LMFDB - Embedded newform 5292.2.l.j.3313.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(361,5292)] 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("5292.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5292, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.l (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.5
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.j.361.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.938454 q^{5} -2.63595 q^{11} +(2.71055 + 4.69481i) q^{13} +(-1.92740 - 3.33835i) q^{17} +(-0.548251 + 0.949599i) q^{19} +6.33440 q^{23} -4.11930 q^{25} +(1.94145 - 3.36268i) q^{29} +(-2.33312 + 4.04109i) q^{31} +(-1.15439 + 1.99946i) q^{37} +(-4.12179 - 7.13915i) q^{41} +(-2.14515 + 3.71551i) q^{43} +(1.32160 + 2.28908i) q^{47} +(-0.637645 - 1.10443i) q^{53} +2.47372 q^{55} +(3.02092 - 5.23239i) q^{59} +(6.71961 + 11.6387i) q^{61} +(-2.54373 - 4.40587i) q^{65} +(3.64378 - 6.31122i) q^{67} -14.7859 q^{71} +(-2.87484 - 4.97937i) q^{73} +(-5.51357 - 9.54979i) q^{79} +(-1.24637 + 2.15877i) q^{83} +(1.80877 + 3.13289i) q^{85} +(-6.75703 + 11.7035i) q^{89} +(0.514508 - 0.891154i) q^{95} +(-1.75370 + 3.03749i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.938454			−0.419689			−0.209845	−	0.977735i	$-0.567296\pi$
$5$	−0.938454			−0.419689			−0.209845	+	0.977735i	$0.567296\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.63595			−0.794768			−0.397384	−	0.917652i	$-0.630082\pi$
$11$	−2.63595			−0.794768			−0.397384	+	0.917652i	$0.630082\pi$
$12$		0			0
$13$	2.71055	+	4.69481i	0.751772	+	1.30211i	0.946963	+	0.321342i	$0.104134\pi$
$13$	2.71055	+	4.69481i	0.751772	+	1.30211i	−0.195191	+	0.980765i	$0.562533\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.92740	−	3.33835i	−0.467463	−	0.809669i	0.531846	−	0.846841i	$-0.321498\pi$
$17$	−1.92740	−	3.33835i	−0.467463	−	0.809669i	−0.999309	+	0.0371717i	$0.988165\pi$
$18$		0			0
$19$	−0.548251	+	0.949599i	−0.125777	+	0.217853i	−0.922037	−	0.387103i	$-0.873476\pi$
$19$	−0.548251	+	0.949599i	−0.125777	+	0.217853i	0.796259	+	0.604956i	$0.206809\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.33440			1.32081			0.660407	−	0.750908i	$-0.270384\pi$
$23$	6.33440			1.32081			0.660407	+	0.750908i	$0.270384\pi$
$24$		0			0
$25$	−4.11930			−0.823861
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.94145	−	3.36268i	0.360517	−	0.624434i	−0.627529	−	0.778593i	$-0.715934\pi$
$29$	1.94145	−	3.36268i	0.360517	−	0.624434i	0.988046	+	0.154159i	$0.0492668\pi$
$30$		0			0
$31$	−2.33312	+	4.04109i	−0.419041	+	0.725801i	−0.995843	−	0.0910831i	$-0.970967\pi$
$31$	−2.33312	+	4.04109i	−0.419041	+	0.725801i	0.576802	+	0.816884i	$0.304300\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.15439	+	1.99946i	−0.189780	+	0.328709i	−0.945177	−	0.326559i	$-0.894111\pi$
$37$	−1.15439	+	1.99946i	−0.189780	+	0.328709i	0.755397	+	0.655268i	$0.227444\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.12179	−	7.13915i	−0.643715	−	1.11495i	−0.984597	−	0.174841i	$-0.944059\pi$
$41$	−4.12179	−	7.13915i	−0.643715	−	1.11495i	0.340882	−	0.940106i	$-0.389274\pi$
$42$		0			0
$43$	−2.14515	+	3.71551i	−0.327132	+	0.566609i	−0.981942	−	0.189185i	$-0.939416\pi$
$43$	−2.14515	+	3.71551i	−0.327132	+	0.566609i	0.654810	+	0.755794i	$0.272749\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.32160	+	2.28908i	0.192775	+	0.333897i	0.946169	−	0.323673i	$-0.104918\pi$
$47$	1.32160	+	2.28908i	0.192775	+	0.333897i	−0.753394	+	0.657570i	$0.771584\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.637645	−	1.10443i	−0.0875873	−	0.151706i	0.818903	−	0.573931i	$-0.194582\pi$
$53$	−0.637645	−	1.10443i	−0.0875873	−	0.151706i	−0.906491	+	0.422226i	$0.861249\pi$
$54$		0			0
$55$	2.47372			0.333556
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.02092	−	5.23239i	0.393291	−	0.681199i	−0.599591	−	0.800307i	$-0.704670\pi$
$59$	3.02092	−	5.23239i	0.393291	−	0.681199i	0.992881	+	0.119107i	$0.0380033\pi$
$60$		0			0
$61$	6.71961	+	11.6387i	0.860357	+	1.49018i	0.871584	+	0.490246i	$0.163093\pi$
$61$	6.71961	+	11.6387i	0.860357	+	1.49018i	−0.0112268	+	0.999937i	$0.503574\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.54373	−	4.40587i	−0.315511	−	0.546480i
$66$		0			0
$67$	3.64378	−	6.31122i	0.445159	−	0.771038i	−0.552904	−	0.833245i	$-0.686480\pi$
$67$	3.64378	−	6.31122i	0.445159	−	0.771038i	0.998063	+	0.0622066i	$0.0198138\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.7859			−1.75476			−0.877382	−	0.479793i	$-0.840712\pi$
$71$	−14.7859			−1.75476			−0.877382	+	0.479793i	$0.840712\pi$
$72$		0			0
$73$	−2.87484	−	4.97937i	−0.336475	−	0.582791i	0.647292	−	0.762242i	$-0.275901\pi$
$73$	−2.87484	−	4.97937i	−0.336475	−	0.582791i	−0.983767	+	0.179451i	$0.942568\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.51357	−	9.54979i	−0.620326	−	1.07444i	−0.989425	−	0.145046i	$-0.953667\pi$
$79$	−5.51357	−	9.54979i	−0.620326	−	1.07444i	0.369099	−	0.929390i	$-0.379666\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.24637	+	2.15877i	−0.136807	+	0.236956i	−0.926286	−	0.376821i	$-0.877017\pi$
$83$	−1.24637	+	2.15877i	−0.136807	+	0.236956i	0.789479	+	0.613777i	$0.210351\pi$
$84$		0			0
$85$	1.80877	+	3.13289i	0.196189	+	0.339810i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.75703	+	11.7035i	−0.716244	+	1.24057i	0.246234	+	0.969210i	$0.420807\pi$
