LMFDB - Embedded newform 448.3.l.b.209.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(209,448)] 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("448.209"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.21
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.21

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.22244 - 2.22244i) q^{3} +(1.66633 + 1.66633i) q^{5} +(-4.68881 - 5.19760i) q^{7} -0.878436i q^{9} +(1.36066 - 1.36066i) q^{11} +(5.09302 - 5.09302i) q^{13} +7.40664 q^{15} -23.0033i q^{17} +(19.4645 - 19.4645i) q^{19} +(-21.9719 - 1.13076i) q^{21} -7.80957i q^{23} -19.4467i q^{25} +(18.0496 + 18.0496i) q^{27} +(-26.2044 - 26.2044i) q^{29} +21.8757i q^{31} -6.04797i q^{33} +(0.847823 - 16.4741i) q^{35} +(-25.0652 + 25.0652i) q^{37} -22.6378i q^{39} +64.9559 q^{41} +(-9.70579 + 9.70579i) q^{43} +(1.46377 - 1.46377i) q^{45} -11.6768i q^{47} +(-5.03016 + 48.7411i) q^{49} +(-51.1233 - 51.1233i) q^{51} +(-15.1568 + 15.1568i) q^{53} +4.53464 q^{55} -86.5172i q^{57} +(-14.7798 - 14.7798i) q^{59} +(-0.766723 + 0.766723i) q^{61} +(-4.56576 + 4.11882i) q^{63} +16.9733 q^{65} +(-50.1767 - 50.1767i) q^{67} +(-17.3563 - 17.3563i) q^{69} +120.845i q^{71} -7.32621 q^{73} +(-43.2189 - 43.2189i) q^{75} +(-13.4521 - 0.692299i) q^{77} -7.61837 q^{79} +88.1343 q^{81} +(95.8459 - 95.8459i) q^{83} +(38.3311 - 38.3311i) q^{85} -116.475 q^{87} +71.9017 q^{89} +(-50.3517 - 2.59130i) q^{91} +(48.6174 + 48.6174i) q^{93} +64.8687 q^{95} +121.785i q^{97} +(-1.19526 - 1.19526i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	2.22244	−	2.22244i	0.740812	−	0.740812i	−0.231923	−	0.972734i	$-0.574502\pi$
$3$	2.22244	−	2.22244i	0.740812	−	0.740812i	0.972734	+	0.231923i	$0.0745016\pi$
$4$		0			0
$5$	1.66633	+	1.66633i	0.333267	+	0.333267i	0.853826	−	0.520559i	$-0.174276\pi$
$5$	1.66633	+	1.66633i	0.333267	+	0.333267i	−0.520559	+	0.853826i	$0.674276\pi$
$6$		0			0
$7$	−4.68881	−	5.19760i	−0.669830	−	0.742515i
$7$	−4.68881	−	5.19760i	−0.669830	−	0.742515i
$8$		0			0
$9$		−	0.878436i		−	0.0976040i
$10$		0			0
$11$	1.36066	−	1.36066i	0.123697	−	0.123697i	−0.642548	−	0.766245i	$-0.722123\pi$
$11$	1.36066	−	1.36066i	0.123697	−	0.123697i	0.766245	+	0.642548i	$0.222123\pi$
$12$		0			0
$13$	5.09302	−	5.09302i	0.391771	−	0.391771i	−0.483548	−	0.875318i	$-0.660652\pi$
$13$	5.09302	−	5.09302i	0.391771	−	0.391771i	0.875318	+	0.483548i	$0.160652\pi$
$14$		0			0
$15$	7.40664			0.493776
$16$		0			0
$17$		−	23.0033i		−	1.35313i	−0.736381	−	0.676567i	$-0.763467\pi$
$17$		−	23.0033i		−	1.35313i	0.736381	−	0.676567i	$-0.236533\pi$
$18$		0			0
$19$	19.4645	−	19.4645i	1.02445	−	1.02445i	0.0247535	−	0.999694i	$-0.492120\pi$
$19$	19.4645	−	19.4645i	1.02445	−	1.02445i	0.999694	−	0.0247535i	$-0.00788009\pi$
$20$		0			0
$21$	−21.9719	−	1.13076i	−1.04628	−	0.0538459i
$22$		0			0
$23$		−	7.80957i		−	0.339546i	−0.985483	−	0.169773i	$-0.945696\pi$
$23$		−	7.80957i		−	0.339546i	0.985483	−	0.169773i	$-0.0543035\pi$
$24$		0			0
$25$		−	19.4467i		−	0.777866i
$26$		0			0
$27$	18.0496	+	18.0496i	0.668506	+	0.668506i
$28$		0			0
$29$	−26.2044	−	26.2044i	−0.903598	−	0.903598i	0.0921470	−	0.995745i	$-0.470627\pi$
$29$	−26.2044	−	26.2044i	−0.903598	−	0.903598i	−0.995745	+	0.0921470i	$0.970627\pi$
$30$		0			0
$31$			21.8757i			0.705669i	0.935686	+	0.352835i	$0.114782\pi$
$31$			21.8757i			0.705669i	−0.935686	+	0.352835i	$0.885218\pi$
$32$		0			0
$33$		−	6.04797i		−	0.183272i
$34$		0			0
$35$	0.847823	−	16.4741i	0.0242235	−	0.470688i
$36$		0			0
$37$	−25.0652	+	25.0652i	−0.677438	+	0.677438i	−0.959420	−	0.281982i	$-0.909008\pi$
$37$	−25.0652	+	25.0652i	−0.677438	+	0.677438i	0.281982	+	0.959420i	$0.409008\pi$
$38$		0			0
$39$		−	22.6378i		−	0.580457i
$40$		0			0
$41$	64.9559			1.58429			0.792145	−	0.610333i	$-0.208964\pi$
$41$	64.9559			1.58429			0.792145	+	0.610333i	$0.208964\pi$
$42$		0			0
$43$	−9.70579	+	9.70579i	−0.225716	+	0.225716i	−0.810900	−	0.585184i	$-0.801022\pi$
$43$	−9.70579	+	9.70579i	−0.225716	+	0.225716i	0.585184	+	0.810900i	$0.301022\pi$
$44$		0			0
$45$	1.46377	−	1.46377i	0.0325282	−	0.0325282i
$46$		0			0
$47$		−	11.6768i		−	0.248444i	−0.992254	−	0.124222i	$-0.960357\pi$
$47$		−	11.6768i		−	0.248444i	0.992254	−	0.124222i	$-0.0396434\pi$
$48$		0			0
$49$	−5.03016	+	48.7411i	−0.102656	+	0.994717i
$50$		0			0
$51$	−51.1233	−	51.1233i	−1.00242	−	1.00242i
$52$		0			0
$53$	−15.1568	+	15.1568i	−0.285978	+	0.285978i	−0.835487	−	0.549510i	$-0.814815\pi$
$53$	−15.1568	+	15.1568i	−0.285978	+	0.285978i	0.549510	+	0.835487i	$0.314815\pi$
$54$		0			0
$55$	4.53464			0.0824480
$56$		0			0
$57$		−	86.5172i		−	1.51784i
$58$		0			0
$59$	−14.7798	−	14.7798i	−0.250505	−	0.250505i	0.570673	−	0.821178i	$-0.306683\pi$
$59$	−14.7798	−	14.7798i	−0.250505	−	0.250505i	−0.821178	+	0.570673i	$0.806683\pi$
$60$		0			0
$61$	−0.766723	+	0.766723i	−0.0125692	+	0.0125692i	−0.713363	−	0.700794i	$-0.752829\pi$
$61$	−0.766723	+	0.766723i	−0.0125692	+	0.0125692i	0.700794	+	0.713363i	$0.252829\pi$
$62$		0			0
$63$	−4.56576	+	4.11882i	−0.0724724	+	0.0653781i
$64$		0			0
$65$	16.9733			0.261128
$66$		0			0
$67$	−50.1767	−	50.1767i	−0.748905	−	0.748905i	0.225368	−	0.974274i	$-0.427641\pi$
$67$	−50.1767	−	50.1767i	−0.748905	−	0.748905i	−0.974274	+	0.225368i	$0.927641\pi$
$68$		0			0
$69$	−17.3563	−	17.3563i	−0.251540	−	0.251540i
$70$		0			0
$71$			120.845i			1.70204i	0.525135	+	0.851019i	$0.324015\pi$
$71$			120.845i			1.70204i	−0.525135	+	0.851019i	$0.675985\pi$
$72$		0			0
$73$	−7.32621			−0.100359			−0.0501795	−	0.998740i	$-0.515979\pi$
$73$	−7.32621			−0.100359			−0.0501795	+	0.998740i	$0.515979\pi$
$74$		0			0
$75$	−43.2189	−	43.2189i	−0.576253	−	0.576253i
$76$		0			0
$77$	−13.4521	−	0.692299i	−0.174702	−	0.00899090i
$78$		0			0
$79$	−7.61837			−0.0964351			−0.0482176	−	0.998837i	$-0.515354\pi$
$79$	−7.61837			−0.0964351			−0.0482176	+	0.998837i	$0.515354\pi$
$80$		0			0
$81$	88.1343			1.08808
$82$		0			0
$83$	95.8459	−	95.8459i	1.15477	−	1.15477i	0.169186	−	0.985584i	$-0.445886\pi$
$83$	95.8459	−	95.8459i	1.15477	−	1.15477i	0.985584	−	0.169186i	$-0.0541140\pi$
