LMFDB - Embedded newform 2013.4.a.e.1.18

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			1.18
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)

$f(q)$	$=$	$q-0.565755 q^{2} -3.00000 q^{3} -7.67992 q^{4} +11.9466 q^{5} +1.69726 q^{6} +27.9962 q^{7} +8.87099 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.565755 q^{2} -3.00000 q^{3} -7.67992 q^{4} +11.9466 q^{5} +1.69726 q^{6} +27.9962 q^{7} +8.87099 q^{8} +9.00000 q^{9} -6.75887 q^{10} -11.0000 q^{11} +23.0398 q^{12} +16.1195 q^{13} -15.8390 q^{14} -35.8399 q^{15} +56.4206 q^{16} +110.577 q^{17} -5.09179 q^{18} -92.4650 q^{19} -91.7493 q^{20} -83.9887 q^{21} +6.22330 q^{22} +143.575 q^{23} -26.6130 q^{24} +17.7223 q^{25} -9.11966 q^{26} -27.0000 q^{27} -215.009 q^{28} +297.633 q^{29} +20.2766 q^{30} +45.1749 q^{31} -102.888 q^{32} +33.0000 q^{33} -62.5595 q^{34} +334.461 q^{35} -69.1193 q^{36} +240.481 q^{37} +52.3125 q^{38} -48.3584 q^{39} +105.979 q^{40} +16.1224 q^{41} +47.5170 q^{42} +224.156 q^{43} +84.4791 q^{44} +107.520 q^{45} -81.2283 q^{46} +454.590 q^{47} -169.262 q^{48} +440.789 q^{49} -10.0265 q^{50} -331.731 q^{51} -123.796 q^{52} -352.514 q^{53} +15.2754 q^{54} -131.413 q^{55} +248.354 q^{56} +277.395 q^{57} -168.387 q^{58} -104.905 q^{59} +275.248 q^{60} -61.0000 q^{61} -25.5579 q^{62} +251.966 q^{63} -393.155 q^{64} +192.573 q^{65} -18.6699 q^{66} +671.206 q^{67} -849.223 q^{68} -430.725 q^{69} -189.223 q^{70} -295.342 q^{71} +79.8389 q^{72} -1063.56 q^{73} -136.053 q^{74} -53.1668 q^{75} +710.124 q^{76} -307.959 q^{77} +27.3590 q^{78} -775.142 q^{79} +674.036 q^{80} +81.0000 q^{81} -9.12130 q^{82} +46.4991 q^{83} +645.027 q^{84} +1321.02 q^{85} -126.817 q^{86} -892.900 q^{87} -97.5809 q^{88} -1045.06 q^{89} -60.8298 q^{90} +451.284 q^{91} -1102.65 q^{92} -135.525 q^{93} -257.186 q^{94} -1104.65 q^{95} +308.664 q^{96} +50.4920 q^{97} -249.379 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.565755	−0.200024	−0.100012	−	0.994986i	$-0.531888\pi$
$2$	−0.565755	−0.200024	−0.100012	+	0.994986i	$0.531888\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.67992	−0.959990
$5$	11.9466	1.06854	0.534270	−	0.845314i	$-0.320587\pi$
$5$	11.9466	1.06854	0.534270	+	0.845314i	$0.320587\pi$
$6$	1.69726	0.115484
$7$	27.9962	1.51165	0.755827	−	0.654771i	$-0.227235\pi$
$7$	27.9962	1.51165	0.755827	+	0.654771i	$0.227235\pi$
$8$	8.87099	0.392046
$9$	9.00000	0.333333
$10$	−6.75887	−0.213734
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	23.0398	0.554251
$13$	16.1195	0.343902	0.171951	−	0.985105i	$-0.444993\pi$
$13$	16.1195	0.343902	0.171951	+	0.985105i	$0.444993\pi$
$14$	−15.8390	−0.302368
$15$	−35.8399	−0.616922
$16$	56.4206	0.881571
$17$	110.577	1.57758	0.788790	−	0.614662i	$-0.210708\pi$
$17$	110.577	1.57758	0.788790	+	0.614662i	$0.210708\pi$
$18$	−5.09179	−0.0666748
$19$	−92.4650	−1.11647	−0.558235	−	0.829683i	$-0.688521\pi$
$19$	−92.4650	−1.11647	−0.558235	+	0.829683i	$0.688521\pi$
$20$	−91.7493	−1.02579
$21$	−83.9887	−0.872754
$22$	6.22330	0.0603097
$23$	143.575	1.30163	0.650815	−	0.759236i	$-0.274427\pi$
$23$	143.575	1.30163	0.650815	+	0.759236i	$0.274427\pi$
$24$	−26.6130	−0.226348
$25$	17.7223	0.141778
$26$	−9.11966	−0.0687889
$27$	−27.0000	−0.192450
$28$	−215.009	−1.45117
$29$	297.633	1.90583	0.952916	−	0.303235i	$-0.0980668\pi$
$29$	297.633	1.90583	0.952916	+	0.303235i	$0.0980668\pi$
$30$	20.2766	0.123400
$31$	45.1749	0.261730	0.130865	−	0.991400i	$-0.458224\pi$
$31$	45.1749	0.261730	0.130865	+	0.991400i	$0.458224\pi$
$32$	−102.888	−0.568382
$33$	33.0000	0.174078
$34$	−62.5595	−0.315555
$35$	334.461	1.61526
$36$	−69.1193	−0.319997
$37$	240.481	1.06851	0.534255	−	0.845323i	$-0.320592\pi$
$37$	240.481	1.06851	0.534255	+	0.845323i	$0.320592\pi$
$38$	52.3125	0.223321
$39$	−48.3584	−0.198552
$40$	105.979	0.418917
$41$	16.1224	0.0614119	0.0307060	−	0.999528i	$-0.490224\pi$
$41$	16.1224	0.0614119	0.0307060	+	0.999528i	$0.490224\pi$
$42$	47.5170	0.174572
$43$	224.156	0.794963	0.397482	−	0.917610i	$-0.369884\pi$
$43$	224.156	0.794963	0.397482	+	0.917610i	$0.369884\pi$
$44$	84.4791	0.289448
$45$	107.520	0.356180
$46$	−81.2283	−0.260358
$47$	454.590	1.41082	0.705412	−	0.708797i	$-0.250762\pi$
$47$	454.590	1.41082	0.705412	+	0.708797i	$0.250762\pi$
$48$	−169.262	−0.508975
$49$	440.789	1.28510
$50$	−10.0265	−0.0283591
$51$	−331.731	−0.910817
$52$	−123.796	−0.330143
$53$	−352.514	−0.913615	−0.456807	−	0.889566i	$-0.651007\pi$
$53$	−352.514	−0.913615	−0.456807	+	0.889566i	$0.651007\pi$
$54$	15.2754	0.0384947
$55$	−131.413	−0.322177
$56$	248.354	0.592638
$57$	277.395	0.644594
$58$	−168.387	−0.381213
$59$	−104.905	−0.231483	−0.115741	−	0.993279i	$-0.536924\pi$
$59$	−104.905	−0.231483	−0.115741	+	0.993279i	$0.536924\pi$
$60$	275.248	0.592239
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−25.5579	−0.0523525
$63$	251.966	0.503885
$64$	−393.155	−0.767881
$65$	192.573	0.367474
$66$	−18.6699	−0.0348198
$67$	671.206	1.22389	0.611947	−	0.790899i	$-0.290387\pi$
$67$	671.206	1.22389	0.611947	+	0.790899i	$0.290387\pi$
$68$	−849.223	−1.51446
$69$	−430.725	−0.751496
$70$	−189.223	−0.323092
$71$	−295.342	−0.493671	−0.246835	−	0.969057i	$-0.579391\pi$
$71$	−295.342	−0.493671	−0.246835	+	0.969057i	$0.579391\pi$
$72$	79.8389	0.130682
$73$	−1063.56	−1.70521	−0.852605	−	0.522557i	$-0.824978\pi$
$73$	−1063.56	−1.70521	−0.852605	+	0.522557i	$0.824978\pi$
$74$	−136.053	−0.213728
$75$	−53.1668	−0.0818557
$76$	710.124	1.07180
$77$	−307.959	−0.455781
$78$	27.3590	0.0397153
$79$	−775.142	−1.10393	−0.551964	−	0.833868i	$-0.686121\pi$
