LMFDB - Embedded newform 668.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.33400685502$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.69135$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} +O(q^{10})$	$q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} -1.62500 q^{11} +3.38270 q^{13} -3.10403 q^{15} -5.30425 q^{17} +6.97394 q^{19} -6.96184 q^{21} +0.132692 q^{23} -1.00000 q^{25} +5.57367 q^{27} -7.86837 q^{29} +10.5868 q^{31} +2.52203 q^{33} +8.97134 q^{35} +1.72133 q^{37} -5.25001 q^{39} +8.40828 q^{41} -6.76540 q^{43} -1.18249 q^{45} +0.208543 q^{47} +13.1212 q^{49} +8.23228 q^{51} +8.68695 q^{53} -3.25001 q^{55} -10.8237 q^{57} +13.8152 q^{59} -3.06741 q^{61} -2.65212 q^{63} +6.76540 q^{65} +7.51019 q^{67} -0.205940 q^{69} +2.06752 q^{71} +5.43694 q^{73} +1.55202 q^{75} -7.28924 q^{77} -17.5256 q^{79} -6.87670 q^{81} -12.1481 q^{83} -10.6085 q^{85} +12.2118 q^{87} -10.8020 q^{89} +15.1737 q^{91} -16.4310 q^{93} +13.9479 q^{95} -4.21809 q^{97} +0.960773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9	$ 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9} + 5 q^{11} - 4 q^{13} + 6 q^{15} - 2 q^{17} + 5 q^{19} - 4 q^{21} + 6 q^{23} - 5 q^{25} + 12 q^{27} - 5 q^{29} + 9 q^{31} - 8 q^{33} + 18 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{45} + 13 q^{47} + 14 q^{49} - 2 q^{53} + 10 q^{55} - 12 q^{57} + 4 q^{59} + 11 q^{61} - q^{63} - 8 q^{65} + 28 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} - 3 q^{75} - 12 q^{77} - 10 q^{79} - 15 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{87} - 17 q^{89} - 12 q^{91} - 12 q^{93} + 10 q^{95} - 27 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9 + 5 * q^11 - 4 * q^13 + 6 * q^15 - 2 * q^17 + 5 * q^19 - 4 * q^21 + 6 * q^23 - 5 * q^25 + 12 * q^27 - 5 * q^29 + 9 * q^31 - 8 * q^33 + 18 * q^35 + 8 * q^37 - 4 * q^41 + 8 * q^43 + 12 * q^45 + 13 * q^47 + 14 * q^49 - 2 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 + 11 * q^61 - q^63 - 8 * q^65 + 28 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 - 3 * q^75 - 12 * q^77 - 10 * q^79 - 15 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^87 - 17 * q^89 - 12 * q^91 - 12 * q^93 + 10 * q^95 - 27 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

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	0
$2$	0	0
$3$	−1.55202	−0.896057	−0.448029	−	0.894019i	$-0.647874\pi$
$3$	−1.55202	−0.896057	−0.448029	+	0.894019i	$0.647874\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	4.48567	1.69542	0.847712	−	0.530457i	$-0.177979\pi$
$7$	4.48567	1.69542	0.847712	+	0.530457i	$0.177979\pi$
$8$	0	0
$9$	−0.591244	−0.197081
$10$	0	0
$11$	−1.62500	−0.489957	−0.244979	−	0.969528i	$-0.578781\pi$
$11$	−1.62500	−0.489957	−0.244979	+	0.969528i	$0.578781\pi$
$12$	0	0
$13$	3.38270	0.938192	0.469096	−	0.883147i	$-0.344580\pi$
$13$	3.38270	0.938192	0.469096	+	0.883147i	$0.344580\pi$
$14$	0	0
$15$	−3.10403	−0.801458
$16$	0	0
$17$	−5.30425	−1.28647	−0.643235	−	0.765669i	$-0.722408\pi$
$17$	−5.30425	−1.28647	−0.643235	+	0.765669i	$0.722408\pi$
$18$	0	0
$19$	6.97394	1.59993	0.799966	−	0.600045i	$-0.204851\pi$
$19$	6.97394	1.59993	0.799966	+	0.600045i	$0.204851\pi$
$20$	0	0
$21$	−6.96184	−1.51920
$22$	0	0
$23$	0.132692	0.0276682	0.0138341	−	0.999904i	$-0.495596\pi$
$23$	0.132692	0.0276682	0.0138341	+	0.999904i	$0.495596\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.57367	1.07265
$28$	0	0
$29$	−7.86837	−1.46112	−0.730560	−	0.682849i	$-0.760741\pi$
$29$	−7.86837	−1.46112	−0.730560	+	0.682849i	$0.760741\pi$
$30$	0	0
$31$	10.5868	1.90145	0.950726	−	0.310031i	$-0.100339\pi$
$31$	10.5868	1.90145	0.950726	+	0.310031i	$0.100339\pi$
$32$	0	0
$33$	2.52203	0.439030
$34$	0	0
$35$	8.97134	1.51643
$36$	0	0
$37$	1.72133	0.282985	0.141493	−	0.989939i	$-0.454810\pi$
$37$	1.72133	0.282985	0.141493	+	0.989939i	$0.454810\pi$
$38$	0	0
$39$	−5.25001	−0.840674
$40$	0	0
$41$	8.40828	1.31315	0.656576	−	0.754259i	$-0.272004\pi$
$41$	8.40828	1.31315	0.656576	+	0.754259i	$0.272004\pi$
$42$	0	0
$43$	−6.76540	−1.03171	−0.515857	−	0.856675i	$-0.672526\pi$
$43$	−6.76540	−1.03171	−0.515857	+	0.856675i	$0.672526\pi$
$44$	0	0
$45$	−1.18249	−0.176275
$46$	0	0
$47$	0.208543	0.0304191	0.0152095	−	0.999884i	$-0.495158\pi$
$47$	0.208543	0.0304191	0.0152095	+	0.999884i	$0.495158\pi$
$48$	0	0
$49$	13.1212	1.87446
$50$	0	0
$51$	8.23228	1.15275
$52$	0	0
$53$	8.68695	1.19324	0.596622	−	0.802522i	$-0.296509\pi$
$53$	8.68695	1.19324	0.596622	+	0.802522i	$0.296509\pi$
$54$	0	0
$55$	−3.25001	−0.438231
$56$	0	0
$57$	−10.8237	−1.43363
$58$	0	0
$59$	13.8152	1.79859	0.899293	−	0.437347i	$-0.144082\pi$
$59$	13.8152	1.79859	0.899293	+	0.437347i	$0.144082\pi$
$60$	0	0
