LMFDB - Embedded newform 8001.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99802 q^{2} +1.99208 q^{4} -1.55092 q^{5} -1.00000 q^{7} +0.0158280 q^{8} +O(q^{10})$	\(q-1.99802 q^{2} +1.99208 q^{4} -1.55092 q^{5} -1.00000 q^{7} +0.0158280 q^{8} +3.09877 q^{10} -5.13227 q^{11} +3.23567 q^{13} +1.99802 q^{14} -4.01578 q^{16} +3.98079 q^{17} +2.86954 q^{19} -3.08956 q^{20} +10.2544 q^{22} -1.67857 q^{23} -2.59464 q^{25} -6.46494 q^{26} -1.99208 q^{28} -7.42810 q^{29} +0.815589 q^{31} +7.99195 q^{32} -7.95370 q^{34} +1.55092 q^{35} +0.873940 q^{37} -5.73339 q^{38} -0.0245479 q^{40} -5.56058 q^{41} +4.95707 q^{43} -10.2239 q^{44} +3.35381 q^{46} -6.85565 q^{47} +1.00000 q^{49} +5.18414 q^{50} +6.44571 q^{52} +8.02140 q^{53} +7.95975 q^{55} -0.0158280 q^{56} +14.8415 q^{58} +9.91489 q^{59} -6.38875 q^{61} -1.62956 q^{62} -7.93650 q^{64} -5.01828 q^{65} -0.256190 q^{67} +7.93005 q^{68} -3.09877 q^{70} -1.58801 q^{71} +9.67421 q^{73} -1.74615 q^{74} +5.71635 q^{76} +5.13227 q^{77} +4.34418 q^{79} +6.22816 q^{80} +11.1101 q^{82} -2.94073 q^{83} -6.17390 q^{85} -9.90432 q^{86} -0.0812334 q^{88} -17.8737 q^{89} -3.23567 q^{91} -3.34384 q^{92} +13.6977 q^{94} -4.45043 q^{95} -3.77305 q^{97} -1.99802 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

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 $

$2$	−1.99802	−1.41281	−0.706406	−	0.707807i	$-0.749685\pi$
$2$	−1.99802	−1.41281	−0.706406	+	0.707807i	$0.749685\pi$
$3$	0	0
$3$	0	0
$4$	1.99208	0.996039
$5$	−1.55092	−0.693593	−0.346797	−	0.937940i	$-0.612731\pi$
$5$	−1.55092	−0.693593	−0.346797	+	0.937940i	$0.612731\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.0158280	0.00559603
$9$	0	0
$10$	3.09877	0.979917
$11$	−5.13227	−1.54744	−0.773719	−	0.633529i	$-0.781606\pi$
$11$	−5.13227	−1.54744	−0.773719	+	0.633529i	$0.781606\pi$
$12$	0	0
$13$	3.23567	0.897414	0.448707	−	0.893679i	$-0.351885\pi$
$13$	3.23567	0.897414	0.448707	+	0.893679i	$0.351885\pi$
$14$	1.99802	0.533993
$15$	0	0
$16$	−4.01578	−1.00395
$17$	3.98079	0.965484	0.482742	−	0.875763i	$-0.339641\pi$
$17$	3.98079	0.965484	0.482742	+	0.875763i	$0.339641\pi$
$18$	0	0
$19$	2.86954	0.658317	0.329159	−	0.944275i	$-0.393235\pi$
$19$	2.86954	0.658317	0.329159	+	0.944275i	$0.393235\pi$
$20$	−3.08956	−0.690846
$21$	0	0
$22$	10.2544	2.18624
$23$	−1.67857	−0.350006	−0.175003	−	0.984568i	$-0.555993\pi$
$23$	−1.67857	−0.350006	−0.175003	+	0.984568i	$0.555993\pi$
$24$	0	0
$25$	−2.59464	−0.518928
$26$	−6.46494	−1.26788
$27$	0	0
$28$	−1.99208	−0.376467
$29$	−7.42810	−1.37936	−0.689682	−	0.724112i	$-0.742250\pi$
$29$	−7.42810	−1.37936	−0.689682	+	0.724112i	$0.742250\pi$
$30$	0	0
$31$	0.815589	0.146484	0.0732420	−	0.997314i	$-0.476665\pi$
$31$	0.815589	0.146484	0.0732420	+	0.997314i	$0.476665\pi$
$32$	7.99195	1.41279
$33$	0	0
$34$	−7.95370	−1.36405
$35$	1.55092	0.262154
$36$	0	0
$37$	0.873940	0.143675	0.0718374	−	0.997416i	$-0.477114\pi$
$37$	0.873940	0.143675	0.0718374	+	0.997416i	$0.477114\pi$
$38$	−5.73339	−0.930079
$39$	0	0
$40$	−0.0245479	−0.00388137
$41$	−5.56058	−0.868417	−0.434208	−	0.900812i	$-0.642972\pi$
$41$	−5.56058	−0.868417	−0.434208	+	0.900812i	$0.642972\pi$
$42$	0	0
$43$	4.95707	0.755946	0.377973	−	0.925817i	$-0.376621\pi$
$43$	4.95707	0.755946	0.377973	+	0.925817i	$0.376621\pi$
$44$	−10.2239	−1.54131
$45$	0	0
$46$	3.35381	0.494493
$47$	−6.85565	−1.00000	−0.500000	−	0.866026i	$-0.666667\pi$
$47$	−6.85565	−1.00000	−0.500000	+	0.866026i	$0.666667\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.18414	0.733148
$51$	0	0
$52$	6.44571	0.893860
$53$	8.02140	1.10182	0.550912	−	0.834563i	$-0.314280\pi$
$53$	8.02140	1.10182	0.550912	+	0.834563i	$0.314280\pi$
$54$	0	0
$55$	7.95975	1.07329
$56$	−0.0158280	−0.00211510
$57$	0	0
$58$	14.8415	1.94878
$59$	9.91489	1.29081	0.645405	−	0.763841i	$-0.276689\pi$
$59$	9.91489	1.29081	0.645405	+	0.763841i	$0.276689\pi$
$60$	0	0
$61$	−6.38875	−0.817996	−0.408998	−	0.912535i	$-0.634122\pi$
$61$	−6.38875	−0.817996	−0.408998	+	0.912535i	$0.634122\pi$
$62$	−1.62956	−0.206955
$63$	0	0
$64$	−7.93650	−0.992063
$65$	−5.01828	−0.622441
$66$	0	0
$67$	−0.256190	−0.0312986	−0.0156493	−	0.999878i	$-0.504982\pi$
$67$	−0.256190	−0.0312986	−0.0156493	+	0.999878i	$0.504982\pi$
$68$	7.93005	0.961660
$69$	0	0
$70$	−3.09877	−0.370374
$71$	−1.58801	−0.188462	−0.0942311	−	0.995550i	$-0.530039\pi$
$71$	−1.58801	−0.188462	−0.0942311	+	0.995550i	$0.530039\pi$
$72$	0	0
$73$	9.67421	1.13228	0.566141	−	0.824309i	$-0.308436\pi$
$73$	9.67421	1.13228	0.566141	+	0.824309i	$0.308436\pi$
$74$	−1.74615	−0.202986
$75$	0	0
$76$	5.71635	0.655710
$77$	5.13227	0.584876
$78$	0	0
$79$	4.34418	0.488758	0.244379	−	0.969680i	$-0.421416\pi$
$79$	4.34418	0.488758	0.244379	+	0.969680i	$0.421416\pi$
$80$	6.22816	0.696330
$81$	0	0
$82$	11.1101	1.22691
