LMFDB - Embedded newform 5292.2.l.f.3313.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(361,5292)]

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, 4]))

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

//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);

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: 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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.3
Root			$0.500000 - 2.05195i$ of defining polynomial
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.f.361.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46050 q^{5} -4.64766 q^{11} +(3.55408 + 6.15585i) q^{13} +(2.25729 + 3.90975i) q^{17} +(2.16372 - 3.74766i) q^{19} -5.86693 q^{23} +1.05408 q^{25} +(-3.48755 + 6.04061i) q^{29} +(-3.69076 + 6.39258i) q^{31} +(0.363327 - 0.629301i) q^{37} +(-0.136673 - 0.236725i) q^{41} +(2.41741 - 4.18708i) q^{43} +(-1.83628 - 3.18054i) q^{47} +(2.52704 + 4.37697i) q^{53} -11.4356 q^{55} +(-4.56654 + 7.90947i) q^{59} +(-6.90856 - 11.9660i) q^{61} +(8.74484 + 15.1465i) q^{65} +(0.663715 - 1.14959i) q^{67} -13.5218 q^{71} +(-2.16372 - 3.74766i) q^{73} +(-3.21780 - 5.57339i) q^{79} +(-0.742705 + 1.28640i) q^{83} +(5.55408 + 9.61996i) q^{85} +(-4.91741 + 8.51721i) q^{89} +(5.32383 - 9.22115i) q^{95} +(-0.246304 + 0.426611i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 4 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{19} - 28 q^{23} - 12 q^{25} + q^{29} - 3 q^{31} + 3 q^{37} - 3 q^{43} - 21 q^{47} + 6 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67}+ \cdots - 9 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 - 4 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^19 - 28 * q^23 - 12 * q^25 + q^29 - 3 * q^31 + 3 * q^37 - 3 * q^43 - 21 * q^47 + 6 * q^53 - 12 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 34 * q^71 - 3 * q^73 + 9 * q^79 - 20 * q^83 + 15 * q^85 - 12 * q^89 + 20 * q^95 - 9 * q^97

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
$5$	2.46050			1.10037			0.550186	−	0.835042i	$-0.314557\pi$
$5$	2.46050			1.10037			0.550186	+	0.835042i	$0.314557\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.64766			−1.40132			−0.700662	−	0.713494i	$-0.747112\pi$
$11$	−4.64766			−1.40132			−0.700662	+	0.713494i	$0.747112\pi$
$12$		0			0
$13$	3.55408	+	6.15585i	0.985726	+	1.70733i	0.638667	+	0.769484i	$0.279486\pi$
$13$	3.55408	+	6.15585i	0.985726	+	1.70733i	0.347059	+	0.937843i	$0.387180\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.25729	+	3.90975i	0.547474	+	0.948253i	0.998447	+	0.0557155i	$0.0177440\pi$
$17$	2.25729	+	3.90975i	0.547474	+	0.948253i	−0.450972	+	0.892538i	$0.648923\pi$
$18$		0			0
$19$	2.16372	−	3.74766i	0.496390	−	0.859773i	−0.503601	−	0.863936i	$-0.667992\pi$
$19$	2.16372	−	3.74766i	0.496390	−	0.859773i	0.999991	+	0.00416311i	$0.00132516\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.86693			−1.22334			−0.611669	−	0.791114i	$-0.709502\pi$
$23$	−5.86693			−1.22334			−0.611669	+	0.791114i	$0.709502\pi$
$24$		0			0
$25$	1.05408			0.210817
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.48755	+	6.04061i	−0.647621	+	1.12171i	0.336068	+	0.941838i	$0.390903\pi$
$29$	−3.48755	+	6.04061i	−0.647621	+	1.12171i	−0.983689	+	0.179875i	$0.942431\pi$
$30$		0			0
$31$	−3.69076	+	6.39258i	−0.662880	+	1.14814i	0.316976	+	0.948434i	$0.397333\pi$
$31$	−3.69076	+	6.39258i	−0.662880	+	1.14814i	−0.979856	+	0.199708i	$0.936001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.363327	−	0.629301i	0.0597306	−	0.103456i	−0.834614	−	0.550835i	$-0.814309\pi$
$37$	0.363327	−	0.629301i	0.0597306	−	0.103456i	0.894344	+	0.447379i	$0.147643\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.136673	−	0.236725i	−0.0213448	−	0.0369702i	0.855156	−	0.518371i	$-0.173461\pi$
$41$	−0.136673	−	0.236725i	−0.0213448	−	0.0369702i	−0.876500	+	0.481401i	$0.840128\pi$
$42$		0			0
$43$	2.41741	−	4.18708i	0.368652	−	0.638524i	−0.620703	−	0.784046i	$-0.713153\pi$
$43$	2.41741	−	4.18708i	0.368652	−	0.638524i	0.989355	+	0.145522i	$0.0464862\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.83628	−	3.18054i	−0.267850	−	0.463929i	0.700457	−	0.713695i	$-0.252980\pi$
$47$	−1.83628	−	3.18054i	−0.267850	−	0.463929i	−0.968306	+	0.249766i	$0.919646\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.52704	+	4.37697i	0.347116	+	0.601222i	0.985736	−	0.168300i	$-0.0538277\pi$
$53$	2.52704	+	4.37697i	0.347116	+	0.601222i	−0.638620	+	0.769522i	$0.720494\pi$
$54$		0			0
$55$	−11.4356			−1.54198
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.56654	+	7.90947i	−0.594513	+	1.02973i	0.399103	+	0.916906i	$0.369322\pi$
$59$	−4.56654	+	7.90947i	−0.594513	+	1.02973i	−0.993615	+	0.112820i	$0.964012\pi$
$60$		0			0
$61$	−6.90856	−	11.9660i	−0.884550	−	1.53209i	−0.846228	−	0.532820i	$-0.821132\pi$
$61$	−6.90856	−	11.9660i	−0.884550	−	1.53209i	−0.0383215	−	0.999265i	$-0.512201\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.74484	+	15.1465i	1.08466	+	1.87869i
$66$		0			0
$67$	0.663715	−	1.14959i	0.0810857	−	0.140445i	−0.822631	−	0.568576i	$-0.807495\pi$
$67$	0.663715	−	1.14959i	0.0810857	−	0.140445i	0.903717	+	0.428131i	$0.140828\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.5218			−1.60474			−0.802370	−	0.596826i	$-0.796428\pi$
$71$	−13.5218			−1.60474			−0.802370	+	0.596826i	$0.796428\pi$
$72$		0			0
$73$	−2.16372	−	3.74766i	−0.253244	−	0.438631i	0.711173	−	0.703017i	$-0.248164\pi$
$73$	−2.16372	−	3.74766i	−0.253244	−	0.438631i	−0.964417	+	0.264386i	$0.914831\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.21780	−	5.57339i	−0.362031	−	0.627056i	0.626264	−	0.779611i	$-0.284583\pi$
