LMFDB - Embedded newform 6223.2.a.o.1.7

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:	$1$
Dimension:	$38$
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	\(q-2.08520 q^{2} +2.29235 q^{3} +2.34805 q^{4} +0.933672 q^{5} -4.78000 q^{6} -0.725759 q^{8} +2.25486 q^{9} -1.94689 q^{10} +2.78681 q^{11} +5.38255 q^{12} -0.193825 q^{13} +2.14030 q^{15} -3.18275 q^{16} -2.51616 q^{17} -4.70182 q^{18} -7.64841 q^{19} +2.19231 q^{20} -5.81105 q^{22} -5.15083 q^{23} -1.66369 q^{24} -4.12826 q^{25} +0.404165 q^{26} -1.70813 q^{27} +5.69651 q^{29} -4.46295 q^{30} +0.00490923 q^{31} +8.08819 q^{32} +6.38833 q^{33} +5.24670 q^{34} +5.29452 q^{36} +6.30470 q^{37} +15.9485 q^{38} -0.444315 q^{39} -0.677621 q^{40} +0.548648 q^{41} +1.75331 q^{43} +6.54357 q^{44} +2.10530 q^{45} +10.7405 q^{46} -11.3206 q^{47} -7.29598 q^{48} +8.60823 q^{50} -5.76792 q^{51} -0.455112 q^{52} +12.5271 q^{53} +3.56179 q^{54} +2.60197 q^{55} -17.5328 q^{57} -11.8784 q^{58} -5.48337 q^{59} +5.02554 q^{60} -9.56485 q^{61} -0.0102367 q^{62} -10.5000 q^{64} -0.180970 q^{65} -13.3209 q^{66} +14.4966 q^{67} -5.90809 q^{68} -11.8075 q^{69} -15.3167 q^{71} -1.63648 q^{72} -6.92829 q^{73} -13.1466 q^{74} -9.46340 q^{75} -17.9589 q^{76} +0.926486 q^{78} -9.73689 q^{79} -2.97165 q^{80} -10.6802 q^{81} -1.14404 q^{82} -0.0567325 q^{83} -2.34927 q^{85} -3.65600 q^{86} +13.0584 q^{87} -2.02255 q^{88} +2.59339 q^{89} -4.38996 q^{90} -12.0944 q^{92} +0.0112537 q^{93} +23.6057 q^{94} -7.14111 q^{95} +18.5409 q^{96} -8.05520 q^{97} +6.28385 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.08520	−1.47446	−0.737229	−	0.675643i	$-0.763866\pi$
$2$	−2.08520	−1.47446	−0.737229	+	0.675643i	$0.763866\pi$
$3$	2.29235	1.32349	0.661744	−	0.749730i	$-0.269817\pi$
$3$	2.29235	1.32349	0.661744	+	0.749730i	$0.269817\pi$
$4$	2.34805	1.17403
$5$	0.933672	0.417551	0.208776	−	0.977964i	$-0.433052\pi$
$5$	0.933672	0.417551	0.208776	+	0.977964i	$0.433052\pi$
$6$	−4.78000	−1.95143
$7$	0	0
$7$	0	0
$8$	−0.725759	−0.256594
$9$	2.25486	0.751619
$10$	−1.94689	−0.615661
$11$	2.78681	0.840255	0.420127	−	0.907465i	$-0.361985\pi$
$11$	2.78681	0.840255	0.420127	+	0.907465i	$0.361985\pi$
$12$	5.38255	1.55381
$13$	−0.193825	−0.0537575	−0.0268788	−	0.999639i	$-0.508557\pi$
$13$	−0.193825	−0.0537575	−0.0268788	+	0.999639i	$0.508557\pi$
$14$	0	0
$15$	2.14030	0.552623
$16$	−3.18275	−0.795689
$17$	−2.51616	−0.610259	−0.305130	−	0.952311i	$-0.598700\pi$
$17$	−2.51616	−0.610259	−0.305130	+	0.952311i	$0.598700\pi$
$18$	−4.70182	−1.10823
$19$	−7.64841	−1.75467	−0.877333	−	0.479881i	$-0.840680\pi$
$19$	−7.64841	−1.75467	−0.877333	+	0.479881i	$0.840680\pi$
$20$	2.19231	0.490216
$21$	0	0
$22$	−5.81105	−1.23892
$23$	−5.15083	−1.07402	−0.537012	−	0.843575i	$-0.680447\pi$
$23$	−5.15083	−1.07402	−0.537012	+	0.843575i	$0.680447\pi$
$24$	−1.66369	−0.339599
$25$	−4.12826	−0.825651
$26$	0.404165	0.0792632
$27$	−1.70813	−0.328729
$28$	0	0
$29$	5.69651	1.05782	0.528908	−	0.848679i	$-0.322602\pi$
$29$	5.69651	1.05782	0.528908	+	0.848679i	$0.322602\pi$
$30$	−4.46295	−0.814820
$31$	0.00490923	0.000881724 0	0.000440862	−	1.00000i	$-0.499860\pi$
$31$	0.00490923	0.000881724 0	0.000440862		1.00000i	$0.499860\pi$
$32$	8.08819	1.42980
$33$	6.38833	1.11207
$34$	5.24670	0.899802
$35$	0	0
$36$	5.29452	0.882420
$37$	6.30470	1.03649	0.518244	−	0.855233i	$-0.326586\pi$
$37$	6.30470	1.03649	0.518244	+	0.855233i	$0.326586\pi$
$38$	15.9485	2.58718
$39$	−0.444315	−0.0711474
$40$	−0.677621	−0.107141
$41$	0.548648	0.0856845	0.0428422	−	0.999082i	$-0.486359\pi$
$41$	0.548648	0.0856845	0.0428422	+	0.999082i	$0.486359\pi$
$42$	0	0
$43$	1.75331	0.267377	0.133689	−	0.991023i	$-0.457318\pi$
$43$	1.75331	0.267377	0.133689	+	0.991023i	$0.457318\pi$
$44$	6.54357	0.986481
$45$	2.10530	0.313839
$46$	10.7405	1.58360
$47$	−11.3206	−1.65128	−0.825639	−	0.564199i	$-0.809185\pi$
$47$	−11.3206	−1.65128	−0.825639	+	0.564199i	$0.809185\pi$
$48$	−7.29598	−1.05308
$49$	0	0
$50$	8.60823	1.21739
$51$	−5.76792	−0.807671
$52$	−0.455112	−0.0631127
$53$	12.5271	1.72074	0.860368	−	0.509673i	$-0.170234\pi$
$53$	12.5271	1.72074	0.860368	+	0.509673i	$0.170234\pi$
$54$	3.56179	0.484698
$55$	2.60197	0.350849
$56$	0	0
$57$	−17.5328	−2.32228
$58$	−11.8784	−1.55970
$59$	−5.48337	−0.713874	−0.356937	−	0.934128i	$-0.616179\pi$
$59$	−5.48337	−0.713874	−0.356937	+	0.934128i	$0.616179\pi$
$60$	5.02554	0.648794
$61$	−9.56485	−1.22465	−0.612327	−	0.790605i	$-0.709766\pi$
$61$	−9.56485	−1.22465	−0.612327	+	0.790605i	$0.709766\pi$
$62$	−0.0102367	−0.00130007
$63$	0	0
$64$	−10.5000	−1.31250
$65$	−0.180970	−0.0224465
$66$	−13.3209	−1.63969
$67$	14.4966	1.77104	0.885519	−	0.464604i	$-0.153803\pi$
$67$	14.4966	1.77104	0.885519	+	0.464604i	$0.153803\pi$
$68$	−5.90809	−0.716461
$69$	−11.8075	−1.42146
$70$	0	0
$71$	−15.3167	−1.81776	−0.908879	−	0.417059i	$-0.863061\pi$
$71$	−15.3167	−1.81776	−0.908879	+	0.417059i	$0.863061\pi$
$72$	−1.63648	−0.192861
$73$	−6.92829	−0.810895	−0.405447	−	0.914118i	$-0.632884\pi$
$73$	−6.92829	−0.810895	−0.405447	+	0.914118i	$0.632884\pi$
$74$	−13.1466	−1.52826
$75$	−9.46340	−1.09274
$76$	−17.9589	−2.06002
$77$	0	0
$78$	0.926486	0.104904
$79$	−9.73689	−1.09549	−0.547743	−	0.836647i	$-0.684513\pi$
$79$	−9.73689	−1.09549	−0.547743	+	0.836647i	$0.684513\pi$
