LMFDB - Embedded newform 2013.4.a.d.1.19

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:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.197712 q^{2} -3.00000 q^{3} -7.96091 q^{4} +13.9720 q^{5} -0.593137 q^{6} +33.5020 q^{7} -3.15567 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.197712 q^{2} -3.00000 q^{3} -7.96091 q^{4} +13.9720 q^{5} -0.593137 q^{6} +33.5020 q^{7} -3.15567 q^{8} +9.00000 q^{9} +2.76244 q^{10} +11.0000 q^{11} +23.8827 q^{12} -56.8451 q^{13} +6.62376 q^{14} -41.9161 q^{15} +63.0634 q^{16} -9.82870 q^{17} +1.77941 q^{18} +46.6182 q^{19} -111.230 q^{20} -100.506 q^{21} +2.17484 q^{22} +63.4685 q^{23} +9.46701 q^{24} +70.2173 q^{25} -11.2390 q^{26} -27.0000 q^{27} -266.706 q^{28} -159.654 q^{29} -8.28733 q^{30} -226.731 q^{31} +37.7138 q^{32} -33.0000 q^{33} -1.94326 q^{34} +468.090 q^{35} -71.6482 q^{36} -145.767 q^{37} +9.21699 q^{38} +170.535 q^{39} -44.0911 q^{40} -496.790 q^{41} -19.8713 q^{42} -447.780 q^{43} -87.5700 q^{44} +125.748 q^{45} +12.5485 q^{46} -388.661 q^{47} -189.190 q^{48} +779.382 q^{49} +13.8828 q^{50} +29.4861 q^{51} +452.539 q^{52} -285.968 q^{53} -5.33824 q^{54} +153.692 q^{55} -105.721 q^{56} -139.854 q^{57} -31.5655 q^{58} +249.983 q^{59} +333.690 q^{60} -61.0000 q^{61} -44.8275 q^{62} +301.518 q^{63} -497.050 q^{64} -794.241 q^{65} -6.52451 q^{66} +396.993 q^{67} +78.2454 q^{68} -190.405 q^{69} +92.5473 q^{70} +145.950 q^{71} -28.4010 q^{72} -1192.69 q^{73} -28.8199 q^{74} -210.652 q^{75} -371.123 q^{76} +368.522 q^{77} +33.7170 q^{78} +981.246 q^{79} +881.122 q^{80} +81.0000 q^{81} -98.2217 q^{82} +281.167 q^{83} +800.119 q^{84} -137.327 q^{85} -88.5316 q^{86} +478.961 q^{87} -34.7124 q^{88} +1187.34 q^{89} +24.8620 q^{90} -1904.42 q^{91} -505.267 q^{92} +680.192 q^{93} -76.8430 q^{94} +651.350 q^{95} -113.141 q^{96} -443.790 q^{97} +154.094 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * 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 $

$2$	0.197712	0.0699019	0.0349510	−	0.999389i	$-0.488872\pi$
$2$	0.197712	0.0699019	0.0349510	+	0.999389i	$0.488872\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.96091	−0.995114
$5$	13.9720	1.24970	0.624848	−	0.780747i	$-0.285161\pi$
$5$	13.9720	1.24970	0.624848	+	0.780747i	$0.285161\pi$
$6$	−0.593137	−0.0403579
$7$	33.5020	1.80894	0.904468	−	0.426541i	$-0.140268\pi$
$7$	33.5020	1.80894	0.904468	+	0.426541i	$0.140268\pi$
$8$	−3.15567	−0.139462
$9$	9.00000	0.333333
$10$	2.76244	0.0873561
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.8827	0.574529
$13$	−56.8451	−1.21277	−0.606385	−	0.795171i	$-0.707381\pi$
$13$	−56.8451	−1.21277	−0.606385	+	0.795171i	$0.707381\pi$
$14$	6.62376	0.126448
$15$	−41.9161	−0.721512
$16$	63.0634	0.985365
$17$	−9.82870	−0.140224	−0.0701121	−	0.997539i	$-0.522336\pi$
$17$	−9.82870	−0.140224	−0.0701121	+	0.997539i	$0.522336\pi$
$18$	1.77941	0.0233006
$19$	46.6182	0.562892	0.281446	−	0.959577i	$-0.409186\pi$
$19$	46.6182	0.562892	0.281446	+	0.959577i	$0.409186\pi$
$20$	−111.230	−1.24359
$21$	−100.506	−1.04439
$22$	2.17484	0.0210762
$23$	63.4685	0.575395	0.287698	−	0.957721i	$-0.407110\pi$
$23$	63.4685	0.575395	0.287698	+	0.957721i	$0.407110\pi$
$24$	9.46701	0.0805186
$25$	70.2173	0.561738
$26$	−11.2390	−0.0847749
$27$	−27.0000	−0.192450
$28$	−266.706	−1.80010
$29$	−159.654	−1.02231	−0.511155	−	0.859489i	$-0.670782\pi$
$29$	−159.654	−1.02231	−0.511155	+	0.859489i	$0.670782\pi$
$30$	−8.28733	−0.0504351
$31$	−226.731	−1.31361	−0.656807	−	0.754059i	$-0.728093\pi$
$31$	−226.731	−1.31361	−0.656807	+	0.754059i	$0.728093\pi$
$32$	37.7138	0.208341
$33$	−33.0000	−0.174078
$34$	−1.94326	−0.00980193
$35$	468.090	2.26062
$36$	−71.6482	−0.331705
$37$	−145.767	−0.647673	−0.323837	−	0.946113i	$-0.604973\pi$
$37$	−145.767	−0.647673	−0.323837	+	0.946113i	$0.604973\pi$
$38$	9.21699	0.0393472
$39$	170.535	0.700193
$40$	−44.0911	−0.174285
$41$	−496.790	−1.89233	−0.946166	−	0.323681i	$-0.895079\pi$
$41$	−496.790	−1.89233	−0.946166	+	0.323681i	$0.895079\pi$
$42$	−19.8713	−0.0730049
$43$	−447.780	−1.58804	−0.794020	−	0.607891i	$-0.792016\pi$
$43$	−447.780	−1.58804	−0.794020	+	0.607891i	$0.792016\pi$
$44$	−87.5700	−0.300038
$45$	125.748	0.416565
$46$	12.5485	0.0402212
$47$	−388.661	−1.20621	−0.603106	−	0.797661i	$-0.706070\pi$
$47$	−388.661	−1.20621	−0.603106	+	0.797661i	$0.706070\pi$
$48$	−189.190	−0.568901
$49$	779.382	2.27225
$50$	13.8828	0.0392666
$51$	29.4861	0.0809584
$52$	452.539	1.20684
$53$	−285.968	−0.741147	−0.370573	−	0.928803i	$-0.620839\pi$
$53$	−285.968	−0.741147	−0.370573	+	0.928803i	$0.620839\pi$
$54$	−5.33824	−0.0134526
$55$	153.692	0.376797
$56$	−105.721	−0.252278
$57$	−139.854	−0.324986
$58$	−31.5655	−0.0714614
$59$	249.983	0.551611	0.275806	−	0.961213i	$-0.411055\pi$
$59$	249.983	0.551611	0.275806	+	0.961213i	$0.411055\pi$
$60$	333.690	0.717986
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−44.8275	−0.0918241
$63$	301.518	0.602979
$64$	−497.050	−0.970802
$65$	−794.241	−1.51559
$66$	−6.52451	−0.0121684
$67$	396.993	0.723887	0.361943	−	0.932200i	$-0.382113\pi$
$67$	396.993	0.723887	0.361943	+	0.932200i	$0.382113\pi$
$68$	78.2454	0.139539
$69$	−190.405	−0.332205
$70$	92.5473	0.158022
$71$	145.950	0.243959	0.121980	−	0.992533i	$-0.461076\pi$
$71$	145.950	0.243959	0.121980	+	0.992533i	$0.461076\pi$
$72$	−28.4010	−0.0464874
$73$	−1192.69	−1.91225	−0.956124	−	0.292963i	$-0.905359\pi$
$73$	−1192.69	−1.91225	−0.956124	+	0.292963i	$0.905359\pi$
$74$	−28.8199	−0.0452736
$75$	−210.652	−0.324320
$76$	−371.123	−0.560141
$77$	368.522	0.545415
