LMFDB - Embedded newform 847.4.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [847,4,Mod(1,847)] 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("847.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.9746177749$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.82843 q^{2} -2.00000 q^{3} +6.65685 q^{4} +6.48528 q^{5} -7.65685 q^{6} -7.00000 q^{7} -5.14214 q^{8} -23.0000 q^{9} +24.8284 q^{10} -13.3137 q^{12} +45.6569 q^{13} -26.7990 q^{14} -12.9706 q^{15} -72.9411 q^{16} +63.6812 q^{17} -88.0538 q^{18} +39.5147 q^{19} +43.1716 q^{20} +14.0000 q^{21} -78.9117 q^{23} +10.2843 q^{24} -82.9411 q^{25} +174.794 q^{26} +100.000 q^{27} -46.5980 q^{28} -256.558 q^{29} -49.6569 q^{30} -170.818 q^{31} -238.113 q^{32} +243.799 q^{34} -45.3970 q^{35} -153.108 q^{36} -223.941 q^{37} +151.279 q^{38} -91.3137 q^{39} -33.3482 q^{40} -307.730 q^{41} +53.5980 q^{42} +316.000 q^{43} -149.161 q^{45} -302.108 q^{46} -576.357 q^{47} +145.882 q^{48} +49.0000 q^{49} -317.534 q^{50} -127.362 q^{51} +303.931 q^{52} +173.951 q^{53} +382.843 q^{54} +35.9949 q^{56} -79.0294 q^{57} -982.215 q^{58} -82.6375 q^{59} -86.3431 q^{60} +63.1472 q^{61} -653.966 q^{62} +161.000 q^{63} -328.068 q^{64} +296.098 q^{65} -349.726 q^{67} +423.917 q^{68} +157.823 q^{69} -173.799 q^{70} -119.050 q^{71} +118.269 q^{72} -573.318 q^{73} -857.342 q^{74} +165.882 q^{75} +263.044 q^{76} -349.588 q^{78} -568.362 q^{79} -473.044 q^{80} +421.000 q^{81} -1178.12 q^{82} -510.447 q^{83} +93.1960 q^{84} +412.991 q^{85} +1209.78 q^{86} +513.117 q^{87} +1090.10 q^{89} -571.054 q^{90} -319.598 q^{91} -525.304 q^{92} +341.637 q^{93} -2206.54 q^{94} +256.264 q^{95} +476.225 q^{96} +396.530 q^{97} +187.593 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} - 14 q^{7} + 18 q^{8} - 46 q^{9} + 44 q^{10} - 4 q^{12} + 80 q^{13} - 14 q^{14} + 8 q^{15} - 78 q^{16} - 48 q^{17} - 46 q^{18} + 96 q^{19} + 92 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 - 14 * q^7 + 18 * q^8 - 46 * q^9 + 44 * q^10 - 4 * q^12 + 80 * q^13 - 14 * q^14 + 8 * q^15 - 78 * q^16 - 48 * q^17 - 46 * q^18 + 96 * q^19 + 92 * q^20 + 28 * q^21 - 56 * q^23 - 36 * q^24 - 98 * q^25 + 112 * q^26 + 200 * q^27 - 14 * q^28 - 4 * q^29 - 88 * q^30 + 60 * q^31 - 414 * q^32 + 448 * q^34 + 28 * q^35 - 46 * q^36 - 380 * q^37 + 48 * q^38 - 160 * q^39 - 276 * q^40 - 112 * q^41 + 28 * q^42 + 632 * q^43 + 92 * q^45 - 344 * q^46 - 604 * q^47 + 156 * q^48 + 98 * q^49 - 290 * q^50 + 96 * q^51 + 144 * q^52 + 676 * q^53 + 200 * q^54 - 126 * q^56 - 192 * q^57 - 1444 * q^58 - 516 * q^59 - 184 * q^60 + 296 * q^61 - 1076 * q^62 + 322 * q^63 + 34 * q^64 - 64 * q^65 - 1152 * q^67 + 944 * q^68 + 112 * q^69 - 308 * q^70 - 1064 * q^71 - 414 * q^72 - 168 * q^73 - 572 * q^74 + 196 * q^75 - 224 * q^78 + 368 * q^79 - 420 * q^80 + 842 * q^81 - 1536 * q^82 - 1728 * q^83 + 28 * q^84 + 1584 * q^85 + 632 * q^86 + 8 * q^87 + 1524 * q^89 - 1012 * q^90 - 560 * q^91 - 632 * q^92 - 120 * q^93 - 2156 * q^94 - 336 * q^95 + 828 * q^96 + 1404 * q^97 + 98 * q^98

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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	6.65685	0.832107
$5$	6.48528	0.580061	0.290031	−	0.957017i	$-0.406335\pi$
$5$	6.48528	0.580061	0.290031	+	0.957017i	$0.406335\pi$
$6$	−7.65685	−0.520983
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−5.14214	−0.227252
$9$	−23.0000	−0.851852
$10$	24.8284	0.785144
$11$	0	0
$11$	0	0
$12$	−13.3137	−0.320278
$13$	45.6569	0.974072	0.487036	−	0.873382i	$-0.338078\pi$
$13$	45.6569	0.974072	0.487036	+	0.873382i	$0.338078\pi$
$14$	−26.7990	−0.511595
$15$	−12.9706	−0.223266
$16$	−72.9411	−1.13971
$17$	63.6812	0.908528	0.454264	−	0.890867i	$-0.349902\pi$
$17$	63.6812	0.908528	0.454264	+	0.890867i	$0.349902\pi$
$18$	−88.0538	−1.15303
$19$	39.5147	0.477121	0.238560	−	0.971128i	$-0.423324\pi$
$19$	39.5147	0.477121	0.238560	+	0.971128i	$0.423324\pi$
$20$	43.1716	0.482673
$21$	14.0000	0.145479
$22$	0	0
$23$	−78.9117	−0.715401	−0.357701	−	0.933836i	$-0.616439\pi$
$23$	−78.9117	−0.715401	−0.357701	+	0.933836i	$0.616439\pi$
$24$	10.2843	0.0874695
$25$	−82.9411	−0.663529
$26$	174.794	1.31846
$27$	100.000	0.712778
$28$	−46.5980	−0.314507
$29$	−256.558	−1.64282	−0.821409	−	0.570340i	$-0.806811\pi$
$29$	−256.558	−1.64282	−0.821409	+	0.570340i	$0.806811\pi$
$30$	−49.6569	−0.302202
$31$	−170.818	−0.989673	−0.494837	−	0.868986i	$-0.664772\pi$
$31$	−170.818	−0.989673	−0.494837	+	0.868986i	$0.664772\pi$
$32$	−238.113	−1.31540
$33$	0	0
$34$	243.799	1.22974
$35$	−45.3970	−0.219243
$36$	−153.108	−0.708832
$37$	−223.941	−0.995019	−0.497509	−	0.867459i	$-0.665752\pi$
$37$	−223.941	−0.995019	−0.497509	+	0.867459i	$0.665752\pi$
$38$	151.279	0.645809
$39$	−91.3137	−0.374920
$40$	−33.3482	−0.131820
$41$	−307.730	−1.17218	−0.586090	−	0.810246i	$-0.699333\pi$
$41$	−307.730	−1.17218	−0.586090	+	0.810246i	$0.699333\pi$
$42$	53.5980	0.196913
$43$	316.000	1.12069	0.560344	−	0.828260i	$-0.310669\pi$
$43$	316.000	1.12069	0.560344	+	0.828260i	$0.310669\pi$
$44$	0	0
$45$	−149.161	−0.494126
$46$	−302.108	−0.968334
$47$	−576.357	−1.78873	−0.894366	−	0.447337i	$-0.852373\pi$
$47$	−576.357	−1.78873	−0.894366	+	0.447337i	$0.852373\pi$
$48$	145.882	0.438673
$49$	49.0000	0.142857
$50$	−317.534	−0.898122
$51$	−127.362	−0.349692
$52$	303.931	0.810532
$53$	173.951	0.450831	0.225415	−	0.974263i	$-0.427626\pi$
$53$	173.951	0.450831	0.225415	+	0.974263i	$0.427626\pi$
$54$	382.843	0.964783
$55$	0	0
$56$	35.9949	0.0858933
$57$	−79.0294	−0.183644
$58$	−982.215	−2.22364
$59$	−82.6375	−0.182347	−0.0911736	−	0.995835i	$-0.529062\pi$
$59$	−82.6375	−0.182347	−0.0911736	+	0.995835i	$0.529062\pi$