$79$	−775.142	−1.10393	−0.551964	+	0.833868i	$0.686121\pi$
$80$	674.036	0.941994
$81$	81.0000	0.111111
$82$	−9.12130	−0.0122839
$83$	46.4991	0.0614932	0.0307466	−	0.999527i	$-0.490212\pi$
$83$	46.4991	0.0614932	0.0307466	+	0.999527i	$0.490212\pi$
$84$	645.027	0.837835
$85$	1321.02	1.68571
$86$	−126.817	−0.159012
$87$	−892.900	−1.10033
$88$	−97.5809	−0.118206
$89$	−1045.06	−1.24468	−0.622341	−	0.782747i	$-0.713818\pi$
$89$	−1045.06	−1.24468	−0.622341	+	0.782747i	$0.713818\pi$
$90$	−60.8298	−0.0712447
$91$	451.284	0.519862
$92$	−1102.65	−1.24955
$93$	−135.525	−0.151110
$94$	−257.186	−0.282199
$95$	−1104.65	−1.19299
$96$	308.664	0.328155
$97$	50.4920	0.0528524	0.0264262	−	0.999651i	$-0.491587\pi$
$97$	50.4920	0.0528524	0.0264262	+	0.999651i	$0.491587\pi$
$98$	−249.379	−0.257051
$99$	−99.0000	−0.100504
$100$	−136.106	−0.136106
$101$	1639.11	1.61483	0.807414	−	0.589985i	$-0.200867\pi$
$101$	1639.11	1.61483	0.807414	+	0.589985i	$0.200867\pi$
$102$	187.678	0.182186
$103$	−401.001	−0.383610	−0.191805	−	0.981433i	$-0.561434\pi$
$103$	−401.001	−0.383610	−0.191805	+	0.981433i	$0.561434\pi$
$104$	142.996	0.134826
$105$	−1003.38	−0.932573
$106$	199.437	0.182745
$107$	1360.27	1.22900	0.614499	−	0.788918i	$-0.289358\pi$
$107$	1360.27	1.22900	0.614499	+	0.788918i	$0.289358\pi$
$108$	207.358	0.184750
$109$	−1946.02	−1.71004	−0.855022	−	0.518592i	$-0.826456\pi$
$109$	−1946.02	−1.71004	−0.855022	+	0.518592i	$0.826456\pi$
$110$	74.3476	0.0644433
$111$	−721.444	−0.616904
$112$	1579.56	1.33263
$113$	−732.357	−0.609684	−0.304842	−	0.952403i	$-0.598604\pi$
$113$	−732.357	−0.609684	−0.304842	+	0.952403i	$0.598604\pi$
$114$	−156.938	−0.128935
$115$	1715.24	1.39084
$116$	−2285.80	−1.82958
$117$	145.075	0.114634
$118$	59.3506	0.0463023
$119$	3095.74	2.38476
$120$	−317.936	−0.241862
$121$	121.000	0.0909091
$122$	34.5110	0.0256105
$123$	−48.3671	−0.0354562
$124$	−346.939	−0.251259
$125$	−1281.61	−0.917045
$126$	−142.551	−0.100789
$127$	−1054.41	−0.736720	−0.368360	−	0.929683i	$-0.620081\pi$
$127$	−1054.41	−0.736720	−0.368360	+	0.929683i	$0.620081\pi$
$128$	1045.53	0.721977
$129$	−672.467	−0.458972
$130$	−108.949	−0.0735037
$131$	−1514.53	−1.01012	−0.505058	−	0.863085i	$-0.668529\pi$
$131$	−1514.53	−1.01012	−0.505058	+	0.863085i	$0.668529\pi$
$132$	−253.437	−0.167113
$133$	−2588.67	−1.68772
$134$	−379.738	−0.244809
$135$	−322.559	−0.205641
$136$	980.928	0.618484
$137$	1955.30	1.21936	0.609682	−	0.792646i	$-0.291297\pi$
$137$	1955.30	1.21936	0.609682	+	0.792646i	$0.291297\pi$
$138$	243.685	0.150318
$139$	1631.24	0.995397	0.497698	−	0.867350i	$-0.334179\pi$
$139$	1631.24	0.995397	0.497698	+	0.867350i	$0.334179\pi$
$140$	−2568.63	−1.55064
$141$	−1363.77	−0.814540
$142$	167.091	0.0987463
$143$	−177.314	−0.103690
$144$	507.785	0.293857
$145$	3555.72	2.03646
$146$	601.714	0.341084
$147$	−1322.37	−0.741953
$148$	−1846.88	−1.02576
$149$	−813.483	−0.447269	−0.223635	−	0.974673i	$-0.571792\pi$
$149$	−813.483	−0.447269	−0.223635	+	0.974673i	$0.571792\pi$
$150$	30.0794	0.0163731
$151$	905.369	0.487933	0.243966	−	0.969784i	$-0.421551\pi$
$151$	905.369	0.487933	0.243966	+	0.969784i	$0.421551\pi$
$152$	−820.256	−0.437708
$153$	995.193	0.525860
$154$	174.229	0.0911674
$155$	539.688	0.279670
$156$	371.388	0.190608
$157$	−1120.76	−0.569721	−0.284860	−	0.958569i	$-0.591947\pi$
$157$	−1120.76	−0.569721	−0.284860	+	0.958569i	$0.591947\pi$
$158$	438.540	0.220813
$159$	1057.54	0.527476
$160$	−1229.17	−0.607339
$161$	4019.56	1.96761
$162$	−45.8261	−0.0222249
$163$	−875.891	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$163$	−875.891	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$164$	−123.818	−0.0589548
$165$	394.239	0.186009
$166$	−26.3071	−0.0123001
$167$	1445.84	0.669955	0.334978	−	0.942226i	$-0.391271\pi$
$167$	1445.84	0.669955	0.334978	+	0.942226i	$0.391271\pi$
$168$	−745.063	−0.342160
$169$	−1937.16	−0.881731
$170$	−747.376	−0.337183
$171$	−832.185	−0.372157
$172$	−1721.50	−0.763157
$173$	−3310.15	−1.45472	−0.727358	−	0.686258i	$-0.759252\pi$
$173$	−3310.15	−1.45472	−0.727358	+	0.686258i	$0.759252\pi$
$174$	505.162	0.220093
$175$	496.157	0.214320
$176$	−620.626	−0.265804
$177$	314.716	0.133647
$178$	591.250	0.248967
$179$	61.6475	0.0257416	0.0128708	−	0.999917i	$-0.495903\pi$
$179$	61.6475	0.0257416	0.0128708	+	0.999917i	$0.495903\pi$
$180$	−825.744	−0.341929
$181$	1986.10	0.815611	0.407805	−	0.913069i	$-0.366294\pi$
$181$	1986.10	0.815611	0.407805	+	0.913069i	$0.366294\pi$
$182$	−255.316	−0.103985
$183$	183.000	0.0739221
$184$	1273.65	0.510299
$185$	2872.94	1.14175
$186$	76.6737	0.0302257
$187$	−1216.35	−0.475658
$188$	−3491.21	−1.35438
$189$	−755.898	−0.290918
$190$	624.959	0.238628
$191$	4376.21	1.65786	0.828930	−	0.559352i	$-0.188950\pi$
$191$	4376.21	1.65786	0.828930	+	0.559352i	$0.188950\pi$
$192$	1179.47	0.443336
$193$	−342.394	−0.127700	−0.0638500	−	0.997960i	$-0.520338\pi$
$193$	−342.394	−0.127700	−0.0638500	+	0.997960i	$0.520338\pi$
$194$	−28.5661	−0.0105718
$195$	−577.720	−0.212161
$196$	−3385.23	−1.23368
$197$	996.062	0.360236	0.180118	−	0.983645i	$-0.442352\pi$
$197$	996.062	0.360236	0.180118	+	0.983645i	$0.442352\pi$
$198$	56.0097	0.0201032
$199$	−3411.88	−1.21538	−0.607692	−	0.794173i	$-0.707905\pi$
$199$	−3411.88	−1.21538	−0.607692	+	0.794173i	$0.707905\pi$
$200$	157.214	0.0555836
$201$	−2013.62	−0.706616
$202$	−927.334	−0.323005
$203$	8332.61	2.88096
$204$	2547.67	0.874375
$205$	192.608	0.0656211
$206$	226.868	0.0767313
$207$	1292.18	0.433877
$208$	909.469	0.303175
