LMFDB - Embedded newform 880.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(1,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.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:	$7.02683537787$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} +5.12311 q^{13} +2.56155 q^{15} +1.43845 q^{17} -6.56155 q^{19} -1.43845 q^{21} +1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -4.56155 q^{29} +3.68466 q^{31} +2.56155 q^{33} -0.561553 q^{35} -10.8078 q^{37} -13.1231 q^{39} -10.0000 q^{41} +3.12311 q^{43} -3.56155 q^{45} -1.12311 q^{47} -6.68466 q^{49} -3.68466 q^{51} -8.56155 q^{53} +1.00000 q^{55} +16.8078 q^{57} -11.3693 q^{59} +0.561553 q^{61} +2.00000 q^{63} -5.12311 q^{65} -2.87689 q^{69} +6.56155 q^{71} +13.1231 q^{73} -2.56155 q^{75} -0.561553 q^{77} -9.12311 q^{79} -7.00000 q^{81} -10.0000 q^{83} -1.43845 q^{85} +11.6847 q^{87} -10.8078 q^{89} +2.87689 q^{91} -9.43845 q^{93} +6.56155 q^{95} +8.87689 q^{97} -3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 - 3 * q^7 + 3 * q^9	$ 2 q - q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + q^{15} + 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} + 2 q^{25} - 7 q^{27} - 5 q^{29} - 5 q^{31} + q^{33} + 3 q^{35} - q^{37} - 18 q^{39} - 20 q^{41} - 2 q^{43} - 3 q^{45} + 6 q^{47} - q^{49} + 5 q^{51} - 13 q^{53} + 2 q^{55} + 13 q^{57} + 2 q^{59} - 3 q^{61} + 4 q^{63} - 2 q^{65} - 14 q^{69} + 9 q^{71} + 18 q^{73} - q^{75} + 3 q^{77} - 10 q^{79} - 14 q^{81} - 20 q^{83} - 7 q^{85} + 11 q^{87} - q^{89} + 14 q^{91} - 23 q^{93} + 9 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + q^15 + 7 * q^17 - 9 * q^19 - 7 * q^21 - 6 * q^23 + 2 * q^25 - 7 * q^27 - 5 * q^29 - 5 * q^31 + q^33 + 3 * q^35 - q^37 - 18 * q^39 - 20 * q^41 - 2 * q^43 - 3 * q^45 + 6 * q^47 - q^49 + 5 * q^51 - 13 * q^53 + 2 * q^55 + 13 * q^57 + 2 * q^59 - 3 * q^61 + 4 * q^63 - 2 * q^65 - 14 * q^69 + 9 * q^71 + 18 * q^73 - q^75 + 3 * q^77 - 10 * q^79 - 14 * q^81 - 20 * q^83 - 7 * q^85 + 11 * q^87 - q^89 + 14 * q^91 - 23 * q^93 + 9 * q^95 + 26 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	1.43845	0.348875	0.174437	−	0.984668i	$-0.444189\pi$
$17$	1.43845	0.348875	0.174437	+	0.984668i	$0.444189\pi$
$18$	0	0
$19$	−6.56155	−1.50532	−0.752662	−	0.658407i	$-0.771230\pi$
$19$	−6.56155	−1.50532	−0.752662	+	0.658407i	$0.771230\pi$
$20$	0	0
$21$	−1.43845	−0.313895
$22$	0	0
$23$	1.12311	0.234184	0.117092	−	0.993121i	$-0.462643\pi$
$23$	1.12311	0.234184	0.117092	+	0.993121i	$0.462643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	−4.56155	−0.847059	−0.423530	−	0.905882i	$-0.639209\pi$
$29$	−4.56155	−0.847059	−0.423530	+	0.905882i	$0.639209\pi$
$30$	0	0
$31$	3.68466	0.661784	0.330892	−	0.943669i	$-0.392650\pi$
$31$	3.68466	0.661784	0.330892	+	0.943669i	$0.392650\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	−0.561553	−0.0949197
$36$	0	0
$37$	−10.8078	−1.77679	−0.888393	−	0.459084i	$-0.848178\pi$
$37$	−10.8078	−1.77679	−0.888393	+	0.459084i	$0.848178\pi$
$38$	0	0
$39$	−13.1231	−2.10138
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	3.12311	0.476269	0.238135	−	0.971232i	$-0.423464\pi$
$43$	3.12311	0.476269	0.238135	+	0.971232i	$0.423464\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	−1.12311	−0.163822	−0.0819109	−	0.996640i	$-0.526102\pi$
$47$	−1.12311	−0.163822	−0.0819109	+	0.996640i	$0.526102\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	0	0
$51$	−3.68466	−0.515955
$52$	0	0
$53$	−8.56155	−1.17602	−0.588010	−	0.808854i	$-0.700088\pi$
$53$	−8.56155	−1.17602	−0.588010	+	0.808854i	$0.700088\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	16.8078	2.22624
$58$	0	0
$59$	−11.3693	−1.48016	−0.740079	−	0.672519i	$-0.765212\pi$
$59$	−11.3693	−1.48016	−0.740079	+	0.672519i	$0.765212\pi$
$60$	0	0
$61$	0.561553	0.0718995	0.0359497	−	0.999354i	$-0.488554\pi$
$61$	0.561553	0.0718995	0.0359497	+	0.999354i	$0.488554\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−5.12311	−0.635443
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−2.87689	−0.346337
$70$	0	0
$71$	6.56155	0.778713	0.389357	−	0.921087i	$-0.372697\pi$
$71$	6.56155	0.778713	0.389357	+	0.921087i	$0.372697\pi$
$72$	0	0
$73$	13.1231	1.53594	0.767972	−	0.640484i	$-0.221266\pi$
$73$	13.1231	1.53594	0.767972	+	0.640484i	$0.221266\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	−0.561553	−0.0639949
$78$	0	0
$79$	−9.12311	−1.02643	−0.513215	−	0.858260i	$-0.671546\pi$
