LMFDB - Embedded newform 1344.2.u.a.1231.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(559,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1344.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.u (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$32$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1231.25
Character	$\chi$	$=$	1344.1231
Dual form			1344.2.u.a.559.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(0.167468 - 0.167468i) q^{5} +(2.61783 + 0.383395i) q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(0.707107 - 0.707107i) q^{3} +(0.167468 - 0.167468i) q^{5} +(2.61783 + 0.383395i) q^{7} -1.00000i q^{9} +(-2.51707 + 2.51707i) q^{11} +(4.28014 + 4.28014i) q^{13} -0.236836i q^{15} +7.14268i q^{17} +(3.61962 - 3.61962i) q^{19} +(2.12218 - 1.57998i) q^{21} -5.62753 q^{23} +4.94391i q^{25} +(-0.707107 - 0.707107i) q^{27} +(0.0732236 - 0.0732236i) q^{29} -2.74337 q^{31} +3.55968i q^{33} +(0.502609 - 0.374196i) q^{35} +(-0.490300 - 0.490300i) q^{37} +6.05303 q^{39} +9.39766 q^{41} +(3.30395 - 3.30395i) q^{43} +(-0.167468 - 0.167468i) q^{45} +0.799500 q^{47} +(6.70602 + 2.00732i) q^{49} +(5.05064 + 5.05064i) q^{51} +(-4.68107 - 4.68107i) q^{53} +0.843058i q^{55} -5.11892i q^{57} +(7.78362 + 7.78362i) q^{59} +(-8.08940 - 8.08940i) q^{61} +(0.383395 - 2.61783i) q^{63} +1.43357 q^{65} +(-9.96374 - 9.96374i) q^{67} +(-3.97926 + 3.97926i) q^{69} -0.235353 q^{71} +11.1990 q^{73} +(3.49587 + 3.49587i) q^{75} +(-7.55428 + 5.62422i) q^{77} +0.211784i q^{79} -1.00000 q^{81} +(8.74602 - 8.74602i) q^{83} +(1.19617 + 1.19617i) q^{85} -0.103554i q^{87} -7.54477 q^{89} +(9.56366 + 12.8456i) q^{91} +(-1.93985 + 1.93985i) q^{93} -1.21234i q^{95} +14.4984i q^{97} +(2.51707 + 2.51707i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q - 8 q^{11} + 16 q^{23} + 16 q^{29} - 24 q^{35} + 16 q^{37} + 8 q^{43} + 16 q^{53} - 56 q^{67} + 128 q^{71} - 64 q^{81} - 8 q^{91} + 8 q^{99}+O(q^{100}) $ 64 * q - 8 * q^11 + 16 * q^23 + 16 * q^29 - 24 * q^35 + 16 * q^37 + 8 * q^43 + 16 * q^53 - 56 * q^67 + 128 * q^71 - 64 * q^81 - 8 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$		0			0
$5$	0.167468	−	0.167468i	0.0748940	−	0.0748940i	−0.668668	−	0.743562i	$-0.733135\pi$
$5$	0.167468	−	0.167468i	0.0748940	−	0.0748940i	0.743562	+	0.668668i	$0.233135\pi$
$6$		0			0
$7$	2.61783	+	0.383395i	0.989445	+	0.144910i
$7$	2.61783	+	0.383395i	0.989445	+	0.144910i
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−2.51707	+	2.51707i	−0.758925	+	0.758925i	−0.976127	−	0.217201i	$-0.930307\pi$
$11$	−2.51707	+	2.51707i	−0.758925	+	0.758925i	0.217201	+	0.976127i	$0.430307\pi$
$12$		0			0
$13$	4.28014	+	4.28014i	1.18710	+	1.18710i	0.977867	+	0.209230i	$0.0670956\pi$
$13$	4.28014	+	4.28014i	1.18710	+	1.18710i	0.209230	+	0.977867i	$0.432904\pi$
$14$		0			0
$15$		−	0.236836i		−	0.0611507i
$16$		0			0
$17$			7.14268i			1.73235i	0.499737	+	0.866177i	$0.333430\pi$
$17$			7.14268i			1.73235i	−0.499737	+	0.866177i	$0.666570\pi$
$18$		0			0
$19$	3.61962	−	3.61962i	0.830398	−	0.830398i	−0.157173	−	0.987571i	$-0.550238\pi$
$19$	3.61962	−	3.61962i	0.830398	−	0.830398i	0.987571	+	0.157173i	$0.0502381\pi$
$20$		0			0
$21$	2.12218	−	1.57998i	0.463098	−	0.344780i
$22$		0			0
$23$	−5.62753			−1.17342			−0.586711	−	0.809797i	$-0.699577\pi$
$23$	−5.62753			−1.17342			−0.586711	+	0.809797i	$0.699577\pi$
$24$		0			0
$25$			4.94391i			0.988782i
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$	0.0732236	−	0.0732236i	0.0135973	−	0.0135973i	−0.700275	−	0.713873i	$-0.746939\pi$
$29$	0.0732236	−	0.0732236i	0.0135973	−	0.0135973i	0.713873	+	0.700275i	$0.246939\pi$
$30$		0			0
$31$	−2.74337			−0.492723			−0.246361	−	0.969178i	$-0.579235\pi$
$31$	−2.74337			−0.492723			−0.246361	+	0.969178i	$0.579235\pi$
$32$		0			0
$33$			3.55968i			0.619660i
$34$		0			0
$35$	0.502609	−	0.374196i	0.0849564	−	0.0632506i
$36$		0			0
$37$	−0.490300	−	0.490300i	−0.0806049	−	0.0806049i	0.665655	−	0.746260i	$-0.268152\pi$
$37$	−0.490300	−	0.490300i	−0.0806049	−	0.0806049i	−0.746260	+	0.665655i	$0.768152\pi$
$38$		0			0
$39$	6.05303			0.969260
$40$		0			0
$41$	9.39766			1.46767			0.733834	−	0.679328i	$-0.237729\pi$
$41$	9.39766			1.46767			0.733834	+	0.679328i	$0.237729\pi$
$42$		0			0
$43$	3.30395	−	3.30395i	0.503847	−	0.503847i	−0.408784	−	0.912631i	$-0.634047\pi$
$43$	3.30395	−	3.30395i	0.503847	−	0.503847i	0.912631	+	0.408784i	$0.134047\pi$
$44$		0			0
$45$	−0.167468	−	0.167468i	−0.0249647	−	0.0249647i
$46$		0			0
$47$	0.799500			0.116619			0.0583095	−	0.998299i	$-0.481429\pi$
$47$	0.799500			0.116619			0.0583095	+	0.998299i	$0.481429\pi$
$48$		0			0
$49$	6.70602	+	2.00732i	0.958002	+	0.286761i
$50$		0			0
$51$	5.05064	+	5.05064i	0.707231	+	0.707231i
$52$		0			0
$53$	−4.68107	−	4.68107i	−0.642994	−	0.642994i	0.308296	−	0.951290i	$-0.400241\pi$
$53$	−4.68107	−	4.68107i	−0.642994	−	0.642994i	−0.951290	+	0.308296i	$0.900241\pi$
$54$		0			0
$55$			0.843058i			0.113678i
$56$		0			0
$57$		−	5.11892i		−	0.678017i
$58$		0			0
$59$	7.78362	+	7.78362i	1.01334	+	1.01334i	0.999910	+	0.0134315i	$0.00427550\pi$
$59$	7.78362	+	7.78362i	1.01334	+	1.01334i	0.0134315	+	0.999910i	$0.495725\pi$
$60$		0			0
$61$	−8.08940	−	8.08940i	−1.03574	−	1.03574i	−0.999337	−	0.0364049i	$-0.988409\pi$
$61$	−8.08940	−	8.08940i	−1.03574	−	1.03574i	−0.0364049	−	0.999337i	$-0.511591\pi$
$62$		0			0
$63$	0.383395	−	2.61783i	0.0483033	−	0.329815i
$64$		0			0
$65$	1.43357			0.177813
$66$		0			0
$67$	−9.96374	−	9.96374i	−1.21726	−	1.21726i	−0.968586	−	0.248678i	$-0.920004\pi$
$67$	−9.96374	−	9.96374i	−1.21726	−	1.21726i	−0.248678	−	0.968586i	$-0.579996\pi$
$68$		0			0
$69$	−3.97926	+	3.97926i	−0.479047	+	0.479047i
$70$		0			0
$71$	−0.235353			−0.0279312			−0.0139656	−	0.999902i	$-0.504446\pi$
$71$	−0.235353			−0.0279312			−0.0139656	+	0.999902i	$0.504446\pi$
$72$		0			0
$73$	11.1990			1.31074			0.655370	−	0.755308i	$-0.272513\pi$
$73$	11.1990			1.31074			0.655370	+	0.755308i	$0.272513\pi$
$74$		0			0
$75$	3.49587	+	3.49587i	0.403668	+	0.403668i
$76$		0			0
$77$	−7.55428	+	5.62422i	−0.860891	+	0.640939i
$78$		0			0
$79$			0.211784i			0.0238276i	0.999929	+	0.0119138i	$0.00379236\pi$
