LMFDB - Embedded newform 405.2.e.l.271.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(136,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.136"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,2,0,-4,-2,0,6] 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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.2.e.l.136.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.366025 - 0.633975i) q^{2} +(0.732051 - 1.26795i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.36603 + 4.09808i) q^{7} -2.53590 q^{8} +0.732051 q^{10} +(2.86603 + 4.96410i) q^{11} +(-0.732051 + 1.26795i) q^{13} +(1.73205 - 3.00000i) q^{14} +(-0.535898 - 0.928203i) q^{16} -2.73205 q^{17} +4.46410 q^{19} +(0.732051 + 1.26795i) q^{20} +(2.09808 - 3.63397i) q^{22} +(1.73205 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.07180 q^{26} +6.92820 q^{28} +(-1.59808 - 2.76795i) q^{29} +(1.50000 - 2.59808i) q^{31} +(-2.92820 + 5.07180i) q^{32} +(1.00000 + 1.73205i) q^{34} -4.73205 q^{35} -2.73205 q^{37} +(-1.63397 - 2.83013i) q^{38} +(1.26795 - 2.19615i) q^{40} +(3.59808 - 6.23205i) q^{41} +(-0.0980762 - 0.169873i) q^{43} +8.39230 q^{44} -2.53590 q^{46} +(4.36603 + 7.56218i) q^{47} +(-7.69615 + 13.3301i) q^{49} +(-0.366025 + 0.633975i) q^{50} +(1.07180 + 1.85641i) q^{52} +6.73205 q^{53} -5.73205 q^{55} +(-6.00000 - 10.3923i) q^{56} +(-1.16987 + 2.02628i) q^{58} +(4.13397 - 7.16025i) q^{59} +(-2.00000 - 3.46410i) q^{61} -2.19615 q^{62} +2.14359 q^{64} +(-0.732051 - 1.26795i) q^{65} +(-1.73205 + 3.00000i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(1.73205 + 3.00000i) q^{70} -3.73205 q^{71} -7.66025 q^{73} +(1.00000 + 1.73205i) q^{74} +(3.26795 - 5.66025i) q^{76} +(-13.5622 + 23.4904i) q^{77} +(-7.73205 - 13.3923i) q^{79} +1.07180 q^{80} -5.26795 q^{82} +(-1.09808 - 1.90192i) q^{83} +(1.36603 - 2.36603i) q^{85} +(-0.0717968 + 0.124356i) q^{86} +(-7.26795 - 12.5885i) q^{88} +5.19615 q^{89} -6.92820 q^{91} +(-2.53590 - 4.39230i) q^{92} +(3.19615 - 5.53590i) q^{94} +(-2.23205 + 3.86603i) q^{95} +(4.83013 + 8.36603i) q^{97} +11.2679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34}+ \cdots + 52 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 - 4 * q^10 + 8 * q^11 + 4 * q^13 - 16 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^25 + 32 * q^26 + 4 * q^29 + 6 * q^31 + 16 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 - 10 * q^38 + 12 * q^40 + 4 * q^41 + 10 * q^43 - 8 * q^44 - 24 * q^46 + 14 * q^47 - 10 * q^49 + 2 * q^50 + 32 * q^52 + 20 * q^53 - 16 * q^55 - 24 * q^56 - 22 * q^58 + 20 * q^59 - 8 * q^61 + 12 * q^62 + 64 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^71 + 4 * q^73 + 4 * q^74 + 20 * q^76 - 30 * q^77 - 24 * q^79 + 32 * q^80 - 28 * q^82 + 6 * q^83 + 2 * q^85 - 28 * q^86 - 36 * q^88 - 24 * q^92 - 8 * q^94 - 2 * q^95 + 2 * q^97 + 52 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.366025	−	0.633975i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$2$	−0.366025	−	0.633975i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$3$		0			0
$3$		0			0
$4$	0.732051	−	1.26795i	0.366025	−	0.633975i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.36603	+	4.09808i	0.894274	+	1.54893i	0.834701	+	0.550703i	$0.185640\pi$
$7$	2.36603	+	4.09808i	0.894274	+	1.54893i	0.0595724	+	0.998224i	$0.481026\pi$
$8$	−2.53590			−0.896575
$9$		0			0
$10$	0.732051			0.231495
$11$	2.86603	+	4.96410i	0.864139	+	1.49673i	0.867899	+	0.496740i	$0.165470\pi$
$11$	2.86603	+	4.96410i	0.864139	+	1.49673i	−0.00376022	+	0.999993i	$0.501197\pi$
$12$		0			0
$13$	−0.732051	+	1.26795i	−0.203034	+	0.351666i	−0.949505	−	0.313753i	$-0.898414\pi$
$13$	−0.732051	+	1.26795i	−0.203034	+	0.351666i	0.746470	+	0.665419i	$0.231747\pi$
$14$	1.73205	−	3.00000i	0.462910	−	0.801784i
$15$		0			0
$16$	−0.535898	−	0.928203i	−0.133975	−	0.232051i
$17$	−2.73205			−0.662620			−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205			−0.662620			−0.331310	+	0.943522i	$0.607491\pi$
$18$		0			0
$19$	4.46410			1.02414			0.512068	−	0.858945i	$-0.328880\pi$
$19$	4.46410			1.02414			0.512068	+	0.858945i	$0.328880\pi$
$20$	0.732051	+	1.26795i	0.163692	+	0.283522i
$21$		0			0
$22$	2.09808	−	3.63397i	0.447311	−	0.774766i
$23$	1.73205	−	3.00000i	0.361158	−	0.625543i	−0.626994	−	0.779024i	$-0.715715\pi$
$23$	1.73205	−	3.00000i	0.361158	−	0.625543i	0.988152	+	0.153481i	$0.0490483\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	1.07180			0.210197
$27$		0			0
$28$	6.92820			1.30931
$29$	−1.59808	−	2.76795i	−0.296755	−	0.513995i	0.678636	−	0.734474i	$-0.262571\pi$
$29$	−1.59808	−	2.76795i	−0.296755	−	0.513995i	−0.975392	+	0.220479i	$0.929238\pi$
$30$		0			0
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	−0.699301	−	0.714827i	$-0.746505\pi$
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	0.968709	+	0.248199i	$0.0798387\pi$
$32$	−2.92820	+	5.07180i	−0.517638	+	0.896575i
$33$		0			0
$34$	1.00000	+	1.73205i	0.171499	+	0.297044i
$35$	−4.73205			−0.799863
$36$		0			0
$37$	−2.73205			−0.449146			−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205			−0.449146			−0.224573	+	0.974457i	$0.572099\pi$
$38$	−1.63397	−	2.83013i	−0.265066	−	0.459107i
$39$		0			0
$40$	1.26795	−	2.19615i	0.200480	−	0.347242i
$41$	3.59808	−	6.23205i	0.561925	−	0.973283i	−0.435403	−	0.900235i	$-0.643394\pi$
$41$	3.59808	−	6.23205i	0.561925	−	0.973283i	0.997328	−	0.0730473i	$-0.0232724\pi$
$42$		0			0
$43$	−0.0980762	−	0.169873i	−0.0149565	−	0.0259054i	0.858450	−	0.512897i	$-0.171428\pi$
$43$	−0.0980762	−	0.169873i	−0.0149565	−	0.0259054i	−0.873407	+	0.486991i	$0.838094\pi$
$44$	8.39230			1.26519
$45$		0			0
$46$	−2.53590			−0.373898
$47$	4.36603	+	7.56218i	0.636850	+	1.10306i	0.986120	+	0.166035i	$0.0530964\pi$
$47$	4.36603	+	7.56218i	0.636850	+	1.10306i	−0.349270	+	0.937022i	$0.613570\pi$
$48$		0			0
$49$	−7.69615	+	13.3301i	−1.09945	+	1.90430i
$50$	−0.366025	+	0.633975i	−0.0517638	+	0.0896575i
$51$		0			0
$52$	1.07180	+	1.85641i	0.148631	+	0.257437i
$53$	6.73205			0.924718			0.462359	−	0.886693i	$-0.347003\pi$
$53$	6.73205			0.924718			0.462359	+	0.886693i	$0.347003\pi$
$54$		0			0
$55$	−5.73205			−0.772910
$56$	−6.00000	−	10.3923i	−0.801784	−	1.38873i
$57$		0			0
$58$	−1.16987	+	2.02628i	−0.153612	+	0.266064i
$59$	4.13397	−	7.16025i	0.538198	−	0.932186i	−0.460804	−	0.887502i	$-0.652439\pi$
$59$	4.13397	−	7.16025i	0.538198	−	0.932186i	0.999001	−	0.0446835i	$-0.0142279\pi$
$60$		0			0
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	0.709113	−	0.705095i	$-0.249096\pi$
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	−0.965187	+	0.261562i	$0.915762\pi$
$62$	−2.19615			−0.278912
$63$		0			0
$64$	2.14359			0.267949
$65$	−0.732051	−	1.26795i	−0.0907997	−	0.157270i
$66$		0			0
$67$	−1.73205	+	3.00000i	−0.211604	+	0.366508i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	−1.73205	+	3.00000i	−0.211604	+	0.366508i	0.740613	+	0.671932i	$0.234535\pi$
$68$	−2.00000	+	3.46410i	−0.242536	+	0.420084i
$69$		0			0
$70$	1.73205	+	3.00000i	0.207020	+	0.358569i
$71$	−3.73205			−0.442913			−0.221456	−	0.975170i	$-0.571081\pi$
