LMFDB - Embedded newform 416.2.bu.a.63.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(63,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 7])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.63"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.bu (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-6,0,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			63.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.63
Dual form			416.2.bu.a.383.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.36603 + 1.36603i) q^{3} +(0.366025 + 0.366025i) q^{5} +(-1.00000 + 0.267949i) q^{7} +(2.23205 - 3.86603i) q^{9} +(-0.732051 + 2.73205i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(-1.36603 - 0.366025i) q^{15} +(-0.401924 - 0.232051i) q^{17} +(-0.901924 - 3.36603i) q^{19} +(2.00000 - 2.00000i) q^{21} +(-4.36603 - 7.56218i) q^{23} -4.73205i q^{25} +4.00000i q^{27} +(-2.23205 - 3.86603i) q^{29} +(2.46410 - 2.46410i) q^{31} +(-2.00000 - 7.46410i) q^{33} +(-0.464102 - 0.267949i) q^{35} +(-3.59808 - 0.964102i) q^{37} +(7.09808 - 6.83013i) q^{39} +(-3.06218 + 11.4282i) q^{41} +(-5.73205 + 9.92820i) q^{43} +(2.23205 - 0.598076i) q^{45} +(2.46410 + 2.46410i) q^{47} +(-5.13397 + 2.96410i) q^{49} +1.26795 q^{51} +1.73205 q^{53} +(-1.26795 + 0.732051i) q^{55} +(6.73205 + 6.73205i) q^{57} +(5.73205 - 1.53590i) q^{59} +(3.33013 - 5.76795i) q^{61} +(-1.19615 + 4.46410i) q^{63} +(-1.59808 - 0.964102i) q^{65} +(-8.83013 - 2.36603i) q^{67} +(20.6603 + 11.9282i) q^{69} +(-2.56218 - 9.56218i) q^{71} +(-9.29423 + 9.29423i) q^{73} +(6.46410 + 11.1962i) q^{75} -2.92820i q^{77} +9.46410i q^{79} +(1.23205 + 2.13397i) q^{81} +(6.19615 - 6.19615i) q^{83} +(-0.0621778 - 0.232051i) q^{85} +(10.5622 + 6.09808i) q^{87} +(14.5622 + 3.90192i) q^{89} +(3.26795 - 1.80385i) q^{91} +(-2.46410 + 9.19615i) q^{93} +(0.901924 - 1.56218i) q^{95} +(-5.83013 + 1.56218i) q^{97} +(8.92820 + 8.92820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 14 q^{13} - 2 q^{15} - 12 q^{17} - 14 q^{19} + 8 q^{21} - 14 q^{23} - 2 q^{29} - 4 q^{31} - 8 q^{33} + 12 q^{35} - 4 q^{37} + 18 q^{39} + 12 q^{41}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 - 14 * q^13 - 2 * q^15 - 12 * q^17 - 14 * q^19 + 8 * q^21 - 14 * q^23 - 2 * q^29 - 4 * q^31 - 8 * q^33 + 12 * q^35 - 4 * q^37 + 18 * q^39 + 12 * q^41 - 16 * q^43 + 2 * q^45 - 4 * q^47 - 24 * q^49 + 12 * q^51 - 12 * q^55 + 20 * q^57 + 16 * q^59 - 4 * q^61 + 16 * q^63 + 4 * q^65 - 18 * q^67 + 48 * q^69 + 14 * q^71 - 6 * q^73 + 12 * q^75 - 2 * q^81 + 4 * q^83 + 24 * q^85 + 18 * q^87 + 34 * q^89 + 20 * q^91 + 4 * q^93 + 14 * q^95 - 6 * q^97 + 8 * q^99

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$e\left(\frac{7}{12}\right)$

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
$3$	−2.36603	+	1.36603i	−1.36603	+	0.788675i	−0.990418	−	0.138104i	$-0.955899\pi$
$3$	−2.36603	+	1.36603i	−1.36603	+	0.788675i	−0.375608	+	0.926779i	$0.622566\pi$
$4$		0			0
$5$	0.366025	+	0.366025i	0.163692	+	0.163692i	0.784200	−	0.620508i	$-0.213074\pi$
$5$	0.366025	+	0.366025i	0.163692	+	0.163692i	−0.620508	+	0.784200i	$0.713074\pi$
$6$		0			0
$7$	−1.00000	+	0.267949i	−0.377964	+	0.101275i	−0.442799	−	0.896621i	$-0.646015\pi$
$7$	−1.00000	+	0.267949i	−0.377964	+	0.101275i	0.0648349	+	0.997896i	$0.479348\pi$
$8$		0			0
$9$	2.23205	−	3.86603i	0.744017	−	1.28868i
$10$		0			0
$11$	−0.732051	+	2.73205i	−0.220722	+	0.823744i	0.763352	+	0.645983i	$0.223552\pi$
$11$	−0.732051	+	2.73205i	−0.220722	+	0.823744i	−0.984074	+	0.177762i	$0.943114\pi$
$12$		0			0
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$14$		0			0
$15$	−1.36603	−	0.366025i	−0.352706	−	0.0945074i
$16$		0			0
$17$	−0.401924	−	0.232051i	−0.0974808	−	0.0562806i	0.450467	−	0.892793i	$-0.351257\pi$
$17$	−0.401924	−	0.232051i	−0.0974808	−	0.0562806i	−0.547948	+	0.836512i	$0.684591\pi$
$18$		0			0
$19$	−0.901924	−	3.36603i	−0.206916	−	0.772219i	−0.988857	−	0.148868i	$-0.952437\pi$
$19$	−0.901924	−	3.36603i	−0.206916	−	0.772219i	0.781942	−	0.623352i	$-0.214229\pi$
$20$		0			0
$21$	2.00000	−	2.00000i	0.436436	−	0.436436i
$22$		0			0
$23$	−4.36603	−	7.56218i	−0.910379	−	1.57682i	−0.813529	−	0.581524i	$-0.802457\pi$
$23$	−4.36603	−	7.56218i	−0.910379	−	1.57682i	−0.0968500	−	0.995299i	$-0.530877\pi$
$24$		0			0
$25$		−	4.73205i		−	0.946410i
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−2.23205	−	3.86603i	−0.414481	−	0.717903i	0.580892	−	0.813980i	$-0.302704\pi$
$29$	−2.23205	−	3.86603i	−0.414481	−	0.717903i	−0.995374	+	0.0960774i	$0.969370\pi$
$30$		0			0
$31$	2.46410	−	2.46410i	0.442566	−	0.442566i	−0.450308	−	0.892873i	$-0.648686\pi$
$31$	2.46410	−	2.46410i	0.442566	−	0.442566i	0.892873	+	0.450308i	$0.148686\pi$
$32$		0			0
$33$	−2.00000	−	7.46410i	−0.348155	−	1.29933i
$34$		0			0
$35$	−0.464102	−	0.267949i	−0.0784475	−	0.0452917i
$36$		0			0
$37$	−3.59808	−	0.964102i	−0.591520	−	0.158497i	−0.0493722	−	0.998780i	$-0.515722\pi$
$37$	−3.59808	−	0.964102i	−0.591520	−	0.158497i	−0.542148	+	0.840283i	$0.682389\pi$
$38$		0			0
$39$	7.09808	−	6.83013i	1.13660	−	1.09370i
$40$		0			0
$41$	−3.06218	+	11.4282i	−0.478232	+	1.78479i	0.130543	+	0.991443i	$0.458328\pi$
$41$	−3.06218	+	11.4282i	−0.478232	+	1.78479i	−0.608775	+	0.793343i	$0.708339\pi$
$42$		0			0
$43$	−5.73205	+	9.92820i	−0.874130	+	1.51404i	−0.0164424	+	0.999865i	$0.505234\pi$
$43$	−5.73205	+	9.92820i	−0.874130	+	1.51404i	−0.857687	+	0.514172i	$0.828099\pi$
$44$		0			0
$45$	2.23205	−	0.598076i	0.332734	−	0.0891559i
$46$		0			0
$47$	2.46410	+	2.46410i	0.359426	+	0.359426i	0.863601	−	0.504175i	$-0.168203\pi$
$47$	2.46410	+	2.46410i	0.359426	+	0.359426i	−0.504175	+	0.863601i	$0.668203\pi$
$48$		0			0
$49$	−5.13397	+	2.96410i	−0.733425	+	0.423443i
$50$		0			0
$51$	1.26795			0.177548
$52$		0			0
$53$	1.73205			0.237915			0.118958	−	0.992899i	$-0.462045\pi$
$53$	1.73205			0.237915			0.118958	+	0.992899i	$0.462045\pi$
$54$		0			0
$55$	−1.26795	+	0.732051i	−0.170970	+	0.0987097i
$56$		0			0
$57$	6.73205	+	6.73205i	0.891682	+	0.891682i
$58$		0			0
$59$	5.73205	−	1.53590i	0.746249	−	0.199957i	0.134396	−	0.990928i	$-0.457091\pi$
$59$	5.73205	−	1.53590i	0.746249	−	0.199957i	0.611854	+	0.790971i	$0.290424\pi$
$60$		0			0
$61$	3.33013	−	5.76795i	0.426379	−	0.738510i	−0.570169	−	0.821527i	$-0.693122\pi$
$61$	3.33013	−	5.76795i	0.426379	−	0.738510i	0.996548	+	0.0830172i	$0.0264556\pi$
$62$		0			0
$63$	−1.19615	+	4.46410i	−0.150701	+	0.562424i
$64$		0			0
$65$	−1.59808	−	0.964102i	−0.198217	−	0.119582i
$66$		0			0
$67$	−8.83013	−	2.36603i	−1.07877	−	0.289056i	−0.324679	−	0.945824i	$-0.605256\pi$
$67$	−8.83013	−	2.36603i	−1.07877	−	0.289056i	−0.754093	+	0.656768i	$0.771923\pi$
$68$		0			0
$69$	20.6603	+	11.9282i	2.48720	+	1.43599i
$70$		0			0
