LMFDB - Embedded newform 5520.2.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5520,2,Mod(1,5520)] 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("5520.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2,0,2,0,2,0,8,0,4,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +5.41421 q^{11} +0.585786 q^{13} -1.00000 q^{15} +8.07107 q^{17} -2.24264 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +9.82843 q^{31} +5.41421 q^{33} -3.82843 q^{35} +3.00000 q^{37} +0.585786 q^{39} -7.58579 q^{41} -6.00000 q^{43} -1.00000 q^{45} +11.4142 q^{47} +7.65685 q^{49} +8.07107 q^{51} -1.24264 q^{53} -5.41421 q^{55} -2.24264 q^{57} -12.8995 q^{59} +2.58579 q^{61} +3.82843 q^{63} -0.585786 q^{65} -5.48528 q^{67} +1.00000 q^{69} +10.8995 q^{71} -1.75736 q^{73} +1.00000 q^{75} +20.7279 q^{77} -7.31371 q^{79} +1.00000 q^{81} +0.0710678 q^{83} -8.07107 q^{85} -6.41421 q^{87} -18.1421 q^{89} +2.24264 q^{91} +9.82843 q^{93} +2.24264 q^{95} -6.82843 q^{97} +5.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 14 q^{31} + 8 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 8 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 14 * q^31 + 8 * q^33 - 2 * q^35 + 6 * q^37 + 4 * q^39 - 18 * q^41 - 12 * q^43 - 2 * q^45 + 20 * q^47 + 4 * q^49 + 2 * q^51 + 6 * q^53 - 8 * q^55 + 4 * q^57 - 6 * q^59 + 8 * q^61 + 2 * q^63 - 4 * q^65 + 6 * q^67 + 2 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 16 * q^77 + 8 * q^79 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 8 * q^89 - 4 * q^91 + 14 * q^93 - 4 * q^95 - 8 * q^97 + 8 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.82843	1.44701	0.723505	−	0.690319i	$-0.242530\pi$
$7$	3.82843	1.44701	0.723505	+	0.690319i	$0.242530\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.41421	1.63245	0.816223	−	0.577736i	$-0.196064\pi$
$11$	5.41421	1.63245	0.816223	+	0.577736i	$0.196064\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	8.07107	1.95752	0.978761	−	0.205006i	$-0.0657214\pi$
$17$	8.07107	1.95752	0.978761	+	0.205006i	$0.0657214\pi$
$18$	0	0
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	0	0
$21$	3.82843	0.835431
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.41421	−1.19109	−0.595545	−	0.803322i	$-0.703064\pi$
$29$	−6.41421	−1.19109	−0.595545	+	0.803322i	$0.703064\pi$
$30$	0	0
$31$	9.82843	1.76524	0.882619	−	0.470089i	$-0.155778\pi$
$31$	9.82843	1.76524	0.882619	+	0.470089i	$0.155778\pi$
$32$	0	0
$33$	5.41421	0.942494
$34$	0	0
$35$	−3.82843	−0.647122
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	0.585786	0.0938009
$40$	0	0
$41$	−7.58579	−1.18470	−0.592350	−	0.805680i	$-0.701800\pi$
$41$	−7.58579	−1.18470	−0.592350	+	0.805680i	$0.701800\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.4142	1.66493	0.832467	−	0.554075i	$-0.186928\pi$
$47$	11.4142	1.66493	0.832467	+	0.554075i	$0.186928\pi$
$48$	0	0
$49$	7.65685	1.09384
$50$	0	0
$51$	8.07107	1.13018
$52$	0	0
$53$	−1.24264	−0.170690	−0.0853449	−	0.996351i	$-0.527199\pi$
$53$	−1.24264	−0.170690	−0.0853449	+	0.996351i	$0.527199\pi$
$54$	0	0
$55$	−5.41421	−0.730052
$56$	0	0
$57$	−2.24264	−0.297045
$58$	0	0
$59$	−12.8995	−1.67937	−0.839686	−	0.543073i	$-0.817261\pi$
$59$	−12.8995	−1.67937	−0.839686	+	0.543073i	$0.817261\pi$
$60$	0	0
$61$	2.58579	0.331076	0.165538	−	0.986203i	$-0.447064\pi$
$61$	2.58579	0.331076	0.165538	+	0.986203i	$0.447064\pi$
$62$	0	0
$63$	3.82843	0.482336
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	−5.48528	−0.670134	−0.335067	−	0.942194i	$-0.608759\pi$
$67$	−5.48528	−0.670134	−0.335067	+	0.942194i	$0.608759\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.8995	1.29353	0.646766	−	0.762688i	$-0.276121\pi$
$71$	10.8995	1.29353	0.646766	+	0.762688i	$0.276121\pi$
$72$	0	0
$73$	−1.75736	−0.205683	−0.102842	−	0.994698i	$-0.532794\pi$
$73$	−1.75736	−0.205683	−0.102842	+	0.994698i	$0.532794\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	20.7279	2.36217
$78$	0	0
$79$	−7.31371	−0.822856	−0.411428	−	0.911442i	$-0.634970\pi$
$79$	−7.31371	−0.822856	−0.411428	+	0.911442i	$0.634970\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0710678	0.00780071	0.00390035	−	0.999992i	$-0.498758\pi$
$83$	0.0710678	0.00780071	0.00390035	+	0.999992i	$0.498758\pi$
$84$	0	0
$85$	−8.07107	−0.875430
$86$	0	0
$87$	−6.41421	−0.687676
