LMFDB - Embedded newform 756.3.p.a.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(397,756)] 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("756.397"); 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, 4, 5])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			577.1
Character	$\chi$	$=$	756.577
Dual form			756.3.p.a.397.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.02073i q^{5} +(-4.77619 - 5.11742i) q^{7} +10.4878 q^{11} +(-20.0242 - 11.5610i) q^{13} +(9.26855 + 5.35120i) q^{17} +(18.7068 - 10.8004i) q^{19} +0.0104756 q^{23} -56.3736 q^{25} +(-8.15959 - 14.1328i) q^{29} +(-24.6144 + 14.2111i) q^{31} +(-46.1629 + 43.0847i) q^{35} +(24.8657 + 43.0686i) q^{37} +(12.6050 + 7.27749i) q^{41} +(-6.83701 - 11.8421i) q^{43} +(-11.9829 - 6.91832i) q^{47} +(-3.37600 + 48.8836i) q^{49} +(-39.1008 + 67.7246i) q^{53} -94.6080i q^{55} +(23.8831 - 13.7889i) q^{59} +(-1.94313 - 1.12187i) q^{61} +(-104.288 + 180.633i) q^{65} +(-27.2436 - 47.1873i) q^{67} -103.268 q^{71} +(-35.1133 - 20.2727i) q^{73} +(-50.0919 - 53.6707i) q^{77} +(10.4854 - 18.1613i) q^{79} +(26.5546 - 15.3313i) q^{83} +(48.2717 - 83.6091i) q^{85} +(136.380 - 78.7389i) q^{89} +(36.4770 + 157.690i) q^{91} +(-97.4271 - 168.749i) q^{95} +(-79.7273 + 46.0306i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) $ 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$		−	9.02073i		−	1.80415i	−0.431583	−	0.902073i	$-0.642045\pi$
$5$		−	9.02073i		−	1.80415i	0.431583	−	0.902073i	$-0.357955\pi$
$6$		0			0
$7$	−4.77619	−	5.11742i	−0.682313	−	0.731060i
$7$	−4.77619	−	5.11742i	−0.682313	−	0.731060i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.4878			0.953440			0.476720	−	0.879055i	$-0.341826\pi$
$11$	10.4878			0.953440			0.476720	+	0.879055i	$0.341826\pi$
$12$		0			0
$13$	−20.0242	−	11.5610i	−1.54032	−	0.889306i	−0.998818	−	0.0486077i	$-0.984522\pi$
$13$	−20.0242	−	11.5610i	−1.54032	−	0.889306i	−0.541504	−	0.840698i	$-0.682145\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	9.26855	+	5.35120i	0.545209	+	0.314776i	0.747187	−	0.664614i	$-0.231404\pi$
$17$	9.26855	+	5.35120i	0.545209	+	0.314776i	−0.201979	+	0.979390i	$0.564737\pi$
$18$		0			0
$19$	18.7068	−	10.8004i	0.984566	−	0.568440i	0.0809206	−	0.996721i	$-0.474214\pi$
$19$	18.7068	−	10.8004i	0.984566	−	0.568440i	0.903646	+	0.428281i	$0.140881\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.0104756			0.000455462			0.000227731	−	1.00000i	$-0.499928\pi$
$23$	0.0104756			0.000455462			0.000227731		1.00000i	$0.499928\pi$
$24$		0			0
$25$	−56.3736			−2.25494
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.15959	−	14.1328i	−0.281365	−	0.487339i	0.690356	−	0.723470i	$-0.257454\pi$
$29$	−8.15959	−	14.1328i	−0.281365	−	0.487339i	−0.971721	+	0.236131i	$0.924121\pi$
$30$		0			0
$31$	−24.6144	+	14.2111i	−0.794014	+	0.458424i	−0.841374	−	0.540454i	$-0.818253\pi$
$31$	−24.6144	+	14.2111i	−0.794014	+	0.458424i	0.0473600	+	0.998878i	$0.484919\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−46.1629	+	43.0847i	−1.31894	+	1.23099i
$36$		0			0
$37$	24.8657	+	43.0686i	0.672046	+	1.16402i	0.977323	+	0.211754i	$0.0679176\pi$
$37$	24.8657	+	43.0686i	0.672046	+	1.16402i	−0.305277	+	0.952264i	$0.598749\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.6050	+	7.27749i	0.307439	+	0.177500i	0.645780	−	0.763524i	$-0.276532\pi$
$41$	12.6050	+	7.27749i	0.307439	+	0.177500i	−0.338341	+	0.941024i	$0.609866\pi$
$42$		0			0
$43$	−6.83701	−	11.8421i	−0.159000	−	0.275397i	0.775508	−	0.631338i	$-0.217494\pi$
$43$	−6.83701	−	11.8421i	−0.159000	−	0.275397i	−0.934508	+	0.355941i	$0.884160\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.9829	−	6.91832i	−0.254955	−	0.147198i	0.367076	−	0.930191i	$-0.380359\pi$
$47$	−11.9829	−	6.91832i	−0.254955	−	0.147198i	−0.622031	+	0.782993i	$0.713692\pi$
$48$		0			0
$49$	−3.37600	+	48.8836i	−0.0688980	+	0.997624i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−39.1008	+	67.7246i	−0.737752	+	1.27782i	0.215754	+	0.976448i	$0.430779\pi$
$53$	−39.1008	+	67.7246i	−0.737752	+	1.27782i	−0.953505	+	0.301376i	$0.902554\pi$
$54$		0			0
$55$		−	94.6080i		−	1.72015i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	23.8831	−	13.7889i	0.404799	−	0.233711i	−0.283754	−	0.958897i	$-0.591580\pi$
$59$	23.8831	−	13.7889i	0.404799	−	0.233711i	0.688553	+	0.725186i	$0.258246\pi$
$60$		0			0
$61$	−1.94313	−	1.12187i	−0.0318546	−	0.0183912i	0.483988	−	0.875075i	$-0.339188\pi$
$61$	−1.94313	−	1.12187i	−0.0318546	−	0.0183912i	−0.515843	+	0.856683i	$0.672521\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−104.288	+	180.633i	−1.60444	+	2.77897i
$66$		0			0
$67$	−27.2436	−	47.1873i	−0.406621	−	0.704288i	0.587888	−	0.808942i	$-0.299960\pi$
$67$	−27.2436	−	47.1873i	−0.406621	−	0.704288i	−0.994509	+	0.104655i	$0.966626\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−103.268			−1.45448			−0.727239	−	0.686384i	$-0.759197\pi$
$71$	−103.268			−1.45448			−0.727239	+	0.686384i	$0.759197\pi$
$72$		0			0
$73$	−35.1133	−	20.2727i	−0.481005	−	0.277708i	0.239830	−	0.970815i	$-0.422908\pi$
$73$	−35.1133	−	20.2727i	−0.481005	−	0.277708i	−0.720835	+	0.693107i	$0.756241\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−50.0919	−	53.6707i	−0.650545	−	0.697022i
$78$		0			0
$79$	10.4854	−	18.1613i	0.132727	−	0.229890i	−0.792000	−	0.610521i	$-0.790960\pi$
$79$	10.4854	−	18.1613i	0.132727	−	0.229890i	0.924727	+	0.380632i	$0.124293\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	26.5546	−	15.3313i	0.319934	−	0.184714i	−0.331429	−	0.943480i	$-0.607531\pi$
$83$	26.5546	−	15.3313i	0.319934	−	0.184714i	0.651363	+	0.758766i	$0.274197\pi$
$84$		0			0
