LMFDB - Embedded newform 4212.2.i.y.2809.3

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 = [12,0,0,0,0,0,-8,0,0,0,0,0,-6,0,0,0,0] 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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 41x^{8} - 152x^{6} + 656x^{4} - 1280x^{2} + 4096 $ x^12 - 5x^10 + 41x^8 - 152x^6 + 656x^4 - 1280*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.3
Root			$1.86751 - 0.715814i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.y.1405.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.313845 - 0.543595i) q^{5} +(1.47522 - 2.55516i) q^{7} +(2.73354 - 4.73463i) q^{11} +(-0.500000 - 0.866025i) q^{13} -0.356762 q^{17} -1.86325 q^{19} +(-1.23983 - 2.14744i) q^{23} +(2.30300 - 3.98892i) q^{25} +(0.373800 - 0.647441i) q^{29} +(-0.456405 - 0.790517i) q^{31} -1.85196 q^{35} +7.60601 q^{37} +(-2.04590 - 3.54359i) q^{41} +(-3.25941 + 5.64546i) q^{43} +(0.925981 - 1.60385i) q^{47} +(-0.852559 - 1.47668i) q^{49} -3.58401 q^{53} -3.43163 q^{55} +(-4.95782 - 8.58719i) q^{59} +(-3.90685 + 6.76686i) q^{61} +(-0.313845 + 0.543595i) q^{65} +(-0.456405 - 0.790517i) q^{67} -7.70691 q^{71} +5.20769 q^{73} +(-8.06516 - 13.9693i) q^{77} +(-1.30896 + 2.26719i) q^{79} +(-1.61363 + 2.79488i) q^{83} +(0.111968 + 0.193934i) q^{85} +15.2110 q^{89} -2.95044 q^{91} +(0.584772 + 1.01286i) q^{95} +(-5.95044 + 10.3065i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7} - 6 q^{13} + 4 q^{19} - 4 q^{25} - 18 q^{31} + 28 q^{37} - 20 q^{43} - 30 q^{49} - 28 q^{55} - 8 q^{61} - 18 q^{67} + 48 q^{73} - 48 q^{79} + 2 q^{85} + 16 q^{91} - 20 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 - 6 * q^13 + 4 * q^19 - 4 * q^25 - 18 * q^31 + 28 * q^37 - 20 * q^43 - 30 * q^49 - 28 * q^55 - 8 * q^61 - 18 * q^67 + 48 * q^73 - 48 * q^79 + 2 * q^85 + 16 * q^91 - 20 * 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
$5$	−0.313845	−	0.543595i	−0.140356	−	0.243103i	0.787275	−	0.616602i	$-0.211491\pi$
$5$	−0.313845	−	0.543595i	−0.140356	−	0.243103i	−0.927631	+	0.373499i	$0.878158\pi$
$6$		0			0
$7$	1.47522	−	2.55516i	0.557581	−	0.965759i	−0.440116	−	0.897941i	$-0.645063\pi$
$7$	1.47522	−	2.55516i	0.557581	−	0.965759i	0.997698	−	0.0678186i	$-0.0216039\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.73354	−	4.73463i	0.824193	−	1.42754i	−0.0783409	−	0.996927i	$-0.524962\pi$
$11$	2.73354	−	4.73463i	0.824193	−	1.42754i	0.902534	−	0.430618i	$-0.141704\pi$
$12$		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
$17$	−0.356762			−0.0865274			−0.0432637	−	0.999064i	$-0.513776\pi$
$17$	−0.356762			−0.0865274			−0.0432637	+	0.999064i	$0.513776\pi$
$18$		0			0
$19$	−1.86325			−0.427460			−0.213730	−	0.976893i	$-0.568561\pi$
$19$	−1.86325			−0.427460			−0.213730	+	0.976893i	$0.568561\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.23983	−	2.14744i	−0.258521	−	0.447772i	0.707325	−	0.706889i	$-0.249902\pi$
$23$	−1.23983	−	2.14744i	−0.258521	−	0.447772i	−0.965846	+	0.259117i	$0.916569\pi$
$24$		0			0
$25$	2.30300	−	3.98892i	0.460601	−	0.797784i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.373800	−	0.647441i	0.0694129	−	0.120227i	−0.829230	−	0.558907i	$-0.811221\pi$
$29$	0.373800	−	0.647441i	0.0694129	−	0.120227i	0.898643	+	0.438681i	$0.144554\pi$
$30$		0			0
$31$	−0.456405	−	0.790517i	−0.0819728	−	0.141981i	0.822124	−	0.569308i	$-0.192789\pi$
$31$	−0.456405	−	0.790517i	−0.0819728	−	0.141981i	−0.904097	+	0.427327i	$0.859455\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.85196			−0.313039
$36$		0			0
$37$	7.60601			1.25042			0.625210	−	0.780457i	$-0.285013\pi$
$37$	7.60601			1.25042			0.625210	+	0.780457i	$0.285013\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.04590	−	3.54359i	−0.319515	−	0.553417i	0.660872	−	0.750499i	$-0.270187\pi$
$41$	−2.04590	−	3.54359i	−0.319515	−	0.553417i	−0.980387	+	0.197082i	$0.936853\pi$
$42$		0			0
$43$	−3.25941	+	5.64546i	−0.497055	+	0.860925i	−0.999994	−	0.00339713i	$-0.998919\pi$
$43$	−3.25941	+	5.64546i	−0.497055	+	0.860925i	0.502939	+	0.864322i	$0.332252\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.925981	−	1.60385i	0.135068	−	0.233945i	−0.790555	−	0.612391i	$-0.790208\pi$
$47$	0.925981	−	1.60385i	0.135068	−	0.233945i	0.925624	+	0.378446i	$0.123541\pi$
$48$		0			0
$49$	−0.852559	−	1.47668i	−0.121794	−	0.210954i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.58401			−0.492302			−0.246151	−	0.969231i	$-0.579166\pi$
$53$	−3.58401			−0.492302			−0.246151	+	0.969231i	$0.579166\pi$
$54$		0			0
$55$	−3.43163			−0.462721
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.95782	−	8.58719i	−0.645453	−	1.11796i	−0.984197	−	0.177078i	$-0.943335\pi$
$59$	−4.95782	−	8.58719i	−0.645453	−	1.11796i	0.338744	−	0.940879i	$-0.389998\pi$
$60$		0			0
$61$	−3.90685	+	6.76686i	−0.500221	+	0.866408i	0.499779	+	0.866153i	$0.333414\pi$
$61$	−3.90685	+	6.76686i	−0.500221	+	0.866408i	−1.00000		0.000254929i	$0.999919\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.313845	+	0.543595i	−0.0389276	+	0.0674246i
$66$		0			0
$67$	−0.456405	−	0.790517i	−0.0557588	−	0.0965770i	0.836799	−	0.547511i	$-0.184424\pi$
$67$	−0.456405	−	0.790517i	−0.0557588	−	0.0965770i	−0.892557	+	0.450934i	$0.851091\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.70691			−0.914642			−0.457321	−	0.889302i	$-0.651191\pi$
$71$	−7.70691			−0.914642			−0.457321	+	0.889302i	$0.651191\pi$
$72$		0			0
$73$	5.20769			0.609514			0.304757	−	0.952430i	$-0.401425\pi$
$73$	5.20769			0.609514			0.304757	+	0.952430i	$0.401425\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.06516	−	13.9693i	−0.919110	−	1.59194i
$78$		0			0
$79$	−1.30896	+	2.26719i	−0.147270	+	0.255079i	−0.930218	−	0.367009i	$-0.880382\pi$
