LMFDB - Embedded newform 19.4.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [19,4,Mod(1,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	19.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.12103629011$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	19.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.89080 q^{2} +2.95388 q^{3} -4.42486 q^{4} -1.51710 q^{5} +5.58521 q^{6} +5.94196 q^{7} -23.4930 q^{8} -18.2746 q^{9} -2.86853 q^{10} +11.5171 q^{11} -13.0705 q^{12} +22.6093 q^{13} +11.2351 q^{14} -4.48133 q^{15} -9.02172 q^{16} +120.560 q^{17} -34.5537 q^{18} -19.0000 q^{19} +6.71294 q^{20} +17.5518 q^{21} +21.7766 q^{22} +63.9160 q^{23} -69.3955 q^{24} -122.698 q^{25} +42.7498 q^{26} -133.736 q^{27} -26.2923 q^{28} -89.7278 q^{29} -8.47331 q^{30} -251.051 q^{31} +170.885 q^{32} +34.0201 q^{33} +227.954 q^{34} -9.01453 q^{35} +80.8625 q^{36} +198.702 q^{37} -35.9253 q^{38} +66.7853 q^{39} +35.6411 q^{40} +373.170 q^{41} +33.1871 q^{42} -448.586 q^{43} -50.9616 q^{44} +27.7243 q^{45} +120.853 q^{46} -186.475 q^{47} -26.6491 q^{48} -307.693 q^{49} -231.999 q^{50} +356.119 q^{51} -100.043 q^{52} +364.882 q^{53} -252.868 q^{54} -17.4726 q^{55} -139.594 q^{56} -56.1238 q^{57} -169.658 q^{58} -376.730 q^{59} +19.8292 q^{60} +816.832 q^{61} -474.689 q^{62} -108.587 q^{63} +395.285 q^{64} -34.3006 q^{65} +64.3254 q^{66} +220.185 q^{67} -533.459 q^{68} +188.800 q^{69} -17.0447 q^{70} +383.466 q^{71} +429.324 q^{72} -537.689 q^{73} +375.706 q^{74} -362.437 q^{75} +84.0724 q^{76} +68.4341 q^{77} +126.278 q^{78} +1062.27 q^{79} +13.6868 q^{80} +98.3741 q^{81} +705.592 q^{82} -616.932 q^{83} -77.6645 q^{84} -182.900 q^{85} -848.189 q^{86} -265.045 q^{87} -270.571 q^{88} -90.2686 q^{89} +52.4212 q^{90} +134.344 q^{91} -282.819 q^{92} -741.576 q^{93} -352.589 q^{94} +28.8248 q^{95} +504.776 q^{96} -524.636 q^{97} -581.787 q^{98} -210.470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 21 q^{4} + 14 q^{5} - 65 q^{6} - 35 q^{7} + 27 q^{8} + 48 q^{9} - 88 q^{10} + 16 q^{11} - 115 q^{12} + 65 q^{13} + 37 q^{14} + 140 q^{15} + 33 q^{16} + 29 q^{17} + 138 q^{18} - 57 q^{19}+ \cdots + 250 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 21 * q^4 + 14 * q^5 - 65 * q^6 - 35 * q^7 + 27 * q^8 + 48 * q^9 - 88 * q^10 + 16 * q^11 - 115 * q^12 + 65 * q^13 + 37 * q^14 + 140 * q^15 + 33 * q^16 + 29 * q^17 + 138 * q^18 - 57 * q^19 + 100 * q^20 - 25 * q^21 + 118 * q^22 - 101 * q^23 - 489 * q^24 - 37 * q^25 + 299 * q^26 - 377 * q^27 - 493 * q^28 + 377 * q^29 - 376 * q^30 - 140 * q^31 + 579 * q^32 - 130 * q^33 + 329 * q^34 - 438 * q^35 + 1074 * q^36 - 290 * q^37 - 57 * q^38 - 371 * q^39 - 72 * q^40 + 956 * q^41 + 1385 * q^42 - 570 * q^43 + 110 * q^44 + 230 * q^45 - 239 * q^46 + 66 * q^47 - 1411 * q^48 - 98 * q^49 - 1639 * q^50 + 187 * q^51 + 697 * q^52 + 817 * q^53 - 947 * q^54 - 198 * q^55 - 915 * q^56 - 19 * q^57 + 647 * q^58 + 265 * q^59 + 1360 * q^60 + 988 * q^61 - 1436 * q^62 - 1304 * q^63 - 159 * q^64 - 240 * q^65 - 274 * q^66 - 207 * q^67 - 1529 * q^68 + 807 * q^69 + 2040 * q^70 + 846 * q^71 + 2826 * q^72 + 627 * q^73 - 770 * q^74 + 2031 * q^75 - 399 * q^76 + 88 * q^77 - 1817 * q^78 + 382 * q^79 - 1472 * q^80 - 501 * q^81 + 872 * q^82 - 766 * q^83 + 907 * q^84 - 1242 * q^85 + 2592 * q^86 - 1671 * q^87 + 342 * q^88 - 172 * q^89 - 1888 * q^90 - 457 * q^91 - 2677 * q^92 + 892 * q^93 - 96 * q^94 - 266 * q^95 + 863 * q^96 - 2450 * q^97 - 2450 * q^98 + 250 * q^99

Coefficient data

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

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

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

$2$	1.89080	0.668500	0.334250	−	0.942484i	$-0.391517\pi$
$2$	1.89080	0.668500	0.334250	+	0.942484i	$0.391517\pi$
$3$	2.95388	0.568475	0.284237	−	0.958754i	$-0.408260\pi$
$3$	2.95388	0.568475	0.284237	+	0.958754i	$0.408260\pi$
$4$	−4.42486	−0.553108
$5$	−1.51710	−0.135693	−0.0678466	−	0.997696i	$-0.521613\pi$
$5$	−1.51710	−0.135693	−0.0678466	+	0.997696i	$0.521613\pi$
$6$	5.58521	0.380025
$7$	5.94196	0.320836	0.160418	−	0.987049i	$-0.448716\pi$
$7$	5.94196	0.320836	0.160418	+	0.987049i	$0.448716\pi$
$8$	−23.4930	−1.03825
$9$	−18.2746	−0.676836
$10$	−2.86853	−0.0907110
$11$	11.5171	0.315685	0.157843	−	0.987464i	$-0.449546\pi$
$11$	11.5171	0.315685	0.157843	+	0.987464i	$0.449546\pi$
$12$	−13.0705	−0.314428
$13$	22.6093	0.482362	0.241181	−	0.970480i	$-0.422465\pi$
$13$	22.6093	0.482362	0.241181	+	0.970480i	$0.422465\pi$
$14$	11.2351	0.214479
$15$	−4.48133	−0.0771382
$16$	−9.02172	−0.140964
$17$	120.560	1.72000	0.859999	−	0.510295i	$-0.170464\pi$
$17$	120.560	1.72000	0.859999	+	0.510295i	$0.170464\pi$
$18$	−34.5537	−0.452465
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	6.71294	0.0750530
$21$	17.5518	0.182387
$22$	21.7766	0.211035
$23$	63.9160	0.579452	0.289726	−	0.957110i	$-0.406436\pi$
$23$	63.9160	0.579452	0.289726	+	0.957110i	$0.406436\pi$
$24$	−69.3955	−0.590220
$25$	−122.698	−0.981587
$26$	42.7498	0.322459
$27$	−133.736	−0.953239
$28$	−26.2923	−0.177457
$29$	−89.7278	−0.574553	−0.287276	−	0.957848i	$-0.592750\pi$
$29$	−89.7278	−0.574553	−0.287276	+	0.957848i	$0.592750\pi$
$30$	−8.47331	−0.0515669
$31$	−251.051	−1.45452	−0.727260	−	0.686362i	$-0.759207\pi$
$31$	−251.051	−1.45452	−0.727260	+	0.686362i	$0.759207\pi$
$32$	170.885	0.944018
$33$	34.0201	0.179459
$34$	227.954	1.14982
$35$	−9.01453	−0.0435352
$36$	80.8625	0.374363
$37$	198.702	0.882875	0.441438	−	0.897292i	$-0.354469\pi$
$37$	198.702	0.882875	0.441438	+	0.897292i	$0.354469\pi$
$38$	−35.9253	−0.153364
$39$	66.7853	0.274210
$40$	35.6411	0.140884
$41$	373.170	1.42145	0.710725	−	0.703470i	$-0.248367\pi$
$41$	373.170	1.42145	0.710725	+	0.703470i	$0.248367\pi$
$42$	33.1871	0.121926
$43$	−448.586	−1.59090	−0.795451	−	0.606018i	$-0.792766\pi$
$43$	−448.586	−1.59090	−0.795451	+	0.606018i	$0.792766\pi$
$44$	−50.9616	−0.174608
$45$	27.7243	0.0918422
$46$	120.853	0.387364
$47$	−186.475	−0.578729	−0.289364	−	0.957219i	$-0.593444\pi$
