LMFDB - Embedded newform 185.2.a.e.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)

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			1.3
Root			$-1.62871$ 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)

$f(q)$	$=$	$q+0.728950 q^{2} +2.62871 q^{3} -1.46863 q^{4} -1.00000 q^{5} +1.91620 q^{6} +2.55244 q^{7} -2.52846 q^{8} +3.91009 q^{9} -0.728950 q^{10} +2.46863 q^{11} -3.86060 q^{12} +1.55854 q^{13} +1.86060 q^{14} -2.62871 q^{15} +1.09414 q^{16} -6.83662 q^{17} +2.85026 q^{18} -7.66011 q^{19} +1.46863 q^{20} +6.70960 q^{21} +1.79951 q^{22} -7.50003 q^{23} -6.64658 q^{24} +1.00000 q^{25} +1.13610 q^{26} +2.39236 q^{27} -3.74859 q^{28} +3.25741 q^{29} -1.91620 q^{30} +0.658785 q^{31} +5.85449 q^{32} +6.48930 q^{33} -4.98356 q^{34} -2.55244 q^{35} -5.74248 q^{36} +1.00000 q^{37} -5.58384 q^{38} +4.09694 q^{39} +2.52846 q^{40} +2.46863 q^{41} +4.89097 q^{42} +10.9579 q^{43} -3.62551 q^{44} -3.91009 q^{45} -5.46715 q^{46} +3.11521 q^{47} +2.87617 q^{48} -0.485072 q^{49} +0.728950 q^{50} -17.9715 q^{51} -2.28892 q^{52} +8.64184 q^{53} +1.74391 q^{54} -2.46863 q^{55} -6.45373 q^{56} -20.1362 q^{57} +2.37449 q^{58} -6.23634 q^{59} +3.86060 q^{60} +3.27808 q^{61} +0.480222 q^{62} +9.98026 q^{63} +2.07935 q^{64} -1.55854 q^{65} +4.73038 q^{66} +1.47764 q^{67} +10.0405 q^{68} -19.7154 q^{69} -1.86060 q^{70} -8.06686 q^{71} -9.88651 q^{72} -4.96199 q^{73} +0.728950 q^{74} +2.62871 q^{75} +11.2499 q^{76} +6.30102 q^{77} +2.98647 q^{78} +12.8206 q^{79} -1.09414 q^{80} -5.44146 q^{81} +1.79951 q^{82} +1.14934 q^{83} -9.85393 q^{84} +6.83662 q^{85} +7.98779 q^{86} +8.56277 q^{87} -6.24184 q^{88} +11.5207 q^{89} -2.85026 q^{90} +3.97807 q^{91} +11.0148 q^{92} +1.73175 q^{93} +2.27083 q^{94} +7.66011 q^{95} +15.3897 q^{96} +17.2929 q^{97} -0.353594 q^{98} +9.65257 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100})$ 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

)

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

$2$	0.728950	0.515446	0.257723	−	0.966219i	$-0.417028\pi$
$2$	0.728950	0.515446	0.257723	+	0.966219i	$0.417028\pi$
$3$	2.62871	1.51768	0.758842	−	0.651275i	$-0.225766\pi$
$3$	2.62871	1.51768	0.758842	+	0.651275i	$0.225766\pi$
$4$	−1.46863	−0.734316
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.91620	0.782284
$7$	2.55244	0.964730	0.482365	−	0.875970i	$-0.339778\pi$
$7$	2.55244	0.964730	0.482365	+	0.875970i	$0.339778\pi$
$8$	−2.52846	−0.893946
$9$	3.91009	1.30336
$10$	−0.728950	−0.230514
$11$	2.46863	0.744320	0.372160	−	0.928169i	$-0.378617\pi$
$11$	2.46863	0.744320	0.372160	+	0.928169i	$0.378617\pi$
$12$	−3.86060	−1.11446
$13$	1.55854	0.432261	0.216131	−	0.976364i	$-0.430656\pi$
$13$	1.55854	0.432261	0.216131	+	0.976364i	$0.430656\pi$
$14$	1.86060	0.497266
$15$	−2.62871	−0.678729
$16$	1.09414	0.273535
$17$	−6.83662	−1.65812	−0.829062	−	0.559156i	$-0.811125\pi$
$17$	−6.83662	−1.65812	−0.829062	+	0.559156i	$0.811125\pi$
$18$	2.85026	0.671813
$19$	−7.66011	−1.75735	−0.878675	−	0.477421i	$-0.841572\pi$
$19$	−7.66011	−1.75735	−0.878675	+	0.477421i	$0.841572\pi$
$20$	1.46863	0.328396
$21$	6.70960	1.46415
$22$	1.79951	0.383657
$23$	−7.50003	−1.56387	−0.781933	−	0.623363i	$-0.785766\pi$
$23$	−7.50003	−1.56387	−0.781933	+	0.623363i	$0.785766\pi$
$24$	−6.64658	−1.35673
$25$	1.00000	0.200000
$26$	1.13610	0.222807
$27$	2.39236	0.460410
$28$	−3.74859	−0.708416
$29$	3.25741	0.604886	0.302443	−	0.953167i	$-0.402198\pi$
$29$	3.25741	0.604886	0.302443	+	0.953167i	$0.402198\pi$
$30$	−1.91620	−0.349848
$31$	0.658785	0.118321	0.0591607	−	0.998248i	$-0.481158\pi$
$31$	0.658785	0.118321	0.0591607	+	0.998248i	$0.481158\pi$
$32$	5.85449	1.03494
$33$	6.48930	1.12964
$34$	−4.98356	−0.854673
$35$	−2.55244	−0.431440
$36$	−5.74248	−0.957080
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−5.58384	−0.905818
$39$	4.09694	0.656036
$40$	2.52846	0.399785
$41$	2.46863	0.385535	0.192768	−	0.981244i	$-0.438254\pi$
$41$	2.46863	0.385535	0.192768	+	0.981244i	$0.438254\pi$
$42$	4.89097	0.754692
$43$	10.9579	1.67107	0.835535	−	0.549438i	$-0.185158\pi$
$43$	10.9579	1.67107	0.835535	+	0.549438i	$0.185158\pi$
$44$	−3.62551	−0.546566
$45$	−3.91009	−0.582882
$46$	−5.46715	−0.806088
$47$	3.11521	0.454400	0.227200	−	0.973848i	$-0.427043\pi$
$47$	3.11521	0.454400	0.227200	+	0.973848i	$0.427043\pi$
$48$	2.87617	0.415140
$49$	−0.485072	−0.0692960
$50$	0.728950	0.103089
$51$	−17.9715	−2.51651
$52$	−2.28892	−0.317416
$53$	8.64184	1.18705	0.593524	−	0.804816i	$-0.297736\pi$
$53$	8.64184	1.18705	0.593524	+	0.804816i	$0.297736\pi$
$54$	1.74391	0.237317
$55$	−2.46863	−0.332870
$56$	−6.45373	−0.862416
$57$	−20.1362	−2.66710
$58$	2.37449	0.311786
$59$	−6.23634	−0.811903	−0.405951	−	0.913895i	$-0.633060\pi$
$59$	−6.23634	−0.811903	−0.405951	+	0.913895i	$0.633060\pi$
$60$	3.86060	0.498401
$61$	3.27808	0.419716	0.209858	−	0.977732i	$-0.432700\pi$
$61$	3.27808	0.419716	0.209858	+	0.977732i	$0.432700\pi$
$62$	0.480222	0.0609882
$63$	9.98026	1.25739
$64$	2.07935	0.259919
$65$	−1.55854	−0.193313
$66$	4.73038	0.582270
$67$	1.47764	0.180523	0.0902615	−	0.995918i	$-0.471230\pi$
$67$	1.47764	0.180523	0.0902615	+	0.995918i	$0.471230\pi$
$68$	10.0405	1.21759
$69$	−19.7154	−2.37345
$70$	−1.86060	−0.222384
$71$	−8.06686	−0.957360	−0.478680	−	0.877989i	$-0.658885\pi$
$71$	−8.06686	−0.957360	−0.478680	+	0.877989i	$0.658885\pi$
