LMFDB - Embedded newform 416.2.k.e.31.4

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			31.4
Root			$2.65328i$ of defining polynomial
Character	$\chi$	$=$	416.31
Dual form			416.2.k.e.255.1

$q$-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 $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.89949i			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	−0.328540	+	0.944490i	$0.606556\pi$
$5$	−2.84658	+	2.84658i	−1.27303	+	1.27303i	−0.944490	+	0.328540i	$0.893444\pi$
$6$		0			0
$7$	−0.193303	+	0.193303i	−0.0730617	+	0.0730617i	−0.742693	−	0.669632i	$-0.766452\pi$
$7$	−0.193303	+	0.193303i	−0.0730617	+	0.0730617i	0.669632	+	0.742693i	$0.266452\pi$
$8$		0			0
$9$	−5.40706			−1.80235
$10$		0			0
$11$	3.65328	−	3.65328i	1.10150	−	1.10150i	0.107275	−	0.994229i	$-0.465787\pi$
$11$	3.65328	−	3.65328i	1.10150	−	1.10150i	0.994229	−	0.107275i	$-0.0342127\pi$
$12$		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	$-0.965210\pi$
$17$		−	0.899494i		−	0.218159i	0.994033	−	0.109080i	$-0.0347903\pi$
$18$		0			0
$19$	0.753785	+	0.753785i	0.172930	+	0.172930i	0.788265	−	0.615335i	$-0.210979\pi$
$19$	0.753785	+	0.753785i	0.172930	+	0.172930i	−0.615335	+	0.788265i	$0.710979\pi$
$20$		0			0
$21$	−0.560481	−	0.560481i	−0.122307	−	0.122307i
$22$		0			0
$23$	−1.89418			−0.394963			−0.197481	−	0.980307i	$-0.563276\pi$
$23$	−1.89418			−0.394963			−0.197481	+	0.980307i	$0.563276\pi$
$24$		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	$-0.114736\pi$
$31$	3.24622	+	3.24622i	0.583038	+	0.583038i	−0.352699	+	0.935737i	$0.614736\pi$
$32$		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	0.906842	−	0.421470i	$-0.138486\pi$
$37$	2.95241	+	2.95241i	0.485373	+	0.485373i	−0.421470	+	0.906842i	$0.638486\pi$
$38$		0			0
$39$	−10.4392	−	0.560481i	−1.67162	−	0.0897488i
$40$		0			0
$41$	−5.79367	+	5.79367i	−0.904819	+	0.904819i	−0.995848	−	0.0910291i	$-0.970984\pi$
$41$	−5.79367	+	5.79367i	−0.904819	+	0.904819i	0.0910291	+	0.995848i	$0.470984\pi$
$42$		0			0
$43$	−2.81972			−0.430004			−0.215002	−	0.976614i	$-0.568976\pi$
$43$	−2.81972			−0.430004			−0.215002	+	0.976614i	$0.568976\pi$
$44$		0			0
$45$	15.3916	−	15.3916i	2.29445	−	2.29445i
$46$		0			0
$47$	7.49986	−	7.49986i	1.09397	−	1.09397i	0.0988662	−	0.995101i	$-0.468478\pi$
$47$	7.49986	−	7.49986i	1.09397	−	1.09397i	0.995101	−	0.0988662i	$-0.0315216\pi$
$48$		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	−0.105253	+	0.994445i	$0.533565\pi$
$59$	−8.44695	+	8.44695i	−1.09970	+	1.09970i	−0.994445	+	0.105253i	$0.966435\pi$
$60$		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	0.996352	+	0.0853373i	$0.0271968\pi$
$67$	8.85401	+	8.85401i	1.08169	+	1.08169i	0.0853373	+	0.996352i	$0.472803\pi$
$68$		0			0
$69$		−	5.49215i		−	0.661177i
$70$		0			0
$71$	−1.91252	−	1.91252i	−0.226975	−	0.226975i	0.584453	−	0.811428i	$-0.301309\pi$
$71$	−1.91252	−	1.91252i	−0.226975	−	0.226975i	−0.811428	+	0.584453i	$0.801309\pi$
$72$		0			0
$73$	5.40706	+	5.40706i	0.632849	+	0.632849i	0.948782	−	0.315933i	$-0.102317\pi$
$73$	5.40706	+	5.40706i	0.632849	+	0.632849i	−0.315933	+	0.948782i	$0.602317\pi$
$74$		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	$-0.196741\pi$
$83$	2.14571	+	2.14571i	0.235522	+	0.235522i	−0.579471	+	0.814993i	$0.696741\pi$