$71$	−2.56218	−	9.56218i	−0.304075	−	1.13482i	−0.933739	−	0.357955i	$-0.883474\pi$
$71$	−2.56218	−	9.56218i	−0.304075	−	1.13482i	0.629664	−	0.776867i	$-0.283192\pi$
$72$		0			0
$73$	−9.29423	+	9.29423i	−1.08781	+	1.08781i	−0.0920531	+	0.995754i	$0.529343\pi$
$73$	−9.29423	+	9.29423i	−1.08781	+	1.08781i	−0.995754	+	0.0920531i	$0.970657\pi$
$74$		0			0
$75$	6.46410	+	11.1962i	0.746410	+	1.29282i
$76$		0			0
$77$		−	2.92820i		−	0.333700i
$78$		0			0
$79$			9.46410i			1.06479i	0.846495	+	0.532397i	$0.178709\pi$
$79$			9.46410i			1.06479i	−0.846495	+	0.532397i	$0.821291\pi$
$80$		0			0
$81$	1.23205	+	2.13397i	0.136895	+	0.237108i
$82$		0			0
$83$	6.19615	−	6.19615i	0.680116	−	0.680116i	−0.279910	−	0.960026i	$-0.590305\pi$
$83$	6.19615	−	6.19615i	0.680116	−	0.680116i	0.960026	+	0.279910i	$0.0903047\pi$
$84$		0			0
$85$	−0.0621778	−	0.232051i	−0.00674413	−	0.0251694i
$86$		0			0
$87$	10.5622	+	6.09808i	1.13238	+	0.653782i
$88$		0			0
$89$	14.5622	+	3.90192i	1.54359	+	0.413603i	0.927423	−	0.374014i	$-0.122019\pi$
$89$	14.5622	+	3.90192i	1.54359	+	0.413603i	0.616165	+	0.787617i	$0.288686\pi$
$90$		0			0
$91$	3.26795	−	1.80385i	0.342574	−	0.189095i
$92$		0			0
$93$	−2.46410	+	9.19615i	−0.255515	+	0.953597i
$94$		0			0
$95$	0.901924	−	1.56218i	0.0925354	−	0.160276i
$96$		0			0
$97$	−5.83013	+	1.56218i	−0.591960	+	0.158615i	−0.542349	−	0.840153i	$-0.682465\pi$
$97$	−5.83013	+	1.56218i	−0.591960	+	0.158615i	−0.0496110	+	0.998769i	$0.515798\pi$
$98$		0			0
$99$	8.92820	+	8.92820i	0.897318	+	0.897318i
$100$		0			0
$101$	−13.7942	+	7.96410i	−1.37258	+	0.792458i	−0.991252	−	0.131983i	$-0.957865\pi$
$101$	−13.7942	+	7.96410i	−1.37258	+	0.792458i	−0.381325	+	0.924441i	$0.624532\pi$
$102$		0			0
$103$	−6.73205			−0.663329			−0.331664	−	0.943397i	$-0.607610\pi$
$103$	−6.73205			−0.663329			−0.331664	+	0.943397i	$0.607610\pi$
$104$		0			0
$105$	1.46410			0.142882
$106$		0			0
$107$	0.633975	−	0.366025i	0.0612886	−	0.0353850i	−0.469043	−	0.883176i	$-0.655401\pi$
$107$	0.633975	−	0.366025i	0.0612886	−	0.0353850i	0.530331	+	0.847791i	$0.322068\pi$
$108$		0			0
$109$	−11.1962	−	11.1962i	−1.07240	−	1.07240i	−0.997166	−	0.0752308i	$-0.976031\pi$
$109$	−11.1962	−	11.1962i	−1.07240	−	1.07240i	−0.0752308	−	0.997166i	$-0.523969\pi$
$110$		0			0
$111$	9.83013	−	2.63397i	0.933034	−	0.250006i
$112$		0			0
$113$	−1.59808	+	2.76795i	−0.150334	+	0.260387i	−0.931350	−	0.364124i	$-0.881368\pi$
$113$	−1.59808	+	2.76795i	−0.150334	+	0.260387i	0.781016	+	0.624511i	$0.214702\pi$
$114$		0			0
$115$	1.16987	−	4.36603i	0.109091	−	0.407134i
$116$		0			0
$117$	−4.46410	+	15.4641i	−0.412706	+	1.42966i
$118$		0			0
$119$	0.464102	+	0.124356i	0.0425441	+	0.0113997i
$120$		0			0
$121$	2.59808	+	1.50000i	0.236189	+	0.136364i
$122$		0			0
$123$	−8.36603	−	31.2224i	−0.754339	−	2.81523i
$124$		0			0
$125$	3.56218	−	3.56218i	0.318611	−	0.318611i
$126$		0			0
$127$	2.90192	+	5.02628i	0.257504	+	0.446010i	0.965573	−	0.260134i	$-0.0837666\pi$
$127$	2.90192	+	5.02628i	0.257504	+	0.446010i	−0.708069	+	0.706144i	$0.750433\pi$
$128$		0			0
$129$		−	31.3205i		−	2.75762i
$130$		0			0
$131$			3.80385i			0.332344i	0.986097	+	0.166172i	$0.0531406\pi$
$131$			3.80385i			0.332344i	−0.986097	+	0.166172i	$0.946859\pi$
$132$		0			0
$133$	1.80385	+	3.12436i	0.156413	+	0.270916i
$134$		0			0
$135$	−1.46410	+	1.46410i	−0.126010	+	0.126010i
$136$		0			0
$137$	−1.50000	−	5.59808i	−0.128154	−	0.478276i	0.871779	−	0.489900i	$-0.162967\pi$
$137$	−1.50000	−	5.59808i	−0.128154	−	0.478276i	−0.999932	+	0.0116238i	$0.996300\pi$
$138$		0			0
$139$	6.12436	+	3.53590i	0.519461	+	0.299911i	0.736714	−	0.676204i	$-0.236376\pi$
$139$	6.12436	+	3.53590i	0.519461	+	0.299911i	−0.217253	+	0.976115i	$0.569710\pi$
$140$		0			0
$141$	−9.19615	−	2.46410i	−0.774456	−	0.207515i
$142$		0			0
$143$	0.196152	−	10.1962i	0.0164031	−	0.852645i
$144$		0			0
$145$	0.598076	−	2.23205i	0.0496675	−	0.185362i
$146$		0			0
$147$	8.09808	−	14.0263i	0.667918	−	1.15687i
$148$		0			0
$149$	5.86603	−	1.57180i	0.480564	−	0.128767i	−0.0104015	−	0.999946i	$-0.503311\pi$
$149$	5.86603	−	1.57180i	0.480564	−	0.128767i	0.490965	+	0.871179i	$0.336644\pi$
$150$		0			0
$151$	0.464102	+	0.464102i	0.0377681	+	0.0377681i	0.725739	−	0.687971i	$-0.241498\pi$
$151$	0.464102	+	0.464102i	0.0377681	+	0.0377681i	−0.687971	+	0.725739i	$0.741498\pi$
$152$		0			0
$153$	−1.79423	+	1.03590i	−0.145055	+	0.0837474i
$154$		0			0
$155$	1.80385			0.144889
$156$		0			0
$157$	−22.4641			−1.79283			−0.896415	−	0.443215i	$-0.853838\pi$
$157$	−22.4641			−1.79283			−0.896415	+	0.443215i	$0.853838\pi$
$158$		0			0
$159$	−4.09808	+	2.36603i	−0.324999	+	0.187638i
$160$		0			0
$161$	6.39230	+	6.39230i	0.503784	+	0.503784i
$162$		0			0
$163$	−7.73205	+	2.07180i	−0.605621	+	0.162276i	−0.548583	−	0.836096i	$-0.684832\pi$
$163$	−7.73205	+	2.07180i	−0.605621	+	0.162276i	−0.0570383	+	0.998372i	$0.518166\pi$
$164$		0			0
$165$	2.00000	−	3.46410i	0.155700	−	0.269680i
$166$		0			0
$167$	−4.19615	+	15.6603i	−0.324708	+	1.21183i	0.589897	+	0.807478i	$0.299168\pi$
$167$	−4.19615	+	15.6603i	−0.324708	+	1.21183i	−0.914605	+	0.404348i	$0.867498\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$	−15.0263	−	4.02628i	−1.14909	−	0.307897i
$172$		0			0
$173$	18.5885	+	10.7321i	1.41325	+	0.815943i	0.995693	−	0.0927063i	$-0.0295518\pi$
$173$	18.5885	+	10.7321i	1.41325	+	0.815943i	0.417561	+	0.908649i	$0.362885\pi$
$174$		0			0
$175$	1.26795	+	4.73205i	0.0958479	+	0.357709i
$176$		0			0
$177$	−11.4641	+	11.4641i	−0.861695	+	0.861695i
$178$		0			0
$179$	2.73205	+	4.73205i	0.204203	+	0.353690i	0.949879	−	0.312619i	$-0.101206\pi$
$179$	2.73205	+	4.73205i	0.204203	+	0.353690i	−0.745675	+	0.666309i	$0.767873\pi$
$180$		0			0
$181$		−	7.19615i		−	0.534886i	−0.963574	−	0.267443i	$-0.913821\pi$
$181$		−	7.19615i		−	0.534886i	0.963574	−	0.267443i	$-0.0861787\pi$
$182$		0			0
$183$			18.1962i			1.34510i
$184$		0			0
$185$	−0.964102	−	1.66987i	−0.0708822	−	0.122772i
$186$		0			0
$187$	0.928203	−	0.928203i	0.0678769	−	0.0678769i
$188$		0			0
$189$	−1.07180	−	4.00000i	−0.0779617	−	0.290957i
$190$		0			0
$191$	−13.9019	−	8.02628i	−1.00591	−	0.580761i	−0.0959170	−	0.995389i	$-0.530578\pi$
$191$	−13.9019	−	8.02628i	−1.00591	−	0.580761i	−0.909991	+	0.414628i	$0.863912\pi$
$192$		0			0
$193$	−7.33013	−	1.96410i	−0.527634	−	0.141379i	−0.0148386	−	0.999890i	$-0.504723\pi$
$193$	−7.33013	−	1.96410i	−0.527634	−	0.141379i	−0.512796	+	0.858511i	$0.671390\pi$
$194$		0			0
$195$	5.09808	+	0.0980762i	0.365081	+	0.00702338i
$196$		0			0
$197$	4.83013	−	18.0263i	0.344132	−	1.28432i	−0.549490	−	0.835500i	$-0.685178\pi$
$197$	4.83013	−	18.0263i	0.344132	−	1.28432i	0.893623	−	0.448819i	$-0.148155\pi$
$198$		0			0
$199$	−8.29423	+	14.3660i	−0.587962	+	1.01838i	0.406537	+	0.913634i	$0.366736\pi$
