LMFDB - Embedded newform 669.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [669,2,Mod(1,669)] 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("669.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 669 = 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	669.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.34199189522$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	669.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.46050 q^{2} -1.00000 q^{3} +0.133074 q^{4} -1.46050 q^{6} -0.866926 q^{7} -2.72665 q^{8} +1.00000 q^{9} -3.18716 q^{11} -0.133074 q^{12} -3.18716 q^{13} -1.26615 q^{14} -4.24844 q^{16} +3.64766 q^{17} +1.46050 q^{18} +0.460505 q^{19} +0.866926 q^{21} -4.65486 q^{22} -7.84202 q^{23} +2.72665 q^{24} -5.00000 q^{25} -4.65486 q^{26} -1.00000 q^{27} -0.115366 q^{28} -8.32743 q^{29} -0.485411 q^{31} -0.751560 q^{32} +3.18716 q^{33} +5.32743 q^{34} +0.133074 q^{36} +9.78794 q^{37} +0.672570 q^{38} +3.18716 q^{39} +8.97509 q^{41} +1.26615 q^{42} -3.51459 q^{43} -0.424129 q^{44} -11.4533 q^{46} -1.53950 q^{47} +4.24844 q^{48} -6.24844 q^{49} -7.30252 q^{50} -3.64766 q^{51} -0.424129 q^{52} +2.72665 q^{53} -1.46050 q^{54} +2.36381 q^{56} -0.460505 q^{57} -12.1623 q^{58} +0.108168 q^{59} +4.54669 q^{61} -0.708945 q^{62} -0.866926 q^{63} +7.39922 q^{64} +4.65486 q^{66} +7.18716 q^{67} +0.485411 q^{68} +7.84202 q^{69} +7.95019 q^{71} -2.72665 q^{72} -9.78074 q^{73} +14.2953 q^{74} +5.00000 q^{75} +0.0612814 q^{76} +2.76303 q^{77} +4.65486 q^{78} -7.73385 q^{79} +1.00000 q^{81} +13.1082 q^{82} -10.5936 q^{83} +0.115366 q^{84} -5.13307 q^{86} +8.32743 q^{87} +8.69028 q^{88} -8.86693 q^{89} +2.76303 q^{91} -1.04357 q^{92} +0.485411 q^{93} -2.24844 q^{94} +0.751560 q^{96} +9.18716 q^{97} -9.12588 q^{98} -3.18716 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 2 q^{6} + q^{7} - 9 q^{8} + 3 q^{9} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 11 q^{14} + 10 q^{16} - q^{17} - 2 q^{18} - 5 q^{19} - q^{21} + 6 q^{22} + 2 q^{23} + 9 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 2 * q^6 + q^7 - 9 * q^8 + 3 * q^9 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 11 * q^14 + 10 * q^16 - q^17 - 2 * q^18 - 5 * q^19 - q^21 + 6 * q^22 + 2 * q^23 + 9 * q^24 - 15 * q^25 + 6 * q^26 - 3 * q^27 + 26 * q^28 - 15 * q^29 - 17 * q^31 - 25 * q^32 + 4 * q^33 + 6 * q^34 + 4 * q^36 + 13 * q^37 + 12 * q^38 + 4 * q^39 + 5 * q^41 + 11 * q^42 + 5 * q^43 - 34 * q^44 - 36 * q^46 - 11 * q^47 - 10 * q^48 + 4 * q^49 + 10 * q^50 + q^51 - 34 * q^52 + 9 * q^53 + 2 * q^54 - 42 * q^56 + 5 * q^57 - 9 * q^58 - 18 * q^59 + 12 * q^61 + 27 * q^62 + q^63 + 33 * q^64 - 6 * q^66 + 16 * q^67 + 17 * q^68 - 2 * q^69 - 20 * q^71 - 9 * q^72 - 21 * q^73 + 19 * q^74 + 15 * q^75 - 17 * q^76 - 30 * q^77 - 6 * q^78 - 16 * q^79 + 3 * q^81 + 21 * q^82 - 29 * q^83 - 26 * q^84 - 19 * q^86 + 15 * q^87 + 66 * q^88 - 23 * q^89 - 30 * q^91 + 44 * q^92 + 17 * q^93 + 16 * q^94 + 25 * q^96 + 22 * q^97 - 39 * q^98 - 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 $

$2$	1.46050	1.03273	0.516366	−	0.856368i	$-0.327284\pi$
$2$	1.46050	1.03273	0.516366	+	0.856368i	$0.327284\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.133074	0.0665372
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.46050	−0.596249
$7$	−0.866926	−0.327667	−0.163834	−	0.986488i	$-0.552386\pi$
$7$	−0.866926	−0.327667	−0.163834	+	0.986488i	$0.552386\pi$
$8$	−2.72665	−0.964018
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.18716	−0.960965	−0.480482	−	0.877004i	$-0.659538\pi$
$11$	−3.18716	−0.960965	−0.480482	+	0.877004i	$0.659538\pi$
$12$	−0.133074	−0.0384153
$13$	−3.18716	−0.883959	−0.441979	−	0.897025i	$-0.645724\pi$
$13$	−3.18716	−0.883959	−0.441979	+	0.897025i	$0.645724\pi$
$14$	−1.26615	−0.338393
$15$	0	0
$16$	−4.24844	−1.06211
$17$	3.64766	0.884688	0.442344	−	0.896845i	$-0.354147\pi$
$17$	3.64766	0.884688	0.442344	+	0.896845i	$0.354147\pi$
$18$	1.46050	0.344244
$19$	0.460505	0.105647	0.0528235	−	0.998604i	$-0.483178\pi$
$19$	0.460505	0.105647	0.0528235	+	0.998604i	$0.483178\pi$
$20$	0	0
$21$	0.866926	0.189179
$22$	−4.65486	−0.992420
$23$	−7.84202	−1.63517	−0.817587	−	0.575805i	$-0.804689\pi$
$23$	−7.84202	−1.63517	−0.817587	+	0.575805i	$0.804689\pi$
$24$	2.72665	0.556576
$25$	−5.00000	−1.00000
$26$	−4.65486	−0.912893
$27$	−1.00000	−0.192450
$28$	−0.115366	−0.0218021
$29$	−8.32743	−1.54637	−0.773183	−	0.634184i	$-0.781336\pi$
$29$	−8.32743	−1.54637	−0.773183	+	0.634184i	$0.781336\pi$
$30$	0	0
$31$	−0.485411	−0.0871824	−0.0435912	−	0.999049i	$-0.513880\pi$
$31$	−0.485411	−0.0871824	−0.0435912	+	0.999049i	$0.513880\pi$
$32$	−0.751560	−0.132858
$33$	3.18716	0.554813
$34$	5.32743	0.913647
$35$	0	0
$36$	0.133074	0.0221791
$37$	9.78794	1.60913	0.804563	−	0.593867i	$-0.202399\pi$
$37$	9.78794	1.60913	0.804563	+	0.593867i	$0.202399\pi$
$38$	0.672570	0.109105
$39$	3.18716	0.510354
$40$	0	0
$41$	8.97509	1.40167	0.700837	−	0.713321i	$-0.252810\pi$
$41$	8.97509	1.40167	0.700837	+	0.713321i	$0.252810\pi$
$42$	1.26615	0.195371
$43$	−3.51459	−0.535970	−0.267985	−	0.963423i	$-0.586358\pi$
$43$	−3.51459	−0.535970	−0.267985	+	0.963423i	$0.586358\pi$
$44$	−0.424129	−0.0639399
$45$	0	0
$46$	−11.4533	−1.68870
$47$	−1.53950	−0.224558	−0.112279	−	0.993677i	$-0.535815\pi$
$47$	−1.53950	−0.224558	−0.112279	+	0.993677i	$0.535815\pi$
$48$	4.24844	0.613210
$49$	−6.24844	−0.892634
$50$	−7.30252	−1.03273
$51$	−3.64766	−0.510775
$52$	−0.424129	−0.0588162
$53$	2.72665	0.374535	0.187267	−	0.982309i	$-0.440037\pi$
