LMFDB - Embedded newform 6223.2.a.t.1.38

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.38
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)

$f(q)$	$=$	\(q+0.269622 q^{2} +0.274466 q^{3} -1.92730 q^{4} +2.02179 q^{5} +0.0740019 q^{6} -1.05889 q^{8} -2.92467 q^{9} +0.545119 q^{10} +1.13000 q^{11} -0.528979 q^{12} +2.62058 q^{13} +0.554913 q^{15} +3.56911 q^{16} +4.97662 q^{17} -0.788554 q^{18} +3.40537 q^{19} -3.89661 q^{20} +0.304673 q^{22} -9.12870 q^{23} -0.290628 q^{24} -0.912359 q^{25} +0.706566 q^{26} -1.62612 q^{27} +2.35080 q^{29} +0.149616 q^{30} +3.33264 q^{31} +3.08008 q^{32} +0.310147 q^{33} +1.34180 q^{34} +5.63673 q^{36} +5.10001 q^{37} +0.918163 q^{38} +0.719260 q^{39} -2.14085 q^{40} -0.641924 q^{41} -11.1780 q^{43} -2.17786 q^{44} -5.91307 q^{45} -2.46129 q^{46} +13.3207 q^{47} +0.979598 q^{48} -0.245992 q^{50} +1.36591 q^{51} -5.05066 q^{52} -1.02638 q^{53} -0.438437 q^{54} +2.28463 q^{55} +0.934659 q^{57} +0.633826 q^{58} +1.14787 q^{59} -1.06949 q^{60} +7.80389 q^{61} +0.898552 q^{62} -6.30776 q^{64} +5.29827 q^{65} +0.0836224 q^{66} -8.89439 q^{67} -9.59145 q^{68} -2.50552 q^{69} -13.6617 q^{71} +3.09689 q^{72} +1.48097 q^{73} +1.37507 q^{74} -0.250411 q^{75} -6.56319 q^{76} +0.193928 q^{78} +5.75021 q^{79} +7.21600 q^{80} +8.32769 q^{81} -0.173077 q^{82} -16.0413 q^{83} +10.0617 q^{85} -3.01382 q^{86} +0.645213 q^{87} -1.19654 q^{88} +9.67695 q^{89} -1.59429 q^{90} +17.5938 q^{92} +0.914696 q^{93} +3.59155 q^{94} +6.88496 q^{95} +0.845377 q^{96} +3.42749 q^{97} -3.30488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Coefficient data

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

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

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

$2$	0.269622	0.190651	0.0953256	−	0.995446i	$-0.469611\pi$
$2$	0.269622	0.190651	0.0953256	+	0.995446i	$0.469611\pi$
$3$	0.274466	0.158463	0.0792314	−	0.996856i	$-0.474753\pi$
$3$	0.274466	0.158463	0.0792314	+	0.996856i	$0.474753\pi$
$4$	−1.92730	−0.963652
$5$	2.02179	0.904173	0.452086	−	0.891974i	$-0.350680\pi$
$5$	2.02179	0.904173	0.452086	+	0.891974i	$0.350680\pi$
$6$	0.0740019	0.0302112
$7$	0	0
$7$	0	0
$8$	−1.05889	−0.374373
$9$	−2.92467	−0.974890
$10$	0.545119	0.172382
$11$	1.13000	0.340709	0.170354	−	0.985383i	$-0.445509\pi$
$11$	1.13000	0.340709	0.170354	+	0.985383i	$0.445509\pi$
$12$	−0.528979	−0.152703
$13$	2.62058	0.726819	0.363409	−	0.931630i	$-0.381613\pi$
$13$	2.62058	0.726819	0.363409	+	0.931630i	$0.381613\pi$
$14$	0	0
$15$	0.554913	0.143278
$16$	3.56911	0.892277
$17$	4.97662	1.20701	0.603503	−	0.797360i	$-0.293771\pi$
$17$	4.97662	1.20701	0.603503	+	0.797360i	$0.293771\pi$
$18$	−0.788554	−0.185864
$19$	3.40537	0.781246	0.390623	−	0.920551i	$-0.372260\pi$
$19$	3.40537	0.781246	0.390623	+	0.920551i	$0.372260\pi$
$20$	−3.89661	−0.871308
$21$	0	0
$22$	0.304673	0.0649565
$23$	−9.12870	−1.90347	−0.951733	−	0.306928i	$-0.900699\pi$
$23$	−9.12870	−1.90347	−0.951733	+	0.306928i	$0.900699\pi$
$24$	−0.290628	−0.0593242
$25$	−0.912359	−0.182472
$26$	0.706566	0.138569
$27$	−1.62612	−0.312947
$28$	0	0
$29$	2.35080	0.436532	0.218266	−	0.975889i	$-0.429960\pi$
$29$	2.35080	0.436532	0.218266	+	0.975889i	$0.429960\pi$
$30$	0.149616	0.0273161
$31$	3.33264	0.598560	0.299280	−	0.954165i	$-0.403254\pi$
$31$	3.33264	0.598560	0.299280	+	0.954165i	$0.403254\pi$
$32$	3.08008	0.544487
$33$	0.310147	0.0539897
$34$	1.34180	0.230117
$35$	0	0
$36$	5.63673	0.939454
$37$	5.10001	0.838436	0.419218	−	0.907886i	$-0.362304\pi$
$37$	5.10001	0.838436	0.419218	+	0.907886i	$0.362304\pi$
$38$	0.918163	0.148946
$39$	0.719260	0.115174
$40$	−2.14085	−0.338498
$41$	−0.641924	−0.100252	−0.0501259	−	0.998743i	$-0.515962\pi$
$41$	−0.641924	−0.100252	−0.0501259	+	0.998743i	$0.515962\pi$
$42$	0	0
$43$	−11.1780	−1.70463	−0.852313	−	0.523033i	$-0.824800\pi$
$43$	−11.1780	−1.70463	−0.852313	+	0.523033i	$0.824800\pi$
$44$	−2.17786	−0.328325
$45$	−5.91307	−0.881468
$46$	−2.46129	−0.362898
$47$	13.3207	1.94303	0.971513	−	0.236987i	$-0.0761600\pi$
$47$	13.3207	1.94303	0.971513	+	0.236987i	$0.0761600\pi$
$48$	0.979598	0.141393
$49$	0	0
$50$	−0.245992	−0.0347885
$51$	1.36591	0.191266
$52$	−5.05066	−0.700400
$53$	−1.02638	−0.140985	−0.0704923	−	0.997512i	$-0.522457\pi$
$53$	−1.02638	−0.140985	−0.0704923	+	0.997512i	$0.522457\pi$
$54$	−0.438437	−0.0596637
$55$	2.28463	0.308059
$56$	0	0
$57$	0.934659	0.123799
$58$	0.633826	0.0832254
$59$	1.14787	0.149440	0.0747198	−	0.997205i	$-0.476194\pi$
$59$	1.14787	0.149440	0.0747198	+	0.997205i	$0.476194\pi$
$60$	−1.06949	−0.138070
$61$	7.80389	0.999186	0.499593	−	0.866260i	$-0.333483\pi$
$61$	7.80389	0.999186	0.499593	+	0.866260i	$0.333483\pi$
$62$	0.898552	0.114116
$63$	0	0
$64$	−6.30776	−0.788470
$65$	5.29827	0.657170
$66$	0.0836224	0.0102932
$67$	−8.89439	−1.08662	−0.543311	−	0.839531i	$-0.682830\pi$
$67$	−8.89439	−1.08662	−0.543311	+	0.839531i	$0.682830\pi$
$68$	−9.59145	−1.16313
$69$	−2.50552	−0.301629
$70$	0	0
$71$	−13.6617	−1.62134	−0.810670	−	0.585503i	$-0.800897\pi$
$71$	−13.6617	−1.62134	−0.810670	+	0.585503i	$0.800897\pi$
$72$	3.09689	0.364972
$73$	1.48097	0.173334	0.0866669	−	0.996237i	$-0.472378\pi$
$73$	1.48097	0.173334	0.0866669	+	0.996237i	$0.472378\pi$
