LMFDB - Embedded newform 4212.2.i.w.2809.4

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 = [8,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0] 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.4
Root			$1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.w.1405.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.09445 + 1.89564i) q^{5} +(1.39564 - 2.41733i) q^{7} +(-1.09445 + 1.89564i) q^{11} +(0.500000 + 0.866025i) q^{13} +1.73205 q^{17} +4.58258 q^{19} +(1.96048 + 3.39564i) q^{23} +(0.104356 - 0.180750i) q^{25} +(-3.74020 + 6.47822i) q^{29} +(-0.895644 - 1.55130i) q^{31} +6.10985 q^{35} +0.791288 q^{37} +(-1.55130 - 2.68693i) q^{41} +(-1.29129 + 2.23658i) q^{43} +(-0.409175 + 0.708712i) q^{47} +(-0.395644 - 0.685275i) q^{49} -6.20520 q^{53} -4.79129 q^{55} +(2.14123 + 3.70871i) q^{59} +(5.18693 - 8.98403i) q^{61} +(-1.09445 + 1.89564i) q^{65} +(6.68693 + 11.5821i) q^{67} +7.02355 q^{71} +0.582576 q^{73} +(3.05493 + 5.29129i) q^{77} +(0.500000 - 0.866025i) q^{79} +(0.504525 - 0.873864i) q^{83} +(1.89564 + 3.28335i) q^{85} +11.7629 q^{89} +2.79129 q^{91} +(5.01540 + 8.68693i) q^{95} +(5.79129 - 10.0308i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{7} + 4 q^{13} + 10 q^{25} + 2 q^{31} - 12 q^{37} + 8 q^{43} + 6 q^{49} - 20 q^{55} + 14 q^{61} + 26 q^{67} - 32 q^{73} + 4 q^{79} + 6 q^{85} + 4 q^{91} + 28 q^{97}+O(q^{100}) $ 8 * q + 2 * q^7 + 4 * q^13 + 10 * q^25 + 2 * q^31 - 12 * q^37 + 8 * q^43 + 6 * q^49 - 20 * q^55 + 14 * q^61 + 26 * q^67 - 32 * q^73 + 4 * q^79 + 6 * q^85 + 4 * q^91 + 28 * 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
$5$	1.09445	+	1.89564i	0.489453	+	0.847758i	0.999926	−	0.0121359i	$-0.00386308\pi$
$5$	1.09445	+	1.89564i	0.489453	+	0.847758i	−0.510473	+	0.859894i	$0.670530\pi$
$6$		0			0
$7$	1.39564	−	2.41733i	0.527504	−	0.913663i	−0.471982	−	0.881608i	$-0.656461\pi$
$7$	1.39564	−	2.41733i	0.527504	−	0.913663i	0.999486	−	0.0320554i	$-0.0102053\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.09445	+	1.89564i	−0.329989	+	0.571558i	−0.982509	−	0.186213i	$-0.940378\pi$
$11$	−1.09445	+	1.89564i	−0.329989	+	0.571558i	0.652520	+	0.757771i	$0.273712\pi$
$12$		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
$17$	1.73205			0.420084			0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205			0.420084			0.210042	+	0.977692i	$0.432640\pi$
$18$		0			0
$19$	4.58258			1.05131			0.525657	−	0.850696i	$-0.323819\pi$
$19$	4.58258			1.05131			0.525657	+	0.850696i	$0.323819\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.96048	+	3.39564i	0.408787	+	0.708041i	0.994754	−	0.102294i	$-0.0326184\pi$
$23$	1.96048	+	3.39564i	0.408787	+	0.708041i	−0.585967	+	0.810335i	$0.699285\pi$
$24$		0			0
$25$	0.104356	−	0.180750i	0.0208712	−	0.0361500i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.74020	+	6.47822i	−0.694538	+	1.20298i	0.275798	+	0.961216i	$0.411058\pi$
$29$	−3.74020	+	6.47822i	−0.694538	+	1.20298i	−0.970336	+	0.241760i	$0.922275\pi$
$30$		0			0
$31$	−0.895644	−	1.55130i	−0.160862	−	0.278622i	0.774316	−	0.632799i	$-0.218094\pi$
$31$	−0.895644	−	1.55130i	−0.160862	−	0.278622i	−0.935178	+	0.354177i	$0.884761\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.10985			1.03275
$36$		0			0
$37$	0.791288			0.130087			0.0650435	−	0.997882i	$-0.479281\pi$
$37$	0.791288			0.130087			0.0650435	+	0.997882i	$0.479281\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.55130	−	2.68693i	−0.242272	−	0.419628i	0.719089	−	0.694918i	$-0.244559\pi$
$41$	−1.55130	−	2.68693i	−0.242272	−	0.419628i	−0.961361	+	0.275290i	$0.911226\pi$
$42$		0			0
$43$	−1.29129	+	2.23658i	−0.196920	+	0.341075i	−0.947528	−	0.319672i	$-0.896427\pi$
$43$	−1.29129	+	2.23658i	−0.196920	+	0.341075i	0.750609	+	0.660747i	$0.229760\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.409175	+	0.708712i	−0.0596843	+	0.103376i	−0.894324	−	0.447421i	$-0.852343\pi$
$47$	−0.409175	+	0.708712i	−0.0596843	+	0.103376i	0.834639	+	0.550797i	$0.185676\pi$
$48$		0			0
$49$	−0.395644	−	0.685275i	−0.0565206	−	0.0978965i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.20520			−0.852350			−0.426175	−	0.904641i	$-0.640139\pi$
$53$	−6.20520			−0.852350			−0.426175	+	0.904641i	$0.640139\pi$
$54$		0			0
$55$	−4.79129			−0.646057
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.14123	+	3.70871i	0.278764	+	0.482833i	0.971078	−	0.238763i	$-0.0767420\pi$
$59$	2.14123	+	3.70871i	0.278764	+	0.482833i	−0.692314	+	0.721596i	$0.743409\pi$
$60$		0			0
$61$	5.18693	−	8.98403i	0.664119	−	1.15029i	−0.315405	−	0.948957i	$-0.602140\pi$
$61$	5.18693	−	8.98403i	0.664119	−	1.15029i	0.979523	−	0.201330i	$-0.0645263\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.09445	+	1.89564i	−0.135750	+	0.235126i
$66$		0			0
$67$	6.68693	+	11.5821i	0.816939	+	1.41498i	0.907927	+	0.419127i	$0.137664\pi$
$67$	6.68693	+	11.5821i	0.816939	+	1.41498i	−0.0909887	+	0.995852i	$0.529003\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.02355			0.833542			0.416771	−	0.909011i	$-0.363162\pi$
$71$	7.02355			0.833542			0.416771	+	0.909011i	$0.363162\pi$
$72$		0			0
$73$	0.582576			0.0681853			0.0340927	−	0.999419i	$-0.489146\pi$
$73$	0.582576			0.0681853			0.0340927	+	0.999419i	$0.489146\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.05493	+	5.29129i	0.348141	+	0.602998i
$78$		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
$83$	0.504525	−	0.873864i	0.0553789	−	0.0959190i	−0.837007	−	0.547192i	$-0.815697\pi$
$83$	0.504525	−	0.873864i	0.0553789	−	0.0959190i	0.892386	+	0.451273i	$0.149030\pi$
$84$		0			0