$60$	−86.3431	−0.185781
$61$	63.1472	0.132544	0.0662719	−	0.997802i	$-0.478890\pi$
$61$	63.1472	0.132544	0.0662719	+	0.997802i	$0.478890\pi$
$62$	−653.966	−1.33958
$63$	161.000	0.321970
$64$	−328.068	−0.640758
$65$	296.098	0.565021
$66$	0	0
$67$	−349.726	−0.637699	−0.318849	−	0.947805i	$-0.603296\pi$
$67$	−349.726	−0.637699	−0.318849	+	0.947805i	$0.603296\pi$
$68$	423.917	0.755992
$69$	157.823	0.275358
$70$	−173.799	−0.296756
$71$	−119.050	−0.198994	−0.0994971	−	0.995038i	$-0.531723\pi$
$71$	−119.050	−0.198994	−0.0994971	+	0.995038i	$0.531723\pi$
$72$	118.269	0.193585
$73$	−573.318	−0.919203	−0.459601	−	0.888125i	$-0.652008\pi$
$73$	−573.318	−0.919203	−0.459601	+	0.888125i	$0.652008\pi$
$74$	−857.342	−1.34681
$75$	165.882	0.255392
$76$	263.044	0.397016
$77$	0	0
$78$	−349.588	−0.507475
$79$	−568.362	−0.809439	−0.404719	−	0.914441i	$-0.632631\pi$
$79$	−568.362	−0.809439	−0.404719	+	0.914441i	$0.632631\pi$
$80$	−473.044	−0.661099
$81$	421.000	0.577503
$82$	−1178.12	−1.58661
$83$	−510.447	−0.675046	−0.337523	−	0.941317i	$-0.609589\pi$
$83$	−510.447	−0.675046	−0.337523	+	0.941317i	$0.609589\pi$
$84$	93.1960	0.121054
$85$	412.991	0.527002
$86$	1209.78	1.51691
$87$	513.117	0.632321
$88$	0	0
$89$	1090.10	1.29832	0.649158	−	0.760654i	$-0.275121\pi$
$89$	1090.10	1.29832	0.649158	+	0.760654i	$0.275121\pi$
$90$	−571.054	−0.668826
$91$	−319.598	−0.368165
$92$	−525.304	−0.595290
$93$	341.637	0.380925
$94$	−2206.54	−2.42114
$95$	256.264	0.276759
$96$	476.225	0.506297
$97$	396.530	0.415067	0.207534	−	0.978228i	$-0.433456\pi$
$97$	396.530	0.415067	0.207534	+	0.978228i	$0.433456\pi$
$98$	187.593	0.193365
$99$	0	0
$100$	−552.127	−0.552127
$101$	−825.844	−0.813609	−0.406804	−	0.913515i	$-0.633357\pi$
$101$	−825.844	−0.813609	−0.406804	+	0.913515i	$0.633357\pi$
$102$	−487.598	−0.473327
$103$	1246.19	1.19214	0.596071	−	0.802932i	$-0.296728\pi$
$103$	1246.19	1.19214	0.596071	+	0.802932i	$0.296728\pi$
$104$	−234.774	−0.221360
$105$	90.7939	0.0843865
$106$	665.960	0.610224
$107$	1368.24	1.23620	0.618099	−	0.786100i	$-0.287903\pi$
$107$	1368.24	1.23620	0.618099	+	0.786100i	$0.287903\pi$
$108$	665.685	0.593107
$109$	1312.80	1.15361	0.576806	−	0.816881i	$-0.304299\pi$
$109$	1312.80	1.15361	0.576806	+	0.816881i	$0.304299\pi$
$110$	0	0
$111$	447.882	0.382983
$112$	510.588	0.430768
$113$	−312.607	−0.260244	−0.130122	−	0.991498i	$-0.541537\pi$
$113$	−312.607	−0.260244	−0.130122	+	0.991498i	$0.541537\pi$
$114$	−302.558	−0.248572
$115$	−511.765	−0.414976
$116$	−1707.87	−1.36700
$117$	−1050.11	−0.829765
$118$	−316.372	−0.246817
$119$	−445.769	−0.343391
$120$	66.6964	0.0507377
$121$	0	0
$122$	241.754	0.179405
$123$	615.460	0.451172
$124$	−1137.11	−0.823514
$125$	−1348.56	−0.964949
$126$	616.377	0.435803
$127$	1592.65	1.11280	0.556398	−	0.830916i	$-0.312183\pi$
$127$	1592.65	1.11280	0.556398	+	0.830916i	$0.312183\pi$
$128$	648.917	0.448099
$129$	−632.000	−0.431353
$130$	1133.59	0.764786
$131$	−1291.43	−0.861321	−0.430661	−	0.902514i	$-0.641719\pi$
$131$	−1291.43	−0.861321	−0.430661	+	0.902514i	$0.641719\pi$
$132$	0	0
$133$	−276.603	−0.180335
$134$	−1338.90	−0.863159
$135$	648.528	0.413455
$136$	−327.458	−0.206465
$137$	−2107.76	−1.31444	−0.657220	−	0.753699i	$-0.728268\pi$
$137$	−2107.76	−1.31444	−0.657220	+	0.753699i	$0.728268\pi$
$138$	604.215	0.372712
$139$	−2318.30	−1.41464	−0.707321	−	0.706892i	$-0.750097\pi$
$139$	−2318.30	−1.41464	−0.707321	+	0.706892i	$0.750097\pi$
$140$	−302.201	−0.182433
$141$	1152.71	0.688483
$142$	−455.773	−0.269349
$143$	0	0
$144$	1677.65	0.970860
$145$	−1663.85	−0.952935
$146$	−2194.91	−1.24419
$147$	−98.0000	−0.0549857
$148$	−1490.74	−0.827962
$149$	−602.061	−0.331025	−0.165513	−	0.986208i	$-0.552928\pi$
$149$	−602.061	−0.331025	−0.165513	+	0.986208i	$0.552928\pi$
$150$	635.068	0.345687
$151$	1928.20	1.03917	0.519584	−	0.854420i	$-0.326087\pi$
$151$	1928.20	1.03917	0.519584	+	0.854420i	$0.326087\pi$
$152$	−203.190	−0.108427
$153$	−1464.67	−0.773931
$154$	0	0
$155$	−1107.80	−0.574071
$156$	−607.862	−0.311974
$157$	2476.45	1.25887	0.629434	−	0.777054i	$-0.283287\pi$
$157$	2476.45	1.25887	0.629434	+	0.777054i	$0.283287\pi$
$158$	−2175.93	−1.09562
$159$	−347.902	−0.173525
$160$	−1544.23	−0.763012
$161$	552.382	0.270396
$162$	1611.77	0.781682
$163$	−1063.06	−0.510829	−0.255415	−	0.966832i	$-0.582212\pi$
$163$	−1063.06	−0.510829	−0.255415	+	0.966832i	$0.582212\pi$
$164$	−2048.51	−0.975378
$165$	0	0
$166$	−1954.21	−0.913710
$167$	3720.65	1.72403	0.862015	−	0.506883i	$-0.169203\pi$
$167$	3720.65	1.72403	0.862015	+	0.506883i	$0.169203\pi$
$168$	−71.9899	−0.0330604
$169$	−112.452	−0.0511842
$170$	1581.11	0.713325
$171$	−908.839	−0.406436
$172$	2103.57	0.932531
$173$	909.421	0.399665	0.199832	−	0.979830i	$-0.435960\pi$
$173$	909.421	0.399665	0.199832	+	0.979830i	$0.435960\pi$
$174$	1964.43	0.855880
$175$	580.588	0.250790
$176$	0	0
$177$	165.275	0.0701855
$178$	4173.36	1.75734
$179$	1349.80	0.563626	0.281813	−	0.959469i	$-0.409064\pi$
$179$	1349.80	0.563626	0.281813	+	0.959469i	$0.409064\pi$
$180$	−992.946	−0.411166
$181$	−4559.47	−1.87239	−0.936196	−	0.351479i	$-0.885679\pi$
$181$	−4559.47	−1.87239	−0.936196	+	0.351479i	$0.885679\pi$
$182$	−1223.56	−0.498330
$183$	−126.294	−0.0510161
$184$	405.775	0.162577
$185$	−1452.32	−0.577172
$186$	1307.93	0.515603
$187$	0	0
$188$	−3836.73	−1.48842
$189$	−700.000	−0.269405
$190$	981.088	0.374609
$191$	−2238.60	−0.848061	−0.424031	−	0.905648i	$-0.639385\pi$
$191$	−2238.60	−0.848061	−0.424031	+	0.905648i	$0.639385\pi$
$192$	656.136	0.246628
$193$	−2966.36	−1.10634	−0.553169	−	0.833069i	$-0.686582\pi$
$193$	−2966.36	−1.10634	−0.553169	+	0.833069i	$0.686582\pi$