$79$	−1.30896	+	2.26719i	−0.147270	+	0.255079i	0.782948	+	0.622088i	$0.213715\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.61363	+	2.79488i	−0.177118	+	0.306778i	−0.940892	−	0.338706i	$-0.890011\pi$
$83$	−1.61363	+	2.79488i	−0.177118	+	0.306778i	0.763774	+	0.645484i	$0.223344\pi$
$84$		0			0
$85$	0.111968	+	0.193934i	0.0121446	+	0.0210351i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.2110			1.61237			0.806184	−	0.591665i	$-0.201529\pi$
$89$	15.2110			1.61237			0.806184	+	0.591665i	$0.201529\pi$
$90$		0			0
$91$	−2.95044			−0.309291
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.584772	+	1.01286i	0.0599964	+	0.103917i
$96$		0			0
$97$	−5.95044	+	10.3065i	−0.604176	+	1.04646i	0.388005	+	0.921657i	$0.373164\pi$
$97$	−5.95044	+	10.3065i	−0.604176	+	1.04646i	−0.992181	+	0.124806i	$0.960169\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.509264	+	0.882071i	−0.0506737	+	0.0877693i	−0.890250	−	0.455473i	$-0.849470\pi$
$101$	−0.509264	+	0.882071i	−0.0506737	+	0.0877693i	0.839576	+	0.543242i	$0.182804\pi$
$102$		0			0
$103$	5.66626	+	9.81425i	0.558313	+	0.967026i	0.997638	+	0.0686980i	$0.0218845\pi$
$103$	5.66626	+	9.81425i	0.558313	+	0.967026i	−0.439325	+	0.898328i	$0.644782\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.5678			−1.50500			−0.752498	−	0.658594i	$-0.771151\pi$
$107$	−15.5678			−1.50500			−0.752498	+	0.658594i	$0.771151\pi$
$108$		0			0
$109$	11.9009			1.13990			0.569949	−	0.821680i	$-0.306963\pi$
$109$	11.9009			1.13990			0.569949	+	0.821680i	$0.306963\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.06293	+	3.57311i	0.194064	+	0.336130i	0.946593	−	0.322430i	$-0.104500\pi$
$113$	2.06293	+	3.57311i	0.194064	+	0.336130i	−0.752529	+	0.658559i	$0.771166\pi$
$114$		0			0
$115$	−0.778225	+	1.34793i	−0.0725699	+	0.125695i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.526302	+	0.911583i	−0.0482461	+	0.0835646i
$120$		0			0
$121$	−9.44448	−	16.3583i	−0.858589	−	1.48712i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.02959			−0.539303
$126$		0			0
$127$	−1.90089			−0.168676			−0.0843382	−	0.996437i	$-0.526878\pi$
$127$	−1.90089			−0.168676			−0.0843382	+	0.996437i	$0.526878\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.15026	−	5.45641i	−0.275239	−	0.476728i	0.694956	−	0.719052i	$-0.255424\pi$
$131$	−3.15026	−	5.45641i	−0.275239	−	0.476728i	−0.970195	+	0.242324i	$0.922090\pi$
$132$		0			0
$133$	−2.74871	+	4.76091i	−0.238344	+	0.412823i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.25760	+	10.8385i	−0.534623	+	0.925994i	0.464559	+	0.885542i	$0.346213\pi$
$137$	−6.25760	+	10.8385i	−0.534623	+	0.925994i	−0.999182	+	0.0404513i	$0.987120\pi$
$138$		0			0
$139$	−6.06241	−	10.5004i	−0.514207	−	0.890633i	−0.999864	−	0.0164833i	$-0.994753\pi$
$139$	−6.06241	−	10.5004i	−0.514207	−	0.890633i	0.485657	−	0.874149i	$-0.338580\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.46708			−0.457180
$144$		0			0
$145$	−0.469261			−0.0389700
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.63067	+	4.55645i	0.215513	+	0.373279i	0.953431	−	0.301611i	$-0.0975244\pi$
$149$	2.63067	+	4.55645i	0.215513	+	0.373279i	−0.737918	+	0.674890i	$0.764191\pi$
$150$		0			0
$151$	−5.07907	+	8.79720i	−0.413329	+	0.715906i	−0.995251	−	0.0973378i	$-0.968967\pi$
$151$	−5.07907	+	8.79720i	−0.413329	+	0.715906i	0.581923	+	0.813244i	$0.302301\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.286481	+	0.496199i	−0.0230107	+	0.0398557i
$156$		0			0
$157$	−4.13078	−	7.15473i	−0.329672	−	0.571009i	0.652774	−	0.757552i	$-0.273605\pi$
$157$	−4.13078	−	7.15473i	−0.329672	−	0.571009i	−0.982447	+	0.186543i	$0.940272\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.31607			−0.576587
$162$		0			0
$163$	20.9385			1.64003			0.820016	−	0.572341i	$-0.193965\pi$
$163$	20.9385			1.64003			0.820016	+	0.572341i	$0.193965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.80978	−	11.7949i	−0.526956	−	0.912715i	−0.999507	−	0.0314115i	$-0.990000\pi$
$167$	−6.80978	−	11.7949i	−0.526956	−	0.912715i	0.472550	−	0.881304i	$-0.343334\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.5478	−	21.7334i	0.953990	−	1.65236i	0.217329	−	0.976098i	$-0.430266\pi$
$173$	12.5478	−	21.7334i	0.953990	−	1.65236i	0.736662	−	0.676262i	$-0.236401\pi$
$174$		0			0
$175$	−6.79488	−	11.7691i	−0.513645	−	0.889659i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.1333			−1.35535			−0.677673	−	0.735363i	$-0.737011\pi$
$179$	−18.1333			−1.35535			−0.677673	+	0.735363i	$0.737011\pi$
$180$		0			0
$181$	21.7641			1.61772			0.808858	−	0.588004i	$-0.200086\pi$
$181$	21.7641			1.61772			0.808858	+	0.588004i	$0.200086\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.38710	−	4.13459i	−0.175503	−	0.303981i
$186$		0			0
$187$	−0.975222	+	1.68913i	−0.0713153	+	0.123522i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.21171	−	9.02694i	0.377106	−	0.653166i	−0.613534	−	0.789668i	$-0.710253\pi$
$191$	5.21171	−	9.02694i	0.377106	−	0.653166i	0.990640	+	0.136502i	$0.0435860\pi$
$192$		0			0
$193$	−6.25941	−	10.8416i	−0.450562	−	0.780396i	0.547859	−	0.836571i	$-0.315443\pi$
$193$	−6.25941	−	10.8416i	−0.450562	−	0.780396i	−0.998421	+	0.0561744i	$0.982110\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.0068			−1.63916			−0.819582	−	0.572962i	$-0.805794\pi$
$197$	−23.0068			−1.63916			−0.819582	+	0.572962i	$0.805794\pi$
$198$		0			0
$199$	−15.8299			−1.12215			−0.561077	−	0.827763i	$-0.689613\pi$
$199$	−15.8299			−1.12215			−0.561077	+	0.827763i	$0.689613\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.10288	−	1.91024i	−0.0774067	−	0.134072i
$204$		0			0
