LMFDB - Embedded newform 185.2.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [185,2,Mod(1,185)] 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("185.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$185 = 5 \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	185.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,2] 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:	$1.47723243739$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.973904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6$ x^5 - 2x^4 - 8x^3 + 6x^2 + 19x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.29298$ of defining polynomial
Character	$\chi$	$=$	185.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.72362 q^{2} -2.29298 q^{3} +5.41809 q^{4} -1.00000 q^{5} -6.24519 q^{6} +3.82710 q^{7} +9.30957 q^{8} +2.25774 q^{9} -2.72362 q^{10} -4.41809 q^{11} -12.4236 q^{12} -3.67583 q^{13} +10.4236 q^{14} +2.29298 q^{15} +14.5195 q^{16} -2.28688 q^{17} +6.14922 q^{18} -2.39037 q^{19} -5.41809 q^{20} -8.77545 q^{21} -12.0332 q^{22} -0.265251 q^{23} -21.3466 q^{24} +1.00000 q^{25} -10.0116 q^{26} +1.70198 q^{27} +20.7356 q^{28} -6.58595 q^{29} +6.24519 q^{30} +2.34076 q^{31} +20.9265 q^{32} +10.1306 q^{33} -6.22860 q^{34} -3.82710 q^{35} +12.2326 q^{36} +1.00000 q^{37} -6.51044 q^{38} +8.42859 q^{39} -9.30957 q^{40} -4.41809 q^{41} -23.9010 q^{42} +7.71249 q^{43} -23.9376 q^{44} -2.25774 q^{45} -0.722443 q^{46} +10.9285 q^{47} -33.2929 q^{48} +7.64669 q^{49} +2.72362 q^{50} +5.24377 q^{51} -19.9160 q^{52} -0.109574 q^{53} +4.63555 q^{54} +4.41809 q^{55} +35.6286 q^{56} +5.48105 q^{57} -17.9376 q^{58} -2.00504 q^{59} +12.4236 q^{60} +3.96271 q^{61} +6.37534 q^{62} +8.64059 q^{63} +27.9567 q^{64} +3.67583 q^{65} +27.5918 q^{66} +6.80664 q^{67} -12.3905 q^{68} +0.608215 q^{69} -10.4236 q^{70} -5.79485 q^{71} +21.0186 q^{72} -0.140654 q^{73} +2.72362 q^{74} -2.29298 q^{75} -12.9512 q^{76} -16.9085 q^{77} +22.9563 q^{78} -6.62418 q^{79} -14.5195 q^{80} -10.6758 q^{81} -12.0332 q^{82} +13.9904 q^{83} -47.5462 q^{84} +2.28688 q^{85} +21.0059 q^{86} +15.1014 q^{87} -41.1305 q^{88} +14.8139 q^{89} -6.14922 q^{90} -14.0678 q^{91} -1.43716 q^{92} -5.36731 q^{93} +29.7651 q^{94} +2.39037 q^{95} -47.9839 q^{96} -8.94394 q^{97} +20.8266 q^{98} -9.97490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100})$ 5 * q + 2 * q^2 + 3 * q^3 + 10 * q^4 - 5 * q^5 - 6 * q^6 + 11 * q^7 + 6 * q^8 + 6 * q^9 - 2 * q^10 - 5 * q^11 - 2 * q^12 + 4 * q^13 - 8 * q^14 - 3 * q^15 + 16 * q^16 + 2 * q^18 - 4 * q^19 - 10 * q^20 + 3 * q^21 - 8 * q^22 + 4 * q^23 - 42 * q^24 + 5 * q^25 - 4 * q^26 + 3 * q^27 + 28 * q^28 - 4 * q^29 + 6 * q^30 + 8 * q^31 + 14 * q^32 + 5 * q^33 - 32 * q^34 - 11 * q^35 + 16 * q^36 + 5 * q^37 - 2 * q^38 + 2 * q^39 - 6 * q^40 - 5 * q^41 - 52 * q^42 + 10 * q^43 - 46 * q^44 - 6 * q^45 + 7 * q^47 - 20 * q^48 + 22 * q^49 + 2 * q^50 - 2 * q^51 - 32 * q^52 - q^53 - 10 * q^54 + 5 * q^55 + 8 * q^56 - 8 * q^57 - 16 * q^58 - 30 * q^59 + 2 * q^60 - 14 * q^61 + 24 * q^62 + 30 * q^63 + 20 * q^64 - 4 * q^65 + 48 * q^66 + 24 * q^67 + 20 * q^68 + 8 * q^69 + 8 * q^70 - 7 * q^71 - 10 * q^72 + 5 * q^73 + 2 * q^74 + 3 * q^75 + 12 * q^76 - 17 * q^77 + 58 * q^78 + 28 * q^79 - 16 * q^80 - 31 * q^81 - 8 * q^82 + 27 * q^83 - 18 * q^84 + 44 * q^86 + 36 * q^87 - 28 * q^88 + 6 * q^89 - 2 * q^90 - 14 * q^91 + 56 * q^92 - 22 * q^93 + 4 * q^94 + 4 * q^95 - 42 * q^96 - 26 * q^97 - 18 * q^98 - 10 * 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

)

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

$2$	2.72362	1.92589	0.962944	−	0.269701i	$-0.0869249\pi$
$2$	2.72362	1.92589	0.962944	+	0.269701i	$0.0869249\pi$
$3$	−2.29298	−1.32385	−0.661925	−	0.749570i	$-0.730260\pi$
$3$	−2.29298	−1.32385	−0.661925	+	0.749570i	$0.730260\pi$
$4$	5.41809	2.70905
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−6.24519	−2.54959
$7$	3.82710	1.44651	0.723254	−	0.690582i	$-0.242646\pi$
$7$	3.82710	1.44651	0.723254	+	0.690582i	$0.242646\pi$
$8$	9.30957	3.29143
$9$	2.25774	0.752580
$10$	−2.72362	−0.861283
$11$	−4.41809	−1.33210	−0.666052	−	0.745905i	$-0.732017\pi$
$11$	−4.41809	−1.33210	−0.666052	+	0.745905i	$0.732017\pi$
$12$	−12.4236	−3.58637
$13$	−3.67583	−1.01949	−0.509746	−	0.860325i	$-0.670261\pi$
$13$	−3.67583	−1.01949	−0.509746	+	0.860325i	$0.670261\pi$
$14$	10.4236	2.78581
$15$	2.29298	0.592044
$16$	14.5195	3.62988
$17$	−2.28688	−0.554651	−0.277325	−	0.960776i	$-0.589448\pi$
$17$	−2.28688	−0.554651	−0.277325	+	0.960776i	$0.589448\pi$
$18$	6.14922	1.44938
$19$	−2.39037	−0.548387	−0.274194	−	0.961674i	$-0.588411\pi$
$19$	−2.39037	−0.548387	−0.274194	+	0.961674i	$0.588411\pi$
$20$	−5.41809	−1.21152
$21$	−8.77545	−1.91496
$22$	−12.0332	−2.56548
$23$	−0.265251	−0.0553087	−0.0276544	−	0.999618i	$-0.508804\pi$
$23$	−0.265251	−0.0553087	−0.0276544	+	0.999618i	$0.508804\pi$
$24$	−21.3466	−4.35736
$25$	1.00000	0.200000
$26$	−10.0116	−1.96343
$27$	1.70198	0.327547
$28$	20.7356	3.91865
$29$	−6.58595	−1.22298	−0.611490	−	0.791252i	$-0.709430\pi$
$29$	−6.58595	−1.22298	−0.611490	+	0.791252i	$0.709430\pi$
$30$	6.24519	1.14021
$31$	2.34076	0.420413	0.210207	−	0.977657i	$-0.432586\pi$
$31$	2.34076	0.420413	0.210207	+	0.977657i	$0.432586\pi$
$32$	20.9265	3.69931
$33$	10.1306	1.76351
$34$	−6.22860	−1.06820
$35$	−3.82710	−0.646898
$36$	12.2326	2.03877
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−6.51044	−1.05613
$39$	8.42859	1.34965
$40$	−9.30957	−1.47197
$41$	−4.41809	−0.689990	−0.344995	−	0.938605i	$-0.612119\pi$