$80$	−2.97165	−0.332241
$81$	−10.6802	−1.18669
$82$	−1.14404	−0.126338
$83$	−0.0567325	−0.00622720	−0.00311360	−	0.999995i	$-0.500991\pi$
$83$	−0.0567325	−0.00622720	−0.00311360	+	0.999995i	$0.500991\pi$
$84$	0	0
$85$	−2.34927	−0.254814
$86$	−3.65600	−0.394237
$87$	13.0584	1.40001
$88$	−2.02255	−0.215605
$89$	2.59339	0.274899	0.137449	−	0.990509i	$-0.456110\pi$
$89$	2.59339	0.274899	0.137449	+	0.990509i	$0.456110\pi$
$90$	−4.38996	−0.462743
$91$	0	0
$92$	−12.0944	−1.26093
$93$	0.0112537	0.00116695
$94$	23.6057	2.43474
$95$	−7.14111	−0.732663
$96$	18.5409	1.89233
$97$	−8.05520	−0.817881	−0.408941	−	0.912561i	$-0.634102\pi$
$97$	−8.05520	−0.817881	−0.408941	+	0.912561i	$0.634102\pi$
$98$	0	0
$99$	6.28385	0.631551
$100$	−9.69336	−0.969336
$101$	−6.45591	−0.642387	−0.321194	−	0.947014i	$-0.604084\pi$
$101$	−6.45591	−0.642387	−0.321194	+	0.947014i	$0.604084\pi$
$102$	12.0273	1.19088
$103$	−12.6560	−1.24704	−0.623518	−	0.781809i	$-0.714297\pi$
$103$	−12.6560	−1.24704	−0.623518	+	0.781809i	$0.714297\pi$
$104$	0.140671	0.0137939
$105$	0	0
$106$	−26.1216	−2.53715
$107$	13.9590	1.34946	0.674732	−	0.738063i	$-0.264259\pi$
$107$	13.9590	1.34946	0.674732	+	0.738063i	$0.264259\pi$
$108$	−4.01077	−0.385937
$109$	−16.2293	−1.55448	−0.777241	−	0.629203i	$-0.783382\pi$
$109$	−16.2293	−1.55448	−0.777241	+	0.629203i	$0.783382\pi$
$110$	−5.42562	−0.517312
$111$	14.4526	1.37178
$112$	0	0
$113$	0.0544202	0.00511942	0.00255971	−	0.999997i	$-0.499185\pi$
$113$	0.0544202	0.00511942	0.00255971	+	0.999997i	$0.499185\pi$
$114$	36.5594	3.42410
$115$	−4.80919	−0.448460
$116$	13.3757	1.24190
$117$	−0.437049	−0.0404052
$118$	11.4339	1.05258
$119$	0	0
$120$	−1.55334	−0.141800
$121$	−3.23369	−0.293972
$122$	19.9446	1.80570
$123$	1.25769	0.113402
$124$	0.0115271	0.00103517
$125$	−8.52280	−0.762302
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	5.71815	0.505418
$129$	4.01920	0.353871
$130$	0.377357	0.0330964
$131$	5.77731	0.504766	0.252383	−	0.967627i	$-0.418786\pi$
$131$	5.77731	0.504766	0.252383	+	0.967627i	$0.418786\pi$
$132$	15.0001	1.30560
$133$	0	0
$134$	−30.2282	−2.61132
$135$	−1.59483	−0.137261
$136$	1.82613	0.156589
$137$	−8.77617	−0.749799	−0.374899	−	0.927065i	$-0.622323\pi$
$137$	−8.77617	−0.749799	−0.374899	+	0.927065i	$0.622323\pi$
$138$	24.6210	2.09588
$139$	11.7256	0.994555	0.497277	−	0.867592i	$-0.334333\pi$
$139$	11.7256	0.994555	0.497277	+	0.867592i	$0.334333\pi$
$140$	0	0
$141$	−25.9507	−2.18544
$142$	31.9384	2.68021
$143$	−0.540155	−0.0451700
$144$	−7.17665	−0.598054
$145$	5.31867	0.441692
$146$	14.4469	1.19563
$147$	0	0
$148$	14.8038	1.21686
$149$	11.2603	0.922479	0.461239	−	0.887276i	$-0.347405\pi$
$149$	11.2603	0.922479	0.461239	+	0.887276i	$0.347405\pi$
$150$	19.7331	1.61120
$151$	12.8444	1.04526	0.522630	−	0.852559i	$-0.324951\pi$
$151$	12.8444	1.04526	0.522630	+	0.852559i	$0.324951\pi$
$152$	5.55090	0.450238
$153$	−5.67359	−0.458682
$154$	0	0
$155$	0.00458361	0.000368165 0
$156$	−1.04328	−0.0835289
$157$	−9.37504	−0.748210	−0.374105	−	0.927386i	$-0.622050\pi$
$157$	−9.37504	−0.748210	−0.374105	+	0.927386i	$0.622050\pi$
$158$	20.3033	1.61525
$159$	28.7166	2.27737
$160$	7.55172	0.597016
$161$	0	0
$162$	22.2703	1.74972
$163$	7.37682	0.577798	0.288899	−	0.957360i	$-0.406711\pi$
$163$	7.37682	0.577798	0.288899	+	0.957360i	$0.406711\pi$
$164$	1.28826	0.100596
$165$	5.96461	0.464344
$166$	0.118299	0.00918175
$167$	−2.44535	−0.189227	−0.0946133	−	0.995514i	$-0.530161\pi$
$167$	−2.44535	−0.189227	−0.0946133	+	0.995514i	$0.530161\pi$
$168$	0	0
$169$	−12.9624	−0.997110
$170$	4.89870	0.375713
$171$	−17.2461	−1.31884
$172$	4.11686	0.313908
$173$	−2.71740	−0.206600	−0.103300	−	0.994650i	$-0.532940\pi$
$173$	−2.71740	−0.206600	−0.103300	+	0.994650i	$0.532940\pi$
$174$	−27.2293	−2.06425
$175$	0	0
$176$	−8.86973	−0.668581
$177$	−12.5698	−0.944803
$178$	−5.40773	−0.405326
$179$	4.73487	0.353901	0.176950	−	0.984220i	$-0.443377\pi$
$179$	4.73487	0.353901	0.176950	+	0.984220i	$0.443377\pi$
$180$	4.94335	0.368455
$181$	−9.91117	−0.736691	−0.368346	−	0.929689i	$-0.620076\pi$
$181$	−9.91117	−0.736691	−0.368346	+	0.929689i	$0.620076\pi$
$182$	0	0
$183$	−21.9260	−1.62081
$184$	3.73826	0.275588
$185$	5.88653	0.432786
$186$	−0.0234661	−0.00172062
$187$	−7.01207	−0.512773
$188$	−26.5813	−1.93864
$189$	0	0
$190$	14.8906	1.08028
$191$	7.35311	0.532052	0.266026	−	0.963966i	$-0.414289\pi$
$191$	7.35311	0.532052	0.266026	+	0.963966i	$0.414289\pi$
$192$	−24.0696	−1.73707
$193$	−22.0783	−1.58923	−0.794616	−	0.607112i	$-0.792328\pi$
$193$	−22.0783	−1.58923	−0.794616	+	0.607112i	$0.792328\pi$
$194$	16.7967	1.20593
$195$	−0.414845	−0.0297077
$196$	0	0
$197$	−15.5041	−1.10462	−0.552310	−	0.833639i	$-0.686253\pi$
$197$	−15.5041	−1.10462	−0.552310	+	0.833639i	$0.686253\pi$
$198$	−13.1031	−0.931196
$199$	21.9047	1.55278	0.776392	−	0.630250i	$-0.217048\pi$
$199$	21.9047	1.55278	0.776392	+	0.630250i	$0.217048\pi$
$200$	2.99612	0.211857
$201$	33.2312	2.34395
$202$	13.4619	0.947173
$203$	0	0
$204$	−13.5434	−0.948226
$205$	0.512258	0.0357776
$206$	26.3903	1.83870
$207$	−11.6144	−0.807256
$208$	0.616899	0.0427742
$209$	−21.3147	−1.47437
$210$	0	0
$211$	−24.5809	−1.69222	−0.846109	−	0.533010i	$-0.821061\pi$
$211$	−24.5809	−1.69222	−0.846109	+	0.533010i	$0.821061\pi$
