LMFDB - Embedded newform 2793.2.a.n.1.2

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 = [2,-2,-2,2,2] 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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} -0.757359 q^{10} +1.82843 q^{11} +1.82843 q^{12} +2.82843 q^{13} +1.82843 q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214 q^{18} +1.00000 q^{19} +3.34315 q^{20} +0.757359 q^{22} +3.82843 q^{23} +1.58579 q^{24} -1.65685 q^{25} +1.17157 q^{26} -1.00000 q^{27} +0.828427 q^{29} +0.757359 q^{30} +8.82843 q^{31} +4.41421 q^{32} -1.82843 q^{33} -3.17157 q^{34} -1.82843 q^{36} +6.82843 q^{37} +0.414214 q^{38} -2.82843 q^{39} +2.89949 q^{40} +3.65685 q^{41} -1.34315 q^{43} -3.34315 q^{44} -1.82843 q^{45} +1.58579 q^{46} -11.8284 q^{47} -3.00000 q^{48} -0.686292 q^{50} +7.65685 q^{51} -5.17157 q^{52} -2.00000 q^{53} -0.414214 q^{54} -3.34315 q^{55} -1.00000 q^{57} +0.343146 q^{58} -5.17157 q^{59} -3.34315 q^{60} +3.34315 q^{61} +3.65685 q^{62} -4.17157 q^{64} -5.17157 q^{65} -0.757359 q^{66} -12.0000 q^{67} +14.0000 q^{68} -3.82843 q^{69} +15.6569 q^{71} -1.58579 q^{72} -12.3137 q^{73} +2.82843 q^{74} +1.65685 q^{75} -1.82843 q^{76} -1.17157 q^{78} +13.6569 q^{79} -5.48528 q^{80} +1.00000 q^{81} +1.51472 q^{82} -9.82843 q^{83} +14.0000 q^{85} -0.556349 q^{86} -0.828427 q^{87} -2.89949 q^{88} -5.17157 q^{89} -0.757359 q^{90} -7.00000 q^{92} -8.82843 q^{93} -4.89949 q^{94} -1.82843 q^{95} -4.41421 q^{96} -11.1716 q^{97} +1.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} - 4 q^{17} - 2 q^{18} + 2 q^{19} + 18 q^{20} + 10 q^{22} + 2 q^{23} + 6 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 - 4 * q^17 - 2 * q^18 + 2 * q^19 + 18 * q^20 + 10 * q^22 + 2 * q^23 + 6 * q^24 + 8 * q^25 + 8 * q^26 - 2 * q^27 - 4 * q^29 + 10 * q^30 + 12 * q^31 + 6 * q^32 + 2 * q^33 - 12 * q^34 + 2 * q^36 + 8 * q^37 - 2 * q^38 - 14 * q^40 - 4 * q^41 - 14 * q^43 - 18 * q^44 + 2 * q^45 + 6 * q^46 - 18 * q^47 - 6 * q^48 - 24 * q^50 + 4 * q^51 - 16 * q^52 - 4 * q^53 + 2 * q^54 - 18 * q^55 - 2 * q^57 + 12 * q^58 - 16 * q^59 - 18 * q^60 + 18 * q^61 - 4 * q^62 - 14 * q^64 - 16 * q^65 - 10 * q^66 - 24 * q^67 + 28 * q^68 - 2 * q^69 + 20 * q^71 - 6 * q^72 - 2 * q^73 - 8 * q^75 + 2 * q^76 - 8 * q^78 + 16 * q^79 + 6 * q^80 + 2 * q^81 + 20 * q^82 - 14 * q^83 + 28 * q^85 + 30 * q^86 + 4 * q^87 + 14 * q^88 - 16 * q^89 - 10 * q^90 - 14 * q^92 - 12 * q^93 + 10 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 2 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.82843	−0.914214
$5$	−1.82843	−0.817697	−0.408849	−	0.912602i	$-0.634070\pi$
$5$	−1.82843	−0.817697	−0.408849	+	0.912602i	$0.634070\pi$
$6$	−0.414214	−0.169102
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	−0.757359	−0.239498
$11$	1.82843	0.551292	0.275646	−	0.961259i	$-0.411108\pi$
$11$	1.82843	0.551292	0.275646	+	0.961259i	$0.411108\pi$
$12$	1.82843	0.527821
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	1.82843	0.472098
$16$	3.00000	0.750000
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0.414214	0.0976311
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.34315	0.747550
$21$	0	0
$22$	0.757359	0.161470
$23$	3.82843	0.798282	0.399141	−	0.916890i	$-0.369308\pi$
$23$	3.82843	0.798282	0.399141	+	0.916890i	$0.369308\pi$
$24$	1.58579	0.323697
$25$	−1.65685	−0.331371
$26$	1.17157	0.229764
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0.757359	0.138274
$31$	8.82843	1.58563	0.792816	−	0.609461i	$-0.208614\pi$
$31$	8.82843	1.58563	0.792816	+	0.609461i	$0.208614\pi$
$32$	4.41421	0.780330
$33$	−1.82843	−0.318288
$34$	−3.17157	−0.543920
$35$	0	0
$36$	−1.82843	−0.304738
$37$	6.82843	1.12259	0.561293	−	0.827617i	$-0.310304\pi$
$37$	6.82843	1.12259	0.561293	+	0.827617i	$0.310304\pi$
$38$	0.414214	0.0671943
$39$	−2.82843	−0.452911
$40$	2.89949	0.458450
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	−1.34315	−0.204828	−0.102414	−	0.994742i	$-0.532657\pi$
$43$	−1.34315	−0.204828	−0.102414	+	0.994742i	$0.532657\pi$
$44$	−3.34315	−0.503998
$45$	−1.82843	−0.272566
$46$	1.58579	0.233811
$47$	−11.8284	−1.72535	−0.862677	−	0.505756i	$-0.831214\pi$
$47$	−11.8284	−1.72535	−0.862677	+	0.505756i	$0.831214\pi$
$48$	−3.00000	−0.433013
$49$	0	0
$50$	−0.686292	−0.0970563
$51$	7.65685	1.07217
$52$	−5.17157	−0.717168
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−0.414214	−0.0563673
$55$	−3.34315	−0.450790
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0.343146	0.0450572
$59$	−5.17157	−0.673281	−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157	−0.673281	−0.336641	+	0.941633i	$0.609291\pi$
$60$	−3.34315	−0.431598
$61$	3.34315	0.428046	0.214023	−	0.976829i	$-0.431343\pi$
$61$	3.34315	0.428046	0.214023	+	0.976829i	$0.431343\pi$
$62$	3.65685	0.464421
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−5.17157	−0.641455
$66$	−0.757359	−0.0932245
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	14.0000	1.69775
$69$	−3.82843	−0.460888
$70$	0	0
$71$	15.6569	1.85813	0.929063	−	0.369921i	$-0.120615\pi$
$71$	15.6569	1.85813	0.929063	+	0.369921i	$0.120615\pi$
$72$	−1.58579	−0.186887
$73$	−12.3137	−1.44121	−0.720605	−	0.693346i	$-0.756136\pi$
$73$	−12.3137	−1.44121	−0.720605	+	0.693346i	$0.756136\pi$
