LMFDB - Embedded newform 550.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [550,2,Mod(1,550)] 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("550.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	550.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-2,1,0,2,-4,-1,1,0,-1,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.39177211117$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	550.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} -7.00000 q^{19} +8.00000 q^{21} +1.00000 q^{22} +3.00000 q^{23} +2.00000 q^{24} -5.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} -10.0000 q^{39} +12.0000 q^{41} -8.00000 q^{42} +5.00000 q^{43} -1.00000 q^{44} -3.00000 q^{46} -2.00000 q^{48} +9.00000 q^{49} +5.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} +4.00000 q^{56} +14.0000 q^{57} -3.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +14.0000 q^{67} -6.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} +4.00000 q^{74} -7.00000 q^{76} +4.00000 q^{77} +10.0000 q^{78} -4.00000 q^{79} -11.0000 q^{81} -12.0000 q^{82} -15.0000 q^{83} +8.00000 q^{84} -5.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +3.00000 q^{89} -20.0000 q^{91} +3.00000 q^{92} -10.0000 q^{93} +2.00000 q^{96} -13.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.00000	−0.577350
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	1.00000	0.213201
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−5.00000	−0.980581
$27$	4.00000	0.769800
$28$	−4.00000	−0.755929
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	7.00000	1.13555
$39$	−10.0000	−1.60128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−8.00000	−1.23443
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.00000	−0.288675
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	5.00000	0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	4.00000	0.534522
$57$	14.0000	1.85435
$58$	−3.00000	−0.393919
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−5.00000	−0.635001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−1.00000	−0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−7.00000	−0.802955
$77$	4.00000	0.455842
$78$	10.0000	1.13228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−12.0000	−1.32518
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	−5.00000	−0.539164
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−20.0000	−2.09657
$92$	3.00000	0.312772
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	2.00000	0.204124
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−9.00000	−0.909137
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	−6.00000	−0.582772
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	4.00000	0.384900
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−14.0000	−1.31122
$115$	0	0
$116$	3.00000	0.278543
$117$	5.00000	0.462250
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	−24.0000	−2.16401
$124$	5.00000	0.449013
$125$	0	0
$126$	4.00000	0.356348
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	2.00000	0.174078
$133$	28.0000	2.42791
$134$	−14.0000	−1.20942
$135$	0	0
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	6.00000	0.510754
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	0	0
$142$	−3.00000	−0.251754
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	−18.0000	−1.48461
$148$	−4.00000	−0.328798
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−10.0000	−0.800641
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	4.00000	0.318223
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−12.0000	−0.945732
$162$	11.0000	0.864242
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	15.0000	1.16423
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	−8.00000	−0.617213
$169$	12.0000	0.923077
$170$	0	0
$171$	−7.00000	−0.535303
$172$	5.00000	0.381246
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−24.0000	−1.80395
$178$	−3.00000	−0.224860
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	20.0000	1.48250
$183$	20.0000	1.47844
$184$	−3.00000	−0.221163
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−2.00000	−0.144338
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	13.0000	0.933346
$195$	0	0
$196$	9.00000	0.642857
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	1.00000	0.0710669
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	−28.0000	−1.97497
$202$	9.00000	0.633238