$41$	−4.41809	−0.689990	−0.344995	+	0.938605i	$0.612119\pi$
$42$	−23.9010	−3.68800
$43$	7.71249	1.17614	0.588072	−	0.808809i	$-0.299887\pi$
$43$	7.71249	1.17614	0.588072	+	0.808809i	$0.299887\pi$
$44$	−23.9376	−3.60873
$45$	−2.25774	−0.336564
$46$	−0.722443	−0.106518
$47$	10.9285	1.59409	0.797045	−	0.603920i	$-0.206395\pi$
$47$	10.9285	1.59409	0.797045	+	0.603920i	$0.206395\pi$
$48$	−33.2929	−4.80542
$49$	7.64669	1.09238
$50$	2.72362	0.385178
$51$	5.24377	0.734275
$52$	−19.9160	−2.76185
$53$	−0.109574	−0.0150512	−0.00752559	−	0.999972i	$-0.502395\pi$
$53$	−0.109574	−0.0150512	−0.00752559	+	0.999972i	$0.502395\pi$
$54$	4.63555	0.630819
$55$	4.41809	0.595735
$56$	35.6286	4.76108
$57$	5.48105	0.725983
$58$	−17.9376	−2.35532
$59$	−2.00504	−0.261034	−0.130517	−	0.991446i	$-0.541664\pi$
$59$	−2.00504	−0.261034	−0.130517	+	0.991446i	$0.541664\pi$
$60$	12.4236	1.60387
$61$	3.96271	0.507374	0.253687	−	0.967286i	$-0.418357\pi$
$61$	3.96271	0.507374	0.253687	+	0.967286i	$0.418357\pi$
$62$	6.37534	0.809669
$63$	8.64059	1.08861
$64$	27.9567	3.49458
$65$	3.67583	0.455931
$66$	27.5918	3.39632
$67$	6.80664	0.831563	0.415782	−	0.909464i	$-0.363508\pi$
$67$	6.80664	0.831563	0.415782	+	0.909464i	$0.363508\pi$
$68$	−12.3905	−1.50257
$69$	0.608215	0.0732205
$70$	−10.4236	−1.24585
$71$	−5.79485	−0.687722	−0.343861	−	0.939020i	$-0.611735\pi$
$71$	−5.79485	−0.687722	−0.343861	+	0.939020i	$0.611735\pi$
$72$	21.0186	2.47706
$73$	−0.140654	−0.0164623	−0.00823116	−	0.999966i	$-0.502620\pi$
$73$	−0.140654	−0.0164623	−0.00823116	+	0.999966i	$0.502620\pi$
$74$	2.72362	0.316614
$75$	−2.29298	−0.264770
$76$	−12.9512	−1.48561
$77$	−16.9085	−1.92690
$78$	22.9563	2.59928
$79$	−6.62418	−0.745278	−0.372639	−	0.927976i	$-0.621547\pi$
$79$	−6.62418	−0.745278	−0.372639	+	0.927976i	$0.621547\pi$
$80$	−14.5195	−1.62333
$81$	−10.6758	−1.18620
$82$	−12.0332	−1.32884
$83$	13.9904	1.53565	0.767825	−	0.640660i	$-0.221339\pi$
$83$	13.9904	1.53565	0.767825	+	0.640660i	$0.221339\pi$
$84$	−47.5462	−5.18771
$85$	2.28688	0.248047
$86$	21.0059	2.26512
$87$	15.1014	1.61904
$88$	−41.1305	−4.38453
$89$	14.8139	1.57027	0.785136	−	0.619323i	$-0.212593\pi$
$89$	14.8139	1.57027	0.785136	+	0.619323i	$0.212593\pi$
$90$	−6.14922	−0.648185
$91$	−14.0678	−1.47470
$92$	−1.43716	−0.149834
$93$	−5.36731	−0.556565
$94$	29.7651	3.07004
$95$	2.39037	0.245246
$96$	−47.9839	−4.89734
$97$	−8.94394	−0.908119	−0.454060	−	0.890971i	$-0.650025\pi$
$97$	−8.94394	−0.908119	−0.454060	+	0.890971i	$0.650025\pi$
$98$	20.8266	2.10381
$99$	−9.97490	−1.00252
$100$	5.41809	0.541809
$101$	−11.5935	−1.15359	−0.576796	−	0.816888i	$-0.695697\pi$
$101$	−11.5935	−1.15359	−0.576796	+	0.816888i	$0.695697\pi$
$102$	14.2820	1.41413
$103$	−13.5918	−1.33924	−0.669620	−	0.742703i	$-0.733543\pi$
$103$	−13.5918	−1.33924	−0.669620	+	0.742703i	$0.733543\pi$
$104$	−34.2204	−3.35559
$105$	8.77545	0.856396
$106$	−0.298438	−0.0289869
$107$	6.66146	0.643988	0.321994	−	0.946742i	$-0.395647\pi$
$107$	6.66146	0.643988	0.321994	+	0.946742i	$0.395647\pi$
$108$	9.22151	0.887340
$109$	−4.85120	−0.464661	−0.232331	−	0.972637i	$-0.574635\pi$
$109$	−4.85120	−0.464661	−0.232331	+	0.972637i	$0.574635\pi$
$110$	12.0332	1.14732
$111$	−2.29298	−0.217640
$112$	55.5676	5.25065
$113$	−2.83965	−0.267132	−0.133566	−	0.991040i	$-0.542643\pi$
$113$	−2.83965	−0.267132	−0.133566	+	0.991040i	$0.542643\pi$
$114$	14.9283	1.39816
$115$	0.265251	0.0247348
$116$	−35.6833	−3.31311
$117$	−8.29907	−0.767249
$118$	−5.46096	−0.502722
$119$	−8.75213	−0.802306
$120$	21.3466	1.94867
$121$	8.51952	0.774502
$122$	10.7929	0.977145
$123$	10.1306	0.913443
$124$	12.6825	1.13892
$125$	−1.00000	−0.0894427
$126$	23.5337	2.09655
$127$	8.81853	0.782518	0.391259	−	0.920281i	$-0.372040\pi$
$127$	8.81853	0.782518	0.391259	+	0.920281i	$0.372040\pi$
$128$	34.2903	3.03087
$129$	−17.6845	−1.55704
$130$	10.0116	0.878071
$131$	4.61549	0.403257	0.201629	−	0.979462i	$-0.435377\pi$
$131$	4.61549	0.403257	0.201629	+	0.979462i	$0.435377\pi$
$132$	54.8884	4.77742
$133$	−9.14817	−0.793247
$134$	18.5387	1.60150
$135$	−1.70198	−0.146484
$136$	−21.2899	−1.82559
$137$	−14.1324	−1.20741	−0.603706	−	0.797207i	$-0.706310\pi$
$137$	−14.1324	−1.20741	−0.603706	+	0.797207i	$0.706310\pi$
$138$	1.65655	0.141015
$139$	13.2152	1.12090	0.560448	−	0.828190i	$-0.310629\pi$
$139$	13.2152	1.12090	0.560448	+	0.828190i	$0.310629\pi$
$140$	−20.7356	−1.75248
$141$	−25.0589	−2.11034
$142$	−15.7830	−1.32448
$143$	16.2402	1.35807
$144$	32.7813	2.73178
$145$	6.58595	0.546934
$146$	−0.383088	−0.0317046
$147$	−17.5337	−1.44615
$148$	5.41809	0.445364
$149$	6.99743	0.573252	0.286626	−	0.958043i	$-0.407466\pi$
$149$	6.99743	0.573252	0.286626	+	0.958043i	$0.407466\pi$
$150$	−6.24519	−0.509918
$151$	19.7185	1.60467	0.802333	−	0.596877i	$-0.203592\pi$
$151$	19.7185	1.60467	0.802333	+	0.596877i	$0.203592\pi$
$152$	−22.2533	−1.80498
$153$	−5.16319	−0.417419
$154$	−46.0522	−3.71099
$155$	−2.34076	−0.188015
$156$	45.6669	3.65628
$157$	12.9768	1.03566	0.517832	−	0.855482i	$-0.326739\pi$
$157$	12.9768	1.03566	0.517832	+	0.855482i	$0.326739\pi$
$158$	−18.0417	−1.43532
$159$	0.251251	0.0199255
$160$	−20.9265	−1.65438
$161$	−1.01514	−0.0800045
$162$	−29.0769	−2.28450
$163$	−2.33572	−0.182948	−0.0914740	−	0.995807i	$-0.529158\pi$
$163$	−2.33572	−0.182948	−0.0914740	+	0.995807i	$0.529158\pi$
$164$	−23.9376	−1.86921
$165$	−10.1306	−0.788664
$166$	38.1046	2.95749
$167$	−22.8816	−1.77063	−0.885314	−	0.464994i	$-0.846056\pi$
$167$	−22.8816	−1.77063	−0.885314	+	0.464994i	$0.846056\pi$
$168$	−81.6956	−6.30295
$169$	0.511728	0.0393637
