LMFDB - Embedded newform 756.3.bk.g.485.8

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)

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);

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")

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			485.8
Root			$-2.92662 + 1.68969i$ of defining polynomial
Character	$\chi$	$=$	756.485
Dual form			756.3.bk.g.53.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.52876 + 4.34673i) q^{5} +(2.97960 - 6.33419i) q^{7} +(-11.7910 + 6.80755i) q^{11} +11.1789 q^{13} +(21.5048 - 12.4158i) q^{17} +(11.6543 - 20.1858i) q^{19} +(8.87573 + 5.12441i) q^{23} +(25.2882 + 43.8004i) q^{25} +14.5093i q^{29} +(-15.7276 - 27.2410i) q^{31} +(49.9658 - 34.7371i) q^{35} +(-22.1093 + 38.2944i) q^{37} +8.92653i q^{41} -56.8453 q^{43} +(29.8937 + 17.2591i) q^{47} +(-31.2439 - 37.7468i) q^{49} +(79.6796 - 46.0030i) q^{53} -118.362 q^{55} +(-3.07301 + 1.77420i) q^{59} +(31.3445 - 54.2903i) q^{61} +(84.1633 + 48.5917i) q^{65} +(58.9819 + 102.160i) q^{67} +56.6034i q^{71} +(32.7066 + 56.6495i) q^{73} +(7.98772 + 94.9704i) q^{77} +(-4.51904 + 7.82720i) q^{79} +39.9286i q^{83} +215.873 q^{85} +(26.5356 + 15.3203i) q^{89} +(33.3087 - 70.8093i) q^{91} +(175.485 - 101.316i) q^{95} -137.668 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	7.52876	+	4.34673i	1.50575	+	0.869347i	0.999978	+	0.00668085i	$0.00212660\pi$
$5$	7.52876	+	4.34673i	1.50575	+	0.869347i	0.505775	+	0.862666i	$0.331207\pi$
$6$		0			0
$7$	2.97960	−	6.33419i	0.425658	−	0.904884i
$7$	2.97960	−	6.33419i	0.425658	−	0.904884i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11.7910	+	6.80755i	−1.07191	+	0.618868i	−0.928703	−	0.370825i	$-0.879075\pi$
$11$	−11.7910	+	6.80755i	−1.07191	+	0.618868i	−0.143208	+	0.989693i	$0.545742\pi$
$12$		0			0
$13$	11.1789			0.859916			0.429958	−	0.902849i	$-0.358528\pi$
$13$	11.1789			0.859916			0.429958	+	0.902849i	$0.358528\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	21.5048	−	12.4158i	1.26499	−	0.730341i	0.290952	−	0.956738i	$-0.406028\pi$
$17$	21.5048	−	12.4158i	1.26499	−	0.730341i	0.974035	+	0.226397i	$0.0726946\pi$
$18$		0			0
$19$	11.6543	−	20.1858i	0.613383	−	1.06241i	−0.377283	−	0.926098i	$-0.623141\pi$
$19$	11.6543	−	20.1858i	0.613383	−	1.06241i	0.990666	−	0.136313i	$-0.0435252\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.87573	+	5.12441i	0.385901	+	0.222800i	0.680383	−	0.732857i	$-0.261814\pi$
$23$	8.87573	+	5.12441i	0.385901	+	0.222800i	−0.294481	+	0.955657i	$0.595147\pi$
$24$		0			0
$25$	25.2882	+	43.8004i	1.01153	+	1.75202i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			14.5093i			0.500321i	0.968204	+	0.250160i	$0.0804833\pi$
$29$			14.5093i			0.500321i	−0.968204	+	0.250160i	$0.919517\pi$
$30$		0			0
$31$	−15.7276	−	27.2410i	−0.507341	−	0.878741i	−0.999964	−	0.00849763i	$-0.997295\pi$
$31$	−15.7276	−	27.2410i	−0.507341	−	0.878741i	0.492623	−	0.870243i	$-0.336038\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	49.9658	−	34.7371i	1.42759	−	0.992487i
$36$		0			0
$37$	−22.1093	+	38.2944i	−0.597547	+	1.03498i	0.395634	+	0.918408i	$0.370525\pi$
$37$	−22.1093	+	38.2944i	−0.597547	+	1.03498i	−0.993182	+	0.116575i	$0.962809\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.92653i			0.217720i	0.994057	+	0.108860i	$0.0347200\pi$
$41$			8.92653i			0.217720i	−0.994057	+	0.108860i	$0.965280\pi$
$42$		0			0
$43$	−56.8453			−1.32198			−0.660992	−	0.750393i	$-0.729864\pi$
$43$	−56.8453			−1.32198			−0.660992	+	0.750393i	$0.729864\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	29.8937	+	17.2591i	0.636036	+	0.367216i	0.783086	−	0.621913i	$-0.213644\pi$
$47$	29.8937	+	17.2591i	0.636036	+	0.367216i	−0.147050	+	0.989129i	$0.546978\pi$
$48$		0			0
$49$	−31.2439	−	37.7468i	−0.637631	−	0.770342i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	79.6796	−	46.0030i	1.50339	−	0.867982i	0.503397	−	0.864055i	$-0.332083\pi$
$53$	79.6796	−	46.0030i	1.50339	−	0.867982i	0.999992	−	0.00392664i	$-0.00124989\pi$
$54$		0			0
$55$	−118.362			−2.15204
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.07301	+	1.77420i	−0.0520849	+	0.0300713i	−0.525816	−	0.850598i	$-0.676240\pi$
$59$	−3.07301	+	1.77420i	−0.0520849	+	0.0300713i	0.473731	+	0.880669i	$0.342907\pi$
$60$		0			0
$61$	31.3445	−	54.2903i	0.513845	−	0.890005i	−0.486026	−	0.873944i	$-0.661554\pi$
$61$	31.3445	−	54.2903i	0.513845	−	0.890005i	0.999871	−	0.0160609i	$-0.00511258\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	84.1633	+	48.5917i	1.29482	+	0.747565i
$66$		0			0
$67$	58.9819	+	102.160i	0.880326	+	1.52477i	0.850978	+	0.525201i	$0.176010\pi$
$67$	58.9819	+	102.160i	0.880326	+	1.52477i	0.0293482	+	0.999569i	$0.490657\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			56.6034i			0.797231i	0.917118	+	0.398616i	$0.130509\pi$
$71$			56.6034i			0.797231i	−0.917118	+	0.398616i	$0.869491\pi$
$72$		0			0
$73$	32.7066	+	56.6495i	0.448035	+	0.776020i	0.998258	−	0.0589978i	$-0.0187905\pi$
$73$	32.7066	+	56.6495i	0.448035	+	0.776020i	−0.550223	+	0.835018i	$0.685457\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.98772	+	94.9704i	0.103737	+	1.23338i
$78$		0			0
$79$	−4.51904	+	7.82720i	−0.0572030	+	0.0990785i	−0.893209	−	0.449642i	$-0.851552\pi$
$79$	−4.51904	+	7.82720i	−0.0572030	+	0.0990785i	0.836006	+	0.548721i	$0.184885\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			39.9286i			0.481067i	0.970641	+	0.240533i	$0.0773224\pi$
$83$			39.9286i			0.481067i	−0.970641	+	0.240533i	$0.922678\pi$
$84$		0			0
$85$	215.873			2.53968
$86$		0			0
$87$		0			0
$88$		0			0
$89$	26.5356	+	15.3203i	0.298153	+	0.172138i	0.641613	−	0.767029i	$-0.278266\pi$
