LMFDB - Embedded newform 8001.2.a.ba.1.1

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:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.76162 q^{2} +5.62653 q^{4} +2.49672 q^{5} +1.00000 q^{7} -10.0151 q^{8} +O(q^{10})$	\(q-2.76162 q^{2} +5.62653 q^{4} +2.49672 q^{5} +1.00000 q^{7} -10.0151 q^{8} -6.89498 q^{10} -2.02342 q^{11} +1.00707 q^{13} -2.76162 q^{14} +16.4048 q^{16} -5.77932 q^{17} +2.08583 q^{19} +14.0479 q^{20} +5.58792 q^{22} +0.561101 q^{23} +1.23359 q^{25} -2.78115 q^{26} +5.62653 q^{28} +7.08501 q^{29} -0.146461 q^{31} -25.2737 q^{32} +15.9603 q^{34} +2.49672 q^{35} +6.93534 q^{37} -5.76028 q^{38} -25.0049 q^{40} +10.9073 q^{41} +9.04653 q^{43} -11.3849 q^{44} -1.54955 q^{46} +0.345998 q^{47} +1.00000 q^{49} -3.40671 q^{50} +5.66632 q^{52} -3.99044 q^{53} -5.05191 q^{55} -10.0151 q^{56} -19.5661 q^{58} +12.6208 q^{59} -1.75192 q^{61} +0.404469 q^{62} +36.9865 q^{64} +2.51437 q^{65} -11.5055 q^{67} -32.5175 q^{68} -6.89498 q^{70} -11.1554 q^{71} -9.76654 q^{73} -19.1528 q^{74} +11.7360 q^{76} -2.02342 q^{77} +13.1891 q^{79} +40.9582 q^{80} -30.1218 q^{82} +5.66568 q^{83} -14.4293 q^{85} -24.9831 q^{86} +20.2648 q^{88} -5.10948 q^{89} +1.00707 q^{91} +3.15706 q^{92} -0.955516 q^{94} +5.20774 q^{95} +17.3076 q^{97} -2.76162 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * 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 $

$2$	−2.76162	−1.95276	−0.976379	−	0.216063i	$-0.930678\pi$
$2$	−2.76162	−1.95276	−0.976379	+	0.216063i	$0.930678\pi$
$3$	0	0
$3$	0	0
$4$	5.62653	2.81327
$5$	2.49672	1.11657	0.558283	−	0.829651i	$-0.311460\pi$
$5$	2.49672	1.11657	0.558283	+	0.829651i	$0.311460\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−10.0151	−3.54087
$9$	0	0
$10$	−6.89498	−2.18038
$11$	−2.02342	−0.610085	−0.305042	−	0.952339i	$-0.598671\pi$
$11$	−2.02342	−0.610085	−0.305042	+	0.952339i	$0.598671\pi$
$12$	0	0
$13$	1.00707	0.279311	0.139656	−	0.990200i	$-0.455400\pi$
$13$	1.00707	0.279311	0.139656	+	0.990200i	$0.455400\pi$
$14$	−2.76162	−0.738074
$15$	0	0
$16$	16.4048	4.10121
$17$	−5.77932	−1.40169	−0.700845	−	0.713313i	$-0.747194\pi$
$17$	−5.77932	−1.40169	−0.700845	+	0.713313i	$0.747194\pi$
$18$	0	0
$19$	2.08583	0.478523	0.239262	−	0.970955i	$-0.423095\pi$
$19$	2.08583	0.478523	0.239262	+	0.970955i	$0.423095\pi$
$20$	14.0479	3.14120
$21$	0	0
$22$	5.58792	1.19135
$23$	0.561101	0.116998	0.0584989	−	0.998287i	$-0.481369\pi$
$23$	0.561101	0.116998	0.0584989	+	0.998287i	$0.481369\pi$
$24$	0	0
$25$	1.23359	0.246718
$26$	−2.78115	−0.545428
$27$	0	0
$28$	5.62653	1.06332
$29$	7.08501	1.31565	0.657827	−	0.753169i	$-0.271476\pi$
$29$	7.08501	1.31565	0.657827	+	0.753169i	$0.271476\pi$
$30$	0	0
$31$	−0.146461	−0.0263052	−0.0131526	−	0.999914i	$-0.504187\pi$
$31$	−0.146461	−0.0263052	−0.0131526	+	0.999914i	$0.504187\pi$
$32$	−25.2737	−4.46779
$33$	0	0
$34$	15.9603	2.73716
$35$	2.49672	0.422022
$36$	0	0
$37$	6.93534	1.14016	0.570081	−	0.821588i	$-0.306912\pi$
$37$	6.93534	1.14016	0.570081	+	0.821588i	$0.306912\pi$
$38$	−5.76028	−0.934441
$39$	0	0
$40$	−25.0049	−3.95362
$41$	10.9073	1.70343	0.851717	−	0.524003i	$-0.175562\pi$
$41$	10.9073	1.70343	0.851717	+	0.524003i	$0.175562\pi$
$42$	0	0
$43$	9.04653	1.37958	0.689791	−	0.724008i	$-0.257702\pi$
$43$	9.04653	1.37958	0.689791	+	0.724008i	$0.257702\pi$
$44$	−11.3849	−1.71633
$45$	0	0
$46$	−1.54955	−0.228468
$47$	0.345998	0.0504691	0.0252345	−	0.999682i	$-0.491967\pi$
$47$	0.345998	0.0504691	0.0252345	+	0.999682i	$0.491967\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.40671	−0.481782
$51$	0	0
$52$	5.66632	0.785777
$53$	−3.99044	−0.548129	−0.274064	−	0.961711i	$-0.588368\pi$
$53$	−3.99044	−0.548129	−0.274064	+	0.961711i	$0.588368\pi$
$54$	0	0
$55$	−5.05191	−0.681200
$56$	−10.0151	−1.33832
$57$	0	0
$58$	−19.5661	−2.56916
$59$	12.6208	1.64309	0.821545	−	0.570143i	$-0.193112\pi$
$59$	12.6208	1.64309	0.821545	+	0.570143i	$0.193112\pi$
$60$	0	0
$61$	−1.75192	−0.224311	−0.112155	−	0.993691i	$-0.535775\pi$
$61$	−1.75192	−0.224311	−0.112155	+	0.993691i	$0.535775\pi$
$62$	0.404469	0.0513676
$63$	0	0
$64$	36.9865	4.62332
$65$	2.51437	0.311869
$66$	0	0
$67$	−11.5055	−1.40562	−0.702809	−	0.711379i	$-0.748071\pi$
$67$	−11.5055	−1.40562	−0.702809	+	0.711379i	$0.748071\pi$
$68$	−32.5175	−3.94333
$69$	0	0
$70$	−6.89498	−0.824107
$71$	−11.1554	−1.32391	−0.661954	−	0.749544i	$-0.730273\pi$
$71$	−11.1554	−1.32391	−0.661954	+	0.749544i	$0.730273\pi$
$72$	0	0
$73$	−9.76654	−1.14309	−0.571543	−	0.820572i	$-0.693655\pi$
$73$	−9.76654	−1.14309	−0.571543	+	0.820572i	$0.693655\pi$
$74$	−19.1528	−2.22646
$75$	0	0
$76$	11.7360	1.34621
$77$	−2.02342	−0.230590
$78$	0	0
$79$	13.1891	1.48389	0.741945	−	0.670461i	$-0.233904\pi$
$79$	13.1891	1.48389	0.741945	+	0.670461i	$0.233904\pi$
