LMFDB - Embedded newform 4600.2.e.n.4049.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,-8,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.n.4049.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +2.56155i q^{7} +0.561553 q^{9} -2.00000 q^{11} -3.56155i q^{13} -1.43845i q^{17} +2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +5.56155i q^{27} +8.12311 q^{29} +0.123106 q^{31} -3.12311i q^{33} -0.561553i q^{37} +5.56155 q^{39} +4.12311 q^{41} +6.24621i q^{43} +8.68466i q^{47} +0.438447 q^{49} +2.24621 q^{51} +8.56155i q^{53} +3.12311i q^{57} +1.43845 q^{59} -0.876894 q^{61} +1.43845i q^{63} +7.43845i q^{67} +1.56155 q^{69} -5.00000 q^{71} +7.56155i q^{73} -5.12311i q^{77} +0.876894 q^{79} -7.00000 q^{81} +7.68466i q^{83} +12.6847i q^{87} -8.00000 q^{89} +9.12311 q^{91} +0.192236i q^{93} -5.12311i q^{97} -1.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 8 q^{11} + 8 q^{19} - 16 q^{21} + 16 q^{29} - 16 q^{31} + 14 q^{39} + 10 q^{49} - 24 q^{51} + 14 q^{59} - 20 q^{61} - 2 q^{69} - 20 q^{71} + 20 q^{79} - 28 q^{81} - 32 q^{89} + 20 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 8 * q^11 + 8 * q^19 - 16 * q^21 + 16 * q^29 - 16 * q^31 + 14 * q^39 + 10 * q^49 - 24 * q^51 + 14 * q^59 - 20 * q^61 - 2 * q^69 - 20 * q^71 + 20 * q^79 - 28 * q^81 - 32 * q^89 + 20 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.56155i	0.968176i	0.875019	+	0.484088i	$0.160849\pi$
$7$	2.56155i	0.968176i	−0.875019	+	0.484088i	$0.839151\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 3.56155i	− 0.987797i	−0.869520	−	0.493899i	$-0.835571\pi$
$13$	− 3.56155i	− 0.987797i	0.869520	−	0.493899i	$-0.164429\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.43845i	− 0.348875i	−0.984668	−	0.174437i	$-0.944189\pi$
$17$	− 1.43845i	− 0.348875i	0.984668	−	0.174437i	$-0.0558106\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	8.12311	1.50842	0.754211	−	0.656632i	$-0.228019\pi$
$29$	8.12311	1.50842	0.754211	+	0.656632i	$0.228019\pi$
$30$	0	0
$31$	0.123106	0.0221104	0.0110552	−	0.999939i	$-0.496481\pi$
$31$	0.123106	0.0221104	0.0110552	+	0.999939i	$0.496481\pi$
$32$	0	0
$33$	− 3.12311i	− 0.543663i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.561553i	− 0.0923187i	−0.998934	−	0.0461594i	$-0.985302\pi$
$37$	− 0.561553i	− 0.0923187i	0.998934	−	0.0461594i	$-0.0146982\pi$
$38$	0	0
$39$	5.56155	0.890561
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	6.24621i	0.952538i	0.879300	+	0.476269i	$0.158011\pi$
$43$	6.24621i	0.952538i	−0.879300	+	0.476269i	$0.841989\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.68466i	1.26679i	0.773830	+	0.633394i	$0.218339\pi$
$47$	8.68466i	1.26679i	−0.773830	+	0.633394i	$0.781661\pi$
$48$	0	0
$49$	0.438447	0.0626353
$50$	0	0
$51$	2.24621	0.314532
$52$	0	0
$53$	8.56155i	1.17602i	0.808854	+	0.588010i	$0.200088\pi$
$53$	8.56155i	1.17602i	−0.808854	+	0.588010i	$0.799912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.12311i	0.413665i
$58$	0	0
$59$	1.43845	0.187270	0.0936349	−	0.995607i	$-0.470151\pi$
$59$	1.43845	0.187270	0.0936349	+	0.995607i	$0.470151\pi$
$60$	0	0
$61$	−0.876894	−0.112275	−0.0561374	−	0.998423i	$-0.517878\pi$
$61$	−0.876894	−0.112275	−0.0561374	+	0.998423i	$0.517878\pi$
$62$	0	0
$63$	1.43845i	0.181227i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.43845i	0.908751i	0.890810	+	0.454375i	$0.150138\pi$
$67$	7.43845i	0.908751i	−0.890810	+	0.454375i	$0.849862\pi$
$68$	0	0
$69$	1.56155	0.187989
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	7.56155i	0.885013i	0.896765	+	0.442506i	$0.145911\pi$
$73$	7.56155i	0.885013i	−0.896765	+	0.442506i	$0.854089\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.12311i	− 0.583832i
$78$	0	0
$79$	0.876894	0.0986583	0.0493292	−	0.998783i	$-0.484292\pi$
$79$	0.876894	0.0986583	0.0493292	+	0.998783i	$0.484292\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	7.68466i	0.843501i	0.906712	+	0.421750i	$0.138584\pi$
$83$	7.68466i	0.843501i	−0.906712	+	0.421750i	$0.861416\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.6847i	1.35994i
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	9.12311	0.956361
$92$	0	0
$93$	0.192236i	0.0199339i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.12311i	− 0.520173i	−0.965585	−	0.260086i	$-0.916249\pi$
$97$	− 5.12311i	− 0.520173i	0.965585	−	0.260086i	$-0.0837510\pi$
$98$	0	0
$99$	−1.12311	−0.112876
