LMFDB - Embedded newform 6223.2.a.p.1.3

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 = [38,-2,11,38,16] 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:	$38$
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	\(q-2.49062 q^{2} +0.897479 q^{3} +4.20317 q^{4} -0.525060 q^{5} -2.23528 q^{6} -5.48724 q^{8} -2.19453 q^{9} +1.30772 q^{10} +2.61358 q^{11} +3.77225 q^{12} +1.45191 q^{13} -0.471231 q^{15} +5.26027 q^{16} -4.55204 q^{17} +5.46573 q^{18} +0.201679 q^{19} -2.20692 q^{20} -6.50942 q^{22} -0.470953 q^{23} -4.92468 q^{24} -4.72431 q^{25} -3.61614 q^{26} -4.66198 q^{27} -1.50355 q^{29} +1.17365 q^{30} -1.80002 q^{31} -2.12683 q^{32} +2.34563 q^{33} +11.3374 q^{34} -9.22398 q^{36} -0.961983 q^{37} -0.502304 q^{38} +1.30306 q^{39} +2.88113 q^{40} +1.83977 q^{41} -6.33513 q^{43} +10.9853 q^{44} +1.15226 q^{45} +1.17296 q^{46} +3.83920 q^{47} +4.72098 q^{48} +11.7664 q^{50} -4.08536 q^{51} +6.10260 q^{52} +3.07544 q^{53} +11.6112 q^{54} -1.37229 q^{55} +0.181002 q^{57} +3.74475 q^{58} +9.37780 q^{59} -1.98066 q^{60} -13.7825 q^{61} +4.48316 q^{62} -5.22342 q^{64} -0.762339 q^{65} -5.84207 q^{66} +13.3810 q^{67} -19.1330 q^{68} -0.422671 q^{69} +10.5072 q^{71} +12.0419 q^{72} -2.94305 q^{73} +2.39593 q^{74} -4.23997 q^{75} +0.847689 q^{76} -3.24541 q^{78} -3.53853 q^{79} -2.76196 q^{80} +2.39956 q^{81} -4.58217 q^{82} +17.4538 q^{83} +2.39010 q^{85} +15.7784 q^{86} -1.34940 q^{87} -14.3413 q^{88} -5.66627 q^{89} -2.86984 q^{90} -1.97950 q^{92} -1.61548 q^{93} -9.56198 q^{94} -0.105893 q^{95} -1.90879 q^{96} +6.82742 q^{97} -5.73558 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) $ 38 * q - 2 * q^2 + 11 * q^3 + 38 * q^4 + 16 * q^5 + 11 * q^6 + 47 * q^9 + 12 * q^10 - 2 * q^11 + 30 * q^12 + 21 * q^13 + 7 * q^15 + 46 * q^16 + 58 * q^17 - 13 * q^18 + 17 * q^19 + 44 * q^20 + 21 * q^22 + 7 * q^23 + 21 * q^24 + 52 * q^25 + 49 * q^26 + 35 * q^27 - 2 * q^29 + 6 * q^30 + 2 * q^31 - 22 * q^32 + 29 * q^33 + 37 * q^34 + 25 * q^36 + 11 * q^37 + 40 * q^38 + 2 * q^39 - 8 * q^40 + 68 * q^41 - 38 * q^43 + 43 * q^44 + 60 * q^45 - 28 * q^46 + 35 * q^47 + 66 * q^48 - 25 * q^50 - 3 * q^51 + 32 * q^52 - 18 * q^53 - 68 * q^54 + 19 * q^55 - 31 * q^57 + 55 * q^58 + 21 * q^59 - 11 * q^60 + 5 * q^61 + 38 * q^62 + 18 * q^64 - 9 * q^65 + 20 * q^66 + 6 * q^67 + 74 * q^68 + 53 * q^69 - 26 * q^71 + 24 * q^72 + 22 * q^73 - 2 * q^74 + 38 * q^75 + 39 * q^76 - 114 * q^78 - 14 * q^79 + 140 * q^80 + 58 * q^81 - 8 * q^82 + 46 * q^83 - 11 * q^85 + 51 * q^86 + 59 * q^87 - 4 * q^88 + 74 * q^89 + q^90 - 81 * q^92 + 92 * q^93 - 36 * q^94 + 20 * q^95 - 108 * q^96 + 66 * q^97 + 16 * 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 $

$2$	−2.49062	−1.76113	−0.880566	−	0.473924i	$-0.842837\pi$
$2$	−2.49062	−1.76113	−0.880566	+	0.473924i	$0.842837\pi$
$3$	0.897479	0.518160	0.259080	−	0.965856i	$-0.416581\pi$
$3$	0.897479	0.518160	0.259080	+	0.965856i	$0.416581\pi$
$4$	4.20317	2.10158
$5$	−0.525060	−0.234814	−0.117407	−	0.993084i	$-0.537458\pi$
$5$	−0.525060	−0.234814	−0.117407	+	0.993084i	$0.537458\pi$
$6$	−2.23528	−0.912547
$7$	0	0
$7$	0	0
$8$	−5.48724	−1.94003
$9$	−2.19453	−0.731511
$10$	1.30772	0.413539
$11$	2.61358	0.788023	0.394012	−	0.919105i	$-0.371087\pi$
$11$	2.61358	0.788023	0.394012	+	0.919105i	$0.371087\pi$
$12$	3.77225	1.08896
$13$	1.45191	0.402686	0.201343	−	0.979521i	$-0.435469\pi$
$13$	1.45191	0.402686	0.201343	+	0.979521i	$0.435469\pi$
$14$	0	0
$15$	−0.471231	−0.121671
$16$	5.26027	1.31507
$17$	−4.55204	−1.10403	−0.552016	−	0.833834i	$-0.686141\pi$
$17$	−4.55204	−1.10403	−0.552016	+	0.833834i	$0.686141\pi$
$18$	5.46573	1.28829
$19$	0.201679	0.0462683	0.0231341	−	0.999732i	$-0.492636\pi$
$19$	0.201679	0.0462683	0.0231341	+	0.999732i	$0.492636\pi$
$20$	−2.20692	−0.493481
$21$	0	0
$22$	−6.50942	−1.38781
$23$	−0.470953	−0.0982006	−0.0491003	−	0.998794i	$-0.515635\pi$
$23$	−0.470953	−0.0982006	−0.0491003	+	0.998794i	$0.515635\pi$
$24$	−4.92468	−1.00525
$25$	−4.72431	−0.944862
$26$	−3.61614	−0.709184
$27$	−4.66198	−0.897199
$28$	0	0
$29$	−1.50355	−0.279201	−0.139601	−	0.990208i	$-0.544582\pi$
$29$	−1.50355	−0.279201	−0.139601	+	0.990208i	$0.544582\pi$
$30$	1.17365	0.214279
$31$	−1.80002	−0.323294	−0.161647	−	0.986849i	$-0.551681\pi$
$31$	−1.80002	−0.323294	−0.161647	+	0.986849i	$0.551681\pi$
$32$	−2.12683	−0.375975
$33$	2.34563	0.408322
$34$	11.3374	1.94434
$35$	0	0
$36$	−9.22398	−1.53733
$37$	−0.961983	−0.158149	−0.0790745	−	0.996869i	$-0.525197\pi$
$37$	−0.961983	−0.158149	−0.0790745	+	0.996869i	$0.525197\pi$
$38$	−0.502304	−0.0814845
$39$	1.30306	0.208656
$40$	2.88113	0.455547
$41$	1.83977	0.287324	0.143662	−	0.989627i	$-0.454112\pi$
$41$	1.83977	0.287324	0.143662	+	0.989627i	$0.454112\pi$
$42$	0	0
$43$	−6.33513	−0.966098	−0.483049	−	0.875593i	$-0.660471\pi$
$43$	−6.33513	−0.966098	−0.483049	+	0.875593i	$0.660471\pi$
$44$	10.9853	1.65610
$45$	1.15226	0.171769
$46$	1.17296	0.172944
$47$	3.83920	0.560005	0.280003	−	0.959999i	$-0.409665\pi$
$47$	3.83920	0.560005	0.280003	+	0.959999i	$0.409665\pi$
$48$	4.72098	0.681415
$49$	0	0
$50$	11.7664	1.66403
$51$	−4.08536	−0.572065
$52$	6.10260	0.846279
$53$	3.07544	0.422444	0.211222	−	0.977438i	$-0.432256\pi$
$53$	3.07544	0.422444	0.211222	+	0.977438i	$0.432256\pi$
$54$	11.6112	1.58009
$55$	−1.37229	−0.185039
$56$	0	0
$57$	0.181002	0.0239743
$58$	3.74475	0.491710
$59$	9.37780	1.22089	0.610443	−	0.792060i	$-0.290992\pi$
$59$	9.37780	1.22089	0.610443	+	0.792060i	$0.290992\pi$
