LMFDB - Embedded newform 640.3.r.b.31.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [640,3,Mod(31,640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("640.31"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	640.r (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.4387369191$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.15
Character	$\chi$	$=$	640.31
Dual form			640.3.r.b.351.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(3.58499 + 3.58499i) q^{3} +(1.58114 + 1.58114i) q^{5} -10.0090 q^{7} +16.7043i q^{9} +(-0.304644 + 0.304644i) q^{11} +(-6.87350 + 6.87350i) q^{13} +11.3367i q^{15} -25.8031 q^{17} +(-10.2562 - 10.2562i) q^{19} +(-35.8821 - 35.8821i) q^{21} +33.7343 q^{23} +5.00000i q^{25} +(-27.6197 + 27.6197i) q^{27} +(-23.1650 + 23.1650i) q^{29} +7.31924i q^{31} -2.18429 q^{33} +(-15.8256 - 15.8256i) q^{35} +(-6.85014 - 6.85014i) q^{37} -49.2828 q^{39} +63.7855i q^{41} +(15.0257 - 15.0257i) q^{43} +(-26.4118 + 26.4118i) q^{45} +41.8744i q^{47} +51.1799 q^{49} +(-92.5039 - 92.5039i) q^{51} +(36.5375 + 36.5375i) q^{53} -0.963370 q^{55} -73.5369i q^{57} +(58.1638 - 58.1638i) q^{59} +(26.2928 - 26.2928i) q^{61} -167.193i q^{63} -21.7359 q^{65} +(-4.48091 - 4.48091i) q^{67} +(120.937 + 120.937i) q^{69} -11.5673 q^{71} +101.723i q^{73} +(-17.9249 + 17.9249i) q^{75} +(3.04918 - 3.04918i) q^{77} -73.1288i q^{79} -47.6943 q^{81} +(39.9292 + 39.9292i) q^{83} +(-40.7983 - 40.7983i) q^{85} -166.093 q^{87} -37.2154i q^{89} +(68.7968 - 68.7968i) q^{91} +(-26.2394 + 26.2394i) q^{93} -32.4331i q^{95} -114.749 q^{97} +(-5.08886 - 5.08886i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 32 q^{11} - 32 q^{19} + 128 q^{23} - 96 q^{27} - 32 q^{29} + 96 q^{37} - 384 q^{39} + 96 q^{43} + 224 q^{49} - 256 q^{51} + 160 q^{53} - 352 q^{59} + 32 q^{61} + 160 q^{67} - 96 q^{69} - 256 q^{71}+ \cdots + 608 q^{99}+O(q^{100}) $ 32 * q + 32 * q^11 - 32 * q^19 + 128 * q^23 - 96 * q^27 - 32 * q^29 + 96 * q^37 - 384 * q^39 + 96 * q^43 + 224 * q^49 - 256 * q^51 + 160 * q^53 - 352 * q^59 + 32 * q^61 + 160 * q^67 - 96 * q^69 - 256 * q^71 - 224 * q^77 - 288 * q^81 - 480 * q^83 - 160 * q^85 - 384 * q^91 - 96 * q^93 + 608 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	3.58499	+	3.58499i	1.19500	+	1.19500i	0.975646	+	0.219350i	$0.0703935\pi$
$3$	3.58499	+	3.58499i	1.19500	+	1.19500i	0.219350	+	0.975646i	$0.429606\pi$
$4$		0			0
$5$	1.58114	+	1.58114i	0.316228	+	0.316228i
$5$	1.58114	+	1.58114i	0.316228	+	0.316228i
$6$		0			0
$7$	−10.0090			−1.42986			−0.714928	−	0.699198i	$-0.753540\pi$
$7$	−10.0090			−1.42986			−0.714928	+	0.699198i	$0.753540\pi$
$8$		0			0
$9$			16.7043i			1.85603i
$10$		0			0
$11$	−0.304644	+	0.304644i	−0.0276949	+	0.0276949i	−0.720819	−	0.693124i	$-0.756234\pi$
$11$	−0.304644	+	0.304644i	−0.0276949	+	0.0276949i	0.693124	+	0.720819i	$0.256234\pi$
$12$		0			0
$13$	−6.87350	+	6.87350i	−0.528731	+	0.528731i	−0.920194	−	0.391463i	$-0.871969\pi$
$13$	−6.87350	+	6.87350i	−0.528731	+	0.528731i	0.391463	+	0.920194i	$0.371969\pi$
$14$		0			0
$15$			11.3367i			0.755782i
$16$		0			0
$17$	−25.8031			−1.51783			−0.758916	−	0.651189i	$-0.774270\pi$
$17$	−25.8031			−1.51783			−0.758916	+	0.651189i	$0.774270\pi$
$18$		0			0
$19$	−10.2562	−	10.2562i	−0.539802	−	0.539802i	0.383669	−	0.923471i	$-0.374660\pi$
$19$	−10.2562	−	10.2562i	−0.539802	−	0.539802i	−0.923471	+	0.383669i	$0.874660\pi$
$20$		0			0
$21$	−35.8821	−	35.8821i	−1.70867	−	1.70867i
$22$		0			0
$23$	33.7343			1.46671			0.733355	−	0.679845i	$-0.237953\pi$
$23$	33.7343			1.46671			0.733355	+	0.679845i	$0.237953\pi$
$24$		0			0
$25$			5.00000i			0.200000i
$26$		0			0
$27$	−27.6197	+	27.6197i	−1.02295	+	1.02295i
$28$		0			0
$29$	−23.1650	+	23.1650i	−0.798794	+	0.798794i	−0.982905	−	0.184111i	$-0.941059\pi$
$29$	−23.1650	+	23.1650i	−0.798794	+	0.798794i	0.184111	+	0.982905i	$0.441059\pi$
$30$		0			0
$31$			7.31924i			0.236104i	0.993007	+	0.118052i	$0.0376650\pi$
$31$			7.31924i			0.236104i	−0.993007	+	0.118052i	$0.962335\pi$
$32$		0			0
$33$	−2.18429			−0.0661907
$34$		0			0
$35$	−15.8256	−	15.8256i	−0.452160	−	0.452160i
$36$		0			0
$37$	−6.85014	−	6.85014i	−0.185139	−	0.185139i	0.608452	−	0.793591i	$-0.291791\pi$
$37$	−6.85014	−	6.85014i	−0.185139	−	0.185139i	−0.793591	+	0.608452i	$0.791791\pi$
$38$		0			0
$39$	−49.2828			−1.26366
$40$		0			0
$41$			63.7855i			1.55574i	0.628423	+	0.777872i	$0.283701\pi$
$41$			63.7855i			1.55574i	−0.628423	+	0.777872i	$0.716299\pi$
$42$		0			0
$43$	15.0257	−	15.0257i	0.349434	−	0.349434i	−0.510465	−	0.859899i	$-0.670527\pi$
$43$	15.0257	−	15.0257i	0.349434	−	0.349434i	0.859899	+	0.510465i	$0.170527\pi$
$44$		0			0
$45$	−26.4118	+	26.4118i	−0.586928	+	0.586928i
$46$		0			0
$47$			41.8744i			0.890944i	0.895296	+	0.445472i	$0.146964\pi$
$47$			41.8744i			0.890944i	−0.895296	+	0.445472i	$0.853036\pi$
$48$		0			0
$49$	51.1799			1.04449
$50$		0			0
$51$	−92.5039	−	92.5039i	−1.81380	−	1.81380i
$52$		0			0
$53$	36.5375	+	36.5375i	0.689388	+	0.689388i	0.962097	−	0.272709i	$-0.0879196\pi$
$53$	36.5375	+	36.5375i	0.689388	+	0.689388i	−0.272709	+	0.962097i	$0.587920\pi$
$54$		0			0
$55$	−0.963370			−0.0175158
$56$		0			0
$57$		−	73.5369i		−	1.29012i
$58$		0			0
$59$	58.1638	−	58.1638i	0.985828	−	0.985828i	−0.0140734	−	0.999901i	$-0.504480\pi$
$59$	58.1638	−	58.1638i	0.985828	−	0.985828i	0.999901	+	0.0140734i	$0.00447984\pi$
$60$		0			0
$61$	26.2928	−	26.2928i	0.431030	−	0.431030i	−0.457949	−	0.888979i	$-0.651416\pi$
$61$	26.2928	−	26.2928i	0.431030	−	0.431030i	0.888979	+	0.457949i	$0.151416\pi$
$62$		0			0
$63$		−	167.193i		−	2.65386i
$64$		0			0
$65$	−21.7359			−0.334399
$66$		0			0
$67$	−4.48091	−	4.48091i	−0.0668793	−	0.0668793i	0.672876	−	0.739755i	$-0.265059\pi$
$67$	−4.48091	−	4.48091i	−0.0668793	−	0.0668793i	−0.739755	+	0.672876i	$0.765059\pi$
$68$		0			0
$69$	120.937	+	120.937i	1.75271	+	1.75271i
$70$		0			0
$71$	−11.5673			−0.162919			−0.0814595	−	0.996677i	$-0.525958\pi$
$71$	−11.5673			−0.162919			−0.0814595	+	0.996677i	$0.525958\pi$
$72$		0			0
$73$			101.723i			1.39346i	0.717332	+	0.696732i	$0.245363\pi$
$73$			101.723i			1.39346i	−0.717332	+	0.696732i	$0.754637\pi$