$83$	−2.94073	−0.322787	−0.161393	−	0.986890i	$-0.551599\pi$
$83$	−2.94073	−0.322787	−0.161393	+	0.986890i	$0.551599\pi$
$84$	0	0
$85$	−6.17390	−0.669654
$86$	−9.90432	−1.06801
$87$	0	0
$88$	−0.0812334	−0.00865950
$89$	−17.8737	−1.89461	−0.947305	−	0.320334i	$-0.896205\pi$
$89$	−17.8737	−1.89461	−0.947305	+	0.320334i	$0.896205\pi$
$90$	0	0
$91$	−3.23567	−0.339191
$92$	−3.34384	−0.348619
$93$	0	0
$94$	13.6977	1.41281
$95$	−4.45043	−0.456605
$96$	0	0
$97$	−3.77305	−0.383095	−0.191547	−	0.981483i	$-0.561351\pi$
$97$	−3.77305	−0.383095	−0.191547	+	0.981483i	$0.561351\pi$
$98$	−1.99802	−0.201830
$99$	0	0
$100$	−5.16873	−0.516873
$101$	−14.2674	−1.41966	−0.709830	−	0.704373i	$-0.751228\pi$
$101$	−14.2674	−1.41966	−0.709830	+	0.704373i	$0.751228\pi$
$102$	0	0
$103$	−10.7257	−1.05683	−0.528417	−	0.848985i	$-0.677214\pi$
$103$	−10.7257	−1.05683	−0.528417	+	0.848985i	$0.677214\pi$
$104$	0.0512141	0.00502196
$105$	0	0
$106$	−16.0269	−1.55667
$107$	6.64559	0.642453	0.321227	−	0.947002i	$-0.395905\pi$
$107$	6.64559	0.642453	0.321227	+	0.947002i	$0.395905\pi$
$108$	0	0
$109$	6.55843	0.628184	0.314092	−	0.949393i	$-0.398300\pi$
$109$	6.55843	0.628184	0.314092	+	0.949393i	$0.398300\pi$
$110$	−15.9037	−1.51636
$111$	0	0
$112$	4.01578	0.379456
$113$	−7.71152	−0.725439	−0.362719	−	0.931898i	$-0.618152\pi$
$113$	−7.71152	−0.725439	−0.362719	+	0.931898i	$0.618152\pi$
$114$	0	0
$115$	2.60333	0.242762
$116$	−14.7974	−1.37390
$117$	0	0
$118$	−19.8101	−1.82367
$119$	−3.98079	−0.364919
$120$	0	0
$121$	15.3402	1.39456
$122$	12.7648	1.15567
$123$	0	0
$124$	1.62472	0.145904
$125$	11.7787	1.05352
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−0.126623	−0.0111920
$129$	0	0
$130$	10.0266	0.879392
$131$	−8.92539	−0.779815	−0.389908	−	0.920854i	$-0.627493\pi$
$131$	−8.92539	−0.779815	−0.389908	+	0.920854i	$0.627493\pi$
$132$	0	0
$133$	−2.86954	−0.248821
$134$	0.511872	0.0442190
$135$	0	0
$136$	0.0630079	0.00540288
$137$	3.87525	0.331085	0.165542	−	0.986203i	$-0.447063\pi$
$137$	3.87525	0.331085	0.165542	+	0.986203i	$0.447063\pi$
$138$	0	0
$139$	−0.869547	−0.0737540	−0.0368770	−	0.999320i	$-0.511741\pi$
$139$	−0.869547	−0.0737540	−0.0368770	+	0.999320i	$0.511741\pi$
$140$	3.08956	0.261115
$141$	0	0
$142$	3.17287	0.266262
$143$	−16.6063	−1.38869
$144$	0	0
$145$	11.5204	0.956718
$146$	−19.3293	−1.59970
$147$	0	0
$148$	1.74096	0.143106
$149$	−6.13788	−0.502835	−0.251417	−	0.967879i	$-0.580897\pi$
$149$	−6.13788	−0.502835	−0.251417	+	0.967879i	$0.580897\pi$
$150$	0	0
$151$	−15.5483	−1.26530	−0.632651	−	0.774437i	$-0.718033\pi$
$151$	−15.5483	−1.26530	−0.632651	+	0.774437i	$0.718033\pi$
$152$	0.0454189	0.00368396
$153$	0	0
$154$	−10.2544	−0.826321
$155$	−1.26491	−0.101600
$156$	0	0
$157$	2.57703	0.205669	0.102835	−	0.994698i	$-0.467209\pi$
$157$	2.57703	0.205669	0.102835	+	0.994698i	$0.467209\pi$
$158$	−8.67975	−0.690524
$159$	0	0
$160$	−12.3949	−0.979902
$161$	1.67857	0.132290
$162$	0	0
$163$	9.26237	0.725485	0.362743	−	0.931889i	$-0.381840\pi$
$163$	9.26237	0.725485	0.362743	+	0.931889i	$0.381840\pi$
$164$	−11.0771	−0.864977
$165$	0	0
$166$	5.87563	0.456037
$167$	−7.38632	−0.571570	−0.285785	−	0.958294i	$-0.592254\pi$
$167$	−7.38632	−0.571570	−0.285785	+	0.958294i	$0.592254\pi$
$168$	0	0
$169$	−2.53042	−0.194648
$170$	12.3356	0.946095
$171$	0	0
$172$	9.87487	0.752952
$173$	22.4222	1.70473	0.852363	−	0.522951i	$-0.175169\pi$
$173$	22.4222	1.70473	0.852363	+	0.522951i	$0.175169\pi$
$174$	0	0
$175$	2.59464	0.196136
$176$	20.6101	1.55354
$177$	0	0
$178$	35.7120	2.67673
$179$	−1.05233	−0.0786550	−0.0393275	−	0.999226i	$-0.512522\pi$
$179$	−1.05233	−0.0786550	−0.0393275	+	0.999226i	$0.512522\pi$
$180$	0	0
$181$	9.92545	0.737753	0.368876	−	0.929478i	$-0.379743\pi$
$181$	9.92545	0.737753	0.368876	+	0.929478i	$0.379743\pi$
$182$	6.46494	0.479213
$183$	0	0
$184$	−0.0265683	−0.00195864
$185$	−1.35541	−0.0996520
$186$	0	0
$187$	−20.4305	−1.49403
$188$	−13.6570	−0.996039
$189$	0	0
$190$	8.89204	0.645097
$191$	−4.83520	−0.349863	−0.174931	−	0.984581i	$-0.555970\pi$
$191$	−4.83520	−0.349863	−0.174931	+	0.984581i	$0.555970\pi$
$192$	0	0
$193$	−3.41421	−0.245760	−0.122880	−	0.992422i	$-0.539213\pi$
$193$	−3.41421	−0.245760	−0.122880	+	0.992422i	$0.539213\pi$
$194$	7.53862	0.541241
$195$	0	0
$196$	1.99208	0.142291
$197$	17.9837	1.28129	0.640643	−	0.767839i	$-0.278668\pi$
$197$	17.9837	1.28129	0.640643	+	0.767839i	$0.278668\pi$
$198$	0	0
$199$	−4.01040	−0.284290	−0.142145	−	0.989846i	$-0.545400\pi$
$199$	−4.01040	−0.284290	−0.142145	+	0.989846i	$0.545400\pi$
$200$	−0.0410679	−0.00290394
$201$	0	0
$202$	28.5065	2.00571
$203$	7.42810	0.521351
$204$	0	0
$205$	8.62403	0.602328
$206$	21.4301	1.49311
$207$	0	0
$208$	−12.9938	−0.900955
$209$	−14.7272	−1.01870
$210$	0	0
$211$	−4.28023	−0.294663	−0.147332	−	0.989087i	$-0.547068\pi$