$170$	6.22860	0.477712
$171$	−5.39682	−0.412705
$172$	41.7869	3.18622
$173$	14.0099	1.06515	0.532577	−	0.846382i	$-0.321224\pi$
$173$	14.0099	1.06515	0.532577	+	0.846382i	$0.321224\pi$
$174$	41.1305	3.11810
$175$	3.82710	0.289301
$176$	−64.1486	−4.83538
$177$	4.59751	0.345570
$178$	40.3474	3.02417
$179$	−14.7815	−1.10482	−0.552412	−	0.833571i	$-0.686292\pi$
$179$	−14.7815	−1.10482	−0.552412	+	0.833571i	$0.686292\pi$
$180$	−12.2326	−0.911767
$181$	−8.18949	−0.608720	−0.304360	−	0.952557i	$-0.598443\pi$
$181$	−8.18949	−0.608720	−0.304360	+	0.952557i	$0.598443\pi$
$182$	−38.3152	−2.84011
$183$	−9.08641	−0.671687
$184$	−2.46938	−0.182045
$185$	−1.00000	−0.0735215
$186$	−14.6185	−1.07188
$187$	10.1037	0.738853
$188$	59.2118	4.31846
$189$	6.51366	0.473799
$190$	6.51044	0.472317
$191$	19.8743	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$191$	19.8743	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$192$	−64.1040	−4.62631
$193$	17.0390	1.22650	0.613249	−	0.789890i	$-0.289862\pi$
$193$	17.0390	1.22650	0.613249	+	0.789890i	$0.289862\pi$
$194$	−24.3599	−1.74894
$195$	−8.42859	−0.603584
$196$	41.4304	2.95932
$197$	−26.5277	−1.89002	−0.945011	−	0.327038i	$-0.893949\pi$
$197$	−26.5277	−1.89002	−0.945011	+	0.327038i	$0.893949\pi$
$198$	−27.1678	−1.93073
$199$	−5.05687	−0.358472	−0.179236	−	0.983806i	$-0.557363\pi$
$199$	−5.05687	−0.358472	−0.179236	+	0.983806i	$0.557363\pi$
$200$	9.30957	0.658286
$201$	−15.6075	−1.10087
$202$	−31.5762	−2.22169
$203$	−25.2051	−1.76905
$204$	28.4112	1.98918
$205$	4.41809	0.308573
$206$	−37.0189	−2.57923
$207$	−0.598869	−0.0416243
$208$	−53.3713	−3.70063
$209$	10.5609	0.730509
$210$	23.9010	1.64932
$211$	−19.6605	−1.35348	−0.676741	−	0.736221i	$-0.736608\pi$
$211$	−19.6605	−1.35348	−0.676741	+	0.736221i	$0.736608\pi$
$212$	−0.593683	−0.0407743
$213$	13.2875	0.910442
$214$	18.1433	1.24025
$215$	−7.71249	−0.525987
$216$	15.8447	1.07810
$217$	8.95833	0.608131
$218$	−13.2128	−0.894885
$219$	0.322517	0.0217937
$220$	23.9376	1.61387
$221$	8.40620	0.565462
$222$	−6.24519	−0.419150
$223$	−24.1933	−1.62010	−0.810052	−	0.586358i	$-0.800561\pi$
$223$	−24.1933	−1.62010	−0.810052	+	0.586358i	$0.800561\pi$
$224$	80.0877	5.35109
$225$	2.25774	0.150516
$226$	−7.73412	−0.514466
$227$	18.1013	1.20143	0.600713	−	0.799465i	$-0.294883\pi$
$227$	18.1013	1.20143	0.600713	+	0.799465i	$0.294883\pi$
$228$	29.6968	1.96672
$229$	19.6182	1.29641	0.648205	−	0.761466i	$-0.275520\pi$
$229$	19.6182	1.29641	0.648205	+	0.761466i	$0.275520\pi$
$230$	0.722443	0.0476365
$231$	38.7707	2.55093
$232$	−61.3124	−4.02536
$233$	−4.67013	−0.305951	−0.152975	−	0.988230i	$-0.548886\pi$
$233$	−4.67013	−0.305951	−0.152975	+	0.988230i	$0.548886\pi$
$234$	−22.6035	−1.47764
$235$	−10.9285	−0.712899
$236$	−10.8635	−0.707152
$237$	15.1891	0.986637
$238$	−23.8375	−1.54515
$239$	−9.64645	−0.623977	−0.311989	−	0.950086i	$-0.600995\pi$
$239$	−9.64645	−0.623977	−0.311989	+	0.950086i	$0.600995\pi$
$240$	33.2929	2.14905
$241$	1.59826	0.102953	0.0514764	−	0.998674i	$-0.483607\pi$
$241$	1.59826	0.102953	0.0514764	+	0.998674i	$0.483607\pi$
$242$	23.2039	1.49160
$243$	19.3735	1.24281
$244$	21.4703	1.37450
$245$	−7.64669	−0.488529
$246$	27.5918	1.75919
$247$	8.78658	0.559077
$248$	21.7915	1.38376
$249$	−32.0797	−2.03297
$250$	−2.72362	−0.172257
$251$	−6.49812	−0.410158	−0.205079	−	0.978745i	$-0.565745\pi$
$251$	−6.49812	−0.410158	−0.205079	+	0.978745i	$0.565745\pi$
$252$	46.8155	2.94910
$253$	1.17190	0.0736770
$254$	24.0183	1.50704
$255$	−5.24377	−0.328378
$256$	37.4804	2.34252
$257$	−4.01155	−0.250234	−0.125117	−	0.992142i	$-0.539931\pi$
$257$	−4.01155	−0.250234	−0.125117	+	0.992142i	$0.539931\pi$
$258$	−48.1659	−2.99868
$259$	3.82710	0.237804
$260$	19.9160	1.23514
$261$	−14.8694	−0.920391
$262$	12.5708	0.776629
$263$	−26.2537	−1.61887	−0.809436	−	0.587208i	$-0.800227\pi$
$263$	−26.2537	−1.61887	−0.809436	+	0.587208i	$0.800227\pi$
$264$	94.3113	5.80446
$265$	0.109574	0.00673109
$266$	−24.9161	−1.52770
$267$	−33.9680	−2.07881
$268$	36.8790	2.25274
$269$	3.00335	0.183117	0.0915587	−	0.995800i	$-0.470815\pi$
$269$	3.00335	0.183117	0.0915587	+	0.995800i	$0.470815\pi$
$270$	−4.63555	−0.282111
$271$	5.91480	0.359298	0.179649	−	0.983731i	$-0.442504\pi$
$271$	5.91480	0.359298	0.179649	+	0.983731i	$0.442504\pi$
$272$	−33.2045	−2.01332
$273$	32.2571	1.95229
$274$	−38.4912	−2.32534
$275$	−4.41809	−0.266421
$276$	3.29537	0.198358
$277$	12.2719	0.737345	0.368672	−	0.929559i	$-0.379812\pi$
$277$	12.2719	0.737345	0.368672	+	0.929559i	$0.379812\pi$
$278$	35.9931	2.15872
$279$	5.28483	0.316395
$280$	−35.6286	−2.12922
$281$	−3.42907	−0.204561	−0.102281	−	0.994756i	$-0.532614\pi$
$281$	−3.42907	−0.204561	−0.102281	+	0.994756i	$0.532614\pi$
$282$	−68.2507	−4.06427
$283$	−21.8283	−1.29756	−0.648780	−	0.760976i	$-0.724720\pi$
$283$	−21.8283	−1.29756	−0.648780	+	0.760976i	$0.724720\pi$
$284$	−31.3970	−1.86307
$285$	−5.48105	−0.324669
$286$	44.2320	2.61549
$287$	−16.9085	−0.998075
$288$	47.2466	2.78403
$289$	−11.7702	−0.692362
$290$	17.9376	1.05333
$291$	20.5082	1.20221
$292$	−0.762077	−0.0445972
$293$	−7.43221	−0.434194	−0.217097	−	0.976150i	$-0.569659\pi$
$293$	−7.43221	−0.434194	−0.217097	+	0.976150i	$0.569659\pi$
$294$	−47.7550	−2.78513
$295$	2.00504	0.116738
$296$	9.30957	0.541108
$297$	−7.51952	−0.436327
$298$	19.0583	1.10402
$299$	0.975019	0.0563868
$300$	−12.4236	−0.717274
$301$	29.5164	1.70130
$302$	53.7055	3.09041
$303$	26.5835	1.52718
$304$	−34.7070	−1.99058
$305$	−3.96271	−0.226904
$306$	−14.0625	−0.803902