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	7.00000	0.487713
$207$	3.00000	0.208514
$208$	5.00000	0.346688
$209$	7.00000	0.484200
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	−6.00000	−0.411113
$214$	−15.0000	−1.02538
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−20.0000	−1.35769
$218$	−5.00000	−0.338643
$219$	−16.0000	−1.08118
$220$	0	0
$221$	0	0
$222$	−8.00000	−0.536925
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	6.00000	0.399114
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	14.0000	0.927173
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	−3.00000	−0.196960
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−5.00000	−0.326860
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−1.00000	−0.0642824
$243$	10.0000	0.641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	24.0000	1.53018
$247$	−35.0000	−2.22700
$248$	−5.00000	−0.317500
$249$	30.0000	1.90117
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−4.00000	−0.251976
$253$	−3.00000	−0.188608
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	10.0000	0.622573
$259$	16.0000	0.994192
$260$	0	0
$261$	3.00000	0.185695
$262$	−21.0000	−1.29738
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	−28.0000	−1.71679
$267$	−6.00000	−0.367194
$268$	14.0000	0.855186
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	40.0000	2.42091
$274$	−3.00000	−0.181237
$275$	0	0
$276$	−6.00000	−0.361158
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	13.0000	0.779688
$279$	5.00000	0.299342
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	5.00000	0.295656
$287$	−48.0000	−2.83335
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	26.0000	1.52415
$292$	8.00000	0.468165
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	4.00000	0.232495
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	15.0000	0.867472
$300$	0	0
$301$	−20.0000	−1.15278
$302$	16.0000	0.920697
$303$	18.0000	1.03407
$304$	−7.00000	−0.401478
$305$	0	0
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	4.00000	0.227921
$309$	14.0000	0.796432
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	10.0000	0.566139
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−4.00000	−0.225018
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	12.0000	0.672927
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−30.0000	−1.67444
$322$	12.0000	0.668734
$323$	0	0
$324$	−11.0000	−0.611111
$325$	0	0
$326$	16.0000	0.886158
$327$	−10.0000	−0.553001
$328$	−12.0000	−0.662589
$329$	0	0
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	−15.0000	−0.823232
$333$	−4.00000	−0.219199
$334$	−24.0000	−1.31322
$335$	0	0
$336$	8.00000	0.436436
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−12.0000	−0.652714
$339$	12.0000	0.651751
$340$	0	0
$341$	−5.00000	−0.270765
$342$	7.00000	0.378517
$343$	−8.00000	−0.431959
$344$	−5.00000	−0.269582
$345$	0	0
$346$	−21.0000	−1.12897
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	1.00000	0.0533002
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	24.0000	1.27559
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−20.0000	−1.05118
$363$	−2.00000	−0.104973
$364$	−20.0000	−1.04828
$365$	0	0
$366$	−20.0000	−1.04542
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	3.00000	0.156386
$369$	12.0000	0.624695
$370$	0	0
$371$	−24.0000	−1.24602
$372$	−10.0000	−0.518476
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.0000	0.772539
$378$	16.0000	0.822951
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	−15.0000	−0.767467
$383$	−3.00000	−0.153293	−0.0766464	−	0.997058i	$-0.524421\pi$
$383$	−3.00000	−0.153293	−0.0766464	+	0.997058i	$0.524421\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	5.00000	0.254164
$388$	−13.0000	−0.659975
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−42.0000	−2.11862
$394$	−3.00000	−0.151138
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−11.0000	−0.551380
$399$	−56.0000	−2.80351
$400$	0	0
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	28.0000	1.39651
$403$	25.0000	1.24534
$404$	−9.00000	−0.447767
$405$	0	0
$406$	12.0000	0.595550
$407$	4.00000	0.198273
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	−7.00000	−0.344865
$413$	−48.0000	−2.36193
$414$	−3.00000	−0.147442
$415$	0	0
$416$	−5.00000	−0.245145
$417$	26.0000	1.27323
$418$	−7.00000	−0.342381
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	6.00000	0.290701
$427$	40.0000	1.93574
