LMFDB - Embedded newform 448.3.l.b.209.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(209,448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			209.22
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.28266 - 2.28266i) q^{3} +(-2.90674 - 2.90674i) q^{5} +(-2.26451 + 6.62359i) q^{7} -1.42104i q^{9} +O(q^{10})$	\(q+(2.28266 - 2.28266i) q^{3} +(-2.90674 - 2.90674i) q^{5} +(-2.26451 + 6.62359i) q^{7} -1.42104i q^{9} +(-9.96314 + 9.96314i) q^{11} +(-15.5346 + 15.5346i) q^{13} -13.2702 q^{15} -32.2316i q^{17} +(-14.4296 + 14.4296i) q^{19} +(9.95030 + 20.2885i) q^{21} +16.0348i q^{23} -8.10171i q^{25} +(17.3002 + 17.3002i) q^{27} +(-10.0884 - 10.0884i) q^{29} +3.11658i q^{31} +45.4848i q^{33} +(25.8354 - 12.6707i) q^{35} +(-36.8766 + 36.8766i) q^{37} +70.9201i q^{39} +39.5924 q^{41} +(19.5881 - 19.5881i) q^{43} +(-4.13060 + 4.13060i) q^{45} -20.9684i q^{47} +(-38.7440 - 29.9984i) q^{49} +(-73.5736 - 73.5736i) q^{51} +(-18.6330 + 18.6330i) q^{53} +57.9205 q^{55} +65.8758i q^{57} +(21.7777 + 21.7777i) q^{59} +(23.5695 - 23.5695i) q^{61} +(9.41240 + 3.21796i) q^{63} +90.3098 q^{65} +(-14.0012 - 14.0012i) q^{67} +(36.6019 + 36.6019i) q^{69} +2.39288i q^{71} -22.2579 q^{73} +(-18.4934 - 18.4934i) q^{75} +(-43.4302 - 88.5534i) q^{77} -67.0638 q^{79} +91.7700 q^{81} +(-82.2869 + 82.2869i) q^{83} +(-93.6888 + 93.6888i) q^{85} -46.0569 q^{87} -103.252 q^{89} +(-67.7165 - 138.073i) q^{91} +(7.11407 + 7.11407i) q^{93} +83.8864 q^{95} +18.8468i q^{97} +(14.1580 + 14.1580i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * 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} + 32 q^{93} + 776 q^{95} - 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

Display 100 coefficients 10 coefficients

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
$3$	2.28266	−	2.28266i	0.760885	−	0.760885i	−0.215597	−	0.976482i	$-0.569170\pi$
$3$	2.28266	−	2.28266i	0.760885	−	0.760885i	0.976482	+	0.215597i	$0.0691697\pi$
$4$		0			0
$5$	−2.90674	−	2.90674i	−0.581348	−	0.581348i	0.353925	−	0.935274i	$-0.384847\pi$
$5$	−2.90674	−	2.90674i	−0.581348	−	0.581348i	−0.935274	+	0.353925i	$0.884847\pi$
$6$		0			0
$7$	−2.26451	+	6.62359i	−0.323501	+	0.946228i
$7$	−2.26451	+	6.62359i	−0.323501	+	0.946228i
$8$		0			0
$9$		−	1.42104i		−	0.157893i
$10$		0			0
$11$	−9.96314	+	9.96314i	−0.905740	+	0.905740i	−0.995925	−	0.0901854i	$-0.971254\pi$
$11$	−9.96314	+	9.96314i	−0.905740	+	0.905740i	0.0901854	+	0.995925i	$0.471254\pi$
$12$		0			0
$13$	−15.5346	+	15.5346i	−1.19497	+	1.19497i	−0.219310	+	0.975655i	$0.570381\pi$
$13$	−15.5346	+	15.5346i	−1.19497	+	1.19497i	−0.975655	+	0.219310i	$0.929619\pi$
$14$		0			0
$15$	−13.2702			−0.884679
$16$		0			0
$17$		−	32.2316i		−	1.89597i	−0.318309	−	0.947987i	$-0.603115\pi$
$17$		−	32.2316i		−	1.89597i	0.318309	−	0.947987i	$-0.396885\pi$
$18$		0			0
$19$	−14.4296	+	14.4296i	−0.759454	+	0.759454i	−0.976223	−	0.216769i	$-0.930448\pi$
$19$	−14.4296	+	14.4296i	−0.759454	+	0.759454i	0.216769	+	0.976223i	$0.430448\pi$
$20$		0			0
$21$	9.95030	+	20.2885i	0.473824	+	0.966118i
$22$		0			0
$23$			16.0348i			0.697165i	0.937278	+	0.348583i	$0.113337\pi$
$23$			16.0348i			0.697165i	−0.937278	+	0.348583i	$0.886663\pi$
$24$		0			0
$25$		−	8.10171i		−	0.324068i
$26$		0			0
$27$	17.3002	+	17.3002i	0.640747	+	0.640747i
$28$		0			0
$29$	−10.0884	−	10.0884i	−0.347877	−	0.347877i	0.511441	−	0.859318i	$-0.329112\pi$
$29$	−10.0884	−	10.0884i	−0.347877	−	0.347877i	−0.859318	+	0.511441i	$0.829112\pi$
$30$		0			0
$31$			3.11658i			0.100535i	0.998736	+	0.0502674i	$0.0160073\pi$
$31$			3.11658i			0.100535i	−0.998736	+	0.0502674i	$0.983993\pi$
$32$		0			0
$33$			45.4848i			1.37833i
$34$		0			0
$35$	25.8354	−	12.6707i	0.738155	−	0.362021i
$36$		0			0
$37$	−36.8766	+	36.8766i	−0.996664	+	0.996664i	−0.999994	−	0.00333020i	$-0.998940\pi$
$37$	−36.8766	+	36.8766i	−0.996664	+	0.996664i	0.00333020	+	0.999994i	$0.498940\pi$
$38$		0			0
$39$			70.9201i			1.81846i
$40$		0			0
$41$	39.5924			0.965668			0.482834	−	0.875712i	$-0.339607\pi$
$41$	39.5924			0.965668			0.482834	+	0.875712i	$0.339607\pi$
$42$		0			0
$43$	19.5881	−	19.5881i	0.455536	−	0.455536i	−0.441651	−	0.897187i	$-0.645607\pi$
$43$	19.5881	−	19.5881i	0.455536	−	0.455536i	0.897187	+	0.441651i	$0.145607\pi$
$44$		0			0
$45$	−4.13060	+	4.13060i	−0.0917910	+	0.0917910i
$46$		0			0
$47$		−	20.9684i		−	0.446136i	−0.974803	−	0.223068i	$-0.928393\pi$
$47$		−	20.9684i		−	0.446136i	0.974803	−	0.223068i	$-0.0716072\pi$
$48$		0			0
$49$	−38.7440	−	29.9984i	−0.790694	−	0.612211i
$50$		0			0
$51$	−73.5736	−	73.5736i	−1.44262	−	1.44262i
$52$		0			0
$53$	−18.6330	+	18.6330i	−0.351567	+	0.351567i	−0.860692	−	0.509126i	$-0.829969\pi$
$53$	−18.6330	+	18.6330i	−0.351567	+	0.351567i	0.509126	+	0.860692i	$0.329969\pi$
$54$		0			0
$55$	57.9205			1.05310
$56$		0			0
$57$			65.8758i			1.15572i
$58$		0			0
$59$	21.7777	+	21.7777i	0.369113	+	0.369113i	0.867154	−	0.498040i	$-0.165947\pi$
$59$	21.7777	+	21.7777i	0.369113	+	0.369113i	−0.498040	+	0.867154i	$0.665947\pi$
$60$		0			0
$61$	23.5695	−	23.5695i	0.386386	−	0.386386i	−0.487010	−	0.873396i	$-0.661913\pi$
$61$	23.5695	−	23.5695i	0.386386	−	0.386386i	0.873396	+	0.487010i	$0.161913\pi$
$62$		0			0
$63$	9.41240	+	3.21796i	0.149403	+	0.0510787i
$64$		0			0
$65$	90.3098			1.38938
$66$		0			0
$67$	−14.0012	−	14.0012i	−0.208974	−	0.208974i	0.594858	−	0.803831i	$-0.297208\pi$
$67$	−14.0012	−	14.0012i	−0.208974	−	0.208974i	−0.803831	+	0.594858i	$0.797208\pi$
$68$		0			0
$69$	36.6019	+	36.6019i	0.530463	+	0.530463i
$70$		0			0
$71$			2.39288i			0.0337026i	0.999858	+	0.0168513i	$0.00536418\pi$
$71$			2.39288i			0.0337026i	−0.999858	+	0.0168513i	$0.994636\pi$
$72$		0			0
$73$	−22.2579			−0.304903			−0.152451	−	0.988311i	$-0.548717\pi$
$73$	−22.2579			−0.304903			−0.152451	+	0.988311i	$0.548717\pi$
$74$		0			0
$75$	−18.4934	−	18.4934i	−0.246579	−	0.246579i
$76$		0			0
$77$	−43.4302	−	88.5534i	−0.564028	−	1.15004i
$78$		0			0
$79$	−67.0638			−0.848908			−0.424454	−	0.905449i	$-0.639534\pi$
$79$	−67.0638			−0.848908			−0.424454	+	0.905449i	$0.639534\pi$
