LMFDB - Embedded newform 756.3.bk.g.485.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(53,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.53"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,24,0,0,0,0,0,88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 46 x^{14} + 1437 x^{12} - 24668 x^{10} + 309582 x^{8} - 2188585 x^{6} + 10478650 x^{4} + \cdots + 194481 $ x^16 - 46x^14 + 1437x^12 - 24668x^10 + 309582x^8 - 2188585x^6 + 10478650x^4 - 1447803*x^2 + 194481
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			485.7
Root			$3.13096 - 1.80766i$ of defining polynomial
Character	$\chi$	$=$	756.485
Dual form			756.3.bk.g.53.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.22199 + 3.01492i) q^{5} +(2.27239 + 6.62090i) q^{7} +(8.17257 - 4.71844i) q^{11} +13.3012 q^{13} +(-9.36705 + 5.40807i) q^{17} +(2.20352 - 3.81661i) q^{19} +(17.2388 + 9.95281i) q^{23} +(5.67943 + 9.83706i) q^{25} +10.6940i q^{29} +(-17.4966 - 30.3049i) q^{31} +(-8.09507 + 41.4253i) q^{35} +(-0.378236 + 0.655125i) q^{37} +32.4157i q^{41} -44.3106 q^{43} +(70.3329 + 40.6067i) q^{47} +(-38.6725 + 30.0905i) q^{49} +(-17.0955 + 9.87011i) q^{53} +56.9027 q^{55} +(38.9892 - 22.5104i) q^{59} +(-20.5078 + 35.5205i) q^{61} +(69.4586 + 40.1020i) q^{65} +(-45.7845 - 79.3010i) q^{67} +11.2890i q^{71} +(10.2690 + 17.7864i) q^{73} +(49.8115 + 43.3876i) q^{77} +(57.5907 - 99.7501i) q^{79} +35.0385i q^{83} -65.2195 q^{85} +(45.2404 + 26.1196i) q^{89} +(30.2254 + 88.0658i) q^{91} +(23.0135 - 13.2869i) q^{95} +182.132 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{7} + 88 q^{13} + 14 q^{19} + 36 q^{25} - 68 q^{31} - 76 q^{37} - 292 q^{43} - 20 q^{49} - 272 q^{55} - 110 q^{61} - 72 q^{67} + 60 q^{73} + 154 q^{79} + 700 q^{85} - 74 q^{91} + 264 q^{97}+O(q^{100}) $ 16 * q + 24 * q^7 + 88 * q^13 + 14 * q^19 + 36 * q^25 - 68 * q^31 - 76 * q^37 - 292 * q^43 - 20 * q^49 - 272 * q^55 - 110 * q^61 - 72 * q^67 + 60 * q^73 + 154 * q^79 + 700 * q^85 - 74 * q^91 + 264 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.22199	+	3.01492i	1.04440	+	0.602983i	0.921076	−	0.389384i	$-0.127312\pi$
$5$	5.22199	+	3.01492i	1.04440	+	0.602983i	0.123322	+	0.992367i	$0.460645\pi$
$6$		0			0
$7$	2.27239	+	6.62090i	0.324627	+	0.945842i
$7$	2.27239	+	6.62090i	0.324627	+	0.945842i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.17257	−	4.71844i	0.742961	−	0.428949i	−0.0801839	−	0.996780i	$-0.525551\pi$
$11$	8.17257	−	4.71844i	0.742961	−	0.428949i	0.823145	+	0.567831i	$0.192217\pi$
$12$		0			0
$13$	13.3012			1.02317			0.511584	−	0.859233i	$-0.329059\pi$
$13$	13.3012			1.02317			0.511584	+	0.859233i	$0.329059\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.36705	+	5.40807i	−0.551003	+	0.318122i	−0.749526	−	0.661975i	$-0.769719\pi$
$17$	−9.36705	+	5.40807i	−0.551003	+	0.318122i	0.198524	+	0.980096i	$0.436385\pi$
$18$		0			0
$19$	2.20352	−	3.81661i	0.115975	−	0.200874i	−0.802194	−	0.597063i	$-0.796334\pi$
$19$	2.20352	−	3.81661i	0.115975	−	0.200874i	0.918169	+	0.396189i	$0.129667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.2388	+	9.95281i	0.749512	+	0.432731i	0.825518	−	0.564376i	$-0.190883\pi$
$23$	17.2388	+	9.95281i	0.749512	+	0.432731i	−0.0760055	+	0.997107i	$0.524217\pi$
$24$		0			0
$25$	5.67943	+	9.83706i	0.227177	+	0.393482i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			10.6940i			0.368758i	0.982855	+	0.184379i	$0.0590274\pi$
$29$			10.6940i			0.368758i	−0.982855	+	0.184379i	$0.940973\pi$
$30$		0			0
$31$	−17.4966	−	30.3049i	−0.564405	−	0.977578i	−0.997105	−	0.0760402i	$-0.975772\pi$
$31$	−17.4966	−	30.3049i	−0.564405	−	0.977578i	0.432700	−	0.901538i	$-0.357561\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.09507	+	41.4253i	−0.231288	+	1.18358i
$36$		0			0
$37$	−0.378236	+	0.655125i	−0.0102226	+	0.0177061i	−0.871091	−	0.491121i	$-0.836587\pi$
$37$	−0.378236	+	0.655125i	−0.0102226	+	0.0177061i	0.860869	+	0.508827i	$0.169921\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			32.4157i			0.790628i	0.918546	+	0.395314i	$0.129364\pi$
$41$			32.4157i			0.790628i	−0.918546	+	0.395314i	$0.870636\pi$
$42$		0			0
$43$	−44.3106			−1.03048			−0.515240	−	0.857046i	$-0.672297\pi$
$43$	−44.3106			−1.03048			−0.515240	+	0.857046i	$0.672297\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	70.3329	+	40.6067i	1.49644	+	0.863972i	0.999992	−	0.00409145i	$-0.00130235\pi$
$47$	70.3329	+	40.6067i	1.49644	+	0.863972i	0.496453	+	0.868064i	$0.334636\pi$
$48$		0			0
$49$	−38.6725	+	30.0905i	−0.789235	+	0.614091i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−17.0955	+	9.87011i	−0.322557	+	0.186229i	−0.652532	−	0.757761i	$-0.726293\pi$
$53$	−17.0955	+	9.87011i	−0.322557	+	0.186229i	0.329975	+	0.943990i	$0.392960\pi$
$54$		0			0
$55$	56.9027			1.03460
$56$		0			0
$57$		0			0
$58$		0			0
$59$	38.9892	−	22.5104i	0.660834	−	0.381533i	−0.131761	−	0.991282i	$-0.542063\pi$
$59$	38.9892	−	22.5104i	0.660834	−	0.381533i	0.792595	+	0.609749i	$0.208730\pi$
$60$		0			0
$61$	−20.5078	+	35.5205i	−0.336193	+	0.582303i	−0.983713	−	0.179745i	$-0.942473\pi$
$61$	−20.5078	+	35.5205i	−0.336193	+	0.582303i	0.647520	+	0.762048i	$0.275806\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	69.4586	+	40.1020i	1.06859	+	0.616953i
$66$		0			0
$67$	−45.7845	−	79.3010i	−0.683350	−	1.18360i	−0.973952	−	0.226753i	$-0.927189\pi$
$67$	−45.7845	−	79.3010i	−0.683350	−	1.18360i	0.290602	−	0.956844i	$-0.406144\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.2890i			0.159000i	0.996835	+	0.0794999i	$0.0253323\pi$
$71$			11.2890i			0.159000i	−0.996835	+	0.0794999i	$0.974668\pi$
$72$		0			0
$73$	10.2690	+	17.7864i	0.140671	+	0.243649i	0.927749	−	0.373204i	$-0.121741\pi$
$73$	10.2690	+	17.7864i	0.140671	+	0.243649i	−0.787079	+	0.616853i	$0.788407\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	49.8115	+	43.3876i	0.646903	+	0.563476i
