LMFDB - Embedded newform 207.8.a.a.1.5

Newspace parameters

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$64.6637002752$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
Defining polynomial:	$ x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 $ x^5 - 455x^3 - 474x^2 + 42284*x + 127016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-17.8260$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

$f(q)$	$=$	$q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +O(q^{10})$	\(q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +553.369 q^{10} +939.114 q^{11} +10151.8 q^{13} -22235.2 q^{14} -4662.78 q^{16} -33965.5 q^{17} +18716.1 q^{19} +5890.88 q^{20} +16740.7 q^{22} -12167.0 q^{23} -77161.3 q^{25} +180965. q^{26} -236704. q^{28} -91722.7 q^{29} -187422. q^{31} -224053. q^{32} -605469. q^{34} -38721.0 q^{35} -112278. q^{37} +333633. q^{38} +34179.6 q^{40} -218882. q^{41} -910269. q^{43} +178212. q^{44} -216889. q^{46} -54041.9 q^{47} +732327. q^{49} -1.37548e6 q^{50} +1.92647e6 q^{52} +135797. q^{53} +29152.7 q^{55} -1.37339e6 q^{56} -1.63505e6 q^{58} +2.26099e6 q^{59} +124600. q^{61} -3.34098e6 q^{62} -3.39714e6 q^{64} +315139. q^{65} +3.30689e6 q^{67} -6.44551e6 q^{68} -690242. q^{70} +740248. q^{71} +2.68349e6 q^{73} -2.00147e6 q^{74} +3.55169e6 q^{76} -1.17140e6 q^{77} -6.55101e6 q^{79} -144746. q^{80} -3.90179e6 q^{82} -3.21765e6 q^{83} -1.05438e6 q^{85} -1.62265e7 q^{86} +1.03401e6 q^{88} +1.14227e7 q^{89} -1.26628e7 q^{91} -2.30889e6 q^{92} -963351. q^{94} +580999. q^{95} -1.41253e7 q^{97} +1.30545e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10}) $ 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8	$ 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100}) $ 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8 + 1452 * q^10 + 1148 * q^11 - 642 * q^13 - 5756 * q^14 - 22606 * q^16 + 5798 * q^17 - 6036 * q^19 + 27376 * q^20 - 97896 * q^22 - 60835 * q^23 - 262477 * q^25 + 355992 * q^26 - 507124 * q^28 + 169162 * q^29 - 199640 * q^31 + 284794 * q^32 - 1027740 * q^34 + 137680 * q^35 - 202002 * q^37 + 554924 * q^38 - 340904 * q^40 - 541282 * q^41 - 909596 * q^43 + 1236032 * q^44 - 80208 * q^47 + 850589 * q^49 + 941416 * q^50 + 146940 * q^52 + 278138 * q^53 - 933560 * q^55 + 539932 * q^56 - 3522712 * q^58 + 3177380 * q^59 + 147782 * q^61 - 4606456 * q^62 - 4142622 * q^64 - 3877332 * q^65 - 464916 * q^67 - 7513072 * q^68 + 2093200 * q^70 - 1576792 * q^71 - 38190 * q^73 - 12164864 * q^74 + 6889436 * q^76 - 10332384 * q^77 - 3913336 * q^79 - 6334776 * q^80 + 6799360 * q^82 - 15774716 * q^83 - 8520740 * q^85 - 24874084 * q^86 + 53216 * q^88 - 1116482 * q^89 - 27369552 * q^91 - 3285090 * q^92 - 7153744 * q^94 + 6067832 * q^95 - 15738566 * q^97 - 11730488 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	17.8260	1.57561	0.787806	−	0.615924i	$-0.211217\pi$
$2$	17.8260	1.57561	0.787806	+	0.615924i	$0.211217\pi$
$3$	0	0
$3$	0	0
$4$	189.767	1.48255
$5$	31.0428	0.111062	0.0555310	−	0.998457i	$-0.482315\pi$
$5$	31.0428	0.111062	0.0555310	+	0.998457i	$0.482315\pi$
$6$	0	0
$7$	−1247.35	−1.37450	−0.687248	−	0.726423i	$-0.741181\pi$
$7$	−1247.35	−1.37450	−0.687248	+	0.726423i	$0.741181\pi$
$8$	1101.05	0.760313
$9$	0	0
$10$	553.369	0.174991
$11$	939.114	0.212737	0.106369	−	0.994327i	$-0.466078\pi$
$11$	939.114	0.212737	0.106369	+	0.994327i	$0.466078\pi$
$12$	0	0
$13$	10151.8	1.28156	0.640781	−	0.767723i	$-0.278610\pi$
$13$	10151.8	1.28156	0.640781	+	0.767723i	$0.278610\pi$
$14$	−22235.2	−2.16567
$15$	0	0
$16$	−4662.78	−0.284594
$17$	−33965.5	−1.67674	−0.838371	−	0.545100i	$-0.816492\pi$
$17$	−33965.5	−1.67674	−0.838371	+	0.545100i	$0.816492\pi$
$18$	0	0
$19$	18716.1	0.626005	0.313002	−	0.949752i	$-0.398665\pi$
$19$	18716.1	0.626005	0.313002	+	0.949752i	$0.398665\pi$
$20$	5890.88	0.164655
$21$	0	0
$22$	16740.7	0.335192
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	0	0
$25$	−77161.3	−0.987665
$26$	180965.	2.01925
$27$	0	0
$28$	−236704.	−2.03776
$29$	−91722.7	−0.698367	−0.349183	−	0.937054i	$-0.613541\pi$
$29$	−91722.7	−0.698367	−0.349183	+	0.937054i	$0.613541\pi$
$30$	0	0
$31$	−187422.	−1.12994	−0.564968	−	0.825113i	$-0.691111\pi$
$31$	−187422.	−1.12994	−0.564968	+	0.825113i	$0.691111\pi$
$32$	−224053.	−1.20872
$33$	0	0
$34$	−605469.	−2.64189
$35$	−38721.0	−0.152654
$36$	0	0
$37$	−112278.	−0.364408	−0.182204	−	0.983261i	$-0.558323\pi$
$37$	−112278.	−0.364408	−0.182204	+	0.983261i	$0.558323\pi$
$38$	333633.	0.986340
$39$	0	0
$40$	34179.6	0.0844418
$41$	−218882.	−0.495983	−0.247991	−	0.968762i	$-0.579770\pi$
$41$	−218882.	−0.495983	−0.247991	+	0.968762i	$0.579770\pi$
$42$	0	0
$43$	−910269.	−1.74594	−0.872971	−	0.487772i	$-0.837810\pi$
$43$	−910269.	−1.74594	−0.872971	+	0.487772i	$0.837810\pi$
$44$	178212.	0.315394
$45$	0	0
$46$	−216889.	−0.328538
$47$	−54041.9	−0.0759256	−0.0379628	−	0.999279i	$-0.512087\pi$
$47$	−54041.9	−0.0759256	−0.0379628	+	0.999279i	$0.512087\pi$
$48$	0	0
$49$	732327.	0.889239
$50$	−1.37548e6	−1.55618
$51$	0	0
$52$	1.92647e6	1.89998
$53$	135797.	0.125292	0.0626462	−	0.998036i	$-0.480046\pi$
$53$	135797.	0.125292	0.0626462	+	0.998036i	$0.480046\pi$
$54$	0	0
$55$	29152.7	0.0236270
$56$	−1.37339e6	−1.04505
$57$	0	0
$58$	−1.63505e6	−1.10035
$59$	2.26099e6	1.43323	0.716616	−	0.697468i	$-0.245690\pi$