$100$	0	0
$101$	10.8078	1.07541	0.537706	−	0.843132i	$-0.319291\pi$
$101$	10.8078	1.07541	0.537706	+	0.843132i	$0.319291\pi$
$102$	0	0
$103$	− 14.2462i	− 1.40372i	−0.712314	−	0.701860i	$-0.752353\pi$
$103$	− 14.2462i	− 1.40372i	0.712314	−	0.701860i	$-0.247647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.68466i	0.162862i	0.996679	+	0.0814310i	$0.0259490\pi$
$107$	1.68466i	0.162862i	−0.996679	+	0.0814310i	$0.974051\pi$
$108$	0	0
$109$	1.12311	0.107574	0.0537870	−	0.998552i	$-0.482871\pi$
$109$	1.12311	0.107574	0.0537870	+	0.998552i	$0.482871\pi$
$110$	0	0
$111$	0.876894	0.0832311
$112$	0	0
$113$	5.43845i	0.511606i	0.966729	+	0.255803i	$0.0823399\pi$
$113$	5.43845i	0.511606i	−0.966729	+	0.255803i	$0.917660\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	3.68466	0.337772
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.43845i	0.580535i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.8078i	1.75765i	0.477140	+	0.878827i	$0.341674\pi$
$127$	19.8078i	1.75765i	−0.477140	+	0.878827i	$0.658326\pi$
$128$	0	0
$129$	−9.75379	−0.858773
$130$	0	0
$131$	−18.9309	−1.65400	−0.826999	−	0.562204i	$-0.809954\pi$
$131$	−18.9309	−1.65400	−0.826999	+	0.562204i	$0.809954\pi$
$132$	0	0
$133$	5.12311i	0.444230i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.6155i	− 1.84674i	−0.383912	−	0.923370i	$-0.625423\pi$
$137$	− 21.6155i	− 1.84674i	0.383912	−	0.923370i	$-0.374577\pi$
$138$	0	0
$139$	10.3693	0.879514	0.439757	−	0.898117i	$-0.355065\pi$
$139$	10.3693	0.879514	0.439757	+	0.898117i	$0.355065\pi$
$140$	0	0
$141$	−13.5616	−1.14209
$142$	0	0
$143$	7.12311i	0.595664i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.684658i	0.0564697i
$148$	0	0
$149$	−0.876894	−0.0718380	−0.0359190	−	0.999355i	$-0.511436\pi$
$149$	−0.876894	−0.0718380	−0.0359190	+	0.999355i	$0.511436\pi$
$150$	0	0
$151$	1.56155	0.127077	0.0635387	−	0.997979i	$-0.479761\pi$
$151$	1.56155	0.127077	0.0635387	+	0.997979i	$0.479761\pi$
$152$	0	0
$153$	− 0.807764i	− 0.0653039i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.6847i	1.25177i	0.779915	+	0.625886i	$0.215262\pi$
$157$	15.6847i	1.25177i	−0.779915	+	0.625886i	$0.784738\pi$
$158$	0	0
$159$	−13.3693	−1.06026
$160$	0	0
$161$	2.56155	0.201879
$162$	0	0
$163$	− 6.68466i	− 0.523583i	−0.965124	−	0.261791i	$-0.915687\pi$
$163$	− 6.68466i	− 0.523583i	0.965124	−	0.261791i	$-0.0843133\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	0.315342	0.0242570
$170$	0	0
$171$	1.12311	0.0858860
$172$	0	0
$173$	− 7.12311i	− 0.541560i	−0.962641	−	0.270780i	$-0.912718\pi$
$173$	− 7.12311i	− 0.541560i	0.962641	−	0.270780i	$-0.0872815\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.24621i	0.168836i
$178$	0	0
$179$	7.31534	0.546774	0.273387	−	0.961904i	$-0.411856\pi$
$179$	7.31534	0.546774	0.273387	+	0.961904i	$0.411856\pi$
$180$	0	0
$181$	−8.87689	−0.659814	−0.329907	−	0.944013i	$-0.607017\pi$
$181$	−8.87689	−0.659814	−0.329907	+	0.944013i	$0.607017\pi$
$182$	0	0
$183$	− 1.36932i	− 0.101223i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.87689i	0.210379i
$188$	0	0
$189$	−14.2462	−1.03626
$190$	0	0
$191$	−3.12311	−0.225980	−0.112990	−	0.993596i	$-0.536043\pi$
$191$	−3.12311	−0.225980	−0.112990	+	0.993596i	$0.536043\pi$
$192$	0	0
$193$	18.9309i	1.36267i	0.731970	+	0.681337i	$0.238601\pi$
$193$	18.9309i	1.36267i	−0.731970	+	0.681337i	$0.761399\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.8078i	− 1.26875i	−0.773025	−	0.634375i	$-0.781257\pi$
$197$	− 17.8078i	− 1.26875i	0.773025	−	0.634375i	$-0.218743\pi$
$198$	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
$201$	−11.6155	−0.819296
$202$	0	0
$203$	20.8078i	1.46042i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.561553i	− 0.0390306i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−9.93087	−0.683669	−0.341835	−	0.939760i	$-0.611048\pi$
$211$	−9.93087	−0.683669	−0.341835	+	0.939760i	$0.611048\pi$
$212$	0	0
$213$	− 7.80776i	− 0.534979i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.315342i	0.0214068i
$218$	0	0
$219$	−11.8078	−0.797895
$220$	0	0
$221$	−5.12311	−0.344617
$222$	0	0
$223$	18.8769i	1.26409i	0.774932	+	0.632045i	$0.217784\pi$
$223$	18.8769i	1.26409i	−0.774932	+	0.632045i	$0.782216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.2462i	− 1.21104i	−0.795829	−	0.605522i	$-0.792964\pi$
$227$	− 18.2462i	− 1.21104i	0.795829	−	0.605522i	$-0.207036\pi$
$228$	0	0