$199$	−8.29423	+	14.3660i	−0.587962	+	1.01838i	−0.994499	+	0.104746i	$0.966597\pi$
$200$		0			0
$201$	24.1244	−	6.46410i	1.70160	−	0.455943i
$202$		0			0
$203$	3.26795	+	3.26795i	0.229365	+	0.229365i
$204$		0			0
$205$	−5.30385	+	3.06218i	−0.370437	+	0.213872i
$206$		0			0
$207$	−38.9808			−2.70935
$208$		0			0
$209$	9.85641			0.681782
$210$		0			0
$211$	−23.3660	+	13.4904i	−1.60858	+	0.928716i	−0.618896	+	0.785473i	$0.712420\pi$
$211$	−23.3660	+	13.4904i	−1.60858	+	0.928716i	−0.989688	+	0.143243i	$0.954247\pi$
$212$		0			0
$213$	19.1244	+	19.1244i	1.31038	+	1.31038i
$214$		0			0
$215$	−5.73205	+	1.53590i	−0.390923	+	0.104747i
$216$		0			0
$217$	−1.80385	+	3.12436i	−0.122453	+	0.212095i
$218$		0			0
$219$	9.29423	−	34.6865i	0.628046	−	2.34390i
$220$		0			0
$221$	1.60770	+	0.464102i	0.108145	+	0.0312189i
$222$		0			0
$223$	4.56218	+	1.22243i	0.305506	+	0.0818601i	0.408315	−	0.912841i	$-0.366116\pi$
$223$	4.56218	+	1.22243i	0.305506	+	0.0818601i	−0.102809	+	0.994701i	$0.532783\pi$
$224$		0			0
$225$	−18.2942	−	10.5622i	−1.21962	−	0.704145i
$226$		0			0
$227$	2.63397	+	9.83013i	0.174823	+	0.652448i	0.996582	+	0.0826128i	$0.0263265\pi$
$227$	2.63397	+	9.83013i	0.174823	+	0.652448i	−0.821759	+	0.569836i	$0.807007\pi$
$228$		0			0
$229$	15.7321	−	15.7321i	1.03960	−	1.03960i	0.0404204	−	0.999183i	$-0.487130\pi$
$229$	15.7321	−	15.7321i	1.03960	−	1.03960i	0.999183	−	0.0404204i	$-0.0128697\pi$
$230$		0			0
$231$	4.00000	+	6.92820i	0.263181	+	0.455842i
$232$		0			0
$233$		−	8.53590i		−	0.559205i	−0.960116	−	0.279603i	$-0.909797\pi$
$233$		−	8.53590i		−	0.559205i	0.960116	−	0.279603i	$-0.0902027\pi$
$234$		0			0
$235$			1.80385i			0.117670i
$236$		0			0
$237$	−12.9282	−	22.3923i	−0.839777	−	1.45454i
$238$		0			0
$239$	10.7321	−	10.7321i	0.694199	−	0.694199i	−0.268954	−	0.963153i	$-0.586678\pi$
$239$	10.7321	−	10.7321i	0.694199	−	0.694199i	0.963153	+	0.268954i	$0.0866781\pi$
$240$		0			0
$241$	−1.50000	−	5.59808i	−0.0966235	−	0.360604i	0.900637	−	0.434572i	$-0.143100\pi$
$241$	−1.50000	−	5.59808i	−0.0966235	−	0.360604i	−0.997261	+	0.0739682i	$0.976434\pi$
$242$		0			0
$243$	−16.2224	−	9.36603i	−1.04067	−	0.600831i
$244$		0			0
$245$	−2.96410	−	0.794229i	−0.189370	−	0.0507414i
$246$		0			0
$247$	6.07180	+	11.0000i	0.386339	+	0.699913i
$248$		0			0
$249$	−6.19615	+	23.1244i	−0.392665	+	1.46545i
$250$		0			0
$251$	6.36603	−	11.0263i	0.401820	−	0.695973i	−0.592126	−	0.805846i	$-0.701711\pi$
$251$	6.36603	−	11.0263i	0.401820	−	0.695973i	0.993946	+	0.109873i	$0.0350444\pi$
$252$		0			0
$253$	23.8564	−	6.39230i	1.49984	−	0.401881i
$254$		0			0
$255$	0.464102	+	0.464102i	0.0290632	+	0.0290632i
$256$		0			0
$257$	0.232051	−	0.133975i	0.0144749	−	0.00835711i	−0.492745	−	0.870174i	$-0.664006\pi$
$257$	0.232051	−	0.133975i	0.0144749	−	0.00835711i	0.507220	+	0.861817i	$0.330673\pi$
$258$		0			0
$259$	3.85641			0.239625
$260$		0			0
$261$	−19.9282			−1.23352
$262$		0			0
$263$	9.58846	−	5.53590i	0.591250	−	0.341358i	−0.174342	−	0.984685i	$-0.555780\pi$
$263$	9.58846	−	5.53590i	0.591250	−	0.341358i	0.765591	+	0.643327i	$0.222446\pi$
$264$		0			0
$265$	0.633975	+	0.633975i	0.0389447	+	0.0389447i
$266$		0			0
$267$	−39.7846	+	10.6603i	−2.43478	+	0.652397i
$268$		0			0
$269$	4.00000	−	6.92820i	0.243884	−	0.422420i	−0.717933	−	0.696112i	$-0.754912\pi$
$269$	4.00000	−	6.92820i	0.243884	−	0.422420i	0.961817	+	0.273692i	$0.0882449\pi$
$270$		0			0
$271$	−6.80385	+	25.3923i	−0.413304	+	1.54247i	0.374904	+	0.927064i	$0.377676\pi$
$271$	−6.80385	+	25.3923i	−0.413304	+	1.54247i	−0.788208	+	0.615409i	$0.788991\pi$
$272$		0			0
$273$	−5.26795	+	8.73205i	−0.318831	+	0.528488i
$274$		0			0
$275$	12.9282	+	3.46410i	0.779600	+	0.208893i
$276$		0			0
$277$	−16.3301	−	9.42820i	−0.981182	−	0.566486i	−0.0785554	−	0.996910i	$-0.525031\pi$
$277$	−16.3301	−	9.42820i	−0.981182	−	0.566486i	−0.902627	+	0.430424i	$0.858364\pi$
$278$		0			0
$279$	−4.02628	−	15.0263i	−0.241047	−	0.899600i
$280$		0			0
$281$	−13.3660	+	13.3660i	−0.797350	+	0.797350i	−0.982677	−	0.185327i	$-0.940666\pi$
$281$	−13.3660	+	13.3660i	−0.797350	+	0.797350i	0.185327	+	0.982677i	$0.440666\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$			4.92820i			0.291922i
$286$		0			0
$287$		−	12.2487i		−	0.723019i
$288$		0			0
$289$	−8.39230	−	14.5359i	−0.493665	−	0.855053i
$290$		0			0
$291$	11.6603	−	11.6603i	0.683536	−	0.683536i
$292$		0			0
$293$	3.23205	+	12.0622i	0.188818	+	0.704680i	0.993781	+	0.111354i	$0.0355188\pi$
$293$	3.23205	+	12.0622i	0.188818	+	0.704680i	−0.804962	+	0.593326i	$0.797815\pi$
$294$		0			0
$295$	2.66025	+	1.53590i	0.154886	+	0.0894235i
$296$		0			0
$297$	−10.9282	−	2.92820i	−0.634119	−	0.169912i
$298$		0			0
$299$	21.8301	+	22.6865i	1.26247	+	1.31200i
$300$		0			0
$301$	3.07180	−	11.4641i	0.177055	−	0.660780i
$302$		0			0
$303$	21.7583	−	37.6865i	1.24998	−	2.16503i
$304$		0			0
$305$	3.33013	−	0.892305i	0.190683	−	0.0510932i
$306$		0			0
$307$	12.2679	+	12.2679i	0.700169	+	0.700169i	0.964447	−	0.264278i	$-0.0851336\pi$
$307$	12.2679	+	12.2679i	0.700169	+	0.700169i	−0.264278	+	0.964447i	$0.585134\pi$
$308$		0			0
$309$	15.9282	−	9.19615i	0.906124	−	0.523151i
$310$		0			0
$311$	−24.5885			−1.39428			−0.697142	−	0.716933i	$-0.745545\pi$
$311$	−24.5885			−1.39428			−0.697142	+	0.716933i	$0.745545\pi$
$312$		0			0
$313$	−11.4641			−0.647989			−0.323995	−	0.946059i	$-0.605026\pi$
$313$	−11.4641			−0.647989			−0.323995	+	0.946059i	$0.605026\pi$
$314$		0			0
$315$	−2.07180	+	1.19615i	−0.116733	+	0.0673956i
$316$		0			0
$317$	−2.09808	−	2.09808i	−0.117840	−	0.117840i	0.645728	−	0.763568i	$-0.276554\pi$
$317$	−2.09808	−	2.09808i	−0.117840	−	0.117840i	−0.763568	+	0.645728i	$0.776554\pi$
$318$		0			0
$319$	12.1962	−	3.26795i	0.682853	−	0.182970i
$320$		0			0
$321$	−1.00000	+	1.73205i	−0.0558146	+	0.0966736i
$322$		0			0
$323$	−0.418584	+	1.56218i	−0.0232907	+	0.0869219i
$324$		0			0
$325$	4.09808	+	16.5622i	0.227320	+	0.918704i
$326$		0			0
$327$	41.7846	+	11.1962i	2.31069	+	0.619149i
$328$		0			0
$329$	−3.12436	−	1.80385i	−0.172251	−	0.0994493i
$330$		0			0
$331$	−2.14359	−	8.00000i	−0.117823	−	0.439720i	0.881660	−	0.471885i	$-0.156426\pi$
$331$	−2.14359	−	8.00000i	−0.117823	−	0.439720i	−0.999483	+	0.0321654i	$0.989760\pi$
$332$		0			0
$333$	−11.7583	+	11.7583i	−0.644353	+	0.644353i
$334$		0			0
$335$	−2.36603	−	4.09808i	−0.129270	−	0.223902i
$336$		0			0
$337$			3.14359i			0.171242i	0.996328	+	0.0856212i	$0.0272875\pi$
$337$			3.14359i			0.171242i	−0.996328	+	0.0856212i	$0.972713\pi$
$338$		0			0
$339$		−	8.73205i		−	0.474260i
$340$		0			0
$341$	4.92820	+	8.53590i	0.266877	+	0.462245i
$342$		0			0
$343$	9.46410	−	9.46410i	0.511013	−	0.511013i
$344$		0			0
