LMFDB - Embedded newform 862.2.a.k.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [862,2,Mod(1,862)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(862, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("862.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$862 = 2 \cdot 431$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	862.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-10,4,10,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.88310465423$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 2x^{9} - 19x^{8} + 46x^{7} + 89x^{6} - 291x^{5} - 10x^{4} + 543x^{3} - 429x^{2} + 64x + 4$ x^10 - 2x^9 - 19x^8 + 46x^7 + 89x^6 - 291x^5 - 10x^4 + 543x^3 - 429x^2 + 64*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.22143$ of defining polynomial
Character	$\chi$	$=$	862.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.12057 q^{3} +1.00000 q^{4} -3.20857 q^{5} -3.12057 q^{6} -3.56348 q^{7} -1.00000 q^{8} +6.73793 q^{9} +3.20857 q^{10} +2.87639 q^{11} +3.12057 q^{12} +5.53503 q^{13} +3.56348 q^{14} -10.0126 q^{15} +1.00000 q^{16} +3.66567 q^{17} -6.73793 q^{18} -1.83231 q^{19} -3.20857 q^{20} -11.1201 q^{21} -2.87639 q^{22} +3.69456 q^{23} -3.12057 q^{24} +5.29492 q^{25} -5.53503 q^{26} +11.6645 q^{27} -3.56348 q^{28} +9.15934 q^{29} +10.0126 q^{30} -7.66238 q^{31} -1.00000 q^{32} +8.97596 q^{33} -3.66567 q^{34} +11.4337 q^{35} +6.73793 q^{36} +8.38816 q^{37} +1.83231 q^{38} +17.2724 q^{39} +3.20857 q^{40} +11.7570 q^{41} +11.1201 q^{42} -9.09492 q^{43} +2.87639 q^{44} -21.6191 q^{45} -3.69456 q^{46} -9.19678 q^{47} +3.12057 q^{48} +5.69839 q^{49} -5.29492 q^{50} +11.4390 q^{51} +5.53503 q^{52} +3.97970 q^{53} -11.6645 q^{54} -9.22909 q^{55} +3.56348 q^{56} -5.71783 q^{57} -9.15934 q^{58} +13.1414 q^{59} -10.0126 q^{60} -14.0376 q^{61} +7.66238 q^{62} -24.0105 q^{63} +1.00000 q^{64} -17.7595 q^{65} -8.97596 q^{66} +0.398780 q^{67} +3.66567 q^{68} +11.5291 q^{69} -11.4337 q^{70} -7.48303 q^{71} -6.73793 q^{72} +10.5899 q^{73} -8.38816 q^{74} +16.5231 q^{75} -1.83231 q^{76} -10.2500 q^{77} -17.2724 q^{78} -3.37728 q^{79} -3.20857 q^{80} +16.1859 q^{81} -11.7570 q^{82} +0.326978 q^{83} -11.1201 q^{84} -11.7616 q^{85} +9.09492 q^{86} +28.5823 q^{87} -2.87639 q^{88} -12.7173 q^{89} +21.6191 q^{90} -19.7240 q^{91} +3.69456 q^{92} -23.9110 q^{93} +9.19678 q^{94} +5.87908 q^{95} -3.12057 q^{96} -0.300716 q^{97} -5.69839 q^{98} +19.3809 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 10 q^{2} + 4 q^{3} + 10 q^{4} + 4 q^{5} - 4 q^{6} - 3 q^{7} - 10 q^{8} + 12 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} + 5 q^{13} + 3 q^{14} - q^{15} + 10 q^{16} + 27 q^{17} - 12 q^{18} - 9 q^{19} + 4 q^{20}+ \cdots + 20 q^{99}+O(q^{100})$ 10 * q - 10 * q^2 + 4 * q^3 + 10 * q^4 + 4 * q^5 - 4 * q^6 - 3 * q^7 - 10 * q^8 + 12 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 + 5 * q^13 + 3 * q^14 - q^15 + 10 * q^16 + 27 * q^17 - 12 * q^18 - 9 * q^19 + 4 * q^20 + 8 * q^21 - 4 * q^22 + 18 * q^23 - 4 * q^24 + 16 * q^25 - 5 * q^26 + 25 * q^27 - 3 * q^28 + 19 * q^29 + q^30 - 20 * q^31 - 10 * q^32 + 25 * q^33 - 27 * q^34 + 22 * q^35 + 12 * q^36 - 3 * q^37 + 9 * q^38 + q^39 - 4 * q^40 + 33 * q^41 - 8 * q^42 - q^43 + 4 * q^44 - 8 * q^45 - 18 * q^46 + 18 * q^47 + 4 * q^48 + 5 * q^49 - 16 * q^50 + 6 * q^51 + 5 * q^52 + 33 * q^53 - 25 * q^54 - 7 * q^55 + 3 * q^56 + 11 * q^57 - 19 * q^58 + 13 * q^59 - q^60 - 2 * q^61 + 20 * q^62 + 6 * q^63 + 10 * q^64 + 28 * q^65 - 25 * q^66 - 2 * q^67 + 27 * q^68 - 2 * q^69 - 22 * q^70 - 6 * q^71 - 12 * q^72 + 19 * q^73 + 3 * q^74 + 15 * q^75 - 9 * q^76 + 39 * q^77 - q^78 - 4 * q^79 + 4 * q^80 + 22 * q^81 - 33 * q^82 + 46 * q^83 + 8 * q^84 - 12 * q^85 + q^86 + 43 * q^87 - 4 * q^88 + 29 * q^89 + 8 * q^90 - 32 * q^91 + 18 * q^92 + 3 * q^93 - 18 * q^94 + 38 * q^95 - 4 * q^96 + 9 * q^97 - 5 * q^98 + 20 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.12057	1.80166	0.900830	−	0.434172i	$-0.142959\pi$
$3$	3.12057	1.80166	0.900830	+	0.434172i	$0.142959\pi$
$4$	1.00000	0.500000
$5$	−3.20857	−1.43492	−0.717458	−	0.696602i	$-0.754694\pi$
$5$	−3.20857	−1.43492	−0.717458	+	0.696602i	$0.754694\pi$
$6$	−3.12057	−1.27397
$7$	−3.56348	−1.34687	−0.673434	−	0.739247i	$-0.735182\pi$
$7$	−3.56348	−1.34687	−0.673434	+	0.739247i	$0.735182\pi$
$8$	−1.00000	−0.353553
$9$	6.73793	2.24598
$10$	3.20857	1.01464
$11$	2.87639	0.867264	0.433632	−	0.901090i	$-0.357232\pi$
$11$	2.87639	0.867264	0.433632	+	0.901090i	$0.357232\pi$
$12$	3.12057	0.900830
$13$	5.53503	1.53514	0.767571	−	0.640964i	$-0.221465\pi$
$13$	5.53503	1.53514	0.767571	+	0.640964i	$0.221465\pi$
$14$	3.56348	0.952380
$15$	−10.0126	−2.58523
$16$	1.00000	0.250000
$17$	3.66567	0.889056	0.444528	−	0.895765i	$-0.353371\pi$
$17$	3.66567	0.889056	0.444528	+	0.895765i	$0.353371\pi$
$18$	−6.73793	−1.58815
$19$	−1.83231	−0.420360	−0.210180	−	0.977663i	$-0.567405\pi$
$19$	−1.83231	−0.420360	−0.210180	+	0.977663i	$0.567405\pi$
$20$	−3.20857	−0.717458
$21$	−11.1201	−2.42660
$22$	−2.87639	−0.613248
$23$	3.69456	0.770369	0.385185	−	0.922839i	$-0.374138\pi$
$23$	3.69456	0.770369	0.385185	+	0.922839i	$0.374138\pi$
$24$	−3.12057	−0.636983
$25$	5.29492	1.05898
$26$	−5.53503	−1.08551
$27$	11.6645	2.24483
$28$	−3.56348	−0.673434
$29$	9.15934	1.70085	0.850424	−	0.526099i	$-0.176346\pi$
$29$	9.15934	1.70085	0.850424	+	0.526099i	$0.176346\pi$
$30$	10.0126	1.82803
$31$	−7.66238	−1.37620	−0.688102	−	0.725614i	$-0.741556\pi$
$31$	−7.66238	−1.37620	−0.688102	+	0.725614i	$0.741556\pi$
$32$	−1.00000	−0.176777
$33$	8.97596	1.56251