$78$	33.7170	0.0489448
$79$	981.246	1.39745	0.698727	−	0.715389i	$-0.253750\pi$
$79$	981.246	1.39745	0.698727	+	0.715389i	$0.253750\pi$
$80$	881.122	1.23141
$81$	81.0000	0.111111
$82$	−98.2217	−0.132278
$83$	281.167	0.371833	0.185916	−	0.982566i	$-0.440475\pi$
$83$	281.167	0.371833	0.185916	+	0.982566i	$0.440475\pi$
$84$	800.119	1.03929
$85$	−137.327	−0.175237
$86$	−88.5316	−0.111007
$87$	478.961	0.590230
$88$	−34.7124	−0.0420495
$89$	1187.34	1.41413	0.707064	−	0.707150i	$-0.250019\pi$
$89$	1187.34	1.41413	0.707064	+	0.707150i	$0.250019\pi$
$90$	24.8620	0.0291187
$91$	−1904.42	−2.19382
$92$	−505.267	−0.572584
$93$	680.192	0.758415
$94$	−76.8430	−0.0843165
$95$	651.350	0.703443
$96$	−113.141	−0.120286
$97$	−443.790	−0.464537	−0.232268	−	0.972652i	$-0.574615\pi$
$97$	−443.790	−0.464537	−0.232268	+	0.972652i	$0.574615\pi$
$98$	154.094	0.158835
$99$	99.0000	0.100504
$100$	−558.993	−0.558993
$101$	−1945.44	−1.91662	−0.958309	−	0.285735i	$-0.907762\pi$
$101$	−1945.44	−1.91662	−0.958309	+	0.285735i	$0.907762\pi$
$102$	5.82977	0.00565915
$103$	−234.188	−0.224031	−0.112016	−	0.993706i	$-0.535731\pi$
$103$	−234.188	−0.224031	−0.112016	+	0.993706i	$0.535731\pi$
$104$	179.385	0.169136
$105$	−1404.27	−1.30517
$106$	−56.5395	−0.0518076
$107$	1495.16	1.35087	0.675433	−	0.737421i	$-0.263957\pi$
$107$	1495.16	1.35087	0.675433	+	0.737421i	$0.263957\pi$
$108$	214.945	0.191510
$109$	−323.291	−0.284088	−0.142044	−	0.989860i	$-0.545368\pi$
$109$	−323.291	−0.284088	−0.142044	+	0.989860i	$0.545368\pi$
$110$	30.3869	0.0263389
$111$	437.300	0.373934
$112$	2112.75	1.78246
$113$	1955.30	1.62778	0.813891	−	0.581018i	$-0.197345\pi$
$113$	1955.30	1.62778	0.813891	+	0.581018i	$0.197345\pi$
$114$	−27.6510	−0.0227171
$115$	886.783	0.719069
$116$	1270.99	1.01731
$117$	−511.606	−0.404257
$118$	49.4248	0.0385587
$119$	−329.281	−0.253657
$120$	132.273	0.100624
$121$	121.000	0.0909091
$122$	−12.0605	−0.00895002
$123$	1490.37	1.09254
$124$	1804.98	1.30719
$125$	−765.425	−0.547694
$126$	59.6138	0.0421494
$127$	−1604.50	−1.12107	−0.560536	−	0.828130i	$-0.689405\pi$
$127$	−1604.50	−1.12107	−0.560536	+	0.828130i	$0.689405\pi$
$128$	−399.983	−0.276202
$129$	1343.34	0.916856
$130$	−157.031	−0.105943
$131$	−1214.43	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$131$	−1214.43	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$132$	262.710	0.173227
$133$	1561.80	1.01824
$134$	78.4905	0.0506011
$135$	−377.244	−0.240504
$136$	31.0161	0.0195560
$137$	−786.580	−0.490526	−0.245263	−	0.969457i	$-0.578874\pi$
$137$	−786.580	−0.490526	−0.245263	+	0.969457i	$0.578874\pi$
$138$	−37.6455	−0.0232217
$139$	2325.14	1.41882	0.709410	−	0.704796i	$-0.248962\pi$
$139$	2325.14	1.41882	0.709410	+	0.704796i	$0.248962\pi$
$140$	−3726.42	−2.24957
$141$	1165.98	0.696407
$142$	28.8562	0.0170532
$143$	−625.297	−0.365664
$144$	567.570	0.328455
$145$	−2230.69	−1.27757
$146$	−235.810	−0.133670
$147$	−2338.15	−1.31188
$148$	1160.44	0.644508
$149$	−1427.45	−0.784838	−0.392419	−	0.919786i	$-0.628362\pi$
$149$	−1427.45	−0.784838	−0.392419	+	0.919786i	$0.628362\pi$
$150$	−41.6485	−0.0226706
$151$	243.694	0.131335	0.0656674	−	0.997842i	$-0.479082\pi$
$151$	243.694	0.131335	0.0656674	+	0.997842i	$0.479082\pi$
$152$	−147.112	−0.0785021
$153$	−88.4583	−0.0467414
$154$	72.8613	0.0381255
$155$	−3167.88	−1.64162
$156$	−1357.62	−0.696771
$157$	−427.635	−0.217382	−0.108691	−	0.994076i	$-0.534666\pi$
$157$	−427.635	−0.217382	−0.108691	+	0.994076i	$0.534666\pi$
$158$	194.005	0.0976847
$159$	857.905	0.427901
$160$	526.938	0.260363
$161$	2126.32	1.04085
$162$	16.0147	0.00776688
$163$	2820.06	1.35512	0.677559	−	0.735468i	$-0.263038\pi$
$163$	2820.06	1.35512	0.677559	+	0.735468i	$0.263038\pi$
$164$	3954.90	1.88309
$165$	−461.077	−0.217544
$166$	55.5903	0.0259918
$167$	−947.676	−0.439122	−0.219561	−	0.975599i	$-0.570462\pi$
$167$	−947.676	−0.439122	−0.219561	+	0.975599i	$0.570462\pi$
$168$	317.164	0.145653
$169$	1034.37	0.470810
$170$	−27.1512	−0.0122494
$171$	419.563	0.187631
$172$	3564.73	1.58028
$173$	−562.336	−0.247131	−0.123565	−	0.992336i	$-0.539433\pi$
$173$	−562.336	−0.247131	−0.123565	+	0.992336i	$0.539433\pi$
$174$	94.6966	0.0412582
$175$	2352.42	1.01615
$176$	693.697	0.297099
$177$	−749.950	−0.318473
$178$	234.751	0.0988502
$179$	−2303.04	−0.961659	−0.480830	−	0.876814i	$-0.659664\pi$
$179$	−2303.04	−0.961659	−0.480830	+	0.876814i	$0.659664\pi$
$180$	−1001.07	−0.414530
$181$	1048.01	0.430374	0.215187	−	0.976573i	$-0.430964\pi$
$181$	1048.01	0.430374	0.215187	+	0.976573i	$0.430964\pi$
$182$	−376.528	−0.153352
$183$	183.000	0.0739221
$184$	−200.286	−0.0802460
$185$	−2036.66	−0.809394
$186$	134.482	0.0530147
$187$	−108.116	−0.0422792
$188$	3094.09	1.20032
$189$	−904.553	−0.348130
$190$	128.780	0.0491720
$191$	−1933.17	−0.732354	−0.366177	−	0.930545i	$-0.619333\pi$
$191$	−1933.17	−0.732354	−0.366177	+	0.930545i	$0.619333\pi$
$192$	1491.15	0.560493
$193$	−1451.90	−0.541504	−0.270752	−	0.962649i	$-0.587272\pi$
$193$	−1451.90	−0.541504	−0.270752	+	0.962649i	$0.587272\pi$
$194$	−87.7429	−0.0324720
$195$	2382.72	0.875028
$196$	−6204.59	−2.26115
$197$	−2876.81	−1.04043	−0.520214	−	0.854036i	$-0.674148\pi$
$197$	−2876.81	−1.04043	−0.520214	+	0.854036i	$0.674148\pi$
$198$	19.5735	0.00702541
$199$	3557.19	1.26715	0.633575	−	0.773681i	$-0.281587\pi$
$199$	3557.19	1.26715	0.633575	+	0.773681i	$0.281587\pi$
$200$	−221.583	−0.0783413
$201$	−1190.98	−0.417936
$202$	−384.637	−0.133975
$203$	−5348.72	−1.84929
$204$	−234.736	−0.0805628
$205$	−6941.16	−2.36484
