LMFDB - Embedded newform 4212.2.i.o.2809.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1405,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1405"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-1,0,3,0,0,0,-1,0,-2,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 4x^{2} + 3x + 9 $ x^4 - x^3 + 4x^2 + 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.1
Root			$1.15139 + 1.99426i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.o.1405.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.15139 - 1.99426i) q^{5} +(1.65139 - 2.86029i) q^{7} +(-1.15139 + 1.99426i) q^{11} +(-0.500000 - 0.866025i) q^{13} -7.60555 q^{17} +3.60555 q^{19} +(1.95416 + 3.38471i) q^{23} +(-0.151388 + 0.262211i) q^{25} +(0.454163 - 0.786634i) q^{29} +(-1.45416 - 2.51868i) q^{31} -7.60555 q^{35} -9.30278 q^{37} +(-1.15139 - 1.99426i) q^{41} +(2.10555 - 3.64692i) q^{43} +(0.802776 - 1.39045i) q^{47} +(-1.95416 - 3.38471i) q^{49} -4.60555 q^{53} +5.30278 q^{55} +(2.19722 + 3.80570i) q^{59} +(-2.25694 + 3.90913i) q^{61} +(-1.15139 + 1.99426i) q^{65} +(-4.45416 - 7.71484i) q^{67} -4.39445 q^{71} -14.8167 q^{73} +(3.80278 + 6.58660i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(8.75694 + 15.1675i) q^{85} +13.1194 q^{89} -3.30278 q^{91} +(-4.15139 - 7.19041i) q^{95} +(6.60555 - 11.4412i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} + 3 q^{7} - q^{11} - 2 q^{13} - 16 q^{17} - 3 q^{23} + 3 q^{25} - 9 q^{29} + 5 q^{31} - 16 q^{35} - 30 q^{37} - q^{41} - 6 q^{43} - 4 q^{47} + 3 q^{49} - 4 q^{53} + 14 q^{55} + 16 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 4 * q - q^5 + 3 * q^7 - q^11 - 2 * q^13 - 16 * q^17 - 3 * q^23 + 3 * q^25 - 9 * q^29 + 5 * q^31 - 16 * q^35 - 30 * q^37 - q^41 - 6 * q^43 - 4 * q^47 + 3 * q^49 - 4 * q^53 + 14 * q^55 + 16 * q^59 + 9 * q^61 - q^65 - 7 * q^67 - 32 * q^71 - 16 * q^73 + 8 * q^77 + 2 * q^79 - 18 * q^83 + 17 * q^85 + 2 * q^89 - 6 * q^91 - 13 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−1.15139	−	1.99426i	−0.514916	−	0.891861i	−0.999850	−	0.0173104i	$-0.994490\pi$
$5$	−1.15139	−	1.99426i	−0.514916	−	0.891861i	0.484934	−	0.874551i	$-0.338844\pi$
$6$		0			0
$7$	1.65139	−	2.86029i	0.624166	−	1.08109i	−0.364536	−	0.931189i	$-0.618772\pi$
$7$	1.65139	−	2.86029i	0.624166	−	1.08109i	0.988702	−	0.149898i	$-0.0478944\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.15139	+	1.99426i	−0.347156	+	0.601293i	−0.985743	−	0.168257i	$-0.946186\pi$
$11$	−1.15139	+	1.99426i	−0.347156	+	0.601293i	0.638587	+	0.769550i	$0.279519\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.60555			−1.84462			−0.922309	−	0.386454i	$-0.873700\pi$
$17$	−7.60555			−1.84462			−0.922309	+	0.386454i	$0.873700\pi$
$18$		0			0
$19$	3.60555			0.827170			0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555			0.827170			0.413585	+	0.910465i	$0.364276\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.95416	+	3.38471i	0.407471	+	0.705761i	0.994606	−	0.103729i	$-0.0330773\pi$
$23$	1.95416	+	3.38471i	0.407471	+	0.705761i	−0.587134	+	0.809489i	$0.699744\pi$
$24$		0			0
$25$	−0.151388	+	0.262211i	−0.0302776	+	0.0524423i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.454163	−	0.786634i	0.0843360	−	0.146074i	−0.820772	−	0.571256i	$-0.806456\pi$
$29$	0.454163	−	0.786634i	0.0843360	−	0.146074i	0.905108	+	0.425182i	$0.139790\pi$
$30$		0			0
$31$	−1.45416	−	2.51868i	−0.261175	−	0.452369i	0.705379	−	0.708830i	$-0.250777\pi$
$31$	−1.45416	−	2.51868i	−0.261175	−	0.452369i	−0.966555	+	0.256461i	$0.917443\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.60555			−1.28557
$36$		0			0
$37$	−9.30278			−1.52937			−0.764683	−	0.644406i	$-0.777105\pi$
$37$	−9.30278			−1.52937			−0.764683	+	0.644406i	$0.777105\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.15139	−	1.99426i	−0.179817	−	0.311451i	0.762001	−	0.647576i	$-0.224217\pi$
$41$	−1.15139	−	1.99426i	−0.179817	−	0.311451i	−0.941818	+	0.336124i	$0.890884\pi$
$42$		0			0
$43$	2.10555	−	3.64692i	0.321094	−	0.556150i	−0.659620	−	0.751599i	$-0.729283\pi$
$43$	2.10555	−	3.64692i	0.321094	−	0.556150i	0.980714	+	0.195449i	$0.0626163\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.802776	−	1.39045i	0.117097	−	0.202818i	−0.801519	−	0.597969i	$-0.795975\pi$
$47$	0.802776	−	1.39045i	0.117097	−	0.202818i	0.918616	+	0.395151i	$0.129308\pi$
$48$		0			0
$49$	−1.95416	−	3.38471i	−0.279166	−	0.483530i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.60555			−0.632621			−0.316311	−	0.948656i	$-0.602444\pi$
$53$	−4.60555			−0.632621			−0.316311	+	0.948656i	$0.602444\pi$
$54$		0			0
$55$	5.30278			0.715026
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.19722	+	3.80570i	0.286054	+	0.495460i	0.972864	−	0.231376i	$-0.0743229\pi$
$59$	2.19722	+	3.80570i	0.286054	+	0.495460i	−0.686810	+	0.726837i	$0.740990\pi$
$60$		0			0
$61$	−2.25694	+	3.90913i	−0.288971	+	0.500513i	−0.973565	−	0.228412i	$-0.926647\pi$
$61$	−2.25694	+	3.90913i	−0.288971	+	0.500513i	0.684593	+	0.728925i	$0.259980\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.15139	+	1.99426i	−0.142812	+	0.247358i
$66$		0			0
$67$	−4.45416	−	7.71484i	−0.544163	−	0.942517i	−0.998659	−	0.0517688i	$-0.983514\pi$
$67$	−4.45416	−	7.71484i	−0.544163	−	0.942517i	0.454496	−	0.890749i	$-0.349819\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.39445			−0.521525			−0.260763	−	0.965403i	$-0.583974\pi$
$71$	−4.39445			−0.521525			−0.260763	+	0.965403i	$0.583974\pi$
$72$		0			0
$73$	−14.8167			−1.73416			−0.867079	−	0.498171i	$-0.834005\pi$