$229$	5.75379	0.380221	0.190111	−	0.981763i	$-0.439115\pi$
$229$	5.75379	0.380221	0.190111	+	0.981763i	$0.439115\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	28.3002i	1.85401i	0.375053	+	0.927003i	$0.377625\pi$
$233$	28.3002i	1.85401i	−0.375053	+	0.927003i	$0.622375\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.36932i	0.0889467i
$238$	0	0
$239$	7.87689	0.509514	0.254757	−	0.967005i	$-0.418005\pi$
$239$	7.87689	0.509514	0.254757	+	0.967005i	$0.418005\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7.12311i	− 0.453232i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	9.12311	0.575845	0.287923	−	0.957654i	$-0.407035\pi$
$251$	9.12311	0.575845	0.287923	+	0.957654i	$0.407035\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.684658i	0.0427078i	0.999772	+	0.0213539i	$0.00679768\pi$
$257$	0.684658i	0.0427078i	−0.999772	+	0.0213539i	$0.993202\pi$
$258$	0	0
$259$	1.43845	0.0893808
$260$	0	0
$261$	4.56155	0.282353
$262$	0	0
$263$	− 14.3153i	− 0.882722i	−0.897330	−	0.441361i	$-0.854496\pi$
$263$	− 14.3153i	− 0.882722i	0.897330	−	0.441361i	$-0.145504\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.4924i	− 0.764524i
$268$	0	0
$269$	27.4924	1.67624	0.838121	−	0.545484i	$-0.183654\pi$
$269$	27.4924	1.67624	0.838121	+	0.545484i	$0.183654\pi$
$270$	0	0
$271$	24.1771	1.46865	0.734327	−	0.678796i	$-0.237498\pi$
$271$	24.1771	1.46865	0.734327	+	0.678796i	$0.237498\pi$
$272$	0	0
$273$	14.2462i	0.862220i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.4384i	1.22803i	0.789295	+	0.614014i	$0.210446\pi$
$277$	20.4384i	1.22803i	−0.789295	+	0.614014i	$0.789554\pi$
$278$	0	0
$279$	0.0691303	0.00413872
$280$	0	0
$281$	−9.12311	−0.544239	−0.272119	−	0.962263i	$-0.587725\pi$
$281$	−9.12311	−0.544239	−0.272119	+	0.962263i	$0.587725\pi$
$282$	0	0
$283$	3.93087i	0.233666i	0.993152	+	0.116833i	$0.0372742\pi$
$283$	3.93087i	0.233666i	−0.993152	+	0.116833i	$0.962726\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.5616i	0.623429i
$288$	0	0
$289$	14.9309	0.878286
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	14.5616i	0.850695i	0.905030	+	0.425347i	$0.139848\pi$
$293$	14.5616i	0.850695i	−0.905030	+	0.425347i	$0.860152\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 11.1231i	− 0.645428i
$298$	0	0
$299$	−3.56155	−0.205970
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	16.8769i	0.969552i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.3693i	0.877173i	0.898689	+	0.438587i	$0.144521\pi$
$307$	15.3693i	0.877173i	−0.898689	+	0.438587i	$0.855479\pi$
$308$	0	0
$309$	22.2462	1.26554
$310$	0	0
$311$	−4.19224	−0.237720	−0.118860	−	0.992911i	$-0.537924\pi$
$311$	−4.19224	−0.237720	−0.118860	+	0.992911i	$0.537924\pi$
$312$	0	0
$313$	− 13.9309i	− 0.787419i	−0.919235	−	0.393710i	$-0.871192\pi$
$313$	− 13.9309i	− 0.787419i	0.919235	−	0.393710i	$-0.128808\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.3693i	− 1.42488i	−0.701732	−	0.712441i	$-0.747589\pi$
$317$	− 25.3693i	− 1.42488i	0.701732	−	0.712441i	$-0.252411\pi$
$318$	0	0
$319$	−16.2462	−0.909613
$320$	0	0
$321$	−2.63068	−0.146830
$322$	0	0
$323$	− 2.87689i	− 0.160075i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.75379i	0.0969847i
$328$	0	0
$329$	−22.2462	−1.22647
$330$	0	0
$331$	−3.49242	−0.191961	−0.0959805	−	0.995383i	$-0.530599\pi$
$331$	−3.49242	−0.191961	−0.0959805	+	0.995383i	$0.530599\pi$
$332$	0	0
$333$	− 0.315342i	− 0.0172806i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.1231i	− 0.714861i	−0.933940	−	0.357431i	$-0.883653\pi$
$337$	− 13.1231i	− 0.714861i	0.933940	−	0.357431i	$-0.116347\pi$
$338$	0	0
$339$	−8.49242	−0.461245
$340$	0	0
$341$	−0.246211	−0.0133331
$342$	0	0
$343$	19.0540i	1.02882i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.2462i	0.764777i	0.924002	+	0.382388i	$0.124898\pi$
$347$	14.2462i	0.764777i	−0.924002	+	0.382388i	$0.875102\pi$
$348$	0	0
$349$	4.12311	0.220705	0.110352	−	0.993893i	$-0.464802\pi$
$349$	4.12311	0.220705	0.110352	+	0.993893i	$0.464802\pi$
$350$	0	0
$351$	19.8078	1.05726
$352$	0	0
$353$	4.19224i	0.223130i	0.993757	+	0.111565i	$0.0355864\pi$
$353$	4.19224i	0.223130i	−0.993757	+	0.111565i	$0.964414\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.75379i	0.304523i
$358$	0	0
$359$	−11.1231	−0.587055	−0.293528	−	0.955951i	$-0.594829\pi$
