LMFDB - Embedded newform 185.2.a.d.1.3

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)

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")

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);

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

Newform invariants

comment:select newform

sage:traces = [5,0] 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:	5.5.368464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4$ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
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			1.3
Root			$0.552543$ 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)

$f(q)$	$=$	$q+0.180152 q^{2} -3.06709 q^{3} -1.96755 q^{4} +1.00000 q^{5} -0.552543 q^{6} +4.41500 q^{7} -0.714762 q^{8} +6.40702 q^{9} +0.180152 q^{10} +4.27171 q^{11} +6.03463 q^{12} -2.79978 q^{13} +0.795373 q^{14} -3.06709 q^{15} +3.80632 q^{16} -4.43948 q^{17} +1.15424 q^{18} +2.43507 q^{19} -1.96755 q^{20} -13.5412 q^{21} +0.769559 q^{22} +5.77387 q^{23} +2.19224 q^{24} +1.00000 q^{25} -0.504387 q^{26} -10.4496 q^{27} -8.68672 q^{28} +0.409254 q^{29} -0.552543 q^{30} +7.79180 q^{31} +2.11524 q^{32} -13.1017 q^{33} -0.799782 q^{34} +4.41500 q^{35} -12.6061 q^{36} -1.00000 q^{37} +0.438683 q^{38} +8.58717 q^{39} -0.714762 q^{40} +0.757374 q^{41} -2.43948 q^{42} +2.19908 q^{43} -8.40479 q^{44} +6.40702 q^{45} +1.04018 q^{46} -4.26487 q^{47} -11.6743 q^{48} +12.4922 q^{49} +0.180152 q^{50} +13.6163 q^{51} +5.50870 q^{52} -0.137540 q^{53} -1.88253 q^{54} +4.27171 q^{55} -3.15568 q^{56} -7.46857 q^{57} +0.0737281 q^{58} -3.07119 q^{59} +6.03463 q^{60} -3.02909 q^{61} +1.40371 q^{62} +28.2870 q^{63} -7.23158 q^{64} -2.79978 q^{65} -2.36030 q^{66} -11.4482 q^{67} +8.73487 q^{68} -17.7090 q^{69} +0.795373 q^{70} -10.7144 q^{71} -4.57950 q^{72} +8.20680 q^{73} -0.180152 q^{74} -3.06709 q^{75} -4.79111 q^{76} +18.8596 q^{77} +1.54700 q^{78} +7.11193 q^{79} +3.80632 q^{80} +12.8289 q^{81} +0.136443 q^{82} -11.3625 q^{83} +26.6429 q^{84} -4.43948 q^{85} +0.396170 q^{86} -1.25522 q^{87} -3.05326 q^{88} -16.2305 q^{89} +1.15424 q^{90} -12.3610 q^{91} -11.3603 q^{92} -23.8981 q^{93} -0.768326 q^{94} +2.43507 q^{95} -6.48763 q^{96} +18.3399 q^{97} +2.25051 q^{98} +27.3690 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} + 7 q^{7} - 6 q^{8} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 4 q^{14} - q^{15} + 8 q^{16} - 8 q^{17} - 6 q^{18} + 14 q^{19} + 6 q^{20} - 9 q^{21} + 2 q^{22} + 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100})$ 5 * q - q^3 + 6 * q^4 + 5 * q^5 - 2 * q^6 + 7 * q^7 - 6 * q^8 + 2 * q^9 + 7 * q^11 + 2 * q^13 + 4 * q^14 - q^15 + 8 * q^16 - 8 * q^17 - 6 * q^18 + 14 * q^19 + 6 * q^20 - 9 * q^21 + 2 * q^22 + 2 * q^23 + 12 * q^24 + 5 * q^25 - 20 * q^26 - 7 * q^27 - 14 * q^28 + 2 * q^29 - 2 * q^30 + 8 * q^31 - 22 * q^32 - 21 * q^33 + 12 * q^34 + 7 * q^35 - 36 * q^36 - 5 * q^37 - 32 * q^38 + 12 * q^39 - 6 * q^40 - 9 * q^41 + 2 * q^42 + 14 * q^43 + 18 * q^44 + 2 * q^45 - 28 * q^46 - 5 * q^47 - 30 * q^48 + 2 * q^49 + 10 * q^51 + 20 * q^52 - 15 * q^53 - 12 * q^54 + 7 * q^55 - 14 * q^56 + 4 * q^57 + 12 * q^59 + 12 * q^61 + 10 * q^62 + 24 * q^63 + 12 * q^64 + 2 * q^65 - 10 * q^66 - 2 * q^67 - 4 * q^68 - 30 * q^69 + 4 * q^70 + 13 * q^71 + 26 * q^72 - 5 * q^73 - q^75 + 18 * q^76 - 7 * q^77 + 10 * q^78 + 36 * q^79 + 8 * q^80 + 21 * q^81 - 2 * q^82 - 9 * q^83 + 30 * q^84 - 8 * q^85 - 8 * q^86 - 6 * q^87 + 8 * q^88 - 16 * q^89 - 6 * q^90 - 4 * q^91 + 12 * q^92 - 16 * q^93 + 20 * q^94 + 14 * q^95 + 30 * q^98 + 18 * 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

)