$212$	29.4144	2.02019
$213$	−35.1112	−2.40578
$214$	−29.1072	−1.98973
$215$	1.63702	0.111644
$216$	1.23969	0.0843501
$217$	0	0
$218$	33.8413	2.29202
$219$	−15.8820	−1.07321
$220$	6.10955	0.411906
$221$	0.487697	0.0328060
$222$	−30.1365	−2.02263
$223$	21.2711	1.42442	0.712210	−	0.701967i	$-0.247694\pi$
$223$	21.2711	1.42442	0.712210	+	0.701967i	$0.247694\pi$
$224$	0	0
$225$	−9.30862	−0.620575
$226$	−0.113477	−0.00754837
$227$	−18.9942	−1.26069	−0.630346	−	0.776314i	$-0.717087\pi$
$227$	−18.9942	−1.26069	−0.630346	+	0.776314i	$0.717087\pi$
$228$	−41.1680	−2.72642
$229$	19.5367	1.29102	0.645509	−	0.763753i	$-0.276645\pi$
$229$	19.5367	1.29102	0.645509	+	0.763753i	$0.276645\pi$
$230$	10.0281	0.661235
$231$	0	0
$232$	−4.13429	−0.271429
$233$	−12.2178	−0.800415	−0.400208	−	0.916425i	$-0.631062\pi$
$233$	−12.2178	−0.800415	−0.400208	+	0.916425i	$0.631062\pi$
$234$	0.911333	0.0595757
$235$	−10.5697	−0.689492
$236$	−12.8752	−0.838107
$237$	−22.3203	−1.44986
$238$	0	0
$239$	13.3637	0.864424	0.432212	−	0.901772i	$-0.357733\pi$
$239$	13.3637	0.864424	0.432212	+	0.901772i	$0.357733\pi$
$240$	−6.81205	−0.439716
$241$	4.66456	0.300471	0.150235	−	0.988650i	$-0.451997\pi$
$241$	4.66456	0.300471	0.150235	+	0.988650i	$0.451997\pi$
$242$	6.74290	0.433450
$243$	−19.3583	−1.24184
$244$	−22.4588	−1.43778
$245$	0	0
$246$	−2.62254	−0.167207
$247$	1.48246	0.0943265
$248$	−0.00356292	−0.000226245 0
$249$	−0.130051	−0.00824163
$250$	17.7717	1.12398
$251$	−5.22865	−0.330030	−0.165015	−	0.986291i	$-0.552767\pi$
$251$	−5.22865	−0.330030	−0.165015	+	0.986291i	$0.552767\pi$
$252$	0	0
$253$	−14.3544	−0.902453
$254$	−2.08520	−0.130837
$255$	−5.38535	−0.337244
$256$	9.07647	0.567280
$257$	−6.58612	−0.410831	−0.205416	−	0.978675i	$-0.565855\pi$
$257$	−6.58612	−0.410831	−0.205416	+	0.978675i	$0.565855\pi$
$258$	−8.38082	−0.521767
$259$	0	0
$260$	−0.424926	−0.0263528
$261$	12.8448	0.795074
$262$	−12.0468	−0.744256
$263$	−13.5267	−0.834092	−0.417046	−	0.908885i	$-0.636935\pi$
$263$	−13.5267	−0.834092	−0.417046	+	0.908885i	$0.636935\pi$
$264$	−4.63639	−0.285350
$265$	11.6963	0.718495
$266$	0	0
$267$	5.94495	0.363825
$268$	34.0387	2.07924
$269$	−2.96880	−0.181011	−0.0905054	−	0.995896i	$-0.528848\pi$
$269$	−2.96880	−0.181011	−0.0905054	+	0.995896i	$0.528848\pi$
$270$	3.32554	0.202386
$271$	15.0827	0.916211	0.458105	−	0.888898i	$-0.348528\pi$
$271$	15.0827	0.916211	0.458105	+	0.888898i	$0.348528\pi$
$272$	8.00833	0.485576
$273$	0	0
$274$	18.3001	1.10555
$275$	−11.5047	−0.693757
$276$	−27.7246	−1.66883
$277$	7.56049	0.454266	0.227133	−	0.973864i	$-0.427065\pi$
$277$	7.56049	0.454266	0.227133	+	0.973864i	$0.427065\pi$
$278$	−24.4503	−1.46643
$279$	0.0110696	0.000662720 0
$280$	0	0
$281$	30.5712	1.82373	0.911864	−	0.410493i	$-0.134643\pi$
$281$	30.5712	1.82373	0.911864	+	0.410493i	$0.134643\pi$
$282$	54.1124	3.22235
$283$	28.7658	1.70995	0.854975	−	0.518669i	$-0.173572\pi$
$283$	28.7658	1.70995	0.854975	+	0.518669i	$0.173572\pi$
$284$	−35.9644	−2.13410
$285$	−16.3699	−0.969670
$286$	1.12633	0.0666013
$287$	0	0
$288$	18.2377	1.07467
$289$	−10.6689	−0.627583
$290$	−11.0905	−0.651256
$291$	−18.4653	−1.08246
$292$	−16.2680	−0.952011
$293$	−10.6589	−0.622697	−0.311349	−	0.950296i	$-0.600781\pi$
$293$	−10.6589	−0.622697	−0.311349	+	0.950296i	$0.600781\pi$
$294$	0	0
$295$	−5.11967	−0.298079
$296$	−4.57569	−0.265957
$297$	−4.76023	−0.276216
$298$	−23.4799	−1.36016
$299$	0.998363	0.0577368
$300$	−22.2206	−1.28290
$301$	0	0
$302$	−26.7831	−1.54119
$303$	−14.7992	−0.850191
$304$	24.3430	1.39617
$305$	−8.93044	−0.511356
$306$	11.8306	0.676308
$307$	−12.1106	−0.691191	−0.345595	−	0.938384i	$-0.612323\pi$
$307$	−12.1106	−0.691191	−0.345595	+	0.938384i	$0.612323\pi$
$308$	0	0
$309$	−29.0120	−1.65044
$310$	−0.00955775	−0.000542843 0
$311$	−18.2756	−1.03631	−0.518156	−	0.855286i	$-0.673381\pi$
$311$	−18.2756	−1.03631	−0.518156	+	0.855286i	$0.673381\pi$
$312$	0.322466	0.0182560
$313$	25.3365	1.43210	0.716052	−	0.698047i	$-0.245947\pi$
$313$	25.3365	1.43210	0.716052	+	0.698047i	$0.245947\pi$
$314$	19.5488	1.10320
$315$	0	0
$316$	−22.8627	−1.28613
$317$	−25.7654	−1.44713	−0.723566	−	0.690255i	$-0.757498\pi$
$317$	−25.7654	−1.44713	−0.723566	+	0.690255i	$0.757498\pi$
$318$	−59.8797	−3.35789
$319$	15.8751	0.888834
$320$	−9.80354	−0.548034
$321$	31.9988	1.78600
$322$	0	0
$323$	19.2447	1.07080
$324$	−25.0777	−1.39320
$325$	0.800161	0.0443850
$326$	−15.3821	−0.851938
$327$	−37.2031	−2.05734
$328$	−0.398186	−0.0219862
$329$	0	0
$330$	−12.4374	−0.684656
$331$	−13.1557	−0.723104	−0.361552	−	0.932352i	$-0.617753\pi$
$331$	−13.1557	−0.723104	−0.361552	+	0.932352i	$0.617753\pi$
$332$	−0.133211	−0.00731090
$333$	14.2162	0.779043
$334$	5.09904	0.279007
$335$	13.5350	0.739498
$336$	0	0
$337$	24.3915	1.32869	0.664346	−	0.747426i	$-0.268710\pi$
$337$	24.3915	1.32869	0.664346	+	0.747426i	$0.268710\pi$
$338$	27.0292	1.47020
$339$	0.124750	0.00677549
$340$	−5.51622	−0.299159
$341$	0.0136811	0.000740873 0
$342$	35.9615	1.94457
$343$	0	0
$344$	−1.27248	−0.0686075
$345$	−11.0243	−0.593530
$346$	5.66632	0.304623
$347$	−3.88385	−0.208496	−0.104248	−	0.994551i	$-0.533244\pi$
$347$	−3.88385	−0.208496	−0.104248	+	0.994551i	$0.533244\pi$
$348$	30.6618	1.64364