$84$		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	0.605623	−	0.795752i	$-0.292924\pi$
$89$	−1.79367	−	1.79367i	−0.190129	−	0.190129i	−0.795752	+	0.605623i	$0.792924\pi$
$90$		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	−0.761014	−	0.648735i	$-0.775298\pi$
$97$	−1.10582	+	1.10582i	−0.112279	+	0.112279i	0.648735	+	0.761014i	$0.275298\pi$
$98$		0			0
$99$	−19.7535	+	19.7535i	−1.98530	+	1.98530i
$100$		0			0
$101$		−	3.22679i		−	0.321077i	−0.987030	−	0.160539i	$-0.948677\pi$
$101$		−	3.22679i		−	0.321077i	0.987030	−	0.160539i	$-0.0513231\pi$
$102$		0			0
$103$	8.46638			0.834217			0.417108	−	0.908857i	$-0.363044\pi$
$103$	8.46638			0.834217			0.417108	+	0.908857i	$0.363044\pi$
$104$		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	0.0292045	−	0.999573i	$-0.490703\pi$
$109$	10.7408	−	10.7408i	1.02878	−	1.02878i	0.999573	−	0.0292045i	$-0.00929739\pi$
$110$		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	$-0.473218\pi$
$127$	1.89418			0.168081			0.0840404	+	0.996462i	$0.473218\pi$
$128$		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	0.914931	−	0.403610i	$-0.132245\pi$
$137$	5.98486	+	5.98486i	0.511321	+	0.511321i	−0.403610	+	0.914931i	$0.632245\pi$
$138$		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	−0.702540	−	0.711645i	$-0.747951\pi$
$149$	0.111143	−	0.111143i	0.00910515	−	0.00910515i	0.711645	+	0.702540i	$0.247951\pi$
$150$		0			0
$151$	6.37890	−	6.37890i	0.519107	−	0.519107i	−0.398194	−	0.917301i	$-0.630363\pi$
$151$	6.37890	−	6.37890i	0.519107	−	0.519107i	0.917301	+	0.398194i	$0.130363\pi$
$152$		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	$-0.616616\pi$
$163$	7.34644	−	7.34644i	0.575418	−	0.575418i	0.933637	+	0.358220i	$0.116616\pi$
$164$		0			0
$165$	−60.3057			−4.69479
$166$		0			0
$167$	2.34112	−	2.34112i	0.181162	−	0.181162i	−0.610700	−	0.791862i	$-0.709112\pi$
$167$	2.34112	−	2.34112i	0.181162	−	0.181162i	0.791862	+	0.610700i	$0.209112\pi$
$168$		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	$0.258296\pi$
$173$			19.0795i			1.45059i	−0.688440	+	0.725293i	$0.741704\pi$
$174$		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	$-0.741460\pi$
$179$	−18.4065			−1.37577			−0.687883	+	0.725821i	$0.741460\pi$
$180$		0			0
$181$			9.98458i			0.742148i	0.928603	+	0.371074i	$0.121010\pi$
$181$			9.98458i			0.742148i	−0.928603	+	0.371074i	$0.878990\pi$
$182$		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	0.999598	+	0.0283344i	$0.00902033\pi$
$193$	14.2805	+	14.2805i	1.02793	+	1.02793i	0.0283344	+	0.999598i	$0.490980\pi$
$194$		0			0
$195$	31.3116	−	28.1207i	2.24227	−	2.01376i
$196$		0			0
$197$	−5.23319	+	5.23319i	−0.372849	+	0.372849i	−0.868514	−	0.495665i	$-0.834924\pi$
$197$	−5.23319	+	5.23319i	−0.372849	+	0.372849i	0.495665	+	0.868514i	$0.334924\pi$
$198$		0			0
$199$	−17.2805			−1.22498			−0.612491	−	0.790477i	$-0.709832\pi$
$199$	−17.2805			−1.22498			−0.612491	+	0.790477i	$0.709832\pi$
$200$		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	$-0.368982\pi$
$223$	−7.71151	−	7.71151i	−0.516401	−	0.516401i	−0.916480	+	0.400080i	$0.868982\pi$
$224$		0			0
$225$			60.5918i			4.03945i
$226$		0			0
$227$	11.5376	+	11.5376i	0.765780	+	0.765780i	0.977360	−	0.211581i	$-0.0678612\pi$
