LMFDB - Embedded newform 5850.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5850,2,Mod(1,5850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5850.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,0,0,-4,-1,0,0,-1,0,-1,4,0,1,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.7124851824$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5850.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} -1.00000 q^{11} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} -3.00000 q^{19} +1.00000 q^{22} +1.00000 q^{26} -4.00000 q^{28} +4.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} -8.00000 q^{37} +3.00000 q^{38} +5.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} -12.0000 q^{47} +9.00000 q^{49} -1.00000 q^{52} +10.0000 q^{53} +4.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} +8.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -9.00000 q^{67} +7.00000 q^{68} +8.00000 q^{71} +13.0000 q^{73} +8.00000 q^{74} -3.00000 q^{76} +4.00000 q^{77} +8.00000 q^{79} -5.00000 q^{82} -3.00000 q^{83} +4.00000 q^{86} +1.00000 q^{88} +11.0000 q^{89} +4.00000 q^{91} +12.0000 q^{94} -10.0000 q^{97} -9.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−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
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	−4.00000	−0.755929
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.00000	−1.20049
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	−1.00000	−0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	7.00000	0.848875
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−3.00000	−0.344124
$77$	4.00000	0.455842
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	−28.0000	−2.56676
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−8.00000	−0.724286
$123$	0	0
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	9.00000	0.777482
$135$	0	0
$136$	−7.00000	−0.600245
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	1.00000	0.0836242
$144$	0	0
$145$	0	0
$146$	−13.0000	−1.07589
$147$	0	0
$148$	−8.00000	−0.657596
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	3.00000	0.232845
$167$	10.0000	0.773823	0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000	0.773823	0.386912	+	0.922117i	$0.373542\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−11.0000	−0.824485
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.00000	−0.511891
$188$	−12.0000	−0.875190
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	8.00000	0.562878
$203$	−16.0000	−1.12298
$204$	0	0
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	3.00000	0.207514
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	3.00000	0.205076
$215$	0	0
$216$	0	0
$217$	−24.0000	−1.62923
$218$	12.0000	0.812743
$219$	0	0
$220$	0	0
$221$	−7.00000	−0.470871
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	11.0000	0.731709
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	−4.00000	−0.262613
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	28.0000	1.81497
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	3.00000	0.190885
$248$	−6.00000	−0.381000
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	0	0
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	0	0
$261$	0	0
$262$	4.00000	0.247121
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	0	0
$268$	−9.00000	−0.549762
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	15.0000	0.906183
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−7.00000	−0.419832
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−31.0000	−1.84276	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000	−1.84276	−0.921379	+	0.388664i	$0.872937\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	−20.0000	−1.18056
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	0	0
$292$	13.0000	0.760767
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	−16.0000	−0.926855
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	18.0000	1.03578
$303$	0	0
$304$	−3.00000	−0.172062
$305$	0	0
$306$	0	0
$307$	−27.0000	−1.54097	−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000	−1.54097	−0.770486	+	0.637457i	$0.779986\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−21.0000	−1.16847
$324$	0	0
$325$	0	0
$326$	−11.0000	−0.609234
$327$	0	0
$328$	−5.00000	−0.276079
$329$	48.0000	2.64633
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−3.00000	−0.164646
$333$	0	0
$334$	−10.0000	−0.547176
$335$	0	0
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−8.00000	−0.431959
$344$	4.00000	0.215666
$345$	0	0
$346$	10.0000	0.537603
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	11.0000	0.582999
$357$	0	0
$358$	23.0000	1.21559
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	20.0000	1.05118
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−40.0000	−2.07670
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	7.00000	0.361961
$375$	0	0
$376$	12.0000	0.618853
$377$	−4.00000	−0.206010
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	0	0
$382$	−18.0000	−0.920960
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	1.00000	0.0508987
$387$	0	0
$388$	−10.0000	−0.507673
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	−9.00000	−0.454569
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	18.0000	0.902258