$359$	−11.1231	−0.587055	−0.293528	+	0.955951i	$0.594829\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 10.9309i	− 0.573722i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.1922i	0.584230i	0.956383	+	0.292115i	$0.0943590\pi$
$367$	11.1922i	0.584230i	−0.956383	+	0.292115i	$0.905641\pi$
$368$	0	0
$369$	2.31534	0.120532
$370$	0	0
$371$	−21.9309	−1.13859
$372$	0	0
$373$	24.7386i	1.28092i	0.767992	+	0.640459i	$0.221256\pi$
$373$	24.7386i	1.28092i	−0.767992	+	0.640459i	$0.778744\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 28.9309i	− 1.49002i
$378$	0	0
$379$	32.9848	1.69432	0.847159	−	0.531340i	$-0.178311\pi$
$379$	32.9848	1.69432	0.847159	+	0.531340i	$0.178311\pi$
$380$	0	0
$381$	−30.9309	−1.58464
$382$	0	0
$383$	− 13.9309i	− 0.711834i	−0.934518	−	0.355917i	$-0.884169\pi$
$383$	− 13.9309i	− 0.711834i	0.934518	−	0.355917i	$-0.115831\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.50758i	0.178300i
$388$	0	0
$389$	18.8769	0.957097	0.478548	−	0.878061i	$-0.341163\pi$
$389$	18.8769	0.957097	0.478548	+	0.878061i	$0.341163\pi$
$390$	0	0
$391$	−1.43845	−0.0727454
$392$	0	0
$393$	− 29.5616i	− 1.49118i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.1771i	1.06285i	0.847106	+	0.531424i	$0.178343\pi$
$397$	21.1771i	1.06285i	−0.847106	+	0.531424i	$0.821657\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−32.7386	−1.63489	−0.817445	−	0.576007i	$-0.804610\pi$
$401$	−32.7386	−1.63489	−0.817445	+	0.576007i	$0.804610\pi$
$402$	0	0
$403$	− 0.438447i	− 0.0218406i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.12311i	0.0556703i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	33.7538	1.66495
$412$	0	0
$413$	3.68466i	0.181310i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.1922i	0.792937i
$418$	0	0
$419$	21.6155	1.05599	0.527994	−	0.849248i	$-0.322944\pi$
$419$	21.6155	1.05599	0.527994	+	0.849248i	$0.322944\pi$
$420$	0	0
$421$	−30.8769	−1.50485	−0.752424	−	0.658679i	$-0.771115\pi$
$421$	−30.8769	−1.50485	−0.752424	+	0.658679i	$0.771115\pi$
$422$	0	0
$423$	4.87689i	0.237123i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.24621i	− 0.108702i
$428$	0	0
$429$	−11.1231	−0.537029
$430$	0	0
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	− 13.6847i	− 0.657643i	−0.944392	−	0.328821i	$-0.893349\pi$
$433$	− 13.6847i	− 0.657643i	0.944392	−	0.328821i	$-0.106651\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	3.80776	0.181735	0.0908673	−	0.995863i	$-0.471036\pi$
$439$	3.80776	0.181735	0.0908673	+	0.995863i	$0.471036\pi$
$440$	0	0
$441$	0.246211	0.0117243
$442$	0	0
$443$	22.0540i	1.04782i	0.851775	+	0.523908i	$0.175526\pi$
$443$	22.0540i	1.04782i	−0.851775	+	0.523908i	$0.824474\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.36932i	− 0.0647665i
$448$	0	0
$449$	−18.3153	−0.864354	−0.432177	−	0.901789i	$-0.642255\pi$
$449$	−18.3153	−0.864354	−0.432177	+	0.901789i	$0.642255\pi$
$450$	0	0
$451$	−8.24621	−0.388299
$452$	0	0
$453$	2.43845i	0.114568i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.9309i	0.838771i	0.907808	+	0.419385i	$0.137754\pi$
$457$	17.9309i	0.838771i	−0.907808	+	0.419385i	$0.862246\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−39.1771	−1.82466	−0.912329	−	0.409457i	$-0.865718\pi$
$461$	−39.1771	−1.82466	−0.912329	+	0.409457i	$0.865718\pi$
$462$	0	0
$463$	− 29.1231i	− 1.35347i	−0.736229	−	0.676733i	$-0.763395\pi$
$463$	− 29.1231i	− 1.35347i	0.736229	−	0.676733i	$-0.236605\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.68466i	0.448153i	0.974572	+	0.224076i	$0.0719365\pi$
$467$	9.68466i	0.448153i	−0.974572	+	0.224076i	$0.928064\pi$
$468$	0	0
$469$	−19.0540	−0.879831
$470$	0	0
$471$	−24.4924	−1.12855
$472$	0	0
$473$	− 12.4924i	− 0.574402i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.80776i	0.220132i
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 0.684658i	− 0.0310248i	−0.999880	−	0.0155124i	$-0.995062\pi$
$487$	− 0.684658i	− 0.0310248i	0.999880	−	0.0155124i	$-0.00493795\pi$
$488$	0	0
$489$	10.4384	0.472043
$490$	0	0
$491$	−12.6155	−0.569331	−0.284665	−	0.958627i	$-0.591882\pi$
$491$	−12.6155	−0.569331	−0.284665	+	0.958627i	$0.591882\pi$
$492$	0	0
$493$	− 11.6847i	− 0.526251i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.8078i	− 0.574507i
$498$	0	0
$499$	39.7386	1.77895	0.889473	−	0.456988i	$-0.151072\pi$
$499$	39.7386	1.77895	0.889473	+	0.456988i	$0.151072\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	15.4384i	0.688366i	0.938903	+	0.344183i	$0.111844\pi$