$209$	1017.12	0.336628
$210$	567.669	0.186537
$211$	213.133	0.0695388	0.0347694	−	0.999395i	$-0.488930\pi$
$211$	213.133	0.0695388	0.0347694	+	0.999395i	$0.488930\pi$
$212$	2707.28	0.877061
$213$	886.026	0.285021
$214$	−769.582	−0.245830
$215$	2677.91	0.849450
$216$	−239.517	−0.0754493
$217$	1264.73	0.395646
$218$	1100.97	0.342051
$219$	3190.68	0.984503
$220$	1009.24	0.309287
$221$	1782.44	0.542534
$222$	408.160	0.123396
$223$	253.977	0.0762672	0.0381336	−	0.999273i	$-0.487859\pi$
$223$	253.977	0.0762672	0.0381336	+	0.999273i	$0.487859\pi$
$224$	−2880.48	−0.859197
$225$	159.500	0.0472594
$226$	414.334	0.121952
$227$	−5149.33	−1.50561	−0.752805	−	0.658244i	$-0.771299\pi$
$227$	−5149.33	−1.50561	−0.752805	+	0.658244i	$0.771299\pi$
$228$	−2130.37	−0.618804
$229$	−4209.91	−1.21484	−0.607421	−	0.794380i	$-0.707796\pi$
$229$	−4209.91	−1.21484	−0.607421	+	0.794380i	$0.707796\pi$
$230$	−970.405	−0.278203
$231$	923.876	0.263145
$232$	2640.30	0.747174
$233$	865.013	0.243214	0.121607	−	0.992578i	$-0.461195\pi$
$233$	865.013	0.243214	0.121607	+	0.992578i	$0.461195\pi$
$234$	−82.0769	−0.0229296
$235$	5430.82	1.50752
$236$	805.664	0.222221
$237$	2325.43	0.637353
$238$	−1751.43	−0.477010
$239$	99.6894	0.0269806	0.0134903	−	0.999909i	$-0.495706\pi$
$239$	99.6894	0.0269806	0.0134903	+	0.999909i	$0.495706\pi$
$240$	−2022.11	−0.543861
$241$	3995.06	1.06782	0.533910	−	0.845542i	$-0.320722\pi$
$241$	3995.06	1.06782	0.533910	+	0.845542i	$0.320722\pi$
$242$	−68.4563	−0.0181840
$243$	−243.000	−0.0641500
$244$	468.475	0.122914
$245$	5265.95	1.37318
$246$	27.3639	0.00709211
$247$	−1490.49	−0.383957
$248$	400.746	0.102610
$249$	−139.497	−0.0355031
$250$	725.076	0.183431
$251$	4518.47	1.13627	0.568134	−	0.822936i	$-0.307666\pi$
$251$	4518.47	1.13627	0.568134	+	0.822936i	$0.307666\pi$
$252$	−1935.08	−0.483725
$253$	−1579.33	−0.392456
$254$	596.536	0.147362
$255$	−3963.07	−0.973244
$256$	2553.73	0.623468
$257$	−6175.89	−1.49899	−0.749497	−	0.662008i	$-0.769705\pi$
$257$	−6175.89	−1.49899	−0.749497	+	0.662008i	$0.769705\pi$
$258$	380.451	0.0918057
$259$	6732.57	1.61522
$260$	−1478.95	−0.352771
$261$	2678.70	0.635277
$262$	856.853	0.202048
$263$	7735.41	1.81363	0.906817	−	0.421525i	$-0.138505\pi$
$263$	7735.41	1.81363	0.906817	+	0.421525i	$0.138505\pi$
$264$	292.743	0.0682465
$265$	−4211.36	−0.976234
$266$	1464.55	0.337585
$267$	3135.19	0.718617
$268$	−5154.81	−1.17493
$269$	4591.26	1.04065	0.520324	−	0.853969i	$-0.325811\pi$
$269$	4591.26	1.04065	0.520324	+	0.853969i	$0.325811\pi$
$270$	182.489	0.0411332
$271$	2974.30	0.666699	0.333350	−	0.942803i	$-0.391821\pi$
$271$	2974.30	0.666699	0.333350	+	0.942803i	$0.391821\pi$
$272$	6238.82	1.39075
$273$	−1353.85	−0.300142
$274$	−1106.22	−0.243903
$275$	−194.945	−0.0427477
$276$	3307.94	0.721429
$277$	4770.76	1.03483	0.517414	−	0.855735i	$-0.326895\pi$
$277$	4770.76	1.03483	0.517414	+	0.855735i	$0.326895\pi$
$278$	−922.883	−0.199104
$279$	406.574	0.0872435
$280$	2967.00	0.633258
$281$	−1853.81	−0.393555	−0.196778	−	0.980448i	$-0.563048\pi$
$281$	−1853.81	−0.393555	−0.196778	+	0.980448i	$0.563048\pi$
$282$	771.559	0.162928
$283$	5039.45	1.05853	0.529265	−	0.848457i	$-0.322468\pi$
$283$	5039.45	1.05853	0.529265	+	0.848457i	$0.322468\pi$
$284$	2268.20	0.473919
$285$	3313.94	0.688775
$286$	100.316	0.0207406
$287$	451.365	0.0928336
$288$	−925.993	−0.189461
$289$	7314.28	1.48876
$290$	−2011.66	−0.407341
$291$	−151.476	−0.0305144
$292$	8168.06	1.63698
$293$	5256.14	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$293$	5256.14	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$294$	748.136	0.148409
$295$	−1253.27	−0.247349
$296$	2133.31	0.418905
$297$	297.000	0.0580259
$298$	460.232	0.0894648
$299$	2314.35	0.447634
$300$	408.317	0.0785807
$301$	6275.52	1.20171
$302$	−512.217	−0.0975985
$303$	−4917.33	−0.932321
$304$	−5216.93	−0.984248
$305$	−728.745	−0.136813
$306$	−563.035	−0.105185
$307$	−1955.45	−0.363530	−0.181765	−	0.983342i	$-0.558181\pi$
$307$	−1955.45	−0.363530	−0.181765	+	0.983342i	$0.558181\pi$
$308$	2365.10	0.437545
$309$	1203.00	0.221477
$310$	−305.331	−0.0559408
$311$	6314.01	1.15124	0.575618	−	0.817718i	$-0.304761\pi$
$311$	6314.01	1.15124	0.575618	+	0.817718i	$0.304761\pi$
$312$	−428.987	−0.0778416
$313$	9413.93	1.70002	0.850011	−	0.526765i	$-0.176595\pi$
$313$	9413.93	1.70002	0.850011	+	0.526765i	$0.176595\pi$
$314$	634.074	0.113958
$315$	3010.15	0.538421
$316$	5953.03	1.05976
$317$	−10109.0	−1.79110	−0.895551	−	0.444959i	$-0.853218\pi$
$317$	−10109.0	−1.79110	−0.895551	+	0.444959i	$0.853218\pi$
$318$	−598.310	−0.105508
$319$	−3273.97	−0.574630
$320$	−4696.88	−0.820512
$321$	−4080.82	−0.709562
$322$	−2274.09	−0.393571
$323$	−10224.5	−1.76132
$324$	−622.074	−0.106666
$325$	285.673	0.0487579
$326$	495.540	0.0841883
$327$	5838.06	0.987294
$328$	143.021	0.0240763
$329$	12726.8	2.13268
$330$	−223.043	−0.0372063
$331$	−763.065	−0.126713	−0.0633563	−	0.997991i	$-0.520180\pi$
$331$	−763.065	−0.126713	−0.0633563	+	0.997991i	$0.520180\pi$
$332$	−357.109	−0.0590329
$333$	2164.33	0.356170
$334$	−817.991	−0.134007
$335$	8018.66	1.30778
$336$	−4738.69	−0.769395
$337$	−4846.88	−0.783460	−0.391730	−	0.920080i	$-0.628123\pi$
$337$	−4846.88	−0.783460	−0.391730	+	0.920080i	$0.628123\pi$
$338$	1095.96	0.176368
$339$	2197.07	0.352001
$340$	−10145.4	−1.61826
$341$	−496.924	−0.0789147
$342$	470.813	0.0744405
$343$	2737.73	0.430972
$344$	1988.48	0.311662
$345$	−5145.72	−0.803004
$346$	1872.73	0.290979
$347$	1101.11	0.170347	0.0851735	−	0.996366i	$-0.472856\pi$
