.223687\pi$
$463$	15204.4	1.52616	0.763078	+	0.646306i	$0.223687\pi$
$464$	−2897.00	−0.289849
$465$	1048.32	0.104548
$466$	1619.46	0.160987
$467$	−5385.73	−0.533666	−0.266833	−	0.963743i	$-0.585977\pi$
$467$	−5385.73	−0.533666	−0.266833	+	0.963743i	$0.585977\pi$
$468$	2364.48	0.233543
$469$	−306.214	−0.0301485
$470$	96.7497	0.00949518
$471$	−4092.93	−0.400408
$472$	−3109.72	−0.303255
$473$	−3127.46	−0.304018
$474$	709.943	0.0687949
$475$	−1348.99	−0.130307
$476$	758.150	0.0730037
$477$	−2391.35	−0.229544
$478$	−643.633	−0.0615881
$479$	15663.6	1.49413	0.747064	−	0.664752i	$-0.231463\pi$
$479$	15663.6	1.49413	0.747064	+	0.664752i	$0.231463\pi$
$480$	1418.79	0.134914
$481$	−13190.8	−1.25041
$482$	556.509	0.0525898
$483$	−37.2491	−0.00350910
$484$	−943.586	−0.0886163
$485$	−8463.73	−0.792409
$486$	109.152	0.0101878
$487$	12176.2	1.13297	0.566483	−	0.824073i	$-0.308304\pi$
$487$	12176.2	1.13297	0.566483	+	0.824073i	$0.308304\pi$
$488$	−432.878	−0.0401546
$489$	−940.870	−0.0870094
$490$	871.638	0.0803604
$491$	−14171.9	−1.30259	−0.651293	−	0.758826i	$-0.725773\pi$
$491$	−14171.9	−1.30259	−0.651293	+	0.758826i	$0.725773\pi$
$492$	6969.65	0.638651
$493$	4859.44	0.443932
$494$	219.946	0.0200320
$495$	−561.649	−0.0509985
$496$	−3646.29	−0.330087
$497$	−906.753	−0.0818379
$498$	−712.692	−0.0641295
$499$	−15576.4	−1.39738	−0.698692	−	0.715423i	$-0.746234\pi$
$499$	−15576.4	−1.39738	−0.698692	+	0.715423i	$0.746234\pi$
$500$	9636.36	0.861902
$501$	−11154.4	−0.994698
$502$	3107.69	0.276301
$503$	13282.6	1.17742	0.588709	−	0.808345i	$-0.299636\pi$
$503$	13282.6	1.17742	0.588709	+	0.808345i	$0.299636\pi$
$504$	62.5303	0.00552643
$505$	1494.50	0.131692
$506$	62.6618	0.00550525
$507$	3186.01	0.279085
$508$	2004.94	0.175108
$509$	−1914.58	−0.166724	−0.0833618	−	0.996519i	$-0.526566\pi$
$509$	−1914.58	−0.166724	−0.0833618	+	0.996519i	$0.526566\pi$
$510$	−759.146	−0.0659129
$511$	−1140.13	−0.0987016
$512$	−8295.62	−0.716051
$513$	−392.424	−0.0337737
$514$	−2619.06	−0.224750
$515$	3080.22	0.263555
$516$	6651.44	0.567468
$517$	417.624	0.0355263
$518$	−172.192	−0.0146056
$519$	−5395.08	−0.456296
$520$	−1356.32	−0.114382
$521$	21702.6	1.82497	0.912483	−	0.409115i	$-0.134163\pi$
$521$	21702.6	1.82497	0.912483	+	0.409115i	$0.134163\pi$
$522$	197.838	0.0165884
$523$	−9230.66	−0.771756	−0.385878	−	0.922550i	$-0.626101\pi$
$523$	−9230.66	−0.771756	−0.385878	+	0.922550i	$0.626101\pi$
$524$	6858.79	0.571808
$525$	272.615	0.0226626
$526$	1362.60	0.112951
$527$	6116.31	0.505561
$528$	1953.54	0.161017
$529$	−12006.2	−0.986781
$530$	677.107	0.0554937
$531$	−3943.91	−0.322319
$532$	−110.969	−0.00904341
$533$	−10036.7	−0.815643
$534$	1577.82	0.127863
$535$	−10388.3	−0.839487
$536$	−2219.46	−0.178855
$537$	−10372.9	−0.833565
$538$	−3642.66	−0.291908
$539$	3762.46	0.300669
$540$	1194.51	0.0951918
$541$	11232.8	0.892674	0.446337	−	0.894865i	$-0.352728\pi$
$541$	11232.8	0.892674	0.446337	+	0.894865i	$0.352728\pi$
$542$	−2982.26	−0.236345
$543$	−6632.08	−0.524143
$544$	8277.80	0.652404
$545$	4613.73	0.362625
$546$	−44.4485	−0.00348392
$547$	−3096.15	−0.242014	−0.121007	−	0.992652i	$-0.538612\pi$
$547$	−3096.15	−0.242014	−0.121007	+	0.992652i	$0.538612\pi$