$84$		0			0
$85$	38.3311	−	38.3311i	0.450954	−	0.450954i
$86$		0			0
$87$	−116.475			−1.33879
$88$		0			0
$89$	71.9017			0.807884			0.403942	−	0.914784i	$-0.367640\pi$
$89$	71.9017			0.807884			0.403942	+	0.914784i	$0.367640\pi$
$90$		0			0
$91$	−50.3517	−	2.59130i	−0.553315	−	0.0284759i
$92$		0			0
$93$	48.6174	+	48.6174i	0.522768	+	0.522768i
$94$		0			0
$95$	64.8687			0.682828
$96$		0			0
$97$			121.785i			1.25551i	0.778410	+	0.627757i	$0.216027\pi$
$97$			121.785i			1.25551i	−0.778410	+	0.627757i	$0.783973\pi$
$98$		0			0
$99$	−1.19526	−	1.19526i	−0.0120733	−	0.0120733i
$100$		0			0
$101$	−63.0038	−	63.0038i	−0.623800	−	0.623800i	0.322701	−	0.946501i	$-0.395409\pi$
$101$	−63.0038	−	63.0038i	−0.623800	−	0.623800i	−0.946501	+	0.322701i	$0.895409\pi$
$102$		0			0
$103$	174.305			1.69228			0.846139	−	0.532962i	$-0.178921\pi$
$103$	174.305			1.69228			0.846139	+	0.532962i	$0.178921\pi$
$104$		0			0
$105$	−34.7283	−	38.4968i	−0.330746	−	0.366636i
$106$		0			0
$107$	−47.0984	+	47.0984i	−0.440172	+	0.440172i	−0.892070	−	0.451898i	$-0.850747\pi$
$107$	−47.0984	+	47.0984i	−0.440172	+	0.440172i	0.451898	+	0.892070i	$0.350747\pi$
$108$		0			0
$109$	86.7341	+	86.7341i	0.795726	+	0.795726i	0.982418	−	0.186693i	$-0.0597769\pi$
$109$	86.7341	+	86.7341i	0.795726	+	0.795726i	−0.186693	+	0.982418i	$0.559777\pi$
$110$		0			0
$111$			111.412i			1.00371i
$112$		0			0
$113$	−49.9675			−0.442191			−0.221095	−	0.975252i	$-0.570963\pi$
$113$	−49.9675			−0.442191			−0.221095	+	0.975252i	$0.570963\pi$
$114$		0			0
$115$	13.0133	−	13.0133i	0.113160	−	0.113160i
$116$		0			0
$117$	−4.47389	−	4.47389i	−0.0382384	−	0.0382384i
$118$		0			0
$119$	−119.562	+	107.858i	−1.00472	+	0.906369i
$120$		0			0
$121$			117.297i			0.969398i
$122$		0			0
$123$	144.360	−	144.360i	1.17366	−	1.17366i
$124$		0			0
$125$	74.0630	−	74.0630i	0.592504	−	0.592504i
$126$		0			0
$127$	235.080			1.85102			0.925512	−	0.378719i	$-0.123635\pi$
$127$	235.080			1.85102			0.925512	+	0.378719i	$0.123635\pi$
$128$		0			0
$129$			43.1410i			0.334426i
$130$		0			0
$131$	−172.394	+	172.394i	−1.31598	+	1.31598i	−0.399058	+	0.916926i	$0.630663\pi$
$131$	−172.394	+	172.394i	−1.31598	+	1.31598i	−0.916926	+	0.399058i	$0.869337\pi$
$132$		0			0
$133$	−192.434	−	9.90345i	−1.44687	−	0.0744620i
$134$		0			0
$135$			60.1535i			0.445581i
$136$		0			0
$137$		−	44.5853i		−	0.325440i	−0.986672	−	0.162720i	$-0.947973\pi$
$137$		−	44.5853i		−	0.325440i	0.986672	−	0.162720i	$-0.0520268\pi$
$138$		0			0
$139$	97.8682	+	97.8682i	0.704088	+	0.704088i	0.965285	−	0.261197i	$-0.0841174\pi$
$139$	97.8682	+	97.8682i	0.704088	+	0.704088i	−0.261197	+	0.965285i	$0.584117\pi$
$140$		0			0
$141$	−25.9510	−	25.9510i	−0.184050	−	0.184050i
$142$		0			0
$143$		−	13.8598i		−	0.0969214i
$144$		0			0
$145$		−	87.3304i		−	0.602279i
$146$		0			0
$147$	97.1448	+	119.503i	0.660849	+	0.812947i
$148$		0			0
$149$	−95.5292	+	95.5292i	−0.641136	+	0.641136i	−0.950835	−	0.309699i	$-0.899772\pi$
$149$	−95.5292	+	95.5292i	−0.641136	+	0.641136i	0.309699	+	0.950835i	$0.399772\pi$
$150$		0			0
$151$		−	165.338i		−	1.09495i	−0.836821	−	0.547476i	$-0.815589\pi$
$151$		−	165.338i		−	1.09495i	0.836821	−	0.547476i	$-0.184411\pi$
$152$		0			0
$153$	−20.2069			−0.132071
$154$		0			0
$155$	−36.4523	+	36.4523i	−0.235176	+	0.235176i
$156$		0			0
$157$	−168.982	+	168.982i	−1.07632	+	1.07632i	−0.0794827	+	0.996836i	$0.525327\pi$
$157$	−168.982	+	168.982i	−1.07632	+	1.07632i	−0.996836	+	0.0794827i	$0.974673\pi$
$158$		0			0
$159$			67.3701i			0.423711i
$160$		0			0
$161$	−40.5910	+	36.6176i	−0.252118	+	0.227438i
$162$		0			0
$163$	−76.2213	−	76.2213i	−0.467616	−	0.467616i	0.433526	−	0.901141i	$-0.357269\pi$
$163$	−76.2213	−	76.2213i	−0.467616	−	0.467616i	−0.901141	+	0.433526i	$0.857269\pi$
$164$		0			0
$165$	10.0779	−	10.0779i	0.0610784	−	0.0610784i
$166$		0			0
$167$	−166.172			−0.995040			−0.497520	−	0.867452i	$-0.665756\pi$
$167$	−166.172			−0.995040			−0.497520	+	0.867452i	$0.665756\pi$
$168$		0			0
$169$			117.122i			0.693032i
$170$		0			0
$171$	−17.0983	−	17.0983i	−0.0999901	−	0.0999901i
$172$		0			0
$173$	−127.346	+	127.346i	−0.736103	+	0.736103i	−0.971821	−	0.235719i	$-0.924256\pi$
$173$	−127.346	+	127.346i	−0.736103	+	0.736103i	0.235719	+	0.971821i	$0.424256\pi$
$174$		0			0
$175$	−101.076	+	91.1817i	−0.577577	+	0.521038i
$176$		0			0
$177$	−65.6943			−0.371154
$178$		0			0
$179$	167.337	+	167.337i	0.934842	+	0.934842i	0.998003	−	0.0631609i	$-0.0201181\pi$
$179$	167.337	+	167.337i	0.934842	+	0.934842i	−0.0631609	+	0.998003i	$0.520118\pi$
$180$		0			0
$181$	11.2916	+	11.2916i	0.0623843	+	0.0623843i	0.737611	−	0.675226i	$-0.235954\pi$
$181$	11.2916	+	11.2916i	0.0623843	+	0.0623843i	−0.675226	+	0.737611i	$0.735954\pi$
$182$		0			0
$183$			3.40798i			0.0186229i
$184$		0			0
$185$	−83.5340			−0.451535
$186$		0			0
$187$	−31.2997	−	31.2997i	−0.167378	−	0.167378i
$188$		0			0
$189$	9.18358	−	178.446i	0.0485904	−	0.944160i
$190$		0			0
$191$	−163.386			−0.855423			−0.427712	−	0.903915i	$-0.640680\pi$
$191$	−163.386			−0.855423			−0.427712	+	0.903915i	$0.640680\pi$
$192$		0			0
$193$	279.119			1.44621			0.723107	−	0.690736i	$-0.242713\pi$
$193$	279.119			1.44621			0.723107	+	0.690736i	$0.242713\pi$
$194$		0			0
$195$	37.7221	−	37.7221i	0.193447	−	0.193447i
$196$		0			0
$197$	147.140	−	147.140i	0.746906	−	0.746906i	−0.226991	−	0.973897i	$-0.572889\pi$
$197$	147.140	−	147.140i	0.746906	−	0.746906i	0.973897	+	0.226991i	$0.0728888\pi$
$198$		0			0
$199$	163.501			0.821612			0.410806	−	0.911723i	$-0.365247\pi$
$199$	163.501			0.821612			0.410806	+	0.911723i	$0.365247\pi$
$200$		0			0
$201$	−223.029			−1.10960
$202$		0			0
$203$	−13.3327	+	259.067i	−0.0656781	+	1.27619i
$204$		0			0
$205$	108.238	+	108.238i	0.527991	+	0.527991i
$206$		0			0
$207$	−6.86020			−0.0331411
$208$		0			0
$209$		−	52.9692i		−	0.253441i
$210$		0			0
$211$	−40.9927	−	40.9927i	−0.194278	−	0.194278i	0.603264	−	0.797542i	$-0.293867\pi$
$211$	−40.9927	−	40.9927i	−0.194278	−	0.194278i	−0.797542	+	0.603264i	$0.793867\pi$