$194$	1518.09	0.561815
$195$	−592.195	−0.217477
$196$	326.186	0.118872
$197$	−3982.18	−1.44020	−0.720099	−	0.693872i	$-0.755903\pi$
$197$	−3982.18	−1.44020	−0.720099	+	0.693872i	$0.755903\pi$
$198$	0	0
$199$	1895.03	0.675052	0.337526	−	0.941316i	$-0.390410\pi$
$199$	1895.03	0.675052	0.337526	+	0.941316i	$0.390410\pi$
$200$	426.495	0.150789
$201$	699.452	0.245450
$202$	−3161.68	−1.10126
$203$	1795.91	0.620927
$204$	−847.833	−0.290981
$205$	−1995.72	−0.679936
$206$	4770.94	1.61363
$207$	1814.97	0.609416
$208$	−3330.26	−1.11015
$209$	0	0
$210$	347.598	0.114222
$211$	1506.79	0.491621	0.245810	−	0.969318i	$-0.420946\pi$
$211$	1506.79	0.491621	0.245810	+	0.969318i	$0.420946\pi$
$212$	1157.97	0.375139
$213$	238.099	0.0765929
$214$	5238.22	1.67326
$215$	2049.35	0.650067
$216$	−514.214	−0.161981
$217$	1195.73	0.374061
$218$	5025.97	1.56147
$219$	1146.64	0.353801
$220$	0	0
$221$	2907.49	0.884971
$222$	1714.68	0.518388
$223$	4289.51	1.28810	0.644051	−	0.764983i	$-0.277252\pi$
$223$	4289.51	1.28810	0.644051	+	0.764983i	$0.277252\pi$
$224$	1666.79	0.497174
$225$	1907.65	0.565228
$226$	−1196.79	−0.352255
$227$	2848.72	0.832934	0.416467	−	0.909151i	$-0.363268\pi$
$227$	2848.72	0.832934	0.416467	+	0.909151i	$0.363268\pi$
$228$	−526.087	−0.152811
$229$	3273.70	0.944681	0.472341	−	0.881416i	$-0.343409\pi$
$229$	3273.70	0.944681	0.472341	+	0.881416i	$0.343409\pi$
$230$	−1959.25	−0.561693
$231$	0	0
$232$	1319.26	0.373334
$233$	−4773.81	−1.34224	−0.671122	−	0.741347i	$-0.734187\pi$
$233$	−4773.81	−1.34224	−0.671122	+	0.741347i	$0.734187\pi$
$234$	−4020.26	−1.12313
$235$	−3737.84	−1.03757
$236$	−550.106	−0.151732
$237$	1136.72	0.311553
$238$	−1706.59	−0.464798
$239$	−1976.50	−0.534934	−0.267467	−	0.963567i	$-0.586187\pi$
$239$	−1976.50	−0.534934	−0.267467	+	0.963567i	$0.586187\pi$
$240$	946.087	0.254457
$241$	2236.73	0.597845	0.298923	−	0.954277i	$-0.403373\pi$
$241$	2236.73	0.597845	0.298923	+	0.954277i	$0.403373\pi$
$242$	0	0
$243$	−3542.00	−0.935059
$244$	420.362	0.110291
$245$	317.779	0.0828659
$246$	2356.24	0.610685
$247$	1804.12	0.464750
$248$	878.371	0.224906
$249$	1020.89	0.259825
$250$	−5162.85	−1.30611
$251$	−6364.56	−1.60051	−0.800254	−	0.599662i	$-0.795302\pi$
$251$	−6364.56	−1.60051	−0.800254	+	0.599662i	$0.795302\pi$
$252$	1071.75	0.267913
$253$	0	0
$254$	6097.36	1.50623
$255$	−825.982	−0.202843
$256$	5108.88	1.24728
$257$	7947.46	1.92898	0.964492	−	0.264110i	$-0.0850783\pi$
$257$	7947.46	1.92898	0.964492	+	0.264110i	$0.0850783\pi$
$258$	−2419.57	−0.583859
$259$	1567.59	0.376082
$260$	1971.08	0.470158
$261$	5900.84	1.39944
$262$	−4944.16	−1.16584
$263$	7330.82	1.71877	0.859387	−	0.511325i	$-0.170845\pi$
$263$	7330.82	1.71877	0.859387	+	0.511325i	$0.170845\pi$
$264$	0	0
$265$	1128.12	0.261510
$266$	−1058.95	−0.244093
$267$	−2180.20	−0.499722
$268$	−2328.07	−0.530633
$269$	−3504.02	−0.794216	−0.397108	−	0.917772i	$-0.629986\pi$
$269$	−3504.02	−0.794216	−0.397108	+	0.917772i	$0.629986\pi$
$270$	2482.84	0.559633
$271$	−36.4996	−0.00818152	−0.00409076	−	0.999992i	$-0.501302\pi$
$271$	−36.4996	−0.00818152	−0.00409076	+	0.999992i	$0.501302\pi$
$272$	−4644.98	−1.03545
$273$	639.196	0.141707
$274$	−8069.42	−1.77917
$275$	0	0
$276$	1050.61	0.229127
$277$	4965.14	1.07699	0.538495	−	0.842629i	$-0.318993\pi$
$277$	4965.14	1.07699	0.538495	+	0.842629i	$0.318993\pi$
$278$	−8875.43	−1.91479
$279$	3928.82	0.843055
$280$	233.437	0.0498234
$281$	4560.42	0.968156	0.484078	−	0.875025i	$-0.339155\pi$
$281$	4560.42	0.968156	0.484078	+	0.875025i	$0.339155\pi$
$282$	4413.08	0.931899
$283$	917.744	0.192771	0.0963855	−	0.995344i	$-0.469272\pi$
$283$	917.744	0.192771	0.0963855	+	0.995344i	$0.469272\pi$
$284$	−792.496	−0.165584
$285$	−512.528	−0.106525
$286$	0	0
$287$	2154.11	0.443042
$288$	5476.59	1.12053
$289$	−857.700	−0.174578
$290$	−6369.94	−1.28985
$291$	−793.060	−0.159759
$292$	−3816.49	−0.764875
$293$	3983.97	0.794355	0.397177	−	0.917742i	$-0.369990\pi$
$293$	3983.97	0.794355	0.397177	+	0.917742i	$0.369990\pi$
$294$	−375.186	−0.0744261
$295$	−535.928	−0.105773
$296$	1151.54	0.226120
$297$	0	0
$298$	−2304.95	−0.448060
$299$	−3602.86	−0.696852
$300$	1104.25	0.212514
$301$	−2212.00	−0.423580
$302$	7381.95	1.40657
$303$	1651.69	0.313158
$304$	−2882.25	−0.543777
$305$	409.527	0.0768835
$306$	−5607.38	−1.04756
$307$	7992.52	1.48585	0.742927	−	0.669372i	$-0.233437\pi$
$307$	7992.52	1.48585	0.742927	+	0.669372i	$0.233437\pi$
$308$	0	0
$309$	−2492.38	−0.458856
$310$	−4241.15	−0.777036
$311$	7217.65	1.31600	0.657999	−	0.753019i	$-0.271403\pi$
$311$	7217.65	1.31600	0.657999	+	0.753019i	$0.271403\pi$
$312$	469.547	0.0852016
$313$	−8415.62	−1.51974	−0.759871	−	0.650074i	$-0.774738\pi$
$313$	−8415.62	−1.51974	−0.759871	+	0.650074i	$0.774738\pi$
$314$	9480.91	1.70395
$315$	1044.13	0.186762
$316$	−3783.50	−0.673540
$317$	632.365	0.112041	0.0560207	−	0.998430i	$-0.482159\pi$
$317$	632.365	0.112041	0.0560207	+	0.998430i	$0.482159\pi$
$318$	−1331.92	−0.234875
$319$	0	0
$320$	−2127.61	−0.371679
$321$	−2736.49	−0.475813
$322$	2114.75	0.365996
$323$	2516.35	0.433478
$324$	2802.54	0.480545
$325$	−3786.83	−0.646325
$326$	−4069.84	−0.691434
$327$	−2625.60	−0.444025
$328$	1582.39	0.266381
$329$	4034.50	0.676077
$330$	0	0
$331$	425.759	0.0707004	0.0353502	−	0.999375i	$-0.488745\pi$
$331$	425.759	0.0707004	0.0353502	+	0.999375i	$0.488745\pi$
$332$	−3397.97	−0.561710
$333$	5150.65	0.847609
$334$	14244.3	2.33357
$335$	−2268.07	−0.369904
$336$	−1021.18	−0.165803
$337$	−1369.56	−0.221379	−0.110689	−	0.993855i	$-0.535306\pi$
$337$	−1369.56	−0.221379	−0.110689	+	0.993855i	$0.535306\pi$
$338$	−430.513	−0.0692805