$79$	−9.12311	−1.02643	−0.513215	+	0.858260i	$0.671546\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−1.43845	−0.156022
$86$	0	0
$87$	11.6847	1.25273
$88$	0	0
$89$	−10.8078	−1.14562	−0.572810	−	0.819688i	$-0.694147\pi$
$89$	−10.8078	−1.14562	−0.572810	+	0.819688i	$0.694147\pi$
$90$	0	0
$91$	2.87689	0.301580
$92$	0	0
$93$	−9.43845	−0.978721
$94$	0	0
$95$	6.56155	0.673201
$96$	0	0
$97$	8.87689	0.901312	0.450656	−	0.892698i	$-0.351190\pi$
$97$	8.87689	0.901312	0.450656	+	0.892698i	$0.351190\pi$
$98$	0	0
$99$	−3.56155	−0.357950
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	1.43845	0.140378
$106$	0	0
$107$	3.12311	0.301922	0.150961	−	0.988540i	$-0.451763\pi$
$107$	3.12311	0.301922	0.150961	+	0.988540i	$0.451763\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	27.6847	2.62771
$112$	0	0
$113$	−7.12311	−0.670085	−0.335043	−	0.942203i	$-0.608751\pi$
$113$	−7.12311	−0.670085	−0.335043	+	0.942203i	$0.608751\pi$
$114$	0	0
$115$	−1.12311	−0.104730
$116$	0	0
$117$	18.2462	1.68686
$118$	0	0
$119$	0.807764	0.0740476
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	25.6155	2.30967
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.3693	−1.54128	−0.770639	−	0.637272i	$-0.780063\pi$
$127$	−17.3693	−1.54128	−0.770639	+	0.637272i	$0.780063\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	11.6847	1.02089	0.510447	−	0.859909i	$-0.329480\pi$
$131$	11.6847	1.02089	0.510447	+	0.859909i	$0.329480\pi$
$132$	0	0
$133$	−3.68466	−0.319500
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	5.36932	0.458732	0.229366	−	0.973340i	$-0.426335\pi$
$137$	5.36932	0.458732	0.229366	+	0.973340i	$0.426335\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	2.87689	0.242278
$142$	0	0
$143$	−5.12311	−0.428416
$144$	0	0
$145$	4.56155	0.378816
$146$	0	0
$147$	17.1231	1.41229
$148$	0	0
$149$	−13.6847	−1.12109	−0.560545	−	0.828124i	$-0.689408\pi$
$149$	−13.6847	−1.12109	−0.560545	+	0.828124i	$0.689408\pi$
$150$	0	0
$151$	9.12311	0.742428	0.371214	−	0.928547i	$-0.378942\pi$
$151$	9.12311	0.742428	0.371214	+	0.928547i	$0.378942\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	−3.68466	−0.295959
$156$	0	0
$157$	15.9309	1.27142	0.635711	−	0.771927i	$-0.280707\pi$
$157$	15.9309	1.27142	0.635711	+	0.771927i	$0.280707\pi$
$158$	0	0
$159$	21.9309	1.73923
$160$	0	0
$161$	0.630683	0.0497048
$162$	0	0
$163$	−20.8078	−1.62979	−0.814895	−	0.579609i	$-0.803205\pi$
$163$	−20.8078	−1.62979	−0.814895	+	0.579609i	$0.803205\pi$
$164$	0	0
$165$	−2.56155	−0.199417
$166$	0	0
$167$	−13.6847	−1.05895	−0.529475	−	0.848325i	$-0.677611\pi$
$167$	−13.6847	−1.05895	−0.529475	+	0.848325i	$0.677611\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−23.3693	−1.78710
$172$	0	0
$173$	3.36932	0.256164	0.128082	−	0.991764i	$-0.459118\pi$
$173$	3.36932	0.256164	0.128082	+	0.991764i	$0.459118\pi$
$174$	0	0
$175$	0.561553	0.0424494
$176$	0	0
$177$	29.1231	2.18903
$178$	0	0
$179$	24.4924	1.83065	0.915325	−	0.402716i	$-0.131934\pi$
$179$	24.4924	1.83065	0.915325	+	0.402716i	$0.131934\pi$
$180$	0	0
$181$	−18.4924	−1.37453	−0.687265	−	0.726406i	$-0.741189\pi$
$181$	−18.4924	−1.37453	−0.687265	+	0.726406i	$0.741189\pi$
$182$	0	0
$183$	−1.43845	−0.106333
$184$	0	0
$185$	10.8078	0.794603
$186$	0	0
$187$	−1.43845	−0.105190
$188$	0	0
$189$	−0.807764	−0.0587562
$190$	0	0
$191$	−18.2462	−1.32025	−0.660125	−	0.751156i	$-0.729497\pi$
$191$	−18.2462	−1.32025	−0.660125	+	0.751156i	$0.729497\pi$
$192$	0	0
$193$	−0.807764	−0.0581441	−0.0290721	−	0.999577i	$-0.509255\pi$
$193$	−0.807764	−0.0581441	−0.0290721	+	0.999577i	$0.509255\pi$
$194$	0	0
$195$	13.1231	0.939765
$196$	0	0
$197$	1.75379	0.124952	0.0624761	−	0.998046i	$-0.480100\pi$
$197$	1.75379	0.124952	0.0624761	+	0.998046i	$0.480100\pi$
$198$	0	0
$199$	−21.9309	−1.55464	−0.777319	−	0.629107i	$-0.783421\pi$
$199$	−21.9309	−1.55464	−0.777319	+	0.629107i	$0.783421\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.56155	−0.179786
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	6.56155	0.453872
$210$	0	0
$211$	11.6847	0.804405	0.402203	−	0.915551i	$-0.368245\pi$
$211$	11.6847	0.804405	0.402203	+	0.915551i	$0.368245\pi$
$212$	0	0
$213$	−16.8078	−1.15165
$214$	0	0
$215$	−3.12311	−0.212994
$216$	0	0
$217$	2.06913	0.140462
$218$	0	0
$219$	−33.6155	−2.27153
$220$	0	0
$221$	7.36932	0.495714
$222$	0	0
$223$	14.8769	0.996231	0.498115	−	0.867111i	$-0.334026\pi$
$223$	14.8769	0.996231	0.498115	+	0.867111i	$0.334026\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	26.4924	1.75067	0.875334	−	0.483518i	$-0.160641\pi$