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

$2$	0.180152	0.127387	0.0636934	−	0.997970i	$-0.479712\pi$
$2$	0.180152	0.127387	0.0636934	+	0.997970i	$0.479712\pi$
$3$	−3.06709	−1.77078	−0.885392	−	0.464846i	$-0.846110\pi$
$3$	−3.06709	−1.77078	−0.885392	+	0.464846i	$0.846110\pi$
$4$	−1.96755	−0.983773
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.552543	−0.225575
$7$	4.41500	1.66871	0.834357	−	0.551224i	$-0.185839\pi$
$7$	4.41500	1.66871	0.834357	+	0.551224i	$0.185839\pi$
$8$	−0.714762	−0.252707
$9$	6.40702	2.13567
$10$	0.180152	0.0569692
$11$	4.27171	1.28797	0.643985	−	0.765038i	$-0.277280\pi$
$11$	4.27171	1.28797	0.643985	+	0.765038i	$0.277280\pi$
$12$	6.03463	1.74205
$13$	−2.79978	−0.776520	−0.388260	−	0.921550i	$-0.626924\pi$
$13$	−2.79978	−0.776520	−0.388260	+	0.921550i	$0.626924\pi$
$14$	0.795373	0.212572
$15$	−3.06709	−0.791918
$16$	3.80632	0.951581
$17$	−4.43948	−1.07673	−0.538366	−	0.842711i	$-0.680958\pi$
$17$	−4.43948	−1.07673	−0.538366	+	0.842711i	$0.680958\pi$
$18$	1.15424	0.272057
$19$	2.43507	0.558643	0.279321	−	0.960198i	$-0.409890\pi$
$19$	2.43507	0.558643	0.279321	+	0.960198i	$0.409890\pi$
$20$	−1.96755	−0.439956
$21$	−13.5412	−2.95493
$22$	0.769559	0.164071
$23$	5.77387	1.20393	0.601967	−	0.798521i	$-0.294384\pi$
$23$	5.77387	1.20393	0.601967	+	0.798521i	$0.294384\pi$
$24$	2.19224	0.447489
$25$	1.00000	0.200000
$26$	−0.504387	−0.0989184
$27$	−10.4496	−2.01103
$28$	−8.68672	−1.64164
$29$	0.409254	0.0759967	0.0379983	−	0.999278i	$-0.487902\pi$
$29$	0.409254	0.0759967	0.0379983	+	0.999278i	$0.487902\pi$
$30$	−0.552543	−0.100880
$31$	7.79180	1.39945	0.699724	−	0.714413i	$-0.253306\pi$
$31$	7.79180	1.39945	0.699724	+	0.714413i	$0.253306\pi$
$32$	2.11524	0.373926
$33$	−13.1017	−2.28072
$34$	−0.799782	−0.137161
$35$	4.41500	0.746272
$36$	−12.6061	−2.10102
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0.438683	0.0711638
$39$	8.58717	1.37505
$40$	−0.714762	−0.113014
$41$	0.757374	0.118282	0.0591410	−	0.998250i	$-0.481164\pi$
$41$	0.757374	0.118282	0.0591410	+	0.998250i	$0.481164\pi$
$42$	−2.43948	−0.376420
$43$	2.19908	0.335357	0.167679	−	0.985842i	$-0.446373\pi$
$43$	2.19908	0.335357	0.167679	+	0.985842i	$0.446373\pi$
$44$	−8.40479	−1.26707
$45$	6.40702	0.955103
$46$	1.04018	0.153366
$47$	−4.26487	−0.622095	−0.311048	−	0.950394i	$-0.600680\pi$
$47$	−4.26487	−0.622095	−0.311048	+	0.950394i	$0.600680\pi$
$48$	−11.6743	−1.68504
$49$	12.4922	1.78461
$50$	0.180152	0.0254774
$51$	13.6163	1.90666
$52$	5.50870	0.763919
$53$	−0.137540	−0.0188926	−0.00944630	−	0.999955i	$-0.503007\pi$
$53$	−0.137540	−0.0188926	−0.00944630	+	0.999955i	$0.503007\pi$
$54$	−1.88253	−0.256179
$55$	4.27171	0.575998
$56$	−3.15568	−0.421695
$57$	−7.46857	−0.989236
$58$	0.0737281	0.00968098
$59$	−3.07119	−0.399835	−0.199918	−	0.979813i	$-0.564067\pi$
$59$	−3.07119	−0.399835	−0.199918	+	0.979813i	$0.564067\pi$
$60$	6.03463	0.779068
$61$	−3.02909	−0.387835	−0.193918	−	0.981018i	$-0.562119\pi$
$61$	−3.02909	−0.387835	−0.193918	+	0.981018i	$0.562119\pi$
$62$	1.40371	0.178271
$63$	28.2870	3.56383
$64$	−7.23158	−0.903948
$65$	−2.79978	−0.347270
$66$	−2.36030	−0.290533
$67$	−11.4482	−1.39862	−0.699310	−	0.714819i	$-0.746509\pi$
$67$	−11.4482	−1.39862	−0.699310	+	0.714819i	$0.746509\pi$
$68$	8.73487	1.05926
$69$	−17.7090	−2.13191
$70$	0.795373	0.0950652
$71$	−10.7144	−1.27156	−0.635781	−	0.771870i	$-0.719322\pi$
$71$	−10.7144	−1.27156	−0.635781	+	0.771870i	$0.719322\pi$
$72$	−4.57950	−0.539699
$73$	8.20680	0.960534	0.480267	−	0.877122i	$-0.340540\pi$
$73$	8.20680	0.960534	0.480267	+	0.877122i	$0.340540\pi$
$74$	−0.180152	−0.0209423
$75$	−3.06709	−0.354157
$76$	−4.79111	−0.549578
$77$	18.8596	2.14925
$78$	1.54700	0.175163
$79$	7.11193	0.800155	0.400077	−	0.916481i	$-0.368983\pi$
$79$	7.11193	0.800155	0.400077	+	0.916481i	$0.368983\pi$
$80$	3.80632	0.425560
$81$	12.8289	1.42543
$82$	0.136443	0.0150676
$83$	−11.3625	−1.24719	−0.623597	−	0.781746i	$-0.714329\pi$
$83$	−11.3625	−1.24719	−0.623597	+	0.781746i	$0.714329\pi$
$84$	26.6429	2.90698
$85$	−4.43948	−0.481529
$86$	0.396170	0.0427201
$87$	−1.25522	−0.134574
$88$	−3.05326	−0.325479
$89$	−16.2305	−1.72043	−0.860214	−	0.509933i	$-0.829670\pi$
$89$	−16.2305	−1.72043	−0.860214	+	0.509933i	$0.829670\pi$
$90$	1.15424	0.121668
$91$	−12.3610	−1.29579
$92$	−11.3603	−1.18440
$93$	−23.8981	−2.47812
$94$	−0.768326	−0.0792468
$95$	2.43507	0.249833
$96$	−6.48763	−0.662141
$97$	18.3399	1.86213	0.931066	−	0.364850i	$-0.118880\pi$
$97$	18.3399	1.86213	0.931066	+	0.364850i	$0.118880\pi$
$98$	2.25051	0.227336
$99$	27.3690	2.75068
$100$	−1.96755	−0.196755
$101$	−11.1319	−1.10767	−0.553835	−	0.832627i	$-0.686836\pi$
$101$	−11.1319	−1.10767	−0.553835	+	0.832627i	$0.686836\pi$
$102$	2.45300	0.242883
$103$	6.78105	0.668157	0.334079	−	0.942545i	$-0.391575\pi$
$103$	6.78105	0.668157	0.334079	+	0.942545i	$0.391575\pi$
$104$	2.00118	0.196232
$105$	−13.5412	−1.32149
$106$	−0.0247782	−0.00240667
$107$	0.536583	0.0518734	0.0259367	−	0.999664i	$-0.491743\pi$
$107$	0.536583	0.0518734	0.0259367	+	0.999664i	$0.491743\pi$
$108$	20.5601	1.97840
$109$	9.53465	0.913254	0.456627	−	0.889658i	$-0.349057\pi$
$109$	9.53465	0.913254	0.456627	+	0.889658i	$0.349057\pi$
$110$	0.769559	0.0733746
$111$	3.06709	0.291115
$112$	16.8049	1.58792
$113$	9.16830	0.862481	0.431240	−	0.902237i	$-0.358076\pi$
$113$	9.16830	0.862481	0.431240	+	0.902237i	$0.358076\pi$
$114$	−1.34548	−0.126016
$115$	5.77387	0.538416
$116$	−0.805227	−0.0747634
$117$	−17.9383	−1.65839
$118$	−0.553282	−0.0509338
$119$	−19.6003	−1.79676
$120$	2.19224	0.200123
$121$	7.24754	0.658867
$122$	−0.545697	−0.0494051
$123$	−2.32293	−0.209452
$124$	−15.3307	−1.37674
$125$	1.00000	0.0894427
$126$	5.09597	0.453985
$127$	9.32621	0.827567	0.413784	−	0.910375i	$-0.364207\pi$
$127$	9.32621	0.827567	0.413784	+	0.910375i	$0.364207\pi$
$128$	−5.53327	−0.489077
