LMFDB - Embedded newform 1344.2.s.d.239.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(239,1344)]

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, 2, 0]))

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

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

chi := DirichletCharacter("1344.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.s (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:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.12
Character	$\chi$	$=$	1344.239
Dual form			1344.2.s.d.911.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.277087 - 1.70974i) q^{3} +(0.459458 + 0.459458i) q^{5} -1.00000 q^{7} +(-2.84645 + 0.947494i) q^{9} +O(q^{10})$	\(q+(-0.277087 - 1.70974i) q^{3} +(0.459458 + 0.459458i) q^{5} -1.00000 q^{7} +(-2.84645 + 0.947494i) q^{9} +(-1.43432 + 1.43432i) q^{11} +(-1.18426 - 1.18426i) q^{13} +(0.658245 - 0.912864i) q^{15} +7.20845i q^{17} +(2.23529 - 2.23529i) q^{19} +(0.277087 + 1.70974i) q^{21} +0.540517i q^{23} -4.57780i q^{25} +(2.40868 + 4.60416i) q^{27} +(-4.14139 + 4.14139i) q^{29} +10.4187i q^{31} +(2.84976 + 2.05490i) q^{33} +(-0.459458 - 0.459458i) q^{35} +(-1.36361 + 1.36361i) q^{37} +(-1.69663 + 2.35292i) q^{39} -2.16181 q^{41} +(6.40813 + 6.40813i) q^{43} +(-1.74315 - 0.872488i) q^{45} -7.63641 q^{47} +1.00000 q^{49} +(12.3246 - 1.99736i) q^{51} +(2.29189 + 2.29189i) q^{53} -1.31802 q^{55} +(-4.44114 - 3.20240i) q^{57} +(6.32068 - 6.32068i) q^{59} +(0.637183 + 0.637183i) q^{61} +(2.84645 - 0.947494i) q^{63} -1.08823i q^{65} +(2.14006 - 2.14006i) q^{67} +(0.924145 - 0.149770i) q^{69} +14.6958i q^{71} +0.233482i q^{73} +(-7.82686 + 1.26845i) q^{75} +(1.43432 - 1.43432i) q^{77} -4.93151i q^{79} +(7.20451 - 5.39398i) q^{81} +(-1.76156 - 1.76156i) q^{83} +(-3.31198 + 3.31198i) q^{85} +(8.22824 + 5.93319i) q^{87} -7.79085 q^{89} +(1.18426 + 1.18426i) q^{91} +(17.8133 - 2.88688i) q^{93} +2.05404 q^{95} -10.0585 q^{97} +(2.72371 - 5.44174i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{7}+O(q^{10}) $ 48 * q - 48 * q^7	$ 48 q - 48 q^{7} - 8 q^{19} + 12 q^{27} + 16 q^{37} + 24 q^{39} + 48 q^{43} + 20 q^{45} + 48 q^{49} + 32 q^{55} + 8 q^{61} + 16 q^{67} - 28 q^{69} + 12 q^{75} - 48 q^{85} - 56 q^{87} - 64 q^{93} - 32 q^{99}+O(q^{100}) $ 48 * q - 48 * q^7 - 8 * q^19 + 12 * q^27 + 16 * q^37 + 24 * q^39 + 48 * q^43 + 20 * q^45 + 48 * q^49 + 32 * q^55 + 8 * q^61 + 16 * q^67 - 28 * q^69 + 12 * q^75 - 48 * q^85 - 56 * q^87 - 64 * q^93 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.277087	−	1.70974i	−0.159976	−	0.987121i
$3$	−0.277087	−	1.70974i	−0.159976	−	0.987121i
$4$		0			0
$5$	0.459458	+	0.459458i	0.205476	+	0.205476i	0.802341	−	0.596866i	$-0.203587\pi$
$5$	0.459458	+	0.459458i	0.205476	+	0.205476i	−0.596866	+	0.802341i	$0.703587\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$	−2.84645	+	0.947494i	−0.948815	+	0.315831i
$10$		0			0
$11$	−1.43432	+	1.43432i	−0.432465	+	0.432465i	−0.889466	−	0.457001i	$-0.848924\pi$
$11$	−1.43432	+	1.43432i	−0.432465	+	0.432465i	0.457001	+	0.889466i	$0.348924\pi$
$12$		0			0
$13$	−1.18426	−	1.18426i	−0.328454	−	0.328454i	0.523545	−	0.851998i	$-0.324609\pi$
$13$	−1.18426	−	1.18426i	−0.328454	−	0.328454i	−0.851998	+	0.523545i	$0.824609\pi$
$14$		0			0
$15$	0.658245	−	0.912864i	0.169958	−	0.235701i
$16$		0			0
$17$			7.20845i			1.74831i	0.485650	+	0.874153i	$0.338583\pi$
$17$			7.20845i			1.74831i	−0.485650	+	0.874153i	$0.661417\pi$
$18$		0			0
$19$	2.23529	−	2.23529i	0.512811	−	0.512811i	−0.402576	−	0.915387i	$-0.631885\pi$
$19$	2.23529	−	2.23529i	0.512811	−	0.512811i	0.915387	+	0.402576i	$0.131885\pi$
$20$		0			0
$21$	0.277087	+	1.70974i	0.0604652	+	0.373097i
$22$		0			0
$23$			0.540517i			0.112706i	0.998411	+	0.0563528i	$0.0179471\pi$
$23$			0.540517i			0.112706i	−0.998411	+	0.0563528i	$0.982053\pi$
$24$		0			0
$25$		−	4.57780i		−	0.915559i
$26$		0			0
$27$	2.40868	+	4.60416i	0.463551	+	0.886070i
$28$		0			0
$29$	−4.14139	+	4.14139i	−0.769037	+	0.769037i	−0.977937	−	0.208900i	$-0.933012\pi$
$29$	−4.14139	+	4.14139i	−0.769037	+	0.769037i	0.208900	+	0.977937i	$0.433012\pi$
$30$		0			0
$31$			10.4187i			1.87125i	0.352992	+	0.935626i	$0.385164\pi$
$31$			10.4187i			1.87125i	−0.352992	+	0.935626i	$0.614836\pi$
$32$		0			0
$33$	2.84976	+	2.05490i	0.496079	+	0.357711i
$34$		0			0
$35$	−0.459458	−	0.459458i	−0.0776625	−	0.0776625i
$36$		0			0
$37$	−1.36361	+	1.36361i	−0.224176	+	0.224176i	−0.810254	−	0.586078i	$-0.800671\pi$
$37$	−1.36361	+	1.36361i	−0.224176	+	0.224176i	0.586078	+	0.810254i	$0.300671\pi$
$38$		0			0
$39$	−1.69663	+	2.35292i	−0.271679	+	0.376768i
$40$		0			0
$41$	−2.16181			−0.337618			−0.168809	−	0.985649i	$-0.553992\pi$
$41$	−2.16181			−0.337618			−0.168809	+	0.985649i	$0.553992\pi$
$42$		0			0
$43$	6.40813	+	6.40813i	0.977231	+	0.977231i	0.999746	−	0.0225153i	$-0.00716745\pi$
$43$	6.40813	+	6.40813i	0.977231	+	0.977231i	−0.0225153	+	0.999746i	$0.507167\pi$
$44$		0			0
$45$	−1.74315	−	0.872488i	−0.259854	−	0.130063i
$46$		0			0
$47$	−7.63641			−1.11388			−0.556942	−	0.830551i	$-0.688026\pi$
$47$	−7.63641			−1.11388			−0.556942	+	0.830551i	$0.688026\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	12.3246	−	1.99736i	1.72579	−	0.279687i
$52$		0			0
$53$	2.29189	+	2.29189i	0.314816	+	0.314816i	0.846772	−	0.531956i	$-0.178543\pi$
$53$	2.29189	+	2.29189i	0.314816	+	0.314816i	−0.531956	+	0.846772i	$0.678543\pi$
$54$		0			0
$55$	−1.31802			−0.177722
$56$		0			0
$57$	−4.44114	−	3.20240i	−0.588243	−	0.424169i
$58$		0			0
$59$	6.32068	−	6.32068i	0.822882	−	0.822882i	−0.163638	−	0.986520i	$-0.552323\pi$
$59$	6.32068	−	6.32068i	0.822882	−	0.822882i	0.986520	+	0.163638i	$0.0523230\pi$
$60$		0			0
$61$	0.637183	+	0.637183i	0.0815829	+	0.0815829i	0.746721	−	0.665138i	$-0.231627\pi$
$61$	0.637183	+	0.637183i	0.0815829	+	0.0815829i	−0.665138	+	0.746721i	$0.731627\pi$
$62$		0			0
$63$	2.84645	−	0.947494i	0.358618	−	0.119373i
$64$		0			0
$65$		−	1.08823i		−	0.134978i
$66$		0			0
$67$	2.14006	−	2.14006i	0.261450	−	0.261450i	−0.564193	−	0.825643i	$-0.690813\pi$
$67$	2.14006	−	2.14006i	0.261450	−	0.261450i	0.825643	+	0.564193i	$0.190813\pi$
$68$		0			0
$69$	0.924145	−	0.149770i	0.111254	−	0.0180302i
$70$		0			0
$71$			14.6958i			1.74408i	0.489438	+	0.872038i	$0.337202\pi$
$71$			14.6958i			1.74408i	−0.489438	+	0.872038i	$0.662798\pi$
$72$		0			0