$85$	48.2717	−	83.6091i	0.567903	−	0.983636i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	136.380	−	78.7389i	1.53236	−	0.884707i	0.533105	−	0.846049i	$-0.321025\pi$
$89$	136.380	−	78.7389i	1.53236	−	0.884707i	0.999253	−	0.0386574i	$-0.0123081\pi$
$90$		0			0
$91$	36.4770	+	157.690i	0.400846	+	1.73285i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−97.4271	−	168.749i	−1.02555	−	1.77630i
$96$		0			0
$97$	−79.7273	+	46.0306i	−0.821931	+	0.474542i	−0.851082	−	0.525033i	$-0.824053\pi$
$97$	−79.7273	+	46.0306i	−0.821931	+	0.474542i	0.0291510	+	0.999575i	$0.490720\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	140.557i		−	1.39165i	−0.718210	−	0.695827i	$-0.755038\pi$
$101$		−	140.557i		−	1.39165i	0.718210	−	0.695827i	$-0.244962\pi$
$102$		0			0
$103$		−	110.411i		−	1.07195i	−0.844234	−	0.535976i	$-0.819944\pi$
$103$		−	110.411i		−	1.07195i	0.844234	−	0.535976i	$-0.180056\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	46.7651	+	80.9996i	0.437057	+	0.757006i	0.997461	−	0.0712141i	$-0.0226873\pi$
$107$	46.7651	+	80.9996i	0.437057	+	0.757006i	−0.560404	+	0.828220i	$0.689354\pi$
$108$		0			0
$109$	64.0675	−	110.968i	0.587775	−	1.01806i	−0.406748	−	0.913540i	$-0.633337\pi$
$109$	64.0675	−	110.968i	0.587775	−	1.01806i	0.994523	−	0.104516i	$-0.0333295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−46.9363	+	81.2961i	−0.415366	+	0.719434i	−0.995467	−	0.0951101i	$-0.969680\pi$
$113$	−46.9363	+	81.2961i	−0.415366	+	0.719434i	0.580101	+	0.814544i	$0.303013\pi$
$114$		0			0
$115$		−	0.0944978i		−	0.000821720i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.8840	−	72.9894i	−0.141882	−	0.613356i
$120$		0			0
$121$	−11.0052			−0.0909520
$122$		0			0
$123$		0			0
$124$		0			0
$125$			283.013i			2.26410i
$126$		0			0
$127$	−181.230			−1.42701			−0.713505	−	0.700651i	$-0.752893\pi$
$127$	−181.230			−1.42701			−0.713505	+	0.700651i	$0.752893\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			47.2091i			0.360375i	0.983632	+	0.180188i	$0.0576705\pi$
$131$			47.2091i			0.360375i	−0.983632	+	0.180188i	$0.942330\pi$
$132$		0			0
$133$	−144.617	−	44.1458i	−1.08735	−	0.331923i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	50.7664			0.370558			0.185279	−	0.982686i	$-0.440681\pi$
$137$	50.7664			0.370558			0.185279	+	0.982686i	$0.440681\pi$
$138$		0			0
$139$	−147.147	−	84.9555i	−1.05861	−	0.611190i	−0.133565	−	0.991040i	$-0.542642\pi$
$139$	−147.147	−	84.9555i	−1.05861	−	0.611190i	−0.925048	+	0.379850i	$0.875976\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−210.011	−	121.250i	−1.46861	−	0.847900i
$144$		0			0
$145$	−127.488	+	73.6055i	−0.879230	+	0.507624i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	105.233			0.706264			0.353132	−	0.935574i	$-0.385117\pi$
$149$	105.233			0.706264			0.353132	+	0.935574i	$0.385117\pi$
$150$		0			0
$151$	213.488			1.41382			0.706912	−	0.707301i	$-0.250087\pi$
$151$	213.488			1.41382			0.706912	+	0.707301i	$0.250087\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	128.195	+	222.040i	0.827064	+	1.43252i
$156$		0			0
$157$	−9.86022	+	5.69280i	−0.0628040	+	0.0362599i	−0.531073	−	0.847326i	$-0.678211\pi$
$157$	−9.86022	+	5.69280i	−0.0628040	+	0.0362599i	0.468269	+	0.883586i	$0.344878\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.0500336	−	0.0536082i	−0.000310768	−	0.000332970i
$162$		0			0
$163$	47.1307	+	81.6328i	0.289145	+	0.500815i	0.973606	−	0.228235i	$-0.0732955\pi$
$163$	47.1307	+	81.6328i	0.289145	+	0.500815i	−0.684461	+	0.729050i	$0.739962\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−82.1245	−	47.4146i	−0.491764	−	0.283920i	0.233542	−	0.972347i	$-0.424968\pi$
$167$	−82.1245	−	47.4146i	−0.491764	−	0.283920i	−0.725306	+	0.688427i	$0.758302\pi$
$168$		0			0
$169$	182.812	+	316.640i	1.08173	+	1.87361i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	261.218	+	150.814i	1.50993	+	0.871758i	0.999933	+	0.0115806i	$0.00368630\pi$
$173$	261.218	+	150.814i	1.50993	+	0.871758i	0.509996	+	0.860177i	$0.329647\pi$
$174$		0			0
$175$	269.251	+	288.487i	1.53858	+	1.64850i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	43.7478	−	75.7733i	0.244401	−	0.423315i	−0.717562	−	0.696495i	$-0.754742\pi$
$179$	43.7478	−	75.7733i	0.244401	−	0.423315i	0.961963	+	0.273180i	$0.0880754\pi$
$180$		0			0
$181$		−	162.562i		−	0.898134i	−0.893498	−	0.449067i	$-0.851756\pi$
$181$		−	162.562i		−	0.898134i	0.893498	−	0.449067i	$-0.148244\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	388.511	−	224.307i	2.10006	−	1.21247i
$186$		0			0
$187$	97.2070	+	56.1225i	0.519824	+	0.300120i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	161.443	−	279.628i	0.845252	−	1.46402i	−0.0401499	−	0.999194i	$-0.512784\pi$
$191$	161.443	−	279.628i	0.845252	−	1.46402i	0.885402	−	0.464826i	$-0.153883\pi$
$192$		0			0
$193$	−3.58006	−	6.20084i	−0.0185495	−	0.0321287i	0.856602	−	0.515978i	$-0.172572\pi$
$193$	−3.58006	−	6.20084i	−0.0185495	−	0.0321287i	−0.875151	+	0.483850i	$0.839238\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−356.459			−1.80944			−0.904718	−	0.426010i	$-0.859919\pi$
$197$	−356.459			−1.80944			−0.904718	+	0.426010i	$0.859919\pi$
$198$		0			0
$199$	−38.3008	−	22.1130i	−0.192466	−	0.111120i	0.400670	−	0.916222i	$-0.368777\pi$
$199$	−38.3008	−	22.1130i	−0.192466	−	0.111120i	−0.593137	+	0.805102i	$0.702111\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−33.3519	+	109.257i	−0.164295	+	0.538212i
$204$		0			0
$205$	65.6483	−	113.706i	0.320236	−	0.554664i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	196.194	−	113.272i	0.938725	−	0.541973i
$210$		0			0
$211$	92.8506	−	160.822i	0.440050	−	0.762189i	−0.557643	−	0.830081i	$-0.688294\pi$
$211$	92.8506	−	160.822i	0.440050	−	0.762189i	0.997693	+	0.0678920i	$0.0216273\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−106.824	+	61.6748i	−0.496856	+	0.286860i
