LMFDB - Embedded newform 6223.2.a.p.1.9

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			1.9
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)

$f(q)$	$=$	\(q-1.80084 q^{2} -3.27481 q^{3} +1.24303 q^{4} +1.98237 q^{5} +5.89741 q^{6} +1.36319 q^{8} +7.72436 q^{9} -3.56994 q^{10} -3.35625 q^{11} -4.07068 q^{12} -2.29226 q^{13} -6.49189 q^{15} -4.94094 q^{16} -0.137620 q^{17} -13.9104 q^{18} +0.990399 q^{19} +2.46415 q^{20} +6.04408 q^{22} -5.97904 q^{23} -4.46417 q^{24} -1.07020 q^{25} +4.12800 q^{26} -15.4714 q^{27} -9.52451 q^{29} +11.6909 q^{30} -9.64350 q^{31} +6.17147 q^{32} +10.9911 q^{33} +0.247832 q^{34} +9.60161 q^{36} -10.0947 q^{37} -1.78355 q^{38} +7.50672 q^{39} +2.70234 q^{40} -4.24513 q^{41} +3.91855 q^{43} -4.17192 q^{44} +15.3126 q^{45} +10.7673 q^{46} -3.08238 q^{47} +16.1806 q^{48} +1.92726 q^{50} +0.450679 q^{51} -2.84935 q^{52} +3.23436 q^{53} +27.8615 q^{54} -6.65334 q^{55} -3.24337 q^{57} +17.1521 q^{58} -7.78466 q^{59} -8.06960 q^{60} -3.32649 q^{61} +17.3664 q^{62} -1.23197 q^{64} -4.54412 q^{65} -19.7932 q^{66} -8.34418 q^{67} -0.171066 q^{68} +19.5802 q^{69} -5.67836 q^{71} +10.5297 q^{72} +7.22014 q^{73} +18.1790 q^{74} +3.50470 q^{75} +1.23109 q^{76} -13.5184 q^{78} -7.90576 q^{79} -9.79477 q^{80} +27.4927 q^{81} +7.64480 q^{82} +17.2797 q^{83} -0.272814 q^{85} -7.05669 q^{86} +31.1909 q^{87} -4.57519 q^{88} +5.07565 q^{89} -27.5755 q^{90} -7.43211 q^{92} +31.5806 q^{93} +5.55087 q^{94} +1.96334 q^{95} -20.2104 q^{96} +14.4942 q^{97} -25.9249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.80084	−1.27339	−0.636693	−	0.771117i	$-0.719698\pi$
$2$	−1.80084	−1.27339	−0.636693	+	0.771117i	$0.719698\pi$
$3$	−3.27481	−1.89071	−0.945356	−	0.326041i	$-0.894285\pi$
$3$	−3.27481	−1.89071	−0.945356	+	0.326041i	$0.894285\pi$
$4$	1.24303	0.621514
$5$	1.98237	0.886544	0.443272	−	0.896387i	$-0.353818\pi$
$5$	1.98237	0.886544	0.443272	+	0.896387i	$0.353818\pi$
$6$	5.89741	2.40761
$7$	0	0
$7$	0	0
$8$	1.36319	0.481959
$9$	7.72436	2.57479
$10$	−3.56994	−1.12891
$11$	−3.35625	−1.01195	−0.505974	−	0.862549i	$-0.668867\pi$
$11$	−3.35625	−1.01195	−0.505974	+	0.862549i	$0.668867\pi$
$12$	−4.07068	−1.17510
$13$	−2.29226	−0.635759	−0.317880	−	0.948131i	$-0.602971\pi$
$13$	−2.29226	−0.635759	−0.317880	+	0.948131i	$0.602971\pi$
$14$	0	0
$15$	−6.49189	−1.67620
$16$	−4.94094	−1.23523
$17$	−0.137620	−0.0333778	−0.0166889	−	0.999861i	$-0.505312\pi$
$17$	−0.137620	−0.0333778	−0.0166889	+	0.999861i	$0.505312\pi$
$18$	−13.9104	−3.27870
$19$	0.990399	0.227213	0.113607	−	0.993526i	$-0.463760\pi$
$19$	0.990399	0.227213	0.113607	+	0.993526i	$0.463760\pi$
$20$	2.46415	0.551000
$21$	0	0
$22$	6.04408	1.28860
$23$	−5.97904	−1.24672	−0.623358	−	0.781937i	$-0.714232\pi$
$23$	−5.97904	−1.24672	−0.623358	+	0.781937i	$0.714232\pi$
$24$	−4.46417	−0.911245
$25$	−1.07020	−0.214040
$26$	4.12800	0.809568
$27$	−15.4714	−2.97747
$28$	0	0
$29$	−9.52451	−1.76866	−0.884329	−	0.466865i	$-0.845383\pi$
$29$	−9.52451	−1.76866	−0.884329	+	0.466865i	$0.845383\pi$
$30$	11.6909	2.13445
$31$	−9.64350	−1.73202	−0.866012	−	0.500024i	$-0.833325\pi$
$31$	−9.64350	−1.73202	−0.866012	+	0.500024i	$0.833325\pi$
$32$	6.17147	1.09097
$33$	10.9911	1.91330
$34$	0.247832	0.0425028
$35$	0	0
$36$	9.60161	1.60027
$37$	−10.0947	−1.65956	−0.829780	−	0.558090i	$-0.811534\pi$
$37$	−10.0947	−1.65956	−0.829780	+	0.558090i	$0.811534\pi$
$38$	−1.78355	−0.289330
$39$	7.50672	1.20204
$40$	2.70234	0.427277
$41$	−4.24513	−0.662978	−0.331489	−	0.943459i	$-0.607551\pi$
$41$	−4.24513	−0.662978	−0.331489	+	0.943459i	$0.607551\pi$
$42$	0	0
$43$	3.91855	0.597573	0.298787	−	0.954320i	$-0.403418\pi$
$43$	3.91855	0.597573	0.298787	+	0.954320i	$0.403418\pi$
$44$	−4.17192	−0.628940
$45$	15.3126	2.28266
$46$	10.7673	1.58755
$47$	−3.08238	−0.449611	−0.224806	−	0.974404i	$-0.572175\pi$
$47$	−3.08238	−0.449611	−0.224806	+	0.974404i	$0.572175\pi$
$48$	16.1806	2.33547
$49$	0	0
$50$	1.92726	0.272556
$51$	0.450679	0.0631077
$52$	−2.84935	−0.395134
$53$	3.23436	0.444273	0.222137	−	0.975016i	$-0.428697\pi$
$53$	3.23436	0.444273	0.222137	+	0.975016i	$0.428697\pi$
$54$	27.8615	3.79147
$55$	−6.65334	−0.897136
$56$	0	0
$57$	−3.24337	−0.429594
$58$	17.1521	2.25218
$59$	−7.78466	−1.01348	−0.506738	−	0.862100i	$-0.669149\pi$
$59$	−7.78466	−1.01348	−0.506738	+	0.862100i	$0.669149\pi$
$60$	−8.06960	−1.04178
$61$	−3.32649	−0.425913	−0.212956	−	0.977062i	$-0.568309\pi$
$61$	−3.32649	−0.425913	−0.212956	+	0.977062i	$0.568309\pi$
$62$	17.3664	2.20554
$63$	0	0
$64$	−1.23197	−0.153996
$65$	−4.54412	−0.563628
$66$	−19.7932	−2.43637
$67$	−8.34418	−1.01940	−0.509702	−	0.860351i	$-0.670244\pi$
$67$	−8.34418	−1.01940	−0.509702	+	0.860351i	$0.670244\pi$
$68$	−0.171066	−0.0207448
$69$	19.5802	2.35718
$70$	0	0
$71$	−5.67836	−0.673898	−0.336949	−	0.941523i	$-0.609395\pi$
$71$	−5.67836	−0.673898	−0.336949	+	0.941523i	$0.609395\pi$
$72$	10.5297	1.24094
$73$	7.22014	0.845053	0.422526	−	0.906351i	$-0.361143\pi$
$73$	7.22014	0.845053	0.422526	+	0.906351i	$0.361143\pi$
$74$	18.1790	2.11326
$75$	3.50470	0.404688
$76$	1.23109	0.141216
$77$	0	0
$78$	−13.5184	−1.53066
$79$	−7.90576	−0.889467	−0.444734	−	0.895663i	$-0.646702\pi$