$339$	625.214	0.100168
$340$	2749.22	0.438522
$341$	0	0
$342$	−3479.42	−0.550133
$343$	−343.000	−0.0539949
$344$	−1624.91	−0.254679
$345$	1023.53	0.159724
$346$	3481.65	0.540968
$347$	5531.76	0.855794	0.427897	−	0.903827i	$-0.359255\pi$
$347$	5531.76	0.855794	0.427897	+	0.903827i	$0.359255\pi$
$348$	3415.74	0.526158
$349$	−10171.7	−1.56011	−0.780055	−	0.625711i	$-0.784809\pi$
$349$	−10171.7	−1.56011	−0.780055	+	0.625711i	$0.784809\pi$
$350$	2222.74	0.339458
$351$	4565.69	0.694297
$352$	0	0
$353$	9626.38	1.45145	0.725723	−	0.687987i	$-0.241505\pi$
$353$	9626.38	1.45145	0.725723	+	0.687987i	$0.241505\pi$
$354$	632.743	0.0949998
$355$	−772.070	−0.115429
$356$	7256.62	1.08034
$357$	891.537	0.132171
$358$	5167.62	0.762898
$359$	−8491.00	−1.24829	−0.624147	−	0.781307i	$-0.714553\pi$
$359$	−8491.00	−1.24829	−0.624147	+	0.781307i	$0.714553\pi$
$360$	767.009	0.112291
$361$	−5297.59	−0.772356
$362$	−17455.6	−2.53438
$363$	0	0
$364$	−2127.52	−0.306352
$365$	−3718.13	−0.533194
$366$	−483.509	−0.0690530
$367$	2948.02	0.419307	0.209653	−	0.977776i	$-0.432766\pi$
$367$	2948.02	0.419307	0.209653	+	0.977776i	$0.432766\pi$
$368$	5755.91	0.815346
$369$	7077.79	0.998523
$370$	−5560.11	−0.781233
$371$	−1217.66	−0.170398
$372$	2274.23	0.316971
$373$	8295.57	1.15155	0.575775	−	0.817608i	$-0.304700\pi$
$373$	8295.57	1.15155	0.575775	+	0.817608i	$0.304700\pi$
$374$	0	0
$375$	2697.11	0.371409
$376$	2963.71	0.406494
$377$	−11713.7	−1.60022
$378$	−2679.90	−0.364654
$379$	−12137.3	−1.64499	−0.822495	−	0.568773i	$-0.807418\pi$
$379$	−12137.3	−1.64499	−0.822495	+	0.568773i	$0.807418\pi$
$380$	1705.91	0.230293
$381$	−3185.31	−0.428316
$382$	−8570.33	−1.14790
$383$	−9053.44	−1.20786	−0.603929	−	0.797038i	$-0.706399\pi$
$383$	−9053.44	−1.20786	−0.603929	+	0.797038i	$0.706399\pi$
$384$	−1297.83	−0.172473
$385$	0	0
$386$	−11356.5	−1.49749
$387$	−7268.00	−0.954659
$388$	2639.64	0.345380
$389$	−1130.94	−0.147406	−0.0737030	−	0.997280i	$-0.523482\pi$
$389$	−1130.94	−0.147406	−0.0737030	+	0.997280i	$0.523482\pi$
$390$	−2267.18	−0.294366
$391$	−5025.19	−0.649962
$392$	−251.965	−0.0324646
$393$	2582.87	0.331523
$394$	−15245.5	−1.94938
$395$	−3685.98	−0.469524
$396$	0	0
$397$	−13113.4	−1.65779	−0.828896	−	0.559403i	$-0.811030\pi$
$397$	−13113.4	−1.65779	−0.828896	+	0.559403i	$0.811030\pi$
$398$	7255.00	0.913719
$399$	553.206	0.0694109
$400$	6049.82	0.756227
$401$	12805.1	1.59465	0.797327	−	0.603548i	$-0.206247\pi$
$401$	12805.1	1.59465	0.797327	+	0.603548i	$0.206247\pi$
$402$	2677.80	0.332230
$403$	−7799.03	−0.964013
$404$	−5497.52	−0.677010
$405$	2730.30	0.334987
$406$	6875.51	0.840457
$407$	0	0
$408$	654.915	0.0794685
$409$	−9184.62	−1.11039	−0.555196	−	0.831720i	$-0.687357\pi$
$409$	−9184.62	−1.11039	−0.555196	+	0.831720i	$0.687357\pi$
$410$	−7640.45	−0.920329
$411$	4215.53	0.505928
$412$	8295.70	0.991990
$413$	578.463	0.0689208
$414$	6948.48	0.824877
$415$	−3310.39	−0.391568
$416$	−10871.5	−1.28129
$417$	4636.59	0.544496
$418$	0	0
$419$	−6249.25	−0.728629	−0.364315	−	0.931276i	$-0.618697\pi$
$419$	−6249.25	−0.728629	−0.364315	+	0.931276i	$0.618697\pi$
$420$	604.402	0.0702186
$421$	4703.21	0.544467	0.272233	−	0.962231i	$-0.412238\pi$
$421$	4703.21	0.544467	0.272233	+	0.962231i	$0.412238\pi$
$422$	5768.65	0.665435
$423$	13256.2	1.52373
$424$	−894.481	−0.102452
$425$	−5281.79	−0.602834
$426$	911.546	0.103673
$427$	−442.030	−0.0500968
$428$	9108.20	1.02865
$429$	0	0
$430$	7845.78	0.879901
$431$	−7554.41	−0.844276	−0.422138	−	0.906532i	$-0.638720\pi$
$431$	−7554.41	−0.844276	−0.422138	+	0.906532i	$0.638720\pi$
$432$	−7294.11	−0.812357
$433$	2631.98	0.292113	0.146057	−	0.989276i	$-0.453342\pi$
$433$	2631.98	0.292113	0.146057	+	0.989276i	$0.453342\pi$
$434$	4577.76	0.506312
$435$	3327.71	0.366785
$436$	8739.13	0.959928
$437$	−3118.17	−0.341333
$438$	4389.81	0.478889
$439$	−12551.2	−1.36455	−0.682274	−	0.731097i	$-0.739009\pi$
$439$	−12551.2	−1.36455	−0.682274	+	0.731097i	$0.739009\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	11131.1	1.19786
$443$	11226.2	1.20400	0.602000	−	0.798496i	$-0.294371\pi$
$443$	11226.2	1.20400	0.602000	+	0.798496i	$0.294371\pi$
$444$	2981.49	0.318683
$445$	7069.59	0.753103
$446$	16422.1	1.74351
$447$	1204.12	0.127412
$448$	2296.48	0.242184
$449$	−11177.4	−1.17482	−0.587410	−	0.809289i	$-0.699853\pi$
$449$	−11177.4	−1.17482	−0.587410	+	0.809289i	$0.699853\pi$
$450$	7303.28	0.765067
$451$	0	0
$452$	−2080.98	−0.216551
$453$	−3856.39	−0.399976
$454$	10906.1	1.12742
$455$	−2072.68	−0.213558
$456$	406.380	0.0417335
$457$	−7040.35	−0.720643	−0.360321	−	0.932828i	$-0.617333\pi$
$457$	−7040.35	−0.720643	−0.360321	+	0.932828i	$0.617333\pi$
$458$	12533.1	1.27868
$459$	6368.12	0.647579
$460$	−3406.74	−0.345305
$461$	1483.35	0.149862	0.0749312	−	0.997189i	$-0.476126\pi$
$461$	1483.35	0.149862	0.0749312	+	0.997189i	$0.476126\pi$
$462$	0	0
$463$	−3106.19	−0.311786	−0.155893	−	0.987774i	$-0.549826\pi$
$463$	−3106.19	−0.311786	−0.155893	+	0.987774i	$0.549826\pi$
$464$	18713.7	1.87233
$465$	2215.61	0.220960
$466$	−18276.2	−1.81680
$467$	−9605.04	−0.951752	−0.475876	−	0.879512i	$-0.657869\pi$
$467$	−9605.04	−0.951752	−0.475876	+	0.879512i	$0.657869\pi$
$468$	−6990.41	−0.690453
$469$	2448.08	0.241027
$470$	−14310.0	−1.40441
$471$	−4952.90	−0.484539
$472$	424.933	0.0414389
$473$	0	0
$474$	4351.86	0.421704
$475$	−3277.40	−0.316584
$476$	−2967.42	−0.285738
$477$	−4000.88	−0.384041
$478$	−7566.88	−0.724061
$479$	−9118.53	−0.869805	−0.434902	−	0.900478i	$-0.643217\pi$
$479$	−9118.53	−0.869805	−0.434902	+	0.900478i	$0.643217\pi$
$480$	3088.46	0.293683
$481$	−10224.4	−0.969220
$482$	8563.17	0.809215
