LMFDB - Embedded newform 1143.3.b.a.890.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.13
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.72

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.99230i q^{2} -4.95386 q^{4} +7.40706i q^{5} -4.26180 q^{7} +2.85423i q^{8} +O(q^{10})$	\(q-2.99230i q^{2} -4.95386 q^{4} +7.40706i q^{5} -4.26180 q^{7} +2.85423i q^{8} +22.1641 q^{10} -11.1948i q^{11} -7.91934 q^{13} +12.7526i q^{14} -11.2747 q^{16} -14.3310i q^{17} +18.0911 q^{19} -36.6935i q^{20} -33.4981 q^{22} +33.5108i q^{23} -29.8645 q^{25} +23.6970i q^{26} +21.1123 q^{28} +23.5050i q^{29} +43.9625 q^{31} +45.1543i q^{32} -42.8826 q^{34} -31.5674i q^{35} +2.75938 q^{37} -54.1340i q^{38} -21.1415 q^{40} +16.3082i q^{41} +48.3645 q^{43} +55.4573i q^{44} +100.274 q^{46} +4.06398i q^{47} -30.8371 q^{49} +89.3635i q^{50} +39.2313 q^{52} +74.2719i q^{53} +82.9203 q^{55} -12.1642i q^{56} +70.3339 q^{58} -55.4624i q^{59} +45.3013 q^{61} -131.549i q^{62} +90.0162 q^{64} -58.6590i q^{65} +124.316 q^{67} +70.9936i q^{68} -94.4591 q^{70} -116.550i q^{71} +93.3854 q^{73} -8.25690i q^{74} -89.6207 q^{76} +47.7098i q^{77} -2.92284 q^{79} -83.5125i q^{80} +48.7991 q^{82} -72.3263i q^{83} +106.150 q^{85} -144.721i q^{86} +31.9525 q^{88} +19.0077i q^{89} +33.7506 q^{91} -166.008i q^{92} +12.1606 q^{94} +134.002i q^{95} +18.2337 q^{97} +92.2738i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 2.99230i	− 1.49615i	−0.663614	−	0.748075i	$-0.730978\pi$
$2$	− 2.99230i	− 1.49615i	0.663614	−	0.748075i	$-0.269022\pi$
$3$	0	0
$3$	0	0
$4$	−4.95386	−1.23846
$5$	7.40706i	1.48141i	0.671830	+	0.740706i	$0.265509\pi$
$5$	7.40706i	1.48141i	−0.671830	+	0.740706i	$0.734491\pi$
$6$	0	0
$7$	−4.26180	−0.608828	−0.304414	−	0.952540i	$-0.598461\pi$
$7$	−4.26180	−0.608828	−0.304414	+	0.952540i	$0.598461\pi$
$8$	2.85423i	0.356779i
$9$	0	0
$10$	22.1641	2.21641
$11$	− 11.1948i	− 1.01771i	−0.860853	−	0.508853i	$-0.830070\pi$
$11$	− 11.1948i	− 1.01771i	0.860853	−	0.508853i	$-0.169930\pi$
$12$	0	0
$13$	−7.91934	−0.609180	−0.304590	−	0.952484i	$-0.598519\pi$
$13$	−7.91934	−0.609180	−0.304590	+	0.952484i	$0.598519\pi$
$14$	12.7526i	0.910899i
$15$	0	0
$16$	−11.2747	−0.704670
$17$	− 14.3310i	− 0.842999i	−0.906829	−	0.421499i	$-0.861504\pi$
$17$	− 14.3310i	− 0.842999i	0.906829	−	0.421499i	$-0.138496\pi$
$18$	0	0
$19$	18.0911	0.952163	0.476082	−	0.879401i	$-0.342057\pi$
$19$	18.0911	0.952163	0.476082	+	0.879401i	$0.342057\pi$
$20$	− 36.6935i	− 1.83468i
$21$	0	0
$22$	−33.4981	−1.52264
$23$	33.5108i	1.45699i	0.685050	+	0.728496i	$0.259780\pi$
$23$	33.5108i	1.45699i	−0.685050	+	0.728496i	$0.740220\pi$
$24$	0	0
$25$	−29.8645	−1.19458
$26$	23.6970i	0.911425i
$27$	0	0
$28$	21.1123	0.754012
$29$	23.5050i	0.810516i	0.914202	+	0.405258i	$0.132818\pi$
$29$	23.5050i	0.810516i	−0.914202	+	0.405258i	$0.867182\pi$
$30$	0	0
$31$	43.9625	1.41814	0.709072	−	0.705136i	$-0.249114\pi$
$31$	43.9625	1.41814	0.709072	+	0.705136i	$0.249114\pi$
$32$	45.1543i	1.41107i
$33$	0	0
$34$	−42.8826	−1.26125
$35$	− 31.5674i	− 0.901925i
$36$	0	0
$37$	2.75938	0.0745779	0.0372890	−	0.999305i	$-0.488128\pi$
$37$	2.75938	0.0745779	0.0372890	+	0.999305i	$0.488128\pi$
$38$	− 54.1340i	− 1.42458i
$39$	0	0
$40$	−21.1415	−0.528536
$41$	16.3082i	0.397761i	0.980024	+	0.198881i	$0.0637306\pi$
$41$	16.3082i	0.397761i	−0.980024	+	0.198881i	$0.936269\pi$
$42$	0	0
$43$	48.3645	1.12476	0.562378	−	0.826880i	$-0.309886\pi$
$43$	48.3645	1.12476	0.562378	+	0.826880i	$0.309886\pi$
$44$	55.4573i	1.26039i
$45$	0	0
$46$	100.274	2.17988
$47$	4.06398i	0.0864677i	0.999065	+	0.0432338i	$0.0137660\pi$
$47$	4.06398i	0.0864677i	−0.999065	+	0.0432338i	$0.986234\pi$
$48$	0	0
$49$	−30.8371	−0.629328
$50$	89.3635i	1.78727i
$51$	0	0
$52$	39.2313	0.754448
$53$	74.2719i	1.40136i	0.713477	+	0.700678i	$0.247119\pi$
$53$	74.2719i	1.40136i	−0.713477	+	0.700678i	$0.752881\pi$
$54$	0	0
$55$	82.9203	1.50764
$56$	− 12.1642i	− 0.217217i
$57$	0	0
$58$	70.3339	1.21265
$59$	− 55.4624i	− 0.940041i	−0.882655	−	0.470021i	$-0.844246\pi$
$59$	− 55.4624i	− 0.940041i	0.882655	−	0.470021i	$-0.155754\pi$
$60$	0	0
$61$	45.3013	0.742644	0.371322	−	0.928504i	$-0.378905\pi$
$61$	45.3013	0.742644	0.371322	+	0.928504i	$0.378905\pi$
$62$	− 131.549i	− 2.12176i
$63$	0	0
$64$	90.0162	1.40650
$65$	− 58.6590i	− 0.902446i
$66$	0	0
$67$	124.316	1.85546	0.927731	−	0.373250i	$-0.121757\pi$
$67$	124.316	1.85546	0.927731	+	0.373250i	$0.121757\pi$
$68$	70.9936i	1.04402i
$69$	0	0
$70$	−94.4591	−1.34942
$71$	− 116.550i	− 1.64155i	−0.571253	−	0.820774i	$-0.693542\pi$
$71$	− 116.550i	− 1.64155i	0.571253	−	0.820774i	$-0.306458\pi$
$72$	0	0
$73$	93.3854	1.27925	0.639626	−	0.768686i	$-0.279089\pi$
$73$	93.3854	1.27925	0.639626	+	0.768686i	$0.279089\pi$
$74$	− 8.25690i	− 0.111580i
$75$	0	0
$76$	−89.6207	−1.17922
$77$	47.7098i	0.619608i
$78$	0	0
$79$	−2.92284	−0.0369980	−0.0184990	−	0.999829i	$-0.505889\pi$
$79$	−2.92284	−0.0369980	−0.0184990	+	0.999829i	$0.505889\pi$