$78$		0			0
$79$	57.5907	−	99.7501i	0.728997	−	1.26266i	−0.228311	−	0.973588i	$-0.573320\pi$
$79$	57.5907	−	99.7501i	0.728997	−	1.26266i	0.957308	−	0.289071i	$-0.0933464\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			35.0385i			0.422151i	0.977470	+	0.211075i	$0.0676966\pi$
$83$			35.0385i			0.422151i	−0.977470	+	0.211075i	$0.932303\pi$
$84$		0			0
$85$	−65.2195			−0.767288
$86$		0			0
$87$		0			0
$88$		0			0
$89$	45.2404	+	26.1196i	0.508319	+	0.293478i	0.732143	−	0.681151i	$-0.238520\pi$
$89$	45.2404	+	26.1196i	0.508319	+	0.293478i	−0.223823	+	0.974630i	$0.571854\pi$
$90$		0			0
$91$	30.2254	+	88.0658i	0.332148	+	0.967756i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	23.0135	−	13.2869i	0.242247	−	0.139862i
$96$		0			0
$97$	182.132			1.87765			0.938827	−	0.344390i	$-0.111914\pi$
$97$	182.132			1.87765			0.938827	+	0.344390i	$0.111914\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.60387	+	3.81275i	−0.0653849	+	0.0377500i	−0.532336	−	0.846533i	$-0.678686\pi$
$101$	−6.60387	+	3.81275i	−0.0653849	+	0.0377500i	0.466951	+	0.884283i	$0.345352\pi$
$102$		0			0
$103$	−33.7607	+	58.4752i	−0.327773	+	0.567720i	−0.982070	−	0.188518i	$-0.939632\pi$
$103$	−33.7607	+	58.4752i	−0.327773	+	0.567720i	0.654296	+	0.756238i	$0.272965\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	127.399	+	73.5538i	1.19064	+	0.687419i	0.958453	−	0.285251i	$-0.0920770\pi$
$107$	127.399	+	73.5538i	1.19064	+	0.687419i	0.232192	+	0.972670i	$0.425410\pi$
$108$		0			0
$109$	−9.72448	−	16.8433i	−0.0892154	−	0.154526i	0.817964	−	0.575269i	$-0.195103\pi$
$109$	−9.72448	−	16.8433i	−0.0892154	−	0.154526i	−0.907180	+	0.420743i	$0.861769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			208.368i			1.84396i	0.387236	+	0.921981i	$0.373430\pi$
$113$			208.368i			1.84396i	−0.387236	+	0.921981i	$0.626570\pi$
$114$		0			0
$115$	60.0138	+	103.947i	0.521859	+	0.903886i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−57.0918	−	49.7290i	−0.479763	−	0.417891i
$120$		0			0
$121$	−15.9727	+	27.6656i	−0.132006	+	0.228641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	82.2538i		−	0.658030i
$126$		0			0
$127$	156.607			1.23312			0.616562	−	0.787307i	$-0.288525\pi$
$127$	156.607			1.23312			0.616562	+	0.787307i	$0.288525\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−217.234	−	125.420i	−1.65827	−	0.957404i	−0.973512	−	0.228636i	$-0.926574\pi$
$131$	−217.234	−	125.420i	−1.65827	−	0.957404i	−0.684760	−	0.728768i	$-0.740093\pi$
$132$		0			0
$133$	30.2766	+	5.91647i	0.227644	+	0.0444847i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−231.846	+	133.857i	−1.69231	+	0.977055i	−0.739661	+	0.672979i	$0.765014\pi$
$137$	−231.846	+	133.857i	−1.69231	+	0.977055i	−0.952648	+	0.304076i	$0.901652\pi$
$138$		0			0
$139$	−204.740			−1.47295			−0.736475	−	0.676465i	$-0.763511\pi$
$139$	−204.740			−1.47295			−0.736475	+	0.676465i	$0.763511\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	108.705	−	62.7608i	0.760174	−	0.438887i
$144$		0			0
$145$	−32.2414	+	55.8438i	−0.222355	+	0.385130i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	93.1095	+	53.7568i	0.624896	+	0.360784i	0.778773	−	0.627306i	$-0.215843\pi$
$149$	93.1095	+	53.7568i	0.624896	+	0.360784i	−0.153877	+	0.988090i	$0.549176\pi$
$150$		0			0
$151$	−63.4771	−	109.946i	−0.420378	−	0.728116i	0.575598	−	0.817733i	$-0.304769\pi$
$151$	−63.4771	−	109.946i	−0.420378	−	0.728116i	−0.995976	+	0.0896164i	$0.971436\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	211.003i		−	1.36131i
$156$		0			0
$157$	−148.868	−	257.846i	−0.948201	−	1.64233i	−0.749211	−	0.662332i	$-0.769567\pi$
$157$	−148.868	−	257.846i	−0.948201	−	1.64233i	−0.198991	−	0.980001i	$-0.563766\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−26.7234	+	136.753i	−0.165984	+	0.849396i
$162$		0			0
$163$	17.4882	−	30.2905i	0.107290	−	0.185831i	−0.807382	−	0.590030i	$-0.799116\pi$
$163$	17.4882	−	30.2905i	0.107290	−	0.185831i	0.914671	+	0.404198i	$0.132449\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	246.347i		−	1.47513i	−0.675276	−	0.737565i	$-0.735975\pi$
$167$		−	246.347i		−	1.47513i	0.675276	−	0.737565i	$-0.264025\pi$
$168$		0			0
$169$	7.92170			0.0468739
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.3331	−	5.96584i	−0.0597291	−	0.0344846i	0.469838	−	0.882753i	$-0.344312\pi$
$173$	−10.3331	−	5.96584i	−0.0597291	−	0.0344846i	−0.529567	+	0.848268i	$0.677646\pi$
$174$		0			0
$175$	−52.2243	+	59.9565i	−0.298424	+	0.342609i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	118.482	−	68.4054i	0.661909	−	0.382153i	−0.131095	−	0.991370i	$-0.541849\pi$
$179$	118.482	−	68.4054i	0.661909	−	0.382153i	0.793004	+	0.609217i	$0.208516\pi$
$180$		0			0
$181$	105.971			0.585474			0.292737	−	0.956193i	$-0.405434\pi$
$181$	105.971			0.585474			0.292737	+	0.956193i	$0.405434\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.95029	+	2.28070i	−0.0213529	+	0.0123281i
$186$		0			0
$187$	−51.0352	+	88.3956i	−0.272916	+	0.472704i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−25.0424	−	14.4582i	−0.131112	−	0.0756974i	0.433010	−	0.901389i	$-0.357452\pi$
$191$	−25.0424	−	14.4582i	−0.131112	−	0.0756974i	−0.564121	+	0.825692i	$0.690785\pi$
$192$		0			0
$193$	−156.164	−	270.484i	−0.809140	−	1.40147i	−0.913460	−	0.406928i	$-0.866600\pi$
$193$	−156.164	−	270.484i	−0.809140	−	1.40147i	0.104320	−	0.994544i	$-0.466733\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	114.463i		−	0.581029i	−0.956871	−	0.290514i	$-0.906174\pi$
$197$		−	114.463i		−	0.581029i	0.956871	−	0.290514i	$-0.0938264\pi$
$198$		0			0
$199$	73.5560	+	127.403i	0.369628	+	0.640215i	0.989507	−	0.144482i	$-0.0461517\pi$