$205$	−1.28419	+	2.22428i	−0.0896915	+	0.155350i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.09328	+	8.82182i	−0.352310	+	0.610218i
$210$		0			0
$211$	5.58123	+	9.66697i	0.384228	+	0.665502i	0.991662	−	0.128868i	$-0.0411345\pi$
$211$	5.58123	+	9.66697i	0.384228	+	0.665502i	−0.607434	+	0.794370i	$0.707801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.09179			0.279058
$216$		0			0
$217$	−2.69320			−0.182826
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.178381	+	0.308965i	0.0119992	+	0.0207832i
$222$		0			0
$223$	7.27823	−	12.6063i	0.487386	−	0.844177i	−0.512509	−	0.858682i	$-0.671284\pi$
$223$	7.27823	−	12.6063i	0.487386	−	0.844177i	0.999895	+	0.0145047i	$0.00461714\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.5174	−	18.2167i	0.698067	−	1.20909i	−0.271068	−	0.962560i	$-0.587377\pi$
$227$	10.5174	−	18.2167i	0.698067	−	1.20909i	0.969136	−	0.246528i	$-0.0792897\pi$
$228$		0			0
$229$	−8.97738	−	15.5493i	−0.593242	−	1.02753i	−0.993792	−	0.111251i	$-0.964514\pi$
$229$	−8.97738	−	15.5493i	−0.593242	−	1.02753i	0.400550	−	0.916275i	$-0.368819\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.57442			0.561729			0.280864	−	0.959747i	$-0.409379\pi$
$233$	8.57442			0.561729			0.280864	+	0.959747i	$0.409379\pi$
$234$		0			0
$235$	−1.16246			−0.0758303
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.13619	−	7.16410i	−0.267548	−	0.463407i	0.700680	−	0.713476i	$-0.252880\pi$
$239$	−4.13619	−	7.16410i	−0.267548	−	0.463407i	−0.968228	+	0.250069i	$0.919547\pi$
$240$		0			0
$241$	0.444482	−	0.769866i	0.0286316	−	0.0495914i	−0.851355	−	0.524591i	$-0.824218\pi$
$241$	0.444482	−	0.769866i	0.0286316	−	0.0495914i	0.879986	+	0.474999i	$0.157552\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.535142	+	0.926894i	−0.0341890	+	0.0592171i
$246$		0			0
$247$	0.931627	+	1.61363i	0.0592780	+	0.102673i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.2170			1.21297			0.606483	−	0.795096i	$-0.292580\pi$
$251$	19.2170			1.21297			0.606483	+	0.795096i	$0.292580\pi$
$252$		0			0
$253$	−13.5565			−0.852287
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.46857	−	11.2039i	−0.403498	−	0.698880i	0.590647	−	0.806930i	$-0.298873\pi$
$257$	−6.46857	−	11.2039i	−0.403498	−	0.698880i	−0.994145	+	0.108050i	$0.965539\pi$
$258$		0			0
$259$	11.2205	−	19.4346i	0.697211	−	1.20760i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.4108	−	21.4962i	0.765285	−	1.32551i	−0.174811	−	0.984602i	$-0.555931\pi$
$263$	12.4108	−	21.4962i	0.765285	−	1.32551i	0.940096	−	0.340910i	$-0.110735\pi$
$264$		0			0
$265$	1.12482	+	1.94825i	0.0690973	+	0.119680i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.8468			−1.14911			−0.574555	−	0.818466i	$-0.694825\pi$
$269$	−18.8468			−1.14911			−0.574555	+	0.818466i	$0.694825\pi$
$270$		0			0
$271$	30.8770			1.87565			0.937823	−	0.347113i	$-0.112838\pi$
$271$	30.8770			1.87565			0.937823	+	0.347113i	$0.112838\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.5907	−	21.8077i	−0.759248	−	1.31506i
$276$		0			0
$277$	−10.6107	+	18.3783i	−0.637538	+	1.10425i	0.348434	+	0.937333i	$0.386714\pi$
$277$	−10.6107	+	18.3783i	−0.637538	+	1.10425i	−0.985971	+	0.166914i	$0.946620\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.221298	+	0.383299i	−0.0132015	+	0.0228657i	−0.872551	−	0.488524i	$-0.837536\pi$
$281$	−0.221298	+	0.383299i	−0.0132015	+	0.0228657i	0.859349	+	0.511389i	$0.170869\pi$
$282$		0			0
$283$	6.55645	+	11.3561i	0.389741	+	0.675050i	0.992414	−	0.122937i	$-0.0392314\pi$
$283$	6.55645	+	11.3561i	0.389741	+	0.675050i	−0.602674	+	0.797988i	$0.705898\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0726			−0.712623
$288$		0			0
$289$	−16.8727			−0.992513
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.61214	+	2.79231i	0.0941822	+	0.163128i	0.909267	−	0.416213i	$-0.136643\pi$
$293$	1.61214	+	2.79231i	0.0941822	+	0.163128i	−0.815085	+	0.579342i	$0.803310\pi$
$294$		0			0
$295$	−3.11197	+	5.39009i	−0.181186	+	0.313823i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.23983	+	2.14744i	−0.0717010	+	0.124190i
$300$		0			0
$301$	9.61670	+	16.6566i	0.554297	+	0.960071i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.90457			0.280835
$306$		0			0
$307$	1.13675			0.0648775			0.0324388	−	0.999474i	$-0.489673\pi$
$307$	1.13675			0.0648775			0.0324388	+	0.999474i	$0.489673\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.86974	−	13.6308i	−0.446252	−	0.772931i	0.551887	−	0.833919i	$-0.313908\pi$
$311$	−7.86974	−	13.6308i	−0.446252	−	0.772931i	−0.998138	+	0.0609883i	$0.980575\pi$
$312$		0			0
$313$	−2.46237	+	4.26494i	−0.139181	+	0.241069i	−0.927187	−	0.374599i	$-0.877780\pi$
$313$	−2.46237	+	4.26494i	−0.139181	+	0.241069i	0.788006	+	0.615668i	$0.211114\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.71872	+	13.3692i	−0.433527	+	0.750890i	−0.997174	−	0.0751254i	$-0.976064\pi$
$317$	−7.71872	+	13.3692i	−0.433527	+	0.750890i	0.563648	+	0.826015i	$0.309398\pi$
$318$		0			0
$319$	−2.04359	−	3.53961i	−0.114419	−	0.198180i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.664737			0.0369870
$324$		0			0
$325$	−4.60601			−0.255495
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.73205	−	4.73206i	−0.150623	−	0.260887i
$330$		0			0
$331$	−5.86541	+	10.1592i	−0.322392	+	0.558400i	−0.980981	−	0.194103i	$-0.937820\pi$
$331$	−5.86541	+	10.1592i	−0.322392	+	0.558400i	0.658589	+	0.752503i	$0.271154\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.286481	+	0.496199i	−0.0156521	+	0.0271102i
$336$		0			0
$337$	−7.38423	−	12.7899i	−0.402245	−	0.696708i	0.591752	−	0.806120i	$-0.298437\pi$
$337$	−7.38423	−	12.7899i	−0.402245	−	0.696708i	−0.993996	+	0.109412i	$0.965103\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.99041			−0.270246