$59$	2.26099e6	1.43323	0.716616	+	0.697468i	$0.245690\pi$
$60$	0	0
$61$	124600.	0.0702851	0.0351426	−	0.999382i	$-0.488811\pi$
$61$	124600.	0.0702851	0.0351426	+	0.999382i	$0.488811\pi$
$62$	−3.34098e6	−1.78034
$63$	0	0
$64$	−3.39714e6	−1.61988
$65$	315139.	0.142333
$66$	0	0
$67$	3.30689e6	1.34325	0.671627	−	0.740890i	$-0.265596\pi$
$67$	3.30689e6	1.34325	0.671627	+	0.740890i	$0.265596\pi$
$68$	−6.44551e6	−2.48586
$69$	0	0
$70$	−690242.	−0.240524
$71$	740248.	0.245456	0.122728	−	0.992440i	$-0.460836\pi$
$71$	740248.	0.245456	0.122728	+	0.992440i	$0.460836\pi$
$72$	0	0
$73$	2.68349e6	0.807365	0.403683	−	0.914899i	$-0.367730\pi$
$73$	2.68349e6	0.807365	0.403683	+	0.914899i	$0.367730\pi$
$74$	−2.00147e6	−0.574166
$75$	0	0
$76$	3.55169e6	0.928084
$77$	−1.17140e6	−0.292407
$78$	0	0
$79$	−6.55101e6	−1.49490	−0.747452	−	0.664315i	$-0.768723\pi$
$79$	−6.55101e6	−1.49490	−0.747452	+	0.664315i	$0.768723\pi$
$80$	−144746.	−0.0316076
$81$	0	0
$82$	−3.90179e6	−0.781476
$83$	−3.21765e6	−0.617682	−0.308841	−	0.951114i	$-0.599941\pi$
$83$	−3.21765e6	−0.617682	−0.308841	+	0.951114i	$0.599941\pi$
$84$	0	0
$85$	−1.05438e6	−0.186222
$86$	−1.62265e7	−2.75093
$87$	0	0
$88$	1.03401e6	0.161747
$89$	1.14227e7	1.71753	0.858765	−	0.512370i	$-0.171232\pi$
$89$	1.14227e7	1.71753	0.858765	+	0.512370i	$0.171232\pi$
$90$	0	0
$91$	−1.26628e7	−1.76150
$92$	−2.30889e6	−0.309133
$93$	0	0
$94$	−963351.	−0.119629
$95$	580999.	0.0695253
$96$	0	0
$97$	−1.41253e7	−1.57143	−0.785717	−	0.618586i	$-0.787706\pi$
$97$	−1.41253e7	−1.57143	−0.785717	+	0.618586i	$0.787706\pi$
$98$	1.30545e7	1.40110
$99$	0	0
$100$	−1.46426e7	−1.46426
$101$	1.20112e7	1.16001	0.580006	−	0.814612i	$-0.303050\pi$
$101$	1.20112e7	1.16001	0.580006	+	0.814612i	$0.303050\pi$
$102$	0	0
$103$	−7.74503e6	−0.698382	−0.349191	−	0.937052i	$-0.613544\pi$
$103$	−7.74503e6	−0.698382	−0.349191	+	0.937052i	$0.613544\pi$
$104$	1.11776e7	0.974388
$105$	0	0
$106$	2.42072e6	0.197412
$107$	−4.90393e6	−0.386991	−0.193495	−	0.981101i	$-0.561982\pi$
$107$	−4.90393e6	−0.386991	−0.193495	+	0.981101i	$0.561982\pi$
$108$	0	0
$109$	2.12306e7	1.57025	0.785127	−	0.619335i	$-0.212598\pi$
$109$	2.12306e7	1.57025	0.785127	+	0.619335i	$0.212598\pi$
$110$	519676.	0.0372270
$111$	0	0
$112$	5.81610e6	0.391173
$113$	−7.90376e6	−0.515299	−0.257649	−	0.966238i	$-0.582948\pi$
$113$	−7.90376e6	−0.515299	−0.257649	+	0.966238i	$0.582948\pi$
$114$	0	0
$115$	−377697.	−0.0231580
$116$	−1.74059e7	−1.03536
$117$	0	0
$118$	4.03044e7	2.25822
$119$	4.23667e7	2.30468
$120$	0	0
$121$	−1.86052e7	−0.954743
$122$	2.22112e6	0.110742
$123$	0	0
$124$	−3.55663e7	−1.67519
$125$	−4.82052e6	−0.220754
$126$	0	0
$127$	3.73010e7	1.61587	0.807936	−	0.589270i	$-0.200585\pi$
$127$	3.73010e7	1.61587	0.807936	+	0.589270i	$0.200585\pi$
$128$	−3.18786e7	−1.34358
$129$	0	0
$130$	5.61767e6	0.224261
$131$	1.67942e6	0.0652694	0.0326347	−	0.999467i	$-0.489610\pi$
$131$	1.67942e6	0.0652694	0.0326347	+	0.999467i	$0.489610\pi$
$132$	0	0
$133$	−2.33454e7	−0.860441
$134$	5.89486e7	2.11644
$135$	0	0
$136$	−3.73977e7	−1.27485
$137$	−8.56278e6	−0.284507	−0.142253	−	0.989830i	$-0.545435\pi$
$137$	−8.56278e6	−0.284507	−0.142253	+	0.989830i	$0.545435\pi$
$138$	0	0
$139$	−1.00144e7	−0.316281	−0.158140	−	0.987417i	$-0.550550\pi$
$139$	−1.00144e7	−0.316281	−0.158140	+	0.987417i	$0.550550\pi$
$140$	−7.34796e6	−0.226318
$141$	0	0
$142$	1.31957e7	0.386743
$143$	9.53367e6	0.272636
$144$	0	0
$145$	−2.84733e6	−0.0775620
$146$	4.78359e7	1.27209
$147$	0	0
$148$	−2.13066e7	−0.540254
$149$	−1.78249e7	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$149$	−1.78249e7	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$150$	0	0
$151$	−1.24711e7	−0.294771	−0.147386	−	0.989079i	$-0.547086\pi$
$151$	−1.24711e7	−0.294771	−0.147386	+	0.989079i	$0.547086\pi$
$152$	2.06074e7	0.475959
$153$	0	0
$154$	−2.08814e7	−0.460719
$155$	−5.81808e6	−0.125493
$156$	0	0
$157$	−2.73688e7	−0.564426	−0.282213	−	0.959352i	$-0.591069\pi$
$157$	−2.73688e7	−0.564426	−0.282213	+	0.959352i	$0.591069\pi$
$158$	−1.16778e8	−2.35539
$159$	0	0
$160$	−6.95523e6	−0.134243
$161$	1.51764e7	0.286602
$162$	0	0
$163$	8.32721e7	1.50606	0.753031	−	0.657985i	$-0.228591\pi$
$163$	8.32721e7	1.50606	0.753031	+	0.657985i	$0.228591\pi$
$164$	−4.15365e7	−0.735320
$165$	0	0
$166$	−5.73578e7	−0.973227
$167$	−9.35501e6	−0.155431	−0.0777153	−	0.996976i	$-0.524763\pi$
$167$	−9.35501e6	−0.155431	−0.0777153	+	0.996976i	$0.524763\pi$
$168$	0	0
$169$	4.03099e7	0.642404
$170$	−1.87954e7	−0.293414
$171$	0	0
$172$	−1.72738e8	−2.58845
$173$	8.62670e7	1.26673	0.633364	−	0.773854i	$-0.281674\pi$
$173$	8.62670e7	1.26673	0.633364	+	0.773854i	$0.281674\pi$
$174$	0	0
$175$	9.62468e7	1.35754
$176$	−4.37889e6	−0.0605438
$177$	0	0
$178$	2.03621e8	2.70616
$179$	9.64475e7	1.25691	0.628456	−	0.777845i	$-0.283687\pi$
$179$	9.64475e7	1.25691	0.628456	+	0.777845i	$0.283687\pi$
$180$	0	0
$181$	−2.62936e7	−0.329591	−0.164796	−	0.986328i	$-0.552696\pi$
$181$	−2.62936e7	−0.329591	−0.164796	+	0.986328i	$0.552696\pi$
$182$	−2.25726e8	−2.77544
$183$	0	0
$184$	−1.33965e7	−0.158536
$185$	−3.48542e6	−0.0404719
$186$	0	0
$187$	−3.18974e7	−0.356706
$188$	−1.02553e7	−0.112564
$189$	0	0
$190$	1.03569e7	0.109545
$191$	−5.41983e7	−0.562819	−0.281410	−	0.959588i	$-0.590802\pi$
$191$	−5.41983e7	−0.562819	−0.281410	+	0.959588i	$0.590802\pi$
$192$	0	0
