LMFDB - Embedded newform 2034.3.d.a.2033.30

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,3,Mod(2033,2034)] 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("2034.2033"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2034.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$55.4224857709$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2033.30
Character	$\chi$	$=$	2034.2033
Dual form			2034.3.d.a.2033.29

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +4.53197 q^{5} -2.50747 q^{7} -2.82843i q^{8} +6.40918i q^{10} -10.4528i q^{11} +10.6901 q^{13} -3.54610i q^{14} +4.00000 q^{16} -23.2826 q^{17} -10.6047i q^{19} -9.06395 q^{20} +14.7825 q^{22} -2.83074 q^{23} -4.46121 q^{25} +15.1181i q^{26} +5.01495 q^{28} +8.27065 q^{29} -36.1400 q^{31} +5.65685i q^{32} -32.9265i q^{34} -11.3638 q^{35} -11.9538i q^{37} +14.9973 q^{38} -12.8184i q^{40} +74.2703i q^{41} +3.48304i q^{43} +20.9056i q^{44} -4.00328i q^{46} -40.4399 q^{47} -42.7126 q^{49} -6.30911i q^{50} -21.3802 q^{52} +2.68929i q^{53} -47.3719i q^{55} +7.09221i q^{56} +11.6965i q^{58} -44.5471 q^{59} -57.4009 q^{61} -51.1097i q^{62} -8.00000 q^{64} +48.4472 q^{65} +23.6418i q^{67} +46.5651 q^{68} -16.0709i q^{70} +92.5662 q^{71} +109.176i q^{73} +16.9052 q^{74} +21.2094i q^{76} +26.2102i q^{77} +74.3503i q^{79} +18.1279 q^{80} -105.034 q^{82} -86.6042i q^{83} -105.516 q^{85} -4.92576 q^{86} -29.5650 q^{88} -120.422 q^{89} -26.8051 q^{91} +5.66149 q^{92} -57.1906i q^{94} -48.0601i q^{95} -5.77622 q^{97} -60.4047i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) $ 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$		0			0
$3$		0			0
$4$	−2.00000			−0.500000
$5$	4.53197			0.906395			0.453197	−	0.891410i	$-0.350283\pi$
$5$	4.53197			0.906395			0.453197	+	0.891410i	$0.350283\pi$
$6$		0			0
$7$	−2.50747			−0.358211			−0.179105	−	0.983830i	$-0.557320\pi$
$7$	−2.50747			−0.358211			−0.179105	+	0.983830i	$0.557320\pi$
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$			6.40918i			0.640918i
$11$		−	10.4528i		−	0.950256i	−0.879917	−	0.475128i	$-0.842402\pi$
$11$		−	10.4528i		−	0.950256i	0.879917	−	0.475128i	$-0.157598\pi$
$12$		0			0
$13$	10.6901			0.822314			0.411157	−	0.911565i	$-0.365125\pi$
$13$	10.6901			0.822314			0.411157	+	0.911565i	$0.365125\pi$
$14$		−	3.54610i		−	0.253293i
$15$		0			0
$16$	4.00000			0.250000
$17$	−23.2826			−1.36956			−0.684781	−	0.728749i	$-0.740102\pi$
$17$	−23.2826			−1.36956			−0.684781	+	0.728749i	$0.740102\pi$
$18$		0			0
$19$		−	10.6047i		−	0.558141i	−0.960271	−	0.279071i	$-0.909974\pi$
$19$		−	10.6047i		−	0.558141i	0.960271	−	0.279071i	$-0.0900263\pi$
$20$	−9.06395			−0.453197
$21$		0			0
$22$	14.7825			0.671932
$23$	−2.83074			−0.123076			−0.0615379	−	0.998105i	$-0.519601\pi$
$23$	−2.83074			−0.123076			−0.0615379	+	0.998105i	$0.519601\pi$
$24$		0			0
$25$	−4.46121			−0.178449
$26$			15.1181i			0.581464i
$27$		0			0
$28$	5.01495			0.179105
$29$	8.27065			0.285195			0.142597	−	0.989781i	$-0.454455\pi$
$29$	8.27065			0.285195			0.142597	+	0.989781i	$0.454455\pi$
$30$		0			0
$31$	−36.1400			−1.16581			−0.582904	−	0.812541i	$-0.698084\pi$
$31$	−36.1400			−1.16581			−0.582904	+	0.812541i	$0.698084\pi$
$32$			5.65685i			0.176777i
$33$		0			0
$34$		−	32.9265i		−	0.968427i
$35$	−11.3638			−0.324680
$36$		0			0
$37$		−	11.9538i		−	0.323076i	−0.986867	−	0.161538i	$-0.948355\pi$
$37$		−	11.9538i		−	0.323076i	0.986867	−	0.161538i	$-0.0516454\pi$
$38$	14.9973			0.394665
$39$		0			0
$40$		−	12.8184i		−	0.320459i
$41$			74.2703i			1.81147i	0.423843	+	0.905735i	$0.360681\pi$
$41$			74.2703i			1.81147i	−0.423843	+	0.905735i	$0.639319\pi$
$42$		0			0
$43$			3.48304i			0.0810009i	0.999180	+	0.0405004i	$0.0128952\pi$
$43$			3.48304i			0.0810009i	−0.999180	+	0.0405004i	$0.987105\pi$
$44$			20.9056i			0.475128i
$45$		0			0
$46$		−	4.00328i		−	0.0870278i
$47$	−40.4399			−0.860423			−0.430211	−	0.902728i	$-0.641561\pi$
$47$	−40.4399			−0.860423			−0.430211	+	0.902728i	$0.641561\pi$
$48$		0			0
$49$	−42.7126			−0.871685
$50$		−	6.30911i		−	0.126182i
$51$		0			0
$52$	−21.3802			−0.411157
$53$			2.68929i			0.0507413i	0.999678	+	0.0253706i	$0.00807659\pi$
$53$			2.68929i			0.0507413i	−0.999678	+	0.0253706i	$0.991923\pi$
$54$		0			0
$55$		−	47.3719i		−	0.861307i
$56$			7.09221i			0.126647i
$57$		0			0
$58$			11.6965i			0.201663i
$59$	−44.5471			−0.755036			−0.377518	−	0.926002i	$-0.623222\pi$
$59$	−44.5471			−0.755036			−0.377518	+	0.926002i	$0.623222\pi$
$60$		0			0
$61$	−57.4009			−0.940998			−0.470499	−	0.882401i	$-0.655926\pi$
$61$	−57.4009			−0.940998			−0.470499	+	0.882401i	$0.655926\pi$
$62$		−	51.1097i		−	0.824350i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	48.4472			0.745341
$66$		0			0
$67$			23.6418i			0.352863i	0.984313	+	0.176431i	$0.0564554\pi$
$67$			23.6418i			0.352863i	−0.984313	+	0.176431i	$0.943545\pi$
$68$	46.5651			0.684781
$69$		0			0
$70$		−	16.0709i		−	0.229584i
$71$	92.5662			1.30375			0.651875	−	0.758327i	$-0.273983\pi$
$71$	92.5662			1.30375			0.651875	+	0.758327i	$0.273983\pi$
$72$		0			0
$73$			109.176i			1.49555i	0.663949	+	0.747777i	$0.268879\pi$
$73$			109.176i			1.49555i	−0.663949	+	0.747777i	$0.731121\pi$
$74$	16.9052			0.228449
$75$		0			0
$76$			21.2094i			0.279071i
$77$			26.2102i			0.340392i
$78$		0			0
$79$			74.3503i			0.941143i	0.882362	+	0.470571i	$0.155952\pi$
$79$			74.3503i			0.941143i	−0.882362	+	0.470571i	$0.844048\pi$
$80$	18.1279			0.226599