$80$	40.9582	4.57927
$81$	0	0
$82$	−30.1218	−3.32639
$83$	5.66568	0.621890	0.310945	−	0.950428i	$-0.399355\pi$
$83$	5.66568	0.621890	0.310945	+	0.950428i	$0.399355\pi$
$84$	0	0
$85$	−14.4293	−1.56508
$86$	−24.9831	−2.69399
$87$	0	0
$88$	20.2648	2.16023
$89$	−5.10948	−0.541604	−0.270802	−	0.962635i	$-0.587289\pi$
$89$	−5.10948	−0.541604	−0.270802	+	0.962635i	$0.587289\pi$
$90$	0	0
$91$	1.00707	0.105570
$92$	3.15706	0.329146
$93$	0	0
$94$	−0.955516	−0.0985539
$95$	5.20774	0.534303
$96$	0	0
$97$	17.3076	1.75732	0.878661	−	0.477446i	$-0.158437\pi$
$97$	17.3076	1.75732	0.878661	+	0.477446i	$0.158437\pi$
$98$	−2.76162	−0.278966
$99$	0	0
$100$	6.94085	0.694085
$101$	−9.69718	−0.964905	−0.482453	−	0.875922i	$-0.660254\pi$
$101$	−9.69718	−0.964905	−0.482453	+	0.875922i	$0.660254\pi$
$102$	0	0
$103$	−0.770553	−0.0759248	−0.0379624	−	0.999279i	$-0.512087\pi$
$103$	−0.770553	−0.0759248	−0.0379624	+	0.999279i	$0.512087\pi$
$104$	−10.0859	−0.989006
$105$	0	0
$106$	11.0201	1.07036
$107$	−8.58511	−0.829954	−0.414977	−	0.909832i	$-0.636210\pi$
$107$	−8.58511	−0.829954	−0.414977	+	0.909832i	$0.636210\pi$
$108$	0	0
$109$	3.97887	0.381107	0.190554	−	0.981677i	$-0.438972\pi$
$109$	3.97887	0.381107	0.190554	+	0.981677i	$0.438972\pi$
$110$	13.9515	1.33022
$111$	0	0
$112$	16.4048	1.55011
$113$	−11.0492	−1.03942	−0.519711	−	0.854342i	$-0.673960\pi$
$113$	−11.0492	−1.03942	−0.519711	+	0.854342i	$0.673960\pi$
$114$	0	0
$115$	1.40091	0.130636
$116$	39.8641	3.70129
$117$	0	0
$118$	−34.8539	−3.20856
$119$	−5.77932	−0.529789
$120$	0	0
$121$	−6.90576	−0.627796
$122$	4.83814	0.438024
$123$	0	0
$124$	−0.824067	−0.0740034
$125$	−9.40365	−0.841088
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−51.5954	−4.56043
$129$	0	0
$130$	−6.94373	−0.609006
$131$	−1.65303	−0.144426	−0.0722129	−	0.997389i	$-0.523006\pi$
$131$	−1.65303	−0.144426	−0.0722129	+	0.997389i	$0.523006\pi$
$132$	0	0
$133$	2.08583	0.180865
$134$	31.7737	2.74483
$135$	0	0
$136$	57.8805	4.96321
$137$	−18.0556	−1.54259	−0.771296	−	0.636477i	$-0.780391\pi$
$137$	−18.0556	−1.54259	−0.771296	+	0.636477i	$0.780391\pi$
$138$	0	0
$139$	−10.6622	−0.904357	−0.452179	−	0.891927i	$-0.649353\pi$
$139$	−10.6622	−0.904357	−0.452179	+	0.891927i	$0.649353\pi$
$140$	14.0479	1.18726
$141$	0	0
$142$	30.8071	2.58527
$143$	−2.03773	−0.170404
$144$	0	0
$145$	17.6893	1.46901
$146$	26.9715	2.23217
$147$	0	0
$148$	39.0219	3.20758
$149$	17.6767	1.44813	0.724065	−	0.689732i	$-0.242272\pi$
$149$	17.6767	1.44813	0.724065	+	0.689732i	$0.242272\pi$
$150$	0	0
$151$	−3.37070	−0.274304	−0.137152	−	0.990550i	$-0.543795\pi$
$151$	−3.37070	−0.274304	−0.137152	+	0.990550i	$0.543795\pi$
$152$	−20.8899	−1.69439
$153$	0	0
$154$	5.58792	0.450288
$155$	−0.365671	−0.0293714
$156$	0	0
$157$	14.4291	1.15157	0.575784	−	0.817602i	$-0.304697\pi$
$157$	14.4291	1.15157	0.575784	+	0.817602i	$0.304697\pi$
$158$	−36.4233	−2.89768
$159$	0	0
$160$	−63.1011	−4.98858
$161$	0.561101	0.0442210
$162$	0	0
$163$	17.8967	1.40177	0.700887	−	0.713272i	$-0.252788\pi$
$163$	17.8967	1.40177	0.700887	+	0.713272i	$0.252788\pi$
$164$	61.3703	4.79221
$165$	0	0
$166$	−15.6465	−1.21440
$167$	−19.3819	−1.49982	−0.749908	−	0.661542i	$-0.769902\pi$
$167$	−19.3819	−1.49982	−0.749908	+	0.661542i	$0.769902\pi$
$168$	0	0
$169$	−11.9858	−0.921985
$170$	39.8483	3.05622
$171$	0	0
$172$	50.9006	3.88113
$173$	15.0125	1.14138	0.570689	−	0.821166i	$-0.306676\pi$
$173$	15.0125	1.14138	0.570689	+	0.821166i	$0.306676\pi$
$174$	0	0
$175$	1.23359	0.0932508
$176$	−33.1939	−2.50208
$177$	0	0
$178$	14.1104	1.05762
$179$	4.76765	0.356351	0.178176	−	0.983999i	$-0.442980\pi$
$179$	4.76765	0.356351	0.178176	+	0.983999i	$0.442980\pi$
$180$	0	0
$181$	1.51902	0.112908	0.0564540	−	0.998405i	$-0.482021\pi$
$181$	1.51902	0.112908	0.0564540	+	0.998405i	$0.482021\pi$
$182$	−2.78115	−0.206152
$183$	0	0
$184$	−5.61949	−0.414274
$185$	17.3156	1.27307
$186$	0	0
$187$	11.6940	0.855150
$188$	1.94677	0.141983
$189$	0	0
$190$	−14.3818	−1.04336
$191$	6.65415	0.481477	0.240739	−	0.970590i	$-0.422610\pi$
$191$	6.65415	0.481477	0.240739	+	0.970590i	$0.422610\pi$
$192$	0	0
$193$	1.72304	0.124027	0.0620137	−	0.998075i	$-0.480248\pi$
$193$	1.72304	0.124027	0.0620137	+	0.998075i	$0.480248\pi$
$194$	−47.7970	−3.43163
$195$	0	0
$196$	5.62653	0.401895
$197$	16.4146	1.16949	0.584745	−	0.811217i	$-0.301195\pi$
$197$	16.4146	1.16949	0.584745	+	0.811217i	$0.301195\pi$
$198$	0	0
$199$	9.63681	0.683135	0.341568	−	0.939857i	$-0.389042\pi$
$199$	9.63681	0.683135	0.341568	+	0.939857i	$0.389042\pi$
$200$	−12.3546	−0.873599
$201$	0	0
$202$	26.7799	1.88423
$203$	7.08501	0.497271
$204$	0	0
$205$	27.2324	1.90199
$206$	2.12797	0.148263
$207$	0	0
$208$	16.5208	1.14551
$209$	−4.22053	−0.291940
$210$	0	0
$211$	21.1116	1.45338	0.726692	−	0.686964i	$-0.241057\pi$