$206$	−46.3018	−0.0156602
$207$	571.216	0.191798
$208$	−3584.85	−1.19502
$209$	512.800	0.169718
$210$	−277.642	−0.0912338
$211$	1375.14	0.448666	0.224333	−	0.974513i	$-0.427980\pi$
$211$	1375.14	0.448666	0.224333	+	0.974513i	$0.427980\pi$
$212$	2276.57	0.737525
$213$	−437.851	−0.140850
$214$	295.612	0.0944282
$215$	−6256.38	−1.98457
$216$	85.2031	0.0268395
$217$	−7595.92	−2.37624
$218$	−63.9186	−0.0198583
$219$	3578.08	1.10404
$220$	−1223.53	−0.374956
$221$	558.714	0.170060
$222$	86.4597	0.0261387
$223$	−381.771	−0.114642	−0.0573212	−	0.998356i	$-0.518256\pi$
$223$	−381.771	−0.114642	−0.0573212	+	0.998356i	$0.518256\pi$
$224$	1263.49	0.376876
$225$	631.955	0.187246
$226$	386.588	0.113785
$227$	−6313.46	−1.84599	−0.922994	−	0.384815i	$-0.874265\pi$
$227$	−6313.46	−1.84599	−0.922994	+	0.384815i	$0.874265\pi$
$228$	1113.37	0.323398
$229$	−4161.29	−1.20081	−0.600406	−	0.799695i	$-0.704994\pi$
$229$	−4161.29	−1.20081	−0.600406	+	0.799695i	$0.704994\pi$
$230$	175.328	0.0502643
$231$	−1105.57	−0.314895
$232$	503.815	0.142574
$233$	−5695.27	−1.60133	−0.800665	−	0.599113i	$-0.795520\pi$
$233$	−5695.27	−1.60133	−0.800665	+	0.599113i	$0.795520\pi$
$234$	−101.151	−0.0282583
$235$	−5430.37	−1.50740
$236$	−1990.10	−0.548916
$237$	−2943.74	−0.806820
$238$	−65.1029	−0.0177311
$239$	2308.56	0.624805	0.312403	−	0.949950i	$-0.398866\pi$
$239$	2308.56	0.624805	0.312403	+	0.949950i	$0.398866\pi$
$240$	−2643.37	−0.710953
$241$	3721.98	0.994829	0.497414	−	0.867513i	$-0.334283\pi$
$241$	3721.98	0.994829	0.497414	+	0.867513i	$0.334283\pi$
$242$	23.9232	0.00635472
$243$	−243.000	−0.0641500
$244$	485.615	0.127411
$245$	10889.5	2.83962
$246$	294.665	0.0763705
$247$	−2650.02	−0.682658
$248$	715.487	0.183199
$249$	−843.502	−0.214678
$250$	−151.334	−0.0382848
$251$	1001.76	0.251915	0.125957	−	0.992036i	$-0.459800\pi$
$251$	1001.76	0.251915	0.125957	+	0.992036i	$0.459800\pi$
$252$	−2400.36	−0.600033
$253$	698.153	0.173488
$254$	−317.229	−0.0783651
$255$	411.980	0.101173
$256$	3897.32	0.951495
$257$	−4452.64	−1.08073	−0.540366	−	0.841430i	$-0.681714\pi$
$257$	−4452.64	−1.08073	−0.540366	+	0.841430i	$0.681714\pi$
$258$	265.595	0.0640900
$259$	−4883.47	−1.17160
$260$	6322.88	1.50819
$261$	−1436.88	−0.340770
$262$	−240.107	−0.0566178
$263$	2753.91	0.645679	0.322840	−	0.946454i	$-0.395363\pi$
$263$	2753.91	0.645679	0.322840	+	0.946454i	$0.395363\pi$
$264$	104.137	0.0242773
$265$	−3995.55	−0.926207
$266$	308.787	0.0711766
$267$	−3562.01	−0.816447
$268$	−3160.43	−0.720350
$269$	5444.36	1.23401	0.617005	−	0.786960i	$-0.288346\pi$
$269$	5444.36	1.23401	0.617005	+	0.786960i	$0.288346\pi$
$270$	−74.5859	−0.0168117
$271$	−5502.47	−1.23340	−0.616699	−	0.787199i	$-0.711531\pi$
$271$	−5502.47	−1.23340	−0.616699	+	0.787199i	$0.711531\pi$
$272$	−619.831	−0.138172
$273$	5713.27	1.26660
$274$	−155.517	−0.0342887
$275$	772.390	0.169370
$276$	1515.80	0.330581
$277$	8435.18	1.82968	0.914839	−	0.403819i	$-0.132318\pi$
$277$	8435.18	1.82968	0.914839	+	0.403819i	$0.132318\pi$
$278$	459.709	0.0991782
$279$	−2040.57	−0.437871
$280$	−1477.14	−0.315271
$281$	1532.14	0.325265	0.162633	−	0.986687i	$-0.448001\pi$
$281$	1532.14	0.325265	0.162633	+	0.986687i	$0.448001\pi$
$282$	230.529	0.0486802
$283$	−8807.30	−1.84996	−0.924982	−	0.380011i	$-0.875920\pi$
$283$	−8807.30	−1.84996	−0.924982	+	0.380011i	$0.875920\pi$
$284$	−1161.90	−0.242767
$285$	−1954.05	−0.406133
$286$	−123.629	−0.0255606
$287$	−16643.5	−3.42311
$288$	339.424	0.0694471
$289$	−4816.40	−0.980337
$290$	−441.034	−0.0893049
$291$	1331.37	0.268200
$292$	9494.92	1.90290
$293$	−6492.19	−1.29446	−0.647232	−	0.762293i	$-0.724073\pi$
$293$	−6492.19	−1.29446	−0.647232	+	0.762293i	$0.724073\pi$
$294$	−462.281	−0.0917033
$295$	3492.77	0.689346
$296$	459.992	0.0903260
$297$	−297.000	−0.0580259
$298$	−282.224	−0.0548617
$299$	−3607.88	−0.697822
$300$	1676.98	0.322735
$301$	−15001.5	−2.87267
$302$	48.1814	0.00918056
$303$	5836.32	1.10656
$304$	2939.90	0.554654
$305$	−852.293	−0.160007
$306$	−17.4893	−0.00326731
$307$	7739.25	1.43877	0.719385	−	0.694612i	$-0.244424\pi$
$307$	7739.25	1.43877	0.719385	+	0.694612i	$0.244424\pi$
$308$	−2933.77	−0.542750
$309$	702.563	0.129344
$310$	−626.330	−0.114752
$311$	7933.13	1.44645	0.723226	−	0.690611i	$-0.242658\pi$
$311$	7933.13	1.44645	0.723226	+	0.690611i	$0.242658\pi$
$312$	−538.154	−0.0976505
$313$	−4686.80	−0.846369	−0.423184	−	0.906044i	$-0.639088\pi$
$313$	−4686.80	−0.846369	−0.423184	+	0.906044i	$0.639088\pi$
$314$	−84.5488	−0.0151954
$315$	4212.81	0.753540
$316$	−7811.61	−1.39063
$317$	5720.67	1.01358	0.506790	−	0.862070i	$-0.330832\pi$
$317$	5720.67	1.01358	0.506790	+	0.862070i	$0.330832\pi$
$318$	169.619	0.0299111
$319$	−1756.19	−0.308238
$320$	−6944.80	−1.21321
$321$	−4485.49	−0.779923
$322$	420.400	0.0727577
$323$	−458.196	−0.0789310
$324$	−644.834	−0.110568
$325$	−3991.51	−0.681259
$326$	557.561	0.0947253
$327$	969.872	0.164018
$328$	1567.71	0.263909
$329$	−13020.9	−2.18196
$330$	−91.1606	−0.0152067
$331$	3366.69	0.559064	0.279532	−	0.960136i	$-0.409821\pi$
$331$	3366.69	0.559064	0.279532	+	0.960136i	$0.409821\pi$
$332$	−2238.35	−0.370016
$333$	−1311.90	−0.215891
$334$	−187.367	−0.0306955
$335$	5546.79	0.904638
$336$	−6338.24	−1.02911
$337$	3367.07	0.544262	0.272131	−	0.962260i	$-0.412272\pi$
$337$	3367.07	0.544262	0.272131	+	0.962260i	$0.412272\pi$
$338$	204.508	0.0329105
$339$	−5865.91	−0.939800
$340$	1093.25	0.174381
$341$	−2494.04	−0.396069
$342$	82.9529	0.0131157
$343$	14619.7	2.30142
$344$	1413.05	0.221472
$345$	−2660.35	−0.415155