$79$			0.211784i			0.0238276i	−0.999929	+	0.0119138i	$0.996208\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	8.74602	−	8.74602i	0.960001	−	0.960001i	−0.0392297	−	0.999230i	$-0.512490\pi$
$83$	8.74602	−	8.74602i	0.960001	−	0.960001i	0.999230	+	0.0392297i	$0.0124904\pi$
$84$		0			0
$85$	1.19617	+	1.19617i	0.129743	+	0.129743i
$86$		0			0
$87$		−	0.103554i		−	0.0111021i
$88$		0			0
$89$	−7.54477			−0.799744			−0.399872	−	0.916571i	$-0.630945\pi$
$89$	−7.54477			−0.799744			−0.399872	+	0.916571i	$0.630945\pi$
$90$		0			0
$91$	9.56366	+	12.8456i	1.00254	+	1.34659i
$92$		0			0
$93$	−1.93985	+	1.93985i	−0.201153	+	0.201153i
$94$		0			0
$95$		−	1.21234i		−	0.124384i
$96$		0			0
$97$			14.4984i			1.47209i	0.676932	+	0.736045i	$0.263309\pi$
$97$			14.4984i			1.47209i	−0.676932	+	0.736045i	$0.736691\pi$
$98$		0			0
$99$	2.51707	+	2.51707i	0.252975	+	0.252975i
$100$		0			0
$101$	4.52552	−	4.52552i	0.450306	−	0.450306i	−0.445150	−	0.895456i	$-0.646850\pi$
$101$	4.52552	−	4.52552i	0.450306	−	0.450306i	0.895456	+	0.445150i	$0.146850\pi$
$102$		0			0
$103$			6.94145i			0.683962i	0.939707	+	0.341981i	$0.111098\pi$
$103$			6.94145i			0.683962i	−0.939707	+	0.341981i	$0.888902\pi$
$104$		0			0
$105$	0.0908017	−	0.619994i	0.00886134	−	0.0605053i
$106$		0			0
$107$	−1.87422	+	1.87422i	−0.181187	+	0.181187i	−0.791873	−	0.610686i	$-0.790894\pi$
$107$	−1.87422	+	1.87422i	−0.181187	+	0.181187i	0.610686	+	0.791873i	$0.290894\pi$
$108$		0			0
$109$	12.6272	−	12.6272i	1.20946	−	1.20946i	0.238261	−	0.971201i	$-0.423423\pi$
$109$	12.6272	−	12.6272i	1.20946	−	1.20946i	0.971201	−	0.238261i	$-0.0765772\pi$
$110$		0			0
$111$	−0.693389			−0.0658136
$112$		0			0
$113$	−7.68573			−0.723012			−0.361506	−	0.932370i	$-0.617737\pi$
$113$	−7.68573			−0.723012			−0.361506	+	0.932370i	$0.617737\pi$
$114$		0			0
$115$	−0.942432	+	0.942432i	−0.0878822	+	0.0878822i
$116$		0			0
$117$	4.28014	−	4.28014i	0.395699	−	0.395699i
$118$		0			0
$119$	−2.73847	+	18.6983i	−0.251035	+	1.71407i
$120$		0			0
$121$		−	1.67129i		−	0.151935i
$122$		0			0
$123$	6.64515	−	6.64515i	0.599173	−	0.599173i
$124$		0			0
$125$	1.66529	+	1.66529i	0.148948	+	0.148948i
$126$		0			0
$127$		−	3.23913i		−	0.287427i	−0.989619	−	0.143713i	$-0.954096\pi$
$127$		−	3.23913i		−	0.287427i	0.989619	−	0.143713i	$-0.0459043\pi$
$128$		0			0
$129$		−	4.67248i		−	0.411389i
$130$		0			0
$131$	0.202227	−	0.202227i	0.0176686	−	0.0176686i	−0.698217	−	0.715886i	$-0.746023\pi$
$131$	0.202227	−	0.202227i	0.0176686	−	0.0176686i	0.715886	+	0.698217i	$0.246023\pi$
$132$		0			0
$133$	10.8633	−	8.08779i	0.941966	−	0.701300i
$134$		0			0
$135$	−0.236836			−0.0203836
$136$		0			0
$137$		−	15.9858i		−	1.36576i	−0.730533	−	0.682878i	$-0.760728\pi$
$137$		−	15.9858i		−	1.36576i	0.730533	−	0.682878i	$-0.239272\pi$
$138$		0			0
$139$	2.02752	+	2.02752i	0.171972	+	0.171972i	0.787845	−	0.615873i	$-0.211197\pi$
$139$	2.02752	+	2.02752i	0.171972	+	0.171972i	−0.615873	+	0.787845i	$0.711197\pi$
$140$		0			0
$141$	0.565332	−	0.565332i	0.0476095	−	0.0476095i
$142$		0			0
$143$	−21.5468			−1.80183
$144$		0			0
$145$		−	0.0245252i		−	0.00203671i
$146$		0			0
$147$	6.16126	−	3.32248i	0.508172	−	0.274033i
$148$		0			0
$149$	6.61834	+	6.61834i	0.542195	+	0.542195i	0.924172	−	0.381977i	$-0.124757\pi$
$149$	6.61834	+	6.61834i	0.542195	+	0.542195i	−0.381977	+	0.924172i	$0.624757\pi$
$150$		0			0
$151$	5.60636			0.456239			0.228120	−	0.973633i	$-0.426742\pi$
$151$	5.60636			0.456239			0.228120	+	0.973633i	$0.426742\pi$
$152$		0			0
$153$	7.14268			0.577451
$154$		0			0
$155$	−0.459426	+	0.459426i	−0.0369020	+	0.0369020i
$156$		0			0
$157$	−11.8394	−	11.8394i	−0.944890	−	0.944890i	0.0536685	−	0.998559i	$-0.482909\pi$
$157$	−11.8394	−	11.8394i	−0.944890	−	0.944890i	−0.998559	+	0.0536685i	$0.982909\pi$
$158$		0			0
$159$	−6.62003			−0.525003
$160$		0			0
$161$	−14.7319	−	2.15757i	−1.16104	−	0.170040i
$162$		0			0
$163$	−0.641460	−	0.641460i	−0.0502430	−	0.0502430i	0.681539	−	0.731782i	$-0.261311\pi$
$163$	−0.641460	−	0.641460i	−0.0502430	−	0.0502430i	−0.731782	+	0.681539i	$0.761311\pi$
$164$		0			0
$165$	0.596132	+	0.596132i	0.0464088	+	0.0464088i
$166$		0			0
$167$		−	8.66571i		−	0.670573i	−0.942116	−	0.335287i	$-0.891167\pi$
$167$		−	8.66571i		−	0.670573i	0.942116	−	0.335287i	$-0.108833\pi$
$168$		0			0
$169$			23.6391i			1.81839i
$170$		0			0
$171$	−3.61962	−	3.61962i	−0.276799	−	0.276799i
$172$		0			0
$173$	−4.80384	−	4.80384i	−0.365229	−	0.365229i	0.500505	−	0.865734i	$-0.333148\pi$
$173$	−4.80384	−	4.80384i	−0.365229	−	0.365229i	−0.865734	+	0.500505i	$0.833148\pi$
$174$		0			0
$175$	−1.89547	+	12.9423i	−0.143284	+	0.978345i
$176$		0			0
$177$	11.0077			0.827390
$178$		0			0
$179$	3.03426	+	3.03426i	0.226792	+	0.226792i	0.811351	−	0.584559i	$-0.198733\pi$
$179$	3.03426	+	3.03426i	0.226792	+	0.226792i	−0.584559	+	0.811351i	$0.698733\pi$
$180$		0			0
$181$	2.63997	−	2.63997i	0.196227	−	0.196227i	−0.602153	−	0.798380i	$-0.705690\pi$
$181$	2.63997	−	2.63997i	0.196227	−	0.196227i	0.798380	+	0.602153i	$0.205690\pi$
$182$		0			0
$183$	−11.4401			−0.845680
$184$		0			0
$185$	−0.164219			−0.0120736
$186$		0			0
$187$	−17.9786	−	17.9786i	−1.31473	−	1.31473i
$188$		0			0
$189$	−1.57998	−	2.12218i	−0.114927	−	0.154366i
$190$		0			0
$191$			12.4428i			0.900328i	0.892946	+	0.450164i	$0.148634\pi$
$191$			12.4428i			0.900328i	−0.892946	+	0.450164i	$0.851366\pi$
$192$		0			0
$193$	6.00200			0.432034			0.216017	−	0.976390i	$-0.430693\pi$
$193$	6.00200			0.432034			0.216017	+	0.976390i	$0.430693\pi$
$194$		0			0
$195$	1.01369	−	1.01369i	0.0725918	−	0.0725918i
$196$		0			0
$197$	−10.8958	−	10.8958i	−0.776297	−	0.776297i	0.202902	−	0.979199i	$-0.434963\pi$
$197$	−10.8958	−	10.8958i	−0.776297	−	0.776297i	−0.979199	+	0.202902i	$0.934963\pi$
$198$		0			0
$199$		−	9.22666i		−	0.654060i	−0.945014	−	0.327030i	$-0.893952\pi$
$199$		−	9.22666i		−	0.654060i	0.945014	−	0.327030i	$-0.106048\pi$
$200$		0			0
$201$	−14.0909			−0.993892
$202$		0			0
$203$	0.219760	−	0.163613i	0.0154241	−	0.0114834i
$204$		0			0
$205$	1.57381	−	1.57381i	0.109920	−	0.109920i
$206$		0			0
$207$			5.62753i			0.391140i
$208$		0			0
$209$			18.2217i			1.26042i
$210$		0			0