$80$		0			0
$81$	91.7700			1.13296
$82$		0			0
$83$	−82.2869	+	82.2869i	−0.991409	+	0.991409i	−0.999963	−	0.00855454i	$-0.997277\pi$
$83$	−82.2869	+	82.2869i	−0.991409	+	0.991409i	0.00855454	+	0.999963i	$0.497277\pi$
$84$		0			0
$85$	−93.6888	+	93.6888i	−1.10222	+	1.10222i
$86$		0			0
$87$	−46.0569			−0.529390
$88$		0			0
$89$	−103.252			−1.16014			−0.580069	−	0.814567i	$-0.696975\pi$
$89$	−103.252			−1.16014			−0.580069	+	0.814567i	$0.696975\pi$
$90$		0			0
$91$	−67.7165	−	138.073i	−0.744137	−	1.51728i
$92$		0			0
$93$	7.11407	+	7.11407i	0.0764954	+	0.0764954i
$94$		0			0
$95$	83.8864			0.883015
$96$		0			0
$97$			18.8468i			0.194297i	0.995270	+	0.0971486i	$0.0309722\pi$
$97$			18.8468i			0.194297i	−0.995270	+	0.0971486i	$0.969028\pi$
$98$		0			0
$99$	14.1580	+	14.1580i	0.143010	+	0.143010i
$100$		0			0
$101$	−17.2491	−	17.2491i	−0.170783	−	0.170783i	0.616540	−	0.787323i	$-0.288534\pi$
$101$	−17.2491	−	17.2491i	−0.170783	−	0.170783i	−0.787323	+	0.616540i	$0.788534\pi$
$102$		0			0
$103$	−52.8655			−0.513258			−0.256629	−	0.966510i	$-0.582612\pi$
$103$	−52.8655			−0.513258			−0.256629	+	0.966510i	$0.582612\pi$
$104$		0			0
$105$	30.0504	−	87.8963i	0.286195	−	0.837108i
$106$		0			0
$107$	−14.7578	+	14.7578i	−0.137923	+	0.137923i	−0.772697	−	0.634774i	$-0.781093\pi$
$107$	−14.7578	+	14.7578i	−0.137923	+	0.137923i	0.634774	+	0.772697i	$0.281093\pi$
$108$		0			0
$109$	16.7521	+	16.7521i	0.153689	+	0.153689i	0.779763	−	0.626075i	$-0.215339\pi$
$109$	16.7521	+	16.7521i	0.153689	+	0.153689i	−0.626075	+	0.779763i	$0.715339\pi$
$110$		0			0
$111$			168.353i			1.51669i
$112$		0			0
$113$	113.623			1.00552			0.502758	−	0.864427i	$-0.332319\pi$
$113$	113.623			1.00552			0.502758	+	0.864427i	$0.332319\pi$
$114$		0			0
$115$	46.6090	−	46.6090i	0.405296	−	0.405296i
$116$		0			0
$117$	22.0752	+	22.0752i	0.188677	+	0.188677i
$118$		0			0
$119$	213.489	+	72.9886i	1.79402	+	0.613350i
$120$		0			0
$121$		−	77.5281i		−	0.640728i
$122$		0			0
$123$	90.3758	−	90.3758i	0.734763	−	0.734763i
$124$		0			0
$125$	−96.2181	+	96.2181i	−0.769745	+	0.769745i
$126$		0			0
$127$	174.458			1.37369			0.686843	−	0.726805i	$-0.258996\pi$
$127$	174.458			1.37369			0.686843	+	0.726805i	$0.258996\pi$
$128$		0			0
$129$		−	89.4256i		−	0.693222i
$130$		0			0
$131$	28.5604	−	28.5604i	0.218018	−	0.218018i	−0.589644	−	0.807663i	$-0.700732\pi$
$131$	28.5604	−	28.5604i	0.218018	−	0.218018i	0.807663	+	0.589644i	$0.200732\pi$
$132$		0			0
$133$	−62.9000	−	128.252i	−0.472932	−	0.964301i
$134$		0			0
$135$		−	100.574i		−	0.744994i
$136$		0			0
$137$			113.638i			0.829475i	0.909941	+	0.414737i	$0.136127\pi$
$137$			113.638i			0.829475i	−0.909941	+	0.414737i	$0.863873\pi$
$138$		0			0
$139$	−103.663	−	103.663i	−0.745780	−	0.745780i	0.227904	−	0.973684i	$-0.426813\pi$
$139$	−103.663	−	103.663i	−0.745780	−	0.745780i	−0.973684	+	0.227904i	$0.926813\pi$
$140$		0			0
$141$	−47.8636	−	47.8636i	−0.339458	−	0.339458i
$142$		0			0
$143$		−	309.546i		−	2.16466i
$144$		0			0
$145$			58.6490i			0.404476i
$146$		0			0
$147$	−156.915	+	19.9633i	−1.06745	+	0.135805i
$148$		0			0
$149$	145.889	−	145.889i	0.979123	−	0.979123i	−0.0206638	−	0.999786i	$-0.506578\pi$
$149$	145.889	−	145.889i	0.979123	−	0.979123i	0.999786	+	0.0206638i	$0.00657795\pi$
$150$		0			0
$151$			138.380i			0.916422i	0.888844	+	0.458211i	$0.151510\pi$
$151$			138.380i			0.916422i	−0.888844	+	0.458211i	$0.848490\pi$
$152$		0			0
$153$	−45.8023			−0.299362
$154$		0			0
$155$	9.05908	−	9.05908i	0.0584457	−	0.0584457i
$156$		0			0
$157$	−3.59938	+	3.59938i	−0.0229260	+	0.0229260i	−0.718477	−	0.695551i	$-0.755160\pi$
$157$	−3.59938	+	3.59938i	−0.0229260	+	0.0229260i	0.695551	+	0.718477i	$0.255160\pi$
$158$		0			0
$159$			85.0656i			0.535004i
$160$		0			0
$161$	−106.208	−	36.3109i	−0.659677	−	0.225534i
$162$		0			0
$163$	−87.4226	−	87.4226i	−0.536335	−	0.536335i	0.386116	−	0.922450i	$-0.373817\pi$
$163$	−87.4226	−	87.4226i	−0.536335	−	0.536335i	−0.922450	+	0.386116i	$0.873817\pi$
$164$		0			0
$165$	132.213	−	132.213i	0.801289	−	0.801289i
$166$		0			0
$167$	−29.9561			−0.179378			−0.0896889	−	0.995970i	$-0.528587\pi$
$167$	−29.9561			−0.179378			−0.0896889	+	0.995970i	$0.528587\pi$
$168$		0			0
$169$		−	313.645i		−	1.85589i
$170$		0			0
$171$	20.5051	+	20.5051i	0.119913	+	0.119913i
$172$		0			0
$173$	191.004	−	191.004i	1.10407	−	1.10407i	0.110158	−	0.993914i	$-0.464864\pi$
$173$	191.004	−	191.004i	1.10407	−	1.10407i	0.993914	−	0.110158i	$-0.0351356\pi$
$174$		0			0
$175$	53.6625	+	18.3464i	0.306643	+	0.104837i
$176$		0			0
$177$	99.4219			0.561706
$178$		0			0
$179$	114.979	+	114.979i	0.642340	+	0.642340i	0.951130	−	0.308790i	$-0.0999240\pi$
$179$	114.979	+	114.979i	0.642340	+	0.642340i	−0.308790	+	0.951130i	$0.599924\pi$
$180$		0			0
$181$	71.2709	+	71.2709i	0.393762	+	0.393762i	0.876026	−	0.482264i	$-0.160185\pi$
$181$	71.2709	+	71.2709i	0.393762	+	0.393762i	−0.482264	+	0.876026i	$0.660185\pi$
$182$		0			0
$183$		−	107.602i		−	0.587991i
$184$		0			0
$185$	214.381			1.15882
$186$		0			0
$187$	321.127	+	321.127i	1.71726	+	1.71726i
$188$		0			0
$189$	−153.766	+	75.4129i	−0.813575	+	0.399010i
$190$		0			0
$191$	−300.815			−1.57495			−0.787474	−	0.616347i	$-0.788612\pi$
$191$	−300.815			−1.57495			−0.787474	+	0.616347i	$0.788612\pi$
$192$		0			0
$193$	−35.0743			−0.181732			−0.0908660	−	0.995863i	$-0.528963\pi$
$193$	−35.0743			−0.181732			−0.0908660	+	0.995863i	$0.528963\pi$
$194$		0			0
$195$	206.146	−	206.146i	1.05716	−	1.05716i
$196$		0			0
$197$	−39.3549	+	39.3549i	−0.199771	+	0.199771i	−0.799902	−	0.600131i	$-0.795115\pi$
$197$	−39.3549	+	39.3549i	−0.199771	+	0.199771i	0.600131	+	0.799902i	$0.295115\pi$
$198$		0			0
$199$	−123.196			−0.619077			−0.309539	−	0.950887i	$-0.600175\pi$
$199$	−123.196			−0.619077			−0.309539	+	0.950887i	$0.600175\pi$
$200$		0			0
$201$	−63.9200			−0.318010
$202$		0			0
$203$	89.6671	−	43.9764i	0.441710	−	0.216633i
$204$		0			0
$205$	−115.085	−	115.085i	−0.561389	−	0.561389i
$206$		0			0
$207$	22.7861			0.110078
$208$		0			0
$209$		−	287.529i		−	1.37574i
