LMFDB - Embedded newform 8330.2.a.cv.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,10,-2,10,-10,-2,0,10,12,-10,-2,-2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 $ x^10 - 2x^9 - 19x^8 + 38x^7 + 100x^6 - 194x^5 - 151x^4 + 282x^3 + 85x^2 - 108*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.34515$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.34515 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.34515 q^{6} +1.00000 q^{8} +2.49972 q^{9} -1.00000 q^{10} +3.95825 q^{11} +2.34515 q^{12} +0.841221 q^{13} -2.34515 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.49972 q^{18} -5.96603 q^{19} -1.00000 q^{20} +3.95825 q^{22} +4.78411 q^{23} +2.34515 q^{24} +1.00000 q^{25} +0.841221 q^{26} -1.17324 q^{27} -2.06380 q^{29} -2.34515 q^{30} +4.26909 q^{31} +1.00000 q^{32} +9.28267 q^{33} -1.00000 q^{34} +2.49972 q^{36} +6.03766 q^{37} -5.96603 q^{38} +1.97279 q^{39} -1.00000 q^{40} +6.18647 q^{41} +6.35786 q^{43} +3.95825 q^{44} -2.49972 q^{45} +4.78411 q^{46} +7.70954 q^{47} +2.34515 q^{48} +1.00000 q^{50} -2.34515 q^{51} +0.841221 q^{52} +10.0922 q^{53} -1.17324 q^{54} -3.95825 q^{55} -13.9912 q^{57} -2.06380 q^{58} -4.36748 q^{59} -2.34515 q^{60} -10.8522 q^{61} +4.26909 q^{62} +1.00000 q^{64} -0.841221 q^{65} +9.28267 q^{66} +2.44661 q^{67} -1.00000 q^{68} +11.2194 q^{69} -16.1776 q^{71} +2.49972 q^{72} +10.1451 q^{73} +6.03766 q^{74} +2.34515 q^{75} -5.96603 q^{76} +1.97279 q^{78} +1.91492 q^{79} -1.00000 q^{80} -10.2506 q^{81} +6.18647 q^{82} -5.72752 q^{83} +1.00000 q^{85} +6.35786 q^{86} -4.83992 q^{87} +3.95825 q^{88} +14.0692 q^{89} -2.49972 q^{90} +4.78411 q^{92} +10.0117 q^{93} +7.70954 q^{94} +5.96603 q^{95} +2.34515 q^{96} +1.10161 q^{97} +9.89449 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.34515	1.35397	0.676986	−	0.735996i	$-0.263286\pi$
$3$	2.34515	1.35397	0.676986	+	0.735996i	$0.263286\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.34515	0.957402
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	2.49972	0.833239
$10$	−1.00000	−0.316228
$11$	3.95825	1.19346	0.596728	−	0.802444i	$-0.296467\pi$
$11$	3.95825	1.19346	0.596728	+	0.802444i	$0.296467\pi$
$12$	2.34515	0.676986
$13$	0.841221	0.233313	0.116656	−	0.993172i	$-0.462782\pi$
$13$	0.841221	0.233313	0.116656	+	0.993172i	$0.462782\pi$
$14$	0	0
$15$	−2.34515	−0.605514
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.49972	0.589189
$19$	−5.96603	−1.36870	−0.684350	−	0.729153i	$-0.739914\pi$
$19$	−5.96603	−1.36870	−0.684350	+	0.729153i	$0.739914\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	3.95825	0.843901
$23$	4.78411	0.997555	0.498778	−	0.866730i	$-0.333782\pi$
$23$	4.78411	0.997555	0.498778	+	0.866730i	$0.333782\pi$
$24$	2.34515	0.478701
$25$	1.00000	0.200000
$26$	0.841221	0.164977
$27$	−1.17324	−0.225790
$28$	0	0
$29$	−2.06380	−0.383239	−0.191619	−	0.981469i	$-0.561374\pi$
$29$	−2.06380	−0.383239	−0.191619	+	0.981469i	$0.561374\pi$
$30$	−2.34515	−0.428163
$31$	4.26909	0.766752	0.383376	−	0.923592i	$-0.374761\pi$
$31$	4.26909	0.766752	0.383376	+	0.923592i	$0.374761\pi$
$32$	1.00000	0.176777
$33$	9.28267	1.61591
$34$	−1.00000	−0.171499
$35$	0	0
$36$	2.49972	0.416619
$37$	6.03766	0.992586	0.496293	−	0.868155i	$-0.334694\pi$
$37$	6.03766	0.992586	0.496293	+	0.868155i	$0.334694\pi$
$38$	−5.96603	−0.967818
$39$	1.97279	0.315899
$40$	−1.00000	−0.158114
$41$	6.18647	0.966164	0.483082	−	0.875575i	$-0.339517\pi$
$41$	6.18647	0.966164	0.483082	+	0.875575i	$0.339517\pi$
$42$	0	0
$43$	6.35786	0.969565	0.484783	−	0.874635i	$-0.338899\pi$
$43$	6.35786	0.969565	0.484783	+	0.874635i	$0.338899\pi$
$44$	3.95825	0.596728
$45$	−2.49972	−0.372636
$46$	4.78411	0.705378
$47$	7.70954	1.12455	0.562276	−	0.826949i	$-0.309926\pi$
$47$	7.70954	1.12455	0.562276	+	0.826949i	$0.309926\pi$
$48$	2.34515	0.338493
$49$	0	0
$50$	1.00000	0.141421
$51$	−2.34515	−0.328386
$52$	0.841221	0.116656
$53$	10.0922	1.38628	0.693138	−	0.720805i	$-0.256228\pi$
$53$	10.0922	1.38628	0.693138	+	0.720805i	$0.256228\pi$
$54$	−1.17324	−0.159658
$55$	−3.95825	−0.533730
$56$	0	0
$57$	−13.9912	−1.85318
$58$	−2.06380	−0.270991
$59$	−4.36748	−0.568597	−0.284299	−	0.958736i	$-0.591761\pi$
$59$	−4.36748	−0.568597	−0.284299	+	0.958736i	$0.591761\pi$
$60$	−2.34515	−0.302757
$61$	−10.8522	−1.38948	−0.694739	−	0.719262i	$-0.744480\pi$
$61$	−10.8522	−1.38948	−0.694739	+	0.719262i	$0.744480\pi$
$62$	4.26909	0.542175
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.841221	−0.104341
$66$	9.28267	1.14262
$67$	2.44661	0.298901	0.149451	−	0.988769i	$-0.452250\pi$
$67$	2.44661	0.298901	0.149451	+	0.988769i	$0.452250\pi$
$68$	−1.00000	−0.121268
$69$	11.2194	1.35066
$70$	0	0
$71$	−16.1776	−1.91992	−0.959962	−	0.280130i	$-0.909622\pi$
$71$	−16.1776	−1.91992	−0.959962	+	0.280130i	$0.909622\pi$
$72$	2.49972	0.294594
$73$	10.1451	1.18739	0.593697	−	0.804689i	$-0.297668\pi$
$73$	10.1451	1.18739	0.593697	+	0.804689i	$0.297668\pi$
$74$	6.03766	0.701864
$75$	2.34515	0.270794
$76$	−5.96603	−0.684350
$77$	0	0
$78$	1.97279	0.223374
$79$	1.91492	0.215445	0.107723	−	0.994181i	$-0.465644\pi$
$79$	1.91492	0.215445	0.107723	+	0.994181i	$0.465644\pi$
$80$	−1.00000	−0.111803
$81$	−10.2506	−1.13895