$211$	21.1116	1.45338	0.726692	+	0.686964i	$0.241057\pi$
$212$	−22.4523	−1.54203
$213$	0	0
$214$	23.7088	1.62070
$215$	22.5866	1.54039
$216$	0	0
$217$	−0.146461	−0.00994242
$218$	−10.9881	−0.744210
$219$	0	0
$220$	−28.4248	−1.91640
$221$	−5.82019	−0.391508
$222$	0	0
$223$	−25.6118	−1.71510	−0.857548	−	0.514404i	$-0.828013\pi$
$223$	−25.6118	−1.71510	−0.857548	+	0.514404i	$0.828013\pi$
$224$	−25.2737	−1.68867
$225$	0	0
$226$	30.5137	2.02974
$227$	21.9818	1.45898	0.729491	−	0.683991i	$-0.239757\pi$
$227$	21.9818	1.45898	0.729491	+	0.683991i	$0.239757\pi$
$228$	0	0
$229$	−19.2856	−1.27443	−0.637213	−	0.770688i	$-0.719913\pi$
$229$	−19.2856	−1.27443	−0.637213	+	0.770688i	$0.719913\pi$
$230$	−3.86878	−0.255100
$231$	0	0
$232$	−70.9572	−4.65857
$233$	6.29010	0.412078	0.206039	−	0.978544i	$-0.433943\pi$
$233$	6.29010	0.412078	0.206039	+	0.978544i	$0.433943\pi$
$234$	0	0
$235$	0.863860	0.0563520
$236$	71.0115	4.62245
$237$	0	0
$238$	15.9603	1.03455
$239$	26.0235	1.68332	0.841662	−	0.540005i	$-0.181578\pi$
$239$	26.0235	1.68332	0.841662	+	0.540005i	$0.181578\pi$
$240$	0	0
$241$	−1.50746	−0.0971042	−0.0485521	−	0.998821i	$-0.515461\pi$
$241$	−1.50746	−0.0971042	−0.0485521	+	0.998821i	$0.515461\pi$
$242$	19.0711	1.22593
$243$	0	0
$244$	−9.85725	−0.631045
$245$	2.49672	0.159509
$246$	0	0
$247$	2.10058	0.133657
$248$	1.46682	0.0931433
$249$	0	0
$250$	25.9693	1.64244
$251$	2.71602	0.171434	0.0857169	−	0.996320i	$-0.472682\pi$
$251$	2.71602	0.171434	0.0857169	+	0.996320i	$0.472682\pi$
$252$	0	0
$253$	−1.13535	−0.0713786
$254$	2.76162	0.173279
$255$	0	0
$256$	68.5136	4.28210
$257$	12.1154	0.755735	0.377867	−	0.925860i	$-0.376658\pi$
$257$	12.1154	0.755735	0.377867	+	0.925860i	$0.376658\pi$
$258$	0	0
$259$	6.93534	0.430941
$260$	14.1472	0.877372
$261$	0	0
$262$	4.56504	0.282029
$263$	18.0698	1.11423	0.557115	−	0.830436i	$-0.311908\pi$
$263$	18.0698	1.11423	0.557115	+	0.830436i	$0.311908\pi$
$264$	0	0
$265$	−9.96299	−0.612022
$266$	−5.76028	−0.353185
$267$	0	0
$268$	−64.7360	−3.95438
$269$	−0.766304	−0.0467224	−0.0233612	−	0.999727i	$-0.507437\pi$
$269$	−0.766304	−0.0467224	−0.0233612	+	0.999727i	$0.507437\pi$
$270$	0	0
$271$	13.1009	0.795826	0.397913	−	0.917423i	$-0.369735\pi$
$271$	13.1009	0.795826	0.397913	+	0.917423i	$0.369735\pi$
$272$	−94.8087	−5.74862
$273$	0	0
$274$	49.8626	3.01231
$275$	−2.49608	−0.150519
$276$	0	0
$277$	4.72067	0.283638	0.141819	−	0.989893i	$-0.454705\pi$
$277$	4.72067	0.283638	0.141819	+	0.989893i	$0.454705\pi$
$278$	29.4450	1.76599
$279$	0	0
$280$	−25.0049	−1.49433
$281$	8.67276	0.517374	0.258687	−	0.965961i	$-0.416710\pi$
$281$	8.67276	0.517374	0.258687	+	0.965961i	$0.416710\pi$
$282$	0	0
$283$	−4.30394	−0.255843	−0.127921	−	0.991784i	$-0.540831\pi$
$283$	−4.30394	−0.255843	−0.127921	+	0.991784i	$0.540831\pi$
$284$	−62.7665	−3.72451
$285$	0	0
$286$	5.62744	0.332757
$287$	10.9073	0.643837
$288$	0	0
$289$	16.4005	0.964737
$290$	−48.8510	−2.86863
$291$	0	0
$292$	−54.9518	−3.21581
$293$	5.96253	0.348335	0.174167	−	0.984716i	$-0.444277\pi$
$293$	5.96253	0.348335	0.174167	+	0.984716i	$0.444277\pi$
$294$	0	0
$295$	31.5106	1.83462
$296$	−69.4581	−4.03717
$297$	0	0
$298$	−48.8162	−2.82785
$299$	0.565069	0.0326788
$300$	0	0
$301$	9.04653	0.521433
$302$	9.30860	0.535649
$303$	0	0
$304$	34.2178	1.96252
$305$	−4.37405	−0.250457
$306$	0	0
$307$	−6.98531	−0.398673	−0.199336	−	0.979931i	$-0.563879\pi$
$307$	−6.98531	−0.398673	−0.199336	+	0.979931i	$0.563879\pi$
$308$	−11.3849	−0.648713
$309$	0	0
$310$	1.00984	0.0573553
$311$	24.2744	1.37648	0.688238	−	0.725485i	$-0.258385\pi$
$311$	24.2744	1.37648	0.688238	+	0.725485i	$0.258385\pi$
$312$	0	0
$313$	−8.66417	−0.489728	−0.244864	−	0.969557i	$-0.578743\pi$
$313$	−8.66417	−0.489728	−0.244864	+	0.969557i	$0.578743\pi$
$314$	−39.8477	−2.24873
$315$	0	0
$316$	74.2090	4.17458
$317$	−6.46017	−0.362839	−0.181420	−	0.983406i	$-0.558069\pi$
$317$	−6.46017	−0.362839	−0.181420	+	0.983406i	$0.558069\pi$
$318$	0	0
$319$	−14.3360	−0.802661
$320$	92.3449	5.16224
$321$	0	0
$322$	−1.54955	−0.0863529
$323$	−12.0547	−0.670742
$324$	0	0
$325$	1.24232	0.0689113
$326$	−49.4237	−2.73733
$327$	0	0
$328$	−109.238	−6.03164
$329$	0.345998	0.0190755
$330$	0	0
$331$	−31.6396	−1.73907	−0.869536	−	0.493870i	$-0.835582\pi$
$331$	−31.6396	−1.73907	−0.869536	+	0.493870i	$0.835582\pi$
$332$	31.8782	1.74954
$333$	0	0
$334$	53.5254	2.92878
$335$	−28.7259	−1.56946
$336$	0	0
$337$	−1.20690	−0.0657439	−0.0328719	−	0.999460i	$-0.510465\pi$
$337$	−1.20690	−0.0657439	−0.0328719	+	0.999460i	$0.510465\pi$
$338$	33.1002	1.80041
$339$	0	0
$340$	−81.1871	−4.40299
$341$	0.296352	0.0160484
$342$	0	0
$343$	1.00000	0.0539949
$344$	−90.6019	−4.88493
$345$	0	0
$346$	−41.4588	−2.22884
$347$	−7.00972	−0.376301	−0.188151	−	0.982140i	$-0.560249\pi$
