LMFDB - Embedded newform 2240.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.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:	$17.8864900528$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} +7.00000 q^{13} +1.00000 q^{15} -5.00000 q^{17} -4.00000 q^{19} +1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +5.00000 q^{29} -2.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} -7.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} +2.00000 q^{45} +9.00000 q^{47} +1.00000 q^{49} +5.00000 q^{51} -3.00000 q^{55} +4.00000 q^{57} -12.0000 q^{59} +6.00000 q^{61} +2.00000 q^{63} -7.00000 q^{65} -16.0000 q^{67} +4.00000 q^{69} -4.00000 q^{71} -14.0000 q^{73} -1.00000 q^{75} -3.00000 q^{77} -9.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} +5.00000 q^{85} -5.00000 q^{87} -6.00000 q^{89} -7.00000 q^{91} +2.00000 q^{93} +4.00000 q^{95} +3.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−7.00000	−1.12090
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−7.00000	−0.868243
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−7.00000	−0.733799
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−15.0000	−1.47799	−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000	−1.47799	−0.738997	+	0.673709i	$0.764700\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−14.0000	−1.29430
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	21.0000	1.75611
$144$	0	0
$145$	−5.00000	−0.415227
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	10.0000	0.808452
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−24.0000	−1.78391	−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000	−1.78391	−0.891953	+	0.452128i	$0.850665\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	7.00000	0.501280
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	−35.0000	−2.35435
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−25.0000	−1.65931	−0.829654	−	0.558278i	$-0.811462\pi$
$227$	−25.0000	−1.65931	−0.829654	+	0.558278i	$0.811462\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	9.00000	0.584613
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−28.0000	−1.78160
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	7.00000	0.423659
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	0	0
$283$	−15.0000	−0.891657	−0.445829	−	0.895118i	$-0.647091\pi$
$283$	−15.0000	−0.891657	−0.445829	+	0.895118i	$0.647091\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−3.00000	−0.175863
$292$	0	0
$293$	25.0000	1.46052	0.730258	−	0.683172i	$-0.239400\pi$
$293$	25.0000	1.46052	0.730258	+	0.683172i	$0.239400\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	−28.0000	−1.61928
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	13.0000	0.741949	0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000	0.741949	0.370975	+	0.928643i	$0.379024\pi$
$308$	0	0
$309$	15.0000	0.853320
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	15.0000	0.839839
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	7.00000	0.388290
$326$	0	0
$327$	9.00000	0.497701
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	35.0000	1.86816
$352$	0	0
$353$	7.00000	0.372572	0.186286	−	0.982496i	$-0.440355\pi$
$353$	7.00000	0.372572	0.186286	+	0.982496i	$0.440355\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	35.0000	1.80259
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.00000	0.452839
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	0	0
$423$	−18.0000	−0.875190
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−21.0000	−1.01389
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	5.00000	0.239732
$436$	0	0
$437$	16.0000	0.765384
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	−15.0000	−0.704761
$454$	0	0
$455$	7.00000	0.328165
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	−25.0000	−1.16690
$460$	0	0
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	−3.00000	−0.136223
$486$	0	0
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	−21.0000	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$499$	−21.0000	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	0	0
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−36.0000	−1.59882
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	15.0000	0.660979
$516$	0	0
$517$	27.0000	1.18746
$518$	0	0
$519$	−21.0000	−0.921798
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	42.0000	1.81922
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	24.0000	1.02994
$544$	0	0
$545$	9.00000	0.385518
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	0	0
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−20.0000	−0.852029
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	0	0
$559$	−56.0000	−2.36855
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	8.00000	0.336563
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	14.0000	0.578829
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−19.0000	−0.780236	−0.390118	−	0.920765i	$-0.627566\pi$
$593$	−19.0000	−0.780236	−0.390118	+	0.920765i	$0.627566\pi$
$594$	0	0
$595$	−5.00000	−0.204980
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	25.0000	1.02147	0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000	1.02147	0.510736	+	0.859738i	$0.329373\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	32.0000	1.30314
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−17.0000	−0.690009	−0.345004	−	0.938601i	$-0.612123\pi$
$607$	−17.0000	−0.690009	−0.345004	+	0.938601i	$0.612123\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	63.0000	2.54871
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	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
$621$	−20.0000	−0.802572