$80$	− 83.5125i	− 1.04391i
$81$	0	0
$82$	48.7991	0.595111
$83$	− 72.3263i	− 0.871401i	−0.900092	−	0.435701i	$-0.856501\pi$
$83$	− 72.3263i	− 0.871401i	0.900092	−	0.435701i	$-0.143499\pi$
$84$	0	0
$85$	106.150	1.24883
$86$	− 144.721i	− 1.68280i
$87$	0	0
$88$	31.9525	0.363096
$89$	19.0077i	0.213570i	0.994282	+	0.106785i	$0.0340556\pi$
$89$	19.0077i	0.213570i	−0.994282	+	0.106785i	$0.965944\pi$
$90$	0	0
$91$	33.7506	0.370886
$92$	− 166.008i	− 1.80443i
$93$	0	0
$94$	12.1606	0.129369
$95$	134.002i	1.41055i
$96$	0	0
$97$	18.2337	0.187977	0.0939883	−	0.995573i	$-0.470038\pi$
$97$	18.2337	0.187977	0.0939883	+	0.995573i	$0.470038\pi$
$98$	92.2738i	0.941569i
$99$	0	0
$100$	147.944	1.47944
$101$	136.191i	1.34843i	0.738536	+	0.674214i	$0.235518\pi$
$101$	136.191i	1.34843i	−0.738536	+	0.674214i	$0.764482\pi$
$102$	0	0
$103$	64.9125	0.630219	0.315109	−	0.949055i	$-0.397959\pi$
$103$	64.9125	0.630219	0.315109	+	0.949055i	$0.397959\pi$
$104$	− 22.6036i	− 0.217343i
$105$	0	0
$106$	222.244	2.09664
$107$	194.211i	1.81505i	0.419995	+	0.907526i	$0.362032\pi$
$107$	194.211i	1.81505i	−0.419995	+	0.907526i	$0.637968\pi$
$108$	0	0
$109$	155.670	1.42816	0.714081	−	0.700063i	$-0.246845\pi$
$109$	155.670	1.42816	0.714081	+	0.700063i	$0.246845\pi$
$110$	− 248.122i	− 2.25566i
$111$	0	0
$112$	48.0506	0.429023
$113$	− 12.0707i	− 0.106820i	−0.998573	−	0.0534100i	$-0.982991\pi$
$113$	− 12.0707i	− 0.106820i	0.998573	−	0.0534100i	$-0.0170090\pi$
$114$	0	0
$115$	−248.217	−2.15840
$116$	− 116.440i	− 1.00380i
$117$	0	0
$118$	−165.960	−1.40644
$119$	61.0758i	0.513242i
$120$	0	0
$121$	−4.32277	−0.0357254
$122$	− 135.555i	− 1.11111i
$123$	0	0
$124$	−217.784	−1.75632
$125$	− 36.0315i	− 0.288252i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	− 88.7385i	− 0.693269i
$129$	0	0
$130$	−175.525	−1.35019
$131$	− 13.1220i	− 0.100168i	−0.998745	−	0.0500839i	$-0.984051\pi$
$131$	− 13.1220i	− 0.100168i	0.998745	−	0.0500839i	$-0.0159489\pi$
$132$	0	0
$133$	−77.1006	−0.579704
$134$	− 371.991i	− 2.77605i
$135$	0	0
$136$	40.9039	0.300764
$137$	− 57.7826i	− 0.421771i	−0.977511	−	0.210885i	$-0.932365\pi$
$137$	− 57.7826i	− 0.421771i	0.977511	−	0.210885i	$-0.0676347\pi$
$138$	0	0
$139$	−91.4008	−0.657560	−0.328780	−	0.944407i	$-0.606637\pi$
$139$	−91.4008	−0.657560	−0.328780	+	0.944407i	$0.606637\pi$
$140$	156.380i	1.11700i
$141$	0	0
$142$	−348.752	−2.45600
$143$	88.6552i	0.619966i
$144$	0	0
$145$	−174.103	−1.20071
$146$	− 279.437i	− 1.91395i
$147$	0	0
$148$	−13.6696	−0.0923621
$149$	− 6.68338i	− 0.0448549i	−0.999748	−	0.0224274i	$-0.992861\pi$
$149$	− 6.68338i	− 0.0448549i	0.999748	−	0.0224274i	$-0.00713948\pi$
$150$	0	0
$151$	246.584	1.63301	0.816504	−	0.577339i	$-0.195909\pi$
$151$	246.584	1.63301	0.816504	+	0.577339i	$0.195909\pi$
$152$	51.6362i	0.339712i
$153$	0	0
$154$	142.762	0.927027
$155$	325.633i	2.10086i
$156$	0	0
$157$	72.7382	0.463300	0.231650	−	0.972799i	$-0.425588\pi$
$157$	72.7382	0.463300	0.231650	+	0.972799i	$0.425588\pi$
$158$	8.74602i	0.0553545i
$159$	0	0
$160$	−334.460	−2.09038
$161$	− 142.816i	− 0.887058i
$162$	0	0
$163$	−165.119	−1.01300	−0.506499	−	0.862241i	$-0.669061\pi$
$163$	−165.119	−1.01300	−0.506499	+	0.862241i	$0.669061\pi$
$164$	− 80.7886i	− 0.492614i
$165$	0	0
$166$	−216.422	−1.30375
$167$	126.920i	0.760001i	0.924986	+	0.380001i	$0.124076\pi$
$167$	126.920i	0.760001i	−0.924986	+	0.380001i	$0.875924\pi$
$168$	0	0
$169$	−106.284	−0.628900
$170$	− 317.634i	− 1.86843i
$171$	0	0
$172$	−239.591	−1.39297
$173$	216.681i	1.25249i	0.779626	+	0.626245i	$0.215409\pi$
$173$	216.681i	1.25249i	−0.779626	+	0.626245i	$0.784591\pi$
$174$	0	0
$175$	127.276	0.727294
$176$	126.218i	0.717147i
$177$	0	0
$178$	56.8767	0.319532
$179$	49.1906i	0.274808i	0.990515	+	0.137404i	$0.0438758\pi$
$179$	49.1906i	0.274808i	−0.990515	+	0.137404i	$0.956124\pi$
$180$	0	0
$181$	−273.724	−1.51229	−0.756143	−	0.654407i	$-0.772918\pi$
$181$	−273.724	−1.51229	−0.756143	+	0.654407i	$0.772918\pi$
$182$	− 100.992i	− 0.554901i
$183$	0	0
$184$	−95.6476	−0.519824
$185$	20.4389i	0.110481i
$186$	0	0
$187$	−160.432	−0.857925
$188$	− 20.1324i	− 0.107087i
$189$	0	0
$190$	400.974	2.11039
$191$	28.6902i	0.150211i	0.997176	+	0.0751054i	$0.0239293\pi$
$191$	28.6902i	0.150211i	−0.997176	+	0.0751054i	$0.976071\pi$
$192$	0	0
$193$	198.332	1.02763	0.513814	−	0.857901i	$-0.328232\pi$
$193$	198.332	1.02763	0.513814	+	0.857901i	$0.328232\pi$
$194$	− 54.5608i	− 0.281241i
$195$	0	0
$196$	152.762	0.779400
$197$	− 236.281i	− 1.19940i	−0.800227	−	0.599698i	$-0.795288\pi$
$197$	− 236.281i	− 1.19940i	0.800227	−	0.599698i	$-0.204712\pi$
$198$	0	0
$199$	143.294	0.720070	0.360035	−	0.932939i	$-0.382765\pi$
$199$	143.294	0.720070	0.360035	+	0.932939i	$0.382765\pi$
$200$	− 85.2401i	− 0.426201i
$201$	0	0
$202$	407.525	2.01745
$203$	− 100.173i	− 0.493465i
$204$	0	0
$205$	−120.796	−0.589248
$206$	− 194.238i	− 0.942901i
$207$	0	0
$208$	89.2883	0.429271
$209$	− 202.526i	− 0.969022i
$210$	0	0