$347$	−7.00972	−0.376301	−0.188151	+	0.982140i	$0.560249\pi$
$348$	0	0
$349$	4.70910	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$349$	4.70910	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$350$	−3.40671	−0.182096
$351$	0	0
$352$	51.1393	2.72573
$353$	−27.1259	−1.44376	−0.721882	−	0.692016i	$-0.756723\pi$
$353$	−27.1259	−1.44376	−0.721882	+	0.692016i	$0.756723\pi$
$354$	0	0
$355$	−27.8520	−1.47823
$356$	−28.7487	−1.52368
$357$	0	0
$358$	−13.1664	−0.695868
$359$	21.1017	1.11371	0.556853	−	0.830611i	$-0.312009\pi$
$359$	21.1017	1.11371	0.556853	+	0.830611i	$0.312009\pi$
$360$	0	0
$361$	−14.6493	−0.771015
$362$	−4.19496	−0.220482
$363$	0	0
$364$	5.66632	0.296996
$365$	−24.3843	−1.27633
$366$	0	0
$367$	8.99114	0.469333	0.234667	−	0.972076i	$-0.424600\pi$
$367$	8.99114	0.469333	0.234667	+	0.972076i	$0.424600\pi$
$368$	9.20477	0.479832
$369$	0	0
$370$	−47.8190	−2.48599
$371$	−3.99044	−0.207173
$372$	0	0
$373$	11.6290	0.602128	0.301064	−	0.953604i	$-0.402658\pi$
$373$	11.6290	0.602128	0.301064	+	0.953604i	$0.402658\pi$
$374$	−32.2944	−1.66990
$375$	0	0
$376$	−3.46521	−0.178705
$377$	7.13512	0.367477
$378$	0	0
$379$	27.3794	1.40639	0.703194	−	0.710998i	$-0.251757\pi$
$379$	27.3794	1.40639	0.703194	+	0.710998i	$0.251757\pi$
$380$	29.3015	1.50314
$381$	0	0
$382$	−18.3762	−0.940209
$383$	−25.2940	−1.29246	−0.646231	−	0.763142i	$-0.723656\pi$
$383$	−25.2940	−1.29246	−0.646231	+	0.763142i	$0.723656\pi$
$384$	0	0
$385$	−5.05191	−0.257469
$386$	−4.75839	−0.242196
$387$	0	0
$388$	97.3819	4.94382
$389$	14.5503	0.737729	0.368864	−	0.929483i	$-0.379747\pi$
$389$	14.5503	0.737729	0.368864	+	0.929483i	$0.379747\pi$
$390$	0	0
$391$	−3.24278	−0.163995
$392$	−10.0151	−0.505839
$393$	0	0
$394$	−45.3308	−2.28373
$395$	32.9295	1.65686
$396$	0	0
$397$	−23.2217	−1.16546	−0.582732	−	0.812665i	$-0.698016\pi$
$397$	−23.2217	−1.16546	−0.582732	+	0.812665i	$0.698016\pi$
$398$	−26.6132	−1.33400
$399$	0	0
$400$	20.2369	1.01184
$401$	−3.03071	−0.151347	−0.0756733	−	0.997133i	$-0.524111\pi$
$401$	−3.03071	−0.151347	−0.0756733	+	0.997133i	$0.524111\pi$
$402$	0	0
$403$	−0.147497	−0.00734733
$404$	−54.5615	−2.71454
$405$	0	0
$406$	−19.5661	−0.971050
$407$	−14.0331	−0.695596
$408$	0	0
$409$	−0.874884	−0.0432602	−0.0216301	−	0.999766i	$-0.506886\pi$
$409$	−0.874884	−0.0432602	−0.0216301	+	0.999766i	$0.506886\pi$
$410$	−75.2055	−3.71414
$411$	0	0
$412$	−4.33554	−0.213597
$413$	12.6208	0.621030
$414$	0	0
$415$	14.1456	0.694380
$416$	−25.4524	−1.24791
$417$	0	0
$418$	11.6555	0.570088
$419$	23.9656	1.17079	0.585397	−	0.810746i	$-0.300938\pi$
$419$	23.9656	1.17079	0.585397	+	0.810746i	$0.300938\pi$
$420$	0	0
$421$	2.43388	0.118620	0.0593100	−	0.998240i	$-0.481110\pi$
$421$	2.43388	0.118620	0.0593100	+	0.998240i	$0.481110\pi$
$422$	−58.3022	−2.83811
$423$	0	0
$424$	39.9647	1.94086
$425$	−7.12932	−0.345823
$426$	0	0
$427$	−1.75192	−0.0847814
$428$	−48.3044	−2.33488
$429$	0	0
$430$	−62.3756	−3.00802
$431$	12.3145	0.593169	0.296585	−	0.955007i	$-0.404152\pi$
$431$	12.3145	0.593169	0.296585	+	0.955007i	$0.404152\pi$
$432$	0	0
$433$	13.1550	0.632191	0.316095	−	0.948727i	$-0.397628\pi$
$433$	13.1550	0.632191	0.316095	+	0.948727i	$0.397628\pi$
$434$	0.404469	0.0194151
$435$	0	0
$436$	22.3873	1.07216
$437$	1.17036	0.0559861
$438$	0	0
$439$	−18.6508	−0.890152	−0.445076	−	0.895493i	$-0.646823\pi$
$439$	−18.6508	−0.890152	−0.445076	+	0.895493i	$0.646823\pi$
$440$	50.5954	2.41204
$441$	0	0
$442$	16.0731	0.764521
$443$	37.1672	1.76587	0.882933	−	0.469500i	$-0.155566\pi$
$443$	37.1672	1.76587	0.882933	+	0.469500i	$0.155566\pi$
$444$	0	0
$445$	−12.7569	−0.604736
$446$	70.7301	3.34917
$447$	0	0
$448$	36.9865	1.74745
$449$	−17.8440	−0.842110	−0.421055	−	0.907035i	$-0.638340\pi$
$449$	−17.8440	−0.842110	−0.421055	+	0.907035i	$0.638340\pi$
$450$	0	0
$451$	−22.0701	−1.03924
$452$	−62.1688	−2.92417
$453$	0	0
$454$	−60.7053	−2.84904
$455$	2.51437	0.117876
$456$	0	0
$457$	−20.3116	−0.950136	−0.475068	−	0.879949i	$-0.657577\pi$
$457$	−20.3116	−0.950136	−0.475068	+	0.879949i	$0.657577\pi$
$458$	53.2594	2.48865
$459$	0	0
$460$	7.88227	0.367513
$461$	−4.06383	−0.189271	−0.0946356	−	0.995512i	$-0.530169\pi$
$461$	−4.06383	−0.189271	−0.0946356	+	0.995512i	$0.530169\pi$
$462$	0	0
$463$	22.2416	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$463$	22.2416	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$464$	116.228	5.39577
$465$	0	0
$466$	−17.3709	−0.804690
$467$	21.1842	0.980289	0.490145	−	0.871641i	$-0.336944\pi$
$467$	21.1842	0.980289	0.490145	+	0.871641i	$0.336944\pi$
$468$	0	0
$469$	−11.5055	−0.531273
$470$	−2.38565	−0.110042
$471$	0	0
$472$	−126.399	−5.81798
$473$	−18.3049	−0.841662
$474$	0	0
$475$	2.57307	0.118061
$476$	−32.5175	−1.49044
$477$	0	0