$72$	−9.88651	−1.16514
$73$	−4.96199	−0.580757	−0.290379	−	0.956912i	$-0.593781\pi$
$73$	−4.96199	−0.580757	−0.290379	+	0.956912i	$0.593781\pi$
$74$	0.728950	0.0847388
$75$	2.62871	0.303537
$76$	11.2499	1.29045
$77$	6.30102	0.718068
$78$	2.98647	0.338151
$79$	12.8206	1.44243	0.721214	−	0.692713i	$-0.243585\pi$
$79$	12.8206	1.44243	0.721214	+	0.692713i	$0.243585\pi$
$80$	−1.09414	−0.122329
$81$	−5.44146	−0.604607
$82$	1.79951	0.198723
$83$	1.14934	0.126157	0.0630784	−	0.998009i	$-0.479908\pi$
$83$	1.14934	0.126157	0.0630784	+	0.998009i	$0.479908\pi$
$84$	−9.85393	−1.07515
$85$	6.83662	0.741536
$86$	7.98779	0.861346
$87$	8.56277	0.918026
$88$	−6.24184	−0.665382
$89$	11.5207	1.22119	0.610596	−	0.791942i	$-0.290930\pi$
$89$	11.5207	1.22119	0.610596	+	0.791942i	$0.290930\pi$
$90$	−2.85026	−0.300444
$91$	3.97807	0.417015
$92$	11.0148	1.14837
$93$	1.73175	0.179574
$94$	2.27083	0.234218
$95$	7.66011	0.785911
$96$	15.3897	1.57071
$97$	17.2929	1.75583	0.877916	−	0.478815i	$-0.158933\pi$
$97$	17.2929	1.75583	0.877916	+	0.478815i	$0.158933\pi$
$98$	−0.353594	−0.0357183
$99$	9.65257	0.970120
$100$	−1.46863	−0.146863
$101$	−7.33253	−0.729614	−0.364807	−	0.931083i	$-0.618865\pi$
$101$	−7.33253	−0.729614	−0.364807	+	0.931083i	$0.618865\pi$
$102$	−13.1003	−1.29712
$103$	9.26962	0.913363	0.456681	−	0.889630i	$-0.349038\pi$
$103$	9.26962	0.913363	0.456681	+	0.889630i	$0.349038\pi$
$104$	−3.94071	−0.386418
$105$	−6.70960	−0.654790
$106$	6.29947	0.611859
$107$	−12.0987	−1.16962	−0.584811	−	0.811170i	$-0.698831\pi$
$107$	−12.0987	−1.16962	−0.584811	+	0.811170i	$0.698831\pi$
$108$	−3.51350	−0.338086
$109$	−2.24262	−0.214804	−0.107402	−	0.994216i	$-0.534253\pi$
$109$	−2.24262	−0.214804	−0.107402	+	0.994216i	$0.534253\pi$
$110$	−1.79951	−0.171577
$111$	2.62871	0.249506
$112$	2.79272	0.263888
$113$	−11.3787	−1.07042	−0.535210	−	0.844719i	$-0.679768\pi$
$113$	−11.3787	−1.07042	−0.535210	+	0.844719i	$0.679768\pi$
$114$	−14.6783	−1.37475
$115$	7.50003	0.699382
$116$	−4.78394	−0.444177
$117$	6.09403	0.563394
$118$	−4.54598	−0.418492
$119$	−17.4500	−1.59964
$120$	6.64658	0.606747
$121$	−4.90586	−0.445987
$122$	2.38956	0.216341
$123$	6.48930	0.585121
$124$	−0.967513	−0.0868852
$125$	−1.00000	−0.0894427
$126$	7.27511	0.648118
$127$	15.6642	1.38997	0.694985	−	0.719024i	$-0.255411\pi$
$127$	15.6642	1.38997	0.694985	+	0.719024i	$0.255411\pi$
$128$	−10.1932	−0.900964
$129$	28.8052	2.53615
$130$	−1.13610	−0.0996424
$131$	−13.6723	−1.19456	−0.597278	−	0.802034i	$-0.703751\pi$
$131$	−13.6723	−1.19456	−0.597278	+	0.802034i	$0.703751\pi$
$132$	−9.53040	−0.829514
$133$	−19.5519	−1.69537
$134$	1.07713	0.0930498
$135$	−2.39236	−0.205902
$136$	17.2861	1.48227
$137$	−14.2031	−1.21346	−0.606728	−	0.794910i	$-0.707518\pi$
$137$	−14.2031	−1.21346	−0.606728	+	0.794910i	$0.707518\pi$
$138$	−14.3715	−1.22339
$139$	−11.8416	−1.00440	−0.502198	−	0.864753i	$-0.667475\pi$
$139$	−11.8416	−1.00440	−0.502198	+	0.864753i	$0.667475\pi$
$140$	3.74859	0.316813
$141$	8.18896	0.689635
$142$	−5.88034	−0.493467
$143$	3.84746	0.321741
$144$	4.27819	0.356516
$145$	−3.25741	−0.270513
$146$	−3.61705	−0.299349
$147$	−1.27511	−0.105169
$148$	−1.46863	−0.120721
$149$	4.11726	0.337299	0.168649	−	0.985676i	$-0.446059\pi$
$149$	4.11726	0.337299	0.168649	+	0.985676i	$0.446059\pi$
$150$	1.91620	0.156457
$151$	16.9092	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$151$	16.9092	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$152$	19.3683	1.57097
$153$	−26.7318	−2.16114
$154$	4.59313	0.370125
$155$	−0.658785	−0.0529149
$156$	−6.01690	−0.481737
$157$	4.02473	0.321208	0.160604	−	0.987019i	$-0.448656\pi$
$157$	4.02473	0.321208	0.160604	+	0.987019i	$0.448656\pi$
$158$	9.34556	0.743493
$159$	22.7169	1.80156
$160$	−5.85449	−0.462838
$161$	−19.1434	−1.50871
$162$	−3.96655	−0.311642
$163$	3.57756	0.280216	0.140108	−	0.990136i	$-0.455255\pi$
$163$	3.57756	0.280216	0.140108	+	0.990136i	$0.455255\pi$
$164$	−3.62551	−0.283105
$165$	−6.48930	−0.505192
$166$	0.837814	0.0650270
$167$	23.6674	1.83144	0.915720	−	0.401816i	$-0.131621\pi$
$167$	23.6674	1.83144	0.915720	+	0.401816i	$0.131621\pi$
$168$	−16.9650	−1.30887
$169$	−10.5710	−0.813150
$170$	4.98356	0.382222
$171$	−29.9517	−2.29047
$172$	−16.0932	−1.22709
$173$	−15.7383	−1.19656	−0.598279	−	0.801288i	$-0.704148\pi$
$173$	−15.7383	−1.19656	−0.598279	+	0.801288i	$0.704148\pi$
$174$	6.24184	0.473192
$175$	2.55244	0.192946
$176$	2.70103	0.203598
$177$	−16.3935	−1.23221
$178$	8.39802	0.629459
$179$	10.1749	0.760510	0.380255	−	0.924882i	$-0.375836\pi$
$179$	10.1749	0.760510	0.380255	+	0.924882i	$0.375836\pi$
$180$	5.74248	0.428019
$181$	−2.54781	−0.189377	−0.0946886	−	0.995507i	$-0.530186\pi$
$181$	−2.54781	−0.189377	−0.0946886	+	0.995507i	$0.530186\pi$
$182$	2.89982	0.214949
$183$	8.61711	0.636995
$184$	18.9635	1.39801
$185$	−1.00000	−0.0735215
$186$	1.26236	0.0925608
$187$	−16.8771	−1.23418
$188$	−4.57509	−0.333673
$189$	6.10635	0.444172
$190$	5.58384	0.405094
$191$	−10.4639	−0.757141	−0.378571	−	0.925572i	$-0.623584\pi$
$191$	−10.4639	−0.757141	−0.378571	+	0.925572i	$0.623584\pi$
$192$	5.46601	0.394475
$193$	−9.81172	−0.706263	−0.353131	−	0.935574i	$-0.614883\pi$
$193$	−9.81172	−0.706263	−0.353131	+	0.935574i	$0.614883\pi$
$194$	12.6057	0.905036
$195$	−4.09694	−0.293388
$196$	0.712392	0.0508852
$197$	13.3945	0.954317	0.477158	−	0.878817i	$-0.341667\pi$