$89$	−6.75703	+	11.7035i	−0.716244	+	1.24057i	−0.962478	+	0.271360i	$0.912526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.514508	−	0.891154i	0.0527874	−	0.0914305i
$96$		0			0
$97$	−1.75370	+	3.03749i	−0.178061	+	0.308410i	−0.941216	−	0.337805i	$-0.890316\pi$
$97$	−1.75370	+	3.03749i	−0.178061	+	0.308410i	0.763155	+	0.646215i	$0.223649\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.09067			0.705548			0.352774	−	0.935708i	$-0.385238\pi$
$101$	7.09067			0.705548			0.352774	+	0.935708i	$0.385238\pi$
$102$		0			0
$103$	11.2462			1.10812			0.554061	−	0.832476i	$-0.313077\pi$
$103$	11.2462			1.10812			0.554061	+	0.832476i	$0.313077\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.42195	−	7.65905i	0.427486	−	0.740428i	−0.569163	−	0.822225i	$-0.692733\pi$
$107$	4.42195	−	7.65905i	0.427486	−	0.740428i	0.996649	+	0.0817968i	$0.0260659\pi$
$108$		0			0
$109$	−9.34721	−	16.1898i	−0.895300	−	1.55071i	−0.833433	−	0.552621i	$-0.813628\pi$
$109$	−9.34721	−	16.1898i	−0.895300	−	1.55071i	−0.0618674	−	0.998084i	$-0.519706\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.99798	−	6.92471i	−0.376099	−	0.651422i	0.614392	−	0.789001i	$-0.289401\pi$
$113$	−3.99798	−	6.92471i	−0.376099	−	0.651422i	−0.990491	+	0.137579i	$0.956068\pi$
$114$		0			0
$115$	−5.94454			−0.554331
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.05178			−0.368343
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.55805			0.765455
$126$		0			0
$127$	3.73961			0.331837			0.165918	−	0.986140i	$-0.446941\pi$
$127$	3.73961			0.331837			0.165918	+	0.986140i	$0.446941\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−21.6344			−1.89021			−0.945104	−	0.326768i	$-0.894040\pi$
$131$	−21.6344			−1.89021			−0.945104	+	0.326768i	$0.894040\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0974			−1.20442			−0.602211	−	0.798337i	$-0.705714\pi$
$137$	−14.0974			−1.20442			−0.602211	+	0.798337i	$0.705714\pi$
$138$		0			0
$139$	−3.46104	−	5.99470i	−0.293562	−	0.508463i	0.681088	−	0.732202i	$-0.261507\pi$
$139$	−3.46104	−	5.99470i	−0.293562	−	0.508463i	−0.974649	+	0.223738i	$0.928174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.14487	−	12.3753i	−0.597484	−	1.03487i
$144$		0			0
$145$	−1.82196	+	3.15572i	−0.151305	+	0.262068i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.4331			−0.936637			−0.468318	−	0.883560i	$-0.655140\pi$
$149$	−11.4331			−0.936637			−0.468318	+	0.883560i	$0.655140\pi$
$150$		0			0
$151$	−10.7735			−0.876739			−0.438369	−	0.898795i	$-0.644444\pi$
$151$	−10.7735			−0.876739			−0.438369	+	0.898795i	$0.644444\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.18953	−	3.79237i	0.175867	−	0.304611i
$156$		0			0
$157$	−8.44500	+	14.6272i	−0.673984	+	1.16737i	0.302780	+	0.953060i	$0.402085\pi$
$157$	−8.44500	+	14.6272i	−0.673984	+	1.16737i	−0.976765	+	0.214315i	$0.931248\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.08879	−	15.7422i	0.711889	−	1.23303i	−0.252258	−	0.967660i	$-0.581173\pi$
$163$	9.08879	−	15.7422i	0.711889	−	1.23303i	0.964147	−	0.265368i	$-0.0854933\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.45398	−	4.25041i	−0.189894	−	0.328907i	0.755320	−	0.655356i	$-0.227481\pi$
$167$	−2.45398	−	4.25041i	−0.189894	−	0.328907i	−0.945215	+	0.326449i	$0.894148\pi$
$168$		0			0
$169$	−8.19418	+	14.1927i	−0.630322	+	1.09175i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.99519	+	10.3840i	0.455806	+	0.789479i	0.998734	−	0.0503002i	$-0.0160178\pi$
$173$	5.99519	+	10.3840i	0.455806	+	0.789479i	−0.542928	+	0.839779i	$0.682684\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.9324	−	20.6675i	−0.891871	−	1.54477i	−0.837631	−	0.546237i	$-0.816060\pi$
$179$	−11.9324	−	20.6675i	−0.891871	−	1.54477i	−0.0542400	−	0.998528i	$-0.517274\pi$
$180$		0			0
$181$	14.0990			1.04797			0.523987	−	0.851726i	$-0.324444\pi$
$181$	14.0990			1.04797			0.523987	+	0.851726i	$0.324444\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.08334	−	1.87640i	0.0796487	−	0.137956i
$186$		0			0
$187$	5.08052	+	8.79972i	0.371525	+	0.643500i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.97457	+	15.5444i	0.649377	+	1.12475i	0.983272	+	0.182144i	$0.0583036\pi$
$191$	8.97457	+	15.5444i	0.649377	+	1.12475i	−0.333895	+	0.942610i	$0.608363\pi$
$192$		0			0
$193$	−10.5119	+	18.2072i	−0.756665	+	1.31058i	0.187878	+	0.982192i	$0.439839\pi$
$193$	−10.5119	+	18.2072i	−0.756665	+	1.31058i	−0.944542	+	0.328389i	$0.893494\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.65707			−0.688038			−0.344019	−	0.938963i	$-0.611788\pi$
$197$	−9.65707			−0.688038			−0.344019	+	0.938963i	$0.611788\pi$
$198$		0			0
$199$	−1.61669	−	2.80018i	−0.114604	−	0.198500i	0.803017	−	0.595955i	$-0.203227\pi$
$199$	−1.61669	−	2.80018i	−0.114604	−	0.198500i	−0.917621	+	0.397456i	$0.869893\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.86811	+	6.69976i	0.270160	+	0.467931i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.44516	−	2.50309i	0.0999639	−	0.173143i
$210$		0			0
$211$	0.853200	+	1.47779i	0.0587367	+	0.101735i	0.893899	−	0.448269i	$-0.147959\pi$
$211$	0.853200	+	1.47779i	0.0587367	+	0.101735i	−0.835162	+	0.550004i	$0.814626\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.01312	−	3.48683i	0.137294	−	0.237800i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.4486	−	18.0976i	0.702851	−	1.21737i
$222$		0			0