$79$	−7.90576	−0.889467	−0.444734	+	0.895663i	$0.646702\pi$
$80$	−9.79477	−1.09509
$81$	27.4927	3.05475
$82$	7.64480	0.844227
$83$	17.2797	1.89669	0.948345	−	0.317242i	$-0.102757\pi$
$83$	17.2797	1.89669	0.948345	+	0.317242i	$0.102757\pi$
$84$	0	0
$85$	−0.272814	−0.0295909
$86$	−7.05669	−0.760942
$87$	31.1909	3.34402
$88$	−4.57519	−0.487717
$89$	5.07565	0.538017	0.269009	−	0.963138i	$-0.413304\pi$
$89$	5.07565	0.538017	0.269009	+	0.963138i	$0.413304\pi$
$90$	−27.5755	−2.90671
$91$	0	0
$92$	−7.43211	−0.774851
$93$	31.5806	3.27476
$94$	5.55087	0.572529
$95$	1.96334	0.201434
$96$	−20.2104	−2.06271
$97$	14.4942	1.47166	0.735831	−	0.677166i	$-0.236792\pi$
$97$	14.4942	1.47166	0.735831	+	0.677166i	$0.236792\pi$
$98$	0	0
$99$	−25.9249	−2.60555
$100$	−1.33029	−0.133029
$101$	9.72113	0.967288	0.483644	−	0.875265i	$-0.339313\pi$
$101$	9.72113	0.967288	0.483644	+	0.875265i	$0.339313\pi$
$102$	−0.811602	−0.0803606
$103$	−9.87020	−0.972540	−0.486270	−	0.873809i	$-0.661643\pi$
$103$	−9.87020	−0.972540	−0.486270	+	0.873809i	$0.661643\pi$
$104$	−3.12478	−0.306410
$105$	0	0
$106$	−5.82457	−0.565732
$107$	−20.3675	−1.96901	−0.984503	−	0.175370i	$-0.943888\pi$
$107$	−20.3675	−1.96901	−0.984503	+	0.175370i	$0.943888\pi$
$108$	−19.2314	−1.85054
$109$	6.19814	0.593674	0.296837	−	0.954928i	$-0.404068\pi$
$109$	6.19814	0.593674	0.296837	+	0.954928i	$0.404068\pi$
$110$	11.9816	1.14240
$111$	33.0582	3.13775
$112$	0	0
$113$	−13.4769	−1.26780	−0.633902	−	0.773413i	$-0.718548\pi$
$113$	−13.4769	−1.26780	−0.633902	+	0.773413i	$0.718548\pi$
$114$	5.84079	0.547040
$115$	−11.8527	−1.10527
$116$	−11.8392	−1.09925
$117$	−17.7063	−1.63695
$118$	14.0189	1.29055
$119$	0	0
$120$	−8.84964	−0.807858
$121$	0.264435	0.0240395
$122$	5.99047	0.542352
$123$	13.9020	1.25350
$124$	−11.9871	−1.07648
$125$	−12.0334	−1.07630
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.1244	−0.894876
$129$	−12.8325	−1.12984
$130$	8.18323	0.717717
$131$	9.60089	0.838833	0.419417	−	0.907794i	$-0.362235\pi$
$131$	9.60089	0.838833	0.419417	+	0.907794i	$0.362235\pi$
$132$	13.6622	1.18914
$133$	0	0
$134$	15.0265	1.29810
$135$	−30.6700	−2.63966
$136$	−0.187602	−0.0160867
$137$	0.675259	0.0576913	0.0288456	−	0.999584i	$-0.490817\pi$
$137$	0.675259	0.0576913	0.0288456	+	0.999584i	$0.490817\pi$
$138$	−35.2608	−3.00160
$139$	−8.99416	−0.762875	−0.381437	−	0.924395i	$-0.624571\pi$
$139$	−8.99416	−0.762875	−0.381437	+	0.924395i	$0.624571\pi$
$140$	0	0
$141$	10.0942	0.850085
$142$	10.2258	0.858133
$143$	7.69342	0.643356
$144$	−38.1656	−3.18047
$145$	−18.8811	−1.56799
$146$	−13.0023	−1.07608
$147$	0	0
$148$	−12.5480	−1.03144
$149$	10.1159	0.828726	0.414363	−	0.910112i	$-0.364004\pi$
$149$	10.1159	0.828726	0.414363	+	0.910112i	$0.364004\pi$
$150$	−6.31142	−0.515325
$151$	17.7508	1.44454	0.722271	−	0.691610i	$-0.243098\pi$
$151$	17.7508	1.44454	0.722271	+	0.691610i	$0.243098\pi$
$152$	1.35010	0.109507
$153$	−1.06303	−0.0859407
$154$	0	0
$155$	−19.1170	−1.53551
$156$	9.33107	0.747083
$157$	5.37588	0.429042	0.214521	−	0.976719i	$-0.431181\pi$
$157$	5.37588	0.429042	0.214521	+	0.976719i	$0.431181\pi$
$158$	14.2370	1.13264
$159$	−10.5919	−0.839993
$160$	12.2342	0.967195
$161$	0	0
$162$	−49.5100	−3.88987
$163$	1.51523	0.118682	0.0593409	−	0.998238i	$-0.481100\pi$
$163$	1.51523	0.118682	0.0593409	+	0.998238i	$0.481100\pi$
$164$	−5.27682	−0.412050
$165$	21.7884	1.69623
$166$	−31.1179	−2.41522
$167$	−1.54155	−0.119289	−0.0596443	−	0.998220i	$-0.518997\pi$
$167$	−1.54155	−0.119289	−0.0596443	+	0.998220i	$0.518997\pi$
$168$	0	0
$169$	−7.74553	−0.595810
$170$	0.491295	0.0376806
$171$	7.65020	0.585026
$172$	4.87087	0.371400
$173$	−6.51043	−0.494979	−0.247490	−	0.968891i	$-0.579606\pi$
$173$	−6.51043	−0.494979	−0.247490	+	0.968891i	$0.579606\pi$
$174$	−56.1699	−4.25823
$175$	0	0
$176$	16.5830	1.24999
$177$	25.4933	1.91619
$178$	−9.14043	−0.685104
$179$	−26.0035	−1.94359	−0.971796	−	0.235823i	$-0.924221\pi$
$179$	−26.0035	−1.94359	−0.971796	+	0.235823i	$0.924221\pi$
$180$	19.0340	1.41871
$181$	−11.6143	−0.863285	−0.431643	−	0.902045i	$-0.642066\pi$
$181$	−11.6143	−0.863285	−0.431643	+	0.902045i	$0.642066\pi$
$182$	0	0
$183$	10.8936	0.805278
$184$	−8.15053	−0.600865
$185$	−20.0115	−1.47127
$186$	−56.8716	−4.17003
$187$	0.461888	0.0337766
$188$	−3.83149	−0.279440
$189$	0	0
$190$	−3.53566	−0.256504
$191$	−0.698264	−0.0505246	−0.0252623	−	0.999681i	$-0.508042\pi$
$191$	−0.698264	−0.0505246	−0.0252623	+	0.999681i	$0.508042\pi$
$192$	4.03445	0.291162
$193$	18.0329	1.29804	0.649018	−	0.760773i	$-0.275180\pi$
$193$	18.0329	1.29804	0.649018	+	0.760773i	$0.275180\pi$
$194$	−26.1017	−1.87399
$195$	14.8811	1.06566
$196$	0	0
$197$	19.0765	1.35914	0.679571	−	0.733609i	$-0.262166\pi$
$197$	19.0765	1.35914	0.679571	+	0.733609i	$0.262166\pi$
$198$	46.6867	3.31788
$199$	−9.17020	−0.650058	−0.325029	−	0.945704i	$-0.605374\pi$
$199$	−9.17020	−0.650058	−0.325029	+	0.945704i	$0.605374\pi$
$200$	−1.45888	−0.103159
$201$	27.3256	1.92740
$202$	−17.5062	−1.23173
$203$	0	0
$204$	0.560207	0.0392224
$205$	−8.41542	−0.587759
$206$	17.7747	1.23842
$207$	−46.1843	−3.21003
$208$	11.3259	0.785312
$209$	−3.32403	−0.229928
$210$	0	0