$307$	0.195164	0.0111386	0.00556930	−	0.999984i	$-0.498227\pi$
$307$	0.195164	0.0111386	0.00556930	+	0.999984i	$0.498227\pi$
$308$	−91.6116	−5.22006
$309$	31.1657	1.77295
$310$	−6.37534	−0.362095
$311$	−9.98641	−0.566277	−0.283139	−	0.959079i	$-0.591376\pi$
$311$	−9.98641	−0.566277	−0.283139	+	0.959079i	$0.591376\pi$
$312$	78.4666	4.44229
$313$	5.33350	0.301467	0.150734	−	0.988574i	$-0.451836\pi$
$313$	5.33350	0.301467	0.150734	+	0.988574i	$0.451836\pi$
$314$	35.3439	1.99457
$315$	−8.64059	−0.486842
$316$	−35.8904	−2.01899
$317$	24.8111	1.39353	0.696765	−	0.717300i	$-0.254622\pi$
$317$	24.8111	1.39353	0.696765	+	0.717300i	$0.254622\pi$
$318$	0.684312	0.0383743
$319$	29.0973	1.62914
$320$	−27.9567	−1.56283
$321$	−15.2746	−0.852544
$322$	−2.76486	−0.154080
$323$	5.46649	0.304164
$324$	−57.8426	−3.21348
$325$	−3.67583	−0.203898
$326$	−6.36162	−0.352337
$327$	11.1237	0.615142
$328$	−41.1305	−2.27105
$329$	41.8246	2.30586
$330$	−27.5918	−1.51888
$331$	−15.8959	−0.873717	−0.436859	−	0.899530i	$-0.643909\pi$
$331$	−15.8959	−0.873717	−0.436859	+	0.899530i	$0.643909\pi$
$332$	75.8015	4.16015
$333$	2.25774	0.123723
$334$	−62.3206	−3.41003
$335$	−6.80664	−0.371886
$336$	−127.415	−6.95107
$337$	25.2470	1.37529	0.687647	−	0.726045i	$-0.258644\pi$
$337$	25.2470	1.37529	0.687647	+	0.726045i	$0.258644\pi$
$338$	1.39375	0.0758101
$339$	6.51125	0.353642
$340$	12.3905	0.671972
$341$	−10.3417	−0.560035
$342$	−14.6989	−0.794824
$343$	2.47493	0.133634
$344$	71.7999	3.87119
$345$	−0.608215	−0.0327452
$346$	38.1576	2.05137
$347$	23.4864	1.26082	0.630408	−	0.776264i	$-0.282888\pi$
$347$	23.4864	1.26082	0.630408	+	0.776264i	$0.282888\pi$
$348$	81.8209	4.38606
$349$	−7.13314	−0.381828	−0.190914	−	0.981607i	$-0.561145\pi$
$349$	−7.13314	−0.381828	−0.190914	+	0.981607i	$0.561145\pi$
$350$	10.4236	0.557162
$351$	−6.25621	−0.333932
$352$	−92.4551	−4.92787
$353$	−1.49954	−0.0798126	−0.0399063	−	0.999203i	$-0.512706\pi$
$353$	−1.49954	−0.0798126	−0.0399063	+	0.999203i	$0.512706\pi$
$354$	12.5218	0.665529
$355$	5.79485	0.307559
$356$	80.2631	4.25394
$357$	20.0684	1.06213
$358$	−40.2593	−2.12777
$359$	−6.47638	−0.341810	−0.170905	−	0.985287i	$-0.554669\pi$
$359$	−6.47638	−0.341810	−0.170905	+	0.985287i	$0.554669\pi$
$360$	−21.0186	−1.10778
$361$	−13.2862	−0.699271
$362$	−22.3050	−1.17233
$363$	−19.5351	−1.02532
$364$	−76.2204	−3.99504
$365$	0.140654	0.00736217
$366$	−24.7479	−1.29359
$367$	24.5122	1.27953	0.639763	−	0.768572i	$-0.279033\pi$
$367$	24.5122	1.27953	0.639763	+	0.768572i	$0.279033\pi$
$368$	−3.85132	−0.200764
$369$	−9.97490	−0.519272
$370$	−2.72362	−0.141594
$371$	−0.419351	−0.0217716
$372$	−29.0806	−1.50776
$373$	−18.9696	−0.982208	−0.491104	−	0.871101i	$-0.663407\pi$
$373$	−18.9696	−0.982208	−0.491104	+	0.871101i	$0.663407\pi$
$374$	27.5185	1.42295
$375$	2.29298	0.118409
$376$	101.740	5.24684
$377$	24.2088	1.24682
$378$	17.7407	0.912485
$379$	23.7286	1.21886	0.609428	−	0.792841i	$-0.291399\pi$
$379$	23.7286	1.21886	0.609428	+	0.792841i	$0.291399\pi$
$380$	12.9512	0.664383
$381$	−20.2207	−1.03594
$382$	54.1299	2.76953
$383$	−12.0392	−0.615173	−0.307586	−	0.951520i	$-0.599521\pi$
$383$	−12.0392	−0.615173	−0.307586	+	0.951520i	$0.599521\pi$
$384$	−78.6269	−4.01241
$385$	16.9085	0.861735
$386$	46.4078	2.36210
$387$	17.4128	0.885142
$388$	−48.4591	−2.46014
$389$	30.3044	1.53649	0.768247	−	0.640153i	$-0.221129\pi$
$389$	30.3044	1.53649	0.768247	+	0.640153i	$0.221129\pi$
$390$	−22.9563	−1.16244
$391$	0.606599	0.0306770
$392$	71.1874	3.59550
$393$	−10.5832	−0.533853
$394$	−72.2514	−3.63997
$395$	6.62418	0.333299
$396$	−54.0449	−2.71586
$397$	−26.5246	−1.33123	−0.665615	−	0.746295i	$-0.731831\pi$
$397$	−26.5246	−1.33123	−0.665615	+	0.746295i	$0.731831\pi$
$398$	−13.7730	−0.690377
$399$	20.9765	1.05014
$400$	14.5195	0.725976
$401$	15.7889	0.788462	0.394231	−	0.919011i	$-0.371011\pi$
$401$	15.7889	0.788462	0.394231	+	0.919011i	$0.371011\pi$
$402$	−42.5088	−2.12014
$403$	−8.60425	−0.428608
$404$	−62.8144	−3.12513
$405$	10.6758	0.530486
$406$	−68.6490	−3.40699
$407$	−4.41809	−0.218997
$408$	48.8172	2.41681
$409$	−9.75340	−0.482275	−0.241137	−	0.970491i	$-0.577520\pi$
$409$	−9.75340	−0.482275	−0.241137	+	0.970491i	$0.577520\pi$
$410$	12.0332	0.594277
$411$	32.4052	1.59843
$412$	−73.6417	−3.62806
$413$	−7.67348	−0.377587
$414$	−1.63109	−0.0801637
$415$	−13.9904	−0.686764
$416$	−76.9222	−3.77142
$417$	−30.3021	−1.48390
$418$	28.7637	1.40688
$419$	−9.64020	−0.470954	−0.235477	−	0.971880i	$-0.575665\pi$
$419$	−9.64020	−0.470954	−0.235477	+	0.971880i	$0.575665\pi$
$420$	47.5462	2.32002
$421$	−17.5464	−0.855161	−0.427581	−	0.903977i	$-0.640634\pi$
$421$	−17.5464	−0.855161	−0.427581	+	0.903977i	$0.640634\pi$
$422$	−53.5476	−2.60666
$423$	24.6738	1.19968
$424$	−1.02009	−0.0495399
$425$	−2.28688	−0.110930
$426$	36.1899	1.75341
$427$	15.1657	0.733920
$428$	36.0924	1.74459
$429$	−37.2383	−1.79788
$430$	−21.0059	−1.01299
$431$	2.48113	0.119512	0.0597558	−	0.998213i	$-0.480968\pi$
$431$	2.48113	0.119512	0.0597558	+	0.998213i	$0.480968\pi$
$432$	24.7120	1.18896
$433$	−18.3658	−0.882603	−0.441302	−	0.897359i	$-0.645483\pi$
$433$	−18.3658	−0.882603	−0.441302	+	0.897359i	$0.645483\pi$
$434$	24.3991	1.17119
$435$	−15.1014	−0.724058
$436$	−26.2843	−1.25879
$437$	0.634048	0.0303306
$438$	0.878412	0.0419721
$439$	23.5970	1.12622	0.563111	−	0.826381i	$-0.309604\pi$
$439$	23.5970	1.12622	0.563111	+	0.826381i	$0.309604\pi$
$440$	41.1305	1.96082
$441$	17.2642	0.822106
$442$	22.8953	1.08902
$443$	−1.45360	−0.0690625	−0.0345313	−	0.999404i	$-0.510994\pi$