$73$			0.233482i			0.0273270i	0.999907	+	0.0136635i	$0.00434936\pi$
$73$			0.233482i			0.0273270i	−0.999907	+	0.0136635i	$0.995651\pi$
$74$		0			0
$75$	−7.82686	+	1.26845i	−0.903768	+	0.146468i
$76$		0			0
$77$	1.43432	−	1.43432i	0.163456	−	0.163456i
$78$		0			0
$79$		−	4.93151i		−	0.554838i	−0.960749	−	0.277419i	$-0.910521\pi$
$79$		−	4.93151i		−	0.554838i	0.960749	−	0.277419i	$-0.0894790\pi$
$80$		0			0
$81$	7.20451	−	5.39398i	0.800501	−	0.599331i
$82$		0			0
$83$	−1.76156	−	1.76156i	−0.193356	−	0.193356i	0.603789	−	0.797145i	$-0.293657\pi$
$83$	−1.76156	−	1.76156i	−0.193356	−	0.193356i	−0.797145	+	0.603789i	$0.793657\pi$
$84$		0			0
$85$	−3.31198	+	3.31198i	−0.359235	+	0.359235i
$86$		0			0
$87$	8.22824	+	5.93319i	0.882160	+	0.636105i
$88$		0			0
$89$	−7.79085			−0.825829			−0.412914	−	0.910770i	$-0.635489\pi$
$89$	−7.79085			−0.825829			−0.412914	+	0.910770i	$0.635489\pi$
$90$		0			0
$91$	1.18426	+	1.18426i	0.124144	+	0.124144i
$92$		0			0
$93$	17.8133	−	2.88688i	1.84715	−	0.299356i
$94$		0			0
$95$	2.05404			0.210740
$96$		0			0
$97$	−10.0585			−1.02128			−0.510641	−	0.859794i	$-0.670592\pi$
$97$	−10.0585			−1.02128			−0.510641	+	0.859794i	$0.670592\pi$
$98$		0			0
$99$	2.72371	−	5.44174i	0.273744	−	0.546916i
$100$		0			0
$101$	6.01152	+	6.01152i	0.598169	+	0.598169i	0.939825	−	0.341656i	$-0.110988\pi$
$101$	6.01152	+	6.01152i	0.598169	+	0.598169i	−0.341656	+	0.939825i	$0.610988\pi$
$102$		0			0
$103$	−16.5787			−1.63355			−0.816775	−	0.576956i	$-0.804240\pi$
$103$	−16.5787			−1.63355			−0.816775	+	0.576956i	$0.804240\pi$
$104$		0			0
$105$	−0.658245	+	0.912864i	−0.0642382	+	0.0890864i
$106$		0			0
$107$	8.62700	−	8.62700i	0.834004	−	0.834004i	−0.154058	−	0.988062i	$-0.549234\pi$
$107$	8.62700	−	8.62700i	0.834004	−	0.834004i	0.988062	+	0.154058i	$0.0492343\pi$
$108$		0			0
$109$	11.8092	+	11.8092i	1.13111	+	1.13111i	0.989992	+	0.141120i	$0.0450705\pi$
$109$	11.8092	+	11.8092i	1.13111	+	1.13111i	0.141120	+	0.989992i	$0.454930\pi$
$110$		0			0
$111$	2.70926	+	1.95358i	0.257151	+	0.185426i
$112$		0			0
$113$			14.6863i			1.38157i	0.723058	+	0.690787i	$0.242736\pi$
$113$			14.6863i			1.38157i	−0.723058	+	0.690787i	$0.757264\pi$
$114$		0			0
$115$	−0.248345	+	0.248345i	−0.0231582	+	0.0231582i
$116$		0			0
$117$	4.49300	+	2.24885i	0.415378	+	0.207906i
$118$		0			0
$119$		−	7.20845i		−	0.660798i
$120$		0			0
$121$			6.88543i			0.625948i
$122$		0			0
$123$	0.599008	+	3.69614i	0.0540107	+	0.333270i
$124$		0			0
$125$	4.40059	−	4.40059i	0.393601	−	0.393601i
$126$		0			0
$127$		−	4.43597i		−	0.393628i	−0.980441	−	0.196814i	$-0.936940\pi$
$127$		−	4.43597i		−	0.393628i	0.980441	−	0.196814i	$-0.0630596\pi$
$128$		0			0
$129$	9.18066	−	12.7319i	0.808312	−	1.12098i
$130$		0			0
$131$	−3.16188	−	3.16188i	−0.276255	−	0.276255i	0.555357	−	0.831612i	$-0.312582\pi$
$131$	−3.16188	−	3.16188i	−0.276255	−	0.276255i	−0.831612	+	0.555357i	$0.812582\pi$
$132$		0			0
$133$	−2.23529	+	2.23529i	−0.193824	+	0.193824i
$134$		0			0
$135$	−1.00873	+	3.22210i	−0.0868173	+	0.277314i
$136$		0			0
$137$	14.2398			1.21659			0.608295	−	0.793711i	$-0.291854\pi$
$137$	14.2398			1.21659			0.608295	+	0.793711i	$0.291854\pi$
$138$		0			0
$139$	5.47134	+	5.47134i	0.464073	+	0.464073i	0.899988	−	0.435915i	$-0.143575\pi$
$139$	5.47134	+	5.47134i	0.464073	+	0.464073i	−0.435915	+	0.899988i	$0.643575\pi$
$140$		0			0
$141$	2.11595	+	13.0563i	0.178195	+	1.09954i
$142$		0			0
$143$	3.39722			0.284089
$144$		0			0
$145$	−3.80559			−0.316037
$146$		0			0
$147$	−0.277087	−	1.70974i	−0.0228537	−	0.141017i
$148$		0			0
$149$	−4.22703	−	4.22703i	−0.346292	−	0.346292i	0.512435	−	0.858726i	$-0.328744\pi$
$149$	−4.22703	−	4.22703i	−0.346292	−	0.346292i	−0.858726	+	0.512435i	$0.828744\pi$
$150$		0			0
$151$	−16.2738			−1.32434			−0.662170	−	0.749353i	$-0.730364\pi$
$151$	−16.2738			−1.32434			−0.662170	+	0.749353i	$0.730364\pi$
$152$		0			0
$153$	−6.82996	−	20.5185i	−0.552170	−	1.65882i
$154$		0			0
$155$	−4.78695	+	4.78695i	−0.384497	+	0.384497i
$156$		0			0
$157$	3.18084	+	3.18084i	0.253859	+	0.253859i	0.822551	−	0.568692i	$-0.192550\pi$
$157$	3.18084	+	3.18084i	0.253859	+	0.253859i	−0.568692	+	0.822551i	$0.692550\pi$
$158$		0			0
$159$	3.28350	−	4.55360i	0.260398	−	0.361124i
$160$		0			0
$161$		−	0.540517i		−	0.0425987i
$162$		0			0
$163$	−14.7921	+	14.7921i	−1.15861	+	1.15861i	−0.173836	+	0.984775i	$0.555616\pi$
$163$	−14.7921	+	14.7921i	−1.15861	+	1.15861i	−0.984775	+	0.173836i	$0.944384\pi$
$164$		0			0
$165$	0.365206	+	2.25348i	0.0284313	+	0.175433i
$166$		0			0
$167$			6.21188i			0.480689i	0.970688	+	0.240345i	$0.0772605\pi$
$167$			6.21188i			0.480689i	−0.970688	+	0.240345i	$0.922740\pi$
$168$		0			0
$169$		−	10.1951i		−	0.784236i
$170$		0			0
$171$	−4.24471	+	8.48055i	−0.324601	+	0.648524i
$172$		0			0
$173$	1.75745	−	1.75745i	0.133617	−	0.133617i	−0.637135	−	0.770752i	$-0.719881\pi$
$173$	1.75745	−	1.75745i	0.133617	−	0.133617i	0.770752	+	0.637135i	$0.219881\pi$
$174$		0			0
$175$			4.57780i			0.346049i
$176$		0			0
$177$	−12.5581	−	9.05536i	−0.943925	−	0.680643i
$178$		0			0
$179$	−6.35744	−	6.35744i	−0.475177	−	0.475177i	0.428408	−	0.903585i	$-0.359075\pi$
$179$	−6.35744	−	6.35744i	−0.475177	−	0.475177i	−0.903585	+	0.428408i	$0.859075\pi$
$180$		0			0
$181$	−9.95720	+	9.95720i	−0.740113	+	0.740113i	−0.972600	−	0.232487i	$-0.925314\pi$
$181$	−9.95720	+	9.95720i	−0.740113	+	0.740113i	0.232487	+	0.972600i	$0.425314\pi$
$182$		0			0
$183$	0.912864	−	1.26597i	0.0674809	−	0.0935835i
$184$		0			0
$185$	−1.25304			−0.0921254
$186$		0			0
$187$	−10.3393	−	10.3393i	−0.756081	−	0.756081i
$188$		0			0
$189$	−2.40868	−	4.60416i	−0.175206	−	0.334903i
$190$		0			0
$191$	−6.17129			−0.446539			−0.223269	−	0.974757i	$-0.571673\pi$
$191$	−6.17129			−0.446539			−0.223269	+	0.974757i	$0.571673\pi$
$192$		0			0
$193$	17.5110			1.26047			0.630234	−	0.776405i	$-0.282959\pi$
$193$	17.5110			1.26047			0.630234	+	0.776405i	$0.282959\pi$
$194$		0			0
$195$	−1.86060	+	0.301534i	−0.133240	+	0.0215933i
$196$		0			0
$197$	−8.25450	−	8.25450i	−0.588109	−	0.588109i	0.349010	−	0.937119i	$-0.386518\pi$
$197$	−8.25450	−	8.25450i	−0.588109	−	0.588109i	−0.937119	+	0.349010i	$0.886518\pi$
$198$		0			0
$199$	8.91908			0.632257			0.316128	−	0.948716i	$-0.397617\pi$
$199$	8.91908			0.632257			0.316128	+	0.948716i	$0.397617\pi$
$200$		0			0
$201$	−4.25193	−	3.06597i	−0.299908	−	0.216257i
