LMFDB - Embedded newform 4600.2.e.n.4049.1

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			4049.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.n.4049.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} -2.00000 q^{11} +0.561553i q^{13} -5.56155i q^{17} +2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +1.43845i q^{27} -0.123106 q^{29} -8.12311 q^{31} +5.12311i q^{33} +3.56155i q^{37} +1.43845 q^{39} -4.12311 q^{41} -10.2462i q^{43} -3.68466i q^{47} +4.56155 q^{49} -14.2462 q^{51} +4.43845i q^{53} -5.12311i q^{57} +5.56155 q^{59} -9.12311 q^{61} +5.56155i q^{63} +11.5616i q^{67} -2.56155 q^{69} -5.00000 q^{71} +3.43845i q^{73} +3.12311i q^{77} +9.12311 q^{79} -7.00000 q^{81} -4.68466i q^{83} +0.315342i q^{87} -8.00000 q^{89} +0.876894 q^{91} +20.8078i q^{93} +3.12311i q^{97} +7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 2.56155i	− 1.47891i	−0.673204	−	0.739457i	$-0.735083\pi$
$3$	− 2.56155i	− 1.47891i	0.673204	−	0.739457i	$-0.264917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.56155i	− 0.590211i	−0.955465	−	0.295106i	$-0.904645\pi$
$7$	− 1.56155i	− 0.590211i	0.955465	−	0.295106i	$-0.0953549\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	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
$13$	0.561553i	0.155747i	0.996963	+	0.0778734i	$0.0248130\pi$
$13$	0.561553i	0.155747i	−0.996963	+	0.0778734i	$0.975187\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.56155i	− 1.34887i	−0.738332	−	0.674437i	$-0.764386\pi$
$17$	− 5.56155i	− 1.34887i	0.738332	−	0.674437i	$-0.235614\pi$
$18$	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
$27$	1.43845i	0.276829i
$28$	0	0
$29$	−0.123106	−0.0228601	−0.0114301	−	0.999935i	$-0.503638\pi$
$29$	−0.123106	−0.0228601	−0.0114301	+	0.999935i	$0.503638\pi$
$30$	0	0
$31$	−8.12311	−1.45895	−0.729476	−	0.684006i	$-0.760236\pi$
$31$	−8.12311	−1.45895	−0.729476	+	0.684006i	$0.760236\pi$
$32$	0	0
$33$	5.12311i	0.891818i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.56155i	0.585516i	0.956187	+	0.292758i	$0.0945730\pi$
$37$	3.56155i	0.585516i	−0.956187	+	0.292758i	$0.905427\pi$
$38$	0	0
$39$	1.43845	0.230336
$40$	0	0
$41$	−4.12311	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$41$	−4.12311	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$42$	0	0
$43$	− 10.2462i	− 1.56253i	−0.624198	−	0.781266i	$-0.714574\pi$
$43$	− 10.2462i	− 1.56253i	0.624198	−	0.781266i	$-0.285426\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.68466i	− 0.537463i	−0.963215	−	0.268731i	$-0.913396\pi$
$47$	− 3.68466i	− 0.537463i	0.963215	−	0.268731i	$-0.0866044\pi$
$48$	0	0
$49$	4.56155	0.651650
$50$	0	0
$51$	−14.2462	−1.99487
$52$	0	0
$53$	4.43845i	0.609668i	0.952406	+	0.304834i	$0.0986009\pi$
$53$	4.43845i	0.609668i	−0.952406	+	0.304834i	$0.901399\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.12311i	− 0.678572i
$58$	0	0
$59$	5.56155	0.724053	0.362026	−	0.932168i	$-0.382085\pi$
$59$	5.56155	0.724053	0.362026	+	0.932168i	$0.382085\pi$
$60$	0	0
$61$	−9.12311	−1.16809	−0.584047	−	0.811720i	$-0.698532\pi$
$61$	−9.12311	−1.16809	−0.584047	+	0.811720i	$0.698532\pi$
$62$	0	0
$63$	5.56155i	0.700690i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.5616i	1.41247i	0.707978	+	0.706234i	$0.249607\pi$
$67$	11.5616i	1.41247i	−0.707978	+	0.706234i	$0.750393\pi$
$68$	0	0
$69$	−2.56155	−0.308375
$70$	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
$73$	3.43845i	0.402440i	0.979546	+	0.201220i	$0.0644906\pi$
$73$	3.43845i	0.402440i	−0.979546	+	0.201220i	$0.935509\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.12311i	0.355911i
$78$	0	0
$79$	9.12311	1.02643	0.513215	−	0.858260i	$-0.328454\pi$
$79$	9.12311	1.02643	0.513215	+	0.858260i	$0.328454\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 4.68466i	− 0.514208i	−0.966384	−	0.257104i	$-0.917232\pi$
$83$	− 4.68466i	− 0.514208i	0.966384	−	0.257104i	$-0.0827683\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.315342i	0.0338082i
$88$	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
$91$	0.876894	0.0919235
$92$	0	0
$93$	20.8078i	2.15766i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.12311i	0.317103i	0.987351	+	0.158552i	$0.0506824\pi$
$97$	3.12311i	0.317103i	−0.987351	+	0.158552i	$0.949318\pi$
$98$	0	0
$99$	7.12311	0.715899
$100$	0	0
$101$	−9.80776	−0.975909	−0.487954	−	0.872869i	$-0.662257\pi$