$478$	−71.8671	−3.28712
$479$	12.4242	0.567676	0.283838	−	0.958872i	$-0.408392\pi$
$479$	12.4242	0.567676	0.283838	+	0.958872i	$0.408392\pi$
$480$	0	0
$481$	6.98438	0.318460
$482$	4.16303	0.189621
$483$	0	0
$484$	−38.8555	−1.76616
$485$	43.2122	1.96217
$486$	0	0
$487$	−1.97044	−0.0892893	−0.0446446	−	0.999003i	$-0.514216\pi$
$487$	−1.97044	−0.0892893	−0.0446446	+	0.999003i	$0.514216\pi$
$488$	17.5457	0.794255
$489$	0	0
$490$	−6.89498	−0.311483
$491$	−20.9162	−0.943934	−0.471967	−	0.881616i	$-0.656456\pi$
$491$	−20.9162	−0.943934	−0.471967	+	0.881616i	$0.656456\pi$
$492$	0	0
$493$	−40.9466	−1.84414
$494$	−5.80101	−0.261000
$495$	0	0
$496$	−2.40267	−0.107883
$497$	−11.1554	−0.500390
$498$	0	0
$499$	15.2674	0.683464	0.341732	−	0.939798i	$-0.388987\pi$
$499$	15.2674	0.683464	0.341732	+	0.939798i	$0.388987\pi$
$500$	−52.9100	−2.36621
$501$	0	0
$502$	−7.50061	−0.334769
$503$	−11.2004	−0.499399	−0.249700	−	0.968323i	$-0.580332\pi$
$503$	−11.2004	−0.499399	−0.249700	+	0.968323i	$0.580332\pi$
$504$	0	0
$505$	−24.2111	−1.07738
$506$	3.13539	0.139385
$507$	0	0
$508$	−5.62653	−0.249637
$509$	33.7808	1.49731	0.748654	−	0.662961i	$-0.230701\pi$
$509$	33.7808	1.49731	0.748654	+	0.662961i	$0.230701\pi$
$510$	0	0
$511$	−9.76654	−0.432046
$512$	−86.0178	−3.80149
$513$	0	0
$514$	−33.4580	−1.47577
$515$	−1.92385	−0.0847750
$516$	0	0
$517$	−0.700101	−0.0307904
$518$	−19.1528	−0.841524
$519$	0	0
$520$	−25.1817	−1.10429
$521$	20.0646	0.879044	0.439522	−	0.898232i	$-0.355148\pi$
$521$	20.0646	0.879044	0.439522	+	0.898232i	$0.355148\pi$
$522$	0	0
$523$	10.6928	0.467562	0.233781	−	0.972289i	$-0.424890\pi$
$523$	10.6928	0.467562	0.233781	+	0.972289i	$0.424890\pi$
$524$	−9.30083	−0.406309
$525$	0	0
$526$	−49.9018	−2.17582
$527$	0.846444	0.0368717
$528$	0	0
$529$	−22.6852	−0.986312
$530$	27.5140	1.19513
$531$	0	0
$532$	11.7360	0.508821
$533$	10.9844	0.475788
$534$	0	0
$535$	−21.4346	−0.926698
$536$	115.229	4.97711
$537$	0	0
$538$	2.11624	0.0912376
$539$	−2.02342	−0.0871550
$540$	0	0
$541$	32.8699	1.41319	0.706594	−	0.707619i	$-0.250231\pi$
$541$	32.8699	1.41319	0.706594	+	0.707619i	$0.250231\pi$
$542$	−36.1798	−1.55406
$543$	0	0
$544$	146.065	6.26246
$545$	9.93412	0.425531
$546$	0	0
$547$	0.419292	0.0179276	0.00896382	−	0.999960i	$-0.497147\pi$
$547$	0.419292	0.0179276	0.00896382	+	0.999960i	$0.497147\pi$
$548$	−101.590	−4.33972
$549$	0	0
$550$	6.89322	0.293928
$551$	14.7782	0.629571
$552$	0	0
$553$	13.1891	0.560858
$554$	−13.0367	−0.553876
$555$	0	0
$556$	−59.9913	−2.54420
$557$	34.4326	1.45896	0.729478	−	0.684004i	$-0.239763\pi$
$557$	34.4326	1.45896	0.729478	+	0.684004i	$0.239763\pi$
$558$	0	0
$559$	9.11050	0.385333
$560$	40.9582	1.73080
$561$	0	0
$562$	−23.9509	−1.01031
$563$	32.8916	1.38621	0.693107	−	0.720834i	$-0.256241\pi$
$563$	32.8916	1.38621	0.693107	+	0.720834i	$0.256241\pi$
$564$	0	0
$565$	−27.5868	−1.16058
$566$	11.8858	0.499599
$567$	0	0
$568$	111.723	4.68779
$569$	44.7581	1.87636	0.938178	−	0.346152i	$-0.112512\pi$
$569$	44.7581	1.87636	0.938178	+	0.346152i	$0.112512\pi$
$570$	0	0
$571$	−30.5380	−1.27797	−0.638987	−	0.769218i	$-0.720646\pi$
$571$	−30.5380	−1.27797	−0.638987	+	0.769218i	$0.720646\pi$
$572$	−11.4654	−0.479391
$573$	0	0
$574$	−30.1218	−1.25726
$575$	0.692170	0.0288655
$576$	0	0
$577$	44.5690	1.85543	0.927717	−	0.373285i	$-0.121769\pi$
$577$	44.5690	1.85543	0.927717	+	0.373285i	$0.121769\pi$
$578$	−45.2920	−1.88390
$579$	0	0
$580$	99.5293	4.13273
$581$	5.66568	0.235052
$582$	0	0
$583$	8.07434	0.334405
$584$	97.8129	4.04753
$585$	0	0
$586$	−16.4662	−0.680214
$587$	47.2633	1.95076	0.975382	−	0.220522i	$-0.0707760\pi$
$587$	47.2633	1.95076	0.975382	+	0.220522i	$0.0707760\pi$
$588$	0	0
$589$	−0.305493	−0.0125876
$590$	−87.0203	−3.58257
$591$	0	0
$592$	113.773	4.67604
$593$	−16.9389	−0.695597	−0.347798	−	0.937569i	$-0.613071\pi$
$593$	−16.9389	−0.695597	−0.347798	+	0.937569i	$0.613071\pi$
$594$	0	0
$595$	−14.4293	−0.591544
$596$	99.4584	4.07398
$597$	0	0
$598$	−1.56051	−0.0638138
$599$	28.8061	1.17699	0.588493	−	0.808502i	$-0.299721\pi$
$599$	28.8061	1.17699	0.588493	+	0.808502i	$0.299721\pi$
$600$	0	0
$601$	17.4877	0.713338	0.356669	−	0.934231i	$-0.383912\pi$
$601$	17.4877	0.713338	0.356669	+	0.934231i	$0.383912\pi$
$602$	−24.9831	−1.01823
$603$	0	0
$604$	−18.9654	−0.771690
$605$	−17.2417	−0.700976
$606$	0	0
$607$	14.1487	0.574276	0.287138	−	0.957889i	$-0.407296\pi$
$607$	14.1487	0.574276	0.287138	+	0.957889i	$0.407296\pi$
$608$	−52.7167	−2.13794
$609$	0	0
$610$	12.0795	0.489083
$611$	0.348445	0.0140966
$612$	0	0
$613$	−3.96514	−0.160151	−0.0800753	−	0.996789i	$-0.525516\pi$
$613$	−3.96514	−0.160151	−0.0800753	+	0.996789i	$0.525516\pi$
$614$	19.2908	0.778512
$615$	0	0
$616$	20.2648	0.816492
$617$	−5.45412	−0.219575	−0.109787	−	0.993955i	$-0.535017\pi$