$223$	1.60137	−	2.77366i	0.107236	−	0.185738i	−0.807414	−	0.589986i	$-0.799133\pi$
$223$	1.60137	−	2.77366i	0.107236	−	0.185738i	0.914649	+	0.404248i	$0.132467\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.0579			−1.53041			−0.765204	−	0.643788i	$-0.777362\pi$
$227$	−23.0579			−1.53041			−0.765204	+	0.643788i	$0.777362\pi$
$228$		0			0
$229$	−4.00303			−0.264528			−0.132264	−	0.991215i	$-0.542225\pi$
$229$	−4.00303			−0.264528			−0.132264	+	0.991215i	$0.542225\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.3624	−	24.8765i	0.940914	−	1.62971i	0.177182	−	0.984178i	$-0.443302\pi$
$233$	14.3624	−	24.8765i	0.940914	−	1.62971i	0.763732	−	0.645534i	$-0.223365\pi$
$234$		0			0
$235$	−1.24026	−	2.14820i	−0.0809057	−	0.140133i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.5967	+	18.3540i	0.685443	+	1.18722i	0.973297	+	0.229548i	$0.0737248\pi$
$239$	10.5967	+	18.3540i	0.685443	+	1.18722i	−0.287854	+	0.957674i	$0.592942\pi$
$240$		0			0
$241$	−18.1185			−1.16712			−0.583558	−	0.812071i	$-0.698340\pi$
$241$	−18.1185			−1.16712			−0.583558	+	0.812071i	$0.698340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.94425			−0.378224
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.04768			−0.255487			−0.127744	−	0.991807i	$-0.540773\pi$
$251$	−4.04768			−0.255487			−0.127744	+	0.991807i	$0.540773\pi$
$252$		0			0
$253$	−16.6971			−1.04974
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−28.7200			−1.79151			−0.895753	−	0.444551i	$-0.853363\pi$
$257$	−28.7200			−1.79151			−0.895753	+	0.444551i	$0.853363\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.12711			0.254488			0.127244	−	0.991871i	$-0.459387\pi$
$263$	4.12711			0.254488			0.127244	+	0.991871i	$0.459387\pi$
$264$		0			0
$265$	0.598401	+	1.03646i	0.0367595	+	0.0636692i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.59348	−	14.8843i	−0.523954	−	0.907514i	−0.999611	−	0.0278838i	$-0.991123\pi$
$269$	−8.59348	−	14.8843i	−0.523954	−	0.907514i	0.475657	−	0.879631i	$-0.342210\pi$
$270$		0			0
$271$	−13.5462	+	23.4628i	−0.822875	+	1.42526i	0.0806581	+	0.996742i	$0.474298\pi$
$271$	−13.5462	+	23.4628i	−0.822875	+	1.42526i	−0.903533	+	0.428519i	$0.859036\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.8583			0.654779
$276$		0			0
$277$	−29.5101			−1.77309			−0.886545	−	0.462642i	$-0.846902\pi$
$277$	−29.5101			−1.77309			−0.886545	+	0.462642i	$0.846902\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.09348	+	5.35807i	−0.184542	+	0.319636i	−0.943422	−	0.331595i	$-0.892413\pi$
$281$	−3.09348	+	5.35807i	−0.184542	+	0.319636i	0.758880	+	0.651230i	$0.225747\pi$
$282$		0			0
$283$	12.1655	−	21.0713i	0.723166	−	1.25256i	−0.236559	−	0.971617i	$-0.576020\pi$
$283$	12.1655	−	21.0713i	0.723166	−	1.25256i	0.959725	−	0.280942i	$-0.0906470\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.07027	−	1.85376i	0.0629570	−	0.109045i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.1615	+	22.7963i	0.768900	+	1.33177i	0.938160	+	0.346203i	$0.112529\pi$
$293$	13.1615	+	22.7963i	0.768900	+	1.33177i	−0.169259	+	0.985572i	$0.554138\pi$
$294$		0			0
$295$	−2.83500	+	4.91036i	−0.165060	+	0.285892i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	17.1697	+	29.7388i	0.992950	+	1.71984i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.30604	−	10.9224i	−0.361083	−	0.625414i
$306$		0			0
$307$	−18.8042			−1.07321			−0.536606	−	0.843833i	$-0.680294\pi$
$307$	−18.8042			−1.07321			−0.536606	+	0.843833i	$0.680294\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.67126	+	11.5550i	−0.378292	+	0.655221i	−0.990814	−	0.135232i	$-0.956822\pi$
$311$	−6.67126	+	11.5550i	−0.378292	+	0.655221i	0.612522	+	0.790454i	$0.290155\pi$
$312$		0			0
$313$	−5.50564	−	9.53604i	−0.311197	−	0.539009i	0.667425	−	0.744677i	$-0.267397\pi$
$313$	−5.50564	−	9.53604i	−0.311197	−	0.539009i	−0.978622	+	0.205668i	$0.934063\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.77717	−	8.27429i	−0.268312	−	0.464731i	0.700114	−	0.714031i	$-0.253133\pi$
$317$	−4.77717	−	8.27429i	−0.268312	−	0.464731i	−0.968426	+	0.249301i	$0.919799\pi$
$318$		0			0
$319$	−5.11755	+	8.86385i	−0.286528	+	0.496281i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.22679			0.235185
$324$		0			0
$325$	−11.1656	−	19.3394i	−0.619355	−	1.07276i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−16.1908	−	28.0433i	−0.889929	−	1.54140i	−0.839958	−	0.542651i	$-0.817421\pi$
$331$	−16.1908	−	28.0433i	−0.889929	−	1.54140i	−0.0499706	−	0.998751i	$-0.515913\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.41952	+	5.92279i	−0.186829	+	0.323596i
$336$		0			0
$337$	−4.33194	−	7.50313i	−0.235976	−	0.408722i	0.723580	−	0.690240i	$-0.242495\pi$
$337$	−4.33194	−	7.50313i	−0.235976	−	0.408722i	−0.959556	+	0.281519i	$0.909162\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.14999	−	10.6521i	0.333041	−	0.576844i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.5569	−	18.2852i	0.566726	−	0.981598i	−0.430161	−	0.902752i	$-0.641543\pi$
$347$	10.5569	−	18.2852i	0.566726	−	0.981598i	0.996887	−	0.0788461i	$-0.0251236\pi$
$348$		0			0
$349$	8.18677	−	14.1799i	0.438228	−	0.759033i	−0.559325	−	0.828948i	$-0.688940\pi$
$349$	8.18677	−	14.1799i	0.438228	−	0.759033i	0.997553	+	0.0699158i	$0.0222731\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	33.5236			1.78428			0.892142	−	0.451755i	$-0.149202\pi$
$353$	33.5236			1.78428			0.892142	+	0.451755i	$0.149202\pi$
