LMFDB - Embedded newform 6223.2.a.t.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, 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("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [74,18,0,86,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$74$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.05093 q^{2} +3.36661 q^{3} +2.20631 q^{4} -4.11869 q^{5} -6.90468 q^{6} -0.423137 q^{8} +8.33406 q^{9} +8.44714 q^{10} +4.89422 q^{11} +7.42780 q^{12} -2.86861 q^{13} -13.8660 q^{15} -3.54481 q^{16} -5.56908 q^{17} -17.0926 q^{18} +1.58477 q^{19} -9.08712 q^{20} -10.0377 q^{22} +5.20345 q^{23} -1.42454 q^{24} +11.9636 q^{25} +5.88332 q^{26} +17.9577 q^{27} -3.57024 q^{29} +28.4382 q^{30} -2.12419 q^{31} +8.11642 q^{32} +16.4769 q^{33} +11.4218 q^{34} +18.3876 q^{36} +4.01087 q^{37} -3.25026 q^{38} -9.65750 q^{39} +1.74277 q^{40} +5.33687 q^{41} -4.34649 q^{43} +10.7982 q^{44} -34.3254 q^{45} -10.6719 q^{46} +3.84000 q^{47} -11.9340 q^{48} -24.5365 q^{50} -18.7489 q^{51} -6.32906 q^{52} +2.13965 q^{53} -36.8300 q^{54} -20.1577 q^{55} +5.33531 q^{57} +7.32232 q^{58} +1.95069 q^{59} -30.5928 q^{60} -12.9121 q^{61} +4.35656 q^{62} -9.55660 q^{64} +11.8149 q^{65} -33.7930 q^{66} +11.8370 q^{67} -12.2871 q^{68} +17.5180 q^{69} -4.52622 q^{71} -3.52645 q^{72} -8.76529 q^{73} -8.22601 q^{74} +40.2767 q^{75} +3.49651 q^{76} +19.8069 q^{78} +12.4735 q^{79} +14.5999 q^{80} +35.4544 q^{81} -10.9456 q^{82} -7.89568 q^{83} +22.9373 q^{85} +8.91434 q^{86} -12.0196 q^{87} -2.07092 q^{88} +0.492530 q^{89} +70.3990 q^{90} +11.4804 q^{92} -7.15130 q^{93} -7.87557 q^{94} -6.52719 q^{95} +27.3248 q^{96} -1.80138 q^{97} +40.7887 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Coefficient data

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

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

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

$2$	−2.05093	−1.45023	−0.725113	−	0.688630i	$-0.758213\pi$
$2$	−2.05093	−1.45023	−0.725113	+	0.688630i	$0.758213\pi$
$3$	3.36661	1.94371	0.971856	−	0.235574i	$-0.0756969\pi$
$3$	3.36661	1.94371	0.971856	+	0.235574i	$0.0756969\pi$
$4$	2.20631	1.10316
$5$	−4.11869	−1.84193	−0.920967	−	0.389641i	$-0.872599\pi$
$5$	−4.11869	−1.84193	−0.920967	+	0.389641i	$0.872599\pi$
$6$	−6.90468	−2.81882
$7$	0	0
$7$	0	0
$8$	−0.423137	−0.149601
$9$	8.33406	2.77802
$10$	8.44714	2.67122
$11$	4.89422	1.47566	0.737831	−	0.674986i	$-0.235850\pi$
$11$	4.89422	1.47566	0.737831	+	0.674986i	$0.235850\pi$
$12$	7.42780	2.14422
$13$	−2.86861	−0.795610	−0.397805	−	0.917470i	$-0.630228\pi$
$13$	−2.86861	−0.795610	−0.397805	+	0.917470i	$0.630228\pi$
$14$	0	0
$15$	−13.8660	−3.58019
$16$	−3.54481	−0.886201
$17$	−5.56908	−1.35070	−0.675350	−	0.737497i	$-0.736007\pi$
$17$	−5.56908	−1.35070	−0.675350	+	0.737497i	$0.736007\pi$
$18$	−17.0926	−4.02876
$19$	1.58477	0.363572	0.181786	−	0.983338i	$-0.441812\pi$
$19$	1.58477	0.363572	0.181786	+	0.983338i	$0.441812\pi$
$20$	−9.08712	−2.03194
$21$	0	0
$22$	−10.0377	−2.14004
$23$	5.20345	1.08499	0.542497	−	0.840058i	$-0.317479\pi$
$23$	5.20345	1.08499	0.542497	+	0.840058i	$0.317479\pi$
$24$	−1.42454	−0.290782
$25$	11.9636	2.39272
$26$	5.88332	1.15381
$27$	17.9577	3.45596
$28$	0	0
$29$	−3.57024	−0.662977	−0.331489	−	0.943459i	$-0.607551\pi$
$29$	−3.57024	−0.662977	−0.331489	+	0.943459i	$0.607551\pi$
$30$	28.4382	5.19209
$31$	−2.12419	−0.381515	−0.190758	−	0.981637i	$-0.561094\pi$
$31$	−2.12419	−0.381515	−0.190758	+	0.981637i	$0.561094\pi$
$32$	8.11642	1.43479
$33$	16.4769	2.86826
$34$	11.4218	1.95882
$35$	0	0
$36$	18.3876	3.06459
$37$	4.01087	0.659383	0.329691	−	0.944089i	$-0.393055\pi$
$37$	4.01087	0.659383	0.329691	+	0.944089i	$0.393055\pi$
$38$	−3.25026	−0.527262
$39$	−9.65750	−1.54644
$40$	1.74277	0.275556
$41$	5.33687	0.833479	0.416740	−	0.909026i	$-0.363173\pi$
$41$	5.33687	0.833479	0.416740	+	0.909026i	$0.363173\pi$
$42$	0	0
$43$	−4.34649	−0.662833	−0.331417	−	0.943485i	$-0.607527\pi$
$43$	−4.34649	−0.662833	−0.331417	+	0.943485i	$0.607527\pi$
$44$	10.7982	1.62789
$45$	−34.3254	−5.11693
$46$	−10.6719	−1.57349
$47$	3.84000	0.560121	0.280061	−	0.959982i	$-0.409645\pi$
$47$	3.84000	0.560121	0.280061	+	0.959982i	$0.409645\pi$
$48$	−11.9340	−1.72252
$49$	0	0
$50$	−24.5365	−3.46998
$51$	−18.7489	−2.62537
$52$	−6.32906	−0.877683
$53$	2.13965	0.293904	0.146952	−	0.989144i	$-0.453054\pi$
$53$	2.13965	0.293904	0.146952	+	0.989144i	$0.453054\pi$
$54$	−36.8300	−5.01193
$55$	−20.1577	−2.71807
$56$	0	0
$57$	5.33531	0.706679
$58$	7.32232	0.961467
$59$	1.95069	0.253958	0.126979	−	0.991905i	$-0.459472\pi$
$59$	1.95069	0.253958	0.126979	+	0.991905i	$0.459472\pi$
$60$	−30.5928	−3.94951
$61$	−12.9121	−1.65323	−0.826614	−	0.562770i	$-0.809736\pi$
$61$	−12.9121	−1.65323	−0.826614	+	0.562770i	$0.809736\pi$
$62$	4.35656	0.553283
$63$	0	0
$64$	−9.55660	−1.19458
$65$	11.8149	1.46546
$66$	−33.7930	−4.15963
$67$	11.8370	1.44612	0.723062	−	0.690783i	$-0.242734\pi$
$67$	11.8370	1.44612	0.723062	+	0.690783i	$0.242734\pi$
$68$	−12.2871	−1.49003
$69$	17.5180	2.10892
$70$	0	0
$71$	−4.52622	−0.537163	−0.268581	−	0.963257i	$-0.586555\pi$
$71$	−4.52622	−0.537163	−0.268581	+	0.963257i	$0.586555\pi$