$443$	−1.45360	−0.0690625	−0.0345313	+	0.999404i	$0.510994\pi$
$444$	−12.4236	−0.589596
$445$	−14.8139	−0.702247
$446$	−65.8933	−3.12014
$447$	−16.0449	−0.758900
$448$	106.993	5.05494
$449$	−38.3957	−1.81200	−0.906001	−	0.423275i	$-0.860881\pi$
$449$	−38.3957	−1.81200	−0.906001	+	0.423275i	$0.860881\pi$
$450$	6.14922	0.289877
$451$	19.5195	0.919138
$452$	−15.3855	−0.723672
$453$	−45.2140	−2.12434
$454$	49.3010	2.31381
$455$	14.0678	0.659507
$456$	51.0262	2.38952
$457$	20.1534	0.942736	0.471368	−	0.881937i	$-0.343760\pi$
$457$	20.1534	0.942736	0.471368	+	0.881937i	$0.343760\pi$
$458$	53.4326	2.49674
$459$	−3.89224	−0.181674
$460$	1.43716	0.0670078
$461$	34.6000	1.61148	0.805742	−	0.592267i	$-0.201767\pi$
$461$	34.6000	1.61148	0.805742	+	0.592267i	$0.201767\pi$
$462$	105.597	4.91280
$463$	33.7416	1.56810	0.784052	−	0.620695i	$-0.213150\pi$
$463$	33.7416	1.56810	0.784052	+	0.620695i	$0.213150\pi$
$464$	−95.6249	−4.43927
$465$	5.36731	0.248903
$466$	−12.7197	−0.589227
$467$	−32.4863	−1.50329	−0.751643	−	0.659570i	$-0.770738\pi$
$467$	−32.4863	−1.50329	−0.751643	+	0.659570i	$0.770738\pi$
$468$	−44.9651	−2.07851
$469$	26.0497	1.20286
$470$	−29.7651	−1.37296
$471$	−29.7556	−1.37106
$472$	−18.6661	−0.859175
$473$	−34.0745	−1.56675
$474$	41.3692	1.90015
$475$	−2.39037	−0.109677
$476$	−47.4198	−2.17348
$477$	−0.247390	−0.0113272
$478$	−26.2732	−1.20171
$479$	19.3799	0.885492	0.442746	−	0.896647i	$-0.354004\pi$
$479$	19.3799	0.885492	0.442746	+	0.896647i	$0.354004\pi$
$480$	47.9839	2.19016
$481$	−3.67583	−0.167603
$482$	4.35304	0.198276
$483$	2.32770	0.105914
$484$	46.1595	2.09816
$485$	8.94394	0.406123
$486$	52.7659	2.39351
$487$	2.36162	0.107015	0.0535076	−	0.998567i	$-0.482960\pi$
$487$	2.36162	0.107015	0.0535076	+	0.998567i	$0.482960\pi$
$488$	36.8912	1.66998
$489$	5.35576	0.242196
$490$	−20.8266	−0.940852
$491$	14.1469	0.638442	0.319221	−	0.947680i	$-0.396579\pi$
$491$	14.1469	0.638442	0.319221	+	0.947680i	$0.396579\pi$
$492$	54.8884	2.47456
$493$	15.0613	0.678327
$494$	23.9313	1.07672
$495$	9.97490	0.448338
$496$	33.9868	1.52605
$497$	−22.1775	−0.994796
$498$	−87.3729	−3.91528
$499$	−31.1547	−1.39468	−0.697338	−	0.716743i	$-0.745632\pi$
$499$	−31.1547	−1.39468	−0.697338	+	0.716743i	$0.745632\pi$
$500$	−5.41809	−0.242304
$501$	52.4669	2.34405
$502$	−17.6984	−0.789918
$503$	−13.3094	−0.593438	−0.296719	−	0.954965i	$-0.595892\pi$
$503$	−13.3094	−0.593438	−0.296719	+	0.954965i	$0.595892\pi$
$504$	80.4402	3.58309
$505$	11.5935	0.515902
$506$	3.19182	0.141894
$507$	−1.17338	−0.0521117
$508$	47.7796	2.11988
$509$	24.4420	1.08337	0.541686	−	0.840581i	$-0.317786\pi$
$509$	24.4420	1.08337	0.541686	+	0.840581i	$0.317786\pi$
$510$	−14.2820	−0.632419
$511$	−0.538297	−0.0238129
$512$	33.5015	1.48057
$513$	−4.06837	−0.179623
$514$	−10.9259	−0.481922
$515$	13.5918	0.598927
$516$	−95.8165	−4.21808
$517$	−48.2832	−2.12349
$518$	10.4236	0.457985
$519$	−32.1244	−1.41010
$520$	34.2204	1.50066
$521$	21.2938	0.932897	0.466448	−	0.884548i	$-0.345533\pi$
$521$	21.2938	0.932897	0.466448	+	0.884548i	$0.345533\pi$
$522$	−40.4985	−1.77257
$523$	−4.62687	−0.202319	−0.101159	−	0.994870i	$-0.532255\pi$
$523$	−4.62687	−0.202319	−0.101159	+	0.994870i	$0.532255\pi$
$524$	25.0072	1.09244
$525$	−8.77545	−0.382992
$526$	−71.5050	−3.11777
$527$	−5.35305	−0.233183
$528$	147.091	6.40132
$529$	−22.9296	−0.996941
$530$	0.298438	0.0129633
$531$	−4.52686	−0.196449
$532$	−49.5656	−2.14894
$533$	16.2402	0.703439
$534$	−92.5157	−4.00355
$535$	−6.66146	−0.288000
$536$	63.3669	2.73703
$537$	33.8937	1.46262
$538$	8.17997	0.352664
$539$	−33.7838	−1.45517
$540$	−9.22151	−0.396830
$541$	−18.9838	−0.816179	−0.408089	−	0.912942i	$-0.633805\pi$
$541$	−18.9838	−0.816179	−0.408089	+	0.912942i	$0.633805\pi$
$542$	16.1096	0.691968
$543$	18.7783	0.805855
$544$	−47.8564	−2.05183
$545$	4.85120	0.207803
$546$	87.8559	3.75988
$547$	3.76479	0.160971	0.0804855	−	0.996756i	$-0.474353\pi$
$547$	3.76479	0.160971	0.0804855	+	0.996756i	$0.474353\pi$
$548$	−76.5706	−3.27093
$549$	8.94678	0.381839
$550$	−12.0332	−0.513097
$551$	15.7428	0.670667
$552$	5.66222	0.241000
$553$	−25.3514	−1.07805
$554$	33.4239	1.42004
$555$	2.29298	0.0973314
$556$	71.6010	3.03656
$557$	−29.2177	−1.23799	−0.618996	−	0.785394i	$-0.712460\pi$
$557$	−29.2177	−1.23799	−0.618996	+	0.785394i	$0.712460\pi$
$558$	14.3939	0.609341
$559$	−28.3498	−1.19907
$560$	−55.5676	−2.34816
$561$	−23.1675	−0.978131
$562$	−9.33948	−0.393962
$563$	−17.2201	−0.725741	−0.362871	−	0.931840i	$-0.618203\pi$
$563$	−17.2201	−0.725741	−0.362871	+	0.931840i	$0.618203\pi$
$564$	−135.771	−5.71700
$565$	2.83965	0.119465
$566$	−59.4520	−2.49895
$567$	−40.8575	−1.71585
$568$	−53.9476	−2.26359
$569$	−28.1495	−1.18009	−0.590045	−	0.807371i	$-0.700890\pi$
$569$	−28.1495	−1.18009	−0.590045	+	0.807371i	$0.700890\pi$
$570$	−14.9283	−0.625277
$571$	29.0284	1.21480	0.607401	−	0.794396i	$-0.292212\pi$
$571$	29.0284	1.21480	0.607401	+	0.794396i	$0.292212\pi$
$572$	87.9906	3.67907
$573$	−45.5713	−1.90377
$574$	−46.0522	−1.92218
$575$	−0.265251	−0.0110617
$576$	63.1189	2.62995
$577$	18.3603	0.764347	0.382174	−	0.924090i	$-0.375176\pi$
$577$	18.3603	0.764347	0.382174	+	0.924090i	$0.375176\pi$
$578$	−32.0574	−1.33341
$579$	−39.0701	−1.62370
$580$	35.6833	1.48167
$581$	53.5428	2.22133
$582$	55.8566	2.31533
$583$	0.484109	0.0200497
$584$	−1.30943	−0.0541846
$585$	8.29907	0.343124
$586$	−20.2425	−0.836210
$587$	−40.9523	−1.69028	−0.845141	−	0.534544i	$-0.820484\pi$
$587$	−40.9523	−1.69028	−0.845141	+	0.534544i	$0.820484\pi$
