LMFDB - Embedded newform 6000.2.a.bl.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6000,2,Mod(1,6000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,3,0,4,0,-3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.47025.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 13x + 109 $ x^4 - x^3 - 22x^2 + 13x + 109
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.11349$ of defining polynomial
Character	$\chi$	$=$	6000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.73152 q^{7} +1.00000 q^{9} +4.34955 q^{11} +2.30621 q^{13} +0.618034 q^{17} +4.41969 q^{19} +4.73152 q^{21} +6.34955 q^{23} +1.00000 q^{27} +7.54227 q^{29} -4.80166 q^{31} +4.34955 q^{33} -10.9620 q^{37} +2.30621 q^{39} -5.20366 q^{41} +6.85410 q^{43} -0.165930 q^{47} +15.3873 q^{49} +0.618034 q^{51} -2.16031 q^{53} +4.41969 q^{57} -12.9387 q^{59} +4.11911 q^{61} +4.73152 q^{63} -14.6611 q^{67} +6.34955 q^{69} +4.92424 q^{71} -7.91514 q^{73} +20.5800 q^{77} -17.0578 q^{79} +1.00000 q^{81} -5.72242 q^{83} +7.54227 q^{87} +0.816376 q^{89} +10.9119 q^{91} -4.80166 q^{93} -7.20366 q^{97} +4.34955 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} - 3 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{21} + 5 q^{23} + 4 q^{27} + 15 q^{29} - 6 q^{31} - 3 q^{33} - 11 q^{37} + 3 q^{39} + 13 q^{41} + 14 q^{43} + 11 q^{47} + 19 q^{49}+ \cdots - 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 - 3 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^21 + 5 * q^23 + 4 * q^27 + 15 * q^29 - 6 * q^31 - 3 * q^33 - 11 * q^37 + 3 * q^39 + 13 * q^41 + 14 * q^43 + 11 * q^47 + 19 * q^49 - 2 * q^51 + 11 * q^53 - 8 * q^59 + 13 * q^61 + 3 * q^63 + 15 * q^67 + 5 * q^69 + 9 * q^71 - 7 * q^73 + 45 * q^77 - 21 * q^79 + 4 * q^81 + 7 * q^83 + 15 * q^87 + 12 * q^89 - 17 * q^91 - 6 * q^93 + 5 * q^97 - 3 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.73152	1.78835	0.894173	−	0.447721i	$-0.147764\pi$
$7$	4.73152	1.78835	0.894173	+	0.447721i	$0.147764\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.34955	1.31144	0.655720	−	0.755004i	$-0.272365\pi$
$11$	4.34955	1.31144	0.655720	+	0.755004i	$0.272365\pi$
$12$	0	0
$13$	2.30621	0.639626	0.319813	−	0.947481i	$-0.396380\pi$
$13$	2.30621	0.639626	0.319813	+	0.947481i	$0.396380\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.618034	0.149895	0.0749476	−	0.997187i	$-0.476121\pi$
$17$	0.618034	0.149895	0.0749476	+	0.997187i	$0.476121\pi$
$18$	0	0
$19$	4.41969	1.01395	0.506973	−	0.861962i	$-0.330764\pi$
$19$	4.41969	1.01395	0.506973	+	0.861962i	$0.330764\pi$
$20$	0	0
$21$	4.73152	1.03250
$22$	0	0
$23$	6.34955	1.32397	0.661987	−	0.749516i	$-0.269714\pi$
$23$	6.34955	1.32397	0.661987	+	0.749516i	$0.269714\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.54227	1.40057	0.700283	−	0.713866i	$-0.253057\pi$
$29$	7.54227	1.40057	0.700283	+	0.713866i	$0.253057\pi$
$30$	0	0
$31$	−4.80166	−0.862403	−0.431202	−	0.902256i	$-0.641910\pi$
$31$	−4.80166	−0.862403	−0.431202	+	0.902256i	$0.641910\pi$
$32$	0	0
$33$	4.34955	0.757160
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.9620	−1.80214	−0.901068	−	0.433678i	$-0.857216\pi$
$37$	−10.9620	−1.80214	−0.901068	+	0.433678i	$0.857216\pi$
$38$	0	0
$39$	2.30621	0.369289
$40$	0	0
$41$	−5.20366	−0.812674	−0.406337	−	0.913723i	$-0.633194\pi$
$41$	−5.20366	−0.812674	−0.406337	+	0.913723i	$0.633194\pi$
$42$	0	0
$43$	6.85410	1.04524	0.522620	−	0.852566i	$-0.324955\pi$
$43$	6.85410	1.04524	0.522620	+	0.852566i	$0.324955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.165930	−0.0242034	−0.0121017	−	0.999927i	$-0.503852\pi$
$47$	−0.165930	−0.0242034	−0.0121017	+	0.999927i	$0.503852\pi$
$48$	0	0
$49$	15.3873	2.19818
$50$	0	0
$51$	0.618034	0.0865421
$52$	0	0
$53$	−2.16031	−0.296741	−0.148371	−	0.988932i	$-0.547403\pi$
$53$	−2.16031	−0.296741	−0.148371	+	0.988932i	$0.547403\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.41969	0.585402
$58$	0	0
$59$	−12.9387	−1.68447	−0.842234	−	0.539112i	$-0.818760\pi$
$59$	−12.9387	−1.68447	−0.842234	+	0.539112i	$0.818760\pi$
$60$	0	0
$61$	4.11911	0.527398	0.263699	−	0.964605i	$-0.415057\pi$
$61$	4.11911	0.527398	0.263699	+	0.964605i	$0.415057\pi$
$62$	0	0
$63$	4.73152	0.596115
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.6611	−1.79114	−0.895568	−	0.444925i	$-0.853230\pi$
$67$	−14.6611	−1.79114	−0.895568	+	0.444925i	$0.853230\pi$
$68$	0	0
$69$	6.34955	0.764396
$70$	0	0
$71$	4.92424	0.584400	0.292200	−	0.956357i	$-0.405613\pi$
$71$	4.92424	0.584400	0.292200	+	0.956357i	$0.405613\pi$
$72$	0	0
$73$	−7.91514	−0.926398	−0.463199	−	0.886254i	$-0.653298\pi$