$81$		0			0
$82$	−105.034			−1.28090
$83$		−	86.6042i		−	1.04342i	−0.853122	−	0.521712i	$-0.825293\pi$
$83$		−	86.6042i		−	1.04342i	0.853122	−	0.521712i	$-0.174707\pi$
$84$		0			0
$85$	−105.516			−1.24136
$86$	−4.92576			−0.0572763
$87$		0			0
$88$	−29.5650			−0.335966
$89$	−120.422			−1.35305			−0.676526	−	0.736419i	$-0.736515\pi$
$89$	−120.422			−1.35305			−0.676526	+	0.736419i	$0.736515\pi$
$90$		0			0
$91$	−26.8051			−0.294562
$92$	5.66149			0.0615379
$93$		0			0
$94$		−	57.1906i		−	0.608411i
$95$		−	48.0601i		−	0.505896i
$96$		0			0
$97$	−5.77622			−0.0595486			−0.0297743	−	0.999557i	$-0.509479\pi$
$97$	−5.77622			−0.0595486			−0.0297743	+	0.999557i	$0.509479\pi$
$98$		−	60.4047i		−	0.616374i
$99$		0			0
$100$	8.92243			0.0892243
$101$	118.153			1.16983			0.584915	−	0.811094i	$-0.301128\pi$
$101$	118.153			1.16983			0.584915	+	0.811094i	$0.301128\pi$
$102$		0			0
$103$			95.5937i			0.928094i	0.885811	+	0.464047i	$0.153603\pi$
$103$			95.5937i			0.928094i	−0.885811	+	0.464047i	$0.846397\pi$
$104$		−	30.2361i		−	0.290732i
$105$		0			0
$106$	−3.80323			−0.0358795
$107$	−191.852			−1.79301			−0.896503	−	0.443037i	$-0.853901\pi$
$107$	−191.852			−1.79301			−0.896503	+	0.443037i	$0.853901\pi$
$108$		0			0
$109$	−80.1645			−0.735454			−0.367727	−	0.929934i	$-0.619864\pi$
$109$	−80.1645			−0.735454			−0.367727	+	0.929934i	$0.619864\pi$
$110$	66.9940			0.609036
$111$		0			0
$112$	−10.0299			−0.0895526
$113$	27.4200	−	109.623i	0.242655	−	0.970113i
$113$	27.4200	−	109.623i	0.242655	−	0.970113i
$114$		0			0
$115$	−12.8289			−0.111555
$116$	−16.5413			−0.142597
$117$		0			0
$118$		−	62.9991i		−	0.533891i
$119$	58.3804			0.490592
$120$		0			0
$121$	11.7387			0.0970139
$122$		−	81.1771i		−	0.665386i
$123$		0			0
$124$	72.2801			0.582904
$125$	−133.517			−1.06814
$126$		0			0
$127$	37.0985			0.292114			0.146057	−	0.989276i	$-0.453342\pi$
$127$	37.0985			0.292114			0.146057	+	0.989276i	$0.453342\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		0			0
$130$			68.5147i			0.527036i
$131$		−	32.0853i		−	0.244926i	−0.992473	−	0.122463i	$-0.960921\pi$
$131$		−	32.0853i		−	0.244926i	0.992473	−	0.122463i	$-0.0390792\pi$
$132$		0			0
$133$			26.5910i			0.199932i
$134$	−33.4346			−0.249512
$135$		0			0
$136$			65.8530i			0.484213i
$137$	123.004			0.897841			0.448921	−	0.893572i	$-0.351809\pi$
$137$	123.004			0.897841			0.448921	+	0.893572i	$0.351809\pi$
$138$		0			0
$139$	−111.579			−0.802725			−0.401362	−	0.915919i	$-0.631463\pi$
$139$	−111.579			−0.802725			−0.401362	+	0.915919i	$0.631463\pi$
$140$	22.7276			0.162340
$141$		0			0
$142$			130.908i			0.921890i
$143$		−	111.741i		−	0.781409i
$144$		0			0
$145$	37.4824			0.258499
$146$	−154.397			−1.05752
$147$		0			0
$148$			23.9076i			0.161538i
$149$			70.2763i			0.471653i	0.971795	+	0.235827i	$0.0757797\pi$
$149$			70.2763i			0.471653i	−0.971795	+	0.235827i	$0.924220\pi$
$150$		0			0
$151$		−	40.4940i		−	0.268172i	−0.990970	−	0.134086i	$-0.957190\pi$
$151$		−	40.4940i		−	0.268172i	0.990970	−	0.134086i	$-0.0428099\pi$
$152$	−29.9946			−0.197333
$153$		0			0
$154$	−37.0668			−0.240693
$155$	−163.786			−1.05668
$156$		0			0
$157$	−275.611			−1.75548			−0.877741	−	0.479135i	$-0.840951\pi$
$157$	−275.611			−1.75548			−0.877741	+	0.479135i	$0.840951\pi$
$158$	−105.147			−0.665488
$159$		0			0
$160$			25.6367i			0.160229i
$161$	7.09802			0.0440871
$162$		0			0
$163$	−261.514			−1.60438			−0.802191	−	0.597068i	$-0.796332\pi$
$163$	−261.514			−1.60438			−0.802191	+	0.597068i	$0.796332\pi$
$164$		−	148.541i		−	0.905735i
$165$		0			0
$166$	122.477			0.737812
$167$	−102.373			−0.613013			−0.306506	−	0.951869i	$-0.599160\pi$
$167$	−102.373			−0.613013			−0.306506	+	0.951869i	$0.599160\pi$
$168$		0			0
$169$	−54.7221			−0.323799
$170$		−	149.222i		−	0.877777i
$171$		0			0
$172$		−	6.96607i		−	0.0405004i
$173$		−	73.0612i		−	0.422319i	−0.977452	−	0.211160i	$-0.932276\pi$
$173$		−	73.0612i		−	0.422319i	0.977452	−	0.211160i	$-0.0677240\pi$
$174$		0			0
$175$	11.1864			0.0639221
$176$		−	41.8113i		−	0.237564i
$177$		0			0
$178$		−	170.302i		−	0.956752i
$179$	332.549			1.85782			0.928908	−	0.370312i	$-0.120749\pi$
$179$	332.549			1.85782			0.928908	+	0.370312i	$0.120749\pi$
$180$		0			0
$181$			188.226i			1.03992i	0.854189	+	0.519962i	$0.174054\pi$
$181$			188.226i			1.03992i	−0.854189	+	0.519962i	$0.825946\pi$
$182$		−	37.9082i		−	0.208287i
$183$		0			0
$184$			8.00655i			0.0435139i
$185$		−	54.1743i		−	0.292834i
$186$		0			0
$187$			243.368i			1.30143i
$188$	80.8797			0.430211
$189$		0			0
$190$	67.9673			0.357723
$191$	230.515			1.20689			0.603443	−	0.797406i	$-0.293795\pi$
$191$	230.515			1.20689			0.603443	+	0.797406i	$0.293795\pi$
$192$		0			0
$193$		−	147.626i		−	0.764903i	−0.923976	−	0.382452i	$-0.875080\pi$
$193$		−	147.626i		−	0.764903i	0.923976	−	0.382452i	$-0.124920\pi$
$194$		−	8.16880i		−	0.0421072i
$195$		0			0
$196$	85.4251			0.435843
$197$	257.345			1.30632			0.653161	−	0.757219i	$-0.273442\pi$
$197$	257.345			1.30632			0.653161	+	0.757219i	$0.273442\pi$
$198$		0			0
$199$		−	372.393i		−	1.87132i	−0.352898	−	0.935662i	$-0.614804\pi$
$199$		−	372.393i		−	1.87132i	0.352898	−	0.935662i	$-0.385196\pi$
$200$			12.6182i			0.0630911i
$201$		0			0
$202$			167.093i			0.827195i
$203$	−20.7384			−0.102160
$204$		0			0
$205$			336.591i			1.64191i
$206$	−135.190			−0.656262
$207$		0			0
$208$	42.7603			0.205579
$209$	−110.849			−0.530377
$210$		0			0