$211$	−4.28023	−0.294663	−0.147332	+	0.989087i	$0.547068\pi$
$212$	15.9793	1.09746
$213$	0	0
$214$	−13.2780	−0.907666
$215$	−7.68803	−0.524319
$216$	0	0
$217$	−0.815589	−0.0553658
$218$	−13.1039	−0.887506
$219$	0	0
$220$	15.8564	1.06904
$221$	12.8805	0.866440
$222$	0	0
$223$	15.6251	1.04633	0.523166	−	0.852231i	$-0.324751\pi$
$223$	15.6251	1.04633	0.523166	+	0.852231i	$0.324751\pi$
$224$	−7.99195	−0.533985
$225$	0	0
$226$	15.4078	1.02491
$227$	15.1144	1.00318	0.501588	−	0.865107i	$-0.332749\pi$
$227$	15.1144	1.00318	0.501588	+	0.865107i	$0.332749\pi$
$228$	0	0
$229$	7.96211	0.526151	0.263075	−	0.964775i	$-0.415263\pi$
$229$	7.96211	0.526151	0.263075	+	0.964775i	$0.415263\pi$
$230$	−5.20150	−0.342977
$231$	0	0
$232$	−0.117572	−0.00771896
$233$	−28.0381	−1.83684	−0.918420	−	0.395608i	$-0.870534\pi$
$233$	−28.0381	−1.83684	−0.918420	+	0.395608i	$0.870534\pi$
$234$	0	0
$235$	10.6326	0.693593
$236$	19.7512	1.28570
$237$	0	0
$238$	7.95370	0.515562
$239$	−20.4983	−1.32592	−0.662961	−	0.748654i	$-0.730700\pi$
$239$	−20.4983	−1.32592	−0.662961	+	0.748654i	$0.730700\pi$
$240$	0	0
$241$	−8.11170	−0.522521	−0.261260	−	0.965268i	$-0.584138\pi$
$241$	−8.11170	−0.522521	−0.261260	+	0.965268i	$0.584138\pi$
$242$	−30.6500	−1.97025
$243$	0	0
$244$	−12.7269	−0.814756
$245$	−1.55092	−0.0990848
$246$	0	0
$247$	9.28489	0.590783
$248$	0.0129091	0.000819729 0
$249$	0	0
$250$	−23.5341	−1.48842
$251$	−6.67584	−0.421376	−0.210688	−	0.977553i	$-0.567570\pi$
$251$	−6.67584	−0.421376	−0.210688	+	0.977553i	$0.567570\pi$
$252$	0	0
$253$	8.61487	0.541612
$254$	−1.99802	−0.125367
$255$	0	0
$256$	16.1260	1.00787
$257$	2.60249	0.162339	0.0811695	−	0.996700i	$-0.474134\pi$
$257$	2.60249	0.162339	0.0811695	+	0.996700i	$0.474134\pi$
$258$	0	0
$259$	−0.873940	−0.0543040
$260$	−9.99680	−0.619975
$261$	0	0
$262$	17.8331	1.10173
$263$	25.9889	1.60255	0.801273	−	0.598299i	$-0.204157\pi$
$263$	25.9889	1.60255	0.801273	+	0.598299i	$0.204157\pi$
$264$	0	0
$265$	−12.4406	−0.764218
$266$	5.73339	0.351537
$267$	0	0
$268$	−0.510350	−0.0311746
$269$	−29.2094	−1.78093	−0.890464	−	0.455053i	$-0.849620\pi$
$269$	−29.2094	−1.78093	−0.890464	+	0.455053i	$0.849620\pi$
$270$	0	0
$271$	−4.63001	−0.281253	−0.140627	−	0.990063i	$-0.544912\pi$
$271$	−4.63001	−0.281253	−0.140627	+	0.990063i	$0.544912\pi$
$272$	−15.9860	−0.969293
$273$	0	0
$274$	−7.74282	−0.467761
$275$	13.3164	0.803009
$276$	0	0
$277$	19.2022	1.15375	0.576873	−	0.816834i	$-0.304273\pi$
$277$	19.2022	1.15375	0.576873	+	0.816834i	$0.304273\pi$
$278$	1.73737	0.104201
$279$	0	0
$280$	0.0245479	0.00146702
$281$	6.10537	0.364216	0.182108	−	0.983279i	$-0.441708\pi$
$281$	6.10537	0.364216	0.182108	+	0.983279i	$0.441708\pi$
$282$	0	0
$283$	−7.84704	−0.466458	−0.233229	−	0.972422i	$-0.574929\pi$
$283$	−7.84704	−0.466458	−0.233229	+	0.972422i	$0.574929\pi$
$284$	−3.16344	−0.187716
$285$	0	0
$286$	33.1798	1.96196
$287$	5.56058	0.328231
$288$	0	0
$289$	−1.15328	−0.0678399
$290$	−23.0180	−1.35166
$291$	0	0
$292$	19.2718	1.12780
$293$	−16.3463	−0.954961	−0.477480	−	0.878642i	$-0.658450\pi$
$293$	−16.3463	−0.954961	−0.477480	+	0.878642i	$0.658450\pi$
$294$	0	0
$295$	−15.3772	−0.895297
$296$	0.0138327	0.000804009 0
$297$	0	0
$298$	12.2636	0.710411
$299$	−5.43130	−0.314100
$300$	0	0
$301$	−4.95707	−0.285721
$302$	31.0658	1.78764
$303$	0	0
$304$	−11.5234	−0.660914
$305$	9.90846	0.567357
$306$	0	0
$307$	0.200702	0.0114547	0.00572733	−	0.999984i	$-0.498177\pi$
$307$	0.200702	0.0114547	0.00572733	+	0.999984i	$0.498177\pi$
$308$	10.2239	0.582560
$309$	0	0
$310$	2.52732	0.143542
$311$	6.82716	0.387133	0.193566	−	0.981087i	$-0.437994\pi$
$311$	6.82716	0.387133	0.193566	+	0.981087i	$0.437994\pi$
$312$	0	0
$313$	−19.0662	−1.07769	−0.538843	−	0.842406i	$-0.681138\pi$
$313$	−19.0662	−1.07769	−0.538843	+	0.842406i	$0.681138\pi$
$314$	−5.14895	−0.290572
$315$	0	0
$316$	8.65394	0.486822
$317$	13.1116	0.736419	0.368209	−	0.929743i	$-0.379971\pi$
$317$	13.1116	0.736419	0.368209	+	0.929743i	$0.379971\pi$
$318$	0	0
$319$	38.1230	2.13448
$320$	12.3089	0.688088
$321$	0	0
$322$	−3.35381	−0.186901
$323$	11.4230	0.635595
$324$	0	0
$325$	−8.39541	−0.465694
$326$	−18.5064	−1.02497
$327$	0	0
$328$	−0.0880127	−0.00485969
$329$	6.85565	0.377964
$330$	0	0
$331$	−22.5040	−1.23693	−0.618467	−	0.785811i	$-0.712246\pi$
$331$	−22.5040	−1.23693	−0.618467	+	0.785811i	$0.712246\pi$
$332$	−5.85816	−0.321508
$333$	0	0
$334$	14.7580	0.807522
$335$	0.397331	0.0217085
$336$	0	0
$337$	2.07864	0.113231	0.0566154	−	0.998396i	$-0.481969\pi$
$337$	2.07864	0.113231	0.0566154	+	0.998396i	$0.481969\pi$
$338$	5.05582	0.275000
$339$	0	0
$340$	−12.2989	−0.667001
$341$	−4.18582	−0.226675
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.0784603	0.00423030
$345$	0	0
$346$	−44.7999	−2.40846