$354$		0			0
$355$	13.8759			0.736456
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.11650	−	10.5941i	0.322816	−	0.559134i	−0.658252	−	0.752798i	$-0.728704\pi$
$359$	6.11650	−	10.5941i	0.322816	−	0.559134i	0.981068	+	0.193664i	$0.0620371\pi$
$360$		0			0
$361$	8.89884	+	15.4132i	0.468360	+	0.811223i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.69791	+	4.67291i	0.141215	+	0.244591i
$366$		0			0
$367$	9.77838			0.510427			0.255214	−	0.966885i	$-0.417854\pi$
$367$	9.77838			0.510427			0.255214	+	0.966885i	$0.417854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−20.3976			−1.05615			−0.528074	−	0.849199i	$-0.677085\pi$
$373$	−20.3976			−1.05615			−0.528074	+	0.849199i	$0.677085\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.0496			1.08411
$378$		0			0
$379$	−9.11318			−0.468113			−0.234056	−	0.972223i	$-0.575200\pi$
$379$	−9.11318			−0.468113			−0.234056	+	0.972223i	$0.575200\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.381668			0.0195023			0.00975117	−	0.999952i	$-0.496896\pi$
$383$	0.381668			0.0195023			0.00975117	+	0.999952i	$0.496896\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.65175			0.438661			0.219331	−	0.975651i	$-0.429613\pi$
$389$	8.65175			0.438661			0.219331	+	0.975651i	$0.429613\pi$
$390$		0			0
$391$	−12.2089	−	21.1464i	−0.617431	−	1.06942i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.17423	+	8.96204i	0.260344	+	0.450929i
$396$		0			0
$397$	−6.83118	+	11.8320i	−0.342847	+	0.593829i	−0.984960	−	0.172781i	$-0.944725\pi$
$397$	−6.83118	+	11.8320i	−0.342847	+	0.593829i	0.642113	+	0.766610i	$0.278058\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.4467			−1.57037			−0.785187	−	0.619258i	$-0.787433\pi$
$401$	−31.4467			−1.57037			−0.785187	+	0.619258i	$0.787433\pi$
$402$		0			0
$403$	−25.2962			−1.26009
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.04290	−	5.27047i	0.150831	−	0.261247i
$408$		0			0
$409$	1.61948	−	2.80503i	0.0800783	−	0.138700i	−0.823205	−	0.567744i	$-0.807816\pi$
$409$	1.61948	−	2.80503i	0.0800783	−	0.138700i	0.903283	+	0.429044i	$0.141150\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.16966	−	2.02591i	0.0574163	−	0.0994480i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.9772	−	19.0130i	−0.536271	−	0.928848i	−0.999101	−	0.0424009i	$-0.986499\pi$
$419$	−10.9772	−	19.0130i	−0.536271	−	0.928848i	0.462830	−	0.886447i	$-0.346834\pi$
$420$		0			0
$421$	11.8273	−	20.4855i	0.576428	−	0.998402i	−0.419457	−	0.907775i	$-0.637780\pi$
$421$	11.8273	−	20.4855i	0.576428	−	0.998402i	0.995885	−	0.0906270i	$-0.0288871\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.93954	+	13.7517i	0.385124	+	0.667055i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.9920	−	24.2349i	−0.673972	−	1.16735i	−0.976768	−	0.214298i	$-0.931253\pi$
$431$	−13.9920	−	24.2349i	−0.673972	−	1.16735i	0.302796	−	0.953055i	$-0.402080\pi$
$432$		0			0
$433$	4.17947			0.200852			0.100426	−	0.994945i	$-0.467979\pi$
$433$	4.17947			0.200852			0.100426	+	0.994945i	$0.467979\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.47284	+	6.01513i	−0.166128	+	0.287743i
$438$		0			0
$439$	16.3653	+	28.3456i	0.781074	+	1.35286i	0.931316	+	0.364211i	$0.118661\pi$
$439$	16.3653	+	28.3456i	0.781074	+	1.35286i	−0.150242	+	0.988649i	$0.548005\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.23770	+	10.8040i	0.296362	+	0.513314i	0.975301	−	0.220881i	$-0.0708931\pi$
$443$	6.23770	+	10.8040i	0.296362	+	0.513314i	−0.678939	+	0.734195i	$0.737560\pi$
$444$		0			0
$445$	6.34116	−	10.9832i	0.300600	−	0.520654i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.8840			−1.83505			−0.917524	−	0.397680i	$-0.869815\pi$
$449$	−38.8840			−1.83505			−0.917524	+	0.397680i	$0.869815\pi$
$450$		0			0
$451$	10.8648	+	18.8184i	0.511604	+	0.886125i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.5977	+	20.0878i	0.542517	+	0.939667i	0.998759	+	0.0498113i	$0.0158620\pi$
$457$	11.5977	+	20.0878i	0.542517	+	0.939667i	−0.456241	+	0.889856i	$0.650805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.96496	+	15.5278i	−0.417540	+	0.723200i	−0.995691	−	0.0927293i	$-0.970441\pi$
$461$	−8.96496	+	15.5278i	−0.417540	+	0.723200i	0.578152	+	0.815929i	$0.303774\pi$
$462$		0			0
$463$	0.481797	+	0.834496i	0.0223910	+	0.0387823i	0.877004	−	0.480483i	$-0.159539\pi$
$463$	0.481797	+	0.834496i	0.0223910	+	0.0387823i	−0.854613	+	0.519266i	$0.826205\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.65095	−	6.32364i	0.168946	−	0.292623i	−0.769104	−	0.639124i	$-0.779297\pi$
$467$	3.65095	−	6.32364i	0.168946	−	0.292623i	0.938050	+	0.346501i	$0.112630\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.65450	−	9.79388i	0.259994	−	0.450323i
$474$		0			0
$475$	2.25841	−	3.91169i	0.103623	−	0.179480i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.5097			−0.617276			−0.308638	−	0.951180i	$-0.599873\pi$
$479$	−13.5097			−0.617276			−0.308638	+	0.951180i	$0.599873\pi$
$480$		0			0
$481$	−12.5161			−0.570685
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.64576	−	2.85054i	0.0747302	−	0.129436i
$486$		0			0
$487$	−8.25241	−	14.2936i	−0.373953	−	0.647705i	0.616217	−	0.787576i	$-0.288664\pi$
$487$	−8.25241	−	14.2936i	−0.373953	−	0.647705i	−0.990170	+	0.139871i	$0.955331\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.04610	+	15.6683i	0.408245	+	0.707101i	0.994693	−	0.102886i	$-0.0328076\pi$