$211$	−4.83013	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$211$	−4.83013	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$212$	4.02040	0.276122
$213$	18.5955	1.27415
$214$	36.6787	2.50731
$215$	7.76802	0.529775
$216$	−21.0904	−1.43502
$217$	0	0
$218$	−11.1619	−0.755976
$219$	−23.6446	−1.59775
$220$	−8.27029	−0.557583
$221$	0.315462	0.0212202
$222$	−59.5327	−3.99557
$223$	5.63995	0.377679	0.188840	−	0.982008i	$-0.439527\pi$
$223$	5.63995	0.377679	0.188840	+	0.982008i	$0.439527\pi$
$224$	0	0
$225$	−8.26663	−0.551109
$226$	24.2698	1.61441
$227$	16.5884	1.10101	0.550505	−	0.834832i	$-0.314435\pi$
$227$	16.5884	1.10101	0.550505	+	0.834832i	$0.314435\pi$
$228$	−4.03160	−0.266999
$229$	4.83012	0.319183	0.159592	−	0.987183i	$-0.448982\pi$
$229$	4.83012	0.319183	0.159592	+	0.987183i	$0.448982\pi$
$230$	21.3448	1.40743
$231$	0	0
$232$	−12.9837	−0.852420
$233$	10.1731	0.666464	0.333232	−	0.942845i	$-0.391861\pi$
$233$	10.1731	0.666464	0.333232	+	0.942845i	$0.391861\pi$
$234$	31.8862	2.08447
$235$	−6.11042	−0.398600
$236$	−9.67656	−0.629890
$237$	25.8898	1.68173
$238$	0	0
$239$	−9.21458	−0.596042	−0.298021	−	0.954559i	$-0.596326\pi$
$239$	−9.21458	−0.596042	−0.298021	+	0.954559i	$0.596326\pi$
$240$	32.0760	2.07050
$241$	−5.67147	−0.365331	−0.182666	−	0.983175i	$-0.558473\pi$
$241$	−5.67147	−0.365331	−0.182666	+	0.983175i	$0.558473\pi$
$242$	−0.476205	−0.0306116
$243$	−43.6192	−2.79817
$244$	−4.13492	−0.264711
$245$	0	0
$246$	−25.0353	−1.59619
$247$	−2.27025	−0.144453
$248$	−13.1459	−0.834764
$249$	−56.5876	−3.58609
$250$	21.6702	1.37055
$251$	−1.26471	−0.0798277	−0.0399139	−	0.999203i	$-0.512708\pi$
$251$	−1.26471	−0.0798277	−0.0399139	+	0.999203i	$0.512708\pi$
$252$	0	0
$253$	20.0672	1.26161
$254$	−1.80084	−0.112995
$255$	0.893414	0.0559478
$256$	20.6963	1.29352
$257$	28.2125	1.75985	0.879924	−	0.475114i	$-0.157593\pi$
$257$	28.2125	1.75985	0.879924	+	0.475114i	$0.157593\pi$
$258$	23.1093	1.43872
$259$	0	0
$260$	−5.64847	−0.350303
$261$	−73.5708	−4.55392
$262$	−17.2897	−1.06816
$263$	−1.93566	−0.119358	−0.0596790	−	0.998218i	$-0.519008\pi$
$263$	−1.93566	−0.119358	−0.0596790	+	0.998218i	$0.519008\pi$
$264$	14.9829	0.922133
$265$	6.41170	0.393868
$266$	0	0
$267$	−16.6218	−1.01724
$268$	−10.3721	−0.633574
$269$	−28.9149	−1.76297	−0.881485	−	0.472212i	$-0.843456\pi$
$269$	−28.9149	−1.76297	−0.881485	+	0.472212i	$0.843456\pi$
$270$	55.2319	3.36130
$271$	15.5401	0.943992	0.471996	−	0.881601i	$-0.343534\pi$
$271$	15.5401	0.943992	0.471996	+	0.881601i	$0.343534\pi$
$272$	0.679972	0.0412294
$273$	0	0
$274$	−1.21603	−0.0734633
$275$	3.59187	0.216598
$276$	24.3387	1.46502
$277$	−2.86619	−0.172213	−0.0861064	−	0.996286i	$-0.527442\pi$
$277$	−2.86619	−0.172213	−0.0861064	+	0.996286i	$0.527442\pi$
$278$	16.1971	0.971435
$279$	−74.4899	−4.45959
$280$	0	0
$281$	−14.3969	−0.858847	−0.429423	−	0.903103i	$-0.641283\pi$
$281$	−14.3969	−0.858847	−0.429423	+	0.903103i	$0.641283\pi$
$282$	−18.1780	−1.08249
$283$	−6.49121	−0.385862	−0.192931	−	0.981212i	$-0.561799\pi$
$283$	−6.49121	−0.385862	−0.192931	+	0.981212i	$0.561799\pi$
$284$	−7.05837	−0.418837
$285$	−6.42956	−0.380854
$286$	−13.8546	−0.819241
$287$	0	0
$288$	47.6707	2.80902
$289$	−16.9811	−0.998886
$290$	34.0019	1.99666
$291$	−47.4657	−2.78249
$292$	8.97484	0.525213
$293$	0.897213	0.0524157	0.0262079	−	0.999657i	$-0.491657\pi$
$293$	0.897213	0.0524157	0.0262079	+	0.999657i	$0.491657\pi$
$294$	0	0
$295$	−15.4321	−0.898491
$296$	−13.7610	−0.799840
$297$	51.9259	3.01305
$298$	−18.2171	−1.05529
$299$	13.7055	0.792611
$300$	4.35645	0.251520
$301$	0	0
$302$	−31.9664	−1.83946
$303$	−31.8348	−1.82886
$304$	−4.89350	−0.280661
$305$	−6.59433	−0.377590
$306$	1.91434	0.109436
$307$	−5.70746	−0.325742	−0.162871	−	0.986647i	$-0.552075\pi$
$307$	−5.70746	−0.325742	−0.162871	+	0.986647i	$0.552075\pi$
$308$	0	0
$309$	32.3230	1.83879
$310$	34.4267	1.95530
$311$	−7.79332	−0.441919	−0.220959	−	0.975283i	$-0.570919\pi$
$311$	−7.79332	−0.441919	−0.220959	+	0.975283i	$0.570919\pi$
$312$	10.2330	0.579332
$313$	−1.91988	−0.108518	−0.0542590	−	0.998527i	$-0.517280\pi$
$313$	−1.91988	−0.108518	−0.0542590	+	0.998527i	$0.517280\pi$
$314$	−9.68111	−0.546337
$315$	0	0
$316$	−9.82708	−0.552817
$317$	11.2521	0.631981	0.315990	−	0.948762i	$-0.397663\pi$
$317$	11.2521	0.631981	0.315990	+	0.948762i	$0.397663\pi$
$318$	19.0743	1.06964
$319$	31.9667	1.78979
$320$	−2.44222	−0.136524
$321$	66.6998	3.72282
$322$	0	0
$323$	−0.136299	−0.00758387
$324$	34.1742	1.89857
$325$	2.45318	0.136078
$326$	−2.72869	−0.151128
$327$	−20.2977	−1.12247
$328$	−5.78690	−0.319528
$329$	0	0
$330$	−39.2375	−2.15995
$331$	−20.3709	−1.11968	−0.559842	−	0.828599i	$-0.689138\pi$
$331$	−20.3709	−1.11968	−0.559842	+	0.828599i	$0.689138\pi$
$332$	21.4791	1.17882
$333$	−77.9753	−4.27302
$334$	2.77608	0.151901
$335$	−16.5413	−0.903746
$336$	0	0
$337$	−22.1231	−1.20512	−0.602561	−	0.798073i	$-0.705853\pi$
$337$	−22.1231	−1.20512	−0.602561	+	0.798073i	$0.705853\pi$
$338$	13.9485	0.758697
$339$	44.1344	2.39705
$340$	−0.339116	−0.0183911
$341$	32.3660	1.75272
$342$	−13.7768	−0.744964
$343$	0	0
$344$	5.34171	0.288006
$345$	38.8152	2.08974
$346$	11.7243	0.630300
$347$	5.07759	0.272579	0.136290	−	0.990669i	$-0.456482\pi$