$211$	−24.6190	−0.116678	−0.0583388	−	0.998297i	$-0.518580\pi$
$211$	−24.6190	−0.116678	−0.0583388	+	0.998297i	$0.518580\pi$
$212$	− 367.933i	− 1.73553i
$213$	0	0
$214$	581.136	2.71559
$215$	358.239i	1.66623i
$216$	0	0
$217$	−187.359	−0.863407
$218$	− 465.810i	− 2.13674i
$219$	0	0
$220$	−410.775	−1.86716
$221$	113.492i	0.513538i
$222$	0	0
$223$	120.070	0.538431	0.269215	−	0.963080i	$-0.413236\pi$
$223$	120.070	0.538431	0.269215	+	0.963080i	$0.413236\pi$
$224$	− 192.438i	− 0.859100i
$225$	0	0
$226$	−36.1190	−0.159819
$227$	− 41.6972i	− 0.183688i	−0.995773	−	0.0918442i	$-0.970724\pi$
$227$	− 41.6972i	− 0.183688i	0.995773	−	0.0918442i	$-0.0292762\pi$
$228$	0	0
$229$	−256.974	−1.12216	−0.561078	−	0.827763i	$-0.689613\pi$
$229$	−256.974	−1.12216	−0.561078	+	0.827763i	$0.689613\pi$
$230$	742.738i	3.22930i
$231$	0	0
$232$	−67.0886	−0.289175
$233$	259.489i	1.11369i	0.830617	+	0.556844i	$0.187988\pi$
$233$	259.489i	1.11369i	−0.830617	+	0.556844i	$0.812012\pi$
$234$	0	0
$235$	−30.1021	−0.128094
$236$	274.753i	1.16421i
$237$	0	0
$238$	182.757	0.767886
$239$	− 180.742i	− 0.756242i	−0.925756	−	0.378121i	$-0.876570\pi$
$239$	− 180.742i	− 0.756242i	0.925756	−	0.378121i	$-0.123430\pi$
$240$	0	0
$241$	−293.575	−1.21816	−0.609078	−	0.793111i	$-0.708460\pi$
$241$	−293.575	−1.21816	−0.609078	+	0.793111i	$0.708460\pi$
$242$	12.9350i	0.0534505i
$243$	0	0
$244$	−224.416	−0.919739
$245$	− 228.412i	− 0.932294i
$246$	0	0
$247$	−143.270	−0.580039
$248$	125.479i	0.505964i
$249$	0	0
$250$	−107.817	−0.431268
$251$	330.675i	1.31743i	0.752393	+	0.658714i	$0.228899\pi$
$251$	330.675i	1.31743i	−0.752393	+	0.658714i	$0.771101\pi$
$252$	0	0
$253$	375.146	1.48279
$254$	− 33.7215i	− 0.132762i
$255$	0	0
$256$	94.5327	0.369268
$257$	− 281.301i	− 1.09456i	−0.836950	−	0.547279i	$-0.815664\pi$
$257$	− 281.301i	− 1.09456i	0.836950	−	0.547279i	$-0.184336\pi$
$258$	0	0
$259$	−11.7599	−0.0454052
$260$	290.588i	1.11765i
$261$	0	0
$262$	−39.2649	−0.149866
$263$	211.737i	0.805085i	0.915401	+	0.402543i	$0.131873\pi$
$263$	211.737i	0.805085i	−0.915401	+	0.402543i	$0.868127\pi$
$264$	0	0
$265$	−550.136	−2.07599
$266$	230.708i	0.867324i
$267$	0	0
$268$	−615.844	−2.29792
$269$	− 72.9266i	− 0.271103i	−0.990770	−	0.135551i	$-0.956719\pi$
$269$	− 72.9266i	− 0.271103i	0.990770	−	0.135551i	$-0.0432806\pi$
$270$	0	0
$271$	−435.095	−1.60552	−0.802759	−	0.596304i	$-0.796635\pi$
$271$	−435.095	−1.60552	−0.802759	+	0.596304i	$0.796635\pi$
$272$	161.578i	0.594036i
$273$	0	0
$274$	−172.903	−0.631032
$275$	334.326i	1.21573i
$276$	0	0
$277$	46.5294	0.167976	0.0839881	−	0.996467i	$-0.473234\pi$
$277$	46.5294	0.167976	0.0839881	+	0.996467i	$0.473234\pi$
$278$	273.499i	0.983808i
$279$	0	0
$280$	90.1006	0.321788
$281$	− 392.596i	− 1.39714i	−0.715543	−	0.698569i	$-0.753821\pi$
$281$	− 392.596i	− 1.39714i	0.715543	−	0.698569i	$-0.246179\pi$
$282$	0	0
$283$	204.724	0.723405	0.361702	−	0.932294i	$-0.382196\pi$
$283$	204.724	0.723405	0.361702	+	0.932294i	$0.382196\pi$
$284$	577.372i	2.03300i
$285$	0	0
$286$	265.283	0.927562
$287$	− 69.5024i	− 0.242168i
$288$	0	0
$289$	83.6230	0.289353
$290$	520.967i	1.79644i
$291$	0	0
$292$	−462.618	−1.58431
$293$	108.950i	0.371843i	0.982565	+	0.185921i	$0.0595270\pi$
$293$	108.950i	0.371843i	−0.982565	+	0.185921i	$0.940473\pi$
$294$	0	0
$295$	410.813	1.39259
$296$	7.87592i	0.0266078i
$297$	0	0
$298$	−19.9987	−0.0671096
$299$	− 265.384i	− 0.887570i
$300$	0	0
$301$	−206.120	−0.684784
$302$	− 737.854i	− 2.44323i
$303$	0	0
$304$	−203.972	−0.670961
$305$	335.549i	1.10016i
$306$	0	0
$307$	−83.1949	−0.270993	−0.135497	−	0.990778i	$-0.543263\pi$
$307$	−83.1949	−0.270993	−0.135497	+	0.990778i	$0.543263\pi$
$308$	− 236.348i	− 0.767363i
$309$	0	0
$310$	974.390	3.14319
$311$	157.785i	0.507347i	0.967290	+	0.253673i	$0.0816389\pi$
$311$	157.785i	0.507347i	−0.967290	+	0.253673i	$0.918361\pi$
$312$	0	0
$313$	522.092	1.66802	0.834012	−	0.551746i	$-0.186038\pi$
$313$	522.092	1.66802	0.834012	+	0.551746i	$0.186038\pi$
$314$	− 217.654i	− 0.693167i
$315$	0	0
$316$	14.4793	0.0458207
$317$	− 524.888i	− 1.65580i	−0.560876	−	0.827900i	$-0.689536\pi$
$317$	− 524.888i	− 1.65580i	0.560876	−	0.827900i	$-0.310464\pi$
$318$	0	0
$319$	263.133	0.824867
$320$	666.755i	2.08361i
$321$	0	0
$322$	−427.349	−1.32717
$323$	− 259.263i	− 0.802672i
$324$	0	0
$325$	236.507	0.727714
$326$	494.084i	1.51560i
$327$	0	0
$328$	−46.5474	−0.141913
$329$	− 17.3199i	− 0.0526440i
$330$	0	0
$331$	65.0345	0.196479	0.0982394	−	0.995163i	$-0.468679\pi$
$331$	65.0345	0.196479	0.0982394	+	0.995163i	$0.468679\pi$
$332$	358.294i	1.07920i
$333$	0	0
$334$	379.783	1.13708
$335$	920.815i	2.74870i
$336$	0	0
$337$	−290.810	−0.862937	−0.431469	−	0.902128i	$-0.642004\pi$
$337$	−290.810	−0.862937	−0.431469	+	0.902128i	$0.642004\pi$
$338$	318.034i	0.940928i
$339$	0	0
$340$	−525.854	−1.54663
$341$	− 492.150i	− 1.44325i
$342$	0	0
$343$	340.250	0.991981
$344$	138.044i	0.401289i
$345$	0	0
$346$	648.374	1.87391
