LMFDB - Embedded newform 2793.2.a.w.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2793,2,Mod(1,2793)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2793.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2793.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,-3,9,0] 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:	$22.3022172845$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	2793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.74483 q^{2} -1.00000 q^{3} +5.53407 q^{4} +2.53407 q^{5} +2.74483 q^{6} -9.70041 q^{8} +1.00000 q^{9} -6.95558 q^{10} +2.42151 q^{11} -5.53407 q^{12} +0.421512 q^{13} -2.53407 q^{15} +15.5578 q^{16} -6.53407 q^{17} -2.74483 q^{18} +1.00000 q^{19} +14.0237 q^{20} -6.64663 q^{22} -1.57849 q^{23} +9.70041 q^{24} +1.42151 q^{25} -1.15698 q^{26} -1.00000 q^{27} +6.02372 q^{29} +6.95558 q^{30} -3.48965 q^{31} -23.3026 q^{32} -2.42151 q^{33} +17.9349 q^{34} +5.53407 q^{36} -2.84302 q^{37} -2.74483 q^{38} -0.421512 q^{39} -24.5815 q^{40} -8.42151 q^{41} +11.4897 q^{43} +13.4008 q^{44} +2.53407 q^{45} +4.33268 q^{46} -8.02372 q^{47} -15.5578 q^{48} -3.90180 q^{50} +6.53407 q^{51} +2.33268 q^{52} -10.8667 q^{53} +2.74483 q^{54} +6.13628 q^{55} -1.00000 q^{57} -16.5341 q^{58} -14.1363 q^{59} -14.0237 q^{60} -6.84302 q^{61} +9.57849 q^{62} +32.8461 q^{64} +1.06814 q^{65} +6.64663 q^{66} -9.91116 q^{67} -36.1600 q^{68} +1.57849 q^{69} -13.3771 q^{71} -9.70041 q^{72} -7.06814 q^{73} +7.80361 q^{74} -1.42151 q^{75} +5.53407 q^{76} +1.15698 q^{78} +14.1363 q^{79} +39.4245 q^{80} +1.00000 q^{81} +23.1156 q^{82} -2.11256 q^{83} -16.5578 q^{85} -31.5371 q^{86} -6.02372 q^{87} -23.4897 q^{88} +9.71477 q^{89} -6.95558 q^{90} -8.73546 q^{92} +3.48965 q^{93} +22.0237 q^{94} +2.53407 q^{95} +23.3026 q^{96} +4.97930 q^{97} +2.42151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} + 9 q^{4} - q^{6} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} - 9 q^{12} - 2 q^{13} + 13 q^{16} - 12 q^{17} + q^{18} + 3 q^{19} + 16 q^{20} - 8 q^{22} - 8 q^{23} + 9 q^{24} + q^{25}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^2 - 3 * q^3 + 9 * q^4 - q^6 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 - 9 * q^12 - 2 * q^13 + 13 * q^16 - 12 * q^17 + q^18 + 3 * q^19 + 16 * q^20 - 8 * q^22 - 8 * q^23 + 9 * q^24 + q^25 - 10 * q^26 - 3 * q^27 - 8 * q^29 + 10 * q^30 + 8 * q^31 - 27 * q^32 - 4 * q^33 + 6 * q^34 + 9 * q^36 - 2 * q^37 + q^38 + 2 * q^39 - 14 * q^40 - 22 * q^41 + 16 * q^43 - 12 * q^46 + 2 * q^47 - 13 * q^48 - 9 * q^50 + 12 * q^51 - 18 * q^52 - q^54 - 12 * q^55 - 3 * q^57 - 42 * q^58 - 12 * q^59 - 16 * q^60 - 14 * q^61 + 32 * q^62 + 29 * q^64 - 12 * q^65 + 8 * q^66 - 8 * q^67 - 52 * q^68 + 8 * q^69 - 26 * q^71 - 9 * q^72 - 6 * q^73 + 18 * q^74 - q^75 + 9 * q^76 + 10 * q^78 + 12 * q^79 + 52 * q^80 + 3 * q^81 + 2 * q^82 - 2 * q^83 - 16 * q^85 - 24 * q^86 + 8 * q^87 - 52 * q^88 + 2 * q^89 - 10 * q^90 - 36 * q^92 - 8 * q^93 + 40 * q^94 + 27 * q^96 - 22 * q^97 + 4 * 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 $

$2$	−2.74483	−1.94089	−0.970443	−	0.241332i	$-0.922416\pi$
$2$	−2.74483	−1.94089	−0.970443	+	0.241332i	$0.922416\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.53407	2.76704
$5$	2.53407	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$5$	2.53407	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$6$	2.74483	1.12057
$7$	0	0
$7$	0	0
$8$	−9.70041	−3.42961
$9$	1.00000	0.333333
$10$	−6.95558	−2.19955
$11$	2.42151	0.730113	0.365057	−	0.930985i	$-0.381050\pi$
$11$	2.42151	0.730113	0.365057	+	0.930985i	$0.381050\pi$
$12$	−5.53407	−1.59755
$13$	0.421512	0.116906	0.0584532	−	0.998290i	$-0.481383\pi$
$13$	0.421512	0.116906	0.0584532	+	0.998290i	$0.481383\pi$
$14$	0	0
$15$	−2.53407	−0.654294
$16$	15.5578	3.88945
$17$	−6.53407	−1.58474	−0.792372	−	0.610038i	$-0.791154\pi$
$17$	−6.53407	−1.58474	−0.792372	+	0.610038i	$0.791154\pi$
$18$	−2.74483	−0.646962
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	14.0237	3.13580
$21$	0	0
$22$	−6.64663	−1.41707
$23$	−1.57849	−0.329138	−0.164569	−	0.986366i	$-0.552623\pi$
$23$	−1.57849	−0.329138	−0.164569	+	0.986366i	$0.552623\pi$
$24$	9.70041	1.98009
$25$	1.42151	0.284302
$26$	−1.15698	−0.226902
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.02372	1.11858	0.559289	−	0.828973i	$-0.311074\pi$
$29$	6.02372	1.11858	0.559289	+	0.828973i	$0.311074\pi$
$30$	6.95558	1.26991
$31$	−3.48965	−0.626760	−0.313380	−	0.949628i	$-0.601461\pi$
$31$	−3.48965	−0.626760	−0.313380	+	0.949628i	$0.601461\pi$
$32$	−23.3026	−4.11936
$33$	−2.42151	−0.421531
$34$	17.9349	3.07581
$35$	0	0
$36$	5.53407	0.922345
$37$	−2.84302	−0.467390	−0.233695	−	0.972310i	$-0.575082\pi$
$37$	−2.84302	−0.467390	−0.233695	+	0.972310i	$0.575082\pi$
$38$	−2.74483	−0.445270
$39$	−0.421512	−0.0674959
$40$	−24.5815	−3.88668
$41$	−8.42151	−1.31522	−0.657610	−	0.753359i	$-0.728432\pi$
$41$	−8.42151	−1.31522	−0.657610	+	0.753359i	$0.728432\pi$
$42$	0	0
$43$	11.4897	1.75216	0.876078	−	0.482170i	$-0.160151\pi$
$43$	11.4897	1.75216	0.876078	+	0.482170i	$0.160151\pi$
$44$	13.4008	2.02025
$45$	2.53407	0.377757
$46$	4.33268	0.638818
$47$	−8.02372	−1.17038	−0.585190	−	0.810896i	$-0.698980\pi$
$47$	−8.02372	−1.17038	−0.585190	+	0.810896i	$0.698980\pi$
$48$	−15.5578	−2.24557
$49$	0	0
$50$	−3.90180	−0.551798
$51$	6.53407	0.914953
$52$	2.33268	0.323484
$53$	−10.8667	−1.49266	−0.746331	−	0.665575i	$-0.768186\pi$
$53$	−10.8667	−1.49266	−0.746331	+	0.665575i	$0.768186\pi$
$54$	2.74483	0.373524
$55$	6.13628	0.827416