$483$	−1104.76	−0.104076
$484$	0	0
$485$	2571.61	0.240764
$486$	−13560.3	−1.26565
$487$	14045.1	1.30687	0.653434	−	0.756983i	$-0.273328\pi$
$487$	14045.1	1.30687	0.653434	+	0.756983i	$0.273328\pi$
$488$	−324.711	−0.0301209
$489$	2126.12	0.196618
$490$	1216.59	0.112163
$491$	1034.26	0.0950620	0.0475310	−	0.998870i	$-0.484865\pi$
$491$	1034.26	0.0950620	0.0475310	+	0.998870i	$0.484865\pi$
$492$	4097.03	0.375423
$493$	−16338.0	−1.49255
$494$	6906.93	0.629064
$495$	0	0
$496$	12459.7	1.12794
$497$	833.347	0.0752128
$498$	3908.42	0.351687
$499$	−14332.3	−1.28577	−0.642886	−	0.765962i	$-0.722263\pi$
$499$	−14332.3	−1.28577	−0.642886	+	0.765962i	$0.722263\pi$
$500$	−8977.15	−0.802940
$501$	−7441.31	−0.663579
$502$	−24366.2	−2.16637
$503$	5007.72	0.443903	0.221951	−	0.975058i	$-0.428757\pi$
$503$	5007.72	0.443903	0.221951	+	0.975058i	$0.428757\pi$
$504$	−827.884	−0.0731684
$505$	−5355.83	−0.471943
$506$	0	0
$507$	224.903	0.0197008
$508$	10602.1	0.925966
$509$	−9234.40	−0.804140	−0.402070	−	0.915609i	$-0.631709\pi$
$509$	−9234.40	−0.804140	−0.402070	+	0.915609i	$0.631709\pi$
$510$	−3162.21	−0.274559
$511$	4013.23	0.347426
$512$	14367.6	1.24017
$513$	3951.47	0.340081
$514$	30426.3	2.61098
$515$	8081.89	0.691516
$516$	−4207.13	−0.358932
$517$	0	0
$518$	6001.40	0.509047
$519$	−1818.84	−0.153831
$520$	−1522.57	−0.128402
$521$	−5141.26	−0.432328	−0.216164	−	0.976357i	$-0.569355\pi$
$521$	−5141.26	−0.432328	−0.216164	+	0.976357i	$0.569355\pi$
$522$	22591.0	1.89421
$523$	19766.3	1.65262	0.826308	−	0.563218i	$-0.190437\pi$
$523$	19766.3	1.65262	0.826308	+	0.563218i	$0.190437\pi$
$524$	−8596.89	−0.716711
$525$	−1161.18	−0.0965293
$526$	28065.5	2.32645
$527$	−10877.9	−0.899146
$528$	0	0
$529$	−5939.95	−0.488201
$530$	4318.94	0.353967
$531$	1900.66	0.155333
$532$	−1841.31	−0.150058
$533$	−14050.0	−1.14179
$534$	−8346.72	−0.676400
$535$	8873.45	0.717070
$536$	1798.34	0.144919
$537$	−2699.61	−0.216940
$538$	−13414.9	−1.07501
$539$	0	0
$540$	4317.16	0.344039
$541$	16805.9	1.33556	0.667782	−	0.744356i	$-0.267244\pi$
$541$	16805.9	1.33556	0.667782	+	0.744356i	$0.267244\pi$
$542$	−139.736	−0.0110741
$543$	9118.95	0.720684
$544$	−15163.3	−1.19508
$545$	8513.89	0.669165
$546$	2447.12	0.191807
$547$	−16284.0	−1.27286	−0.636428	−	0.771336i	$-0.719589\pi$
$547$	−16284.0	−1.27286	−0.636428	+	0.771336i	$0.719589\pi$
$548$	−14031.1	−1.09375
$549$	−1452.39	−0.112908
$550$	0	0
$551$	−10137.8	−0.783823
$552$	−811.549	−0.0625758
$553$	3978.53	0.305939
$554$	19008.7	1.45776
$555$	2904.64	0.222154
$556$	−15432.6	−1.17713
$557$	13972.6	1.06291	0.531454	−	0.847087i	$-0.321646\pi$
$557$	13972.6	1.06291	0.531454	+	0.847087i	$0.321646\pi$
$558$	15041.2	1.14112
$559$	14427.6	1.09163
$560$	3311.31	0.249872
$561$	0	0
$562$	17459.2	1.31045
$563$	−6396.33	−0.478816	−0.239408	−	0.970919i	$-0.576953\pi$
$563$	−6396.33	−0.478816	−0.239408	+	0.970919i	$0.576953\pi$
$564$	7673.45	0.572891
$565$	−2027.35	−0.150958
$566$	3513.51	0.260926
$567$	−2947.00	−0.218276
$568$	612.169	0.0452219
$569$	15320.3	1.12876	0.564378	−	0.825517i	$-0.309116\pi$
$569$	15320.3	1.12876	0.564378	+	0.825517i	$0.309116\pi$
$570$	−1962.18	−0.144187
$571$	14347.1	1.05150	0.525750	−	0.850639i	$-0.323785\pi$
$571$	14347.1	1.05150	0.525750	+	0.850639i	$0.323785\pi$
$572$	0	0
$573$	4477.21	0.326419
$574$	8246.85	0.599681
$575$	6545.02	0.474689
$576$	7545.57	0.545831
$577$	−19882.3	−1.43451	−0.717256	−	0.696810i	$-0.754602\pi$
$577$	−19882.3	−1.43451	−0.717256	+	0.696810i	$0.754602\pi$
$578$	−3283.64	−0.236300
$579$	5932.72	0.425829
$580$	−11076.0	−0.792943
$581$	3573.13	0.255143
$582$	−3036.17	−0.216243
$583$	0	0
$584$	2948.08	0.208891
$585$	−6810.24	−0.481314
$586$	15252.3	1.07520
$587$	17868.5	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$587$	17868.5	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$588$	−652.372	−0.0457540
$589$	−6749.84	−0.472194
$590$	−2051.76	−0.143169
$591$	7964.37	0.554332
$592$	16334.5	1.13403
$593$	−24594.0	−1.70312	−0.851562	−	0.524253i	$-0.824344\pi$
$593$	−24594.0	−1.70312	−0.851562	+	0.524253i	$0.824344\pi$
$594$	0	0
$595$	−2890.94	−0.199188
$596$	−4007.83	−0.275448
$597$	−3790.07	−0.259828
$598$	−13793.3	−0.943226
$599$	13645.6	0.930789	0.465394	−	0.885103i	$-0.345913\pi$
$599$	13645.6	0.930789	0.465394	+	0.885103i	$0.345913\pi$
$600$	−852.989	−0.0580386
$601$	23143.9	1.57081	0.785407	−	0.618980i	$-0.212454\pi$
$601$	23143.9	1.57081	0.785407	+	0.618980i	$0.212454\pi$
$602$	−8468.48	−0.573338
$603$	8043.69	0.543225
$604$	12835.7	0.864698
$605$	0	0
$606$	6323.36	0.423876
$607$	11968.7	0.800321	0.400160	−	0.916445i	$-0.368954\pi$
$607$	11968.7	0.800321	0.400160	+	0.916445i	$0.368954\pi$
$608$	−9408.96	−0.627605
$609$	−3591.82	−0.238995
$610$	1567.85	0.104066
$611$	−26314.7	−1.74235
$612$	−9750.08	−0.643993
$613$	−7079.21	−0.466438	−0.233219	−	0.972424i	$-0.574926\pi$
$613$	−7079.21	−0.466438	−0.233219	+	0.972424i	$0.574926\pi$
$614$	30598.8	2.01118
$615$	3991.43	0.261707
$616$	0	0
$617$	−24592.0	−1.60460	−0.802300	−	0.596921i	$-0.796390\pi$
$617$	−24592.0	−1.60460	−0.802300	+	0.596921i	$0.796390\pi$
$618$	−9541.89	−0.621086
$619$	−5460.23	−0.354548	−0.177274	−	0.984162i	$-0.556728\pi$
$619$	−5460.23	−0.354548	−0.177274	+	0.984162i	$0.556728\pi$
$620$	−7374.50	−0.477689
$621$	−7891.17	−0.509922
$622$	27632.3	1.78127
$623$	−7630.68	−0.490717
$624$	6660.52	0.427299
$625$	1621.87	0.103800
$626$	−32218.6	−2.05705
$627$	0	0
$628$	16485.4	1.04751
$629$	−14260.8	−0.904002
$630$	3997.38	0.252793
$631$	−23358.4	−1.47367	−0.736833	−	0.676075i	$-0.763680\pi$
$631$	−23358.4	−1.47367	−0.736833	+	0.676075i	$0.763680\pi$
$632$	2922.59	0.183947
