LMFDB - Embedded newform 668.2.a.c.1.6

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
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			1.6
Root			$-0.721798$ 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)

$f(q)$	$=$	$q+0.678805 q^{3} -2.77086 q^{5} +1.71364 q^{7} -2.53922 q^{9} +O(q^{10})$	$q+0.678805 q^{3} -2.77086 q^{5} +1.71364 q^{7} -2.53922 q^{9} +0.585637 q^{11} -6.56764 q^{13} -1.88087 q^{15} +0.228214 q^{17} -5.57385 q^{19} +1.16323 q^{21} +4.63541 q^{23} +2.67765 q^{25} -3.76005 q^{27} -6.25628 q^{29} +7.04697 q^{31} +0.397533 q^{33} -4.74825 q^{35} -11.6932 q^{37} -4.45815 q^{39} +2.26860 q^{41} -3.41325 q^{43} +7.03583 q^{45} -6.83524 q^{47} -4.06344 q^{49} +0.154913 q^{51} +9.69168 q^{53} -1.62272 q^{55} -3.78356 q^{57} +9.29439 q^{59} -2.23577 q^{61} -4.35131 q^{63} +18.1980 q^{65} -5.40600 q^{67} +3.14654 q^{69} -10.7081 q^{71} +14.0249 q^{73} +1.81761 q^{75} +1.00357 q^{77} -9.29974 q^{79} +5.06533 q^{81} +14.0341 q^{83} -0.632348 q^{85} -4.24680 q^{87} +2.45479 q^{89} -11.2546 q^{91} +4.78352 q^{93} +15.4444 q^{95} +4.76719 q^{97} -1.48706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * 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
$3$	0.678805	0.391908	0.195954	−	0.980613i	$-0.437220\pi$
$3$	0.678805	0.391908	0.195954	+	0.980613i	$0.437220\pi$
$4$	0	0
$5$	−2.77086	−1.23917	−0.619583	−	0.784931i	$-0.712698\pi$
$5$	−2.77086	−1.23917	−0.619583	+	0.784931i	$0.712698\pi$
$6$	0	0
$7$	1.71364	0.647695	0.323847	−	0.946109i	$-0.395024\pi$
$7$	1.71364	0.647695	0.323847	+	0.946109i	$0.395024\pi$
$8$	0	0
$9$	−2.53922	−0.846408
$10$	0	0
$11$	0.585637	0.176576	0.0882881	−	0.996095i	$-0.471860\pi$
$11$	0.585637	0.176576	0.0882881	+	0.996095i	$0.471860\pi$
$12$	0	0
$13$	−6.56764	−1.82154	−0.910768	−	0.412919i	$-0.864509\pi$
$13$	−6.56764	−1.82154	−0.910768	+	0.412919i	$0.864509\pi$
$14$	0	0
$15$	−1.88087	−0.485639
$16$	0	0
$17$	0.228214	0.0553500	0.0276750	−	0.999617i	$-0.491190\pi$
$17$	0.228214	0.0553500	0.0276750	+	0.999617i	$0.491190\pi$
$18$	0	0
$19$	−5.57385	−1.27873	−0.639365	−	0.768903i	$-0.720803\pi$
$19$	−5.57385	−1.27873	−0.639365	+	0.768903i	$0.720803\pi$
$20$	0	0
$21$	1.16323	0.253837
$22$	0	0
$23$	4.63541	0.966550	0.483275	−	0.875469i	$-0.339447\pi$
$23$	4.63541	0.966550	0.483275	+	0.875469i	$0.339447\pi$
$24$	0	0
$25$	2.67765	0.535531
$26$	0	0
$27$	−3.76005	−0.723622
$28$	0	0
$29$	−6.25628	−1.16176	−0.580881	−	0.813988i	$-0.697292\pi$
$29$	−6.25628	−1.16176	−0.580881	+	0.813988i	$0.697292\pi$
$30$	0	0
$31$	7.04697	1.26567	0.632837	−	0.774285i	$-0.281890\pi$
$31$	7.04697	1.26567	0.632837	+	0.774285i	$0.281890\pi$
$32$	0	0
$33$	0.397533	0.0692017
$34$	0	0
$35$	−4.74825	−0.802601
$36$	0	0
$37$	−11.6932	−1.92235	−0.961173	−	0.275947i	$-0.911009\pi$
$37$	−11.6932	−1.92235	−0.961173	+	0.275947i	$0.911009\pi$
$38$	0	0
$39$	−4.45815	−0.713875
$40$	0	0
$41$	2.26860	0.354296	0.177148	−	0.984184i	$-0.443313\pi$
$41$	2.26860	0.354296	0.177148	+	0.984184i	$0.443313\pi$
$42$	0	0
$43$	−3.41325	−0.520515	−0.260258	−	0.965539i	$-0.583808\pi$
$43$	−3.41325	−0.520515	−0.260258	+	0.965539i	$0.583808\pi$
$44$	0	0
$45$	7.03583	1.04884
$46$	0	0
$47$	−6.83524	−0.997022	−0.498511	−	0.866883i	$-0.666120\pi$
$47$	−6.83524	−0.997022	−0.498511	+	0.866883i	$0.666120\pi$
$48$	0	0
$49$	−4.06344	−0.580492
$50$	0	0
$51$	0.154913	0.0216921
$52$	0	0
$53$	9.69168	1.33126	0.665628	−	0.746284i	$-0.268164\pi$
$53$	9.69168	1.33126	0.665628	+	0.746284i	$0.268164\pi$
$54$	0	0
$55$	−1.62272	−0.218807
$56$	0	0
$57$	−3.78356	−0.501145
$58$	0	0
$59$	9.29439	1.21003	0.605013	−	0.796215i	$-0.293168\pi$
$59$	9.29439	1.21003	0.605013	+	0.796215i	$0.293168\pi$
$60$	0	0
$61$	−2.23577	−0.286261	−0.143130	−	0.989704i	$-0.545717\pi$
$61$	−2.23577	−0.286261	−0.143130	+	0.989704i	$0.545717\pi$
$62$	0	0
$63$	−4.35131	−0.548214
$64$	0	0
$65$	18.1980	2.25718
$66$	0	0
$67$	−5.40600	−0.660448	−0.330224	−	0.943903i	$-0.607124\pi$
$67$	−5.40600	−0.660448	−0.330224	+	0.943903i	$0.607124\pi$
$68$	0	0
$69$	3.14654	0.378799
$70$	0	0
$71$	−10.7081	−1.27082	−0.635411	−	0.772174i	$-0.719169\pi$
$71$	−10.7081	−1.27082	−0.635411	+	0.772174i	$0.719169\pi$
$72$	0	0
$73$	14.0249	1.64149	0.820743	−	0.571298i	$-0.193560\pi$
$73$	14.0249	1.64149	0.820743	+	0.571298i	$0.193560\pi$
$74$	0	0
$75$	1.81761	0.209879
$76$	0	0
$77$	1.00357	0.114368
$78$	0	0
$79$	−9.29974	−1.04630	−0.523151	−	0.852240i	$-0.675244\pi$
$79$	−9.29974	−1.04630	−0.523151	+	0.852240i	$0.675244\pi$