$193$	−1.54438e8	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$193$	−1.54438e8	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$194$	−2.51797e8	−2.47597
$195$	0	0
$196$	1.38971e8	1.31834
$197$	−1.17189e8	−1.09208	−0.546041	−	0.837759i	$-0.683866\pi$
$197$	−1.17189e8	−1.09208	−0.546041	+	0.837759i	$0.683866\pi$
$198$	0	0
$199$	−9.17056e7	−0.824917	−0.412458	−	0.910976i	$-0.635330\pi$
$199$	−9.17056e7	−0.824917	−0.412458	+	0.910976i	$0.635330\pi$
$200$	−8.49585e7	−0.750934
$201$	0	0
$202$	2.14112e8	1.82773
$203$	1.14410e8	0.959903
$204$	0	0
$205$	−6.79470e6	−0.0550848
$206$	−1.38063e8	−1.10038
$207$	0	0
$208$	−4.73355e7	−0.364725
$209$	1.75766e7	0.133175
$210$	0	0
$211$	3.23106e7	0.236786	0.118393	−	0.992967i	$-0.462226\pi$
$211$	3.23106e7	0.236786	0.118393	+	0.992967i	$0.462226\pi$
$212$	2.57697e7	0.185752
$213$	0	0
$214$	−8.74174e7	−0.609747
$215$	−2.82573e7	−0.193908
$216$	0	0
$217$	2.33779e8	1.55309
$218$	3.78457e8	2.47411
$219$	0	0
$220$	5.53221e6	0.0350283
$221$	−3.44809e8	−2.14885
$222$	0	0
$223$	1.25273e6	0.00756468	0.00378234	−	0.999993i	$-0.498796\pi$
$223$	1.25273e6	0.00756468	0.00378234	+	0.999993i	$0.498796\pi$
$224$	2.79472e8	1.66138
$225$	0	0
$226$	−1.40892e8	−0.811911
$227$	1.41797e8	0.804594	0.402297	−	0.915509i	$-0.368212\pi$
$227$	1.41797e8	0.804594	0.402297	+	0.915509i	$0.368212\pi$
$228$	0	0
$229$	−7.55158e7	−0.415541	−0.207770	−	0.978178i	$-0.566621\pi$
$229$	−7.55158e7	−0.415541	−0.207770	+	0.978178i	$0.566621\pi$
$230$	−6.73284e6	−0.0364880
$231$	0	0
$232$	−1.00991e8	−0.530977
$233$	2.66330e8	1.37935	0.689674	−	0.724120i	$-0.257754\pi$
$233$	2.66330e8	1.37935	0.689674	+	0.724120i	$0.257754\pi$
$234$	0	0
$235$	−1.67761e6	−0.00843244
$236$	4.29060e8	2.12484
$237$	0	0
$238$	7.55228e8	3.63127
$239$	3.47869e8	1.64825	0.824125	−	0.566408i	$-0.191667\pi$
$239$	3.47869e8	1.64825	0.824125	+	0.566408i	$0.191667\pi$
$240$	0	0
$241$	−1.34331e8	−0.618184	−0.309092	−	0.951032i	$-0.600025\pi$
$241$	−1.34331e8	−0.618184	−0.309092	+	0.951032i	$0.600025\pi$
$242$	−3.31657e8	−1.50430
$243$	0	0
$244$	2.36449e7	0.104201
$245$	2.27335e7	0.0987607
$246$	0	0
$247$	1.90001e8	0.802265
$248$	−2.06361e8	−0.859104
$249$	0	0
$250$	−8.59306e7	−0.347823
$251$	1.25191e8	0.499707	0.249854	−	0.968284i	$-0.419618\pi$
$251$	1.25191e8	0.499707	0.249854	+	0.968284i	$0.419618\pi$
$252$	0	0
$253$	−1.14262e7	−0.0443588
$254$	6.64927e8	2.54599
$255$	0	0
$256$	−1.33434e8	−0.497082
$257$	3.83946e8	1.41093	0.705463	−	0.708746i	$-0.250739\pi$
$257$	3.83946e8	1.41093	0.705463	+	0.708746i	$0.250739\pi$
$258$	0	0
$259$	1.40049e8	0.500878
$260$	5.98028e7	0.211016
$261$	0	0
$262$	2.99373e7	0.102839
$263$	5.17704e8	1.75483	0.877417	−	0.479729i	$-0.159265\pi$
$263$	5.17704e8	1.75483	0.877417	+	0.479729i	$0.159265\pi$
$264$	0	0
$265$	4.21552e6	0.0139152
$266$	−4.16156e8	−1.35572
$267$	0	0
$268$	6.27537e8	1.99144
$269$	2.04806e8	0.641518	0.320759	−	0.947161i	$-0.396062\pi$
$269$	2.04806e8	0.641518	0.320759	+	0.947161i	$0.396062\pi$
$270$	0	0
$271$	−2.30872e8	−0.704659	−0.352329	−	0.935876i	$-0.614610\pi$
$271$	−2.30872e8	−0.704659	−0.352329	+	0.935876i	$0.614610\pi$
$272$	1.58374e8	0.477190
$273$	0	0
$274$	−1.52640e8	−0.448272
$275$	−7.24633e7	−0.210113
$276$	0	0
$277$	−3.59723e8	−1.01693	−0.508463	−	0.861084i	$-0.669786\pi$
$277$	−3.59723e8	−1.01693	−0.508463	+	0.861084i	$0.669786\pi$
$278$	−1.78517e8	−0.498335
$279$	0	0
$280$	−4.26338e7	−0.116065
$281$	−5.59198e8	−1.50346	−0.751732	−	0.659469i	$-0.770781\pi$
$281$	−5.59198e8	−1.50346	−0.751732	+	0.659469i	$0.770781\pi$
$282$	0	0
$283$	−2.74952e8	−0.721114	−0.360557	−	0.932737i	$-0.617413\pi$
$283$	−2.74952e8	−0.721114	−0.360557	+	0.932737i	$0.617413\pi$
$284$	1.40474e8	0.363900
$285$	0	0
$286$	1.69947e8	0.429569
$287$	2.73021e8	0.681726
$288$	0	0
$289$	7.43314e8	1.81146
$290$	−5.07564e7	−0.122208
$291$	0	0
$292$	5.09237e8	1.19696
$293$	−2.01642e8	−0.468321	−0.234161	−	0.972198i	$-0.575234\pi$
$293$	−2.01642e8	−0.468321	−0.234161	+	0.972198i	$0.575234\pi$
$294$	0	0
$295$	7.01873e7	0.159178
$296$	−1.23624e8	−0.277064
$297$	0	0
$298$	−3.17747e8	−0.695543
$299$	−1.23517e8	−0.267224
$300$	0	0
$301$	1.13542e9	2.39979
$302$	−2.22310e8	−0.464445
$303$	0	0
$304$	−8.72691e7	−0.178157
$305$	3.86793e6	0.00780600
$306$	0	0
$307$	−4.38553e8	−0.865043	−0.432521	−	0.901624i	$-0.642376\pi$
$307$	−4.38553e8	−0.865043	−0.432521	+	0.901624i	$0.642376\pi$
$308$	−2.22292e8	−0.433508
$309$	0	0
$310$	−1.03713e8	−0.197728
$311$	−6.05423e8	−1.14129	−0.570647	−	0.821196i	$-0.693307\pi$
$311$	−6.05423e8	−1.14129	−0.570647	+	0.821196i	$0.693307\pi$
$312$	0	0
$313$	−1.07052e7	−0.0197329	−0.00986647	−	0.999951i	$-0.503141\pi$
$313$	−1.07052e7	−0.0197329	−0.00986647	+	0.999951i	$0.503141\pi$
$314$	−4.87877e8	−0.889316
$315$	0	0
$316$	−1.24316e9	−2.21627
$317$	−9.99712e8	−1.76266	−0.881328	−	0.472505i	$-0.843350\pi$
$317$	−9.99712e8	−1.76266	−0.881328	+	0.472505i	$0.843350\pi$
$318$	0	0
$319$	−8.61381e7	−0.148569
$320$	−1.05457e8	−0.179907
$321$	0	0
$322$	2.70535e8	0.451574
$323$	−6.35701e8	−1.04965
$324$	0	0
$325$	−7.83324e8	−1.26576
$326$	1.48441e9	2.37297
$327$	0	0
$328$	−2.41000e8	−0.377102
$329$	6.74089e7	0.104359
$330$	0	0
$331$	−8.39788e8	−1.27283	−0.636417	−	0.771345i	$-0.719584\pi$
$331$	−8.39788e8	−1.27283	−0.636417	+	0.771345i	$0.719584\pi$
$332$	−6.10602e8	−0.915745