$53$	2.72665	0.374535	0.187267	+	0.982309i	$0.440037\pi$
$54$	−1.46050	−0.198750
$55$	0	0
$56$	2.36381	0.315877
$57$	−0.460505	−0.0609954
$58$	−12.1623	−1.59698
$59$	0.108168	0.0140823	0.00704117	−	0.999975i	$-0.497759\pi$
$59$	0.108168	0.0140823	0.00704117	+	0.999975i	$0.497759\pi$
$60$	0	0
$61$	4.54669	0.582144	0.291072	−	0.956701i	$-0.405988\pi$
$61$	4.54669	0.582144	0.291072	+	0.956701i	$0.405988\pi$
$62$	−0.708945	−0.0900361
$63$	−0.866926	−0.109222
$64$	7.39922	0.924903
$65$	0	0
$66$	4.65486	0.572974
$67$	7.18716	0.878051	0.439026	−	0.898475i	$-0.355324\pi$
$67$	7.18716	0.878051	0.439026	+	0.898475i	$0.355324\pi$
$68$	0.485411	0.0588647
$69$	7.84202	0.944068
$70$	0	0
$71$	7.95019	0.943514	0.471757	−	0.881729i	$-0.343620\pi$
$71$	7.95019	0.943514	0.471757	+	0.881729i	$0.343620\pi$
$72$	−2.72665	−0.321339
$73$	−9.78074	−1.14475	−0.572374	−	0.819992i	$-0.693978\pi$
$73$	−9.78074	−1.14475	−0.572374	+	0.819992i	$0.693978\pi$
$74$	14.2953	1.66180
$75$	5.00000	0.577350
$76$	0.0612814	0.00702946
$77$	2.76303	0.314876
$78$	4.65486	0.527059
$79$	−7.73385	−0.870126	−0.435063	−	0.900400i	$-0.643274\pi$
$79$	−7.73385	−0.870126	−0.435063	+	0.900400i	$0.643274\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	13.1082	1.44756
$83$	−10.5936	−1.16280	−0.581398	−	0.813619i	$-0.697494\pi$
$83$	−10.5936	−1.16280	−0.581398	+	0.813619i	$0.697494\pi$
$84$	0.115366	0.0125874
$85$	0	0
$86$	−5.13307	−0.553514
$87$	8.32743	0.892794
$88$	8.69028	0.926387
$89$	−8.86693	−0.939892	−0.469946	−	0.882695i	$-0.655727\pi$
$89$	−8.86693	−0.939892	−0.469946	+	0.882695i	$0.655727\pi$
$90$	0	0
$91$	2.76303	0.289644
$92$	−1.04357	−0.108800
$93$	0.485411	0.0503348
$94$	−2.24844	−0.231909
$95$	0	0
$96$	0.751560	0.0767058
$97$	9.18716	0.932815	0.466407	−	0.884570i	$-0.345548\pi$
$97$	9.18716	0.932815	0.466407	+	0.884570i	$0.345548\pi$
$98$	−9.12588	−0.921853
$99$	−3.18716	−0.320322
$100$	−0.665372	−0.0665372
$101$	−10.7089	−1.06558	−0.532790	−	0.846248i	$-0.678856\pi$
$101$	−10.7089	−1.06558	−0.532790	+	0.846248i	$0.678856\pi$
$102$	−5.32743	−0.527494
$103$	12.2163	1.20371	0.601856	−	0.798605i	$-0.294428\pi$
$103$	12.2163	1.20371	0.601856	+	0.798605i	$0.294428\pi$
$104$	8.69028	0.852152
$105$	0	0
$106$	3.98229	0.386794
$107$	−18.4825	−1.78677	−0.893385	−	0.449293i	$-0.851676\pi$
$107$	−18.4825	−1.78677	−0.893385	+	0.449293i	$0.851676\pi$
$108$	−0.133074	−0.0128051
$109$	17.1154	1.63935	0.819677	−	0.572825i	$-0.194153\pi$
$109$	17.1154	1.63935	0.819677	+	0.572825i	$0.194153\pi$
$110$	0	0
$111$	−9.78794	−0.929030
$112$	3.68308	0.348018
$113$	19.5759	1.84154	0.920771	−	0.390102i	$-0.127560\pi$
$113$	19.5759	1.84154	0.920771	+	0.390102i	$0.127560\pi$
$114$	−0.672570	−0.0629919
$115$	0	0
$116$	−1.10817	−0.102891
$117$	−3.18716	−0.294653
$118$	0.157981	0.0145433
$119$	−3.16225	−0.289883
$120$	0	0
$121$	−0.842019	−0.0765472
$122$	6.64047	0.601200
$123$	−8.97509	−0.809257
$124$	−0.0645958	−0.00580087
$125$	0	0
$126$	−1.26615	−0.112798
$127$	13.5979	1.20661	0.603307	−	0.797509i	$-0.293849\pi$
$127$	13.5979	1.20661	0.603307	+	0.797509i	$0.293849\pi$
$128$	12.3097	1.08804
$129$	3.51459	0.309442
$130$	0	0
$131$	−12.0541	−1.05317	−0.526585	−	0.850122i	$-0.676528\pi$
$131$	−12.0541	−1.05317	−0.526585	+	0.850122i	$0.676528\pi$
$132$	0.424129	0.0369157
$133$	−0.399223	−0.0346171
$134$	10.4969	0.906792
$135$	0	0
$136$	−9.94592	−0.852855
$137$	−1.84202	−0.157374	−0.0786872	−	0.996899i	$-0.525073\pi$
$137$	−1.84202	−0.157374	−0.0786872	+	0.996899i	$0.525073\pi$
$138$	11.4533	0.974970
$139$	−8.86693	−0.752083	−0.376041	−	0.926603i	$-0.622715\pi$
$139$	−8.86693	−0.752083	−0.376041	+	0.926603i	$0.622715\pi$
$140$	0	0
$141$	1.53950	0.129649
$142$	11.6113	0.974398
$143$	10.1580	0.849453
$144$	−4.24844	−0.354037
$145$	0	0
$146$	−14.2848	−1.18222
$147$	6.24844	0.515363
$148$	1.30252	0.107067
$149$	9.29533	0.761503	0.380751	−	0.924677i	$-0.375665\pi$
$149$	9.29533	0.761503	0.380751	+	0.924677i	$0.375665\pi$
$150$	7.30252	0.596249
$151$	14.2163	1.15691	0.578455	−	0.815715i	$-0.303656\pi$
$151$	14.2163	1.15691	0.578455	+	0.815715i	$0.303656\pi$
$152$	−1.25564	−0.101846
$153$	3.64766	0.294896
$154$	4.03542	0.325183
$155$	0	0
$156$	0.424129	0.0339575
$157$	16.7630	1.33784	0.668918	−	0.743336i	$-0.266758\pi$
$157$	16.7630	1.33784	0.668918	+	0.743336i	$0.266758\pi$
$158$	−11.2953	−0.898608
$159$	−2.72665	−0.216238
$160$	0	0
$161$	6.79845	0.535793
$162$	1.46050	0.114748
$163$	−7.13735	−0.559040	−0.279520	−	0.960140i	$-0.590175\pi$
$163$	−7.13735	−0.559040	−0.279520	+	0.960140i	$0.590175\pi$
$164$	1.19436	0.0932635
$165$	0	0
$166$	−15.4720	−1.20086
$167$	−8.81284	−0.681958	−0.340979	−	0.940071i	$-0.610759\pi$
$167$	−8.81284	−0.681958	−0.340979	+	0.940071i	$0.610759\pi$
$168$	−2.36381	−0.182372
$169$	−2.84202	−0.218617
$170$	0	0
$171$	0.460505	0.0352157
$172$	−0.467702	−0.0356620
$173$	−21.8420	−1.66062	−0.830309	−	0.557303i	$-0.811836\pi$
$173$	−21.8420	−1.66062	−0.830309	+	0.557303i	$0.811836\pi$
$174$	12.1623	0.922018
$175$	4.33463	0.327667
$176$	13.5405	1.02065
$177$	−0.108168	−0.00813044
$178$	−12.9502	−0.970658
$179$	−17.8961	−1.33762	−0.668809	−	0.743434i	$-0.733196\pi$
$179$	−17.8961	−1.33762	−0.668809	+	0.743434i	$0.733196\pi$
$180$	0	0
$181$	−14.9679	−1.11256	−0.556278	−	0.830997i	$-0.687771\pi$
$181$	−14.9679	−1.11256	−0.556278	+	0.830997i	$0.687771\pi$
$182$	4.03542	0.299125
$183$	−4.54669	−0.336101
$184$	21.3825	1.57634
$185$	0	0
$186$	0.708945	0.0519824
$187$	−11.6257	−0.850154
$188$	−0.204868	−0.0149415