$491$	9.04610	+	15.6683i	0.408245	+	0.707101i	−0.586448	+	0.809987i	$0.699474\pi$
$492$		0			0
$493$	−14.9678			−0.674114
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	30.9204			1.38419			0.692093	−	0.721808i	$-0.256689\pi$
$499$	30.9204			1.38419			0.692093	+	0.721808i	$0.256689\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.3747			1.30976			0.654878	−	0.755735i	$-0.272720\pi$
$503$	29.3747			1.30976			0.654878	+	0.755735i	$0.272720\pi$
$504$		0			0
$505$	−6.65427			−0.296111
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.6041			−0.514344			−0.257172	−	0.966366i	$-0.582791\pi$
$509$	−11.6041			−0.514344			−0.257172	+	0.966366i	$0.582791\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.5540			−0.465067
$516$		0			0
$517$	−3.48367	−	6.03390i	−0.153212	−	0.265371i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.51960	−	11.2923i	−0.285629	−	0.494724i	0.687133	−	0.726532i	$-0.258869\pi$
$521$	−6.51960	−	11.2923i	−0.285629	−	0.494724i	−0.972762	+	0.231808i	$0.925536\pi$
$522$		0			0
$523$	20.8474	−	36.1088i	0.911595	−	1.57893i	0.0997839	−	0.995009i	$-0.468185\pi$
$523$	20.8474	−	36.1088i	0.911595	−	1.57893i	0.811811	−	0.583920i	$-0.198482\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.9874			0.783545
$528$		0			0
$529$	17.1246			0.744547
$530$		0			0
$531$		0			0
$532$		0			0
$533$	22.3446	−	38.7021i	0.967854	−	1.67637i
$534$		0			0
$535$	−4.14980	+	7.18766i	−0.179411	+	0.310750i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.14656	+	8.91410i	−0.221268	+	0.383247i	−0.955193	−	0.295983i	$-0.904353\pi$
$541$	−5.14656	+	8.91410i	−0.221268	+	0.383247i	0.733925	+	0.679230i	$0.237686\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.77192	+	15.1934i	0.375748	+	0.650814i
$546$		0			0
$547$	−4.05659	+	7.02622i	−0.173447	+	0.300420i	−0.939623	−	0.342212i	$-0.888824\pi$
$547$	−4.05659	+	7.02622i	−0.173447	+	0.300420i	0.766176	+	0.642631i	$0.222157\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.12880	+	3.68719i	0.0906899	+	0.157079i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.8085	+	18.7208i	0.457969	+	0.793225i	0.998854	−	0.0478714i	$-0.0152438\pi$
$557$	10.8085	+	18.7208i	0.457969	+	0.793225i	−0.540885	+	0.841097i	$0.681910\pi$
$558$		0			0
$559$	−23.2581			−0.983715
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.55754	+	9.62593i	−0.234222	+	0.405685i	−0.959046	−	0.283249i	$-0.908588\pi$
$563$	−5.55754	+	9.62593i	−0.234222	+	0.405685i	0.724824	+	0.688934i	$0.241921\pi$
$564$		0			0
$565$	3.75192	+	6.49852i	0.157845	+	0.273395i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.85754	+	8.41350i	0.203639	+	0.352712i	0.949698	−	0.313167i	$-0.101390\pi$
$569$	4.85754	+	8.41350i	0.203639	+	0.352712i	−0.746059	+	0.665879i	$0.768057\pi$
$570$		0			0
$571$	−6.10695	+	10.5775i	−0.255568	+	0.442656i	−0.965050	−	0.262067i	$-0.915596\pi$
$571$	−6.10695	+	10.5775i	−0.255568	+	0.442656i	0.709482	+	0.704724i	$0.248929\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−26.0933			−1.08817
$576$		0			0
$577$	2.25670	+	3.90872i	0.0939477	+	0.162722i	0.909169	−	0.416427i	$-0.136718\pi$
$577$	2.25670	+	3.90872i	0.0939477	+	0.162722i	−0.815221	+	0.579150i	$0.803385\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.68080	+	2.91123i	0.0696116	+	0.120571i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.0425	+	26.0545i	−0.620872	+	1.07538i	0.368451	+	0.929647i	$0.379888\pi$
$587$	−15.0425	+	26.0545i	−0.620872	+	1.07538i	−0.989324	+	0.145735i	$0.953445\pi$
$588$		0			0
$589$	−2.55828	−	4.43106i	−0.105412	−	0.182579i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.46894	−	6.00838i	0.142452	−	0.246735i	−0.785967	−	0.618268i	$-0.787835\pi$
$593$	3.46894	−	6.00838i	0.142452	−	0.246735i	0.928420	+	0.371533i	$0.121168\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.77649	+	6.54108i	−0.154303	+	0.267261i	−0.932805	−	0.360381i	$-0.882647\pi$
$599$	−3.77649	+	6.54108i	−0.154303	+	0.267261i	0.778502	+	0.627642i	$0.215980\pi$
$600$		0			0
$601$	−2.43406	+	4.21591i	−0.0992873	+	0.171971i	−0.911390	−	0.411544i	$-0.864990\pi$
$601$	−2.43406	+	4.21591i	−0.0992873	+	0.171971i	0.812103	+	0.583515i	$0.198323\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.80240			0.154590
$606$		0			0
$607$	−21.6943			−0.880546			−0.440273	−	0.897864i	$-0.645118\pi$
$607$	−21.6943			−0.880546			−0.440273	+	0.897864i	$0.645118\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.16454	+	12.4093i	−0.289846	+	0.502028i
$612$		0			0
$613$	6.17923	+	10.7027i	0.249577	+	0.432280i	0.963408	−	0.268038i	$-0.0863752\pi$
$613$	6.17923	+	10.7027i	0.249577	+	0.432280i	−0.713832	+	0.700317i	$0.753042\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.9992	+	22.5152i	0.523327	+	0.906428i	0.999631	+	0.0271481i	$0.00864258\pi$
$617$	12.9992	+	22.5152i	0.523327	+	0.906428i	−0.476305	+	0.879280i	$0.658024\pi$
$618$		0			0
$619$	1.17410			0.0471909			0.0235954	−	0.999722i	$-0.492489\pi$
$619$	1.17410			0.0471909			0.0235954	+	0.999722i	$0.492489\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	12.5652			0.502608
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.89986			0.354861
$630$		0			0
$631$	42.2661			1.68259			0.841294	−	0.540579i	$-0.181795\pi$
$631$	42.2661			1.68259			0.841294	+	0.540579i	$0.181795\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.50945			−0.139268