$216$		0			0
$217$	190.288	+	58.0872i	0.876901	+	0.267683i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−123.730	−	214.307i	−0.559865	−	0.969714i
$222$		0			0
$223$	3.51982	−	2.03217i	0.0157840	−	0.00911287i	−0.492087	−	0.870546i	$-0.663766\pi$
$223$	3.51982	−	2.03217i	0.0157840	−	0.00911287i	0.507871	+	0.861433i	$0.330433\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			301.496i			1.32818i	0.747654	+	0.664089i	$0.231180\pi$
$227$			301.496i			1.32818i	−0.747654	+	0.664089i	$0.768820\pi$
$228$		0			0
$229$			215.751i			0.942145i	0.882095	+	0.471072i	$0.156133\pi$
$229$			215.751i			0.942145i	−0.882095	+	0.471072i	$0.843867\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0845	−	24.3951i	−0.0604486	−	0.104700i	0.834217	−	0.551436i	$-0.185920\pi$
$233$	−14.0845	−	24.3951i	−0.0604486	−	0.104700i	−0.894666	+	0.446736i	$0.852586\pi$
$234$		0			0
$235$	−62.4083	+	108.094i	−0.265567	+	0.459976i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	30.4914	−	52.8126i	0.127579	−	0.220973i	−0.795159	−	0.606401i	$-0.792613\pi$
$239$	30.4914	−	52.8126i	0.127579	−	0.220973i	0.922738	+	0.385428i	$0.125946\pi$
$240$		0			0
$241$			3.92969i			0.0163058i	0.999967	+	0.00815288i	$0.00259517\pi$
$241$			3.92969i			0.0163058i	−0.999967	+	0.00815288i	$0.997405\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	440.965	+	30.4540i	1.79986	+	0.124302i
$246$		0			0
$247$	−499.450			−2.02207
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	370.568i		−	1.47637i	−0.674601	−	0.738183i	$-0.735684\pi$
$251$		−	370.568i		−	1.47637i	0.674601	−	0.738183i	$-0.264316\pi$
$252$		0			0
$253$	0.109867			0.000434256
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	368.771i		−	1.43491i	−0.696607	−	0.717453i	$-0.745308\pi$
$257$		−	368.771i		−	1.43491i	0.696607	−	0.717453i	$-0.254692\pi$
$258$		0			0
$259$	101.637	−	332.952i	0.392421	−	1.28553i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−39.1616			−0.148903			−0.0744517	−	0.997225i	$-0.523721\pi$
$263$	−39.1616			−0.148903			−0.0744517	+	0.997225i	$0.523721\pi$
$264$		0			0
$265$	610.926	+	352.718i	2.30538	+	1.33101i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	407.431	+	235.230i	1.51461	+	0.874461i	0.999853	+	0.0171262i	$0.00545171\pi$
$269$	407.431	+	235.230i	1.51461	+	0.874461i	0.514758	+	0.857335i	$0.327882\pi$
$270$		0			0
$271$	−238.810	+	137.877i	−0.881218	+	0.508772i	−0.871060	−	0.491177i	$-0.836567\pi$
$271$	−238.810	+	137.877i	−0.881218	+	0.508772i	−0.0101585	+	0.999948i	$0.503234\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−591.237			−2.14995
$276$		0			0
$277$	−122.059			−0.440646			−0.220323	−	0.975427i	$-0.570711\pi$
$277$	−122.059			−0.440646			−0.220323	+	0.975427i	$0.570711\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	121.284	+	210.071i	0.431617	+	0.747583i	0.997013	−	0.0772371i	$-0.0246099\pi$
$281$	121.284	+	210.071i	0.431617	+	0.747583i	−0.565396	+	0.824820i	$0.691277\pi$
$282$		0			0
$283$	423.344	−	244.418i	1.49591	−	0.863667i	0.495925	−	0.868365i	$-0.334829\pi$
$283$	423.344	−	244.418i	1.49591	−	0.863667i	0.999989	+	0.00469851i	$0.00149559\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−22.9618	−	99.2637i	−0.0800063	−	0.345867i
$288$		0			0
$289$	−87.2294	−	151.086i	−0.301832	−	0.522788i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	46.1025	+	26.6173i	0.157346	+	0.0908440i	0.576606	−	0.817023i	$-0.304377\pi$
$293$	46.1025	+	26.6173i	0.157346	+	0.0908440i	−0.419259	+	0.907866i	$0.637710\pi$
$294$		0			0
$295$	−124.386	−	215.443i	−0.421649	−	0.730317i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.209766	−	0.121108i	−0.000701559	−	0.000405045i
$300$		0			0
$301$	−27.9459	+	91.5478i	−0.0928435	+	0.304145i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.1200	+	17.5284i	−0.0331805	+	0.0574703i
$306$		0			0
$307$		−	514.866i		−	1.67709i	−0.544834	−	0.838544i	$-0.683407\pi$
$307$		−	514.866i		−	1.67709i	0.544834	−	0.838544i	$-0.316593\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	198.831	−	114.795i	0.639329	−	0.369117i	−0.145027	−	0.989428i	$-0.546327\pi$
$311$	198.831	−	114.795i	0.639329	−	0.369117i	0.784356	+	0.620311i	$0.212993\pi$
$312$		0			0
$313$	54.3358	+	31.3708i	0.173597	+	0.100226i	0.584281	−	0.811552i	$-0.301377\pi$
$313$	54.3358	+	31.3708i	0.173597	+	0.100226i	−0.410684	+	0.911778i	$0.634710\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	154.536	−	267.665i	0.487497	−	0.844369i	−0.512400	−	0.858747i	$-0.671243\pi$
$317$	154.536	−	267.665i	0.487497	−	0.844369i	0.999897	+	0.0143778i	$0.00457675\pi$
$318$		0			0
$319$	−85.5765	−	148.223i	−0.268265	−	0.464648i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	231.179			0.715725
$324$		0			0
$325$	1128.84	+	651.733i	3.47334	+	2.00533i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	21.8286	+	94.3647i	0.0663482	+	0.286823i
$330$		0			0
$331$	271.081	−	469.526i	0.818976	−	1.41851i	−0.0874620	−	0.996168i	$-0.527876\pi$
$331$	271.081	−	469.526i	0.818976	−	1.41851i	0.906438	−	0.422340i	$-0.138791\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−425.664	+	245.757i	−1.27064	+	0.733603i
$336$		0			0
$337$	−214.898	+	372.213i	−0.637678	+	1.10449i	0.348263	+	0.937397i	$0.386772\pi$
$337$	−214.898	+	372.213i	−0.637678	+	1.10449i	−0.985941	+	0.167094i	$0.946562\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−258.152	+	149.044i	−0.757044	+	0.437080i
$342$		0			0
$343$	266.282	−	216.201i	0.776333	−	0.630323i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−85.6830	−	148.407i	−0.246925	−	0.427687i	0.715746	−	0.698361i	$-0.246087\pi$