$342$		0			0
$343$	15.6222			0.843522
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.82682	−	13.5565i	−0.420166	−	0.727748i	0.575790	−	0.817598i	$-0.304695\pi$
$347$	−7.82682	−	13.5565i	−0.420166	−	0.727748i	−0.995955	+	0.0898495i	$0.971361\pi$
$348$		0			0
$349$	0.199157	−	0.344950i	0.0106606	−	0.0184648i	−0.860646	−	0.509204i	$-0.829940\pi$
$349$	0.199157	−	0.344950i	0.0106606	−	0.0184648i	0.871307	+	0.490739i	$0.163273\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.75941	−	3.04740i	0.0936442	−	0.162197i	−0.815398	−	0.578901i	$-0.803482\pi$
$353$	1.75941	−	3.04740i	0.0936442	−	0.162197i	0.909042	+	0.416705i	$0.136815\pi$
$354$		0			0
$355$	2.41877	+	4.18944i	0.128375	+	0.222352i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	35.1105			1.85306			0.926530	−	0.376222i	$-0.122777\pi$
$359$	35.1105			1.85306			0.926530	+	0.376222i	$0.122777\pi$
$360$		0			0
$361$	−15.5283			−0.817278
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.63441	−	2.83087i	−0.0855487	−	0.148175i
$366$		0			0
$367$	−3.46710	+	6.00519i	−0.180981	+	0.313469i	−0.942215	−	0.335009i	$-0.891261\pi$
$367$	−3.46710	+	6.00519i	−0.180981	+	0.313469i	0.761234	+	0.648478i	$0.224594\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.28721	+	9.15772i	−0.274498	+	0.475445i
$372$		0			0
$373$	11.1684	+	19.3443i	0.578279	+	1.00161i	0.995677	+	0.0928845i	$0.0296087\pi$
$373$	11.1684	+	19.3443i	0.578279	+	1.00161i	−0.417398	+	0.908724i	$0.637058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.747600			−0.0385034
$378$		0			0
$379$	−28.9009			−1.48454			−0.742269	−	0.670102i	$-0.766251\pi$
$379$	−28.9009			−1.48454			−0.742269	+	0.670102i	$0.766251\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.59808	−	4.50000i	−0.132755	−	0.229939i	0.791982	−	0.610544i	$-0.209049\pi$
$383$	−2.59808	−	4.50000i	−0.132755	−	0.229939i	−0.924738	+	0.380605i	$0.875716\pi$
$384$		0			0
$385$	−5.06241	+	8.76835i	−0.258004	+	0.446877i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.34717	+	7.52951i	−0.220410	+	0.381761i	−0.954933	−	0.296823i	$-0.904073\pi$
$389$	−4.34717	+	7.52951i	−0.220410	+	0.381761i	0.734522	+	0.678584i	$0.237406\pi$
$390$		0			0
$391$	0.442322	+	0.766124i	0.0223692	+	0.0387446i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.64325			0.0826807
$396$		0			0
$397$	30.5941			1.53547			0.767737	−	0.640766i	$-0.221383\pi$
$397$	30.5941			1.53547			0.767737	+	0.640766i	$0.221383\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.02373	+	15.6296i	0.450624	+	0.780503i	0.998425	−	0.0561056i	$-0.0178684\pi$
$401$	9.02373	+	15.6296i	0.450624	+	0.780503i	−0.547801	+	0.836608i	$0.684535\pi$
$402$		0			0
$403$	−0.456405	+	0.790517i	−0.0227352	+	0.0393785i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.7913	−	36.0116i	1.03059	−	1.78503i
$408$		0			0
$409$	0.865415	+	1.49894i	0.0427920	+	0.0741179i	0.886628	−	0.462483i	$-0.153041\pi$
$409$	0.865415	+	1.49894i	0.0427920	+	0.0741179i	−0.843836	+	0.536601i	$0.819708\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−29.2555			−1.43957
$414$		0			0
$415$	2.02571			0.0994382
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.43597	−	9.41538i	−0.265565	−	0.459972i	0.702147	−	0.712032i	$-0.252225\pi$
$419$	−5.43597	−	9.41538i	−0.265565	−	0.459972i	−0.967711	+	0.252061i	$0.918892\pi$
$420$		0			0
$421$	−9.56837	+	16.5729i	−0.466334	+	0.807714i	−0.999261	−	0.0384473i	$-0.987759\pi$
$421$	−9.56837	+	16.5729i	−0.466334	+	0.807714i	0.532927	+	0.846161i	$0.321092\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.821623	+	1.42309i	−0.0398546	+	0.0690301i
$426$		0			0
$427$	11.5269	+	19.9652i	0.557828	+	0.966186i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.4404			0.840075			0.420038	−	0.907507i	$-0.362017\pi$
$431$	17.4404			0.840075			0.420038	+	0.907507i	$0.362017\pi$
$432$		0			0
$433$	8.03763			0.386264			0.193132	−	0.981173i	$-0.438135\pi$
$433$	8.03763			0.386264			0.193132	+	0.981173i	$0.438135\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.31011	+	4.00123i	0.110508	+	0.191405i
$438$		0			0
$439$	12.6791	−	21.9609i	0.605141	−	1.04814i	−0.386888	−	0.922127i	$-0.626450\pi$
$439$	12.6791	−	21.9609i	0.605141	−	1.04814i	0.992029	−	0.126008i	$-0.0402166\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15.4753	+	26.8039i	−0.735252	+	1.27349i	0.219361	+	0.975644i	$0.429603\pi$
$443$	−15.4753	+	26.8039i	−0.735252	+	1.27349i	−0.954613	+	0.297850i	$0.903731\pi$
$444$		0			0
$445$	−4.77390	−	8.26864i	−0.226305	−	0.391971i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.02959			−0.284554			−0.142277	−	0.989827i	$-0.545442\pi$
$449$	−6.02959			−0.284554			−0.142277	+	0.989827i	$0.545442\pi$
$450$		0			0
$451$	−22.3701			−1.05337
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.925981	+	1.60385i	0.0434107	+	0.0751895i
$456$		0			0
$457$	3.95044	−	6.84237i	0.184794	−	0.320073i	−0.758713	−	0.651425i	$-0.774172\pi$
$457$	3.95044	−	6.84237i	0.184794	−	0.320073i	0.943507	+	0.331352i	$0.107505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.43376	−	2.48334i	0.0667768	−	0.115661i	−0.830704	−	0.556714i	$-0.812062\pi$
$461$	1.43376	−	2.48334i	0.0667768	−	0.115661i	0.897481	+	0.441054i	$0.145395\pi$
$462$		0			0
$463$	−11.6979	−	20.2614i	−0.543649	−	0.941627i	−0.998691	−	0.0511574i	$-0.983709\pi$
$463$	−11.6979	−	20.2614i	−0.543649	−	0.941627i	0.455042	−	0.890470i	$-0.349624\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.7099			0.541869			0.270935	−	0.962598i	$-0.412667\pi$
$467$	11.7099			0.541869			0.270935	+	0.962598i	$0.412667\pi$
$468$		0			0
$469$	−2.69320			−0.124360
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.8194	+	30.8642i	0.819339	+	1.41914i