$347$	1101.11	0.170347	0.0851735	+	0.996366i	$0.472856\pi$
$348$	6857.40	1.05631
$349$	−2260.58	−0.346722	−0.173361	−	0.984858i	$-0.555463\pi$
$349$	−2260.58	−0.346722	−0.173361	+	0.984858i	$0.555463\pi$
$350$	−280.703	−0.0428692
$351$	−435.225	−0.0661841
$352$	1131.77	0.171374
$353$	−8111.83	−1.22309	−0.611543	−	0.791211i	$-0.709451\pi$
$353$	−8111.83	−1.22309	−0.611543	+	0.791211i	$0.709451\pi$
$354$	−178.052	−0.0267326
$355$	−3528.34	−0.527507
$356$	8026.01	1.19488
$357$	−9287.22	−1.37684
$358$	−34.8773	−0.00514895
$359$	−5411.63	−0.795585	−0.397792	−	0.917475i	$-0.630224\pi$
$359$	−5411.63	−0.795585	−0.397792	+	0.917475i	$0.630224\pi$
$360$	953.807	0.139639
$361$	1690.78	0.246506
$362$	−1123.64	−0.163142
$363$	−363.000	−0.0524864
$364$	−3465.83	−0.499062
$365$	−12706.0	−1.82208
$366$	−103.533	−0.0147862
$367$	5338.26	0.759277	0.379639	−	0.925135i	$-0.376048\pi$
$367$	5338.26	0.759277	0.379639	+	0.925135i	$0.376048\pi$
$368$	8100.59	1.14748
$369$	145.101	0.0204706
$370$	−1625.38	−0.228377
$371$	−9869.08	−1.38107
$372$	1040.82	0.145064
$373$	−3038.43	−0.421780	−0.210890	−	0.977510i	$-0.567636\pi$
$373$	−3038.43	−0.421780	−0.210890	+	0.977510i	$0.567636\pi$
$374$	688.154	0.0951433
$375$	3844.83	0.529456
$376$	4032.66	0.553108
$377$	4797.69	0.655420
$378$	427.653	0.0581907
$379$	1638.48	0.222066	0.111033	−	0.993817i	$-0.464584\pi$
$379$	1638.48	0.222066	0.111033	+	0.993817i	$0.464584\pi$
$380$	8483.60	1.14526
$381$	3163.22	0.425346
$382$	−2475.86	−0.331613
$383$	3744.32	0.499545	0.249773	−	0.968305i	$-0.419644\pi$
$383$	3744.32	0.499545	0.249773	+	0.968305i	$0.419644\pi$
$384$	−3136.60	−0.416834
$385$	−3679.07	−0.487020
$386$	193.711	0.0255431
$387$	2017.40	0.264988
$388$	−387.775	−0.0507378
$389$	−8540.84	−1.11321	−0.556604	−	0.830778i	$-0.687896\pi$
$389$	−8540.84	−1.11321	−0.556604	+	0.830778i	$0.687896\pi$
$390$	326.848	0.0424374
$391$	15876.1	2.05343
$392$	3910.24	0.503818
$393$	4543.59	0.583191
$394$	−563.527	−0.0720560
$395$	−9260.35	−1.17959
$396$	760.312	0.0964826
$397$	4540.78	0.574043	0.287022	−	0.957924i	$-0.407335\pi$
$397$	4540.78	0.574043	0.287022	+	0.957924i	$0.407335\pi$
$398$	1930.29	0.243107
$399$	7766.02	0.974404
$400$	999.901	0.124988
$401$	−4308.74	−0.536579	−0.268290	−	0.963338i	$-0.586458\pi$
$401$	−4308.74	−0.536579	−0.268290	+	0.963338i	$0.586458\pi$
$402$	1139.21	0.141340
$403$	728.194	0.0900098
$404$	−12588.2	−1.55022
$405$	967.678	0.118727
$406$	−4714.21	−0.576262
$407$	−2645.29	−0.322168
$408$	−2942.78	−0.357082
$409$	10055.0	1.21562	0.607809	−	0.794084i	$-0.292049\pi$
$409$	10055.0	1.21562	0.607809	+	0.794084i	$0.292049\pi$
$410$	−108.969	−0.0131258
$411$	−5865.91	−0.704000
$412$	3079.65	0.368261
$413$	−2936.95	−0.349922
$414$	−731.055	−0.0867859
$415$	555.508	0.0657080
$416$	−1658.50	−0.195468
$417$	−4893.73	−0.574693
$418$	−575.438	−0.0673339
$419$	10139.6	1.18222	0.591111	−	0.806590i	$-0.298690\pi$
$419$	10139.6	1.18222	0.591111	+	0.806590i	$0.298690\pi$
$420$	7705.90	0.895261
$421$	−7377.31	−0.854033	−0.427017	−	0.904244i	$-0.640435\pi$
$421$	−7377.31	−0.854033	−0.427017	+	0.904244i	$0.640435\pi$
$422$	−120.581	−0.0139095
$423$	4091.31	0.470275
$424$	−3127.15	−0.358179
$425$	1959.68	0.223667
$426$	−501.273	−0.0570112
$427$	−1707.77	−0.193548
$428$	−10446.8	−1.17983
$429$	531.942	0.0598657
$430$	−1515.04	−0.169911
$431$	−5155.57	−0.576183	−0.288092	−	0.957603i	$-0.593021\pi$
$431$	−5155.57	−0.576183	−0.288092	+	0.957603i	$0.593021\pi$
$432$	−1523.36	−0.169658
$433$	−15582.0	−1.72939	−0.864693	−	0.502301i	$-0.832487\pi$
$433$	−15582.0	−1.72939	−0.864693	+	0.502301i	$0.832487\pi$
$434$	−715.525	−0.0791389
$435$	−10667.2	−1.17575
$436$	14945.3	1.64163
$437$	−13275.7	−1.45323
$438$	−1805.14	−0.196925
$439$	3326.11	0.361609	0.180805	−	0.983519i	$-0.442130\pi$
$439$	3326.11	0.361609	0.180805	+	0.983519i	$0.442130\pi$
$440$	−1165.76	−0.126308
$441$	3967.10	0.428367
$442$	−1008.42	−0.108520
$443$	−404.276	−0.0433583	−0.0216792	−	0.999765i	$-0.506901\pi$
$443$	−404.276	−0.0433583	−0.0216792	+	0.999765i	$0.506901\pi$
$444$	5540.63	0.592222
$445$	−12485.0	−1.32999
$446$	−143.689	−0.0152553
$447$	2440.45	0.258231
$448$	−11006.9	−1.16077
$449$	−8938.23	−0.939468	−0.469734	−	0.882808i	$-0.655650\pi$
$449$	−8938.23	−0.939468	−0.469734	+	0.882808i	$0.655650\pi$
$450$	−90.2381	−0.00945304
$451$	−177.346	−0.0185164
$452$	5624.44	0.585291
$453$	−2716.11	−0.281708
$454$	2913.26	0.301159
$455$	5391.33	0.555493
$456$	2460.77	0.252711
$457$	4815.56	0.492915	0.246458	−	0.969154i	$-0.420733\pi$
$457$	4815.56	0.492915	0.246458	+	0.969154i	$0.420733\pi$
$458$	2381.78	0.242998
$459$	−2985.58	−0.303606
$460$	−13172.9	−1.33520
$461$	12659.2	1.27895	0.639475	−	0.768812i	$-0.279152\pi$
$461$	12659.2	1.27895	0.639475	+	0.768812i	$0.279152\pi$
$462$	−522.687	−0.0526355
$463$	−14299.8	−1.43535	−0.717677	−	0.696376i	$-0.754794\pi$
$463$	−14299.8	−1.43535	−0.717677	+	0.696376i	$0.754794\pi$
$464$	16792.6	1.68013
$465$	−1619.06	−0.161467
$466$	−489.385	−0.0486488
$467$	11160.7	1.10590	0.552950	−	0.833215i	$-0.313502\pi$
$467$	11160.7	1.10590	0.552950	+	0.833215i	$0.313502\pi$
$468$	−1114.17	−0.110048
$469$	18791.3	1.85011
$470$	−3072.51	−0.301541
$471$	3362.27	0.328928
$472$	−930.613	−0.0907520
$473$	−2465.71	−0.239690
$474$	−1315.62	−0.127486
$475$	−1638.69	−0.158291
$476$	−23775.0	−2.28934
$477$	−3172.63	−0.304538
$478$	−56.3997	−0.00539679
$479$	16007.5	1.52694	0.763469	−	0.645845i	$-0.223495\pi$
$479$	16007.5	1.52694	0.763469	+	0.645845i	$0.223495\pi$
$480$	3687.50	0.350647