$211$	−7.53917	−	7.53917i	−0.519018	−	0.519018i	0.398256	−	0.917274i	$-0.369615\pi$
$211$	−7.53917	−	7.53917i	−0.519018	−	0.519018i	−0.917274	+	0.398256i	$0.869615\pi$
$212$		0			0
$213$	−0.166420	+	0.166420i	−0.0114029	+	0.0114029i
$214$		0			0
$215$		−	1.10661i		−	0.0754702i
$216$		0			0
$217$	−7.18165	−	1.05179i	−0.487522	−	0.0714004i
$218$		0			0
$219$	7.91887	−	7.91887i	0.535108	−	0.535108i
$220$		0			0
$221$	−30.5716	+	30.5716i	−2.05647	+	2.05647i
$222$		0			0
$223$	−0.834795			−0.0559020			−0.0279510	−	0.999609i	$-0.508898\pi$
$223$	−0.834795			−0.0559020			−0.0279510	+	0.999609i	$0.508898\pi$
$224$		0			0
$225$	4.94391			0.329594
$226$		0			0
$227$	−3.92781	+	3.92781i	−0.260698	+	0.260698i	−0.825337	−	0.564640i	$-0.809015\pi$
$227$	−3.92781	+	3.92781i	−0.260698	+	0.260698i	0.564640	+	0.825337i	$0.309015\pi$
$228$		0			0
$229$	−1.17563	+	1.17563i	−0.0776881	+	0.0776881i	−0.744883	−	0.667195i	$-0.767495\pi$
$229$	−1.17563	+	1.17563i	−0.0776881	+	0.0776881i	0.667195	+	0.744883i	$0.267495\pi$
$230$		0			0
$231$	−1.36476	+	9.31861i	−0.0897948	+	0.613119i
$232$		0			0
$233$			10.6073i			0.694905i	0.937698	+	0.347452i	$0.112953\pi$
$233$			10.6073i			0.694905i	−0.937698	+	0.347452i	$0.887047\pi$
$234$		0			0
$235$	0.133891	−	0.133891i	0.00873407	−	0.00873407i
$236$		0			0
$237$	0.149754	+	0.149754i	0.00972756	+	0.00972756i
$238$		0			0
$239$		−	7.16271i		−	0.463317i	−0.972797	−	0.231659i	$-0.925585\pi$
$239$		−	7.16271i		−	0.463317i	0.972797	−	0.231659i	$-0.0744152\pi$
$240$		0			0
$241$		−	13.4240i		−	0.864716i	−0.901702	−	0.432358i	$-0.857682\pi$
$241$		−	13.4240i		−	0.864716i	0.901702	−	0.432358i	$-0.142318\pi$
$242$		0			0
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$		0			0
$245$	1.45921	−	0.786881i	0.0932253	−	0.0502720i
$246$		0			0
$247$	30.9849			1.97152
$248$		0			0
$249$		−	12.3687i		−	0.783837i
$250$		0			0
$251$	5.28753	+	5.28753i	0.333746	+	0.333746i	0.854007	−	0.520261i	$-0.174165\pi$
$251$	5.28753	+	5.28753i	0.333746	+	0.333746i	−0.520261	+	0.854007i	$0.674165\pi$
$252$		0			0
$253$	14.1649	−	14.1649i	0.890539	−	0.890539i
$254$		0			0
$255$	1.69164			0.105935
$256$		0			0
$257$		−	0.425017i		−	0.0265118i	−0.999912	−	0.0132559i	$-0.995780\pi$
$257$		−	0.425017i		−	0.0265118i	0.999912	−	0.0132559i	$-0.00421961\pi$
$258$		0			0
$259$	−1.09554	−	1.47150i	−0.0680736	−	0.0914345i
$260$		0			0
$261$	−0.0732236	−	0.0732236i	−0.00453243	−	0.00453243i
$262$		0			0
$263$	−26.6012			−1.64030			−0.820151	−	0.572147i	$-0.806111\pi$
$263$	−26.6012			−1.64030			−0.820151	+	0.572147i	$0.806111\pi$
$264$		0			0
$265$	−1.56786			−0.0963128
$266$		0			0
$267$	−5.33496	+	5.33496i	−0.326494	+	0.326494i
$268$		0			0
$269$	−15.3116	−	15.3116i	−0.933565	−	0.933565i	0.0643617	−	0.997927i	$-0.479499\pi$
$269$	−15.3116	−	15.3116i	−0.933565	−	0.933565i	−0.997927	+	0.0643617i	$0.979499\pi$
$270$		0			0
$271$	−20.1296			−1.22278			−0.611392	−	0.791328i	$-0.709390\pi$
$271$	−20.1296			−1.22278			−0.611392	+	0.791328i	$0.709390\pi$
$272$		0			0
$273$	15.8458	+	2.32070i	0.959029	+	0.140455i
$274$		0			0
$275$	−12.4442	−	12.4442i	−0.750412	−	0.750412i
$276$		0			0
$277$	16.8102	+	16.8102i	1.01003	+	1.01003i	0.999949	+	0.0100769i	$0.00320764\pi$
$277$	16.8102	+	16.8102i	1.01003	+	1.01003i	0.0100769	+	0.999949i	$0.496792\pi$
$278$		0			0
$279$			2.74337i			0.164241i
$280$		0			0
$281$		−	4.47706i		−	0.267079i	−0.991044	−	0.133539i	$-0.957366\pi$
$281$		−	4.47706i		−	0.267079i	0.991044	−	0.133539i	$-0.0426343\pi$
$282$		0			0
$283$	9.67731	+	9.67731i	0.575256	+	0.575256i	0.933593	−	0.358336i	$-0.116656\pi$
$283$	9.67731	+	9.67731i	0.575256	+	0.575256i	−0.358336	+	0.933593i	$0.616656\pi$
$284$		0			0
$285$	−0.857255	−	0.857255i	−0.0507794	−	0.0507794i
$286$		0			0
$287$	24.6014	+	3.60302i	1.45218	+	0.212680i
$288$		0			0
$289$	−34.0179			−2.00105
$290$		0			0
$291$	10.2519	+	10.2519i	0.600978	+	0.600978i
$292$		0			0
$293$	−8.48795	+	8.48795i	−0.495871	+	0.495871i	−0.910150	−	0.414279i	$-0.864034\pi$
$293$	−8.48795	+	8.48795i	−0.495871	+	0.495871i	0.414279	+	0.910150i	$0.364034\pi$
$294$		0			0
$295$	2.60702			0.151786
$296$		0			0
$297$	3.55968			0.206553
$298$		0			0
$299$	−24.0866	−	24.0866i	−1.39296	−	1.39296i
$300$		0			0
$301$	9.91587	−	7.38243i	0.571541	−	0.425516i
$302$		0			0
$303$		−	6.40005i		−	0.367673i
$304$		0			0
$305$	−2.70943			−0.155142
$306$		0			0
$307$	9.08093	−	9.08093i	0.518276	−	0.518276i	−0.398774	−	0.917049i	$-0.630564\pi$
$307$	9.08093	−	9.08093i	0.518276	−	0.518276i	0.917049	+	0.398774i	$0.130564\pi$
$308$		0			0
$309$	4.90835	+	4.90835i	0.279226	+	0.279226i
$310$		0			0
$311$		−	17.7083i		−	1.00415i	−0.864825	−	0.502074i	$-0.832571\pi$
$311$		−	17.7083i		−	1.00415i	0.864825	−	0.502074i	$-0.167429\pi$
$312$		0			0
$313$	−14.1377			−0.799112			−0.399556	−	0.916709i	$-0.630836\pi$
$313$	−14.1377			−0.799112			−0.399556	+	0.916709i	$0.630836\pi$
$314$		0			0
$315$	−0.374196	−	0.502609i	−0.0210835	−	0.0283188i
$316$		0			0
$317$	6.54573	−	6.54573i	0.367645	−	0.367645i	−0.498973	−	0.866618i	$-0.666289\pi$
$317$	6.54573	−	6.54573i	0.367645	−	0.367645i	0.866618	+	0.498973i	$0.166289\pi$
$318$		0			0
$319$			0.368618i			0.0206386i
$320$		0			0
$321$			2.65054i			0.147939i
$322$		0			0
$323$	25.8538	+	25.8538i	1.43854	+	1.43854i
$324$		0			0
$325$	−21.1606	+	21.1606i	−1.17378	+	1.17378i
$326$		0			0
$327$		−	17.8575i		−	0.987521i
$328$		0			0
$329$	2.09295	+	0.306525i	0.115388	+	0.0168993i
$330$		0			0
$331$	−9.17452	+	9.17452i	−0.504277	+	0.504277i	−0.912764	−	0.408487i	$-0.866056\pi$
$331$	−9.17452	+	9.17452i	−0.504277	+	0.504277i	0.408487	+	0.912764i	$0.366056\pi$
$332$		0			0
$333$	−0.490300	+	0.490300i	−0.0268683	+	0.0268683i
$334$		0			0
$335$	−3.33722			−0.182332
$336$		0			0
$337$	−6.41647			−0.349528			−0.174764	−	0.984610i	$-0.555916\pi$
$337$	−6.41647			−0.349528			−0.174764	+	0.984610i	$0.555916\pi$
$338$		0			0
$339$	−5.43463	+	5.43463i	−0.295169	+	0.295169i
$340$		0			0
$341$	6.90524	−	6.90524i	0.373940	−	0.373940i
$342$		0			0
$343$	16.7856	+	7.82588i	0.906336	+	0.422558i
$344$		0			0
$345$			1.33280i			0.0717555i
$346$		0			0
$347$	−4.80880	+	4.80880i	−0.258150	+	0.258150i	−0.824301	−	0.566151i	$-0.808432\pi$