$202$		0			0
$203$	4.14139	−	4.14139i	0.290669	−	0.290669i
$204$		0			0
$205$	−0.993260	−	0.993260i	−0.0693723	−	0.0693723i
$206$		0			0
$207$	−0.512136	−	1.53855i	−0.0355959	−	0.106937i
$208$		0			0
$209$			6.41226i			0.443545i
$210$		0			0
$211$	3.36181	−	3.36181i	0.231436	−	0.231436i	−0.581856	−	0.813292i	$-0.697673\pi$
$211$	3.36181	−	3.36181i	0.231436	−	0.231436i	0.813292	+	0.581856i	$0.197673\pi$
$212$		0			0
$213$	25.1261	−	4.07202i	1.72161	−	0.279010i
$214$		0			0
$215$			5.88853i			0.401595i
$216$		0			0
$217$		−	10.4187i		−	0.707267i
$218$		0			0
$219$	0.399194	−	0.0646946i	0.0269750	−	0.00437166i
$220$		0			0
$221$	8.53665	−	8.53665i	0.574238	−	0.574238i
$222$		0			0
$223$		−	22.5878i		−	1.51259i	−0.654231	−	0.756295i	$-0.727007\pi$
$223$		−	22.5878i		−	1.51259i	0.654231	−	0.756295i	$-0.272993\pi$
$224$		0			0
$225$	4.33743	+	13.0305i	0.289162	+	0.868697i
$226$		0			0
$227$	−13.6329	−	13.6329i	−0.904847	−	0.904847i	0.0910037	−	0.995851i	$-0.470992\pi$
$227$	−13.6329	−	13.6329i	−0.904847	−	0.904847i	−0.995851	+	0.0910037i	$0.970992\pi$
$228$		0			0
$229$	−4.15041	+	4.15041i	−0.274267	+	0.274267i	−0.830815	−	0.556549i	$-0.812125\pi$
$229$	−4.15041	+	4.15041i	−0.274267	+	0.274267i	0.556549	+	0.830815i	$0.312125\pi$
$230$		0			0
$231$	−2.84976	−	2.05490i	−0.187500	−	0.135202i
$232$		0			0
$233$	3.93645			0.257886			0.128943	−	0.991652i	$-0.458842\pi$
$233$	3.93645			0.257886			0.128943	+	0.991652i	$0.458842\pi$
$234$		0			0
$235$	−3.50861	−	3.50861i	−0.228876	−	0.228876i
$236$		0			0
$237$	−8.43161	+	1.36645i	−0.547692	+	0.0887608i
$238$		0			0
$239$	−16.3555			−1.05795			−0.528976	−	0.848637i	$-0.677424\pi$
$239$	−16.3555			−1.05795			−0.528976	+	0.848637i	$0.677424\pi$
$240$		0			0
$241$	−24.3675			−1.56965			−0.784826	−	0.619716i	$-0.787248\pi$
$241$	−24.3675			−1.56965			−0.784826	+	0.619716i	$0.787248\pi$
$242$		0			0
$243$	−11.2186	−	10.8233i	−0.719673	−	0.694313i
$244$		0			0
$245$	0.459458	+	0.459458i	0.0293537	+	0.0293537i
$246$		0			0
$247$	−5.29431			−0.336869
$248$		0			0
$249$	−2.52371	+	3.49991i	−0.159933	+	0.221798i
$250$		0			0
$251$	18.5928	−	18.5928i	1.17357	−	1.17357i	0.192213	−	0.981353i	$-0.438434\pi$
$251$	18.5928	−	18.5928i	1.17357	−	1.17357i	0.981353	−	0.192213i	$-0.0615664\pi$
$252$		0			0
$253$	−0.775276	−	0.775276i	−0.0487412	−	0.0487412i
$254$		0			0
$255$	6.58034	+	4.74493i	0.412077	+	0.297139i
$256$		0			0
$257$		−	29.3016i		−	1.82778i	−0.405961	−	0.913890i	$-0.633063\pi$
$257$		−	29.3016i		−	1.82778i	0.405961	−	0.913890i	$-0.366937\pi$
$258$		0			0
$259$	1.36361	−	1.36361i	0.0847305	−	0.0847305i
$260$		0			0
$261$	7.86430	−	15.7122i	0.486788	−	0.972560i
$262$		0			0
$263$			14.8023i			0.912747i	0.889788	+	0.456373i	$0.150852\pi$
$263$			14.8023i			0.912747i	−0.889788	+	0.456373i	$0.849148\pi$
$264$		0			0
$265$			2.10606i			0.129374i
$266$		0			0
$267$	2.15874	+	13.3204i	0.132113	+	0.815193i
$268$		0			0
$269$	−13.8770	+	13.8770i	−0.846095	+	0.846095i	−0.989643	−	0.143548i	$-0.954149\pi$
$269$	−13.8770	+	13.8770i	−0.846095	+	0.846095i	0.143548	+	0.989643i	$0.454149\pi$
$270$		0			0
$271$			20.1033i			1.22119i	0.791943	+	0.610595i	$0.209070\pi$
$271$			20.1033i			1.22119i	−0.791943	+	0.610595i	$0.790930\pi$
$272$		0			0
$273$	1.69663	−	2.35292i	0.102685	−	0.142405i
$274$		0			0
$275$	6.56605	+	6.56605i	0.395948	+	0.395948i
$276$		0			0
$277$	−19.8547	+	19.8547i	−1.19295	+	1.19295i	−0.216719	+	0.976234i	$0.569536\pi$
$277$	−19.8547	+	19.8547i	−1.19295	+	1.19295i	−0.976234	+	0.216719i	$0.930464\pi$
$278$		0			0
$279$	−9.87165	−	29.6563i	−0.591000	−	1.77547i
$280$		0			0
$281$	−1.87453			−0.111825			−0.0559126	−	0.998436i	$-0.517807\pi$
$281$	−1.87453			−0.111825			−0.0559126	+	0.998436i	$0.517807\pi$
$282$		0			0
$283$	−17.9733	−	17.9733i	−1.06840	−	1.06840i	−0.997482	−	0.0709207i	$-0.977406\pi$
$283$	−17.9733	−	17.9733i	−1.06840	−	1.06840i	−0.0709207	−	0.997482i	$-0.522594\pi$
$284$		0			0
$285$	−0.569147	−	3.51189i	−0.0337134	−	0.208026i
$286$		0			0
$287$	2.16181			0.127608
$288$		0			0
$289$	−34.9618			−2.05657
$290$		0			0
$291$	2.78707	+	17.1974i	0.163381	+	1.00813i
$292$		0			0
$293$	−17.4851	−	17.4851i	−1.02149	−	1.02149i	−0.999764	−	0.0217268i	$-0.993084\pi$
$293$	−17.4851	−	17.4851i	−1.02149	−	1.02149i	−0.0217268	−	0.999764i	$-0.506916\pi$
$294$		0			0
$295$	5.80817			0.338165
$296$		0			0
$297$	−10.0587	−	3.14902i	−0.583664	−	0.182725i
$298$		0			0
$299$	0.640110	−	0.640110i	0.0370185	−	0.0370185i
$300$		0			0
$301$	−6.40813	−	6.40813i	−0.369359	−	0.369359i
$302$		0			0
$303$	8.61245	−	11.9439i	0.494772	−	0.686158i
$304$		0			0
$305$			0.585517i			0.0335266i
$306$		0			0
$307$	−1.90363	+	1.90363i	−0.108646	+	0.108646i	−0.759340	−	0.650694i	$-0.774478\pi$
$307$	−1.90363	+	1.90363i	−0.108646	+	0.108646i	0.650694	+	0.759340i	$0.274478\pi$
$308$		0			0
$309$	4.59374	+	28.3454i	0.261329	+	1.61251i
$310$		0			0
$311$			3.81852i			0.216529i	0.994122	+	0.108264i	$0.0345293\pi$
$311$			3.81852i			0.216529i	−0.994122	+	0.108264i	$0.965471\pi$
$312$		0			0
$313$		−	2.80396i		−	0.158489i	−0.996855	−	0.0792446i	$-0.974749\pi$
$313$		−	2.80396i		−	0.158489i	0.996855	−	0.0792446i	$-0.0252508\pi$
$314$		0			0
$315$	1.74315	+	0.872488i	0.0982157	+	0.0491591i
$316$		0			0
$317$	12.6164	−	12.6164i	0.708609	−	0.708609i	−0.257633	−	0.966243i	$-0.582943\pi$
$317$	12.6164	−	12.6164i	0.708609	−	0.708609i	0.966243	+	0.257633i	$0.0829427\pi$
$318$		0			0
$319$		−	11.8802i		−	0.665163i
$320$		0			0
$321$	−17.1404	−	12.3595i	−0.956683	−	0.689842i
$322$		0			0
$323$	16.1130	+	16.1130i	0.896550	+	0.896550i
$324$		0			0
$325$	−5.42129	+	5.42129i	−0.300719	+	0.300719i
$326$		0			0
$327$	16.9185	−	23.4628i	0.935594	−	1.29750i
$328$		0			0
$329$	7.63641			0.421009
$330$		0			0
$331$	3.56268	+	3.56268i	0.195823	+	0.195823i	0.798207	−	0.602384i	$-0.205782\pi$
$331$	3.56268	+	3.56268i	0.195823	+	0.195823i	−0.602384	+	0.798207i	$0.705782\pi$
$332$		0			0
$333$	2.58943	−	5.17345i	0.141900	−	0.283503i
$334$		0			0
$335$	1.96653			0.107443
$336$		0			0
$337$	20.7354			1.12953			0.564765	−	0.825252i	$-0.308967\pi$
$337$	20.7354			1.12953			0.564765	+	0.825252i	$0.308967\pi$
$338$		0			0
$339$	25.1099	−	4.06939i	1.36378	−	0.221019i
$340$		0			0
$341$	−14.9438	−	14.9438i	−0.809252	−	0.809252i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$	0.493418	+	0.355793i	0.0265648	+	0.0191552i