$227$	11.5376	+	11.5376i	0.765780	+	0.765780i	−0.211581	+	0.977360i	$0.567861\pi$
$228$		0			0
$229$	7.13800	+	7.13800i	0.471692	+	0.471692i	0.902462	−	0.430770i	$-0.141758\pi$
$229$	7.13800	+	7.13800i	0.471692	+	0.471692i	−0.430770	+	0.902462i	$0.641758\pi$
$230$		0			0
$231$	−4.09519			−0.269444
$232$		0			0
$233$		−	15.6338i		−	1.02421i	−0.858924	−	0.512104i	$-0.828866\pi$
$233$		−	15.6338i		−	1.02421i	0.858924	−	0.512104i	$-0.171134\pi$
$234$		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	0.983631	+	0.180193i	$0.0576723\pi$
$239$	17.9923	+	17.9923i	1.16382	+	1.16382i	0.180193	+	0.983631i	$0.442328\pi$
$240$		0			0
$241$	−2.56688	−	2.56688i	−0.165347	−	0.165347i	0.619583	−	0.784931i	$-0.287302\pi$
$241$	−2.56688	−	2.56688i	−0.165347	−	0.165347i	−0.784931	+	0.619583i	$0.787302\pi$
$242$		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	$-0.649723\pi$
$251$	−14.3606			−0.906430			−0.453215	+	0.891401i	$0.649723\pi$
$252$		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	$0.0495272\pi$
$257$			4.96863i			0.309934i	−0.987920	+	0.154967i	$0.950473\pi$
$258$		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	$-0.860005\pi$
$271$	−7.88647	+	7.88647i	−0.479069	+	0.479069i	0.425765	+	0.904834i	$0.360005\pi$
$272$		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	$0.101273\pi$
$277$			10.4127i			0.625636i	−0.949813	+	0.312818i	$0.898727\pi$
$278$		0			0
$279$	−17.5525	−	17.5525i	−1.05084	−	1.05084i
$280$		0			0
$281$	13.8734	+	13.8734i	0.827620	+	0.827620i	0.987187	−	0.159567i	$-0.0510099\pi$
$281$	13.8734	+	13.8734i	0.827620	+	0.827620i	−0.159567	+	0.987187i	$0.551010\pi$
$282$		0			0
$283$	20.0000			1.18888			0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000			1.18888			0.594438	+	0.804141i	$0.297374\pi$
$284$		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	−0.905348	−	0.424671i	$-0.860390\pi$
$293$	−22.7662	−	22.7662i	−1.33002	−	1.33002i	−0.424671	−	0.905348i	$-0.639610\pi$
$294$		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	$-0.584436\pi$
$307$	12.3151	−	12.3151i	0.702858	−	0.702858i	0.965023	+	0.262165i	$0.0844365\pi$
$308$		0			0
$309$			24.5482i			1.39650i
$310$		0			0
$311$	21.0258			1.19226			0.596131	−	0.802887i	$-0.296704\pi$
$311$	21.0258			1.19226			0.596131	+	0.802887i	$0.296704\pi$
$312$		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	−0.170286	−	0.985395i	$-0.554469\pi$
$317$	14.5126	−	14.5126i	0.815109	−	0.815109i	0.985395	+	0.170286i	$0.0544690\pi$
$318$		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	$-0.172705\pi$
$331$	6.18662	+	6.18662i	0.340047	+	0.340047i	−0.516338	+	0.856385i	$0.672705\pi$
$332$		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	$0.0602684\pi$
$337$			6.91013i			0.376419i	−0.982129	+	0.188209i	$0.939732\pi$
$338$		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	0.269636	−	0.962962i	$-0.413097\pi$
$349$	−12.9524	−	12.9524i	−0.693326	−	0.693326i	−0.962962	+	0.269636i	$0.913097\pi$
$350$		0			0
$351$	25.1279	+	1.34911i	1.34123	+	0.0720104i
$352$		0			0
$353$	−14.7341	+	14.7341i	−0.784216	+	0.784216i	−0.980539	−	0.196323i	$-0.937100\pi$
$353$	−14.7341	+	14.7341i	−0.784216	+	0.784216i	0.196323	+	0.980539i	$0.437100\pi$
$354$		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	$-0.631601\pi$