$82$	6.18647	0.683181
$83$	−5.72752	−0.628677	−0.314338	−	0.949311i	$-0.601783\pi$
$83$	−5.72752	−0.628677	−0.314338	+	0.949311i	$0.601783\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	6.35786	0.685586
$87$	−4.83992	−0.518894
$88$	3.95825	0.421950
$89$	14.0692	1.49134	0.745668	−	0.666317i	$-0.232130\pi$
$89$	14.0692	1.49134	0.745668	+	0.666317i	$0.232130\pi$
$90$	−2.49972	−0.263493
$91$	0	0
$92$	4.78411	0.498778
$93$	10.0117	1.03816
$94$	7.70954	0.795179
$95$	5.96603	0.612102
$96$	2.34515	0.239351
$97$	1.10161	0.111852	0.0559258	−	0.998435i	$-0.482189\pi$
$97$	1.10161	0.111852	0.0559258	+	0.998435i	$0.482189\pi$
$98$	0	0
$99$	9.89449	0.994434
$100$	1.00000	0.100000
$101$	−8.17811	−0.813752	−0.406876	−	0.913483i	$-0.633382\pi$
$101$	−8.17811	−0.813752	−0.406876	+	0.913483i	$0.633382\pi$
$102$	−2.34515	−0.232204
$103$	12.8886	1.26995	0.634976	−	0.772531i	$-0.281010\pi$
$103$	12.8886	1.26995	0.634976	+	0.772531i	$0.281010\pi$
$104$	0.841221	0.0824885
$105$	0	0
$106$	10.0922	0.980244
$107$	6.14230	0.593798	0.296899	−	0.954909i	$-0.404047\pi$
$107$	6.14230	0.593798	0.296899	+	0.954909i	$0.404047\pi$
$108$	−1.17324	−0.112895
$109$	0.718456	0.0688156	0.0344078	−	0.999408i	$-0.489045\pi$
$109$	0.718456	0.0688156	0.0344078	+	0.999408i	$0.489045\pi$
$110$	−3.95825	−0.377404
$111$	14.1592	1.34393
$112$	0	0
$113$	15.9603	1.50142	0.750710	−	0.660632i	$-0.229712\pi$
$113$	15.9603	1.50142	0.750710	+	0.660632i	$0.229712\pi$
$114$	−13.9912	−1.31040
$115$	−4.78411	−0.446120
$116$	−2.06380	−0.191619
$117$	2.10281	0.194405
$118$	−4.36748	−0.402059
$119$	0	0
$120$	−2.34515	−0.214082
$121$	4.66771	0.424338
$122$	−10.8522	−0.982509
$123$	14.5082	1.30816
$124$	4.26909	0.383376
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.04237	−0.358702	−0.179351	−	0.983785i	$-0.557400\pi$
$127$	−4.04237	−0.358702	−0.179351	+	0.983785i	$0.557400\pi$
$128$	1.00000	0.0883883
$129$	14.9101	1.31276
$130$	−0.841221	−0.0737799
$131$	11.5774	1.01152	0.505762	−	0.862673i	$-0.331212\pi$
$131$	11.5774	1.01152	0.505762	+	0.862673i	$0.331212\pi$
$132$	9.28267	0.807953
$133$	0	0
$134$	2.44661	0.211355
$135$	1.17324	0.100976
$136$	−1.00000	−0.0857493
$137$	1.69633	0.144927	0.0724634	−	0.997371i	$-0.476914\pi$
$137$	1.69633	0.144927	0.0724634	+	0.997371i	$0.476914\pi$
$138$	11.2194	0.955062
$139$	−13.1117	−1.11212	−0.556061	−	0.831141i	$-0.687688\pi$
$139$	−13.1117	−1.11212	−0.556061	+	0.831141i	$0.687688\pi$
$140$	0	0
$141$	18.0800	1.52261
$142$	−16.1776	−1.35759
$143$	3.32976	0.278448
$144$	2.49972	0.208310
$145$	2.06380	0.171390
$146$	10.1451	0.839614
$147$	0	0
$148$	6.03766	0.496293
$149$	−7.05512	−0.577978	−0.288989	−	0.957332i	$-0.593319\pi$
$149$	−7.05512	−0.577978	−0.288989	+	0.957332i	$0.593319\pi$
$150$	2.34515	0.191480
$151$	21.4470	1.74533	0.872667	−	0.488316i	$-0.162389\pi$
$151$	21.4470	1.74533	0.872667	+	0.488316i	$0.162389\pi$
$152$	−5.96603	−0.483909
$153$	−2.49972	−0.202090
$154$	0	0
$155$	−4.26909	−0.342902
$156$	1.97279	0.157949
$157$	−21.8380	−1.74286	−0.871431	−	0.490518i	$-0.836808\pi$
$157$	−21.8380	−1.74286	−0.871431	+	0.490518i	$0.836808\pi$
$158$	1.91492	0.152343
$159$	23.6678	1.87698
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−10.2506	−0.805361
$163$	−9.80190	−0.767744	−0.383872	−	0.923386i	$-0.625410\pi$
$163$	−9.80190	−0.767744	−0.383872	+	0.923386i	$0.625410\pi$
$164$	6.18647	0.483082
$165$	−9.28267	−0.722655
$166$	−5.72752	−0.444541
$167$	−3.49467	−0.270426	−0.135213	−	0.990817i	$-0.543172\pi$
$167$	−3.49467	−0.270426	−0.135213	+	0.990817i	$0.543172\pi$
$168$	0	0
$169$	−12.2923	−0.945565
$170$	1.00000	0.0766965
$171$	−14.9134	−1.14045
$172$	6.35786	0.484783
$173$	6.28337	0.477716	0.238858	−	0.971055i	$-0.423227\pi$
$173$	6.28337	0.477716	0.238858	+	0.971055i	$0.423227\pi$
$174$	−4.83992	−0.366914
$175$	0	0
$176$	3.95825	0.298364
$177$	−10.2424	−0.769865
$178$	14.0692	1.05453
$179$	3.76919	0.281722	0.140861	−	0.990029i	$-0.455013\pi$
$179$	3.76919	0.281722	0.140861	+	0.990029i	$0.455013\pi$
$180$	−2.49972	−0.186318
$181$	0.786659	0.0584719	0.0292359	−	0.999573i	$-0.490693\pi$
$181$	0.786659	0.0584719	0.0292359	+	0.999573i	$0.490693\pi$
$182$	0	0
$183$	−25.4499	−1.88131
$184$	4.78411	0.352689
$185$	−6.03766	−0.443898
$186$	10.0117	0.734090
$187$	−3.95825	−0.289456
$188$	7.70954	0.562276
$189$	0	0
$190$	5.96603	0.432821
$191$	−24.8380	−1.79721	−0.898606	−	0.438755i	$-0.855419\pi$
$191$	−24.8380	−1.79721	−0.898606	+	0.438755i	$0.855419\pi$
$192$	2.34515	0.169246
$193$	19.5742	1.40898	0.704489	−	0.709715i	$-0.251176\pi$
$193$	19.5742	1.40898	0.704489	+	0.709715i	$0.251176\pi$
$194$	1.10161	0.0790911
$195$	−1.97279	−0.141274
$196$	0	0
$197$	1.70467	0.121452	0.0607262	−	0.998154i	$-0.480658\pi$
$197$	1.70467	0.121452	0.0607262	+	0.998154i	$0.480658\pi$
$198$	9.89449	0.703171
$199$	8.43343	0.597830	0.298915	−	0.954280i	$-0.403375\pi$
$199$	8.43343	0.597830	0.298915	+	0.954280i	$0.403375\pi$
$200$	1.00000	0.0707107
$201$	5.73766	0.404704
$202$	−8.17811	−0.575409
$203$	0	0
$204$	−2.34515	−0.164193
$205$	−6.18647	−0.432082
$206$	12.8886	0.897992
$207$	11.9589	0.831201
$208$	0.841221	0.0583282
$209$	−23.6150	−1.63348
$210$	0	0
$211$	−21.5843	−1.48593	−0.742963	−	0.669333i	$-0.766580\pi$