$349$	6.73196	0.360353	0.180177	−	0.983634i	$-0.442333\pi$
$349$	6.73196	0.360353	0.180177	+	0.983634i	$0.442333\pi$
$350$	0	0
$351$	0.331079	0.0176717
$352$	22.5402	1.20140
$353$	13.2967	0.707711	0.353855	−	0.935300i	$-0.384870\pi$
$353$	13.2967	0.707711	0.353855	+	0.935300i	$0.384870\pi$
$354$	26.2105	1.39307
$355$	−14.3008	−0.759007
$356$	6.08941	0.322738
$357$	0	0
$358$	−9.87314	−0.521811
$359$	6.68500	0.352821	0.176410	−	0.984317i	$-0.443551\pi$
$359$	6.68500	0.352821	0.176410	+	0.984317i	$0.443551\pi$
$360$	−1.52794	−0.0805294
$361$	39.4983	2.07886
$362$	20.6668	1.08622
$363$	−7.41275	−0.389069
$364$	0	0
$365$	−6.46875	−0.338590
$366$	45.7200	2.38982
$367$	11.2531	0.587405	0.293702	−	0.955897i	$-0.405113\pi$
$367$	11.2531	0.587405	0.293702	+	0.955897i	$0.405113\pi$
$368$	16.3938	0.854588
$369$	1.23712	0.0644021
$370$	−12.2746	−0.638125
$371$	0	0
$372$	0.0264242	0.00137003
$373$	2.79234	0.144582	0.0722910	−	0.997384i	$-0.476969\pi$
$373$	2.79234	0.144582	0.0722910	+	0.997384i	$0.476969\pi$
$374$	14.6216	0.756063
$375$	−19.5372	−1.00890
$376$	8.21601	0.423708
$377$	−1.10413	−0.0568655
$378$	0	0
$379$	16.7969	0.862797	0.431399	−	0.902161i	$-0.358020\pi$
$379$	16.7969	0.862797	0.431399	+	0.902161i	$0.358020\pi$
$380$	−16.7677	−0.860165
$381$	2.29235	0.117441
$382$	−15.3327	−0.784489
$383$	3.42162	0.174836	0.0874182	−	0.996172i	$-0.472138\pi$
$383$	3.42162	0.174836	0.0874182	+	0.996172i	$0.472138\pi$
$384$	13.1080	0.668914
$385$	0	0
$386$	46.0377	2.34326
$387$	3.95346	0.200966
$388$	−18.9140	−0.960214
$389$	−8.81296	−0.446835	−0.223417	−	0.974723i	$-0.571721\pi$
$389$	−8.81296	−0.446835	−0.223417	+	0.974723i	$0.571721\pi$
$390$	0.865034	0.0438027
$391$	12.9603	0.655433
$392$	0	0
$393$	13.2436	0.668051
$394$	32.3291	1.62872
$395$	−9.09106	−0.457421
$396$	14.7548	0.741458
$397$	−10.5507	−0.529522	−0.264761	−	0.964314i	$-0.585293\pi$
$397$	−10.5507	−0.529522	−0.264761	+	0.964314i	$0.585293\pi$
$398$	−45.6757	−2.28952
$399$	0	0
$400$	13.1392	0.656961
$401$	−18.2483	−0.911277	−0.455639	−	0.890165i	$-0.650589\pi$
$401$	−18.2483	−0.911277	−0.455639	+	0.890165i	$0.650589\pi$
$402$	−69.2936	−3.45605
$403$	−0.000951534 0	−4.73993e−5 0
$404$	−15.1588	−0.754179
$405$	−9.97180	−0.495503
$406$	0	0
$407$	17.5700	0.870913
$408$	4.18612	0.207244
$409$	−31.0961	−1.53760	−0.768801	−	0.639488i	$-0.779146\pi$
$409$	−31.0961	−1.53760	−0.768801	+	0.639488i	$0.779146\pi$
$410$	−1.06816	−0.0527526
$411$	−20.1180	−0.992349
$412$	−29.7170	−1.46405
$413$	0	0
$414$	24.2183	1.19027
$415$	−0.0529696	−0.00260018
$416$	−1.56770	−0.0768627
$417$	26.8792	1.31628
$418$	44.4453	2.17389
$419$	−25.8715	−1.26390	−0.631952	−	0.775007i	$-0.717746\pi$
$419$	−25.8715	−1.26390	−0.631952	+	0.775007i	$0.717746\pi$
$420$	0	0
$421$	−2.01873	−0.0983869	−0.0491934	−	0.998789i	$-0.515665\pi$
$421$	−2.01873	−0.0983869	−0.0491934	+	0.998789i	$0.515665\pi$
$422$	51.2560	2.49510
$423$	−25.5263	−1.24113
$424$	−9.09168	−0.441531
$425$	10.3874	0.503861
$426$	73.2138	3.54722
$427$	0	0
$428$	32.7764	1.58431
$429$	−1.23822	−0.0597819
$430$	−3.41351	−0.164614
$431$	16.6506	0.802031	0.401016	−	0.916071i	$-0.368657\pi$
$431$	16.6506	0.802031	0.401016	+	0.916071i	$0.368657\pi$
$432$	5.43655	0.261566
$433$	1.22543	0.0588905	0.0294452	−	0.999566i	$-0.490626\pi$
$433$	1.22543	0.0588905	0.0294452	+	0.999566i	$0.490626\pi$
$434$	0	0
$435$	12.1922	0.584574
$436$	−38.1072	−1.82500
$437$	39.3957	1.88455
$438$	33.1172	1.58240
$439$	20.5160	0.979177	0.489589	−	0.871953i	$-0.337147\pi$
$439$	20.5160	0.979177	0.489589	+	0.871953i	$0.337147\pi$
$440$	−1.88840	−0.0900259
$441$	0	0
$442$	−1.01694	−0.0483711
$443$	−27.2725	−1.29576	−0.647878	−	0.761744i	$-0.724344\pi$
$443$	−27.2725	−1.29576	−0.647878	+	0.761744i	$0.724344\pi$
$444$	33.9354	1.61050
$445$	2.42138	0.114784
$446$	−44.3545	−2.10025
$447$	25.8125	1.22089
$448$	0	0
$449$	5.50357	0.259730	0.129865	−	0.991532i	$-0.458546\pi$
$449$	5.50357	0.259730	0.129865	+	0.991532i	$0.458546\pi$
$450$	19.4103	0.915012
$451$	1.52898	0.0719968
$452$	0.127782	0.00601034
$453$	29.4438	1.38339
$454$	39.6068	1.85884
$455$	0	0
$456$	12.7246	0.595884
$457$	10.1859	0.476475	0.238237	−	0.971207i	$-0.423430\pi$
$457$	10.1859	0.476475	0.238237	+	0.971207i	$0.423430\pi$
$458$	−40.7378	−1.90355
$459$	4.29793	0.200610
$460$	−11.2922	−0.526503
$461$	−26.3546	−1.22746	−0.613728	−	0.789518i	$-0.710331\pi$
$461$	−26.3546	−1.22746	−0.613728	+	0.789518i	$0.710331\pi$
$462$	0	0
$463$	−11.3537	−0.527652	−0.263826	−	0.964570i	$-0.584985\pi$
$463$	−11.3537	−0.527652	−0.263826	+	0.964570i	$0.584985\pi$
$464$	−18.1306	−0.841692
$465$	0.0105072	0.000487261 0
$466$	25.4766	1.18018
$467$	−29.9352	−1.38523	−0.692617	−	0.721305i	$-0.743542\pi$
$467$	−29.9352	−1.38523	−0.692617	+	0.721305i	$0.743542\pi$
$468$	−1.02621	−0.0474367
$469$	0	0
$470$	22.0400	1.01663
$471$	−21.4909	−0.990246
$472$	3.97960	0.183176
$473$	4.88614	0.224665
$474$	46.5423	2.13776
$475$	31.5746	1.44874
$476$	0	0
$477$	28.2469	1.29334
$478$	−27.8659	−1.27456
$479$	−12.7527	−0.582685	−0.291342	−	0.956619i	$-0.594102\pi$
$479$	−12.7527	−0.582685	−0.291342	+	0.956619i	$0.594102\pi$
$480$	17.3112	0.790143
$481$	−1.22201	−0.0557190
$482$	−9.72654	−0.443032
$483$	0	0
$484$	−7.59289	−0.345131
$485$	−7.52092	−0.341507
$486$	40.3659	1.83104