$60$	−1.98066	−0.255702
$61$	−13.7825	−1.76467	−0.882334	−	0.470623i	$-0.844029\pi$
$61$	−13.7825	−1.76467	−0.882334	+	0.470623i	$0.844029\pi$
$62$	4.48316	0.569362
$63$	0	0
$64$	−5.22342	−0.652927
$65$	−0.762339	−0.0945565
$66$	−5.84207	−0.719109
$67$	13.3810	1.63475	0.817377	−	0.576103i	$-0.195427\pi$
$67$	13.3810	1.63475	0.817377	+	0.576103i	$0.195427\pi$
$68$	−19.1330	−2.32021
$69$	−0.422671	−0.0508836
$70$	0	0
$71$	10.5072	1.24698	0.623490	−	0.781832i	$-0.285714\pi$
$71$	10.5072	1.24698	0.623490	+	0.781832i	$0.285714\pi$
$72$	12.0419	1.41915
$73$	−2.94305	−0.344458	−0.172229	−	0.985057i	$-0.555097\pi$
$73$	−2.94305	−0.344458	−0.172229	+	0.985057i	$0.555097\pi$
$74$	2.39593	0.278521
$75$	−4.23997	−0.489590
$76$	0.847689	0.0972366
$77$	0	0
$78$	−3.24541	−0.367470
$79$	−3.53853	−0.398116	−0.199058	−	0.979988i	$-0.563788\pi$
$79$	−3.53853	−0.398116	−0.199058	+	0.979988i	$0.563788\pi$
$80$	−2.76196	−0.308797
$81$	2.39956	0.266618
$82$	−4.58217	−0.506016
$83$	17.4538	1.91580	0.957902	−	0.287094i	$-0.0926893\pi$
$83$	17.4538	1.91580	0.957902	+	0.287094i	$0.0926893\pi$
$84$	0	0
$85$	2.39010	0.259242
$86$	15.7784	1.70143
$87$	−1.34940	−0.144671
$88$	−14.3413	−1.52879
$89$	−5.66627	−0.600623	−0.300312	−	0.953841i	$-0.597091\pi$
$89$	−5.66627	−0.600623	−0.300312	+	0.953841i	$0.597091\pi$
$90$	−2.86984	−0.302508
$91$	0	0
$92$	−1.97950	−0.206377
$93$	−1.61548	−0.167518
$94$	−9.56198	−0.986243
$95$	−0.105893	−0.0108644
$96$	−1.90879	−0.194815
$97$	6.82742	0.693219	0.346610	−	0.938009i	$-0.387333\pi$
$97$	6.82742	0.693219	0.346610	+	0.938009i	$0.387333\pi$
$98$	0	0
$99$	−5.73558	−0.576447
$100$	−19.8571	−1.98571
$101$	6.99039	0.695570	0.347785	−	0.937574i	$-0.386934\pi$
$101$	6.99039	0.695570	0.347785	+	0.937574i	$0.386934\pi$
$102$	10.1751	1.00748
$103$	4.00812	0.394932	0.197466	−	0.980310i	$-0.436729\pi$
$103$	4.00812	0.394932	0.197466	+	0.980310i	$0.436729\pi$
$104$	−7.96696	−0.781224
$105$	0	0
$106$	−7.65974	−0.743980
$107$	−6.11377	−0.591040	−0.295520	−	0.955336i	$-0.595493\pi$
$107$	−6.11377	−0.591040	−0.295520	+	0.955336i	$0.595493\pi$
$108$	−19.5951	−1.88554
$109$	−4.49826	−0.430855	−0.215428	−	0.976520i	$-0.569115\pi$
$109$	−4.49826	−0.430855	−0.215428	+	0.976520i	$0.569115\pi$
$110$	3.41784	0.325878
$111$	−0.863360	−0.0819465
$112$	0	0
$113$	−7.47274	−0.702977	−0.351488	−	0.936192i	$-0.614324\pi$
$113$	−7.47274	−0.702977	−0.351488	+	0.936192i	$0.614324\pi$
$114$	−0.450807	−0.0422220
$115$	0.247279	0.0230589
$116$	−6.31965	−0.586765
$117$	−3.18625	−0.294569
$118$	−23.3565	−2.15014
$119$	0	0
$120$	2.58576	0.236046
$121$	−4.16921	−0.379019
$122$	34.3269	3.10781
$123$	1.65116	0.148880
$124$	−7.56579	−0.679428
$125$	5.10585	0.456681
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	17.2632	1.52586
$129$	−5.68564	−0.500593
$130$	1.89869	0.166526
$131$	0.0107038	0.000935197 0	0.000467599	−	1.00000i	$-0.499851\pi$
$131$	0.0107038	0.000935197 0	0.000467599		1.00000i	$0.499851\pi$
$132$	9.85908	0.858122
$133$	0	0
$134$	−33.3270	−2.87902
$135$	2.44782	0.210675
$136$	24.9781	2.14186
$137$	−2.47578	−0.211520	−0.105760	−	0.994392i	$-0.533728\pi$
$137$	−2.47578	−0.211520	−0.105760	+	0.994392i	$0.533728\pi$
$138$	1.05271	0.0896127
$139$	−5.74197	−0.487028	−0.243514	−	0.969897i	$-0.578300\pi$
$139$	−5.74197	−0.487028	−0.243514	+	0.969897i	$0.578300\pi$
$140$	0	0
$141$	3.44560	0.290172
$142$	−26.1695	−2.19609
$143$	3.79467	0.317326
$144$	−11.5438	−0.961986
$145$	0.789452	0.0655604
$146$	7.33000	0.606635
$147$	0	0
$148$	−4.04338	−0.332363
$149$	5.95505	0.487857	0.243928	−	0.969793i	$-0.421564\pi$
$149$	5.95505	0.487857	0.243928	+	0.969793i	$0.421564\pi$
$150$	10.5601	0.862231
$151$	15.7901	1.28498	0.642488	−	0.766295i	$-0.277902\pi$
$151$	15.7901	1.28498	0.642488	+	0.766295i	$0.277902\pi$
$152$	−1.10666	−0.0897619
$153$	9.98959	0.807611
$154$	0	0
$155$	0.945120	0.0759139
$156$	5.47696	0.438508
$157$	−6.54627	−0.522449	−0.261224	−	0.965278i	$-0.584126\pi$
$157$	−6.54627	−0.522449	−0.261224	+	0.965278i	$0.584126\pi$
$158$	8.81313	0.701135
$159$	2.76014	0.218894
$160$	1.11672	0.0882842
$161$	0	0
$162$	−5.97639	−0.469549
$163$	6.30025	0.493474	0.246737	−	0.969082i	$-0.420642\pi$
$163$	6.30025	0.493474	0.246737	+	0.969082i	$0.420642\pi$
$164$	7.73287	0.603836
$165$	−1.23160	−0.0958798
$166$	−43.4707	−3.37398
$167$	16.4914	1.27614	0.638072	−	0.769976i	$-0.279732\pi$
$167$	16.4914	1.27614	0.638072	+	0.769976i	$0.279732\pi$
$168$	0	0
$169$	−10.8920	−0.837844
$170$	−5.95281	−0.456560
$171$	−0.442590	−0.0338457
$172$	−26.6276	−2.03034
$173$	21.1960	1.61150	0.805751	−	0.592255i	$-0.201762\pi$
$173$	21.1960	1.61150	0.805751	+	0.592255i	$0.201762\pi$
$174$	3.36084	0.254784
$175$	0	0
$176$	13.7481	1.03630
$177$	8.41638	0.632614
$178$	14.1125	1.05778
$179$	−6.25695	−0.467666	−0.233833	−	0.972277i	$-0.575127\pi$
$179$	−6.25695	−0.467666	−0.233833	+	0.972277i	$0.575127\pi$
$180$	4.84315	0.360987
$181$	7.21573	0.536341	0.268171	−	0.963371i	$-0.413581\pi$
$181$	7.21573	0.536341	0.268171	+	0.963371i	$0.413581\pi$
$182$	0	0
$183$	−12.3695	−0.914380
$184$	2.58423	0.190512
$185$	0.505099	0.0371356
$186$	4.02354	0.295021
$187$	−11.8971	−0.870003
$188$	16.1368	1.17690
$189$	0	0
$190$	0.263740	0.0191337
$191$	23.0114	1.66505	0.832524	−	0.553989i	$-0.186895\pi$
$191$	23.0114	1.66505	0.832524	+	0.553989i	$0.186895\pi$
$192$	−4.68791	−0.338320
$193$	13.4101	0.965282	0.482641	−	0.875818i	$-0.339678\pi$
$193$	13.4101	0.965282	0.482641	+	0.875818i	$0.339678\pi$