$61$	−3.06741	−0.392742	−0.196371	−	0.980530i	$-0.562916\pi$
$61$	−3.06741	−0.392742	−0.196371	+	0.980530i	$0.562916\pi$
$62$	0	0
$63$	−2.65212	−0.334136
$64$	0	0
$65$	6.76540	0.839145
$66$	0	0
$67$	7.51019	0.917515	0.458758	−	0.888561i	$-0.348295\pi$
$67$	7.51019	0.917515	0.458758	+	0.888561i	$0.348295\pi$
$68$	0	0
$69$	−0.205940	−0.0247923
$70$	0	0
$71$	2.06752	0.245370	0.122685	−	0.992446i	$-0.460850\pi$
$71$	2.06752	0.245370	0.122685	+	0.992446i	$0.460850\pi$
$72$	0	0
$73$	5.43694	0.636346	0.318173	−	0.948033i	$-0.396931\pi$
$73$	5.43694	0.636346	0.318173	+	0.948033i	$0.396931\pi$
$74$	0	0
$75$	1.55202	0.179211
$76$	0	0
$77$	−7.28924	−0.830686
$78$	0	0
$79$	−17.5256	−1.97178	−0.985892	−	0.167383i	$-0.946468\pi$
$79$	−17.5256	−1.97178	−0.985892	+	0.167383i	$0.946468\pi$
$80$	0	0
$81$	−6.87670	−0.764078
$82$	0	0
$83$	−12.1481	−1.33343	−0.666714	−	0.745314i	$-0.732300\pi$
$83$	−12.1481	−1.33343	−0.666714	+	0.745314i	$0.732300\pi$
$84$	0	0
$85$	−10.6085	−1.15065
$86$	0	0
$87$	12.2118	1.30925
$88$	0	0
$89$	−10.8020	−1.14501	−0.572506	−	0.819900i	$-0.694029\pi$
$89$	−10.8020	−1.14501	−0.572506	+	0.819900i	$0.694029\pi$
$90$	0	0
$91$	15.1737	1.59063
$92$	0	0
$93$	−16.4310	−1.70381
$94$	0	0
$95$	13.9479	1.43102
$96$	0	0
$97$	−4.21809	−0.428282	−0.214141	−	0.976803i	$-0.568695\pi$
$97$	−4.21809	−0.428282	−0.214141	+	0.976803i	$0.568695\pi$
$98$	0	0
$99$	0.960773	0.0965614
$100$	0	0
$101$	−2.33291	−0.232133	−0.116066	−	0.993241i	$-0.537029\pi$
$101$	−2.33291	−0.232133	−0.116066	+	0.993241i	$0.537029\pi$
$102$	0	0
$103$	−11.3121	−1.11461	−0.557307	−	0.830306i	$-0.688165\pi$
$103$	−11.3121	−1.11461	−0.557307	+	0.830306i	$0.688165\pi$
$104$	0	0
$105$	−13.9237	−1.35881
$106$	0	0
$107$	11.0942	1.07251	0.536257	−	0.844055i	$-0.319838\pi$
$107$	11.0942	1.07251	0.536257	+	0.844055i	$0.319838\pi$
$108$	0	0
$109$	−2.77633	−0.265924	−0.132962	−	0.991121i	$-0.542449\pi$
$109$	−2.77633	−0.265924	−0.132962	+	0.991121i	$0.542449\pi$
$110$	0	0
$111$	−2.67154	−0.253571
$112$	0	0
$113$	−17.2579	−1.62348	−0.811742	−	0.584017i	$-0.801480\pi$
$113$	−17.2579	−1.62348	−0.811742	+	0.584017i	$0.801480\pi$
$114$	0	0
$115$	0.265384	0.0247472
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−23.7931	−2.18111
$120$	0	0
$121$	−8.35936	−0.759942
$122$	0	0
$123$	−13.0498	−1.17666
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−8.31768	−0.738075	−0.369037	−	0.929415i	$-0.620313\pi$
$127$	−8.31768	−0.738075	−0.369037	+	0.929415i	$0.620313\pi$
$128$	0	0
$129$	10.5000	0.924475
$130$	0	0
$131$	−5.84309	−0.510513	−0.255257	−	0.966873i	$-0.582160\pi$
$131$	−5.84309	−0.510513	−0.255257	+	0.966873i	$0.582160\pi$
$132$	0	0
$133$	31.2828	2.71256
$134$	0	0
$135$	11.1473	0.959410
$136$	0	0
$137$	13.6762	1.16843	0.584217	−	0.811598i	$-0.301402\pi$
$137$	13.6762	1.16843	0.584217	+	0.811598i	$0.301402\pi$
$138$	0	0
$139$	10.2998	0.873618	0.436809	−	0.899554i	$-0.356109\pi$
$139$	10.2998	0.873618	0.436809	+	0.899554i	$0.356109\pi$
$140$	0	0
$141$	−0.323662	−0.0272572
$142$	0	0
$143$	−5.49690	−0.459674
$144$	0	0
$145$	−15.7367	−1.30687
$146$	0	0
$147$	−20.3644	−1.67963
$148$	0	0
$149$	−7.84309	−0.642531	−0.321266	−	0.946989i	$-0.604108\pi$
$149$	−7.84309	−0.642531	−0.321266	+	0.946989i	$0.604108\pi$
$150$	0	0
$151$	−15.5232	−1.26326	−0.631632	−	0.775268i	$-0.717615\pi$
$151$	−15.5232	−1.26326	−0.631632	+	0.775268i	$0.717615\pi$
$152$	0	0
$153$	3.13610	0.253539
$154$	0	0
$155$	21.1737	1.70071
$156$	0	0
$157$	17.4174	1.39006	0.695030	−	0.718980i	$-0.255391\pi$
$157$	17.4174	1.39006	0.695030	+	0.718980i	$0.255391\pi$
$158$	0	0
$159$	−13.4823	−1.06922
$160$	0	0
$161$	0.595212	0.0469093
$162$	0	0
$163$	23.9912	1.87914	0.939568	−	0.342363i	$-0.111227\pi$
$163$	23.9912	1.87914	0.939568	+	0.342363i	$0.111227\pi$
$164$	0	0
$165$	5.04407	0.392680
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−1.55733	−0.119795
$170$	0	0
$171$	−4.12330	−0.315317
$172$	0	0
$173$	−1.68232	−0.127904	−0.0639522	−	0.997953i	$-0.520371\pi$
$173$	−1.68232	−0.127904	−0.0639522	+	0.997953i	$0.520371\pi$
$174$	0	0
$175$	−4.48567	−0.339085
$176$	0	0
$177$	−21.4414	−1.61164
$178$	0	0
$179$	−0.760672	−0.0568553	−0.0284276	−	0.999596i	$-0.509050\pi$
$179$	−0.760672	−0.0568553	−0.0284276	+	0.999596i	$0.509050\pi$
$180$	0	0
$181$	−6.35665	−0.472486	−0.236243	−	0.971694i	$-0.575916\pi$
$181$	−6.35665	−0.472486	−0.236243	+	0.971694i	$0.575916\pi$
$182$	0	0
$183$	4.76067	0.351919
$184$	0	0
$185$	3.44267	0.253110
$186$	0	0
$187$	8.61943	0.630315
$188$	0	0
$189$	25.0017	1.81860
$190$	0	0
$191$	11.1989	0.810327	0.405163	−	0.914244i	$-0.367215\pi$
$191$	11.1989	0.810327	0.405163	+	0.914244i	$0.367215\pi$
$192$	0	0
$193$	−14.9471	−1.07592	−0.537959	−	0.842971i	$-0.680804\pi$