$229$	26.4924	1.75067	0.875334	+	0.483518i	$0.160641\pi$
$230$	0	0
$231$	1.43845	0.0946429
$232$	0	0
$233$	23.6847	1.55163	0.775817	−	0.630958i	$-0.217338\pi$
$233$	23.6847	1.55163	0.775817	+	0.630958i	$0.217338\pi$
$234$	0	0
$235$	1.12311	0.0732633
$236$	0	0
$237$	23.3693	1.51800
$238$	0	0
$239$	−9.12311	−0.590125	−0.295062	−	0.955478i	$-0.595340\pi$
$239$	−9.12311	−0.590125	−0.295062	+	0.955478i	$0.595340\pi$
$240$	0	0
$241$	20.7386	1.33589	0.667946	−	0.744209i	$-0.267174\pi$
$241$	20.7386	1.33589	0.667946	+	0.744209i	$0.267174\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	6.68466	0.427067
$246$	0	0
$247$	−33.6155	−2.13890
$248$	0	0
$249$	25.6155	1.62332
$250$	0	0
$251$	−14.8769	−0.939021	−0.469511	−	0.882927i	$-0.655570\pi$
$251$	−14.8769	−0.939021	−0.469511	+	0.882927i	$0.655570\pi$
$252$	0	0
$253$	−1.12311	−0.0706090
$254$	0	0
$255$	3.68466	0.230742
$256$	0	0
$257$	−24.2462	−1.51244	−0.756219	−	0.654319i	$-0.772955\pi$
$257$	−24.2462	−1.51244	−0.756219	+	0.654319i	$0.772955\pi$
$258$	0	0
$259$	−6.06913	−0.377117
$260$	0	0
$261$	−16.2462	−1.00562
$262$	0	0
$263$	−28.5616	−1.76118	−0.880590	−	0.473878i	$-0.842854\pi$
$263$	−28.5616	−1.76118	−0.880590	+	0.473878i	$0.842854\pi$
$264$	0	0
$265$	8.56155	0.525932
$266$	0	0
$267$	27.6847	1.69427
$268$	0	0
$269$	−31.1231	−1.89761	−0.948805	−	0.315864i	$-0.897706\pi$
$269$	−31.1231	−1.89761	−0.948805	+	0.315864i	$0.897706\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−7.36932	−0.446011
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−30.7386	−1.84691	−0.923453	−	0.383712i	$-0.874645\pi$
$277$	−30.7386	−1.84691	−0.923453	+	0.383712i	$0.874645\pi$
$278$	0	0
$279$	13.1231	0.785660
$280$	0	0
$281$	−14.4924	−0.864545	−0.432273	−	0.901743i	$-0.642288\pi$
$281$	−14.4924	−0.864545	−0.432273	+	0.901743i	$0.642288\pi$
$282$	0	0
$283$	−14.4924	−0.861485	−0.430743	−	0.902475i	$-0.641748\pi$
$283$	−14.4924	−0.861485	−0.430743	+	0.902475i	$0.641748\pi$
$284$	0	0
$285$	−16.8078	−0.995606
$286$	0	0
$287$	−5.61553	−0.331474
$288$	0	0
$289$	−14.9309	−0.878286
$290$	0	0
$291$	−22.7386	−1.33296
$292$	0	0
$293$	−2.87689	−0.168070	−0.0840350	−	0.996463i	$-0.526781\pi$
$293$	−2.87689	−0.168070	−0.0840350	+	0.996463i	$0.526781\pi$
$294$	0	0
$295$	11.3693	0.661947
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	5.75379	0.332750
$300$	0	0
$301$	1.75379	0.101087
$302$	0	0
$303$	−0.630683	−0.0362318
$304$	0	0
$305$	−0.561553	−0.0321544
$306$	0	0
$307$	28.7386	1.64020	0.820100	−	0.572220i	$-0.193918\pi$
$307$	28.7386	1.64020	0.820100	+	0.572220i	$0.193918\pi$
$308$	0	0
$309$	5.75379	0.327322
$310$	0	0
$311$	24.1771	1.37096	0.685478	−	0.728093i	$-0.259593\pi$
$311$	24.1771	1.37096	0.685478	+	0.728093i	$0.259593\pi$
$312$	0	0
$313$	−0.246211	−0.0139167	−0.00695834	−	0.999976i	$-0.502215\pi$
$313$	−0.246211	−0.0139167	−0.00695834	+	0.999976i	$0.502215\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	0.561553	0.0315399	0.0157700	−	0.999876i	$-0.494980\pi$
$317$	0.561553	0.0315399	0.0157700	+	0.999876i	$0.494980\pi$
$318$	0	0
$319$	4.56155	0.255398
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−9.43845	−0.525169
$324$	0	0
$325$	5.12311	0.284179
$326$	0	0
$327$	5.12311	0.283308
$328$	0	0
$329$	−0.630683	−0.0347707
$330$	0	0
$331$	−14.2462	−0.783043	−0.391521	−	0.920169i	$-0.628051\pi$
$331$	−14.2462	−0.783043	−0.391521	+	0.920169i	$0.628051\pi$
$332$	0	0
$333$	−38.4924	−2.10937
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.9309	−1.19465	−0.597325	−	0.801999i	$-0.703770\pi$
$337$	−21.9309	−1.19465	−0.597325	+	0.801999i	$0.703770\pi$
$338$	0	0
$339$	18.2462	0.990998
$340$	0	0
$341$	−3.68466	−0.199535
$342$	0	0
$343$	−7.68466	−0.414933
$344$	0	0
$345$	2.87689	0.154887
$346$	0	0
$347$	−7.75379	−0.416245	−0.208123	−	0.978103i	$-0.566735\pi$
$347$	−7.75379	−0.416245	−0.208123	+	0.978103i	$0.566735\pi$
$348$	0	0
$349$	24.7386	1.32423	0.662114	−	0.749403i	$-0.269659\pi$
$349$	24.7386	1.32423	0.662114	+	0.749403i	$0.269659\pi$
$350$	0	0
$351$	−7.36932	−0.393345
$352$	0	0
$353$	20.2462	1.07760	0.538799	−	0.842435i	$-0.318878\pi$
$353$	20.2462	1.07760	0.538799	+	0.842435i	$0.318878\pi$
$354$	0	0
$355$	−6.56155	−0.348251
$356$	0	0
$357$	−2.06913	−0.109510
$358$	0	0
$359$	−24.4924	−1.29266	−0.646330	−	0.763058i	$-0.723697\pi$
$359$	−24.4924	−1.29266	−0.646330	+	0.763058i	$0.723697\pi$
$360$	0	0
$361$	24.0540	1.26600
$362$	0	0
$363$	−2.56155	−0.134447
$364$	0	0