$212$		0			0
$213$	268.569	+	268.569i	1.26089	+	1.26089i
$214$		0			0
$215$	−32.3462			−0.150447
$216$		0			0
$217$	113.701	−	102.571i	0.523970	−	0.472678i
$218$		0			0
$219$	−16.2820	+	16.2820i	−0.0743472	+	0.0743472i
$220$		0			0
$221$	−117.156	−	117.156i	−0.530118	−	0.530118i
$222$		0			0
$223$		−	197.114i		−	0.883917i	−0.897035	−	0.441959i	$-0.854284\pi$
$223$		−	197.114i		−	0.883917i	0.897035	−	0.441959i	$-0.145716\pi$
$224$		0			0
$225$	−17.0826			−0.0759229
$226$		0			0
$227$	−50.7474	+	50.7474i	−0.223557	+	0.223557i	−0.809994	−	0.586438i	$-0.800530\pi$
$227$	−50.7474	+	50.7474i	−0.223557	+	0.223557i	0.586438	+	0.809994i	$0.300530\pi$
$228$		0			0
$229$	276.035	+	276.035i	1.20539	+	1.20539i	0.972504	+	0.232887i	$0.0748172\pi$
$229$	276.035	+	276.035i	1.20539	+	1.20539i	0.232887	+	0.972504i	$0.425183\pi$
$230$		0			0
$231$	−31.4350	+	28.3578i	−0.136082	+	0.122761i
$232$		0			0
$233$			416.018i			1.78549i	0.450566	+	0.892743i	$0.351222\pi$
$233$			416.018i			1.78549i	−0.450566	+	0.892743i	$0.648778\pi$
$234$		0			0
$235$	19.4575	−	19.4575i	0.0827980	−	0.0827980i
$236$		0			0
$237$	−16.9313	+	16.9313i	−0.0714403	+	0.0714403i
$238$		0			0
$239$	−35.9992			−0.150624			−0.0753121	−	0.997160i	$-0.523995\pi$
$239$	−35.9992			−0.150624			−0.0753121	+	0.997160i	$0.523995\pi$
$240$		0			0
$241$		−	277.725i		−	1.15239i	−0.817313	−	0.576194i	$-0.804537\pi$
$241$		−	277.725i		−	1.15239i	0.817313	−	0.576194i	$-0.195463\pi$
$242$		0			0
$243$	33.4259	−	33.4259i	0.137555	−	0.137555i
$244$		0			0
$245$	−89.6009	+	72.8371i	−0.365718	+	0.297294i
$246$		0			0
$247$		−	198.266i		−	0.802696i
$248$		0			0
$249$		−	426.023i		−	1.71093i
$250$		0			0
$251$	−177.118	−	177.118i	−0.705651	−	0.705651i	0.259967	−	0.965618i	$-0.416288\pi$
$251$	−177.118	−	177.118i	−0.705651	−	0.705651i	−0.965618	+	0.259967i	$0.916288\pi$
$252$		0			0
$253$	−10.6262	−	10.6262i	−0.0420007	−	0.0420007i
$254$		0			0
$255$		−	170.377i		−	0.668145i
$256$		0			0
$257$		−	289.567i		−	1.12672i	−0.826211	−	0.563361i	$-0.809508\pi$
$257$		−	289.567i		−	1.12672i	0.826211	−	0.563361i	$-0.190492\pi$
$258$		0			0
$259$	247.805	+	12.7531i	0.956775	+	0.0492396i
$260$		0			0
$261$	−23.0188	+	23.0188i	−0.0881948	+	0.0881948i
$262$		0			0
$263$		−	216.416i		−	0.822875i	−0.911438	−	0.411438i	$-0.865027\pi$
$263$		−	216.416i		−	0.822875i	0.911438	−	0.411438i	$-0.134973\pi$
$264$		0			0
$265$	−50.5127			−0.190614
$266$		0			0
$267$	159.797	−	159.797i	0.598490	−	0.598490i
$268$		0			0
$269$	161.704	−	161.704i	0.601131	−	0.601131i	−0.339482	−	0.940613i	$-0.610252\pi$
$269$	161.704	−	161.704i	0.601131	−	0.601131i	0.940613	+	0.339482i	$0.110252\pi$
$270$		0			0
$271$		−	98.2578i		−	0.362575i	−0.983430	−	0.181287i	$-0.941974\pi$
$271$		−	98.2578i		−	0.362575i	0.983430	−	0.181287i	$-0.0580264\pi$
$272$		0			0
$273$	−117.662	+	106.144i	−0.430998	+	0.388807i
$274$		0			0
$275$	−26.4604	−	26.4604i	−0.0962195	−	0.0962195i
$276$		0			0
$277$	380.735	−	380.735i	1.37449	−	1.37449i	0.520839	−	0.853655i	$-0.325619\pi$
$277$	380.735	−	380.735i	1.37449	−	1.37449i	0.853655	−	0.520839i	$-0.174381\pi$
$278$		0			0
$279$	19.2164			0.0688761
$280$		0			0
$281$			264.950i			0.942884i	0.881897	+	0.471442i	$0.156266\pi$
$281$			264.950i			0.942884i	−0.881897	+	0.471442i	$0.843734\pi$
$282$		0			0
$283$	50.3482	+	50.3482i	0.177909	+	0.177909i	0.790444	−	0.612535i	$-0.209850\pi$
$283$	50.3482	+	50.3482i	0.177909	+	0.177909i	−0.612535	+	0.790444i	$0.709850\pi$
$284$		0			0
$285$	144.166	−	144.166i	0.505847	−	0.505847i
$286$		0			0
$287$	−304.566	−	337.615i	−1.06120	−	1.17636i
$288$		0			0
$289$	−240.150			−0.830969
$290$		0			0
$291$	270.659	+	270.659i	0.930099	+	0.930099i
$292$		0			0
$293$	−244.020	−	244.020i	−0.832834	−	0.832834i	0.155070	−	0.987904i	$-0.450440\pi$
$293$	−244.020	−	244.020i	−0.832834	−	0.832834i	−0.987904	+	0.155070i	$0.950440\pi$
$294$		0			0
$295$		−	49.2562i		−	0.166970i
$296$		0			0
$297$	49.1190			0.165384
$298$		0			0
$299$	−39.7743	−	39.7743i	−0.133024	−	0.133024i
$300$		0			0
$301$	95.9554	+	4.93826i	0.318789	+	0.0164062i
$302$		0			0
$303$	−280.044			−0.924236
$304$		0			0
$305$	−2.55523			−0.00837781
$306$		0			0
$307$	−206.183	+	206.183i	−0.671605	+	0.671605i	−0.958086	−	0.286481i	$-0.907515\pi$
$307$	−206.183	+	206.183i	−0.671605	+	0.671605i	0.286481	+	0.958086i	$0.407515\pi$
$308$		0			0
$309$	387.381	−	387.381i	1.25366	−	1.25366i
$310$		0			0
$311$	311.503			1.00162			0.500809	−	0.865558i	$-0.333036\pi$
$311$	311.503			1.00162			0.500809	+	0.865558i	$0.333036\pi$
$312$		0			0
$313$	150.068			0.479450			0.239725	−	0.970841i	$-0.422943\pi$
$313$	150.068			0.479450			0.239725	+	0.970841i	$0.422943\pi$
$314$		0			0
$315$	−14.4714	−	0.744759i	−0.0459410	−	0.00236431i
$316$		0			0
$317$	−255.523	−	255.523i	−0.806066	−	0.806066i	0.177970	−	0.984036i	$-0.443047\pi$
$317$	−255.523	−	255.523i	−0.806066	−	0.806066i	−0.984036	+	0.177970i	$0.943047\pi$
$318$		0			0
$319$	−71.3106			−0.223544
$320$		0			0
$321$			209.346i			0.652168i
$322$		0			0
$323$	−447.747	−	447.747i	−1.38621	−	1.38621i
$324$		0			0
$325$	−99.0422	−	99.0422i	−0.304745	−	0.304745i
$326$		0			0
$327$	385.522			1.17897
$328$		0			0
$329$	−60.6916	+	54.7505i	−0.184473	+	0.166415i
$330$		0			0
$331$	−207.933	+	207.933i	−0.628197	+	0.628197i	−0.947614	−	0.319417i	$-0.896513\pi$
$331$	−207.933	+	207.933i	−0.628197	+	0.628197i	0.319417	+	0.947614i	$0.396513\pi$
$332$		0			0
$333$	22.0182	+	22.0182i	0.0661206	+	0.0661206i
$334$		0			0
$335$		−	167.222i		−	0.499171i
$336$		0			0
$337$	−498.996			−1.48070			−0.740350	−	0.672221i	$-0.765340\pi$
$337$	−498.996			−1.48070			−0.740350	+	0.672221i	$0.765340\pi$
$338$		0			0
$339$	−111.050	+	111.050i	−0.327580	+	0.327580i
$340$		0			0
$341$	29.7655	+	29.7655i	0.0872889	+	0.0872889i
$342$		0			0
$343$	276.923	−	202.393i	0.807354	−	0.590067i
$344$		0			0
$345$		−	57.8426i		−	0.167660i
$346$		0			0
$347$	266.997	−	266.997i	0.769444	−	0.769444i	−0.208565	−	0.978009i	$-0.566879\pi$
$347$	266.997	−	266.997i	0.769444	−	0.769444i	0.978009	+	0.208565i	$0.0668793\pi$