$211$	−188.536			−0.893536			−0.446768	−	0.894650i	$-0.647425\pi$
$211$	−188.536			−0.893536			−0.446768	+	0.894650i	$0.647425\pi$
$212$		−	5.37857i		−	0.0253706i
$213$		0			0
$214$		−	271.319i		−	1.26785i
$215$			15.7850i			0.0734188i
$216$		0			0
$217$	90.6202			0.417605
$218$		−	113.370i		−	0.520045i
$219$		0			0
$220$			94.7438i			0.430653i
$221$	−248.893			−1.12621
$222$		0			0
$223$			11.5760i			0.0519101i	0.999663	+	0.0259551i	$0.00826268\pi$
$223$			11.5760i			0.0519101i	−0.999663	+	0.0259551i	$0.991737\pi$
$224$		−	14.1844i		−	0.0633233i
$225$		0			0
$226$	155.030	+	38.7777i	0.685973	+	0.171583i
$227$		−	292.117i		−	1.28686i	−0.765505	−	0.643430i	$-0.777511\pi$
$227$		−	292.117i		−	1.28686i	0.765505	−	0.643430i	$-0.222489\pi$
$228$		0			0
$229$		−	142.048i		−	0.620297i	−0.950688	−	0.310149i	$-0.899621\pi$
$229$		−	142.048i		−	0.620297i	0.950688	−	0.310149i	$-0.100379\pi$
$230$		−	18.1427i		−	0.0788815i
$231$		0			0
$232$		−	23.3929i		−	0.100832i
$233$			316.029i			1.35635i	0.734902	+	0.678173i	$0.237228\pi$
$233$			316.029i			1.35635i	−0.734902	+	0.678173i	$0.762772\pi$
$234$		0			0
$235$	−183.272			−0.779883
$236$	89.0942			0.377518
$237$		0			0
$238$			82.5624i			0.346901i
$239$		−	202.335i		−	0.846589i	−0.905992	−	0.423295i	$-0.860874\pi$
$239$		−	202.335i		−	0.846589i	0.905992	−	0.423295i	$-0.139126\pi$
$240$		0			0
$241$	−428.030			−1.77606			−0.888030	−	0.459786i	$-0.847926\pi$
$241$	−428.030			−1.77606			−0.888030	+	0.459786i	$0.847926\pi$
$242$			16.6010i			0.0685992i
$243$		0			0
$244$	114.802			0.470499
$245$	−193.572			−0.790091
$246$		0			0
$247$		−	113.365i		−	0.458967i
$248$			102.219i			0.412175i
$249$		0			0
$250$		−	188.822i		−	0.755289i
$251$		−	82.2970i		−	0.327876i	−0.986471	−	0.163938i	$-0.947580\pi$
$251$		−	82.2970i		−	0.327876i	0.986471	−	0.163938i	$-0.0524198\pi$
$252$		0			0
$253$			29.5892i			0.116954i
$254$			52.4651i			0.206556i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	223.093i		−	0.868065i	−0.900897	−	0.434032i	$-0.857090\pi$
$257$		−	223.093i		−	0.868065i	0.900897	−	0.434032i	$-0.142910\pi$
$258$		0			0
$259$			29.9739i			0.115729i
$260$	−96.8944			−0.372671
$261$		0			0
$262$	45.3754			0.173189
$263$	312.042			1.18647			0.593236	−	0.805028i	$-0.297850\pi$
$263$	312.042			1.18647			0.593236	+	0.805028i	$0.297850\pi$
$264$		0			0
$265$			12.1878i			0.0459916i
$266$	−37.6053			−0.141373
$267$		0			0
$268$		−	47.2836i		−	0.176431i
$269$	−327.667			−1.21809			−0.609047	−	0.793134i	$-0.708448\pi$
$269$	−327.667			−1.21809			−0.609047	+	0.793134i	$0.708448\pi$
$270$		0			0
$271$		−	230.322i		−	0.849898i	−0.905217	−	0.424949i	$-0.860292\pi$
$271$		−	230.322i		−	0.849898i	0.905217	−	0.424949i	$-0.139708\pi$
$272$	−93.1302			−0.342391
$273$		0			0
$274$			173.954i			0.634870i
$275$			46.6322i			0.169572i
$276$		0			0
$277$	−265.431			−0.958235			−0.479118	−	0.877751i	$-0.659043\pi$
$277$	−265.431			−0.958235			−0.479118	+	0.877751i	$0.659043\pi$
$278$		−	157.796i		−	0.567612i
$279$		0			0
$280$			32.1417i			0.114792i
$281$	24.3973			0.0868232			0.0434116	−	0.999057i	$-0.486177\pi$
$281$	24.3973			0.0868232			0.0434116	+	0.999057i	$0.486177\pi$
$282$		0			0
$283$	−136.388			−0.481935			−0.240968	−	0.970533i	$-0.577465\pi$
$283$	−136.388			−0.481935			−0.240968	+	0.970533i	$0.577465\pi$
$284$	−185.132			−0.651875
$285$		0			0
$286$	158.026			0.552539
$287$		−	186.231i		−	0.648888i
$288$		0			0
$289$	253.078			0.875701
$290$			53.0081i			0.182787i
$291$		0			0
$292$		−	218.351i		−	0.747777i
$293$	−174.136			−0.594322			−0.297161	−	0.954827i	$-0.596040\pi$
$293$	−174.136			−0.594322			−0.297161	+	0.954827i	$0.596040\pi$
$294$		0			0
$295$	−201.886			−0.684360
$296$	−33.8105			−0.114225
$297$		0			0
$298$	−99.3857			−0.333509
$299$	−30.2609			−0.101207
$300$		0			0
$301$		−	8.73363i		−	0.0290154i
$302$	57.2671			0.189626
$303$		0			0
$304$		−	42.4187i		−	0.139535i
$305$	−260.139			−0.852915
$306$		0			0
$307$	222.583			0.725026			0.362513	−	0.931979i	$-0.381919\pi$
$307$	222.583			0.725026			0.362513	+	0.931979i	$0.381919\pi$
$308$		−	52.4203i		−	0.170196i
$309$		0			0
$310$		−	231.628i		−	0.747187i
$311$		−	340.171i		−	1.09380i	−0.837199	−	0.546899i	$-0.815808\pi$
$311$		−	340.171i		−	1.09380i	0.837199	−	0.546899i	$-0.184192\pi$
$312$		0			0
$313$	−355.514			−1.13583			−0.567914	−	0.823088i	$-0.692249\pi$
$313$	−355.514			−1.13583			−0.567914	+	0.823088i	$0.692249\pi$
$314$		−	389.773i		−	1.24131i
$315$		0			0
$316$		−	148.701i		−	0.470571i
$317$		−	170.673i		−	0.538402i	−0.963084	−	0.269201i	$-0.913240\pi$
$317$		−	170.673i		−	0.538402i	0.963084	−	0.269201i	$-0.0867596\pi$
$318$		0			0
$319$		−	86.4516i		−	0.271008i
$320$	−36.2558			−0.113299
$321$		0			0
$322$			10.0381i			0.0311743i
$323$			246.904i			0.764409i
$324$		0			0
$325$	−47.6907			−0.146741
$326$		−	369.837i		−	1.13447i
$327$		0			0
$328$	210.068			0.640452
$329$	101.402			0.308213
$330$		0			0
$331$	230.899			0.697579			0.348789	−	0.937201i	$-0.386593\pi$
$331$	230.899			0.697579			0.348789	+	0.937201i	$0.386593\pi$
$332$			173.208i			0.521712i
$333$		0			0
$334$		−	144.777i		−	0.433465i
$335$			107.144i			0.319833i
$336$		0			0
$337$	284.567			0.844413			0.422207	−	0.906500i	$-0.361256\pi$
$337$	284.567			0.844413			0.422207	+	0.906500i	$0.361256\pi$
$338$		−	77.3887i		−	0.228961i
$339$		0			0
$340$	211.032			0.620682
$341$			377.765i			1.10782i
$342$		0			0
$343$	229.967			0.670457
$344$	9.85152			0.0286381