$194$	−17.0045	−1.22085
$195$	−0.684183	−0.0489954
$196$	0	0
$197$	4.15719	0.296187	0.148094	−	0.988973i	$-0.452686\pi$
$197$	4.15719	0.296187	0.148094	+	0.988973i	$0.452686\pi$
$198$	14.2851	1.01520
$199$	−4.75536	−0.337099	−0.168549	−	0.985693i	$-0.553908\pi$
$199$	−4.75536	−0.337099	−0.168549	+	0.985693i	$0.553908\pi$
$200$	25.9234	1.83306
$201$	12.0092	0.847064
$202$	−17.4104	−1.22499
$203$	0	0
$204$	−17.1714	−1.20224
$205$	−0.965992	−0.0674678
$206$	−9.98269	−0.695527
$207$	1.03352	0.0718348
$208$	7.63742	0.529560
$209$	0.527103	0.0364605
$210$	0	0
$211$	27.7227	1.90851	0.954254	−	0.298997i	$-0.0966520\pi$
$211$	27.7227	1.90851	0.954254	+	0.298997i	$0.0966520\pi$
$212$	12.9266	0.887801
$213$	9.43002	0.646134
$214$	15.2270	1.04090
$215$	3.32633	0.226854
$216$	25.5814	1.74059
$217$	0	0
$218$	11.2034	0.758793
$219$	−2.64132	−0.178484
$220$	−5.76795	−0.388875
$221$	−6.60913	−0.444579
$222$	2.15030	0.144319
$223$	19.5825	1.31134	0.655671	−	0.755047i	$-0.272386\pi$
$223$	19.5825	1.31134	0.655671	+	0.755047i	$0.272386\pi$
$224$	0	0
$225$	10.3677	0.691177
$226$	18.6117	1.23803
$227$	9.51389	0.631459	0.315730	−	0.948849i	$-0.397751\pi$
$227$	9.51389	0.631459	0.315730	+	0.948849i	$0.397751\pi$
$228$	0.760783	0.0503841
$229$	7.96207	0.526149	0.263074	−	0.964776i	$-0.415264\pi$
$229$	7.96207	0.526149	0.263074	+	0.964776i	$0.415264\pi$
$230$	−0.615877	−0.0406097
$231$	0	0
$232$	8.25031	0.541660
$233$	2.76312	0.181018	0.0905091	−	0.995896i	$-0.471151\pi$
$233$	2.76312	0.181018	0.0905091	+	0.995896i	$0.471151\pi$
$234$	7.93574	0.518775
$235$	−2.01581	−0.131497
$236$	39.4164	2.56579
$237$	−3.17576	−0.206288
$238$	0	0
$239$	−29.2782	−1.89385	−0.946924	−	0.321458i	$-0.895827\pi$
$239$	−29.2782	−1.89385	−0.946924	+	0.321458i	$0.895827\pi$
$240$	−2.47880	−0.160006
$241$	−1.43755	−0.0926006	−0.0463003	−	0.998928i	$-0.514743\pi$
$241$	−1.43755	−0.0926006	−0.0463003	+	0.998928i	$0.514743\pi$
$242$	10.3839	0.667503
$243$	16.1395	1.03535
$244$	−57.9301	−3.70860
$245$	0	0
$246$	−4.11240	−0.262197
$247$	0.292819	0.0186316
$248$	9.87715	0.627200
$249$	15.6644	0.992693
$250$	−12.7167	−0.804275
$251$	−9.26401	−0.584739	−0.292370	−	0.956305i	$-0.594444\pi$
$251$	−9.26401	−0.584739	−0.292370	+	0.956305i	$0.594444\pi$
$252$	0	0
$253$	−1.23087	−0.0773844
$254$	−2.49062	−0.156275
$255$	2.14506	0.134329
$256$	−32.5491	−2.03432
$257$	−10.6958	−0.667183	−0.333591	−	0.942718i	$-0.608261\pi$
$257$	−10.6958	−0.667183	−0.333591	+	0.942718i	$0.608261\pi$
$258$	14.1608	0.881610
$259$	0	0
$260$	−3.20424	−0.198718
$261$	3.29958	0.204239
$262$	−0.0266591	−0.00164701
$263$	−21.5917	−1.33140	−0.665701	−	0.746219i	$-0.731867\pi$
$263$	−21.5917	−1.33140	−0.665701	+	0.746219i	$0.731867\pi$
$264$	−12.8710	−0.792158
$265$	−1.61479	−0.0991959
$266$	0	0
$267$	−5.08536	−0.311219
$268$	56.2427	3.43557
$269$	−6.15178	−0.375081	−0.187540	−	0.982257i	$-0.560052\pi$
$269$	−6.15178	−0.375081	−0.187540	+	0.982257i	$0.560052\pi$
$270$	−6.09659	−0.371026
$271$	−0.236792	−0.0143841	−0.00719205	−	0.999974i	$-0.502289\pi$
$271$	−0.236792	−0.0143841	−0.00719205	+	0.999974i	$0.502289\pi$
$272$	−23.9450	−1.45188
$273$	0	0
$274$	6.16621	0.372515
$275$	−12.3474	−0.744574
$276$	−1.77656	−0.106936
$277$	6.47769	0.389206	0.194603	−	0.980882i	$-0.437658\pi$
$277$	6.47769	0.389206	0.194603	+	0.980882i	$0.437658\pi$
$278$	14.3010	0.857719
$279$	3.95021	0.236493
$280$	0	0
$281$	8.39461	0.500781	0.250390	−	0.968145i	$-0.419441\pi$
$281$	8.39461	0.500781	0.250390	+	0.968145i	$0.419441\pi$
$282$	−8.58167	−0.511031
$283$	−23.2781	−1.38374	−0.691871	−	0.722021i	$-0.743213\pi$
$283$	−23.2781	−1.38374	−0.691871	+	0.722021i	$0.743213\pi$
$284$	44.1636	2.62063
$285$	−0.0950372	−0.00562952
$286$	−9.45107	−0.558853
$287$	0	0
$288$	4.66740	0.275029
$289$	3.72106	0.218886
$290$	−1.96622	−0.115461
$291$	6.12746	0.359198
$292$	−12.3701	−0.723907
$293$	5.84226	0.341308	0.170654	−	0.985331i	$-0.445412\pi$
$293$	5.84226	0.341308	0.170654	+	0.985331i	$0.445412\pi$
$294$	0	0
$295$	−4.92391	−0.286681
$296$	5.27863	0.306814
$297$	−12.1845	−0.707014
$298$	−14.8317	−0.859180
$299$	−0.683780	−0.0395440
$300$	−17.8213	−1.02891
$301$	0	0
$302$	−39.3270	−2.26301
$303$	6.27373	0.360416
$304$	1.06088	0.0608459
$305$	7.23665	0.414369
$306$	−24.8802	−1.42231
$307$	6.39402	0.364926	0.182463	−	0.983213i	$-0.441593\pi$
$307$	6.39402	0.364926	0.182463	+	0.983213i	$0.441593\pi$
$308$	0	0
$309$	3.59721	0.204638
$310$	−2.35393	−0.133694
$311$	15.4301	0.874962	0.437481	−	0.899228i	$-0.355871\pi$
$311$	15.4301	0.874962	0.437481	+	0.899228i	$0.355871\pi$
$312$	−7.15018	−0.404799
$313$	−19.7355	−1.11552	−0.557758	−	0.830004i	$-0.688338\pi$
$313$	−19.7355	−1.11552	−0.557758	+	0.830004i	$0.688338\pi$
$314$	16.3042	0.920101
$315$	0	0
$316$	−14.8730	−0.836674
$317$	−11.8507	−0.665601	−0.332800	−	0.942997i	$-0.607994\pi$
$317$	−11.8507	−0.665601	−0.332800	+	0.942997i	$0.607994\pi$
$318$	−6.87445	−0.385500
$319$	−3.92963	−0.220017
$320$	2.74261	0.153316
$321$	−5.48698	−0.306253
$322$	0	0
$323$	−0.918049	−0.0510816
$324$	10.0858	0.560320
$325$	−6.85926	−0.380483
$326$	−15.6915	−0.869072
$327$	−4.03710	−0.223252
$328$	−10.0953	−0.557418
$329$	0	0
$330$	3.06744	0.168857
$331$	−9.73295	−0.534971	−0.267486	−	0.963562i	$-0.586193\pi$
$331$	−9.73295	−0.534971	−0.267486	+	0.963562i	$0.586193\pi$
$332$	73.3613	4.02622
$333$	2.11110	0.115688
$334$	−41.0738	−2.24746
$335$	−7.02585	−0.383863
$336$	0	0