$588$	−94.9990	−3.91769
$589$	−5.59528	−0.230549
$590$	5.46096	0.224824
$591$	60.8275	2.50211
$592$	14.5195	0.596749
$593$	12.6520	0.519557	0.259778	−	0.965668i	$-0.416350\pi$
$593$	12.6520	0.519557	0.259778	+	0.965668i	$0.416350\pi$
$594$	−20.4803	−0.840317
$595$	8.75213	0.358802
$596$	37.9127	1.55297
$597$	11.5953	0.474563
$598$	2.65558	0.108595
$599$	−22.0493	−0.900910	−0.450455	−	0.892799i	$-0.648738\pi$
$599$	−22.0493	−0.900910	−0.450455	+	0.892799i	$0.648738\pi$
$600$	−21.3466	−0.871472
$601$	0.498980	0.0203538	0.0101769	−	0.999948i	$-0.496761\pi$
$601$	0.498980	0.0203538	0.0101769	+	0.999948i	$0.496761\pi$
$602$	80.3915	3.27651
$603$	15.3676	0.625818
$604$	106.836	4.34711
$605$	−8.51952	−0.346368
$606$	72.4034	2.94119
$607$	41.2564	1.67455	0.837274	−	0.546784i	$-0.184148\pi$
$607$	41.2564	1.67455	0.837274	+	0.546784i	$0.184148\pi$
$608$	−50.0219	−2.02866
$609$	57.7947	2.34196
$610$	−10.7929	−0.436992
$611$	−40.1714	−1.62516
$612$	−27.9746	−1.13081
$613$	−4.13184	−0.166883	−0.0834417	−	0.996513i	$-0.526591\pi$
$613$	−4.13184	−0.166883	−0.0834417	+	0.996513i	$0.526591\pi$
$614$	0.531552	0.0214517
$615$	−10.1306	−0.408504
$616$	−157.411	−6.34225
$617$	37.7095	1.51813	0.759063	−	0.651017i	$-0.225657\pi$
$617$	37.7095	1.51813	0.759063	+	0.651017i	$0.225657\pi$
$618$	84.8834	3.41451
$619$	−23.7226	−0.953493	−0.476746	−	0.879041i	$-0.658184\pi$
$619$	−23.7226	−0.953493	−0.476746	+	0.879041i	$0.658184\pi$
$620$	−12.6825	−0.509340
$621$	−0.451454	−0.0181162
$622$	−27.1992	−1.09059
$623$	56.6943	2.27141
$624$	122.379	4.89909
$625$	1.00000	0.0400000
$626$	14.5264	0.580592
$627$	−24.2158	−0.967085
$628$	70.3097	2.80566
$629$	−2.28688	−0.0911840
$630$	−23.5337	−0.937604
$631$	21.4175	0.852615	0.426308	−	0.904578i	$-0.359814\pi$
$631$	21.4175	0.852615	0.426308	+	0.904578i	$0.359814\pi$
$632$	−61.6682	−2.45303
$633$	45.0810	1.79181
$634$	67.5759	2.68378
$635$	−8.81853	−0.349953
$636$	1.36130	0.0539791
$637$	−28.1079	−1.11368
$638$	79.2500	3.13754
$639$	−13.0833	−0.517566
$640$	−34.2903	−1.35544
$641$	−36.6406	−1.44722	−0.723608	−	0.690211i	$-0.757518\pi$
$641$	−36.6406	−1.44722	−0.723608	+	0.690211i	$0.757518\pi$
$642$	−41.6021	−1.64190
$643$	13.5452	0.534170	0.267085	−	0.963673i	$-0.413940\pi$
$643$	13.5452	0.534170	0.267085	+	0.963673i	$0.413940\pi$
$644$	−5.50014	−0.216736
$645$	17.6845	0.696328
$646$	14.8886	0.585785
$647$	18.8904	0.742657	0.371329	−	0.928502i	$-0.378902\pi$
$647$	18.8904	0.742657	0.371329	+	0.928502i	$0.378902\pi$
$648$	−99.3874	−3.90431
$649$	8.85844	0.347724
$650$	−10.0116	−0.392685
$651$	−20.5412	−0.805075
$652$	−12.6552	−0.495614
$653$	−7.44489	−0.291341	−0.145670	−	0.989333i	$-0.546534\pi$
$653$	−7.44489	−0.291341	−0.145670	+	0.989333i	$0.546534\pi$
$654$	30.2967	1.18469
$655$	−4.61549	−0.180342
$656$	−64.1486	−2.50458
$657$	−0.317560	−0.0123892
$658$	113.914	4.44083
$659$	5.00068	0.194799	0.0973993	−	0.995245i	$-0.468948\pi$
$659$	5.00068	0.194799	0.0973993	+	0.995245i	$0.468948\pi$
$660$	−54.8884	−2.13653
$661$	−7.47493	−0.290741	−0.145371	−	0.989377i	$-0.546437\pi$
$661$	−7.47493	−0.290741	−0.145371	+	0.989377i	$0.546437\pi$
$662$	−43.2943	−1.68268
$663$	−19.2752	−0.748587
$664$	130.245	5.05448
$665$	9.14817	0.354751
$666$	6.14922	0.238277
$667$	1.74693	0.0676415
$668$	−123.974	−4.79671
$669$	55.4747	2.14477
$670$	−18.5387	−0.716212
$671$	−17.5076	−0.675875
$672$	−183.639	−7.08404
$673$	−38.6412	−1.48951	−0.744755	−	0.667338i	$-0.767434\pi$
$673$	−38.6412	−1.48951	−0.744755	+	0.667338i	$0.767434\pi$
$674$	68.7632	2.64866
$675$	1.70198	0.0655094
$676$	2.77259	0.106638
$677$	43.8791	1.68641	0.843206	−	0.537590i	$-0.180665\pi$
$677$	43.8791	1.68641	0.843206	+	0.537590i	$0.180665\pi$
$678$	17.7341	0.681076
$679$	−34.2293	−1.31360
$680$	21.2899	0.816431
$681$	−41.5059	−1.59051
$682$	−28.1668	−1.07856
$683$	18.8393	0.720867	0.360433	−	0.932785i	$-0.382629\pi$
$683$	18.8393	0.720867	0.360433	+	0.932785i	$0.382629\pi$
$684$	−29.2405	−1.11804
$685$	14.1324	0.539971
$686$	6.74077	0.257364
$687$	−44.9842	−1.71625
$688$	111.982	4.26926
$689$	0.402776	0.0153445
$690$	−1.65655	−0.0630636
$691$	−42.3770	−1.61210	−0.806048	−	0.591850i	$-0.798398\pi$
$691$	−42.3770	−1.61210	−0.806048	+	0.591850i	$0.798398\pi$
$692$	75.9069	2.88555
$693$	−38.1749	−1.45015
$694$	63.9680	2.42819
$695$	−13.2152	−0.501280
$696$	140.588	5.32897
$697$	10.1037	0.382703
$698$	−19.4279	−0.735359
$699$	10.7085	0.405033
$700$	20.7356	0.783731
$701$	16.1386	0.609546	0.304773	−	0.952425i	$-0.401419\pi$
$701$	16.1386	0.609546	0.304773	+	0.952425i	$0.401419\pi$
$702$	−17.0395	−0.643115
$703$	−2.39037	−0.0901543
$704$	−123.515	−4.65515
$705$	25.0589	0.943771
$706$	−4.08418	−0.153710
$707$	−44.3693	−1.66868
$708$	24.9097	0.936164
$709$	−20.5619	−0.772218	−0.386109	−	0.922453i	$-0.626181\pi$
$709$	−20.5619	−0.772218	−0.386109	+	0.922453i	$0.626181\pi$
$710$	15.7830	0.592324
$711$	−14.9557	−0.560881
$712$	137.911	5.16844
$713$	−0.620891	−0.0232525
$714$	54.6587	2.04555
$715$	−16.2402	−0.607347
$716$	−80.0877	−2.99302
$717$	22.1191	0.826052
$718$	−17.6392	−0.658288
$719$	5.93966	0.221512	0.110756	−	0.993848i	$-0.464673\pi$
$719$	5.93966	0.221512	0.110756	+	0.993848i	$0.464673\pi$
$720$	−32.7813	−1.22169
$721$	−52.0172	−1.93722
$722$	−36.1864	−1.34672
$723$	−3.66477	−0.136294
$724$	−44.3714	−1.64905
$725$	−6.58595	−0.244596
$726$	−53.2060	−1.97466
$727$	−24.8976	−0.923401	−0.461701	−	0.887036i	$-0.652761\pi$
$727$	−24.8976	−0.923401	−0.461701	+	0.887036i	$0.652761\pi$
$728$	−130.965	−4.85388
$729$	−12.3954	−0.459089