$88$	0	0
$89$	−18.1421	−1.92306	−0.961531	−	0.274696i	$-0.911423\pi$
$89$	−18.1421	−1.92306	−0.961531	+	0.274696i	$0.911423\pi$
$90$	0	0
$91$	2.24264	0.235093
$92$	0	0
$93$	9.82843	1.01916
$94$	0	0
$95$	2.24264	0.230090
$96$	0	0
$97$	−6.82843	−0.693322	−0.346661	−	0.937991i	$-0.612685\pi$
$97$	−6.82843	−0.693322	−0.346661	+	0.937991i	$0.612685\pi$
$98$	0	0
$99$	5.41421	0.544149
$100$	0	0
$101$	−18.5563	−1.84643	−0.923213	−	0.384289i	$-0.874447\pi$
$101$	−18.5563	−1.84643	−0.923213	+	0.384289i	$0.874447\pi$
$102$	0	0
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	0	0
$105$	−3.82843	−0.373616
$106$	0	0
$107$	1.58579	0.153304	0.0766519	−	0.997058i	$-0.475577\pi$
$107$	1.58579	0.153304	0.0766519	+	0.997058i	$0.475577\pi$
$108$	0	0
$109$	5.75736	0.551455	0.275728	−	0.961236i	$-0.411081\pi$
$109$	5.75736	0.551455	0.275728	+	0.961236i	$0.411081\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−7.58579	−0.713611	−0.356805	−	0.934179i	$-0.616134\pi$
$113$	−7.58579	−0.713611	−0.356805	+	0.934179i	$0.616134\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	30.8995	2.83255
$120$	0	0
$121$	18.3137	1.66488
$122$	0	0
$123$	−7.58579	−0.683987
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.24264	−0.731416	−0.365708	−	0.930730i	$-0.619173\pi$
$127$	−8.24264	−0.731416	−0.365708	+	0.930730i	$0.619173\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	2.34315	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$131$	2.34315	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$132$	0	0
$133$	−8.58579	−0.744482
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	11.4142	0.961250
$142$	0	0
$143$	3.17157	0.265220
$144$	0	0
$145$	6.41421	0.532671
$146$	0	0
$147$	7.65685	0.631527
$148$	0	0
$149$	−5.07107	−0.415438	−0.207719	−	0.978189i	$-0.566604\pi$
$149$	−5.07107	−0.415438	−0.207719	+	0.978189i	$0.566604\pi$
$150$	0	0
$151$	−13.3137	−1.08345	−0.541727	−	0.840554i	$-0.682229\pi$
$151$	−13.3137	−1.08345	−0.541727	+	0.840554i	$0.682229\pi$
$152$	0	0
$153$	8.07107	0.652507
$154$	0	0
$155$	−9.82843	−0.789438
$156$	0	0
$157$	15.4853	1.23586	0.617930	−	0.786233i	$-0.287972\pi$
$157$	15.4853	1.23586	0.617930	+	0.786233i	$0.287972\pi$
$158$	0	0
$159$	−1.24264	−0.0985478
$160$	0	0
$161$	3.82843	0.301722
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	−5.41421	−0.421496
$166$	0	0
$167$	−18.7279	−1.44921	−0.724605	−	0.689164i	$-0.757978\pi$
$167$	−18.7279	−1.44921	−0.724605	+	0.689164i	$0.757978\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−2.24264	−0.171499
$172$	0	0
$173$	2.34315	0.178146	0.0890730	−	0.996025i	$-0.471610\pi$
$173$	2.34315	0.178146	0.0890730	+	0.996025i	$0.471610\pi$
$174$	0	0
$175$	3.82843	0.289402
$176$	0	0
$177$	−12.8995	−0.969585
$178$	0	0
$179$	−3.65685	−0.273326	−0.136663	−	0.990618i	$-0.543638\pi$
$179$	−3.65685	−0.273326	−0.136663	+	0.990618i	$0.543638\pi$
$180$	0	0
$181$	−1.17157	−0.0870823	−0.0435412	−	0.999052i	$-0.513864\pi$
$181$	−1.17157	−0.0870823	−0.0435412	+	0.999052i	$0.513864\pi$
$182$	0	0
$183$	2.58579	0.191147
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	43.6985	3.19555
$188$	0	0
$189$	3.82843	0.278477
$190$	0	0
$191$	−1.75736	−0.127158	−0.0635790	−	0.997977i	$-0.520251\pi$
$191$	−1.75736	−0.127158	−0.0635790	+	0.997977i	$0.520251\pi$
$192$	0	0
$193$	8.48528	0.610784	0.305392	−	0.952227i	$-0.401213\pi$
$193$	8.48528	0.610784	0.305392	+	0.952227i	$0.401213\pi$
$194$	0	0
$195$	−0.585786	−0.0419490
$196$	0	0
$197$	11.1716	0.795942	0.397971	−	0.917398i	$-0.369715\pi$
$197$	11.1716	0.795942	0.397971	+	0.917398i	$0.369715\pi$
$198$	0	0
$199$	6.48528	0.459729	0.229865	−	0.973223i	$-0.426172\pi$
$199$	6.48528	0.459729	0.229865	+	0.973223i	$0.426172\pi$
$200$	0	0
$201$	−5.48528	−0.386902
$202$	0	0
$203$	−24.5563	−1.72352
$204$	0	0
$205$	7.58579	0.529814
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−12.1421	−0.839889
$210$	0	0
$211$	10.6569	0.733648	0.366824	−	0.930290i	$-0.380445\pi$
$211$	10.6569	0.733648	0.366824	+	0.930290i	$0.380445\pi$
$212$	0	0
$213$	10.8995	0.746821
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	37.6274	2.55432
$218$	0	0
$219$	−1.75736	−0.118751
$220$	0	0
$221$	4.72792	0.318034
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	0	0