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	20.0000	0.793676
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	0	0
$663$	35.0000	1.35929
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−20.0000	−0.774403
$668$	0	0
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	18.0000	0.694882
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−21.0000	−0.807096	−0.403548	−	0.914959i	$-0.632223\pi$
$677$	−21.0000	−0.807096	−0.403548	+	0.914959i	$0.632223\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	0	0
$681$	25.0000	0.958002
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	0	0
$708$	0	0
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	0	0
$711$	18.0000	0.675053
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−21.0000	−0.785355
$716$	0	0
$717$	9.00000	0.336111
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	15.0000	0.558629
$722$	0	0
$723$	−28.0000	−1.04133
$724$	0	0
$725$	5.00000	0.185695
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	28.0000	1.02861
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−16.0000	−0.585409
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	19.0000	0.693320	0.346660	−	0.937991i	$-0.387316\pi$
$751$	19.0000	0.693320	0.346660	+	0.937991i	$0.387316\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−15.0000	−0.545906
$756$	0	0
$757$	52.0000	1.88997	0.944986	−	0.327111i	$-0.106075\pi$
$757$	52.0000	1.88997	0.944986	+	0.327111i	$0.106075\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	9.00000	0.325822
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	−84.0000	−3.03306
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	25.0000	0.893427
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−49.0000	−1.74666	−0.873331	−	0.487128i	$-0.838045\pi$
$787$	−49.0000	−1.74666	−0.873331	+	0.487128i	$0.838045\pi$
$788$	0	0
$789$	10.0000	0.356009
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	42.0000	1.49146
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.0000	−0.673015	−0.336507	−	0.941681i	$-0.609246\pi$
$797$	−19.0000	−0.673015	−0.336507	+	0.941681i	$0.609246\pi$
$798$	0	0
$799$	−45.0000	−1.59199
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−42.0000	−1.48215
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	0	0
$813$	−14.0000	−0.491001
$814$	0	0
$815$	−10.0000	−0.350285
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	14.0000	0.489200
$820$	0	0
$821$	43.0000	1.50071	0.750355	−	0.661035i	$-0.229882\pi$
$821$	43.0000	1.50071	0.750355	+	0.661035i	$0.229882\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	15.0000	0.519096
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	9.00000	0.309976
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	15.0000	0.514799
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	58.0000	1.98124	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	58.0000	1.98124	0.990621	+	0.136637i	$0.0436295\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	2.00000	0.0680808	0.0340404	−	0.999420i	$-0.489163\pi$
$863$	2.00000	0.0680808	0.0340404	+	0.999420i	$0.489163\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−27.0000	−0.915912
$870$	0	0
$871$	−112.000	−3.79498
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	−25.0000	−0.843229
$880$	0	0
$881$	8.00000	0.269527	0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000	0.269527	0.134763	+	0.990878i	$0.456973\pi$
$882$	0	0
$883$	−58.0000	−1.95186	−0.975928	−	0.218094i	$-0.930016\pi$
$883$	−58.0000	−1.95186	−0.975928	+	0.218094i	$0.930016\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	−36.0000	−1.20469
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	28.0000	0.934893
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35.0000	1.15454	0.577272	−	0.816552i	$-0.304117\pi$
$919$	35.0000	1.15454	0.577272	+	0.816552i	$0.304117\pi$
$920$	0	0
$921$	−13.0000	−0.428365
$922$	0	0
$923$	−28.0000	−0.921631
$924$	0	0
$925$	0	0
$926$	0	0
$927$	30.0000	0.985329
$928$	0	0
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	15.0000	0.490552
$936$	0	0
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	−98.0000	−3.18121
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	11.0000	0.355952
$956$	0	0
$957$	−15.0000	−0.484881
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	20.0000	0.644491
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	−42.0000	−1.35063	−0.675314	−	0.737530i	$-0.735992\pi$
$967$	−42.0000	−1.35063	−0.675314	+	0.737530i	$0.735992\pi$
$968$	0	0
$969$	−20.0000	−0.642493
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	6.00000	0.192351
$974$	0	0
$975$	−7.00000	−0.224179
$976$	0	0
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	−45.0000	−1.43528	−0.717639	−	0.696416i	$-0.754777\pi$
$983$	−45.0000	−1.43528	−0.717639	+	0.696416i	$0.754777\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	9.00000	0.286473
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	41.0000	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$997$	41.0000	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.f.1.1		1
4.3	odd	2		2240.2.a.s.1.1		1
8.3	odd	2		1120.2.a.f.1.1	✓	1
8.5	even	2		1120.2.a.m.1.1	yes	1
40.19	odd	2		5600.2.a.s.1.1		1
40.29	even	2		5600.2.a.e.1.1		1
56.13	odd	2		7840.2.a.f.1.1		1
56.27	even	2		7840.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.f.1.1	✓	1	8.3	odd	2
1120.2.a.m.1.1	yes	1	8.5	even	2
2240.2.a.f.1.1		1	1.1	even	1	trivial
2240.2.a.s.1.1		1	4.3	odd	2
5600.2.a.e.1.1		1	40.29	even	2
5600.2.a.s.1.1		1	40.19	odd	2
7840.2.a.f.1.1		1	56.13	odd	2
7840.2.a.r.1.1		1	56.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

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