$101$	−9.80776	−0.975909	−0.487954	+	0.872869i	$0.662257\pi$
$102$	0	0
$103$	2.24621i	0.221326i	0.993858	+	0.110663i	$0.0352974\pi$
$103$	2.24621i	0.221326i	−0.993858	+	0.110663i	$0.964703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.6847i	− 1.03292i	−0.856310	−	0.516462i	$-0.827249\pi$
$107$	− 10.6847i	− 1.03292i	0.856310	−	0.516462i	$-0.172751\pi$
$108$	0	0
$109$	−7.12311	−0.682270	−0.341135	−	0.940014i	$-0.610811\pi$
$109$	−7.12311	−0.682270	−0.341135	+	0.940014i	$0.610811\pi$
$110$	0	0
$111$	9.12311	0.865927
$112$	0	0
$113$	9.56155i	0.899475i	0.893161	+	0.449738i	$0.148483\pi$
$113$	9.56155i	0.899475i	−0.893161	+	0.449738i	$0.851517\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−8.68466	−0.796121
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.5616i	0.952303i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 0.807764i	− 0.0716775i	−0.999358	−	0.0358387i	$-0.988590\pi$
$127$	− 0.807764i	− 0.0716775i	0.999358	−	0.0358387i	$-0.0114103\pi$
$128$	0	0
$129$	−26.2462	−2.31085
$130$	0	0
$131$	9.93087	0.867664	0.433832	−	0.900994i	$-0.357161\pi$
$131$	9.93087	0.867664	0.433832	+	0.900994i	$0.357161\pi$
$132$	0	0
$133$	− 3.12311i	− 0.270808i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.6155i	1.67587i	0.545772	+	0.837934i	$0.316237\pi$
$137$	19.6155i	1.67587i	−0.545772	+	0.837934i	$0.683763\pi$
$138$	0	0
$139$	−14.3693	−1.21879	−0.609395	−	0.792867i	$-0.708588\pi$
$139$	−14.3693	−1.21879	−0.609395	+	0.792867i	$0.708588\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	0	0
$143$	− 1.12311i	− 0.0939188i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.6847i	− 0.963734i
$148$	0	0
$149$	−9.12311	−0.747394	−0.373697	−	0.927551i	$-0.621910\pi$
$149$	−9.12311	−0.747394	−0.373697	+	0.927551i	$0.621910\pi$
$150$	0	0
$151$	−2.56155	−0.208456	−0.104228	−	0.994553i	$-0.533237\pi$
$151$	−2.56155	−0.208456	−0.104228	+	0.994553i	$0.533237\pi$
$152$	0	0
$153$	19.8078i	1.60136i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.31534i	0.264593i	0.991210	+	0.132297i	$0.0422351\pi$
$157$	3.31534i	0.264593i	−0.991210	+	0.132297i	$0.957765\pi$
$158$	0	0
$159$	11.3693	0.901645
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	5.68466i	0.445257i	0.974903	+	0.222628i	$0.0714637\pi$
$163$	5.68466i	0.445257i	−0.974903	+	0.222628i	$0.928536\pi$
$164$	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
$169$	12.6847	0.975743
$170$	0	0
$171$	−7.12311	−0.544718
$172$	0	0
$173$	1.12311i	0.0853881i	0.999088	+	0.0426941i	$0.0135941\pi$
$173$	1.12311i	0.0853881i	−0.999088	+	0.0426941i	$0.986406\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 14.2462i	− 1.07081i
$178$	0	0
$179$	19.6847	1.47130	0.735650	−	0.677362i	$-0.236877\pi$
$179$	19.6847	1.47130	0.735650	+	0.677362i	$0.236877\pi$
$180$	0	0
$181$	−17.1231	−1.27275	−0.636375	−	0.771380i	$-0.719567\pi$
$181$	−17.1231	−1.27275	−0.636375	+	0.771380i	$0.719567\pi$
$182$	0	0
$183$	23.3693i	1.72751i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.1231i	0.813402i
$188$	0	0
$189$	2.24621	0.163388
$190$	0	0
$191$	5.12311	0.370695	0.185347	−	0.982673i	$-0.440659\pi$
$191$	5.12311	0.370695	0.185347	+	0.982673i	$0.440659\pi$
$192$	0	0
$193$	− 9.93087i	− 0.714840i	−0.933944	−	0.357420i	$-0.883657\pi$
$193$	− 9.93087i	− 0.714840i	0.933944	−	0.357420i	$-0.116343\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.80776i	0.200045i	0.994985	+	0.100022i	$0.0318915\pi$
$197$	2.80776i	0.200045i	−0.994985	+	0.100022i	$0.968109\pi$
$198$	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
$201$	29.6155	2.08892
$202$	0	0
$203$	0.192236i	0.0134923i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.56155i	0.247545i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	18.9309	1.30325	0.651627	−	0.758539i	$-0.274087\pi$
$211$	18.9309	1.30325	0.651627	+	0.758539i	$0.274087\pi$
$212$	0	0
$213$	12.8078i	0.877574i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.6847i	0.861091i
$218$	0	0
$219$	8.80776	0.595174
$220$	0	0
$221$	3.12311	0.210083
$222$	0	0
$223$	27.1231i	1.81630i	0.418648	+	0.908149i	$0.362504\pi$
$223$	27.1231i	1.81630i	−0.418648	+	0.908149i	$0.637496\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.75379i	− 0.116403i	−0.998305	−	0.0582015i	$-0.981463\pi$
$227$	− 1.75379i	− 0.116403i	0.998305	−	0.0582015i	$-0.0185366\pi$
$228$	0	0
$229$	22.2462	1.47007	0.735036	−	0.678029i	$-0.237165\pi$