$80$	0	0
$81$	5.06533	0.562814
$82$	0	0
$83$	14.0341	1.54044	0.770221	−	0.637777i	$-0.220146\pi$
$83$	14.0341	1.54044	0.770221	+	0.637777i	$0.220146\pi$
$84$	0	0
$85$	−0.632348	−0.0685878
$86$	0	0
$87$	−4.24680	−0.455304
$88$	0	0
$89$	2.45479	0.260207	0.130104	−	0.991500i	$-0.458469\pi$
$89$	2.45479	0.260207	0.130104	+	0.991500i	$0.458469\pi$
$90$	0	0
$91$	−11.2546	−1.17980
$92$	0	0
$93$	4.78352	0.496028
$94$	0	0
$95$	15.4444	1.58456
$96$	0	0
$97$	4.76719	0.484034	0.242017	−	0.970272i	$-0.422191\pi$
$97$	4.76719	0.484034	0.242017	+	0.970272i	$0.422191\pi$
$98$	0	0
$99$	−1.48706	−0.149456
$100$	0	0
$101$	−7.13677	−0.710135	−0.355068	−	0.934841i	$-0.615542\pi$
$101$	−7.13677	−0.710135	−0.355068	+	0.934841i	$0.615542\pi$
$102$	0	0
$103$	−12.6116	−1.24266	−0.621329	−	0.783549i	$-0.713407\pi$
$103$	−12.6116	−1.24266	−0.621329	+	0.783549i	$0.713407\pi$
$104$	0	0
$105$	−3.22314	−0.314546
$106$	0	0
$107$	10.1707	0.983239	0.491620	−	0.870810i	$-0.336405\pi$
$107$	10.1707	0.983239	0.491620	+	0.870810i	$0.336405\pi$
$108$	0	0
$109$	4.27498	0.409469	0.204734	−	0.978818i	$-0.434367\pi$
$109$	4.27498	0.409469	0.204734	+	0.978818i	$0.434367\pi$
$110$	0	0
$111$	−7.93738	−0.753383
$112$	0	0
$113$	6.69541	0.629851	0.314925	−	0.949116i	$-0.398020\pi$
$113$	6.69541	0.629851	0.314925	+	0.949116i	$0.398020\pi$
$114$	0	0
$115$	−12.8441	−1.19772
$116$	0	0
$117$	16.6767	1.54176
$118$	0	0
$119$	0.391076	0.0358499
$120$	0	0
$121$	−10.6570	−0.968821
$122$	0	0
$123$	1.53994	0.138852
$124$	0	0
$125$	6.43489	0.575554
$126$	0	0
$127$	−6.60420	−0.586028	−0.293014	−	0.956108i	$-0.594658\pi$
$127$	−6.60420	−0.586028	−0.293014	+	0.956108i	$0.594658\pi$
$128$	0	0
$129$	−2.31693	−0.203994
$130$	0	0
$131$	17.9966	1.57237	0.786187	−	0.617989i	$-0.212052\pi$
$131$	17.9966	1.57237	0.786187	+	0.617989i	$0.212052\pi$
$132$	0	0
$133$	−9.55157	−0.828227
$134$	0	0
$135$	10.4186	0.896688
$136$	0	0
$137$	−18.2765	−1.56147	−0.780734	−	0.624863i	$-0.785155\pi$
$137$	−18.2765	−1.56147	−0.780734	+	0.624863i	$0.785155\pi$
$138$	0	0
$139$	12.3236	1.04527	0.522637	−	0.852555i	$-0.324948\pi$
$139$	12.3236	1.04527	0.522637	+	0.852555i	$0.324948\pi$
$140$	0	0
$141$	−4.63980	−0.390741
$142$	0	0
$143$	−3.84625	−0.321640
$144$	0	0
$145$	17.3353	1.43962
$146$	0	0
$147$	−2.75828	−0.227499
$148$	0	0
$149$	−13.5994	−1.11411	−0.557054	−	0.830476i	$-0.688068\pi$
$149$	−13.5994	−1.11411	−0.557054	+	0.830476i	$0.688068\pi$
$150$	0	0
$151$	9.79260	0.796910	0.398455	−	0.917188i	$-0.369546\pi$
$151$	9.79260	0.796910	0.398455	+	0.917188i	$0.369546\pi$
$152$	0	0
$153$	−0.579486	−0.0468487
$154$	0	0
$155$	−19.5262	−1.56838
$156$	0	0
$157$	4.33227	0.345753	0.172876	−	0.984944i	$-0.444694\pi$
$157$	4.33227	0.345753	0.172876	+	0.984944i	$0.444694\pi$
$158$	0	0
$159$	6.57876	0.521730
$160$	0	0
$161$	7.94342	0.626030
$162$	0	0
$163$	−17.0361	−1.33437	−0.667185	−	0.744892i	$-0.732501\pi$
$163$	−17.0361	−1.33437	−0.667185	+	0.744892i	$0.732501\pi$
$164$	0	0
$165$	−1.10151	−0.0857523
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	30.1339	2.31799
$170$	0	0
$171$	14.1533	1.08233
$172$	0	0
$173$	9.05336	0.688314	0.344157	−	0.938912i	$-0.388165\pi$
$173$	9.05336	0.688314	0.344157	+	0.938912i	$0.388165\pi$
$174$	0	0
$175$	4.58853	0.346861
$176$	0	0
$177$	6.30908	0.474219
$178$	0	0
$179$	−23.6332	−1.76643	−0.883213	−	0.468972i	$-0.844624\pi$
$179$	−23.6332	−1.76643	−0.883213	+	0.468972i	$0.844624\pi$
$180$	0	0
$181$	10.4755	0.778638	0.389319	−	0.921103i	$-0.372710\pi$
$181$	10.4755	0.778638	0.389319	+	0.921103i	$0.372710\pi$
$182$	0	0
$183$	−1.51765	−0.112188
$184$	0	0
$185$	32.4001	2.38210
$186$	0	0
$187$	0.133651	0.00977350
$188$	0	0
$189$	−6.44337	−0.468686
$190$	0	0
$191$	4.29746	0.310953	0.155476	−	0.987840i	$-0.450309\pi$
$191$	4.29746	0.310953	0.155476	+	0.987840i	$0.450309\pi$
$192$	0	0
$193$	10.0045	0.720143	0.360072	−	0.932925i	$-0.382752\pi$
$193$	10.0045	0.720143	0.360072	+	0.932925i	$0.382752\pi$
$194$	0	0
$195$	12.3529	0.884609
$196$	0	0
$197$	−3.91352	−0.278827	−0.139413	−	0.990234i	$-0.544522\pi$
$197$	−3.91352	−0.278827	−0.139413	+	0.990234i	$0.544522\pi$
$198$	0	0
$199$	14.7037	1.04232	0.521158	−	0.853460i	$-0.325500\pi$
$199$	14.7037	1.04232	0.521158	+	0.853460i	$0.325500\pi$
$200$	0	0
$201$	−3.66962	−0.258835
$202$	0	0
$203$	−10.7210	−0.752468
$204$	0	0
$205$	−6.28598	−0.439032
$206$	0	0
$207$	−11.7703	−0.818096
$208$	0	0
$209$	−3.26426	−0.225793
$210$	0	0