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−16.5341	−2.17103
$59$	−14.1363	−1.84039	−0.920194	−	0.391464i	$-0.871969\pi$
$59$	−14.1363	−1.84039	−0.920194	+	0.391464i	$0.871969\pi$
$60$	−14.0237	−1.81045
$61$	−6.84302	−0.876159	−0.438080	−	0.898936i	$-0.644341\pi$
$61$	−6.84302	−0.876159	−0.438080	+	0.898936i	$0.644341\pi$
$62$	9.57849	1.21647
$63$	0	0
$64$	32.8461	4.10576
$65$	1.06814	0.132487
$66$	6.64663	0.818143
$67$	−9.91116	−1.21084	−0.605421	−	0.795906i	$-0.706995\pi$
$67$	−9.91116	−1.21084	−0.605421	+	0.795906i	$0.706995\pi$
$68$	−36.1600	−4.38504
$69$	1.57849	0.190028
$70$	0	0
$71$	−13.3771	−1.58757	−0.793784	−	0.608199i	$-0.791892\pi$
$71$	−13.3771	−1.58757	−0.793784	+	0.608199i	$0.791892\pi$
$72$	−9.70041	−1.14320
$73$	−7.06814	−0.827263	−0.413632	−	0.910444i	$-0.635740\pi$
$73$	−7.06814	−0.827263	−0.413632	+	0.910444i	$0.635740\pi$
$74$	7.80361	0.907151
$75$	−1.42151	−0.164142
$76$	5.53407	0.634801
$77$	0	0
$78$	1.15698	0.131002
$79$	14.1363	1.59046	0.795228	−	0.606311i	$-0.207351\pi$
$79$	14.1363	1.59046	0.795228	+	0.606311i	$0.207351\pi$
$80$	39.4245	4.40780
$81$	1.00000	0.111111
$82$	23.1156	2.55269
$83$	−2.11256	−0.231883	−0.115942	−	0.993256i	$-0.536989\pi$
$83$	−2.11256	−0.231883	−0.115942	+	0.993256i	$0.536989\pi$
$84$	0	0
$85$	−16.5578	−1.79594
$86$	−31.5371	−3.40073
$87$	−6.02372	−0.645811
$88$	−23.4897	−2.50401
$89$	9.71477	1.02976	0.514882	−	0.857261i	$-0.327836\pi$
$89$	9.71477	1.02976	0.514882	+	0.857261i	$0.327836\pi$
$90$	−6.95558	−0.733183
$91$	0	0
$92$	−8.73546	−0.910735
$93$	3.48965	0.361860
$94$	22.0237	2.27157
$95$	2.53407	0.259990
$96$	23.3026	2.37831
$97$	4.97930	0.505572	0.252786	−	0.967522i	$-0.418653\pi$
$97$	4.97930	0.505572	0.252786	+	0.967522i	$0.418653\pi$
$98$	0	0
$99$	2.42151	0.243371
$100$	7.86675	0.786675
$101$	1.69105	0.168265	0.0841327	−	0.996455i	$-0.473188\pi$
$101$	1.69105	0.168265	0.0841327	+	0.996455i	$0.473188\pi$
$102$	−17.9349	−1.77582
$103$	−3.15698	−0.311066	−0.155533	−	0.987831i	$-0.549710\pi$
$103$	−3.15698	−0.311066	−0.155533	+	0.987831i	$0.549710\pi$
$104$	−4.08884	−0.400943
$105$	0	0
$106$	29.8273	2.89709
$107$	−16.5341	−1.59841	−0.799204	−	0.601059i	$-0.794746\pi$
$107$	−16.5341	−1.59841	−0.799204	+	0.601059i	$0.794746\pi$
$108$	−5.53407	−0.532516
$109$	1.77488	0.170003	0.0850015	−	0.996381i	$-0.472910\pi$
$109$	1.77488	0.170003	0.0850015	+	0.996381i	$0.472910\pi$
$110$	−16.8430	−1.60592
$111$	2.84302	0.269848
$112$	0	0
$113$	10.0237	0.942952	0.471476	−	0.881879i	$-0.343721\pi$
$113$	10.0237	0.942952	0.471476	+	0.881879i	$0.343721\pi$
$114$	2.74483	0.257077
$115$	−4.00000	−0.373002
$116$	33.3357	3.09514
$117$	0.421512	0.0389688
$118$	38.8016	3.57198
$119$	0	0
$120$	24.5815	2.24398
$121$	−5.13628	−0.466935
$122$	18.7829	1.70052
$123$	8.42151	0.759342
$124$	−19.3120	−1.73427
$125$	−9.06814	−0.811079
$126$	0	0
$127$	2.08884	0.185354	0.0926771	−	0.995696i	$-0.470458\pi$
$127$	2.08884	0.185354	0.0926771	+	0.995696i	$0.470458\pi$
$128$	−43.5515	−3.84944
$129$	−11.4897	−1.01161
$130$	−2.93186	−0.257141
$131$	13.9349	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$131$	13.9349	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$132$	−13.4008	−1.16639
$133$	0	0
$134$	27.2044	2.35010
$135$	−2.53407	−0.218098
$136$	63.3831	5.43506
$137$	16.9793	1.45064	0.725320	−	0.688412i	$-0.241692\pi$
$137$	16.9793	1.45064	0.725320	+	0.688412i	$0.241692\pi$
$138$	−4.33268	−0.368822
$139$	1.06814	0.0905985	0.0452992	−	0.998973i	$-0.485576\pi$
$139$	1.06814	0.0905985	0.0452992	+	0.998973i	$0.485576\pi$
$140$	0	0
$141$	8.02372	0.675719
$142$	36.7178	3.08129
$143$	1.02070	0.0853549
$144$	15.5578	1.29648
$145$	15.2645	1.26765
$146$	19.4008	1.60562
$147$	0	0
$148$	−15.7335	−1.29329
$149$	14.0474	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$149$	14.0474	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$150$	3.90180	0.318581
$151$	−14.7542	−1.20068	−0.600339	−	0.799745i	$-0.704968\pi$
$151$	−14.7542	−1.20068	−0.600339	+	0.799745i	$0.704968\pi$
$152$	−9.70041	−0.786807
$153$	−6.53407	−0.528248
$154$	0	0
$155$	−8.84302	−0.710289
$156$	−2.33268	−0.186764
$157$	16.1363	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$157$	16.1363	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$158$	−38.8016	−3.08689
$159$	10.8667	0.861789
$160$	−59.0505	−4.66835
$161$	0	0
$162$	−2.74483	−0.215654
$163$	−6.64663	−0.520604	−0.260302	−	0.965527i	$-0.583822\pi$
$163$	−6.64663	−0.520604	−0.260302	+	0.965527i	$0.583822\pi$
$164$	−46.6052	−3.63926
$165$	−6.13628	−0.477709
$166$	5.79861	0.450059
$167$	14.3614	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$167$	14.3614	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$168$	0	0
$169$	−12.8223	−0.986333
$170$	45.4483	3.48572
$171$	1.00000	0.0764719
$172$	63.5845	4.84828
$173$	5.71477	0.434486	0.217243	−	0.976118i	$-0.430294\pi$
$173$	5.71477	0.434486	0.217243	+	0.976118i	$0.430294\pi$
$174$	16.5341	1.25344
$175$	0	0
$176$	37.6734	2.83974
$177$	14.1363	1.06255
$178$	−26.6654	−1.99865
$179$	−9.60221	−0.717703	−0.358851	−	0.933395i	$-0.616832\pi$
$179$	−9.60221	−0.717703	−0.358851	+	0.933395i	$0.616832\pi$
$180$	14.0237	1.04527
$181$	0.979304	0.0727911	0.0363956	−	0.999337i	$-0.488412\pi$
$181$	0.979304	0.0727911	0.0363956	+	0.999337i	$0.488412\pi$
$182$	0	0
$183$	6.84302	0.505851
$184$	15.3120	1.12881
$185$	−7.20442	−0.529680
$186$	−9.57849	−0.702329
$187$	−15.8223	−1.15704
$188$	−44.4038	−3.23848
$189$	0	0
$190$	−6.95558	−0.504611
$191$	−11.7148	−0.847651	−0.423825	−	0.905744i	$-0.639313\pi$