$210$		0			0
$211$	270.885	+	270.885i	1.28381	+	1.28381i	0.938479	+	0.345335i	$0.112235\pi$
$211$	270.885	+	270.885i	1.28381	+	1.28381i	0.345335	+	0.938479i	$0.387765\pi$
$212$		0			0
$213$	5.46213	+	5.46213i	0.0256438	+	0.0256438i
$214$		0			0
$215$	−113.875			−0.529651
$216$		0			0
$217$	−20.6429	−	7.05751i	−0.0951287	−	0.0325231i
$218$		0			0
$219$	−50.8071	+	50.8071i	−0.231996	+	0.231996i
$220$		0			0
$221$	500.703	+	500.703i	2.26562	+	2.26562i
$222$		0			0
$223$		−	152.745i		−	0.684956i	−0.939526	−	0.342478i	$-0.888734\pi$
$223$		−	152.745i		−	0.684956i	0.939526	−	0.342478i	$-0.111266\pi$
$224$		0			0
$225$	−11.5129			−0.0511683
$226$		0			0
$227$	−123.736	+	123.736i	−0.545094	+	0.545094i	−0.925018	−	0.379924i	$-0.875950\pi$
$227$	−123.736	+	123.736i	−0.545094	+	0.545094i	0.379924	+	0.925018i	$0.375950\pi$
$228$		0			0
$229$	−111.719	−	111.719i	−0.487855	−	0.487855i	0.419774	−	0.907629i	$-0.362109\pi$
$229$	−111.719	−	111.719i	−0.487855	−	0.487855i	−0.907629	+	0.419774i	$0.862109\pi$
$230$		0			0
$231$	−301.273	−	103.001i	−1.30421	−	0.445891i
$232$		0			0
$233$			68.8807i			0.295625i	0.989015	+	0.147813i	$0.0472232\pi$
$233$			68.8807i			0.295625i	−0.989015	+	0.147813i	$0.952777\pi$
$234$		0			0
$235$	−60.9497	+	60.9497i	−0.259360	+	0.259360i
$236$		0			0
$237$	−153.084	+	153.084i	−0.645922	+	0.645922i
$238$		0			0
$239$	32.0400			0.134059			0.0670293	−	0.997751i	$-0.478648\pi$
$239$	32.0400			0.134059			0.0670293	+	0.997751i	$0.478648\pi$
$240$		0			0
$241$			288.635i			1.19765i	0.800878	+	0.598827i	$0.204366\pi$
$241$			288.635i			1.19765i	−0.800878	+	0.598827i	$0.795634\pi$
$242$		0			0
$243$	53.7779	−	53.7779i	0.221308	−	0.221308i
$244$		0			0
$245$	25.4213	+	199.816i	0.103761	+	0.815577i
$246$		0			0
$247$		−	448.316i		−	1.81504i
$248$		0			0
$249$			375.666i			1.50870i
$250$		0			0
$251$	−18.3592	−	18.3592i	−0.0731441	−	0.0731441i	0.669588	−	0.742732i	$-0.266471\pi$
$251$	−18.3592	−	18.3592i	−0.0731441	−	0.0731441i	−0.742732	+	0.669588i	$0.766471\pi$
$252$		0			0
$253$	−159.757	−	159.757i	−0.631450	−	0.631450i
$254$		0			0
$255$			427.719i			1.67733i
$256$		0			0
$257$			334.673i			1.30223i	0.758980	+	0.651114i	$0.225698\pi$
$257$			334.673i			1.30223i	−0.758980	+	0.651114i	$0.774302\pi$
$258$		0			0
$259$	−160.748	−	327.763i	−0.620649	−	1.26549i
$260$		0			0
$261$	−14.3361	+	14.3361i	−0.0549275	+	0.0549275i
$262$		0			0
$263$			157.697i			0.599607i	0.954001	+	0.299803i	$0.0969210\pi$
$263$			157.697i			0.599607i	−0.954001	+	0.299803i	$0.903079\pi$
$264$		0			0
$265$	108.323			0.408765
$266$		0			0
$267$	−235.690	+	235.690i	−0.882732	+	0.882732i
$268$		0			0
$269$	−36.7111	+	36.7111i	−0.136473	+	0.136473i	−0.772043	−	0.635570i	$-0.780765\pi$
$269$	−36.7111	+	36.7111i	−0.136473	+	0.136473i	0.635570	+	0.772043i	$0.280765\pi$
$270$		0			0
$271$		−	520.390i		−	1.92026i	−0.279558	−	0.960129i	$-0.590188\pi$
$271$		−	520.390i		−	1.92026i	0.279558	−	0.960129i	$-0.409812\pi$
$272$		0			0
$273$	−469.746	−	160.599i	−1.72068	−	0.588275i
$274$		0			0
$275$	80.7185	+	80.7185i	0.293522	+	0.293522i
$276$		0			0
$277$	27.7640	−	27.7640i	0.100231	−	0.100231i	−0.655213	−	0.755444i	$-0.727421\pi$
$277$	27.7640	−	27.7640i	0.100231	−	0.100231i	0.755444	+	0.655213i	$0.227421\pi$
$278$		0			0
$279$	4.42878			0.0158738
$280$		0			0
$281$			383.880i			1.36612i	0.730361	+	0.683061i	$0.239352\pi$
$281$			383.880i			1.36612i	−0.730361	+	0.683061i	$0.760648\pi$
$282$		0			0
$283$	41.4810	+	41.4810i	0.146576	+	0.146576i	0.776587	−	0.630011i	$-0.216949\pi$
$283$	41.4810	+	41.4810i	0.146576	+	0.146576i	−0.630011	+	0.776587i	$0.716949\pi$
$284$		0			0
$285$	191.484	−	191.484i	0.671873	−	0.671873i
$286$		0			0
$287$	−89.6572	+	262.244i	−0.312395	+	0.913742i
$288$		0			0
$289$	−749.873			−2.59472
$290$		0			0
$291$	43.0208	+	43.0208i	0.147838	+	0.147838i
$292$		0			0
$293$	−213.203	−	213.203i	−0.727655	−	0.727655i	0.242497	−	0.970152i	$-0.422034\pi$
$293$	−213.203	−	213.203i	−0.727655	−	0.727655i	−0.970152	+	0.242497i	$0.922034\pi$
$294$		0			0
$295$		−	126.604i		−	0.429167i
$296$		0			0
$297$	−344.728			−1.16070
$298$		0			0
$299$	−249.093	−	249.093i	−0.833089	−	0.833089i
$300$		0			0
$301$	85.3861	+	174.101i	0.283675	+	0.578408i
$302$		0			0
$303$	−78.7473			−0.259892
$304$		0			0
$305$	−137.021			−0.449250
$306$		0			0
$307$	−326.956	+	326.956i	−1.06500	+	1.06500i	−0.0672699	+	0.997735i	$0.521429\pi$
$307$	−326.956	+	326.956i	−1.06500	+	1.06500i	−0.997735	+	0.0672699i	$0.978571\pi$
$308$		0			0
$309$	−120.674	+	120.674i	−0.390530	+	0.390530i
$310$		0			0
$311$	221.386			0.711851			0.355926	−	0.934514i	$-0.384166\pi$
$311$	221.386			0.711851			0.355926	+	0.934514i	$0.384166\pi$
$312$		0			0
$313$	195.538			0.624722			0.312361	−	0.949963i	$-0.398880\pi$
$313$	195.538			0.624722			0.312361	+	0.949963i	$0.398880\pi$
$314$		0			0
$315$	−18.0056	−	36.7132i	−0.0571607	−	0.116550i
$316$		0			0
$317$	32.4658	+	32.4658i	0.102416	+	0.102416i	0.756458	−	0.654042i	$-0.226928\pi$
$317$	32.4658	+	32.4658i	0.102416	+	0.102416i	−0.654042	+	0.756458i	$0.726928\pi$
$318$		0			0
$319$	201.025			0.630173
$320$		0			0
$321$			67.3738i			0.209887i
$322$		0			0
$323$	465.089	+	465.089i	1.43991	+	1.43991i
$324$		0			0
$325$	125.856	+	125.856i	0.387251	+	0.387251i
$326$		0			0
$327$	76.4784			0.233879
$328$		0			0
$329$	138.886	+	47.4831i	0.422146	+	0.144325i
$330$		0			0
$331$	−198.058	+	198.058i	−0.598364	+	0.598364i	−0.939877	−	0.341513i	$-0.889061\pi$
$331$	−198.058	+	198.058i	−0.598364	+	0.598364i	0.341513	+	0.939877i	$0.389061\pi$
$332$		0			0
$333$	52.4031	+	52.4031i	0.157367	+	0.157367i
$334$		0			0
$335$			81.3959i			0.242973i
$336$		0			0
$337$	306.746			0.910225			0.455112	−	0.890434i	$-0.349599\pi$
$337$	306.746			0.910225			0.455112	+	0.890434i	$0.349599\pi$
$338$		0			0
$339$	259.363	−	259.363i	0.765083	−	0.765083i
$340$		0			0
$341$	−31.0509	−	31.0509i	−0.0910583	−	0.0910583i
$342$		0			0
$343$	286.433	−	188.693i	0.835082	−	0.550126i
$344$		0			0
$345$		−	212.785i		−	0.616767i
$346$		0			0