$211$	−6.89868	−0.474924	−0.237462	−	0.971397i	$-0.576316\pi$
$211$	−6.89868	−0.474924	−0.237462	+	0.971397i	$0.576316\pi$
$212$	0	0
$213$	−7.26873	−0.498045
$214$	0	0
$215$	9.45763	0.645005
$216$	0	0
$217$	12.0760	0.819770
$218$	0	0
$219$	9.52014	0.643312
$220$	0	0
$221$	−1.49883	−0.100822
$222$	0	0
$223$	−8.86291	−0.593504	−0.296752	−	0.954955i	$-0.595904\pi$
$223$	−8.86291	−0.593504	−0.296752	+	0.954955i	$0.595904\pi$
$224$	0	0
$225$	−6.79916	−0.453278
$226$	0	0
$227$	−8.60440	−0.571094	−0.285547	−	0.958365i	$-0.592175\pi$
$227$	−8.60440	−0.571094	−0.285547	+	0.958365i	$0.592175\pi$
$228$	0	0
$229$	−18.7420	−1.23851	−0.619253	−	0.785191i	$-0.712565\pi$
$229$	−18.7420	−1.23851	−0.619253	+	0.785191i	$0.712565\pi$
$230$	0	0
$231$	0.681229	0.0448216
$232$	0	0
$233$	11.2199	0.735041	0.367521	−	0.930015i	$-0.380207\pi$
$233$	11.2199	0.735041	0.367521	+	0.930015i	$0.380207\pi$
$234$	0	0
$235$	18.9395	1.23548
$236$	0	0
$237$	−6.31271	−0.410055
$238$	0	0
$239$	−14.9142	−0.964720	−0.482360	−	0.875973i	$-0.660220\pi$
$239$	−14.9142	−0.964720	−0.482360	+	0.875973i	$0.660220\pi$
$240$	0	0
$241$	11.1684	0.719421	0.359711	−	0.933064i	$-0.382875\pi$
$241$	11.1684	0.719421	0.359711	+	0.933064i	$0.382875\pi$
$242$	0	0
$243$	14.7185	0.944194
$244$	0	0
$245$	11.2592	0.719325
$246$	0	0
$247$	36.6071	2.32925
$248$	0	0
$249$	9.52641	0.603712
$250$	0	0
$251$	20.5130	1.29477	0.647383	−	0.762165i	$-0.275863\pi$
$251$	20.5130	1.29477	0.647383	+	0.762165i	$0.275863\pi$
$252$	0	0
$253$	2.71467	0.170670
$254$	0	0
$255$	−0.429241	−0.0268801
$256$	0	0
$257$	−7.64794	−0.477066	−0.238533	−	0.971134i	$-0.576666\pi$
$257$	−7.64794	−0.477066	−0.238533	+	0.971134i	$0.576666\pi$
$258$	0	0
$259$	−20.0379	−1.24509
$260$	0	0
$261$	15.8861	0.983325
$262$	0	0
$263$	−16.6200	−1.02483	−0.512416	−	0.858737i	$-0.671249\pi$
$263$	−16.6200	−1.02483	−0.512416	+	0.858737i	$0.671249\pi$
$264$	0	0
$265$	−26.8543	−1.64965
$266$	0	0
$267$	1.66632	0.101977
$268$	0	0
$269$	−0.367120	−0.0223837	−0.0111919	−	0.999937i	$-0.503563\pi$
$269$	−0.367120	−0.0223837	−0.0111919	+	0.999937i	$0.503563\pi$
$270$	0	0
$271$	−9.58213	−0.582073	−0.291037	−	0.956712i	$-0.594000\pi$
$271$	−9.58213	−0.582073	−0.291037	+	0.956712i	$0.594000\pi$
$272$	0	0
$273$	−7.63965	−0.462373
$274$	0	0
$275$	1.56813	0.0945621
$276$	0	0
$277$	−25.8556	−1.55351	−0.776756	−	0.629801i	$-0.783136\pi$
$277$	−25.8556	−1.55351	−0.776756	+	0.629801i	$0.783136\pi$
$278$	0	0
$279$	−17.8938	−1.07128
$280$	0	0
$281$	−33.2164	−1.98153	−0.990763	−	0.135608i	$-0.956701\pi$
$281$	−33.2164	−1.98153	−0.990763	+	0.135608i	$0.956701\pi$
$282$	0	0
$283$	−23.2682	−1.38315	−0.691576	−	0.722304i	$-0.743083\pi$
$283$	−23.2682	−1.38315	−0.691576	+	0.722304i	$0.743083\pi$
$284$	0	0
$285$	10.4837	0.621001
$286$	0	0
$287$	3.88757	0.229476
$288$	0	0
$289$	−16.9479	−0.996936
$290$	0	0
$291$	3.23599	0.189697
$292$	0	0
$293$	−18.0611	−1.05514	−0.527572	−	0.849511i	$-0.676897\pi$
$293$	−18.0611	−1.05514	−0.527572	+	0.849511i	$0.676897\pi$
$294$	0	0
$295$	−25.7534	−1.49942
$296$	0	0
$297$	−2.20203	−0.127775
$298$	0	0
$299$	−30.4437	−1.76061
$300$	0	0
$301$	−5.84908	−0.337135
$302$	0	0
$303$	−4.84447	−0.278308
$304$	0	0
$305$	6.19500	0.354725
$306$	0	0
$307$	−7.98089	−0.455493	−0.227747	−	0.973720i	$-0.573136\pi$
$307$	−7.98089	−0.455493	−0.227747	+	0.973720i	$0.573136\pi$
$308$	0	0
$309$	−8.56082	−0.487008
$310$	0	0
$311$	3.36404	0.190757	0.0953785	−	0.995441i	$-0.469594\pi$
$311$	3.36404	0.190757	0.0953785	+	0.995441i	$0.469594\pi$
$312$	0	0
$313$	−19.2582	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$313$	−19.2582	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$314$	0	0
$315$	12.0569	0.679328
$316$	0	0
$317$	−7.65560	−0.429981	−0.214991	−	0.976616i	$-0.568972\pi$
$317$	−7.65560	−0.429981	−0.214991	+	0.976616i	$0.568972\pi$
$318$	0	0
$319$	−3.66391	−0.205140
$320$	0	0
$321$	6.90392	0.385339
$322$	0	0
$323$	−1.27203	−0.0707777
$324$	0	0
$325$	−17.5859	−0.975489
$326$	0	0
$327$	2.90188	0.160474
$328$	0	0
$329$	−11.7131	−0.645766
$330$	0	0
$331$	7.38974	0.406177	0.203089	−	0.979160i	$-0.434902\pi$
$331$	7.38974	0.406177	0.203089	+	0.979160i	$0.434902\pi$
$332$	0	0
$333$	29.6916	1.62709
$334$	0	0
$335$	14.9793	0.818404
$336$	0	0
$337$	10.1292	0.551774	0.275887	−	0.961190i	$-0.411028\pi$
$337$	10.1292	0.551774	0.275887	+	0.961190i	$0.411028\pi$
$338$	0	0
$339$	4.54487	0.246844
$340$	0	0
$341$	4.12697	0.223488
$342$	0	0
$343$	−18.9587	−1.02368