$347$	−4.80880	+	4.80880i	−0.258150	+	0.258150i	0.566151	+	0.824301i	$0.308432\pi$
$348$		0			0
$349$	6.99706	+	6.99706i	0.374544	+	0.374544i	0.869129	−	0.494585i	$-0.164680\pi$
$349$	6.99706	+	6.99706i	0.374544	+	0.374544i	−0.494585	+	0.869129i	$0.664680\pi$
$350$		0			0
$351$		−	6.05303i		−	0.323087i
$352$		0			0
$353$		−	27.6603i		−	1.47221i	−0.676869	−	0.736104i	$-0.736663\pi$
$353$		−	27.6603i		−	1.47221i	0.676869	−	0.736104i	$-0.263337\pi$
$354$		0			0
$355$	−0.0394141	+	0.0394141i	−0.00209188	+	0.00209188i
$356$		0			0
$357$	11.2853	+	15.1581i	0.597281	+	0.802250i
$358$		0			0
$359$	−29.1366			−1.53777			−0.768884	−	0.639388i	$-0.779188\pi$
$359$	−29.1366			−1.53777			−0.768884	+	0.639388i	$0.779188\pi$
$360$		0			0
$361$		−	7.20331i		−	0.379121i
$362$		0			0
$363$	−1.18178	−	1.18178i	−0.0620274	−	0.0620274i
$364$		0			0
$365$	1.87547	−	1.87547i	0.0981666	−	0.0981666i
$366$		0			0
$367$	−0.411143			−0.0214615			−0.0107307	−	0.999942i	$-0.503416\pi$
$367$	−0.411143			−0.0214615			−0.0107307	+	0.999942i	$0.503416\pi$
$368$		0			0
$369$		−	9.39766i		−	0.489223i
$370$		0			0
$371$	−10.4595	−	14.0489i	−0.543031	−	0.729383i
$372$		0			0
$373$	−13.7349	−	13.7349i	−0.711165	−	0.711165i	0.255614	−	0.966779i	$-0.417722\pi$
$373$	−13.7349	−	13.7349i	−0.711165	−	0.711165i	−0.966779	+	0.255614i	$0.917722\pi$
$374$		0			0
$375$	2.35507			0.121615
$376$		0			0
$377$	0.626814			0.0322826
$378$		0			0
$379$	−5.47899	+	5.47899i	−0.281437	+	0.281437i	−0.833682	−	0.552245i	$-0.813771\pi$
$379$	−5.47899	+	5.47899i	−0.281437	+	0.281437i	0.552245	+	0.833682i	$0.313771\pi$
$380$		0			0
$381$	−2.29041	−	2.29041i	−0.117341	−	0.117341i
$382$		0			0
$383$	−28.4801			−1.45527			−0.727633	−	0.685967i	$-0.759379\pi$
$383$	−28.4801			−1.45527			−0.727633	+	0.685967i	$0.759379\pi$
$384$		0			0
$385$	−0.323225	+	2.20698i	−0.0164731	+	0.112478i
$386$		0			0
$387$	−3.30395	−	3.30395i	−0.167949	−	0.167949i
$388$		0			0
$389$	−8.44152	−	8.44152i	−0.428002	−	0.428002i	0.459945	−	0.887947i	$-0.347869\pi$
$389$	−8.44152	−	8.44152i	−0.428002	−	0.428002i	−0.887947	+	0.459945i	$0.847869\pi$
$390$		0			0
$391$		−	40.1956i		−	2.03278i
$392$		0			0
$393$		−	0.285992i		−	0.0144264i
$394$		0			0
$395$	0.0354671	+	0.0354671i	0.00178454	+	0.00178454i
$396$		0			0
$397$	−3.19995	−	3.19995i	−0.160601	−	0.160601i	0.622232	−	0.782833i	$-0.286226\pi$
$397$	−3.19995	−	3.19995i	−0.160601	−	0.160601i	−0.782833	+	0.622232i	$0.786226\pi$
$398$		0			0
$399$	1.96257	−	13.4004i	0.0982514	−	0.670861i
$400$		0			0
$401$	19.1019			0.953904			0.476952	−	0.878929i	$-0.341742\pi$
$401$	19.1019			0.953904			0.476952	+	0.878929i	$0.341742\pi$
$402$		0			0
$403$	−11.7420	−	11.7420i	−0.584909	−	0.584909i
$404$		0			0
$405$	−0.167468	+	0.167468i	−0.00832156	+	0.00832156i
$406$		0			0
$407$	2.46824			0.122346
$408$		0			0
$409$	9.89111			0.489084			0.244542	−	0.969639i	$-0.421362\pi$
$409$	9.89111			0.489084			0.244542	+	0.969639i	$0.421362\pi$
$410$		0			0
$411$	−11.3036	−	11.3036i	−0.557567	−	0.557567i
$412$		0			0
$413$	17.3920	+	23.3604i	0.855802	+	1.14949i
$414$		0			0
$415$		−	2.92936i		−	0.143797i
$416$		0			0
$417$	2.86734			0.140414
$418$		0			0
$419$	−14.3138	+	14.3138i	−0.699276	+	0.699276i	−0.964254	−	0.264978i	$-0.914635\pi$
$419$	−14.3138	+	14.3138i	−0.699276	+	0.699276i	0.264978	+	0.964254i	$0.414635\pi$
$420$		0			0
$421$	−16.6566	−	16.6566i	−0.811792	−	0.811792i	0.173110	−	0.984902i	$-0.444618\pi$
$421$	−16.6566	−	16.6566i	−0.811792	−	0.811792i	−0.984902	+	0.173110i	$0.944618\pi$
$422$		0			0
$423$		−	0.799500i		−	0.0388730i
$424$		0			0
$425$	−35.3128			−1.71292
$426$		0			0
$427$	−18.0752	−	24.2781i	−0.874720	−	1.17490i
$428$		0			0
$429$	−15.2359	+	15.2359i	−0.735596	+	0.735596i
$430$		0			0
$431$		−	15.3363i		−	0.738721i	−0.929286	−	0.369361i	$-0.879577\pi$
$431$		−	15.3363i		−	0.738721i	0.929286	−	0.369361i	$-0.120423\pi$
$432$		0			0
$433$		−	4.47978i		−	0.215285i	−0.994190	−	0.107642i	$-0.965670\pi$
$433$		−	4.47978i		−	0.215285i	0.994190	−	0.107642i	$-0.0343301\pi$
$434$		0			0
$435$	−0.0173420	−	0.0173420i	−0.000831484	−	0.000831484i
$436$		0			0
$437$	−20.3695	+	20.3695i	−0.974406	+	0.974406i
$438$		0			0
$439$		−	0.171353i		−	0.00817822i	−0.999992	−	0.00408911i	$-0.998698\pi$
$439$		−	0.171353i		−	0.00817822i	0.999992	−	0.00408911i	$-0.00130161\pi$
$440$		0			0
$441$	2.00732	−	6.70602i	0.0955869	−	0.319334i
$442$		0			0
$443$	0.195944	−	0.195944i	0.00930960	−	0.00930960i	−0.702437	−	0.711746i	$-0.747905\pi$
$443$	0.195944	−	0.195944i	0.00930960	−	0.00930960i	0.711746	+	0.702437i	$0.247905\pi$
$444$		0			0
$445$	−1.26351	+	1.26351i	−0.0598961	+	0.0598961i
$446$		0			0
$447$	9.35974			0.442701
$448$		0			0
$449$	24.7162			1.16643			0.583215	−	0.812318i	$-0.301794\pi$
$449$	24.7162			1.16643			0.583215	+	0.812318i	$0.301794\pi$
$450$		0			0
$451$	−23.6546	+	23.6546i	−1.11385	+	1.11385i
$452$		0			0
$453$	3.96430	−	3.96430i	0.186259	−	0.186259i
$454$		0			0
$455$	3.75284	+	0.549625i	0.175936	+	0.0257668i
$456$		0			0
$457$			23.1197i			1.08149i	0.841186	+	0.540746i	$0.181858\pi$
$457$			23.1197i			1.08149i	−0.841186	+	0.540746i	$0.818142\pi$
$458$		0			0
$459$	5.05064	−	5.05064i	0.235744	−	0.235744i
$460$		0			0
$461$	17.4475	+	17.4475i	0.812613	+	0.812613i	0.985025	−	0.172412i	$-0.0551561\pi$
$461$	17.4475	+	17.4475i	0.812613	+	0.812613i	−0.172412	+	0.985025i	$0.555156\pi$
$462$		0			0
$463$			31.6552i			1.47114i	0.677448	+	0.735570i	$0.263086\pi$
$463$			31.6552i			1.47114i	−0.677448	+	0.735570i	$0.736914\pi$
$464$		0			0
$465$			0.649727i			0.0301304i
$466$		0			0
$467$	9.32784	−	9.32784i	0.431641	−	0.431641i	−0.457545	−	0.889186i	$-0.651271\pi$
$467$	9.32784	−	9.32784i	0.431641	−	0.431641i	0.889186	+	0.457545i	$0.151271\pi$
$468$		0			0
$469$	−22.2633	−	29.9034i	−1.02802	−	1.38081i
$470$		0			0
$471$	−16.7435			−0.771500
$472$		0			0
$473$			16.6325i			0.764765i
$474$		0			0
$475$	17.8951	+	17.8951i	0.821082	+	0.821082i
$476$		0			0
$477$	−4.68107	+	4.68107i	−0.214331	+	0.214331i
$478$		0			0
$479$	36.7307			1.67827			0.839134	−	0.543925i	$-0.183063\pi$
$479$	36.7307			1.67827			0.839134	+	0.543925i	$0.183063\pi$
$480$		0			0