$474$		0			0
$475$	−4.29108	+	7.43237i	−0.196888	+	0.341020i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.04441	−	1.80896i	0.0477201	−	0.0826537i	−0.841179	−	0.540757i	$-0.818138\pi$
$479$	1.04441	−	1.80896i	0.0477201	−	0.0826537i	0.888899	+	0.458104i	$0.151471\pi$
$480$		0			0
$481$	−3.80300	−	6.58699i	−0.173402	−	0.300341i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.47006			0.339198
$486$		0			0
$487$	18.0138			0.816283			0.408141	−	0.912919i	$-0.366177\pi$
$487$	18.0138			0.816283			0.408141	+	0.912919i	$0.366177\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.16878	+	7.22055i	0.188135	+	0.325859i	0.944628	−	0.328142i	$-0.106423\pi$
$491$	4.16878	+	7.22055i	0.188135	+	0.325859i	−0.756494	+	0.654001i	$0.773089\pi$
$492$		0			0
$493$	−0.133357	+	0.230982i	−0.00600612	+	0.0104029i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.3694	+	19.6924i	−0.509987	+	0.883324i
$498$		0			0
$499$	−12.0650	−	20.8972i	−0.540103	−	0.935486i	−0.998898	−	0.0469433i	$-0.985052\pi$
$499$	−12.0650	−	20.8972i	−0.540103	−	0.935486i	0.458795	−	0.888542i	$-0.348281\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.6521			0.965420			0.482710	−	0.875780i	$-0.339652\pi$
$503$	21.6521			0.965420			0.482710	+	0.875780i	$0.339652\pi$
$504$		0			0
$505$	0.639319			0.0284493
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.68838	−	9.85256i	−0.252133	−	0.436707i	0.711980	−	0.702200i	$-0.247799\pi$
$509$	−5.68838	−	9.85256i	−0.252133	−	0.436707i	−0.964113	+	0.265493i	$0.914465\pi$
$510$		0			0
$511$	7.68250	−	13.3065i	0.339854	−	0.588644i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.55665	−	6.16030i	0.156725	−	0.271455i
$516$		0			0
$517$	−5.06241	−	8.76835i	−0.222645	−	0.385632i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.2185			−1.28008			−0.640042	−	0.768340i	$-0.721083\pi$
$521$	−29.2185			−1.28008			−0.640042	+	0.768340i	$0.721083\pi$
$522$		0			0
$523$	9.01379			0.394145			0.197073	−	0.980389i	$-0.436856\pi$
$523$	9.01379			0.394145			0.197073	+	0.980389i	$0.436856\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.162828	+	0.282026i	0.00709289	+	0.0122852i
$528$		0			0
$529$	8.42567	−	14.5937i	0.366333	−	0.634508i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.04590	+	3.54359i	−0.0886176	+	0.153490i
$534$		0			0
$535$	4.88587	+	8.46258i	0.211235	+	0.365869i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.32202			−0.401528
$540$		0			0
$541$	19.7027			0.847084			0.423542	−	0.905877i	$-0.360787\pi$
$541$	19.7027			0.847084			0.423542	+	0.905877i	$0.360787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.73503	−	6.46926i	−0.159991	−	0.277113i
$546$		0			0
$547$	13.1744	−	22.8187i	0.563296	−	0.975657i	−0.433910	−	0.900956i	$-0.642866\pi$
$547$	13.1744	−	22.8187i	0.563296	−	0.975657i	0.997206	−	0.0747008i	$-0.0238002\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.696485	+	1.20635i	−0.0296712	+	0.0513921i
$552$		0			0
$553$	3.86203	+	6.68923i	0.164230	+	0.284455i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.40785			0.398624			0.199312	−	0.979936i	$-0.436129\pi$
$557$	9.40785			0.398624			0.199312	+	0.979936i	$0.436129\pi$
$558$		0			0
$559$	6.51882			0.275717
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.59295	+	16.6155i	0.404295	+	0.700259i	0.994239	−	0.107185i	$-0.0341837\pi$
$563$	9.59295	+	16.6155i	0.404295	+	0.700259i	−0.589945	+	0.807444i	$0.700850\pi$
$564$		0			0
$565$	1.29488	−	2.24280i	0.0544761	−	0.0943553i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−23.4338	+	40.5885i	−0.982396	+	1.70156i	−0.329416	+	0.944185i	$0.606852\pi$
$569$	−23.4338	+	40.5885i	−0.982396	+	1.70156i	−0.652980	+	0.757375i	$0.726482\pi$
$570$		0			0
$571$	15.2099	+	26.3442i	0.636513	+	1.10247i	0.986193	+	0.165603i	$0.0529570\pi$
$571$	15.2099	+	26.3442i	0.636513	+	1.10247i	−0.349680	+	0.936869i	$0.613710\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.4213			−0.476301
$576$		0			0
$577$	20.7685			0.864602			0.432301	−	0.901729i	$-0.357702\pi$
$577$	20.7685			0.864602			0.432301	+	0.901729i	$0.357702\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.76091	+	8.24614i	0.197516	+	0.342108i
$582$		0			0
$583$	−9.79704	+	16.9690i	−0.405752	+	0.702783i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.2006	−	28.0603i	0.668670	−	1.15817i	−0.309606	−	0.950865i	$-0.600197\pi$
$587$	16.2006	−	28.0603i	0.668670	−	1.15817i	0.978276	−	0.207306i	$-0.0664695\pi$
$588$		0			0
$589$	0.850399	+	1.47293i	0.0350401	+	0.0606912i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.67282			−0.150825			−0.0754123	−	0.997152i	$-0.524027\pi$
$593$	−3.67282			−0.150825			−0.0754123	+	0.997152i	$0.524027\pi$
$594$		0			0
$595$	0.660709			0.0270864
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.8165	+	25.6629i	0.605384	+	1.04856i	0.991991	+	0.126312i	$0.0403139\pi$
$599$	14.8165	+	25.6629i	0.605384	+	1.04856i	−0.386606	+	0.922245i	$0.626353\pi$
$600$		0			0
$601$	17.1038	−	29.6247i	0.697680	−	1.20842i	−0.271588	−	0.962414i	$-0.587549\pi$
$601$	17.1038	−	29.6247i	0.697680	−	1.20842i	0.969269	−	0.246004i	$-0.0791178\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.92820	+	10.2679i	−0.241016	+	0.417451i
$606$		0			0
$607$	−15.8783	−	27.5020i	−0.644479	−	1.11627i	−0.984422	−	0.175824i	$-0.943741\pi$
$607$	−15.8783	−	27.5020i	−0.644479	−	1.11627i	0.339942	−	0.940446i	$-0.389592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.85196			−0.0749224
$612$		0			0
$613$	8.22394			0.332162			0.166081	−	0.986112i	$-0.446889\pi$
$613$	8.22394			0.332162			0.166081	+	0.986112i	$0.446889\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.5086	+	35.5219i	0.825645	+	1.43006i	0.901426	+	0.432934i	$0.142522\pi$
