LMFDB - Embedded newform 5292.2.j.d.3529.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1765,5292)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5292.1765");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3529.3
Root			$0.500000 - 0.224437i$ of defining polynomial
Character	$\chi$	$=$	5292.3529
Dual form			5292.2.j.d.1765.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.849814 + 1.47192i) q^{5} +O(q^{10})$	$q+(0.849814 + 1.47192i) q^{5} +(1.23855 - 2.14523i) q^{11} +(0.388736 + 0.673310i) q^{13} +2.81089 q^{17} +4.98762 q^{19} +(0.356004 + 0.616617i) q^{23} +(1.05563 - 1.82841i) q^{25} +(2.25526 - 3.90623i) q^{29} +(2.54944 + 4.41576i) q^{31} -6.87636 q^{37} +(2.93818 + 5.08907i) q^{41} +(2.32691 - 4.03033i) q^{43} +(-6.49381 + 11.2476i) q^{47} -1.88874 q^{53} +4.21015 q^{55} +(-7.14400 - 12.3738i) q^{59} +(7.15452 - 12.3920i) q^{61} +(-0.660706 + 1.14438i) q^{65} +(-3.99381 - 6.91748i) q^{67} +10.2632 q^{71} -4.98762 q^{73} +(4.60507 - 7.97622i) q^{79} +(-4.40545 + 7.63046i) q^{83} +(2.38874 + 4.13741i) q^{85} +9.65383 q^{89} +(4.23855 + 7.34138i) q^{95} +(4.32072 - 7.48371i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5}+O(q^{10}) $ 6 * q - q^5	$ 6 q - q^{5} + 2 q^{11} + 3 q^{13} + 4 q^{17} - 6 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} - 3 q^{31} - 6 q^{37} - 3 q^{43} - 21 q^{47} - 12 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 34 q^{71} + 6 q^{73} + 9 q^{79} - 20 q^{83} + 15 q^{85} + 24 q^{89} + 20 q^{95} - 9 q^{97}+O(q^{100}) $ 6 * q - q^5 + 2 * q^11 + 3 * q^13 + 4 * q^17 - 6 * q^19 + 14 * q^23 + 6 * q^25 + q^29 - 3 * q^31 - 6 * q^37 - 3 * q^43 - 21 * q^47 - 12 * q^53 - 12 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 34 * q^71 + 6 * q^73 + 9 * q^79 - 20 * q^83 + 15 * q^85 + 24 * q^89 + 20 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.849814	+	1.47192i	0.380048	+	0.658263i	0.991069	−	0.133352i	$-0.0425740\pi$
$5$	0.849814	+	1.47192i	0.380048	+	0.658263i	−0.611020	+	0.791615i	$0.709241\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.23855	−	2.14523i	0.373437	−	0.646812i	−0.616655	−	0.787234i	$-0.711513\pi$
$11$	1.23855	−	2.14523i	0.373437	−	0.646812i	0.990092	+	0.140422i	$0.0448459\pi$
$12$		0			0
$13$	0.388736	+	0.673310i	0.107816	+	0.186743i	0.914885	−	0.403714i	$-0.132281\pi$
$13$	0.388736	+	0.673310i	0.107816	+	0.186743i	−0.807069	+	0.590457i	$0.798948\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.81089			0.681742			0.340871	−	0.940110i	$-0.389278\pi$
$17$	2.81089			0.681742			0.340871	+	0.940110i	$0.389278\pi$
$18$		0			0
$19$	4.98762			1.14424			0.572119	−	0.820170i	$-0.306121\pi$
$19$	4.98762			1.14424			0.572119	+	0.820170i	$0.306121\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.356004	+	0.616617i	0.0742320	+	0.128574i	0.900752	−	0.434334i	$-0.143016\pi$
$23$	0.356004	+	0.616617i	0.0742320	+	0.128574i	−0.826520	+	0.562907i	$0.809683\pi$
$24$		0			0
$25$	1.05563	−	1.82841i	0.211126	−	0.365682i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.25526	−	3.90623i	0.418791	−	0.725368i	−0.577027	−	0.816725i	$-0.695787\pi$
$29$	2.25526	−	3.90623i	0.418791	−	0.725368i	0.995818	+	0.0913573i	$0.0291205\pi$
$30$		0			0
$31$	2.54944	+	4.41576i	0.457893	+	0.793095i	0.998849	−	0.0479563i	$-0.0152708\pi$
$31$	2.54944	+	4.41576i	0.457893	+	0.793095i	−0.540956	+	0.841051i	$0.681937\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.87636			−1.13047			−0.565233	−	0.824931i	$-0.691214\pi$
$37$	−6.87636			−1.13047			−0.565233	+	0.824931i	$0.691214\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.93818	+	5.08907i	0.458866	+	0.794780i	0.998901	−	0.0468628i	$-0.0149223\pi$
$41$	2.93818	+	5.08907i	0.458866	+	0.794780i	−0.540035	+	0.841643i	$0.681589\pi$
$42$		0			0
$43$	2.32691	−	4.03033i	0.354851	−	0.614620i	−0.632241	−	0.774771i	$-0.717865\pi$
$43$	2.32691	−	4.03033i	0.354851	−	0.614620i	0.987092	+	0.160151i	$0.0511982\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.49381	+	11.2476i	−0.947220	+	1.64063i	−0.195975	+	0.980609i	$0.562787\pi$
$47$	−6.49381	+	11.2476i	−0.947220	+	1.64063i	−0.751245	+	0.660023i	$0.770546\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.88874			−0.259438			−0.129719	−	0.991551i	$-0.541407\pi$
$53$	−1.88874			−0.259438			−0.129719	+	0.991551i	$0.541407\pi$
$54$		0			0
$55$	4.21015			0.567696
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.14400	−	12.3738i	−0.930069	−	1.61093i	−0.783199	−	0.621771i	$-0.786413\pi$
$59$	−7.14400	−	12.3738i	−0.930069	−	1.61093i	−0.146870	−	0.989156i	$-0.546920\pi$
$60$		0			0
$61$	7.15452	−	12.3920i	0.916042	−	1.58663i	0.110673	−	0.993857i	$-0.464699\pi$
$61$	7.15452	−	12.3920i	0.916042	−	1.58663i	0.805369	−	0.592774i	$-0.201967\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.660706	+	1.14438i	−0.0819505	+	0.141942i
$66$		0			0
$67$	−3.99381	−	6.91748i	−0.487922	−	0.845105i	0.511982	−	0.858996i	$-0.328911\pi$
$67$	−3.99381	−	6.91748i	−0.487922	−	0.845105i	−0.999904	+	0.0138913i	$0.995578\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.2632			1.21802			0.609011	−	0.793162i	$-0.291567\pi$
$71$	10.2632			1.21802			0.609011	+	0.793162i	$0.291567\pi$
$72$		0			0
$73$	−4.98762			−0.583757			−0.291878	−	0.956455i	$-0.594280\pi$
$73$	−4.98762			−0.583757			−0.291878	+	0.956455i	$0.594280\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.60507	−	7.97622i	0.518111	−	0.897395i	−0.481667	−	0.876354i	$-0.659969\pi$
$79$	4.60507	−	7.97622i	0.518111	−	0.897395i	0.999779	−	0.0210410i	$-0.00669805\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.40545	+	7.63046i	−0.483561	+	0.837551i	−0.999822	−	0.0188798i	$-0.993990\pi$