$229$	22.2462	1.47007	0.735036	+	0.678029i	$0.237165\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	− 25.3002i	− 1.65747i	−0.559641	−	0.828735i	$-0.689061\pi$
$233$	− 25.3002i	− 1.65747i	0.559641	−	0.828735i	$-0.310939\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 23.3693i	− 1.51800i
$238$	0	0
$239$	16.1231	1.04292	0.521459	−	0.853277i	$-0.325388\pi$
$239$	16.1231	1.04292	0.521459	+	0.853277i	$0.325388\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	22.2462i	1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.12311i	0.0714615i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0.876894	0.0553491	0.0276745	−	0.999617i	$-0.491190\pi$
$251$	0.876894	0.0553491	0.0276745	+	0.999617i	$0.491190\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.6847i	− 0.728869i	−0.931229	−	0.364434i	$-0.881262\pi$
$257$	− 11.6847i	− 0.728869i	0.931229	−	0.364434i	$-0.118738\pi$
$258$	0	0
$259$	5.56155	0.345578
$260$	0	0
$261$	0.438447	0.0271392
$262$	0	0
$263$	− 26.6847i	− 1.64545i	−0.568442	−	0.822723i	$-0.692454\pi$
$263$	− 26.6847i	− 1.64545i	0.568442	−	0.822723i	$-0.307546\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.4924i	1.25412i
$268$	0	0
$269$	−5.49242	−0.334879	−0.167439	−	0.985882i	$-0.553550\pi$
$269$	−5.49242	−0.334879	−0.167439	+	0.985882i	$0.553550\pi$
$270$	0	0
$271$	−21.1771	−1.28642	−0.643208	−	0.765691i	$-0.722397\pi$
$271$	−21.1771	−1.28642	−0.643208	+	0.765691i	$0.722397\pi$
$272$	0	0
$273$	− 2.24621i	− 0.135947i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.5616i	1.47576i	0.674932	+	0.737880i	$0.264173\pi$
$277$	24.5616i	1.47576i	−0.674932	+	0.737880i	$0.735827\pi$
$278$	0	0
$279$	28.9309	1.73205
$280$	0	0
$281$	−0.876894	−0.0523111	−0.0261556	−	0.999658i	$-0.508327\pi$
$281$	−0.876894	−0.0523111	−0.0261556	+	0.999658i	$0.508327\pi$
$282$	0	0
$283$	− 24.9309i	− 1.48199i	−0.671513	−	0.740993i	$-0.734355\pi$
$283$	− 24.9309i	− 1.48199i	0.671513	−	0.740993i	$-0.265645\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.43845i	0.380050i
$288$	0	0
$289$	−13.9309	−0.819463
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	10.4384i	0.609821i	0.952381	+	0.304910i	$0.0986265\pi$
$293$	10.4384i	0.609821i	−0.952381	+	0.304910i	$0.901374\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.87689i	− 0.166934i
$298$	0	0
$299$	0.561553	0.0324754
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	25.1231i	1.44328i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 9.36932i	− 0.534735i	−0.963595	−	0.267368i	$-0.913846\pi$
$307$	− 9.36932i	− 0.534735i	0.963595	−	0.267368i	$-0.0861538\pi$
$308$	0	0
$309$	5.75379	0.327322
$310$	0	0
$311$	−24.8078	−1.40672	−0.703360	−	0.710834i	$-0.748318\pi$
$311$	−24.8078	−1.40672	−0.703360	+	0.710834i	$0.748318\pi$
$312$	0	0
$313$	14.9309i	0.843943i	0.906609	+	0.421971i	$0.138662\pi$
$313$	14.9309i	0.843943i	−0.906609	+	0.421971i	$0.861338\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.630683i	− 0.0354227i	−0.999843	−	0.0177113i	$-0.994362\pi$
$317$	− 0.630683i	− 0.0354227i	0.999843	−	0.0177113i	$-0.00563799\pi$
$318$	0	0
$319$	0.246211	0.0137852
$320$	0	0
$321$	−27.3693	−1.52761
$322$	0	0
$323$	− 11.1231i	− 0.618906i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.2462i	1.00902i
$328$	0	0
$329$	−5.75379	−0.317217
$330$	0	0
$331$	29.4924	1.62105	0.810525	−	0.585704i	$-0.199182\pi$
$331$	29.4924	1.62105	0.810525	+	0.585704i	$0.199182\pi$
$332$	0	0
$333$	− 12.6847i	− 0.695115i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.87689i	− 0.265661i	−0.991139	−	0.132831i	$-0.957593\pi$
$337$	− 4.87689i	− 0.265661i	0.991139	−	0.132831i	$-0.0424067\pi$
$338$	0	0
$339$	24.4924	1.33025
$340$	0	0
$341$	16.2462	0.879782
$342$	0	0
$343$	− 18.0540i	− 0.974823i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.24621i	− 0.120583i	−0.998181	−	0.0602915i	$-0.980797\pi$
$347$	− 2.24621i	− 0.120583i	0.998181	−	0.0602915i	$-0.0192030\pi$
$348$	0	0
$349$	−4.12311	−0.220705	−0.110352	−	0.993893i	$-0.535198\pi$
$349$	−4.12311	−0.220705	−0.110352	+	0.993893i	$0.535198\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	24.8078i	1.32038i	0.751097	+	0.660192i	$0.229525\pi$
$353$	24.8078i	1.32038i	−0.751097	+	0.660192i	$0.770475\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	22.2462i	1.17739i
$358$	0	0
$359$	−2.87689	−0.151837	−0.0759183	−	0.997114i	$-0.524189\pi$