$548$	−18362.3	−1.43138
$549$	−549.000	−0.0426790
$550$	−458.602	−0.0355543
$551$	−711.265	−0.0549925
$552$	−269.985	−0.0208176
$553$	515.807	0.0396643
$554$	841.399	0.0645264
$555$	−6663.85	−0.509666
$556$	11140.8	0.849776
$557$	−17189.4	−1.30761	−0.653806	−	0.756662i	$-0.726829\pi$
$557$	−17189.4	−1.30761	−0.653806	+	0.756662i	$0.726829\pi$
$558$	249.007	0.0188912
$559$	−9578.46	−0.724733
$560$	328.815	0.0248124
$561$	−3276.88	−0.246613
$562$	−114.150	−0.00856787
$563$	17148.3	1.28368	0.641841	−	0.766837i	$-0.278171\pi$
$563$	17148.3	1.28368	0.641841	+	0.766837i	$0.278171\pi$
$564$	−888.198	−0.0663119
$565$	8824.58	0.657084
$566$	−1057.29	−0.0785180
$567$	79.3044	0.00587385
$568$	−6572.22	−0.485501
$569$	1406.20	0.103605	0.0518024	−	0.998657i	$-0.483503\pi$
$569$	1406.20	0.103605	0.0518024	+	0.998657i	$0.483503\pi$
$570$	111.114	0.00816503
$571$	−3009.53	−0.220569	−0.110284	−	0.993900i	$-0.535176\pi$
$571$	−3009.53	−0.220569	−0.110284	+	0.993900i	$0.535176\pi$
$572$	−2889.92	−0.211248
$573$	6007.02	0.437953
$574$	−131.019	−0.00952720
$575$	−1177.06	−0.0853684
$576$	−3925.27	−0.283946
$577$	9540.47	0.688345	0.344173	−	0.938906i	$-0.388159\pi$
$577$	9540.47	0.688345	0.344173	+	0.938906i	$0.388159\pi$
$578$	−2222.30	−0.159923
$579$	2635.86	0.189193
$580$	2165.04	0.154997
$581$	−517.805	−0.0369744
$582$	−2010.39	−0.143184
$583$	2922.76	0.207630
$584$	−8263.79	−0.585544
$585$	−1720.16	−0.121573
$586$	663.254	0.0467556
$587$	26054.1	1.83197	0.915986	−	0.401210i	$-0.131410\pi$
$587$	26054.1	1.83197	0.915986	+	0.401210i	$0.131410\pi$
$588$	−8001.95	−0.561216
$589$	−895.228	−0.0626269
$590$	1116.71	0.0779227
$591$	−1326.86	−0.0923515
$592$	23178.4	1.60916
$593$	−18593.4	−1.28759	−0.643793	−	0.765200i	$-0.722640\pi$
$593$	−18593.4	−1.28759	−0.643793	+	0.765200i	$0.722640\pi$
$594$	−133.409	−0.00921518
$595$	−551.556	−0.0380027
$596$	−7487.57	−0.514602
$597$	−12754.2	−0.874360
$598$	191.914	0.0131237
$599$	−5547.67	−0.378417	−0.189208	−	0.981937i	$-0.560592\pi$
$599$	−5547.67	−0.378417	−0.189208	+	0.981937i	$0.560592\pi$
$600$	1975.93	0.134445
$601$	−11715.8	−0.795169	−0.397584	−	0.917566i	$-0.630151\pi$
$601$	−11715.8	−0.795169	−0.397584	+	0.917566i	$0.630151\pi$
$602$	−125.037	−0.00846532
$603$	−2814.85	−0.190099
$604$	6508.23	0.438437
$605$	686.460	0.0461299
$606$	354.988	0.0237961
$607$	21661.1	1.44843	0.724214	−	0.689575i	$-0.242203\pi$
$607$	21661.1	1.44843	0.724214	+	0.689575i	$0.242203\pi$
$608$	−1211.60	−0.0808172
$609$	143.739	0.00956417
$610$	155.449	0.0103179
$611$	1279.06	0.0846891
$612$	6969.24	0.460319
$613$	23674.2	1.55986	0.779929	−	0.625868i	$-0.215255\pi$
$613$	23674.2	1.55986	0.779929	+	0.625868i	$0.215255\pi$
$614$	−1963.63	−0.129065
$615$	−5070.43	−0.332455
$616$	−76.4259	−0.00499884
$617$	−17102.1	−1.11589	−0.557946	−	0.829877i	$-0.688410\pi$
$617$	−17102.1	−1.11589	−0.557946	+	0.829877i	$0.688410\pi$
$618$	731.645	0.0476231
$619$	3379.24	0.219423	0.109712	−	0.993963i	$-0.465007\pi$
$619$	3379.24	0.219423	0.109712	+	0.993963i	$0.465007\pi$
$620$	2725.01	0.176515
$621$	−342.410	−0.0221263
$622$	783.626	0.0505153
$623$	1146.36	0.0737207
$624$	5983.11	0.383840
$625$	4591.34	0.293846