$346$		0			0
$347$	0.00985354	−	0.00985354i	0.000528966	−	0.000528966i	−0.706842	−	0.707371i	$-0.749881\pi$
$347$	0.00985354	−	0.00985354i	0.000528966	−	0.000528966i	0.707371	+	0.706842i	$0.249881\pi$
$348$		0			0
$349$	0.770588	+	0.770588i	0.0412486	+	0.0412486i	0.727430	−	0.686182i	$-0.240714\pi$
$349$	0.770588	+	0.770588i	0.0412486	+	0.0412486i	−0.686182	+	0.727430i	$0.740714\pi$
$350$		0			0
$351$	2.60000	−	8.30500i	0.138778	−	0.443288i
$352$		0			0
$353$		−	2.34275i		−	0.124692i	−0.998055	−	0.0623460i	$-0.980142\pi$
$353$		−	2.34275i		−	0.124692i	0.998055	−	0.0623460i	$-0.0198582\pi$
$354$		0			0
$355$	−6.75212	+	6.75212i	−0.358365	+	0.358365i
$356$		0			0
$357$	−12.3246	+	1.99736i	−0.652287	+	0.105712i
$358$		0			0
$359$		−	14.4079i		−	0.760420i	−0.924900	−	0.380210i	$-0.875852\pi$
$359$		−	14.4079i		−	0.760420i	0.924900	−	0.380210i	$-0.124148\pi$
$360$		0			0
$361$			9.00696i			0.474051i
$362$		0			0
$363$	11.7723	−	1.90786i	0.617886	−	0.100137i
$364$		0			0
$365$	−0.107275	+	0.107275i	−0.00561503	+	0.00561503i
$366$		0			0
$367$			27.8523i			1.45388i	0.686703	+	0.726938i	$0.259057\pi$
$367$			27.8523i			1.45388i	−0.686703	+	0.726938i	$0.740943\pi$
$368$		0			0
$369$	6.15347	−	2.04830i	0.320337	−	0.106630i
$370$		0			0
$371$	−2.29189	−	2.29189i	−0.118989	−	0.118989i
$372$		0			0
$373$	21.3363	−	21.3363i	1.10475	−	1.10475i	0.110924	−	0.993829i	$-0.464619\pi$
$373$	21.3363	−	21.3363i	1.10475	−	1.10475i	0.993829	−	0.110924i	$-0.0353810\pi$
$374$		0			0
$375$	−8.74323	−	6.30454i	−0.451498	−	0.325565i
$376$		0			0
$377$	9.80894			0.505186
$378$		0			0
$379$	24.6296	+	24.6296i	1.26514	+	1.26514i	0.948570	+	0.316567i	$0.102530\pi$
$379$	24.6296	+	24.6296i	1.26514	+	1.26514i	0.316567	+	0.948570i	$0.397470\pi$
$380$		0			0
$381$	−7.58436	+	1.22915i	−0.388559	+	0.0629711i
$382$		0			0
$383$	20.0598			1.02501			0.512503	−	0.858685i	$-0.328718\pi$
$383$	20.0598			1.02501			0.512503	+	0.858685i	$0.328718\pi$
$384$		0			0
$385$	1.31802			0.0671727
$386$		0			0
$387$	−24.3121	−	12.1687i	−1.23585	−	0.618572i
$388$		0			0
$389$	−9.20544	−	9.20544i	−0.466735	−	0.466735i	0.434120	−	0.900855i	$-0.357059\pi$
$389$	−9.20544	−	9.20544i	−0.466735	−	0.466735i	−0.900855	+	0.434120i	$0.857059\pi$
$390$		0			0
$391$	−3.89629			−0.197044
$392$		0			0
$393$	−4.52989	+	6.28213i	−0.228503	+	0.316891i
$394$		0			0
$395$	2.26582	−	2.26582i	0.114006	−	0.114006i
$396$		0			0
$397$	−25.2512	−	25.2512i	−1.26732	−	1.26732i	−0.947468	−	0.319852i	$-0.896367\pi$
$397$	−25.2512	−	25.2512i	−1.26732	−	1.26732i	−0.319852	−	0.947468i	$-0.603633\pi$
$398$		0			0
$399$	4.44114	+	3.20240i	0.222335	+	0.160321i
$400$		0			0
$401$			32.1746i			1.60673i	0.595490	+	0.803363i	$0.296958\pi$
$401$			32.1746i			1.60673i	−0.595490	+	0.803363i	$0.703042\pi$
$402$		0			0
$403$	12.3384	−	12.3384i	0.614620	−	0.614620i
$404$		0			0
$405$	5.78847	+	0.831862i	0.287632	+	0.0413356i
$406$		0			0
$407$		−	3.91172i		−	0.193897i
$408$		0			0
$409$			37.8980i			1.87393i	0.349417	+	0.936967i	$0.386380\pi$
$409$			37.8980i			1.87393i	−0.349417	+	0.936967i	$0.613620\pi$
$410$		0			0
$411$	−3.94566	−	24.3464i	−0.194625	−	1.20092i
$412$		0			0
$413$	−6.32068	+	6.32068i	−0.311020	+	0.311020i
$414$		0			0
$415$		−	1.61872i		−	0.0794599i
$416$		0			0
$417$	7.83855	−	10.8706i	0.383855	−	0.532336i
$418$		0			0
$419$	−4.83167	−	4.83167i	−0.236043	−	0.236043i	0.579167	−	0.815209i	$-0.303378\pi$
$419$	−4.83167	−	4.83167i	−0.236043	−	0.236043i	−0.815209	+	0.579167i	$0.803378\pi$
$420$		0			0
$421$	24.5684	−	24.5684i	1.19739	−	1.19739i	0.222444	−	0.974945i	$-0.428597\pi$
$421$	24.5684	−	24.5684i	1.19739	−	1.19739i	0.974945	−	0.222444i	$-0.0714035\pi$
$422$		0			0
$423$	21.7366	−	7.23545i	1.05687	−	0.351800i
$424$		0			0
$425$	32.9988			1.60068
$426$		0			0
$427$	−0.637183	−	0.637183i	−0.0308354	−	0.0308354i
$428$		0			0
$429$	−0.941323	−	5.80837i	−0.0454475	−	0.280431i
$430$		0			0
$431$	4.29534			0.206899			0.103450	−	0.994635i	$-0.467012\pi$
$431$	4.29534			0.206899			0.103450	+	0.994635i	$0.467012\pi$
$432$		0			0
$433$	−1.69551			−0.0814809			−0.0407404	−	0.999170i	$-0.512972\pi$
$433$	−1.69551			−0.0814809			−0.0407404	+	0.999170i	$0.512972\pi$
$434$		0			0
$435$	1.05448	+	6.50658i	0.0505583	+	0.311967i
$436$		0			0
$437$	1.20821	+	1.20821i	0.0577966	+	0.0577966i
$438$		0			0
$439$	−6.92612			−0.330565			−0.165283	−	0.986246i	$-0.552854\pi$
$439$	−6.92612			−0.330565			−0.165283	+	0.986246i	$0.552854\pi$
$440$		0			0
$441$	−2.84645	+	0.947494i	−0.135545	+	0.0451188i
$442$		0			0
$443$	24.7883	−	24.7883i	1.17773	−	1.17773i	0.197404	−	0.980322i	$-0.436749\pi$
$443$	24.7883	−	24.7883i	1.17773	−	1.17773i	0.980322	−	0.197404i	$-0.0632510\pi$
$444$		0			0
$445$	−3.57957	−	3.57957i	−0.169688	−	0.169688i
$446$		0			0
$447$	−6.05588	+	8.39839i	−0.286433	+	0.397230i
$448$		0			0
$449$			7.33520i			0.346169i	0.984907	+	0.173085i	$0.0553734\pi$
$449$			7.33520i			0.346169i	−0.984907	+	0.173085i	$0.944627\pi$
$450$		0			0
$451$	3.10074	−	3.10074i	0.146008	−	0.146008i
$452$		0			0
$453$	4.50924	+	27.8240i	0.211863	+	1.30728i
$454$		0			0
$455$			1.08823i			0.0510171i
$456$		0			0
$457$		−	9.97328i		−	0.466531i	−0.972413	−	0.233265i	$-0.925059\pi$
$457$		−	9.97328i		−	0.466531i	0.972413	−	0.233265i	$-0.0749411\pi$
$458$		0			0
$459$	−33.1888	+	17.3629i	−1.54912	+	0.810430i
$460$		0			0
$461$	11.6080	−	11.6080i	0.540638	−	0.540638i	−0.383078	−	0.923716i	$-0.625136\pi$
$461$	11.6080	−	11.6080i	0.540638	−	0.540638i	0.923716	+	0.383078i	$0.125136\pi$
$462$		0			0
$463$			16.8223i			0.781801i	0.920433	+	0.390900i	$0.127836\pi$
$463$			16.8223i			0.781801i	−0.920433	+	0.390900i	$0.872164\pi$
$464$		0			0
$465$	9.51086	+	6.85806i	0.441055	+	0.318035i
$466$		0			0
$467$	14.4026	+	14.4026i	0.666474	+	0.666474i	0.956898	−	0.290424i	$-0.0937962\pi$
$467$	14.4026	+	14.4026i	0.666474	+	0.666474i	−0.290424	+	0.956898i	$0.593796\pi$
$468$		0			0
$469$	−2.14006	+	2.14006i	−0.0988187	+	0.0988187i
$470$		0			0
$471$	4.55706	−	6.31980i	0.209978	−	0.291201i
$472$		0			0
$473$	−18.3827			−0.845237
$474$		0			0
$475$	−10.2327	−	10.2327i	−0.469509	−	0.469509i
$476$		0			0
$477$	−8.69531	−	4.35220i	−0.398131	−	0.199273i
$478$		0			0
$479$	4.11467			0.188004			0.0940021	−	0.995572i	$-0.470034\pi$
$479$	4.11467			0.188004			0.0940021	+	0.995572i	$0.470034\pi$