$359$	−2.87689	−0.151837	−0.0759183	+	0.997114i	$0.524189\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	17.9309i	0.941127i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.8078i	1.66035i	0.557502	+	0.830176i	$0.311760\pi$
$367$	31.8078i	1.66035i	−0.557502	+	0.830176i	$0.688240\pi$
$368$	0	0
$369$	14.6847	0.764453
$370$	0	0
$371$	6.93087	0.359833
$372$	0	0
$373$	− 24.7386i	− 1.28092i	−0.767992	−	0.640459i	$-0.778744\pi$
$373$	− 24.7386i	− 1.28092i	0.767992	−	0.640459i	$-0.221256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.0691303i	− 0.00356039i
$378$	0	0
$379$	−32.9848	−1.69432	−0.847159	−	0.531340i	$-0.821689\pi$
$379$	−32.9848	−1.69432	−0.847159	+	0.531340i	$0.821689\pi$
$380$	0	0
$381$	−2.06913	−0.106005
$382$	0	0
$383$	14.9309i	0.762932i	0.924383	+	0.381466i	$0.124581\pi$
$383$	14.9309i	0.762932i	−0.924383	+	0.381466i	$0.875419\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	36.4924i	1.85501i
$388$	0	0
$389$	27.1231	1.37520	0.687598	−	0.726092i	$-0.258665\pi$
$389$	27.1231	1.37520	0.687598	+	0.726092i	$0.258665\pi$
$390$	0	0
$391$	−5.56155	−0.281260
$392$	0	0
$393$	− 25.4384i	− 1.28320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 24.1771i	− 1.21341i	−0.794926	−	0.606706i	$-0.792490\pi$
$397$	− 24.1771i	− 1.21341i	0.794926	−	0.606706i	$-0.207510\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	16.7386	0.835887	0.417944	−	0.908473i	$-0.362751\pi$
$401$	16.7386	0.835887	0.417944	+	0.908473i	$0.362751\pi$
$402$	0	0
$403$	− 4.56155i	− 0.227227i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.12311i	− 0.353079i
$408$	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
$411$	50.2462	2.47846
$412$	0	0
$413$	− 8.68466i	− 0.427344i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	36.8078i	1.80248i
$418$	0	0
$419$	−19.6155	−0.958281	−0.479141	−	0.877738i	$-0.659052\pi$
$419$	−19.6155	−0.958281	−0.479141	+	0.877738i	$0.659052\pi$
$420$	0	0
$421$	−39.1231	−1.90674	−0.953372	−	0.301798i	$-0.902413\pi$
$421$	−39.1231	−1.90674	−0.953372	+	0.301798i	$0.902413\pi$
$422$	0	0
$423$	13.1231i	0.638067i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.2462i	0.689422i
$428$	0	0
$429$	−2.87689	−0.138898
$430$	0	0
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	− 1.31534i	− 0.0632113i	−0.999500	−	0.0316056i	$-0.989938\pi$
$433$	− 1.31534i	− 0.0632113i	0.999500	−	0.0316056i	$-0.0100621\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	−16.8078	−0.802191	−0.401095	−	0.916036i	$-0.631370\pi$
$439$	−16.8078	−0.802191	−0.401095	+	0.916036i	$0.631370\pi$
$440$	0	0
$441$	−16.2462	−0.773629
$442$	0	0
$443$	− 15.0540i	− 0.715236i	−0.933868	−	0.357618i	$-0.883589\pi$
$443$	− 15.0540i	− 0.715236i	0.933868	−	0.357618i	$-0.116411\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.3693i	1.10533i
$448$	0	0
$449$	−30.6847	−1.44810	−0.724049	−	0.689748i	$-0.757721\pi$
$449$	−30.6847	−1.44810	−0.724049	+	0.689748i	$0.757721\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	0	0
$453$	6.56155i	0.308289i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.9309i	− 0.511325i	−0.966766	−	0.255662i	$-0.917706\pi$
$457$	− 10.9309i	− 0.511325i	0.966766	−	0.255662i	$-0.0822935\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	6.17708	0.287695	0.143848	−	0.989600i	$-0.454052\pi$
$461$	6.17708	0.287695	0.143848	+	0.989600i	$0.454052\pi$
$462$	0	0
$463$	− 20.8769i	− 0.970232i	−0.874450	−	0.485116i	$-0.838777\pi$
$463$	− 20.8769i	− 0.970232i	0.874450	−	0.485116i	$-0.161223\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.68466i	− 0.124231i	−0.998069	−	0.0621156i	$-0.980215\pi$
$467$	− 2.68466i	− 0.124231i	0.998069	−	0.0621156i	$-0.0197847\pi$
$468$	0	0
$469$	18.0540	0.833655
$470$	0	0
$471$	8.49242	0.391310
$472$	0	0
$473$	20.4924i	0.942243i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 15.8078i	− 0.723788i
$478$	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
$487$	11.6847i	0.529482i	0.964319	+	0.264741i	$0.0852865\pi$
$487$	11.6847i	0.529482i	−0.964319	+	0.264741i	$0.914713\pi$
$488$	0	0
$489$	14.5616	0.658496
$490$	0	0
$491$	28.6155	1.29140	0.645700	−	0.763591i	$-0.276566\pi$
$491$	28.6155	1.29140	0.645700	+	0.763591i	$0.276566\pi$
$492$	0	0
$493$	0.684658i	0.0308355i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.80776i	0.350226i
$498$	0	0
$499$	−9.73863	−0.435961	−0.217981	−	0.975953i	$-0.569947\pi$