$333$	0	0
$334$	−1.66762e8	−0.244898
$335$	1.02655e8	0.149184
$336$	0	0
$337$	−5.53660e8	−0.788022	−0.394011	−	0.919106i	$-0.628913\pi$
$337$	−5.53660e8	−0.788022	−0.394011	+	0.919106i	$0.628913\pi$
$338$	7.18564e8	1.01218
$339$	0	0
$340$	−2.00086e8	−0.276084
$341$	−1.76010e8	−0.240380
$342$	0	0
$343$	1.13778e8	0.152240
$344$	−1.00225e9	−1.32746
$345$	0	0
$346$	1.53780e9	1.99587
$347$	−8.05434e8	−1.03485	−0.517424	−	0.855729i	$-0.673109\pi$
$347$	−8.05434e8	−1.03485	−0.517424	+	0.855729i	$0.673109\pi$
$348$	0	0
$349$	6.32931e8	0.797017	0.398509	−	0.917165i	$-0.369528\pi$
$349$	6.32931e8	0.797017	0.398509	+	0.917165i	$0.369528\pi$
$350$	1.71570e9	2.13896
$351$	0	0
$352$	−2.10412e8	−0.257140
$353$	−3.46589e8	−0.419376	−0.209688	−	0.977768i	$-0.567245\pi$
$353$	−3.46589e8	−0.419376	−0.209688	+	0.977768i	$0.567245\pi$
$354$	0	0
$355$	2.29793e7	0.0272608
$356$	2.16765e9	2.54633
$357$	0	0
$358$	1.71927e9	1.98041
$359$	2.14794e8	0.245014	0.122507	−	0.992468i	$-0.460907\pi$
$359$	2.14794e8	0.245014	0.122507	+	0.992468i	$0.460907\pi$
$360$	0	0
$361$	−5.43580e8	−0.608118
$362$	−4.68710e8	−0.519307
$363$	0	0
$364$	−2.40297e9	−2.61152
$365$	8.33030e7	0.0896676
$366$	0	0
$367$	1.71463e9	1.81067	0.905335	−	0.424699i	$-0.139620\pi$
$367$	1.71463e9	1.81067	0.905335	+	0.424699i	$0.139620\pi$
$368$	5.67321e7	0.0593419
$369$	0	0
$370$	−6.21311e7	−0.0637680
$371$	−1.69386e8	−0.172214
$372$	0	0
$373$	−8.64659e8	−0.862709	−0.431354	−	0.902183i	$-0.641964\pi$
$373$	−8.64659e8	−0.862709	−0.431354	+	0.902183i	$0.641964\pi$
$374$	−5.68604e8	−0.562030
$375$	0	0
$376$	−5.95028e7	−0.0577272
$377$	−9.31147e8	−0.895001
$378$	0	0
$379$	−5.13537e7	−0.0484546	−0.0242273	−	0.999706i	$-0.507713\pi$
$379$	−5.13537e7	−0.0484546	−0.0242273	+	0.999706i	$0.507713\pi$
$380$	1.10254e8	0.103075
$381$	0	0
$382$	−9.66139e8	−0.886784
$383$	−2.01952e9	−1.83676	−0.918379	−	0.395703i	$-0.870501\pi$
$383$	−2.01952e9	−1.83676	−0.918379	+	0.395703i	$0.870501\pi$
$384$	0	0
$385$	−3.63635e7	−0.0324753
$386$	−2.75301e9	−2.43642
$387$	0	0
$388$	−2.68051e9	−2.32973
$389$	−1.39868e9	−1.20474	−0.602370	−	0.798217i	$-0.705777\pi$
$389$	−1.39868e9	−1.20474	−0.602370	+	0.798217i	$0.705777\pi$
$390$	0	0
$391$	4.13258e8	0.349625
$392$	8.06329e8	0.676100
$393$	0	0
$394$	−2.08901e9	−1.72070
$395$	−2.03361e8	−0.166027
$396$	0	0
$397$	−1.96246e9	−1.57411	−0.787055	−	0.616883i	$-0.788395\pi$
$397$	−1.96246e9	−1.57411	−0.787055	+	0.616883i	$0.788395\pi$
$398$	−1.63474e9	−1.29975
$399$	0	0
$400$	3.59787e8	0.281083
$401$	2.26678e9	1.75552	0.877758	−	0.479104i	$-0.159038\pi$
$401$	2.26678e9	1.75552	0.877758	+	0.479104i	$0.159038\pi$
$402$	0	0
$403$	−1.90266e9	−1.44808
$404$	2.27933e9	1.71978
$405$	0	0
$406$	2.03947e9	1.51243
$407$	−1.05442e8	−0.0775233
$408$	0	0
$409$	2.28154e9	1.64891	0.824453	−	0.565931i	$-0.191483\pi$
$409$	2.28154e9	1.64891	0.824453	+	0.565931i	$0.191483\pi$
$410$	−1.21122e8	−0.0867923
$411$	0	0
$412$	−1.46975e9	−1.03539
$413$	−2.82023e9	−1.96997
$414$	0	0
$415$	−9.98847e7	−0.0686010
$416$	−2.27454e9	−1.54905
$417$	0	0
$418$	3.13320e8	0.209832
$419$	−6.99241e7	−0.0464384	−0.0232192	−	0.999730i	$-0.507392\pi$
$419$	−6.99241e7	−0.0464384	−0.0232192	+	0.999730i	$0.507392\pi$
$420$	0	0
$421$	−1.08544e9	−0.708952	−0.354476	−	0.935065i	$-0.615341\pi$
$421$	−1.08544e9	−0.708952	−0.354476	+	0.935065i	$0.615341\pi$
$422$	5.75969e8	0.373083
$423$	0	0
$424$	1.49519e8	0.0952614
$425$	2.62082e9	1.65606
$426$	0	0
$427$	−1.55419e8	−0.0966066
$428$	−9.30601e8	−0.573734
$429$	0	0
$430$	−5.03714e8	−0.305523
$431$	3.16872e9	1.90640	0.953198	−	0.302348i	$-0.0977705\pi$
$431$	3.16872e9	1.90640	0.953198	+	0.302348i	$0.0977705\pi$
$432$	0	0
$433$	3.26047e8	0.193007	0.0965033	−	0.995333i	$-0.469234\pi$
$433$	3.26047e8	0.193007	0.0965033	+	0.995333i	$0.469234\pi$
$434$	4.16735e9	2.44707
$435$	0	0
$436$	4.02886e9	2.32798
$437$	−2.27719e8	−0.130531
$438$	0	0
$439$	−2.63160e9	−1.48455	−0.742273	−	0.670097i	$-0.766252\pi$
$439$	−2.63160e9	−1.48455	−0.742273	+	0.670097i	$0.766252\pi$
$440$	3.20986e7	0.0179639
$441$	0	0
$442$	−6.14658e9	−3.38575
$443$	3.10154e8	0.169498	0.0847489	−	0.996402i	$-0.472991\pi$
$443$	3.10154e8	0.169498	0.0847489	+	0.996402i	$0.472991\pi$
$444$	0	0
$445$	3.54593e8	0.190752
$446$	2.23312e7	0.0119190
$447$	0	0
$448$	4.23740e9	2.22652
$449$	1.98075e9	1.03268	0.516341	−	0.856383i	$-0.327294\pi$
$449$	1.98075e9	1.03268	0.516341	+	0.856383i	$0.327294\pi$
$450$	0	0
$451$	−2.05555e8	−0.105514
$452$	−1.49987e9	−0.763957
$453$	0	0
$454$	2.52767e9	1.26773
$455$	−3.93087e8	−0.195636
$456$	0	0
$457$	−1.57211e9	−0.770504	−0.385252	−	0.922811i	$-0.625885\pi$
$457$	−1.57211e9	−0.770504	−0.385252	+	0.922811i	$0.625885\pi$
$458$	−1.34615e9	−0.654731
$459$	0	0
$460$	−7.16743e7	−0.0343329
$461$	−2.80297e9	−1.33249	−0.666247	−	0.745731i	$-0.732101\pi$
$461$	−2.80297e9	−1.33249	−0.666247	+	0.745731i	$0.732101\pi$
$462$	0	0
$463$	3.29626e9	1.54343	0.771717	−	0.635966i	$-0.219398\pi$
$463$	3.29626e9	1.54343	0.771717	+	0.635966i	$0.219398\pi$
$464$	4.27683e8	0.198751
$465$	0	0
$466$	4.74759e9	2.17332
$467$	2.22099e9	1.00911	0.504554	−	0.863380i	$-0.331657\pi$
$467$	2.22099e9	1.00911	0.504554	+	0.863380i	$0.331657\pi$
$468$	0	0
$469$	−4.12483e9	−1.84630
$470$	−2.99051e7	−0.0132863
$471$	0	0
$472$	2.48946e9	1.08970
$473$	−8.54846e8	−0.371427