$211$	−21.5843	−1.48593	−0.742963	+	0.669333i	$0.766580\pi$
$212$	10.0922	0.693138
$213$	−37.9388	−2.59952
$214$	6.14230	0.419879
$215$	−6.35786	−0.433603
$216$	−1.17324	−0.0798289
$217$	0	0
$218$	0.718456	0.0486600
$219$	23.7917	1.60770
$220$	−3.95825	−0.266865
$221$	−0.841221	−0.0565866
$222$	14.1592	0.950304
$223$	2.67003	0.178798	0.0893992	−	0.995996i	$-0.471505\pi$
$223$	2.67003	0.178798	0.0893992	+	0.995996i	$0.471505\pi$
$224$	0	0
$225$	2.49972	0.166648
$226$	15.9603	1.06166
$227$	−0.283479	−0.0188152	−0.00940759	−	0.999956i	$-0.502995\pi$
$227$	−0.283479	−0.0188152	−0.00940759	+	0.999956i	$0.502995\pi$
$228$	−13.9912	−0.926591
$229$	7.18681	0.474918	0.237459	−	0.971398i	$-0.423685\pi$
$229$	7.18681	0.474918	0.237459	+	0.971398i	$0.423685\pi$
$230$	−4.78411	−0.315455
$231$	0	0
$232$	−2.06380	−0.135495
$233$	−1.45602	−0.0953872	−0.0476936	−	0.998862i	$-0.515187\pi$
$233$	−1.45602	−0.0953872	−0.0476936	+	0.998862i	$0.515187\pi$
$234$	2.10281	0.137465
$235$	−7.70954	−0.502915
$236$	−4.36748	−0.284299
$237$	4.49076	0.291707
$238$	0	0
$239$	−0.187743	−0.0121441	−0.00607205	−	0.999982i	$-0.501933\pi$
$239$	−0.187743	−0.0121441	−0.00607205	+	0.999982i	$0.501933\pi$
$240$	−2.34515	−0.151379
$241$	−13.6144	−0.876978	−0.438489	−	0.898737i	$-0.644486\pi$
$241$	−13.6144	−0.876978	−0.438489	+	0.898737i	$0.644486\pi$
$242$	4.66771	0.300052
$243$	−20.5194	−1.31632
$244$	−10.8522	−0.694739
$245$	0	0
$246$	14.5082	0.925007
$247$	−5.01875	−0.319335
$248$	4.26909	0.271088
$249$	−13.4319	−0.851210
$250$	−1.00000	−0.0632456
$251$	−14.4128	−0.909726	−0.454863	−	0.890561i	$-0.650312\pi$
$251$	−14.4128	−0.909726	−0.454863	+	0.890561i	$0.650312\pi$
$252$	0	0
$253$	18.9367	1.19054
$254$	−4.04237	−0.253641
$255$	2.34515	0.146859
$256$	1.00000	0.0625000
$257$	3.67957	0.229525	0.114763	−	0.993393i	$-0.463389\pi$
$257$	3.67957	0.229525	0.114763	+	0.993393i	$0.463389\pi$
$258$	14.9101	0.928264
$259$	0	0
$260$	−0.841221	−0.0521703
$261$	−5.15892	−0.319329
$262$	11.5774	0.715255
$263$	26.7164	1.64740	0.823702	−	0.567023i	$-0.191905\pi$
$263$	26.7164	1.64740	0.823702	+	0.567023i	$0.191905\pi$
$264$	9.28267	0.571309
$265$	−10.0922	−0.619961
$266$	0	0
$267$	32.9944	2.01923
$268$	2.44661	0.149451
$269$	−15.6583	−0.954702	−0.477351	−	0.878713i	$-0.658403\pi$
$269$	−15.6583	−0.954702	−0.477351	+	0.878713i	$0.658403\pi$
$270$	1.17324	0.0714011
$271$	−4.22991	−0.256949	−0.128474	−	0.991713i	$-0.541008\pi$
$271$	−4.22991	−0.256949	−0.128474	+	0.991713i	$0.541008\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	1.69633	0.102479
$275$	3.95825	0.238691
$276$	11.2194	0.675331
$277$	−0.167793	−0.0100817	−0.00504086	−	0.999987i	$-0.501605\pi$
$277$	−0.167793	−0.0100817	−0.00504086	+	0.999987i	$0.501605\pi$
$278$	−13.1117	−0.786390
$279$	10.6715	0.638887
$280$	0	0
$281$	15.9964	0.954268	0.477134	−	0.878831i	$-0.341676\pi$
$281$	15.9964	0.954268	0.477134	+	0.878831i	$0.341676\pi$
$282$	18.0800	1.07665
$283$	−30.2665	−1.79916	−0.899578	−	0.436761i	$-0.856126\pi$
$283$	−30.2665	−1.79916	−0.899578	+	0.436761i	$0.856126\pi$
$284$	−16.1776	−0.959962
$285$	13.9912	0.828768
$286$	3.32976	0.196893
$287$	0	0
$288$	2.49972	0.147297
$289$	1.00000	0.0588235
$290$	2.06380	0.121191
$291$	2.58344	0.151444
$292$	10.1451	0.593697
$293$	−5.53049	−0.323095	−0.161547	−	0.986865i	$-0.551648\pi$
$293$	−5.53049	−0.323095	−0.161547	+	0.986865i	$0.551648\pi$
$294$	0	0
$295$	4.36748	0.254284
$296$	6.03766	0.350932
$297$	−4.64397	−0.269471
$298$	−7.05512	−0.408692
$299$	4.02449	0.232742
$300$	2.34515	0.135397
$301$	0	0
$302$	21.4470	1.23414
$303$	−19.1789	−1.10180
$304$	−5.96603	−0.342175
$305$	10.8522	0.621393
$306$	−2.49972	−0.142899
$307$	18.9207	1.07986	0.539932	−	0.841709i	$-0.318450\pi$
$307$	18.9207	1.07986	0.539932	+	0.841709i	$0.318450\pi$
$308$	0	0
$309$	30.2257	1.71948
$310$	−4.26909	−0.242468
$311$	16.6685	0.945183	0.472591	−	0.881282i	$-0.343319\pi$
$311$	16.6685	0.945183	0.472591	+	0.881282i	$0.343319\pi$
$312$	1.97279	0.111687
$313$	−23.3668	−1.32077	−0.660384	−	0.750928i	$-0.729606\pi$
$313$	−23.3668	−1.32077	−0.660384	+	0.750928i	$0.729606\pi$
$314$	−21.8380	−1.23239
$315$	0	0
$316$	1.91492	0.107723
$317$	18.7350	1.05226	0.526132	−	0.850403i	$-0.323642\pi$
$317$	18.7350	1.05226	0.526132	+	0.850403i	$0.323642\pi$
$318$	23.6678	1.32722
$319$	−8.16905	−0.457379
$320$	−1.00000	−0.0559017
$321$	14.4046	0.803986
$322$	0	0
$323$	5.96603	0.331959
$324$	−10.2506	−0.569476
$325$	0.841221	0.0466625
$326$	−9.80190	−0.542877
$327$	1.68488	0.0931743
$328$	6.18647	0.341590
$329$	0	0
$330$	−9.28267	−0.510994
$331$	7.37951	0.405615	0.202807	−	0.979219i	$-0.434993\pi$
$331$	7.37951	0.405615	0.202807	+	0.979219i	$0.434993\pi$
$332$	−5.72752	−0.314338
$333$	15.0924	0.827061
$334$	−3.49467	−0.191220
$335$	−2.44661	−0.133673
$336$	0	0
$337$	−22.6963	−1.23635	−0.618173	−	0.786042i	$-0.712127\pi$
$337$	−22.6963	−1.23635	−0.618173	+	0.786042i	$0.712127\pi$
$338$	−12.2923	−0.668616
$339$	37.4293	2.03288
$340$	1.00000	0.0542326
$341$	16.8981	0.915085
$342$	−14.9134	−0.806423
$343$	0	0
$344$	6.35786	0.342793
$345$	−11.2194	−0.604034
$346$	6.28337	0.337796
$347$	−24.0599	−1.29160	−0.645801	−	0.763506i	$-0.723476\pi$