$345$		0			0
$346$	103.324			0.298625
$347$		−	225.262i		−	0.649169i	−0.945857	−	0.324585i	$-0.894776\pi$
$347$		−	225.262i		−	0.649169i	0.945857	−	0.324585i	$-0.105224\pi$
$348$		0			0
$349$		−	230.146i		−	0.659445i	−0.944078	−	0.329722i	$-0.893045\pi$
$349$		−	230.146i		−	0.659445i	0.944078	−	0.329722i	$-0.106955\pi$
$350$			15.8199i			0.0451998i
$351$		0			0
$352$	59.1300			0.167983
$353$			116.697i			0.330586i	0.986245	+	0.165293i	$0.0528569\pi$
$353$			116.697i			0.330586i	−0.986245	+	0.165293i	$0.947143\pi$
$354$		0			0
$355$	419.508			1.18171
$356$	240.843			0.676526
$357$		0			0
$358$			470.295i			1.31367i
$359$	320.214			0.891961			0.445981	−	0.895043i	$-0.352855\pi$
$359$	320.214			0.891961			0.445981	+	0.895043i	$0.352855\pi$
$360$		0			0
$361$	248.541			0.688478
$362$	−266.192			−0.735338
$363$		0			0
$364$	53.6102			0.147281
$365$			494.781i			1.35556i
$366$		0			0
$367$	−171.409			−0.467055			−0.233527	−	0.972350i	$-0.575027\pi$
$367$	−171.409			−0.467055			−0.233527	+	0.972350i	$0.575027\pi$
$368$	−11.3230			−0.0307690
$369$		0			0
$370$	76.6141			0.207065
$371$		−	6.74332i		−	0.0181761i
$372$		0			0
$373$			78.5907i			0.210699i	0.994435	+	0.105349i	$0.0335961\pi$
$373$			78.5907i			0.210699i	−0.994435	+	0.105349i	$0.966404\pi$
$374$	−344.175			−0.920253
$375$		0			0
$376$			114.381i			0.304205i
$377$	88.4140			0.234520
$378$		0			0
$379$			90.6148i			0.239089i	0.992829	+	0.119545i	$0.0381434\pi$
$379$			90.6148i			0.239089i	−0.992829	+	0.119545i	$0.961857\pi$
$380$			96.1203i			0.252948i
$381$		0			0
$382$			325.998i			0.853397i
$383$		−	152.480i		−	0.398121i	−0.979987	−	0.199060i	$-0.936211\pi$
$383$		−	152.480i		−	0.398121i	0.979987	−	0.199060i	$-0.0637890\pi$
$384$		0			0
$385$			118.784i			0.308529i
$386$	208.775			0.540868
$387$		0			0
$388$	11.5524			0.0297743
$389$		−	373.242i		−	0.959490i	−0.877408	−	0.479745i	$-0.840729\pi$
$389$		−	373.242i		−	0.959490i	0.877408	−	0.479745i	$-0.159271\pi$
$390$		0			0
$391$	65.9070			0.168560
$392$			120.809i			0.308187i
$393$		0			0
$394$			363.941i			0.923709i
$395$			336.954i			0.853047i
$396$		0			0
$397$		−	356.580i		−	0.898186i	−0.893485	−	0.449093i	$-0.851747\pi$
$397$		−	356.580i		−	0.898186i	0.893485	−	0.449093i	$-0.148253\pi$
$398$	526.644			1.32323
$399$		0			0
$400$	−17.8449			−0.0446121
$401$			208.230i			0.519278i	0.965706	+	0.259639i	$0.0836035\pi$
$401$			208.230i			0.519278i	−0.965706	+	0.259639i	$0.916396\pi$
$402$		0			0
$403$	−386.340			−0.958660
$404$	−236.306			−0.584915
$405$		0			0
$406$		−	29.3286i		−	0.0722379i
$407$	−124.951			−0.307005
$408$		0			0
$409$			297.771i			0.728046i	0.931390	+	0.364023i	$0.118597\pi$
$409$			297.771i			0.728046i	−0.931390	+	0.364023i	$0.881403\pi$
$410$	−476.012			−1.16100
$411$		0			0
$412$		−	191.187i		−	0.464047i
$413$	111.701			0.270462
$414$		0			0
$415$		−	392.488i		−	0.945754i
$416$			60.4722i			0.145366i
$417$		0			0
$418$		−	156.764i		−	0.375033i
$419$	−78.5814			−0.187545			−0.0937725	−	0.995594i	$-0.529893\pi$
$419$	−78.5814			−0.187545			−0.0937725	+	0.995594i	$0.529893\pi$
$420$		0			0
$421$	−141.234			−0.335472			−0.167736	−	0.985832i	$-0.553646\pi$
$421$	−141.234			−0.335472			−0.167736	+	0.985832i	$0.553646\pi$
$422$		−	266.630i		−	0.631825i
$423$		0			0
$424$	7.60645			0.0179397
$425$	103.868			0.244396
$426$		0			0
$427$	143.931			0.337075
$428$	383.703			0.896503
$429$		0			0
$430$	−22.3234			−0.0519149
$431$	361.441			0.838610			0.419305	−	0.907845i	$-0.362274\pi$
$431$	361.441			0.838610			0.419305	+	0.907845i	$0.362274\pi$
$432$		0			0
$433$			722.880i			1.66947i	0.550653	+	0.834734i	$0.314379\pi$
$433$			722.880i			1.66947i	−0.550653	+	0.834734i	$0.685621\pi$
$434$			128.156i			0.295291i
$435$		0			0
$436$	160.329			0.367727
$437$			30.0191i			0.0686937i
$438$		0			0
$439$	191.501			0.436220			0.218110	−	0.975924i	$-0.430011\pi$
$439$	191.501			0.436220			0.218110	+	0.975924i	$0.430011\pi$
$440$	−133.988			−0.304518
$441$		0			0
$442$		−	351.987i		−	0.796351i
$443$			767.622i			1.73278i	0.499367	+	0.866391i	$0.333566\pi$
$443$			767.622i			1.73278i	−0.499367	+	0.866391i	$0.666434\pi$
$444$		0			0
$445$	−545.748			−1.22640
$446$	−16.3709			−0.0367060
$447$		0			0
$448$	20.0598			0.0447763
$449$	233.838			0.520798			0.260399	−	0.965501i	$-0.416146\pi$
$449$	233.838			0.520798			0.260399	+	0.965501i	$0.416146\pi$
$450$		0			0
$451$	776.334			1.72136
$452$	−54.8400	+	219.245i	−0.121327	+	0.485056i
$453$		0			0
$454$	413.116			0.909947
$455$	−121.480			−0.266989
$456$		0			0
$457$			331.622i			0.725649i	0.931858	+	0.362824i	$0.118188\pi$
$457$			331.622i			0.725649i	−0.931858	+	0.362824i	$0.881812\pi$
$458$	200.886			0.438617
$459$		0			0
$460$	25.6577			0.0557776
$461$		−	315.892i		−	0.685232i	−0.939475	−	0.342616i	$-0.888687\pi$
$461$		−	315.892i		−	0.685232i	0.939475	−	0.342616i	$-0.111313\pi$
$462$		0			0
$463$	362.231			0.782355			0.391178	−	0.920315i	$-0.372068\pi$
$463$	362.231			0.782355			0.391178	+	0.920315i	$0.372068\pi$
$464$	33.0826			0.0712987
$465$		0			0
$466$	−446.932			−0.959082
$467$			114.462i			0.245101i	0.992462	+	0.122550i	$0.0391073\pi$
$467$			114.462i			0.245101i	−0.992462	+	0.122550i	$0.960893\pi$
$468$		0			0
$469$		−	59.2812i		−	0.126399i
$470$		−	259.186i		−	0.551460i
$471$		0			0
$472$			125.998i			0.266945i
$473$	36.4075			0.0769715
$474$		0			0
$475$			47.3097i			0.0995995i
$476$	−116.761			−0.245296
$477$		0			0
$478$	286.145			0.598629