$71$	−3.73205			−0.442913			−0.221456	+	0.975170i	$0.571081\pi$
$72$		0			0
$73$	−7.66025			−0.896565			−0.448282	−	0.893892i	$-0.647964\pi$
$73$	−7.66025			−0.896565			−0.448282	+	0.893892i	$0.647964\pi$
$74$	1.00000	+	1.73205i	0.116248	+	0.201347i
$75$		0			0
$76$	3.26795	−	5.66025i	0.374859	−	0.649276i
$77$	−13.5622	+	23.4904i	−1.54555	+	2.67698i
$78$		0			0
$79$	−7.73205	−	13.3923i	−0.869924	−	1.50675i	−0.862074	−	0.506783i	$-0.830835\pi$
$79$	−7.73205	−	13.3923i	−0.869924	−	1.50675i	−0.00784992	−	0.999969i	$-0.502499\pi$
$80$	1.07180			0.119831
$81$		0			0
$82$	−5.26795			−0.581748
$83$	−1.09808	−	1.90192i	−0.120530	−	0.208763i	0.799447	−	0.600737i	$-0.205126\pi$
$83$	−1.09808	−	1.90192i	−0.120530	−	0.208763i	−0.919977	+	0.391973i	$0.871793\pi$
$84$		0			0
$85$	1.36603	−	2.36603i	0.148166	−	0.256631i
$86$	−0.0717968	+	0.124356i	−0.00774204	+	0.0134096i
$87$		0			0
$88$	−7.26795	−	12.5885i	−0.774766	−	1.34193i
$89$	5.19615			0.550791			0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615			0.550791			0.275396	+	0.961331i	$0.411191\pi$
$90$		0			0
$91$	−6.92820			−0.726273
$92$	−2.53590	−	4.39230i	−0.264386	−	0.457929i
$93$		0			0
$94$	3.19615	−	5.53590i	0.329658	−	0.570984i
$95$	−2.23205	+	3.86603i	−0.229004	+	0.396646i
$96$		0			0
$97$	4.83013	+	8.36603i	0.490425	+	0.849441i	0.999939	−	0.0110211i	$-0.00350819\pi$
$97$	4.83013	+	8.36603i	0.490425	+	0.849441i	−0.509514	+	0.860462i	$0.670175\pi$
$98$	11.2679			1.13823
$99$		0			0
$100$	−1.46410			−0.146410
$101$	1.33013	+	2.30385i	0.132353	+	0.229241i	0.924583	−	0.380981i	$-0.124414\pi$
$101$	1.33013	+	2.30385i	0.132353	+	0.229241i	−0.792230	+	0.610222i	$0.791080\pi$
$102$		0			0
$103$	0.267949	−	0.464102i	0.0264018	−	0.0457293i	−0.852523	−	0.522690i	$-0.824928\pi$
$103$	0.267949	−	0.464102i	0.0264018	−	0.0457293i	0.878924	+	0.476961i	$0.158262\pi$
$104$	1.85641	−	3.21539i	0.182036	−	0.315295i
$105$		0			0
$106$	−2.46410	−	4.26795i	−0.239335	−	0.414540i
$107$	−8.53590			−0.825196			−0.412598	−	0.910913i	$-0.635379\pi$
$107$	−8.53590			−0.825196			−0.412598	+	0.910913i	$0.635379\pi$
$108$		0			0
$109$	−6.07180			−0.581573			−0.290786	−	0.956788i	$-0.593917\pi$
$109$	−6.07180			−0.581573			−0.290786	+	0.956788i	$0.593917\pi$
$110$	2.09808	+	3.63397i	0.200044	+	0.346486i
$111$		0			0
$112$	2.53590	−	4.39230i	0.239620	−	0.415034i
$113$	9.56218	−	16.5622i	0.899534	−	1.55804i	0.0714432	−	0.997445i	$-0.477240\pi$
$113$	9.56218	−	16.5622i	0.899534	−	1.55804i	0.828091	−	0.560594i	$-0.189427\pi$
$114$		0			0
$115$	1.73205	+	3.00000i	0.161515	+	0.279751i
$116$	−4.67949			−0.434480
$117$		0			0
$118$	−6.05256			−0.557183
$119$	−6.46410	−	11.1962i	−0.592563	−	1.02635i
$120$		0			0
$121$	−10.9282	+	18.9282i	−0.993473	+	1.72075i
$122$	−1.46410	+	2.53590i	−0.132554	+	0.229589i
$123$		0			0
$124$	−2.19615	−	3.80385i	−0.197220	−	0.341596i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−14.5885			−1.29452			−0.647258	−	0.762271i	$-0.724084\pi$
$127$	−14.5885			−1.29452			−0.647258	+	0.762271i	$0.724084\pi$
$128$	5.07180	+	8.78461i	0.448288	+	0.776457i
$129$		0			0
$130$	−0.535898	+	0.928203i	−0.0470014	+	0.0814088i
$131$	−7.79423	+	13.5000i	−0.680985	+	1.17950i	0.293696	+	0.955899i	$0.405115\pi$
$131$	−7.79423	+	13.5000i	−0.680985	+	1.17950i	−0.974681	+	0.223602i	$0.928219\pi$
$132$		0			0
$133$	10.5622	+	18.2942i	0.915857	+	1.58631i
$134$	2.53590			0.219068
$135$		0			0
$136$	6.92820			0.594089
$137$	−1.26795	−	2.19615i	−0.108328	−	0.187630i	0.806765	−	0.590873i	$-0.201216\pi$
$137$	−1.26795	−	2.19615i	−0.108328	−	0.187630i	−0.915093	+	0.403243i	$0.867883\pi$
$138$		0			0
$139$	−0.303848	+	0.526279i	−0.0257720	+	0.0446384i	−0.878624	−	0.477515i	$-0.841538\pi$
$139$	−0.303848	+	0.526279i	−0.0257720	+	0.0446384i	0.852852	+	0.522153i	$0.174871\pi$
$140$	−3.46410	+	6.00000i	−0.292770	+	0.507093i
$141$		0			0
$142$	1.36603	+	2.36603i	0.114634	+	0.198552i
$143$	−8.39230			−0.701800
$144$		0			0
$145$	3.19615			0.265426
$146$	2.80385	+	4.85641i	0.232048	+	0.401919i
$147$		0			0
$148$	−2.00000	+	3.46410i	−0.164399	+	0.284747i
$149$	4.00000	−	6.92820i	0.327693	−	0.567581i	−0.654361	−	0.756182i	$-0.727062\pi$
$149$	4.00000	−	6.92820i	0.327693	−	0.567581i	0.982054	+	0.188602i	$0.0603956\pi$
$150$		0			0
$151$	−2.69615	−	4.66987i	−0.219410	−	0.380029i	0.735218	−	0.677831i	$-0.237080\pi$
$151$	−2.69615	−	4.66987i	−0.219410	−	0.380029i	−0.954628	+	0.297802i	$0.903746\pi$
$152$	−11.3205			−0.918214
$153$		0			0
$154$	19.8564			1.60007
$155$	1.50000	+	2.59808i	0.120483	+	0.208683i
$156$		0			0
$157$	9.56218	−	16.5622i	0.763145	−	1.32181i	−0.178077	−	0.984017i	$-0.556988\pi$
$157$	9.56218	−	16.5622i	0.763145	−	1.32181i	0.941222	−	0.337789i	$-0.109679\pi$
$158$	−5.66025	+	9.80385i	−0.450306	+	0.779952i
$159$		0			0
$160$	−2.92820	−	5.07180i	−0.231495	−	0.400961i
$161$	16.3923			1.29189
$162$		0			0
$163$	12.7321			0.997251			0.498626	−	0.866817i	$-0.333838\pi$
$163$	12.7321			0.997251			0.498626	+	0.866817i	$0.333838\pi$
$164$	−5.26795	−	9.12436i	−0.411358	−	0.712492i
$165$		0			0
$166$	−0.803848	+	1.39230i	−0.0623907	+	0.108064i
$167$	8.83013	−	15.2942i	0.683296	−	1.18350i	−0.290673	−	0.956822i	$-0.593879\pi$
$167$	8.83013	−	15.2942i	0.683296	−	1.18350i	0.973969	−	0.226681i	$-0.0727874\pi$
$168$		0			0
$169$	5.42820	+	9.40192i	0.417554	+	0.723225i
$170$	−2.00000			−0.153393
$171$		0			0
$172$	−0.287187			−0.0218978
$173$	−4.26795	−	7.39230i	−0.324486	−	0.562027i	0.656922	−	0.753958i	$-0.271858\pi$
$173$	−4.26795	−	7.39230i	−0.324486	−	0.562027i	−0.981408	+	0.191932i	$0.938525\pi$
$174$		0			0
$175$	2.36603	−	4.09808i	0.178855	−	0.309785i
$176$	3.07180	−	5.32051i	0.231545	−	0.401048i
$177$		0			0
$178$	−1.90192	−	3.29423i	−0.142555	−	0.246913i
$179$	8.12436			0.607243			0.303621	−	0.952793i	$-0.401804\pi$
$179$	8.12436			0.607243			0.303621	+	0.952793i	$0.401804\pi$
$180$		0			0
$181$	26.4641			1.96706			0.983531	−	0.180742i	$-0.0578498\pi$
$181$	26.4641			1.96706			0.983531	+	0.180742i	$0.0578498\pi$
$182$	2.53590	+	4.39230i	0.187973	+	0.325579i
$183$		0			0
$184$	−4.39230	+	7.60770i	−0.323805	+	0.560847i
$185$	1.36603	−	2.36603i	0.100432	−	0.173954i
$186$		0			0
$187$	−7.83013	−	13.5622i	−0.572596	−	0.991765i
$188$	12.7846			0.932413
$189$		0			0
$190$	3.26795			0.237082
$191$	4.06218	+	7.03590i	0.293929	+	0.509100i	0.974735	−	0.223364i	$-0.0717038\pi$
$191$	4.06218	+	7.03590i	0.293929	+	0.509100i	−0.680806	+	0.732464i	$0.738370\pi$
$192$		0			0
$193$	2.63397	−	4.56218i	0.189598	−	0.328393i	−0.755519	−	0.655127i	$-0.772615\pi$
$193$	2.63397	−	4.56218i	0.189598	−	0.328393i	0.945116	+	0.326735i	$0.105948\pi$
$194$	3.53590	−	6.12436i	0.253863	−	0.439703i
$195$		0			0
$196$	11.2679	+	19.5167i	0.804854	+	1.39405i
$197$	−13.8564			−0.987228			−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564			−0.987228			−0.493614	+	0.869681i	$0.664324\pi$
$198$		0			0
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$	1.26795	+	2.19615i	0.0896575	+	0.155291i
$201$		0			0
$202$	0.973721	−	1.68653i	0.0685107	−	0.118664i
$203$	7.56218	−	13.0981i	0.530761	−	0.919305i
$204$		0			0