$79$	−3.21780	−	5.57339i	−0.362031	−	0.627056i	−0.988295	+	0.152555i	$0.951250\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.742705	+	1.28640i	−0.0815225	+	0.141201i	−0.903904	−	0.427735i	$-0.859312\pi$
$83$	−0.742705	+	1.28640i	−0.0815225	+	0.141201i	0.822382	+	0.568936i	$0.192645\pi$
$84$		0			0
$85$	5.55408	+	9.61996i	0.602425	+	1.04343i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.91741	+	8.51721i	−0.521245	+	0.902822i	0.478450	+	0.878115i	$0.341199\pi$
$89$	−4.91741	+	8.51721i	−0.521245	+	0.902822i	−0.999695	+	0.0247073i	$0.992135\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.32383	−	9.22115i	0.546214	−	0.946070i
$96$		0			0
$97$	−0.246304	+	0.426611i	−0.0250084	+	0.0433158i	−0.878259	−	0.478186i	$-0.841295\pi$
$97$	−0.246304	+	0.426611i	−0.0250084	+	0.0433158i	0.853250	+	0.521502i	$0.174628\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.40642			0.338952			0.169476	−	0.985534i	$-0.445793\pi$
$101$	3.40642			0.338952			0.169476	+	0.985534i	$0.445793\pi$
$102$		0			0
$103$	−5.16225			−0.508652			−0.254326	−	0.967119i	$-0.581854\pi$
$103$	−5.16225			−0.508652			−0.254326	+	0.967119i	$0.581854\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.88151	+	4.99093i	−0.278567	+	0.482491i	−0.971029	−	0.238963i	$-0.923193\pi$
$107$	−2.88151	+	4.99093i	−0.278567	+	0.482491i	0.692462	+	0.721454i	$0.256526\pi$
$108$		0			0
$109$	4.49115	+	7.77889i	0.430174	+	0.745083i	0.996888	−	0.0788317i	$-0.0251190\pi$
$109$	4.49115	+	7.77889i	0.430174	+	0.745083i	−0.566714	+	0.823914i	$0.691786\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.679767	−	1.17739i	−0.0639471	−	0.110760i	0.832279	−	0.554356i	$-0.187036\pi$
$113$	−0.679767	−	1.17739i	−0.0639471	−	0.110760i	−0.896226	+	0.443597i	$0.853702\pi$
$114$		0			0
$115$	−14.4356			−1.34613
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.6008			0.963707
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.70895			−0.868394
$126$		0			0
$127$	−0.820039			−0.0727667			−0.0363833	−	0.999338i	$-0.511584\pi$
$127$	−0.820039			−0.0727667			−0.0363833	+	0.999338i	$0.511584\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.78794			0.680435			0.340218	−	0.940347i	$-0.389499\pi$
$131$	7.78794			0.680435			0.340218	+	0.940347i	$0.389499\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.99280			0.255692			0.127846	−	0.991794i	$-0.459194\pi$
$137$	2.99280			0.255692			0.127846	+	0.991794i	$0.459194\pi$
$138$		0			0
$139$	3.16372	+	5.47972i	0.268343	+	0.464783i	0.968434	−	0.249270i	$-0.0801907\pi$
$139$	3.16372	+	5.47972i	0.268343	+	0.464783i	−0.700091	+	0.714053i	$0.746857\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.5182	−	28.6103i	−1.38132	−	2.39252i
$144$		0			0
$145$	−8.58113	+	14.8629i	−0.712624	+	1.23430i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.38151			0.358948			0.179474	−	0.983763i	$-0.442560\pi$
$149$	4.38151			0.358948			0.179474	+	0.983763i	$0.442560\pi$
$150$		0			0
$151$	6.60078			0.537164			0.268582	−	0.963257i	$-0.413445\pi$
$151$	6.60078			0.537164			0.268582	+	0.963257i	$0.413445\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.08113	+	15.7290i	−0.729414	+	1.26338i
$156$		0			0
$157$	−2.89037	+	5.00627i	−0.230677	+	0.399544i	−0.958007	−	0.286743i	$-0.907427\pi$
$157$	−2.89037	+	5.00627i	−0.230677	+	0.399544i	0.727331	+	0.686287i	$0.240761\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.66372	+	6.34574i	−0.286964	+	0.497037i	−0.973084	−	0.230452i	$-0.925979\pi$
$163$	−3.66372	+	6.34574i	−0.286964	+	0.497037i	0.686119	+	0.727489i	$0.259313\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.01459	+	10.4176i	0.465423	+	0.806136i	0.999221	−	0.0394762i	$-0.0125689\pi$
$167$	6.01459	+	10.4176i	0.465423	+	0.806136i	−0.533798	+	0.845612i	$0.679236\pi$
$168$		0			0
$169$	−18.7630	+	32.4985i	−1.44331	+	2.49989i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.44951	−	4.24268i	−0.186233	−	0.322565i	0.757758	−	0.652535i	$-0.226295\pi$
$173$	−2.44951	−	4.24268i	−0.186233	−	0.322565i	−0.943991	+	0.329970i	$0.892961\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.890369	−	1.54216i	−0.0665493	−	0.115267i	0.830831	−	0.556525i	$-0.187866\pi$
$179$	−0.890369	−	1.54216i	−0.0665493	−	0.115267i	−0.897380	+	0.441258i	$0.854532\pi$
$180$		0			0
$181$	16.9430			1.25936			0.629681	−	0.776854i	$-0.283185\pi$
$181$	16.9430			1.25936			0.629681	+	0.776854i	$0.283185\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.893968	−	1.54840i	0.0657258	−	0.113840i
$186$		0			0
$187$	−10.4911	−	18.1712i	−0.767189	−	1.32881i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.74484	−	4.75420i	−0.198610	−	0.344002i	0.749468	−	0.662040i	$-0.230309\pi$
$191$	−2.74484	−	4.75420i	−0.198610	−	0.344002i	−0.948078	+	0.318038i	$0.896976\pi$
$192$		0			0
$193$	2.75370	−	4.76954i	0.198215	−	0.343319i	−0.749734	−	0.661739i	$-0.769819\pi$
$193$	2.75370	−	4.76954i	0.198215	−	0.343319i	0.947950	+	0.318420i	$0.103152\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.6300			−0.828600			−0.414300	−	0.910140i	$-0.635974\pi$
$197$	−11.6300			−0.828600			−0.414300	+	0.910140i	$0.635974\pi$
$198$		0			0
$199$	−2.07373	−	3.59181i	−0.147003	−	0.254617i	0.783115	−	0.621876i	$-0.213629\pi$
$199$	−2.07373	−	3.59181i	−0.147003	−	0.254617i	−0.930118	+	0.367260i	$0.880296\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.336285	−	0.582462i	−0.0234871	−	0.0406809i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.0562	+	17.4179i	−0.695603	+	1.20482i
$210$		0			0
$211$	13.6082	+	23.5700i	0.936825	+	1.62263i	0.771347	+	0.636415i	$0.219583\pi$