$73$	−14.8167			−1.73416			−0.867079	+	0.498171i	$0.834005\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.80278	+	6.58660i	0.433367	+	0.750613i
$78$		0			0
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	0.892781	+	0.450490i	$0.148751\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		0			0
$85$	8.75694	+	15.1675i	0.949823	+	1.64514i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.1194			1.39066			0.695328	−	0.718692i	$-0.255259\pi$
$89$	13.1194			1.39066			0.695328	+	0.718692i	$0.255259\pi$
$90$		0			0
$91$	−3.30278			−0.346225
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.15139	−	7.19041i	−0.425923	−	0.737721i
$96$		0			0
$97$	6.60555	−	11.4412i	0.670692	−	1.16167i	−0.307016	−	0.951704i	$-0.599331\pi$
$97$	6.60555	−	11.4412i	0.670692	−	1.16167i	0.977708	−	0.209968i	$-0.0673361\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.80278	+	11.7828i	−0.676901	+	1.17243i	0.299008	+	0.954251i	$0.403344\pi$
$101$	−6.80278	+	11.7828i	−0.676901	+	1.17243i	−0.975909	+	0.218177i	$0.929989\pi$
$102$		0			0
$103$	8.55971	+	14.8259i	0.843414	+	1.46084i	0.886992	+	0.461785i	$0.152791\pi$
$103$	8.55971	+	14.8259i	0.843414	+	1.46084i	−0.0435779	+	0.999050i	$0.513876\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.09167			−0.492231			−0.246115	−	0.969241i	$-0.579154\pi$
$107$	−5.09167			−0.492231			−0.246115	+	0.969241i	$0.579154\pi$
$108$		0			0
$109$	−5.39445			−0.516694			−0.258347	−	0.966052i	$-0.583178\pi$
$109$	−5.39445			−0.516694			−0.258347	+	0.966052i	$0.583178\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$	4.50000	−	7.79423i	0.419627	−	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.5597	+	21.7541i	−1.15135	+	1.99419i
$120$		0			0
$121$	2.84861	+	4.93394i	0.258965	+	0.448540i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.8167			−0.967471
$126$		0			0
$127$	−8.60555			−0.763619			−0.381810	−	0.924241i	$-0.624699\pi$
$127$	−8.60555			−0.763619			−0.381810	+	0.924241i	$0.624699\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.0597	+	19.1560i	0.966292	+	1.67367i	0.706104	+	0.708108i	$0.250451\pi$
$131$	11.0597	+	19.1560i	0.966292	+	1.67367i	0.260188	+	0.965558i	$0.416215\pi$
$132$		0			0
$133$	5.95416	−	10.3129i	0.516291	−	0.894243i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.15139	+	1.99426i	−0.0983697	+	0.170381i	−0.911010	−	0.412384i	$-0.864696\pi$
$137$	−1.15139	+	1.99426i	−0.0983697	+	0.170381i	0.812640	+	0.582766i	$0.198029\pi$
$138$		0			0
$139$	−2.15139	−	3.72631i	−0.182478	−	0.316062i	0.760246	−	0.649636i	$-0.225079\pi$
$139$	−2.15139	−	3.72631i	−0.182478	−	0.316062i	−0.942724	+	0.333574i	$0.891745\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.30278			0.192568
$144$		0			0
$145$	−2.09167			−0.173704
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.90833	−	6.76942i	−0.320183	−	0.554573i	0.660343	−	0.750964i	$-0.270411\pi$
$149$	−3.90833	−	6.76942i	−0.320183	−	0.554573i	−0.980526	+	0.196392i	$0.937078\pi$
$150$		0			0
$151$	−3.05971	+	5.29958i	−0.248996	+	0.431274i	−0.963248	−	0.268615i	$-0.913434\pi$
$151$	−3.05971	+	5.29958i	−0.248996	+	0.431274i	0.714251	+	0.699889i	$0.246767\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.34861	+	5.79997i	−0.268967	+	0.465865i
$156$		0			0
$157$	9.95416	+	17.2411i	0.794429	+	1.37599i	0.923201	+	0.384317i	$0.125563\pi$
$157$	9.95416	+	17.2411i	0.794429	+	1.37599i	−0.128773	+	0.991674i	$0.541104\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.9083			1.01732
$162$		0			0
$163$	23.6333			1.85110			0.925552	−	0.378621i	$-0.123602\pi$
$163$	23.6333			1.85110			0.925552	+	0.378621i	$0.123602\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4083	−	19.7598i	−0.882803	−	1.52906i	−0.848211	−	0.529658i	$-0.822320\pi$
$167$	−11.4083	−	19.7598i	−0.882803	−	1.52906i	−0.0345914	−	0.999402i	$-0.511013\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.71110	−	13.3560i	0.586264	−	1.01544i	−0.408452	−	0.912780i	$-0.633931\pi$
$173$	7.71110	−	13.3560i	0.586264	−	1.01544i	0.994717	−	0.102660i	$-0.0327353\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.0377964	+	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.90833			0.292122			0.146061	−	0.989276i	$-0.453340\pi$
$179$	3.90833			0.292122			0.146061	+	0.989276i	$0.453340\pi$
$180$		0			0
$181$	−8.81665			−0.655337			−0.327668	−	0.944793i	$-0.606263\pi$
$181$	−8.81665			−0.655337			−0.327668	+	0.944793i	$0.606263\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.7111	+	18.5522i	0.787496	+	1.36398i
$186$		0			0
$187$	8.75694	−	15.1675i	0.640371	−	1.10915i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.34861	+	5.79997i	−0.242297	+	0.419671i	−0.961368	−	0.275266i	$-0.911234\pi$
$191$	−3.34861	+	5.79997i	−0.242297	+	0.419671i	0.719071	+	0.694936i	$0.244568\pi$
$192$		0			0
$193$	5.10555	+	8.84307i	0.367506	+	0.636538i	0.989175	−	0.146741i	$-0.0468785\pi$
$193$	5.10555	+	8.84307i	0.367506	+	0.636538i	−0.621669	+	0.783280i	$0.713545\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$		0			0
$199$	−10.2111			−0.723846			−0.361923	−	0.932208i	$-0.617880\pi$
$199$	−10.2111			−0.723846			−0.361923	+	0.932208i	$0.617880\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.50000	−	2.59808i	−0.105279	−	0.182349i
$204$		0			0
$205$	−2.65139	+	4.59234i	−0.185181	+	0.320743i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.15139	+	7.19041i	−0.287157	+	0.497371i
$210$		0			0
$211$	3.84861	+	6.66599i	0.264949	+	0.458906i	0.967550	−	0.252678i	$-0.0813114\pi$
$211$	3.84861	+	6.66599i	0.264949	+	0.458906i	−0.702601	+	0.711584i	$0.747978\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.69722			−0.661345