$205$	3.59808	+	6.23205i	0.251301	+	0.435265i
$206$	−0.392305			−0.0273332
$207$		0			0
$208$	1.56922			0.108806
$209$	12.7942	+	22.1603i	0.884995	+	1.53286i
$210$		0			0
$211$	4.42820	−	7.66987i	0.304850	−	0.528016i	−0.672378	−	0.740208i	$-0.734727\pi$
$211$	4.42820	−	7.66987i	0.304850	−	0.528016i	0.977228	+	0.212192i	$0.0680603\pi$
$212$	4.92820	−	8.53590i	0.338470	−	0.586248i
$213$		0			0
$214$	3.12436	+	5.41154i	0.213577	+	0.369925i
$215$	0.196152			0.0133775
$216$		0			0
$217$	14.1962			0.963698
$218$	2.22243	+	3.84936i	0.150522	+	0.260712i
$219$		0			0
$220$	−4.19615	+	7.26795i	−0.282905	+	0.490005i
$221$	2.00000	−	3.46410i	0.134535	−	0.233021i
$222$		0			0
$223$	−8.39230	−	14.5359i	−0.561990	−	0.973396i	−0.997323	−	0.0731260i	$-0.976702\pi$
$223$	−8.39230	−	14.5359i	−0.561990	−	0.973396i	0.435332	−	0.900270i	$-0.356631\pi$
$224$	−27.7128			−1.85164
$225$		0			0
$226$	−14.0000			−0.931266
$227$	9.02628	+	15.6340i	0.599095	+	1.03766i	0.992955	+	0.118493i	$0.0378062\pi$
$227$	9.02628	+	15.6340i	0.599095	+	1.03766i	−0.393860	+	0.919171i	$0.628860\pi$
$228$		0			0
$229$	6.00000	−	10.3923i	0.396491	−	0.686743i	−0.596799	−	0.802391i	$-0.703561\pi$
$229$	6.00000	−	10.3923i	0.396491	−	0.686743i	0.993290	+	0.115648i	$0.0368944\pi$
$230$	1.26795	−	2.19615i	0.0836061	−	0.144810i
$231$		0			0
$232$	4.05256	+	7.01924i	0.266064	+	0.460836i
$233$	−28.0526			−1.83778			−0.918892	−	0.394509i	$-0.870915\pi$
$233$	−28.0526			−1.83778			−0.918892	+	0.394509i	$0.870915\pi$
$234$		0			0
$235$	−8.73205			−0.569616
$236$	−6.05256	−	10.4833i	−0.393988	−	0.682407i
$237$		0			0
$238$	−4.73205	+	8.19615i	−0.306733	+	0.531278i
$239$	0.267949	−	0.464102i	0.0173322	−	0.0300202i	−0.857229	−	0.514935i	$-0.827816\pi$
$239$	0.267949	−	0.464102i	0.0173322	−	0.0300202i	0.874561	+	0.484915i	$0.161149\pi$
$240$		0			0
$241$	8.16025	+	14.1340i	0.525648	+	0.910449i	0.999554	+	0.0298736i	$0.00951049\pi$
$241$	8.16025	+	14.1340i	0.525648	+	0.910449i	−0.473906	+	0.880576i	$0.657156\pi$
$242$	16.0000			1.02852
$243$		0			0
$244$	−5.85641			−0.374918
$245$	−7.69615	−	13.3301i	−0.491689	−	0.851631i
$246$		0			0
$247$	−3.26795	+	5.66025i	−0.207935	+	0.360153i
$248$	−3.80385	+	6.58846i	−0.241545	+	0.418367i
$249$		0			0
$250$	−0.366025	−	0.633975i	−0.0231495	−	0.0400961i
$251$	−10.3923			−0.655956			−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923			−0.655956			−0.327978	+	0.944685i	$0.606367\pi$
$252$		0			0
$253$	19.8564			1.24836
$254$	5.33975	+	9.24871i	0.335045	+	0.580316i
$255$		0			0
$256$	5.85641	−	10.1436i	0.366025	−	0.633975i
$257$	−5.19615	+	9.00000i	−0.324127	+	0.561405i	−0.981335	−	0.192304i	$-0.938404\pi$
$257$	−5.19615	+	9.00000i	−0.324127	+	0.561405i	0.657208	+	0.753709i	$0.271737\pi$
$258$		0			0
$259$	−6.46410	−	11.1962i	−0.401660	−	0.695695i
$260$	−2.14359			−0.132940
$261$		0			0
$262$	11.4115			0.705007
$263$	−10.6603	−	18.4641i	−0.657339	−	1.13855i	−0.981302	−	0.192475i	$-0.938348\pi$
$263$	−10.6603	−	18.4641i	−0.657339	−	1.13855i	0.323962	−	0.946070i	$-0.394985\pi$
$264$		0			0
$265$	−3.36603	+	5.83013i	−0.206773	+	0.358142i
$266$	7.73205	−	13.3923i	0.474082	−	0.821135i
$267$		0			0
$268$	2.53590	+	4.39230i	0.154905	+	0.268303i
$269$	10.6603			0.649967			0.324984	−	0.945720i	$-0.394641\pi$
$269$	10.6603			0.649967			0.324984	+	0.945720i	$0.394641\pi$
$270$		0			0
$271$	−2.92820			−0.177876			−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820			−0.177876			−0.0889378	+	0.996037i	$0.528347\pi$
$272$	1.46410	+	2.53590i	0.0887742	+	0.153761i
$273$		0			0
$274$	−0.928203	+	1.60770i	−0.0560748	+	0.0971244i
$275$	2.86603	−	4.96410i	0.172828	−	0.299347i
$276$		0			0
$277$	−1.90192	−	3.29423i	−0.114276	−	0.197931i	0.803214	−	0.595690i	$-0.203121\pi$
$277$	−1.90192	−	3.29423i	−0.114276	−	0.197931i	−0.917490	+	0.397759i	$0.869788\pi$
$278$	0.444864			0.0266812
$279$		0			0
$280$	12.0000			0.717137
$281$	−7.73205	−	13.3923i	−0.461255	−	0.798918i	0.537768	−	0.843093i	$-0.319268\pi$
$281$	−7.73205	−	13.3923i	−0.461255	−	0.798918i	−0.999024	+	0.0441747i	$0.985934\pi$
$282$		0			0
$283$	14.6603	−	25.3923i	0.871462	−	1.50942i	0.0109768	−	0.999940i	$-0.496506\pi$
$283$	14.6603	−	25.3923i	0.871462	−	1.50942i	0.860485	−	0.509476i	$-0.170161\pi$
$284$	−2.73205	+	4.73205i	−0.162117	+	0.280796i
$285$		0			0
$286$	3.07180	+	5.32051i	0.181639	+	0.314608i
$287$	34.0526			2.01006
$288$		0			0
$289$	−9.53590			−0.560935
$290$	−1.16987	−	2.02628i	−0.0686973	−	0.118987i
$291$		0			0
$292$	−5.60770	+	9.71281i	−0.328166	+	0.568399i
$293$	−14.3660	+	24.8827i	−0.839272	+	1.45366i	0.0512319	+	0.998687i	$0.483685\pi$
$293$	−14.3660	+	24.8827i	−0.839272	+	1.45366i	−0.890504	+	0.454975i	$0.849648\pi$
$294$		0			0
$295$	4.13397	+	7.16025i	0.240689	+	0.416886i
$296$	6.92820			0.402694
$297$		0			0
$298$	−5.85641			−0.339253
$299$	2.53590	+	4.39230i	0.146655	+	0.254014i
$300$		0			0
$301$	0.464102	−	0.803848i	0.0267504	−	0.0463330i
$302$	−1.97372	+	3.41858i	−0.113575	+	0.196717i
$303$		0			0
$304$	−2.39230	−	4.14359i	−0.137208	−	0.237651i
$305$	4.00000			0.229039
$306$		0			0
$307$	14.0526			0.802022			0.401011	−	0.916073i	$-0.368659\pi$
$307$	14.0526			0.802022			0.401011	+	0.916073i	$0.368659\pi$
$308$	19.8564	+	34.3923i	1.13142	+	1.95968i
$309$		0			0
$310$	1.09808	−	1.90192i	0.0623665	−	0.108022i
$311$	−9.86603	+	17.0885i	−0.559451	+	0.968998i	0.438091	+	0.898930i	$0.355655\pi$
$311$	−9.86603	+	17.0885i	−0.559451	+	0.968998i	−0.997542	+	0.0700670i	$0.977679\pi$
$312$		0			0
$313$	4.53590	+	7.85641i	0.256384	+	0.444070i	0.965271	−	0.261252i	$-0.0841355\pi$
$313$	4.53590	+	7.85641i	0.256384	+	0.444070i	−0.708886	+	0.705323i	$0.750802\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	−22.6410			−1.27366
$317$	−3.09808	−	5.36603i	−0.174005	−	0.301386i	0.765811	−	0.643065i	$-0.222338\pi$
$317$	−3.09808	−	5.36603i	−0.174005	−	0.301386i	−0.939817	+	0.341679i	$0.889004\pi$
$318$		0			0
$319$	9.16025	−	15.8660i	0.512876	−	0.888327i
$320$	−1.07180	+	1.85641i	−0.0599153	+	0.103776i
$321$		0			0
$322$	−6.00000	−	10.3923i	−0.334367	−	0.579141i
$323$	−12.1962			−0.678612
$324$		0			0
$325$	1.46410			0.0812137
$326$	−4.66025	−	8.07180i	−0.258108	−	0.447055i
$327$		0			0
$328$	−9.12436	+	15.8038i	−0.503808	+	0.872622i
$329$	−20.6603	+	35.7846i	−1.13904	+	1.97287i
$330$		0			0
$331$	0.232051	+	0.401924i	0.0127547	+	0.0220917i	0.872332	−	0.488913i	$-0.162607\pi$
$331$	0.232051	+	0.401924i	0.0127547	+	0.0220917i	−0.859578	+	0.511005i	$0.829273\pi$
$332$	−3.21539			−0.176467
$333$		0			0
$334$	−12.9282			−0.707400
$335$	−1.73205	−	3.00000i	−0.0946320	−	0.163908i
$336$		0			0
$337$	−13.6603	+	23.6603i	−0.744121	+	1.28886i	0.206483	+	0.978450i	$0.433798\pi$
$337$	−13.6603	+	23.6603i	−0.744121	+	1.28886i	−0.950604	+	0.310406i	$0.899535\pi$
$338$	3.97372	−	6.88269i	0.216142	−	0.374369i
$339$		0			0
$340$	−2.00000	−	3.46410i	−0.108465	−	0.187867i
$341$	17.1962			0.931224
$342$		0			0
$343$	−39.7128			−2.14429
$344$	0.248711	+	0.430781i	0.0134096	+	0.0232261i
$345$		0			0
$346$	−3.12436	+	5.41154i	−0.167966	+	0.290926i
$347$	14.2942	−	24.7583i	0.767354	−	1.32910i	−0.171638	−	0.985160i	$-0.554906\pi$