$344$	0	0
$345$	−8.71862	−0.469395
$346$	0	0
$347$	5.54348	0.297589	0.148795	−	0.988868i	$-0.452461\pi$
$347$	5.54348	0.297589	0.148795	+	0.988868i	$0.452461\pi$
$348$	0	0
$349$	−30.6005	−1.63801	−0.819005	−	0.573787i	$-0.805474\pi$
$349$	−30.6005	−1.63801	−0.819005	+	0.573787i	$0.805474\pi$
$350$	0	0
$351$	24.6947	1.31810
$352$	0	0
$353$	−22.0676	−1.17454	−0.587270	−	0.809391i	$-0.699797\pi$
$353$	−22.0676	−1.17454	−0.587270	+	0.809391i	$0.699797\pi$
$354$	0	0
$355$	29.6707	1.57476
$356$	0	0
$357$	0.265465	0.0140499
$358$	0	0
$359$	32.2627	1.70276	0.851381	−	0.524548i	$-0.175766\pi$
$359$	32.2627	1.70276	0.851381	+	0.524548i	$0.175766\pi$
$360$	0	0
$361$	12.0679	0.635150
$362$	0	0
$363$	−7.23404	−0.379689
$364$	0	0
$365$	−38.8609	−2.03407
$366$	0	0
$367$	−31.7573	−1.65772	−0.828860	−	0.559457i	$-0.811010\pi$
$367$	−31.7573	−1.65772	−0.828860	+	0.559457i	$0.811010\pi$
$368$	0	0
$369$	−5.76049	−0.299879
$370$	0	0
$371$	16.6081	0.862247
$372$	0	0
$373$	22.7618	1.17856	0.589280	−	0.807929i	$-0.299412\pi$
$373$	22.7618	1.17856	0.589280	+	0.807929i	$0.299412\pi$
$374$	0	0
$375$	4.36804	0.225564
$376$	0	0
$377$	41.0890	2.11619
$378$	0	0
$379$	11.4274	0.586985	0.293492	−	0.955961i	$-0.405182\pi$
$379$	11.4274	0.586985	0.293492	+	0.955961i	$0.405182\pi$
$380$	0	0
$381$	−4.48296	−0.229669
$382$	0	0
$383$	−10.3950	−0.531161	−0.265580	−	0.964089i	$-0.585564\pi$
$383$	−10.3950	−0.531161	−0.265580	+	0.964089i	$0.585564\pi$
$384$	0	0
$385$	−2.78075	−0.141720
$386$	0	0
$387$	8.66700	0.440568
$388$	0	0
$389$	−3.59726	−0.182388	−0.0911940	−	0.995833i	$-0.529068\pi$
$389$	−3.59726	−0.182388	−0.0911940	+	0.995833i	$0.529068\pi$
$390$	0	0
$391$	1.05787	0.0534986
$392$	0	0
$393$	12.2162	0.616226
$394$	0	0
$395$	25.7683	1.29654
$396$	0	0
$397$	29.5512	1.48313	0.741567	−	0.670879i	$-0.234083\pi$
$397$	29.5512	1.48313	0.741567	+	0.670879i	$0.234083\pi$
$398$	0	0
$399$	−6.48366	−0.324589
$400$	0	0
$401$	0.592119	0.0295690	0.0147845	−	0.999891i	$-0.495294\pi$
$401$	0.592119	0.0295690	0.0147845	+	0.999891i	$0.495294\pi$
$402$	0	0
$403$	−46.2820	−2.30547
$404$	0	0
$405$	−14.0353	−0.697420
$406$	0	0
$407$	−6.84796	−0.339441
$408$	0	0
$409$	5.86082	0.289799	0.144899	−	0.989446i	$-0.453714\pi$
$409$	5.86082	0.289799	0.144899	+	0.989446i	$0.453714\pi$
$410$	0	0
$411$	−12.4062	−0.611952
$412$	0	0
$413$	15.9272	0.783728
$414$	0	0
$415$	−38.8865	−1.90886
$416$	0	0
$417$	8.36532	0.409652
$418$	0	0
$419$	20.3752	0.995393	0.497696	−	0.867351i	$-0.334179\pi$
$419$	20.3752	0.995393	0.497696	+	0.867351i	$0.334179\pi$
$420$	0	0
$421$	−17.2938	−0.842851	−0.421425	−	0.906863i	$-0.638470\pi$
$421$	−17.2938	−0.842851	−0.421425	+	0.906863i	$0.638470\pi$
$422$	0	0
$423$	17.3562	0.843888
$424$	0	0
$425$	0.611078	0.0296416
$426$	0	0
$427$	−3.83130	−0.185410
$428$	0	0
$429$	−2.61086	−0.126053
$430$	0	0
$431$	−28.1523	−1.35605	−0.678024	−	0.735039i	$-0.737164\pi$
$431$	−28.1523	−1.35605	−0.678024	+	0.735039i	$0.737164\pi$
$432$	0	0
$433$	−4.45580	−0.214132	−0.107066	−	0.994252i	$-0.534146\pi$
$433$	−4.45580	−0.214132	−0.107066	+	0.994252i	$0.534146\pi$
$434$	0	0
$435$	11.7673	0.564197
$436$	0	0
$437$	−25.8371	−1.23596
$438$	0	0
$439$	14.1663	0.676122	0.338061	−	0.941124i	$-0.390229\pi$
$439$	14.1663	0.676122	0.338061	+	0.941124i	$0.390229\pi$
$440$	0	0
$441$	10.3180	0.491333
$442$	0	0
$443$	30.1172	1.43091	0.715457	−	0.698657i	$-0.246219\pi$
$443$	30.1172	1.43091	0.715457	+	0.698657i	$0.246219\pi$
$444$	0	0
$445$	−6.80187	−0.322440
$446$	0	0
$447$	−9.23136	−0.436628
$448$	0	0
$449$	6.16186	0.290796	0.145398	−	0.989373i	$-0.453554\pi$
$449$	6.16186	0.290796	0.145398	+	0.989373i	$0.453554\pi$
$450$	0	0
$451$	1.32858	0.0625603
$452$	0	0
$453$	6.64726	0.312316
$454$	0	0
$455$	31.1848	1.46197
$456$	0	0
$457$	−16.5182	−0.772691	−0.386345	−	0.922354i	$-0.626263\pi$
$457$	−16.5182	−0.772691	−0.386345	+	0.922354i	$0.626263\pi$
$458$	0	0
$459$	−0.858096	−0.0400525
$460$	0	0
$461$	−33.1479	−1.54385	−0.771925	−	0.635714i	$-0.780706\pi$
$461$	−33.1479	−1.54385	−0.771925	+	0.635714i	$0.780706\pi$
$462$	0	0
$463$	−31.9393	−1.48435	−0.742173	−	0.670208i	$-0.766205\pi$
$463$	−31.9393	−1.48435	−0.742173	+	0.670208i	$0.766205\pi$
$464$	0	0
$465$	−13.2545	−0.614661
$466$	0	0
$467$	−23.7351	−1.09833	−0.549166	−	0.835713i	$-0.685054\pi$
$467$	−23.7351	−1.09833	−0.549166	+	0.835713i	$0.685054\pi$
$468$	0	0
$469$	−9.26393	−0.427769
$470$	0	0
$471$	2.94077	0.135503
$472$	0	0