$359$	9.73865	−	9.73865i	0.513986	−	0.513986i	0.915745	+	0.401759i	$0.131601\pi$
$360$		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	$-0.136292\pi$
$379$	9.62722	+	9.62722i	0.494517	+	0.494517i	−0.415209	+	0.909726i	$0.636292\pi$
$380$		0			0
$381$			5.49215i			0.281371i
$382$		0			0
$383$	20.5102	+	20.5102i	1.04802	+	1.04802i	0.998787	+	0.0492358i	$0.0156786\pi$
$383$	20.5102	+	20.5102i	1.04802	+	1.04802i	0.0492358	+	0.998787i	$0.484321\pi$
$384$		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	$0.230812\pi$
$389$			26.1616i			1.32645i	−0.748422	+	0.663223i	$0.769188\pi$
$390$		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	0.914976	−	0.403509i	$-0.132209\pi$
$397$	10.1909	+	10.1909i	0.511467	+	0.511467i	−0.403509	+	0.914976i	$0.632209\pi$
$398$		0			0
$399$		−	0.844964i		−	0.0423011i
$400$		0			0
$401$	−1.85298	−	1.85298i	−0.0925335	−	0.0925335i	0.659325	−	0.751858i	$-0.270842\pi$
$401$	−1.85298	−	1.85298i	−0.0925335	−	0.0925335i	−0.751858	+	0.659325i	$0.770842\pi$
$402$		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	−0.906979	−	0.421176i	$-0.861618\pi$
$409$	−9.82476	+	9.82476i	−0.485803	+	0.485803i	0.421176	+	0.906979i	$0.361618\pi$
$410$		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	−0.755235	−	0.655455i	$-0.772477\pi$
$421$	−2.04731	+	2.04731i	−0.0997799	+	0.0997799i	0.655455	+	0.755235i	$0.272477\pi$
$422$		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	$-0.269716\pi$
$431$	−1.81733	−	1.81733i	−0.0875378	−	0.0875378i	−0.749520	+	0.661982i	$0.769716\pi$
$432$		0			0
$433$			8.36615i			0.402052i	0.979586	+	0.201026i	$0.0644275\pi$
$433$			8.36615i			0.402052i	−0.979586	+	0.201026i	$0.935573\pi$
$434$		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	$-0.213531\pi$
$439$	32.8242			1.56661			0.783307	+	0.621635i	$0.213531\pi$
$440$		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	0.663109	−	0.748523i	$-0.269237\pi$
$449$	−1.80990	−	1.80990i	−0.0854147	−	0.0854147i	−0.748523	+	0.663109i	$0.769237\pi$
$450$		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	−0.930813	−	0.365496i	$-0.880899\pi$
$457$	−12.0851	+	12.0851i	−0.565317	+	0.565317i	0.365496	+	0.930813i	$0.380899\pi$
$458$		0			0
$459$	−6.27780			−0.293023
$460$		0			0
$461$	8.14250	−	8.14250i	0.379234	−	0.379234i	−0.491592	−	0.870826i	$-0.663585\pi$
$461$	8.14250	−	8.14250i	0.379234	−	0.379234i	0.870826	+	0.491592i	$0.163585\pi$
$462$		0			0
$463$	−15.2350	+	15.2350i	−0.708031	+	0.708031i	−0.966121	−	0.258090i	$-0.916907\pi$
$463$	−15.2350	+	15.2350i	−0.708031	+	0.708031i	0.258090	+	0.966121i	$0.416907\pi$
$464$		0			0
$465$		−	53.5862i		−	2.48500i
$466$		0			0
$467$	39.1847			1.81325			0.906627	−	0.421933i	$-0.138648\pi$
$467$	39.1847			1.81325			0.906627	+	0.421933i	$0.138648\pi$
$468$		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	0.450968	−	0.892540i	$-0.351079\pi$
$479$	29.4041	−	29.4041i	1.34351	−	1.34351i	0.892540	−	0.450968i	$-0.148921\pi$
$480$		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	$-0.401978\pi$
$487$	−14.3411	−	14.3411i	−0.649858	−	0.649858i	−0.952959	+	0.303100i	$0.901978\pi$
$488$		0			0
$489$	21.3010	+	21.3010i	0.963263	+	0.963263i
$490$		0			0
$491$	−22.1949			−1.00164			−0.500820	−	0.865552i	$-0.666968\pi$
$491$	−22.1949			−1.00164			−0.500820	+	0.865552i	$0.666968\pi$