$47$	−186.475	−0.578729	−0.289364	+	0.957219i	$0.593444\pi$
$48$	−26.6491	−0.0801347
$49$	−307.693	−0.897065
$50$	−231.999	−0.656191
$51$	356.119	0.977776
$52$	−100.043	−0.266798
$53$	364.882	0.945669	0.472834	−	0.881151i	$-0.343231\pi$
$53$	364.882	0.945669	0.472834	+	0.881151i	$0.343231\pi$
$54$	−252.868	−0.637241
$55$	−17.4726	−0.0428363
$56$	−139.594	−0.333108
$57$	−56.1238	−0.130417
$58$	−169.658	−0.384089
$59$	−376.730	−0.831290	−0.415645	−	0.909527i	$-0.636444\pi$
$59$	−376.730	−0.831290	−0.415645	+	0.909527i	$0.636444\pi$
$60$	19.8292	0.0426657
$61$	816.832	1.71450	0.857251	−	0.514899i	$-0.172171\pi$
$61$	816.832	1.71450	0.857251	+	0.514899i	$0.172171\pi$
$62$	−474.689	−0.972347
$63$	−108.587	−0.217153
$64$	395.285	0.772040
$65$	−34.3006	−0.0654532
$66$	64.3254	0.119968
$67$	220.185	0.401490	0.200745	−	0.979644i	$-0.435664\pi$
$67$	220.185	0.401490	0.200745	+	0.979644i	$0.435664\pi$
$68$	−533.459	−0.951344
$69$	188.800	0.329404
$70$	−17.0447	−0.0291033
$71$	383.466	0.640973	0.320486	−	0.947253i	$-0.396154\pi$
$71$	383.466	0.640973	0.320486	+	0.947253i	$0.396154\pi$
$72$	429.324	0.702727
$73$	−537.689	−0.862078	−0.431039	−	0.902333i	$-0.641853\pi$
$73$	−537.689	−0.862078	−0.431039	+	0.902333i	$0.641853\pi$
$74$	375.706	0.590202
$75$	−362.437	−0.558008
$76$	84.0724	0.126892
$77$	68.4341	0.101283
$78$	126.278	0.183310
$79$	1062.27	1.51284	0.756421	−	0.654085i	$-0.226946\pi$
$79$	1062.27	1.51284	0.756421	+	0.654085i	$0.226946\pi$
$80$	13.6868	0.0191279
$81$	98.3741	0.134944
$82$	705.592	0.950239
$83$	−616.932	−0.815869	−0.407935	−	0.913011i	$-0.633751\pi$
$83$	−616.932	−0.815869	−0.407935	+	0.913011i	$0.633751\pi$
$84$	−77.6645	−0.100880
$85$	−182.900	−0.233392
$86$	−848.189	−1.06352
$87$	−265.045	−0.326619
$88$	−270.571	−0.327761
$89$	−90.2686	−0.107511	−0.0537554	−	0.998554i	$-0.517119\pi$
$89$	−90.2686	−0.107511	−0.0537554	+	0.998554i	$0.517119\pi$
$90$	52.4212	0.0613965
$91$	134.344	0.154759
$92$	−282.819	−0.320499
$93$	−741.576	−0.826858
$94$	−352.589	−0.386880
$95$	28.8248	0.0311302
$96$	504.776	0.536650
$97$	−524.636	−0.549162	−0.274581	−	0.961564i	$-0.588539\pi$
$97$	−524.636	−0.549162	−0.274581	+	0.961564i	$0.588539\pi$
$98$	−581.787	−0.599688
$99$	−210.470	−0.213667
$100$	542.923	0.542923
$101$	−336.523	−0.331538	−0.165769	−	0.986165i	$-0.553011\pi$
$101$	−336.523	−0.331538	−0.165769	+	0.986165i	$0.553011\pi$
$102$	673.350	0.653643
$103$	−718.709	−0.687539	−0.343770	−	0.939054i	$-0.611704\pi$
$103$	−718.709	−0.687539	−0.343770	+	0.939054i	$0.611704\pi$
$104$	−531.161	−0.500813
$105$	−26.6278	−0.0247487
$106$	689.921	0.632180
$107$	1321.73	1.19417	0.597087	−	0.802177i	$-0.296325\pi$
$107$	1321.73	1.19417	0.597087	+	0.802177i	$0.296325\pi$
$108$	591.762	0.527244
$109$	472.039	0.414799	0.207400	−	0.978256i	$-0.433500\pi$
$109$	472.039	0.414799	0.207400	+	0.978256i	$0.433500\pi$
$110$	−33.0372	−0.0286361
$111$	586.942	0.501892
$112$	−53.6067	−0.0452264
$113$	442.672	0.368523	0.184261	−	0.982877i	$-0.441011\pi$
$113$	442.672	0.368523	0.184261	+	0.982877i	$0.441011\pi$
$114$	−106.119	−0.0871838
$115$	−96.9668	−0.0786278
$116$	397.033	0.317790
$117$	−413.176	−0.326480
$118$	−712.323	−0.555717
$119$	716.360	0.551837
$120$	105.280	0.0800890
$121$	−1198.36	−0.900343
$122$	1544.47	1.14614
$123$	1102.30	0.808058
$124$	1110.87	0.804506
$125$	375.783	0.268888
$126$	−205.316	−0.145167
$127$	−1482.52	−1.03585	−0.517923	−	0.855427i	$-0.673295\pi$
$127$	−1482.52	−1.03585	−0.517923	+	0.855427i	$0.673295\pi$
$128$	−619.678	−0.427909
$129$	−1325.07	−0.904387
$130$	−64.8556	−0.0437555
$131$	785.710	0.524029	0.262015	−	0.965064i	$-0.415613\pi$
$131$	785.710	0.524029	0.262015	+	0.965064i	$0.415613\pi$
$132$	−150.534	−0.0992601
$133$	−112.897	−0.0736047
$134$	416.326	0.268396
$135$	202.890	0.129348
$136$	−2832.30	−1.78579
$137$	−1423.76	−0.887886	−0.443943	−	0.896055i	$-0.646421\pi$
$137$	−1423.76	−0.887886	−0.443943	+	0.896055i	$0.646421\pi$
$138$	356.984	0.220207
$139$	−1416.42	−0.864311	−0.432155	−	0.901799i	$-0.642247\pi$
$139$	−1416.42	−0.864311	−0.432155	+	0.901799i	$0.642247\pi$
$140$	39.8880	0.0240797
$141$	−550.827	−0.328993
$142$	725.059	0.428490
$143$	260.394	0.152274
$144$	164.868	0.0954098
$145$	136.126	0.0779630
$146$	−1016.66	−0.576299
$147$	−908.889	−0.509959
$148$	−879.228	−0.488325
$149$	3342.97	1.83803	0.919016	−	0.394221i	$-0.128986\pi$
$149$	3342.97	1.83803	0.919016	+	0.394221i	$0.128986\pi$
$150$	−685.297	−0.373028
$151$	2233.19	1.20354	0.601769	−	0.798670i	$-0.294463\pi$
$151$	2233.19	1.20354	0.601769	+	0.798670i	$0.294463\pi$
$152$	446.367	0.238191
$153$	−2203.17	−1.16416
$154$	129.395	0.0677077
$155$	380.869	0.197369
$156$	−295.516	−0.151668
$157$	1748.03	0.888588	0.444294	−	0.895881i	$-0.353455\pi$
$157$	1748.03	0.888588	0.444294	+	0.895881i	$0.353455\pi$
$158$	2008.54	1.01134
$159$	1077.82	0.537589
$160$	−259.250	−0.128097
$161$	379.786	0.185909
$162$	186.006	0.0902100
$163$	3442.43	1.65418	0.827092	−	0.562067i	$-0.189994\pi$
$163$	3442.43	1.65418	0.827092	+	0.562067i	$0.189994\pi$
$164$	−1651.23	−0.786215
$165$	−51.6119	−0.0243514
$166$	−1166.50	−0.545409
$167$	−3440.98	−1.59444	−0.797218	−	0.603692i	$-0.793696\pi$
$167$	−3440.98	−1.59444	−0.797218	+	0.603692i	$0.793696\pi$
$168$	−412.345	−0.189364
$169$	−1685.82	−0.767327
$170$	−345.829	−0.156023
$171$	347.217	0.155277
$172$	1984.93	0.879940
$173$	3159.81	1.38865	0.694324	−	0.719663i	$-0.255704\pi$
$173$	3159.81	1.38865	0.694324	+	0.719663i	$0.255704\pi$
$174$	−501.149	−0.218345
$175$	−729.069	−0.314928
$176$	−103.904	−0.0445003
$177$	−1112.82	−0.472567
$178$	−170.680	−0.0718709
$179$	2056.18	0.858580	0.429290	−	0.903167i	$-0.358764\pi$
$179$	2056.18	0.858580	0.429290	+	0.903167i	$0.358764\pi$
$180$	−122.676	−0.0507986
$181$	−123.527	−0.0507277	−0.0253638	−	0.999678i	$-0.508074\pi$
$181$	−123.527	−0.0507277	−0.0253638	+	0.999678i	$0.508074\pi$
$182$	254.018	0.103456
$183$	2412.82	0.974651
$184$	−1501.58	−0.601618
$185$	−301.450	−0.119800
$186$	−1402.17	−0.552755
$187$	1388.50	0.542978