$191$	−11.7148	−0.847651	−0.423825	+	0.905744i	$0.639313\pi$
$192$	−32.8461	−2.37046
$193$	17.8223	1.28288	0.641440	−	0.767174i	$-0.278337\pi$
$193$	17.8223	1.28288	0.641440	+	0.767174i	$0.278337\pi$
$194$	−13.6673	−0.981257
$195$	−1.06814	−0.0764911
$196$	0	0
$197$	−19.1156	−1.36193	−0.680965	−	0.732316i	$-0.738439\pi$
$197$	−19.1156	−1.36193	−0.680965	+	0.732316i	$0.738439\pi$
$198$	−6.64663	−0.472355
$199$	−16.0474	−1.13757	−0.568787	−	0.822485i	$-0.692587\pi$
$199$	−16.0474	−1.13757	−0.568787	+	0.822485i	$0.692587\pi$
$200$	−13.7892	−0.975047
$201$	9.91116	0.699080
$202$	−4.64163	−0.326584
$203$	0	0
$204$	36.1600	2.53171
$205$	−21.3407	−1.49050
$206$	8.66535	0.603744
$207$	−1.57849	−0.109713
$208$	6.55779	0.454701
$209$	2.42151	0.167499
$210$	0	0
$211$	−9.29326	−0.639774	−0.319887	−	0.947456i	$-0.603645\pi$
$211$	−9.29326	−0.639774	−0.319887	+	0.947456i	$0.603645\pi$
$212$	−60.1373	−4.13025
$213$	13.3771	0.916583
$214$	45.3831	3.10233
$215$	29.1156	1.98567
$216$	9.70041	0.660029
$217$	0	0
$218$	−4.87175	−0.329956
$219$	7.06814	0.477621
$220$	33.9586	2.28949
$221$	−2.75419	−0.185267
$222$	−7.80361	−0.523744
$223$	8.33268	0.557997	0.278999	−	0.960292i	$-0.409998\pi$
$223$	8.33268	0.557997	0.278999	+	0.960292i	$0.409998\pi$
$224$	0	0
$225$	1.42151	0.0947675
$226$	−27.5134	−1.83016
$227$	−16.8905	−1.12106	−0.560530	−	0.828134i	$-0.689403\pi$
$227$	−16.8905	−1.12106	−0.560530	+	0.828134i	$0.689403\pi$
$228$	−5.53407	−0.366503
$229$	−6.04744	−0.399626	−0.199813	−	0.979834i	$-0.564034\pi$
$229$	−6.04744	−0.399626	−0.199813	+	0.979834i	$0.564034\pi$
$230$	10.9793	0.723954
$231$	0	0
$232$	−58.4326	−3.83629
$233$	21.2044	1.38915	0.694574	−	0.719421i	$-0.255593\pi$
$233$	21.2044	1.38915	0.694574	+	0.719421i	$0.255593\pi$
$234$	−1.15698	−0.0756339
$235$	−20.3327	−1.32636
$236$	−78.2312	−5.09242
$237$	−14.1363	−0.918250
$238$	0	0
$239$	−5.80361	−0.375404	−0.187702	−	0.982226i	$-0.560104\pi$
$239$	−5.80361	−0.375404	−0.187702	+	0.982226i	$0.560104\pi$
$240$	−39.4245	−2.54484
$241$	12.9793	0.836070	0.418035	−	0.908431i	$-0.362719\pi$
$241$	12.9793	0.836070	0.418035	+	0.908431i	$0.362719\pi$
$242$	14.0982	0.906266
$243$	−1.00000	−0.0641500
$244$	−37.8698	−2.42436
$245$	0	0
$246$	−23.1156	−1.47380
$247$	0.421512	0.0268202
$248$	33.8510	2.14954
$249$	2.11256	0.133878
$250$	24.8905	1.57421
$251$	−10.3377	−0.652508	−0.326254	−	0.945282i	$-0.605787\pi$
$251$	−10.3377	−0.652508	−0.326254	+	0.945282i	$0.605787\pi$
$252$	0	0
$253$	−3.82233	−0.240308
$254$	−5.73349	−0.359751
$255$	16.5578	1.03689
$256$	53.8491	3.36557
$257$	15.1757	0.946634	0.473317	−	0.880892i	$-0.343056\pi$
$257$	15.1757	0.946634	0.473317	+	0.880892i	$0.343056\pi$
$258$	31.5371	1.96341
$259$	0	0
$260$	5.91116	0.366595
$261$	6.02372	0.372859
$262$	−38.2488	−2.36302
$263$	−8.51035	−0.524771	−0.262385	−	0.964963i	$-0.584509\pi$
$263$	−8.51035	−0.524771	−0.262385	+	0.964963i	$0.584509\pi$
$264$	23.4897	1.44569
$265$	−27.5371	−1.69159
$266$	0	0
$267$	−9.71477	−0.594534
$268$	−54.8491	−3.35044
$269$	−15.1757	−0.925279	−0.462639	−	0.886547i	$-0.653098\pi$
$269$	−15.1757	−0.925279	−0.462639	+	0.886547i	$0.653098\pi$
$270$	6.95558	0.423303
$271$	−0.843024	−0.0512100	−0.0256050	−	0.999672i	$-0.508151\pi$
$271$	−0.843024	−0.0512100	−0.0256050	+	0.999672i	$0.508151\pi$
$272$	−101.656	−6.16378
$273$	0	0
$274$	−46.6052	−2.81553
$275$	3.44221	0.207573
$276$	8.73546	0.525813
$277$	10.5578	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$277$	10.5578	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$278$	−2.93186	−0.175841
$279$	−3.48965	−0.208920
$280$	0	0
$281$	15.0444	0.897475	0.448737	−	0.893664i	$-0.351874\pi$
$281$	15.0444	0.897475	0.448737	+	0.893664i	$0.351874\pi$
$282$	−22.0237	−1.31149
$283$	−1.24581	−0.0740559	−0.0370279	−	0.999314i	$-0.511789\pi$
$283$	−1.24581	−0.0740559	−0.0370279	+	0.999314i	$0.511789\pi$
$284$	−74.0298	−4.39286
$285$	−2.53407	−0.150105
$286$	−2.80163	−0.165664
$287$	0	0
$288$	−23.3026	−1.37312
$289$	25.6941	1.51142
$290$	−41.8985	−2.46036
$291$	−4.97930	−0.291892
$292$	−39.1156	−2.28907
$293$	−20.4215	−1.19304	−0.596519	−	0.802599i	$-0.703450\pi$
$293$	−20.4215	−1.19304	−0.596519	+	0.802599i	$0.703450\pi$
$294$	0	0
$295$	−35.8223	−2.08566
$296$	27.5785	1.60297
$297$	−2.42151	−0.140510
$298$	−38.5578	−2.23359
$299$	−0.665351	−0.0384783
$300$	−7.86675	−0.454187
$301$	0	0
$302$	40.4977	2.33038
$303$	−1.69105	−0.0971481
$304$	15.5578	0.892301
$305$	−17.3407	−0.992926
$306$	17.9349	1.02527
$307$	2.64663	0.151051	0.0755255	−	0.997144i	$-0.475937\pi$
$307$	2.64663	0.151051	0.0755255	+	0.997144i	$0.475937\pi$
$308$	0	0
$309$	3.15698	0.179594
$310$	24.2726	1.37859
$311$	5.09186	0.288733	0.144367	−	0.989524i	$-0.453886\pi$
$311$	5.09186	0.288733	0.144367	+	0.989524i	$0.453886\pi$
$312$	4.08884	0.231485
$313$	−19.9586	−1.12813	−0.564064	−	0.825731i	$-0.690763\pi$
$313$	−19.9586	−1.12813	−0.564064	+	0.825731i	$0.690763\pi$
$314$	−44.2913	−2.49950
$315$	0	0
$316$	78.2312	4.40085
$317$	−6.81930	−0.383010	−0.191505	−	0.981492i	$-0.561337\pi$
$317$	−6.81930	−0.383010	−0.191505	+	0.981492i	$0.561337\pi$
$318$	−29.8273	−1.67263
$319$	14.5865	0.816688
$320$	83.2342	4.65293
$321$	16.5341	0.922842
$322$	0	0
$323$	−6.53407	−0.363565
$324$	5.53407	0.307448
$325$	0.599184	0.0332367
$326$	18.2438	1.01043
$327$	−1.77488	−0.0981513
$328$	81.6921	4.51069
$329$	0	0
$330$	16.8430	0.927178
$331$	−5.91116	−0.324907	−0.162453	−	0.986716i	$-0.551941\pi$
$331$	−5.91116	−0.324907	−0.162453	+	0.986716i	$0.551941\pi$
$332$	−11.6910	−0.641630
$333$	−2.84302	−0.155797