$492$		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	$-0.403293\pi$
$499$	−14.6325	−	14.6325i	−0.655042	−	0.655042i	−0.954203	+	0.299160i	$0.903293\pi$
$500$		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	−0.157570	−	0.987508i	$-0.550366\pi$
$509$	18.7243	−	18.7243i	0.829938	−	0.829938i	0.987508	+	0.157570i	$0.0503659\pi$
$510$		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	−0.999366	−	0.0356048i	$-0.988664\pi$
$541$	−24.0728	−	24.0728i	−1.03497	−	1.03497i	−0.0356048	−	0.999366i	$-0.511336\pi$
$542$		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	0.557268	−	0.830333i	$-0.311850\pi$
$557$	−6.44456	−	6.44456i	−0.273065	−	0.273065i	−0.830333	+	0.557268i	$0.811850\pi$
$558$		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	$-0.831747\pi$
$563$	−40.9787			−1.72705			−0.863523	+	0.504309i	$0.831747\pi$
$564$		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	$0.167431\pi$
$569$			23.9529i			1.00416i	−0.864822	+	0.502079i	$0.832569\pi$
$570$		0			0
$571$	22.6181			0.946540			0.473270	−	0.880917i	$-0.343073\pi$
$571$	22.6181			0.946540			0.473270	+	0.880917i	$0.343073\pi$
$572$		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	−0.995682	−	0.0928287i	$-0.970409\pi$
$577$	−21.6873	+	21.6873i	−0.902853	+	0.902853i	0.0928287	+	0.995682i	$0.470409\pi$
$578$		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	$-0.204265\pi$
$587$	4.90612	+	4.90612i	0.202497	+	0.202497i	−0.598572	+	0.801069i	$0.704265\pi$
$588$		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	0.781739	−	0.623605i	$-0.214333\pi$
$593$	3.85082	+	3.85082i	0.158134	+	0.158134i	−0.623605	+	0.781739i	$0.714333\pi$
$594$		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	−0.0736188	−	0.997286i	$-0.523455\pi$
$613$	22.8689	−	22.8689i	0.923668	−	0.923668i	0.997286	+	0.0736188i	$0.0234548\pi$
$614$		0			0
$615$	95.6378			3.85649
$616$		0			0
$617$	8.80459	−	8.80459i	0.354459	−	0.354459i	−0.507306	−	0.861766i	$-0.669359\pi$
$617$	8.80459	−	8.80459i	0.354459	−	0.354459i	0.861766	+	0.507306i	$0.169359\pi$
$618$		0			0
$619$	17.6690	−	17.6690i	0.710175	−	0.710175i	−0.256397	−	0.966572i	$-0.582535\pi$
$619$	17.6690	−	17.6690i	0.710175	−	0.710175i	0.966572	+	0.256397i	$0.0825353\pi$
$620$		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	$-0.822725\pi$
$631$	−8.04600	+	8.04600i	−0.320306	+	0.320306i	0.528578	+	0.848885i	$0.322725\pi$
$632$		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	$-0.743737\pi$
$641$		−	36.5025i		−	1.44176i	0.693059	−	0.720881i	$-0.256263\pi$
$642$		0			0
$643$	−31.9489	−	31.9489i	−1.25994	−	1.25994i	−0.951120	−	0.308823i	$-0.900065\pi$
$643$	−31.9489	−	31.9489i	−1.25994	−	1.25994i	−0.308823	−	0.951120i	$-0.599935\pi$
$644$		0			0
$645$	23.2730	+	23.2730i	0.916374	+	0.916374i
$646$		0			0
$647$	5.41660			0.212949			0.106474	−	0.994315i	$-0.466044\pi$
$647$	5.41660			0.212949			0.106474	+	0.994315i	$0.466044\pi$
$648$		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	0.399125	−	0.916897i	$-0.369314\pi$
$661$	−13.3119	−	13.3119i	−0.517772	−	0.517772i	−0.916897	+	0.399125i	$0.869314\pi$
$662$		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	$0.119930\pi$
$673$			19.0893i			0.735839i	−0.929858	+	0.367919i	$0.880070\pi$