$481$	3876.43	0.367463
$482$	−2260.22	−0.213590
$483$	−12058.7	−1.13600
$484$	−929.271	−0.0872718
$485$	603.210	0.0564749
$486$	137.478	0.0128316
$487$	7509.99	0.698789	0.349394	−	0.936976i	$-0.386387\pi$
$487$	7509.99	0.698789	0.349394	+	0.936976i	$0.386387\pi$
$488$	−541.130	−0.0501964
$489$	2627.67	0.243001
$490$	−2979.24	−0.274670
$491$	2995.01	0.275281	0.137641	−	0.990482i	$-0.456048\pi$
$491$	2995.01	0.275281	0.137641	+	0.990482i	$0.456048\pi$
$492$	371.455	0.0340376
$493$	32911.4	3.00660
$494$	843.249	0.0768008
$495$	−1182.72	−0.107392
$496$	2548.79	0.230734
$497$	−8268.46	−0.746260
$498$	78.9212	0.00710149
$499$	357.476	0.0320698	0.0160349	−	0.999871i	$-0.494896\pi$
$499$	357.476	0.0320698	0.0160349	+	0.999871i	$0.494896\pi$
$500$	9842.65	0.880354
$501$	−4337.52	−0.386799
$502$	−2556.34	−0.227281
$503$	−19511.3	−1.72955	−0.864775	−	0.502159i	$-0.832539\pi$
$503$	−19511.3	−1.72955	−0.864775	+	0.502159i	$0.832539\pi$
$504$	2235.19	0.197546
$505$	19581.9	1.72551
$506$	893.511	0.0785008
$507$	5811.49	0.509068
$508$	8097.76	0.707244
$509$	11970.9	1.04244	0.521220	−	0.853422i	$-0.325477\pi$
$509$	11970.9	1.04244	0.521220	+	0.853422i	$0.325477\pi$
$510$	2242.13	0.194673
$511$	−29775.7	−2.57769
$512$	−9809.06	−0.846686
$513$	2496.56	0.214865
$514$	3494.04	0.299836
$515$	−4790.61	−0.409902
$516$	5164.49	0.440609
$517$	−5000.49	−0.425380
$518$	−3808.98	−0.323083
$519$	9930.44	0.839881
$520$	1708.32	0.144067
$521$	−7447.93	−0.626295	−0.313148	−	0.949704i	$-0.601383\pi$
$521$	−7447.93	−0.626295	−0.313148	+	0.949704i	$0.601383\pi$
$522$	−1515.49	−0.127071
$523$	19991.7	1.67146	0.835732	−	0.549137i	$-0.185044\pi$
$523$	19991.7	1.67146	0.835732	+	0.549137i	$0.185044\pi$
$524$	11631.5	0.969702
$525$	−1488.47	−0.123738
$526$	−4376.34	−0.362771
$527$	4995.30	0.412901
$528$	1861.88	0.153462
$529$	8446.82	0.694240
$530$	2382.60	0.195271
$531$	−944.147	−0.0771610
$532$	19880.8	1.62019
$533$	259.884	0.0211197
$534$	−1773.75	−0.143741
$535$	16250.7	1.31323
$536$	5954.26	0.479823
$537$	−184.942	−0.0148619
$538$	−2597.53	−0.208155
$539$	−4848.68	−0.387472
$540$	2477.23	0.197413
$541$	14394.0	1.14390	0.571948	−	0.820290i	$-0.306188\pi$
$541$	14394.0	1.14390	0.571948	+	0.820290i	$0.306188\pi$
$542$	−1682.72	−0.133356
$543$	−5958.29	−0.470893
$544$	−11377.1	−0.896668
$545$	−23248.4	−1.82725
$546$	765.948	0.0600358
$547$	−115.772	−0.00904948	−0.00452474	−	0.999990i	$-0.501440\pi$
$547$	−115.772	−0.00904948	−0.00452474	+	0.999990i	$0.501440\pi$
$548$	−15016.6	−1.17058
$549$	−549.000	−0.0426790
$550$	110.291	0.00855059
$551$	−27520.7	−2.12780
$552$	−3820.96	−0.294621
$553$	−21701.1	−1.66876
$554$	−2699.08	−0.206991
$555$	−8618.83	−0.659187
$556$	−12527.8	−0.955571
$557$	−14063.2	−1.06980	−0.534899	−	0.844916i	$-0.679651\pi$
$557$	−14063.2	−1.06980	−0.534899	+	0.844916i	$0.679651\pi$
$558$	−230.021	−0.0174508
$559$	3613.27	0.273390
$560$	18870.5	1.42397
$561$	3649.04	0.274622
$562$	1048.80	0.0787207
$563$	−23824.8	−1.78347	−0.891737	−	0.452555i	$-0.850513\pi$
$563$	−23824.8	−1.78347	−0.891737	+	0.452555i	$0.850513\pi$
$564$	10473.6	0.781950
$565$	−8749.21	−0.651472
$566$	−2851.09	−0.211732
$567$	2267.70	0.167962
$568$	−2619.97	−0.193542
$569$	20618.5	1.51911	0.759553	−	0.650445i	$-0.225418\pi$
$569$	20618.5	1.51911	0.759553	+	0.650445i	$0.225418\pi$
$570$	−1874.88	−0.137772
$571$	1756.15	0.128708	0.0643542	−	0.997927i	$-0.479501\pi$
$571$	1756.15	0.128708	0.0643542	+	0.997927i	$0.479501\pi$
$572$	1361.76	0.0995419
$573$	−13128.6	−0.957166
$574$	−255.362	−0.0185690
$575$	2544.48	0.184543
$576$	−3538.40	−0.255960
$577$	−5241.71	−0.378189	−0.189095	−	0.981959i	$-0.560555\pi$
$577$	−5241.71	−0.378189	−0.189095	+	0.981959i	$0.560555\pi$
$578$	−4138.09	−0.297789
$579$	1027.18	0.0737276
$580$	−27307.6	−1.95498
$581$	1301.80	0.0929565
$582$	85.6982	0.00610362
$583$	3877.66	0.275465
$584$	−9434.83	−0.668520
$585$	1733.16	0.122491
$586$	−2973.69	−0.209628
$587$	−13341.9	−0.938126	−0.469063	−	0.883165i	$-0.655408\pi$
$587$	−13341.9	−0.938126	−0.469063	+	0.883165i	$0.655408\pi$
$588$	10155.7	0.712267
$589$	−4177.10	−0.292214
$590$	709.041	0.0494758
$591$	−2988.19	−0.207982
$592$	13568.1	0.941968
$593$	−359.561	−0.0248995	−0.0124497	−	0.999922i	$-0.503963\pi$
$593$	−359.561	−0.0248995	−0.0124497	+	0.999922i	$0.503963\pi$
$594$	−168.029	−0.0116066
$595$	36983.7	2.54821
$596$	6247.49	0.429374
$597$	10235.6	0.701703
$598$	−1309.36	−0.0895377
$599$	−9858.97	−0.672498	−0.336249	−	0.941773i	$-0.609158\pi$
$599$	−9858.97	−0.672498	−0.336249	+	0.941773i	$0.609158\pi$
$600$	−471.642	−0.0320912
$601$	22293.1	1.51307	0.756533	−	0.653955i	$-0.226891\pi$
$601$	22293.1	1.51307	0.756533	+	0.653955i	$0.226891\pi$
$602$	−3550.40	−0.240371
$603$	6040.86	0.407965
$604$	−6953.16	−0.468411
$605$	1445.54	0.0971400
$606$	2782.00	0.186487
$607$	1052.00	0.0703449	0.0351725	−	0.999381i	$-0.488802\pi$
$607$	1052.00	0.0703449	0.0351725	+	0.999381i	$0.488802\pi$
$608$	9513.55	0.634581
$609$	−24997.8	−1.66332
$610$	412.291	0.0273659
$611$	7327.74	0.485186
$612$	−7643.01	−0.504821
$613$	17156.8	1.13044	0.565218	−	0.824942i	$-0.308792\pi$
$613$	17156.8	1.13044	0.565218	+	0.824942i	$0.308792\pi$
$614$	1106.31	0.0727149
$615$	−577.824	−0.0378864
$616$	−2731.90	−0.178687
$617$	−8268.83	−0.539531	−0.269766	−	0.962926i	$-0.586946\pi$
$617$	−8268.83	−0.539531	−0.269766	+	0.962926i	$0.586946\pi$
$618$	−680.604	−0.0443008
$619$	21493.9	1.39566	0.697828	−	0.716265i	$-0.254150\pi$
$619$	21493.9	1.39566	0.697828	+	0.716265i	$0.254150\pi$