$189$	0.866926	0.0630596
$190$	0	0
$191$	−10.2661	−0.742832	−0.371416	−	0.928466i	$-0.621128\pi$
$191$	−10.2661	−0.742832	−0.371416	+	0.928466i	$0.621128\pi$
$192$	−7.39922	−0.533993
$193$	11.5615	0.832213	0.416107	−	0.909316i	$-0.363394\pi$
$193$	11.5615	0.832213	0.416107	+	0.909316i	$0.363394\pi$
$194$	13.4179	0.963348
$195$	0	0
$196$	−0.831508	−0.0593934
$197$	−25.8319	−1.84045	−0.920223	−	0.391394i	$-0.871993\pi$
$197$	−25.8319	−1.84045	−0.920223	+	0.391394i	$0.871993\pi$
$198$	−4.65486	−0.330807
$199$	−7.73093	−0.548031	−0.274015	−	0.961725i	$-0.588352\pi$
$199$	−7.73093	−0.548031	−0.274015	+	0.961725i	$0.588352\pi$
$200$	13.6333	0.964018
$201$	−7.18716	−0.506943
$202$	−15.6405	−1.10046
$203$	7.21926	0.506693
$204$	−0.485411	−0.0339856
$205$	0	0
$206$	17.8420	1.24311
$207$	−7.84202	−0.545058
$208$	13.5405	0.938861
$209$	−1.46770	−0.101523
$210$	0	0
$211$	4.46050	0.307074	0.153537	−	0.988143i	$-0.450934\pi$
$211$	4.46050	0.307074	0.153537	+	0.988143i	$0.450934\pi$
$212$	0.362848	0.0249205
$213$	−7.95019	−0.544738
$214$	−26.9938	−1.84526
$215$	0	0
$216$	2.72665	0.185525
$217$	0.420815	0.0285668
$218$	24.9971	1.69302
$219$	9.78074	0.660921
$220$	0	0
$221$	−11.6257	−0.782028
$222$	−14.2953	−0.959440
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	0.651546	0.0435333
$225$	−5.00000	−0.333333
$226$	28.5907	1.90182
$227$	−16.2163	−1.07632	−0.538158	−	0.842844i	$-0.680880\pi$
$227$	−16.2163	−1.07632	−0.538158	+	0.842844i	$0.680880\pi$
$228$	−0.0612814	−0.00405846
$229$	−22.4825	−1.48568	−0.742842	−	0.669467i	$-0.766523\pi$
$229$	−22.4825	−1.48568	−0.742842	+	0.669467i	$0.766523\pi$
$230$	0	0
$231$	−2.76303	−0.181794
$232$	22.7060	1.49072
$233$	−21.5759	−1.41348	−0.706741	−	0.707472i	$-0.749835\pi$
$233$	−21.5759	−1.41348	−0.706741	+	0.707472i	$0.749835\pi$
$234$	−4.65486	−0.304298
$235$	0	0
$236$	0.0143945	0.000936999 0
$237$	7.73385	0.502368
$238$	−4.61849	−0.299372
$239$	5.84202	0.377889	0.188944	−	0.981988i	$-0.439493\pi$
$239$	5.84202	0.377889	0.188944	+	0.981988i	$0.439493\pi$
$240$	0	0
$241$	9.64766	0.621461	0.310730	−	0.950498i	$-0.399426\pi$
$241$	9.64766	0.621461	0.310730	+	0.950498i	$0.399426\pi$
$242$	−1.22977	−0.0790528
$243$	−1.00000	−0.0641500
$244$	0.605049	0.0387343
$245$	0	0
$246$	−13.1082	−0.835746
$247$	−1.46770	−0.0933876
$248$	1.32355	0.0840454
$249$	10.5936	0.671341
$250$	0	0
$251$	−16.8741	−1.06509	−0.532543	−	0.846403i	$-0.678763\pi$
$251$	−16.8741	−1.06509	−0.532543	+	0.846403i	$0.678763\pi$
$252$	−0.115366	−0.00726735
$253$	24.9938	1.57134
$254$	19.8597	1.24611
$255$	0	0
$256$	3.17996	0.198748
$257$	8.32743	0.519451	0.259725	−	0.965683i	$-0.416368\pi$
$257$	8.32743	0.519451	0.259725	+	0.965683i	$0.416368\pi$
$258$	5.13307	0.319571
$259$	−8.48541	−0.527258
$260$	0	0
$261$	−8.32743	−0.515455
$262$	−17.6050	−1.08764
$263$	−2.37432	−0.146407	−0.0732033	−	0.997317i	$-0.523322\pi$
$263$	−2.37432	−0.146407	−0.0732033	+	0.997317i	$0.523322\pi$
$264$	−8.69028	−0.534850
$265$	0	0
$266$	−0.583068	−0.0357502
$267$	8.86693	0.542647
$268$	0.956427	0.0584231
$269$	5.78366	0.352636	0.176318	−	0.984333i	$-0.443581\pi$
$269$	5.78366	0.352636	0.176318	+	0.984333i	$0.443581\pi$
$270$	0	0
$271$	−5.66964	−0.344406	−0.172203	−	0.985061i	$-0.555089\pi$
$271$	−5.66964	−0.344406	−0.172203	+	0.985061i	$0.555089\pi$
$272$	−15.4969	−0.939636
$273$	−2.76303	−0.167226
$274$	−2.69028	−0.162526
$275$	15.9358	0.960965
$276$	1.04357	0.0628157
$277$	12.8568	0.772490	0.386245	−	0.922396i	$-0.373772\pi$
$277$	12.8568	0.772490	0.386245	+	0.922396i	$0.373772\pi$
$278$	−12.9502	−0.776701
$279$	−0.485411	−0.0290608
$280$	0	0
$281$	−15.5729	−0.929004	−0.464502	−	0.885572i	$-0.653767\pi$
$281$	−15.5729	−0.929004	−0.464502	+	0.885572i	$0.653767\pi$
$282$	2.24844	0.133893
$283$	−7.02491	−0.417587	−0.208794	−	0.977960i	$-0.566954\pi$
$283$	−7.02491	−0.417587	−0.208794	+	0.977960i	$0.566954\pi$
$284$	1.05797	0.0627788
$285$	0	0
$286$	14.8358	0.877258
$287$	−7.78074	−0.459282
$288$	−0.751560	−0.0442861
$289$	−3.69455	−0.217327
$290$	0	0
$291$	−9.18716	−0.538561
$292$	−1.30157	−0.0761684
$293$	20.2163	1.18105	0.590526	−	0.807019i	$-0.298921\pi$
$293$	20.2163	1.18105	0.590526	+	0.807019i	$0.298921\pi$
$294$	9.12588	0.532232
$295$	0	0
$296$	−26.6883	−1.55123
$297$	3.18716	0.184938
$298$	13.5759	0.786429
$299$	24.9938	1.44543
$300$	0.665372	0.0384153
$301$	3.04689	0.175620
$302$	20.7630	1.19478
$303$	10.7089	0.615213
$304$	−1.95643	−0.112209
$305$	0	0
$306$	5.32743	0.304549
$307$	−17.2455	−0.984254	−0.492127	−	0.870524i	$-0.663780\pi$
$307$	−17.2455	−0.984254	−0.492127	+	0.870524i	$0.663780\pi$
$308$	0.367689	0.0209510
$309$	−12.2163	−0.694963
$310$	0	0
$311$	−26.1226	−1.48127	−0.740637	−	0.671905i	$-0.765476\pi$
$311$	−26.1226	−1.48127	−0.740637	+	0.671905i	$0.765476\pi$
$312$	−8.69028	−0.491990
$313$	14.2019	0.802741	0.401371	−	0.915916i	$-0.368534\pi$
$313$	14.2019	0.802741	0.401371	+	0.915916i	$0.368534\pi$
$314$	24.4825	1.38163
$315$	0	0
$316$	−1.02918	−0.0578958
$317$	9.51459	0.534393	0.267196	−	0.963642i	$-0.413903\pi$
$317$	9.51459	0.534393	0.267196	+	0.963642i	$0.413903\pi$
$318$	−3.98229	−0.223316
$319$	26.5408	1.48600
$320$	0	0
$321$	18.4825	1.03159
$322$	9.92916	0.553331
$323$	1.67977	0.0934647
$324$	0.133074	0.00739303
$325$	15.9358	0.883959
$326$	−10.4241	−0.577339
$327$	−17.1154	−0.946482
$328$	−24.4720	−1.35124
$329$	1.33463	0.0735804
$330$	0	0
$331$	−12.6050	−0.692836	−0.346418	−	0.938080i	$-0.612602\pi$
$331$	−12.6050	−0.692836	−0.346418	+	0.938080i	$0.612602\pi$