$674$		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	$-0.713169\pi$
$683$	4.26695	−	4.26695i	0.163270	−	0.163270i	0.784014	+	0.620743i	$0.213169\pi$
$684$		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	−0.999938	−	0.0111470i	$-0.996452\pi$
$691$	−26.5783	−	26.5783i	−1.01108	−	1.01108i	−0.0111470	−	0.999938i	$-0.503548\pi$
$692$		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	$-0.639576\pi$
$701$		−	47.9431i		−	1.81079i	0.424574	−	0.905393i	$-0.360424\pi$
$702$		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	0.984229	−	0.176896i	$-0.0566057\pi$
$709$	21.4969	+	21.4969i	0.807333	+	0.807333i	−0.176896	+	0.984229i	$0.556606\pi$
$710$		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	$-0.328457\pi$
$719$	27.5224			1.02641			0.513207	+	0.858265i	$0.328457\pi$
$720$		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	$-0.207006\pi$
$727$	42.9188			1.59177			0.795886	+	0.605447i	$0.207006\pi$
$728$		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	−0.991749	−	0.128192i	$-0.959083\pi$
$733$	−23.3799	+	23.3799i	−0.863557	+	0.863557i	0.128192	+	0.991749i	$0.459083\pi$
$734$		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	$-0.501504\pi$
$739$	27.0558	−	27.0558i	0.995265	−	0.995265i	0.999989	+	0.00472373i	$0.00150362\pi$
$740$		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	$-0.288248\pi$
$743$	−4.62082	−	4.62082i	−0.169522	−	0.169522i	−0.786769	+	0.617248i	$0.788248\pi$
$744$		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	$-0.561913\pi$
$751$	−10.5935			−0.386561			−0.193281	+	0.981144i	$0.561913\pi$
$752$		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	0.782044	−	0.623223i	$-0.214177\pi$
$761$	4.38129	+	4.38129i	0.158822	+	0.158822i	−0.623223	+	0.782044i	$0.714177\pi$
$762$		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	−0.500892	−	0.865510i	$-0.666995\pi$
$769$	10.1111	−	10.1111i	0.364617	−	0.364617i	0.865510	+	0.500892i	$0.166995\pi$
$770$		0			0
$771$	−14.4065			−0.518837
$772$		0			0
$773$	−1.84208	+	1.84208i	−0.0662550	+	0.0662550i	−0.739458	−	0.673203i	$-0.764918\pi$
$773$	−1.84208	+	1.84208i	−0.0662550	+	0.0662550i	0.673203	+	0.739458i	$0.264918\pi$
$774$		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	0.461083	−	0.887357i	$-0.347461\pi$
$787$	37.8285	−	37.8285i	1.34844	−	1.34844i	0.887357	−	0.461083i	$-0.152539\pi$
$788$		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	$0.104947\pi$
$797$			18.2802i			0.647519i	−0.946139	+	0.323759i	$0.895053\pi$
$798$		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	$-0.414254\pi$
$811$	−19.8721	−	19.8721i	−0.697805	−	0.697805i	−0.963937	+	0.266132i	$0.914254\pi$
$812$		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	−0.0587062	−	0.998275i	$-0.518697\pi$
$821$	26.9216	−	26.9216i	0.939569	−	0.939569i	0.998275	+	0.0587062i	$0.0186975\pi$
$822$		0			0
$823$	−48.2105			−1.68051			−0.840257	−	0.542189i	$-0.817596\pi$
$823$	−48.2105			−1.68051			−0.840257	+	0.542189i	$0.817596\pi$
$824$		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	$-0.880250\pi$
$827$	−16.1810	+	16.1810i	−0.562669	+	0.562669i	0.367396	+	0.930065i	$0.380250\pi$
$828$		0			0
$829$			40.8476i			1.41870i	0.704858	+	0.709349i	$0.251011\pi$
$829$			40.8476i			1.41870i	−0.704858	+	0.709349i	$0.748989\pi$