$216$		0			0
$217$	−9.60555			−0.652067
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.80278	+	6.58660i	0.255802	+	0.443063i
$222$		0			0
$223$	−10.9083	+	18.8938i	−0.730476	+	1.26522i	0.226205	+	0.974080i	$0.427368\pi$
$223$	−10.9083	+	18.8938i	−0.730476	+	1.26522i	−0.956680	+	0.291141i	$0.905965\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.69722	+	6.40378i	−0.245393	+	0.425034i	−0.962242	−	0.272195i	$-0.912250\pi$
$227$	−3.69722	+	6.40378i	−0.245393	+	0.425034i	0.716849	+	0.697229i	$0.245584\pi$
$228$		0			0
$229$	11.5597	+	20.0220i	0.763887	+	1.32309i	0.940833	+	0.338871i	$0.110045\pi$
$229$	11.5597	+	20.0220i	0.763887	+	1.32309i	−0.176945	+	0.984221i	$0.556622\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0917			−0.923176			−0.461588	−	0.887094i	$-0.652720\pi$
$233$	−14.0917			−0.923176			−0.461588	+	0.887094i	$0.652720\pi$
$234$		0			0
$235$	−3.69722			−0.241180
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.86249	−	10.1541i	−0.379213	−	0.656816i	0.611735	−	0.791063i	$-0.290472\pi$
$239$	−5.86249	−	10.1541i	−0.379213	−	0.656816i	−0.990948	+	0.134247i	$0.957139\pi$
$240$		0			0
$241$	−0.756939	+	1.31106i	−0.0487587	+	0.0844526i	−0.889375	−	0.457179i	$-0.848860\pi$
$241$	−0.756939	+	1.31106i	−0.0487587	+	0.0844526i	0.840616	+	0.541632i	$0.182193\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	+	7.79423i	−0.287494	+	0.497955i
$246$		0			0
$247$	−1.80278	−	3.12250i	−0.114708	−	0.198680i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3028			−1.09214			−0.546071	−	0.837739i	$-0.683877\pi$
$251$	−17.3028			−1.09214			−0.546071	+	0.837739i	$0.683877\pi$
$252$		0			0
$253$	−9.00000			−0.565825
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.54584	+	9.60567i	0.345940	+	0.599185i	0.985524	−	0.169536i	$-0.0542268\pi$
$257$	5.54584	+	9.60567i	0.345940	+	0.599185i	−0.639584	+	0.768721i	$0.720893\pi$
$258$		0			0
$259$	−15.3625	+	26.6086i	−0.954579	+	1.65338i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.4680	+	23.3273i	−0.830475	+	1.43842i	0.0671870	+	0.997740i	$0.478598\pi$
$263$	−13.4680	+	23.3273i	−0.830475	+	1.43842i	−0.897662	+	0.440685i	$0.854736\pi$
$264$		0			0
$265$	5.30278	+	9.18468i	0.325747	+	0.564210i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.1194			−0.799906			−0.399953	−	0.916536i	$-0.630974\pi$
$269$	−13.1194			−0.799906			−0.399953	+	0.916536i	$0.630974\pi$
$270$		0			0
$271$	−14.8167			−0.900048			−0.450024	−	0.893017i	$-0.648585\pi$
$271$	−14.8167			−0.900048			−0.450024	+	0.893017i	$0.648585\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.348612	−	0.603814i	−0.0210221	−	0.0364114i
$276$		0			0
$277$	8.69722	−	15.0640i	0.522566	−	0.905110i	−0.477090	−	0.878855i	$-0.658308\pi$
$277$	8.69722	−	15.0640i	0.522566	−	0.905110i	0.999655	−	0.0262555i	$-0.00835835\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.55971	−	11.3618i	0.391320	−	0.677786i	−0.601304	−	0.799020i	$-0.705352\pi$
$281$	6.55971	−	11.3618i	0.391320	−	0.677786i	0.992624	+	0.121235i	$0.0386853\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.60555			−0.448941
$288$		0			0
$289$	40.8444			2.40261
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.697224	−	1.20763i	−0.0407323	−	0.0705504i	0.844941	−	0.534860i	$-0.179636\pi$
$293$	−0.697224	−	1.20763i	−0.0407323	−	0.0705504i	−0.885673	+	0.464310i	$0.846302\pi$
$294$		0			0
$295$	5.05971	−	8.76368i	0.294588	−	0.510241i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.95416	−	3.38471i	0.113012	−	0.195743i
$300$		0			0
$301$	−6.95416	−	12.0450i	−0.400831	−	0.694260i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.3944			0.595184
$306$		0			0
$307$	−19.2111			−1.09644			−0.548218	−	0.836336i	$-0.684694\pi$
$307$	−19.2111			−1.09644			−0.548218	+	0.836336i	$0.684694\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.4542	−	26.7674i	−0.876325	−	1.51784i	−0.855344	−	0.518060i	$-0.826654\pi$
$311$	−15.4542	−	26.7674i	−0.876325	−	1.51784i	−0.0209809	−	0.999780i	$-0.506679\pi$
$312$		0			0
$313$	−11.9222	+	20.6499i	−0.673883	+	1.16720i	0.302911	+	0.953019i	$0.402041\pi$
$313$	−11.9222	+	20.6499i	−0.673883	+	1.16720i	−0.976794	+	0.214181i	$0.931292\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.36249	+	12.7522i	−0.413519	+	0.716235i	−0.995272	−	0.0971302i	$-0.969034\pi$
$317$	−7.36249	+	12.7522i	−0.413519	+	0.716235i	0.581753	+	0.813365i	$0.302367\pi$
$318$		0			0
$319$	1.04584	+	1.81144i	0.0585556	+	0.101421i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−27.4222			−1.52581
$324$		0			0
$325$	0.302776			0.0167950
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.65139	−	4.59234i	−0.146176	−	0.253184i
$330$		0			0
$331$	14.3167	−	24.7972i	0.786914	−	1.36298i	−0.140934	−	0.990019i	$-0.545011\pi$
$331$	14.3167	−	24.7972i	0.786914	−	1.36298i	0.927849	−	0.372957i	$-0.121656\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.2569	+	17.7655i	−0.560396	+	0.970635i
$336$		0			0
$337$	−17.0139	−	29.4689i	−0.926805	−	1.60527i	−0.788632	−	0.614865i	$-0.789210\pi$
$337$	−17.0139	−	29.4689i	−0.926805	−	1.60527i	−0.138173	−	0.990408i	$-0.544123\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.69722			0.362675
$342$		0			0
$343$	10.2111			0.551348
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	5.66527	−	9.81253i	0.303255	−	0.525253i	−0.673616	−	0.739081i	$-0.735260\pi$
$349$	5.66527	−	9.81253i	0.303255	−	0.525253i	0.976871	+	0.213828i	$0.0685934\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.71110	−	8.15987i	0.250747	−	0.434306i	−0.712985	−	0.701179i	$-0.752657\pi$