$129$	−6.74478	−0.593845
$130$	−0.504387	−0.0442377
$131$	15.6006	1.36303	0.681515	−	0.731804i	$-0.261321\pi$
$131$	15.6006	1.36303	0.681515	+	0.731804i	$0.261321\pi$
$132$	25.7782	2.24371
$133$	10.7508	0.932215
$134$	−2.06242	−0.178166
$135$	−10.4496	−0.899361
$136$	3.17317	0.272097
$137$	−1.01986	−0.0871326	−0.0435663	−	0.999051i	$-0.513872\pi$
$137$	−1.01986	−0.0871326	−0.0435663	+	0.999051i	$0.513872\pi$
$138$	−3.19031	−0.271577
$139$	7.08913	0.601292	0.300646	−	0.953736i	$-0.402798\pi$
$139$	7.08913	0.601292	0.300646	+	0.953736i	$0.402798\pi$
$140$	−8.68672	−0.734162
$141$	13.0807	1.10160
$142$	−1.93022	−0.161980
$143$	−11.9599	−1.00013
$144$	24.3872	2.03227
$145$	0.409254	0.0339867
$146$	1.47847	0.122359
$147$	−38.3148	−3.16015
$148$	1.96755	0.161731
$149$	−15.5572	−1.27449	−0.637246	−	0.770660i	$-0.719926\pi$
$149$	−15.5572	−1.27449	−0.637246	+	0.770660i	$0.719926\pi$
$150$	−0.552543	−0.0451149
$151$	−4.00882	−0.326233	−0.163117	−	0.986607i	$-0.552155\pi$
$151$	−4.00882	−0.326233	−0.163117	+	0.986607i	$0.552155\pi$
$152$	−1.74049	−0.141173
$153$	−28.4438	−2.29955
$154$	3.39761	0.273787
$155$	7.79180	0.625853
$156$	−16.8957	−1.35273
$157$	3.06585	0.244682	0.122341	−	0.992488i	$-0.460960\pi$
$157$	3.06585	0.244682	0.122341	+	0.992488i	$0.460960\pi$
$158$	1.28123	0.101929
$159$	0.421848	0.0334547
$160$	2.11524	0.167225
$161$	25.4916	2.00902
$162$	2.31115	0.181581
$163$	2.90600	0.227616	0.113808	−	0.993503i	$-0.463695\pi$
$163$	2.90600	0.227616	0.113808	+	0.993503i	$0.463695\pi$
$164$	−1.49017	−0.116363
$165$	−13.1017	−1.01997
$166$	−2.04698	−0.158876
$167$	−23.5324	−1.82099	−0.910495	−	0.413519i	$-0.864299\pi$
$167$	−23.5324	−1.82099	−0.910495	+	0.413519i	$0.864299\pi$
$168$	9.67874	0.746731
$169$	−5.16122	−0.397017
$170$	−0.799782	−0.0613405
$171$	15.6015	1.19308
$172$	−4.32680	−0.329915
$173$	−18.6902	−1.42099	−0.710495	−	0.703703i	$-0.751529\pi$
$173$	−18.6902	−1.42099	−0.710495	+	0.703703i	$0.751529\pi$
$174$	−0.226131	−0.0171429
$175$	4.41500	0.333743
$176$	16.2595	1.22561
$177$	9.41962	0.708022
$178$	−2.92396	−0.219160
$179$	−1.22059	−0.0912310	−0.0456155	−	0.998959i	$-0.514525\pi$
$179$	−1.22059	−0.0912310	−0.0456155	+	0.998959i	$0.514525\pi$
$180$	−12.6061	−0.939604
$181$	−23.8124	−1.76996	−0.884982	−	0.465626i	$-0.845829\pi$
$181$	−23.8124	−1.76996	−0.884982	+	0.465626i	$0.845829\pi$
$182$	−2.22687	−0.165067
$183$	9.29048	0.686772
$184$	−4.12694	−0.304242
$185$	−1.00000	−0.0735215
$186$	−4.30530	−0.315680
$187$	−18.9642	−1.38680
$188$	8.39132	0.612000
$189$	−46.1352	−3.35584
$190$	0.438683	0.0318254
$191$	14.9462	1.08147	0.540736	−	0.841192i	$-0.318146\pi$
$191$	14.9462	1.08147	0.540736	+	0.841192i	$0.318146\pi$
$192$	22.1799	1.60070
$193$	−18.9075	−1.36099	−0.680495	−	0.732753i	$-0.738235\pi$
$193$	−18.9075	−1.36099	−0.680495	+	0.732753i	$0.738235\pi$
$194$	3.30397	0.237211
$195$	8.58717	0.614940
$196$	−24.5791	−1.75565
$197$	3.72602	0.265468	0.132734	−	0.991152i	$-0.457624\pi$
$197$	3.72602	0.265468	0.132734	+	0.991152i	$0.457624\pi$
$198$	4.93058	0.350401
$199$	−18.7237	−1.32729	−0.663645	−	0.748048i	$-0.730991\pi$
$199$	−18.7237	−1.32729	−0.663645	+	0.748048i	$0.730991\pi$
$200$	−0.714762	−0.0505413
$201$	35.1126	2.47665
$202$	−2.00544	−0.141103
$203$	1.80686	0.126817
$204$	−26.7906	−1.87572
$205$	0.757374	0.0528973
$206$	1.22162	0.0851145
$207$	36.9933	2.57121
$208$	−10.6569	−0.738922
$209$	10.4019	0.719516
$210$	−2.43948	−0.168340
$211$	−13.3971	−0.922295	−0.461148	−	0.887323i	$-0.652562\pi$
$211$	−13.3971	−0.922295	−0.461148	+	0.887323i	$0.652562\pi$
$212$	0.270617	0.0185860
$213$	32.8619	2.25166
$214$	0.0966666	0.00660800
$215$	2.19908	0.149976
$216$	7.46900	0.508201
$217$	34.4008	2.33528
$218$	1.71769	0.116337
$219$	−25.1710	−1.70090
$220$	−8.40479	−0.566651
$221$	12.4296	0.836103
$222$	0.552543	0.0370842
$223$	−17.4471	−1.16834	−0.584172	−	0.811630i	$-0.698581\pi$
$223$	−17.4471	−1.16834	−0.584172	+	0.811630i	$0.698581\pi$
$224$	9.33880	0.623975
$225$	6.40702	0.427135
$226$	1.65169	0.109869
$227$	20.4988	1.36056	0.680278	−	0.732954i	$-0.261859\pi$
$227$	20.4988	1.36056	0.680278	+	0.732954i	$0.261859\pi$
$228$	14.6947	0.973183
$229$	−14.0698	−0.929756	−0.464878	−	0.885375i	$-0.653902\pi$
$229$	−14.0698	−0.929756	−0.464878	+	0.885375i	$0.653902\pi$
$230$	1.04018	0.0685872
$231$	−57.8441	−3.80586
$232$	−0.292520	−0.0192049
$233$	15.1114	0.989983	0.494992	−	0.868898i	$-0.335171\pi$
$233$	15.1114	0.989983	0.494992	+	0.868898i	$0.335171\pi$
$234$	−3.23162	−0.211258
$235$	−4.26487	−0.278209
$236$	6.04271	0.393347
$237$	−21.8129	−1.41690
$238$	−3.53104	−0.228883
$239$	0.0747636	0.00483606	0.00241803	−	0.999997i	$-0.499230\pi$
$239$	0.0747636	0.00483606	0.00241803	+	0.999997i	$0.499230\pi$
$240$	−11.6743	−0.753575
$241$	20.4873	1.31971	0.659853	−	0.751395i	$-0.270619\pi$
$241$	20.4873	1.31971	0.659853	+	0.751395i	$0.270619\pi$
$242$	1.30566	0.0839311
$243$	−7.99836	−0.513095
$244$	5.95987	0.381541
$245$	12.4922	0.798100
$246$	−0.418482	−0.0266814
$247$	−6.81766	−0.433797
$248$	−5.56929	−0.353650
$249$	34.8497	2.20851
$250$	0.180152	0.0113938
$251$	−6.85355	−0.432592	−0.216296	−	0.976328i	$-0.569398\pi$
$251$	−6.85355	−0.432592	−0.216296	+	0.976328i	$0.569398\pi$
$252$	−55.6560	−3.50600
$253$	24.6643	1.55063
$254$	1.68014	0.105421
$255$	13.6163	0.852683
$256$	13.4663	0.841646
$257$	−10.4751	−0.653422	−0.326711	−	0.945124i	$-0.605940\pi$
$257$	−10.4751	−0.653422	−0.326711	+	0.945124i	$0.605940\pi$
$258$	−1.21509	−0.0756481
$259$	−4.41500	−0.274335
$260$	5.50870	0.341635
$261$	2.62210	0.162304
$262$	2.81048	0.173632
$263$	5.99787	0.369844	0.184922	−	0.982753i	$-0.440797\pi$
$263$	5.99787	0.369844	0.184922	+	0.982753i	$0.440797\pi$
$264$	9.36461	0.576352
$265$	−0.137540	−0.00844903
$266$	1.93679	0.118752
$267$	49.7803	3.04651
$268$	22.5248	1.37592
$269$	−2.32781	−0.141929	−0.0709646	−	0.997479i	$-0.522608\pi$