$347$	−24.0599	−1.29160	−0.645801	+	0.763506i	$0.723476\pi$
$348$	−4.83992	−0.259447
$349$	−0.252751	−0.0135294	−0.00676472	−	0.999977i	$-0.502153\pi$
$349$	−0.252751	−0.0135294	−0.00676472	+	0.999977i	$0.502153\pi$
$350$	0	0
$351$	−0.986954	−0.0526797
$352$	3.95825	0.210975
$353$	−31.6691	−1.68557	−0.842787	−	0.538247i	$-0.819087\pi$
$353$	−31.6691	−1.68557	−0.842787	+	0.538247i	$0.819087\pi$
$354$	−10.2424	−0.544376
$355$	16.1776	0.858616
$356$	14.0692	0.745668
$357$	0	0
$358$	3.76919	0.199208
$359$	25.7300	1.35798	0.678990	−	0.734148i	$-0.262418\pi$
$359$	25.7300	1.35798	0.678990	+	0.734148i	$0.262418\pi$
$360$	−2.49972	−0.131747
$361$	16.5935	0.873342
$362$	0.786659	0.0413459
$363$	10.9465	0.574541
$364$	0	0
$365$	−10.1451	−0.531018
$366$	−25.4499	−1.33029
$367$	−24.9879	−1.30436	−0.652180	−	0.758064i	$-0.726145\pi$
$367$	−24.9879	−1.30436	−0.652180	+	0.758064i	$0.726145\pi$
$368$	4.78411	0.249389
$369$	15.4644	0.805045
$370$	−6.03766	−0.313883
$371$	0	0
$372$	10.0117	0.519080
$373$	−0.702850	−0.0363922	−0.0181961	−	0.999834i	$-0.505792\pi$
$373$	−0.702850	−0.0363922	−0.0181961	+	0.999834i	$0.505792\pi$
$374$	−3.95825	−0.204676
$375$	−2.34515	−0.121103
$376$	7.70954	0.397589
$377$	−1.73611	−0.0894145
$378$	0	0
$379$	−33.3675	−1.71397	−0.856987	−	0.515338i	$-0.827666\pi$
$379$	−33.3675	−1.71397	−0.856987	+	0.515338i	$0.827666\pi$
$380$	5.96603	0.306051
$381$	−9.47995	−0.485673
$382$	−24.8380	−1.27082
$383$	−26.1379	−1.33558	−0.667791	−	0.744349i	$-0.732760\pi$
$383$	−26.1379	−1.33558	−0.667791	+	0.744349i	$0.732760\pi$
$384$	2.34515	0.119675
$385$	0	0
$386$	19.5742	0.996298
$387$	15.8929	0.807879
$388$	1.10161	0.0559258
$389$	−28.2167	−1.43064	−0.715322	−	0.698794i	$-0.753720\pi$
$389$	−28.2167	−1.43064	−0.715322	+	0.698794i	$0.753720\pi$
$390$	−1.97279	−0.0998959
$391$	−4.78411	−0.241943
$392$	0	0
$393$	27.1507	1.36957
$394$	1.70467	0.0858799
$395$	−1.91492	−0.0963500
$396$	9.89449	0.497217
$397$	−5.44020	−0.273036	−0.136518	−	0.990638i	$-0.543591\pi$
$397$	−5.44020	−0.273036	−0.136518	+	0.990638i	$0.543591\pi$
$398$	8.43343	0.422730
$399$	0	0
$400$	1.00000	0.0500000
$401$	−4.32597	−0.216029	−0.108014	−	0.994149i	$-0.534449\pi$
$401$	−4.32597	−0.216029	−0.108014	+	0.994149i	$0.534449\pi$
$402$	5.73766	0.286169
$403$	3.59125	0.178893
$404$	−8.17811	−0.406876
$405$	10.2506	0.509355
$406$	0	0
$407$	23.8986	1.18461
$408$	−2.34515	−0.116102
$409$	25.8813	1.27975	0.639873	−	0.768481i	$-0.278987\pi$
$409$	25.8813	1.27975	0.639873	+	0.768481i	$0.278987\pi$
$410$	−6.18647	−0.305528
$411$	3.97813	0.196227
$412$	12.8886	0.634976
$413$	0	0
$414$	11.9589	0.587748
$415$	5.72752	0.281153
$416$	0.841221	0.0412442
$417$	−30.7489	−1.50578
$418$	−23.6150	−1.15505
$419$	1.23007	0.0600927	0.0300464	−	0.999549i	$-0.490435\pi$
$419$	1.23007	0.0600927	0.0300464	+	0.999549i	$0.490435\pi$
$420$	0	0
$421$	18.7083	0.911788	0.455894	−	0.890034i	$-0.349320\pi$
$421$	18.7083	0.911788	0.455894	+	0.890034i	$0.349320\pi$
$422$	−21.5843	−1.05071
$423$	19.2717	0.937021
$424$	10.0922	0.490122
$425$	−1.00000	−0.0485071
$426$	−37.9388	−1.83814
$427$	0	0
$428$	6.14230	0.296899
$429$	7.80877	0.377011
$430$	−6.35786	−0.306603
$431$	9.70967	0.467698	0.233849	−	0.972273i	$-0.424868\pi$
$431$	9.70967	0.467698	0.233849	+	0.972273i	$0.424868\pi$
$432$	−1.17324	−0.0564476
$433$	−8.57171	−0.411930	−0.205965	−	0.978559i	$-0.566033\pi$
$433$	−8.57171	−0.411930	−0.205965	+	0.978559i	$0.566033\pi$
$434$	0	0
$435$	4.83992	0.232057
$436$	0.718456	0.0344078
$437$	−28.5421	−1.36535
$438$	23.7917	1.13681
$439$	26.5905	1.26909	0.634547	−	0.772884i	$-0.281187\pi$
$439$	26.5905	1.26909	0.634547	+	0.772884i	$0.281187\pi$
$440$	−3.95825	−0.188702
$441$	0	0
$442$	−0.841221	−0.0400128
$443$	1.02060	0.0484901	0.0242451	−	0.999706i	$-0.492282\pi$
$443$	1.02060	0.0484901	0.0242451	+	0.999706i	$0.492282\pi$
$444$	14.1592	0.671966
$445$	−14.0692	−0.666946
$446$	2.67003	0.126430
$447$	−16.5453	−0.782566
$448$	0	0
$449$	4.39147	0.207246	0.103623	−	0.994617i	$-0.466956\pi$
$449$	4.39147	0.207246	0.103623	+	0.994617i	$0.466956\pi$
$450$	2.49972	0.117838
$451$	24.4876	1.15307
$452$	15.9603	0.750710
$453$	50.2964	2.36313
$454$	−0.283479	−0.0133043
$455$	0	0
$456$	−13.9912	−0.655199
$457$	28.3785	1.32749	0.663744	−	0.747960i	$-0.268966\pi$
$457$	28.3785	1.32749	0.663744	+	0.747960i	$0.268966\pi$
$458$	7.18681	0.335818
$459$	1.17324	0.0547622
$460$	−4.78411	−0.223060
$461$	6.71669	0.312828	0.156414	−	0.987692i	$-0.450007\pi$
$461$	6.71669	0.312828	0.156414	+	0.987692i	$0.450007\pi$
$462$	0	0
$463$	7.53393	0.350131	0.175066	−	0.984557i	$-0.443986\pi$
$463$	7.53393	0.350131	0.175066	+	0.984557i	$0.443986\pi$
$464$	−2.06380	−0.0958097
$465$	−10.0117	−0.464279
$466$	−1.45602	−0.0674489
$467$	−9.39180	−0.434601	−0.217300	−	0.976105i	$-0.569725\pi$
$467$	−9.39180	−0.434601	−0.217300	+	0.976105i	$0.569725\pi$
$468$	2.10281	0.0972025
$469$	0	0
$470$	−7.70954	−0.355615
$471$	−51.2133	−2.35979
$472$	−4.36748	−0.201030
$473$	25.1660	1.15713
$474$	4.49076	0.206268
$475$	−5.96603	−0.273740
$476$	0	0
$477$	25.2277	1.15510
$478$	−0.187743	−0.00858718
$479$	−3.94238	−0.180132	−0.0900659	−	0.995936i	$-0.528708\pi$