$617$	−5.45412	−0.219575	−0.109787	+	0.993955i	$0.535017\pi$
$618$	0	0
$619$	0.640385	0.0257392	0.0128696	−	0.999917i	$-0.495903\pi$
$619$	0.640385	0.0257392	0.0128696	+	0.999917i	$0.495903\pi$
$620$	−2.05746	−0.0826297
$621$	0	0
$622$	−67.0367	−2.68793
$623$	−5.10948	−0.204707
$624$	0	0
$625$	−29.6462	−1.18585
$626$	23.9271	0.956320
$627$	0	0
$628$	81.1858	3.23967
$629$	−40.0815	−1.59816
$630$	0	0
$631$	−39.4169	−1.56916	−0.784581	−	0.620027i	$-0.787122\pi$
$631$	−39.4169	−1.56916	−0.784581	+	0.620027i	$0.787122\pi$
$632$	−132.090	−5.25427
$633$	0	0
$634$	17.8405	0.708538
$635$	−2.49672	−0.0990792
$636$	0	0
$637$	1.00707	0.0399016
$638$	39.5905	1.56740
$639$	0	0
$640$	−128.819	−5.09202
$641$	−32.3949	−1.27952	−0.639761	−	0.768574i	$-0.720966\pi$
$641$	−32.3949	−1.27952	−0.639761	+	0.768574i	$0.720966\pi$
$642$	0	0
$643$	8.57819	0.338291	0.169145	−	0.985591i	$-0.445899\pi$
$643$	8.57819	0.338291	0.169145	+	0.985591i	$0.445899\pi$
$644$	3.15706	0.124405
$645$	0	0
$646$	33.2905	1.30980
$647$	−42.1220	−1.65599	−0.827993	−	0.560738i	$-0.810518\pi$
$647$	−42.1220	−1.65599	−0.827993	+	0.560738i	$0.810518\pi$
$648$	0	0
$649$	−25.5373	−1.00242
$650$	−3.43080	−0.134567
$651$	0	0
$652$	100.696	3.94357
$653$	25.1570	0.984471	0.492235	−	0.870462i	$-0.336180\pi$
$653$	25.1570	0.984471	0.492235	+	0.870462i	$0.336180\pi$
$654$	0	0
$655$	−4.12715	−0.161261
$656$	178.932	6.98613
$657$	0	0
$658$	−0.955516	−0.0372499
$659$	−21.7398	−0.846860	−0.423430	−	0.905929i	$-0.639174\pi$
$659$	−21.7398	−0.846860	−0.423430	+	0.905929i	$0.639174\pi$
$660$	0	0
$661$	−23.9766	−0.932583	−0.466292	−	0.884631i	$-0.654410\pi$
$661$	−23.9766	−0.932583	−0.466292	+	0.884631i	$0.654410\pi$
$662$	87.3766	3.39599
$663$	0	0
$664$	−56.7424	−2.20203
$665$	5.20774	0.201947
$666$	0	0
$667$	3.97541	0.153929
$668$	−109.053	−4.21938
$669$	0	0
$670$	79.3300	3.06478
$671$	3.54488	0.136848
$672$	0	0
$673$	22.5678	0.869925	0.434962	−	0.900449i	$-0.356762\pi$
$673$	22.5678	0.869925	0.434962	+	0.900449i	$0.356762\pi$
$674$	3.33299	0.128382
$675$	0	0
$676$	−67.4386	−2.59379
$677$	−6.11393	−0.234978	−0.117489	−	0.993074i	$-0.537484\pi$
$677$	−6.11393	−0.234978	−0.117489	+	0.993074i	$0.537484\pi$
$678$	0	0
$679$	17.3076	0.664205
$680$	144.511	5.54175
$681$	0	0
$682$	−0.818412	−0.0313386
$683$	−11.4313	−0.437406	−0.218703	−	0.975791i	$-0.570183\pi$
$683$	−11.4313	−0.437406	−0.218703	+	0.975791i	$0.570183\pi$
$684$	0	0
$685$	−45.0796	−1.72240
$686$	−2.76162	−0.105439
$687$	0	0
$688$	148.407	5.65795
$689$	−4.01866	−0.153099
$690$	0	0
$691$	13.0739	0.497355	0.248677	−	0.968586i	$-0.420004\pi$
$691$	13.0739	0.497355	0.248677	+	0.968586i	$0.420004\pi$
$692$	84.4683	3.21100
$693$	0	0
$694$	19.3582	0.734826
$695$	−26.6205	−1.00977
$696$	0	0
$697$	−63.0367	−2.38769
$698$	−13.0047	−0.492236
$699$	0	0
$700$	6.94085	0.262339
$701$	36.3959	1.37465	0.687327	−	0.726348i	$-0.258784\pi$
$701$	36.3959	1.37465	0.687327	+	0.726348i	$0.258784\pi$
$702$	0	0
$703$	14.4660	0.545594
$704$	−74.8394	−2.82062
$705$	0	0
$706$	74.9113	2.81932
$707$	−9.69718	−0.364700
$708$	0	0
$709$	18.0378	0.677425	0.338713	−	0.940890i	$-0.390009\pi$
$709$	18.0378	0.677425	0.338713	+	0.940890i	$0.390009\pi$
$710$	76.9166	2.88663
$711$	0	0
$712$	51.1720	1.91775
$713$	−0.0821794	−0.00307764
$714$	0	0
$715$	−5.08764	−0.190267
$716$	26.8254	1.00251
$717$	0	0
$718$	−58.2748	−2.17480
$719$	27.7677	1.03556	0.517781	−	0.855513i	$-0.326758\pi$
$719$	27.7677	1.03556	0.517781	+	0.855513i	$0.326758\pi$
$720$	0	0
$721$	−0.770553	−0.0286969
$722$	40.4558	1.50561
$723$	0	0
$724$	8.54683	0.317641
$725$	8.74002	0.324596
$726$	0	0
$727$	4.73166	0.175488	0.0877438	−	0.996143i	$-0.472034\pi$
$727$	4.73166	0.175488	0.0877438	+	0.996143i	$0.472034\pi$
$728$	−10.0859	−0.373809
$729$	0	0
$730$	67.3401	2.49237
$731$	−52.2828	−1.93375
$732$	0	0
$733$	30.8404	1.13912	0.569559	−	0.821951i	$-0.307114\pi$
$733$	30.8404	1.13912	0.569559	+	0.821951i	$0.307114\pi$
$734$	−24.8301	−0.916495
$735$	0	0
$736$	−14.1811	−0.522722
$737$	23.2804	0.857546
$738$	0	0
$739$	−21.3933	−0.786967	−0.393483	−	0.919332i	$-0.628730\pi$
$739$	−21.3933	−0.786967	−0.393483	+	0.919332i	$0.628730\pi$
$740$	97.4267	3.58148
$741$	0	0
$742$	11.0201	0.404559
$743$	−23.1866	−0.850633	−0.425316	−	0.905045i	$-0.639837\pi$
$743$	−23.1866	−0.850633	−0.425316	+	0.905045i	$0.639837\pi$
$744$	0	0
$745$	44.1336	1.61693
$746$	−32.1149	−1.17581
$747$	0	0
$748$	65.7967	2.40577
$749$	−8.58511	−0.313693
$750$	0	0
$751$	−52.8404	−1.92817	−0.964087	−	0.265586i	$-0.914435\pi$
$751$	−52.8404	−1.92817	−0.964087	+	0.265586i	$0.914435\pi$
$752$	5.67604	0.206984
$753$	0	0
$754$	−19.7045	−0.717594
$755$	−8.41569	−0.306278
$756$	0	0
$757$	−24.1344	−0.877181	−0.438591	−	0.898687i	$-0.644522\pi$