$269$	−2.32781	−0.141929	−0.0709646	+	0.997479i	$0.522608\pi$
$270$	−1.88253	−0.114567
$271$	−6.13081	−0.372420	−0.186210	−	0.982510i	$-0.559620\pi$
$271$	−6.13081	−0.372420	−0.186210	+	0.982510i	$0.559620\pi$
$272$	−16.8981	−1.02460
$273$	37.9124	2.29456
$274$	−0.183730	−0.0110996
$275$	4.27171	0.257594
$276$	34.8432	2.09731
$277$	−8.30966	−0.499279	−0.249639	−	0.968339i	$-0.580312\pi$
$277$	−8.30966	−0.499279	−0.249639	+	0.968339i	$0.580312\pi$
$278$	1.27712	0.0765967
$279$	49.9223	2.98877
$280$	−3.15568	−0.188588
$281$	−5.01313	−0.299058	−0.149529	−	0.988757i	$-0.547776\pi$
$281$	−5.01313	−0.299058	−0.149529	+	0.988757i	$0.547776\pi$
$282$	2.35652	0.140329
$283$	12.5308	0.744876	0.372438	−	0.928057i	$-0.378522\pi$
$283$	12.5308	0.744876	0.372438	+	0.928057i	$0.378522\pi$
$284$	21.0810	1.25093
$285$	−7.46857	−0.442400
$286$	−2.15460	−0.127404
$287$	3.34381	0.197379
$288$	13.5524	0.798583
$289$	2.70896	0.159351
$290$	0.0737281	0.00432946
$291$	−56.2500	−3.29743
$292$	−16.1473	−0.944947
$293$	−6.13130	−0.358194	−0.179097	−	0.983831i	$-0.557318\pi$
$293$	−6.13130	−0.358194	−0.179097	+	0.983831i	$0.557318\pi$
$294$	−6.90250	−0.402562
$295$	−3.07119	−0.178812
$296$	0.714762	0.0415447
$297$	−44.6379	−2.59015
$298$	−2.80266	−0.162354
$299$	−16.1656	−0.934879
$300$	6.03463	0.348410
$301$	9.70896	0.559615
$302$	−0.722198	−0.0415578
$303$	34.1426	1.96144
$304$	9.26866	0.531594
$305$	−3.02909	−0.173445
$306$	−5.12422	−0.292932
$307$	12.6426	0.721552	0.360776	−	0.932652i	$-0.382512\pi$
$307$	12.6426	0.721552	0.360776	+	0.932652i	$0.382512\pi$
$308$	−37.1072	−2.11438
$309$	−20.7981	−1.18316
$310$	1.40371	0.0797254
$311$	8.05133	0.456549	0.228275	−	0.973597i	$-0.426692\pi$
$311$	8.05133	0.456549	0.228275	+	0.973597i	$0.426692\pi$
$312$	−6.13779	−0.347484
$313$	−2.99549	−0.169315	−0.0846576	−	0.996410i	$-0.526980\pi$
$313$	−2.99549	−0.169315	−0.0846576	+	0.996410i	$0.526980\pi$
$314$	0.552321	0.0311693
$315$	28.2870	1.59379
$316$	−13.9930	−0.787170
$317$	2.24377	0.126023	0.0630113	−	0.998013i	$-0.479930\pi$
$317$	2.24377	0.126023	0.0630113	+	0.998013i	$0.479930\pi$
$318$	0.0759969	0.00426169
$319$	1.74822	0.0978814
$320$	−7.23158	−0.404258
$321$	−1.64575	−0.0918566
$322$	4.59238	0.255923
$323$	−10.8104	−0.601508
$324$	−25.2414	−1.40230
$325$	−2.79978	−0.155304
$326$	0.523523	0.0289952
$327$	−29.2436	−1.61718
$328$	−0.541343	−0.0298906
$329$	−18.8294	−1.03810
$330$	−2.36030	−0.129930
$331$	1.73477	0.0953518	0.0476759	−	0.998863i	$-0.484819\pi$
$331$	1.73477	0.0953518	0.0476759	+	0.998863i	$0.484819\pi$
$332$	22.3562	1.22696
$333$	−6.40702	−0.351103
$334$	−4.23941	−0.231970
$335$	−11.4482	−0.625482
$336$	−51.5422	−2.81186
$337$	−29.9565	−1.63183	−0.815917	−	0.578169i	$-0.803768\pi$
$337$	−29.9565	−1.63183	−0.815917	+	0.578169i	$0.803768\pi$
$338$	−0.929806	−0.0505748
$339$	−28.1200	−1.52727
$340$	8.73487	0.473715
$341$	33.2844	1.80245
$342$	2.81065	0.151983
$343$	24.2483	1.30928
$344$	−1.57182	−0.0847470
$345$	−17.7090	−0.953418
$346$	−3.36708	−0.181015
$347$	−14.9978	−0.805123	−0.402561	−	0.915393i	$-0.631880\pi$
$347$	−14.9978	−0.805123	−0.402561	+	0.915393i	$0.631880\pi$
$348$	2.46970	0.132390
$349$	−7.29913	−0.390714	−0.195357	−	0.980732i	$-0.562586\pi$
$349$	−7.29913	−0.390714	−0.195357	+	0.980732i	$0.562586\pi$
$350$	0.795373	0.0425145
$351$	29.2567	1.56161
$352$	9.03571	0.481605
$353$	−12.6061	−0.670957	−0.335479	−	0.942048i	$-0.608898\pi$
$353$	−12.6061	−0.670957	−0.335479	+	0.942048i	$0.608898\pi$
$354$	1.69697	0.0901927
$355$	−10.7144	−0.568660
$356$	31.9342	1.69251
$357$	60.1158	3.18167
$358$	−0.219892	−0.0116216
$359$	21.5358	1.13661	0.568307	−	0.822817i	$-0.307599\pi$
$359$	21.5358	1.13661	0.568307	+	0.822817i	$0.307599\pi$
$360$	−4.57950	−0.241361
$361$	−13.0704	−0.687918
$362$	−4.28986	−0.225470
$363$	−22.2288	−1.16671
$364$	24.3209	1.27476
$365$	8.20680	0.429564
$366$	1.67370	0.0874857
$367$	20.1463	1.05163	0.525814	−	0.850600i	$-0.323761\pi$
$367$	20.1463	1.05163	0.525814	+	0.850600i	$0.323761\pi$
$368$	21.9772	1.14564
$369$	4.85251	0.252612
$370$	−0.180152	−0.00936567
$371$	−0.607241	−0.0315264
$372$	47.0207	2.43791
$373$	9.88395	0.511772	0.255886	−	0.966707i	$-0.417633\pi$
$373$	9.88395	0.511772	0.255886	+	0.966707i	$0.417633\pi$
$374$	−3.41644	−0.176660
$375$	−3.06709	−0.158384
$376$	3.04837	0.157208
$377$	−1.14582	−0.0590129
$378$	−8.31135	−0.427490
$379$	26.5438	1.36346	0.681732	−	0.731602i	$-0.261227\pi$
$379$	26.5438	1.36346	0.681732	+	0.731602i	$0.261227\pi$
$380$	−4.79111	−0.245779
$381$	−28.6043	−1.46544
$382$	2.69260	0.137765
$383$	1.43834	0.0734959	0.0367479	−	0.999325i	$-0.488300\pi$
$383$	1.43834	0.0734959	0.0367479	+	0.999325i	$0.488300\pi$
$384$	16.9710	0.866049
$385$	18.8596	0.961176
$386$	−3.40623	−0.173372
$387$	14.0896	0.716214
$388$	−36.0845	−1.83192
$389$	28.7600	1.45819	0.729094	−	0.684414i	$-0.239942\pi$
$389$	28.7600	1.45819	0.729094	+	0.684414i	$0.239942\pi$
$390$	1.54700	0.0783353
$391$	−25.6330	−1.29631
$392$	−8.92899	−0.450982
$393$	−47.8484	−2.41363
$394$	0.671250	0.0338171
$395$	7.11193	0.357840
$396$	−53.8497	−2.70605
$397$	33.8486	1.69881	0.849406	−	0.527740i	$-0.176960\pi$
$397$	33.8486	1.69881	0.849406	+	0.527740i	$0.176960\pi$
$398$	−3.37312	−0.169079
$399$	−32.9737	−1.65075
$400$	3.80632	0.190316
$401$	−20.7788	−1.03764	−0.518822	−	0.854883i	$-0.673629\pi$
$401$	−20.7788	−1.03764	−0.518822	+	0.854883i	$0.673629\pi$
$402$	6.32562	0.315493
$403$	−21.8153	−1.08670
$404$	21.9026	1.08970
$405$	12.8289	0.637472
$406$	0.325510	0.0161548
$407$	−4.27171	−0.211741
$408$	−9.73239	−0.481825
$409$	−3.46191	−0.171180	−0.0855902	−	0.996330i	$-0.527278\pi$
$409$	−3.46191	−0.171180	−0.0855902	+	0.996330i	$0.527278\pi$
$410$	0.136443	0.00673843
$411$	3.12800	0.154293
$412$	−13.3420	−0.657315
$413$	−13.5593	−0.667211
$414$	6.66443	0.327539
$415$	−11.3625	−0.557762
$416$	−5.92222	−0.290361