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−35.2805			−1.39350			−0.696748	−	0.717316i	$-0.745370\pi$
$641$	−35.2805			−1.39350			−0.696748	+	0.717316i	$0.745370\pi$
$642$		0			0
$643$	7.98321	+	13.8273i	0.314827	+	0.545296i	0.979401	−	0.201926i	$-0.0647202\pi$
$643$	7.98321	+	13.8273i	0.314827	+	0.545296i	−0.664574	+	0.747223i	$0.731387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.0390	−	17.3881i	−0.394674	−	0.683596i	0.598385	−	0.801208i	$-0.295809\pi$
$647$	−10.0390	−	17.3881i	−0.394674	−	0.683596i	−0.993060	+	0.117613i	$0.962476\pi$
$648$		0			0
$649$	−7.96300	+	13.7923i	−0.312575	+	0.541396i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.4700			−0.879318			−0.439659	−	0.898165i	$-0.644901\pi$
$653$	−22.4700			−0.879318			−0.439659	+	0.898165i	$0.644901\pi$
$654$		0			0
$655$	20.3029			0.793300
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.25173	−	14.2924i	0.321442	−	0.556754i	−0.659344	−	0.751842i	$-0.729166\pi$
$659$	8.25173	−	14.2924i	0.321442	−	0.556754i	0.980786	+	0.195088i	$0.0624992\pi$
$660$		0			0
$661$	6.95039	−	12.0384i	0.270339	−	0.468241i	−0.698610	−	0.715503i	$-0.746198\pi$
$661$	6.95039	−	12.0384i	0.270339	−	0.468241i	0.968949	+	0.247262i	$0.0795309\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.2979	−	21.3006i	0.476176	−	0.824761i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.7125	−	30.6790i	−0.683785	−	1.18435i
$672$		0			0
$673$	1.03684	−	1.79586i	0.0399672	−	0.0692252i	−0.845350	−	0.534213i	$-0.820608\pi$
$673$	1.03684	−	1.79586i	0.0399672	−	0.0692252i	0.885317	+	0.464988i	$0.153941\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.48092	−	2.56504i	−0.0569166	−	0.0985824i	0.836163	−	0.548481i	$-0.184794\pi$
$677$	−1.48092	−	2.56504i	−0.0569166	−	0.0985824i	−0.893080	+	0.449898i	$0.851460\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.2670	+	43.7637i	0.966814	+	1.67457i	0.704659	+	0.709546i	$0.251100\pi$
$683$	25.2670	+	43.7637i	0.966814	+	1.67457i	0.262156	+	0.965026i	$0.415567\pi$
$684$		0			0
$685$	13.2298			0.505483
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.45674	−	5.98725i	0.131691	−	0.228096i
$690$		0			0
$691$	−14.0134	−	24.2720i	−0.533096	−	0.923350i	−0.999253	−	0.0386479i	$-0.987695\pi$
$691$	−14.0134	−	24.2720i	−0.533096	−	0.923350i	0.466156	−	0.884702i	$-0.345638\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.24803	+	5.62574i	0.123205	+	0.213397i
$696$		0			0
$697$	−15.8887	+	27.5200i	−0.601826	+	1.04239i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.60621			−0.173974			−0.0869871	−	0.996209i	$-0.527724\pi$
$701$	−4.60621			−0.173974			−0.0869871	+	0.996209i	$0.527724\pi$
$702$		0			0
$703$	−1.26579	−	2.19241i	−0.0477401	−	0.0826883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.65298	−	11.5233i	−0.249858	−	0.432767i	0.713628	−	0.700525i	$-0.247051\pi$
$709$	−6.65298	−	11.5233i	−0.249858	−	0.432767i	−0.963486	+	0.267758i	$0.913717\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−14.7789	+	25.5979i	−0.553475	+	0.958647i
$714$		0			0
$715$	6.70513	+	11.6136i	0.250758	+	0.434325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.6002	−	39.1447i	0.842845	−	1.45985i	−0.0446355	−	0.999003i	$-0.514213\pi$
$719$	22.6002	−	39.1447i	0.842845	−	1.45985i	0.887480	−	0.460846i	$-0.152454\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.99740	+	13.8519i	−0.297016	+	0.514447i
$726$		0			0
$727$	−12.4702	+	21.5989i	−0.462492	+	0.801060i	−0.999084	−	0.0427815i	$-0.986378\pi$
$727$	−12.4702	+	21.5989i	−0.462492	+	0.801060i	0.536592	+	0.843842i	$0.319711\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.5382			0.611688
$732$		0			0
$733$	11.7526			0.434092			0.217046	−	0.976161i	$-0.430358\pi$
$733$	11.7526			0.434092			0.217046	+	0.976161i	$0.430358\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.60483	+	16.6361i	−0.353798	+	0.612797i
$738$		0			0
$739$	−9.91252	−	17.1690i	−0.364638	−	0.631571i	0.624080	−	0.781360i	$-0.285474\pi$
$739$	−9.91252	−	17.1690i	−0.364638	−	0.631571i	−0.988718	+	0.149789i	$0.952141\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00263	−	3.46866i	−0.0734695	−	0.127253i	0.826950	−	0.562275i	$-0.190074\pi$
$743$	−2.00263	−	3.46866i	−0.0734695	−	0.127253i	−0.900420	+	0.435022i	$0.856740\pi$
$744$		0			0
$745$	10.7294			0.393096
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	22.8743			0.834694			0.417347	−	0.908747i	$-0.362960\pi$
$751$	22.8743			0.834694			0.417347	+	0.908747i	$0.362960\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.1105			0.367958
$756$		0			0
$757$	−26.4149			−0.960064			−0.480032	−	0.877251i	$-0.659375\pi$
$757$	−26.4149			−0.960064			−0.480032	+	0.877251i	$0.659375\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.4735			−0.633412			−0.316706	−	0.948524i	$-0.602577\pi$
$761$	−17.4735			−0.633412			−0.316706	+	0.948524i	$0.602577\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.7535			1.18266
$768$		0			0
$769$	16.0849	+	27.8598i	0.580035	+	1.00465i	0.995475	+	0.0950290i	$0.0302944\pi$
$769$	16.0849	+	27.8598i	0.580035	+	1.00465i	−0.415440	+	0.909621i	$0.636372\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.12986	+	1.95697i	0.0406382	+	0.0703875i	0.885629	−	0.464393i	$-0.153728\pi$
$773$	1.12986	+	1.95697i	0.0406382	+	0.0703875i	−0.844991	+	0.534781i	$0.820394\pi$
$774$		0			0