$74$		0			0
$75$	−17.9249	+	17.9249i	−0.238999	+	0.238999i
$76$		0			0
$77$	3.04918	−	3.04918i	0.0395998	−	0.0395998i
$78$		0			0
$79$		−	73.1288i		−	0.925681i	−0.886442	−	0.462840i	$-0.846830\pi$
$79$		−	73.1288i		−	0.925681i	0.886442	−	0.462840i	$-0.153170\pi$
$80$		0			0
$81$	−47.6943			−0.588819
$82$		0			0
$83$	39.9292	+	39.9292i	0.481075	+	0.481075i	0.905475	−	0.424400i	$-0.139515\pi$
$83$	39.9292	+	39.9292i	0.481075	+	0.481075i	−0.424400	+	0.905475i	$0.639515\pi$
$84$		0			0
$85$	−40.7983	−	40.7983i	−0.479980	−	0.479980i
$86$		0			0
$87$	−166.093			−1.90911
$88$		0			0
$89$		−	37.2154i		−	0.418151i	−0.977899	−	0.209076i	$-0.932955\pi$
$89$		−	37.2154i		−	0.418151i	0.977899	−	0.209076i	$-0.0670455\pi$
$90$		0			0
$91$	68.7968	−	68.7968i	0.756008	−	0.756008i
$92$		0			0
$93$	−26.2394	+	26.2394i	−0.282144	+	0.282144i
$94$		0			0
$95$		−	32.4331i		−	0.341401i
$96$		0			0
$97$	−114.749			−1.18298			−0.591490	−	0.806312i	$-0.701460\pi$
$97$	−114.749			−1.18298			−0.591490	+	0.806312i	$0.701460\pi$
$98$		0			0
$99$	−5.08886	−	5.08886i	−0.0514026	−	0.0514026i
$100$		0			0
$101$	−3.86189	−	3.86189i	−0.0382366	−	0.0382366i	0.687730	−	0.725967i	$-0.258607\pi$
$101$	−3.86189	−	3.86189i	−0.0382366	−	0.0382366i	−0.725967	+	0.687730i	$0.758607\pi$
$102$		0			0
$103$	−82.3074			−0.799101			−0.399551	−	0.916711i	$-0.630834\pi$
$103$	−82.3074			−0.799101			−0.399551	+	0.916711i	$0.630834\pi$
$104$		0			0
$105$		−	113.469i		−	1.08066i
$106$		0			0
$107$	−109.631	+	109.631i	−1.02459	+	1.02459i	−0.0248967	+	0.999690i	$0.507926\pi$
$107$	−109.631	+	109.631i	−1.02459	+	1.02459i	−0.999690	+	0.0248967i	$0.992074\pi$
$108$		0			0
$109$	2.99231	−	2.99231i	0.0274524	−	0.0274524i	−0.693247	−	0.720700i	$-0.743821\pi$
$109$	2.99231	−	2.99231i	0.0274524	−	0.0274524i	0.720700	+	0.693247i	$0.243821\pi$
$110$		0			0
$111$		−	49.1153i		−	0.442480i
$112$		0			0
$113$	73.3488			0.649105			0.324552	−	0.945868i	$-0.394786\pi$
$113$	73.3488			0.649105			0.324552	+	0.945868i	$0.394786\pi$
$114$		0			0
$115$	53.3387	+	53.3387i	0.463815	+	0.463815i
$116$		0			0
$117$	−114.817	−	114.817i	−0.981340	−	0.981340i
$118$		0			0
$119$	258.263			2.17028
$120$		0			0
$121$			120.814i			0.998466i
$122$		0			0
$123$	−228.670	+	228.670i	−1.85911	+	1.85911i
$124$		0			0
$125$	−7.90569	+	7.90569i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			212.909i			1.67645i	0.545323	+	0.838226i	$0.316407\pi$
$127$			212.909i			1.67645i	−0.545323	+	0.838226i	$0.683593\pi$
$128$		0			0
$129$	107.734			0.835145
$130$		0			0
$131$	34.5741	+	34.5741i	0.263924	+	0.263924i	0.826646	−	0.562722i	$-0.190246\pi$
$131$	34.5741	+	34.5741i	0.263924	+	0.263924i	−0.562722	+	0.826646i	$0.690246\pi$
$132$		0			0
$133$	102.655	+	102.655i	0.771838	+	0.771838i
$134$		0			0
$135$	−87.3413			−0.646972
$136$		0			0
$137$			156.675i			1.14362i	0.820388	+	0.571808i	$0.193758\pi$
$137$			156.675i			1.14362i	−0.820388	+	0.571808i	$0.806242\pi$
$138$		0			0
$139$	150.961	−	150.961i	1.08605	−	1.08605i	0.0901190	−	0.995931i	$-0.471275\pi$
$139$	150.961	−	150.961i	1.08605	−	1.08605i	0.995931	−	0.0901190i	$-0.0287247\pi$
$140$		0			0
$141$	−150.119	+	150.119i	−1.06467	+	1.06467i
$142$		0			0
$143$		−	4.18794i		−	0.0292863i
$144$		0			0
$145$	−73.2543			−0.505202
$146$		0			0
$147$	183.479	+	183.479i	1.24816	+	1.24816i
$148$		0			0
$149$	−96.4520	−	96.4520i	−0.647329	−	0.647329i	0.305018	−	0.952347i	$-0.401338\pi$
$149$	−96.4520	−	96.4520i	−0.647329	−	0.647329i	−0.952347	+	0.305018i	$0.901338\pi$
$150$		0			0
$151$	−29.4114			−0.194777			−0.0973886	−	0.995246i	$-0.531049\pi$
$151$	−29.4114			−0.194777			−0.0973886	+	0.995246i	$0.531049\pi$
$152$		0			0
$153$		−	431.023i		−	2.81714i
$154$		0			0
$155$	−11.5727	+	11.5727i	−0.0746628	+	0.0746628i
$156$		0			0
$157$	117.658	−	117.658i	0.749415	−	0.749415i	−0.224954	−	0.974369i	$-0.572223\pi$
$157$	117.658	−	117.658i	0.749415	−	0.749415i	0.974369	+	0.224954i	$0.0722232\pi$
$158$		0			0
$159$			261.973i			1.64763i
$160$		0			0
$161$	−337.647			−2.09718
$162$		0			0
$163$	−7.81267	−	7.81267i	−0.0479305	−	0.0479305i	0.682735	−	0.730666i	$-0.260790\pi$
$163$	−7.81267	−	7.81267i	−0.0479305	−	0.0479305i	−0.730666	+	0.682735i	$0.760790\pi$
$164$		0			0
$165$	−3.45367	−	3.45367i	−0.0209313	−	0.0209313i
$166$		0			0
$167$	255.372			1.52918			0.764588	−	0.644520i	$-0.222943\pi$
$167$	255.372			1.52918			0.764588	+	0.644520i	$0.222943\pi$
$168$		0			0
$169$			74.5100i			0.440888i
$170$		0			0
$171$	171.323	−	171.323i	1.00189	−	1.00189i
$172$		0			0
$173$	−27.0161	+	27.0161i	−0.156162	+	0.156162i	−0.780864	−	0.624701i	$-0.785221\pi$
$173$	−27.0161	+	27.0161i	−0.156162	+	0.156162i	0.624701	+	0.780864i	$0.285221\pi$
$174$		0			0
$175$		−	50.0449i		−	0.285971i
$176$		0			0
$177$	417.033			2.35612
$178$		0			0
$179$	50.8893	+	50.8893i	0.284298	+	0.284298i	0.834820	−	0.550523i	$-0.185572\pi$
$179$	50.8893	+	50.8893i	0.284298	+	0.284298i	−0.550523	+	0.834820i	$0.685572\pi$
$180$		0			0
$181$	13.6488	+	13.6488i	0.0754079	+	0.0754079i	0.743805	−	0.668397i	$-0.233019\pi$
$181$	13.6488	+	13.6488i	0.0754079	+	0.0754079i	−0.668397	+	0.743805i	$0.733019\pi$
$182$		0			0
$183$	188.519			1.03016
$184$		0			0
$185$		−	21.6620i		−	0.117092i
$186$		0			0
$187$	7.86078	−	7.86078i	0.0420362	−	0.0420362i
$188$		0			0
$189$	276.446	−	276.446i	1.46268	−	1.46268i
$190$		0			0
$191$		−	263.019i		−	1.37706i	−0.725206	−	0.688532i	$-0.758255\pi$
$191$		−	263.019i		−	1.37706i	0.725206	−	0.688532i	$-0.241745\pi$
$192$		0			0
$193$	−58.9456			−0.305418			−0.152709	−	0.988271i	$-0.548800\pi$
$193$	−58.9456			−0.305418			−0.152709	+	0.988271i	$0.548800\pi$
$194$		0			0
$195$	−77.9230	−	77.9230i	−0.399605	−	0.399605i
$196$		0			0
$197$	−9.54461	−	9.54461i	−0.0484498	−	0.0484498i	0.682467	−	0.730917i	$-0.260907\pi$
$197$	−9.54461	−	9.54461i	−0.0484498	−	0.0484498i	−0.730917	+	0.682467i	$0.760907\pi$
$198$		0			0
$199$	−103.151			−0.518348			−0.259174	−	0.965831i	$-0.583450\pi$
$199$	−103.151			−0.518348			−0.259174	+	0.965831i	$0.583450\pi$
$200$		0			0
$201$		−	32.1280i		−	0.159841i
$202$		0			0
$203$	231.859	−	231.859i	1.14216	−	1.14216i