$474$	0	0
$475$	−1.44416e9	−0.618283
$476$	8.03977e9	3.41680
$477$	0	0
$478$	6.20111e9	2.59700
$479$	−4.08881e9	−1.69990	−0.849948	−	0.526867i	$-0.823367\pi$
$479$	−4.08881e9	−1.69990	−0.849948	+	0.526867i	$0.823367\pi$
$480$	0	0
$481$	−1.13982e9	−0.467012
$482$	−2.39459e9	−0.974017
$483$	0	0
$484$	−3.53065e9	−1.41545
$485$	−4.38488e8	−0.174527
$486$	0	0
$487$	−2.20516e9	−0.865144	−0.432572	−	0.901599i	$-0.642394\pi$
$487$	−2.20516e9	−0.865144	−0.432572	+	0.901599i	$0.642394\pi$
$488$	1.37191e8	0.0534387
$489$	0	0
$490$	4.05247e8	0.155608
$491$	2.44960e9	0.933921	0.466961	−	0.884278i	$-0.345349\pi$
$491$	2.44960e9	0.933921	0.466961	+	0.884278i	$0.345349\pi$
$492$	0	0
$493$	3.11540e9	1.17098
$494$	3.38697e9	1.26406
$495$	0	0
$496$	8.73906e8	0.321573
$497$	−9.23345e8	−0.337378
$498$	0	0
$499$	−4.09114e9	−1.47398	−0.736991	−	0.675903i	$-0.763754\pi$
$499$	−4.09114e9	−1.47398	−0.736991	+	0.675903i	$0.763754\pi$
$500$	−9.14773e8	−0.327279
$501$	0	0
$502$	2.23166e9	0.787344
$503$	−4.53054e8	−0.158731	−0.0793655	−	0.996846i	$-0.525289\pi$
$503$	−4.53054e8	−0.158731	−0.0793655	+	0.996846i	$0.525289\pi$
$504$	0	0
$505$	3.72862e8	0.128833
$506$	−2.03684e8	−0.0698923
$507$	0	0
$508$	7.07847e9	2.39561
$509$	2.23184e9	0.750156	0.375078	−	0.926993i	$-0.377616\pi$
$509$	2.23184e9	0.750156	0.375078	+	0.926993i	$0.377616\pi$
$510$	0	0
$511$	−3.34724e9	−1.10972
$512$	1.70186e9	0.560375
$513$	0	0
$514$	6.84423e9	2.22307
$515$	−2.40427e8	−0.0775636
$516$	0	0
$517$	−5.07515e7	−0.0161522
$518$	2.49652e9	0.789189
$519$	0	0
$520$	3.46984e8	0.108218
$521$	−3.73130e9	−1.15592	−0.577960	−	0.816065i	$-0.696151\pi$
$521$	−3.73130e9	−1.15592	−0.577960	+	0.816065i	$0.696151\pi$
$522$	0	0
$523$	−3.65832e9	−1.11822	−0.559108	−	0.829095i	$-0.688856\pi$
$523$	−3.65832e9	−1.11822	−0.559108	+	0.829095i	$0.688856\pi$
$524$	3.18698e8	0.0967652
$525$	0	0
$526$	9.22859e9	2.76494
$527$	6.36586e9	1.89461
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	7.51458e7	0.0219250
$531$	0	0
$532$	−4.43018e9	−1.27565
$533$	−2.22204e9	−0.635633
$534$	0	0
$535$	−1.52231e8	−0.0429800
$536$	3.64105e9	1.02129
$537$	0	0
$538$	3.65087e9	1.01078
$539$	6.87739e8	0.189175
$540$	0	0
$541$	2.44725e9	0.664490	0.332245	−	0.943193i	$-0.392194\pi$
$541$	2.44725e9	0.664490	0.332245	+	0.943193i	$0.392194\pi$
$542$	−4.11553e9	−1.11027
$543$	0	0
$544$	7.61007e9	2.02671
$545$	6.59057e8	0.174396
$546$	0	0
$547$	3.76239e9	0.982897	0.491449	−	0.870907i	$-0.336468\pi$
$547$	3.76239e9	0.982897	0.491449	+	0.870907i	$0.336468\pi$
$548$	−1.62493e9	−0.421796
$549$	0	0
$550$	−1.29173e9	−0.331057
$551$	−1.71669e9	−0.437181
$552$	0	0
$553$	8.17137e9	2.05474
$554$	−6.41242e9	−1.60228
$555$	0	0
$556$	−1.90040e9	−0.468902
$557$	−6.50679e9	−1.59541	−0.797707	−	0.603045i	$-0.793954\pi$
$557$	−6.50679e9	−1.59541	−0.797707	+	0.603045i	$0.793954\pi$
$558$	0	0
$559$	−9.24084e9	−2.23754
$560$	1.80548e8	0.0434445
$561$	0	0
$562$	−9.96826e9	−2.36888
$563$	−2.23762e9	−0.528453	−0.264226	−	0.964461i	$-0.585117\pi$
$563$	−2.23762e9	−0.528453	−0.264226	+	0.964461i	$0.585117\pi$
$564$	0	0
$565$	−2.45355e8	−0.0572301
$566$	−4.90129e9	−1.13620
$567$	0	0
$568$	8.15050e8	0.186623
$569$	−3.55887e9	−0.809877	−0.404939	−	0.914344i	$-0.632707\pi$
$569$	−3.55887e9	−0.809877	−0.404939	+	0.914344i	$0.632707\pi$
$570$	0	0
$571$	−2.69760e9	−0.606389	−0.303194	−	0.952929i	$-0.598053\pi$
$571$	−2.69760e9	−0.606389	−0.303194	+	0.952929i	$0.598053\pi$
$572$	1.80917e9	0.404197
$573$	0	0
$574$	4.86688e9	1.07414
$575$	9.38822e8	0.205942
$576$	0	0
$577$	−2.73131e9	−0.591910	−0.295955	−	0.955202i	$-0.595638\pi$
$577$	−2.73131e9	−0.591910	−0.295955	+	0.955202i	$0.595638\pi$
$578$	1.32503e10	2.85416
$579$	0	0
$580$	−5.40327e8	−0.114990
$581$	4.01352e9	0.849002
$582$	0	0
$583$	1.27529e8	0.0266544
$584$	2.95466e9	0.613850
$585$	0	0
$586$	−3.59447e9	−0.737892
$587$	4.25360e9	0.868008	0.434004	−	0.900911i	$-0.357100\pi$
$587$	4.25360e9	0.868008	0.434004	+	0.900911i	$0.357100\pi$
$588$	0	0
$589$	−3.50780e9	−0.707345
$590$	1.25116e9	0.250802
$591$	0	0
$592$	5.23528e8	0.103708
$593$	−8.85467e9	−1.74374	−0.871869	−	0.489740i	$-0.837092\pi$
$593$	−8.85467e9	−1.74374	−0.871869	+	0.489740i	$0.837092\pi$
$594$	0	0
$595$	1.31518e9	0.255962
$596$	−3.38257e9	−0.654462
$597$	0	0
$598$	−2.20181e9	−0.421042
$599$	8.77964e9	1.66910	0.834551	−	0.550930i	$-0.185727\pi$
$599$	8.77964e9	1.66910	0.834551	+	0.550930i	$0.185727\pi$
$600$	0	0
$601$	4.36354e9	0.819933	0.409967	−	0.912101i	$-0.365540\pi$
$601$	4.36354e9	0.819933	0.409967	+	0.912101i	$0.365540\pi$
$602$	2.02400e10	3.78114
$603$	0	0
$604$	−2.36659e9	−0.437013
$605$	−5.77558e8	−0.106036
$606$	0	0
$607$	−1.00264e10	−1.81963	−0.909815	−	0.415015i	$-0.863776\pi$
$607$	−1.00264e10	−1.81963	−0.909815	+	0.415015i	$0.863776\pi$
$608$	−4.19340e9	−0.756666
$609$	0	0
$610$	6.89497e7	0.0122992
$611$	−5.48621e8	−0.0973034
$612$	0	0
$613$	−1.06422e10	−1.86603	−0.933013	−	0.359842i	$-0.882830\pi$
$613$	−1.06422e10	−1.86603	−0.933013	+	0.359842i	$0.882830\pi$
$614$	−7.81765e9	−1.36297
$615$	0	0
$616$	−1.28977e9	−0.222321
$617$	1.31952e9	0.226161	0.113080	−	0.993586i	$-0.463928\pi$
$617$	1.31952e9	0.226161	0.113080	+	0.993586i	$0.463928\pi$
$618$	0	0
$619$	8.51725e9	1.44339	0.721693	−	0.692214i	$-0.243364\pi$
$619$	8.51725e9	1.44339	0.721693	+	0.692214i	$0.243364\pi$