$620$	−4144.76	−0.268480
$621$	−3876.53	−0.250499
$622$	−3572.18	−0.230276
$623$	−29257.9	−1.88153
$624$	−2728.41	−0.175038
$625$	−17526.2	−1.12168
$626$	−5325.98	−0.340046
$627$	−3051.35	−0.194353
$628$	8607.33	0.546926
$629$	26591.7	1.68566
$630$	−1703.01	−0.107697
$631$	−8800.13	−0.555194	−0.277597	−	0.960698i	$-0.589538\pi$
$631$	−8800.13	−0.555194	−0.277597	+	0.960698i	$0.589538\pi$
$632$	−6876.28	−0.432791
$633$	−639.399	−0.0401482
$634$	5719.23	0.358264
$635$	−12596.6	−0.787215
$636$	−8121.85	−0.506371
$637$	7105.28	0.441949
$638$	1852.26	0.114940
$639$	−2658.08	−0.164557
$640$	12490.6	0.771461
$641$	−2615.53	−0.161166	−0.0805829	−	0.996748i	$-0.525678\pi$
$641$	−2615.53	−0.161166	−0.0805829	+	0.996748i	$0.525678\pi$
$642$	2308.75	0.141930
$643$	−11167.0	−0.684889	−0.342445	−	0.939538i	$-0.611255\pi$
$643$	−11167.0	−0.684889	−0.342445	+	0.939538i	$0.611255\pi$
$644$	−30869.9	−1.88889
$645$	−8033.72	−0.490430
$646$	5784.56	0.352308
$647$	7727.39	0.469544	0.234772	−	0.972050i	$-0.424566\pi$
$647$	7727.39	0.469544	0.234772	+	0.972050i	$0.424566\pi$
$648$	718.550	0.0435607
$649$	1153.96	0.0697947
$650$	−161.621	−0.00975277
$651$	−3794.18	−0.228426
$652$	6726.78	0.404050
$653$	5022.25	0.300974	0.150487	−	0.988612i	$-0.451916\pi$
$653$	5022.25	0.300974	0.150487	+	0.988612i	$0.451916\pi$
$654$	−3302.91	−0.197483
$655$	−18093.6	−1.07935
$656$	909.632	0.0541390
$657$	−9572.04	−0.568403
$658$	−7200.25	−0.426588
$659$	−21939.7	−1.29689	−0.648445	−	0.761261i	$-0.724580\pi$
$659$	−21939.7	−1.29689	−0.648445	+	0.761261i	$0.724580\pi$
$660$	−3027.73	−0.178567
$661$	−16085.2	−0.946505	−0.473253	−	0.880927i	$-0.656920\pi$
$661$	−16085.2	−0.946505	−0.473253	+	0.880927i	$0.656920\pi$
$662$	431.708	0.0253456
$663$	−5347.32	−0.313232
$664$	412.493	0.0241082
$665$	−30926.0	−1.80339
$666$	−1224.48	−0.0712427
$667$	42732.7	2.48069
$668$	−11103.9	−0.643150
$669$	−761.932	−0.0440329
$670$	−4536.60	−0.261588
$671$	671.000	0.0386046
$672$	8641.44	0.496058
$673$	−9730.27	−0.557317	−0.278659	−	0.960390i	$-0.589890\pi$
$673$	−9730.27	−0.557317	−0.278659	+	0.960390i	$0.589890\pi$
$674$	2742.14	0.156711
$675$	−478.501	−0.0272852
$676$	14877.3	0.846453
$677$	19575.1	1.11127	0.555637	−	0.831425i	$-0.312474\pi$
$677$	19575.1	1.11127	0.555637	+	0.831425i	$0.312474\pi$
$678$	−1243.00	−0.0704089
$679$	1413.59	0.0798946
$680$	11718.8	0.660875
$681$	15448.0	0.869264
$682$	281.137	0.0157849
$683$	−5506.91	−0.308515	−0.154258	−	0.988031i	$-0.549299\pi$
$683$	−5506.91	−0.308515	−0.154258	+	0.988031i	$0.549299\pi$
$684$	6391.12	0.357267
$685$	23359.3	1.30294
$686$	−1548.88	−0.0862050
$687$	12629.7	0.701389
$688$	12647.0	0.700817
$689$	−5682.34	−0.314194
$690$	2911.22	0.160620
$691$	24893.4	1.37046	0.685231	−	0.728326i	$-0.259701\pi$
$691$	24893.4	1.37046	0.685231	+	0.728326i	$0.259701\pi$
$692$	25421.7	1.39651
$693$	−2771.63	−0.151927
$694$	−622.956	−0.0340736
$695$	19487.9	1.06362
$696$	−7920.90	−0.431381
$697$	1782.76	0.0968823
$698$	1278.93	0.0693529
$699$	−2595.04	−0.140420
$700$	−3810.45	−0.205745
$701$	−16858.3	−0.908315	−0.454158	−	0.890921i	$-0.650060\pi$
$701$	−16858.3	−0.908315	−0.454158	+	0.890921i	$0.650060\pi$
$702$	246.231	0.0132384
$703$	−22236.1	−1.19296
$704$	4324.71	0.231525
$705$	−16292.5	−0.870369
$706$	4589.31	0.244647
$707$	45888.9	2.44106
$708$	−2416.99	−0.128300
$709$	29657.1	1.57094	0.785469	−	0.618902i	$-0.212422\pi$
$709$	29657.1	1.57094	0.785469	+	0.618902i	$0.212422\pi$
$710$	1996.18	0.105514
$711$	−6976.28	−0.367976
$712$	−9270.76	−0.487972
$713$	6485.99	0.340676
$714$	5254.29	0.275402
$715$	−2118.31	−0.110797
$716$	−473.448	−0.0247117
$717$	−299.068	−0.0155773
$718$	3061.66	0.159136
$719$	−2117.71	−0.109843	−0.0549217	−	0.998491i	$-0.517491\pi$
$719$	−2117.71	−0.109843	−0.0549217	+	0.998491i	$0.517491\pi$
$720$	6066.33	0.313998
$721$	−11226.5	−0.579885
$722$	−956.568	−0.0493072
$723$	−11985.2	−0.616506
$724$	−15253.1	−0.782978
$725$	5274.74	0.270205
$726$	205.369	0.0104986
$727$	−26848.6	−1.36968	−0.684841	−	0.728692i	$-0.740128\pi$
$727$	−26848.6	−1.36968	−0.684841	+	0.728692i	$0.740128\pi$
$728$	4003.34	0.203810
$729$	729.000	0.0370370
$730$	7188.46	0.364462
$731$	24786.5	1.25412
$732$	−1405.43	−0.0709645
$733$	21470.2	1.08188	0.540941	−	0.841061i	$-0.318068\pi$
$733$	21470.2	1.08188	0.540941	+	0.841061i	$0.318068\pi$
$734$	−3020.14	−0.151874
$735$	−15797.9	−0.792806
$736$	−14772.2	−0.739823
$737$	−7383.27	−0.369018
$738$	−82.0917	−0.00409463
$739$	16371.7	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$739$	16371.7	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$740$	−22064.0	−1.09606
$741$	4471.46	0.221678
$742$	5583.48	0.276248
$743$	−22285.5	−1.10037	−0.550186	−	0.835042i	$-0.685443\pi$
$743$	−22285.5	−1.10037	−0.550186	+	0.835042i	$0.685443\pi$
$744$	−1202.24	−0.0592421
$745$	−9718.39	−0.477925
$746$	1719.00	0.0843663
$747$	418.492	0.0204977
$748$	9341.45	0.456627
$749$	38082.6	1.85782
$750$	−2175.23	−0.105904
$751$	36906.1	1.79324	0.896621	−	0.442800i	$-0.146015\pi$
$751$	36906.1	1.79324	0.896621	+	0.442800i	$0.146015\pi$
$752$	25648.2	1.24374
$753$	−13555.4	−0.656024
$754$	−2714.31	−0.131100
$755$	10816.1	0.521376
$756$	5805.24	0.279278
$757$	−7002.83	−0.336225	−0.168112	−	0.985768i	$-0.553767\pi$
$757$	−7002.83	−0.336225	−0.168112	+	0.985768i	$0.553767\pi$
$758$	−926.978	−0.0444187
$759$	4737.98	0.226585
$760$	−9799.31	−0.467708
$761$	−27829.2	−1.32563	−0.662817	−	0.748781i	$-0.730639\pi$
$761$	−27829.2	−1.32563	−0.662817	+	0.748781i	$0.730639\pi$