$499$	−9.73863	−0.435961	−0.217981	+	0.975953i	$0.569947\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	19.5616i	0.872207i	0.899897	+	0.436103i	$0.143642\pi$
$503$	19.5616i	0.872207i	−0.899897	+	0.436103i	$0.856358\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 32.4924i	− 1.44304i
$508$	0	0
$509$	28.5616	1.26597	0.632984	−	0.774165i	$-0.281830\pi$
$509$	28.5616	1.26597	0.632984	+	0.774165i	$0.281830\pi$
$510$	0	0
$511$	5.36932	0.237525
$512$	0	0
$513$	2.87689i	0.127018i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.36932i	0.324102i
$518$	0	0
$519$	2.87689	0.126282
$520$	0	0
$521$	−33.8617	−1.48351	−0.741755	−	0.670671i	$-0.766006\pi$
$521$	−33.8617	−1.48351	−0.741755	+	0.670671i	$0.766006\pi$
$522$	0	0
$523$	− 6.24621i	− 0.273128i	−0.990631	−	0.136564i	$-0.956394\pi$
$523$	− 6.24621i	− 0.273128i	0.990631	−	0.136564i	$-0.0436059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	45.1771i	1.96794i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−19.8078	−0.859584
$532$	0	0
$533$	− 2.31534i	− 0.100289i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 50.4233i	− 2.17593i
$538$	0	0
$539$	−9.12311	−0.392960
$540$	0	0
$541$	−21.0540	−0.905181	−0.452591	−	0.891718i	$-0.649500\pi$
$541$	−21.0540	−0.905181	−0.452591	+	0.891718i	$0.649500\pi$
$542$	0	0
$543$	43.8617i	1.88229i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.6847i	0.841655i	0.907141	+	0.420828i	$0.138260\pi$
$547$	19.6847i	0.841655i	−0.907141	+	0.420828i	$0.861740\pi$
$548$	0	0
$549$	32.4924	1.38674
$550$	0	0
$551$	−0.246211	−0.0104890
$552$	0	0
$553$	− 14.2462i	− 0.605811i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.3002i	− 1.36860i	−0.729199	−	0.684301i	$-0.760107\pi$
$557$	− 32.3002i	− 1.36860i	0.729199	−	0.684301i	$-0.239893\pi$
$558$	0	0
$559$	5.75379	0.243359
$560$	0	0
$561$	28.4924	1.20295
$562$	0	0
$563$	− 31.8078i	− 1.34054i	−0.742118	−	0.670269i	$-0.766179\pi$
$563$	− 31.8078i	− 1.34054i	0.742118	−	0.670269i	$-0.233821\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.9309i	0.459053i
$568$	0	0
$569$	4.73863	0.198654	0.0993269	−	0.995055i	$-0.468331\pi$
$569$	4.73863	0.198654	0.0993269	+	0.995055i	$0.468331\pi$
$570$	0	0
$571$	18.7386	0.784187	0.392094	−	0.919925i	$-0.371751\pi$
$571$	18.7386	0.784187	0.392094	+	0.919925i	$0.371751\pi$
$572$	0	0
$573$	− 13.1231i	− 0.548226i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.6847i	0.819483i	0.912202	+	0.409742i	$0.134381\pi$
$577$	19.6847i	0.819483i	−0.912202	+	0.409742i	$0.865619\pi$
$578$	0	0
$579$	−25.4384	−1.05719
$580$	0	0
$581$	−7.31534	−0.303492
$582$	0	0
$583$	− 8.87689i	− 0.367643i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 43.3002i	− 1.78719i	−0.448874	−	0.893595i	$-0.648175\pi$
$587$	− 43.3002i	− 1.78719i	0.448874	−	0.893595i	$-0.351825\pi$
$588$	0	0
$589$	−16.2462	−0.669413
$590$	0	0
$591$	7.19224	0.295849
$592$	0	0
$593$	48.3542i	1.98567i	0.119504	+	0.992834i	$0.461870\pi$
$593$	48.3542i	1.98567i	−0.119504	+	0.992834i	$0.538130\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	46.1080i	1.88707i
$598$	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
$601$	−40.8617	−1.66679	−0.833393	−	0.552682i	$-0.813605\pi$
$601$	−40.8617	−1.66679	−0.833393	+	0.552682i	$0.813605\pi$
$602$	0	0
$603$	− 41.1771i	− 1.67686i
$604$	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
$609$	0.492423	0.0199540
$610$	0	0
$611$	2.06913	0.0837081
$612$	0	0
$613$	31.6155i	1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$613$	31.6155i	1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 46.3002i	− 1.86398i	−0.362490	−	0.931988i	$-0.618073\pi$
$617$	− 46.3002i	− 1.86398i	0.362490	−	0.931988i	$-0.381927\pi$
$618$	0	0
$619$	−11.5076	−0.462529	−0.231264	−	0.972891i	$-0.574286\pi$
$619$	−11.5076	−0.462529	−0.231264	+	0.972891i	$0.574286\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	12.4924i	0.500498i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.2462i	0.409194i
$628$	0	0
$629$	19.8078	0.789787
$630$	0	0
$631$	−34.7386	−1.38292	−0.691462	−	0.722413i	$-0.743033\pi$
$631$	−34.7386	−1.38292	−0.691462	+	0.722413i	$0.743033\pi$
$632$	0	0
$633$	− 48.4924i	− 1.92740i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.56155i	0.101492i
$638$	0	0
$639$	17.8078	0.704464
$640$	0	0
$641$	11.1231	0.439336	0.219668	−	0.975575i	$-0.429503\pi$