$428$	15.0000	0.725052
$429$	10.0000	0.482805
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	4.00000	0.192450
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	5.00000	0.239457
$437$	−21.0000	−1.00457
$438$	16.0000	0.764510
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−8.00000	−0.378811
$447$	12.0000	0.567581
$448$	−4.00000	−0.188982
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−6.00000	−0.282216
$453$	32.0000	1.50349
$454$	3.00000	0.140797
$455$	0	0
$456$	−14.0000	−0.655610
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	8.00000	0.372194
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	18.0000	0.833834
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	5.00000	0.231125
$469$	−56.0000	−2.58584
$470$	0	0
$471$	−28.0000	−1.29017
$472$	−12.0000	−0.552345
$473$	−5.00000	−0.229900
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	6.00000	0.274434
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	10.0000	0.455488
$483$	24.0000	1.09204
$484$	1.00000	0.0454545
$485$	0	0
$486$	−10.0000	−0.453609
$487$	29.0000	1.31412	0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000	1.31412	0.657058	+	0.753840i	$0.271801\pi$
$488$	10.0000	0.452679
$489$	32.0000	1.44709
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	−24.0000	−1.08200
$493$	0	0
$494$	35.0000	1.57472
$495$	0	0
$496$	5.00000	0.224507
$497$	−12.0000	−0.538274
$498$	−30.0000	−1.34433
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	−24.0000	−1.07117
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	3.00000	0.133366
$507$	−24.0000	−1.06588
$508$	−16.0000	−0.709885
$509$	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
$511$	−32.0000	−1.41560
$512$	−1.00000	−0.0441942
$513$	−28.0000	−1.23623
$514$	9.00000	0.396973
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	−16.0000	−0.703000
$519$	−42.0000	−1.84360
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	−3.00000	−0.131306
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	21.0000	0.917389
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.00000	0.0870388
$529$	−14.0000	−0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	28.0000	1.21395
$533$	60.0000	2.59889
$534$	6.00000	0.259645
$535$	0	0
$536$	−14.0000	−0.604708
$537$	−24.0000	−1.03568
$538$	−30.0000	−1.29339
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	−2.00000	−0.0859074
$543$	−40.0000	−1.71656
$544$	0	0
$545$	0	0
$546$	−40.0000	−1.71184
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	3.00000	0.128154
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−21.0000	−0.894630
$552$	6.00000	0.255377
$553$	16.0000	0.680389
$554$	−26.0000	−1.10463
$555$	0	0
$556$	−13.0000	−0.551323
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	−5.00000	−0.211667
$559$	25.0000	1.05739
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	44.0000	1.84783
$568$	−3.00000	−0.125877
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	−5.00000	−0.209061
$573$	−30.0000	−1.25327
$574$	48.0000	2.00348
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	17.0000	0.707107
$579$	8.00000	0.332469
$580$	0	0
$581$	60.0000	2.48922
$582$	−26.0000	−1.07773
$583$	−6.00000	−0.248495
$584$	−8.00000	−0.331042
$585$	0	0
$586$	18.0000	0.743573
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−18.0000	−0.742307
$589$	−35.0000	−1.44215
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−4.00000	−0.164399
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−22.0000	−0.900400
$598$	−15.0000	−0.613396
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	20.0000	0.815139
$603$	14.0000	0.570124
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−18.0000	−0.731200
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	7.00000	0.283887
$609$	24.0000	0.972529
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	−14.0000	−0.563163
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	12.0000	0.481543
$622$	−3.00000	−0.120289
$623$	−12.0000	−0.480770
$624$	−10.0000	−0.400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	−14.0000	−0.559106
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	4.00000	0.159111
$633$	8.00000	0.317971
$634$	−12.0000	−0.476581
$635$	0	0
$636$	−12.0000	−0.475831
$637$	45.0000	1.78296
$638$	3.00000	0.118771
$639$	3.00000	0.118678
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	30.0000	1.18401
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	11.0000	0.432121