$204$		0			0
$205$	−100.854	+	100.854i	−0.491969	+	0.491969i
$206$		0			0
$207$			563.508i			2.72226i
$208$		0			0
$209$	6.24900			0.0298995
$210$		0			0
$211$	4.93870	+	4.93870i	0.0234062	+	0.0234062i	0.718713	−	0.695307i	$-0.244732\pi$
$211$	4.93870	+	4.93870i	0.0234062	+	0.0234062i	−0.695307	+	0.718713i	$0.744732\pi$
$212$		0			0
$213$	−41.4685	−	41.4685i	−0.194688	−	0.194688i
$214$		0			0
$215$	47.5153			0.221002
$216$		0			0
$217$		−	73.2582i		−	0.337595i
$218$		0			0
$219$	−364.675	+	364.675i	−1.66518	+	1.66518i
$220$		0			0
$221$	177.358	−	177.358i	0.802524	−	0.802524i
$222$		0			0
$223$			177.151i			0.794401i	0.917732	+	0.397200i	$0.130018\pi$
$223$			177.151i			0.794401i	−0.917732	+	0.397200i	$0.869982\pi$
$224$		0			0
$225$	−83.5214			−0.371206
$226$		0			0
$227$	−239.268	−	239.268i	−1.05404	−	1.05404i	−0.998454	−	0.0555894i	$-0.982296\pi$
$227$	−239.268	−	239.268i	−1.05404	−	1.05404i	−0.0555894	−	0.998454i	$-0.517704\pi$
$228$		0			0
$229$	106.699	+	106.699i	0.465933	+	0.465933i	0.900594	−	0.434661i	$-0.143132\pi$
$229$	106.699	+	106.699i	0.465933	+	0.465933i	−0.434661	+	0.900594i	$0.643132\pi$
$230$		0			0
$231$	21.8626			0.0946431
$232$		0			0
$233$		−	132.961i		−	0.570649i	−0.958431	−	0.285324i	$-0.907899\pi$
$233$		−	132.961i		−	0.570649i	0.958431	−	0.285324i	$-0.0921013\pi$
$234$		0			0
$235$	−66.2092	+	66.2092i	−0.281741	+	0.281741i
$236$		0			0
$237$	262.166	−	262.166i	1.10618	−	1.10618i
$238$		0			0
$239$		−	89.6649i		−	0.375167i	−0.982249	−	0.187584i	$-0.939934\pi$
$239$		−	89.6649i		−	0.375167i	0.982249	−	0.187584i	$-0.0600655\pi$
$240$		0			0
$241$	259.019			1.07477			0.537383	−	0.843338i	$-0.319413\pi$
$241$	259.019			1.07477			0.537383	+	0.843338i	$0.319413\pi$
$242$		0			0
$243$	77.5940	+	77.5940i	0.319317	+	0.319317i
$244$		0			0
$245$	80.9225	+	80.9225i	0.330296	+	0.330296i
$246$		0			0
$247$	140.992			0.570819
$248$		0			0
$249$			286.292i			1.14977i
$250$		0			0
$251$	85.2338	−	85.2338i	0.339577	−	0.339577i	−0.516631	−	0.856208i	$-0.672814\pi$
$251$	85.2338	−	85.2338i	0.339577	−	0.339577i	0.856208	+	0.516631i	$0.172814\pi$
$252$		0			0
$253$	−10.2770	+	10.2770i	−0.0406205	+	0.0406205i
$254$		0			0
$255$		−	292.523i		−	1.14715i
$256$		0			0
$257$	451.683			1.75752			0.878760	−	0.477264i	$-0.158372\pi$
$257$	451.683			1.75752			0.878760	+	0.477264i	$0.158372\pi$
$258$		0			0
$259$	68.5629	+	68.5629i	0.264722	+	0.264722i
$260$		0			0
$261$	−386.955	−	386.955i	−1.48259	−	1.48259i
$262$		0			0
$263$	0.939393			0.00357184			0.00178592	−	0.999998i	$-0.499432\pi$
$263$	0.939393			0.00357184			0.00178592	+	0.999998i	$0.499432\pi$
$264$		0			0
$265$			115.542i			0.436007i
$266$		0			0
$267$	133.417	−	133.417i	0.499689	−	0.499689i
$268$		0			0
$269$	−141.098	+	141.098i	−0.524527	+	0.524527i	−0.918935	−	0.394408i	$-0.870950\pi$
$269$	−141.098	+	141.098i	−0.524527	+	0.524527i	0.394408	+	0.918935i	$0.370950\pi$
$270$		0			0
$271$			237.677i			0.877037i	0.898722	+	0.438518i	$0.144497\pi$
$271$			237.677i			0.877037i	−0.898722	+	0.438518i	$0.855503\pi$
$272$		0			0
$273$	493.271			1.80685
$274$		0			0
$275$	−1.52322	−	1.52322i	−0.00553899	−	0.00553899i
$276$		0			0
$277$	301.681	+	301.681i	1.08910	+	1.08910i	0.995621	+	0.0934800i	$0.0297991\pi$
$277$	301.681	+	301.681i	1.08910	+	1.08910i	0.0934800	+	0.995621i	$0.470201\pi$
$278$		0			0
$279$	−122.263			−0.438217
$280$		0			0
$281$		−	209.865i		−	0.746850i	−0.927660	−	0.373425i	$-0.878183\pi$
$281$		−	209.865i		−	0.746850i	0.927660	−	0.373425i	$-0.121817\pi$
$282$		0			0
$283$	87.1403	−	87.1403i	0.307916	−	0.307916i	−0.536184	−	0.844101i	$-0.680135\pi$
$283$	87.1403	−	87.1403i	0.307916	−	0.307916i	0.844101	+	0.536184i	$0.180135\pi$
$284$		0			0
$285$	116.272	−	116.272i	0.407972	−	0.407972i
$286$		0			0
$287$		−	638.428i		−	2.22449i
$288$		0			0
$289$	376.801			1.30381
$290$		0			0
$291$	−411.374	−	411.374i	−1.41366	−	1.41366i
$292$		0			0
$293$	−210.545	−	210.545i	−0.718583	−	0.718583i	0.249732	−	0.968315i	$-0.419657\pi$
$293$	−210.545	−	210.545i	−0.718583	−	0.718583i	−0.968315	+	0.249732i	$0.919657\pi$
$294$		0			0
$295$	183.930			0.623492
$296$		0			0
$297$		−	16.8284i		−	0.0566612i
$298$		0			0
$299$	−231.873	+	231.873i	−0.775495	+	0.775495i
$300$		0			0
$301$	−150.392	+	150.392i	−0.499640	+	0.499640i
$302$		0			0
$303$		−	27.6897i		−	0.0913851i
$304$		0			0
$305$	83.1452			0.272607
$306$		0			0
$307$	345.339	+	345.339i	1.12488	+	1.12488i	0.990996	+	0.133888i	$0.0427462\pi$
$307$	345.339	+	345.339i	1.12488	+	1.12488i	0.133888	+	0.990996i	$0.457254\pi$
$308$		0			0
$309$	−295.071	−	295.071i	−0.954923	−	0.954923i
$310$		0			0
$311$	99.7096			0.320610			0.160305	−	0.987068i	$-0.448752\pi$
$311$	99.7096			0.320610			0.160305	+	0.987068i	$0.448752\pi$
$312$		0			0
$313$			311.334i			0.994678i	0.867556	+	0.497339i	$0.165689\pi$
$313$			311.334i			0.994678i	−0.867556	+	0.497339i	$0.834311\pi$
$314$		0			0
$315$	264.355	−	264.355i	0.839223	−	0.839223i
$316$		0			0
$317$	−19.2419	+	19.2419i	−0.0607000	+	0.0607000i	−0.736805	−	0.676105i	$-0.763666\pi$
$317$	−19.2419	+	19.2419i	−0.0607000	+	0.0607000i	0.676105	+	0.736805i	$0.263666\pi$
$318$		0			0
$319$		−	14.1142i		−	0.0442451i
$320$		0			0
$321$	−786.050			−2.44875
$322$		0			0
$323$	264.643	+	264.643i	0.819328	+	0.819328i
$324$		0			0
$325$	−34.3675	−	34.3675i	−0.105746	−	0.105746i
$326$		0			0
$327$	21.4548			0.0656110
$328$		0			0
$329$		−	419.120i		−	1.27392i
$330$		0			0
$331$	194.214	−	194.214i	0.586750	−	0.586750i	−0.350000	−	0.936750i	$-0.613818\pi$
$331$	194.214	−	194.214i	0.586750	−	0.586750i	0.936750	+	0.350000i	$0.113818\pi$
$332$		0			0
$333$	114.427	−	114.427i	0.343623	−	0.343623i
$334$		0			0
$335$		−	14.1699i		−	0.0422982i
$336$		0			0
$337$	−552.303			−1.63888			−0.819441	−	0.573164i	$-0.805716\pi$
$337$	−552.303			−1.63888			−0.819441	+	0.573164i	$0.805716\pi$
$338$		0			0
$339$	262.955	+	262.955i	0.775677	+	0.775677i
$340$		0			0
$341$	−2.22976	−	2.22976i	−0.00653890	−	0.00653890i
$342$		0			0
$343$	−21.8182			−0.0636099
$344$		0			0
$345$			382.437i			1.10851i
$346$		0			0
$347$	119.162	−	119.162i	0.343407	−	0.343407i	−0.514239	−	0.857647i	$-0.671926\pi$