$641$	11.1231	0.439336	0.219668	+	0.975575i	$0.429503\pi$
$642$	0	0
$643$	23.1771i	0.914015i	0.889463	+	0.457007i	$0.151079\pi$
$643$	23.1771i	0.914015i	−0.889463	+	0.457007i	$0.848921\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.1771i	1.34364i	0.740715	+	0.671820i	$0.234487\pi$
$647$	34.1771i	1.34364i	−0.740715	+	0.671820i	$0.765513\pi$
$648$	0	0
$649$	−11.1231	−0.436620
$650$	0	0
$651$	32.4924	1.27348
$652$	0	0
$653$	− 32.4233i	− 1.26882i	−0.772996	−	0.634411i	$-0.781243\pi$
$653$	− 32.4233i	− 1.26882i	0.772996	−	0.634411i	$-0.218757\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 12.2462i	− 0.477770i
$658$	0	0
$659$	−11.6155	−0.452477	−0.226238	−	0.974072i	$-0.572643\pi$
$659$	−11.6155	−0.452477	−0.226238	+	0.974072i	$0.572643\pi$
$660$	0	0
$661$	44.2462	1.72098	0.860489	−	0.509469i	$-0.170158\pi$
$661$	44.2462	1.72098	0.860489	+	0.509469i	$0.170158\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.123106i	0.00476667i
$668$	0	0
$669$	69.4773	2.68615
$670$	0	0
$671$	18.2462	0.704387
$672$	0	0
$673$	− 36.8078i	− 1.41884i	−0.704788	−	0.709418i	$-0.748958\pi$
$673$	− 36.8078i	− 1.41884i	0.704788	−	0.709418i	$-0.251042\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.68466i	0.256912i	0.991715	+	0.128456i	$0.0410022\pi$
$677$	6.68466i	0.256912i	−0.991715	+	0.128456i	$0.958998\pi$
$678$	0	0
$679$	4.87689	0.187158
$680$	0	0
$681$	−4.49242	−0.172150
$682$	0	0
$683$	− 21.9309i	− 0.839161i	−0.907718	−	0.419581i	$-0.862177\pi$
$683$	− 21.9309i	− 0.839161i	0.907718	−	0.419581i	$-0.137823\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 56.9848i	− 2.17411i
$688$	0	0
$689$	−2.49242	−0.0949537
$690$	0	0
$691$	32.4924	1.23607	0.618035	−	0.786151i	$-0.287929\pi$
$691$	32.4924	1.23607	0.618035	+	0.786151i	$0.287929\pi$
$692$	0	0
$693$	− 11.1231i	− 0.422532i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22.9309i	0.868569i
$698$	0	0
$699$	−64.8078	−2.45125
$700$	0	0
$701$	26.2462	0.991306	0.495653	−	0.868521i	$-0.334929\pi$
$701$	26.2462	0.991306	0.495653	+	0.868521i	$0.334929\pi$
$702$	0	0
$703$	7.12311i	0.268653i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.3153i	0.575993i
$708$	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
$711$	−32.4924	−1.21856
$712$	0	0
$713$	8.12311i	0.304213i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 41.3002i	− 1.54238i
$718$	0	0
$719$	−8.68466	−0.323883	−0.161942	−	0.986800i	$-0.551776\pi$
$719$	−8.68466	−0.323883	−0.161942	+	0.986800i	$0.551776\pi$
$720$	0	0
$721$	3.50758	0.130629
$722$	0	0
$723$	− 37.1231i	− 1.38062i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15.5616i	− 0.577146i	−0.957458	−	0.288573i	$-0.906819\pi$
$727$	− 15.5616i	− 0.577146i	0.957458	−	0.288573i	$-0.0931808\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	−56.9848	−2.10766
$732$	0	0
$733$	− 5.17708i	− 0.191220i	−0.995419	−	0.0956099i	$-0.969520\pi$
$733$	− 5.17708i	− 0.191220i	0.995419	−	0.0956099i	$-0.0304801\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 23.1231i	− 0.851751i
$738$	0	0
$739$	31.4924	1.15847	0.579234	−	0.815161i	$-0.303352\pi$
$739$	31.4924	1.15847	0.579234	+	0.815161i	$0.303352\pi$
$740$	0	0
$741$	2.87689	0.105685
$742$	0	0
$743$	− 29.7538i	− 1.09156i	−0.837928	−	0.545780i	$-0.816233\pi$
$743$	− 29.7538i	− 1.09156i	0.837928	−	0.545780i	$-0.183767\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.6847i	0.610460i
$748$	0	0
$749$	−16.6847	−0.609644
$750$	0	0
$751$	−46.3542	−1.69149	−0.845744	−	0.533589i	$-0.820843\pi$
$751$	−46.3542	−1.69149	−0.845744	+	0.533589i	$0.820843\pi$
$752$	0	0
$753$	− 2.24621i	− 0.0818565i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 44.5464i	− 1.61907i	−0.587074	−	0.809533i	$-0.699720\pi$
$757$	− 44.5464i	− 1.61907i	0.587074	−	0.809533i	$-0.300280\pi$
$758$	0	0
$759$	5.12311	0.185957
$760$	0	0
$761$	−43.9848	−1.59445	−0.797225	−	0.603683i	$-0.793699\pi$
$761$	−43.9848	−1.59445	−0.797225	+	0.603683i	$0.793699\pi$
$762$	0	0
$763$	11.1231i	0.402683i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.12311i	0.112769i
$768$	0	0
$769$	−33.6155	−1.21221	−0.606103	−	0.795386i	$-0.707268\pi$
$769$	−33.6155	−1.21221	−0.606103	+	0.795386i	$0.707268\pi$
$770$	0	0
$771$	−29.9309	−1.07793
$772$	0	0
$773$	− 20.8769i	− 0.750890i	−0.926845	−	0.375445i	$-0.877490\pi$