$346$	−111.181	−0.0172749
$347$	−1256.91	−0.194452	−0.0972258	−	0.995262i	$-0.530997\pi$
$347$	−1256.91	−0.194452	−0.0972258	+	0.995262i	$0.530997\pi$
$348$	−3812.97	−0.587346
$349$	−11678.4	−1.79121	−0.895605	−	0.444849i	$-0.853257\pi$
$349$	−11678.4	−1.79121	−0.895605	+	0.444849i	$0.853257\pi$
$350$	465.102	0.0710307
$351$	1534.82	0.233398
$352$	414.852	0.0628172
$353$	−1799.77	−0.271366	−0.135683	−	0.990752i	$-0.543323\pi$
$353$	−1799.77	−0.271366	−0.135683	+	0.990752i	$0.543323\pi$
$354$	−148.275	−0.0222619
$355$	2039.22	0.304875
$356$	−9452.28	−1.40722
$357$	987.843	0.146449
$358$	−455.339	−0.0672218
$359$	−11861.9	−1.74386	−0.871932	−	0.489628i	$-0.837133\pi$
$359$	−11861.9	−1.74386	−0.871932	+	0.489628i	$0.837133\pi$
$360$	−396.820	−0.0580951
$361$	−4685.75	−0.683153
$362$	207.204	0.0300840
$363$	−363.000	−0.0524864
$364$	15160.9	2.18310
$365$	−16664.3	−2.38973
$366$	36.1814	0.00516730
$367$	−767.937	−0.109226	−0.0546131	−	0.998508i	$-0.517393\pi$
$367$	−767.937	−0.109226	−0.0546131	+	0.998508i	$0.517393\pi$
$368$	4002.54	0.566975
$369$	−4471.11	−0.630777
$370$	−402.672	−0.0565782
$371$	−9580.50	−1.34069
$372$	−5414.94	−0.754709
$373$	−3400.84	−0.472088	−0.236044	−	0.971742i	$-0.575851\pi$
$373$	−3400.84	−0.472088	−0.236044	+	0.971742i	$0.575851\pi$
$374$	−21.3758	−0.00295539
$375$	2296.28	0.316211
$376$	1226.48	0.168221
$377$	9075.54	1.23983
$378$	−178.841	−0.0243350
$379$	8951.11	1.21316	0.606580	−	0.795022i	$-0.292541\pi$
$379$	8951.11	1.21316	0.606580	+	0.795022i	$0.292541\pi$
$380$	−5185.34	−0.700006
$381$	4813.49	0.647251
$382$	−382.213	−0.0511929
$383$	8481.85	1.13160	0.565799	−	0.824543i	$-0.308568\pi$
$383$	8481.85	1.13160	0.565799	+	0.824543i	$0.308568\pi$
$384$	1199.95	0.159465
$385$	5148.99	0.681602
$386$	−287.059	−0.0378522
$387$	−4030.02	−0.529347
$388$	3532.97	0.462267
$389$	964.417	0.125702	0.0628508	−	0.998023i	$-0.479981\pi$
$389$	964.417	0.125702	0.0628508	+	0.998023i	$0.479981\pi$
$390$	471.094	0.0611661
$391$	−623.813	−0.0806843
$392$	−2459.47	−0.316893
$393$	3643.28	0.467631
$394$	−568.781	−0.0727278
$395$	13710.0	1.74639
$396$	−788.130	−0.100013
$397$	−9838.61	−1.24379	−0.621896	−	0.783099i	$-0.713638\pi$
$397$	−9838.61	−1.24379	−0.621896	+	0.783099i	$0.713638\pi$
$398$	703.302	0.0885762
$399$	−4685.40	−0.587878
$400$	4428.14	0.553517
$401$	9724.56	1.21103	0.605513	−	0.795835i	$-0.292968\pi$
$401$	9724.56	1.21103	0.605513	+	0.795835i	$0.292968\pi$
$402$	−235.471	−0.0292145
$403$	12888.5	1.59311
$404$	15487.5	1.90725
$405$	1131.73	0.138855
$406$	−1057.51	−0.129269
$407$	−1603.43	−0.195281
$408$	−93.0484	−0.0112906
$409$	10264.7	1.24097	0.620483	−	0.784220i	$-0.286937\pi$
$409$	10264.7	1.24097	0.620483	+	0.784220i	$0.286937\pi$
$410$	−1372.35	−0.165307
$411$	2359.74	0.283205
$412$	1864.35	0.222936
$413$	8374.94	0.997830
$414$	112.937	0.0134071
$415$	3928.48	0.464678
$416$	−2143.85	−0.252670
$417$	−6975.42	−0.819156
$418$	101.387	0.0118636
$419$	−10454.3	−1.21891	−0.609456	−	0.792820i	$-0.708612\pi$
$419$	−10454.3	−1.21891	−0.609456	+	0.792820i	$0.708612\pi$
$420$	11179.3	1.29879
$421$	3535.08	0.409239	0.204619	−	0.978842i	$-0.434404\pi$
$421$	3535.08	0.409239	0.204619	+	0.978842i	$0.434404\pi$
$422$	271.882	0.0313626
$423$	−3497.95	−0.402071
$424$	902.422	0.103362
$425$	−690.144	−0.0787692
$426$	−86.5685	−0.00984568
$427$	−2043.62	−0.231611
$428$	−11902.8	−1.34427
$429$	1875.89	0.211116
$430$	−1236.97	−0.138725
$431$	14217.8	1.58897	0.794485	−	0.607284i	$-0.207741\pi$
$431$	14217.8	1.58897	0.794485	+	0.607284i	$0.207741\pi$
$432$	−1702.71	−0.189634
$433$	−3561.36	−0.395261	−0.197631	−	0.980277i	$-0.563325\pi$
$433$	−3561.36	−0.395261	−0.197631	+	0.980277i	$0.563325\pi$
$434$	−1501.81	−0.166104
$435$	6692.06	0.737608
$436$	2573.69	0.282700
$437$	2958.78	0.323885
$438$	707.431	0.0771743
$439$	−1173.13	−0.127541	−0.0637707	−	0.997965i	$-0.520313\pi$
$439$	−1173.13	−0.127541	−0.0637707	+	0.997965i	$0.520313\pi$
$440$	−485.002	−0.0525490
$441$	7014.44	0.757417
$442$	110.465	0.0118875
$443$	6875.78	0.737422	0.368711	−	0.929544i	$-0.379799\pi$
$443$	6875.78	0.737422	0.368711	+	0.929544i	$0.379799\pi$
$444$	−3481.31	−0.372107
$445$	16589.5	1.76723
$446$	−75.4808	−0.00801373
$447$	4282.34	0.453127
$448$	−16652.2	−1.75612
$449$	11253.3	1.18280	0.591398	−	0.806380i	$-0.298576\pi$
$449$	11253.3	1.18280	0.591398	+	0.806380i	$0.298576\pi$
$450$	124.945	0.0130889
$451$	−5464.69	−0.570560
$452$	−15566.0	−1.61983
$453$	−731.083	−0.0758262
$454$	−1248.25	−0.129038
$455$	−26608.6	−2.74161
$456$	441.335	0.0453232
$457$	5157.09	0.527874	0.263937	−	0.964540i	$-0.414979\pi$
$457$	5157.09	0.527874	0.263937	+	0.964540i	$0.414979\pi$
$458$	−822.740	−0.0839391
$459$	265.375	0.0269861
$460$	−7059.60	−0.715555
$461$	13850.7	1.39933	0.699666	−	0.714470i	$-0.253332\pi$
$461$	13850.7	1.39933	0.699666	+	0.714470i	$0.253332\pi$
$462$	−218.584	−0.0220118
$463$	−19267.7	−1.93401	−0.967006	−	0.254753i	$-0.918006\pi$
$463$	−19267.7	−1.93401	−0.967006	+	0.254753i	$0.918006\pi$
$464$	−10068.3	−1.00735
$465$	9503.65	0.947787
$466$	−1126.03	−0.111936
$467$	−6619.73	−0.655941	−0.327971	−	0.944688i	$-0.606365\pi$
$467$	−6619.73	−0.655941	−0.327971	+	0.944688i	$0.606365\pi$
$468$	4072.85	0.402281
$469$	13300.0	1.30947
$470$	−1073.65	−0.105370
$471$	1282.90	0.125506
$472$	−788.865	−0.0769290
$473$	−4925.58	−0.478812
$474$	−582.014	−0.0563983
$475$	3273.40	0.316198
$476$	2621.38	0.252417
$477$	−2573.72	−0.247049
$478$	456.432	0.0436751
$479$	1262.99	0.120475	0.0602374	−	0.998184i	$-0.480814\pi$