$617$	20.5086	+	35.5219i	0.825645	+	1.43006i	−0.0757812	+	0.997124i	$0.524145\pi$
$618$		0			0
$619$	−2.19104	+	3.79498i	−0.0880652	+	0.152533i	−0.906693	−	0.421791i	$-0.861402\pi$
$619$	−2.19104	+	3.79498i	−0.0880652	+	0.152533i	0.818628	+	0.574324i	$0.194735\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22.4397	−	38.8666i	0.899026	−	1.55716i
$624$		0			0
$625$	−9.62266	−	16.6669i	−0.384907	−	0.666678i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.71353			−0.108196
$630$		0			0
$631$	−41.2754			−1.64315			−0.821573	−	0.570103i	$-0.806903\pi$
$631$	−41.2754			−1.64315			−0.821573	+	0.570103i	$0.806903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.596583	+	1.03331i	0.0236747	+	0.0410058i
$636$		0			0
$637$	−0.852559	+	1.47668i	−0.0337796	+	0.0585080i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.42566	+	12.8616i	−0.293296	+	0.508003i	−0.974587	−	0.224009i	$-0.928085\pi$
$641$	−7.42566	+	12.8616i	−0.293296	+	0.508003i	0.681291	+	0.732012i	$0.261419\pi$
$642$		0			0
$643$	8.17818	+	14.1650i	0.322516	+	0.558614i	0.981006	−	0.193975i	$-0.0621381\pi$
$643$	8.17818	+	14.1650i	0.322516	+	0.558614i	−0.658491	+	0.752589i	$0.728805\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.8608			0.938067			0.469033	−	0.883180i	$-0.344602\pi$
$647$	23.8608			0.938067			0.469033	+	0.883180i	$0.344602\pi$
$648$		0			0
$649$	−54.2096			−2.12791
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.7247	+	20.3077i	0.458822	+	0.794703i	0.998899	−	0.0469125i	$-0.0149382\pi$
$653$	11.7247	+	20.3077i	0.458822	+	0.794703i	−0.540077	+	0.841616i	$0.681605\pi$
$654$		0			0
$655$	−1.97738	+	3.42493i	−0.0772627	+	0.133823i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.55665	+	7.89235i	−0.177502	+	0.307442i	−0.941024	−	0.338339i	$-0.890135\pi$
$659$	−4.55665	+	7.89235i	−0.177502	+	0.307442i	0.763522	+	0.645781i	$0.223468\pi$
$660$		0			0
$661$	−15.6274	−	27.0674i	−0.607835	−	1.05280i	−0.991596	−	0.129369i	$-0.958705\pi$
$661$	−15.6274	−	27.0674i	−0.607835	−	1.05280i	0.383761	−	0.923432i	$-0.374629\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.45068			0.133811
$666$		0			0
$667$	−1.85379			−0.0717789
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.3591	+	36.9950i	0.824557	+	1.42818i
$672$		0			0
$673$	−8.06837	+	13.9748i	−0.311013	+	0.538690i	−0.978582	−	0.205858i	$-0.934002\pi$
$673$	−8.06837	+	13.9748i	−0.311013	+	0.538690i	0.667569	+	0.744548i	$0.267335\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.8328	+	36.0834i	−0.800668	+	1.38680i	0.118509	+	0.992953i	$0.462188\pi$
$677$	−20.8328	+	36.0834i	−0.800668	+	1.38680i	−0.919177	+	0.393844i	$0.871145\pi$
$678$		0			0
$679$	17.5565	+	30.4087i	0.673755	+	1.16698i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.1369			−0.426143			−0.213071	−	0.977037i	$-0.568347\pi$
$683$	−11.1369			−0.426143			−0.213071	+	0.977037i	$0.568347\pi$
$684$		0			0
$685$	7.85565			0.300149
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.79201	+	3.10385i	0.0682700	+	0.118247i
$690$		0			0
$691$	11.4611	−	19.8513i	0.436002	−	0.755178i	−0.561375	−	0.827562i	$-0.689727\pi$
$691$	11.4611	−	19.8513i	0.436002	−	0.755178i	0.997377	+	0.0723837i	$0.0230606\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.80531	+	6.59099i	−0.144344	+	0.250011i
$696$		0			0
$697$	0.729897	+	1.26422i	0.0276468	+	0.0478857i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.1635			0.497180			0.248590	−	0.968609i	$-0.420033\pi$
$701$	13.1635			0.497180			0.248590	+	0.968609i	$0.420033\pi$
$702$		0			0
$703$	−14.1719			−0.534504
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.50255	+	2.60250i	0.0565094	+	0.0978771i
$708$		0			0
$709$	3.19916	−	5.54110i	0.120147	−	0.208100i	−0.799679	−	0.600428i	$-0.794997\pi$
$709$	3.19916	−	5.54110i	0.120147	−	0.208100i	0.919825	+	0.392328i	$0.128330\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.13173	+	1.96021i	−0.0423835	+	0.0734103i
$714$		0			0
$715$	1.71581	+	2.97188i	0.0641678	+	0.111142i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21.5116			0.802246			0.401123	−	0.916024i	$-0.368620\pi$
$719$	21.5116			0.802246			0.401123	+	0.916024i	$0.368620\pi$
$720$		0			0
$721$	33.4359			1.24522
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.72173	−	2.98212i	−0.0639433	−	0.110753i
$726$		0			0
$727$	−2.01286	+	3.48637i	−0.0746527	+	0.129302i	−0.900935	−	0.433954i	$-0.857118\pi$
$727$	−2.01286	+	3.48637i	−0.0746527	+	0.129302i	0.826283	+	0.563256i	$0.190451\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.16283	−	2.01408i	0.0430089	−	0.0744936i
$732$		0			0
$733$	13.9740	+	24.2037i	0.516141	+	0.893983i	0.999824	+	0.0187397i	$0.00596538\pi$
$733$	13.9740	+	24.2037i	0.516141	+	0.893983i	−0.483683	+	0.875243i	$0.660701\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.99041			−0.183824
$738$		0			0
$739$	−30.8299			−1.13410			−0.567049	−	0.823684i	$-0.691915\pi$
$739$	−30.8299			−1.13410			−0.567049	+	0.823684i	$0.691915\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.80978	−	11.7949i	−0.249827	−	0.432712i	0.713651	−	0.700501i	$-0.247040\pi$
$743$	−6.80978	−	11.7949i	−0.249827	−	0.432712i	−0.963478	+	0.267789i	$0.913707\pi$
$744$		0			0
$745$	1.65124	−	2.86003i	0.0604968	−	0.104784i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.9660	+	39.7782i	−0.839158	+	1.45346i
$750$		0			0
$751$	3.47049	+	6.01106i	0.126640	+	0.219347i	0.922373	−	0.386301i	$-0.126247\pi$
$751$	3.47049	+	6.01106i	0.126640	+	0.219347i	−0.795733	+	0.605648i	$0.792914\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.37615			0.232052
$756$		0			0
$757$	−17.3701			−0.631329			−0.315664	−	0.948871i	$-0.602227\pi$