$503$	15.4384i	0.688366i	−0.938903	+	0.344183i	$0.888156\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.492423i	0.0218693i
$508$	0	0
$509$	24.4384	1.08322	0.541608	−	0.840631i	$-0.317816\pi$
$509$	24.4384	1.08322	0.541608	+	0.840631i	$0.317816\pi$
$510$	0	0
$511$	−19.3693	−0.856848
$512$	0	0
$513$	11.1231i	0.491097i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 17.3693i	− 0.763902i
$518$	0	0
$519$	11.1231	0.488250
$520$	0	0
$521$	23.8617	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$521$	23.8617	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$522$	0	0
$523$	10.2462i	0.448036i	0.974585	+	0.224018i	$0.0719174\pi$
$523$	10.2462i	0.448036i	−0.974585	+	0.224018i	$0.928083\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.177081i	− 0.00771377i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0.807764	0.0350540
$532$	0	0
$533$	− 14.6847i	− 0.636063i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.4233i	0.492951i
$538$	0	0
$539$	−0.876894	−0.0377705
$540$	0	0
$541$	16.0540	0.690214	0.345107	−	0.938563i	$-0.387843\pi$
$541$	16.0540	0.690214	0.345107	+	0.938563i	$0.387843\pi$
$542$	0	0
$543$	− 13.8617i	− 0.594864i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.31534i	0.312781i	0.987695	+	0.156391i	$0.0499859\pi$
$547$	7.31534i	0.312781i	−0.987695	+	0.156391i	$0.950014\pi$
$548$	0	0
$549$	−0.492423	−0.0210161
$550$	0	0
$551$	16.2462	0.692112
$552$	0	0
$553$	2.24621i	0.0955186i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.3002i	0.902518i	0.892393	+	0.451259i	$0.149025\pi$
$557$	21.3002i	0.902518i	−0.892393	+	0.451259i	$0.850975\pi$
$558$	0	0
$559$	22.2462	0.940914
$560$	0	0
$561$	−4.49242	−0.189670
$562$	0	0
$563$	− 11.1922i	− 0.471697i	−0.971790	−	0.235848i	$-0.924213\pi$
$563$	− 11.1922i	− 0.471697i	0.971790	−	0.235848i	$-0.0757868\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17.9309i	− 0.753026i
$568$	0	0
$569$	−44.7386	−1.87554	−0.937771	−	0.347256i	$-0.887114\pi$
$569$	−44.7386	−1.87554	−0.937771	+	0.347256i	$0.887114\pi$
$570$	0	0
$571$	−30.7386	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$571$	−30.7386	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$572$	0	0
$573$	− 4.87689i	− 0.203735i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.31534i	0.304542i	0.988339	+	0.152271i	$0.0486586\pi$
$577$	7.31534i	0.304542i	−0.988339	+	0.152271i	$0.951341\pi$
$578$	0	0
$579$	−29.5616	−1.22854
$580$	0	0
$581$	−19.6847	−0.816657
$582$	0	0
$583$	− 17.1231i	− 0.709167i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.3002i	0.425134i	0.977146	+	0.212567i	$0.0681824\pi$
$587$	10.3002i	0.425134i	−0.977146	+	0.212567i	$0.931818\pi$
$588$	0	0
$589$	0.246211	0.0101450
$590$	0	0
$591$	27.8078	1.14386
$592$	0	0
$593$	− 42.3542i	− 1.73928i	−0.493689	−	0.869638i	$-0.664352\pi$
$593$	− 42.3542i	− 1.73928i	0.493689	−	0.869638i	$-0.335648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 28.1080i	− 1.15038i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	16.8617	0.687805	0.343902	−	0.939005i	$-0.388251\pi$
$601$	16.8617	0.687805	0.343902	+	0.939005i	$0.388251\pi$
$602$	0	0
$603$	4.17708i	0.170104i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 36.0000i	− 1.46119i	−0.682808	−	0.730597i	$-0.739242\pi$
$607$	− 36.0000i	− 1.46119i	0.682808	−	0.730597i	$-0.260758\pi$
$608$	0	0
$609$	−32.4924	−1.31666
$610$	0	0
$611$	30.9309	1.25133
$612$	0	0
$613$	− 9.61553i	− 0.388368i	−0.980965	−	0.194184i	$-0.937794\pi$
$613$	− 9.61553i	− 0.388368i	0.980965	−	0.194184i	$-0.0622058\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.30019i	0.293894i	0.989144	+	0.146947i	$0.0469447\pi$
$617$	7.30019i	0.293894i	−0.989144	+	0.146947i	$0.953055\pi$
$618$	0	0
$619$	−44.4924	−1.78830	−0.894151	−	0.447766i	$-0.852220\pi$
$619$	−44.4924	−1.78830	−0.894151	+	0.447766i	$0.852220\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	− 20.4924i	− 0.821012i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.24621i	− 0.249450i
$628$	0	0
$629$	−0.807764	−0.0322077
$630$	0	0
$631$	14.7386	0.586736	0.293368	−	0.956000i	$-0.405224\pi$
$631$	14.7386	0.586736	0.293368	+	0.956000i	$0.405224\pi$
$632$	0	0
$633$	− 15.5076i	− 0.616371i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.56155i	− 0.0618710i
$638$	0	0
$639$	−2.80776	−0.111073
$640$	0	0
$641$	2.87689	0.113630	0.0568152	−	0.998385i	$-0.481905\pi$