$481$		−	4.19710i		−	0.191371i
$482$		0			0
$483$	−11.9427	+	8.89139i	−0.543409	+	0.404572i
$484$		0			0
$485$	2.42802	+	2.42802i	0.110251	+	0.110251i
$486$		0			0
$487$	23.4740			1.06371			0.531854	−	0.846836i	$-0.321495\pi$
$487$	23.4740			1.06371			0.531854	+	0.846836i	$0.321495\pi$
$488$		0			0
$489$	−0.907161			−0.0410233
$490$		0			0
$491$	19.6670	−	19.6670i	0.887559	−	0.887559i	−0.106730	−	0.994288i	$-0.534038\pi$
$491$	19.6670	−	19.6670i	0.887559	−	0.887559i	0.994288	+	0.106730i	$0.0340379\pi$
$492$		0			0
$493$	0.523013	+	0.523013i	0.0235553	+	0.0235553i
$494$		0			0
$495$	0.843058			0.0378926
$496$		0			0
$497$	−0.616113	−	0.0902332i	−0.0276364	−	0.00404751i
$498$		0			0
$499$	23.7424	+	23.7424i	1.06286	+	1.06286i	0.997887	+	0.0649694i	$0.0206950\pi$
$499$	23.7424	+	23.7424i	1.06286	+	1.06286i	0.0649694	+	0.997887i	$0.479305\pi$
$500$		0			0
$501$	−6.12758	−	6.12758i	−0.273760	−	0.273760i
$502$		0			0
$503$			28.1022i			1.25301i	0.779416	+	0.626507i	$0.215516\pi$
$503$			28.1022i			1.25301i	−0.779416	+	0.626507i	$0.784484\pi$
$504$		0			0
$505$		−	1.51576i		−	0.0674505i
$506$		0			0
$507$	16.7154	+	16.7154i	0.742356	+	0.742356i
$508$		0			0
$509$	5.21671	+	5.21671i	0.231227	+	0.231227i	0.813205	−	0.581978i	$-0.197721\pi$
$509$	5.21671	+	5.21671i	0.231227	+	0.231227i	−0.581978	+	0.813205i	$0.697721\pi$
$510$		0			0
$511$	29.3170	+	4.29364i	1.29691	+	0.189939i
$512$		0			0
$513$	−5.11892			−0.226006
$514$		0			0
$515$	1.16247	+	1.16247i	0.0512246	+	0.0512246i
$516$		0			0
$517$	−2.01240	+	2.01240i	−0.0885052	+	0.0885052i
$518$		0			0
$519$	−6.79365			−0.298208
$520$		0			0
$521$	19.5127			0.854867			0.427433	−	0.904047i	$-0.359418\pi$
$521$	19.5127			0.854867			0.427433	+	0.904047i	$0.359418\pi$
$522$		0			0
$523$	−21.8246	−	21.8246i	−0.954322	−	0.954322i	0.0446791	−	0.999001i	$-0.485773\pi$
$523$	−21.8246	−	21.8246i	−0.954322	−	0.954322i	−0.999001	+	0.0446791i	$0.985773\pi$
$524$		0			0
$525$	7.81128	+	10.4919i	0.340912	+	0.457903i
$526$		0			0
$527$		−	19.5950i		−	0.853570i
$528$		0			0
$529$	8.66909			0.376917
$530$		0			0
$531$	7.78362	−	7.78362i	0.337780	−	0.337780i
$532$		0			0
$533$	40.2233	+	40.2233i	1.74226	+	1.74226i
$534$		0			0
$535$			0.627743i			0.0271397i
$536$		0			0
$537$	4.29110			0.185174
$538$		0			0
$539$	−21.9321	+	11.8269i	−0.944682	+	0.509422i
$540$		0			0
$541$	20.4164	−	20.4164i	0.877769	−	0.877769i	−0.115534	−	0.993303i	$-0.536858\pi$
$541$	20.4164	−	20.4164i	0.877769	−	0.877769i	0.993303	+	0.115534i	$0.0368580\pi$
$542$		0			0
$543$		−	3.73348i		−	0.160219i
$544$		0			0
$545$		−	4.22929i		−	0.181163i
$546$		0			0
$547$	−17.2377	−	17.2377i	−0.737030	−	0.737030i	0.234972	−	0.972002i	$-0.424500\pi$
$547$	−17.2377	−	17.2377i	−0.737030	−	0.737030i	−0.972002	+	0.234972i	$0.924500\pi$
$548$		0			0
$549$	−8.08940	+	8.08940i	−0.345247	+	0.345247i
$550$		0			0
$551$		−	0.530083i		−	0.0225823i
$552$		0			0
$553$	−0.0811970	+	0.554414i	−0.00345285	+	0.0235761i
$554$		0			0
$555$	−0.116121	+	0.116121i	−0.00492904	+	0.00492904i
$556$		0			0
$557$	22.7060	−	22.7060i	0.962084	−	0.962084i	−0.0372230	−	0.999307i	$-0.511851\pi$
$557$	22.7060	−	22.7060i	0.962084	−	0.962084i	0.999307	+	0.0372230i	$0.0118512\pi$
$558$		0			0
$559$	28.2827			1.19623
$560$		0			0
$561$	−25.4256			−1.07347
$562$		0			0
$563$	−17.0537	+	17.0537i	−0.718726	+	0.718726i	−0.968344	−	0.249618i	$-0.919695\pi$
$563$	−17.0537	+	17.0537i	−0.718726	+	0.718726i	0.249618	+	0.968344i	$0.419695\pi$
$564$		0			0
$565$	−1.28711	+	1.28711i	−0.0541493	+	0.0541493i
$566$		0			0
$567$	−2.61783	−	0.383395i	−0.109938	−	0.0161011i
$568$		0			0
$569$		−	22.0290i		−	0.923502i	−0.887009	−	0.461751i	$-0.847221\pi$
$569$		−	22.0290i		−	0.923502i	0.887009	−	0.461751i	$-0.152779\pi$
$570$		0			0
$571$	−5.13880	+	5.13880i	−0.215052	+	0.215052i	−0.806410	−	0.591357i	$-0.798592\pi$
$571$	−5.13880	+	5.13880i	−0.215052	+	0.215052i	0.591357	+	0.806410i	$0.298592\pi$
$572$		0			0
$573$	8.79837	+	8.79837i	0.367557	+	0.367557i
$574$		0			0
$575$		−	27.8220i		−	1.16026i
$576$		0			0
$577$		−	13.7657i		−	0.573074i	−0.958069	−	0.286537i	$-0.907496\pi$
$577$		−	13.7657i		−	0.573074i	0.958069	−	0.286537i	$-0.0925042\pi$
$578$		0			0
$579$	4.24406	−	4.24406i	0.176377	−	0.176377i
$580$		0			0
$581$	26.2487	−	19.5424i	1.08898	−	0.810754i
$582$		0			0
$583$	23.5652			0.975969
$584$		0			0
$585$		−	1.43357i		−	0.0592709i
$586$		0			0
$587$	10.4983	+	10.4983i	0.433312	+	0.433312i	0.889753	−	0.456442i	$-0.150876\pi$
$587$	10.4983	+	10.4983i	0.433312	+	0.433312i	−0.456442	+	0.889753i	$0.650876\pi$
$588$		0			0
$589$	−9.92994	+	9.92994i	−0.409156	+	0.409156i
$590$		0			0
$591$	−15.4091			−0.633844
$592$		0			0
$593$		−	16.4073i		−	0.673768i	−0.941546	−	0.336884i	$-0.890627\pi$
$593$		−	16.4073i		−	0.673768i	0.941546	−	0.336884i	$-0.109373\pi$
$594$		0			0
$595$	2.67276	+	3.58997i	0.109572	+	0.147175i
$596$		0			0
$597$	−6.52423	−	6.52423i	−0.267019	−	0.267019i
$598$		0			0
$599$	21.9084			0.895154			0.447577	−	0.894245i	$-0.352287\pi$
$599$	21.9084			0.895154			0.447577	+	0.894245i	$0.352287\pi$
$600$		0			0
$601$	41.7949			1.70485			0.852425	−	0.522849i	$-0.175131\pi$
$601$	41.7949			1.70485			0.852425	+	0.522849i	$0.175131\pi$
$602$		0			0
$603$	−9.96374	+	9.96374i	−0.405755	+	0.405755i
$604$		0			0
$605$	−0.279888	−	0.279888i	−0.0113791	−	0.0113791i
$606$		0			0
$607$	42.9099			1.74166			0.870830	−	0.491584i	$-0.163582\pi$
$607$	42.9099			1.74166			0.870830	+	0.491584i	$0.163582\pi$
$608$		0			0
$609$	0.0397021	−	0.271086i	0.00160881	−	0.0109850i
$610$		0			0
$611$	3.42197	+	3.42197i	0.138438	+	0.138438i
$612$		0			0
$613$	−3.30257	−	3.30257i	−0.133390	−	0.133390i	0.637260	−	0.770649i	$-0.280068\pi$
$613$	−3.30257	−	3.30257i	−0.133390	−	0.133390i	−0.770649	+	0.637260i	$0.780068\pi$
$614$		0			0
$615$		−	2.22570i		−	0.0897490i
$616$		0			0
$617$		−	20.7360i		−	0.834800i	−0.908723	−	0.417400i	$-0.862941\pi$
$617$		−	20.7360i		−	0.834800i	0.908723	−	0.417400i	$-0.137059\pi$
$618$		0			0
$619$	16.9754	+	16.9754i	0.682298	+	0.682298i	0.960518	−	0.278219i	$-0.0897442\pi$
$619$	16.9754	+	16.9754i	0.682298	+	0.682298i	−0.278219	+	0.960518i	$0.589744\pi$