$332$	−1.40974	−0.0773693
$333$	9.78794	0.536376
$334$	−12.8712	−0.704281
$335$	0	0
$336$	−3.68308	−0.200929
$337$	15.6696	0.853580	0.426790	−	0.904351i	$-0.359644\pi$
$337$	15.6696	0.853580	0.426790	+	0.904351i	$0.359644\pi$
$338$	−4.15078	−0.225773
$339$	−19.5759	−1.06322
$340$	0	0
$341$	1.54708	0.0837792
$342$	0.672570	0.0363684
$343$	11.4854	0.620154
$344$	9.58307	0.516684
$345$	0	0
$346$	−31.9004	−1.71497
$347$	2.14454	0.115125	0.0575626	−	0.998342i	$-0.481667\pi$
$347$	2.14454	0.115125	0.0575626	+	0.998342i	$0.481667\pi$
$348$	1.10817	0.0594041
$349$	5.25895	0.281505	0.140753	−	0.990045i	$-0.455048\pi$
$349$	5.25895	0.281505	0.140753	+	0.990045i	$0.455048\pi$
$350$	6.33074	0.338393
$351$	3.18716	0.170118
$352$	2.39534	0.127672
$353$	−19.7951	−1.05359	−0.526794	−	0.849993i	$-0.676606\pi$
$353$	−19.7951	−1.05359	−0.526794	+	0.849993i	$0.676606\pi$
$354$	−0.157981	−0.00839657
$355$	0	0
$356$	−1.17996	−0.0625378
$357$	3.16225	0.167364
$358$	−26.1373	−1.38140
$359$	27.5979	1.45656	0.728279	−	0.685280i	$-0.240320\pi$
$359$	27.5979	1.45656	0.728279	+	0.685280i	$0.240320\pi$
$360$	0	0
$361$	−18.7879	−0.988839
$362$	−21.8607	−1.14897
$363$	0.842019	0.0441946
$364$	0.367689	0.0192721
$365$	0	0
$366$	−6.64047	−0.347103
$367$	−32.4327	−1.69297	−0.846486	−	0.532411i	$-0.821286\pi$
$367$	−32.4327	−1.69297	−0.846486	+	0.532411i	$0.821286\pi$
$368$	33.3164	1.73673
$369$	8.97509	0.467225
$370$	0	0
$371$	−2.36381	−0.122723
$372$	0.0645958	0.00334914
$373$	−22.5408	−1.16712	−0.583560	−	0.812070i	$-0.698341\pi$
$373$	−22.5408	−1.16712	−0.583560	+	0.812070i	$0.698341\pi$
$374$	−16.9794	−0.877982
$375$	0	0
$376$	4.19767	0.216478
$377$	26.5408	1.36692
$378$	1.26615	0.0651237
$379$	26.2278	1.34723	0.673616	−	0.739082i	$-0.264740\pi$
$379$	26.2278	1.34723	0.673616	+	0.739082i	$0.264740\pi$
$380$	0	0
$381$	−13.5979	−0.696639
$382$	−14.9938	−0.767148
$383$	33.6840	1.72117	0.860587	−	0.509303i	$-0.170097\pi$
$383$	33.6840	1.72117	0.860587	+	0.509303i	$0.170097\pi$
$384$	−12.3097	−0.628178
$385$	0	0
$386$	16.8856	0.859454
$387$	−3.51459	−0.178657
$388$	1.22258	0.0620669
$389$	−32.8755	−1.66685	−0.833426	−	0.552631i	$-0.813624\pi$
$389$	−32.8755	−1.66685	−0.833426	+	0.552631i	$0.813624\pi$
$390$	0	0
$391$	−28.6050	−1.44662
$392$	17.0373	0.860515
$393$	12.0541	0.608048
$394$	−37.7276	−1.90069
$395$	0	0
$396$	−0.424129	−0.0213133
$397$	5.23697	0.262836	0.131418	−	0.991327i	$-0.458047\pi$
$397$	5.23697	0.262836	0.131418	+	0.991327i	$0.458047\pi$
$398$	−11.2911	−0.565969
$399$	0.399223	0.0199862
$400$	21.2422	1.06211
$401$	14.9430	0.746217	0.373109	−	0.927788i	$-0.378292\pi$
$401$	14.9430	0.746217	0.373109	+	0.927788i	$0.378292\pi$
$402$	−10.4969	−0.523537
$403$	1.54708	0.0770656
$404$	−1.42509	−0.0709007
$405$	0	0
$406$	10.5438	0.523278
$407$	−31.1957	−1.54631
$408$	9.94592	0.492396
$409$	−4.81284	−0.237980	−0.118990	−	0.992895i	$-0.537966\pi$
$409$	−4.81284	−0.237980	−0.118990	+	0.992895i	$0.537966\pi$
$410$	0	0
$411$	1.84202	0.0908601
$412$	1.62568	0.0800916
$413$	−0.0937740	−0.00461432
$414$	−11.4533	−0.562899
$415$	0	0
$416$	2.39534	0.117441
$417$	8.86693	0.434215
$418$	−2.14359	−0.104846
$419$	8.21634	0.401394	0.200697	−	0.979653i	$-0.435679\pi$
$419$	8.21634	0.401394	0.200697	+	0.979653i	$0.435679\pi$
$420$	0	0
$421$	36.3829	1.77319	0.886596	−	0.462544i	$-0.153063\pi$
$421$	36.3829	1.77319	0.886596	+	0.462544i	$0.153063\pi$
$422$	6.51459	0.317125
$423$	−1.53950	−0.0748528
$424$	−7.43464	−0.361058
$425$	−18.2383	−0.884688
$426$	−11.6113	−0.562569
$427$	−3.94164	−0.190750
$428$	−2.45955	−0.118887
$429$	−10.1580	−0.490432
$430$	0	0
$431$	−2.76303	−0.133090	−0.0665452	−	0.997783i	$-0.521198\pi$
$431$	−2.76303	−0.133090	−0.0665452	+	0.997783i	$0.521198\pi$
$432$	4.24844	0.204403
$433$	37.3638	1.79559	0.897795	−	0.440414i	$-0.145168\pi$
$433$	37.3638	1.79559	0.897795	+	0.440414i	$0.145168\pi$
$434$	0.614603	0.0295019
$435$	0	0
$436$	2.27762	0.109078
$437$	−3.61129	−0.172751
$438$	14.2848	0.682555
$439$	−1.34514	−0.0642000	−0.0321000	−	0.999485i	$-0.510220\pi$
$439$	−1.34514	−0.0642000	−0.0321000	+	0.999485i	$0.510220\pi$
$440$	0	0
$441$	−6.24844	−0.297545
$442$	−16.9794	−0.807626
$443$	7.41362	0.352232	0.176116	−	0.984369i	$-0.443647\pi$
$443$	7.41362	0.352232	0.176116	+	0.984369i	$0.443647\pi$
$444$	−1.30252	−0.0618151
$445$	0	0
$446$	−1.46050	−0.0691569
$447$	−9.29533	−0.439654
$448$	−6.41458	−0.303060
$449$	0.546692	0.0258000	0.0129000	−	0.999917i	$-0.495894\pi$
$449$	0.546692	0.0258000	0.0129000	+	0.999917i	$0.495894\pi$
$450$	−7.30252	−0.344244
$451$	−28.6050	−1.34696
$452$	2.60505	0.122531
$453$	−14.2163	−0.667942
$454$	−23.6840	−1.11155
$455$	0	0
$456$	1.25564	0.0588006
$457$	12.7047	0.594300	0.297150	−	0.954831i	$-0.403964\pi$
$457$	12.7047	0.594300	0.297150	+	0.954831i	$0.403964\pi$
$458$	−32.8358	−1.53432
$459$	−3.64766	−0.170258
$460$	0	0
$461$	35.3566	1.64672	0.823361	−	0.567518i	$-0.192096\pi$
$461$	35.3566	1.64672	0.823361	+	0.567518i	$0.192096\pi$
$462$	−4.03542	−0.187745
$463$	−18.6375	−0.866160	−0.433080	−	0.901355i	$-0.642573\pi$
$463$	−18.6375	−0.866160	−0.433080	+	0.901355i	$0.642573\pi$
$464$	35.3786	1.64241
$465$	0	0
$466$	−31.5117	−1.45975
$467$	32.2163	1.49079	0.745397	−	0.666621i	$-0.232260\pi$
$467$	32.2163	1.49079	0.745397	+	0.666621i	$0.232260\pi$
$468$	−0.424129	−0.0196054
$469$	−6.23073	−0.287708
$470$	0	0
$471$	−16.7630	−0.772400
$472$	−0.294938	−0.0135756
$473$	11.2016	0.515048
$474$	11.2953	0.518812
$475$	−2.30252	−0.105647
$476$	−0.420815	−0.0192880