$188$	825.128	0.320099
$189$	−794.652	−0.305833
$190$	54.5021	0.0208105
$191$	−3379.82	−1.28039	−0.640197	−	0.768210i	$-0.721147\pi$
$191$	−3379.82	−1.28039	−0.640197	+	0.768210i	$0.721147\pi$
$192$	1167.62	0.438886
$193$	−386.034	−0.143976	−0.0719880	−	0.997405i	$-0.522934\pi$
$193$	−386.034	−0.143976	−0.0719880	+	0.997405i	$0.522934\pi$
$194$	−991.983	−0.367115
$195$	−101.320	−0.0372085
$196$	1361.50	0.496173
$197$	−3242.43	−1.17266	−0.586328	−	0.810073i	$-0.699427\pi$
$197$	−3242.43	−1.17266	−0.586328	+	0.810073i	$0.699427\pi$
$198$	−397.958	−0.142836
$199$	−752.507	−0.268059	−0.134030	−	0.990977i	$-0.542792\pi$
$199$	−752.507	−0.268059	−0.134030	+	0.990977i	$0.542792\pi$
$200$	2882.55	1.01914
$201$	650.400	0.228237
$202$	−636.300	−0.221633
$203$	−533.159	−0.184337
$204$	−1575.77	−0.540815
$205$	−566.136	−0.192881
$206$	−1358.94	−0.459620
$207$	−1168.04	−0.392194
$208$	−203.975	−0.0679958
$209$	−218.825	−0.0724231
$210$	−50.3480	−0.0165445
$211$	−3996.39	−1.30390	−0.651949	−	0.758263i	$-0.726048\pi$
$211$	−3996.39	−1.30390	−0.651949	+	0.758263i	$0.726048\pi$
$212$	−1614.55	−0.523057
$213$	1132.71	0.364377
$214$	2499.13	0.798305
$215$	680.549	0.215875
$216$	3141.85	0.989703
$217$	−1491.74	−0.466662
$218$	892.533	0.277293
$219$	−1588.27	−0.490070
$220$	77.3136	0.0236931
$221$	2725.77	0.829661
$222$	1109.79	0.335515
$223$	5465.82	1.64134	0.820669	−	0.571404i	$-0.193601\pi$
$223$	5465.82	1.64134	0.820669	+	0.571404i	$0.193601\pi$
$224$	1015.39	0.302875
$225$	2242.26	0.664374
$226$	837.006	0.246358
$227$	−1866.81	−0.545836	−0.272918	−	0.962037i	$-0.587989\pi$
$227$	−1866.81	−0.545836	−0.272918	+	0.962037i	$0.587989\pi$
$228$	248.340	0.0721347
$229$	456.437	0.131713	0.0658563	−	0.997829i	$-0.479022\pi$
$229$	456.437	0.131713	0.0658563	+	0.997829i	$0.479022\pi$
$230$	−183.345	−0.0525627
$231$	202.146	0.0575768
$232$	2107.97	0.596531
$233$	−1268.44	−0.356645	−0.178323	−	0.983972i	$-0.557067\pi$
$233$	−1268.44	−0.356645	−0.178323	+	0.983972i	$0.557067\pi$
$234$	−781.235	−0.218252
$235$	282.901	0.0785296
$236$	1666.98	0.459793
$237$	3137.82	0.860013
$238$	1354.50	0.368903
$239$	−5115.26	−1.38443	−0.692214	−	0.721692i	$-0.743365\pi$
$239$	−5115.26	−1.38443	−0.692214	+	0.721692i	$0.743365\pi$
$240$	40.4293	0.0108737
$241$	−1509.15	−0.403372	−0.201686	−	0.979450i	$-0.564642\pi$
$241$	−1509.15	−0.403372	−0.201686	+	0.979450i	$0.564642\pi$
$242$	−2265.86	−0.601879
$243$	3901.45	1.02995
$244$	−3614.37	−0.948304
$245$	466.800	0.121726
$246$	2084.24	0.540187
$247$	−429.577	−0.110661
$248$	5897.94	1.51016
$249$	−1822.35	−0.463801
$250$	710.531	0.179752
$251$	−4695.51	−1.18079	−0.590394	−	0.807115i	$-0.701028\pi$
$251$	−4695.51	−1.18079	−0.590394	+	0.807115i	$0.701028\pi$
$252$	480.481	0.120109
$253$	736.127	0.182924
$254$	−2803.16	−0.692464
$255$	−540.266	−0.132678
$256$	−4333.97	−1.05810
$257$	−6383.64	−1.54942	−0.774710	−	0.632317i	$-0.782104\pi$
$257$	−6383.64	−1.54942	−0.774710	+	0.632317i	$0.782104\pi$
$258$	−2505.45	−0.604583
$259$	1180.68	0.283258
$260$	151.775	0.0362027
$261$	1639.74	0.388878
$262$	1485.62	0.350314
$263$	−3805.82	−0.892308	−0.446154	−	0.894956i	$-0.647207\pi$
$263$	−3805.82	−0.892308	−0.446154	+	0.894956i	$0.647207\pi$
$264$	−799.234	−0.186324
$265$	−553.562	−0.128321
$266$	−213.466	−0.0492048
$267$	−266.643	−0.0611171
$268$	−974.287	−0.222067
$269$	6390.72	1.44851	0.724255	−	0.689533i	$-0.242184\pi$
$269$	6390.72	1.44851	0.724255	+	0.689533i	$0.242184\pi$
$270$	383.625	0.0864693
$271$	−2526.33	−0.566286	−0.283143	−	0.959078i	$-0.591377\pi$
$271$	−2526.33	−0.566286	−0.283143	+	0.959078i	$0.591377\pi$
$272$	−1087.65	−0.242459
$273$	396.835	0.0879765
$274$	−2692.06	−0.593552
$275$	−1413.13	−0.309872
$276$	−835.415	−0.182196
$277$	3156.63	0.684706	0.342353	−	0.939571i	$-0.388776\pi$
$277$	3156.63	0.684706	0.342353	+	0.939571i	$0.388776\pi$
$278$	−2678.17	−0.577792
$279$	4587.86	0.984472
$280$	211.778	0.0452006
$281$	8144.80	1.72910	0.864552	−	0.502544i	$-0.167602\pi$
$281$	8144.80	1.72910	0.864552	+	0.502544i	$0.167602\pi$
$282$	−1041.50	−0.219932
$283$	−3378.88	−0.709729	−0.354865	−	0.934918i	$-0.615473\pi$
$283$	−3378.88	−0.709729	−0.354865	+	0.934918i	$0.615473\pi$
$284$	−1696.78	−0.354527
$285$	85.1452	0.0176967
$286$	492.354	0.101795
$287$	2217.36	0.456052
$288$	−3122.86	−0.638946
$289$	9621.60	1.95840
$290$	257.387	0.0521183
$291$	−1549.71	−0.312185
$292$	2379.20	0.476822
$293$	3191.30	0.636307	0.318154	−	0.948039i	$-0.396937\pi$
$293$	3191.30	0.636307	0.318154	+	0.948039i	$0.396937\pi$
$294$	−1718.53	−0.340907
$295$	571.536	0.112800
$296$	−4668.10	−0.916647
$297$	−1540.25	−0.300923
$298$	6320.90	1.22872
$299$	1445.10	0.279506
$300$	1603.73	0.308638
$301$	−2665.48	−0.510418
$302$	4222.52	0.804566
$303$	−994.050	−0.188471
$304$	171.413	0.0323394
$305$	−1239.21	−0.232646
$306$	−4165.77	−0.778240
$307$	4313.64	0.801931	0.400965	−	0.916093i	$-0.368675\pi$
$307$	4313.64	0.801931	0.400965	+	0.916093i	$0.368675\pi$
$308$	−302.811	−0.0560204
$309$	−2122.98	−0.390849
$310$	720.149	0.131941
$311$	−296.966	−0.0541459	−0.0270730	−	0.999633i	$-0.508619\pi$
$311$	−296.966	−0.0541459	−0.0270730	+	0.999633i	$0.508619\pi$
$312$	−1568.99	−0.284700
$313$	−74.6344	−0.0134779	−0.00673895	−	0.999977i	$-0.502145\pi$
$313$	−74.6344	−0.0134779	−0.00673895	+	0.999977i	$0.502145\pi$
$314$	3305.19	0.594021
$315$	164.737	0.0294662
$316$	−4700.39	−0.836764
$317$	−634.636	−0.112444	−0.0562220	−	0.998418i	$-0.517905\pi$
$317$	−634.636	−0.112444	−0.0562220	+	0.998418i	$0.517905\pi$
$318$	2037.94	0.359378
$319$	−1033.40	−0.181378
$320$	−599.685	−0.104761
$321$	3904.24	0.678858
$322$	718.101	0.124280
$323$	−2290.63	−0.394595
$324$	−435.292	−0.0746385
$325$	−2774.13	−0.473480
$326$	6508.96	1.10582
$327$	1394.35	0.235803
$328$	−8766.88	−1.47582
$329$	−1108.03	−0.185677
$330$	−97.5879	−0.0162789
$331$	−1010.00	−0.167718	−0.0838588	−	0.996478i	$-0.526724\pi$
$331$	−1010.00	−0.167718	−0.0838588	+	0.996478i	$0.526724\pi$
$332$	2729.84	0.451263