$347$	−85.6830	−	148.407i	−0.246925	−	0.427687i	−0.962671	+	0.270674i	$0.912753\pi$
$348$		0			0
$349$	−69.0793	+	39.8829i	−0.197935	+	0.114278i	−0.595692	−	0.803213i	$-0.703122\pi$
$349$	−69.0793	+	39.8829i	−0.197935	+	0.114278i	0.397757	+	0.917491i	$0.369789\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	468.865i		−	1.32823i	−0.747631	−	0.664114i	$-0.768809\pi$
$353$		−	468.865i		−	1.32823i	0.747631	−	0.664114i	$-0.231191\pi$
$354$		0			0
$355$			931.552i			2.62409i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−113.648	−	196.844i	−0.316568	−	0.548313i	0.663201	−	0.748441i	$-0.269197\pi$
$359$	−113.648	−	196.844i	−0.316568	−	0.548313i	−0.979770	+	0.200129i	$0.935864\pi$
$360$		0			0
$361$	52.7952	−	91.4440i	0.146247	−	0.253308i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−182.875	+	316.748i	−0.501026	+	0.867803i
$366$		0			0
$367$		−	59.0917i		−	0.161013i	−0.996754	−	0.0805064i	$-0.974346\pi$
$367$		−	59.0917i		−	0.161013i	0.996754	−	0.0805064i	$-0.0256537\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	533.329	−	123.370i	1.43754	−	0.332535i
$372$		0			0
$373$	−258.346			−0.692617			−0.346308	−	0.938121i	$-0.612565\pi$
$373$	−258.346			−0.692617			−0.346308	+	0.938121i	$0.612565\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			377.331i			1.00088i
$378$		0			0
$379$	−618.172			−1.63106			−0.815530	−	0.578715i	$-0.803554\pi$
$379$	−618.172			−1.63106			−0.815530	+	0.578715i	$0.803554\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			392.164i			1.02393i	0.859007	+	0.511964i	$0.171082\pi$
$383$			392.164i			1.02393i	−0.859007	+	0.511964i	$0.828918\pi$
$384$		0			0
$385$	−484.149	+	451.866i	−1.25753	+	1.17368i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−569.268			−1.46341			−0.731707	−	0.681620i	$-0.761276\pi$
$389$	−569.268			−1.46341			−0.731707	+	0.681620i	$0.761276\pi$
$390$		0			0
$391$	0.0970939	+	0.0560572i	0.000248322	+	0.000143369i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−163.828	−	94.5861i	−0.414754	−	0.239458i
$396$		0			0
$397$	−162.553	+	93.8503i	−0.409454	+	0.236399i	−0.690555	−	0.723280i	$-0.742634\pi$
$397$	−162.553	+	93.8503i	−0.409454	+	0.236399i	0.281101	+	0.959678i	$0.409300\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−304.320			−0.758902			−0.379451	−	0.925212i	$-0.623887\pi$
$401$	−304.320			−0.758902			−0.379451	+	0.925212i	$0.623887\pi$
$402$		0			0
$403$	657.179			1.63072
$404$		0			0
$405$		0			0
$406$		0			0
$407$	260.787	+	451.697i	0.640755	+	1.10982i
$408$		0			0
$409$	20.3869	−	11.7704i	0.0498458	−	0.0287785i	−0.474870	−	0.880056i	$-0.657505\pi$
$409$	20.3869	−	11.7704i	0.0498458	−	0.0287785i	0.524716	+	0.851277i	$0.324172\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−184.634	−	56.3615i	−0.447056	−	0.136469i
$414$		0			0
$415$	−138.299	−	239.542i	−0.333251	−	0.577208i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	420.858	+	242.983i	1.00443	+	0.579911i	0.909557	−	0.415578i	$-0.136421\pi$
$419$	420.858	+	242.983i	1.00443	+	0.579911i	0.0948772	+	0.995489i	$0.469754\pi$
$420$		0			0
$421$	185.968	+	322.106i	0.441730	+	0.765099i	0.997818	−	0.0660248i	$-0.0210316\pi$
$421$	185.968	+	322.106i	0.441730	+	0.765099i	−0.556088	+	0.831123i	$0.687698\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−522.501	−	301.666i	−1.22941	−	0.709803i
$426$		0			0
$427$	3.53969	+	15.3021i	0.00828968	+	0.0358362i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−55.5725	+	96.2544i	−0.128939	+	0.223328i	−0.923266	−	0.384162i	$-0.874490\pi$
$431$	−55.5725	+	96.2544i	−0.128939	+	0.223328i	0.794327	+	0.607490i	$0.207824\pi$
$432$		0			0
$433$			307.470i			0.710091i	0.934849	+	0.355046i	$0.115535\pi$
$433$			307.470i			0.710091i	−0.934849	+	0.355046i	$0.884465\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.195965	−	0.113140i	0.000448433	−	0.000258903i
$438$		0			0
$439$	−533.664	−	308.111i	−1.21564	−	0.701848i	−0.251655	−	0.967817i	$-0.580975\pi$
$439$	−533.664	−	308.111i	−1.21564	−	0.701848i	−0.963981	+	0.265969i	$0.914308\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−99.8992	+	173.031i	−0.225506	+	0.390588i	−0.956471	−	0.291827i	$-0.905737\pi$
$443$	−99.8992	+	173.031i	−0.225506	+	0.390588i	0.730965	+	0.682415i	$0.239070\pi$
$444$		0			0
$445$	−710.282	−	1230.25i	−1.59614	−	2.76460i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	351.847			0.783623			0.391812	−	0.920045i	$-0.371848\pi$
$449$	351.847			0.783623			0.391812	+	0.920045i	$0.371848\pi$
$450$		0			0
$451$	132.199	+	76.3252i	0.293124	+	0.169235i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1422.48	−	329.049i	3.12632	−	0.723185i
$456$		0			0
$457$	−85.9428	+	148.857i	−0.188059	+	0.325727i	−0.944603	−	0.328215i	$-0.893553\pi$
$457$	−85.9428	+	148.857i	−0.188059	+	0.325727i	0.756544	+	0.653942i	$0.226886\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−352.003	+	203.229i	−0.763564	+	0.440844i	−0.830574	−	0.556908i	$-0.811987\pi$
$461$	−352.003	+	203.229i	−0.763564	+	0.440844i	0.0670098	+	0.997752i	$0.478654\pi$
$462$		0			0
$463$	216.398	−	374.812i	0.467382	−	0.809529i	−0.531924	−	0.846792i	$-0.678531\pi$
$463$	216.398	−	374.812i	0.467382	−	0.809529i	0.999305	+	0.0372633i	$0.0118640\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−320.910	+	185.277i	−0.687173	+	0.396739i	−0.802552	−	0.596582i	$-0.796525\pi$
$467$	−320.910	+	185.277i	−0.687173	+	0.396739i	0.115379	+	0.993321i	$0.463192\pi$
$468$		0			0
$469$	−111.357	+	364.792i	−0.237434	+	0.777809i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−71.7055	−	124.198i	−0.151597	−	0.262574i
$474$		0			0
$475$	−1054.57	+	608.855i	−2.22014	+	1.28180i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	300.722i		−	0.627813i	−0.949454	−	0.313906i	$-0.898362\pi$