$74$	2.82843	0.328798
$75$	1.65685	0.191317
$76$	−1.82843	−0.209735
$77$	0	0
$78$	−1.17157	−0.132655
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	−5.48528	−0.613273
$81$	1.00000	0.111111
$82$	1.51472	0.167273
$83$	−9.82843	−1.07881	−0.539405	−	0.842046i	$-0.681351\pi$
$83$	−9.82843	−1.07881	−0.539405	+	0.842046i	$0.681351\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	−0.556349	−0.0599927
$87$	−0.828427	−0.0888167
$88$	−2.89949	−0.309087
$89$	−5.17157	−0.548186	−0.274093	−	0.961703i	$-0.588378\pi$
$89$	−5.17157	−0.548186	−0.274093	+	0.961703i	$0.588378\pi$
$90$	−0.757359	−0.0798327
$91$	0	0
$92$	−7.00000	−0.729800
$93$	−8.82843	−0.915465
$94$	−4.89949	−0.505344
$95$	−1.82843	−0.187593
$96$	−4.41421	−0.450524
$97$	−11.1716	−1.13430	−0.567151	−	0.823614i	$-0.691954\pi$
$97$	−11.1716	−1.13430	−0.567151	+	0.823614i	$0.691954\pi$
$98$	0	0
$99$	1.82843	0.183764
$100$	3.02944	0.302944
$101$	−13.4853	−1.34184	−0.670918	−	0.741532i	$-0.734100\pi$
$101$	−13.4853	−1.34184	−0.670918	+	0.741532i	$0.734100\pi$
$102$	3.17157	0.314033
$103$	7.65685	0.754452	0.377226	−	0.926121i	$-0.376878\pi$
$103$	7.65685	0.754452	0.377226	+	0.926121i	$0.376878\pi$
$104$	−4.48528	−0.439818
$105$	0	0
$106$	−0.828427	−0.0804640
$107$	2.82843	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$107$	2.82843	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$108$	1.82843	0.175940
$109$	−10.1421	−0.971440	−0.485720	−	0.874114i	$-0.661443\pi$
$109$	−10.1421	−0.971440	−0.485720	+	0.874114i	$0.661443\pi$
$110$	−1.38478	−0.132033
$111$	−6.82843	−0.648126
$112$	0	0
$113$	−7.17157	−0.674645	−0.337322	−	0.941389i	$-0.609521\pi$
$113$	−7.17157	−0.674645	−0.337322	+	0.941389i	$0.609521\pi$
$114$	−0.414214	−0.0387947
$115$	−7.00000	−0.652753
$116$	−1.51472	−0.140638
$117$	2.82843	0.261488
$118$	−2.14214	−0.197200
$119$	0	0
$120$	−2.89949	−0.264686
$121$	−7.65685	−0.696078
$122$	1.38478	0.125372
$123$	−3.65685	−0.329727
$124$	−16.1421	−1.44961
$125$	12.1716	1.08866
$126$	0	0
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	−10.5563	−0.933058
$129$	1.34315	0.118257
$130$	−2.14214	−0.187878
$131$	−21.6569	−1.89217	−0.946084	−	0.323921i	$-0.894999\pi$
$131$	−21.6569	−1.89217	−0.946084	+	0.323921i	$0.894999\pi$
$132$	3.34315	0.290983
$133$	0	0
$134$	−4.97056	−0.429391
$135$	1.82843	0.157366
$136$	12.1421	1.04118
$137$	−8.17157	−0.698145	−0.349072	−	0.937096i	$-0.613503\pi$
$137$	−8.17157	−0.698145	−0.349072	+	0.937096i	$0.613503\pi$
$138$	−1.58579	−0.134991
$139$	−4.65685	−0.394989	−0.197495	−	0.980304i	$-0.563280\pi$
$139$	−4.65685	−0.394989	−0.197495	+	0.980304i	$0.563280\pi$
$140$	0	0
$141$	11.8284	0.996133
$142$	6.48528	0.544233
$143$	5.17157	0.432469
$144$	3.00000	0.250000
$145$	−1.51472	−0.125791
$146$	−5.10051	−0.422121
$147$	0	0
$148$	−12.4853	−1.02628
$149$	−1.82843	−0.149791	−0.0748953	−	0.997191i	$-0.523862\pi$
$149$	−1.82843	−0.149791	−0.0748953	+	0.997191i	$0.523862\pi$
$150$	0.686292	0.0560355
$151$	13.3137	1.08345	0.541727	−	0.840554i	$-0.317771\pi$
$151$	13.3137	1.08345	0.541727	+	0.840554i	$0.317771\pi$
$152$	−1.58579	−0.128624
$153$	−7.65685	−0.619020
$154$	0	0
$155$	−16.1421	−1.29657
$156$	5.17157	0.414057
$157$	0.656854	0.0524227	0.0262113	−	0.999656i	$-0.491656\pi$
$157$	0.656854	0.0524227	0.0262113	+	0.999656i	$0.491656\pi$
$158$	5.65685	0.450035
$159$	2.00000	0.158610
$160$	−8.07107	−0.638074
$161$	0	0
$162$	0.414214	0.0325437
$163$	−11.9706	−0.937607	−0.468803	−	0.883303i	$-0.655315\pi$
$163$	−11.9706	−0.937607	−0.468803	+	0.883303i	$0.655315\pi$
$164$	−6.68629	−0.522112
$165$	3.34315	0.260264
$166$	−4.07107	−0.315976
$167$	−3.17157	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$167$	−3.17157	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	5.79899	0.444762
$171$	1.00000	0.0764719
$172$	2.45584	0.187256
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	−0.343146	−0.0260138
$175$	0	0
$176$	5.48528	0.413469
$177$	5.17157	0.388719
$178$	−2.14214	−0.160560
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	3.34315	0.249183
$181$	0.343146	0.0255058	0.0127529	−	0.999919i	$-0.495941\pi$
$181$	0.343146	0.0255058	0.0127529	+	0.999919i	$0.495941\pi$
$182$	0	0
$183$	−3.34315	−0.247132
$184$	−6.07107	−0.447565
$185$	−12.4853	−0.917936
$186$	−3.65685	−0.268134
$187$	−14.0000	−1.02378
$188$	21.6274	1.57734
$189$	0	0
$190$	−0.757359	−0.0549446
$191$	6.17157	0.446559	0.223280	−	0.974754i	$-0.428324\pi$
$191$	6.17157	0.446559	0.223280	+	0.974754i	$0.428324\pi$
$192$	4.17157	0.301057
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−4.62742	−0.332229
$195$	5.17157	0.370344
$196$	0	0
$197$	−0.514719	−0.0366722	−0.0183361	−	0.999832i	$-0.505837\pi$
$197$	−0.514719	−0.0366722	−0.0183361	+	0.999832i	$0.505837\pi$
$198$	0.757359	0.0538232
$199$	−14.6569	−1.03900	−0.519498	−	0.854471i	$-0.673881\pi$
$199$	−14.6569	−1.03900	−0.519498	+	0.854471i	$0.673881\pi$
$200$	2.62742	0.185786
$201$	12.0000	0.846415
$202$	−5.58579	−0.393015
$203$	0	0
$204$	−14.0000	−0.980196
$205$	−6.68629	−0.466991
$206$	3.17157	0.220974
$207$	3.82843	0.266094
$208$	8.48528	0.588348
$209$	1.82843	0.126475
$210$	0	0
$211$	−0.485281	−0.0334081	−0.0167041	−	0.999860i	$-0.505317\pi$