$34$	−3.66567	−0.628657
$35$	11.4337	1.93264
$36$	6.73793	1.12299
$37$	8.38816	1.37901	0.689503	−	0.724283i	$-0.257829\pi$
$37$	8.38816	1.37901	0.689503	+	0.724283i	$0.257829\pi$
$38$	1.83231	0.297239
$39$	17.2724	2.76580
$40$	3.20857	0.507319
$41$	11.7570	1.83614	0.918068	−	0.396422i	$-0.129748\pi$
$41$	11.7570	1.83614	0.918068	+	0.396422i	$0.129748\pi$
$42$	11.1201	1.71586
$43$	−9.09492	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$43$	−9.09492	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$44$	2.87639	0.433632
$45$	−21.6191	−3.22279
$46$	−3.69456	−0.544733
$47$	−9.19678	−1.34149	−0.670744	−	0.741689i	$-0.734025\pi$
$47$	−9.19678	−1.34149	−0.670744	+	0.741689i	$0.734025\pi$
$48$	3.12057	0.450415
$49$	5.69839	0.814055
$50$	−5.29492	−0.748814
$51$	11.4390	1.60178
$52$	5.53503	0.767571
$53$	3.97970	0.546654	0.273327	−	0.961921i	$-0.411876\pi$
$53$	3.97970	0.546654	0.273327	+	0.961921i	$0.411876\pi$
$54$	−11.6645	−1.58733
$55$	−9.22909	−1.24445
$56$	3.56348	0.476190
$57$	−5.71783	−0.757345
$58$	−9.15934	−1.20268
$59$	13.1414	1.71087	0.855433	−	0.517914i	$-0.173291\pi$
$59$	13.1414	1.71087	0.855433	+	0.517914i	$0.173291\pi$
$60$	−10.0126	−1.29261
$61$	−14.0376	−1.79733	−0.898664	−	0.438638i	$-0.855461\pi$
$61$	−14.0376	−1.79733	−0.898664	+	0.438638i	$0.855461\pi$
$62$	7.66238	0.973123
$63$	−24.0105	−3.02504
$64$	1.00000	0.125000
$65$	−17.7595	−2.20280
$66$	−8.97596	−1.10486
$67$	0.398780	0.0487188	0.0243594	−	0.999703i	$-0.492245\pi$
$67$	0.398780	0.0487188	0.0243594	+	0.999703i	$0.492245\pi$
$68$	3.66567	0.444528
$69$	11.5291	1.38794
$70$	−11.4337	−1.36659
$71$	−7.48303	−0.888072	−0.444036	−	0.896009i	$-0.646454\pi$
$71$	−7.48303	−0.888072	−0.444036	+	0.896009i	$0.646454\pi$
$72$	−6.73793	−0.794073
$73$	10.5899	1.23946	0.619728	−	0.784817i	$-0.287243\pi$
$73$	10.5899	1.23946	0.619728	+	0.784817i	$0.287243\pi$
$74$	−8.38816	−0.975104
$75$	16.5231	1.90793
$76$	−1.83231	−0.210180
$77$	−10.2500	−1.16809
$78$	−17.2724	−1.95572
$79$	−3.37728	−0.379974	−0.189987	−	0.981787i	$-0.560845\pi$
$79$	−3.37728	−0.379974	−0.189987	+	0.981787i	$0.560845\pi$
$80$	−3.20857	−0.358729
$81$	16.1859	1.79844
$82$	−11.7570	−1.29834
$83$	0.326978	0.0358905	0.0179452	−	0.999839i	$-0.494288\pi$
$83$	0.326978	0.0358905	0.0179452	+	0.999839i	$0.494288\pi$
$84$	−11.1201	−1.21330
$85$	−11.7616	−1.27572
$86$	9.09492	0.980731
$87$	28.5823	3.06435
$88$	−2.87639	−0.306624
$89$	−12.7173	−1.34803	−0.674017	−	0.738716i	$-0.735433\pi$
$89$	−12.7173	−1.34803	−0.674017	+	0.738716i	$0.735433\pi$
$90$	21.6191	2.27886
$91$	−19.7240	−2.06764
$92$	3.69456	0.385185
$93$	−23.9110	−2.47945
$94$	9.19678	0.948575
$95$	5.87908	0.603181
$96$	−3.12057	−0.318491
$97$	−0.300716	−0.0305331	−0.0152665	−	0.999883i	$-0.504860\pi$
$97$	−0.300716	−0.0305331	−0.0152665	+	0.999883i	$0.504860\pi$
$98$	−5.69839	−0.575624
$99$	19.3809	1.94785
$100$	5.29492	0.529492
$101$	4.90531	0.488097	0.244048	−	0.969763i	$-0.421524\pi$
$101$	4.90531	0.488097	0.244048	+	0.969763i	$0.421524\pi$
$102$	−11.4390	−1.13263
$103$	8.69745	0.856985	0.428492	−	0.903545i	$-0.359045\pi$
$103$	8.69745	0.856985	0.428492	+	0.903545i	$0.359045\pi$
$104$	−5.53503	−0.542755
$105$	35.6795	3.48197
$106$	−3.97970	−0.386543
$107$	−3.09984	−0.299673	−0.149837	−	0.988711i	$-0.547875\pi$
$107$	−3.09984	−0.299673	−0.149837	+	0.988711i	$0.547875\pi$
$108$	11.6645	1.12241
$109$	−5.69670	−0.545645	−0.272822	−	0.962064i	$-0.587957\pi$
$109$	−5.69670	−0.545645	−0.272822	+	0.962064i	$0.587957\pi$
$110$	9.22909	0.879959
$111$	26.1758	2.48450
$112$	−3.56348	−0.336717
$113$	−19.9262	−1.87449	−0.937247	−	0.348665i	$-0.886635\pi$
$113$	−19.9262	−1.87449	−0.937247	+	0.348665i	$0.886635\pi$
$114$	5.71783	0.535524
$115$	−11.8543	−1.10542
$116$	9.15934	0.850424
$117$	37.2947	3.44790
$118$	−13.1414	−1.20976
$119$	−13.0625	−1.19744
$120$	10.0126	0.914017
$121$	−2.72639	−0.247854
$122$	14.0376	1.27090
$123$	36.6885	3.30809
$124$	−7.66238	−0.688102
$125$	−0.946260	−0.0846360
$126$	24.0105	2.13902
$127$	5.38823	0.478128	0.239064	−	0.971004i	$-0.423159\pi$
$127$	5.38823	0.478128	0.239064	+	0.971004i	$0.423159\pi$
$128$	−1.00000	−0.0883883
$129$	−28.3813	−2.49883
$130$	17.7595	1.55761
$131$	−3.12698	−0.273205	−0.136603	−	0.990626i	$-0.543618\pi$
$131$	−3.12698	−0.273205	−0.136603	+	0.990626i	$0.543618\pi$
$132$	8.97596	0.781257
$133$	6.52938	0.566169
$134$	−0.398780	−0.0344494
$135$	−37.4263	−3.22114
$136$	−3.66567	−0.314329
$137$	−5.16238	−0.441052	−0.220526	−	0.975381i	$-0.570777\pi$
$137$	−5.16238	−0.441052	−0.220526	+	0.975381i	$0.570777\pi$
$138$	−11.5291	−0.981424
$139$	−1.07965	−0.0915747	−0.0457873	−	0.998951i	$-0.514580\pi$
$139$	−1.07965	−0.0915747	−0.0457873	+	0.998951i	$0.514580\pi$
$140$	11.4337	0.966322
$141$	−28.6992	−2.41691
$142$	7.48303	0.627962
$143$	15.9209	1.33137
$144$	6.73793	0.561494
$145$	−29.3884	−2.44057
$146$	−10.5899	−0.876427
$147$	17.7822	1.46665
$148$	8.38816	0.689503
$149$	−1.31649	−0.107851	−0.0539255	−	0.998545i	$-0.517173\pi$
$149$	−1.31649	−0.107851	−0.0539255	+	0.998545i	$0.517173\pi$
$150$	−16.5231	−1.34911
$151$	−2.16535	−0.176214	−0.0881070	−	0.996111i	$-0.528082\pi$
$151$	−2.16535	−0.176214	−0.0881070	+	0.996111i	$0.528082\pi$
$152$	1.83231	0.148620
$153$	24.6990	1.99680
$154$	10.2500	0.825965
$155$	24.5853	1.97474
$156$	17.2724	1.38290
$157$	7.67225	0.612312	0.306156	−	0.951981i	$-0.400957\pi$
$157$	7.67225	0.612312	0.306156	+	0.951981i	$0.400957\pi$
$158$	3.37728	0.268682
$159$	12.4189	0.984885
$160$	3.20857	0.253660
$161$	−13.1655	−1.03759
$162$	−16.1859	−1.27169