$347$	28.1043	1.50872	0.754360	−	0.656461i	$-0.227947\pi$
$347$	28.1043	1.50872	0.754360	+	0.656461i	$0.227947\pi$
$348$	0	0
$349$	5.51546	0.295236	0.147618	−	0.989044i	$-0.452839\pi$
$349$	5.51546	0.295236	0.147618	+	0.989044i	$0.452839\pi$
$350$	−5.18414	−0.277104
$351$	0	0
$352$	−41.0168	−2.18620
$353$	6.64296	0.353569	0.176784	−	0.984250i	$-0.443430\pi$
$353$	6.64296	0.353569	0.176784	+	0.984250i	$0.443430\pi$
$354$	0	0
$355$	2.46288	0.130716
$356$	−35.6058	−1.88710
$357$	0	0
$358$	2.10258	0.111125
$359$	31.7116	1.67368	0.836838	−	0.547451i	$-0.184402\pi$
$359$	31.7116	1.67368	0.836838	+	0.547451i	$0.184402\pi$
$360$	0	0
$361$	−10.7657	−0.566618
$362$	−19.8312	−1.04231
$363$	0	0
$364$	−6.44571	−0.337847
$365$	−15.0040	−0.785343
$366$	0	0
$367$	37.4792	1.95640	0.978198	−	0.207673i	$-0.0665889\pi$
$367$	37.4792	1.95640	0.978198	+	0.207673i	$0.0665889\pi$
$368$	6.74076	0.351387
$369$	0	0
$370$	2.70814	0.140790
$371$	−8.02140	−0.416451
$372$	0	0
$373$	0.523333	0.0270972	0.0135486	−	0.999908i	$-0.495687\pi$
$373$	0.523333	0.0270972	0.0135486	+	0.999908i	$0.495687\pi$
$374$	40.8205	2.11078
$375$	0	0
$376$	−0.108511	−0.00559603
$377$	−24.0349	−1.23786
$378$	0	0
$379$	27.0604	1.39000	0.695000	−	0.719009i	$-0.255404\pi$
$379$	27.0604	1.39000	0.695000	+	0.719009i	$0.255404\pi$
$380$	−8.86561	−0.454796
$381$	0	0
$382$	9.66082	0.494291
$383$	−3.70250	−0.189189	−0.0945945	−	0.995516i	$-0.530155\pi$
$383$	−3.70250	−0.189189	−0.0945945	+	0.995516i	$0.530155\pi$
$384$	0	0
$385$	−7.95975	−0.405666
$386$	6.82165	0.347213
$387$	0	0
$388$	−7.51621	−0.381578
$389$	11.6605	0.591209	0.295604	−	0.955310i	$-0.404479\pi$
$389$	11.6605	0.591209	0.295604	+	0.955310i	$0.404479\pi$
$390$	0	0
$391$	−6.68204	−0.337925
$392$	0.0158280	0.000799433 0
$393$	0	0
$394$	−35.9318	−1.81022
$395$	−6.73748	−0.338999
$396$	0	0
$397$	−22.9316	−1.15090	−0.575451	−	0.817836i	$-0.695174\pi$
$397$	−22.9316	−1.15090	−0.575451	+	0.817836i	$0.695174\pi$
$398$	8.01286	0.401648
$399$	0	0
$400$	10.4195	0.520975
$401$	−18.0860	−0.903173	−0.451586	−	0.892227i	$-0.649142\pi$
$401$	−18.0860	−0.903173	−0.451586	+	0.892227i	$0.649142\pi$
$402$	0	0
$403$	2.63898	0.131457
$404$	−28.4218	−1.41404
$405$	0	0
$406$	−14.8415	−0.736571
$407$	−4.48530	−0.222328
$408$	0	0
$409$	−3.52170	−0.174137	−0.0870684	−	0.996202i	$-0.527750\pi$
$409$	−3.52170	−0.174137	−0.0870684	+	0.996202i	$0.527750\pi$
$410$	−17.2310	−0.850977
$411$	0	0
$412$	−21.3664	−1.05265
$413$	−9.91489	−0.487880
$414$	0	0
$415$	4.56084	0.223883
$416$	25.8593	1.26786
$417$	0	0
$418$	29.4253	1.43924
$419$	−0.101725	−0.00496960	−0.00248480	−	0.999997i	$-0.500791\pi$
$419$	−0.101725	−0.00496960	−0.00248480	+	0.999997i	$0.500791\pi$
$420$	0	0
$421$	−12.6284	−0.615472	−0.307736	−	0.951472i	$-0.599571\pi$
$421$	−12.6284	−0.615472	−0.307736	+	0.951472i	$0.599571\pi$
$422$	8.55198	0.416304
$423$	0	0
$424$	0.126962	0.00616584
$425$	−10.3287	−0.501017
$426$	0	0
$427$	6.38875	0.309173
$428$	13.2385	0.639909
$429$	0	0
$430$	15.3608	0.740765
$431$	15.8922	0.765500	0.382750	−	0.923852i	$-0.374977\pi$
$431$	15.8922	0.765500	0.382750	+	0.923852i	$0.374977\pi$
$432$	0	0
$433$	−8.32317	−0.399986	−0.199993	−	0.979797i	$-0.564092\pi$
$433$	−8.32317	−0.399986	−0.199993	+	0.979797i	$0.564092\pi$
$434$	1.62956	0.0782215
$435$	0	0
$436$	13.0649	0.625696
$437$	−4.81672	−0.230415
$438$	0	0
$439$	16.3660	0.781105	0.390552	−	0.920581i	$-0.372284\pi$
$439$	16.3660	0.781105	0.390552	+	0.920581i	$0.372284\pi$
$440$	0.125987	0.00600618
$441$	0	0
$442$	−25.7356	−1.22412
$443$	25.5970	1.21615	0.608074	−	0.793880i	$-0.291942\pi$
$443$	25.5970	1.21615	0.608074	+	0.793880i	$0.291942\pi$
$444$	0	0
$445$	27.7207	1.31409
$446$	−31.2192	−1.47827
$447$	0	0
$448$	7.93650	0.374964
$449$	18.6338	0.879386	0.439693	−	0.898148i	$-0.355087\pi$
$449$	18.6338	0.879386	0.439693	+	0.898148i	$0.355087\pi$
$450$	0	0
$451$	28.5384	1.34382
$452$	−15.3620	−0.722566
$453$	0	0
$454$	−30.1988	−1.41730
$455$	5.01828	0.235260
$456$	0	0
$457$	14.2257	0.665451	0.332726	−	0.943024i	$-0.392032\pi$
$457$	14.2257	0.665451	0.332726	+	0.943024i	$0.392032\pi$
$458$	−15.9084	−0.743353
$459$	0	0
$460$	5.18604	0.241800
$461$	1.81891	0.0847151	0.0423575	−	0.999103i	$-0.486513\pi$
$461$	1.81891	0.0847151	0.0423575	+	0.999103i	$0.486513\pi$
$462$	0	0
$463$	29.9051	1.38981	0.694904	−	0.719103i	$-0.255447\pi$
$463$	29.9051	1.38981	0.694904	+	0.719103i	$0.255447\pi$
$464$	29.8296	1.38481
$465$	0	0
$466$	56.0207	2.59511
$467$	−9.18692	−0.425120	−0.212560	−	0.977148i	$-0.568180\pi$
$467$	−9.18692	−0.425120	−0.212560	+	0.977148i	$0.568180\pi$
$468$	0	0
$469$	0.256190	0.0118298
$470$	−21.2441	−0.979917
$471$	0	0
$472$	0.156933	0.00722341
$473$	−25.4410	−1.16978
$474$	0	0
$475$	−7.44542	−0.341619
$476$	−7.93005	−0.363473
$477$	0	0