$199$	73.5560	+	127.403i	0.369628	+	0.640215i	−0.619879	+	0.784697i	$0.712818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−70.8037	+	24.3009i	−0.348787	+	0.119709i
$204$		0			0
$205$	−97.7307	+	169.275i	−0.476735	+	0.825729i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	41.5887i		−	0.198989i
$210$		0			0
$211$	−252.521			−1.19678			−0.598390	−	0.801205i	$-0.704193\pi$
$211$	−252.521			−1.19678			−0.598390	+	0.801205i	$0.704193\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−231.389	−	133.593i	−1.07623	−	0.621362i
$216$		0			0
$217$	160.887	−	184.707i	0.741414	−	0.851186i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−124.593	+	71.9337i	−0.563769	+	0.325492i
$222$		0			0
$223$	44.0347			0.197465			0.0987326	−	0.995114i	$-0.468521\pi$
$223$	44.0347			0.197465			0.0987326	+	0.995114i	$0.468521\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	307.463	−	177.514i	1.35446	−	0.781998i	0.365590	−	0.930776i	$-0.380867\pi$
$227$	307.463	−	177.514i	1.35446	−	0.781998i	0.988871	+	0.148778i	$0.0475339\pi$
$228$		0			0
$229$	27.9688	−	48.4434i	0.122134	−	0.211543i	−0.798475	−	0.602028i	$-0.794359\pi$
$229$	27.9688	−	48.4434i	0.122134	−	0.211543i	0.920609	+	0.390485i	$0.127693\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−87.2545	−	50.3764i	−0.374483	−	0.216208i	0.300932	−	0.953645i	$-0.402702\pi$
$233$	−87.2545	−	50.3764i	−0.374483	−	0.216208i	−0.675415	+	0.737438i	$0.736035\pi$
$234$		0			0
$235$	244.852	+	424.095i	1.04192	+	1.80466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	254.342i		−	1.06419i	−0.846684	−	0.532096i	$-0.821404\pi$
$239$		−	254.342i		−	1.06419i	0.846684	−	0.532096i	$-0.178596\pi$
$240$		0			0
$241$	8.91574	+	15.4425i	0.0369948	+	0.0640768i	0.883930	−	0.467619i	$-0.154888\pi$
$241$	8.91574	+	15.4425i	0.0369948	+	0.0640768i	−0.846935	+	0.531696i	$0.821555\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−292.668	+	40.5376i	−1.19456	+	0.165460i
$246$		0			0
$247$	29.3095	−	50.7655i	0.118662	−	0.205528i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	444.838i		−	1.77226i	−0.463436	−	0.886130i	$-0.653384\pi$
$251$		−	444.838i		−	1.77226i	0.463436	−	0.886130i	$-0.346616\pi$
$252$		0			0
$253$	187.847			0.742478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−204.837	−	118.263i	−0.797030	−	0.460166i	0.0454014	−	0.998969i	$-0.485543\pi$
$257$	−204.837	−	118.263i	−0.797030	−	0.460166i	−0.842432	+	0.538803i	$0.818877\pi$
$258$		0			0
$259$	−5.19701	−	1.01557i	−0.0200657	−	0.00392111i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−364.086	+	210.205i	−1.38436	+	0.799258i	−0.992672	−	0.120841i	$-0.961441\pi$
$263$	−364.086	+	210.205i	−1.38436	+	0.799258i	−0.391684	+	0.920100i	$0.628107\pi$
$264$		0			0
$265$	−119.030			−0.449171
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−371.562	+	214.521i	−1.38127	+	0.797477i	−0.992310	−	0.123777i	$-0.960499\pi$
$269$	−371.562	+	214.521i	−1.38127	+	0.797477i	−0.388961	+	0.921254i	$0.627166\pi$
$270$		0			0
$271$	205.651	−	356.198i	0.758859	−	1.31438i	−0.184574	−	0.982819i	$-0.559091\pi$
$271$	205.651	−	356.198i	0.758859	−	1.31438i	0.943433	−	0.331563i	$-0.107576\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	92.8311	+	53.5960i	0.337567	+	0.194895i
$276$		0			0
$277$	207.078	+	358.669i	0.747574	+	1.29484i	0.948983	+	0.315328i	$0.102115\pi$
$277$	207.078	+	358.669i	0.747574	+	1.29484i	−0.201409	+	0.979507i	$0.564552\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	441.381i		−	1.57075i	−0.619021	−	0.785375i	$-0.712470\pi$
$281$		−	441.381i		−	1.57075i	0.619021	−	0.785375i	$-0.287530\pi$
$282$		0			0
$283$	70.2240	+	121.632i	0.248141	+	0.429793i	0.963010	−	0.269465i	$-0.0868469\pi$
$283$	70.2240	+	121.632i	0.248141	+	0.429793i	−0.714869	+	0.699259i	$0.753514\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−214.621	+	73.6611i	−0.747809	+	0.256659i
$288$		0			0
$289$	−86.0056	+	148.966i	−0.297597	+	0.515454i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	469.357i		−	1.60190i	−0.598730	−	0.800951i	$-0.704328\pi$
$293$		−	469.357i		−	1.60190i	0.598730	−	0.800951i	$-0.295672\pi$
$294$		0			0
$295$	271.468			0.920231
$296$		0			0
$297$		0			0
$298$		0			0
$299$	229.296	+	132.384i	0.766877	+	0.442757i
$300$		0			0
$301$	−100.691	−	293.376i	−0.334521	−	0.974671i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−214.183	+	123.658i	−0.702238	+	0.405437i
$306$		0			0
$307$	−356.418			−1.16097			−0.580485	−	0.814271i	$-0.697137\pi$
$307$	−356.418			−1.16097			−0.580485	+	0.814271i	$0.697137\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.8828	+	6.28317i	−0.0349928	+	0.0202031i	−0.517394	−	0.855747i	$-0.673098\pi$
$311$	−10.8828	+	6.28317i	−0.0349928	+	0.0202031i	0.482402	+	0.875950i	$0.339765\pi$
$312$		0			0
$313$	79.4386	−	137.592i	0.253797	−	0.439590i	−0.710771	−	0.703424i	$-0.751654\pi$
$313$	79.4386	−	137.592i	0.253797	−	0.439590i	0.964568	+	0.263834i	$0.0849870\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	495.809	+	286.255i	1.56407	+	0.903014i	0.996839	+	0.0794531i	$0.0253174\pi$
$317$	495.809	+	286.255i	1.56407	+	0.903014i	0.567228	+	0.823561i	$0.308016\pi$
$318$		0			0
$319$	50.4589	+	87.3973i	0.158178	+	0.273973i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			47.6672i			0.147576i
$324$		0			0
$325$	75.5432	+	130.845i	0.232440	+	0.402599i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−109.029	+	557.941i	−0.331396	+	1.69587i
$330$		0			0
$331$	177.627	−	307.658i	0.536636	−	0.929481i	−0.462446	−	0.886648i	$-0.653028\pi$
$331$	177.627	−	307.658i	0.536636	−	0.929481i	0.999082	−	0.0428339i	$-0.0136386\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	552.145i		−	1.64819i
$336$		0			0
$337$	−351.090			−1.04181			−0.520905	−	0.853615i	$-0.674405\pi$