$337$	27.8574	1.51749	0.758746	−	0.651387i	$-0.225812\pi$
$337$	27.8574	1.51749	0.758746	+	0.651387i	$0.225812\pi$
$338$	27.1277	1.47555
$339$	−6.70663	−0.364254
$340$	10.0460	0.544819
$341$	−4.70450	−0.254763
$342$	1.10232	0.0596067
$343$	0	0
$344$	34.7624	1.87426
$345$	0.221928	0.0119482
$346$	−52.7911	−2.83807
$347$	15.7198	0.843882	0.421941	−	0.906623i	$-0.361349\pi$
$347$	15.7198	0.843882	0.421941	+	0.906623i	$0.361349\pi$
$348$	−5.67175	−0.304038
$349$	−25.6094	−1.37084	−0.685419	−	0.728149i	$-0.740381\pi$
$349$	−25.6094	−1.37084	−0.685419	+	0.728149i	$0.740381\pi$
$350$	0	0
$351$	−6.76876	−0.361290
$352$	−5.55865	−0.296277
$353$	−15.6692	−0.833987	−0.416993	−	0.908910i	$-0.636916\pi$
$353$	−15.6692	−0.833987	−0.416993	+	0.908910i	$0.636916\pi$
$354$	−20.9620	−1.11412
$355$	−5.51693	−0.292808
$356$	−23.8163	−1.26226
$357$	0	0
$358$	15.5837	0.823622
$359$	9.97457	0.526438	0.263219	−	0.964736i	$-0.415216\pi$
$359$	9.97457	0.526438	0.263219	+	0.964736i	$0.415216\pi$
$360$	−6.32273	−0.333237
$361$	−18.9593	−0.997859
$362$	−17.9716	−0.944567
$363$	−3.74178	−0.196392
$364$	0	0
$365$	1.54528	0.0808836
$366$	30.8077	1.61034
$367$	7.83802	0.409141	0.204571	−	0.978852i	$-0.434420\pi$
$367$	7.83802	0.409141	0.204571	+	0.978852i	$0.434420\pi$
$368$	−2.47734	−0.129140
$369$	−4.03744	−0.210181
$370$	−1.25801	−0.0654007
$371$	0	0
$372$	−6.79014	−0.352052
$373$	−19.2278	−0.995578	−0.497789	−	0.867298i	$-0.665855\pi$
$373$	−19.2278	−0.995578	−0.497789	+	0.867298i	$0.665855\pi$
$374$	29.6311	1.53219
$375$	4.58239	0.236634
$376$	−21.0666	−1.08643
$377$	−2.18301	−0.112431
$378$	0	0
$379$	34.6924	1.78203	0.891014	−	0.453976i	$-0.149995\pi$
$379$	34.6924	1.78203	0.891014	+	0.453976i	$0.149995\pi$
$380$	−0.445088	−0.0228325
$381$	0.897479	0.0459792
$382$	−57.3126	−2.93237
$383$	11.6275	0.594137	0.297069	−	0.954856i	$-0.403991\pi$
$383$	11.6275	0.594137	0.297069	+	0.954856i	$0.403991\pi$
$384$	15.4933	0.790642
$385$	0	0
$386$	−33.3995	−1.69999
$387$	13.9026	0.706711
$388$	28.6968	1.45686
$389$	6.71525	0.340477	0.170238	−	0.985403i	$-0.445546\pi$
$389$	6.71525	0.340477	0.170238	+	0.985403i	$0.445546\pi$
$390$	1.70404	0.0862872
$391$	2.14380	0.108417
$392$	0	0
$393$	0.00960646	0.000484582 0
$394$	−10.3540	−0.521625
$395$	1.85794	0.0934833
$396$	−24.1076	−1.21145
$397$	18.5309	0.930038	0.465019	−	0.885301i	$-0.346047\pi$
$397$	18.5309	0.930038	0.465019	+	0.885301i	$0.346047\pi$
$398$	11.8438	0.593675
$399$	0	0
$400$	−24.8512	−1.24256
$401$	3.91734	0.195622	0.0978112	−	0.995205i	$-0.468816\pi$
$401$	3.91734	0.195622	0.0978112	+	0.995205i	$0.468816\pi$
$402$	−29.9103	−1.49179
$403$	−2.61346	−0.130186
$404$	29.3818	1.46180
$405$	−1.25992	−0.0626057
$406$	0	0
$407$	−2.51422	−0.124625
$408$	22.4173	1.10982
$409$	−23.9698	−1.18523	−0.592614	−	0.805487i	$-0.701904\pi$
$409$	−23.9698	−1.18523	−0.592614	+	0.805487i	$0.701904\pi$
$410$	2.40591	0.118820
$411$	−2.22196	−0.109601
$412$	16.8468	0.829982
$413$	0	0
$414$	−2.57411	−0.126510
$415$	−9.16431	−0.449858
$416$	−3.08796	−0.151400
$417$	−5.15330	−0.252358
$418$	−1.31281	−0.0642117
$419$	16.6001	0.810970	0.405485	−	0.914102i	$-0.367103\pi$
$419$	16.6001	0.810970	0.405485	+	0.914102i	$0.367103\pi$
$420$	0	0
$421$	24.9623	1.21659	0.608294	−	0.793712i	$-0.291854\pi$
$421$	24.9623	1.21659	0.608294	+	0.793712i	$0.291854\pi$
$422$	−69.0465	−3.36113
$423$	−8.42525	−0.409650
$424$	−16.8757	−0.819555
$425$	21.5052	1.04316
$426$	−23.4866	−1.13793
$427$	0	0
$428$	−25.6972	−1.24212
$429$	3.40564	0.164426
$430$	−8.28460	−0.399519
$431$	−23.7386	−1.14345	−0.571723	−	0.820447i	$-0.693725\pi$
$431$	−23.7386	−1.14345	−0.571723	+	0.820447i	$0.693725\pi$
$432$	−24.5233	−1.17988
$433$	26.6967	1.28296	0.641481	−	0.767139i	$-0.278320\pi$
$433$	26.6967	1.28296	0.641481	+	0.767139i	$0.278320\pi$
$434$	0	0
$435$	0.708517	0.0339708
$436$	−18.9069	−0.905478
$437$	−0.0949813	−0.00454357
$438$	6.57852	0.314334
$439$	0.762459	0.0363902	0.0181951	−	0.999834i	$-0.494208\pi$
$439$	0.762459	0.0363902	0.0181951	+	0.999834i	$0.494208\pi$
$440$	7.53006	0.358982
$441$	0	0
$442$	16.4608	0.782961
$443$	−12.0862	−0.574235	−0.287117	−	0.957895i	$-0.592697\pi$
$443$	−12.0862	−0.574235	−0.287117	+	0.957895i	$0.592697\pi$
$444$	−3.62884	−0.172217
$445$	2.97513	0.141035
$446$	−48.7725	−2.30944
$447$	5.34453	0.252788
$448$	0	0
$449$	−1.21186	−0.0571911	−0.0285955	−	0.999591i	$-0.509103\pi$
$449$	−1.21186	−0.0571911	−0.0285955	+	0.999591i	$0.509103\pi$
$450$	−25.8218	−1.21725
$451$	4.80839	0.226418
$452$	−31.4092	−1.47736
$453$	14.1712	0.665823
$454$	−23.6954	−1.11208
$455$	0	0
$456$	−0.993203	−0.0465110
$457$	−26.9089	−1.25874	−0.629372	−	0.777104i	$-0.716688\pi$
$457$	−26.9089	−1.25874	−0.629372	+	0.777104i	$0.716688\pi$
$458$	−19.8305	−0.926617
$459$	21.2215	0.990536
$460$	1.03935	0.0484602
$461$	−5.84995	−0.272459	−0.136230	−	0.990677i	$-0.543499\pi$
$461$	−5.84995	−0.272459	−0.136230	+	0.990677i	$0.543499\pi$
$462$	0	0
$463$	6.82196	0.317043	0.158522	−	0.987355i	$-0.449327\pi$
$463$	6.82196	0.317043	0.158522	+	0.987355i	$0.449327\pi$
$464$	−7.90906	−0.367169
$465$	0.848226	0.0393355
$466$	−6.88188	−0.318797
$467$	32.5375	1.50565	0.752827	−	0.658218i	$-0.228690\pi$
$467$	32.5375	1.50565	0.752827	+	0.658218i	$0.228690\pi$
$468$	−13.3924	−0.619062
$469$	0	0
$470$	5.02062	0.231584
$471$	−5.87514	−0.270712
$472$	−51.4582	−2.36856
$473$	−16.5574	−0.761308
$474$	7.90960	0.363300
$475$	−0.952793	−0.0437171
$476$	0	0
$477$	−6.74915	−0.309022