$479$		−	300.722i		−	0.627813i	0.949454	−	0.313906i	$-0.101638\pi$
$480$		0			0
$481$		−	1149.89i		−	2.39062i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	415.229	+	719.198i	0.856143	+	1.48288i
$486$		0			0
$487$	391.682	−	678.413i	0.804275	−	1.39304i	−0.112505	−	0.993651i	$-0.535887\pi$
$487$	391.682	−	678.413i	0.804275	−	1.39304i	0.916780	−	0.399393i	$-0.130779\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−66.8034	+	115.707i	−0.136056	+	0.235656i	−0.926000	−	0.377523i	$-0.876776\pi$
$491$	−66.8034	+	115.707i	−0.136056	+	0.235656i	0.789944	+	0.613178i	$0.210109\pi$
$492$		0			0
$493$		−	174.654i		−	0.354268i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	493.227	+	528.466i	0.992409	+	1.06331i
$498$		0			0
$499$	−580.230			−1.16279			−0.581393	−	0.813623i	$-0.697492\pi$
$499$	−580.230			−1.16279			−0.581393	+	0.813623i	$0.697492\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	849.632i		−	1.68913i	−0.535453	−	0.844565i	$-0.679859\pi$
$503$		−	849.632i		−	1.68913i	0.535453	−	0.844565i	$-0.320141\pi$
$504$		0			0
$505$	−1267.93			−2.51075
$506$		0			0
$507$		0			0
$508$		0			0
$509$			313.979i			0.616855i	0.951248	+	0.308428i	$0.0998027\pi$
$509$			313.979i			0.616855i	−0.951248	+	0.308428i	$0.900197\pi$
$510$		0			0
$511$	63.9641	+	276.516i	0.125174	+	0.541127i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−995.987			−1.93396
$516$		0			0
$517$	−125.675	−	72.5582i	−0.243084	−	0.140345i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.6375	−	23.4621i	−0.0779991	−	0.0450328i	0.460493	−	0.887663i	$-0.347673\pi$
$521$	−40.6375	−	23.4621i	−0.0779991	−	0.0450328i	−0.538492	+	0.842630i	$0.681006\pi$
$522$		0			0
$523$	142.311	−	82.1630i	0.272104	−	0.157100i	−0.357739	−	0.933822i	$-0.616452\pi$
$523$	142.311	−	82.1630i	0.272104	−	0.157100i	0.629844	+	0.776722i	$0.283119\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−304.187			−0.577204
$528$		0			0
$529$	−529.000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−168.270	−	291.452i	−0.315703	−	0.546814i
$534$		0			0
$535$	730.675	−	421.856i	1.36575	−	0.788515i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−35.4070	+	512.683i	−0.0656901	+	0.951174i
$540$		0			0
$541$	40.6441	+	70.3977i	0.0751278	+	0.130125i	0.901142	−	0.433524i	$-0.142730\pi$
$541$	40.6441	+	70.3977i	0.0751278	+	0.130125i	−0.826014	+	0.563650i	$0.809397\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1001.01	−	577.936i	−1.83672	−	1.06043i
$546$		0			0
$547$	−132.040	−	228.700i	−0.241390	−	0.418099i	0.719721	−	0.694264i	$-0.244270\pi$
$547$	−132.040	−	228.700i	−0.241390	−	0.418099i	−0.961110	+	0.276164i	$0.910937\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−305.279	−	176.253i	−0.554045	−	0.319878i
$552$		0			0
$553$	−143.019	+	33.0834i	−0.258624	+	0.0598253i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	150.745	−	261.098i	0.270637	−	0.468757i	−0.698388	−	0.715719i	$-0.746099\pi$
$557$	150.745	−	261.098i	0.270637	−	0.468757i	0.969025	+	0.246962i	$0.0794323\pi$
$558$		0			0
$559$			316.170i			0.565599i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	138.100	−	79.7321i	0.245293	−	0.141620i	−0.372314	−	0.928107i	$-0.621436\pi$
$563$	138.100	−	79.7321i	0.245293	−	0.141620i	0.617607	+	0.786487i	$0.288102\pi$
$564$		0			0
$565$	733.350	+	423.400i	1.29796	+	0.749380i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	198.694	−	344.149i	0.349199	−	0.604831i	−0.636908	−	0.770940i	$-0.719787\pi$
$569$	198.694	−	344.149i	0.349199	−	0.604831i	0.986107	+	0.166109i	$0.0531203\pi$
$570$		0			0
$571$	−74.5288	−	129.088i	−0.130523	−	0.226073i	0.793355	−	0.608759i	$-0.208332\pi$
$571$	−74.5288	−	129.088i	−0.130523	−	0.226073i	−0.923878	+	0.382686i	$0.874999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.590549			−0.00102704
$576$		0			0
$577$	−326.436	−	188.468i	−0.565746	−	0.326634i	0.189702	−	0.981842i	$-0.439248\pi$
$577$	−326.436	−	188.468i	−0.565746	−	0.326634i	−0.755449	+	0.655208i	$0.772581\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−205.286	−	62.6657i	−0.353333	−	0.107858i
$582$		0			0
$583$	−410.083	+	710.285i	−0.703402	+	1.21833i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−203.516	+	117.500i	−0.346705	+	0.200170i	−0.663233	−	0.748413i	$-0.730816\pi$
$587$	−203.516	+	117.500i	−0.346705	+	0.200170i	0.316528	+	0.948583i	$0.397483\pi$
$588$		0			0
$589$	−306.971	+	531.689i	−0.521173	+	0.902698i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	521.241	−	300.939i	0.878990	−	0.507485i	0.00866477	−	0.999962i	$-0.497242\pi$
$593$	521.241	−	300.939i	0.878990	−	0.507485i	0.870325	+	0.492477i	$0.163909\pi$
$594$		0			0
$595$	−658.418	+	152.306i	−1.10658	+	0.255977i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−170.436	−	295.204i	−0.284534	−	0.492828i	0.687962	−	0.725747i	$-0.258506\pi$
$599$	−170.436	−	295.204i	−0.284534	−	0.492828i	−0.972496	+	0.232919i	$0.925172\pi$
$600$		0			0
$601$	602.938	−	348.106i	1.00322	−	0.579212i	0.0940235	−	0.995570i	$-0.470027\pi$
$601$	602.938	−	348.106i	1.00322	−	0.579212i	0.909201	+	0.416358i	$0.136694\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			99.2749i			0.164091i
$606$		0			0
$607$		−	548.762i		−	0.904057i	−0.892004	−	0.452028i	$-0.850701\pi$
$607$		−	548.762i		−	0.904057i	0.892004	−	0.452028i	$-0.149299\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	159.965	+	277.068i	0.261809	+	0.453466i
$612$		0			0
$613$	156.668	−	271.356i	0.255575	−	0.442669i	−0.709476	−	0.704729i	$-0.751068\pi$
$613$	156.668	−	271.356i	0.255575	−	0.442669i	0.965052	+	0.262060i	$0.0844018\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−218.859	+	379.075i	−0.354715	+	0.614385i	−0.987069	−	0.160295i	$-0.948756\pi$
