LMFDB - Embedded newform 416.2.k.f.255.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(31,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.31"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,2,0,-20,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16 $ x^8 + 17x^6 + 84x^4 + 100*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			255.4
Root			$2.65328i$ of defining polynomial
Character	$\chi$	$=$	416.255
Dual form			416.2.k.f.31.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.89949i q^{3} +(-2.84658 - 2.84658i) q^{5} +(0.193303 + 0.193303i) q^{7} -5.40706 q^{9} +(-3.65328 - 3.65328i) q^{11} +(-0.193303 - 3.60037i) q^{13} +(8.25364 - 8.25364i) q^{15} +0.899494i q^{17} +(-0.753785 + 0.753785i) q^{19} +(-0.560481 + 0.560481i) q^{21} +1.89418 q^{23} +11.2060i q^{25} -6.97926i q^{27} -5.41238 q^{29} +(-3.24622 + 3.24622i) q^{31} +(10.5927 - 10.5927i) q^{33} -1.10051i q^{35} +(2.95241 - 2.95241i) q^{37} +(10.4392 - 0.560481i) q^{39} +(-5.79367 - 5.79367i) q^{41} +2.81972 q^{43} +(15.3916 + 15.3916i) q^{45} +(-7.49986 - 7.49986i) q^{47} -6.92527i q^{49} -2.60808 q^{51} -5.69316 q^{53} +20.7987i q^{55} +(-2.18559 - 2.18559i) q^{57} +(8.44695 + 8.44695i) q^{59} -2.98486 q^{61} +(-1.04520 - 1.04520i) q^{63} +(-9.69848 + 10.7990i) q^{65} +(-8.85401 + 8.85401i) q^{67} +5.49215i q^{69} +(1.91252 - 1.91252i) q^{71} +(5.40706 - 5.40706i) q^{73} -32.4919 q^{75} -1.41238i q^{77} +1.20073i q^{79} +4.01514 q^{81} +(-2.14571 + 2.14571i) q^{83} +(2.56048 - 2.56048i) q^{85} -15.6932i q^{87} +(-1.79367 + 1.79367i) q^{89} +(0.658596 - 0.733328i) q^{91} +(-9.41238 - 9.41238i) q^{93} +4.29142 q^{95} +(-1.10582 - 1.10582i) q^{97} +(19.7535 + 19.7535i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{7} - 20 q^{9} - 6 q^{11} - 2 q^{13} + 20 q^{15} - 6 q^{19} - 4 q^{21} + 16 q^{23} + 4 q^{29} - 26 q^{31} + 16 q^{33} - 8 q^{39} - 24 q^{41} + 44 q^{43} + 8 q^{45} - 14 q^{47} - 44 q^{51} + 28 q^{57}+ \cdots + 66 q^{99}+O(q^{100}) $ 8 * q + 2 * q^7 - 20 * q^9 - 6 * q^11 - 2 * q^13 + 20 * q^15 - 6 * q^19 - 4 * q^21 + 16 * q^23 + 4 * q^29 - 26 * q^31 + 16 * q^33 - 8 * q^39 - 24 * q^41 + 44 * q^43 + 8 * q^45 - 14 * q^47 - 44 * q^51 + 28 * q^57 + 22 * q^59 - 24 * q^61 + 38 * q^63 - 8 * q^65 - 2 * q^67 + 14 * q^71 + 20 * q^73 - 76 * q^75 + 32 * q^81 + 6 * q^83 + 20 * q^85 + 8 * q^89 - 42 * q^91 - 28 * q^93 - 12 * q^95 - 8 * q^97 + 66 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$			2.89949i			1.67402i	0.547185	+	0.837012i	$0.315699\pi$
$3$			2.89949i			1.67402i	−0.547185	+	0.837012i	$0.684301\pi$
$4$		0			0
$5$	−2.84658	−	2.84658i	−1.27303	−	1.27303i	−0.944490	−	0.328540i	$-0.893444\pi$
$5$	−2.84658	−	2.84658i	−1.27303	−	1.27303i	−0.328540	−	0.944490i	$-0.606556\pi$
$6$		0			0
$7$	0.193303	+	0.193303i	0.0730617	+	0.0730617i	0.742693	−	0.669632i	$-0.233548\pi$
$7$	0.193303	+	0.193303i	0.0730617	+	0.0730617i	−0.669632	+	0.742693i	$0.733548\pi$
$8$		0			0
$9$	−5.40706			−1.80235
$10$		0			0
$11$	−3.65328	−	3.65328i	−1.10150	−	1.10150i	−0.994229	−	0.107275i	$-0.965787\pi$
$11$	−3.65328	−	3.65328i	−1.10150	−	1.10150i	−0.107275	−	0.994229i	$-0.534213\pi$
$12$		0			0
$13$	−0.193303	−	3.60037i	−0.0536127	−	0.998562i
$13$	−0.193303	−	3.60037i	−0.0536127	−	0.998562i
$14$		0			0
$15$	8.25364	−	8.25364i	2.13108	−	2.13108i
$16$		0			0
$17$			0.899494i			0.218159i	0.994033	+	0.109080i	$0.0347903\pi$
$17$			0.899494i			0.218159i	−0.994033	+	0.109080i	$0.965210\pi$
$18$		0			0
$19$	−0.753785	+	0.753785i	−0.172930	+	0.172930i	−0.788265	−	0.615335i	$-0.789021\pi$
$19$	−0.753785	+	0.753785i	−0.172930	+	0.172930i	0.615335	+	0.788265i	$0.289021\pi$
$20$		0			0
$21$	−0.560481	+	0.560481i	−0.122307	+	0.122307i
$22$		0			0
$23$	1.89418			0.394963			0.197481	−	0.980307i	$-0.436724\pi$
$23$	1.89418			0.394963			0.197481	+	0.980307i	$0.436724\pi$
$24$		0			0
$25$			11.2060i			2.24121i
$26$		0			0
$27$		−	6.97926i		−	1.34316i
$28$		0			0
$29$	−5.41238			−1.00505			−0.502527	−	0.864562i	$-0.667596\pi$
$29$	−5.41238			−1.00505			−0.502527	+	0.864562i	$0.667596\pi$
$30$		0			0
$31$	−3.24622	+	3.24622i	−0.583038	+	0.583038i	−0.935737	−	0.352699i	$-0.885264\pi$
$31$	−3.24622	+	3.24622i	−0.583038	+	0.583038i	0.352699	+	0.935737i	$0.385264\pi$
$32$		0			0
$33$	10.5927	−	10.5927i	1.84394	−	1.84394i
$34$		0			0
$35$		−	1.10051i		−	0.186020i
$36$		0			0
$37$	2.95241	−	2.95241i	0.485373	−	0.485373i	−0.421470	−	0.906842i	$-0.638486\pi$
$37$	2.95241	−	2.95241i	0.485373	−	0.485373i	0.906842	+	0.421470i	$0.138486\pi$
$38$		0			0
$39$	10.4392	−	0.560481i	1.67162	−	0.0897488i
$40$		0			0
$41$	−5.79367	−	5.79367i	−0.904819	−	0.904819i	0.0910291	−	0.995848i	$-0.470984\pi$
$41$	−5.79367	−	5.79367i	−0.904819	−	0.904819i	−0.995848	+	0.0910291i	$0.970984\pi$
$42$		0			0
$43$	2.81972			0.430004			0.215002	−	0.976614i	$-0.431024\pi$
$43$	2.81972			0.430004			0.215002	+	0.976614i	$0.431024\pi$
$44$		0			0
$45$	15.3916	+	15.3916i	2.29445	+	2.29445i
$46$		0			0
$47$	−7.49986	−	7.49986i	−1.09397	−	1.09397i	−0.995101	−	0.0988662i	$-0.968478\pi$
$47$	−7.49986	−	7.49986i	−1.09397	−	1.09397i	−0.0988662	−	0.995101i	$-0.531522\pi$
$48$		0			0
$49$		−	6.92527i		−	0.989324i
$50$		0			0
$51$	−2.60808			−0.365204
$52$		0			0
$53$	−5.69316			−0.782016			−0.391008	−	0.920387i	$-0.627874\pi$
$53$	−5.69316			−0.782016			−0.391008	+	0.920387i	$0.627874\pi$
$54$		0			0
$55$			20.7987i			2.80450i
$56$		0			0
$57$	−2.18559	−	2.18559i	−0.289489	−	0.289489i
$58$		0			0
$59$	8.44695	+	8.44695i	1.09970	+	1.09970i	0.994445	+	0.105253i	$0.0335654\pi$
$59$	8.44695	+	8.44695i	1.09970	+	1.09970i	0.105253	+	0.994445i	$0.466435\pi$
$60$		0			0
$61$	−2.98486			−0.382172			−0.191086	−	0.981573i	$-0.561201\pi$
$61$	−2.98486			−0.382172			−0.191086	+	0.981573i	$0.561201\pi$
$62$		0			0
$63$	−1.04520	−	1.04520i	−0.131683	−	0.131683i
$64$		0			0
$65$	−9.69848	+	10.7990i	−1.20295	+	1.33945i
$66$		0			0
$67$	−8.85401	+	8.85401i	−1.08169	+	1.08169i	−0.0853373	+	0.996352i	$0.527197\pi$
$67$	−8.85401	+	8.85401i	−1.08169	+	1.08169i	−0.996352	+	0.0853373i	$0.972803\pi$
$68$		0			0
$69$			5.49215i			0.661177i
$70$		0			0
$71$	1.91252	−	1.91252i	0.226975	−	0.226975i	−0.584453	−	0.811428i	$-0.698691\pi$