$211$	13.6082	+	23.5700i	0.936825	+	1.62263i	0.165478	+	0.986213i	$0.447083\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.94805	−	10.3023i	0.405654	−	0.702613i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.0452	+	27.7912i	−1.07932	+	1.86944i
$222$		0			0
$223$	−1.60817	+	2.78543i	−0.107691	+	0.186526i	−0.914834	−	0.403829i	$-0.867679\pi$
$223$	−1.60817	+	2.78543i	−0.107691	+	0.186526i	0.807144	+	0.590355i	$0.201012\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.9459			1.05837			0.529184	−	0.848507i	$-0.322498\pi$
$227$	15.9459			1.05837			0.529184	+	0.848507i	$0.322498\pi$
$228$		0			0
$229$	1.21634			0.0803778			0.0401889	−	0.999192i	$-0.487204\pi$
$229$	1.21634			0.0803778			0.0401889	+	0.999192i	$0.487204\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.98608	−	17.2964i	0.654210	−	1.13313i	−0.327881	−	0.944719i	$-0.606335\pi$
$233$	9.98608	−	17.2964i	0.654210	−	1.13313i	0.982091	−	0.188406i	$-0.0603321\pi$
$234$		0			0
$235$	−4.51819	−	7.82573i	−0.294734	−	0.510494i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00739	+	5.20896i	0.194532	+	0.336939i	0.946747	−	0.321978i	$-0.104348\pi$
$239$	3.00739	+	5.20896i	0.194532	+	0.336939i	−0.752215	+	0.658918i	$0.771015\pi$
$240$		0			0
$241$	18.6156			1.19913			0.599567	−	0.800325i	$-0.295340\pi$
$241$	18.6156			1.19913			0.599567	+	0.800325i	$0.295340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	30.7601			1.95722
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.99707			0.441651			0.220826	−	0.975313i	$-0.429125\pi$
$251$	6.99707			0.441651			0.220826	+	0.975313i	$0.429125\pi$
$252$		0			0
$253$	27.2675			1.71429
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.7778			−1.10895			−0.554475	−	0.832201i	$-0.687081\pi$
$257$	−17.7778			−1.10895			−0.554475	+	0.832201i	$0.687081\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	27.1986			1.67714			0.838570	−	0.544794i	$-0.183392\pi$
$263$	27.1986			1.67714			0.838570	+	0.544794i	$0.183392\pi$
$264$		0			0
$265$	6.21780	+	10.7695i	0.381956	+	0.661568i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.9481	−	20.6946i	−0.728486	−	1.26177i	−0.957523	−	0.288356i	$-0.906891\pi$
$269$	−11.9481	−	20.6946i	−0.728486	−	1.26177i	0.229038	−	0.973418i	$-0.426442\pi$
$270$		0			0
$271$	6.13667	−	10.6290i	0.372776	−	0.645668i	−0.617215	−	0.786794i	$-0.711739\pi$
$271$	6.13667	−	10.6290i	0.372776	−	0.645668i	0.989992	+	0.141127i	$0.0450725\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.89903			−0.295423
$276$		0			0
$277$	12.7807			0.767920			0.383960	−	0.923350i	$-0.374560\pi$
$277$	12.7807			0.767920			0.383960	+	0.923350i	$0.374560\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.2573	+	24.6944i	−0.850519	+	1.47314i	0.0302219	+	0.999543i	$0.490379\pi$
$281$	−14.2573	+	24.6944i	−0.850519	+	1.47314i	−0.880741	+	0.473599i	$0.842955\pi$
$282$		0			0
$283$	0.363327	−	0.629301i	0.0215975	−	0.0374080i	−0.855025	−	0.518587i	$-0.826458\pi$
$283$	0.363327	−	0.629301i	0.0215975	−	0.0374080i	0.876622	+	0.481179i	$0.159791\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.69076	+	2.92848i	−0.0994563	+	0.172263i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.7901	−	22.1531i	−0.747204	−	1.29420i	−0.949158	−	0.314800i	$-0.898062\pi$
$293$	−12.7901	−	22.1531i	−0.747204	−	1.29420i	0.201954	−	0.979395i	$-0.435271\pi$
$294$		0			0
$295$	−11.2360	+	19.4613i	−0.654184	+	1.13308i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.8515	−	36.1159i	−1.20588	−	2.08864i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.9985	−	29.4423i	−0.973333	−	1.68586i
$306$		0			0
$307$	6.23405			0.355796			0.177898	−	0.984049i	$-0.443070\pi$
$307$	6.23405			0.355796			0.177898	+	0.984049i	$0.443070\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−14.6192	+	25.3211i	−0.828976	+	1.43583i	0.0698655	+	0.997556i	$0.477743\pi$
$311$	−14.6192	+	25.3211i	−0.828976	+	1.43583i	−0.898842	+	0.438273i	$0.855590\pi$
$312$		0			0
$313$	14.2434	+	24.6703i	0.805083	+	1.39445i	0.916235	+	0.400642i	$0.131213\pi$
$313$	14.2434	+	24.6703i	0.805083	+	1.39445i	−0.111151	+	0.993803i	$0.535454\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.809243	+	1.40165i	0.0454516	+	0.0787245i	0.887856	−	0.460121i	$-0.152194\pi$
$317$	0.809243	+	1.40165i	0.0454516	+	0.0787245i	−0.842405	+	0.538846i	$0.818861\pi$
$318$		0			0
$319$	16.2089	−	28.0747i	0.907527	−	1.57188i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	19.5366			1.08704
$324$		0			0
$325$	3.74630	+	6.48879i	0.207808	+	0.359933i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.99115	+	12.1090i	0.384268	+	0.665572i	0.991667	−	0.128825i	$-0.0411205\pi$
$331$	6.99115	+	12.1090i	0.384268	+	0.665572i	−0.607399	+	0.794397i	$0.707787\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.63307	−	2.82857i	0.0892244	−	0.154541i
$336$		0			0
$337$	−13.8619	−	24.0095i	−0.755104	−	1.30788i	−0.945323	−	0.326137i	$-0.894253\pi$
$337$	−13.8619	−	24.0095i	−0.755104	−	1.30788i	0.190219	−	0.981742i	$-0.439080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.1534	−	29.7106i	0.928909	−	1.60892i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.76449	−	6.52029i	0.202089	−	0.350028i	−0.747113	−	0.664697i	$-0.768560\pi$
$347$	3.76449	−	6.52029i	0.202089	−	0.350028i	0.949201	+	0.314670i	$0.101894\pi$
$348$		0			0
$349$	−15.0541	+	26.0744i	−0.805827	+	1.39573i	0.109905	+	0.993942i	$0.464945\pi$
$349$	−15.0541	+	26.0744i	−0.805827	+	1.39573i	−0.915732	+	0.401791i	$0.868388\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.3638			−1.08386			−0.541928	−	0.840425i	$-0.682305\pi$