$477$	2.72665	0.124845
$478$	8.53230	0.390258
$479$	−8.21634	−0.375414	−0.187707	−	0.982225i	$-0.560106\pi$
$479$	−8.21634	−0.375414	−0.187707	+	0.982225i	$0.560106\pi$
$480$	0	0
$481$	−31.1957	−1.42240
$482$	14.0905	0.641803
$483$	−6.79845	−0.309340
$484$	−0.112051	−0.00509324
$485$	0	0
$486$	−1.46050	−0.0662498
$487$	−19.4399	−0.880905	−0.440452	−	0.897776i	$-0.645182\pi$
$487$	−19.4399	−0.880905	−0.440452	+	0.897776i	$0.645182\pi$
$488$	−12.3973	−0.561197
$489$	7.13735	0.322762
$490$	0	0
$491$	−6.54669	−0.295448	−0.147724	−	0.989029i	$-0.547195\pi$
$491$	−6.54669	−0.295448	−0.147724	+	0.989029i	$0.547195\pi$
$492$	−1.19436	−0.0538457
$493$	−30.3757	−1.36805
$494$	−2.14359	−0.0964445
$495$	0	0
$496$	2.06224	0.0925973
$497$	−6.89222	−0.309158
$498$	15.4720	0.693316
$499$	−14.7089	−0.658463	−0.329231	−	0.944249i	$-0.606790\pi$
$499$	−14.7089	−0.658463	−0.329231	+	0.944249i	$0.606790\pi$
$500$	0	0
$501$	8.81284	0.393729
$502$	−24.6447	−1.09995
$503$	39.5700	1.76434	0.882170	−	0.470931i	$-0.156082\pi$
$503$	39.5700	1.76434	0.882170	+	0.470931i	$0.156082\pi$
$504$	2.36381	0.105292
$505$	0	0
$506$	36.5035	1.62278
$507$	2.84202	0.126219
$508$	1.80953	0.0802848
$509$	−14.0220	−0.621513	−0.310757	−	0.950489i	$-0.600582\pi$
$509$	−14.0220	−0.621513	−0.310757	+	0.950489i	$0.600582\pi$
$510$	0	0
$511$	8.47917	0.375096
$512$	−19.9751	−0.882783
$513$	−0.460505	−0.0203318
$514$	12.1623	0.536454
$515$	0	0
$516$	0.467702	0.0205894
$517$	4.90662	0.215793
$518$	−12.3930	−0.544516
$519$	21.8420	0.958758
$520$	0	0
$521$	−34.8856	−1.52837	−0.764183	−	0.645000i	$-0.776857\pi$
$521$	−34.8856	−1.52837	−0.764183	+	0.645000i	$0.776857\pi$
$522$	−12.1623	−0.532327
$523$	1.35953	0.0594483	0.0297241	−	0.999558i	$-0.490537\pi$
$523$	1.35953	0.0594483	0.0297241	+	0.999558i	$0.490537\pi$
$524$	−1.60409	−0.0700750
$525$	−4.33463	−0.189179
$526$	−3.46770	−0.151199
$527$	−1.77062	−0.0771292
$528$	−13.5405	−0.589273
$529$	38.4973	1.67379
$530$	0	0
$531$	0.108168	0.00469411
$532$	−0.0531264	−0.00230332
$533$	−28.6050	−1.23902
$534$	12.9502	0.560409
$535$	0	0
$536$	−19.5969	−0.846457
$537$	17.8961	0.772274
$538$	8.44707	0.364179
$539$	19.9148	0.857790
$540$	0	0
$541$	−26.2661	−1.12927	−0.564635	−	0.825341i	$-0.690983\pi$
$541$	−26.2661	−1.12927	−0.564635	+	0.825341i	$0.690983\pi$
$542$	−8.28054	−0.355680
$543$	14.9679	0.642334
$544$	−2.74144	−0.117538
$545$	0	0
$546$	−4.03542	−0.172700
$547$	−27.1230	−1.15969	−0.579847	−	0.814725i	$-0.696888\pi$
$547$	−27.1230	−1.15969	−0.579847	+	0.814725i	$0.696888\pi$
$548$	−0.245126	−0.0104713
$549$	4.54669	0.194048
$550$	23.2743	0.992420
$551$	−3.83482	−0.163369
$552$	−21.3825	−0.910098
$553$	6.70467	0.285112
$554$	18.7774	0.797776
$555$	0	0
$556$	−1.17996	−0.0500415
$557$	−0.532298	−0.0225542	−0.0112771	−	0.999936i	$-0.503590\pi$
$557$	−0.532298	−0.0225542	−0.0112771	+	0.999936i	$0.503590\pi$
$558$	−0.708945	−0.0300120
$559$	11.2016	0.473775
$560$	0	0
$561$	11.6257	0.490837
$562$	−22.7444	−0.959413
$563$	−6.49688	−0.273811	−0.136905	−	0.990584i	$-0.543716\pi$
$563$	−6.49688	−0.273811	−0.136905	+	0.990584i	$0.543716\pi$
$564$	0.204868	0.00862648
$565$	0	0
$566$	−10.2599	−0.431256
$567$	−0.866926	−0.0364074
$568$	−21.6774	−0.909564
$569$	17.6257	0.738907	0.369454	−	0.929249i	$-0.379545\pi$
$569$	17.6257	0.738907	0.369454	+	0.929249i	$0.379545\pi$
$570$	0	0
$571$	13.8420	0.579270	0.289635	−	0.957137i	$-0.406466\pi$
$571$	13.8420	0.579270	0.289635	+	0.957137i	$0.406466\pi$
$572$	1.35177	0.0565203
$573$	10.2661	0.428875
$574$	−11.3638	−0.474316
$575$	39.2101	1.63517
$576$	7.39922	0.308301
$577$	27.7060	1.15342	0.576708	−	0.816950i	$-0.304337\pi$
$577$	27.7060	1.15342	0.576708	+	0.816950i	$0.304337\pi$
$578$	−5.39591	−0.224440
$579$	−11.5615	−0.480479
$580$	0	0
$581$	9.18384	0.381010
$582$	−13.4179	−0.556189
$583$	−8.69028	−0.359915
$584$	26.6687	1.10356
$585$	0	0
$586$	29.5261	1.21971
$587$	−8.93540	−0.368804	−0.184402	−	0.982851i	$-0.559035\pi$
$587$	−8.93540	−0.368804	−0.184402	+	0.982851i	$0.559035\pi$
$588$	0.831508	0.0342908
$589$	−0.223534	−0.00921056
$590$	0	0
$591$	25.8319	1.06258
$592$	−41.5835	−1.70907
$593$	4.71322	0.193549	0.0967743	−	0.995306i	$-0.469147\pi$
$593$	4.71322	0.193549	0.0967743	+	0.995306i	$0.469147\pi$
$594$	4.65486	0.190991
$595$	0	0
$596$	1.23697	0.0506683
$597$	7.73093	0.316406
$598$	36.5035	1.49274
$599$	44.8113	1.83094	0.915469	−	0.402388i	$-0.131820\pi$
$599$	44.8113	1.83094	0.915469	+	0.402388i	$0.131820\pi$
$600$	−13.6333	−0.556576
$601$	−15.1373	−0.617465	−0.308733	−	0.951149i	$-0.599905\pi$
$601$	−15.1373	−0.617465	−0.308733	+	0.951149i	$0.599905\pi$
$602$	4.44999	0.181368
$603$	7.18716	0.292684
$604$	1.89183	0.0769775
$605$	0	0
$606$	15.6405	0.635351
$607$	−3.07899	−0.124972	−0.0624862	−	0.998046i	$-0.519903\pi$
$607$	−3.07899	−0.124972	−0.0624862	+	0.998046i	$0.519903\pi$
$608$	−0.346097	−0.0140361
$609$	−7.21926	−0.292539
$610$	0	0
$611$	4.90662	0.198500
$612$	0.485411	0.0196216
$613$	31.0790	1.25527	0.627634	−	0.778508i	$-0.284023\pi$
$613$	31.0790	1.25527	0.627634	+	0.778508i	$0.284023\pi$
$614$	−25.1872	−1.01647
$615$	0	0
$616$	−7.53382	−0.303546
$617$	0.758757	0.0305464	0.0152732	−	0.999883i	$-0.495138\pi$
$617$	0.758757	0.0305464	0.0152732	+	0.999883i	$0.495138\pi$
$618$	−17.8420	−0.717711
$619$	5.67549	0.228117	0.114059	−	0.993474i	$-0.463615\pi$
$619$	5.67549	0.228117	0.114059	+	0.993474i	$0.463615\pi$
$620$	0	0
$621$	7.84202	0.314689
$622$	−38.1521	−1.52976
$623$	7.68696	0.307972
$624$	−13.5405	−0.542052