$775$	9.61085	−	16.6465i	0.345232	−	0.597959i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.03910			0.323859
$780$		0			0
$781$	38.9749			1.39463
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.92524	−	13.7269i	0.282864	−	0.489935i
$786$		0			0
$787$	−7.93595	+	13.7455i	−0.282886	+	0.489973i	−0.972094	−	0.234590i	$-0.924625\pi$
$787$	−7.93595	+	13.7455i	−0.282886	+	0.489973i	0.689208	+	0.724563i	$0.257959\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−36.4277	+	63.0946i	−1.29358	+	2.24055i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.3376	−	31.7616i	−0.649550	−	1.12505i	−0.983231	−	0.182367i	$-0.941624\pi$
$797$	−18.3376	−	31.7616i	−0.649550	−	1.12505i	0.333681	−	0.942686i	$-0.391709\pi$
$798$		0			0
$799$	5.09450	−	8.82394i	0.180231	−	0.312169i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.57793	+	13.1254i	0.267419	+	0.463184i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.1800	+	34.9529i	0.709492	+	1.22888i	0.965046	+	0.262082i	$0.0844091\pi$
$809$	20.1800	+	34.9529i	0.709492	+	1.22888i	−0.255553	+	0.966795i	$0.582258\pi$
$810$		0			0
$811$	6.44178			0.226202			0.113101	−	0.993584i	$-0.463922\pi$
$811$	6.44178			0.226202			0.113101	+	0.993584i	$0.463922\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.52941	+	14.7734i	−0.298772	+	0.517488i
$816$		0			0
$817$	−2.35216	−	4.07406i	−0.0822916	−	0.142533i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.47086	−	9.47581i	−0.190934	−	0.330708i	0.754626	−	0.656155i	$-0.227818\pi$
$821$	−5.47086	−	9.47581i	−0.190934	−	0.330708i	−0.945560	+	0.325447i	$0.894485\pi$
$822$		0			0
$823$	28.1738	−	48.7985i	0.982079	−	1.70101i	0.327818	−	0.944741i	$-0.393687\pi$
$823$	28.1738	−	48.7985i	0.982079	−	1.70101i	0.654261	−	0.756269i	$-0.272980\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.4286			1.37107			0.685533	−	0.728042i	$-0.259569\pi$
$827$	39.4286			1.37107			0.685533	+	0.728042i	$0.259569\pi$
$828$		0			0
$829$	10.0453	+	17.3989i	0.348887	+	0.604290i	0.986052	−	0.166437i	$-0.0532263\pi$
$829$	10.0453	+	17.3989i	0.348887	+	0.604290i	−0.637165	+	0.770728i	$0.719893\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	2.30294	+	3.98881i	0.0796966	+	0.138039i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.7318	−	41.1047i	0.819312	−	1.41909i	−0.0868775	−	0.996219i	$-0.527689\pi$
$839$	23.7318	−	41.1047i	0.819312	−	1.41909i	0.906190	−	0.422871i	$-0.138978\pi$
$840$		0			0
$841$	6.96158	+	12.0578i	0.240055	+	0.415787i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.68986	−	13.3192i	0.264539	−	0.458196i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.31235	+	12.6654i	−0.250664	+	0.434163i
$852$		0			0
$853$	−3.02004	+	5.23086i	−0.103404	+	0.179101i	−0.913085	−	0.407769i	$-0.866307\pi$
$853$	−3.02004	+	5.23086i	−0.103404	+	0.179101i	0.809681	+	0.586870i	$0.199640\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.9325			0.851678			0.425839	−	0.904799i	$-0.359979\pi$
$857$	24.9325			0.851678			0.425839	+	0.904799i	$0.359979\pi$
$858$		0			0
$859$	9.06753			0.309380			0.154690	−	0.987963i	$-0.450562\pi$
$859$	9.06753			0.309380			0.154690	+	0.987963i	$0.450562\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.9801	−	44.9989i	0.884374	−	1.53178i	0.0379438	−	0.999280i	$-0.487919\pi$
$863$	25.9801	−	44.9989i	0.884374	−	1.53178i	0.846430	−	0.532500i	$-0.178747\pi$
$864$		0			0
$865$	−5.62621	−	9.74488i	−0.191297	−	0.331336i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.5335	+	25.1728i	0.493015	+	0.853927i
$870$		0			0
$871$	39.5067			1.33863
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−47.6003			−1.60735			−0.803674	−	0.595070i	$-0.797124\pi$
$877$	−47.6003			−1.60735			−0.803674	+	0.595070i	$0.797124\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−44.3793			−1.49518			−0.747588	−	0.664163i	$-0.768788\pi$
$881$	−44.3793			−1.49518			−0.747588	+	0.664163i	$0.768788\pi$
$882$		0			0
$883$	−2.68771			−0.0904488			−0.0452244	−	0.998977i	$-0.514400\pi$
$883$	−2.68771			−0.0904488			−0.0452244	+	0.998977i	$0.514400\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.50614			0.184878			0.0924391	−	0.995718i	$-0.470534\pi$
$887$	5.50614			0.184878			0.0924391	+	0.995718i	$0.470534\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.89828			−0.0969871
$894$		0			0
$895$	11.1980	+	19.3955i	0.374308	+	0.648321i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.05926	+	15.6911i	0.302143	+	0.523328i
$900$		0			0
$901$	−2.45799	+	4.25737i	−0.0818876	+	0.141834i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.2313			−0.439823
$906$		0			0
$907$	53.8249			1.78723			0.893613	−	0.448837i	$-0.148162\pi$
$907$	53.8249			1.78723			0.893613	+	0.448837i	$0.148162\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.48779	+	11.2372i	−0.214950	+	0.372305i	−0.953257	−	0.302160i	$-0.902292\pi$
$911$	−6.48779	+	11.2372i	−0.214950	+	0.372305i	0.738307	+	0.674465i	$0.235626\pi$
$912$		0			0
$913$	3.28536	−	5.69042i	0.108730	−	0.188325i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.839322	−	1.45375i	0.0276867	−	0.0479547i	−0.851850	−	0.523786i	$-0.824519\pi$
$919$	0.839322	−	1.45375i	0.0276867	−	0.0479547i	0.879537	+	0.475831i	$0.157853\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40.0780	−	69.4171i	−1.31918	−	2.28489i
$924$		0			0