$478$	72.9207	3.33531
$479$	15.2882	0.698535	0.349268	−	0.937023i	$-0.386430\pi$
$479$	15.2882	0.698535	0.349268	+	0.937023i	$0.386430\pi$
$480$	1.00223	0.0457453
$481$	−1.39671	−0.0636845
$482$	3.58038	0.163082
$483$	0	0
$484$	−17.5239	−0.796540
$485$	−3.58481	−0.162778
$486$	−40.1973	−1.82339
$487$	−23.2820	−1.05501	−0.527504	−	0.849553i	$-0.676872\pi$
$487$	−23.2820	−1.05501	−0.527504	+	0.849553i	$0.676872\pi$
$488$	75.6279	3.42351
$489$	5.65434	0.255698
$490$	0	0
$491$	16.4779	0.743639	0.371820	−	0.928305i	$-0.378734\pi$
$491$	16.4779	0.743639	0.371820	+	0.928305i	$0.378734\pi$
$492$	6.94009	0.312883
$493$	6.84420	0.308247
$494$	−0.729298	−0.0328127
$495$	3.01153	0.135358
$496$	−9.46860	−0.425153
$497$	0	0
$498$	−39.0141	−1.74826
$499$	−25.9745	−1.16278	−0.581390	−	0.813625i	$-0.697491\pi$
$499$	−25.9745	−1.16278	−0.581390	+	0.813625i	$0.697491\pi$
$500$	21.4607	0.959753
$501$	14.8007	0.661247
$502$	23.0731	1.02980
$503$	−22.6548	−1.01013	−0.505063	−	0.863082i	$-0.668531\pi$
$503$	−22.6548	−1.01013	−0.505063	+	0.863082i	$0.668531\pi$
$504$	0	0
$505$	−3.67038	−0.163330
$506$	3.06563	0.136284
$507$	−9.77531	−0.434137
$508$	4.20317	0.186485
$509$	21.1127	0.935803	0.467901	−	0.883781i	$-0.345010\pi$
$509$	21.1127	0.935803	0.467901	+	0.883781i	$0.345010\pi$
$510$	−5.34252	−0.236571
$511$	0	0
$512$	46.5410	2.05684
$513$	−0.940222	−0.0415118
$514$	26.6390	1.17500
$515$	−2.10451	−0.0927356
$516$	−23.8977	−1.05204
$517$	10.0341	0.441297
$518$	0	0
$519$	19.0230	0.835015
$520$	4.18313	0.183443
$521$	11.8646	0.519798	0.259899	−	0.965636i	$-0.416311\pi$
$521$	11.8646	0.519798	0.259899	+	0.965636i	$0.416311\pi$
$522$	−8.21798	−0.359691
$523$	39.4678	1.72581	0.862903	−	0.505370i	$-0.168644\pi$
$523$	39.4678	1.72581	0.862903	+	0.505370i	$0.168644\pi$
$524$	0.0449899	0.00196539
$525$	0	0
$526$	53.7766	2.34477
$527$	8.19377	0.356926
$528$	12.3387	0.536971
$529$	−22.7782	−0.990357
$530$	4.02183	0.174697
$531$	−20.5799	−0.893090
$532$	0	0
$533$	2.67118	0.115702
$534$	12.6657	0.548097
$535$	3.21010	0.138785
$536$	−73.4249	−3.17147
$537$	−5.61548	−0.242326
$538$	15.3217	0.660566
$539$	0	0
$540$	10.2886	0.442751
$541$	−18.5264	−0.796511	−0.398255	−	0.917275i	$-0.630384\pi$
$541$	−18.5264	−0.796511	−0.398255	+	0.917275i	$0.630384\pi$
$542$	0.589758	0.0253323
$543$	6.47597	0.277910
$544$	9.68143	0.415088
$545$	2.36186	0.101171
$546$	0	0
$547$	19.8599	0.849148	0.424574	−	0.905393i	$-0.360424\pi$
$547$	19.8599	0.849148	0.424574	+	0.905393i	$0.360424\pi$
$548$	−10.4061	−0.444527
$549$	30.2461	1.29087
$550$	30.7525	1.31129
$551$	−0.303233	−0.0129182
$552$	2.31930	0.0987158
$553$	0	0
$554$	−16.1334	−0.685444
$555$	0.453316	0.0192422
$556$	−24.1345	−1.02353
$557$	−16.2691	−0.689343	−0.344672	−	0.938723i	$-0.612010\pi$
$557$	−16.2691	−0.689343	−0.344672	+	0.938723i	$0.612010\pi$
$558$	−9.83844	−0.416495
$559$	−9.19801	−0.389035
$560$	0	0
$561$	−10.6774	−0.450800
$562$	−20.9077	−0.881940
$563$	31.9205	1.34529	0.672645	−	0.739965i	$-0.265158\pi$
$563$	31.9205	1.34529	0.672645	+	0.739965i	$0.265158\pi$
$564$	14.4824	0.609821
$565$	3.92364	0.165069
$566$	57.9769	2.43695
$567$	0	0
$568$	−57.6557	−2.41918
$569$	−18.6740	−0.782857	−0.391428	−	0.920209i	$-0.628019\pi$
$569$	−18.6740	−0.782857	−0.391428	+	0.920209i	$0.628019\pi$
$570$	0.236701	0.00991431
$571$	30.6429	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$571$	30.6429	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$572$	15.9496	0.666887
$573$	20.6523	0.862761
$574$	0	0
$575$	2.22493	0.0927860
$576$	11.4629	0.477623
$577$	−13.7392	−0.571969	−0.285984	−	0.958234i	$-0.592321\pi$
$577$	−13.7392	−0.571969	−0.285984	+	0.958234i	$0.592321\pi$
$578$	−9.26772	−0.385487
$579$	12.0353	0.500170
$580$	3.31820	0.137781
$581$	0	0
$582$	−15.2612	−0.632595
$583$	8.03790	0.332896
$584$	16.1492	0.668259
$585$	1.67298	0.0691691
$586$	−14.5508	−0.601089
$587$	43.1552	1.78121	0.890603	−	0.454782i	$-0.150283\pi$
$587$	43.1552	1.78121	0.890603	+	0.454782i	$0.150283\pi$
$588$	0	0
$589$	−0.363026	−0.0149582
$590$	12.2636	0.504883
$591$	3.73099	0.153472
$592$	−5.06029	−0.207977
$593$	37.8643	1.55490	0.777450	−	0.628945i	$-0.216513\pi$
$593$	37.8643	1.55490	0.777450	+	0.628945i	$0.216513\pi$
$594$	30.3468	1.24514
$595$	0	0
$596$	25.0301	1.02527
$597$	−4.26784	−0.174671
$598$	1.70303	0.0696422
$599$	43.4643	1.77590	0.887952	−	0.459936i	$-0.152128\pi$
$599$	43.4643	1.77590	0.887952	+	0.459936i	$0.152128\pi$
$600$	23.2657	0.949819
$601$	−6.30594	−0.257225	−0.128612	−	0.991695i	$-0.541052\pi$
$601$	−6.30594	−0.257225	−0.128612	+	0.991695i	$0.541052\pi$
$602$	0	0
$603$	−29.3651	−1.19584
$604$	66.3682	2.70049
$605$	2.18909	0.0889991
$606$	−15.6255	−0.634741
$607$	34.6545	1.40658	0.703291	−	0.710902i	$-0.251713\pi$
$607$	34.6545	1.40658	0.703291	+	0.710902i	$0.251713\pi$
$608$	−0.428937	−0.0173957
$609$	0	0
$610$	−18.0237	−0.729759
$611$	5.57416	0.225506
$612$	41.9879	1.69726
$613$	−20.4108	−0.824385	−0.412192	−	0.911097i	$-0.635237\pi$
$613$	−20.4108	−0.824385	−0.412192	+	0.911097i	$0.635237\pi$
$614$	−15.9250	−0.642682
$615$	−0.866957	−0.0349591
$616$	0	0
$617$	−7.50281	−0.302052	−0.151026	−	0.988530i	$-0.548258\pi$
$617$	−7.50281	−0.302052	−0.151026	+	0.988530i	$0.548258\pi$
$618$	−8.95926	−0.360394
$619$	17.1606	0.689741	0.344871	−	0.938650i	$-0.387923\pi$
$619$	17.1606	0.689741	0.344871	+	0.938650i	$0.387923\pi$
$620$	3.97250	0.159539
$621$	2.19558	0.0881055
$622$	−38.4305	−1.54092
$623$	0	0