$620$		0			0
$621$	3.97926	+	3.97926i	0.159682	+	0.159682i
$622$		0			0
$623$	−19.7509	−	2.89263i	−0.791303	−	0.115891i
$624$		0			0
$625$	−24.1618			−0.966471
$626$		0			0
$627$	12.8847	+	12.8847i	0.514564	+	0.514564i
$628$		0			0
$629$	3.50206	−	3.50206i	0.139636	−	0.139636i
$630$		0			0
$631$	−41.1032			−1.63629			−0.818146	−	0.575010i	$-0.804998\pi$
$631$	−41.1032			−1.63629			−0.818146	+	0.575010i	$0.804998\pi$
$632$		0			0
$633$	−10.6620			−0.423776
$634$		0			0
$635$	−0.542451	−	0.542451i	−0.0215265	−	0.0215265i
$636$		0			0
$637$	20.1110	+	37.2943i	0.796828	+	1.47765i
$638$		0			0
$639$			0.235353i			0.00931041i
$640$		0			0
$641$	−7.71535			−0.304738			−0.152369	−	0.988324i	$-0.548690\pi$
$641$	−7.71535			−0.304738			−0.152369	+	0.988324i	$0.548690\pi$
$642$		0			0
$643$	−32.7634	+	32.7634i	−1.29206	+	1.29206i	−0.358555	+	0.933509i	$0.616730\pi$
$643$	−32.7634	+	32.7634i	−1.29206	+	1.29206i	−0.933509	+	0.358555i	$0.883270\pi$
$644$		0			0
$645$	−0.782492	−	0.782492i	−0.0308106	−	0.0308106i
$646$		0			0
$647$		−	17.2904i		−	0.679756i	−0.940469	−	0.339878i	$-0.889614\pi$
$647$		−	17.2904i		−	0.679756i	0.940469	−	0.339878i	$-0.110386\pi$
$648$		0			0
$649$	−39.1839			−1.53810
$650$		0			0
$651$	−5.82192	+	4.33446i	−0.228179	+	0.169881i
$652$		0			0
$653$	2.03773	−	2.03773i	0.0797425	−	0.0797425i	−0.666111	−	0.745853i	$-0.732042\pi$
$653$	2.03773	−	2.03773i	0.0797425	−	0.0797425i	0.745853	+	0.666111i	$0.232042\pi$
$654$		0			0
$655$		−	0.0677331i		−	0.00264655i
$656$		0			0
$657$		−	11.1990i		−	0.436914i
$658$		0			0
$659$	17.0779	+	17.0779i	0.665259	+	0.665259i	0.956615	−	0.291356i	$-0.0941064\pi$
$659$	17.0779	+	17.0779i	0.665259	+	0.665259i	−0.291356	+	0.956615i	$0.594106\pi$
$660$		0			0
$661$	2.96315	−	2.96315i	0.115253	−	0.115253i	−0.647128	−	0.762381i	$-0.724030\pi$
$661$	2.96315	−	2.96315i	0.115253	−	0.115253i	0.762381	+	0.647128i	$0.224030\pi$
$662$		0			0
$663$			43.2348i			1.67910i
$664$		0			0
$665$	0.464806	−	3.17370i	0.0180244	−	0.123071i
$666$		0			0
$667$	−0.412068	+	0.412068i	−0.0159553	+	0.0159553i
$668$		0			0
$669$	−0.590289	+	0.590289i	−0.0228219	+	0.0228219i
$670$		0			0
$671$	40.7232			1.57210
$672$		0			0
$673$	−8.94408			−0.344769			−0.172384	−	0.985030i	$-0.555147\pi$
$673$	−8.94408			−0.344769			−0.172384	+	0.985030i	$0.555147\pi$
$674$		0			0
$675$	3.49587	−	3.49587i	0.134556	−	0.134556i
$676$		0			0
$677$	7.48867	−	7.48867i	0.287813	−	0.287813i	−0.548402	−	0.836215i	$-0.684764\pi$
$677$	7.48867	−	7.48867i	0.287813	−	0.287813i	0.836215	+	0.548402i	$0.184764\pi$
$678$		0			0
$679$	−5.55862	+	37.9543i	−0.213320	+	1.45655i
$680$		0			0
$681$			5.55476i			0.212859i
$682$		0			0
$683$	−19.2209	+	19.2209i	−0.735467	+	0.735467i	−0.971697	−	0.236230i	$-0.924088\pi$
$683$	−19.2209	+	19.2209i	−0.735467	+	0.735467i	0.236230	+	0.971697i	$0.424088\pi$
$684$		0			0
$685$	−2.67710	−	2.67710i	−0.102287	−	0.102287i
$686$		0			0
$687$			1.66260i			0.0634321i
$688$		0			0
$689$		−	40.0712i		−	1.52659i
$690$		0			0
$691$	−6.36522	+	6.36522i	−0.242144	+	0.242144i	−0.817737	−	0.575592i	$-0.804771\pi$
$691$	−6.36522	+	6.36522i	−0.242144	+	0.242144i	0.575592	+	0.817737i	$0.304771\pi$
$692$		0			0
$693$	5.62422	+	7.55428i	0.213646	+	0.286964i
$694$		0			0
$695$	0.679089			0.0257593
$696$		0			0
$697$			67.1245i			2.54252i
$698$		0			0
$699$	7.50047	+	7.50047i	0.283694	+	0.283694i
$700$		0			0
$701$	2.92182	−	2.92182i	0.110356	−	0.110356i	−0.649773	−	0.760128i	$-0.725136\pi$
$701$	2.92182	−	2.92182i	0.110356	−	0.110356i	0.760128	+	0.649773i	$0.225136\pi$
$702$		0			0
$703$	−3.54940			−0.133868
$704$		0			0
$705$		−	0.189350i		−	0.00713134i
$706$		0			0
$707$	13.5821	−	10.1120i	0.510807	−	0.380299i
$708$		0			0
$709$	20.3789	+	20.3789i	0.765346	+	0.765346i	0.977283	−	0.211937i	$-0.0679773\pi$
$709$	20.3789	+	20.3789i	0.765346	+	0.765346i	−0.211937	+	0.977283i	$0.567977\pi$
$710$		0			0
$711$	0.211784			0.00794252
$712$		0			0
$713$	15.4384			0.578171
$714$		0			0
$715$	−3.60840	+	3.60840i	−0.134947	+	0.134947i
$716$		0			0
$717$	−5.06480	−	5.06480i	−0.189148	−	0.189148i
$718$		0			0
$719$	28.7270			1.07134			0.535668	−	0.844429i	$-0.320060\pi$
$719$	28.7270			1.07134			0.535668	+	0.844429i	$0.320060\pi$
$720$		0			0
$721$	−2.66132	+	18.1715i	−0.0991128	+	0.676742i
$722$		0			0
$723$	−9.49221	−	9.49221i	−0.353019	−	0.353019i
$724$		0			0
$725$	0.362011	+	0.362011i	0.0134447	+	0.0134447i
$726$		0			0
$727$			53.1288i			1.97044i	0.171302	+	0.985219i	$0.445203\pi$
$727$			53.1288i			1.97044i	−0.171302	+	0.985219i	$0.554797\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	23.5990	+	23.5990i	0.872841	+	0.872841i
$732$		0			0
$733$	22.8737	+	22.8737i	0.844860	+	0.844860i	0.989486	−	0.144627i	$-0.0461981\pi$
$733$	22.8737	+	22.8737i	0.844860	+	0.844860i	−0.144627	+	0.989486i	$0.546198\pi$
$734$		0			0
$735$	0.475406	−	1.58822i	0.0175356	−	0.0585825i
$736$		0			0
$737$	50.1589			1.84763
$738$		0			0
$739$	−17.1191	−	17.1191i	−0.629735	−	0.629735i	0.318266	−	0.948001i	$-0.396899\pi$
$739$	−17.1191	−	17.1191i	−0.629735	−	0.629735i	−0.948001	+	0.318266i	$0.896899\pi$
$740$		0			0
$741$	21.9097	−	21.9097i	0.804871	−	0.804871i
$742$		0			0
$743$	52.4170			1.92299			0.961496	−	0.274818i	$-0.0886175\pi$
$743$	52.4170			1.92299			0.961496	+	0.274818i	$0.0886175\pi$
$744$		0			0
$745$	2.21672			0.0812143
$746$		0			0
$747$	−8.74602	−	8.74602i	−0.320000	−	0.320000i
$748$		0			0
$749$	−5.62494	+	4.18781i	−0.205531	+	0.153019i
$750$		0			0
$751$			5.04763i			0.184191i	0.995750	+	0.0920953i	$0.0293564\pi$
$751$			5.04763i			0.184191i	−0.995750	+	0.0920953i	$0.970644\pi$
$752$		0			0
$753$	7.47769			0.272502
$754$		0			0
$755$	0.938887	−	0.938887i	0.0341696	−	0.0341696i
$756$		0			0
$757$	−19.3984	−	19.3984i	−0.705045	−	0.705045i	0.260444	−	0.965489i	$-0.416131\pi$
$757$	−19.3984	−	19.3984i	−0.705045	−	0.705045i	−0.965489	+	0.260444i	$0.916131\pi$
$758$		0			0
$759$		−	20.0322i		−	0.727122i
$760$		0			0
$761$	−17.5182			−0.635034			−0.317517	−	0.948253i	$-0.602849\pi$
$761$	−17.5182			−0.635034			−0.317517	+	0.948253i	$0.602849\pi$
$762$		0			0
$763$	37.8969	−	28.2145i	1.37196	−	1.02143i
$764$		0			0