$163$	−5.77376	−0.452236	−0.226118	−	0.974100i	$-0.572604\pi$
$163$	−5.77376	−0.452236	−0.226118	+	0.974100i	$0.572604\pi$
$164$	11.7570	0.918068
$165$	−28.8000	−2.24208
$166$	−0.326978	−0.0253784
$167$	−3.58322	−0.277278	−0.138639	−	0.990343i	$-0.544273\pi$
$167$	−3.58322	−0.277278	−0.138639	+	0.990343i	$0.544273\pi$
$168$	11.1201	0.857932
$169$	17.6366	1.35666
$170$	11.7616	0.902070
$171$	−12.3459	−0.944118
$172$	−9.09492	−0.693481
$173$	18.5271	1.40859	0.704295	−	0.709907i	$-0.251263\pi$
$173$	18.5271	1.40859	0.704295	+	0.709907i	$0.251263\pi$
$174$	−28.5823	−2.16682
$175$	−18.8683	−1.42631
$176$	2.87639	0.216816
$177$	41.0086	3.08240
$178$	12.7173	0.953204
$179$	7.39440	0.552684	0.276342	−	0.961059i	$-0.410878\pi$
$179$	7.39440	0.552684	0.276342	+	0.961059i	$0.410878\pi$
$180$	−21.6191	−1.61139
$181$	−7.34217	−0.545739	−0.272870	−	0.962051i	$-0.587973\pi$
$181$	−7.34217	−0.545739	−0.272870	+	0.962051i	$0.587973\pi$
$182$	19.7240	1.46204
$183$	−43.8052	−3.23817
$184$	−3.69456	−0.272367
$185$	−26.9140	−1.97876
$186$	23.9110	1.75324
$187$	10.5439	0.771046
$188$	−9.19678	−0.670744
$189$	−41.5661	−3.02349
$190$	−5.87908	−0.426513
$191$	−3.32848	−0.240840	−0.120420	−	0.992723i	$-0.538424\pi$
$191$	−3.32848	−0.240840	−0.120420	+	0.992723i	$0.538424\pi$
$192$	3.12057	0.225207
$193$	−7.71797	−0.555552	−0.277776	−	0.960646i	$-0.589597\pi$
$193$	−7.71797	−0.555552	−0.277776	+	0.960646i	$0.589597\pi$
$194$	0.300716	0.0215901
$195$	−55.4198	−3.96870
$196$	5.69839	0.407028
$197$	−5.18348	−0.369308	−0.184654	−	0.982804i	$-0.559116\pi$
$197$	−5.18348	−0.369308	−0.184654	+	0.982804i	$0.559116\pi$
$198$	−19.3809	−1.37734
$199$	−13.5710	−0.962022	−0.481011	−	0.876715i	$-0.659730\pi$
$199$	−13.5710	−0.962022	−0.481011	+	0.876715i	$0.659730\pi$
$200$	−5.29492	−0.374407
$201$	1.24442	0.0877747
$202$	−4.90531	−0.345136
$203$	−32.6391	−2.29082
$204$	11.4390	0.800888
$205$	−37.7232	−2.63470
$206$	−8.69745	−0.605980
$207$	24.8937	1.73023
$208$	5.53503	0.383786
$209$	−5.27042	−0.364563
$210$	−35.6795	−2.46212
$211$	−15.5181	−1.06831	−0.534154	−	0.845387i	$-0.679370\pi$
$211$	−15.5181	−1.06831	−0.534154	+	0.845387i	$0.679370\pi$
$212$	3.97970	0.273327
$213$	−23.3513	−1.60000
$214$	3.09984	0.211901
$215$	29.1817	1.99017
$216$	−11.6645	−0.793667
$217$	27.3047	1.85357
$218$	5.69670	0.385829
$219$	33.0465	2.23308
$220$	−9.22909	−0.622225
$221$	20.2896	1.36483
$222$	−26.1758	−1.75681
$223$	−7.29073	−0.488223	−0.244112	−	0.969747i	$-0.578496\pi$
$223$	−7.29073	−0.488223	−0.244112	+	0.969747i	$0.578496\pi$
$224$	3.56348	0.238095
$225$	35.6768	2.37845
$226$	19.9262	1.32547
$227$	24.9134	1.65356	0.826780	−	0.562525i	$-0.190170\pi$
$227$	24.9134	1.65356	0.826780	+	0.562525i	$0.190170\pi$
$228$	−5.71783	−0.378672
$229$	−2.62325	−0.173349	−0.0866747	−	0.996237i	$-0.527624\pi$
$229$	−2.62325	−0.173349	−0.0866747	+	0.996237i	$0.527624\pi$
$230$	11.8543	0.781647
$231$	−31.9856	−2.10450
$232$	−9.15934	−0.601340
$233$	−10.8332	−0.709706	−0.354853	−	0.934922i	$-0.615469\pi$
$233$	−10.8332	−0.709706	−0.354853	+	0.934922i	$0.615469\pi$
$234$	−37.2947	−2.43803
$235$	29.5085	1.92492
$236$	13.1414	0.855433
$237$	−10.5390	−0.684584
$238$	13.0625	0.846719
$239$	−5.86356	−0.379282	−0.189641	−	0.981854i	$-0.560732\pi$
$239$	−5.86356	−0.379282	−0.189641	+	0.981854i	$0.560732\pi$
$240$	−10.0126	−0.646307
$241$	12.7523	0.821447	0.410724	−	0.911760i	$-0.365276\pi$
$241$	12.7523	0.821447	0.410724	+	0.911760i	$0.365276\pi$
$242$	2.72639	0.175259
$243$	15.5159	0.995346
$244$	−14.0376	−0.898664
$245$	−18.2837	−1.16810
$246$	−36.6885	−2.33918
$247$	−10.1419	−0.645312
$248$	7.66238	0.486562
$249$	1.02036	0.0646624
$250$	0.946260	0.0598467
$251$	7.29556	0.460491	0.230246	−	0.973133i	$-0.426047\pi$
$251$	7.29556	0.460491	0.230246	+	0.973133i	$0.426047\pi$
$252$	−24.0105	−1.51252
$253$	10.6270	0.668113
$254$	−5.38823	−0.338088
$255$	−36.7027	−2.29841
$256$	1.00000	0.0625000
$257$	18.6064	1.16063	0.580317	−	0.814391i	$-0.302929\pi$
$257$	18.6064	1.16063	0.580317	+	0.814391i	$0.302929\pi$
$258$	28.3813	1.76694
$259$	−29.8910	−1.85734
$260$	−17.7595	−1.10140
$261$	61.7150	3.82006
$262$	3.12698	0.193185
$263$	−6.66943	−0.411255	−0.205627	−	0.978630i	$-0.565923\pi$
$263$	−6.66943	−0.411255	−0.205627	+	0.978630i	$0.565923\pi$
$264$	−8.97596	−0.552432
$265$	−12.7692	−0.784403
$266$	−6.52938	−0.400342
$267$	−39.6853	−2.42870
$268$	0.398780	0.0243594
$269$	−4.92623	−0.300358	−0.150179	−	0.988659i	$-0.547985\pi$
$269$	−4.92623	−0.300358	−0.150179	+	0.988659i	$0.547985\pi$
$270$	37.4263	2.27769
$271$	2.38538	0.144901	0.0724507	−	0.997372i	$-0.476918\pi$
$271$	2.38538	0.144901	0.0724507	+	0.997372i	$0.476918\pi$
$272$	3.66567	0.222264
$273$	−61.5500	−3.72517
$274$	5.16238	0.311871
$275$	15.2302	0.918418
$276$	11.5291	0.693972
$277$	21.0933	1.26737	0.633687	−	0.773590i	$-0.281541\pi$
$277$	21.0933	1.26737	0.633687	+	0.773590i	$0.281541\pi$
$278$	1.07965	0.0647531
$279$	−51.6286	−3.09092
$280$	−11.4337	−0.683293
$281$	24.2208	1.44489	0.722444	−	0.691429i	$-0.243019\pi$
$281$	24.2208	1.44489	0.722444	+	0.691429i	$0.243019\pi$
$282$	28.6992	1.70901
$283$	6.07871	0.361342	0.180671	−	0.983544i	$-0.442173\pi$
$283$	6.07871	0.361342	0.180671	+	0.983544i	$0.442173\pi$
$284$	−7.48303	−0.444036
$285$	18.3461	1.08673
$286$	−15.9209	−0.941423
$287$	−41.8959	−2.47304
$288$	−6.73793	−0.397037
$289$	−3.56286	−0.209580
$290$	29.3884	1.72575
$291$	−0.938404	−0.0550102
$292$	10.5899	0.619728
$293$	−0.494993	−0.0289178	−0.0144589	−	0.999895i	$-0.504603\pi$