$73$	−7.91514	−0.926398	−0.463199	+	0.886254i	$0.653298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.5800	2.34531
$78$	0	0
$79$	−17.0578	−1.91915	−0.959574	−	0.281457i	$-0.909182\pi$
$79$	−17.0578	−1.91915	−0.959574	+	0.281457i	$0.909182\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.72242	−0.628118	−0.314059	−	0.949404i	$-0.601689\pi$
$83$	−5.72242	−0.628118	−0.314059	+	0.949404i	$0.601689\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.54227	0.808617
$88$	0	0
$89$	0.816376	0.0865357	0.0432678	−	0.999064i	$-0.486223\pi$
$89$	0.816376	0.0865357	0.0432678	+	0.999064i	$0.486223\pi$
$90$	0	0
$91$	10.9119	1.14387
$92$	0	0
$93$	−4.80166	−0.497909
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.20366	−0.731420	−0.365710	−	0.930729i	$-0.619174\pi$
$97$	−7.20366	−0.731420	−0.365710	+	0.930729i	$0.619174\pi$
$98$	0	0
$99$	4.34955	0.437147
$100$	0	0
$101$	−8.47745	−0.843538	−0.421769	−	0.906703i	$-0.638591\pi$
$101$	−8.47745	−0.843538	−0.421769	+	0.906703i	$0.638591\pi$
$102$	0	0
$103$	1.93334	0.190497	0.0952486	−	0.995454i	$-0.469635\pi$
$103$	1.93334	0.190497	0.0952486	+	0.995454i	$0.469635\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.6324	−1.22122	−0.610612	−	0.791930i	$-0.709077\pi$
$107$	−12.6324	−1.22122	−0.610612	+	0.791930i	$0.709077\pi$
$108$	0	0
$109$	−3.79604	−0.363594	−0.181797	−	0.983336i	$-0.558191\pi$
$109$	−3.79604	−0.363594	−0.181797	+	0.983336i	$0.558191\pi$
$110$	0	0
$111$	−10.9620	−1.04046
$112$	0	0
$113$	16.7692	1.57752	0.788759	−	0.614703i	$-0.210724\pi$
$113$	16.7692	1.57752	0.788759	+	0.614703i	$0.210724\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.30621	0.213209
$118$	0	0
$119$	2.92424	0.268065
$120$	0	0
$121$	7.91862	0.719874
$122$	0	0
$123$	−5.20366	−0.469198
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.89183	−0.611551	−0.305775	−	0.952104i	$-0.598916\pi$
$127$	−6.89183	−0.611551	−0.305775	+	0.952104i	$0.598916\pi$
$128$	0	0
$129$	6.85410	0.603470
$130$	0	0
$131$	16.2882	1.42311	0.711553	−	0.702632i	$-0.247992\pi$
$131$	16.2882	1.42311	0.711553	+	0.702632i	$0.247992\pi$
$132$	0	0
$133$	20.9119	1.81329
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.641350	0.0547942	0.0273971	−	0.999625i	$-0.491278\pi$
$137$	0.641350	0.0547942	0.0273971	+	0.999625i	$0.491278\pi$
$138$	0	0
$139$	−16.5133	−1.40064	−0.700321	−	0.713828i	$-0.746960\pi$
$139$	−16.5133	−1.40064	−0.700321	+	0.713828i	$0.746960\pi$
$140$	0	0
$141$	−0.165930	−0.0139738
$142$	0	0
$143$	10.0310	0.838832
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.3873	1.26912
$148$	0	0
$149$	−5.88273	−0.481932	−0.240966	−	0.970534i	$-0.577464\pi$
$149$	−5.88273	−0.481932	−0.240966	+	0.970534i	$0.577464\pi$
$150$	0	0
$151$	−18.2971	−1.48900	−0.744499	−	0.667624i	$-0.767312\pi$
$151$	−18.2971	−1.48900	−0.744499	+	0.667624i	$0.767312\pi$
$152$	0	0
$153$	0.618034	0.0499651
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.37868	−0.110031	−0.0550154	−	0.998486i	$-0.517521\pi$
$157$	−1.37868	−0.110031	−0.0550154	+	0.998486i	$0.517521\pi$
$158$	0	0
$159$	−2.16031	−0.171324
$160$	0	0
$161$	30.0430	2.36772
$162$	0	0
$163$	0.598002	0.0468391	0.0234196	−	0.999726i	$-0.492545\pi$
$163$	0.598002	0.0468391	0.0234196	+	0.999726i	$0.492545\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.06104	0.236871	0.118435	−	0.992962i	$-0.462212\pi$
$167$	3.06104	0.236871	0.118435	+	0.992962i	$0.462212\pi$
$168$	0	0
$169$	−7.68141	−0.590878
$170$	0	0
$171$	4.41969	0.337982
$172$	0	0
$173$	11.2685	0.856727	0.428363	−	0.903607i	$-0.359090\pi$
$173$	11.2685	0.856727	0.428363	+	0.903607i	$0.359090\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.9387	−0.972528
$178$	0	0
$179$	−19.9876	−1.49394	−0.746972	−	0.664855i	$-0.768493\pi$
$179$	−19.9876	−1.49394	−0.746972	+	0.664855i	$0.768493\pi$
$180$	0	0
$181$	−7.61456	−0.565986	−0.282993	−	0.959122i	$-0.591327\pi$
$181$	−7.61456	−0.565986	−0.282993	+	0.959122i	$0.591327\pi$
$182$	0	0
$183$	4.11911	0.304493
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.68817	0.196579
$188$	0	0
$189$	4.73152	0.344167
$190$	0	0
$191$	−19.3458	−1.39981	−0.699905	−	0.714236i	$-0.746775\pi$
$191$	−19.3458	−1.39981	−0.699905	+	0.714236i	$0.746775\pi$
$192$	0	0
$193$	−0.863008	−0.0621207	−0.0310603	−	0.999518i	$-0.509888\pi$
$193$	−0.863008	−0.0621207	−0.0310603	+	0.999518i	$0.509888\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0524	1.14369	0.571845	−	0.820362i	$-0.306228\pi$
$197$	16.0524	1.14369	0.571845	+	0.820362i	$0.306228\pi$
$198$	0	0
$199$	−5.30835	−0.376299	−0.188150	−	0.982140i	$-0.560249\pi$