$347$	5.07759	0.272579	0.136290	+	0.990669i	$0.456482\pi$
$348$	38.7712	2.07836
$349$	0.944947	0.0505818	0.0252909	−	0.999680i	$-0.491949\pi$
$349$	0.944947	0.0505818	0.0252909	+	0.999680i	$0.491949\pi$
$350$	0	0
$351$	35.4645	1.89295
$352$	−20.7130	−1.10401
$353$	−7.50900	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$353$	−7.50900	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$354$	−45.9093	−2.44005
$355$	−11.2566	−0.597440
$356$	6.30917	0.334386
$357$	0	0
$358$	46.8281	2.47494
$359$	16.1613	0.852962	0.426481	−	0.904497i	$-0.359753\pi$
$359$	16.1613	0.852962	0.426481	+	0.904497i	$0.359753\pi$
$360$	20.8739	1.10015
$361$	−18.0191	−0.948374
$362$	20.9155	1.09930
$363$	−0.865974	−0.0454518
$364$	0	0
$365$	14.3130	0.749176
$366$	−19.6176	−1.02543
$367$	29.2953	1.52920	0.764601	−	0.644504i	$-0.222936\pi$
$367$	29.2953	1.52920	0.764601	+	0.644504i	$0.222936\pi$
$368$	29.5420	1.53999
$369$	−32.7909	−1.70703
$370$	36.0375	1.87350
$371$	0	0
$372$	39.2556	2.03531
$373$	10.4385	0.540487	0.270243	−	0.962792i	$-0.412896\pi$
$373$	10.4385	0.540487	0.270243	+	0.962792i	$0.412896\pi$
$374$	−0.831787	−0.0430107
$375$	39.4071	2.03497
$376$	−4.20185	−0.216694
$377$	21.8327	1.12444
$378$	0	0
$379$	26.5965	1.36617	0.683085	−	0.730339i	$-0.260638\pi$
$379$	26.5965	1.36617	0.683085	+	0.730339i	$0.260638\pi$
$380$	2.44049	0.125194
$381$	−3.27481	−0.167773
$382$	1.25746	0.0643374
$383$	36.7529	1.87799	0.938993	−	0.343937i	$-0.111761\pi$
$383$	36.7529	1.87799	0.938993	+	0.343937i	$0.111761\pi$
$384$	33.1554	1.69195
$385$	0	0
$386$	−32.4744	−1.65290
$387$	30.2683	1.53862
$388$	18.0167	0.914659
$389$	−22.8557	−1.15883	−0.579414	−	0.815033i	$-0.696719\pi$
$389$	−22.8557	−1.15883	−0.579414	+	0.815033i	$0.696719\pi$
$390$	−26.7985	−1.35700
$391$	0.822836	0.0416126
$392$	0	0
$393$	−31.4411	−1.58599
$394$	−34.3537	−1.73071
$395$	−15.6722	−0.788552
$396$	−32.2254	−1.61939
$397$	9.08520	0.455973	0.227987	−	0.973664i	$-0.426786\pi$
$397$	9.08520	0.455973	0.227987	+	0.973664i	$0.426786\pi$
$398$	16.5141	0.827775
$399$	0	0
$400$	5.28780	0.264390
$401$	32.8290	1.63940	0.819701	−	0.572792i	$-0.194140\pi$
$401$	32.8290	1.63940	0.819701	+	0.572792i	$0.194140\pi$
$402$	−49.2091	−2.45432
$403$	22.1054	1.10115
$404$	12.0836	0.601183
$405$	54.5008	2.70817
$406$	0	0
$407$	33.8804	1.67939
$408$	0.614360	0.0304153
$409$	29.5854	1.46290	0.731452	−	0.681893i	$-0.238843\pi$
$409$	29.5854	1.46290	0.731452	+	0.681893i	$0.238843\pi$
$410$	15.1548	0.748444
$411$	−2.21134	−0.109078
$412$	−12.2689	−0.604447
$413$	0	0
$414$	83.1705	4.08761
$415$	34.2547	1.68150
$416$	−14.1466	−0.693596
$417$	29.4542	1.44238
$418$	5.98605	0.292787
$419$	1.88338	0.0920092	0.0460046	−	0.998941i	$-0.485351\pi$
$419$	1.88338	0.0920092	0.0460046	+	0.998941i	$0.485351\pi$
$420$	0	0
$421$	−4.88170	−0.237920	−0.118960	−	0.992899i	$-0.537956\pi$
$421$	−4.88170	−0.237920	−0.118960	+	0.992899i	$0.537956\pi$
$422$	8.69830	0.423426
$423$	−23.8094	−1.15765
$424$	4.40903	0.214121
$425$	0.147281	0.00714419
$426$	−33.4876	−1.62248
$427$	0	0
$428$	−25.3174	−1.22376
$429$	−25.1945	−1.21640
$430$	−13.9890	−0.674608
$431$	−14.4183	−0.694506	−0.347253	−	0.937771i	$-0.612885\pi$
$431$	−14.4183	−0.694506	−0.347253	+	0.937771i	$0.612885\pi$
$432$	76.4431	3.67787
$433$	−4.06302	−0.195256	−0.0976280	−	0.995223i	$-0.531126\pi$
$433$	−4.06302	−0.195256	−0.0976280	+	0.995223i	$0.531126\pi$
$434$	0	0
$435$	61.8320	2.96462
$436$	7.70446	0.368977
$437$	−5.92163	−0.283270
$438$	42.5801	2.03456
$439$	10.8643	0.518526	0.259263	−	0.965807i	$-0.416520\pi$
$439$	10.8643	0.518526	0.259263	+	0.965807i	$0.416520\pi$
$440$	−9.06974	−0.432383
$441$	0	0
$442$	−0.568096	−0.0270216
$443$	−9.86391	−0.468649	−0.234324	−	0.972158i	$-0.575288\pi$
$443$	−9.86391	−0.468649	−0.234324	+	0.972158i	$0.575288\pi$
$444$	41.0923	1.95016
$445$	10.0618	0.476976
$446$	−10.1567	−0.480932
$447$	−33.1276	−1.56688
$448$	0	0
$449$	30.0837	1.41974	0.709869	−	0.704334i	$-0.248754\pi$
$449$	30.0837	1.41974	0.709869	+	0.704334i	$0.248754\pi$
$450$	14.8869	0.701774
$451$	14.2477	0.670899
$452$	−16.7522	−0.787959
$453$	−58.1305	−2.73121
$454$	−29.8730	−1.40201
$455$	0	0
$456$	−4.42131	−0.207047
$457$	−10.6100	−0.496317	−0.248158	−	0.968719i	$-0.579825\pi$
$457$	−10.6100	−0.496317	−0.248158	+	0.968719i	$0.579825\pi$
$458$	−8.69827	−0.406444
$459$	2.12917	0.0993813
$460$	−14.7332	−0.686940
$461$	29.4065	1.36960	0.684799	−	0.728732i	$-0.259890\pi$
$461$	29.4065	1.36960	0.684799	+	0.728732i	$0.259890\pi$
$462$	0	0
$463$	−14.6163	−0.679278	−0.339639	−	0.940556i	$-0.610305\pi$
$463$	−14.6163	−0.679278	−0.339639	+	0.940556i	$0.610305\pi$
$464$	47.0600	2.18471
$465$	62.6045	2.90321
$466$	−18.3202	−0.848666
$467$	−10.3293	−0.477982	−0.238991	−	0.971022i	$-0.576817\pi$
$467$	−10.3293	−0.477982	−0.238991	+	0.971022i	$0.576817\pi$
$468$	−22.0094	−1.01739
$469$	0	0
$470$	11.0039	0.507572
$471$	−17.6050	−0.811195
$472$	−10.6119	−0.488454
$473$	−13.1516	−0.604713
$474$	−46.6235	−2.14149
$475$	−1.05993	−0.0486328
$476$	0	0
$477$	24.9834	1.14391
$478$	16.5940	0.758992
$479$	8.63906	0.394729	0.197364	−	0.980330i	$-0.436762\pi$
$479$	8.63906	0.394729	0.197364	+	0.980330i	$0.436762\pi$