$71$	1.91252	−	1.91252i	0.226975	−	0.226975i	0.811428	+	0.584453i	$0.198691\pi$
$72$		0			0
$73$	5.40706	−	5.40706i	0.632849	−	0.632849i	−0.315933	−	0.948782i	$-0.602317\pi$
$73$	5.40706	−	5.40706i	0.632849	−	0.632849i	0.948782	+	0.315933i	$0.102317\pi$
$74$		0			0
$75$	−32.4919			−3.75184
$76$		0			0
$77$		−	1.41238i		−	0.160956i
$78$		0			0
$79$			1.20073i			0.135093i	0.997716	+	0.0675464i	$0.0215171\pi$
$79$			1.20073i			0.135093i	−0.997716	+	0.0675464i	$0.978483\pi$
$80$		0			0
$81$	4.01514			0.446126
$82$		0			0
$83$	−2.14571	+	2.14571i	−0.235522	+	0.235522i	−0.814993	−	0.579471i	$-0.803259\pi$
$83$	−2.14571	+	2.14571i	−0.235522	+	0.235522i	0.579471	+	0.814993i	$0.303259\pi$
$84$		0			0
$85$	2.56048	−	2.56048i	0.277723	−	0.277723i
$86$		0			0
$87$		−	15.6932i		−	1.68248i
$88$		0			0
$89$	−1.79367	+	1.79367i	−0.190129	+	0.190129i	−0.795752	−	0.605623i	$-0.792924\pi$
$89$	−1.79367	+	1.79367i	−0.190129	+	0.190129i	0.605623	+	0.795752i	$0.292924\pi$
$90$		0			0
$91$	0.658596	−	0.733328i	0.0690396	−	0.0768737i
$92$		0			0
$93$	−9.41238	−	9.41238i	−0.976018	−	0.976018i
$94$		0			0
$95$	4.29142			0.440290
$96$		0			0
$97$	−1.10582	−	1.10582i	−0.112279	−	0.112279i	0.648735	−	0.761014i	$-0.275298\pi$
$97$	−1.10582	−	1.10582i	−0.112279	−	0.112279i	−0.761014	+	0.648735i	$0.775298\pi$
$98$		0			0
$99$	19.7535	+	19.7535i	1.98530	+	1.98530i
$100$		0			0
$101$			3.22679i			0.321077i	0.987030	+	0.160539i	$0.0513231\pi$
$101$			3.22679i			0.321077i	−0.987030	+	0.160539i	$0.948677\pi$
$102$		0			0
$103$	−8.46638			−0.834217			−0.417108	−	0.908857i	$-0.636956\pi$
$103$	−8.46638			−0.834217			−0.417108	+	0.908857i	$0.636956\pi$
$104$		0			0
$105$	3.19091			0.311401
$106$		0			0
$107$			8.29142i			0.801562i	0.916174	+	0.400781i	$0.131261\pi$
$107$			8.29142i			0.801562i	−0.916174	+	0.400781i	$0.868739\pi$
$108$		0			0
$109$	10.7408	+	10.7408i	1.02878	+	1.02878i	0.999573	+	0.0292045i	$0.00929739\pi$
$109$	10.7408	+	10.7408i	1.02878	+	1.02878i	0.0292045	+	0.999573i	$0.490703\pi$
$110$		0			0
$111$	8.56048	+	8.56048i	0.812525	+	0.812525i
$112$		0			0
$113$	16.4919			1.55142			0.775712	−	0.631087i	$-0.217391\pi$
$113$	16.4919			1.55142			0.775712	+	0.631087i	$0.217391\pi$
$114$		0			0
$115$	−5.39192	−	5.39192i	−0.502800	−	0.502800i
$116$		0			0
$117$	1.04520	+	19.4674i	0.0966290	+	1.79976i
$118$		0			0
$119$	−0.173875	+	0.173875i	−0.0159391	+	0.0159391i
$120$		0			0
$121$			15.6929i			1.42663i
$122$		0			0
$123$	16.7987	−	16.7987i	1.51469	−	1.51469i
$124$		0			0
$125$	17.6660	−	17.6660i	1.58010	−	1.58010i
$126$		0			0
$127$	−1.89418			−0.168081			−0.0840404	−	0.996462i	$-0.526782\pi$
$127$	−1.89418			−0.168081			−0.0840404	+	0.996462i	$0.526782\pi$
$128$		0			0
$129$			8.17577i			0.719837i
$130$		0			0
$131$		−	18.3267i		−	1.60121i	−0.599190	−	0.800607i	$-0.704511\pi$
$131$		−	18.3267i		−	1.60121i	0.599190	−	0.800607i	$-0.295489\pi$
$132$		0			0
$133$	−0.291418			−0.0252691
$134$		0			0
$135$	−19.8670	+	19.8670i	−1.70988	+	1.70988i
$136$		0			0
$137$	5.98486	−	5.98486i	0.511321	−	0.511321i	−0.403610	−	0.914931i	$-0.632245\pi$
$137$	5.98486	−	5.98486i	0.511321	−	0.511321i	0.914931	+	0.403610i	$0.132245\pi$
$138$		0			0
$139$		−	1.31216i		−	0.111296i	−0.998450	−	0.0556478i	$-0.982278\pi$
$139$		−	1.31216i		−	0.111296i	0.998450	−	0.0556478i	$-0.0177224\pi$
$140$		0			0
$141$	21.7458	−	21.7458i	1.83133	−	1.83133i
$142$		0			0
$143$	−12.4469	+	13.8593i	−1.04087	+	1.15898i
$144$		0			0
$145$	15.4068	+	15.4068i	1.27946	+	1.27946i
$146$		0			0
$147$	20.0798			1.65615
$148$		0			0
$149$	0.111143	+	0.111143i	0.00910515	+	0.00910515i	0.711645	−	0.702540i	$-0.247951\pi$
$149$	0.111143	+	0.111143i	0.00910515	+	0.00910515i	−0.702540	+	0.711645i	$0.747951\pi$
$150$		0			0
$151$	−6.37890	−	6.37890i	−0.519107	−	0.519107i	0.398194	−	0.917301i	$-0.369637\pi$
$151$	−6.37890	−	6.37890i	−0.519107	−	0.519107i	−0.917301	+	0.398194i	$0.869637\pi$
$152$		0			0
$153$		−	4.86362i		−	0.393200i
$154$		0			0
$155$	18.4812			1.48445
$156$		0			0
$157$	−6.49243			−0.518152			−0.259076	−	0.965857i	$-0.583418\pi$
$157$	−6.49243			−0.518152			−0.259076	+	0.965857i	$0.583418\pi$
$158$		0			0
$159$		−	16.5073i		−	1.30911i
$160$		0			0
$161$	0.366150	+	0.366150i	0.0288567	+	0.0288567i
$162$		0			0
$163$	−7.34644	−	7.34644i	−0.575418	−	0.575418i	0.358220	−	0.933637i	$-0.383384\pi$
$163$	−7.34644	−	7.34644i	−0.575418	−	0.575418i	−0.933637	+	0.358220i	$0.883384\pi$
$164$		0			0
$165$	−60.3057			−4.69479
$166$		0			0
$167$	−2.34112	−	2.34112i	−0.181162	−	0.181162i	0.610700	−	0.791862i	$-0.290888\pi$
$167$	−2.34112	−	2.34112i	−0.181162	−	0.181162i	−0.791862	+	0.610700i	$0.790888\pi$
$168$		0			0
$169$	−12.9253	+	1.39192i	−0.994251	+	0.107071i
$170$		0			0
$171$	4.07576	−	4.07576i	0.311681	−	0.311681i
$172$		0			0
$173$		−	19.0795i		−	1.45059i	−0.688440	−	0.725293i	$-0.741704\pi$
$173$		−	19.0795i		−	1.45059i	0.688440	−	0.725293i	$-0.258296\pi$
$174$		0			0
$175$	−2.16617	+	2.16617i	−0.163747	+	0.163747i
$176$		0			0
$177$	−24.4919	+	24.4919i	−1.84092	+	1.84092i
$178$		0			0
$179$	18.4065			1.37577			0.687883	−	0.725821i	$-0.258540\pi$
$179$	18.4065			1.37577			0.687883	+	0.725821i	$0.258540\pi$
$180$		0			0
$181$		−	9.98458i		−	0.742148i	−0.928603	−	0.371074i	$-0.878990\pi$
$181$		−	9.98458i		−	0.742148i	0.928603	−	0.371074i	$-0.121010\pi$
$182$		0			0
$183$		−	8.65459i		−	0.639766i
$184$		0			0
$185$	−16.8085			−1.23579
$186$		0			0
$187$	3.28610	−	3.28610i	0.240303	−	0.240303i
$188$		0			0
$189$	1.34911	−	1.34911i	0.0981336	−	0.0981336i
$190$		0			0
$191$			4.79871i			0.347222i	0.984814	+	0.173611i	$0.0555436\pi$
$191$			4.79871i			0.347222i	−0.984814	+	0.173611i	$0.944456\pi$
$192$		0			0
$193$	14.2805	−	14.2805i	1.02793	−	1.02793i	0.0283344	−	0.999598i	$-0.490980\pi$
$193$	14.2805	−	14.2805i	1.02793	−	1.02793i	0.999598	−	0.0283344i	$-0.00902033\pi$
$194$		0			0
$195$	−31.3116	−	28.1207i	−2.24227	−	2.01376i
$196$		0			0
$197$	−5.23319	−	5.23319i	−0.372849	−	0.372849i	0.495665	−	0.868514i	$-0.334924\pi$
$197$	−5.23319	−	5.23319i	−0.372849	−	0.372849i	−0.868514	+	0.495665i	$0.834924\pi$
$198$		0			0
$199$	17.2805			1.22498			0.612491	−	0.790477i	$-0.290168\pi$