$72$	−3.52645	−0.415596
$73$	−8.76529	−1.02590	−0.512950	−	0.858419i	$-0.671447\pi$
$73$	−8.76529	−1.02590	−0.512950	+	0.858419i	$0.671447\pi$
$74$	−8.22601	−0.956254
$75$	40.2767	4.65076
$76$	3.49651	0.401077
$77$	0	0
$78$	19.8069	2.24268
$79$	12.4735	1.40337	0.701687	−	0.712485i	$-0.252430\pi$
$79$	12.4735	1.40337	0.701687	+	0.712485i	$0.252430\pi$
$80$	14.5999	1.63232
$81$	35.4544	3.93937
$82$	−10.9456	−1.20873
$83$	−7.89568	−0.866664	−0.433332	−	0.901234i	$-0.642662\pi$
$83$	−7.89568	−0.866664	−0.433332	+	0.901234i	$0.642662\pi$
$84$	0	0
$85$	22.9373	2.48790
$86$	8.91434	0.961258
$87$	−12.0196	−1.28864
$88$	−2.07092	−0.220761
$89$	0.492530	0.0522081	0.0261040	−	0.999659i	$-0.491690\pi$
$89$	0.492530	0.0522081	0.0261040	+	0.999659i	$0.491690\pi$
$90$	70.3990	7.42070
$91$	0	0
$92$	11.4804	1.19692
$93$	−7.15130	−0.741556
$94$	−7.87557	−0.812303
$95$	−6.52719	−0.669675
$96$	27.3248	2.78883
$97$	−1.80138	−0.182903	−0.0914514	−	0.995810i	$-0.529151\pi$
$97$	−1.80138	−0.182903	−0.0914514	+	0.995810i	$0.529151\pi$
$98$	0	0
$99$	40.7887	4.09942
$100$	26.3954	2.63954
$101$	14.7763	1.47030	0.735150	−	0.677904i	$-0.237111\pi$
$101$	14.7763	1.47030	0.735150	+	0.677904i	$0.237111\pi$
$102$	38.4527	3.80739
$103$	−6.30141	−0.620897	−0.310448	−	0.950590i	$-0.600479\pi$
$103$	−6.30141	−0.620897	−0.310448	+	0.950590i	$0.600479\pi$
$104$	1.21382	0.119024
$105$	0	0
$106$	−4.38827	−0.426227
$107$	−11.5285	−1.11450	−0.557249	−	0.830345i	$-0.688143\pi$
$107$	−11.5285	−1.11450	−0.557249	+	0.830345i	$0.688143\pi$
$108$	39.6203	3.81247
$109$	−12.1560	−1.16433	−0.582167	−	0.813069i	$-0.697795\pi$
$109$	−12.1560	−1.16433	−0.582167	+	0.813069i	$0.697795\pi$
$110$	41.3421	3.94182
$111$	13.5030	1.28165
$112$	0	0
$113$	16.4357	1.54614	0.773072	−	0.634318i	$-0.218719\pi$
$113$	16.4357	1.54614	0.773072	+	0.634318i	$0.218719\pi$
$114$	−10.9424	−1.02485
$115$	−21.4314	−1.99849
$116$	−7.87708	−0.731368
$117$	−23.9072	−2.21022
$118$	−4.00073	−0.368297
$119$	0	0
$120$	5.86722	0.535601
$121$	12.9533	1.17758
$122$	26.4819	2.39755
$123$	17.9672	1.62004
$124$	−4.68662	−0.420871
$125$	−28.6809	−2.56529
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	3.36708	0.297611
$129$	−14.6329	−1.28836
$130$	−24.2316	−2.12525
$131$	5.80520	0.507202	0.253601	−	0.967309i	$-0.418385\pi$
$131$	5.80520	0.507202	0.253601	+	0.967309i	$0.418385\pi$
$132$	36.3532	3.16414
$133$	0	0
$134$	−24.2769	−2.09721
$135$	−73.9621	−6.36565
$136$	2.35648	0.202067
$137$	19.9322	1.70293	0.851463	−	0.524414i	$-0.175716\pi$
$137$	19.9322	1.70293	0.851463	+	0.524414i	$0.175716\pi$
$138$	−35.9282	−3.05841
$139$	11.7412	0.995875	0.497937	−	0.867213i	$-0.334091\pi$
$139$	11.7412	0.995875	0.497937	+	0.867213i	$0.334091\pi$
$140$	0	0
$141$	12.9278	1.08872
$142$	9.28295	0.779008
$143$	−14.0396	−1.17405
$144$	−29.5426	−2.46188
$145$	14.7047	1.22116
$146$	17.9770	1.48779
$147$	0	0
$148$	8.84924	0.727403
$149$	18.6037	1.52407	0.762036	−	0.647535i	$-0.224200\pi$
$149$	18.6037	1.52407	0.762036	+	0.647535i	$0.224200\pi$
$150$	−82.6048	−6.74465
$151$	−17.4023	−1.41618	−0.708088	−	0.706125i	$-0.750442\pi$
$151$	−17.4023	−1.41618	−0.708088	+	0.706125i	$0.750442\pi$
$152$	−0.670576	−0.0543909
$153$	−46.4130	−3.75227
$154$	0	0
$155$	8.74886	0.702725
$156$	−21.3075	−1.70596
$157$	4.94866	0.394946	0.197473	−	0.980308i	$-0.436726\pi$
$157$	4.94866	0.394946	0.197473	+	0.980308i	$0.436726\pi$
$158$	−25.5822	−2.03521
$159$	7.20337	0.571264
$160$	−33.4290	−2.64280
$161$	0	0
$162$	−72.7144	−5.71299
$163$	14.9866	1.17384	0.586919	−	0.809646i	$-0.300341\pi$
$163$	14.9866	1.17384	0.586919	+	0.809646i	$0.300341\pi$
$164$	11.7748	0.919459
$165$	−67.8633	−5.28315
$166$	16.1935	1.25686
$167$	−1.48923	−0.115240	−0.0576200	−	0.998339i	$-0.518351\pi$
$167$	−1.48923	−0.115240	−0.0576200	+	0.998339i	$0.518351\pi$
$168$	0	0
$169$	−4.77106	−0.367005
$170$	−47.0428	−3.60802
$171$	13.2076	1.01001
$172$	−9.58972	−0.731209
$173$	21.8498	1.66121	0.830603	−	0.556864i	$-0.187996\pi$
$173$	21.8498	1.66121	0.830603	+	0.556864i	$0.187996\pi$
$174$	24.6514	1.86882
$175$	0	0
$176$	−17.3490	−1.30773
$177$	6.56721	0.493621
$178$	−1.01014	−0.0757135
$179$	14.8249	1.10806	0.554031	−	0.832496i	$-0.313089\pi$
$179$	14.8249	1.10806	0.554031	+	0.832496i	$0.313089\pi$
$180$	−75.7326	−5.64478
$181$	10.6273	0.789919	0.394960	−	0.918699i	$-0.370759\pi$
$181$	10.6273	0.789919	0.394960	+	0.918699i	$0.370759\pi$
$182$	0	0
$183$	−43.4701	−3.21340
$184$	−2.20177	−0.162317
$185$	−16.5195	−1.21454
$186$	14.6668	1.07542
$187$	−27.2563	−1.99318
$188$	8.47224	0.617902
$189$	0	0
$190$	13.3868	0.971181
$191$	−1.42657	−0.103223	−0.0516114	−	0.998667i	$-0.516436\pi$
$191$	−1.42657	−0.103223	−0.0516114	+	0.998667i	$0.516436\pi$
$192$	−32.1734	−2.32191
$193$	−11.8975	−0.856400	−0.428200	−	0.903684i	$-0.640852\pi$
$193$	−11.8975	−0.856400	−0.428200	+	0.903684i	$0.640852\pi$
$194$	3.69451	0.265250
$195$	39.7762	2.84843
$196$	0	0
$197$	−2.27786	−0.162291	−0.0811453	−	0.996702i	$-0.525858\pi$
$197$	−2.27786	−0.162291	−0.0811453	+	0.996702i	$0.525858\pi$
$198$	−83.6547	−5.94508
$199$	19.6158	1.39053	0.695265	−	0.718753i	$-0.255287\pi$
$199$	19.6158	1.39053	0.695265	+	0.718753i	$0.255287\pi$