$478$	40.9559	1.87328
$479$	16.0675	0.734142	0.367071	−	0.930193i	$-0.380361\pi$
$479$	16.0675	0.734142	0.367071	+	0.930193i	$0.380361\pi$
$480$	0	0
$481$	2.82779	0.128936
$482$	16.2073	0.738224
$483$	0	0
$484$	30.5588	1.38904
$485$	5.85170	0.265712
$486$	0	0
$487$	27.6113	1.25119	0.625594	−	0.780149i	$-0.284857\pi$
$487$	27.6113	1.25119	0.625594	+	0.780149i	$0.284857\pi$
$488$	−0.101121	−0.00457753
$489$	0	0
$490$	3.09877	0.139988
$491$	−17.9258	−0.808982	−0.404491	−	0.914542i	$-0.632551\pi$
$491$	−17.9258	−0.808982	−0.404491	+	0.914542i	$0.632551\pi$
$492$	0	0
$493$	−29.5697	−1.33175
$494$	−18.5514	−0.834666
$495$	0	0
$496$	−3.27523	−0.147062
$497$	1.58801	0.0712320
$498$	0	0
$499$	−3.73826	−0.167347	−0.0836737	−	0.996493i	$-0.526665\pi$
$499$	−3.73826	−0.167347	−0.0836737	+	0.996493i	$0.526665\pi$
$500$	23.4641	1.04935
$501$	0	0
$502$	13.3385	0.595325
$503$	10.8739	0.484843	0.242422	−	0.970171i	$-0.422058\pi$
$503$	10.8739	0.484843	0.242422	+	0.970171i	$0.422058\pi$
$504$	0	0
$505$	22.1276	0.984667
$506$	−17.2127	−0.765196
$507$	0	0
$508$	1.99208	0.0883842
$509$	2.99993	0.132970	0.0664849	−	0.997787i	$-0.478822\pi$
$509$	2.99993	0.132970	0.0664849	+	0.997787i	$0.478822\pi$
$510$	0	0
$511$	−9.67421	−0.427962
$512$	−31.9668	−1.41275
$513$	0	0
$514$	−5.19983	−0.229355
$515$	16.6347	0.733013
$516$	0	0
$517$	35.1850	1.54744
$518$	1.74615	0.0767214
$519$	0	0
$520$	−0.0794291	−0.00348320
$521$	11.3414	0.496877	0.248439	−	0.968648i	$-0.420083\pi$
$521$	11.3414	0.496877	0.248439	+	0.968648i	$0.420083\pi$
$522$	0	0
$523$	−21.8401	−0.954999	−0.477500	−	0.878632i	$-0.658457\pi$
$523$	−21.8401	−0.954999	−0.477500	+	0.878632i	$0.658457\pi$
$524$	−17.7801	−0.776727
$525$	0	0
$526$	−51.9263	−2.26410
$527$	3.24669	0.141428
$528$	0	0
$529$	−20.1824	−0.877496
$530$	24.8565	1.07970
$531$	0	0
$532$	−5.71635	−0.247835
$533$	−17.9922	−0.779330
$534$	0	0
$535$	−10.3068	−0.445601
$536$	−0.00405497	−0.000175148 0
$537$	0	0
$538$	58.3609	2.51612
$539$	−5.13227	−0.221062
$540$	0	0
$541$	−25.1107	−1.07959	−0.539796	−	0.841796i	$-0.681499\pi$
$541$	−25.1107	−1.07959	−0.539796	+	0.841796i	$0.681499\pi$
$542$	9.25085	0.397358
$543$	0	0
$544$	31.8143	1.36403
$545$	−10.1716	−0.435704
$546$	0	0
$547$	−33.0426	−1.41280	−0.706399	−	0.707813i	$-0.749682\pi$
$547$	−33.0426	−1.41280	−0.706399	+	0.707813i	$0.749682\pi$
$548$	7.71980	0.329774
$549$	0	0
$550$	−26.6064	−1.13450
$551$	−21.3152	−0.908059
$552$	0	0
$553$	−4.34418	−0.184733
$554$	−38.3663	−1.63003
$555$	0	0
$556$	−1.73221	−0.0734619
$557$	−24.9659	−1.05784	−0.528919	−	0.848672i	$-0.677402\pi$
$557$	−24.9659	−1.05784	−0.528919	+	0.848672i	$0.677402\pi$
$558$	0	0
$559$	16.0395	0.678397
$560$	−6.22816	−0.263188
$561$	0	0
$562$	−12.1986	−0.514568
$563$	−17.1234	−0.721667	−0.360834	−	0.932630i	$-0.617508\pi$
$563$	−17.1234	−0.721667	−0.360834	+	0.932630i	$0.617508\pi$
$564$	0	0
$565$	11.9600	0.503160
$566$	15.6785	0.659018
$567$	0	0
$568$	−0.0251350	−0.00105464
$569$	32.4727	1.36132	0.680662	−	0.732597i	$-0.261692\pi$
$569$	32.4727	1.36132	0.680662	+	0.732597i	$0.261692\pi$
$570$	0	0
$571$	−22.4947	−0.941375	−0.470688	−	0.882300i	$-0.655994\pi$
$571$	−22.4947	−0.941375	−0.470688	+	0.882300i	$0.655994\pi$
$572$	−33.0811	−1.38319
$573$	0	0
$574$	−11.1101	−0.463728
$575$	4.35528	0.181628
$576$	0	0
$577$	2.42913	0.101126	0.0505630	−	0.998721i	$-0.483898\pi$
$577$	2.42913	0.101126	0.0505630	+	0.998721i	$0.483898\pi$
$578$	2.30427	0.0958451
$579$	0	0
$580$	22.9496	0.952928
$581$	2.94073	0.122002
$582$	0	0
$583$	−41.1680	−1.70500
$584$	0.153123	0.00633628
$585$	0	0
$586$	32.6602	1.34918
$587$	−11.9885	−0.494818	−0.247409	−	0.968911i	$-0.579579\pi$
$587$	−11.9885	−0.494818	−0.247409	+	0.968911i	$0.579579\pi$
$588$	0	0
$589$	2.34036	0.0964330
$590$	30.7240	1.26489
$591$	0	0
$592$	−3.50955	−0.144242
$593$	10.8653	0.446185	0.223092	−	0.974797i	$-0.428385\pi$
$593$	10.8653	0.446185	0.223092	+	0.974797i	$0.428385\pi$
$594$	0	0
$595$	6.17390	0.253105
$596$	−12.2271	−0.500843
$597$	0	0
$598$	10.8518	0.443765
$599$	40.4780	1.65389	0.826943	−	0.562285i	$-0.190078\pi$
$599$	40.4780	1.65389	0.826943	+	0.562285i	$0.190078\pi$
$600$	0	0
$601$	24.6000	1.00345	0.501727	−	0.865026i	$-0.332698\pi$
$601$	24.6000	1.00345	0.501727	+	0.865026i	$0.332698\pi$
$602$	9.90432	0.403670
$603$	0	0
$604$	−30.9734	−1.26029
$605$	−23.7914	−0.967259
$606$	0	0
$607$	−22.7065	−0.921629	−0.460815	−	0.887496i	$-0.652443\pi$
$607$	−22.7065	−0.921629	−0.460815	+	0.887496i	$0.652443\pi$
$608$	22.9332	0.930064
$609$	0	0
$610$	−19.7973	−0.801569
$611$	−22.1826	−0.897414
$612$	0	0
$613$	−33.3737	−1.34795	−0.673975	−	0.738754i	$-0.735414\pi$
$613$	−33.3737	−1.34795	−0.673975	+	0.738754i	$0.735414\pi$
$614$	−0.401006	−0.0161833
$615$	0	0
$616$	0.0812334	0.00327299