$83$	−4.40545	+	7.63046i	−0.483561	+	0.837551i	0.516261	+	0.856431i	$0.327323\pi$
$84$		0			0
$85$	2.38874	+	4.13741i	0.259095	+	0.448765i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.65383			1.02330			0.511652	−	0.859193i	$-0.329034\pi$
$89$	9.65383			1.02330			0.511652	+	0.859193i	$0.329034\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.23855	+	7.34138i	0.434866	+	0.753210i
$96$		0			0
$97$	4.32072	−	7.48371i	0.438703	−	0.759856i	−0.558887	−	0.829244i	$-0.688771\pi$
$97$	4.32072	−	7.48371i	0.438703	−	0.759856i	0.997590	+	0.0693880i	$0.0221047\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.20582	+	2.08854i	−0.119983	+	0.207817i	−0.919761	−	0.392479i	$-0.871617\pi$
$101$	−1.20582	+	2.08854i	−0.119983	+	0.207817i	0.799777	+	0.600297i	$0.204951\pi$
$102$		0			0
$103$	−2.16690	−	3.75317i	−0.213511	−	0.369811i	0.739300	−	0.673376i	$-0.235156\pi$
$103$	−2.16690	−	3.75317i	−0.213511	−	0.369811i	−0.952811	+	0.303565i	$0.901823\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.1978			−1.85592			−0.927959	−	0.372682i	$-0.878438\pi$
$107$	−19.1978			−1.85592			−0.927959	+	0.372682i	$0.878438\pi$
$108$		0			0
$109$	18.9629			1.81631			0.908156	−	0.418631i	$-0.137490\pi$
$109$	18.9629			1.81631			0.908156	+	0.418631i	$0.137490\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.46472	+	11.1972i	0.608150	+	1.05335i	0.991545	+	0.129762i	$0.0414213\pi$
$113$	6.46472	+	11.1972i	0.608150	+	1.05335i	−0.383395	+	0.923584i	$0.625245\pi$
$114$		0			0
$115$	−0.605074	+	1.04802i	−0.0564235	+	0.0977283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.43199	+	4.21233i	0.221090	+	0.382939i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0865			1.08105
$126$		0			0
$127$	17.6291			1.56433			0.782163	−	0.623073i	$-0.214116\pi$
$127$	17.6291			1.56433			0.782163	+	0.623073i	$0.214116\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.84362	+	4.92530i	0.248449	+	0.430326i	0.963096	−	0.269160i	$-0.0867460\pi$
$131$	2.84362	+	4.92530i	0.248449	+	0.430326i	−0.714647	+	0.699485i	$0.753413\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.72617	+	16.8462i	−0.830963	+	1.43927i	0.0663128	+	0.997799i	$0.478876\pi$
$137$	−9.72617	+	16.8462i	−0.830963	+	1.43927i	−0.897276	+	0.441471i	$0.854457\pi$
$138$		0			0
$139$	−1.49381	−	2.58736i	−0.126703	−	0.219457i	0.795694	−	0.605699i	$-0.207106\pi$
$139$	−1.49381	−	2.58736i	−0.126703	−	0.219457i	−0.922397	+	0.386242i	$0.873773\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.92587			0.161050
$144$		0			0
$145$	7.66621			0.636644
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.04944	+	7.01384i	0.331743	+	0.574596i	0.982854	−	0.184387i	$-0.0590299\pi$
$149$	4.04944	+	7.01384i	0.331743	+	0.574596i	−0.651111	+	0.758983i	$0.725697\pi$
$150$		0			0
$151$	4.43199	−	7.67643i	0.360670	−	0.624699i	−0.627401	−	0.778696i	$-0.715881\pi$
$151$	4.43199	−	7.67643i	0.360670	−	0.624699i	0.988071	+	0.153997i	$0.0492147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.33310	+	7.50516i	−0.348043	+	0.602829i
$156$		0			0
$157$	−4.38255	−	7.59079i	−0.349765	−	0.605811i	0.636442	−	0.771324i	$-0.280405\pi$
$157$	−4.38255	−	7.59079i	−0.349765	−	0.605811i	−0.986208	+	0.165513i	$0.947072\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.98762			−0.155682			−0.0778412	−	0.996966i	$-0.524803\pi$
$163$	−1.98762			−0.155682			−0.0778412	+	0.996966i	$0.524803\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.31089	−	2.27053i	−0.101440	−	0.175699i	0.810838	−	0.585270i	$-0.199012\pi$
$167$	−1.31089	−	2.27053i	−0.101440	−	0.175699i	−0.912278	+	0.409571i	$0.865678\pi$
$168$		0			0
$169$	6.19777	−	10.7349i	0.476751	−	0.825758i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.61491	−	4.52915i	0.198808	−	0.344345i	−0.749334	−	0.662192i	$-0.769626\pi$
$173$	2.61491	−	4.52915i	0.198808	−	0.344345i	0.948142	+	0.317847i	$0.102960\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.76509			0.356160			0.178080	−	0.984016i	$-0.443011\pi$
$179$	4.76509			0.356160			0.178080	+	0.984016i	$0.443011\pi$
$180$		0			0
$181$	10.4313			0.775352			0.387676	−	0.921796i	$-0.373278\pi$
$181$	10.4313			0.775352			0.387676	+	0.921796i	$0.373278\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.84362	−	10.1215i	−0.429632	−	0.744144i
$186$		0			0
$187$	3.48143	−	6.03001i	0.254587	−	0.440958i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.66071	−	11.5367i	0.481952	−	0.834765i	−0.517834	−	0.855481i	$-0.673261\pi$
$191$	6.66071	−	11.5367i	0.481952	−	0.834765i	0.999785	+	0.0207164i	$0.00659470\pi$
$192$		0			0
$193$	7.32072	+	12.6799i	0.526957	+	0.912717i	0.999507	+	0.0314125i	$0.0100005\pi$
$193$	7.32072	+	12.6799i	0.526957	+	0.912717i	−0.472549	+	0.881304i	$0.656666\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.4858			1.31706			0.658528	−	0.752556i	$-0.271179\pi$
$197$	18.4858			1.31706			0.658528	+	0.752556i	$0.271179\pi$
$198$		0			0
$199$	−23.6167			−1.67414			−0.837071	−	0.547094i	$-0.815734\pi$
$199$	−23.6167			−1.67414			−0.837071	+	0.547094i	$0.815734\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.99381	+	8.64953i	−0.348783	+	0.604110i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.17742	−	10.6996i	0.427301	−	0.740107i
$210$		0			0
$211$	7.27747	+	12.6050i	0.501002	+	0.867761i	0.999999	+	0.00115718i	$0.000368342\pi$
$211$	7.27747	+	12.6050i	0.501002	+	0.867761i	−0.498998	+	0.866603i	$0.666298\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.90978			0.539442
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.09269	+	1.89260i	0.0735026	+	0.127310i