$365$	−13.1231	−0.686895
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−35.6155	−1.85407
$370$	0	0
$371$	−4.80776	−0.249607
$372$	0	0
$373$	−4.63068	−0.239768	−0.119884	−	0.992788i	$-0.538252\pi$
$373$	−4.63068	−0.239768	−0.119884	+	0.992788i	$0.538252\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	−23.3693	−1.20358
$378$	0	0
$379$	13.6155	0.699383	0.349691	−	0.936865i	$-0.386286\pi$
$379$	13.6155	0.699383	0.349691	+	0.936865i	$0.386286\pi$
$380$	0	0
$381$	44.4924	2.27942
$382$	0	0
$383$	−27.8617	−1.42367	−0.711834	−	0.702348i	$-0.752135\pi$
$383$	−27.8617	−1.42367	−0.711834	+	0.702348i	$0.752135\pi$
$384$	0	0
$385$	0.561553	0.0286194
$386$	0	0
$387$	11.1231	0.565419
$388$	0	0
$389$	−15.7538	−0.798749	−0.399374	−	0.916788i	$-0.630773\pi$
$389$	−15.7538	−0.798749	−0.399374	+	0.916788i	$0.630773\pi$
$390$	0	0
$391$	1.61553	0.0817008
$392$	0	0
$393$	−29.9309	−1.50981
$394$	0	0
$395$	9.12311	0.459033
$396$	0	0
$397$	38.4924	1.93188	0.965940	−	0.258767i	$-0.0833163\pi$
$397$	38.4924	1.93188	0.965940	+	0.258767i	$0.0833163\pi$
$398$	0	0
$399$	9.43845	0.472513
$400$	0	0
$401$	−1.19224	−0.0595374	−0.0297687	−	0.999557i	$-0.509477\pi$
$401$	−1.19224	−0.0595374	−0.0297687	+	0.999557i	$0.509477\pi$
$402$	0	0
$403$	18.8769	0.940325
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	10.8078	0.535721
$408$	0	0
$409$	−8.87689	−0.438934	−0.219467	−	0.975620i	$-0.570432\pi$
$409$	−8.87689	−0.438934	−0.219467	+	0.975620i	$0.570432\pi$
$410$	0	0
$411$	−13.7538	−0.678424
$412$	0	0
$413$	−6.38447	−0.314159
$414$	0	0
$415$	10.0000	0.490881
$416$	0	0
$417$	−42.2462	−2.06881
$418$	0	0
$419$	22.2462	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$419$	22.2462	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$420$	0	0
$421$	12.8769	0.627581	0.313791	−	0.949492i	$-0.398401\pi$
$421$	12.8769	0.627581	0.313791	+	0.949492i	$0.398401\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	1.43845	0.0697749
$426$	0	0
$427$	0.315342	0.0152604
$428$	0	0
$429$	13.1231	0.633590
$430$	0	0
$431$	12.4924	0.601739	0.300869	−	0.953665i	$-0.402723\pi$
$431$	12.4924	0.601739	0.300869	+	0.953665i	$0.402723\pi$
$432$	0	0
$433$	33.8617	1.62729	0.813646	−	0.581361i	$-0.197480\pi$
$433$	33.8617	1.62729	0.813646	+	0.581361i	$0.197480\pi$
$434$	0	0
$435$	−11.6847	−0.560236
$436$	0	0
$437$	−7.36932	−0.352522
$438$	0	0
$439$	−40.9848	−1.95610	−0.978050	−	0.208371i	$-0.933184\pi$
$439$	−40.9848	−1.95610	−0.978050	+	0.208371i	$0.933184\pi$
$440$	0	0
$441$	−23.8078	−1.13370
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	10.8078	0.512337
$446$	0	0
$447$	35.0540	1.65800
$448$	0	0
$449$	20.7386	0.978717	0.489358	−	0.872083i	$-0.337231\pi$
$449$	20.7386	0.978717	0.489358	+	0.872083i	$0.337231\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	−23.3693	−1.09799
$454$	0	0
$455$	−2.87689	−0.134871
$456$	0	0
$457$	−0.946025	−0.0442532	−0.0221266	−	0.999755i	$-0.507044\pi$
$457$	−0.946025	−0.0442532	−0.0221266	+	0.999755i	$0.507044\pi$
$458$	0	0
$459$	−2.06913	−0.0965787
$460$	0	0
$461$	0.0691303	0.00321972	0.00160986	−	0.999999i	$-0.499488\pi$
$461$	0.0691303	0.00321972	0.00160986	+	0.999999i	$0.499488\pi$
$462$	0	0
$463$	9.12311	0.423987	0.211993	−	0.977271i	$-0.432004\pi$
$463$	9.12311	0.423987	0.211993	+	0.977271i	$0.432004\pi$
$464$	0	0
$465$	9.43845	0.437698
$466$	0	0
$467$	3.68466	0.170506	0.0852528	−	0.996359i	$-0.472830\pi$
$467$	3.68466	0.170506	0.0852528	+	0.996359i	$0.472830\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−40.8078	−1.88032
$472$	0	0
$473$	−3.12311	−0.143601
$474$	0	0
$475$	−6.56155	−0.301065
$476$	0	0
$477$	−30.4924	−1.39615
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−55.3693	−2.52462
$482$	0	0
$483$	−1.61553	−0.0735091
$484$	0	0
$485$	−8.87689	−0.403079
$486$	0	0
$487$	17.7538	0.804501	0.402250	−	0.915530i	$-0.368228\pi$
$487$	17.7538	0.804501	0.402250	+	0.915530i	$0.368228\pi$
$488$	0	0
$489$	53.3002	2.41032
$490$	0	0
$491$	−4.80776	−0.216971	−0.108486	−	0.994098i	$-0.534600\pi$
$491$	−4.80776	−0.216971	−0.108486	+	0.994098i	$0.534600\pi$
$492$	0	0
$493$	−6.56155	−0.295517
$494$	0	0
$495$	3.56155	0.160080
$496$	0	0
$497$	3.68466	0.165280
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	35.0540	1.56610
$502$	0	0
$503$	−20.8769	−0.930855	−0.465427	−	0.885086i	$-0.654099\pi$
$503$	−20.8769	−0.930855	−0.465427	+	0.885086i	$0.654099\pi$
$504$	0	0
$505$	−0.246211	−0.0109563
$506$	0	0
$507$	−33.9309	−1.50692
$508$	0	0