$353$	4.71110	−	8.15987i	0.250747	−	0.434306i	0.963732	+	0.266873i	$0.0859905\pi$
$354$		0			0
$355$	5.05971	+	8.76368i	0.268542	+	0.465128i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.09167			−0.110394			−0.0551971	−	0.998475i	$-0.517579\pi$
$359$	−2.09167			−0.110394			−0.0551971	+	0.998475i	$0.517579\pi$
$360$		0			0
$361$	−6.00000			−0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	17.0597	+	29.5483i	0.892946	+	1.54663i
$366$		0			0
$367$	12.9222	−	22.3819i	0.674534	−	1.16833i	−0.302071	−	0.953285i	$-0.597678\pi$
$367$	12.9222	−	22.3819i	0.674534	−	1.16833i	0.976605	−	0.215041i	$-0.0689886\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.60555	+	13.1732i	−0.394861	+	0.683919i
$372$		0			0
$373$	1.44029	+	2.49465i	0.0745751	+	0.129168i	0.900901	−	0.434024i	$-0.142907\pi$
$373$	1.44029	+	2.49465i	0.0745751	+	0.129168i	−0.826326	+	0.563192i	$0.809573\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.908327			−0.0467812
$378$		0			0
$379$	30.3944			1.56126			0.780629	−	0.624995i	$-0.214899\pi$
$379$	30.3944			1.56126			0.780629	+	0.624995i	$0.214899\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.591673	−	1.02481i	−0.0302331	−	0.0523652i	0.850513	−	0.525954i	$-0.176292\pi$
$383$	−0.591673	−	1.02481i	−0.0302331	−	0.0523652i	−0.880746	+	0.473589i	$0.842958\pi$
$384$		0			0
$385$	8.75694	−	15.1675i	0.446295	−	0.773006i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.2569	+	17.7655i	−0.520047	+	0.900749i	0.479681	+	0.877443i	$0.340752\pi$
$389$	−10.2569	+	17.7655i	−0.520047	+	0.900749i	−0.999728	+	0.0233056i	$0.992581\pi$
$390$		0			0
$391$	−14.8625	−	25.7426i	−0.751628	−	1.30186i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.30278			−0.115865
$396$		0			0
$397$	−27.2389			−1.36708			−0.683540	−	0.729913i	$-0.739560\pi$
$397$	−27.2389			−1.36708			−0.683540	+	0.729913i	$0.739560\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.3167	−	21.3331i	−0.615064	−	1.06532i	−0.990373	−	0.138423i	$-0.955797\pi$
$401$	−12.3167	−	21.3331i	−0.615064	−	1.06532i	0.375309	−	0.926900i	$-0.377537\pi$
$402$		0			0
$403$	−1.45416	+	2.51868i	−0.0724370	+	0.125465i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.7111	−	18.5522i	0.530930	−	0.919597i
$408$		0			0
$409$	−4.31665	−	7.47666i	−0.213445	−	0.369697i	0.739346	−	0.673326i	$-0.235135\pi$
$409$	−4.31665	−	7.47666i	−0.213445	−	0.369697i	−0.952790	+	0.303629i	$0.901802\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.5139			0.714181
$414$		0			0
$415$	20.7250			1.01735
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.2111	−	21.1503i	−0.596551	−	1.03326i	−0.993326	−	0.115341i	$-0.963204\pi$
$419$	−12.2111	−	21.1503i	−0.596551	−	1.03326i	0.396775	−	0.917916i	$-0.370129\pi$
$420$		0			0
$421$	9.60555	−	16.6373i	0.468146	−	0.810853i	−0.531191	−	0.847252i	$-0.678255\pi$
$421$	9.60555	−	16.6373i	0.468146	−	0.810853i	0.999337	+	0.0363993i	$0.0115888\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.15139	−	1.99426i	0.0558505	−	0.0967359i
$426$		0			0
$427$	7.45416	+	12.9110i	0.360732	+	0.624807i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.3944			−0.645188			−0.322594	−	0.946537i	$-0.604555\pi$
$431$	−13.3944			−0.645188			−0.322594	+	0.946537i	$0.604555\pi$
$432$		0			0
$433$	−31.4222			−1.51005			−0.755027	−	0.655693i	$-0.772376\pi$
$433$	−31.4222			−1.51005			−0.755027	+	0.655693i	$0.772376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.04584	+	12.2037i	0.337048	+	0.583784i
$438$		0			0
$439$	−2.71110	+	4.69577i	−0.129394	+	0.224117i	−0.923442	−	0.383738i	$-0.874636\pi$
$439$	−2.71110	+	4.69577i	−0.129394	+	0.224117i	0.794048	+	0.607855i	$0.207970\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.6194	−	25.3216i	0.694590	−	1.20307i	−0.275729	−	0.961236i	$-0.588919\pi$
$443$	14.6194	−	25.3216i	0.694590	−	1.20307i	0.970319	−	0.241830i	$-0.0777475\pi$
$444$		0			0
$445$	−15.1056	−	26.1636i	−0.716072	−	1.24027i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000			1.27421			0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000			1.27421			0.637104	+	0.770778i	$0.280132\pi$
$450$		0			0
$451$	5.30278			0.249698
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.80278	+	6.58660i	0.178277	+	0.308785i
$456$		0			0
$457$	1.30278	−	2.25647i	0.0609413	−	0.105553i	−0.833945	−	0.551847i	$-0.813923\pi$
$457$	1.30278	−	2.25647i	0.0609413	−	0.105553i	0.894886	+	0.446294i	$0.147256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.59167	−	6.22096i	0.167281	−	0.289739i	−0.770182	−	0.637824i	$-0.779835\pi$
$461$	3.59167	−	6.22096i	0.167281	−	0.289739i	0.937463	+	0.348085i	$0.113168\pi$
$462$		0			0
$463$	−1.21110	−	2.09769i	−0.0562847	−	0.0974880i	0.836510	−	0.547951i	$-0.184592\pi$
$463$	−1.21110	−	2.09769i	−0.0562847	−	0.0974880i	−0.892795	+	0.450463i	$0.851259\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.4222			−1.68542			−0.842709	−	0.538369i	$-0.819041\pi$
$467$	−36.4222			−1.68542			−0.842709	+	0.538369i	$0.819041\pi$
$468$		0			0
$469$	−29.4222			−1.35859
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.84861	+	8.39804i	0.222939	+	0.386142i
$474$		0			0
$475$	−0.545837	+	0.945417i	−0.0250447	+	0.0433787i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.1194	−	33.1158i	0.873589	−	1.51310i	0.0153308	−	0.999882i	$-0.495120\pi$
$479$	19.1194	−	33.1158i	0.873589	−	1.51310i	0.858258	−	0.513218i	$-0.171547\pi$
$480$		0			0
$481$	4.65139	+	8.05644i	0.212085	+	0.367342i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.4222			−1.38140