$347$	−91.7162	+	91.7162i	−0.264312	+	0.264312i	−0.826803	−	0.562491i	$-0.809843\pi$
$347$	−91.7162	+	91.7162i	−0.264312	+	0.264312i	0.562491	+	0.826803i	$0.309843\pi$
$348$		0			0
$349$	395.328	−	395.328i	1.13274	−	1.13274i	0.143025	−	0.989719i	$-0.454317\pi$
$349$	395.328	−	395.328i	1.13274	−	1.13274i	0.989719	−	0.143025i	$-0.0456828\pi$
$350$		0			0
$351$	−537.500			−1.53134
$352$		0			0
$353$			164.022i			0.464650i	0.972638	+	0.232325i	$0.0746333\pi$
$353$			164.022i			0.464650i	−0.972638	+	0.232325i	$0.925367\pi$
$354$		0			0
$355$	6.95549	−	6.95549i	0.0195929	−	0.0195929i
$356$		0			0
$357$	653.929	−	320.714i	1.83173	−	0.898357i
$358$		0			0
$359$			93.5104i			0.260475i	0.991483	+	0.130237i	$0.0415739\pi$
$359$			93.5104i			0.260475i	−0.991483	+	0.130237i	$0.958426\pi$
$360$		0			0
$361$		−	55.4285i		−	0.153541i
$362$		0			0
$363$	−176.970	−	176.970i	−0.487521	−	0.487521i
$364$		0			0
$365$	64.6979	+	64.6979i	0.177255	+	0.177255i
$366$		0			0
$367$			412.826i			1.12487i	0.826843	+	0.562433i	$0.190134\pi$
$367$			412.826i			1.12487i	−0.826843	+	0.562433i	$0.809866\pi$
$368$		0			0
$369$		−	56.2624i		−	0.152473i
$370$		0			0
$371$	−81.2230	−	165.612i	−0.218930	−	0.446394i
$372$		0			0
$373$	−340.213	+	340.213i	−0.912099	+	0.912099i	−0.996437	−	0.0843379i	$-0.973122\pi$
$373$	−340.213	+	340.213i	−0.912099	+	0.912099i	0.0843379	+	0.996437i	$0.473122\pi$
$374$		0			0
$375$			439.266i			1.17138i
$376$		0			0
$377$	313.439			0.831403
$378$		0			0
$379$	−450.626	+	450.626i	−1.18899	+	1.18899i	−0.211640	+	0.977348i	$0.567881\pi$
$379$	−450.626	+	450.626i	−1.18899	+	1.18899i	−0.977348	+	0.211640i	$0.932119\pi$
$380$		0			0
$381$	398.228	−	398.228i	1.04522	−	1.04522i
$382$		0			0
$383$			374.094i			0.976745i	0.872635	+	0.488373i	$0.162409\pi$
$383$			374.094i			0.976745i	−0.872635	+	0.488373i	$0.837591\pi$
$384$		0			0
$385$	−131.161	+	383.642i	−0.340679	+	0.996473i
$386$		0			0
$387$	−27.8354	−	27.8354i	−0.0719262	−	0.0719262i
$388$		0			0
$389$	295.277	−	295.277i	0.759067	−	0.759067i	−0.217085	−	0.976153i	$-0.569655\pi$
$389$	295.277	−	295.277i	0.759067	−	0.759067i	0.976153	+	0.217085i	$0.0696550\pi$
$390$		0			0
$391$	516.827			1.32181
$392$		0			0
$393$		−	130.387i		−	0.331774i
$394$		0			0
$395$	194.937	+	194.937i	0.493511	+	0.493511i
$396$		0			0
$397$	−531.600	+	531.600i	−1.33904	+	1.33904i	−0.442053	+	0.896989i	$0.645750\pi$
$397$	−531.600	+	531.600i	−1.33904	+	1.33904i	−0.896989	+	0.442053i	$0.854250\pi$
$398$		0			0
$399$	−436.334	−	149.176i	−1.09357	−	0.373875i
$400$		0			0
$401$	671.156			1.67371			0.836853	−	0.547428i	$-0.184393\pi$
$401$	671.156			1.67371			0.836853	+	0.547428i	$0.184393\pi$
$402$		0			0
$403$	−48.4146	−	48.4146i	−0.120136	−	0.120136i
$404$		0			0
$405$	−266.752	−	266.752i	−0.658646	−	0.658646i
$406$		0			0
$407$		−	734.813i		−	1.80544i
$408$		0			0
$409$	−379.027			−0.926717			−0.463358	−	0.886171i	$-0.653356\pi$
$409$	−379.027			−0.926717			−0.463358	+	0.886171i	$0.653356\pi$
$410$		0			0
$411$	259.397	+	259.397i	0.631135	+	0.631135i
$412$		0			0
$413$	−193.562	+	94.9308i	−0.468674	+	0.229857i
$414$		0			0
$415$	478.374			1.15271
$416$		0			0
$417$	−473.256			−1.13491
$418$		0			0
$419$	426.017	−	426.017i	1.01675	−	1.01675i	0.0168902	−	0.999857i	$-0.494623\pi$
$419$	426.017	−	426.017i	1.01675	−	1.01675i	0.999857	−	0.0168902i	$-0.00537657\pi$
$420$		0			0
$421$	−8.05175	+	8.05175i	−0.0191253	+	0.0191253i	−0.716605	−	0.697479i	$-0.754305\pi$
$421$	−8.05175	+	8.05175i	−0.0191253	+	0.0191253i	0.697479	+	0.716605i	$0.254305\pi$
$422$		0			0
$423$	−29.7969			−0.0704419
$424$		0			0
$425$	−261.131			−0.614425
$426$		0			0
$427$	102.742	+	209.489i	0.240613	+	0.490605i
$428$		0			0
$429$	−706.586	−	706.586i	−1.64705	−	1.64705i
$430$		0			0
$431$	389.916			0.904677			0.452338	−	0.891846i	$-0.350590\pi$
$431$	389.916			0.904677			0.452338	+	0.891846i	$0.350590\pi$
$432$		0			0
$433$			158.384i			0.365782i	0.983133	+	0.182891i	$0.0585456\pi$
$433$			158.384i			0.365782i	−0.983133	+	0.182891i	$0.941454\pi$
$434$		0			0
$435$	133.875	+	133.875i	0.307760	+	0.307760i
$436$		0			0
$437$	−231.376	−	231.376i	−0.529465	−	0.529465i
$438$		0			0
$439$	−764.403			−1.74124			−0.870618	−	0.491960i	$-0.836281\pi$
$439$	−764.403			−1.74124			−0.870618	+	0.491960i	$0.836281\pi$
$440$		0			0
$441$	−42.6289	+	55.0568i	−0.0966641	+	0.124845i
$442$		0			0
$443$	119.551	−	119.551i	0.269867	−	0.269867i	−0.559180	−	0.829047i	$-0.688884\pi$
$443$	119.551	−	119.551i	0.269867	−	0.269867i	0.829047	+	0.559180i	$0.188884\pi$
$444$		0			0
$445$	300.128	+	300.128i	0.674444	+	0.674444i
$446$		0			0
$447$		−	666.030i		−	1.49000i
$448$		0			0
$449$	652.600			1.45345			0.726726	−	0.686927i	$-0.241041\pi$
$449$	652.600			1.45345			0.726726	+	0.686927i	$0.241041\pi$
$450$		0			0
$451$	−394.464	+	394.464i	−0.874643	+	0.874643i
$452$		0			0
$453$	315.873	+	315.873i	0.697292	+	0.697292i
$454$		0			0
$455$	−204.507	+	598.176i	−0.449467	+	1.31467i
$456$		0			0
$457$		−	719.580i		−	1.57457i	−0.616588	−	0.787286i	$-0.711485\pi$
$457$		−	719.580i		−	1.57457i	0.616588	−	0.787286i	$-0.288515\pi$
$458$		0			0
$459$	557.611	−	557.611i	1.21484	−	1.21484i
$460$		0			0
$461$	323.632	−	323.632i	0.702022	−	0.702022i	−0.262822	−	0.964844i	$-0.584653\pi$
$461$	323.632	−	323.632i	0.702022	−	0.702022i	0.964844	+	0.262822i	$0.0846532\pi$
$462$		0			0
$463$	65.5977			0.141680			0.0708399	−	0.997488i	$-0.477432\pi$
$463$	65.5977			0.141680			0.0708399	+	0.997488i	$0.477432\pi$
$464$		0			0
$465$		−	41.3575i		−	0.0889409i
$466$		0			0
$467$	11.7847	−	11.7847i	0.0252349	−	0.0252349i	−0.694377	−	0.719612i	$-0.744320\pi$
$467$	11.7847	−	11.7847i	0.0252349	−	0.0252349i	0.719612	+	0.694377i	$0.244320\pi$
$468$		0			0
$469$	124.444	−	61.0326i	0.265340	−	0.130133i
$470$		0			0
$471$			16.4323i			0.0348881i
$472$		0			0
$473$			390.317i			0.825195i
$474$		0			0
$475$	116.905	+	116.905i	0.246115	+	0.246115i
$476$		0			0
$477$	26.4783	+	26.4783i	0.0555101	+	0.0555101i
$478$		0			0
$479$		−	475.079i		−	0.991814i	−0.868376	−	0.495907i	$-0.834836\pi$
$479$		−	475.079i		−	0.991814i	0.868376	−	0.495907i	$-0.165164\pi$