$199$	17.2805			1.22498			0.612491	+	0.790477i	$0.290168\pi$
$200$		0			0
$201$	−25.6721	−	25.6721i	−1.81077	−	1.81077i
$202$		0			0
$203$	−1.04623	−	1.04623i	−0.0734310	−	0.0734310i
$204$		0			0
$205$			32.9843i			2.30372i
$206$		0			0
$207$	−10.2419			−0.711863
$208$		0			0
$209$	5.50757			0.380967
$210$		0			0
$211$			7.30152i			0.502657i	0.967902	+	0.251329i	$0.0808675\pi$
$211$			7.30152i			0.502657i	−0.967902	+	0.251329i	$0.919133\pi$
$212$		0			0
$213$	5.54534	+	5.54534i	0.379961	+	0.379961i
$214$		0			0
$215$	−8.02658	−	8.02658i	−0.547408	−	0.547408i
$216$		0			0
$217$	−1.25501			−0.0851955
$218$		0			0
$219$	15.6777	+	15.6777i	1.05940	+	1.05940i
$220$		0			0
$221$	3.23851	−	0.173875i	0.217845	−	0.0116961i
$222$		0			0
$223$	7.71151	−	7.71151i	0.516401	−	0.516401i	−0.400080	−	0.916480i	$-0.631018\pi$
$223$	7.71151	−	7.71151i	0.516401	−	0.516401i	0.916480	+	0.400080i	$0.131018\pi$
$224$		0			0
$225$		−	60.5918i		−	4.03945i
$226$		0			0
$227$	−11.5376	+	11.5376i	−0.765780	+	0.765780i	−0.977360	−	0.211581i	$-0.932139\pi$
$227$	−11.5376	+	11.5376i	−0.765780	+	0.765780i	0.211581	+	0.977360i	$0.432139\pi$
$228$		0			0
$229$	7.13800	−	7.13800i	0.471692	−	0.471692i	−0.430770	−	0.902462i	$-0.641758\pi$
$229$	7.13800	−	7.13800i	0.471692	−	0.471692i	0.902462	+	0.430770i	$0.141758\pi$
$230$		0			0
$231$	4.09519			0.269444
$232$		0			0
$233$			15.6338i			1.02421i	0.858924	+	0.512104i	$0.171134\pi$
$233$			15.6338i			1.02421i	−0.858924	+	0.512104i	$0.828866\pi$
$234$		0			0
$235$			42.6979i			2.78531i
$236$		0			0
$237$	−3.48151			−0.226149
$238$		0			0
$239$	−17.9923	+	17.9923i	−1.16382	+	1.16382i	−0.180193	+	0.983631i	$0.557672\pi$
$239$	−17.9923	+	17.9923i	−1.16382	+	1.16382i	−0.983631	+	0.180193i	$0.942328\pi$
$240$		0			0
$241$	−2.56688	+	2.56688i	−0.165347	+	0.165347i	−0.784931	−	0.619583i	$-0.787302\pi$
$241$	−2.56688	+	2.56688i	−0.165347	+	0.165347i	0.619583	+	0.784931i	$0.287302\pi$
$242$		0			0
$243$		−	9.29592i		−	0.596334i
$244$		0			0
$245$	−19.7133	+	19.7133i	−1.25944	+	1.25944i
$246$		0			0
$247$	2.85961	+	2.56819i	0.181953	+	0.163410i
$248$		0			0
$249$	−6.22147	−	6.22147i	−0.394270	−	0.394270i
$250$		0			0
$251$	14.3606			0.906430			0.453215	−	0.891401i	$-0.350277\pi$
$251$	14.3606			0.906430			0.453215	+	0.891401i	$0.350277\pi$
$252$		0			0
$253$	−6.91995	−	6.91995i	−0.435054	−	0.435054i
$254$		0			0
$255$	7.42410	+	7.42410i	0.464915	+	0.464915i
$256$		0			0
$257$		−	4.96863i		−	0.309934i	−0.987920	−	0.154967i	$-0.950473\pi$
$257$		−	4.96863i		−	0.309934i	0.987920	−	0.154967i	$-0.0495272\pi$
$258$		0			0
$259$	1.14142			0.0709243
$260$		0			0
$261$	29.2651			1.81146
$262$		0			0
$263$		−	19.6777i		−	1.21338i	−0.794938	−	0.606691i	$-0.792497\pi$
$263$		−	19.6777i		−	1.21338i	0.794938	−	0.606691i	$-0.207503\pi$
$264$		0			0
$265$	16.2060	+	16.2060i	0.995530	+	0.995530i
$266$		0			0
$267$	−5.20073	−	5.20073i	−0.318280	−	0.318280i
$268$		0			0
$269$	11.2544			0.686196			0.343098	−	0.939300i	$-0.388524\pi$
$269$	11.2544			0.686196			0.343098	+	0.939300i	$0.388524\pi$
$270$		0			0
$271$	7.88647	+	7.88647i	0.479069	+	0.479069i	0.904834	−	0.425765i	$-0.139995\pi$
$271$	7.88647	+	7.88647i	0.479069	+	0.479069i	−0.425765	+	0.904834i	$0.639995\pi$
$272$		0			0
$273$	2.12628	+	1.90960i	0.128688	+	0.115574i
$274$		0			0
$275$	40.9388	−	40.9388i	2.46870	−	2.46870i
$276$		0			0
$277$		−	10.4127i		−	0.625636i	−0.949813	−	0.312818i	$-0.898727\pi$
$277$		−	10.4127i		−	0.625636i	0.949813	−	0.312818i	$-0.101273\pi$
$278$		0			0
$279$	17.5525	−	17.5525i	1.05084	−	1.05084i
$280$		0			0
$281$	13.8734	−	13.8734i	0.827620	−	0.827620i	−0.159567	−	0.987187i	$-0.551010\pi$
$281$	13.8734	−	13.8734i	0.827620	−	0.827620i	0.987187	+	0.159567i	$0.0510099\pi$
$282$		0			0
$283$	−20.0000			−1.18888			−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000			−1.18888			−0.594438	+	0.804141i	$0.702626\pi$
$284$		0			0
$285$			12.4429i			0.737056i
$286$		0			0
$287$		−	2.23987i		−	0.132215i
$288$		0			0
$289$	16.1909			0.952407
$290$		0			0
$291$	3.20633	−	3.20633i	0.187958	−	0.187958i
$292$		0			0
$293$	−22.7662	+	22.7662i	−1.33002	+	1.33002i	−0.424671	+	0.905348i	$0.639610\pi$
$293$	−22.7662	+	22.7662i	−1.33002	+	1.33002i	−0.905348	+	0.424671i	$0.860390\pi$
$294$		0			0
$295$		−	48.0898i		−	2.79990i
$296$		0			0
$297$	−25.4972	+	25.4972i	−1.47950	+	1.47950i
$298$		0			0
$299$	−0.366150	−	6.81972i	−0.0211750	−	0.394395i
$300$		0			0
$301$	0.545062	+	0.545062i	0.0314168	+	0.0314168i
$302$		0			0
$303$	−9.35605			−0.537491
$304$		0			0
$305$	8.49665	+	8.49665i	0.486517	+	0.486517i
$306$		0			0
$307$	−12.3151	−	12.3151i	−0.702858	−	0.702858i	0.262165	−	0.965023i	$-0.415564\pi$
$307$	−12.3151	−	12.3151i	−0.702858	−	0.702858i	−0.965023	+	0.262165i	$0.915564\pi$
$308$		0			0
$309$		−	24.5482i		−	1.39650i
$310$		0			0
$311$	−21.0258			−1.19226			−0.596131	−	0.802887i	$-0.703296\pi$
$311$	−21.0258			−1.19226			−0.596131	+	0.802887i	$0.703296\pi$
$312$		0			0
$313$	−0.498030			−0.0281503			−0.0140752	−	0.999901i	$-0.504480\pi$
$313$	−0.498030			−0.0281503			−0.0140752	+	0.999901i	$0.504480\pi$
$314$		0			0
$315$			5.95051i			0.335273i
$316$		0			0
$317$	14.5126	+	14.5126i	0.815109	+	0.815109i	0.985395	−	0.170286i	$-0.0544690\pi$
$317$	14.5126	+	14.5126i	0.815109	+	0.815109i	−0.170286	+	0.985395i	$0.554469\pi$
$318$		0			0
$319$	19.7729	+	19.7729i	1.10707	+	1.10707i
$320$		0			0
$321$	−24.0409			−1.34183
$322$		0			0
$323$	−0.678024	−	0.678024i	−0.0377263	−	0.0377263i
$324$		0			0
$325$	40.3459	−	2.16617i	2.23799	−	0.120157i
$326$		0			0
$327$	−31.1428	+	31.1428i	−1.72220	+	1.72220i
$328$		0			0
$329$		−	2.89949i		−	0.159854i
$330$		0			0
$331$	−6.18662	+	6.18662i	−0.340047	+	0.340047i	−0.856385	−	0.516338i	$-0.827295\pi$
$331$	−6.18662	+	6.18662i	−0.340047	+	0.340047i	0.516338	+	0.856385i	$0.327295\pi$
$332$		0			0
$333$	−15.9638	+	15.9638i	−0.874813	+	0.874813i
$334$		0			0
$335$	50.4073			2.75405
$336$		0			0
$337$		−	6.91013i		−	0.376419i	−0.982129	−	0.188209i	$-0.939732\pi$
$337$		−	6.91013i		−	0.376419i	0.982129	−	0.188209i	$-0.0602684\pi$
$338$		0			0
$339$			47.8181i			2.59712i
$340$		0			0
$341$	23.7187			1.28444