$486$		0			0
$487$	−10.2111			−0.462709			−0.231355	−	0.972869i	$-0.574316\pi$
$487$	−10.2111			−0.462709			−0.231355	+	0.972869i	$0.574316\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.2708	+	19.5216i	0.508645	+	0.880999i	0.999950	+	0.0100112i	$0.00318670\pi$
$491$	11.2708	+	19.5216i	0.508645	+	0.880999i	−0.491305	+	0.870988i	$0.663480\pi$
$492$		0			0
$493$	−3.45416	+	5.98279i	−0.155568	+	0.269451i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.25694	+	12.5694i	−0.325518	+	0.563814i
$498$		0			0
$499$	4.16527	+	7.21445i	0.186463	+	0.322963i	0.944069	−	0.329749i	$-0.106964\pi$
$499$	4.16527	+	7.21445i	0.186463	+	0.322963i	−0.757606	+	0.652713i	$0.773631\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.1472			1.21043			0.605217	−	0.796061i	$-0.293087\pi$
$503$	27.1472			1.21043			0.605217	+	0.796061i	$0.293087\pi$
$504$		0			0
$505$	31.3305			1.39419
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.6791	+	28.8891i	0.739290	+	1.28049i	0.952815	+	0.303550i	$0.0981721\pi$
$509$	16.6791	+	28.8891i	0.739290	+	1.28049i	−0.213525	+	0.976938i	$0.568495\pi$
$510$		0			0
$511$	−24.4680	+	42.3799i	−1.08240	+	1.87478i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.7111	−	34.1406i	0.868575	−	1.50442i
$516$		0			0
$517$	1.84861	+	3.20189i	0.0813019	+	0.140819i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	40.6056			1.77896			0.889481	−	0.456973i	$-0.151066\pi$
$521$	40.6056			1.77896			0.889481	+	0.456973i	$0.151066\pi$
$522$		0			0
$523$	−13.2111			−0.577681			−0.288841	−	0.957377i	$-0.593270\pi$
$523$	−13.2111			−0.577681			−0.288841	+	0.957377i	$0.593270\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11.0597	+	19.1560i	0.481769	+	0.834448i
$528$		0			0
$529$	3.86249	−	6.69003i	0.167934	−	0.290871i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.15139	+	1.99426i	−0.0498721	+	0.0863811i
$534$		0			0
$535$	5.86249	+	10.1541i	0.253458	+	0.439001i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.00000			0.387657
$540$		0			0
$541$	−8.18335			−0.351830			−0.175915	−	0.984405i	$-0.556288\pi$
$541$	−8.18335			−0.351830			−0.175915	+	0.984405i	$0.556288\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.21110	+	10.7579i	0.266054	+	0.460820i
$546$		0			0
$547$	−4.90833	+	8.50147i	−0.209865	+	0.363497i	−0.951672	−	0.307117i	$-0.900636\pi$
$547$	−4.90833	+	8.50147i	−0.209865	+	0.363497i	0.741807	+	0.670614i	$0.233969\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.63751	−	2.83625i	0.0697603	−	0.120828i
$552$		0			0
$553$	−1.65139	−	2.86029i	−0.0702242	−	0.121632i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.0917			0.597083			0.298542	−	0.954397i	$-0.403500\pi$
$557$	14.0917			0.597083			0.298542	+	0.954397i	$0.403500\pi$
$558$		0			0
$559$	−4.21110			−0.178111
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.2111	−	36.7387i	−0.893941	−	1.54835i	−0.835109	−	0.550084i	$-0.814596\pi$
$563$	−21.2111	−	36.7387i	−0.893941	−	1.54835i	−0.0588321	−	0.998268i	$-0.518738\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.04584	−	7.00759i	0.169610	−	0.293774i	−0.768673	−	0.639642i	$-0.779082\pi$
$569$	4.04584	−	7.00759i	0.169610	−	0.293774i	0.938283	+	0.345869i	$0.112416\pi$
$570$		0			0
$571$	9.01388	+	15.6125i	0.377219	+	0.653363i	0.990656	−	0.136381i	$-0.0435471\pi$
$571$	9.01388	+	15.6125i	0.377219	+	0.653363i	−0.613437	+	0.789743i	$0.710214\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.18335			−0.0493489
$576$		0			0
$577$	16.0278			0.667244			0.333622	−	0.942707i	$-0.391729\pi$
$577$	16.0278			0.667244			0.333622	+	0.942707i	$0.391729\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	14.8625	+	25.7426i	0.616600	+	1.06798i
$582$		0			0
$583$	5.30278	−	9.18468i	0.219619	−	0.380390i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.2569	+	17.7655i	−0.423349	+	0.733262i	−0.996265	−	0.0863521i	$-0.972479\pi$
$587$	−10.2569	+	17.7655i	−0.423349	+	0.733262i	0.572915	+	0.819614i	$0.305812\pi$
$588$		0			0
$589$	−5.24306	−	9.08125i	−0.216037	−	0.374186i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	27.9083			1.14606			0.573029	−	0.819535i	$-0.305768\pi$
$593$	27.9083			1.14606			0.573029	+	0.819535i	$0.305768\pi$
$594$		0			0
$595$	57.8444			2.37139
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.31665	−	10.9408i	−0.258091	−	0.447028i	0.707639	−	0.706574i	$-0.249760\pi$
$599$	−6.31665	−	10.9408i	−0.258091	−	0.447028i	−0.965731	+	0.259546i	$0.916427\pi$
$600$		0			0
$601$	2.90833	−	5.03737i	0.118633	−	0.205479i	−0.800593	−	0.599208i	$-0.795482\pi$
$601$	2.90833	−	5.03737i	0.118633	−	0.205479i	0.919226	+	0.393730i	$0.128815\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.55971	−	11.3618i	0.266690	−	0.461921i
$606$		0			0
$607$	−5.04584	−	8.73965i	−0.204804	−	0.354731i	0.745266	−	0.666767i	$-0.232322\pi$
$607$	−5.04584	−	8.73965i	−0.204804	−	0.354731i	−0.950070	+	0.312036i	$0.898989\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.60555			−0.0649537
$612$		0			0
$613$	11.6972			0.472446			0.236223	−	0.971699i	$-0.424090\pi$
$613$	11.6972			0.472446			0.236223	+	0.971699i	$0.424090\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.95416	−	3.38471i	−0.0786717	−	0.136263i	0.824005	−	0.566582i	$-0.191735\pi$
$617$	−1.95416	−	3.38471i	−0.0786717	−	0.136263i	−0.902677	+	0.430319i	$0.858401\pi$
$618$		0			0
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	−0.971718	−	0.236143i	$-0.924117\pi$