$417$	−21.7430	−1.06476
$418$	1.87393	0.0916569
$419$	−18.7259	−0.914818	−0.457409	−	0.889256i	$-0.651223\pi$
$419$	−18.7259	−0.914818	−0.457409	+	0.889256i	$0.651223\pi$
$420$	26.6429	1.30004
$421$	1.91518	0.0933404	0.0466702	−	0.998910i	$-0.485139\pi$
$421$	1.91518	0.0933404	0.0466702	+	0.998910i	$0.485139\pi$
$422$	−2.41352	−0.117488
$423$	−27.3251	−1.32859
$424$	0.0983086	0.00477429
$425$	−4.43948	−0.215346
$426$	5.92015	0.286832
$427$	−13.3734	−0.647186
$428$	−1.05575	−0.0510317
$429$	36.6820	1.77102
$430$	0.396170	0.0191050
$431$	6.17915	0.297639	0.148820	−	0.988864i	$-0.452453\pi$
$431$	6.17915	0.297639	0.148820	+	0.988864i	$0.452453\pi$
$432$	−39.7747	−1.91366
$433$	−11.7275	−0.563588	−0.281794	−	0.959475i	$-0.590929\pi$
$433$	−11.7275	−0.563588	−0.281794	+	0.959475i	$0.590929\pi$
$434$	6.19739	0.297484
$435$	−1.25522	−0.0601832
$436$	−18.7599	−0.898434
$437$	14.0598	0.672570
$438$	−4.53461	−0.216672
$439$	−18.4724	−0.881640	−0.440820	−	0.897595i	$-0.645312\pi$
$439$	−18.4724	−0.881640	−0.440820	+	0.897595i	$0.645312\pi$
$440$	−3.05326	−0.145558
$441$	80.0381	3.81134
$442$	2.23922	0.106509
$443$	3.81261	0.181142	0.0905712	−	0.995890i	$-0.471131\pi$
$443$	3.81261	0.181142	0.0905712	+	0.995890i	$0.471131\pi$
$444$	−6.03463	−0.286391
$445$	−16.2305	−0.769399
$446$	−3.14313	−0.148832
$447$	47.7152	2.25685
$448$	−31.9275	−1.50843
$449$	−4.81563	−0.227264	−0.113632	−	0.993523i	$-0.536248\pi$
$449$	−4.81563	−0.227264	−0.113632	+	0.993523i	$0.536248\pi$
$450$	1.15424	0.0544114
$451$	3.23529	0.152344
$452$	−18.0390	−0.848485
$453$	12.2954	0.577688
$454$	3.69291	0.173317
$455$	−12.3610	−0.579495
$456$	5.33825	0.249986
$457$	−27.2012	−1.27242	−0.636210	−	0.771516i	$-0.719499\pi$
$457$	−27.2012	−1.27242	−0.636210	+	0.771516i	$0.719499\pi$
$458$	−2.53470	−0.118439
$459$	46.3909	2.16534
$460$	−11.3603	−0.529679
$461$	−40.8432	−1.90226	−0.951128	−	0.308796i	$-0.900074\pi$
$461$	−40.8432	−1.90226	−0.951128	+	0.308796i	$0.900074\pi$
$462$	−10.4208	−0.484817
$463$	34.8233	1.61837	0.809187	−	0.587551i	$-0.199908\pi$
$463$	34.8233	1.61837	0.809187	+	0.587551i	$0.199908\pi$
$464$	1.55776	0.0723170
$465$	−23.8981	−1.10825
$466$	2.72236	0.126111
$467$	−9.38939	−0.434489	−0.217245	−	0.976117i	$-0.569707\pi$
$467$	−9.38939	−0.434489	−0.217245	+	0.976117i	$0.569707\pi$
$468$	35.2943	1.63148
$469$	−50.5438	−2.33390
$470$	−0.768326	−0.0354402
$471$	−9.40324	−0.433278
$472$	2.19517	0.101041
$473$	9.39386	0.431930
$474$	−3.92965	−0.180495
$475$	2.43507	0.111729
$476$	38.5645	1.76760
$477$	−0.881224	−0.0403485
$478$	0.0134688	0.000616050 0
$479$	21.0154	0.960217	0.480109	−	0.877209i	$-0.340597\pi$
$479$	21.0154	0.960217	0.480109	+	0.877209i	$0.340597\pi$
$480$	−6.48763	−0.296119
$481$	2.79978	0.127659
$482$	3.69084	0.168113
$483$	−78.1851	−3.55755
$484$	−14.2599	−0.648176
$485$	18.3399	0.832771
$486$	−1.44092	−0.0653616
$487$	−35.7633	−1.62059	−0.810294	−	0.586023i	$-0.800693\pi$
$487$	−35.7633	−1.62059	−0.810294	+	0.586023i	$0.800693\pi$
$488$	2.16508	0.0980085
$489$	−8.91296	−0.403058
$490$	2.25051	0.101668
$491$	15.0742	0.680288	0.340144	−	0.940373i	$-0.389524\pi$
$491$	15.0742	0.680288	0.340144	+	0.940373i	$0.389524\pi$
$492$	4.57048	0.206053
$493$	−1.81688	−0.0818280
$494$	−1.22822	−0.0552601
$495$	27.3690	1.23014
$496$	29.6581	1.33169
$497$	−47.3040	−2.12187
$498$	6.27826	0.281335
$499$	32.9424	1.47470	0.737352	−	0.675509i	$-0.236076\pi$
$499$	32.9424	1.47470	0.737352	+	0.675509i	$0.236076\pi$
$500$	−1.96755	−0.0879913
$501$	72.1759	3.22458
$502$	−1.23468	−0.0551066
$503$	−8.61983	−0.384339	−0.192170	−	0.981362i	$-0.561552\pi$
$503$	−8.61983	−0.384339	−0.192170	+	0.981362i	$0.561552\pi$
$504$	−20.2185	−0.900603
$505$	−11.1319	−0.495365
$506$	4.44333	0.197530
$507$	15.8299	0.703031
$508$	−18.3497	−0.814138
$509$	37.3100	1.65374	0.826869	−	0.562395i	$-0.190120\pi$
$509$	37.3100	1.65374	0.826869	+	0.562395i	$0.190120\pi$
$510$	2.45300	0.108621
$511$	36.2331	1.60286
$512$	13.4925	0.596291
$513$	−25.4456	−1.12345
$514$	−1.88712	−0.0832374
$515$	6.78105	0.298809
$516$	13.2707	0.584208
$517$	−18.2183	−0.801240
$518$	−0.795373	−0.0349467
$519$	57.3245	2.51626
$520$	2.00118	0.0877575
$521$	−18.6417	−0.816710	−0.408355	−	0.912823i	$-0.633897\pi$
$521$	−18.6417	−0.816710	−0.408355	+	0.912823i	$0.633897\pi$
$522$	0.472378	0.0206754
$523$	6.56533	0.287082	0.143541	−	0.989644i	$-0.454151\pi$
$523$	6.56533	0.287082	0.143541	+	0.989644i	$0.454151\pi$
$524$	−30.6949	−1.34091
$525$	−13.5412	−0.590986
$526$	1.08053	0.0471133
$527$	−34.5915	−1.50683
$528$	−49.8694	−2.17029
$529$	10.3376	0.449459
$530$	−0.0247782	−0.00107630
$531$	−19.6772	−0.853918
$532$	−21.1527	−0.917088
$533$	−2.12048	−0.0918483
$534$	8.96804	0.388085
$535$	0.536583	0.0231985
$536$	8.18274	0.353441
$537$	3.74365	0.161550
$538$	−0.419361	−0.0180799
$539$	53.3633	2.29852
$540$	20.5601	0.884767
$541$	5.25295	0.225842	0.112921	−	0.993604i	$-0.463979\pi$
$541$	5.25295	0.225842	0.112921	+	0.993604i	$0.463979\pi$
$542$	−1.10448	−0.0474414
$543$	73.0348	3.13422
$544$	−9.39057	−0.402617
$545$	9.53465	0.408420
$546$	6.83000	0.292297
$547$	37.4845	1.60272	0.801361	−	0.598181i	$-0.204110\pi$
$547$	37.4845	1.60272	0.801361	+	0.598181i	$0.204110\pi$
$548$	2.00662	0.0857187
$549$	−19.4074	−0.828289
$550$	0.769559	0.0328141
$551$	0.996563	0.0424550
$552$	12.6577	0.538747
$553$	31.3992	1.33523
$554$	−1.49700	−0.0636016
$555$	3.06709	0.130191
$556$	−13.9482	−0.591534
$557$	12.2789	0.520275	0.260138	−	0.965572i	$-0.416232\pi$
$557$	12.2789	0.520275	0.260138	+	0.965572i	$0.416232\pi$
$558$	8.99361	0.380730
$559$	−6.15695	−0.260411
$560$	16.8049	0.710138
$561$	58.1648	2.45572
$562$	−0.903127	−0.0380961
$563$	20.4994	0.863948	0.431974	−	0.901886i	$-0.357817\pi$
$563$	20.4994	0.863948	0.431974	+	0.901886i	$0.357817\pi$
$564$	−25.7369	−1.08372
$565$	9.16830	0.385713
$566$	2.25744	0.0948874
$567$	56.6395	2.37864