$347$	− 80.9729i	− 0.233351i	−0.993170	−	0.116676i	$-0.962776\pi$
$347$	− 80.9729i	− 0.233351i	0.993170	−	0.116676i	$-0.0372238\pi$
$348$	0	0
$349$	388.559	1.11335	0.556675	−	0.830730i	$-0.312077\pi$
$349$	388.559	1.11335	0.556675	+	0.830730i	$0.312077\pi$
$350$	− 380.849i	− 1.08814i
$351$	0	0
$352$	505.491	1.43606
$353$	− 529.417i	− 1.49977i	−0.661570	−	0.749883i	$-0.730110\pi$
$353$	− 529.417i	− 1.49977i	0.661570	−	0.749883i	$-0.269890\pi$
$354$	0	0
$355$	863.292	2.43181
$356$	− 94.1614i	− 0.264498i
$357$	0	0
$358$	147.193	0.411154
$359$	488.615i	1.36105i	0.732727	+	0.680523i	$0.238247\pi$
$359$	488.615i	1.36105i	−0.732727	+	0.680523i	$0.761753\pi$
$360$	0	0
$361$	−33.7122	−0.0933855
$362$	819.063i	2.26261i
$363$	0	0
$364$	−167.196	−0.459329
$365$	691.711i	1.89510i
$366$	0	0
$367$	−289.731	−0.789458	−0.394729	−	0.918798i	$-0.629162\pi$
$367$	−289.731	−0.789458	−0.394729	+	0.918798i	$0.629162\pi$
$368$	− 377.825i	− 1.02670i
$369$	0	0
$370$	61.1593	0.165296
$371$	− 316.532i	− 0.853186i
$372$	0	0
$373$	523.958	1.40471	0.702356	−	0.711826i	$-0.252131\pi$
$373$	523.958	1.40471	0.702356	+	0.711826i	$0.252131\pi$
$374$	480.061i	1.28358i
$375$	0	0
$376$	−11.5995	−0.0308498
$377$	− 186.144i	− 0.493750i
$378$	0	0
$379$	−716.027	−1.88925	−0.944626	−	0.328148i	$-0.893576\pi$
$379$	−716.027	−1.88925	−0.944626	+	0.328148i	$0.893576\pi$
$380$	− 663.826i	− 1.74691i
$381$	0	0
$382$	85.8498	0.224738
$383$	364.557i	0.951847i	0.879487	+	0.475923i	$0.157886\pi$
$383$	364.557i	0.951847i	−0.879487	+	0.475923i	$0.842114\pi$
$384$	0	0
$385$	−353.389	−0.917895
$386$	− 593.470i	− 1.53749i
$387$	0	0
$388$	−90.3273	−0.232802
$389$	13.8459i	0.0355937i	0.999842	+	0.0177968i	$0.00566521\pi$
$389$	13.8459i	0.0355937i	−0.999842	+	0.0177968i	$0.994335\pi$
$390$	0	0
$391$	480.243	1.22824
$392$	− 88.0161i	− 0.224531i
$393$	0	0
$394$	−707.023	−1.79447
$395$	− 21.6496i	− 0.0548092i
$396$	0	0
$397$	−370.702	−0.933757	−0.466879	−	0.884321i	$-0.654622\pi$
$397$	−370.702	−0.933757	−0.466879	+	0.884321i	$0.654622\pi$
$398$	− 428.779i	− 1.07733i
$399$	0	0
$400$	336.714	0.841784
$401$	− 95.5302i	− 0.238230i	−0.992880	−	0.119115i	$-0.961994\pi$
$401$	− 95.5302i	− 0.238230i	0.992880	−	0.119115i	$-0.0380057\pi$
$402$	0	0
$403$	−348.154	−0.863905
$404$	− 674.672i	− 1.66998i
$405$	0	0
$406$	−299.749	−0.738298
$407$	− 30.8906i	− 0.0758984i
$408$	0	0
$409$	182.733	0.446779	0.223390	−	0.974729i	$-0.428288\pi$
$409$	182.733	0.446779	0.223390	+	0.974729i	$0.428288\pi$
$410$	361.458i	0.881604i
$411$	0	0
$412$	−321.567	−0.780503
$413$	236.370i	0.572324i
$414$	0	0
$415$	535.725	1.29090
$416$	− 357.592i	− 0.859596i
$417$	0	0
$418$	−606.017	−1.44980
$419$	811.040i	1.93566i	0.251611	+	0.967829i	$0.419040\pi$
$419$	811.040i	1.93566i	−0.251611	+	0.967829i	$0.580960\pi$
$420$	0	0
$421$	−448.621	−1.06561	−0.532804	−	0.846239i	$-0.678862\pi$
$421$	−448.621	−1.06561	−0.532804	+	0.846239i	$0.678862\pi$
$422$	73.6673i	0.174567i
$423$	0	0
$424$	−211.989	−0.499975
$425$	427.987i	1.00703i
$426$	0	0
$427$	−193.065	−0.452143
$428$	− 962.092i	− 2.24788i
$429$	0	0
$430$	1071.96	2.49293
$431$	− 205.939i	− 0.477816i	−0.971042	−	0.238908i	$-0.923211\pi$
$431$	− 205.939i	− 0.477816i	0.971042	−	0.238908i	$-0.0767895\pi$
$432$	0	0
$433$	427.090	0.986350	0.493175	−	0.869930i	$-0.335836\pi$
$433$	427.090	0.986350	0.493175	+	0.869930i	$0.335836\pi$
$434$	560.635i	1.29179i
$435$	0	0
$436$	−771.165	−1.76873
$437$	606.247i	1.38729i
$438$	0	0
$439$	−675.407	−1.53851	−0.769256	−	0.638941i	$-0.779373\pi$
$439$	−675.407	−1.53851	−0.769256	+	0.638941i	$0.779373\pi$
$440$	236.674i	0.537895i
$441$	0	0
$442$	339.602	0.768330
$443$	− 85.6791i	− 0.193406i	−0.995313	−	0.0967032i	$-0.969170\pi$
$443$	− 85.6791i	− 0.193406i	0.995313	−	0.0967032i	$-0.0308298\pi$
$444$	0	0
$445$	−140.791	−0.316384
$446$	− 359.285i	− 0.805573i
$447$	0	0
$448$	−383.631	−0.856319
$449$	416.580i	0.927795i	0.885889	+	0.463897i	$0.153549\pi$
$449$	416.580i	0.927795i	−0.885889	+	0.463897i	$0.846451\pi$
$450$	0	0
$451$	182.567	0.404804
$452$	59.7963i	0.132293i
$453$	0	0
$454$	−124.771	−0.274825
$455$	249.993i	0.549435i
$456$	0	0
$457$	−185.518	−0.405947	−0.202974	−	0.979184i	$-0.565061\pi$
$457$	−185.518	−0.405947	−0.202974	+	0.979184i	$0.565061\pi$
$458$	768.942i	1.67891i
$459$	0	0
$460$	1229.63	2.67311
$461$	− 334.030i	− 0.724578i	−0.932066	−	0.362289i	$-0.881995\pi$
$461$	− 334.030i	− 0.724578i	0.932066	−	0.362289i	$-0.118005\pi$
$462$	0	0
$463$	490.710	1.05985	0.529924	−	0.848045i	$-0.322220\pi$
$463$	490.710	1.05985	0.529924	+	0.848045i	$0.322220\pi$
$464$	− 265.012i	− 0.571146i
$465$	0	0
$466$	776.470	1.66624
$467$	656.431i	1.40563i	0.711371	+	0.702817i	$0.248075\pi$
$467$	656.431i	1.40563i	−0.711371	+	0.702817i	$0.751925\pi$
$468$	0	0
$469$	−529.810	−1.12966
$470$	90.0746i	0.191648i
$471$	0	0
$472$	158.303	0.335387
$473$	− 541.430i	− 1.14467i
$474$	0	0
$475$	−540.281	−1.13743
$476$	− 302.561i	− 0.635632i
$477$	0	0