$74$	1.37507	0.159849
$75$	−0.250411	−0.0289150
$76$	−6.56319	−0.752850
$77$	0	0
$78$	0.193928	0.0219580
$79$	5.75021	0.646949	0.323475	−	0.946237i	$-0.395149\pi$
$79$	5.75021	0.646949	0.323475	+	0.946237i	$0.395149\pi$
$80$	7.21600	0.806773
$81$	8.32769	0.925299
$82$	−0.173077	−0.0191131
$83$	−16.0413	−1.76076	−0.880382	−	0.474265i	$-0.842714\pi$
$83$	−16.0413	−1.76076	−0.880382	+	0.474265i	$0.842714\pi$
$84$	0	0
$85$	10.0617	1.09134
$86$	−3.01382	−0.324989
$87$	0.645213	0.0691741
$88$	−1.19654	−0.127552
$89$	9.67695	1.02575	0.512877	−	0.858462i	$-0.328580\pi$
$89$	9.67695	1.02575	0.512877	+	0.858462i	$0.328580\pi$
$90$	−1.59429	−0.168053
$91$	0	0
$92$	17.5938	1.83428
$93$	0.914696	0.0948495
$94$	3.59155	0.370440
$95$	6.88496	0.706382
$96$	0.845377	0.0862809
$97$	3.42749	0.348009	0.174004	−	0.984745i	$-0.444329\pi$
$97$	3.42749	0.348009	0.174004	+	0.984745i	$0.444329\pi$
$98$	0	0
$99$	−3.30488	−0.332153
$100$	1.75839	0.175839
$101$	3.88185	0.386258	0.193129	−	0.981173i	$-0.438136\pi$
$101$	3.88185	0.386258	0.193129	+	0.981173i	$0.438136\pi$
$102$	0.368279	0.0364651
$103$	16.0280	1.57928	0.789641	−	0.613569i	$-0.210267\pi$
$103$	16.0280	1.57928	0.789641	+	0.613569i	$0.210267\pi$
$104$	−2.77490	−0.272101
$105$	0	0
$106$	−0.276735	−0.0268789
$107$	−2.15195	−0.208037	−0.104018	−	0.994575i	$-0.533170\pi$
$107$	−2.15195	−0.208037	−0.104018	+	0.994575i	$0.533170\pi$
$108$	3.13403	0.301572
$109$	11.9139	1.14115	0.570574	−	0.821246i	$-0.306721\pi$
$109$	11.9139	1.14115	0.570574	+	0.821246i	$0.306721\pi$
$110$	0.615986	0.0587319
$111$	1.39978	0.132861
$112$	0	0
$113$	15.1243	1.42277	0.711387	−	0.702800i	$-0.248067\pi$
$113$	15.1243	1.42277	0.711387	+	0.702800i	$0.248067\pi$
$114$	0.252004	0.0236024
$115$	−18.4563	−1.72106
$116$	−4.53070	−0.420665
$117$	−7.66434	−0.708568
$118$	0.309490	0.0284909
$119$	0	0
$120$	−0.587589	−0.0536393
$121$	−9.72309	−0.883918
$122$	2.10410	0.190496
$123$	−0.176186	−0.0158862
$124$	−6.42301	−0.576803
$125$	−11.9536	−1.06916
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−7.86087	−0.694810
$129$	−3.06797	−0.270120
$130$	1.42853	0.125290
$131$	12.3055	1.07514	0.537570	−	0.843219i	$-0.319343\pi$
$131$	12.3055	1.07514	0.537570	+	0.843219i	$0.319343\pi$
$132$	−0.597748	−0.0520273
$133$	0	0
$134$	−2.39812	−0.207166
$135$	−3.28767	−0.282958
$136$	−5.26967	−0.451870
$137$	18.7668	1.60335	0.801677	−	0.597757i	$-0.203941\pi$
$137$	18.7668	1.60335	0.801677	+	0.597757i	$0.203941\pi$
$138$	−0.675541	−0.0575059
$139$	−14.1460	−1.19985	−0.599925	−	0.800056i	$-0.704803\pi$
$139$	−14.1460	−1.19985	−0.599925	+	0.800056i	$0.704803\pi$
$140$	0	0
$141$	3.65608	0.307897
$142$	−3.68348	−0.309111
$143$	2.96127	0.247633
$144$	−10.4385	−0.869872
$145$	4.75282	0.394700
$146$	0.399300	0.0330463
$147$	0	0
$148$	−9.82927	−0.807961
$149$	0.401780	0.0329151	0.0164575	−	0.999865i	$-0.494761\pi$
$149$	0.401780	0.0329151	0.0164575	+	0.999865i	$0.494761\pi$
$150$	−0.0675163	−0.00551268
$151$	−10.0522	−0.818038	−0.409019	−	0.912526i	$-0.634129\pi$
$151$	−10.0522	−0.818038	−0.409019	+	0.912526i	$0.634129\pi$
$152$	−3.60590	−0.292477
$153$	−14.5550	−1.17670
$154$	0	0
$155$	6.73790	0.541201
$156$	−1.38623	−0.110987
$157$	10.2515	0.818158	0.409079	−	0.912499i	$-0.365850\pi$
$157$	10.2515	0.818158	0.409079	+	0.912499i	$0.365850\pi$
$158$	1.55038	0.123342
$159$	−0.281707	−0.0223408
$160$	6.22728	0.492310
$161$	0	0
$162$	2.24533	0.176409
$163$	13.3515	1.04577	0.522886	−	0.852403i	$-0.324855\pi$
$163$	13.3515	1.04577	0.522886	+	0.852403i	$0.324855\pi$
$164$	1.23718	0.0966078
$165$	0.627053	0.0488160
$166$	−4.32509	−0.335692
$167$	17.6154	1.36312	0.681560	−	0.731763i	$-0.261302\pi$
$167$	17.6154	1.36312	0.681560	+	0.731763i	$0.261302\pi$
$168$	0	0
$169$	−6.13255	−0.471734
$170$	2.71285	0.208066
$171$	−9.95959	−0.761629
$172$	21.5434	1.64267
$173$	21.2168	1.61308	0.806542	−	0.591177i	$-0.201337\pi$
$173$	21.2168	1.61308	0.806542	+	0.591177i	$0.201337\pi$
$174$	0.173964	0.0131881
$175$	0	0
$176$	4.03310	0.304007
$177$	0.315050	0.0236806
$178$	2.60911	0.195561
$179$	−1.20573	−0.0901207	−0.0450604	−	0.998984i	$-0.514348\pi$
$179$	−1.20573	−0.0901207	−0.0450604	+	0.998984i	$0.514348\pi$
$180$	11.3963	0.849429
$181$	−0.613845	−0.0456267	−0.0228134	−	0.999740i	$-0.507262\pi$
$181$	−0.613845	−0.0456267	−0.0228134	+	0.999740i	$0.507262\pi$
$182$	0	0
$183$	2.14190	0.158334
$184$	9.66625	0.712606
$185$	10.3112	0.758091
$186$	0.246622	0.0180832
$187$	5.62359	0.411238
$188$	−25.6731	−1.87240
$189$	0	0
$190$	1.85633	0.134673
$191$	12.8600	0.930514	0.465257	−	0.885176i	$-0.345962\pi$
$191$	12.8600	0.930514	0.465257	+	0.885176i	$0.345962\pi$
$192$	−1.73126	−0.124943
$193$	−10.4070	−0.749116	−0.374558	−	0.927204i	$-0.622205\pi$
$193$	−10.4070	−0.749116	−0.374558	+	0.927204i	$0.622205\pi$
$194$	0.924125	0.0663483
$195$	1.45419	0.104137
$196$	0	0
$197$	5.16874	0.368257	0.184129	−	0.982902i	$-0.441054\pi$
$197$	5.16874	0.368257	0.184129	+	0.982902i	$0.441054\pi$
$198$	−0.891068	−0.0633255
$199$	16.2637	1.15291	0.576453	−	0.817131i	$-0.304437\pi$
$199$	16.2637	1.15291	0.576453	+	0.817131i	$0.304437\pi$
$200$	0.966084	0.0683125
$201$	−2.44120	−0.172189
$202$	1.04663	0.0736407
$203$	0	0
$204$	−2.63253	−0.184314
$205$	−1.29784	−0.0906449