$193$	−14.9471	−1.07592	−0.537959	+	0.842971i	$0.680804\pi$
$194$	0	0
$195$	−10.5000	−0.751922
$196$	0	0
$197$	18.0775	1.28797	0.643984	−	0.765039i	$-0.277280\pi$
$197$	18.0775	1.28797	0.643984	+	0.765039i	$0.277280\pi$
$198$	0	0
$199$	−3.67617	−0.260596	−0.130298	−	0.991475i	$-0.541593\pi$
$199$	−3.67617	−0.260596	−0.130298	+	0.991475i	$0.541593\pi$
$200$	0	0
$201$	−11.6559	−0.822147
$202$	0	0
$203$	−35.2949	−2.47722
$204$	0	0
$205$	16.8166	1.17452
$206$	0	0
$207$	−0.0784532	−0.00545288
$208$	0	0
$209$	−11.3327	−0.783899
$210$	0	0
$211$	−10.2297	−0.704243	−0.352122	−	0.935954i	$-0.614540\pi$
$211$	−10.2297	−0.704243	−0.352122	+	0.935954i	$0.614540\pi$
$212$	0	0
$213$	−3.20883	−0.219865
$214$	0	0
$215$	−13.5308	−0.922793
$216$	0	0
$217$	47.4891	3.22377
$218$	0	0
$219$	−8.43822	−0.570202
$220$	0	0
$221$	−17.9427	−1.20696
$222$	0	0
$223$	−22.1032	−1.48014	−0.740070	−	0.672530i	$-0.765208\pi$
$223$	−22.1032	−1.48014	−0.740070	+	0.672530i	$0.765208\pi$
$224$	0	0
$225$	0.591244	0.0394162
$226$	0	0
$227$	22.2050	1.47380	0.736898	−	0.676003i	$-0.236290\pi$
$227$	22.2050	1.47380	0.736898	+	0.676003i	$0.236290\pi$
$228$	0	0
$229$	1.66030	0.109716	0.0548580	−	0.998494i	$-0.482529\pi$
$229$	1.66030	0.109716	0.0548580	+	0.998494i	$0.482529\pi$
$230$	0	0
$231$	11.3130	0.744342
$232$	0	0
$233$	−19.0524	−1.24817	−0.624083	−	0.781358i	$-0.714527\pi$
$233$	−19.0524	−1.24817	−0.624083	+	0.781358i	$0.714527\pi$
$234$	0	0
$235$	0.417085	0.0272076
$236$	0	0
$237$	27.2000	1.76683
$238$	0	0
$239$	5.68706	0.367865	0.183933	−	0.982939i	$-0.441117\pi$
$239$	5.68706	0.367865	0.183933	+	0.982939i	$0.441117\pi$
$240$	0	0
$241$	−15.9721	−1.02885	−0.514427	−	0.857534i	$-0.671995\pi$
$241$	−15.9721	−1.02885	−0.514427	+	0.857534i	$0.671995\pi$
$242$	0	0
$243$	−6.04826	−0.387996
$244$	0	0
$245$	26.2425	1.67657
$246$	0	0
$247$	23.5908	1.50104
$248$	0	0
$249$	18.8541	1.19483
$250$	0	0
$251$	−30.9878	−1.95593	−0.977967	−	0.208760i	$-0.933057\pi$
$251$	−30.9878	−1.95593	−0.977967	+	0.208760i	$0.933057\pi$
$252$	0	0
$253$	−0.215625	−0.0135562
$254$	0	0
$255$	16.4646	1.03105
$256$	0	0
$257$	−3.32611	−0.207477	−0.103739	−	0.994605i	$-0.533081\pi$
$257$	−3.32611	−0.207477	−0.103739	+	0.994605i	$0.533081\pi$
$258$	0	0
$259$	7.72133	0.479780
$260$	0	0
$261$	4.65212	0.287959
$262$	0	0
$263$	8.47433	0.522549	0.261275	−	0.965265i	$-0.415857\pi$
$263$	8.47433	0.522549	0.261275	+	0.965265i	$0.415857\pi$
$264$	0	0
$265$	17.3739	1.06727
$266$	0	0
$267$	16.7649	1.02600
$268$	0	0
$269$	−2.94140	−0.179340	−0.0896702	−	0.995972i	$-0.528581\pi$
$269$	−2.94140	−0.179340	−0.0896702	+	0.995972i	$0.528581\pi$
$270$	0	0
$271$	12.7654	0.775443	0.387721	−	0.921777i	$-0.373262\pi$
$271$	12.7654	0.775443	0.387721	+	0.921777i	$0.373262\pi$
$272$	0	0
$273$	−23.5498	−1.42530
$274$	0	0
$275$	1.62500	0.0979915
$276$	0	0
$277$	−6.54097	−0.393009	−0.196505	−	0.980503i	$-0.562959\pi$
$277$	−6.54097	−0.393009	−0.196505	+	0.980503i	$0.562959\pi$
$278$	0	0
$279$	−6.25940	−0.374741
$280$	0	0
$281$	5.83281	0.347956	0.173978	−	0.984750i	$-0.444338\pi$
$281$	5.83281	0.347956	0.173978	+	0.984750i	$0.444338\pi$
$282$	0	0
$283$	6.70270	0.398434	0.199217	−	0.979955i	$-0.436160\pi$
$283$	6.70270	0.398434	0.199217	+	0.979955i	$0.436160\pi$
$284$	0	0
$285$	−21.6474	−1.28228
$286$	0	0
$287$	37.7168	2.22635
$288$	0	0
$289$	11.1350	0.655003
$290$	0	0
$291$	6.54655	0.383765
$292$	0	0
$293$	−0.113390	−0.00662433	−0.00331217	−	0.999995i	$-0.501054\pi$
$293$	−0.113390	−0.00662433	−0.00331217	+	0.999995i	$0.501054\pi$
$294$	0	0
$295$	27.6304	1.60870
$296$	0	0
$297$	−9.05724	−0.525554
$298$	0	0
$299$	0.448857	0.0259581
$300$	0	0
$301$	−30.3474	−1.74919
$302$	0	0
$303$	3.62071	0.208004
$304$	0	0
$305$	−6.13482	−0.351279
$306$	0	0
$307$	−19.7204	−1.12550	−0.562751	−	0.826627i	$-0.690257\pi$
$307$	−19.7204	−1.12550	−0.562751	+	0.826627i	$0.690257\pi$
$308$	0	0
$309$	17.5566	0.998759
$310$	0	0
$311$	−28.2858	−1.60394	−0.801970	−	0.597364i	$-0.796215\pi$
$311$	−28.2858	−1.60394	−0.801970	+	0.597364i	$0.796215\pi$
$312$	0	0
$313$	−2.95285	−0.166905	−0.0834526	−	0.996512i	$-0.526595\pi$
$313$	−2.95285	−0.166905	−0.0834526	+	0.996512i	$0.526595\pi$
$314$	0	0
$315$	−5.30425	−0.298861
$316$	0	0
$317$	−23.5101	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$317$	−23.5101	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$318$	0	0
$319$	12.7861	0.715886
$320$	0	0
$321$	−17.2183	−0.961034
$322$	0	0
$323$	−36.9915	−2.05826
$324$	0	0
$325$	−3.38270	−0.187638
$326$	0	0
$327$	4.30892	0.238284
$328$	0	0
$329$	0.935453	0.0515732
$330$	0	0
$331$	−3.53388	−0.194240	−0.0971198	−	0.995273i	$-0.530963\pi$
$331$	−3.53388	−0.194240	−0.0971198	+	0.995273i	$0.530963\pi$
$332$	0	0
$333$	−1.01773	−0.0557711