$568$	7.65823	0.321332
$569$	1.60123	0.0671269	0.0335634	−	0.999437i	$-0.489314\pi$
$569$	1.60123	0.0671269	0.0335634	+	0.999437i	$0.489314\pi$
$570$	−1.34548	−0.0563559
$571$	−41.8238	−1.75027	−0.875136	−	0.483877i	$-0.839228\pi$
$571$	−41.8238	−1.75027	−0.875136	+	0.483877i	$0.839228\pi$
$572$	23.5316	0.983905
$573$	−45.8414	−1.91505
$574$	0.602395	0.0251435
$575$	5.77387	0.240787
$576$	−46.3329	−1.93054
$577$	−14.9405	−0.621981	−0.310991	−	0.950413i	$-0.600661\pi$
$577$	−14.9405	−0.621981	−0.310991	+	0.950413i	$0.600661\pi$
$578$	0.488025	0.0202992
$579$	57.9909	2.41002
$580$	−0.805227	−0.0334352
$581$	−50.1654	−2.08121
$582$	−10.1336	−0.420050
$583$	−0.587533	−0.0243331
$584$	−5.86591	−0.242733
$585$	−17.9383	−0.741656
$586$	−1.10457	−0.0456293
$587$	−32.8499	−1.35586	−0.677930	−	0.735126i	$-0.737123\pi$
$587$	−32.8499	−1.35586	−0.677930	+	0.735126i	$0.737123\pi$
$588$	75.3861	3.10887
$589$	18.9736	0.781792
$590$	−0.553282	−0.0227783
$591$	−11.4280	−0.470086
$592$	−3.80632	−0.156439
$593$	39.7259	1.63135	0.815674	−	0.578512i	$-0.196366\pi$
$593$	39.7259	1.63135	0.815674	+	0.578512i	$0.196366\pi$
$594$	−8.04161	−0.329951
$595$	−19.6003	−0.803534
$596$	30.6094	1.25381
$597$	57.4273	2.35034
$598$	−2.91227	−0.119091
$599$	−10.4669	−0.427668	−0.213834	−	0.976870i	$-0.568595\pi$
$599$	−10.4669	−0.427668	−0.213834	+	0.976870i	$0.568595\pi$
$600$	2.19224	0.0894977
$601$	−8.56503	−0.349375	−0.174687	−	0.984624i	$-0.555891\pi$
$601$	−8.56503	−0.349375	−0.174687	+	0.984624i	$0.555891\pi$
$602$	1.74909	0.0712876
$603$	−73.3489	−2.98700
$604$	7.88753	0.320939
$605$	7.24754	0.294655
$606$	6.15087	0.249862
$607$	8.90888	0.361600	0.180800	−	0.983520i	$-0.442131\pi$
$607$	8.90888	0.361600	0.180800	+	0.983520i	$0.442131\pi$
$608$	5.15076	0.208891
$609$	−5.54180	−0.224565
$610$	−0.545697	−0.0220946
$611$	11.9407	0.483069
$612$	55.9645	2.26223
$613$	−10.5868	−0.427598	−0.213799	−	0.976878i	$-0.568584\pi$
$613$	−10.5868	−0.427598	−0.213799	+	0.976878i	$0.568584\pi$
$614$	2.27760	0.0919163
$615$	−2.32293	−0.0936697
$616$	−13.4802	−0.543131
$617$	−6.83972	−0.275357	−0.137678	−	0.990477i	$-0.543964\pi$
$617$	−6.83972	−0.275357	−0.137678	+	0.990477i	$0.543964\pi$
$618$	−3.74682	−0.150719
$619$	−15.5577	−0.625318	−0.312659	−	0.949865i	$-0.601220\pi$
$619$	−15.5577	−0.625318	−0.312659	+	0.949865i	$0.601220\pi$
$620$	−15.3307	−0.615697
$621$	−60.3348	−2.42115
$622$	1.45047	0.0581584
$623$	−71.6576	−2.87090
$624$	32.6856	1.30847
$625$	1.00000	0.0400000
$626$	−0.539645	−0.0215685
$627$	−31.9036	−1.27411
$628$	−6.03221	−0.240711
$629$	4.43948	0.177014
$630$	5.09597	0.203028
$631$	12.0975	0.481596	0.240798	−	0.970575i	$-0.422591\pi$
$631$	12.0975	0.481596	0.240798	+	0.970575i	$0.422591\pi$
$632$	−5.08334	−0.202204
$633$	41.0901	1.63319
$634$	0.404220	0.0160536
$635$	9.32621	0.370099
$636$	−0.830005	−0.0329118
$637$	−34.9756	−1.38578
$638$	0.314946	0.0124688
$639$	−68.6472	−2.71564
$640$	−5.53327	−0.218722
$641$	−39.4179	−1.55691	−0.778457	−	0.627698i	$-0.783997\pi$
$641$	−39.4179	−1.55691	−0.778457	+	0.627698i	$0.783997\pi$
$642$	−0.296485	−0.0117013
$643$	−2.80810	−0.110741	−0.0553704	−	0.998466i	$-0.517634\pi$
$643$	−2.80810	−0.110741	−0.0553704	+	0.998466i	$0.517634\pi$
$644$	−50.1560	−1.97642
$645$	−6.74478	−0.265575
$646$	−1.94752	−0.0766243
$647$	−47.0232	−1.84867	−0.924337	−	0.381577i	$-0.875381\pi$
$647$	−47.0232	−1.84867	−0.924337	+	0.381577i	$0.875381\pi$
$648$	−9.16959	−0.360216
$649$	−13.1193	−0.514976
$650$	−0.504387	−0.0197837
$651$	−105.510	−4.13528
$652$	−5.71769	−0.223922
$653$	21.7426	0.850852	0.425426	−	0.904993i	$-0.360124\pi$
$653$	21.7426	0.850852	0.425426	+	0.904993i	$0.360124\pi$
$654$	−5.26830	−0.206007
$655$	15.6006	0.609566
$656$	2.88281	0.112555
$657$	52.5812	2.05139
$658$	−3.39216	−0.132240
$659$	−26.3436	−1.02620	−0.513100	−	0.858329i	$-0.671503\pi$
$659$	−26.3436	−1.02620	−0.513100	+	0.858329i	$0.671503\pi$
$660$	25.7782	1.00342
$661$	20.5259	0.798364	0.399182	−	0.916872i	$-0.369294\pi$
$661$	20.5259	0.798364	0.399182	+	0.916872i	$0.369294\pi$
$662$	0.312523	0.0121466
$663$	−38.1226	−1.48056
$664$	8.12147	0.315174
$665$	10.7508	0.416899
$666$	−1.15424	−0.0447259
$667$	2.36298	0.0914950
$668$	46.3010	1.79144
$669$	53.5118	2.06888
$670$	−2.06242	−0.0796782
$671$	−12.9394	−0.499520
$672$	−28.6429	−1.10492
$673$	−5.66357	−0.218315	−0.109157	−	0.994024i	$-0.534815\pi$
$673$	−5.66357	−0.218315	−0.109157	+	0.994024i	$0.534815\pi$
$674$	−5.39673	−0.207874
$675$	−10.4496	−0.402207
$676$	10.1549	0.390574
$677$	6.93823	0.266658	0.133329	−	0.991072i	$-0.457433\pi$
$677$	6.93823	0.266658	0.133329	+	0.991072i	$0.457433\pi$
$678$	−5.06588	−0.194554
$679$	80.9706	3.10737
$680$	3.17317	0.121686
$681$	−62.8717	−2.40925
$682$	5.99625	0.229608
$683$	19.4849	0.745570	0.372785	−	0.927918i	$-0.378403\pi$
$683$	19.4849	0.745570	0.372785	+	0.927918i	$0.378403\pi$
$684$	−30.6967	−1.17372
$685$	−1.01986	−0.0389669
$686$	4.36838	0.166786
$687$	43.1532	1.64640
$688$	8.37043	0.319119
$689$	0.385083	0.0146705
$690$	−3.19031	−0.121453
$691$	−19.9927	−0.760557	−0.380278	−	0.924872i	$-0.624172\pi$
$691$	−19.9927	−0.760557	−0.380278	+	0.924872i	$0.624172\pi$
$692$	36.7738	1.39793
$693$	120.834	4.59011
$694$	−2.70188	−0.102562
$695$	7.08913	0.268906
$696$	0.897183	0.0340076
$697$	−3.36235	−0.127358
$698$	−1.31496	−0.0497718
$699$	−46.3481	−1.75305
$700$	−8.68672	−0.328327
$701$	−25.5726	−0.965863	−0.482932	−	0.875658i	$-0.660428\pi$
$701$	−25.5726	−0.965863	−0.482932	+	0.875658i	$0.660428\pi$
$702$	5.27066	0.198928
$703$	−2.43507	−0.0918403
$704$	−30.8913	−1.16426
$705$	13.0807	0.492649
$706$	−2.27103	−0.0854712
$707$	−49.1476	−1.84838
$708$	−18.5335	−0.696532
$709$	−36.7156	−1.37888	−0.689441	−	0.724342i	$-0.742144\pi$
$709$	−36.7156	−1.37888	−0.689441	+	0.724342i	$0.742144\pi$
$710$	−1.93022	−0.0724398
$711$	45.5663	1.70887
$712$	11.6009	0.434764