$206$	4.32149	0.301092
$207$	26.6984	1.85567
$208$	9.35315	0.648524
$209$	3.84808	0.266177
$210$	0	0
$211$	17.0106	1.17106	0.585529	−	0.810652i	$-0.300887\pi$
$211$	17.0106	1.17106	0.585529	+	0.810652i	$0.300887\pi$
$212$	1.97815	0.135860
$213$	−3.74966	−0.256922
$214$	−0.580212	−0.0396625
$215$	−22.5995	−1.54128
$216$	1.72187	0.117159
$217$	0	0
$218$	3.21225	0.217561
$219$	0.406474	0.0274670
$220$	−4.40318	−0.296862
$221$	13.0416	0.877275
$222$	0.377410	0.0253301
$223$	12.1608	0.814346	0.407173	−	0.913351i	$-0.366515\pi$
$223$	12.1608	0.814346	0.407173	+	0.913351i	$0.366515\pi$
$224$	0	0
$225$	2.66835	0.177890
$226$	4.07784	0.271254
$227$	−24.3957	−1.61920	−0.809599	−	0.586983i	$-0.800316\pi$
$227$	−24.3957	−1.61920	−0.809599	+	0.586983i	$0.800316\pi$
$228$	−1.80137	−0.119299
$229$	−1.07917	−0.0713132	−0.0356566	−	0.999364i	$-0.511352\pi$
$229$	−1.07917	−0.0713132	−0.0356566	+	0.999364i	$0.511352\pi$
$230$	−4.97622	−0.328123
$231$	0	0
$232$	−2.48923	−0.163426
$233$	−12.7061	−0.832401	−0.416201	−	0.909273i	$-0.636639\pi$
$233$	−12.7061	−0.832401	−0.416201	+	0.909273i	$0.636639\pi$
$234$	−2.06647	−0.135089
$235$	26.9317	1.75683
$236$	−2.21229	−0.144008
$237$	1.57824	0.102517
$238$	0	0
$239$	21.9779	1.42163	0.710817	−	0.703377i	$-0.248325\pi$
$239$	21.9779	1.42163	0.710817	+	0.703377i	$0.248325\pi$
$240$	1.98054	0.127844
$241$	−6.16488	−0.397115	−0.198557	−	0.980089i	$-0.563626\pi$
$241$	−6.16488	−0.397115	−0.198557	+	0.980089i	$0.563626\pi$
$242$	−2.62156	−0.168520
$243$	7.16402	0.459572
$244$	−15.0405	−0.962868
$245$	0	0
$246$	−0.0475036	−0.00302872
$247$	8.92406	0.567825
$248$	−3.52889	−0.224085
$249$	−4.40279	−0.279016
$250$	−3.22294	−0.203836
$251$	12.1683	0.768057	0.384029	−	0.923321i	$-0.374536\pi$
$251$	12.1683	0.768057	0.384029	+	0.923321i	$0.374536\pi$
$252$	0	0
$253$	−10.3155	−0.648527
$254$	0.269622	0.0169176
$255$	2.76159	0.172937
$256$	10.4961	0.656004
$257$	−26.9996	−1.68419	−0.842094	−	0.539331i	$-0.818677\pi$
$257$	−26.9996	−1.68419	−0.842094	+	0.539331i	$0.818677\pi$
$258$	−0.827191	−0.0514987
$259$	0	0
$260$	−10.2114	−0.633283
$261$	−6.87530	−0.425571
$262$	3.31784	0.204977
$263$	20.6781	1.27506	0.637532	−	0.770424i	$-0.279955\pi$
$263$	20.6781	1.27506	0.637532	+	0.770424i	$0.279955\pi$
$264$	−0.328410	−0.0202123
$265$	−2.07513	−0.127474
$266$	0	0
$267$	2.65599	0.162544
$268$	17.1422	1.04713
$269$	14.5531	0.887321	0.443660	−	0.896195i	$-0.353680\pi$
$269$	14.5531	0.887321	0.443660	+	0.896195i	$0.353680\pi$
$270$	−0.886428	−0.0539463
$271$	2.51912	0.153025	0.0765127	−	0.997069i	$-0.475621\pi$
$271$	2.51912	0.153025	0.0765127	+	0.997069i	$0.475621\pi$
$272$	17.7621	1.07698
$273$	0	0
$274$	5.05993	0.305682
$275$	−1.03097	−0.0621697
$276$	4.82889	0.290665
$277$	21.5468	1.29462	0.647310	−	0.762227i	$-0.275894\pi$
$277$	21.5468	1.29462	0.647310	+	0.762227i	$0.275894\pi$
$278$	−3.81407	−0.228753
$279$	−9.74687	−0.583530
$280$	0	0
$281$	−9.99798	−0.596429	−0.298215	−	0.954499i	$-0.596391\pi$
$281$	−9.99798	−0.596429	−0.298215	+	0.954499i	$0.596391\pi$
$282$	0.985758	0.0587010
$283$	−13.4458	−0.799267	−0.399634	−	0.916675i	$-0.630863\pi$
$283$	−13.4458	−0.799267	−0.399634	+	0.916675i	$0.630863\pi$
$284$	26.3302	1.56241
$285$	1.88968	0.111935
$286$	0.798421	0.0472116
$287$	0	0
$288$	−9.00822	−0.530814
$289$	7.76671	0.456865
$290$	1.28146	0.0752502
$291$	0.940728	0.0551465
$292$	−2.85427	−0.167034
$293$	−0.126038	−0.00736321	−0.00368161	−	0.999993i	$-0.501172\pi$
$293$	−0.126038	−0.00736321	−0.00368161	+	0.999993i	$0.501172\pi$
$294$	0	0
$295$	2.32075	0.135119
$296$	−5.40033	−0.313888
$297$	−1.83752	−0.106624
$298$	0.108329	0.00627530
$299$	−23.9225	−1.38347
$300$	0.482619	0.0278640
$301$	0	0
$302$	−2.71030	−0.155960
$303$	1.06543	0.0612076
$304$	12.1542	0.697089
$305$	15.7778	0.903437
$306$	−3.92433	−0.224339
$307$	−12.1097	−0.691135	−0.345567	−	0.938394i	$-0.612313\pi$
$307$	−12.1097	−0.691135	−0.345567	+	0.938394i	$0.612313\pi$
$308$	0	0
$309$	4.39913	0.250258
$310$	1.81668	0.103181
$311$	31.2146	1.77002	0.885008	−	0.465576i	$-0.154153\pi$
$311$	31.2146	1.77002	0.885008	+	0.465576i	$0.154153\pi$
$312$	−0.761615	−0.0431179
$313$	17.7500	1.00329	0.501646	−	0.865073i	$-0.332728\pi$
$313$	17.7500	1.00329	0.501646	+	0.865073i	$0.332728\pi$
$314$	2.76402	0.155983
$315$	0	0
$316$	−11.0824	−0.623434
$317$	25.1186	1.41080	0.705402	−	0.708808i	$-0.250767\pi$
$317$	25.1186	1.41080	0.705402	+	0.708808i	$0.250767\pi$
$318$	−0.0759544	−0.00425931
$319$	2.65641	0.148730
$320$	−12.7530	−0.712913
$321$	−0.590636	−0.0329661
$322$	0	0
$323$	16.9472	0.942970
$324$	−16.0500	−0.891666
$325$	−2.39091	−0.132624
$326$	3.59986	0.199378
$327$	3.26997	0.180830
$328$	0.679725	0.0375315
$329$	0	0
$330$	0.169067	0.00930683
$331$	−31.9329	−1.75519	−0.877594	−	0.479404i	$-0.840853\pi$
$331$	−31.9329	−1.75519	−0.877594	+	0.479404i	$0.840853\pi$
$332$	30.9165	1.69676
$333$	−14.9158	−0.817383
$334$	4.74949	0.259880
$335$	−17.9826	−0.982494
$336$	0	0
$337$	35.3257	1.92432	0.962158	−	0.272494i	$-0.0878484\pi$
$337$	35.3257	1.92432	0.962158	+	0.272494i	$0.0878484\pi$
$338$	−1.65347	−0.0899368
$339$	4.15110	0.225457
$340$	−19.3919	−1.05167
$341$	3.76589	0.203935
$342$	−2.68532	−0.145206
$343$	0	0
$344$	11.8362	0.638165