$197$	13.3945	0.954317	0.477158	+	0.878817i	$0.341667\pi$
$198$	7.03625	0.500044
$199$	4.20221	0.297887	0.148943	−	0.988846i	$-0.452413\pi$
$199$	4.20221	0.297887	0.148943	+	0.988846i	$0.452413\pi$
$200$	−2.52846	−0.178789
$201$	3.88429	0.273977
$202$	−5.34505	−0.376077
$203$	8.31433	0.583552
$204$	26.3935	1.84791
$205$	−2.46863	−0.172417
$206$	6.75709	0.470789
$207$	−29.3258	−2.03829
$208$	1.70526	0.118239
$209$	−18.9100	−1.30803
$210$	−4.89097	−0.337509
$211$	20.0610	1.38106	0.690530	−	0.723304i	$-0.257377\pi$
$211$	20.0610	1.38106	0.690530	+	0.723304i	$0.257377\pi$
$212$	−12.6917	−0.871668
$213$	−21.2054	−1.45297
$214$	−8.81932	−0.602876
$215$	−10.9579	−0.747325
$216$	−6.04899	−0.411582
$217$	1.68151	0.114148
$218$	−1.63476	−0.110720
$219$	−13.0436	−0.881406
$220$	3.62551	0.244432
$221$	−10.6552	−0.716743
$222$	1.91620	0.128607
$223$	1.72511	0.115522	0.0577610	−	0.998330i	$-0.481604\pi$
$223$	1.72511	0.115522	0.0577610	+	0.998330i	$0.481604\pi$
$224$	14.9432	0.998436
$225$	3.91009	0.260673
$226$	−8.29452	−0.551744
$227$	4.59930	0.305266	0.152633	−	0.988283i	$-0.451225\pi$
$227$	4.59930	0.305266	0.152633	+	0.988283i	$0.451225\pi$
$228$	29.5726	1.95849
$229$	−23.4902	−1.55227	−0.776137	−	0.630565i	$-0.782823\pi$
$229$	−23.4902	−1.55227	−0.776137	+	0.630565i	$0.782823\pi$
$230$	5.46715	0.360493
$231$	16.5635	1.08980
$232$	−8.23623	−0.540735
$233$	2.43465	0.159499	0.0797496	−	0.996815i	$-0.474588\pi$
$233$	2.43465	0.159499	0.0797496	+	0.996815i	$0.474588\pi$
$234$	4.44225	0.290399
$235$	−3.11521	−0.203214
$236$	9.15889	0.596193
$237$	33.7015	2.18915
$238$	−12.7202	−0.824529
$239$	2.03195	0.131436	0.0657180	−	0.997838i	$-0.479066\pi$
$239$	2.03195	0.131436	0.0657180	+	0.997838i	$0.479066\pi$
$240$	−2.87617	−0.185656
$241$	−20.2246	−1.30278	−0.651390	−	0.758743i	$-0.725814\pi$
$241$	−20.2246	−1.30278	−0.651390	+	0.758743i	$0.725814\pi$
$242$	−3.57613	−0.229882
$243$	−21.4811	−1.37801
$244$	−4.81430	−0.308204
$245$	0.485072	0.0309901
$246$	4.73038	0.301598
$247$	−11.9386	−0.759634
$248$	−1.66571	−0.105773
$249$	3.02128	0.191466
$250$	−0.728950	−0.0461029
$251$	0.695568	0.0439038	0.0219519	−	0.999759i	$-0.493012\pi$
$251$	0.695568	0.0439038	0.0219519	+	0.999759i	$0.493012\pi$
$252$	−14.6573	−0.923324
$253$	−18.5148	−1.16402
$254$	11.4184	0.716454
$255$	17.9715	1.12542
$256$	−11.5891	−0.724317
$257$	7.13610	0.445138	0.222569	−	0.974917i	$-0.428556\pi$
$257$	7.13610	0.445138	0.222569	+	0.974917i	$0.428556\pi$
$258$	20.9975	1.30725
$259$	2.55244	0.158601
$260$	2.28892	0.141953
$261$	12.7368	0.788386
$262$	−9.96644	−0.615729
$263$	21.2754	1.31190	0.655948	−	0.754806i	$-0.272269\pi$
$263$	21.2754	1.31190	0.655948	+	0.754806i	$0.272269\pi$
$264$	−16.4079	−1.00984
$265$	−8.64184	−0.530864
$266$	−14.2524	−0.873870
$267$	30.2845	1.85338
$268$	−2.17011	−0.132561
$269$	18.3525	1.11897	0.559486	−	0.828840i	$-0.310998\pi$
$269$	18.3525	1.11897	0.559486	+	0.828840i	$0.310998\pi$
$270$	−1.74391	−0.106131
$271$	−23.2195	−1.41048	−0.705241	−	0.708967i	$-0.749161\pi$
$271$	−23.2195	−1.41048	−0.705241	+	0.708967i	$0.749161\pi$
$272$	−7.48023	−0.453555
$273$	10.4572	0.632897
$274$	−10.3534	−0.625471
$275$	2.46863	0.148864
$276$	28.9546	1.74286
$277$	5.65674	0.339880	0.169940	−	0.985454i	$-0.445643\pi$
$277$	5.65674	0.339880	0.169940	+	0.985454i	$0.445643\pi$
$278$	−8.63197	−0.517711
$279$	2.57591	0.154216
$280$	6.45373	0.385684
$281$	−24.4373	−1.45781	−0.728903	−	0.684616i	$-0.759970\pi$
$281$	−24.4373	−1.45781	−0.728903	+	0.684616i	$0.759970\pi$
$282$	5.96935	0.355469
$283$	20.8498	1.23939	0.619697	−	0.784841i	$-0.287256\pi$
$283$	20.8498	1.23939	0.619697	+	0.784841i	$0.287256\pi$
$284$	11.8472	0.703005
$285$	20.1362	1.19276
$286$	2.80461	0.165840
$287$	6.30102	0.371938
$288$	22.8916	1.34890
$289$	29.7394	1.74938
$290$	−2.37449	−0.139435
$291$	45.4580	2.66480
$292$	7.28734	0.426459
$293$	7.72199	0.451123	0.225562	−	0.974229i	$-0.427578\pi$
$293$	7.72199	0.451123	0.225562	+	0.974229i	$0.427578\pi$
$294$	−0.929493	−0.0542091
$295$	6.23634	0.363094
$296$	−2.52846	−0.146964
$297$	5.90586	0.342693
$298$	3.00128	0.173859
$299$	−11.6891	−0.675998
$300$	−3.86060	−0.222892
$301$	27.9694	1.61213
$302$	12.3260	0.709280
$303$	−19.2751	−1.10732
$304$	−8.38124	−0.480697
$305$	−3.27808	−0.187703
$306$	−19.4862	−1.11395
$307$	−9.01041	−0.514251	−0.257126	−	0.966378i	$-0.582775\pi$
$307$	−9.01041	−0.514251	−0.257126	+	0.966378i	$0.582775\pi$
$308$	−9.25388	−0.527289
$309$	24.3671	1.38620
$310$	−0.480222	−0.0272748
$311$	−32.1220	−1.82147	−0.910735	−	0.412991i	$-0.864484\pi$
$311$	−32.1220	−1.82147	−0.910735	+	0.412991i	$0.864484\pi$
$312$	−10.3590	−0.586460
$313$	19.8623	1.12268	0.561342	−	0.827584i	$-0.310285\pi$
$313$	19.8623	1.12268	0.561342	+	0.827584i	$0.310285\pi$
$314$	2.93383	0.165565
$315$	−9.98026	−0.562324
$316$	−18.8287	−1.05920
$317$	−8.58984	−0.482453	−0.241227	−	0.970469i	$-0.577550\pi$
$317$	−8.58984	−0.482453	−0.241227	+	0.970469i	$0.577550\pi$
$318$	16.5595	0.928609
$319$	8.04135	0.450229
$320$	−2.07935	−0.116239
$321$	−31.8038	−1.77512
$322$	−13.9546	−0.777657
$323$	52.3693	2.91390
$324$	7.99150	0.443972
$325$	1.55854	0.0864522
$326$	2.60786	0.144436
$327$	−5.89520	−0.326005
$328$	−6.24184	−0.344648
$329$	7.95137	0.438373
$330$	−4.73038	−0.260399
$331$	−22.9711	−1.26260	−0.631302	−	0.775537i	$-0.717479\pi$