$333$	−3631.19	−0.597562
$334$	−6506.21	−1.06588
$335$	−334.042	−0.0544795
$336$	−158.348	−0.0257101
$337$	−3136.03	−0.506916	−0.253458	−	0.967346i	$-0.581568\pi$
$337$	−3136.03	−0.506916	−0.253458	+	0.967346i	$0.581568\pi$
$338$	−3187.55	−0.512958
$339$	1307.60	0.209496
$340$	809.309	0.129091
$341$	−2891.38	−0.459170
$342$	656.519	0.103803
$343$	−3866.39	−0.608646
$344$	10538.6	1.65176
$345$	−286.428	−0.0446979
$346$	5974.58	0.928311
$347$	−11450.8	−1.77150	−0.885749	−	0.464164i	$-0.846355\pi$
$347$	−11450.8	−1.77150	−0.885749	+	0.464164i	$0.846355\pi$
$348$	1172.79	0.180655
$349$	4403.78	0.675442	0.337721	−	0.941246i	$-0.390344\pi$
$349$	4403.78	0.675442	0.337721	+	0.941246i	$0.390344\pi$
$350$	−1378.53	−0.210529
$351$	−3023.68	−0.459806
$352$	1968.10	0.298012
$353$	601.300	0.0906629	0.0453314	−	0.998972i	$-0.485566\pi$
$353$	601.300	0.0906629	0.0453314	+	0.998972i	$0.485566\pi$
$354$	−2104.12	−0.315911
$355$	−581.755	−0.0869757
$356$	399.426	0.0594650
$357$	2116.04	0.313705
$358$	3887.83	0.573961
$359$	7924.45	1.16500	0.582502	−	0.812829i	$-0.302074\pi$
$359$	7924.45	1.16500	0.582502	+	0.812829i	$0.302074\pi$
$360$	−651.327	−0.0953554
$361$	361.000	0.0526316
$362$	−233.566	−0.0339115
$363$	−3539.80	−0.511822
$364$	−594.452	−0.0855982
$365$	815.726	0.116978
$366$	4562.18	0.651554
$367$	−2232.61	−0.317552	−0.158776	−	0.987315i	$-0.550755\pi$
$367$	−2232.61	−0.317552	−0.158776	+	0.987315i	$0.550755\pi$
$368$	−576.632	−0.0816821
$369$	−6819.53	−0.962089
$370$	−569.983	−0.0800865
$371$	2168.12	0.303404
$372$	3281.37	0.457341
$373$	−9994.88	−1.38744	−0.693720	−	0.720245i	$-0.744030\pi$
$373$	−9994.88	−1.38744	−0.693720	+	0.720245i	$0.744030\pi$
$374$	2625.37	0.362981
$375$	1110.02	0.152856
$376$	4380.86	0.600867
$377$	−2028.69	−0.277142
$378$	−1502.53	−0.204449
$379$	13486.3	1.82782	0.913912	−	0.405913i	$-0.133046\pi$
$379$	13486.3	1.82782	0.913912	+	0.405913i	$0.133046\pi$
$380$	−127.546	−0.0172183
$381$	−4379.19	−0.588853
$382$	−6390.58	−0.855944
$383$	5181.58	0.691296	0.345648	−	0.938364i	$-0.387659\pi$
$383$	5181.58	0.691296	0.345648	+	0.938364i	$0.387659\pi$
$384$	−1830.46	−0.243255
$385$	−103.821	−0.0137434
$386$	−729.915	−0.0962480
$387$	8197.73	1.07678
$388$	2321.44	0.303745
$389$	−371.727	−0.0484507	−0.0242253	−	0.999707i	$-0.507712\pi$
$389$	−371.727	−0.0484507	−0.0242253	+	0.999707i	$0.507712\pi$
$390$	−191.576	−0.0248739
$391$	7705.68	0.996657
$392$	7228.63	0.931380
$393$	2320.90	0.297897
$394$	−6130.79	−0.783921
$395$	−1611.56	−0.205283
$396$	931.301	0.118181
$397$	−4571.57	−0.577936	−0.288968	−	0.957339i	$-0.593312\pi$
$397$	−4571.57	−0.577936	−0.288968	+	0.957339i	$0.593312\pi$
$398$	−1422.84	−0.179198
$399$	−333.485	−0.0418424
$400$	1106.95	0.138369
$401$	13606.8	1.69449	0.847245	−	0.531203i	$-0.178260\pi$
$401$	13606.8	1.69449	0.847245	+	0.531203i	$0.178260\pi$
$402$	1229.78	0.152577
$403$	−5676.10	−0.701605
$404$	1489.07	0.183376
$405$	−149.243	−0.0183110
$406$	−1008.10	−0.123229
$407$	2288.47	0.278710
$408$	−8366.28	−1.01518
$409$	−3182.34	−0.384735	−0.192368	−	0.981323i	$-0.561617\pi$
$409$	−3182.34	−0.384735	−0.192368	+	0.981323i	$0.561617\pi$
$410$	−1070.45	−0.128941
$411$	−4205.63	−0.504741
$412$	3180.19	0.380283
$413$	−2238.51	−0.266707
$414$	−2208.53	−0.262182
$415$	935.947	0.110708
$416$	3863.61	0.455358
$417$	−4183.94	−0.491339
$418$	−413.755	−0.0484149
$419$	7452.18	0.868885	0.434442	−	0.900700i	$-0.356945\pi$
$419$	7452.18	0.868885	0.434442	+	0.900700i	$0.356945\pi$
$420$	117.825	0.0136887
$421$	−9082.58	−1.05144	−0.525722	−	0.850656i	$-0.676205\pi$
$421$	−9082.58	−1.05144	−0.525722	+	0.850656i	$0.676205\pi$
$422$	−7556.38	−0.871656
$423$	3407.76	0.391705
$424$	−8572.17	−0.981843
$425$	−14792.5	−1.68833
$426$	2141.74	0.243586
$427$	4853.58	0.550073
$428$	−5848.48	−0.660507
$429$	769.173	0.0865641
$430$	1286.78	0.144312
$431$	−10010.1	−1.11872	−0.559362	−	0.828923i	$-0.688954\pi$
$431$	−10010.1	−1.11872	−0.559362	+	0.828923i	$0.688954\pi$
$432$	1206.53	0.134373
$433$	−8137.16	−0.903111	−0.451555	−	0.892243i	$-0.649131\pi$
$433$	−8137.16	−0.903111	−0.451555	+	0.892243i	$0.649131\pi$
$434$	−2820.58	−0.311963
$435$	402.100	0.0443200
$436$	−2088.71	−0.229429
$437$	−1214.40	−0.132935
$438$	−3003.10	−0.327612
$439$	−5582.62	−0.606933	−0.303467	−	0.952842i	$-0.598144\pi$
$439$	−5582.62	−0.606933	−0.303467	+	0.952842i	$0.598144\pi$
$440$	410.482	0.0444749
$441$	5622.96	0.607166
$442$	5153.90	0.554629
$443$	5823.52	0.624568	0.312284	−	0.949989i	$-0.398906\pi$
$443$	5823.52	0.624568	0.312284	+	0.949989i	$0.398906\pi$
$444$	−2597.14	−0.277600
$445$	136.946	0.0145885
$446$	10334.8	1.09723
$447$	9874.73	1.04487
$448$	2348.77	0.247698
$449$	3999.42	0.420366	0.210183	−	0.977662i	$-0.432594\pi$
$449$	3999.42	0.420366	0.210183	+	0.977662i	$0.432594\pi$
$450$	4239.68	0.444134
$451$	4297.84	0.448730
$452$	−1958.76	−0.203833
$453$	6596.58	0.684182
$454$	−3529.78	−0.364892
$455$	−203.812	−0.0209997
$456$	1318.51	0.135406
$457$	10800.0	1.10548	0.552739	−	0.833354i	$-0.313583\pi$
$457$	10800.0	1.10548	0.552739	+	0.833354i	$0.313583\pi$
$458$	863.033	0.0880499
$459$	−16123.1	−1.63957
$460$	429.064	0.0434896
$461$	11879.6	1.20019	0.600096	−	0.799928i	$-0.295129\pi$
$461$	11879.6	1.20019	0.600096	+	0.799928i	$0.295129\pi$
$462$	382.219	0.0384901
$463$	10252.9	1.02914	0.514569	−	0.857449i	$-0.327952\pi$
$463$	10252.9	1.02914	0.514569	+	0.857449i	$0.327952\pi$
$464$	809.499	0.0809915
$465$	1125.04	0.112199
$466$	−2398.37	−0.238417
$467$	−4190.63	−0.415245	−0.207622	−	0.978209i	$-0.566573\pi$
$467$	−4190.63	−0.415245	−0.207622	+	0.978209i	$0.566573\pi$
$468$	1828.25	0.180579
$469$	1308.33	0.128812
$470$	534.911	0.0524970
$471$	5163.48	0.505140
$472$	8850.51	0.863089
$473$	−5166.41	−0.502224
$474$	5932.99	0.574919
$475$	2331.27	0.225192
$476$	−3169.79	−0.305225
$477$	−6668.07	−0.640063
$478$	−9671.94	−0.925490
$479$	−11777.5	−1.12344	−0.561721	−	0.827327i	$-0.689861\pi$
$479$	−11777.5	−1.12344	−0.561721	+	0.827327i	$0.689861\pi$
$480$	−765.794	−0.0728199