$222$		0			0
$223$	4.72253	−	8.17966i	0.316244	−	0.547750i	−0.663457	−	0.748214i	$-0.730912\pi$
$223$	4.72253	−	8.17966i	0.316244	−	0.547750i	0.979701	+	0.200464i	$0.0642449\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.55563	+	16.5508i	−0.634230	+	1.09852i	0.352448	+	0.935831i	$0.385349\pi$
$227$	−9.55563	+	16.5508i	−0.634230	+	1.09852i	−0.986678	+	0.162687i	$0.947984\pi$
$228$		0			0
$229$	5.72253	+	9.91171i	0.378155	+	0.654984i	0.990794	−	0.135379i	$-0.0432252\pi$
$229$	5.72253	+	9.91171i	0.378155	+	0.654984i	−0.612639	+	0.790363i	$0.709892\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.19049			0.0779913			0.0389956	−	0.999239i	$-0.487584\pi$
$233$	1.19049			0.0779913			0.0389956	+	0.999239i	$0.487584\pi$
$234$		0			0
$235$	−22.0741			−1.43996
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.1414	+	21.0296i	0.785365	+	1.36029i	0.928781	+	0.370630i	$0.120858\pi$
$239$	12.1414	+	21.0296i	0.785365	+	1.36029i	−0.143416	+	0.989663i	$0.545809\pi$
$240$		0			0
$241$	−10.7095	+	18.5493i	−0.689857	+	1.19487i	0.282027	+	0.959406i	$0.408993\pi$
$241$	−10.7095	+	18.5493i	−0.689857	+	1.19487i	−0.971884	+	0.235461i	$0.924340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.93887	+	3.35822i	0.123367	+	0.213678i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.67996			−0.169158			−0.0845789	−	0.996417i	$-0.526955\pi$
$251$	−2.67996			−0.169158			−0.0845789	+	0.996417i	$0.526955\pi$
$252$		0			0
$253$	1.76371			0.110884
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.54256	+	9.60000i	0.345736	+	0.598832i	0.985487	−	0.169750i	$-0.0542961\pi$
$257$	5.54256	+	9.60000i	0.345736	+	0.598832i	−0.639752	+	0.768582i	$0.720963\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.70396	−	11.6116i	0.413384	−	0.716002i	−0.581873	−	0.813279i	$-0.697680\pi$
$263$	6.70396	−	11.6116i	0.413384	−	0.716002i	0.995257	+	0.0972776i	$0.0310135\pi$
$264$		0			0
$265$	−1.60507	−	2.78007i	−0.0985989	−	0.170778i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.09022			0.249385			0.124693	−	0.992195i	$-0.460206\pi$
$269$	4.09022			0.249385			0.124693	+	0.992195i	$0.460206\pi$
$270$		0			0
$271$	−6.12364			−0.371985			−0.185992	−	0.982551i	$-0.559550\pi$
$271$	−6.12364			−0.371985			−0.185992	+	0.982551i	$0.559550\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.61491	−	4.52915i	−0.157685	−	0.273118i
$276$		0			0
$277$	−7.88255	+	13.6530i	−0.473616	+	0.820327i	−0.999544	−	0.0302019i	$-0.990385\pi$
$277$	−7.88255	+	13.6530i	−0.473616	+	0.820327i	0.525928	+	0.850529i	$0.323718\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.5946	+	18.3503i	−0.632018	+	1.09469i	0.355120	+	0.934821i	$0.384440\pi$
$281$	−10.5946	+	18.3503i	−0.632018	+	1.09469i	−0.987139	+	0.159867i	$0.948893\pi$
$282$		0			0
$283$	3.43818	+	5.95510i	0.204378	+	0.353994i	0.949935	−	0.312449i	$-0.101149\pi$
$283$	3.43818	+	5.95510i	0.204378	+	0.353994i	−0.745556	+	0.666443i	$0.767816\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.09888			−0.535228
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.7534	+	23.8216i	0.803482	+	1.39167i	0.917311	+	0.398172i	$0.130355\pi$
$293$	13.7534	+	23.8216i	0.803482	+	1.39167i	−0.113829	+	0.993500i	$0.536311\pi$
$294$		0			0
$295$	12.1421	−	21.0308i	0.706943	−	1.22446i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.276783	+	0.479402i	−0.0160068	+	0.0277245i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.3200			1.39256
$306$		0			0
$307$	21.5178			1.22809			0.614043	−	0.789273i	$-0.289542\pi$
$307$	21.5178			1.22809			0.614043	+	0.789273i	$0.289542\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.19275	−	15.9223i	−0.521273	−	0.902871i	−0.999694	−	0.0247407i	$-0.992124\pi$
$311$	−9.19275	−	15.9223i	−0.521273	−	0.902871i	0.478421	−	0.878131i	$-0.341209\pi$
$312$		0			0
$313$	−0.000688709		0.00119288i	−3.89281e−5		6.74255e-5i	−0.866045	−	0.499966i	$-0.833346\pi$
$313$	−0.000688709		0.00119288i	−3.89281e−5		6.74255e-5i	0.866006	+	0.500034i	$0.166679\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.04944	−	12.2100i	0.395936	−	0.685781i	−0.597284	−	0.802030i	$-0.703753\pi$
$317$	7.04944	−	12.2100i	0.395936	−	0.685781i	0.993220	+	0.116248i	$0.0370868\pi$
$318$		0			0
$319$	−5.58650	−	9.67611i	−0.312784	−	0.541758i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.0197			0.780075
$324$		0			0
$325$	1.64145			0.0910512
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.98143	+	12.0922i	−0.383734	+	0.664647i	−0.991593	−	0.129398i	$-0.958695\pi$
$331$	−6.98143	+	12.0922i	−0.383734	+	0.664647i	0.607859	+	0.794045i	$0.292029\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.78799	−	11.7571i	0.370868	−	0.642362i
$336$		0			0
$337$	−12.0982	−	20.9547i	−0.659031	−	1.14147i	−0.980867	−	0.194679i	$-0.937633\pi$
$337$	−12.0982	−	20.9547i	−0.659031	−	1.14147i	0.321836	−	0.946795i	$-0.395700\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.6304			0.683977
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.3578	−	28.3325i	−0.878132	−	1.52097i	−0.853389	−	0.521275i	$-0.825456\pi$
$347$	−16.3578	−	28.3325i	−0.878132	−	1.52097i	−0.0247435	−	0.999694i	$-0.507877\pi$
$348$		0			0
$349$	−11.8887	+	20.5919i	−0.636389	+	1.10226i	0.349830	+	0.936813i	$0.386240\pi$
$349$	−11.8887	+	20.5919i	−0.636389	+	1.10226i	−0.986219	+	0.165445i	$0.947094\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.0309	+	17.3740i	−0.533889	+	0.924724i	0.465327	+	0.885139i	$0.345937\pi$
$353$	−10.0309	+	17.3740i	−0.533889	+	0.924724i	−0.999216	+	0.0395847i	$0.987396\pi$
$354$		0			0