$479$	625.572			1.30600			0.652998	−	0.757359i	$-0.273511\pi$
$479$	625.572			1.30600			0.652998	+	0.757359i	$0.273511\pi$
$480$		0			0
$481$		−	127.787i		−	0.265670i
$482$		−	605.326i		−	1.25586i
$483$		0			0
$484$	−23.4774			−0.0485069
$485$	−26.1777			−0.0539746
$486$		0			0
$487$			123.163i			0.252901i	0.991973	+	0.126451i	$0.0403586\pi$
$487$			123.163i			0.252901i	−0.991973	+	0.126451i	$0.959641\pi$
$488$			162.354i			0.332693i
$489$		0			0
$490$		−	273.753i		−	0.558679i
$491$	−460.379			−0.937636			−0.468818	−	0.883295i	$-0.655320\pi$
$491$	−460.379			−0.937636			−0.468818	+	0.883295i	$0.655320\pi$
$492$		0			0
$493$	−192.562			−0.390592
$494$	160.322			0.324539
$495$		0			0
$496$	−144.560			−0.291452
$497$	−232.107			−0.467017
$498$		0			0
$499$		−	218.951i		−	0.438780i	−0.975637	−	0.219390i	$-0.929593\pi$
$499$		−	218.951i		−	0.438780i	0.975637	−	0.219390i	$-0.0704068\pi$
$500$	267.035			0.534070
$501$		0			0
$502$	116.386			0.231844
$503$			746.918i			1.48493i	0.669887	+	0.742463i	$0.266343\pi$
$503$			746.918i			1.48493i	−0.669887	+	0.742463i	$0.733657\pi$
$504$		0			0
$505$	535.466			1.06033
$506$	−41.8455			−0.0826986
$507$		0			0
$508$	−74.1969			−0.146057
$509$		−	850.423i		−	1.67077i	−0.549663	−	0.835386i	$-0.685244\pi$
$509$		−	850.423i		−	1.67077i	0.549663	−	0.835386i	$-0.314756\pi$
$510$		0			0
$511$		−	273.755i		−	0.535724i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	315.501			0.613815
$515$			433.228i			0.841220i
$516$		0			0
$517$			422.710i			0.817622i
$518$	−42.3894			−0.0818329
$519$		0			0
$520$		−	137.029i		−	0.263518i
$521$			324.251i			0.622364i	0.950350	+	0.311182i	$0.100725\pi$
$521$			324.251i			0.622364i	−0.950350	+	0.311182i	$0.899275\pi$
$522$		0			0
$523$			576.554i			1.10240i	0.834374	+	0.551199i	$0.185830\pi$
$523$			576.554i			1.10240i	−0.834374	+	0.551199i	$0.814170\pi$
$524$			64.1705i			0.122463i
$525$		0			0
$526$			441.294i			0.838963i
$527$	841.432			1.59665
$528$		0			0
$529$	−520.987			−0.984852
$530$	−17.2361			−0.0325210
$531$		0			0
$532$		−	53.1819i		−	0.0999660i
$533$			793.956i			1.48960i
$534$		0			0
$535$	−869.467			−1.62517
$536$	66.8691			0.124756
$537$		0			0
$538$		−	463.392i		−	0.861323i
$539$			446.467i			0.828324i
$540$		0			0
$541$			322.630i			0.596359i	0.954510	+	0.298180i	$0.0963795\pi$
$541$			322.630i			0.596359i	−0.954510	+	0.298180i	$0.903621\pi$
$542$	325.725			0.600969
$543$		0			0
$544$		−	131.706i		−	0.242107i
$545$	−363.303			−0.666612
$546$		0			0
$547$	−269.141			−0.492031			−0.246015	−	0.969266i	$-0.579121\pi$
$547$	−269.141			−0.492031			−0.246015	+	0.969266i	$0.579121\pi$
$548$	−246.009			−0.448921
$549$		0			0
$550$	−65.9479			−0.119905
$551$		−	87.7077i		−	0.159179i
$552$		0			0
$553$		−	186.431i		−	0.337127i
$554$		−	375.376i		−	0.677575i
$555$		0			0
$556$	223.158			0.401362
$557$		−	477.847i		−	0.857894i	−0.903330	−	0.428947i	$-0.858885\pi$
$557$		−	477.847i		−	0.857894i	0.903330	−	0.428947i	$-0.141115\pi$
$558$		0			0
$559$			37.2340i			0.0666082i
$560$	−45.4552			−0.0811701
$561$		0			0
$562$			34.5030i			0.0613933i
$563$		−	291.541i		−	0.517834i	−0.965900	−	0.258917i	$-0.916634\pi$
$563$		−	291.541i		−	0.517834i	0.965900	−	0.258917i	$-0.0833657\pi$
$564$		0			0
$565$	124.267	−	496.807i	0.219941	−	0.879305i
$566$		−	192.881i		−	0.340780i
$567$		0			0
$568$		−	261.817i		−	0.460945i
$569$			340.585i			0.598568i	0.954164	+	0.299284i	$0.0967479\pi$
$569$			340.585i			0.598568i	−0.954164	+	0.299284i	$0.903252\pi$
$570$		0			0
$571$		−	516.530i		−	0.904606i	−0.891864	−	0.452303i	$-0.850603\pi$
$571$		−	516.530i		−	0.904606i	0.891864	−	0.452303i	$-0.149397\pi$
$572$			223.483i			0.390704i
$573$		0			0
$574$	263.370			0.458833
$575$	12.6286			0.0219627
$576$		0			0
$577$			14.9577i			0.0259232i	0.999916	+	0.0129616i	$0.00412593\pi$
$577$			14.9577i			0.0259232i	−0.999916	+	0.0129616i	$0.995874\pi$
$578$			357.906i			0.619214i
$579$		0			0
$580$	−74.9648			−0.129250
$581$			217.158i			0.373766i
$582$		0			0
$583$	28.1106			0.0482172
$584$	308.795			0.528758
$585$		0			0
$586$		−	246.266i		−	0.420249i
$587$			578.114i			0.984861i	0.870352	+	0.492431i	$0.163892\pi$
$587$			578.114i			0.984861i	−0.870352	+	0.492431i	$0.836108\pi$
$588$		0			0
$589$			383.254i			0.650685i
$590$		−	285.510i		−	0.483916i
$591$		0			0
$592$		−	47.8152i		−	0.0807690i
$593$			1183.03i			1.99499i	0.0707239	+	0.997496i	$0.477469\pi$
$593$			1183.03i			1.99499i	−0.0707239	+	0.997496i	$0.522531\pi$
$594$		0			0
$595$	264.579			0.444670
$596$		−	140.553i		−	0.235827i
$597$		0			0
$598$		−	42.7954i		−	0.0715642i
$599$	−785.388			−1.31116			−0.655582	−	0.755124i	$-0.727577\pi$
$599$	−785.388			−1.31116			−0.655582	+	0.755124i	$0.727577\pi$
$600$		0			0
$601$	346.207			0.576051			0.288026	−	0.957623i	$-0.407001\pi$
$601$	346.207			0.576051			0.288026	+	0.957623i	$0.407001\pi$
$602$	12.3512			0.0205170
$603$		0			0
$604$			80.9880i			0.134086i
$605$	53.1994			0.0879329
$606$		0			0
$607$		−	640.284i		−	1.05483i	−0.849607	−	0.527417i	$-0.823161\pi$
$607$		−	640.284i		−	1.05483i	0.849607	−	0.527417i	$-0.176839\pi$
$608$	59.9891			0.0986664
$609$		0			0
$610$		−	367.892i		−	0.603102i
$611$	−432.306			−0.707538
$612$		0			0
$613$		−	55.2509i		−	0.0901320i	−0.998984	−	0.0450660i	$-0.985650\pi$
$613$		−	55.2509i		−	0.0901320i	0.998984	−	0.0450660i	$-0.0143498\pi$
$614$			314.780i			0.512671i
$615$		0			0
$616$	74.1335			0.120347