$487$	18.2165	0.825467	0.412734	−	0.910852i	$-0.364574\pi$
$487$	18.2165	0.825467	0.412734	+	0.910852i	$0.364574\pi$
$488$	6.94177	0.314239
$489$	16.9102	0.764708
$490$	0	0
$491$	−16.9535	−0.765100	−0.382550	−	0.923935i	$-0.624954\pi$
$491$	−16.9535	−0.765100	−0.382550	+	0.923935i	$0.624954\pi$
$492$	2.95313	0.133137
$493$	−14.3334	−0.645542
$494$	−3.09122	−0.139080
$495$	5.86706	0.263705
$496$	−0.0156249	−0.000701578 0
$497$	0	0
$498$	0.271181	0.0121519
$499$	21.9257	0.981529	0.490764	−	0.871292i	$-0.336718\pi$
$499$	21.9257	0.981529	0.490764	+	0.871292i	$0.336718\pi$
$500$	−20.0120	−0.894963
$501$	−5.60559	−0.250439
$502$	10.9028	0.486615
$503$	−26.9633	−1.20223	−0.601117	−	0.799161i	$-0.705277\pi$
$503$	−26.9633	−1.20223	−0.601117	+	0.799161i	$0.705277\pi$
$504$	0	0
$505$	−6.02771	−0.268229
$506$	29.9318	1.33063
$507$	−29.7144	−1.31966
$508$	2.34805	0.104178
$509$	−21.5261	−0.954128	−0.477064	−	0.878869i	$-0.658299\pi$
$509$	−21.5261	−0.954128	−0.477064	+	0.878869i	$0.658299\pi$
$510$	11.2295	0.497252
$511$	0	0
$512$	−30.3626	−1.34185
$513$	13.0645	0.576811
$514$	13.7334	0.605753
$515$	−11.8166	−0.520701
$516$	9.43728	0.415453
$517$	−31.5483	−1.38749
$518$	0	0
$519$	−6.22923	−0.273433
$520$	0.131340	0.00575965
$521$	12.9036	0.565316	0.282658	−	0.959221i	$-0.408784\pi$
$521$	12.9036	0.565316	0.282658	+	0.959221i	$0.408784\pi$
$522$	−26.7840	−1.17230
$523$	21.6556	0.946931	0.473466	−	0.880812i	$-0.343003\pi$
$523$	21.6556	0.946931	0.473466	+	0.880812i	$0.343003\pi$
$524$	13.5654	0.592608
$525$	0	0
$526$	28.2058	1.22983
$527$	−0.0123524	−0.000538080 0
$528$	−20.3325	−0.884858
$529$	3.53110	0.153526
$530$	−24.3890	−1.05939
$531$	−12.3642	−0.536561
$532$	0	0
$533$	−0.106342	−0.00460618
$534$	−12.3964	−0.536444
$535$	13.0331	0.563470
$536$	−10.5210	−0.454438
$537$	10.8540	0.468383
$538$	6.19053	0.266893
$539$	0	0
$540$	−3.74475	−0.161148
$541$	−3.63466	−0.156266	−0.0781330	−	0.996943i	$-0.524896\pi$
$541$	−3.63466	−0.156266	−0.0781330	+	0.996943i	$0.524896\pi$
$542$	−31.4505	−1.35091
$543$	−22.7198	−0.975002
$544$	−20.3512	−0.872551
$545$	−15.1528	−0.649076
$546$	0	0
$547$	−10.3932	−0.444381	−0.222191	−	0.975003i	$-0.571321\pi$
$547$	−10.3932	−0.444381	−0.222191	+	0.975003i	$0.571321\pi$
$548$	−20.6069	−0.880284
$549$	−21.5674	−0.920473
$550$	23.9895	1.02292
$551$	−43.5693	−1.85611
$552$	8.56940	0.364738
$553$	0	0
$554$	−15.7651	−0.669796
$555$	13.4940	0.572787
$556$	27.5324	1.16763
$557$	22.4511	0.951284	0.475642	−	0.879639i	$-0.342216\pi$
$557$	22.4511	0.951284	0.475642	+	0.879639i	$0.342216\pi$
$558$	−0.0230823	−0.000977153 0
$559$	−0.339836	−0.0143735
$560$	0	0
$561$	−16.0741	−0.678649
$562$	−63.7471	−2.68901
$563$	30.2139	1.27336	0.636682	−	0.771127i	$-0.280307\pi$
$563$	30.2139	1.27336	0.636682	+	0.771127i	$0.280307\pi$
$564$	−60.9336	−2.56577
$565$	0.0508107	0.00213762
$566$	−59.9824	−2.52125
$567$	0	0
$568$	11.1162	0.466427
$569$	47.0362	1.97186	0.985931	−	0.167155i	$-0.0534582\pi$
$569$	47.0362	1.97186	0.985931	+	0.167155i	$0.0534582\pi$
$570$	34.1345	1.42974
$571$	−27.5118	−1.15133	−0.575666	−	0.817685i	$-0.695257\pi$
$571$	−27.5118	−1.15133	−0.575666	+	0.817685i	$0.695257\pi$
$572$	−1.26831	−0.0530308
$573$	16.8559	0.704165
$574$	0	0
$575$	21.2640	0.886769
$576$	−23.6759	−0.986497
$577$	−16.8224	−0.700327	−0.350164	−	0.936689i	$-0.613874\pi$
$577$	−16.8224	−0.700327	−0.350164	+	0.936689i	$0.613874\pi$
$578$	22.2468	0.925345
$579$	−50.6112	−2.10333
$580$	12.4885	0.518558
$581$	0	0
$582$	38.5038	1.59604
$583$	34.9108	1.44586
$584$	5.02826	0.208071
$585$	−0.408060	−0.0168712
$586$	22.2258	0.918141
$587$	−8.96942	−0.370208	−0.185104	−	0.982719i	$-0.559262\pi$
$587$	−8.96942	−0.370208	−0.185104	+	0.982719i	$0.559262\pi$
$588$	0	0
$589$	−0.0375478	−0.00154713
$590$	10.6755	0.439505
$591$	−35.5407	−1.46195
$592$	−20.0663	−0.824721
$593$	−39.7027	−1.63040	−0.815198	−	0.579182i	$-0.803372\pi$
$593$	−39.7027	−1.63040	−0.815198	+	0.579182i	$0.803372\pi$
$594$	9.92602	0.407269
$595$	0	0
$596$	26.4397	1.08301
$597$	50.2132	2.05509
$598$	−2.08179	−0.0851305
$599$	−47.1062	−1.92471	−0.962354	−	0.271799i	$-0.912381\pi$
$599$	−47.1062	−1.92471	−0.962354	+	0.271799i	$0.912381\pi$
$600$	6.86814	0.280391
$601$	−13.9775	−0.570154	−0.285077	−	0.958505i	$-0.592019\pi$
$601$	−13.9775	−0.570154	−0.285077	+	0.958505i	$0.592019\pi$
$602$	0	0
$603$	32.6877	1.33114
$604$	30.1593	1.22716
$605$	−3.01921	−0.122748
$606$	30.8593	1.25357
$607$	22.5493	0.915249	0.457625	−	0.889146i	$-0.348700\pi$
$607$	22.5493	0.915249	0.457625	+	0.889146i	$0.348700\pi$
$608$	−61.8618	−2.50883
$609$	0	0
$610$	18.6217	0.753972
$611$	2.19422	0.0887686
$612$	−13.3219	−0.538505
$613$	43.6758	1.76405	0.882025	−	0.471202i	$-0.156180\pi$
$613$	43.6758	1.76405	0.882025	+	0.471202i	$0.156180\pi$
$614$	25.2531	1.01913
$615$	1.17427	0.0473513
$616$	0	0
$617$	2.20142	0.0886258	0.0443129	−	0.999018i	$-0.485890\pi$
$617$	2.20142	0.0886258	0.0443129	+	0.999018i	$0.485890\pi$
$618$	60.4958	2.43350
$619$	42.2248	1.69716	0.848580	−	0.529068i	$-0.177458\pi$
$619$	42.2248	1.69716	0.848580	+	0.529068i	$0.177458\pi$
$620$	0.0107626	0.000432235 0
$621$	8.79829	0.353063
$622$	38.1082	1.52800
$623$	0	0
$624$	1.41415	0.0566112
$625$	12.6838	0.507351
$626$	−52.8316	−2.11158