$479$	1262.99	0.120475	0.0602374	+	0.998184i	$0.480814\pi$
$480$	−1580.81	−0.150321
$481$	8286.13	0.785478
$482$	735.882	0.0695404
$483$	−6378.96	−0.600937
$484$	−963.270	−0.0904649
$485$	−6200.64	−0.580529
$486$	−48.0441	−0.00448421
$487$	10820.1	1.00679	0.503394	−	0.864057i	$-0.332084\pi$
$487$	10820.1	1.00679	0.503394	+	0.864057i	$0.332084\pi$
$488$	192.496	0.0178563
$489$	−8460.18	−0.782378
$490$	2153.00	0.198495
$491$	10896.5	1.00153	0.500767	−	0.865582i	$-0.333051\pi$
$491$	10896.5	1.00153	0.500767	+	0.865582i	$0.333051\pi$
$492$	−11864.7	−1.08720
$493$	1569.19	0.143352
$494$	−523.941	−0.0477191
$495$	1383.23	0.125599
$496$	−14298.4	−1.29439
$497$	4889.62	0.441307
$498$	−166.771	−0.0150064
$499$	−12763.4	−1.14503	−0.572514	−	0.819895i	$-0.694032\pi$
$499$	−12763.4	−1.14503	−0.572514	+	0.819895i	$0.694032\pi$
$500$	6093.48	0.545018
$501$	2843.03	0.253527
$502$	198.061	0.0176093
$503$	21730.5	1.92627	0.963135	−	0.269018i	$-0.0866991\pi$
$503$	21730.5	1.92627	0.963135	+	0.269018i	$0.0866991\pi$
$504$	−951.491	−0.0840928
$505$	−27181.7	−2.39519
$506$	138.034	0.0121272
$507$	−3103.11	−0.271822
$508$	12773.3	1.11559
$509$	−3996.95	−0.348059	−0.174029	−	0.984740i	$-0.555679\pi$
$509$	−3996.95	−0.348059	−0.174029	+	0.984740i	$0.555679\pi$
$510$	81.4536	0.00707221
$511$	−39957.6	−3.45914
$512$	3970.42	0.342713
$513$	−1258.69	−0.108329
$514$	−880.343	−0.0755453
$515$	−3272.08	−0.279971
$516$	−10694.2	−0.912376
$517$	−4275.27	−0.363687
$518$	−965.524	−0.0818970
$519$	1687.01	0.142681
$520$	2506.36	0.211368
$521$	−2080.98	−0.174989	−0.0874946	−	0.996165i	$-0.527886\pi$
$521$	−2080.98	−0.174989	−0.0874946	+	0.996165i	$0.527886\pi$
$522$	−284.090	−0.0238205
$523$	−11259.0	−0.941341	−0.470671	−	0.882309i	$-0.655988\pi$
$523$	−11259.0	−0.941341	−0.470671	+	0.882309i	$0.655988\pi$
$524$	9667.93	0.806003
$525$	−7057.25	−0.586674
$526$	544.483	0.0451342
$527$	2228.47	0.184200
$528$	−2081.09	−0.171530
$529$	−8138.75	−0.668920
$530$	−789.971	−0.0647437
$531$	2249.85	0.183870
$532$	−12433.4	−1.01326
$533$	28240.1	2.29496
$534$	−704.253	−0.0570712
$535$	20890.4	1.68817
$536$	−1252.78	−0.100955
$537$	6909.11	0.555214
$538$	1076.42	0.0862596
$539$	8573.20	0.685110
$540$	3003.21	0.239329
$541$	−18942.2	−1.50534	−0.752671	−	0.658397i	$-0.771235\pi$
$541$	−18942.2	−1.50534	−0.752671	+	0.658397i	$0.771235\pi$
$542$	−1087.91	−0.0862169
$543$	−3144.02	−0.248477
$544$	−370.677	−0.0292145
$545$	−4517.02	−0.355024
$546$	1129.59	0.0885381
$547$	20731.3	1.62049	0.810245	−	0.586092i	$-0.199334\pi$
$547$	20731.3	1.62049	0.810245	+	0.586092i	$0.199334\pi$
$548$	6261.89	0.488129
$549$	−549.000	−0.0426790
$550$	152.711	0.0118393
$551$	−7442.77	−0.575449
$552$	600.857	0.0463300
$553$	32873.7	2.52791
$554$	1667.74	0.127898
$555$	6109.97	0.467304
$556$	−18510.2	−1.41189
$557$	17594.8	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$557$	17594.8	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$558$	−403.447	−0.0306080
$559$	25454.1	1.92593
$560$	29519.3	2.22754
$561$	324.347	0.0244099
$562$	302.922	0.0227367
$563$	12562.0	0.940362	0.470181	−	0.882570i	$-0.344189\pi$
$563$	12562.0	0.940362	0.470181	+	0.882570i	$0.344189\pi$
$564$	−9282.28	−0.693004
$565$	27319.5	2.03423
$566$	−1741.31	−0.129316
$567$	2713.66	0.200993
$568$	−460.571	−0.0340231
$569$	−375.491	−0.0276650	−0.0138325	−	0.999904i	$-0.504403\pi$
$569$	−375.491	−0.0276650	−0.0138325	+	0.999904i	$0.504403\pi$
$570$	−386.340	−0.0283895
$571$	1540.81	0.112926	0.0564630	−	0.998405i	$-0.482018\pi$
$571$	1540.81	0.112926	0.0564630	+	0.998405i	$0.482018\pi$
$572$	4977.93	0.363877
$573$	5799.52	0.422825
$574$	−3290.62	−0.239282
$575$	4456.58	0.323222
$576$	−4473.45	−0.323601
$577$	5287.00	0.381457	0.190728	−	0.981643i	$-0.438915\pi$
$577$	5287.00	0.381457	0.190728	+	0.981643i	$0.438915\pi$
$578$	−952.262	−0.0685274
$579$	4355.71	0.312637
$580$	17758.3	1.27133
$581$	9419.66	0.672622
$582$	263.229	0.0187477
$583$	−3145.65	−0.223464
$584$	3763.74	0.266686
$585$	−7148.17	−0.505197
$586$	−1283.59	−0.0904855
$587$	−9132.34	−0.642133	−0.321066	−	0.947057i	$-0.604041\pi$
$587$	−9132.34	−0.642133	−0.321066	+	0.947057i	$0.604041\pi$
$588$	18613.8	1.30547
$589$	−10569.8	−0.739422
$590$	690.565	0.0481866
$591$	8630.43	0.600691
$592$	−9192.54	−0.638194
$593$	−22308.2	−1.54483	−0.772417	−	0.635116i	$-0.780952\pi$
$593$	−22308.2	−1.54483	−0.772417	+	0.635116i	$0.780952\pi$
$594$	−58.7206	−0.00405612
$595$	−4600.72	−0.316993
$596$	11363.8	0.781004
$597$	−10671.6	−0.731589
$598$	−713.322	−0.0487791
$599$	−5560.79	−0.379312	−0.189656	−	0.981851i	$-0.560737\pi$
$599$	−5560.79	−0.379312	−0.189656	+	0.981851i	$0.560737\pi$
$600$	664.748	0.0452304
$601$	−10430.2	−0.707917	−0.353959	−	0.935261i	$-0.615165\pi$
$601$	−10430.2	−0.707917	−0.353959	+	0.935261i	$0.615165\pi$
$602$	−2965.98	−0.200805
$603$	3572.94	0.241296
$604$	−1940.03	−0.130693
$605$	1690.61	0.113609
$606$	1153.91	0.0773506
$607$	−16816.3	−1.12447	−0.562234	−	0.826978i	$-0.690058\pi$
$607$	−16816.3	−1.12447	−0.562234	+	0.826978i	$0.690058\pi$
$608$	1758.15	0.117273
$609$	16046.2	1.06769
$610$	−168.509	−0.0111848
$611$	22093.5	1.46286
$612$	704.209	0.0465130
$613$	28143.8	1.85435	0.927174	−	0.374630i	$-0.122230\pi$
$613$	28143.8	1.85435	0.927174	+	0.374630i	$0.122230\pi$
$614$	1530.15	0.100573
$615$	20823.5	1.36534
$616$	−1162.93	−0.0760648
$617$	2046.92	0.133559	0.0667794	−	0.997768i	$-0.478728\pi$
$617$	2046.92	0.133559	0.0667794	+	0.997768i	$0.478728\pi$
$618$	138.906	0.00904142
$619$	−17825.9	−1.15749	−0.578743	−	0.815510i	$-0.696456\pi$