$617$		−	92.6432i		−	0.150151i	−0.997178	−	0.0750755i	$-0.976080\pi$
$617$		−	92.6432i		−	0.150151i	0.997178	−	0.0750755i	$-0.0239198\pi$
$618$		0			0
$619$			843.835i			1.36322i	0.731714	+	0.681612i	$0.238721\pi$
$619$			843.835i			1.36322i	−0.731714	+	0.681612i	$0.761279\pi$
$620$	327.571			0.528341
$621$		0			0
$622$	481.075			0.773432
$623$	301.954			0.484678
$624$		0			0
$625$	−493.567			−0.789708
$626$		−	502.773i		−	0.803152i
$627$		0			0
$628$	551.222			0.877741
$629$			278.315i			0.442473i
$630$		0			0
$631$			506.192i			0.802206i	0.916033	+	0.401103i	$0.131373\pi$
$631$			506.192i			0.802206i	−0.916033	+	0.401103i	$0.868627\pi$
$632$	210.294			0.332744
$633$		0			0
$634$	241.369			0.380708
$635$	168.129			0.264770
$636$		0			0
$637$	−456.601			−0.716799
$638$	122.261			0.191632
$639$		0			0
$640$		−	51.2734i		−	0.0801147i
$641$	1233.98			1.92508			0.962541	−	0.271136i	$-0.0873992\pi$
$641$	1233.98			1.92508			0.962541	+	0.271136i	$0.0873992\pi$
$642$		0			0
$643$			830.851i			1.29215i	0.763275	+	0.646074i	$0.223590\pi$
$643$			830.851i			1.29215i	−0.763275	+	0.646074i	$0.776410\pi$
$644$	−14.1960			−0.0220435
$645$		0			0
$646$	−349.175			−0.540519
$647$		−	804.573i		−	1.24354i	−0.783198	−	0.621772i	$-0.786413\pi$
$647$		−	804.573i		−	1.24354i	0.783198	−	0.621772i	$-0.213587\pi$
$648$		0			0
$649$			465.643i			0.717477i
$650$		−	67.4449i		−	0.103761i
$651$		0			0
$652$	523.028			0.802191
$653$			516.423i			0.790847i	0.918499	+	0.395424i	$0.129402\pi$
$653$			516.423i			0.790847i	−0.918499	+	0.395424i	$0.870598\pi$
$654$		0			0
$655$		−	145.410i		−	0.221999i
$656$			297.081i			0.452868i
$657$		0			0
$658$			143.404i			0.217939i
$659$	−604.207			−0.916854			−0.458427	−	0.888732i	$-0.651587\pi$
$659$	−604.207			−0.916854			−0.458427	+	0.888732i	$0.651587\pi$
$660$		0			0
$661$			515.621i			0.780062i	0.920802	+	0.390031i	$0.127536\pi$
$661$			515.621i			0.780062i	−0.920802	+	0.390031i	$0.872464\pi$
$662$			326.540i			0.493263i
$663$		0			0
$664$	−244.954			−0.368906
$665$			120.510i			0.181217i
$666$		0			0
$667$	−23.4121			−0.0351006
$668$	204.746			0.306506
$669$		0			0
$670$	−151.525			−0.226156
$671$			600.001i			0.894189i
$672$		0			0
$673$		−	313.504i		−	0.465831i	−0.972497	−	0.232915i	$-0.925173\pi$
$673$		−	313.504i		−	0.465831i	0.972497	−	0.232915i	$-0.0748265\pi$
$674$			402.439i			0.597090i
$675$		0			0
$676$	109.444			0.161900
$677$		−	90.9903i		−	0.134402i	−0.997739	−	0.0672011i	$-0.978593\pi$
$677$		−	90.9903i		−	0.134402i	0.997739	−	0.0672011i	$-0.0214069\pi$
$678$		0			0
$679$	14.4837			0.0213309
$680$			298.444i			0.438889i
$681$		0			0
$682$	−534.240			−0.783344
$683$	−1208.48			−1.76937			−0.884683	−	0.466193i	$-0.845625\pi$
$683$	−1208.48			−1.76937			−0.884683	+	0.466193i	$0.845625\pi$
$684$		0			0
$685$	557.452			0.813799
$686$			325.222i			0.474085i
$687$		0			0
$688$			13.9321i			0.0202502i
$689$			28.7487i			0.0417253i
$690$		0			0
$691$	293.476			0.424712			0.212356	−	0.977192i	$-0.431886\pi$
$691$	293.476			0.424712			0.212356	+	0.977192i	$0.431886\pi$
$692$			146.122i			0.211160i
$693$		0			0
$694$	318.568			0.459032
$695$	−505.672			−0.727586
$696$		0			0
$697$		−	1729.20i		−	2.48092i
$698$	325.476			0.466298
$699$		0			0
$700$	−22.3728			−0.0319611
$701$	765.869			1.09254			0.546269	−	0.837610i	$-0.316048\pi$
$701$	765.869			1.09254			0.546269	+	0.837610i	$0.316048\pi$
$702$		0			0
$703$	−126.766			−0.180322
$704$			83.6225i			0.118782i
$705$		0			0
$706$	−165.034			−0.233759
$707$	−296.265			−0.419046
$708$		0			0
$709$	−956.994			−1.34978			−0.674890	−	0.737918i	$-0.735809\pi$
$709$	−956.994			−1.34978			−0.674890	+	0.737918i	$0.735809\pi$
$710$			593.273i			0.835596i
$711$		0			0
$712$			340.604i			0.478376i
$713$	102.303			0.143483
$714$		0			0
$715$		−	506.409i		−	0.708265i
$716$	−665.098			−0.928908
$717$		0			0
$718$			452.851i			0.630712i
$719$			747.120i			1.03911i	0.854437	+	0.519555i	$0.173902\pi$
$719$			747.120i			1.03911i	−0.854437	+	0.519555i	$0.826098\pi$
$720$		0			0
$721$		−	239.699i		−	0.332453i
$722$			351.490i			0.486828i
$723$		0			0
$724$		−	376.453i		−	0.519962i
$725$	−36.8971			−0.0508926
$726$		0			0
$727$	−503.837			−0.693036			−0.346518	−	0.938043i	$-0.612636\pi$
$727$	−503.837			−0.693036			−0.346518	+	0.938043i	$0.612636\pi$
$728$			75.8163i			0.104143i
$729$		0			0
$730$	−699.725			−0.958528
$731$		−	81.0940i		−	0.110936i
$732$		0			0
$733$		−	1311.04i		−	1.78859i	−0.447480	−	0.894294i	$-0.647678\pi$
$733$		−	1311.04i		−	1.78859i	0.447480	−	0.894294i	$-0.352322\pi$
$734$		−	242.409i		−	0.330258i
$735$		0			0
$736$		−	16.0131i		−	0.0217569i
$737$	247.123			0.335310
$738$		0			0
$739$	864.763			1.17018			0.585090	−	0.810968i	$-0.301059\pi$
$739$	864.763			1.17018			0.585090	+	0.810968i	$0.301059\pi$
$740$			108.349i			0.146417i
$741$		0			0
$742$	9.53649			0.0128524
$743$	−280.253			−0.377191			−0.188595	−	0.982055i	$-0.560393\pi$
$743$	−280.253			−0.377191			−0.188595	+	0.982055i	$0.560393\pi$
$744$		0			0
$745$			318.490i			0.427504i
$746$	−111.144			−0.148987
$747$		0			0
$748$		−	486.737i		−	0.650717i
$749$	481.063			0.642274
$750$		0			0
$751$			854.087i			1.13727i	0.822591	+	0.568633i	$0.192527\pi$
$751$			854.087i			1.13727i	−0.822591	+	0.568633i	$0.807473\pi$
$752$	−161.759			−0.215106
$753$		0			0
$754$			125.036i			0.165831i
$755$		−	183.518i		−	0.243070i
$756$		0			0
$757$			1313.66i			1.73534i	0.497136	+	0.867672i	$0.334385\pi$