$347$	14.2942	−	24.7583i	0.767354	−	1.32910i	0.938993	−	0.343937i	$-0.111761\pi$
$348$		0			0
$349$	9.42820	+	16.3301i	0.504680	+	0.874132i	0.999985	+	0.00541263i	$0.00172290\pi$
$349$	9.42820	+	16.3301i	0.504680	+	0.874132i	−0.495305	+	0.868719i	$0.664944\pi$
$350$	−3.46410			−0.185164
$351$		0			0
$352$	−33.5692			−1.78925
$353$	−12.7583	−	22.0981i	−0.679057	−	1.17616i	−0.975265	−	0.221038i	$-0.929055\pi$
$353$	−12.7583	−	22.0981i	−0.679057	−	1.17616i	0.296208	−	0.955124i	$-0.404278\pi$
$354$		0			0
$355$	1.86603	−	3.23205i	0.0990383	−	0.171539i
$356$	3.80385	−	6.58846i	0.201604	−	0.349188i
$357$		0			0
$358$	−2.97372	−	5.15064i	−0.157166	−	0.272220i
$359$	−6.12436			−0.323231			−0.161616	−	0.986854i	$-0.551670\pi$
$359$	−6.12436			−0.323231			−0.161616	+	0.986854i	$0.551670\pi$
$360$		0			0
$361$	0.928203			0.0488528
$362$	−9.68653	−	16.7776i	−0.509113	−	0.881809i
$363$		0			0
$364$	−5.07180	+	8.78461i	−0.265834	+	0.460439i
$365$	3.83013	−	6.63397i	0.200478	−	0.347238i
$366$		0			0
$367$	−15.5885	−	27.0000i	−0.813711	−	1.40939i	−0.910250	−	0.414059i	$-0.864111\pi$
$367$	−15.5885	−	27.0000i	−0.813711	−	1.40939i	0.0965390	−	0.995329i	$-0.469223\pi$
$368$	−3.71281			−0.193544
$369$		0			0
$370$	−2.00000			−0.103975
$371$	15.9282	+	27.5885i	0.826951	+	1.43232i
$372$		0			0
$373$	−10.0263	+	17.3660i	−0.519141	+	0.899179i	0.480611	+	0.876934i	$0.340415\pi$
$373$	−10.0263	+	17.3660i	−0.519141	+	0.899179i	−0.999753	+	0.0222451i	$0.992919\pi$
$374$	−5.73205	+	9.92820i	−0.296397	+	0.513375i
$375$		0			0
$376$	−11.0718	−	19.1769i	−0.570984	−	0.988974i
$377$	4.67949			0.241006
$378$		0			0
$379$	2.39230			0.122884			0.0614422	−	0.998111i	$-0.480430\pi$
$379$	2.39230			0.122884			0.0614422	+	0.998111i	$0.480430\pi$
$380$	3.26795	+	5.66025i	0.167642	+	0.290365i
$381$		0			0
$382$	2.97372	−	5.15064i	0.152149	−	0.263529i
$383$	−1.26795	+	2.19615i	−0.0647892	+	0.112218i	−0.896600	−	0.442840i	$-0.853971\pi$
$383$	−1.26795	+	2.19615i	−0.0647892	+	0.112218i	0.831811	+	0.555059i	$0.187304\pi$
$384$		0			0
$385$	−13.5622	−	23.4904i	−0.691193	−	1.19718i
$386$	−3.85641			−0.196286
$387$		0			0
$388$	14.1436			0.718032
$389$	−13.7321	−	23.7846i	−0.696243	−	1.20593i	−0.969760	−	0.244061i	$-0.921520\pi$
$389$	−13.7321	−	23.7846i	−0.696243	−	1.20593i	0.273517	−	0.961867i	$-0.411813\pi$
$390$		0			0
$391$	−4.73205	+	8.19615i	−0.239310	+	0.414497i
$392$	19.5167	−	33.8038i	0.985740	−	1.70735i
$393$		0			0
$394$	5.07180	+	8.78461i	0.255513	+	0.442562i
$395$	15.4641			0.778083
$396$		0			0
$397$	14.3923			0.722329			0.361165	−	0.932502i	$-0.382379\pi$
$397$	14.3923			0.722329			0.361165	+	0.932502i	$0.382379\pi$
$398$	0.732051	+	1.26795i	0.0366944	+	0.0635566i
$399$		0			0
$400$	−0.535898	+	0.928203i	−0.0267949	+	0.0464102i
$401$	−12.4641	+	21.5885i	−0.622428	+	1.07808i	0.366605	+	0.930377i	$0.380520\pi$
$401$	−12.4641	+	21.5885i	−0.622428	+	1.07808i	−0.989032	+	0.147699i	$0.952813\pi$
$402$		0			0
$403$	2.19615	+	3.80385i	0.109398	+	0.189483i
$404$	3.89488			0.193778
$405$		0			0
$406$	−11.0718			−0.549484
$407$	−7.83013	−	13.5622i	−0.388125	−	0.672252i
$408$		0			0
$409$	8.92820	−	15.4641i	0.441471	−	0.764651i	−0.556328	−	0.830963i	$-0.687790\pi$
$409$	8.92820	−	15.4641i	0.441471	−	0.764651i	0.997799	+	0.0663124i	$0.0211234\pi$
$410$	2.63397	−	4.56218i	0.130083	−	0.225310i
$411$		0			0
$412$	−0.392305	−	0.679492i	−0.0193275	−	0.0334762i
$413$	39.1244			1.92518
$414$		0			0
$415$	2.19615			0.107805
$416$	−4.28719	−	7.42563i	−0.210197	−	0.364071i
$417$		0			0
$418$	9.36603	−	16.2224i	0.458107	−	0.793465i
$419$	−10.1962	+	17.6603i	−0.498115	+	0.862760i	−0.999998	−	0.00217566i	$-0.999307\pi$
$419$	−10.1962	+	17.6603i	−0.498115	+	0.862760i	0.501883	+	0.864936i	$0.332641\pi$
$420$		0			0
$421$	−16.8923	−	29.2583i	−0.823281	−	1.42596i	−0.903226	−	0.429165i	$-0.858808\pi$
$421$	−16.8923	−	29.2583i	−0.823281	−	1.42596i	0.0799458	−	0.996799i	$-0.474525\pi$
$422$	−6.48334			−0.315604
$423$		0			0
$424$	−17.0718			−0.829080
$425$	1.36603	+	2.36603i	0.0662620	+	0.114769i
$426$		0			0
$427$	9.46410	−	16.3923i	0.458000	−	0.793279i
$428$	−6.24871	+	10.8231i	−0.302043	+	0.523154i
$429$		0			0
$430$	−0.0717968	−	0.124356i	−0.00346235	−	0.00599696i
$431$	−21.3397			−1.02790			−0.513950	−	0.857820i	$-0.671818\pi$
$431$	−21.3397			−1.02790			−0.513950	+	0.857820i	$0.671818\pi$
$432$		0			0
$433$	−35.4641			−1.70430			−0.852148	−	0.523301i	$-0.824700\pi$
$433$	−35.4641			−1.70430			−0.852148	+	0.523301i	$0.824700\pi$
$434$	−5.19615	−	9.00000i	−0.249423	−	0.432014i
$435$		0			0
$436$	−4.44486	+	7.69873i	−0.212870	+	0.368702i
$437$	7.73205	−	13.3923i	0.369874	−	0.640641i
$438$		0			0
$439$	−2.69615	−	4.66987i	−0.128680	−	0.222881i	0.794485	−	0.607284i	$-0.207741\pi$
$439$	−2.69615	−	4.66987i	−0.128680	−	0.222881i	−0.923166	+	0.384403i	$0.874407\pi$
$440$	14.5359			0.692972
$441$		0			0
$442$	−2.92820			−0.139280
$443$	0.169873	+	0.294229i	0.00807091	+	0.0139792i	0.870033	−	0.492994i	$-0.164098\pi$
$443$	0.169873	+	0.294229i	0.00807091	+	0.0139792i	−0.861962	+	0.506973i	$0.830764\pi$
$444$		0			0
$445$	−2.59808	+	4.50000i	−0.123161	+	0.213320i
$446$	−6.14359	+	10.6410i	−0.290908	+	0.503867i
$447$		0			0
$448$	5.07180	+	8.78461i	0.239620	+	0.415034i
$449$	8.12436			0.383412			0.191706	−	0.981452i	$-0.438598\pi$
$449$	8.12436			0.383412			0.191706	+	0.981452i	$0.438598\pi$
$450$		0			0
$451$	41.2487			1.94233
$452$	−14.0000	−	24.2487i	−0.658505	−	1.14056i
$453$		0			0
$454$	6.60770	−	11.4449i	0.310114	−	0.537134i
$455$	3.46410	−	6.00000i	0.162400	−	0.281284i
$456$		0			0
$457$	−1.36603	−	2.36603i	−0.0639000	−	0.110678i	0.832306	−	0.554317i	$-0.187021\pi$
$457$	−1.36603	−	2.36603i	−0.0639000	−	0.110678i	−0.896206	+	0.443639i	$0.853687\pi$
$458$	−8.78461			−0.410478
$459$		0			0
$460$	5.07180			0.236474
$461$	18.5263	+	32.0885i	0.862855	+	1.49451i	0.869162	+	0.494528i	$0.164659\pi$
$461$	18.5263	+	32.0885i	0.862855	+	1.49451i	−0.00630665	+	0.999980i	$0.502007\pi$
$462$		0			0
$463$	5.19615	−	9.00000i	0.241486	−	0.418265i	−0.719652	−	0.694335i	$-0.755699\pi$
$463$	5.19615	−	9.00000i	0.241486	−	0.418265i	0.961138	+	0.276069i	$0.0890320\pi$
$464$	−1.71281	+	2.96668i	−0.0795153	+	0.137725i
$465$		0			0
$466$	10.2679	+	17.7846i	0.475654	+	0.823856i
$467$	37.3731			1.72942			0.864710	−	0.502272i	$-0.167502\pi$
$467$	37.3731			1.72942			0.864710	+	0.502272i	$0.167502\pi$
$468$		0			0
$469$	−16.3923			−0.756926
$470$	3.19615	+	5.53590i	0.147428	+	0.255352i
$471$		0			0
$472$	−10.4833	+	18.1577i	−0.482535	+	0.835775i
$473$	0.562178	−	0.973721i	0.0258490	−	0.0447717i
$474$		0			0
$475$	−2.23205	−	3.86603i	−0.102414	−	0.177385i
$476$	−18.9282			−0.867573
$477$		0			0
$478$	−0.392305			−0.0179436
$479$	5.93782	+	10.2846i	0.271306	+	0.469916i	0.969197	−	0.246289i	$-0.0792111\pi$
$479$	5.93782	+	10.2846i	0.271306	+	0.469916i	−0.697890	+	0.716204i	$0.745878\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$	5.97372	−	10.3468i	0.272096	−	0.471283i
$483$		0			0
$484$	16.0000	+	27.7128i	0.727273	+	1.25967i
$485$	−9.66025			−0.438650