$627$	−48.8606	−1.95131
$628$	−22.0131	−0.878418
$629$	−15.8637	−0.632526
$630$	0	0
$631$	30.8546	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$631$	30.8546	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$632$	7.06663	0.281095
$633$	−56.3479	−2.23963
$634$	53.7261	2.13373
$635$	0.933672	0.0370517
$636$	67.4280	2.67369
$637$	0	0
$638$	−33.1027	−1.31055
$639$	−34.5370	−1.36626
$640$	5.33888	0.211038
$641$	−35.0810	−1.38562	−0.692808	−	0.721122i	$-0.743626\pi$
$641$	−35.0810	−1.38562	−0.692808	+	0.721122i	$0.743626\pi$
$642$	−66.7239	−2.63338
$643$	3.80378	0.150007	0.0750033	−	0.997183i	$-0.476103\pi$
$643$	3.80378	0.150007	0.0750033	+	0.997183i	$0.476103\pi$
$644$	0	0
$645$	3.75261	0.147759
$646$	−40.1289	−1.57885
$647$	25.7342	1.01172	0.505858	−	0.862617i	$-0.331176\pi$
$647$	25.7342	1.01172	0.505858	+	0.862617i	$0.331176\pi$
$648$	7.75124	0.304497
$649$	−15.2811	−0.599836
$650$	−1.66849	−0.0654437
$651$	0	0
$652$	17.3212	0.678349
$653$	0.517721	0.0202600	0.0101300	−	0.999949i	$-0.496775\pi$
$653$	0.517721	0.0202600	0.0101300	+	0.999949i	$0.496775\pi$
$654$	77.5759	3.03346
$655$	5.39412	0.210766
$656$	−1.74621	−0.0681782
$657$	−15.6223	−0.609484
$658$	0	0
$659$	1.28916	0.0502187	0.0251093	−	0.999685i	$-0.492007\pi$
$659$	1.28916	0.0502187	0.0251093	+	0.999685i	$0.492007\pi$
$660$	14.0052	0.545153
$661$	−9.00670	−0.350320	−0.175160	−	0.984540i	$-0.556044\pi$
$661$	−9.00670	−0.350320	−0.175160	+	0.984540i	$0.556044\pi$
$662$	27.4323	1.06619
$663$	1.11797	0.0434184
$664$	0.0411741	0.00159787
$665$	0	0
$666$	−29.6436	−1.14867
$667$	−29.3418	−1.13612
$668$	−5.74180	−0.222157
$669$	48.7608	1.88520
$670$	−28.2233	−1.09036
$671$	−26.6554	−1.02902
$672$	0	0
$673$	22.2110	0.856173	0.428086	−	0.903738i	$-0.359188\pi$
$673$	22.2110	0.856173	0.428086	+	0.903738i	$0.359188\pi$
$674$	−50.8612	−1.95910
$675$	7.05159	0.271416
$676$	−30.4365	−1.17063
$677$	42.3852	1.62900	0.814499	−	0.580166i	$-0.197012\pi$
$677$	42.3852	1.62900	0.814499	+	0.580166i	$0.197012\pi$
$678$	−0.260129	−0.00999018
$679$	0	0
$680$	1.70500	0.0653840
$681$	−43.5414	−1.66851
$682$	−0.0285278	−0.00109239
$683$	18.2898	0.699839	0.349920	−	0.936780i	$-0.386209\pi$
$683$	18.2898	0.699839	0.349920	+	0.936780i	$0.386209\pi$
$684$	−40.4947	−1.54835
$685$	−8.19407	−0.313079
$686$	0	0
$687$	44.7848	1.70865
$688$	−5.58036	−0.212749
$689$	−2.42808	−0.0925025
$690$	22.9879	0.875136
$691$	−31.8326	−1.21097	−0.605484	−	0.795858i	$-0.707020\pi$
$691$	−31.8326	−1.21097	−0.605484	+	0.795858i	$0.707020\pi$
$692$	−6.38060	−0.242554
$693$	0	0
$694$	8.09860	0.307419
$695$	10.9479	0.415277
$696$	−9.47723	−0.359233
$697$	−1.38049	−0.0522898
$698$	−14.0375	−0.531326
$699$	−28.0075	−1.05934
$700$	0	0
$701$	22.4553	0.848127	0.424063	−	0.905633i	$-0.360603\pi$
$701$	22.4553	0.848127	0.424063	+	0.905633i	$0.360603\pi$
$702$	−0.690365	−0.0260561
$703$	−48.2210	−1.81869
$704$	−29.2614	−1.10283
$705$	−24.2295	−0.912534
$706$	−27.7262	−1.04349
$707$	0	0
$708$	−29.5145	−1.10922
$709$	−23.9767	−0.900465	−0.450233	−	0.892911i	$-0.648659\pi$
$709$	−23.9767	−0.900465	−0.450233	+	0.892911i	$0.648659\pi$
$710$	29.8200	1.11912
$711$	−21.9553	−0.823387
$712$	−1.88217	−0.0705374
$713$	−0.0252866	−0.000946992 0
$714$	0	0
$715$	−0.504327	−0.0188608
$716$	11.1177	0.415489
$717$	30.6342	1.14405
$718$	−13.9396	−0.520220
$719$	−32.0616	−1.19570	−0.597848	−	0.801609i	$-0.703978\pi$
$719$	−32.0616	−1.19570	−0.597848	+	0.801609i	$0.703978\pi$
$720$	−6.70064	−0.249718
$721$	0	0
$722$	−82.3617	−3.06518
$723$	10.6928	0.397669
$724$	−23.2719	−0.864895
$725$	−23.5166	−0.873386
$726$	15.4571	0.573665
$727$	15.4885	0.574438	0.287219	−	0.957865i	$-0.407269\pi$
$727$	15.4885	0.574438	0.287219	+	0.957865i	$0.407269\pi$
$728$	0	0
$729$	−12.3354	−0.456868
$730$	13.4886	0.499236
$731$	−4.41162	−0.163170
$732$	−51.4833	−1.90288
$733$	−45.9771	−1.69820	−0.849101	−	0.528230i	$-0.822856\pi$
$733$	−45.9771	−1.69820	−0.849101	+	0.528230i	$0.822856\pi$
$734$	−23.4648	−0.866103
$735$	0	0
$736$	−41.6609	−1.53564
$737$	40.3992	1.48812
$738$	−2.57965	−0.0949581
$739$	15.5562	0.572245	0.286122	−	0.958193i	$-0.407634\pi$
$739$	15.5562	0.572245	0.286122	+	0.958193i	$0.407634\pi$
$740$	13.8219	0.508102
$741$	3.39831	0.124840
$742$	0	0
$743$	7.57725	0.277982	0.138991	−	0.990294i	$-0.455614\pi$
$743$	7.57725	0.277982	0.138991	+	0.990294i	$0.455614\pi$
$744$	−0.00816744	−0.000299433 0
$745$	10.5134	0.385182
$746$	−5.82259	−0.213180
$747$	−0.127924	−0.00468048
$748$	−16.4647	−0.602009
$749$	0	0
$750$	40.7390	1.48758
$751$	−46.6503	−1.70229	−0.851146	−	0.524928i	$-0.824092\pi$
$751$	−46.6503	−1.70229	−0.851146	+	0.524928i	$0.824092\pi$
$752$	36.0306	1.31390
$753$	−11.9859	−0.436790
$754$	2.30233	0.0838458
$755$	11.9924	0.436450
$756$	0	0
$757$	13.7608	0.500145	0.250073	−	0.968227i	$-0.419545\pi$
$757$	13.7608	0.500145	0.250073	+	0.968227i	$0.419545\pi$
$758$	−35.0248	−1.27216
$759$	−32.9053	−1.19439
$760$	5.18272	0.187997
$761$	−0.805405	−0.0291959	−0.0145980	−	0.999893i	$-0.504647\pi$
$761$	−0.805405	−0.0291959	−0.0145980	+	0.999893i	$0.504647\pi$
$762$	−4.78000	−0.173161
$763$	0	0
$764$	17.2655	0.624644
$765$	−5.29727	−0.191523
$766$	−7.13475	−0.257789
$767$	1.06282	0.0383761
$768$	20.8064	0.750787