$481$	4492.52	0.425865
$482$	−2853.50	−0.269654
$483$	1121.84	0.105685
$484$	5302.56	0.497987
$485$	795.923	0.0745176
$486$	7376.88	0.688523
$487$	17288.2	1.60863	0.804315	−	0.594203i	$-0.202532\pi$
$487$	17288.2	1.60863	0.804315	+	0.594203i	$0.202532\pi$
$488$	−19189.8	−1.78009
$489$	10168.5	0.940361
$490$	882.628	0.0813736
$491$	−4426.69	−0.406872	−0.203436	−	0.979088i	$-0.565211\pi$
$491$	−4426.69	−0.406872	−0.203436	+	0.979088i	$0.565211\pi$
$492$	−4877.53	−0.446943
$493$	−10817.5	−0.988230
$494$	−812.246	−0.0739771
$495$	319.304	0.0289932
$496$	2264.91	0.205036
$497$	2278.54	0.205647
$498$	−3445.70	−0.310051
$499$	12876.9	1.15521	0.577606	−	0.816316i	$-0.303987\pi$
$499$	12876.9	1.15521	0.577606	+	0.816316i	$0.303987\pi$
$500$	−1662.79	−0.148724
$501$	−10164.2	−0.906396
$502$	−8878.28	−0.789357
$503$	−3852.90	−0.341535	−0.170768	−	0.985311i	$-0.554625\pi$
$503$	−3852.90	−0.341535	−0.170768	+	0.985311i	$0.554625\pi$
$504$	2551.03	0.225460
$505$	510.539	0.0449875
$506$	1391.87	0.122285
$507$	−4979.71	−0.436206
$508$	6559.95	0.572935
$509$	−20087.1	−1.74920	−0.874601	−	0.484844i	$-0.838876\pi$
$509$	−20087.1	−1.74920	−0.874601	+	0.484844i	$0.838876\pi$
$510$	−1021.54	−0.0886950
$511$	−3194.92	−0.276585
$512$	−3237.26	−0.279430
$513$	2540.98	0.218688
$514$	−12070.2	−1.03579
$515$	1090.35	0.0932945
$516$	5863.25	0.500224
$517$	−2147.66	−0.182696
$518$	2232.43	0.189358
$519$	9333.71	0.789411
$520$	805.822	0.0679570
$521$	21715.8	1.82608	0.913038	−	0.407874i	$-0.133730\pi$
$521$	21715.8	1.82608	0.913038	+	0.407874i	$0.133730\pi$
$522$	3100.42	0.259965
$523$	3451.54	0.288576	0.144288	−	0.989536i	$-0.453911\pi$
$523$	3451.54	0.288576	0.144288	+	0.989536i	$0.453911\pi$
$524$	−3476.66	−0.289845
$525$	−2153.58	−0.179029
$526$	−7196.06	−0.596508
$527$	−30266.6	−2.50177
$528$	−306.920	−0.0252973
$529$	−8081.75	−0.664235
$530$	−1046.68	−0.0857825
$531$	6884.59	0.562647
$532$	499.554	0.0407113
$533$	8437.13	0.685653
$534$	−504.169	−0.0408568
$535$	−2005.19	−0.162041
$536$	−5172.79	−0.416848
$537$	6073.70	0.488081
$538$	12083.6	0.968329
$539$	−3543.73	−0.283190
$540$	−897.761	−0.0715435
$541$	−5540.30	−0.440288	−0.220144	−	0.975467i	$-0.570653\pi$
$541$	−5540.30	−0.440288	−0.220144	+	0.975467i	$0.570653\pi$
$542$	−4776.79	−0.378562
$543$	−364.885	−0.0288374
$544$	20601.9	1.62371
$545$	−716.129	−0.0562855
$546$	750.338	0.0588123
$547$	−463.137	−0.0362016	−0.0181008	−	0.999836i	$-0.505762\pi$
$547$	−463.137	−0.0362016	−0.0181008	+	0.999836i	$0.505762\pi$
$548$	6299.96	0.491096
$549$	−14927.3	−1.16044
$550$	−2671.95	−0.207150
$551$	1704.83	0.131811
$552$	−4435.48	−0.342005
$553$	6311.95	0.485374
$554$	5968.57	0.457726
$555$	−890.448	−0.0681034
$556$	6267.46	0.478057
$557$	2283.84	0.173733	0.0868667	−	0.996220i	$-0.472315\pi$
$557$	2283.84	0.173733	0.0868667	+	0.996220i	$0.472315\pi$
$558$	8674.73	0.658120
$559$	−10142.2	−0.767390
$560$	81.3266	0.00613692
$561$	4101.45	0.308669
$562$	15400.2	1.15591
$563$	12065.7	0.903211	0.451606	−	0.892218i	$-0.350851\pi$
$563$	12065.7	0.903211	0.451606	+	0.892218i	$0.350851\pi$
$564$	2437.33	0.181968
$565$	−671.577	−0.0500061
$566$	−6388.79	−0.474454
$567$	584.535	0.0432948
$568$	−9008.76	−0.665492
$569$	4496.79	0.331309	0.165655	−	0.986184i	$-0.447026\pi$
$569$	4496.79	0.331309	0.165655	+	0.986184i	$0.447026\pi$
$570$	160.993	0.0118303
$571$	−8884.67	−0.651159	−0.325580	−	0.945515i	$-0.605559\pi$
$571$	−8884.67	−0.651159	−0.325580	+	0.945515i	$0.605559\pi$
$572$	−1152.21	−0.0842241
$573$	−9983.60	−0.727872
$574$	4192.60	0.304871
$575$	−7842.39	−0.568783
$576$	−7223.66	−0.522545
$577$	8760.77	0.632089	0.316045	−	0.948744i	$-0.397645\pi$
$577$	8760.77	0.632089	0.316045	+	0.948744i	$0.397645\pi$
$578$	18192.6	1.30919
$579$	−1140.30	−0.0818467
$580$	−602.338	−0.0431219
$581$	−3665.79	−0.261760
$582$	−2930.20	−0.208695
$583$	4202.39	0.298533
$584$	12631.9	0.895055
$585$	626.828	0.0443011
$586$	6034.13	0.425371
$587$	4691.92	0.329909	0.164954	−	0.986301i	$-0.447252\pi$
$587$	4691.92	0.329909	0.164954	+	0.986301i	$0.447252\pi$
$588$	4021.71	0.282062
$589$	4769.97	0.333690
$590$	1080.66	0.0754071
$591$	−9577.75	−0.666626
$592$	−1792.63	−0.124454
$593$	9399.23	0.650894	0.325447	−	0.945560i	$-0.394485\pi$
$593$	9399.23	0.650894	0.325447	+	0.945560i	$0.394485\pi$
$594$	−2912.31	−0.201167
$595$	−1086.79	−0.0748805
$596$	−14792.2	−1.01663
$597$	−2222.82	−0.152385
$598$	2732.40	0.186850
$599$	15981.9	1.09016	0.545078	−	0.838386i	$-0.316500\pi$
$599$	15981.9	1.09016	0.545078	+	0.838386i	$0.316500\pi$
$600$	8514.71	0.579353
$601$	−24969.1	−1.69470	−0.847348	−	0.531038i	$-0.821802\pi$
$601$	−24969.1	−1.69470	−0.847348	+	0.531038i	$0.821802\pi$
$602$	−5039.90	−0.341214
$603$	−4023.78	−0.271743
$604$	−9881.55	−0.665687
$605$	1818.02	0.122171
$606$	−1879.55	−0.125993
$607$	−12651.2	−0.845956	−0.422978	−	0.906140i	$-0.639015\pi$
$607$	−12651.2	−0.845956	−0.422978	+	0.906140i	$0.639015\pi$
$608$	−3246.82	−0.216573
$609$	−1574.89	−0.104791
$610$	−2343.11	−0.155524
$611$	−4216.09	−0.279156
$612$	9748.74	0.643904
$613$	11851.0	0.780847	0.390424	−	0.920635i	$-0.372329\pi$
$613$	11851.0	0.780847	0.390424	+	0.920635i	$0.372329\pi$
$614$	8156.26	0.536091
$615$	−1672.30	−0.109648
$616$	−1607.72	−0.105157
$617$	19473.2	1.27060	0.635301	−	0.772265i	$-0.280876\pi$
$617$	19473.2	1.27060	0.635301	+	0.772265i	$0.280876\pi$
$618$	−4014.14	−0.261282
$619$	−25664.7	−1.66648	−0.833240	−	0.552912i	$-0.813517\pi$
$619$	−25664.7	−1.66648	−0.833240	+	0.552912i	$0.813517\pi$
$620$	−1685.29	−0.109166
$621$	−8547.85	−0.552357
$622$	−561.504	−0.0361966
$623$	−536.372	−0.0344933
$624$	−602.518	−0.0386539
$625$	14767.2	0.945101
$626$	−141.119	−0.00900998
$627$	−646.383	−0.0411707
$628$	−7734.80	−0.491485
$629$	23955.4	1.51854
$630$	311.485	0.0196982
$631$	−19618.1	−1.23769	−0.618846	−	0.785512i	$-0.712399\pi$
$631$	−19618.1	−1.23769	−0.618846	+	0.785512i	$0.712399\pi$
$632$	−24955.8	−1.57071
$633$	−11804.8	−0.741233
$634$	−1199.97	−0.0751688