$85$	1.89564	+	3.28335i	0.205611	+	0.356129i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.7629			1.24686			0.623430	−	0.781879i	$-0.285739\pi$
$89$	11.7629			1.24686			0.623430	+	0.781879i	$0.285739\pi$
$90$		0			0
$91$	2.79129			0.292606
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.01540	+	8.68693i	0.514569	+	0.891260i
$96$		0			0
$97$	5.79129	−	10.0308i	0.588016	−	1.01847i	−0.406476	−	0.913662i	$-0.633242\pi$
$97$	5.79129	−	10.0308i	0.588016	−	1.01847i	0.994492	−	0.104812i	$-0.0334242\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.51178	+	6.08258i	−0.349435	+	0.605239i	−0.986149	−	0.165861i	$-0.946960\pi$
$101$	−3.51178	+	6.08258i	−0.349435	+	0.605239i	0.636714	+	0.771100i	$0.280293\pi$
$102$		0			0
$103$	0.604356	+	1.04678i	0.0595490	+	0.103142i	0.894263	−	0.447542i	$-0.147700\pi$
$103$	0.604356	+	1.04678i	0.0595490	+	0.103142i	−0.834714	+	0.550684i	$0.814367\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.73930			0.458166			0.229083	−	0.973407i	$-0.426427\pi$
$107$	4.73930			0.458166			0.229083	+	0.973407i	$0.426427\pi$
$108$		0			0
$109$	−18.7477			−1.79571			−0.897853	−	0.440295i	$-0.854874\pi$
$109$	−18.7477			−1.79571			−0.897853	+	0.440295i	$0.854874\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.55945	−	6.16515i	−0.334845	−	0.579969i	0.648610	−	0.761121i	$-0.275351\pi$
$113$	−3.55945	−	6.16515i	−0.334845	−	0.579969i	−0.983455	+	0.181152i	$0.942017\pi$
$114$		0			0
$115$	−4.29129	+	7.43273i	−0.400165	+	0.693106i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.41733	−	4.18693i	0.221596	−	0.383815i
$120$		0			0
$121$	3.10436	+	5.37690i	0.282214	+	0.488809i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.4014			1.01977
$126$		0			0
$127$	−9.58258			−0.850316			−0.425158	−	0.905119i	$-0.639781\pi$
$127$	−9.58258			−0.850316			−0.425158	+	0.905119i	$0.639781\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.02265	+	13.8956i	0.700943	+	1.21407i	0.968136	+	0.250425i	$0.0805704\pi$
$131$	8.02265	+	13.8956i	0.700943	+	1.21407i	−0.267193	+	0.963643i	$0.586096\pi$
$132$		0			0
$133$	6.39564	−	11.0776i	0.554573	−	0.960548i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.55130	−	2.68693i	0.132537	−	0.229560i	−0.792117	−	0.610369i	$-0.791021\pi$
$137$	1.55130	−	2.68693i	0.132537	−	0.229560i	0.924654	+	0.380809i	$0.124354\pi$
$138$		0			0
$139$	3.68693	+	6.38595i	0.312721	+	0.541649i	0.978951	−	0.204098i	$-0.0654260\pi$
$139$	3.68693	+	6.38595i	0.312721	+	0.541649i	−0.666229	+	0.745747i	$0.732093\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.18890			−0.183045
$144$		0			0
$145$	−16.3739			−1.35978
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.46410	+	6.00000i	0.283790	+	0.491539i	0.972315	−	0.233674i	$-0.0750747\pi$
$149$	3.46410	+	6.00000i	0.283790	+	0.491539i	−0.688525	+	0.725213i	$0.741741\pi$
$150$		0			0
$151$	−2.68693	+	4.65390i	−0.218659	+	0.378729i	−0.954398	−	0.298536i	$-0.903502\pi$
$151$	−2.68693	+	4.65390i	−0.218659	+	0.378729i	0.735739	+	0.677265i	$0.236835\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.96048	−	3.39564i	0.157469	−	0.272745i
$156$		0			0
$157$	6.97822	+	12.0866i	0.556923	+	0.964618i	0.997751	+	0.0670269i	$0.0213513\pi$
$157$	6.97822	+	12.0866i	0.556923	+	0.964618i	−0.440829	+	0.897591i	$0.645315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.9445			0.862548
$162$		0			0
$163$	−5.58258			−0.437261			−0.218631	−	0.975808i	$-0.570159\pi$
$163$	−5.58258			−0.437261			−0.218631	+	0.975808i	$0.570159\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.16478	+	15.8739i	0.709192	+	1.22836i	0.965157	+	0.261670i	$0.0842734\pi$
$167$	9.16478	+	15.8739i	0.709192	+	1.22836i	−0.255965	+	0.966686i	$0.582393\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.8105	−	20.4564i	0.897938	−	1.55527i	0.0678122	−	0.997698i	$-0.478398\pi$
$173$	11.8105	−	20.4564i	0.897938	−	1.55527i	0.830126	−	0.557576i	$-0.188269\pi$
$174$		0			0
$175$	−0.291288	−	0.504525i	−0.0220193	−	0.0381385i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.7810			−1.03004			−0.515019	−	0.857179i	$-0.672215\pi$
$179$	−13.7810			−1.03004			−0.515019	+	0.857179i	$0.672215\pi$
$180$		0			0
$181$	−15.0000			−1.11494			−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000			−1.11494			−0.557471	+	0.830197i	$0.688228\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.866025	+	1.50000i	0.0636715	+	0.110282i
$186$		0			0
$187$	−1.89564	+	3.28335i	−0.138623	+	0.240102i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.14213	−	1.97822i	0.0826413	−	0.143139i	−0.821742	−	0.569859i	$-0.806998\pi$
$191$	1.14213	−	1.97822i	0.0826413	−	0.143139i	0.904384	+	0.426720i	$0.140331\pi$
$192$		0			0
$193$	−4.87386	−	8.44178i	−0.350828	−	0.607653i	0.635566	−	0.772046i	$-0.280767\pi$
$193$	−4.87386	−	8.44178i	−0.350828	−	0.607653i	−0.986395	+	0.164394i	$0.947433\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.45505			0.174915			0.0874576	−	0.996168i	$-0.472126\pi$
$197$	2.45505			0.174915			0.0874576	+	0.996168i	$0.472126\pi$
$198$		0			0
$199$	15.7477			1.11633			0.558163	−	0.829731i	$-0.311506\pi$
$199$	15.7477			1.11633			0.558163	+	0.829731i	$0.311506\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.4400	+	18.0826i	0.732743	+	1.26915i
$204$		0			0
$205$	3.39564	−	5.88143i	0.237162	−	0.410777i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.01540	+	8.68693i	−0.346923	+	0.600888i
$210$		0			0
$211$	−11.2695	−	19.5194i	−0.775825	−	1.34377i	−0.934330	−	0.356410i	$-0.884001\pi$
$211$	−11.2695	−	19.5194i	−0.775825	−	1.34377i	0.158505	−	0.987358i	$-0.449333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.65300			−0.385532