$353$	−20.3638			−1.08386			−0.541928	+	0.840425i	$0.682305\pi$
$354$		0			0
$355$	−33.2704			−1.76581
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.01313	+	13.8791i	−0.422917	+	0.732513i	−0.996223	−	0.0868277i	$-0.972327\pi$
$359$	−8.01313	+	13.8791i	−0.422917	+	0.732513i	0.573307	+	0.819341i	$0.305660\pi$
$360$		0			0
$361$	0.136673	+	0.236725i	0.00719332	+	0.0124592i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.32383	−	9.22115i	−0.278662	−	0.482657i
$366$		0			0
$367$	13.5979			0.709802			0.354901	−	0.934904i	$-0.384515\pi$
$367$	13.5979			0.709802			0.354901	+	0.934904i	$0.384515\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−21.9282			−1.13540			−0.567700	−	0.823236i	$-0.692167\pi$
$373$	−21.9282			−1.13540			−0.567700	+	0.823236i	$0.692167\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−49.5801			−2.55351
$378$		0			0
$379$	−29.7965			−1.53054			−0.765271	−	0.643708i	$-0.777395\pi$
$379$	−29.7965			−1.53054			−0.765271	+	0.643708i	$0.777395\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.0219809			−0.00112317			−0.000561587	−	1.00000i	$-0.500179\pi$
$383$	−0.0219809			−0.00112317			−0.000561587		1.00000i	$0.500179\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	35.3566			1.79265			0.896326	−	0.443396i	$-0.146227\pi$
$389$	35.3566			1.79265			0.896326	+	0.443396i	$0.146227\pi$
$390$		0			0
$391$	−13.2434	−	22.9382i	−0.669746	−	1.16003i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.91741	−	13.7134i	−0.398368	−	0.689994i
$396$		0			0
$397$	−8.47150	+	14.6731i	−0.425172	+	0.736420i	−0.996436	−	0.0843464i	$-0.973120\pi$
$397$	−8.47150	+	14.6731i	−0.425172	+	0.736420i	0.571264	+	0.820766i	$0.306453\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.96362			0.147996			0.0739982	−	0.997258i	$-0.476424\pi$
$401$	2.96362			0.147996			0.0739982	+	0.997258i	$0.476424\pi$
$402$		0			0
$403$	−52.4690			−2.61367
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.68862	+	2.92478i	−0.0837018	+	0.144976i
$408$		0			0
$409$	−7.32743	+	12.6915i	−0.362318	+	0.627553i	−0.988342	−	0.152251i	$-0.951348\pi$
$409$	−7.32743	+	12.6915i	−0.362318	+	0.627553i	0.626024	+	0.779804i	$0.284681\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.82743	+	3.16520i	−0.0897050	+	0.155374i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.6352	+	21.8848i	0.617270	+	1.06914i	0.989982	+	0.141196i	$0.0450949\pi$
$419$	12.6352	+	21.8848i	0.617270	+	1.06914i	−0.372711	+	0.927947i	$0.621572\pi$
$420$		0			0
$421$	7.99854	−	13.8539i	0.389825	−	0.675196i	−0.602601	−	0.798043i	$-0.705869\pi$
$421$	7.99854	−	13.8539i	0.389825	−	0.675196i	0.992426	+	0.122846i	$0.0392022\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.37938	+	4.12120i	0.115417	+	0.199908i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.51673	+	11.2873i	0.313900	+	0.543690i	0.979203	−	0.202883i	$-0.0650311\pi$
$431$	6.51673	+	11.2873i	0.313900	+	0.543690i	−0.665303	+	0.746573i	$0.731698\pi$
$432$		0			0
$433$	−23.5467			−1.13158			−0.565791	−	0.824549i	$-0.691429\pi$
$433$	−23.5467			−1.13158			−0.565791	+	0.824549i	$0.691429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.6944	+	21.9873i	−0.607253	+	1.05179i
$438$		0			0
$439$	3.35447	+	5.81012i	0.160100	+	0.277302i	0.934904	−	0.354900i	$-0.115485\pi$
$439$	3.35447	+	5.81012i	0.160100	+	0.277302i	−0.774804	+	0.632201i	$0.782152\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.6228	+	30.5235i	0.837282	+	1.45022i	0.892158	+	0.451723i	$0.149190\pi$
$443$	17.6228	+	30.5235i	0.837282	+	1.45022i	−0.0548760	+	0.998493i	$0.517476\pi$
$444$		0			0
$445$	−12.0993	+	20.9566i	−0.573562	+	0.993439i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.9387			0.610616			0.305308	−	0.952254i	$-0.401241\pi$
$449$	12.9387			0.610616			0.305308	+	0.952254i	$0.401241\pi$
$450$		0			0
$451$	0.635211	+	1.10022i	0.0299109	+	0.0518072i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.5993	+	25.2868i	0.682927	+	1.18286i	0.974083	+	0.226189i	$0.0726267\pi$
$457$	14.5993	+	25.2868i	0.682927	+	1.18286i	−0.291156	+	0.956675i	$0.594040\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.34348	−	16.1834i	0.435169	−	0.753735i	−0.562140	−	0.827042i	$-0.690022\pi$
$461$	9.34348	−	16.1834i	0.435169	−	0.753735i	0.997309	+	0.0733066i	$0.0233552\pi$
$462$		0			0
$463$	19.1249	+	33.1253i	0.888809	+	1.53946i	0.841285	+	0.540593i	$0.181800\pi$
$463$	19.1249	+	33.1253i	0.888809	+	1.53946i	0.0475247	+	0.998870i	$0.484867\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.64387	−	13.2396i	0.353716	−	0.612654i	−0.633181	−	0.774004i	$-0.718251\pi$
$467$	7.64387	−	13.2396i	0.353716	−	0.612654i	0.986897	+	0.161349i	$0.0515846\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.2353	+	19.4601i	−0.516600	+	0.894778i
$474$		0			0
$475$	2.28074	−	3.95035i	0.104647	−	0.181255i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.0321			0.504070			0.252035	−	0.967718i	$-0.418900\pi$
$479$	11.0321			0.504070			0.252035	+	0.967718i	$0.418900\pi$
$480$		0			0
$481$	5.16518			0.235512
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.606032	+	1.04968i	−0.0275185	+	0.0476635i
$486$		0			0
$487$	−8.30039	−	14.3767i	−0.376126	−	0.651470i	0.614368	−	0.789019i	$-0.289411\pi$
$487$	−8.30039	−	14.3767i	−0.376126	−	0.651470i	−0.990495	+	0.137549i	$0.956078\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.3633	−	23.1460i	−0.603079	−	1.04456i	−0.992352	−	0.123440i	$-0.960607\pi$
$491$	−13.3633	−	23.1460i	−0.603079	−	1.04456i	0.389273	−	0.921122i	$-0.372726\pi$
$492$		0			0
$493$	−31.4897			−1.41822