$620$	−1.10408e9	−0.186050
$621$	0	0
$622$	−1.07923e10	−1.79823
$623$	−1.42481e10	−2.36074
$624$	0	0
$625$	5.87859e9	0.963148
$626$	−1.90832e8	−0.0310914
$627$	0	0
$628$	−5.19368e9	−0.836791
$629$	3.81357e9	0.611019
$630$	0	0
$631$	1.32684e9	0.210240	0.105120	−	0.994460i	$-0.466477\pi$
$631$	1.32684e9	0.210240	0.105120	+	0.994460i	$0.466477\pi$
$632$	−7.21299e9	−1.13659
$633$	0	0
$634$	−1.78209e10	−2.77726
$635$	1.15792e9	0.179462
$636$	0	0
$637$	7.43441e9	1.13962
$638$	−1.53550e9	−0.234087
$639$	0	0
$640$	−9.89600e8	−0.149221
$641$	−3.00273e9	−0.450312	−0.225156	−	0.974323i	$-0.572289\pi$
$641$	−3.00273e9	−0.450312	−0.225156	+	0.974323i	$0.572289\pi$
$642$	0	0
$643$	−1.08587e10	−1.61080	−0.805398	−	0.592734i	$-0.798049\pi$
$643$	−1.08587e10	−1.61080	−0.805398	+	0.592734i	$0.798049\pi$
$644$	2.87998e9	0.424902
$645$	0	0
$646$	−1.13320e10	−1.65384
$647$	8.53415e9	1.23878	0.619392	−	0.785082i	$-0.287379\pi$
$647$	8.53415e9	1.23878	0.619392	+	0.785082i	$0.287379\pi$
$648$	0	0
$649$	2.12333e9	0.304902
$650$	−1.39635e10	−1.99434
$651$	0	0
$652$	1.58022e10	2.23281
$653$	2.44692e9	0.343893	0.171947	−	0.985106i	$-0.444994\pi$
$653$	2.44692e9	0.343893	0.171947	+	0.985106i	$0.444994\pi$
$654$	0	0
$655$	5.21338e7	0.00724895
$656$	1.02060e9	0.141154
$657$	0	0
$658$	1.20163e9	0.164430
$659$	−7.66941e9	−1.04391	−0.521955	−	0.852973i	$-0.674797\pi$
$659$	−7.66941e9	−1.04391	−0.521955	+	0.852973i	$0.674797\pi$
$660$	0	0
$661$	1.06623e10	1.43597	0.717987	−	0.696057i	$-0.245064\pi$
$661$	1.06623e10	1.43597	0.717987	+	0.696057i	$0.245064\pi$
$662$	−1.49701e10	−2.00549
$663$	0	0
$664$	−3.54279e9	−0.469632
$665$	−7.24707e8	−0.0955623
$666$	0	0
$667$	1.11599e9	0.145620
$668$	−1.77527e9	−0.230434
$669$	0	0
$670$	1.82993e9	0.235057
$671$	1.17014e8	0.0149523
$672$	0	0
$673$	1.13404e10	1.43408	0.717042	−	0.697030i	$-0.245495\pi$
$673$	1.13404e10	1.43408	0.717042	+	0.697030i	$0.245495\pi$
$674$	−9.86954e9	−1.24162
$675$	0	0
$676$	7.64946e9	0.952396
$677$	2.59963e9	0.321996	0.160998	−	0.986955i	$-0.448529\pi$
$677$	2.59963e9	0.321996	0.160998	+	0.986955i	$0.448529\pi$
$678$	0	0
$679$	1.76191e10	2.15993
$680$	−1.16093e9	−0.141587
$681$	0	0
$682$	−3.13756e9	−0.378745
$683$	−1.25557e10	−1.50789	−0.753943	−	0.656940i	$-0.771851\pi$
$683$	−1.25557e10	−1.50789	−0.753943	+	0.656940i	$0.771851\pi$
$684$	0	0
$685$	−2.65812e8	−0.0315979
$686$	2.02821e9	0.239871
$687$	0	0
$688$	4.24439e9	0.496884
$689$	1.37858e9	0.160570
$690$	0	0
$691$	4.66084e9	0.537392	0.268696	−	0.963225i	$-0.413407\pi$
$691$	4.66084e9	0.537392	0.268696	+	0.963225i	$0.413407\pi$
$692$	1.63706e10	1.87799
$693$	0	0
$694$	−1.43577e10	−1.63052
$695$	−3.10874e8	−0.0351268
$696$	0	0
$697$	7.43443e9	0.831635
$698$	1.12826e10	1.25579
$699$	0	0
$700$	1.82644e10	2.01263
$701$	4.37132e9	0.479292	0.239646	−	0.970860i	$-0.422969\pi$
$701$	4.37132e9	0.479292	0.239646	+	0.970860i	$0.422969\pi$
$702$	0	0
$703$	−2.10140e9	−0.228121
$704$	−3.19030e9	−0.344610
$705$	0	0
$706$	−6.17830e9	−0.660774
$707$	−1.49822e10	−1.59443
$708$	0	0
$709$	−1.06398e10	−1.12117	−0.560585	−	0.828097i	$-0.689424\pi$
$709$	−1.06398e10	−1.12117	−0.560585	+	0.828097i	$0.689424\pi$
$710$	4.09630e8	0.0429524
$711$	0	0
$712$	1.25770e10	1.30586
$713$	2.28036e9	0.235608
$714$	0	0
$715$	2.95951e8	0.0302795
$716$	1.83025e10	1.86344
$717$	0	0
$718$	3.82892e9	0.386048
$719$	1.04952e10	1.05302	0.526512	−	0.850168i	$-0.323499\pi$
$719$	1.04952e10	1.05302	0.526512	+	0.850168i	$0.323499\pi$
$720$	0	0
$721$	9.66073e9	0.959923
$722$	−9.68985e9	−0.958158
$723$	0	0
$724$	−4.98965e9	−0.488635
$725$	7.07744e9	0.689753
$726$	0	0
$727$	1.98822e9	0.191908	0.0959541	−	0.995386i	$-0.469410\pi$
$727$	1.98822e9	0.191908	0.0959541	+	0.995386i	$0.469410\pi$
$728$	−1.39423e10	−1.33929
$729$	0	0
$730$	1.48496e9	0.141281
$731$	3.09177e10	2.92750
$732$	0	0
$733$	2.85161e9	0.267440	0.133720	−	0.991019i	$-0.457308\pi$
$733$	2.85161e9	0.267440	0.133720	+	0.991019i	$0.457308\pi$
$734$	3.05650e10	2.85291
$735$	0	0
$736$	2.72606e9	0.252036
$737$	3.10555e9	0.285760
$738$	0	0
$739$	−1.35553e10	−1.23553	−0.617766	−	0.786362i	$-0.711962\pi$
$739$	−1.35553e10	−1.23553	−0.617766	+	0.786362i	$0.711962\pi$
$740$	−6.61416e8	−0.0600017
$741$	0	0
$742$	−3.01947e9	−0.271342
$743$	6.48538e9	0.580062	0.290031	−	0.957017i	$-0.406334\pi$
$743$	6.48538e9	0.580062	0.290031	+	0.957017i	$0.406334\pi$
$744$	0	0
$745$	−5.53334e8	−0.0490276
$746$	−1.54134e10	−1.35929
$747$	0	0
$748$	−6.05307e9	−0.528835
$749$	6.11689e9	0.531917
$750$	0	0
$751$	8.10260e9	0.698047	0.349024	−	0.937114i	$-0.386513\pi$
$751$	8.10260e9	0.698047	0.349024	+	0.937114i	$0.386513\pi$
$752$	2.51986e8	0.0216079
$753$	0	0
$754$	−1.65986e10	−1.41017
$755$	−3.87137e8	−0.0327379
$756$	0	0
$757$	3.72306e9	0.311935	0.155968	−	0.987762i	$-0.450150\pi$
$757$	3.72306e9	0.311935	0.155968	+	0.987762i	$0.450150\pi$
$758$	−9.15432e8	−0.0763456
$759$	0	0
$760$	6.39709e8	0.0528610
$761$	−1.48338e10	−1.22013	−0.610065	−	0.792351i	$-0.708857\pi$
$761$	−1.48338e10	−1.22013	−0.610065	+	0.792351i	$0.708857\pi$
$762$	0	0
$763$	−2.64819e10	−2.15831
$764$	−1.02850e10	−0.834408
$765$	0	0
$766$	−3.59999e10	−2.89402
$767$	2.29530e10	1.83678
$768$	0	0
$769$	9.76344e9	0.774213	0.387107	−	0.922035i	$-0.373475\pi$
$769$	9.76344e9	0.774213	0.387107	+	0.922035i	$0.373475\pi$