$624$	6.85442	0.274397
$625$	20.9407	0.837627
$626$	49.1535	1.96457
$627$	0.473064	0.0188923
$628$	−27.5150	−1.09797
$629$	4.37899	0.174602
$630$	0	0
$631$	8.55084	0.340403	0.170202	−	0.985409i	$-0.445558\pi$
$631$	8.55084	0.340403	0.170202	+	0.985409i	$0.445558\pi$
$632$	19.4168	0.772358
$633$	24.8805	0.988912
$634$	29.5155	1.17221
$635$	−0.525060	−0.0208364
$636$	11.6013	0.460023
$637$	0	0
$638$	9.78720	0.387479
$639$	−23.0585	−0.912178
$640$	−9.06422	−0.358295
$641$	22.7012	0.896644	0.448322	−	0.893872i	$-0.352022\pi$
$641$	22.7012	0.896644	0.448322	+	0.893872i	$0.352022\pi$
$642$	13.6660	0.539352
$643$	10.9385	0.431374	0.215687	−	0.976463i	$-0.430801\pi$
$643$	10.9385	0.431374	0.215687	+	0.976463i	$0.430801\pi$
$644$	0	0
$645$	2.98531	0.117546
$646$	2.28651	0.0899614
$647$	−2.75442	−0.108287	−0.0541436	−	0.998533i	$-0.517243\pi$
$647$	−2.75442	−0.108287	−0.0541436	+	0.998533i	$0.517243\pi$
$648$	−13.1670	−0.517248
$649$	24.5096	0.962086
$650$	17.0838	0.670081
$651$	0	0
$652$	26.4810	1.03708
$653$	−35.0485	−1.37156	−0.685778	−	0.727811i	$-0.740538\pi$
$653$	−35.0485	−1.37156	−0.685778	+	0.727811i	$0.740538\pi$
$654$	10.0549	0.393176
$655$	−0.00562015	−0.000219598 0
$656$	9.67770	0.377851
$657$	6.45861	0.251975
$658$	0	0
$659$	−44.9909	−1.75260	−0.876299	−	0.481767i	$-0.839995\pi$
$659$	−44.9909	−1.75260	−0.876299	+	0.481767i	$0.839995\pi$
$660$	−5.17661	−0.201499
$661$	14.8057	0.575876	0.287938	−	0.957649i	$-0.407030\pi$
$661$	14.8057	0.575876	0.287938	+	0.957649i	$0.407030\pi$
$662$	24.2410	0.942154
$663$	−5.93156	−0.230363
$664$	−95.7732	−3.71672
$665$	0	0
$666$	−5.25795	−0.203741
$667$	0.708100	0.0274177
$668$	69.3162	2.68192
$669$	17.5749	0.679484
$670$	17.4987	0.676034
$671$	−36.0216	−1.39060
$672$	0	0
$673$	−30.9268	−1.19214	−0.596070	−	0.802932i	$-0.703272\pi$
$673$	−30.9268	−1.19214	−0.596070	+	0.802932i	$0.703272\pi$
$674$	−69.3822	−2.67250
$675$	22.0247	0.847730
$676$	−45.7807	−1.76080
$677$	51.3587	1.97388	0.986938	−	0.161103i	$-0.0515051\pi$
$677$	51.3587	1.97388	0.986938	+	0.161103i	$0.0515051\pi$
$678$	16.7036	0.641499
$679$	0	0
$680$	−13.1150	−0.502938
$681$	8.53852	0.327197
$682$	11.7171	0.448671
$683$	−14.7793	−0.565515	−0.282757	−	0.959191i	$-0.591249\pi$
$683$	−14.7793	−0.565515	−0.282757	+	0.959191i	$0.591249\pi$
$684$	−1.86028	−0.0711296
$685$	1.29993	0.0496679
$686$	0	0
$687$	7.14579	0.272629
$688$	−33.3245	−1.27048
$689$	4.46525	0.170113
$690$	−0.552737	−0.0210423
$691$	−32.6442	−1.24184	−0.620922	−	0.783873i	$-0.713242\pi$
$691$	−32.6442	−1.24184	−0.620922	+	0.783873i	$0.713242\pi$
$692$	89.0903	3.38670
$693$	0	0
$694$	−39.1519	−1.48619
$695$	3.01488	0.114361
$696$	7.40448	0.280666
$697$	−8.37472	−0.317215
$698$	63.7831	2.41422
$699$	2.47984	0.0937963
$700$	0	0
$701$	−28.6064	−1.08045	−0.540224	−	0.841521i	$-0.681661\pi$
$701$	−28.6064	−1.08045	−0.540224	+	0.841521i	$0.681661\pi$
$702$	16.8584	0.636279
$703$	−0.194011	−0.00731728
$704$	−13.6518	−0.514522
$705$	−1.80915	−0.0681365
$706$	39.0259	1.46876
$707$	0	0
$708$	35.3754	1.32949
$709$	−4.01954	−0.150957	−0.0754785	−	0.997147i	$-0.524048\pi$
$709$	−4.01954	−0.150957	−0.0754785	+	0.997147i	$0.524048\pi$
$710$	13.7406	0.515674
$711$	7.76543	0.291226
$712$	31.0922	1.16523
$713$	0.847727	0.0317476
$714$	0	0
$715$	−1.99243	−0.0745127
$716$	−26.2990	−0.982840
$717$	−26.2766	−0.981316
$718$	−24.8428	−0.927126
$719$	2.37429	0.0885462	0.0442731	−	0.999019i	$-0.485903\pi$
$719$	2.37429	0.0885462	0.0442731	+	0.999019i	$0.485903\pi$
$720$	6.06121	0.225888
$721$	0	0
$722$	47.2204	1.75736
$723$	−1.29017	−0.0479819
$724$	30.3289	1.12717
$725$	7.10322	0.263807
$726$	9.31933	0.345873
$727$	24.6101	0.912737	0.456368	−	0.889791i	$-0.349150\pi$
$727$	24.6101	0.912737	0.456368	+	0.889791i	$0.349150\pi$
$728$	0	0
$729$	7.28618	0.269858
$730$	−3.84869	−0.142447
$731$	28.8378	1.06660
$732$	−51.9911	−1.92165
$733$	19.0043	0.701939	0.350970	−	0.936387i	$-0.385852\pi$
$733$	19.0043	0.701939	0.350970	+	0.936387i	$0.385852\pi$
$734$	−19.5215	−0.720552
$735$	0	0
$736$	1.00164	0.0369209
$737$	34.9724	1.28822
$738$	10.0557	0.370156
$739$	35.1007	1.29120	0.645600	−	0.763676i	$-0.276607\pi$
$739$	35.1007	1.29120	0.645600	+	0.763676i	$0.276607\pi$
$740$	2.12302	0.0780436
$741$	0.262798	0.00965414
$742$	0	0
$743$	21.0754	0.773180	0.386590	−	0.922252i	$-0.373653\pi$
$743$	21.0754	0.773180	0.386590	+	0.922252i	$0.373653\pi$
$744$	8.86454	0.324990
$745$	−3.12676	−0.114556
$746$	47.8891	1.75334
$747$	−38.3029	−1.40143
$748$	−50.0055	−1.82838
$749$	0	0
$750$	−11.4130	−0.416743
$751$	18.0271	0.657817	0.328908	−	0.944362i	$-0.393319\pi$
$751$	18.0271	0.657817	0.328908	+	0.944362i	$0.393319\pi$
$752$	20.1952	0.736445
$753$	−8.31425	−0.302988
$754$	5.43703	0.198005
$755$	−8.29074	−0.301731
$756$	0	0
$757$	4.30504	0.156469	0.0782347	−	0.996935i	$-0.475072\pi$
$757$	4.30504	0.156469	0.0782347	+	0.996935i	$0.475072\pi$
$758$	−86.4054	−3.13839
$759$	−1.10468	−0.0400975
$760$	0.581063	0.0210774
$761$	17.6944	0.641422	0.320711	−	0.947177i	$-0.396078\pi$
$761$	17.6944	0.641422	0.320711	+	0.947177i	$0.396078\pi$
$762$	−2.23528	−0.0809755
$763$	0	0
$764$	96.7208	3.49924
$765$	−5.24514	−0.189638
$766$	−28.9596	−1.04635
$767$	13.6157	0.491634
$768$	−29.2122	−1.05410
$769$	30.3350	1.09391	0.546954	−	0.837163i	$-0.315787\pi$
$769$	30.3350	1.09391	0.546954	+	0.837163i	$0.315787\pi$
$770$	0	0
$771$	−9.59921	−0.345707
$772$	56.3650	2.02862
$773$	−38.1914	−1.37365	−0.686825	−	0.726823i	$-0.740996\pi$