$347$	119.162	−	119.162i	0.343407	−	0.343407i	0.857647	+	0.514239i	$0.171926\pi$
$348$		0			0
$349$	170.249	−	170.249i	0.487821	−	0.487821i	−0.419797	−	0.907618i	$-0.637899\pi$
$349$	170.249	−	170.249i	0.487821	−	0.487821i	0.907618	+	0.419797i	$0.137899\pi$
$350$		0			0
$351$		−	379.688i		−	1.08173i
$352$		0			0
$353$	−551.816			−1.56322			−0.781609	−	0.623769i	$-0.785601\pi$
$353$	−551.816			−1.56322			−0.781609	+	0.623769i	$0.785601\pi$
$354$		0			0
$355$	−18.2894	−	18.2894i	−0.0515195	−	0.0515195i
$356$		0			0
$357$	925.871	+	925.871i	2.59347	+	2.59347i
$358$		0			0
$359$	−432.275			−1.20411			−0.602054	−	0.798456i	$-0.705651\pi$
$359$	−432.275			−1.20411			−0.602054	+	0.798456i	$0.705651\pi$
$360$		0			0
$361$		−	150.619i		−	0.417228i
$362$		0			0
$363$	−433.118	+	433.118i	−1.19316	+	1.19316i
$364$		0			0
$365$	−160.838	+	160.838i	−0.440652	+	0.440652i
$366$		0			0
$367$			223.982i			0.610306i	0.952303	+	0.305153i	$0.0987076\pi$
$367$			223.982i			0.610306i	−0.952303	+	0.305153i	$0.901292\pi$
$368$		0			0
$369$	−1065.49			−2.88751
$370$		0			0
$371$	−365.704	−	365.704i	−0.985725	−	0.985725i
$372$		0			0
$373$	348.716	+	348.716i	0.934897	+	0.934897i	0.998007	−	0.0631100i	$-0.0201019\pi$
$373$	348.716	+	348.716i	0.934897	+	0.934897i	−0.0631100	+	0.998007i	$0.520102\pi$
$374$		0			0
$375$	−56.6836			−0.151156
$376$		0			0
$377$		−	318.450i		−	0.844694i
$378$		0			0
$379$	−374.578	+	374.578i	−0.988332	+	0.988332i	−0.999933	−	0.0116005i	$-0.996307\pi$
$379$	−374.578	+	374.578i	−0.988332	+	0.988332i	0.0116005	+	0.999933i	$0.496307\pi$
$380$		0			0
$381$	−763.278	+	763.278i	−2.00335	+	2.00335i
$382$		0			0
$383$			431.347i			1.12623i	0.826378	+	0.563116i	$0.190398\pi$
$383$			431.347i			1.12623i	−0.826378	+	0.563116i	$0.809602\pi$
$384$		0			0
$385$	9.64236			0.0250451
$386$		0			0
$387$	250.993	+	250.993i	0.648560	+	0.648560i
$388$		0			0
$389$	−55.8528	−	55.8528i	−0.143580	−	0.143580i	0.631663	−	0.775243i	$-0.282373\pi$
$389$	−55.8528	−	55.8528i	−0.143580	−	0.143580i	−0.775243	+	0.631663i	$0.782373\pi$
$390$		0			0
$391$	−870.452			−2.22622
$392$		0			0
$393$			247.895i			0.630777i
$394$		0			0
$395$	115.627	−	115.627i	0.292726	−	0.292726i
$396$		0			0
$397$	104.083	−	104.083i	0.262173	−	0.262173i	−0.563763	−	0.825936i	$-0.690647\pi$
$397$	104.083	−	104.083i	0.262173	−	0.262173i	0.825936	+	0.563763i	$0.190647\pi$
$398$		0			0
$399$			736.030i			1.84469i
$400$		0			0
$401$	32.9517			0.0821738			0.0410869	−	0.999156i	$-0.486918\pi$
$401$	32.9517			0.0821738			0.0410869	+	0.999156i	$0.486918\pi$
$402$		0			0
$403$	−50.3088	−	50.3088i	−0.124836	−	0.124836i
$404$		0			0
$405$	−75.4114	−	75.4114i	−0.186201	−	0.186201i
$406$		0			0
$407$	4.17371			0.0102548
$408$		0			0
$409$		−	559.822i		−	1.36876i	−0.729126	−	0.684379i	$-0.760073\pi$
$409$		−	559.822i		−	1.36876i	0.729126	−	0.684379i	$-0.239927\pi$
$410$		0			0
$411$	−561.679	+	561.679i	−1.36662	+	1.36662i
$412$		0			0
$413$	−582.161	+	582.161i	−1.40959	+	1.40959i
$414$		0			0
$415$			126.267i			0.304259i
$416$		0			0
$417$	1082.39			2.59565
$418$		0			0
$419$	168.614	+	168.614i	0.402420	+	0.402420i	0.879085	−	0.476665i	$-0.158155\pi$
$419$	168.614	+	168.614i	0.402420	+	0.402420i	−0.476665	+	0.879085i	$0.658155\pi$
$420$		0			0
$421$	−409.607	−	409.607i	−0.972937	−	0.972937i	0.0267062	−	0.999643i	$-0.491498\pi$
$421$	−409.607	−	409.607i	−0.972937	−	0.972937i	−0.999643	+	0.0267062i	$0.991498\pi$
$422$		0			0
$423$	−699.481			−1.65362
$424$		0			0
$425$		−	129.016i		−	0.303566i
$426$		0			0
$427$	−263.164	+	263.164i	−0.616310	+	0.616310i
$428$		0			0
$429$	15.0137	−	15.0137i	0.0349970	−	0.0349970i
$430$		0			0
$431$		−	307.273i		−	0.712930i	−0.934309	−	0.356465i	$-0.883982\pi$
$431$		−	307.273i		−	0.712930i	0.934309	−	0.356465i	$-0.116018\pi$
$432$		0			0
$433$	−573.906			−1.32542			−0.662709	−	0.748877i	$-0.730593\pi$
$433$	−573.906			−1.32542			−0.662709	+	0.748877i	$0.730593\pi$
$434$		0			0
$435$	−262.616	−	262.616i	−0.603714	−	0.603714i
$436$		0			0
$437$	−345.987	−	345.987i	−0.791733	−	0.791733i
$438$		0			0
$439$	491.449			1.11947			0.559737	−	0.828670i	$-0.310902\pi$
$439$	491.449			1.11947			0.559737	+	0.828670i	$0.310902\pi$
$440$		0			0
$441$			854.922i			1.93860i
$442$		0			0
$443$	343.974	−	343.974i	0.776466	−	0.776466i	−0.202762	−	0.979228i	$-0.564992\pi$
$443$	343.974	−	343.974i	0.776466	−	0.776466i	0.979228	+	0.202762i	$0.0649919\pi$
$444$		0			0
$445$	58.8428	−	58.8428i	0.132231	−	0.132231i
$446$		0			0
$447$		−	691.558i		−	1.54711i
$448$		0			0
$449$	−153.260			−0.341337			−0.170668	−	0.985329i	$-0.554593\pi$
$449$	−153.260			−0.341337			−0.170668	+	0.985329i	$0.554593\pi$
$450$		0			0
$451$	−19.4319	−	19.4319i	−0.0430862	−	0.0430862i
$452$		0			0
$453$	−105.439	−	105.439i	−0.232758	−	0.232758i
$454$		0			0
$455$	217.554			0.478142
$456$		0			0
$457$			708.281i			1.54985i	0.632054	+	0.774924i	$0.282212\pi$
$457$			708.281i			1.54985i	−0.632054	+	0.774924i	$0.717788\pi$
$458$		0			0
$459$	712.676	−	712.676i	1.55267	−	1.55267i
$460$		0			0
$461$	−20.0249	+	20.0249i	−0.0434379	+	0.0434379i	−0.728492	−	0.685054i	$-0.759779\pi$
$461$	−20.0249	+	20.0249i	−0.0434379	+	0.0434379i	0.685054	+	0.728492i	$0.259779\pi$
$462$		0			0
$463$		−	30.9338i		−	0.0668117i	−0.999442	−	0.0334059i	$-0.989365\pi$
$463$		−	30.9338i		−	0.0668117i	0.999442	−	0.0334059i	$-0.0106354\pi$
$464$		0			0
$465$	−82.9762			−0.178443
$466$		0			0
$467$	89.9400	+	89.9400i	0.192591	+	0.192591i	0.796815	−	0.604224i	$-0.206517\pi$
$467$	89.9400	+	89.9400i	0.192591	+	0.192591i	−0.604224	+	0.796815i	$0.706517\pi$
$468$		0			0
$469$	44.8494	+	44.8494i	0.0956277	+	0.0956277i
$470$		0			0
$471$	843.606			1.79110
$472$		0			0
$473$			9.15497i			0.0193551i
$474$		0			0
$475$	51.2812	−	51.2812i	0.107960	−	0.107960i
$476$		0			0
$477$	−610.333	+	610.333i	−1.27952	+	1.27952i
$478$		0			0
$479$		−	424.069i		−	0.885322i	−0.896689	−	0.442661i	$-0.854035\pi$
$479$		−	424.069i		−	0.885322i	0.896689	−	0.442661i	$-0.145965\pi$
$480$		0			0
$481$	94.1688			0.195777
$482$		0			0
$483$	−1210.46	−	1210.46i	−2.50613	−	2.50613i
$484$		0			0