$617$	−218.859	+	379.075i	−0.354715	+	0.614385i	0.632354	+	0.774680i	$0.282089\pi$
$618$		0			0
$619$			365.173i			0.589941i	0.955506	+	0.294970i	$0.0953098\pi$
$619$			365.173i			0.589941i	−0.955506	+	0.294970i	$0.904690\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1054.32	−	321.841i	−1.69232	−	0.516598i
$624$		0			0
$625$	1143.64			1.82983
$626$		0			0
$627$		0			0
$628$		0			0
$629$			532.245i			0.846177i
$630$		0			0
$631$	776.357			1.23036			0.615180	−	0.788387i	$-0.289083\pi$
$631$	776.357			1.23036			0.615180	+	0.788387i	$0.289083\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1634.83i			2.57453i
$636$		0			0
$637$	632.743	−	939.824i	0.993317	−	1.47539i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.63339			−0.00410825			−0.00205413	−	0.999998i	$-0.500654\pi$
$641$	−2.63339			−0.00410825			−0.00205413	+	0.999998i	$0.500654\pi$
$642$		0			0
$643$	165.599	+	95.6084i	0.257541	+	0.148691i	0.623212	−	0.782053i	$-0.285827\pi$
$643$	165.599	+	95.6084i	0.257541	+	0.148691i	−0.365671	+	0.930744i	$0.619161\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−563.620	−	325.406i	−0.871129	−	0.502946i	−0.00340561	−	0.999994i	$-0.501084\pi$
$647$	−563.620	−	325.406i	−0.871129	−	0.502946i	−0.867723	+	0.497048i	$0.834417\pi$
$648$		0			0
$649$	250.483	−	144.616i	0.385952	−	0.222829i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−410.792			−0.629085			−0.314542	−	0.949243i	$-0.601851\pi$
$653$	−410.792			−0.629085			−0.314542	+	0.949243i	$0.601851\pi$
$654$		0			0
$655$	425.861			0.650169
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−115.724	−	200.439i	−0.175605	−	0.304157i	0.764765	−	0.644309i	$-0.222855\pi$
$659$	−115.724	−	200.439i	−0.175605	−	0.304157i	−0.940371	+	0.340152i	$0.889522\pi$
$660$		0			0
$661$	−484.932	+	279.975i	−0.733633	+	0.423563i	−0.819750	−	0.572722i	$-0.805888\pi$
$661$	−484.932	+	279.975i	−0.733633	+	0.423563i	0.0861166	+	0.996285i	$0.472554\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−398.228	+	1304.55i	−0.598838	+	1.96173i
$666$		0			0
$667$	−0.0854768	−	0.148050i	−0.000128151	−	0.000221964i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20.3792	−	11.7659i	−0.0303714	−	0.0175349i
$672$		0			0
$673$	144.487	+	250.259i	0.214691	+	0.371856i	0.953177	−	0.302413i	$-0.0977922\pi$
$673$	144.487	+	250.259i	0.214691	+	0.371856i	−0.738486	+	0.674269i	$0.764459\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	414.313	+	239.204i	0.611983	+	0.353329i	0.773741	−	0.633502i	$-0.218383\pi$
$677$	414.313	+	239.204i	0.611983	+	0.353329i	−0.161758	+	0.986830i	$0.551716\pi$
$678$		0			0
$679$	616.351	+	188.147i	0.907733	+	0.277095i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	110.124	−	190.741i	0.161236	−	0.279269i	−0.774076	−	0.633092i	$-0.781785\pi$
$683$	110.124	−	190.741i	0.161236	−	0.279269i	0.935312	+	0.353823i	$0.115119\pi$
$684$		0			0
$685$		−	457.950i		−	0.668540i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1565.93	−	904.088i	2.27275	−	1.31217i
$690$		0			0
$691$	−446.326	−	257.686i	−0.645913	−	0.372918i	0.140976	−	0.990013i	$-0.454976\pi$
$691$	−446.326	−	257.686i	−0.645913	−	0.372918i	−0.786889	+	0.617095i	$0.788309\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−766.360	+	1327.37i	−1.10268	+	1.90989i
$696$		0			0
$697$	77.8866	+	134.904i	0.111745	+	0.193549i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	186.358			0.265846			0.132923	−	0.991126i	$-0.457564\pi$
$701$	186.358			0.265846			0.132923	+	0.991126i	$0.457564\pi$
$702$		0			0
$703$	930.313	+	537.117i	1.32335	+	0.764035i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−719.289	+	671.327i	−1.01738	+	0.949543i
$708$		0			0
$709$	315.605	−	546.644i	0.445141	−	0.771007i	−0.552921	−	0.833234i	$-0.686487\pi$
$709$	315.605	−	546.644i	0.445141	−	0.771007i	0.998062	+	0.0622270i	$0.0198203\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.257852	+	0.148871i	−0.000361643	+	0.000208795i
$714$		0			0
$715$	−1093.76	+	1894.45i	−1.52973	+	2.64958i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	972.708	−	561.593i	1.35286	−	0.781075i	0.364213	−	0.931316i	$-0.381338\pi$
$719$	972.708	−	561.593i	1.35286	−	0.781075i	0.988649	+	0.150240i	$0.0480048\pi$
$720$		0			0
$721$	−565.019	+	527.344i	−0.783661	+	0.731406i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	459.985	+	796.718i	0.634462	+	1.09892i
$726$		0			0
$727$	243.180	−	140.400i	0.334497	−	0.193122i	−0.323339	−	0.946283i	$-0.604805\pi$
$727$	243.180	−	140.400i	0.334497	−	0.193122i	0.657836	+	0.753161i	$0.271472\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	146.345i		−	0.200198i
$732$		0			0
$733$			180.872i			0.246756i	0.992360	+	0.123378i	$0.0393727\pi$
$733$			180.872i			0.246756i	−0.992360	+	0.123378i	$0.960627\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−285.726	−	494.893i	−0.387689	−	0.671496i
$738$		0			0
$739$	−322.671	+	558.883i	−0.436633	+	0.756270i	−0.997427	−	0.0716851i	$-0.977162\pi$
$739$	−322.671	+	558.883i	−0.436633	+	0.756270i	0.560795	+	0.827955i	$0.310496\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	207.964	−	360.204i	0.279897	−	0.484797i	−0.691462	−	0.722413i	$-0.743033\pi$
$743$	207.964	−	360.204i	0.279897	−	0.484797i	0.971359	+	0.237617i	$0.0763662\pi$
$744$		0			0
$745$		−	949.281i		−	1.27420i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	191.150	−	626.186i	0.255207	−	0.836030i
$750$		0			0
$751$	113.313			0.150882			0.0754411	−	0.997150i	$-0.475964\pi$
$751$	113.313			0.150882			0.0754411	+	0.997150i	$0.475964\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1925.81i		−	2.55075i
$756$		0			0
$757$	−697.964			−0.922013			−0.461006	−	0.887397i	$-0.652512\pi$