$619$	−17825.9	−1.15749	−0.578743	+	0.815510i	$0.696456\pi$
$620$	25219.2	1.63359
$621$	−1713.65	−0.110735
$622$	1568.48	0.101110
$623$	39778.1	2.55807
$624$	10754.5	0.689946
$625$	−19471.7	−1.24619
$626$	−926.638	−0.0591628
$627$	−1538.40	−0.0979868
$628$	3404.36	0.216320
$629$	1432.70	0.0908194
$630$	832.925	0.0526739
$631$	14971.7	0.944556	0.472278	−	0.881450i	$-0.343432\pi$
$631$	14971.7	0.944556	0.472278	+	0.881450i	$0.343432\pi$
$632$	−3096.49	−0.194892
$633$	−4125.42	−0.259037
$634$	1131.05	0.0708512
$635$	−22418.1	−1.40100
$636$	−6829.70	−0.425810
$637$	−44304.1	−2.75572
$638$	−347.221	−0.0215464
$639$	1313.55	0.0813197
$640$	−5588.57	−0.345168
$641$	−6392.43	−0.393893	−0.196947	−	0.980414i	$-0.563103\pi$
$641$	−6392.43	−0.393893	−0.196947	+	0.980414i	$0.563103\pi$
$642$	−886.837	−0.0545181
$643$	−5758.67	−0.353188	−0.176594	−	0.984284i	$-0.556508\pi$
$643$	−5758.67	−0.353188	−0.176594	+	0.984284i	$0.556508\pi$
$644$	−16927.4	−1.03577
$645$	18769.2	1.14579
$646$	−90.5910	−0.00551743
$647$	−9592.90	−0.582899	−0.291450	−	0.956586i	$-0.594138\pi$
$647$	−9592.90	−0.582899	−0.291450	+	0.956586i	$0.594138\pi$
$648$	−255.609	−0.0154958
$649$	2749.82	0.166317
$650$	−789.171	−0.0476213
$651$	22787.8	1.37192
$652$	−22450.3	−1.34850
$653$	9114.85	0.546235	0.273118	−	0.961981i	$-0.411945\pi$
$653$	9114.85	0.546235	0.273118	+	0.961981i	$0.411945\pi$
$654$	191.756	0.0114652
$655$	−16968.0	−1.01220
$656$	−31329.3	−1.86464
$657$	−10734.2	−0.637416
$658$	−2574.39	−0.152523
$659$	−16541.0	−0.977761	−0.488881	−	0.872351i	$-0.662595\pi$
$659$	−16541.0	−0.977761	−0.488881	+	0.872351i	$0.662595\pi$
$660$	3670.59	0.216481
$661$	6257.95	0.368239	0.184120	−	0.982904i	$-0.441057\pi$
$661$	6257.95	0.368239	0.184120	+	0.982904i	$0.441057\pi$
$662$	665.637	0.0390796
$663$	−1676.14	−0.0981839
$664$	−887.272	−0.0518567
$665$	21821.5	1.27248
$666$	−259.379	−0.0150912
$667$	−10133.0	−0.588232
$668$	7544.36	0.436976
$669$	1145.31	0.0661888
$670$	1096.67	0.0632359
$671$	−671.000	−0.0386046
$672$	−3790.46	−0.217589
$673$	−9232.56	−0.528810	−0.264405	−	0.964412i	$-0.585176\pi$
$673$	−9232.56	−0.528810	−0.264405	+	0.964412i	$0.585176\pi$
$674$	665.713	0.0380449
$675$	−1895.87	−0.108107
$676$	−8234.52	−0.468510
$677$	11142.3	0.632544	0.316272	−	0.948669i	$-0.397569\pi$
$677$	11142.3	0.632544	0.316272	+	0.948669i	$0.397569\pi$
$678$	−1159.76	−0.0656938
$679$	−14867.8	−0.840318
$680$	433.358	0.0244390
$681$	18940.4	1.06578
$682$	−493.102	−0.0276860
$683$	−5410.54	−0.303116	−0.151558	−	0.988448i	$-0.548429\pi$
$683$	−5410.54	−0.303116	−0.151558	+	0.988448i	$0.548429\pi$
$684$	−3340.11	−0.186714
$685$	−10990.1	−0.613008
$686$	2890.49	0.160874
$687$	12483.9	0.693289
$688$	−28238.5	−1.56480
$689$	16255.9	0.898840
$690$	−525.984	−0.0290201
$691$	−19998.3	−1.10097	−0.550486	−	0.834844i	$-0.685558\pi$
$691$	−19998.3	−1.10097	−0.550486	+	0.834844i	$0.685558\pi$
$692$	4476.71	0.245923
$693$	3316.70	0.181805
$694$	−248.508	−0.0135925
$695$	32486.9	1.77309
$696$	−1511.44	−0.0823149
$697$	4882.80	0.265351
$698$	−2308.97	−0.125209
$699$	17085.8	0.924528
$700$	−18727.4	−1.01118
$701$	32480.0	1.75000	0.875002	−	0.484120i	$-0.160860\pi$
$701$	32480.0	1.75000	0.875002	+	0.484120i	$0.160860\pi$
$702$	303.453	0.0163149
$703$	−6795.38	−0.364570
$704$	−5467.55	−0.292708
$705$	16291.1	0.870296
$706$	−355.837	−0.0189690
$707$	−65176.0	−3.46704
$708$	5970.29	0.316917
$709$	−20177.7	−1.06881	−0.534406	−	0.845228i	$-0.679465\pi$
$709$	−20177.7	−1.06881	−0.534406	+	0.845228i	$0.679465\pi$
$710$	403.179	0.0213113
$711$	8831.22	0.465818
$712$	−3746.84	−0.197217
$713$	−14390.2	−0.755847
$714$	195.309	0.0102370
$715$	−8736.65	−0.456968
$716$	18334.3	0.956960
$717$	−6925.69	−0.360732
$718$	−2345.24	−0.121899
$719$	−6697.90	−0.347412	−0.173706	−	0.984798i	$-0.555574\pi$
$719$	−6697.90	−0.347412	−0.173706	+	0.984798i	$0.555574\pi$
$720$	7930.10	0.410469
$721$	−7845.75	−0.405258
$722$	−926.431	−0.0477537
$723$	−11165.9	−0.574365
$724$	−8343.09	−0.428271
$725$	−11210.5	−0.574270
$726$	−71.7696	−0.00366890
$727$	283.617	0.0144687	0.00723437	−	0.999974i	$-0.497697\pi$
$727$	283.617	0.0144687	0.00723437	+	0.999974i	$0.497697\pi$
$728$	6009.74	0.305956
$729$	729.000	0.0370370
$730$	−3294.74	−0.167046
$731$	4401.09	0.222682
$732$	−1456.85	−0.0735609
$733$	−18675.5	−0.941057	−0.470528	−	0.882385i	$-0.655937\pi$
$733$	−18675.5	−0.941057	−0.470528	+	0.882385i	$0.655937\pi$
$734$	−151.831	−0.00763512
$735$	−32668.6	−1.63946
$736$	2393.64	0.119879
$737$	4366.92	0.218260
$738$	−883.995	−0.0440926
$739$	−36449.6	−1.81437	−0.907185	−	0.420731i	$-0.861773\pi$
$739$	−36449.6	−1.81437	−0.907185	+	0.420731i	$0.861773\pi$
$740$	16213.6	0.805439
$741$	7950.05	0.394133
$742$	−1894.19	−0.0937166
$743$	−3131.69	−0.154630	−0.0773152	−	0.997007i	$-0.524635\pi$
$743$	−3131.69	−0.154630	−0.0773152	+	0.997007i	$0.524635\pi$
$744$	−2146.46	−0.105770
$745$	−19944.3	−0.980809
$746$	−672.389	−0.0329999
$747$	2530.51	0.123944
$748$	860.699	0.0420726
$749$	50090.9	2.44363
$750$	454.002	0.0221038
$751$	12188.8	0.592245	0.296122	−	0.955150i	$-0.404306\pi$
$751$	12188.8	0.592245	0.296122	+	0.955150i	$0.404306\pi$
$752$	−24510.2	−1.18856
$753$	−3005.28	−0.145443
$754$	1794.35	0.0866662
$755$	3404.90	0.164129
$756$	7201.07	0.346429
$757$	1752.43	0.0841389	0.0420695	−	0.999115i	$-0.486605\pi$
$757$	1752.43	0.0841389	0.0420695	+	0.999115i	$0.486605\pi$
$758$	1769.75	0.0848022
$759$	−2094.46	−0.100163
$760$	−2055.45	−0.0981037