$485$	−181.434	−	181.434i	−0.374091	−	0.374091i
$486$		0			0
$487$	−509.544			−1.04629			−0.523146	−	0.852243i	$-0.675242\pi$
$487$	−509.544			−1.04629			−0.523146	+	0.852243i	$0.675242\pi$
$488$		0			0
$489$		−	56.0166i		−	0.114553i
$490$		0			0
$491$	232.469	−	232.469i	0.473461	−	0.473461i	−0.429572	−	0.903033i	$-0.641335\pi$
$491$	232.469	−	232.469i	0.473461	−	0.473461i	0.903033	+	0.429572i	$0.141335\pi$
$492$		0			0
$493$	597.730	−	597.730i	1.21243	−	1.21243i
$494$		0			0
$495$		−	16.0924i		−	0.0325099i
$496$		0			0
$497$	115.777			0.232951
$498$		0			0
$499$	−609.443	−	609.443i	−1.22133	−	1.22133i	−0.967160	−	0.254170i	$-0.918198\pi$
$499$	−609.443	−	609.443i	−1.22133	−	1.22133i	−0.254170	−	0.967160i	$-0.581802\pi$
$500$		0			0
$501$	915.506	+	915.506i	1.82736	+	1.82736i
$502$		0			0
$503$	83.3154			0.165637			0.0828185	−	0.996565i	$-0.473608\pi$
$503$	83.3154			0.165637			0.0828185	+	0.996565i	$0.473608\pi$
$504$		0			0
$505$		−	12.2124i		−	0.0241829i
$506$		0			0
$507$	−267.118	+	267.118i	−0.526859	+	0.526859i
$508$		0			0
$509$	−534.566	+	534.566i	−1.05023	+	1.05023i	−0.0515585	+	0.998670i	$0.516419\pi$
$509$	−534.566	+	534.566i	−1.05023	+	1.05023i	−0.998670	+	0.0515585i	$0.983581\pi$
$510$		0			0
$511$		−	1018.14i		−	1.99245i
$512$		0			0
$513$	566.549			1.10438
$514$		0			0
$515$	−130.139	−	130.139i	−0.252698	−	0.252698i
$516$		0			0
$517$	−12.7568	−	12.7568i	−0.0246746	−	0.0246746i
$518$		0			0
$519$	−193.705			−0.373227
$520$		0			0
$521$		−	513.800i		−	0.986181i	−0.869978	−	0.493091i	$-0.835867\pi$
$521$		−	513.800i		−	0.986181i	0.869978	−	0.493091i	$-0.164133\pi$
$522$		0			0
$523$	−146.299	+	146.299i	−0.279731	+	0.279731i	−0.833001	−	0.553271i	$-0.813379\pi$
$523$	−146.299	+	146.299i	−0.279731	+	0.279731i	0.553271	+	0.833001i	$0.313379\pi$
$524$		0			0
$525$	179.411	−	179.411i	0.341734	−	0.341734i
$526$		0			0
$527$		−	188.859i		−	0.358367i
$528$		0			0
$529$	609.006			1.15124
$530$		0			0
$531$	971.585	+	971.585i	1.82973	+	1.82973i
$532$		0			0
$533$	−438.429	−	438.429i	−0.822569	−	0.822569i
$534$		0			0
$535$	−346.683			−0.648006
$536$		0			0
$537$			364.875i			0.679469i
$538$		0			0
$539$	−15.5917	+	15.5917i	−0.0289270	+	0.0289270i
$540$		0			0
$541$	610.827	−	610.827i	1.12907	−	1.12907i	0.138743	−	0.990328i	$-0.455694\pi$
$541$	610.827	−	610.827i	1.12907	−	1.12907i	0.990328	−	0.138743i	$-0.0443060\pi$
$542$		0			0
$543$			97.8618i			0.180224i
$544$		0			0
$545$	9.46252			0.0173624
$546$		0			0
$547$	−392.410	−	392.410i	−0.717386	−	0.717386i	0.250683	−	0.968069i	$-0.419345\pi$
$547$	−392.410	−	392.410i	−0.717386	−	0.717386i	−0.968069	+	0.250683i	$0.919345\pi$
$548$		0			0
$549$	439.202	+	439.202i	0.800004	+	0.800004i
$550$		0			0
$551$	475.172			0.862381
$552$		0			0
$553$			731.945i			1.32359i
$554$		0			0
$555$	77.6581	−	77.6581i	0.139925	−	0.139925i
$556$		0			0
$557$	587.051	−	587.051i	1.05395	−	1.05395i	0.0554921	−	0.998459i	$-0.482327\pi$
$557$	587.051	−	587.051i	1.05395	−	1.05395i	0.998459	−	0.0554921i	$-0.0176728\pi$
$558$		0			0
$559$			206.558i			0.369513i
$560$		0			0
$561$	56.3616			0.100466
$562$		0			0
$563$	168.514	+	168.514i	0.299314	+	0.299314i	0.840745	−	0.541431i	$-0.182117\pi$
$563$	168.514	+	168.514i	0.299314	+	0.299314i	−0.541431	+	0.840745i	$0.682117\pi$
$564$		0			0
$565$	115.975	+	115.975i	0.205265	+	0.205265i
$566$		0			0
$567$	477.372			0.841926
$568$		0			0
$569$			550.627i			0.967711i	0.875148	+	0.483855i	$0.160764\pi$
$569$			550.627i			0.967711i	−0.875148	+	0.483855i	$0.839236\pi$
$570$		0			0
$571$	−163.518	+	163.518i	−0.286371	+	0.286371i	−0.835644	−	0.549272i	$-0.814905\pi$
$571$	−163.518	+	163.518i	−0.286371	+	0.286371i	0.549272	+	0.835644i	$0.314905\pi$
$572$		0			0
$573$	942.921	−	942.921i	1.64559	−	1.64559i
$574$		0			0
$575$			168.672i			0.293342i
$576$		0			0
$577$	15.6965			0.0272037			0.0136018	−	0.999907i	$-0.495670\pi$
$577$	15.6965			0.0272037			0.0136018	+	0.999907i	$0.495670\pi$
$578$		0			0
$579$	−211.319	−	211.319i	−0.364973	−	0.364973i
$580$		0			0
$581$	−399.651	−	399.651i	−0.687868	−	0.687868i
$582$		0			0
$583$	−22.2619			−0.0381851
$584$		0			0
$585$		−	363.083i		−	0.620654i
$586$		0			0
$587$	715.538	−	715.538i	1.21897	−	1.21897i	0.250983	−	0.967992i	$-0.419246\pi$
$587$	715.538	−	715.538i	1.21897	−	1.21897i	0.967992	−	0.250983i	$-0.0807538\pi$
$588$		0			0
$589$	75.0678	−	75.0678i	0.127450	−	0.127450i
$590$		0			0
$591$		−	68.4346i		−	0.115795i
$592$		0			0
$593$	−557.100			−0.939460			−0.469730	−	0.882810i	$-0.655649\pi$
$593$	−557.100			−0.939460			−0.469730	+	0.882810i	$0.655649\pi$
$594$		0			0
$595$	408.350	+	408.350i	0.686303	+	0.686303i
$596$		0			0
$597$	−369.796	−	369.796i	−0.619424	−	0.619424i
$598$		0			0
$599$	−524.845			−0.876202			−0.438101	−	0.898926i	$-0.644349\pi$
$599$	−524.845			−0.876202			−0.438101	+	0.898926i	$0.644349\pi$
$600$		0			0
$601$			182.301i			0.303330i	0.988432	+	0.151665i	$0.0484635\pi$
$601$			182.301i			0.303330i	−0.988432	+	0.151665i	$0.951537\pi$
$602$		0			0
$603$	74.8504	−	74.8504i	0.124130	−	0.124130i
$604$		0			0
$605$	−191.024	+	191.024i	−0.315743	+	0.315743i
$606$		0			0
$607$			987.505i			1.62686i	0.581662	+	0.813431i	$0.302403\pi$
$607$			987.505i			1.62686i	−0.581662	+	0.813431i	$0.697597\pi$
$608$		0			0
$609$	1662.42			2.72975
$610$		0			0
$611$	−287.823	−	287.823i	−0.471069	−	0.471069i
$612$		0			0
$613$	−768.517	−	768.517i	−1.25370	−	1.25370i	−0.954050	−	0.299648i	$-0.903131\pi$
$613$	−768.517	−	768.517i	−1.25370	−	1.25370i	−0.299648	−	0.954050i	$-0.596869\pi$
$614$		0			0
$615$	−723.119			−1.17580
$616$		0			0
$617$		−	467.462i		−	0.757636i	−0.925471	−	0.378818i	$-0.876331\pi$
$617$		−	467.462i		−	0.757636i	0.925471	−	0.378818i	$-0.123669\pi$
$618$		0			0
$619$	−272.667	+	272.667i	−0.440496	+	0.440496i	−0.892179	−	0.451682i	$-0.850824\pi$
$619$	−272.667	+	272.667i	−0.440496	+	0.440496i	0.451682	+	0.892179i	$0.350824\pi$
$620$		0			0
$621$	−931.734	+	931.734i	−1.50038	+	1.50038i
$622$		0			0
$623$			372.489i			0.597896i
$624$		0			0
$625$	−25.0000			−0.0400000
$626$		0			0
$627$	22.4026	+	22.4026i	0.0357298	+	0.0357298i
$628$		0			0