$479$	−3.94238	−0.180132	−0.0900659	+	0.995936i	$0.528708\pi$
$480$	−2.34515	−0.107041
$481$	5.07901	0.231583
$482$	−13.6144	−0.620117
$483$	0	0
$484$	4.66771	0.212169
$485$	−1.10161	−0.0500216
$486$	−20.5194	−0.930778
$487$	−38.9841	−1.76654	−0.883269	−	0.468867i	$-0.844662\pi$
$487$	−38.9841	−1.76654	−0.883269	+	0.468867i	$0.844662\pi$
$488$	−10.8522	−0.491255
$489$	−22.9869	−1.03950
$490$	0	0
$491$	30.9826	1.39823	0.699114	−	0.715011i	$-0.253578\pi$
$491$	30.9826	1.39823	0.699114	+	0.715011i	$0.253578\pi$
$492$	14.5082	0.654079
$493$	2.06380	0.0929491
$494$	−5.01875	−0.225804
$495$	−9.89449	−0.444724
$496$	4.26909	0.191688
$497$	0	0
$498$	−13.4319	−0.601896
$499$	−11.7545	−0.526205	−0.263102	−	0.964768i	$-0.584746\pi$
$499$	−11.7545	−0.526205	−0.263102	+	0.964768i	$0.584746\pi$
$500$	−1.00000	−0.0447214
$501$	−8.19552	−0.366149
$502$	−14.4128	−0.643274
$503$	26.2755	1.17157	0.585783	−	0.810468i	$-0.300787\pi$
$503$	26.2755	1.17157	0.585783	+	0.810468i	$0.300787\pi$
$504$	0	0
$505$	8.17811	0.363921
$506$	18.9367	0.841838
$507$	−28.8274	−1.28027
$508$	−4.04237	−0.179351
$509$	−36.1557	−1.60257	−0.801287	−	0.598280i	$-0.795851\pi$
$509$	−36.1557	−1.60257	−0.801287	+	0.598280i	$0.795851\pi$
$510$	2.34515	0.103845
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.99959	0.309039
$514$	3.67957	0.162299
$515$	−12.8886	−0.567940
$516$	14.9101	0.656382
$517$	30.5163	1.34210
$518$	0	0
$519$	14.7354	0.646813
$520$	−0.841221	−0.0368900
$521$	−17.9950	−0.788376	−0.394188	−	0.919030i	$-0.628974\pi$
$521$	−17.9950	−0.788376	−0.394188	+	0.919030i	$0.628974\pi$
$522$	−5.15892	−0.225800
$523$	12.4407	0.543995	0.271997	−	0.962298i	$-0.412316\pi$
$523$	12.4407	0.543995	0.271997	+	0.962298i	$0.412316\pi$
$524$	11.5774	0.505762
$525$	0	0
$526$	26.7164	1.16489
$527$	−4.26909	−0.185965
$528$	9.28267	0.403976
$529$	−0.112322	−0.00488358
$530$	−10.0922	−0.438379
$531$	−10.9175	−0.473777
$532$	0	0
$533$	5.20418	0.225418
$534$	32.9944	1.42781
$535$	−6.14230	−0.265555
$536$	2.44661	0.105678
$537$	8.83930	0.381444
$538$	−15.6583	−0.675076
$539$	0	0
$540$	1.17324	0.0504882
$541$	14.2241	0.611540	0.305770	−	0.952105i	$-0.401086\pi$
$541$	14.2241	0.611540	0.305770	+	0.952105i	$0.401086\pi$
$542$	−4.22991	−0.181690
$543$	1.84483	0.0791693
$544$	−1.00000	−0.0428746
$545$	−0.718456	−0.0307753
$546$	0	0
$547$	5.56031	0.237742	0.118871	−	0.992910i	$-0.462073\pi$
$547$	5.56031	0.237742	0.118871	+	0.992910i	$0.462073\pi$
$548$	1.69633	0.0724634
$549$	−27.1273	−1.15777
$550$	3.95825	0.168780
$551$	12.3127	0.524539
$552$	11.2194	0.477531
$553$	0	0
$554$	−0.167793	−0.00712885
$555$	−14.1592	−0.601025
$556$	−13.1117	−0.556061
$557$	−27.2329	−1.15389	−0.576947	−	0.816782i	$-0.695756\pi$
$557$	−27.2329	−1.15389	−0.576947	+	0.816782i	$0.695756\pi$
$558$	10.6715	0.451761
$559$	5.34837	0.226212
$560$	0	0
$561$	−9.28267	−0.391915
$562$	15.9964	0.674769
$563$	−20.9554	−0.883165	−0.441582	−	0.897221i	$-0.645583\pi$
$563$	−20.9554	−0.883165	−0.441582	+	0.897221i	$0.645583\pi$
$564$	18.0800	0.761306
$565$	−15.9603	−0.671455
$566$	−30.2665	−1.27219
$567$	0	0
$568$	−16.1776	−0.678796
$569$	26.0694	1.09288	0.546442	−	0.837497i	$-0.315982\pi$
$569$	26.0694	1.09288	0.546442	+	0.837497i	$0.315982\pi$
$570$	13.9912	0.586028
$571$	21.4688	0.898441	0.449220	−	0.893421i	$-0.351702\pi$
$571$	21.4688	0.898441	0.449220	+	0.893421i	$0.351702\pi$
$572$	3.32976	0.139224
$573$	−58.2487	−2.43337
$574$	0	0
$575$	4.78411	0.199511
$576$	2.49972	0.104155
$577$	−43.1450	−1.79615	−0.898075	−	0.439843i	$-0.855034\pi$
$577$	−43.1450	−1.79615	−0.898075	+	0.439843i	$0.855034\pi$
$578$	1.00000	0.0415945
$579$	45.9043	1.90772
$580$	2.06380	0.0856948
$581$	0	0
$582$	2.58344	0.107087
$583$	39.9475	1.65446
$584$	10.1451	0.419807
$585$	−2.10281	−0.0869406
$586$	−5.53049	−0.228463
$587$	26.2900	1.08511	0.542553	−	0.840022i	$-0.317458\pi$
$587$	26.2900	1.08511	0.542553	+	0.840022i	$0.317458\pi$
$588$	0	0
$589$	−25.4695	−1.04945
$590$	4.36748	0.179806
$591$	3.99769	0.164443
$592$	6.03766	0.248146
$593$	40.7431	1.67312	0.836559	−	0.547876i	$-0.184564\pi$
$593$	40.7431	1.67312	0.836559	+	0.547876i	$0.184564\pi$
$594$	−4.64397	−0.190545
$595$	0	0
$596$	−7.05512	−0.288989
$597$	19.7776	0.809445
$598$	4.02449	0.164574
$599$	−4.32455	−0.176696	−0.0883481	−	0.996090i	$-0.528159\pi$
$599$	−4.32455	−0.176696	−0.0883481	+	0.996090i	$0.528159\pi$
$600$	2.34515	0.0957402
$601$	−5.33141	−0.217473	−0.108736	−	0.994071i	$-0.534680\pi$
$601$	−5.33141	−0.217473	−0.108736	+	0.994071i	$0.534680\pi$
$602$	0	0
$603$	6.11583	0.249056
$604$	21.4470	0.872667
$605$	−4.66771	−0.189770
$606$	−19.1789	−0.779088
$607$	15.7578	0.639590	0.319795	−	0.947487i	$-0.396386\pi$
$607$	15.7578	0.639590	0.319795	+	0.947487i	$0.396386\pi$
$608$	−5.96603	−0.241954
$609$	0	0
$610$	10.8522	0.439391
$611$	6.48543	0.262372
$612$	−2.49972	−0.101045
$613$	21.5993	0.872388	0.436194	−	0.899853i	$-0.356326\pi$
$613$	21.5993	0.872388	0.436194	+	0.899853i	$0.356326\pi$
$614$	18.9207	0.763579
$615$	−14.5082	−0.585026
$616$	0	0
$617$	−18.6327	−0.750125	−0.375063	−	0.927000i	$-0.622379\pi$
$617$	−18.6327	−0.750125	−0.375063	+	0.927000i	$0.622379\pi$
$618$	30.2257	1.21586