$342$		0			0
$343$	2.69180	−	2.69180i	0.145343	−	0.145343i
$344$		0			0
$345$	15.6338	−	15.6338i	0.841698	−	0.841698i
$346$		0			0
$347$		−	28.2965i		−	1.51903i	−0.650488	−	0.759517i	$-0.725436\pi$
$347$		−	28.2965i		−	1.51903i	0.650488	−	0.759517i	$-0.274564\pi$
$348$		0			0
$349$	−12.9524	+	12.9524i	−0.693326	+	0.693326i	−0.962962	−	0.269636i	$-0.913097\pi$
$349$	−12.9524	+	12.9524i	−0.693326	+	0.693326i	0.269636	+	0.962962i	$0.413097\pi$
$350$		0			0
$351$	−25.1279	+	1.34911i	−1.34123	+	0.0720104i
$352$		0			0
$353$	−14.7341	−	14.7341i	−0.784216	−	0.784216i	0.196323	−	0.980539i	$-0.437100\pi$
$353$	−14.7341	−	14.7341i	−0.784216	−	0.784216i	−0.980539	+	0.196323i	$0.937100\pi$
$354$		0			0
$355$	−10.8883			−0.577891
$356$		0			0
$357$	−0.504149	−	0.504149i	−0.0266824	−	0.0266824i
$358$		0			0
$359$	−9.73865	−	9.73865i	−0.513986	−	0.513986i	0.401759	−	0.915745i	$-0.368399\pi$
$359$	−9.73865	−	9.73865i	−0.513986	−	0.513986i	−0.915745	+	0.401759i	$0.868399\pi$
$360$		0			0
$361$			17.8636i			0.940190i
$362$		0			0
$363$	−45.5014			−2.38820
$364$		0			0
$365$	−30.7833			−1.61127
$366$		0			0
$367$		−	11.2114i		−	0.585229i	−0.956231	−	0.292614i	$-0.905475\pi$
$367$		−	11.2114i		−	0.585229i	0.956231	−	0.292614i	$-0.0945252\pi$
$368$		0			0
$369$	31.3267	+	31.3267i	1.63080	+	1.63080i
$370$		0			0
$371$	−1.10051	−	1.10051i	−0.0571355	−	0.0571355i
$372$		0			0
$373$	15.3475			0.794662			0.397331	−	0.917675i	$-0.369937\pi$
$373$	15.3475			0.794662			0.397331	+	0.917675i	$0.369937\pi$
$374$		0			0
$375$	51.2225	+	51.2225i	2.64512	+	2.64512i
$376$		0			0
$377$	1.04623	+	19.4866i	0.0538836	+	1.00361i
$378$		0			0
$379$	−9.62722	+	9.62722i	−0.494517	+	0.494517i	−0.909726	−	0.415209i	$-0.863708\pi$
$379$	−9.62722	+	9.62722i	−0.494517	+	0.494517i	0.415209	+	0.909726i	$0.363708\pi$
$380$		0			0
$381$		−	5.49215i		−	0.281371i
$382$		0			0
$383$	−20.5102	+	20.5102i	−1.04802	+	1.04802i	−0.0492358	+	0.998787i	$0.515679\pi$
$383$	−20.5102	+	20.5102i	−1.04802	+	1.04802i	−0.998787	+	0.0492358i	$0.984321\pi$
$384$		0			0
$385$	−4.02046	+	4.02046i	−0.204901	+	0.204901i
$386$		0			0
$387$	−15.2464			−0.775019
$388$		0			0
$389$		−	26.1616i		−	1.32645i	−0.748422	−	0.663223i	$-0.769188\pi$
$389$		−	26.1616i		−	1.32645i	0.748422	−	0.663223i	$-0.230812\pi$
$390$		0			0
$391$			1.70380i			0.0861648i
$392$		0			0
$393$	53.1382			2.68047
$394$		0			0
$395$	3.41798	−	3.41798i	0.171977	−	0.171977i
$396$		0			0
$397$	10.1909	−	10.1909i	0.511467	−	0.511467i	−0.403509	−	0.914976i	$-0.632209\pi$
$397$	10.1909	−	10.1909i	0.511467	−	0.511467i	0.914976	+	0.403509i	$0.132209\pi$
$398$		0			0
$399$		−	0.844964i		−	0.0423011i
$400$		0			0
$401$	−1.85298	+	1.85298i	−0.0925335	+	0.0925335i	−0.751858	−	0.659325i	$-0.770842\pi$
$401$	−1.85298	+	1.85298i	−0.0925335	+	0.0925335i	0.659325	+	0.751858i	$0.270842\pi$
$402$		0			0
$403$	12.3151	+	11.0601i	0.613457	+	0.550941i
$404$		0			0
$405$	−11.4294	−	11.4294i	−0.567932	−	0.567932i
$406$		0			0
$407$	−21.5719			−1.06928
$408$		0			0
$409$	−9.82476	−	9.82476i	−0.485803	−	0.485803i	0.421176	−	0.906979i	$-0.361618\pi$
$409$	−9.82476	−	9.82476i	−0.485803	−	0.485803i	−0.906979	+	0.421176i	$0.861618\pi$
$410$		0			0
$411$	17.3531	+	17.3531i	0.855964	+	0.855964i
$412$		0			0
$413$			3.26564i			0.160692i
$414$		0			0
$415$	12.2159			0.599653
$416$		0			0
$417$	3.80459			0.186311
$418$		0			0
$419$		−	21.7243i		−	1.06130i	−0.847591	−	0.530650i	$-0.821948\pi$
$419$		−	21.7243i		−	1.06130i	0.847591	−	0.530650i	$-0.178052\pi$
$420$		0			0
$421$	−2.04731	−	2.04731i	−0.0997799	−	0.0997799i	0.655455	−	0.755235i	$-0.272477\pi$
$421$	−2.04731	−	2.04731i	−0.0997799	−	0.0997799i	−0.755235	+	0.655455i	$0.772477\pi$
$422$		0			0
$423$	40.5522	+	40.5522i	1.97172	+	1.97172i
$424$		0			0
$425$	−10.0798			−0.488941
$426$		0			0
$427$	−0.576983	−	0.576983i	−0.0279222	−	0.0279222i
$428$		0			0
$429$	−40.1850	−	36.0898i	−1.94015	−	1.74243i
$430$		0			0
$431$	1.81733	−	1.81733i	0.0875378	−	0.0875378i	−0.661982	−	0.749520i	$-0.730284\pi$
$431$	1.81733	−	1.81733i	0.0875378	−	0.0875378i	0.749520	+	0.661982i	$0.230284\pi$
$432$		0			0
$433$		−	8.36615i		−	0.402052i	−0.979586	−	0.201026i	$-0.935573\pi$
$433$		−	8.36615i		−	0.402052i	0.979586	−	0.201026i	$-0.0644275\pi$
$434$		0			0
$435$	−44.6719	+	44.6719i	−2.14185	+	2.14185i
$436$		0			0
$437$	−1.42780	+	1.42780i	−0.0683009	+	0.0683009i
$438$		0			0
$439$	−32.8242			−1.56661			−0.783307	−	0.621635i	$-0.786469\pi$
$439$	−32.8242			−1.56661			−0.783307	+	0.621635i	$0.786469\pi$
$440$		0			0
$441$			37.4454i			1.78311i
$442$		0			0
$443$			14.9208i			0.708907i	0.935074	+	0.354453i	$0.115333\pi$
$443$			14.9208i			0.708907i	−0.935074	+	0.354453i	$0.884667\pi$
$444$		0			0
$445$	10.2116			0.484079
$446$		0			0
$447$	−0.322257	+	0.322257i	−0.0152422	+	0.0152422i
$448$		0			0
$449$	−1.80990	+	1.80990i	−0.0854147	+	0.0854147i	−0.748523	−	0.663109i	$-0.769237\pi$
$449$	−1.80990	+	1.80990i	−0.0854147	+	0.0854147i	0.663109	+	0.748523i	$0.269237\pi$
$450$		0			0
$451$			42.3318i			1.99333i
$452$		0			0
$453$	18.4956	−	18.4956i	0.868998	−	0.868998i
$454$		0			0
$455$	−3.96223	+	0.212731i	−0.185752	+	0.00997300i
$456$		0			0
$457$	−12.0851	−	12.0851i	−0.565317	−	0.565317i	0.365496	−	0.930813i	$-0.380899\pi$
$457$	−12.0851	−	12.0851i	−0.565317	−	0.565317i	−0.930813	+	0.365496i	$0.880899\pi$
$458$		0			0
$459$	6.27780			0.293023
$460$		0			0
$461$	8.14250	+	8.14250i	0.379234	+	0.379234i	0.870826	−	0.491592i	$-0.163585\pi$
$461$	8.14250	+	8.14250i	0.379234	+	0.379234i	−0.491592	+	0.870826i	$0.663585\pi$
$462$		0			0
$463$	15.2350	+	15.2350i	0.708031	+	0.708031i	0.966121	−	0.258090i	$-0.0830930\pi$
$463$	15.2350	+	15.2350i	0.708031	+	0.708031i	−0.258090	+	0.966121i	$0.583093\pi$
$464$		0			0
$465$			53.5862i			2.48500i
$466$		0			0
$467$	−39.1847			−1.81325			−0.906627	−	0.421933i	$-0.861352\pi$
$467$	−39.1847			−1.81325			−0.906627	+	0.421933i	$0.861352\pi$
$468$		0			0
$469$	−3.42302			−0.158060
$470$		0			0
$471$		−	18.8248i		−	0.867399i
$472$		0			0
$473$	−10.3012	−	10.3012i	−0.473651	−	0.473651i
$474$		0			0
$475$	−8.44695	−	8.44695i	−0.387573	−	0.387573i
$476$		0			0
$477$	30.7833			1.40947