$293$	−0.494993	−0.0289178	−0.0144589	+	0.999895i	$0.504603\pi$
$294$	−17.7822	−1.03708
$295$	−42.1651	−2.45495
$296$	−8.38816	−0.487552
$297$	33.5515	1.94686
$298$	1.31649	0.0762621
$299$	20.4495	1.18263
$300$	16.5231	0.953964
$301$	32.4096	1.86806
$302$	2.16535	0.124602
$303$	15.3073	0.879384
$304$	−1.83231	−0.105090
$305$	45.0405	2.57901
$306$	−24.6990	−1.41195
$307$	−17.8896	−1.02101	−0.510506	−	0.859874i	$-0.670542\pi$
$307$	−17.8896	−1.02101	−0.510506	+	0.859874i	$0.670542\pi$
$308$	−10.2500	−0.584045
$309$	27.1410	1.54399
$310$	−24.5853	−1.39635
$311$	−29.3209	−1.66264	−0.831318	−	0.555796i	$-0.812413\pi$
$311$	−29.3209	−1.66264	−0.831318	+	0.555796i	$0.812413\pi$
$312$	−17.2724	−0.977859
$313$	−5.90097	−0.333543	−0.166771	−	0.985996i	$-0.553334\pi$
$313$	−5.90097	−0.333543	−0.166771	+	0.985996i	$0.553334\pi$
$314$	−7.67225	−0.432970
$315$	77.0393	4.34067
$316$	−3.37728	−0.189987
$317$	−10.2514	−0.575775	−0.287887	−	0.957664i	$-0.592953\pi$
$317$	−10.2514	−0.575775	−0.287887	+	0.957664i	$0.592953\pi$
$318$	−12.4189	−0.696419
$319$	26.3458	1.47508
$320$	−3.20857	−0.179364
$321$	−9.67326	−0.539909
$322$	13.1655	0.733684
$323$	−6.71663	−0.373723
$324$	16.1859	0.899219
$325$	29.3075	1.62569
$326$	5.77376	0.319779
$327$	−17.7769	−0.983066
$328$	−11.7570	−0.649172
$329$	32.7725	1.80681
$330$	28.8000	1.58539
$331$	−15.0238	−0.825784	−0.412892	−	0.910780i	$-0.635481\pi$
$331$	−15.0238	−0.825784	−0.412892	+	0.910780i	$0.635481\pi$
$332$	0.326978	0.0179452
$333$	56.5189	3.09721
$334$	3.58322	0.196065
$335$	−1.27951	−0.0699074
$336$	−11.1201	−0.606650
$337$	12.3998	0.675460	0.337730	−	0.941243i	$-0.390341\pi$
$337$	12.3998	0.675460	0.337730	+	0.941243i	$0.390341\pi$
$338$	−17.6366	−0.959305
$339$	−62.1809	−3.37720
$340$	−11.7616	−0.637860
$341$	−22.0400	−1.19353
$342$	12.3459	0.667592
$343$	4.63827	0.250443
$344$	9.09492	0.490365
$345$	−36.9920	−1.99158
$346$	−18.5271	−0.996024
$347$	−1.39426	−0.0748477	−0.0374238	−	0.999299i	$-0.511915\pi$
$347$	−1.39426	−0.0748477	−0.0374238	+	0.999299i	$0.511915\pi$
$348$	28.5823	1.53217
$349$	0.0613943	0.00328636	0.00164318	−	0.999999i	$-0.499477\pi$
$349$	0.0613943	0.00328636	0.00164318	+	0.999999i	$0.499477\pi$
$350$	18.8683	1.00855
$351$	64.5632	3.44613
$352$	−2.87639	−0.153312
$353$	−27.3083	−1.45347	−0.726737	−	0.686916i	$-0.758964\pi$
$353$	−27.3083	−1.45347	−0.726737	+	0.686916i	$0.758964\pi$
$354$	−41.0086	−2.17958
$355$	24.0098	1.27431
$356$	−12.7173	−0.674017
$357$	−40.7625	−2.15738
$358$	−7.39440	−0.390806
$359$	27.0449	1.42737	0.713687	−	0.700465i	$-0.247024\pi$
$359$	27.0449	1.42737	0.713687	+	0.700465i	$0.247024\pi$
$360$	21.6191	1.13943
$361$	−15.6427	−0.823298
$362$	7.34217	0.385896
$363$	−8.50788	−0.446548
$364$	−19.7240	−1.03382
$365$	−33.9785	−1.77851
$366$	43.8052	2.28973
$367$	24.5553	1.28177	0.640887	−	0.767635i	$-0.278567\pi$
$367$	24.5553	1.28177	0.640887	+	0.767635i	$0.278567\pi$
$368$	3.69456	0.192592
$369$	79.2180	4.12392
$370$	26.9140	1.39919
$371$	−14.1816	−0.736271
$372$	−23.9110	−1.23973
$373$	13.9864	0.724189	0.362094	−	0.932141i	$-0.382062\pi$
$373$	13.9864	0.724189	0.362094	+	0.932141i	$0.382062\pi$
$374$	−10.5439	−0.545212
$375$	−2.95287	−0.152485
$376$	9.19678	0.474288
$377$	50.6973	2.61104
$378$	41.5661	2.13793
$379$	−26.0430	−1.33774	−0.668869	−	0.743380i	$-0.733221\pi$
$379$	−26.0430	−1.33774	−0.668869	+	0.743380i	$0.733221\pi$
$380$	5.87908	0.301590
$381$	16.8143	0.861424
$382$	3.32848	0.170300
$383$	−11.8191	−0.603928	−0.301964	−	0.953319i	$-0.597642\pi$
$383$	−11.8191	−0.603928	−0.301964	+	0.953319i	$0.597642\pi$
$384$	−3.12057	−0.159246
$385$	32.8877	1.67611
$386$	7.71797	0.392834
$387$	−61.2810	−3.11509
$388$	−0.300716	−0.0152665
$389$	−32.8956	−1.66787	−0.833937	−	0.551859i	$-0.813919\pi$
$389$	−32.8956	−1.66787	−0.833937	+	0.551859i	$0.813919\pi$
$390$	55.4198	2.80629
$391$	13.5430	0.684901
$392$	−5.69839	−0.287812
$393$	−9.75794	−0.492223
$394$	5.18348	0.261140
$395$	10.8362	0.545231
$396$	19.3809	0.973927
$397$	17.9175	0.899251	0.449626	−	0.893217i	$-0.351557\pi$
$397$	17.9175	0.899251	0.449626	+	0.893217i	$0.351557\pi$
$398$	13.5710	0.680252
$399$	20.3754	1.02004
$400$	5.29492	0.264746
$401$	−6.27747	−0.313482	−0.156741	−	0.987640i	$-0.550099\pi$
$401$	−6.27747	−0.313482	−0.156741	+	0.987640i	$0.550099\pi$
$402$	−1.24442	−0.0620661
$403$	−42.4115	−2.11267
$404$	4.90531	0.244048
$405$	−51.9337	−2.58061
$406$	32.6391	1.61985
$407$	24.1276	1.19596
$408$	−11.4390	−0.566313
$409$	13.9011	0.687364	0.343682	−	0.939086i	$-0.388326\pi$
$409$	13.9011	0.687364	0.343682	+	0.939086i	$0.388326\pi$
$410$	37.7232	1.86302
$411$	−16.1096	−0.794626
$412$	8.69745	0.428492
$413$	−46.8291	−2.30431
$414$	−24.8937	−1.22346
$415$	−1.04913	−0.0514998
$416$	−5.53503	−0.271377
$417$	−3.36912	−0.164986
$418$	5.27042	0.257785
$419$	13.9559	0.681792	0.340896	−	0.940101i	$-0.389270\pi$
$419$	13.9559	0.681792	0.340896	+	0.940101i	$0.389270\pi$
$420$	35.6795	1.74098
$421$	−21.9631	−1.07042	−0.535208	−	0.844720i	$-0.679767\pi$
$421$	−21.9631	−1.07042	−0.535208	+	0.844720i	$0.679767\pi$
$422$	15.5181	0.755408
$423$	−61.9673	−3.01295
$424$	−3.97970	−0.193271
$425$	19.4094	0.941495
$426$	23.3513	1.13137
$427$	50.0226	2.42076
$428$	−3.09984	−0.149837
$429$	49.6822	2.39868
$430$	−29.1817	−1.40727
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	11.6645	0.561207
$433$	−27.1869	−1.30652	−0.653260	−	0.757133i	$-0.726599\pi$
$433$	−27.1869	−1.30652	−0.653260	+	0.757133i	$0.726599\pi$
$434$	−27.3047	−1.31067
$435$	−91.7084	−4.39708
$436$	−5.69670	−0.272822