$345$	3.19615	+	11.9282i	0.172075	+	0.642193i
$346$		0			0
$347$	8.24167	+	4.75833i	0.442436	+	0.255441i	0.704630	−	0.709575i	$-0.251113\pi$
$347$	8.24167	+	4.75833i	0.442436	+	0.255441i	−0.262194	+	0.965015i	$0.584446\pi$
$348$		0			0
$349$	−4.36603	−	1.16987i	−0.233708	−	0.0626219i	0.140064	−	0.990142i	$-0.455269\pi$
$349$	−4.36603	−	1.16987i	−0.233708	−	0.0626219i	−0.373772	+	0.927521i	$0.621936\pi$
$350$		0			0
$351$	−3.46410	−	14.0000i	−0.184900	−	0.747265i
$352$		0			0
$353$	6.64359	−	24.7942i	0.353603	−	1.31966i	−0.528631	−	0.848852i	$-0.677294\pi$
$353$	6.64359	−	24.7942i	0.353603	−	1.31966i	0.882234	−	0.470812i	$-0.156039\pi$
$354$		0			0
$355$	2.56218	−	4.43782i	0.135986	−	0.235535i
$356$		0			0
$357$	−1.26795	+	0.339746i	−0.0671070	+	0.0179813i
$358$		0			0
$359$	−22.9282	−	22.9282i	−1.21010	−	1.21010i	−0.970992	−	0.239112i	$-0.923144\pi$
$359$	−22.9282	−	22.9282i	−1.21010	−	1.21010i	−0.239112	−	0.970992i	$-0.576856\pi$
$360$		0			0
$361$	5.93782	−	3.42820i	0.312517	−	0.180432i
$362$		0			0
$363$	−8.19615			−0.430186
$364$		0			0
$365$	−6.80385			−0.356130
$366$		0			0
$367$	−5.49038	+	3.16987i	−0.286596	+	0.165466i	−0.636406	−	0.771355i	$-0.719579\pi$
$367$	−5.49038	+	3.16987i	−0.286596	+	0.165466i	0.349810	+	0.936821i	$0.386246\pi$
$368$		0			0
$369$	37.3468	+	37.3468i	1.94420	+	1.94420i
$370$		0			0
$371$	−1.73205	+	0.464102i	−0.0899236	+	0.0240950i
$372$		0			0
$373$	−11.9641	+	20.7224i	−0.619478	+	1.07297i	0.370103	+	0.928991i	$0.379322\pi$
$373$	−11.9641	+	20.7224i	−0.619478	+	1.07297i	−0.989581	+	0.143976i	$0.954011\pi$
$374$		0			0
$375$	−3.56218	+	13.2942i	−0.183950	+	0.686511i
$376$		0			0
$377$	11.1603	+	11.5981i	0.574782	+	0.597331i
$378$		0			0
$379$	18.0263	+	4.83013i	0.925948	+	0.248107i	0.690126	−	0.723689i	$-0.257555\pi$
$379$	18.0263	+	4.83013i	0.925948	+	0.248107i	0.235822	+	0.971796i	$0.424222\pi$
$380$		0			0
$381$	−13.7321	−	7.92820i	−0.703514	−	0.406174i
$382$		0			0
$383$	6.92820	+	25.8564i	0.354015	+	1.32120i	0.881720	+	0.471773i	$0.156386\pi$
$383$	6.92820	+	25.8564i	0.354015	+	1.32120i	−0.527705	+	0.849428i	$0.676947\pi$
$384$		0			0
$385$	1.07180	−	1.07180i	0.0546238	−	0.0546238i
$386$		0			0
$387$	25.5885	+	44.3205i	1.30073	+	2.25294i
$388$		0			0
$389$			6.66025i			0.337688i	0.985643	+	0.168844i	$0.0540035\pi$
$389$			6.66025i			0.337688i	−0.985643	+	0.168844i	$0.945997\pi$
$390$		0			0
$391$			4.05256i			0.204947i
$392$		0			0
$393$	−5.19615	−	9.00000i	−0.262111	−	0.453990i
$394$		0			0
$395$	−3.46410	+	3.46410i	−0.174298	+	0.174298i
$396$		0			0
$397$	7.56218	+	28.2224i	0.379535	+	1.41644i	0.846604	+	0.532223i	$0.178643\pi$
$397$	7.56218	+	28.2224i	0.379535	+	1.41644i	−0.467070	+	0.884221i	$0.654690\pi$
$398$		0			0
$399$	−8.53590	−	4.92820i	−0.427329	−	0.246719i
$400$		0			0
$401$	9.96410	+	2.66987i	0.497583	+	0.133327i	0.498878	−	0.866672i	$-0.333745\pi$
$401$	9.96410	+	2.66987i	0.497583	+	0.133327i	−0.00129478	+	0.999999i	$0.500412\pi$
$402$		0			0
$403$	−6.49038	+	10.7583i	−0.323309	+	0.535911i
$404$		0			0
$405$	−0.330127	+	1.23205i	−0.0164041	+	0.0612211i
$406$		0			0
$407$	5.26795	−	9.12436i	0.261123	−	0.452278i
$408$		0			0
$409$	21.2583	−	5.69615i	1.05116	−	0.281657i	0.308427	−	0.951248i	$-0.400197\pi$
$409$	21.2583	−	5.69615i	1.05116	−	0.281657i	0.742730	+	0.669591i	$0.233531\pi$
$410$		0			0
$411$	11.1962	+	11.1962i	0.552265	+	0.552265i
$412$		0			0
$413$	−5.32051	+	3.07180i	−0.261805	+	0.151153i
$414$		0			0
$415$	4.53590			0.222658
$416$		0			0
$417$	−19.3205			−0.946129
$418$		0			0
$419$	−25.0981	+	14.4904i	−1.22612	+	0.707901i	−0.966216	−	0.257733i	$-0.917025\pi$
$419$	−25.0981	+	14.4904i	−1.22612	+	0.707901i	−0.259905	+	0.965634i	$0.583691\pi$
$420$		0			0
$421$	3.70577	+	3.70577i	0.180608	+	0.180608i	0.791621	−	0.611013i	$-0.209238\pi$
$421$	3.70577	+	3.70577i	0.180608	+	0.180608i	−0.611013	+	0.791621i	$0.709238\pi$
$422$		0			0
$423$	15.0263	−	4.02628i	0.730603	−	0.195764i
$424$		0			0
$425$	−1.09808	+	1.90192i	−0.0532645	+	0.0922569i
$426$		0			0
$427$	−1.78461	+	6.66025i	−0.0863633	+	0.322312i
$428$		0			0
$429$	13.4641	+	24.3923i	0.650053	+	1.17767i
$430$		0			0
$431$	25.1244	+	6.73205i	1.21020	+	0.324271i	0.806839	−	0.590771i	$-0.201176\pi$
$431$	25.1244	+	6.73205i	1.21020	+	0.324271i	0.403358	+	0.915042i	$0.367843\pi$
$432$		0			0
$433$	13.2846	+	7.66987i	0.638418	+	0.368591i	0.784005	−	0.620755i	$-0.213174\pi$
$433$	13.2846	+	7.66987i	0.638418	+	0.368591i	−0.145587	+	0.989345i	$0.546507\pi$
$434$		0			0
$435$	1.63397	+	6.09808i	0.0783431	+	0.292380i
$436$		0			0
$437$	−21.5167	+	21.5167i	−1.02928	+	1.02928i
$438$		0			0
$439$	−8.46410	−	14.6603i	−0.403970	−	0.699696i	0.590231	−	0.807234i	$-0.299036\pi$
$439$	−8.46410	−	14.6603i	−0.403970	−	0.699696i	−0.994201	+	0.107538i	$0.965703\pi$
$440$		0			0
$441$			26.4641i			1.26020i
$442$		0			0
$443$			9.85641i			0.468292i	0.972201	+	0.234146i	$0.0752294\pi$
$443$			9.85641i			0.468292i	−0.972201	+	0.234146i	$0.924771\pi$
$444$		0			0
$445$	3.90192	+	6.75833i	0.184969	+	0.320376i
$446$		0			0
$447$	−11.7321	+	11.7321i	−0.554907	+	0.554907i
$448$		0			0
$449$	−1.95448	−	7.29423i	−0.0922377	−	0.344236i	0.904348	−	0.426795i	$-0.140357\pi$
$449$	−1.95448	−	7.29423i	−0.0922377	−	0.344236i	−0.996586	+	0.0825590i	$0.973691\pi$
$450$		0			0
$451$	−28.9808	−	16.7321i	−1.36465	−	0.787882i
$452$		0			0
$453$	−1.73205	−	0.464102i	−0.0813788	−	0.0218054i
$454$		0			0
$455$	1.85641	+	0.535898i	0.0870297	+	0.0251233i
$456$		0			0
$457$	8.25833	−	30.8205i	0.386308	−	1.44172i	−0.449786	−	0.893137i	$-0.648500\pi$
$457$	8.25833	−	30.8205i	0.386308	−	1.44172i	0.836094	−	0.548586i	$-0.184834\pi$
$458$		0			0
$459$	0.928203	−	1.60770i	0.0433248	−	0.0750408i
$460$		0			0
$461$	17.3301	−	4.64359i	0.807144	−	0.216274i	0.168426	−	0.985714i	$-0.446132\pi$
$461$	17.3301	−	4.64359i	0.807144	−	0.216274i	0.638718	+	0.769441i	$0.279465\pi$
$462$		0			0
$463$	7.26795	+	7.26795i	0.337770	+	0.337770i	0.855528	−	0.517757i	$-0.173233\pi$
$463$	7.26795	+	7.26795i	0.337770	+	0.337770i	−0.517757	+	0.855528i	$0.673233\pi$
$464$		0			0
$465$	−4.26795	+	2.46410i	−0.197921	+	0.114270i
$466$		0			0
$467$	13.8564			0.641198			0.320599	−	0.947215i	$-0.396116\pi$
$467$	13.8564			0.641198			0.320599	+	0.947215i	$0.396116\pi$
$468$		0			0
$469$	9.46410			0.437012
$470$		0			0
$471$	53.1506	−	30.6865i	2.44905	−	1.41396i
$472$		0			0
$473$	−22.9282	−	22.9282i	−1.05424	−	1.05424i
$474$		0			0
$475$	−15.9282	+	4.26795i	−0.730836	+	0.195827i
$476$		0			0
$477$	3.86603	−	6.69615i	0.177013	−	0.306596i
$478$		0			0
$479$	7.36603	−	27.4904i	0.336562	−	1.25607i	−0.565603	−	0.824678i	$-0.691357\pi$
$479$	7.36603	−	27.4904i	0.336562	−	1.25607i	0.902166	−	0.431390i	$-0.141977\pi$
$480$		0			0