$199$	−5.30835	−0.376299	−0.188150	+	0.982140i	$0.560249\pi$
$200$	0	0
$201$	−14.6611	−1.03411
$202$	0	0
$203$	35.6864	2.50470
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.34955	0.441324
$208$	0	0
$209$	19.2237	1.32973
$210$	0	0
$211$	25.0952	1.72762	0.863812	−	0.503815i	$-0.168071\pi$
$211$	25.0952	1.72762	0.863812	+	0.503815i	$0.168071\pi$
$212$	0	0
$213$	4.92424	0.337403
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−22.7191	−1.54228
$218$	0	0
$219$	−7.91514	−0.534856
$220$	0	0
$221$	1.42531	0.0958770
$222$	0	0
$223$	−2.75155	−0.184258	−0.0921288	−	0.995747i	$-0.529367\pi$
$223$	−2.75155	−0.184258	−0.0921288	+	0.995747i	$0.529367\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2157	0.678042	0.339021	−	0.940779i	$-0.389904\pi$
$227$	10.2157	0.678042	0.339021	+	0.940779i	$0.389904\pi$
$228$	0	0
$229$	−15.2179	−1.00563	−0.502813	−	0.864395i	$-0.667701\pi$
$229$	−15.2179	−1.00563	−0.502813	+	0.864395i	$0.667701\pi$
$230$	0	0
$231$	20.5800	1.35406
$232$	0	0
$233$	0.443506	0.0290551	0.0145275	−	0.999894i	$-0.495376\pi$
$233$	0.443506	0.0290551	0.0145275	+	0.999894i	$0.495376\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.0578	−1.10802
$238$	0	0
$239$	0.749712	0.0484949	0.0242474	−	0.999706i	$-0.492281\pi$
$239$	0.749712	0.0484949	0.0242474	+	0.999706i	$0.492281\pi$
$240$	0	0
$241$	−24.2826	−1.56418	−0.782089	−	0.623166i	$-0.785846\pi$
$241$	−24.2826	−1.56418	−0.782089	+	0.623166i	$0.785846\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.1927	0.648547
$248$	0	0
$249$	−5.72242	−0.362644
$250$	0	0
$251$	−19.0377	−1.20165	−0.600825	−	0.799380i	$-0.705161\pi$
$251$	−19.0377	−1.20165	−0.600825	+	0.799380i	$0.705161\pi$
$252$	0	0
$253$	27.6177	1.73631
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.4662	−0.839999	−0.419999	−	0.907524i	$-0.637970\pi$
$257$	−13.4662	−0.839999	−0.419999	+	0.907524i	$0.637970\pi$
$258$	0	0
$259$	−51.8668	−3.22284
$260$	0	0
$261$	7.54227	0.466855
$262$	0	0
$263$	−23.6844	−1.46044	−0.730221	−	0.683211i	$-0.760583\pi$
$263$	−23.6844	−1.46044	−0.730221	+	0.683211i	$0.760583\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.816376	0.0499614
$268$	0	0
$269$	22.6319	1.37989	0.689947	−	0.723860i	$-0.257634\pi$
$269$	22.6319	1.37989	0.689947	+	0.723860i	$0.257634\pi$
$270$	0	0
$271$	4.32043	0.262447	0.131224	−	0.991353i	$-0.458109\pi$
$271$	4.32043	0.262447	0.131224	+	0.991353i	$0.458109\pi$
$272$	0	0
$273$	10.9119	0.660416
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.69942	0.582781	0.291391	−	0.956604i	$-0.405882\pi$
$277$	9.69942	0.582781	0.291391	+	0.956604i	$0.405882\pi$
$278$	0	0
$279$	−4.80166	−0.287468
$280$	0	0
$281$	−1.61272	−0.0962068	−0.0481034	−	0.998842i	$-0.515318\pi$
$281$	−1.61272	−0.0962068	−0.0481034	+	0.998842i	$0.515318\pi$
$282$	0	0
$283$	21.6723	1.28829	0.644143	−	0.764905i	$-0.277214\pi$
$283$	21.6723	1.28829	0.644143	+	0.764905i	$0.277214\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.6212	−1.45334
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	−7.20366	−0.422286
$292$	0	0
$293$	24.2270	1.41535	0.707677	−	0.706536i	$-0.249743\pi$
$293$	24.2270	1.41535	0.707677	+	0.706536i	$0.249743\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.34955	0.252387
$298$	0	0
$299$	14.6434	0.846849
$300$	0	0
$301$	32.4303	1.86925
$302$	0	0
$303$	−8.47745	−0.487017
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.59472	0.490526	0.245263	−	0.969457i	$-0.421126\pi$
$307$	8.59472	0.490526	0.245263	+	0.969457i	$0.421126\pi$
$308$	0	0
$309$	1.93334	0.109984
$310$	0	0
$311$	−25.0933	−1.42291	−0.711456	−	0.702730i	$-0.751964\pi$
$311$	−25.0933	−1.42291	−0.711456	+	0.702730i	$0.751964\pi$
$312$	0	0
$313$	−0.399660	−0.0225901	−0.0112951	−	0.999936i	$-0.503595\pi$
$313$	−0.399660	−0.0225901	−0.0112951	+	0.999936i	$0.503595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.0722	1.52052	0.760262	−	0.649616i	$-0.225070\pi$
$317$	27.0722	1.52052	0.760262	+	0.649616i	$0.225070\pi$
$318$	0	0
$319$	32.8055	1.83676
$320$	0	0
$321$	−12.6324	−0.705074
$322$	0	0
$323$	2.73152	0.151986
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.79604	−0.209921
$328$	0	0
$329$	−0.785101	−0.0432840
$330$	0	0
$331$	7.41457	0.407542	0.203771	−	0.979019i	$-0.434680\pi$
$331$	7.41457	0.407542	0.203771	+	0.979019i	$0.434680\pi$
$332$	0	0
$333$	−10.9620	−0.600712
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.24731	−0.503733	−0.251867	−	0.967762i	$-0.581044\pi$
$337$	−9.24731	−0.503733	−0.251867	+	0.967762i	$0.581044\pi$
$338$	0	0
$339$	16.7692	0.910780
$340$	0	0
$341$	−20.8851	−1.13099
$342$	0	0
$343$	39.6846	2.14277