$635$	2249.13	0.140557
$636$	−4769.20	−0.297344
$637$	−6956.74	−0.432709
$638$	−1953.96	−0.121251
$639$	−7007.68	−0.433834
$640$	940.112	0.0580644
$641$	−683.277	−0.0421027	−0.0210513	−	0.999778i	$-0.506701\pi$
$641$	−683.277	−0.0421027	−0.0210513	+	0.999778i	$0.506701\pi$
$642$	7382.15	0.453816
$643$	7107.06	0.435886	0.217943	−	0.975961i	$-0.430065\pi$
$643$	7107.06	0.435886	0.217943	+	0.975961i	$0.430065\pi$
$644$	−1680.50	−0.102828
$645$	2010.26	0.122719
$646$	−4331.13	−0.263787
$647$	23626.1	1.43561	0.717805	−	0.696244i	$-0.245147\pi$
$647$	23626.1	1.43561	0.717805	+	0.696244i	$0.245147\pi$
$648$	−2311.10	−0.140106
$649$	−4338.84	−0.262426
$650$	−5245.33	−0.316521
$651$	−4406.41	−0.265285
$652$	−15232.3	−0.914941
$653$	1155.57	0.0692508	0.0346254	−	0.999400i	$-0.488976\pi$
$653$	1155.57	0.0692508	0.0346254	+	0.999400i	$0.488976\pi$
$654$	2636.44	0.157634
$655$	−1192.00	−0.0711073
$656$	−3366.64	−0.200374
$657$	9826.04	0.583486
$658$	−2095.07	−0.124125
$659$	10881.9	0.643246	0.321623	−	0.946868i	$-0.395772\pi$
$659$	10881.9	0.643246	0.321623	+	0.946868i	$0.395772\pi$
$660$	228.375	0.0134689
$661$	18028.3	1.06085	0.530425	−	0.847732i	$-0.322032\pi$
$661$	18028.3	1.06085	0.530425	+	0.847732i	$0.322032\pi$
$662$	−1909.71	−0.112119
$663$	8051.60	0.471642
$664$	14493.6	0.847078
$665$	171.276	0.00998767
$666$	−6865.87	−0.399470
$667$	−5735.04	−0.332926
$668$	15225.8	0.881894
$669$	16145.4	0.933059
$670$	−631.607	−0.0364196
$671$	9407.53	0.541242
$672$	2999.36	0.172177
$673$	−8571.78	−0.490963	−0.245481	−	0.969401i	$-0.578946\pi$
$673$	−8571.78	−0.490963	−0.245481	+	0.969401i	$0.578946\pi$
$674$	−5929.62	−0.338873
$675$	16409.2	0.935688
$676$	7459.51	0.424415
$677$	−19800.9	−1.12409	−0.562045	−	0.827107i	$-0.689985\pi$
$677$	−19800.9	−1.12409	−0.562045	+	0.827107i	$0.689985\pi$
$678$	2472.42	0.140048
$679$	−3117.36	−0.176191
$680$	4296.88	0.242320
$681$	−5514.35	−0.310294
$682$	−5467.03	−0.306955
$683$	11953.7	0.669686	0.334843	−	0.942274i	$-0.391317\pi$
$683$	11953.7	0.669686	0.334843	+	0.942274i	$0.391317\pi$
$684$	−1536.39	−0.0858848
$685$	2159.99	0.120480
$686$	−7310.59	−0.406880
$687$	1348.26	0.0748753
$688$	4047.02	0.224260
$689$	8249.75	0.456154
$690$	−541.580	−0.0298806
$691$	15664.0	0.862355	0.431177	−	0.902267i	$-0.358098\pi$
$691$	15664.0	0.862355	0.431177	+	0.902267i	$0.358098\pi$
$692$	−13981.7	−0.768071
$693$	−1250.60	−0.0685520
$694$	−21651.2	−1.18425
$695$	2148.85	0.117281
$696$	6226.70	0.339113
$697$	44989.2	2.44489
$698$	8326.69	0.451533
$699$	−3746.82	−0.202744
$700$	3226.03	0.174189
$701$	27240.6	1.46771	0.733855	−	0.679306i	$-0.237719\pi$
$701$	27240.6	1.46771	0.733855	+	0.679306i	$0.237719\pi$
$702$	−5717.18	−0.307380
$703$	−3775.33	−0.202545
$704$	4552.53	0.243722
$705$	835.657	0.0446421
$706$	1136.94	0.0606081
$707$	−1999.61	−0.106369
$708$	4924.06	0.261381
$709$	−30706.2	−1.62651	−0.813256	−	0.581906i	$-0.802307\pi$
$709$	−30706.2	−1.62651	−0.813256	+	0.581906i	$0.802307\pi$
$710$	−1099.99	−0.0581433
$711$	−19412.5	−1.02395
$712$	2120.68	0.111623
$713$	−16046.2	−0.842825
$714$	4001.02	0.209712
$715$	−395.043	−0.0206626
$716$	−9098.30	−0.474887
$717$	−15109.9	−0.787013
$718$	14983.6	0.778806
$719$	−10948.3	−0.567874	−0.283937	−	0.958843i	$-0.591641\pi$
$719$	−10948.3	−0.567874	−0.283937	+	0.958843i	$0.591641\pi$
$720$	−250.121	−0.0129465
$721$	−4270.54	−0.220587
$722$	682.580	0.0351842
$723$	−4457.84	−0.229307
$724$	546.591	0.0280579
$725$	11009.5	0.563974
$726$	−6693.07	−0.342153
$727$	−32291.1	−1.64733	−0.823666	−	0.567076i	$-0.808075\pi$
$727$	−32291.1	−1.64733	−0.823666	+	0.567076i	$0.808075\pi$
$728$	−3156.13	−0.160679
$729$	8868.32	0.450558
$730$	1542.38	0.0782000
$731$	−54081.3	−2.73635
$732$	−10676.4	−0.539087
$733$	−12538.5	−0.631813	−0.315907	−	0.948790i	$-0.602309\pi$
$733$	−12538.5	−0.631813	−0.315907	+	0.948790i	$0.602309\pi$
$734$	−4221.43	−0.212283
$735$	1378.87	0.0691980
$736$	10922.3	0.547013
$737$	2535.89	0.126744
$738$	−12894.4	−0.643156
$739$	7341.82	0.365458	0.182729	−	0.983163i	$-0.441507\pi$
$739$	7341.82	0.365458	0.182729	+	0.983163i	$0.441507\pi$
$740$	1333.87	0.0662624
$741$	−1268.92	−0.0629082
$742$	4099.48	0.202826
$743$	10142.9	0.500819	0.250410	−	0.968140i	$-0.419435\pi$
$743$	10142.9	0.500819	0.250410	+	0.968140i	$0.419435\pi$
$744$	17421.8	0.858488
$745$	−5071.61	−0.249409
$746$	−18898.4	−0.927504
$747$	11274.2	0.552210
$748$	−6143.90	−0.300325
$749$	7853.67	0.383133
$750$	2098.82	0.102184
$751$	17843.5	0.867000	0.433500	−	0.901153i	$-0.357278\pi$
$751$	17843.5	0.867000	0.433500	+	0.901153i	$0.357278\pi$
$752$	1682.33	0.0815801
$753$	−13870.0	−0.671248
$754$	−3835.85	−0.185270
$755$	−3387.97	−0.163312
$756$	3516.23	0.169159
$757$	−8510.80	−0.408627	−0.204313	−	0.978906i	$-0.565496\pi$
$757$	−8510.80	−0.408627	−0.204313	+	0.978906i	$0.565496\pi$
$758$	25500.0	1.22190
$759$	2174.43	0.103988
$760$	−677.181	−0.0323210
$761$	11227.3	0.534809	0.267404	−	0.963584i	$-0.413834\pi$
$761$	11227.3	0.534809	0.267404	+	0.963584i	$0.413834\pi$
$762$	−8280.20	−0.393648
$763$	2804.84	0.133082
$764$	14955.2	0.708196
$765$	3342.43	0.157968
$766$	9797.35	0.462131
$767$	−8517.62	−0.400982
$768$	−12802.0	−0.601502
$769$	7372.60	0.345725	0.172863	−	0.984946i	$-0.444698\pi$
$769$	7372.60	0.345725	0.172863	+	0.984946i	$0.444698\pi$
$770$	−196.306	−0.00918748
$771$	−18856.5	−0.880806
$772$	1708.15	0.0796342
$773$	2922.48	0.135982	0.0679910	−	0.997686i	$-0.478341\pi$
$773$	2922.48	0.135982	0.0679910	+	0.997686i	$0.478341\pi$
$774$	15500.3	0.719827
$775$	30803.6	1.42774
$776$	12325.3	0.570169
$777$	3487.58	0.161025
$778$	−702.863	−0.0323893
$779$	−7090.24	−0.326103
$780$	448.326	0.0205803
$781$	4416.42	0.202345
$782$	14569.9	0.666265
$783$	11999.8	0.547686
$784$	2775.92	0.126454
$785$	−2651.94	−0.120575
$786$	4388.36	0.199144
$787$	−7067.94	−0.320133	−0.160067	−	0.987106i	$-0.551171\pi$
$787$	−7067.94	−0.320133	−0.160067	+	0.987106i	$0.551171\pi$
$788$	14347.3	0.648605
$789$	−11241.9	−0.507255
$790$	−3047.15	−0.137231