$649$	−12.0000	−0.471041
$650$	0	0
$651$	40.0000	1.56772
$652$	−16.0000	−0.626608
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	12.0000	0.468521
$657$	8.00000	0.312110
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	−32.0000	−1.24372
$663$	0	0
$664$	15.0000	0.582113
$665$	0	0
$666$	4.00000	0.154997
$667$	9.00000	0.348481
$668$	24.0000	0.928588
$669$	−16.0000	−0.618596
$670$	0	0
$671$	10.0000	0.386046
$672$	−8.00000	−0.308607
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	12.0000	0.461538
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	−12.0000	−0.460857
$679$	52.0000	1.99558
$680$	0	0
$681$	6.00000	0.229920
$682$	5.00000	0.191460
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	8.00000	0.305441
$687$	−4.00000	−0.152610
$688$	5.00000	0.190623
$689$	30.0000	1.14291
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	21.0000	0.798300
$693$	4.00000	0.151947
$694$	12.0000	0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	−11.0000	−0.416356
$699$	36.0000	1.36165
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	−20.0000	−0.754851
$703$	28.0000	1.05604
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	9.00000	0.338719
$707$	36.0000	1.35392
$708$	−24.0000	−0.901975
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−3.00000	−0.112430
$713$	15.0000	0.561754
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	12.0000	0.448148
$718$	24.0000	0.895672
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	−30.0000	−1.11648
$723$	20.0000	0.743808
$724$	20.0000	0.743294
$725$	0	0
$726$	2.00000	0.0742270
$727$	5.00000	0.185440	0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000	0.185440	0.0927199	+	0.995692i	$0.470444\pi$
$728$	20.0000	0.741249
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	20.0000	0.739221
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	−3.00000	−0.110581
$737$	−14.0000	−0.515697
$738$	−12.0000	−0.441726
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	70.0000	2.57151
$742$	24.0000	0.881068
$743$	54.0000	1.98107	0.990534	−	0.137268i	$-0.0438322\pi$
$743$	54.0000	1.98107	0.990534	+	0.137268i	$0.0438322\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−15.0000	−0.548821
$748$	0	0
$749$	−60.0000	−2.19235
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	−15.0000	−0.546268
$755$	0	0
$756$	−16.0000	−0.581914
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	−2.00000	−0.0726433
$759$	6.00000	0.217786
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−32.0000	−1.15924
$763$	−20.0000	−0.724049
$764$	15.0000	0.542681
$765$	0	0
$766$	3.00000	0.108394
$767$	60.0000	2.16647
$768$	−2.00000	−0.0721688
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−4.00000	−0.143963
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−5.00000	−0.179721
$775$	0	0
$776$	13.0000	0.466673
$777$	−32.0000	−1.14799
$778$	24.0000	0.860442
$779$	−84.0000	−3.00961
$780$	0	0
$781$	−3.00000	−0.107348
$782$	0	0
$783$	12.0000	0.428845
$784$	9.00000	0.321429
$785$	0	0
$786$	42.0000	1.49809
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	3.00000	0.106871
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	1.00000	0.0355335
$793$	−50.0000	−1.77555
$794$	−14.0000	−0.496841
$795$	0	0
$796$	11.0000	0.389885
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	56.0000	1.98238
$799$	0	0
$800$	0	0
$801$	3.00000	0.106000
$802$	9.00000	0.317801
$803$	−8.00000	−0.282314
$804$	−28.0000	−0.987484
$805$	0	0
$806$	−25.0000	−0.880587
$807$	−60.0000	−2.11210
$808$	9.00000	0.316619
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−12.0000	−0.421117
$813$	−4.00000	−0.140286
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	−35.0000	−1.22449
$818$	4.00000	0.139857
$819$	−20.0000	−0.698857
$820$	0	0
$821$	−9.00000	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$821$	−9.00000	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$822$	6.00000	0.209274
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	48.0000	1.67013
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	3.00000	0.104257
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−52.0000	−1.80386
$832$	5.00000	0.173344
$833$	0	0
$834$	−26.0000	−0.900306
$835$	0	0
$836$	7.00000	0.242100
$837$	20.0000	0.691301
$838$	−30.0000	−1.03633
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−26.0000	−0.896019
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	−4.00000	−0.137442