$509$	−10.6307	−0.471197	−0.235598	−	0.971851i	$-0.575705\pi$
$509$	−10.6307	−0.471197	−0.235598	+	0.971851i	$0.575705\pi$
$510$	0	0
$511$	7.36932	0.325999
$512$	0	0
$513$	9.43845	0.416718
$514$	0	0
$515$	2.24621	0.0989799
$516$	0	0
$517$	1.12311	0.0493941
$518$	0	0
$519$	−8.63068	−0.378845
$520$	0	0
$521$	42.4924	1.86163	0.930813	−	0.365495i	$-0.119100\pi$
$521$	42.4924	1.86163	0.930813	+	0.365495i	$0.119100\pi$
$522$	0	0
$523$	1.36932	0.0598760	0.0299380	−	0.999552i	$-0.490469\pi$
$523$	1.36932	0.0598760	0.0299380	+	0.999552i	$0.490469\pi$
$524$	0	0
$525$	−1.43845	−0.0627790
$526$	0	0
$527$	5.30019	0.230880
$528$	0	0
$529$	−21.7386	−0.945158
$530$	0	0
$531$	−40.4924	−1.75722
$532$	0	0
$533$	−51.2311	−2.21906
$534$	0	0
$535$	−3.12311	−0.135024
$536$	0	0
$537$	−62.7386	−2.70737
$538$	0	0
$539$	6.68466	0.287929
$540$	0	0
$541$	7.43845	0.319804	0.159902	−	0.987133i	$-0.448882\pi$
$541$	7.43845	0.319804	0.159902	+	0.987133i	$0.448882\pi$
$542$	0	0
$543$	47.3693	2.03281
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	4.87689	0.208521	0.104260	−	0.994550i	$-0.466752\pi$
$547$	4.87689	0.208521	0.104260	+	0.994550i	$0.466752\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	29.9309	1.27510
$552$	0	0
$553$	−5.12311	−0.217857
$554$	0	0
$555$	−27.6847	−1.17515
$556$	0	0
$557$	−1.12311	−0.0475875	−0.0237938	−	0.999717i	$-0.507575\pi$
$557$	−1.12311	−0.0475875	−0.0237938	+	0.999717i	$0.507575\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	3.68466	0.155566
$562$	0	0
$563$	−8.24621	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$563$	−8.24621	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$564$	0	0
$565$	7.12311	0.299671
$566$	0	0
$567$	−3.93087	−0.165081
$568$	0	0
$569$	−11.1231	−0.466305	−0.233152	−	0.972440i	$-0.574904\pi$
$569$	−11.1231	−0.466305	−0.233152	+	0.972440i	$0.574904\pi$
$570$	0	0
$571$	−11.1922	−0.468380	−0.234190	−	0.972191i	$-0.575244\pi$
$571$	−11.1922	−0.468380	−0.234190	+	0.972191i	$0.575244\pi$
$572$	0	0
$573$	46.7386	1.95253
$574$	0	0
$575$	1.12311	0.0468367
$576$	0	0
$577$	2.49242	0.103761	0.0518805	−	0.998653i	$-0.483479\pi$
$577$	2.49242	0.103761	0.0518805	+	0.998653i	$0.483479\pi$
$578$	0	0
$579$	2.06913	0.0859901
$580$	0	0
$581$	−5.61553	−0.232971
$582$	0	0
$583$	8.56155	0.354583
$584$	0	0
$585$	−18.2462	−0.754388
$586$	0	0
$587$	−17.4384	−0.719762	−0.359881	−	0.932998i	$-0.617183\pi$
$587$	−17.4384	−0.719762	−0.359881	+	0.932998i	$0.617183\pi$
$588$	0	0
$589$	−24.1771	−0.996199
$590$	0	0
$591$	−4.49242	−0.184794
$592$	0	0
$593$	33.6155	1.38042	0.690212	−	0.723607i	$-0.257517\pi$
$593$	33.6155	1.38042	0.690212	+	0.723607i	$0.257517\pi$
$594$	0	0
$595$	−0.807764	−0.0331151
$596$	0	0
$597$	56.1771	2.29917
$598$	0	0
$599$	27.6847	1.13116	0.565582	−	0.824692i	$-0.308651\pi$
$599$	27.6847	1.13116	0.565582	+	0.824692i	$0.308651\pi$
$600$	0	0
$601$	−45.2311	−1.84501	−0.922507	−	0.385981i	$-0.873863\pi$
$601$	−45.2311	−1.84501	−0.922507	+	0.385981i	$0.873863\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−22.3153	−0.905752	−0.452876	−	0.891574i	$-0.649602\pi$
$607$	−22.3153	−0.905752	−0.452876	+	0.891574i	$0.649602\pi$
$608$	0	0
$609$	6.56155	0.265888
$610$	0	0
$611$	−5.75379	−0.232773
$612$	0	0
$613$	−13.6155	−0.549926	−0.274963	−	0.961455i	$-0.588666\pi$
$613$	−13.6155	−0.549926	−0.274963	+	0.961455i	$0.588666\pi$
$614$	0	0
$615$	−25.6155	−1.03292
$616$	0	0
$617$	11.1231	0.447799	0.223900	−	0.974612i	$-0.428121\pi$
$617$	11.1231	0.447799	0.223900	+	0.974612i	$0.428121\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−1.61553	−0.0648289
$622$	0	0
$623$	−6.06913	−0.243155
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.8078	−0.671237
$628$	0	0
$629$	−15.5464	−0.619875
$630$	0	0
$631$	6.56155	0.261211	0.130606	−	0.991434i	$-0.458308\pi$
$631$	6.56155	0.261211	0.130606	+	0.991434i	$0.458308\pi$
$632$	0	0
$633$	−29.9309	−1.18965
$634$	0	0
$635$	17.3693	0.689280
$636$	0	0
$637$	−34.2462	−1.35688
$638$	0	0
$639$	23.3693	0.924476
$640$	0	0
$641$	41.0540	1.62153	0.810767	−	0.585369i	$-0.199050\pi$
$641$	41.0540	1.62153	0.810767	+	0.585369i	$0.199050\pi$
$642$	0	0
$643$	31.6847	1.24952	0.624760	−	0.780816i	$-0.285197\pi$
$643$	31.6847	1.24952	0.624760	+	0.780816i	$0.285197\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−15.3693	−0.604230	−0.302115	−	0.953271i	$-0.597693\pi$
$647$	−15.3693	−0.604230	−0.302115	+	0.953271i	$0.597693\pi$
$648$	0	0
$649$	11.3693	0.446285
$650$	0	0
$651$	−5.30019	−0.207731
$652$	0	0