$757$	−17.3701			−0.631329			−0.315664	+	0.948871i	$0.602227\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.2798	+	40.3218i	0.843893	+	1.46166i	0.886579	+	0.462576i	$0.153075\pi$
$761$	23.2798	+	40.3218i	0.843893	+	1.46166i	−0.0426869	+	0.999088i	$0.513592\pi$
$762$		0			0
$763$	17.5565	−	30.4087i	0.635586	−	1.10087i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.95782	+	8.58719i	−0.179016	+	0.310065i
$768$		0			0
$769$	22.6436	+	39.2199i	0.816551	+	1.41431i	0.908209	+	0.418517i	$0.137450\pi$
$769$	22.6436	+	39.2199i	0.816551	+	1.41431i	−0.0916587	+	0.995790i	$0.529217\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	32.6233			1.17338			0.586690	−	0.809812i	$-0.300431\pi$
$773$	32.6233			1.17338			0.586690	+	0.809812i	$0.300431\pi$
$774$		0			0
$775$	−4.20441			−0.151027
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.81202	+	6.60262i	0.136580	+	0.236563i
$780$		0			0
$781$	−21.0671	+	36.4894i	−0.753842	+	1.30569i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.59285	+	4.49095i	−0.0925427	+	0.160289i
$786$		0			0
$787$	−3.59788	−	6.23172i	−0.128251	−	0.222137i	0.794748	−	0.606939i	$-0.207603\pi$
$787$	−3.59788	−	6.23172i	−0.128251	−	0.222137i	−0.922999	+	0.384802i	$0.874270\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.1731			0.432827
$792$		0			0
$793$	7.81370			0.277473
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.3472	−	33.5104i	−0.685314	−	1.18700i	−0.973338	−	0.229376i	$-0.926331\pi$
$797$	−19.3472	−	33.5104i	−0.685314	−	1.18700i	0.288024	−	0.957623i	$-0.407002\pi$
$798$		0			0
$799$	−0.330354	+	0.572190i	−0.0116871	+	0.0202426i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.2354	−	24.6565i	0.502358	−	0.870109i
$804$		0			0
$805$	2.29611	+	3.97698i	0.0809272	+	0.140170i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.7855			0.836252			0.418126	−	0.908389i	$-0.362687\pi$
$809$	23.7855			0.836252			0.418126	+	0.908389i	$0.362687\pi$
$810$		0			0
$811$	−36.9266			−1.29667			−0.648334	−	0.761356i	$-0.724534\pi$
$811$	−36.9266			−1.29667			−0.648334	+	0.761356i	$0.724534\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.57144	−	11.3821i	−0.230188	−	0.398697i
$816$		0			0
$817$	6.07311	−	10.5189i	0.212471	−	0.368011i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.5108	−	44.1860i	0.890333	−	1.54210i	0.0508569	−	0.998706i	$-0.483805\pi$
$821$	25.5108	−	44.1860i	0.890333	−	1.54210i	0.839476	−	0.543396i	$-0.182862\pi$
$822$		0			0
$823$	7.28080	+	12.6107i	0.253793	+	0.439582i	0.964567	−	0.263838i	$-0.0849886\pi$
$823$	7.28080	+	12.6107i	0.253793	+	0.439582i	−0.710774	+	0.703420i	$0.751655\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.78016			0.305316			0.152658	−	0.988279i	$-0.451217\pi$
$827$	8.78016			0.305316			0.152658	+	0.988279i	$0.451217\pi$
$828$		0			0
$829$	9.98808			0.346900			0.173450	−	0.984843i	$-0.444508\pi$
$829$	9.98808			0.346900			0.173450	+	0.984843i	$0.444508\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.304160	+	0.526821i	0.0105385	+	0.0182533i
$834$		0			0
$835$	−4.27442	+	7.40352i	−0.147923	+	0.256209i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.72321	−	4.71675i	0.0940158	−	0.162840i	−0.815182	−	0.579205i	$-0.803363\pi$
$839$	2.72321	−	4.71675i	0.0940158	−	0.162840i	0.909197	+	0.416365i	$0.136696\pi$
$840$		0			0
$841$	14.2205	+	24.6307i	0.490364	+	0.849335i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.627689			0.0215932
$846$		0			0
$847$	−55.7308			−1.91493
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.43012	−	16.3334i	−0.323260	−	0.559903i
$852$		0			0
$853$	−0.333743	+	0.578059i	−0.0114271	+	0.0197924i	−0.871682	−	0.490071i	$-0.836971\pi$
$853$	−0.333743	+	0.578059i	−0.0114271	+	0.0197924i	0.860255	+	0.509864i	$0.170304\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.7988	−	44.6848i	0.881269	−	1.52640i	0.0313382	−	0.999509i	$-0.490023\pi$
$857$	25.7988	−	44.6848i	0.881269	−	1.52640i	0.849931	−	0.526894i	$-0.176644\pi$
$858$		0			0
$859$	11.8739	+	20.5663i	0.405134	+	0.701713i	0.994337	−	0.106272i	$-0.0338916\pi$
$859$	11.8739	+	20.5663i	0.405134	+	0.701713i	−0.589203	+	0.807985i	$0.700558\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.41982			−0.0483313			−0.0241656	−	0.999708i	$-0.507693\pi$
$863$	−1.41982			−0.0483313			−0.0241656	+	0.999708i	$0.507693\pi$
$864$		0			0
$865$	−15.7522			−0.535592
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.15621	+	12.3949i	0.242758	+	0.420469i
$870$		0			0
$871$	−0.456405	+	0.790517i	−0.0154647	+	0.0267856i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.89498	+	15.4066i	−0.300705	+	0.520837i
$876$		0			0
$877$	2.61886	+	4.53600i	0.0884327	+	0.153170i	0.906849	−	0.421456i	$-0.138481\pi$
$877$	2.61886	+	4.53600i	0.0884327	+	0.153170i	−0.818416	+	0.574626i	$0.805148\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	28.9816			0.976416			0.488208	−	0.872727i	$-0.337651\pi$
$881$	28.9816			0.976416			0.488208	+	0.872727i	$0.337651\pi$
$882$		0			0
$883$	−47.9095			−1.61228			−0.806142	−	0.591722i	$-0.798448\pi$
$883$	−47.9095			−1.61228			−0.806142	+	0.591722i	$0.798448\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.5686	−	20.0373i	−0.388434	−	0.672788i	0.603805	−	0.797132i	$-0.293651\pi$
$887$	−11.5686	−	20.0373i	−0.388434	−	0.672788i	−0.992239	+	0.124344i	$0.960317\pi$
$888$		0			0
$889$	−2.80423	+	4.85707i	−0.0940509	+	0.162901i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.72534	+	2.98837i	−0.0577362	+	0.100002i
$894$		0			0
$895$	5.69104	+	9.85716i	0.190230	+	0.329489i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.682417			−0.0227599
$900$		0			0
$901$	1.27864			0.0425976