$629$	176.755	+	176.755i	0.281009	+	0.281009i
$630$		0			0
$631$	−1107.18			−1.75465			−0.877324	−	0.479898i	$-0.840674\pi$
$631$	−1107.18			−1.75465			−0.877324	+	0.479898i	$0.840674\pi$
$632$		0			0
$633$			35.4104i			0.0559405i
$634$		0			0
$635$	−336.639	+	336.639i	−0.530141	+	0.530141i
$636$		0			0
$637$	−351.785	+	351.785i	−0.552252	+	0.552252i
$638$		0			0
$639$		−	193.223i		−	0.302383i
$640$		0			0
$641$	−645.436			−1.00692			−0.503460	−	0.864018i	$-0.667940\pi$
$641$	−645.436			−1.00692			−0.503460	+	0.864018i	$0.667940\pi$
$642$		0			0
$643$	397.860	+	397.860i	0.618756	+	0.618756i	0.945212	−	0.326456i	$-0.105855\pi$
$643$	397.860	+	397.860i	0.618756	+	0.618756i	−0.326456	+	0.945212i	$0.605855\pi$
$644$		0			0
$645$	170.342	+	170.342i	0.264096	+	0.264096i
$646$		0			0
$647$	386.850			0.597913			0.298956	−	0.954267i	$-0.403361\pi$
$647$	386.850			0.597913			0.298956	+	0.954267i	$0.403361\pi$
$648$		0			0
$649$			35.4386i			0.0546049i
$650$		0			0
$651$	262.630	−	262.630i	0.403425	−	0.403425i
$652$		0			0
$653$	557.600	−	557.600i	0.853905	−	0.853905i	−0.136707	−	0.990612i	$-0.543652\pi$
$653$	557.600	−	557.600i	0.853905	−	0.853905i	0.990612	+	0.136707i	$0.0436517\pi$
$654$		0			0
$655$			109.333i			0.166920i
$656$		0			0
$657$	−1699.21			−2.58631
$658$		0			0
$659$	85.5304	+	85.5304i	0.129788	+	0.129788i	0.769017	−	0.639229i	$-0.220746\pi$
$659$	85.5304	+	85.5304i	0.129788	+	0.129788i	−0.639229	+	0.769017i	$0.720746\pi$
$660$		0			0
$661$	677.148	+	677.148i	1.02443	+	1.02443i	0.999694	+	0.0247351i	$0.00787424\pi$
$661$	677.148	+	677.148i	1.02443	+	1.02443i	0.0247351	+	0.999694i	$0.492126\pi$
$662$		0			0
$663$	1271.65			1.91803
$664$		0			0
$665$			324.622i			0.488153i
$666$		0			0
$667$	−781.457	+	781.457i	−1.17160	+	1.17160i
$668$		0			0
$669$	−635.086	+	635.086i	−0.949306	+	0.949306i
$670$		0			0
$671$			16.0199i			0.0238747i
$672$		0			0
$673$	1077.92			1.60167			0.800836	−	0.598884i	$-0.204389\pi$
$673$	1077.92			1.60167			0.800836	+	0.598884i	$0.204389\pi$
$674$		0			0
$675$	−138.099	−	138.099i	−0.204591	−	0.204591i
$676$		0			0
$677$	−926.183	−	926.183i	−1.36807	−	1.36807i	−0.863189	−	0.504881i	$-0.831536\pi$
$677$	−926.183	−	926.183i	−1.36807	−	1.36807i	−0.504881	−	0.863189i	$-0.668464\pi$
$678$		0			0
$679$	1148.52			1.69149
$680$		0			0
$681$		−	1715.54i		−	2.51915i
$682$		0			0
$683$	548.522	−	548.522i	0.803107	−	0.803107i	−0.180473	−	0.983580i	$-0.557763\pi$
$683$	548.522	−	548.522i	0.803107	−	0.803107i	0.983580	+	0.180473i	$0.0577629\pi$
$684$		0			0
$685$	−247.725	+	247.725i	−0.361643	+	0.361643i
$686$		0			0
$687$			765.027i			1.11358i
$688$		0			0
$689$	−502.281			−0.729001
$690$		0			0
$691$	−21.6062	−	21.6062i	−0.0312680	−	0.0312680i	0.691300	−	0.722568i	$-0.257038\pi$
$691$	−21.6062	−	21.6062i	−0.0312680	−	0.0312680i	−0.722568	+	0.691300i	$0.757038\pi$
$692$		0			0
$693$	50.9344	+	50.9344i	0.0734984	+	0.0734984i
$694$		0			0
$695$	477.380			0.686878
$696$		0			0
$697$		−	1645.86i		−	2.36136i
$698$		0			0
$699$	476.664	−	476.664i	0.681923	−	0.681923i
$700$		0			0
$701$	462.868	−	462.868i	0.660296	−	0.660296i	−0.295153	−	0.955450i	$-0.595371\pi$
$701$	462.868	−	462.868i	0.660296	−	0.660296i	0.955450	+	0.295153i	$0.0953708\pi$
$702$		0			0
$703$			140.513i			0.199877i
$704$		0			0
$705$	−474.718			−0.673359
$706$		0			0
$707$	38.6536	+	38.6536i	0.0546728	+	0.0546728i
$708$		0			0
$709$	91.9564	+	91.9564i	0.129699	+	0.129699i	0.768976	−	0.639277i	$-0.220766\pi$
$709$	91.9564	+	91.9564i	0.129699	+	0.129699i	−0.639277	+	0.768976i	$0.720766\pi$
$710$		0			0
$711$	1221.56			1.71809
$712$		0			0
$713$			246.910i			0.346297i
$714$		0			0
$715$	6.62172	−	6.62172i	0.00926115	−	0.00926115i
$716$		0			0
$717$	321.448	−	321.448i	0.448323	−	0.448323i
$718$		0			0
$719$			463.016i			0.643972i	0.946744	+	0.321986i	$0.104350\pi$
$719$			463.016i			0.643972i	−0.946744	+	0.321986i	$0.895650\pi$
$720$		0			0
$721$	823.814			1.14260
$722$		0			0
$723$	928.579	+	928.579i	1.28434	+	1.28434i
$724$		0			0
$725$	−115.825	−	115.825i	−0.159759	−	0.159759i
$726$		0			0
$727$	233.171			0.320730			0.160365	−	0.987058i	$-0.448733\pi$
$727$	233.171			0.320730			0.160365	+	0.987058i	$0.448733\pi$
$728$		0			0
$729$			985.596i			1.35198i
$730$		0			0
$731$	−387.709	+	387.709i	−0.530382	+	0.530382i
$732$		0			0
$733$	−873.035	+	873.035i	−1.19104	+	1.19104i	−0.214269	+	0.976775i	$0.568737\pi$
$733$	−873.035	+	873.035i	−1.19104	+	1.19104i	−0.976775	+	0.214269i	$0.931263\pi$
$734$		0			0
$735$			580.212i			0.789404i
$736$		0			0
$737$	2.73017			0.00370443
$738$		0			0
$739$	−127.699	−	127.699i	−0.172799	−	0.172799i	0.615409	−	0.788208i	$-0.288991\pi$
$739$	−127.699	−	127.699i	−0.172799	−	0.172799i	−0.788208	+	0.615409i	$0.788991\pi$
$740$		0			0
$741$	505.456	+	505.456i	0.682127	+	0.682127i
$742$		0			0
$743$	−226.592			−0.304968			−0.152484	−	0.988306i	$-0.548727\pi$
$743$	−226.592			−0.304968			−0.152484	+	0.988306i	$0.548727\pi$
$744$		0			0
$745$		−	305.008i		−	0.409407i
$746$		0			0
$747$	−666.989	+	666.989i	−0.892890	+	0.892890i
$748$		0			0
$749$	1097.29	−	1097.29i	1.46501	−	1.46501i
$750$		0			0
$751$			516.811i			0.688164i	0.938940	+	0.344082i	$0.111810\pi$
$751$			516.811i			0.688164i	−0.938940	+	0.344082i	$0.888190\pi$
$752$		0			0
$753$	611.124			0.811586
$754$		0			0
$755$	−46.5034	−	46.5034i	−0.0615940	−	0.0615940i
$756$		0			0
$757$	−135.597	−	135.597i	−0.179124	−	0.179124i	0.611850	−	0.790974i	$-0.290426\pi$
$757$	−135.597	−	135.597i	−0.179124	−	0.179124i	−0.790974	+	0.611850i	$0.790426\pi$
$758$		0			0
$759$	−73.6857			−0.0970826
$760$		0			0
$761$		−	268.435i		−	0.352739i	−0.984324	−	0.176370i	$-0.943565\pi$
$761$		−	268.435i		−	0.352739i	0.984324	−	0.176370i	$-0.0564354\pi$
$762$		0			0
$763$	−29.9500	+	29.9500i	−0.0392530	+	0.0392530i
$764$		0			0
$765$	681.507	−	681.507i	0.890858	−	0.890858i
$766$		0			0
$767$			799.578i			1.04247i
$768$		0			0
$769$	−826.424			−1.07467			−0.537337	−	0.843368i	$-0.680570\pi$
$769$	−826.424			−1.07467			−0.537337	+	0.843368i	$0.680570\pi$
$770$		0			0
$771$	1619.28	+	1619.28i	2.10023	+	2.10023i
$772$		0			0