$437$	−6.76957	−0.323832
$438$	−33.0465	−1.57902
$439$	2.72928	0.130261	0.0651306	−	0.997877i	$-0.479254\pi$
$439$	2.72928	0.130261	0.0651306	+	0.997877i	$0.479254\pi$
$440$	9.22909	0.439980
$441$	38.3953	1.82835
$442$	−20.2896	−0.965079
$443$	12.7310	0.604867	0.302433	−	0.953171i	$-0.402201\pi$
$443$	12.7310	0.604867	0.302433	+	0.953171i	$0.402201\pi$
$444$	26.1758	1.24225
$445$	40.8044	1.93432
$446$	7.29073	0.345226
$447$	−4.10819	−0.194311
$448$	−3.56348	−0.168359
$449$	−2.49955	−0.117961	−0.0589805	−	0.998259i	$-0.518785\pi$
$449$	−2.49955	−0.117961	−0.0589805	+	0.998259i	$0.518785\pi$
$450$	−35.6768	−1.68182
$451$	33.8177	1.59241
$452$	−19.9262	−0.937247
$453$	−6.75713	−0.317478
$454$	−24.9134	−1.16924
$455$	63.2858	2.96688
$456$	5.71783	0.267762
$457$	−35.4681	−1.65913	−0.829563	−	0.558413i	$-0.811411\pi$
$457$	−35.4681	−1.65913	−0.829563	+	0.558413i	$0.811411\pi$
$458$	2.62325	0.122577
$459$	42.7581	1.99578
$460$	−11.8543	−0.552708
$461$	6.53689	0.304453	0.152227	−	0.988346i	$-0.451356\pi$
$461$	6.53689	0.304453	0.152227	+	0.988346i	$0.451356\pi$
$462$	31.9856	1.48811
$463$	29.2282	1.35835	0.679176	−	0.733976i	$-0.262337\pi$
$463$	29.2282	1.35835	0.679176	+	0.733976i	$0.262337\pi$
$464$	9.15934	0.425212
$465$	76.7200	3.55780
$466$	10.8332	0.501838
$467$	−7.26969	−0.336401	−0.168201	−	0.985753i	$-0.553796\pi$
$467$	−7.26969	−0.336401	−0.168201	+	0.985753i	$0.553796\pi$
$468$	37.2947	1.72395
$469$	−1.42105	−0.0656178
$470$	−29.5085	−1.36113
$471$	23.9418	1.10318
$472$	−13.1414	−0.604882
$473$	−26.1605	−1.20286
$474$	10.5390	0.484074
$475$	−9.70190	−0.445154
$476$	−13.0625	−0.598721
$477$	26.8150	1.22777
$478$	5.86356	0.268193
$479$	27.9334	1.27631	0.638155	−	0.769908i	$-0.279698\pi$
$479$	27.9334	1.27631	0.638155	+	0.769908i	$0.279698\pi$
$480$	10.0126	0.457008
$481$	46.4288	2.11697
$482$	−12.7523	−0.580851
$483$	−41.0838	−1.86938
$484$	−2.72639	−0.123927
$485$	0.964868	0.0438124
$486$	−15.5159	−0.703816
$487$	−29.2096	−1.32361	−0.661807	−	0.749674i	$-0.730210\pi$
$487$	−29.2096	−1.32361	−0.661807	+	0.749674i	$0.730210\pi$
$488$	14.0376	0.635451
$489$	−18.0174	−0.814775
$490$	18.2837	0.825972
$491$	7.36315	0.332294	0.166147	−	0.986101i	$-0.446867\pi$
$491$	7.36315	0.332294	0.166147	+	0.986101i	$0.446867\pi$
$492$	36.6885	1.65405
$493$	33.5751	1.51215
$494$	10.1419	0.456304
$495$	−62.1850	−2.79501
$496$	−7.66238	−0.344051
$497$	26.6656	1.19612
$498$	−1.02036	−0.0457232
$499$	3.84359	0.172063	0.0860314	−	0.996292i	$-0.472581\pi$
$499$	3.84359	0.172063	0.0860314	+	0.996292i	$0.472581\pi$
$500$	−0.946260	−0.0423180
$501$	−11.1817	−0.499560
$502$	−7.29556	−0.325617
$503$	4.08957	0.182345	0.0911725	−	0.995835i	$-0.470939\pi$
$503$	4.08957	0.182345	0.0911725	+	0.995835i	$0.470939\pi$
$504$	24.0105	1.06951
$505$	−15.7390	−0.700377
$506$	−10.6270	−0.472428
$507$	55.0362	2.44424
$508$	5.38823	0.239064
$509$	5.44950	0.241545	0.120773	−	0.992680i	$-0.461463\pi$
$509$	5.44950	0.241545	0.120773	+	0.992680i	$0.461463\pi$
$510$	36.7027	1.62522
$511$	−37.7369	−1.66938
$512$	−1.00000	−0.0441942
$513$	−21.3729	−0.943635
$514$	−18.6064	−0.820692
$515$	−27.9064	−1.22970
$516$	−28.3813	−1.24942
$517$	−26.4535	−1.16342
$518$	29.8910	1.31334
$519$	57.8151	2.53780
$520$	17.7595	0.778807
$521$	−22.1226	−0.969207	−0.484604	−	0.874734i	$-0.661036\pi$
$521$	−22.1226	−0.969207	−0.484604	+	0.874734i	$0.661036\pi$
$522$	−61.7150	−2.70119
$523$	−15.4728	−0.676578	−0.338289	−	0.941042i	$-0.609848\pi$
$523$	−15.4728	−0.676578	−0.338289	+	0.941042i	$0.609848\pi$
$524$	−3.12698	−0.136603
$525$	−58.8799	−2.56973
$526$	6.66943	0.290801
$527$	−28.0878	−1.22352
$528$	8.97596	0.390629
$529$	−9.35021	−0.406531
$530$	12.7692	0.554657
$531$	88.5459	3.84257
$532$	6.52938	0.283085
$533$	65.0755	2.81873
$534$	39.6853	1.71735
$535$	9.94606	0.430006
$536$	−0.398780	−0.0172247
$537$	23.0747	0.995748
$538$	4.92623	0.212385
$539$	16.3908	0.706000
$540$	−37.4263	−1.61057
$541$	−36.5353	−1.57078	−0.785388	−	0.619004i	$-0.787537\pi$
$541$	−36.5353	−1.57078	−0.785388	+	0.619004i	$0.787537\pi$
$542$	−2.38538	−0.102461
$543$	−22.9117	−0.983236
$544$	−3.66567	−0.157164
$545$	18.2783	0.782954
$546$	61.5500	2.63410
$547$	−31.8512	−1.36186	−0.680929	−	0.732349i	$-0.738424\pi$
$547$	−31.8512	−1.36186	−0.680929	+	0.732349i	$0.738424\pi$
$548$	−5.16238	−0.220526
$549$	−94.5843	−4.03676
$550$	−15.2302	−0.649419
$551$	−16.7827	−0.714967
$552$	−11.5291	−0.490712
$553$	12.0349	0.511775
$554$	−21.0933	−0.896168
$555$	−83.9869	−3.56505
$556$	−1.07965	−0.0457873
$557$	−22.4877	−0.952835	−0.476417	−	0.879219i	$-0.658065\pi$
$557$	−22.4877	−0.952835	−0.476417	+	0.879219i	$0.658065\pi$
$558$	51.6286	2.18561
$559$	−50.3407	−2.12918
$560$	11.4337	0.483161
$561$	32.9029	1.38916
$562$	−24.2208	−1.02169
$563$	39.2872	1.65576	0.827879	−	0.560907i	$-0.189548\pi$
$563$	39.2872	1.65576	0.827879	+	0.560907i	$0.189548\pi$
$564$	−28.6992	−1.20845
$565$	63.9344	2.68974
$566$	−6.07871	−0.255507
$567$	−57.6783	−2.42226
$568$	7.48303	0.313981
$569$	13.9330	0.584103	0.292051	−	0.956403i	$-0.405662\pi$
$569$	13.9330	0.584103	0.292051	+	0.956403i	$0.405662\pi$
$570$	−18.3461	−0.768432
$571$	18.2680	0.764491	0.382246	−	0.924061i	$-0.375151\pi$
$571$	18.2680	0.764491	0.382246	+	0.924061i	$0.375151\pi$
$572$	15.9209	0.665687
$573$	−10.3867	−0.433912
$574$	41.8959	1.74870
$575$	19.5624	0.815808
$576$	6.73793	0.280747
$577$	39.3576	1.63848	0.819240	−	0.573451i	$-0.194396\pi$
$577$	39.3576	1.63848	0.819240	+	0.573451i	$0.194396\pi$
$578$	3.56286	0.148195
$579$	−24.0844	−1.00091