$653$	30.8078	1.20560	0.602800	−	0.797892i	$-0.294051\pi$
$653$	30.8078	1.20560	0.602800	+	0.797892i	$0.294051\pi$
$654$	0	0
$655$	−11.6847	−0.456557
$656$	0	0
$657$	46.7386	1.82345
$658$	0	0
$659$	−33.9309	−1.32176	−0.660880	−	0.750492i	$-0.729817\pi$
$659$	−33.9309	−1.32176	−0.660880	+	0.750492i	$0.729817\pi$
$660$	0	0
$661$	18.4924	0.719272	0.359636	−	0.933093i	$-0.382901\pi$
$661$	18.4924	0.719272	0.359636	+	0.933093i	$0.382901\pi$
$662$	0	0
$663$	−18.8769	−0.733118
$664$	0	0
$665$	3.68466	0.142885
$666$	0	0
$667$	−5.12311	−0.198367
$668$	0	0
$669$	−38.1080	−1.47334
$670$	0	0
$671$	−0.561553	−0.0216785
$672$	0	0
$673$	24.1771	0.931958	0.465979	−	0.884796i	$-0.345702\pi$
$673$	24.1771	0.931958	0.465979	+	0.884796i	$0.345702\pi$
$674$	0	0
$675$	−1.43845	−0.0553659
$676$	0	0
$677$	10.2462	0.393794	0.196897	−	0.980424i	$-0.436914\pi$
$677$	10.2462	0.393794	0.196897	+	0.980424i	$0.436914\pi$
$678$	0	0
$679$	4.98485	0.191301
$680$	0	0
$681$	−46.1080	−1.76686
$682$	0	0
$683$	16.8078	0.643131	0.321566	−	0.946887i	$-0.395791\pi$
$683$	16.8078	0.643131	0.321566	+	0.946887i	$0.395791\pi$
$684$	0	0
$685$	−5.36932	−0.205151
$686$	0	0
$687$	−67.8617	−2.58909
$688$	0	0
$689$	−43.8617	−1.67100
$690$	0	0
$691$	−5.61553	−0.213625	−0.106812	−	0.994279i	$-0.534064\pi$
$691$	−5.61553	−0.213625	−0.106812	+	0.994279i	$0.534064\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−16.4924	−0.625593
$696$	0	0
$697$	−14.3845	−0.544851
$698$	0	0
$699$	−60.6695	−2.29473
$700$	0	0
$701$	33.5464	1.26703	0.633515	−	0.773730i	$-0.281611\pi$
$701$	33.5464	1.26703	0.633515	+	0.773730i	$0.281611\pi$
$702$	0	0
$703$	70.9157	2.67464
$704$	0	0
$705$	−2.87689	−0.108350
$706$	0	0
$707$	0.138261	0.00519983
$708$	0	0
$709$	31.7538	1.19254	0.596269	−	0.802784i	$-0.296649\pi$
$709$	31.7538	1.19254	0.596269	+	0.802784i	$0.296649\pi$
$710$	0	0
$711$	−32.4924	−1.21856
$712$	0	0
$713$	4.13826	0.154979
$714$	0	0
$715$	5.12311	0.191593
$716$	0	0
$717$	23.3693	0.872743
$718$	0	0
$719$	4.94602	0.184456	0.0922278	−	0.995738i	$-0.470601\pi$
$719$	4.94602	0.184456	0.0922278	+	0.995738i	$0.470601\pi$
$720$	0	0
$721$	−1.26137	−0.0469757
$722$	0	0
$723$	−53.1231	−1.97567
$724$	0	0
$725$	−4.56155	−0.169412
$726$	0	0
$727$	17.6155	0.653324	0.326662	−	0.945141i	$-0.394076\pi$
$727$	17.6155	0.653324	0.326662	+	0.945141i	$0.394076\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	4.49242	0.166158
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	−17.1231	−0.631595
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−2.73863	−0.100742	−0.0503711	−	0.998731i	$-0.516040\pi$
$739$	−2.73863	−0.100742	−0.0503711	+	0.998731i	$0.516040\pi$
$740$	0	0
$741$	86.1080	3.16325
$742$	0	0
$743$	32.5616	1.19457	0.597284	−	0.802030i	$-0.296247\pi$
$743$	32.5616	1.19457	0.597284	+	0.802030i	$0.296247\pi$
$744$	0	0
$745$	13.6847	0.501367
$746$	0	0
$747$	−35.6155	−1.30310
$748$	0	0
$749$	1.75379	0.0640820
$750$	0	0
$751$	−22.5616	−0.823283	−0.411641	−	0.911346i	$-0.635044\pi$
$751$	−22.5616	−0.823283	−0.411641	+	0.911346i	$0.635044\pi$
$752$	0	0
$753$	38.1080	1.38873
$754$	0	0
$755$	−9.12311	−0.332024
$756$	0	0
$757$	6.49242	0.235971	0.117986	−	0.993015i	$-0.462356\pi$
$757$	6.49242	0.235971	0.117986	+	0.993015i	$0.462356\pi$
$758$	0	0
$759$	2.87689	0.104425
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	−1.12311	−0.0406592
$764$	0	0
$765$	−5.12311	−0.185226
$766$	0	0
$767$	−58.2462	−2.10315
$768$	0	0
$769$	−34.9848	−1.26159	−0.630793	−	0.775951i	$-0.717270\pi$
$769$	−34.9848	−1.26159	−0.630793	+	0.775951i	$0.717270\pi$
$770$	0	0
$771$	62.1080	2.23676
$772$	0	0
$773$	46.1771	1.66087	0.830437	−	0.557112i	$-0.188091\pi$
$773$	46.1771	1.66087	0.830437	+	0.557112i	$0.188091\pi$
$774$	0	0
$775$	3.68466	0.132357
$776$	0	0
$777$	15.5464	0.557724
$778$	0	0
$779$	65.6155	2.35092
$780$	0	0
$781$	−6.56155	−0.234791
$782$	0	0
$783$	6.56155	0.234491
$784$	0	0
$785$	−15.9309	−0.568597
$786$	0	0
$787$	35.1231	1.25200	0.626002	−	0.779822i	$-0.284690\pi$
$787$	35.1231	1.25200	0.626002	+	0.779822i	$0.284690\pi$
$788$	0	0
$789$	73.1619	2.60463
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	2.87689	0.102162
$794$	0	0
$795$	−21.9309	−0.777808
$796$	0	0
$797$	−36.7386	−1.30135	−0.650675	−	0.759357i	$-0.725514\pi$
$797$	−36.7386	−1.30135	−0.650675	+	0.759357i	$0.725514\pi$
$798$	0	0
$799$	−1.61553	−0.0571533
$800$	0	0
$801$	−38.4924	−1.36006
$802$	0	0
$803$	−13.1231	−0.463104
$804$	0	0
$805$	−0.630683	−0.0222287