$480$		0			0
$481$	3.22972			0.147263
$482$		0			0
$483$	−0.924145	+	0.149770i	−0.0420500	+	0.00681477i
$484$		0			0
$485$	−4.62144	−	4.62144i	−0.209849	−	0.209849i
$486$		0			0
$487$	25.8876			1.17308			0.586540	−	0.809920i	$-0.300490\pi$
$487$	25.8876			1.17308			0.586540	+	0.809920i	$0.300490\pi$
$488$		0			0
$489$	29.3895	+	21.1921i	1.32904	+	0.958339i
$490$		0			0
$491$	−19.4155	+	19.4155i	−0.876207	+	0.876207i	−0.993140	−	0.116933i	$-0.962694\pi$
$491$	−19.4155	+	19.4155i	−0.876207	+	0.876207i	0.116933	+	0.993140i	$0.462694\pi$
$492$		0			0
$493$	−29.8530	−	29.8530i	−1.34451	−	1.34451i
$494$		0			0
$495$	3.75168	−	1.24882i	0.168626	−	0.0561302i
$496$		0			0
$497$		−	14.6958i		−	0.659199i
$498$		0			0
$499$	−18.9116	+	18.9116i	−0.846599	+	0.846599i	−0.989707	−	0.143108i	$-0.954290\pi$
$499$	−18.9116	+	18.9116i	−0.846599	+	0.846599i	0.143108	+	0.989707i	$0.454290\pi$
$500$		0			0
$501$	10.6207	−	1.72123i	0.474498	−	0.0768988i
$502$		0			0
$503$		−	6.29527i		−	0.280692i	−0.990102	−	0.140346i	$-0.955178\pi$
$503$		−	6.29527i		−	0.280692i	0.990102	−	0.140346i	$-0.0448215\pi$
$504$		0			0
$505$			5.52408i			0.245818i
$506$		0			0
$507$	−17.4310	+	2.82492i	−0.774136	+	0.125459i
$508$		0			0
$509$	9.53952	−	9.53952i	0.422832	−	0.422832i	−0.463346	−	0.886178i	$-0.653351\pi$
$509$	9.53952	−	9.53952i	0.422832	−	0.422832i	0.886178	+	0.463346i	$0.153351\pi$
$510$		0			0
$511$		−	0.233482i		−	0.0103286i
$512$		0			0
$513$	15.6757	+	4.90751i	0.692100	+	0.216672i
$514$		0			0
$515$	−7.61722	−	7.61722i	−0.335655	−	0.335655i
$516$		0			0
$517$	10.9531	−	10.9531i	0.481716	−	0.481716i
$518$		0			0
$519$	−3.49176	−	2.51783i	−0.153271	−	0.110520i
$520$		0			0
$521$	−10.4082			−0.455991			−0.227995	−	0.973662i	$-0.573217\pi$
$521$	−10.4082			−0.455991			−0.227995	+	0.973662i	$0.573217\pi$
$522$		0			0
$523$	−13.2569	−	13.2569i	−0.579685	−	0.579685i	0.355131	−	0.934816i	$-0.384436\pi$
$523$	−13.2569	−	13.2569i	−0.579685	−	0.579685i	−0.934816	+	0.355131i	$0.884436\pi$
$524$		0			0
$525$	7.82686	−	1.26845i	0.341592	−	0.0553595i
$526$		0			0
$527$	−75.1027			−3.27152
$528$		0			0
$529$	22.7078			0.987297
$530$		0			0
$531$	−12.0027	+	23.9803i	−0.520871	+	1.04065i
$532$		0			0
$533$	2.56014	+	2.56014i	0.110892	+	0.110892i
$534$		0			0
$535$	7.92748			0.342735
$536$		0			0
$537$	−9.10803	+	12.6311i	−0.393040	+	0.545074i
$538$		0			0
$539$	−1.43432	+	1.43432i	−0.0617807	+	0.0617807i
$540$		0			0
$541$	10.4371	+	10.4371i	0.448725	+	0.448725i	0.894930	−	0.446206i	$-0.147225\pi$
$541$	10.4371	+	10.4371i	0.448725	+	0.448725i	−0.446206	+	0.894930i	$0.647225\pi$
$542$		0			0
$543$	19.7833	+	14.2653i	0.848981	+	0.612181i
$544$		0			0
$545$			10.8516i			0.464832i
$546$		0			0
$547$	−0.334525	+	0.334525i	−0.0143032	+	0.0143032i	−0.714222	−	0.699919i	$-0.753219\pi$
$547$	−0.334525	+	0.334525i	−0.0143032	+	0.0143032i	0.699919	+	0.714222i	$0.253219\pi$
$548$		0			0
$549$	−2.41743	−	1.20998i	−0.103174	−	0.0516407i
$550$		0			0
$551$			18.5144i			0.788741i
$552$		0			0
$553$			4.93151i			0.209709i
$554$		0			0
$555$	0.347201	+	2.14238i	0.0147379	+	0.0909389i
$556$		0			0
$557$	−22.8719	+	22.8719i	−0.969113	+	0.969113i	−0.999537	−	0.0304239i	$-0.990314\pi$
$557$	−22.8719	+	22.8719i	−0.969113	+	0.969113i	0.0304239	+	0.999537i	$0.490314\pi$
$558$		0			0
$559$		−	15.1777i		−	0.641950i
$560$		0			0
$561$	−14.8126	+	20.5424i	−0.625389	+	0.867299i
$562$		0			0
$563$	23.5403	+	23.5403i	0.992106	+	0.992106i	0.999969	−	0.00786281i	$-0.00250284\pi$
$563$	23.5403	+	23.5403i	0.992106	+	0.992106i	−0.00786281	+	0.999969i	$0.502503\pi$
$564$		0			0
$565$	−6.74775	+	6.74775i	−0.283880	+	0.283880i
$566$		0			0
$567$	−7.20451	+	5.39398i	−0.302561	+	0.226526i
$568$		0			0
$569$	31.9154			1.33797			0.668983	−	0.743278i	$-0.266730\pi$
$569$	31.9154			1.33797			0.668983	+	0.743278i	$0.266730\pi$
$570$		0			0
$571$	2.02190	+	2.02190i	0.0846137	+	0.0846137i	0.748147	−	0.663533i	$-0.230944\pi$
$571$	2.02190	+	2.02190i	0.0846137	+	0.0846137i	−0.663533	+	0.748147i	$0.730944\pi$
$572$		0			0
$573$	1.70998	+	10.5513i	0.0714355	+	0.440788i
$574$		0			0
$575$	2.47438			0.103189
$576$		0			0
$577$	43.1754			1.79742			0.898708	−	0.438548i	$-0.144507\pi$
$577$	43.1754			1.79742			0.898708	+	0.438548i	$0.144507\pi$
$578$		0			0
$579$	−4.85206	−	29.9393i	−0.201645	−	1.24424i
$580$		0			0
$581$	1.76156	+	1.76156i	0.0730817	+	0.0730817i
$582$		0			0
$583$	−6.57464			−0.272294
$584$		0			0
$585$	1.03109	+	3.09759i	0.0426304	+	0.128070i
$586$		0			0
$587$	−11.8925	+	11.8925i	−0.490854	+	0.490854i	−0.908575	−	0.417721i	$-0.862829\pi$
$587$	−11.8925	+	11.8925i	−0.490854	+	0.490854i	0.417721	+	0.908575i	$0.362829\pi$
$588$		0			0
$589$	23.2888	+	23.2888i	0.959598	+	0.959598i
$590$		0			0
$591$	−11.8259	+	16.4003i	−0.486451	+	0.674618i
$592$		0			0
$593$		−	36.6876i		−	1.50658i	−0.657688	−	0.753291i	$-0.728465\pi$
$593$		−	36.6876i		−	1.50658i	0.657688	−	0.753291i	$-0.271535\pi$
$594$		0			0
$595$	3.31198	−	3.31198i	0.135778	−	0.135778i
$596$		0			0
$597$	−2.47136	−	15.2493i	−0.101146	−	0.624114i
$598$		0			0
$599$		−	14.1269i		−	0.577211i	−0.957448	−	0.288606i	$-0.906808\pi$
$599$		−	14.1269i		−	0.577211i	0.957448	−	0.288606i	$-0.0931916\pi$
$600$		0			0
$601$			6.81447i			0.277968i	0.990295	+	0.138984i	$0.0443836\pi$
$601$			6.81447i			0.277968i	−0.990295	+	0.138984i	$0.955616\pi$
$602$		0			0
$603$	−4.06387	+	8.11926i	−0.165494	+	0.330642i
$604$		0			0
$605$	−3.16356	+	3.16356i	−0.128617	+	0.128617i
$606$		0			0
$607$			5.05289i			0.205091i	0.994728	+	0.102545i	$0.0326987\pi$
$607$			5.05289i			0.205091i	−0.994728	+	0.102545i	$0.967301\pi$
$608$		0			0
$609$	−8.22824	−	5.93319i	−0.333425	−	0.240425i
$610$		0			0
$611$	9.04347	+	9.04347i	0.365860	+	0.365860i
$612$		0			0
$613$	24.4036	−	24.4036i	0.985651	−	0.985651i	−0.0142478	−	0.999898i	$-0.504535\pi$
$613$	24.4036	−	24.4036i	0.985651	−	0.985651i	0.999898	+	0.0142478i	$0.00453537\pi$
$614$		0			0
$615$	−1.42300	+	1.97344i	−0.0573809	+	0.0795767i
$616$		0			0
$617$	10.5641			0.425294			0.212647	−	0.977129i	$-0.431792\pi$
$617$	10.5641			0.425294			0.212647	+	0.977129i	$0.431792\pi$
$618$		0			0
$619$	−10.2400	−	10.2400i	−0.411581	−	0.411581i	0.470708	−	0.882289i	$-0.343999\pi$
$619$	−10.2400	−	10.2400i	−0.411581	−	0.411581i	−0.882289	+	0.470708i	$0.843999\pi$
$620$		0			0