$334$	0	0
$335$	15.0204	0.820651
$336$	0	0
$337$	−30.7676	−1.67602	−0.838008	−	0.545657i	$-0.816280\pi$
$337$	−30.7676	−1.67602	−0.838008	+	0.545657i	$0.816280\pi$
$338$	0	0
$339$	26.7845	1.45473
$340$	0	0
$341$	−17.2037	−0.931631
$342$	0	0
$343$	27.4579	1.48259
$344$	0	0
$345$	−0.411880	−0.0221749
$346$	0	0
$347$	8.95468	0.480713	0.240356	−	0.970685i	$-0.422736\pi$
$347$	8.95468	0.480713	0.240356	+	0.970685i	$0.422736\pi$
$348$	0	0
$349$	3.10711	0.166320	0.0831599	−	0.996536i	$-0.473499\pi$
$349$	3.10711	0.166320	0.0831599	+	0.996536i	$0.473499\pi$
$350$	0	0
$351$	18.8541	1.00636
$352$	0	0
$353$	−12.0170	−0.639600	−0.319800	−	0.947485i	$-0.603616\pi$
$353$	−12.0170	−0.639600	−0.319800	+	0.947485i	$0.603616\pi$
$354$	0	0
$355$	4.13504	0.219465
$356$	0	0
$357$	36.9273	1.95440
$358$	0	0
$359$	−28.7949	−1.51974	−0.759869	−	0.650077i	$-0.774737\pi$
$359$	−28.7949	−1.51974	−0.759869	+	0.650077i	$0.774737\pi$
$360$	0	0
$361$	29.6359	1.55978
$362$	0	0
$363$	12.9739	0.680951
$364$	0	0
$365$	10.8739	0.569165
$366$	0	0
$367$	7.28255	0.380146	0.190073	−	0.981770i	$-0.439128\pi$
$367$	7.28255	0.380146	0.190073	+	0.981770i	$0.439128\pi$
$368$	0	0
$369$	−4.97134	−0.258798
$370$	0	0
$371$	38.9668	2.02306
$372$	0	0
$373$	25.2500	1.30740	0.653698	−	0.756756i	$-0.273217\pi$
$373$	25.2500	1.30740	0.653698	+	0.756756i	$0.273217\pi$
$374$	0	0
$375$	18.6242	0.961750
$376$	0	0
$377$	−26.6163	−1.37081
$378$	0	0
$379$	−15.2733	−0.784535	−0.392268	−	0.919851i	$-0.628309\pi$
$379$	−15.2733	−0.784535	−0.392268	+	0.919851i	$0.628309\pi$
$380$	0	0
$381$	12.9092	0.661357
$382$	0	0
$383$	2.83494	0.144859	0.0724293	−	0.997374i	$-0.476925\pi$
$383$	2.83494	0.144859	0.0724293	+	0.997374i	$0.476925\pi$
$384$	0	0
$385$	−14.5785	−0.742988
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−17.8121	−0.903110	−0.451555	−	0.892243i	$-0.649131\pi$
$389$	−17.8121	−0.903110	−0.451555	+	0.892243i	$0.649131\pi$
$390$	0	0
$391$	−0.703831	−0.0355942
$392$	0	0
$393$	9.06858	0.457449
$394$	0	0
$395$	−35.0512	−1.76362
$396$	0	0
$397$	18.5585	0.931425	0.465712	−	0.884936i	$-0.345798\pi$
$397$	18.5585	0.931425	0.465712	+	0.884936i	$0.345798\pi$
$398$	0	0
$399$	−48.5515	−2.43061
$400$	0	0
$401$	19.8040	0.988967	0.494483	−	0.869187i	$-0.335357\pi$
$401$	19.8040	0.988967	0.494483	+	0.869187i	$0.335357\pi$
$402$	0	0
$403$	35.8121	1.78393
$404$	0	0
$405$	−13.7534	−0.683412
$406$	0	0
$407$	−2.79717	−0.138651
$408$	0	0
$409$	24.2604	1.19960	0.599799	−	0.800151i	$-0.295247\pi$
$409$	24.2604	1.19960	0.599799	+	0.800151i	$0.295247\pi$
$410$	0	0
$411$	−21.2256	−1.04698
$412$	0	0
$413$	61.9704	3.04937
$414$	0	0
$415$	−24.2962	−1.19265
$416$	0	0
$417$	−15.9855	−0.782812
$418$	0	0
$419$	−16.5947	−0.810704	−0.405352	−	0.914161i	$-0.632851\pi$
$419$	−16.5947	−0.810704	−0.405352	+	0.914161i	$0.632851\pi$
$420$	0	0
$421$	−24.1760	−1.17827	−0.589134	−	0.808035i	$-0.700531\pi$
$421$	−24.1760	−1.17827	−0.589134	+	0.808035i	$0.700531\pi$
$422$	0	0
$423$	−0.123299	−0.00599502
$424$	0	0
$425$	5.30425	0.257294
$426$	0	0
$427$	−13.7594	−0.665864
$428$	0	0
$429$	8.53129	0.411894
$430$	0	0
$431$	0.233761	0.0112598	0.00562992	−	0.999984i	$-0.498208\pi$
$431$	0.233761	0.0112598	0.00562992	+	0.999984i	$0.498208\pi$
$432$	0	0
$433$	31.0595	1.49263	0.746313	−	0.665595i	$-0.231822\pi$
$433$	31.0595	1.49263	0.746313	+	0.665595i	$0.231822\pi$
$434$	0	0
$435$	24.4237	1.17103
$436$	0	0
$437$	0.925386	0.0442672
$438$	0	0
$439$	−20.3994	−0.973610	−0.486805	−	0.873511i	$-0.661838\pi$
$439$	−20.3994	−0.973610	−0.486805	+	0.873511i	$0.661838\pi$
$440$	0	0
$441$	−7.75785	−0.369421
$442$	0	0
$443$	0.445779	0.0211796	0.0105898	−	0.999944i	$-0.496629\pi$
$443$	0.445779	0.0211796	0.0105898	+	0.999944i	$0.496629\pi$
$444$	0	0
$445$	−21.6041	−1.02413
$446$	0	0
$447$	12.1726	0.575745
$448$	0	0
$449$	23.8874	1.12732	0.563659	−	0.826007i	$-0.309393\pi$
$449$	23.8874	1.12732	0.563659	+	0.826007i	$0.309393\pi$
$450$	0	0
$451$	−13.6635	−0.643389
$452$	0	0
$453$	24.0923	1.13196
$454$	0	0
$455$	30.3474	1.42271
$456$	0	0
$457$	11.8064	0.552280	0.276140	−	0.961117i	$-0.410945\pi$
$457$	11.8064	0.552280	0.276140	+	0.961117i	$0.410945\pi$
$458$	0	0
$459$	−29.5641	−1.37994
$460$	0	0
$461$	14.0913	0.656295	0.328148	−	0.944626i	$-0.393576\pi$
$461$	14.0913	0.656295	0.328148	+	0.944626i	$0.393576\pi$
$462$	0	0
$463$	14.6663	0.681602	0.340801	−	0.940135i	$-0.389302\pi$
$463$	14.6663	0.681602	0.340801	+	0.940135i	$0.389302\pi$
$464$	0	0
$465$	−32.8619	−1.52393
$466$	0	0
$467$	30.8445	1.42731	0.713656	−	0.700496i	$-0.247038\pi$
$467$	30.8445	1.42731	0.713656	+	0.700496i	$0.247038\pi$
$468$	0	0
$469$	33.6882	1.55558
$470$	0	0
$471$	−27.0321	−1.24557
$472$	0	0
$473$	10.9938	0.505496
$474$	0	0