$89$	26.5356	+	15.3203i	0.298153	+	0.172138i	−0.343460	+	0.939167i	$0.611599\pi$
$90$		0			0
$91$	33.3087	−	70.8093i	0.366030	−	0.778125i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	175.485	−	101.316i	1.84721	−	1.06649i
$96$		0			0
$97$	−137.668			−1.41926			−0.709630	−	0.704575i	$-0.751138\pi$
$97$	−137.668			−1.41926			−0.709630	+	0.704575i	$0.751138\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	109.220	−	63.0580i	1.08138	−	0.624336i	0.150113	−	0.988669i	$-0.452036\pi$
$101$	109.220	−	63.0580i	1.08138	−	0.624336i	0.931269	+	0.364332i	$0.118703\pi$
$102$		0			0
$103$	93.8955	−	162.632i	0.911607	−	1.57895i	0.0998115	−	0.995006i	$-0.468176\pi$
$103$	93.8955	−	162.632i	0.911607	−	1.57895i	0.811795	−	0.583942i	$-0.198491\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−90.3695	−	52.1749i	−0.844575	−	0.487615i	0.0142419	−	0.999899i	$-0.495467\pi$
$107$	−90.3695	−	52.1749i	−0.844575	−	0.487615i	−0.858817	+	0.512283i	$0.828800\pi$
$108$		0			0
$109$	2.30011	+	3.98391i	0.0211019	+	0.0365496i	0.876384	−	0.481614i	$-0.159949\pi$
$109$	2.30011	+	3.98391i	0.0211019	+	0.0365496i	−0.855282	+	0.518163i	$0.826616\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	41.2575i		−	0.365110i	−0.983196	−	0.182555i	$-0.941563\pi$
$113$		−	41.2575i		−	0.365110i	0.983196	−	0.182555i	$-0.0584368\pi$
$114$		0			0
$115$	44.5488	+	77.1608i	0.387381	+	0.670964i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.5682	−	173.210i	−0.122422	−	1.45554i
$120$		0			0
$121$	32.1854	−	55.7468i	0.265995	−	0.460717i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			222.347i			1.77878i
$126$		0			0
$127$	5.70178			0.0448959			0.0224480	−	0.999748i	$-0.492854\pi$
$127$	5.70178			0.0448959			0.0224480	+	0.999748i	$0.492854\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	111.208	+	64.2061i	0.848917	+	0.490123i	0.860285	−	0.509813i	$-0.170285\pi$
$131$	111.208	+	64.2061i	0.848917	+	0.490123i	−0.0113680	+	0.999935i	$0.503619\pi$
$132$		0			0
$133$	−93.1356	−	133.966i	−0.700267	−	1.00726i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−159.139	+	91.8790i	−1.16160	+	0.670649i	−0.951687	−	0.307071i	$-0.900651\pi$
$137$	−159.139	+	91.8790i	−1.16160	+	0.670649i	−0.209912	+	0.977720i	$0.567318\pi$
$138$		0			0
$139$	−43.9291			−0.316037			−0.158018	−	0.987436i	$-0.550511\pi$
$139$	−43.9291			−0.316037			−0.158018	+	0.987436i	$0.550511\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−131.811	+	76.1010i	−0.921753	+	0.532175i
$144$		0			0
$145$	−63.0680	+	109.237i	−0.434952	+	0.753359i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−151.850	−	87.6704i	−1.01913	−	0.588392i	−0.105275	−	0.994443i	$-0.533572\pi$
$149$	−151.850	−	87.6704i	−1.01913	−	0.588392i	−0.913850	+	0.406051i	$0.866906\pi$
$150$		0			0
$151$	−8.69643	−	15.0627i	−0.0575922	−	0.0997527i	0.835792	−	0.549046i	$-0.185009\pi$
$151$	−8.69643	−	15.0627i	−0.0575922	−	0.0997527i	−0.893384	+	0.449294i	$0.851676\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	273.454i		−	1.76422i
$156$		0			0
$157$	72.9478	+	126.349i	0.464636	+	0.804773i	0.999185	−	0.0403645i	$-0.0128519\pi$
$157$	72.9478	+	126.349i	0.464636	+	0.804773i	−0.534549	+	0.845137i	$0.679519\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	58.9051	−	40.9519i	0.365870	−	0.254359i
$162$		0			0
$163$	−134.374	+	232.743i	−0.824383	+	1.42787i	0.0780063	+	0.996953i	$0.475145\pi$
$163$	−134.374	+	232.743i	−0.824383	+	1.42787i	−0.902390	+	0.430921i	$0.858189\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	181.784i		−	1.08853i	−0.838914	−	0.544264i	$-0.816809\pi$
$167$		−	181.784i		−	1.08853i	0.838914	−	0.544264i	$-0.183191\pi$
$168$		0			0
$169$	−44.0320			−0.260544
$170$		0			0
$171$		0			0
$172$		0			0
$173$	256.882	+	148.311i	1.48487	+	0.857288i	0.999852	−	0.0172178i	$-0.00548086\pi$
$173$	256.882	+	148.311i	1.48487	+	0.857288i	0.485015	+	0.874506i	$0.338814\pi$
$174$		0			0
$175$	352.789	−	29.6722i	2.01594	−	0.169555i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−37.6828	+	21.7562i	−0.210519	+	0.121543i	−0.601552	−	0.798833i	$-0.705451\pi$
$179$	−37.6828	+	21.7562i	−0.210519	+	0.121543i	0.391034	+	0.920376i	$0.372118\pi$
$180$		0			0
$181$	−292.976			−1.61865			−0.809327	−	0.587358i	$-0.800168\pi$
$181$	−292.976			−1.61865			−0.809327	+	0.587358i	$0.800168\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−332.911	+	192.206i	−1.79952	+	1.03895i
$186$		0			0
$187$	−169.042	+	292.790i	−0.903969	+	1.56572i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−241.060	−	139.176i	−1.26209	−	0.728670i	−0.288615	−	0.957445i	$-0.593195\pi$
$191$	−241.060	−	139.176i	−1.26209	−	0.728670i	−0.973479	+	0.228775i	$0.926528\pi$
$192$		0			0
$193$	−103.146	−	178.654i	−0.534436	−	0.925670i	−0.999190	−	0.0402304i	$-0.987191\pi$
$193$	−103.146	−	178.654i	−0.534436	−	0.925670i	0.464755	−	0.885439i	$-0.346143\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			346.492i			1.75884i	0.476042	+	0.879422i	$0.342071\pi$
$197$			346.492i			1.75884i	−0.476042	+	0.879422i	$0.657929\pi$
$198$		0			0
$199$	−175.902	−	304.671i	−0.883929	−	1.53101i	−0.846936	−	0.531695i	$-0.821555\pi$
$199$	−175.902	−	304.671i	−0.883929	−	1.53101i	−0.0369937	−	0.999315i	$-0.511778\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	91.9047	+	43.2320i	0.452732	+	0.212965i
$204$		0			0
$205$	−38.8012	+	67.2057i	−0.189274	+	0.327833i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			317.348i			1.51841i
$210$		0			0
$211$	−18.2223			−0.0863616			−0.0431808	−	0.999067i	$-0.513749\pi$
$211$	−18.2223			−0.0863616			−0.0431808	+	0.999067i	$0.513749\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−427.975	−	247.091i	−1.99058	−	1.14926i