$617$	36.6644	1.47605	0.738026	−	0.674772i	$-0.235758\pi$
$617$	36.6644	1.47605	0.738026	+	0.674772i	$0.235758\pi$
$618$	0	0
$619$	49.0015	1.96954	0.984768	−	0.173875i	$-0.0556288\pi$
$619$	49.0015	1.96954	0.984768	+	0.173875i	$0.0556288\pi$
$620$	−2.51981	−0.101198
$621$	0	0
$622$	−13.6408	−0.546946
$623$	17.8737	0.716095
$624$	0	0
$625$	−5.29464	−0.211785
$626$	38.0947	1.52257
$627$	0	0
$628$	5.13364	0.204854
$629$	3.47898	0.138716
$630$	0	0
$631$	32.2194	1.28263	0.641316	−	0.767277i	$-0.278389\pi$
$631$	32.2194	1.28263	0.641316	+	0.767277i	$0.278389\pi$
$632$	0.0687595	0.00273511
$633$	0	0
$634$	−26.1972	−1.04042
$635$	−1.55092	−0.0615465
$636$	0	0
$637$	3.23567	0.128202
$638$	−76.1705	−3.01562
$639$	0	0
$640$	0.196382	0.00776268
$641$	44.1868	1.74527	0.872637	−	0.488369i	$-0.162408\pi$
$641$	44.1868	1.74527	0.872637	+	0.488369i	$0.162408\pi$
$642$	0	0
$643$	44.9872	1.77412	0.887062	−	0.461651i	$-0.152743\pi$
$643$	44.9872	1.77412	0.887062	+	0.461651i	$0.152743\pi$
$644$	3.34384	0.131766
$645$	0	0
$646$	−22.8234	−0.897977
$647$	12.3832	0.486836	0.243418	−	0.969922i	$-0.421731\pi$
$647$	12.3832	0.486836	0.243418	+	0.969922i	$0.421731\pi$
$648$	0	0
$649$	−50.8859	−1.99745
$650$	16.7742	0.657938
$651$	0	0
$652$	18.4514	0.722612
$653$	13.5523	0.530342	0.265171	−	0.964201i	$-0.414572\pi$
$653$	13.5523	0.530342	0.265171	+	0.964201i	$0.414572\pi$
$654$	0	0
$655$	13.8426	0.540875
$656$	22.3301	0.871843
$657$	0	0
$658$	−13.6977	−0.533993
$659$	−12.9849	−0.505819	−0.252910	−	0.967490i	$-0.581387\pi$
$659$	−12.9849	−0.505819	−0.252910	+	0.967490i	$0.581387\pi$
$660$	0	0
$661$	2.31636	0.0900961	0.0450481	−	0.998985i	$-0.485656\pi$
$661$	2.31636	0.0900961	0.0450481	+	0.998985i	$0.485656\pi$
$662$	44.9635	1.74756
$663$	0	0
$664$	−0.0465458	−0.00180633
$665$	4.45043	0.172580
$666$	0	0
$667$	12.4686	0.482785
$668$	−14.7141	−0.569306
$669$	0	0
$670$	−0.793874	−0.0306700
$671$	32.7888	1.26580
$672$	0	0
$673$	5.50998	0.212394	0.106197	−	0.994345i	$-0.466133\pi$
$673$	5.50998	0.212394	0.106197	+	0.994345i	$0.466133\pi$
$674$	−4.15316	−0.159974
$675$	0	0
$676$	−5.04079	−0.193877
$677$	37.0593	1.42430	0.712152	−	0.702026i	$-0.247721\pi$
$677$	37.0593	1.42430	0.712152	+	0.702026i	$0.247721\pi$
$678$	0	0
$679$	3.77305	0.144796
$680$	−0.0977203	−0.00374740
$681$	0	0
$682$	8.36335	0.320249
$683$	31.3466	1.19945	0.599723	−	0.800208i	$-0.295278\pi$
$683$	31.3466	1.19945	0.599723	+	0.800208i	$0.295278\pi$
$684$	0	0
$685$	−6.01021	−0.229638
$686$	1.99802	0.0762847
$687$	0	0
$688$	−19.9065	−0.758929
$689$	25.9546	0.988793
$690$	0	0
$691$	27.3304	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$691$	27.3304	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$692$	44.6667	1.69797
$693$	0	0
$694$	−56.1530	−2.13154
$695$	1.34860	0.0511553
$696$	0	0
$697$	−22.1355	−0.838443
$698$	−11.0200	−0.417112
$699$	0	0
$700$	5.16873	0.195360
$701$	27.0821	1.02288	0.511438	−	0.859320i	$-0.329113\pi$
$701$	27.0821	1.02288	0.511438	+	0.859320i	$0.329113\pi$
$702$	0	0
$703$	2.50781	0.0945837
$704$	40.7323	1.53515
$705$	0	0
$706$	−13.2727	−0.499526
$707$	14.2674	0.536581
$708$	0	0
$709$	−6.36902	−0.239193	−0.119597	−	0.992823i	$-0.538160\pi$
$709$	−6.36902	−0.239193	−0.119597	+	0.992823i	$0.538160\pi$
$710$	−4.92088	−0.184677
$711$	0	0
$712$	−0.282904	−0.0106023
$713$	−1.36902	−0.0512703
$714$	0	0
$715$	25.7551	0.963188
$716$	−2.09633	−0.0783434
$717$	0	0
$718$	−63.3604	−2.36459
$719$	11.6862	0.435822	0.217911	−	0.975969i	$-0.430076\pi$
$719$	11.6862	0.435822	0.217911	+	0.975969i	$0.430076\pi$
$720$	0	0
$721$	10.7257	0.399446
$722$	21.5102	0.800525
$723$	0	0
$724$	19.7723	0.734831
$725$	19.2733	0.715791
$726$	0	0
$727$	15.8229	0.586840	0.293420	−	0.955984i	$-0.405207\pi$
$727$	15.8229	0.586840	0.293420	+	0.955984i	$0.405207\pi$
$728$	−0.0512141	−0.00189812
$729$	0	0
$730$	29.9782	1.10954
$731$	19.7331	0.729854
$732$	0	0
$733$	−45.0289	−1.66318	−0.831589	−	0.555391i	$-0.812569\pi$
$733$	−45.0289	−1.66318	−0.831589	+	0.555391i	$0.812569\pi$
$734$	−74.8841	−2.76402
$735$	0	0
$736$	−13.4150	−0.494485
$737$	1.31484	0.0484326
$738$	0	0
$739$	14.6395	0.538522	0.269261	−	0.963067i	$-0.413221\pi$
$739$	14.6395	0.538522	0.269261	+	0.963067i	$0.413221\pi$
$740$	−2.70009	−0.0992572
$741$	0	0
$742$	16.0269	0.588367
$743$	13.0736	0.479622	0.239811	−	0.970820i	$-0.422914\pi$
$743$	13.0736	0.479622	0.239811	+	0.970820i	$0.422914\pi$
$744$	0	0
$745$	9.51938	0.348763
$746$	−1.04563	−0.0382832
$747$	0	0
$748$	−40.6992	−1.48811
$749$	−6.64559	−0.242825
$750$	0	0
$751$	−10.5692	−0.385676	−0.192838	−	0.981231i	$-0.561769\pi$
$751$	−10.5692	−0.385676	−0.192838	+	0.981231i	$0.561769\pi$
$752$	27.5308	1.00394
$753$	0	0
$754$	48.0222	1.74887
$755$	24.1142	0.877606
$756$	0	0
$757$	−41.3384	−1.50247	−0.751234	−	0.660036i	$-0.770541\pi$