$830$		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	$-0.873511\pi$
$839$	−15.4988	+	15.4988i	−0.535077	+	0.535077i	0.387002	+	0.922079i	$0.373511\pi$
$840$		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	0.669743	−	0.742593i	$-0.266404\pi$
$853$	−2.12764	−	2.12764i	−0.0728492	−	0.0728492i	−0.742593	+	0.669743i	$0.766404\pi$
$854$		0			0
$855$	23.2040			0.793559
$856$		0			0
$857$			27.4076i			0.936226i	0.883669	+	0.468113i	$0.155066\pi$
$857$			27.4076i			0.936226i	−0.883669	+	0.468113i	$0.844934\pi$
$858$		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	$-0.268636\pi$
$863$	−2.43101	−	2.43101i	−0.0827525	−	0.0827525i	−0.747271	+	0.664519i	$0.768636\pi$
$864$		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	−0.743147	−	0.669128i	$-0.766668\pi$
$877$	−2.19199	+	2.19199i	−0.0740184	+	0.0740184i	0.669128	+	0.743147i	$0.266668\pi$
$878$		0			0
$879$	66.0106	−	66.0106i	2.22648	−	2.22648i
$880$		0			0
$881$		−	2.92715i		−	0.0986182i	−0.998784	−	0.0493091i	$-0.984298\pi$
$881$		−	2.92715i		−	0.0986182i	0.998784	−	0.0493091i	$-0.0157019\pi$
$882$		0			0
$883$	−6.53632			−0.219965			−0.109982	−	0.993934i	$-0.535079\pi$
$883$	−6.53632			−0.219965			−0.109982	+	0.993934i	$0.535079\pi$
$884$		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	$-0.773214\pi$
$907$	−45.5812			−1.51350			−0.756749	+	0.653705i	$0.773214\pi$
$908$		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	−0.753301	−	0.657675i	$-0.771540\pi$
$929$	−2.91463	+	2.91463i	−0.0956260	+	0.0956260i	0.657675	+	0.753301i	$0.271540\pi$
$930$		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	−0.276065	+	0.961139i	$0.589030\pi$
$941$	−37.9521	+	37.9521i	−1.23720	+	1.23720i	−0.961139	+	0.276065i	$0.910970\pi$
$942$		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	$-0.613660\pi$
$947$	18.0760	−	18.0760i	0.587392	−	0.587392i	0.936924	+	0.349533i	$0.113660\pi$
$948$		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	$-0.838883\pi$
$953$		−	29.9338i		−	0.969652i	0.874611	−	0.484826i	$-0.161117\pi$
$954$		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	0.959945	+	0.280187i	$0.0903964\pi$
$967$	38.5639	+	38.5639i	1.24013	+	1.24013i	0.280187	+	0.959945i	$0.409604\pi$
$968$		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	−0.935491	−	0.353351i	$-0.885042\pi$
$977$	−18.1959	+	18.1959i	−0.582140	+	0.582140i	0.353351	+	0.935491i	$0.385042\pi$
$978$		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	$-0.600379\pi$
$983$	20.0827	−	20.0827i	0.640538	−	0.640538i	0.950688	+	0.310149i	$0.100379\pi$
$984$		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.e.31.4	✓	8
4.3	odd	2		416.2.k.f.31.1	yes	8
8.3	odd	2		832.2.k.i.447.4		8
8.5	even	2		832.2.k.g.447.1		8
13.8	odd	4		416.2.k.f.255.4	yes	8
52.47	even	4	inner	416.2.k.e.255.1	yes	8
104.21	odd	4		832.2.k.i.255.1		8
104.99	even	4		832.2.k.g.255.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.k.e.31.4	✓	8	1.1	even	1	trivial
416.2.k.e.255.1	yes	8	52.47	even	4	inner
416.2.k.f.31.1	yes	8	4.3	odd	2
416.2.k.f.255.4	yes	8	13.8	odd	4
832.2.k.g.255.4		8	104.99	even	4
832.2.k.g.447.1		8	8.5	even	2
832.2.k.i.255.1		8	104.21	odd	4
832.2.k.i.447.4		8	8.3	odd	2

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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