$713$	44.9888	1.68485
$714$	10.8300	0.405303
$715$	−11.9599	−0.447274
$716$	2.40156	0.0897505
$717$	−0.229307	−0.00856361
$718$	3.87971	0.144790
$719$	4.11485	0.153458	0.0767290	−	0.997052i	$-0.475552\pi$
$719$	4.11485	0.153458	0.0767290	+	0.997052i	$0.475552\pi$
$720$	24.3872	0.908857
$721$	29.9384	1.11496
$722$	−2.35467	−0.0876317
$723$	−62.8364	−2.33691
$724$	46.8520	1.74124
$725$	0.409254	0.0151993
$726$	−4.00458	−0.148624
$727$	27.2507	1.01067	0.505336	−	0.862923i	$-0.331369\pi$
$727$	27.2507	1.01067	0.505336	+	0.862923i	$0.331369\pi$
$728$	8.83521	0.327455
$729$	−13.9550	−0.516850
$730$	1.47847	0.0547208
$731$	−9.76278	−0.361090
$732$	−18.2794	−0.675627
$733$	27.7634	1.02546	0.512732	−	0.858549i	$-0.328634\pi$
$733$	27.7634	1.02546	0.512732	+	0.858549i	$0.328634\pi$
$734$	3.62940	0.133964
$735$	−38.3148	−1.41326
$736$	12.2131	0.450182
$737$	−48.9034	−1.80138
$738$	0.874192	0.0321794
$739$	−41.6263	−1.53125	−0.765624	−	0.643288i	$-0.777570\pi$
$739$	−41.6263	−1.53125	−0.765624	+	0.643288i	$0.777570\pi$
$740$	1.96755	0.0723284
$741$	20.9104	0.768161
$742$	−0.109396	−0.00401605
$743$	−48.9645	−1.79633	−0.898166	−	0.439657i	$-0.855100\pi$
$743$	−48.9645	−1.79633	−0.898166	+	0.439657i	$0.855100\pi$
$744$	17.0815	0.626238
$745$	−15.5572	−0.569970
$746$	1.78062	0.0651930
$747$	−72.7997	−2.66360
$748$	37.3129	1.36429
$749$	2.36902	0.0865619
$750$	−0.552543	−0.0201760
$751$	7.34688	0.268091	0.134046	−	0.990975i	$-0.457203\pi$
$751$	7.34688	0.268091	0.134046	+	0.990975i	$0.457203\pi$
$752$	−16.2335	−0.591974
$753$	21.0204	0.766027
$754$	−0.206423	−0.00751747
$755$	−4.00882	−0.145896
$756$	90.7730	3.30138
$757$	−4.47852	−0.162775	−0.0813873	−	0.996683i	$-0.525935\pi$
$757$	−4.47852	−0.162775	−0.0813873	+	0.996683i	$0.525935\pi$
$758$	4.78193	0.173687
$759$	−75.6476	−2.74583
$760$	−1.74049	−0.0631344
$761$	33.7349	1.22289	0.611445	−	0.791287i	$-0.290589\pi$
$761$	33.7349	1.22289	0.611445	+	0.791287i	$0.290589\pi$
$762$	−5.15313	−0.186678
$763$	42.0955	1.52396
$764$	−29.4074	−1.06392
$765$	−28.4438	−1.02839
$766$	0.259121	0.00936241
$767$	8.59867	0.310480
$768$	−41.3024	−1.49037
$769$	25.1523	0.907014	0.453507	−	0.891253i	$-0.350173\pi$
$769$	25.1523	0.907014	0.453507	+	0.891253i	$0.350173\pi$
$770$	3.39761	0.122441
$771$	32.1282	1.15707
$772$	37.2013	1.33890
$773$	13.1660	0.473549	0.236774	−	0.971565i	$-0.423910\pi$
$773$	13.1660	0.473549	0.236774	+	0.971565i	$0.423910\pi$
$774$	2.53827	0.0912362
$775$	7.79180	0.279890
$776$	−13.1087	−0.470573
$777$	13.5412	0.485788
$778$	5.18117	0.185754
$779$	1.84426	0.0660774
$780$	−16.8957	−0.604961
$781$	−45.7687	−1.63773
$782$	−4.61784	−0.165134
$783$	−4.27656	−0.152832
$784$	47.5495	1.69820
$785$	3.06585	0.109425
$786$	−8.62000	−0.307465
$787$	19.5552	0.697066	0.348533	−	0.937296i	$-0.386680\pi$
$787$	19.5552	0.697066	0.348533	+	0.937296i	$0.386680\pi$
$788$	−7.33111	−0.261160
$789$	−18.3960	−0.654914
$790$	1.28123	0.0455841
$791$	40.4781	1.43923
$792$	−19.5623	−0.695116
$793$	8.48079	0.301162
$794$	6.09790	0.216406
$795$	0.421848	0.0149614
$796$	36.8398	1.30575
$797$	51.4656	1.82300	0.911502	−	0.411295i	$-0.134923\pi$
$797$	51.4656	1.82300	0.911502	+	0.411295i	$0.134923\pi$
$798$	−5.94029	−0.210284
$799$	18.9338	0.669829
$800$	2.11524	0.0747851
$801$	−103.989	−3.67427
$802$	−3.74335	−0.132182
$803$	35.0571	1.23714
$804$	−69.0857	−2.43646
$805$	25.4916	0.898463
$806$	−3.93008	−0.138431
$807$	7.13960	0.251326
$808$	7.95669	0.279915
$809$	42.6242	1.49859	0.749293	−	0.662238i	$-0.230393\pi$
$809$	42.6242	1.49859	0.749293	+	0.662238i	$0.230393\pi$
$810$	2.31115	0.0812055
$811$	−26.7451	−0.939147	−0.469574	−	0.882893i	$-0.655592\pi$
$811$	−26.7451	−0.939147	−0.469574	+	0.882893i	$0.655592\pi$
$812$	−3.55508	−0.124759
$813$	18.8037	0.659475
$814$	−0.769559	−0.0269730
$815$	2.90600	0.101793
$816$	51.8279	1.81434
$817$	5.35492	0.187345
$818$	−0.623671	−0.0218061
$819$	−79.1975	−2.76738
$820$	−1.49017	−0.0520389
$821$	43.8050	1.52880	0.764402	−	0.644740i	$-0.223034\pi$
$821$	43.8050	1.52880	0.764402	+	0.644740i	$0.223034\pi$
$822$	0.563517	0.0196549
$823$	−5.72621	−0.199603	−0.0998016	−	0.995007i	$-0.531821\pi$
$823$	−5.72621	−0.199603	−0.0998016	+	0.995007i	$0.531821\pi$
$824$	−4.84684	−0.168848
$825$	−13.1017	−0.456143
$826$	−2.44274	−0.0849939
$827$	−35.4306	−1.23204	−0.616021	−	0.787730i	$-0.711256\pi$
$827$	−35.4306	−1.23204	−0.616021	+	0.787730i	$0.711256\pi$
$828$	−72.7860	−2.52949
$829$	−20.7040	−0.719081	−0.359541	−	0.933129i	$-0.617067\pi$
$829$	−20.7040	−0.719081	−0.359541	+	0.933129i	$0.617067\pi$
$830$	−2.04698	−0.0710516
$831$	25.4864	0.884115
$832$	20.2469	0.701933
$833$	−55.4590	−1.92154
$834$	−3.91704	−0.135636
$835$	−23.5324	−0.814372
$836$	−20.4662	−0.707840
$837$	−81.4215	−2.81434
$838$	−3.37351	−0.116536
$839$	10.9086	0.376607	0.188303	−	0.982111i	$-0.439701\pi$
$839$	10.9086	0.376607	0.188303	+	0.982111i	$0.439701\pi$
$840$	9.67874	0.333948
$841$	−28.8325	−0.994225
$842$	0.345025	0.0118903
$843$	15.3757	0.529567
$844$	26.3594	0.907329
$845$	−5.16122	−0.177551
$846$	−4.92268	−0.169245
$847$	31.9979	1.09946
$848$	−0.523523	−0.0179778
$849$	−38.4329	−1.31901
$850$	−0.799782	−0.0274323
$851$	−5.77387	−0.197926
$852$	−64.6573	−2.21512
$853$	−38.8833	−1.33134	−0.665670	−	0.746247i	$-0.731854\pi$
$853$	−38.8833	−1.33134	−0.665670	+	0.746247i	$0.731854\pi$
$854$	−2.40925	−0.0824430
$855$	15.6015	0.533561
$856$	−0.383529	−0.0131088
$857$	40.6975	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$857$	40.6975	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$858$	6.60834	0.225605
$859$	28.1532	0.960574	0.480287	−	0.877111i	$-0.340533\pi$
$859$	28.1532	0.960574	0.480287	+	0.877111i	$0.340533\pi$
$860$	−4.32680	−0.147543
$861$	−10.2558	−0.349515
$862$	1.11319	0.0379154
$863$	18.5236	0.630552	0.315276	−	0.949000i	$-0.397903\pi$
$863$	18.5236	0.630552	0.315276	+	0.949000i	$0.397903\pi$