$580$	−29.3884	−1.22029
$581$	−1.16518	−0.0483397
$582$	0.938404	0.0388981
$583$	11.4472	0.474093
$584$	−10.5899	−0.438214
$585$	−119.663	−4.94744
$586$	0.494993	0.0204480
$587$	−27.4225	−1.13185	−0.565924	−	0.824458i	$-0.691480\pi$
$587$	−27.4225	−1.13185	−0.565924	+	0.824458i	$0.691480\pi$
$588$	17.7822	0.733325
$589$	14.0398	0.578501
$590$	42.1651	1.73591
$591$	−16.1754	−0.665367
$592$	8.38816	0.344751
$593$	−38.9146	−1.59803	−0.799015	−	0.601311i	$-0.794645\pi$
$593$	−38.9146	−1.59803	−0.799015	+	0.601311i	$0.794645\pi$
$594$	−33.5515	−1.37664
$595$	41.9121	1.71823
$596$	−1.31649	−0.0539255
$597$	−42.3492	−1.73324
$598$	−20.4495	−0.836243
$599$	−23.1879	−0.947431	−0.473715	−	0.880678i	$-0.657087\pi$
$599$	−23.1879	−0.947431	−0.473715	+	0.880678i	$0.657087\pi$
$600$	−16.5231	−0.674554
$601$	17.4362	0.711236	0.355618	−	0.934631i	$-0.384270\pi$
$601$	17.4362	0.711236	0.355618	+	0.934631i	$0.384270\pi$
$602$	−32.4096	−1.32092
$603$	2.68696	0.109421
$604$	−2.16535	−0.0881070
$605$	8.74781	0.355649
$606$	−15.3073	−0.621818
$607$	6.11449	0.248179	0.124090	−	0.992271i	$-0.460399\pi$
$607$	6.11449	0.248179	0.124090	+	0.992271i	$0.460399\pi$
$608$	1.83231	0.0743098
$609$	−101.853	−4.12727
$610$	−45.0405	−1.82364
$611$	−50.9045	−2.05938
$612$	24.6990	0.998400
$613$	5.83033	0.235485	0.117742	−	0.993044i	$-0.462434\pi$
$613$	5.83033	0.235485	0.117742	+	0.993044i	$0.462434\pi$
$614$	17.8896	0.721964
$615$	−117.718	−4.74684
$616$	10.2500	0.412982
$617$	−0.923392	−0.0371744	−0.0185872	−	0.999827i	$-0.505917\pi$
$617$	−0.923392	−0.0371744	−0.0185872	+	0.999827i	$0.505917\pi$
$618$	−27.1410	−1.09177
$619$	−9.98614	−0.401377	−0.200688	−	0.979655i	$-0.564318\pi$
$619$	−9.98614	−0.401377	−0.200688	+	0.979655i	$0.564318\pi$
$620$	24.5853	0.987368
$621$	43.0951	1.72935
$622$	29.3209	1.17566
$623$	45.3179	1.81562
$624$	17.2724	0.691451
$625$	−23.4384	−0.937538
$626$	5.90097	0.235850
$627$	−16.4467	−0.656818
$628$	7.67225	0.306156
$629$	30.7482	1.22601
$630$	−77.0393	−3.06932
$631$	6.32745	0.251892	0.125946	−	0.992037i	$-0.459803\pi$
$631$	6.32745	0.251892	0.125946	+	0.992037i	$0.459803\pi$
$632$	3.37728	0.134341
$633$	−48.4252	−1.92473
$634$	10.2514	0.407134
$635$	−17.2885	−0.686074
$636$	12.4189	0.492442
$637$	31.5408	1.24969
$638$	−26.3458	−1.04304
$639$	−50.4202	−1.99459
$640$	3.20857	0.126830
$641$	−38.4143	−1.51727	−0.758636	−	0.651515i	$-0.774134\pi$
$641$	−38.4143	−1.51727	−0.758636	+	0.651515i	$0.774134\pi$
$642$	9.67326	0.381773
$643$	33.6227	1.32595	0.662974	−	0.748642i	$-0.269294\pi$
$643$	33.6227	1.32595	0.662974	+	0.748642i	$0.269294\pi$
$644$	−13.1655	−0.518793
$645$	91.0634	3.58562
$646$	6.71663	0.264262
$647$	−29.0243	−1.14106	−0.570532	−	0.821275i	$-0.693263\pi$
$647$	−29.0243	−1.14106	−0.570532	+	0.821275i	$0.693263\pi$
$648$	−16.1859	−0.635844
$649$	37.7998	1.48377
$650$	−29.3075	−1.14954
$651$	85.2062	3.33949
$652$	−5.77376	−0.226118
$653$	−1.82119	−0.0712685	−0.0356342	−	0.999365i	$-0.511345\pi$
$653$	−1.82119	−0.0712685	−0.0356342	+	0.999365i	$0.511345\pi$
$654$	17.7769	0.695133
$655$	10.0331	0.392027
$656$	11.7570	0.459034
$657$	71.3541	2.78379
$658$	−32.7725	−1.27761
$659$	−31.4954	−1.22688	−0.613442	−	0.789739i	$-0.710216\pi$
$659$	−31.4954	−1.22688	−0.613442	+	0.789739i	$0.710216\pi$
$660$	−28.8000	−1.12104
$661$	38.1820	1.48511	0.742554	−	0.669786i	$-0.233614\pi$
$661$	38.1820	1.48511	0.742554	+	0.669786i	$0.233614\pi$
$662$	15.0238	0.583917
$663$	63.3151	2.45895
$664$	−0.326978	−0.0126892
$665$	−20.9500	−0.812405
$666$	−56.5189	−2.19006
$667$	33.8398	1.31028
$668$	−3.58322	−0.138639
$669$	−22.7512	−0.879612
$670$	1.27951	0.0494320
$671$	−40.3775	−1.55876
$672$	11.1201	0.428966
$673$	−7.98325	−0.307732	−0.153866	−	0.988092i	$-0.549172\pi$
$673$	−7.98325	−0.307732	−0.153866	+	0.988092i	$0.549172\pi$
$674$	−12.3998	−0.477622
$675$	61.7624	2.37724
$676$	17.6366	0.678331
$677$	35.4169	1.36118	0.680590	−	0.732664i	$-0.261723\pi$
$677$	35.4169	1.36118	0.680590	+	0.732664i	$0.261723\pi$
$678$	62.1809	2.38804
$679$	1.07159	0.0411240
$680$	11.7616	0.451035
$681$	77.7439	2.97915
$682$	22.0400	0.843954
$683$	24.2299	0.927130	0.463565	−	0.886063i	$-0.346570\pi$
$683$	24.2299	0.927130	0.463565	+	0.886063i	$0.346570\pi$
$684$	−12.3459	−0.472059
$685$	16.5639	0.632873
$686$	−4.63827	−0.177090
$687$	−8.18603	−0.312317
$688$	−9.09492	−0.346741
$689$	22.0278	0.839192
$690$	36.9920	1.40826
$691$	−35.9371	−1.36711	−0.683556	−	0.729898i	$-0.739567\pi$
$691$	−35.9371	−1.36711	−0.683556	+	0.729898i	$0.739567\pi$
$692$	18.5271	0.704295
$693$	−69.0635	−2.62350
$694$	1.39426	0.0529253
$695$	3.46413	0.131402
$696$	−28.5823	−1.08341
$697$	43.0973	1.63243
$698$	−0.0613943	−0.00232381
$699$	−33.8057	−1.27865
$700$	−18.8683	−0.713156
$701$	−32.6912	−1.23473	−0.617365	−	0.786677i	$-0.711800\pi$
$701$	−32.6912	−1.23473	−0.617365	+	0.786677i	$0.711800\pi$
$702$	−64.5632	−2.43678
$703$	−15.3697	−0.579678
$704$	2.87639	0.108408
$705$	92.0832	3.46806
$706$	27.3083	1.02776
$707$	−17.4800	−0.657402
$708$	41.0086	1.54120
$709$	−20.2019	−0.758698	−0.379349	−	0.925254i	$-0.623852\pi$
$709$	−20.2019	−0.758698	−0.379349	+	0.925254i	$0.623852\pi$
$710$	−24.0098	−0.901073
$711$	−22.7559	−0.853413
$712$	12.7173	0.476602
$713$	−28.3091	−1.06019
$714$	40.7625	1.52550
$715$	−51.0833	−1.91041
$716$	7.39440	0.276342
$717$	−18.2976	−0.683337
$718$	−27.0449	−1.00931
$719$	−9.27692	−0.345971	−0.172985	−	0.984924i	$-0.555341\pi$
$719$	−9.27692	−0.345971	−0.172985	+	0.984924i	$0.555341\pi$
$720$	−21.6191	−0.805697
$721$	−30.9932	−1.15425