$216$		0			0
$217$	−219.411	+	18.4541i	−1.01111	+	0.0850421i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	240.400	−	138.795i	1.08778	−	0.628032i
$222$		0			0
$223$	298.635			1.33917			0.669586	−	0.742734i	$-0.266471\pi$
$223$	298.635			1.33917			0.669586	+	0.742734i	$0.266471\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−71.5442	+	41.3061i	−0.315173	+	0.181965i	−0.649239	−	0.760585i	$-0.724912\pi$
$227$	−71.5442	+	41.3061i	−0.315173	+	0.181965i	0.334066	+	0.942550i	$0.391579\pi$
$228$		0			0
$229$	114.635	−	198.554i	0.500591	−	0.867049i	−0.499409	−	0.866366i	$-0.666449\pi$
$229$	114.635	−	198.554i	0.500591	−	0.867049i	1.00000		0.000682554i	$-0.000217264\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−186.088	−	107.438i	−0.798661	−	0.461107i	0.0443419	−	0.999016i	$-0.485881\pi$
$233$	−186.088	−	107.438i	−0.798661	−	0.461107i	−0.843003	+	0.537909i	$0.819214\pi$
$234$		0			0
$235$	150.042	+	259.880i	0.638475	+	1.10587i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			249.380i			1.04343i	0.853120	+	0.521715i	$0.174707\pi$
$239$			249.380i			1.04343i	−0.853120	+	0.521715i	$0.825293\pi$
$240$		0			0
$241$	−32.4265	−	56.1644i	−0.134550	−	0.233047i	0.790876	−	0.611977i	$-0.209626\pi$
$241$	−32.4265	−	56.1644i	−0.134550	−	0.233047i	−0.925425	+	0.378930i	$0.876292\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−71.1529	−	419.995i	−0.290420	−	1.71427i
$246$		0			0
$247$	130.282	−	225.655i	0.527458	−	0.913584i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	345.782i		−	1.37762i	−0.724943	−	0.688809i	$-0.758134\pi$
$251$		−	345.782i		−	1.37762i	0.724943	−	0.688809i	$-0.241866\pi$
$252$		0			0
$253$	−139.539			−0.551536
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−394.776	−	227.924i	−1.53609	−	0.886865i	−0.999062	−	0.0433032i	$-0.986212\pi$
$257$	−394.776	−	227.924i	−1.53609	−	0.886865i	−0.537033	−	0.843561i	$-0.680455\pi$
$258$		0			0
$259$	176.687	+	254.146i	0.682189	+	0.981260i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−299.848	+	173.117i	−1.14011	+	0.658241i	−0.946457	−	0.322830i	$-0.895366\pi$
$263$	−299.848	+	173.117i	−1.14011	+	0.658241i	−0.193650	+	0.981071i	$0.562033\pi$
$264$		0			0
$265$	799.852			3.01831
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.2236	+	12.8308i	−0.0826155	+	0.0476981i	−0.540739	−	0.841191i	$-0.681855\pi$
$269$	−22.2236	+	12.8308i	−0.0826155	+	0.0476981i	0.458123	+	0.888889i	$0.348522\pi$
$270$		0			0
$271$	8.00466	−	13.8645i	0.0295375	−	0.0511604i	−0.850879	−	0.525362i	$-0.823930\pi$
$271$	8.00466	−	13.8645i	0.0295375	−	0.0511604i	0.880416	+	0.474202i	$0.157263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−596.347	−	344.301i	−2.16853	−	1.25200i
$276$		0			0
$277$	−165.286	−	286.284i	−0.596700	−	1.03352i	−0.993305	−	0.115526i	$-0.963145\pi$
$277$	−165.286	−	286.284i	−0.596700	−	1.03352i	0.396604	−	0.917990i	$-0.370189\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			112.448i			0.400171i	0.979778	+	0.200086i	$0.0641220\pi$
$281$			112.448i			0.400171i	−0.979778	+	0.200086i	$0.935878\pi$
$282$		0			0
$283$	−96.7479	−	167.572i	−0.341865	−	0.592128i	0.642914	−	0.765939i	$-0.277725\pi$
$283$	−96.7479	−	167.572i	−0.341865	−	0.592128i	−0.984779	+	0.173810i	$0.944392\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	56.5423	+	26.5975i	0.197012	+	0.0926743i
$288$		0			0
$289$	163.804	−	283.717i	0.566795	−	0.981718i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			527.275i			1.79957i	0.436332	+	0.899786i	$0.356277\pi$
$293$			527.275i			1.79957i	−0.436332	+	0.899786i	$0.643723\pi$
$294$		0			0
$295$	−30.8480			−0.104569
$296$		0			0
$297$		0			0
$298$		0			0
$299$	99.2210	+	57.2853i	0.331843	+	0.191590i
$300$		0			0
$301$	−169.377	+	360.069i	−0.562713	+	1.19624i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	471.971	−	272.493i	1.54745	−	0.893418i
$306$		0			0
$307$	246.199			0.801952			0.400976	−	0.916089i	$-0.368671\pi$
$307$	246.199			0.801952			0.400976	+	0.916089i	$0.368671\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−166.859	+	96.3362i	−0.536525	+	0.309763i	−0.743669	−	0.668548i	$-0.766916\pi$
$311$	−166.859	+	96.3362i	−0.536525	+	0.309763i	0.207145	+	0.978310i	$0.433583\pi$
$312$		0			0
$313$	−20.4157	+	35.3611i	−0.0652259	+	0.112975i	−0.896794	−	0.442448i	$-0.854110\pi$
$313$	−20.4157	+	35.3611i	−0.0652259	+	0.112975i	0.831568	+	0.555423i	$0.187443\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−424.474	−	245.070i	−1.33903	−	0.773091i	−0.352370	−	0.935861i	$-0.614624\pi$
$317$	−424.474	−	245.070i	−1.33903	−	0.773091i	−0.986664	+	0.162769i	$0.947957\pi$
$318$		0			0
$319$	−98.7728	−	171.079i	−0.309632	−	0.536299i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	578.789i		−	1.79191i
$324$		0			0
$325$	282.694	+	489.641i	0.869828	+	1.50659i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	198.394	−	137.927i	0.603021	−	0.419231i
$330$		0			0
$331$	10.3922	−	17.9999i	0.0313964	−	0.0543802i	−0.849900	−	0.526944i	$-0.823338\pi$
$331$	10.3922	−	17.9999i	0.0313964	−	0.0543802i	0.881297	+	0.472563i	$0.156671\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1025.51i			3.06123i
$336$		0			0
$337$	39.2512			0.116472			0.0582362	−	0.998303i	$-0.481452\pi$
$337$	39.2512			0.116472			0.0582362	+	0.998303i	$0.481452\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	370.888	+	214.132i	1.08765	+	0.627954i
$342$		0			0
$343$	−332.190	+	85.4344i	−0.968483	+	0.249080i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	337.526	−	194.871i	0.972697	−	0.561587i	0.0726398	−	0.997358i	$-0.476858\pi$
$347$	337.526	−	194.871i	0.972697	−	0.561587i	0.900057	+	0.435771i	$0.143524\pi$