$229$	−15.7990	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$229$	−15.7990	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$230$	0	0
$231$	20.7279	1.36380
$232$	0	0
$233$	−1.65685	−0.108544	−0.0542721	−	0.998526i	$-0.517284\pi$
$233$	−1.65685	−0.108544	−0.0542721	+	0.998526i	$0.517284\pi$
$234$	0	0
$235$	−11.4142	−0.744581
$236$	0	0
$237$	−7.31371	−0.475076
$238$	0	0
$239$	4.41421	0.285532	0.142766	−	0.989756i	$-0.454400\pi$
$239$	4.41421	0.285532	0.142766	+	0.989756i	$0.454400\pi$
$240$	0	0
$241$	20.5858	1.32605	0.663024	−	0.748599i	$-0.269273\pi$
$241$	20.5858	1.32605	0.663024	+	0.748599i	$0.269273\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−7.65685	−0.489178
$246$	0	0
$247$	−1.31371	−0.0835893
$248$	0	0
$249$	0.0710678	0.00450374
$250$	0	0
$251$	−24.6274	−1.55447	−0.777234	−	0.629211i	$-0.783378\pi$
$251$	−24.6274	−1.55447	−0.777234	+	0.629211i	$0.783378\pi$
$252$	0	0
$253$	5.41421	0.340389
$254$	0	0
$255$	−8.07107	−0.505430
$256$	0	0
$257$	−18.0416	−1.12541	−0.562703	−	0.826659i	$-0.690239\pi$
$257$	−18.0416	−1.12541	−0.562703	+	0.826659i	$0.690239\pi$
$258$	0	0
$259$	11.4853	0.713661
$260$	0	0
$261$	−6.41421	−0.397030
$262$	0	0
$263$	21.7279	1.33980	0.669901	−	0.742451i	$-0.266337\pi$
$263$	21.7279	1.33980	0.669901	+	0.742451i	$0.266337\pi$
$264$	0	0
$265$	1.24264	0.0763348
$266$	0	0
$267$	−18.1421	−1.11028
$268$	0	0
$269$	4.55635	0.277806	0.138903	−	0.990306i	$-0.455642\pi$
$269$	4.55635	0.277806	0.138903	+	0.990306i	$0.455642\pi$
$270$	0	0
$271$	23.4853	1.42663	0.713315	−	0.700844i	$-0.247193\pi$
$271$	23.4853	1.42663	0.713315	+	0.700844i	$0.247193\pi$
$272$	0	0
$273$	2.24264	0.135731
$274$	0	0
$275$	5.41421	0.326489
$276$	0	0
$277$	2.68629	0.161404	0.0807018	−	0.996738i	$-0.474284\pi$
$277$	2.68629	0.161404	0.0807018	+	0.996738i	$0.474284\pi$
$278$	0	0
$279$	9.82843	0.588413
$280$	0	0
$281$	14.5858	0.870115	0.435058	−	0.900403i	$-0.356728\pi$
$281$	14.5858	0.870115	0.435058	+	0.900403i	$0.356728\pi$
$282$	0	0
$283$	17.3431	1.03094	0.515472	−	0.856907i	$-0.327617\pi$
$283$	17.3431	1.03094	0.515472	+	0.856907i	$0.327617\pi$
$284$	0	0
$285$	2.24264	0.132843
$286$	0	0
$287$	−29.0416	−1.71427
$288$	0	0
$289$	48.1421	2.83189
$290$	0	0
$291$	−6.82843	−0.400289
$292$	0	0
$293$	−22.8995	−1.33780	−0.668901	−	0.743351i	$-0.733235\pi$
$293$	−22.8995	−1.33780	−0.668901	+	0.743351i	$0.733235\pi$
$294$	0	0
$295$	12.8995	0.751038
$296$	0	0
$297$	5.41421	0.314165
$298$	0	0
$299$	0.585786	0.0338769
$300$	0	0
$301$	−22.9706	−1.32400
$302$	0	0
$303$	−18.5563	−1.06603
$304$	0	0
$305$	−2.58579	−0.148062
$306$	0	0
$307$	15.2132	0.868263	0.434132	−	0.900849i	$-0.357055\pi$
$307$	15.2132	0.868263	0.434132	+	0.900849i	$0.357055\pi$
$308$	0	0
$309$	4.82843	0.274680
$310$	0	0
$311$	−6.34315	−0.359687	−0.179843	−	0.983695i	$-0.557559\pi$
$311$	−6.34315	−0.359687	−0.179843	+	0.983695i	$0.557559\pi$
$312$	0	0
$313$	−7.97056	−0.450523	−0.225261	−	0.974298i	$-0.572324\pi$
$313$	−7.97056	−0.450523	−0.225261	+	0.974298i	$0.572324\pi$
$314$	0	0
$315$	−3.82843	−0.215707
$316$	0	0
$317$	1.55635	0.0874133	0.0437066	−	0.999044i	$-0.486083\pi$
$317$	1.55635	0.0874133	0.0437066	+	0.999044i	$0.486083\pi$
$318$	0	0
$319$	−34.7279	−1.94439
$320$	0	0
$321$	1.58579	0.0885100
$322$	0	0
$323$	−18.1005	−1.00714
$324$	0	0
$325$	0.585786	0.0324936
$326$	0	0
$327$	5.75736	0.318383
$328$	0	0
$329$	43.6985	2.40918
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	5.48528	0.299693
$336$	0	0
$337$	−2.97056	−0.161817	−0.0809084	−	0.996722i	$-0.525782\pi$
$337$	−2.97056	−0.161817	−0.0809084	+	0.996722i	$0.525782\pi$
$338$	0	0
$339$	−7.58579	−0.412003
$340$	0	0
$341$	53.2132	2.88166
$342$	0	0
$343$	2.51472	0.135782
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−18.8284	−1.01076	−0.505381	−	0.862896i	$-0.668648\pi$
$347$	−18.8284	−1.01076	−0.505381	+	0.862896i	$0.668648\pi$
$348$	0	0
$349$	30.1127	1.61190	0.805948	−	0.591986i	$-0.201656\pi$
$349$	30.1127	1.61190	0.805948	+	0.591986i	$0.201656\pi$
$350$	0	0
$351$	0.585786	0.0312670
$352$	0	0
$353$	30.5269	1.62478	0.812392	−	0.583112i	$-0.198165\pi$
$353$	30.5269	1.62478	0.812392	+	0.583112i	$0.198165\pi$
$354$	0	0
$355$	−10.8995	−0.578485
$356$	0	0
$357$	30.8995	1.63537
$358$	0	0
$359$	25.0711	1.32320	0.661600	−	0.749857i	$-0.269878\pi$