$621$	−2.48862	+	1.30193i	−0.0998650	+	0.0522448i
$622$		0			0
$623$	7.79085			0.312134
$624$		0			0
$625$	−18.8452			−0.753809
$626$		0			0
$627$	10.9633	−	1.77675i	0.437833	−	0.0709566i
$628$		0			0
$629$	−9.82951	−	9.82951i	−0.391928	−	0.391928i
$630$		0			0
$631$	28.8196			1.14729			0.573645	−	0.819104i	$-0.305529\pi$
$631$	28.8196			1.14729			0.573645	+	0.819104i	$0.305529\pi$
$632$		0			0
$633$	−6.67934	−	4.81631i	−0.265480	−	0.191431i
$634$		0			0
$635$	2.03814	−	2.03814i	0.0808811	−	0.0808811i
$636$		0			0
$637$	−1.18426	−	1.18426i	−0.0469219	−	0.0469219i
$638$		0			0
$639$	−13.9242	−	41.8309i	−0.550834	−	1.65481i
$640$		0			0
$641$			33.6392i			1.32867i	0.747435	+	0.664335i	$0.231285\pi$
$641$			33.6392i			1.32867i	−0.747435	+	0.664335i	$0.768715\pi$
$642$		0			0
$643$	−2.90938	+	2.90938i	−0.114735	+	0.114735i	−0.762143	−	0.647409i	$-0.775853\pi$
$643$	−2.90938	+	2.90938i	−0.114735	+	0.114735i	0.647409	+	0.762143i	$0.275853\pi$
$644$		0			0
$645$	10.0679	−	1.63163i	0.396422	−	0.0642455i
$646$		0			0
$647$			29.6246i			1.16466i	0.812952	+	0.582331i	$0.197859\pi$
$647$			29.6246i			1.16466i	−0.812952	+	0.582331i	$0.802141\pi$
$648$		0			0
$649$			18.1318i			0.711736i
$650$		0			0
$651$	−17.8133	+	2.88688i	−0.698158	+	0.113146i
$652$		0			0
$653$	−20.6380	+	20.6380i	−0.807627	+	0.807627i	−0.984274	−	0.176647i	$-0.943475\pi$
$653$	−20.6380	+	20.6380i	−0.807627	+	0.807627i	0.176647	+	0.984274i	$0.443475\pi$
$654$		0			0
$655$		−	2.90550i		−	0.113527i
$656$		0			0
$657$	−0.221222	−	0.664593i	−0.00863071	−	0.0259282i
$658$		0			0
$659$	23.8523	+	23.8523i	0.929153	+	0.929153i	0.997651	−	0.0684985i	$-0.0218208\pi$
$659$	23.8523	+	23.8523i	0.929153	+	0.929153i	−0.0684985	+	0.997651i	$0.521821\pi$
$660$		0			0
$661$	−11.8802	+	11.8802i	−0.462085	+	0.462085i	−0.899338	−	0.437254i	$-0.855951\pi$
$661$	−11.8802	+	11.8802i	−0.462085	+	0.462085i	0.437254	+	0.899338i	$0.355951\pi$
$662$		0			0
$663$	−16.9609	−	12.2301i	−0.658706	−	0.474978i
$664$		0			0
$665$	−2.05404			−0.0796523
$666$		0			0
$667$	−2.23849	−	2.23849i	−0.0866747	−	0.0866747i
$668$		0			0
$669$	−38.6193	+	6.25877i	−1.49311	+	0.241978i
$670$		0			0
$671$	−1.82785			−0.0705635
$672$		0			0
$673$	13.0244			0.502052			0.251026	−	0.967980i	$-0.419232\pi$
$673$	13.0244			0.502052			0.251026	+	0.967980i	$0.419232\pi$
$674$		0			0
$675$	21.0769	−	11.0265i	0.811250	−	0.424409i
$676$		0			0
$677$	−21.3792	−	21.3792i	−0.821668	−	0.821668i	0.164679	−	0.986347i	$-0.447341\pi$
$677$	−21.3792	−	21.3792i	−0.821668	−	0.821668i	−0.986347	+	0.164679i	$0.947341\pi$
$678$		0			0
$679$	10.0585			0.386009
$680$		0			0
$681$	−19.5313	+	27.0862i	−0.748439	+	1.03795i
$682$		0			0
$683$	−14.3476	+	14.3476i	−0.548996	+	0.548996i	−0.926150	−	0.377154i	$-0.876903\pi$
$683$	−14.3476	+	14.3476i	−0.548996	+	0.548996i	0.377154	+	0.926150i	$0.376903\pi$
$684$		0			0
$685$	6.54260	+	6.54260i	0.249980	+	0.249980i
$686$		0			0
$687$	8.24615	+	5.94611i	0.314610	+	0.226858i
$688$		0			0
$689$		−	5.42838i		−	0.206805i
$690$		0			0
$691$	6.59157	−	6.59157i	0.250755	−	0.250755i	−0.570525	−	0.821280i	$-0.693260\pi$
$691$	6.59157	−	6.59157i	0.250755	−	0.250755i	0.821280	+	0.570525i	$0.193260\pi$
$692$		0			0
$693$	−2.72371	+	5.44174i	−0.103465	+	0.206715i
$694$		0			0
$695$			5.02769i			0.190711i
$696$		0			0
$697$		−	15.5833i		−	0.590259i
$698$		0			0
$699$	−1.09074	−	6.73032i	−0.0412555	−	0.254564i
$700$		0			0
$701$	−27.9140	+	27.9140i	−1.05430	+	1.05430i	−0.0558581	+	0.998439i	$0.517789\pi$
$701$	−27.9140	+	27.9140i	−1.05430	+	1.05430i	−0.998439	+	0.0558581i	$0.982211\pi$
$702$		0			0
$703$			6.09612i			0.229920i
$704$		0			0
$705$	−5.02663	+	6.97101i	−0.189314	+	0.262543i
$706$		0			0
$707$	−6.01152	−	6.01152i	−0.226087	−	0.226087i
$708$		0			0
$709$	7.14776	−	7.14776i	0.268440	−	0.268440i	−0.560031	−	0.828471i	$-0.689211\pi$
$709$	7.14776	−	7.14776i	0.268440	−	0.268440i	0.828471	+	0.560031i	$0.189211\pi$
$710$		0			0
$711$	4.67257	+	14.0373i	0.175235	+	0.526439i
$712$		0			0
$713$	−5.63148			−0.210901
$714$		0			0
$715$	1.56088	+	1.56088i	0.0583735	+	0.0583735i
$716$		0			0
$717$	4.53190	+	27.9638i	0.169247	+	1.04433i
$718$		0			0
$719$	16.0695			0.599292			0.299646	−	0.954050i	$-0.403131\pi$
$719$	16.0695			0.599292			0.299646	+	0.954050i	$0.403131\pi$
$720$		0			0
$721$	16.5787			0.617424
$722$		0			0
$723$	6.75192	+	41.6623i	0.251107	+	1.54944i
$724$		0			0
$725$	18.9585	+	18.9585i	0.704099	+	0.704099i
$726$		0			0
$727$	14.7667			0.547667			0.273833	−	0.961777i	$-0.411708\pi$
$727$	14.7667			0.547667			0.273833	+	0.961777i	$0.411708\pi$
$728$		0			0
$729$	−15.3965	+	22.1799i	−0.570240	+	0.821478i
$730$		0			0
$731$	−46.1927	+	46.1927i	−1.70850	+	1.70850i
$732$		0			0
$733$	14.1137	+	14.1137i	0.521303	+	0.521303i	0.917965	−	0.396662i	$-0.129831\pi$
$733$	14.1137	+	14.1137i	0.521303	+	0.521303i	−0.396662	+	0.917965i	$0.629831\pi$
$734$		0			0
$735$	0.658245	−	0.912864i	0.0242797	−	0.0336715i
$736$		0			0
$737$			6.13908i			0.226136i
$738$		0			0
$739$	6.96635	−	6.96635i	0.256261	−	0.256261i	−0.567270	−	0.823532i	$-0.692001\pi$
$739$	6.96635	−	6.96635i	0.256261	−	0.256261i	0.823532	+	0.567270i	$0.192001\pi$
$740$		0			0
$741$	1.46698	+	9.05192i	0.0538910	+	0.332530i
$742$		0			0
$743$			26.0074i			0.954117i	0.878871	+	0.477059i	$0.158297\pi$
$743$			26.0074i			0.954117i	−0.878871	+	0.477059i	$0.841703\pi$
$744$		0			0
$745$		−	3.88428i		−	0.142309i
$746$		0			0
$747$	6.68324	+	3.34511i	0.244527	+	0.122391i
$748$		0			0
$749$	−8.62700	+	8.62700i	−0.315224	+	0.315224i
$750$		0			0
$751$		−	24.2030i		−	0.883181i	−0.897217	−	0.441590i	$-0.854414\pi$
$751$		−	24.2030i		−	0.883181i	0.897217	−	0.441590i	$-0.145586\pi$
$752$		0			0
$753$	−36.9407	−	26.6371i	−1.34619	−	0.970709i
$754$		0			0
$755$	−7.47711	−	7.47711i	−0.272120	−	0.272120i
$756$		0			0
$757$	10.9658	−	10.9658i	0.398560	−	0.398560i	−0.479165	−	0.877725i	$-0.659060\pi$
$757$	10.9658	−	10.9658i	0.398560	−	0.398560i	0.877725	+	0.479165i	$0.159060\pi$
$758$		0			0
$759$	−1.11071	+	1.54034i	−0.0403160	+	0.0559109i
$760$		0			0
$761$	15.7528			0.571040			0.285520	−	0.958373i	$-0.407834\pi$
$761$	15.7528			0.571040			0.285520	+	0.958373i	$0.407834\pi$
$762$		0			0
$763$	−11.8092	−	11.8092i	−0.427520	−	0.427520i
$764$		0			0
$765$	6.28929	−	12.5654i	0.227390	−	0.454305i
$766$		0			0
$767$	−14.9706			−0.540557