$200$	−5.06223	−0.357954
$201$	39.8507	2.81085
$202$	−30.3052	−2.13227
$203$	0	0
$204$	−41.3660	−2.89620
$205$	−21.9809	−1.53521
$206$	12.9238	0.900441
$207$	43.3659	3.01414
$208$	10.1687	0.705071
$209$	7.75622	0.536509
$210$	0	0
$211$	−11.5484	−0.795023	−0.397511	−	0.917597i	$-0.630126\pi$
$211$	−11.5484	−0.795023	−0.397511	+	0.917597i	$0.630126\pi$
$212$	4.72074	0.324222
$213$	−15.2380	−1.04409
$214$	23.6441	1.61628
$215$	17.9018	1.22089
$216$	−7.59856	−0.517016
$217$	0	0
$218$	24.9311	1.68855
$219$	−29.5093	−1.99405
$220$	−44.4743	−2.99846
$221$	15.9755	1.07463
$222$	−27.6938	−1.85868
$223$	−6.93950	−0.464703	−0.232352	−	0.972632i	$-0.574642\pi$
$223$	−6.93950	−0.464703	−0.232352	+	0.972632i	$0.574642\pi$
$224$	0	0
$225$	99.7053	6.64702
$226$	−33.7086	−2.24226
$227$	9.02761	0.599184	0.299592	−	0.954067i	$-0.403150\pi$
$227$	9.02761	0.599184	0.299592	+	0.954067i	$0.403150\pi$
$228$	11.7714	0.779579
$229$	−24.8645	−1.64309	−0.821547	−	0.570141i	$-0.806889\pi$
$229$	−24.8645	−1.64309	−0.821547	+	0.570141i	$0.806889\pi$
$230$	43.9543	2.89826
$231$	0	0
$232$	1.51070	0.0991823
$233$	−11.6893	−0.765794	−0.382897	−	0.923791i	$-0.625074\pi$
$233$	−11.6893	−0.765794	−0.382897	+	0.923791i	$0.625074\pi$
$234$	49.0320	3.20532
$235$	−15.8158	−1.03171
$236$	4.30383	0.280156
$237$	41.9933	2.72776
$238$	0	0
$239$	−17.9805	−1.16306	−0.581531	−	0.813524i	$-0.697546\pi$
$239$	−17.9805	−1.16306	−0.581531	+	0.813524i	$0.697546\pi$
$240$	49.1523	3.17277
$241$	−5.37930	−0.346511	−0.173256	−	0.984877i	$-0.555429\pi$
$241$	−5.37930	−0.346511	−0.173256	+	0.984877i	$0.555429\pi$
$242$	−26.5664	−1.70775
$243$	65.4879	4.20105
$244$	−28.4882	−1.82377
$245$	0	0
$246$	−36.8494	−2.34943
$247$	−4.54610	−0.289261
$248$	0.898821	0.0570752
$249$	−26.5817	−1.68455
$250$	58.8224	3.72026
$251$	11.6135	0.733039	0.366520	−	0.930410i	$-0.380549\pi$
$251$	11.6135	0.733039	0.366520	+	0.930410i	$0.380549\pi$
$252$	0	0
$253$	25.4668	1.60108
$254$	−2.05093	−0.128687
$255$	77.2209	4.83576
$256$	12.2076	0.762972
$257$	13.1919	0.822887	0.411444	−	0.911435i	$-0.365025\pi$
$257$	13.1919	0.822887	0.411444	+	0.911435i	$0.365025\pi$
$258$	30.0111	1.86841
$259$	0	0
$260$	26.0674	1.61663
$261$	−29.7546	−1.84176
$262$	−11.9061	−0.735558
$263$	21.8413	1.34679	0.673396	−	0.739282i	$-0.264835\pi$
$263$	21.8413	1.34679	0.673396	+	0.739282i	$0.264835\pi$
$264$	−6.97199	−0.429096
$265$	−8.81255	−0.541351
$266$	0	0
$267$	1.65816	0.101477
$268$	26.1162	1.59530
$269$	−4.39470	−0.267950	−0.133975	−	0.990985i	$-0.542774\pi$
$269$	−4.39470	−0.267950	−0.133975	+	0.990985i	$0.542774\pi$
$270$	151.691	9.23163
$271$	−4.66803	−0.283563	−0.141781	−	0.989898i	$-0.545283\pi$
$271$	−4.66803	−0.283563	−0.141781	+	0.989898i	$0.545283\pi$
$272$	19.7413	1.19699
$273$	0	0
$274$	−40.8796	−2.46963
$275$	58.5524	3.53084
$276$	38.6502	2.32647
$277$	2.38523	0.143314	0.0716572	−	0.997429i	$-0.477171\pi$
$277$	2.38523	0.143314	0.0716572	+	0.997429i	$0.477171\pi$
$278$	−24.0804	−1.44424
$279$	−17.7031	−1.05986
$280$	0	0
$281$	−22.7315	−1.35605	−0.678025	−	0.735039i	$-0.737164\pi$
$281$	−22.7315	−1.35605	−0.678025	+	0.735039i	$0.737164\pi$
$282$	−26.5140	−1.57888
$283$	31.7184	1.88547	0.942733	−	0.333548i	$-0.108246\pi$
$283$	31.7184	1.88547	0.942733	+	0.333548i	$0.108246\pi$
$284$	−9.98625	−0.592575
$285$	−21.9745	−1.30166
$286$	28.7943	1.70264
$287$	0	0
$288$	67.6427	3.98589
$289$	14.0146	0.824390
$290$	−30.1583	−1.77096
$291$	−6.06455	−0.355510
$292$	−19.3390	−1.13173
$293$	2.45935	0.143677	0.0718384	−	0.997416i	$-0.477113\pi$
$293$	2.45935	0.143677	0.0718384	+	0.997416i	$0.477113\pi$
$294$	0	0
$295$	−8.03428	−0.467774
$296$	−1.69715	−0.0986446
$297$	87.8888	5.09983
$298$	−38.1548	−2.21025
$299$	−14.9267	−0.863232
$300$	88.8632	5.13052
$301$	0	0
$302$	35.6908	2.05378
$303$	49.7462	2.85784
$304$	−5.61771	−0.322198
$305$	53.1810	3.04513
$306$	95.1899	5.44164
$307$	−6.17340	−0.352335	−0.176167	−	0.984360i	$-0.556370\pi$
$307$	−6.17340	−0.352335	−0.176167	+	0.984360i	$0.556370\pi$
$308$	0	0
$309$	−21.2144	−1.20684
$310$	−17.9433	−1.01911
$311$	9.43715	0.535131	0.267566	−	0.963540i	$-0.413781\pi$
$311$	9.43715	0.535131	0.267566	+	0.963540i	$0.413781\pi$
$312$	4.08644	0.231349
$313$	1.60837	0.0909104	0.0454552	−	0.998966i	$-0.485526\pi$
$313$	1.60837	0.0909104	0.0454552	+	0.998966i	$0.485526\pi$
$314$	−10.1494	−0.572762
$315$	0	0
$316$	27.5204	1.54814
$317$	−8.77562	−0.492888	−0.246444	−	0.969157i	$-0.579262\pi$
$317$	−8.77562	−0.492888	−0.246444	+	0.969157i	$0.579262\pi$
$318$	−14.7736	−0.828462
$319$	−17.4735	−0.978330
$320$	39.3607	2.20033
$321$	−38.8118	−2.16627
$322$	0	0
$323$	−8.82573	−0.491077
$324$	78.2235	4.34575
$325$	−34.3189	−1.90367
$326$	−30.7364	−1.70233
$327$	−40.9245	−2.26313
$328$	−2.25823	−0.124690
$329$	0	0
$330$	139.183	7.66176
$331$	16.4521	0.904289	0.452145	−	0.891945i	$-0.350659\pi$
$331$	16.4521	0.904289	0.452145	+	0.891945i	$0.350659\pi$
$332$	−17.4204	−0.956066
$333$	33.4268	1.83178
$334$	3.05430	0.167124
$335$	−48.7531	−2.66366
$336$	0	0
$337$	26.1327	1.42354	0.711770	−	0.702413i	$-0.247894\pi$
$337$	26.1327	1.42354	0.711770	+	0.702413i	$0.247894\pi$
$338$	9.78511	0.532240