$211$	−0.485281	−0.0334081	−0.0167041	+	0.999860i	$0.505317\pi$
$212$	3.65685	0.251154
$213$	−15.6569	−1.07279
$214$	1.17157	0.0800871
$215$	2.45584	0.167487
$216$	1.58579	0.107899
$217$	0	0
$218$	−4.20101	−0.284528
$219$	12.3137	0.832083
$220$	6.11270	0.412118
$221$	−21.6569	−1.45680
$222$	−2.82843	−0.189832
$223$	17.7990	1.19191	0.595954	−	0.803018i	$-0.296774\pi$
$223$	17.7990	1.19191	0.595954	+	0.803018i	$0.296774\pi$
$224$	0	0
$225$	−1.65685	−0.110457
$226$	−2.97056	−0.197599
$227$	17.6569	1.17193	0.585963	−	0.810338i	$-0.300716\pi$
$227$	17.6569	1.17193	0.585963	+	0.810338i	$0.300716\pi$
$228$	1.82843	0.121091
$229$	4.34315	0.287003	0.143502	−	0.989650i	$-0.454164\pi$
$229$	4.34315	0.287003	0.143502	+	0.989650i	$0.454164\pi$
$230$	−2.89949	−0.191187
$231$	0	0
$232$	−1.31371	−0.0862492
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	1.17157	0.0765881
$235$	21.6274	1.41082
$236$	9.45584	0.615523
$237$	−13.6569	−0.887108
$238$	0	0
$239$	13.6569	0.883388	0.441694	−	0.897166i	$-0.354378\pi$
$239$	13.6569	0.883388	0.441694	+	0.897166i	$0.354378\pi$
$240$	5.48528	0.354073
$241$	19.7990	1.27537	0.637683	−	0.770299i	$-0.279893\pi$
$241$	19.7990	1.27537	0.637683	+	0.770299i	$0.279893\pi$
$242$	−3.17157	−0.203876
$243$	−1.00000	−0.0641500
$244$	−6.11270	−0.391325
$245$	0	0
$246$	−1.51472	−0.0965749
$247$	2.82843	0.179969
$248$	−14.0000	−0.889001
$249$	9.82843	0.622851
$250$	5.04163	0.318861
$251$	−3.48528	−0.219989	−0.109995	−	0.993932i	$-0.535083\pi$
$251$	−3.48528	−0.219989	−0.109995	+	0.993932i	$0.535083\pi$
$252$	0	0
$253$	7.00000	0.440086
$254$	−7.02944	−0.441066
$255$	−14.0000	−0.876714
$256$	3.97056	0.248160
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0.556349	0.0346368
$259$	0	0
$260$	9.45584	0.586427
$261$	0.828427	0.0512784
$262$	−8.97056	−0.554203
$263$	−11.3137	−0.697633	−0.348817	−	0.937191i	$-0.613416\pi$
$263$	−11.3137	−0.697633	−0.348817	+	0.937191i	$0.613416\pi$
$264$	2.89949	0.178452
$265$	3.65685	0.224639
$266$	0	0
$267$	5.17157	0.316495
$268$	21.9411	1.34027
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0.757359	0.0460914
$271$	−25.6274	−1.55675	−0.778377	−	0.627797i	$-0.783957\pi$
$271$	−25.6274	−1.55675	−0.778377	+	0.627797i	$0.783957\pi$
$272$	−22.9706	−1.39279
$273$	0	0
$274$	−3.38478	−0.204482
$275$	−3.02944	−0.182682
$276$	7.00000	0.421350
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−1.92893	−0.115690
$279$	8.82843	0.528544
$280$	0	0
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	4.89949	0.291761
$283$	13.6274	0.810066	0.405033	−	0.914302i	$-0.367260\pi$
$283$	13.6274	0.810066	0.405033	+	0.914302i	$0.367260\pi$
$284$	−28.6274	−1.69872
$285$	1.82843	0.108307
$286$	2.14214	0.126667
$287$	0	0
$288$	4.41421	0.260110
$289$	41.6274	2.44867
$290$	−0.627417	−0.0368432
$291$	11.1716	0.654889
$292$	22.5147	1.31757
$293$	−4.82843	−0.282080	−0.141040	−	0.990004i	$-0.545045\pi$
$293$	−4.82843	−0.282080	−0.141040	+	0.990004i	$0.545045\pi$
$294$	0	0
$295$	9.45584	0.550541
$296$	−10.8284	−0.629390
$297$	−1.82843	−0.106096
$298$	−0.757359	−0.0438726
$299$	10.8284	0.626224
$300$	−3.02944	−0.174905
$301$	0	0
$302$	5.51472	0.317336
$303$	13.4853	0.774709
$304$	3.00000	0.172062
$305$	−6.11270	−0.350012
$306$	−3.17157	−0.181307
$307$	14.4853	0.826719	0.413359	−	0.910568i	$-0.364355\pi$
$307$	14.4853	0.826719	0.413359	+	0.910568i	$0.364355\pi$
$308$	0	0
$309$	−7.65685	−0.435583
$310$	−6.68629	−0.379756
$311$	22.6274	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$311$	22.6274	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$312$	4.48528	0.253929
$313$	25.2843	1.42915	0.714576	−	0.699558i	$-0.246620\pi$
$313$	25.2843	1.42915	0.714576	+	0.699558i	$0.246620\pi$
$314$	0.272078	0.0153542
$315$	0	0
$316$	−24.9706	−1.40470
$317$	−9.79899	−0.550366	−0.275183	−	0.961392i	$-0.588738\pi$
$317$	−9.79899	−0.550366	−0.275183	+	0.961392i	$0.588738\pi$
$318$	0.828427	0.0464559
$319$	1.51472	0.0848080
$320$	7.62742	0.426386
$321$	−2.82843	−0.157867
$322$	0	0
$323$	−7.65685	−0.426039
$324$	−1.82843	−0.101579
$325$	−4.68629	−0.259949
$326$	−4.95837	−0.274619
$327$	10.1421	0.560861
$328$	−5.79899	−0.320196
$329$	0	0
$330$	1.38478	0.0762294
$331$	−22.4853	−1.23590	−0.617951	−	0.786216i	$-0.712037\pi$
$331$	−22.4853	−1.23590	−0.617951	+	0.786216i	$0.712037\pi$
$332$	17.9706	0.986263
$333$	6.82843	0.374196
$334$	−1.31371	−0.0718829
$335$	21.9411	1.19877
$336$	0	0
$337$	10.4853	0.571170	0.285585	−	0.958353i	$-0.407812\pi$
$337$	10.4853	0.571170	0.285585	+	0.958353i	$0.407812\pi$
$338$	−2.07107	−0.112651
$339$	7.17157	0.389506
$340$	−25.5980	−1.38825
$341$	16.1421	0.874146
$342$	0.414214	0.0223981
$343$	0	0
$344$	2.12994	0.114839
$345$	7.00000	0.376867
$346$	1.17157	0.0629841
$347$	−30.4558	−1.63496	−0.817478	−	0.575960i	$-0.804628\pi$
$347$	−30.4558	−1.63496	−0.817478	+	0.575960i	$0.804628\pi$
$348$	1.51472	0.0811974
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−2.82843	−0.150970
$352$	8.07107	0.430189
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	2.14214	0.113853
$355$	−28.6274	−1.51939
$356$	9.45584	0.501159
$357$	0	0
$358$	2.68629	0.141975
$359$	−11.8284	−0.624281	−0.312140	−	0.950036i	$-0.601046\pi$