$478$	−540.834	−1.13145
$479$	− 376.859i	− 0.786762i	−0.919375	−	0.393381i	$-0.871305\pi$
$479$	− 376.859i	− 0.786762i	0.919375	−	0.393381i	$-0.128695\pi$
$480$	0	0
$481$	−21.8525	−0.0454314
$482$	878.466i	1.82254i
$483$	0	0
$484$	21.4144	0.0442446
$485$	135.058i	0.278471i
$486$	0	0
$487$	138.419	0.284228	0.142114	−	0.989850i	$-0.454610\pi$
$487$	138.419	0.284228	0.142114	+	0.989850i	$0.454610\pi$
$488$	129.300i	0.264960i
$489$	0	0
$490$	−683.477	−1.39485
$491$	833.517i	1.69759i	0.528722	+	0.848795i	$0.322671\pi$
$491$	833.517i	1.69759i	−0.528722	+	0.848795i	$0.677329\pi$
$492$	0	0
$493$	336.849	0.683264
$494$	428.705i	0.867825i
$495$	0	0
$496$	−495.665	−0.999324
$497$	496.712i	0.999422i
$498$	0	0
$499$	−557.277	−1.11679	−0.558394	−	0.829576i	$-0.688582\pi$
$499$	−557.277	−1.11679	−0.558394	+	0.829576i	$0.688582\pi$
$500$	178.495i	0.356990i
$501$	0	0
$502$	989.478	1.97107
$503$	32.4506i	0.0645140i	0.999480	+	0.0322570i	$0.0102695\pi$
$503$	32.4506i	0.0645140i	−0.999480	+	0.0322570i	$0.989730\pi$
$504$	0	0
$505$	−1008.78	−1.99758
$506$	− 1122.55i	− 2.21848i
$507$	0	0
$508$	−55.8272	−0.109896
$509$	411.171i	0.807802i	0.914803	+	0.403901i	$0.132346\pi$
$509$	411.171i	0.807802i	−0.914803	+	0.403901i	$0.867654\pi$
$510$	0	0
$511$	−397.990	−0.778845
$512$	− 637.824i	− 1.24575i
$513$	0	0
$514$	−841.738	−1.63762
$515$	480.811i	0.933613i
$516$	0	0
$517$	45.4953	0.0879987
$518$	35.1893i	0.0679329i
$519$	0	0
$520$	167.426	0.321974
$521$	− 740.959i	− 1.42219i	−0.703098	−	0.711093i	$-0.748200\pi$
$521$	− 740.959i	− 1.42219i	0.703098	−	0.711093i	$-0.251800\pi$
$522$	0	0
$523$	398.655	0.762248	0.381124	−	0.924524i	$-0.375537\pi$
$523$	398.655	0.762248	0.381124	+	0.924524i	$0.375537\pi$
$524$	65.0044i	0.124054i
$525$	0	0
$526$	633.582	1.20453
$527$	− 630.025i	− 1.19549i
$528$	0	0
$529$	−593.975	−1.12283
$530$	1646.17i	3.10599i
$531$	0	0
$532$	381.946	0.717943
$533$	− 129.150i	− 0.242308i
$534$	0	0
$535$	−1438.53	−2.68884
$536$	354.826i	0.661990i
$537$	0	0
$538$	−218.218	−0.405610
$539$	345.214i	0.640471i
$540$	0	0
$541$	164.327	0.303747	0.151873	−	0.988400i	$-0.451469\pi$
$541$	164.327	0.303747	0.151873	+	0.988400i	$0.451469\pi$
$542$	1301.94i	2.40209i
$543$	0	0
$544$	647.105	1.18953
$545$	1153.05i	2.11569i
$546$	0	0
$547$	−503.882	−0.921174	−0.460587	−	0.887614i	$-0.652361\pi$
$547$	−503.882	−0.921174	−0.460587	+	0.887614i	$0.652361\pi$
$548$	286.247i	0.522348i
$549$	0	0
$550$	1000.40	1.81891
$551$	425.231i	0.771743i
$552$	0	0
$553$	12.4566	0.0225254
$554$	− 139.230i	− 0.251318i
$555$	0	0
$556$	452.787	0.814365
$557$	− 217.504i	− 0.390492i	−0.980754	−	0.195246i	$-0.937450\pi$
$557$	− 217.504i	− 0.390492i	0.980754	−	0.195246i	$-0.0625505\pi$
$558$	0	0
$559$	−383.015	−0.685179
$560$	355.913i	0.635560i
$561$	0	0
$562$	−1174.76	−2.09033
$563$	− 522.367i	− 0.927827i	−0.885880	−	0.463914i	$-0.846445\pi$
$563$	− 522.367i	− 0.927827i	0.885880	−	0.463914i	$-0.153555\pi$
$564$	0	0
$565$	89.4080	0.158244
$566$	− 612.594i	− 1.08232i
$567$	0	0
$568$	332.661	0.585670
$569$	− 28.4538i	− 0.0500066i	−0.999687	−	0.0250033i	$-0.992040\pi$
$569$	− 28.4538i	− 0.0500066i	0.999687	−	0.0250033i	$-0.00795963\pi$
$570$	0	0
$571$	−302.891	−0.530457	−0.265228	−	0.964186i	$-0.585447\pi$
$571$	−302.891	−0.530457	−0.265228	+	0.964186i	$0.585447\pi$
$572$	− 439.185i	− 0.767806i
$573$	0	0
$574$	−207.972	−0.362320
$575$	− 1000.78i	− 1.74049i
$576$	0	0
$577$	1061.82	1.84024	0.920121	−	0.391634i	$-0.128090\pi$
$577$	1061.82	1.84024	0.920121	+	0.391634i	$0.128090\pi$
$578$	− 250.225i	− 0.432915i
$579$	0	0
$580$	862.480	1.48703
$581$	308.240i	0.530534i
$582$	0	0
$583$	831.457	1.42617
$584$	266.544i	0.456410i
$585$	0	0
$586$	326.011	0.556333
$587$	− 718.632i	− 1.22425i	−0.790763	−	0.612123i	$-0.790316\pi$
$587$	− 718.632i	− 1.22425i	0.790763	−	0.612123i	$-0.209684\pi$
$588$	0	0
$589$	795.330	1.35030
$590$	− 1229.28i	− 2.08352i
$591$	0	0
$592$	−31.1113	−0.0525528
$593$	− 88.2527i	− 0.148824i	−0.997228	−	0.0744121i	$-0.976292\pi$
$593$	− 88.2527i	− 0.148824i	0.997228	−	0.0744121i	$-0.0237080\pi$
$594$	0	0
$595$	−452.392	−0.760322
$596$	33.1085i	0.0555512i
$597$	0	0
$598$	−794.107	−1.32794
$599$	719.971i	1.20196i	0.799266	+	0.600978i	$0.205222\pi$
$599$	719.971i	1.20196i	−0.799266	+	0.600978i	$0.794778\pi$
$600$	0	0
$601$	251.466	0.418413	0.209206	−	0.977872i	$-0.432912\pi$
$601$	251.466	0.418413	0.209206	+	0.977872i	$0.432912\pi$
$602$	616.773i	1.02454i
$603$	0	0
$604$	−1221.54	−2.02242
$605$	− 32.0190i	− 0.0529239i
$606$	0	0
$607$	−124.195	−0.204605	−0.102302	−	0.994753i	$-0.532621\pi$
$607$	−124.195	−0.204605	−0.102302	+	0.994753i	$0.532621\pi$
$608$	816.890i	1.34357i
$609$	0	0
$610$	1004.06	1.64601
$611$	− 32.1840i	− 0.0526744i
$612$	0	0
$613$	1001.06	1.63305	0.816525	−	0.577310i	$-0.195898\pi$
$613$	1001.06	1.63305	0.816525	+	0.577310i	$0.195898\pi$
$614$	248.944i	0.405447i
$615$	0	0
$616$	−136.175	−0.221063
$617$	− 222.659i	− 0.360873i	−0.983587	−	0.180437i	$-0.942249\pi$