$757$	−41.3384	−1.50247	−0.751234	+	0.660036i	$0.770541\pi$
$758$	−54.0672	−1.96381
$759$	0	0
$760$	−0.0704412	−0.00255517
$761$	−20.7335	−0.751588	−0.375794	−	0.926703i	$-0.622630\pi$
$761$	−20.7335	−0.751588	−0.375794	+	0.926703i	$0.622630\pi$
$762$	0	0
$763$	−6.55843	−0.237431
$764$	−9.63210	−0.348477
$765$	0	0
$766$	7.39766	0.267288
$767$	32.0814	1.15839
$768$	0	0
$769$	20.0797	0.724092	0.362046	−	0.932160i	$-0.382078\pi$
$769$	20.0797	0.724092	0.362046	+	0.932160i	$0.382078\pi$
$770$	15.9037	0.573130
$771$	0	0
$772$	−6.80136	−0.244786
$773$	21.8177	0.784727	0.392363	−	0.919810i	$-0.371658\pi$
$773$	21.8177	0.784727	0.392363	+	0.919810i	$0.371658\pi$
$774$	0	0
$775$	−2.11616	−0.0760147
$776$	−0.0597197	−0.00214381
$777$	0	0
$778$	−23.2978	−0.835267
$779$	−15.9563	−0.571694
$780$	0	0
$781$	8.15009	0.291633
$782$	13.3508	0.477425
$783$	0	0
$784$	−4.01578	−0.143421
$785$	−3.99677	−0.142651
$786$	0	0
$787$	−37.1914	−1.32573	−0.662865	−	0.748739i	$-0.730660\pi$
$787$	−37.1914	−1.32573	−0.662865	+	0.748739i	$0.730660\pi$
$788$	35.8249	1.27621
$789$	0	0
$790$	13.4616	0.478943
$791$	7.71152	0.274190
$792$	0	0
$793$	−20.6719	−0.734081
$794$	45.8177	1.62601
$795$	0	0
$796$	−7.98903	−0.283164
$797$	45.6760	1.61793	0.808963	−	0.587860i	$-0.200029\pi$
$797$	45.6760	1.61793	0.808963	+	0.587860i	$0.200029\pi$
$798$	0	0
$799$	−27.2909	−0.965484
$800$	−20.7362	−0.733137
$801$	0	0
$802$	36.1362	1.27601
$803$	−49.6507	−1.75213
$804$	0	0
$805$	−2.60333	−0.0917553
$806$	−5.27273	−0.185724
$807$	0	0
$808$	−0.225824	−0.00794446
$809$	15.7808	0.554822	0.277411	−	0.960751i	$-0.410524\pi$
$809$	15.7808	0.554822	0.277411	+	0.960751i	$0.410524\pi$
$810$	0	0
$811$	37.0351	1.30048	0.650239	−	0.759730i	$-0.274669\pi$
$811$	37.0351	1.30048	0.650239	+	0.759730i	$0.274669\pi$
$812$	14.7974	0.519286
$813$	0	0
$814$	8.96170	0.314108
$815$	−14.3652	−0.503192
$816$	0	0
$817$	14.2245	0.497652
$818$	7.03642	0.246023
$819$	0	0
$820$	17.1797	0.599942
$821$	42.9138	1.49770	0.748851	−	0.662738i	$-0.230606\pi$
$821$	42.9138	1.49770	0.748851	+	0.662738i	$0.230606\pi$
$822$	0	0
$823$	3.42351	0.119336	0.0596681	−	0.998218i	$-0.480996\pi$
$823$	3.42351	0.119336	0.0596681	+	0.998218i	$0.480996\pi$
$824$	−0.169766	−0.00591407
$825$	0	0
$826$	19.8101	0.689283
$827$	43.0420	1.49672	0.748359	−	0.663294i	$-0.230842\pi$
$827$	43.0420	1.49672	0.748359	+	0.663294i	$0.230842\pi$
$828$	0	0
$829$	40.7436	1.41508	0.707542	−	0.706671i	$-0.249804\pi$
$829$	40.7436	1.41508	0.707542	+	0.706671i	$0.249804\pi$
$830$	−9.11265	−0.316305
$831$	0	0
$832$	−25.6799	−0.890291
$833$	3.98079	0.137926
$834$	0	0
$835$	11.4556	0.396437
$836$	−29.3378	−1.01467
$837$	0	0
$838$	0.203249	0.00702111
$839$	40.7380	1.40643	0.703215	−	0.710977i	$-0.251747\pi$
$839$	40.7380	1.40643	0.703215	+	0.710977i	$0.251747\pi$
$840$	0	0
$841$	26.1767	0.902645
$842$	25.2319	0.869547
$843$	0	0
$844$	−8.52656	−0.293496
$845$	3.92448	0.135006
$846$	0	0
$847$	−15.3402	−0.527095
$848$	−32.2122	−1.10617
$849$	0	0
$850$	20.6370	0.707843
$851$	−1.46697	−0.0502870
$852$	0	0
$853$	−35.0623	−1.20051	−0.600255	−	0.799809i	$-0.704934\pi$
$853$	−35.0623	−1.20051	−0.600255	+	0.799809i	$0.704934\pi$
$854$	−12.7648	−0.436804
$855$	0	0
$856$	0.105186	0.00359519
$857$	10.4998	0.358666	0.179333	−	0.983788i	$-0.442606\pi$
$857$	10.4998	0.358666	0.179333	+	0.983788i	$0.442606\pi$
$858$	0	0
$859$	−24.9157	−0.850112	−0.425056	−	0.905167i	$-0.639746\pi$
$859$	−24.9157	−0.850112	−0.425056	+	0.905167i	$0.639746\pi$
$860$	−15.3152	−0.522243
$861$	0	0
$862$	−31.7529	−1.08151
$863$	−1.65282	−0.0562628	−0.0281314	−	0.999604i	$-0.508956\pi$
$863$	−1.65282	−0.0562628	−0.0281314	+	0.999604i	$0.508956\pi$
$864$	0	0
$865$	−34.7750	−1.18239
$866$	16.6299	0.565105
$867$	0	0
$868$	−1.62472	−0.0551465
$869$	−22.2955	−0.756323
$870$	0	0
$871$	−0.828947	−0.0280878
$872$	0.103807	0.00351534
$873$	0	0
$874$	9.62389	0.325533
$875$	−11.7787	−0.398193
$876$	0	0
$877$	26.1172	0.881915	0.440957	−	0.897528i	$-0.354639\pi$
$877$	26.1172	0.881915	0.440957	+	0.897528i	$0.354639\pi$
$878$	−32.6995	−1.10355
$879$	0	0
$880$	−31.9646	−1.07753
$881$	44.7117	1.50637	0.753187	−	0.657806i	$-0.228515\pi$
$881$	44.7117	1.50637	0.753187	+	0.657806i	$0.228515\pi$
$882$	0	0
$883$	−43.3655	−1.45936	−0.729682	−	0.683787i	$-0.760332\pi$
$883$	−43.3655	−1.45936	−0.729682	+	0.683787i	$0.760332\pi$
$884$	25.6591	0.863008
$885$	0	0
$886$	−51.1432	−1.71819
$887$	−26.6987	−0.896455	−0.448227	−	0.893920i	$-0.647945\pi$
$887$	−26.6987	−0.896455	−0.448227	+	0.893920i	$0.647945\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−55.3865	−1.85656
$891$	0	0
$892$	31.1263	1.04219
$893$	−19.6726	−0.658317
$894$	0	0
$895$	1.63208	0.0545546
$896$	0.126623	0.00423017
$897$	0	0
$898$	−37.2308	−1.24241
$899$	−6.05828	−0.202055
$900$	0	0