$399$	0	0
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−8.00000	−0.398015
$405$	0	0
$406$	16.0000	0.794067
$407$	8.00000	0.396545
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	0	0
$412$	−6.00000	−0.295599
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	−3.00000	−0.146735
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	23.0000	1.11962
$423$	0	0
$424$	−10.0000	−0.485643
$425$	0	0
$426$	0	0
$427$	−32.0000	−1.54859
$428$	−3.00000	−0.145010
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	24.0000	1.15204
$435$	0	0
$436$	−12.0000	−0.574696
$437$	0	0
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	7.00000	0.332956
$443$	11.0000	0.522626	0.261313	−	0.965254i	$-0.415845\pi$
$443$	11.0000	0.522626	0.261313	+	0.965254i	$0.415845\pi$
$444$	0	0
$445$	0	0
$446$	26.0000	1.23114
$447$	0	0
$448$	−4.00000	−0.188982
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	−11.0000	−0.517396
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	−28.0000	−1.28338
$477$	0	0
$478$	−20.0000	−0.914779
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	25.0000	1.13872
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−8.00000	−0.362143
$489$	0	0
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	−3.00000	−0.134976
$495$	0	0
$496$	6.00000	0.269408
$497$	−32.0000	−1.43540
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	9.00000	0.401690
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	10.0000	0.443678
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	−52.0000	−2.30034
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	12.0000	0.527759
$518$	−32.0000	−1.40600
$519$	0	0
$520$	0	0
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	0	0
$523$	15.0000	0.655904	0.327952	−	0.944694i	$-0.393642\pi$
$523$	15.0000	0.655904	0.327952	+	0.944694i	$0.393642\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−28.0000	−1.22086
$527$	42.0000	1.82955
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	12.0000	0.520266
$533$	−5.00000	−0.216574
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	24.0000	1.03471
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−7.00000	−0.300123
$545$	0	0
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	−15.0000	−0.640768
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−32.0000	−1.36078
$554$	22.0000	0.934690
$555$	0	0
$556$	7.00000	0.296866
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	31.0000	1.30303
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	20.0000	0.834784
$575$	0	0
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−10.0000	−0.414158
$584$	−13.0000	−0.537944
$585$	0	0
$586$	−26.0000	−1.07405
$587$	9.00000	0.371470	0.185735	−	0.982600i	$-0.440533\pi$
$587$	9.00000	0.371470	0.185735	+	0.982600i	$0.440533\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−37.0000	−1.51941	−0.759704	−	0.650269i	$-0.774656\pi$
$593$	−37.0000	−1.51941	−0.759704	+	0.650269i	$0.774656\pi$
$594$	0	0
$595$	0	0
$596$	16.0000	0.655386
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	31.0000	1.26452	0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000	1.26452	0.632258	+	0.774758i	$0.282128\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	−18.0000	−0.732410
$605$	0	0
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	27.0000	1.08963
$615$	0	0
$616$	−4.00000	−0.161165
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.0000	−1.76282
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	16.0000	0.638470
$629$	−56.0000	−2.23287
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	−9.00000	−0.356593
$638$	4.00000	0.158362
$639$	0	0
$640$	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
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	21.0000	0.826234
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	11.0000	0.430793
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	−48.0000	−1.87123
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−13.0000	−0.505259
$663$	0	0
$664$	3.00000	0.116423
$665$	0	0
$666$	0	0
$667$	0	0
$668$	10.0000	0.386912
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	40.0000	1.53506
$680$	0	0
$681$	0	0
$682$	6.00000	0.229752
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−10.0000	−0.380970
$690$	0	0
$691$	13.0000	0.494543	0.247272	−	0.968946i	$-0.420466\pi$
$691$	13.0000	0.494543	0.247272	+	0.968946i	$0.420466\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	35.0000	1.32572
$698$	34.0000	1.28692
$699$	0	0
$700$	0	0
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−14.0000	−0.526897
$707$	32.0000	1.20348
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	−11.0000	−0.412242
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−23.0000	−0.859550
$717$	0	0
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	10.0000	0.372161
$723$	0	0
$724$	−20.0000	−0.743294
$725$	0	0
$726$	0	0
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	−4.00000	−0.148250
$729$	0	0
$730$	0	0
$731$	−28.0000	−1.03562
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	9.00000	0.331519
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	40.0000	1.46845
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	0	0
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−7.00000	−0.255945