$617$	− 222.659i	− 0.360873i	0.983587	−	0.180437i	$-0.0577511\pi$
$618$	0	0
$619$	1015.62	1.64074	0.820372	−	0.571830i	$-0.193766\pi$
$619$	1015.62	1.64074	0.820372	+	0.571830i	$0.193766\pi$
$620$	− 1613.14i	− 2.60183i
$621$	0	0
$622$	472.140	0.759067
$623$	− 81.0070i	− 0.130027i
$624$	0	0
$625$	−479.725	−0.767560
$626$	− 1562.25i	− 2.49561i
$627$	0	0
$628$	−360.335	−0.573781
$629$	− 39.5447i	− 0.0628691i
$630$	0	0
$631$	1155.66	1.83148	0.915740	−	0.401772i	$-0.131605\pi$
$631$	1155.66	1.83148	0.915740	+	0.401772i	$0.131605\pi$
$632$	− 8.34247i	− 0.0132001i
$633$	0	0
$634$	−1570.62	−2.47732
$635$	83.4733i	0.131454i
$636$	0	0
$637$	244.209	0.383374
$638$	− 787.372i	− 1.23412i
$639$	0	0
$640$	657.291	1.02702
$641$	805.647i	1.25686i	0.777867	+	0.628430i	$0.216302\pi$
$641$	805.647i	1.25686i	−0.777867	+	0.628430i	$0.783698\pi$
$642$	0	0
$643$	−949.145	−1.47612	−0.738060	−	0.674736i	$-0.764258\pi$
$643$	−949.145	−1.47612	−0.738060	+	0.674736i	$0.764258\pi$
$644$	707.492i	1.09859i
$645$	0	0
$646$	−775.793	−1.20092
$647$	− 816.742i	− 1.26235i	−0.775639	−	0.631176i	$-0.782572\pi$
$647$	− 816.742i	− 1.26235i	0.775639	−	0.631176i	$-0.217428\pi$
$648$	0	0
$649$	−620.889	−0.956686
$650$	− 707.700i	− 1.08877i
$651$	0	0
$652$	817.974	1.25456
$653$	− 1086.25i	− 1.66347i	−0.555174	−	0.831735i	$-0.687348\pi$
$653$	− 1086.25i	− 1.66347i	0.555174	−	0.831735i	$-0.312652\pi$
$654$	0	0
$655$	97.1952	0.148390
$656$	− 183.871i	− 0.280291i
$657$	0	0
$658$	−51.8262	−0.0787633
$659$	− 277.890i	− 0.421685i	−0.977520	−	0.210842i	$-0.932379\pi$
$659$	− 277.890i	− 0.421685i	0.977520	−	0.210842i	$-0.0676207\pi$
$660$	0	0
$661$	161.745	0.244697	0.122349	−	0.992487i	$-0.460957\pi$
$661$	161.745	0.244697	0.122349	+	0.992487i	$0.460957\pi$
$662$	− 194.603i	− 0.293962i
$663$	0	0
$664$	206.436	0.310898
$665$	− 571.089i	− 0.858780i
$666$	0	0
$667$	−787.671	−1.18092
$668$	− 628.745i	− 0.941235i
$669$	0	0
$670$	2755.36	4.11247
$671$	− 507.138i	− 0.755794i
$672$	0	0
$673$	−1234.51	−1.83434	−0.917168	−	0.398501i	$-0.869531\pi$
$673$	−1234.51	−1.83434	−0.917168	+	0.398501i	$0.869531\pi$
$674$	870.190i	1.29108i
$675$	0	0
$676$	526.516	0.778870
$677$	1105.68i	1.63321i	0.577196	+	0.816606i	$0.304147\pi$
$677$	1105.68i	1.63321i	−0.577196	+	0.816606i	$0.695853\pi$
$678$	0	0
$679$	−77.7085	−0.114445
$680$	302.978i	0.445555i
$681$	0	0
$682$	−1472.66	−2.15932
$683$	− 276.581i	− 0.404950i	−0.979287	−	0.202475i	$-0.935101\pi$
$683$	− 276.581i	− 0.404950i	0.979287	−	0.202475i	$-0.0648985\pi$
$684$	0	0
$685$	427.999	0.624816
$686$	− 1018.13i	− 1.48415i
$687$	0	0
$688$	−545.297	−0.792582
$689$	− 588.185i	− 0.853679i
$690$	0	0
$691$	−540.511	−0.782215	−0.391108	−	0.920345i	$-0.627908\pi$
$691$	−540.511	−0.782215	−0.391108	+	0.920345i	$0.627908\pi$
$692$	− 1073.41i	− 1.55117i
$693$	0	0
$694$	−242.295	−0.349128
$695$	− 677.011i	− 0.974117i
$696$	0	0
$697$	233.713	0.335312
$698$	− 1162.69i	− 1.66574i
$699$	0	0
$700$	−630.509	−0.900728
$701$	133.789i	0.190855i	0.995436	+	0.0954276i	$0.0304218\pi$
$701$	133.789i	0.190855i	−0.995436	+	0.0954276i	$0.969578\pi$
$702$	0	0
$703$	49.9203	0.0710103
$704$	− 1007.71i	− 1.43141i
$705$	0	0
$706$	−1584.18	−2.24388
$707$	− 580.420i	− 0.820962i
$708$	0	0
$709$	376.567	0.531124	0.265562	−	0.964094i	$-0.414443\pi$
$709$	376.567	0.531124	0.265562	+	0.964094i	$0.414443\pi$
$710$	− 2583.23i	− 3.63835i
$711$	0	0
$712$	−54.2523	−0.0761971
$713$	1473.22i	2.06623i
$714$	0	0
$715$	−656.674	−0.918425
$716$	− 243.683i	− 0.340340i
$717$	0	0
$718$	1462.08	2.03633
$719$	57.6348i	0.0801596i	0.999196	+	0.0400798i	$0.0127612\pi$
$719$	57.6348i	0.0801596i	−0.999196	+	0.0400798i	$0.987239\pi$
$720$	0	0
$721$	−276.644	−0.383695
$722$	100.877i	0.139719i
$723$	0	0
$724$	1355.99	1.87291
$725$	− 701.964i	− 0.968226i
$726$	0	0
$727$	341.495	0.469732	0.234866	−	0.972028i	$-0.424535\pi$
$727$	341.495	0.469732	0.234866	+	0.972028i	$0.424535\pi$
$728$	96.3321i	0.132324i
$729$	0	0
$730$	2069.81	2.83535
$731$	− 693.111i	− 0.948169i
$732$	0	0
$733$	−442.745	−0.604018	−0.302009	−	0.953305i	$-0.597657\pi$
$733$	−442.745	−0.604018	−0.302009	+	0.953305i	$0.597657\pi$
$734$	866.962i	1.18115i
$735$	0	0
$736$	−1513.16	−2.05592
$737$	− 1391.69i	− 1.88831i
$738$	0	0
$739$	−152.397	−0.206220	−0.103110	−	0.994670i	$-0.532879\pi$
$739$	−152.397	−0.206220	−0.103110	+	0.994670i	$0.532879\pi$
$740$	− 101.251i	− 0.136826i
$741$	0	0
$742$	−947.159	−1.27649
$743$	− 143.763i	− 0.193490i	−0.995309	−	0.0967449i	$-0.969157\pi$
$743$	− 143.763i	− 0.193490i	0.995309	−	0.0967449i	$-0.0308431\pi$
$744$	0	0
$745$	49.5042	0.0664485
$746$	− 1567.84i	− 2.10166i
$747$	0	0
$748$	794.757	1.06251
$749$	− 827.687i	− 1.10506i
$750$	0	0
$751$	−721.144	−0.960244	−0.480122	−	0.877202i	$-0.659408\pi$
$751$	−721.144	−0.960244	−0.480122	+	0.877202i	$0.659408\pi$
$752$	− 45.8202i	− 0.0609312i
$753$	0	0
$754$	−556.998	−0.738724
$755$	1826.46i	2.41916i
$756$	0	0