$359$	25.0711	1.32320	0.661600	+	0.749857i	$0.269878\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	0	0
$363$	18.3137	0.961220
$364$	0	0
$365$	1.75736	0.0919844
$366$	0	0
$367$	−19.9706	−1.04245	−0.521227	−	0.853418i	$-0.674526\pi$
$367$	−19.9706	−1.04245	−0.521227	+	0.853418i	$0.674526\pi$
$368$	0	0
$369$	−7.58579	−0.394900
$370$	0	0
$371$	−4.75736	−0.246990
$372$	0	0
$373$	20.9706	1.08581	0.542907	−	0.839793i	$-0.317323\pi$
$373$	20.9706	1.08581	0.542907	+	0.839793i	$0.317323\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−3.75736	−0.193514
$378$	0	0
$379$	4.97056	0.255321	0.127660	−	0.991818i	$-0.459253\pi$
$379$	4.97056	0.255321	0.127660	+	0.991818i	$0.459253\pi$
$380$	0	0
$381$	−8.24264	−0.422283
$382$	0	0
$383$	−11.5858	−0.592006	−0.296003	−	0.955187i	$-0.595654\pi$
$383$	−11.5858	−0.592006	−0.296003	+	0.955187i	$0.595654\pi$
$384$	0	0
$385$	−20.7279	−1.05639
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	−2.48528	−0.126009	−0.0630044	−	0.998013i	$-0.520068\pi$
$389$	−2.48528	−0.126009	−0.0630044	+	0.998013i	$0.520068\pi$
$390$	0	0
$391$	8.07107	0.408171
$392$	0	0
$393$	2.34315	0.118196
$394$	0	0
$395$	7.31371	0.367993
$396$	0	0
$397$	7.31371	0.367065	0.183532	−	0.983014i	$-0.441247\pi$
$397$	7.31371	0.367065	0.183532	+	0.983014i	$0.441247\pi$
$398$	0	0
$399$	−8.58579	−0.429827
$400$	0	0
$401$	−3.85786	−0.192653	−0.0963263	−	0.995350i	$-0.530709\pi$
$401$	−3.85786	−0.192653	−0.0963263	+	0.995350i	$0.530709\pi$
$402$	0	0
$403$	5.75736	0.286794
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	16.2426	0.805118
$408$	0	0
$409$	−15.4853	−0.765698	−0.382849	−	0.923811i	$-0.625057\pi$
$409$	−15.4853	−0.765698	−0.382849	+	0.923811i	$0.625057\pi$
$410$	0	0
$411$	−2.82843	−0.139516
$412$	0	0
$413$	−49.3848	−2.43007
$414$	0	0
$415$	−0.0710678	−0.00348858
$416$	0	0
$417$	11.0000	0.538672
$418$	0	0
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	−10.2426	−0.499196	−0.249598	−	0.968350i	$-0.580298\pi$
$421$	−10.2426	−0.499196	−0.249598	+	0.968350i	$0.580298\pi$
$422$	0	0
$423$	11.4142	0.554978
$424$	0	0
$425$	8.07107	0.391504
$426$	0	0
$427$	9.89949	0.479070
$428$	0	0
$429$	3.17157	0.153125
$430$	0	0
$431$	−11.7990	−0.568337	−0.284169	−	0.958774i	$-0.591718\pi$
$431$	−11.7990	−0.568337	−0.284169	+	0.958774i	$0.591718\pi$
$432$	0	0
$433$	3.82843	0.183982	0.0919912	−	0.995760i	$-0.470677\pi$
$433$	3.82843	0.183982	0.0919912	+	0.995760i	$0.470677\pi$
$434$	0	0
$435$	6.41421	0.307538
$436$	0	0
$437$	−2.24264	−0.107280
$438$	0	0
$439$	1.51472	0.0722936	0.0361468	−	0.999346i	$-0.488492\pi$
$439$	1.51472	0.0722936	0.0361468	+	0.999346i	$0.488492\pi$
$440$	0	0
$441$	7.65685	0.364612
$442$	0	0
$443$	34.5269	1.64042	0.820212	−	0.572060i	$-0.193856\pi$
$443$	34.5269	1.64042	0.820212	+	0.572060i	$0.193856\pi$
$444$	0	0
$445$	18.1421	0.860020
$446$	0	0
$447$	−5.07107	−0.239853
$448$	0	0
$449$	−4.75736	−0.224514	−0.112257	−	0.993679i	$-0.535808\pi$
$449$	−4.75736	−0.224514	−0.112257	+	0.993679i	$0.535808\pi$
$450$	0	0
$451$	−41.0711	−1.93396
$452$	0	0
$453$	−13.3137	−0.625533
$454$	0	0
$455$	−2.24264	−0.105137
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	8.07107	0.376725
$460$	0	0
$461$	−26.4853	−1.23354	−0.616771	−	0.787142i	$-0.711560\pi$
$461$	−26.4853	−1.23354	−0.616771	+	0.787142i	$0.711560\pi$
$462$	0	0
$463$	−2.92893	−0.136119	−0.0680595	−	0.997681i	$-0.521681\pi$
$463$	−2.92893	−0.136119	−0.0680595	+	0.997681i	$0.521681\pi$
$464$	0	0
$465$	−9.82843	−0.455782
$466$	0	0
$467$	−6.41421	−0.296814	−0.148407	−	0.988926i	$-0.547415\pi$
$467$	−6.41421	−0.296814	−0.148407	+	0.988926i	$0.547415\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	0	0
$471$	15.4853	0.713524
$472$	0	0
$473$	−32.4853	−1.49367
$474$	0	0
$475$	−2.24264	−0.102899
$476$	0	0
$477$	−1.24264	−0.0568966
$478$	0	0
$479$	−6.72792	−0.307407	−0.153703	−	0.988117i	$-0.549120\pi$
$479$	−6.72792	−0.307407	−0.153703	+	0.988117i	$0.549120\pi$
$480$	0	0
$481$	1.75736	0.0801287
$482$	0	0
$483$	3.82843	0.174199
$484$	0	0
$485$	6.82843	0.310063
$486$	0	0
$487$	26.3848	1.19561	0.597804	−	0.801642i	$-0.296040\pi$
$487$	26.3848	1.19561	0.597804	+	0.801642i	$0.296040\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	12.0711	0.544760	0.272380	−	0.962190i	$-0.412189\pi$
$491$	12.0711	0.544760	0.272380	+	0.962190i	$0.412189\pi$