$762$	−1789.61	−0.0850796
$763$	−54481.2	−2.58500
$764$	−33608.9	−1.59153
$765$	11889.2	0.561903
$766$	−2118.37	−0.0999213
$767$	−1691.01	−0.0796076
$768$	−7661.18	−0.359959
$769$	−19864.6	−0.931517	−0.465758	−	0.884912i	$-0.654218\pi$
$769$	−19864.6	−0.931517	−0.465758	+	0.884912i	$0.654218\pi$
$770$	2081.45	0.0974160
$771$	18527.7	0.865445
$772$	2629.56	0.122591
$773$	11259.6	0.523908	0.261954	−	0.965080i	$-0.415633\pi$
$773$	11259.6	0.523908	0.261954	+	0.965080i	$0.415633\pi$
$774$	−1141.35	−0.0530040
$775$	800.601	0.0371077
$776$	447.914	0.0207206
$777$	−20197.7	−0.932547
$778$	4832.02	0.222669
$779$	−1490.75	−0.0685646
$780$	4436.85	0.203672
$781$	3248.76	0.148847
$782$	−8981.98	−0.410735
$783$	−8036.10	−0.366777
$784$	24869.6	1.13291
$785$	−13389.3	−0.608770
$786$	−2570.56	−0.116652
$787$	12758.2	0.577866	0.288933	−	0.957349i	$-0.406699\pi$
$787$	12758.2	0.577866	0.288933	+	0.957349i	$0.406699\pi$
$788$	−7649.68	−0.345823
$789$	−23206.2	−1.04710
$790$	5239.09	0.235947
$791$	−20503.2	−0.921632
$792$	−878.228	−0.0394021
$793$	−983.287	−0.0440322
$794$	−2568.97	−0.114823
$795$	12634.1	0.563629
$796$	26202.9	1.16676
$797$	−3283.24	−0.145920	−0.0729600	−	0.997335i	$-0.523245\pi$
$797$	−3283.24	−0.145920	−0.0729600	+	0.997335i	$0.523245\pi$
$798$	−4393.66	−0.194905
$799$	50267.2	2.22569
$800$	−1823.41	−0.0805842
$801$	−9405.58	−0.414894
$802$	2437.69	0.107329
$803$	11699.2	0.514140
$804$	15464.4	0.678344
$805$	48020.3	2.10248
$806$	−411.979	−0.0180042
$807$	−13773.8	−0.600818
$808$	14540.5	0.633087
$809$	−44601.3	−1.93832	−0.969158	−	0.246441i	$-0.920739\pi$
$809$	−44601.3	−1.93832	−0.969158	+	0.246441i	$0.920739\pi$
$810$	−547.468	−0.0237482
$811$	24415.3	1.05714	0.528568	−	0.848891i	$-0.322729\pi$
$811$	24415.3	1.05714	0.528568	+	0.848891i	$0.322729\pi$
$812$	−63993.8	−2.76569
$813$	−8922.89	−0.384919
$814$	1496.59	0.0644415
$815$	−10464.0	−0.449738
$816$	−18716.5	−0.802950
$817$	−20726.6	−0.887553
$818$	−5688.66	−0.243153
$819$	4061.56	0.173287
$820$	−1479.21	−0.0629956
$821$	5527.83	0.234985	0.117492	−	0.993074i	$-0.462514\pi$
$821$	5527.83	0.234985	0.117492	+	0.993074i	$0.462514\pi$
$822$	3318.67	0.140817
$823$	−22876.0	−0.968901	−0.484451	−	0.874819i	$-0.660980\pi$
$823$	−22876.0	−0.968901	−0.484451	+	0.874819i	$0.660980\pi$
$824$	−3557.27	−0.150393
$825$	584.835	0.0246804
$826$	1661.59	0.0699930
$827$	12820.0	0.539051	0.269525	−	0.962993i	$-0.413133\pi$
$827$	12820.0	0.539051	0.269525	+	0.962993i	$0.413133\pi$
$828$	−9923.81	−0.416517
$829$	−17042.3	−0.713998	−0.356999	−	0.934105i	$-0.616200\pi$
$829$	−17042.3	−0.713998	−0.356999	+	0.934105i	$0.616200\pi$
$830$	−314.281	−0.0131432
$831$	−14312.3	−0.597458
$832$	−6337.45	−0.264076
$833$	48741.2	2.02735
$834$	2768.65	0.114953
$835$	17272.9	0.715874
$836$	−7811.37	−0.323160
$837$	−1219.72	−0.0503701
$838$	−5736.51	−0.236473
$839$	−28853.3	−1.18728	−0.593640	−	0.804731i	$-0.702310\pi$
$839$	−28853.3	−1.18728	−0.593640	+	0.804731i	$0.702310\pi$
$840$	−8901.00	−0.365612
$841$	64196.6	2.63219
$842$	4173.75	0.170828
$843$	5561.43	0.227219
$844$	−1636.85	−0.0667566
$845$	−23142.6	−0.942165
$846$	−2314.68	−0.0940665
$847$	3387.54	0.137423
$848$	−19889.1	−0.805417
$849$	−15118.3	−0.611143
$850$	−1108.70	−0.0447388
$851$	34527.1	1.39080
$852$	−6804.61	−0.273617
$853$	−35050.9	−1.40694	−0.703470	−	0.710725i	$-0.748367\pi$
$853$	−35050.9	−1.40694	−0.703470	+	0.710725i	$0.748367\pi$
$854$	966.179	0.0387142
$855$	−9941.82	−0.397664
$856$	12067.0	0.481824
$857$	35541.5	1.41666	0.708328	−	0.705883i	$-0.249450\pi$
$857$	35541.5	1.41666	0.708328	+	0.705883i	$0.249450\pi$
$858$	−300.949	−0.0119746
$859$	10647.0	0.422902	0.211451	−	0.977389i	$-0.432181\pi$
$859$	10647.0	0.422902	0.211451	+	0.977389i	$0.432181\pi$
$860$	−20566.1	−0.815464
$861$	−1354.10	−0.0535975
$862$	2916.79	0.115251
$863$	7136.98	0.281513	0.140756	−	0.990044i	$-0.455047\pi$
$863$	7136.98	0.281513	0.140756	+	0.990044i	$0.455047\pi$
$864$	2777.98	0.109385
$865$	−39545.2	−1.55442
$866$	8815.60	0.345920
$867$	−21942.8	−0.859536
$868$	−9713.00	−0.379816
$869$	8526.57	0.332847
$870$	6034.99	0.235179
$871$	10819.5	0.420900
$872$	−17263.1	−0.670416
$873$	454.428	0.0176175
$874$	7510.78	0.290682
$875$	−35880.2	−1.38625
$876$	−24504.2	−0.945113
$877$	29158.1	1.12269	0.561346	−	0.827582i	$-0.310284\pi$
$877$	29158.1	1.12269	0.561346	+	0.827582i	$0.310284\pi$
$878$	−1881.76	−0.0723307
$879$	−15768.4	−0.605069
$880$	−7414.40	−0.284022
$881$	19248.9	0.736110	0.368055	−	0.929804i	$-0.380024\pi$
$881$	19248.9	0.736110	0.368055	+	0.929804i	$0.380024\pi$
$882$	−2244.41	−0.0856838
$883$	−42577.6	−1.62271	−0.811353	−	0.584556i	$-0.801269\pi$
$883$	−42577.6	−1.62271	−0.811353	+	0.584556i	$0.801269\pi$
$884$	−13689.0	−0.520827
$885$	3759.80	0.142807
$886$	228.721	0.00867273
$887$	−34429.0	−1.30328	−0.651642	−	0.758527i	$-0.725919\pi$
$887$	−34429.0	−1.30328	−0.651642	+	0.758527i	$0.725919\pi$
$888$	−6399.92	−0.241855
$889$	−29519.4	−1.11367
$890$	7063.45	0.266031
$891$	−891.000	−0.0335013
$892$	−1950.53	−0.0732158
$893$	−42033.7	−1.57514
$894$	−1380.70	−0.0516525
$895$	736.480	0.0275059
$896$	29271.0	1.09138
$897$	−6943.06	−0.258441
$898$	5056.85	0.187917
$899$	13445.5	0.498814
$900$	−1224.95	−0.0453686
$901$	−38980.0	−1.44130
$902$	100.334	0.00370373
$903$	−18826.5	−0.693808
$904$	−6496.73	−0.239024
$905$	23727.2	0.871513
$906$	1536.65	0.0563485
$907$	23656.5	0.866042	0.433021	−	0.901384i	$-0.357448\pi$
$907$	23656.5	0.866042	0.433021	+	0.901384i	$0.357448\pi$
$908$	39546.5	1.44537