$216$		0			0
$217$	−5.00000			−0.339422
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.866025	+	1.50000i	0.0582552	+	0.100901i
$222$		0			0
$223$	11.3739	−	19.7001i	0.761650	−	1.31922i	−0.180349	−	0.983603i	$-0.557723\pi$
$223$	11.3739	−	19.7001i	0.761650	−	1.31922i	0.942000	−	0.335614i	$-0.108944\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0308	+	17.3739i	−0.665768	+	1.15314i	0.313309	+	0.949651i	$0.398563\pi$
$227$	−10.0308	+	17.3739i	−0.665768	+	1.15314i	−0.979077	+	0.203492i	$0.934771\pi$
$228$		0			0
$229$	2.60436	+	4.51088i	0.172101	+	0.298087i	0.939154	−	0.343496i	$-0.111611\pi$
$229$	2.60436	+	4.51088i	0.172101	+	0.298087i	−0.767053	+	0.641583i	$0.778278\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.9536			−0.783103			−0.391552	−	0.920156i	$-0.628062\pi$
$233$	−11.9536			−0.783103			−0.391552	+	0.920156i	$0.628062\pi$
$234$		0			0
$235$	−1.79129			−0.116851
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.40828	+	2.43920i	0.0910938	+	0.157779i	0.907972	−	0.419032i	$-0.137630\pi$
$239$	1.40828	+	2.43920i	0.0910938	+	0.157779i	−0.816878	+	0.576811i	$0.804297\pi$
$240$		0			0
$241$	−3.47822	+	6.02445i	−0.224052	+	0.388069i	−0.956035	−	0.293254i	$-0.905262\pi$
$241$	−3.47822	+	6.02445i	−0.224052	+	0.388069i	0.731983	+	0.681323i	$0.238595\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.866025	−	1.50000i	0.0553283	−	0.0958315i
$246$		0			0
$247$	2.29129	+	3.96863i	0.145791	+	0.252518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.6374			1.74445			0.872227	−	0.489100i	$-0.162675\pi$
$251$	27.6374			1.74445			0.872227	+	0.489100i	$0.162675\pi$
$252$		0			0
$253$	−8.58258			−0.539582
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.55130	−	2.68693i	−0.0967675	−	0.167606i	0.813577	−	0.581457i	$-0.197517\pi$
$257$	−1.55130	−	2.68693i	−0.0967675	−	0.167606i	−0.910345	+	0.413850i	$0.864184\pi$
$258$		0			0
$259$	1.10436	−	1.91280i	0.0686213	−	0.118856i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.98403	+	15.5608i	−0.553979	+	0.959520i	0.444003	+	0.896025i	$0.353558\pi$
$263$	−8.98403	+	15.5608i	−0.553979	+	0.959520i	−0.997982	+	0.0634945i	$0.979775\pi$
$264$		0			0
$265$	−6.79129	−	11.7629i	−0.417185	−	0.722586i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.09355			0.127646			0.0638230	−	0.997961i	$-0.479671\pi$
$269$	2.09355			0.127646			0.0638230	+	0.997961i	$0.479671\pi$
$270$		0			0
$271$	−5.00000			−0.303728			−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000			−0.303728			−0.151864	+	0.988401i	$0.548528\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.228425	+	0.395644i	0.0137746	+	0.0238582i
$276$		0			0
$277$	3.00000	−	5.19615i	0.180253	−	0.312207i	−0.761714	−	0.647913i	$-0.775642\pi$
$277$	3.00000	−	5.19615i	0.180253	−	0.312207i	0.941966	+	0.335707i	$0.108975\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.97678	−	10.3521i	0.356545	−	0.617554i	−0.630836	−	0.775916i	$-0.717288\pi$
$281$	5.97678	−	10.3521i	0.356545	−	0.617554i	0.987381	+	0.158362i	$0.0506214\pi$
$282$		0			0
$283$	13.1652	+	22.8027i	0.782587	+	1.35548i	0.930430	+	0.366470i	$0.119434\pi$
$283$	13.1652	+	22.8027i	0.782587	+	1.35548i	−0.147843	+	0.989011i	$0.547233\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.66025			−0.511199
$288$		0			0
$289$	−14.0000			−0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.4877	+	18.1652i	0.612695	+	1.06122i	0.990784	+	0.135450i	$0.0432481\pi$
$293$	10.4877	+	18.1652i	0.612695	+	1.06122i	−0.378089	+	0.925769i	$0.623419\pi$
$294$		0			0
$295$	−4.68693	+	8.11800i	−0.272884	+	0.472648i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.96048	+	3.39564i	−0.113377	+	0.196375i
$300$		0			0
$301$	3.60436	+	6.24293i	0.207752	+	0.359836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.7074			1.30022
$306$		0			0
$307$	21.5826			1.23178			0.615891	−	0.787831i	$-0.288796\pi$
$307$	21.5826			1.23178			0.615891	+	0.787831i	$0.288796\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.38595	+	11.0608i	0.362114	+	0.627200i	0.988309	−	0.152467i	$-0.0487217\pi$
$311$	6.38595	+	11.0608i	0.362114	+	0.627200i	−0.626194	+	0.779667i	$0.715388\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.6847	+	25.4347i	−0.824775	+	1.42855i	0.0773156	+	0.997007i	$0.475365\pi$
$317$	−14.6847	+	25.4347i	−0.824775	+	1.42855i	−0.902091	+	0.431546i	$0.857968\pi$
$318$		0			0
$319$	−8.18693	−	14.1802i	−0.458380	−	0.793938i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.93725			0.441641
$324$		0			0
$325$	0.208712			0.0115773
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.14213	+	1.97822i	0.0629674	+	0.109063i
$330$		0			0
$331$	−10.0826	+	17.4635i	−0.554188	+	0.959883i	0.443778	+	0.896137i	$0.353638\pi$
$331$	−10.0826	+	17.4635i	−0.554188	+	0.959883i	−0.997966	+	0.0637457i	$0.979695\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.6370	+	25.3521i	−0.799706	+	1.38513i
$336$		0			0
$337$	14.0826	+	24.3917i	0.767127	+	1.32870i	0.939115	+	0.343604i	$0.111648\pi$
$337$	14.0826	+	24.3917i	0.767127	+	1.32870i	−0.171988	+	0.985099i	$0.555019\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.92095			0.212331
$342$		0			0
$343$	17.3303			0.935748
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.4014	+	19.7477i	0.612057	+	1.06011i	0.990893	+	0.134650i	$0.0429910\pi$
$347$	11.4014	+	19.7477i	0.612057	+	1.06011i	−0.378836	+	0.925464i	$0.623676\pi$
$348$		0			0
$349$	−2.47822	+	4.29240i	−0.132656	+	0.229767i	−0.924700	−	0.380698i	$-0.875684\pi$