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.0285	−1.66570	−0.832849	−	0.553500i	$-0.813292\pi$
$347$	−31.0285	−1.66570	−0.832849	+	0.553500i	$0.813292\pi$
$348$	0	0
$349$	29.5939	1.58413	0.792063	−	0.610440i	$-0.209007\pi$
$349$	29.5939	1.58413	0.792063	+	0.610440i	$0.209007\pi$
$350$	0	0
$351$	2.30621	0.123096
$352$	0	0
$353$	16.8771	0.898278	0.449139	−	0.893462i	$-0.351731\pi$
$353$	16.8771	0.898278	0.449139	+	0.893462i	$0.351731\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.92424	0.154767
$358$	0	0
$359$	35.6788	1.88305	0.941527	−	0.336938i	$-0.109391\pi$
$359$	35.6788	1.88305	0.941527	+	0.336938i	$0.109391\pi$
$360$	0	0
$361$	0.533676	0.0280882
$362$	0	0
$363$	7.91862	0.415620
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.4070	1.95263	0.976315	−	0.216355i	$-0.0694167\pi$
$367$	37.4070	1.95263	0.976315	+	0.216355i	$0.0694167\pi$
$368$	0	0
$369$	−5.20366	−0.270891
$370$	0	0
$371$	−10.2215	−0.530676
$372$	0	0
$373$	19.0195	0.984794	0.492397	−	0.870371i	$-0.336121\pi$
$373$	19.0195	0.984794	0.492397	+	0.870371i	$0.336121\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.3940	0.895839
$378$	0	0
$379$	6.08967	0.312805	0.156403	−	0.987693i	$-0.450010\pi$
$379$	6.08967	0.312805	0.156403	+	0.987693i	$0.450010\pi$
$380$	0	0
$381$	−6.89183	−0.353079
$382$	0	0
$383$	23.1458	1.18269	0.591347	−	0.806417i	$-0.298596\pi$
$383$	23.1458	1.18269	0.591347	+	0.806417i	$0.298596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.85410	0.348414
$388$	0	0
$389$	0.133518	0.00676961	0.00338481	−	0.999994i	$-0.498923\pi$
$389$	0.133518	0.00676961	0.00338481	+	0.999994i	$0.498923\pi$
$390$	0	0
$391$	3.92424	0.198457
$392$	0	0
$393$	16.2882	0.821631
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.3115	1.47110	0.735552	−	0.677469i	$-0.236923\pi$
$397$	29.3115	1.47110	0.735552	+	0.677469i	$0.236923\pi$
$398$	0	0
$399$	20.9119	1.04690
$400$	0	0
$401$	5.08998	0.254181	0.127091	−	0.991891i	$-0.459436\pi$
$401$	5.08998	0.254181	0.127091	+	0.991891i	$0.459436\pi$
$402$	0	0
$403$	−11.0736	−0.551616
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−47.6797	−2.36339
$408$	0	0
$409$	−23.3817	−1.15615	−0.578074	−	0.815984i	$-0.696196\pi$
$409$	−23.3817	−1.15615	−0.578074	+	0.815984i	$0.696196\pi$
$410$	0	0
$411$	0.641350	0.0316355
$412$	0	0
$413$	−61.2195	−3.01241
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.5133	−0.808662
$418$	0	0
$419$	27.7571	1.35602	0.678010	−	0.735052i	$-0.262842\pi$
$419$	27.7571	1.35602	0.678010	+	0.735052i	$0.262842\pi$
$420$	0	0
$421$	−26.3672	−1.28506	−0.642531	−	0.766260i	$-0.722115\pi$
$421$	−26.3672	−1.28506	−0.642531	+	0.766260i	$0.722115\pi$
$422$	0	0
$423$	−0.165930	−0.00806779
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.4896	0.943170
$428$	0	0
$429$	10.0310	0.484300
$430$	0	0
$431$	24.7982	1.19449	0.597243	−	0.802060i	$-0.296263\pi$
$431$	24.7982	1.19449	0.597243	+	0.802060i	$0.296263\pi$
$432$	0	0
$433$	19.7483	0.949041	0.474521	−	0.880244i	$-0.342621\pi$
$433$	19.7483	0.949041	0.474521	+	0.880244i	$0.342621\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.0631	1.34244
$438$	0	0
$439$	−20.8247	−0.993907	−0.496954	−	0.867777i	$-0.665548\pi$
$439$	−20.8247	−0.993907	−0.496954	+	0.867777i	$0.665548\pi$
$440$	0	0
$441$	15.3873	0.732728
$442$	0	0
$443$	−15.2321	−0.723699	−0.361849	−	0.932237i	$-0.617855\pi$
$443$	−15.2321	−0.723699	−0.361849	+	0.932237i	$0.617855\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.88273	−0.278244
$448$	0	0
$449$	14.7545	0.696309	0.348155	−	0.937437i	$-0.386808\pi$
$449$	14.7545	0.696309	0.348155	+	0.937437i	$0.386808\pi$
$450$	0	0
$451$	−22.6336	−1.06577
$452$	0	0
$453$	−18.2971	−0.859673
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.4889	−1.19232	−0.596160	−	0.802866i	$-0.703308\pi$
$457$	−25.4889	−1.19232	−0.596160	+	0.802866i	$0.703308\pi$
$458$	0	0
$459$	0.618034	0.0288474
$460$	0	0
$461$	12.2708	0.571509	0.285754	−	0.958303i	$-0.407756\pi$
$461$	12.2708	0.571509	0.285754	+	0.958303i	$0.407756\pi$
$462$	0	0
$463$	12.7782	0.593851	0.296926	−	0.954901i	$-0.404039\pi$
$463$	12.7782	0.593851	0.296926	+	0.954901i	$0.404039\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.46335	−0.113990	−0.0569951	−	0.998374i	$-0.518152\pi$
$467$	−2.46335	−0.113990	−0.0569951	+	0.998374i	$0.518152\pi$
$468$	0	0
$469$	−69.3692	−3.20317
$470$	0	0
$471$	−1.37868	−0.0635263
$472$	0	0
$473$	29.8123	1.37077
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.16031	−0.0989137
$478$	0	0
$479$	−12.9330	−0.590925	−0.295463	−	0.955354i	$-0.595474\pi$