$909$	14752.0	0.538276
$910$	−3050.17	−0.111112
$911$	8008.73	0.291264	0.145632	−	0.989339i	$-0.453479\pi$
$911$	8008.73	0.291264	0.145632	+	0.989339i	$0.453479\pi$
$912$	15650.8	0.568256
$913$	−511.490	−0.0185409
$914$	−2724.42	−0.0985951
$915$	2186.24	0.0789888
$916$	32331.8	1.16624
$917$	−42401.2	−1.52695
$918$	1689.11	0.0607285
$919$	6290.04	0.225777	0.112889	−	0.993608i	$-0.463990\pi$
$919$	6290.04	0.225777	0.112889	+	0.993608i	$0.463990\pi$
$920$	15215.9	0.545275
$921$	5866.36	0.209884
$922$	−7161.98	−0.255821
$923$	−4760.75	−0.169775
$924$	−7095.29	−0.252617
$925$	4261.87	0.151491
$926$	8090.19	0.287106
$927$	−3609.01	−0.127870
$928$	−30622.9	−1.08324
$929$	21656.3	0.764821	0.382411	−	0.923993i	$-0.375094\pi$
$929$	21656.3	0.764821	0.382411	+	0.923993i	$0.375094\pi$
$930$	915.993	0.0322974
$931$	−40757.6	−1.43478
$932$	−6643.23	−0.233483
$933$	−18942.0	−0.664667
$934$	−6314.21	−0.221207
$935$	−14531.3	−0.508260
$936$	1286.96	0.0449419
$937$	−54222.4	−1.89047	−0.945233	−	0.326396i	$-0.894166\pi$
$937$	−54222.4	−1.89047	−0.945233	+	0.326396i	$0.894166\pi$
$938$	−10631.2	−0.370066
$939$	−28241.8	−0.981508
$940$	−41708.3	−1.44721
$941$	49346.2	1.70950	0.854750	−	0.519040i	$-0.173711\pi$
$941$	49346.2	1.70950	0.854750	+	0.519040i	$0.173711\pi$
$942$	−1902.22	−0.0657937
$943$	2314.77	0.0799356
$944$	−5918.81	−0.204069
$945$	−9030.45	−0.310858
$946$	1394.99	0.0479440
$947$	−44425.9	−1.52444	−0.762222	−	0.647316i	$-0.775891\pi$
$947$	−44425.9	−1.52444	−0.762222	+	0.647316i	$0.775891\pi$
$948$	−17859.1	−0.611853
$949$	−17144.0	−0.586426
$950$	927.097	0.0316621
$951$	30327.1	1.03409
$952$	27462.3	0.934935
$953$	−19703.0	−0.669720	−0.334860	−	0.942268i	$-0.608689\pi$
$953$	−19703.0	−0.669720	−0.334860	+	0.942268i	$0.608689\pi$
$954$	1794.93	0.0609151
$955$	52281.0	1.77149
$956$	−765.606	−0.0259011
$957$	9821.90	0.331763
$958$	−9056.34	−0.305425
$959$	54741.1	1.84326
$960$	14090.7	0.473723
$961$	−27750.2	−0.931497
$962$	−2193.11	−0.0735016
$963$	12242.5	0.409666
$964$	−30681.7	−1.02510
$965$	−4090.46	−0.136453
$966$	6822.26	0.227228
$967$	−9814.88	−0.326396	−0.163198	−	0.986593i	$-0.552181\pi$
$967$	−9814.88	−0.326396	−0.163198	+	0.986593i	$0.552181\pi$
$968$	1073.39	0.0356405
$969$	30673.5	1.01690
$970$	−341.269	−0.0112964
$971$	−36719.1	−1.21357	−0.606783	−	0.794867i	$-0.707540\pi$
$971$	−36719.1	−1.21357	−0.606783	+	0.794867i	$0.707540\pi$
$972$	1866.22	0.0615834
$973$	45668.6	1.50470
$974$	−4248.81	−0.139775
$975$	−857.020	−0.0281504
$976$	−3441.65	−0.112874
$977$	21179.9	0.693557	0.346779	−	0.937947i	$-0.387276\pi$
$977$	21179.9	0.693557	0.346779	+	0.937947i	$0.387276\pi$
$978$	−1486.62	−0.0486062
$979$	11495.7	0.375285
$980$	−40442.1	−1.31824
$981$	−17514.2	−0.570015
$982$	−1694.44	−0.0550630
$983$	−26165.6	−0.848986	−0.424493	−	0.905431i	$-0.639548\pi$
$983$	−26165.6	−0.848986	−0.424493	+	0.905431i	$0.639548\pi$
$984$	−429.064	−0.0139005
$985$	11899.6	0.384927
$986$	−18619.8	−0.601394
$987$	−38180.4	−1.23130
$988$	11446.8	0.368595
$989$	32183.2	1.03475
$990$	669.128	0.0214811
$991$	39683.1	1.27202	0.636011	−	0.771680i	$-0.280583\pi$
$991$	39683.1	1.27202	0.636011	+	0.771680i	$0.280583\pi$
$992$	−4647.96	−0.148763
$993$	2289.20	0.0731575
$994$	4677.92	0.149270
$995$	−40760.5	−1.29869
$996$	1071.33	0.0340826
$997$	43229.9	1.37322	0.686612	−	0.727024i	$-0.259097\pi$
$997$	43229.9	1.37322	0.686612	+	0.727024i	$0.259097\pi$
$998$	−202.244	−0.00641475
$999$	−6492.99	−0.205635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.e.1.18	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.e.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-0.565755 q^{2} -3.00000 q^{3} -7.67992 q^{4} +11.9466 q^{5} +1.69726 q^{6} +27.9962 q^{7} +8.87099 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.565755 q^{2} -3.00000 q^{3} -7.67992 q^{4} +11.9466 q^{5} +1.69726 q^{6} +27.9962 q^{7} +8.87099 q^{8} +9.00000 q^{9} -6.75887 q^{10} -11.0000 q^{11} +23.0398 q^{12} +16.1195 q^{13} -15.8390 q^{14} -35.8399 q^{15} +56.4206 q^{16} +110.577 q^{17} -5.09179 q^{18} -92.4650 q^{19} -91.7493 q^{20} -83.9887 q^{21} +6.22330 q^{22} +143.575 q^{23} -26.6130 q^{24} +17.7223 q^{25} -9.11966 q^{26} -27.0000 q^{27} -215.009 q^{28} +297.633 q^{29} +20.2766 q^{30} +45.1749 q^{31} -102.888 q^{32} +33.0000 q^{33} -62.5595 q^{34} +334.461 q^{35} -69.1193 q^{36} +240.481 q^{37} +52.3125 q^{38} -48.3584 q^{39} +105.979 q^{40} +16.1224 q^{41} +47.5170 q^{42} +224.156 q^{43} +84.4791 q^{44} +107.520 q^{45} -81.2283 q^{46} +454.590 q^{47} -169.262 q^{48} +440.789 q^{49} -10.0265 q^{50} -331.731 q^{51} -123.796 q^{52} -352.514 q^{53} +15.2754 q^{54} -131.413 q^{55} +248.354 q^{56} +277.395 q^{57} -168.387 q^{58} -104.905 q^{59} +275.248 q^{60} -61.0000 q^{61} -25.5579 q^{62} +251.966 q^{63} -393.155 q^{64} +192.573 q^{65} -18.6699 q^{66} +671.206 q^{67} -849.223 q^{68} -430.725 q^{69} -189.223 q^{70} -295.342 q^{71} +79.8389 q^{72} -1063.56 q^{73} -136.053 q^{74} -53.1668 q^{75} +710.124 q^{76} -307.959 q^{77} +27.3590 q^{78} -775.142 q^{79} +674.036 q^{80} +81.0000 q^{81} -9.12130 q^{82} +46.4991 q^{83} +645.027 q^{84} +1321.02 q^{85} -126.817 q^{86} -892.900 q^{87} -97.5809 q^{88} -1045.06 q^{89} -60.8298 q^{90} +451.284 q^{91} -1102.65 q^{92} -135.525 q^{93} -257.186 q^{94} -1104.65 q^{95} +308.664 q^{96} +50.4920 q^{97} -249.379 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9}+O(q^{10}) \) 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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