$348$		0			0
$349$	−30.6346	+	30.6346i	−0.0877782	+	0.0877782i	−0.749633	−	0.661854i	$-0.769770\pi$
$349$	−30.6346	+	30.6346i	−0.0877782	+	0.0877782i	0.661854	+	0.749633i	$0.269770\pi$
$350$		0			0
$351$	183.854			0.523802
$352$		0			0
$353$			219.589i			0.622066i	0.950399	+	0.311033i	$0.100675\pi$
$353$			219.589i			0.622066i	−0.950399	+	0.311033i	$0.899325\pi$
$354$		0			0
$355$	−201.368	+	201.368i	−0.567233	+	0.567233i
$356$		0			0
$357$	−26.0113	+	505.426i	−0.0728607	+	1.41576i
$358$		0			0
$359$			401.224i			1.11761i	0.829298	+	0.558807i	$0.188741\pi$
$359$			401.224i			1.11761i	−0.829298	+	0.558807i	$0.811259\pi$
$360$		0			0
$361$		−	396.733i		−	1.09898i
$362$		0			0
$363$	260.685	+	260.685i	0.718142	+	0.718142i
$364$		0			0
$365$	−12.2079	−	12.2079i	−0.0334463	−	0.0334463i
$366$		0			0
$367$			240.624i			0.655652i	0.944738	+	0.327826i	$0.106316\pi$
$367$			240.624i			0.655652i	−0.944738	+	0.327826i	$0.893684\pi$
$368$		0			0
$369$		−	57.0596i		−	0.154633i
$370$		0			0
$371$	149.847	+	7.71172i	0.403899	+	0.0207863i
$372$		0			0
$373$	178.908	−	178.908i	0.479646	−	0.479646i	−0.425372	−	0.905018i	$-0.639857\pi$
$373$	178.908	−	178.908i	0.479646	−	0.479646i	0.905018	+	0.425372i	$0.139857\pi$
$374$		0			0
$375$		−	329.200i		−	0.877868i
$376$		0			0
$377$	−266.918			−0.708007
$378$		0			0
$379$	40.4554	−	40.4554i	0.106742	−	0.106742i	−0.651718	−	0.758461i	$-0.725952\pi$
$379$	40.4554	−	40.4554i	0.106742	−	0.106742i	0.758461	+	0.651718i	$0.225952\pi$
$380$		0			0
$381$	522.450	−	522.450i	1.37126	−	1.37126i
$382$		0			0
$383$		−	339.537i		−	0.886521i	−0.896393	−	0.443260i	$-0.853822\pi$
$383$		−	339.537i		−	0.886521i	0.896393	−	0.443260i	$-0.146178\pi$
$384$		0			0
$385$	−21.2621	−	23.5693i	−0.0552261	−	0.0612188i
$386$		0			0
$387$	8.52591	+	8.52591i	0.0220308	+	0.0220308i
$388$		0			0
$389$	−281.781	+	281.781i	−0.724374	+	0.724374i	−0.969493	−	0.245119i	$-0.921173\pi$
$389$	−281.781	+	281.781i	−0.724374	+	0.724374i	0.245119	+	0.969493i	$0.421173\pi$
$390$		0			0
$391$	−179.646			−0.459451
$392$		0			0
$393$			766.268i			1.94979i
$394$		0			0
$395$	−12.6948	−	12.6948i	−0.0321386	−	0.0321386i
$396$		0			0
$397$	−405.318	+	405.318i	−1.02095	+	1.02095i	−0.0211753	+	0.999776i	$0.506741\pi$
$397$	−405.318	+	405.318i	−1.02095	+	1.02095i	−0.999776	+	0.0211753i	$0.993259\pi$
$398$		0			0
$399$	−449.682	+	405.662i	−1.12702	+	1.01670i
$400$		0			0
$401$	−691.547			−1.72456			−0.862278	−	0.506435i	$-0.830963\pi$
$401$	−691.547			−1.72456			−0.862278	+	0.506435i	$0.830963\pi$
$402$		0			0
$403$	111.414	+	111.414i	0.276460	+	0.276460i
$404$		0			0
$405$	146.861	+	146.861i	0.362620	+	0.362620i
$406$		0			0
$407$			68.2106i			0.167594i
$408$		0			0
$409$	398.371			0.974012			0.487006	−	0.873399i	$-0.338089\pi$
$409$	398.371			0.974012			0.487006	+	0.873399i	$0.338089\pi$
$410$		0			0
$411$	−99.0880	−	99.0880i	−0.241090	−	0.241090i
$412$		0			0
$413$	−7.51990	+	146.119i	−0.0182080	+	0.353800i
$414$		0			0
$415$	319.423			0.769693
$416$		0			0
$417$	435.012			1.04319
$418$		0			0
$419$	−72.7602	+	72.7602i	−0.173652	+	0.173652i	−0.788582	−	0.614930i	$-0.789184\pi$
$419$	−72.7602	+	72.7602i	−0.173652	+	0.173652i	0.614930	+	0.788582i	$0.289184\pi$
$420$		0			0
$421$	228.343	−	228.343i	0.542382	−	0.542382i	−0.381844	−	0.924227i	$-0.624711\pi$
$421$	228.343	−	228.343i	0.542382	−	0.542382i	0.924227	+	0.381844i	$0.124711\pi$
$422$		0			0
$423$	−10.2574			−0.0242491
$424$		0			0
$425$	−447.337			−1.05256
$426$		0			0
$427$	7.58014	+	0.390105i	0.0177521	+	0.000913595i
$428$		0			0
$429$	−30.8024	−	30.8024i	−0.0718005	−	0.0718005i
$430$		0			0
$431$	−125.697			−0.291640			−0.145820	−	0.989311i	$-0.546582\pi$
$431$	−125.697			−0.291640			−0.145820	+	0.989311i	$0.546582\pi$
$432$		0			0
$433$		−	466.729i		−	1.07790i	−0.842339	−	0.538948i	$-0.818822\pi$
$433$		−	466.729i		−	1.07790i	0.842339	−	0.538948i	$-0.181178\pi$
$434$		0			0
$435$	−194.086	−	194.086i	−0.446175	−	0.446175i
$436$		0			0
$437$	−152.009	−	152.009i	−0.347847	−	0.347847i
$438$		0			0
$439$	−293.932			−0.669549			−0.334774	−	0.942298i	$-0.608660\pi$
$439$	−293.932			−0.669549			−0.334774	+	0.942298i	$0.608660\pi$
$440$		0			0
$441$	42.8160	+	4.41867i	0.0970884	+	0.0100197i
$442$		0			0
$443$	−119.753	+	119.753i	−0.270322	+	0.270322i	−0.829230	−	0.558908i	$-0.811221\pi$
$443$	−119.753	+	119.753i	−0.270322	+	0.270322i	0.558908	+	0.829230i	$0.311221\pi$
$444$		0			0
$445$	119.812	+	119.812i	0.269241	+	0.269241i
$446$		0			0
$447$			424.615i			0.949922i
$448$		0			0
$449$	213.568			0.475653			0.237827	−	0.971308i	$-0.423565\pi$
$449$	213.568			0.475653			0.237827	+	0.971308i	$0.423565\pi$
$450$		0			0
$451$	88.3831	−	88.3831i	0.195971	−	0.195971i
$452$		0			0
$453$	−367.453	−	367.453i	−0.811153	−	0.811153i
$454$		0			0
$455$	−79.5847	−	88.2207i	−0.174911	−	0.193892i
$456$		0			0
$457$		−	200.183i		−	0.438037i	−0.975721	−	0.219019i	$-0.929714\pi$
$457$		−	200.183i		−	0.438037i	0.975721	−	0.219019i	$-0.0702856\pi$
$458$		0			0
$459$	415.201	−	415.201i	0.904577	−	0.904577i
$460$		0			0
$461$	−18.3869	+	18.3869i	−0.0398848	+	0.0398848i	−0.726768	−	0.686883i	$-0.758978\pi$
$461$	−18.3869	+	18.3869i	−0.0398848	+	0.0398848i	0.686883	+	0.726768i	$0.258978\pi$
$462$		0			0
$463$	865.534			1.86940			0.934702	−	0.355433i	$-0.115667\pi$
$463$	865.534			1.86940			0.934702	+	0.355433i	$0.115667\pi$
$464$		0			0
$465$			162.026i			0.348442i
$466$		0			0
$467$	256.219	−	256.219i	0.548649	−	0.548649i	−0.377401	−	0.926050i	$-0.623182\pi$
$467$	256.219	−	256.219i	0.548649	−	0.548649i	0.926050	+	0.377401i	$0.123182\pi$
$468$		0			0
$469$	−25.5297	+	496.067i	−0.0544342	+	1.05771i
$470$		0			0
$471$			751.103i			1.59470i
$472$		0			0
$473$			26.4126i			0.0558406i
$474$		0			0
$475$	−378.519	−	378.519i	−0.796883	−	0.796883i
$476$		0			0
$477$	13.3143	+	13.3143i	0.0279126	+	0.0279126i
$478$		0			0
$479$			81.5854i			0.170324i	0.996367	+	0.0851622i	$0.0271408\pi$
$479$			81.5854i			0.170324i	−0.996367	+	0.0851622i	$0.972859\pi$
$480$		0			0