$355$	8.72184	+	15.1067i	0.462907	+	0.801779i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.30175			−0.438150			−0.219075	−	0.975708i	$-0.570304\pi$
$359$	−8.30175			−0.438150			−0.219075	+	0.975708i	$0.570304\pi$
$360$		0			0
$361$	5.87636			0.309282
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.23855	−	7.34138i	−0.221856	−	0.384266i
$366$		0			0
$367$	5.77197	−	9.99735i	0.301294	−	0.521857i	−0.675135	−	0.737694i	$-0.735915\pi$
$367$	5.77197	−	9.99735i	0.301294	−	0.521857i	0.976429	+	0.215837i	$0.0692480\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.42580	−	2.46956i	−0.0738250	−	0.127869i	0.826750	−	0.562570i	$-0.190187\pi$
$373$	−1.42580	−	2.46956i	−0.0738250	−	0.127869i	−0.900575	+	0.434701i	$0.856854\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.50680			0.180609
$378$		0			0
$379$	35.9519			1.84672			0.923361	−	0.383932i	$-0.125430\pi$
$379$	35.9519			1.84672			0.923361	+	0.383932i	$0.125430\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.915278	+	1.58531i	0.0467685	+	0.0810054i	0.888462	−	0.458950i	$-0.151774\pi$
$383$	0.915278	+	1.58531i	0.0467685	+	0.0810054i	−0.841694	+	0.539956i	$0.818441\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.69530	+	9.86454i	−0.288763	+	0.500152i	−0.973515	−	0.228624i	$-0.926577\pi$
$389$	−5.69530	+	9.86454i	−0.288763	+	0.500152i	0.684752	+	0.728776i	$0.259911\pi$
$390$		0			0
$391$	1.00069	+	1.73324i	0.0506070	+	0.0876539i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.6538			0.787630
$396$		0			0
$397$	10.4313			0.523532			0.261766	−	0.965131i	$-0.415695\pi$
$397$	10.4313			0.523532			0.261766	+	0.965131i	$0.415695\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.0371	−	29.5091i	−0.850790	−	1.47361i	−0.880496	−	0.474053i	$-0.842790\pi$
$401$	−17.0371	−	29.5091i	−0.850790	−	1.47361i	0.0297058	−	0.999559i	$-0.490543\pi$
$402$		0			0
$403$	−1.98212	+	3.43313i	−0.0987364	+	0.171016i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.51671	+	14.7514i	−0.422158	+	0.731199i
$408$		0			0
$409$	1.98762	+	3.44266i	0.0982815	+	0.170229i	0.910973	−	0.412465i	$-0.135332\pi$
$409$	1.98762	+	3.44266i	0.0982815	+	0.170229i	−0.812692	+	0.582694i	$0.801999\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−14.9752			−0.735106
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.72184	+	8.17847i	0.230677	+	0.399544i	0.958008	−	0.286743i	$-0.0925726\pi$
$419$	4.72184	+	8.17847i	0.230677	+	0.399544i	−0.727331	+	0.686287i	$0.759239\pi$
$420$		0			0
$421$	3.16002	−	5.47331i	0.154010	−	0.266753i	−0.778688	−	0.627411i	$-0.784115\pi$
$421$	3.16002	−	5.47331i	0.154010	−	0.266753i	0.932698	+	0.360658i	$0.117448\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.96727	−	5.13946i	0.143934	−	0.249300i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.7541			1.33687			0.668434	−	0.743772i	$-0.266965\pi$
$431$	27.7541			1.33687			0.668434	+	0.743772i	$0.266965\pi$
$432$		0			0
$433$	−11.2473			−0.540510			−0.270255	−	0.962789i	$-0.587108\pi$
$433$	−11.2473			−0.540510			−0.270255	+	0.962789i	$0.587108\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.77561	+	3.07545i	0.0849391	+	0.147119i
$438$		0			0
$439$	−7.54325	+	13.0653i	−0.360020	+	0.623573i	−0.987964	−	0.154686i	$-0.950563\pi$
$439$	−7.54325	+	13.0653i	−0.360020	+	0.623573i	0.627944	+	0.778259i	$0.283897\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.96658	−	6.87032i	0.188458	−	0.326419i	−0.756278	−	0.654250i	$-0.772984\pi$
$443$	3.96658	−	6.87032i	0.188458	−	0.326419i	0.944736	+	0.327831i	$0.106318\pi$
$444$		0			0
$445$	8.20396	+	14.2097i	0.388905	+	0.673603i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.5636			1.53677			0.768386	−	0.639987i	$-0.221060\pi$
$449$	32.5636			1.53677			0.768386	+	0.639987i	$0.221060\pi$
$450$		0			0
$451$	14.5563			0.685430
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.70396	+	9.87955i	−0.266820	+	0.462146i	−0.968039	−	0.250800i	$-0.919306\pi$
$457$	−5.70396	+	9.87955i	−0.266820	+	0.462146i	0.701219	+	0.712946i	$0.252640\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.45853	+	4.25830i	−0.114505	+	0.198329i	−0.917582	−	0.397547i	$-0.869862\pi$
$461$	−2.45853	+	4.25830i	−0.114505	+	0.198329i	0.803077	+	0.595876i	$0.203195\pi$
$462$		0			0
$463$	−7.59957	−	13.1628i	−0.353182	−	0.611729i	0.633623	−	0.773642i	$-0.281567\pi$
$463$	−7.59957	−	13.1628i	−0.353182	−	0.611729i	−0.986805	+	0.161913i	$0.948234\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.7810			1.10046			0.550228	−	0.835015i	$-0.314541\pi$
$467$	23.7810			1.10046			0.550228	+	0.835015i	$0.314541\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.76400	−	9.98354i	−0.265029	−	0.459044i
$474$		0			0
$475$	5.26509	−	9.11941i	0.241579	−	0.418427i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.02909	+	5.24654i	−0.138403	+	0.239720i	−0.926892	−	0.375328i	$-0.877530\pi$
$479$	−3.02909	+	5.24654i	−0.138403	+	0.239720i	0.788489	+	0.615048i	$0.210863\pi$
$480$		0			0
$481$	−2.67309	−	4.62992i	−0.121882	−	0.211106i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.6872			0.666914
$486$		0			0
$487$	1.13602			0.0514781			0.0257391	−	0.999669i	$-0.491806\pi$
$487$	1.13602			0.0514781			0.0257391	+	0.999669i	$0.491806\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.4382	−	28.4718i	−0.741845	−	1.28491i	−0.951655	−	0.307170i	$-0.900618\pi$
$491$	−16.4382	−	28.4718i	−0.741845	−	1.28491i	0.209810	−	0.977742i	$-0.432715\pi$
$492$		0			0
$493$	6.33929	−	10.9800i	0.285507	−	0.494513i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.0989	−	22.6879i	−0.586387	−	1.01565i	−0.994701	−	0.102810i	$-0.967217\pi$