$773$	− 20.8769i	− 0.750890i	0.926845	−	0.375445i	$-0.122510\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 14.2462i	− 0.511080i
$778$	0	0
$779$	−8.24621	−0.295451
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	− 0.177081i	− 0.00632836i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 20.3002i	− 0.723624i	−0.932251	−	0.361812i	$-0.882158\pi$
$787$	− 20.3002i	− 0.723624i	0.932251	−	0.361812i	$-0.117842\pi$
$788$	0	0
$789$	−68.3542	−2.43347
$790$	0	0
$791$	14.9309	0.530881
$792$	0	0
$793$	− 5.12311i	− 0.181927i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 45.8078i	− 1.62259i	−0.584634	−	0.811297i	$-0.698762\pi$
$797$	− 45.8078i	− 1.62259i	0.584634	−	0.811297i	$-0.301238\pi$
$798$	0	0
$799$	−20.4924	−0.724970
$800$	0	0
$801$	28.4924	1.00673
$802$	0	0
$803$	− 6.87689i	− 0.242680i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0691i	0.495257i
$808$	0	0
$809$	−30.1922	−1.06150	−0.530751	−	0.847528i	$-0.678090\pi$
$809$	−30.1922	−1.06150	−0.530751	+	0.847528i	$0.678090\pi$
$810$	0	0
$811$	−10.3693	−0.364116	−0.182058	−	0.983288i	$-0.558276\pi$
$811$	−10.3693	−0.364116	−0.182058	+	0.983288i	$0.558276\pi$
$812$	0	0
$813$	54.2462i	1.90250i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20.4924i	− 0.716939i
$818$	0	0
$819$	−3.12311	−0.109130
$820$	0	0
$821$	5.50758	0.192216	0.0961079	−	0.995371i	$-0.469361\pi$
$821$	5.50758	0.192216	0.0961079	+	0.995371i	$0.469361\pi$
$822$	0	0
$823$	12.1771i	0.424466i	0.977219	+	0.212233i	$0.0680736\pi$
$823$	12.1771i	0.424466i	−0.977219	+	0.212233i	$0.931926\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 40.6847i	− 1.41474i	−0.706841	−	0.707372i	$-0.749881\pi$
$827$	− 40.6847i	− 1.41474i	0.706841	−	0.707372i	$-0.250119\pi$
$828$	0	0
$829$	16.4384	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$829$	16.4384	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$830$	0	0
$831$	62.9157	2.18252
$832$	0	0
$833$	− 25.3693i	− 0.878995i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 11.6847i	− 0.403881i
$838$	0	0
$839$	−7.61553	−0.262917	−0.131459	−	0.991322i	$-0.541966\pi$
$839$	−7.61553	−0.262917	−0.131459	+	0.991322i	$0.541966\pi$
$840$	0	0
$841$	−28.9848	−0.999477
$842$	0	0
$843$	2.24621i	0.0773636i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.9309i	0.375589i
$848$	0	0
$849$	−63.8617	−2.19173
$850$	0	0
$851$	3.56155	0.122088
$852$	0	0
$853$	18.4924i	0.633168i	0.948564	+	0.316584i	$0.102536\pi$
$853$	18.4924i	0.633168i	−0.948564	+	0.316584i	$0.897464\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29.5464i	− 1.00929i	−0.863328	−	0.504643i	$-0.831624\pi$
$857$	− 29.5464i	− 1.00929i	0.863328	−	0.504643i	$-0.168376\pi$
$858$	0	0
$859$	−24.3693	−0.831470	−0.415735	−	0.909486i	$-0.636476\pi$
$859$	−24.3693	−0.831470	−0.415735	+	0.909486i	$0.636476\pi$
$860$	0	0
$861$	16.4924	0.562060
$862$	0	0
$863$	21.6847i	0.738154i	0.929399	+	0.369077i	$0.120326\pi$
$863$	21.6847i	0.738154i	−0.929399	+	0.369077i	$0.879674\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35.6847i	1.21191i
$868$	0	0
$869$	−18.2462	−0.618960
$870$	0	0
$871$	−6.49242	−0.219987
$872$	0	0
$873$	− 11.1231i	− 0.376460i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 51.4773i	− 1.73826i	−0.494580	−	0.869132i	$-0.664678\pi$
$877$	− 51.4773i	− 1.73826i	0.494580	−	0.869132i	$-0.335322\pi$
$878$	0	0
$879$	26.7386	0.901872
$880$	0	0
$881$	−26.7386	−0.900847	−0.450424	−	0.892815i	$-0.648727\pi$
$881$	−26.7386	−0.900847	−0.450424	+	0.892815i	$0.648727\pi$
$882$	0	0
$883$	− 20.9848i	− 0.706196i	−0.935586	−	0.353098i	$-0.885128\pi$
$883$	− 20.9848i	− 0.706196i	0.935586	−	0.353098i	$-0.114872\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.17708i	0.0730992i	0.999332	+	0.0365496i	$0.0116367\pi$
$887$	2.17708i	0.0730992i	−0.999332	+	0.0365496i	$0.988363\pi$
$888$	0	0
$889$	−1.26137	−0.0423049
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	− 7.36932i	− 0.246605i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.43845i	− 0.0480284i
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	24.6847	0.822365
$902$	0	0
$903$	40.9848i	1.36389i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37.4233i	− 1.24262i	−0.783565	−	0.621310i	$-0.786601\pi$
$907$	− 37.4233i	− 1.24262i	0.783565	−	0.621310i	$-0.213399\pi$
$908$	0	0
$909$	34.9309	1.15858
$910$	0	0
$911$	14.8769	0.492894	0.246447	−	0.969156i	$-0.420737\pi$
$911$	14.8769	0.492894	0.246447	+	0.969156i	$0.420737\pi$