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	0.690365	+	0.723462i	$0.257450\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.6653	−	37.5253i	0.868001	−	1.50342i
$624$		0			0
$625$	13.2111	+	22.8823i	0.528444	+	0.915292i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	70.7527			2.82110
$630$		0			0
$631$	−44.3305			−1.76477			−0.882385	−	0.470528i	$-0.844064\pi$
$631$	−44.3305			−1.76477			−0.882385	+	0.470528i	$0.844064\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.90833	+	17.1617i	0.393200	+	0.681042i
$636$		0			0
$637$	−1.95416	+	3.38471i	−0.0774268	+	0.134107i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.2569	−	17.7655i	0.405125	−	0.701697i	−0.589211	−	0.807979i	$-0.700561\pi$
$641$	10.2569	−	17.7655i	0.405125	−	0.701697i	0.994336	+	0.106282i	$0.0338948\pi$
$642$		0			0
$643$	3.36249	+	5.82400i	0.132604	+	0.229676i	0.924679	−	0.380746i	$-0.124333\pi$
$643$	3.36249	+	5.82400i	0.132604	+	0.229676i	−0.792076	+	0.610423i	$0.791000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−48.3583			−1.90116			−0.950580	−	0.310479i	$-0.899511\pi$
$647$	−48.3583			−1.90116			−0.950580	+	0.310479i	$0.899511\pi$
$648$		0			0
$649$	−10.1194			−0.397222
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.7847	−	39.4643i	−0.891634	−	1.54436i	−0.837916	−	0.545799i	$-0.816226\pi$
$653$	−22.7847	−	39.4643i	−0.891634	−	1.54436i	−0.0537181	−	0.998556i	$-0.517107\pi$
$654$		0			0
$655$	25.4680	−	44.1119i	0.995119	−	1.72360i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.45416	−	11.1789i	0.251419	−	0.435470i	−0.712498	−	0.701674i	$-0.752436\pi$
$659$	6.45416	−	11.1789i	0.251419	−	0.435470i	0.963917	+	0.266204i	$0.0857696\pi$
$660$		0			0
$661$	2.69722	+	4.67173i	0.104910	+	0.181709i	0.913701	−	0.406386i	$-0.133211\pi$
$661$	2.69722	+	4.67173i	0.104910	+	0.181709i	−0.808792	+	0.588095i	$0.799878\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.4222			−1.06339
$666$		0			0
$667$	3.55004			0.137458
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.19722	−	9.00186i	−0.200637	−	0.347513i
$672$		0			0
$673$	2.10555	−	3.64692i	0.0811630	−	0.140579i	−0.822587	−	0.568640i	$-0.807470\pi$
$673$	2.10555	−	3.64692i	0.0811630	−	0.140579i	0.903750	+	0.428061i	$0.140803\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−24.8764	+	43.0871i	−0.956077	+	1.65597i	−0.224191	+	0.974545i	$0.571974\pi$
$677$	−24.8764	+	43.0871i	−0.956077	+	1.65597i	−0.731885	+	0.681428i	$0.761359\pi$
$678$		0			0
$679$	−21.8167	−	37.7876i	−0.837246	−	1.45015i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27.4861			1.05173			0.525864	−	0.850569i	$-0.323742\pi$
$683$	27.4861			1.05173			0.525864	+	0.850569i	$0.323742\pi$
$684$		0			0
$685$	5.30278			0.202609
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.30278	+	3.98852i	0.0877288	+	0.151951i
$690$		0			0
$691$	−21.7569	+	37.6841i	−0.827673	+	1.43357i	0.0721862	+	0.997391i	$0.477002\pi$
$691$	−21.7569	+	37.6841i	−0.827673	+	1.43357i	−0.899859	+	0.436180i	$0.856331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.95416	+	8.58086i	−0.187922	+	0.325491i
$696$		0			0
$697$	8.75694	+	15.1675i	0.331693	+	0.574509i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.5139			0.548182			0.274091	−	0.961704i	$-0.411623\pi$
$701$	14.5139			0.548182			0.274091	+	0.961704i	$0.411623\pi$
$702$		0			0
$703$	−33.5416			−1.26505
$704$		0			0
$705$		0			0
$706$		0			0
$707$	22.4680	+	38.9158i	0.844998	+	1.46358i
$708$		0			0
$709$	6.84861	−	11.8621i	0.257205	−	0.445492i	−0.708287	−	0.705925i	$-0.750532\pi$
$709$	6.84861	−	11.8621i	0.257205	−	0.445492i	0.965492	+	0.260432i	$0.0838651\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.68335	−	9.84384i	0.212843	−	0.368655i
$714$		0			0
$715$	−2.65139	−	4.59234i	−0.0991563	−	0.171744i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.81665			0.179631			0.0898154	−	0.995958i	$-0.471372\pi$
$719$	4.81665			0.179631			0.0898154	+	0.995958i	$0.471372\pi$
$720$		0			0
$721$	56.5416			2.10572
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.137510	+	0.238174i	0.00510698	+	0.00884555i
$726$		0			0
$727$	−16.4542	+	28.4994i	−0.610251	+	1.05699i	0.380947	+	0.924597i	$0.375598\pi$
$727$	−16.4542	+	28.4994i	−0.610251	+	1.05699i	−0.991198	+	0.132389i	$0.957735\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.0139	+	27.7369i	−0.592295	+	1.02588i
$732$		0			0
$733$	12.2889	+	21.2850i	0.453901	+	0.786179i	0.998624	−	0.0524365i	$-0.0166987\pi$
$733$	12.2889	+	21.2850i	0.453901	+	0.786179i	−0.544723	+	0.838616i	$0.683365\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20.5139			0.755638
$738$		0			0
$739$	26.4222			0.971957			0.485978	−	0.873971i	$-0.338463\pi$
$739$	26.4222			0.971957			0.485978	+	0.873971i	$0.338463\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.5000	−	38.9711i	−0.825445	−	1.42971i	−0.901579	−	0.432615i	$-0.857591\pi$
$743$	−22.5000	−	38.9711i	−0.825445	−	1.42971i	0.0761338	−	0.997098i	$-0.475742\pi$
$744$		0			0
$745$	−9.00000	+	15.5885i	−0.329734	+	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.40833	+	14.5636i	−0.307234	+	0.532144i
$750$		0			0
$751$	6.04584	+	10.4717i	0.220616	+	0.382118i	0.954995	−	0.296622i	$-0.0958600\pi$
$751$	6.04584	+	10.4717i	0.220616	+	0.382118i	−0.734379	+	0.678739i	$0.762527\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.0917			0.512848
$756$		0			0
$757$	31.9361			1.16074			0.580368	−	0.814354i	$-0.302909\pi$
$757$	31.9361			1.16074			0.580368	+	0.814354i	$0.302909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.07359	−	15.7159i	−0.328917	−	0.569702i	0.653380	−	0.757030i	$-0.273350\pi$