$761$	4534.97	0.216022	0.108011	−	0.994150i	$-0.465552\pi$
$761$	4534.97	0.216022	0.108011	+	0.994150i	$0.465552\pi$
$762$	951.688	0.0452441
$763$	−10830.9	−0.513898
$764$	15389.8	0.728775
$765$	−1235.94	−0.0584125
$766$	1676.97	0.0791009
$767$	−14210.3	−0.668977
$768$	−11692.0	−0.549346
$769$	−26354.7	−1.23586	−0.617930	−	0.786233i	$-0.712028\pi$
$769$	−26354.7	−1.23586	−0.617930	+	0.786233i	$0.712028\pi$
$770$	1018.02	0.0476453
$771$	13357.9	0.623961
$772$	11558.5	0.538858
$773$	−3743.67	−0.174192	−0.0870961	−	0.996200i	$-0.527759\pi$
$773$	−3743.67	−0.174192	−0.0870961	+	0.996200i	$0.527759\pi$
$774$	−796.785	−0.0370024
$775$	−15920.4	−0.737906
$776$	1400.46	0.0647854
$777$	14650.4	0.676423
$778$	190.677	0.00878678
$779$	−23159.5	−1.06518
$780$	−18968.6	−0.870752
$781$	1605.45	0.0735565
$782$	−123.336	−0.00563999
$783$	4310.65	0.196743
$784$	49150.5	2.23900
$785$	−5974.92	−0.271661
$786$	720.321	0.0326883
$787$	6818.07	0.308816	0.154408	−	0.988007i	$-0.450653\pi$
$787$	6818.07	0.308816	0.154408	+	0.988007i	$0.450653\pi$
$788$	22902.0	1.03534
$789$	−8261.74	−0.372783
$790$	2710.64	0.122076
$791$	65506.5	2.94455
$792$	−312.411	−0.0140165
$793$	3467.55	0.155279
$794$	−1945.22	−0.0869435
$795$	11986.7	0.534746
$796$	−28318.5	−1.26096
$797$	−19020.3	−0.845336	−0.422668	−	0.906285i	$-0.638906\pi$
$797$	−19020.3	−0.845336	−0.422668	+	0.906285i	$0.638906\pi$
$798$	−926.362	−0.0410938
$799$	3820.03	0.169140
$800$	2648.16	0.117033
$801$	10686.0	0.471376
$802$	1922.67	0.0846530
$803$	−13119.6	−0.576564
$804$	9481.28	0.415894
$805$	29709.0	1.30075
$806$	2548.22	0.111361
$807$	−16333.1	−0.712455
$808$	6139.16	0.267296
$809$	8434.96	0.366573	0.183286	−	0.983060i	$-0.441326\pi$
$809$	8434.96	0.366573	0.183286	+	0.983060i	$0.441326\pi$
$810$	223.758	0.00970623
$811$	−26238.6	−1.13608	−0.568041	−	0.823000i	$-0.692299\pi$
$811$	−26238.6	−1.13608	−0.568041	+	0.823000i	$0.692299\pi$
$812$	42580.7	1.84026
$813$	16507.4	0.712103
$814$	−317.019	−0.0136505
$815$	39401.9	1.69348
$816$	1859.49	0.0797736
$817$	−20874.7	−0.893895
$818$	2029.45	0.0867458
$819$	−17139.8	−0.731274
$820$	55258.0	2.35328
$821$	3323.46	0.141279	0.0706393	−	0.997502i	$-0.477496\pi$
$821$	3323.46	0.141279	0.0706393	+	0.997502i	$0.477496\pi$
$822$	466.550	0.0197966
$823$	−6656.11	−0.281917	−0.140958	−	0.990016i	$-0.545018\pi$
$823$	−6656.11	−0.281917	−0.140958	+	0.990016i	$0.545018\pi$
$824$	739.020	0.0312439
$825$	−2317.17	−0.0977860
$826$	1655.83	0.0697502
$827$	6934.84	0.291594	0.145797	−	0.989315i	$-0.453425\pi$
$827$	6934.84	0.291594	0.145797	+	0.989315i	$0.453425\pi$
$828$	−4547.40	−0.190861
$829$	−10187.8	−0.426825	−0.213412	−	0.976962i	$-0.568458\pi$
$829$	−10187.8	−0.426825	−0.213412	+	0.976962i	$0.568458\pi$
$830$	776.709	0.0324819
$831$	−25305.5	−1.05636
$832$	28254.9	1.17736
$833$	−7660.31	−0.318624
$834$	−1379.13	−0.0572605
$835$	−13240.9	−0.548769
$836$	−4082.35	−0.168889
$837$	6121.72	0.252805
$838$	−2066.94	−0.0852043
$839$	20691.4	0.851425	0.425713	−	0.904858i	$-0.360023\pi$
$839$	20691.4	0.851425	0.425713	+	0.904858i	$0.360023\pi$
$840$	4431.42	0.182022
$841$	1100.33	0.0451160
$842$	698.930	0.0286066
$843$	−4596.41	−0.187792
$844$	−10947.4	−0.446474
$845$	14452.2	0.588369
$846$	−691.587	−0.0281055
$847$	4053.74	0.164449
$848$	−18034.1	−0.730300
$849$	26421.9	1.06808
$850$	−136.450	−0.00550612
$851$	−9251.59	−0.372668
$852$	3485.69	0.140162
$853$	19547.7	0.784642	0.392321	−	0.919828i	$-0.371672\pi$
$853$	19547.7	0.784642	0.392321	+	0.919828i	$0.371672\pi$
$854$	−404.049	−0.0161900
$855$	5862.15	0.234481
$856$	−4718.24	−0.188395
$857$	−29946.6	−1.19365	−0.596823	−	0.802373i	$-0.703571\pi$
$857$	−29946.6	−1.19365	−0.596823	+	0.802373i	$0.703571\pi$
$858$	370.887	0.0147574
$859$	−17501.7	−0.695169	−0.347584	−	0.937649i	$-0.612998\pi$
$859$	−17501.7	−0.695169	−0.347584	+	0.937649i	$0.612998\pi$
$860$	49806.5	1.97487
$861$	49930.4	1.97633
$862$	2811.03	0.111072
$863$	22892.8	0.902988	0.451494	−	0.892274i	$-0.350891\pi$
$863$	22892.8	0.902988	0.451494	+	0.892274i	$0.350891\pi$
$864$	−1018.27	−0.0400953
$865$	−7856.97	−0.308838
$866$	−704.126	−0.0276295
$867$	14449.2	0.565998
$868$	60470.4	2.36463
$869$	10793.7	0.421348
$870$	1323.10	0.0515602
$871$	−22567.1	−0.877908
$872$	1020.20	0.0396196
$873$	−3994.11	−0.154846
$874$	584.989	0.0226402
$875$	−25643.3	−0.990743
$876$	−28484.8	−1.09864
$877$	−26657.2	−1.02640	−0.513198	−	0.858270i	$-0.671539\pi$
$877$	−26657.2	−1.02640	−0.513198	+	0.858270i	$0.671539\pi$
$878$	−231.943	−0.00891539
$879$	19476.6	0.747359
$880$	9692.35	0.371283
$881$	−39468.6	−1.50934	−0.754671	−	0.656104i	$-0.772203\pi$
$881$	−39468.6	−1.50934	−0.754671	+	0.656104i	$0.772203\pi$
$882$	1386.84	0.0529449
$883$	23168.0	0.882973	0.441487	−	0.897268i	$-0.354451\pi$
$883$	23168.0	0.882973	0.441487	+	0.897268i	$0.354451\pi$
$884$	−4447.87	−0.169229
$885$	−10478.3	−0.397994
$886$	1359.43	0.0515472
$887$	−19700.9	−0.745764	−0.372882	−	0.927879i	$-0.621630\pi$
$887$	−19700.9	−0.745764	−0.372882	+	0.927879i	$0.621630\pi$
$888$	−1379.98	−0.0521497
$889$	−53753.9	−2.02795
$890$	3279.95	0.123533
$891$	891.000	0.0335013
$892$	3039.24	0.114082
$893$	−18118.6	−0.678967
$894$	846.672	0.0316744
$895$	−32178.1	−1.20178
$896$	−13400.2	−0.499632
$897$	10823.6	0.402888
$898$	2224.92	0.0826797
$899$	36198.4	1.34292
$900$	−5030.94	−0.186331
$901$	2810.70	0.103927
$902$	−1080.44	−0.0398832
$903$	45004.5	1.65853
$904$	−6170.29	−0.227014
$905$	14642.8	0.537837
$906$	−144.544	−0.00530040
$907$	10658.1	0.390182	0.195091	−	0.980785i	$-0.437500\pi$