$925$	4.75527	−	8.23637i	0.156352	−	0.270810i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.66371	+	13.2739i	0.251438	+	0.435504i	0.963922	−	0.266185i	$-0.0857632\pi$
$929$	7.66371	+	13.2739i	0.251438	+	0.435504i	−0.712484	+	0.701688i	$0.752430\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.76784	−	8.25813i	−0.155925	−	0.270070i
$936$		0			0
$937$	38.8140			1.26800			0.633999	−	0.773334i	$-0.281412\pi$
$937$	38.8140			1.26800			0.633999	+	0.773334i	$0.281412\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.26496	+	12.5833i	−0.236831	+	0.410203i	−0.959803	−	0.280674i	$-0.909442\pi$
$941$	−7.26496	+	12.5833i	−0.236831	+	0.410203i	0.722972	+	0.690877i	$0.242775\pi$
$942$		0			0
$943$	−26.1090	−	45.2222i	−0.850227	−	1.47264i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.6532	+	34.0403i	0.638642	+	1.10616i	0.985731	+	0.168328i	$0.0538367\pi$
$947$	19.6532	+	34.0403i	0.638642	+	1.10616i	−0.347089	+	0.937832i	$0.612830\pi$
$948$		0			0
$949$	15.5848	−	26.9937i	0.505905	−	0.876252i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−35.9078			−1.16317			−0.581583	−	0.813487i	$-0.697567\pi$
$953$	−35.9078			−1.16317			−0.581583	+	0.813487i	$0.697567\pi$
$954$		0			0
$955$	−8.42222	−	14.5877i	−0.272537	−	0.472047i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.61307	+	7.99007i	0.148809	+	0.257744i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.86495	−	17.0866i	0.317564	−	0.550037i
$966$		0			0
$967$	9.80707	+	16.9863i	0.315374	+	0.546244i	0.979517	−	0.201362i	$-0.0645366\pi$
$967$	9.80707	+	16.9863i	0.315374	+	0.546244i	−0.664143	+	0.747606i	$0.731203\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.4938	−	44.1566i	0.818136	−	1.41705i	−0.0889186	−	0.996039i	$-0.528341\pi$
$971$	25.4938	−	44.1566i	0.818136	−	1.41705i	0.907054	−	0.421014i	$-0.138326\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.9989	+	36.3712i	−0.671815	+	1.16362i	0.305574	+	0.952168i	$0.401152\pi$
$977$	−20.9989	+	36.3712i	−0.671815	+	1.16362i	−0.977389	+	0.211450i	$0.932182\pi$
$978$		0			0
$979$	17.8112	−	30.8499i	0.569248	−	0.985966i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	58.8956			1.87848			0.939239	−	0.343263i	$-0.111532\pi$
$983$	58.8956			1.87848			0.939239	+	0.343263i	$0.111532\pi$
$984$		0			0
$985$	9.06272			0.288762
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.5882	+	23.5355i	−0.432080	+	0.748385i
$990$		0			0
$991$	16.1041	+	27.8931i	0.511563	+	0.886053i	0.999910	+	0.0134034i	$0.00426656\pi$
$991$	16.1041	+	27.8931i	0.511563	+	0.886053i	−0.488347	+	0.872649i	$0.662400\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.51718	+	2.62784i	0.0480980	+	0.0833082i
$996$		0			0
$997$	13.2030			0.418143			0.209071	−	0.977900i	$-0.432956\pi$
$997$	13.2030			0.418143			0.209071	+	0.977900i	$0.432956\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.j.3313.5		24
3.2	odd	2		1764.2.l.j.961.6		24
7.2	even	3		5292.2.j.i.3529.8		24
7.3	odd	6		5292.2.i.j.2125.5		24
7.4	even	3		5292.2.i.j.2125.8		24
7.5	odd	6		5292.2.j.i.3529.5		24
7.6	odd	2	inner	5292.2.l.j.3313.8		24
9.4	even	3		5292.2.i.j.1549.8		24
9.5	odd	6		1764.2.i.j.373.3		24
21.2	odd	6		1764.2.j.i.1177.11	yes	24
21.5	even	6		1764.2.j.i.1177.2	yes	24
21.11	odd	6		1764.2.i.j.1537.3		24
21.17	even	6		1764.2.i.j.1537.10		24
21.20	even	2		1764.2.l.j.961.7		24
63.4	even	3	inner	5292.2.l.j.361.5		24
63.5	even	6		1764.2.j.i.589.2	✓	24
63.13	odd	6		5292.2.i.j.1549.5		24
63.23	odd	6		1764.2.j.i.589.11	yes	24
63.31	odd	6	inner	5292.2.l.j.361.8		24
63.32	odd	6		1764.2.l.j.949.6		24
63.40	odd	6		5292.2.j.i.1765.5		24
63.41	even	6		1764.2.i.j.373.10		24
63.58	even	3		5292.2.j.i.1765.8		24
63.59	even	6		1764.2.l.j.949.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.3		24	9.5	odd	6
1764.2.i.j.373.10		24	63.41	even	6
1764.2.i.j.1537.3		24	21.11	odd	6
1764.2.i.j.1537.10		24	21.17	even	6
1764.2.j.i.589.2	✓	24	63.5	even	6
1764.2.j.i.589.11	yes	24	63.23	odd	6
1764.2.j.i.1177.2	yes	24	21.5	even	6
1764.2.j.i.1177.11	yes	24	21.2	odd	6
1764.2.l.j.949.6		24	63.32	odd	6
1764.2.l.j.949.7		24	63.59	even	6
1764.2.l.j.961.6		24	3.2	odd	2
1764.2.l.j.961.7		24	21.20	even	2
5292.2.i.j.1549.5		24	63.13	odd	6
5292.2.i.j.1549.8		24	9.4	even	3
5292.2.i.j.2125.5		24	7.3	odd	6
5292.2.i.j.2125.8		24	7.4	even	3
5292.2.j.i.1765.5		24	63.40	odd	6
5292.2.j.i.1765.8		24	63.58	even	3
5292.2.j.i.3529.5		24	7.5	odd	6
5292.2.j.i.3529.8		24	7.2	even	3
5292.2.l.j.361.5		24	63.4	even	3	inner
5292.2.l.j.361.8		24	63.31	odd	6	inner
5292.2.l.j.3313.5		24	1.1	even	1	trivial
5292.2.l.j.3313.8		24	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-0.938454 q^{5} -2.63595 q^{11} +(2.71055 + 4.69481i) q^{13} +(-1.92740 - 3.33835i) q^{17} +(-0.548251 + 0.949599i) q^{19} +6.33440 q^{23} -4.11930 q^{25} +(1.94145 - 3.36268i) q^{29} +(-2.33312 + 4.04109i) q^{31} +(-1.15439 + 1.99946i) q^{37} +(-4.12179 - 7.13915i) q^{41} +(-2.14515 + 3.71551i) q^{43} +(1.32160 + 2.28908i) q^{47} +(-0.637645 - 1.10443i) q^{53} +2.47372 q^{55} +(3.02092 - 5.23239i) q^{59} +(6.71961 + 11.6387i) q^{61} +(-2.54373 - 4.40587i) q^{65} +(3.64378 - 6.31122i) q^{67} -14.7859 q^{71} +(-2.87484 - 4.97937i) q^{73} +(-5.51357 - 9.54979i) q^{79} +(-1.24637 + 2.15877i) q^{83} +(1.80877 + 3.13289i) q^{85} +(-6.75703 + 11.7035i) q^{89} +(0.514508 - 0.891154i) q^{95} +(-1.75370 + 3.03749i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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