$499$	−13.0989	−	22.6879i	−0.586387	−	1.01565i	0.408314	−	0.912841i	$-0.366117\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.8516			−1.15267			−0.576333	−	0.817215i	$-0.695517\pi$
$503$	−25.8516			−1.15267			−0.576333	+	0.817215i	$0.695517\pi$
$504$		0			0
$505$	−4.09888			−0.182398
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.5858	−	30.4595i	−0.779478	−	1.35009i	−0.932243	−	0.361832i	$-0.882151\pi$
$509$	−17.5858	−	30.4595i	−0.779478	−	1.35009i	0.152766	−	0.988262i	$-0.451182\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.68292	−	6.37900i	0.162289	−	0.281092i
$516$		0			0
$517$	16.0858	+	27.8615i	0.707453	+	1.22535i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.8626			−0.782575			−0.391287	−	0.920269i	$-0.627970\pi$
$521$	−17.8626			−0.782575			−0.391287	+	0.920269i	$0.627970\pi$
$522$		0			0
$523$	22.8640			0.999772			0.499886	−	0.866091i	$-0.333375\pi$
$523$	22.8640			0.999772			0.499886	+	0.866091i	$0.333375\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.16621	+	12.4122i	0.312165	+	0.540685i
$528$		0			0
$529$	11.2465	−	19.4795i	0.488979	−	0.846937i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.28435	+	3.95661i	−0.0989462	+	0.171380i
$534$		0			0
$535$	−16.3145	−	28.2576i	−0.705339	−	1.22168i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	22.3077			0.959081			0.479541	−	0.877520i	$-0.340803\pi$
$541$	22.3077			0.959081			0.479541	+	0.877520i	$0.340803\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.1149	+	27.9118i	0.690287	+	1.19561i
$546$		0			0
$547$	−10.8083	+	18.7206i	−0.462131	+	0.800435i	−0.999067	−	0.0431882i	$-0.986249\pi$
$547$	−10.8083	+	18.7206i	−0.462131	+	0.800435i	0.536936	+	0.843623i	$0.319582\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.2484	−	19.4828i	0.479197	−	0.829994i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.17535			−0.134544			−0.0672720	−	0.997735i	$-0.521430\pi$
$557$	−3.17535			−0.134544			−0.0672720	+	0.997735i	$0.521430\pi$
$558$		0			0
$559$	3.61822			0.153034
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.8814	−	37.8997i	−0.922190	−	1.59728i	−0.796019	−	0.605271i	$-0.793065\pi$
$563$	−21.8814	−	37.8997i	−0.922190	−	1.59728i	−0.126171	−	0.992009i	$-0.540269\pi$
$564$		0			0
$565$	−10.9876	+	19.0311i	−0.462253	+	0.800645i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.9313	+	20.6656i	−0.500186	+	0.866348i	0.499814	+	0.866133i	$0.333402\pi$
$569$	−11.9313	+	20.6656i	−0.500186	+	0.866348i	−1.00000		0.000214897i	$0.999932\pi$
$570$		0			0
$571$	−5.11058	−	8.85178i	−0.213871	−	0.370435i	0.739052	−	0.673649i	$-0.235274\pi$
$571$	−5.11058	−	8.85178i	−0.213871	−	0.370435i	−0.952923	+	0.303213i	$0.901941\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.50324			0.0626893
$576$		0			0
$577$	36.0370			1.50024			0.750120	−	0.661302i	$-0.229996\pi$
$577$	36.0370			1.50024			0.750120	+	0.661302i	$0.229996\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.33929	+	4.05178i	−0.0968836	+	0.167807i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.5142	−	18.2111i	0.433966	−	0.751651i	−0.563245	−	0.826290i	$-0.690447\pi$
$587$	10.5142	−	18.2111i	0.433966	−	0.751651i	0.997211	+	0.0746391i	$0.0237805\pi$
$588$		0			0
$589$	12.7156	+	22.0242i	0.523939	+	0.907489i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−25.1606			−1.03322			−0.516612	−	0.856220i	$-0.672807\pi$
$593$	−25.1606			−1.03322			−0.516612	+	0.856220i	$0.672807\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.11126	−	1.92477i	−0.0454050	−	0.0786438i	0.842430	−	0.538806i	$-0.181125\pi$
$599$	−1.11126	−	1.92477i	−0.0454050	−	0.0786438i	−0.887835	+	0.460162i	$0.847791\pi$
$600$		0			0
$601$	14.0494	−	24.3343i	0.573089	−	0.992619i	−0.423158	−	0.906056i	$-0.639078\pi$
$601$	14.0494	−	24.3343i	0.573089	−	0.992619i	0.996246	−	0.0865627i	$-0.0275883\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.13348	+	7.15939i	−0.168050	+	0.291071i
$606$		0			0
$607$	−3.26509	−	5.65531i	−0.132526	−	0.229542i	0.792124	−	0.610361i	$-0.208975\pi$
$607$	−3.26509	−	5.65531i	−0.132526	−	0.229542i	−0.924650	+	0.380819i	$0.875642\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.0975			−0.408501
$612$		0			0
$613$	10.7280			0.433298			0.216649	−	0.976250i	$-0.430487\pi$
$613$	10.7280			0.433298			0.216649	+	0.976250i	$0.430487\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.5265	−	26.8928i	−0.625075	−	1.08266i	−0.988526	−	0.151049i	$-0.951735\pi$
$617$	−15.5265	−	26.8928i	−0.625075	−	1.08266i	0.363451	−	0.931613i	$-0.381598\pi$
$618$		0			0
$619$	−0.723217	+	1.25265i	−0.0290685	+	0.0503482i	−0.880194	−	0.474615i	$-0.842587\pi$
$619$	−0.723217	+	1.25265i	−0.0290685	+	0.0503482i	0.851125	+	0.524963i	$0.175921\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.99312	+	8.64834i	0.199725	+	0.345934i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.3287			−0.770686
$630$		0			0
$631$	0.0741250			0.00295087			0.00147544	−	0.999999i	$-0.499530\pi$
$631$	0.0741250			0.00295087			0.00147544	+	0.999999i	$0.499530\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9814	+	25.9486i	0.594520	+	1.02974i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.5204	+	40.7384i	−0.928998	+	1.60907i	−0.143996	+	0.989578i	$0.545995\pi$
$641$	−23.5204	+	40.7384i	−0.928998	+	1.60907i	−0.785002	+	0.619494i	$0.787338\pi$
$642$		0			0
$643$	−16.8647	−	29.2105i	−0.665077	−	1.15195i	−0.979264	−	0.202587i	$-0.935065\pi$
$643$	−16.8647	−	29.2105i	−0.665077	−	1.15195i	0.314187	−	0.949361i	$-0.398268\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	44.9629			1.76767			0.883836	−	0.467796i	$-0.154952\pi$