$473$	−1.99893	−0.0919107
$474$	0	0
$475$	−14.9249	−0.684799
$476$	0	0
$477$	−24.6094	−1.12679
$478$	0	0
$479$	−13.2636	−0.606031	−0.303015	−	0.952986i	$-0.597993\pi$
$479$	−13.2636	−0.606031	−0.303015	+	0.952986i	$0.597993\pi$
$480$	0	0
$481$	76.7966	3.50162
$482$	0	0
$483$	5.39204	0.245346
$484$	0	0
$485$	−13.2092	−0.599799
$486$	0	0
$487$	19.2898	0.874102	0.437051	−	0.899437i	$-0.356023\pi$
$487$	19.2898	0.874102	0.437051	+	0.899437i	$0.356023\pi$
$488$	0	0
$489$	−11.5642	−0.522950
$490$	0	0
$491$	−31.9967	−1.44399	−0.721995	−	0.691899i	$-0.756775\pi$
$491$	−31.9967	−1.44399	−0.721995	+	0.691899i	$0.756775\pi$
$492$	0	0
$493$	−1.42777	−0.0643036
$494$	0	0
$495$	4.12044	0.185200
$496$	0	0
$497$	−18.3499	−0.823104
$498$	0	0
$499$	6.70719	0.300255	0.150128	−	0.988667i	$-0.452032\pi$
$499$	6.70719	0.300255	0.150128	+	0.988667i	$0.452032\pi$
$500$	0	0
$501$	0.678805	0.0303268
$502$	0	0
$503$	23.5455	1.04984	0.524922	−	0.851151i	$-0.324095\pi$
$503$	23.5455	1.04984	0.524922	+	0.851151i	$0.324095\pi$
$504$	0	0
$505$	19.7750	0.879975
$506$	0	0
$507$	20.4550	0.908440
$508$	0	0
$509$	10.4379	0.462650	0.231325	−	0.972877i	$-0.425694\pi$
$509$	10.4379	0.462650	0.231325	+	0.972877i	$0.425694\pi$
$510$	0	0
$511$	24.0335	1.06318
$512$	0	0
$513$	20.9580	0.925318
$514$	0	0
$515$	34.9450	1.53986
$516$	0	0
$517$	−4.00297	−0.176050
$518$	0	0
$519$	6.14547	0.269756
$520$	0	0
$521$	16.7107	0.732109	0.366054	−	0.930593i	$-0.380708\pi$
$521$	16.7107	0.732109	0.366054	+	0.930593i	$0.380708\pi$
$522$	0	0
$523$	−23.7005	−1.03635	−0.518176	−	0.855274i	$-0.673389\pi$
$523$	−23.7005	−1.03635	−0.518176	+	0.855274i	$0.673389\pi$
$524$	0	0
$525$	3.11472	0.135937
$526$	0	0
$527$	1.60822	0.0700550
$528$	0	0
$529$	−1.51295	−0.0657805
$530$	0	0
$531$	−23.6005	−1.02418
$532$	0	0
$533$	−14.8994	−0.645363
$534$	0	0
$535$	−28.1816	−1.21840
$536$	0	0
$537$	−16.0423	−0.692277
$538$	0	0
$539$	−2.37970	−0.102501
$540$	0	0
$541$	27.1775	1.16845	0.584225	−	0.811592i	$-0.301399\pi$
$541$	27.1775	1.16845	0.584225	+	0.811592i	$0.301399\pi$
$542$	0	0
$543$	7.11083	0.305155
$544$	0	0
$545$	−11.8454	−0.507399
$546$	0	0
$547$	−0.234751	−0.0100372	−0.00501861	−	0.999987i	$-0.501597\pi$
$547$	−0.234751	−0.0100372	−0.00501861	+	0.999987i	$0.501597\pi$
$548$	0	0
$549$	5.67712	0.242294
$550$	0	0
$551$	34.8716	1.48558
$552$	0	0
$553$	−15.9364	−0.677685
$554$	0	0
$555$	21.9934	0.933566
$556$	0	0
$557$	22.3980	0.949034	0.474517	−	0.880246i	$-0.342623\pi$
$557$	22.3980	0.949034	0.474517	+	0.880246i	$0.342623\pi$
$558$	0	0
$559$	22.4170	0.948138
$560$	0	0
$561$	0.0907227	0.00383031
$562$	0	0
$563$	32.4021	1.36559	0.682793	−	0.730612i	$-0.260765\pi$
$563$	32.4021	1.36559	0.682793	+	0.730612i	$0.260765\pi$
$564$	0	0
$565$	−18.5520	−0.780489
$566$	0	0
$567$	8.68015	0.364532
$568$	0	0
$569$	41.1964	1.72704	0.863522	−	0.504311i	$-0.168254\pi$
$569$	41.1964	1.72704	0.863522	+	0.504311i	$0.168254\pi$
$570$	0	0
$571$	−20.0207	−0.837841	−0.418920	−	0.908023i	$-0.637591\pi$
$571$	−20.0207	−0.837841	−0.418920	+	0.908023i	$0.637591\pi$
$572$	0	0
$573$	2.91713	0.121865
$574$	0	0
$575$	12.4120	0.517618
$576$	0	0
$577$	−37.4095	−1.55738	−0.778689	−	0.627410i	$-0.784115\pi$
$577$	−37.4095	−1.55738	−0.778689	+	0.627410i	$0.784115\pi$
$578$	0	0
$579$	6.79114	0.282230
$580$	0	0
$581$	24.0494	0.997736
$582$	0	0
$583$	5.67581	0.235068
$584$	0	0
$585$	−46.2088	−1.91050
$586$	0	0
$587$	−12.1522	−0.501576	−0.250788	−	0.968042i	$-0.580690\pi$
$587$	−12.1522	−0.501576	−0.250788	+	0.968042i	$0.580690\pi$
$588$	0	0
$589$	−39.2788	−1.61845
$590$	0	0
$591$	−2.65652	−0.109275
$592$	0	0
$593$	−24.5616	−1.00862	−0.504311	−	0.863522i	$-0.668254\pi$
$593$	−24.5616	−1.00862	−0.504311	+	0.863522i	$0.668254\pi$
$594$	0	0
$595$	−1.08362	−0.0444240
$596$	0	0
$597$	9.98093	0.408492
$598$	0	0
$599$	15.2317	0.622351	0.311176	−	0.950352i	$-0.399277\pi$
$599$	15.2317	0.622351	0.311176	+	0.950352i	$0.399277\pi$
$600$	0	0
$601$	37.1414	1.51503	0.757515	−	0.652817i	$-0.226413\pi$
$601$	37.1414	1.51503	0.757515	+	0.652817i	$0.226413\pi$
$602$	0	0
$603$	13.7270	0.559008
$604$	0	0
$605$	29.5291	1.20053
$606$	0	0
$607$	−11.3279	−0.459783	−0.229892	−	0.973216i	$-0.573837\pi$
$607$	−11.3279	−0.459783	−0.229892	+	0.973216i	$0.573837\pi$
$608$	0	0
$609$	−7.27748	−0.294898
$610$	0	0
$611$	44.8914	1.81611
$612$	0	0
$613$	23.3329	0.942406	0.471203	−	0.882025i	$-0.343820\pi$
$613$	23.3329	0.942406	0.471203	+	0.882025i	$0.343820\pi$
$614$	0	0