$757$	483.778	0.639073	0.319536	−	0.947574i	$-0.396473\pi$
$757$	483.778	0.639073	0.319536	+	0.947574i	$0.396473\pi$
$758$	2142.57i	2.82661i
$759$	0	0
$760$	−382.472	−0.503253
$761$	− 418.522i	− 0.549963i	−0.961450	−	0.274981i	$-0.911328\pi$
$761$	− 418.522i	− 0.549963i	0.961450	−	0.274981i	$-0.0886717\pi$
$762$	0	0
$763$	−663.433	−0.869505
$764$	− 142.127i	− 0.186031i
$765$	0	0
$766$	1090.86	1.42411
$767$	439.226i	0.572654i
$768$	0	0
$769$	−617.073	−0.802436	−0.401218	−	0.915983i	$-0.631413\pi$
$769$	−617.073	−0.802436	−0.401218	+	0.915983i	$0.631413\pi$
$770$	1057.45i	1.37331i
$771$	0	0
$772$	−982.510	−1.27268
$773$	82.0130i	0.106097i	0.998592	+	0.0530485i	$0.0168938\pi$
$773$	82.0130i	0.106097i	−0.998592	+	0.0530485i	$0.983106\pi$
$774$	0	0
$775$	−1312.92	−1.69409
$776$	52.0433i	0.0670661i
$777$	0	0
$778$	41.4312	0.0532535
$779$	295.034i	0.378734i
$780$	0	0
$781$	−1304.75	−1.67061
$782$	− 1437.03i	− 1.83764i
$783$	0	0
$784$	347.679	0.443468
$785$	538.776i	0.686339i
$786$	0	0
$787$	−384.669	−0.488779	−0.244389	−	0.969677i	$-0.578588\pi$
$787$	−384.669	−0.488779	−0.244389	+	0.969677i	$0.578588\pi$
$788$	1170.50i	1.48541i
$789$	0	0
$790$	−64.7822	−0.0820028
$791$	51.4427i	0.0650350i
$792$	0	0
$793$	−358.756	−0.452404
$794$	1109.25i	1.39704i
$795$	0	0
$796$	−709.858	−0.891782
$797$	− 4.35189i	− 0.00546033i	−0.999996	−	0.00273017i	$-0.999131\pi$
$797$	− 4.35189i	− 0.00546033i	0.999996	−	0.00273017i	$-0.000869040\pi$
$798$	0	0
$799$	58.2408	0.0728921
$800$	− 1348.51i	− 1.68564i
$801$	0	0
$802$	−285.855	−0.356428
$803$	− 1045.43i	− 1.30190i
$804$	0	0
$805$	1057.85	1.31410
$806$	1041.78i	1.29253i
$807$	0	0
$808$	−388.721	−0.481091
$809$	− 147.197i	− 0.181949i	−0.995853	−	0.0909745i	$-0.971002\pi$
$809$	− 147.197i	− 0.181949i	0.995853	−	0.0909745i	$-0.0289982\pi$
$810$	0	0
$811$	428.815	0.528749	0.264375	−	0.964420i	$-0.414835\pi$
$811$	428.815	0.528749	0.264375	+	0.964420i	$0.414835\pi$
$812$	496.245i	0.611139i
$813$	0	0
$814$	−92.4341	−0.113555
$815$	− 1223.04i	− 1.50067i
$816$	0	0
$817$	874.968	1.07095
$818$	− 546.791i	− 0.668449i
$819$	0	0
$820$	598.406	0.729763
$821$	− 1041.06i	− 1.26803i	−0.773319	−	0.634017i	$-0.781405\pi$
$821$	− 1041.06i	− 1.26803i	0.773319	−	0.634017i	$-0.218595\pi$
$822$	0	0
$823$	688.322	0.836357	0.418178	−	0.908365i	$-0.362669\pi$
$823$	688.322	0.836357	0.418178	+	0.908365i	$0.362669\pi$
$824$	185.275i	0.224849i
$825$	0	0
$826$	707.289	0.856282
$827$	1196.75i	1.44710i	0.690274	+	0.723548i	$0.257490\pi$
$827$	1196.75i	1.44710i	−0.690274	+	0.723548i	$0.742510\pi$
$828$	0	0
$829$	534.150	0.644330	0.322165	−	0.946683i	$-0.395589\pi$
$829$	534.150	0.644330	0.322165	+	0.946683i	$0.395589\pi$
$830$	− 1603.05i	− 1.93139i
$831$	0	0
$832$	−712.869	−0.856814
$833$	441.925i	0.530523i
$834$	0	0
$835$	−940.105	−1.12587
$836$	1003.28i	1.20010i
$837$	0	0
$838$	2426.88	2.89603
$839$	1019.14i	1.21471i	0.794430	+	0.607356i	$0.207770\pi$
$839$	1019.14i	1.21471i	−0.794430	+	0.607356i	$0.792230\pi$
$840$	0	0
$841$	288.517	0.343064
$842$	1342.41i	1.59431i
$843$	0	0
$844$	121.959	0.144501
$845$	− 787.252i	− 0.931659i
$846$	0	0
$847$	18.4228	0.0217506
$848$	− 837.395i	− 0.987494i
$849$	0	0
$850$	1280.67	1.50667
$851$	92.4692i	0.108659i
$852$	0	0
$853$	1551.01	1.81830	0.909152	−	0.416465i	$-0.136731\pi$
$853$	1551.01	1.81830	0.909152	+	0.416465i	$0.136731\pi$
$854$	577.709i	0.676474i
$855$	0	0
$856$	−554.322	−0.647573
$857$	59.4182i	0.0693328i	0.999399	+	0.0346664i	$0.0110369\pi$
$857$	59.4182i	0.0693328i	−0.999399	+	0.0346664i	$0.988963\pi$
$858$	0	0
$859$	1306.85	1.52136	0.760681	−	0.649126i	$-0.224865\pi$
$859$	1306.85	1.52136	0.760681	+	0.649126i	$0.224865\pi$
$860$	− 1774.66i	− 2.06356i
$861$	0	0
$862$	−616.231	−0.714885
$863$	− 183.653i	− 0.212807i	−0.994323	−	0.106404i	$-0.966066\pi$
$863$	− 183.653i	− 0.212807i	0.994323	−	0.106404i	$-0.0339335\pi$
$864$	0	0
$865$	−1604.97	−1.85545
$866$	− 1277.98i	− 1.47573i
$867$	0	0
$868$	928.151	1.06930
$869$	32.7205i	0.0376531i
$870$	0	0
$871$	−984.500	−1.13031
$872$	444.317i	0.509538i
$873$	0	0
$874$	1814.07	2.07560
$875$	153.559i	0.175496i
$876$	0	0
$877$	−313.408	−0.357364	−0.178682	−	0.983907i	$-0.557183\pi$
$877$	−313.408	−0.357364	−0.178682	+	0.983907i	$0.557183\pi$
$878$	2021.02i	2.30184i
$879$	0	0
$880$	−934.903	−1.06239
$881$	− 1370.61i	− 1.55575i	−0.628421	−	0.777874i	$-0.716298\pi$
$881$	− 1370.61i	− 1.55575i	0.628421	−	0.777874i	$-0.283702\pi$
$882$	0	0
$883$	1403.11	1.58903	0.794514	−	0.607246i	$-0.207726\pi$
$883$	1403.11	1.58903	0.794514	+	0.607246i	$0.207726\pi$
$884$	− 562.223i	− 0.635999i
$885$	0	0
$886$	−256.377	−0.289365
$887$	− 1565.83i	− 1.76531i	−0.470018	−	0.882657i	$-0.655752\pi$
$887$	− 1565.83i	− 1.76531i	0.470018	−	0.882657i	$-0.344248\pi$
$888$	0	0
$889$	−48.0280	−0.0540248
$890$	421.289i	0.473358i
$891$	0	0
$892$	−594.810	−0.666827
$893$	73.5219i	0.0823313i
$894$	0	0
$895$	−364.357	−0.407103
$896$	378.186i	0.422082i
$897$	0	0
$898$	1246.53	1.38812