$768$		0			0
$769$	−12.6179			−0.455012			−0.227506	−	0.973777i	$-0.573057\pi$
$769$	−12.6179			−0.455012			−0.227506	+	0.973777i	$0.573057\pi$
$770$		0			0
$771$	−50.0981	+	8.11907i	−1.80424	+	0.292401i
$772$		0			0
$773$	3.02531	+	3.02531i	0.108813	+	0.108813i	0.759417	−	0.650604i	$-0.225484\pi$
$773$	3.02531	+	3.02531i	0.108813	+	0.108813i	−0.650604	+	0.759417i	$0.725484\pi$
$774$		0			0
$775$	47.6947			1.71324
$776$		0			0
$777$	−2.70926	−	1.95358i	−0.0971941	−	0.0700844i
$778$		0			0
$779$	−4.83227	+	4.83227i	−0.173134	+	0.173134i
$780$		0			0
$781$	−21.0786	−	21.0786i	−0.754252	−	0.754252i
$782$		0			0
$783$	−29.0429	−	9.09231i	−1.03791	−	0.324933i
$784$		0			0
$785$			2.92293i			0.104324i
$786$		0			0
$787$	−21.7502	+	21.7502i	−0.775311	+	0.775311i	−0.979029	−	0.203719i	$-0.934697\pi$
$787$	−21.7502	+	21.7502i	−0.775311	+	0.775311i	0.203719	+	0.979029i	$0.434697\pi$
$788$		0			0
$789$	25.3081	−	4.10151i	0.900991	−	0.146018i
$790$		0			0
$791$		−	14.6863i		−	0.522186i
$792$		0			0
$793$		−	1.50918i		−	0.0535924i
$794$		0			0
$795$	3.60082	−	0.583560i	0.127708	−	0.0206967i
$796$		0			0
$797$	35.4008	−	35.4008i	1.25396	−	1.25396i	0.300030	−	0.953930i	$-0.403003\pi$
$797$	35.4008	−	35.4008i	1.25396	−	1.25396i	0.953930	−	0.300030i	$-0.0969968\pi$
$798$		0			0
$799$		−	55.0467i		−	1.94741i
$800$		0			0
$801$	22.1762	−	7.38178i	0.783559	−	0.260822i
$802$		0			0
$803$	−0.334888	−	0.334888i	−0.0118180	−	0.0118180i
$804$		0			0
$805$	0.248345	−	0.248345i	0.00875299	−	0.00875299i
$806$		0			0
$807$	27.5712	+	19.8810i	0.970553	+	0.699843i
$808$		0			0
$809$	−8.99677			−0.316310			−0.158155	−	0.987414i	$-0.550555\pi$
$809$	−8.99677			−0.316310			−0.158155	+	0.987414i	$0.550555\pi$
$810$		0			0
$811$	20.1943	+	20.1943i	0.709119	+	0.709119i	0.966350	−	0.257231i	$-0.0828102\pi$
$811$	20.1943	+	20.1943i	0.709119	+	0.709119i	−0.257231	+	0.966350i	$0.582810\pi$
$812$		0			0
$813$	34.3716	−	5.57037i	1.20546	−	0.195361i
$814$		0			0
$815$	−13.5927			−0.476133
$816$		0			0
$817$	28.6481			1.00227
$818$		0			0
$819$	−4.49300	−	2.24885i	−0.156998	−	0.0785810i
$820$		0			0
$821$	−19.3687	−	19.3687i	−0.675974	−	0.675974i	0.283113	−	0.959087i	$-0.408633\pi$
$821$	−19.3687	−	19.3687i	−0.675974	−	0.675974i	−0.959087	+	0.283113i	$0.908633\pi$
$822$		0			0
$823$	29.7220			1.03604			0.518022	−	0.855368i	$-0.326669\pi$
$823$	29.7220			1.03604			0.518022	+	0.855368i	$0.326669\pi$
$824$		0			0
$825$	9.40689	−	13.0456i	0.327506	−	0.454190i
$826$		0			0
$827$	30.0621	−	30.0621i	1.04536	−	1.04536i	0.0464408	−	0.998921i	$-0.485212\pi$
$827$	30.0621	−	30.0621i	1.04536	−	1.04536i	0.998921	−	0.0464408i	$-0.0147879\pi$
$828$		0			0
$829$	8.43540	+	8.43540i	0.292973	+	0.292973i	0.838254	−	0.545280i	$-0.183577\pi$
$829$	8.43540	+	8.43540i	0.292973	+	0.292973i	−0.545280	+	0.838254i	$0.683577\pi$
$830$		0			0
$831$	39.4479	+	28.4450i	1.36843	+	0.986745i
$832$		0			0
$833$			7.20845i			0.249758i
$834$		0			0
$835$	−2.85409	+	2.85409i	−0.0987700	+	0.0987700i
$836$		0			0
$837$	−47.9693	+	25.0953i	−1.65806	+	0.867422i
$838$		0			0
$839$			11.9278i			0.411792i	0.978574	+	0.205896i	$0.0660109\pi$
$839$			11.9278i			0.411792i	−0.978574	+	0.205896i	$0.933989\pi$
$840$		0			0
$841$		−	5.30225i		−	0.182836i
$842$		0			0
$843$	0.519407	+	3.20497i	0.0178893	+	0.110385i
$844$		0			0
$845$	4.68421	−	4.68421i	0.161142	−	0.161142i
$846$		0			0
$847$		−	6.88543i		−	0.236586i
$848$		0			0
$849$	−25.7496	+	35.7099i	−0.883724	+	1.22556i
$850$		0			0
$851$	−0.737053	−	0.737053i	−0.0252659	−	0.0252659i
$852$		0			0
$853$	35.5832	−	35.5832i	1.21835	−	1.21835i	0.250136	−	0.968211i	$-0.419525\pi$
$853$	35.5832	−	35.5832i	1.21835	−	1.21835i	0.968211	−	0.250136i	$-0.0804752\pi$
$854$		0			0
$855$	−5.84672	+	1.94619i	−0.199954	+	0.0665584i
$856$		0			0
$857$	8.13947			0.278039			0.139019	−	0.990290i	$-0.455605\pi$
$857$	8.13947			0.278039			0.139019	+	0.990290i	$0.455605\pi$
$858$		0			0
$859$	13.9483	+	13.9483i	0.475912	+	0.475912i	0.903821	−	0.427910i	$-0.140750\pi$
$859$	13.9483	+	13.9483i	0.475912	+	0.475912i	−0.427910	+	0.903821i	$0.640750\pi$
$860$		0			0
$861$	−0.599008	−	3.69614i	−0.0204141	−	0.125964i
$862$		0			0
$863$	41.5365			1.41392			0.706959	−	0.707254i	$-0.250066\pi$
$863$	41.5365			1.41392			0.706959	+	0.707254i	$0.250066\pi$
$864$		0			0
$865$	1.61495			0.0549100
$866$		0			0
$867$	9.68744	+	59.7757i	0.329003	+	2.03009i
$868$		0			0
$869$	7.07338	+	7.07338i	0.239948	+	0.239948i
$870$		0			0
$871$	−5.06876			−0.171748
$872$		0			0
$873$	28.6309	−	9.53034i	0.969009	−	0.322553i
$874$		0			0
$875$	−4.40059	+	4.40059i	−0.148767	+	0.148767i
$876$		0			0
$877$	31.1750	+	31.1750i	1.05270	+	1.05270i	0.998532	+	0.0541722i	$0.0172520\pi$
$877$	31.1750	+	31.1750i	1.05270	+	1.05270i	0.0541722	+	0.998532i	$0.482748\pi$
$878$		0			0
$879$	−25.0502	+	34.7399i	−0.844921	+	1.17175i
$880$		0			0
$881$			15.8809i			0.535041i	0.963552	+	0.267521i	$0.0862044\pi$
$881$			15.8809i			0.535041i	−0.963552	+	0.267521i	$0.913796\pi$
$882$		0			0
$883$	20.2365	−	20.2365i	0.681014	−	0.681014i	−0.279215	−	0.960229i	$-0.590074\pi$
$883$	20.2365	−	20.2365i	0.681014	−	0.681014i	0.960229	+	0.279215i	$0.0900741\pi$
$884$		0			0
$885$	−1.60936	−	9.93048i	−0.0540982	−	0.333809i
$886$		0			0
$887$		−	48.4641i		−	1.62727i	−0.581379	−	0.813633i	$-0.697487\pi$
$887$		−	48.4641i		−	1.62727i	0.581379	−	0.813633i	$-0.302513\pi$
$888$		0			0
$889$			4.43597i			0.148777i
$890$		0			0
$891$	−2.59689	+	18.0703i	−0.0869990	+	0.605379i
$892$		0			0
$893$	−17.0696	+	17.0696i	−0.571212	+	0.571212i
$894$		0			0
$895$		−	5.84195i		−	0.195275i
$896$		0			0
$897$	−1.27179	−	0.917059i	−0.0424638	−	0.0306197i
$898$		0			0
$899$	−43.1479	−	43.1479i	−1.43906	−	1.43906i
$900$		0			0
$901$	−16.5210	+	16.5210i	−0.550395	+	0.550395i
$902$		0			0
$903$	−9.18066	+	12.7319i	−0.305513	+	0.423690i
$904$		0			0
$905$	−9.14983			−0.304151
$906$		0			0
$907$	16.8309	+	16.8309i	0.558861	+	0.558861i	0.928983	−	0.370122i	$-0.120684\pi$
$907$	16.8309	+	16.8309i	0.558861	+	0.558861i	−0.370122	+	0.928983i	$0.620684\pi$
$908$		0			0
$909$	−22.8074	−	11.4156i	−0.756472	−	0.378631i
$910$		0			0
$911$	−2.53307			−0.0839242			−0.0419621	−	0.999119i	$-0.513361\pi$
$911$	−2.53307			−0.0839242			−0.0419621	+	0.999119i	$0.513361\pi$
$912$		0			0
$913$	5.05329			0.167239
$914$		0			0
$915$	1.00108	−	0.162239i	0.0330948	−	0.00536345i