$625$	25.0000	1.00000
$626$	20.7420	0.829017
$627$	1.46770	0.0586144
$628$	2.23073	0.0890159
$629$	35.7031	1.42358
$630$	0	0
$631$	17.4179	0.693395	0.346698	−	0.937977i	$-0.387303\pi$
$631$	17.4179	0.693395	0.346698	+	0.937977i	$0.387303\pi$
$632$	21.0875	0.838817
$633$	−4.46050	−0.177289
$634$	13.8961	0.551885
$635$	0	0
$636$	−0.362848	−0.0143879
$637$	19.9148	0.789052
$638$	38.7630	1.53464
$639$	7.95019	0.314505
$640$	0	0
$641$	5.28093	0.208584	0.104292	−	0.994547i	$-0.466742\pi$
$641$	5.28093	0.208584	0.104292	+	0.994547i	$0.466742\pi$
$642$	26.9938	1.06536
$643$	−40.7496	−1.60701	−0.803504	−	0.595300i	$-0.797033\pi$
$643$	−40.7496	−1.60701	−0.803504	+	0.595300i	$0.797033\pi$
$644$	0.904700	0.0356502
$645$	0	0
$646$	2.45331	0.0965241
$647$	17.1475	0.674137	0.337068	−	0.941480i	$-0.390565\pi$
$647$	17.1475	0.674137	0.337068	+	0.941480i	$0.390565\pi$
$648$	−2.72665	−0.107113
$649$	−0.344750	−0.0135326
$650$	23.2743	0.912893
$651$	−0.420815	−0.0164930
$652$	−0.949799	−0.0371970
$653$	−22.9938	−0.899815	−0.449908	−	0.893075i	$-0.648543\pi$
$653$	−22.9938	−0.899815	−0.449908	+	0.893075i	$0.648543\pi$
$654$	−24.9971	−0.977463
$655$	0	0
$656$	−38.1301	−1.48873
$657$	−9.78074	−0.381583
$658$	1.94923	0.0759889
$659$	7.09766	0.276485	0.138243	−	0.990398i	$-0.455855\pi$
$659$	7.09766	0.276485	0.138243	+	0.990398i	$0.455855\pi$
$660$	0	0
$661$	−41.8506	−1.62780	−0.813899	−	0.581006i	$-0.802659\pi$
$661$	−41.8506	−1.62780	−0.813899	+	0.581006i	$0.802659\pi$
$662$	−18.4097	−0.715515
$663$	11.6257	0.451504
$664$	28.8850	1.12096
$665$	0	0
$666$	14.2953	0.553933
$667$	65.3039	2.52858
$668$	−1.17276	−0.0453756
$669$	1.00000	0.0386622
$670$	0	0
$671$	−14.4910	−0.559420
$672$	−0.651546	−0.0251339
$673$	31.2058	1.20290	0.601448	−	0.798912i	$-0.294591\pi$
$673$	31.2058	1.20290	0.601448	+	0.798912i	$0.294591\pi$
$674$	22.8856	0.881520
$675$	5.00000	0.192450
$676$	−0.378200	−0.0145462
$677$	32.0364	1.23126	0.615629	−	0.788036i	$-0.288902\pi$
$677$	32.0364	1.23126	0.615629	+	0.788036i	$0.288902\pi$
$678$	−28.5907	−1.09802
$679$	−7.96458	−0.305653
$680$	0	0
$681$	16.2163	0.621411
$682$	2.25952	0.0865215
$683$	−44.3753	−1.69797	−0.848986	−	0.528415i	$-0.822787\pi$
$683$	−44.3753	−1.69797	−0.848986	+	0.528415i	$0.822787\pi$
$684$	0.0612814	0.00234315
$685$	0	0
$686$	16.7745	0.640453
$687$	22.4825	0.857760
$688$	14.9315	0.569259
$689$	−8.69028	−0.331073
$690$	0	0
$691$	−3.73385	−0.142042	−0.0710212	−	0.997475i	$-0.522626\pi$
$691$	−3.73385	−0.142042	−0.0710212	+	0.997475i	$0.522626\pi$
$692$	−2.90662	−0.110493
$693$	2.76303	0.104959
$694$	3.13212	0.118894
$695$	0	0
$696$	−22.7060	−0.860669
$697$	32.7381	1.24004
$698$	7.68072	0.290720
$699$	21.5759	0.816074
$700$	0.576828	0.0218021
$701$	29.1010	1.09913	0.549564	−	0.835451i	$-0.314794\pi$
$701$	29.1010	1.09913	0.549564	+	0.835451i	$0.314794\pi$
$702$	4.65486	0.175686
$703$	4.50739	0.170000
$704$	−23.5825	−0.888799
$705$	0	0
$706$	−28.9109	−1.08808
$707$	9.28386	0.349155
$708$	−0.0143945	−0.000540977 0
$709$	6.76303	0.253991	0.126995	−	0.991903i	$-0.459467\pi$
$709$	6.76303	0.253991	0.126995	+	0.991903i	$0.459467\pi$
$710$	0	0
$711$	−7.73385	−0.290042
$712$	24.1770	0.906073
$713$	3.80660	0.142558
$714$	4.61849	0.172842
$715$	0	0
$716$	−2.38151	−0.0890014
$717$	−5.84202	−0.218174
$718$	40.3068	1.50424
$719$	−18.1158	−0.675604	−0.337802	−	0.941217i	$-0.609683\pi$
$719$	−18.1158	−0.675604	−0.337802	+	0.941217i	$0.609683\pi$
$720$	0	0
$721$	−10.5907	−0.394417
$722$	−27.4399	−1.02121
$723$	−9.64766	−0.358800
$724$	−1.99185	−0.0740263
$725$	41.6372	1.54637
$726$	1.22977	0.0456412
$727$	−20.9836	−0.778240	−0.389120	−	0.921187i	$-0.627221\pi$
$727$	−20.9836	−0.778240	−0.389120	+	0.921187i	$0.627221\pi$
$728$	−7.53382	−0.279222
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.8200	−0.474166
$732$	−0.605049	−0.0223632
$733$	19.8391	0.732774	0.366387	−	0.930463i	$-0.380595\pi$
$733$	19.8391	0.732774	0.366387	+	0.930463i	$0.380595\pi$
$734$	−47.3681	−1.74839
$735$	0	0
$736$	5.89375	0.217246
$737$	−22.9066	−0.843776
$738$	13.1082	0.482518
$739$	−25.2370	−0.928357	−0.464178	−	0.885742i	$-0.653650\pi$
$739$	−25.2370	−0.928357	−0.464178	+	0.885742i	$0.653650\pi$
$740$	0	0
$741$	1.46770	0.0539174
$742$	−3.45235	−0.126740
$743$	18.2881	0.670926	0.335463	−	0.942053i	$-0.391107\pi$
$743$	18.2881	0.670926	0.335463	+	0.942053i	$0.391107\pi$
$744$	−1.32355	−0.0485236
$745$	0	0
$746$	−32.9210	−1.20532
$747$	−10.5936	−0.387599
$748$	−1.54708	−0.0565669
$749$	16.0229	0.585465
$750$	0	0
$751$	−15.9722	−0.582833	−0.291416	−	0.956596i	$-0.594127\pi$
$751$	−15.9722	−0.582833	−0.291416	+	0.956596i	$0.594127\pi$
$752$	6.54045	0.238506
$753$	16.8741	0.614927
$754$	38.7630	1.41167
$755$	0	0
$756$	0.115366	0.00419581
$757$	6.81284	0.247617	0.123808	−	0.992306i	$-0.460489\pi$
$757$	6.81284	0.247617	0.123808	+	0.992306i	$0.460489\pi$
$758$	38.3058	1.39133
$759$	−24.9938	−0.907216
$760$	0	0
$761$	−5.84202	−0.211773	−0.105887	−	0.994378i	$-0.533768\pi$
$761$	−5.84202	−0.211773	−0.105887	+	0.994378i	$0.533768\pi$
$762$	−19.8597	−0.719442
$763$	−14.8377	−0.537163
$764$	−1.36616	−0.0494260
$765$	0	0
$766$	49.1957	1.77751
$767$	−0.344750	−0.0124482
$768$	−3.17996	−0.114747
$769$	17.5873	0.634216	0.317108	−	0.948389i	$-0.397288\pi$
$769$	17.5873	0.634216	0.317108	+	0.948389i	$0.397288\pi$
$770$	0	0
$771$	−8.32743	−0.299905
$772$	1.53854	0.0553732
$773$	49.3183	1.77385	0.886927	−	0.461909i	$-0.152835\pi$
$773$	49.3183	1.77385	0.886927	+	0.461909i	$0.152835\pi$
$774$	−5.13307	−0.184505