$761$	−9.07359	−	15.7159i	−0.328917	−	0.569702i	−0.982297	+	0.187328i	$0.940017\pi$
$762$		0			0
$763$	−8.90833	+	15.4297i	−0.322503	+	0.558592i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.19722	−	3.80570i	0.0793372	−	0.137416i
$768$		0			0
$769$	−16.0000	−	27.7128i	−0.576975	−	0.999350i	−0.995824	−	0.0912938i	$-0.970900\pi$
$769$	−16.0000	−	27.7128i	−0.576975	−	0.999350i	0.418849	−	0.908056i	$-0.362434\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.81665			0.173243			0.0866215	−	0.996241i	$-0.472393\pi$
$773$	4.81665			0.173243			0.0866215	+	0.996241i	$0.472393\pi$
$774$		0			0
$775$	0.880571			0.0316310
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.15139	−	7.19041i	−0.148739	−	0.257623i
$780$		0			0
$781$	5.05971	−	8.76368i	0.181051	−	0.313589i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.9222	−	39.7024i	0.818129	−	1.41704i
$786$		0			0
$787$	−18.7569	−	32.4880i	−0.668613	−	1.15807i	−0.978292	−	0.207230i	$-0.933555\pi$
$787$	−18.7569	−	32.4880i	−0.668613	−	1.15807i	0.309680	−	0.950841i	$-0.399778\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.51388			0.160293
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.44029	−	14.6190i	−0.298970	−	0.517832i	0.676930	−	0.736047i	$-0.263310\pi$
$797$	−8.44029	−	14.6190i	−0.298970	−	0.517832i	−0.975901	+	0.218215i	$0.929976\pi$
$798$		0			0
$799$	−6.10555	+	10.5751i	−0.215999	+	0.374121i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.0597	−	29.5483i	0.602024	−	1.04274i
$804$		0			0
$805$	−14.8625	−	25.7426i	−0.523834	−	0.907307i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.8444			1.50633			0.753165	−	0.657832i	$-0.228526\pi$
$809$	42.8444			1.50633			0.753165	+	0.657832i	$0.228526\pi$
$810$		0			0
$811$	−10.6972			−0.375630			−0.187815	−	0.982204i	$-0.560141\pi$
$811$	−10.6972			−0.375630			−0.187815	+	0.982204i	$0.560141\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.2111	−	47.1310i	−0.953163	−	1.65093i
$816$		0			0
$817$	7.59167	−	13.1492i	0.265599	−	0.460031i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.59167	−	16.6133i	0.334752	−	0.579807i	−0.648685	−	0.761057i	$-0.724681\pi$
$821$	9.59167	−	16.6133i	0.334752	−	0.579807i	0.983437	+	0.181250i	$0.0580142\pi$
$822$		0			0
$823$	−18.4083	−	31.8842i	−0.641674	−	1.11141i	−0.985059	−	0.172217i	$-0.944907\pi$
$823$	−18.4083	−	31.8842i	−0.641674	−	1.11141i	0.343385	−	0.939195i	$-0.388426\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.97224			0.138128			0.0690642	−	0.997612i	$-0.477999\pi$
$827$	3.97224			0.138128			0.0690642	+	0.997612i	$0.477999\pi$
$828$		0			0
$829$	28.5139			0.990328			0.495164	−	0.868800i	$-0.335108\pi$
$829$	28.5139			0.990328			0.495164	+	0.868800i	$0.335108\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.8625	+	25.7426i	0.514955	+	0.891928i
$834$		0			0
$835$	−26.2708	+	45.5024i	−0.909139	+	1.57468i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.80278	−	16.9789i	0.338429	−	0.586177i	−0.645708	−	0.763584i	$-0.723438\pi$
$839$	9.80278	−	16.9789i	0.338429	−	0.586177i	0.984137	+	0.177407i	$0.0567710\pi$
$840$		0			0
$841$	14.0875	+	24.4002i	0.485775	+	0.841387i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.30278			0.0792179
$846$		0			0
$847$	18.8167			0.646548
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.1791	−	31.4872i	−0.623173	−	1.07937i
$852$		0			0
$853$	−16.0736	+	27.8403i	−0.550349	+	0.953233i	0.447900	+	0.894084i	$0.352172\pi$
$853$	−16.0736	+	27.8403i	−0.550349	+	0.953233i	−0.998249	+	0.0591493i	$0.981161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.86249	+	10.1541i	−0.200259	+	0.346859i	−0.948612	−	0.316442i	$-0.897512\pi$
$857$	−5.86249	+	10.1541i	−0.200259	+	0.346859i	0.748353	+	0.663301i	$0.230845\pi$
$858$		0			0
$859$	−1.34861	−	2.33586i	−0.0460141	−	0.0796987i	0.842101	−	0.539320i	$-0.181319\pi$
$859$	−1.34861	−	2.33586i	−0.0460141	−	0.0796987i	−0.888115	+	0.459621i	$0.847985\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.8806			0.676742			0.338371	−	0.941013i	$-0.390124\pi$
$863$	19.8806			0.676742			0.338371	+	0.941013i	$0.390124\pi$
$864$		0			0
$865$	−35.5139			−1.20751
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.15139	+	1.99426i	0.0390582	+	0.0676507i
$870$		0			0
$871$	−4.45416	+	7.71484i	−0.150924	+	0.261407i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.8625	+	30.9387i	−0.603862	+	1.04592i
$876$		0			0
$877$	14.8764	+	25.7666i	0.502339	+	0.870077i	0.999996	+	0.00270327i	$0.000860478\pi$
$877$	14.8764	+	25.7666i	0.502339	+	0.870077i	−0.497657	+	0.867374i	$0.665806\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.2111			0.613548			0.306774	−	0.951782i	$-0.400750\pi$
$881$	18.2111			0.613548			0.306774	+	0.951782i	$0.400750\pi$
$882$		0			0
$883$	34.7889			1.17074			0.585370	−	0.810766i	$-0.300949\pi$
$883$	34.7889			1.17074			0.585370	+	0.810766i	$0.300949\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.5000	−	44.1673i	−0.856206	−	1.48299i	−0.875521	−	0.483179i	$-0.839482\pi$
$887$	−25.5000	−	44.1673i	−0.856206	−	1.48299i	0.0193153	−	0.999813i	$-0.493851\pi$
$888$		0			0
$889$	−14.2111	+	24.6144i	−0.476625	+	0.825539i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.89445	−	5.01333i	0.0968590	−	0.167765i
$894$		0			0
$895$	−4.50000	−	7.79423i	−0.150418	−	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.64171			−0.0881060
$900$		0			0
$901$	35.0278			1.16694
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.1514	+	17.5827i	0.337444	+	0.584469i
$906$		0			0
$907$	−26.2569	+	45.4784i	−0.871847	+	1.51008i	−0.0117637	+	0.999931i	$0.503745\pi$