$722$	15.6427	0.582159
$723$	39.7944	1.47997
$724$	−7.34217	−0.272870
$725$	48.4980	1.80117
$726$	8.50788	0.315757
$727$	−49.0183	−1.81799	−0.908994	−	0.416809i	$-0.863149\pi$
$727$	−49.0183	−1.81799	−0.908994	+	0.416809i	$0.863149\pi$
$728$	19.7240	0.731019
$729$	−0.139421	−0.00516374
$730$	33.9785	1.25760
$731$	−33.3390	−1.23309
$732$	−43.8052	−1.61909
$733$	40.2474	1.48657	0.743285	−	0.668975i	$-0.233266\pi$
$733$	40.2474	1.48657	0.743285	+	0.668975i	$0.233266\pi$
$734$	−24.5553	−0.906351
$735$	−57.0554	−2.10452
$736$	−3.69456	−0.136183
$737$	1.14705	0.0422520
$738$	−79.2180	−2.91605
$739$	18.3815	0.676176	0.338088	−	0.941115i	$-0.390220\pi$
$739$	18.3815	0.676176	0.338088	+	0.941115i	$0.390220\pi$
$740$	−26.9140	−0.989378
$741$	−31.6484	−1.16263
$742$	14.1816	0.520623
$743$	28.9777	1.06309	0.531544	−	0.847030i	$-0.321612\pi$
$743$	28.9777	1.06309	0.531544	+	0.847030i	$0.321612\pi$
$744$	23.9110	0.876618
$745$	4.22404	0.154757
$746$	−13.9864	−0.512079
$747$	2.20315	0.0806092
$748$	10.5439	0.385523
$749$	11.0462	0.403620
$750$	2.95287	0.107823
$751$	48.2362	1.76016	0.880082	−	0.474821i	$-0.157487\pi$
$751$	48.2362	1.76016	0.880082	+	0.474821i	$0.157487\pi$
$752$	−9.19678	−0.335372
$753$	22.7663	0.829649
$754$	−50.6973	−1.84629
$755$	6.94769	0.252852
$756$	−41.5661	−1.51174
$757$	11.8816	0.431844	0.215922	−	0.976411i	$-0.430724\pi$
$757$	11.8816	0.431844	0.215922	+	0.976411i	$0.430724\pi$
$758$	26.0430	0.945924
$759$	33.1622	1.20371
$760$	−5.87908	−0.213257
$761$	24.5967	0.891631	0.445815	−	0.895125i	$-0.352914\pi$
$761$	24.5967	0.891631	0.445815	+	0.895125i	$0.352914\pi$
$762$	−16.8143	−0.609119
$763$	20.3001	0.734912
$764$	−3.32848	−0.120420
$765$	−79.2486	−2.86524
$766$	11.8191	0.427041
$767$	72.7382	2.62642
$768$	3.12057	0.112604
$769$	−40.3764	−1.45601	−0.728005	−	0.685572i	$-0.759552\pi$
$769$	−40.3764	−1.45601	−0.728005	+	0.685572i	$0.759552\pi$
$770$	−32.8877	−1.18519
$771$	58.0624	2.09107
$772$	−7.71797	−0.277776
$773$	35.5118	1.27727	0.638635	−	0.769510i	$-0.279499\pi$
$773$	35.5118	1.27727	0.638635	+	0.769510i	$0.279499\pi$
$774$	61.2810	2.20270
$775$	−40.5717	−1.45738
$776$	0.300716	0.0107951
$777$	−93.2770	−3.34629
$778$	32.8956	1.17937
$779$	−21.5424	−0.771838
$780$	−55.4198	−1.98435
$781$	−21.5241	−0.770193
$782$	−13.5430	−0.484298
$783$	106.839	3.81811
$784$	5.69839	0.203514
$785$	−24.6170	−0.878617
$786$	9.75794	0.348054
$787$	6.60881	0.235579	0.117789	−	0.993039i	$-0.462419\pi$
$787$	6.60881	0.235579	0.117789	+	0.993039i	$0.462419\pi$
$788$	−5.18348	−0.184654
$789$	−20.8124	−0.740941
$790$	−10.8362	−0.385536
$791$	71.0064	2.52470
$792$	−19.3809	−0.688671
$793$	−77.6985	−2.75915
$794$	−17.9175	−0.635867
$795$	−39.8470	−1.41323
$796$	−13.5710	−0.481011
$797$	27.4362	0.971841	0.485921	−	0.874003i	$-0.338484\pi$
$797$	27.4362	0.971841	0.485921	+	0.874003i	$0.338484\pi$
$798$	−20.3754	−0.721280
$799$	−33.7124	−1.19266
$800$	−5.29492	−0.187204
$801$	−85.6885	−3.02765
$802$	6.27747	0.221665
$803$	30.4607	1.07493
$804$	1.24442	0.0438873
$805$	42.2424	1.48885
$806$	42.4115	1.49388
$807$	−15.3726	−0.541142
$808$	−4.90531	−0.172568
$809$	39.7409	1.39722	0.698609	−	0.715504i	$-0.253803\pi$
$809$	39.7409	1.39722	0.698609	+	0.715504i	$0.253803\pi$
$810$	51.9337	1.82476
$811$	−7.88798	−0.276985	−0.138492	−	0.990364i	$-0.544226\pi$
$811$	−7.88798	−0.276985	−0.138492	+	0.990364i	$0.544226\pi$
$812$	−32.6391	−1.14541
$813$	7.44373	0.261063
$814$	−24.1276	−0.845672
$815$	18.5255	0.648921
$816$	11.4390	0.400444
$817$	16.6647	0.583023
$818$	−13.9011	−0.486040
$819$	−132.899	−4.64386
$820$	−37.7232	−1.31735
$821$	5.31774	0.185590	0.0927951	−	0.995685i	$-0.470420\pi$
$821$	5.31774	0.185590	0.0927951	+	0.995685i	$0.470420\pi$
$822$	16.1096	0.561885
$823$	−12.2961	−0.428616	−0.214308	−	0.976766i	$-0.568750\pi$
$823$	−12.2961	−0.428616	−0.214308	+	0.976766i	$0.568750\pi$
$824$	−8.69745	−0.302990
$825$	47.5270	1.65468
$826$	46.8291	1.62939
$827$	11.7753	0.409467	0.204734	−	0.978818i	$-0.434367\pi$
$827$	11.7753	0.409467	0.204734	+	0.978818i	$0.434367\pi$
$828$	24.8937	0.865116
$829$	−9.47466	−0.329069	−0.164534	−	0.986371i	$-0.552612\pi$
$829$	−9.47466	−0.329069	−0.164534	+	0.986371i	$0.552612\pi$
$830$	1.04913	0.0364159
$831$	65.8230	2.28338
$832$	5.53503	0.191893
$833$	20.8884	0.723740
$834$	3.36912	0.116663
$835$	11.4970	0.397870
$836$	−5.27042	−0.182281
$837$	−89.3776	−3.08934
$838$	−13.9559	−0.482099
$839$	−45.1458	−1.55861	−0.779303	−	0.626647i	$-0.784427\pi$
$839$	−45.1458	−1.55861	−0.779303	+	0.626647i	$0.784427\pi$
$840$	−35.6795	−1.23106
$841$	54.8935	1.89288
$842$	21.9631	0.756898
$843$	75.5825	2.60320
$844$	−15.5181	−0.534154
$845$	−56.5883	−1.94670
$846$	61.9673	2.13048
$847$	9.71544	0.333826
$848$	3.97970	0.136664
$849$	18.9690	0.651015
$850$	−19.4094	−0.665738
$851$	30.9906	1.06234
$852$	−23.3513	−0.800002
$853$	−17.6085	−0.602904	−0.301452	−	0.953481i	$-0.597471\pi$
$853$	−17.6085	−0.602904	−0.301452	+	0.953481i	$0.597471\pi$
$854$	−50.0226	−1.71174
$855$	39.6128	1.35473
$856$	3.09984	0.105950
$857$	−23.9488	−0.818077	−0.409038	−	0.912517i	$-0.634136\pi$
$857$	−23.9488	−0.818077	−0.409038	+	0.912517i	$0.634136\pi$
$858$	−49.6822	−1.69612
$859$	−23.3114	−0.795375	−0.397687	−	0.917521i	$-0.630187\pi$
$859$	−23.3114	−0.795375	−0.397687	+	0.917521i	$0.630187\pi$
$860$	29.1817	0.995087
$861$	−130.739	−4.45557
$862$	−1.00000	−0.0340601
$863$	54.7118	1.86241	0.931205	−	0.364495i	$-0.118758\pi$
$863$	54.7118	1.86241	0.931205	+	0.364495i	$0.118758\pi$
$864$	−11.6645	−0.396833