$478$		0			0
$479$	−29.4041	−	29.4041i	−1.34351	−	1.34351i	−0.892540	−	0.450968i	$-0.851079\pi$
$479$	−29.4041	−	29.4041i	−1.34351	−	1.34351i	−0.450968	−	0.892540i	$-0.648921\pi$
$480$		0			0
$481$	−11.2005	−	10.0590i	−0.510697	−	0.458652i
$482$		0			0
$483$	−1.06165	+	1.06165i	−0.0483067	+	0.0483067i
$484$		0			0
$485$			6.29564i			0.285870i
$486$		0			0
$487$	14.3411	−	14.3411i	0.649858	−	0.649858i	−0.303100	−	0.952959i	$-0.598022\pi$
$487$	14.3411	−	14.3411i	0.649858	−	0.649858i	0.952959	+	0.303100i	$0.0980217\pi$
$488$		0			0
$489$	21.3010	−	21.3010i	0.963263	−	0.963263i
$490$		0			0
$491$	22.1949			1.00164			0.500820	−	0.865552i	$-0.333032\pi$
$491$	22.1949			1.00164			0.500820	+	0.865552i	$0.333032\pi$
$492$		0			0
$493$		−	4.86840i		−	0.219262i
$494$		0			0
$495$		−	112.460i		−	5.05470i
$496$		0			0
$497$	0.739393			0.0331663
$498$		0			0
$499$	14.6325	−	14.6325i	0.655042	−	0.655042i	−0.299160	−	0.954203i	$-0.596707\pi$
$499$	14.6325	−	14.6325i	0.655042	−	0.655042i	0.954203	+	0.299160i	$0.0967066\pi$
$500$		0			0
$501$	6.78807	−	6.78807i	0.303269	−	0.303269i
$502$		0			0
$503$			39.8978i			1.77896i	0.456978	+	0.889478i	$0.348932\pi$
$503$			39.8978i			1.77896i	−0.456978	+	0.889478i	$0.651068\pi$
$504$		0			0
$505$	9.18531	−	9.18531i	0.408741	−	0.408741i
$506$		0			0
$507$	−4.03588	−	37.4767i	−0.179240	−	1.66440i
$508$		0			0
$509$	18.7243	+	18.7243i	0.829938	+	0.829938i	0.987508	−	0.157570i	$-0.0503659\pi$
$509$	18.7243	+	18.7243i	0.829938	+	0.829938i	−0.157570	+	0.987508i	$0.550366\pi$
$510$		0			0
$511$	2.09040			0.0924741
$512$		0			0
$513$	5.26086	+	5.26086i	0.232273	+	0.232273i
$514$		0			0
$515$	24.1002	+	24.1002i	1.06198	+	1.06198i
$516$		0			0
$517$			54.7981i			2.41002i
$518$		0			0
$519$	55.3209			2.42832
$520$		0			0
$521$	−29.1106			−1.27536			−0.637679	−	0.770302i	$-0.720105\pi$
$521$	−29.1106			−1.27536			−0.637679	+	0.770302i	$0.720105\pi$
$522$		0			0
$523$			4.69344i			0.205230i	0.994721	+	0.102615i	$0.0327210\pi$
$523$			4.69344i			0.205230i	−0.994721	+	0.102615i	$0.967279\pi$
$524$		0			0
$525$	−6.28078	−	6.28078i	−0.274116	−	0.274116i
$526$		0			0
$527$	−2.91995	−	2.91995i	−0.127195	−	0.127195i
$528$		0			0
$529$	−19.4121			−0.844004
$530$		0			0
$531$	−45.6732	−	45.6732i	−1.98205	−	1.98205i
$532$		0			0
$533$	−19.7394	+	21.9793i	−0.855008	+	0.952028i
$534$		0			0
$535$	23.6022	−	23.6022i	1.02041	−	1.02041i
$536$		0			0
$537$			53.3695i			2.30307i
$538$		0			0
$539$	−25.2999	+	25.2999i	−1.08975	+	1.08975i
$540$		0			0
$541$	−24.0728	+	24.0728i	−1.03497	+	1.03497i	−0.0356048	+	0.999366i	$0.511336\pi$
$541$	−24.0728	+	24.0728i	−1.03497	+	1.03497i	−0.999366	+	0.0356048i	$0.988664\pi$
$542$		0			0
$543$	28.9502			1.24237
$544$		0			0
$545$		−	61.1489i		−	2.61933i
$546$		0			0
$547$		−	28.5687i		−	1.22151i	−0.791821	−	0.610754i	$-0.790867\pi$
$547$		−	28.5687i		−	1.22151i	0.791821	−	0.610754i	$-0.209133\pi$
$548$		0			0
$549$	16.1393			0.688810
$550$		0			0
$551$	4.07977	−	4.07977i	0.173804	−	0.173804i
$552$		0			0
$553$	−0.232105	+	0.232105i	−0.00987012	+	0.00987012i
$554$		0			0
$555$		−	48.7362i		−	2.06874i
$556$		0			0
$557$	−6.44456	+	6.44456i	−0.273065	+	0.273065i	−0.830333	−	0.557268i	$-0.811850\pi$
$557$	−6.44456	+	6.44456i	−0.273065	+	0.273065i	0.557268	+	0.830333i	$0.311850\pi$
$558$		0			0
$559$	−0.545062	−	10.1520i	−0.0230537	−	0.429386i
$560$		0			0
$561$	9.52803	+	9.52803i	0.402274	+	0.402274i
$562$		0			0
$563$	40.9787			1.72705			0.863523	−	0.504309i	$-0.168253\pi$
$563$	40.9787			1.72705			0.863523	+	0.504309i	$0.168253\pi$
$564$		0			0
$565$	−46.9454	−	46.9454i	−1.97501	−	1.97501i
$566$		0			0
$567$	0.776139	+	0.776139i	0.0325948	+	0.0325948i
$568$		0			0
$569$		−	23.9529i		−	1.00416i	−0.864822	−	0.502079i	$-0.832569\pi$
$569$		−	23.9529i		−	1.00416i	0.864822	−	0.502079i	$-0.167431\pi$
$570$		0			0
$571$	−22.6181			−0.946540			−0.473270	−	0.880917i	$-0.656927\pi$
$571$	−22.6181			−0.946540			−0.473270	+	0.880917i	$0.656927\pi$
$572$		0			0
$573$	−13.9138			−0.581258
$574$		0			0
$575$			21.2262i			0.885195i
$576$		0			0
$577$	−21.6873	−	21.6873i	−0.902853	−	0.902853i	0.0928287	−	0.995682i	$-0.470409\pi$
$577$	−21.6873	−	21.6873i	−0.902853	−	0.902853i	−0.995682	+	0.0928287i	$0.970409\pi$
$578$		0			0
$579$	41.4062	+	41.4062i	1.72078	+	1.72078i
$580$		0			0
$581$	−0.829545			−0.0344153
$582$		0			0
$583$	20.7987	+	20.7987i	0.861394	+	0.861394i
$584$		0			0
$585$	52.4403	−	58.3908i	2.16814	−	2.41416i
$586$		0			0
$587$	−4.90612	+	4.90612i	−0.202497	+	0.202497i	−0.801069	−	0.598572i	$-0.795735\pi$
$587$	−4.90612	+	4.90612i	−0.202497	+	0.202497i	0.598572	+	0.801069i	$0.295735\pi$
$588$		0			0
$589$		−	4.89389i		−	0.201649i
$590$		0			0
$591$	15.1736	−	15.1736i	0.624158	−	0.624158i
$592$		0			0
$593$	3.85082	−	3.85082i	0.158134	−	0.158134i	−0.623605	−	0.781739i	$-0.714333\pi$
$593$	3.85082	−	3.85082i	0.158134	−	0.158134i	0.781739	+	0.623605i	$0.214333\pi$
$594$		0			0
$595$	0.989898			0.0405819
$596$		0			0
$597$			50.1047i			2.05065i
$598$		0			0
$599$		−	1.08427i		−	0.0443021i	−0.999755	−	0.0221511i	$-0.992949\pi$
$599$		−	1.08427i		−	0.0443021i	0.999755	−	0.0221511i	$-0.00705148\pi$
$600$		0			0
$601$	20.3273			0.829168			0.414584	−	0.910011i	$-0.363927\pi$
$601$	20.3273			0.829168			0.414584	+	0.910011i	$0.363927\pi$
$602$		0			0
$603$	47.8742	−	47.8742i	1.94959	−	1.94959i
$604$		0			0
$605$	44.6711	−	44.6711i	1.81614	−	1.81614i
$606$		0			0
$607$		−	4.68225i		−	0.190046i	−0.995475	−	0.0950232i	$-0.969707\pi$
$607$		−	4.68225i		−	0.190046i	0.995475	−	0.0950232i	$-0.0302925\pi$
$608$		0			0
$609$	3.03354	−	3.03354i	0.122925	−	0.122925i
$610$		0			0
$611$	−25.5525	+	28.4520i	−1.03374	+	1.15104i
$612$		0			0
$613$	22.8689	+	22.8689i	0.923668	+	0.923668i	0.997286	−	0.0736188i	$-0.0234548\pi$
$613$	22.8689	+	22.8689i	0.923668	+	0.923668i	−0.0736188	+	0.997286i	$0.523455\pi$
$614$		0			0
$615$	−95.6378			−3.85649
$616$		0			0
$617$	8.80459	+	8.80459i	0.354459	+	0.354459i	0.861766	−	0.507306i	$-0.169359\pi$
$617$	8.80459	+	8.80459i	0.354459	+	0.354459i	−0.507306	+	0.861766i	$0.669359\pi$
$618$		0			0
$619$	−17.6690	−	17.6690i	−0.710175	−	0.710175i	0.256397	−	0.966572i	$-0.417465\pi$