$641$	2.87689	0.113630	0.0568152	+	0.998385i	$0.481905\pi$
$642$	0	0
$643$	− 22.1771i	− 0.874579i	−0.899321	−	0.437289i	$-0.855939\pi$
$643$	− 22.1771i	− 0.874579i	0.899321	−	0.437289i	$-0.144061\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 11.1771i	− 0.439416i	−0.975566	−	0.219708i	$-0.929489\pi$
$647$	− 11.1771i	− 0.439416i	0.975566	−	0.219708i	$-0.0705105\pi$
$648$	0	0
$649$	−2.87689	−0.112928
$650$	0	0
$651$	−0.492423	−0.0192996
$652$	0	0
$653$	29.4233i	1.15142i	0.817653	+	0.575711i	$0.195275\pi$
$653$	29.4233i	1.15142i	−0.817653	+	0.575711i	$0.804725\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.24621i	0.165660i
$658$	0	0
$659$	29.6155	1.15366	0.576829	−	0.816865i	$-0.304290\pi$
$659$	29.6155	1.15366	0.576829	+	0.816865i	$0.304290\pi$
$660$	0	0
$661$	27.7538	1.07950	0.539749	−	0.841826i	$-0.318519\pi$
$661$	27.7538	1.07950	0.539749	+	0.841826i	$0.318519\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.12311i	− 0.314528i
$668$	0	0
$669$	−29.4773	−1.13966
$670$	0	0
$671$	1.75379	0.0677043
$672$	0	0
$673$	− 16.1922i	− 0.624165i	−0.950055	−	0.312082i	$-0.898973\pi$
$673$	− 16.1922i	− 0.624165i	0.950055	−	0.312082i	$-0.101027\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.68466i	− 0.218479i	−0.994015	−	0.109240i	$-0.965158\pi$
$677$	− 5.68466i	− 0.218479i	0.994015	−	0.109240i	$-0.0348416\pi$
$678$	0	0
$679$	13.1231	0.503619
$680$	0	0
$681$	28.4924	1.09183
$682$	0	0
$683$	6.93087i	0.265202i	0.991169	+	0.132601i	$0.0423329\pi$
$683$	6.93087i	0.265202i	−0.991169	+	0.132601i	$0.957667\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.98485i	0.342793i
$688$	0	0
$689$	30.4924	1.16167
$690$	0	0
$691$	−0.492423	−0.0187326	−0.00936632	−	0.999956i	$-0.502981\pi$
$691$	−0.492423	−0.0187326	−0.00936632	+	0.999956i	$0.502981\pi$
$692$	0	0
$693$	− 2.87689i	− 0.109284i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.93087i	− 0.224648i
$698$	0	0
$699$	−44.1922	−1.67150
$700$	0	0
$701$	9.75379	0.368396	0.184198	−	0.982889i	$-0.441031\pi$
$701$	9.75379	0.368396	0.184198	+	0.982889i	$0.441031\pi$
$702$	0	0
$703$	− 1.12311i	− 0.0423587i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.6847i	1.04119i
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	0.492423	0.0184673
$712$	0	0
$713$	− 0.123106i	− 0.00461034i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.3002i	0.459359i
$718$	0	0
$719$	3.68466	0.137415	0.0687073	−	0.997637i	$-0.478113\pi$
$719$	3.68466	0.137415	0.0687073	+	0.997637i	$0.478113\pi$
$720$	0	0
$721$	36.4924	1.35905
$722$	0	0
$723$	− 28.8769i	− 1.07394i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 11.4384i	− 0.424229i	−0.977245	−	0.212114i	$-0.931965\pi$
$727$	− 11.4384i	− 0.424229i	0.977245	−	0.212114i	$-0.0680349\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	8.98485	0.332316
$732$	0	0
$733$	40.1771i	1.48397i	0.670414	+	0.741987i	$0.266116\pi$
$733$	40.1771i	1.48397i	−0.670414	+	0.741987i	$0.733884\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.8769i	− 0.547997i
$738$	0	0
$739$	−1.49242	−0.0548996	−0.0274498	−	0.999623i	$-0.508739\pi$
$739$	−1.49242	−0.0548996	−0.0274498	+	0.999623i	$0.508739\pi$
$740$	0	0
$741$	11.1231	0.408617
$742$	0	0
$743$	− 46.2462i	− 1.69661i	−0.529509	−	0.848304i	$-0.677624\pi$
$743$	− 46.2462i	− 1.69661i	0.529509	−	0.848304i	$-0.322376\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.31534i	0.157890i
$748$	0	0
$749$	−4.31534	−0.157679
$750$	0	0
$751$	44.3542	1.61851	0.809253	−	0.587460i	$-0.199872\pi$
$751$	44.3542	1.61851	0.809253	+	0.587460i	$0.199872\pi$
$752$	0	0
$753$	14.2462i	0.519161i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.5464i	0.928500i	0.885704	+	0.464250i	$0.153676\pi$
$757$	25.5464i	0.928500i	−0.885704	+	0.464250i	$0.846324\pi$
$758$	0	0
$759$	−3.12311	−0.113362
$760$	0	0
$761$	21.9848	0.796950	0.398475	−	0.917179i	$-0.369540\pi$
$761$	21.9848	0.796950	0.398475	+	0.917179i	$0.369540\pi$
$762$	0	0
$763$	2.87689i	0.104151i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.12311i	− 0.184985i
$768$	0	0
$769$	7.61553	0.274623	0.137311	−	0.990528i	$-0.456154\pi$
$769$	7.61553	0.274623	0.137311	+	0.990528i	$0.456154\pi$
$770$	0	0
$771$	−1.06913	−0.0385038
$772$	0	0
$773$	− 29.1231i	− 1.04749i	−0.851877	−	0.523743i	$-0.824535\pi$
$773$	− 29.1231i	− 1.04749i	0.851877	−	0.523743i	$-0.175465\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.24621i	0.0805824i