$486$		0			0
$487$	24.3923			1.10532			0.552660	−	0.833407i	$-0.313613\pi$
$487$	24.3923			1.10532			0.552660	+	0.833407i	$0.313613\pi$
$488$	5.07180	+	8.78461i	0.229589	+	0.397661i
$489$		0			0
$490$	−5.63397	+	9.75833i	−0.254517	+	0.440836i
$491$	6.93782	−	12.0167i	0.313100	−	0.542304i	−0.665932	−	0.746012i	$-0.731966\pi$
$491$	6.93782	−	12.0167i	0.313100	−	0.542304i	0.979032	+	0.203708i	$0.0652993\pi$
$492$		0			0
$493$	4.36603	+	7.56218i	0.196636	+	0.340583i
$494$	4.78461			0.215270
$495$		0			0
$496$	−3.21539			−0.144375
$497$	−8.83013	−	15.2942i	−0.396085	−	0.686040i
$498$		0			0
$499$	−12.1603	+	21.0622i	−0.544368	+	0.942873i	0.454279	+	0.890860i	$0.349897\pi$
$499$	−12.1603	+	21.0622i	−0.544368	+	0.942873i	−0.998646	+	0.0520129i	$0.983436\pi$
$500$	0.732051	−	1.26795i	0.0327383	−	0.0567044i
$501$		0			0
$502$	3.80385	+	6.58846i	0.169774	+	0.294057i
$503$	−7.32051			−0.326405			−0.163203	−	0.986593i	$-0.552182\pi$
$503$	−7.32051			−0.326405			−0.163203	+	0.986593i	$0.552182\pi$
$504$		0			0
$505$	−2.66025			−0.118380
$506$	−7.26795	−	12.5885i	−0.323100	−	0.559625i
$507$		0			0
$508$	−10.6795	+	18.4974i	−0.473826	+	0.820690i
$509$	3.39230	−	5.87564i	0.150361	−	0.260433i	−0.780999	−	0.624532i	$-0.785290\pi$
$509$	3.39230	−	5.87564i	0.150361	−	0.260433i	0.931360	+	0.364099i	$0.118623\pi$
$510$		0			0
$511$	−18.1244	−	31.3923i	−0.801774	−	1.38871i
$512$	11.7128			0.517638
$513$		0			0
$514$	7.60770			0.335561
$515$	0.267949	+	0.464102i	0.0118073	+	0.0204508i
$516$		0			0
$517$	−25.0263	+	43.3468i	−1.10065	+	1.90639i
$518$	−4.73205	+	8.19615i	−0.207914	+	0.360118i
$519$		0			0
$520$	1.85641	+	3.21539i	0.0814088	+	0.141004i
$521$	19.4641			0.852738			0.426369	−	0.904549i	$-0.359793\pi$
$521$	19.4641			0.852738			0.426369	+	0.904549i	$0.359793\pi$
$522$		0			0
$523$	22.2487			0.972868			0.486434	−	0.873717i	$-0.338297\pi$
$523$	22.2487			0.972868			0.486434	+	0.873717i	$0.338297\pi$
$524$	11.4115	+	19.7654i	0.498516	+	0.863454i
$525$		0			0
$526$	−7.80385	+	13.5167i	−0.340264	+	0.589354i
$527$	−4.09808	+	7.09808i	−0.178515	+	0.309197i
$528$		0			0
$529$	5.50000	+	9.52628i	0.239130	+	0.414186i
$530$	4.92820			0.214067
$531$		0			0
$532$	30.9282			1.34091
$533$	5.26795	+	9.12436i	0.228180	+	0.395220i
$534$		0			0
$535$	4.26795	−	7.39230i	0.184520	−	0.319597i
$536$	4.39230	−	7.60770i	0.189719	−	0.328602i
$537$		0			0
$538$	−3.90192	−	6.75833i	−0.168224	−	0.291372i
$539$	−88.2295			−3.80031
$540$		0			0
$541$	−24.4641			−1.05179			−0.525897	−	0.850548i	$-0.676270\pi$
$541$	−24.4641			−1.05179			−0.525897	+	0.850548i	$0.676270\pi$
$542$	1.07180	+	1.85641i	0.0460376	+	0.0797395i
$543$		0			0
$544$	8.00000	−	13.8564i	0.342997	−	0.594089i
$545$	3.03590	−	5.25833i	0.130044	−	0.225242i
$546$		0			0
$547$	16.9282	+	29.3205i	0.723798	+	1.25365i	0.959467	+	0.281821i	$0.0909385\pi$
$547$	16.9282	+	29.3205i	0.723798	+	1.25365i	−0.235669	+	0.971833i	$0.575728\pi$
$548$	−3.71281			−0.158604
$549$		0			0
$550$	−4.19615			−0.178925
$551$	−7.13397	−	12.3564i	−0.303918	−	0.526401i
$552$		0			0
$553$	36.5885	−	63.3731i	1.55590	−	2.69490i
$554$	−1.39230	+	2.41154i	−0.0591534	+	0.102457i
$555$		0			0
$556$	0.444864	+	0.770527i	0.0188664	+	0.0326776i
$557$	−2.53590			−0.107449			−0.0537247	−	0.998556i	$-0.517109\pi$
$557$	−2.53590			−0.107449			−0.0537247	+	0.998556i	$0.517109\pi$
$558$		0			0
$559$	0.287187			0.0121467
$560$	2.53590	+	4.39230i	0.107161	+	0.185609i
$561$		0			0
$562$	−5.66025	+	9.80385i	−0.238763	+	0.413550i
$563$	8.36603	−	14.4904i	0.352586	−	0.610697i	−0.634116	−	0.773238i	$-0.718636\pi$
$563$	8.36603	−	14.4904i	0.352586	−	0.610697i	0.986702	+	0.162541i	$0.0519691\pi$
$564$		0			0
$565$	9.56218	+	16.5622i	0.402284	+	0.696776i
$566$	−21.4641			−0.902203
$567$		0			0
$568$	9.46410			0.397105
$569$	16.4545	+	28.5000i	0.689808	+	1.19478i	0.971900	+	0.235395i	$0.0756383\pi$
$569$	16.4545	+	28.5000i	0.689808	+	1.19478i	−0.282092	+	0.959387i	$0.591028\pi$
$570$		0			0
$571$	19.8923	−	34.4545i	0.832467	−	1.44188i	−0.0636091	−	0.997975i	$-0.520261\pi$
$571$	19.8923	−	34.4545i	0.832467	−	1.44188i	0.896076	−	0.443900i	$-0.146406\pi$
$572$	−6.14359	+	10.6410i	−0.256877	+	0.444923i
$573$		0			0
$574$	−12.4641	−	21.5885i	−0.520242	−	0.901085i
$575$	−3.46410			−0.144463
$576$		0			0
$577$	15.2679			0.635613			0.317807	−	0.948156i	$-0.397054\pi$
$577$	15.2679			0.635613			0.317807	+	0.948156i	$0.397054\pi$
$578$	3.49038	+	6.04552i	0.145181	+	0.251460i
$579$		0			0
$580$	2.33975	−	4.05256i	0.0971527	−	0.168273i
$581$	5.19615	−	9.00000i	0.215573	−	0.373383i
$582$		0			0
$583$	19.2942	+	33.4186i	0.799085	+	1.38406i
$584$	19.4256			0.803838
$585$		0			0
$586$	21.0333			0.868878
$587$	5.83013	+	10.0981i	0.240635	+	0.416792i	0.960895	−	0.276912i	$-0.0893110\pi$
$587$	5.83013	+	10.0981i	0.240635	+	0.416792i	−0.720260	+	0.693704i	$0.755978\pi$
$588$		0			0
$589$	6.69615	−	11.5981i	0.275910	−	0.477890i
$590$	3.02628	−	5.24167i	0.124590	−	0.215796i
$591$		0			0
$592$	1.46410	+	2.53590i	0.0601742	+	0.104225i
$593$	−0.143594			−0.00589668			−0.00294834	−	0.999996i	$-0.500938\pi$
$593$	−0.143594			−0.00589668			−0.00294834	+	0.999996i	$0.500938\pi$
$594$		0			0
$595$	12.9282			0.530005
$596$	−5.85641	−	10.1436i	−0.239888	−	0.415498i
$597$		0			0
$598$	1.85641	−	3.21539i	0.0759141	−	0.131487i
$599$	−13.5981	+	23.5526i	−0.555602	+	0.962331i	0.442254	+	0.896890i	$0.354179\pi$
$599$	−13.5981	+	23.5526i	−0.555602	+	0.962331i	−0.997856	+	0.0654417i	$0.979154\pi$
$600$		0			0
$601$	15.6244	+	27.0622i	0.637331	+	1.10389i	0.986016	+	0.166649i	$0.0532948\pi$
$601$	15.6244	+	27.0622i	0.637331	+	1.10389i	−0.348685	+	0.937240i	$0.613372\pi$
$602$	−0.679492			−0.0276940
$603$		0			0
$604$	−7.89488			−0.321238
$605$	−10.9282	−	18.9282i	−0.444295	−	0.769541i
$606$		0			0
$607$	8.09808	−	14.0263i	0.328691	−	0.569309i	−0.653562	−	0.756873i	$-0.726726\pi$
$607$	8.09808	−	14.0263i	0.328691	−	0.569309i	0.982252	+	0.187564i	$0.0600593\pi$
$608$	−13.0718	+	22.6410i	−0.530131	+	0.918214i
$609$		0			0
$610$	−1.46410	−	2.53590i	−0.0592797	−	0.102676i
$611$	−12.7846			−0.517210
$612$		0			0
$613$	1.46410			0.0591345			0.0295673	−	0.999563i	$-0.490587\pi$
$613$	1.46410			0.0591345			0.0295673	+	0.999563i	$0.490587\pi$
$614$	−5.14359	−	8.90897i	−0.207579	−	0.359537i
$615$		0			0
$616$	34.3923	−	59.5692i	1.38571	−	2.40011i
$617$	3.46410	−	6.00000i	0.139459	−	0.241551i	−0.787833	−	0.615889i	$-0.788797\pi$
$617$	3.46410	−	6.00000i	0.139459	−	0.241551i	0.927292	+	0.374338i	$0.122130\pi$
$618$		0			0
$619$	5.92820	+	10.2679i	0.238275	+	0.412704i	0.960219	−	0.279247i	$-0.0900848\pi$
$619$	5.92820	+	10.2679i	0.238275	+	0.412704i	−0.721945	+	0.691951i	$0.756752\pi$
$620$	4.39230			0.176399
$621$		0			0
$622$	14.4449			0.579186
$623$	12.2942	+	21.2942i	0.492558	+	0.853135i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	3.32051	−	5.75129i	0.132714	−	0.229868i
$627$		0			0
$628$	−14.0000	−	24.2487i	−0.558661	−	0.967629i
$629$	7.46410			0.297613
$630$		0			0
$631$	−32.7128			−1.30228			−0.651138	−	0.758959i	$-0.725708\pi$
$631$	−32.7128			−1.30228			−0.651138	+	0.758959i	$0.725708\pi$