$480$	−40.0645	−1.82869
$481$	23.1397	1.05508
$482$	10.2134	0.465208
$483$	0	0
$484$	0.328700	0.0149409
$485$	28.7329	1.30469
$486$	78.5513	3.56316
$487$	−2.28698	−0.103633	−0.0518165	−	0.998657i	$-0.516501\pi$
$487$	−2.28698	−0.103633	−0.0518165	+	0.998657i	$0.516501\pi$
$488$	−4.53461	−0.205272
$489$	−4.96208	−0.224393
$490$	0	0
$491$	−25.1332	−1.13425	−0.567123	−	0.823633i	$-0.691944\pi$
$491$	−25.1332	−1.13425	−0.567123	+	0.823633i	$0.691944\pi$
$492$	17.2806	0.779068
$493$	1.31076	0.0590339
$494$	4.08837	0.183944
$495$	−51.3928	−2.30994
$496$	47.6479	2.13945
$497$	0	0
$498$	101.905	4.56648
$499$	−13.5966	−0.608666	−0.304333	−	0.952566i	$-0.598434\pi$
$499$	−13.5966	−0.608666	−0.304333	+	0.952566i	$0.598434\pi$
$500$	−14.9579	−0.668936
$501$	5.04827	0.225540
$502$	2.27754	0.101652
$503$	−12.2279	−0.545217	−0.272609	−	0.962125i	$-0.587886\pi$
$503$	−12.2279	−0.545217	−0.272609	+	0.962125i	$0.587886\pi$
$504$	0	0
$505$	19.2709	0.857543
$506$	−36.1378	−1.60652
$507$	25.3651	1.12650
$508$	1.24303	0.0551505
$509$	−15.6079	−0.691808	−0.345904	−	0.938270i	$-0.612428\pi$
$509$	−15.6079	−0.691808	−0.345904	+	0.938270i	$0.612428\pi$
$510$	−1.60890	−0.0712432
$511$	0	0
$512$	−17.0220	−0.752275
$513$	−15.3228	−0.676520
$514$	−50.8063	−2.24097
$515$	−19.5664	−0.862199
$516$	−15.9512	−0.702211
$517$	10.3452	0.454983
$518$	0	0
$519$	21.3204	0.935862
$520$	−6.19447	−0.271646
$521$	40.5639	1.77714	0.888569	−	0.458743i	$-0.151700\pi$
$521$	40.5639	1.77714	0.888569	+	0.458743i	$0.151700\pi$
$522$	132.489	5.79890
$523$	−11.4330	−0.499929	−0.249964	−	0.968255i	$-0.580419\pi$
$523$	−11.4330	−0.499929	−0.249964	+	0.968255i	$0.580419\pi$
$524$	11.9342	0.521347
$525$	0	0
$526$	3.48582	0.151989
$527$	1.32714	0.0578111
$528$	−54.3063	−2.36338
$529$	12.7489	0.554299
$530$	−11.5465	−0.501546
$531$	−60.1316	−2.60949
$532$	0	0
$533$	9.73095	0.421494
$534$	29.9332	1.29533
$535$	−40.3761	−1.74561
$536$	−11.3747	−0.491311
$537$	85.1564	3.67477
$538$	52.0711	2.24494
$539$	0	0
$540$	−38.1237	−1.64058
$541$	7.22979	0.310833	0.155416	−	0.987849i	$-0.450328\pi$
$541$	7.22979	0.310833	0.155416	+	0.987849i	$0.450328\pi$
$542$	−27.9852	−1.20207
$543$	38.0346	1.63222
$544$	−0.849319	−0.0364142
$545$	12.2870	0.526318
$546$	0	0
$547$	28.3399	1.21173	0.605863	−	0.795569i	$-0.292828\pi$
$547$	28.3399	1.21173	0.605863	+	0.795569i	$0.292828\pi$
$548$	0.839366	0.0358559
$549$	−25.6950	−1.09664
$550$	−6.46838	−0.275813
$551$	−9.43306	−0.401862
$552$	26.6914	1.13606
$553$	0	0
$554$	5.16156	0.219293
$555$	65.5337	2.78175
$556$	−11.1800	−0.474138
$557$	21.2960	0.902341	0.451171	−	0.892438i	$-0.351007\pi$
$557$	21.2960	0.902341	0.451171	+	0.892438i	$0.351007\pi$
$558$	134.144	5.67879
$559$	−8.98235	−0.379913
$560$	0	0
$561$	−1.51259	−0.0638618
$562$	25.9265	1.09364
$563$	34.6578	1.46065	0.730326	−	0.683099i	$-0.239368\pi$
$563$	34.6578	1.46065	0.730326	+	0.683099i	$0.239368\pi$
$564$	12.5474	0.528340
$565$	−26.7163	−1.12396
$566$	11.6896	0.491352
$567$	0	0
$568$	−7.74066	−0.324791
$569$	38.0599	1.59555	0.797777	−	0.602952i	$-0.206009\pi$
$569$	38.0599	1.59555	0.797777	+	0.602952i	$0.206009\pi$
$570$	11.5786	0.484975
$571$	−46.9386	−1.96432	−0.982160	−	0.188049i	$-0.939784\pi$
$571$	−46.9386	−1.96432	−0.982160	+	0.188049i	$0.939784\pi$
$572$	9.56314	0.399855
$573$	2.28668	0.0955274
$574$	0	0
$575$	6.39877	0.266847
$576$	−9.51616	−0.396507
$577$	2.55401	0.106325	0.0531624	−	0.998586i	$-0.483070\pi$
$577$	2.55401	0.106325	0.0531624	+	0.998586i	$0.483070\pi$
$578$	30.5802	1.27197
$579$	−59.0542	−2.45421
$580$	−23.4698	−0.974529
$581$	0	0
$582$	85.4781	3.54318
$583$	−10.8553	−0.449582
$584$	9.84238	0.407281
$585$	−35.1004	−1.45122
$586$	−1.61574	−0.0667455
$587$	−39.0979	−1.61374	−0.806871	−	0.590727i	$-0.798841\pi$
$587$	−39.0979	−1.61374	−0.806871	+	0.590727i	$0.798841\pi$
$588$	0	0
$589$	−9.55091	−0.393538
$590$	27.7907	1.14413
$591$	−62.4718	−2.56975
$592$	49.8773	2.04995
$593$	22.4591	0.922283	0.461141	−	0.887327i	$-0.347440\pi$
$593$	22.4591	0.922283	0.461141	+	0.887327i	$0.347440\pi$
$594$	−93.5103	−3.83677
$595$	0	0
$596$	12.5743	0.515065
$597$	30.0306	1.22907
$598$	−24.6815	−1.00930
$599$	−23.0291	−0.940942	−0.470471	−	0.882415i	$-0.655916\pi$
$599$	−23.0291	−0.940942	−0.470471	+	0.882415i	$0.655916\pi$
$600$	4.77756	0.195043
$601$	−19.0265	−0.776106	−0.388053	−	0.921637i	$-0.626852\pi$
$601$	−19.0265	−0.776106	−0.388053	+	0.921637i	$0.626852\pi$
$602$	0	0
$603$	−64.4535	−2.62475
$604$	22.0648	0.897804
$605$	0.524208	0.0213121
$606$	57.3294	2.32885
$607$	−3.24590	−0.131747	−0.0658735	−	0.997828i	$-0.520983\pi$
$607$	−3.24590	−0.131747	−0.0658735	+	0.997828i	$0.520983\pi$
$608$	6.11222	0.247883
$609$	0	0
$610$	11.8753	0.480819
$611$	7.06562	0.285845
$612$	−1.32137	−0.0534134
$613$	18.9340	0.764735	0.382368	−	0.924010i	$-0.375109\pi$
$613$	18.9340	0.764735	0.382368	+	0.924010i	$0.375109\pi$
$614$	10.2782	0.414796
$615$	27.5589	1.11128
$616$	0	0
$617$	−5.25658	−0.211622	−0.105811	−	0.994386i	$-0.533744\pi$
$617$	−5.25658	−0.211622	−0.105811	+	0.994386i	$0.533744\pi$
$618$	−58.2086	−2.34149
$619$	−14.8394	−0.596445	−0.298222	−	0.954496i	$-0.596394\pi$
$619$	−14.8394	−0.596445	−0.298222	+	0.954496i	$0.596394\pi$