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.83056	−	11.8309i	−0.227055	−	0.393272i
$906$		0			0
$907$	−11.5671	+	20.0349i	−0.384081	+	0.665247i	−0.991641	−	0.129026i	$-0.958815\pi$
$907$	−11.5671	+	20.0349i	−0.384081	+	0.665247i	0.607560	+	0.794273i	$0.292148\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.13845	−	5.43595i	0.103981	−	0.180101i	−0.809340	−	0.587340i	$-0.800175\pi$
$911$	3.13845	−	5.43595i	0.103981	−	0.180101i	0.913322	+	0.407239i	$0.133508\pi$
$912$		0			0
$913$	8.82182	+	15.2798i	0.291960	+	0.505689i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.5893			−0.613873
$918$		0			0
$919$	−2.14621			−0.0707970			−0.0353985	−	0.999373i	$-0.511270\pi$
$919$	−2.14621			−0.0707970			−0.0353985	+	0.999373i	$0.511270\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.85345	+	6.67438i	0.126838	+	0.219690i
$924$		0			0
$925$	17.5167	−	30.3397i	0.575944	−	0.997564i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.8550	+	49.9783i	−0.946702	+	1.63974i	−0.194394	+	0.980924i	$0.562274\pi$
$929$	−28.8550	+	49.9783i	−0.946702	+	1.63974i	−0.752308	+	0.658812i	$0.771059\pi$
$930$		0			0
$931$	1.58853	+	2.75142i	0.0520621	+	0.0901742i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.22427			0.0400380
$936$		0			0
$937$	54.8822			1.79292			0.896461	−	0.443122i	$-0.146129\pi$
$937$	54.8822			1.79292			0.896461	+	0.443122i	$0.146129\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.4516	+	51.0116i	0.960094	+	1.66293i	0.722255	+	0.691627i	$0.243106\pi$
$941$	29.4516	+	51.0116i	0.960094	+	1.66293i	0.237839	+	0.971305i	$0.423561\pi$
$942$		0			0
$943$	−5.07311	+	8.78688i	−0.165203	+	0.286140i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.5456	−	39.0502i	0.732635	−	1.26896i	−0.223118	−	0.974791i	$-0.571624\pi$
$947$	22.5456	−	39.0502i	0.732635	−	1.26896i	0.955753	−	0.294169i	$-0.0950430\pi$
$948$		0			0
$949$	−2.60385	−	4.50999i	−0.0845244	−	0.146401i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	52.2636			1.69298			0.846491	−	0.532402i	$-0.178711\pi$
$953$	52.2636			1.69298			0.846491	+	0.532402i	$0.178711\pi$
$954$		0			0
$955$	−6.54266			−0.211716
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.4627	+	31.9783i	0.596191	+	1.03263i
$960$		0			0
$961$	15.0834	−	26.1252i	0.486561	−	0.842748i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.92896	+	6.80516i	−0.126478	+	0.219066i
$966$		0			0
$967$	−8.22867	−	14.2525i	−0.264616	−	0.458329i	0.702847	−	0.711341i	$-0.251912\pi$
$967$	−8.22867	−	14.2525i	−0.264616	−	0.458329i	−0.967463	+	0.253013i	$0.918579\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	42.8380			1.37474			0.687369	−	0.726309i	$-0.258766\pi$
$971$	42.8380			1.37474			0.687369	+	0.726309i	$0.258766\pi$
$972$		0			0
$973$	−35.7736			−1.14685
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.1895	+	21.1129i	0.389978	+	0.675462i	0.992446	−	0.122681i	$-0.0391493\pi$
$977$	12.1895	+	21.1129i	0.389978	+	0.675462i	−0.602468	+	0.798143i	$0.705816\pi$
$978$		0			0
$979$	41.5800	−	72.0187i	1.32890	−	2.30173i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.29317	+	16.0962i	−0.296406	+	0.513390i	−0.975311	−	0.220836i	$-0.929122\pi$
$983$	−9.29317	+	16.0962i	−0.296406	+	0.513390i	0.678905	+	0.734226i	$0.262455\pi$
$984$		0			0
$985$	7.22055	+	12.5064i	0.230066	+	0.398486i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.1644			0.513998
$990$		0			0
$991$	47.2634			1.50137			0.750686	−	0.660659i	$-0.229723\pi$
$991$	47.2634			1.50137			0.750686	+	0.660659i	$0.229723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.96814	+	8.60507i	0.157501	+	0.272799i
$996$		0			0
$997$	−22.0197	+	38.1393i	−0.697372	+	1.20788i	0.272002	+	0.962297i	$0.412314\pi$
$997$	−22.0197	+	38.1393i	−0.697372	+	1.20788i	−0.969374	+	0.245588i	$0.921019\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.y.2809.3		12
3.2	odd	2	inner	4212.2.i.y.2809.4		12
9.2	odd	6		4212.2.a.l.1.3	✓	6
9.4	even	3	inner	4212.2.i.y.1405.3		12
9.5	odd	6	inner	4212.2.i.y.1405.4		12
9.7	even	3		4212.2.a.l.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4212.2.a.l.1.3	✓	6	9.2	odd	6
4212.2.a.l.1.4	yes	6	9.7	even	3
4212.2.i.y.1405.3		12	9.4	even	3	inner
4212.2.i.y.1405.4		12	9.5	odd	6	inner
4212.2.i.y.2809.3		12	1.1	even	1	trivial
4212.2.i.y.2809.4		12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.313845 - 0.543595i) q^{5} +(1.47522 - 2.55516i) q^{7} +(2.73354 - 4.73463i) q^{11} +(-0.500000 - 0.866025i) q^{13} -0.356762 q^{17} -1.86325 q^{19} +(-1.23983 - 2.14744i) q^{23} +(2.30300 - 3.98892i) q^{25} +(0.373800 - 0.647441i) q^{29} +(-0.456405 - 0.790517i) q^{31} -1.85196 q^{35} +7.60601 q^{37} +(-2.04590 - 3.54359i) q^{41} +(-3.25941 + 5.64546i) q^{43} +(0.925981 - 1.60385i) q^{47} +(-0.852559 - 1.47668i) q^{49} -3.58401 q^{53} -3.43163 q^{55} +(-4.95782 - 8.58719i) q^{59} +(-3.90685 + 6.76686i) q^{61} +(-0.313845 + 0.543595i) q^{65} +(-0.456405 - 0.790517i) q^{67} -7.70691 q^{71} +5.20769 q^{73} +(-8.06516 - 13.9693i) q^{77} +(-1.30896 + 2.26719i) q^{79} +(-1.61363 + 2.79488i) q^{83} +(0.111968 + 0.193934i) q^{85} +15.2110 q^{89} -2.95044 q^{91} +(0.584772 + 1.01286i) q^{95} +(-5.95044 + 10.3065i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7} - 6 q^{13} + 4 q^{19} - 4 q^{25} - 18 q^{31} + 28 q^{37} - 20 q^{43} - 30 q^{49} - 28 q^{55} - 8 q^{61} - 18 q^{67} + 48 q^{73} - 48 q^{79} + 2 q^{85} + 16 q^{91} - 20 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 - 6 * q^13 + 4 * q^19 - 4 * q^25 - 18 * q^31 + 28 * q^37 - 20 * q^43 - 30 * q^49 - 28 * q^55 - 8 * q^61 - 18 * q^67 + 48 * q^73 - 48 * q^79 + 2 * q^85 + 16 * q^91 - 20 * q^97

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