$615$	−4.26695	−0.172060
$616$	0	0
$617$	−1.96533	−0.0791213	−0.0395606	−	0.999217i	$-0.512596\pi$
$617$	−1.96533	−0.0791213	−0.0395606	+	0.999217i	$0.512596\pi$
$618$	0	0
$619$	−13.4982	−0.542540	−0.271270	−	0.962503i	$-0.587444\pi$
$619$	−13.4982	−0.542540	−0.271270	+	0.962503i	$0.587444\pi$
$620$	0	0
$621$	−17.4294	−0.699418
$622$	0	0
$623$	4.20662	0.168535
$624$	0	0
$625$	−31.2184	−1.24874
$626$	0	0
$627$	−2.21579	−0.0884903
$628$	0	0
$629$	−2.66854	−0.106402
$630$	0	0
$631$	48.5568	1.93301	0.966507	−	0.256640i	$-0.0826155\pi$
$631$	48.5568	1.93301	0.966507	+	0.256640i	$0.0826155\pi$
$632$	0	0
$633$	−4.68286	−0.186127
$634$	0	0
$635$	18.2993	0.726185
$636$	0	0
$637$	26.6872	1.05739
$638$	0	0
$639$	27.1903	1.07563
$640$	0	0
$641$	9.23627	0.364811	0.182405	−	0.983223i	$-0.441612\pi$
$641$	9.23627	0.364811	0.182405	+	0.983223i	$0.441612\pi$
$642$	0	0
$643$	1.04809	0.0413325	0.0206662	−	0.999786i	$-0.493421\pi$
$643$	1.04809	0.0413325	0.0206662	+	0.999786i	$0.493421\pi$
$644$	0	0
$645$	6.41988	0.252783
$646$	0	0
$647$	13.5506	0.532728	0.266364	−	0.963872i	$-0.414178\pi$
$647$	13.5506	0.532728	0.266364	+	0.963872i	$0.414178\pi$
$648$	0	0
$649$	5.44314	0.213662
$650$	0	0
$651$	8.19723	0.321275
$652$	0	0
$653$	−32.0960	−1.25601	−0.628007	−	0.778208i	$-0.716129\pi$
$653$	−32.0960	−1.25601	−0.628007	+	0.778208i	$0.716129\pi$
$654$	0	0
$655$	−49.8661	−1.94843
$656$	0	0
$657$	−35.6123	−1.38937
$658$	0	0
$659$	36.6960	1.42947	0.714736	−	0.699394i	$-0.246547\pi$
$659$	36.6960	1.42947	0.714736	+	0.699394i	$0.246547\pi$
$660$	0	0
$661$	25.1573	0.978507	0.489253	−	0.872142i	$-0.337269\pi$
$661$	25.1573	0.978507	0.489253	+	0.872142i	$0.337269\pi$
$662$	0	0
$663$	−1.01741	−0.0395130
$664$	0	0
$665$	26.4661	1.02631
$666$	0	0
$667$	−29.0005	−1.12290
$668$	0	0
$669$	−6.01619	−0.232599
$670$	0	0
$671$	−1.30935	−0.0505469
$672$	0	0
$673$	−2.07481	−0.0799780	−0.0399890	−	0.999200i	$-0.512732\pi$
$673$	−2.07481	−0.0799780	−0.0399890	+	0.999200i	$0.512732\pi$
$674$	0	0
$675$	−10.0681	−0.387522
$676$	0	0
$677$	3.38656	0.130156	0.0650780	−	0.997880i	$-0.479270\pi$
$677$	3.38656	0.130156	0.0650780	+	0.997880i	$0.479270\pi$
$678$	0	0
$679$	8.16924	0.313506
$680$	0	0
$681$	−5.84071	−0.223817
$682$	0	0
$683$	−23.7305	−0.908021	−0.454010	−	0.890996i	$-0.650007\pi$
$683$	−23.7305	−0.908021	−0.454010	+	0.890996i	$0.650007\pi$
$684$	0	0
$685$	50.6417	1.93492
$686$	0	0
$687$	−12.7222	−0.485381
$688$	0	0
$689$	−63.6515	−2.42493
$690$	0	0
$691$	0.116707	0.00443975	0.00221988	−	0.999998i	$-0.499293\pi$
$691$	0.116707	0.00443975	0.00221988	+	0.999998i	$0.499293\pi$
$692$	0	0
$693$	−2.54829	−0.0968016
$694$	0	0
$695$	−34.1469	−1.29527
$696$	0	0
$697$	0.517727	0.0196103
$698$	0	0
$699$	7.61613	0.288069
$700$	0	0
$701$	6.10099	0.230431	0.115216	−	0.993341i	$-0.463244\pi$
$701$	6.10099	0.230431	0.115216	+	0.993341i	$0.463244\pi$
$702$	0	0
$703$	65.1760	2.45816
$704$	0	0
$705$	12.8562	0.484193
$706$	0	0
$707$	−12.2298	−0.459951
$708$	0	0
$709$	26.7128	1.00322	0.501610	−	0.865094i	$-0.332741\pi$
$709$	26.7128	1.00322	0.501610	+	0.865094i	$0.332741\pi$
$710$	0	0
$711$	23.6141	0.885599
$712$	0	0
$713$	32.6656	1.22334
$714$	0	0
$715$	10.6574	0.398565
$716$	0	0
$717$	−10.1238	−0.378082
$718$	0	0
$719$	1.51903	0.0566502	0.0283251	−	0.999599i	$-0.490983\pi$
$719$	1.51903	0.0566502	0.0283251	+	0.999599i	$0.490983\pi$
$720$	0	0
$721$	−21.6118	−0.804864
$722$	0	0
$723$	7.58118	0.281947
$724$	0	0
$725$	−16.7522	−0.622160
$726$	0	0
$727$	13.3806	0.496258	0.248129	−	0.968727i	$-0.420184\pi$
$727$	13.3806	0.496258	0.248129	+	0.968727i	$0.420184\pi$
$728$	0	0
$729$	−5.20498	−0.192777
$730$	0	0
$731$	−0.778951	−0.0288105
$732$	0	0
$733$	−22.2289	−0.821045	−0.410522	−	0.911851i	$-0.634654\pi$
$733$	−22.2289	−0.821045	−0.410522	+	0.911851i	$0.634654\pi$
$734$	0	0
$735$	7.64281	0.281909
$736$	0	0
$737$	−3.16596	−0.116619
$738$	0	0
$739$	10.7007	0.393632	0.196816	−	0.980440i	$-0.436940\pi$
$739$	10.7007	0.393632	0.196816	+	0.980440i	$0.436940\pi$
$740$	0	0
$741$	24.8491	0.912853
$742$	0	0
$743$	18.8163	0.690302	0.345151	−	0.938547i	$-0.387828\pi$
$743$	18.8163	0.690302	0.345151	+	0.938547i	$0.387828\pi$
$744$	0	0
$745$	37.6821	1.38056
$746$	0	0
$747$	−35.6357	−1.30384
$748$	0	0
$749$	17.4289	0.636839
$750$	0	0
$751$	−25.8900	−0.944740	−0.472370	−	0.881400i	$-0.656601\pi$
$751$	−25.8900	−0.944740	−0.472370	+	0.881400i	$0.656601\pi$
$752$	0	0
$753$	13.9243	0.507430
$754$	0	0
$755$	−27.1339	−0.987504