$770$	−6.48216e8	−0.0511684
$771$	0	0
$772$	−2.93071e10	−2.29252
$773$	−1.51981e10	−1.18348	−0.591740	−	0.806129i	$-0.701559\pi$
$773$	−1.51981e10	−1.18348	−0.591740	+	0.806129i	$0.701559\pi$
$774$	0	0
$775$	1.44617e10	1.11600
$776$	−1.55526e10	−1.19478
$777$	0	0
$778$	−2.49328e10	−1.89820
$779$	−4.09662e9	−0.310488
$780$	0	0
$781$	6.95177e8	0.0522176
$782$	7.36674e9	0.550873
$783$	0	0
$784$	−3.41468e9	−0.253072
$785$	−8.49604e8	−0.0626863
$786$	0	0
$787$	−5.97907e8	−0.0437242	−0.0218621	−	0.999761i	$-0.506959\pi$
$787$	−5.97907e8	−0.0437242	−0.0218621	+	0.999761i	$0.506959\pi$
$788$	−2.22385e10	−1.61907
$789$	0	0
$790$	−3.62512e9	−0.261594
$791$	9.85872e9	0.708276
$792$	0	0
$793$	1.26491e9	0.0900748
$794$	−3.49829e10	−2.48018
$795$	0	0
$796$	−1.74026e10	−1.22298
$797$	6.05571e9	0.423703	0.211851	−	0.977302i	$-0.432051\pi$
$797$	6.05571e9	0.423703	0.211851	+	0.977302i	$0.432051\pi$
$798$	0	0
$799$	1.83556e9	0.127308
$800$	1.72883e10	1.19381
$801$	0	0
$802$	4.04077e10	2.76601
$803$	2.52010e9	0.171757
$804$	0	0
$805$	4.71119e8	0.0318306
$806$	−3.39168e10	−2.28162
$807$	0	0
$808$	1.32250e10	0.881972
$809$	7.82042e9	0.519290	0.259645	−	0.965704i	$-0.416394\pi$
$809$	7.82042e9	0.519290	0.259645	+	0.965704i	$0.416394\pi$
$810$	0	0
$811$	2.15298e10	1.41731	0.708657	−	0.705553i	$-0.249301\pi$
$811$	2.15298e10	1.41731	0.708657	+	0.705553i	$0.249301\pi$
$812$	2.17112e10	1.42310
$813$	0	0
$814$	−1.87961e9	−0.122147
$815$	2.58500e9	0.167266
$816$	0	0
$817$	−1.70367e10	−1.09297
$818$	4.06707e10	2.59803
$819$	0	0
$820$	−1.28941e9	−0.0816661
$821$	2.41869e10	1.52539	0.762693	−	0.646761i	$-0.223877\pi$
$821$	2.41869e10	1.52539	0.762693	+	0.646761i	$0.223877\pi$
$822$	0	0
$823$	−3.08772e8	−0.0193081	−0.00965403	−	0.999953i	$-0.503073\pi$
$823$	−3.08772e8	−0.0193081	−0.00965403	+	0.999953i	$0.503073\pi$
$824$	−8.52767e9	−0.530988
$825$	0	0
$826$	−5.02735e10	−3.10391
$827$	1.13854e10	0.699967	0.349984	−	0.936756i	$-0.386187\pi$
$827$	1.13854e10	0.699967	0.349984	+	0.936756i	$0.386187\pi$
$828$	0	0
$829$	−5.67599e9	−0.346019	−0.173010	−	0.984920i	$-0.555349\pi$
$829$	−5.67599e9	−0.346019	−0.173010	+	0.984920i	$0.555349\pi$
$830$	−1.78054e9	−0.108089
$831$	0	0
$832$	−3.44870e10	−2.07598
$833$	−2.48738e10	−1.49103
$834$	0	0
$835$	−2.90405e8	−0.0172624
$836$	3.33544e9	0.197438
$837$	0	0
$838$	−1.24647e9	−0.0731689
$839$	−2.26083e10	−1.32160	−0.660801	−	0.750561i	$-0.729783\pi$
$839$	−2.26083e10	−1.32160	−0.660801	+	0.750561i	$0.729783\pi$
$840$	0	0
$841$	−8.83683e9	−0.512284
$842$	−1.93490e10	−1.11703
$843$	0	0
$844$	6.13147e9	0.351047
$845$	1.25133e9	0.0713466
$846$	0	0
$847$	2.32072e10	1.31229
$848$	−6.33193e8	−0.0356575
$849$	0	0
$850$	4.67188e10	2.60931
$851$	1.36609e9	0.0759844
$852$	0	0
$853$	−3.13840e10	−1.73136	−0.865679	−	0.500599i	$-0.833113\pi$
$853$	−3.13840e10	−1.73136	−0.865679	+	0.500599i	$0.833113\pi$
$854$	−2.77050e9	−0.152214
$855$	0	0
$856$	−5.39947e9	−0.294234
$857$	−6.44481e9	−0.349766	−0.174883	−	0.984589i	$-0.555955\pi$
$857$	−6.44481e9	−0.349766	−0.174883	+	0.984589i	$0.555955\pi$
$858$	0	0
$859$	−2.36238e10	−1.27167	−0.635834	−	0.771826i	$-0.719344\pi$
$859$	−2.36238e10	−1.27167	−0.635834	+	0.771826i	$0.719344\pi$
$860$	−5.36228e9	−0.287478
$861$	0	0
$862$	5.64856e10	3.00374
$863$	1.31698e10	0.697496	0.348748	−	0.937217i	$-0.386607\pi$
$863$	1.31698e10	0.697496	0.348748	+	0.937217i	$0.386607\pi$
$864$	0	0
$865$	2.67797e9	0.140685
$866$	5.81211e9	0.304103
$867$	0	0
$868$	4.43635e10	2.30254
$869$	−6.15215e9	−0.318022
$870$	0	0
$871$	3.35708e10	1.72146
$872$	2.33760e10	1.19388
$873$	0	0
$874$	−4.05932e9	−0.205666
$875$	6.01285e9	0.303426
$876$	0	0
$877$	−1.72997e10	−0.866046	−0.433023	−	0.901383i	$-0.642553\pi$
$877$	−1.72997e10	−0.866046	−0.433023	+	0.901383i	$0.642553\pi$
$878$	−4.69109e10	−2.33907
$879$	0	0
$880$	−1.35933e8	−0.00672411
$881$	−2.51040e10	−1.23688	−0.618439	−	0.785832i	$-0.712235\pi$
$881$	−2.51040e10	−1.23688	−0.618439	+	0.785832i	$0.712235\pi$
$882$	0	0
$883$	2.96678e10	1.45018	0.725091	−	0.688653i	$-0.241797\pi$
$883$	2.96678e10	1.45018	0.725091	+	0.688653i	$0.241797\pi$
$884$	−6.54333e10	−3.18578
$885$	0	0
$886$	5.52881e9	0.267063
$887$	−1.46350e10	−0.704140	−0.352070	−	0.935974i	$-0.614522\pi$
$887$	−1.46350e10	−0.704140	−0.352070	+	0.935974i	$0.614522\pi$
$888$	0	0
$889$	−4.65272e10	−2.22101
$890$	6.32097e9	0.300551
$891$	0	0
$892$	2.37726e8	0.0112150
$893$	−1.01145e9	−0.0475298
$894$	0	0
$895$	2.99400e9	0.139595
$896$	3.97636e10	1.84675
$897$	0	0
$898$	3.53088e10	1.62711
$899$	1.71908e10	0.789110
$900$	0	0
$901$	−4.61241e9	−0.210083
$902$	−3.66423e9	−0.166249
$903$	0	0
$904$	−8.70244e9	−0.391788
$905$	−8.16227e8	−0.0366050
$906$	0	0
$907$	1.17379e10	0.522353	0.261177	−	0.965291i	$-0.415889\pi$
$907$	1.17379e10	0.522353	0.261177	+	0.965291i	$0.415889\pi$
$908$	2.69083e10	1.19285
$909$	0	0
$910$	−7.00717e9	−0.308246
$911$	3.54807e9	0.155481	0.0777405	−	0.996974i	$-0.475229\pi$
$911$	3.54807e9	0.155481	0.0777405	+	0.996974i	$0.475229\pi$
$912$	0	0
$913$	−3.02174e9	−0.131404
$914$	−2.80244e10	−1.21401
$915$	0	0
$916$	−1.43304e10	−0.616060
$917$	−2.09482e9	−0.0897126
$918$	0	0
$919$	3.70231e10	1.57350	0.786752	−	0.617269i	$-0.211761\pi$
$919$	3.70231e10	1.57350	0.786752	+	0.617269i	$0.211761\pi$
$920$	−4.15864e8	−0.0176073
$921$	0	0
$922$	−4.99658e10	−2.09949