$632$	19.6077	+	33.9615i	0.779952	+	1.35092i
$633$		0			0
$634$	−2.26795	+	3.92820i	−0.0900718	+	0.156009i
$635$	7.29423	−	12.6340i	0.289463	−	0.501364i
$636$		0			0
$637$	−11.2679	−	19.5167i	−0.446452	−	0.773278i
$638$	−13.4115			−0.530968
$639$		0			0
$640$	−10.1436			−0.400961
$641$	−4.66987	−	8.08846i	−0.184449	−	0.319475i	0.758942	−	0.651158i	$-0.225717\pi$
$641$	−4.66987	−	8.08846i	−0.184449	−	0.319475i	−0.943391	+	0.331684i	$0.892383\pi$
$642$		0			0
$643$	−20.8301	+	36.0788i	−0.821460	+	1.42281i	0.0831349	+	0.996538i	$0.473507\pi$
$643$	−20.8301	+	36.0788i	−0.821460	+	1.42281i	−0.904595	+	0.426272i	$0.859827\pi$
$644$	12.0000	−	20.7846i	0.472866	−	0.819028i
$645$		0			0
$646$	4.46410	+	7.73205i	0.175638	+	0.304213i
$647$	11.4641			0.450700			0.225350	−	0.974278i	$-0.427647\pi$
$647$	11.4641			0.450700			0.225350	+	0.974278i	$0.427647\pi$
$648$		0			0
$649$	47.3923			1.86031
$650$	−0.535898	−	0.928203i	−0.0210197	−	0.0364071i
$651$		0			0
$652$	9.32051	−	16.1436i	0.365019	−	0.632232i
$653$	8.73205	−	15.1244i	0.341712	−	0.591862i	−0.643039	−	0.765833i	$-0.722327\pi$
$653$	8.73205	−	15.1244i	0.341712	−	0.591862i	0.984751	+	0.173972i	$0.0556601\pi$
$654$		0			0
$655$	−7.79423	−	13.5000i	−0.304546	−	0.527489i
$656$	−7.71281			−0.301135
$657$		0			0
$658$	30.2487			1.17922
$659$	0.732051	+	1.26795i	0.0285167	+	0.0493923i	0.879932	−	0.475101i	$-0.157588\pi$
$659$	0.732051	+	1.26795i	0.0285167	+	0.0493923i	−0.851415	+	0.524493i	$0.824255\pi$
$660$		0			0
$661$	−9.16025	+	15.8660i	−0.356293	+	0.617117i	−0.987338	−	0.158629i	$-0.949293\pi$
$661$	−9.16025	+	15.8660i	−0.356293	+	0.617117i	0.631046	+	0.775746i	$0.282626\pi$
$662$	0.169873	−	0.294229i	0.00660230	−	0.0114355i
$663$		0			0
$664$	2.78461	+	4.82309i	0.108064	+	0.187172i
$665$	−21.1244			−0.819167
$666$		0			0
$667$	−11.0718			−0.428702
$668$	−12.9282	−	22.3923i	−0.500207	−	0.866384i
$669$		0			0
$670$	−1.26795	+	2.19615i	−0.0489852	+	0.0848448i
$671$	11.4641	−	19.8564i	0.442567	−	0.766548i
$672$		0			0
$673$	5.19615	+	9.00000i	0.200297	+	0.346925i	0.948624	−	0.316405i	$-0.102476\pi$
$673$	5.19615	+	9.00000i	0.200297	+	0.346925i	−0.748327	+	0.663330i	$0.769143\pi$
$674$	20.0000			0.770371
$675$		0			0
$676$	15.8949			0.611342
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	0.985517	−	0.169580i	$-0.0542410\pi$
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	−0.639618	+	0.768693i	$0.720908\pi$
$678$		0			0
$679$	−22.8564	+	39.5885i	−0.877148	+	1.51927i
$680$	−3.46410	+	6.00000i	−0.132842	+	0.230089i
$681$		0			0
$682$	−6.29423	−	10.9019i	−0.241018	−	0.417456i
$683$	−19.6077			−0.750268			−0.375134	−	0.926971i	$-0.622403\pi$
$683$	−19.6077			−0.750268			−0.375134	+	0.926971i	$0.622403\pi$
$684$		0			0
$685$	2.53590			0.0968917
$686$	14.5359	+	25.1769i	0.554983	+	0.961259i
$687$		0			0
$688$	−0.105118	+	0.182069i	−0.00400758	+	0.00694133i
$689$	−4.92820	+	8.53590i	−0.187750	+	0.325192i
$690$		0			0
$691$	8.85641	+	15.3397i	0.336914	+	0.583551i	0.983851	−	0.178992i	$-0.0572836\pi$
$691$	8.85641	+	15.3397i	0.336914	+	0.583551i	−0.646937	+	0.762544i	$0.723950\pi$
$692$	−12.4974			−0.475081
$693$		0			0
$694$	−20.9282			−0.794424
$695$	−0.303848	−	0.526279i	−0.0115256	−	0.0199629i
$696$		0			0
$697$	−9.83013	+	17.0263i	−0.372343	+	0.644916i
$698$	6.90192	−	11.9545i	0.261242	−	0.452484i
$699$		0			0
$700$	−3.46410	−	6.00000i	−0.130931	−	0.226779i
$701$	−20.8038			−0.785750			−0.392875	−	0.919592i	$-0.628520\pi$
$701$	−20.8038			−0.785750			−0.392875	+	0.919592i	$0.628520\pi$
$702$		0			0
$703$	−12.1962			−0.459987
$704$	6.14359	+	10.6410i	0.231545	+	0.401048i
$705$		0			0
$706$	−9.33975	+	16.1769i	−0.351506	+	0.608826i
$707$	−6.29423	+	10.9019i	−0.236719	+	0.410009i
$708$		0			0
$709$	11.2679	+	19.5167i	0.423177	+	0.732964i	0.996248	−	0.0865418i	$-0.0275816\pi$
$709$	11.2679	+	19.5167i	0.423177	+	0.732964i	−0.573072	+	0.819505i	$0.694248\pi$
$710$	−2.73205			−0.102532
$711$		0			0
$712$	−13.1769			−0.493826
$713$	−5.19615	−	9.00000i	−0.194597	−	0.337053i
$714$		0			0
$715$	4.19615	−	7.26795i	0.156927	−	0.271806i
$716$	5.94744	−	10.3013i	0.222266	−	0.384977i
$717$		0			0
$718$	2.24167	+	3.88269i	0.0836584	+	0.144901i
$719$	8.41154			0.313698			0.156849	−	0.987623i	$-0.449866\pi$
$719$	8.41154			0.313698			0.156849	+	0.987623i	$0.449866\pi$
$720$		0			0
$721$	2.53590			0.0944418
$722$	−0.339746	−	0.588457i	−0.0126440	−	0.0219001i
$723$		0			0
$724$	19.3731	−	33.5551i	0.719994	−	1.24707i
$725$	−1.59808	+	2.76795i	−0.0593511	+	0.102799i
$726$		0			0
$727$	4.19615	+	7.26795i	0.155627	+	0.269553i	0.933287	−	0.359131i	$-0.116927\pi$
$727$	4.19615	+	7.26795i	0.155627	+	0.269553i	−0.777660	+	0.628685i	$0.783594\pi$
$728$	17.5692			0.651159
$729$		0			0
$730$	−5.60770			−0.207550
$731$	0.267949	+	0.464102i	0.00991046	+	0.0171654i
$732$		0			0
$733$	−17.3923	+	30.1244i	−0.642399	+	1.11267i	0.342496	+	0.939519i	$0.388728\pi$
$733$	−17.3923	+	30.1244i	−0.642399	+	1.11267i	−0.984896	+	0.173149i	$0.944606\pi$
$734$	−11.4115	+	19.7654i	−0.421208	+	0.729553i
$735$		0			0
$736$	10.1436	+	17.5692i	0.373898	+	0.647610i
$737$	−19.8564			−0.731420
$738$		0			0
$739$	22.4641			0.826355			0.413178	−	0.910650i	$-0.364419\pi$
$739$	22.4641			0.826355			0.413178	+	0.910650i	$0.364419\pi$
$740$	−2.00000	−	3.46410i	−0.0735215	−	0.127343i
$741$		0			0
$742$	11.6603	−	20.1962i	0.428061	−	0.741424i
$743$	−9.95448	+	17.2417i	−0.365195	+	0.632536i	−0.988807	−	0.149198i	$-0.952331\pi$
$743$	−9.95448	+	17.2417i	−0.365195	+	0.632536i	0.623613	+	0.781733i	$0.285664\pi$
$744$		0			0
$745$	4.00000	+	6.92820i	0.146549	+	0.253830i
$746$	14.6795			0.537454
$747$		0			0
$748$	−22.9282			−0.838338
$749$	−20.1962	−	34.9808i	−0.737951	−	1.27817i
$750$		0			0
$751$	−24.3923	+	42.2487i	−0.890088	+	1.54168i	−0.0503191	+	0.998733i	$0.516024\pi$
$751$	−24.3923	+	42.2487i	−0.890088	+	1.54168i	−0.839769	+	0.542944i	$0.817310\pi$
$752$	4.67949	−	8.10512i	0.170644	−	0.295563i
$753$		0			0
$754$	−1.71281	−	2.96668i	−0.0623770	−	0.108040i
$755$	5.39230			0.196246
$756$		0			0
$757$	−9.17691			−0.333541			−0.166770	−	0.985996i	$-0.553334\pi$
$757$	−9.17691			−0.333541			−0.166770	+	0.985996i	$0.553334\pi$
$758$	−0.875644	−	1.51666i	−0.0318048	−	0.0550876i
$759$		0			0
$760$	5.66025	−	9.80385i	0.205319	−	0.355623i
$761$	−11.7224	+	20.3038i	−0.424938	+	0.736014i	−0.996415	−	0.0846043i	$-0.973037\pi$
$761$	−11.7224	+	20.3038i	−0.424938	+	0.736014i	0.571477	+	0.820618i	$0.306371\pi$
$762$		0			0
$763$	−14.3660	−	24.8827i	−0.520085	−	0.900814i
$764$	11.8949			0.430342
$765$		0			0
$766$	1.85641			0.0670747
$767$	6.05256	+	10.4833i	0.218545	+	0.378531i
$768$		0			0
$769$	4.76795	−	8.25833i	0.171937	−	0.297803i	−0.767160	−	0.641456i	$-0.778331\pi$
$769$	4.76795	−	8.25833i	0.171937	−	0.297803i	0.939097	+	0.343653i	$0.111664\pi$
$770$	−9.92820	+	17.1962i	−0.357788	+	0.619706i
$771$		0			0
$772$	−3.85641	−	6.67949i	−0.138795	−	0.240400i
$773$	1.51666			0.0545505			0.0272752	−	0.999628i	$-0.491317\pi$
$773$	1.51666			0.0545505			0.0272752	+	0.999628i	$0.491317\pi$