$349$	−2.47822	+	4.29240i	−0.132656	+	0.229767i	0.792044	+	0.610465i	$0.209017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.77793	−	6.54356i	0.201079	−	0.348279i	−0.747798	−	0.663927i	$-0.768889\pi$
$353$	3.77793	−	6.54356i	0.201079	−	0.348279i	0.948876	+	0.315648i	$0.102222\pi$
$354$		0			0
$355$	7.68693	+	13.3142i	0.407980	+	0.706642i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.83465			−0.255163			−0.127582	−	0.991828i	$-0.540721\pi$
$359$	−4.83465			−0.255163			−0.127582	+	0.991828i	$0.540721\pi$
$360$		0			0
$361$	2.00000			0.105263
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.637600	+	1.10436i	0.0333735	+	0.0578046i
$366$		0			0
$367$	11.8739	−	20.5661i	0.619811	−	1.07354i	−0.369709	−	0.929147i	$-0.620543\pi$
$367$	11.8739	−	20.5661i	0.619811	−	1.07354i	0.989520	−	0.144396i	$-0.0461239\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.66025	+	15.0000i	−0.449618	+	0.778761i
$372$		0			0
$373$	4.60436	+	7.97498i	0.238405	+	0.412929i	0.960257	−	0.279119i	$-0.0900423\pi$
$373$	4.60436	+	7.97498i	0.238405	+	0.412929i	−0.721852	+	0.692047i	$0.756709\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.48040			−0.385260
$378$		0			0
$379$	−6.58258			−0.338124			−0.169062	−	0.985605i	$-0.554074\pi$
$379$	−6.58258			−0.338124			−0.169062	+	0.985605i	$0.554074\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.5353	−	18.2477i	−0.538330	−	0.932415i	−0.998994	−	0.0448407i	$-0.985722\pi$
$383$	−10.5353	−	18.2477i	−0.538330	−	0.932415i	0.460664	−	0.887575i	$-0.347611\pi$
$384$		0			0
$385$	−6.68693	+	11.5821i	−0.340798	+	0.590279i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.1711	−	22.8131i	0.667803	−	1.15667i	−0.310714	−	0.950503i	$-0.600568\pi$
$389$	13.1711	−	22.8131i	0.667803	−	1.15667i	0.978517	−	0.206165i	$-0.0660984\pi$
$390$		0			0
$391$	3.39564	+	5.88143i	0.171725	+	0.297437i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.18890			0.110136
$396$		0			0
$397$	−8.58258			−0.430747			−0.215374	−	0.976532i	$-0.569097\pi$
$397$	−8.58258			−0.430747			−0.215374	+	0.976532i	$0.569097\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.59808	−	4.50000i	−0.129742	−	0.224719i	0.793835	−	0.608134i	$-0.208081\pi$
$401$	−2.59808	−	4.50000i	−0.129742	−	0.224719i	−0.923576	+	0.383414i	$0.874748\pi$
$402$		0			0
$403$	0.895644	−	1.55130i	0.0446152	−	0.0772758i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.866025	+	1.50000i	−0.0429273	+	0.0743522i
$408$		0			0
$409$	−15.6652	−	27.1328i	−0.774592	−	1.34163i	−0.935024	−	0.354586i	$-0.884622\pi$
$409$	−15.6652	−	27.1328i	−0.774592	−	1.34163i	0.160432	−	0.987047i	$-0.448711\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.9536			0.588196
$414$		0			0
$415$	2.20871			0.108421
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.6066	−	30.4955i	−0.860137	−	1.48980i	−0.871796	−	0.489868i	$-0.837045\pi$
$419$	−17.6066	−	30.4955i	−0.860137	−	1.48980i	0.0116597	−	0.999932i	$-0.496289\pi$
$420$		0			0
$421$	10.9564	−	18.9771i	0.533984	−	0.924888i	−0.465228	−	0.885191i	$-0.654028\pi$
$421$	10.9564	−	18.9771i	0.533984	−	0.924888i	0.999212	−	0.0396966i	$-0.0126391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.180750	−	0.313068i	0.00876766	−	0.0151860i
$426$		0			0
$427$	−14.4782	−	25.0770i	−0.700650	−	1.21356i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−32.4720			−1.56412			−0.782061	−	0.623202i	$-0.785831\pi$
$431$	−32.4720			−1.56412			−0.782061	+	0.623202i	$0.785831\pi$
$432$		0			0
$433$	22.4955			1.08106			0.540531	−	0.841324i	$-0.318223\pi$
$433$	22.4955			1.08106			0.540531	+	0.841324i	$0.318223\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.98403	+	15.5608i	0.429764	+	0.744374i
$438$		0			0
$439$	2.87386	−	4.97768i	0.137162	−	0.237572i	−0.789259	−	0.614060i	$-0.789535\pi$
$439$	2.87386	−	4.97768i	0.137162	−	0.237572i	0.926421	+	0.376489i	$0.122869\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.2583	+	19.5000i	−0.534899	+	0.926473i	0.464269	+	0.885694i	$0.346317\pi$
$443$	−11.2583	+	19.5000i	−0.534899	+	0.926473i	−0.999168	+	0.0407786i	$0.987016\pi$
$444$		0			0
$445$	12.8739	+	22.2982i	0.610280	+	1.05704i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.74655			0.365582			0.182791	−	0.983152i	$-0.441487\pi$
$449$	7.74655			0.365582			0.182791	+	0.983152i	$0.441487\pi$
$450$		0			0
$451$	6.79129			0.319789
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.05493	+	5.29129i	0.143217	+	0.248059i
$456$		0			0
$457$	6.20871	−	10.7538i	0.290431	−	0.503042i	−0.683480	−	0.729969i	$-0.739535\pi$
$457$	6.20871	−	10.7538i	0.290431	−	0.503042i	0.973912	+	0.226927i	$0.0728679\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.5353	−	18.2477i	0.490679	−	0.849881i	−0.509263	−	0.860611i	$-0.670082\pi$
$461$	10.5353	−	18.2477i	0.490679	−	0.849881i	0.999942	+	0.0107294i	$0.00341536\pi$
$462$		0			0
$463$	7.95644	+	13.7810i	0.369767	+	0.640455i	0.989529	−	0.144335i	$-0.0461041\pi$
$463$	7.95644	+	13.7810i	0.369767	+	0.640455i	−0.619762	+	0.784790i	$0.712771\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.4394			1.13092			0.565461	−	0.824775i	$-0.308698\pi$
$467$	24.4394			1.13092			0.565461	+	0.824775i	$0.308698\pi$
$468$		0			0
$469$	37.3303			1.72375
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.82650	−	4.89564i	−0.129963	−	0.225102i
$474$		0			0
$475$	0.478220	−	0.828301i	0.0219422	−	0.0380050i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.4104	−	21.4955i	0.567046	−	0.982152i	−0.429810	−	0.902919i	$-0.641420\pi$
$479$	12.4104	−	21.4955i	0.567046	−	0.982152i	0.996856	−	0.0792331i	$-0.0252471\pi$
$480$		0			0
$481$	0.395644	+	0.685275i	0.0180398	+	0.0312459i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	25.3531			1.15123