$757$	−24.1344	−0.877181	−0.438591	+	0.898687i	$0.644522\pi$
$758$	−75.6116	−2.74634
$759$	0	0
$760$	−52.1560	−1.89190
$761$	36.8841	1.33705	0.668524	−	0.743691i	$-0.266926\pi$
$761$	36.8841	1.33705	0.668524	+	0.743691i	$0.266926\pi$
$762$	0	0
$763$	3.97887	0.144045
$764$	37.4398	1.35452
$765$	0	0
$766$	69.8523	2.52387
$767$	12.7101	0.458934
$768$	0	0
$769$	46.3235	1.67047	0.835235	−	0.549893i	$-0.185332\pi$
$769$	46.3235	1.67047	0.835235	+	0.549893i	$0.185332\pi$
$770$	13.9515	0.502776
$771$	0	0
$772$	9.69477	0.348922
$773$	−7.47980	−0.269030	−0.134515	−	0.990912i	$-0.542948\pi$
$773$	−7.47980	−0.269030	−0.134515	+	0.990912i	$0.542948\pi$
$774$	0	0
$775$	−0.180673	−0.00648997
$776$	−173.338	−6.22246
$777$	0	0
$778$	−40.1823	−1.44061
$779$	22.7508	0.815133
$780$	0	0
$781$	22.5722	0.807697
$782$	8.95533	0.320242
$783$	0	0
$784$	16.4048	0.585887
$785$	36.0254	1.28580
$786$	0	0
$787$	52.1283	1.85817	0.929086	−	0.369863i	$-0.120596\pi$
$787$	52.1283	1.85817	0.929086	+	0.369863i	$0.120596\pi$
$788$	92.3572	3.29009
$789$	0	0
$790$	−90.9386	−3.23545
$791$	−11.0492	−0.392865
$792$	0	0
$793$	−1.76431	−0.0626525
$794$	64.1294	2.27587
$795$	0	0
$796$	54.2218	1.92184
$797$	−3.40826	−0.120727	−0.0603634	−	0.998176i	$-0.519226\pi$
$797$	−3.40826	−0.120727	−0.0603634	+	0.998176i	$0.519226\pi$
$798$	0	0
$799$	−1.99964	−0.0707420
$800$	−31.1774	−1.10229
$801$	0	0
$802$	8.36968	0.295544
$803$	19.7618	0.697380
$804$	0	0
$805$	1.40091	0.0493756
$806$	0.407329	0.0143476
$807$	0	0
$808$	97.1183	3.41661
$809$	15.4635	0.543669	0.271834	−	0.962344i	$-0.412370\pi$
$809$	15.4635	0.543669	0.271834	+	0.962344i	$0.412370\pi$
$810$	0	0
$811$	28.4531	0.999125	0.499563	−	0.866278i	$-0.333494\pi$
$811$	28.4531	0.999125	0.499563	+	0.866278i	$0.333494\pi$
$812$	39.8641	1.39896
$813$	0	0
$814$	38.7541	1.35833
$815$	44.6829	1.56517
$816$	0	0
$817$	18.8696	0.660162
$818$	2.41610	0.0844768
$819$	0	0
$820$	153.224	5.35082
$821$	−20.6925	−0.722173	−0.361087	−	0.932532i	$-0.617594\pi$
$821$	−20.6925	−0.722173	−0.361087	+	0.932532i	$0.617594\pi$
$822$	0	0
$823$	0.581900	0.0202838	0.0101419	−	0.999949i	$-0.496772\pi$
$823$	0.581900	0.0202838	0.0101419	+	0.999949i	$0.496772\pi$
$824$	7.71716	0.268840
$825$	0	0
$826$	−34.8539	−1.21272
$827$	−5.43301	−0.188924	−0.0944622	−	0.995528i	$-0.530113\pi$
$827$	−5.43301	−0.188924	−0.0944622	+	0.995528i	$0.530113\pi$
$828$	0	0
$829$	23.0859	0.801807	0.400904	−	0.916120i	$-0.368696\pi$
$829$	23.0859	0.801807	0.400904	+	0.916120i	$0.368696\pi$
$830$	−39.0648	−1.35596
$831$	0	0
$832$	37.2481	1.29134
$833$	−5.77932	−0.200242
$834$	0	0
$835$	−48.3911	−1.67464
$836$	−23.7469	−0.821305
$837$	0	0
$838$	−66.1838	−2.28628
$839$	−1.02328	−0.0353274	−0.0176637	−	0.999844i	$-0.505623\pi$
$839$	−1.02328	−0.0353274	−0.0176637	+	0.999844i	$0.505623\pi$
$840$	0	0
$841$	21.1974	0.730946
$842$	−6.72145	−0.231636
$843$	0	0
$844$	118.785	4.08876
$845$	−29.9252	−1.02946
$846$	0	0
$847$	−6.90576	−0.237285
$848$	−65.4624	−2.24799
$849$	0	0
$850$	19.6885	0.675309
$851$	3.89143	0.133396
$852$	0	0
$853$	40.7286	1.39452	0.697260	−	0.716818i	$-0.254402\pi$
$853$	40.7286	1.39452	0.697260	+	0.716818i	$0.254402\pi$
$854$	4.83814	0.165558
$855$	0	0
$856$	85.9808	2.93876
$857$	−33.4961	−1.14420	−0.572102	−	0.820183i	$-0.693872\pi$
$857$	−33.4961	−1.14420	−0.572102	+	0.820183i	$0.693872\pi$
$858$	0	0
$859$	−14.3089	−0.488214	−0.244107	−	0.969748i	$-0.578495\pi$
$859$	−14.3089	−0.488214	−0.244107	+	0.969748i	$0.578495\pi$
$860$	127.084	4.33354
$861$	0	0
$862$	−34.0080	−1.15832
$863$	−26.5119	−0.902475	−0.451237	−	0.892404i	$-0.649017\pi$
$863$	−26.5119	−0.902475	−0.451237	+	0.892404i	$0.649017\pi$
$864$	0	0
$865$	37.4819	1.27442
$866$	−36.3292	−1.23452
$867$	0	0
$868$	−0.824067	−0.0279707
$869$	−26.6871	−0.905299
$870$	0	0
$871$	−11.5868	−0.392605
$872$	−39.8488	−1.34945
$873$	0	0
$874$	−3.23210	−0.109327
$875$	−9.40365	−0.317901
$876$	0	0
$877$	−2.22639	−0.0751798	−0.0375899	−	0.999293i	$-0.511968\pi$
$877$	−2.22639	−0.0751798	−0.0375899	+	0.999293i	$0.511968\pi$
$878$	51.5063	1.73825
$879$	0	0
$880$	−82.8758	−2.79374
$881$	9.40194	0.316759	0.158380	−	0.987378i	$-0.449373\pi$
$881$	9.40194	0.316759	0.158380	+	0.987378i	$0.449373\pi$
$882$	0	0
$883$	17.4194	0.586209	0.293105	−	0.956080i	$-0.405312\pi$
$883$	17.4194	0.586209	0.293105	+	0.956080i	$0.405312\pi$
$884$	−32.7475	−1.10142
$885$	0	0
$886$	−102.642	−3.44831
$887$	−37.1034	−1.24581	−0.622905	−	0.782297i	$-0.714048\pi$
$887$	−37.1034	−1.24581	−0.622905	+	0.782297i	$0.714048\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	35.2298	1.18090
$891$	0	0
$892$	−144.106	−4.82502
$893$	0.721696	0.0241506
$894$	0	0
$895$	11.9035	0.397889
$896$	−51.5954	−1.72368
$897$	0	0
$898$	49.2783	1.64444
$899$	−1.03768	−0.0346085