$480$		0			0
$481$		−	1145.72i		−	2.38196i
$482$		0			0
$483$	−325.322	+	159.551i	−0.673544	+	0.330333i
$484$		0			0
$485$	54.7829	−	54.7829i	0.112954	−	0.112954i
$486$		0			0
$487$			524.310i			1.07661i	0.842750	+	0.538306i	$0.180935\pi$
$487$			524.310i			1.07661i	−0.842750	+	0.538306i	$0.819065\pi$
$488$		0			0
$489$	−399.111			−0.816179
$490$		0			0
$491$	335.136	−	335.136i	0.682558	−	0.682558i	−0.278018	−	0.960576i	$-0.589677\pi$
$491$	335.136	−	335.136i	0.682558	−	0.682558i	0.960576	+	0.278018i	$0.0896774\pi$
$492$		0			0
$493$	−325.166	+	325.166i	−0.659566	+	0.659566i
$494$		0			0
$495$		−	82.3074i		−	0.166278i
$496$		0			0
$497$	−15.8495	−	5.41870i	−0.0318903	−	0.0109028i
$498$		0			0
$499$	−56.5874	−	56.5874i	−0.113402	−	0.113402i	0.648129	−	0.761531i	$-0.275552\pi$
$499$	−56.5874	−	56.5874i	−0.113402	−	0.113402i	−0.761531	+	0.648129i	$0.775552\pi$
$500$		0			0
$501$	−68.3795	+	68.3795i	−0.136486	+	0.136486i
$502$		0			0
$503$	−458.739			−0.912006			−0.456003	−	0.889978i	$-0.650719\pi$
$503$	−458.739			−0.912006			−0.456003	+	0.889978i	$0.650719\pi$
$504$		0			0
$505$			100.277i			0.198568i
$506$		0			0
$507$	−715.943	−	715.943i	−1.41212	−	1.41212i
$508$		0			0
$509$	−154.189	+	154.189i	−0.302926	+	0.302926i	−0.842157	−	0.539232i	$-0.818715\pi$
$509$	−154.189	+	154.189i	−0.302926	+	0.302926i	0.539232	+	0.842157i	$0.318715\pi$
$510$		0			0
$511$	50.4032	−	147.427i	0.0986363	−	0.288507i
$512$		0			0
$513$	−499.270			−0.973236
$514$		0			0
$515$	153.666	+	153.666i	0.298381	+	0.298381i
$516$		0			0
$517$	208.911	+	208.911i	0.404083	+	0.404083i
$518$		0			0
$519$		−	871.995i		−	1.68014i
$520$		0			0
$521$	328.373			0.630274			0.315137	−	0.949046i	$-0.397950\pi$
$521$	328.373			0.630274			0.315137	+	0.949046i	$0.397950\pi$
$522$		0			0
$523$	−285.071	−	285.071i	−0.545068	−	0.545068i	0.379942	−	0.925010i	$-0.375944\pi$
$523$	−285.071	−	285.071i	−0.545068	−	0.545068i	−0.925010	+	0.379942i	$0.875944\pi$
$524$		0			0
$525$	164.371	−	80.6144i	0.313088	−	0.153551i
$526$		0			0
$527$	100.452			0.190611
$528$		0			0
$529$	271.885			0.513960
$530$		0			0
$531$	30.9470	−	30.9470i	0.0582805	−	0.0582805i
$532$		0			0
$533$	−615.050	+	615.050i	−1.15394	+	1.15394i
$534$		0			0
$535$	85.7940			0.160363
$536$		0			0
$537$	524.915			0.977495
$538$		0			0
$539$	684.890	−	87.1341i	1.27067	−	0.161659i
$540$		0			0
$541$	434.228	+	434.228i	0.802640	+	0.802640i	0.983507	−	0.180868i	$-0.0578906\pi$
$541$	434.228	+	434.228i	0.802640	+	0.802640i	−0.180868	+	0.983507i	$0.557891\pi$
$542$		0			0
$543$	325.374			0.599215
$544$		0			0
$545$		−	97.3878i		−	0.178693i
$546$		0			0
$547$	−693.945	−	693.945i	−1.26864	−	1.26864i	−0.946792	−	0.321846i	$-0.895697\pi$
$547$	−693.945	−	693.945i	−1.26864	−	1.26864i	−0.321846	−	0.946792i	$-0.604303\pi$
$548$		0			0
$549$	−33.4933	−	33.4933i	−0.0610078	−	0.0610078i
$550$		0			0
$551$	291.145			0.528394
$552$		0			0
$553$	151.866	−	444.203i	0.274623	−	0.803261i
$554$		0			0
$555$	489.359	−	489.359i	0.881728	−	0.881728i
$556$		0			0
$557$	−257.121	−	257.121i	−0.461618	−	0.461618i	0.437568	−	0.899186i	$-0.355840\pi$
$557$	−257.121	−	257.121i	−0.461618	−	0.461618i	−0.899186	+	0.437568i	$0.855840\pi$
$558$		0			0
$559$			608.584i			1.08870i
$560$		0			0
$561$	1466.05			2.61327
$562$		0			0
$563$	−62.8920	+	62.8920i	−0.111709	+	0.111709i	−0.760752	−	0.649043i	$-0.775170\pi$
$563$	−62.8920	+	62.8920i	−0.111709	+	0.111709i	0.649043	+	0.760752i	$0.275170\pi$
$564$		0			0
$565$	−330.274	−	330.274i	−0.584555	−	0.584555i
$566$		0			0
$567$	−207.814	+	607.847i	−0.366515	+	1.07204i
$568$		0			0
$569$			333.590i			0.586274i	0.956070	+	0.293137i	$0.0946992\pi$
$569$			333.590i			0.586274i	−0.956070	+	0.293137i	$0.905301\pi$
$570$		0			0
$571$	221.941	−	221.941i	0.388688	−	0.388688i	−0.485531	−	0.874219i	$-0.661374\pi$
$571$	221.941	−	221.941i	0.388688	−	0.388688i	0.874219	+	0.485531i	$0.161374\pi$
$572$		0			0
$573$	−686.658	+	686.658i	−1.19836	+	1.19836i
$574$		0			0
$575$	129.909			0.225929
$576$		0			0
$577$		−	73.1448i		−	0.126767i	−0.997989	−	0.0633837i	$-0.979811\pi$
$577$		−	73.1448i		−	0.126767i	0.997989	−	0.0633837i	$-0.0201892\pi$
$578$		0			0
$579$	−80.0625	+	80.0625i	−0.138277	+	0.138277i
$580$		0			0
$581$	−358.696	−	731.375i	−0.617377	−	1.25882i
$582$		0			0
$583$		−	371.287i		−	0.636856i
$584$		0			0
$585$		−	128.334i		−	0.219374i
$586$		0			0
$587$	339.824	+	339.824i	0.578917	+	0.578917i	0.934605	−	0.355688i	$-0.115753\pi$
$587$	339.824	+	339.824i	0.578917	+	0.578917i	−0.355688	+	0.934605i	$0.615753\pi$
$588$		0			0
$589$	−44.9710	−	44.9710i	−0.0763515	−	0.0763515i
$590$		0			0
$591$			179.667i			0.304006i
$592$		0			0
$593$			96.6120i			0.162921i	0.996677	+	0.0814604i	$0.0259584\pi$
$593$			96.6120i			0.162921i	−0.996677	+	0.0814604i	$0.974042\pi$
$594$		0			0
$595$	−408.398	−	832.716i	−0.686382	−	1.39952i
$596$		0			0
$597$	−281.215	+	281.215i	−0.471047	+	0.471047i
$598$		0			0
$599$			754.853i			1.26019i	0.776519	+	0.630094i	$0.216984\pi$
$599$			754.853i			1.26019i	−0.776519	+	0.630094i	$0.783016\pi$
$600$		0			0
$601$	−678.470			−1.12890			−0.564451	−	0.825467i	$-0.690912\pi$
$601$	−678.470			−1.12890			−0.564451	+	0.825467i	$0.690912\pi$
$602$		0			0
$603$	−19.8963	+	19.8963i	−0.0329955	+	0.0329955i
$604$		0			0
$605$	−225.354	+	225.354i	−0.372486	+	0.372486i
$606$		0			0
$607$		−	10.2945i		−	0.0169596i	−0.999964	−	0.00847980i	$-0.997301\pi$
$607$		−	10.2945i		−	0.0169596i	0.999964	−	0.00847980i	$-0.00269924\pi$
$608$		0			0
$609$	104.296	−	305.062i	0.171258	−	0.500923i
$610$		0			0
$611$	325.734	+	325.734i	0.533117	+	0.533117i
$612$		0			0
$613$	442.295	−	442.295i	0.721525	−	0.721525i	−0.247390	−	0.968916i	$-0.579573\pi$
$613$	442.295	−	442.295i	0.721525	−	0.721525i	0.968916	+	0.247390i	$0.0795730\pi$
$614$		0			0
$615$	−525.398			−0.854306
$616$		0			0
$617$		−	531.027i		−	0.860660i	−0.902672	−	0.430330i	$-0.858397\pi$
$617$		−	531.027i		−	0.860660i	0.902672	−	0.430330i	$-0.141603\pi$
$618$		0			0
$619$	207.632	+	207.632i	0.335432	+	0.335432i	0.854645	−	0.519213i	$-0.173775\pi$