$479$	−12.9330	−0.590925	−0.295463	+	0.955354i	$0.595474\pi$
$480$	0	0
$481$	−25.2806	−1.15269
$482$	0	0
$483$	30.0430	1.36701
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−28.1155	−1.27404	−0.637018	−	0.770849i	$-0.719832\pi$
$487$	−28.1155	−1.27404	−0.637018	+	0.770849i	$0.719832\pi$
$488$	0	0
$489$	0.598002	0.0270426
$490$	0	0
$491$	−3.86320	−0.174344	−0.0871718	−	0.996193i	$-0.527783\pi$
$491$	−3.86320	−0.174344	−0.0871718	+	0.996193i	$0.527783\pi$
$492$	0	0
$493$	4.66138	0.209938
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.2991	1.04511
$498$	0	0
$499$	14.6000	0.653587	0.326794	−	0.945096i	$-0.394032\pi$
$499$	14.6000	0.653587	0.326794	+	0.945096i	$0.394032\pi$
$500$	0	0
$501$	3.06104	0.136757
$502$	0	0
$503$	2.55699	0.114011	0.0570053	−	0.998374i	$-0.481845\pi$
$503$	2.55699	0.114011	0.0570053	+	0.998374i	$0.481845\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.68141	−0.341144
$508$	0	0
$509$	32.4897	1.44008	0.720041	−	0.693932i	$-0.244123\pi$
$509$	32.4897	1.44008	0.720041	+	0.693932i	$0.244123\pi$
$510$	0	0
$511$	−37.4507	−1.65672
$512$	0	0
$513$	4.41969	0.195134
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.721721	−0.0317413
$518$	0	0
$519$	11.2685	0.494631
$520$	0	0
$521$	13.1913	0.577920	0.288960	−	0.957341i	$-0.406690\pi$
$521$	13.1913	0.577920	0.288960	+	0.957341i	$0.406690\pi$
$522$	0	0
$523$	−10.9416	−0.478444	−0.239222	−	0.970965i	$-0.576892\pi$
$523$	−10.9416	−0.478444	−0.239222	+	0.970965i	$0.576892\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.96759	−0.129270
$528$	0	0
$529$	17.3168	0.752906
$530$	0	0
$531$	−12.9387	−0.561490
$532$	0	0
$533$	−12.0007	−0.519808
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.9876	−0.862529
$538$	0	0
$539$	66.9278	2.88278
$540$	0	0
$541$	−22.1946	−0.954219	−0.477109	−	0.878844i	$-0.658315\pi$
$541$	−22.1946	−0.954219	−0.477109	+	0.878844i	$0.658315\pi$
$542$	0	0
$543$	−7.61456	−0.326772
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.2057	1.24874	0.624372	−	0.781127i	$-0.285355\pi$
$547$	29.2057	1.24874	0.624372	+	0.781127i	$0.285355\pi$
$548$	0	0
$549$	4.11911	0.175799
$550$	0	0
$551$	33.3345	1.42010
$552$	0	0
$553$	−80.7091	−3.43210
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.3043	−0.945062	−0.472531	−	0.881314i	$-0.656660\pi$
$557$	−22.3043	−0.945062	−0.472531	+	0.881314i	$0.656660\pi$
$558$	0	0
$559$	15.8070	0.668564
$560$	0	0
$561$	2.68817	0.113495
$562$	0	0
$563$	−10.0736	−0.424552	−0.212276	−	0.977210i	$-0.568088\pi$
$563$	−10.0736	−0.424552	−0.212276	+	0.977210i	$0.568088\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.73152	0.198705
$568$	0	0
$569$	−36.3949	−1.52575	−0.762877	−	0.646543i	$-0.776214\pi$
$569$	−36.3949	−1.52575	−0.762877	+	0.646543i	$0.776214\pi$
$570$	0	0
$571$	−37.9546	−1.58835	−0.794175	−	0.607689i	$-0.792097\pi$
$571$	−37.9546	−1.58835	−0.794175	+	0.607689i	$0.792097\pi$
$572$	0	0
$573$	−19.3458	−0.808181
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.5155	−0.479396	−0.239698	−	0.970848i	$-0.577048\pi$
$577$	−11.5155	−0.479396	−0.239698	+	0.970848i	$0.577048\pi$
$578$	0	0
$579$	−0.863008	−0.0358654
$580$	0	0
$581$	−27.0758	−1.12329
$582$	0	0
$583$	−9.39638	−0.389158
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.39935	0.387953	0.193976	−	0.981006i	$-0.437861\pi$
$587$	9.39935	0.387953	0.193976	+	0.981006i	$0.437861\pi$
$588$	0	0
$589$	−21.2218	−0.874431
$590$	0	0
$591$	16.0524	0.660309
$592$	0	0
$593$	15.0133	0.616521	0.308261	−	0.951302i	$-0.400253\pi$
$593$	15.0133	0.616521	0.308261	+	0.951302i	$0.400253\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.30835	−0.217256
$598$	0	0
$599$	17.4488	0.712939	0.356470	−	0.934307i	$-0.383980\pi$
$599$	17.4488	0.712939	0.356470	+	0.934307i	$0.383980\pi$
$600$	0	0
$601$	11.5390	0.470685	0.235343	−	0.971912i	$-0.424379\pi$
$601$	11.5390	0.470685	0.235343	+	0.971912i	$0.424379\pi$
$602$	0	0
$603$	−14.6611	−0.597045
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.0952	1.83036	0.915178	−	0.403049i	$-0.132050\pi$
$607$	45.0952	1.83036	0.915178	+	0.403049i	$0.132050\pi$
$608$	0	0
$609$	35.6864	1.44609
$610$	0	0
$611$	−0.382669	−0.0154811
$612$	0	0
$613$	0.314804	0.0127148	0.00635740	−	0.999980i	$-0.497976\pi$
$613$	0.314804	0.0127148	0.00635740	+	0.999980i	$0.497976\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.7312	−0.955383	−0.477691	−	0.878528i	$-0.658526\pi$
$617$	−23.7312	−0.955383	−0.477691	+	0.878528i	$0.658526\pi$
$618$	0	0
$619$	−20.7676	−0.834721	−0.417360	−	0.908741i	$-0.637045\pi$
$619$	−20.7676	−0.834721	−0.417360	+	0.908741i	$0.637045\pi$
$620$	0	0
$621$	6.34955	0.254799