$337$	−351.090			−1.04181			−0.520905	+	0.853615i	$0.674405\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−285.984	−	165.113i	−0.838662	−	0.484202i
$342$		0			0
$343$	−287.105	−	187.670i	−0.837040	−	0.547142i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	431.379	−	249.057i	1.24317	−	0.717743i	0.273430	−	0.961892i	$-0.411842\pi$
$347$	431.379	−	249.057i	1.24317	−	0.717743i	0.969738	+	0.244149i	$0.0785086\pi$
$348$		0			0
$349$	−170.749			−0.489252			−0.244626	−	0.969617i	$-0.578665\pi$
$349$	−170.749			−0.489252			−0.244626	+	0.969617i	$0.578665\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	245.387	−	141.674i	0.695148	−	0.401344i	−0.110390	−	0.993888i	$-0.535210\pi$
$353$	245.387	−	141.674i	0.695148	−	0.401344i	0.805538	+	0.592544i	$0.201877\pi$
$354$		0			0
$355$	−34.0353	+	58.9509i	−0.0958742	+	0.166059i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	127.691	+	73.7227i	0.355686	+	0.205356i	0.667187	−	0.744890i	$-0.267498\pi$
$359$	127.691	+	73.7227i	0.355686	+	0.205356i	−0.311501	+	0.950246i	$0.600832\pi$
$360$		0			0
$361$	170.789	+	295.815i	0.473100	+	0.819433i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			123.840i			0.339289i
$366$		0			0
$367$	−142.805	−	247.345i	−0.389113	−	0.673964i	0.603217	−	0.797577i	$-0.293885\pi$
$367$	−142.805	−	247.345i	−0.389113	−	0.673964i	−0.992331	+	0.123613i	$0.960552\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−104.197	−	90.7590i	−0.280853	−	0.244634i
$372$		0			0
$373$	−124.758	+	216.087i	−0.334472	+	0.579322i	−0.983383	−	0.181542i	$-0.941891\pi$
$373$	−124.758	+	216.087i	−0.334472	+	0.579322i	0.648911	+	0.760864i	$0.275225\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			142.243i			0.377301i
$378$		0			0
$379$	50.2895			0.132690			0.0663450	−	0.997797i	$-0.478866\pi$
$379$	50.2895			0.132690			0.0663450	+	0.997797i	$0.478866\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−89.3333	−	51.5766i	−0.233246	−	0.134665i	0.378822	−	0.925469i	$-0.376329\pi$
$383$	−89.3333	−	51.5766i	−0.233246	−	0.134665i	−0.612069	+	0.790805i	$0.709662\pi$
$384$		0			0
$385$	129.305	+	376.747i	0.335857	+	0.978564i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−514.322	+	296.944i	−1.32217	+	0.763353i	−0.984074	−	0.177761i	$-0.943115\pi$
$389$	−514.322	+	296.944i	−1.32217	+	0.763353i	−0.338092	+	0.941113i	$0.609781\pi$
$390$		0			0
$391$	−215.302			−0.550644
$392$		0			0
$393$		0			0
$394$		0			0
$395$	601.476	−	347.262i	1.52272	−	0.879145i
$396$		0			0
$397$	132.372	−	229.275i	0.333431	−	0.577519i	−0.649752	−	0.760147i	$-0.725127\pi$
$397$	132.372	−	229.275i	0.333431	−	0.577519i	0.983182	+	0.182628i	$0.0584604\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	370.025	+	213.634i	0.922755	+	0.532753i	0.884513	−	0.466516i	$-0.154491\pi$
$401$	370.025	+	213.634i	0.922755	+	0.532753i	0.0382419	+	0.999269i	$0.487824\pi$
$402$		0			0
$403$	−232.725	−	403.092i	−0.577482	−	1.00023i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.13874i			0.0175399i
$408$		0			0
$409$	−89.8497	−	155.624i	−0.219681	−	0.380499i	0.735029	−	0.678035i	$-0.237168\pi$
$409$	−89.8497	−	155.624i	−0.219681	−	0.380499i	−0.954711	+	0.297536i	$0.903835\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	237.638	+	206.991i	0.575394	+	0.501189i
$414$		0			0
$415$	−105.638	+	182.971i	−0.254550	+	0.440893i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	171.206i		−	0.408606i	−0.978908	−	0.204303i	$-0.934507\pi$
$419$		−	171.206i		−	0.408606i	0.978908	−	0.204303i	$-0.0654928\pi$
$420$		0			0
$421$	33.6353			0.0798938			0.0399469	−	0.999202i	$-0.487281\pi$
$421$	33.6353			0.0798938			0.0399469	+	0.999202i	$0.487281\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−106.399	−	61.4295i	−0.250350	−	0.144540i
$426$		0			0
$427$	−281.779	−	55.0635i	−0.659904	−	0.128954i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−364.783	+	210.607i	−0.846363	+	0.488648i	−0.859422	−	0.511267i	$-0.829176\pi$
$431$	−364.783	+	210.607i	−0.846363	+	0.488648i	0.0130590	+	0.999915i	$0.495843\pi$
$432$		0			0
$433$	−390.359			−0.901521			−0.450760	−	0.892645i	$-0.648847\pi$
$433$	−390.359			−0.901521			−0.450760	+	0.892645i	$0.648847\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	75.9720	−	43.8625i	0.173849	−	0.100372i
$438$		0			0
$439$	196.067	−	339.599i	0.446623	−	0.773573i	−0.551541	−	0.834148i	$-0.685960\pi$
$439$	196.067	−	339.599i	0.446623	−	0.773573i	0.998164	+	0.0605748i	$0.0192934\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	505.888	+	292.074i	1.14196	+	0.659310i	0.946915	−	0.321485i	$-0.104182\pi$
$443$	505.888	+	292.074i	1.14196	+	0.659310i	0.195044	+	0.980795i	$0.437515\pi$
$444$		0			0
$445$	157.497	+	272.792i	0.353925	+	0.613016i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			333.245i			0.742193i	0.928594	+	0.371097i	$0.121018\pi$
$449$			333.245i			0.742193i	−0.928594	+	0.371097i	$0.878982\pi$
$450$		0			0
$451$	152.952	+	264.920i	0.339139	+	0.587406i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−107.674	+	551.006i	−0.236646	+	1.21100i
$456$		0			0
$457$	−192.583	+	333.564i	−0.421407	+	0.729899i	−0.996077	−	0.0884861i	$-0.971797\pi$
$457$	−192.583	+	333.564i	−0.421407	+	0.729899i	0.574670	+	0.818385i	$0.305130\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			603.468i			1.30904i	0.756044	+	0.654521i	$0.227130\pi$
$461$			603.468i			1.30904i	−0.756044	+	0.654521i	$0.772870\pi$
$462$		0			0
$463$	578.006			1.24839			0.624197	−	0.781267i	$-0.285426\pi$
$463$	578.006			1.24839			0.624197	+	0.781267i	$0.285426\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.88677	−	2.82138i	−0.0104642	−	0.00604150i	0.494759	−	0.869030i	$-0.335256\pi$
$467$	−4.88677	−	2.82138i	−0.0104642	−	0.00604150i	−0.505223	+	0.862989i	$0.668590\pi$
$468$		0			0