$773$	1042.45	+	1042.45i	1.34857	+	1.34857i	0.887213	+	0.461359i	$0.152638\pi$
$773$	1042.45	+	1042.45i	1.34857	+	1.34857i	0.461359	+	0.887213i	$0.347362\pi$
$774$		0			0
$775$	−36.5962			−0.0472209
$776$		0			0
$777$			491.595i			0.632683i
$778$		0			0
$779$	654.199	−	654.199i	0.839793	−	0.839793i
$780$		0			0
$781$	3.52390	−	3.52390i	0.00451203	−	0.00451203i
$782$		0			0
$783$		−	1279.62i		−	1.63426i
$784$		0			0
$785$	372.068			0.473972
$786$		0			0
$787$	−924.222	−	924.222i	−1.17436	−	1.17436i	−0.981159	−	0.193201i	$-0.938113\pi$
$787$	−924.222	−	924.222i	−1.17436	−	1.17436i	−0.193201	−	0.981159i	$-0.561887\pi$
$788$		0			0
$789$	3.36771	+	3.36771i	0.00426833	+	0.00426833i
$790$		0			0
$791$	−734.148			−0.928126
$792$		0			0
$793$			361.447i			0.455797i
$794$		0			0
$795$	−414.216	+	414.216i	−0.521027	+	0.521027i
$796$		0			0
$797$	−761.055	+	761.055i	−0.954899	+	0.954899i	−0.999026	−	0.0441268i	$-0.985949\pi$
$797$	−761.055	+	761.055i	−0.954899	+	0.954899i	0.0441268	+	0.999026i	$0.485949\pi$
$798$		0			0
$799$		−	1080.49i		−	1.35230i
$800$		0			0
$801$	621.657			0.776101
$802$		0			0
$803$	−30.9893	−	30.9893i	−0.0385919	−	0.0385919i
$804$		0			0
$805$	−533.866	−	533.866i	−0.663188	−	0.663188i
$806$		0			0
$807$	−1011.67			−1.25362
$808$		0			0
$809$			106.311i			0.131410i	0.997839	+	0.0657052i	$0.0209297\pi$
$809$			106.311i			0.131410i	−0.997839	+	0.0657052i	$0.979070\pi$
$810$		0			0
$811$	−602.747	+	602.747i	−0.743214	+	0.743214i	−0.973195	−	0.229981i	$-0.926134\pi$
$811$	−602.747	+	602.747i	−0.743214	+	0.743214i	0.229981	+	0.973195i	$0.426134\pi$
$812$		0			0
$813$	−852.069	+	852.069i	−1.04806	+	1.04806i
$814$		0			0
$815$		−	24.7058i		−	0.0303139i
$816$		0			0
$817$	−308.213			−0.377250
$818$		0			0
$819$	1149.20	+	1149.20i	1.40317	+	1.40317i
$820$		0			0
$821$	−236.461	−	236.461i	−0.288016	−	0.288016i	0.548280	−	0.836295i	$-0.315283\pi$
$821$	−236.461	−	236.461i	−0.288016	−	0.288016i	−0.836295	+	0.548280i	$0.815283\pi$
$822$		0			0
$823$	412.888			0.501687			0.250843	−	0.968028i	$-0.419292\pi$
$823$	412.888			0.501687			0.250843	+	0.968028i	$0.419292\pi$
$824$		0			0
$825$		−	10.9215i		−	0.0132381i
$826$		0			0
$827$	−772.140	+	772.140i	−0.933664	+	0.933664i	−0.997933	−	0.0642682i	$-0.979529\pi$
$827$	−772.140	+	772.140i	−0.933664	+	0.933664i	0.0642682	+	0.997933i	$0.479529\pi$
$828$		0			0
$829$	163.360	−	163.360i	0.197056	−	0.197056i	−0.601681	−	0.798737i	$-0.705502\pi$
$829$	163.360	−	163.360i	0.197056	−	0.197056i	0.798737	+	0.601681i	$0.205502\pi$
$830$		0			0
$831$			2163.05i			2.60294i
$832$		0			0
$833$	−1320.60			−1.58535
$834$		0			0
$835$	403.779	+	403.779i	0.483568	+	0.483568i
$836$		0			0
$837$	−202.155	−	202.155i	−0.241524	−	0.241524i
$838$		0			0
$839$	−156.751			−0.186831			−0.0934154	−	0.995627i	$-0.529778\pi$
$839$	−156.751			−0.186831			−0.0934154	+	0.995627i	$0.529778\pi$
$840$		0			0
$841$		−	232.237i		−	0.276144i
$842$		0			0
$843$	752.363	−	752.363i	0.892482	−	0.892482i
$844$		0			0
$845$	−117.811	+	117.811i	−0.139421	+	0.139421i
$846$		0			0
$847$		−	1209.23i		−	1.42766i
$848$		0			0
$849$	624.794			0.735917
$850$		0			0
$851$	−231.085	−	231.085i	−0.271545	−	0.271545i
$852$		0			0
$853$	933.500	+	933.500i	1.09437	+	1.09437i	0.995056	+	0.0993169i	$0.0316657\pi$
$853$	933.500	+	933.500i	1.09437	+	1.09437i	0.0993169	+	0.995056i	$0.468334\pi$
$854$		0			0
$855$	541.771			0.633650
$856$		0			0
$857$		−	988.694i		−	1.15367i	−0.816861	−	0.576834i	$-0.804288\pi$
$857$		−	988.694i		−	1.15367i	0.816861	−	0.576834i	$-0.195712\pi$
$858$		0			0
$859$	868.729	−	868.729i	1.01133	−	1.01133i	0.0113913	−	0.999935i	$-0.496374\pi$
$859$	868.729	−	868.729i	1.01133	−	1.01133i	0.999935	−	0.0113913i	$-0.00362604\pi$
$860$		0			0
$861$	2288.76	−	2288.76i	2.65825	−	2.65825i
$862$		0			0
$863$		−	1065.18i		−	1.23428i	−0.786854	−	0.617139i	$-0.788291\pi$
$863$		−	1065.18i		−	1.23428i	0.786854	−	0.617139i	$-0.211709\pi$
$864$		0			0
$865$	−85.4324			−0.0987658
$866$		0			0
$867$	1350.83	+	1350.83i	1.55805	+	1.55805i
$868$		0			0
$869$	22.2783	+	22.2783i	0.0256367	+	0.0256367i
$870$		0			0
$871$	61.5991			0.0707222
$872$		0			0
$873$		−	1916.80i		−	2.19565i
$874$		0			0
$875$	79.1280	−	79.1280i	0.0904320	−	0.0904320i
$876$		0			0
$877$	107.637	−	107.637i	0.122733	−	0.122733i	−0.643072	−	0.765805i	$-0.722341\pi$
$877$	107.637	−	107.637i	0.122733	−	0.122733i	0.765805	+	0.643072i	$0.222341\pi$
$878$		0			0
$879$		−	1509.60i		−	1.71741i
$880$		0			0
$881$	1289.21			1.46334			0.731672	−	0.681657i	$-0.238740\pi$
$881$	1289.21			1.46334			0.731672	+	0.681657i	$0.238740\pi$
$882$		0			0
$883$	−89.1493	−	89.1493i	−0.100962	−	0.100962i	0.654822	−	0.755783i	$-0.272744\pi$
$883$	−89.1493	−	89.1493i	−0.100962	−	0.100962i	−0.755783	+	0.654822i	$0.772744\pi$
$884$		0			0
$885$	659.387	+	659.387i	0.745071	+	0.745071i
$886$		0			0
$887$	−1235.62			−1.39303			−0.696516	−	0.717541i	$-0.745268\pi$
$887$	−1235.62			−1.39303			−0.696516	+	0.717541i	$0.745268\pi$
$888$		0			0
$889$		−	2131.01i		−	2.39708i
$890$		0			0
$891$	14.5298	−	14.5298i	0.0163073	−	0.0163073i
$892$		0			0
$893$	429.473	−	429.473i	0.480933	−	0.480933i
$894$		0			0
$895$			160.926i			0.179806i
$896$		0			0
$897$	−1662.52			−1.85343
$898$		0			0
$899$	−169.550	−	169.550i	−0.188599	−	0.188599i
$900$		0			0
$901$	−942.783	−	942.783i	−1.04637	−	1.04637i
$902$		0			0
$903$	−1078.30			−1.19414
$904$		0			0
$905$			43.1614i			0.0476921i
$906$		0			0
$907$	1170.07	−	1170.07i	1.29004	−	1.29004i	0.355287	−	0.934757i	$-0.384383\pi$
$907$	1170.07	−	1170.07i	1.29004	−	1.29004i	0.934757	−	0.355287i	$-0.115617\pi$
$908$		0			0
$909$	64.5101	−	64.5101i	0.0709682	−	0.0709682i
$910$		0			0
$911$			174.325i			0.191356i	0.995412	+	0.0956781i	$0.0305019\pi$
$911$			174.325i			0.191356i	−0.995412	+	0.0956781i	$0.969498\pi$
$912$		0			0
$913$	−24.3284			−0.0266467
$914$		0			0
$915$	298.074	+	298.074i	0.325764	+	0.325764i
$916$		0			0
$917$	−346.052	−	346.052i	−0.377374	−	0.377374i
$918$		0			0
$919$	1108.52			1.20622			0.603110	−	0.797658i	$-0.293928\pi$
$919$	1108.52			1.20622			0.603110	+	0.797658i	$0.293928\pi$
$920$		0			0
$921$			2476.08i			2.68846i