$757$			1313.66i			1.73534i	−0.497136	+	0.867672i	$0.665615\pi$
$758$	−128.149			−0.169062
$759$		0			0
$760$	−135.935			−0.178861
$761$			1040.57i			1.36737i	0.729778	+	0.683684i	$0.239623\pi$
$761$			1040.57i			1.36737i	−0.729778	+	0.683684i	$0.760377\pi$
$762$		0			0
$763$	201.010			0.263447
$764$	−461.030			−0.603443
$765$		0			0
$766$	215.640			0.281514
$767$	−476.212			−0.620877
$768$		0			0
$769$	−264.999			−0.344602			−0.172301	−	0.985044i	$-0.555120\pi$
$769$	−264.999			−0.344602			−0.172301	+	0.985044i	$0.555120\pi$
$770$	−167.986			−0.218163
$771$		0			0
$772$			295.253i			0.382452i
$773$			617.295i			0.798571i	0.916827	+	0.399286i	$0.130742\pi$
$773$			617.295i			0.798571i	−0.916827	+	0.399286i	$0.869258\pi$
$774$		0			0
$775$	161.228			0.208037
$776$			16.3376i			0.0210536i
$777$		0			0
$778$	527.844			0.678462
$779$	787.613			1.01106
$780$		0			0
$781$		−	967.577i		−	1.23890i
$782$			93.2065i			0.119190i
$783$		0			0
$784$	−170.850			−0.217921
$785$	−1249.06			−1.59116
$786$		0			0
$787$	172.487			0.219170			0.109585	−	0.993977i	$-0.465048\pi$
$787$	172.487			0.219170			0.109585	+	0.993977i	$0.465048\pi$
$788$	−514.691			−0.653161
$789$		0			0
$790$	−476.524			−0.603195
$791$	−68.7549	+	274.876i	−0.0869215	+	0.347505i
$792$		0			0
$793$	−613.620			−0.773796
$794$	504.280			0.635113
$795$		0			0
$796$			744.787i			0.935662i
$797$	118.504			0.148687			0.0743436	−	0.997233i	$-0.476314\pi$
$797$	118.504			0.148687			0.0743436	+	0.997233i	$0.476314\pi$
$798$		0			0
$799$	941.544			1.17840
$800$		−	25.2364i		−	0.0315455i
$801$		0			0
$802$	−294.482			−0.367185
$803$	1141.19			1.42116
$804$		0			0
$805$	32.1680			0.0399603
$806$		−	546.367i		−	0.677875i
$807$		0			0
$808$		−	334.187i		−	0.413598i
$809$			489.657i			0.605262i	0.953108	+	0.302631i	$0.0978650\pi$
$809$			489.657i			0.605262i	−0.953108	+	0.302631i	$0.902135\pi$
$810$		0			0
$811$		−	1279.07i		−	1.57716i	−0.614935	−	0.788578i	$-0.710818\pi$
$811$		−	1279.07i		−	1.57716i	0.614935	−	0.788578i	$-0.289182\pi$
$812$	41.4769			0.0510799
$813$		0			0
$814$		−	176.707i		−	0.217085i
$815$	−1185.18			−1.45420
$816$		0			0
$817$	36.9365			0.0452099
$818$	−421.111			−0.514806
$819$		0			0
$820$		−	673.182i		−	0.820954i
$821$			661.166i			0.805318i	0.915350	+	0.402659i	$0.131914\pi$
$821$			661.166i			0.805318i	−0.915350	+	0.402659i	$0.868086\pi$
$822$		0			0
$823$	553.375			0.672388			0.336194	−	0.941793i	$-0.390860\pi$
$823$	553.375			0.672388			0.336194	+	0.941793i	$0.390860\pi$
$824$	270.380			0.328131
$825$		0			0
$826$			157.969i			0.191245i
$827$		−	905.937i		−	1.09545i	−0.836658	−	0.547725i	$-0.815494\pi$
$827$		−	905.937i		−	1.09545i	0.836658	−	0.547725i	$-0.184506\pi$
$828$		0			0
$829$		−	784.873i		−	0.946771i	−0.880855	−	0.473386i	$-0.843032\pi$
$829$		−	784.873i		−	0.946771i	0.880855	−	0.473386i	$-0.156968\pi$
$830$	555.062			0.668749
$831$		0			0
$832$	−85.5207			−0.102789
$833$	994.458			1.19383
$834$		0			0
$835$	−463.952			−0.555631
$836$	221.698			0.265188
$837$		0			0
$838$		−	111.131i		−	0.132614i
$839$	441.773			0.526547			0.263273	−	0.964721i	$-0.415198\pi$
$839$	441.773			0.526547			0.263273	+	0.964721i	$0.415198\pi$
$840$		0			0
$841$	−772.596			−0.918664
$842$		−	199.735i		−	0.237214i
$843$		0			0
$844$	377.072			0.446768
$845$	−247.999			−0.293490
$846$		0			0
$847$	−29.4344			−0.0347514
$848$			10.7571i			0.0126853i
$849$		0			0
$850$			146.892i			0.172814i
$851$			33.8382i			0.0397628i
$852$		0			0
$853$	361.609			0.423926			0.211963	−	0.977278i	$-0.432014\pi$
$853$	361.609			0.423926			0.211963	+	0.977278i	$0.432014\pi$
$854$			203.549i			0.238348i
$855$		0			0
$856$			542.638i			0.633923i
$857$	−1168.59			−1.36359			−0.681794	−	0.731545i	$-0.738800\pi$
$857$	−1168.59			−1.36359			−0.681794	+	0.731545i	$0.738800\pi$
$858$		0			0
$859$		−	1038.31i		−	1.20874i	−0.796704	−	0.604369i	$-0.793425\pi$
$859$		−	1038.31i		−	1.20874i	0.796704	−	0.604369i	$-0.206575\pi$
$860$		−	31.5701i		−	0.0367094i
$861$		0			0
$862$			511.155i			0.592987i
$863$		−	1004.91i		−	1.16444i	−0.813032	−	0.582219i	$-0.802185\pi$
$863$		−	1004.91i		−	1.16444i	0.813032	−	0.582219i	$-0.197815\pi$
$864$		0			0
$865$		−	331.112i		−	0.382788i
$866$	−1022.31			−1.18049
$867$		0			0
$868$	−181.240			−0.208802
$869$	777.170			0.894326
$870$		0			0
$871$			252.733i			0.290164i
$872$			226.739i			0.260022i
$873$		0			0
$874$	−42.4535			−0.0485738
$875$	334.792			0.382619
$876$		0			0
$877$			1544.85i			1.76152i	0.473563	+	0.880760i	$0.342968\pi$
$877$			1544.85i			1.76152i	−0.473563	+	0.880760i	$0.657032\pi$
$878$			270.823i			0.308454i
$879$		0			0
$880$		−	189.488i		−	0.215327i
$881$	1444.64			1.63978			0.819888	−	0.572524i	$-0.194036\pi$
$881$	1444.64			1.63978			0.819888	+	0.572524i	$0.194036\pi$
$882$		0			0
$883$		−	207.853i		−	0.235394i	−0.993050	−	0.117697i	$-0.962449\pi$
$883$		−	207.853i		−	0.235394i	0.993050	−	0.117697i	$-0.0375512\pi$
$884$	497.785			0.563105
$885$		0			0
$886$	−1085.58			−1.22526
$887$	1688.88			1.90404			0.952020	−	0.306037i	$-0.0990030\pi$
$887$	1688.88			1.90404			0.952020	+	0.306037i	$0.0990030\pi$
$888$		0			0
$889$	−93.0234			−0.104638
$890$		−	771.804i		−	0.867195i
$891$		0			0
$892$		−	23.1519i		−	0.0259551i
$893$			428.852i			0.480237i
$894$		0			0
$895$	1507.10			1.68391
$896$			28.3688i			0.0316616i
$897$		0			0
$898$			330.697i			0.368260i
$899$	−298.902			−0.332482
$900$		0			0