$647$	44.9629			1.76767			0.883836	+	0.467796i	$0.154952\pi$
$648$		0			0
$649$	−35.3928			−1.38929
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.8578	+	36.1267i	0.816228	+	1.41375i	0.908443	+	0.418010i	$0.137272\pi$
$653$	20.8578	+	36.1267i	0.816228	+	1.41375i	−0.0922143	+	0.995739i	$0.529394\pi$
$654$		0			0
$655$	−4.83310	+	8.37118i	−0.188845	+	0.327089i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.5259	+	18.2313i	−0.410029	+	0.710191i	−0.994892	−	0.100941i	$-0.967815\pi$
$659$	−10.5259	+	18.2313i	−0.410029	+	0.710191i	0.584863	+	0.811132i	$0.301148\pi$
$660$		0			0
$661$	11.2218	+	19.4368i	0.436479	+	0.756004i	0.997415	−	0.0718553i	$-0.0228920\pi$
$661$	11.2218	+	19.4368i	0.436479	+	0.756004i	−0.560936	+	0.827859i	$0.689559\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3.21153			0.124351
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.7225	−	30.6962i	−0.684168	−	1.18501i
$672$		0			0
$673$	5.83929	−	10.1140i	0.225088	−	0.389864i	−0.731258	−	0.682101i	$-0.761066\pi$
$673$	5.83929	−	10.1140i	0.225088	−	0.389864i	0.956346	+	0.292237i	$0.0943996\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.23422	+	9.06593i	−0.201167	+	0.348432i	−0.948905	−	0.315562i	$-0.897807\pi$
$677$	−5.23422	+	9.06593i	−0.201167	+	0.348432i	0.747737	+	0.663995i	$0.231140\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−32.8158			−1.25566			−0.627832	−	0.778349i	$-0.716057\pi$
$683$	−32.8158			−1.25566			−0.627832	+	0.778349i	$0.716057\pi$
$684$		0			0
$685$	−33.0617			−1.26322
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.734219	−	1.27171i	−0.0279715	−	0.0484481i
$690$		0			0
$691$	2.95056	−	5.11052i	0.112245	−	0.194413i	−0.804430	−	0.594047i	$-0.797529\pi$
$691$	2.95056	−	5.11052i	0.112245	−	0.194413i	0.916675	+	0.399634i	$0.130863\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.53892	−	4.39754i	0.0963068	−	0.166808i
$696$		0			0
$697$	8.25890	+	14.3048i	0.312828	+	0.541834i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.3782			0.467519			0.233759	−	0.972294i	$-0.424897\pi$
$701$	12.3782			0.467519			0.233759	+	0.972294i	$0.424897\pi$
$702$		0			0
$703$	−34.2967			−1.29352
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.64145	−	11.5033i	0.249425	−	0.432016i	−0.713942	−	0.700205i	$-0.753092\pi$
$709$	6.64145	−	11.5033i	0.249425	−	0.432016i	0.963366	+	0.268189i	$0.0864251\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.81522	+	3.14406i	−0.0679806	+	0.117746i
$714$		0			0
$715$	1.63664	+	2.83474i	0.0612067	+	0.106013i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.3694			−0.908825			−0.454413	−	0.890791i	$-0.650151\pi$
$719$	−24.3694			−0.908825			−0.454413	+	0.890791i	$0.650151\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.76145	−	8.24707i	−0.176836	−	0.306289i
$726$		0			0
$727$	−7.99450	+	13.8469i	−0.296500	+	0.513552i	−0.975333	−	0.220740i	$-0.929153\pi$
$727$	−7.99450	+	13.8469i	−0.296500	+	0.513552i	0.678833	+	0.734293i	$0.262486\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.54070	−	11.3288i	0.241917	−	0.419012i
$732$		0			0
$733$	−21.1414	−	36.6181i	−0.780877	−	1.35252i	−0.931431	−	0.363917i	$-0.881439\pi$
$733$	−21.1414	−	36.6181i	−0.780877	−	1.35252i	0.150554	−	0.988602i	$-0.451894\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.7861			−0.728832
$738$		0			0
$739$	−3.08650			−0.113539			−0.0567695	−	0.998387i	$-0.518080\pi$
$739$	−3.08650			−0.113539			−0.0567695	+	0.998387i	$0.518080\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.31522	+	5.74213i	0.121624	+	0.210658i	0.920408	−	0.390959i	$-0.127857\pi$
$743$	3.31522	+	5.74213i	0.121624	+	0.210658i	−0.798784	+	0.601617i	$0.794523\pi$
$744$		0			0
$745$	−6.88255	+	11.9209i	−0.252157	+	0.436749i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.3702	+	37.0142i	0.779808	+	1.35067i	0.932052	+	0.362325i	$0.118017\pi$
$751$	21.3702	+	37.0142i	0.779808	+	1.35067i	−0.152243	+	0.988343i	$0.548650\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.0655			0.548288
$756$		0			0
$757$	−31.0232			−1.12756			−0.563779	−	0.825926i	$-0.690653\pi$
$757$	−31.0232			−1.12756			−0.563779	+	0.825926i	$0.690653\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.8182	−	20.4697i	−0.428409	−	0.742025i	0.568323	−	0.822805i	$-0.307592\pi$
$761$	−11.8182	−	20.4697i	−0.428409	−	0.742025i	−0.996732	+	0.0807799i	$0.974259\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.55425	−	9.62025i	0.200553	−	0.347367i
$768$		0			0
$769$	−1.73422	−	3.00376i	−0.0625375	−	0.108318i	0.833061	−	0.553180i	$-0.186586\pi$
$769$	−1.73422	−	3.00376i	−0.0625375	−	0.108318i	−0.895599	+	0.444862i	$0.853253\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	34.5970			1.24437			0.622184	−	0.782871i	$-0.286245\pi$
$773$	34.5970			1.24437			0.622184	+	0.782871i	$0.286245\pi$
$774$		0			0
$775$	10.7651			0.386694
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.6545	+	25.3824i	0.525053	+	0.909418i
$780$		0			0
$781$	12.7115	−	22.0170i	0.454854	−	0.787831i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.44870	−	12.9015i	0.265855	−	0.460475i
$786$		0			0
$787$	−6.07963	−	10.5302i	−0.216715	−	0.375362i	0.737087	−	0.675798i	$-0.236201\pi$
$787$	−6.07963	−	10.5302i	−0.216715	−	0.375362i	−0.953802	+	0.300437i	$0.902868\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	11.1249			0.395056
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.89493	−	5.01416i	−0.102544	−	0.177611i	0.810188	−	0.586170i	$-0.199365\pi$
$797$	−2.89493	−	5.01416i	−0.102544	−	0.177611i	−0.912732	+	0.408559i	$0.866031\pi$