$619$	−17.6690	−	17.6690i	−0.710175	−	0.710175i	−0.966572	+	0.256397i	$0.917465\pi$
$620$		0			0
$621$		−	13.2199i		−	0.530498i
$622$		0			0
$623$	−0.693444			−0.0277822
$624$		0			0
$625$	−44.5453			−1.78181
$626$		0			0
$627$			15.9692i			0.637747i
$628$		0			0
$629$	2.65567	+	2.65567i	0.105888	+	0.105888i
$630$		0			0
$631$	8.04600	+	8.04600i	0.320306	+	0.320306i	0.848885	−	0.528578i	$-0.177275\pi$
$631$	8.04600	+	8.04600i	0.320306	+	0.320306i	−0.528578	+	0.848885i	$0.677275\pi$
$632$		0			0
$633$	−21.1707			−0.841460
$634$		0			0
$635$	5.39192	+	5.39192i	0.213972	+	0.213972i
$636$		0			0
$637$	−24.9335	+	1.33868i	−0.987901	+	0.0530403i
$638$		0			0
$639$	−10.3411	+	10.3411i	−0.409088	+	0.409088i
$640$		0			0
$641$			36.5025i			1.44176i	0.693059	+	0.720881i	$0.256263\pi$
$641$			36.5025i			1.44176i	−0.693059	+	0.720881i	$0.743737\pi$
$642$		0			0
$643$	31.9489	−	31.9489i	1.25994	−	1.25994i	0.308823	−	0.951120i	$-0.400065\pi$
$643$	31.9489	−	31.9489i	1.25994	−	1.25994i	0.951120	−	0.308823i	$-0.0999350\pi$
$644$		0			0
$645$	23.2730	−	23.2730i	0.916374	−	0.916374i
$646$		0			0
$647$	−5.41660			−0.212949			−0.106474	−	0.994315i	$-0.533956\pi$
$647$	−5.41660			−0.212949			−0.106474	+	0.994315i	$0.533956\pi$
$648$		0			0
$649$		−	61.7181i		−	2.42265i
$650$		0			0
$651$		−	3.63889i		−	0.142619i
$652$		0			0
$653$	−0.543979			−0.0212876			−0.0106438	−	0.999943i	$-0.503388\pi$
$653$	−0.543979			−0.0212876			−0.0106438	+	0.999943i	$0.503388\pi$
$654$		0			0
$655$	−52.1685	+	52.1685i	−2.03839	+	2.03839i
$656$		0			0
$657$	−29.2363	+	29.2363i	−1.14062	+	1.14062i
$658$		0			0
$659$			21.4358i			0.835021i	0.908672	+	0.417510i	$0.137097\pi$
$659$			21.4358i			0.835021i	−0.908672	+	0.417510i	$0.862903\pi$
$660$		0			0
$661$	−13.3119	+	13.3119i	−0.517772	+	0.517772i	−0.916897	−	0.399125i	$-0.869314\pi$
$661$	−13.3119	+	13.3119i	−0.517772	+	0.517772i	0.399125	+	0.916897i	$0.369314\pi$
$662$		0			0
$663$	0.504149	+	9.39003i	0.0195795	+	0.364678i
$664$		0			0
$665$	0.829545	+	0.829545i	0.0321684	+	0.0321684i
$666$		0			0
$667$	−10.2520			−0.396959
$668$		0			0
$669$	22.3595	+	22.3595i	0.864467	+	0.864467i
$670$		0			0
$671$	10.9045	+	10.9045i	0.420965	+	0.420965i
$672$		0			0
$673$		−	19.0893i		−	0.735839i	−0.929858	−	0.367919i	$-0.880070\pi$
$673$		−	19.0893i		−	0.735839i	0.929858	−	0.367919i	$-0.119930\pi$
$674$		0			0
$675$	78.2100			3.01030
$676$		0			0
$677$	−11.8421			−0.455128			−0.227564	−	0.973763i	$-0.573076\pi$
$677$	−11.8421			−0.455128			−0.227564	+	0.973763i	$0.573076\pi$
$678$		0			0
$679$		−	0.427519i		−	0.0164067i
$680$		0			0
$681$	−33.4533	−	33.4533i	−1.28193	−	1.28193i
$682$		0			0
$683$	−4.26695	−	4.26695i	−0.163270	−	0.163270i	0.620743	−	0.784014i	$-0.286831\pi$
$683$	−4.26695	−	4.26695i	−0.163270	−	0.163270i	−0.784014	+	0.620743i	$0.786831\pi$
$684$		0			0
$685$	−34.0728			−1.30185
$686$		0			0
$687$	20.6966	+	20.6966i	0.789624	+	0.789624i
$688$		0			0
$689$	1.10051	+	20.4975i	0.0419260	+	0.780891i
$690$		0			0
$691$	26.5783	−	26.5783i	1.01108	−	1.01108i	0.0111470	−	0.999938i	$-0.496452\pi$
$691$	26.5783	−	26.5783i	1.01108	−	1.01108i	0.999938	−	0.0111470i	$-0.00354826\pi$
$692$		0			0
$693$			7.63683i			0.290099i
$694$		0			0
$695$	−3.73516	+	3.73516i	−0.141683	+	0.141683i
$696$		0			0
$697$	5.21137	−	5.21137i	0.197395	−	0.197395i
$698$		0			0
$699$	−45.3302			−1.71455
$700$		0			0
$701$			47.9431i			1.81079i	0.424574	+	0.905393i	$0.360424\pi$
$701$			47.9431i			1.81079i	−0.424574	+	0.905393i	$0.639576\pi$
$702$		0			0
$703$			4.45096i			0.167871i
$704$		0			0
$705$	−123.802			−4.66267
$706$		0			0
$707$	−0.623748	+	0.623748i	−0.0234585	+	0.0234585i
$708$		0			0
$709$	21.4969	−	21.4969i	0.807333	−	0.807333i	−0.176896	−	0.984229i	$-0.556606\pi$
$709$	21.4969	−	21.4969i	0.807333	−	0.807333i	0.984229	+	0.176896i	$0.0566057\pi$
$710$		0			0
$711$		−	6.49243i		−	0.243485i
$712$		0			0
$713$	−6.14890	+	6.14890i	−0.230278	+	0.230278i
$714$		0			0
$715$	74.8829	−	4.02046i	2.80046	−	0.150357i
$716$		0			0
$717$	−52.1685	−	52.1685i	−1.94827	−	1.94827i
$718$		0			0
$719$	−27.5224			−1.02641			−0.513207	−	0.858265i	$-0.671543\pi$
$719$	−27.5224			−1.02641			−0.513207	+	0.858265i	$0.671543\pi$
$720$		0			0
$721$	−1.63658	−	1.63658i	−0.0609493	−	0.0609493i
$722$		0			0
$723$	−7.44266	−	7.44266i	−0.276795	−	0.276795i
$724$		0			0
$725$		−	60.6514i		−	2.25254i
$726$		0			0
$727$	−42.9188			−1.59177			−0.795886	−	0.605447i	$-0.792994\pi$
$727$	−42.9188			−1.59177			−0.795886	+	0.605447i	$0.792994\pi$
$728$		0			0
$729$	38.9989			1.44440
$730$		0			0
$731$			2.53632i			0.0938093i
$732$		0			0
$733$	−23.3799	−	23.3799i	−0.863557	−	0.863557i	0.128192	−	0.991749i	$-0.459083\pi$
$733$	−23.3799	−	23.3799i	−0.863557	−	0.863557i	−0.991749	+	0.128192i	$0.959083\pi$
$734$		0			0
$735$	−57.1587	−	57.1587i	−2.10833	−	2.10833i
$736$		0			0
$737$	64.6923			2.38297
$738$		0			0
$739$	−27.0558	−	27.0558i	−0.995265	−	0.995265i	0.00472373	−	0.999989i	$-0.498496\pi$
$739$	−27.0558	−	27.0558i	−0.995265	−	0.995265i	−0.999989	+	0.00472373i	$0.998496\pi$
$740$		0			0
$741$	−7.44645	+	8.29142i	−0.273552	+	0.304593i
$742$		0			0
$743$	4.62082	−	4.62082i	0.169522	−	0.169522i	−0.617248	−	0.786769i	$-0.711752\pi$
$743$	4.62082	−	4.62082i	0.169522	−	0.169522i	0.786769	+	0.617248i	$0.211752\pi$
$744$		0			0
$745$		−	0.632753i		−	0.0231823i
$746$		0			0
$747$	11.6020	−	11.6020i	0.424494	−	0.424494i
$748$		0			0
$749$	−1.60276	+	1.60276i	−0.0585635	+	0.0585635i
$750$		0			0
$751$	10.5935			0.386561			0.193281	−	0.981144i	$-0.438087\pi$
$751$	10.5935			0.386561			0.193281	+	0.981144i	$0.438087\pi$
$752$		0			0
$753$			41.6383i			1.51738i
$754$		0			0
$755$			36.3161i			1.32168i
$756$		0			0
$757$	−21.3124			−0.774613			−0.387306	−	0.921951i	$-0.626594\pi$
$757$	−21.3124			−0.774613			−0.387306	+	0.921951i	$0.626594\pi$
$758$		0			0
$759$	20.0643	−	20.0643i	0.728290	−	0.728290i
$760$		0			0
$761$	4.38129	−	4.38129i	0.158822	−	0.158822i	−0.623223	−	0.782044i	$-0.714177\pi$
$761$	4.38129	−	4.38129i	0.158822	−	0.158822i	0.782044	+	0.623223i	$0.214177\pi$
$762$		0			0
$763$			4.15245i			0.150329i
$764$		0			0