$626$	−1604.63	−0.102451
$627$	479.629	0.0305495
$628$	−10639.2	−0.676035
$629$	−38879.5	−2.46459
$630$	−22.4549	−0.00142004
$631$	21308.2	1.34432	0.672160	−	0.740406i	$-0.265367\pi$
$631$	21308.2	1.34432	0.672160	+	0.740406i	$0.265367\pi$
$632$	3738.61	0.235307
$633$	1436.77	0.0902155
$634$	−622.405	−0.0389887
$635$	−1458.60	−0.0911538
$636$	−6216.09	−0.387554
$637$	11523.3	0.716748
$638$	−241.802	−0.0150047
$639$	−8335.26	−0.516022
$640$	4894.89	0.302324
$641$	4652.39	0.286674	0.143337	−	0.989674i	$-0.454217\pi$
$641$	4652.39	0.286674	0.143337	+	0.989674i	$0.454217\pi$
$642$	−2467.53	−0.151691
$643$	6187.10	0.379464	0.189732	−	0.981836i	$-0.439238\pi$
$643$	6187.10	0.379464	0.189732	+	0.981836i	$0.439238\pi$
$644$	−96.8258	−0.00592465
$645$	−4838.94	−0.295400
$646$	648.284	0.0394836
$647$	−1554.96	−0.0944852	−0.0472426	−	0.998883i	$-0.515043\pi$
$647$	−1554.96	−0.0944852	−0.0472426	+	0.998883i	$0.515043\pi$
$648$	574.805	0.0348464
$649$	4820.34	0.291548
$650$	−1404.56	−0.0847559
$651$	180.915	0.0108919
$652$	−2445.71	−0.146904
$653$	−14933.2	−0.894918	−0.447459	−	0.894304i	$-0.647671\pi$
$653$	−14933.2	−0.894918	−0.447459	+	0.894304i	$0.647671\pi$
$654$	1095.90	0.0655246
$655$	−4989.78	−0.297659
$656$	17636.1	1.04966
$657$	−10480.6	−0.622355
$658$	16.6967	0.000989220 0
$659$	−10393.9	−0.614397	−0.307198	−	0.951645i	$-0.599392\pi$
$659$	−10393.9	−0.614397	−0.307198	+	0.951645i	$0.599392\pi$
$660$	−1459.96	−0.0861042
$661$	5615.46	0.330433	0.165216	−	0.986257i	$-0.447168\pi$
$661$	5615.46	0.330433	0.165216	+	0.986257i	$0.447168\pi$
$662$	−2760.44	−0.162066
$663$	−10036.1	−0.587888
$664$	−3753.09	−0.219350
$665$	80.7298	0.00470762
$666$	−1582.86	−0.0920941
$667$	−620.615	−0.0360275
$668$	−28995.0	−1.67942
$669$	−11279.6	−0.651861
$670$	797.021	0.0459576
$671$	671.000	0.0386046
$672$	244.851	0.0140555
$673$	−15575.2	−0.892094	−0.446047	−	0.895010i	$-0.647169\pi$
$673$	−15575.2	−0.892094	−0.446047	+	0.895010i	$0.647169\pi$
$674$	−4303.37	−0.245934
$675$	2505.99	0.142897
$676$	8281.76	0.471197
$677$	13848.3	0.786166	0.393083	−	0.919503i	$-0.371409\pi$
$677$	13848.3	0.786166	0.393083	+	0.919503i	$0.371409\pi$
$678$	2096.10	0.118732
$679$	−1460.64	−0.0825542
$680$	−3997.72	−0.225450
$681$	−12153.9	−0.683902
$682$	−304.342	−0.0170878
$683$	−24442.1	−1.36933	−0.684663	−	0.728860i	$-0.740051\pi$
$683$	−24442.1	−1.36933	−0.684663	+	0.728860i	$0.740051\pi$
$684$	−1020.07	−0.0570225
$685$	13358.6	0.745117
$686$	301.270	0.0167676
$687$	13865.7	0.770026
$688$	16830.9	0.932663
$689$	8951.53	0.494958
$690$	96.9530	0.00534919
$691$	10829.1	0.596175	0.298088	−	0.954539i	$-0.403651\pi$
$691$	10829.1	0.596175	0.298088	+	0.954539i	$0.403651\pi$
$692$	−14024.0	−0.770395
$693$	−96.9276	−0.00531309
$694$	−583.049	−0.0318908
$695$	−8104.96	−0.442358
$696$	1041.83	0.0567391
$697$	−29582.9	−1.60765
$698$	−1401.47	−0.0759977
$699$	10815.9	0.585259
$700$	708.638	0.0382628
$701$	−34575.5	−1.86291	−0.931455	−	0.363856i	$-0.881460\pi$
$701$	−34575.5	−1.86291	−0.931455	+	0.363856i	$0.881460\pi$
$702$	−408.590	−0.0219676
$703$	5690.69	0.305304
$704$	4797.55	0.256839
$705$	646.166	0.0345192
$706$	−109.977	−0.00586264
$707$	257.916	0.0137198