$922$		0			0
$923$	79.5075	−	79.5075i	0.0861403	−	0.0861403i
$924$		0			0
$925$	34.2507	−	34.2507i	0.0370278	−	0.0370278i
$926$		0			0
$927$		−	1374.89i		−	1.48316i
$928$		0			0
$929$	−410.945			−0.442352			−0.221176	−	0.975234i	$-0.570990\pi$
$929$	−410.945			−0.442352			−0.221176	+	0.975234i	$0.570990\pi$
$930$		0			0
$931$	−524.912	−	524.912i	−0.563816	−	0.563816i
$932$		0			0
$933$	357.458	+	357.458i	0.383127	+	0.383127i
$934$		0			0
$935$	24.8580			0.0265860
$936$		0			0
$937$			181.492i			0.193694i	0.995299	+	0.0968472i	$0.0308758\pi$
$937$			181.492i			0.193694i	−0.995299	+	0.0968472i	$0.969124\pi$
$938$		0			0
$939$	−1116.13	+	1116.13i	−1.18864	+	1.18864i
$940$		0			0
$941$	363.737	−	363.737i	0.386543	−	0.386543i	−0.486910	−	0.873452i	$-0.661876\pi$
$941$	363.737	−	363.737i	0.386543	−	0.386543i	0.873452	+	0.486910i	$0.161876\pi$
$942$		0			0
$943$			2151.76i			2.28183i
$944$		0			0
$945$	874.198			0.925077
$946$		0			0
$947$	−616.971	−	616.971i	−0.651500	−	0.651500i	0.301854	−	0.953354i	$-0.402395\pi$
$947$	−616.971	−	616.971i	−0.651500	−	0.651500i	−0.953354	+	0.301854i	$0.902395\pi$
$948$		0			0
$949$	−699.192	−	699.192i	−0.736767	−	0.736767i
$950$		0			0
$951$	−137.964			−0.145072
$952$		0			0
$953$			602.419i			0.632129i	0.948738	+	0.316064i	$0.102362\pi$
$953$			602.419i			0.632129i	−0.948738	+	0.316064i	$0.897638\pi$
$954$		0			0
$955$	415.870	−	415.870i	0.435466	−	0.435466i
$956$		0			0
$957$	50.5992	−	50.5992i	0.0528727	−	0.0528727i
$958$		0			0
$959$		−	1568.16i		−	1.63521i
$960$		0			0
$961$	907.429			0.944255
$962$		0			0
$963$	−1831.30	−	1831.30i	−1.90166	−	1.90166i
$964$		0			0
$965$	−93.2012	−	93.2012i	−0.0965815	−	0.0965815i
$966$		0			0
$967$	−1277.51			−1.32111			−0.660553	−	0.750779i	$-0.729678\pi$
$967$	−1277.51			−1.32111			−0.660553	+	0.750779i	$0.729678\pi$
$968$		0			0
$969$			1897.48i			1.95819i
$970$		0			0
$971$	122.966	−	122.966i	0.126639	−	0.126639i	−0.640947	−	0.767585i	$-0.721458\pi$
$971$	122.966	−	122.966i	0.126639	−	0.126639i	0.767585	+	0.640947i	$0.221458\pi$
$972$		0			0
$973$	−1510.97	+	1510.97i	−1.55289	+	1.55289i
$974$		0			0
$975$		−	246.414i		−	0.252732i
$976$		0			0
$977$	809.129			0.828177			0.414088	−	0.910237i	$-0.364100\pi$
$977$	809.129			0.828177			0.414088	+	0.910237i	$0.364100\pi$
$978$		0			0
$979$	11.3375	+	11.3375i	0.0115807	+	0.0115807i
$980$		0			0
$981$	49.9844	+	49.9844i	0.0509525	+	0.0509525i
$982$		0			0
$983$	986.542			1.00360			0.501802	−	0.864983i	$-0.332671\pi$
$983$	986.542			1.00360			0.501802	+	0.864983i	$0.332671\pi$
$984$		0			0
$985$		−	30.1827i		−	0.0306423i
$986$		0			0
$987$	1502.54	−	1502.54i	1.52233	−	1.52233i
$988$		0			0
$989$	506.881	−	506.881i	0.512519	−	0.512519i
$990$		0			0
$991$		−	674.433i		−	0.680558i	−0.940325	−	0.340279i	$-0.889479\pi$
$991$		−	674.433i		−	0.680558i	0.940325	−	0.340279i	$-0.110521\pi$
$992$		0			0
$993$	1392.51			1.40233
$994$		0			0
$995$	−163.097	−	163.097i	−0.163916	−	0.163916i
$996$		0			0
$997$	1315.09	+	1315.09i	1.31905	+	1.31905i	0.914530	+	0.404519i	$0.132561\pi$
$997$	1315.09	+	1315.09i	1.31905	+	1.31905i	0.404519	+	0.914530i	$0.367439\pi$
$998$		0			0
$999$	378.398			0.378777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.3.r.b.31.15		32
4.3	odd	2		640.3.r.a.31.2		32
8.3	odd	2		80.3.r.a.11.7	✓	32
8.5	even	2		320.3.r.a.271.2		32
16.3	odd	4	inner	640.3.r.b.351.15		32
16.5	even	4		80.3.r.a.51.7	yes	32
16.11	odd	4		320.3.r.a.111.2		32
16.13	even	4		640.3.r.a.351.2		32
40.3	even	4		400.3.k.g.299.1		32
40.19	odd	2		400.3.r.f.251.10		32
40.27	even	4		400.3.k.h.299.16		32
80.37	odd	4		400.3.k.g.99.1		32
80.53	odd	4		400.3.k.h.99.16		32
80.69	even	4		400.3.r.f.51.10		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.r.a.11.7	✓	32	8.3	odd	2
80.3.r.a.51.7	yes	32	16.5	even	4
320.3.r.a.111.2		32	16.11	odd	4
320.3.r.a.271.2		32	8.5	even	2
400.3.k.g.99.1		32	80.37	odd	4
400.3.k.g.299.1		32	40.3	even	4
400.3.k.h.99.16		32	80.53	odd	4
400.3.k.h.299.16		32	40.27	even	4
400.3.r.f.51.10		32	80.69	even	4
400.3.r.f.251.10		32	40.19	odd	2
640.3.r.a.31.2		32	4.3	odd	2
640.3.r.a.351.2		32	16.13	even	4
640.3.r.b.31.15		32	1.1	even	1	trivial
640.3.r.b.351.15		32	16.3	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:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	640.r (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.58499 + 3.58499i) q^{3} +(1.58114 + 1.58114i) q^{5} -10.0090 q^{7} +16.7043i q^{9} +(-0.304644 + 0.304644i) q^{11} +(-6.87350 + 6.87350i) q^{13} +11.3367i q^{15} -25.8031 q^{17} +(-10.2562 - 10.2562i) q^{19} +(-35.8821 - 35.8821i) q^{21} +33.7343 q^{23} +5.00000i q^{25} +(-27.6197 + 27.6197i) q^{27} +(-23.1650 + 23.1650i) q^{29} +7.31924i q^{31} -2.18429 q^{33} +(-15.8256 - 15.8256i) q^{35} +(-6.85014 - 6.85014i) q^{37} -49.2828 q^{39} +63.7855i q^{41} +(15.0257 - 15.0257i) q^{43} +(-26.4118 + 26.4118i) q^{45} +41.8744i q^{47} +51.1799 q^{49} +(-92.5039 - 92.5039i) q^{51} +(36.5375 + 36.5375i) q^{53} -0.963370 q^{55} -73.5369i q^{57} +(58.1638 - 58.1638i) q^{59} +(26.2928 - 26.2928i) q^{61} -167.193i q^{63} -21.7359 q^{65} +(-4.48091 - 4.48091i) q^{67} +(120.937 + 120.937i) q^{69} -11.5673 q^{71} +101.723i q^{73} +(-17.9249 + 17.9249i) q^{75} +(3.04918 - 3.04918i) q^{77} -73.1288i q^{79} -47.6943 q^{81} +(39.9292 + 39.9292i) q^{83} +(-40.7983 - 40.7983i) q^{85} -166.093 q^{87} -37.2154i q^{89} +(68.7968 - 68.7968i) q^{91} +(-26.2394 + 26.2394i) q^{93} -32.4331i q^{95} -114.749 q^{97} +(-5.08886 - 5.08886i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{11} - 32 q^{19} + 128 q^{23} - 96 q^{27} - 32 q^{29} + 96 q^{37} - 384 q^{39} + 96 q^{43} + 224 q^{49} - 256 q^{51} + 160 q^{53} - 352 q^{59} + 32 q^{61} + 160 q^{67} - 96 q^{69} - 256 q^{71}+ \cdots + 608 q^{99}+O(q^{100}) \) 32 * q + 32 * q^11 - 32 * q^19 + 128 * q^23 - 96 * q^27 - 32 * q^29 + 96 * q^37 - 384 * q^39 + 96 * q^43 + 224 * q^49 - 256 * q^51 + 160 * q^53 - 352 * q^59 + 32 * q^61 + 160 * q^67 - 96 * q^69 - 256 * q^71 - 224 * q^77 - 288 * q^81 - 480 * q^83 - 160 * q^85 - 384 * q^91 - 96 * q^93 + 608 * q^99

Introduction

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