$486$		0			0
$487$	−11.7477			−0.532340			−0.266170	−	0.963926i	$-0.585758\pi$
$487$	−11.7477			−0.532340			−0.266170	+	0.963926i	$0.585758\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.56760	−	9.64337i	−0.251262	−	0.435199i	0.712611	−	0.701559i	$-0.247512\pi$
$491$	−5.56760	−	9.64337i	−0.251262	−	0.435199i	−0.963874	+	0.266360i	$0.914179\pi$
$492$		0			0
$493$	−6.47822	+	11.2206i	−0.291764	+	0.505351i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.80238	−	16.9782i	0.439697	−	0.761577i
$498$		0			0
$499$	−20.5608	−	35.6123i	−0.920428	−	1.59423i	−0.798754	−	0.601657i	$-0.794507\pi$
$499$	−20.5608	−	35.6123i	−0.920428	−	1.59423i	−0.121673	−	0.992570i	$-0.538826\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.6675			−0.520228			−0.260114	−	0.965578i	$-0.583760\pi$
$503$	−11.6675			−0.520228			−0.260114	+	0.965578i	$0.583760\pi$
$504$		0			0
$505$	−15.3739			−0.684128
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.1874	−	29.7695i	−0.761820	−	1.31951i	−0.941912	−	0.335860i	$-0.890973\pi$
$509$	−17.1874	−	29.7695i	−0.761820	−	1.31951i	0.180092	−	0.983650i	$-0.442360\pi$
$510$		0			0
$511$	0.813068	−	1.40828i	0.0359680	−	0.0622984i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.32288	+	2.29129i	−0.0582929	+	0.100966i
$516$		0			0
$517$	−0.895644	−	1.55130i	−0.0393904	−	0.0682261i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.1950			−1.45430			−0.727150	−	0.686479i	$-0.759156\pi$
$521$	−33.1950			−1.45430			−0.727150	+	0.686479i	$0.759156\pi$
$522$		0			0
$523$	−13.9129			−0.608368			−0.304184	−	0.952613i	$-0.598384\pi$
$523$	−13.9129			−0.608368			−0.304184	+	0.952613i	$0.598384\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.55130	−	2.68693i	−0.0675757	−	0.117045i
$528$		0			0
$529$	3.81307	−	6.60443i	0.165786	−	0.287149i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.55130	−	2.68693i	0.0671943	−	0.116384i
$534$		0			0
$535$	5.18693	+	8.98403i	0.224251	+	0.388413i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.73205			0.0746047
$540$		0			0
$541$	33.9129			1.45803			0.729014	−	0.684499i	$-0.239979\pi$
$541$	33.9129			1.45803			0.729014	+	0.684499i	$0.239979\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.5185	−	35.5390i	−0.878914	−	1.52232i
$546$		0			0
$547$	−9.41742	+	16.3115i	−0.402660	+	0.697428i	−0.994046	−	0.108961i	$-0.965248\pi$
$547$	−9.41742	+	16.3115i	−0.402660	+	0.697428i	0.591386	+	0.806389i	$0.298581\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−17.1398	+	29.6869i	−0.730178	+	1.26471i
$552$		0			0
$553$	−1.39564	−	2.41733i	−0.0593488	−	0.102795i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.9144			1.14040			0.570199	−	0.821507i	$-0.306866\pi$
$557$	26.9144			1.14040			0.570199	+	0.821507i	$0.306866\pi$
$558$		0			0
$559$	−2.58258			−0.109231
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.11710	−	15.7913i	−0.384240	−	0.665523i	0.607423	−	0.794378i	$-0.292203\pi$
$563$	−9.11710	−	15.7913i	−0.384240	−	0.665523i	−0.991663	+	0.128855i	$0.958870\pi$
$564$		0			0
$565$	7.79129	−	13.4949i	0.327782	−	0.567735i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.3691	−	28.3521i	0.686228	−	1.18858i	−0.286822	−	0.957984i	$-0.592599\pi$
$569$	16.3691	−	28.3521i	0.686228	−	1.18858i	0.973049	−	0.230597i	$-0.0740680\pi$
$570$		0			0
$571$	10.6652	+	18.4726i	0.446323	+	0.773054i	0.998143	−	0.0609093i	$-0.0194000\pi$
$571$	10.6652	+	18.4726i	0.446323	+	0.773054i	−0.551821	+	0.833963i	$0.686067\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.818350			0.0341276
$576$		0			0
$577$	11.8348			0.492691			0.246346	−	0.969182i	$-0.420770\pi$
$577$	11.8348			0.492691			0.246346	+	0.969182i	$0.420770\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.40828	−	2.43920i	−0.0584251	−	0.101195i
$582$		0			0
$583$	6.79129	−	11.7629i	0.281266	−	0.487168i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.15118	+	3.72595i	−0.0887885	+	0.153786i	−0.906999	−	0.421132i	$-0.861633\pi$
$587$	−2.15118	+	3.72595i	−0.0887885	+	0.153786i	0.818211	+	0.574918i	$0.194966\pi$
$588$		0			0
$589$	−4.10436	−	7.10895i	−0.169117	−	0.292919i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.2415			−0.872282			−0.436141	−	0.899878i	$-0.643655\pi$
$593$	−21.2415			−0.872282			−0.436141	+	0.899878i	$0.643655\pi$
$594$		0			0
$595$	10.5826			0.433843
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.3537	+	19.6652i	0.463899	+	0.803496i	0.999151	−	0.0411959i	$-0.0131168\pi$
$599$	11.3537	+	19.6652i	0.463899	+	0.803496i	−0.535252	+	0.844692i	$0.679783\pi$
$600$		0			0
$601$	3.37386	−	5.84370i	0.137623	−	0.238370i	−0.788974	−	0.614427i	$-0.789387\pi$
$601$	3.37386	−	5.84370i	0.137623	−	0.238370i	0.926596	+	0.376058i	$0.122721\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.79513	+	11.7695i	−0.276261	+	0.478499i
$606$		0			0
$607$	−1.18693	−	2.05583i	−0.0481761	−	0.0834434i	0.840932	−	0.541141i	$-0.182008\pi$
$607$	−1.18693	−	2.05583i	−0.0481761	−	0.0834434i	−0.889108	+	0.457698i	$0.848674\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.818350			−0.0331069
$612$		0			0
$613$	−24.2087			−0.977781			−0.488890	−	0.872345i	$-0.662598\pi$
$613$	−24.2087			−0.977781			−0.488890	+	0.872345i	$0.662598\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.24473	+	7.35208i	0.170886	+	0.295984i	0.938730	−	0.344654i	$-0.112004\pi$
$617$	4.24473	+	7.35208i	0.170886	+	0.295984i	−0.767844	+	0.640637i	$0.778670\pi$
$618$		0			0
$619$	−4.83485	+	8.37420i	−0.194329	+	0.336588i	−0.946680	−	0.322174i	$-0.895586\pi$