$348$		0			0
$349$	−231.687			−0.663859			−0.331930	−	0.943304i	$-0.607700\pi$
$349$	−231.687			−0.663859			−0.331930	+	0.943304i	$0.607700\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−209.073	+	120.709i	−0.592276	+	0.341950i	−0.765997	−	0.642844i	$-0.777754\pi$
$353$	−209.073	+	120.709i	−0.592276	+	0.341950i	0.173721	+	0.984795i	$0.444421\pi$
$354$		0			0
$355$	−246.040	+	426.154i	−0.693070	+	1.20043i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	44.6293	+	25.7668i	0.124316	+	0.0717737i	0.560868	−	0.827905i	$-0.310467\pi$
$359$	44.6293	+	25.7668i	0.124316	+	0.0717737i	−0.436553	+	0.899679i	$0.643801\pi$
$360$		0			0
$361$	−91.1445	−	157.867i	−0.252478	−	0.437305i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			568.667i			1.55799i
$366$		0			0
$367$	−298.159	−	516.426i	−0.812421	−	1.40716i	−0.911165	−	0.412042i	$-0.864816\pi$
$367$	−298.159	−	516.426i	−0.812421	−	1.40716i	0.0987433	−	0.995113i	$-0.468518\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−53.9782	−	641.777i	−0.145494	−	1.72986i
$372$		0			0
$373$	177.554	−	307.533i	0.476017	−	0.824486i	−0.523606	−	0.851961i	$-0.675413\pi$
$373$	177.554	−	307.533i	0.476017	−	0.824486i	0.999622	+	0.0274753i	$0.00874676\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			162.198i			0.430234i
$378$		0			0
$379$	−688.685			−1.81711			−0.908555	−	0.417765i	$-0.862813\pi$
$379$	−688.685			−1.81711			−0.908555	+	0.417765i	$0.862813\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−277.588	−	160.266i	−0.724774	−	0.418449i	0.0917332	−	0.995784i	$-0.470759\pi$
$383$	−277.588	−	160.266i	−0.724774	−	0.418449i	−0.816507	+	0.577335i	$0.804093\pi$
$384$		0			0
$385$	−352.673	+	749.730i	−0.916034	+	1.94735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−358.901	+	207.212i	−0.922626	+	0.532678i	−0.884472	−	0.466594i	$-0.845481\pi$
$389$	−358.901	+	207.212i	−0.922626	+	0.532678i	−0.0381540	+	0.999272i	$0.512148\pi$
$390$		0			0
$391$	254.494			0.650880
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−68.0455	+	39.2861i	−0.172267	+	0.0994585i
$396$		0			0
$397$	104.723	−	181.386i	0.263786	−	0.456891i	−0.703459	−	0.710736i	$-0.748362\pi$
$397$	104.723	−	181.386i	0.263786	−	0.456891i	0.967245	+	0.253845i	$0.0816953\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	281.142	+	162.317i	0.701101	+	0.404781i	0.807757	−	0.589515i	$-0.200681\pi$
$401$	281.142	+	162.317i	0.701101	+	0.404781i	−0.106656	+	0.994296i	$0.534014\pi$
$402$		0			0
$403$	−175.817	−	304.524i	−0.436271	−	0.755643i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	602.039i		−	1.47921i
$408$		0			0
$409$	213.424	+	369.662i	0.521820	+	0.903818i	0.999678	+	0.0253811i	$0.00807994\pi$
$409$	213.424	+	369.662i	0.521820	+	0.903818i	−0.477858	+	0.878437i	$0.658587\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.08178	+	24.7515i	0.00504064	+	0.0599309i
$414$		0			0
$415$	−173.559	+	300.613i	−0.418214	+	0.724368i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			75.6074i			0.180447i	0.995922	+	0.0902236i	$0.0287582\pi$
$419$			75.6074i			0.180447i	−0.995922	+	0.0902236i	$0.971242\pi$
$420$		0			0
$421$	−235.309			−0.558928			−0.279464	−	0.960156i	$-0.590157\pi$
$421$	−235.309			−0.558928			−0.279464	+	0.960156i	$0.590157\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1087.63	+	627.945i	2.55914	+	1.47752i
$426$		0			0
$427$	−250.491	−	360.306i	−0.586630	−	0.843808i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	185.155	−	106.900i	0.429595	−	0.248027i	−0.269579	−	0.962978i	$-0.586885\pi$
$431$	185.155	−	106.900i	0.429595	−	0.248027i	0.699174	+	0.714951i	$0.253551\pi$
$432$		0			0
$433$	−194.582			−0.449382			−0.224691	−	0.974430i	$-0.572137\pi$
$433$	−194.582			−0.449382			−0.224691	+	0.974430i	$0.572137\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	206.881	−	119.443i	0.473411	−	0.273324i
$438$		0			0
$439$	−125.056	+	216.604i	−0.284866	+	0.493403i	−0.972577	−	0.232582i	$-0.925283\pi$
$439$	−125.056	+	216.604i	−0.284866	+	0.493403i	0.687710	+	0.725985i	$0.258616\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.1511	−	17.9851i	−0.0703184	−	0.0405984i	0.464429	−	0.885611i	$-0.346260\pi$
$443$	−31.1511	−	17.9851i	−0.0703184	−	0.0405984i	−0.534747	+	0.845012i	$0.679593\pi$
$444$		0			0
$445$	133.187	+	230.686i	0.299296	+	0.518396i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			488.642i			1.08829i	0.838991	+	0.544145i	$0.183146\pi$
$449$			488.642i			1.08829i	−0.838991	+	0.544145i	$0.816854\pi$
$450$		0			0
$451$	−60.7678	−	105.253i	−0.134740	−	0.233377i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	558.563	−	388.322i	1.22761	−	0.853456i
$456$		0			0
$457$	396.542	−	686.831i	0.867707	−	1.50291i	0.00337398	−	0.999994i	$-0.498926\pi$
$457$	396.542	−	686.831i	0.867707	−	1.50291i	0.864333	−	0.502919i	$-0.167741\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	562.398i		−	1.21995i	−0.792420	−	0.609976i	$-0.791179\pi$
$461$		−	562.398i		−	1.21995i	0.792420	−	0.609976i	$-0.208821\pi$
$462$		0			0
$463$	23.3103			0.0503463			0.0251731	−	0.999683i	$-0.491986\pi$
$463$	23.3103			0.0503463			0.0251731	+	0.999683i	$0.491986\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	449.648	+	259.604i	0.962844	+	0.555898i	0.897047	−	0.441935i	$-0.145708\pi$
$467$	449.648	+	259.604i	0.962844	+	0.555898i	0.0657967	+	0.997833i	$0.479041\pi$
$468$		0			0
$469$	822.841	−	69.2071i	1.75446	−	0.147563i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	670.264	−	386.977i	1.41705	−	0.818133i
$474$		0			0
$475$	1178.86			2.48181
$476$		0			0
$477$		0			0
$478$		0			0
$479$	60.4640	−	34.9089i	0.126230	−	0.0728787i	−0.435556	−	0.900162i	$-0.643448\pi$