$481$			255.315i			0.530800i
$482$		0			0
$483$	−8.83078	+	171.591i	−0.0182832	+	0.355261i
$484$		0			0
$485$	−202.934	+	202.934i	−0.418421	+	0.418421i
$486$		0			0
$487$		−	446.445i		−	0.916725i	−0.888765	−	0.458362i	$-0.848436\pi$
$487$		−	446.445i		−	0.916725i	0.888765	−	0.458362i	$-0.151564\pi$
$488$		0			0
$489$	−338.794			−0.692830
$490$		0			0
$491$	−285.566	+	285.566i	−0.581601	+	0.581601i	−0.935343	−	0.353742i	$-0.884909\pi$
$491$	−285.566	+	285.566i	−0.581601	+	0.581601i	0.353742	+	0.935343i	$0.384909\pi$
$492$		0			0
$493$	−602.786	+	602.786i	−1.22269	+	1.22269i
$494$		0			0
$495$		−	3.98339i		−	0.00804725i
$496$		0			0
$497$	628.103	−	566.617i	1.26379	−	1.14008i
$498$		0			0
$499$	349.704	+	349.704i	0.700810	+	0.700810i	0.964584	−	0.263774i	$-0.0849674\pi$
$499$	349.704	+	349.704i	0.700810	+	0.700810i	−0.263774	+	0.964584i	$0.584967\pi$
$500$		0			0
$501$	−369.306	+	369.306i	−0.737137	+	0.737137i
$502$		0			0
$503$	−44.1832			−0.0878393			−0.0439196	−	0.999035i	$-0.513985\pi$
$503$	−44.1832			−0.0878393			−0.0439196	+	0.999035i	$0.513985\pi$
$504$		0			0
$505$		−	209.971i		−	0.415783i
$506$		0			0
$507$	260.297	+	260.297i	0.513406	+	0.513406i
$508$		0			0
$509$	432.447	−	432.447i	0.849601	−	0.849601i	−0.140482	−	0.990083i	$-0.544865\pi$
$509$	432.447	−	432.447i	0.849601	−	0.849601i	0.990083	+	0.140482i	$0.0448653\pi$
$510$		0			0
$511$	34.3512	+	38.0787i	0.0672235	+	0.0745181i
$512$		0			0
$513$	702.655			1.36970
$514$		0			0
$515$	290.450	+	290.450i	0.563980	+	0.563980i
$516$		0			0
$517$	−15.8883	−	15.8883i	−0.0307316	−	0.0307316i
$518$		0			0
$519$			566.035i			1.09063i
$520$		0			0
$521$	−288.769			−0.554258			−0.277129	−	0.960833i	$-0.589383\pi$
$521$	−288.769			−0.554258			−0.277129	+	0.960833i	$0.589383\pi$
$522$		0			0
$523$	292.459	+	292.459i	0.559195	+	0.559195i	0.929078	−	0.369884i	$-0.120602\pi$
$523$	292.459	+	292.459i	0.559195	+	0.559195i	−0.369884	+	0.929078i	$0.620602\pi$
$524$		0			0
$525$	−21.9896	+	427.280i	−0.0418850	+	0.813867i
$526$		0			0
$527$	503.214			0.954864
$528$		0			0
$529$	468.011			0.884708
$530$		0			0
$531$	−12.9831	+	12.9831i	−0.0244503	+	0.0244503i
$532$		0			0
$533$	330.822	−	330.822i	0.620678	−	0.620678i
$534$		0			0
$535$	−156.963			−0.293389
$536$		0			0
$537$	743.790			1.38508
$538$		0			0
$539$	59.4759	+	73.1646i	0.110345	+	0.135741i
$540$		0			0
$541$	−275.144	−	275.144i	−0.508584	−	0.508584i	0.405507	−	0.914092i	$-0.367095\pi$
$541$	−275.144	−	275.144i	−0.508584	−	0.508584i	−0.914092	+	0.405507i	$0.867095\pi$
$542$		0			0
$543$	50.1895			0.0924301
$544$		0			0
$545$			289.056i			0.530378i
$546$		0			0
$547$	−264.248	−	264.248i	−0.483087	−	0.483087i	0.423029	−	0.906116i	$-0.360967\pi$
$547$	−264.248	−	264.248i	−0.483087	−	0.483087i	−0.906116	+	0.423029i	$0.860967\pi$
$548$		0			0
$549$	0.673517	+	0.673517i	0.00122681	+	0.00122681i
$550$		0			0
$551$	−1020.11			−1.85138
$552$		0			0
$553$	35.7211	+	39.5973i	0.0645951	+	0.0716045i
$554$		0			0
$555$	−185.649	+	185.649i	−0.334502	+	0.334502i
$556$		0			0
$557$	332.593	+	332.593i	0.597116	+	0.597116i	0.939544	−	0.342428i	$-0.111249\pi$
$557$	332.593	+	332.593i	0.597116	+	0.597116i	−0.342428	+	0.939544i	$0.611249\pi$
$558$		0			0
$559$			98.8635i			0.176858i
$560$		0			0
$561$	−139.123			−0.247991
$562$		0			0
$563$	7.10518	−	7.10518i	0.0126202	−	0.0126202i	−0.700769	−	0.713389i	$-0.747159\pi$
$563$	7.10518	−	7.10518i	0.0126202	−	0.0126202i	0.713389	+	0.700769i	$0.247159\pi$
$564$		0			0
$565$	−83.2626	−	83.2626i	−0.147367	−	0.147367i
$566$		0			0
$567$	−413.245	−	458.087i	−0.728827	−	0.807914i
$568$		0			0
$569$		−	511.170i		−	0.898366i	−0.893440	−	0.449183i	$-0.851715\pi$
$569$		−	511.170i		−	0.898366i	0.893440	−	0.449183i	$-0.148285\pi$
$570$		0			0
$571$	−520.460	+	520.460i	−0.911488	+	0.911488i	−0.996389	−	0.0849014i	$-0.972942\pi$
$571$	−520.460	+	520.460i	−0.911488	+	0.911488i	0.0849014	+	0.996389i	$0.472942\pi$
$572$		0			0
$573$	−363.114	+	363.114i	−0.633707	+	0.633707i
$574$		0			0
$575$	−151.870			−0.264122
$576$		0			0
$577$			212.128i			0.367640i	0.982960	+	0.183820i	$0.0588463\pi$
$577$			212.128i			0.367640i	−0.982960	+	0.183820i	$0.941154\pi$
$578$		0			0
$579$	620.325	−	620.325i	1.07137	−	1.07137i
$580$		0			0
$581$	−947.572	−	48.7660i	−1.63093	−	0.0839346i
$582$		0			0
$583$			41.2467i			0.0707490i
$584$		0			0
$585$		−	14.9100i		−	0.0254872i
$586$		0			0
$587$	−334.289	−	334.289i	−0.569487	−	0.569487i	0.362498	−	0.931985i	$-0.381924\pi$
$587$	−334.289	−	334.289i	−0.569487	−	0.569487i	−0.931985	+	0.362498i	$0.881924\pi$
$588$		0			0
$589$	425.800	+	425.800i	0.722921	+	0.722921i
$590$		0			0
$591$		−	654.020i		−	1.10663i
$592$		0			0
$593$			855.444i			1.44257i	0.692639	+	0.721285i	$0.256448\pi$
$593$			855.444i			1.44257i	−0.692639	+	0.721285i	$0.743552\pi$
$594$		0			0
$595$	−378.957	−	19.5027i	−0.636903	−	0.0327777i
$596$		0			0
$597$	363.370	−	363.370i	0.608660	−	0.608660i
$598$		0			0
$599$			434.765i			0.725819i	0.931824	+	0.362909i	$0.118217\pi$
$599$			434.765i			0.725819i	−0.931824	+	0.362909i	$0.881783\pi$
$600$		0			0
$601$	1059.09			1.76221			0.881105	−	0.472922i	$-0.156801\pi$
$601$	1059.09			1.76221			0.881105	+	0.472922i	$0.156801\pi$
$602$		0			0
$603$	−44.0770	+	44.0770i	−0.0730961	+	0.0730961i
$604$		0			0
$605$	−195.456	+	195.456i	−0.323068	+	0.323068i
$606$		0			0
$607$			1153.55i			1.90041i	0.311622	+	0.950206i	$0.399128\pi$
$607$			1153.55i			1.90041i	−0.311622	+	0.950206i	$0.600872\pi$
$608$		0			0
$609$	546.129	+	605.391i	0.896763	+	0.994073i
$610$		0			0
$611$	−59.4704	−	59.4704i	−0.0973329	−	0.0973329i
$612$		0			0
$613$	−431.336	+	431.336i	−0.703648	+	0.703648i	−0.965192	−	0.261544i	$-0.915768\pi$
$613$	−431.336	+	431.336i	−0.703648	+	0.703648i	0.261544	+	0.965192i	$0.415768\pi$
$614$		0			0
$615$	481.105			0.782284
$616$		0			0
$617$			307.446i			0.498292i	0.968466	+	0.249146i	$0.0801500\pi$
$617$			307.446i			0.498292i	−0.968466	+	0.249146i	$0.919850\pi$
$618$		0			0
$619$	−713.160	−	713.160i	−1.15212	−	1.15212i	−0.986128	−	0.165988i	$-0.946919\pi$