$708$	−10251.9	−0.544192
$709$	16774.0	0.888520	0.444260	−	0.895898i	$-0.353467\pi$
$709$	16774.0	0.888520	0.444260	+	0.895898i	$0.353467\pi$
$710$	2360.12	0.124752
$711$	4741.52	0.250100
$712$	8308.92	0.437346
$713$	−781.133	−0.0410290
$714$	−131.011	−0.00686689
$715$	2102.42	0.109967
$716$	−26963.5	−1.40736
$717$	−4298.66	−0.223900
$718$	−1001.05	−0.0520317
$719$	24400.2	1.26561	0.632805	−	0.774311i	$-0.281903\pi$
$719$	24400.2	1.26561	0.632805	+	0.774311i	$0.281903\pi$
$720$	3022.61	0.156453
$721$	531.574	0.0274575
$722$	2986.09	0.153920
$723$	3716.78	0.191187
$724$	−17239.5	−0.884946
$725$	4542.09	0.232674
$726$	163.055	0.00833545
$727$	27451.8	1.40046	0.700228	−	0.713920i	$-0.253082\pi$
$727$	27451.8	1.40046	0.700228	+	0.713920i	$0.253082\pi$
$728$	−234.069	−0.0119165
$729$	729.000	0.0370370
$730$	2967.57	0.150458
$731$	−28232.3	−1.42846
$732$	−1427.08	−0.0720577
$733$	1014.81	0.0511363	0.0255681	−	0.999673i	$-0.491861\pi$
$733$	1014.81	0.0511363	0.0255681	+	0.999673i	$0.491861\pi$
$734$	105.487	0.00530461
$735$	5821.44	0.292145
$736$	−1057.18	−0.0529461
$737$	3440.37	0.171951
$738$	−1204.38	−0.0600729
$739$	26047.9	1.29660	0.648299	−	0.761386i	$-0.275481\pi$
$739$	26047.9	1.29660	0.648299	+	0.761386i	$0.275481\pi$
$740$	−17322.1	−0.860503
$741$	1468.96	0.0728253
$742$	116.853	0.00578141
$743$	−4333.90	−0.213991	−0.106996	−	0.994260i	$-0.534123\pi$
$743$	−4333.90	−0.213991	−0.106996	+	0.994260i	$0.534123\pi$
$744$	1311.29	0.0646159
$745$	5447.22	0.267880
$746$	−5767.73	−0.283072
$747$	−4759.88	−0.233139
$748$	−8517.96	−0.416374
$749$	−1792.78	−0.0874589
$750$	−1665.20	−0.0810725
$751$	22566.2	1.09647	0.548236	−	0.836323i	$-0.315299\pi$
$751$	22566.2	1.09647	0.548236	+	0.836323i	$0.315299\pi$
$752$	−2247.51	−0.108987
$753$	20755.5	1.00448
$754$	−740.566	−0.0357690
$755$	−4734.75	−0.228232
$756$	206.145	0.00991721
$757$	−10707.0	−0.514072	−0.257036	−	0.966402i	$-0.582746\pi$
$757$	−10707.0	−0.514072	−0.257036	+	0.966402i	$0.582746\pi$
$758$	−1683.51	−0.0806700
$759$	418.501	0.0200140
$760$	585.136	0.0279278
$761$	26005.8	1.23877	0.619387	−	0.785085i	$-0.287381\pi$
$761$	26005.8	1.23877	0.619387	+	0.785085i	$0.287381\pi$
$762$	−346.461	−0.0164711
$763$	796.223	0.0377788
$764$	15614.7	0.739425
$765$	−5070.14	−0.239622
$766$	−1643.68	−0.0775310
$767$	14763.3	0.695007
$768$	−9304.70	−0.437180
$769$	12389.4	0.580979	0.290489	−	0.956878i	$-0.406182\pi$
$769$	12389.4	0.580979	0.290489	+	0.956878i	$0.406182\pi$
$770$	27.4449	0.00128448
$771$	−17492.0	−0.817067
$772$	6851.69	0.319427
$773$	−41121.9	−1.91339	−0.956696	−	0.291090i	$-0.905982\pi$
$773$	−41121.9	−1.91339	−0.956696	+	0.291090i	$0.905982\pi$
$774$	−1149.39	−0.0533773
$775$	5716.87	0.264975
$776$	−10586.9	−0.489750
$777$	−1150.02	−0.0530977
$778$	5116.83	0.235793
$779$	4329.97	0.199149
$780$	−4471.41	−0.205259
$781$	10187.5	0.466759
$782$	565.662	0.0258670
$783$	1321.31	0.0603060
$784$	−20248.3	−0.922388
$785$	7740.03	0.351915
$786$	−1185.22	−0.0537856
$787$	25690.7	1.16363	0.581814	−	0.813322i	$-0.302343\pi$
$787$	25690.7	1.16363	0.581814	+	0.813322i	$0.302343\pi$
$788$	−3449.05	−0.155923
$789$	9100.43	0.410626
$790$	−1342.56	−0.0604633