$775$	2.42705	0.0871824
$776$	−25.0502	−0.899250
$777$	8.48541	0.304412
$778$	−48.0148	−1.72141
$779$	4.13307	0.148083
$780$	0	0
$781$	−25.3385	−0.906683
$782$	−41.7778	−1.49397
$783$	8.32743	0.297598
$784$	26.5461	0.948076
$785$	0	0
$786$	17.6050	0.627951
$787$	52.5054	1.87162	0.935808	−	0.352510i	$-0.114672\pi$
$787$	52.5054	1.87162	0.935808	+	0.352510i	$0.114672\pi$
$788$	−3.43757	−0.122458
$789$	2.37432	0.0845279
$790$	0	0
$791$	−16.9708	−0.603413
$792$	8.69028	0.308796
$793$	−14.4910	−0.514592
$794$	7.64862	0.271439
$795$	0	0
$796$	−1.02879	−0.0364645
$797$	−13.0833	−0.463433	−0.231716	−	0.972783i	$-0.574434\pi$
$797$	−13.0833	−0.463433	−0.231716	+	0.972783i	$0.574434\pi$
$798$	0.583068	0.0206404
$799$	−5.61556	−0.198664
$800$	3.75780	0.132858
$801$	−8.86693	−0.313297
$802$	21.8243	0.770643
$803$	31.1728	1.10006
$804$	−0.956427	−0.0337306
$805$	0	0
$806$	2.25952	0.0795882
$807$	−5.78366	−0.203595
$808$	29.1996	1.02724
$809$	−26.2163	−0.921717	−0.460859	−	0.887474i	$-0.652458\pi$
$809$	−26.2163	−0.921717	−0.460859	+	0.887474i	$0.652458\pi$
$810$	0	0
$811$	6.31011	0.221578	0.110789	−	0.993844i	$-0.464662\pi$
$811$	6.31011	0.221578	0.110789	+	0.993844i	$0.464662\pi$
$812$	0.960699	0.0337139
$813$	5.66964	0.198843
$814$	−45.5615	−1.59693
$815$	0	0
$816$	15.4969	0.542499
$817$	−1.61849	−0.0566236
$818$	−7.02918	−0.245769
$819$	2.76303	0.0965481
$820$	0	0
$821$	−1.63327	−0.0570015	−0.0285007	−	0.999594i	$-0.509073\pi$
$821$	−1.63327	−0.0570015	−0.0285007	+	0.999594i	$0.509073\pi$
$822$	2.69028	0.0938342
$823$	2.59065	0.0903045	0.0451523	−	0.998980i	$-0.485623\pi$
$823$	2.59065	0.0903045	0.0451523	+	0.998980i	$0.485623\pi$
$824$	−33.3097	−1.16040
$825$	−15.9358	−0.554813
$826$	−0.136957	−0.00476536
$827$	−36.7132	−1.27664	−0.638322	−	0.769770i	$-0.720371\pi$
$827$	−36.7132	−1.27664	−0.638322	+	0.769770i	$0.720371\pi$
$828$	−1.04357	−0.0362667
$829$	−22.8128	−0.792323	−0.396161	−	0.918181i	$-0.629658\pi$
$829$	−22.8128	−0.792323	−0.396161	+	0.918181i	$0.629658\pi$
$830$	0	0
$831$	−12.8568	−0.445998
$832$	−23.5825	−0.817576
$833$	−22.7922	−0.789703
$834$	12.9502	0.448428
$835$	0	0
$836$	−0.195314	−0.00675507
$837$	0.485411	0.0167783
$838$	12.0000	0.414533
$839$	23.8860	0.824636	0.412318	−	0.911040i	$-0.364719\pi$
$839$	23.8860	0.824636	0.412318	+	0.911040i	$0.364719\pi$
$840$	0	0
$841$	40.3461	1.39124
$842$	53.1373	1.83123
$843$	15.5729	0.536361
$844$	0.593579	0.0204318
$845$	0	0
$846$	−2.24844	−0.0773030
$847$	0.729968	0.0250820
$848$	−11.5840	−0.397797
$849$	7.02491	0.241094
$850$	−26.6372	−0.913647
$851$	−76.7572	−2.63120
$852$	−1.05797	−0.0362454
$853$	−42.0440	−1.43956	−0.719779	−	0.694203i	$-0.755757\pi$
$853$	−42.0440	−1.43956	−0.719779	+	0.694203i	$0.755757\pi$
$854$	−5.75679	−0.196993
$855$	0	0
$856$	50.3953	1.72248
$857$	−32.8755	−1.12300	−0.561502	−	0.827475i	$-0.689776\pi$
$857$	−32.8755	−1.12300	−0.561502	+	0.827475i	$0.689776\pi$
$858$	−14.8358	−0.506485
$859$	7.73385	0.263876	0.131938	−	0.991258i	$-0.457880\pi$
$859$	7.73385	0.263876	0.131938	+	0.991258i	$0.457880\pi$
$860$	0	0
$861$	7.78074	0.265167
$862$	−4.03542	−0.137447
$863$	20.9296	0.712450	0.356225	−	0.934400i	$-0.384064\pi$
$863$	20.9296	0.712450	0.356225	+	0.934400i	$0.384064\pi$
$864$	0.751560	0.0255686
$865$	0	0
$866$	54.5700	1.85436
$867$	3.69455	0.125474
$868$	0.0559998	0.00190076
$869$	24.6490	0.836160
$870$	0	0
$871$	−22.9066	−0.776161
$872$	−46.6677	−1.58037
$873$	9.18716	0.310938
$874$	−5.27430	−0.178406
$875$	0	0
$876$	1.30157	0.0439759
$877$	9.07314	0.306378	0.153189	−	0.988197i	$-0.451046\pi$
$877$	9.07314	0.306378	0.153189	+	0.988197i	$0.451046\pi$
$878$	−1.96458	−0.0663014
$879$	−20.2163	−0.681880
$880$	0	0
$881$	21.1986	0.714200	0.357100	−	0.934066i	$-0.383766\pi$
$881$	21.1986	0.714200	0.357100	+	0.934066i	$0.383766\pi$
$882$	−9.12588	−0.307284
$883$	37.2599	1.25390	0.626948	−	0.779061i	$-0.284304\pi$
$883$	37.2599	1.25390	0.626948	+	0.779061i	$0.284304\pi$
$884$	−1.54708	−0.0520340
$885$	0	0
$886$	10.8276	0.363761
$887$	−14.1609	−0.475477	−0.237738	−	0.971329i	$-0.576406\pi$
$887$	−14.1609	−0.475477	−0.237738	+	0.971329i	$0.576406\pi$
$888$	26.6883	0.895601
$889$	−11.7883	−0.395368
$890$	0	0
$891$	−3.18716	−0.106774
$892$	−0.133074	−0.00445566
$893$	−0.708945	−0.0237239
$894$	−13.5759	−0.454045
$895$	0	0
$896$	−10.6716	−0.356514
$897$	−24.9938	−0.834517
$898$	0.798447	0.0266445
$899$	4.04223	0.134816
$900$	−0.665372	−0.0221791
$901$	9.94592	0.331346
$902$	−41.7778	−1.39105
$903$	−3.04689	−0.101394
$904$	−53.3766	−1.77528
$905$	0	0
$906$	−20.7630	−0.689805
$907$	2.76010	0.0916478	0.0458239	−	0.998950i	$-0.485409\pi$
$907$	2.76010	0.0916478	0.0458239	+	0.998950i	$0.485409\pi$
$908$	−2.15798	−0.0716151
$909$	−10.7089	−0.355193
$910$	0	0
$911$	−8.48676	−0.281179	−0.140589	−	0.990068i	$-0.544900\pi$
$911$	−8.48676	−0.281179	−0.140589	+	0.990068i	$0.544900\pi$
$912$	1.95643	0.0647838
$913$	33.7634	1.11741
$914$	18.5552	0.613753
$915$	0	0
$916$	−2.99185	−0.0988533
$917$	10.4500	0.345089
$918$	−5.32743	−0.175831
$919$	−24.3245	−0.802391	−0.401196	−	0.915992i	$-0.631405\pi$
$919$	−24.3245	−0.802391	−0.401196	+	0.915992i	$0.631405\pi$
$920$	0	0
$921$	17.2455	0.568259
$922$	51.6385	1.70062
$923$	−25.3385	−0.834027
$924$	−0.367689	−0.0120961
$925$	−48.9397	−1.60913
$926$	−27.2202	−0.894512
$927$	12.2163	0.401237
$928$	6.25856	0.205447
$929$	−1.37297	−0.0450457	−0.0225228	−	0.999746i	$-0.507170\pi$
$929$	−1.37297	−0.0450457	−0.0225228	+	0.999746i	$0.507170\pi$
$930$	0	0