$475$	−6.97394	−0.319987
$476$	0	0
$477$	−5.13610	−0.235166
$478$	0	0
$479$	4.40615	0.201322	0.100661	−	0.994921i	$-0.467904\pi$
$479$	4.40615	0.201322	0.100661	+	0.994921i	$0.467904\pi$
$480$	0	0
$481$	5.82275	0.265495
$482$	0	0
$483$	−0.923779	−0.0420334
$484$	0	0
$485$	−8.43618	−0.383067
$486$	0	0
$487$	24.2128	1.09719	0.548594	−	0.836089i	$-0.315163\pi$
$487$	24.2128	1.09719	0.548594	+	0.836089i	$0.315163\pi$
$488$	0	0
$489$	−37.2347	−1.68381
$490$	0	0
$491$	−32.9516	−1.48709	−0.743543	−	0.668689i	$-0.766856\pi$
$491$	−32.9516	−1.48709	−0.743543	+	0.668689i	$0.766856\pi$
$492$	0	0
$493$	41.7358	1.87969
$494$	0	0
$495$	1.92155	0.0863671
$496$	0	0
$497$	9.27422	0.416006
$498$	0	0
$499$	9.79619	0.438538	0.219269	−	0.975664i	$-0.429633\pi$
$499$	9.79619	0.438538	0.219269	+	0.975664i	$0.429633\pi$
$500$	0	0
$501$	1.55202	0.0693390
$502$	0	0
$503$	14.0414	0.626075	0.313038	−	0.949741i	$-0.398653\pi$
$503$	14.0414	0.626075	0.313038	+	0.949741i	$0.398653\pi$
$504$	0	0
$505$	−4.66581	−0.207626
$506$	0	0
$507$	2.41701	0.107343
$508$	0	0
$509$	−30.9992	−1.37402	−0.687008	−	0.726650i	$-0.741076\pi$
$509$	−30.9992	−1.37402	−0.687008	+	0.726650i	$0.741076\pi$
$510$	0	0
$511$	24.3883	1.07888
$512$	0	0
$513$	38.8705	1.71617
$514$	0	0
$515$	−22.6242	−0.996941
$516$	0	0
$517$	−0.338883	−0.0149040
$518$	0	0
$519$	2.61099	0.114610
$520$	0	0
$521$	23.3877	1.02463	0.512317	−	0.858796i	$-0.328787\pi$
$521$	23.3877	1.02463	0.512317	+	0.858796i	$0.328787\pi$
$522$	0	0
$523$	−14.2130	−0.621490	−0.310745	−	0.950493i	$-0.600579\pi$
$523$	−14.2130	−0.621490	−0.310745	+	0.950493i	$0.600579\pi$
$524$	0	0
$525$	6.96184	0.303839
$526$	0	0
$527$	−56.1552	−2.44616
$528$	0	0
$529$	−22.9824	−0.999234
$530$	0	0
$531$	−8.16815	−0.354467
$532$	0	0
$533$	28.4427	1.23199
$534$	0	0
$535$	22.1883	0.959285
$536$	0	0
$537$	1.18058	0.0509456
$538$	0	0
$539$	−21.3221	−0.918407
$540$	0	0
$541$	−3.61867	−0.155579	−0.0777893	−	0.996970i	$-0.524786\pi$
$541$	−3.61867	−0.155579	−0.0777893	+	0.996970i	$0.524786\pi$
$542$	0	0
$543$	9.86562	0.423374
$544$	0	0
$545$	−5.55267	−0.237850
$546$	0	0
$547$	16.1945	0.692426	0.346213	−	0.938156i	$-0.387467\pi$
$547$	16.1945	0.692426	0.346213	+	0.938156i	$0.387467\pi$
$548$	0	0
$549$	1.81359	0.0774020
$550$	0	0
$551$	−54.8736	−2.33769
$552$	0	0
$553$	−78.6141	−3.34301
$554$	0	0
$555$	−5.34308	−0.226801
$556$	0	0
$557$	−5.39689	−0.228674	−0.114337	−	0.993442i	$-0.536474\pi$
$557$	−5.39689	−0.228674	−0.114337	+	0.993442i	$0.536474\pi$
$558$	0	0
$559$	−22.8853	−0.967946
$560$	0	0
$561$	−13.3775	−0.564798
$562$	0	0
$563$	38.3990	1.61833	0.809163	−	0.587585i	$-0.199921\pi$
$563$	38.3990	1.61833	0.809163	+	0.587585i	$0.199921\pi$
$564$	0	0
$565$	−34.5157	−1.45209
$566$	0	0
$567$	−30.8466	−1.29544
$568$	0	0
$569$	−7.14674	−0.299607	−0.149803	−	0.988716i	$-0.547864\pi$
$569$	−7.14674	−0.299607	−0.149803	+	0.988716i	$0.547864\pi$
$570$	0	0
$571$	9.09003	0.380406	0.190203	−	0.981745i	$-0.439085\pi$
$571$	9.09003	0.380406	0.190203	+	0.981745i	$0.439085\pi$
$572$	0	0
$573$	−17.3809	−0.726099
$574$	0	0
$575$	−0.132692	−0.00553363
$576$	0	0
$577$	−5.92103	−0.246496	−0.123248	−	0.992376i	$-0.539331\pi$
$577$	−5.92103	−0.246496	−0.123248	+	0.992376i	$0.539331\pi$
$578$	0	0
$579$	23.1982	0.964084
$580$	0	0
$581$	−54.4924	−2.26073
$582$	0	0
$583$	−14.1163	−0.584639
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	27.2003	1.12267	0.561337	−	0.827587i	$-0.310287\pi$
$587$	27.2003	1.12267	0.561337	+	0.827587i	$0.310287\pi$
$588$	0	0
$589$	73.8320	3.04220
$590$	0	0
$591$	−28.0566	−1.15409
$592$	0	0
$593$	16.3314	0.670648	0.335324	−	0.942103i	$-0.391154\pi$
$593$	16.3314	0.670648	0.335324	+	0.942103i	$0.391154\pi$
$594$	0	0
$595$	−47.5862	−1.95084
$596$	0	0
$597$	5.70547	0.233509
$598$	0	0
$599$	−14.6823	−0.599902	−0.299951	−	0.953955i	$-0.596970\pi$
$599$	−14.6823	−0.599902	−0.299951	+	0.953955i	$0.596970\pi$
$600$	0	0
$601$	−20.3773	−0.831206	−0.415603	−	0.909546i	$-0.636430\pi$
$601$	−20.3773	−0.831206	−0.415603	+	0.909546i	$0.636430\pi$
$602$	0	0
$603$	−4.44035	−0.180825
$604$	0	0
$605$	−16.7187	−0.679713
$606$	0	0
$607$	−1.72706	−0.0700992	−0.0350496	−	0.999386i	$-0.511159\pi$
$607$	−1.72706	−0.0700992	−0.0350496	+	0.999386i	$0.511159\pi$
$608$	0	0
$609$	54.7783	2.21973
$610$	0	0
$611$	0.705437	0.0285389
$612$	0	0
$613$	−9.43357	−0.381018	−0.190509	−	0.981685i	$-0.561014\pi$
$613$	−9.43357	−0.381018	−0.190509	+	0.981685i	$0.561014\pi$
$614$	0	0
$615$	−26.0996	−1.05244
$616$	0	0
$617$	−19.2934	−0.776724	−0.388362	−	0.921507i	$-0.626959\pi$
$617$	−19.2934	−0.776724	−0.388362	+	0.921507i	$0.626959\pi$
$618$	0	0
$619$	−11.7698	−0.473067	−0.236533	−	0.971623i	$-0.576011\pi$