$774$		0			0
$775$	−3.00000			−0.107763
$776$	−12.2487	−	21.2154i	−0.439703	−	0.761588i
$777$		0			0
$778$	−10.0526	+	17.4115i	−0.360402	+	0.624234i
$779$	16.0622	−	27.8205i	0.575487	−	0.996773i
$780$		0			0
$781$	−10.6962	−	18.5263i	−0.382738	−	0.662922i
$782$	6.92820			0.247752
$783$		0			0
$784$	16.4974			0.589194
$785$	9.56218	+	16.5622i	0.341289	+	0.591129i
$786$		0			0
$787$	24.0263	−	41.6147i	0.856444	−	1.48341i	−0.0188542	−	0.999822i	$-0.506002\pi$
$787$	24.0263	−	41.6147i	0.856444	−	1.48341i	0.875299	−	0.483583i	$-0.160665\pi$
$788$	−10.1436	+	17.5692i	−0.361351	+	0.625878i
$789$		0			0
$790$	−5.66025	−	9.80385i	−0.201383	−	0.348805i
$791$	90.4974			3.21772
$792$		0			0
$793$	5.85641			0.207967
$794$	−5.26795	−	9.12436i	−0.186953	−	0.323811i
$795$		0			0
$796$	−1.46410	+	2.53590i	−0.0518937	+	0.0898825i
$797$	7.80385	−	13.5167i	0.276426	−	0.478785i	−0.694068	−	0.719910i	$-0.744183\pi$
$797$	7.80385	−	13.5167i	0.276426	−	0.478785i	0.970494	+	0.241125i	$0.0775164\pi$
$798$		0			0
$799$	−11.9282	−	20.6603i	−0.421989	−	0.730907i
$800$	5.85641			0.207055
$801$		0			0
$802$	18.2487			0.644384
$803$	−21.9545	−	38.0263i	−0.774757	−	1.34192i
$804$		0			0
$805$	−8.19615	+	14.1962i	−0.288876	+	0.500349i
$806$	1.60770	−	2.78461i	0.0566286	−	0.0980837i
$807$		0			0
$808$	−3.37307	−	5.84232i	−0.118664	−	0.205532i
$809$	45.4449			1.59776			0.798878	−	0.601493i	$-0.205427\pi$
$809$	45.4449			1.59776			0.798878	+	0.601493i	$0.205427\pi$
$810$		0			0
$811$	−18.4641			−0.648362			−0.324181	−	0.945995i	$-0.605089\pi$
$811$	−18.4641			−0.648362			−0.324181	+	0.945995i	$0.605089\pi$
$812$	−11.0718	−	19.1769i	−0.388544	−	0.672978i
$813$		0			0
$814$	−5.73205	+	9.92820i	−0.200908	+	0.347983i
$815$	−6.36603	+	11.0263i	−0.222992	+	0.386234i
$816$		0			0
$817$	−0.437822	−	0.758330i	−0.0153175	−	0.0265306i
$818$	−13.0718			−0.457045
$819$		0			0
$820$	10.5359			0.367930
$821$	18.8660	+	32.6769i	0.658429	+	1.14043i	0.981022	+	0.193895i	$0.0621120\pi$
$821$	18.8660	+	32.6769i	0.658429	+	1.14043i	−0.322594	+	0.946538i	$0.604555\pi$
$822$		0			0
$823$	−1.92820	+	3.33975i	−0.0672129	+	0.116416i	−0.897673	−	0.440661i	$-0.854744\pi$
$823$	−1.92820	+	3.33975i	−0.0672129	+	0.116416i	0.830461	+	0.557077i	$0.188077\pi$
$824$	−0.679492	+	1.17691i	−0.0236712	+	0.0409998i
$825$		0			0
$826$	−14.3205	−	24.8038i	−0.498274	−	0.863036i
$827$	−11.6077			−0.403639			−0.201820	−	0.979423i	$-0.564685\pi$
$827$	−11.6077			−0.403639			−0.201820	+	0.979423i	$0.564685\pi$
$828$		0			0
$829$	−23.7846			−0.826074			−0.413037	−	0.910714i	$-0.635532\pi$
$829$	−23.7846			−0.826074			−0.413037	+	0.910714i	$0.635532\pi$
$830$	−0.803848	−	1.39230i	−0.0279020	−	0.0483276i
$831$		0			0
$832$	−1.56922	+	2.71797i	−0.0544029	+	0.0942286i
$833$	21.0263	−	36.4186i	0.728517	−	1.26183i
$834$		0			0
$835$	8.83013	+	15.2942i	0.305579	+	0.529279i
$836$	37.4641			1.29572
$837$		0			0
$838$	14.9282			0.515686
$839$	−16.5981	−	28.7487i	−0.573029	−	0.992516i	−0.996253	−	0.0864904i	$-0.972435\pi$
$839$	−16.5981	−	28.7487i	−0.573029	−	0.992516i	0.423223	−	0.906025i	$-0.360899\pi$
$840$		0			0
$841$	9.39230	−	16.2679i	0.323873	−	0.560964i
$842$	−12.3660	+	21.4186i	−0.426161	+	0.738133i
$843$		0			0
$844$	−6.48334	−	11.2295i	−0.223166	−	0.386534i
$845$	−10.8564			−0.373472
$846$		0			0
$847$	−103.426			−3.55375
$848$	−3.60770	−	6.24871i	−0.123889	−	0.214582i
$849$		0			0
$850$	1.00000	−	1.73205i	0.0342997	−	0.0594089i
$851$	−4.73205	+	8.19615i	−0.162213	+	0.280960i
$852$		0			0
$853$	−5.24167	−	9.07884i	−0.179471	−	0.310854i	0.762228	−	0.647308i	$-0.224105\pi$
$853$	−5.24167	−	9.07884i	−0.179471	−	0.310854i	−0.941700	+	0.336455i	$0.890772\pi$
$854$	−13.8564			−0.474156
$855$		0			0
$856$	21.6462			0.739851
$857$	−22.2487	−	38.5359i	−0.760002	−	1.31636i	−0.942849	−	0.333220i	$-0.891865\pi$
$857$	−22.2487	−	38.5359i	−0.760002	−	1.31636i	0.182848	−	0.983141i	$-0.441469\pi$
$858$		0			0
$859$	5.50000	−	9.52628i	0.187658	−	0.325032i	−0.756811	−	0.653633i	$-0.773244\pi$
$859$	5.50000	−	9.52628i	0.187658	−	0.325032i	0.944469	+	0.328601i	$0.106577\pi$
$860$	0.143594	−	0.248711i	0.00489650	−	0.00848099i
$861$		0			0
$862$	7.81089	+	13.5289i	0.266040	+	0.460795i
$863$	−23.1244			−0.787162			−0.393581	−	0.919290i	$-0.628764\pi$
$863$	−23.1244			−0.787162			−0.393581	+	0.919290i	$0.628764\pi$
$864$		0			0
$865$	8.53590			0.290229
$866$	12.9808	+	22.4833i	0.441104	+	0.764015i
$867$		0			0
$868$	10.3923	−	18.0000i	0.352738	−	0.610960i
$869$	44.3205	−	76.7654i	1.50347	−	2.60409i
$870$		0			0
$871$	−2.53590	−	4.39230i	−0.0859256	−	0.148828i
$872$	15.3975			0.521424
$873$		0			0
$874$	−11.3205			−0.382922
$875$	2.36603	+	4.09808i	0.0799863	+	0.138540i
$876$		0			0
$877$	−16.7321	+	28.9808i	−0.565001	+	0.978611i	0.432048	+	0.901850i	$0.357791\pi$
$877$	−16.7321	+	28.9808i	−0.565001	+	0.978611i	−0.997050	+	0.0767604i	$0.975542\pi$
$878$	−1.97372	+	3.41858i	−0.0666098	+	0.115372i
$879$		0			0
$880$	3.07180	+	5.32051i	0.103550	+	0.179354i
$881$	−47.0526			−1.58524			−0.792620	−	0.609715i	$-0.791284\pi$
$881$	−47.0526			−1.58524			−0.792620	+	0.609715i	$0.791284\pi$
$882$		0			0
$883$	−30.1962			−1.01618			−0.508091	−	0.861304i	$-0.669649\pi$
$883$	−30.1962			−1.01618			−0.508091	+	0.861304i	$0.669649\pi$
$884$	−2.92820	−	5.07180i	−0.0984861	−	0.170583i
$885$		0			0
$886$	0.124356	−	0.215390i	0.00417781	−	0.00723618i
$887$	−16.9019	+	29.2750i	−0.567511	+	0.982958i	0.429300	+	0.903162i	$0.358760\pi$
$887$	−16.9019	+	29.2750i	−0.567511	+	0.982958i	−0.996811	+	0.0797961i	$0.974573\pi$
$888$		0			0
$889$	−34.5167	−	59.7846i	−1.15765	−	2.00511i
$890$	3.80385			0.127505
$891$		0			0
$892$	−24.5744			−0.822811
$893$	19.4904	+	33.7583i	0.652221	+	1.12968i
$894$		0			0
$895$	−4.06218	+	7.03590i	−0.135784	+	0.235184i
$896$	−24.0000	+	41.5692i	−0.801784	+	1.38873i
$897$		0			0
$898$	−2.97372	−	5.15064i	−0.0992343	−	0.171879i
$899$	−9.58846			−0.319793
$900$		0			0
$901$	−18.3923			−0.612737
$902$	−15.0981	−	26.1506i	−0.502711	−	0.870721i
$903$		0			0
$904$	−24.2487	+	42.0000i	−0.806500	+	1.39690i
$905$	−13.2321	+	22.9186i	−0.439848	+	0.761840i
$906$		0			0
$907$	15.0000	+	25.9808i	0.498067	+	0.862677i	0.999998	−	0.00223080i	$-0.000710087\pi$
$907$	15.0000	+	25.9808i	0.498067	+	0.862677i	−0.501931	+	0.864908i	$0.667377\pi$
$908$	26.4308			0.877136
$909$		0			0
$910$	−5.07180			−0.168128
$911$	−11.7942	−	20.4282i	−0.390760	−	0.676817i	0.601790	−	0.798655i	$-0.294455\pi$
$911$	−11.7942	−	20.4282i	−0.390760	−	0.676817i	−0.992550	+	0.121838i	$0.961121\pi$
$912$		0			0
$913$	6.29423	−	10.9019i	0.208309	−	0.360801i
$914$	−1.00000	+	1.73205i	−0.0330771	+	0.0572911i
$915$		0			0
$916$	−8.78461	−	15.2154i	−0.290252	−	0.502731i
$917$	−73.7654			−2.43595
$918$		0			0
$919$	−56.9615			−1.87899			−0.939494	−	0.342566i	$-0.888704\pi$
$919$	−56.9615			−1.87899			−0.939494	+	0.342566i	$0.888704\pi$
$920$	−4.39230	−	7.60770i	−0.144810	−	0.250818i
$921$		0			0
$922$	13.5622	−	23.4904i	0.446647	−	0.773615i
$923$	2.73205	−	4.73205i	0.0899265	−	0.155757i
$924$		0			0
$925$	1.36603	+	2.36603i	0.0449146	+	0.0777944i