$798$		0			0
$799$	−18.2534	+	31.6158i	−0.645759	+	1.11849i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.17742	+	10.6996i	−0.217996	+	0.377581i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.1817			1.72914			0.864568	−	0.502516i	$-0.167592\pi$
$809$	49.1817			1.72914			0.864568	+	0.502516i	$0.167592\pi$
$810$		0			0
$811$	−40.7266			−1.43010			−0.715052	−	0.699072i	$-0.753597\pi$
$811$	−40.7266			−1.43010			−0.715052	+	0.699072i	$0.753597\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.68911	−	2.92562i	−0.0591669	−	0.102480i
$816$		0			0
$817$	11.6058	−	20.1018i	0.406034	−	0.703272i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.54689	−	13.0716i	0.263388	−	0.456202i	−0.703752	−	0.710446i	$-0.748493\pi$
$821$	7.54689	−	13.0716i	0.263388	−	0.456202i	0.967140	+	0.254244i	$0.0818266\pi$
$822$		0			0
$823$	−8.00000	−	13.8564i	−0.278862	−	0.483004i	0.692240	−	0.721668i	$-0.256624\pi$
$823$	−8.00000	−	13.8564i	−0.278862	−	0.483004i	−0.971102	+	0.238664i	$0.923291\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.2348			−1.22523			−0.612616	−	0.790381i	$-0.709883\pi$
$827$	−35.2348			−1.22523			−0.612616	+	0.790381i	$0.709883\pi$
$828$		0			0
$829$	3.23491			0.112353			0.0561765	−	0.998421i	$-0.482109\pi$
$829$	3.23491			0.112353			0.0561765	+	0.998421i	$0.482109\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	2.22803	−	3.85906i	0.0771041	−	0.133548i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.5197	−	26.8808i	0.535798	−	0.928030i	−0.463326	−	0.886188i	$-0.653344\pi$
$839$	15.5197	−	26.8808i	0.535798	−	0.928030i	0.999124	−	0.0418419i	$-0.0133226\pi$
$840$		0			0
$841$	4.32760	+	7.49563i	0.149228	+	0.258470i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	21.0678			0.724755
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.44801	−	4.24008i	−0.0839167	−	0.145348i
$852$		0			0
$853$	−8.03637	+	13.9194i	−0.275160	+	0.476591i	−0.970176	−	0.242403i	$-0.922064\pi$
$853$	−8.03637	+	13.9194i	−0.275160	+	0.476591i	0.695015	+	0.718995i	$0.255398\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.61058	+	16.6460i	−0.328291	+	0.568617i	−0.982173	−	0.187980i	$-0.939806\pi$
$857$	−9.61058	+	16.6460i	−0.328291	+	0.568617i	0.653882	+	0.756597i	$0.273139\pi$
$858$		0			0
$859$	−7.40112	−	12.8191i	−0.252523	−	0.437382i	0.711697	−	0.702487i	$-0.247927\pi$
$859$	−7.40112	−	12.8191i	−0.252523	−	0.437382i	−0.964220	+	0.265104i	$0.914594\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.7688			−0.502736			−0.251368	−	0.967892i	$-0.580880\pi$
$863$	−14.7688			−0.502736			−0.251368	+	0.967892i	$0.580880\pi$
$864$		0			0
$865$	8.88874			0.302226
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.4072	−	19.7579i	−0.386964	−	0.670241i
$870$		0			0
$871$	3.10507	−	5.37815i	0.105211	−	0.182232i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−26.1916	−	45.3651i	−0.884427	−	1.53187i	−0.846369	−	0.532597i	$-0.821216\pi$
$877$	−26.1916	−	45.3651i	−0.884427	−	1.53187i	−0.0380575	−	0.999276i	$-0.512117\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.3214			−1.05525			−0.527623	−	0.849479i	$-0.676916\pi$
$881$	−31.3214			−1.05525			−0.527623	+	0.849479i	$0.676916\pi$
$882$		0			0
$883$	−43.0494			−1.44873			−0.724363	−	0.689419i	$-0.757866\pi$
$883$	−43.0494			−1.44873			−0.724363	+	0.689419i	$0.757866\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.48831	+	12.9701i	0.251433	+	0.435494i	0.963921	−	0.266190i	$-0.0857649\pi$
$887$	7.48831	+	12.9701i	0.251433	+	0.435494i	−0.712488	+	0.701685i	$0.752432\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.3887	+	56.0988i	−1.08385	+	1.87727i
$894$		0			0
$895$	4.04944	+	7.01384i	0.135358	+	0.234447i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	22.9986			0.767047
$900$		0			0
$901$	−5.30903			−0.176870
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.86467	+	15.3541i	0.294671	+	0.510386i
$906$		0			0
$907$	−15.2280	+	26.3756i	−0.505636	+	0.875787i	0.494343	+	0.869267i	$0.335409\pi$
$907$	−15.2280	+	26.3756i	−0.505636	+	0.875787i	−0.999979	+	0.00652002i	$0.997925\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.97593	+	17.2788i	−0.330517	+	0.572473i	−0.982613	−	0.185664i	$-0.940557\pi$
$911$	−9.97593	+	17.2788i	−0.330517	+	0.572473i	0.652096	+	0.758136i	$0.273890\pi$
$912$		0			0
$913$	10.9127	+	18.9014i	0.361159	+	0.625545i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	45.6291			1.50516			0.752582	−	0.658498i	$-0.228808\pi$
$919$	45.6291			1.50516			0.752582	+	0.658498i	$0.228808\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.98969	+	6.91034i	0.131322	+	0.227457i
$924$		0			0
$925$	−7.25890	+	12.5728i	−0.238671	+	0.413391i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.1861	+	48.8197i	−0.924755	+	1.60172i	−0.132801	+	0.991143i	$0.542397\pi$
$929$	−28.1861	+	48.8197i	−0.924755	+	1.60172i	−0.791954	+	0.610580i	$0.790936\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.8343			0.387022
$936$		0			0
$937$	36.8530			1.20393			0.601967	−	0.798521i	$-0.294384\pi$
$937$	36.8530			1.20393			0.601967	+	0.798521i	$0.294384\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.38000	+	7.58638i	0.142784	+	0.247309i	0.928544	−	0.371223i	$-0.121061\pi$
$941$	4.38000	+	7.58638i	0.142784	+	0.247309i	−0.785760	+	0.618531i	$0.787728\pi$
$942$		0			0
$943$	−2.09201	+	3.62346i	−0.0681251	+	0.117996i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.3226	−	23.0754i	0.432926	−	0.749849i	−0.564198	−	0.825640i	$-0.690815\pi$
$947$	13.3226	−	23.0754i	0.432926	−	0.749849i	0.997124	+	0.0757901i	$0.0241479\pi$