$469$	421.004	−	483.337i	0.897663	−	1.03057i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−362.132	+	209.077i	−0.765606	+	0.442023i
$474$		0			0
$475$	50.0589			0.105387
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.38949	−	1.95692i	0.00707618	−	0.00408544i	−0.496458	−	0.868061i	$-0.665366\pi$
$479$	3.38949	−	1.95692i	0.00707618	−	0.00408544i	0.503534	+	0.863975i	$0.332033\pi$
$480$		0			0
$481$	−5.03099	+	8.71394i	−0.0104594	+	0.0181163i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	951.093	+	549.114i	1.96102	+	1.13219i
$486$		0			0
$487$	48.7855	+	84.4989i	0.100175	+	0.173509i	0.911757	−	0.410730i	$-0.134726\pi$
$487$	48.7855	+	84.4989i	0.100175	+	0.173509i	−0.811581	+	0.584239i	$0.801393\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			250.072i			0.509313i	0.967032	+	0.254656i	$0.0819623\pi$
$491$			250.072i			0.509313i	−0.967032	+	0.254656i	$0.918038\pi$
$492$		0			0
$493$	−57.8338	−	100.171i	−0.117310	−	0.203187i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−74.7432	+	25.6529i	−0.150389	+	0.0516156i
$498$		0			0
$499$	210.025	−	363.774i	0.420891	−	0.729005i	−0.575135	−	0.818058i	$-0.695051\pi$
$499$	210.025	−	363.774i	0.420891	−	0.729005i	0.996027	+	0.0890528i	$0.0283840\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	490.340i		−	0.974830i	−0.873170	−	0.487415i	$-0.837940\pi$
$503$		−	490.340i		−	0.974830i	0.873170	−	0.487415i	$-0.162060\pi$
$504$		0			0
$505$	−45.9804			−0.0910504
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.2098	+	7.04932i	0.0239878	+	0.0138494i	0.511946	−	0.859018i	$-0.328925\pi$
$509$	12.2098	+	7.04932i	0.0239878	+	0.0138494i	−0.487958	+	0.872867i	$0.662258\pi$
$510$		0			0
$511$	−94.4267	+	108.407i	−0.184788	+	0.212147i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−352.595	+	203.571i	−0.684651	+	0.395284i
$516$		0			0
$517$	766.401			1.48240
$518$		0			0
$519$		0			0
$520$		0			0
$521$	507.374	−	292.933i	0.973847	−	0.562251i	0.0734401	−	0.997300i	$-0.476602\pi$
$521$	507.374	−	292.933i	0.973847	−	0.562251i	0.900407	+	0.435049i	$0.143269\pi$
$522$		0			0
$523$	−83.6465	+	144.880i	−0.159936	+	0.277017i	−0.934845	−	0.355055i	$-0.884462\pi$
$523$	−83.6465	+	144.880i	−0.159936	+	0.277017i	0.774909	+	0.632072i	$0.217795\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	327.782	+	189.245i	0.621978	+	0.359099i
$528$		0			0
$529$	−66.3830	−	114.979i	−0.125488	−	0.217351i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			431.168i			0.808945i
$534$		0			0
$535$	443.517	+	768.194i	0.829004	+	1.43588i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−174.074	+	428.390i	−0.322957	+	0.794787i
$540$		0			0
$541$	−32.7910	+	56.7957i	−0.0606119	+	0.104983i	−0.894739	−	0.446589i	$-0.852639\pi$
$541$	−32.7910	+	56.7957i	−0.0606119	+	0.104983i	0.834127	+	0.551572i	$0.185972\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	117.274i		−	0.215182i
$546$		0			0
$547$	7.07747			0.0129387			0.00646935	−	0.999979i	$-0.497941\pi$
$547$	7.07747			0.0129387			0.00646935	+	0.999979i	$0.497941\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	40.8147	+	23.5644i	0.0740739	+	0.0427666i
$552$		0			0
$553$	791.303	+	154.632i	1.43093	+	0.279623i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	381.297	−	220.142i	0.684555	−	0.395228i	−0.117014	−	0.993130i	$-0.537332\pi$
$557$	381.297	−	220.142i	0.684555	−	0.395228i	0.801569	+	0.597902i	$0.203999\pi$
$558$		0			0
$559$	−589.384			−1.05435
$560$		0			0
$561$		0			0
$562$		0			0
$563$	291.898	−	168.528i	0.518469	−	0.299338i	−0.217839	−	0.975985i	$-0.569901\pi$
$563$	291.898	−	168.528i	0.518469	−	0.299338i	0.736308	+	0.676646i	$0.236567\pi$
$564$		0			0
$565$	−628.211	+	1088.09i	−1.11188	+	1.92583i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	837.837	+	483.726i	1.47247	+	0.850133i	0.999521	−	0.0309548i	$-0.00985481\pi$
$569$	837.837	+	483.726i	1.47247	+	0.850133i	0.472953	+	0.881088i	$0.343188\pi$
$570$		0			0
$571$	−84.5926	−	146.519i	−0.148148	−	0.256600i	0.782395	−	0.622783i	$-0.213998\pi$
$571$	−84.5926	−	146.519i	−0.148148	−	0.256600i	−0.930543	+	0.366182i	$0.880665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			226.105i			0.393226i
$576$		0			0
$577$	−296.356	−	513.303i	−0.513615	−	0.889607i	−0.999875	−	0.0157933i	$-0.994973\pi$
$577$	−296.356	−	513.303i	−0.513615	−	0.889607i	0.486260	−	0.873814i	$-0.338361\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−231.986	+	79.6211i	−0.399288	+	0.137041i
$582$		0			0
$583$	−93.1430	+	161.328i	−0.159765	+	0.276721i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	621.150i		−	1.05818i	−0.848567	−	0.529088i	$-0.822534\pi$
$587$		−	621.150i		−	1.05818i	0.848567	−	0.529088i	$-0.177466\pi$
$588$		0			0
$589$	−154.216			−0.261827
$590$		0			0
$591$		0			0
$592$		0			0
$593$	102.660	+	59.2709i	0.173120	+	0.0999510i	0.584056	−	0.811713i	$-0.301465\pi$
$593$	102.660	+	59.2709i	0.173120	+	0.0999510i	−0.410936	+	0.911664i	$0.634798\pi$
$594$		0			0
$595$	−148.204	−	431.811i	−0.249082	−	0.725733i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−510.471	+	294.721i	−0.852206	+	0.492021i	−0.861395	−	0.507936i	$-0.830408\pi$
$599$	−510.471	+	294.721i	−0.852206	+	0.492021i	0.00918855	+	0.999958i	$0.497075\pi$
$600$		0			0
$601$	982.153			1.63420			0.817099	−	0.576497i	$-0.195581\pi$
$601$	982.153			1.63420			0.817099	+	0.576497i	$0.195581\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−166.819	+	96.3128i	−0.275733	+	0.159195i
$606$		0			0
$607$	−287.364	+	497.729i	−0.473417	+	0.819982i	−0.999537	−	0.0304281i	$-0.990313\pi$
$607$	−287.364	+	497.729i	−0.473417	+	0.819982i	0.526120	+	0.850410i	$0.323646\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	935.511	+	540.118i	1.53111	+	0.883989i
$612$		0			0
$613$	−182.662	−	316.380i	−0.297981	−	0.516117i	0.677693	−	0.735345i	$-0.262980\pi$