$481$	13.4282	+	0.258330i	0.612273	+	0.0117788i
$482$		0			0
$483$	−23.8564	−	6.39230i	−1.08550	−	0.290860i
$484$		0			0
$485$	−2.70577	−	1.56218i	−0.122863	−	0.0709348i
$486$		0			0
$487$	−5.00000	−	18.6603i	−0.226572	−	0.845577i	−0.981769	−	0.190079i	$-0.939126\pi$
$487$	−5.00000	−	18.6603i	−0.226572	−	0.845577i	0.755197	−	0.655498i	$-0.227541\pi$
$488$		0			0
$489$	15.4641	−	15.4641i	0.699311	−	0.699311i
$490$		0			0
$491$	12.4641	+	21.5885i	0.562497	+	0.974273i	0.997278	+	0.0737371i	$0.0234926\pi$
$491$	12.4641	+	21.5885i	0.562497	+	0.974273i	−0.434781	+	0.900536i	$0.643174\pi$
$492$		0			0
$493$			2.07180i			0.0933090i
$494$		0			0
$495$			6.53590i			0.293767i
$496$		0			0
$497$	5.12436	+	8.87564i	0.229859	+	0.398127i
$498$		0			0
$499$	13.3923	−	13.3923i	0.599522	−	0.599522i	−0.340663	−	0.940185i	$-0.610652\pi$
$499$	13.3923	−	13.3923i	0.599522	−	0.599522i	0.940185	+	0.340663i	$0.110652\pi$
$500$		0			0
$501$	−11.4641	−	42.7846i	−0.512178	−	1.91148i
$502$		0			0
$503$	−7.60770	−	4.39230i	−0.339210	−	0.195843i	0.320712	−	0.947177i	$-0.396078\pi$
$503$	−7.60770	−	4.39230i	−0.339210	−	0.195843i	−0.659923	+	0.751333i	$0.729411\pi$
$504$		0			0
$505$	−7.96410	−	2.13397i	−0.354398	−	0.0949606i
$506$		0			0
$507$	−18.9282	+	30.0526i	−0.840631	+	1.33468i
$508$		0			0
$509$	−2.96410	+	11.0622i	−0.131381	+	0.490322i	−0.999987	−	0.00518261i	$-0.998350\pi$
$509$	−2.96410	+	11.0622i	−0.131381	+	0.490322i	0.868605	+	0.495505i	$0.165017\pi$
$510$		0			0
$511$	6.80385	−	11.7846i	0.300984	−	0.521320i
$512$		0			0
$513$	13.4641	−	3.60770i	0.594455	−	0.159284i
$514$		0			0
$515$	−2.46410	−	2.46410i	−0.108581	−	0.108581i
$516$		0			0
$517$	−8.53590	+	4.92820i	−0.375408	+	0.216742i
$518$		0			0
$519$	−58.6410			−2.57405
$520$		0			0
$521$	−21.3397			−0.934911			−0.467456	−	0.884017i	$-0.654829\pi$
$521$	−21.3397			−0.934911			−0.467456	+	0.884017i	$0.654829\pi$
$522$		0			0
$523$	26.3205	−	15.1962i	1.15092	−	0.664481i	0.201805	−	0.979426i	$-0.435319\pi$
$523$	26.3205	−	15.1962i	1.15092	−	0.664481i	0.949110	+	0.314944i	$0.101986\pi$
$524$		0			0
$525$	−9.46410	−	9.46410i	−0.413047	−	0.413047i
$526$		0			0
$527$	−1.56218	+	0.418584i	−0.0680495	+	0.0182338i
$528$		0			0
$529$	−26.6244	+	46.1147i	−1.15758	+	2.00499i
$530$		0			0
$531$	6.85641	−	25.5885i	0.297543	−	1.11044i
$532$		0			0
$533$	0.820508	−	42.6506i	0.0355401	−	1.84740i
$534$		0			0
$535$	0.366025	+	0.0980762i	0.0158247	+	0.00424020i
$536$		0			0
$537$	−12.9282	−	7.46410i	−0.557893	−	0.322100i
$538$		0			0
$539$	−4.33975	−	16.1962i	−0.186926	−	0.697618i
$540$		0			0
$541$	−19.4378	+	19.4378i	−0.835697	+	0.835697i	−0.988289	−	0.152592i	$-0.951238\pi$
$541$	−19.4378	+	19.4378i	−0.835697	+	0.835697i	0.152592	+	0.988289i	$0.451238\pi$
$542$		0			0
$543$	9.83013	+	17.0263i	0.421851	+	0.730668i
$544$		0			0
$545$		−	8.19615i		−	0.351085i
$546$		0			0
$547$		−	18.1962i		−	0.778011i	−0.921235	−	0.389006i	$-0.872819\pi$
$547$		−	18.1962i		−	0.778011i	0.921235	−	0.389006i	$-0.127181\pi$
$548$		0			0
$549$	−14.8660	−	25.7487i	−0.634467	−	1.09893i
$550$		0			0
$551$	−11.0000	+	11.0000i	−0.468616	+	0.468616i
$552$		0			0
$553$	−2.53590	−	9.46410i	−0.107837	−	0.402455i
$554$		0			0
$555$	4.56218	+	2.63397i	0.193654	+	0.111806i
$556$		0			0
$557$	9.96410	+	2.66987i	0.422193	+	0.113126i	0.463659	−	0.886014i	$-0.346536\pi$
$557$	9.96410	+	2.66987i	0.422193	+	0.113126i	−0.0414664	+	0.999140i	$0.513203\pi$
$558$		0			0
$559$	11.4641	−	39.7128i	0.484880	−	1.67967i
$560$		0			0
$561$	−0.928203	+	3.46410i	−0.0391888	+	0.146254i
$562$		0			0
$563$	0.464102	−	0.803848i	0.0195596	−	0.0338781i	−0.856080	−	0.516843i	$-0.827107\pi$
$563$	0.464102	−	0.803848i	0.0195596	−	0.0338781i	0.875640	+	0.482965i	$0.160440\pi$
$564$		0			0
$565$	−1.59808	+	0.428203i	−0.0672316	+	0.0180146i
$566$		0			0
$567$	−1.80385	−	1.80385i	−0.0757545	−	0.0757545i
$568$		0			0
$569$	−15.0000	+	8.66025i	−0.628833	+	0.363057i	−0.780300	−	0.625406i	$-0.784934\pi$
$569$	−15.0000	+	8.66025i	−0.628833	+	0.363057i	0.151467	+	0.988462i	$0.451600\pi$
$570$		0			0
$571$	8.67949			0.363225			0.181613	−	0.983370i	$-0.441868\pi$
$571$	8.67949			0.363225			0.181613	+	0.983370i	$0.441868\pi$
$572$		0			0
$573$	43.8564			1.83213
$574$		0			0
$575$	−35.7846	+	20.6603i	−1.49232	+	0.861592i
$576$		0			0
$577$	−24.1506	−	24.1506i	−1.00540	−	1.00540i	−0.999985	−	0.00541878i	$-0.998275\pi$
$577$	−24.1506	−	24.1506i	−1.00540	−	1.00540i	−0.00541878	−	0.999985i	$-0.501725\pi$
$578$		0			0
$579$	20.0263	−	5.36603i	0.832264	−	0.223004i
$580$		0			0
$581$	−4.53590	+	7.85641i	−0.188181	+	0.325939i
$582$		0			0
$583$	−1.26795	+	4.73205i	−0.0525131	+	0.195982i
$584$		0			0
$585$	−7.29423	+	4.02628i	−0.301579	+	0.166466i
$586$		0			0
$587$	6.26795	+	1.67949i	0.258706	+	0.0693201i	0.385841	−	0.922565i	$-0.373911\pi$
$587$	6.26795	+	1.67949i	0.258706	+	0.0693201i	−0.127135	+	0.991885i	$0.540578\pi$
$588$		0			0
$589$	−10.5167	−	6.07180i	−0.433331	−	0.250184i
$590$		0			0
$591$	13.1962	+	49.2487i	0.542817	+	2.02582i
$592$		0			0
$593$	12.1699	−	12.1699i	0.499757	−	0.499757i	−0.411605	−	0.911362i	$-0.635032\pi$
$593$	12.1699	−	12.1699i	0.499757	−	0.499757i	0.911362	+	0.411605i	$0.135032\pi$
$594$		0			0
$595$	0.124356	+	0.215390i	0.00509808	+	0.00883014i
$596$		0			0
$597$		−	45.3205i		−	1.85484i
$598$		0			0
$599$		−	16.0526i		−	0.655890i	−0.944697	−	0.327945i	$-0.893644\pi$
$599$		−	16.0526i		−	0.655890i	0.944697	−	0.327945i	$-0.106356\pi$
$600$		0			0
$601$	8.33013	+	14.4282i	0.339793	+	0.588539i	0.984394	−	0.175981i	$-0.0563097\pi$
$601$	8.33013	+	14.4282i	0.339793	+	0.588539i	−0.644601	+	0.764519i	$0.722976\pi$
$602$		0			0
$603$	−28.8564	+	28.8564i	−1.17512	+	1.17512i
$604$		0			0
$605$	0.401924	+	1.50000i	0.0163405	+	0.0609837i
$606$		0			0
$607$	18.8827	+	10.9019i	0.766425	+	0.442495i	0.831598	−	0.555379i	$-0.187427\pi$
$607$	18.8827	+	10.9019i	0.766425	+	0.442495i	−0.0651731	+	0.997874i	$0.520760\pi$
$608$		0			0
$609$	−12.1962	−	3.26795i	−0.494213	−	0.132424i
$610$		0			0
$611$	−10.7583	−	6.49038i	−0.435235	−	0.262573i
$612$		0			0
$613$	7.91858	−	29.5526i	0.319829	−	1.19362i	−0.599581	−	0.800314i	$-0.704666\pi$
$613$	7.91858	−	29.5526i	0.319829	−	1.19362i	0.919409	−	0.393302i	$-0.128667\pi$
$614$		0			0
$615$	8.36603	−	14.4904i	0.337351	−	0.584309i
$616$		0			0
$617$	−27.6244	+	7.40192i	−1.11211	+	0.297990i	−0.767689	−	0.640823i	$-0.778593\pi$
$617$	−27.6244	+	7.40192i	−1.11211	+	0.297990i	−0.344426	+	0.938813i	$0.611927\pi$
$618$		0			0
$619$	−13.2679	−	13.2679i	−0.533284	−	0.533284i	0.388264	−	0.921548i	$-0.373075\pi$
$619$	−13.2679	−	13.2679i	−0.533284	−	0.533284i	−0.921548	+	0.388264i	$0.873075\pi$
$620$		0			0
$621$	30.2487	−	17.4641i	1.21384	−	0.700810i
$622$		0			0
$623$	−15.6077			−0.625309