$622$	0	0
$623$	3.86270	0.154756
$624$	0	0
$625$	0	0
$626$	0	0
$627$	19.2237	0.767720
$628$	0	0
$629$	−6.77487	−0.270132
$630$	0	0
$631$	6.60925	0.263110	0.131555	−	0.991309i	$-0.458003\pi$
$631$	6.60925	0.263110	0.131555	+	0.991309i	$0.458003\pi$
$632$	0	0
$633$	25.0952	0.997444
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35.4862	1.40602
$638$	0	0
$639$	4.92424	0.194800
$640$	0	0
$641$	−21.0379	−0.830948	−0.415474	−	0.909605i	$-0.636384\pi$
$641$	−21.0379	−0.830948	−0.415474	+	0.909605i	$0.636384\pi$
$642$	0	0
$643$	8.71199	0.343567	0.171784	−	0.985135i	$-0.445047\pi$
$643$	8.71199	0.343567	0.171784	+	0.985135i	$0.445047\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.95318	0.116101	0.0580507	−	0.998314i	$-0.481511\pi$
$647$	2.95318	0.116101	0.0580507	+	0.998314i	$0.481511\pi$
$648$	0	0
$649$	−56.2774	−2.20908
$650$	0	0
$651$	−22.7191	−0.890433
$652$	0	0
$653$	1.84848	0.0723366	0.0361683	−	0.999346i	$-0.488485\pi$
$653$	1.84848	0.0723366	0.0361683	+	0.999346i	$0.488485\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.91514	−0.308799
$658$	0	0
$659$	−16.4035	−0.638991	−0.319495	−	0.947588i	$-0.603513\pi$
$659$	−16.4035	−0.638991	−0.319495	+	0.947588i	$0.603513\pi$
$660$	0	0
$661$	49.0916	1.90944	0.954721	−	0.297504i	$-0.0961541\pi$
$661$	49.0916	1.90944	0.954721	+	0.297504i	$0.0961541\pi$
$662$	0	0
$663$	1.42531	0.0553546
$664$	0	0
$665$	0	0
$666$	0	0
$667$	47.8901	1.85431
$668$	0	0
$669$	−2.75155	−0.106381
$670$	0	0
$671$	17.9163	0.691650
$672$	0	0
$673$	−18.5588	−0.715390	−0.357695	−	0.933838i	$-0.616437\pi$
$673$	−18.5588	−0.715390	−0.357695	+	0.933838i	$0.616437\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.67182	−0.141119	−0.0705597	−	0.997508i	$-0.522479\pi$
$677$	−3.67182	−0.141119	−0.0705597	+	0.997508i	$0.522479\pi$
$678$	0	0
$679$	−34.0842	−1.30803
$680$	0	0
$681$	10.2157	0.391468
$682$	0	0
$683$	35.4795	1.35758	0.678792	−	0.734330i	$-0.262504\pi$
$683$	35.4795	1.35758	0.678792	+	0.734330i	$0.262504\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.2179	−0.580598
$688$	0	0
$689$	−4.98212	−0.189803
$690$	0	0
$691$	34.0164	1.29405	0.647023	−	0.762470i	$-0.276014\pi$
$691$	34.0164	1.29405	0.647023	+	0.762470i	$0.276014\pi$
$692$	0	0
$693$	20.5800	0.781770
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.21604	−0.121816
$698$	0	0
$699$	0.443506	0.0167750
$700$	0	0
$701$	−18.9201	−0.714604	−0.357302	−	0.933989i	$-0.616303\pi$
$701$	−18.9201	−0.714604	−0.357302	+	0.933989i	$0.616303\pi$
$702$	0	0
$703$	−48.4485	−1.82727
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.1112	−1.50854
$708$	0	0
$709$	8.81576	0.331083	0.165541	−	0.986203i	$-0.447063\pi$
$709$	8.81576	0.331083	0.165541	+	0.986203i	$0.447063\pi$
$710$	0	0
$711$	−17.0578	−0.639716
$712$	0	0
$713$	−30.4884	−1.14180
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.749712	0.0279985
$718$	0	0
$719$	4.68734	0.174808	0.0874042	−	0.996173i	$-0.472143\pi$
$719$	4.68734	0.174808	0.0874042	+	0.996173i	$0.472143\pi$
$720$	0	0
$721$	9.14762	0.340675
$722$	0	0
$723$	−24.2826	−0.903079
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−36.5994	−1.35740	−0.678699	−	0.734417i	$-0.737456\pi$
$727$	−36.5994	−1.35740	−0.678699	+	0.734417i	$0.737456\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.23607	0.156677
$732$	0	0
$733$	34.4593	1.27278	0.636391	−	0.771367i	$-0.280427\pi$
$733$	34.4593	1.27278	0.636391	+	0.771367i	$0.280427\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−63.7691	−2.34897
$738$	0	0
$739$	22.8606	0.840939	0.420470	−	0.907307i	$-0.361865\pi$
$739$	22.8606	0.840939	0.420470	+	0.907307i	$0.361865\pi$
$740$	0	0
$741$	10.1927	0.374439
$742$	0	0
$743$	39.5223	1.44993	0.724967	−	0.688784i	$-0.241855\pi$
$743$	39.5223	1.44993	0.724967	+	0.688784i	$0.241855\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.72242	−0.209373
$748$	0	0
$749$	−59.7707	−2.18397
$750$	0	0
$751$	47.3004	1.72601	0.863007	−	0.505191i	$-0.168578\pi$
$751$	47.3004	1.72601	0.863007	+	0.505191i	$0.168578\pi$
$752$	0	0
$753$	−19.0377	−0.693773
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.54392	0.274189	0.137094	−	0.990558i	$-0.456224\pi$
$757$	7.54392	0.274189	0.137094	+	0.990558i	$0.456224\pi$
$758$	0	0
$759$	27.6177	1.00246
$760$	0	0
$761$	−12.7152	−0.460924	−0.230462	−	0.973081i	$-0.574024\pi$
$761$	−12.7152	−0.460924	−0.230462	+	0.973081i	$0.574024\pi$
$762$	0	0
$763$	−17.9610	−0.650233
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29.8392	−1.07743
$768$	0	0
$769$	−49.0399	−1.76843	−0.884213	−	0.467084i	$-0.845304\pi$