$331$	−22.9711	−1.26260	−0.631302	+	0.775537i	$0.717479\pi$
$332$	−1.68796	−0.0926389
$333$	3.91009	0.214272
$334$	17.2524	0.944008
$335$	−1.47764	−0.0807323
$336$	7.34125	0.400498
$337$	26.2320	1.42895	0.714473	−	0.699663i	$-0.246666\pi$
$337$	26.2320	1.42895	0.714473	+	0.699663i	$0.246666\pi$
$338$	−7.70570	−0.419135
$339$	−29.9113	−1.62456
$340$	−10.0405	−0.544521
$341$	1.62630	0.0880690
$342$	−21.8333	−1.18061
$343$	−19.1052	−1.03158
$344$	−27.7067	−1.49385
$345$	19.7154	1.06144
$346$	−11.4724	−0.616760
$347$	−0.390348	−0.0209550	−0.0104775	−	0.999945i	$-0.503335\pi$
$347$	−0.390348	−0.0209550	−0.0104775	+	0.999945i	$0.503335\pi$
$348$	−12.5756	−0.674121
$349$	−6.37205	−0.341088	−0.170544	−	0.985350i	$-0.554552\pi$
$349$	−6.37205	−0.341088	−0.170544	+	0.985350i	$0.554552\pi$
$350$	1.86060	0.0994532
$351$	3.72859	0.199017
$352$	14.4526	0.770326
$353$	−9.35970	−0.498167	−0.249083	−	0.968482i	$-0.580129\pi$
$353$	−9.35970	−0.498167	−0.249083	+	0.968482i	$0.580129\pi$
$354$	−11.9501	−0.635138
$355$	8.06686	0.428145
$356$	−16.9197	−0.896741
$357$	−45.8710	−2.42775
$358$	7.41702	0.392002
$359$	−5.38443	−0.284179	−0.142090	−	0.989854i	$-0.545382\pi$
$359$	−5.38443	−0.284179	−0.142090	+	0.989854i	$0.545382\pi$
$360$	9.88651	0.521065
$361$	39.6773	2.08828
$362$	−1.85723	−0.0976137
$363$	−12.8961	−0.676868
$364$	−5.84232	−0.306221
$365$	4.96199	0.259722
$366$	6.28145	0.328337
$367$	−22.1964	−1.15864	−0.579321	−	0.815099i	$-0.696682\pi$
$367$	−22.1964	−1.15864	−0.579321	+	0.815099i	$0.696682\pi$
$368$	−8.20609	−0.427772
$369$	9.65257	0.502493
$370$	−0.728950	−0.0378963
$371$	22.0577	1.14518
$372$	−2.54331	−0.131864
$373$	−31.6626	−1.63943	−0.819713	−	0.572774i	$-0.805867\pi$
$373$	−31.6626	−1.63943	−0.819713	+	0.572774i	$0.805867\pi$
$374$	−12.3026	−0.636151
$375$	−2.62871	−0.135746
$376$	−7.87668	−0.406209
$377$	5.07680	0.261469
$378$	4.45123	0.228946
$379$	−15.6622	−0.804515	−0.402257	−	0.915527i	$-0.631774\pi$
$379$	−15.6622	−0.804515	−0.402257	+	0.915527i	$0.631774\pi$
$380$	−11.2499	−0.577106
$381$	41.1765	2.10953
$382$	−7.62766	−0.390265
$383$	7.84825	0.401027	0.200513	−	0.979691i	$-0.435739\pi$
$383$	7.84825	0.401027	0.200513	+	0.979691i	$0.435739\pi$
$384$	−26.7950	−1.36738
$385$	−6.30102	−0.321130
$386$	−7.15226	−0.364040
$387$	42.8465	2.17801
$388$	−25.3970	−1.28933
$389$	17.6518	0.894981	0.447491	−	0.894289i	$-0.352318\pi$
$389$	17.6518	0.894981	0.447491	+	0.894289i	$0.352318\pi$
$390$	−2.98647	−0.151226
$391$	51.2749	2.59308
$392$	1.22649	0.0619469
$393$	−35.9405	−1.81296
$394$	9.76391	0.491899
$395$	−12.8206	−0.645073
$396$	−14.1761	−0.712374
$397$	−22.7648	−1.14253	−0.571267	−	0.820765i	$-0.693548\pi$
$397$	−22.7648	−1.14253	−0.571267	+	0.820765i	$0.693548\pi$
$398$	3.06320	0.153544
$399$	−51.3963	−2.57303
$400$	1.09414	0.0547070
$401$	−0.168397	−0.00840934	−0.00420467	−	0.999991i	$-0.501338\pi$
$401$	−0.168397	−0.00840934	−0.00420467	+	0.999991i	$0.501338\pi$
$402$	2.83146	0.141220
$403$	1.02674	0.0511457
$404$	10.7688	0.535767
$405$	5.44146	0.270388
$406$	6.06073	0.300789
$407$	2.46863	0.122366
$408$	45.4401	2.24962
$409$	−21.1075	−1.04370	−0.521850	−	0.853037i	$-0.674758\pi$
$409$	−21.1075	−1.04370	−0.521850	+	0.853037i	$0.674758\pi$
$410$	−1.79951	−0.0888715
$411$	−37.3359	−1.84164
$412$	−13.6137	−0.670697
$413$	−15.9179	−0.783267
$414$	−21.3771	−1.05063
$415$	−1.14934	−0.0564190
$416$	9.12446	0.447364
$417$	−31.1282	−1.52435
$418$	−13.7844	−0.674219
$419$	−22.3217	−1.09049	−0.545243	−	0.838278i	$-0.683563\pi$
$419$	−22.3217	−1.09049	−0.545243	+	0.838278i	$0.683563\pi$
$420$	9.85393	0.480823
$421$	−27.4605	−1.33835	−0.669173	−	0.743107i	$-0.733351\pi$
$421$	−27.4605	−1.33835	−0.669173	+	0.743107i	$0.733351\pi$
$422$	14.6235	0.711861
$423$	12.1807	0.592248
$424$	−21.8506	−1.06116
$425$	−6.83662	−0.331625
$426$	−15.4577	−0.748927
$427$	8.36710	0.404912
$428$	17.7685	0.858871
$429$	10.1138	0.488301
$430$	−7.98779	−0.385205
$431$	36.7380	1.76960	0.884802	−	0.465966i	$-0.154293\pi$
$431$	36.7380	1.76960	0.884802	+	0.465966i	$0.154293\pi$
$432$	2.61758	0.125938
$433$	0.370435	0.0178020	0.00890098	−	0.999960i	$-0.497167\pi$
$433$	0.370435	0.0178020	0.00890098	+	0.999960i	$0.497167\pi$
$434$	1.22574	0.0588372
$435$	−8.56277	−0.410553
$436$	3.29359	0.157734
$437$	57.4511	2.74826
$438$	−9.50815	−0.454317
$439$	30.1768	1.44026	0.720130	−	0.693839i	$-0.244082\pi$
$439$	30.1768	1.44026	0.720130	+	0.693839i	$0.244082\pi$
$440$	6.24184	0.297568
$441$	−1.89668	−0.0903179
$442$	−7.76708	−0.369442
$443$	38.1789	1.81393	0.906966	−	0.421203i	$-0.138392\pi$
$443$	38.1789	1.81393	0.906966	+	0.421203i	$0.138392\pi$
$444$	−3.86060	−0.183216
$445$	−11.5207	−0.546134
$446$	1.25752	0.0595454
$447$	10.8231	0.511913
$448$	5.30742	0.250752
$449$	−3.77840	−0.178314	−0.0891569	−	0.996018i	$-0.528417\pi$
$449$	−3.77840	−0.178314	−0.0891569	+	0.996018i	$0.528417\pi$
$450$	2.85026	0.134363
$451$	6.09414	0.286962
$452$	16.7111	0.786026
$453$	44.4493	2.08841
$454$	3.35266	0.157348
$455$	−3.97807	−0.186495
$456$	50.9135	2.38424
$457$	25.3343	1.18509	0.592544	−	0.805538i	$-0.298124\pi$
$457$	25.3343	1.18509	0.592544	+	0.805538i	$0.298124\pi$
$458$	−17.1232	−0.800113
$459$	−16.3557	−0.763418
$460$	−11.0148	−0.513567
$461$	−14.7582	−0.687359	−0.343680	−	0.939087i	$-0.611673\pi$