$864$	−22.1035	−0.751977
$865$	−18.6902	−0.635486
$866$	−2.11274	−0.0717937
$867$	−8.30862	−0.282175
$868$	−67.6852	−2.29738
$869$	30.3801	1.03058
$870$	−0.226131	−0.00766654
$871$	32.0525	1.08606
$872$	−6.81501	−0.230785
$873$	117.504	3.97691
$874$	2.53290	0.0856766
$875$	4.41500	0.149254
$876$	49.5250	1.67330
$877$	14.0422	0.474172	0.237086	−	0.971489i	$-0.423808\pi$
$877$	14.0422	0.474172	0.237086	+	0.971489i	$0.423808\pi$
$878$	−3.32785	−0.112309
$879$	18.8052	0.634285
$880$	16.2595	0.548109
$881$	7.68268	0.258836	0.129418	−	0.991590i	$-0.458689\pi$
$881$	7.68268	0.258836	0.129418	+	0.991590i	$0.458689\pi$
$882$	14.4190	0.485515
$883$	35.0479	1.17945	0.589727	−	0.807602i	$-0.299235\pi$
$883$	35.0479	1.17945	0.589727	+	0.807602i	$0.299235\pi$
$884$	−24.4557	−0.822535
$885$	9.41962	0.316637
$886$	0.686850	0.0230752
$887$	−9.30606	−0.312467	−0.156233	−	0.987720i	$-0.549935\pi$
$887$	−9.30606	−0.312467	−0.156233	+	0.987720i	$0.549935\pi$
$888$	−2.19224	−0.0735667
$889$	41.1752	1.38097
$890$	−2.92396	−0.0980113
$891$	54.8013	1.83591
$892$	34.3280	1.14938
$893$	−10.3852	−0.347529
$894$	8.59599	0.287493
$895$	−1.22059	−0.0407997
$896$	−24.4294	−0.816129
$897$	49.5812	1.65547
$898$	−0.867547	−0.0289504
$899$	3.18883	0.106353
$900$	−12.6061	−0.420204
$901$	0.610607	0.0203423
$902$	0.582844	0.0194066
$903$	−29.7782	−0.990957
$904$	−6.55315	−0.217955
$905$	−23.8124	−0.791552
$906$	2.21504	0.0735899
$907$	−4.56415	−0.151550	−0.0757751	−	0.997125i	$-0.524143\pi$
$907$	−4.56415	−0.151550	−0.0757751	+	0.997125i	$0.524143\pi$
$908$	−40.3324	−1.33848
$909$	−71.3226	−2.36562
$910$	−2.22687	−0.0738200
$911$	−0.294602	−0.00976060	−0.00488030	−	0.999988i	$-0.501553\pi$
$911$	−0.294602	−0.00976060	−0.00488030	+	0.999988i	$0.501553\pi$
$912$	−28.4278	−0.941338
$913$	−48.5373	−1.60635
$914$	−4.90036	−0.162090
$915$	9.29048	0.307134
$916$	27.6829	0.914668
$917$	68.8767	2.27451
$918$	8.35743	0.275836
$919$	33.9110	1.11862	0.559311	−	0.828958i	$-0.311066\pi$
$919$	33.9110	1.11862	0.559311	+	0.828958i	$0.311066\pi$
$920$	−4.12694	−0.136061
$921$	−38.7760	−1.27771
$922$	−7.35799	−0.242323
$923$	29.9979	0.987393
$924$	113.811	3.74410
$925$	−1.00000	−0.0328798
$926$	6.27349	0.206160
$927$	43.4464	1.42697
$928$	0.865673	0.0284171
$929$	22.4955	0.738055	0.369028	−	0.929418i	$-0.379691\pi$
$929$	22.4955	0.738055	0.369028	+	0.929418i	$0.379691\pi$
$930$	−4.30530	−0.141176
$931$	30.4195	0.996958
$932$	−29.7324	−0.973918
$933$	−24.6941	−0.808450
$934$	−1.69152	−0.0553482
$935$	−18.9642	−0.620195
$936$	12.8216	0.419087
$937$	25.4994	0.833028	0.416514	−	0.909129i	$-0.363252\pi$
$937$	25.4994	0.833028	0.416514	+	0.909129i	$0.363252\pi$
$938$	−9.10558	−0.297308
$939$	9.18743	0.299821
$940$	8.39132	0.273695
$941$	52.3734	1.70732	0.853662	−	0.520827i	$-0.174376\pi$
$941$	52.3734	1.70732	0.853662	+	0.520827i	$0.174376\pi$
$942$	−1.69402	−0.0551940
$943$	4.37298	0.142404
$944$	−11.6900	−0.380476
$945$	−46.1352	−1.50078
$946$	1.69232	0.0550222
$947$	−50.7001	−1.64753	−0.823766	−	0.566931i	$-0.808131\pi$
$947$	−50.7001	−1.64753	−0.823766	+	0.566931i	$0.808131\pi$
$948$	42.9179	1.39391
$949$	−22.9773	−0.745874
$950$	0.438683	0.0142328
$951$	−6.88183	−0.223159
$952$	14.0096	0.454052
$953$	18.5912	0.602228	0.301114	−	0.953588i	$-0.402641\pi$
$953$	18.5912	0.602228	0.301114	+	0.953588i	$0.402641\pi$
$954$	−0.158754	−0.00513986
$955$	14.9462	0.483649
$956$	−0.147101	−0.00475758
$957$	−5.36194	−0.173327
$958$	3.78597	0.122319
$959$	−4.50269	−0.145399
$960$	22.1799	0.715853
$961$	29.7122	0.958457
$962$	0.504387	0.0162621
$963$	3.43790	0.110785
$964$	−40.3098	−1.29829
$965$	−18.9075	−0.608653
$966$	−14.0852	−0.453185
$967$	−20.5566	−0.661056	−0.330528	−	0.943796i	$-0.607227\pi$
$967$	−20.5566	−0.661056	−0.330528	+	0.943796i	$0.607227\pi$
$968$	−5.18027	−0.166500
$969$	33.1565	1.06514
$970$	3.30397	0.106084
$971$	40.7627	1.30814	0.654069	−	0.756435i	$-0.273061\pi$
$971$	40.7627	1.30814	0.654069	+	0.756435i	$0.273061\pi$
$972$	15.7371	0.504769
$973$	31.2985	1.00338
$974$	−6.44284	−0.206442
$975$	8.58717	0.275010
$976$	−11.5297	−0.369056
$977$	5.60203	0.179225	0.0896124	−	0.995977i	$-0.471437\pi$
$977$	5.60203	0.179225	0.0896124	+	0.995977i	$0.471437\pi$
$978$	−1.60569	−0.0513443
$979$	−69.3320	−2.21586
$980$	−24.5791	−0.785149
$981$	61.0887	1.95041
$982$	2.71565	0.0866598
$983$	16.0493	0.511894	0.255947	−	0.966691i	$-0.417613\pi$
$983$	16.0493	0.511894	0.255947	+	0.966691i	$0.417613\pi$
$984$	1.66034	0.0529299
$985$	3.72602	0.118721
$986$	−0.327314	−0.0104238
$987$	57.7514	1.83825
$988$	13.4141	0.426758
$989$	12.6972	0.403748
$990$	4.93058	0.156704
$991$	−36.8868	−1.17175	−0.585874	−	0.810402i	$-0.699249\pi$
$991$	−36.8868	−1.17175	−0.585874	+	0.810402i	$0.699249\pi$
$992$	16.4816	0.523290
$993$	−5.32070	−0.168847
$994$	−8.52192	−0.270299
$995$	−18.7237	−0.593582
$996$	−68.5684	−2.17267
$997$	−0.491984	−0.0155813	−0.00779064	−	0.999970i	$-0.502480\pi$
$997$	−0.491984	−0.0155813	−0.00779064	+	0.999970i	$0.502480\pi$
$998$	5.93465	0.187858
$999$	10.4496	0.330612

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.d.1.3	✓	5
3.2	odd	2		1665.2.a.q.1.3		5
4.3	odd	2		2960.2.a.ba.1.5		5
5.2	odd	4		925.2.b.g.149.6		10
5.3	odd	4		925.2.b.g.149.5		10
5.4	even	2		925.2.a.h.1.3		5
7.6	odd	2		9065.2.a.j.1.3		5
15.14	odd	2		8325.2.a.cc.1.3		5
37.36	even	2		6845.2.a.g.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.d.1.3	✓	5	1.1	even	1	trivial
925.2.a.h.1.3		5	5.4	even	2
925.2.b.g.149.5		10	5.3	odd	4
925.2.b.g.149.6		10	5.2	odd	4
1665.2.a.q.1.3		5	3.2	odd	2
2960.2.a.ba.1.5		5	4.3	odd	2
6845.2.a.g.1.3		5	37.36	even	2
8325.2.a.cc.1.3		5	15.14	odd	2
9065.2.a.j.1.3		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

Representations

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.d.1.3

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