$345$	−5.06563	−0.272724
$346$	5.72051	0.307536
$347$	18.2778	0.981205	0.490603	−	0.871383i	$-0.336777\pi$
$347$	18.2778	0.981205	0.490603	+	0.871383i	$0.336777\pi$
$348$	−1.24352	−0.0666598
$349$	−28.0916	−1.50371	−0.751856	−	0.659328i	$-0.770841\pi$
$349$	−28.0916	−1.50371	−0.751856	+	0.659328i	$0.770841\pi$
$350$	0	0
$351$	−4.26138	−0.227456
$352$	3.48050	0.185511
$353$	4.77508	0.254152	0.127076	−	0.991893i	$-0.459441\pi$
$353$	4.77508	0.254152	0.127076	+	0.991893i	$0.459441\pi$
$354$	0.0849444	0.00451474
$355$	−27.6210	−1.46597
$356$	−18.6504	−0.988470
$357$	0	0
$358$	−0.325092	−0.0171816
$359$	11.1728	0.589679	0.294840	−	0.955547i	$-0.404734\pi$
$359$	11.1728	0.589679	0.294840	+	0.955547i	$0.404734\pi$
$360$	6.26127	0.329998
$361$	−7.40342	−0.389654
$362$	−0.165506	−0.00869880
$363$	−2.66866	−0.140068
$364$	0	0
$365$	2.99420	0.156724
$366$	0.577503	0.0301866
$367$	6.97390	0.364034	0.182017	−	0.983295i	$-0.441737\pi$
$367$	6.97390	0.364034	0.182017	+	0.983295i	$0.441737\pi$
$368$	−32.5813	−1.69842
$369$	1.87742	0.0977344
$370$	2.78011	0.144531
$371$	0	0
$372$	−1.76290	−0.0914019
$373$	5.85705	0.303266	0.151633	−	0.988437i	$-0.451547\pi$
$373$	5.85705	0.303266	0.151633	+	0.988437i	$0.451547\pi$
$374$	1.51624	0.0784030
$375$	−3.28084	−0.169422
$376$	−14.1051	−0.727416
$377$	6.16046	0.317280
$378$	0	0
$379$	−12.7386	−0.654339	−0.327170	−	0.944966i	$-0.606095\pi$
$379$	−12.7386	−0.654339	−0.327170	+	0.944966i	$0.606095\pi$
$380$	−13.2694	−0.680706
$381$	0.274466	0.0140613
$382$	3.46732	0.177404
$383$	−27.6769	−1.41422	−0.707111	−	0.707102i	$-0.750002\pi$
$383$	−27.6769	−1.41422	−0.707111	+	0.707102i	$0.750002\pi$
$384$	−2.15754	−0.110102
$385$	0	0
$386$	−2.80597	−0.142820
$387$	32.6919	1.66182
$388$	−6.60581	−0.335359
$389$	−29.2578	−1.48343	−0.741715	−	0.670715i	$-0.765987\pi$
$389$	−29.2578	−1.48343	−0.741715	+	0.670715i	$0.765987\pi$
$390$	0.392082	0.0198539
$391$	−45.4300	−2.29750
$392$	0	0
$393$	3.37745	0.170370
$394$	1.39360	0.0702087
$395$	11.6257	0.584954
$396$	6.36952	0.320080
$397$	−19.6387	−0.985638	−0.492819	−	0.870132i	$-0.664033\pi$
$397$	−19.6387	−0.985638	−0.492819	+	0.870132i	$0.664033\pi$
$398$	4.38505	0.219803
$399$	0	0
$400$	−3.25631	−0.162815
$401$	−21.7714	−1.08721	−0.543607	−	0.839340i	$-0.682942\pi$
$401$	−21.7714	−1.08721	−0.543607	+	0.839340i	$0.682942\pi$
$402$	−0.658202	−0.0328281
$403$	8.73346	0.435045
$404$	−7.48150	−0.372219
$405$	16.8369	0.836630
$406$	0	0
$407$	5.76302	0.285662
$408$	−1.44634	−0.0716047
$409$	36.5311	1.80635	0.903174	−	0.429275i	$-0.141231\pi$
$409$	36.5311	1.80635	0.903174	+	0.429275i	$0.141231\pi$
$410$	−0.349925	−0.0172816
$411$	5.15084	0.254072
$412$	−30.8908	−1.52188
$413$	0	0
$414$	7.19847	0.353786
$415$	−32.4322	−1.59203
$416$	8.07161	0.395743
$417$	−3.88260	−0.190132
$418$	1.03753	0.0507471
$419$	27.3459	1.33593	0.667967	−	0.744191i	$-0.267165\pi$
$419$	27.3459	1.33593	0.667967	+	0.744191i	$0.267165\pi$
$420$	0	0
$421$	−28.3835	−1.38333	−0.691663	−	0.722220i	$-0.743122\pi$
$421$	−28.3835	−1.38333	−0.691663	+	0.722220i	$0.743122\pi$
$422$	4.58642	0.223264
$423$	−38.9587	−1.89424
$424$	1.08682	0.0527808
$425$	−4.54046	−0.220245
$426$	−1.01099	−0.0489826
$427$	0	0
$428$	4.14746	0.200475
$429$	0.812766	0.0392407
$430$	−6.09332	−0.293846
$431$	−39.3867	−1.89719	−0.948595	−	0.316493i	$-0.897495\pi$
$431$	−39.3867	−1.89719	−0.948595	+	0.316493i	$0.897495\pi$
$432$	−5.80380	−0.279235
$433$	32.0013	1.53788	0.768942	−	0.639319i	$-0.220783\pi$
$433$	32.0013	1.53788	0.768942	+	0.639319i	$0.220783\pi$
$434$	0	0
$435$	1.30449	0.0625454
$436$	−22.9618	−1.09967
$437$	−31.0866	−1.48708
$438$	0.109594	0.00523662
$439$	20.8928	0.997158	0.498579	−	0.866844i	$-0.333855\pi$
$439$	20.8928	0.997158	0.498579	+	0.866844i	$0.333855\pi$
$440$	−2.41916	−0.115329
$441$	0	0
$442$	3.51631	0.167254
$443$	−15.2798	−0.725964	−0.362982	−	0.931796i	$-0.618241\pi$
$443$	−15.2798	−0.725964	−0.362982	+	0.931796i	$0.618241\pi$
$444$	−2.69780	−0.128032
$445$	19.5648	0.927459
$446$	3.27881	0.155256
$447$	0.110275	0.00521582
$448$	0	0
$449$	4.20626	0.198506	0.0992528	−	0.995062i	$-0.468355\pi$
$449$	4.20626	0.198506	0.0992528	+	0.995062i	$0.468355\pi$
$450$	0.719444	0.0339149
$451$	−0.725376	−0.0341566
$452$	−29.1491	−1.37106
$453$	−2.75899	−0.129629
$454$	−6.57761	−0.308702
$455$	0	0
$456$	−0.989697	−0.0463468
$457$	15.4413	0.722313	0.361156	−	0.932505i	$-0.382382\pi$
$457$	15.4413	0.722313	0.361156	+	0.932505i	$0.382382\pi$
$458$	−0.290966	−0.0135960
$459$	−8.09257	−0.377729
$460$	35.5710	1.65850
$461$	−15.5490	−0.724189	−0.362095	−	0.932141i	$-0.617938\pi$
$461$	−15.5490	−0.724189	−0.362095	+	0.932141i	$0.617938\pi$
$462$	0	0
$463$	−5.76567	−0.267953	−0.133977	−	0.990984i	$-0.542775\pi$
$463$	−5.76567	−0.267953	−0.133977	+	0.990984i	$0.542775\pi$
$464$	8.39025	0.389508
$465$	1.84932	0.0857603
$466$	−3.42583	−0.158698
$467$	−14.0411	−0.649742	−0.324871	−	0.945758i	$-0.605321\pi$
$467$	−14.0411	−0.649742	−0.324871	+	0.945758i	$0.605321\pi$
$468$	14.7715	0.682813
$469$	0	0
$470$	7.26137	0.334942
$471$	2.81368	0.129648
$472$	−1.21546	−0.0559461
$473$	−12.6311	−0.580780
$474$	0.425527	0.0195451
$475$	−3.10692	−0.142555
$476$	0	0
$477$	3.00183	0.137444
$478$	5.92573	0.271036