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.23697			0.0553744			0.0276872	−	0.999617i	$-0.491186\pi$
$499$	1.23697			0.0553744			0.0276872	+	0.999617i	$0.491186\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.07179			−0.0477889			−0.0238944	−	0.999714i	$-0.507607\pi$
$503$	−1.07179			−0.0477889			−0.0238944	+	0.999714i	$0.507607\pi$
$504$		0			0
$505$	8.38151			0.372973
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.0689			0.889537			0.444768	−	0.895646i	$-0.353286\pi$
$509$	20.0689			0.889537			0.444768	+	0.895646i	$0.353286\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.7017			−0.559706
$516$		0			0
$517$	8.53443	+	14.7821i	0.375344	+	0.650115i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.4430	+	26.7480i	0.676570	+	1.17185i	0.976007	+	0.217737i	$0.0698676\pi$
$521$	15.4430	+	26.7480i	0.676570	+	1.17185i	−0.299438	+	0.954116i	$0.596799\pi$
$522$		0			0
$523$	−3.69961	+	6.40792i	−0.161773	+	0.280199i	−0.935505	−	0.353315i	$-0.885054\pi$
$523$	−3.69961	+	6.40792i	−0.161773	+	0.280199i	0.773732	+	0.633513i	$0.218388\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−33.3245			−1.45164
$528$		0			0
$529$	11.4208			0.496557
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.971495	−	1.68268i	0.0420801	−	0.0728849i
$534$		0			0
$535$	−7.08998	+	12.2802i	−0.306527	+	0.530920i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11.3348	+	19.6325i	−0.487322	+	0.844067i	−0.999894	−	0.0145779i	$-0.995360\pi$
$541$	−11.3348	+	19.6325i	−0.487322	+	0.844067i	0.512572	+	0.858644i	$0.328693\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.0505	+	19.1400i	0.473351	+	0.819868i
$546$		0			0
$547$	3.07373	−	5.32386i	0.131423	−	0.227632i	−0.792802	−	0.609479i	$-0.791379\pi$
$547$	3.07373	−	5.32386i	0.131423	−	0.227632i	0.924225	+	0.381847i	$0.124712\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.0921	+	26.1403i	0.642946	+	1.11361i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.8370	+	25.6984i	0.628662	+	1.08887i	0.987820	+	0.155598i	$0.0497305\pi$
$557$	14.8370	+	25.6984i	0.628662	+	1.08887i	−0.359158	+	0.933277i	$0.616936\pi$
$558$		0			0
$559$	34.3667			1.45356
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.6555	−	25.3841i	0.617657	−	1.06981i	−0.372255	−	0.928131i	$-0.621415\pi$
$563$	14.6555	−	25.3841i	0.617657	−	1.06981i	0.989912	−	0.141683i	$-0.0452514\pi$
$564$		0			0
$565$	−1.67257	−	2.89698i	−0.0703655	−	0.121877i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.4430	−	31.9442i	−0.773170	−	1.33917i	−0.935817	−	0.352486i	$-0.885336\pi$
$569$	−18.4430	−	31.9442i	−0.773170	−	1.33917i	0.162647	−	0.986684i	$-0.447997\pi$
$570$		0			0
$571$	−16.1893	+	28.0407i	−0.677501	+	1.17347i	0.298230	+	0.954494i	$0.403604\pi$
$571$	−16.1893	+	28.0407i	−0.677501	+	1.17347i	−0.975731	+	0.218972i	$0.929730\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.18423			−0.257900
$576$		0			0
$577$	11.5093	+	19.9348i	0.479140	+	0.829895i	0.999714	−	0.0239220i	$-0.00761535\pi$
$577$	11.5093	+	19.9348i	0.479140	+	0.829895i	−0.520574	+	0.853817i	$0.674282\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−11.7448	−	20.3427i	−0.486422	−	0.842507i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.87052	+	4.97189i	−0.118479	+	0.205212i	−0.919165	−	0.393872i	$-0.871135\pi$
$587$	−2.87052	+	4.97189i	−0.118479	+	0.205212i	0.800686	+	0.599084i	$0.204469\pi$
$588$		0			0
$589$	15.9715	+	27.6634i	0.658094	+	1.13985i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.8727	+	24.0282i	−0.569682	+	0.986718i	0.426915	+	0.904292i	$0.359600\pi$
$593$	−13.8727	+	24.0282i	−0.569682	+	0.986718i	−0.996597	+	0.0824263i	$0.973733\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.05408	−	3.55778i	0.0839276	−	0.145367i	−0.821006	−	0.570919i	$-0.806587\pi$
$599$	2.05408	−	3.55778i	0.0839276	−	0.145367i	0.904934	+	0.425552i	$0.139920\pi$
$600$		0			0
$601$	7.80924	−	13.5260i	0.318546	−	0.551737i	−0.661639	−	0.749822i	$-0.730139\pi$
$601$	7.80924	−	13.5260i	0.318546	−	0.551737i	0.980185	+	0.198085i	$0.0634723\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	26.0833			1.06044
$606$		0			0
$607$	0.561476			0.0227896			0.0113948	−	0.999935i	$-0.496373\pi$
$607$	0.561476			0.0227896			0.0113948	+	0.999935i	$0.496373\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.0526	−	22.6078i	0.528053	−	0.914614i
$612$		0			0
$613$	10.1008	+	17.4951i	0.407967	+	0.706619i	0.994662	−	0.103189i	$-0.0329047\pi$
$613$	10.1008	+	17.4951i	0.407967	+	0.706619i	−0.586695	+	0.809808i	$0.699571\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.4569	−	19.8439i	−0.461238	−	0.798887i	0.537785	−	0.843082i	$-0.319261\pi$
$617$	−11.4569	−	19.8439i	−0.461238	−	0.798887i	−0.999023	+	0.0441948i	$0.985928\pi$
$618$		0			0
$619$	−39.7031			−1.59580			−0.797901	−	0.602788i	$-0.794056\pi$
$619$	−39.7031			−1.59580			−0.797901	+	0.602788i	$0.794056\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−29.1593			−1.16637
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.28054			0.130804
$630$		0			0
$631$	−31.0364			−1.23554			−0.617769	−	0.786359i	$-0.711963\pi$
$631$	−31.0364			−1.23554			−0.617769	+	0.786359i	$0.711963\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.01771			−0.0800703
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	29.5864			1.16859			0.584296	−	0.811541i	$-0.301371\pi$
$641$	29.5864			1.16859			0.584296	+	0.811541i	$0.301371\pi$
$642$		0			0
$643$	12.8442	+	22.2467i	0.506524	+	0.877325i	0.999972	+	0.00754978i	$0.00240319\pi$
$643$	12.8442	+	22.2467i	0.506524	+	0.877325i	−0.493447	+	0.869776i	$0.664263\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.50885	−	14.7378i	−0.334518	−	0.579401i	0.648874	−	0.760895i	$-0.275240\pi$