$359$	−11.8284	−0.624281	−0.312140	+	0.950036i	$0.601046\pi$
$360$	2.89949	0.152817
$361$	1.00000	0.0526316
$362$	0.142136	0.00747048
$363$	7.65685	0.401881
$364$	0	0
$365$	22.5147	1.17847
$366$	−1.38478	−0.0723834
$367$	−18.3431	−0.957504	−0.478752	−	0.877950i	$-0.658911\pi$
$367$	−18.3431	−0.957504	−0.478752	+	0.877950i	$0.658911\pi$
$368$	11.4853	0.598712
$369$	3.65685	0.190368
$370$	−5.17157	−0.268857
$371$	0	0
$372$	16.1421	0.836931
$373$	−36.2843	−1.87873	−0.939364	−	0.342921i	$-0.888584\pi$
$373$	−36.2843	−1.87873	−0.939364	+	0.342921i	$0.888584\pi$
$374$	−5.79899	−0.299859
$375$	−12.1716	−0.628537
$376$	18.7574	0.967337
$377$	2.34315	0.120678
$378$	0	0
$379$	0.485281	0.0249272	0.0124636	−	0.999922i	$-0.496033\pi$
$379$	0.485281	0.0249272	0.0124636	+	0.999922i	$0.496033\pi$
$380$	3.34315	0.171500
$381$	16.9706	0.869428
$382$	2.55635	0.130794
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	10.5563	0.538701
$385$	0	0
$386$	1.65685	0.0843317
$387$	−1.34315	−0.0682759
$388$	20.4264	1.03699
$389$	15.6569	0.793834	0.396917	−	0.917855i	$-0.370080\pi$
$389$	15.6569	0.793834	0.396917	+	0.917855i	$0.370080\pi$
$390$	2.14214	0.108471
$391$	−29.3137	−1.48246
$392$	0	0
$393$	21.6569	1.09244
$394$	−0.213203	−0.0107410
$395$	−24.9706	−1.25641
$396$	−3.34315	−0.167999
$397$	17.3137	0.868950	0.434475	−	0.900684i	$-0.356934\pi$
$397$	17.3137	0.868950	0.434475	+	0.900684i	$0.356934\pi$
$398$	−6.07107	−0.304315
$399$	0	0
$400$	−4.97056	−0.248528
$401$	−23.3137	−1.16423	−0.582116	−	0.813106i	$-0.697775\pi$
$401$	−23.3137	−1.16423	−0.582116	+	0.813106i	$0.697775\pi$
$402$	4.97056	0.247909
$403$	24.9706	1.24387
$404$	24.6569	1.22672
$405$	−1.82843	−0.0908553
$406$	0	0
$407$	12.4853	0.618872
$408$	−12.1421	−0.601125
$409$	17.3137	0.856108	0.428054	−	0.903753i	$-0.359199\pi$
$409$	17.3137	0.856108	0.428054	+	0.903753i	$0.359199\pi$
$410$	−2.76955	−0.136778
$411$	8.17157	0.403074
$412$	−14.0000	−0.689730
$413$	0	0
$414$	1.58579	0.0779372
$415$	17.9706	0.882140
$416$	12.4853	0.612141
$417$	4.65685	0.228047
$418$	0.757359	0.0370437
$419$	15.8284	0.773269	0.386635	−	0.922233i	$-0.373637\pi$
$419$	15.8284	0.773269	0.386635	+	0.922233i	$0.373637\pi$
$420$	0	0
$421$	24.8284	1.21006	0.605032	−	0.796201i	$-0.293160\pi$
$421$	24.8284	1.21006	0.605032	+	0.796201i	$0.293160\pi$
$422$	−0.201010	−0.00978502
$423$	−11.8284	−0.575118
$424$	3.17157	0.154025
$425$	12.6863	0.615376
$426$	−6.48528	−0.314213
$427$	0	0
$428$	−5.17157	−0.249977
$429$	−5.17157	−0.249686
$430$	1.01724	0.0490559
$431$	−28.8284	−1.38862	−0.694308	−	0.719678i	$-0.744290\pi$
$431$	−28.8284	−1.38862	−0.694308	+	0.719678i	$0.744290\pi$
$432$	−3.00000	−0.144338
$433$	8.48528	0.407777	0.203888	−	0.978994i	$-0.434642\pi$
$433$	8.48528	0.407777	0.203888	+	0.978994i	$0.434642\pi$
$434$	0	0
$435$	1.51472	0.0726252
$436$	18.5442	0.888104
$437$	3.82843	0.183139
$438$	5.10051	0.243712
$439$	−19.7990	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$439$	−19.7990	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$440$	5.30152	0.252740
$441$	0	0
$442$	−8.97056	−0.426686
$443$	−22.3431	−1.06155	−0.530777	−	0.847511i	$-0.678100\pi$
$443$	−22.3431	−1.06155	−0.530777	+	0.847511i	$0.678100\pi$
$444$	12.4853	0.592525
$445$	9.45584	0.448250
$446$	7.37258	0.349102
$447$	1.82843	0.0864816
$448$	0	0
$449$	−2.14214	−0.101094	−0.0505468	−	0.998722i	$-0.516096\pi$
$449$	−2.14214	−0.101094	−0.0505468	+	0.998722i	$0.516096\pi$
$450$	−0.686292	−0.0323521
$451$	6.68629	0.314845
$452$	13.1127	0.616769
$453$	−13.3137	−0.625533
$454$	7.31371	0.343249
$455$	0	0
$456$	1.58579	0.0742613
$457$	−3.34315	−0.156386	−0.0781929	−	0.996938i	$-0.524915\pi$
$457$	−3.34315	−0.156386	−0.0781929	+	0.996938i	$0.524915\pi$
$458$	1.79899	0.0840613
$459$	7.65685	0.357391
$460$	12.7990	0.596756
$461$	28.4558	1.32532	0.662660	−	0.748920i	$-0.269427\pi$
$461$	28.4558	1.32532	0.662660	+	0.748920i	$0.269427\pi$
$462$	0	0
$463$	−9.34315	−0.434213	−0.217106	−	0.976148i	$-0.569662\pi$
$463$	−9.34315	−0.434213	−0.217106	+	0.976148i	$0.569662\pi$
$464$	2.48528	0.115376
$465$	16.1421	0.748574
$466$	2.48528	0.115128
$467$	−8.51472	−0.394014	−0.197007	−	0.980402i	$-0.563122\pi$
$467$	−8.51472	−0.394014	−0.197007	+	0.980402i	$0.563122\pi$
$468$	−5.17157	−0.239056
$469$	0	0
$470$	8.95837	0.413219
$471$	−0.656854	−0.0302662
$472$	8.20101	0.377482
$473$	−2.45584	−0.112920
$474$	−5.65685	−0.259828
$475$	−1.65685	−0.0760217
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	5.65685	0.258738
$479$	−17.8284	−0.814602	−0.407301	−	0.913294i	$-0.633530\pi$
$479$	−17.8284	−0.814602	−0.407301	+	0.913294i	$0.633530\pi$
$480$	8.07107	0.368392
$481$	19.3137	0.880629
$482$	8.20101	0.373546
$483$	0	0
$484$	14.0000	0.636364
$485$	20.4264	0.927515
$486$	−0.414214	−0.0187891
$487$	−11.6569	−0.528222	−0.264111	−	0.964492i	$-0.585079\pi$
$487$	−11.6569	−0.528222	−0.264111	+	0.964492i	$0.585079\pi$
$488$	−5.30152	−0.239988
$489$	11.9706	0.541328
$490$	0	0
$491$	−25.4853	−1.15013	−0.575067	−	0.818106i	$-0.695024\pi$
$491$	−25.4853	−1.15013	−0.575067	+	0.818106i	$0.695024\pi$
$492$	6.68629	0.301441
$493$	−6.34315	−0.285681
$494$	1.17157	0.0527116
$495$	−3.34315	−0.150263
$496$	26.4853	1.18922
$497$	0	0