$479$	3.09421	0.141378	0.0706890	−	0.997498i	$-0.477480\pi$
$479$	3.09421	0.141378	0.0706890	+	0.997498i	$0.477480\pi$
$480$	1.70918	0.0780129
$481$	13.3650	0.609391
$482$	−1.66218	−0.0757104
$483$	0	0
$484$	18.7394	0.851789
$485$	6.92967	0.314660
$486$	1.93158	0.0876180
$487$	4.67605	0.211892	0.105946	−	0.994372i	$-0.466213\pi$
$487$	4.67605	0.211892	0.105946	+	0.994372i	$0.466213\pi$
$488$	−8.26343	−0.374068
$489$	3.66454	0.165716
$490$	0	0
$491$	−1.24709	−0.0562803	−0.0281401	−	0.999604i	$-0.508958\pi$
$491$	−1.24709	−0.0562803	−0.0281401	+	0.999604i	$0.508958\pi$
$492$	0.339564	0.0153087
$493$	11.6990	0.526897
$494$	2.40612	0.108257
$495$	−6.68179	−0.300324
$496$	11.8946	0.534081
$497$	0	0
$498$	−1.18709	−0.0531947
$499$	14.9481	0.669167	0.334583	−	0.942366i	$-0.391404\pi$
$499$	14.9481	0.669167	0.334583	+	0.942366i	$0.391404\pi$
$500$	23.0381	1.03030
$501$	4.83482	0.216004
$502$	3.28084	0.146431
$503$	−19.5895	−0.873455	−0.436727	−	0.899594i	$-0.643863\pi$
$503$	−19.5895	−0.873455	−0.436727	+	0.899594i	$0.643863\pi$
$504$	0	0
$505$	7.84829	0.349244
$506$	−2.78127	−0.123643
$507$	−1.68317	−0.0747524
$508$	−1.92730	−0.0855103
$509$	1.73063	0.0767090	0.0383545	−	0.999264i	$-0.487788\pi$
$509$	1.73063	0.0767090	0.0383545	+	0.999264i	$0.487788\pi$
$510$	0.744584	0.0329707
$511$	0	0
$512$	18.5517	0.819878
$513$	−5.53754	−0.244488
$514$	−7.27967	−0.321093
$515$	32.4052	1.42794
$516$	5.91291	0.260301
$517$	15.0524	0.662006
$518$	0	0
$519$	5.82329	0.255614
$520$	−5.61027	−0.246026
$521$	10.7776	0.472175	0.236088	−	0.971732i	$-0.424135\pi$
$521$	10.7776	0.472175	0.236088	+	0.971732i	$0.424135\pi$
$522$	−1.85373	−0.0811356
$523$	45.6385	1.99563	0.997817	−	0.0660391i	$-0.0210362\pi$
$523$	45.6385	1.99563	0.997817	+	0.0660391i	$0.0210362\pi$
$524$	−23.7165	−1.03606
$525$	0	0
$526$	5.57525	0.243093
$527$	16.5853	0.722466
$528$	1.10695	0.0481738
$529$	60.3331	2.62318
$530$	−0.559501	−0.0243032
$531$	−3.35713	−0.145687
$532$	0	0
$533$	−1.68222	−0.0728649
$534$	0.716113	0.0309892
$535$	−4.35079	−0.188101
$536$	9.41814	0.406802
$537$	−0.330932	−0.0142808
$538$	3.92384	0.169169
$539$	0	0
$540$	6.33635	0.272673
$541$	39.0400	1.67846	0.839231	−	0.543775i	$-0.183006\pi$
$541$	39.0400	1.67846	0.839231	+	0.543775i	$0.183006\pi$
$542$	0.679208	0.0291745
$543$	−0.168479	−0.00723014
$544$	15.3284	0.657199
$545$	24.0875	1.03179
$546$	0	0
$547$	−12.7969	−0.547157	−0.273578	−	0.961850i	$-0.588207\pi$
$547$	−12.7969	−0.547157	−0.273578	+	0.961850i	$0.588207\pi$
$548$	−36.1693	−1.54508
$549$	−22.8238	−0.974096
$550$	−0.277971	−0.0118527
$551$	8.00535	0.341039
$552$	2.65306	0.112922
$553$	0	0
$554$	5.80948	0.246821
$555$	2.83006	0.120129
$556$	27.2637	1.15624
$557$	−3.07139	−0.130139	−0.0650694	−	0.997881i	$-0.520727\pi$
$557$	−3.07139	−0.130139	−0.0650694	+	0.997881i	$0.520727\pi$
$558$	−2.62797	−0.111251
$559$	−29.2928	−1.23895
$560$	0	0
$561$	1.54348	0.0651659
$562$	−2.69567	−0.113710
$563$	−29.7463	−1.25366	−0.626828	−	0.779157i	$-0.715647\pi$
$563$	−29.7463	−1.25366	−0.626828	+	0.779157i	$0.715647\pi$
$564$	−7.04638	−0.296706
$565$	30.5782	1.28643
$566$	−3.62527	−0.152381
$567$	0	0
$568$	14.4661	0.606986
$569$	−11.0802	−0.464506	−0.232253	−	0.972655i	$-0.574610\pi$
$569$	−11.0802	−0.464506	−0.232253	+	0.972655i	$0.574610\pi$
$570$	0.509500	0.0213406
$571$	35.3770	1.48048	0.740240	−	0.672343i	$-0.234712\pi$
$571$	35.3770	1.48048	0.740240	+	0.672343i	$0.234712\pi$
$572$	−5.70726	−0.238632
$573$	3.52962	0.147452
$574$	0	0
$575$	8.32865	0.347329
$576$	18.4481	0.768671
$577$	−23.8020	−0.990892	−0.495446	−	0.868639i	$-0.664995\pi$
$577$	−23.8020	−0.990892	−0.495446	+	0.868639i	$0.664995\pi$
$578$	2.09407	0.0871019
$579$	−2.85638	−0.118707
$580$	−9.16014	−0.380354
$581$	0	0
$582$	0.253641	0.0105137
$583$	−1.15982	−0.0480347
$584$	−1.56817	−0.0648915
$585$	−15.4957	−0.640668
$586$	−0.0339825	−0.00140381
$587$	−12.1858	−0.502961	−0.251481	−	0.967862i	$-0.580917\pi$
$587$	−12.1858	−0.502961	−0.251481	+	0.967862i	$0.580917\pi$
$588$	0	0
$589$	11.3489	0.467623
$590$	0.625724	0.0257607
$591$	1.41864	0.0583551
$592$	18.2025	0.748118
$593$	35.8720	1.47309	0.736543	−	0.676391i	$-0.236457\pi$
$593$	35.8720	1.47309	0.736543	+	0.676391i	$0.236457\pi$
$594$	−0.495435	−0.0203279
$595$	0	0
$596$	−0.774352	−0.0317187
$597$	4.46384	0.182693
$598$	−6.45003	−0.263761
$599$	−14.6586	−0.598934	−0.299467	−	0.954107i	$-0.596809\pi$
$599$	−14.6586	−0.598934	−0.299467	+	0.954107i	$0.596809\pi$
$600$	0.265157	0.0108250
$601$	15.1302	0.617175	0.308587	−	0.951196i	$-0.400144\pi$
$601$	15.1302	0.617175	0.308587	+	0.951196i	$0.400144\pi$
$602$	0	0
$603$	26.0131	1.05934
$604$	19.3737	0.788304
$605$	−19.6581	−0.799214
$606$	0.287264	0.0116693
$607$	18.8277	0.764194	0.382097	−	0.924122i	$-0.375202\pi$
$607$	18.8277	0.764194	0.382097	+	0.924122i	$0.375202\pi$
$608$	10.4888	0.425378
$609$	0	0
$610$	4.25405	0.172241
$611$	34.9080	1.41223
$612$	28.0518	1.13393
$613$	−33.3581	−1.34732	−0.673660	−	0.739041i	$-0.735279\pi$
$613$	−33.3581	−1.34732	−0.673660	+	0.739041i	$0.735279\pi$
$614$	−3.26503	−0.131766
$615$	−0.356212	−0.0143638
$616$	0	0
$617$	−15.4849	−0.623399	−0.311700	−	0.950181i	$-0.600898\pi$
$617$	−15.4849	−0.623399	−0.311700	+	0.950181i	$0.600898\pi$