$647$	−8.50885	−	14.7378i	−0.334518	−	0.579401i	−0.983392	+	0.181494i	$0.941907\pi$
$648$		0			0
$649$	21.2237	−	36.7606i	0.833104	−	1.44298i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.47102			−0.0575653			−0.0287827	−	0.999586i	$-0.509163\pi$
$653$	−1.47102			−0.0575653			−0.0287827	+	0.999586i	$0.509163\pi$
$654$		0			0
$655$	19.1623			0.748731
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.7003	+	35.8539i	−0.806369	+	1.39667i	0.108995	+	0.994042i	$0.465237\pi$
$659$	−20.7003	+	35.8539i	−0.806369	+	1.39667i	−0.915363	+	0.402629i	$0.868096\pi$
$660$		0			0
$661$	19.1352	−	33.1432i	0.744273	−	1.28912i	−0.206260	−	0.978497i	$-0.566129\pi$
$661$	19.1352	−	33.1432i	0.744273	−	1.28912i	0.950533	−	0.310622i	$-0.100537\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	20.4612	−	35.4398i	0.792260	−	1.37223i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	32.1086	+	55.6138i	1.23954	+	2.14695i
$672$		0			0
$673$	15.2448	−	26.4048i	0.587645	−	1.01783i	−0.406894	−	0.913475i	$-0.633388\pi$
$673$	15.2448	−	26.4048i	0.587645	−	1.01783i	0.994540	−	0.104357i	$-0.0332783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.4626	−	38.9064i	−0.863309	−	1.49530i	−0.868716	−	0.495310i	$-0.835054\pi$
$677$	−22.4626	−	38.9064i	−0.863309	−	1.49530i	0.00540665	−	0.999985i	$-0.498279\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.1986	−	41.9133i	−0.925935	−	1.60377i	−0.790051	−	0.613041i	$-0.789946\pi$
$683$	−24.1986	−	41.9133i	−0.925935	−	1.60377i	−0.135884	−	0.990725i	$-0.543387\pi$
$684$		0			0
$685$	7.36381			0.281357
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.9626	+	31.1122i	−0.684322	+	1.18528i
$690$		0			0
$691$	9.19076	+	15.9189i	0.349633	+	0.605582i	0.986184	−	0.165652i	$-0.0529730\pi$
$691$	9.19076	+	15.9189i	0.349633	+	0.605582i	−0.636551	+	0.771234i	$0.719640\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.78434	+	13.4829i	0.295277	+	0.511434i
$696$		0			0
$697$	0.617023	−	1.06871i	0.0233714	−	0.0404805i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.0292			1.02088			0.510439	−	0.859914i	$-0.329483\pi$
$701$	27.0292			1.02088			0.510439	+	0.859914i	$0.329483\pi$
$702$		0			0
$703$	−1.57227	−	2.72325i	−0.0592994	−	0.102710i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.49261	−	4.31732i	−0.0936119	−	0.162141i	0.815417	−	0.578875i	$-0.196508\pi$
$709$	−2.49261	−	4.31732i	−0.0936119	−	0.162141i	−0.909028	+	0.416734i	$0.863175\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.6534	−	37.5048i	0.810926	−	1.40457i
$714$		0			0
$715$	−40.6431	−	70.3959i	−1.51997	−	2.63266i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.84708	+	13.5915i	−0.292647	+	0.506879i	−0.974435	−	0.224671i	$-0.927869\pi$
$719$	−7.84708	+	13.5915i	−0.292647	+	0.506879i	0.681788	+	0.731550i	$0.261203\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.67617	+	6.36731i	−0.136529	+	0.236476i
$726$		0			0
$727$	10.9071	−	18.8916i	0.404522	−	0.700652i	−0.589744	−	0.807590i	$-0.700771\pi$
$727$	10.9071	−	18.8916i	0.404522	−	0.700652i	0.994266	+	0.106938i	$0.0341047\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.8272			0.807309
$732$		0			0
$733$	24.0148			0.887006			0.443503	−	0.896273i	$-0.353736\pi$
$733$	24.0148			0.887006			0.443503	+	0.896273i	$0.353736\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.08472	+	5.34290i	−0.113627	+	0.196808i
$738$		0			0
$739$	−9.35447	−	16.2024i	−0.344110	−	0.596016i	0.641082	−	0.767473i	$-0.278486\pi$
$739$	−9.35447	−	16.2024i	−0.344110	−	0.596016i	−0.985192	+	0.171457i	$0.945153\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.1534	−	34.9067i	−0.739356	−	1.28060i	−0.952785	−	0.303644i	$-0.901797\pi$
$743$	−20.1534	−	34.9067i	−0.739356	−	1.28060i	0.213429	−	0.976959i	$-0.431537\pi$
$744$		0			0
$745$	10.7807			0.394976
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−21.1259			−0.770894			−0.385447	−	0.922730i	$-0.625953\pi$
$751$	−21.1259			−0.770894			−0.385447	+	0.922730i	$0.625953\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.2412			0.591079
$756$		0			0
$757$	8.85934			0.321998			0.160999	−	0.986955i	$-0.448528\pi$
$757$	8.85934			0.321998			0.160999	+	0.986955i	$0.448528\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.38910			0.0503549			0.0251774	−	0.999683i	$-0.491985\pi$
$761$	1.38910			0.0503549			0.0251774	+	0.999683i	$0.491985\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−64.9194			−2.34410
$768$		0			0
$769$	−18.9626	−	32.8443i	−0.683810	−	1.18439i	−0.973809	−	0.227367i	$-0.926988\pi$
$769$	−18.9626	−	32.8443i	−0.683810	−	1.18439i	0.289999	−	0.957027i	$-0.406345\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.657981	−	1.13966i	−0.0236659	−	0.0409906i	0.853950	−	0.520355i	$-0.174200\pi$
$773$	−0.657981	−	1.13966i	−0.0236659	−	0.0409906i	−0.877616	+	0.479365i	$0.840867\pi$
$774$		0			0
$775$	−3.89037	+	6.73832i	−0.139746	+	0.242047i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.18289			−0.0423813
$780$		0			0
$781$	62.8447			2.24876
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.11177	+	12.3179i	−0.253830	+	0.439646i
$786$		0			0
$787$	6.12928	−	10.6162i	0.218485	−	0.378428i	−0.735860	−	0.677134i	$-0.763222\pi$
$787$	6.12928	−	10.6162i	0.218485	−	0.378428i	0.954345	+	0.298706i	$0.0965551\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	49.1072	−	85.0561i	1.74385	−	3.02043i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.7178	−	18.5638i	−0.379644	−	0.657563i	0.611366	−	0.791348i	$-0.290620\pi$