$620$	−23.7630	−0.954344
$621$	92.5040	3.71206
$622$	14.0345	0.562734
$623$	0	0
$624$	−37.0902	−1.48480
$625$	−18.5037	−0.740146
$626$	3.45740	0.138185
$627$	10.8856	0.434727
$628$	6.68238	0.266656
$629$	1.38924	0.0553925
$630$	0	0
$631$	−22.6218	−0.900558	−0.450279	−	0.892888i	$-0.648675\pi$
$631$	−22.6218	−0.900558	−0.450279	+	0.892888i	$0.648675\pi$
$632$	−10.7770	−0.428686
$633$	15.8177	0.628699
$634$	−20.2632	−0.804756
$635$	1.98237	0.0786680
$636$	−13.1660	−0.522067
$637$	0	0
$638$	−57.5669	−2.27909
$639$	−43.8617	−1.73514
$640$	−20.0703	−0.793347
$641$	−7.00436	−0.276656	−0.138328	−	0.990386i	$-0.544173\pi$
$641$	−7.00436	−0.276656	−0.138328	+	0.990386i	$0.544173\pi$
$642$	−120.116	−4.74059
$643$	−32.6067	−1.28588	−0.642940	−	0.765916i	$-0.722286\pi$
$643$	−32.6067	−1.28588	−0.642940	+	0.765916i	$0.722286\pi$
$644$	0	0
$645$	−25.4388	−1.00165
$646$	0.245452	0.00965720
$647$	31.6983	1.24619	0.623095	−	0.782146i	$-0.285875\pi$
$647$	31.6983	1.24619	0.623095	+	0.782146i	$0.285875\pi$
$648$	37.4777	1.47226
$649$	26.1273	1.02559
$650$	−4.41779	−0.173280
$651$	0	0
$652$	1.88347	0.0737625
$653$	−30.3739	−1.18862	−0.594312	−	0.804235i	$-0.702575\pi$
$653$	−30.3739	−1.18862	−0.594312	+	0.804235i	$0.702575\pi$
$654$	36.5529	1.42933
$655$	19.0325	0.743663
$656$	20.9749	0.818933
$657$	55.7710	2.17583
$658$	0	0
$659$	0.151867	0.00591590	0.00295795	−	0.999996i	$-0.499058\pi$
$659$	0.151867	0.00591590	0.00295795	+	0.999996i	$0.499058\pi$
$660$	27.0836	1.05423
$661$	−48.2442	−1.87648	−0.938242	−	0.345981i	$-0.887546\pi$
$661$	−48.2442	−1.87648	−0.938242	+	0.345981i	$0.887546\pi$
$662$	36.6847	1.42579
$663$	−1.03308	−0.0401213
$664$	23.5554	0.914126
$665$	0	0
$666$	140.421	5.44121
$667$	56.9474	2.20501
$668$	−1.91619	−0.0741396
$669$	−18.4698	−0.714082
$670$	29.7882	1.15082
$671$	11.1645	0.431002
$672$	0	0
$673$	−17.1107	−0.659570	−0.329785	−	0.944056i	$-0.606976\pi$
$673$	−17.1107	−0.659570	−0.329785	+	0.944056i	$0.606976\pi$
$674$	39.8402	1.53459
$675$	16.5575	0.637299
$676$	−9.62792	−0.370304
$677$	1.94027	0.0745706	0.0372853	−	0.999305i	$-0.488129\pi$
$677$	1.94027	0.0745706	0.0372853	+	0.999305i	$0.488129\pi$
$678$	−79.4790	−3.05237
$679$	0	0
$680$	−0.371896	−0.0142616
$681$	−54.3237	−2.08169
$682$	−58.2860	−2.23189
$683$	12.4756	0.477365	0.238683	−	0.971098i	$-0.423284\pi$
$683$	12.4756	0.477365	0.238683	+	0.971098i	$0.423284\pi$
$684$	9.50942	0.363602
$685$	1.33861	0.0511458
$686$	0	0
$687$	−15.8177	−0.603483
$688$	−19.3613	−0.738143
$689$	−7.41400	−0.282451
$690$	−69.9001	−2.66105
$691$	−3.87191	−0.147295	−0.0736473	−	0.997284i	$-0.523464\pi$
$691$	−3.87191	−0.147295	−0.0736473	+	0.997284i	$0.523464\pi$
$692$	−8.09265	−0.307637
$693$	0	0
$694$	−9.14392	−0.347099
$695$	−17.8298	−0.676322
$696$	42.5190	1.61168
$697$	0.584215	0.0221287
$698$	−1.70170	−0.0644103
$699$	−33.3150	−1.26009
$700$	0	0
$701$	−32.6013	−1.23133	−0.615666	−	0.788007i	$-0.711113\pi$
$701$	−32.6013	−1.23133	−0.615666	+	0.788007i	$0.711113\pi$
$702$	−63.8659	−2.41046
$703$	−9.99779	−0.377074
$704$	4.13479	0.155836
$705$	20.0105	0.753637
$706$	13.5225	0.508927
$707$	0	0
$708$	31.6889	1.19094
$709$	−13.4445	−0.504918	−0.252459	−	0.967608i	$-0.581239\pi$
$709$	−13.4445	−0.504918	−0.252459	+	0.967608i	$0.581239\pi$
$710$	20.2714	0.760772
$711$	−61.0670	−2.29019
$712$	6.91905	0.259302
$713$	57.6588	2.15934
$714$	0	0
$715$	15.2512	0.570363
$716$	−32.3231	−1.20797
$717$	30.1760	1.12694
$718$	−29.1040	−1.08615
$719$	−12.1949	−0.454792	−0.227396	−	0.973802i	$-0.573021\pi$
$719$	−12.1949	−0.454792	−0.227396	+	0.973802i	$0.573021\pi$
$720$	−75.6584	−2.81962
$721$	0	0
$722$	32.4496	1.20765
$723$	18.5730	0.690736
$724$	−14.4369	−0.536544
$725$	10.1931	0.378564
$726$	1.55948	0.0578778
$727$	−10.5423	−0.390994	−0.195497	−	0.980704i	$-0.562632\pi$
$727$	−10.5423	−0.390994	−0.195497	+	0.980704i	$0.562632\pi$
$728$	0	0
$729$	60.3663	2.23579
$730$	−25.7754	−0.953991
$731$	−0.539271	−0.0199457
$732$	13.5411	0.500492
$733$	−52.6760	−1.94563	−0.972816	−	0.231581i	$-0.925610\pi$
$733$	−52.6760	−1.94563	−0.972816	+	0.231581i	$0.925610\pi$
$734$	−52.7562	−1.94727
$735$	0	0
$736$	−36.8995	−1.36013
$737$	28.0052	1.03158
$738$	59.0512	2.17371
$739$	23.8910	0.878844	0.439422	−	0.898281i	$-0.355183\pi$
$739$	23.8910	0.878844	0.439422	+	0.898281i	$0.355183\pi$
$740$	−24.8748	−0.914417
$741$	7.43465	0.273119
$742$	0	0
$743$	−15.2274	−0.558641	−0.279320	−	0.960198i	$-0.590109\pi$
$743$	−15.2274	−0.558641	−0.279320	+	0.960198i	$0.590109\pi$
$744$	43.0502	1.57830
$745$	20.0534	0.734701
$746$	−18.7981	−0.688249
$747$	133.474	4.88357
$748$	0.574140	0.0209926
$749$	0	0
$750$	−70.9659	−2.59131
$751$	−15.6849	−0.572349	−0.286175	−	0.958178i	$-0.592384\pi$
$751$	−15.6849	−0.572349	−0.286175	+	0.958178i	$0.592384\pi$
$752$	15.2298	0.555375
$753$	4.14168	0.150931
$754$	−39.3172	−1.43185
$755$	35.1887	1.28065
$756$	0	0
$757$	−48.0778	−1.74742	−0.873709	−	0.486449i	$-0.838292\pi$
$757$	−48.0778	−1.74742	−0.873709	+	0.486449i	$0.838292\pi$
$758$	−47.8960	−1.73966
$759$	−65.7161	−2.38534
$760$	2.67639	0.0970830
$761$	29.1430	1.05643	0.528217	−	0.849110i	$-0.322861\pi$