$900$	0	0
$901$	23.0620	0.768307
$902$	60.9491	2.02938
$903$	0	0
$904$	110.659	3.68046
$905$	3.79257	0.126069
$906$	0	0
$907$	27.9132	0.926843	0.463422	−	0.886138i	$-0.346622\pi$
$907$	27.9132	0.926843	0.463422	+	0.886138i	$0.346622\pi$
$908$	123.681	4.10450
$909$	0	0
$910$	−6.94373	−0.230183
$911$	8.65606	0.286788	0.143394	−	0.989666i	$-0.454198\pi$
$911$	8.65606	0.286788	0.143394	+	0.989666i	$0.454198\pi$
$912$	0	0
$913$	−11.4641	−0.379405
$914$	56.0929	1.85539
$915$	0	0
$916$	−108.511	−3.58530
$917$	−1.65303	−0.0545878
$918$	0	0
$919$	23.2094	0.765608	0.382804	−	0.923830i	$-0.374958\pi$
$919$	23.2094	0.765608	0.382804	+	0.923830i	$0.374958\pi$
$920$	−14.0303	−0.462564
$921$	0	0
$922$	11.2227	0.369601
$923$	−11.2343	−0.369783
$924$	0	0
$925$	8.55538	0.281299
$926$	−61.4229	−2.01848
$927$	0	0
$928$	−179.064	−5.87807
$929$	48.5051	1.59140	0.795700	−	0.605691i	$-0.207103\pi$
$929$	48.5051	1.59140	0.795700	+	0.605691i	$0.207103\pi$
$930$	0	0
$931$	2.08583	0.0683605
$932$	35.3915	1.15929
$933$	0	0
$934$	−58.5028	−1.91427
$935$	29.1966	0.954832
$936$	0	0
$937$	−58.4810	−1.91049	−0.955245	−	0.295816i	$-0.904408\pi$
$937$	−58.4810	−1.91049	−0.955245	+	0.295816i	$0.904408\pi$
$938$	31.7737	1.03745
$939$	0	0
$940$	4.86054	0.158533
$941$	−41.4180	−1.35019	−0.675093	−	0.737732i	$-0.735897\pi$
$941$	−41.4180	−1.35019	−0.675093	+	0.737732i	$0.735897\pi$
$942$	0	0
$943$	6.12010	0.199298
$944$	207.042	6.73865
$945$	0	0
$946$	50.5513	1.64356
$947$	−4.25540	−0.138282	−0.0691409	−	0.997607i	$-0.522026\pi$
$947$	−4.25540	−0.138282	−0.0691409	+	0.997607i	$0.522026\pi$
$948$	0	0
$949$	−9.83560	−0.319277
$950$	−7.10584	−0.230544
$951$	0	0
$952$	57.8805	1.87592
$953$	−38.6431	−1.25177	−0.625886	−	0.779915i	$-0.715262\pi$
$953$	−38.6431	−1.25177	−0.625886	+	0.779915i	$0.715262\pi$
$954$	0	0
$955$	16.6135	0.537601
$956$	146.422	4.73564
$957$	0	0
$958$	−34.3109	−1.10853
$959$	−18.0556	−0.583045
$960$	0	0
$961$	−30.9785	−0.999308
$962$	−19.2882	−0.621876
$963$	0	0
$964$	−8.48179	−0.273180
$965$	4.30195	0.138485
$966$	0	0
$967$	−11.3857	−0.366140	−0.183070	−	0.983100i	$-0.558603\pi$
$967$	−11.3857	−0.366140	−0.183070	+	0.983100i	$0.558603\pi$
$968$	69.1619	2.22295
$969$	0	0
$970$	−119.336	−3.83164
$971$	23.0973	0.741226	0.370613	−	0.928787i	$-0.379148\pi$
$971$	23.0973	0.741226	0.370613	+	0.928787i	$0.379148\pi$
$972$	0	0
$973$	−10.6622	−0.341815
$974$	5.44161	0.174360
$975$	0	0
$976$	−28.7400	−0.919944
$977$	5.74259	0.183722	0.0918609	−	0.995772i	$-0.470718\pi$
$977$	5.74259	0.183722	0.0918609	+	0.995772i	$0.470718\pi$
$978$	0	0
$979$	10.3386	0.330424
$980$	14.0479	0.448742
$981$	0	0
$982$	57.7625	1.84328
$983$	0.0836981	0.00266956	0.00133478	−	0.999999i	$-0.499575\pi$
$983$	0.0836981	0.00266956	0.00133478	+	0.999999i	$0.499575\pi$
$984$	0	0
$985$	40.9825	1.30581
$986$	113.079	3.60116
$987$	0	0
$988$	11.8190	0.376013
$989$	5.07602	0.161408
$990$	0	0
$991$	−48.0556	−1.52654	−0.763268	−	0.646082i	$-0.776406\pi$
$991$	−48.0556	−1.52654	−0.763268	+	0.646082i	$0.776406\pi$
$992$	3.70160	0.117526
$993$	0	0
$994$	30.8071	0.977142
$995$	24.0604	0.762765
$996$	0	0
$997$	32.9825	1.04457	0.522283	−	0.852772i	$-0.325080\pi$
$997$	32.9825	1.04457	0.522283	+	0.852772i	$0.325080\pi$
$998$	−42.1628	−1.33464
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.1	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.1	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.40	yes	40	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)

\(f(q)\)	\(=\)	\(q-2.76162 q^{2} +5.62653 q^{4} +2.49672 q^{5} +1.00000 q^{7} -10.0151 q^{8} +O(q^{10})\)	\(q-2.76162 q^{2} +5.62653 q^{4} +2.49672 q^{5} +1.00000 q^{7} -10.0151 q^{8} -6.89498 q^{10} -2.02342 q^{11} +1.00707 q^{13} -2.76162 q^{14} +16.4048 q^{16} -5.77932 q^{17} +2.08583 q^{19} +14.0479 q^{20} +5.58792 q^{22} +0.561101 q^{23} +1.23359 q^{25} -2.78115 q^{26} +5.62653 q^{28} +7.08501 q^{29} -0.146461 q^{31} -25.2737 q^{32} +15.9603 q^{34} +2.49672 q^{35} +6.93534 q^{37} -5.76028 q^{38} -25.0049 q^{40} +10.9073 q^{41} +9.04653 q^{43} -11.3849 q^{44} -1.54955 q^{46} +0.345998 q^{47} +1.00000 q^{49} -3.40671 q^{50} +5.66632 q^{52} -3.99044 q^{53} -5.05191 q^{55} -10.0151 q^{56} -19.5661 q^{58} +12.6208 q^{59} -1.75192 q^{61} +0.404469 q^{62} +36.9865 q^{64} +2.51437 q^{65} -11.5055 q^{67} -32.5175 q^{68} -6.89498 q^{70} -11.1554 q^{71} -9.76654 q^{73} -19.1528 q^{74} +11.7360 q^{76} -2.02342 q^{77} +13.1891 q^{79} +40.9582 q^{80} -30.1218 q^{82} +5.66568 q^{83} -14.4293 q^{85} -24.9831 q^{86} +20.2648 q^{88} -5.10948 q^{89} +1.00707 q^{91} +3.15706 q^{92} -0.955516 q^{94} +5.20774 q^{95} +17.3076 q^{97} -2.76162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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