$797$	−10.7178	−	18.5638i	−0.379644	−	0.657563i	−0.991010	+	0.133785i	$0.957287\pi$
$798$		0			0
$799$	8.29007	−	14.3588i	0.293282	−	0.507979i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.0562	+	17.4179i	0.354876	+	0.614664i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.3478	+	23.1190i	0.469282	+	0.812820i	0.999383	−	0.0351140i	$-0.0111794\pi$
$809$	13.3478	+	23.1190i	0.469282	+	0.812820i	−0.530101	+	0.847934i	$0.677846\pi$
$810$		0			0
$811$	−38.2852			−1.34438			−0.672188	−	0.740381i	$-0.734645\pi$
$811$	−38.2852			−1.34438			−0.672188	+	0.740381i	$0.734645\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.01459	+	15.6137i	−0.315767	+	0.546925i
$816$		0			0
$817$	−10.4612	−	18.1193i	−0.365990	−	0.633914i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.24990	−	9.09310i	−0.183223	−	0.317351i	0.759753	−	0.650211i	$-0.225320\pi$
$821$	−5.24990	−	9.09310i	−0.183223	−	0.317351i	−0.942976	+	0.332860i	$0.891986\pi$
$822$		0			0
$823$	−8.00000	+	13.8564i	−0.278862	+	0.483004i	−0.971102	−	0.238664i	$-0.923291\pi$
$823$	−8.00000	+	13.8564i	−0.278862	+	0.483004i	0.692240	+	0.721668i	$0.256624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	48.7817			1.69631			0.848153	−	0.529752i	$-0.177715\pi$
$827$	48.7817			1.69631			0.848153	+	0.529752i	$0.177715\pi$
$828$		0			0
$829$	−3.10963	−	5.38604i	−0.108002	−	0.187065i	0.806959	−	0.590608i	$-0.201112\pi$
$829$	−3.10963	−	5.38604i	−0.108002	−	0.187065i	−0.914961	+	0.403543i	$0.867779\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	14.7989	+	25.6325i	0.512138	+	0.887049i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.0366	−	36.4364i	0.726263	−	1.25792i	−0.232189	−	0.972671i	$-0.574589\pi$
$839$	21.0366	−	36.4364i	0.726263	−	1.25792i	0.958452	−	0.285254i	$-0.0920779\pi$
$840$		0			0
$841$	−9.82597	−	17.0191i	−0.338826	−	0.586865i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−46.1665	+	79.9628i	−1.58818	+	2.75080i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.13161	+	3.69206i	−0.0730707	+	0.126562i
$852$		0			0
$853$	−6.72519	+	11.6484i	−0.230266	+	0.398833i	−0.957886	−	0.287147i	$-0.907293\pi$
$853$	−6.72519	+	11.6484i	−0.230266	+	0.398833i	0.727620	+	0.685980i	$0.240626\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.3786			1.41347			0.706733	−	0.707481i	$-0.250168\pi$
$857$	41.3786			1.41347			0.706733	+	0.707481i	$0.250168\pi$
$858$		0			0
$859$	39.7630			1.35670			0.678349	−	0.734740i	$-0.262696\pi$
$859$	39.7630			1.35670			0.678349	+	0.734740i	$0.262696\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.6929	−	46.2334i	0.908637	−	1.57380i	0.0926768	−	0.995696i	$-0.470458\pi$
$863$	26.6929	−	46.2334i	0.908637	−	1.57380i	0.815960	−	0.578109i	$-0.196209\pi$
$864$		0			0
$865$	−6.02704	−	10.4391i	−0.204926	−	0.354942i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.9552	+	25.9033i	0.507322	+	0.878708i
$870$		0			0
$871$	9.43560			0.319713
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.85349			−0.231426			−0.115713	−	0.993283i	$-0.536915\pi$
$877$	−6.85349			−0.231426			−0.115713	+	0.993283i	$0.536915\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.5103			−0.421483			−0.210742	−	0.977542i	$-0.567588\pi$
$881$	−12.5103			−0.421483			−0.210742	+	0.977542i	$0.567588\pi$
$882$		0			0
$883$	6.69124			0.225178			0.112589	−	0.993642i	$-0.464086\pi$
$883$	6.69124			0.225178			0.112589	+	0.993642i	$0.464086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	32.1416			1.07921			0.539605	−	0.841918i	$-0.318574\pi$
$887$	32.1416			1.07921			0.539605	+	0.841918i	$0.318574\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.8928			−0.531832
$894$		0			0
$895$	−2.19076	−	3.79450i	−0.0732289	−	0.126836i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−25.7434	−	44.5888i	−0.858590	−	1.48712i
$900$		0			0
$901$	−11.4086	+	19.7602i	−0.380074	+	0.658308i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41.6883			1.38577
$906$		0			0
$907$	−31.4031			−1.04272			−0.521362	−	0.853336i	$-0.674576\pi$
$907$	−31.4031			−1.04272			−0.521362	+	0.853336i	$0.674576\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.8982	−	39.6609i	0.758653	−	1.31402i	−0.184885	−	0.982760i	$-0.559191\pi$
$911$	22.8982	−	39.6609i	0.758653	−	1.31402i	0.943538	−	0.331265i	$-0.107475\pi$
$912$		0			0
$913$	3.45185	−	5.97877i	0.114239	−	0.197868i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−13.5900	+	23.5385i	−0.448292	+	0.776465i	−0.998275	−	0.0587112i	$-0.981301\pi$
$919$	−13.5900	+	23.5385i	−0.448292	+	0.776465i	0.549983	+	0.835176i	$0.314634\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48.0576	−	83.2381i	−1.58183	−	2.73982i
$924$		0			0
$925$	0.382977	−	0.663336i	0.0125922	−	0.0218104i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.3338	+	35.2192i	0.667132	+	1.15551i	0.978703	+	0.205283i	$0.0658115\pi$
$929$	20.3338	+	35.2192i	0.667132	+	1.15551i	−0.311571	+	0.950223i	$0.600855\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−25.8135	−	44.7103i	−0.844192	−	1.46218i
$936$		0			0
$937$	−16.4150			−0.536254			−0.268127	−	0.963384i	$-0.586405\pi$
$937$	−16.4150			−0.536254			−0.268127	+	0.963384i	$0.586405\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.66878	+	6.35451i	−0.119599	+	0.207151i	−0.919609	−	0.392836i	$-0.871494\pi$
$941$	−3.66878	+	6.35451i	−0.119599	+	0.207151i	0.800010	+	0.599987i	$0.204827\pi$
$942$		0			0
$943$	0.801851	+	1.38885i	0.0261119	+	0.0452271i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29.5562	+	51.1929i	0.960448	+	1.66354i	0.721377	+	0.692543i	$0.243510\pi$
$947$	29.5562	+	51.1929i	0.960448	+	1.66354i	0.239071	+	0.971002i	$0.423157\pi$