$769$	−49.0399	−1.76843	−0.884213	+	0.467084i	$0.845304\pi$
$770$	0	0
$771$	−13.4662	−0.484974
$772$	0	0
$773$	32.9203	1.18406	0.592030	−	0.805916i	$-0.298327\pi$
$773$	32.9203	1.18406	0.592030	+	0.805916i	$0.298327\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−51.8668	−1.86071
$778$	0	0
$779$	−22.9986	−0.824009
$780$	0	0
$781$	21.4182	0.766405
$782$	0	0
$783$	7.54227	0.269539
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−46.8396	−1.66965	−0.834825	−	0.550515i	$-0.814431\pi$
$787$	−46.8396	−1.66965	−0.834825	+	0.550515i	$0.814431\pi$
$788$	0	0
$789$	−23.6844	−0.843187
$790$	0	0
$791$	79.3440	2.82115
$792$	0	0
$793$	9.49951	0.337338
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.94673	0.175222	0.0876110	−	0.996155i	$-0.472077\pi$
$797$	4.94673	0.175222	0.0876110	+	0.996155i	$0.472077\pi$
$798$	0	0
$799$	−0.102550	−0.00362797
$800$	0	0
$801$	0.816376	0.0288452
$802$	0	0
$803$	−34.4273	−1.21491
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.6319	0.796682
$808$	0	0
$809$	−17.6464	−0.620413	−0.310206	−	0.950669i	$-0.600398\pi$
$809$	−17.6464	−0.620413	−0.310206	+	0.950669i	$0.600398\pi$
$810$	0	0
$811$	−25.8503	−0.907727	−0.453864	−	0.891071i	$-0.649955\pi$
$811$	−25.8503	−0.907727	−0.453864	+	0.891071i	$0.649955\pi$
$812$	0	0
$813$	4.32043	0.151524
$814$	0	0
$815$	0	0
$816$	0	0
$817$	30.2930	1.05982
$818$	0	0
$819$	10.9119	0.381291
$820$	0	0
$821$	−6.11563	−0.213437	−0.106719	−	0.994289i	$-0.534034\pi$
$821$	−6.11563	−0.213437	−0.106719	+	0.994289i	$0.534034\pi$
$822$	0	0
$823$	47.8125	1.66664	0.833319	−	0.552792i	$-0.186438\pi$
$823$	47.8125	1.66664	0.833319	+	0.552792i	$0.186438\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.2754	0.461632	0.230816	−	0.972997i	$-0.425860\pi$
$827$	13.2754	0.461632	0.230816	+	0.972997i	$0.425860\pi$
$828$	0	0
$829$	−5.32276	−0.184867	−0.0924336	−	0.995719i	$-0.529465\pi$
$829$	−5.32276	−0.184867	−0.0924336	+	0.995719i	$0.529465\pi$
$830$	0	0
$831$	9.69942	0.336469
$832$	0	0
$833$	9.50986	0.329497
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.80166	−0.165970
$838$	0	0
$839$	28.1680	0.972466	0.486233	−	0.873829i	$-0.338371\pi$
$839$	28.1680	0.972466	0.486233	+	0.873829i	$0.338371\pi$
$840$	0	0
$841$	27.8859	0.961583
$842$	0	0
$843$	−1.61272	−0.0555450
$844$	0	0
$845$	0	0
$846$	0	0
$847$	37.4671	1.28738
$848$	0	0
$849$	21.6723	0.743792
$850$	0	0
$851$	−69.6036	−2.38598
$852$	0	0
$853$	35.0599	1.20043	0.600214	−	0.799839i	$-0.295082\pi$
$853$	35.0599	1.20043	0.600214	+	0.799839i	$0.295082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.2517	1.64825	0.824123	−	0.566410i	$-0.191668\pi$
$857$	48.2517	1.64825	0.824123	+	0.566410i	$0.191668\pi$
$858$	0	0
$859$	−26.1213	−0.891248	−0.445624	−	0.895220i	$-0.647018\pi$
$859$	−26.1213	−0.891248	−0.445624	+	0.895220i	$0.647018\pi$
$860$	0	0
$861$	−24.6212	−0.839088
$862$	0	0
$863$	−40.7860	−1.38837	−0.694186	−	0.719796i	$-0.744235\pi$
$863$	−40.7860	−1.38837	−0.694186	+	0.719796i	$0.744235\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.6180	−0.564378
$868$	0	0
$869$	−74.1936	−2.51685
$870$	0	0
$871$	−33.8115	−1.14566
$872$	0	0
$873$	−7.20366	−0.243807
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.5142	1.26677	0.633383	−	0.773839i	$-0.281666\pi$
$877$	37.5142	1.26677	0.633383	+	0.773839i	$0.281666\pi$
$878$	0	0
$879$	24.2270	0.817155
$880$	0	0
$881$	−23.9170	−0.805784	−0.402892	−	0.915248i	$-0.631995\pi$
$881$	−23.9170	−0.805784	−0.402892	+	0.915248i	$0.631995\pi$
$882$	0	0
$883$	−5.24086	−0.176369	−0.0881845	−	0.996104i	$-0.528107\pi$
$883$	−5.24086	−0.176369	−0.0881845	+	0.996104i	$0.528107\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.6016	0.859616	0.429808	−	0.902920i	$-0.358581\pi$
$887$	25.6016	0.859616	0.429808	+	0.902920i	$0.358581\pi$
$888$	0	0
$889$	−32.6088	−1.09366
$890$	0	0
$891$	4.34955	0.145716
$892$	0	0
$893$	−0.733359	−0.0245409
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.6434	0.488928
$898$	0	0
$899$	−36.2154	−1.20785
$900$	0	0
$901$	−1.33514	−0.0444801
$902$	0	0
$903$	32.4303	1.07921
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−15.2035	−0.504823	−0.252411	−	0.967620i	$-0.581224\pi$
$907$	−15.2035	−0.504823	−0.252411	+	0.967620i	$0.581224\pi$
$908$	0	0
$909$	−8.47745	−0.281179
$910$	0	0
$911$	−2.95849	−0.0980192	−0.0490096	−	0.998798i	$-0.515606\pi$
$911$	−2.95849	−0.0980192	−0.0490096	+	0.998798i	$0.515606\pi$
$912$	0	0
$913$	−24.8900	−0.823738
$914$	0	0
$915$	0	0
$916$	0	0
$917$	77.0680	2.54501
$918$	0	0
$919$	−14.7727	−0.487307	−0.243654	−	0.969862i	$-0.578346\pi$
$919$	−14.7727	−0.487307	−0.243654	+	0.969862i	$0.578346\pi$