$618$	1.18610	0.0477119
$619$	12.9113	0.518947	0.259474	−	0.965750i	$-0.416451\pi$
$619$	12.9113	0.518947	0.259474	+	0.965750i	$0.416451\pi$
$620$	−12.9860	−0.521530
$621$	14.8443	0.595683
$622$	8.41612	0.337456
$623$	0	0
$624$	2.56712	0.102767
$625$	−19.6058	−0.784232
$626$	4.78580	0.191279
$627$	1.05617	0.0421792
$628$	−19.7577	−0.788420
$629$	25.3808	1.01200
$630$	0	0
$631$	−17.9656	−0.715201	−0.357600	−	0.933875i	$-0.616405\pi$
$631$	−17.9656	−0.715201	−0.357600	+	0.933875i	$0.616405\pi$
$632$	−6.08882	−0.242200
$633$	4.66882	0.185569
$634$	6.77253	0.268971
$635$	2.02179	0.0802324
$636$	0.542935	0.0215288
$637$	0	0
$638$	0.716225	0.0283556
$639$	39.9558	1.58063
$640$	−15.8930	−0.628228
$641$	−19.5113	−0.770650	−0.385325	−	0.922781i	$-0.625911\pi$
$641$	−19.5113	−0.770650	−0.385325	+	0.922781i	$0.625911\pi$
$642$	−0.159248	−0.00628503
$643$	30.0465	1.18492	0.592458	−	0.805601i	$-0.298157\pi$
$643$	30.0465	1.18492	0.592458	+	0.805601i	$0.298157\pi$
$644$	0	0
$645$	−6.20280	−0.244235
$646$	4.56934	0.179778
$647$	−7.90500	−0.310778	−0.155389	−	0.987853i	$-0.549663\pi$
$647$	−7.90500	−0.310778	−0.155389	+	0.987853i	$0.549663\pi$
$648$	−8.81808	−0.346407
$649$	1.29709	0.0509154
$650$	−0.644642	−0.0252849
$651$	0	0
$652$	−25.7324	−1.00776
$653$	31.2160	1.22158	0.610788	−	0.791794i	$-0.290853\pi$
$653$	31.2160	1.22158	0.610788	+	0.791794i	$0.290853\pi$
$654$	0.881654	0.0344754
$655$	24.8792	0.972111
$656$	−2.29110	−0.0894524
$657$	−4.33133	−0.168981
$658$	0	0
$659$	−32.3112	−1.25866	−0.629332	−	0.777137i	$-0.716671\pi$
$659$	−32.3112	−1.25866	−0.629332	+	0.777137i	$0.716671\pi$
$660$	−1.20852	−0.0470416
$661$	33.3796	1.29831	0.649157	−	0.760654i	$-0.275122\pi$
$661$	33.3796	1.29831	0.649157	+	0.760654i	$0.275122\pi$
$662$	−8.60979	−0.334629
$663$	3.57948	0.139016
$664$	16.9859	0.659182
$665$	0	0
$666$	−4.02163	−0.155835
$667$	−21.4597	−0.830924
$668$	−33.9502	−1.31357
$669$	3.33772	0.129044
$670$	−4.84850	−0.187314
$671$	8.81842	0.340431
$672$	0	0
$673$	−28.2936	−1.09064	−0.545318	−	0.838229i	$-0.683591\pi$
$673$	−28.2936	−1.09064	−0.545318	+	0.838229i	$0.683591\pi$
$674$	9.52458	0.366873
$675$	1.48360	0.0571039
$676$	11.8193	0.454588
$677$	0.357613	0.0137442	0.00687209	−	0.999976i	$-0.497813\pi$
$677$	0.357613	0.0137442	0.00687209	+	0.999976i	$0.497813\pi$
$678$	1.11923	0.0429837
$679$	0	0
$680$	−10.6542	−0.408569
$681$	−6.69578	−0.256583
$682$	1.01537	0.0388804
$683$	−21.2948	−0.814823	−0.407411	−	0.913245i	$-0.633569\pi$
$683$	−21.2948	−0.814823	−0.407411	+	0.913245i	$0.633569\pi$
$684$	19.1952	0.733945
$685$	37.9425	1.44971
$686$	0	0
$687$	−0.296194	−0.0113005
$688$	−39.8954	−1.52100
$689$	−2.68972	−0.102470
$690$	−1.36580	−0.0519952
$691$	5.71204	0.217296	0.108648	−	0.994080i	$-0.465348\pi$
$691$	5.71204	0.217296	0.108648	+	0.994080i	$0.465348\pi$
$692$	−40.8912	−1.55445
$693$	0	0
$694$	4.92810	0.187068
$695$	−28.6003	−1.08487
$696$	−0.683208	−0.0258969
$697$	−3.19461	−0.121005
$698$	−7.57412	−0.286684
$699$	−3.48738	−0.131905
$700$	0	0
$701$	−13.8087	−0.521546	−0.260773	−	0.965400i	$-0.583977\pi$
$701$	−13.8087	−0.521546	−0.260773	+	0.965400i	$0.583977\pi$
$702$	−1.14896	−0.0433647
$703$	17.3674	0.655025
$704$	−7.12779	−0.268639
$705$	7.39183	0.278392
$706$	1.28746	0.0484543
$707$	0	0
$708$	−0.607198	−0.0228199
$709$	24.6224	0.924712	0.462356	−	0.886694i	$-0.347004\pi$
$709$	24.6224	0.924712	0.462356	+	0.886694i	$0.347004\pi$
$710$	−7.44722	−0.279489
$711$	−16.8175	−0.630704
$712$	−10.2468	−0.384015
$713$	−30.4227	−1.13934
$714$	0	0
$715$	5.98706	0.223903
$716$	2.32381	0.0868450
$717$	6.03219	0.225276
$718$	3.01244	0.112423
$719$	22.8206	0.851065	0.425533	−	0.904943i	$-0.360087\pi$
$719$	22.8206	0.851065	0.425533	+	0.904943i	$0.360087\pi$
$720$	−21.1044	−0.786514
$721$	0	0
$722$	−1.99612	−0.0742880
$723$	−1.69205	−0.0629279
$724$	1.18307	0.0439683
$725$	−2.14477	−0.0796548
$726$	−0.719527	−0.0267042
$727$	−38.1114	−1.41348	−0.706738	−	0.707476i	$-0.749834\pi$
$727$	−38.1114	−1.41348	−0.706738	+	0.707476i	$0.749834\pi$
$728$	0	0
$729$	−23.0168	−0.852474
$730$	0.807302	0.0298796
$731$	−55.6285	−2.05749
$732$	−4.12810	−0.152579
$733$	−16.9697	−0.626792	−0.313396	−	0.949623i	$-0.601467\pi$
$733$	−16.9697	−0.626792	−0.313396	+	0.949623i	$0.601467\pi$
$734$	1.88031	0.0694036
$735$	0	0
$736$	−28.1171	−1.03641
$737$	−10.0507	−0.370222
$738$	0.506192	0.0186332
$739$	−8.28513	−0.304774	−0.152387	−	0.988321i	$-0.548696\pi$
$739$	−8.28513	−0.304774	−0.152387	+	0.988321i	$0.548696\pi$
$740$	−19.8727	−0.730536
$741$	2.44935	0.0899791
$742$	0	0
$743$	40.9820	1.50349	0.751743	−	0.659457i	$-0.229214\pi$
$743$	40.9820	1.50349	0.751743	+	0.659457i	$0.229214\pi$
$744$	−0.968559	−0.0355091
$745$	0.812315	0.0297609
$746$	1.57919	0.0578181
$747$	46.9155	1.71655
$748$	−10.8384	−0.396290
$749$	0	0
$750$	−0.884586	−0.0323005
$751$	−11.3363	−0.413668	−0.206834	−	0.978376i	$-0.566316\pi$
$751$	−11.3363	−0.413668	−0.206834	+	0.978376i	$0.566316\pi$
$752$	47.5431	1.73372
$753$	3.33979	0.121709
$754$	1.66099	0.0604898
$755$	−20.3235	−0.739648
$756$	0	0
$757$	−49.9464	−1.81533	−0.907666	−	0.419692i	$-0.862138\pi$
$757$	−49.9464	−1.81533	−0.907666	+	0.419692i	$0.862138\pi$
$758$	−3.43461	−0.124751
$759$	−2.83124	−0.102767