$492$	0	0
$493$	−51.7696	−2.33158
$494$	0	0
$495$	−5.41421	−0.243351
$496$	0	0
$497$	41.7279	1.87175
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	−18.7279	−0.836702
$502$	0	0
$503$	39.7279	1.77138	0.885690	−	0.464277i	$-0.153686\pi$
$503$	39.7279	1.77138	0.885690	+	0.464277i	$0.153686\pi$
$504$	0	0
$505$	18.5563	0.825747
$506$	0	0
$507$	−12.6569	−0.562111
$508$	0	0
$509$	7.79899	0.345684	0.172842	−	0.984950i	$-0.444705\pi$
$509$	7.79899	0.345684	0.172842	+	0.984950i	$0.444705\pi$
$510$	0	0
$511$	−6.72792	−0.297626
$512$	0	0
$513$	−2.24264	−0.0990150
$514$	0	0
$515$	−4.82843	−0.212766
$516$	0	0
$517$	61.7990	2.71792
$518$	0	0
$519$	2.34315	0.102853
$520$	0	0
$521$	−31.5563	−1.38251	−0.691254	−	0.722612i	$-0.742942\pi$
$521$	−31.5563	−1.38251	−0.691254	+	0.722612i	$0.742942\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	3.82843	0.167086
$526$	0	0
$527$	79.3259	3.45549
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.8995	−0.559790
$532$	0	0
$533$	−4.44365	−0.192476
$534$	0	0
$535$	−1.58579	−0.0685595
$536$	0	0
$537$	−3.65685	−0.157805
$538$	0	0
$539$	41.4558	1.78563
$540$	0	0
$541$	5.85786	0.251849	0.125925	−	0.992040i	$-0.459810\pi$
$541$	5.85786	0.251849	0.125925	+	0.992040i	$0.459810\pi$
$542$	0	0
$543$	−1.17157	−0.0502770
$544$	0	0
$545$	−5.75736	−0.246618
$546$	0	0
$547$	−12.9706	−0.554581	−0.277291	−	0.960786i	$-0.589436\pi$
$547$	−12.9706	−0.554581	−0.277291	+	0.960786i	$0.589436\pi$
$548$	0	0
$549$	2.58579	0.110359
$550$	0	0
$551$	14.3848	0.612812
$552$	0	0
$553$	−28.0000	−1.19068
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−6.89949	−0.292341	−0.146170	−	0.989259i	$-0.546695\pi$
$557$	−6.89949	−0.292341	−0.146170	+	0.989259i	$0.546695\pi$
$558$	0	0
$559$	−3.51472	−0.148657
$560$	0	0
$561$	43.6985	1.84495
$562$	0	0
$563$	−20.8995	−0.880809	−0.440404	−	0.897800i	$-0.645165\pi$
$563$	−20.8995	−0.880809	−0.440404	+	0.897800i	$0.645165\pi$
$564$	0	0
$565$	7.58579	0.319136
$566$	0	0
$567$	3.82843	0.160779
$568$	0	0
$569$	12.1421	0.509025	0.254512	−	0.967070i	$-0.418085\pi$
$569$	12.1421	0.509025	0.254512	+	0.967070i	$0.418085\pi$
$570$	0	0
$571$	−28.7279	−1.20223	−0.601113	−	0.799164i	$-0.705276\pi$
$571$	−28.7279	−1.20223	−0.601113	+	0.799164i	$0.705276\pi$
$572$	0	0
$573$	−1.75736	−0.0734147
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−3.02944	−0.126117	−0.0630586	−	0.998010i	$-0.520085\pi$
$577$	−3.02944	−0.126117	−0.0630586	+	0.998010i	$0.520085\pi$
$578$	0	0
$579$	8.48528	0.352636
$580$	0	0
$581$	0.272078	0.0112877
$582$	0	0
$583$	−6.72792	−0.278642
$584$	0	0
$585$	−0.585786	−0.0242193
$586$	0	0
$587$	4.62742	0.190994	0.0954970	−	0.995430i	$-0.469556\pi$
$587$	4.62742	0.190994	0.0954970	+	0.995430i	$0.469556\pi$
$588$	0	0
$589$	−22.0416	−0.908210
$590$	0	0
$591$	11.1716	0.459537
$592$	0	0
$593$	−33.5563	−1.37799	−0.688997	−	0.724764i	$-0.741949\pi$
$593$	−33.5563	−1.37799	−0.688997	+	0.724764i	$0.741949\pi$
$594$	0	0
$595$	−30.8995	−1.26676
$596$	0	0
$597$	6.48528	0.265425
$598$	0	0
$599$	−38.8284	−1.58649	−0.793243	−	0.608905i	$-0.791609\pi$
$599$	−38.8284	−1.58649	−0.793243	+	0.608905i	$0.791609\pi$
$600$	0	0
$601$	4.31371	0.175960	0.0879799	−	0.996122i	$-0.471959\pi$
$601$	4.31371	0.175960	0.0879799	+	0.996122i	$0.471959\pi$
$602$	0	0
$603$	−5.48528	−0.223378
$604$	0	0
$605$	−18.3137	−0.744558
$606$	0	0
$607$	1.61522	0.0655599	0.0327800	−	0.999463i	$-0.489564\pi$
$607$	1.61522	0.0655599	0.0327800	+	0.999463i	$0.489564\pi$
$608$	0	0
$609$	−24.5563	−0.995073
$610$	0	0
$611$	6.68629	0.270498
$612$	0	0
$613$	12.6274	0.510017	0.255008	−	0.966939i	$-0.417922\pi$
$613$	12.6274	0.510017	0.255008	+	0.966939i	$0.417922\pi$
$614$	0	0
$615$	7.58579	0.305888
$616$	0	0
$617$	−14.8995	−0.599831	−0.299916	−	0.953966i	$-0.596959\pi$
$617$	−14.8995	−0.599831	−0.299916	+	0.953966i	$0.596959\pi$
$618$	0	0
$619$	−8.28427	−0.332973	−0.166486	−	0.986044i	$-0.553242\pi$
$619$	−8.28427	−0.332973	−0.166486	+	0.986044i	$0.553242\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−69.4558	−2.78269
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.1421	−0.484910
$628$	0	0
$629$	24.2132	0.965444
$630$	0	0
$631$	9.21320	0.366772	0.183386	−	0.983041i	$-0.441294\pi$
$631$	9.21320	0.366772	0.183386	+	0.983041i	$0.441294\pi$
$632$	0	0
$633$	10.6569	0.423572