$461$	−14.7582	−0.687359	−0.343680	+	0.939087i	$0.611673\pi$
$462$	12.0740	0.561733
$463$	8.63700	0.401395	0.200698	−	0.979653i	$-0.435679\pi$
$463$	8.63700	0.401395	0.200698	+	0.979653i	$0.435679\pi$
$464$	3.56407	0.165458
$465$	−1.73175	−0.0803081
$466$	1.77474	0.0822132
$467$	−1.64618	−0.0761762	−0.0380881	−	0.999274i	$-0.512127\pi$
$467$	−1.64618	−0.0761762	−0.0380881	+	0.999274i	$0.512127\pi$
$468$	−8.94989	−0.413709
$469$	3.77159	0.174156
$470$	−2.27083	−0.104746
$471$	10.5798	0.487493
$472$	15.7683	0.725797
$473$	27.0511	1.24381
$474$	24.5667	1.12839
$475$	−7.66011	−0.351470
$476$	25.6277	1.17464
$477$	33.7904	1.54716
$478$	1.48119	0.0677482
$479$	16.0571	0.733669	0.366835	−	0.930286i	$-0.380442\pi$
$479$	16.0571	0.733669	0.366835	+	0.930286i	$0.380442\pi$
$480$	−15.3897	−0.702442
$481$	1.55854	0.0710633
$482$	−14.7427	−0.671513
$483$	−50.3222	−2.28974
$484$	7.20490	0.327495
$485$	−17.2929	−0.785232
$486$	−15.6586	−0.710290
$487$	−6.60786	−0.299431	−0.149715	−	0.988729i	$-0.547836\pi$
$487$	−6.60786	−0.299431	−0.149715	+	0.988729i	$0.547836\pi$
$488$	−8.28850	−0.375203
$489$	9.40435	0.425279
$490$	0.353594	0.0159737
$491$	−18.2039	−0.821532	−0.410766	−	0.911741i	$-0.634739\pi$
$491$	−18.2039	−0.821532	−0.410766	+	0.911741i	$0.634739\pi$
$492$	−9.53040	−0.429663
$493$	−22.2697	−1.00298
$494$	−8.70264	−0.391550
$495$	−9.65257	−0.433851
$496$	0.720804	0.0323650
$497$	−20.5901	−0.923594
$498$	2.20237	0.0986904
$499$	−26.1795	−1.17196	−0.585978	−	0.810327i	$-0.699289\pi$
$499$	−26.1795	−1.17196	−0.585978	+	0.810327i	$0.699289\pi$
$500$	1.46863	0.0656792
$501$	62.2147	2.77955
$502$	0.507034	0.0226301
$503$	0.546197	0.0243537	0.0121769	−	0.999926i	$-0.496124\pi$
$503$	0.546197	0.0243537	0.0121769	+	0.999926i	$0.496124\pi$
$504$	−25.2347	−1.12404
$505$	7.33253	0.326293
$506$	−13.4964	−0.599988
$507$	−27.7879	−1.23410
$508$	−23.0049	−1.02068
$509$	−27.4237	−1.21553	−0.607767	−	0.794115i	$-0.707935\pi$
$509$	−27.4237	−1.21553	−0.607767	+	0.794115i	$0.707935\pi$
$510$	13.1003	0.580091
$511$	−12.6652	−0.560274
$512$	11.9386	0.527618
$513$	−18.3258	−0.809102
$514$	5.20186	0.229444
$515$	−9.26962	−0.408468
$516$	−42.3042	−1.86234
$517$	7.69030	0.338219
$518$	1.86060	0.0817500
$519$	−41.3712	−1.81600
$520$	3.94071	0.172811
$521$	39.2211	1.71831	0.859154	−	0.511717i	$-0.170990\pi$
$521$	39.2211	1.71831	0.859154	+	0.511717i	$0.170990\pi$
$522$	9.28448	0.406370
$523$	−2.89217	−0.126466	−0.0632329	−	0.997999i	$-0.520141\pi$
$523$	−2.89217	−0.126466	−0.0632329	+	0.997999i	$0.520141\pi$
$524$	20.0796	0.877181
$525$	6.70960	0.292831
$526$	15.5087	0.676212
$527$	−4.50387	−0.196191
$528$	7.10021	0.308997
$529$	33.2505	1.44567
$530$	−6.29947	−0.273632
$531$	−24.3847	−1.05820
$532$	28.7146	1.24494
$533$	3.84746	0.166652
$534$	22.0759	0.955319
$535$	12.0987	0.523071
$536$	−3.73616	−0.161378
$537$	26.7469	1.15421
$538$	13.3781	0.576770
$539$	−1.19746	−0.0515784
$540$	3.51350	0.151197
$541$	−30.3727	−1.30582	−0.652912	−	0.757434i	$-0.726453\pi$
$541$	−30.3727	−1.30582	−0.652912	+	0.757434i	$0.726453\pi$
$542$	−16.9258	−0.727027
$543$	−6.69744	−0.287415
$544$	−40.0250	−1.71606
$545$	2.24262	0.0960635
$546$	7.62277	0.326224
$547$	18.8597	0.806384	0.403192	−	0.915115i	$-0.367901\pi$
$547$	18.8597	0.806384	0.403192	+	0.915115i	$0.367901\pi$
$548$	20.8592	0.891060
$549$	12.8176	0.547042
$550$	1.79951	0.0767314
$551$	−24.9521	−1.06300
$552$	49.8496	2.12174
$553$	32.7237	1.39155
$554$	4.12348	0.175190
$555$	−2.62871	−0.111582
$556$	17.3910	0.737543
$557$	24.2064	1.02566	0.512828	−	0.858491i	$-0.328598\pi$
$557$	24.2064	1.02566	0.512828	+	0.858491i	$0.328598\pi$
$558$	1.87771	0.0794898
$559$	17.0784	0.722339
$560$	−2.79272	−0.118014
$561$	−44.3649	−1.87309
$562$	−17.8136	−0.751420
$563$	5.13444	0.216391	0.108196	−	0.994130i	$-0.465493\pi$
$563$	5.13444	0.216391	0.108196	+	0.994130i	$0.465493\pi$
$564$	−12.0266	−0.506410
$565$	11.3787	0.478706
$566$	15.1985	0.638840
$567$	−13.8890	−0.583282
$568$	20.3967	0.855828
$569$	−11.9797	−0.502214	−0.251107	−	0.967959i	$-0.580795\pi$
$569$	−11.9797	−0.502214	−0.251107	+	0.967959i	$0.580795\pi$
$570$	14.6783	0.614805
$571$	−25.5874	−1.07080	−0.535399	−	0.844599i	$-0.679839\pi$
$571$	−25.5874	−1.07080	−0.535399	+	0.844599i	$0.679839\pi$
$572$	−5.65050	−0.236259
$573$	−27.5065	−1.14910
$574$	4.59313	0.191714
$575$	−7.50003	−0.312773
$576$	8.13047	0.338769
$577$	−40.3272	−1.67884	−0.839421	−	0.543481i	$-0.817106\pi$
$577$	−40.3272	−1.67884	−0.839421	+	0.543481i	$0.817106\pi$
$578$	21.6786	0.901709
$579$	−25.7921	−1.07188
$580$	4.78394	0.198642
$581$	2.93362	0.121707
$582$	33.1367	1.37356
$583$	21.3335	0.883544
$584$	12.5462	0.519165
$585$	−6.09403	−0.251957
$586$	5.62894	0.232530
$587$	−31.6751	−1.30737	−0.653686	−	0.756766i	$-0.726778\pi$
$587$	−31.6751	−1.30737	−0.653686	+	0.756766i	$0.726778\pi$
$588$	1.87267	0.0772276
$589$	−5.04637	−0.207932
$590$	4.54598	0.187155
$591$	35.2101	1.44835
$592$	1.09414	0.0449689
$593$	6.26075	0.257098	0.128549	−	0.991703i	$-0.458968\pi$
$593$	6.26075	0.257098	0.128549	+	0.991703i	$0.458968\pi$
$594$	4.30508	0.176640
$595$	17.4500	0.715382
$596$	−6.04673	−0.247684
$597$	11.0464	0.452098
$598$	−8.52078	−0.348440
$599$	34.4197	1.40635	0.703175	−	0.711017i	$-0.251765\pi$
$599$	34.4197	1.40635	0.703175	+	0.711017i	$0.251765\pi$
$600$	−6.64658	−0.271345
$601$	12.2491	0.499650	0.249825	−	0.968291i	$-0.419627\pi$