$899$	1033.34i	1.14943i
$900$	0	0
$901$	1064.39	1.18134
$902$	− 546.294i	− 0.605648i
$903$	0	0
$904$	34.4524	0.0381111
$905$	− 2027.49i	− 2.24032i
$906$	0	0
$907$	1160.24	1.27921	0.639606	−	0.768703i	$-0.279098\pi$
$907$	1160.24	1.27921	0.639606	+	0.768703i	$0.279098\pi$
$908$	206.562i	0.227491i
$909$	0	0
$910$	748.054	0.822037
$911$	1150.44i	1.26283i	0.775445	+	0.631415i	$0.217525\pi$
$911$	1150.44i	1.26283i	−0.775445	+	0.631415i	$0.782475\pi$
$912$	0	0
$913$	−809.676	−0.886830
$914$	555.125i	0.607358i
$915$	0	0
$916$	1273.01	1.38975
$917$	55.9232i	0.0609850i
$918$	0	0
$919$	−950.545	−1.03433	−0.517163	−	0.855887i	$-0.673012\pi$
$919$	−950.545	−1.03433	−0.517163	+	0.855887i	$0.673012\pi$
$920$	− 708.467i	− 0.770073i
$921$	0	0
$922$	−999.519	−1.08408
$923$	922.999i	0.999999i
$924$	0	0
$925$	−82.4076	−0.0890892
$926$	− 1468.35i	− 1.58569i
$927$	0	0
$928$	−1061.35	−1.14370
$929$	40.1479i	0.0432163i	0.999767	+	0.0216082i	$0.00687862\pi$
$929$	40.1479i	0.0432163i	−0.999767	+	0.0216082i	$0.993121\pi$
$930$	0	0
$931$	−557.876	−0.599223
$932$	− 1285.47i	− 1.37926i
$933$	0	0
$934$	1964.24	2.10304
$935$	− 1188.33i	− 1.27094i
$936$	0	0
$937$	−392.262	−0.418637	−0.209318	−	0.977848i	$-0.567124\pi$
$937$	−392.262	−0.418637	−0.209318	+	0.977848i	$0.567124\pi$
$938$	1585.35i	1.69014i
$939$	0	0
$940$	149.122	0.158640
$941$	− 409.879i	− 0.435578i	−0.975996	−	0.217789i	$-0.930116\pi$
$941$	− 409.879i	− 0.435578i	0.975996	−	0.217789i	$-0.0698845\pi$
$942$	0	0
$943$	−546.502	−0.579535
$944$	625.323i	0.662419i
$945$	0	0
$946$	−1620.12	−1.71260
$947$	− 54.8522i	− 0.0579221i	−0.999581	−	0.0289610i	$-0.990780\pi$
$947$	− 54.8522i	− 0.0579221i	0.999581	−	0.0289610i	$-0.00921988\pi$
$948$	0	0
$949$	−739.551	−0.779295
$950$	1616.68i	1.70177i
$951$	0	0
$952$	−174.324	−0.183114
$953$	1108.85i	1.16354i	0.813354	+	0.581770i	$0.197640\pi$
$953$	1108.85i	1.16354i	−0.813354	+	0.581770i	$0.802360\pi$
$954$	0	0
$955$	−212.510	−0.222524
$956$	895.370i	0.936579i
$957$	0	0
$958$	−1127.68	−1.17711
$959$	246.258i	0.256786i
$960$	0	0
$961$	971.700	1.01113
$962$	65.3892i	0.0679722i
$963$	0	0
$964$	1454.33	1.50864
$965$	1469.06i	1.52234i
$966$	0	0
$967$	1039.03	1.07449	0.537243	−	0.843427i	$-0.319466\pi$
$967$	1039.03	1.07449	0.537243	+	0.843427i	$0.319466\pi$
$968$	− 12.3382i	− 0.0127461i
$969$	0	0
$970$	404.135	0.416634
$971$	− 1405.38i	− 1.44735i	−0.690140	−	0.723676i	$-0.742451\pi$
$971$	− 1405.38i	− 1.44735i	0.690140	−	0.723676i	$-0.257549\pi$
$972$	0	0
$973$	389.532	0.400341
$974$	− 414.191i	− 0.425247i
$975$	0	0
$976$	−510.760	−0.523319
$977$	385.735i	0.394816i	0.980321	+	0.197408i	$0.0632524\pi$
$977$	385.735i	0.394816i	−0.980321	+	0.197408i	$0.936748\pi$
$978$	0	0
$979$	212.787	0.217351
$980$	1131.52i	1.15461i
$981$	0	0
$982$	2494.13	2.53985
$983$	− 504.637i	− 0.513364i	−0.966496	−	0.256682i	$-0.917371\pi$
$983$	− 504.637i	− 0.513364i	0.966496	−	0.256682i	$-0.0826293\pi$
$984$	0	0
$985$	1750.15	1.77680
$986$	− 1007.95i	− 1.02227i
$987$	0	0
$988$	709.737	0.718357
$989$	1620.74i	1.63876i
$990$	0	0
$991$	−1595.30	−1.60978	−0.804892	−	0.593421i	$-0.797777\pi$
$991$	−1595.30	−1.60978	−0.804892	+	0.593421i	$0.797777\pi$
$992$	1985.09i	2.00110i
$993$	0	0
$994$	1486.31	1.49528
$995$	1061.39i	1.06672i
$996$	0	0
$997$	461.034	0.462421	0.231210	−	0.972904i	$-0.425731\pi$
$997$	461.034	0.462421	0.231210	+	0.972904i	$0.425731\pi$
$998$	1667.54i	1.67088i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.13	✓	84
3.2	odd	2	inner	1143.3.b.a.890.72	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.13	✓	84	1.1	even	1	trivial
1143.3.b.a.890.72	yes	84	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.99230i q^{2} -4.95386 q^{4} +7.40706i q^{5} -4.26180 q^{7} +2.85423i q^{8} +O(q^{10})\)	\(q-2.99230i q^{2} -4.95386 q^{4} +7.40706i q^{5} -4.26180 q^{7} +2.85423i q^{8} +22.1641 q^{10} -11.1948i q^{11} -7.91934 q^{13} +12.7526i q^{14} -11.2747 q^{16} -14.3310i q^{17} +18.0911 q^{19} -36.6935i q^{20} -33.4981 q^{22} +33.5108i q^{23} -29.8645 q^{25} +23.6970i q^{26} +21.1123 q^{28} +23.5050i q^{29} +43.9625 q^{31} +45.1543i q^{32} -42.8826 q^{34} -31.5674i q^{35} +2.75938 q^{37} -54.1340i q^{38} -21.1415 q^{40} +16.3082i q^{41} +48.3645 q^{43} +55.4573i q^{44} +100.274 q^{46} +4.06398i q^{47} -30.8371 q^{49} +89.3635i q^{50} +39.2313 q^{52} +74.2719i q^{53} +82.9203 q^{55} -12.1642i q^{56} +70.3339 q^{58} -55.4624i q^{59} +45.3013 q^{61} -131.549i q^{62} +90.0162 q^{64} -58.6590i q^{65} +124.316 q^{67} +70.9936i q^{68} -94.4591 q^{70} -116.550i q^{71} +93.3854 q^{73} -8.25690i q^{74} -89.6207 q^{76} +47.7098i q^{77} -2.92284 q^{79} -83.5125i q^{80} +48.7991 q^{82} -72.3263i q^{83} +106.150 q^{85} -144.721i q^{86} +31.9525 q^{88} +19.0077i q^{89} +33.7506 q^{91} -166.008i q^{92} +12.1606 q^{94} +134.002i q^{95} +18.2337 q^{97} +92.2738i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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