$806$	0	0
$807$	79.7235	2.80640
$808$	0	0
$809$	−3.26137	−0.114664	−0.0573318	−	0.998355i	$-0.518259\pi$
$809$	−3.26137	−0.114664	−0.0573318	+	0.998355i	$0.518259\pi$
$810$	0	0
$811$	50.5616	1.77546	0.887728	−	0.460368i	$-0.152283\pi$
$811$	50.5616	1.77546	0.887728	+	0.460368i	$0.152283\pi$
$812$	0	0
$813$	30.7386	1.07805
$814$	0	0
$815$	20.8078	0.728864
$816$	0	0
$817$	−20.4924	−0.716939
$818$	0	0
$819$	10.2462	0.358032
$820$	0	0
$821$	36.2462	1.26500	0.632501	−	0.774560i	$-0.282029\pi$
$821$	36.2462	1.26500	0.632501	+	0.774560i	$0.282029\pi$
$822$	0	0
$823$	30.7386	1.07148	0.535741	−	0.844383i	$-0.320032\pi$
$823$	30.7386	1.07148	0.535741	+	0.844383i	$0.320032\pi$
$824$	0	0
$825$	2.56155	0.0891818
$826$	0	0
$827$	−51.6155	−1.79485	−0.897424	−	0.441169i	$-0.854564\pi$
$827$	−51.6155	−1.79485	−0.897424	+	0.441169i	$0.854564\pi$
$828$	0	0
$829$	−9.86174	−0.342512	−0.171256	−	0.985227i	$-0.554783\pi$
$829$	−9.86174	−0.342512	−0.171256	+	0.985227i	$0.554783\pi$
$830$	0	0
$831$	78.7386	2.73141
$832$	0	0
$833$	−9.61553	−0.333158
$834$	0	0
$835$	13.6847	0.473577
$836$	0	0
$837$	−5.30019	−0.183201
$838$	0	0
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−8.19224	−0.282491
$842$	0	0
$843$	37.1231	1.27859
$844$	0	0
$845$	−13.2462	−0.455684
$846$	0	0
$847$	0.561553	0.0192952
$848$	0	0
$849$	37.1231	1.27406
$850$	0	0
$851$	−12.1383	−0.416094
$852$	0	0
$853$	34.1080	1.16783	0.583917	−	0.811813i	$-0.301519\pi$
$853$	34.1080	1.16783	0.583917	+	0.811813i	$0.301519\pi$
$854$	0	0
$855$	23.3693	0.799214
$856$	0	0
$857$	12.9460	0.442228	0.221114	−	0.975248i	$-0.429031\pi$
$857$	12.9460	0.442228	0.221114	+	0.975248i	$0.429031\pi$
$858$	0	0
$859$	3.36932	0.114960	0.0574798	−	0.998347i	$-0.481694\pi$
$859$	3.36932	0.114960	0.0574798	+	0.998347i	$0.481694\pi$
$860$	0	0
$861$	14.3845	0.490221
$862$	0	0
$863$	15.3693	0.523178	0.261589	−	0.965179i	$-0.415754\pi$
$863$	15.3693	0.523178	0.261589	+	0.965179i	$0.415754\pi$
$864$	0	0
$865$	−3.36932	−0.114560
$866$	0	0
$867$	38.2462	1.29891
$868$	0	0
$869$	9.12311	0.309480
$870$	0	0
$871$	0	0
$872$	0	0
$873$	31.6155	1.07002
$874$	0	0
$875$	−0.561553	−0.0189839
$876$	0	0
$877$	30.8769	1.04264	0.521319	−	0.853362i	$-0.325440\pi$
$877$	30.8769	1.04264	0.521319	+	0.853362i	$0.325440\pi$
$878$	0	0
$879$	7.36932	0.248561
$880$	0	0
$881$	−1.50758	−0.0507916	−0.0253958	−	0.999677i	$-0.508085\pi$
$881$	−1.50758	−0.0507916	−0.0253958	+	0.999677i	$0.508085\pi$
$882$	0	0
$883$	55.0540	1.85271	0.926357	−	0.376647i	$-0.122923\pi$
$883$	55.0540	1.85271	0.926357	+	0.376647i	$0.122923\pi$
$884$	0	0
$885$	−29.1231	−0.978962
$886$	0	0
$887$	30.6307	1.02848	0.514239	−	0.857647i	$-0.328074\pi$
$887$	30.6307	1.02848	0.514239	+	0.857647i	$0.328074\pi$
$888$	0	0
$889$	−9.75379	−0.327132
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	7.36932	0.246605
$894$	0	0
$895$	−24.4924	−0.818691
$896$	0	0
$897$	−14.7386	−0.492109
$898$	0	0
$899$	−16.8078	−0.560570
$900$	0	0
$901$	−12.3153	−0.410284
$902$	0	0
$903$	−4.49242	−0.149498
$904$	0	0
$905$	18.4924	0.614709
$906$	0	0
$907$	0.177081	0.00587988	0.00293994	−	0.999996i	$-0.499064\pi$
$907$	0.177081	0.00587988	0.00293994	+	0.999996i	$0.499064\pi$
$908$	0	0
$909$	0.876894	0.0290848
$910$	0	0
$911$	−36.6695	−1.21491	−0.607457	−	0.794352i	$-0.707810\pi$
$911$	−36.6695	−1.21491	−0.607457	+	0.794352i	$0.707810\pi$
$912$	0	0
$913$	10.0000	0.330952
$914$	0	0
$915$	1.43845	0.0475536
$916$	0	0
$917$	6.56155	0.216682
$918$	0	0
$919$	26.2462	0.865783	0.432891	−	0.901446i	$-0.357493\pi$
$919$	26.2462	0.865783	0.432891	+	0.901446i	$0.357493\pi$
$920$	0	0
$921$	−73.6155	−2.42571
$922$	0	0
$923$	33.6155	1.10647
$924$	0	0
$925$	−10.8078	−0.355357
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−27.9309	−0.916382	−0.458191	−	0.888854i	$-0.651502\pi$
$929$	−27.9309	−0.916382	−0.458191	+	0.888854i	$0.651502\pi$
$930$	0	0
$931$	43.8617	1.43751
$932$	0	0
$933$	−61.9309	−2.02753
$934$	0	0
$935$	1.43845	0.0470423
$936$	0	0
$937$	−4.63068	−0.151278	−0.0756389	−	0.997135i	$-0.524100\pi$
$937$	−4.63068	−0.151278	−0.0756389	+	0.997135i	$0.524100\pi$
$938$	0	0
$939$	0.630683	0.0205816
$940$	0	0
$941$	−32.5616	−1.06148	−0.530738	−	0.847536i	$-0.678085\pi$
$941$	−32.5616	−1.06148	−0.530738	+	0.847536i	$0.678085\pi$
$942$	0	0
$943$	−11.2311	−0.365734
$944$	0	0
$945$	0.807764	0.0262766
$946$	0	0
$947$	−12.9460	−0.420689	−0.210345	−	0.977627i	$-0.567459\pi$
$947$	−12.9460	−0.420689	−0.210345	+	0.977627i	$0.567459\pi$