$498$	4.07107	0.182429
$499$	−13.9706	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$499$	−13.9706	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$500$	−22.2548	−0.995266
$501$	3.17157	0.141695
$502$	−1.44365	−0.0644333
$503$	42.1127	1.87771	0.938856	−	0.344309i	$-0.111887\pi$
$503$	42.1127	1.87771	0.938856	+	0.344309i	$0.111887\pi$
$504$	0	0
$505$	24.6569	1.09722
$506$	2.89949	0.128898
$507$	5.00000	0.222058
$508$	31.0294	1.37671
$509$	−6.48528	−0.287455	−0.143728	−	0.989617i	$-0.545909\pi$
$509$	−6.48528	−0.287455	−0.143728	+	0.989617i	$0.545909\pi$
$510$	−5.79899	−0.256784
$511$	0	0
$512$	22.7574	1.00574
$513$	−1.00000	−0.0441511
$514$	−10.7696	−0.475025
$515$	−14.0000	−0.616914
$516$	−2.45584	−0.108113
$517$	−21.6274	−0.951173
$518$	0	0
$519$	−2.82843	−0.124154
$520$	8.20101	0.359638
$521$	7.02944	0.307965	0.153983	−	0.988074i	$-0.450790\pi$
$521$	7.02944	0.307965	0.153983	+	0.988074i	$0.450790\pi$
$522$	0.343146	0.0150191
$523$	−1.79899	−0.0786643	−0.0393322	−	0.999226i	$-0.512523\pi$
$523$	−1.79899	−0.0786643	−0.0393322	+	0.999226i	$0.512523\pi$
$524$	39.5980	1.72985
$525$	0	0
$526$	−4.68629	−0.204332
$527$	−67.5980	−2.94461
$528$	−5.48528	−0.238716
$529$	−8.34315	−0.362745
$530$	1.51472	0.0657952
$531$	−5.17157	−0.224427
$532$	0	0
$533$	10.3431	0.448011
$534$	2.14214	0.0926993
$535$	−5.17157	−0.223587
$536$	19.0294	0.821946
$537$	−6.48528	−0.279861
$538$	3.31371	0.142864
$539$	0	0
$540$	−3.34315	−0.143866
$541$	23.0000	0.988847	0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000	0.988847	0.494424	+	0.869221i	$0.335379\pi$
$542$	−10.6152	−0.455963
$543$	−0.343146	−0.0147258
$544$	−33.7990	−1.44912
$545$	18.5442	0.794344
$546$	0	0
$547$	33.7990	1.44514	0.722570	−	0.691298i	$-0.242961\pi$
$547$	33.7990	1.44514	0.722570	+	0.691298i	$0.242961\pi$
$548$	14.9411	0.638253
$549$	3.34315	0.142682
$550$	−1.25483	−0.0535063
$551$	0.828427	0.0352922
$552$	6.07107	0.258402
$553$	0	0
$554$	5.38478	0.228777
$555$	12.4853	0.529971
$556$	8.51472	0.361105
$557$	−36.4558	−1.54468	−0.772342	−	0.635207i	$-0.780915\pi$
$557$	−36.4558	−1.54468	−0.772342	+	0.635207i	$0.780915\pi$
$558$	3.65685	0.154807
$559$	−3.79899	−0.160680
$560$	0	0
$561$	14.0000	0.591080
$562$	−11.1716	−0.471244
$563$	16.4853	0.694772	0.347386	−	0.937722i	$-0.387069\pi$
$563$	16.4853	0.694772	0.347386	+	0.937722i	$0.387069\pi$
$564$	−21.6274	−0.910679
$565$	13.1127	0.551655
$566$	5.64466	0.237263
$567$	0	0
$568$	−24.8284	−1.04178
$569$	−6.20101	−0.259960	−0.129980	−	0.991517i	$-0.541491\pi$
$569$	−6.20101	−0.259960	−0.129980	+	0.991517i	$0.541491\pi$
$570$	0.757359	0.0317223
$571$	−29.2843	−1.22551	−0.612754	−	0.790273i	$-0.709938\pi$
$571$	−29.2843	−1.22551	−0.612754	+	0.790273i	$0.709938\pi$
$572$	−9.45584	−0.395369
$573$	−6.17157	−0.257821
$574$	0	0
$575$	−6.34315	−0.264527
$576$	−4.17157	−0.173816
$577$	4.31371	0.179582	0.0897910	−	0.995961i	$-0.471380\pi$
$577$	4.31371	0.179582	0.0897910	+	0.995961i	$0.471380\pi$
$578$	17.2426	0.717199
$579$	−4.00000	−0.166234
$580$	2.76955	0.114999
$581$	0	0
$582$	4.62742	0.191813
$583$	−3.65685	−0.151451
$584$	19.5269	0.808029
$585$	−5.17157	−0.213818
$586$	−2.00000	−0.0826192
$587$	−18.6274	−0.768836	−0.384418	−	0.923159i	$-0.625598\pi$
$587$	−18.6274	−0.768836	−0.384418	+	0.923159i	$0.625598\pi$
$588$	0	0
$589$	8.82843	0.363769
$590$	3.91674	0.161250
$591$	0.514719	0.0211727
$592$	20.4853	0.841940
$593$	−10.5147	−0.431788	−0.215894	−	0.976417i	$-0.569267\pi$
$593$	−10.5147	−0.431788	−0.215894	+	0.976417i	$0.569267\pi$
$594$	−0.757359	−0.0310748
$595$	0	0
$596$	3.34315	0.136941
$597$	14.6569	0.599865
$598$	4.48528	0.183417
$599$	33.1127	1.35295	0.676474	−	0.736466i	$-0.263507\pi$
$599$	33.1127	1.35295	0.676474	+	0.736466i	$0.263507\pi$
$600$	−2.62742	−0.107264
$601$	31.3137	1.27731	0.638656	−	0.769492i	$-0.279491\pi$
$601$	31.3137	1.27731	0.638656	+	0.769492i	$0.279491\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−24.3431	−0.990509
$605$	14.0000	0.569181
$606$	5.58579	0.226907
$607$	−42.1421	−1.71050	−0.855248	−	0.518219i	$-0.826595\pi$
$607$	−42.1421	−1.71050	−0.855248	+	0.518219i	$0.826595\pi$
$608$	4.41421	0.179020
$609$	0	0
$610$	−2.53196	−0.102516
$611$	−33.4558	−1.35348
$612$	14.0000	0.565916
$613$	8.62742	0.348458	0.174229	−	0.984705i	$-0.444257\pi$
$613$	8.62742	0.348458	0.174229	+	0.984705i	$0.444257\pi$
$614$	6.00000	0.242140
$615$	6.68629	0.269617
$616$	0	0
$617$	−37.8284	−1.52292	−0.761458	−	0.648215i	$-0.775516\pi$
$617$	−37.8284	−1.52292	−0.761458	+	0.648215i	$0.775516\pi$
$618$	−3.17157	−0.127579
$619$	−20.6569	−0.830269	−0.415135	−	0.909760i	$-0.636265\pi$
$619$	−20.6569	−0.830269	−0.415135	+	0.909760i	$0.636265\pi$
$620$	29.5147	1.18534
$621$	−3.82843	−0.153629
$622$	9.37258	0.375806
$623$	0	0
$624$	−8.48528	−0.339683
$625$	−13.9706	−0.558823
$626$	10.4731	0.418589
$627$	−1.82843	−0.0730203
$628$	−1.20101	−0.0479255
$629$	−52.2843	−2.08471
$630$	0	0
$631$	−5.34315	−0.212707	−0.106354	−	0.994328i	$-0.533918\pi$
$631$	−5.34315	−0.212707	−0.106354	+	0.994328i	$0.533918\pi$
$632$	−21.6569	−0.861463
$633$	0.485281	0.0192882
$634$	−4.05887	−0.161198
$635$	31.0294	1.23137
$636$	−3.65685	−0.145004
$637$	0	0
$638$	0.627417	0.0248397