$773$	−38.1914	−1.37365	−0.686825	+	0.726823i	$0.740996\pi$
$774$	−34.6261	−1.24461
$775$	8.50387	0.305468
$776$	−37.4637	−1.34487
$777$	0	0
$778$	−16.7251	−0.599624
$779$	0.371043	0.0132940
$780$	−2.87573	−0.102968
$781$	27.4615	0.982649
$782$	−5.33938	−0.190936
$783$	7.00950	0.250499
$784$	0	0
$785$	3.43719	0.122678
$786$	−0.0239260	−0.000853412 0
$787$	23.0664	0.822230	0.411115	−	0.911584i	$-0.365140\pi$
$787$	23.0664	0.822230	0.411115	+	0.911584i	$0.365140\pi$
$788$	17.4734	0.622462
$789$	−19.3781	−0.689879
$790$	−4.62743	−0.164636
$791$	0	0
$792$	31.4725	1.11833
$793$	−20.0109	−0.710608
$794$	−46.1533	−1.63792
$795$	−1.44924	−0.0513993
$796$	−19.9876	−0.708441
$797$	−40.2087	−1.42427	−0.712133	−	0.702044i	$-0.752271\pi$
$797$	−40.2087	−1.42427	−0.712133	+	0.702044i	$0.752271\pi$
$798$	0	0
$799$	−17.4762	−0.618263
$800$	10.0478	0.355244
$801$	12.4348	0.439362
$802$	−9.75658	−0.344517
$803$	−7.69189	−0.271441
$804$	50.4766	1.78017
$805$	0	0
$806$	6.50913	0.229274
$807$	−5.52110	−0.194352
$808$	−38.3580	−1.34943
$809$	10.1096	0.355434	0.177717	−	0.984082i	$-0.443129\pi$
$809$	10.1096	0.355434	0.177717	+	0.984082i	$0.443129\pi$
$810$	3.13797	0.110257
$811$	30.6004	1.07452	0.537262	−	0.843416i	$-0.319459\pi$
$811$	30.6004	1.07452	0.537262	+	0.843416i	$0.319459\pi$
$812$	0	0
$813$	−0.212516	−0.00745326
$814$	6.26195	0.219481
$815$	−3.30801	−0.115875
$816$	−21.4901	−0.752304
$817$	−1.27766	−0.0446997
$818$	59.6994	2.08734
$819$	0	0
$820$	−4.06022	−0.141789
$821$	6.80440	0.237475	0.118738	−	0.992926i	$-0.462115\pi$
$821$	6.80440	0.237475	0.118738	+	0.992926i	$0.462115\pi$
$822$	5.53405	0.193022
$823$	37.1405	1.29464	0.647318	−	0.762220i	$-0.275891\pi$
$823$	37.1405	1.29464	0.647318	+	0.762220i	$0.275891\pi$
$824$	−21.9935	−0.766181
$825$	−11.0815	−0.385808
$826$	0	0
$827$	24.5185	0.852591	0.426296	−	0.904584i	$-0.359818\pi$
$827$	24.5185	0.852591	0.426296	+	0.904584i	$0.359818\pi$
$828$	4.34407	0.150967
$829$	−44.0647	−1.53043	−0.765215	−	0.643775i	$-0.777367\pi$
$829$	−44.0647	−1.53043	−0.765215	+	0.643775i	$0.777367\pi$
$830$	22.8248	0.792259
$831$	5.81359	0.201671
$832$	−7.58391	−0.262925
$833$	0	0
$834$	12.8349	0.444436
$835$	−8.65900	−0.299657
$836$	2.21550	0.0766247
$837$	8.39167	0.290059
$838$	−41.3446	−1.42822
$839$	4.05620	0.140036	0.0700178	−	0.997546i	$-0.477694\pi$
$839$	4.05620	0.140036	0.0700178	+	0.997546i	$0.477694\pi$
$840$	0	0
$841$	−26.7394	−0.922047
$842$	−62.1715	−2.14257
$843$	7.53399	0.259484
$844$	116.523	4.01089
$845$	5.71894	0.196738
$846$	20.9841	0.721447
$847$	0	0
$848$	16.1776	0.555543
$849$	−20.8916	−0.716999
$850$	−53.5613	−1.83714
$851$	0.453049	0.0155303
$852$	39.6359	1.35790
$853$	50.0794	1.71469	0.857343	−	0.514745i	$-0.172113\pi$
$853$	50.0794	1.71469	0.857343	+	0.514745i	$0.172113\pi$
$854$	0	0
$855$	0.232387	0.00794745
$856$	33.5477	1.14664
$857$	13.2766	0.453520	0.226760	−	0.973951i	$-0.427187\pi$
$857$	13.2766	0.453520	0.226760	+	0.973951i	$0.427187\pi$
$858$	−8.48213	−0.289575
$859$	30.0133	1.02404	0.512021	−	0.858973i	$-0.328897\pi$
$859$	30.0133	1.02404	0.512021	+	0.858973i	$0.328897\pi$
$860$	13.9811	0.476751
$861$	0	0
$862$	59.1236	2.01376
$863$	45.0695	1.53418	0.767091	−	0.641538i	$-0.221703\pi$
$863$	45.0695	1.53418	0.767091	+	0.641538i	$0.221703\pi$
$864$	9.91526	0.337324
$865$	−11.1292	−0.378403
$866$	−66.4913	−2.25947
$867$	3.33957	0.113418
$868$	0	0
$869$	−9.24824	−0.313725
$870$	−1.76464	−0.0598270
$871$	19.4280	0.658293
$872$	24.6830	0.835873
$873$	−14.9830	−0.507097
$874$	0.236562	0.00800182
$875$	0	0
$876$	−11.1019	−0.375099
$877$	−28.9548	−0.977734	−0.488867	−	0.872358i	$-0.662590\pi$
$877$	−28.9548	−0.977734	−0.488867	+	0.872358i	$0.662590\pi$
$878$	−1.89899	−0.0640878
$879$	5.24330	0.176852
$880$	−7.21860	−0.243339
$881$	43.4223	1.46294	0.731468	−	0.681876i	$-0.238836\pi$
$881$	43.4223	1.46294	0.731468	+	0.681876i	$0.238836\pi$
$882$	0	0
$883$	28.5142	0.959578	0.479789	−	0.877384i	$-0.340713\pi$
$883$	28.5142	0.959578	0.479789	+	0.877384i	$0.340713\pi$
$884$	−27.7793	−0.934319
$885$	−4.41911	−0.148547
$886$	30.1022	1.01130
$887$	48.4818	1.62786	0.813929	−	0.580964i	$-0.197324\pi$
$887$	48.4818	1.62786	0.813929	+	0.580964i	$0.197324\pi$
$888$	4.73746	0.158979
$889$	0	0
$890$	−7.40991	−0.248381
$891$	6.27144	0.210101
$892$	82.3085	2.75589
$893$	0.774285	0.0259105
$894$	−13.3112	−0.445192
$895$	3.28528	0.109815
$896$	0	0
$897$	−0.613679	−0.0204901
$898$	3.01827	0.100721
$899$	2.70641	0.0902640
$900$	43.5770	1.45257
$901$	−13.9995	−0.466392
$902$	−11.9758	−0.398752
$903$	0	0
$904$	41.0047	1.36380
$905$	−3.78869	−0.125940
$906$	−35.2951	−1.17260
$907$	−46.8426	−1.55538	−0.777692	−	0.628646i	$-0.783609\pi$
$907$	−46.8426	−1.55538	−0.777692	+	0.628646i	$0.783609\pi$
$908$	39.9885	1.32706
$909$	−15.3406	−0.508817
$910$	0	0
$911$	10.1028	0.334722	0.167361	−	0.985896i	$-0.446476\pi$
$911$	10.1028	0.334722	0.167361	+	0.985896i	$0.446476\pi$
$912$	0.952121	0.0315279
$913$	45.6169	1.50970
$914$	67.0197	2.21681
$915$	6.49474	0.214709
$916$	33.4659	1.10575
$917$	0	0
$918$	−52.8547	−1.74446
$919$	−16.0034	−0.527904	−0.263952	−	0.964536i	$-0.585026\pi$
$919$	−16.0034	−0.527904	−0.263952	+	0.964536i	$0.585026\pi$
$920$	−1.35688	−0.0447350
$921$	5.73850	0.189090
$922$	14.5700	0.479837
$923$	15.2555	0.502142
$924$	0	0
$925$	4.54471	0.149429
$926$	−16.9909	−0.558355
$927$	−8.79595	−0.288897
$928$	3.19779	0.104973
$929$	18.3956	0.603539	0.301769	−	0.953381i	$-0.402423\pi$