$601$	12.2491	0.499650	0.249825	+	0.968291i	$0.419627\pi$
$602$	20.3883	0.830966
$603$	5.77772	0.235287
$604$	−24.8334	−1.01046
$605$	4.90586	0.199452
$606$	−14.0506	−0.570765
$607$	−31.0932	−1.26204	−0.631018	−	0.775768i	$-0.717363\pi$
$607$	−31.0932	−1.26204	−0.631018	+	0.775768i	$0.717363\pi$
$608$	−44.8461	−1.81875
$609$	21.8559	0.885647
$610$	−2.38956	−0.0967505
$611$	4.85518	0.196419
$612$	39.2592	1.58696
$613$	15.0998	0.609876	0.304938	−	0.952372i	$-0.401364\pi$
$613$	15.0998	0.609876	0.304938	+	0.952372i	$0.401364\pi$
$614$	−6.56814	−0.265069
$615$	−6.48930	−0.261674
$616$	−15.9319	−0.641914
$617$	−27.3635	−1.10161	−0.550807	−	0.834632i	$-0.685680\pi$
$617$	−27.3635	−1.10161	−0.550807	+	0.834632i	$0.685680\pi$
$618$	17.7624	0.714509
$619$	−11.1466	−0.448021	−0.224010	−	0.974587i	$-0.571915\pi$
$619$	−11.1466	−0.448021	−0.224010	+	0.974587i	$0.571915\pi$
$620$	0.967513	0.0388562
$621$	−17.9428	−0.720020
$622$	−23.4153	−0.938869
$623$	29.4059	1.17812
$624$	4.48263	0.179449
$625$	1.00000	0.0400000
$626$	14.4786	0.578683
$627$	−49.7088	−1.98518
$628$	−5.91084	−0.235868
$629$	−6.83662	−0.272594
$630$	−7.27511	−0.289847
$631$	11.9459	0.475558	0.237779	−	0.971319i	$-0.423581\pi$
$631$	11.9459	0.475558	0.237779	+	0.971319i	$0.423581\pi$
$632$	−32.4163	−1.28945
$633$	52.7346	2.09601
$634$	−6.26156	−0.248679
$635$	−15.6642	−0.621613
$636$	−33.3627	−1.32292
$637$	−0.756004	−0.0299540
$638$	5.86174	0.232069
$639$	−31.5422	−1.24779
$640$	10.1932	0.402923
$641$	27.9313	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$641$	27.9313	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$642$	−23.1834	−0.914976
$643$	39.0497	1.53997	0.769984	−	0.638063i	$-0.220264\pi$
$643$	39.0497	1.53997	0.769984	+	0.638063i	$0.220264\pi$
$644$	28.1145	1.10787
$645$	−28.8052	−1.13420
$646$	38.1746	1.50196
$647$	−3.60562	−0.141752	−0.0708759	−	0.997485i	$-0.522579\pi$
$647$	−3.60562	−0.141752	−0.0708759	+	0.997485i	$0.522579\pi$
$648$	13.7585	0.540485
$649$	−15.3952	−0.604316
$650$	1.13610	0.0445614
$651$	4.42019	0.173241
$652$	−5.25411	−0.205767
$653$	−16.9343	−0.662691	−0.331345	−	0.943510i	$-0.607503\pi$
$653$	−16.9343	−0.662691	−0.331345	+	0.943510i	$0.607503\pi$
$654$	−4.29731	−0.168038
$655$	13.6723	0.534222
$656$	2.70103	0.105458
$657$	−19.4018	−0.756938
$658$	5.79615	0.225958
$659$	31.2664	1.21797	0.608983	−	0.793183i	$-0.291578\pi$
$659$	31.2664	1.21797	0.608983	+	0.793183i	$0.291578\pi$
$660$	9.53040	0.370970
$661$	−39.2941	−1.52836	−0.764181	−	0.645002i	$-0.776857\pi$
$661$	−39.2941	−1.52836	−0.764181	+	0.645002i	$0.776857\pi$
$662$	−16.7448	−0.650804
$663$	−28.0093	−1.08779
$664$	−2.90607	−0.112777
$665$	19.5519	0.758192
$666$	2.85026	0.110445
$667$	−24.4307	−0.945960
$668$	−34.7587	−1.34486
$669$	4.53481	0.175326
$670$	−1.07713	−0.0416131
$671$	8.09238	0.312403
$672$	39.2813	1.51531
$673$	33.7258	1.30004	0.650018	−	0.759919i	$-0.274762\pi$
$673$	33.7258	1.30004	0.650018	+	0.759919i	$0.274762\pi$
$674$	19.1218	0.736544
$675$	2.39236	0.0920820
$676$	15.5248	0.597109
$677$	−20.4385	−0.785515	−0.392758	−	0.919642i	$-0.628479\pi$
$677$	−20.4385	−0.785515	−0.392758	+	0.919642i	$0.628479\pi$
$678$	−21.8039	−0.837372
$679$	44.1391	1.69390
$680$	−17.2861	−0.662893
$681$	12.0902	0.463298
$682$	1.18549	0.0453948
$683$	−31.0965	−1.18988	−0.594938	−	0.803771i	$-0.702823\pi$
$683$	−31.0965	−1.18988	−0.594938	+	0.803771i	$0.702823\pi$
$684$	43.9880	1.68192
$685$	14.2031	0.542674
$686$	−13.9267	−0.531725
$687$	−61.7487	−2.35586
$688$	11.9895	0.457096
$689$	13.4687	0.513115
$690$	14.3715	0.547115
$691$	10.8292	0.411960	0.205980	−	0.978556i	$-0.433962\pi$
$691$	10.8292	0.411960	0.205980	+	0.978556i	$0.433962\pi$
$692$	23.1137	0.878651
$693$	24.6376	0.935904
$694$	−0.284545	−0.0108012
$695$	11.8416	0.449179
$696$	−21.6506	−0.820665
$697$	−16.8771	−0.639266
$698$	−4.64491	−0.175812
$699$	6.39998	0.242069
$700$	−3.74859	−0.141683
$701$	3.32122	0.125441	0.0627204	−	0.998031i	$-0.480022\pi$
$701$	3.32122	0.125441	0.0627204	+	0.998031i	$0.480022\pi$
$702$	2.71796	0.102583
$703$	−7.66011	−0.288906
$704$	5.13316	0.193463
$705$	−8.18896	−0.308414
$706$	−6.82276	−0.256778
$707$	−18.7158	−0.703881
$708$	24.0760	0.904832
$709$	17.6659	0.663458	0.331729	−	0.943375i	$-0.392368\pi$
$709$	17.6659	0.663458	0.331729	+	0.943375i	$0.392368\pi$
$710$	5.88034	0.220685
$711$	50.1296	1.88001
$712$	−29.1296	−1.09168
$713$	−4.94091	−0.185039
$714$	−33.4377	−1.25137
$715$	−3.84746	−0.143887
$716$	−14.9432	−0.558454
$717$	5.34140	0.199478
$718$	−3.92498	−0.146479
$719$	−17.4938	−0.652410	−0.326205	−	0.945299i	$-0.605770\pi$
$719$	−17.4938	−0.652410	−0.326205	+	0.945299i	$0.605770\pi$
$720$	−4.27819	−0.159439
$721$	23.6601	0.881148
$722$	28.9228	1.07639
$723$	−53.1645	−1.97721
$724$	3.74179	0.139063
$725$	3.25741	0.120977
$726$	−9.40059	−0.348888
$727$	19.2435	0.713701	0.356851	−	0.934161i	$-0.383851\pi$
$727$	19.2435	0.713701	0.356851	+	0.934161i	$0.383851\pi$
$728$	−10.0584	−0.372789
$729$	−40.1430	−1.48678
$730$	3.61705	0.133873
$731$	−74.9153	−2.77084
$732$	−12.6554	−0.467756
$733$	−17.2213	−0.636082	−0.318041	−	0.948077i	$-0.603025\pi$
$733$	−17.2213	−0.636082	−0.318041	+	0.948077i	$0.603025\pi$
$734$	−16.1801	−0.597217
$735$	1.27511	0.0470332
$736$	−43.9089	−1.61850
$737$	3.64776	0.134367
$738$	7.03625	0.259008