$639$	15.6569	0.619376
$640$	19.3015	0.762959
$641$	−10.8284	−0.427697	−0.213849	−	0.976867i	$-0.568600\pi$
$641$	−10.8284	−0.427697	−0.213849	+	0.976867i	$0.568600\pi$
$642$	−1.17157	−0.0462383
$643$	−46.6274	−1.83881	−0.919403	−	0.393317i	$-0.871327\pi$
$643$	−46.6274	−1.83881	−0.919403	+	0.393317i	$0.871327\pi$
$644$	0	0
$645$	−2.45584	−0.0966988
$646$	−3.17157	−0.124784
$647$	−35.1421	−1.38158	−0.690790	−	0.723055i	$-0.742737\pi$
$647$	−35.1421	−1.38158	−0.690790	+	0.723055i	$0.742737\pi$
$648$	−1.58579	−0.0622956
$649$	−9.45584	−0.371174
$650$	−1.94113	−0.0761372
$651$	0	0
$652$	21.8873	0.857173
$653$	−27.9411	−1.09342	−0.546710	−	0.837322i	$-0.684120\pi$
$653$	−27.9411	−1.09342	−0.546710	+	0.837322i	$0.684120\pi$
$654$	4.20101	0.164272
$655$	39.5980	1.54722
$656$	10.9706	0.428329
$657$	−12.3137	−0.480404
$658$	0	0
$659$	−38.4853	−1.49917	−0.749587	−	0.661906i	$-0.769748\pi$
$659$	−38.4853	−1.49917	−0.749587	+	0.661906i	$0.769748\pi$
$660$	−6.11270	−0.237936
$661$	44.2843	1.72246	0.861229	−	0.508217i	$-0.169695\pi$
$661$	44.2843	1.72246	0.861229	+	0.508217i	$0.169695\pi$
$662$	−9.31371	−0.361988
$663$	21.6569	0.841083
$664$	15.5858	0.604846
$665$	0	0
$666$	2.82843	0.109599
$667$	3.17157	0.122804
$668$	5.79899	0.224370
$669$	−17.7990	−0.688149
$670$	9.08831	0.351112
$671$	6.11270	0.235978
$672$	0	0
$673$	−2.82843	−0.109028	−0.0545139	−	0.998513i	$-0.517361\pi$
$673$	−2.82843	−0.109028	−0.0545139	+	0.998513i	$0.517361\pi$
$674$	4.34315	0.167292
$675$	1.65685	0.0637723
$676$	9.14214	0.351621
$677$	16.0000	0.614930	0.307465	−	0.951559i	$-0.400519\pi$
$677$	16.0000	0.614930	0.307465	+	0.951559i	$0.400519\pi$
$678$	2.97056	0.114084
$679$	0	0
$680$	−22.2010	−0.851370
$681$	−17.6569	−0.676612
$682$	6.68629	0.256031
$683$	38.9706	1.49117	0.745584	−	0.666412i	$-0.232171\pi$
$683$	38.9706	1.49117	0.745584	+	0.666412i	$0.232171\pi$
$684$	−1.82843	−0.0699117
$685$	14.9411	0.570871
$686$	0	0
$687$	−4.34315	−0.165701
$688$	−4.02944	−0.153621
$689$	−5.65685	−0.215509
$690$	2.89949	0.110382
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	−5.17157	−0.196594
$693$	0	0
$694$	−12.6152	−0.478867
$695$	8.51472	0.322982
$696$	1.31371	0.0497960
$697$	−28.0000	−1.06058
$698$	−5.79899	−0.219495
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−48.4558	−1.83015	−0.915076	−	0.403281i	$-0.867870\pi$
$701$	−48.4558	−1.83015	−0.915076	+	0.403281i	$0.867870\pi$
$702$	−1.17157	−0.0442182
$703$	6.82843	0.257539
$704$	−7.62742	−0.287469
$705$	−21.6274	−0.814536
$706$	−8.82843	−0.332262
$707$	0	0
$708$	−9.45584	−0.355372
$709$	17.9706	0.674899	0.337449	−	0.941344i	$-0.390436\pi$
$709$	17.9706	0.674899	0.337449	+	0.941344i	$0.390436\pi$
$710$	−11.8579	−0.445018
$711$	13.6569	0.512172
$712$	8.20101	0.307346
$713$	33.7990	1.26578
$714$	0	0
$715$	−9.45584	−0.353629
$716$	−11.8579	−0.443149
$717$	−13.6569	−0.510025
$718$	−4.89949	−0.182848
$719$	−32.9706	−1.22959	−0.614797	−	0.788685i	$-0.710762\pi$
$719$	−32.9706	−1.22959	−0.614797	+	0.788685i	$0.710762\pi$
$720$	−5.48528	−0.204424
$721$	0	0
$722$	0.414214	0.0154154
$723$	−19.7990	−0.736332
$724$	−0.627417	−0.0233178
$725$	−1.37258	−0.0509765
$726$	3.17157	0.117708
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	9.32590	0.345167
$731$	10.2843	0.380378
$732$	6.11270	0.225932
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−7.59798	−0.280447
$735$	0	0
$736$	16.8995	0.622924
$737$	−21.9411	−0.808212
$738$	1.51472	0.0557576
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	22.8284	0.839190
$741$	−2.82843	−0.103905
$742$	0	0
$743$	38.8284	1.42448	0.712238	−	0.701938i	$-0.247681\pi$
$743$	38.8284	1.42448	0.712238	+	0.701938i	$0.247681\pi$
$744$	14.0000	0.513265
$745$	3.34315	0.122483
$746$	−15.0294	−0.550267
$747$	−9.82843	−0.359603
$748$	25.5980	0.935955
$749$	0	0
$750$	−5.04163	−0.184094
$751$	−10.6274	−0.387800	−0.193900	−	0.981021i	$-0.562114\pi$
$751$	−10.6274	−0.387800	−0.193900	+	0.981021i	$0.562114\pi$
$752$	−35.4853	−1.29402
$753$	3.48528	0.127011
$754$	0.970563	0.0353458
$755$	−24.3431	−0.885938
$756$	0	0
$757$	20.6569	0.750786	0.375393	−	0.926866i	$-0.377508\pi$
$757$	20.6569	0.750786	0.375393	+	0.926866i	$0.377508\pi$
$758$	0.201010	0.00730102
$759$	−7.00000	−0.254084
$760$	2.89949	0.105176
$761$	−20.7990	−0.753963	−0.376981	−	0.926221i	$-0.623038\pi$
$761$	−20.7990	−0.753963	−0.376981	+	0.926221i	$0.623038\pi$
$762$	7.02944	0.254650
$763$	0	0
$764$	−11.2843	−0.408251
$765$	14.0000	0.506171
$766$	−6.62742	−0.239458
$767$	−14.6274	−0.528165
$768$	−3.97056	−0.143275
$769$	8.31371	0.299800	0.149900	−	0.988701i	$-0.452105\pi$
$769$	8.31371	0.299800	0.149900	+	0.988701i	$0.452105\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	−7.31371	−0.263226
$773$	2.82843	0.101731	0.0508657	−	0.998706i	$-0.483802\pi$
$773$	2.82843	0.101731	0.0508657	+	0.998706i	$0.483802\pi$
$774$	−0.556349	−0.0199976
$775$	−14.6274	−0.525432
$776$	17.7157	0.635958
$777$	0	0
$778$	6.48528	0.232509
$779$	3.65685	0.131020
$780$	−9.45584	−0.338574
$781$	28.6274	1.02437
$782$	−12.1421	−0.434202
$783$	−0.828427	−0.0296056
$784$	0	0
$785$	−1.20101	−0.0428659
$786$	8.97056	0.319969
$787$	40.2843	1.43598	0.717990	−	0.696054i	$-0.245063\pi$