$619$	−2.87958	−0.115740	−0.0578700	−	0.998324i	$-0.518431\pi$
$619$	−2.87958	−0.115740	−0.0578700	+	0.998324i	$0.518431\pi$
$620$	−4.26909	−0.171451
$621$	−5.61291	−0.225238
$622$	16.6685	0.668345
$623$	0	0
$624$	1.97279	0.0789747
$625$	1.00000	0.0400000
$626$	−23.3668	−0.933923
$627$	−55.3807	−2.21169
$628$	−21.8380	−0.871431
$629$	−6.03766	−0.240737
$630$	0	0
$631$	36.1097	1.43750	0.718752	−	0.695267i	$-0.244714\pi$
$631$	36.1097	1.43750	0.718752	+	0.695267i	$0.244714\pi$
$632$	1.91492	0.0761713
$633$	−50.6184	−2.01190
$634$	18.7350	0.744062
$635$	4.04237	0.160417
$636$	23.6678	0.938488
$637$	0	0
$638$	−8.16905	−0.323416
$639$	−40.4393	−1.59975
$640$	−1.00000	−0.0395285
$641$	18.6963	0.738460	0.369230	−	0.929338i	$-0.379621\pi$
$641$	18.6963	0.738460	0.369230	+	0.929338i	$0.379621\pi$
$642$	14.4046	0.568504
$643$	4.20180	0.165703	0.0828514	−	0.996562i	$-0.473597\pi$
$643$	4.20180	0.165703	0.0828514	+	0.996562i	$0.473597\pi$
$644$	0	0
$645$	−14.9101	−0.587086
$646$	5.96603	0.234730
$647$	1.65210	0.0649508	0.0324754	−	0.999473i	$-0.489661\pi$
$647$	1.65210	0.0649508	0.0324754	+	0.999473i	$0.489661\pi$
$648$	−10.2506	−0.402680
$649$	−17.2876	−0.678596
$650$	0.841221	0.0329954
$651$	0	0
$652$	−9.80190	−0.383872
$653$	44.5448	1.74317	0.871586	−	0.490242i	$-0.163092\pi$
$653$	44.5448	1.74317	0.871586	+	0.490242i	$0.163092\pi$
$654$	1.68488	0.0658842
$655$	−11.5774	−0.452367
$656$	6.18647	0.241541
$657$	25.3598	0.989382
$658$	0	0
$659$	−37.6758	−1.46764	−0.733820	−	0.679343i	$-0.762265\pi$
$659$	−37.6758	−1.46764	−0.733820	+	0.679343i	$0.762265\pi$
$660$	−9.28267	−0.361327
$661$	−45.6855	−1.77696	−0.888479	−	0.458917i	$-0.848238\pi$
$661$	−45.6855	−1.77696	−0.888479	+	0.458917i	$0.848238\pi$
$662$	7.37951	0.286813
$663$	−1.97279	−0.0766167
$664$	−5.72752	−0.222271
$665$	0	0
$666$	15.0924	0.584820
$667$	−9.87346	−0.382302
$668$	−3.49467	−0.135213
$669$	6.26161	0.242088
$670$	−2.44661	−0.0945208
$671$	−42.9556	−1.65828
$672$	0	0
$673$	−22.4151	−0.864038	−0.432019	−	0.901865i	$-0.642199\pi$
$673$	−22.4151	−0.864038	−0.432019	+	0.901865i	$0.642199\pi$
$674$	−22.6963	−0.874229
$675$	−1.17324	−0.0451580
$676$	−12.2923	−0.472783
$677$	−24.3187	−0.934643	−0.467321	−	0.884087i	$-0.654781\pi$
$677$	−24.3187	−0.934643	−0.467321	+	0.884087i	$0.654781\pi$
$678$	37.4293	1.43746
$679$	0	0
$680$	1.00000	0.0383482
$681$	−0.664801	−0.0254752
$682$	16.8981	0.647063
$683$	−29.7327	−1.13769	−0.568846	−	0.822444i	$-0.692610\pi$
$683$	−29.7327	−1.13769	−0.568846	+	0.822444i	$0.692610\pi$
$684$	−14.9134	−0.570227
$685$	−1.69633	−0.0648133
$686$	0	0
$687$	16.8541	0.643025
$688$	6.35786	0.242391
$689$	8.48980	0.323435
$690$	−11.2194	−0.427117
$691$	−38.2211	−1.45400	−0.727000	−	0.686638i	$-0.759086\pi$
$691$	−38.2211	−1.45400	−0.727000	+	0.686638i	$0.759086\pi$
$692$	6.28337	0.238858
$693$	0	0
$694$	−24.0599	−0.913301
$695$	13.1117	0.497356
$696$	−4.83992	−0.183457
$697$	−6.18647	−0.234329
$698$	−0.252751	−0.00956676
$699$	−3.41459	−0.129151
$700$	0	0
$701$	22.6906	0.857010	0.428505	−	0.903539i	$-0.359040\pi$
$701$	22.6906	0.857010	0.428505	+	0.903539i	$0.359040\pi$
$702$	−0.986954	−0.0372502
$703$	−36.0209	−1.35855
$704$	3.95825	0.149182
$705$	−18.0800	−0.680933
$706$	−31.6691	−1.19188
$707$	0	0
$708$	−10.2424	−0.384932
$709$	−10.0719	−0.378257	−0.189128	−	0.981952i	$-0.560566\pi$
$709$	−10.0719	−0.378257	−0.189128	+	0.981952i	$0.560566\pi$
$710$	16.1776	0.607133
$711$	4.78675	0.179517
$712$	14.0692	0.527267
$713$	20.4238	0.764877
$714$	0	0
$715$	−3.32976	−0.124526
$716$	3.76919	0.140861
$717$	−0.440286	−0.0164428
$718$	25.7300	0.960237
$719$	10.2907	0.383777	0.191888	−	0.981417i	$-0.438539\pi$
$719$	10.2907	0.383777	0.191888	+	0.981417i	$0.438539\pi$
$720$	−2.49972	−0.0931589
$721$	0	0
$722$	16.5935	0.617546
$723$	−31.9277	−1.18740
$724$	0.786659	0.0292359
$725$	−2.06380	−0.0766478
$726$	10.9465	0.406262
$727$	−19.0749	−0.707448	−0.353724	−	0.935350i	$-0.615085\pi$
$727$	−19.0749	−0.707448	−0.353724	+	0.935350i	$0.615085\pi$
$728$	0	0
$729$	−17.3692	−0.643305
$730$	−10.1451	−0.375487
$731$	−6.35786	−0.235154
$732$	−25.4499	−0.940657
$733$	−25.0820	−0.926426	−0.463213	−	0.886247i	$-0.653304\pi$
$733$	−25.0820	−0.926426	−0.463213	+	0.886247i	$0.653304\pi$
$734$	−24.9879	−0.922322
$735$	0	0
$736$	4.78411	0.176345
$737$	9.68429	0.356725
$738$	15.4644	0.569253
$739$	−43.5788	−1.60307	−0.801536	−	0.597947i	$-0.795984\pi$
$739$	−43.5788	−1.60307	−0.801536	+	0.597947i	$0.795984\pi$
$740$	−6.03766	−0.221949
$741$	−11.7697	−0.432371
$742$	0	0
$743$	−22.4398	−0.823238	−0.411619	−	0.911356i	$-0.635037\pi$
$743$	−22.4398	−0.823238	−0.411619	+	0.911356i	$0.635037\pi$
$744$	10.0117	0.367045
$745$	7.05512	0.258480
$746$	−0.702850	−0.0257332
$747$	−14.3172	−0.523838
$748$	−3.95825	−0.144728
$749$	0	0
$750$	−2.34515	−0.0856327
$751$	25.6671	0.936604	0.468302	−	0.883568i	$-0.344866\pi$
$751$	25.6671	0.936604	0.468302	+	0.883568i	$0.344866\pi$
$752$	7.70954	0.281138
$753$	−33.8001	−1.23174
$754$	−1.73611	−0.0632256
$755$	−21.4470	−0.780537
$756$	0	0
$757$	−18.5832	−0.675417	−0.337708	−	0.941251i	$-0.609652\pi$
$757$	−18.5832	−0.675417	−0.337708	+	0.941251i	$0.609652\pi$
$758$	−33.3675	−1.21196