$778$	0	0
$779$	8.24621	0.295451
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	45.1771i	1.61450i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.3002i	1.18702i	0.804825	+	0.593512i	$0.202259\pi$
$787$	33.3002i	1.18702i	−0.804825	+	0.593512i	$0.797741\pi$
$788$	0	0
$789$	22.3542	0.795829
$790$	0	0
$791$	−13.9309	−0.495325
$792$	0	0
$793$	3.12311i	0.110905i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 25.1922i	− 0.892355i	−0.894945	−	0.446177i	$-0.852785\pi$
$797$	− 25.1922i	− 0.892355i	0.894945	−	0.446177i	$-0.147215\pi$
$798$	0	0
$799$	12.4924	0.441950
$800$	0	0
$801$	−4.49242	−0.158732
$802$	0	0
$803$	− 15.1231i	− 0.533683i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.9309i	1.51124i
$808$	0	0
$809$	−50.8078	−1.78631	−0.893153	−	0.449753i	$-0.851512\pi$
$809$	−50.8078	−1.78631	−0.893153	+	0.449753i	$0.851512\pi$
$810$	0	0
$811$	14.3693	0.504575	0.252287	−	0.967652i	$-0.418817\pi$
$811$	14.3693	0.504575	0.252287	+	0.967652i	$0.418817\pi$
$812$	0	0
$813$	37.7538i	1.32408i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.4924i	0.437055i
$818$	0	0
$819$	5.12311	0.179016
$820$	0	0
$821$	38.4924	1.34339	0.671697	−	0.740826i	$-0.265566\pi$
$821$	38.4924	1.34339	0.671697	+	0.740826i	$0.265566\pi$
$822$	0	0
$823$	− 33.1771i	− 1.15648i	−0.815867	−	0.578240i	$-0.803740\pi$
$823$	− 33.1771i	− 1.15648i	0.815867	−	0.578240i	$-0.196260\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.3153i	− 0.984621i	−0.870420	−	0.492310i	$-0.836152\pi$
$827$	− 28.3153i	− 0.984621i	0.870420	−	0.492310i	$-0.163848\pi$
$828$	0	0
$829$	20.5616	0.714132	0.357066	−	0.934079i	$-0.383777\pi$
$829$	20.5616	0.714132	0.357066	+	0.934079i	$0.383777\pi$
$830$	0	0
$831$	−31.9157	−1.10714
$832$	0	0
$833$	− 0.630683i	− 0.0218519i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.684658i	0.0236653i
$838$	0	0
$839$	33.6155	1.16054	0.580268	−	0.814425i	$-0.302948\pi$
$839$	33.6155	1.16054	0.580268	+	0.814425i	$0.302948\pi$
$840$	0	0
$841$	36.9848	1.27534
$842$	0	0
$843$	− 14.2462i	− 0.490666i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 17.9309i	− 0.616112i
$848$	0	0
$849$	−6.13826	−0.210665
$850$	0	0
$851$	−0.561553	−0.0192498
$852$	0	0
$853$	− 14.4924i	− 0.496211i	−0.968733	−	0.248106i	$-0.920192\pi$
$853$	− 14.4924i	− 0.496211i	0.968733	−	0.248106i	$-0.0798080\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.5464i	1.38504i	0.721399	+	0.692519i	$0.243499\pi$
$857$	40.5464i	1.38504i	−0.721399	+	0.692519i	$0.756501\pi$
$858$	0	0
$859$	0.369317	0.0126009	0.00630046	−	0.999980i	$-0.497994\pi$
$859$	0.369317	0.0126009	0.00630046	+	0.999980i	$0.497994\pi$
$860$	0	0
$861$	−16.4924	−0.562060
$862$	0	0
$863$	9.31534i	0.317098i	0.987351	+	0.158549i	$0.0506816\pi$
$863$	9.31534i	0.317098i	−0.987351	+	0.158549i	$0.949318\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23.3153i	0.791831i
$868$	0	0
$869$	−1.75379	−0.0594932
$870$	0	0
$871$	26.4924	0.897661
$872$	0	0
$873$	− 2.87689i	− 0.0973681i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.4773i	1.60319i	0.597865	+	0.801597i	$0.296016\pi$
$877$	47.4773i	1.60319i	−0.597865	+	0.801597i	$0.703984\pi$
$878$	0	0
$879$	−22.7386	−0.766955
$880$	0	0
$881$	22.7386	0.766084	0.383042	−	0.923731i	$-0.374876\pi$
$881$	22.7386	0.766084	0.383042	+	0.923731i	$0.374876\pi$
$882$	0	0
$883$	44.9848i	1.51386i	0.653496	+	0.756930i	$0.273302\pi$
$883$	44.9848i	1.51386i	−0.653496	+	0.756930i	$0.726698\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 43.1771i	− 1.44974i	−0.688883	−	0.724872i	$-0.741899\pi$
$887$	− 43.1771i	− 1.44974i	0.688883	−	0.724872i	$-0.258101\pi$
$888$	0	0
$889$	−50.7386	−1.70172
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	17.3693i	0.581242i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.56155i	− 0.185695i
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	12.3153	0.410284
$902$	0	0
$903$	− 24.9848i	− 0.831444i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.4233i	0.810962i	0.914103	+	0.405481i	$0.132896\pi$
$907$	24.4233i	0.810962i	−0.914103	+	0.405481i	$0.867104\pi$
$908$	0	0
$909$	6.06913	0.201300
$910$	0	0
$911$	23.1231	0.766103	0.383051	−	0.923727i	$-0.374873\pi$
$911$	23.1231	0.766103	0.383051	+	0.923727i	$0.374873\pi$
$912$	0	0
$913$	− 15.3693i	− 0.508650i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.4924i	− 1.60136i
$918$	0	0