$334$	−39.4195	−2.15694
$335$	−25.1156	−1.37221
$336$	0	0
$337$	4.08884	0.222733	0.111367	−	0.993779i	$-0.464477\pi$
$337$	4.08884	0.222733	0.111367	+	0.993779i	$0.464477\pi$
$338$	35.1951	1.91436
$339$	−10.0237	−0.544414
$340$	−91.6320	−4.96944
$341$	−8.45023	−0.457606
$342$	−2.74483	−0.148423
$343$	0	0
$344$	−111.454	−6.00921
$345$	4.00000	0.215353
$346$	−15.6860	−0.843287
$347$	−17.3534	−0.931578	−0.465789	−	0.884896i	$-0.654229\pi$
$347$	−17.3534	−0.931578	−0.465789	+	0.884896i	$0.654229\pi$
$348$	−33.3357	−1.78698
$349$	4.70674	0.251946	0.125973	−	0.992034i	$-0.459795\pi$
$349$	4.70674	0.251946	0.125973	+	0.992034i	$0.459795\pi$
$350$	0	0
$351$	−0.421512	−0.0224986
$352$	−56.4276	−3.00760
$353$	−20.6704	−1.10017	−0.550086	−	0.835108i	$-0.685405\pi$
$353$	−20.6704	−1.10017	−0.550086	+	0.835108i	$0.685405\pi$
$354$	−38.8016	−2.06228
$355$	−33.8985	−1.79915
$356$	53.7622	2.84939
$357$	0	0
$358$	26.3564	1.39298
$359$	8.10756	0.427901	0.213950	−	0.976845i	$-0.431367\pi$
$359$	8.10756	0.427901	0.213950	+	0.976845i	$0.431367\pi$
$360$	−24.5815	−1.29556
$361$	1.00000	0.0526316
$362$	−2.68802	−0.141279
$363$	5.13628	0.269585
$364$	0	0
$365$	−17.9112	−0.937513
$366$	−18.7829	−0.981798
$367$	−28.8905	−1.50807	−0.754035	−	0.656834i	$-0.771895\pi$
$367$	−28.8905	−1.50807	−0.754035	+	0.656834i	$0.771895\pi$
$368$	−24.5578	−1.28016
$369$	−8.42151	−0.438406
$370$	19.7749	1.02805
$371$	0	0
$372$	19.3120	1.00128
$373$	11.9586	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$373$	11.9586	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$374$	43.4295	2.24569
$375$	9.06814	0.468277
$376$	77.8334	4.01395
$377$	2.53907	0.130769
$378$	0	0
$379$	−15.3821	−0.790125	−0.395063	−	0.918654i	$-0.629277\pi$
$379$	−15.3821	−0.790125	−0.395063	+	0.918654i	$0.629277\pi$
$380$	14.0237	0.719402
$381$	−2.08884	−0.107014
$382$	32.1550	1.64519
$383$	0.617907	0.0315736	0.0157868	−	0.999875i	$-0.494975\pi$
$383$	0.617907	0.0315736	0.0157868	+	0.999875i	$0.494975\pi$
$384$	43.5515	2.22248
$385$	0	0
$386$	−48.9192	−2.48992
$387$	11.4897	0.584052
$388$	27.5558	1.39893
$389$	−11.0681	−0.561177	−0.280588	−	0.959828i	$-0.590530\pi$
$389$	−11.0681	−0.561177	−0.280588	+	0.959828i	$0.590530\pi$
$390$	2.93186	0.148460
$391$	10.3140	0.521599
$392$	0	0
$393$	−13.9349	−0.702922
$394$	52.4690	2.64335
$395$	35.8223	1.80242
$396$	13.4008	0.673416
$397$	10.4502	0.524482	0.262241	−	0.965002i	$-0.415538\pi$
$397$	10.4502	0.524482	0.262241	+	0.965002i	$0.415538\pi$
$398$	44.0474	2.20790
$399$	0	0
$400$	22.1156	1.10578
$401$	3.26953	0.163273	0.0816364	−	0.996662i	$-0.473985\pi$
$401$	3.26953	0.163273	0.0816364	+	0.996662i	$0.473985\pi$
$402$	−27.2044	−1.35683
$403$	−1.47093	−0.0732722
$404$	9.35837	0.465596
$405$	2.53407	0.125919
$406$	0	0
$407$	−6.88441	−0.341248
$408$	−63.3831	−3.13793
$409$	−7.40082	−0.365947	−0.182973	−	0.983118i	$-0.558572\pi$
$409$	−7.40082	−0.365947	−0.182973	+	0.983118i	$0.558572\pi$
$410$	58.5765	2.89289
$411$	−16.9793	−0.837527
$412$	−17.4709	−0.860731
$413$	0	0
$414$	4.33268	0.212939
$415$	−5.35337	−0.262787
$416$	−9.82233	−0.481579
$417$	−1.06814	−0.0523071
$418$	−6.64663	−0.325097
$419$	31.8461	1.55578	0.777891	−	0.628400i	$-0.216290\pi$
$419$	31.8461	1.55578	0.777891	+	0.628400i	$0.216290\pi$
$420$	0	0
$421$	40.0061	1.94978	0.974888	−	0.222696i	$-0.0714858\pi$
$421$	40.0061	1.94978	0.974888	+	0.222696i	$0.0714858\pi$
$422$	25.5084	1.24173
$423$	−8.02372	−0.390127
$424$	105.412	5.11925
$425$	−9.28826	−0.450547
$426$	−36.7178	−1.77898
$427$	0	0
$428$	−91.5007	−4.42285
$429$	−1.02070	−0.0492797
$430$	−79.9172	−3.85395
$431$	3.91616	0.188635	0.0943175	−	0.995542i	$-0.469933\pi$
$431$	3.91616	0.188635	0.0943175	+	0.995542i	$0.469933\pi$
$432$	−15.5578	−0.748525
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	−15.2645	−0.731878
$436$	9.82233	0.470404
$437$	−1.57849	−0.0755093
$438$	−19.4008	−0.927007
$439$	−17.4483	−0.832760	−0.416380	−	0.909191i	$-0.636701\pi$
$439$	−17.4483	−0.832760	−0.416380	+	0.909191i	$0.636701\pi$
$440$	−59.5244	−2.83772
$441$	0	0
$442$	7.55977	0.359581
$443$	−3.03942	−0.144407	−0.0722036	−	0.997390i	$-0.523003\pi$
$443$	−3.03942	−0.144407	−0.0722036	+	0.997390i	$0.523003\pi$
$444$	15.7335	0.746678
$445$	24.6179	1.16700
$446$	−22.8717	−1.08301
$447$	−14.0474	−0.664421
$448$	0	0
$449$	35.9349	1.69587	0.847936	−	0.530099i	$-0.177845\pi$
$449$	35.9349	1.69587	0.847936	+	0.530099i	$0.177845\pi$
$450$	−3.90180	−0.183933
$451$	−20.3928	−0.960259
$452$	55.4720	2.60918
$453$	14.7542	0.693212
$454$	46.3614	2.17585
$455$	0	0
$456$	9.70041	0.454263
$457$	−24.4215	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$457$	−24.4215	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$458$	16.5992	0.775629
$459$	6.53407	0.304984
$460$	−22.1363	−1.03211
$461$	−10.1313	−0.471861	−0.235930	−	0.971770i	$-0.575814\pi$
$461$	−10.1313	−0.471861	−0.235930	+	0.971770i	$0.575814\pi$
$462$	0	0
$463$	−5.86372	−0.272510	−0.136255	−	0.990674i	$-0.543507\pi$
$463$	−5.86372	−0.272510	−0.136255	+	0.990674i	$0.543507\pi$
$464$	93.7158	4.35065
$465$	8.84302	0.410085
$466$	−58.2024	−2.69618
$467$	14.9556	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$467$	14.9556	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$468$	2.33268	0.107828
$469$	0	0
$470$	55.8097	2.57431
$471$	−16.1363	−0.743521
$472$	137.128	6.31181
$473$	27.8223	1.27927
$474$	38.8016	1.78222
$475$	1.42151	0.0652234
$476$	0	0
$477$	−10.8667	−0.497554
$478$	15.9299	0.728616
$479$	33.7098	1.54024	0.770119	−	0.637900i	$-0.220197\pi$