$791$	1522.92	0.0684559
$792$	−702.540	−0.0315198
$793$	2055.07	0.0920274
$794$	5478.51	0.244868
$795$	4522.22	0.201744
$796$	−33153.3	−1.47624
$797$	18805.3	0.835781	0.417890	−	0.908497i	$-0.362770\pi$
$797$	18805.3	0.835781	0.417890	+	0.908497i	$0.362770\pi$
$798$	19.1757	0.000850644 0
$799$	3769.98	0.166924
$800$	7737.20	0.341939
$801$	10537.8	0.464839
$802$	−598.870	−0.0263676
$803$	12809.6	0.562941
$804$	−7316.95	−0.320956
$805$	70.4410	0.00308412
$806$	−932.108	−0.0407346
$807$	−24328.4	−1.06121
$808$	1869.40	0.0813925
$809$	−13590.4	−0.590624	−0.295312	−	0.955401i	$-0.595424\pi$
$809$	−13590.4	−0.590624	−0.295312	+	0.955401i	$0.595424\pi$
$810$	−206.415	−0.00895395
$811$	3809.27	0.164934	0.0824670	−	0.996594i	$-0.473720\pi$
$811$	3809.27	0.164934	0.0824670	+	0.996594i	$0.473720\pi$
$812$	373.635	0.0161478
$813$	−19917.7	−0.859219
$814$	1934.61	0.0833023
$815$	1779.26	0.0764719
$816$	17635.1	0.756557
$817$	4132.28	0.176953
$818$	2765.78	0.118219
$819$	−296.860	−0.0126656
$820$	−13180.1	−0.561305
$821$	24914.8	1.05911	0.529556	−	0.848275i	$-0.322358\pi$
$821$	24914.8	1.05911	0.529556	+	0.848275i	$0.322358\pi$
$822$	3173.06	0.134639
$823$	−37180.8	−1.57477	−0.787387	−	0.616458i	$-0.788567\pi$
$823$	−37180.8	−1.57477	−0.787387	+	0.616458i	$0.788567\pi$
$824$	3852.90	0.162891
$825$	−3062.88	−0.129255
$826$	192.719	0.00811810
$827$	−24098.8	−1.01330	−0.506649	−	0.862153i	$-0.669116\pi$
$827$	−24098.8	−1.01330	−0.506649	+	0.862153i	$0.669116\pi$
$828$	−890.064	−0.0373573
$829$	−36998.5	−1.55008	−0.775038	−	0.631915i	$-0.782269\pi$
$829$	−36998.5	−1.55008	−0.775038	+	0.631915i	$0.782269\pi$
$830$	1347.75	0.0563630
$831$	5619.48	0.234582
$832$	14693.5	0.612264
$833$	33964.5	1.41273
$834$	−1925.17	−0.0799319
$835$	21093.9	0.874233
$836$	1246.75	0.0515788
$837$	1663.05	0.0686780
$838$	837.100	0.0345073
$839$	26525.4	1.09149	0.545745	−	0.837951i	$-0.316247\pi$
$839$	26525.4	1.09149	0.545745	+	0.837951i	$0.316247\pi$
$840$	−118.249	−0.00485714
$841$	−21994.1	−0.901806
$842$	−6449.28	−0.263963
$843$	−762.380	−0.0311480
$844$	3734.75	0.152317
$845$	−6024.99	−0.245285
$846$	153.484	0.00623744
$847$	118.467	0.00480588
$848$	−15729.3	−0.636965
$849$	−7061.36	−0.285448
$850$	−4139.90	−0.167056
$851$	4965.43	0.200015
$852$	−21666.8	−0.871234
$853$	6412.16	0.257383	0.128692	−	0.991685i	$-0.458922\pi$
$853$	6412.16	0.257383	0.128692	+	0.991685i	$0.458922\pi$
$854$	26.8268	0.00107494
$855$	742.103	0.0296835
$856$	−12994.2	−0.518847
$857$	−15296.6	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$857$	−15296.6	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$858$	499.388	0.0198704
$859$	43157.8	1.71423	0.857116	−	0.515123i	$-0.172254\pi$
$859$	43157.8	1.71423	0.857116	+	0.515123i	$0.172254\pi$
$860$	−12578.4	−0.498743
$861$	−875.039	−0.0346356
$862$	−3611.58	−0.142704
$863$	−23603.4	−0.931020	−0.465510	−	0.885043i	$-0.654129\pi$
$863$	−23603.4	−0.931020	−0.465510	+	0.885043i	$0.654129\pi$
$864$	2250.77	0.0886260
$865$	10202.5	0.401035
$866$	−4535.33	−0.177964
$867$	−14842.2	−0.581391
$868$	470.274	0.0183895
$869$	−5795.19	−0.226224
$870$	−374.126	−0.0145794
$871$	10536.8	0.409904
$872$	5771.09	0.224121