$919$	20.4924	0.675983	0.337991	−	0.941149i	$-0.390253\pi$
$919$	20.4924	0.675983	0.337991	+	0.941149i	$0.390253\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	17.8078i	0.586150i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−0.261366	−0.00857515	−0.00428757	−	0.999991i	$-0.501365\pi$
$929$	−0.261366	−0.00857515	−0.00428757	+	0.999991i	$0.501365\pi$
$930$	0	0
$931$	0.876894	0.0287391
$932$	0	0
$933$	− 6.54640i	− 0.214319i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	0	0
$939$	21.7538	0.709908
$940$	0	0
$941$	32.4924	1.05922	0.529611	−	0.848240i	$-0.322338\pi$
$941$	32.4924	1.05922	0.529611	+	0.848240i	$0.322338\pi$
$942$	0	0
$943$	− 4.12311i	− 0.134267i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.5616i	− 0.570674i	−0.958427	−	0.285337i	$-0.907895\pi$
$947$	− 17.5616i	− 0.570674i	0.958427	−	0.285337i	$-0.0921055\pi$
$948$	0	0
$949$	26.9309	0.874213
$950$	0	0
$951$	39.6155	1.28462
$952$	0	0
$953$	− 44.2462i	− 1.43328i	−0.697446	−	0.716638i	$-0.745680\pi$
$953$	− 44.2462i	− 1.43328i	0.697446	−	0.716638i	$-0.254320\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 25.3693i	− 0.820074i
$958$	0	0
$959$	55.3693	1.78797
$960$	0	0
$961$	−30.9848	−0.999511
$962$	0	0
$963$	0.946025i	0.0304852i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 9.17708i	− 0.295115i	−0.989053	−	0.147558i	$-0.952859\pi$
$967$	− 9.17708i	− 0.295115i	0.989053	−	0.147558i	$-0.0471412\pi$
$968$	0	0
$969$	4.49242	0.144317
$970$	0	0
$971$	−40.1080	−1.28713	−0.643563	−	0.765393i	$-0.722544\pi$
$971$	−40.1080	−1.28713	−0.643563	+	0.765393i	$0.722544\pi$
$972$	0	0
$973$	26.5616i	0.851524i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 53.6847i	− 1.71752i	−0.512374	−	0.858762i	$-0.671234\pi$
$977$	− 53.6847i	− 1.71752i	0.512374	−	0.858762i	$-0.328766\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	0.630683	0.0201362
$982$	0	0
$983$	62.0388i	1.97873i	0.145450	+	0.989366i	$0.453537\pi$
$983$	62.0388i	1.97873i	−0.145450	+	0.989366i	$0.546463\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 34.7386i	− 1.10574i
$988$	0	0
$989$	6.24621	0.198618
$990$	0	0
$991$	22.5616	0.716691	0.358346	−	0.933589i	$-0.383341\pi$
$991$	22.5616	0.716691	0.358346	+	0.933589i	$0.383341\pi$
$992$	0	0
$993$	− 5.45360i	− 0.173065i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 28.1080i	− 0.890188i	−0.895484	−	0.445094i	$-0.853170\pi$
$997$	− 28.1080i	− 0.890188i	0.895484	−	0.445094i	$-0.146830\pi$
$998$	0	0
$999$	3.12311	0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.n.4049.3		4
5.2	odd	4		920.2.a.e.1.2	✓	2
5.3	odd	4		4600.2.a.t.1.1		2
5.4	even	2	inner	4600.2.e.n.4049.2		4
15.2	even	4		8280.2.a.bf.1.1		2
20.3	even	4		9200.2.a.bq.1.2		2
20.7	even	4		1840.2.a.o.1.1		2
40.27	even	4		7360.2.a.bl.1.2		2
40.37	odd	4		7360.2.a.bp.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.e.1.2	✓	2	5.2	odd	4
1840.2.a.o.1.1		2	20.7	even	4
4600.2.a.t.1.1		2	5.3	odd	4
4600.2.e.n.4049.2		4	5.4	even	2	inner
4600.2.e.n.4049.3		4	1.1	even	1	trivial
7360.2.a.bl.1.2		2	40.27	even	4
7360.2.a.bp.1.1		2	40.37	odd	4
8280.2.a.bf.1.1		2	15.2	even	4
9200.2.a.bq.1.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +2.56155i q^{7} +0.561553 q^{9} -2.00000 q^{11} -3.56155i q^{13} -1.43845i q^{17} +2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +5.56155i q^{27} +8.12311 q^{29} +0.123106 q^{31} -3.12311i q^{33} -0.561553i q^{37} +5.56155 q^{39} +4.12311 q^{41} +6.24621i q^{43} +8.68466i q^{47} +0.438447 q^{49} +2.24621 q^{51} +8.56155i q^{53} +3.12311i q^{57} +1.43845 q^{59} -0.876894 q^{61} +1.43845i q^{63} +7.43845i q^{67} +1.56155 q^{69} -5.00000 q^{71} +7.56155i q^{73} -5.12311i q^{77} +0.876894 q^{79} -7.00000 q^{81} +7.68466i q^{83} +12.6847i q^{87} -8.00000 q^{89} +9.12311 q^{91} +0.192236i q^{93} -5.12311i q^{97} -1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 8 q^{11} + 8 q^{19} - 16 q^{21} + 16 q^{29} - 16 q^{31} + 14 q^{39} + 10 q^{49} - 24 q^{51} + 14 q^{59} - 20 q^{61} - 2 q^{69} - 20 q^{71} + 20 q^{79} - 28 q^{81} - 32 q^{89} + 20 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 8 * q^11 + 8 * q^19 - 16 * q^21 + 16 * q^29 - 16 * q^31 + 14 * q^39 + 10 * q^49 - 24 * q^51 + 14 * q^59 - 20 * q^61 - 2 * q^69 - 20 * q^71 + 20 * q^79 - 28 * q^81 - 32 * q^89 + 20 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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