$479$	33.7098	1.54024	0.770119	+	0.637900i	$0.220197\pi$
$480$	59.0505	2.69527
$481$	−1.19837	−0.0546409
$482$	−35.6259	−1.62272
$483$	0	0
$484$	−28.4245	−1.29202
$485$	12.6179	0.572950
$486$	2.74483	0.124508
$487$	21.5084	0.974637	0.487319	−	0.873224i	$-0.337975\pi$
$487$	21.5084	0.974637	0.487319	+	0.873224i	$0.337975\pi$
$488$	66.3801	3.00489
$489$	6.64663	0.300571
$490$	0	0
$491$	−17.3534	−0.783147	−0.391573	−	0.920147i	$-0.628069\pi$
$491$	−17.3534	−0.783147	−0.391573	+	0.920147i	$0.628069\pi$
$492$	46.6052	2.10113
$493$	−39.3594	−1.77266
$494$	−1.15698	−0.0520548
$495$	6.13628	0.275805
$496$	−54.2913	−2.43775
$497$	0	0
$498$	−5.79861	−0.259842
$499$	−32.9379	−1.47450	−0.737252	−	0.675618i	$-0.763877\pi$
$499$	−32.9379	−1.47450	−0.737252	+	0.675618i	$0.763877\pi$
$500$	−50.1837	−2.24428
$501$	−14.3614	−0.641620
$502$	28.3751	1.26644
$503$	38.2074	1.70359	0.851793	−	0.523879i	$-0.175515\pi$
$503$	38.2074	1.70359	0.851793	+	0.523879i	$0.175515\pi$
$504$	0	0
$505$	4.28523	0.190690
$506$	10.4916	0.466410
$507$	12.8223	0.569460
$508$	11.5598	0.512882
$509$	8.82430	0.391130	0.195565	−	0.980691i	$-0.437346\pi$
$509$	8.82430	0.391130	0.195565	+	0.980691i	$0.437346\pi$
$510$	−45.4483	−2.01248
$511$	0	0
$512$	−60.7034	−2.68274
$513$	−1.00000	−0.0441511
$514$	−41.6547	−1.83731
$515$	−8.00000	−0.352522
$516$	−63.5845	−2.79915
$517$	−19.4295	−0.854510
$518$	0	0
$519$	−5.71477	−0.250851
$520$	−10.3614	−0.454377
$521$	24.2913	1.06422	0.532110	−	0.846675i	$-0.321399\pi$
$521$	24.2913	1.06422	0.532110	+	0.846675i	$0.321399\pi$
$522$	−16.5341	−0.723677
$523$	−19.0969	−0.835047	−0.417524	−	0.908666i	$-0.637102\pi$
$523$	−19.0969	−0.835047	−0.417524	+	0.908666i	$0.637102\pi$
$524$	77.1166	3.36886
$525$	0	0
$526$	23.3594	1.01852
$527$	22.8016	0.993255
$528$	−37.6734	−1.63952
$529$	−20.5084	−0.891668
$530$	75.5845	3.28318
$531$	−14.1363	−0.613462
$532$	0	0
$533$	−3.54977	−0.153757
$534$	26.6654	1.15392
$535$	−41.8985	−1.81143
$536$	96.1423	4.15272
$537$	9.60221	0.414366
$538$	41.6547	1.79586
$539$	0	0
$540$	−14.0237	−0.603485
$541$	−36.2312	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$541$	−36.2312	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$542$	2.31395	0.0993928
$543$	−0.979304	−0.0420260
$544$	152.261	6.52813
$545$	4.49768	0.192659
$546$	0	0
$547$	−29.9586	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$547$	−29.9586	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$548$	93.9647	4.01397
$549$	−6.84302	−0.292053
$550$	−9.44826	−0.402875
$551$	6.02372	0.256619
$552$	−15.3120	−0.651721
$553$	0	0
$554$	−28.9793	−1.23121
$555$	7.20442	0.305811
$556$	5.91116	0.250689
$557$	−14.4402	−0.611852	−0.305926	−	0.952055i	$-0.598966\pi$
$557$	−14.4402	−0.611852	−0.305926	+	0.952055i	$0.598966\pi$
$558$	9.57849	0.405490
$559$	4.84302	0.204838
$560$	0	0
$561$	15.8223	0.668019
$562$	−41.2943	−1.74190
$563$	17.9112	0.754866	0.377433	−	0.926037i	$-0.376807\pi$
$563$	17.9112	0.754866	0.377433	+	0.926037i	$0.376807\pi$
$564$	44.4038	1.86974
$565$	25.4008	1.06862
$566$	3.41954	0.143734
$567$	0	0
$568$	129.763	5.44475
$569$	−8.50535	−0.356563	−0.178281	−	0.983980i	$-0.557054\pi$
$569$	−8.50535	−0.356563	−0.178281	+	0.983980i	$0.557054\pi$
$570$	6.95558	0.291337
$571$	−6.80163	−0.284639	−0.142320	−	0.989821i	$-0.545456\pi$
$571$	−6.80163	−0.284639	−0.142320	+	0.989821i	$0.545456\pi$
$572$	5.64860	0.236180
$573$	11.7148	0.489391
$574$	0	0
$575$	−2.24384	−0.0935746
$576$	32.8461	1.36859
$577$	14.2726	0.594175	0.297087	−	0.954850i	$-0.403985\pi$
$577$	14.2726	0.594175	0.297087	+	0.954850i	$0.403985\pi$
$578$	−70.5258	−2.93348
$579$	−17.8223	−0.740671
$580$	84.4750	3.50763
$581$	0	0
$582$	13.6673	0.566529
$583$	−26.3140	−1.08981
$584$	68.5638	2.83719
$585$	1.06814	0.0441622
$586$	56.0535	2.31555
$587$	−40.2963	−1.66321	−0.831603	−	0.555371i	$-0.812576\pi$
$587$	−40.2963	−1.66321	−0.831603	+	0.555371i	$0.812576\pi$
$588$	0	0
$589$	−3.48965	−0.143789
$590$	98.3261	4.04802
$591$	19.1156	0.786310
$592$	−44.2312	−1.81789
$593$	−26.1313	−1.07308	−0.536542	−	0.843874i	$-0.680270\pi$
$593$	−26.1313	−1.07308	−0.536542	+	0.843874i	$0.680270\pi$
$594$	6.64663	0.272714
$595$	0	0
$596$	77.7395	3.18434
$597$	16.0474	0.656778
$598$	1.82627	0.0746819
$599$	3.06314	0.125157	0.0625783	−	0.998040i	$-0.480068\pi$
$599$	3.06314	0.125157	0.0625783	+	0.998040i	$0.480068\pi$
$600$	13.7892	0.562944
$601$	−28.1363	−1.14770	−0.573851	−	0.818959i	$-0.694551\pi$
$601$	−28.1363	−1.14770	−0.573851	+	0.818959i	$0.694551\pi$
$602$	0	0
$603$	−9.91116	−0.403614
$604$	−81.6507	−3.32232
$605$	−13.0157	−0.529163
$606$	4.64163	0.188553
$607$	−28.8430	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$607$	−28.8430	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$608$	−23.3026	−0.945046
$609$	0	0
$610$	47.5972	1.92715
$611$	−3.38209	−0.136825
$612$	−36.1600	−1.46168
$613$	22.2852	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$613$	22.2852	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$614$	−7.26454	−0.293173
$615$	21.3407	0.860540
$616$	0	0
$617$	4.13628	0.166520	0.0832602	−	0.996528i	$-0.473467\pi$
$617$	4.13628	0.166520	0.0832602	+	0.996528i	$0.473467\pi$
$618$	−8.66535	−0.348572
$619$	9.06814	0.364479	0.182240	−	0.983254i	$-0.441665\pi$
$619$	9.06814	0.364479	0.182240	+	0.983254i	$0.441665\pi$
$620$	−48.9379	−1.96539
$621$	1.57849	0.0633426
$622$	−13.9763	−0.560398
$623$	0	0
$624$	−6.55779	−0.262522
$625$	−30.0869	−1.20347
$626$	54.7829	2.18957
$627$	−2.42151	−0.0967059
$628$	89.2993	3.56343
$629$	18.5765	0.740694