$633$	−3013.59	−0.189225
$634$	2420.96	0.151654
$635$	10328.8	0.645490
$636$	−2315.94	−0.144391
$637$	2237.19	0.139153
$638$	0	0
$639$	2738.14	0.169514
$640$	4208.41	0.259925
$641$	9328.68	0.574822	0.287411	−	0.957807i	$-0.407205\pi$
$641$	9328.68	0.574822	0.287411	+	0.957807i	$0.407205\pi$
$642$	−10476.4	−0.644038
$643$	−267.859	−0.0164282	−0.00821409	−	0.999966i	$-0.502615\pi$
$643$	−267.859	−0.0164282	−0.00821409	+	0.999966i	$0.502615\pi$
$644$	3677.13	0.224998
$645$	−4098.70	−0.250211
$646$	9633.65	0.586735
$647$	11518.0	0.699877	0.349939	−	0.936773i	$-0.386202\pi$
$647$	11518.0	0.699877	0.349939	+	0.936773i	$0.386202\pi$
$648$	−2164.84	−0.131239
$649$	0	0
$650$	−14497.6	−0.874835
$651$	−2391.46	−0.143976
$652$	−7076.62	−0.425064
$653$	−12565.0	−0.752994	−0.376497	−	0.926418i	$-0.622871\pi$
$653$	−12565.0	−0.752994	−0.376497	+	0.926418i	$0.622871\pi$
$654$	−10051.9	−0.601012
$655$	−8375.31	−0.499619
$656$	22446.2	1.33594
$657$	13186.3	0.783024
$658$	15445.8	0.915106
$659$	4020.26	0.237643	0.118822	−	0.992916i	$-0.462088\pi$
$659$	4020.26	0.237643	0.118822	+	0.992916i	$0.462088\pi$
$660$	0	0
$661$	−2080.20	−0.122406	−0.0612031	−	0.998125i	$-0.519494\pi$
$661$	−2080.20	−0.122406	−0.0612031	+	0.998125i	$0.519494\pi$
$662$	1629.99	0.0956968
$663$	−5814.97	−0.340626
$664$	2624.79	0.153406
$665$	−1793.85	−0.104605
$666$	19718.9	1.14728
$667$	20245.5	1.17527
$668$	24767.9	1.43458
$669$	−8579.02	−0.495791
$670$	−8683.14	−0.500685
$671$	0	0
$672$	−3333.58	−0.191362
$673$	17535.3	1.00436	0.502182	−	0.864762i	$-0.332531\pi$
$673$	17535.3	1.00436	0.502182	+	0.864762i	$0.332531\pi$
$674$	−5243.26	−0.299648
$675$	−8294.11	−0.472949
$676$	−748.574	−0.0425907
$677$	−12560.8	−0.713074	−0.356537	−	0.934281i	$-0.616043\pi$
$677$	−12560.8	−0.713074	−0.356537	+	0.934281i	$0.616043\pi$
$678$	2393.59	0.135583
$679$	−2775.71	−0.156881
$680$	−2123.65	−0.119762
$681$	−5697.43	−0.320596
$682$	0	0
$683$	−4961.51	−0.277961	−0.138980	−	0.990295i	$-0.544382\pi$
$683$	−4961.51	−0.277961	−0.138980	+	0.990295i	$0.544382\pi$
$684$	−6050.01	−0.338198
$685$	−13669.4	−0.762456
$686$	−1313.15	−0.0730850
$687$	−6547.40	−0.363608
$688$	−23049.4	−1.27725
$689$	7942.07	0.439142
$690$	3918.51	0.216196
$691$	−15429.4	−0.849441	−0.424720	−	0.905325i	$-0.639628\pi$
$691$	−15429.4	−0.849441	−0.424720	+	0.905325i	$0.639628\pi$
$692$	6053.89	0.332564
$693$	0	0
$694$	21178.0	1.15836
$695$	−15034.8	−0.820579
$696$	−2638.52	−0.143696
$697$	−19596.6	−1.06496
$698$	−38941.6	−2.11169
$699$	9547.62	0.516630
$700$	3864.89	0.208684
$701$	−20446.3	−1.10163	−0.550817	−	0.834626i	$-0.685684\pi$
$701$	−20446.3	−1.10163	−0.550817	+	0.834626i	$0.685684\pi$
$702$	17479.4	0.939768
$703$	−8848.97	−0.474744
$704$	0	0
$705$	7475.68	0.399362
$706$	36853.9	1.96461
$707$	5780.91	0.307515
$708$	1100.21	0.0584018
$709$	−28090.1	−1.48794	−0.743968	−	0.668216i	$-0.767058\pi$
$709$	−28090.1	−1.48794	−0.743968	+	0.668216i	$0.767058\pi$
$710$	−2955.82	−0.156239
$711$	13072.3	0.689522
$712$	−5605.43	−0.295045
$713$	13479.6	0.708013
$714$	3413.19	0.178901
$715$	0	0
$716$	8985.44	0.468997
$717$	3953.00	0.205896
$718$	−32507.2	−1.68963
$719$	24923.7	1.29277	0.646383	−	0.763013i	$-0.276281\pi$
$719$	24923.7	1.29277	0.646383	+	0.763013i	$0.276281\pi$
$720$	10880.0	0.563158
$721$	−8723.32	−0.450587
$722$	−20281.4	−1.04542
$723$	−4473.47	−0.230111
$724$	−30351.7	−1.55803
$725$	21279.2	1.09006
$726$	0	0
$727$	−24025.9	−1.22568	−0.612842	−	0.790205i	$-0.709974\pi$
$727$	−24025.9	−1.22568	−0.612842	+	0.790205i	$0.709974\pi$
$728$	1643.42	0.0836663
$729$	−4283.00	−0.217599
$730$	−14234.6	−0.721706
$731$	20123.3	1.01818
$732$	−840.723	−0.0424509
$733$	−31004.0	−1.56229	−0.781145	−	0.624350i	$-0.785364\pi$
$733$	−31004.0	−1.56229	−0.781145	+	0.624350i	$0.785364\pi$
$734$	11286.3	0.567554
$735$	−635.558	−0.0318951
$736$	18789.9	0.941038
$737$	0	0
$738$	27096.8	1.35155
$739$	5542.14	0.275874	0.137937	−	0.990441i	$-0.455953\pi$
$739$	5542.14	0.275874	0.137937	+	0.990441i	$0.455953\pi$
$740$	−9667.89	−0.480269
$741$	−3608.24	−0.178882
$742$	−4661.72	−0.230643
$743$	−35227.3	−1.73939	−0.869694	−	0.493591i	$-0.835684\pi$
$743$	−35227.3	−1.73939	−0.869694	+	0.493591i	$0.835684\pi$
$744$	−1756.74	−0.0865662
$745$	−3904.53	−0.192015
$746$	31759.0	1.55868
$747$	11740.3	0.575039
$748$	0	0
$749$	−9577.71	−0.467239
$750$	10325.7	0.502722
$751$	19148.6	0.930415	0.465207	−	0.885202i	$-0.345980\pi$
$751$	19148.6	0.930415	0.465207	+	0.885202i	$0.345980\pi$
$752$	42040.2	2.03863
$753$	12729.1	0.616036
$754$	−44844.9	−2.16599
$755$	12504.9	0.602781
$756$	−4659.80	−0.224174
$757$	12899.6	0.619345	0.309673	−	0.950843i	$-0.399781\pi$
$757$	12899.6	0.619345	0.309673	+	0.950843i	$0.399781\pi$
$758$	−46466.7	−2.22658
$759$	0	0
$760$	−1317.74	−0.0628942
$761$	−4568.70	−0.217628	−0.108814	−	0.994062i	$-0.534705\pi$
$761$	−4568.70	−0.217628	−0.108814	+	0.994062i	$0.534705\pi$
$762$	−12194.7	−0.579748
$763$	−9189.62	−0.436024
$764$	−14902.1	−0.705677
$765$	−9498.79	−0.448927
$766$	−34660.4	−1.63490
$767$	−3772.97	−0.177619
$768$	−10217.8	−0.480080
$769$	−6294.43	−0.295166	−0.147583	−	0.989050i	$-0.547149\pi$
$769$	−6294.43	−0.295166	−0.147583	+	0.989050i	$0.547149\pi$
$770$	0	0
$771$	−15894.9	−0.742467
$772$	−19746.6	−0.920591
$773$	−4218.70	−0.196295	−0.0981475	−	0.995172i	$-0.531292\pi$
$773$	−4218.70	−0.196295	−0.0981475	+	0.995172i	$0.531292\pi$
$774$	−27825.0	−1.29218
$775$	14167.9	0.656677
$776$	−2039.01	−0.0943250
$777$	−3135.18	−0.144754
$778$	−4329.72	−0.199522
$779$	−12159.9	−0.559271
$780$	−3942.16	−0.180964
$781$	0	0
$782$	−19238.6	−0.879758
$783$	−25655.8	−1.17096