$920$	0	0
$921$	8.59472	0.283206
$922$	0	0
$923$	11.3563	0.373798
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.93334	0.0634991
$928$	0	0
$929$	35.8915	1.17756	0.588781	−	0.808292i	$-0.299608\pi$
$929$	35.8915	1.17756	0.588781	+	0.808292i	$0.299608\pi$
$930$	0	0
$931$	68.0070	2.22884
$932$	0	0
$933$	−25.0933	−0.821519
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.1290	−0.722924	−0.361462	−	0.932387i	$-0.617722\pi$
$937$	−22.1290	−0.722924	−0.361462	+	0.932387i	$0.617722\pi$
$938$	0	0
$939$	−0.399660	−0.0130424
$940$	0	0
$941$	45.2047	1.47363	0.736815	−	0.676094i	$-0.236329\pi$
$941$	45.2047	1.47363	0.736815	+	0.676094i	$0.236329\pi$
$942$	0	0
$943$	−33.0409	−1.07596
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.8853	0.938645	0.469322	−	0.883027i	$-0.344498\pi$
$947$	28.8853	0.938645	0.469322	+	0.883027i	$0.344498\pi$
$948$	0	0
$949$	−18.2540	−0.592548
$950$	0	0
$951$	27.0722	0.877875
$952$	0	0
$953$	22.9455	0.743277	0.371639	−	0.928377i	$-0.378796\pi$
$953$	22.9455	0.743277	0.371639	+	0.928377i	$0.378796\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.8055	1.06045
$958$	0	0
$959$	3.03456	0.0979910
$960$	0	0
$961$	−7.94408	−0.256261
$962$	0	0
$963$	−12.6324	−0.407075
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.1830	−1.26004	−0.630020	−	0.776579i	$-0.716953\pi$
$967$	−39.1830	−1.26004	−0.630020	+	0.776579i	$0.716953\pi$
$968$	0	0
$969$	2.73152	0.0877491
$970$	0	0
$971$	−21.3489	−0.685120	−0.342560	−	0.939496i	$-0.611294\pi$
$971$	−21.3489	−0.685120	−0.342560	+	0.939496i	$0.611294\pi$
$972$	0	0
$973$	−78.1332	−2.50483
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.86435	0.219610	0.109805	−	0.993953i	$-0.464977\pi$
$977$	6.86435	0.219610	0.109805	+	0.993953i	$0.464977\pi$
$978$	0	0
$979$	3.55087	0.113486
$980$	0	0
$981$	−3.79604	−0.121198
$982$	0	0
$983$	−19.5704	−0.624199	−0.312099	−	0.950049i	$-0.601032\pi$
$983$	−19.5704	−0.624199	−0.312099	+	0.950049i	$0.601032\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.785101	−0.0249900
$988$	0	0
$989$	43.5205	1.38387
$990$	0	0
$991$	−3.28617	−0.104389	−0.0521944	−	0.998637i	$-0.516622\pi$
$991$	−3.28617	−0.104389	−0.0521944	+	0.998637i	$0.516622\pi$
$992$	0	0
$993$	7.41457	0.235294
$994$	0	0
$995$	0	0
$996$	0	0
$997$	27.1168	0.858797	0.429398	−	0.903115i	$-0.358726\pi$
$997$	27.1168	0.858797	0.429398	+	0.903115i	$0.358726\pi$
$998$	0	0
$999$	−10.9620	−0.346821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.a.bl.1.4		4
4.3	odd	2		3000.2.a.j.1.1	✓	4
5.2	odd	4		6000.2.f.p.1249.4		8
5.3	odd	4		6000.2.f.p.1249.5		8
5.4	even	2		6000.2.a.bc.1.1		4
12.11	even	2		9000.2.a.t.1.1		4
20.3	even	4		3000.2.f.h.1249.4		8
20.7	even	4		3000.2.f.h.1249.5		8
20.19	odd	2		3000.2.a.o.1.4	yes	4
60.59	even	2		9000.2.a.y.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.j.1.1	✓	4	4.3	odd	2
3000.2.a.o.1.4	yes	4	20.19	odd	2
3000.2.f.h.1249.4		8	20.3	even	4
3000.2.f.h.1249.5		8	20.7	even	4
6000.2.a.bc.1.1		4	5.4	even	2
6000.2.a.bl.1.4		4	1.1	even	1	trivial
6000.2.f.p.1249.4		8	5.2	odd	4
6000.2.f.p.1249.5		8	5.3	odd	4
9000.2.a.t.1.1		4	12.11	even	2
9000.2.a.y.1.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.73152 q^{7} +1.00000 q^{9} +4.34955 q^{11} +2.30621 q^{13} +0.618034 q^{17} +4.41969 q^{19} +4.73152 q^{21} +6.34955 q^{23} +1.00000 q^{27} +7.54227 q^{29} -4.80166 q^{31} +4.34955 q^{33} -10.9620 q^{37} +2.30621 q^{39} -5.20366 q^{41} +6.85410 q^{43} -0.165930 q^{47} +15.3873 q^{49} +0.618034 q^{51} -2.16031 q^{53} +4.41969 q^{57} -12.9387 q^{59} +4.11911 q^{61} +4.73152 q^{63} -14.6611 q^{67} +6.34955 q^{69} +4.92424 q^{71} -7.91514 q^{73} +20.5800 q^{77} -17.0578 q^{79} +1.00000 q^{81} -5.72242 q^{83} +7.54227 q^{87} +0.816376 q^{89} +10.9119 q^{91} -4.80166 q^{93} -7.20366 q^{97} +4.34955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} - 3 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{21} + 5 q^{23} + 4 q^{27} + 15 q^{29} - 6 q^{31} - 3 q^{33} - 11 q^{37} + 3 q^{39} + 13 q^{41} + 14 q^{43} + 11 q^{47} + 19 q^{49}+ \cdots - 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 - 3 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^21 + 5 * q^23 + 4 * q^27 + 15 * q^29 - 6 * q^31 - 3 * q^33 - 11 * q^37 + 3 * q^39 + 13 * q^41 + 14 * q^43 + 11 * q^47 + 19 * q^49 - 2 * q^51 + 11 * q^53 - 8 * q^59 + 13 * q^61 + 3 * q^63 + 15 * q^67 + 5 * q^69 + 9 * q^71 - 7 * q^73 + 45 * q^77 - 21 * q^79 + 4 * q^81 + 7 * q^83 + 15 * q^87 + 12 * q^89 - 17 * q^91 - 6 * q^93 + 5 * q^97 - 3 * q^99

Introduction

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