$619$	−4.83485	+	8.37420i	−0.194329	+	0.336588i	0.752351	+	0.658762i	$0.228920\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.4168	−	28.4347i	0.657724	−	1.13921i
$624$		0			0
$625$	11.9564	+	20.7092i	0.478258	+	0.828366i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.37055			0.0546474
$630$		0			0
$631$	1.62614			0.0647355			0.0323677	−	0.999476i	$-0.489695\pi$
$631$	1.62614			0.0647355			0.0323677	+	0.999476i	$0.489695\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.4877	−	18.1652i	−0.416190	−	0.720862i
$636$		0			0
$637$	0.395644	−	0.685275i	0.0156760	−	0.0271516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.6588	−	40.9782i	0.934466	−	1.61854i	0.158882	−	0.987298i	$-0.449211\pi$
$641$	23.6588	−	40.9782i	0.934466	−	1.61854i	0.775584	−	0.631245i	$-0.217456\pi$
$642$		0			0
$643$	−11.8956	−	20.6039i	−0.469118	−	0.812537i	0.530258	−	0.847836i	$-0.322095\pi$
$643$	−11.8956	−	20.6039i	−0.469118	−	0.812537i	−0.999377	+	0.0352992i	$0.988762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28.2650			−1.11121			−0.555606	−	0.831446i	$-0.687514\pi$
$647$	−28.2650			−1.11121			−0.555606	+	0.831446i	$0.687514\pi$
$648$		0			0
$649$	−9.37386			−0.367956
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.20250	−	15.9392i	−0.360122	−	0.623749i	0.627859	−	0.778327i	$-0.283932\pi$
$653$	−9.20250	−	15.9392i	−0.360122	−	0.623749i	−0.987981	+	0.154578i	$0.950598\pi$
$654$		0			0
$655$	−17.5608	+	30.4162i	−0.686157	+	1.18846i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.18980	+	2.06080i	−0.0463481	+	0.0802772i	−0.888269	−	0.459324i	$-0.848092\pi$
$659$	−1.18980	+	2.06080i	−0.0463481	+	0.0802772i	0.841921	+	0.539601i	$0.181425\pi$
$660$		0			0
$661$	9.58258	+	16.5975i	0.372719	+	0.645568i	0.989983	−	0.141188i	$-0.0450922\pi$
$661$	9.58258	+	16.5975i	0.372719	+	0.645568i	−0.617264	+	0.786756i	$0.711759\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	27.9989			1.08575
$666$		0			0
$667$	−29.3303			−1.13567
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.3537	+	19.6652i	0.438304	+	0.759165i
$672$		0			0
$673$	9.29129	−	16.0930i	0.358153	−	0.620339i	−0.629499	−	0.777001i	$-0.716740\pi$
$673$	9.29129	−	16.0930i	0.358153	−	0.620339i	0.987652	+	0.156662i	$0.0500733\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.75650	−	13.4347i	0.298107	−	0.516336i	−0.677596	−	0.735434i	$-0.736978\pi$
$677$	7.75650	−	13.4347i	0.298107	−	0.516336i	0.975703	+	0.219098i	$0.0703116\pi$
$678$		0			0
$679$	−16.1652	−	27.9989i	−0.620362	−	1.07450i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.2023			−1.38524			−0.692621	−	0.721302i	$-0.743544\pi$
$683$	−36.2023			−1.38524			−0.692621	+	0.721302i	$0.743544\pi$
$684$		0			0
$685$	6.79129			0.259482
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.10260	−	5.37386i	−0.118200	−	0.204728i
$690$		0			0
$691$	−4.06080	+	7.03350i	−0.154480	+	0.267567i	−0.932870	−	0.360214i	$-0.882704\pi$
$691$	−4.06080	+	7.03350i	−0.154480	+	0.267567i	0.778390	+	0.627782i	$0.216037\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.07033	+	13.9782i	−0.306125	+	0.530224i
$696$		0			0
$697$	−2.68693	−	4.65390i	−0.101775	−	0.176279i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.4322			−0.809482			−0.404741	−	0.914431i	$-0.632638\pi$
$701$	−21.4322			−0.809482			−0.404741	+	0.914431i	$0.632638\pi$
$702$		0			0
$703$	3.62614			0.136762
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.80238	+	16.9782i	0.368656	+	0.638532i
$708$		0			0
$709$	−17.8956	+	30.9962i	−0.672085	+	1.16409i	0.305227	+	0.952280i	$0.401268\pi$
$709$	−17.8956	+	30.9962i	−0.672085	+	1.16409i	−0.977312	+	0.211806i	$0.932066\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.51178	−	6.08258i	0.131517	−	0.227794i
$714$		0			0
$715$	−2.39564	−	4.14938i	−0.0895920	−	0.155178i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.9518			−0.520313			−0.260156	−	0.965567i	$-0.583774\pi$
$719$	−13.9518			−0.520313			−0.260156	+	0.965567i	$0.583774\pi$
$720$		0			0
$721$	3.37386			0.125649
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.780626	+	1.35208i	0.0289917	+	0.0502151i
$726$		0			0
$727$	22.6434	−	39.2195i	0.839796	−	1.45457i	−0.0502687	−	0.998736i	$-0.516008\pi$
$727$	22.6434	−	39.2195i	0.839796	−	1.45457i	0.890065	−	0.455834i	$-0.150659\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.23658	+	3.87386i	−0.0827228	+	0.143280i
$732$		0			0
$733$	−13.0826	−	22.6597i	−0.483216	−	0.836955i	0.516598	−	0.856228i	$-0.327198\pi$
$733$	−13.0826	−	22.6597i	−0.483216	−	0.836955i	−0.999814	+	0.0192733i	$0.993865\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−29.2741			−1.07832
$738$		0			0
$739$	25.5826			0.941070			0.470535	−	0.882381i	$-0.344061\pi$
$739$	25.5826			0.941070			0.470535	+	0.882381i	$0.344061\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.86423	−	4.96099i	−0.105078	−	0.182001i	0.808692	−	0.588232i	$-0.200176\pi$
$743$	−2.86423	−	4.96099i	−0.105078	−	0.182001i	−0.913770	+	0.406232i	$0.866843\pi$
$744$		0			0
$745$	−7.58258	+	13.1334i	−0.277804	+	0.481171i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.61438	−	11.4564i	0.241684	−	0.418609i
$750$		0			0
$751$	−16.1869	−	28.0366i	−0.590670	−	1.02307i	−0.994142	−	0.108078i	$-0.965530\pi$
$751$	−16.1869	−	28.0366i	−0.590670	−	1.02307i	0.403473	−	0.914992i	$-0.367803\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.7629			−0.428094
$756$		0			0
$757$	−41.7042			−1.51576			−0.757882	−	0.652392i	$-0.773766\pi$
$757$	−41.7042			−1.51576			−0.757882	+	0.652392i	$0.773766\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.0148	−	32.9347i	−0.689287	−	1.19388i	−0.972069	−	0.234696i	$-0.924590\pi$