$730$	0.383088	0.0141787
$731$	−17.6376	−0.652349
$732$	−49.2310	−1.81963
$733$	20.8362	0.769604	0.384802	−	0.922999i	$-0.374270\pi$
$733$	20.8362	0.769604	0.384802	+	0.922999i	$0.374270\pi$
$734$	66.7618	2.46422
$735$	17.5337	0.646739
$736$	−5.55078	−0.204604
$737$	−30.0724	−1.10773
$738$	−27.1678	−1.00006
$739$	−39.3500	−1.44751	−0.723756	−	0.690056i	$-0.757586\pi$
$739$	−39.3500	−1.44751	−0.723756	+	0.690056i	$0.757586\pi$
$740$	−5.41809	−0.199173
$741$	−20.1474	−0.740134
$742$	−1.14215	−0.0419297
$743$	3.25437	0.119391	0.0596955	−	0.998217i	$-0.480987\pi$
$743$	3.25437	0.119391	0.0596955	+	0.998217i	$0.480987\pi$
$744$	−49.9674	−1.83189
$745$	−6.99743	−0.256366
$746$	−51.6659	−1.89162
$747$	31.5868	1.15570
$748$	54.7425	2.00159
$749$	25.4941	0.931533
$750$	6.24519	0.228042
$751$	45.4895	1.65994	0.829968	−	0.557811i	$-0.188358\pi$
$751$	45.4895	1.65994	0.829968	+	0.557811i	$0.188358\pi$
$752$	158.677	5.78636
$753$	14.9000	0.542988
$754$	65.9356	2.40123
$755$	−19.7185	−0.717628
$756$	35.2916	1.28354
$757$	31.5740	1.14758	0.573788	−	0.819004i	$-0.305473\pi$
$757$	31.5740	1.14758	0.573788	+	0.819004i	$0.305473\pi$
$758$	64.6276	2.34738
$759$	−2.68715	−0.0975374
$760$	22.2533	0.807211
$761$	41.8552	1.51725	0.758626	−	0.651527i	$-0.225871\pi$
$761$	41.8552	1.51725	0.758626	+	0.651527i	$0.225871\pi$
$762$	−55.0734	−1.99510
$763$	−18.5660	−0.672136
$764$	107.681	3.89575
$765$	5.16319	0.186675
$766$	−32.7901	−1.18475
$767$	7.37018	0.266122
$768$	−85.9416	−3.10115
$769$	−22.1619	−0.799180	−0.399590	−	0.916694i	$-0.630848\pi$
$769$	−22.1619	−0.799180	−0.399590	+	0.916694i	$0.630848\pi$
$770$	46.0522	1.65961
$771$	9.19840	0.331272
$772$	92.3191	3.32264
$773$	19.3276	0.695165	0.347582	−	0.937649i	$-0.387003\pi$
$773$	19.3276	0.695165	0.347582	+	0.937649i	$0.387003\pi$
$774$	47.4258	1.70468
$775$	2.34076	0.0840827
$776$	−83.2642	−2.98901
$777$	−8.77545	−0.314817
$778$	82.5376	2.95912
$779$	10.5609	0.378382
$780$	−45.6669	−1.63514
$781$	25.6022	0.916118
$782$	1.65214	0.0590806
$783$	−11.2092	−0.400584
$784$	111.026	3.96522
$785$	−12.9768	−0.463163
$786$	−28.8246	−1.02814
$787$	19.5111	0.695496	0.347748	−	0.937588i	$-0.386946\pi$
$787$	19.5111	0.695496	0.347748	+	0.937588i	$0.386946\pi$
$788$	−143.730	−5.12016
$789$	60.1991	2.14314
$790$	18.0417	0.641896
$791$	−10.8676	−0.386408
$792$	−92.8620	−3.29971
$793$	−14.5663	−0.517263
$794$	−72.2428	−2.56380
$795$	−0.251251	−0.00891096
$796$	−27.3986	−0.971117
$797$	−22.2367	−0.787664	−0.393832	−	0.919183i	$-0.628851\pi$
$797$	−22.2367	−0.787664	−0.393832	+	0.919183i	$0.628851\pi$
$798$	57.1320	2.02245
$799$	−24.9923	−0.884163
$800$	20.9265	0.739863
$801$	33.4460	1.18176
$802$	43.0030	1.51849
$803$	0.621423	0.0219295
$804$	−84.5627	−2.98229
$805$	1.01514	0.0357791
$806$	−23.4347	−0.825451
$807$	−6.88661	−0.242420
$808$	−107.930	−3.79697
$809$	12.4896	0.439110	0.219555	−	0.975600i	$-0.429539\pi$
$809$	12.4896	0.439110	0.219555	+	0.975600i	$0.429539\pi$
$810$	29.0769	1.02166
$811$	−3.99535	−0.140296	−0.0701478	−	0.997537i	$-0.522347\pi$
$811$	−3.99535	−0.140296	−0.0701478	+	0.997537i	$0.522347\pi$
$812$	−136.563	−4.79244
$813$	−13.5625	−0.475657
$814$	−12.0332	−0.421763
$815$	2.33572	0.0818168
$816$	76.1370	2.66533
$817$	−18.4357	−0.644982
$818$	−26.5645	−0.928807
$819$	−31.7614	−1.10983
$820$	23.9376	0.835938
$821$	−1.51952	−0.0530317	−0.0265159	−	0.999648i	$-0.508441\pi$
$821$	−1.51952	−0.0530317	−0.0265159	+	0.999648i	$0.508441\pi$
$822$	88.2595	3.07840
$823$	46.9850	1.63779	0.818897	−	0.573940i	$-0.194586\pi$
$823$	46.9850	1.63779	0.818897	+	0.573940i	$0.194586\pi$
$824$	−126.534	−4.40802
$825$	10.1306	0.352701
$826$	−20.8996	−0.727191
$827$	−10.5199	−0.365814	−0.182907	−	0.983130i	$-0.558551\pi$
$827$	−10.5199	−0.365814	−0.182907	+	0.983130i	$0.558551\pi$
$828$	−3.24473	−0.112762
$829$	−53.8789	−1.87129	−0.935645	−	0.352942i	$-0.885181\pi$
$829$	−53.8789	−1.87129	−0.935645	+	0.352942i	$0.885181\pi$
$830$	−38.1046	−1.32263
$831$	−28.1391	−0.976134
$832$	−102.764	−3.56270
$833$	−17.4871	−0.605892
$834$	−82.5312	−2.85782
$835$	22.8816	0.791849
$836$	57.2196	1.97898
$837$	3.98394	0.137705
$838$	−26.2562	−0.907006
$839$	12.0120	0.414699	0.207349	−	0.978267i	$-0.433516\pi$
$839$	12.0120	0.414699	0.207349	+	0.978267i	$0.433516\pi$
$840$	81.6956	2.81877
$841$	14.3748	0.495682
$842$	−47.7898	−1.64694
$843$	7.86278	0.270808
$844$	−106.522	−3.66665
$845$	−0.511728	−0.0176040
$846$	67.2019	2.31045
$847$	32.6051	1.12032
$848$	−1.59096	−0.0546340
$849$	50.0518	1.71777
$850$	−6.22860	−0.213639
$851$	−0.265251	−0.00909270
$852$	71.9926	2.46643
$853$	−0.967727	−0.0331343	−0.0165672	−	0.999863i	$-0.505274\pi$
$853$	−0.967727	−0.0331343	−0.0165672	+	0.999863i	$0.505274\pi$
$854$	41.3056	1.41345
$855$	5.39682	0.184567
$856$	62.0154	2.11964
$857$	0.683897	0.0233615	0.0116807	−	0.999932i	$-0.496282\pi$
$857$	0.683897	0.0233615	0.0116807	+	0.999932i	$0.496282\pi$
$858$	−101.423	−3.46252
$859$	23.9043	0.815605	0.407803	−	0.913070i	$-0.366295\pi$
$859$	23.9043	0.815605	0.407803	+	0.913070i	$0.366295\pi$
$860$	−41.7869	−1.42492
$861$	38.7707	1.32130
$862$	6.75764	0.230166
$863$	51.4456	1.75123	0.875614	−	0.483012i	$-0.160457\pi$
$863$	51.4456	1.75123	0.875614	+	0.483012i	$0.160457\pi$
$864$	35.6166	1.21170
$865$	−14.0099	−0.476351
$866$	−50.0214	−1.69979
$867$	26.9887	0.916584
$868$	48.5371	1.64745
$869$	29.2662	0.992788
$870$	−41.1305	−1.39446
$871$	−25.0201	−0.847772
$872$	−45.1626	−1.52940
$873$	−20.1931	−0.683433
$874$	1.72690	0.0584134
$875$	−3.82710	−0.129380
$876$	1.74742	0.0590400