$765$	1.19617	−	1.19617i	0.0432476	−	0.0432476i
$766$		0			0
$767$			66.6299i			2.40587i
$768$		0			0
$769$		−	31.6824i		−	1.14250i	−0.820777	−	0.571249i	$-0.806459\pi$
$769$		−	31.6824i		−	1.14250i	0.820777	−	0.571249i	$-0.193541\pi$
$770$		0			0
$771$	−0.300532	−	0.300532i	−0.0108234	−	0.0108234i
$772$		0			0
$773$	−34.9011	+	34.9011i	−1.25531	+	1.25531i	−0.301998	+	0.953308i	$0.597654\pi$
$773$	−34.9011	+	34.9011i	−1.25531	+	1.25531i	−0.953308	+	0.301998i	$0.902346\pi$
$774$		0			0
$775$		−	13.5629i		−	0.487195i
$776$		0			0
$777$	−1.81517	−	0.265842i	−0.0651189	−	0.00953704i
$778$		0			0
$779$	34.0160	−	34.0160i	1.21875	−	1.21875i
$780$		0			0
$781$	0.592400	−	0.592400i	0.0211977	−	0.0211977i
$782$		0			0
$783$	−0.103554			−0.00370071
$784$		0			0
$785$	−3.96546			−0.141533
$786$		0			0
$787$	20.4440	−	20.4440i	0.728749	−	0.728749i	−0.241622	−	0.970371i	$-0.577679\pi$
$787$	20.4440	−	20.4440i	0.728749	−	0.728749i	0.970371	+	0.241622i	$0.0776793\pi$
$788$		0			0
$789$	−18.8099	+	18.8099i	−0.669651	+	0.669651i
$790$		0			0
$791$	−20.1199	−	2.94667i	−0.715381	−	0.104772i
$792$		0			0
$793$		−	69.2475i		−	2.45905i
$794$		0			0
$795$	−1.10864	+	1.10864i	−0.0393195	+	0.0393195i
$796$		0			0
$797$	−8.54875	−	8.54875i	−0.302812	−	0.302812i	0.539301	−	0.842113i	$-0.318689\pi$
$797$	−8.54875	−	8.54875i	−0.302812	−	0.302812i	−0.842113	+	0.539301i	$0.818689\pi$
$798$		0			0
$799$			5.71057i			0.202026i
$800$		0			0
$801$			7.54477i			0.266581i
$802$		0			0
$803$	−28.1886	+	28.1886i	−0.994755	+	0.994755i
$804$		0			0
$805$	−2.82845	+	2.10580i	−0.0996896	+	0.0742196i
$806$		0			0
$807$	−21.6539			−0.762253
$808$		0			0
$809$		−	13.2816i		−	0.466956i	−0.972362	−	0.233478i	$-0.924989\pi$
$809$		−	13.2816i		−	0.466956i	0.972362	−	0.233478i	$-0.0750108\pi$
$810$		0			0
$811$	2.35686	+	2.35686i	0.0827606	+	0.0827606i	0.747275	−	0.664515i	$-0.231362\pi$
$811$	2.35686	+	2.35686i	0.0827606	+	0.0827606i	−0.664515	+	0.747275i	$0.731362\pi$
$812$		0			0
$813$	−14.2338	+	14.2338i	−0.499200	+	0.499200i
$814$		0			0
$815$	−0.214848			−0.00752580
$816$		0			0
$817$		−	23.9181i		−	0.836787i
$818$		0			0
$819$	12.8456	−	9.56366i	0.448863	−	0.334181i
$820$		0			0
$821$	2.18290	+	2.18290i	0.0761836	+	0.0761836i	0.744172	−	0.667988i	$-0.232844\pi$
$821$	2.18290	+	2.18290i	0.0761836	+	0.0761836i	−0.667988	+	0.744172i	$0.732844\pi$
$822$		0			0
$823$	−9.24257			−0.322176			−0.161088	−	0.986940i	$-0.551500\pi$
$823$	−9.24257			−0.322176			−0.161088	+	0.986940i	$0.551500\pi$
$824$		0			0
$825$	−17.5987			−0.612708
$826$		0			0
$827$	25.0240	−	25.0240i	0.870171	−	0.870171i	−0.122319	−	0.992491i	$-0.539033\pi$
$827$	25.0240	−	25.0240i	0.870171	−	0.870171i	0.992491	+	0.122319i	$0.0390332\pi$
$828$		0			0
$829$	−9.38010	−	9.38010i	−0.325784	−	0.325784i	0.525197	−	0.850981i	$-0.323992\pi$
$829$	−9.38010	−	9.38010i	−0.325784	−	0.325784i	−0.850981	+	0.525197i	$0.823992\pi$
$830$		0			0
$831$	23.7732			0.824683
$832$		0			0
$833$	−14.3377	+	47.8989i	−0.496771	+	1.65960i
$834$		0			0
$835$	−1.45123	−	1.45123i	−0.0502219	−	0.0502219i
$836$		0			0
$837$	1.93985	+	1.93985i	0.0670511	+	0.0670511i
$838$		0			0
$839$		−	20.9881i		−	0.724591i	−0.932063	−	0.362296i	$-0.881993\pi$
$839$		−	20.9881i		−	0.724591i	0.932063	−	0.362296i	$-0.118007\pi$
$840$		0			0
$841$			28.9893i			0.999630i
$842$		0			0
$843$	−3.16576	−	3.16576i	−0.109034	−	0.109034i
$844$		0			0
$845$	3.95880	+	3.95880i	0.136187	+	0.136187i
$846$		0			0
$847$	0.640765	−	4.37514i	0.0220169	−	0.150332i
$848$		0			0
$849$	13.6858			0.469695
$850$		0			0
$851$	2.75918	+	2.75918i	0.0945835	+	0.0945835i
$852$		0			0
$853$	−29.8015	+	29.8015i	−1.02038	+	1.02038i	−0.0205966	+	0.999788i	$0.506557\pi$
$853$	−29.8015	+	29.8015i	−1.02038	+	1.02038i	−0.999788	+	0.0205966i	$0.993443\pi$
$854$		0			0
$855$	−1.21234			−0.0414612
$856$		0			0
$857$	51.8982			1.77281			0.886405	−	0.462910i	$-0.153195\pi$
$857$	51.8982			1.77281			0.886405	+	0.462910i	$0.153195\pi$
$858$		0			0
$859$	10.4965	+	10.4965i	0.358138	+	0.358138i	0.863126	−	0.504988i	$-0.168503\pi$
$859$	10.4965	+	10.4965i	0.358138	+	0.358138i	−0.504988	+	0.863126i	$0.668503\pi$
$860$		0			0
$861$	19.9436	−	14.8481i	0.679675	−	0.506023i
$862$		0			0
$863$		−	21.5284i		−	0.732836i	−0.930450	−	0.366418i	$-0.880584\pi$
$863$		−	21.5284i		−	0.732836i	0.930450	−	0.366418i	$-0.119416\pi$
$864$		0			0
$865$	−1.60898			−0.0547070
$866$		0			0
$867$	−24.0543	+	24.0543i	−0.816925	+	0.816925i
$868$		0			0
$869$	−0.533075	−	0.533075i	−0.0180833	−	0.0180833i
$870$		0			0
$871$		−	85.2923i		−	2.89002i
$872$		0			0
$873$	14.4984			0.490697
$874$		0			0
$875$	3.72097	+	4.99790i	0.125792	+	0.168960i
$876$		0			0
$877$	−6.07418	+	6.07418i	−0.205110	+	0.205110i	−0.802185	−	0.597075i	$-0.796329\pi$
$877$	−6.07418	+	6.07418i	−0.205110	+	0.205110i	0.597075	+	0.802185i	$0.296329\pi$
$878$		0			0
$879$			12.0038i			0.404877i
$880$		0			0
$881$		−	22.7089i		−	0.765081i	−0.923939	−	0.382541i	$-0.875049\pi$
$881$		−	22.7089i		−	0.765081i	0.923939	−	0.382541i	$-0.124951\pi$
$882$		0			0
$883$	−27.9984	−	27.9984i	−0.942222	−	0.942222i	0.0561975	−	0.998420i	$-0.482102\pi$
$883$	−27.9984	−	27.9984i	−0.942222	−	0.942222i	−0.998420	+	0.0561975i	$0.982102\pi$
$884$		0			0
$885$	1.84344	−	1.84344i	0.0619665	−	0.0619665i
$886$		0			0
$887$			33.0274i			1.10895i	0.832200	+	0.554475i	$0.187081\pi$
$887$			33.0274i			1.10895i	−0.832200	+	0.554475i	$0.812919\pi$
$888$		0			0
$889$	1.24187	−	8.47948i	0.0416509	−	0.284393i
$890$		0			0
$891$	2.51707	−	2.51707i	0.0843250	−	0.0843250i
$892$		0			0
$893$	2.89389	−	2.89389i	0.0968402	−	0.0968402i
$894$		0			0
$895$	1.01628			0.0339707
$896$		0			0
$897$	−34.0636			−1.13735
$898$		0			0
$899$	−0.200879	+	0.200879i	−0.00669969	+	0.00669969i
$900$		0			0
$901$	33.4354	−	33.4354i	1.11389	−	1.11389i
$902$		0			0
$903$	1.79141	−	12.2317i	0.0596144	−	0.407047i
$904$		0			0
$905$		−	0.884220i		−	0.0293925i
$906$		0			0
$907$	−6.57527	+	6.57527i	−0.218328	+	0.218328i	−0.807794	−	0.589465i	$-0.799338\pi$
$907$	−6.57527	+	6.57527i	−0.218328	+	0.218328i	0.589465	+	0.807794i	$0.299338\pi$
$908$		0			0
$909$	−4.52552	−	4.52552i	−0.150102	−	0.150102i
$910$		0			0