$479$	60.4640	−	34.9089i	0.126230	−	0.0728787i	0.561785	+	0.827283i	$0.310115\pi$
$480$		0			0
$481$	−247.157	+	428.089i	−0.513841	+	0.889998i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1036.47	−	598.407i	−2.13705	−	1.23383i
$486$		0			0
$487$	51.2075	+	88.6940i	0.105149	+	0.182123i	0.913799	−	0.406167i	$-0.133135\pi$
$487$	51.2075	+	88.6940i	0.105149	+	0.182123i	−0.808650	+	0.588290i	$0.799801\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			297.180i			0.605254i	0.953109	+	0.302627i	$0.0978637\pi$
$491$			297.180i			0.605254i	−0.953109	+	0.302627i	$0.902136\pi$
$492$		0			0
$493$	180.144	+	312.019i	0.365405	+	0.632899i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	358.537	+	168.656i	0.721402	+	0.339348i
$498$		0			0
$499$	−39.3143	+	68.0943i	−0.0787861	+	0.136461i	−0.902726	−	0.430215i	$-0.858438\pi$
$499$	−39.3143	+	68.0943i	−0.0787861	+	0.136461i	0.823940	+	0.566677i	$0.191771\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			666.828i			1.32570i	0.748752	+	0.662851i	$0.230654\pi$
$503$			666.828i			1.32570i	−0.748752	+	0.662851i	$0.769346\pi$
$504$		0			0
$505$	1096.38			2.17106
$506$		0			0
$507$		0			0
$508$		0			0
$509$	335.646	+	193.785i	0.659422	+	0.380717i	0.792057	−	0.610448i	$-0.209010\pi$
$509$	335.646	+	193.785i	0.659422	+	0.380717i	−0.132635	+	0.991165i	$0.542344\pi$
$510$		0			0
$511$	456.281	−	38.3767i	0.892918	−	0.0751011i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1413.83	−	816.277i	2.74531	−	1.58500i
$516$		0			0
$517$	−469.970			−0.909032
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−711.180	+	410.600i	−1.36503	+	0.788100i	−0.990288	−	0.139029i	$-0.955602\pi$
$521$	−711.180	+	410.600i	−1.36503	+	0.788100i	−0.374741	+	0.927129i	$0.622268\pi$
$522$		0			0
$523$	236.710	−	409.993i	0.452599	−	0.783925i	−0.545947	−	0.837820i	$-0.683830\pi$
$523$	236.710	−	409.993i	0.452599	−	0.783925i	0.998547	+	0.0538944i	$0.0171634\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−676.436	−	390.541i	−1.28356	−	0.741064i
$528$		0			0
$529$	−211.981	−	367.162i	−0.400720	−	0.694068i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			99.7888i			0.187221i
$534$		0			0
$535$	−453.580	−	785.624i	−0.847814	−	1.46846i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	625.360	+	232.378i	1.16022	+	0.431129i
$540$		0			0
$541$	−168.645	+	292.102i	−0.311729	+	0.539930i	−0.978737	−	0.205121i	$-0.934241\pi$
$541$	−168.645	+	292.102i	−0.311729	+	0.539930i	0.667008	+	0.745051i	$0.267575\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			39.9919i			0.0733796i
$546$		0			0
$547$	381.134			0.696772			0.348386	−	0.937351i	$-0.386730\pi$
$547$	381.134			0.696772			0.348386	+	0.937351i	$0.386730\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	292.882	+	169.095i	0.531546	+	0.306888i
$552$		0			0
$553$	36.1140	+	51.9464i	0.0653057	+	0.0939356i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	458.708	−	264.835i	0.823533	−	0.475467i	−0.0281002	−	0.999605i	$-0.508946\pi$
$557$	458.708	−	264.835i	0.823533	−	0.475467i	0.851633	+	0.524138i	$0.175612\pi$
$558$		0			0
$559$	−635.468			−1.13679
$560$		0			0
$561$		0			0
$562$		0			0
$563$	58.7210	−	33.9026i	0.104300	−	0.0602177i	−0.446943	−	0.894563i	$-0.647487\pi$
$563$	58.7210	−	33.9026i	0.104300	−	0.0602177i	0.551243	+	0.834345i	$0.314154\pi$
$564$		0			0
$565$	179.335	−	310.618i	0.317407	−	0.549766i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	235.234	+	135.812i	0.413416	+	0.238686i	0.692256	−	0.721652i	$-0.256617\pi$
$569$	235.234	+	135.812i	0.413416	+	0.238686i	−0.278841	+	0.960337i	$0.589950\pi$
$570$		0			0
$571$	−259.979	−	450.297i	−0.455305	−	0.788611i	0.543401	−	0.839473i	$-0.317136\pi$
$571$	−259.979	−	450.297i	−0.455305	−	0.788611i	−0.998706	+	0.0508620i	$0.983803\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			518.347i			0.901473i
$576$		0			0
$577$	−122.138	−	211.549i	−0.211678	−	0.366637i	0.740562	−	0.671988i	$-0.234559\pi$
$577$	−122.138	−	211.549i	−0.211678	−	0.366637i	−0.952240	+	0.305351i	$0.901226\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	252.915	+	118.971i	0.435310	+	0.204770i
$582$		0			0
$583$	−626.336	+	1084.85i	−1.07433	+	1.86080i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			444.765i			0.757691i	0.925460	+	0.378845i	$0.123679\pi$
$587$			444.765i			0.757691i	−0.925460	+	0.378845i	$0.876321\pi$
$588$		0			0
$589$	−733.174			−1.24478
$590$		0			0
$591$		0			0
$592$		0			0
$593$	229.701	+	132.618i	0.387354	+	0.223639i	0.681013	−	0.732271i	$-0.261540\pi$
$593$	229.701	+	132.618i	0.387354	+	0.223639i	−0.293659	+	0.955910i	$0.594873\pi$
$594$		0			0
$595$	643.215	−	1367.38i	1.08103	−	2.29811i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−279.461	+	161.347i	−0.466547	+	0.269361i	−0.714793	−	0.699336i	$-0.753479\pi$
$599$	−279.461	+	161.347i	−0.466547	+	0.269361i	0.248246	+	0.968697i	$0.420146\pi$
$600$		0			0
$601$	−269.269			−0.448035			−0.224018	−	0.974585i	$-0.571917\pi$
$601$	−269.269			−0.448035			−0.224018	+	0.974585i	$0.571917\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	484.633	−	279.803i	0.801046	−	0.462484i
$606$		0			0
$607$	−284.822	+	493.327i	−0.469230	+	0.812730i	−0.999381	−	0.0351731i	$-0.988802\pi$
$607$	−284.822	+	493.327i	−0.469230	+	0.812730i	0.530151	+	0.847903i	$0.322135\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	334.179	+	192.938i	0.546938	+	0.315775i
$612$		0			0
$613$	263.939	+	457.156i	0.430570	+	0.745769i	0.996922	−	0.0783940i	$-0.0249792\pi$
$613$	263.939	+	457.156i	0.430570	+	0.745769i	−0.566352	+	0.824163i	$0.691646\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	391.961i		−	0.635270i	−0.948213	−	0.317635i	$-0.897111\pi$