$948$	0	0
$949$	67.2311	2.18241
$950$	0	0
$951$	−1.43845	−0.0466448
$952$	0	0
$953$	−40.6695	−1.31741	−0.658707	−	0.752399i	$-0.728896\pi$
$953$	−40.6695	−1.31741	−0.658707	+	0.752399i	$0.728896\pi$
$954$	0	0
$955$	18.2462	0.590434
$956$	0	0
$957$	−11.6847	−0.377711
$958$	0	0
$959$	3.01515	0.0973644
$960$	0	0
$961$	−17.4233	−0.562042
$962$	0	0
$963$	11.1231	0.358437
$964$	0	0
$965$	0.807764	0.0260028
$966$	0	0
$967$	−29.0540	−0.934313	−0.467156	−	0.884175i	$-0.654722\pi$
$967$	−29.0540	−0.934313	−0.467156	+	0.884175i	$0.654722\pi$
$968$	0	0
$969$	24.1771	0.776680
$970$	0	0
$971$	14.2462	0.457183	0.228591	−	0.973522i	$-0.426588\pi$
$971$	14.2462	0.457183	0.228591	+	0.973522i	$0.426588\pi$
$972$	0	0
$973$	9.26137	0.296906
$974$	0	0
$975$	−13.1231	−0.420276
$976$	0	0
$977$	−26.6307	−0.851991	−0.425996	−	0.904725i	$-0.640076\pi$
$977$	−26.6307	−0.851991	−0.425996	+	0.904725i	$0.640076\pi$
$978$	0	0
$979$	10.8078	0.345418
$980$	0	0
$981$	−7.12311	−0.227423
$982$	0	0
$983$	−29.1231	−0.928883	−0.464441	−	0.885604i	$-0.653745\pi$
$983$	−29.1231	−0.928883	−0.464441	+	0.885604i	$0.653745\pi$
$984$	0	0
$985$	−1.75379	−0.0558804
$986$	0	0
$987$	1.61553	0.0514228
$988$	0	0
$989$	3.50758	0.111534
$990$	0	0
$991$	19.5076	0.619679	0.309839	−	0.950789i	$-0.399725\pi$
$991$	19.5076	0.619679	0.309839	+	0.950789i	$0.399725\pi$
$992$	0	0
$993$	36.4924	1.15805
$994$	0	0
$995$	21.9309	0.695255
$996$	0	0
$997$	−52.9848	−1.67805	−0.839023	−	0.544095i	$-0.816873\pi$
$997$	−52.9848	−1.67805	−0.839023	+	0.544095i	$0.816873\pi$
$998$	0	0
$999$	15.5464	0.491866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.a.l.1.1		2
3.2	odd	2		7920.2.a.ca.1.2		2
4.3	odd	2		440.2.a.f.1.2	✓	2
5.2	odd	4		4400.2.b.u.4049.4		4
5.3	odd	4		4400.2.b.u.4049.1		4
5.4	even	2		4400.2.a.br.1.2		2
8.3	odd	2		3520.2.a.bl.1.1		2
8.5	even	2		3520.2.a.bs.1.2		2
11.10	odd	2		9680.2.a.bl.1.1		2
12.11	even	2		3960.2.a.be.1.1		2
20.3	even	4		2200.2.b.h.1849.4		4
20.7	even	4		2200.2.b.h.1849.1		4
20.19	odd	2		2200.2.a.m.1.1		2
44.43	even	2		4840.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.f.1.2	✓	2	4.3	odd	2
880.2.a.l.1.1		2	1.1	even	1	trivial
2200.2.a.m.1.1		2	20.19	odd	2
2200.2.b.h.1849.1		4	20.7	even	4
2200.2.b.h.1849.4		4	20.3	even	4
3520.2.a.bl.1.1		2	8.3	odd	2
3520.2.a.bs.1.2		2	8.5	even	2
3960.2.a.be.1.1		2	12.11	even	2
4400.2.a.br.1.2		2	5.4	even	2
4400.2.b.u.4049.1		4	5.3	odd	4
4400.2.b.u.4049.4		4	5.2	odd	4
4840.2.a.n.1.2		2	44.43	even	2
7920.2.a.ca.1.2		2	3.2	odd	2
9680.2.a.bl.1.1		2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} -1.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} +5.12311 q^{13} +2.56155 q^{15} +1.43845 q^{17} -6.56155 q^{19} -1.43845 q^{21} +1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -4.56155 q^{29} +3.68466 q^{31} +2.56155 q^{33} -0.561553 q^{35} -10.8078 q^{37} -13.1231 q^{39} -10.0000 q^{41} +3.12311 q^{43} -3.56155 q^{45} -1.12311 q^{47} -6.68466 q^{49} -3.68466 q^{51} -8.56155 q^{53} +1.00000 q^{55} +16.8078 q^{57} -11.3693 q^{59} +0.561553 q^{61} +2.00000 q^{63} -5.12311 q^{65} -2.87689 q^{69} +6.56155 q^{71} +13.1231 q^{73} -2.56155 q^{75} -0.561553 q^{77} -9.12311 q^{79} -7.00000 q^{81} -10.0000 q^{83} -1.43845 q^{85} +11.6847 q^{87} -10.8078 q^{89} +2.87689 q^{91} -9.43845 q^{93} +6.56155 q^{95} +8.87689 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 - 3 * q^7 + 3 * q^9	\( 2 q - q^{3} - 2 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + q^{15} + 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} + 2 q^{25} - 7 q^{27} - 5 q^{29} - 5 q^{31} + q^{33} + 3 q^{35} - q^{37} - 18 q^{39} - 20 q^{41} - 2 q^{43} - 3 q^{45} + 6 q^{47} - q^{49} + 5 q^{51} - 13 q^{53} + 2 q^{55} + 13 q^{57} + 2 q^{59} - 3 q^{61} + 4 q^{63} - 2 q^{65} - 14 q^{69} + 9 q^{71} + 18 q^{73} - q^{75} + 3 q^{77} - 10 q^{79} - 14 q^{81} - 20 q^{83} - 7 q^{85} + 11 q^{87} - q^{89} + 14 q^{91} - 23 q^{93} + 9 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + q^15 + 7 * q^17 - 9 * q^19 - 7 * q^21 - 6 * q^23 + 2 * q^25 - 7 * q^27 - 5 * q^29 - 5 * q^31 + q^33 + 3 * q^35 - q^37 - 18 * q^39 - 20 * q^41 - 2 * q^43 - 3 * q^45 + 6 * q^47 - q^49 + 5 * q^51 - 13 * q^53 + 2 * q^55 + 13 * q^57 + 2 * q^59 - 3 * q^61 + 4 * q^63 - 2 * q^65 - 14 * q^69 + 9 * q^71 + 18 * q^73 - q^75 + 3 * q^77 - 10 * q^79 - 14 * q^81 - 20 * q^83 - 7 * q^85 + 11 * q^87 - q^89 + 14 * q^91 - 23 * q^93 + 9 * q^95 + 26 * q^97 - 3 * q^99

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