$931$	−2.87744	−0.0943042
$932$	−2.87120	−0.0940492
$933$	26.1226	0.855214
$934$	47.0521	1.53959
$935$	0	0
$936$	8.69028	0.284051
$937$	−45.9502	−1.50113	−0.750564	−	0.660798i	$-0.770218\pi$
$937$	−45.9502	−1.50113	−0.750564	+	0.660798i	$0.770218\pi$
$938$	−9.10001	−0.297126
$939$	−14.2019	−0.463463
$940$	0	0
$941$	−34.4615	−1.12341	−0.561706	−	0.827337i	$-0.689855\pi$
$941$	−34.4615	−1.12341	−0.561706	+	0.827337i	$0.689855\pi$
$942$	−24.4825	−0.797682
$943$	−70.3829	−2.29198
$944$	−0.459547	−0.0149570
$945$	0	0
$946$	16.3599	0.531907
$947$	−15.7263	−0.511035	−0.255517	−	0.966804i	$-0.582246\pi$
$947$	−15.7263	−0.511035	−0.255517	+	0.966804i	$0.582246\pi$
$948$	1.02918	0.0334262
$949$	31.1728	1.01191
$950$	−3.36285	−0.109105
$951$	−9.51459	−0.308532
$952$	8.62237	0.279453
$953$	−38.7926	−1.25662	−0.628308	−	0.777965i	$-0.716252\pi$
$953$	−38.7926	−1.25662	−0.628308	+	0.777965i	$0.716252\pi$
$954$	3.98229	0.128931
$955$	0	0
$956$	0.777424	0.0251437
$957$	−26.5408	−0.857944
$958$	−12.0000	−0.387702
$959$	1.59689	0.0515664
$960$	0	0
$961$	−30.7644	−0.992399
$962$	−45.5615	−1.46896
$963$	−18.4825	−0.595590
$964$	1.28386	0.0413503
$965$	0	0
$966$	−9.92916	−0.319466
$967$	15.8420	0.509445	0.254723	−	0.967014i	$-0.418016\pi$
$967$	15.8420	0.509445	0.254723	+	0.967014i	$0.418016\pi$
$968$	2.29590	0.0737929
$969$	−1.67977	−0.0539619
$970$	0	0
$971$	25.3097	0.812228	0.406114	−	0.913823i	$-0.366884\pi$
$971$	25.3097	0.812228	0.406114	+	0.913823i	$0.366884\pi$
$972$	−0.133074	−0.00426837
$973$	7.68696	0.246433
$974$	−28.3920	−0.909739
$975$	−15.9358	−0.510354
$976$	−19.3164	−0.618301
$977$	3.90623	0.124971	0.0624856	−	0.998046i	$-0.480097\pi$
$977$	3.90623	0.124971	0.0624856	+	0.998046i	$0.480097\pi$
$978$	10.4241	0.333327
$979$	28.2603	0.903203
$980$	0	0
$981$	17.1154	0.546452
$982$	−9.56148	−0.305119
$983$	2.59650	0.0828156	0.0414078	−	0.999142i	$-0.486816\pi$
$983$	2.59650	0.0828156	0.0414078	+	0.999142i	$0.486816\pi$
$984$	24.4720	0.780138
$985$	0	0
$986$	−44.3638	−1.41283
$987$	−1.33463	−0.0424817
$988$	−0.195314	−0.00621376
$989$	27.5615	0.876404
$990$	0	0
$991$	18.6903	0.593716	0.296858	−	0.954922i	$-0.404061\pi$
$991$	18.6903	0.593716	0.296858	+	0.954922i	$0.404061\pi$
$992$	0.364815	0.0115829
$993$	12.6050	0.400009
$994$	−10.0661	−0.319278
$995$	0	0
$996$	1.40974	0.0446692
$997$	42.4297	1.34376	0.671882	−	0.740658i	$-0.265486\pi$
$997$	42.4297	1.34376	0.671882	+	0.740658i	$0.265486\pi$
$998$	−21.4825	−0.680016
$999$	−9.78794	−0.309677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.2.a.e.1.3	✓	3
3.2	odd	2		2007.2.a.h.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.2.a.e.1.3	✓	3	1.1	even	1	trivial
2007.2.a.h.1.1		3	3.2	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:	\( N \)	\(=\)	\( 669 = 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	669.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46050 q^{2} -1.00000 q^{3} +0.133074 q^{4} -1.46050 q^{6} -0.866926 q^{7} -2.72665 q^{8} +1.00000 q^{9} -3.18716 q^{11} -0.133074 q^{12} -3.18716 q^{13} -1.26615 q^{14} -4.24844 q^{16} +3.64766 q^{17} +1.46050 q^{18} +0.460505 q^{19} +0.866926 q^{21} -4.65486 q^{22} -7.84202 q^{23} +2.72665 q^{24} -5.00000 q^{25} -4.65486 q^{26} -1.00000 q^{27} -0.115366 q^{28} -8.32743 q^{29} -0.485411 q^{31} -0.751560 q^{32} +3.18716 q^{33} +5.32743 q^{34} +0.133074 q^{36} +9.78794 q^{37} +0.672570 q^{38} +3.18716 q^{39} +8.97509 q^{41} +1.26615 q^{42} -3.51459 q^{43} -0.424129 q^{44} -11.4533 q^{46} -1.53950 q^{47} +4.24844 q^{48} -6.24844 q^{49} -7.30252 q^{50} -3.64766 q^{51} -0.424129 q^{52} +2.72665 q^{53} -1.46050 q^{54} +2.36381 q^{56} -0.460505 q^{57} -12.1623 q^{58} +0.108168 q^{59} +4.54669 q^{61} -0.708945 q^{62} -0.866926 q^{63} +7.39922 q^{64} +4.65486 q^{66} +7.18716 q^{67} +0.485411 q^{68} +7.84202 q^{69} +7.95019 q^{71} -2.72665 q^{72} -9.78074 q^{73} +14.2953 q^{74} +5.00000 q^{75} +0.0612814 q^{76} +2.76303 q^{77} +4.65486 q^{78} -7.73385 q^{79} +1.00000 q^{81} +13.1082 q^{82} -10.5936 q^{83} +0.115366 q^{84} -5.13307 q^{86} +8.32743 q^{87} +8.69028 q^{88} -8.86693 q^{89} +2.76303 q^{91} -1.04357 q^{92} +0.485411 q^{93} -2.24844 q^{94} +0.751560 q^{96} +9.18716 q^{97} -9.12588 q^{98} -3.18716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 2 q^{6} + q^{7} - 9 q^{8} + 3 q^{9} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 11 q^{14} + 10 q^{16} - q^{17} - 2 q^{18} - 5 q^{19} - q^{21} + 6 q^{22} + 2 q^{23} + 9 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 2 * q^6 + q^7 - 9 * q^8 + 3 * q^9 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 11 * q^14 + 10 * q^16 - q^17 - 2 * q^18 - 5 * q^19 - q^21 + 6 * q^22 + 2 * q^23 + 9 * q^24 - 15 * q^25 + 6 * q^26 - 3 * q^27 + 26 * q^28 - 15 * q^29 - 17 * q^31 - 25 * q^32 + 4 * q^33 + 6 * q^34 + 4 * q^36 + 13 * q^37 + 12 * q^38 + 4 * q^39 + 5 * q^41 + 11 * q^42 + 5 * q^43 - 34 * q^44 - 36 * q^46 - 11 * q^47 - 10 * q^48 + 4 * q^49 + 10 * q^50 + q^51 - 34 * q^52 + 9 * q^53 + 2 * q^54 - 42 * q^56 + 5 * q^57 - 9 * q^58 - 18 * q^59 + 12 * q^61 + 27 * q^62 + q^63 + 33 * q^64 - 6 * q^66 + 16 * q^67 + 17 * q^68 - 2 * q^69 - 20 * q^71 - 9 * q^72 - 21 * q^73 + 19 * q^74 + 15 * q^75 - 17 * q^76 - 30 * q^77 - 6 * q^78 - 16 * q^79 + 3 * q^81 + 21 * q^82 - 29 * q^83 - 26 * q^84 - 19 * q^86 + 15 * q^87 + 66 * q^88 - 23 * q^89 - 30 * q^91 + 44 * q^92 + 17 * q^93 + 16 * q^94 + 25 * q^96 + 22 * q^97 - 39 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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