$769$	48.5248	1.74985	0.874925	−	0.484258i	$-0.160911\pi$
$769$	48.5248	1.74985	0.874925	+	0.484258i	$0.160911\pi$
$770$	0	0
$771$	−15.0977	−0.543730
$772$	−51.8411	−1.86580
$773$	−18.6869	−0.672121	−0.336060	−	0.941840i	$-0.609095\pi$
$773$	−18.6869	−0.672121	−0.336060	+	0.941840i	$0.609095\pi$
$774$	−8.24375	−0.296316
$775$	−0.0202666	−0.000727996 0
$776$	5.84613	0.209864
$777$	0	0
$778$	18.3768	0.658839
$779$	−4.19629	−0.150348
$780$	−0.974078	−0.0348776
$781$	−42.6847	−1.52738
$782$	−27.0249	−0.966408
$783$	−9.73037	−0.347735
$784$	0	0
$785$	−8.75322	−0.312416
$786$	−27.6155	−0.985014
$787$	−6.71735	−0.239448	−0.119724	−	0.992807i	$-0.538201\pi$
$787$	−6.71735	−0.239448	−0.119724	+	0.992807i	$0.538201\pi$
$788$	−36.4044	−1.29685
$789$	−31.0079	−1.10391
$790$	18.9567	0.674448
$791$	0	0
$792$	−4.56056	−0.162052
$793$	1.85391	0.0658344
$794$	22.0002	0.780758
$795$	26.8119	0.950919
$796$	51.4334	1.82301
$797$	−23.0778	−0.817459	−0.408730	−	0.912655i	$-0.634028\pi$
$797$	−23.0778	−0.817459	−0.408730	+	0.912655i	$0.634028\pi$
$798$	0	0
$799$	28.4844	1.00771
$800$	−33.3901	−1.18052
$801$	5.84772	0.206619
$802$	38.0513	1.34364
$803$	−19.3078	−0.681358
$804$	78.0285	2.75185
$805$	0	0
$806$	0.00198414	6.98883e−5 0
$807$	−6.80552	−0.239566
$808$	4.68543	0.164833
$809$	53.9924	1.89827	0.949135	−	0.314869i	$-0.101961\pi$
$809$	53.9924	1.89827	0.949135	+	0.314869i	$0.101961\pi$
$810$	20.7932	0.730598
$811$	−27.2849	−0.958103	−0.479051	−	0.877787i	$-0.659019\pi$
$811$	−27.2849	−0.958103	−0.479051	+	0.877787i	$0.659019\pi$
$812$	0	0
$813$	34.5749	1.21259
$814$	−36.6370	−1.28412
$815$	6.88754	0.241260
$816$	18.3579	0.642654
$817$	−13.4100	−0.469158
$818$	64.8415	2.26713
$819$	0	0
$820$	1.20281	0.0420039
$821$	−0.485767	−0.0169534	−0.00847669	−	0.999964i	$-0.502698\pi$
$821$	−0.485767	−0.0169534	−0.00847669	+	0.999964i	$0.502698\pi$
$822$	41.9501	1.46318
$823$	−19.5296	−0.680760	−0.340380	−	0.940288i	$-0.610556\pi$
$823$	−19.5296	−0.680760	−0.340380	+	0.940288i	$0.610556\pi$
$824$	9.18522	0.319982
$825$	−26.3727	−0.918179
$826$	0	0
$827$	−33.0521	−1.14933	−0.574667	−	0.818387i	$-0.694868\pi$
$827$	−33.0521	−1.14933	−0.574667	+	0.818387i	$0.694868\pi$
$828$	−27.2712	−0.947740
$829$	−1.53452	−0.0532959	−0.0266480	−	0.999645i	$-0.508483\pi$
$829$	−1.53452	−0.0532959	−0.0266480	+	0.999645i	$0.508483\pi$
$830$	0.110452	0.00383385
$831$	17.3313	0.601215
$832$	2.03516	0.0705566
$833$	0	0
$834$	−56.0485	−1.94080
$835$	−2.28315	−0.0790118
$836$	−50.0480	−1.73095
$837$	−0.00838560	−0.000289849 0
$838$	53.9472	1.86357
$839$	−3.69432	−0.127542	−0.0637710	−	0.997965i	$-0.520313\pi$
$839$	−3.69432	−0.127542	−0.0637710	+	0.997965i	$0.520313\pi$
$840$	0	0
$841$	3.45022	0.118973
$842$	4.20945	0.145067
$843$	70.0799	2.41368
$844$	−57.7172	−1.98671
$845$	−12.1027	−0.416344
$846$	53.2274	1.83000
$847$	0	0
$848$	−39.8708	−1.36917
$849$	65.9412	2.26310
$850$	−21.6597	−0.742922
$851$	−32.4745	−1.11321
$852$	−82.4430	−2.82445
$853$	−13.0036	−0.445235	−0.222618	−	0.974906i	$-0.571460\pi$
$853$	−13.0036	−0.445235	−0.222618	+	0.974906i	$0.571460\pi$
$854$	0	0
$855$	−16.1022	−0.550683
$856$	−10.1308	−0.346265
$857$	8.79211	0.300333	0.150166	−	0.988661i	$-0.452019\pi$
$857$	8.79211	0.300333	0.150166	+	0.988661i	$0.452019\pi$
$858$	2.58194	0.0881459
$859$	28.3314	0.966655	0.483328	−	0.875440i	$-0.339428\pi$
$859$	28.3314	0.966655	0.483328	+	0.875440i	$0.339428\pi$
$860$	3.84380	0.131073
$861$	0	0
$862$	−34.7198	−1.18256
$863$	−17.6452	−0.600648	−0.300324	−	0.953837i	$-0.597095\pi$
$863$	−17.6452	−0.600648	−0.300324	+	0.953837i	$0.597095\pi$
$864$	−13.8157	−0.470019
$865$	−2.53716	−0.0862662
$866$	−2.55527	−0.0868315
$867$	−24.4569	−0.830599
$868$	0	0
$869$	−27.1348	−0.920487
$870$	−25.4233	−0.861929
$871$	−2.80980	−0.0952066
$872$	11.7785	0.398872
$873$	−18.1633	−0.614735
$874$	−82.1479	−2.77869
$875$	0	0
$876$	−37.2919	−1.25998
$877$	−12.6152	−0.425986	−0.212993	−	0.977054i	$-0.568321\pi$
$877$	−12.6152	−0.425986	−0.212993	+	0.977054i	$0.568321\pi$
$878$	−42.7800	−1.44376
$879$	−24.4338	−0.824132
$880$	−8.28142	−0.279167
$881$	34.6918	1.16880	0.584398	−	0.811467i	$-0.301331\pi$
$881$	34.6918	1.16880	0.584398	+	0.811467i	$0.301331\pi$
$882$	0	0
$883$	−1.63980	−0.0551838	−0.0275919	−	0.999619i	$-0.508784\pi$
$883$	−1.63980	−0.0551838	−0.0275919	+	0.999619i	$0.508784\pi$
$884$	1.14514	0.0385151
$885$	−11.7361	−0.394504
$886$	56.8686	1.91054
$887$	46.9777	1.57736	0.788679	−	0.614806i	$-0.210766\pi$
$887$	46.9777	1.57736	0.788679	+	0.614806i	$0.210766\pi$
$888$	−10.4891	−0.351990
$889$	0	0
$890$	−5.04905	−0.169244
$891$	−29.7637	−0.997120
$892$	49.9457	1.67231
$893$	86.5845	2.89744
$894$	−53.8242	−1.80015
$895$	4.42081	0.147772
$896$	0	0
$897$	2.28859	0.0764140
$898$	−11.4760	−0.382961
$899$	0.0279655	0.000932701 0
$900$	−21.8571	−0.728571
$901$	−31.5204	−1.05010
$902$	−3.18822	−0.106156
$903$	0	0
$904$	−0.0394959	−0.00131362
$905$	−9.25379	−0.307606
$906$	−61.3961	−2.03975
$907$	−17.9824	−0.597096	−0.298548	−	0.954395i	$-0.596502\pi$
$907$	−17.9824	−0.597096	−0.298548	+	0.954395i	$0.596502\pi$
$908$	−44.5995	−1.48009
$909$	−14.5572	−0.482830
$910$	0	0
$911$	−5.69883	−0.188811	−0.0944053	−	0.995534i	$-0.530095\pi$
$911$	−5.69883	−0.188811	−0.0944053	+	0.995534i	$0.530095\pi$