$619$	−713.160	−	713.160i	−1.15212	−	1.15212i	−0.165988	−	0.986128i	$-0.553081\pi$
$620$		0			0
$621$	140.960	−	140.960i	0.226989	−	0.226989i
$622$		0			0
$623$	−337.133	−	373.717i	−0.541145	−	0.599866i
$624$		0			0
$625$	−239.339			−0.382943
$626$		0			0
$627$	−117.721	−	117.721i	−0.187752	−	0.187752i
$628$		0			0
$629$	576.581	+	576.581i	0.916664	+	0.916664i
$630$		0			0
$631$			605.748i			0.959981i	0.877274	+	0.479991i	$0.159360\pi$
$631$			605.748i			0.959981i	−0.877274	+	0.479991i	$0.840640\pi$
$632$		0			0
$633$	−182.207			−0.287847
$634$		0			0
$635$	391.722	+	391.722i	0.616885	+	0.616885i
$636$		0			0
$637$	222.621	+	273.858i	0.349483	+	0.429919i
$638$		0			0
$639$	106.154			0.166126
$640$		0			0
$641$	320.432			0.499895			0.249947	−	0.968259i	$-0.419587\pi$
$641$	320.432			0.499895			0.249947	+	0.968259i	$0.419587\pi$
$642$		0			0
$643$	192.773	−	192.773i	0.299803	−	0.299803i	−0.541134	−	0.840937i	$-0.682005\pi$
$643$	192.773	−	192.773i	0.299803	−	0.299803i	0.840937	+	0.541134i	$0.182005\pi$
$644$		0			0
$645$	−71.8873	+	71.8873i	−0.111453	+	0.111453i
$646$		0			0
$647$	−684.599			−1.05811			−0.529056	−	0.848587i	$-0.677454\pi$
$647$	−684.599			−1.05811			−0.529056	+	0.848587i	$0.677454\pi$
$648$		0			0
$649$	−40.2207			−0.0619733
$650$		0			0
$651$	24.7363	−	480.652i	0.0379974	−	0.738328i
$652$		0			0
$653$	−54.1559	−	54.1559i	−0.0829340	−	0.0829340i	0.664423	−	0.747357i	$-0.268677\pi$
$653$	−54.1559	−	54.1559i	−0.0829340	−	0.0829340i	−0.747357	+	0.664423i	$0.768677\pi$
$654$		0			0
$655$	−574.531			−0.877147
$656$		0			0
$657$			6.43561i			0.00979544i
$658$		0			0
$659$	−823.988	−	823.988i	−1.25036	−	1.25036i	−0.955558	−	0.294804i	$-0.904746\pi$
$659$	−823.988	−	823.988i	−1.25036	−	1.25036i	−0.294804	−	0.955558i	$-0.595254\pi$
$660$		0			0
$661$	−246.912	−	246.912i	−0.373544	−	0.373544i	0.495223	−	0.868766i	$-0.335086\pi$
$661$	−246.912	−	246.912i	−0.373544	−	0.373544i	−0.868766	+	0.495223i	$0.835086\pi$
$662$		0			0
$663$	−520.743			−0.785435
$664$		0			0
$665$	−304.157	−	337.162i	−0.457379	−	0.507010i
$666$		0			0
$667$	−204.645	+	204.645i	−0.306814	+	0.306814i
$668$		0			0
$669$	−438.072	−	438.072i	−0.654816	−	0.654816i
$670$		0			0
$671$			2.08650i			0.00310954i
$672$		0			0
$673$	−380.314			−0.565102			−0.282551	−	0.959252i	$-0.591181\pi$
$673$	−380.314			−0.565102			−0.282551	+	0.959252i	$0.591181\pi$
$674$		0			0
$675$	351.005	−	351.005i	0.520008	−	0.520008i
$676$		0			0
$677$	−291.402	−	291.402i	−0.430431	−	0.430431i	0.458344	−	0.888775i	$-0.348443\pi$
$677$	−291.402	−	291.402i	−0.430431	−	0.430431i	−0.888775	+	0.458344i	$0.848443\pi$
$678$		0			0
$679$	632.989	−	571.025i	0.932237	−	0.840980i
$680$		0			0
$681$			225.566i			0.331227i
$682$		0			0
$683$	28.3409	−	28.3409i	0.0414947	−	0.0414947i	−0.686055	−	0.727550i	$-0.740659\pi$
$683$	28.3409	−	28.3409i	0.0414947	−	0.0414947i	0.727550	+	0.686055i	$0.240659\pi$
$684$		0			0
$685$	74.2941	−	74.2941i	0.108458	−	0.108458i
$686$		0			0
$687$	1226.94			1.78594
$688$		0			0
$689$			154.388i			0.224075i
$690$		0			0
$691$	42.2644	−	42.2644i	0.0611642	−	0.0611642i	−0.675863	−	0.737027i	$-0.736229\pi$
$691$	42.2644	−	42.2644i	0.0611642	−	0.0611642i	0.737027	+	0.675863i	$0.236229\pi$
$692$		0			0
$693$	−0.608141	+	11.8168i	−0.000877548	+	0.0170516i
$694$		0			0
$695$			326.162i			0.469298i
$696$		0			0
$697$		−	1494.20i		−	2.14376i
$698$		0			0
$699$	924.574	+	924.574i	1.32271	+	1.32271i
$700$		0			0
$701$	−649.896	−	649.896i	−0.927098	−	0.927098i	0.0704192	−	0.997517i	$-0.477566\pi$
$701$	−649.896	−	649.896i	−0.927098	−	0.927098i	−0.997517	+	0.0704192i	$0.977566\pi$
$702$		0			0
$703$			975.763i			1.38800i
$704$		0			0
$705$		−	86.4862i		−	0.122675i
$706$		0			0
$707$	−32.0560	+	622.881i	−0.0453409	+	0.881020i
$708$		0			0
$709$	−142.807	+	142.807i	−0.201421	+	0.201421i	−0.800609	−	0.599188i	$-0.795490\pi$
$709$	−142.807	+	142.807i	−0.201421	+	0.201421i	0.599188	+	0.800609i	$0.295490\pi$
$710$		0			0
$711$			6.69225i			0.00941245i
$712$		0			0
$713$	170.840			0.239607
$714$		0			0
$715$	23.0950	−	23.0950i	0.0323007	−	0.0323007i
$716$		0			0
$717$	−80.0058	+	80.0058i	−0.111584	+	0.111584i
$718$		0			0
$719$			648.272i			0.901630i	0.892617	+	0.450815i	$0.148867\pi$
$719$			648.272i			0.901630i	−0.892617	+	0.450815i	$0.851133\pi$
$720$		0			0
$721$	−817.281	−	905.967i	−1.13354	−	1.25654i
$722$		0			0
$723$	−617.227	−	617.227i	−0.853702	−	0.853702i
$724$		0			0
$725$	−509.587	+	509.587i	−0.702879	+	0.702879i
$726$		0			0
$727$	1370.10			1.88459			0.942295	−	0.334784i	$-0.108663\pi$
$727$	1370.10			1.88459			0.942295	+	0.334784i	$0.108663\pi$
$728$		0			0
$729$			644.635i			0.884273i
$730$		0			0
$731$	223.265	+	223.265i	0.305424	+	0.305424i
$732$		0			0
$733$	525.966	−	525.966i	0.717552	−	0.717552i	−0.250551	−	0.968103i	$-0.580612\pi$
$733$	525.966	−	525.966i	0.717552	−	0.717552i	0.968103	+	0.250551i	$0.0806118\pi$
$734$		0			0
$735$	−37.2566	+	361.008i	−0.0506892	+	0.491167i
$736$		0			0
$737$	−136.547			−0.185274
$738$		0			0
$739$	311.891	+	311.891i	0.422045	+	0.422045i	0.885907	−	0.463862i	$-0.153537\pi$
$739$	311.891	+	311.891i	0.422045	+	0.422045i	−0.463862	+	0.885907i	$0.653537\pi$
$740$		0			0
$741$	−440.633	−	440.633i	−0.594647	−	0.594647i
$742$		0			0
$743$			118.889i			0.160012i	0.996794	+	0.0800059i	$0.0254939\pi$
$743$			118.889i			0.160012i	−0.996794	+	0.0800059i	$0.974506\pi$
$744$		0			0
$745$	−318.367			−0.427339
$746$		0			0
$747$	−84.1945	−	84.1945i	−0.112710	−	0.112710i
$748$		0			0
$749$	465.634	+	23.9634i	0.621674	+	0.0319939i
$750$		0			0
$751$	742.079			0.988121			0.494061	−	0.869427i	$-0.335512\pi$
$751$	742.079			0.988121			0.494061	+	0.869427i	$0.335512\pi$
$752$		0			0
$753$	−787.268			−1.04551
$754$		0			0
$755$	275.508	−	275.508i	0.364911	−	0.364911i
$756$		0			0
$757$	511.624	−	511.624i	0.675858	−	0.675858i	−0.283202	−	0.959060i	$-0.591397\pi$
$757$	511.624	−	511.624i	0.675858	−	0.675858i	0.959060	+	0.283202i	$0.0913968\pi$
$758$		0			0
$759$	−47.2320			−0.0622293