$619$	−11.7698	−0.473067	−0.236533	+	0.971623i	$0.576011\pi$
$620$	0	0
$621$	0.739581	0.0296784
$622$	0	0
$623$	−48.4543	−1.94128
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	17.5885	0.702418
$628$	0	0
$629$	−9.13038	−0.364052
$630$	0	0
$631$	35.7072	1.42148	0.710742	−	0.703453i	$-0.248360\pi$
$631$	35.7072	1.42148	0.710742	+	0.703453i	$0.248360\pi$
$632$	0	0
$633$	15.8767	0.631042
$634$	0	0
$635$	−16.6354	−0.660154
$636$	0	0
$637$	44.3852	1.75861
$638$	0	0
$639$	−1.22241	−0.0483578
$640$	0	0
$641$	−32.4396	−1.28129	−0.640644	−	0.767838i	$-0.721333\pi$
$641$	−32.4396	−1.28129	−0.640644	+	0.767838i	$0.721333\pi$
$642$	0	0
$643$	24.8629	0.980496	0.490248	−	0.871583i	$-0.336906\pi$
$643$	24.8629	0.980496	0.490248	+	0.871583i	$0.336906\pi$
$644$	0	0
$645$	21.0000	0.826876
$646$	0	0
$647$	−22.5888	−0.888056	−0.444028	−	0.896013i	$-0.646451\pi$
$647$	−22.5888	−0.888056	−0.444028	+	0.896013i	$0.646451\pi$
$648$	0	0
$649$	−22.4498	−0.881230
$650$	0	0
$651$	−73.7039	−2.88868
$652$	0	0
$653$	22.9597	0.898481	0.449241	−	0.893411i	$-0.351695\pi$
$653$	22.9597	0.898481	0.449241	+	0.893411i	$0.351695\pi$
$654$	0	0
$655$	−11.6862	−0.456617
$656$	0	0
$657$	−3.21456	−0.125412
$658$	0	0
$659$	−1.23673	−0.0481760	−0.0240880	−	0.999710i	$-0.507668\pi$
$659$	−1.23673	−0.0481760	−0.0240880	+	0.999710i	$0.507668\pi$
$660$	0	0
$661$	12.5706	0.488940	0.244470	−	0.969657i	$-0.421386\pi$
$661$	12.5706	0.488940	0.244470	+	0.969657i	$0.421386\pi$
$662$	0	0
$663$	27.8473	1.08150
$664$	0	0
$665$	62.5656	2.42619
$666$	0	0
$667$	−1.04407	−0.0404265
$668$	0	0
$669$	34.3045	1.32629
$670$	0	0
$671$	4.98455	0.192427
$672$	0	0
$673$	−48.5661	−1.87208	−0.936042	−	0.351888i	$-0.885540\pi$
$673$	−48.5661	−1.87208	−0.936042	+	0.351888i	$0.885540\pi$
$674$	0	0
$675$	−5.57367	−0.214531
$676$	0	0
$677$	3.90041	0.149905	0.0749525	−	0.997187i	$-0.476119\pi$
$677$	3.90041	0.149905	0.0749525	+	0.997187i	$0.476119\pi$
$678$	0	0
$679$	−18.9210	−0.726120
$680$	0	0
$681$	−34.4625	−1.32061
$682$	0	0
$683$	−17.2389	−0.659626	−0.329813	−	0.944046i	$-0.606986\pi$
$683$	−17.2389	−0.659626	−0.329813	+	0.944046i	$0.606986\pi$
$684$	0	0
$685$	27.3523	1.04508
$686$	0	0
$687$	−2.57682	−0.0983118
$688$	0	0
$689$	29.3853	1.11949
$690$	0	0
$691$	−26.1641	−0.995330	−0.497665	−	0.867369i	$-0.665809\pi$
$691$	−26.1641	−0.995330	−0.497665	+	0.867369i	$0.665809\pi$
$692$	0	0
$693$	4.30971	0.163712
$694$	0	0
$695$	20.5996	0.781388
$696$	0	0
$697$	−44.5996	−1.68933
$698$	0	0
$699$	29.5697	1.11843
$700$	0	0
$701$	31.0982	1.17456	0.587282	−	0.809382i	$-0.300198\pi$
$701$	31.0982	1.17456	0.587282	+	0.809382i	$0.300198\pi$
$702$	0	0
$703$	12.0045	0.452758
$704$	0	0
$705$	−0.647323	−0.0243796
$706$	0	0
$707$	−10.4646	−0.393564
$708$	0	0
$709$	−9.23408	−0.346793	−0.173397	−	0.984852i	$-0.555474\pi$
$709$	−9.23408	−0.346793	−0.173397	+	0.984852i	$0.555474\pi$
$710$	0	0
$711$	10.3619	0.388601
$712$	0	0
$713$	1.40479	0.0526097
$714$	0	0
$715$	−10.9938	−0.411145
$716$	0	0
$717$	−8.82641	−0.329629
$718$	0	0
$719$	33.3348	1.24318	0.621590	−	0.783343i	$-0.286487\pi$
$719$	33.3348	1.24318	0.621590	+	0.783343i	$0.286487\pi$
$720$	0	0
$721$	−50.7424	−1.88974
$722$	0	0
$723$	24.7890	0.921912
$724$	0	0
$725$	7.86837	0.292224
$726$	0	0
$727$	2.10327	0.0780061	0.0390030	−	0.999239i	$-0.487582\pi$
$727$	2.10327	0.0780061	0.0390030	+	0.999239i	$0.487582\pi$
$728$	0	0
$729$	30.0171	1.11174
$730$	0	0
$731$	35.8854	1.32727
$732$	0	0
$733$	−34.0965	−1.25938	−0.629691	−	0.776846i	$-0.716818\pi$
$733$	−34.0965	−1.25938	−0.629691	+	0.776846i	$0.716818\pi$
$734$	0	0
$735$	−40.7288	−1.50230
$736$	0	0
$737$	−12.2041	−0.449543
$738$	0	0
$739$	−11.1479	−0.410081	−0.205040	−	0.978754i	$-0.565733\pi$
$739$	−11.1479	−0.410081	−0.205040	+	0.978754i	$0.565733\pi$
$740$	0	0
$741$	−36.6133	−1.34502
$742$	0	0
$743$	−23.4849	−0.861579	−0.430789	−	0.902452i	$-0.641765\pi$
$743$	−23.4849	−0.861579	−0.430789	+	0.902452i	$0.641765\pi$
$744$	0	0
$745$	−15.6862	−0.574697
$746$	0	0
$747$	7.18249	0.262793
$748$	0	0
$749$	49.7648	1.81837
$750$	0	0
$751$	31.1954	1.13834	0.569169	−	0.822221i	$-0.307265\pi$
$751$	31.1954	1.13834	0.569169	+	0.822221i	$0.307265\pi$
$752$	0	0
$753$	48.0936	1.75263
$754$	0	0
$755$	−31.0465	−1.12990
$756$	0	0
$757$	−19.3753	−0.704208	−0.352104	−	0.935961i	$-0.614534\pi$
$757$	−19.3753	−0.704208	−0.352104	+	0.935961i	$0.614534\pi$
$758$	0	0
$759$	0.334653	0.0121472
$760$	0	0
$761$	34.7008	1.25790	0.628951	−	0.777445i	$-0.283485\pi$
$761$	34.7008	1.25790	0.628951	+	0.777445i	$0.283485\pi$
$762$	0	0
$763$	−12.4537	−0.450855
$764$	0	0
$765$	6.27220	0.226772
$766$	0	0
$767$	46.7327	1.68742
$768$	0	0
$769$	−25.6173	−0.923785	−0.461892	−	0.886936i	$-0.652829\pi$