$630$	0	0
$631$	−41.6259	−1.65710	−0.828551	−	0.559913i	$-0.810834\pi$
$631$	−41.6259	−1.65710	−0.828551	+	0.559913i	$0.810834\pi$
$632$	−137.128	−5.45465
$633$	9.29326	0.369374
$634$	18.7178	0.743379
$635$	5.29326	0.210057
$636$	60.1373	2.38460
$637$	0	0
$638$	−40.0374	−1.58510
$639$	−13.3771	−0.529190
$640$	−110.362	−4.36246
$641$	3.09186	0.122121	0.0610606	−	0.998134i	$-0.480552\pi$
$641$	3.09186	0.122121	0.0610606	+	0.998134i	$0.480552\pi$
$642$	−45.3831	−1.79113
$643$	−2.93186	−0.115621	−0.0578106	−	0.998328i	$-0.518412\pi$
$643$	−2.93186	−0.115621	−0.0578106	+	0.998328i	$0.518412\pi$
$644$	0	0
$645$	−29.1156	−1.14643
$646$	17.9349	0.705639
$647$	5.93489	0.233324	0.116662	−	0.993172i	$-0.462781\pi$
$647$	5.93489	0.233324	0.116662	+	0.993172i	$0.462781\pi$
$648$	−9.70041	−0.381068
$649$	−34.2312	−1.34369
$650$	−1.64466	−0.0645087
$651$	0	0
$652$	−36.7829	−1.44053
$653$	1.82233	0.0713132	0.0356566	−	0.999364i	$-0.488648\pi$
$653$	1.82233	0.0713132	0.0356566	+	0.999364i	$0.488648\pi$
$654$	4.87175	0.190500
$655$	35.3120	1.37975
$656$	−131.020	−5.11548
$657$	−7.06814	−0.275754
$658$	0	0
$659$	10.3978	0.405040	0.202520	−	0.979278i	$-0.435087\pi$
$659$	10.3978	0.405040	0.202520	+	0.979278i	$0.435087\pi$
$660$	−33.9586	−1.32184
$661$	−23.2231	−0.903276	−0.451638	−	0.892201i	$-0.649160\pi$
$661$	−23.2231	−0.903276	−0.451638	+	0.892201i	$0.649160\pi$
$662$	16.2251	0.630607
$663$	2.75419	0.106964
$664$	20.4927	0.795270
$665$	0	0
$666$	7.80361	0.302384
$667$	−9.50837	−0.368166
$668$	79.4770	3.07506
$669$	−8.33268	−0.322160
$670$	68.9379	2.66330
$671$	−16.5705	−0.639696
$672$	0	0
$673$	13.5498	0.522305	0.261153	−	0.965298i	$-0.415897\pi$
$673$	13.5498	0.522305	0.261153	+	0.965298i	$0.415897\pi$
$674$	−11.2231	−0.432299
$675$	−1.42151	−0.0547140
$676$	−70.9597	−2.72922
$677$	12.6466	0.486049	0.243025	−	0.970020i	$-0.421860\pi$
$677$	12.6466	0.486049	0.243025	+	0.970020i	$0.421860\pi$
$678$	27.5134	1.05664
$679$	0	0
$680$	160.617	6.15939
$681$	16.8905	0.647244
$682$	23.1944	0.888160
$683$	−13.7799	−0.527273	−0.263636	−	0.964622i	$-0.584922\pi$
$683$	−13.7799	−0.527273	−0.263636	+	0.964622i	$0.584922\pi$
$684$	5.53407	0.211600
$685$	43.0267	1.64397
$686$	0	0
$687$	6.04744	0.230724
$688$	178.754	6.81492
$689$	−4.58046	−0.174502
$690$	−10.9793	−0.417975
$691$	−1.11559	−0.0424389	−0.0212194	−	0.999775i	$-0.506755\pi$
$691$	−1.11559	−0.0424389	−0.0212194	+	0.999775i	$0.506755\pi$
$692$	31.6259	1.20224
$693$	0	0
$694$	47.6320	1.80809
$695$	2.70674	0.102673
$696$	58.4326	2.21488
$697$	55.0267	2.08429
$698$	−12.9192	−0.488999
$699$	−21.2044	−0.802025
$700$	0	0
$701$	16.7067	0.631005	0.315502	−	0.948925i	$-0.397827\pi$
$701$	16.7067	0.631005	0.315502	+	0.948925i	$0.397827\pi$
$702$	1.15698	0.0436673
$703$	−2.84302	−0.107227
$704$	79.5371	2.99767
$705$	20.3327	0.765773
$706$	56.7365	2.13531
$707$	0	0
$708$	78.2312	2.94011
$709$	−25.1443	−0.944314	−0.472157	−	0.881514i	$-0.656525\pi$
$709$	−25.1443	−0.944314	−0.472157	+	0.881514i	$0.656525\pi$
$710$	93.0455	3.49193
$711$	14.1363	0.530152
$712$	−94.2372	−3.53169
$713$	5.50837	0.206290
$714$	0	0
$715$	2.58651	0.0967302
$716$	−53.1393	−1.98591
$717$	5.80361	0.216740
$718$	−22.2538	−0.830506
$719$	−32.2488	−1.20268	−0.601339	−	0.798994i	$-0.705366\pi$
$719$	−32.2488	−1.20268	−0.601339	+	0.798994i	$0.705366\pi$
$720$	39.4245	1.46927
$721$	0	0
$722$	−2.74483	−0.102152
$723$	−12.9793	−0.482706
$724$	5.41954	0.201416
$725$	8.56279	0.318014
$726$	−14.0982	−0.523233
$727$	30.2312	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$727$	30.2312	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	49.1630	1.81961
$731$	−75.0742	−2.77672
$732$	37.8698	1.39971
$733$	−25.0267	−0.924384	−0.462192	−	0.886780i	$-0.652937\pi$
$733$	−25.0267	−0.924384	−0.462192	+	0.886780i	$0.652937\pi$
$734$	79.2993	2.92699
$735$	0	0
$736$	36.7829	1.35584
$737$	−24.0000	−0.884051
$738$	23.1156	0.850896
$739$	5.53104	0.203463	0.101731	−	0.994812i	$-0.467562\pi$
$739$	5.53104	0.203463	0.101731	+	0.994812i	$0.467562\pi$
$740$	−39.8698	−1.46564
$741$	−0.421512	−0.0154846
$742$	0	0
$743$	21.6971	0.795989	0.397995	−	0.917388i	$-0.369706\pi$
$743$	21.6971	0.795989	0.397995	+	0.917388i	$0.369706\pi$
$744$	−33.8510	−1.24104
$745$	35.5972	1.30418
$746$	−32.8243	−1.20178
$747$	−2.11256	−0.0772945
$748$	−87.5619	−3.20158
$749$	0	0
$750$	−24.8905	−0.908871
$751$	−48.9854	−1.78750	−0.893751	−	0.448564i	$-0.851935\pi$
$751$	−48.9854	−1.78750	−0.893751	+	0.448564i	$0.851935\pi$
$752$	−124.831	−4.55213
$753$	10.3377	0.376726
$754$	−6.96931	−0.253807
$755$	−37.3881	−1.36069
$756$	0	0
$757$	−4.04139	−0.146887	−0.0734434	−	0.997299i	$-0.523399\pi$
$757$	−4.04139	−0.146887	−0.0734434	+	0.997299i	$0.523399\pi$
$758$	42.2212	1.53354
$759$	3.82233	0.138742
$760$	−24.5815	−0.891665
$761$	36.9429	1.33918	0.669590	−	0.742731i	$-0.266470\pi$
$761$	36.9429	1.33918	0.669590	+	0.742731i	$0.266470\pi$
$762$	5.73349	0.207702
$763$	0	0
$764$	−64.8304	−2.34548
$765$	−16.5578	−0.598648
$766$	−1.69605	−0.0612806
$767$	−5.95861	−0.215153
$768$	−53.8491	−1.94311
$769$	−24.1363	−0.870377	−0.435188	−	0.900339i	$-0.643318\pi$
$769$	−24.1363	−0.870377	−0.435188	+	0.900339i	$0.643318\pi$
$770$	0	0
$771$	−15.1757	−0.546539
$772$	98.6300	3.54977
$773$	25.5371	0.918506	0.459253	−	0.888306i	$-0.348117\pi$
$773$	25.5371	0.918506	0.459253	+	0.888306i	$0.348117\pi$
$774$	−31.5371	−1.13358
$775$	−4.96058	−0.178189
$776$	−48.3013	−1.73392
$777$	0	0
$778$	30.3801	1.08918
$779$	−8.42151	−0.301732
$780$	−5.91116	−0.211654
$781$	−32.3928	−1.15911