$760$		0			0
$761$	97.6670			0.128340			0.0641702	−	0.997939i	$-0.479560\pi$
$761$	97.6670			0.128340			0.0641702	+	0.997939i	$0.479560\pi$
$762$		0			0
$763$	44.1299	−	857.489i	0.0578374	−	1.12384i
$764$		0			0
$765$	−33.6714	−	33.6714i	−0.0440150	−	0.0440150i
$766$		0			0
$767$	−150.548			−0.196281
$768$		0			0
$769$		−	491.325i		−	0.638914i	−0.947601	−	0.319457i	$-0.896499\pi$
$769$		−	491.325i		−	0.638914i	0.947601	−	0.319457i	$-0.103501\pi$
$770$		0			0
$771$	−643.545	−	643.545i	−0.834688	−	0.834688i
$772$		0			0
$773$	−223.072	−	223.072i	−0.288580	−	0.288580i	0.547939	−	0.836518i	$-0.315413\pi$
$773$	−223.072	−	223.072i	−0.288580	−	0.288580i	−0.836518	+	0.547939i	$0.815413\pi$
$774$		0			0
$775$	425.410			0.548916
$776$		0			0
$777$	579.073	−	522.387i	0.745268	−	0.672313i
$778$		0			0
$779$	1264.33	−	1264.33i	1.62302	−	1.62302i
$780$		0			0
$781$	164.429	+	164.429i	0.210536	+	0.210536i
$782$		0			0
$783$		−	945.959i		−	1.20812i
$784$		0			0
$785$	−563.161			−0.717403
$786$		0			0
$787$	−76.7048	+	76.7048i	−0.0974647	+	0.0974647i	−0.754158	−	0.656693i	$-0.771955\pi$
$787$	−76.7048	+	76.7048i	−0.0974647	+	0.0974647i	0.656693	+	0.754158i	$0.271955\pi$
$788$		0			0
$789$	−480.971	−	480.971i	−0.609596	−	0.609596i
$790$		0			0
$791$	234.288	+	259.711i	0.296192	+	0.328333i
$792$		0			0
$793$			7.80986i			0.00984850i
$794$		0			0
$795$	−112.261	+	112.261i	−0.141209	+	0.141209i
$796$		0			0
$797$	712.807	−	712.807i	0.894363	−	0.894363i	−0.100567	−	0.994930i	$-0.532066\pi$
$797$	712.807	−	712.807i	0.894363	−	0.894363i	0.994930	+	0.100567i	$0.0320657\pi$
$798$		0			0
$799$	−268.606			−0.336177
$800$		0			0
$801$		−	63.1611i		−	0.0788528i
$802$		0			0
$803$	−9.96850	+	9.96850i	−0.0124141	+	0.0124141i
$804$		0			0
$805$	−128.655	−	6.62113i	−0.159820	−	0.00822501i
$806$		0			0
$807$		−	718.754i		−	0.890649i
$808$		0			0
$809$			527.771i			0.652374i	0.945305	+	0.326187i	$0.105764\pi$
$809$			527.771i			0.652374i	−0.945305	+	0.326187i	$0.894236\pi$
$810$		0			0
$811$	−793.267	−	793.267i	−0.978134	−	0.978134i	0.0216320	−	0.999766i	$-0.493114\pi$
$811$	−793.267	−	793.267i	−0.978134	−	0.978134i	−0.999766	+	0.0216320i	$0.993114\pi$
$812$		0			0
$813$	−218.372	−	218.372i	−0.268600	−	0.268600i
$814$		0			0
$815$		−	254.020i		−	0.311682i
$816$		0			0
$817$			377.836i			0.462468i
$818$		0			0
$819$	−2.27630	+	44.2307i	−0.00277936	+	0.0540058i
$820$		0			0
$821$	383.964	−	383.964i	0.467678	−	0.467678i	−0.433484	−	0.901161i	$-0.642716\pi$
$821$	383.964	−	383.964i	0.467678	−	0.467678i	0.901161	+	0.433484i	$0.142716\pi$
$822$		0			0
$823$			213.017i			0.258830i	0.991590	+	0.129415i	$0.0413100\pi$
$823$			213.017i			0.258830i	−0.991590	+	0.129415i	$0.958690\pi$
$824$		0			0
$825$	−117.613			−0.142561
$826$		0			0
$827$	−134.908	+	134.908i	−0.163129	+	0.163129i	−0.783951	−	0.620822i	$-0.786799\pi$
$827$	−134.908	+	134.908i	−0.163129	+	0.163129i	0.620822	+	0.783951i	$0.286799\pi$
$828$		0			0
$829$	824.011	−	824.011i	0.993982	−	0.993982i	−0.00599965	−	0.999982i	$-0.501910\pi$
$829$	824.011	−	824.011i	0.993982	−	0.993982i	0.999982	+	0.00599965i	$0.00190976\pi$
$830$		0			0
$831$		−	1692.32i		−	2.03648i
$832$		0			0
$833$	1121.21	+	115.710i	1.34598	+	0.138908i
$834$		0			0
$835$	−276.898	−	276.898i	−0.331614	−	0.331614i
$836$		0			0
$837$	−394.849	+	394.849i	−0.471744	+	0.471744i
$838$		0			0
$839$	212.466			0.253238			0.126619	−	0.991951i	$-0.459588\pi$
$839$	212.466			0.253238			0.126619	+	0.991951i	$0.459588\pi$
$840$		0			0
$841$			532.336i			0.632980i
$842$		0			0
$843$	588.835	+	588.835i	0.698500	+	0.698500i
$844$		0			0
$845$	−195.165	+	195.165i	−0.230964	+	0.230964i
$846$		0			0
$847$	609.664	−	549.984i	0.719793	−	0.649332i
$848$		0			0
$849$	223.791			0.263594
$850$		0			0
$851$	195.748	+	195.748i	0.230022	+	0.230022i
$852$		0			0
$853$	362.999	+	362.999i	0.425555	+	0.425555i	0.887111	−	0.461556i	$-0.152709\pi$
$853$	362.999	+	362.999i	0.425555	+	0.425555i	−0.461556	+	0.887111i	$0.652709\pi$
$854$		0			0
$855$		−	56.9830i		−	0.0666468i
$856$		0			0
$857$	1437.25			1.67707			0.838533	−	0.544851i	$-0.183414\pi$
$857$	1437.25			1.67707			0.838533	+	0.544851i	$0.183414\pi$
$858$		0			0
$859$	−112.226	−	112.226i	−0.130647	−	0.130647i	0.638759	−	0.769407i	$-0.279448\pi$
$859$	−112.226	−	112.226i	−0.130647	−	0.130647i	−0.769407	+	0.638759i	$0.779448\pi$
$860$		0			0
$861$	−1427.21	−	73.4498i	−1.65761	−	0.0853076i
$862$		0			0
$863$	−528.342			−0.612216			−0.306108	−	0.951997i	$-0.599027\pi$
$863$	−528.342			−0.612216			−0.306108	+	0.951997i	$0.599027\pi$
$864$		0			0
$865$	−424.401			−0.490637
$866$		0			0
$867$	−533.718	+	533.718i	−0.615592	+	0.615592i
$868$		0			0
$869$	−10.3660	+	10.3660i	−0.0119287	+	0.0119287i
$870$		0			0
$871$	−511.101			−0.586798
$872$		0			0
$873$	106.980			0.122543
$874$		0			0
$875$	−732.217	−	37.6829i	−0.836820	−	0.0430662i
$876$		0			0
$877$	575.334	+	575.334i	0.656025	+	0.656025i	0.954437	−	0.298412i	$-0.0964569\pi$
$877$	575.334	+	575.334i	0.656025	+	0.656025i	−0.298412	+	0.954437i	$0.596457\pi$
$878$		0			0
$879$	−1084.64			−1.23395
$880$		0			0
$881$		−	1300.14i		−	1.47575i	−0.674936	−	0.737877i	$-0.735829\pi$
$881$		−	1300.14i		−	1.47575i	0.674936	−	0.737877i	$-0.264171\pi$
$882$		0			0
$883$	197.002	+	197.002i	0.223105	+	0.223105i	0.809805	−	0.586700i	$-0.199573\pi$
$883$	197.002	+	197.002i	0.223105	+	0.223105i	−0.586700	+	0.809805i	$0.699573\pi$
$884$		0			0
$885$	−109.469	−	109.469i	−0.123693	−	0.123693i
$886$		0			0
$887$	1296.24			1.46137			0.730686	−	0.682714i	$-0.239200\pi$
$887$	1296.24			1.46137			0.730686	+	0.682714i	$0.239200\pi$
$888$		0			0
$889$	−1102.24	−	1221.85i	−1.23987	−	1.37441i
$890$		0			0
$891$	119.921	−	119.921i	0.134592	−	0.134592i
$892$		0			0
$893$	−227.284	−	227.284i	−0.254517	−	0.254517i
$894$		0			0
$895$			557.678i			0.623104i
$896$		0			0
$897$	−176.791			−0.197092
$898$		0			0
$899$	573.240	−	573.240i	0.637642	−	0.637642i
$900$		0			0
$901$	348.656	+	348.656i	0.386966	+	0.386966i
$902$		0			0
$903$	224.230	−	202.280i	0.248316	−	0.224009i