$613$	−182.662	−	316.380i	−0.297981	−	0.516117i	−0.975674	+	0.219227i	$0.929646\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	962.980i		−	1.56075i	−0.625315	−	0.780373i	$-0.715029\pi$
$617$		−	962.980i		−	1.56075i	0.625315	−	0.780373i	$-0.284971\pi$
$618$		0			0
$619$	−157.488	−	272.778i	−0.254424	−	0.440675i	0.710315	−	0.703884i	$-0.248553\pi$
$619$	−157.488	−	272.778i	−0.254424	−	0.440675i	−0.964739	+	0.263209i	$0.915219\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−70.1312	+	358.886i	−0.112570	+	0.576061i
$624$		0			0
$625$	389.974	−	675.455i	0.623958	−	1.08073i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8.18211i		−	0.0130081i
$630$		0			0
$631$	−451.196			−0.715048			−0.357524	−	0.933904i	$-0.616379\pi$
$631$	−451.196			−0.715048			−0.357524	+	0.933904i	$0.616379\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	817.798	+	472.156i	1.28787	+	0.743552i
$636$		0			0
$637$	−514.391	+	400.239i	−0.807521	+	0.628319i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	362.856	−	209.495i	0.566078	−	0.326825i	−0.189503	−	0.981880i	$-0.560688\pi$
$641$	362.856	−	209.495i	0.566078	−	0.326825i	0.755581	+	0.655055i	$0.227354\pi$
$642$		0			0
$643$	1118.12			1.73891			0.869457	−	0.494008i	$-0.164469\pi$
$643$	1118.12			1.73891			0.869457	+	0.494008i	$0.164469\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	38.6214	−	22.2981i	0.0596931	−	0.0344638i	−0.469856	−	0.882743i	$-0.655694\pi$
$647$	38.6214	−	22.2981i	0.0596931	−	0.0344638i	0.529550	+	0.848279i	$0.322361\pi$
$648$		0			0
$649$	212.428	−	367.936i	0.327316	−	0.566928i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	207.252	+	119.657i	0.317385	+	0.183242i	0.650226	−	0.759741i	$-0.274674\pi$
$653$	207.252	+	119.657i	0.317385	+	0.183242i	−0.332841	+	0.942983i	$0.608007\pi$
$654$		0			0
$655$	−756.261	−	1309.88i	−1.15460	−	1.99982i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			548.456i			0.832255i	0.909306	+	0.416128i	$0.136613\pi$
$659$			548.456i			0.832255i	−0.909306	+	0.416128i	$0.863387\pi$
$660$		0			0
$661$	−336.318	−	582.519i	−0.508801	−	0.881270i	−0.999948	−	0.0101928i	$-0.996755\pi$
$661$	−336.318	−	582.519i	−0.508801	−	0.881270i	0.491147	−	0.871077i	$-0.336578\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	140.266	+	122.177i	0.210927	+	0.183725i
$666$		0			0
$667$	−106.435	+	184.351i	−0.159573	+	0.276388i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			387.058i			0.576838i
$672$		0			0
$673$	−1188.64			−1.76618			−0.883088	−	0.469206i	$-0.844540\pi$
$673$	−1188.64			−1.76618			−0.883088	+	0.469206i	$0.844540\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−531.239	−	306.711i	−0.784696	−	0.453045i	0.0533958	−	0.998573i	$-0.482996\pi$
$677$	−531.239	−	306.711i	−0.784696	−	0.453045i	−0.838092	+	0.545529i	$0.816329\pi$
$678$		0			0
$679$	413.875	+	1205.88i	0.609536	+	1.77596i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−713.111	+	411.715i	−1.04409	+	0.602803i	−0.920988	−	0.389591i	$-0.872616\pi$
$683$	−713.111	+	411.715i	−1.04409	+	0.602803i	−0.123098	+	0.992395i	$0.539283\pi$
$684$		0			0
$685$	−1614.26			−2.35659
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−227.391	+	131.284i	−0.330030	+	0.190543i
$690$		0			0
$691$	−125.117	+	216.709i	−0.181067	+	0.313617i	−0.942244	−	0.334927i	$-0.891288\pi$
$691$	−125.117	+	216.709i	−0.181067	+	0.313617i	0.761177	+	0.648544i	$0.224622\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1069.15	−	617.274i	−1.53834	−	0.888164i
$696$		0			0
$697$	−175.306	−	303.640i	−0.251516	−	0.435638i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	647.904i		−	0.924257i	−0.886813	−	0.462129i	$-0.847086\pi$
$701$		−	647.904i		−	0.924257i	0.886813	−	0.462129i	$-0.152914\pi$
$702$		0			0
$703$	1.66690	+	2.88716i	0.00237113	+	0.00410691i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−40.2504	−	35.0595i	−0.0569312	−	0.0495891i
$708$		0			0
$709$	470.015	−	814.090i	0.662927	−	1.14822i	−0.316916	−	0.948454i	$-0.602647\pi$
$709$	470.015	−	814.090i	0.662927	−	1.14822i	0.979843	−	0.199770i	$-0.0640194\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	696.560i		−	0.976942i
$714$		0			0
$715$	756.874			1.05857
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−444.866	−	256.843i	−0.618729	−	0.357223i	0.157645	−	0.987496i	$-0.449610\pi$
$719$	−444.866	−	256.843i	−0.618729	−	0.357223i	−0.776374	+	0.630273i	$0.782943\pi$
$720$		0			0
$721$	−463.875	−	90.6476i	−0.643378	−	0.125725i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−105.197	+	60.7357i	−0.145100	+	0.0837734i
$726$		0			0
$727$	314.268			0.432281			0.216141	−	0.976362i	$-0.430653\pi$
$727$	314.268			0.432281			0.216141	+	0.976362i	$0.430653\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	415.060	−	239.635i	0.567797	−	0.327818i
$732$		0			0
$733$	85.0062	−	147.235i	0.115970	−	0.200866i	−0.802197	−	0.597060i	$-0.796336\pi$
$733$	85.0062	−	147.235i	0.115970	−	0.200866i	0.918167	+	0.396193i	$0.129669\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−748.354	−	432.062i	−1.01541	−	0.586244i
$738$		0			0
$739$	323.911	+	561.030i	0.438310	+	0.759175i	0.997559	−	0.0698245i	$-0.0222439\pi$
$739$	323.911	+	561.030i	0.438310	+	0.759175i	−0.559249	+	0.828999i	$0.688911\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			604.982i			0.814243i	0.913374	+	0.407121i	$0.133467\pi$
$743$			604.982i			0.814243i	−0.913374	+	0.407121i	$0.866533\pi$
$744$		0			0
$745$	324.144	+	561.434i	0.435093	+	0.753603i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−197.493	+	1010.64i	−0.263675	+	1.34932i
$750$		0			0
$751$	−434.913	+	753.292i	−0.579112	+	1.00305i	0.416469	+	0.909150i	$0.363267\pi$
$751$	−434.913	+	753.292i	−0.579112	+	1.00305i	−0.995581	+	0.0939022i	$0.970066\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	765.512i		−	1.01392i