$339$	55.3327	3.00526
$340$	50.6069	2.74454
$341$	−10.3962	−0.562987
$342$	−27.0879	−1.46474
$343$	0	0
$344$	1.83916	0.0991608
$345$	−72.1511	−3.88448
$346$	−44.8123	−2.40913
$347$	2.69696	0.144781	0.0723903	−	0.997376i	$-0.476937\pi$
$347$	2.69696	0.144781	0.0723903	+	0.997376i	$0.476937\pi$
$348$	−26.5190	−1.42157
$349$	13.7988	0.738634	0.369317	−	0.929303i	$-0.379592\pi$
$349$	13.7988	0.738634	0.369317	+	0.929303i	$0.379592\pi$
$350$	0	0
$351$	−51.5137	−2.74960
$352$	39.7235	2.11727
$353$	−2.35503	−0.125346	−0.0626728	−	0.998034i	$-0.519962\pi$
$353$	−2.35503	−0.125346	−0.0626728	+	0.998034i	$0.519962\pi$
$354$	−13.4689	−0.715863
$355$	18.6421	0.989418
$356$	1.08668	0.0575937
$357$	0	0
$358$	−30.4048	−1.60694
$359$	7.45264	0.393335	0.196668	−	0.980470i	$-0.436988\pi$
$359$	7.45264	0.393335	0.196668	+	0.980470i	$0.436988\pi$
$360$	14.5243	0.765499
$361$	−16.4885	−0.867815
$362$	−21.7958	−1.14556
$363$	43.6088	2.28887
$364$	0	0
$365$	36.1015	1.88964
$366$	89.1541	4.66016
$367$	23.1277	1.20725	0.603627	−	0.797267i	$-0.293721\pi$
$367$	23.1277	1.20725	0.603627	+	0.797267i	$0.293721\pi$
$368$	−18.4452	−0.961523
$369$	44.4778	2.31542
$370$	33.8804	1.76136
$371$	0	0
$372$	−15.7780	−0.818053
$373$	−0.199225	−0.0103155	−0.00515775	−	0.999987i	$-0.501642\pi$
$373$	−0.199225	−0.0103155	−0.00515775	+	0.999987i	$0.501642\pi$
$374$	55.9007	2.89056
$375$	−96.5572	−4.98619
$376$	−1.62484	−0.0837949
$377$	10.2416	0.527471
$378$	0	0
$379$	−22.0311	−1.13166	−0.565830	−	0.824522i	$-0.691444\pi$
$379$	−22.0311	−1.13166	−0.565830	+	0.824522i	$0.691444\pi$
$380$	−14.4010	−0.738757
$381$	3.36661	0.172477
$382$	2.92579	0.149697
$383$	−30.0986	−1.53797	−0.768983	−	0.639269i	$-0.779237\pi$
$383$	−30.0986	−1.53797	−0.768983	+	0.639269i	$0.779237\pi$
$384$	11.3357	0.578470
$385$	0	0
$386$	24.4009	1.24197
$387$	−36.2239	−1.84136
$388$	−3.97442	−0.201770
$389$	31.1697	1.58037	0.790184	−	0.612869i	$-0.209985\pi$
$389$	31.1697	1.58037	0.790184	+	0.612869i	$0.209985\pi$
$390$	−81.5783	−4.13088
$391$	−28.9784	−1.46550
$392$	0	0
$393$	19.5438	0.985856
$394$	4.67173	0.235358
$395$	−51.3743	−2.58492
$396$	89.9927	4.52230
$397$	22.8555	1.14709	0.573543	−	0.819176i	$-0.305569\pi$
$397$	22.8555	1.14709	0.573543	+	0.819176i	$0.305569\pi$
$398$	−40.2307	−2.01658
$399$	0	0
$400$	−42.4086	−2.12043
$401$	10.3378	0.516244	0.258122	−	0.966112i	$-0.416896\pi$
$401$	10.3378	0.516244	0.258122	+	0.966112i	$0.416896\pi$
$402$	−81.7310	−4.07637
$403$	6.09347	0.303537
$404$	32.6013	1.62197
$405$	−146.025	−7.25607
$406$	0	0
$407$	19.6301	0.973026
$408$	7.93335	0.392759
$409$	−12.9534	−0.640505	−0.320253	−	0.947332i	$-0.603768\pi$
$409$	−12.9534	−0.640505	−0.320253	+	0.947332i	$0.603768\pi$
$410$	45.0813	2.22641
$411$	67.1041	3.31000
$412$	−13.9029	−0.684946
$413$	0	0
$414$	−88.9403	−4.37118
$415$	32.5198	1.59634
$416$	−23.2829	−1.14154
$417$	39.5280	1.93569
$418$	−15.9075	−0.778060
$419$	−32.7486	−1.59987	−0.799937	−	0.600085i	$-0.795134\pi$
$419$	−32.7486	−1.59987	−0.799937	+	0.600085i	$0.795134\pi$
$420$	0	0
$421$	−29.9675	−1.46053	−0.730264	−	0.683165i	$-0.760603\pi$
$421$	−29.9675	−1.46053	−0.730264	+	0.683165i	$0.760603\pi$
$422$	23.6849	1.15296
$423$	32.0028	1.55603
$424$	−0.905364	−0.0439684
$425$	−66.6262	−3.23184
$426$	31.2521	1.51417
$427$	0	0
$428$	−25.4354	−1.22947
$429$	−47.2659	−2.28202
$430$	−36.7154	−1.77057
$431$	35.9923	1.73369	0.866843	−	0.498581i	$-0.166145\pi$
$431$	35.9923	1.73369	0.866843	+	0.498581i	$0.166145\pi$
$432$	−63.6565	−3.06268
$433$	6.31077	0.303276	0.151638	−	0.988436i	$-0.451545\pi$
$433$	6.31077	0.303276	0.151638	+	0.988436i	$0.451545\pi$
$434$	0	0
$435$	49.5050	2.37358
$436$	−26.8200	−1.28444
$437$	8.24629	0.394473
$438$	60.5215	2.89183
$439$	7.65105	0.365165	0.182582	−	0.983191i	$-0.441554\pi$
$439$	7.65105	0.365165	0.182582	+	0.983191i	$0.441554\pi$
$440$	8.52948	0.406627
$441$	0	0
$442$	−32.7647	−1.55846
$443$	−7.36887	−0.350106	−0.175053	−	0.984559i	$-0.556010\pi$
$443$	−7.36887	−0.350106	−0.175053	+	0.984559i	$0.556010\pi$
$444$	29.7919	1.41386
$445$	−2.02858	−0.0961638
$446$	14.2324	0.673925
$447$	62.6313	2.96236
$448$	0	0
$449$	26.4113	1.24643	0.623213	−	0.782052i	$-0.285827\pi$
$449$	26.4113	1.24643	0.623213	+	0.782052i	$0.285827\pi$
$450$	−204.489	−9.63968
$451$	26.1198	1.22993
$452$	36.2624	1.70564
$453$	−58.5866	−2.75264
$454$	−18.5150	−0.868952
$455$	0	0
$456$	−2.25757	−0.105720
$457$	30.9520	1.44787	0.723937	−	0.689866i	$-0.242330\pi$
$457$	30.9520	1.44787	0.723937	+	0.689866i	$0.242330\pi$
$458$	50.9954	2.38286
$459$	−100.008	−4.66797
$460$	−47.2844	−2.20464
$461$	38.2261	1.78037	0.890183	−	0.455603i	$-0.150576\pi$
$461$	38.2261	1.78037	0.890183	+	0.455603i	$0.150576\pi$
$462$	0	0
$463$	−23.0513	−1.07128	−0.535641	−	0.844446i	$-0.679930\pi$
$463$	−23.0513	−1.07128	−0.535641	+	0.844446i	$0.679930\pi$
$464$	12.6558	0.587531
$465$	29.4540	1.36590
$466$	23.9740	1.11058
$467$	32.8437	1.51982	0.759912	−	0.650026i	$-0.225242\pi$
$467$	32.8437	1.51982	0.759912	+	0.650026i	$0.225242\pi$
$468$	−52.7468	−2.43822
$469$	0	0
$470$	32.4370	1.49621
$471$	16.6602	0.767662
$472$	−0.825408	−0.0379925
$473$	−21.2726	−0.978117
$474$	−86.1253	−3.95587