$739$	44.1590	1.62442	0.812208	−	0.583368i	$-0.198265\pi$
$739$	44.1590	1.62442	0.812208	+	0.583368i	$0.198265\pi$
$740$	1.46863	0.0539880
$741$	−31.3830	−1.15288
$742$	16.0790	0.590279
$743$	−15.8768	−0.582461	−0.291231	−	0.956653i	$-0.594065\pi$
$743$	−15.8768	−0.582461	−0.291231	+	0.956653i	$0.594065\pi$
$744$	−4.37867	−0.160530
$745$	−4.11726	−0.150845
$746$	−23.0805	−0.845035
$747$	4.49404	0.164428
$748$	24.7862	0.906275
$749$	−30.8811	−1.12837
$750$	−1.91620	−0.0699696
$751$	20.6030	0.751815	0.375907	−	0.926657i	$-0.377331\pi$
$751$	20.6030	0.751815	0.375907	+	0.926657i	$0.377331\pi$
$752$	3.40848	0.124294
$753$	1.82844	0.0666322
$754$	3.70074	0.134773
$755$	−16.9092	−0.615389
$756$	−8.96798	−0.326162
$757$	−21.6914	−0.788387	−0.394194	−	0.919027i	$-0.628976\pi$
$757$	−21.6914	−0.788387	−0.394194	+	0.919027i	$0.628976\pi$
$758$	−11.4170	−0.414684
$759$	−48.6700	−1.76661
$760$	−19.3683	−0.702561
$761$	22.5166	0.816225	0.408113	−	0.912932i	$-0.366187\pi$
$761$	22.5166	0.816225	0.408113	+	0.912932i	$0.366187\pi$
$762$	30.0156	1.08735
$763$	−5.72415	−0.207228
$764$	15.3676	0.555981
$765$	26.7318	0.966491
$766$	5.72098	0.206708
$767$	−9.71959	−0.350954
$768$	−30.4643	−1.09928
$769$	−0.975961	−0.0351941	−0.0175970	−	0.999845i	$-0.505602\pi$
$769$	−0.975961	−0.0351941	−0.0175970	+	0.999845i	$0.505602\pi$
$770$	−4.59313	−0.165525
$771$	18.7587	0.675578
$772$	14.4098	0.518620
$773$	15.6271	0.562066	0.281033	−	0.959698i	$-0.409323\pi$
$773$	15.6271	0.562066	0.281033	+	0.959698i	$0.409323\pi$
$774$	31.2330	1.12265
$775$	0.658785	0.0236643
$776$	−43.7245	−1.56962
$777$	6.70960	0.240706
$778$	12.8673	0.461314
$779$	−18.9100	−0.677521
$780$	6.01690	0.215440
$781$	−19.9141	−0.712583
$782$	37.3769	1.33659
$783$	7.79290	0.278496
$784$	−0.530737	−0.0189549
$785$	−4.02473	−0.143649
$786$	−26.1988	−0.934482
$787$	44.8503	1.59874	0.799370	−	0.600840i	$-0.205167\pi$
$787$	44.8503	1.59874	0.799370	+	0.600840i	$0.205167\pi$
$788$	−19.6715	−0.700770
$789$	55.9267	1.99104
$790$	−9.34556	−0.332500
$791$	−29.0435	−1.03267
$792$	−24.4061	−0.867235
$793$	5.10902	0.181427
$794$	−16.5944	−0.588914
$795$	−22.7169	−0.805684
$796$	−6.17149	−0.218743
$797$	12.4685	0.441658	0.220829	−	0.975313i	$-0.429124\pi$
$797$	12.4685	0.441658	0.220829	+	0.975313i	$0.429124\pi$
$798$	−37.4653	−1.32626
$799$	−21.2975	−0.753451
$800$	5.85449	0.206988
$801$	45.0470	1.59166
$802$	−0.122753	−0.00433456
$803$	−12.2493	−0.432269
$804$	−5.70459	−0.201185
$805$	19.1434	0.674715
$806$	0.748445	0.0263628
$807$	48.2434	1.69825
$808$	18.5400	0.652236
$809$	18.8505	0.662748	0.331374	−	0.943500i	$-0.392488\pi$
$809$	18.8505	0.662748	0.331374	+	0.943500i	$0.392488\pi$
$810$	3.96655	0.139371
$811$	29.8244	1.04728	0.523638	−	0.851941i	$-0.324575\pi$
$811$	29.8244	1.04728	0.523638	+	0.851941i	$0.324575\pi$
$812$	−12.2107	−0.428511
$813$	−61.0371	−2.14067
$814$	1.79951	0.0630728
$815$	−3.57756	−0.125316
$816$	−19.6633	−0.688354
$817$	−83.9390	−2.93665
$818$	−15.3863	−0.537971
$819$	15.5546	0.543523
$820$	3.62551	0.126608
$821$	11.9059	0.415517	0.207759	−	0.978180i	$-0.433383\pi$
$821$	11.9059	0.415517	0.207759	+	0.978180i	$0.433383\pi$
$822$	−27.2160	−0.949267
$823$	41.2544	1.43804	0.719020	−	0.694990i	$-0.244591\pi$
$823$	41.2544	1.43804	0.719020	+	0.694990i	$0.244591\pi$
$824$	−23.4379	−0.816497
$825$	6.48930	0.225929
$826$	−11.6033	−0.403732
$827$	27.0596	0.940953	0.470477	−	0.882412i	$-0.344082\pi$
$827$	27.0596	0.940953	0.470477	+	0.882412i	$0.344082\pi$
$828$	43.0688	1.49674
$829$	−20.6787	−0.718200	−0.359100	−	0.933299i	$-0.616916\pi$
$829$	−20.6787	−0.718200	−0.359100	+	0.933299i	$0.616916\pi$
$830$	−0.837814	−0.0290810
$831$	14.8699	0.515831
$832$	3.24076	0.112353
$833$	3.31626	0.114901
$834$	−22.6909	−0.785722
$835$	−23.6674	−0.819045
$836$	27.7718	0.960508
$837$	1.57605	0.0544763
$838$	−16.2714	−0.562086
$839$	−0.0974789	−0.00336534	−0.00168267	−	0.999999i	$-0.500536\pi$
$839$	−0.0974789	−0.00336534	−0.00168267	+	0.999999i	$0.500536\pi$
$840$	16.9650	0.585347
$841$	−18.3893	−0.634113
$842$	−20.0174	−0.689844
$843$	−64.2385	−2.21249
$844$	−29.4623	−1.01413
$845$	10.5710	0.363652
$846$	8.87916	0.305272
$847$	−12.5219	−0.430257
$848$	9.45539	0.324700
$849$	54.8080	1.88101
$850$	−4.98356	−0.170935
$851$	−7.50003	−0.257098
$852$	31.1429	1.06694
$853$	0.919899	0.0314967	0.0157484	−	0.999876i	$-0.494987\pi$
$853$	0.919899	0.0314967	0.0157484	+	0.999876i	$0.494987\pi$
$854$	6.09920	0.208710
$855$	29.9517	1.02433
$856$	30.5910	1.04558
$857$	20.3344	0.694608	0.347304	−	0.937753i	$-0.387097\pi$
$857$	20.3344	0.694608	0.347304	+	0.937753i	$0.387097\pi$
$858$	7.37249	0.251693
$859$	15.8959	0.542360	0.271180	−	0.962529i	$-0.412586\pi$
$859$	15.8959	0.542360	0.271180	+	0.962529i	$0.412586\pi$
$860$	16.0932	0.548772
$861$	16.5635	0.564484
$862$	26.7801	0.912135
$863$	−37.0558	−1.26139	−0.630697	−	0.776029i	$-0.717231\pi$
$863$	−37.0558	−1.26139	−0.630697	+	0.776029i	$0.717231\pi$
$864$	14.0061	0.476496
$865$	15.7383	0.535117
$866$	0.270029	0.00917595
$867$	78.1762	2.65500
$868$	−2.46951	−0.0838208
$869$	31.6493	1.07363
$870$	−6.24184	−0.211618
$871$	2.30297	0.0780331
$872$	5.67039	0.192024
$873$	67.6170	2.28849
$874$	41.8790	1.41658
$875$	−2.55244	−0.0862881
$876$	19.1563	0.647230
$877$	−25.4103	−0.858045	−0.429022	−	0.903294i	$-0.641142\pi$