$760$	−7.29039	−0.264450
$761$	−22.8650	−0.828856	−0.414428	−	0.910082i	$-0.636018\pi$
$761$	−22.8650	−0.828856	−0.414428	+	0.910082i	$0.636018\pi$
$762$	0.0740019	0.00268081
$763$	0	0
$764$	−24.7851	−0.896692
$765$	−29.4271	−1.06394
$766$	−7.46229	−0.269623
$767$	3.00808	0.108616
$768$	2.88081	0.103952
$769$	13.3784	0.482438	0.241219	−	0.970471i	$-0.422453\pi$
$769$	13.3784	0.482438	0.241219	+	0.970471i	$0.422453\pi$
$770$	0	0
$771$	−7.41046	−0.266881
$772$	20.0575	0.721887
$773$	30.3610	1.09201	0.546005	−	0.837782i	$-0.316148\pi$
$773$	30.3610	1.09201	0.546005	+	0.837782i	$0.316148\pi$
$774$	8.81444	0.316828
$775$	−3.04056	−0.109220
$776$	−3.62932	−0.130285
$777$	0	0
$778$	−7.88854	−0.282818
$779$	−2.18599	−0.0783213
$780$	−2.80267	−0.100352
$781$	−15.4377	−0.552405
$782$	−12.2489	−0.438020
$783$	−3.82268	−0.136611
$784$	0	0
$785$	20.7264	0.739756
$786$	0.910633	0.0324812
$787$	4.90786	0.174946	0.0874731	−	0.996167i	$-0.472121\pi$
$787$	4.90786	0.174946	0.0874731	+	0.996167i	$0.472121\pi$
$788$	−9.96173	−0.354872
$789$	5.67542	0.202050
$790$	3.13455	0.111522
$791$	0	0
$792$	3.49950	0.124349
$793$	20.4507	0.726227
$794$	−5.29502	−0.187913
$795$	−0.569553	−0.0202000
$796$	−31.3452	−1.11100
$797$	39.1775	1.38774	0.693869	−	0.720101i	$-0.255905\pi$
$797$	39.1775	1.38774	0.693869	+	0.720101i	$0.255905\pi$
$798$	0	0
$799$	66.2921	2.34524
$800$	−2.81014	−0.0993534
$801$	−28.3019	−0.999997
$802$	−5.87005	−0.207279
$803$	1.67350	0.0590564
$804$	4.70494	0.165931
$805$	0	0
$806$	2.35473	0.0829418
$807$	3.99434	0.140607
$808$	−4.11044	−0.144605
$809$	−1.01667	−0.0357441	−0.0178721	−	0.999840i	$-0.505689\pi$
$809$	−1.01667	−0.0357441	−0.0178721	+	0.999840i	$0.505689\pi$
$810$	4.53958	0.159505
$811$	−21.3848	−0.750922	−0.375461	−	0.926838i	$-0.622516\pi$
$811$	−21.3848	−0.750922	−0.375461	+	0.926838i	$0.622516\pi$
$812$	0	0
$813$	0.691411	0.0242488
$814$	1.55384	0.0544619
$815$	26.9940	0.945558
$816$	4.87509	0.170662
$817$	−38.0652	−1.33173
$818$	9.84958	0.344383
$819$	0	0
$820$	2.50133	0.0873501
$821$	4.95024	0.172765	0.0863823	−	0.996262i	$-0.472469\pi$
$821$	4.95024	0.172765	0.0863823	+	0.996262i	$0.472469\pi$
$822$	1.38878	0.0484392
$823$	40.4461	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$823$	40.4461	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$824$	−16.9718	−0.591240
$825$	−0.282965	−0.00985159
$826$	0	0
$827$	8.59870	0.299006	0.149503	−	0.988761i	$-0.452233\pi$
$827$	8.59870	0.299006	0.149503	+	0.988761i	$0.452233\pi$
$828$	−51.4560	−1.78822
$829$	54.5933	1.89610	0.948051	−	0.318118i	$-0.103051\pi$
$829$	54.5933	1.89610	0.948051	+	0.318118i	$0.103051\pi$
$830$	−8.74443	−0.303523
$831$	5.91385	0.205149
$832$	−16.5300	−0.573075
$833$	0	0
$834$	−1.04683	−0.0362488
$835$	35.6146	1.23250
$836$	−7.41643	−0.256502
$837$	−5.41927	−0.187317
$838$	7.37304	0.254697
$839$	−11.9896	−0.413928	−0.206964	−	0.978349i	$-0.566358\pi$
$839$	−11.9896	−0.413928	−0.206964	+	0.978349i	$0.566358\pi$
$840$	0	0
$841$	−23.4738	−0.809440
$842$	−7.65280	−0.263733
$843$	−2.74410	−0.0945119
$844$	−32.7846	−1.12849
$845$	−12.3987	−0.426529
$846$	−10.5041	−0.361138
$847$	0	0
$848$	−3.66328	−0.125797
$849$	−3.69040	−0.126654
$850$	−1.22421	−0.0419899
$851$	−46.5564	−1.59593
$852$	7.22673	0.247584
$853$	−4.71104	−0.161303	−0.0806515	−	0.996742i	$-0.525700\pi$
$853$	−4.71104	−0.161303	−0.0806515	+	0.996742i	$0.525700\pi$
$854$	0	0
$855$	−20.1362	−0.688644
$856$	2.27867	0.0778833
$857$	15.3452	0.524181	0.262090	−	0.965043i	$-0.415588\pi$
$857$	15.3452	0.524181	0.262090	+	0.965043i	$0.415588\pi$
$858$	0.219139	0.00748129
$859$	11.6869	0.398751	0.199376	−	0.979923i	$-0.436109\pi$
$859$	11.6869	0.398751	0.199376	+	0.979923i	$0.436109\pi$
$860$	43.5562	1.48525
$861$	0	0
$862$	−10.6195	−0.361702
$863$	−41.1279	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$863$	−41.1279	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$864$	−5.00858	−0.170395
$865$	42.8960	1.45851
$866$	8.62824	0.293200
$867$	2.13170	0.0723962
$868$	0	0
$869$	6.49776	0.220421
$870$	0.351718	0.0119244
$871$	−23.3085	−0.789778
$872$	−12.6155	−0.427215
$873$	−10.0243	−0.339270
$874$	−8.38163	−0.283513
$875$	0	0
$876$	−0.783400	−0.0264686
$877$	9.69819	0.327485	0.163742	−	0.986503i	$-0.447643\pi$
$877$	9.69819	0.327485	0.163742	+	0.986503i	$0.447643\pi$
$878$	5.63314	0.190109
$879$	−0.0345931	−0.00116680
$880$	8.15410	0.274874
$881$	30.8120	1.03808	0.519041	−	0.854749i	$-0.326289\pi$
$881$	30.8120	1.03808	0.519041	+	0.854749i	$0.326289\pi$
$882$	0	0
$883$	−3.92355	−0.132038	−0.0660190	−	0.997818i	$-0.521030\pi$
$883$	−3.92355	−0.132038	−0.0660190	+	0.997818i	$0.521030\pi$
$884$	−25.1352	−0.845388
$885$	0.636966	0.0214114
$886$	−4.11976	−0.138406
$887$	−14.2323	−0.477875	−0.238937	−	0.971035i	$-0.576799\pi$
$887$	−14.2323	−0.477875	−0.238937	+	0.971035i	$0.576799\pi$
$888$	−1.48221	−0.0497395
$889$	0	0
$890$	5.27509	0.176821
$891$	9.41031	0.315257
$892$	−23.4375	−0.784746
$893$	45.3620	1.51798
$894$	0.0297325	0.000994402 0
$895$	−2.43774	−0.0814847
$896$	0	0
$897$	−6.56591	−0.219229
$898$	1.13410	0.0378453
$899$	7.83436	0.261291
$900$	−5.14272	−0.171424
$901$	−5.10792	−0.170169
$902$	−0.195577	−0.00651201
$903$	0	0
$904$	−16.0149	−0.532648