$475$	18.9596	0.869925
$476$	0	0
$477$	17.8320	0.816470
$478$	36.8768	1.68670
$479$	16.4098	0.749784	0.374892	−	0.927068i	$-0.377680\pi$
$479$	16.4098	0.749784	0.374892	+	0.927068i	$0.377680\pi$
$480$	−112.542	−5.13683
$481$	−11.5056	−0.524612
$482$	11.0326	0.502520
$483$	0	0
$484$	28.5791	1.29905
$485$	7.41933	0.336895
$486$	−134.311	−6.09248
$487$	12.2478	0.555000	0.277500	−	0.960726i	$-0.410494\pi$
$487$	12.2478	0.555000	0.277500	+	0.960726i	$0.410494\pi$
$488$	5.46359	0.247325
$489$	50.4539	2.28160
$490$	0	0
$491$	−3.67455	−0.165830	−0.0829151	−	0.996557i	$-0.526423\pi$
$491$	−3.67455	−0.165830	−0.0829151	+	0.996557i	$0.526423\pi$
$492$	39.6412	1.78716
$493$	19.8830	0.895483
$494$	9.32373	0.419495
$495$	−167.996	−7.55085
$496$	7.52983	0.338099
$497$	0	0
$498$	54.5172	2.44297
$499$	−33.8264	−1.51428	−0.757140	−	0.653253i	$-0.773404\pi$
$499$	−33.8264	−1.51428	−0.757140	+	0.653253i	$0.773404\pi$
$500$	−63.2790	−2.82992
$501$	−5.01365	−0.223993
$502$	−23.8185	−1.06307
$503$	14.7656	0.658364	0.329182	−	0.944266i	$-0.393227\pi$
$503$	14.7656	0.658364	0.329182	+	0.944266i	$0.393227\pi$
$504$	0	0
$505$	−60.8591	−2.70820
$506$	−52.2306	−2.32193
$507$	−16.0623	−0.713352
$508$	2.20631	0.0978894
$509$	−22.8105	−1.01106	−0.505529	−	0.862810i	$-0.668703\pi$
$509$	−22.8105	−1.01106	−0.505529	+	0.862810i	$0.668703\pi$
$510$	−158.375	−7.01295
$511$	0	0
$512$	−31.7710	−1.40409
$513$	28.4589	1.25649
$514$	−27.0556	−1.19337
$515$	25.9535	1.14365
$516$	−32.2848	−1.42126
$517$	18.7938	0.826549
$518$	0	0
$519$	73.5596	3.22891
$520$	−4.99933	−0.219235
$521$	34.5325	1.51289	0.756447	−	0.654055i	$-0.226933\pi$
$521$	34.5325	1.51289	0.756447	+	0.654055i	$0.226933\pi$
$522$	61.0246	2.67098
$523$	−4.62166	−0.202091	−0.101045	−	0.994882i	$-0.532219\pi$
$523$	−4.62166	−0.202091	−0.101045	+	0.994882i	$0.532219\pi$
$524$	12.8081	0.559524
$525$	0	0
$526$	−44.7950	−1.95315
$527$	11.8298	0.515312
$528$	−58.4074	−2.54186
$529$	4.07588	0.177212
$530$	18.0739	0.785081
$531$	16.2572	0.705500
$532$	0	0
$533$	−15.3094	−0.663125
$534$	−3.40076	−0.147165
$535$	47.4821	2.05283
$536$	−5.00868	−0.216342
$537$	49.9095	2.15376
$538$	9.01322	0.388588
$539$	0	0
$540$	−163.184	−7.02231
$541$	9.91005	0.426066	0.213033	−	0.977045i	$-0.431666\pi$
$541$	9.91005	0.426066	0.213033	+	0.977045i	$0.431666\pi$
$542$	9.57381	0.411230
$543$	35.7779	1.53538
$544$	−45.2010	−1.93798
$545$	50.0668	2.14462
$546$	0	0
$547$	0.114979	0.00491615	0.00245807	−	0.999997i	$-0.499218\pi$
$547$	0.114979	0.00491615	0.00245807	+	0.999997i	$0.499218\pi$
$548$	43.9768	1.87860
$549$	−107.610	−4.59270
$550$	−120.087	−5.12052
$551$	−5.65802	−0.241040
$552$	−7.41250	−0.315497
$553$	0	0
$554$	−4.89193	−0.207838
$555$	−55.6148	−2.36072
$556$	25.9048	1.09861
$557$	2.37326	0.100558	0.0502791	−	0.998735i	$-0.483989\pi$
$557$	2.37326	0.100558	0.0502791	+	0.998735i	$0.483989\pi$
$558$	36.3078	1.53703
$559$	12.4684	0.527357
$560$	0	0
$561$	−91.7612	−3.87416
$562$	46.6208	1.96658
$563$	−2.63294	−0.110965	−0.0554825	−	0.998460i	$-0.517670\pi$
$563$	−2.63294	−0.110965	−0.0554825	+	0.998460i	$0.517670\pi$
$564$	28.5227	1.20102
$565$	−67.6937	−2.84790
$566$	−65.0523	−2.73435
$567$	0	0
$568$	1.91521	0.0803603
$569$	−21.7705	−0.912666	−0.456333	−	0.889809i	$-0.650837\pi$
$569$	−21.7705	−0.912666	−0.456333	+	0.889809i	$0.650837\pi$
$570$	45.0681	1.88770
$571$	37.3131	1.56151	0.780753	−	0.624840i	$-0.214836\pi$
$571$	37.3131	1.56151	0.780753	+	0.624840i	$0.214836\pi$
$572$	−30.9758	−1.29516
$573$	−4.80270	−0.200636
$574$	0	0
$575$	62.2519	2.59608
$576$	−79.6453	−3.31855
$577$	8.99963	0.374659	0.187330	−	0.982297i	$-0.440017\pi$
$577$	8.99963	0.374659	0.187330	+	0.982297i	$0.440017\pi$
$578$	−28.7430	−1.19555
$579$	−40.0542	−1.66460
$580$	32.4432	1.34713
$581$	0	0
$582$	12.4380	0.515571
$583$	10.4719	0.433702
$584$	3.70891	0.153476
$585$	98.4663	4.07108
$586$	−5.04395	−0.208364
$587$	31.7546	1.31065	0.655326	−	0.755346i	$-0.272531\pi$
$587$	31.7546	1.31065	0.655326	+	0.755346i	$0.272531\pi$
$588$	0	0
$589$	−3.36635	−0.138708
$590$	16.4777	0.678378
$591$	−7.66866	−0.315446
$592$	−14.2177	−0.584346
$593$	39.5540	1.62429	0.812144	−	0.583456i	$-0.198300\pi$
$593$	39.5540	1.62429	0.812144	+	0.583456i	$0.198300\pi$
$594$	−180.254	−7.39591
$595$	0	0
$596$	41.0455	1.68129
$597$	66.0389	2.70279
$598$	30.6136	1.25188
$599$	−30.0971	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$599$	−30.0971	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$600$	−17.0426	−0.695760
$601$	9.05936	0.369539	0.184769	−	0.982782i	$-0.440846\pi$
$601$	9.05936	0.369539	0.184769	+	0.982782i	$0.440846\pi$
$602$	0	0
$603$	98.6506	4.01736
$604$	−38.3948	−1.56226
$605$	−53.3508	−2.16902
$606$	−102.026	−4.14452
$607$	−13.5579	−0.550300	−0.275150	−	0.961401i	$-0.588727\pi$
$607$	−13.5579	−0.550300	−0.275150	+	0.961401i	$0.588727\pi$
$608$	12.8627	0.521651
$609$	0	0
$610$	−109.070	−4.41614
$611$	−11.0155	−0.445638
$612$	−102.402	−4.13935
$613$	4.60692	0.186072	0.0930358	−	0.995663i	$-0.470343\pi$
$613$	4.60692	0.186072	0.0930358	+	0.995663i	$0.470343\pi$
$614$	12.6612	0.510965
$615$	−74.0012	−2.98401
$616$	0	0
$617$	27.2211	1.09588	0.547940	−	0.836518i	$-0.315412\pi$