$624$		0			0
$625$	−21.0526			−0.842102
$626$		0			0
$627$	−23.3205	+	13.4641i	−0.931331	+	0.537704i
$628$		0			0
$629$	1.22243	+	1.22243i	0.0487416	+	0.0487416i
$630$		0			0
$631$	−31.3205	+	8.39230i	−1.24685	+	0.334092i	−0.821119	−	0.570757i	$-0.806650\pi$
$631$	−31.3205	+	8.39230i	−1.24685	+	0.334092i	−0.425731	+	0.904850i	$0.639983\pi$
$632$		0			0
$633$	36.8564	−	63.8372i	1.46491	−	2.53730i
$634$		0			0
$635$	−0.777568	+	2.90192i	−0.0308569	+	0.115159i
$636$		0			0
$637$	15.4019	−	14.8205i	0.610246	−	0.587210i
$638$		0			0
$639$	−42.6865	−	11.4378i	−1.68865	−	0.452473i
$640$		0			0
$641$	34.3301	+	19.8205i	1.35596	+	0.782863i	0.989076	−	0.147404i	$-0.0470919\pi$
$641$	34.3301	+	19.8205i	1.35596	+	0.782863i	0.366882	+	0.930267i	$0.380425\pi$
$642$		0			0
$643$	−3.80385	−	14.1962i	−0.150009	−	0.559842i	−0.999481	−	0.0322080i	$-0.989746\pi$
$643$	−3.80385	−	14.1962i	−0.150009	−	0.559842i	0.849472	−	0.527634i	$-0.176921\pi$
$644$		0			0
$645$	11.4641	−	11.4641i	0.451399	−	0.451399i
$646$		0			0
$647$	−16.4904	−	28.5622i	−0.648304	−	1.12290i	−0.983528	−	0.180757i	$-0.942145\pi$
$647$	−16.4904	−	28.5622i	−0.648304	−	1.12290i	0.335224	−	0.942138i	$-0.391188\pi$
$648$		0			0
$649$			16.7846i			0.658854i
$650$		0			0
$651$		−	9.85641i		−	0.386303i
$652$		0			0
$653$	15.4641	+	26.7846i	0.605157	+	1.04816i	0.992027	+	0.126028i	$0.0402230\pi$
$653$	15.4641	+	26.7846i	0.605157	+	1.04816i	−0.386870	+	0.922134i	$0.626444\pi$
$654$		0			0
$655$	−1.39230	+	1.39230i	−0.0544019	+	0.0544019i
$656$		0			0
$657$	15.1865	+	56.6769i	0.592483	+	2.21118i
$658$		0			0
$659$	−22.0526	−	12.7321i	−0.859046	−	0.495970i	0.00464688	−	0.999989i	$-0.498521\pi$
$659$	−22.0526	−	12.7321i	−0.859046	−	0.495970i	−0.863693	+	0.504019i	$0.831854\pi$
$660$		0			0
$661$	37.4545	+	10.0359i	1.45681	+	0.390351i	0.898387	−	0.439205i	$-0.144740\pi$
$661$	37.4545	+	10.0359i	1.45681	+	0.390351i	0.558424	+	0.829556i	$0.311406\pi$
$662$		0			0
$663$	−4.43782	+	1.09808i	−0.172351	+	0.0426457i
$664$		0			0
$665$	−0.483340	+	1.80385i	−0.0187431	+	0.0699502i
$666$		0			0
$667$	−19.4904	+	33.7583i	−0.754671	+	1.30713i
$668$		0			0
$669$	−12.4641	+	3.33975i	−0.481890	+	0.129122i
$670$		0			0
$671$	13.3205	+	13.3205i	0.514233	+	0.514233i
$672$		0			0
$673$	12.6506	−	7.30385i	0.487646	−	0.281543i	−0.235951	−	0.971765i	$-0.575821\pi$
$673$	12.6506	−	7.30385i	0.487646	−	0.281543i	0.723597	+	0.690222i	$0.242487\pi$
$674$		0			0
$675$	18.9282			0.728547
$676$		0			0
$677$	−18.9282			−0.727470			−0.363735	−	0.931502i	$-0.618499\pi$
$677$	−18.9282			−0.727470			−0.363735	+	0.931502i	$0.618499\pi$
$678$		0			0
$679$	5.41154	−	3.12436i	0.207676	−	0.119902i
$680$		0			0
$681$	−19.6603	−	19.6603i	−0.753383	−	0.753383i
$682$		0			0
$683$	−12.3660	+	3.31347i	−0.473173	+	0.126786i	−0.487522	−	0.873111i	$-0.662099\pi$
$683$	−12.3660	+	3.31347i	−0.473173	+	0.126786i	0.0143491	+	0.999897i	$0.495432\pi$
$684$		0			0
$685$	1.50000	−	2.59808i	0.0573121	−	0.0992674i
$686$		0			0
$687$	−15.7321	+	58.7128i	−0.600215	+	2.24003i
$688$		0			0
$689$	−6.06218	+	1.50000i	−0.230951	+	0.0571454i
$690$		0			0
$691$	−37.3205	−	10.0000i	−1.41974	−	0.380418i	−0.534348	−	0.845265i	$-0.679443\pi$
$691$	−37.3205	−	10.0000i	−1.41974	−	0.380418i	−0.885391	+	0.464847i	$0.846109\pi$
$692$		0			0
$693$	−11.3205	−	6.53590i	−0.430031	−	0.248278i
$694$		0			0
$695$	0.947441	+	3.53590i	0.0359385	+	0.134124i
$696$		0			0
$697$	3.88269	−	3.88269i	0.147067	−	0.147067i
$698$		0			0
$699$	11.6603	+	20.1962i	0.441031	+	0.763889i
$700$		0			0
$701$		−	24.9282i		−	0.941525i	−0.882260	−	0.470763i	$-0.843979\pi$
$701$		−	24.9282i		−	0.941525i	0.882260	−	0.470763i	$-0.156021\pi$
$702$		0			0
$703$			12.9808i			0.489579i
$704$		0			0
$705$	−2.46410	−	4.26795i	−0.0928034	−	0.160740i
$706$		0			0
$707$	11.6603	−	11.6603i	0.438529	−	0.438529i
$708$		0			0
$709$	7.47372	+	27.8923i	0.280681	+	1.04752i	0.951938	+	0.306292i	$0.0990884\pi$
$709$	7.47372	+	27.8923i	0.280681	+	1.04752i	−0.671256	+	0.741225i	$0.734245\pi$
$710$		0			0
$711$	36.5885	+	21.1244i	1.37217	+	0.792225i
$712$		0			0
$713$	−29.3923	−	7.87564i	−1.10075	−	0.294945i
$714$		0			0
$715$	3.80385	−	3.66025i	0.142256	−	0.136886i
$716$		0			0
$717$	−10.7321	+	40.0526i	−0.400796	+	1.49579i
$718$		0			0
$719$	−12.4641	+	21.5885i	−0.464833	+	0.805114i	−0.999194	−	0.0401425i	$-0.987219\pi$
$719$	−12.4641	+	21.5885i	−0.464833	+	0.805114i	0.534361	+	0.845256i	$0.320552\pi$
$720$		0			0
$721$	6.73205	−	1.80385i	0.250715	−	0.0671788i
$722$		0			0
$723$	11.1962	+	11.1962i	0.416389	+	0.416389i
$724$		0			0
$725$	−18.2942	+	10.5622i	−0.679431	+	0.392269i
$726$		0			0
$727$	44.7846			1.66097			0.830485	−	0.557042i	$-0.188064\pi$
$727$	44.7846			1.66097			0.830485	+	0.557042i	$0.188064\pi$
$728$		0			0
$729$	43.7846			1.62165
$730$		0			0
$731$	4.60770	−	2.66025i	0.170422	−	0.0983930i
$732$		0			0
$733$	26.6147	+	26.6147i	0.983038	+	0.983038i	0.999859	−	0.0168208i	$-0.00535448\pi$
$733$	26.6147	+	26.6147i	0.983038	+	0.983038i	−0.0168208	+	0.999859i	$0.505354\pi$
$734$		0			0
$735$	8.09808	−	2.16987i	0.298702	−	0.0800370i
$736$		0			0
$737$	12.9282	−	22.3923i	0.476216	−	0.824831i
$738$		0			0
$739$	7.66025	−	28.5885i	0.281787	−	1.05164i	−0.669368	−	0.742931i	$-0.733435\pi$
$739$	7.66025	−	28.5885i	0.281787	−	1.05164i	0.951155	−	0.308713i	$-0.0998982\pi$
$740$		0			0
$741$	−29.3923	−	17.7321i	−1.07975	−	0.651403i
$742$		0			0
$743$	15.3923	+	4.12436i	0.564689	+	0.151308i	0.529860	−	0.848085i	$-0.322245\pi$
$743$	15.3923	+	4.12436i	0.564689	+	0.151308i	0.0348292	+	0.999393i	$0.488911\pi$
$744$		0			0
$745$	2.72243	+	1.57180i	0.0997422	+	0.0575862i
$746$		0			0
$747$	−10.1244	−	37.7846i	−0.370431	−	1.38247i
$748$		0			0
$749$	−0.535898	+	0.535898i	−0.0195813	+	0.0195813i
$750$		0			0
$751$	−24.1244	−	41.7846i	−0.880310	−	1.52474i	−0.850996	−	0.525172i	$-0.824001\pi$
$751$	−24.1244	−	41.7846i	−0.880310	−	1.52474i	−0.0293140	−	0.999570i	$-0.509332\pi$
$752$		0			0
$753$			34.7846i			1.26762i
$754$		0			0
$755$			0.339746i			0.0123646i
$756$		0			0
$757$	1.39230	+	2.41154i	0.0506042	+	0.0876490i	0.890218	−	0.455535i	$-0.150552\pi$
$757$	1.39230	+	2.41154i	0.0506042	+	0.0876490i	−0.839614	+	0.543184i	$0.817219\pi$
$758$		0			0
$759$	−47.7128	+	47.7128i	−1.73187	+	1.73187i
$760$		0			0
$761$	−4.83013	−	18.0263i	−0.175092	−	0.653452i	−0.996536	−	0.0831629i	$-0.973498\pi$
$761$	−4.83013	−	18.0263i	−0.175092	−	0.653452i	0.821444	−	0.570289i	$-0.193169\pi$
$762$		0			0
$763$	14.1962	+	8.19615i	0.513935	+	0.296721i
$764$		0			0
$765$	−1.03590	−	0.277568i	−0.0374530	−	0.0100355i
$766$		0			0
$767$	−18.7321	+	10.3397i	−0.676375	+	0.373347i
$768$		0			0
$769$	2.09808	−	7.83013i	0.0756586	−	0.282362i	−0.917723	−	0.397220i	$-0.869975\pi$