$782$	−28.3100	−1.01236
$783$	−6.02372	−0.215270
$784$	0	0
$785$	40.8905	1.45944
$786$	38.2488	1.36429
$787$	19.0969	0.680730	0.340365	−	0.940293i	$-0.389449\pi$
$787$	19.0969	0.680730	0.340365	+	0.940293i	$0.389449\pi$
$788$	−105.787	−3.76851
$789$	8.51035	0.302976
$790$	−98.3261	−3.49828
$791$	0	0
$792$	−23.4897	−0.834668
$793$	−2.88441	−0.102429
$794$	−28.6841	−1.01796
$795$	27.5371	0.976640
$796$	−88.8077	−3.14770
$797$	−8.02872	−0.284392	−0.142196	−	0.989839i	$-0.545416\pi$
$797$	−8.02872	−0.284392	−0.142196	+	0.989839i	$0.545416\pi$
$798$	0	0
$799$	52.4276	1.85475
$800$	−33.1249	−1.17114
$801$	9.71477	0.343254
$802$	−8.97430	−0.316894
$803$	−17.1156	−0.603996
$804$	54.8491	1.93438
$805$	0	0
$806$	4.03745	0.142213
$807$	15.1757	0.534210
$808$	−16.4038	−0.577085
$809$	−29.0742	−1.02219	−0.511097	−	0.859523i	$-0.670761\pi$
$809$	−29.0742	−1.02219	−0.511097	+	0.859523i	$0.670761\pi$
$810$	−6.95558	−0.244394
$811$	−4.66535	−0.163823	−0.0819113	−	0.996640i	$-0.526102\pi$
$811$	−4.66535	−0.163823	−0.0819113	+	0.996640i	$0.526102\pi$
$812$	0	0
$813$	0.843024	0.0295661
$814$	18.8965	0.662323
$815$	−16.8430	−0.589985
$816$	101.656	3.55866
$817$	11.4897	0.401972
$818$	20.3140	0.710261
$819$	0	0
$820$	−118.101	−4.12426
$821$	−17.5972	−0.614147	−0.307073	−	0.951686i	$-0.599350\pi$
$821$	−17.5972	−0.614147	−0.307073	+	0.951686i	$0.599350\pi$
$822$	46.6052	1.62554
$823$	−7.93989	−0.276767	−0.138384	−	0.990379i	$-0.544191\pi$
$823$	−7.93989	−0.276767	−0.138384	+	0.990379i	$0.544191\pi$
$824$	30.6240	1.06684
$825$	−3.44221	−0.119842
$826$	0	0
$827$	−31.7859	−1.10531	−0.552653	−	0.833412i	$-0.686384\pi$
$827$	−31.7859	−1.10531	−0.552653	+	0.833412i	$0.686384\pi$
$828$	−8.73546	−0.303578
$829$	48.9793	1.70112	0.850561	−	0.525877i	$-0.176263\pi$
$829$	48.9793	1.70112	0.850561	+	0.525877i	$0.176263\pi$
$830$	14.6941	0.510039
$831$	−10.5578	−0.366246
$832$	13.8450	0.479989
$833$	0	0
$834$	2.93186	0.101522
$835$	36.3928	1.25942
$836$	13.4008	0.463477
$837$	3.48965	0.120620
$838$	−87.4119	−3.01959
$839$	−36.2726	−1.25227	−0.626134	−	0.779716i	$-0.715364\pi$
$839$	−36.2726	−1.25227	−0.626134	+	0.779716i	$0.715364\pi$
$840$	0	0
$841$	7.28523	0.251215
$842$	−109.810	−3.78429
$843$	−15.0444	−0.518157
$844$	−51.4295	−1.77028
$845$	−32.4927	−1.11778
$846$	22.0237	0.757191
$847$	0	0
$848$	−169.063	−5.80563
$849$	1.24581	0.0427562
$850$	25.4947	0.874459
$851$	4.48768	0.153836
$852$	74.0298	2.53622
$853$	34.6654	1.18692	0.593460	−	0.804864i	$-0.297762\pi$
$853$	34.6654	1.18692	0.593460	+	0.804864i	$0.297762\pi$
$854$	0	0
$855$	2.53407	0.0866634
$856$	160.387	5.48192
$857$	4.87175	0.166416	0.0832078	−	0.996532i	$-0.473483\pi$
$857$	4.87175	0.166416	0.0832078	+	0.996532i	$0.473483\pi$
$858$	2.80163	0.0956461
$859$	37.1156	1.26637	0.633184	−	0.774002i	$-0.281748\pi$
$859$	37.1156	1.26637	0.633184	+	0.774002i	$0.281748\pi$
$860$	161.128	5.49441
$861$	0	0
$862$	−10.7492	−0.366119
$863$	−29.1520	−0.992345	−0.496172	−	0.868224i	$-0.665262\pi$
$863$	−29.1520	−0.992345	−0.496172	+	0.868224i	$0.665262\pi$
$864$	23.3026	0.792771
$865$	14.4816	0.492390
$866$	49.4069	1.67891
$867$	−25.6941	−0.872616
$868$	0	0
$869$	34.2312	1.16121
$870$	41.8985	1.42049
$871$	−4.17767	−0.141555
$872$	−17.2171	−0.583044
$873$	4.97930	0.168524
$874$	4.33268	0.146555
$875$	0	0
$876$	39.1156	1.32159
$877$	14.4502	0.487950	0.243975	−	0.969782i	$-0.421549\pi$
$877$	14.4502	0.487950	0.243975	+	0.969782i	$0.421549\pi$
$878$	47.8924	1.61629
$879$	20.4215	0.688800
$880$	95.4670	3.21819
$881$	−10.5341	−0.354902	−0.177451	−	0.984130i	$-0.556785\pi$
$881$	−10.5341	−0.354902	−0.177451	+	0.984130i	$0.556785\pi$
$882$	0	0
$883$	−5.11559	−0.172153	−0.0860766	−	0.996289i	$-0.527433\pi$
$883$	−5.11559	−0.172153	−0.0860766	+	0.996289i	$0.527433\pi$
$884$	−15.2419	−0.512639
$885$	35.8223	1.20415
$886$	8.34267	0.280278
$887$	0.617907	0.0207473	0.0103736	−	0.999946i	$-0.496698\pi$
$887$	0.617907	0.0207473	0.0103736	+	0.999946i	$0.496698\pi$
$888$	−27.5785	−0.925473
$889$	0	0
$890$	−67.5719	−2.26501
$891$	2.42151	0.0811237
$892$	46.1136	1.54400
$893$	−8.02372	−0.268504
$894$	38.5578	1.28957
$895$	−24.3327	−0.813352
$896$	0	0
$897$	0.665351	0.0222154
$898$	−98.6350	−3.29149
$899$	−21.0207	−0.701079
$900$	7.86675	0.262225
$901$	71.0041	2.36549
$902$	55.9747	1.86375
$903$	0	0
$904$	−97.2342	−3.23396
$905$	2.48163	0.0824920
$906$	−40.4977	−1.34545
$907$	−8.27256	−0.274686	−0.137343	−	0.990524i	$-0.543856\pi$
$907$	−8.27256	−0.274686	−0.137343	+	0.990524i	$0.543856\pi$
$908$	−93.4730	−3.10201
$909$	1.69105	0.0560885
$910$	0	0
$911$	−5.77988	−0.191496	−0.0957480	−	0.995406i	$-0.530524\pi$
$911$	−5.77988	−0.191496	−0.0957480	+	0.995406i	$0.530524\pi$
$912$	−15.5578	−0.515170
$913$	−5.11559	−0.169301
$914$	67.0328	2.21725
$915$	17.3407	0.573266
$916$	−33.4670	−1.10578
$917$	0	0
$918$	−17.9349	−0.591939
$919$	4.62791	0.152661	0.0763303	−	0.997083i	$-0.475680\pi$
$919$	4.62791	0.152661	0.0763303	+	0.997083i	$0.475680\pi$
$920$	38.8016	1.27925
$921$	−2.64663	−0.0872094
$922$	27.8086	0.915828
$923$	−5.63860	−0.185597
$924$	0	0
$925$	−4.04139	−0.132880
$926$	16.0949	0.528911
$927$	−3.15698	−0.103689
$928$	−140.369	−4.60782
$929$	29.9536	0.982746	0.491373	−	0.870949i	$-0.336495\pi$
$929$	29.9536	0.982746	0.491373	+	0.870949i	$0.336495\pi$
$930$	−24.2726	−0.795929
$931$	0	0
$932$	117.347	3.84382
$933$	−5.09186	−0.166700
$934$	−41.0505	−1.34321
$935$	−40.0949	−1.31124
$936$	−4.08884	−0.133648
$937$	−4.58651	−0.149835	−0.0749174	−	0.997190i	$-0.523869\pi$