$765$	−13.8447	+	13.8447i	−0.500556	+	0.500556i
$766$		0			0
$767$	28.7793	−	32.0449i	1.03916	−	1.15708i
$768$		0			0
$769$	10.1111	+	10.1111i	0.364617	+	0.364617i	0.865510	−	0.500892i	$-0.166995\pi$
$769$	10.1111	+	10.1111i	0.364617	+	0.364617i	−0.500892	+	0.865510i	$0.666995\pi$
$770$		0			0
$771$	14.4065			0.518837
$772$		0			0
$773$	−1.84208	−	1.84208i	−0.0662550	−	0.0662550i	0.673203	−	0.739458i	$-0.264918\pi$
$773$	−1.84208	−	1.84208i	−0.0662550	−	0.0662550i	−0.739458	+	0.673203i	$0.764918\pi$
$774$		0			0
$775$	−36.3773	−	36.3773i	−1.30671	−	1.30671i
$776$		0			0
$777$			3.30954i			0.118729i
$778$		0			0
$779$	8.73436			0.312941
$780$		0			0
$781$	−13.9739			−0.500027
$782$		0			0
$783$			37.7744i			1.34995i
$784$		0			0
$785$	18.4812	+	18.4812i	0.659623	+	0.659623i
$786$		0			0
$787$	−37.8285	−	37.8285i	−1.34844	−	1.34844i	−0.887357	−	0.461083i	$-0.847461\pi$
$787$	−37.8285	−	37.8285i	−1.34844	−	1.34844i	−0.461083	−	0.887357i	$-0.652539\pi$
$788$		0			0
$789$	57.0555			2.03123
$790$		0			0
$791$	3.18793	+	3.18793i	0.113350	+	0.113350i
$792$		0			0
$793$	0.576983	+	10.7466i	0.0204893	+	0.381623i
$794$		0			0
$795$	−46.9893	+	46.9893i	−1.66654	+	1.66654i
$796$		0			0
$797$		−	18.2802i		−	0.647519i	−0.946139	−	0.323759i	$-0.895053\pi$
$797$		−	18.2802i		−	0.647519i	0.946139	−	0.323759i	$-0.104947\pi$
$798$		0			0
$799$	6.74607	−	6.74607i	0.238659	−	0.238659i
$800$		0			0
$801$	9.69848	−	9.69848i	0.342679	−	0.342679i
$802$		0			0
$803$	−39.5070			−1.39417
$804$		0			0
$805$		−	2.08455i		−	0.0734708i
$806$		0			0
$807$			32.6322i			1.14871i
$808$		0			0
$809$	17.1302			0.602266			0.301133	−	0.953582i	$-0.402635\pi$
$809$	17.1302			0.602266			0.301133	+	0.953582i	$0.402635\pi$
$810$		0			0
$811$	19.8721	−	19.8721i	0.697805	−	0.697805i	−0.266132	−	0.963937i	$-0.585746\pi$
$811$	19.8721	−	19.8721i	0.697805	−	0.697805i	0.963937	+	0.266132i	$0.0857457\pi$
$812$		0			0
$813$	−22.8668	+	22.8668i	−0.801972	+	0.801972i
$814$		0			0
$815$			41.8245i			1.46505i
$816$		0			0
$817$	−2.12546	+	2.12546i	−0.0743606	+	0.0743606i
$818$		0			0
$819$	−3.56107	+	3.96515i	−0.124434	+	0.138554i
$820$		0			0
$821$	26.9216	+	26.9216i	0.939569	+	0.939569i	0.998275	−	0.0587062i	$-0.0186975\pi$
$821$	26.9216	+	26.9216i	0.939569	+	0.939569i	−0.0587062	+	0.998275i	$0.518697\pi$
$822$		0			0
$823$	48.2105			1.68051			0.840257	−	0.542189i	$-0.182404\pi$
$823$	48.2105			1.68051			0.840257	+	0.542189i	$0.182404\pi$
$824$		0			0
$825$	118.702	+	118.702i	4.13267	+	4.13267i
$826$		0			0
$827$	16.1810	+	16.1810i	0.562669	+	0.562669i	0.930065	−	0.367396i	$-0.119750\pi$
$827$	16.1810	+	16.1810i	0.562669	+	0.562669i	−0.367396	+	0.930065i	$0.619750\pi$
$828$		0			0
$829$		−	40.8476i		−	1.41870i	−0.704858	−	0.709349i	$-0.748989\pi$
$829$		−	40.8476i		−	1.41870i	0.704858	−	0.709349i	$-0.251011\pi$
$830$		0			0
$831$	30.1914			1.04733
$832$		0			0
$833$	6.22923			0.215830
$834$		0			0
$835$			13.3284i			0.461248i
$836$		0			0
$837$	22.6562	+	22.6562i	0.783112	+	0.783112i
$838$		0			0
$839$	15.4988	+	15.4988i	0.535077	+	0.535077i	0.922079	−	0.387002i	$-0.126489\pi$
$839$	15.4988	+	15.4988i	0.535077	+	0.535077i	−0.387002	+	0.922079i	$0.626489\pi$
$840$		0			0
$841$	0.293864			0.0101332
$842$		0			0
$843$	40.2259	+	40.2259i	1.38546	+	1.38546i
$844$		0			0
$845$	40.7551	+	32.8306i	1.40202	+	1.12941i
$846$		0			0
$847$	−3.03348	+	3.03348i	−0.104232	+	0.104232i
$848$		0			0
$849$		−	57.9899i		−	1.99021i
$850$		0			0
$851$	5.59237	−	5.59237i	0.191704	−	0.191704i
$852$		0			0
$853$	−2.12764	+	2.12764i	−0.0728492	+	0.0728492i	−0.742593	−	0.669743i	$-0.766404\pi$
$853$	−2.12764	+	2.12764i	−0.0728492	+	0.0728492i	0.669743	+	0.742593i	$0.266404\pi$
$854$		0			0
$855$	−23.2040			−0.793559
$856$		0			0
$857$		−	27.4076i		−	0.936226i	−0.883669	−	0.468113i	$-0.844934\pi$
$857$		−	27.4076i		−	0.936226i	0.883669	−	0.468113i	$-0.155066\pi$
$858$		0			0
$859$			25.9197i			0.884368i	0.896924	+	0.442184i	$0.145796\pi$
$859$			25.9197i			0.884368i	−0.896924	+	0.442184i	$0.854204\pi$
$860$		0			0
$861$	6.49449			0.221332
$862$		0			0
$863$	2.43101	−	2.43101i	0.0827525	−	0.0827525i	−0.664519	−	0.747271i	$-0.731364\pi$
$863$	2.43101	−	2.43101i	0.0827525	−	0.0827525i	0.747271	+	0.664519i	$0.231364\pi$
$864$		0			0
$865$	−54.3113	+	54.3113i	−1.84664	+	1.84664i
$866$		0			0
$867$			46.9454i			1.59435i
$868$		0			0
$869$	4.38661	−	4.38661i	0.148805	−	0.148805i
$870$		0			0
$871$	33.5892	+	30.1662i	1.13813	+	1.02214i
$872$		0			0
$873$	5.97926	+	5.97926i	0.202367	+	0.202367i
$874$		0			0
$875$	6.82980			0.230889
$876$		0			0
$877$	−2.19199	−	2.19199i	−0.0740184	−	0.0740184i	0.669128	−	0.743147i	$-0.266668\pi$
$877$	−2.19199	−	2.19199i	−0.0740184	−	0.0740184i	−0.743147	+	0.669128i	$0.766668\pi$
$878$		0			0
$879$	−66.0106	−	66.0106i	−2.22648	−	2.22648i
$880$		0			0
$881$			2.92715i			0.0986182i	0.998784	+	0.0493091i	$0.0157019\pi$
$881$			2.92715i			0.0986182i	−0.998784	+	0.0493091i	$0.984298\pi$
$882$		0			0
$883$	6.53632			0.219965			0.109982	−	0.993934i	$-0.464921\pi$
$883$	6.53632			0.219965			0.109982	+	0.993934i	$0.464921\pi$
$884$		0			0
$885$	139.436			4.68710
$886$		0			0
$887$			6.92580i			0.232546i	0.993217	+	0.116273i	$0.0370947\pi$
$887$			6.92580i			0.232546i	−0.993217	+	0.116273i	$0.962905\pi$
$888$		0			0
$889$	−0.366150	−	0.366150i	−0.0122803	−	0.0122803i
$890$		0			0
$891$	−14.6684	−	14.6684i	−0.491410	−	0.491410i
$892$		0			0
$893$	11.3066			0.378359
$894$		0			0
$895$	−52.3956	−	52.3956i	−1.75139	−	1.75139i
$896$		0			0
$897$	19.7737	−	1.06165i	0.660226	−	0.0354475i
$898$		0			0
$899$	17.5698	−	17.5698i	0.585984	−	0.585984i
$900$		0			0
$901$		−	5.12096i		−	0.170604i
$902$		0			0
$903$	−1.58040	+	1.58040i	−0.0525925	+	0.0525925i
$904$		0			0
$905$	−28.4219	+	28.4219i	−0.944777	+	0.944777i
$906$		0			0
$907$	45.5812			1.51350			0.756749	−	0.653705i	$-0.226786\pi$
$907$	45.5812			1.51350			0.756749	+	0.653705i	$0.226786\pi$
$908$		0			0
$909$		−	17.4474i		−	0.578695i
$910$		0			0
$911$			43.3384i			1.43587i	0.696112	+	0.717933i	$0.254912\pi$
$911$			43.3384i			1.43587i	−0.696112	+	0.717933i	$0.745088\pi$
$912$		0			0
$913$	15.6777			0.518858
$914$		0			0