$948$		0			0
$949$	−1.93887	−	3.35822i	−0.0629383	−	0.109012i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.3039			−0.787282			−0.393641	−	0.919264i	$-0.628785\pi$
$953$	−24.3039			−0.787282			−0.393641	+	0.919264i	$0.628785\pi$
$954$		0			0
$955$	22.6414			0.732660
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.50069	−	4.33132i	0.0806674	−	0.139720i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.4425	+	21.5511i	−0.400539	+	0.693753i
$966$		0			0
$967$	5.22872	+	9.05641i	0.168144	+	0.291234i	0.937767	−	0.347264i	$-0.112889\pi$
$967$	5.22872	+	9.05641i	0.168144	+	0.291234i	−0.769623	+	0.638498i	$0.779556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.7156			−1.33872			−0.669358	−	0.742940i	$-0.733431\pi$
$971$	−41.7156			−1.33872			−0.669358	+	0.742940i	$0.733431\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.94506	+	5.10099i	0.0942207	+	0.163195i	0.909283	−	0.416178i	$-0.136631\pi$
$977$	2.94506	+	5.10099i	0.0942207	+	0.163195i	−0.815062	+	0.579373i	$0.803297\pi$
$978$		0			0
$979$	11.9567	−	20.7097i	0.382139	−	0.661885i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.9196	+	36.2338i	−0.667232	+	1.15568i	0.311443	+	0.950265i	$0.399188\pi$
$983$	−20.9196	+	36.2338i	−0.667232	+	1.15568i	−0.978675	+	0.205415i	$0.934146\pi$
$984$		0			0
$985$	15.7095	+	27.2096i	0.500545	+	0.866969i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.31356			0.105365
$990$		0			0
$991$	−54.7156			−1.73810			−0.869049	−	0.494726i	$-0.835268\pi$
$991$	−54.7156			−1.73810			−0.869049	+	0.494726i	$0.835268\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.0698	−	34.7619i	−0.636255	−	1.10203i
$996$		0			0
$997$	−9.02476	+	15.6313i	−0.285817	+	0.495050i	−0.972807	−	0.231617i	$-0.925598\pi$
$997$	−9.02476	+	15.6313i	−0.285817	+	0.495050i	0.686990	+	0.726667i	$0.258932\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.j.d.3529.3		6
3.2	odd	2		1764.2.j.e.1177.1		6
7.2	even	3		5292.2.i.e.2125.3		6
7.3	odd	6		5292.2.l.e.3313.3		6
7.4	even	3		5292.2.l.f.3313.1		6
7.5	odd	6		5292.2.i.f.2125.1		6
7.6	odd	2		756.2.j.b.505.1		6
9.4	even	3	inner	5292.2.j.d.1765.3		6
9.5	odd	6		1764.2.j.e.589.1		6
21.2	odd	6		1764.2.i.d.1537.3		6
21.5	even	6		1764.2.i.g.1537.1		6
21.11	odd	6		1764.2.l.f.961.3		6
21.17	even	6		1764.2.l.e.961.1		6
21.20	even	2		252.2.j.a.169.3	yes	6
28.27	even	2		3024.2.r.j.2017.1		6
63.4	even	3		5292.2.i.e.1549.3		6
63.5	even	6		1764.2.l.e.949.1		6
63.13	odd	6		756.2.j.b.253.1		6
63.20	even	6		2268.2.a.i.1.1		3
63.23	odd	6		1764.2.l.f.949.3		6
63.31	odd	6		5292.2.i.f.1549.1		6
63.32	odd	6		1764.2.i.d.373.3		6
63.34	odd	6		2268.2.a.h.1.3		3
63.40	odd	6		5292.2.l.e.361.3		6
63.41	even	6		252.2.j.a.85.3	✓	6
63.58	even	3		5292.2.l.f.361.1		6
63.59	even	6		1764.2.i.g.373.1		6
84.83	odd	2		1008.2.r.j.673.1		6
252.83	odd	6		9072.2.a.by.1.1		3
252.139	even	6		3024.2.r.j.1009.1		6
252.167	odd	6		1008.2.r.j.337.1		6
252.223	even	6		9072.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.3	✓	6	63.41	even	6
252.2.j.a.169.3	yes	6	21.20	even	2
756.2.j.b.253.1		6	63.13	odd	6
756.2.j.b.505.1		6	7.6	odd	2
1008.2.r.j.337.1		6	252.167	odd	6
1008.2.r.j.673.1		6	84.83	odd	2
1764.2.i.d.373.3		6	63.32	odd	6
1764.2.i.d.1537.3		6	21.2	odd	6
1764.2.i.g.373.1		6	63.59	even	6
1764.2.i.g.1537.1		6	21.5	even	6
1764.2.j.e.589.1		6	9.5	odd	6
1764.2.j.e.1177.1		6	3.2	odd	2
1764.2.l.e.949.1		6	63.5	even	6
1764.2.l.e.961.1		6	21.17	even	6
1764.2.l.f.949.3		6	63.23	odd	6
1764.2.l.f.961.3		6	21.11	odd	6
2268.2.a.h.1.3		3	63.34	odd	6
2268.2.a.i.1.1		3	63.20	even	6
3024.2.r.j.1009.1		6	252.139	even	6
3024.2.r.j.2017.1		6	28.27	even	2
5292.2.i.e.1549.3		6	63.4	even	3
5292.2.i.e.2125.3		6	7.2	even	3
5292.2.i.f.1549.1		6	63.31	odd	6
5292.2.i.f.2125.1		6	7.5	odd	6
5292.2.j.d.1765.3		6	9.4	even	3	inner
5292.2.j.d.3529.3		6	1.1	even	1	trivial
5292.2.l.e.361.3		6	63.40	odd	6
5292.2.l.e.3313.3		6	7.3	odd	6
5292.2.l.f.361.1		6	63.58	even	3
5292.2.l.f.3313.1		6	7.4	even	3
9072.2.a.bv.1.3		3	252.223	even	6
9072.2.a.by.1.1		3	252.83	odd	6

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.849814 + 1.47192i) q^{5} +O(q^{10})\)	\(q+(0.849814 + 1.47192i) q^{5} +(1.23855 - 2.14523i) q^{11} +(0.388736 + 0.673310i) q^{13} +2.81089 q^{17} +4.98762 q^{19} +(0.356004 + 0.616617i) q^{23} +(1.05563 - 1.82841i) q^{25} +(2.25526 - 3.90623i) q^{29} +(2.54944 + 4.41576i) q^{31} -6.87636 q^{37} +(2.93818 + 5.08907i) q^{41} +(2.32691 - 4.03033i) q^{43} +(-6.49381 + 11.2476i) q^{47} -1.88874 q^{53} +4.21015 q^{55} +(-7.14400 - 12.3738i) q^{59} +(7.15452 - 12.3920i) q^{61} +(-0.660706 + 1.14438i) q^{65} +(-3.99381 - 6.91748i) q^{67} +10.2632 q^{71} -4.98762 q^{73} +(4.60507 - 7.97622i) q^{79} +(-4.40545 + 7.63046i) q^{83} +(2.38874 + 4.13741i) q^{85} +9.65383 q^{89} +(4.23855 + 7.34138i) q^{95} +(4.32072 - 7.48371i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5}+O(q^{10}) \) 6 * q - q^5	\( 6 q - q^{5} + 2 q^{11} + 3 q^{13} + 4 q^{17} - 6 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} - 3 q^{31} - 6 q^{37} - 3 q^{43} - 21 q^{47} - 12 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 34 q^{71} + 6 q^{73} + 9 q^{79} - 20 q^{83} + 15 q^{85} + 24 q^{89} + 20 q^{95} - 9 q^{97}+O(q^{100}) \) 6 * q - q^5 + 2 * q^11 + 3 * q^13 + 4 * q^17 - 6 * q^19 + 14 * q^23 + 6 * q^25 + q^29 - 3 * q^31 - 6 * q^37 - 3 * q^43 - 21 * q^47 - 12 * q^53 - 12 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 34 * q^71 + 6 * q^73 + 9 * q^79 - 20 * q^83 + 15 * q^85 + 24 * q^89 + 20 * q^95 - 9 * q^97

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