$901$		−	62.6135i		−	0.0694933i
$902$			1097.90i			1.21719i
$903$		0			0
$904$	−310.060	−	77.5554i	−0.342987	−	0.0857914i
$905$			853.037i			0.942582i
$906$		0			0
$907$		−	833.634i		−	0.919112i	−0.888149	−	0.459556i	$-0.848009\pi$
$907$		−	833.634i		−	0.919112i	0.888149	−	0.459556i	$-0.151991\pi$
$908$			584.234i			0.643430i
$909$		0			0
$910$		−	171.799i		−	0.188790i
$911$		−	451.928i		−	0.496079i	−0.968750	−	0.248039i	$-0.920214\pi$
$911$		−	451.928i		−	0.496079i	0.968750	−	0.248039i	$-0.0797863\pi$
$912$		0			0
$913$	−905.258			−0.991520
$914$	−468.984			−0.513111
$915$		0			0
$916$			284.096i			0.310149i
$917$			80.4530i			0.0877350i
$918$		0			0
$919$	337.761			0.367531			0.183766	−	0.982970i	$-0.441171\pi$
$919$	337.761			0.367531			0.183766	+	0.982970i	$0.441171\pi$
$920$			36.2855i			0.0394408i
$921$		0			0
$922$	446.739			0.484532
$923$	989.540			1.07209
$924$		0			0
$925$			53.3285i			0.0576524i
$926$			512.271i			0.553209i
$927$		0			0
$928$			46.7859i			0.0504158i
$929$		−	1269.43i		−	1.36645i	−0.730207	−	0.683226i	$-0.760576\pi$
$929$		−	1269.43i		−	1.36645i	0.730207	−	0.683226i	$-0.239424\pi$
$930$		0			0
$931$			452.953i			0.486523i
$932$		−	632.057i		−	0.678173i
$933$		0			0
$934$	−161.874			−0.173312
$935$			1102.94i			1.17961i
$936$		0			0
$937$		−	127.691i		−	0.136277i	−0.997676	−	0.0681383i	$-0.978294\pi$
$937$		−	127.691i		−	0.136277i	0.997676	−	0.0681383i	$-0.0217059\pi$
$938$	83.8363			0.0893777
$939$		0			0
$940$	366.545			0.389941
$941$	−84.4296			−0.0897233			−0.0448616	−	0.998993i	$-0.514285\pi$
$941$	−84.4296			−0.0897233			−0.0448616	+	0.998993i	$0.514285\pi$
$942$		0			0
$943$		−	210.240i		−	0.222948i
$944$	−178.188			−0.188759
$945$		0			0
$946$			51.4880i			0.0544271i
$947$	−255.008			−0.269280			−0.134640	−	0.990895i	$-0.542988\pi$
$947$	−255.008			−0.269280			−0.134640	+	0.990895i	$0.542988\pi$
$948$		0			0
$949$			1167.10i			1.22982i
$950$	−66.9061			−0.0704275
$951$		0			0
$952$		−	165.125i		−	0.173450i
$953$			1535.63i			1.61136i	0.592350	+	0.805681i	$0.298200\pi$
$953$			1535.63i			1.61136i	−0.592350	+	0.805681i	$0.701800\pi$
$954$		0			0
$955$	1044.69			1.09391
$956$			404.670i			0.423295i
$957$		0			0
$958$			884.693i			0.923479i
$959$	−308.430			−0.321616
$960$		0			0
$961$	345.102			0.359107
$962$	180.718			0.187857
$963$		0			0
$964$	856.061			0.888030
$965$		−	669.039i		−	0.693304i
$966$		0			0
$967$	914.969			0.946193			0.473097	−	0.881011i	$-0.343136\pi$
$967$	914.969			0.946193			0.473097	+	0.881011i	$0.343136\pi$
$968$		−	33.2020i		−	0.0342996i
$969$		0			0
$970$		−	37.0208i		−	0.0381658i
$971$	−281.015			−0.289407			−0.144704	−	0.989475i	$-0.546223\pi$
$971$	−281.015			−0.289407			−0.144704	+	0.989475i	$0.546223\pi$
$972$		0			0
$973$	279.781			0.287545
$974$	−174.179			−0.178828
$975$		0			0
$976$	−229.603			−0.235249
$977$	523.821			0.536153			0.268076	−	0.963398i	$-0.413612\pi$
$977$	523.821			0.536153			0.268076	+	0.963398i	$0.413612\pi$
$978$		0			0
$979$			1258.74i			1.28575i
$980$	387.145			0.395045
$981$		0			0
$982$		−	651.075i		−	0.663009i
$983$	−394.148			−0.400964			−0.200482	−	0.979697i	$-0.564251\pi$
$983$	−394.148			−0.400964			−0.200482	+	0.979697i	$0.564251\pi$
$984$		0			0
$985$	1166.28			1.18404
$986$		−	272.324i		−	0.276190i
$987$		0			0
$988$			226.730i			0.229484i
$989$		−	9.85959i		−	0.00996925i
$990$		0			0
$991$	201.396			0.203225			0.101613	−	0.994824i	$-0.467600\pi$
$991$	201.396			0.203225			0.101613	+	0.994824i	$0.467600\pi$
$992$		−	204.439i		−	0.206088i
$993$		0			0
$994$		−	328.249i		−	0.330231i
$995$		−	1687.68i		−	1.69616i
$996$		0			0
$997$		−	1462.47i		−	1.46687i	−0.679760	−	0.733435i	$-0.737916\pi$
$997$		−	1462.47i		−	1.46687i	0.679760	−	0.733435i	$-0.262084\pi$
$998$	309.644			0.310265
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.3.d.a.2033.30	yes	36
3.2	odd	2	inner	2034.3.d.a.2033.21	✓	36
113.112	even	2	inner	2034.3.d.a.2033.22	yes	36
339.338	odd	2	inner	2034.3.d.a.2033.29	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2034.3.d.a.2033.21	✓	36	3.2	odd	2	inner
2034.3.d.a.2033.22	yes	36	113.112	even	2	inner
2034.3.d.a.2033.29	yes	36	339.338	odd	2	inner
2034.3.d.a.2033.30	yes	36	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2034.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.53197 q^{5} -2.50747 q^{7} -2.82843i q^{8} +6.40918i q^{10} -10.4528i q^{11} +10.6901 q^{13} -3.54610i q^{14} +4.00000 q^{16} -23.2826 q^{17} -10.6047i q^{19} -9.06395 q^{20} +14.7825 q^{22} -2.83074 q^{23} -4.46121 q^{25} +15.1181i q^{26} +5.01495 q^{28} +8.27065 q^{29} -36.1400 q^{31} +5.65685i q^{32} -32.9265i q^{34} -11.3638 q^{35} -11.9538i q^{37} +14.9973 q^{38} -12.8184i q^{40} +74.2703i q^{41} +3.48304i q^{43} +20.9056i q^{44} -4.00328i q^{46} -40.4399 q^{47} -42.7126 q^{49} -6.30911i q^{50} -21.3802 q^{52} +2.68929i q^{53} -47.3719i q^{55} +7.09221i q^{56} +11.6965i q^{58} -44.5471 q^{59} -57.4009 q^{61} -51.1097i q^{62} -8.00000 q^{64} +48.4472 q^{65} +23.6418i q^{67} +46.5651 q^{68} -16.0709i q^{70} +92.5662 q^{71} +109.176i q^{73} +16.9052 q^{74} +21.2094i q^{76} +26.2102i q^{77} +74.3503i q^{79} +18.1279 q^{80} -105.034 q^{82} -86.6042i q^{83} -105.516 q^{85} -4.92576 q^{86} -29.5650 q^{88} -120.422 q^{89} -26.8051 q^{91} +5.66149 q^{92} -57.1906i q^{94} -48.0601i q^{95} -5.77622 q^{97} -60.4047i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) \) 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more