$877$	−25.4103	−0.858045	−0.429022	+	0.903294i	$0.641142\pi$
$878$	21.9974	0.742376
$879$	20.2988	0.684662
$880$	−2.70103	−0.0910517
$881$	1.37793	0.0464238	0.0232119	−	0.999731i	$-0.492611\pi$
$881$	1.37793	0.0464238	0.0232119	+	0.999731i	$0.492611\pi$
$882$	−1.38258	−0.0465540
$883$	−54.5280	−1.83501	−0.917506	−	0.397722i	$-0.869801\pi$
$883$	−54.5280	−1.83501	−0.917506	+	0.397722i	$0.869801\pi$
$884$	15.6485	0.526316
$885$	16.3935	0.551062
$886$	27.8305	0.934984
$887$	−6.43212	−0.215969	−0.107985	−	0.994153i	$-0.534440\pi$
$887$	−6.43212	−0.215969	−0.107985	+	0.994153i	$0.534440\pi$
$888$	−6.64658	−0.223045
$889$	39.9818	1.34095
$890$	−8.39802	−0.281502
$891$	−13.4330	−0.450021
$892$	−2.53355	−0.0848297
$893$	−23.8628	−0.798539
$894$	7.88947	0.263863
$895$	−10.1749	−0.340110
$896$	−26.0176	−0.869187
$897$	−30.7272	−1.02595
$898$	−2.75427	−0.0919111
$899$	2.14593	0.0715709
$900$	−5.74248	−0.191416
$901$	−59.0810	−1.96827
$902$	4.44233	0.147913
$903$	73.5234	2.44670
$904$	28.7706	0.956897
$905$	2.54781	0.0846921
$906$	32.4013	1.07646
$907$	−33.4197	−1.10968	−0.554841	−	0.831957i	$-0.687221\pi$
$907$	−33.4197	−1.10968	−0.554841	+	0.831957i	$0.687221\pi$
$908$	−6.75468	−0.224162
$909$	−28.6709	−0.950953
$910$	−2.89982	−0.0961280
$911$	52.4122	1.73649	0.868247	−	0.496132i	$-0.165247\pi$
$911$	52.4122	1.73649	0.868247	+	0.496132i	$0.165247\pi$
$912$	−22.0318	−0.729546
$913$	2.83730	0.0939011
$914$	18.4674	0.610849
$915$	−8.61711	−0.284873
$916$	34.4984	1.13986
$917$	−34.8977	−1.15242
$918$	−11.9225	−0.393500
$919$	−11.6967	−0.385838	−0.192919	−	0.981215i	$-0.561795\pi$
$919$	−11.6967	−0.385838	−0.192919	+	0.981215i	$0.561795\pi$
$920$	−18.9635	−0.625209
$921$	−23.6857	−0.780470
$922$	−10.7580	−0.354297
$923$	−12.5725	−0.413830
$924$	−24.3257	−0.800257
$925$	1.00000	0.0328798
$926$	6.29594	0.206898
$927$	36.2451	1.19044
$928$	19.0705	0.626020
$929$	−10.4868	−0.344059	−0.172030	−	0.985092i	$-0.555032\pi$
$929$	−10.4868	−0.344059	−0.172030	+	0.985092i	$0.555032\pi$
$930$	−1.26236	−0.0413945
$931$	3.71571	0.121777
$932$	−3.57560	−0.117123
$933$	−84.4392	−2.76442
$934$	−1.19999	−0.0392647
$935$	16.8771	0.551940
$936$	−15.4085	−0.503643
$937$	38.3244	1.25200	0.626002	−	0.779821i	$-0.284690\pi$
$937$	38.3244	1.25200	0.626002	+	0.779821i	$0.284690\pi$
$938$	2.74930	0.0897679
$939$	52.2122	1.70388
$940$	4.57509	0.149223
$941$	33.4866	1.09163	0.545816	−	0.837905i	$-0.316220\pi$
$941$	33.4866	1.09163	0.545816	+	0.837905i	$0.316220\pi$
$942$	7.71217	0.251276
$943$	−18.5148	−0.602926
$944$	−6.82344	−0.222084
$945$	−6.10635	−0.198640
$946$	19.7189	0.641117
$947$	−40.2997	−1.30956	−0.654782	−	0.755818i	$-0.727240\pi$
$947$	−40.2997	−1.30956	−0.654782	+	0.755818i	$0.727240\pi$
$948$	−49.4951	−1.60753
$949$	−7.73346	−0.251039
$950$	−5.58384	−0.181164
$951$	−22.5801	−0.732211
$952$	44.1217	1.42999
$953$	10.8028	0.349939	0.174969	−	0.984574i	$-0.444017\pi$
$953$	10.8028	0.349939	0.174969	+	0.984574i	$0.444017\pi$
$954$	24.6315	0.797475
$955$	10.4639	0.338604
$956$	−2.98419	−0.0965156
$957$	21.1383	0.683305
$958$	11.7049	0.378167
$959$	−36.2526	−1.17066
$960$	−5.46601	−0.176415
$961$	−30.5660	−0.986000
$962$	1.13610	0.0366293
$963$	−47.3069	−1.52444
$964$	29.7025	0.956652
$965$	9.81172	0.315850
$966$	−36.6824	−1.18024
$967$	46.0540	1.48100	0.740499	−	0.672058i	$-0.234589\pi$
$967$	46.0540	1.48100	0.740499	+	0.672058i	$0.234589\pi$
$968$	12.4043	0.398688
$969$	137.663	4.42239
$970$	−12.6057	−0.404744
$971$	−0.231513	−0.00742961	−0.00371481	−	0.999993i	$-0.501182\pi$
$971$	−0.231513	−0.00742961	−0.00371481	+	0.999993i	$0.501182\pi$
$972$	31.5478	1.01190
$973$	−30.2250	−0.968970
$974$	−4.81680	−0.154340
$975$	4.09694	0.131207
$976$	3.58668	0.114807
$977$	8.81121	0.281896	0.140948	−	0.990017i	$-0.454985\pi$
$977$	8.81121	0.281896	0.140948	+	0.990017i	$0.454985\pi$
$978$	6.85530	0.219208
$979$	28.4404	0.908958
$980$	−0.712392	−0.0227565
$981$	−8.76887	−0.279968
$982$	−13.2698	−0.423455
$983$	18.8943	0.602634	0.301317	−	0.953524i	$-0.402574\pi$
$983$	18.8943	0.602634	0.301317	+	0.953524i	$0.402574\pi$
$984$	−16.4079	−0.523066
$985$	−13.3945	−0.426783
$986$	−16.2335	−0.516980
$987$	20.9018	0.665312
$988$	17.5334	0.557811
$989$	−82.1849	−2.61333
$990$	−7.03625	−0.223627
$991$	30.3969	0.965589	0.482794	−	0.875734i	$-0.339622\pi$
$991$	30.3969	0.965589	0.482794	+	0.875734i	$0.339622\pi$
$992$	3.85685	0.122455
$993$	−60.3842	−1.91623
$994$	−15.0092	−0.476063
$995$	−4.20221	−0.133219
$996$	−4.43715	−0.140597
$997$	35.2291	1.11572	0.557859	−	0.829936i	$-0.311623\pi$
$997$	35.2291	1.11572	0.557859	+	0.829936i	$0.311623\pi$
$998$	−19.0836	−0.604079
$999$	2.39236	0.0756910

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.1.3	✓	5
3.2	odd	2		1665.2.a.p.1.3		5
4.3	odd	2		2960.2.a.w.1.1		5
5.2	odd	4		925.2.b.f.149.6		10
5.3	odd	4		925.2.b.f.149.5		10
5.4	even	2		925.2.a.f.1.3		5
7.6	odd	2		9065.2.a.k.1.3		5
15.14	odd	2		8325.2.a.ch.1.3		5
37.36	even	2		6845.2.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.3	✓	5	1.1	even	1	trivial
925.2.a.f.1.3		5	5.4	even	2
925.2.b.f.149.5		10	5.3	odd	4
925.2.b.f.149.6		10	5.2	odd	4
1665.2.a.p.1.3		5	3.2	odd	2
2960.2.a.w.1.1		5	4.3	odd	2
6845.2.a.f.1.3		5	37.36	even	2
8325.2.a.ch.1.3		5	15.14	odd	2
9065.2.a.k.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.e.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	$1$