$865$	−59.4455	−2.02121
$866$	27.1869	0.923850
$867$	−11.1181	−0.377592
$868$	27.3047	0.926783
$869$	−9.71438	−0.329538
$870$	91.7084	3.10921
$871$	2.20726	0.0747903
$872$	5.69670	0.192915
$873$	−2.02620	−0.0685766
$874$	6.76957	0.228984
$875$	3.37198	0.113994
$876$	33.0465	1.11654
$877$	10.6128	0.358370	0.179185	−	0.983815i	$-0.442654\pi$
$877$	10.6128	0.358370	0.179185	+	0.983815i	$0.442654\pi$
$878$	−2.72928	−0.0921086
$879$	−1.54466	−0.0521001
$880$	−9.22909	−0.311113
$881$	50.5021	1.70146	0.850730	−	0.525603i	$-0.176160\pi$
$881$	50.5021	1.70146	0.850730	+	0.525603i	$0.176160\pi$
$882$	−38.3953	−1.29284
$883$	25.4836	0.857592	0.428796	−	0.903401i	$-0.358938\pi$
$883$	25.4836	0.857592	0.428796	+	0.903401i	$0.358938\pi$
$884$	20.2896	0.682414
$885$	−131.579	−4.42298
$886$	−12.7310	−0.427705
$887$	5.68533	0.190895	0.0954474	−	0.995434i	$-0.469572\pi$
$887$	5.68533	0.190895	0.0954474	+	0.995434i	$0.469572\pi$
$888$	−26.1758	−0.878403
$889$	−19.2008	−0.643976
$890$	−40.8044	−1.36777
$891$	46.5571	1.55972
$892$	−7.29073	−0.244112
$893$	16.8513	0.563907
$894$	4.10819	0.137398
$895$	−23.7255	−0.793054
$896$	3.56348	0.119047
$897$	63.8141	2.13069
$898$	2.49955	0.0834111
$899$	−70.1823	−2.34071
$900$	35.6768	1.18923
$901$	14.5883	0.486006
$902$	−33.8177	−1.12601
$903$	101.136	3.36560
$904$	19.9262	0.662734
$905$	23.5579	0.783090
$906$	6.75713	0.224491
$907$	15.4507	0.513032	0.256516	−	0.966540i	$-0.417425\pi$
$907$	15.4507	0.513032	0.256516	+	0.966540i	$0.417425\pi$
$908$	24.9134	0.826780
$909$	33.0516	1.09625
$910$	−63.2858	−2.09790
$911$	−20.1186	−0.666559	−0.333279	−	0.942828i	$-0.608155\pi$
$911$	−20.1186	−0.666559	−0.333279	+	0.942828i	$0.608155\pi$
$912$	−5.71783	−0.189336
$913$	0.940515	0.0311265
$914$	35.4681	1.17318
$915$	140.552	4.64651
$916$	−2.62325	−0.0866747
$917$	11.1429	0.367972
$918$	−42.7581	−1.41123
$919$	32.0332	1.05668	0.528338	−	0.849034i	$-0.322815\pi$
$919$	32.0332	1.05668	0.528338	+	0.849034i	$0.322815\pi$
$920$	11.8543	0.390823
$921$	−55.8256	−1.83951
$922$	−6.53689	−0.215281
$923$	−41.4188	−1.36332
$924$	−31.9856	−1.05225
$925$	44.4146	1.46034
$926$	−29.2282	−0.960499
$927$	58.6028	1.92477
$928$	−9.15934	−0.300670
$929$	42.4240	1.39189	0.695944	−	0.718096i	$-0.254986\pi$
$929$	42.4240	1.39189	0.695944	+	0.718096i	$0.254986\pi$
$930$	−76.7200	−2.51575
$931$	−10.4412	−0.342196
$932$	−10.8332	−0.354853
$933$	−91.4979	−2.99551
$934$	7.26969	0.237871
$935$	−33.8308	−1.10639
$936$	−37.2947	−1.21902
$937$	8.36566	0.273294	0.136647	−	0.990620i	$-0.456367\pi$
$937$	8.36566	0.273294	0.136647	+	0.990620i	$0.456367\pi$
$938$	1.42105	0.0463988
$939$	−18.4144	−0.600930
$940$	29.5085	0.962461
$941$	−45.6000	−1.48652	−0.743259	−	0.669004i	$-0.766721\pi$
$941$	−45.6000	−1.48652	−0.743259	+	0.669004i	$0.766721\pi$
$942$	−23.9418	−0.780065
$943$	43.4370	1.41450
$944$	13.1414	0.427716
$945$	133.368	4.33845
$946$	26.1605	0.850552
$947$	52.8756	1.71823	0.859113	−	0.511785i	$-0.171016\pi$
$947$	52.8756	1.71823	0.859113	+	0.511785i	$0.171016\pi$
$948$	−10.5390	−0.342292
$949$	58.6155	1.90274
$950$	9.70190	0.314771
$951$	−31.9901	−1.03735
$952$	13.0625	0.423359
$953$	0.851000	0.0275666	0.0137833	−	0.999905i	$-0.495613\pi$
$953$	0.851000	0.0275666	0.0137833	+	0.999905i	$0.495613\pi$
$954$	−26.8150	−0.868167
$955$	10.6796	0.345585
$956$	−5.86356	−0.189641
$957$	82.2139	2.65760
$958$	−27.9334	−0.902488
$959$	18.3960	0.594039
$960$	−10.0126	−0.323154
$961$	27.7120	0.893937
$962$	−46.4288	−1.49692
$963$	−20.8865	−0.673059
$964$	12.7523	0.410724
$965$	24.7636	0.797170
$966$	41.0838	1.32185
$967$	23.6991	0.762112	0.381056	−	0.924552i	$-0.375560\pi$
$967$	23.6991	0.762112	0.381056	+	0.924552i	$0.375560\pi$
$968$	2.72639	0.0876295
$969$	−20.9597	−0.673322
$970$	−0.964868	−0.0309800
$971$	−48.8946	−1.56910	−0.784551	−	0.620065i	$-0.787106\pi$
$971$	−48.8946	−1.56910	−0.784551	+	0.620065i	$0.787106\pi$
$972$	15.5159	0.497673
$973$	3.84731	0.123339
$974$	29.2096	0.935937
$975$	91.4561	2.92894
$976$	−14.0376	−0.449332
$977$	−37.8227	−1.21006	−0.605028	−	0.796204i	$-0.706838\pi$
$977$	−37.8227	−1.21006	−0.605028	+	0.796204i	$0.706838\pi$
$978$	18.0174	0.576133
$979$	−36.5800	−1.16910
$980$	−18.2837	−0.584050
$981$	−38.3840	−1.22551
$982$	−7.36315	−0.234967
$983$	43.4739	1.38660	0.693300	−	0.720649i	$-0.256156\pi$
$983$	43.4739	1.38660	0.693300	+	0.720649i	$0.256156\pi$
$984$	−36.6885	−1.16959
$985$	16.6316	0.529926
$986$	−33.5751	−1.06925
$987$	102.269	3.25525
$988$	−10.1419	−0.322656
$989$	−33.6018	−1.06847
$990$	62.1850	1.97637
$991$	−50.7542	−1.61226	−0.806130	−	0.591738i	$-0.798442\pi$
$991$	−50.7542	−1.61226	−0.806130	+	0.591738i	$0.798442\pi$
$992$	7.66238	0.243281
$993$	−46.8828	−1.48778
$994$	−26.6656	−0.845782
$995$	43.5434	1.38042
$996$	1.02036	0.0323312
$997$	13.3258	0.422031	0.211016	−	0.977483i	$-0.432323\pi$
$997$	13.3258	0.422031	0.211016	+	0.977483i	$0.432323\pi$
$998$	−3.84359	−0.121667
$999$	97.8434	3.09563

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	862.2.a.k.1.10	✓	10
3.2	odd	2		7758.2.a.v.1.9		10
4.3	odd	2		6896.2.a.t.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
862.2.a.k.1.10	✓	10	1.1	even	1	trivial
6896.2.a.t.1.1		10	4.3	odd	2
7758.2.a.v.1.9		10	3.2	odd	2

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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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	862.2.a.k.1.10

Level	$862$
Weight	$2$
Character	862.1
Self dual	yes
Analytic conductor	$6.883$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$1$