$617$		−	391.961i		−	0.635270i	0.948213	−	0.317635i	$-0.102889\pi$
$618$		0			0
$619$	329.659	+	570.986i	0.532567	+	0.922432i	0.999277	+	0.0380221i	$0.0121057\pi$
$619$	329.659	+	570.986i	0.532567	+	0.922432i	−0.466710	+	0.884410i	$0.654561\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	176.107	−	122.433i	0.282676	−	0.196521i
$624$		0			0
$625$	−334.279	+	578.988i	−0.534846	+	0.926380i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1098.02i			1.74565i
$630$		0			0
$631$	924.173			1.46462			0.732308	−	0.680973i	$-0.238443\pi$
$631$	924.173			1.46462			0.732308	+	0.680973i	$0.238443\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	42.9273	+	24.7841i	0.0676021	+	0.0390301i
$636$		0			0
$637$	−349.273	−	421.968i	−0.548309	−	0.662430i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−301.965	+	174.340i	−0.471084	+	0.271981i	−0.716694	−	0.697388i	$-0.754345\pi$
$641$	−301.965	+	174.340i	−0.471084	+	0.271981i	0.245609	+	0.969369i	$0.421012\pi$
$642$		0			0
$643$	−639.644			−0.994781			−0.497390	−	0.867527i	$-0.665708\pi$
$643$	−639.644			−0.994781			−0.497390	+	0.867527i	$0.665708\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−209.912	+	121.193i	−0.324439	+	0.187315i	−0.653369	−	0.757039i	$-0.726645\pi$
$647$	−209.912	+	121.193i	−0.324439	+	0.187315i	0.328931	+	0.944354i	$0.393312\pi$
$648$		0			0
$649$	24.1560	−	41.8394i	0.0372203	−	0.0644674i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	470.794	+	271.813i	0.720970	+	0.416252i	0.815110	−	0.579307i	$-0.196677\pi$
$653$	470.794	+	271.813i	0.720970	+	0.416252i	−0.0941394	+	0.995559i	$0.530010\pi$
$654$		0			0
$655$	558.173	+	966.784i	0.852173	+	1.47601i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			31.2961i			0.0474902i	0.999718	+	0.0237451i	$0.00755902\pi$
$659$			31.2961i			0.0474902i	−0.999718	+	0.0237451i	$0.992441\pi$
$660$		0			0
$661$	135.736	+	235.101i	0.205349	+	0.355675i	0.950244	−	0.311507i	$-0.100834\pi$
$661$	135.736	+	235.101i	0.205349	+	0.355675i	−0.744895	+	0.667182i	$0.767500\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−118.880	−	1413.43i	−0.178768	−	2.12547i
$666$		0			0
$667$	−74.3515	+	128.781i	−0.111472	+	0.193074i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			853.518i			1.27201i
$672$		0			0
$673$	−216.682			−0.321964			−0.160982	−	0.986957i	$-0.551466\pi$
$673$	−216.682			−0.321964			−0.160982	+	0.986957i	$0.551466\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1068.34	−	616.805i	−1.57805	−	0.911085i	−0.995132	−	0.0985510i	$-0.968579\pi$
$677$	−1068.34	−	616.805i	−1.57805	−	0.911085i	−0.582914	−	0.812534i	$-0.698087\pi$
$678$		0			0
$679$	−410.197	+	872.017i	−0.604119	+	1.28427i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	401.841	−	232.003i	0.588347	−	0.339682i	−0.176097	−	0.984373i	$-0.556347\pi$
$683$	401.841	−	232.003i	0.588347	−	0.339682i	0.764443	+	0.644691i	$0.223014\pi$
$684$		0			0
$685$	−1597.49			−2.33211
$686$		0			0
$687$		0			0
$688$		0			0
$689$	890.731	−	514.264i	1.29279	−	0.746392i
$690$		0			0
$691$	229.489	−	397.487i	0.332112	−	0.575235i	−0.650814	−	0.759237i	$-0.725572\pi$
$691$	229.489	−	397.487i	0.332112	−	0.575235i	0.982926	+	0.184002i	$0.0589054\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−330.732	−	190.948i	−0.475873	−	0.274745i
$696$		0			0
$697$	110.830	+	191.963i	0.159010	+	0.275413i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	530.208i		−	0.756359i	−0.925732	−	0.378180i	$-0.876550\pi$
$701$		−	530.208i		−	0.756359i	0.925732	−	0.378180i	$-0.123450\pi$
$702$		0			0
$703$	515.335	+	892.586i	0.733051	+	1.26968i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−73.9898	−	879.706i	−0.104653	−	1.24428i
$708$		0			0
$709$	492.976	−	853.860i	0.695312	−	1.20432i	−0.274764	−	0.961512i	$-0.588600\pi$
$709$	492.976	−	853.860i	0.695312	−	1.20432i	0.970075	−	0.242804i	$-0.0780671\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	322.378i		−	0.452143i
$714$		0			0
$715$	−1323.16			−1.85058
$716$		0			0
$717$		0			0
$718$		0			0
$719$	335.054	+	193.443i	0.466000	+	0.269045i	0.714564	−	0.699570i	$-0.246625\pi$
$719$	335.054	+	193.443i	0.466000	+	0.269045i	−0.248564	+	0.968615i	$0.579959\pi$
$720$		0			0
$721$	−750.369	−	1079.33i	−1.04073	−	1.49699i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−635.513	+	366.914i	−0.876570	+	0.506088i
$726$		0			0
$727$	271.922			0.374033			0.187017	−	0.982357i	$-0.440118\pi$
$727$	271.922			0.374033			0.187017	+	0.982357i	$0.440118\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1222.45	+	705.779i	−1.67229	+	0.965498i
$732$		0			0
$733$	376.261	−	651.703i	0.513317	−	0.889091i	−0.486564	−	0.873645i	$-0.661750\pi$
$733$	376.261	−	651.703i	0.513317	−	0.889091i	0.999881	−	0.0154457i	$-0.00491671\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1390.91	−	803.044i	−1.88726	−	1.08961i
$738$		0			0
$739$	638.228	+	1105.44i	0.863638	+	1.49586i	0.868394	+	0.495875i	$0.165153\pi$
$739$	638.228	+	1105.44i	0.863638	+	1.49586i	−0.00475611	+	0.999989i	$0.501514\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			135.770i			0.182733i	0.995817	+	0.0913664i	$0.0291234\pi$
$743$			135.770i			0.182733i	−0.995817	+	0.0913664i	$0.970877\pi$
$744$		0			0
$745$	−762.160	−	1320.10i	−1.02303	−	1.77195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−599.751	+	416.957i	−0.800735	+	0.556685i
$750$		0			0
$751$	−410.744	+	711.430i	−0.546930	+	0.947310i	0.451553	+	0.892244i	$0.350870\pi$
$751$	−410.744	+	711.430i	−0.546930	+	0.947310i	−0.998483	+	0.0550655i	$0.982463\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	151.204i		−	0.200270i
$756$		0			0