$769$	2.09808	−	7.83013i	0.0756586	−	0.282362i	0.993382	+	0.114859i	$0.0366415\pi$
$770$		0			0
$771$	−0.366025	+	0.633975i	−0.0131821	+	0.0228320i
$772$		0			0
$773$	−27.2224	+	7.29423i	−0.979123	+	0.262355i	−0.712675	−	0.701494i	$-0.752517\pi$
$773$	−27.2224	+	7.29423i	−0.979123	+	0.262355i	−0.266448	+	0.963849i	$0.585850\pi$
$774$		0			0
$775$	−11.6603	−	11.6603i	−0.418849	−	0.418849i
$776$		0			0
$777$	−9.12436	+	5.26795i	−0.327334	+	0.188987i
$778$		0			0
$779$	41.2295			1.47720
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$	15.4641	−	8.92820i	0.552642	−	0.319068i
$784$		0			0
$785$	−8.22243	−	8.22243i	−0.293471	−	0.293471i
$786$		0			0
$787$	35.9545	−	9.63397i	1.28164	−	0.343414i	0.447160	−	0.894454i	$-0.352435\pi$
$787$	35.9545	−	9.63397i	1.28164	−	0.343414i	0.834479	+	0.551040i	$0.185769\pi$
$788$		0			0
$789$	−15.1244	+	26.1962i	−0.538441	+	0.932608i
$790$		0			0
$791$	0.856406	−	3.19615i	0.0304503	−	0.113642i
$792$		0			0
$793$	−6.66025	+	23.0718i	−0.236513	+	0.819304i
$794$		0			0
$795$	−2.36603	−	0.633975i	−0.0839143	−	0.0224848i
$796$		0			0
$797$	−22.8564	−	13.1962i	−0.809615	−	0.467432i	0.0372069	−	0.999308i	$-0.488154\pi$
$797$	−22.8564	−	13.1962i	−0.809615	−	0.467432i	−0.846822	+	0.531876i	$0.821487\pi$
$798$		0			0
$799$	−0.418584	−	1.56218i	−0.0148084	−	0.0552659i
$800$		0			0
$801$	47.5885	−	47.5885i	1.68146	−	1.68146i
$802$		0			0
$803$	−18.5885	−	32.1962i	−0.655972	−	1.13618i
$804$		0			0
$805$			4.67949i			0.164930i
$806$		0			0
$807$			21.8564i			0.769382i
$808$		0			0
$809$	−10.7942	−	18.6962i	−0.379505	−	0.657322i	0.611485	−	0.791256i	$-0.290572\pi$
$809$	−10.7942	−	18.6962i	−0.379505	−	0.657322i	−0.990990	+	0.133934i	$0.957239\pi$
$810$		0			0
$811$	31.3923	−	31.3923i	1.10233	−	1.10233i	0.108204	−	0.994129i	$-0.465490\pi$
$811$	31.3923	−	31.3923i	1.10233	−	1.10233i	0.994129	−	0.108204i	$-0.0345100\pi$
$812$		0			0
$813$	−18.5885	−	69.3731i	−0.651926	−	2.43302i
$814$		0			0
$815$	−3.58846	−	2.07180i	−0.125698	−	0.0725719i
$816$		0			0
$817$	38.5885	+	10.3397i	1.35004	+	0.361742i
$818$		0			0
$819$	0.320508	−	16.6603i	0.0111995	−	0.582156i
$820$		0			0
$821$	−11.9545	+	44.6147i	−0.417214	+	1.55707i	0.363144	+	0.931733i	$0.381703\pi$
$821$	−11.9545	+	44.6147i	−0.417214	+	1.55707i	−0.780359	+	0.625332i	$0.784963\pi$
$822$		0			0
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	0.317712	+	0.948187i	$0.397086\pi$
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	−0.980010	+	0.198947i	$0.936248\pi$
$824$		0			0
$825$	−35.3205	+	9.46410i	−1.22970	+	0.329498i
$826$		0			0
$827$	−11.3205	−	11.3205i	−0.393653	−	0.393653i	0.482334	−	0.875987i	$-0.339789\pi$
$827$	−11.3205	−	11.3205i	−0.393653	−	0.393653i	−0.875987	+	0.482334i	$0.839789\pi$
$828$		0			0
$829$	−29.0429	+	16.7679i	−1.00870	+	0.582375i	−0.910812	−	0.412822i	$-0.864543\pi$
$829$	−29.0429	+	16.7679i	−1.00870	+	0.582375i	−0.0978913	+	0.995197i	$0.531210\pi$
$830$		0			0
$831$	51.5167			1.78709
$832$		0			0
$833$	2.75129			0.0953265
$834$		0			0
$835$	−7.26795	+	4.19615i	−0.251518	+	0.145214i
$836$		0			0
$837$	9.85641	+	9.85641i	0.340687	+	0.340687i
$838$		0			0
$839$	−9.09808	+	2.43782i	−0.314100	+	0.0841630i	−0.412425	−	0.910991i	$-0.635318\pi$
$839$	−9.09808	+	2.43782i	−0.314100	+	0.0841630i	0.0983249	+	0.995154i	$0.468652\pi$
$840$		0			0
$841$	4.53590	−	7.85641i	0.156410	−	0.270911i
$842$		0			0
$843$	13.3660	−	49.8827i	0.460350	−	1.71805i
$844$		0			0
$845$	6.42820	+	1.99038i	0.221137	+	0.0684712i
$846$		0			0
$847$	−3.00000	−	0.803848i	−0.103081	−	0.0276205i
$848$		0			0
$849$	33.1244	+	19.1244i	1.13682	+	0.656346i
$850$		0			0
$851$	8.41858	+	31.4186i	0.288585	+	1.07702i
$852$		0			0
$853$	−9.70577	+	9.70577i	−0.332319	+	0.332319i	−0.853467	−	0.521147i	$-0.825504\pi$
$853$	−9.70577	+	9.70577i	−0.332319	+	0.332319i	0.521147	+	0.853467i	$0.325504\pi$
$854$		0			0
$855$	−4.02628	−	6.97372i	−0.137696	−	0.238496i
$856$		0			0
$857$		−	8.94744i		−	0.305639i	−0.988254	−	0.152819i	$-0.951165\pi$
$857$		−	8.94744i		−	0.305639i	0.988254	−	0.152819i	$-0.0488353\pi$
$858$		0			0
$859$		−	16.7321i		−	0.570890i	−0.958395	−	0.285445i	$-0.907859\pi$
$859$		−	16.7321i		−	0.570890i	0.958395	−	0.285445i	$-0.0921414\pi$
$860$		0			0
$861$	16.7321	+	28.9808i	0.570227	+	0.987662i
$862$		0			0
$863$	40.1962	−	40.1962i	1.36829	−	1.36829i	0.505419	−	0.862874i	$-0.331338\pi$
$863$	40.1962	−	40.1962i	1.36829	−	1.36829i	0.862874	−	0.505419i	$-0.168662\pi$
$864$		0			0
$865$	2.87564	+	10.7321i	0.0977748	+	0.364901i
$866$		0			0
$867$	39.7128	+	22.9282i	1.34872	+	0.778683i
$868$		0			0
$869$	−25.8564	−	6.92820i	−0.877119	−	0.235023i
$870$		0			0
$871$	32.9545	+	0.633975i	1.11662	+	0.0214814i
$872$		0			0
$873$	−6.97372	+	26.0263i	−0.236025	+	0.880856i
$874$		0			0
$875$	−2.60770	+	4.51666i	−0.0881562	+	0.152691i
$876$		0			0
$877$	−19.1603	+	5.13397i	−0.646996	+	0.173362i	−0.567370	−	0.823463i	$-0.692039\pi$
$877$	−19.1603	+	5.13397i	−0.646996	+	0.173362i	−0.0796256	+	0.996825i	$0.525372\pi$
$878$		0			0
$879$	−24.1244	−	24.1244i	−0.813694	−	0.813694i
$880$		0			0
$881$	8.89230	−	5.13397i	0.299589	−	0.172968i	−0.342669	−	0.939456i	$-0.611331\pi$
$881$	8.89230	−	5.13397i	0.299589	−	0.172968i	0.642258	+	0.766488i	$0.277998\pi$
$882$		0			0
$883$	41.6603			1.40198			0.700990	−	0.713172i	$-0.252742\pi$
$883$	41.6603			1.40198			0.700990	+	0.713172i	$0.252742\pi$
$884$		0			0
$885$	−8.39230			−0.282104
$886$		0			0
$887$	4.39230	−	2.53590i	0.147479	−	0.0851471i	−0.424445	−	0.905454i	$-0.639531\pi$
$887$	4.39230	−	2.53590i	0.147479	−	0.0851471i	0.571924	+	0.820307i	$0.306197\pi$
$888$		0			0
$889$	−4.24871	−	4.24871i	−0.142497	−	0.142497i
$890$		0			0
$891$	−6.73205	+	1.80385i	−0.225532	+	0.0604312i
$892$		0			0
$893$	6.07180	−	10.5167i	0.203185	−	0.351927i
$894$		0			0
$895$	−0.732051	+	2.73205i	−0.0244698	+	0.0913224i
$896$		0			0
$897$	−82.6410	−	23.8564i	−2.75930	−	0.796542i
$898$		0			0
$899$	−15.0263	−	4.02628i	−0.501154	−	0.134284i
$900$		0			0
$901$	−0.696152	−	0.401924i	−0.0231922	−	0.0133900i
$902$		0			0
$903$	8.39230	+	31.3205i	0.279278	+	1.04228i
$904$		0			0
$905$	2.63397	−	2.63397i	0.0875563	−	0.0875563i
$906$		0			0
$907$	−23.0526	−	39.9282i	−0.765448	−	1.32579i	−0.940010	−	0.341148i	$-0.889184\pi$
$907$	−23.0526	−	39.9282i	−0.765448	−	1.32579i	0.174562	−	0.984646i	$-0.444149\pi$
$908$		0			0
$909$			71.1051i			2.35841i
$910$		0			0
$911$			19.1769i			0.635360i	0.948198	+	0.317680i	$0.102904\pi$
$911$			19.1769i			0.635360i	−0.948198	+	0.317680i	$0.897096\pi$
$912$		0			0
$913$	12.3923	+	21.4641i	0.410125	+	0.710358i
$914$		0			0
$915$	−6.66025	+	6.66025i	−0.220181	+	0.220181i
$916$		0			0
$917$	−1.01924	−	3.80385i	−0.0336582	−	0.125614i
$918$		0			0