$619$	207.632	+	207.632i	0.335432	+	0.335432i	−0.519213	+	0.854645i	$0.673775\pi$
$620$		0			0
$621$	−277.405	+	277.405i	−0.446706	+	0.446706i
$622$		0			0
$623$	233.816	−	683.901i	0.375306	−	1.09776i
$624$		0			0
$625$	356.819			0.570911
$626$		0			0
$627$	−656.329	−	656.329i	−1.04678	−	1.04678i
$628$		0			0
$629$	1188.59	+	1188.59i	1.88965	+	1.88965i
$630$		0			0
$631$		−	25.0278i		−	0.0396637i	−0.999803	−	0.0198319i	$-0.993687\pi$
$631$		−	25.0278i		−	0.0396637i	0.999803	−	0.0198319i	$-0.00631309\pi$
$632$		0			0
$633$	1236.67			1.95367
$634$		0			0
$635$	−507.105	−	507.105i	−0.798590	−	0.798590i
$636$		0			0
$637$	1067.88	−	135.860i	1.67642	−	0.213281i
$638$		0			0
$639$	3.40038			0.00532141
$640$		0			0
$641$	−702.170			−1.09543			−0.547715	−	0.836665i	$-0.684502\pi$
$641$	−702.170			−1.09543			−0.547715	+	0.836665i	$0.684502\pi$
$642$		0			0
$643$	−356.339	+	356.339i	−0.554182	+	0.554182i	−0.927645	−	0.373463i	$-0.878170\pi$
$643$	−356.339	+	356.339i	−0.554182	+	0.554182i	0.373463	+	0.927645i	$0.378170\pi$
$644$		0			0
$645$	−259.937	+	259.937i	−0.403003	+	0.403003i
$646$		0			0
$647$	663.547			1.02557			0.512787	−	0.858516i	$-0.328613\pi$
$647$	663.547			1.02557			0.512787	+	0.858516i	$0.328613\pi$
$648$		0			0
$649$	−433.948			−0.668641
$650$		0			0
$651$	−63.2306	+	31.0109i	−0.0971284	+	0.0476357i
$652$		0			0
$653$	−749.238	−	749.238i	−1.14738	−	1.14738i	−0.987066	−	0.160311i	$-0.948750\pi$
$653$	−749.238	−	749.238i	−1.14738	−	1.14738i	−0.160311	−	0.987066i	$-0.551250\pi$
$654$		0			0
$655$	−166.035			−0.253489
$656$		0			0
$657$			31.6294i			0.0481421i
$658$		0			0
$659$	183.052	+	183.052i	0.277773	+	0.277773i	0.832219	−	0.554446i	$-0.187070\pi$
$659$	183.052	+	183.052i	0.277773	+	0.277773i	−0.554446	+	0.832219i	$0.687070\pi$
$660$		0			0
$661$	488.925	+	488.925i	0.739674	+	0.739674i	0.972515	−	0.232841i	$-0.0748021\pi$
$661$	488.925	+	488.925i	0.739674	+	0.739674i	−0.232841	+	0.972515i	$0.574802\pi$
$662$		0			0
$663$	2285.86			3.44776
$664$		0			0
$665$	−189.961	+	555.629i	−0.285656	+	0.835533i
$666$		0			0
$667$	161.766	−	161.766i	0.242528	−	0.242528i
$668$		0			0
$669$	−348.665	−	348.665i	−0.521173	−	0.521173i
$670$		0			0
$671$			469.653i			0.699930i
$672$		0			0
$673$	722.532			1.07360			0.536800	−	0.843710i	$-0.319633\pi$
$673$	722.532			1.07360			0.536800	+	0.843710i	$0.319633\pi$
$674$		0			0
$675$	140.161	−	140.161i	0.207646	−	0.207646i
$676$		0			0
$677$	85.6013	+	85.6013i	0.126442	+	0.126442i	0.767496	−	0.641054i	$-0.221502\pi$
$677$	85.6013	+	85.6013i	0.126442	+	0.126442i	−0.641054	+	0.767496i	$0.721502\pi$
$678$		0			0
$679$	−124.834	−	42.6788i	−0.183849	−	0.0628554i
$680$		0			0
$681$			564.895i			0.829508i
$682$		0			0
$683$	−118.436	+	118.436i	−0.173406	+	0.173406i	−0.788474	−	0.615068i	$-0.789129\pi$
$683$	−118.436	+	118.436i	−0.173406	+	0.173406i	0.615068	+	0.788474i	$0.289129\pi$
$684$		0			0
$685$	330.316	−	330.316i	0.482214	−	0.482214i
$686$		0			0
$687$	−510.031			−0.742403
$688$		0			0
$689$		−	578.912i		−	0.840220i
$690$		0			0
$691$	−412.838	+	412.838i	−0.597450	+	0.597450i	−0.939633	−	0.342183i	$-0.888834\pi$
$691$	−412.838	+	412.838i	−0.597450	+	0.597450i	0.342183	+	0.939633i	$0.388834\pi$
$692$		0			0
$693$	−125.838	+	61.7160i	−0.181584	+	0.0890563i
$694$		0			0
$695$			602.645i			0.867115i
$696$		0			0
$697$		−	1276.12i		−	1.83088i
$698$		0			0
$699$	157.231	+	157.231i	0.224937	+	0.224937i
$700$		0			0
$701$	−264.725	−	264.725i	−0.377639	−	0.377639i	0.492611	−	0.870250i	$-0.336043\pi$
$701$	−264.725	−	264.725i	−0.377639	−	0.377639i	−0.870250	+	0.492611i	$0.836043\pi$
$702$		0			0
$703$		−	1064.23i		−	1.51384i
$704$		0			0
$705$			278.254i			0.394687i
$706$		0			0
$707$	153.311	−	75.1901i	0.216848	−	0.106351i
$708$		0			0
$709$	−416.810	+	416.810i	−0.587885	+	0.587885i	−0.937058	−	0.349173i	$-0.886462\pi$
$709$	−416.810	+	416.810i	−0.587885	+	0.587885i	0.349173	+	0.937058i	$0.386462\pi$
$710$		0			0
$711$			95.3003i			0.134037i
$712$		0			0
$713$	−49.9737			−0.0700893
$714$		0			0
$715$	−899.769	+	899.769i	−1.25842	+	1.25842i
$716$		0			0
$717$	73.1363	−	73.1363i	0.102003	−	0.102003i
$718$		0			0
$719$			1253.20i			1.74298i	0.490412	+	0.871491i	$0.336846\pi$
$719$			1253.20i			1.74298i	−0.490412	+	0.871491i	$0.663154\pi$
$720$		0			0
$721$	119.714	−	350.160i	0.166039	−	0.485659i
$722$		0			0
$723$	658.854	+	658.854i	0.911278	+	0.911278i
$724$		0			0
$725$	−81.7337	+	81.7337i	−0.112736	+	0.112736i
$726$		0			0
$727$	−673.130			−0.925901			−0.462950	−	0.886384i	$-0.653209\pi$
$727$	−673.130			−0.925901			−0.462950	+	0.886384i	$0.653209\pi$
$728$		0			0
$729$			580.417i			0.796182i
$730$		0			0
$731$	−631.354	−	631.354i	−0.863685	−	0.863685i
$732$		0			0
$733$	495.647	−	495.647i	0.676189	−	0.676189i	−0.282946	−	0.959136i	$-0.591312\pi$
$733$	495.647	−	495.647i	0.676189	−	0.676189i	0.959136	+	0.282946i	$0.0913119\pi$
$734$		0			0
$735$	514.140	+	398.084i	0.699510	+	0.541611i
$736$		0			0
$737$	278.992			0.378551
$738$		0			0
$739$	−730.041	−	730.041i	−0.987876	−	0.987876i	0.0120510	−	0.999927i	$-0.496164\pi$
$739$	−730.041	−	730.041i	−0.987876	−	0.987876i	−0.999927	+	0.0120510i	$0.996164\pi$
$740$		0			0
$741$	−1023.35	−	1023.35i	−1.38104	−	1.38104i
$742$		0			0
$743$			362.785i			0.488271i	0.969741	+	0.244135i	$0.0785041\pi$
$743$			362.785i			0.488271i	−0.969741	+	0.244135i	$0.921496\pi$
$744$		0			0
$745$	−848.125			−1.13842
$746$		0			0
$747$	116.933	+	116.933i	0.156537	+	0.156537i
$748$		0			0
$749$	−64.3304	−	131.168i	−0.0858883	−	0.175125i
$750$		0			0
$751$	352.201			0.468976			0.234488	−	0.972119i	$-0.424659\pi$
$751$	352.201			0.468976			0.234488	+	0.972119i	$0.424659\pi$
$752$		0			0
$753$	−83.8153			−0.111308
$754$		0			0
$755$	402.234	−	402.234i	0.532760	−	0.532760i
$756$		0			0
$757$	−1010.20	+	1010.20i	−1.33447	+	1.33447i	−0.433151	+	0.901321i	$0.642598\pi$
$757$	−1010.20	+	1010.20i	−1.33447	+	1.33447i	−0.901321	+	0.433151i	$0.857402\pi$
$758$		0			0
$759$	−729.340			−0.960923
$760$		0			0