$873$	−13426.9	−0.520539
$874$	−82.7945	−0.00320431
$875$	−1209.84	−0.0467431
$876$	−27243.4	−1.05076
$877$	−18669.8	−0.718853	−0.359426	−	0.933173i	$-0.617028\pi$
$877$	−18669.8	−0.718853	−0.359426	+	0.933173i	$0.617028\pi$
$878$	1857.95	0.0714155
$879$	4429.70	0.169977
$880$	−3694.30	−0.141517
$881$	14791.8	0.565661	0.282830	−	0.959170i	$-0.408727\pi$
$881$	14791.8	0.565661	0.282830	+	0.959170i	$0.408727\pi$
$882$	1382.76	0.0527892
$883$	381.551	0.0145416	0.00727078	−	0.999974i	$-0.497686\pi$
$883$	381.551	0.0145416	0.00727078	+	0.999974i	$0.497686\pi$
$884$	−26088.0	−0.992571
$885$	7458.24	0.283284
$886$	2107.25	0.0799033
$887$	29579.9	1.11973	0.559863	−	0.828585i	$-0.310854\pi$
$887$	29579.9	1.11973	0.559863	+	0.828585i	$0.310854\pi$
$888$	−8335.48	−0.315000
$889$	−251.720	−0.00949653
$890$	−2983.78	−0.112378
$891$	−891.000	−0.0335013
$892$	−29320.3	−1.10058
$893$	−551.803	−0.0206779
$894$	1293.88	0.0484046
$895$	19616.0	0.732614
$896$	844.743	0.0314965
$897$	1281.74	0.0477103
$898$	684.407	0.0254331
$899$	3014.27	0.111826
$900$	6514.10	0.241263
$901$	26384.4	0.975574
$902$	1472.02	0.0543380
$903$	−835.088	−0.0307752
$904$	11038.2	0.406113
$905$	12541.8	0.460665
$906$	−1124.64	−0.0412404
$907$	43320.5	1.58592	0.792962	−	0.609272i	$-0.208538\pi$
$907$	43320.5	1.58592	0.792962	+	0.609272i	$0.208538\pi$
$908$	−31592.9	−1.15468
$909$	2370.87	0.0865093
$910$	84.0556	0.00306199
$911$	−23772.6	−0.864567	−0.432284	−	0.901738i	$-0.642292\pi$
$911$	−23772.6	−0.864567	−0.432284	+	0.901738i	$0.642292\pi$
$912$	−2581.20	−0.0937194
$913$	5817.63	0.210882
$914$	−2914.75	−0.105483
$915$	1038.20	0.0375102
$916$	36042.6	1.30009
$917$	−861.119	−0.0310105
$918$	−1204.31	−0.0432986
$919$	25979.5	0.932517	0.466259	−	0.884648i	$-0.345602\pi$
$919$	25979.5	0.932517	0.466259	+	0.884648i	$0.345602\pi$
$920$	510.562	0.0182964
$921$	−13114.6	−0.469207
$922$	6566.85	0.234564
$923$	31201.4	1.11268
$924$	−251.955	−0.00897045
$925$	−36340.4	−1.29175
$926$	−6829.64	−0.242371
$927$	4886.46	0.173131
$928$	4079.51	0.144306
$929$	11774.2	0.415821	0.207910	−	0.978148i	$-0.433334\pi$
$929$	11774.2	0.415821	0.207910	+	0.978148i	$0.433334\pi$
$930$	−470.891	−0.0166034
$931$	−4971.30	−0.175003
$932$	28115.1	0.988132
$933$	5233.63	0.183645
$934$	2419.20	0.0847524
$935$	6196.83	0.216747
$936$	−2151.67	−0.0751383
$937$	−48878.4	−1.70415	−0.852075	−	0.523420i	$-0.824656\pi$
$937$	−48878.4	−1.70415	−0.852075	+	0.523420i	$0.824656\pi$
$938$	137.547	0.00478793
$939$	−10716.9	−0.372453
$940$	1679.65	0.0582810
$941$	−36678.4	−1.27065	−0.635325	−	0.772245i	$-0.719134\pi$
$941$	−36678.4	−1.27065	−0.635325	+	0.772245i	$0.719134\pi$
$942$	1838.49	0.0635894
$943$	3778.13	0.130470
$944$	−25941.4	−0.894409
$945$	−149.971	−0.00516248
$946$	1404.81	0.0482816
$947$	−4740.15	−0.162655	−0.0813275	−	0.996687i	$-0.525916\pi$
$947$	−4740.15	−0.162655	−0.0813275	+	0.996687i	$0.525916\pi$
$948$	12325.2	0.422260
$949$	39232.0	1.34197
$950$	605.947	0.0206942
$951$	−4156.87	−0.141741
$952$	−689.914	−0.0234876
$953$	−26542.2	−0.902189	−0.451094	−	0.892476i	$-0.648966\pi$
$953$	−26542.2	−0.902189	−0.451094	+	0.892476i	$0.648966\pi$
$954$	1074.16	0.0364542
$955$	−11359.7	−0.384913