$787$	40.2843	1.43598	0.717990	+	0.696054i	$0.245063\pi$
$788$	0.941125	0.0335262
$789$	11.3137	0.402779
$790$	−10.3431	−0.367993
$791$	0	0
$792$	−2.89949	−0.103029
$793$	9.45584	0.335787
$794$	7.17157	0.254510
$795$	−3.65685	−0.129695
$796$	26.7990	0.949865
$797$	−24.4853	−0.867313	−0.433657	−	0.901078i	$-0.642777\pi$
$797$	−24.4853	−0.867313	−0.433657	+	0.901078i	$0.642777\pi$
$798$	0	0
$799$	90.5685	3.20408
$800$	−7.31371	−0.258579
$801$	−5.17157	−0.182729
$802$	−9.65685	−0.340995
$803$	−22.5147	−0.794527
$804$	−21.9411	−0.773804
$805$	0	0
$806$	10.3431	0.364322
$807$	−8.00000	−0.281613
$808$	21.3848	0.752314
$809$	14.7990	0.520305	0.260152	−	0.965568i	$-0.416227\pi$
$809$	14.7990	0.520305	0.260152	+	0.965568i	$0.416227\pi$
$810$	−0.757359	−0.0266109
$811$	−16.1421	−0.566827	−0.283414	−	0.958998i	$-0.591467\pi$
$811$	−16.1421	−0.566827	−0.283414	+	0.958998i	$0.591467\pi$
$812$	0	0
$813$	25.6274	0.898793
$814$	5.17157	0.181264
$815$	21.8873	0.766679
$816$	22.9706	0.804131
$817$	−1.34315	−0.0469907
$818$	7.17157	0.250748
$819$	0	0
$820$	12.2254	0.426929
$821$	35.1421	1.22647	0.613234	−	0.789901i	$-0.289868\pi$
$821$	35.1421	1.22647	0.613234	+	0.789901i	$0.289868\pi$
$822$	3.38478	0.118058
$823$	−1.97056	−0.0686895	−0.0343447	−	0.999410i	$-0.510934\pi$
$823$	−1.97056	−0.0686895	−0.0343447	+	0.999410i	$0.510934\pi$
$824$	−12.1421	−0.422991
$825$	3.02944	0.105471
$826$	0	0
$827$	47.1127	1.63827	0.819135	−	0.573601i	$-0.194454\pi$
$827$	47.1127	1.63827	0.819135	+	0.573601i	$0.194454\pi$
$828$	−7.00000	−0.243267
$829$	9.85786	0.342378	0.171189	−	0.985238i	$-0.445239\pi$
$829$	9.85786	0.342378	0.171189	+	0.985238i	$0.445239\pi$
$830$	7.44365	0.258373
$831$	−13.0000	−0.450965
$832$	−11.7990	−0.409056
$833$	0	0
$834$	1.92893	0.0667935
$835$	5.79899	0.200682
$836$	−3.34315	−0.115625
$837$	−8.82843	−0.305155
$838$	6.55635	0.226485
$839$	−45.2548	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$839$	−45.2548	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	10.2843	0.354419
$843$	26.9706	0.928916
$844$	0.887302	0.0305422
$845$	9.14214	0.314499
$846$	−4.89949	−0.168448
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−13.6274	−0.467692
$850$	5.25483	0.180239
$851$	26.1421	0.896141
$852$	28.6274	0.980759
$853$	51.6274	1.76769	0.883845	−	0.467781i	$-0.154946\pi$
$853$	51.6274	1.76769	0.883845	+	0.467781i	$0.154946\pi$
$854$	0	0
$855$	−1.82843	−0.0625309
$856$	−4.48528	−0.153304
$857$	21.1716	0.723207	0.361604	−	0.932332i	$-0.382229\pi$
$857$	21.1716	0.723207	0.361604	+	0.932332i	$0.382229\pi$
$858$	−2.14214	−0.0731313
$859$	36.9411	1.26041	0.630207	−	0.776427i	$-0.282970\pi$
$859$	36.9411	1.26041	0.630207	+	0.776427i	$0.282970\pi$
$860$	−4.49033	−0.153119
$861$	0	0
$862$	−11.9411	−0.406716
$863$	12.8284	0.436685	0.218342	−	0.975872i	$-0.429935\pi$
$863$	12.8284	0.436685	0.218342	+	0.975872i	$0.429935\pi$
$864$	−4.41421	−0.150175
$865$	−5.17157	−0.175839
$866$	3.51472	0.119435
$867$	−41.6274	−1.41374
$868$	0	0
$869$	24.9706	0.847068
$870$	0.627417	0.0212714
$871$	−33.9411	−1.15005
$872$	16.0833	0.544648
$873$	−11.1716	−0.378100
$874$	1.58579	0.0536400
$875$	0	0
$876$	−22.5147	−0.760702
$877$	10.3431	0.349263	0.174632	−	0.984634i	$-0.444127\pi$
$877$	10.3431	0.349263	0.174632	+	0.984634i	$0.444127\pi$
$878$	−8.20101	−0.276771
$879$	4.82843	0.162859
$880$	−10.0294	−0.338092
$881$	−40.6274	−1.36877	−0.684386	−	0.729120i	$-0.739930\pi$
$881$	−40.6274	−1.36877	−0.684386	+	0.729120i	$0.739930\pi$
$882$	0	0
$883$	−11.0294	−0.371170	−0.185585	−	0.982628i	$-0.559418\pi$
$883$	−11.0294	−0.371170	−0.185585	+	0.982628i	$0.559418\pi$
$884$	39.5980	1.33182
$885$	−9.45584	−0.317855
$886$	−9.25483	−0.310922
$887$	−35.4558	−1.19049	−0.595245	−	0.803544i	$-0.702945\pi$
$887$	−35.4558	−1.19049	−0.595245	+	0.803544i	$0.702945\pi$
$888$	10.8284	0.363378
$889$	0	0
$890$	3.91674	0.131289
$891$	1.82843	0.0612546
$892$	−32.5442	−1.08966
$893$	−11.8284	−0.395823
$894$	0.757359	0.0253299
$895$	−11.8579	−0.396365
$896$	0	0
$897$	−10.8284	−0.361551
$898$	−0.887302	−0.0296096
$899$	7.31371	0.243926
$900$	3.02944	0.100981
$901$	15.3137	0.510174
$902$	2.76955	0.0922160
$903$	0	0
$904$	11.3726	0.378246
$905$	−0.627417	−0.0208560
$906$	−5.51472	−0.183214
$907$	−8.34315	−0.277030	−0.138515	−	0.990360i	$-0.544233\pi$
$907$	−8.34315	−0.277030	−0.138515	+	0.990360i	$0.544233\pi$
$908$	−32.2843	−1.07139
$909$	−13.4853	−0.447279
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−3.00000	−0.0993399
$913$	−17.9706	−0.594739
$914$	−1.38478	−0.0458043
$915$	6.11270	0.202080
$916$	−7.94113	−0.262382
$917$	0	0
$918$	3.17157	0.104678
$919$	−54.6569	−1.80296	−0.901482	−	0.432817i	$-0.857519\pi$
$919$	−54.6569	−1.80296	−0.901482	+	0.432817i	$0.857519\pi$
$920$	11.1005	0.365973
$921$	−14.4853	−0.477306
$922$	11.7868	0.388177
$923$	44.2843	1.45763
$924$	0	0
$925$	−11.3137	−0.371992
$926$	−3.87006	−0.127178
$927$	7.65685	0.251484
$928$	3.65685	0.120042
$929$	34.4558	1.13046	0.565230	−	0.824934i	$-0.308788\pi$
$929$	34.4558	1.13046	0.565230	+	0.824934i	$0.308788\pi$
$930$	6.68629	0.219252
$931$	0	0
$932$	−10.9706	−0.359353
$933$	−22.6274	−0.740788
$934$	−3.52691	−0.115404
$935$	25.5980	0.837143
$936$	−4.48528	−0.146606