$634$	0	0
$635$	8.24264	0.327099
$636$	0	0
$637$	4.48528	0.177713
$638$	0	0
$639$	10.8995	0.431177
$640$	0	0
$641$	19.0711	0.753262	0.376631	−	0.926363i	$-0.377082\pi$
$641$	19.0711	0.753262	0.376631	+	0.926363i	$0.377082\pi$
$642$	0	0
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	21.2132	0.833977	0.416989	−	0.908912i	$-0.363086\pi$
$647$	21.2132	0.833977	0.416989	+	0.908912i	$0.363086\pi$
$648$	0	0
$649$	−69.8406	−2.74148
$650$	0	0
$651$	37.6274	1.47473
$652$	0	0
$653$	−5.41421	−0.211875	−0.105937	−	0.994373i	$-0.533784\pi$
$653$	−5.41421	−0.211875	−0.105937	+	0.994373i	$0.533784\pi$
$654$	0	0
$655$	−2.34315	−0.0915543
$656$	0	0
$657$	−1.75736	−0.0685611
$658$	0	0
$659$	34.3848	1.33944	0.669720	−	0.742613i	$-0.266414\pi$
$659$	34.3848	1.33944	0.669720	+	0.742613i	$0.266414\pi$
$660$	0	0
$661$	−39.4558	−1.53465	−0.767327	−	0.641256i	$-0.778414\pi$
$661$	−39.4558	−1.53465	−0.767327	+	0.641256i	$0.778414\pi$
$662$	0	0
$663$	4.72792	0.183617
$664$	0	0
$665$	8.58579	0.332943
$666$	0	0
$667$	−6.41421	−0.248359
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	−17.6985	−0.682226	−0.341113	−	0.940022i	$-0.610804\pi$
$673$	−17.6985	−0.682226	−0.341113	+	0.940022i	$0.610804\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−10.7574	−0.413439	−0.206719	−	0.978400i	$-0.566279\pi$
$677$	−10.7574	−0.413439	−0.206719	+	0.978400i	$0.566279\pi$
$678$	0	0
$679$	−26.1421	−1.00324
$680$	0	0
$681$	−6.34315	−0.243070
$682$	0	0
$683$	−44.1838	−1.69064	−0.845322	−	0.534257i	$-0.820592\pi$
$683$	−44.1838	−1.69064	−0.845322	+	0.534257i	$0.820592\pi$
$684$	0	0
$685$	2.82843	0.108069
$686$	0	0
$687$	−15.7990	−0.602769
$688$	0	0
$689$	−0.727922	−0.0277316
$690$	0	0
$691$	4.34315	0.165221	0.0826105	−	0.996582i	$-0.473674\pi$
$691$	4.34315	0.165221	0.0826105	+	0.996582i	$0.473674\pi$
$692$	0	0
$693$	20.7279	0.787389
$694$	0	0
$695$	−11.0000	−0.417254
$696$	0	0
$697$	−61.2254	−2.31908
$698$	0	0
$699$	−1.65685	−0.0626680
$700$	0	0
$701$	0.384776	0.0145328	0.00726640	−	0.999974i	$-0.497687\pi$
$701$	0.384776	0.0145328	0.00726640	+	0.999974i	$0.497687\pi$
$702$	0	0
$703$	−6.72792	−0.253748
$704$	0	0
$705$	−11.4142	−0.429884
$706$	0	0
$707$	−71.0416	−2.67180
$708$	0	0
$709$	−8.72792	−0.327784	−0.163892	−	0.986478i	$-0.552405\pi$
$709$	−8.72792	−0.327784	−0.163892	+	0.986478i	$0.552405\pi$
$710$	0	0
$711$	−7.31371	−0.274285
$712$	0	0
$713$	9.82843	0.368077
$714$	0	0
$715$	−3.17157	−0.118610
$716$	0	0
$717$	4.41421	0.164852
$718$	0	0
$719$	6.21320	0.231713	0.115857	−	0.993266i	$-0.463039\pi$
$719$	6.21320	0.231713	0.115857	+	0.993266i	$0.463039\pi$
$720$	0	0
$721$	18.4853	0.688428
$722$	0	0
$723$	20.5858	0.765594
$724$	0	0
$725$	−6.41421	−0.238218
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.4264	−1.79112
$732$	0	0
$733$	−0.514719	−0.0190116	−0.00950578	−	0.999955i	$-0.503026\pi$
$733$	−0.514719	−0.0190116	−0.00950578	+	0.999955i	$0.503026\pi$
$734$	0	0
$735$	−7.65685	−0.282427
$736$	0	0
$737$	−29.6985	−1.09396
$738$	0	0
$739$	8.37258	0.307990	0.153995	−	0.988072i	$-0.450786\pi$
$739$	8.37258	0.307990	0.153995	+	0.988072i	$0.450786\pi$
$740$	0	0
$741$	−1.31371	−0.0482603
$742$	0	0
$743$	14.3431	0.526199	0.263099	−	0.964769i	$-0.415255\pi$
$743$	14.3431	0.526199	0.263099	+	0.964769i	$0.415255\pi$
$744$	0	0
$745$	5.07107	0.185790
$746$	0	0
$747$	0.0710678	0.00260024
$748$	0	0
$749$	6.07107	0.221832
$750$	0	0
$751$	34.7279	1.26724	0.633620	−	0.773644i	$-0.281568\pi$
$751$	34.7279	1.26724	0.633620	+	0.773644i	$0.281568\pi$
$752$	0	0
$753$	−24.6274	−0.897473
$754$	0	0
$755$	13.3137	0.484535
$756$	0	0
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	0	0
$759$	5.41421	0.196524
$760$	0	0
$761$	−34.0711	−1.23508	−0.617538	−	0.786541i	$-0.711870\pi$
$761$	−34.0711	−1.23508	−0.617538	+	0.786541i	$0.711870\pi$
$762$	0	0
$763$	22.0416	0.797961
$764$	0	0
$765$	−8.07107	−0.291810
$766$	0	0
$767$	−7.55635	−0.272844
$768$	0	0
$769$	−28.0416	−1.01121	−0.505604	−	0.862766i	$-0.668730\pi$
$769$	−28.0416	−1.01121	−0.505604	+	0.862766i	$0.668730\pi$
$770$	0	0
$771$	−18.0416	−0.649753
$772$	0	0
$773$	−8.82843	−0.317536	−0.158768	−	0.987316i	$-0.550752\pi$
$773$	−8.82843	−0.317536	−0.158768	+	0.987316i	$0.550752\pi$
$774$	0	0
$775$	9.82843	0.353048
$776$	0	0
$777$	11.4853	0.412032
$778$	0	0