$923$	7.51482e9	0.314567
$924$	0	0
$925$	8.66352e9	0.359914
$926$	5.87592e10	2.43185
$927$	0	0
$928$	2.05508e10	0.844131
$929$	−2.67577e10	−1.09495	−0.547475	−	0.836822i	$-0.684411\pi$
$929$	−2.67577e10	−1.09495	−0.547475	+	0.836822i	$0.684411\pi$
$930$	0	0
$931$	1.37063e10	0.556668
$932$	5.05405e10	2.04495
$933$	0	0
$934$	3.95914e10	1.58996
$935$	−9.90185e8	−0.0396165
$936$	0	0
$937$	−3.93013e9	−0.156069	−0.0780347	−	0.996951i	$-0.524865\pi$
$937$	−3.93013e9	−0.156069	−0.0780347	+	0.996951i	$0.524865\pi$
$938$	−7.35293e10	−2.90904
$939$	0	0
$940$	−3.18354e8	−0.0125015
$941$	2.54150e10	0.994320	0.497160	−	0.867659i	$-0.334376\pi$
$941$	2.54150e10	0.994320	0.497160	+	0.867659i	$0.334376\pi$
$942$	0	0
$943$	2.66314e9	0.103420
$944$	−1.05425e10	−0.407889
$945$	0	0
$946$	−1.52385e10	−0.585225
$947$	2.76072e10	1.05632	0.528161	−	0.849144i	$-0.322882\pi$
$947$	2.76072e10	1.05632	0.528161	+	0.849144i	$0.322882\pi$
$948$	0	0
$949$	2.72422e10	1.03469
$950$	−2.57436e10	−0.974174
$951$	0	0
$952$	4.66478e10	1.75227
$953$	2.15615e10	0.806962	0.403481	−	0.914988i	$-0.367800\pi$
$953$	2.15615e10	0.806962	0.403481	+	0.914988i	$0.367800\pi$
$954$	0	0
$955$	−1.68246e9	−0.0625078
$956$	6.60139e10	2.44361
$957$	0	0
$958$	−7.28871e10	−2.67837
$959$	1.06807e10	0.391053
$960$	0	0
$961$	7.61423e9	0.276754
$962$	−2.03184e10	−0.735830
$963$	0	0
$964$	−2.54916e10	−0.916489
$965$	−4.79418e9	−0.171739
$966$	0	0
$967$	8.89528e9	0.316349	0.158175	−	0.987411i	$-0.449439\pi$
$967$	8.89528e9	0.316349	0.158175	+	0.987411i	$0.449439\pi$
$968$	−2.04853e10	−0.725903
$969$	0	0
$970$	−7.81649e9	−0.274986
$971$	9.39030e9	0.329164	0.164582	−	0.986363i	$-0.447372\pi$
$971$	9.39030e9	0.329164	0.164582	+	0.986363i	$0.447372\pi$
$972$	0	0
$973$	1.24914e10	0.434727
$974$	−3.93092e10	−1.36313
$975$	0	0
$976$	−5.80983e8	−0.0200027
$977$	−4.04777e10	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$977$	−4.04777e10	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$978$	0	0
$979$	1.07272e10	0.365383
$980$	4.31405e9	0.146418
$981$	0	0
$982$	4.36666e10	1.47150
$983$	−2.17203e10	−0.729336	−0.364668	−	0.931138i	$-0.618818\pi$
$983$	−2.17203e10	−0.729336	−0.364668	+	0.931138i	$0.618818\pi$
$984$	0	0
$985$	−3.63787e9	−0.121289
$986$	5.55352e10	1.84501
$987$	0	0
$988$	3.60559e10	1.18940
$989$	1.10752e10	0.364054
$990$	0	0
$991$	4.02205e10	1.31277	0.656387	−	0.754425i	$-0.272084\pi$
$991$	4.02205e10	1.31277	0.656387	+	0.754425i	$0.272084\pi$
$992$	4.19924e10	1.36578
$993$	0	0
$994$	−1.64595e10	−0.531576
$995$	−2.84680e9	−0.0916169
$996$	0	0
$997$	1.02432e10	0.327342	0.163671	−	0.986515i	$-0.447666\pi$
$997$	1.02432e10	0.327342	0.163671	+	0.986515i	$0.447666\pi$
$998$	−7.29287e10	−2.32242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.a.1.5		5
3.2	odd	2		69.8.a.a.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.1	✓	5	3.2	odd	2
207.8.a.a.1.5		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +O(q^{10})\)	\(q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +553.369 q^{10} +939.114 q^{11} +10151.8 q^{13} -22235.2 q^{14} -4662.78 q^{16} -33965.5 q^{17} +18716.1 q^{19} +5890.88 q^{20} +16740.7 q^{22} -12167.0 q^{23} -77161.3 q^{25} +180965. q^{26} -236704. q^{28} -91722.7 q^{29} -187422. q^{31} -224053. q^{32} -605469. q^{34} -38721.0 q^{35} -112278. q^{37} +333633. q^{38} +34179.6 q^{40} -218882. q^{41} -910269. q^{43} +178212. q^{44} -216889. q^{46} -54041.9 q^{47} +732327. q^{49} -1.37548e6 q^{50} +1.92647e6 q^{52} +135797. q^{53} +29152.7 q^{55} -1.37339e6 q^{56} -1.63505e6 q^{58} +2.26099e6 q^{59} +124600. q^{61} -3.34098e6 q^{62} -3.39714e6 q^{64} +315139. q^{65} +3.30689e6 q^{67} -6.44551e6 q^{68} -690242. q^{70} +740248. q^{71} +2.68349e6 q^{73} -2.00147e6 q^{74} +3.55169e6 q^{76} -1.17140e6 q^{77} -6.55101e6 q^{79} -144746. q^{80} -3.90179e6 q^{82} -3.21765e6 q^{83} -1.05438e6 q^{85} -1.62265e7 q^{86} +1.03401e6 q^{88} +1.14227e7 q^{89} -1.26628e7 q^{91} -2.30889e6 q^{92} -963351. q^{94} +580999. q^{95} -1.41253e7 q^{97} +1.30545e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10}) \) 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8	\( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100}) \) 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8 + 1452 * q^10 + 1148 * q^11 - 642 * q^13 - 5756 * q^14 - 22606 * q^16 + 5798 * q^17 - 6036 * q^19 + 27376 * q^20 - 97896 * q^22 - 60835 * q^23 - 262477 * q^25 + 355992 * q^26 - 507124 * q^28 + 169162 * q^29 - 199640 * q^31 + 284794 * q^32 - 1027740 * q^34 + 137680 * q^35 - 202002 * q^37 + 554924 * q^38 - 340904 * q^40 - 541282 * q^41 - 909596 * q^43 + 1236032 * q^44 - 80208 * q^47 + 850589 * q^49 + 941416 * q^50 + 146940 * q^52 + 278138 * q^53 - 933560 * q^55 + 539932 * q^56 - 3522712 * q^58 + 3177380 * q^59 + 147782 * q^61 - 4606456 * q^62 - 4142622 * q^64 - 3877332 * q^65 - 464916 * q^67 - 7513072 * q^68 + 2093200 * q^70 - 1576792 * q^71 - 38190 * q^73 - 12164864 * q^74 + 6889436 * q^76 - 10332384 * q^77 - 3913336 * q^79 - 6334776 * q^80 + 6799360 * q^82 - 15774716 * q^83 - 8520740 * q^85 - 24874084 * q^86 + 53216 * q^88 - 1116482 * q^89 - 27369552 * q^91 - 3285090 * q^92 - 7153744 * q^94 + 6067832 * q^95 - 15738566 * q^97 - 11730488 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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