$784$	−3574.12	−0.162815
$785$	16060.5	0.730221
$786$	9888.32	0.448734
$787$	−10453.6	−0.473482	−0.236741	−	0.971573i	$-0.576079\pi$
$787$	−10453.6	−0.473482	−0.236741	+	0.971573i	$0.576079\pi$
$788$	−26508.8	−1.19840
$789$	−14661.6	−0.661557
$790$	−14111.5	−0.635526
$791$	2188.25	0.0983631
$792$	0	0
$793$	2883.10	0.129107
$794$	−50203.7	−2.24391
$795$	−2256.25	−0.100655
$796$	12615.0	0.561715
$797$	2131.75	0.0947436	0.0473718	−	0.998877i	$-0.484915\pi$
$797$	2131.75	0.0947436	0.0473718	+	0.998877i	$0.484915\pi$
$798$	2117.91	0.0939513
$799$	−36703.2	−1.62511
$800$	19749.3	0.872806
$801$	−25072.2	−1.10597
$802$	49023.4	2.15845
$803$	0	0
$804$	4656.15	0.204241
$805$	3582.35	0.156846
$806$	−29858.0	−1.30484
$807$	7008.05	0.305694
$808$	4246.60	0.184895
$809$	35641.7	1.54894	0.774472	−	0.632608i	$-0.218016\pi$
$809$	35641.7	1.54894	0.774472	+	0.632608i	$0.218016\pi$
$810$	10452.8	0.453423
$811$	8034.96	0.347898	0.173949	−	0.984755i	$-0.444347\pi$
$811$	8034.96	0.347898	0.173949	+	0.984755i	$0.444347\pi$
$812$	11955.1	0.516677
$813$	72.9991	0.00314907
$814$	0	0
$815$	−6894.23	−0.296312
$816$	9289.96	0.398546
$817$	12486.7	0.534703
$818$	−35162.6	−1.50297
$819$	7350.75	0.313622
$820$	−13285.2	−0.565779
$821$	−37824.5	−1.60790	−0.803950	−	0.594697i	$-0.797272\pi$
$821$	−37824.5	−1.60790	−0.803950	+	0.594697i	$0.797272\pi$
$822$	16138.8	0.684801
$823$	19245.1	0.815118	0.407559	−	0.913179i	$-0.366380\pi$
$823$	19245.1	0.815118	0.407559	+	0.913179i	$0.366380\pi$
$824$	−6408.07	−0.270917
$825$	0	0
$826$	2214.60	0.0932880
$827$	31294.1	1.31584	0.657921	−	0.753087i	$-0.271436\pi$
$827$	31294.1	1.31584	0.657921	+	0.753087i	$0.271436\pi$
$828$	12082.0	0.507099
$829$	−13121.7	−0.549740	−0.274870	−	0.961481i	$-0.588635\pi$
$829$	−13121.7	−0.549740	−0.274870	+	0.961481i	$0.588635\pi$
$830$	−12673.6	−0.530008
$831$	−9930.27	−0.414533
$832$	−14978.6	−0.624144
$833$	3120.38	0.129790
$834$	17750.9	0.737005
$835$	24129.5	1.00004
$836$	0	0
$837$	−17081.8	−0.705418
$838$	−23924.8	−0.986239
$839$	−5784.11	−0.238009	−0.119005	−	0.992894i	$-0.537970\pi$
$839$	−5784.11	−0.238009	−0.119005	+	0.992894i	$0.537970\pi$
$840$	−466.875	−0.0191770
$841$	41433.2	1.69885
$842$	18005.9	0.736965
$843$	−9120.84	−0.372643
$844$	10030.5	0.409081
$845$	−729.281	−0.0296900
$846$	50750.5	2.06246
$847$	0	0
$848$	−12688.2	−0.513814
$849$	−1835.49	−0.0741976
$850$	−20221.0	−0.815969
$851$	17671.6	0.711837
$852$	1584.99	0.0637335
$853$	−39103.1	−1.56959	−0.784797	−	0.619753i	$-0.787233\pi$
$853$	−39103.1	−1.56959	−0.784797	+	0.619753i	$0.787233\pi$
$854$	−1692.28	−0.0678087
$855$	−5894.07	−0.235758
$856$	−7035.70	−0.280929
$857$	−5793.14	−0.230910	−0.115455	−	0.993313i	$-0.536833\pi$
$857$	−5793.14	−0.230910	−0.115455	+	0.993313i	$0.536833\pi$
$858$	0	0
$859$	16365.8	0.650049	0.325025	−	0.945706i	$-0.394627\pi$
$859$	16365.8	0.650049	0.325025	+	0.945706i	$0.394627\pi$
$860$	13642.2	0.540925
$861$	−4308.22	−0.170527
$862$	−28921.5	−1.14277
$863$	−20999.3	−0.828303	−0.414151	−	0.910208i	$-0.635922\pi$
$863$	−20999.3	−0.828303	−0.414151	+	0.910208i	$0.635922\pi$
$864$	−23811.3	−0.937588
$865$	5897.85	0.231830
$866$	10076.3	0.395391
$867$	1715.40	0.0671949
$868$	7959.79	0.311259
$869$	0	0
$870$	12739.9	0.496463
$871$	−15967.4	−0.621164
$872$	−6750.61	−0.262161
$873$	−9120.19	−0.353576
$874$	−11937.7	−0.462012
$875$	9439.90	0.364716
$876$	7632.99	0.294400
$877$	−22042.8	−0.848727	−0.424364	−	0.905492i	$-0.639502\pi$
$877$	−22042.8	−0.848727	−0.424364	+	0.905492i	$0.639502\pi$
$878$	−48051.4	−1.84699
$879$	−7967.94	−0.305747
$880$	0	0
$881$	−7746.91	−0.296254	−0.148127	−	0.988968i	$-0.547324\pi$
$881$	−7746.91	−0.296254	−0.148127	+	0.988968i	$0.547324\pi$
$882$	−4314.64	−0.164718
$883$	−12901.9	−0.491715	−0.245857	−	0.969306i	$-0.579070\pi$
$883$	−12901.9	−0.491715	−0.245857	+	0.969306i	$0.579070\pi$
$884$	19354.7	0.736390
$885$	1071.86	0.0407119
$886$	42978.6	1.62968
$887$	22230.9	0.841532	0.420766	−	0.907169i	$-0.361761\pi$
$887$	22230.9	0.841532	0.420766	+	0.907169i	$0.361761\pi$
$888$	−2303.07	−0.0870338
$889$	−11148.6	−0.420598
$890$	27065.4	1.01936
$891$	0	0
$892$	28554.6	1.07184
$893$	−22774.6	−0.853441
$894$	4609.89	0.172458
$895$	8753.85	0.326937
$896$	−4542.42	−0.169366
$897$	7205.72	0.268218
$898$	−42791.9	−1.59018
$899$	43824.9	1.62585
$900$	12698.9	0.470330
$901$	11077.4	0.409592
$902$	0	0
$903$	4424.00	0.163036
$904$	1607.47	0.0591412
$905$	−29569.5	−1.08610
$906$	−14763.9	−0.541389
$907$	−35025.3	−1.28225	−0.641123	−	0.767438i	$-0.721531\pi$
$907$	−35025.3	−1.28225	−0.641123	+	0.767438i	$0.721531\pi$
$908$	18963.5	0.693090
$909$	18994.4	0.693074
$910$	−7935.12	−0.289062
$911$	9605.98	0.349353	0.174676	−	0.984626i	$-0.444112\pi$
$911$	9605.98	0.349353	0.174676	+	0.984626i	$0.444112\pi$
$912$	5764.50	0.209300
$913$	0	0
$914$	−26953.5	−0.975428
$915$	−819.055	−0.0295925
$916$	21792.5	0.786076
$917$	9040.04	0.325549
$918$	24379.9	0.876532
$919$	791.576	0.0284132	0.0142066	−	0.999899i	$-0.495478\pi$
$919$	791.576	0.0284132	0.0142066	+	0.999899i	$0.495478\pi$
$920$	2631.56	0.0943044
$921$	−15985.0	−0.571906
$922$	5678.90	0.202847
$923$	−5435.43	−0.193835
$924$	0	0
$925$	18573.9	0.660224
$926$	−11891.8	−0.422019
$927$	−28662.4	−1.01553
$928$	61089.8	2.16096
$929$	25050.6	0.884695	0.442348	−	0.896844i	$-0.354146\pi$
$929$	25050.6	0.884695	0.442348	+	0.896844i	$0.354146\pi$
$930$	8482.30	0.299081
$931$	1936.22	0.0681601
$932$	−31778.6	−1.11689
$933$	−14435.3	−0.506528
$934$	−36772.2	−1.28825
$935$	0	0
$936$	5399.80	0.188566
$937$	−14769.7	−0.514948	−0.257474	−	0.966285i	$-0.582890\pi$
$937$	−14769.7	−0.514948	−0.257474	+	0.966285i	$0.582890\pi$