$761$	29.1430	1.05643	0.528217	+	0.849110i	$0.322861\pi$
$762$	5.89741	0.213641
$763$	0	0
$764$	−0.867962	−0.0314018
$765$	−2.10732	−0.0761902
$766$	−66.1861	−2.39140
$767$	17.8445	0.644327
$768$	−67.7764	−2.44567
$769$	−5.94776	−0.214482	−0.107241	−	0.994233i	$-0.534202\pi$
$769$	−5.94776	−0.214482	−0.107241	+	0.994233i	$0.534202\pi$
$770$	0	0
$771$	−92.3906	−3.32737
$772$	22.4154	0.806748
$773$	33.0476	1.18864	0.594321	−	0.804228i	$-0.297421\pi$
$773$	33.0476	1.18864	0.594321	+	0.804228i	$0.297421\pi$
$774$	−54.5084	−1.95926
$775$	10.3205	0.370723
$776$	19.7583	0.709280
$777$	0	0
$778$	41.1594	1.47564
$779$	−4.20437	−0.150637
$780$	18.4976	0.662322
$781$	19.0580	0.681950
$782$	−1.48180	−0.0529889
$783$	147.357	5.26612
$784$	0	0
$785$	10.6570	0.380365
$786$	56.6204	2.01958
$787$	−49.3581	−1.75943	−0.879713	−	0.475506i	$-0.842265\pi$
$787$	−49.3581	−1.75943	−0.879713	+	0.475506i	$0.842265\pi$
$788$	23.7126	0.844727
$789$	6.33892	0.225671
$790$	28.2231	1.00413
$791$	0	0
$792$	−35.3405	−1.25577
$793$	7.62518	0.270778
$794$	−16.3610	−0.580630
$795$	−20.9971	−0.744690
$796$	−11.3988	−0.404020
$797$	4.85688	0.172040	0.0860198	−	0.996293i	$-0.472585\pi$
$797$	4.85688	0.172040	0.0860198	+	0.996293i	$0.472585\pi$
$798$	0	0
$799$	0.424197	0.0150070
$800$	−6.60472	−0.233512
$801$	39.2061	1.38528
$802$	−59.1198	−2.08759
$803$	−24.2326	−0.855150
$804$	33.9665	1.19791
$805$	0	0
$806$	−39.8084	−1.40219
$807$	94.6906	3.33327
$808$	13.2517	0.466193
$809$	−20.7623	−0.729964	−0.364982	−	0.931015i	$-0.618925\pi$
$809$	−20.7623	−0.729964	−0.364982	+	0.931015i	$0.618925\pi$
$810$	−98.1473	−3.44854
$811$	6.06265	0.212888	0.106444	−	0.994319i	$-0.466053\pi$
$811$	6.06265	0.212888	0.106444	+	0.994319i	$0.466053\pi$
$812$	0	0
$813$	−50.8907	−1.78482
$814$	−61.0132	−2.13851
$815$	3.00375	0.105217
$816$	−2.22678	−0.0779528
$817$	3.88093	0.135776
$818$	−53.2786	−1.86284
$819$	0	0
$820$	−10.4606	−0.365300
$821$	−49.7361	−1.73580	−0.867901	−	0.496736i	$-0.834532\pi$
$821$	−49.7361	−1.73580	−0.867901	+	0.496736i	$0.834532\pi$
$822$	3.98228	0.138898
$823$	−9.35498	−0.326094	−0.163047	−	0.986618i	$-0.552132\pi$
$823$	−9.35498	−0.326094	−0.163047	+	0.986618i	$0.552132\pi$
$824$	−13.4549	−0.468724
$825$	−11.7627	−0.409524
$826$	0	0
$827$	22.4842	0.781852	0.390926	−	0.920422i	$-0.372155\pi$
$827$	22.4842	0.781852	0.390926	+	0.920422i	$0.372155\pi$
$828$	−57.4084	−1.99508
$829$	−10.2099	−0.354606	−0.177303	−	0.984156i	$-0.556737\pi$
$829$	−10.2099	−0.354606	−0.177303	+	0.984156i	$0.556737\pi$
$830$	−61.6873	−2.14120
$831$	9.38623	0.325605
$832$	2.82399	0.0979043
$833$	0	0
$834$	−53.0422	−1.83670
$835$	−3.05592	−0.105755
$836$	−4.13186	−0.142903
$837$	149.198	5.15705
$838$	−3.39167	−0.117163
$839$	−31.9667	−1.10361	−0.551806	−	0.833972i	$-0.686061\pi$
$839$	−31.9667	−1.10361	−0.551806	+	0.833972i	$0.686061\pi$
$840$	0	0
$841$	61.7163	2.12815
$842$	8.79117	0.302964
$843$	47.1471	1.62383
$844$	−6.00399	−0.206666
$845$	−15.3545	−0.528212
$846$	42.8770	1.47414
$847$	0	0
$848$	−15.9808	−0.548782
$849$	21.2575	0.729554
$850$	−0.265230	−0.00909732
$851$	60.3567	2.06900
$852$	23.1148	0.791900
$853$	49.9952	1.71180	0.855902	−	0.517138i	$-0.173003\pi$
$853$	49.9952	1.71180	0.855902	+	0.517138i	$0.173003\pi$
$854$	0	0
$855$	15.1655	0.518651
$856$	−27.7647	−0.948979
$857$	42.1916	1.44124	0.720618	−	0.693332i	$-0.243858\pi$
$857$	42.1916	1.44124	0.720618	+	0.693332i	$0.243858\pi$
$858$	45.3712	1.54895
$859$	−53.7062	−1.83243	−0.916216	−	0.400685i	$-0.868772\pi$
$859$	−53.7062	−1.83243	−0.916216	+	0.400685i	$0.868772\pi$
$860$	9.65587	0.329263
$861$	0	0
$862$	25.9651	0.884375
$863$	2.67063	0.0909093	0.0454547	−	0.998966i	$-0.485526\pi$
$863$	2.67063	0.0909093	0.0454547	+	0.998966i	$0.485526\pi$
$864$	−95.4812	−3.24834
$865$	−12.9061	−0.438821
$866$	7.31685	0.248637
$867$	55.6097	1.88860
$868$	0	0
$869$	26.5337	0.900095
$870$	−111.350	−3.77511
$871$	19.1271	0.648096
$872$	8.44921	0.286126
$873$	111.958	3.78922
$874$	10.6639	0.360712
$875$	0	0
$876$	−29.3909	−0.993025
$877$	39.5491	1.33548	0.667739	−	0.744395i	$-0.267262\pi$
$877$	39.5491	1.33548	0.667739	+	0.744395i	$0.267262\pi$
$878$	−19.5649	−0.660284
$879$	−2.93820	−0.0991030
$880$	32.8737	1.10817
$881$	−1.82159	−0.0613709	−0.0306854	−	0.999529i	$-0.509769\pi$
$881$	−1.82159	−0.0613709	−0.0306854	+	0.999529i	$0.509769\pi$
$882$	0	0
$883$	14.6049	0.491493	0.245746	−	0.969334i	$-0.420967\pi$
$883$	14.6049	0.491493	0.245746	+	0.969334i	$0.420967\pi$
$884$	0.392128	0.0131887
$885$	50.5371	1.69879
$886$	17.7633	0.596771
$887$	24.3065	0.816132	0.408066	−	0.912952i	$-0.366203\pi$
$887$	24.3065	0.816132	0.408066	+	0.912952i	$0.366203\pi$
$888$	45.0645	1.51227
$889$	0	0
$890$	−18.1197	−0.607375
$891$	−92.2725	−3.09125
$892$	7.01062	0.234733
$893$	−3.05278	−0.102158
$894$	59.6575	1.99525
$895$	−51.5486	−1.72308
$896$	0	0
$897$	−44.8830	−1.49860
$898$	−54.1760	−1.80788
$899$	91.8496	3.06335
$900$	−10.2757	−0.342522
$901$	−0.445113	−0.0148289
$902$	−25.6579	−0.854314
$903$	0	0
$904$	−18.3716	−0.611029
$905$	−23.0239	−0.765340
$906$	104.684	3.47789
$907$	−32.8383	−1.09038	−0.545190	−	0.838313i	$-0.683542\pi$
$907$	−32.8383	−1.09038	−0.545190	+	0.838313i	$0.683542\pi$