$911$			12.7400i			0.422095i	0.977476	+	0.211047i	$0.0676875\pi$
$911$			12.7400i			0.422095i	−0.977476	+	0.211047i	$0.932313\pi$
$912$		0			0
$913$			44.0287i			1.45714i
$914$		0			0
$915$	−1.91586	+	1.91586i	−0.0633364	+	0.0633364i
$916$		0			0
$917$	0.606927	−	0.451861i	0.0200425	−	0.0149218i
$918$		0			0
$919$	−22.9394			−0.756700			−0.378350	−	0.925663i	$-0.623508\pi$
$919$	−22.9394			−0.756700			−0.378350	+	0.925663i	$0.623508\pi$
$920$		0			0
$921$		−	12.8424i		−	0.423170i
$922$		0			0
$923$	−1.00734	−	1.00734i	−0.0331571	−	0.0331571i
$924$		0			0
$925$	2.42400	−	2.42400i	0.0797006	−	0.0797006i
$926$		0			0
$927$	6.94145			0.227987
$928$		0			0
$929$			29.6249i			0.971963i	0.873969	+	0.485981i	$0.161538\pi$
$929$			29.6249i			0.971963i	−0.873969	+	0.485981i	$0.838462\pi$
$930$		0			0
$931$	31.5390	−	17.0075i	1.03365	−	0.557398i
$932$		0			0
$933$	−12.5217	−	12.5217i	−0.409942	−	0.409942i
$934$		0			0
$935$	−6.02169			−0.196930
$936$		0			0
$937$	27.4992			0.898361			0.449181	−	0.893441i	$-0.351716\pi$
$937$	27.4992			0.898361			0.449181	+	0.893441i	$0.351716\pi$
$938$		0			0
$939$	−9.99689	+	9.99689i	−0.326236	+	0.326236i
$940$		0			0
$941$	27.7697	+	27.7697i	0.905265	+	0.905265i	0.995886	−	0.0906204i	$-0.0288850\pi$
$941$	27.7697	+	27.7697i	0.905265	+	0.905265i	−0.0906204	+	0.995886i	$0.528885\pi$
$942$		0			0
$943$	−52.8856			−1.72219
$944$		0			0
$945$	−0.619994	−	0.0908017i	−0.0201684	−	0.00295378i
$946$		0			0
$947$	9.00930	+	9.00930i	0.292763	+	0.292763i	0.838171	−	0.545408i	$-0.183625\pi$
$947$	9.00930	+	9.00930i	0.292763	+	0.292763i	−0.545408	+	0.838171i	$0.683625\pi$
$948$		0			0
$949$	47.9331	+	47.9331i	1.55598	+	1.55598i
$950$		0			0
$951$		−	9.25707i		−	0.300181i
$952$		0			0
$953$			19.0905i			0.618402i	0.950997	+	0.309201i	$0.100062\pi$
$953$			19.0905i			0.618402i	−0.950997	+	0.309201i	$0.899938\pi$
$954$		0			0
$955$	2.08377	+	2.08377i	0.0674291	+	0.0674291i
$956$		0			0
$957$	0.260652	+	0.260652i	0.00842569	+	0.00842569i
$958$		0			0
$959$	6.12887	−	41.8479i	0.197911	−	1.35134i
$960$		0			0
$961$	−23.4739			−0.757224
$962$		0			0
$963$	1.87422	+	1.87422i	0.0603958	+	0.0603958i
$964$		0			0
$965$	1.00514	−	1.00514i	0.0323567	−	0.0323567i
$966$		0			0
$967$	−26.0264			−0.836951			−0.418476	−	0.908228i	$-0.637435\pi$
$967$	−26.0264			−0.836951			−0.418476	+	0.908228i	$0.637435\pi$
$968$		0			0
$969$	36.5628			1.17457
$970$		0			0
$971$	−28.6108	−	28.6108i	−0.918165	−	0.918165i	0.0787312	−	0.996896i	$-0.474913\pi$
$971$	−28.6108	−	28.6108i	−0.918165	−	0.918165i	−0.996896	+	0.0787312i	$0.974913\pi$
$972$		0			0
$973$	4.53034	+	6.08502i	0.145236	+	0.195077i
$974$		0			0
$975$			29.9256i			0.958387i
$976$		0			0
$977$	−56.1767			−1.79725			−0.898626	−	0.438716i	$-0.855433\pi$
$977$	−56.1767			−1.79725			−0.898626	+	0.438716i	$0.855433\pi$
$978$		0			0
$979$	18.9907	−	18.9907i	0.606946	−	0.606946i
$980$		0			0
$981$	−12.6272	−	12.6272i	−0.403154	−	0.403154i
$982$		0			0
$983$		−	32.0551i		−	1.02240i	−0.859462	−	0.511200i	$-0.829201\pi$
$983$		−	32.0551i		−	1.02240i	0.859462	−	0.511200i	$-0.170799\pi$
$984$		0			0
$985$	−3.64941			−0.116280
$986$		0			0
$987$	1.69669	−	1.26319i	0.0540061	−	0.0402079i
$988$		0			0
$989$	−18.5931	+	18.5931i	−0.591225	+	0.591225i
$990$		0			0
$991$		−	54.6720i		−	1.73671i	−0.495940	−	0.868357i	$-0.665177\pi$
$991$		−	54.6720i		−	1.73671i	0.495940	−	0.868357i	$-0.334823\pi$
$992$		0			0
$993$			12.9747i			0.411740i
$994$		0			0
$995$	−1.54517	−	1.54517i	−0.0489852	−	0.0489852i
$996$		0			0
$997$	−5.64016	+	5.64016i	−0.178626	+	0.178626i	−0.790756	−	0.612131i	$-0.790313\pi$
$997$	−5.64016	+	5.64016i	−0.178626	+	0.178626i	0.612131	+	0.790756i	$0.290313\pi$
$998$		0			0
$999$			0.693389i			0.0219379i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.u.a.1231.25		64
4.3	odd	2		336.2.u.a.139.7	✓	64
7.6	odd	2	inner	1344.2.u.a.1231.8		64
16.3	odd	4	inner	1344.2.u.a.559.8		64
16.13	even	4		336.2.u.a.307.8	yes	64
28.27	even	2		336.2.u.a.139.8	yes	64
112.13	odd	4		336.2.u.a.307.7	yes	64
112.83	even	4	inner	1344.2.u.a.559.25		64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.u.a.139.7	✓	64	4.3	odd	2
336.2.u.a.139.8	yes	64	28.27	even	2
336.2.u.a.307.7	yes	64	112.13	odd	4
336.2.u.a.307.8	yes	64	16.13	even	4
1344.2.u.a.559.8		64	16.3	odd	4	inner
1344.2.u.a.559.25		64	112.83	even	4	inner
1344.2.u.a.1231.8		64	7.6	odd	2	inner
1344.2.u.a.1231.25		64	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.u (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.167468 - 0.167468i) q^{5} +(2.61783 + 0.383395i) q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.167468 - 0.167468i) q^{5} +(2.61783 + 0.383395i) q^{7} -1.00000i q^{9} +(-2.51707 + 2.51707i) q^{11} +(4.28014 + 4.28014i) q^{13} -0.236836i q^{15} +7.14268i q^{17} +(3.61962 - 3.61962i) q^{19} +(2.12218 - 1.57998i) q^{21} -5.62753 q^{23} +4.94391i q^{25} +(-0.707107 - 0.707107i) q^{27} +(0.0732236 - 0.0732236i) q^{29} -2.74337 q^{31} +3.55968i q^{33} +(0.502609 - 0.374196i) q^{35} +(-0.490300 - 0.490300i) q^{37} +6.05303 q^{39} +9.39766 q^{41} +(3.30395 - 3.30395i) q^{43} +(-0.167468 - 0.167468i) q^{45} +0.799500 q^{47} +(6.70602 + 2.00732i) q^{49} +(5.05064 + 5.05064i) q^{51} +(-4.68107 - 4.68107i) q^{53} +0.843058i q^{55} -5.11892i q^{57} +(7.78362 + 7.78362i) q^{59} +(-8.08940 - 8.08940i) q^{61} +(0.383395 - 2.61783i) q^{63} +1.43357 q^{65} +(-9.96374 - 9.96374i) q^{67} +(-3.97926 + 3.97926i) q^{69} -0.235353 q^{71} +11.1990 q^{73} +(3.49587 + 3.49587i) q^{75} +(-7.55428 + 5.62422i) q^{77} +0.211784i q^{79} -1.00000 q^{81} +(8.74602 - 8.74602i) q^{83} +(1.19617 + 1.19617i) q^{85} -0.103554i q^{87} -7.54477 q^{89} +(9.56366 + 12.8456i) q^{91} +(-1.93985 + 1.93985i) q^{93} -1.21234i q^{95} +14.4984i q^{97} +(2.51707 + 2.51707i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q - 8 q^{11} + 16 q^{23} + 16 q^{29} - 24 q^{35} + 16 q^{37} + 8 q^{43} + 16 q^{53} - 56 q^{67} + 128 q^{71} - 64 q^{81} - 8 q^{91} + 8 q^{99}+O(q^{100}) \) 64 * q - 8 * q^11 + 16 * q^23 + 16 * q^29 - 24 * q^35 + 16 * q^37 + 8 * q^43 + 16 * q^53 - 56 * q^67 + 128 * q^71 - 64 * q^81 - 8 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more