$761$	−19.0148	−	32.9347i	−0.689287	−	1.19388i	0.282782	−	0.959184i	$-0.408743\pi$
$762$		0			0
$763$	−26.1652	+	45.3194i	−0.947242	+	1.64067i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.14123	+	3.70871i	−0.0773152	+	0.133914i
$768$		0			0
$769$	5.20871	+	9.02175i	0.187831	+	0.325333i	0.944527	−	0.328434i	$-0.106521\pi$
$769$	5.20871	+	9.02175i	0.187831	+	0.325333i	−0.756696	+	0.653767i	$0.773188\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.9518			−0.501810			−0.250905	−	0.968012i	$-0.580728\pi$
$773$	−13.9518			−0.501810			−0.250905	+	0.968012i	$0.580728\pi$
$774$		0			0
$775$	−0.373864			−0.0134296
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.10895	−	12.3131i	−0.254705	−	0.441161i
$780$		0			0
$781$	−7.68693	+	13.3142i	−0.275060	+	0.476418i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.2746	+	26.4564i	−0.545175	+	0.944271i
$786$		0			0
$787$	−15.6869	−	27.1706i	−0.559179	−	0.968526i	−0.997565	−	0.0697394i	$-0.977783\pi$
$787$	−15.6869	−	27.1706i	−0.559179	−	0.968526i	0.438387	−	0.898787i	$-0.355550\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−19.8709			−0.706528
$792$		0			0
$793$	10.3739			0.368387
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.70703	−	16.8131i	−0.343841	−	0.595549i	0.641302	−	0.767289i	$-0.278395\pi$
$797$	−9.70703	−	16.8131i	−0.343841	−	0.595549i	−0.985142	+	0.171739i	$0.945061\pi$
$798$		0			0
$799$	−0.708712	+	1.22753i	−0.0250724	+	0.0434267i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.637600	+	1.10436i	−0.0225004	+	0.0389719i
$804$		0			0
$805$	11.9782	+	20.7469i	0.422177	+	0.731232i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.1334			0.461746			0.230873	−	0.972984i	$-0.425842\pi$
$809$	13.1334			0.461746			0.230873	+	0.972984i	$0.425842\pi$
$810$		0			0
$811$	2.46099			0.0864169			0.0432084	−	0.999066i	$-0.486242\pi$
$811$	2.46099			0.0864169			0.0432084	+	0.999066i	$0.486242\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.10985	−	10.5826i	−0.214019	−	0.370691i
$816$		0			0
$817$	−5.91742	+	10.2493i	−0.207024	+	0.358577i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26.0284	−	45.0826i	0.908399	−	1.57339i	0.0921098	−	0.995749i	$-0.470639\pi$
$821$	26.0284	−	45.0826i	0.908399	−	1.57339i	0.816289	−	0.577644i	$-0.196028\pi$
$822$		0			0
$823$	7.33485	+	12.7043i	0.255677	+	0.442845i	0.965079	−	0.261959i	$-0.0843684\pi$
$823$	7.33485	+	12.7043i	0.255677	+	0.442845i	−0.709402	+	0.704804i	$0.751035\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.93725			0.276005			0.138003	−	0.990432i	$-0.455932\pi$
$827$	7.93725			0.276005			0.138003	+	0.990432i	$0.455932\pi$
$828$		0			0
$829$	−17.9564			−0.623653			−0.311826	−	0.950139i	$-0.600941\pi$
$829$	−17.9564			−0.623653			−0.311826	+	0.950139i	$0.600941\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.685275	−	1.18693i	−0.0237434	−	0.0411247i
$834$		0			0
$835$	−20.0608	+	34.7463i	−0.694232	+	1.20245i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.8486	+	43.0390i	−0.857868	+	1.48587i	0.0160896	+	0.999871i	$0.494878\pi$
$839$	−24.8486	+	43.0390i	−0.857868	+	1.48587i	−0.873958	+	0.486001i	$0.838455\pi$
$840$		0			0
$841$	−13.4782	−	23.3450i	−0.464766	−	0.804999i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.18890			−0.0753005
$846$		0			0
$847$	17.3303			0.595476
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.55130	+	2.68693i	0.0531779	+	0.0921068i
$852$		0			0
$853$	−26.0998	+	45.2062i	−0.893640	+	1.54783i	−0.0581623	+	0.998307i	$0.518524\pi$
$853$	−26.0998	+	45.2062i	−0.893640	+	1.54783i	−0.835478	+	0.549524i	$0.814809\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.1947	−	34.9782i	0.689837	−	1.19483i	−0.282053	−	0.959399i	$-0.591015\pi$
$857$	20.1947	−	34.9782i	0.689837	−	1.19483i	0.971890	−	0.235434i	$-0.0756513\pi$
$858$		0			0
$859$	−10.2305	−	17.7197i	−0.349060	−	0.604589i	0.637023	−	0.770845i	$-0.280166\pi$
$859$	−10.2305	−	17.7197i	−0.349060	−	0.604589i	−0.986083	+	0.166255i	$0.946832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−35.9162			−1.22260			−0.611301	−	0.791398i	$-0.709354\pi$
$863$	−35.9162			−1.22260			−0.611301	+	0.791398i	$0.709354\pi$
$864$		0			0
$865$	51.7042			1.75799
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.09445	+	1.89564i	0.0371267	+	0.0643053i
$870$		0			0
$871$	−6.68693	+	11.5821i	−0.226578	+	0.392445i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.9122	−	27.5608i	0.537932	−	0.931725i
$876$		0			0
$877$	5.68693	+	9.85005i	0.192034	+	0.332613i	0.945924	−	0.324388i	$-0.105158\pi$
$877$	5.68693	+	9.85005i	0.192034	+	0.332613i	−0.753890	+	0.657000i	$0.771825\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−41.1323			−1.38578			−0.692891	−	0.721043i	$-0.743663\pi$
$881$	−41.1323			−1.38578			−0.692891	+	0.721043i	$0.743663\pi$
$882$		0			0
$883$	8.74773			0.294384			0.147192	−	0.989108i	$-0.452976\pi$
$883$	8.74773			0.294384			0.147192	+	0.989108i	$0.452976\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.0067	−	29.4564i	−0.571029	−	0.989050i	−0.996461	−	0.0840601i	$-0.973211\pi$
$887$	−17.0067	−	29.4564i	−0.571029	−	0.989050i	0.425432	−	0.904990i	$-0.360122\pi$
$888$		0			0
$889$	−13.3739	+	23.1642i	−0.448545	+	0.776903i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.87508	+	3.24773i	−0.0627470	+	0.108681i
$894$		0			0
$895$	−15.0826	−	26.1238i	−0.504155	−	0.873222i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.3996			0.446900
$900$		0			0
$901$	−10.7477			−0.358059
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.4168	−	28.4347i	−0.545711	−	0.945200i