$791$	2630.34	0.118235
$792$	4944.57	0.221840
$793$	18468.0	0.827010
$794$	−8643.94	−0.386350
$795$	−1635.16	−0.0729472
$796$	3329.74	0.148266
$797$	−41550.9	−1.84669	−0.923343	−	0.383975i	$-0.874555\pi$
$797$	−41550.9	−1.84669	−0.923343	+	0.383975i	$0.874555\pi$
$798$	−630.555	−0.0279717
$799$	−22481.4	−0.995412
$800$	−20967.4	−0.926636
$801$	1649.62	0.0727672
$802$	25727.8	1.13277
$803$	−6192.61	−0.272145
$804$	−2877.93	−0.126240
$805$	−576.172	−0.0252266
$806$	−10732.4	−0.469023
$807$	18877.4	0.823441
$808$	7905.94	0.344220
$809$	−15664.4	−0.680753	−0.340377	−	0.940289i	$-0.610555\pi$
$809$	−15664.4	−0.680753	−0.340377	+	0.940289i	$0.610555\pi$
$810$	−282.189	−0.0122409
$811$	16913.6	0.732329	0.366164	−	0.930550i	$-0.380671\pi$
$811$	16913.6	0.732329	0.366164	+	0.930550i	$0.380671\pi$
$812$	2359.15	0.101958
$813$	−7462.47	−0.321919
$814$	4327.04	0.186318
$815$	−5222.50	−0.224462
$816$	−3212.80	−0.137832
$817$	8523.14	0.364978
$818$	−6017.18	−0.257195
$819$	−2455.08	−0.104746
$820$	2505.07	0.106684
$821$	17022.9	0.723632	0.361816	−	0.932249i	$-0.382157\pi$
$821$	17022.9	0.723632	0.361816	+	0.932249i	$0.382157\pi$
$822$	−7952.02	−0.337419
$823$	22976.8	0.973173	0.486586	−	0.873632i	$-0.338242\pi$
$823$	22976.8	0.973173	0.486586	+	0.873632i	$0.338242\pi$
$824$	16884.6	0.713839
$825$	−4174.22	−0.176155
$826$	−4232.59	−0.178294
$827$	597.155	0.0251090	0.0125545	−	0.999921i	$-0.496004\pi$
$827$	597.155	0.0251090	0.0125545	+	0.999921i	$0.496004\pi$
$828$	5168.41	0.216926
$829$	−24800.2	−1.03902	−0.519510	−	0.854464i	$-0.673885\pi$
$829$	−24800.2	−1.03902	−0.519510	+	0.854464i	$0.673885\pi$
$830$	1769.69	0.0740083
$831$	9324.32	0.389238
$832$	8937.12	0.372403
$833$	−37095.3	−1.54295
$834$	−7911.00	−0.328460
$835$	5220.29	0.216354
$836$	968.269	0.0400578
$837$	33574.5	1.38651
$838$	14090.6	0.580850
$839$	−8479.08	−0.348904	−0.174452	−	0.984666i	$-0.555815\pi$
$839$	−8479.08	−0.348904	−0.174452	+	0.984666i	$0.555815\pi$
$840$	625.567	0.0256954
$841$	−16337.9	−0.669889
$842$	−17173.4	−0.702890
$843$	24058.8	0.982952
$844$	17683.4	0.721196
$845$	2557.55	0.104121
$846$	6443.41	0.261855
$847$	−7120.58	−0.288862
$848$	−3291.87	−0.133306
$849$	−9980.80	−0.403463
$850$	−27969.6	−1.12865
$851$	12700.2	0.511584
$852$	−5012.10	−0.201540
$853$	−26344.7	−1.05748	−0.528738	−	0.848785i	$-0.677334\pi$
$853$	−26344.7	−1.05748	−0.528738	+	0.848785i	$0.677334\pi$
$854$	9177.17	0.367724
$855$	−526.762	−0.0210700
$856$	−31051.4	−1.23985
$857$	−22575.2	−0.899829	−0.449915	−	0.893072i	$-0.648546\pi$
$857$	−22575.2	−0.899829	−0.449915	+	0.893072i	$0.648546\pi$
$858$	1454.35	0.0578681
$859$	−6824.44	−0.271067	−0.135534	−	0.990773i	$-0.543275\pi$
$859$	−6824.44	−0.271067	−0.135534	+	0.990773i	$0.543275\pi$
$860$	−3011.33	−0.119402
$861$	6549.83	0.259254
$862$	−18927.2	−0.747867
$863$	11178.9	0.440944	0.220472	−	0.975393i	$-0.429240\pi$
$863$	11178.9	0.440944	0.220472	+	0.975393i	$0.429240\pi$
$864$	−22853.5	−0.899875
$865$	−4793.74	−0.188430
$866$	−15385.8	−0.603730
$867$	28421.1	1.11330
$868$	6600.72	0.258114
$869$	12234.2	0.477582
$870$	760.291	0.0296279
$871$	4978.23	0.193663
$872$	−11089.6	−0.430667
$873$	9587.50	0.371693
$874$	−2296.20	−0.0888674
$875$	2232.88	0.0862689
$876$	7027.87	0.271061
$877$	29614.6	1.14027	0.570133	−	0.821552i	$-0.306892\pi$
$877$	29614.6	1.14027	0.570133	+	0.821552i	$0.306892\pi$
$878$	−10555.6	−0.405735
$879$	9426.74	0.361725
$880$	157.633	0.00603840
$881$	−44074.5	−1.68548	−0.842739	−	0.538322i	$-0.819059\pi$
$881$	−44074.5	−1.68548	−0.842739	+	0.538322i	$0.819059\pi$
$882$	10631.9	0.405890
$883$	−11301.5	−0.430720	−0.215360	−	0.976535i	$-0.569093\pi$
$883$	−11301.5	−0.430720	−0.215360	+	0.976535i	$0.569093\pi$
$884$	−12061.2	−0.458892
$885$	1688.25	0.0641242
$886$	11011.1	0.417524
$887$	−11839.8	−0.448187	−0.224093	−	0.974568i	$-0.571942\pi$
$887$	−11839.8	−0.448187	−0.224093	+	0.974568i	$0.571942\pi$
$888$	−13789.0	−0.521091
$889$	−8809.08	−0.332336
$890$	258.938	0.00975240
$891$	1132.98	0.0425998
$892$	−24185.5	−0.907837
$893$	3543.03	0.132769
$894$	18671.2	0.698499
$895$	−3119.42	−0.116504
$896$	−3682.10	−0.137288
$897$	4268.65	0.158892
$898$	7562.13	0.281015
$899$	22526.3	0.835699
$900$	−9921.70	−0.367470
$901$	43990.0	1.62655
$902$	8126.37	0.299976
$903$	−7873.51	−0.290160
$904$	−10399.7	−0.382620
$905$	187.403	0.00688340
$906$	12472.8	0.457375
$907$	−30147.6	−1.10368	−0.551838	−	0.833952i	$-0.686073\pi$
$907$	−30147.6	−1.10368	−0.551838	+	0.833952i	$0.686073\pi$
$908$	8260.40	0.301906
$909$	6149.82	0.224397
$910$	−385.369	−0.0140383
$911$	42005.9	1.52768	0.763841	−	0.645405i	$-0.223311\pi$
$911$	42005.9	1.52768	0.763841	+	0.645405i	$0.223311\pi$
$912$	506.333	0.0183842
$913$	−7105.27	−0.257558
$914$	20420.7	0.739012
$915$	−3660.49	−0.132254
$916$	−2019.67	−0.0728513
$917$	4668.66	0.168127
$918$	−30485.7	−1.09605
$919$	41588.7	1.49280	0.746401	−	0.665497i	$-0.231780\pi$
$919$	41588.7	1.49280	0.746401	+	0.665497i	$0.231780\pi$
$920$	2278.04	0.0816355
$921$	12742.0	0.455877
$922$	22462.0	0.802328
$923$	8669.91	0.309181
$924$	−894.469	−0.0318462
$925$	−24380.4	−0.866619
$926$	19386.2	0.687979
$927$	13134.1	0.465352
$928$	−15333.2	−0.542388
$929$	−32440.9	−1.14570	−0.572848	−	0.819662i	$-0.694161\pi$
$929$	−32440.9	−1.14570	−0.572848	+	0.819662i	$0.694161\pi$
$930$	2127.23	0.0750051
$931$	5846.17	0.205801
$932$	5612.67	0.197263
$933$	−877.202	−0.0307806
$934$	−7923.66	−0.277591
$935$	−2106.48	−0.0736785
$936$	9706.74	0.338969
$937$	3765.99	0.131302	0.0656508	−	0.997843i	$-0.479088\pi$
$937$	3765.99	0.131302	0.0656508	+	0.997843i	$0.479088\pi$
$938$	2473.79	0.0861111
$939$	−220.461	−0.00766185
$940$	−1251.80	−0.0434353
$941$	6311.38	0.218645	0.109323	−	0.994006i	$-0.465132\pi$
$941$	6311.38	0.218645	0.109323	+	0.994006i	$0.465132\pi$
$942$	9763.14	0.337686
$943$	23851.6	0.823662
$944$	3398.75	0.117182
$945$	1205.56	0.0414995
$946$	−9768.67	−0.335737
$947$	29630.9	1.01676	0.508381	−	0.861132i	$-0.330244\pi$
$947$	29630.9	1.01676	0.508381	+	0.861132i	$0.330244\pi$