$938$	9372.30	0.326244
$939$	16831.2	0.584949
$940$	−24882.3	−0.863372
$941$	−678.921	−0.0235199	−0.0117599	−	0.999931i	$-0.503743\pi$
$941$	−678.921	−0.0235199	−0.0117599	+	0.999931i	$0.503743\pi$
$942$	−18961.8	−0.655849
$943$	24283.5	0.838578
$944$	6027.67	0.207822
$945$	−4539.70	−0.156271
$946$	0	0
$947$	−21354.7	−0.732772	−0.366386	−	0.930463i	$-0.619405\pi$
$947$	−21354.7	−0.732772	−0.366386	+	0.930463i	$0.619405\pi$
$948$	7567.00	0.259246
$949$	−26175.9	−0.895369
$950$	−12547.3	−0.428513
$951$	−1264.73	−0.0431248
$952$	2292.20	0.0780365
$953$	3942.06	0.133994	0.0669968	−	0.997753i	$-0.478658\pi$
$953$	3942.06	0.133994	0.0669968	+	0.997753i	$0.478658\pi$
$954$	−15317.1	−0.519820
$955$	−14518.0	−0.491927
$956$	−13157.3	−0.445122
$957$	0	0
$958$	−34909.6	−1.17733
$959$	14754.3	0.496812
$960$	4255.23	0.143059
$961$	−612.100	−0.0205465
$962$	−39143.6	−1.31189
$963$	−31469.6	−1.05306
$964$	14889.6	0.497471
$965$	−19237.7	−0.641743
$966$	−4229.51	−0.140872
$967$	27117.9	0.901813	0.450907	−	0.892571i	$-0.351101\pi$
$967$	27117.9	0.901813	0.450907	+	0.892571i	$0.351101\pi$
$968$	0	0
$969$	−5032.69	−0.166846
$970$	9845.21	0.325887
$971$	−28536.2	−0.943120	−0.471560	−	0.881834i	$-0.656309\pi$
$971$	−28536.2	−0.943120	−0.471560	+	0.881834i	$0.656309\pi$
$972$	−23578.6	−0.778069
$973$	16228.1	0.534685
$974$	53770.7	1.76892
$975$	7573.66	0.248771
$976$	−4606.03	−0.151061
$977$	24692.4	0.808577	0.404288	−	0.914632i	$-0.367519\pi$
$977$	24692.4	0.808577	0.404288	+	0.914632i	$0.367519\pi$
$978$	8139.68	0.266133
$979$	0	0
$980$	2115.41	0.0689533
$981$	−30194.5	−0.982706
$982$	3959.58	0.128672
$983$	26814.7	0.870047	0.435024	−	0.900419i	$-0.356740\pi$
$983$	26814.7	0.870047	0.435024	+	0.900419i	$0.356740\pi$
$984$	−3164.78	−0.102530
$985$	−25825.6	−0.835402
$986$	−62548.7	−2.02024
$987$	−8069.00	−0.260222
$988$	12009.7	0.386722
$989$	−24936.1	−0.801741
$990$	0	0
$991$	46160.9	1.47967	0.739833	−	0.672790i	$-0.234904\pi$
$991$	46160.9	1.47967	0.739833	+	0.672790i	$0.234904\pi$
$992$	40674.0	1.30182
$993$	−851.519	−0.0272126
$994$	3190.41	0.101804
$995$	12289.8	0.391572
$996$	6795.94	0.216202
$997$	43200.4	1.37229	0.686143	−	0.727467i	$-0.259302\pi$
$997$	43200.4	1.37229	0.686143	+	0.727467i	$0.259302\pi$
$998$	−54870.0	−1.74036
$999$	−22394.1	−0.709228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.4.a.c.1.2		2
11.10	odd	2		77.4.a.b.1.1	✓	2
33.32	even	2		693.4.a.i.1.2		2
44.43	even	2		1232.4.a.m.1.2		2
55.54	odd	2		1925.4.a.l.1.2		2
77.76	even	2		539.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.b.1.1	✓	2	11.10	odd	2
539.4.a.d.1.1		2	77.76	even	2
693.4.a.i.1.2		2	33.32	even	2
847.4.a.c.1.2		2	1.1	even	1	trivial
1232.4.a.m.1.2		2	44.43	even	2
1925.4.a.l.1.2		2	55.54	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} -2.00000 q^{3} +6.65685 q^{4} +6.48528 q^{5} -7.65685 q^{6} -7.00000 q^{7} -5.14214 q^{8} -23.0000 q^{9} +24.8284 q^{10} -13.3137 q^{12} +45.6569 q^{13} -26.7990 q^{14} -12.9706 q^{15} -72.9411 q^{16} +63.6812 q^{17} -88.0538 q^{18} +39.5147 q^{19} +43.1716 q^{20} +14.0000 q^{21} -78.9117 q^{23} +10.2843 q^{24} -82.9411 q^{25} +174.794 q^{26} +100.000 q^{27} -46.5980 q^{28} -256.558 q^{29} -49.6569 q^{30} -170.818 q^{31} -238.113 q^{32} +243.799 q^{34} -45.3970 q^{35} -153.108 q^{36} -223.941 q^{37} +151.279 q^{38} -91.3137 q^{39} -33.3482 q^{40} -307.730 q^{41} +53.5980 q^{42} +316.000 q^{43} -149.161 q^{45} -302.108 q^{46} -576.357 q^{47} +145.882 q^{48} +49.0000 q^{49} -317.534 q^{50} -127.362 q^{51} +303.931 q^{52} +173.951 q^{53} +382.843 q^{54} +35.9949 q^{56} -79.0294 q^{57} -982.215 q^{58} -82.6375 q^{59} -86.3431 q^{60} +63.1472 q^{61} -653.966 q^{62} +161.000 q^{63} -328.068 q^{64} +296.098 q^{65} -349.726 q^{67} +423.917 q^{68} +157.823 q^{69} -173.799 q^{70} -119.050 q^{71} +118.269 q^{72} -573.318 q^{73} -857.342 q^{74} +165.882 q^{75} +263.044 q^{76} -349.588 q^{78} -568.362 q^{79} -473.044 q^{80} +421.000 q^{81} -1178.12 q^{82} -510.447 q^{83} +93.1960 q^{84} +412.991 q^{85} +1209.78 q^{86} +513.117 q^{87} +1090.10 q^{89} -571.054 q^{90} -319.598 q^{91} -525.304 q^{92} +341.637 q^{93} -2206.54 q^{94} +256.264 q^{95} +476.225 q^{96} +396.530 q^{97} +187.593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} - 14 q^{7} + 18 q^{8} - 46 q^{9} + 44 q^{10} - 4 q^{12} + 80 q^{13} - 14 q^{14} + 8 q^{15} - 78 q^{16} - 48 q^{17} - 46 q^{18} + 96 q^{19} + 92 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 - 14 * q^7 + 18 * q^8 - 46 * q^9 + 44 * q^10 - 4 * q^12 + 80 * q^13 - 14 * q^14 + 8 * q^15 - 78 * q^16 - 48 * q^17 - 46 * q^18 + 96 * q^19 + 92 * q^20 + 28 * q^21 - 56 * q^23 - 36 * q^24 - 98 * q^25 + 112 * q^26 + 200 * q^27 - 14 * q^28 - 4 * q^29 - 88 * q^30 + 60 * q^31 - 414 * q^32 + 448 * q^34 + 28 * q^35 - 46 * q^36 - 380 * q^37 + 48 * q^38 - 160 * q^39 - 276 * q^40 - 112 * q^41 + 28 * q^42 + 632 * q^43 + 92 * q^45 - 344 * q^46 - 604 * q^47 + 156 * q^48 + 98 * q^49 - 290 * q^50 + 96 * q^51 + 144 * q^52 + 676 * q^53 + 200 * q^54 - 126 * q^56 - 192 * q^57 - 1444 * q^58 - 516 * q^59 - 184 * q^60 + 296 * q^61 - 1076 * q^62 + 322 * q^63 + 34 * q^64 - 64 * q^65 - 1152 * q^67 + 944 * q^68 + 112 * q^69 - 308 * q^70 - 1064 * q^71 - 414 * q^72 - 168 * q^73 - 572 * q^74 + 196 * q^75 - 224 * q^78 + 368 * q^79 - 420 * q^80 + 842 * q^81 - 1536 * q^82 - 1728 * q^83 + 28 * q^84 + 1584 * q^85 + 632 * q^86 + 8 * q^87 + 1524 * q^89 - 1012 * q^90 - 560 * q^91 - 632 * q^92 - 120 * q^93 - 2156 * q^94 - 336 * q^95 + 828 * q^96 + 1404 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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