$779$	17.0122	0.609525
$780$	0	0
$781$	59.0122	2.11162
$782$	0	0
$783$	−6.41421	−0.229225
$784$	0	0
$785$	−15.4853	−0.552693
$786$	0	0
$787$	41.6274	1.48386	0.741929	−	0.670479i	$-0.233911\pi$
$787$	41.6274	1.48386	0.741929	+	0.670479i	$0.233911\pi$
$788$	0	0
$789$	21.7279	0.773535
$790$	0	0
$791$	−29.0416	−1.03260
$792$	0	0
$793$	1.51472	0.0537892
$794$	0	0
$795$	1.24264	0.0440719
$796$	0	0
$797$	25.2426	0.894140	0.447070	−	0.894499i	$-0.352467\pi$
$797$	25.2426	0.894140	0.447070	+	0.894499i	$0.352467\pi$
$798$	0	0
$799$	92.1249	3.25914
$800$	0	0
$801$	−18.1421	−0.641021
$802$	0	0
$803$	−9.51472	−0.335767
$804$	0	0
$805$	−3.82843	−0.134934
$806$	0	0
$807$	4.55635	0.160391
$808$	0	0
$809$	47.8701	1.68302	0.841511	−	0.540240i	$-0.181667\pi$
$809$	47.8701	1.68302	0.841511	+	0.540240i	$0.181667\pi$
$810$	0	0
$811$	17.2843	0.606933	0.303466	−	0.952842i	$-0.401856\pi$
$811$	17.2843	0.606933	0.303466	+	0.952842i	$0.401856\pi$
$812$	0	0
$813$	23.4853	0.823665
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	13.4558	0.470760
$818$	0	0
$819$	2.24264	0.0783642
$820$	0	0
$821$	−4.20101	−0.146616	−0.0733081	−	0.997309i	$-0.523356\pi$
$821$	−4.20101	−0.146616	−0.0733081	+	0.997309i	$0.523356\pi$
$822$	0	0
$823$	−34.3431	−1.19713	−0.598563	−	0.801075i	$-0.704262\pi$
$823$	−34.3431	−1.19713	−0.598563	+	0.801075i	$0.704262\pi$
$824$	0	0
$825$	5.41421	0.188499
$826$	0	0
$827$	−32.0122	−1.11317	−0.556587	−	0.830790i	$-0.687889\pi$
$827$	−32.0122	−1.11317	−0.556587	+	0.830790i	$0.687889\pi$
$828$	0	0
$829$	6.51472	0.226266	0.113133	−	0.993580i	$-0.463911\pi$
$829$	6.51472	0.226266	0.113133	+	0.993580i	$0.463911\pi$
$830$	0	0
$831$	2.68629	0.0931864
$832$	0	0
$833$	61.7990	2.14121
$834$	0	0
$835$	18.7279	0.648106
$836$	0	0
$837$	9.82843	0.339720
$838$	0	0
$839$	−40.6274	−1.40261	−0.701307	−	0.712859i	$-0.747400\pi$
$839$	−40.6274	−1.40261	−0.701307	+	0.712859i	$0.747400\pi$
$840$	0	0
$841$	12.1421	0.418694
$842$	0	0
$843$	14.5858	0.502361
$844$	0	0
$845$	12.6569	0.435409
$846$	0	0
$847$	70.1127	2.40910
$848$	0	0
$849$	17.3431	0.595215
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	19.4558	0.666155	0.333078	−	0.942899i	$-0.391913\pi$
$853$	19.4558	0.666155	0.333078	+	0.942899i	$0.391913\pi$
$854$	0	0
$855$	2.24264	0.0766967
$856$	0	0
$857$	−57.5980	−1.96751	−0.983755	−	0.179518i	$-0.942546\pi$
$857$	−57.5980	−1.96751	−0.983755	+	0.179518i	$0.942546\pi$
$858$	0	0
$859$	57.0000	1.94481	0.972407	−	0.233289i	$-0.0749488\pi$
$859$	57.0000	1.94481	0.972407	+	0.233289i	$0.0749488\pi$
$860$	0	0
$861$	−29.0416	−0.989736
$862$	0	0
$863$	−16.8284	−0.572846	−0.286423	−	0.958103i	$-0.592466\pi$
$863$	−16.8284	−0.572846	−0.286423	+	0.958103i	$0.592466\pi$
$864$	0	0
$865$	−2.34315	−0.0796693
$866$	0	0
$867$	48.1421	1.63499
$868$	0	0
$869$	−39.5980	−1.34327
$870$	0	0
$871$	−3.21320	−0.108875
$872$	0	0
$873$	−6.82843	−0.231107
$874$	0	0
$875$	−3.82843	−0.129424
$876$	0	0
$877$	−18.4853	−0.624204	−0.312102	−	0.950049i	$-0.601033\pi$
$877$	−18.4853	−0.624204	−0.312102	+	0.950049i	$0.601033\pi$
$878$	0	0
$879$	−22.8995	−0.772381
$880$	0	0
$881$	16.5858	0.558789	0.279395	−	0.960176i	$-0.409866\pi$
$881$	16.5858	0.558789	0.279395	+	0.960176i	$0.409866\pi$
$882$	0	0
$883$	−17.8995	−0.602366	−0.301183	−	0.953566i	$-0.597381\pi$
$883$	−17.8995	−0.602366	−0.301183	+	0.953566i	$0.597381\pi$
$884$	0	0
$885$	12.8995	0.433612
$886$	0	0
$887$	17.7990	0.597632	0.298816	−	0.954311i	$-0.403408\pi$
$887$	17.7990	0.597632	0.298816	+	0.954311i	$0.403408\pi$
$888$	0	0
$889$	−31.5563	−1.05837
$890$	0	0
$891$	5.41421	0.181383
$892$	0	0
$893$	−25.5980	−0.856604
$894$	0	0
$895$	3.65685	0.122235
$896$	0	0
$897$	0.585786	0.0195588
$898$	0	0
$899$	−63.0416	−2.10256
$900$	0	0
$901$	−10.0294	−0.334129
$902$	0	0
$903$	−22.9706	−0.764412
$904$	0	0
$905$	1.17157	0.0389444
$906$	0	0
$907$	−2.85786	−0.0948938	−0.0474469	−	0.998874i	$-0.515108\pi$
$907$	−2.85786	−0.0948938	−0.0474469	+	0.998874i	$0.515108\pi$
$908$	0	0
$909$	−18.5563	−0.615475
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	0.384776	0.0127342
$914$	0	0
$915$	−2.58579	−0.0854835
$916$	0	0
$917$	8.97056	0.296234
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	15.2132	0.501292
$922$	0	0
$923$	6.38478	0.210157
$924$	0	0
$925$	3.00000	0.0986394