$617$	27.2211	1.09588	0.547940	+	0.836518i	$0.315412\pi$
$618$	43.5092	1.75020
$619$	−28.4442	−1.14327	−0.571634	−	0.820509i	$-0.693690\pi$
$619$	−28.4442	−1.14327	−0.571634	+	0.820509i	$0.693690\pi$
$620$	19.3027	0.775216
$621$	93.4419	3.74970
$622$	−19.3549	−0.776062
$623$	0	0
$624$	34.2340	1.37045
$625$	58.3095	2.33238
$626$	−3.29865	−0.131841
$627$	26.1122	1.04282
$628$	10.9183	0.435688
$629$	−22.3368	−0.890628
$630$	0	0
$631$	26.8998	1.07086	0.535431	−	0.844579i	$-0.320149\pi$
$631$	26.8998	1.07086	0.535431	+	0.844579i	$0.320149\pi$
$632$	−5.27798	−0.209947
$633$	−38.8789	−1.54530
$634$	17.9982	0.714799
$635$	−4.11869	−0.163445
$636$	15.8929	0.630194
$637$	0	0
$638$	35.8370	1.41880
$639$	−37.7218	−1.49225
$640$	−13.8680	−0.548179
$641$	−15.4637	−0.610778	−0.305389	−	0.952228i	$-0.598787\pi$
$641$	−15.4637	−0.610778	−0.305389	+	0.952228i	$0.598787\pi$
$642$	79.6004	3.14158
$643$	14.9711	0.590402	0.295201	−	0.955435i	$-0.404613\pi$
$643$	14.9711	0.590402	0.295201	+	0.955435i	$0.404613\pi$
$644$	0	0
$645$	60.2685	2.37307
$646$	18.1009	0.712172
$647$	25.3349	0.996019	0.498010	−	0.867172i	$-0.334064\pi$
$647$	25.3349	0.996019	0.498010	+	0.867172i	$0.334064\pi$
$648$	−15.0020	−0.589336
$649$	9.54709	0.374756
$650$	70.3857	2.76075
$651$	0	0
$652$	33.0651	1.29493
$653$	−9.66195	−0.378102	−0.189051	−	0.981967i	$-0.560541\pi$
$653$	−9.66195	−0.378102	−0.189051	+	0.981967i	$0.560541\pi$
$654$	83.9333	3.28205
$655$	−23.9098	−0.934233
$656$	−18.9182	−0.738631
$657$	−73.0504	−2.84997
$658$	0	0
$659$	16.8558	0.656607	0.328304	−	0.944572i	$-0.393523\pi$
$659$	16.8558	0.656607	0.328304	+	0.944572i	$0.393523\pi$
$660$	−149.728	−5.82814
$661$	25.5996	0.995709	0.497855	−	0.867261i	$-0.334121\pi$
$661$	25.5996	0.995709	0.497855	+	0.867261i	$0.334121\pi$
$662$	−33.7421	−1.31142
$663$	53.7834	2.08877
$664$	3.34095	0.129654
$665$	0	0
$666$	−68.5561	−2.65649
$667$	−18.5776	−0.719326
$668$	−3.28571	−0.127128
$669$	−23.3626	−0.903249
$670$	99.9891	3.86292
$671$	−63.1947	−2.43960
$672$	0	0
$673$	−21.7472	−0.838292	−0.419146	−	0.907919i	$-0.637670\pi$
$673$	−21.7472	−0.838292	−0.419146	+	0.907919i	$0.637670\pi$
$674$	−53.5964	−2.06446
$675$	214.839	8.26914
$676$	−10.5265	−0.404864
$677$	−14.6170	−0.561779	−0.280889	−	0.959740i	$-0.590629\pi$
$677$	−14.6170	−0.561779	−0.280889	+	0.959740i	$0.590629\pi$
$678$	−113.484	−4.35831
$679$	0	0
$680$	−9.70561	−0.372193
$681$	30.3924	1.16464
$682$	21.3219	0.816459
$683$	−28.0331	−1.07266	−0.536328	−	0.844010i	$-0.680189\pi$
$683$	−28.0331	−1.07266	−0.536328	+	0.844010i	$0.680189\pi$
$684$	29.1401	1.11420
$685$	−82.0947	−3.13668
$686$	0	0
$687$	−83.7091	−3.19370
$688$	15.4075	0.587404
$689$	−6.13783	−0.233833
$690$	147.977	5.63338
$691$	−8.96411	−0.341011	−0.170505	−	0.985357i	$-0.554540\pi$
$691$	−8.96411	−0.341011	−0.170505	+	0.985357i	$0.554540\pi$
$692$	48.2075	1.83257
$693$	0	0
$694$	−5.53129	−0.209965
$695$	−48.3583	−1.83433
$696$	5.08594	0.192782
$697$	−29.7215	−1.12578
$698$	−28.3004	−1.07119
$699$	−39.3535	−1.48848
$700$	0	0
$701$	−38.4950	−1.45394	−0.726968	−	0.686672i	$-0.759071\pi$
$701$	−38.4950	−1.45394	−0.726968	+	0.686672i	$0.759071\pi$
$702$	105.651	3.98754
$703$	6.35632	0.239733
$704$	−46.7721	−1.76279
$705$	−53.2455	−2.00534
$706$	4.83000	0.181780
$707$	0	0
$708$	14.4893	0.544542
$709$	−52.3540	−1.96620	−0.983098	−	0.183078i	$-0.941394\pi$
$709$	−52.3540	−1.96620	−0.983098	+	0.183078i	$0.941394\pi$
$710$	−38.2336	−1.43488
$711$	103.955	3.89860
$712$	−0.208407	−0.00781040
$713$	−11.0531	−0.413942
$714$	0	0
$715$	57.8248	2.16252
$716$	32.7083	1.22237
$717$	−60.5334	−2.26066
$718$	−15.2848	−0.570425
$719$	44.2758	1.65121	0.825605	−	0.564249i	$-0.190834\pi$
$719$	44.2758	1.65121	0.825605	+	0.564249i	$0.190834\pi$
$720$	121.677	4.53463
$721$	0	0
$722$	33.8167	1.25853
$723$	−18.1100	−0.673518
$724$	23.4471	0.871405
$725$	−42.7129	−1.58632
$726$	−89.4387	−3.31938
$727$	9.80050	0.363480	0.181740	−	0.983347i	$-0.441827\pi$
$727$	9.80050	0.363480	0.181740	+	0.983347i	$0.441827\pi$
$728$	0	0
$729$	114.109	4.22627
$730$	−74.0416	−2.74040
$731$	24.2059	0.895289
$732$	−95.9086	−3.54489
$733$	31.9796	1.18119	0.590597	−	0.806967i	$-0.298892\pi$
$733$	31.9796	1.18119	0.590597	+	0.806967i	$0.298892\pi$
$734$	−47.4333	−1.75079
$735$	0	0
$736$	42.2334	1.55674
$737$	57.9330	2.13399
$738$	−91.2209	−3.35789
$739$	−20.7323	−0.762650	−0.381325	−	0.924441i	$-0.624532\pi$
$739$	−20.7323	−0.762650	−0.381325	+	0.924441i	$0.624532\pi$
$740$	−36.4473	−1.33983
$741$	−15.3049	−0.562241
$742$	0	0
$743$	15.1605	0.556186	0.278093	−	0.960554i	$-0.410298\pi$
$743$	15.1605	0.556186	0.278093	+	0.960554i	$0.410298\pi$
$744$	3.02598	0.110938
$745$	−76.6227	−2.80724
$746$	0.408597	0.0149598
$747$	−65.8031	−2.40761
$748$	−60.1359	−2.19879
$749$	0	0
$750$	198.032	7.23111
$751$	25.5705	0.933081	0.466540	−	0.884500i	$-0.345500\pi$
$751$	25.5705	0.933081	0.466540	+	0.884500i	$0.345500\pi$
$752$	−13.6120	−0.496380
$753$	39.0982	1.42482
$754$	−21.0049	−0.764953
$755$	71.6744	2.60850
$756$	0	0
$757$	10.3797	0.377257	0.188629	−	0.982049i	$-0.439596\pi$
$757$	10.3797	0.377257	0.188629	+	0.982049i	$0.439596\pi$