$757$	1299.84			1.71710			0.858550	−	0.512730i	$-0.171366\pi$
$757$	1299.84			1.71710			0.858550	+	0.512730i	$0.171366\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	369.670	+	213.429i	0.485769	+	0.280459i	0.722817	−	0.691039i	$-0.242847\pi$
$761$	369.670	+	213.429i	0.485769	+	0.280459i	−0.237049	+	0.971498i	$0.576180\pi$
$762$		0			0
$763$	32.0882	−	2.69886i	0.0420554	−	0.00353717i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−34.3529	+	19.8337i	−0.0447887	+	0.0258588i
$768$		0			0
$769$	−1130.15			−1.46963			−0.734817	−	0.678265i	$-0.762732\pi$
$769$	−1130.15			−1.46963			−0.734817	+	0.678265i	$0.762732\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	642.713	−	371.070i	0.831452	−	0.480039i	−0.0228973	−	0.999738i	$-0.507289\pi$
$773$	642.713	−	371.070i	0.831452	−	0.480039i	0.854350	+	0.519699i	$0.173956\pi$
$774$		0			0
$775$	795.443	−	1377.75i	1.02638	−	1.77774i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	180.189	+	104.032i	0.231308	+	0.133546i
$780$		0			0
$781$	−385.330	−	667.412i	−0.493381	−	0.854561i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1268.34i			1.61572i
$786$		0			0
$787$	365.724	+	633.453i	0.464707	+	0.804896i	0.999188	−	0.0402843i	$-0.0128264\pi$
$787$	365.724	+	633.453i	0.464707	+	0.804896i	−0.534481	+	0.845180i	$0.679493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−261.333	−	122.931i	−0.330383	−	0.155412i
$792$		0			0
$793$	350.398	−	606.906i	0.441863	−	0.765330i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	337.677i		−	0.423685i	−0.977304	−	0.211842i	$-0.932054\pi$
$797$		−	337.677i		−	0.423685i	0.977304	−	0.211842i	$-0.0679464\pi$
$798$		0			0
$799$	857.143			1.07277
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−771.288	−	445.303i	−0.960508	−	0.554550i
$804$		0			0
$805$	621.489	−	52.2719i	0.772036	−	0.0649341i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	419.756	−	242.346i	0.518858	−	0.299563i	−0.217609	−	0.976036i	$-0.569826\pi$
$809$	419.756	−	242.346i	0.518858	−	0.299563i	0.736467	+	0.676473i	$0.236493\pi$
$810$		0			0
$811$	−312.764			−0.385652			−0.192826	−	0.981233i	$-0.561765\pi$
$811$	−312.764			−0.385652			−0.192826	+	0.981233i	$0.561765\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2023.35	+	1168.18i	−2.48263	+	1.43335i
$816$		0			0
$817$	−662.491	+	1147.47i	−0.810882	+	1.40449i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1002.76	−	578.942i	−1.22138	−	0.705167i	−0.256171	−	0.966631i	$-0.582461\pi$
$821$	−1002.76	−	578.942i	−1.22138	−	0.705167i	−0.965213	+	0.261465i	$0.915794\pi$
$822$		0			0
$823$	−181.483	−	314.337i	−0.220514	−	0.381941i	0.734450	−	0.678662i	$-0.237440\pi$
$823$	−181.483	−	314.337i	−0.220514	−	0.381941i	−0.954964	+	0.296722i	$0.904107\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1185.32i		−	1.43327i	−0.697448	−	0.716635i	$-0.745681\pi$
$827$		−	1185.32i		−	1.43327i	0.697448	−	0.716635i	$-0.254319\pi$
$828$		0			0
$829$	458.115	+	793.478i	0.552611	+	0.957151i	0.998085	+	0.0618557i	$0.0197019\pi$
$829$	458.115	+	793.478i	0.552611	+	0.957151i	−0.445474	+	0.895295i	$0.646965\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1140.55	−	423.818i	−1.36921	−	0.508785i
$834$		0			0
$835$	790.167	−	1368.61i	0.946308	−	1.63905i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			854.517i			1.01849i	0.860620	+	0.509247i	$0.170076\pi$
$839$			854.517i			1.01849i	−0.860620	+	0.509247i	$0.829924\pi$
$840$		0			0
$841$	630.480			0.749679
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−331.506	−	191.395i	−0.392315	−	0.226503i
$846$		0			0
$847$	−257.211	−	369.972i	−0.303673	−	0.436803i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−392.472	+	226.594i	−0.461189	+	0.266267i
$852$		0			0
$853$	1245.78			1.46047			0.730233	−	0.683198i	$-0.239411\pi$
$853$	1245.78			1.46047			0.730233	+	0.683198i	$0.239411\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	426.637	−	246.319i	0.497826	−	0.287420i	−0.229989	−	0.973193i	$-0.573869\pi$
$857$	426.637	−	246.319i	0.497826	−	0.287420i	0.727815	+	0.685773i	$0.240536\pi$
$858$		0			0
$859$	617.200	−	1069.02i	0.718510	−	1.24450i	−0.243081	−	0.970006i	$-0.578158\pi$
$859$	617.200	−	1069.02i	0.718510	−	1.24450i	0.961590	−	0.274489i	$-0.0885087\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	913.600	+	527.467i	1.05863	+	0.611202i	0.925054	−	0.379836i	$-0.124020\pi$
$863$	913.600	+	527.467i	1.05863	+	0.611202i	0.133579	+	0.991038i	$0.457353\pi$
$864$		0			0
$865$	1289.34	+	2233.19i	1.49056	+	2.58173i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	123.054i		−	0.141604i
$870$		0			0
$871$	659.353	+	1142.03i	0.757007	+	1.31117i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1408.39	+	662.506i	1.60959	+	0.757150i
$876$		0			0
$877$	−461.795	+	799.852i	−0.526562	+	0.912032i	0.472959	+	0.881084i	$0.343186\pi$
$877$	−461.795	+	799.852i	−0.526562	+	0.912032i	−0.999521	+	0.0309473i	$0.990148\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	128.732i		−	0.146121i	−0.997328	−	0.0730603i	$-0.976723\pi$
$881$		−	128.732i		−	0.146121i	0.997328	−	0.0730603i	$-0.0232766\pi$
$882$		0			0
$883$	153.879			0.174268			0.0871340	−	0.996197i	$-0.472229\pi$
$883$	153.879			0.174268			0.0871340	+	0.996197i	$0.472229\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	345.601	+	199.533i	0.389629	+	0.224953i	0.681999	−	0.731353i	$-0.261111\pi$
$887$	345.601	+	199.533i	0.389629	+	0.224953i	−0.292370	+	0.956305i	$0.594444\pi$
$888$		0			0
$889$	16.9891	−	36.1162i	0.0191103	−	0.0406256i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	696.779	−	402.286i	0.780268	−	0.450488i
$894$		0			0
$895$	−378.273			−0.422652
$896$		0			0
$897$		0			0