$926$	0	0
$927$	4.82843	0.158586
$928$	0	0
$929$	−11.8701	−0.389444	−0.194722	−	0.980858i	$-0.562380\pi$
$929$	−11.8701	−0.389444	−0.194722	+	0.980858i	$0.562380\pi$
$930$	0	0
$931$	−17.1716	−0.562776
$932$	0	0
$933$	−6.34315	−0.207665
$934$	0	0
$935$	−43.6985	−1.42909
$936$	0	0
$937$	−20.2010	−0.659938	−0.329969	−	0.943992i	$-0.607038\pi$
$937$	−20.2010	−0.659938	−0.329969	+	0.943992i	$0.607038\pi$
$938$	0	0
$939$	−7.97056	−0.260109
$940$	0	0
$941$	−4.58579	−0.149492	−0.0747462	−	0.997203i	$-0.523815\pi$
$941$	−4.58579	−0.149492	−0.0747462	+	0.997203i	$0.523815\pi$
$942$	0	0
$943$	−7.58579	−0.247027
$944$	0	0
$945$	−3.82843	−0.124539
$946$	0	0
$947$	−35.7990	−1.16331	−0.581655	−	0.813435i	$-0.697595\pi$
$947$	−35.7990	−1.16331	−0.581655	+	0.813435i	$0.697595\pi$
$948$	0	0
$949$	−1.02944	−0.0334169
$950$	0	0
$951$	1.55635	0.0504681
$952$	0	0
$953$	52.6274	1.70477	0.852385	−	0.522915i	$-0.175156\pi$
$953$	52.6274	1.70477	0.852385	+	0.522915i	$0.175156\pi$
$954$	0	0
$955$	1.75736	0.0568668
$956$	0	0
$957$	−34.7279	−1.12259
$958$	0	0
$959$	−10.8284	−0.349668
$960$	0	0
$961$	65.5980	2.11606
$962$	0	0
$963$	1.58579	0.0511013
$964$	0	0
$965$	−8.48528	−0.273151
$966$	0	0
$967$	−31.5563	−1.01478	−0.507392	−	0.861715i	$-0.669390\pi$
$967$	−31.5563	−1.01478	−0.507392	+	0.861715i	$0.669390\pi$
$968$	0	0
$969$	−18.1005	−0.581472
$970$	0	0
$971$	54.4264	1.74663	0.873313	−	0.487159i	$-0.161967\pi$
$971$	54.4264	1.74663	0.873313	+	0.487159i	$0.161967\pi$
$972$	0	0
$973$	42.1127	1.35007
$974$	0	0
$975$	0.585786	0.0187602
$976$	0	0
$977$	−18.4142	−0.589123	−0.294561	−	0.955633i	$-0.595174\pi$
$977$	−18.4142	−0.589123	−0.294561	+	0.955633i	$0.595174\pi$
$978$	0	0
$979$	−98.2254	−3.13930
$980$	0	0
$981$	5.75736	0.183818
$982$	0	0
$983$	58.0122	1.85030	0.925151	−	0.379600i	$-0.123938\pi$
$983$	58.0122	1.85030	0.925151	+	0.379600i	$0.123938\pi$
$984$	0	0
$985$	−11.1716	−0.355956
$986$	0	0
$987$	43.6985	1.39094
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−12.6569	−0.402058	−0.201029	−	0.979585i	$-0.564429\pi$
$991$	−12.6569	−0.402058	−0.201029	+	0.979585i	$0.564429\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	−6.48528	−0.205597
$996$	0	0
$997$	−5.85786	−0.185520	−0.0927602	−	0.995688i	$-0.529569\pi$
$997$	−5.85786	−0.185520	−0.0927602	+	0.995688i	$0.529569\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bm.1.2		2
4.3	odd	2		345.2.a.h.1.2	✓	2
12.11	even	2		1035.2.a.j.1.1		2
20.3	even	4		1725.2.b.s.1174.1		4
20.7	even	4		1725.2.b.s.1174.4		4
20.19	odd	2		1725.2.a.z.1.1		2
60.59	even	2		5175.2.a.bj.1.2		2
92.91	even	2		7935.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.2	✓	2	4.3	odd	2
1035.2.a.j.1.1		2	12.11	even	2
1725.2.a.z.1.1		2	20.19	odd	2
1725.2.b.s.1174.1		4	20.3	even	4
1725.2.b.s.1174.4		4	20.7	even	4
5175.2.a.bj.1.2		2	60.59	even	2
5520.2.a.bm.1.2		2	1.1	even	1	trivial
7935.2.a.q.1.2		2	92.91	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +5.41421 q^{11} +0.585786 q^{13} -1.00000 q^{15} +8.07107 q^{17} -2.24264 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +9.82843 q^{31} +5.41421 q^{33} -3.82843 q^{35} +3.00000 q^{37} +0.585786 q^{39} -7.58579 q^{41} -6.00000 q^{43} -1.00000 q^{45} +11.4142 q^{47} +7.65685 q^{49} +8.07107 q^{51} -1.24264 q^{53} -5.41421 q^{55} -2.24264 q^{57} -12.8995 q^{59} +2.58579 q^{61} +3.82843 q^{63} -0.585786 q^{65} -5.48528 q^{67} +1.00000 q^{69} +10.8995 q^{71} -1.75736 q^{73} +1.00000 q^{75} +20.7279 q^{77} -7.31371 q^{79} +1.00000 q^{81} +0.0710678 q^{83} -8.07107 q^{85} -6.41421 q^{87} -18.1421 q^{89} +2.24264 q^{91} +9.82843 q^{93} +2.24264 q^{95} -6.82843 q^{97} +5.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 14 q^{31} + 8 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 8 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 14 * q^31 + 8 * q^33 - 2 * q^35 + 6 * q^37 + 4 * q^39 - 18 * q^41 - 12 * q^43 - 2 * q^45 + 20 * q^47 + 4 * q^49 + 2 * q^51 + 6 * q^53 - 8 * q^55 + 4 * q^57 - 6 * q^59 + 8 * q^61 + 2 * q^63 - 4 * q^65 + 6 * q^67 + 2 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 16 * q^77 + 8 * q^79 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 8 * q^89 - 4 * q^91 + 14 * q^93 - 4 * q^95 - 8 * q^97 + 8 * q^99

Introduction

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