$916$		0			0
$917$	3.16188	+	3.16188i	0.104415	+	0.104415i
$918$		0			0
$919$	−34.0974			−1.12477			−0.562384	−	0.826876i	$-0.690116\pi$
$919$	−34.0974			−1.12477			−0.562384	+	0.826876i	$0.690116\pi$
$920$		0			0
$921$	3.78218	+	2.72724i	0.124627	+	0.0898658i
$922$		0			0
$923$	17.4037	−	17.4037i	0.572848	−	0.572848i
$924$		0			0
$925$	6.24232	+	6.24232i	0.205246	+	0.205246i
$926$		0			0
$927$	47.1904	−	15.7082i	1.54994	−	0.515926i
$928$		0			0
$929$		−	10.3387i		−	0.339201i	−0.985513	−	0.169600i	$-0.945752\pi$
$929$		−	10.3387i		−	0.339201i	0.985513	−	0.169600i	$-0.0542477\pi$
$930$		0			0
$931$	2.23529	−	2.23529i	0.0732587	−	0.0732587i
$932$		0			0
$933$	6.52870	−	1.05806i	0.213740	−	0.0346394i
$934$		0			0
$935$		−	9.50090i		−	0.310713i
$936$		0			0
$937$		−	12.2190i		−	0.399178i	−0.979880	−	0.199589i	$-0.936039\pi$
$937$		−	12.2190i		−	0.399178i	0.979880	−	0.199589i	$-0.0639606\pi$
$938$		0			0
$939$	−4.79405	+	0.776939i	−0.156448	+	0.0253545i
$940$		0			0
$941$	8.49227	−	8.49227i	0.276840	−	0.276840i	−0.555006	−	0.831846i	$-0.687284\pi$
$941$	8.49227	−	8.49227i	0.276840	−	0.276840i	0.831846	+	0.555006i	$0.187284\pi$
$942$		0			0
$943$		−	1.16849i		−	0.0380514i
$944$		0			0
$945$	1.00873	−	3.22210i	0.0328139	−	0.104815i
$946$		0			0
$947$	−6.61991	−	6.61991i	−0.215118	−	0.215118i	0.591319	−	0.806437i	$-0.298607\pi$
$947$	−6.61991	−	6.61991i	−0.215118	−	0.215118i	−0.806437	+	0.591319i	$0.798607\pi$
$948$		0			0
$949$	0.276502	−	0.276502i	0.00897564	−	0.00897564i
$950$		0			0
$951$	−25.0667	−	18.0750i	−0.812844	−	0.586123i
$952$		0			0
$953$	40.5837			1.31463			0.657317	−	0.753614i	$-0.271691\pi$
$953$	40.5837			1.31463			0.657317	+	0.753614i	$0.271691\pi$
$954$		0			0
$955$	−2.83545	−	2.83545i	−0.0917529	−	0.0917529i
$956$		0			0
$957$	−20.3121	+	3.29184i	−0.656597	+	0.106410i
$958$		0			0
$959$	−14.2398			−0.459828
$960$		0			0
$961$	−77.5492			−2.50159
$962$		0			0
$963$	−16.3823	+	32.7303i	−0.527911	+	1.05472i
$964$		0			0
$965$	8.04556	+	8.04556i	0.258996	+	0.258996i
$966$		0			0
$967$	30.2460			0.972646			0.486323	−	0.873779i	$-0.338338\pi$
$967$	30.2460			0.972646			0.486323	+	0.873779i	$0.338338\pi$
$968$		0			0
$969$	23.0844	−	32.0137i	0.741577	−	1.02843i
$970$		0			0
$971$	7.00612	−	7.00612i	0.224837	−	0.224837i	−0.585695	−	0.810532i	$-0.699178\pi$
$971$	7.00612	−	7.00612i	0.224837	−	0.224837i	0.810532	+	0.585695i	$0.199178\pi$
$972$		0			0
$973$	−5.47134	−	5.47134i	−0.175403	−	0.175403i
$974$		0			0
$975$	10.7712	+	7.76684i	0.344954	+	0.248738i
$976$		0			0
$977$			31.7366i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$977$			31.7366i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$978$		0			0
$979$	11.1746	−	11.1746i	0.357142	−	0.357142i
$980$		0			0
$981$	−44.8033	−	22.4250i	−1.43046	−	0.715976i
$982$		0			0
$983$		−	8.56246i		−	0.273100i	−0.990633	−	0.136550i	$-0.956399\pi$
$983$		−	8.56246i		−	0.273100i	0.990633	−	0.136550i	$-0.0436014\pi$
$984$		0			0
$985$		−	7.58519i		−	0.241684i
$986$		0			0
$987$	−2.11595	−	13.0563i	−0.0673513	−	0.415587i
$988$		0			0
$989$	−3.46370	+	3.46370i	−0.110139	+	0.110139i
$990$		0			0
$991$			50.7710i			1.61279i	0.591374	+	0.806397i	$0.298586\pi$
$991$			50.7710i			1.61279i	−0.591374	+	0.806397i	$0.701414\pi$
$992$		0			0
$993$	5.10410	−	7.07845i	0.161974	−	0.224628i
$994$		0			0
$995$	4.09794	+	4.09794i	0.129913	+	0.129913i
$996$		0			0
$997$	−18.1206	+	18.1206i	−0.573886	+	0.573886i	−0.933212	−	0.359326i	$-0.883006\pi$
$997$	−18.1206	+	18.1206i	−0.573886	+	0.573886i	0.359326	+	0.933212i	$0.383006\pi$
$998$		0			0
$999$	−9.56277	−	2.99376i	−0.302553	−	0.0947185i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.s.d.239.12		48
3.2	odd	2	inner	1344.2.s.d.239.23		48
4.3	odd	2		336.2.s.d.323.24	yes	48
12.11	even	2		336.2.s.d.323.1	yes	48
16.5	even	4		336.2.s.d.155.1	✓	48
16.11	odd	4	inner	1344.2.s.d.911.23		48
48.5	odd	4		336.2.s.d.155.24	yes	48
48.11	even	4	inner	1344.2.s.d.911.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.s.d.155.1	✓	48	16.5	even	4
336.2.s.d.155.24	yes	48	48.5	odd	4
336.2.s.d.323.1	yes	48	12.11	even	2
336.2.s.d.323.24	yes	48	4.3	odd	2
1344.2.s.d.239.12		48	1.1	even	1	trivial
1344.2.s.d.239.23		48	3.2	odd	2	inner
1344.2.s.d.911.12		48	48.11	even	4	inner
1344.2.s.d.911.23		48	16.11	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.277087 - 1.70974i) q^{3} +(0.459458 + 0.459458i) q^{5} -1.00000 q^{7} +(-2.84645 + 0.947494i) q^{9} +O(q^{10})\)	\(q+(-0.277087 - 1.70974i) q^{3} +(0.459458 + 0.459458i) q^{5} -1.00000 q^{7} +(-2.84645 + 0.947494i) q^{9} +(-1.43432 + 1.43432i) q^{11} +(-1.18426 - 1.18426i) q^{13} +(0.658245 - 0.912864i) q^{15} +7.20845i q^{17} +(2.23529 - 2.23529i) q^{19} +(0.277087 + 1.70974i) q^{21} +0.540517i q^{23} -4.57780i q^{25} +(2.40868 + 4.60416i) q^{27} +(-4.14139 + 4.14139i) q^{29} +10.4187i q^{31} +(2.84976 + 2.05490i) q^{33} +(-0.459458 - 0.459458i) q^{35} +(-1.36361 + 1.36361i) q^{37} +(-1.69663 + 2.35292i) q^{39} -2.16181 q^{41} +(6.40813 + 6.40813i) q^{43} +(-1.74315 - 0.872488i) q^{45} -7.63641 q^{47} +1.00000 q^{49} +(12.3246 - 1.99736i) q^{51} +(2.29189 + 2.29189i) q^{53} -1.31802 q^{55} +(-4.44114 - 3.20240i) q^{57} +(6.32068 - 6.32068i) q^{59} +(0.637183 + 0.637183i) q^{61} +(2.84645 - 0.947494i) q^{63} -1.08823i q^{65} +(2.14006 - 2.14006i) q^{67} +(0.924145 - 0.149770i) q^{69} +14.6958i q^{71} +0.233482i q^{73} +(-7.82686 + 1.26845i) q^{75} +(1.43432 - 1.43432i) q^{77} -4.93151i q^{79} +(7.20451 - 5.39398i) q^{81} +(-1.76156 - 1.76156i) q^{83} +(-3.31198 + 3.31198i) q^{85} +(8.22824 + 5.93319i) q^{87} -7.79085 q^{89} +(1.18426 + 1.18426i) q^{91} +(17.8133 - 2.88688i) q^{93} +2.05404 q^{95} -10.0585 q^{97} +(2.72371 - 5.44174i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{7}+O(q^{10}) \) 48 * q - 48 * q^7	\( 48 q - 48 q^{7} - 8 q^{19} + 12 q^{27} + 16 q^{37} + 24 q^{39} + 48 q^{43} + 20 q^{45} + 48 q^{49} + 32 q^{55} + 8 q^{61} + 16 q^{67} - 28 q^{69} + 12 q^{75} - 48 q^{85} - 56 q^{87} - 64 q^{93} - 32 q^{99}+O(q^{100}) \) 48 * q - 48 * q^7 - 8 * q^19 + 12 * q^27 + 16 * q^37 + 24 * q^39 + 48 * q^43 + 20 * q^45 + 48 * q^49 + 32 * q^55 + 8 * q^61 + 16 * q^67 - 28 * q^69 + 12 * q^75 - 48 * q^85 - 56 * q^87 - 64 * q^93 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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