$907$	10658.1	0.390182	0.195091	+	0.980785i	$0.437500\pi$
$908$	50260.9	1.83697
$909$	−17508.9	−0.638872
$910$	−5260.86	−0.191644
$911$	37431.4	1.36132	0.680658	−	0.732601i	$-0.261694\pi$
$911$	37431.4	1.36132	0.680658	+	0.732601i	$0.261694\pi$
$912$	−8819.69	−0.320229
$913$	3092.84	0.112112
$914$	1019.62	0.0368994
$915$	2556.88	0.0923801
$916$	33127.7	1.19494
$917$	−40685.6	−1.46517
$918$	52.4679	0.00188638
$919$	17295.8	0.620822	0.310411	−	0.950602i	$-0.399533\pi$
$919$	17295.8	0.620822	0.310411	+	0.950602i	$0.399533\pi$
$920$	−2798.39	−0.100283
$921$	−23217.8	−0.830674
$922$	2738.46	0.0978160
$923$	−8296.56	−0.295866
$924$	8801.30	0.313357
$925$	−10235.3	−0.363823
$926$	−3809.47	−0.135191
$927$	−2107.69	−0.0746770
$928$	−6021.15	−0.212989
$929$	11775.7	0.415874	0.207937	−	0.978142i	$-0.433325\pi$
$929$	11775.7	0.415874	0.207937	+	0.978142i	$0.433325\pi$
$930$	1878.99	0.0662522
$931$	36333.4	1.27903
$932$	45339.5	1.59350
$933$	−23799.4	−0.835110
$934$	−1308.80	−0.0458516
$935$	−1510.59	−0.0528361
$936$	1614.46	0.0563785
$937$	51037.5	1.77943	0.889713	−	0.456520i	$-0.150904\pi$
$937$	51037.5	1.77943	0.889713	+	0.456520i	$0.150904\pi$
$938$	2629.59	0.0915341
$939$	14060.4	0.488651
$940$	43230.7	1.50003
$941$	−13563.3	−0.469872	−0.234936	−	0.972011i	$-0.575488\pi$
$941$	−13563.3	−0.469872	−0.234936	+	0.972011i	$0.575488\pi$
$942$	253.646	0.00877308
$943$	−31530.5	−1.08884
$944$	15764.8	0.543538
$945$	−12638.4	−0.435056
$946$	−973.848	−0.0334699
$947$	−29710.6	−1.01950	−0.509750	−	0.860323i	$-0.670262\pi$
$947$	−29710.6	−1.01950	−0.509750	+	0.860323i	$0.670262\pi$
$948$	23434.8	0.802878
$949$	67798.8	2.31912
$950$	647.192	0.0221028
$951$	−17162.0	−0.585190
$952$	1039.10	0.0353755
$953$	2921.21	0.0992940	0.0496470	−	0.998767i	$-0.484190\pi$
$953$	2921.21	0.0992940	0.0496470	+	0.998767i	$0.484190\pi$
$954$	−508.856	−0.0172692
$955$	−27010.3	−0.915219
$956$	−18378.3	−0.621752
$957$	5268.58	0.177961
$958$	249.709	0.00842142
$959$	−26352.0	−0.887331
$960$	20834.4	0.700445
$961$	21615.7	0.725579
$962$	1638.27	0.0549064
$963$	13456.5	0.450289
$964$	−29630.3	−0.989968
$965$	−20286.0	−0.676715
$966$	−1261.20	−0.0420067
$967$	−51462.2	−1.71139	−0.855694	−	0.517482i	$-0.826869\pi$
$967$	−51462.2	−1.71139	−0.855694	+	0.517482i	$0.826869\pi$
$968$	−381.836	−0.0126784
$969$	1374.59	0.0455708
$970$	−1225.94	−0.0405801
$971$	1043.96	0.0345027	0.0172514	−	0.999851i	$-0.494508\pi$
$971$	1043.96	0.0345027	0.0172514	+	0.999851i	$0.494508\pi$
$972$	1934.50	0.0638366
$973$	77896.8	2.56655
$974$	2139.27	0.0703764
$975$	11974.5	0.393325
$976$	−3846.87	−0.126163
$977$	−44311.0	−1.45101	−0.725504	−	0.688218i	$-0.758393\pi$
$977$	−44311.0	−1.45101	−0.725504	+	0.688218i	$0.758393\pi$
$978$	−1672.68	−0.0546897
$979$	13060.7	0.426376
$980$	−86690.7	−2.82575
$981$	−2909.62	−0.0946961
$982$	2154.38	0.0700092
$983$	14551.4	0.472144	0.236072	−	0.971736i	$-0.424140\pi$
$983$	14551.4	0.472144	0.236072	+	0.971736i	$0.424140\pi$
$984$	−4703.12	−0.152368
$985$	−40194.8	−1.30022
$986$	310.248	0.0100206
$987$	39062.7	1.25976
$988$	21096.5	0.679322
$989$	−28419.9	−0.913751
$990$	273.482	0.00877962
$991$	−3313.85	−0.106224	−0.0531120	−	0.998589i	$-0.516914\pi$
$991$	−3313.85	−0.106224	−0.0531120	+	0.998589i	$0.516914\pi$
$992$	−8550.87	−0.273680
$993$	−10100.1	−0.322776
$994$	966.739	0.0308482
$995$	49701.2	1.58355
$996$	6715.05	0.213629
$997$	−24007.0	−0.762597	−0.381298	−	0.924452i	$-0.624523\pi$
$997$	−24007.0	−0.762597	−0.381298	+	0.924452i	$0.624523\pi$
$998$	−2523.49	−0.0800397
$999$	3935.70	0.124645

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.d.1.19	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.d.1.19	✓	37	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)

\(f(q)\)	\(=\)	\(q+0.197712 q^{2} -3.00000 q^{3} -7.96091 q^{4} +13.9720 q^{5} -0.593137 q^{6} +33.5020 q^{7} -3.15567 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.197712 q^{2} -3.00000 q^{3} -7.96091 q^{4} +13.9720 q^{5} -0.593137 q^{6} +33.5020 q^{7} -3.15567 q^{8} +9.00000 q^{9} +2.76244 q^{10} +11.0000 q^{11} +23.8827 q^{12} -56.8451 q^{13} +6.62376 q^{14} -41.9161 q^{15} +63.0634 q^{16} -9.82870 q^{17} +1.77941 q^{18} +46.6182 q^{19} -111.230 q^{20} -100.506 q^{21} +2.17484 q^{22} +63.4685 q^{23} +9.46701 q^{24} +70.2173 q^{25} -11.2390 q^{26} -27.0000 q^{27} -266.706 q^{28} -159.654 q^{29} -8.28733 q^{30} -226.731 q^{31} +37.7138 q^{32} -33.0000 q^{33} -1.94326 q^{34} +468.090 q^{35} -71.6482 q^{36} -145.767 q^{37} +9.21699 q^{38} +170.535 q^{39} -44.0911 q^{40} -496.790 q^{41} -19.8713 q^{42} -447.780 q^{43} -87.5700 q^{44} +125.748 q^{45} +12.5485 q^{46} -388.661 q^{47} -189.190 q^{48} +779.382 q^{49} +13.8828 q^{50} +29.4861 q^{51} +452.539 q^{52} -285.968 q^{53} -5.33824 q^{54} +153.692 q^{55} -105.721 q^{56} -139.854 q^{57} -31.5655 q^{58} +249.983 q^{59} +333.690 q^{60} -61.0000 q^{61} -44.8275 q^{62} +301.518 q^{63} -497.050 q^{64} -794.241 q^{65} -6.52451 q^{66} +396.993 q^{67} +78.2454 q^{68} -190.405 q^{69} +92.5473 q^{70} +145.950 q^{71} -28.4010 q^{72} -1192.69 q^{73} -28.8199 q^{74} -210.652 q^{75} -371.123 q^{76} +368.522 q^{77} +33.7170 q^{78} +981.246 q^{79} +881.122 q^{80} +81.0000 q^{81} -98.2217 q^{82} +281.167 q^{83} +800.119 q^{84} -137.327 q^{85} -88.5316 q^{86} +478.961 q^{87} -34.7124 q^{88} +1187.34 q^{89} +24.8620 q^{90} -1904.42 q^{91} -505.267 q^{92} +680.192 q^{93} -76.8430 q^{94} +651.350 q^{95} -113.141 q^{96} -443.790 q^{97} +154.094 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more