$756$		0			0
$757$	864.656			1.14221			0.571107	−	0.820876i	$-0.306514\pi$
$757$	864.656			1.14221			0.571107	+	0.820876i	$0.306514\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−366.131	−	211.386i	−0.481119	−	0.277774i	0.239764	−	0.970831i	$-0.422930\pi$
$761$	−366.131	−	211.386i	−0.481119	−	0.277774i	−0.720883	+	0.693057i	$0.756263\pi$
$762$		0			0
$763$	89.4199	−	102.659i	0.117195	−	0.134547i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	518.603	−	299.415i	0.676144	−	0.390372i
$768$		0			0
$769$	1052.61			1.36881			0.684405	−	0.729102i	$-0.260062\pi$
$769$	1052.61			1.36881			0.684405	+	0.729102i	$0.260062\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−722.654	+	417.225i	−0.934869	+	0.539747i	−0.888348	−	0.459170i	$-0.848147\pi$
$773$	−722.654	+	417.225i	−0.934869	+	0.539747i	−0.0465211	+	0.998917i	$0.514813\pi$
$774$		0			0
$775$	198.741	−	344.229i	0.256440	−	0.444167i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	123.718	+	71.4288i	0.158817	+	0.0916929i
$780$		0			0
$781$	53.2664	+	92.2601i	0.0682028	+	0.118131i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1795.29i		−	2.28700i
$786$		0			0
$787$	484.207	+	838.672i	0.615257	+	1.06566i	0.990339	+	0.138665i	$0.0442811\pi$
$787$	484.207	+	838.672i	0.615257	+	1.06566i	−0.375082	+	0.926991i	$0.622386\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1379.58	+	473.492i	−1.74410	+	0.598599i
$792$		0			0
$793$	−272.778	+	472.465i	−0.343982	+	0.595794i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1445.50i		−	1.81368i	−0.421476	−	0.906840i	$-0.638488\pi$
$797$		−	1445.50i		−	1.81368i	0.421476	−	0.906840i	$-0.361512\pi$
$798$		0			0
$799$	−878.415			−1.09939
$800$		0			0
$801$		0			0
$802$		0			0
$803$	167.848	+	96.9070i	0.209026	+	0.120681i
$804$		0			0
$805$	−551.847	+	633.552i	−0.685524	+	0.787022i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−593.386	+	342.591i	−0.733480	+	0.423475i	−0.819694	−	0.572802i	$-0.805857\pi$
$809$	−593.386	+	342.591i	−0.733480	+	0.423475i	0.0862138	+	0.996277i	$0.472523\pi$
$810$		0			0
$811$	236.166			0.291204			0.145602	−	0.989343i	$-0.453488\pi$
$811$	236.166			0.291204			0.145602	+	0.989343i	$0.453488\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	182.647	−	105.451i	0.224106	−	0.129388i
$816$		0			0
$817$	−97.6394	+	169.116i	−0.119510	+	0.206997i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	344.957	+	199.161i	0.420167	+	0.242583i	0.695149	−	0.718866i	$-0.255339\pi$
$821$	344.957	+	199.161i	0.420167	+	0.242583i	−0.274982	+	0.961449i	$0.588672\pi$
$822$		0			0
$823$	225.410	+	390.422i	0.273888	+	0.474388i	0.969854	−	0.243687i	$-0.0783568\pi$
$823$	225.410	+	390.422i	0.273888	+	0.474388i	−0.695966	+	0.718075i	$0.745023\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	647.284i		−	0.782689i	−0.920244	−	0.391344i	$-0.872010\pi$
$827$		−	647.284i		−	0.782689i	0.920244	−	0.391344i	$-0.127990\pi$
$828$		0			0
$829$	−339.722	−	588.415i	−0.409797	−	0.709789i	0.585070	−	0.810983i	$-0.301067\pi$
$829$	−339.722	−	588.415i	−0.409797	−	0.709789i	−0.994867	+	0.101194i	$0.967734\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	199.516	−	491.002i	0.239515	−	0.589439i
$834$		0			0
$835$	742.715	−	1286.42i	0.889479	−	1.54062i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			488.101i			0.581765i	0.956759	+	0.290882i	$0.0939488\pi$
$839$			488.101i			0.581765i	−0.956759	+	0.290882i	$0.906051\pi$
$840$		0			0
$841$	726.639			0.864018
$842$		0			0
$843$		0			0
$844$		0			0
$845$	41.3670	+	23.8832i	0.0489550	+	0.0282642i
$846$		0			0
$847$	−219.467	−	42.8869i	−0.259111	−	0.0506339i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.0407	+	7.52903i	−0.0153239	+	0.00884727i
$852$		0			0
$853$	1166.17			1.36714			0.683568	−	0.729887i	$-0.260427\pi$
$853$	1166.17			1.36714			0.683568	+	0.729887i	$0.260427\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	524.207	−	302.651i	0.611677	−	0.353152i	−0.161945	−	0.986800i	$-0.551777\pi$
$857$	524.207	−	302.651i	0.611677	−	0.353152i	0.773621	+	0.633648i	$0.218443\pi$
$858$		0			0
$859$	−130.430	+	225.911i	−0.151839	+	0.262993i	−0.931904	−	0.362706i	$-0.881853\pi$
$859$	−130.430	+	225.911i	−0.151839	+	0.262993i	0.780064	+	0.625699i	$0.215186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−192.049	−	110.879i	−0.222536	−	0.128481i	0.384588	−	0.923088i	$-0.374344\pi$
$863$	−192.049	−	110.879i	−0.222536	−	0.128481i	−0.607124	+	0.794607i	$0.707677\pi$
$864$		0			0
$865$	−35.9730	−	62.3071i	−0.0415873	−	0.0720313i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1086.95i		−	1.25081i
$870$		0			0
$871$	−608.988	−	1054.80i	−0.699183	−	1.21102i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	544.594	−	186.912i	0.622393	−	0.213614i
$876$		0			0
$877$	432.414	−	748.962i	0.493060	−	0.854005i	−0.506908	−	0.862000i	$-0.669212\pi$
$877$	432.414	−	748.962i	0.493060	−	0.854005i	0.999968	+	0.00799526i	$0.00254500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			131.618i			0.149396i	0.997206	+	0.0746981i	$0.0237993\pi$
$881$			131.618i			0.149396i	−0.997206	+	0.0746981i	$0.976201\pi$
$882$		0			0
$883$	−374.847			−0.424516			−0.212258	−	0.977214i	$-0.568082\pi$
$883$	−374.847			−0.424516			−0.212258	+	0.977214i	$0.568082\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−997.904	−	576.140i	−1.12503	−	0.649538i	−0.182352	−	0.983233i	$-0.558371\pi$
$887$	−997.904	−	576.140i	−1.12503	−	0.649538i	−0.942681	+	0.333696i	$0.891704\pi$
$888$		0			0
$889$	355.871	+	1036.88i	0.400305	+	1.16634i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	309.960	−	178.955i	0.347100	−	0.200398i
$894$		0			0
$895$	824.946			0.921728
$896$		0			0
$897$		0			0
$898$		0			0