$912$	55.8027	1.84781
$913$	−0.158103	−0.00523244
$914$	−21.2396	−0.702542
$915$	−20.4717	−0.676773
$916$	45.8731	1.51569
$917$	0	0
$918$	−8.96204	−0.295791
$919$	30.9677	1.02153	0.510765	−	0.859720i	$-0.329362\pi$
$919$	30.9677	1.02153	0.510765	+	0.859720i	$0.329362\pi$
$920$	3.49031	0.115072
$921$	−27.7618	−0.914783
$922$	54.9545	1.80983
$923$	2.96877	0.0977182
$924$	0	0
$925$	−26.0274	−0.855777
$926$	23.6748	0.778001
$927$	−28.5375	−0.937295
$928$	46.0745	1.51247
$929$	−9.62268	−0.315710	−0.157855	−	0.987462i	$-0.550458\pi$
$929$	−9.62268	−0.315710	−0.157855	+	0.987462i	$0.550458\pi$
$930$	−0.0219097	−0.000718446 0
$931$	0	0
$932$	−28.6881	−0.939708
$933$	−41.8939	−1.37155
$934$	62.4208	2.04247
$935$	−6.54698	−0.214109
$936$	0.317192	0.0103677
$937$	−27.5737	−0.900795	−0.450397	−	0.892828i	$-0.648718\pi$
$937$	−27.5737	−0.900795	−0.450397	+	0.892828i	$0.648718\pi$
$938$	0	0
$939$	58.0801	1.89537
$940$	−24.8183	−0.809482
$941$	−48.4998	−1.58105	−0.790524	−	0.612431i	$-0.790192\pi$
$941$	−48.4998	−1.58105	−0.790524	+	0.612431i	$0.790192\pi$
$942$	44.8127	1.46008
$943$	−2.82600	−0.0920271
$944$	17.4522	0.568021
$945$	0	0
$946$	−10.1886	−0.331259
$947$	27.5695	0.895889	0.447945	−	0.894061i	$-0.352156\pi$
$947$	27.5695	0.895889	0.447945	+	0.894061i	$0.352156\pi$
$948$	−52.4093	−1.70217
$949$	1.34288	0.0435917
$950$	−65.8393	−2.13611
$951$	−59.0634	−1.91526
$952$	0	0
$953$	−6.77555	−0.219482	−0.109741	−	0.993960i	$-0.535002\pi$
$953$	−6.77555	−0.219482	−0.109741	+	0.993960i	$0.535002\pi$
$954$	−58.9004	−1.90697
$955$	6.86540	0.222159
$956$	31.3786	1.01486
$957$	36.3912	1.17636
$958$	26.5919	0.859145
$959$	0	0
$960$	−22.4731	−0.725317
$961$	−31.0000	−0.999999
$962$	2.54814	0.0821553
$963$	31.4755	1.01428
$964$	10.9526	0.352761
$965$	−20.6139	−0.663586
$966$	0	0
$967$	21.4389	0.689430	0.344715	−	0.938707i	$-0.387976\pi$
$967$	21.4389	0.689430	0.344715	+	0.938707i	$0.387976\pi$
$968$	2.34688	0.0754316
$969$	44.1155	1.41719
$970$	15.6826	0.503538
$971$	12.5837	0.403829	0.201915	−	0.979403i	$-0.435284\pi$
$971$	12.5837	0.403829	0.201915	+	0.979403i	$0.435284\pi$
$972$	−45.4544	−1.45795
$973$	0	0
$974$	−37.9850	−1.21712
$975$	1.83425	0.0587429
$976$	30.4426	0.974443
$977$	−16.0849	−0.514601	−0.257301	−	0.966331i	$-0.582833\pi$
$977$	−16.0849	−0.514601	−0.257301	+	0.966331i	$0.582833\pi$
$978$	−35.2612	−1.12753
$979$	7.22728	0.230985
$980$	0	0
$981$	−36.5947	−1.16838
$982$	35.3514	1.12811
$983$	−48.8867	−1.55924	−0.779621	−	0.626251i	$-0.784588\pi$
$983$	−48.8867	−1.55924	−0.779621	+	0.626251i	$0.784588\pi$
$984$	−0.912781	−0.0290984
$985$	−14.4757	−0.461235
$986$	29.8879	0.951824
$987$	0	0
$988$	3.48089	0.110742
$989$	−9.03101	−0.287169
$990$	−12.2340	−0.388822
$991$	−41.3328	−1.31298	−0.656490	−	0.754335i	$-0.727960\pi$
$991$	−41.3328	−1.31298	−0.656490	+	0.754335i	$0.727960\pi$
$992$	0.0397068	0.00126069
$993$	−30.1575	−0.957019
$994$	0	0
$995$	20.4518	0.648367
$996$	−0.305366	−0.00967589
$997$	−32.7780	−1.03809	−0.519045	−	0.854747i	$-0.673712\pi$
$997$	−32.7780	−1.03809	−0.519045	+	0.854747i	$0.673712\pi$
$998$	−45.7194	−1.44722
$999$	−10.7692	−0.340724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.o.1.7		38
7.3	odd	6		889.2.f.c.128.32	✓	76
7.5	odd	6		889.2.f.c.382.32	yes	76
7.6	odd	2		6223.2.a.p.1.7		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.f.c.128.32	✓	76	7.3	odd	6
889.2.f.c.382.32	yes	76	7.5	odd	6
6223.2.a.o.1.7		38	1.1	even	1	trivial
6223.2.a.p.1.7		38	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.08520 q^{2} +2.29235 q^{3} +2.34805 q^{4} +0.933672 q^{5} -4.78000 q^{6} -0.725759 q^{8} +2.25486 q^{9} -1.94689 q^{10} +2.78681 q^{11} +5.38255 q^{12} -0.193825 q^{13} +2.14030 q^{15} -3.18275 q^{16} -2.51616 q^{17} -4.70182 q^{18} -7.64841 q^{19} +2.19231 q^{20} -5.81105 q^{22} -5.15083 q^{23} -1.66369 q^{24} -4.12826 q^{25} +0.404165 q^{26} -1.70813 q^{27} +5.69651 q^{29} -4.46295 q^{30} +0.00490923 q^{31} +8.08819 q^{32} +6.38833 q^{33} +5.24670 q^{34} +5.29452 q^{36} +6.30470 q^{37} +15.9485 q^{38} -0.444315 q^{39} -0.677621 q^{40} +0.548648 q^{41} +1.75331 q^{43} +6.54357 q^{44} +2.10530 q^{45} +10.7405 q^{46} -11.3206 q^{47} -7.29598 q^{48} +8.60823 q^{50} -5.76792 q^{51} -0.455112 q^{52} +12.5271 q^{53} +3.56179 q^{54} +2.60197 q^{55} -17.5328 q^{57} -11.8784 q^{58} -5.48337 q^{59} +5.02554 q^{60} -9.56485 q^{61} -0.0102367 q^{62} -10.5000 q^{64} -0.180970 q^{65} -13.3209 q^{66} +14.4966 q^{67} -5.90809 q^{68} -11.8075 q^{69} -15.3167 q^{71} -1.63648 q^{72} -6.92829 q^{73} -13.1466 q^{74} -9.46340 q^{75} -17.9589 q^{76} +0.926486 q^{78} -9.73689 q^{79} -2.97165 q^{80} -10.6802 q^{81} -1.14404 q^{82} -0.0567325 q^{83} -2.34927 q^{85} -3.65600 q^{86} +13.0584 q^{87} -2.02255 q^{88} +2.59339 q^{89} -4.38996 q^{90} -12.0944 q^{92} +0.0112537 q^{93} +23.6057 q^{94} -7.14111 q^{95} +18.5409 q^{96} -8.05520 q^{97} +6.28385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 11 q^{3} + 38 q^{4} - 16 q^{5} - 11 q^{6} + 47 q^{9} - 12 q^{10} - 2 q^{11} - 30 q^{12} - 21 q^{13} + 7 q^{15} + 46 q^{16} - 58 q^{17} - 13 q^{18} - 17 q^{19} - 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 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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