$919$	−4.56218	−	2.63397i	−0.150492	−	0.0868868i	0.422863	−	0.906194i	$-0.361025\pi$
$919$	−4.56218	−	2.63397i	−0.150492	−	0.0868868i	−0.573355	+	0.819307i	$0.694359\pi$
$920$		0			0
$921$	−45.7846	−	12.2679i	−1.50865	−	0.404243i
$922$		0			0
$923$	17.2487	+	31.2487i	0.567748	+	1.02856i
$924$		0			0
$925$	−4.56218	+	17.0263i	−0.150003	+	0.559821i
$926$		0			0
$927$	−15.0263	+	26.0263i	−0.493528	+	0.854815i
$928$		0			0
$929$	−41.6506	+	11.1603i	−1.36651	+	0.366156i	−0.866204	−	0.499691i	$-0.833447\pi$
$929$	−41.6506	+	11.1603i	−1.36651	+	0.366156i	−0.500309	+	0.865847i	$0.666780\pi$
$930$		0			0
$931$	14.6077	+	14.6077i	0.478748	+	0.478748i
$932$		0			0
$933$	58.1769	−	33.5885i	1.90463	−	1.09964i
$934$		0			0
$935$	0.679492			0.0222218
$936$		0			0
$937$	31.0000			1.01273			0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000			1.01273			0.506363	+	0.862320i	$0.330990\pi$
$938$		0			0
$939$	27.1244	−	15.6603i	0.885170	−	0.511053i
$940$		0			0
$941$	−1.78461	−	1.78461i	−0.0581766	−	0.0581766i	0.677420	−	0.735597i	$-0.263098\pi$
$941$	−1.78461	−	1.78461i	−0.0581766	−	0.0581766i	−0.735597	+	0.677420i	$0.763098\pi$
$942$		0			0
$943$	99.7917	−	26.7391i	3.24966	−	0.870745i
$944$		0			0
$945$	1.07180	−	1.85641i	0.0348656	−	0.0603889i
$946$		0			0
$947$	−10.8301	+	40.4186i	−0.351932	+	1.31343i	0.532370	+	0.846512i	$0.321302\pi$
$947$	−10.8301	+	40.4186i	−0.351932	+	1.31343i	−0.884302	+	0.466916i	$0.845365\pi$
$948$		0			0
$949$	24.4808	−	40.5788i	0.794679	−	1.31724i
$950$		0			0
$951$	7.83013	+	2.09808i	0.253909	+	0.0680348i
$952$		0			0
$953$	40.6410	+	23.4641i	1.31649	+	0.760077i	0.983162	−	0.182734i	$-0.0584946\pi$
$953$	40.6410	+	23.4641i	1.31649	+	0.760077i	0.333329	+	0.942811i	$0.391828\pi$
$954$		0			0
$955$	−2.15064	−	8.02628i	−0.0695929	−	0.259724i
$956$		0			0
$957$	−24.3923	+	24.3923i	−0.788491	+	0.788491i
$958$		0			0
$959$	3.00000	+	5.19615i	0.0968751	+	0.167793i
$960$		0			0
$961$			18.8564i			0.608271i
$962$		0			0
$963$		−	3.26795i		−	0.105308i
$964$		0			0
$965$	−1.96410	−	3.40192i	−0.0632267	−	0.109512i
$966$		0			0
$967$	3.33975	−	3.33975i	0.107399	−	0.107399i	−0.651365	−	0.758764i	$-0.725803\pi$
$967$	3.33975	−	3.33975i	0.107399	−	0.107399i	0.758764	+	0.651365i	$0.225803\pi$
$968$		0			0
$969$	−1.14359	−	4.26795i	−0.0367375	−	0.137106i
$970$		0			0
$971$	−22.0070	−	12.7058i	−0.706240	−	0.407748i	0.103428	−	0.994637i	$-0.467019\pi$
$971$	−22.0070	−	12.7058i	−0.706240	−	0.407748i	−0.809667	+	0.586889i	$0.800352\pi$
$972$		0			0
$973$	−7.07180	−	1.89488i	−0.226711	−	0.0607471i
$974$		0			0
$975$	−32.3205	−	33.5885i	−1.03508	−	1.07569i
$976$		0			0
$977$	−7.47372	+	27.8923i	−0.239106	+	0.892354i	0.737150	+	0.675730i	$0.236171\pi$
$977$	−7.47372	+	27.8923i	−0.239106	+	0.892354i	−0.976255	+	0.216624i	$0.930495\pi$
$978$		0			0
$979$	−21.3205	+	36.9282i	−0.681406	+	1.18023i
$980$		0			0
$981$	−68.2750	+	18.2942i	−2.17985	+	0.584090i
$982$		0			0
$983$	23.7846	+	23.7846i	0.758611	+	0.758611i	0.976070	−	0.217458i	$-0.0697766\pi$
$983$	23.7846	+	23.7846i	0.758611	+	0.758611i	−0.217458	+	0.976070i	$0.569777\pi$
$984$		0			0
$985$	8.36603	−	4.83013i	0.266564	−	0.153901i
$986$		0			0
$987$	9.85641			0.313733
$988$		0			0
$989$	100.105			3.18316
$990$		0			0
$991$	−18.1699	+	10.4904i	−0.577185	+	0.333238i	−0.760014	−	0.649907i	$-0.774808\pi$
$991$	−18.1699	+	10.4904i	−0.577185	+	0.333238i	0.182829	+	0.983145i	$0.441475\pi$
$992$		0			0
$993$	16.0000	+	16.0000i	0.507745	+	0.507745i
$994$		0			0
$995$	−8.29423	+	2.22243i	−0.262945	+	0.0704558i
$996$		0			0
$997$	13.5263	−	23.4282i	0.428382	−	0.741979i	−0.568348	−	0.822788i	$-0.692417\pi$
$997$	13.5263	−	23.4282i	0.428382	−	0.741979i	0.996730	+	0.0808096i	$0.0257506\pi$
$998$		0			0
$999$	3.85641	−	14.3923i	0.122011	−	0.455352i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bu.a.63.1	✓	4
4.3	odd	2		416.2.bu.b.63.1	yes	4
8.3	odd	2		832.2.bu.b.63.1		4
8.5	even	2		832.2.bu.g.63.1		4
13.6	odd	12		416.2.bu.b.383.1	yes	4
52.19	even	12	inner	416.2.bu.a.383.1	yes	4
104.19	even	12		832.2.bu.g.383.1		4
104.45	odd	12		832.2.bu.b.383.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.bu.a.63.1	✓	4	1.1	even	1	trivial
416.2.bu.a.383.1	yes	4	52.19	even	12	inner
416.2.bu.b.63.1	yes	4	4.3	odd	2
416.2.bu.b.383.1	yes	4	13.6	odd	12
832.2.bu.b.63.1		4	8.3	odd	2
832.2.bu.b.383.1		4	104.45	odd	12
832.2.bu.g.63.1		4	8.5	even	2
832.2.bu.g.383.1		4	104.19	even	12

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.bu (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.36603 + 1.36603i) q^{3} +(0.366025 + 0.366025i) q^{5} +(-1.00000 + 0.267949i) q^{7} +(2.23205 - 3.86603i) q^{9} +(-0.732051 + 2.73205i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(-1.36603 - 0.366025i) q^{15} +(-0.401924 - 0.232051i) q^{17} +(-0.901924 - 3.36603i) q^{19} +(2.00000 - 2.00000i) q^{21} +(-4.36603 - 7.56218i) q^{23} -4.73205i q^{25} +4.00000i q^{27} +(-2.23205 - 3.86603i) q^{29} +(2.46410 - 2.46410i) q^{31} +(-2.00000 - 7.46410i) q^{33} +(-0.464102 - 0.267949i) q^{35} +(-3.59808 - 0.964102i) q^{37} +(7.09808 - 6.83013i) q^{39} +(-3.06218 + 11.4282i) q^{41} +(-5.73205 + 9.92820i) q^{43} +(2.23205 - 0.598076i) q^{45} +(2.46410 + 2.46410i) q^{47} +(-5.13397 + 2.96410i) q^{49} +1.26795 q^{51} +1.73205 q^{53} +(-1.26795 + 0.732051i) q^{55} +(6.73205 + 6.73205i) q^{57} +(5.73205 - 1.53590i) q^{59} +(3.33013 - 5.76795i) q^{61} +(-1.19615 + 4.46410i) q^{63} +(-1.59808 - 0.964102i) q^{65} +(-8.83013 - 2.36603i) q^{67} +(20.6603 + 11.9282i) q^{69} +(-2.56218 - 9.56218i) q^{71} +(-9.29423 + 9.29423i) q^{73} +(6.46410 + 11.1962i) q^{75} -2.92820i q^{77} +9.46410i q^{79} +(1.23205 + 2.13397i) q^{81} +(6.19615 - 6.19615i) q^{83} +(-0.0621778 - 0.232051i) q^{85} +(10.5622 + 6.09808i) q^{87} +(14.5622 + 3.90192i) q^{89} +(3.26795 - 1.80385i) q^{91} +(-2.46410 + 9.19615i) q^{93} +(0.901924 - 1.56218i) q^{95} +(-5.83013 + 1.56218i) q^{97} +(8.92820 + 8.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 14 q^{13} - 2 q^{15} - 12 q^{17} - 14 q^{19} + 8 q^{21} - 14 q^{23} - 2 q^{29} - 4 q^{31} - 8 q^{33} + 12 q^{35} - 4 q^{37} + 18 q^{39} + 12 q^{41}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 - 14 * q^13 - 2 * q^15 - 12 * q^17 - 14 * q^19 + 8 * q^21 - 14 * q^23 - 2 * q^29 - 4 * q^31 - 8 * q^33 + 12 * q^35 - 4 * q^37 + 18 * q^39 + 12 * q^41 - 16 * q^43 + 2 * q^45 - 4 * q^47 - 24 * q^49 + 12 * q^51 - 12 * q^55 + 20 * q^57 + 16 * q^59 - 4 * q^61 + 16 * q^63 + 4 * q^65 - 18 * q^67 + 48 * q^69 + 14 * q^71 - 6 * q^73 + 12 * q^75 - 2 * q^81 + 4 * q^83 + 24 * q^85 + 18 * q^87 + 34 * q^89 + 20 * q^91 + 4 * q^93 + 14 * q^95 - 6 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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