$912$	0	0
$913$	9.36932i	0.310079i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 15.5076i	− 0.512105i
$918$	0	0
$919$	−12.4924	−0.412087	−0.206043	−	0.978543i	$-0.566059\pi$
$919$	−12.4924	−0.412087	−0.206043	+	0.978543i	$0.566059\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	− 2.80776i	− 0.0924187i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−49.7386	−1.63187	−0.815936	−	0.578142i	$-0.803778\pi$
$929$	−49.7386	−1.63187	−0.815936	+	0.578142i	$0.803778\pi$
$930$	0	0
$931$	9.12311	0.298998
$932$	0	0
$933$	63.5464i	2.08042i
$934$	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
$939$	38.2462	1.24812
$940$	0	0
$941$	−0.492423	−0.0160525	−0.00802626	−	0.999968i	$-0.502555\pi$
$941$	−0.492423	−0.0160525	−0.00802626	+	0.999968i	$0.502555\pi$
$942$	0	0
$943$	4.12311i	0.134267i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.4384i	− 0.436691i	−0.975872	−	0.218345i	$-0.929934\pi$
$947$	− 13.4384i	− 0.436691i	0.975872	−	0.218345i	$-0.0700659\pi$
$948$	0	0
$949$	−1.93087	−0.0626787
$950$	0	0
$951$	−1.61553	−0.0523871
$952$	0	0
$953$	− 27.7538i	− 0.899033i	−0.893272	−	0.449517i	$-0.851596\pi$
$953$	− 27.7538i	− 0.899033i	0.893272	−	0.449517i	$-0.148404\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 0.630683i	− 0.0203871i
$958$	0	0
$959$	30.6307	0.989116
$960$	0	0
$961$	34.9848	1.12854
$962$	0	0
$963$	38.0540i	1.22627i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	36.1771i	1.16338i	0.813412	+	0.581688i	$0.197608\pi$
$967$	36.1771i	1.16338i	−0.813412	+	0.581688i	$0.802392\pi$
$968$	0	0
$969$	−28.4924	−0.915308
$970$	0	0
$971$	34.1080	1.09458	0.547288	−	0.836944i	$-0.315660\pi$
$971$	34.1080	1.09458	0.547288	+	0.836944i	$0.315660\pi$
$972$	0	0
$973$	22.4384i	0.719344i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 41.3153i	− 1.32179i	−0.750476	−	0.660897i	$-0.770176\pi$
$977$	− 41.3153i	− 1.32179i	0.750476	−	0.660897i	$-0.229824\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	25.3693	0.809980
$982$	0	0
$983$	− 41.0388i	− 1.30894i	−0.756090	−	0.654468i	$-0.772893\pi$
$983$	− 41.0388i	− 1.30894i	0.756090	−	0.654468i	$-0.227107\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14.7386i	0.469136i
$988$	0	0
$989$	−10.2462	−0.325811
$990$	0	0
$991$	18.4384	0.585717	0.292858	−	0.956156i	$-0.405394\pi$
$991$	18.4384	0.585717	0.292858	+	0.956156i	$0.405394\pi$
$992$	0	0
$993$	− 75.5464i	− 2.39739i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.1080i	1.46025i	0.683312	+	0.730127i	$0.260539\pi$
$997$	46.1080i	1.46025i	−0.683312	+	0.730127i	$0.739461\pi$
$998$	0	0
$999$	−5.12311	−0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.n.4049.1		4
5.2	odd	4		920.2.a.e.1.1	✓	2
5.3	odd	4		4600.2.a.t.1.2		2
5.4	even	2	inner	4600.2.e.n.4049.4		4
15.2	even	4		8280.2.a.bf.1.2		2
20.3	even	4		9200.2.a.bq.1.1		2
20.7	even	4		1840.2.a.o.1.2		2
40.27	even	4		7360.2.a.bl.1.1		2
40.37	odd	4		7360.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.e.1.1	✓	2	5.2	odd	4
1840.2.a.o.1.2		2	20.7	even	4
4600.2.a.t.1.2		2	5.3	odd	4
4600.2.e.n.4049.1		4	1.1	even	1	trivial
4600.2.e.n.4049.4		4	5.4	even	2	inner
7360.2.a.bl.1.1		2	40.27	even	4
7360.2.a.bp.1.2		2	40.37	odd	4
8280.2.a.bf.1.2		2	15.2	even	4
9200.2.a.bq.1.1		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)

\(f(q)\)	\(=\)	\(q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} -2.00000 q^{11} +0.561553i q^{13} -5.56155i q^{17} +2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +1.43845i q^{27} -0.123106 q^{29} -8.12311 q^{31} +5.12311i q^{33} +3.56155i q^{37} +1.43845 q^{39} -4.12311 q^{41} -10.2462i q^{43} -3.68466i q^{47} +4.56155 q^{49} -14.2462 q^{51} +4.43845i q^{53} -5.12311i q^{57} +5.56155 q^{59} -9.12311 q^{61} +5.56155i q^{63} +11.5616i q^{67} -2.56155 q^{69} -5.00000 q^{71} +3.43845i q^{73} +3.12311i q^{77} +9.12311 q^{79} -7.00000 q^{81} -4.68466i q^{83} +0.315342i q^{87} -8.00000 q^{89} +0.876894 q^{91} +20.8078i q^{93} +3.12311i q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 8 q^{11} + 8 q^{19} - 16 q^{21} + 16 q^{29} - 16 q^{31} + 14 q^{39} + 10 q^{49} - 24 q^{51} + 14 q^{59} - 20 q^{61} - 2 q^{69} - 20 q^{71} + 20 q^{79} - 28 q^{81} - 32 q^{89} + 20 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 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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