$758$	45.1842	1.64116
$759$	85.7368	3.11205
$760$	2.76189	0.100184
$761$	−29.1038	−1.05501	−0.527507	−	0.849551i	$-0.676873\pi$
$761$	−29.1038	−1.05501	−0.527507	+	0.849551i	$0.676873\pi$
$762$	−6.90468	−0.250130
$763$	0	0
$764$	−3.14746	−0.113871
$765$	191.161	6.91143
$766$	61.7301	2.23040
$767$	−5.59577	−0.202052
$768$	41.0981	1.48300
$769$	17.1929	0.619994	0.309997	−	0.950738i	$-0.399672\pi$
$769$	17.1929	0.619994	0.309997	+	0.950738i	$0.399672\pi$
$770$	0	0
$771$	44.4119	1.59946
$772$	−26.2496	−0.944744
$773$	−4.46087	−0.160446	−0.0802231	−	0.996777i	$-0.525563\pi$
$773$	−4.46087	−0.160446	−0.0802231	+	0.996777i	$0.525563\pi$
$774$	74.2927	2.67039
$775$	−25.4129	−0.912858
$776$	0.762231	0.0273625
$777$	0	0
$778$	−63.9270	−2.29189
$779$	8.45773	0.303030
$780$	87.7589	3.14227
$781$	−22.1523	−0.792671
$782$	59.4327	2.12531
$783$	−64.1133	−2.29122
$784$	0	0
$785$	−20.3820	−0.727465
$786$	−40.0830	−1.42971
$787$	−20.3039	−0.723756	−0.361878	−	0.932225i	$-0.617864\pi$
$787$	−20.3039	−0.723756	−0.361878	+	0.932225i	$0.617864\pi$
$788$	−5.02567	−0.179032
$789$	73.5311	2.61778
$790$	105.365	3.74872
$791$	0	0
$792$	−17.2592	−0.613278
$793$	37.0399	1.31532
$794$	−46.8750	−1.66353
$795$	−29.6684	−1.05223
$796$	43.2787	1.53397
$797$	−19.0172	−0.673623	−0.336811	−	0.941572i	$-0.609348\pi$
$797$	−19.0172	−0.673623	−0.336811	+	0.941572i	$0.609348\pi$
$798$	0	0
$799$	−21.3853	−0.756556
$800$	97.1015	3.43306
$801$	4.10477	0.145035
$802$	−21.2021	−0.748671
$803$	−42.8992	−1.51388
$804$	87.9231	3.10081
$805$	0	0
$806$	−12.4973	−0.440198
$807$	−14.7952	−0.520817
$808$	−6.25241	−0.219959
$809$	−8.26981	−0.290751	−0.145376	−	0.989377i	$-0.546439\pi$
$809$	−8.26981	−0.290751	−0.145376	+	0.989377i	$0.546439\pi$
$810$	299.488	10.5229
$811$	5.29940	0.186087	0.0930436	−	0.995662i	$-0.470340\pi$
$811$	5.29940	0.186087	0.0930436	+	0.995662i	$0.470340\pi$
$812$	0	0
$813$	−15.7154	−0.551165
$814$	−40.2599	−1.41111
$815$	−61.7250	−2.16213
$816$	66.4612	2.32661
$817$	−6.88820	−0.240988
$818$	26.5666	0.928878
$819$	0	0
$820$	−48.4968	−1.69358
$821$	17.0332	0.594464	0.297232	−	0.954805i	$-0.403937\pi$
$821$	17.0332	0.594464	0.297232	+	0.954805i	$0.403937\pi$
$822$	−137.626	−4.80025
$823$	6.99492	0.243828	0.121914	−	0.992541i	$-0.461097\pi$
$823$	6.99492	0.243828	0.121914	+	0.992541i	$0.461097\pi$
$824$	2.66636	0.0928870
$825$	197.123	6.86294
$826$	0	0
$827$	−4.65051	−0.161714	−0.0808571	−	0.996726i	$-0.525766\pi$
$827$	−4.65051	−0.161714	−0.0808571	+	0.996726i	$0.525766\pi$
$828$	95.6787	3.32506
$829$	−20.0100	−0.694976	−0.347488	−	0.937685i	$-0.612965\pi$
$829$	−20.0100	−0.694976	−0.347488	+	0.937685i	$0.612965\pi$
$830$	−66.6959	−2.31505
$831$	8.03012	0.278562
$832$	27.4142	0.950416
$833$	0	0
$834$	−81.0692	−2.80720
$835$	6.13367	0.212264
$836$	17.1127	0.591854
$837$	−38.1455	−1.31850
$838$	67.1651	2.32018
$839$	−1.79994	−0.0621409	−0.0310704	−	0.999517i	$-0.509892\pi$
$839$	−1.79994	−0.0621409	−0.0310704	+	0.999517i	$0.509892\pi$
$840$	0	0
$841$	−16.2534	−0.560461
$842$	61.4613	2.11810
$843$	−76.5282	−2.63577
$844$	−25.4793	−0.877035
$845$	19.6505	0.675998
$846$	−65.6355	−2.25659
$847$	0	0
$848$	−7.58464	−0.260458
$849$	106.784	3.66480
$850$	136.646	4.68691
$851$	20.8703	0.715426
$852$	−33.6198	−1.15180
$853$	−18.8135	−0.644161	−0.322081	−	0.946712i	$-0.604382\pi$
$853$	−18.8135	−0.644161	−0.322081	+	0.946712i	$0.604382\pi$
$854$	0	0
$855$	−54.3980	−1.86037
$856$	4.87812	0.166731
$857$	−35.7199	−1.22017	−0.610084	−	0.792337i	$-0.708864\pi$
$857$	−35.7199	−1.22017	−0.610084	+	0.792337i	$0.708864\pi$
$858$	96.9390	3.30944
$859$	0.393194	0.0134156	0.00670780	−	0.999978i	$-0.497865\pi$
$859$	0.393194	0.0134156	0.00670780	+	0.999978i	$0.497865\pi$
$860$	39.4971	1.34684
$861$	0	0
$862$	−73.8176	−2.51424
$863$	37.9269	1.29105	0.645524	−	0.763740i	$-0.276639\pi$
$863$	37.9269	1.29105	0.645524	+	0.763740i	$0.276639\pi$
$864$	145.752	4.95859
$865$	−89.9924	−3.05983
$866$	−12.9429	−0.439819
$867$	47.1818	1.60238
$868$	0	0
$869$	61.0478	2.07091
$870$	−101.531	−3.44224
$871$	−33.9559	−1.15055
$872$	5.14365	0.174186
$873$	−15.0128	−0.508107
$874$	−16.9126	−0.572076
$875$	0	0
$876$	−65.1068	−2.19975
$877$	−23.3730	−0.789251	−0.394625	−	0.918842i	$-0.629126\pi$
$877$	−23.3730	−0.789251	−0.394625	+	0.918842i	$0.629126\pi$
$878$	−15.6918	−0.529572
$879$	8.27967	0.279266
$880$	71.4553	2.40876
$881$	27.2506	0.918095	0.459047	−	0.888412i	$-0.348191\pi$
$881$	27.2506	0.918095	0.459047	+	0.888412i	$0.348191\pi$
$882$	0	0
$883$	39.6048	1.33281	0.666404	−	0.745590i	$-0.267832\pi$
$883$	39.6048	1.33281	0.666404	+	0.745590i	$0.267832\pi$
$884$	35.2470	1.18549
$885$	−27.0483	−0.909218
$886$	15.1130	0.507733
$887$	−4.13182	−0.138733	−0.0693665	−	0.997591i	$-0.522098\pi$
$887$	−4.13182	−0.138733	−0.0693665	+	0.997591i	$0.522098\pi$
$888$	−5.71363	−0.191737
$889$	0	0
$890$	4.16047	0.139459
$891$	173.521	5.81318
$892$	−15.3107	−0.512641
$893$	6.08553	0.203644
$894$	−128.452	−4.29609
$895$	−61.0590	−2.04098
$896$	0	0
$897$	−50.2523	−1.67788
$898$	−54.1677	−1.80760
$899$	7.58386	0.252936
$900$	219.981	7.33271
$901$	−11.9159	−0.396975
$902$	−53.5699	−1.78368
$903$	0	0
$904$	−6.95457	−0.231305