$898$		0			0
$899$	395.247	−	228.196i	0.439652	−	0.253833i
$900$		0			0
$901$	1142.33	−	1978.57i	1.26785	−	2.19597i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2205.75	−	1273.49i	−2.43729	−	1.40717i
$906$		0			0
$907$	20.1149	+	34.8400i	0.0221774	+	0.0384123i	0.876901	−	0.480671i	$-0.159607\pi$
$907$	20.1149	+	34.8400i	0.0221774	+	0.0384123i	−0.854724	+	0.519083i	$0.826273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1702.97i			1.86934i	0.355511	+	0.934672i	$0.384307\pi$
$911$			1702.97i			1.86934i	−0.355511	+	0.934672i	$0.615693\pi$
$912$		0			0
$913$	−271.816	−	470.798i	−0.297717	−	0.515661i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	738.050	−	513.105i	0.804853	−	0.559547i
$918$		0			0
$919$	−145.450	+	251.927i	−0.158270	+	0.274131i	−0.934245	−	0.356632i	$-0.883925\pi$
$919$	−145.450	+	251.927i	−0.158270	+	0.274131i	0.775975	+	0.630764i	$0.217258\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			632.764i			0.685552i
$924$		0			0
$925$	−2236.41			−2.41774
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1490.97	−	860.811i	−1.60492	−	0.926600i	−0.990483	−	0.137632i	$-0.956051\pi$
$929$	−1490.97	−	860.811i	−1.60492	−	0.926600i	−0.614434	−	0.788968i	$-0.710616\pi$
$930$		0			0
$931$	−1126.07	+	190.772i	−1.20953	+	0.204911i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2545.36	+	1469.56i	−2.72231	+	1.57172i
$936$		0			0
$937$	295.592			0.315466			0.157733	−	0.987482i	$-0.449581\pi$
$937$	295.592			0.315466			0.157733	+	0.987482i	$0.449581\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−637.151	+	367.860i	−0.677100	+	0.390924i	−0.798762	−	0.601648i	$-0.794511\pi$
$941$	−637.151	+	367.860i	−0.677100	+	0.390924i	0.121661	+	0.992572i	$0.461178\pi$
$942$		0			0
$943$	−45.7431	+	79.2294i	−0.0485081	+	0.0840185i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−197.842	−	114.224i	−0.208915	−	0.120617i	0.391892	−	0.920011i	$-0.371821\pi$
$947$	−197.842	−	114.224i	−0.208915	−	0.120617i	−0.600807	+	0.799394i	$0.705154\pi$
$948$		0			0
$949$	365.624	+	633.279i	0.385273	+	0.667312i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1260.44i			1.32260i	0.750123	+	0.661299i	$0.229994\pi$
$953$			1260.44i			1.32260i	−0.750123	+	0.661299i	$0.770006\pi$
$954$		0			0
$955$	−1209.92	−	2095.65i	−1.26693	−	2.19439i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	107.807	+	1281.78i	0.112416	+	1.33658i
$960$		0			0
$961$	−14.2132	+	24.6180i	−0.0147900	+	0.0256170i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1793.39i		−	1.85844i
$966$		0			0
$967$	−511.535			−0.528992			−0.264496	−	0.964387i	$-0.585206\pi$
$967$	−511.535			−0.528992			−0.264496	+	0.964387i	$0.585206\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−938.990	−	542.126i	−0.967034	−	0.558317i	−0.0687030	−	0.997637i	$-0.521886\pi$
$971$	−938.990	−	542.126i	−0.967034	−	0.558317i	−0.898331	+	0.439320i	$0.855219\pi$
$972$		0			0
$973$	−130.891	+	278.255i	−0.134524	+	0.285977i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−153.444	+	88.5907i	−0.157056	+	0.0906763i	−0.576468	−	0.817120i	$-0.695570\pi$
$977$	−153.444	+	88.5907i	−0.157056	+	0.0906763i	0.419412	+	0.907796i	$0.362236\pi$
$978$		0			0
$979$	−417.175			−0.426124
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1293.47	+	746.784i	−1.31584	+	0.759699i	−0.983056	−	0.183305i	$-0.941320\pi$
$983$	−1293.47	+	746.784i	−1.31584	+	0.759699i	−0.332781	+	0.943004i	$0.607987\pi$
$984$		0			0
$985$	−1506.11	+	2608.66i	−1.52905	+	2.64838i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−504.543	−	291.298i	−0.510155	−	0.294538i
$990$		0			0
$991$	−182.943	−	316.866i	−0.184604	−	0.319744i	0.758839	−	0.651278i	$-0.225767\pi$
$991$	−182.943	−	316.866i	−0.184604	−	0.319744i	−0.943443	+	0.331535i	$0.892434\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	3058.40i		−	3.07376i
$996$		0			0
$997$	−271.714	−	470.622i	−0.272531	−	0.472038i	0.696978	−	0.717092i	$-0.254527\pi$
$997$	−271.714	−	470.622i	−0.272531	−	0.472038i	−0.969509	+	0.245054i	$0.921194\pi$
$998$		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.485.8	yes	16
3.2	odd	2	inner	756.3.bk.g.485.1	yes	16
7.4	even	3	inner	756.3.bk.g.53.1	✓	16
21.11	odd	6	inner	756.3.bk.g.53.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.g.53.1	✓	16	7.4	even	3	inner
756.3.bk.g.53.8	yes	16	21.11	odd	6	inner
756.3.bk.g.485.1	yes	16	3.2	odd	2	inner
756.3.bk.g.485.8	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)

\(f(q)\)	\(=\)	\(q+(7.52876 + 4.34673i) q^{5} +(2.97960 - 6.33419i) q^{7} +(-11.7910 + 6.80755i) q^{11} +11.1789 q^{13} +(21.5048 - 12.4158i) q^{17} +(11.6543 - 20.1858i) q^{19} +(8.87573 + 5.12441i) q^{23} +(25.2882 + 43.8004i) q^{25} +14.5093i q^{29} +(-15.7276 - 27.2410i) q^{31} +(49.9658 - 34.7371i) q^{35} +(-22.1093 + 38.2944i) q^{37} +8.92653i q^{41} -56.8453 q^{43} +(29.8937 + 17.2591i) q^{47} +(-31.2439 - 37.7468i) q^{49} +(79.6796 - 46.0030i) q^{53} -118.362 q^{55} +(-3.07301 + 1.77420i) q^{59} +(31.3445 - 54.2903i) q^{61} +(84.1633 + 48.5917i) q^{65} +(58.9819 + 102.160i) q^{67} +56.6034i q^{71} +(32.7066 + 56.6495i) q^{73} +(7.98772 + 94.9704i) q^{77} +(-4.51904 + 7.82720i) q^{79} +39.9286i q^{83} +215.873 q^{85} +(26.5356 + 15.3203i) q^{89} +(33.3087 - 70.8093i) q^{91} +(175.485 - 101.316i) q^{95} -137.668 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{7} + 88 q^{13} + 14 q^{19} + 36 q^{25} - 68 q^{31} - 76 q^{37} - 292 q^{43} - 20 q^{49} - 272 q^{55} - 110 q^{61} - 72 q^{67} + 60 q^{73} + 154 q^{79} + 700 q^{85} - 74 q^{91} + 264 q^{97}+O(q^{100}) \) 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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