$956$	−11174.0	−0.378025
$957$	−1614.93	−0.0545489
$958$	−7035.87	−0.237285
$959$	2305.38	0.0776273
$960$	7422.98	0.249558
$961$	−25997.1	−0.872650
$962$	5925.13	0.198580
$963$	−16480.0	−0.551464
$964$	9661.43	0.322794
$965$	−4984.62	−0.166280
$966$	16.7318	0.000557285 0
$967$	−22189.9	−0.737930	−0.368965	−	0.929443i	$-0.620288\pi$
$967$	−22189.9	−0.737930	−0.368965	+	0.929443i	$0.620288\pi$
$968$	858.659	0.0285107
$969$	4329.72	0.143540
$970$	3801.80	0.125844
$971$	−2839.27	−0.0938377	−0.0469189	−	0.998899i	$-0.514940\pi$
$971$	−2839.27	−0.0938377	−0.0469189	+	0.998899i	$0.514940\pi$
$972$	1894.97	0.0625321
$973$	−1398.73	−0.0460854
$974$	−5469.37	−0.179928
$975$	−9380.67	−0.308125
$976$	−3611.09	−0.118431
$977$	−40992.9	−1.34235	−0.671176	−	0.741298i	$-0.734210\pi$
$977$	−40992.9	−1.34235	−0.671176	+	0.741298i	$0.734210\pi$
$978$	422.626	0.0138181
$979$	−12879.6	−0.420463
$980$	15132.3	0.493249
$981$	7319.22	0.238211
$982$	6365.84	0.206866
$983$	−40555.3	−1.31588	−0.657942	−	0.753069i	$-0.728573\pi$
$983$	−40555.3	−1.31588	−0.657942	+	0.753069i	$0.728573\pi$
$984$	−6342.35	−0.205474
$985$	2509.19	0.0811671
$986$	−2182.80	−0.0705015
$987$	111.513	0.00359625
$988$	3818.43	0.122956
$989$	3605.63	0.115928
$990$	252.286	0.00809915
$991$	−12739.5	−0.408360	−0.204180	−	0.978933i	$-0.565453\pi$
$991$	−12739.5	−0.408360	−0.204180	+	0.978933i	$0.565453\pi$
$992$	5134.64	0.164340
$993$	−18436.2	−0.589181
$994$	407.302	0.0129968
$995$	24119.1	0.768468
$996$	−12372.9	−0.393624
$997$	−5997.83	−0.190525	−0.0952624	−	0.995452i	$-0.530369\pi$
$997$	−5997.83	−0.190525	−0.0952624	+	0.995452i	$0.530369\pi$
$998$	6996.70	0.221921
$999$	−10571.5	−0.334803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.e.1.20	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.e.1.20	✓	38	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-0.449187 q^{2} -3.00000 q^{3} -7.79823 q^{4} +5.67323 q^{5} +1.34756 q^{6} +0.979067 q^{7} +7.09636 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.449187 q^{2} -3.00000 q^{3} -7.79823 q^{4} +5.67323 q^{5} +1.34756 q^{6} +0.979067 q^{7} +7.09636 q^{8} +9.00000 q^{9} -2.54834 q^{10} -11.0000 q^{11} +23.3947 q^{12} -33.6897 q^{13} -0.439784 q^{14} -17.0197 q^{15} +59.1983 q^{16} -99.2995 q^{17} -4.04268 q^{18} +14.5342 q^{19} -44.2411 q^{20} -2.93720 q^{21} +4.94106 q^{22} +12.6819 q^{23} -21.2891 q^{24} -92.8145 q^{25} +15.1330 q^{26} -27.0000 q^{27} -7.63499 q^{28} -48.9373 q^{29} +7.64502 q^{30} -61.5945 q^{31} -83.3620 q^{32} +33.0000 q^{33} +44.6040 q^{34} +5.55447 q^{35} -70.1841 q^{36} +391.538 q^{37} -6.52858 q^{38} +101.069 q^{39} +40.2593 q^{40} +297.916 q^{41} +1.31935 q^{42} +284.314 q^{43} +85.7805 q^{44} +51.0590 q^{45} -5.69652 q^{46} -37.9658 q^{47} -177.595 q^{48} -342.041 q^{49} +41.6911 q^{50} +297.898 q^{51} +262.720 q^{52} -265.705 q^{53} +12.1280 q^{54} -62.4055 q^{55} +6.94781 q^{56} -43.6026 q^{57} +21.9820 q^{58} -438.213 q^{59} +132.723 q^{60} -61.0000 q^{61} +27.6675 q^{62} +8.81160 q^{63} -436.141 q^{64} -191.129 q^{65} -14.8232 q^{66} -312.761 q^{67} +774.360 q^{68} -38.0456 q^{69} -2.49499 q^{70} -926.140 q^{71} +63.8672 q^{72} -1164.51 q^{73} -175.874 q^{74} +278.443 