$761$	1065.18			1.39971			0.699855	−	0.714285i	$-0.253248\pi$
$761$	1065.18			1.39971			0.699855	+	0.714285i	$0.253248\pi$
$762$		0			0
$763$	−148.894	+	73.0237i	−0.195143	+	0.0957060i
$764$		0			0
$765$	133.136	+	133.136i	0.174033	+	0.174033i
$766$		0			0
$767$	−676.613			−0.882155
$768$		0			0
$769$			1415.77i			1.84105i	0.390680	+	0.920527i	$0.372240\pi$
$769$			1415.77i			1.84105i	−0.390680	+	0.920527i	$0.627760\pi$
$770$		0			0
$771$	763.943	+	763.943i	0.990847	+	0.990847i
$772$		0			0
$773$	−363.001	−	363.001i	−0.469600	−	0.469600i	0.432185	−	0.901785i	$-0.357743\pi$
$773$	−363.001	−	363.001i	−0.469600	−	0.469600i	−0.901785	+	0.432185i	$0.857743\pi$
$774$		0			0
$775$	25.2496			0.0325801
$776$		0			0
$777$	−1115.10	−	381.237i	−1.43514	−	0.490652i
$778$		0			0
$779$	−571.303	+	571.303i	−0.733380	+	0.733380i
$780$		0			0
$781$	−23.8406	−	23.8406i	−0.0305257	−	0.0305257i
$782$		0			0
$783$		−	349.063i		−	0.445802i
$784$		0			0
$785$	20.9249			0.0266560
$786$		0			0
$787$	464.183	−	464.183i	0.589814	−	0.589814i	−0.347767	−	0.937581i	$-0.613060\pi$
$787$	464.183	−	464.183i	0.589814	−	0.589814i	0.937581	+	0.347767i	$0.113060\pi$
$788$		0			0
$789$	359.967	+	359.967i	0.456232	+	0.456232i
$790$		0			0
$791$	−257.301	+	752.595i	−0.325286	+	0.951448i
$792$		0			0
$793$			732.285i			0.923436i
$794$		0			0
$795$	247.264	−	247.264i	0.311024	−	0.311024i
$796$		0			0
$797$	252.417	−	252.417i	0.316708	−	0.316708i	−0.530793	−	0.847501i	$-0.678106\pi$
$797$	252.417	−	252.417i	0.316708	−	0.316708i	0.847501	+	0.530793i	$0.178106\pi$
$798$		0			0
$799$	−675.844			−0.845862
$800$		0			0
$801$			146.726i			0.183178i
$802$		0			0
$803$	221.758	−	221.758i	0.276162	−	0.276162i
$804$		0			0
$805$	203.173	+	414.266i	0.252389	+	0.514616i
$806$		0			0
$807$			167.598i			0.207680i
$808$		0			0
$809$			332.184i			0.410611i	0.978698	+	0.205306i	$0.0658188\pi$
$809$			332.184i			0.410611i	−0.978698	+	0.205306i	$0.934181\pi$
$810$		0			0
$811$	566.768	+	566.768i	0.698851	+	0.698851i	0.964163	−	0.265311i	$-0.0854748\pi$
$811$	566.768	+	566.768i	0.698851	+	0.698851i	−0.265311	+	0.964163i	$0.585475\pi$
$812$		0			0
$813$	−1187.87	−	1187.87i	−1.46110	−	1.46110i
$814$		0			0
$815$			508.230i			0.623595i
$816$		0			0
$817$			565.297i			0.691918i
$818$		0			0
$819$	−196.207	+	96.2278i	−0.239569	+	0.117494i
$820$		0			0
$821$	380.824	−	380.824i	0.463854	−	0.463854i	−0.436063	−	0.899916i	$-0.643627\pi$
$821$	380.824	−	380.824i	0.463854	−	0.463854i	0.899916	+	0.436063i	$0.143627\pi$
$822$		0			0
$823$			275.483i			0.334730i	0.985895	+	0.167365i	$0.0535259\pi$
$823$			275.483i			0.334730i	−0.985895	+	0.167365i	$0.946474\pi$
$824$		0			0
$825$	368.505			0.446673
$826$		0			0
$827$	13.5616	−	13.5616i	0.0163986	−	0.0163986i	−0.698860	−	0.715259i	$-0.746309\pi$
$827$	13.5616	−	13.5616i	0.0163986	−	0.0163986i	0.715259	+	0.698860i	$0.246309\pi$
$828$		0			0
$829$	260.673	−	260.673i	0.314443	−	0.314443i	−0.532185	−	0.846628i	$-0.678629\pi$
$829$	260.673	−	260.673i	0.314443	−	0.314443i	0.846628	+	0.532185i	$0.178629\pi$
$830$		0			0
$831$		−	126.751i		−	0.152528i
$832$		0			0
$833$	−966.894	+	1248.78i	−1.16074	+	1.49914i
$834$		0			0
$835$	87.0746	+	87.0746i	0.104281	+	0.104281i
$836$		0			0
$837$	−53.9173	+	53.9173i	−0.0644173	+	0.0644173i
$838$		0			0
$839$	795.591			0.948261			0.474130	−	0.880455i	$-0.342763\pi$
$839$	795.591			0.948261			0.474130	+	0.880455i	$0.342763\pi$
$840$		0			0
$841$		−	637.447i		−	0.757963i
$842$		0			0
$843$	876.267	+	876.267i	1.03946	+	1.03946i
$844$		0			0
$845$	−911.684	+	911.684i	−1.07892	+	1.07892i
$846$		0			0
$847$	513.515	+	175.563i	0.606275	+	0.207276i
$848$		0			0
$849$	189.374			0.223055
$850$		0			0
$851$	−591.309	−	591.309i	−0.694840	−	0.694840i
$852$		0			0
$853$	−397.199	−	397.199i	−0.465650	−	0.465650i	0.434852	−	0.900502i	$-0.356801\pi$
$853$	−397.199	−	397.199i	−0.465650	−	0.465650i	−0.900502	+	0.434852i	$0.856801\pi$
$854$		0			0
$855$		−	119.206i		−	0.139422i
$856$		0			0
$857$	−1328.76			−1.55048			−0.775239	−	0.631668i	$-0.782371\pi$
$857$	−1328.76			−1.55048			−0.775239	+	0.631668i	$0.782371\pi$
$858$		0			0
$859$	−889.815	−	889.815i	−1.03587	−	1.03587i	−0.999332	−	0.0365413i	$-0.988366\pi$
$859$	−889.815	−	889.815i	−1.03587	−	1.03587i	−0.0365413	−	0.999332i	$-0.511634\pi$
$860$		0			0
$861$	393.956	+	803.269i	0.457556	+	0.932949i
$862$		0			0
$863$	−951.400			−1.10243			−0.551216	−	0.834362i	$-0.685836\pi$
$863$	−951.400			−1.10243			−0.551216	+	0.834362i	$0.685836\pi$
$864$		0			0
$865$	−1110.40			−1.28370
$866$		0			0
$867$	−1711.70	+	1711.70i	−1.97428	+	1.97428i
$868$		0			0
$869$	668.165	−	668.165i	0.768890	−	0.768890i
$870$		0			0
$871$	435.006			0.499432
$872$		0			0
$873$	26.7821			0.0306782
$874$		0			0
$875$	−419.423	−	855.196i	−0.479341	−	0.977367i
$876$		0			0
$877$	1165.54	+	1165.54i	1.32901	+	1.32901i	0.906239	+	0.422767i	$0.138941\pi$
$877$	1165.54	+	1165.54i	1.32901	+	1.32901i	0.422767	+	0.906239i	$0.361059\pi$
$878$		0			0
$879$	−973.338			−1.10732
$880$		0			0
$881$			107.367i			0.121870i	0.998142	+	0.0609349i	$0.0194082\pi$
$881$			107.367i			0.121870i	−0.998142	+	0.0609349i	$0.980592\pi$
$882$		0			0
$883$	473.801	+	473.801i	0.536581	+	0.536581i	0.922523	−	0.385942i	$-0.126123\pi$
$883$	473.801	+	473.801i	0.536581	+	0.536581i	−0.385942	+	0.922523i	$0.626123\pi$
$884$		0			0
$885$	−288.994	−	288.994i	−0.326547	−	0.326547i
$886$		0			0
$887$	−410.161			−0.462414			−0.231207	−	0.972905i	$-0.574267\pi$
$887$	−410.161			−0.462414			−0.231207	+	0.972905i	$0.574267\pi$
$888$		0			0
$889$	−395.062	+	1155.54i	−0.444389	+	1.29982i
$890$		0			0
$891$	−914.317	+	914.317i	−1.02617	+	1.02617i
$892$		0			0
$893$	302.566	+	302.566i	0.338820	+	0.338820i
$894$		0			0
$895$		−	668.428i		−	0.746847i
$896$		0			0
$897$	−1137.19			−1.26777
$898$		0			0
$899$	31.4414	−	31.4414i	0.0349737	−	0.0349737i
$900$		0			0
$901$	600.572	+	600.572i	0.666561	+	0.666561i
$902$		0			0
$903$	592.319	+	202.505i	0.655946	+	0.224258i
$904$		0			0
$905$		−	414.332i		−	0.457826i
$906$		0			0