$948$		0			0
$949$	15.3801	−	26.6390i	0.499258	−	0.864740i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.9354			0.548592			0.274296	−	0.961645i	$-0.411555\pi$
$953$	16.9354			0.548592			0.274296	+	0.961645i	$0.411555\pi$
$954$		0			0
$955$	−6.75370	−	11.6977i	−0.218544	−	0.378530i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−11.7434	−	20.3401i	−0.378819	−	0.656133i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.77548	−	11.7355i	0.218110	−	0.377778i
$966$		0			0
$967$	3.55555	+	6.15839i	0.114339	+	0.198040i	0.917515	−	0.397701i	$-0.130192\pi$
$967$	3.55555	+	6.15839i	0.114339	+	0.198040i	−0.803177	+	0.595741i	$0.796858\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.735508	−	1.27394i	0.0236036	−	0.0408826i	−0.853982	−	0.520302i	$-0.825819\pi$
$971$	0.735508	−	1.27394i	0.0236036	−	0.0408826i	0.877586	+	0.479419i	$0.159153\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.71634	+	16.8292i	−0.310853	+	0.538413i	−0.978547	−	0.206022i	$-0.933948\pi$
$977$	−9.71634	+	16.8292i	−0.310853	+	0.538413i	0.667694	+	0.744436i	$0.267281\pi$
$978$		0			0
$979$	22.8545	−	39.5851i	0.730432	−	1.26515i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.74436			0.247007			0.123503	−	0.992344i	$-0.460587\pi$
$983$	7.74436			0.247007			0.123503	+	0.992344i	$0.460587\pi$
$984$		0			0
$985$	−28.6156			−0.911768
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.1828	+	24.5653i	−0.450986	+	0.781130i
$990$		0			0
$991$	7.23551	+	12.5323i	0.229843	+	0.398101i	0.957762	−	0.287563i	$-0.0928452\pi$
$991$	7.23551	+	12.5323i	0.229843	+	0.398101i	−0.727918	+	0.685664i	$0.759512\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.10243	−	8.83767i	−0.161758	−	0.280173i
$996$		0			0
$997$	55.3097			1.75168			0.875838	−	0.482605i	$-0.160309\pi$
$997$	55.3097			1.75168			0.875838	+	0.482605i	$0.160309\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.f.3313.3		6
3.2	odd	2		1764.2.l.f.961.2		6
7.2	even	3		5292.2.j.d.3529.1		6
7.3	odd	6		5292.2.i.f.2125.3		6
7.4	even	3		5292.2.i.e.2125.1		6
7.5	odd	6		756.2.j.b.505.3		6
7.6	odd	2		5292.2.l.e.3313.1		6
9.4	even	3		5292.2.i.e.1549.1		6
9.5	odd	6		1764.2.i.d.373.1		6
21.2	odd	6		1764.2.j.e.1177.2		6
21.5	even	6		252.2.j.a.169.2	yes	6
21.11	odd	6		1764.2.i.d.1537.1		6
21.17	even	6		1764.2.i.g.1537.3		6
21.20	even	2		1764.2.l.e.961.2		6
28.19	even	6		3024.2.r.j.2017.3		6
63.4	even	3	inner	5292.2.l.f.361.3		6
63.5	even	6		252.2.j.a.85.2	✓	6
63.13	odd	6		5292.2.i.f.1549.3		6
63.23	odd	6		1764.2.j.e.589.2		6
63.31	odd	6		5292.2.l.e.361.1		6
63.32	odd	6		1764.2.l.f.949.2		6
63.40	odd	6		756.2.j.b.253.3		6
63.41	even	6		1764.2.i.g.373.3		6
63.47	even	6		2268.2.a.i.1.3		3
63.58	even	3		5292.2.j.d.1765.1		6
63.59	even	6		1764.2.l.e.949.2		6
63.61	odd	6		2268.2.a.h.1.1		3
84.47	odd	6		1008.2.r.j.673.2		6
252.47	odd	6		9072.2.a.by.1.3		3
252.103	even	6		3024.2.r.j.1009.3		6
252.131	odd	6		1008.2.r.j.337.2		6
252.187	even	6		9072.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.2	✓	6	63.5	even	6
252.2.j.a.169.2	yes	6	21.5	even	6
756.2.j.b.253.3		6	63.40	odd	6
756.2.j.b.505.3		6	7.5	odd	6
1008.2.r.j.337.2		6	252.131	odd	6
1008.2.r.j.673.2		6	84.47	odd	6
1764.2.i.d.373.1		6	9.5	odd	6
1764.2.i.d.1537.1		6	21.11	odd	6
1764.2.i.g.373.3		6	63.41	even	6
1764.2.i.g.1537.3		6	21.17	even	6
1764.2.j.e.589.2		6	63.23	odd	6
1764.2.j.e.1177.2		6	21.2	odd	6
1764.2.l.e.949.2		6	63.59	even	6
1764.2.l.e.961.2		6	21.20	even	2
1764.2.l.f.949.2		6	63.32	odd	6
1764.2.l.f.961.2		6	3.2	odd	2
2268.2.a.h.1.1		3	63.61	odd	6
2268.2.a.i.1.3		3	63.47	even	6
3024.2.r.j.1009.3		6	252.103	even	6
3024.2.r.j.2017.3		6	28.19	even	6
5292.2.i.e.1549.1		6	9.4	even	3
5292.2.i.e.2125.1		6	7.4	even	3
5292.2.i.f.1549.3		6	63.13	odd	6
5292.2.i.f.2125.3		6	7.3	odd	6
5292.2.j.d.1765.1		6	63.58	even	3
5292.2.j.d.3529.1		6	7.2	even	3
5292.2.l.e.361.1		6	63.31	odd	6
5292.2.l.e.3313.1		6	7.6	odd	2
5292.2.l.f.361.3		6	63.4	even	3	inner
5292.2.l.f.3313.3		6	1.1	even	1	trivial
9072.2.a.bv.1.1		3	252.187	even	6
9072.2.a.by.1.3		3	252.47	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.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+2.46050 q^{5} -4.64766 q^{11} +(3.55408 + 6.15585i) q^{13} +(2.25729 + 3.90975i) q^{17} +(2.16372 - 3.74766i) q^{19} -5.86693 q^{23} +1.05408 q^{25} +(-3.48755 + 6.04061i) q^{29} +(-3.69076 + 6.39258i) q^{31} +(0.363327 - 0.629301i) q^{37} +(-0.136673 - 0.236725i) q^{41} +(2.41741 - 4.18708i) q^{43} +(-1.83628 - 3.18054i) q^{47} +(2.52704 + 4.37697i) q^{53} -11.4356 q^{55} +(-4.56654 + 7.90947i) q^{59} +(-6.90856 - 11.9660i) q^{61} +(8.74484 + 15.1465i) q^{65} +(0.663715 - 1.14959i) q^{67} -13.5218 q^{71} +(-2.16372 - 3.74766i) q^{73} +(-3.21780 - 5.57339i) q^{79} +(-0.742705 + 1.28640i) q^{83} +(5.55408 + 9.61996i) q^{85} +(-4.91741 + 8.51721i) q^{89} +(5.32383 - 9.22115i) q^{95} +(-0.246304 + 0.426611i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 4 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{19} - 28 q^{23} - 12 q^{25} + q^{29} - 3 q^{31} + 3 q^{37} - 3 q^{43} - 21 q^{47} + 6 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67}+ \cdots - 9 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 - 4 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^19 - 28 * q^23 - 12 * q^25 + q^29 - 3 * q^31 + 3 * q^37 - 3 * q^43 - 21 * q^47 + 6 * q^53 - 12 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 34 * q^71 - 3 * q^73 + 9 * q^79 - 20 * q^83 + 15 * q^85 - 12 * q^89 + 20 * q^95 - 9 * q^97

Introduction

L-functions

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