$926$	−7.60770			−0.250004
$927$		0			0
$928$	18.7180			0.614447
$929$	22.1865	+	38.4282i	0.727917	+	1.26079i	0.957762	+	0.287561i	$0.0928445\pi$
$929$	22.1865	+	38.4282i	0.727917	+	1.26079i	−0.229846	+	0.973227i	$0.573822\pi$
$930$		0			0
$931$	−34.3564	+	59.5070i	−1.12599	+	1.95026i
$932$	−20.5359	+	35.5692i	−0.672676	+	1.16511i
$933$		0			0
$934$	−13.6795	−	23.6936i	−0.447607	−	0.775277i
$935$	15.6603			0.512145
$936$		0			0
$937$	−4.14359			−0.135365			−0.0676827	−	0.997707i	$-0.521561\pi$
$937$	−4.14359			−0.135365			−0.0676827	+	0.997707i	$0.521561\pi$
$938$	6.00000	+	10.3923i	0.195907	+	0.339321i
$939$		0			0
$940$	−6.39230	+	11.0718i	−0.208494	+	0.361122i
$941$	−17.5885	+	30.4641i	−0.573367	+	0.993101i	0.422850	+	0.906200i	$0.361030\pi$
$941$	−17.5885	+	30.4641i	−0.573367	+	0.993101i	−0.996217	+	0.0869015i	$0.972303\pi$
$942$		0			0
$943$	−12.4641	−	21.5885i	−0.405887	−	0.703017i
$944$	−8.86156			−0.288419
$945$		0			0
$946$	−0.823085			−0.0267608
$947$	28.8564	+	49.9808i	0.937707	+	1.62416i	0.769734	+	0.638365i	$0.220389\pi$
$947$	28.8564	+	49.9808i	0.937707	+	1.62416i	0.167973	+	0.985792i	$0.446278\pi$
$948$		0			0
$949$	5.60770	−	9.71281i	0.182033	−	0.315291i
$950$	−1.63397	+	2.83013i	−0.0530131	+	0.0918214i
$951$		0			0
$952$	16.3923	+	28.3923i	0.531278	+	0.920200i
$953$	36.3923			1.17886			0.589431	−	0.807819i	$-0.299352\pi$
$953$	36.3923			1.17886			0.589431	+	0.807819i	$0.299352\pi$
$954$		0			0
$955$	−8.12436			−0.262898
$956$	−0.392305	−	0.679492i	−0.0126880	−	0.0219763i
$957$		0			0
$958$	4.34679	−	7.52886i	0.140438	−	0.243246i
$959$	6.00000	−	10.3923i	0.193750	−	0.335585i
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$	−2.92820			−0.0944091
$963$		0			0
$964$	23.8949			0.769602
$965$	2.63397	+	4.56218i	0.0847906	+	0.146862i
$966$		0			0
$967$	12.9019	−	22.3468i	0.414898	−	0.718624i	−0.580520	−	0.814246i	$-0.697151\pi$
$967$	12.9019	−	22.3468i	0.414898	−	0.718624i	0.995418	+	0.0956219i	$0.0304840\pi$
$968$	27.7128	−	48.0000i	0.890724	−	1.54278i
$969$		0			0
$970$	3.53590	+	6.12436i	0.113531	+	0.196641i
$971$	17.4449			0.559832			0.279916	−	0.960024i	$-0.409693\pi$
$971$	17.4449			0.559832			0.279916	+	0.960024i	$0.409693\pi$
$972$		0			0
$973$	−2.87564			−0.0921889
$974$	−8.92820	−	15.4641i	−0.286078	−	0.495502i
$975$		0			0
$976$	−2.14359	+	3.71281i	−0.0686148	+	0.118844i
$977$	−2.73205	+	4.73205i	−0.0874060	+	0.151392i	−0.906414	−	0.422391i	$-0.861191\pi$
$977$	−2.73205	+	4.73205i	−0.0874060	+	0.151392i	0.819008	+	0.573782i	$0.194524\pi$
$978$		0			0
$979$	14.8923	+	25.7942i	0.475960	+	0.824387i
$980$	−22.5359			−0.719883
$981$		0			0
$982$	−10.1577			−0.324144
$983$	−24.2942	−	42.0788i	−0.774866	−	1.34211i	−0.934870	−	0.354990i	$-0.884484\pi$
$983$	−24.2942	−	42.0788i	−0.774866	−	1.34211i	0.160004	−	0.987116i	$-0.448849\pi$
$984$		0			0
$985$	6.92820	−	12.0000i	0.220751	−	0.382352i
$986$	3.19615	−	5.53590i	0.101786	−	0.176299i
$987$		0			0
$988$	4.78461	+	8.28719i	0.152219	+	0.263651i
$989$	−0.679492			−0.0216066
$990$		0			0
$991$	30.8564			0.980186			0.490093	−	0.871670i	$-0.336963\pi$
$991$	30.8564			0.980186			0.490093	+	0.871670i	$0.336963\pi$
$992$	8.78461	+	15.2154i	0.278912	+	0.483089i
$993$		0			0
$994$	−6.46410	+	11.1962i	−0.205029	+	0.355120i
$995$	1.00000	−	1.73205i	0.0317021	−	0.0549097i
$996$		0			0
$997$	−12.7583	−	22.0981i	−0.404060	−	0.699853i	0.590151	−	0.807293i	$-0.299068\pi$
$997$	−12.7583	−	22.0981i	−0.404060	−	0.699853i	−0.994212	+	0.107440i	$0.965735\pi$
$998$	17.8038			0.563571
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.l.271.1		4
3.2	odd	2		405.2.e.i.271.2		4
9.2	odd	6		405.2.e.i.136.2		4
9.4	even	3		405.2.a.g.1.2	✓	2
9.5	odd	6		405.2.a.h.1.1	yes	2
9.7	even	3	inner	405.2.e.l.136.1		4
36.23	even	6		6480.2.a.bi.1.2		2
36.31	odd	6		6480.2.a.br.1.2		2
45.4	even	6		2025.2.a.m.1.1		2
45.13	odd	12		2025.2.b.g.649.2		4
45.14	odd	6		2025.2.a.g.1.2		2
45.22	odd	12		2025.2.b.g.649.3		4
45.23	even	12		2025.2.b.h.649.3		4
45.32	even	12		2025.2.b.h.649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.2	✓	2	9.4	even	3
405.2.a.h.1.1	yes	2	9.5	odd	6
405.2.e.i.136.2		4	9.2	odd	6
405.2.e.i.271.2		4	3.2	odd	2
405.2.e.l.136.1		4	9.7	even	3	inner
405.2.e.l.271.1		4	1.1	even	1	trivial
2025.2.a.g.1.2		2	45.14	odd	6
2025.2.a.m.1.1		2	45.4	even	6
2025.2.b.g.649.2		4	45.13	odd	12
2025.2.b.g.649.3		4	45.22	odd	12
2025.2.b.h.649.2		4	45.32	even	12
2025.2.b.h.649.3		4	45.23	even	12
6480.2.a.bi.1.2		2	36.23	even	6
6480.2.a.br.1.2		2	36.31	odd	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 - 0.633975i) q^{2} +(0.732051 - 1.26795i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.36603 + 4.09808i) q^{7} -2.53590 q^{8} +0.732051 q^{10} +(2.86603 + 4.96410i) q^{11} +(-0.732051 + 1.26795i) q^{13} +(1.73205 - 3.00000i) q^{14} +(-0.535898 - 0.928203i) q^{16} -2.73205 q^{17} +4.46410 q^{19} +(0.732051 + 1.26795i) q^{20} +(2.09808 - 3.63397i) q^{22} +(1.73205 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.07180 q^{26} +6.92820 q^{28} +(-1.59808 - 2.76795i) q^{29} +(1.50000 - 2.59808i) q^{31} +(-2.92820 + 5.07180i) q^{32} +(1.00000 + 1.73205i) q^{34} -4.73205 q^{35} -2.73205 q^{37} +(-1.63397 - 2.83013i) q^{38} +(1.26795 - 2.19615i) q^{40} +(3.59808 - 6.23205i) q^{41} +(-0.0980762 - 0.169873i) q^{43} +8.39230 q^{44} -2.53590 q^{46} +(4.36603 + 7.56218i) q^{47} +(-7.69615 + 13.3301i) q^{49} +(-0.366025 + 0.633975i) q^{50} +(1.07180 + 1.85641i) q^{52} +6.73205 q^{53} -5.73205 q^{55} +(-6.00000 - 10.3923i) q^{56} +(-1.16987 + 2.02628i) q^{58} +(4.13397 - 7.16025i) q^{59} +(-2.00000 - 3.46410i) q^{61} -2.19615 q^{62} +2.14359 q^{64} +(-0.732051 - 1.26795i) q^{65} +(-1.73205 + 3.00000i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(1.73205 + 3.00000i) q^{70} -3.73205 q^{71} -7.66025 q^{73} +(1.00000 + 1.73205i) q^{74} +(3.26795 - 5.66025i) q^{76} +(-13.5622 + 23.4904i) q^{77} +(-7.73205 - 13.3923i) q^{79} +1.07180 q^{80} -5.26795 q^{82} +(-1.09808 - 1.90192i) q^{83} +(1.36603 - 2.36603i) q^{85} +(-0.0717968 + 0.124356i) q^{86} +(-7.26795 - 12.5885i) q^{88} +5.19615 q^{89} -6.92820 q^{91} +(-2.53590 - 4.39230i) q^{92} +(3.19615 - 5.53590i) q^{94} +(-2.23205 + 3.86603i) q^{95} +(4.83013 + 8.36603i) q^{97} +11.2679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34}+ \cdots + 52 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 - 4 * q^10 + 8 * q^11 + 4 * q^13 - 16 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^25 + 32 * q^26 + 4 * q^29 + 6 * q^31 + 16 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 - 10 * q^38 + 12 * q^40 + 4 * q^41 + 10 * q^43 - 8 * q^44 - 24 * q^46 + 14 * q^47 - 10 * q^49 + 2 * q^50 + 32 * q^52 + 20 * q^53 - 16 * q^55 - 24 * q^56 - 22 * q^58 + 20 * q^59 - 8 * q^61 + 12 * q^62 + 64 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^71 + 4 * q^73 + 4 * q^74 + 20 * q^76 - 30 * q^77 - 24 * q^79 + 32 * q^80 - 28 * q^82 + 6 * q^83 + 2 * q^85 - 28 * q^86 - 36 * q^88 - 24 * q^92 - 8 * q^94 - 2 * q^95 + 2 * q^97 + 52 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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