$905$	−1.24107	−0.0412544
$906$	−0.743883	−0.0247139
$907$	−57.8175	−1.91980	−0.959900	−	0.280344i	$-0.909552\pi$
$907$	−57.8175	−1.91980	−0.959900	+	0.280344i	$0.909552\pi$
$908$	47.0179	1.56034
$909$	−11.3531	−0.376559
$910$	0	0
$911$	−4.69276	−0.155478	−0.0777390	−	0.996974i	$-0.524770\pi$
$911$	−4.69276	−0.155478	−0.0777390	+	0.996974i	$0.524770\pi$
$912$	3.33590	0.110463
$913$	−18.1267	−0.599907
$914$	4.16330	0.137710
$915$	4.33048	0.143161
$916$	2.07988	0.0687211
$917$	0	0
$918$	−2.18193	−0.0720145
$919$	−47.5378	−1.56813	−0.784064	−	0.620681i	$-0.786856\pi$
$919$	−47.5378	−1.56813	−0.784064	+	0.620681i	$0.786856\pi$
$920$	19.5431	0.644319
$921$	−3.32369	−0.109519
$922$	−4.19235	−0.138068
$923$	−35.8015	−1.17842
$924$	0	0
$925$	−4.65304	−0.152991
$926$	−1.55455	−0.0510857
$927$	−46.8765	−1.53963
$928$	7.24065	0.237686
$929$	−47.2665	−1.55076	−0.775382	−	0.631492i	$-0.782443\pi$
$929$	−47.2665	−1.55076	−0.775382	+	0.631492i	$0.782443\pi$
$930$	0.498618	0.0163503
$931$	0	0
$932$	24.4884	0.802145
$933$	8.56733	0.280482
$934$	−3.78577	−0.123874
$935$	11.3697	0.371830
$936$	8.11566	0.265269
$937$	8.54328	0.279097	0.139548	−	0.990215i	$-0.455435\pi$
$937$	8.54328	0.279097	0.139548	+	0.990215i	$0.455435\pi$
$938$	0	0
$939$	4.87178	0.158985
$940$	−51.9056	−1.69297
$941$	−38.6407	−1.25965	−0.629826	−	0.776736i	$-0.716874\pi$
$941$	−38.6407	−1.25965	−0.629826	+	0.776736i	$0.716874\pi$
$942$	0.758630	0.0247175
$943$	5.85993	0.190826
$944$	4.09687	0.133342
$945$	0	0
$946$	−3.40563	−0.110727
$947$	−6.62023	−0.215129	−0.107564	−	0.994198i	$-0.534305\pi$
$947$	−6.62023	−0.215129	−0.107564	+	0.994198i	$0.534305\pi$
$948$	−3.04174	−0.0987912
$949$	3.88099	0.125982
$950$	−0.837694	−0.0271784
$951$	6.89420	0.223560
$952$	0	0
$953$	−23.5338	−0.762335	−0.381167	−	0.924506i	$-0.624478\pi$
$953$	−23.5338	−0.762335	−0.381167	+	0.924506i	$0.624478\pi$
$954$	0.809359	0.0262040
$955$	26.0002	0.841346
$956$	−42.3582	−1.36996
$957$	0.729093	0.0235682
$958$	0.834266	0.0269539
$959$	0	0
$960$	−3.50026	−0.112970
$961$	−19.8935	−0.641726
$962$	3.60349	0.116181
$963$	6.29374	0.202813
$964$	11.8816	0.382680
$965$	−21.0409	−0.677330
$966$	0	0
$967$	−15.1190	−0.486194	−0.243097	−	0.970002i	$-0.578163\pi$
$967$	−15.1190	−0.486194	−0.243097	+	0.970002i	$0.578163\pi$
$968$	10.2956	0.330915
$969$	4.65144	0.149426
$970$	1.86839	0.0599903
$971$	−14.3014	−0.458953	−0.229476	−	0.973314i	$-0.573701\pi$
$971$	−14.3014	−0.458953	−0.229476	+	0.973314i	$0.573701\pi$
$972$	−13.8072	−0.442868
$973$	0	0
$974$	1.26076	0.0403975
$975$	−0.656223	−0.0210160
$976$	27.8529	0.891551
$977$	−26.5923	−0.850763	−0.425381	−	0.905014i	$-0.639860\pi$
$977$	−26.5923	−0.850763	−0.425381	+	0.905014i	$0.639860\pi$
$978$	0.988038	0.0315940
$979$	10.9350	0.349483
$980$	0	0
$981$	−34.8443	−1.11249
$982$	−0.336242	−0.0107299
$983$	44.0252	1.40419	0.702093	−	0.712085i	$-0.252249\pi$
$983$	44.0252	1.40419	0.702093	+	0.712085i	$0.252249\pi$
$984$	0.186561	0.00594735
$985$	10.4501	0.332968
$986$	3.15431	0.100454
$987$	0	0
$988$	−17.1994	−0.547185
$989$	102.040	3.24469
$990$	−1.80155	−0.0572571
$991$	−23.3522	−0.741808	−0.370904	−	0.928671i	$-0.620952\pi$
$991$	−23.3522	−0.741808	−0.370904	+	0.928671i	$0.620952\pi$
$992$	10.2648	0.325908
$993$	−8.76447	−0.278132
$994$	0	0
$995$	32.8819	1.04243
$996$	8.48552	0.268874
$997$	−37.0417	−1.17312	−0.586561	−	0.809905i	$-0.699519\pi$
$997$	−37.0417	−1.17312	−0.586561	+	0.809905i	$0.699519\pi$
$998$	4.03032	0.127577
$999$	−8.29322	−0.262386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.t.1.38	yes	74
7.6	odd	2	inner	6223.2.a.t.1.37	✓	74

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.t.1.37	✓	74	7.6	odd	2	inner
6223.2.a.t.1.38	yes	74	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.269622 q^{2} +0.274466 q^{3} -1.92730 q^{4} +2.02179 q^{5} +0.0740019 q^{6} -1.05889 q^{8} -2.92467 q^{9} +0.545119 q^{10} +1.13000 q^{11} -0.528979 q^{12} +2.62058 q^{13} +0.554913 q^{15} +3.56911 q^{16} +4.97662 q^{17} -0.788554 q^{18} +3.40537 q^{19} -3.89661 q^{20} +0.304673 q^{22} -9.12870 q^{23} -0.290628 q^{24} -0.912359 q^{25} +0.706566 q^{26} -1.62612 q^{27} +2.35080 q^{29} +0.149616 q^{30} +3.33264 q^{31} +3.08008 q^{32} +0.310147 q^{33} +1.34180 q^{34} +5.63673 q^{36} +5.10001 q^{37} +0.918163 q^{38} +0.719260 q^{39} -2.14085 q^{40} -0.641924 q^{41} -11.1780 q^{43} -2.17786 q^{44} -5.91307 q^{45} -2.46129 q^{46} +13.3207 q^{47} +0.979598 q^{48} -0.245992 q^{50} +1.36591 q^{51} -5.05066 q^{52} -1.02638 q^{53} -0.438437 q^{54} +2.28463 q^{55} +0.934659 q^{57} +0.633826 q^{58} +1.14787 q^{59} -1.06949 q^{60} +7.80389 q^{61} +0.898552 q^{62} -6.30776 q^{64} +5.29827 q^{65} +0.0836224 q^{66} -8.89439 q^{67} -9.59145 q^{68} -2.50552 q^{69} -13.6617 q^{71} +3.09689 q^{72} +1.48097 q^{73} +1.37507 q^{74} -0.250411 q^{75} -6.56319 q^{76} +0.193928 q^{78} +5.75021 q^{79} +7.21600 q^{80} +8.32769 q^{81} -0.173077 q^{82} -16.0413 q^{83} +10.0617 q^{85} -3.01382 q^{86} +0.645213 q^{87} -1.19654 q^{88} +9.67695 q^{89} -1.59429 q^{90} +17.5938 q^{92} +0.914696 q^{93} +3.59155 q^{94} +6.88496 q^{95} +0.845377 q^{96} +3.42749 q^{97} -3.30488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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