$757$	−697.964			−0.922013			−0.461006	+	0.887397i	$0.652512\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1346.02i			1.76875i	0.466781	+	0.884373i	$0.345414\pi$
$761$			1346.02i			1.76875i	−0.466781	+	0.884373i	$0.654586\pi$
$762$		0			0
$763$	−873.870	+	202.145i	−1.14531	+	0.264934i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−637.654			−0.831361
$768$		0			0
$769$	−289.467	−	167.124i	−0.376420	−	0.217326i	0.299840	−	0.953990i	$-0.403067\pi$
$769$	−289.467	−	167.124i	−0.376420	−	0.217326i	−0.676259	+	0.736664i	$0.736400\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1078.42	−	622.629i	−1.39512	−	0.805470i	−0.401240	−	0.915973i	$-0.631421\pi$
$773$	−1078.42	−	622.629i	−1.39512	−	0.805470i	−0.993876	+	0.110503i	$0.964754\pi$
$774$		0			0
$775$	1387.60	−	801.133i	1.79046	−	1.03372i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	314.398			0.403592
$780$		0			0
$781$	−1083.06			−1.38676
$782$		0			0
$783$		0			0
$784$		0			0
$785$	51.3532	+	88.9464i	0.0654181	+	0.113308i
$786$		0			0
$787$	−279.629	+	161.444i	−0.355310	+	0.205139i	−0.667022	−	0.745038i	$-0.732431\pi$
$787$	−279.629	+	161.444i	−0.355310	+	0.205139i	0.311711	+	0.950177i	$0.399098\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	640.203	−	148.093i	0.809359	−	0.187222i
$792$		0			0
$793$	25.9397	+	44.9289i	0.0327109	+	0.0566569i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	160.864	+	92.8747i	0.201836	+	0.116530i	0.597512	−	0.801860i	$-0.296156\pi$
$797$	160.864	+	92.8747i	0.201836	+	0.116530i	−0.395675	+	0.918390i	$0.629489\pi$
$798$		0			0
$799$	−74.0426	−	128.246i	−0.0926691	−	0.160508i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−368.263	−	212.617i	−0.458609	−	0.264778i
$804$		0			0
$805$	−0.483585	+	0.451340i	−0.000600727	+	0.000560670i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	315.257	−	546.041i	0.389687	−	0.674958i	−0.602720	−	0.797953i	$-0.705916\pi$
$809$	315.257	−	546.041i	0.389687	−	0.674958i	0.992407	+	0.122994i	$0.0392497\pi$
$810$		0			0
$811$			491.571i			0.606129i	0.952970	+	0.303065i	$0.0980098\pi$
$811$			491.571i			0.606129i	−0.952970	+	0.303065i	$0.901990\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	736.387	−	425.153i	0.903543	−	0.521661i
$816$		0			0
$817$	−255.797	−	147.684i	−0.313093	−	0.180764i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	478.233	−	828.323i	0.582500	−	1.00892i	−0.412682	−	0.910875i	$-0.635408\pi$
$821$	478.233	−	828.323i	0.582500	−	1.00892i	0.995182	−	0.0980445i	$-0.0312588\pi$
$822$		0			0
$823$	297.172	+	514.718i	0.361084	+	0.625416i	0.988140	−	0.153558i	$-0.0490733\pi$
$823$	297.172	+	514.718i	0.361084	+	0.625416i	−0.627055	+	0.778975i	$0.715740\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	599.763			0.725228			0.362614	−	0.931940i	$-0.381884\pi$
$827$	599.763			0.725228			0.362614	+	0.931940i	$0.381884\pi$
$828$		0			0
$829$	411.247	+	237.434i	0.496076	+	0.286410i	0.727092	−	0.686540i	$-0.240872\pi$
$829$	411.247	+	237.434i	0.496076	+	0.286410i	−0.231016	+	0.972950i	$0.574205\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−292.876	+	435.014i	−0.351592	+	0.522226i
$834$		0			0
$835$	−427.715	+	740.823i	−0.512233	+	0.887214i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	775.765	−	447.888i	0.924631	−	0.533836i	0.0395213	−	0.999219i	$-0.487417\pi$
$839$	775.765	−	447.888i	0.924631	−	0.533836i	0.885109	+	0.465383i	$0.154083\pi$
$840$		0			0
$841$	287.342	−	497.691i	0.341667	−	0.591785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2856.32	−	1649.10i	3.38026	−	1.95160i
$846$		0			0
$847$	52.5629	+	56.3182i	0.0620578	+	0.0664914i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.260484	+	0.451171i	0.000306091	+	0.000530166i
$852$		0			0
$853$	1388.59	−	801.700i	1.62788	−	0.939860i	0.643162	−	0.765730i	$-0.277622\pi$
$853$	1388.59	−	801.700i	1.62788	−	0.939860i	0.984723	−	0.174130i	$-0.0557112\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	535.909i		−	0.625331i	−0.949863	−	0.312665i	$-0.898778\pi$
$857$		−	535.909i		−	0.625331i	0.949863	−	0.312665i	$-0.101222\pi$
$858$		0			0
$859$		−	616.375i		−	0.717550i	−0.933424	−	0.358775i	$-0.883195\pi$
$859$		−	616.375i		−	0.717550i	0.933424	−	0.358775i	$-0.116805\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	697.416	+	1207.96i	0.808129	+	1.39972i	0.914158	+	0.405358i	$0.132853\pi$
$863$	697.416	+	1207.96i	0.808129	+	1.39972i	−0.106029	+	0.994363i	$0.533814\pi$
$864$		0			0
$865$	1360.45	−	2356.37i	1.57278	−	2.72413i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	109.969	−	190.473i	0.126547	−	0.219186i
$870$		0			0
$871$			1259.85i			1.44644i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1448.29	−	1351.72i	1.65519	−	1.54483i
$876$		0			0
$877$	557.282			0.635441			0.317721	−	0.948184i	$-0.397083\pi$
$877$	557.282			0.635441			0.317721	+	0.948184i	$0.397083\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	691.635i		−	0.785057i	−0.919740	−	0.392528i	$-0.871601\pi$
$881$		−	691.635i		−	0.785057i	0.919740	−	0.392528i	$-0.128399\pi$
$882$		0			0
$883$	171.979			0.194766			0.0973831	−	0.995247i	$-0.468953\pi$
$883$	171.979			0.194766			0.0973831	+	0.995247i	$0.468953\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	361.006i		−	0.406997i	−0.979075	−	0.203499i	$-0.934769\pi$
$887$		−	361.006i		−	0.406997i	0.979075	−	0.203499i	$-0.0652312\pi$
$888$		0			0
$889$	865.590	+	927.431i	0.973667	+	1.04323i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−298.881			−0.334693
$894$		0			0
$895$	−683.531	−	394.637i	−0.763722	−	0.440935i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	401.687	+	231.914i	0.446816	+	0.257969i
$900$		0			0