$901$	31.9316	1.06379
$902$	−57.0202	−1.89857
$903$	0	0
$904$	−0.122058	−0.00405958
$905$	−15.3936	−0.511701
$906$	0	0
$907$	20.7281	0.688266	0.344133	−	0.938921i	$-0.388173\pi$
$907$	20.7281	0.688266	0.344133	+	0.938921i	$0.388173\pi$
$908$	30.1090	0.999202
$909$	0	0
$910$	−10.0266	−0.332379
$911$	−40.0495	−1.32690	−0.663450	−	0.748220i	$-0.730909\pi$
$911$	−40.0495	−1.32690	−0.663450	+	0.748220i	$0.730909\pi$
$912$	0	0
$913$	15.0926	0.499493
$914$	−28.4233	−0.940158
$915$	0	0
$916$	15.8611	0.524067
$917$	8.92539	0.294743
$918$	0	0
$919$	−16.4606	−0.542984	−0.271492	−	0.962441i	$-0.587517\pi$
$919$	−16.4606	−0.542984	−0.271492	+	0.962441i	$0.587517\pi$
$920$	0.0412054	0.00135850
$921$	0	0
$922$	−3.63422	−0.119687
$923$	−5.13828	−0.169129
$924$	0	0
$925$	−2.26756	−0.0745569
$926$	−59.7510	−1.96354
$927$	0	0
$928$	−59.3650	−1.94875
$929$	12.0241	0.394498	0.197249	−	0.980353i	$-0.436799\pi$
$929$	12.0241	0.394498	0.197249	+	0.980353i	$0.436799\pi$
$930$	0	0
$931$	2.86954	0.0940453
$932$	−55.8542	−1.82956
$933$	0	0
$934$	18.3556	0.600615
$935$	31.6861	1.03625
$936$	0	0
$937$	38.4952	1.25758	0.628791	−	0.777574i	$-0.283550\pi$
$937$	38.4952	1.25758	0.628791	+	0.777574i	$0.283550\pi$
$938$	−0.511872	−0.0167132
$939$	0	0
$940$	21.1809	0.690846
$941$	−3.32398	−0.108359	−0.0541794	−	0.998531i	$-0.517254\pi$
$941$	−3.32398	−0.108359	−0.0541794	+	0.998531i	$0.517254\pi$
$942$	0	0
$943$	9.33382	0.303951
$944$	−39.8160	−1.29590
$945$	0	0
$946$	50.8316	1.65268
$947$	−15.6069	−0.507157	−0.253578	−	0.967315i	$-0.581608\pi$
$947$	−15.6069	−0.507157	−0.253578	+	0.967315i	$0.581608\pi$
$948$	0	0
$949$	31.3026	1.01613
$950$	14.8761	0.482644
$951$	0	0
$952$	−0.0630079	−0.00204210
$953$	−13.0607	−0.423078	−0.211539	−	0.977370i	$-0.567847\pi$
$953$	−13.0607	−0.423078	−0.211539	+	0.977370i	$0.567847\pi$
$954$	0	0
$955$	7.49902	0.242663
$956$	−40.8341	−1.32067
$957$	0	0
$958$	−32.1031	−1.03720
$959$	−3.87525	−0.125138
$960$	0	0
$961$	−30.3348	−0.978542
$962$	−5.64997	−0.182162
$963$	0	0
$964$	−16.1591	−0.520451
$965$	5.29517	0.170457
$966$	0	0
$967$	−16.0766	−0.516988	−0.258494	−	0.966013i	$-0.583226\pi$
$967$	−16.0766	−0.516988	−0.258494	+	0.966013i	$0.583226\pi$
$968$	0.242804	0.00780401
$969$	0	0
$970$	−11.6918	−0.375401
$971$	−16.3263	−0.523937	−0.261968	−	0.965076i	$-0.584372\pi$
$971$	−16.3263	−0.523937	−0.261968	+	0.965076i	$0.584372\pi$
$972$	0	0
$973$	0.869547	0.0278764
$974$	−55.1679	−1.76769
$975$	0	0
$976$	25.6558	0.821223
$977$	10.7672	0.344474	0.172237	−	0.985056i	$-0.444901\pi$
$977$	10.7672	0.344474	0.172237	+	0.985056i	$0.444901\pi$
$978$	0	0
$979$	91.7327	2.93179
$980$	−3.08956	−0.0986923
$981$	0	0
$982$	35.8161	1.14294
$983$	15.5930	0.497338	0.248669	−	0.968588i	$-0.420007\pi$
$983$	15.5930	0.497338	0.248669	+	0.968588i	$0.420007\pi$
$984$	0	0
$985$	−27.8913	−0.888692
$986$	59.0809	1.88152
$987$	0	0
$988$	18.4962	0.588443
$989$	−8.32078	−0.264586
$990$	0	0
$991$	6.87536	0.218403	0.109202	−	0.994020i	$-0.465171\pi$
$991$	6.87536	0.218403	0.109202	+	0.994020i	$0.465171\pi$
$992$	6.51814	0.206951
$993$	0	0
$994$	−3.17287	−0.100637
$995$	6.21982	0.197182
$996$	0	0
$997$	44.2889	1.40264	0.701322	−	0.712845i	$-0.252594\pi$
$997$	44.2889	1.40264	0.701322	+	0.712845i	$0.252594\pi$
$998$	7.46911	0.236431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.6	✓	28
3.2	odd	2	inner	8001.2.a.y.1.23	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.6	✓	28	1.1	even	1	trivial
8001.2.a.y.1.23	yes	28	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99802 q^{2} +1.99208 q^{4} -1.55092 q^{5} -1.00000 q^{7} +0.0158280 q^{8} +O(q^{10})\)	\(q-1.99802 q^{2} +1.99208 q^{4} -1.55092 q^{5} -1.00000 q^{7} +0.0158280 q^{8} +3.09877 q^{10} -5.13227 q^{11} +3.23567 q^{13} +1.99802 q^{14} -4.01578 q^{16} +3.98079 q^{17} +2.86954 q^{19} -3.08956 q^{20} +10.2544 q^{22} -1.67857 q^{23} -2.59464 q^{25} -6.46494 q^{26} -1.99208 q^{28} -7.42810 q^{29} +0.815589 q^{31} +7.99195 q^{32} -7.95370 q^{34} +1.55092 q^{35} +0.873940 q^{37} -5.73339 q^{38} -0.0245479 q^{40} -5.56058 q^{41} +4.95707 q^{43} -10.2239 q^{44} +3.35381 q^{46} -6.85565 q^{47} +1.00000 q^{49} +5.18414 q^{50} +6.44571 q^{52} +8.02140 q^{53} +7.95975 q^{55} -0.0158280 q^{56} +14.8415 q^{58} +9.91489 q^{59} -6.38875 q^{61} -1.62956 q^{62} -7.93650 q^{64} -5.01828 q^{65} -0.256190 q^{67} +7.93005 q^{68} -3.09877 q^{70} -1.58801 q^{71} +9.67421 q^{73} -1.74615 q^{74} +5.71635 q^{76} +5.13227 q^{77} +4.34418 q^{79} +6.22816 q^{80} +11.1101 q^{82} -2.94073 q^{83} -6.17390 q^{85} -9.90432 q^{86} -0.0812334 q^{88} -17.8737 q^{89} -3.23567 q^{91} -3.34384 q^{92} +13.6977 q^{94} -4.45043 q^{95} -3.77305 q^{97} -1.99802 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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