$907$	−725.905	+	725.905i	−0.800337	+	0.800337i	−0.983148	−	0.182811i	$-0.941480\pi$
$907$	−725.905	+	725.905i	−0.800337	+	0.800337i	0.182811	+	0.983148i	$0.441480\pi$
$908$		0			0
$909$	−24.5116	+	24.5116i	−0.0269655	+	0.0269655i
$910$		0			0
$911$	1068.94			1.17337			0.586684	−	0.809816i	$-0.300433\pi$
$911$	1068.94			1.17337			0.586684	+	0.809816i	$0.300433\pi$
$912$		0			0
$913$		−	1639.67i		−	1.79592i
$914$		0			0
$915$	−312.772	+	312.772i	−0.341828	+	0.341828i
$916$		0			0
$917$	124.497	+	253.848i	0.135766	+	0.276824i
$918$		0			0
$919$		−	166.789i		−	0.181489i	−0.995874	−	0.0907447i	$-0.971075\pi$
$919$		−	166.789i		−	0.181489i	0.995874	−	0.0907447i	$-0.0289247\pi$
$920$		0			0
$921$			1492.66i			1.62069i
$922$		0			0
$923$	−37.1723	−	37.1723i	−0.0402734	−	0.0402734i
$924$		0			0
$925$	298.763	+	298.763i	0.322987	+	0.322987i
$926$		0			0
$927$			75.1241i			0.0810400i
$928$		0			0
$929$		−	1609.29i		−	1.73228i	−0.499798	−	0.866142i	$-0.666593\pi$
$929$		−	1609.29i		−	1.73228i	0.499798	−	0.866142i	$-0.333407\pi$
$930$		0			0
$931$	991.927	−	126.196i	1.06544	−	0.135549i
$932$		0			0
$933$	505.348	−	505.348i	0.541637	−	0.541637i
$934$		0			0
$935$		−	1866.87i		−	1.99665i
$936$		0			0
$937$	1185.02			1.26470			0.632348	−	0.774684i	$-0.282091\pi$
$937$	1185.02			1.26470			0.632348	+	0.774684i	$0.282091\pi$
$938$		0			0
$939$	446.346	−	446.346i	0.475342	−	0.475342i
$940$		0			0
$941$	−8.79618	+	8.79618i	−0.00934770	+	0.00934770i	−0.711765	−	0.702417i	$-0.752104\pi$
$941$	−8.79618	+	8.79618i	−0.00934770	+	0.00934770i	0.702417	+	0.711765i	$0.252104\pi$
$942$		0			0
$943$			634.856i			0.673230i
$944$		0			0
$945$	666.163	+	227.751i	0.704934	+	0.241006i
$946$		0			0
$947$	−670.227	−	670.227i	−0.707737	−	0.707737i	0.258322	−	0.966059i	$-0.416831\pi$
$947$	−670.227	−	670.227i	−0.707737	−	0.707737i	−0.966059	+	0.258322i	$0.916831\pi$
$948$		0			0
$949$	345.766	−	345.766i	0.364348	−	0.364348i
$950$		0			0
$951$	148.217			0.155853
$952$		0			0
$953$		−	777.795i		−	0.816154i	−0.912947	−	0.408077i	$-0.866199\pi$
$953$		−	777.795i		−	0.816154i	0.912947	−	0.408077i	$-0.133801\pi$
$954$		0			0
$955$	874.392	+	874.392i	0.915594	+	0.915594i
$956$		0			0
$957$	458.871	−	458.871i	0.479489	−	0.479489i
$958$		0			0
$959$	−752.692	−	257.334i	−0.784872	−	0.268336i
$960$		0			0
$961$	951.287			0.989893
$962$		0			0
$963$	20.9714	+	20.9714i	0.0217771	+	0.0217771i
$964$		0			0
$965$	101.952	+	101.952i	0.105650	+	0.105650i
$966$		0			0
$967$		−	980.179i		−	1.01363i	−0.862055	−	0.506815i	$-0.830823\pi$
$967$		−	980.179i		−	1.01363i	0.862055	−	0.506815i	$-0.169177\pi$
$968$		0			0
$969$	2123.28			2.19121
$970$		0			0
$971$	631.690	+	631.690i	0.650556	+	0.650556i	0.953127	−	0.302571i	$-0.0978449\pi$
$971$	631.690	+	631.690i	0.650556	+	0.650556i	−0.302571	+	0.953127i	$0.597845\pi$
$972$		0			0
$973$	921.370	−	451.878i	0.946938	−	0.464417i
$974$		0			0
$975$	574.574			0.589307
$976$		0			0
$977$	−1150.81			−1.17790			−0.588949	−	0.808170i	$-0.700458\pi$
$977$	−1150.81			−1.17790			−0.588949	+	0.808170i	$0.700458\pi$
$978$		0			0
$979$	1028.72	−	1028.72i	1.05078	−	1.05078i
$980$		0			0
$981$	23.8053	−	23.8053i	0.0242664	−	0.0242664i
$982$		0			0
$983$	−1733.07			−1.76304			−0.881520	−	0.472147i	$-0.843479\pi$
$983$	−1733.07			−1.76304			−0.881520	+	0.472147i	$0.843479\pi$
$984$		0			0
$985$	228.789			0.232273
$986$		0			0
$987$	425.417	−	208.642i	0.431020	−	0.211390i
$988$		0			0
$989$	314.091	+	314.091i	0.317584	+	0.317584i
$990$		0			0
$991$	918.372			0.926713			0.463356	−	0.886172i	$-0.346645\pi$
$991$	918.372			0.926713			0.463356	+	0.886172i	$0.346645\pi$
$992$		0			0
$993$			904.198i			0.910572i
$994$		0			0
$995$	358.100	+	358.100i	0.359900	+	0.359900i
$996$		0			0
$997$	505.584	+	505.584i	0.507105	+	0.507105i	0.913637	−	0.406532i	$-0.133262\pi$
$997$	505.584	+	505.584i	0.507105	+	0.507105i	−0.406532	+	0.913637i	$0.633262\pi$
$998$		0			0
$999$	−1275.94			−1.27722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.22		56
4.3	odd	2		112.3.l.b.13.9	✓	56
7.6	odd	2	inner	448.3.l.b.209.7		56
16.5	even	4	inner	448.3.l.b.433.7		56
16.11	odd	4		112.3.l.b.69.10	yes	56
28.27	even	2		112.3.l.b.13.10	yes	56
112.27	even	4		112.3.l.b.69.9	yes	56
112.69	odd	4	inner	448.3.l.b.433.22		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.9	✓	56	4.3	odd	2
112.3.l.b.13.10	yes	56	28.27	even	2
112.3.l.b.69.9	yes	56	112.27	even	4
112.3.l.b.69.10	yes	56	16.11	odd	4
448.3.l.b.209.7		56	7.6	odd	2	inner
448.3.l.b.209.22		56	1.1	even	1	trivial
448.3.l.b.433.7		56	16.5	even	4	inner
448.3.l.b.433.22		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(2.28266 - 2.28266i) q^{3} +(-2.90674 - 2.90674i) q^{5} +(-2.26451 + 6.62359i) q^{7} -1.42104i q^{9} +O(q^{10})\)	\(q+(2.28266 - 2.28266i) q^{3} +(-2.90674 - 2.90674i) q^{5} +(-2.26451 + 6.62359i) q^{7} -1.42104i q^{9} +(-9.96314 + 9.96314i) q^{11} +(-15.5346 + 15.5346i) q^{13} -13.2702 q^{15} -32.2316i q^{17} +(-14.4296 + 14.4296i) q^{19} +(9.95030 + 20.2885i) q^{21} +16.0348i q^{23} -8.10171i q^{25} +(17.3002 + 17.3002i) q^{27} +(-10.0884 - 10.0884i) q^{29} +3.11658i q^{31} +45.4848i q^{33} +(25.8354 - 12.6707i) q^{35} +(-36.8766 + 36.8766i) q^{37} +70.9201i q^{39} +39.5924 q^{41} +(19.5881 - 19.5881i) q^{43} +(-4.13060 + 4.13060i) q^{45} -20.9684i q^{47} +(-38.7440 - 29.9984i) q^{49} +(-73.5736 - 73.5736i) q^{51} +(-18.6330 + 18.6330i) q^{53} +57.9205 q^{55} +65.8758i q^{57} +(21.7777 + 21.7777i) q^{59} +(23.5695 - 23.5695i) q^{61} +(9.41240 + 3.21796i) q^{63} +90.3098 q^{65} +(-14.0012 - 14.0012i) q^{67} +(36.6019 + 36.6019i) q^{69} +2.39288i q^{71} -22.2579 q^{73} +(-18.4934 - 18.4934i) q^{75} +(-43.4302 - 88.5534i) q^{77} -67.0638 q^{79} +91.7700 q^{81} +(-82.2869 + 82.2869i) q^{83} +(-93.6888 + 93.6888i) q^{85} -46.0569 q^{87} -103.252 q^{89} +(-67.7165 - 138.073i) q^{91} +(7.11407 + 7.11407i) q^{93} +83.8864 q^{95} +18.8468i q^{97} +(14.1580 + 14.1580i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 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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