$937$	27.0000	0.882052	0.441026	−	0.897494i	$-0.354615\pi$
$937$	27.0000	0.882052	0.441026	+	0.897494i	$0.354615\pi$
$938$	0	0
$939$	−25.2843	−0.825121
$940$	−39.5442	−1.28979
$941$	7.94113	0.258873	0.129437	−	0.991588i	$-0.458683\pi$
$941$	7.94113	0.258873	0.129437	+	0.991588i	$0.458683\pi$
$942$	−0.272078	−0.00886478
$943$	14.0000	0.455903
$944$	−15.5147	−0.504961
$945$	0	0
$946$	−1.01724	−0.0330735
$947$	−39.3137	−1.27752	−0.638762	−	0.769404i	$-0.720553\pi$
$947$	−39.3137	−1.27752	−0.638762	+	0.769404i	$0.720553\pi$
$948$	24.9706	0.811006
$949$	−34.8284	−1.13058
$950$	−0.686292	−0.0222662
$951$	9.79899	0.317754
$952$	0	0
$953$	18.6274	0.603401	0.301701	−	0.953403i	$-0.402446\pi$
$953$	18.6274	0.603401	0.301701	+	0.953403i	$0.402446\pi$
$954$	−0.828427	−0.0268213
$955$	−11.2843	−0.365150
$956$	−24.9706	−0.807606
$957$	−1.51472	−0.0489639
$958$	−7.38478	−0.238591
$959$	0	0
$960$	−7.62742	−0.246174
$961$	46.9411	1.51423
$962$	8.00000	0.257930
$963$	2.82843	0.0911448
$964$	−36.2010	−1.16596
$965$	−7.31371	−0.235437
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	12.1421	0.390263
$969$	7.65685	0.245974
$970$	8.46089	0.271663
$971$	0.201010	0.00645072	0.00322536	−	0.999995i	$-0.498973\pi$
$971$	0.201010	0.00645072	0.00322536	+	0.999995i	$0.498973\pi$
$972$	1.82843	0.0586468
$973$	0	0
$974$	−4.82843	−0.154713
$975$	4.68629	0.150081
$976$	10.0294	0.321034
$977$	−39.1127	−1.25133	−0.625663	−	0.780093i	$-0.715171\pi$
$977$	−39.1127	−1.25133	−0.625663	+	0.780093i	$0.715171\pi$
$978$	4.95837	0.158551
$979$	−9.45584	−0.302210
$980$	0	0
$981$	−10.1421	−0.323813
$982$	−10.5563	−0.336867
$983$	47.4558	1.51361	0.756803	−	0.653643i	$-0.226760\pi$
$983$	47.4558	1.51361	0.756803	+	0.653643i	$0.226760\pi$
$984$	5.79899	0.184865
$985$	0.941125	0.0299868
$986$	−2.62742	−0.0836740
$987$	0	0
$988$	−5.17157	−0.164530
$989$	−5.14214	−0.163510
$990$	−1.38478	−0.0440111
$991$	−37.2548	−1.18344	−0.591719	−	0.806144i	$-0.701551\pi$
$991$	−37.2548	−1.18344	−0.591719	+	0.806144i	$0.701551\pi$
$992$	38.9706	1.23732
$993$	22.4853	0.713549
$994$	0	0
$995$	26.7990	0.849585
$996$	−17.9706	−0.569419
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	−5.78680	−0.183178
$999$	−6.82843	−0.216042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.2.a.n.1.2		2
3.2	odd	2		8379.2.a.bh.1.1		2
7.3	odd	6		399.2.j.c.58.1	✓	4
7.5	odd	6		399.2.j.c.172.1	yes	4
7.6	odd	2		2793.2.a.o.1.2		2
21.5	even	6		1197.2.j.d.172.2		4
21.17	even	6		1197.2.j.d.856.2		4
21.20	even	2		8379.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.c.58.1	✓	4	7.3	odd	6
399.2.j.c.172.1	yes	4	7.5	odd	6
1197.2.j.d.172.2		4	21.5	even	6
1197.2.j.d.856.2		4	21.17	even	6
2793.2.a.n.1.2		2	1.1	even	1	trivial
2793.2.a.o.1.2		2	7.6	odd	2
8379.2.a.bh.1.1		2	3.2	odd	2
8379.2.a.bm.1.1		2	21.20	even	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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} -0.757359 q^{10} +1.82843 q^{11} +1.82843 q^{12} +2.82843 q^{13} +1.82843 q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214 q^{18} +1.00000 q^{19} +3.34315 q^{20} +0.757359 q^{22} +3.82843 q^{23} +1.58579 q^{24} -1.65685 q^{25} +1.17157 q^{26} -1.00000 q^{27} +0.828427 q^{29} +0.757359 q^{30} +8.82843 q^{31} +4.41421 q^{32} -1.82843 q^{33} -3.17157 q^{34} -1.82843 q^{36} +6.82843 q^{37} +0.414214 q^{38} -2.82843 q^{39} +2.89949 q^{40} +3.65685 q^{41} -1.34315 q^{43} -3.34315 q^{44} -1.82843 q^{45} +1.58579 q^{46} -11.8284 q^{47} -3.00000 q^{48} -0.686292 q^{50} +7.65685 q^{51} -5.17157 q^{52} -2.00000 q^{53} -0.414214 q^{54} -3.34315 q^{55} -1.00000 q^{57} +0.343146 q^{58} -5.17157 q^{59} -3.34315 q^{60} +3.34315 q^{61} +3.65685 q^{62} -4.17157 q^{64} -5.17157 q^{65} -0.757359 q^{66} -12.0000 q^{67} +14.0000 q^{68} -3.82843 q^{69} +15.6569 q^{71} -1.58579 q^{72} -12.3137 q^{73} +2.82843 q^{74} +1.65685 q^{75} -1.82843 q^{76} -1.17157 q^{78} +13.6569 q^{79} -5.48528 q^{80} +1.00000 q^{81} +1.51472 q^{82} -9.82843 q^{83} +14.0000 q^{85} -0.556349 q^{86} -0.828427 q^{87} -2.89949 q^{88} -5.17157 q^{89} -0.757359 q^{90} -7.00000 q^{92} -8.82843 q^{93} -4.89949 q^{94} -1.82843 q^{95} -4.41421 q^{96} -11.1716 q^{97} +1.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} - 4 q^{17} - 2 q^{18} + 2 q^{19} + 18 q^{20} + 10 q^{22} + 2 q^{23} + 6 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 - 4 * q^17 - 2 * q^18 + 2 * q^19 + 18 * q^20 + 10 * q^22 + 2 * q^23 + 6 * q^24 + 8 * q^25 + 8 * q^26 - 2 * q^27 - 4 * q^29 + 10 * q^30 + 12 * q^31 + 6 * q^32 + 2 * q^33 - 12 * q^34 + 2 * q^36 + 8 * q^37 - 2 * q^38 - 14 * q^40 - 4 * q^41 - 14 * q^43 - 18 * q^44 + 2 * q^45 + 6 * q^46 - 18 * q^47 - 6 * q^48 - 24 * q^50 + 4 * q^51 - 16 * q^52 - 4 * q^53 + 2 * q^54 - 18 * q^55 - 2 * q^57 + 12 * q^58 - 16 * q^59 - 18 * q^60 + 18 * q^61 - 4 * q^62 - 14 * q^64 - 16 * q^65 - 10 * q^66 - 24 * q^67 + 28 * q^68 - 2 * q^69 + 20 * q^71 - 6 * q^72 - 2 * q^73 - 8 * q^75 + 2 * q^76 - 8 * q^78 + 16 * q^79 + 6 * q^80 + 2 * q^81 + 20 * q^82 - 14 * q^83 + 28 * q^85 + 30 * q^86 + 4 * q^87 + 14 * q^88 - 16 * q^89 - 10 * q^90 - 14 * q^92 - 12 * q^93 + 10 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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