$848$	6.00000	0.206041
$849$	8.00000	0.274559
$850$	0	0
$851$	−12.0000	−0.411355
$852$	−6.00000	−0.205557
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	−15.0000	−0.512689
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	−10.0000	−0.341394
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	96.0000	3.27167
$862$	−12.0000	−0.408722
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−11.0000	−0.373795
$867$	34.0000	1.15470
$868$	−20.0000	−0.678844
$869$	4.00000	0.135691
$870$	0	0
$871$	70.0000	2.37186
$872$	−5.00000	−0.169321
$873$	−13.0000	−0.439983
$874$	21.0000	0.710336
$875$	0	0
$876$	−16.0000	−0.540590
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	−20.0000	−0.674967
$879$	36.0000	1.21425
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	−9.00000	−0.303046
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	−8.00000	−0.268462
$889$	64.0000	2.14649
$890$	0	0
$891$	11.0000	0.368514
$892$	8.00000	0.267860
$893$	0	0
$894$	−12.0000	−0.401340
$895$	0	0
$896$	4.00000	0.133631
$897$	−30.0000	−1.00167
$898$	27.0000	0.901002
$899$	15.0000	0.500278
$900$	0	0
$901$	0	0
$902$	12.0000	0.399556
$903$	40.0000	1.33112
$904$	6.00000	0.199557
$905$	0	0
$906$	−32.0000	−1.06313
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	−3.00000	−0.0995585
$909$	−9.00000	−0.298511
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	14.0000	0.463586
$913$	15.0000	0.496428
$914$	28.0000	0.926158
$915$	0	0
$916$	2.00000	0.0660819
$917$	−84.0000	−2.77392
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	6.00000	0.197599
$923$	15.0000	0.493731
$924$	−8.00000	−0.263181
$925$	0	0
$926$	13.0000	0.427207
$927$	−7.00000	−0.229910
$928$	−3.00000	−0.0984798
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	−63.0000	−2.06474
$932$	−18.0000	−0.589610
$933$	−6.00000	−0.196431
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−5.00000	−0.163430
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	56.0000	1.82846
$939$	−52.0000	−1.69696
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	28.0000	0.912289
$943$	36.0000	1.17232
$944$	12.0000	0.390567
$945$	0	0
$946$	5.00000	0.162564
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	8.00000	0.259828
$949$	40.0000	1.29845
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−6.00000	−0.194054
$957$	6.00000	0.193952
$958$	−6.00000	−0.193851
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	20.0000	0.644826
$963$	15.0000	0.483368
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−24.0000	−0.772187
$967$	26.0000	0.836104	0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000	0.836104	0.418052	+	0.908423i	$0.362713\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	10.0000	0.320750
$973$	52.0000	1.66704
$974$	−29.0000	−0.929220
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−32.0000	−1.02325
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	5.00000	0.159638
$982$	−9.00000	−0.287202
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	24.0000	0.765092
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−35.0000	−1.11350
$989$	15.0000	0.476972
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−5.00000	−0.158750
$993$	−64.0000	−2.03098
$994$	12.0000	0.380617
$995$	0	0
$996$	30.0000	0.950586
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	10.0000	0.316544
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.2.a.a.1.1	✓	1
3.2	odd	2		4950.2.a.y.1.1		1
4.3	odd	2		4400.2.a.bc.1.1		1
5.2	odd	4		550.2.b.d.199.1		2
5.3	odd	4		550.2.b.d.199.2		2
5.4	even	2		550.2.a.m.1.1	yes	1
11.10	odd	2		6050.2.a.bb.1.1		1
15.2	even	4		4950.2.c.ba.199.2		2
15.8	even	4		4950.2.c.ba.199.1		2
15.14	odd	2		4950.2.a.u.1.1		1
20.3	even	4		4400.2.b.e.4049.2		2
20.7	even	4		4400.2.b.e.4049.1		2
20.19	odd	2		4400.2.a.d.1.1		1
55.54	odd	2		6050.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
550.2.a.a.1.1	✓	1	1.1	even	1	trivial
550.2.a.m.1.1	yes	1	5.4	even	2
550.2.b.d.199.1		2	5.2	odd	4
550.2.b.d.199.2		2	5.3	odd	4
4400.2.a.d.1.1		1	20.19	odd	2
4400.2.a.bc.1.1		1	4.3	odd	2
4400.2.b.e.4049.1		2	20.7	even	4
4400.2.b.e.4049.2		2	20.3	even	4
4950.2.a.u.1.1		1	15.14	odd	2
4950.2.a.y.1.1		1	3.2	odd	2
4950.2.c.ba.199.1		2	15.8	even	4
4950.2.c.ba.199.2		2	15.2	even	4
6050.2.a.n.1.1		1	55.54	odd	2
6050.2.a.bb.1.1		1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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