$908$	20.6198	0.684293
$909$	75.0895	2.49056
$910$	0	0
$911$	43.3639	1.43671	0.718355	−	0.695676i	$-0.244895\pi$
$911$	43.3639	1.43671	0.718355	+	0.695676i	$0.244895\pi$
$912$	16.0253	0.530650
$913$	−57.9949	−1.91935
$914$	19.1070	0.632003
$915$	21.5952	0.713914
$916$	6.00397	0.198377
$917$	0	0
$918$	−3.83430	−0.126551
$919$	−15.3957	−0.507858	−0.253929	−	0.967223i	$-0.581723\pi$
$919$	−15.3957	−0.507858	−0.253929	+	0.967223i	$0.581723\pi$
$920$	−16.1574	−0.532693
$921$	18.6908	0.615884
$922$	−52.9565	−1.74403
$923$	13.0163	0.428437
$924$	0	0
$925$	10.8034	0.355213
$926$	26.3217	0.864984
$927$	−76.2410	−2.50408
$928$	−58.7803	−1.92956
$929$	−29.4087	−0.964869	−0.482435	−	0.875932i	$-0.660247\pi$
$929$	−29.4087	−0.964869	−0.482435	+	0.875932i	$0.660247\pi$
$930$	−112.741	−3.69691
$931$	0	0
$932$	12.6455	0.414217
$933$	25.5216	0.835541
$934$	18.6014	0.608656
$935$	0.915634	0.0299444
$936$	−24.1369	−0.788940
$937$	−25.1984	−0.823196	−0.411598	−	0.911365i	$-0.635029\pi$
$937$	−25.1984	−0.823196	−0.411598	+	0.911365i	$0.635029\pi$
$938$	0	0
$939$	6.28724	0.205176
$940$	−7.59543	−0.247736
$941$	−49.1919	−1.60361	−0.801805	−	0.597586i	$-0.796126\pi$
$941$	−49.1919	−1.60361	−0.801805	+	0.597586i	$0.796126\pi$
$942$	31.7038	1.03297
$943$	25.3818	0.826544
$944$	38.4635	1.25188
$945$	0	0
$946$	23.6840	0.770034
$947$	33.0148	1.07284	0.536419	−	0.843952i	$-0.319777\pi$
$947$	33.0148	1.07284	0.536419	+	0.843952i	$0.319777\pi$
$948$	32.1818	1.04522
$949$	−16.5505	−0.537250
$950$	1.90876	0.0619283
$951$	−36.8485	−1.19489
$952$	0	0
$953$	35.8422	1.16104	0.580522	−	0.814244i	$-0.302848\pi$
$953$	35.8422	1.16104	0.580522	+	0.814244i	$0.302848\pi$
$954$	−44.9911	−1.45664
$955$	−1.38422	−0.0447923
$956$	−11.4540	−0.370448
$957$	−104.685	−3.38398
$958$	−15.5576	−0.502642
$959$	0	0
$960$	7.99779	0.258128
$961$	61.9970	1.99990
$962$	−41.6710	−1.34353
$963$	−157.326	−5.06977
$964$	−7.04979	−0.227059
$965$	35.7479	1.15077
$966$	0	0
$967$	35.5185	1.14220	0.571100	−	0.820881i	$-0.306517\pi$
$967$	35.5185	1.14220	0.571100	+	0.820881i	$0.306517\pi$
$968$	0.360474	0.0115861
$969$	0.446352	0.0143389
$970$	−51.7433	−1.66138
$971$	27.3670	0.878248	0.439124	−	0.898427i	$-0.355289\pi$
$971$	27.3670	0.878248	0.439124	+	0.898427i	$0.355289\pi$
$972$	−54.2199	−1.73910
$973$	0	0
$974$	4.11849	0.131965
$975$	−8.03370	−0.257284
$976$	16.4360	0.526102
$977$	−48.2170	−1.54260	−0.771299	−	0.636472i	$-0.780393\pi$
$977$	−48.2170	−1.54260	−0.771299	+	0.636472i	$0.780393\pi$
$978$	8.93592	0.285739
$979$	−17.0352	−0.544446
$980$	0	0
$981$	47.8767	1.52858
$982$	45.2610	1.44434
$983$	−26.1721	−0.834762	−0.417381	−	0.908732i	$-0.637052\pi$
$983$	−26.1721	−0.834762	−0.417381	+	0.908732i	$0.637052\pi$
$984$	18.9510	0.604135
$985$	37.8167	1.20494
$986$	−2.36048	−0.0751729
$987$	0	0
$988$	−2.82199	−0.0897795
$989$	−23.4292	−0.745004
$990$	92.5503	2.94144
$991$	−29.0448	−0.922639	−0.461319	−	0.887234i	$-0.652624\pi$
$991$	−29.0448	−0.922639	−0.461319	+	0.887234i	$0.652624\pi$
$992$	−59.5146	−1.88959
$993$	66.7107	2.11700
$994$	0	0
$995$	−18.1787	−0.576305
$996$	−70.3400	−2.22881
$997$	−18.4810	−0.585299	−0.292649	−	0.956220i	$-0.594537\pi$
$997$	−18.4810	−0.585299	−0.292649	+	0.956220i	$0.594537\pi$
$998$	24.4853	0.775068
$999$	156.179	4.94129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.p.1.9		38
7.2	even	3		889.2.f.c.382.30	yes	76
7.4	even	3		889.2.f.c.128.30	✓	76
7.6	odd	2		6223.2.a.o.1.9		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.f.c.128.30	✓	76	7.4	even	3
889.2.f.c.382.30	yes	76	7.2	even	3
6223.2.a.o.1.9		38	7.6	odd	2
6223.2.a.p.1.9		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)

\(f(q)\)	\(=\)	\(q-1.80084 q^{2} -3.27481 q^{3} +1.24303 q^{4} +1.98237 q^{5} +5.89741 q^{6} +1.36319 q^{8} +7.72436 q^{9} -3.56994 q^{10} -3.35625 q^{11} -4.07068 q^{12} -2.29226 q^{13} -6.49189 q^{15} -4.94094 q^{16} -0.137620 q^{17} -13.9104 q^{18} +0.990399 q^{19} +2.46415 q^{20} +6.04408 q^{22} -5.97904 q^{23} -4.46417 q^{24} -1.07020 q^{25} +4.12800 q^{26} -15.4714 q^{27} -9.52451 q^{29} +11.6909 q^{30} -9.64350 q^{31} +6.17147 q^{32} +10.9911 q^{33} +0.247832 q^{34} +9.60161 q^{36} -10.0947 q^{37} -1.78355 q^{38} +7.50672 q^{39} +2.70234 q^{40} -4.24513 q^{41} +3.91855 q^{43} -4.17192 q^{44} +15.3126 q^{45} +10.7673 q^{46} -3.08238 q^{47} +16.1806 q^{48} +1.92726 q^{50} +0.450679 q^{51} -2.84935 q^{52} +3.23436 q^{53} +27.8615 q^{54} -6.65334 q^{55} -3.24337 q^{57} +17.1521 q^{58} -7.78466 q^{59} -8.06960 q^{60} -3.32649 q^{61} +17.3664 q^{62} -1.23197 q^{64} -4.54412 q^{65} -19.7932 q^{66} -8.34418 q^{67} -0.171066 q^{68} +19.5802 q^{69} -5.67836 q^{71} +10.5297 q^{72} +7.22014 q^{73} +18.1790 q^{74} +3.50470 q^{75} +1.23109 q^{76} -13.5184 q^{78} -7.90576 q^{79} -9.79477 q^{80} +27.4927 q^{81} +7.64480 q^{82} +17.2797 q^{83} -0.272814 q^{85} -7.05669 q^{86} +31.1909 q^{87} -4.57519 q^{88} +5.07565 q^{89} -27.5755 q^{90} -7.43211 q^{92} +31.5806 q^{93} +5.55087 q^{94} +1.96334 q^{95} -20.2104 q^{96} +14.4942 q^{97} -25.9249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 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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