$907$	−26.2569	+	45.4784i	−0.871847	+	1.51008i	−0.860084	+	0.510153i	$0.829589\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	28.6056	−	49.5463i	0.947744	−	1.64154i	0.197583	−	0.980286i	$-0.436691\pi$
$911$	28.6056	−	49.5463i	0.947744	−	1.64154i	0.750161	−	0.661255i	$-0.229976\pi$
$912$		0			0
$913$	−10.3625	−	17.9484i	−0.342948	−	0.594004i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	73.0555			2.41251
$918$		0			0
$919$	11.0000			0.362857			0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000			0.362857			0.181428	+	0.983404i	$0.441928\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.19722	+	3.80570i	0.0723225	+	0.125266i
$924$		0			0
$925$	1.40833	−	2.43929i	0.0463055	−	0.0802035i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.9222	−	34.5063i	0.653626	−	1.13211i	−0.328610	−	0.944466i	$-0.606580\pi$
$929$	19.9222	−	34.5063i	0.653626	−	1.13211i	0.982236	−	0.187648i	$-0.0600865\pi$
$930$		0			0
$931$	−7.04584	−	12.2037i	−0.230918	−	0.399962i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40.3305			−1.31895
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.89445	+	10.2095i	0.192154	+	0.332820i	0.945964	−	0.324273i	$-0.105120\pi$
$941$	5.89445	+	10.2095i	0.192154	+	0.332820i	−0.753810	+	0.657092i	$0.771786\pi$
$942$		0			0
$943$	4.50000	−	7.79423i	0.146540	−	0.253815i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.5597	+	32.1464i	−0.603110	+	1.04462i	0.389237	+	0.921137i	$0.372739\pi$
$947$	−18.5597	+	32.1464i	−0.603110	+	1.04462i	−0.992347	+	0.123479i	$0.960595\pi$
$948$		0			0
$949$	7.40833	+	12.8316i	0.240484	+	0.416531i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.4500			−1.34270			−0.671348	−	0.741142i	$-0.734284\pi$
$953$	−41.4500			−1.34270			−0.671348	+	0.741142i	$0.734284\pi$
$954$		0			0
$955$	15.4222			0.499051
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.80278	+	6.58660i	0.122798	+	0.212692i
$960$		0			0
$961$	11.2708	−	19.5216i	0.363575	−	0.629730i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.7569	−	20.3636i	0.378469	−	0.655528i
$966$		0			0
$967$	4.78890	+	8.29461i	0.154001	+	0.266737i	0.932695	−	0.360667i	$-0.117451\pi$
$967$	4.78890	+	8.29461i	0.154001	+	0.266737i	−0.778694	+	0.627404i	$0.784118\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−47.2389			−1.51597			−0.757984	−	0.652274i	$-0.773815\pi$
$971$	−47.2389			−1.51597			−0.757984	+	0.652274i	$0.773815\pi$
$972$		0			0
$973$	−14.2111			−0.455587
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.8167	−	34.3235i	−0.633991	−	1.09810i	−0.986728	−	0.162382i	$-0.948082\pi$
$977$	−19.8167	−	34.3235i	−0.633991	−	1.09810i	0.352737	−	0.935722i	$-0.385251\pi$
$978$		0			0
$979$	−15.1056	+	26.1636i	−0.482776	+	0.836192i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.18335	−	12.4419i	0.229113	−	0.396836i	−0.728432	−	0.685118i	$-0.759751\pi$
$983$	7.18335	−	12.4419i	0.229113	−	0.396836i	0.957546	+	0.288282i	$0.0930841\pi$
$984$		0			0
$985$	17.2708	+	29.9139i	0.550294	+	0.953137i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.4584			0.523346
$990$		0			0
$991$	−16.4222			−0.521669			−0.260834	−	0.965384i	$-0.583998\pi$
$991$	−16.4222			−0.521669			−0.260834	+	0.965384i	$0.583998\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.7569	+	20.3636i	0.372720	+	0.645570i
$996$		0			0
$997$	−22.0736	+	38.2326i	−0.699078	+	1.21084i	0.269709	+	0.962942i	$0.413073\pi$
$997$	−22.0736	+	38.2326i	−0.699078	+	1.21084i	−0.968787	+	0.247896i	$0.920261\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.o.2809.1		4
3.2	odd	2		4212.2.i.t.2809.2		4
9.2	odd	6		4212.2.a.c.1.1	✓	2
9.4	even	3	inner	4212.2.i.o.1405.1		4
9.5	odd	6		4212.2.i.t.1405.2		4
9.7	even	3		4212.2.a.e.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4212.2.a.c.1.1	✓	2	9.2	odd	6
4212.2.a.e.1.2	yes	2	9.7	even	3
4212.2.i.o.1405.1		4	9.4	even	3	inner
4212.2.i.o.2809.1		4	1.1	even	1	trivial
4212.2.i.t.1405.2		4	9.5	odd	6
4212.2.i.t.2809.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.15139 - 1.99426i) q^{5} +(1.65139 - 2.86029i) q^{7} +(-1.15139 + 1.99426i) q^{11} +(-0.500000 - 0.866025i) q^{13} -7.60555 q^{17} +3.60555 q^{19} +(1.95416 + 3.38471i) q^{23} +(-0.151388 + 0.262211i) q^{25} +(0.454163 - 0.786634i) q^{29} +(-1.45416 - 2.51868i) q^{31} -7.60555 q^{35} -9.30278 q^{37} +(-1.15139 - 1.99426i) q^{41} +(2.10555 - 3.64692i) q^{43} +(0.802776 - 1.39045i) q^{47} +(-1.95416 - 3.38471i) q^{49} -4.60555 q^{53} +5.30278 q^{55} +(2.19722 + 3.80570i) q^{59} +(-2.25694 + 3.90913i) q^{61} +(-1.15139 + 1.99426i) q^{65} +(-4.45416 - 7.71484i) q^{67} -4.39445 q^{71} -14.8167 q^{73} +(3.80278 + 6.58660i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(8.75694 + 15.1675i) q^{85} +13.1194 q^{89} -3.30278 q^{91} +(-4.15139 - 7.19041i) q^{95} +(6.60555 - 11.4412i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} + 3 q^{7} - q^{11} - 2 q^{13} - 16 q^{17} - 3 q^{23} + 3 q^{25} - 9 q^{29} + 5 q^{31} - 16 q^{35} - 30 q^{37} - q^{41} - 6 q^{43} - 4 q^{47} + 3 q^{49} - 4 q^{53} + 14 q^{55} + 16 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 4 * q - q^5 + 3 * q^7 - q^11 - 2 * q^13 - 16 * q^17 - 3 * q^23 + 3 * q^25 - 9 * q^29 + 5 * q^31 - 16 * q^35 - 30 * q^37 - q^41 - 6 * q^43 - 4 * q^47 + 3 * q^49 - 4 * q^53 + 14 * q^55 + 16 * q^59 + 9 * q^61 - q^65 - 7 * q^67 - 32 * q^71 - 16 * q^73 + 8 * q^77 + 2 * q^79 - 18 * q^83 + 17 * q^85 + 2 * q^89 - 6 * q^91 - 13 * q^95 + 12 * q^97

Introduction

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