$769$	−25.6173	−0.923785	−0.461892	+	0.886936i	$0.652829\pi$
$770$	0	0
$771$	5.16218	0.185911
$772$	0	0
$773$	−25.4065	−0.913808	−0.456904	−	0.889516i	$-0.651042\pi$
$773$	−25.4065	−0.913808	−0.456904	+	0.889516i	$0.651042\pi$
$774$	0	0
$775$	−10.5868	−0.380291
$776$	0	0
$777$	−11.9836	−0.429911
$778$	0	0
$779$	58.6389	2.10096
$780$	0	0
$781$	−3.35973	−0.120221
$782$	0	0
$783$	−43.8557	−1.56728
$784$	0	0
$785$	34.8348	1.24331
$786$	0	0
$787$	34.5483	1.23151	0.615757	−	0.787936i	$-0.288850\pi$
$787$	34.5483	1.23151	0.615757	+	0.787936i	$0.288850\pi$
$788$	0	0
$789$	−13.1523	−0.468234
$790$	0	0
$791$	−77.4131	−2.75249
$792$	0	0
$793$	−10.3761	−0.368467
$794$	0	0
$795$	−26.9646	−0.956335
$796$	0	0
$797$	43.1106	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$797$	43.1106	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$798$	0	0
$799$	−1.10616	−0.0391332
$800$	0	0
$801$	6.38663	0.225660
$802$	0	0
$803$	−8.83505	−0.311782
$804$	0	0
$805$	1.19042	0.0419569
$806$	0	0
$807$	4.56510	0.160699
$808$	0	0
$809$	24.4150	0.858386	0.429193	−	0.903213i	$-0.358798\pi$
$809$	24.4150	0.858386	0.429193	+	0.903213i	$0.358798\pi$
$810$	0	0
$811$	−39.1406	−1.37441	−0.687206	−	0.726463i	$-0.741163\pi$
$811$	−39.1406	−1.37441	−0.687206	+	0.726463i	$0.741163\pi$
$812$	0	0
$813$	−19.8121	−0.694841
$814$	0	0
$815$	47.9824	1.68075
$816$	0	0
$817$	−47.1815	−1.65067
$818$	0	0
$819$	−8.97134	−0.313484
$820$	0	0
$821$	26.3557	0.919822	0.459911	−	0.887965i	$-0.347881\pi$
$821$	26.3557	0.919822	0.459911	+	0.887965i	$0.347881\pi$
$822$	0	0
$823$	11.0365	0.384709	0.192354	−	0.981326i	$-0.438388\pi$
$823$	11.0365	0.384709	0.192354	+	0.981326i	$0.438388\pi$
$824$	0	0
$825$	−2.52203	−0.0878060
$826$	0	0
$827$	−11.5066	−0.400123	−0.200062	−	0.979783i	$-0.564114\pi$
$827$	−11.5066	−0.400123	−0.200062	+	0.979783i	$0.564114\pi$
$828$	0	0
$829$	−17.9151	−0.622217	−0.311108	−	0.950374i	$-0.600700\pi$
$829$	−17.9151	−0.622217	−0.311108	+	0.950374i	$0.600700\pi$
$830$	0	0
$831$	10.1517	0.352159
$832$	0	0
$833$	−69.5983	−2.41144
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	59.0076	2.03960
$838$	0	0
$839$	46.8888	1.61878	0.809390	−	0.587271i	$-0.199798\pi$
$839$	46.8888	1.61878	0.809390	+	0.587271i	$0.199798\pi$
$840$	0	0
$841$	32.9113	1.13487
$842$	0	0
$843$	−9.05262	−0.311789
$844$	0	0
$845$	−3.11467	−0.107148
$846$	0	0
$847$	−37.4973	−1.28842
$848$	0	0
$849$	−10.4027	−0.357020
$850$	0	0
$851$	0.228407	0.00782969
$852$	0	0
$853$	−7.97457	−0.273044	−0.136522	−	0.990637i	$-0.543592\pi$
$853$	−7.97457	−0.273044	−0.136522	+	0.990637i	$0.543592\pi$
$854$	0	0
$855$	−8.24660	−0.282028
$856$	0	0
$857$	38.3453	1.30985	0.654925	−	0.755694i	$-0.272700\pi$
$857$	38.3453	1.30985	0.654925	+	0.755694i	$0.272700\pi$
$858$	0	0
$859$	55.7156	1.90099	0.950497	−	0.310735i	$-0.100575\pi$
$859$	55.7156	1.90099	0.950497	+	0.310735i	$0.100575\pi$
$860$	0	0
$861$	−58.5371	−1.99494
$862$	0	0
$863$	35.5096	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$863$	35.5096	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$864$	0	0
$865$	−3.36464	−0.114401
$866$	0	0
$867$	−17.2818	−0.586920
$868$	0	0
$869$	28.4792	0.966090
$870$	0	0
$871$	25.4047	0.860806
$872$	0	0
$873$	2.49392	0.0844063
$874$	0	0
$875$	−53.8281	−1.81972
$876$	0	0
$877$	1.96237	0.0662646	0.0331323	−	0.999451i	$-0.489452\pi$
$877$	1.96237	0.0662646	0.0331323	+	0.999451i	$0.489452\pi$
$878$	0	0
$879$	0.175984	0.00593578
$880$	0	0
$881$	−25.9608	−0.874642	−0.437321	−	0.899305i	$-0.644073\pi$
$881$	−25.9608	−0.874642	−0.437321	+	0.899305i	$0.644073\pi$
$882$	0	0
$883$	32.1139	1.08072	0.540360	−	0.841434i	$-0.318288\pi$
$883$	32.1139	1.08072	0.540360	+	0.841434i	$0.318288\pi$
$884$	0	0
$885$	−42.8828	−1.44149
$886$	0	0
$887$	6.96553	0.233880	0.116940	−	0.993139i	$-0.462692\pi$
$887$	6.96553	0.233880	0.116940	+	0.993139i	$0.462692\pi$
$888$	0	0
$889$	−37.3104	−1.25135
$890$	0	0
$891$	11.1747	0.374366
$892$	0	0
$893$	1.45436	0.0486684
$894$	0	0
$895$	−1.52134	−0.0508529
$896$	0	0
$897$	−0.696634	−0.0232599
$898$	0	0
$899$	−83.3012	−2.77825
$900$	0	0
$901$	−46.0777	−1.53507
$902$	0	0
$903$	47.0996	1.56738
$904$	0	0
$905$	−12.7133	−0.422604
$906$	0	0
$907$	34.2026	1.13568	0.567839	−	0.823140i	$-0.307780\pi$
$907$	34.2026	1.13568	0.567839	+	0.823140i	$0.307780\pi$
$908$	0	0
$909$	1.37932	0.0457490
$910$	0	0
$911$	−33.5867	−1.11278	−0.556388	−	0.830922i	$-0.687813\pi$
$911$	−33.5867	−1.11278	−0.556388	+	0.830922i	$0.687813\pi$
$912$	0	0
$913$	19.7407	0.653323
$914$	0	0
$915$	9.52134	0.314766
$916$	0	0
$917$	−26.2102	−0.865537
$918$	0	0
$919$	44.8672	1.48003	0.740017	−	0.672588i	$-0.234817\pi$
$919$	44.8672	1.48003	0.740017	+	0.672588i	$0.234817\pi$
$920$	0	0
$921$	30.6064	1.00851
$922$	0	0