$906$		0			0
$907$	−11.7695	+	20.3854i	−0.390800	+	0.676886i	−0.992555	−	0.121795i	$-0.961135\pi$
$907$	−11.7695	+	20.3854i	−0.390800	+	0.676886i	0.601755	+	0.798681i	$0.294468\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.8872	+	41.3739i	−0.791419	+	1.37078i	0.133670	+	0.991026i	$0.457324\pi$
$911$	−23.8872	+	41.3739i	−0.791419	+	1.37078i	−0.925089	+	0.379751i	$0.876010\pi$
$912$		0			0
$913$	1.10436	+	1.91280i	0.0365489	+	0.0633045i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	44.7871			1.47900
$918$		0			0
$919$	33.3303			1.09947			0.549733	−	0.835341i	$-0.314730\pi$
$919$	33.3303			1.09947			0.549733	+	0.835341i	$0.314730\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.51178	+	6.08258i	0.115592	+	0.200210i
$924$		0			0
$925$	0.0825757	−	0.143025i	0.00271507	−	0.00470264i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.81233	+	16.9955i	−0.321932	+	0.557603i	−0.980887	−	0.194580i	$-0.937666\pi$
$929$	−9.81233	+	16.9955i	−0.321932	+	0.557603i	0.658955	+	0.752183i	$0.270999\pi$
$930$		0			0
$931$	−1.81307	−	3.14033i	−0.0594209	−	0.102920i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.29875			−0.271398
$936$		0			0
$937$	−4.74773			−0.155101			−0.0775507	−	0.996988i	$-0.524710\pi$
$937$	−4.74773			−0.155101			−0.0775507	+	0.996988i	$0.524710\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.42548	+	7.66515i	0.144266	+	0.249877i	0.929099	−	0.369831i	$-0.120584\pi$
$941$	4.42548	+	7.66515i	0.144266	+	0.249877i	−0.784833	+	0.619708i	$0.787251\pi$
$942$		0			0
$943$	6.08258	−	10.5353i	0.198076	−	0.343078i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.99308	+	17.3085i	−0.324731	+	0.562451i	−0.981458	−	0.191677i	$-0.938607\pi$
$947$	−9.99308	+	17.3085i	−0.324731	+	0.562451i	0.656727	+	0.754129i	$0.271941\pi$
$948$		0			0
$949$	0.291288	+	0.504525i	0.00945560	+	0.0163776i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−53.6380			−1.73751			−0.868753	−	0.495246i	$-0.835078\pi$
$953$	−53.6380			−1.73751			−0.868753	+	0.495246i	$0.835078\pi$
$954$		0			0
$955$	5.00000			0.161796
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.33013	−	7.50000i	−0.139827	−	0.242188i
$960$		0			0
$961$	13.8956	−	24.0680i	0.448247	−	0.776386i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.6684	−	18.4782i	0.343428	−	0.594835i
$966$		0			0
$967$	−19.0000	−	32.9090i	−0.610999	−	1.05828i	−0.991072	−	0.133325i	$-0.957435\pi$
$967$	−19.0000	−	32.9090i	−0.610999	−	1.05828i	0.380074	−	0.924956i	$-0.375899\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.2578			0.810560			0.405280	−	0.914193i	$-0.367174\pi$
$971$	25.2578			0.810560			0.405280	+	0.914193i	$0.367174\pi$
$972$		0			0
$973$	20.5826			0.659847
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.1153	−	19.2523i	−0.355610	−	0.615935i	0.631612	−	0.775285i	$-0.282394\pi$
$977$	−11.1153	−	19.2523i	−0.355610	−	0.615935i	−0.987222	+	0.159350i	$0.949060\pi$
$978$		0			0
$979$	−12.8739	+	22.2982i	−0.411450	+	0.712653i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.8800	+	36.1652i	−0.665967	+	1.15349i	0.313055	+	0.949735i	$0.398648\pi$
$983$	−20.8800	+	36.1652i	−0.665967	+	1.15349i	−0.979022	+	0.203754i	$0.934686\pi$
$984$		0			0
$985$	2.68693	+	4.65390i	0.0856128	+	0.148286i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10.1262			−0.321993
$990$		0			0
$991$	3.49545			0.111037			0.0555184	−	0.998458i	$-0.482319\pi$
$991$	3.49545			0.111037			0.0555184	+	0.998458i	$0.482319\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.2351	+	29.8521i	0.546390	+	0.946375i
$996$		0			0
$997$	21.3521	−	36.9829i	0.676227	−	1.17126i	−0.299881	−	0.953977i	$-0.596947\pi$
$997$	21.3521	−	36.9829i	0.676227	−	1.17126i	0.976109	−	0.217283i	$-0.0697196\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.w.2809.4		8
3.2	odd	2	inner	4212.2.i.w.2809.1		8
9.2	odd	6		4212.2.a.g.1.4	yes	4
9.4	even	3	inner	4212.2.i.w.1405.4		8
9.5	odd	6	inner	4212.2.i.w.1405.1		8
9.7	even	3		4212.2.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4212.2.a.g.1.1	✓	4	9.7	even	3
4212.2.a.g.1.4	yes	4	9.2	odd	6
4212.2.i.w.1405.1		8	9.5	odd	6	inner
4212.2.i.w.1405.4		8	9.4	even	3	inner
4212.2.i.w.2809.1		8	3.2	odd	2	inner
4212.2.i.w.2809.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.09445 + 1.89564i) q^{5} +(1.39564 - 2.41733i) q^{7} +(-1.09445 + 1.89564i) q^{11} +(0.500000 + 0.866025i) q^{13} +1.73205 q^{17} +4.58258 q^{19} +(1.96048 + 3.39564i) q^{23} +(0.104356 - 0.180750i) q^{25} +(-3.74020 + 6.47822i) q^{29} +(-0.895644 - 1.55130i) q^{31} +6.10985 q^{35} +0.791288 q^{37} +(-1.55130 - 2.68693i) q^{41} +(-1.29129 + 2.23658i) q^{43} +(-0.409175 + 0.708712i) q^{47} +(-0.395644 - 0.685275i) q^{49} -6.20520 q^{53} -4.79129 q^{55} +(2.14123 + 3.70871i) q^{59} +(5.18693 - 8.98403i) q^{61} +(-1.09445 + 1.89564i) q^{65} +(6.68693 + 11.5821i) q^{67} +7.02355 q^{71} +0.582576 q^{73} +(3.05493 + 5.29129i) q^{77} +(0.500000 - 0.866025i) q^{79} +(0.504525 - 0.873864i) q^{83} +(1.89564 + 3.28335i) q^{85} +11.7629 q^{89} +2.79129 q^{91} +(5.01540 + 8.68693i) q^{95} +(5.79129 - 10.0308i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{7} + 4 q^{13} + 10 q^{25} + 2 q^{31} - 12 q^{37} + 8 q^{43} + 6 q^{49} - 20 q^{55} + 14 q^{61} + 26 q^{67} - 32 q^{73} + 4 q^{79} + 6 q^{85} + 4 q^{91} + 28 q^{97}+O(q^{100}) \) 8 * q + 2 * q^7 + 4 * q^13 + 10 * q^25 + 2 * q^31 - 12 * q^37 + 8 * q^43 + 6 * q^49 - 20 * q^55 + 14 * q^61 + 26 * q^67 - 32 * q^73 + 4 * q^79 + 6 * q^85 + 4 * q^91 + 28 * q^97

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