$877$	−29.8721	−1.00871	−0.504354	−	0.863497i	$-0.668269\pi$
$877$	−29.8721	−1.00871	−0.504354	+	0.863497i	$0.668269\pi$
$878$	64.2691	2.16898
$879$	17.0419	0.574808
$880$	64.1486	2.16245
$881$	25.1190	0.846279	0.423140	−	0.906064i	$-0.360928\pi$
$881$	25.1190	0.846279	0.423140	+	0.906064i	$0.360928\pi$
$882$	47.0211	1.58328
$883$	39.3961	1.32578	0.662892	−	0.748715i	$-0.269329\pi$
$883$	39.3961	1.32578	0.662892	+	0.748715i	$0.269329\pi$
$884$	45.5455	1.53186
$885$	−4.59751	−0.154543
$886$	−3.95904	−0.133007
$887$	18.7079	0.628148	0.314074	−	0.949398i	$-0.398306\pi$
$887$	18.7079	0.628148	0.314074	+	0.949398i	$0.398306\pi$
$888$	−21.3466	−0.716346
$889$	33.7494	1.13192
$890$	−40.3474	−1.35245
$891$	47.1668	1.58015
$892$	−131.082	−4.38893
$893$	−26.1232	−0.874179
$894$	−43.7003	−1.46156
$895$	14.7815	0.494092
$896$	131.232	4.38417
$897$	−2.23570	−0.0746477
$898$	−104.575	−3.48971
$899$	−15.4162	−0.514158
$900$	12.2326	0.407755
$901$	0.250583	0.00834815
$902$	53.1637	1.77016
$903$	−67.6805	−2.25227
$904$	−26.4359	−0.879245
$905$	8.18949	0.272228
$906$	−123.146	−4.09124
$907$	16.8702	0.560164	0.280082	−	0.959976i	$-0.409638\pi$
$907$	16.8702	0.560164	0.280082	+	0.959976i	$0.409638\pi$
$908$	98.0745	3.25472
$909$	−26.1750	−0.868171
$910$	38.3152	1.27014
$911$	32.4128	1.07388	0.536941	−	0.843620i	$-0.319580\pi$
$911$	32.4128	1.07388	0.536941	+	0.843620i	$0.319580\pi$
$912$	79.5823	2.63523
$913$	−61.8110	−2.04565
$914$	54.8901	1.81560
$915$	9.08641	0.300387
$916$	106.293	3.51203
$917$	17.6639	0.583315
$918$	−10.6010	−0.349884
$919$	58.0857	1.91607	0.958035	−	0.286650i	$-0.0925418\pi$
$919$	58.0857	1.91607	0.958035	+	0.286650i	$0.0925418\pi$
$920$	2.46938	0.0814129
$921$	−0.447506	−0.0147458
$922$	94.2372	3.10354
$923$	21.3009	0.701127
$924$	210.063	6.91057
$925$	1.00000	0.0328798
$926$	91.8991	3.01999
$927$	−30.6868	−1.00789
$928$	−137.821	−4.52419
$929$	−7.38274	−0.242220	−0.121110	−	0.992639i	$-0.538645\pi$
$929$	−7.38274	−0.242220	−0.121110	+	0.992639i	$0.538645\pi$
$930$	14.6185	0.479360
$931$	−18.2784	−0.599050
$932$	−25.3032	−0.828834
$933$	22.8986	0.749666
$934$	−88.4802	−2.89516
$935$	−10.1037	−0.330425
$936$	−77.2608	−2.52535
$937$	−34.2781	−1.11982	−0.559908	−	0.828555i	$-0.689163\pi$
$937$	−34.2781	−1.11982	−0.559908	+	0.828555i	$0.689163\pi$
$938$	70.9494	2.31658
$939$	−12.2296	−0.399097
$940$	−59.2118	−1.93127
$941$	8.74328	0.285023	0.142511	−	0.989793i	$-0.454482\pi$
$941$	8.74328	0.285023	0.142511	+	0.989793i	$0.454482\pi$
$942$	−81.0428	−2.64052
$943$	1.17190	0.0381625
$944$	−29.1122	−0.947522
$945$	−6.51366	−0.211890
$946$	−92.8058	−3.01738
$947$	41.2549	1.34060	0.670302	−	0.742089i	$-0.266165\pi$
$947$	41.2549	1.34060	0.670302	+	0.742089i	$0.266165\pi$
$948$	82.2958	2.67284
$949$	0.517021	0.0167832
$950$	−6.51044	−0.211227
$951$	−56.8912	−1.84482
$952$	−81.4786	−2.64074
$953$	13.2048	0.427745	0.213873	−	0.976862i	$-0.431392\pi$
$953$	13.2048	0.427745	0.213873	+	0.976862i	$0.431392\pi$
$954$	−0.673796	−0.0218149
$955$	−19.8743	−0.643117
$956$	−52.2654	−1.69038
$957$	−66.7195	−2.15673
$958$	52.7835	1.70536
$959$	−54.0861	−1.74653
$960$	64.1040	2.06895
$961$	−25.5208	−0.823253
$962$	−10.0116	−0.322785
$963$	15.0399	0.484652
$964$	8.65950	0.278904
$965$	−17.0390	−0.548506
$966$	6.33976	0.203979
$967$	12.1924	0.392082	0.196041	−	0.980596i	$-0.437191\pi$
$967$	12.1924	0.392082	0.196041	+	0.980596i	$0.437191\pi$
$968$	79.3131	2.54922
$969$	−12.5345	−0.402667
$970$	24.3599	0.782148
$971$	6.80253	0.218303	0.109152	−	0.994025i	$-0.465187\pi$
$971$	6.80253	0.218303	0.109152	+	0.994025i	$0.465187\pi$
$972$	104.967	3.36682
$973$	50.5758	1.62138
$974$	6.43214	0.206099
$975$	8.42859	0.269931
$976$	57.5367	1.84171
$977$	55.4080	1.77266	0.886329	−	0.463056i	$-0.153247\pi$
$977$	55.4080	1.77266	0.886329	+	0.463056i	$0.153247\pi$
$978$	14.5870	0.466442
$979$	−65.4492	−2.09177
$980$	−41.4304	−1.32345
$981$	−10.9528	−0.349695
$982$	38.5308	1.22957
$983$	−13.4921	−0.430331	−0.215165	−	0.976578i	$-0.569029\pi$
$983$	−13.4921	−0.430331	−0.215165	+	0.976578i	$0.569029\pi$
$984$	94.3113	3.00653
$985$	26.5277	0.845244
$986$	41.0212	1.30638
$987$	−95.9027	−3.05262
$988$	47.6065	1.51456
$989$	−2.04575	−0.0650510
$990$	27.1678	0.863450
$991$	−2.47900	−0.0787481	−0.0393740	−	0.999225i	$-0.512536\pi$
$991$	−2.47900	−0.0787481	−0.0393740	+	0.999225i	$0.512536\pi$
$992$	48.9839	1.55524
$993$	36.4489	1.15667
$994$	−60.4029	−1.91587
$995$	5.05687	0.160314
$996$	−173.811	−5.50741
$997$	−13.6685	−0.432885	−0.216443	−	0.976295i	$-0.569445\pi$
$997$	−13.6685	−0.432885	−0.216443	+	0.976295i	$0.569445\pi$
$998$	−84.8534	−2.68599
$999$	1.70198	0.0538484

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.a.e.1.5	✓	5
3.2	odd	2		1665.2.a.p.1.1		5
4.3	odd	2		2960.2.a.w.1.5		5
5.2	odd	4		925.2.b.f.149.10		10
5.3	odd	4		925.2.b.f.149.1		10
5.4	even	2		925.2.a.f.1.1		5
7.6	odd	2		9065.2.a.k.1.5		5
15.14	odd	2		8325.2.a.ch.1.5		5
37.36	even	2		6845.2.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.5	✓	5	1.1	even	1	trivial
925.2.a.f.1.1		5	5.4	even	2
925.2.b.f.149.1		10	5.3	odd	4
925.2.b.f.149.10		10	5.2	odd	4
1665.2.a.p.1.1		5	3.2	odd	2
2960.2.a.w.1.5		5	4.3	odd	2
6845.2.a.f.1.1		5	37.36	even	2
8325.2.a.ch.1.5		5	15.14	odd	2
9065.2.a.k.1.5		5	7.6	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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	185.2.a.e.1.5

Level	$185$
Weight	$2$
Character	185.1
Self dual	yes
Analytic conductor	$1.477$
Analytic rank	$0$
Dimension	$5$
CM	no
Inner twists	$1$