q^{75} -113.341 q^{76} -10.7697 q^{77} -45.3989 q^{78} +526.836 q^{79} +335.845 q^{80} +81.0000 q^{81} -133.820 q^{82} -528.876 q^{83} +22.9050 q^{84} -563.348 q^{85} -127.710 q^{86} +146.812 q^{87} -78.0599 q^{88} +1170.87 q^{89} -22.9351 q^{90} -32.9844 q^{91} -98.8961 q^{92} +184.784 q^{93} +17.0537 q^{94} +82.4559 q^{95} +250.086 q^{96} -1491.87 q^{97} +153.641 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9}+O(q^{10}) \) 38 * q - 2 * q^2 - 114 * q^3 + 142 * q^4 + 15 * q^5 + 6 * q^6 + 63 * q^7 - 45 * q^8 + 342 * q^9	\( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9} + 95 q^{10} - 418 q^{11} - 426 q^{12} + 13 q^{13} + 26 q^{14} - 45 q^{15} + 486 q^{16} - 224 q^{17} - 18 q^{18} + 367 q^{19} + 18 q^{20} - 189 q^{21} + 22 q^{22} + 51 q^{23} + 135 q^{24} + 773 q^{25} - 439 q^{26} - 1026 q^{27} + 22 q^{28} - 462 q^{29} - 285 q^{30} + 234 q^{31} - 597 q^{32} + 1254 q^{33} + 956 q^{34} - 522 q^{35} + 1278 q^{36} + 954 q^{37} + 705 q^{38} - 39 q^{39} + 1495 q^{40} - 740 q^{41} - 78 q^{42} + 1441 q^{43} - 1562 q^{44} + 135 q^{45} + 581 q^{46} + 1003 q^{47} - 1458 q^{48} + 2707 q^{49} + 388 q^{50} + 672 q^{51} + 788 q^{52} + 735 q^{53} + 54 q^{54} - 165 q^{55} + 1059 q^{56} - 1101 q^{57} + 177 q^{58} + 261 q^{59} - 54 q^{60} - 2318 q^{61} + 1251 q^{62} + 567 q^{63} + 5571 q^{64} - 1354 q^{65} - 66 q^{66} + 3495 q^{67} - 1856 q^{68} - 153 q^{69} + 542 q^{70} - 873 q^{71} - 405 q^{72} + 989 q^{73} - 3406 q^{74} - 2319 q^{75} + 1712 q^{76} - 693 q^{77} + 1317 q^{78} + 2313 q^{79} + 1593 q^{80} + 3078 q^{81} + 5170 q^{82} + 569 q^{83} - 66 q^{84} - 1271 q^{85} + 3065 q^{86} + 1386 q^{87} + 495 q^{88} - 2917 q^{89} + 855 q^{90} + 2740 q^{91} + 1083 q^{92} - 702 q^{93} + 3272 q^{94} + 2696 q^{95} + 1791 q^{96} + 4250 q^{97} + 5952 q^{98} - 3762 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 - 114 * q^3 + 142 * q^4 + 15 * q^5 + 6 * q^6 + 63 * q^7 - 45 * q^8 + 342 * q^9 + 95 * q^10 - 418 * q^11 - 426 * q^12 + 13 * q^13 + 26 * q^14 - 45 * q^15 + 486 * q^16 - 224 * q^17 - 18 * q^18 + 367 * q^19 + 18 * q^20 - 189 * q^21 + 22 * q^22 + 51 * q^23 + 135 * q^24 + 773 * q^25 - 439 * q^26 - 1026 * q^27 + 22 * q^28 - 462 * q^29 - 285 * q^30 + 234 * q^31 - 597 * q^32 + 1254 * q^33 + 956 * q^34 - 522 * q^35 + 1278 * q^36 + 954 * q^37 + 705 * q^38 - 39 * q^39 + 1495 * q^40 - 740 * q^41 - 78 * q^42 + 1441 * q^43 - 1562 * q^44 + 135 * q^45 + 581 * q^46 + 1003 * q^47 - 1458 * q^48 + 2707 * q^49 + 388 * q^50 + 672 * q^51 + 788 * q^52 + 735 * q^53 + 54 * q^54 - 165 * q^55 + 1059 * q^56 - 1101 * q^57 + 177 * q^58 + 261 * q^59 - 54 * q^60 - 2318 * q^61 + 1251 * q^62 + 567 * q^63 + 5571 * q^64 - 1354 * q^65 - 66 * q^66 + 3495 * q^67 - 1856 * q^68 - 153 * q^69 + 542 * q^70 - 873 * q^71 - 405 * q^72 + 989 * q^73 - 3406 * q^74 - 2319 * q^75 + 1712 * q^76 - 693 * q^77 + 1317 * q^78 + 2313 * q^79 + 1593 * q^80 + 3078 * q^81 + 5170 * q^82 + 569 * q^83 - 66 * q^84 - 1271 * q^85 + 3065 * q^86 + 1386 * q^87 + 495 * q^88 - 2917 * q^89 + 855 * q^90 + 2740 * q^91 + 1083 * q^92 - 702 * q^93 + 3272 * q^94 + 2696 * q^95 + 1791 * q^96 + 4250 * q^97 + 5952 * q^98 - 3762 * q^99

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