$948$	−13884.4	−0.475680
$949$	−12156.8	−0.415833
$950$	4407.97	0.150541
$951$	−1874.64	−0.0639216
$952$	−16829.4	−0.572946
$953$	−23155.3	−0.787065	−0.393532	−	0.919311i	$-0.628747\pi$
$953$	−23155.3	−0.787065	−0.393532	+	0.919311i	$0.628747\pi$
$954$	−12608.0	−0.427882
$955$	5127.52	0.173741
$956$	22634.3	0.765738
$957$	−3052.55	−0.103109
$958$	−22269.0	−0.751021
$959$	−8459.95	−0.284865
$960$	−1771.40	−0.0595538
$961$	33235.7	1.11563
$962$	8494.47	0.284691
$963$	−24154.1	−0.808260
$964$	6677.76	0.223108
$965$	585.652	0.0195366
$966$	2121.19	0.0706501
$967$	16357.8	0.543984	0.271992	−	0.962300i	$-0.412318\pi$
$967$	16357.8	0.543984	0.271992	+	0.962300i	$0.412318\pi$
$968$	28153.0	0.934783
$969$	−6766.25	−0.224317
$970$	1504.93	0.0498150
$971$	−32741.7	−1.08211	−0.541055	−	0.840987i	$-0.681975\pi$
$971$	−32741.7	−1.08211	−0.541055	+	0.840987i	$0.681975\pi$
$972$	−17263.4	−0.569674
$973$	−8416.31	−0.277302
$974$	32688.6	1.07537
$975$	−8194.45	−0.269161
$976$	−7369.23	−0.241684
$977$	−8985.92	−0.294253	−0.147126	−	0.989118i	$-0.547002\pi$
$977$	−8985.92	−0.294253	−0.147126	+	0.989118i	$0.547002\pi$
$978$	19226.7	0.628632
$979$	−1039.63	−0.0339395
$980$	−2065.53	−0.0673274
$981$	−8626.31	−0.280751
$982$	−8370.01	−0.271994
$983$	−10051.2	−0.326128	−0.163064	−	0.986615i	$-0.552138\pi$
$983$	−10051.2	−0.326128	−0.163064	+	0.986615i	$0.552138\pi$
$984$	−25896.3	−0.838969
$985$	4919.08	0.159122
$986$	−20453.8	−0.660632
$987$	−3272.99	−0.105553
$988$	1900.82	0.0612076
$989$	−28671.8	−0.921852
$990$	603.741	0.0193820
$991$	−33527.0	−1.07469	−0.537346	−	0.843362i	$-0.680573\pi$
$991$	−33527.0	−1.07469	−0.537346	+	0.843362i	$0.680573\pi$
$992$	−42901.0	−1.37309
$993$	−2983.41	−0.0953432
$994$	4308.27	0.137475
$995$	1141.63	0.0363739
$996$	8063.63	0.256532
$997$	38204.5	1.21359	0.606795	−	0.794858i	$-0.292455\pi$
$997$	38204.5	1.21359	0.606795	+	0.794858i	$0.292455\pi$
$998$	24347.8	0.772260
$999$	−26573.5	−0.841591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.a.b.1.2	✓	3
3.2	odd	2		171.4.a.f.1.2		3
4.3	odd	2		304.4.a.i.1.2		3
5.2	odd	4		475.4.b.f.324.4		6
5.3	odd	4		475.4.b.f.324.3		6
5.4	even	2		475.4.a.f.1.2		3
7.6	odd	2		931.4.a.c.1.2		3
8.3	odd	2		1216.4.a.u.1.2		3
8.5	even	2		1216.4.a.s.1.2		3
11.10	odd	2		2299.4.a.h.1.2		3
19.18	odd	2		361.4.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.2	✓	3	1.1	even	1	trivial
171.4.a.f.1.2		3	3.2	odd	2
304.4.a.i.1.2		3	4.3	odd	2
361.4.a.i.1.2		3	19.18	odd	2
475.4.a.f.1.2		3	5.4	even	2
475.4.b.f.324.3		6	5.3	odd	4
475.4.b.f.324.4		6	5.2	odd	4
931.4.a.c.1.2		3	7.6	odd	2
1216.4.a.s.1.2		3	8.5	even	2
1216.4.a.u.1.2		3	8.3	odd	2
2299.4.a.h.1.2		3	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	19.a (trivial)

\(f(q)\)	\(=\)	\(q+1.89080 q^{2} +2.95388 q^{3} -4.42486 q^{4} -1.51710 q^{5} +5.58521 q^{6} +5.94196 q^{7} -23.4930 q^{8} -18.2746 q^{9} -2.86853 q^{10} +11.5171 q^{11} -13.0705 q^{12} +22.6093 q^{13} +11.2351 q^{14} -4.48133 q^{15} -9.02172 q^{16} +120.560 q^{17} -34.5537 q^{18} -19.0000 q^{19} +6.71294 q^{20} +17.5518 q^{21} +21.7766 q^{22} +63.9160 q^{23} -69.3955 q^{24} -122.698 q^{25} +42.7498 q^{26} -133.736 q^{27} -26.2923 q^{28} -89.7278 q^{29} -8.47331 q^{30} -251.051 q^{31} +170.885 q^{32} +34.0201 q^{33} +227.954 q^{34} -9.01453 q^{35} +80.8625 q^{36} +198.702 q^{37} -35.9253 q^{38} +66.7853 q^{39} +35.6411 q^{40} +373.170 q^{41} +33.1871 q^{42} -448.586 q^{43} -50.9616 q^{44} +27.7243 q^{45} +120.853 q^{46} -186.475 q^{47} -26.6491 q^{48} -307.693 q^{49} -231.999 q^{50} +356.119 q^{51} -100.043 q^{52} +364.882 q^{53} -252.868 q^{54} -17.4726 q^{55} -139.594 q^{56} -56.1238 q^{57} -169.658 q^{58} -376.730 q^{59} +19.8292 q^{60} +816.832 q^{61} -474.689 q^{62} -108.587 q^{63} +395.285 q^{64} -34.3006 q^{65} +64.3254 q^{66} +220.185 q^{67} -533.459 q^{68} +188.800 q^{69} -17.0447 q^{70} +383.466 q^{71} +429.324 q^{72} -537.689 q^{73} +375.706 q^{74} -362.437 q^{75} +84.0724 q^{76} +68.4341 q^{77} +126.278 q^{78} +1062.27 q^{79} +13.6868 q^{80} +98.3741 q^{81} +705.592 q^{82} -616.932 q^{83} -77.6645 q^{84} -182.900 q^{85} -848.189 q^{86} -265.045 q^{87} -270.571 q^{88} -90.2686 q^{89} +52.4212 q^{90} +134.344 q^{91} -282.819 q^{92} -741.576 q^{93} -352.589 q^{94} +28.8248 q^{95} +504.776 q^{96} -524.636 q^{97} -581.787 q^{98} -210.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 21 q^{4} + 14 q^{5} - 65 q^{6} - 35 q^{7} + 27 q^{8} + 48 q^{9} - 88 q^{10} + 16 q^{11} - 115 q^{12} + 65 q^{13} + 37 q^{14} + 140 q^{15} + 33 q^{16} + 29 q^{17} + 138 q^{18} - 57 q^{19}+ \cdots + 250 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 21 * q^4 + 14 * q^5 - 65 * q^6 - 35 * q^7 + 27 * q^8 + 48 * q^9 - 88 * q^10 + 16 * q^11 - 115 * q^12 + 65 * q^13 + 37 * q^14 + 140 * q^15 + 33 * q^16 + 29 * q^17 + 138 * q^18 - 57 * q^19 + 100 * q^20 - 25 * q^21 + 118 * q^22 - 101 * q^23 - 489 * q^24 - 37 * q^25 + 299 * q^26 - 377 * q^27 - 493 * q^28 + 377 * q^29 - 376 * q^30 - 140 * q^31 + 579 * q^32 - 130 * q^33 + 329 * q^34 - 438 * q^35 + 1074 * q^36 - 290 * q^37 - 57 * q^38 - 371 * q^39 - 72 * q^40 + 956 * q^41 + 1385 * q^42 - 570 * q^43 + 110 * q^44 + 230 * q^45 - 239 * q^46 + 66 * q^47 - 1411 * q^48 - 98 * q^49 - 1639 * q^50 + 187 * q^51 + 697 * q^52 + 817 * q^53 - 947 * q^54 - 198 * q^55 - 915 * q^56 - 19 * q^57 + 647 * q^58 + 265 * q^59 + 1360 * q^60 + 988 * q^61 - 1436 * q^62 - 1304 * q^63 - 159 * q^64 - 240 * q^65 - 274 * q^66 - 207 * q^67 - 1529 * q^68 + 807 * q^69 + 2040 * q^70 + 846 * q^71 + 2826 * q^72 + 627 * q^73 - 770 * q^74 + 2031 * q^75 - 399 * q^76 + 88 * q^77 - 1817 * q^78 + 382 * q^79 - 1472 * q^80 - 501 * q^81 + 872 * q^82 - 766 * q^83 + 907 * q^84 - 1242 * q^85 + 2592 * q^86 - 1671 * q^87 + 342 * q^88 - 172 * q^89 - 1888 * q^90 - 457 * q^91 - 2677 * q^92 + 892 * q^93 - 96 * q^94 - 266 * q^95 + 863 * q^96 - 2450 * q^97 - 2450 * q^98 + 250 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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