$901$	−724.816	+	418.473i	−0.804457	+	0.464454i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1466.43			−1.62037
$906$		0			0
$907$	530.828			0.585257			0.292628	−	0.956226i	$-0.405470\pi$
$907$	530.828			0.585257			0.292628	+	0.956226i	$0.405470\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	432.297	+	748.761i	0.474531	+	0.821911i	0.999575	−	0.0291639i	$-0.00928448\pi$
$911$	432.297	+	748.761i	0.474531	+	0.821911i	−0.525044	+	0.851075i	$0.675951\pi$
$912$		0			0
$913$	278.500	−	160.792i	0.305038	−	0.176114i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	241.589	−	225.480i	0.263456	−	0.245889i
$918$		0			0
$919$	489.835	+	848.419i	0.533009	+	0.923198i	0.999257	+	0.0385444i	$0.0122721\pi$
$919$	489.835	+	848.419i	0.533009	+	0.923198i	−0.466248	+	0.884654i	$0.654395\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2067.86	+	1193.88i	2.24037	+	1.29348i
$924$		0			0
$925$	−1401.77	−	2427.93i	−1.51543	−	2.62479i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−422.919	−	244.172i	−0.455241	−	0.262834i	0.254800	−	0.966994i	$-0.417990\pi$
$929$	−422.919	−	244.172i	−0.455241	−	0.262834i	−0.710041	+	0.704160i	$0.751324\pi$
$930$		0			0
$931$	464.806	+	950.915i	0.499254	+	1.02139i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	506.266	−	876.879i	0.541461	−	0.937838i
$936$		0			0
$937$			173.479i			0.185143i	0.995706	+	0.0925714i	$0.0295086\pi$
$937$			173.479i			0.185143i	−0.995706	+	0.0925714i	$0.970491\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1285.82	+	742.370i	−1.36644	+	0.788916i	−0.990472	−	0.137715i	$-0.956024\pi$
$941$	−1285.82	+	742.370i	−1.36644	+	0.788916i	−0.375971	+	0.926631i	$0.622691\pi$
$942$		0			0
$943$	0.132045	+	0.0762363i	0.000140027	+	8.08444e-5i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	121.700	−	210.791i	0.128511	−	0.222588i	−0.794589	−	0.607148i	$-0.792313\pi$
$947$	121.700	−	210.791i	0.128511	−	0.222588i	0.923100	+	0.384560i	$0.125647\pi$
$948$		0			0
$949$	468.744	+	811.889i	0.493935	+	0.855520i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1555.73			−1.63245			−0.816226	−	0.577733i	$-0.803937\pi$
$953$	−1555.73			−1.63245			−0.816226	+	0.577733i	$0.803937\pi$
$954$		0			0
$955$	−2522.45	−	1456.34i	−2.64131	−	1.52496i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−242.470	−	259.793i	−0.252836	−	0.270900i
$960$		0			0
$961$	−76.5868	+	132.652i	−0.0796949	+	0.138036i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−55.9361	+	32.2947i	−0.0579649	+	0.0334660i
$966$		0			0
$967$	−183.047	+	317.047i	−0.189294	+	0.327867i	−0.945015	−	0.327027i	$-0.893953\pi$
$967$	−183.047	+	317.047i	−0.189294	+	0.327867i	0.755721	+	0.654894i	$0.227287\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−976.771	+	563.939i	−1.00594	+	0.580782i	−0.910001	−	0.414605i	$-0.863920\pi$
$971$	−976.771	+	563.939i	−1.00594	+	0.580782i	−0.0959419	+	0.995387i	$0.530586\pi$
$972$		0			0
$973$	268.050	+	1158.78i	0.275488	+	1.19093i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−282.673	−	489.604i	−0.289327	−	0.501130i	0.684322	−	0.729180i	$-0.260098\pi$
$977$	−282.673	−	489.604i	−0.289327	−	0.501130i	−0.973649	+	0.228050i	$0.926765\pi$
$978$		0			0
$979$	1430.33	−	825.801i	1.46101	−	0.843515i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			474.866i			0.483079i	0.970391	+	0.241539i	$0.0776523\pi$
$983$			474.866i			0.483079i	−0.970391	+	0.241539i	$0.922348\pi$
$984$		0			0
$985$			3215.52i			3.26449i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.0716220	−	0.124053i	−7.24186e−5	−	0.000125433i
$990$		0			0
$991$	−299.289	+	518.383i	−0.302007	+	0.523091i	−0.976590	−	0.215107i	$-0.930990\pi$
$991$	−299.289	+	518.383i	−0.302007	+	0.523091i	0.674584	+	0.738198i	$0.264323\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−199.475	+	345.501i	−0.200477	+	0.347237i
$996$		0			0
$997$			1347.04i			1.35109i	0.737319	+	0.675545i	$0.236092\pi$
$997$			1347.04i			1.35109i	−0.737319	+	0.675545i	$0.763908\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.p.a.577.1		32
3.2	odd	2		252.3.p.a.157.9	yes	32
7.5	odd	6		756.3.bd.a.145.16		32
9.2	odd	6		252.3.bd.a.241.4	yes	32
9.7	even	3		756.3.bd.a.73.16		32
21.5	even	6		252.3.bd.a.229.4	yes	32
63.47	even	6		252.3.p.a.61.9	✓	32
63.61	odd	6	inner	756.3.p.a.397.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.p.a.61.9	✓	32	63.47	even	6
252.3.p.a.157.9	yes	32	3.2	odd	2
252.3.bd.a.229.4	yes	32	21.5	even	6
252.3.bd.a.241.4	yes	32	9.2	odd	6
756.3.p.a.397.16		32	63.61	odd	6	inner
756.3.p.a.577.1		32	1.1	even	1	trivial
756.3.bd.a.73.16		32	9.7	even	3
756.3.bd.a.145.16		32	7.5	odd	6

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.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-9.02073i q^{5} +(-4.77619 - 5.11742i) q^{7} +10.4878 q^{11} +(-20.0242 - 11.5610i) q^{13} +(9.26855 + 5.35120i) q^{17} +(18.7068 - 10.8004i) q^{19} +0.0104756 q^{23} -56.3736 q^{25} +(-8.15959 - 14.1328i) q^{29} +(-24.6144 + 14.2111i) q^{31} +(-46.1629 + 43.0847i) q^{35} +(24.8657 + 43.0686i) q^{37} +(12.6050 + 7.27749i) q^{41} +(-6.83701 - 11.8421i) q^{43} +(-11.9829 - 6.91832i) q^{47} +(-3.37600 + 48.8836i) q^{49} +(-39.1008 + 67.7246i) q^{53} -94.6080i q^{55} +(23.8831 - 13.7889i) q^{59} +(-1.94313 - 1.12187i) q^{61} +(-104.288 + 180.633i) q^{65} +(-27.2436 - 47.1873i) q^{67} -103.268 q^{71} +(-35.1133 - 20.2727i) q^{73} +(-50.0919 - 53.6707i) q^{77} +(10.4854 - 18.1613i) q^{79} +(26.5546 - 15.3313i) q^{83} +(48.2717 - 83.6091i) q^{85} +(136.380 - 78.7389i) q^{89} +(36.4770 + 157.690i) q^{91} +(-97.4271 - 168.749i) q^{95} +(-79.7273 + 46.0306i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) \) 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

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