$905$	−43.7704	−1.45498
$906$	120.157	3.99195
$907$	−8.13062	−0.269973	−0.134986	−	0.990847i	$-0.543099\pi$
$907$	−8.13062	−0.269973	−0.134986	+	0.990847i	$0.543099\pi$
$908$	19.9177	0.660994
$909$	123.147	4.08453
$910$	0	0
$911$	−14.0255	−0.464685	−0.232343	−	0.972634i	$-0.574639\pi$
$911$	−14.0255	−0.464685	−0.232343	+	0.972634i	$0.574639\pi$
$912$	−18.9126	−0.626260
$913$	−38.6432	−1.27890
$914$	−63.4804	−2.09975
$915$	179.040	5.91887
$916$	−54.8590	−1.81259
$917$	0	0
$918$	205.109	6.76961
$919$	32.7303	1.07967	0.539836	−	0.841770i	$-0.318486\pi$
$919$	32.7303	1.07967	0.539836	+	0.841770i	$0.318486\pi$
$920$	9.06840	0.298976
$921$	−20.7834	−0.684838
$922$	−78.3990	−2.58193
$923$	12.9840	0.427372
$924$	0	0
$925$	47.9844	1.57772
$926$	47.2765	1.55360
$927$	−52.5163	−1.72486
$928$	−28.9776	−0.951236
$929$	−16.5866	−0.544189	−0.272094	−	0.962271i	$-0.587716\pi$
$929$	−16.5866	−0.544189	−0.272094	+	0.962271i	$0.587716\pi$
$930$	−60.4081	−1.98086
$931$	0	0
$932$	−25.7904	−0.844792
$933$	31.7712	1.04014
$934$	−67.3601	−2.20409
$935$	112.260	3.67130
$936$	10.1160	0.330652
$937$	−52.3826	−1.71126	−0.855632	−	0.517584i	$-0.826832\pi$
$937$	−52.3826	−1.71126	−0.855632	+	0.517584i	$0.826832\pi$
$938$	0	0
$939$	5.41475	0.176704
$940$	−34.8945	−1.13813
$941$	12.7563	0.415844	0.207922	−	0.978145i	$-0.433330\pi$
$941$	12.7563	0.415844	0.207922	+	0.978145i	$0.433330\pi$
$942$	−34.1689	−1.11328
$943$	27.7701	0.904320
$944$	−6.91481	−0.225058
$945$	0	0
$946$	43.6287	1.41849
$947$	−46.2884	−1.50417	−0.752085	−	0.659066i	$-0.770952\pi$
$947$	−46.2884	−1.50417	−0.752085	+	0.659066i	$0.770952\pi$
$948$	92.6504	3.00915
$949$	25.1442	0.816216
$950$	−38.8848	−1.26159
$951$	−29.5441	−0.958032
$952$	0	0
$953$	−9.09754	−0.294698	−0.147349	−	0.989085i	$-0.547074\pi$
$953$	−9.09754	−0.294698	−0.147349	+	0.989085i	$0.547074\pi$
$954$	−36.5721	−1.18407
$955$	5.87559	0.190130
$956$	−39.6707	−1.28304
$957$	−58.8266	−1.90159
$958$	−33.6554	−1.08736
$959$	0	0
$960$	132.512	4.27681
$961$	−26.4878	−0.854446
$962$	23.5972	0.760806
$963$	−96.0789	−3.09610
$964$	−11.8684	−0.382256
$965$	49.0020	1.57743
$966$	0	0
$967$	−14.7706	−0.474991	−0.237495	−	0.971389i	$-0.576326\pi$
$967$	−14.7706	−0.474991	−0.237495	+	0.971389i	$0.576326\pi$
$968$	−5.48103	−0.176167
$969$	−29.7128	−0.954512
$970$	−15.2165	−0.488573
$971$	−33.4545	−1.07361	−0.536803	−	0.843708i	$-0.680368\pi$
$971$	−33.4545	−1.07361	−0.536803	+	0.843708i	$0.680368\pi$
$972$	144.487	4.63442
$973$	0	0
$974$	−25.1193	−0.804875
$975$	−115.538	−3.70019
$976$	45.7709	1.46509
$977$	30.6859	0.981728	0.490864	−	0.871236i	$-0.336681\pi$
$977$	30.6859	0.981728	0.490864	+	0.871236i	$0.336681\pi$
$978$	−103.477	−3.30884
$979$	2.41055	0.0770414
$980$	0	0
$981$	−101.309	−3.23454
$982$	7.53625	0.240491
$983$	47.4874	1.51461	0.757307	−	0.653059i	$-0.226515\pi$
$983$	47.4874	1.51461	0.757307	+	0.653059i	$0.226515\pi$
$984$	−7.60257	−0.242361
$985$	9.38179	0.298929
$986$	−40.7786	−1.29865
$987$	0	0
$988$	−10.0301	−0.319101
$989$	−22.6167	−0.719170
$990$	344.548	10.9504
$991$	−24.5775	−0.780728	−0.390364	−	0.920661i	$-0.627651\pi$
$991$	−24.5775	−0.780728	−0.390364	+	0.920661i	$0.627651\pi$
$992$	−17.2408	−0.547396
$993$	55.3878	1.75768
$994$	0	0
$995$	−80.7916	−2.56126
$996$	−58.6475	−1.85832
$997$	−38.9650	−1.23403	−0.617016	−	0.786950i	$-0.711659\pi$
$997$	−38.9650	−1.23403	−0.617016	+	0.786950i	$0.711659\pi$
$998$	69.3757	2.19605
$999$	72.0260	2.27880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.t.1.10	yes	74
7.6	odd	2	inner	6223.2.a.t.1.9	✓	74

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.t.1.9	✓	74	7.6	odd	2	inner
6223.2.a.t.1.10	yes	74	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05093 q^{2} +3.36661 q^{3} +2.20631 q^{4} -4.11869 q^{5} -6.90468 q^{6} -0.423137 q^{8} +8.33406 q^{9} +8.44714 q^{10} +4.89422 q^{11} +7.42780 q^{12} -2.86861 q^{13} -13.8660 q^{15} -3.54481 q^{16} -5.56908 q^{17} -17.0926 q^{18} +1.58477 q^{19} -9.08712 q^{20} -10.0377 q^{22} +5.20345 q^{23} -1.42454 q^{24} +11.9636 q^{25} +5.88332 q^{26} +17.9577 q^{27} -3.57024 q^{29} +28.4382 q^{30} -2.12419 q^{31} +8.11642 q^{32} +16.4769 q^{33} +11.4218 q^{34} +18.3876 q^{36} +4.01087 q^{37} -3.25026 q^{38} -9.65750 q^{39} +1.74277 q^{40} +5.33687 q^{41} -4.34649 q^{43} +10.7982 q^{44} -34.3254 q^{45} -10.6719 q^{46} +3.84000 q^{47} -11.9340 q^{48} -24.5365 q^{50} -18.7489 q^{51} -6.32906 q^{52} +2.13965 q^{53} -36.8300 q^{54} -20.1577 q^{55} +5.33531 q^{57} +7.32232 q^{58} +1.95069 q^{59} -30.5928 q^{60} -12.9121 q^{61} +4.35656 q^{62} -9.55660 q^{64} +11.8149 q^{65} -33.7930 q^{66} +11.8370 q^{67} -12.2871 q^{68} +17.5180 q^{69} -4.52622 q^{71} -3.52645 q^{72} -8.76529 q^{73} -8.22601 q^{74} +40.2767 q^{75} +3.49651 q^{76} +19.8069 q^{78} +12.4735 q^{79} +14.5999 q^{80} +35.4544 q^{81} -10.9456 q^{82} -7.89568 q^{83} +22.9373 q^{85} +8.91434 q^{86} -12.0196 q^{87} -2.07092 q^{88} +0.492530 q^{89} +70.3990 q^{90} +11.4804 q^{92} -7.15130 q^{93} -7.87557 q^{94} -6.52719 q^{95} +27.3248 q^{96} -1.80138 q^{97} +40.7887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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