$756$	0	0
$757$	−2.95205	−0.107294	−0.0536470	−	0.998560i	$-0.517085\pi$
$757$	−2.95205	−0.107294	−0.0536470	+	0.998560i	$0.517085\pi$
$758$	0	0
$759$	1.84273	0.0668869
$760$	0	0
$761$	6.00010	0.217503	0.108752	−	0.994069i	$-0.465315\pi$
$761$	6.00010	0.217503	0.108752	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	7.32577	0.265211
$764$	0	0
$765$	1.60567	0.0580533
$766$	0	0
$767$	−61.0422	−2.20411
$768$	0	0
$769$	37.1318	1.33901	0.669503	−	0.742809i	$-0.266507\pi$
$769$	37.1318	1.33901	0.669503	+	0.742809i	$0.266507\pi$
$770$	0	0
$771$	−5.19146	−0.186966
$772$	0	0
$773$	−54.8700	−1.97354	−0.986768	−	0.162141i	$-0.948160\pi$
$773$	−54.8700	−1.97354	−0.986768	+	0.162141i	$0.948160\pi$
$774$	0	0
$775$	18.8694	0.677807
$776$	0	0
$777$	−13.6018	−0.487962
$778$	0	0
$779$	−12.6449	−0.453049
$780$	0	0
$781$	−6.27108	−0.224397
$782$	0	0
$783$	23.5240	0.840678
$784$	0	0
$785$	−12.0041	−0.428445
$786$	0	0
$787$	36.9629	1.31758	0.658792	−	0.752325i	$-0.271068\pi$
$787$	36.9629	1.31758	0.658792	+	0.752325i	$0.271068\pi$
$788$	0	0
$789$	−11.2817	−0.401640
$790$	0	0
$791$	11.4735	0.407951
$792$	0	0
$793$	14.6837	0.521435
$794$	0	0
$795$	−18.2288	−0.646510
$796$	0	0
$797$	−31.0729	−1.10066	−0.550330	−	0.834947i	$-0.685498\pi$
$797$	−31.0729	−1.10066	−0.550330	+	0.834947i	$0.685498\pi$
$798$	0	0
$799$	−1.55990	−0.0551852
$800$	0	0
$801$	−6.23326	−0.220241
$802$	0	0
$803$	8.21348	0.289847
$804$	0	0
$805$	−22.0101	−0.775754
$806$	0	0
$807$	−0.249203	−0.00877237
$808$	0	0
$809$	22.4559	0.789508	0.394754	−	0.918787i	$-0.370830\pi$
$809$	22.4559	0.789508	0.394754	+	0.918787i	$0.370830\pi$
$810$	0	0
$811$	−16.9050	−0.593615	−0.296808	−	0.954937i	$-0.595922\pi$
$811$	−16.9050	−0.593615	−0.296808	+	0.954937i	$0.595922\pi$
$812$	0	0
$813$	−6.50440	−0.228119
$814$	0	0
$815$	47.2046	1.65350
$816$	0	0
$817$	19.0249	0.665599
$818$	0	0
$819$	28.5779	0.998591
$820$	0	0
$821$	53.4021	1.86375	0.931874	−	0.362783i	$-0.118173\pi$
$821$	53.4021	1.86375	0.931874	+	0.362783i	$0.118173\pi$
$822$	0	0
$823$	−26.5265	−0.924655	−0.462328	−	0.886709i	$-0.652986\pi$
$823$	−26.5265	−0.924655	−0.462328	+	0.886709i	$0.652986\pi$
$824$	0	0
$825$	1.06446	0.0370596
$826$	0	0
$827$	−25.5972	−0.890101	−0.445051	−	0.895505i	$-0.646814\pi$
$827$	−25.5972	−0.890101	−0.445051	+	0.895505i	$0.646814\pi$
$828$	0	0
$829$	18.8085	0.653248	0.326624	−	0.945154i	$-0.394089\pi$
$829$	18.8085	0.653248	0.326624	+	0.945154i	$0.394089\pi$
$830$	0	0
$831$	−17.5509	−0.608834
$832$	0	0
$833$	−0.927334	−0.0321302
$834$	0	0
$835$	−2.77086	−0.0958895
$836$	0	0
$837$	−26.4970	−0.915870
$838$	0	0
$839$	8.57098	0.295903	0.147952	−	0.988995i	$-0.452732\pi$
$839$	8.57098	0.295903	0.147952	+	0.988995i	$0.452732\pi$
$840$	0	0
$841$	10.1411	0.349692
$842$	0	0
$843$	−22.5475	−0.776576
$844$	0	0
$845$	−83.4968	−2.87238
$846$	0	0
$847$	−18.2623	−0.627500
$848$	0	0
$849$	−15.7946	−0.542069
$850$	0	0
$851$	−54.2027	−1.85804
$852$	0	0
$853$	−7.49983	−0.256789	−0.128395	−	0.991723i	$-0.540982\pi$
$853$	−7.49983	−0.256789	−0.128395	+	0.991723i	$0.540982\pi$
$854$	0	0
$855$	−39.2167	−1.34118
$856$	0	0
$857$	−22.6640	−0.774187	−0.387093	−	0.922041i	$-0.626521\pi$
$857$	−22.6640	−0.774187	−0.387093	+	0.922041i	$0.626521\pi$
$858$	0	0
$859$	29.3254	1.00057	0.500285	−	0.865861i	$-0.333229\pi$
$859$	29.3254	1.00057	0.500285	+	0.865861i	$0.333229\pi$
$860$	0	0
$861$	2.63890	0.0899335
$862$	0	0
$863$	−6.92832	−0.235843	−0.117921	−	0.993023i	$-0.537623\pi$
$863$	−6.92832	−0.235843	−0.117921	+	0.993023i	$0.537623\pi$
$864$	0	0
$865$	−25.0856	−0.852935
$866$	0	0
$867$	−11.5043	−0.390708
$868$	0	0
$869$	−5.44628	−0.184752
$870$	0	0
$871$	35.5047	1.20303
$872$	0	0
$873$	−12.1050	−0.409691
$874$	0	0
$875$	11.0271	0.372783
$876$	0	0
$877$	13.2962	0.448981	0.224490	−	0.974476i	$-0.427928\pi$
$877$	13.2962	0.448981	0.224490	+	0.974476i	$0.427928\pi$
$878$	0	0
$879$	−12.2600	−0.413519
$880$	0	0
$881$	39.4145	1.32791	0.663953	−	0.747774i	$-0.268877\pi$
$881$	39.4145	1.32791	0.663953	+	0.747774i	$0.268877\pi$
$882$	0	0
$883$	−53.5642	−1.80258	−0.901289	−	0.433219i	$-0.857378\pi$
$883$	−53.5642	−1.80258	−0.901289	+	0.433219i	$0.857378\pi$
$884$	0	0
$885$	−17.4816	−0.587636
$886$	0	0
$887$	16.6031	0.557479	0.278739	−	0.960367i	$-0.410083\pi$
$887$	16.6031	0.557479	0.278739	+	0.960367i	$0.410083\pi$
$888$	0	0
$889$	−11.3172	−0.379567
$890$	0	0
$891$	2.96645	0.0993797
$892$	0	0
$893$	38.0986	1.27492
$894$	0	0