$759$	44.4093	1.61195
$760$	5.96603	0.216411
$761$	−24.2461	−0.878921	−0.439461	−	0.898262i	$-0.644830\pi$
$761$	−24.2461	−0.878921	−0.439461	+	0.898262i	$0.644830\pi$
$762$	−9.47995	−0.343422
$763$	0	0
$764$	−24.8380	−0.898606
$765$	2.49972	0.0903774
$766$	−26.1379	−0.944399
$767$	−3.67401	−0.132661
$768$	2.34515	0.0846232
$769$	−40.0898	−1.44567	−0.722837	−	0.691018i	$-0.757162\pi$
$769$	−40.0898	−1.44567	−0.722837	+	0.691018i	$0.757162\pi$
$770$	0	0
$771$	8.62912	0.310770
$772$	19.5742	0.704489
$773$	−40.1190	−1.44298	−0.721491	−	0.692424i	$-0.756543\pi$
$773$	−40.1190	−1.44298	−0.721491	+	0.692424i	$0.756543\pi$
$774$	15.8929	0.571257
$775$	4.26909	0.153350
$776$	1.10161	0.0395455
$777$	0	0
$778$	−28.2167	−1.01162
$779$	−36.9086	−1.32239
$780$	−1.97279	−0.0706371
$781$	−64.0348	−2.29134
$782$	−4.78411	−0.171079
$783$	2.42134	0.0865316
$784$	0	0
$785$	21.8380	0.779432
$786$	27.1507	0.968435
$787$	50.5986	1.80364	0.901822	−	0.432107i	$-0.142230\pi$
$787$	50.5986	1.80364	0.901822	+	0.432107i	$0.142230\pi$
$788$	1.70467	0.0607262
$789$	62.6539	2.23054
$790$	−1.91492	−0.0681297
$791$	0	0
$792$	9.89449	0.351585
$793$	−9.12907	−0.324183
$794$	−5.44020	−0.193066
$795$	−23.6678	−0.839410
$796$	8.43343	0.298915
$797$	−38.8101	−1.37473	−0.687363	−	0.726314i	$-0.741232\pi$
$797$	−38.8101	−1.37473	−0.687363	+	0.726314i	$0.741232\pi$
$798$	0	0
$799$	−7.70954	−0.272744
$800$	1.00000	0.0353553
$801$	35.1691	1.24264
$802$	−4.32597	−0.152755
$803$	40.1568	1.41710
$804$	5.73766	0.202352
$805$	0	0
$806$	3.59125	0.126496
$807$	−36.7210	−1.29264
$808$	−8.17811	−0.287705
$809$	−22.6318	−0.795691	−0.397845	−	0.917452i	$-0.630242\pi$
$809$	−22.6318	−0.795691	−0.397845	+	0.917452i	$0.630242\pi$
$810$	10.2506	0.360168
$811$	16.6394	0.584288	0.292144	−	0.956374i	$-0.405631\pi$
$811$	16.6394	0.584288	0.292144	+	0.956374i	$0.405631\pi$
$812$	0	0
$813$	−9.91976	−0.347901
$814$	23.8986	0.837644
$815$	9.80190	0.343345
$816$	−2.34515	−0.0820966
$817$	−37.9312	−1.32704
$818$	25.8813	0.904917
$819$	0	0
$820$	−6.18647	−0.216041
$821$	−27.7222	−0.967513	−0.483757	−	0.875203i	$-0.660728\pi$
$821$	−27.7222	−0.967513	−0.483757	+	0.875203i	$0.660728\pi$
$822$	3.97813	0.138753
$823$	46.6224	1.62515	0.812577	−	0.582854i	$-0.198064\pi$
$823$	46.6224	1.62515	0.812577	+	0.582854i	$0.198064\pi$
$824$	12.8886	0.448996
$825$	9.28267	0.323181
$826$	0	0
$827$	−17.5503	−0.610282	−0.305141	−	0.952307i	$-0.598704\pi$
$827$	−17.5503	−0.610282	−0.305141	+	0.952307i	$0.598704\pi$
$828$	11.9589	0.415601
$829$	35.6445	1.23798	0.618992	−	0.785397i	$-0.287541\pi$
$829$	35.6445	1.23798	0.618992	+	0.785397i	$0.287541\pi$
$830$	5.72752	0.198805
$831$	−0.393500	−0.0136504
$832$	0.841221	0.0291641
$833$	0	0
$834$	−30.7489	−1.06475
$835$	3.49467	0.120938
$836$	−23.6150	−0.816742
$837$	−5.00867	−0.173125
$838$	1.23007	0.0424920
$839$	31.4080	1.08432	0.542162	−	0.840274i	$-0.317606\pi$
$839$	31.4080	1.08432	0.542162	+	0.840274i	$0.317606\pi$
$840$	0	0
$841$	−24.7407	−0.853128
$842$	18.7083	0.644731
$843$	37.5140	1.29205
$844$	−21.5843	−0.742963
$845$	12.2923	0.422870
$846$	19.2717	0.662574
$847$	0	0
$848$	10.0922	0.346569
$849$	−70.9793	−2.43600
$850$	−1.00000	−0.0342997
$851$	28.8848	0.990159
$852$	−37.9388	−1.29976
$853$	17.0649	0.584293	0.292146	−	0.956374i	$-0.405631\pi$
$853$	17.0649	0.584293	0.292146	+	0.956374i	$0.405631\pi$
$854$	0	0
$855$	14.9134	0.510027
$856$	6.14230	0.209939
$857$	46.0441	1.57284	0.786419	−	0.617694i	$-0.211933\pi$
$857$	46.0441	1.57284	0.786419	+	0.617694i	$0.211933\pi$
$858$	7.80877	0.266587
$859$	11.5903	0.395456	0.197728	−	0.980257i	$-0.436644\pi$
$859$	11.5903	0.395456	0.197728	+	0.980257i	$0.436644\pi$
$860$	−6.35786	−0.216801
$861$	0	0
$862$	9.70967	0.330713
$863$	24.8201	0.844884	0.422442	−	0.906390i	$-0.361173\pi$
$863$	24.8201	0.844884	0.422442	+	0.906390i	$0.361173\pi$
$864$	−1.17324	−0.0399145
$865$	−6.28337	−0.213641
$866$	−8.57171	−0.291279
$867$	2.34515	0.0796454
$868$	0	0
$869$	7.57972	0.257124
$870$	4.83992	0.164089
$871$	2.05814	0.0697374
$872$	0.718456	0.0243300
$873$	2.75371	0.0931991
$874$	−28.5421	−0.965452
$875$	0	0
$876$	23.7917	0.803848
$877$	7.05497	0.238229	0.119115	−	0.992881i	$-0.461994\pi$
$877$	7.05497	0.238229	0.119115	+	0.992881i	$0.461994\pi$
$878$	26.5905	0.897385
$879$	−12.9698	−0.437461
$880$	−3.95825	−0.133432
$881$	−32.6312	−1.09937	−0.549687	−	0.835371i	$-0.685253\pi$
$881$	−32.6312	−1.09937	−0.549687	+	0.835371i	$0.685253\pi$
$882$	0	0
$883$	−27.5638	−0.927596	−0.463798	−	0.885941i	$-0.653514\pi$
$883$	−27.5638	−0.927596	−0.463798	+	0.885941i	$0.653514\pi$
$884$	−0.841221	−0.0282933
$885$	10.2424	0.344294
$886$	1.02060	0.0342877
$887$	27.3877	0.919588	0.459794	−	0.888026i	$-0.347923\pi$
$887$	27.3877	0.919588	0.459794	+	0.888026i	$0.347923\pi$
$888$	14.1592	0.475152
$889$	0	0
$890$	−14.0692	−0.471602
$891$	−40.5743	−1.35929
$892$	2.67003	0.0893992
$893$	−45.9954	−1.53918
$894$	−16.5453	−0.553358
$895$	−3.76919	−0.125990
$896$	0	0
$897$	9.43802	0.315126
$898$	4.39147	0.146545
$899$	−8.81057	−0.293849
$900$	2.49972	0.0833239
$901$	−10.0922	−0.336221
$902$	24.4876	0.815347
$903$	0	0
$904$	15.9603	0.530832
$905$	−0.786659	−0.0261494