$929$	18.3956	0.603539	0.301769	+	0.953381i	$0.402423\pi$
$930$	−2.11260	−0.0692750
$931$	0	0
$932$	11.6139	0.380425
$933$	13.8482	0.453370
$934$	−81.0383	−2.65165
$935$	6.24670	0.204289
$936$	17.4837	0.571474
$937$	−24.7977	−0.810105	−0.405053	−	0.914293i	$-0.632747\pi$
$937$	−24.7977	−0.810105	−0.405053	+	0.914293i	$0.632747\pi$
$938$	0	0
$939$	−17.7122	−0.578016
$940$	−8.47280	−0.276352
$941$	45.0547	1.46874	0.734371	−	0.678748i	$-0.237477\pi$
$941$	45.0547	1.46874	0.734371	+	0.678748i	$0.237477\pi$
$942$	14.6327	0.476759
$943$	−0.866447	−0.0282154
$944$	49.3298	1.60555
$945$	0	0
$946$	41.2380	1.34076
$947$	50.2871	1.63411	0.817056	−	0.576559i	$-0.195605\pi$
$947$	50.2871	1.63411	0.817056	+	0.576559i	$0.195605\pi$
$948$	−13.3482	−0.433531
$949$	−4.27303	−0.138708
$950$	2.37304	0.0769916
$951$	−10.6357	−0.344888
$952$	0	0
$953$	22.8566	0.740398	0.370199	−	0.928952i	$-0.379289\pi$
$953$	22.8566	0.740398	0.370199	+	0.928952i	$0.379289\pi$
$954$	16.8095	0.544229
$955$	−12.0824	−0.390977
$956$	−123.061	−3.98008
$957$	−3.52676	−0.114004
$958$	−38.0770	−1.23021
$959$	0	0
$960$	2.46143	0.0794424
$961$	−27.7599	−0.895481
$962$	3.47867	0.112157
$963$	13.4169	0.432352
$964$	−6.04225	−0.194608
$965$	−7.04113	−0.226662
$966$	0	0
$967$	−32.5270	−1.04600	−0.522998	−	0.852334i	$-0.675187\pi$
$967$	−32.5270	−1.04600	−0.522998	+	0.852334i	$0.675187\pi$
$968$	22.8775	0.735309
$969$	−0.823930	−0.0264684
$970$	8.92837	0.286673
$971$	−55.6335	−1.78536	−0.892682	−	0.450687i	$-0.851179\pi$
$971$	−55.6335	−1.78536	−0.892682	+	0.450687i	$0.851179\pi$
$972$	67.8370	2.17587
$973$	0	0
$974$	57.9865	1.85801
$975$	−6.15604	−0.197151
$976$	−72.4997	−2.32066
$977$	−46.5136	−1.48810	−0.744051	−	0.668123i	$-0.767098\pi$
$977$	−46.5136	−1.48810	−0.744051	+	0.668123i	$0.767098\pi$
$978$	−14.0828	−0.450318
$979$	−14.8092	−0.473305
$980$	0	0
$981$	9.87158	0.315175
$982$	−41.0402	−1.30965
$983$	22.7831	0.726669	0.363335	−	0.931659i	$-0.381638\pi$
$983$	22.7831	0.726669	0.363335	+	0.931659i	$0.381638\pi$
$984$	−9.06029	−0.288832
$985$	−2.18278	−0.0695490
$986$	−17.0463	−0.542864
$987$	0	0
$988$	1.23076	0.0391558
$989$	2.98355	0.0948714
$990$	−7.50055	−0.238383
$991$	25.9446	0.824158	0.412079	−	0.911148i	$-0.364803\pi$
$991$	25.9446	0.824158	0.412079	+	0.911148i	$0.364803\pi$
$992$	3.82835	0.121550
$993$	−8.73512	−0.277201
$994$	0	0
$995$	2.49685	0.0791556
$996$	65.8402	2.08623
$997$	40.6587	1.28767	0.643837	−	0.765163i	$-0.277342\pi$
$997$	40.6587	1.28767	0.643837	+	0.765163i	$0.277342\pi$
$998$	64.6925	2.04781
$999$	4.48475	0.141891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.p.1.3		38
7.2	even	3		889.2.f.c.382.36	yes	76
7.4	even	3		889.2.f.c.128.36	✓	76
7.6	odd	2		6223.2.a.o.1.3		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.f.c.128.36	✓	76	7.4	even	3
889.2.f.c.382.36	yes	76	7.2	even	3
6223.2.a.o.1.3		38	7.6	odd	2
6223.2.a.p.1.3		38	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)

\(f(q)\)	\(=\)	\(q-2.49062 q^{2} +0.897479 q^{3} +4.20317 q^{4} -0.525060 q^{5} -2.23528 q^{6} -5.48724 q^{8} -2.19453 q^{9} +1.30772 q^{10} +2.61358 q^{11} +3.77225 q^{12} +1.45191 q^{13} -0.471231 q^{15} +5.26027 q^{16} -4.55204 q^{17} +5.46573 q^{18} +0.201679 q^{19} -2.20692 q^{20} -6.50942 q^{22} -0.470953 q^{23} -4.92468 q^{24} -4.72431 q^{25} -3.61614 q^{26} -4.66198 q^{27} -1.50355 q^{29} +1.17365 q^{30} -1.80002 q^{31} -2.12683 q^{32} +2.34563 q^{33} +11.3374 q^{34} -9.22398 q^{36} -0.961983 q^{37} -0.502304 q^{38} +1.30306 q^{39} +2.88113 q^{40} +1.83977 q^{41} -6.33513 q^{43} +10.9853 q^{44} +1.15226 q^{45} +1.17296 q^{46} +3.83920 q^{47} +4.72098 q^{48} +11.7664 q^{50} -4.08536 q^{51} +6.10260 q^{52} +3.07544 q^{53} +11.6112 q^{54} -1.37229 q^{55} +0.181002 q^{57} +3.74475 q^{58} +9.37780 q^{59} -1.98066 q^{60} -13.7825 q^{61} +4.48316 q^{62} -5.22342 q^{64} -0.762339 q^{65} -5.84207 q^{66} +13.3810 q^{67} -19.1330 q^{68} -0.422671 q^{69} +10.5072 q^{71} +12.0419 q^{72} -2.94305 q^{73} +2.39593 q^{74} -4.23997 q^{75} +0.847689 q^{76} -3.24541 q^{78} -3.53853 q^{79} -2.76196 q^{80} +2.39956 q^{81} -4.58217 q^{82} +17.4538 q^{83} +2.39010 q^{85} +15.7784 q^{86} -1.34940 q^{87} -14.3413 q^{88} -5.66627 q^{89} -2.86984 q^{90} -1.97950 q^{92} -1.61548 q^{93} -9.56198 q^{94} -0.105893 q^{95} -1.90879 q^{96} +6.82742 q^{97} -5.73558 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 + 11 * q^3 + 38 * q^4 + 16 * q^5 + 11 * q^6 + 47 * q^9 + 12 * q^10 - 2 * q^11 + 30 * q^12 + 21 * q^13 + 7 * q^15 + 46 * q^16 + 58 * q^17 - 13 * q^18 + 17 * q^19 + 44 * q^20 + 21 * q^22 + 7 * q^23 + 21 * q^24 + 52 * q^25 + 49 * q^26 + 35 * q^27 - 2 * q^29 + 6 * q^30 + 2 * q^31 - 22 * q^32 + 29 * q^33 + 37 * q^34 + 25 * q^36 + 11 * q^37 + 40 * q^38 + 2 * q^39 - 8 * q^40 + 68 * q^41 - 38 * q^43 + 43 * q^44 + 60 * q^45 - 28 * q^46 + 35 * q^47 + 66 * q^48 - 25 * q^50 - 3 * q^51 + 32 * q^52 - 18 * q^53 - 68 * q^54 + 19 * q^55 - 31 * q^57 + 55 * q^58 + 21 * q^59 - 11 * q^60 + 5 * q^61 + 38 * q^62 + 18 * q^64 - 9 * q^65 + 20 * q^66 + 6 * q^67 + 74 * q^68 + 53 * q^69 - 26 * q^71 + 24 * q^72 + 22 * q^73 - 2 * q^74 + 38 * q^75 + 39 * q^76 - 114 * q^78 - 14 * q^79 + 140 * q^80 + 58 * q^81 - 8 * q^82 + 46 * q^83 - 11 * q^85 + 51 * q^86 + 59 * q^87 - 4 * q^88 + 74 * q^89 + q^90 - 81 * q^92 + 92 * q^93 - 36 * q^94 + 20 * q^95 - 108 * q^96 + 66 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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