$895$	65.4842	2.18889
$896$	0	0
$897$	−20.6653	−0.689996
$898$	0	0
$899$	−44.0879	−1.47041
$900$	0	0
$901$	2.21178	0.0736850
$902$	0	0
$903$	−3.97038	−0.132126
$904$	0	0
$905$	−29.0261	−0.964862
$906$	0	0
$907$	36.3655	1.20750	0.603748	−	0.797175i	$-0.293673\pi$
$907$	36.3655	1.20750	0.603748	+	0.797175i	$0.293673\pi$
$908$	0	0
$909$	18.1219	0.601064
$910$	0	0
$911$	7.51194	0.248882	0.124441	−	0.992227i	$-0.460286\pi$
$911$	7.51194	0.248882	0.124441	+	0.992227i	$0.460286\pi$
$912$	0	0
$913$	8.21889	0.272005
$914$	0	0
$915$	4.20520	0.139020
$916$	0	0
$917$	30.8397	1.01842
$918$	0	0
$919$	−31.8502	−1.05064	−0.525321	−	0.850904i	$-0.676055\pi$
$919$	−31.8502	−1.05064	−0.525321	+	0.850904i	$0.676055\pi$
$920$	0	0
$921$	−5.41747	−0.178512
$922$	0	0
$923$	70.3272	2.31485
$924$	0	0
$925$	−31.3103	−1.02948
$926$	0	0
$927$	32.0237	1.05180
$928$	0	0
$929$	9.78115	0.320909	0.160455	−	0.987043i	$-0.448704\pi$
$929$	9.78115	0.320909	0.160455	+	0.987043i	$0.448704\pi$
$930$	0	0
$931$	22.6490	0.742292
$932$	0	0
$933$	2.28352	0.0747592
$934$	0	0
$935$	−0.370327	−0.0121110
$936$	0	0
$937$	−10.7495	−0.351170	−0.175585	−	0.984464i	$-0.556182\pi$
$937$	−10.7495	−0.351170	−0.175585	+	0.984464i	$0.556182\pi$
$938$	0	0
$939$	−13.0726	−0.426607
$940$	0	0
$941$	42.0724	1.37152	0.685760	−	0.727828i	$-0.259470\pi$
$941$	42.0724	1.37152	0.685760	+	0.727828i	$0.259470\pi$
$942$	0	0
$943$	10.5159	0.342445
$944$	0	0
$945$	17.8537	0.580780
$946$	0	0
$947$	−44.5792	−1.44863	−0.724315	−	0.689470i	$-0.757844\pi$
$947$	−44.5792	−1.44863	−0.724315	+	0.689470i	$0.757844\pi$
$948$	0	0
$949$	−92.1102	−2.99002
$950$	0	0
$951$	−5.19666	−0.168513
$952$	0	0
$953$	−7.05877	−0.228656	−0.114328	−	0.993443i	$-0.536471\pi$
$953$	−7.05877	−0.228656	−0.114328	+	0.993443i	$0.536471\pi$
$954$	0	0
$955$	−11.9076	−0.385322
$956$	0	0
$957$	−2.48708	−0.0803959
$958$	0	0
$959$	−31.3194	−1.01136
$960$	0	0
$961$	18.6598	0.601930
$962$	0	0
$963$	−25.8257	−0.832221
$964$	0	0
$965$	−27.7212	−0.892376
$966$	0	0
$967$	23.7072	0.762371	0.381186	−	0.924499i	$-0.375516\pi$
$967$	23.7072	0.762371	0.381186	+	0.924499i	$0.375516\pi$
$968$	0	0
$969$	−0.863461	−0.0277384
$970$	0	0
$971$	42.0876	1.35065	0.675327	−	0.737518i	$-0.264003\pi$
$971$	42.0876	1.35065	0.675327	+	0.737518i	$0.264003\pi$
$972$	0	0
$973$	21.1182	0.677019
$974$	0	0
$975$	−11.9374	−0.382302
$976$	0	0
$977$	30.2508	0.967808	0.483904	−	0.875121i	$-0.339218\pi$
$977$	30.2508	0.967808	0.483904	+	0.875121i	$0.339218\pi$
$978$	0	0
$979$	1.43762	0.0459464
$980$	0	0
$981$	−10.8551	−0.346578
$982$	0	0
$983$	−4.78021	−0.152465	−0.0762325	−	0.997090i	$-0.524289\pi$
$983$	−4.78021	−0.152465	−0.0762325	+	0.997090i	$0.524289\pi$
$984$	0	0
$985$	10.8438	0.345513
$986$	0	0
$987$	−7.95093	−0.253081
$988$	0	0
$989$	−15.8218	−0.503104
$990$	0	0
$991$	−25.1676	−0.799476	−0.399738	−	0.916630i	$-0.630899\pi$
$991$	−25.1676	−0.799476	−0.399738	+	0.916630i	$0.630899\pi$
$992$	0	0
$993$	5.01619	0.159184
$994$	0	0
$995$	−40.7418	−1.29160
$996$	0	0
$997$	3.20041	0.101358	0.0506790	−	0.998715i	$-0.483861\pi$
$997$	3.20041	0.101358	0.0506790	+	0.998715i	$0.483861\pi$
$998$	0	0
$999$	43.9669	1.39105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.c.1.6	✓	7
3.2	odd	2		6012.2.a.g.1.6		7
4.3	odd	2		2672.2.a.k.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.6	✓	7	1.1	even	1	trivial
2672.2.a.k.1.2		7	4.3	odd	2
6012.2.a.g.1.6		7	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)

\(f(q)\)	\(=\)	\(q+0.678805 q^{3} -2.77086 q^{5} +1.71364 q^{7} -2.53922 q^{9} +O(q^{10})\)	\(q+0.678805 q^{3} -2.77086 q^{5} +1.71364 q^{7} -2.53922 q^{9} +0.585637 q^{11} -6.56764 q^{13} -1.88087 q^{15} +0.228214 q^{17} -5.57385 q^{19} +1.16323 q^{21} +4.63541 q^{23} +2.67765 q^{25} -3.76005 q^{27} -6.25628 q^{29} +7.04697 q^{31} +0.397533 q^{33} -4.74825 q^{35} -11.6932 q^{37} -4.45815 q^{39} +2.26860 q^{41} -3.41325 q^{43} +7.03583 q^{45} -6.83524 q^{47} -4.06344 q^{49} +0.154913 q^{51} +9.69168 q^{53} -1.62272 q^{55} -3.78356 q^{57} +9.29439 q^{59} -2.23577 q^{61} -4.35131 q^{63} +18.1980 q^{65} -5.40600 q^{67} +3.14654 q^{69} -10.7081 q^{71} +14.0249 q^{73} +1.81761 q^{75} +1.00357 q^{77} -9.29974 q^{79} +5.06533 q^{81} +14.0341 q^{83} -0.632348 q^{85} -4.24680 q^{87} +2.45479 q^{89} -11.2546 q^{91} +4.78352 q^{93} +15.4444 q^{95} +4.76719 q^{97} -1.48706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * q^99

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