$904$		0			0
$905$			37.6310i			0.0415813i
$906$		0			0
$907$	−1233.16	+	1233.16i	−1.35960	+	1.35960i	−0.485189	+	0.874410i	$0.661249\pi$
$907$	−1233.16	+	1233.16i	−1.35960	+	1.35960i	−0.874410	+	0.485189i	$0.838751\pi$
$908$		0			0
$909$	−55.3448	+	55.3448i	−0.0608853	+	0.0608853i
$910$		0			0
$911$	−557.307			−0.611753			−0.305877	−	0.952071i	$-0.598949\pi$
$911$	−557.307			−0.611753			−0.305877	+	0.952071i	$0.598949\pi$
$912$		0			0
$913$		−	260.828i		−	0.285682i
$914$		0			0
$915$	−5.67884	+	5.67884i	−0.00620638	+	0.00620638i
$916$		0			0
$917$	1704.36	+	87.7132i	1.85862	+	0.0956523i
$918$		0			0
$919$		−	1222.59i		−	1.33035i	−0.746690	−	0.665173i	$-0.768358\pi$
$919$		−	1222.59i		−	1.33035i	0.746690	−	0.665173i	$-0.231642\pi$
$920$		0			0
$921$			916.456i			0.995066i
$922$		0			0
$923$	615.464	+	615.464i	0.666808	+	0.666808i
$924$		0			0
$925$	487.434	+	487.434i	0.526956	+	0.526956i
$926$		0			0
$927$		−	153.116i		−	0.165173i
$928$		0			0
$929$			1513.72i			1.62941i	0.579874	+	0.814706i	$0.303102\pi$
$929$			1513.72i			1.62941i	−0.579874	+	0.814706i	$0.696898\pi$
$930$		0			0
$931$	850.812	+	1046.63i	0.913869	+	1.12420i
$932$		0			0
$933$	692.295	−	692.295i	0.742010	−	0.742010i
$934$		0			0
$935$		−	104.311i		−	0.111563i
$936$		0			0
$937$	513.107			0.547606			0.273803	−	0.961786i	$-0.411718\pi$
$937$	513.107			0.547606			0.273803	+	0.961786i	$0.411718\pi$
$938$		0			0
$939$	333.516	−	333.516i	0.355182	−	0.355182i
$940$		0			0
$941$	−509.156	+	509.156i	−0.541080	+	0.541080i	−0.923845	−	0.382766i	$-0.874972\pi$
$941$	−509.156	+	509.156i	−0.541080	+	0.541080i	0.382766	+	0.923845i	$0.374972\pi$
$942$		0			0
$943$		−	507.277i		−	0.537940i
$944$		0			0
$945$	312.654	−	282.048i	0.330851	−	0.298464i
$946$		0			0
$947$	16.1826	+	16.1826i	0.0170883	+	0.0170883i	0.715599	−	0.698511i	$-0.246154\pi$
$947$	16.1826	+	16.1826i	0.0170883	+	0.0170883i	−0.698511	+	0.715599i	$0.746154\pi$
$948$		0			0
$949$	−37.3125	+	37.3125i	−0.0393177	+	0.0393177i
$950$		0			0
$951$	−1135.77			−1.19429
$952$		0			0
$953$		−	57.1945i		−	0.0600152i	−0.999550	−	0.0300076i	$-0.990447\pi$
$953$		−	57.1945i		−	0.0600152i	0.999550	−	0.0300076i	$-0.00955316\pi$
$954$		0			0
$955$	−272.255	−	272.255i	−0.285084	−	0.285084i
$956$		0			0
$957$	−158.483	+	158.483i	−0.165604	+	0.165604i
$958$		0			0
$959$	−231.737	+	209.052i	−0.241644	+	0.217990i
$960$		0			0
$961$	482.452			0.502031
$962$		0			0
$963$	41.3729	+	41.3729i	0.0429625	+	0.0429625i
$964$		0			0
$965$	465.106	+	465.106i	0.481975	+	0.481975i
$966$		0			0
$967$			127.870i			0.132234i	0.997812	+	0.0661169i	$0.0210610\pi$
$967$			127.870i			0.132234i	−0.997812	+	0.0661169i	$0.978939\pi$
$968$		0			0
$969$	−1990.18			−2.05385
$970$		0			0
$971$	521.789	+	521.789i	0.537373	+	0.537373i	0.922757	−	0.385383i	$-0.125931\pi$
$971$	521.789	+	521.789i	0.537373	+	0.537373i	−0.385383	+	0.922757i	$0.625931\pi$
$972$		0			0
$973$	49.7949	−	967.565i	0.0511767	−	0.994415i
$974$		0			0
$975$	−440.230			−0.451518
$976$		0			0
$977$	−448.769			−0.459333			−0.229667	−	0.973269i	$-0.573764\pi$
$977$	−448.769			−0.459333			−0.229667	+	0.973269i	$0.573764\pi$
$978$		0			0
$979$	97.8340	−	97.8340i	0.0999326	−	0.0999326i
$980$		0			0
$981$	76.1904	−	76.1904i	0.0776660	−	0.0776660i
$982$		0			0
$983$	−1.44681			−0.00147183			−0.000735915	−	1.00000i	$-0.500234\pi$
$983$	−1.44681			−0.00147183			−0.000735915		1.00000i	$0.500234\pi$
$984$		0			0
$985$	490.370			0.497838
$986$		0			0
$987$	−13.2038	+	256.563i	−0.0133777	+	0.259942i
$988$		0			0
$989$	75.7980	+	75.7980i	0.0766410	+	0.0766410i
$990$		0			0
$991$	−1468.27			−1.48161			−0.740803	−	0.671722i	$-0.765555\pi$
$991$	−1468.27			−1.48161			−0.740803	+	0.671722i	$0.765555\pi$
$992$		0			0
$993$			924.237i			0.930752i
$994$		0			0
$995$	272.447	+	272.447i	0.273816	+	0.273816i
$996$		0			0
$997$	889.450	+	889.450i	0.892127	+	0.892127i	0.994723	−	0.102597i	$-0.0327151\pi$
$997$	889.450	+	889.450i	0.892127	+	0.892127i	−0.102597	+	0.994723i	$0.532715\pi$
$998$		0			0
$999$	−904.836			−0.905742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.21		56
4.3	odd	2		112.3.l.b.13.27	✓	56
7.6	odd	2	inner	448.3.l.b.209.8		56
16.5	even	4	inner	448.3.l.b.433.8		56
16.11	odd	4		112.3.l.b.69.28	yes	56
28.27	even	2		112.3.l.b.13.28	yes	56
112.27	even	4		112.3.l.b.69.27	yes	56
112.69	odd	4	inner	448.3.l.b.433.21		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.27	✓	56	4.3	odd	2
112.3.l.b.13.28	yes	56	28.27	even	2
112.3.l.b.69.27	yes	56	112.27	even	4
112.3.l.b.69.28	yes	56	16.11	odd	4
448.3.l.b.209.8		56	7.6	odd	2	inner
448.3.l.b.209.21		56	1.1	even	1	trivial
448.3.l.b.433.8		56	16.5	even	4	inner
448.3.l.b.433.21		56	112.69	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.22244 - 2.22244i) q^{3} +(1.66633 + 1.66633i) q^{5} +(-4.68881 - 5.19760i) q^{7} -0.878436i q^{9} +(1.36066 - 1.36066i) q^{11} +(5.09302 - 5.09302i) q^{13} +7.40664 q^{15} -23.0033i q^{17} +(19.4645 - 19.4645i) q^{19} +(-21.9719 - 1.13076i) q^{21} -7.80957i q^{23} -19.4467i q^{25} +(18.0496 + 18.0496i) q^{27} +(-26.2044 - 26.2044i) q^{29} +21.8757i q^{31} -6.04797i q^{33} +(0.847823 - 16.4741i) q^{35} +(-25.0652 + 25.0652i) q^{37} -22.6378i q^{39} +64.9559 q^{41} +(-9.70579 + 9.70579i) q^{43} +(1.46377 - 1.46377i) q^{45} -11.6768i q^{47} +(-5.03016 + 48.7411i) q^{49} +(-51.1233 - 51.1233i) q^{51} +(-15.1568 + 15.1568i) q^{53} +4.53464 q^{55} -86.5172i q^{57} +(-14.7798 - 14.7798i) q^{59} +(-0.766723 + 0.766723i) q^{61} +(-4.56576 + 4.11882i) q^{63} +16.9733 q^{65} +(-50.1767 - 50.1767i) q^{67} +(-17.3563 - 17.3563i) q^{69} +120.845i q^{71} -7.32621 q^{73} +(-43.2189 - 43.2189i) q^{75} +(-13.4521 - 0.692299i) q^{77} -7.61837 q^{79} +88.1343 q^{81} +(95.8459 - 95.8459i) q^{83} +(38.3311 - 38.3311i) q^{85} -116.475 q^{87} +71.9017 q^{89} +(-50.3517 - 2.59130i) q^{91} +(48.6174 + 48.6174i) q^{93} +64.8687 q^{95} +121.785i q^{97} +(-1.19526 - 1.19526i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

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