$906$	50.2964	1.67099
$907$	−15.3826	−0.510771	−0.255386	−	0.966839i	$-0.582202\pi$
$907$	−15.3826	−0.510771	−0.255386	+	0.966839i	$0.582202\pi$
$908$	−0.283479	−0.00940759
$909$	−20.4429	−0.678049
$910$	0	0
$911$	−40.7773	−1.35101	−0.675506	−	0.737354i	$-0.736075\pi$
$911$	−40.7773	−1.35101	−0.675506	+	0.737354i	$0.736075\pi$
$912$	−13.9912	−0.463295
$913$	−22.6709	−0.750298
$914$	28.3785	0.938676
$915$	25.4499	0.841349
$916$	7.18681	0.237459
$917$	0	0
$918$	1.17324	0.0387227
$919$	25.5639	0.843276	0.421638	−	0.906764i	$-0.361455\pi$
$919$	25.5639	0.843276	0.421638	+	0.906764i	$0.361455\pi$
$920$	−4.78411	−0.157727
$921$	44.3719	1.46210
$922$	6.71669	0.221203
$923$	−13.6089	−0.447943
$924$	0	0
$925$	6.03766	0.198517
$926$	7.53393	0.247580
$927$	32.2179	1.05817
$928$	−2.06380	−0.0677477
$929$	0.675279	0.0221552	0.0110776	−	0.999939i	$-0.496474\pi$
$929$	0.675279	0.0221552	0.0110776	+	0.999939i	$0.496474\pi$
$930$	−10.0117	−0.328295
$931$	0	0
$932$	−1.45602	−0.0476936
$933$	39.0900	1.27975
$934$	−9.39180	−0.307309
$935$	3.95825	0.129448
$936$	2.10281	0.0687326
$937$	36.6305	1.19667	0.598333	−	0.801248i	$-0.295830\pi$
$937$	36.6305	1.19667	0.598333	+	0.801248i	$0.295830\pi$
$938$	0	0
$939$	−54.7985	−1.78828
$940$	−7.70954	−0.251458
$941$	5.32981	0.173747	0.0868734	−	0.996219i	$-0.472312\pi$
$941$	5.32981	0.173747	0.0868734	+	0.996219i	$0.472312\pi$
$942$	−51.2133	−1.66862
$943$	29.5967	0.963802
$944$	−4.36748	−0.142149
$945$	0	0
$946$	25.1660	0.818217
$947$	−48.2283	−1.56721	−0.783605	−	0.621260i	$-0.786621\pi$
$947$	−48.2283	−1.56721	−0.783605	+	0.621260i	$0.786621\pi$
$948$	4.49076	0.145853
$949$	8.53426	0.277034
$950$	−5.96603	−0.193564
$951$	43.9364	1.42473
$952$	0	0
$953$	11.8350	0.383372	0.191686	−	0.981456i	$-0.438604\pi$
$953$	11.8350	0.383372	0.191686	+	0.981456i	$0.438604\pi$
$954$	25.2277	0.816777
$955$	24.8380	0.803738
$956$	−0.187743	−0.00607205
$957$	−19.1576	−0.619278
$958$	−3.94238	−0.127372
$959$	0	0
$960$	−2.34515	−0.0756893
$961$	−12.7748	−0.412092
$962$	5.07901	0.163754
$963$	15.3540	0.494776
$964$	−13.6144	−0.438489
$965$	−19.5742	−0.630114
$966$	0	0
$967$	−3.41017	−0.109664	−0.0548318	−	0.998496i	$-0.517462\pi$
$967$	−3.41017	−0.109664	−0.0548318	+	0.998496i	$0.517462\pi$
$968$	4.66771	0.150026
$969$	13.9912	0.449463
$970$	−1.10161	−0.0353706
$971$	−4.30726	−0.138227	−0.0691133	−	0.997609i	$-0.522017\pi$
$971$	−4.30726	−0.138227	−0.0691133	+	0.997609i	$0.522017\pi$
$972$	−20.5194	−0.658159
$973$	0	0
$974$	−38.9841	−1.24913
$975$	1.97279	0.0631797
$976$	−10.8522	−0.347369
$977$	−9.68799	−0.309946	−0.154973	−	0.987919i	$-0.549529\pi$
$977$	−9.68799	−0.309946	−0.154973	+	0.987919i	$0.549529\pi$
$978$	−22.9869	−0.735040
$979$	55.6895	1.77984
$980$	0	0
$981$	1.79593	0.0573398
$982$	30.9826	0.988696
$983$	−18.2687	−0.582681	−0.291341	−	0.956619i	$-0.594101\pi$
$983$	−18.2687	−0.582681	−0.291341	+	0.956619i	$0.594101\pi$
$984$	14.5082	0.462504
$985$	−1.70467	−0.0543152
$986$	2.06380	0.0657249
$987$	0	0
$988$	−5.01875	−0.159668
$989$	30.4167	0.967195
$990$	−9.89449	−0.314468
$991$	−16.5573	−0.525958	−0.262979	−	0.964801i	$-0.584705\pi$
$991$	−16.5573	−0.525958	−0.262979	+	0.964801i	$0.584705\pi$
$992$	4.26909	0.135544
$993$	17.3060	0.549191
$994$	0	0
$995$	−8.43343	−0.267358
$996$	−13.4319	−0.425605
$997$	28.6155	0.906262	0.453131	−	0.891444i	$-0.350307\pi$
$997$	28.6155	0.906262	0.453131	+	0.891444i	$0.350307\pi$
$998$	−11.7545	−0.372083
$999$	−7.08363	−0.224116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cv.1.9	✓	10
7.6	odd	2		8330.2.a.cw.1.2	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cv.1.9	✓	10	1.1	even	1	trivial
8330.2.a.cw.1.2	yes	10	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.34515 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.34515 q^{6} +1.00000 q^{8} +2.49972 q^{9} -1.00000 q^{10} +3.95825 q^{11} +2.34515 q^{12} +0.841221 q^{13} -2.34515 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.49972 q^{18} -5.96603 q^{19} -1.00000 q^{20} +3.95825 q^{22} +4.78411 q^{23} +2.34515 q^{24} +1.00000 q^{25} +0.841221 q^{26} -1.17324 q^{27} -2.06380 q^{29} -2.34515 q^{30} +4.26909 q^{31} +1.00000 q^{32} +9.28267 q^{33} -1.00000 q^{34} +2.49972 q^{36} +6.03766 q^{37} -5.96603 q^{38} +1.97279 q^{39} -1.00000 q^{40} +6.18647 q^{41} +6.35786 q^{43} +3.95825 q^{44} -2.49972 q^{45} +4.78411 q^{46} +7.70954 q^{47} +2.34515 q^{48} +1.00000 q^{50} -2.34515 q^{51} +0.841221 q^{52} +10.0922 q^{53} -1.17324 q^{54} -3.95825 q^{55} -13.9912 q^{57} -2.06380 q^{58} -4.36748 q^{59} -2.34515 q^{60} -10.8522 q^{61} +4.26909 q^{62} +1.00000 q^{64} -0.841221 q^{65} +9.28267 q^{66} +2.44661 q^{67} -1.00000 q^{68} +11.2194 q^{69} -16.1776 q^{71} +2.49972 q^{72} +10.1451 q^{73} +6.03766 q^{74} +2.34515 q^{75} -5.96603 q^{76} +1.97279 q^{78} +1.91492 q^{79} -1.00000 q^{80} -10.2506 q^{81} +6.18647 q^{82} -5.72752 q^{83} +1.00000 q^{85} +6.35786 q^{86} -4.83992 q^{87} +3.95825 q^{88} +14.0692 q^{89} -2.49972 q^{90} +4.78411 q^{92} +10.0117 q^{93} +7.70954 q^{94} +5.96603 q^{95} +2.34515 q^{96} +1.10161 q^{97} +9.89449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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