$937$	−4.58651	−0.149835	−0.0749174	+	0.997190i	$0.523869\pi$
$938$	0	0
$939$	19.9586	0.651325
$940$	−112.522	−3.67008
$941$	−7.41082	−0.241586	−0.120793	−	0.992678i	$-0.538544\pi$
$941$	−7.41082	−0.241586	−0.120793	+	0.992678i	$0.538544\pi$
$942$	44.2913	1.44309
$943$	13.2933	0.432888
$944$	−219.929	−7.15809
$945$	0	0
$946$	−76.3675	−2.48292
$947$	59.6320	1.93778	0.968890	−	0.247493i	$-0.0796068\pi$
$947$	59.6320	1.93778	0.968890	+	0.247493i	$0.0796068\pi$
$948$	−78.2312	−2.54083
$949$	−2.97930	−0.0967123
$950$	−3.90180	−0.126591
$951$	6.81930	0.221131
$952$	0	0
$953$	25.7986	0.835699	0.417849	−	0.908516i	$-0.362784\pi$
$953$	25.7986	0.835699	0.417849	+	0.908516i	$0.362784\pi$
$954$	29.8273	0.965695
$955$	−29.6860	−0.960618
$956$	−32.1176	−1.03876
$957$	−14.5865	−0.471515
$958$	−92.5275	−2.98943
$959$	0	0
$960$	−83.2342	−2.68637
$961$	−18.8223	−0.607172
$962$	3.28931	0.106052
$963$	−16.5341	−0.532803
$964$	71.8284	2.31344
$965$	45.1630	1.45385
$966$	0	0
$967$	55.5845	1.78748	0.893739	−	0.448587i	$-0.148073\pi$
$967$	55.5845	1.78748	0.893739	+	0.448587i	$0.148073\pi$
$968$	49.8240	1.60140
$969$	6.53407	0.209905
$970$	−34.6340	−1.11203
$971$	48.4877	1.55604	0.778022	−	0.628237i	$-0.216223\pi$
$971$	48.4877	1.55604	0.778022	+	0.628237i	$0.216223\pi$
$972$	−5.53407	−0.177505
$973$	0	0
$974$	−59.0367	−1.89166
$975$	−0.599184	−0.0191892
$976$	−106.462	−3.40778
$977$	53.0505	1.69723	0.848617	−	0.529007i	$-0.177435\pi$
$977$	53.0505	1.69723	0.848617	+	0.529007i	$0.177435\pi$
$978$	−18.2438	−0.583374
$979$	23.5244	0.751844
$980$	0	0
$981$	1.77488	0.0566677
$982$	47.6320	1.52000
$983$	−12.1777	−0.388407	−0.194204	−	0.980961i	$-0.562212\pi$
$983$	−12.1777	−0.388407	−0.194204	+	0.980961i	$0.562212\pi$
$984$	−81.6921	−2.60425
$985$	−48.4402	−1.54343
$986$	108.035	3.44053
$987$	0	0
$988$	2.33268	0.0742123
$989$	−18.1363	−0.576700
$990$	−16.8430	−0.535306
$991$	−2.04139	−0.0648469	−0.0324235	−	0.999474i	$-0.510323\pi$
$991$	−2.04139	−0.0648469	−0.0324235	+	0.999474i	$0.510323\pi$
$992$	81.3180	2.58185
$993$	5.91116	0.187585
$994$	0	0
$995$	−40.6654	−1.28918
$996$	11.6910	0.370445
$997$	45.6447	1.44558	0.722790	−	0.691067i	$-0.242859\pi$
$997$	45.6447	1.44558	0.722790	+	0.691067i	$0.242859\pi$
$998$	90.4088	2.86184
$999$	2.84302	0.0899493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.2.a.w.1.1		3
3.2	odd	2		8379.2.a.bq.1.3		3
7.6	odd	2		399.2.a.e.1.1	✓	3
21.20	even	2		1197.2.a.m.1.3		3
28.27	even	2		6384.2.a.bu.1.1		3
35.34	odd	2		9975.2.a.x.1.3		3
133.132	even	2		7581.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.1	✓	3	7.6	odd	2
1197.2.a.m.1.3		3	21.20	even	2
2793.2.a.w.1.1		3	1.1	even	1	trivial
6384.2.a.bu.1.1		3	28.27	even	2
7581.2.a.l.1.3		3	133.132	even	2
8379.2.a.bq.1.3		3	3.2	odd	2
9975.2.a.x.1.3		3	35.34	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:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2793.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74483 q^{2} -1.00000 q^{3} +5.53407 q^{4} +2.53407 q^{5} +2.74483 q^{6} -9.70041 q^{8} +1.00000 q^{9} -6.95558 q^{10} +2.42151 q^{11} -5.53407 q^{12} +0.421512 q^{13} -2.53407 q^{15} +15.5578 q^{16} -6.53407 q^{17} -2.74483 q^{18} +1.00000 q^{19} +14.0237 q^{20} -6.64663 q^{22} -1.57849 q^{23} +9.70041 q^{24} +1.42151 q^{25} -1.15698 q^{26} -1.00000 q^{27} +6.02372 q^{29} +6.95558 q^{30} -3.48965 q^{31} -23.3026 q^{32} -2.42151 q^{33} +17.9349 q^{34} +5.53407 q^{36} -2.84302 q^{37} -2.74483 q^{38} -0.421512 q^{39} -24.5815 q^{40} -8.42151 q^{41} +11.4897 q^{43} +13.4008 q^{44} +2.53407 q^{45} +4.33268 q^{46} -8.02372 q^{47} -15.5578 q^{48} -3.90180 q^{50} +6.53407 q^{51} +2.33268 q^{52} -10.8667 q^{53} +2.74483 q^{54} +6.13628 q^{55} -1.00000 q^{57} -16.5341 q^{58} -14.1363 q^{59} -14.0237 q^{60} -6.84302 q^{61} +9.57849 q^{62} +32.8461 q^{64} +1.06814 q^{65} +6.64663 q^{66} -9.91116 q^{67} -36.1600 q^{68} +1.57849 q^{69} -13.3771 q^{71} -9.70041 q^{72} -7.06814 q^{73} +7.80361 q^{74} -1.42151 q^{75} +5.53407 q^{76} +1.15698 q^{78} +14.1363 q^{79} +39.4245 q^{80} +1.00000 q^{81} +23.1156 q^{82} -2.11256 q^{83} -16.5578 q^{85} -31.5371 q^{86} -6.02372 q^{87} -23.4897 q^{88} +9.71477 q^{89} -6.95558 q^{90} -8.73546 q^{92} +3.48965 q^{93} +22.0237 q^{94} +2.53407 q^{95} +23.3026 q^{96} +4.97930 q^{97} +2.42151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} + 9 q^{4} - q^{6} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} - 9 q^{12} - 2 q^{13} + 13 q^{16} - 12 q^{17} + q^{18} + 3 q^{19} + 16 q^{20} - 8 q^{22} - 8 q^{23} + 9 q^{24} + q^{25}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + 9 * q^4 - q^6 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 - 9 * q^12 - 2 * q^13 + 13 * q^16 - 12 * q^17 + q^18 + 3 * q^19 + 16 * q^20 - 8 * q^22 - 8 * q^23 + 9 * q^24 + q^25 - 10 * q^26 - 3 * q^27 - 8 * q^29 + 10 * q^30 + 8 * q^31 - 27 * q^32 - 4 * q^33 + 6 * q^34 + 9 * q^36 - 2 * q^37 + q^38 + 2 * q^39 - 14 * q^40 - 22 * q^41 + 16 * q^43 - 12 * q^46 + 2 * q^47 - 13 * q^48 - 9 * q^50 + 12 * q^51 - 18 * q^52 - q^54 - 12 * q^55 - 3 * q^57 - 42 * q^58 - 12 * q^59 - 16 * q^60 - 14 * q^61 + 32 * q^62 + 29 * q^64 - 12 * q^65 + 8 * q^66 - 8 * q^67 - 52 * q^68 + 8 * q^69 - 26 * q^71 - 9 * q^72 - 6 * q^73 + 18 * q^74 - q^75 + 9 * q^76 + 10 * q^78 + 12 * q^79 + 52 * q^80 + 3 * q^81 + 2 * q^82 - 2 * q^83 - 16 * q^85 - 24 * q^86 + 8 * q^87 - 52 * q^88 + 2 * q^89 - 10 * q^90 - 36 * q^92 - 8 * q^93 + 40 * q^94 + 27 * q^96 - 22 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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