$899$	324.080	−	187.108i	0.360490	−	0.208129i
$900$		0			0
$901$	106.756	−	184.908i	0.118487	−	0.205225i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	553.378	+	319.493i	0.611467	+	0.353031i
$906$		0			0
$907$	−375.097	−	649.686i	−0.413557	−	0.716302i	0.581718	−	0.813390i	$-0.302381\pi$
$907$	−375.097	−	649.686i	−0.413557	−	0.716302i	−0.995276	+	0.0970878i	$0.969047\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1726.98i			1.89570i	0.318718	+	0.947849i	$0.396748\pi$
$911$			1726.98i			1.89570i	−0.318718	+	0.947849i	$0.603252\pi$
$912$		0			0
$913$	165.327	+	286.355i	0.181081	+	0.313642i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	336.753	−	1723.28i	0.367234	−	1.87926i
$918$		0			0
$919$	−394.899	+	683.985i	−0.429705	+	0.744271i	−0.996847	−	0.0793489i	$-0.974716\pi$
$919$	−394.899	+	683.985i	−0.429705	+	0.744271i	0.567142	+	0.823620i	$0.308049\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			150.157i			0.162684i
$924$		0			0
$925$	−8.59266			−0.00928937
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1135.58	−	655.625i	−1.22236	−	0.705732i	−0.256942	−	0.966427i	$-0.582715\pi$
$929$	−1135.58	−	655.625i	−1.22236	−	0.705732i	−0.965421	+	0.260695i	$0.916048\pi$
$930$		0			0
$931$	29.6279	+	213.903i	0.0318237	+	0.229756i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−533.011	+	307.734i	−0.570065	+	0.329127i
$936$		0			0
$937$	−1490.93			−1.59118			−0.795588	−	0.605838i	$-0.792838\pi$
$937$	−1490.93			−1.59118			−0.795588	+	0.605838i	$0.792838\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	176.666	−	101.998i	0.187743	−	0.108394i	−0.403182	−	0.915120i	$-0.632096\pi$
$941$	176.666	−	101.998i	0.187743	−	0.108394i	0.590926	+	0.806726i	$0.298763\pi$
$942$		0			0
$943$	−322.628	+	558.808i	−0.342129	+	0.592585i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−728.251	−	420.456i	−0.769009	−	0.443987i	0.0635121	−	0.997981i	$-0.479770\pi$
$947$	−728.251	−	420.456i	−0.769009	−	0.443987i	−0.832521	+	0.553994i	$0.813103\pi$
$948$		0			0
$949$	136.590	+	236.580i	0.143930	+	0.249294i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	676.046i		−	0.709387i	−0.934983	−	0.354693i	$-0.884585\pi$
$953$		−	676.046i		−	0.709387i	0.934983	−	0.354693i	$-0.115415\pi$
$954$		0			0
$955$	−87.1805	−	151.001i	−0.0912885	−	0.158116i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1413.09	−	1230.86i	−1.47351	−	1.28348i
$960$		0			0
$961$	−131.759	+	228.214i	−0.137106	+	0.237475i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1883.29i		−	1.95159i
$966$		0			0
$967$	−513.914			−0.531452			−0.265726	−	0.964049i	$-0.585612\pi$
$967$	−513.914			−0.531452			−0.265726	+	0.964049i	$0.585612\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	491.968	+	284.038i	0.506661	+	0.292521i	0.731460	−	0.681884i	$-0.238839\pi$
$971$	491.968	+	284.038i	0.506661	+	0.292521i	−0.224799	+	0.974405i	$0.572173\pi$
$972$		0			0
$973$	−465.248	−	1355.56i	−0.478159	−	1.39318i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−609.956	+	352.158i	−0.624315	+	0.360448i	−0.778547	−	0.627586i	$-0.784043\pi$
$977$	−609.956	+	352.158i	−0.624315	+	0.360448i	0.154232	+	0.988035i	$0.450710\pi$
$978$		0			0
$979$	492.974			0.503549
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1536.87	+	887.310i	−1.56344	+	0.902655i	−0.566540	+	0.824034i	$0.691718\pi$
$983$	−1536.87	+	887.310i	−1.56344	+	0.902655i	−0.996905	+	0.0786212i	$0.974948\pi$
$984$		0			0
$985$	345.095	−	597.722i	0.350350	−	0.606825i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−763.861	−	441.015i	−0.772357	−	0.445920i
$990$		0			0
$991$	599.026	+	1037.54i	0.604466	+	1.04697i	0.992136	+	0.125167i	$0.0399468\pi$
$991$	599.026	+	1037.54i	0.604466	+	1.04697i	−0.387670	+	0.921798i	$0.626720\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			887.061i			0.891519i
$996$		0			0
$997$	343.411	+	594.805i	0.344444	+	0.596595i	0.985253	−	0.171106i	$-0.0547340\pi$
$997$	343.411	+	594.805i	0.344444	+	0.596595i	−0.640808	+	0.767701i	$0.721401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.bk.g.485.7	yes	16
3.2	odd	2	inner	756.3.bk.g.485.2	yes	16
7.4	even	3	inner	756.3.bk.g.53.2	✓	16
21.11	odd	6	inner	756.3.bk.g.53.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.g.53.2	✓	16	7.4	even	3	inner
756.3.bk.g.53.7	yes	16	21.11	odd	6	inner
756.3.bk.g.485.2	yes	16	3.2	odd	2	inner
756.3.bk.g.485.7	yes	16	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(5.22199 + 3.01492i) q^{5} +(2.27239 + 6.62090i) q^{7} +(8.17257 - 4.71844i) q^{11} +13.3012 q^{13} +(-9.36705 + 5.40807i) q^{17} +(2.20352 - 3.81661i) q^{19} +(17.2388 + 9.95281i) q^{23} +(5.67943 + 9.83706i) q^{25} +10.6940i q^{29} +(-17.4966 - 30.3049i) q^{31} +(-8.09507 + 41.4253i) q^{35} +(-0.378236 + 0.655125i) q^{37} +32.4157i q^{41} -44.3106 q^{43} +(70.3329 + 40.6067i) q^{47} +(-38.6725 + 30.0905i) q^{49} +(-17.0955 + 9.87011i) q^{53} +56.9027 q^{55} +(38.9892 - 22.5104i) q^{59} +(-20.5078 + 35.5205i) q^{61} +(69.4586 + 40.1020i) q^{65} +(-45.7845 - 79.3010i) q^{67} +11.2890i q^{71} +(10.2690 + 17.7864i) q^{73} +(49.8115 + 43.3876i) q^{77} +(57.5907 - 99.7501i) q^{79} +35.0385i q^{83} -65.2195 q^{85} +(45.2404 + 26.1196i) q^{89} +(30.2254 + 88.0658i) q^{91} +(23.0135 - 13.2869i) q^{95} +182.132 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{7} + 88 q^{13} + 14 q^{19} + 36 q^{25} - 68 q^{31} - 76 q^{37} - 292 q^{43} - 20 q^{49} - 272 q^{55} - 110 q^{61} - 72 q^{67} + 60 q^{73} + 154 q^{79} + 700 q^{85} - 74 q^{91} + 264 q^{97}+O(q^{100}) \) 16 * q + 24 * q^7 + 88 * q^13 + 14 * q^19 + 36 * q^25 - 68 * q^31 - 76 * q^37 - 292 * q^43 - 20 * q^49 - 272 * q^55 - 110 * q^61 - 72 * q^67 + 60 * q^73 + 154 * q^79 + 700 * q^85 - 74 * q^91 + 264 * q^97

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