$749$	12.0000	0.438470
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	−15.0000	−0.544825
$759$	0	0
$760$	0	0
$761$	−19.0000	−0.688749	−0.344375	−	0.938832i	$-0.611909\pi$
$761$	−19.0000	−0.688749	−0.344375	+	0.938832i	$0.611909\pi$
$762$	0	0
$763$	48.0000	1.73772
$764$	18.0000	0.651217
$765$	0	0
$766$	−6.00000	−0.216789
$767$	4.00000	0.144432
$768$	0	0
$769$	−53.0000	−1.91123	−0.955614	−	0.294620i	$-0.904807\pi$
$769$	−53.0000	−1.91123	−0.955614	+	0.294620i	$0.904807\pi$
$770$	0	0
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	−16.0000	−0.573628
$779$	−15.0000	−0.537431
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	0	0
$791$	44.0000	1.56446
$792$	0	0
$793$	−8.00000	−0.284088
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−18.0000	−0.637993
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	−84.0000	−2.97171
$800$	0	0
$801$	0	0
$802$	−9.00000	−0.317801
$803$	−13.0000	−0.458760
$804$	0	0
$805$	0	0
$806$	6.00000	0.211341
$807$	0	0
$808$	8.00000	0.281439
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−16.0000	−0.561490
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	12.0000	0.419827
$818$	−7.00000	−0.244749
$819$	0	0
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−16.0000	−0.556711
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	63.0000	2.18282
$834$	0	0
$835$	0	0
$836$	3.00000	0.103757
$837$	0	0
$838$	19.0000	0.656344
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	8.00000	0.275698
$843$	0	0
$844$	−23.0000	−0.791693
$845$	0	0
$846$	0	0
$847$	40.0000	1.37442
$848$	10.0000	0.343401
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	3.00000	0.102538
$857$	−37.0000	−1.26390	−0.631948	−	0.775011i	$-0.717744\pi$
$857$	−37.0000	−1.26390	−0.631948	+	0.775011i	$0.717744\pi$
$858$	0	0
$859$	−9.00000	−0.307076	−0.153538	−	0.988143i	$-0.549067\pi$
$859$	−9.00000	−0.307076	−0.153538	+	0.988143i	$0.549067\pi$
$860$	0	0
$861$	0	0
$862$	12.0000	0.408722
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	0	0
$866$	−25.0000	−0.849535
$867$	0	0
$868$	−24.0000	−0.814613
$869$	−8.00000	−0.271381
$870$	0	0
$871$	9.00000	0.304953
$872$	12.0000	0.406371
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	14.0000	0.472477
$879$	0	0
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	−7.00000	−0.235435
$885$	0	0
$886$	−11.0000	−0.369552
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	0	0
$891$	0	0
$892$	−26.0000	−0.870544
$893$	36.0000	1.20469
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	−9.00000	−0.300334
$899$	24.0000	0.800445
$900$	0	0
$901$	70.0000	2.33204
$902$	5.00000	0.166482
$903$	0	0
$904$	11.0000	0.365855
$905$	0	0
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	3.00000	0.0992855
$914$	25.0000	0.826927
$915$	0	0
$916$	−8.00000	−0.264327
$917$	16.0000	0.528367
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−8.00000	−0.263323
$924$	0	0
$925$	0	0
$926$	−10.0000	−0.328620
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−27.0000	−0.884889
$932$	18.0000	0.589610
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	−21.0000	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$938$	−36.0000	−1.17544
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	56.0000	1.81976	0.909878	−	0.414876i	$-0.136175\pi$
$947$	56.0000	1.81976	0.909878	+	0.414876i	$0.136175\pi$
$948$	0	0
$949$	−13.0000	−0.421998
$950$	0	0
$951$	0	0
$952$	28.0000	0.907485
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	0	0
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	14.0000	0.452319
$959$	60.0000	1.93750
$960$	0	0
$961$	5.00000	0.161290
$962$	−8.00000	−0.257930
$963$	0	0
$964$	−25.0000	−0.805196
$965$	0	0
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	0	0
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	0	0
$973$	−28.0000	−0.897639
$974$	4.00000	0.128168
$975$	0	0
$976$	8.00000	0.256074
$977$	−39.0000	−1.24772	−0.623860	−	0.781536i	$-0.714437\pi$
$977$	−39.0000	−1.24772	−0.623860	+	0.781536i	$0.714437\pi$
$978$	0	0
$979$	−11.0000	−0.351562
$980$	0	0
$981$	0	0
$982$	36.0000	1.14881
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	0	0
$986$	−28.0000	−0.891702
$987$	0	0
$988$	3.00000	0.0954427
$989$	0	0
$990$	0	0
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	−6.00000	−0.190500
$993$	0	0
$994$	32.0000	1.01498
$995$	0	0
$996$	0	0
$997$	16.0000	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.a.b.1.1		1
3.2	odd	2		650.2.a.i.1.1	yes	1
5.2	odd	4		5850.2.e.l.5149.1		2
5.3	odd	4		5850.2.e.l.5149.2		2
5.4	even	2		5850.2.a.bz.1.1		1
12.11	even	2		5200.2.a.ba.1.1		1
15.2	even	4		650.2.b.c.599.2		2
15.8	even	4		650.2.b.c.599.1		2
15.14	odd	2		650.2.a.e.1.1	✓	1
39.38	odd	2		8450.2.a.e.1.1		1
60.59	even	2		5200.2.a.j.1.1		1
195.194	odd	2		8450.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.a.e.1.1	✓	1	15.14	odd	2
650.2.a.i.1.1	yes	1	3.2	odd	2
650.2.b.c.599.1		2	15.8	even	4
650.2.b.c.599.2		2	15.2	even	4
5200.2.a.j.1.1		1	60.59	even	2
5200.2.a.ba.1.1		1	12.11	even	2
5850.2.a.b.1.1		1	1.1	even	1	trivial
5850.2.a.bz.1.1		1	5.4	even	2
5850.2.e.l.5149.1		2	5.2	odd	4
5850.2.e.l.5149.2		2	5.3	odd	4
8450.2.a.e.1.1		1	39.38	odd	2
8450.2.a.t.1.1		1	195.194	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more