$915$	−24.6360	+	24.6360i	−0.814441	+	0.814441i
$916$		0			0
$917$	3.54262	−	3.54262i	0.116987	−	0.116987i
$918$		0			0
$919$		−	50.6508i		−	1.67082i	−0.549629	−	0.835409i	$-0.685231\pi$
$919$		−	50.6508i		−	1.67082i	0.549629	−	0.835409i	$-0.314769\pi$
$920$		0			0
$921$	35.7075	−	35.7075i	1.17660	−	1.17660i
$922$		0			0
$923$	−7.25547	−	6.51608i	−0.238817	−	0.214479i
$924$		0			0
$925$	33.0848	+	33.0848i	1.08782	+	1.08782i
$926$		0			0
$927$	45.7782			1.50355
$928$		0			0
$929$	−2.91463	−	2.91463i	−0.0956260	−	0.0956260i	0.657675	−	0.753301i	$-0.271540\pi$
$929$	−2.91463	−	2.91463i	−0.0956260	−	0.0956260i	−0.753301	+	0.657675i	$0.771540\pi$
$930$		0			0
$931$	5.22016	+	5.22016i	0.171084	+	0.171084i
$932$		0			0
$933$		−	60.9641i		−	1.99587i
$934$		0			0
$935$	−18.7083			−0.611827
$936$		0			0
$937$	−1.02634			−0.0335290			−0.0167645	−	0.999859i	$-0.505337\pi$
$937$	−1.02634			−0.0335290			−0.0167645	+	0.999859i	$0.505337\pi$
$938$		0			0
$939$		−	1.44404i		−	0.0471243i
$940$		0			0
$941$	−37.9521	−	37.9521i	−1.23720	−	1.23720i	−0.961139	−	0.276065i	$-0.910970\pi$
$941$	−37.9521	−	37.9521i	−1.23720	−	1.23720i	−0.276065	−	0.961139i	$-0.589030\pi$
$942$		0			0
$943$	−10.9742	−	10.9742i	−0.357370	−	0.357370i
$944$		0			0
$945$	−7.68072			−0.249854
$946$		0			0
$947$	−18.0760	−	18.0760i	−0.587392	−	0.587392i	0.349533	−	0.936924i	$-0.386340\pi$
$947$	−18.0760	−	18.0760i	−0.587392	−	0.587392i	−0.936924	+	0.349533i	$0.886340\pi$
$948$		0			0
$949$	−20.5126	−	18.4222i	−0.665867	−	0.598010i
$950$		0			0
$951$	−42.0792	+	42.0792i	−1.36451	+	1.36451i
$952$		0			0
$953$			29.9338i			0.969652i	0.874611	+	0.484826i	$0.161117\pi$
$953$			29.9338i			0.969652i	−0.874611	+	0.484826i	$0.838883\pi$
$954$		0			0
$955$	13.6599	−	13.6599i	0.442024	−	0.442024i
$956$		0			0
$957$	−57.3315	+	57.3315i	−1.85326	+	1.85326i
$958$		0			0
$959$	2.31379			0.0747160
$960$		0			0
$961$			9.92417i			0.320135i
$962$		0			0
$963$		−	44.8322i		−	1.44470i
$964$		0			0
$965$	−81.3012			−2.61718
$966$		0			0
$967$	−38.5639	+	38.5639i	−1.24013	+	1.24013i	−0.280187	+	0.959945i	$0.590396\pi$
$967$	−38.5639	+	38.5639i	−1.24013	+	1.24013i	−0.959945	+	0.280187i	$0.909604\pi$
$968$		0			0
$969$	1.96593	−	1.96593i	0.0631547	−	0.0631547i
$970$		0			0
$971$			4.40912i			0.141495i	0.997494	+	0.0707477i	$0.0225385\pi$
$971$			4.40912i			0.141495i	−0.997494	+	0.0707477i	$0.977461\pi$
$972$		0			0
$973$	0.253644	−	0.253644i	0.00813145	−	0.00813145i
$974$		0			0
$975$	6.28078	+	116.983i	0.201146	+	3.74644i
$976$		0			0
$977$	−18.1959	−	18.1959i	−0.582140	−	0.582140i	0.353351	−	0.935491i	$-0.385042\pi$
$977$	−18.1959	−	18.1959i	−0.582140	−	0.582140i	−0.935491	+	0.353351i	$0.885042\pi$
$978$		0			0
$979$	13.1055			0.418855
$980$		0			0
$981$	−58.0759	−	58.0759i	−1.85422	−	1.85422i
$982$		0			0
$983$	−20.0827	−	20.0827i	−0.640538	−	0.640538i	0.310149	−	0.950688i	$-0.399621\pi$
$983$	−20.0827	−	20.0827i	−0.640538	−	0.640538i	−0.950688	+	0.310149i	$0.899621\pi$
$984$		0			0
$985$			29.7934i			0.949296i
$986$		0			0
$987$	8.40706			0.267600
$988$		0			0
$989$	5.34105			0.169836
$990$		0			0
$991$			32.6626i			1.03756i	0.854907	+	0.518781i	$0.173614\pi$
$991$			32.6626i			1.03756i	−0.854907	+	0.518781i	$0.826386\pi$
$992$		0			0
$993$	−17.9381	−	17.9381i	−0.569247	−	0.569247i
$994$		0			0
$995$	−49.1903	−	49.1903i	−1.55944	−	1.55944i
$996$		0			0
$997$	40.6365			1.28697			0.643486	−	0.765458i	$-0.277487\pi$
$997$	40.6365			1.28697			0.643486	+	0.765458i	$0.277487\pi$
$998$		0			0
$999$	−20.6056	−	20.6056i	−0.651933	−	0.651933i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.k.f.255.4	yes	8
4.3	odd	2		416.2.k.e.255.1	yes	8
8.3	odd	2		832.2.k.g.255.4		8
8.5	even	2		832.2.k.i.255.1		8
13.5	odd	4		416.2.k.e.31.4	✓	8
52.31	even	4	inner	416.2.k.f.31.1	yes	8
104.5	odd	4		832.2.k.g.447.1		8
104.83	even	4		832.2.k.i.447.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.k.e.31.4	✓	8	13.5	odd	4
416.2.k.e.255.1	yes	8	4.3	odd	2
416.2.k.f.31.1	yes	8	52.31	even	4	inner
416.2.k.f.255.4	yes	8	1.1	even	1	trivial
832.2.k.g.255.4		8	8.3	odd	2
832.2.k.g.447.1		8	104.5	odd	4
832.2.k.i.255.1		8	8.5	even	2
832.2.k.i.447.4		8	104.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.89949i q^{3} +(-2.84658 - 2.84658i) q^{5} +(0.193303 + 0.193303i) q^{7} -5.40706 q^{9} +(-3.65328 - 3.65328i) q^{11} +(-0.193303 - 3.60037i) q^{13} +(8.25364 - 8.25364i) q^{15} +0.899494i q^{17} +(-0.753785 + 0.753785i) q^{19} +(-0.560481 + 0.560481i) q^{21} +1.89418 q^{23} +11.2060i q^{25} -6.97926i q^{27} -5.41238 q^{29} +(-3.24622 + 3.24622i) q^{31} +(10.5927 - 10.5927i) q^{33} -1.10051i q^{35} +(2.95241 - 2.95241i) q^{37} +(10.4392 - 0.560481i) q^{39} +(-5.79367 - 5.79367i) q^{41} +2.81972 q^{43} +(15.3916 + 15.3916i) q^{45} +(-7.49986 - 7.49986i) q^{47} -6.92527i q^{49} -2.60808 q^{51} -5.69316 q^{53} +20.7987i q^{55} +(-2.18559 - 2.18559i) q^{57} +(8.44695 + 8.44695i) q^{59} -2.98486 q^{61} +(-1.04520 - 1.04520i) q^{63} +(-9.69848 + 10.7990i) q^{65} +(-8.85401 + 8.85401i) q^{67} +5.49215i q^{69} +(1.91252 - 1.91252i) q^{71} +(5.40706 - 5.40706i) q^{73} -32.4919 q^{75} -1.41238i q^{77} +1.20073i q^{79} +4.01514 q^{81} +(-2.14571 + 2.14571i) q^{83} +(2.56048 - 2.56048i) q^{85} -15.6932i q^{87} +(-1.79367 + 1.79367i) q^{89} +(0.658596 - 0.733328i) q^{91} +(-9.41238 - 9.41238i) q^{93} +4.29142 q^{95} +(-1.10582 - 1.10582i) q^{97} +(19.7535 + 19.7535i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{7} - 20 q^{9} - 6 q^{11} - 2 q^{13} + 20 q^{15} - 6 q^{19} - 4 q^{21} + 16 q^{23} + 4 q^{29} - 26 q^{31} + 16 q^{33} - 8 q^{39} - 24 q^{41} + 44 q^{43} + 8 q^{45} - 14 q^{47} - 44 q^{51} + 28 q^{57}+ \cdots + 66 q^{99}+O(q^{100}) \) 8 * q + 2 * q^7 - 20 * q^9 - 6 * q^11 - 2 * q^13 + 20 * q^15 - 6 * q^19 - 4 * q^21 + 16 * q^23 + 4 * q^29 - 26 * q^31 + 16 * q^33 - 8 * q^39 - 24 * q^41 + 44 * q^43 + 8 * q^45 - 14 * q^47 - 44 * q^51 + 28 * q^57 + 22 * q^59 - 24 * q^61 + 38 * q^63 - 8 * q^65 - 2 * q^67 + 14 * q^71 + 20 * q^73 - 76 * q^75 + 32 * q^81 + 6 * q^83 + 20 * q^85 + 8 * q^89 - 42 * q^91 - 28 * q^93 - 12 * q^95 - 8 * q^97 + 66 * q^99

Introduction

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