$923$	6.99381	0.230204
$924$	0	0
$925$	−1.72133	−0.0565971
$926$	0	0
$927$	6.68821	0.219670
$928$	0	0
$929$	−12.6760	−0.415887	−0.207943	−	0.978141i	$-0.566677\pi$
$929$	−12.6760	−0.415887	−0.207943	+	0.978141i	$0.566677\pi$
$930$	0	0
$931$	91.5068	2.99902
$932$	0	0
$933$	43.9000	1.43722
$934$	0	0
$935$	17.2389	0.563771
$936$	0	0
$937$	3.52060	0.115013	0.0575065	−	0.998345i	$-0.481685\pi$
$937$	3.52060	0.115013	0.0575065	+	0.998345i	$0.481685\pi$
$938$	0	0
$939$	4.58288	0.149557
$940$	0	0
$941$	54.2981	1.77007	0.885034	−	0.465526i	$-0.154135\pi$
$941$	54.2981	1.77007	0.885034	+	0.465526i	$0.154135\pi$
$942$	0	0
$943$	1.11571	0.0363325
$944$	0	0
$945$	50.0033	1.62661
$946$	0	0
$947$	35.3015	1.14715	0.573573	−	0.819155i	$-0.305557\pi$
$947$	35.3015	1.14715	0.573573	+	0.819155i	$0.305557\pi$
$948$	0	0
$949$	18.3915	0.597015
$950$	0	0
$951$	36.4881	1.18321
$952$	0	0
$953$	41.2048	1.33475	0.667377	−	0.744720i	$-0.267417\pi$
$953$	41.2048	1.33475	0.667377	+	0.744720i	$0.267417\pi$
$954$	0	0
$955$	22.3979	0.724778
$956$	0	0
$957$	−19.8443	−0.641475
$958$	0	0
$959$	61.3468	1.98099
$960$	0	0
$961$	81.0812	2.61552
$962$	0	0
$963$	−6.55935	−0.211372
$964$	0	0
$965$	−29.8943	−0.962330
$966$	0	0
$967$	−16.5718	−0.532913	−0.266457	−	0.963847i	$-0.585853\pi$
$967$	−16.5718	−0.532913	−0.266457	+	0.963847i	$0.585853\pi$
$968$	0	0
$969$	57.4115	1.84432
$970$	0	0
$971$	−41.8901	−1.34432	−0.672158	−	0.740408i	$-0.734632\pi$
$971$	−41.8901	−1.34432	−0.672158	+	0.740408i	$0.734632\pi$
$972$	0	0
$973$	46.2015	1.48115
$974$	0	0
$975$	5.25001	0.168135
$976$	0	0
$977$	54.2586	1.73589	0.867944	−	0.496663i	$-0.165441\pi$
$977$	54.2586	1.73589	0.867944	+	0.496663i	$0.165441\pi$
$978$	0	0
$979$	17.5533	0.561007
$980$	0	0
$981$	1.64149	0.0524087
$982$	0	0
$983$	−40.0351	−1.27692	−0.638460	−	0.769655i	$-0.720428\pi$
$983$	−40.0351	−1.27692	−0.638460	+	0.769655i	$0.720428\pi$
$984$	0	0
$985$	36.1550	1.15199
$986$	0	0
$987$	−1.45184	−0.0462126
$988$	0	0
$989$	−0.897714	−0.0285456
$990$	0	0
$991$	6.86359	0.218029	0.109015	−	0.994040i	$-0.465230\pi$
$991$	6.86359	0.218029	0.109015	+	0.994040i	$0.465230\pi$
$992$	0	0
$993$	5.48464	0.174050
$994$	0	0
$995$	−7.35233	−0.233085
$996$	0	0
$997$	26.6578	0.844260	0.422130	−	0.906535i	$-0.361283\pi$
$997$	26.6578	0.844260	0.422130	+	0.906535i	$0.361283\pi$
$998$	0	0
$999$	9.59414	0.303545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.b.1.2	✓	5
3.2	odd	2		6012.2.a.d.1.4		5
4.3	odd	2		2672.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.2	✓	5	1.1	even	1	trivial
2672.2.a.j.1.4		5	4.3	odd	2
6012.2.a.d.1.4		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} +O(q^{10})\)	\(q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} -1.62500 q^{11} +3.38270 q^{13} -3.10403 q^{15} -5.30425 q^{17} +6.97394 q^{19} -6.96184 q^{21} +0.132692 q^{23} -1.00000 q^{25} +5.57367 q^{27} -7.86837 q^{29} +10.5868 q^{31} +2.52203 q^{33} +8.97134 q^{35} +1.72133 q^{37} -5.25001 q^{39} +8.40828 q^{41} -6.76540 q^{43} -1.18249 q^{45} +0.208543 q^{47} +13.1212 q^{49} +8.23228 q^{51} +8.68695 q^{53} -3.25001 q^{55} -10.8237 q^{57} +13.8152 q^{59} -3.06741 q^{61} -2.65212 q^{63} +6.76540 q^{65} +7.51019 q^{67} -0.205940 q^{69} +2.06752 q^{71} +5.43694 q^{73} +1.55202 q^{75} -7.28924 q^{77} -17.5256 q^{79} -6.87670 q^{81} -12.1481 q^{83} -10.6085 q^{85} +12.2118 q^{87} -10.8020 q^{89} +15.1737 q^{91} -16.4310 q^{93} +13.9479 q^{95} -4.21809 q^{97} +0.960773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9	\( 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9} + 5 q^{11} - 4 q^{13} + 6 q^{15} - 2 q^{17} + 5 q^{19} - 4 q^{21} + 6 q^{23} - 5 q^{25} + 12 q^{27} - 5 q^{29} + 9 q^{31} - 8 q^{33} + 18 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{45} + 13 q^{47} + 14 q^{49} - 2 q^{53} + 10 q^{55} - 12 q^{57} + 4 q^{59} + 11 q^{61} - q^{63} - 8 q^{65} + 28 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} - 3 q^{75} - 12 q^{77} - 10 q^{79} - 15 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{87} - 17 q^{89} - 12 q^{91} - 12 q^{93} + 10 q^{95} - 27 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9 + 5 * q^11 - 4 * q^13 + 6 * q^15 - 2 * q^17 + 5 * q^19 - 4 * q^21 + 6 * q^23 - 5 * q^25 + 12 * q^27 - 5 * q^29 + 9 * q^31 - 8 * q^33 + 18 * q^35 + 8 * q^37 - 4 * q^41 + 8 * q^43 + 12 * q^45 + 13 * q^47 + 14 * q^49 - 2 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 + 11 * q^61 - q^63 - 8 * q^65 + 28 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 - 3 * q^75 - 12 * q^77 - 10 * q^79 - 15 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^87 - 17 * q^89 - 12 * q^91 - 12 * q^93 + 10 * q^95 - 27 * q^97 + 3 * q^99

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