LMFDB - Embedded newform 1680.2.dx.h.31.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(31,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.dx (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,6,0,0,0,-2,0,-6,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 328 x^{8} - 658 x^{7} + 1045 x^{6} - 1270 x^{5} + 1183 x^{4} + \cdots + 13 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 328x^8 - 658x^7 + 1045x^6 - 1270x^5 + 1183x^4 - 810x^3 + 376x^2 - 104*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.4
Root			$0.500000 + 0.280541i$ of defining polynomial
Character	$\chi$	$=$	1680.31
Dual form			1680.2.dx.h.271.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 - 0.866025i) q^{3} +(0.866025 - 0.500000i) q^{5} +(-2.52722 + 0.783034i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.57422 - 2.06357i) q^{11} +6.61553i q^{13} -1.00000i q^{15} +(0.894746 + 0.516582i) q^{17} +(2.61021 + 4.52102i) q^{19} +(-0.585484 + 2.58016i) q^{21} +(2.85625 - 1.64906i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +2.30501 q^{29} +(-1.56951 + 2.71848i) q^{31} +(-3.57422 + 2.06357i) q^{33} +(-1.79712 + 1.94174i) q^{35} +(4.14373 + 7.17715i) q^{37} +(5.72922 + 3.30776i) q^{39} +7.90476i q^{41} -0.130144i q^{43} +(-0.866025 - 0.500000i) q^{45} +(5.85920 + 10.1484i) q^{47} +(5.77372 - 3.95780i) q^{49} +(0.894746 - 0.516582i) q^{51} +(0.829031 - 1.43592i) q^{53} -4.12715 q^{55} +5.22043 q^{57} +(2.28498 - 3.95771i) q^{59} +(8.22919 - 4.75113i) q^{61} +(1.94174 + 1.79712i) q^{63} +(3.30776 + 5.72922i) q^{65} +(-5.66625 - 3.27141i) q^{67} -3.29812i q^{69} -5.00499i q^{71} +(-0.175670 - 0.101423i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(10.6487 + 2.41638i) q^{77} +(-7.06366 + 4.07821i) q^{79} +(-0.500000 + 0.866025i) q^{81} -10.0544 q^{83} +1.03316 q^{85} +(1.15251 - 1.99620i) q^{87} +(-13.0711 + 7.54663i) q^{89} +(-5.18018 - 16.7189i) q^{91} +(1.56951 + 2.71848i) q^{93} +(4.52102 + 2.61021i) q^{95} +8.29245i q^{97} +4.12715i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} - 2 q^{7} - 6 q^{9} + 10 q^{19} - 4 q^{21} + 12 q^{23} + 6 q^{25} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 4 q^{35} - 10 q^{37} + 6 q^{39} + 2 q^{49} + 16 q^{53} + 20 q^{57} + 24 q^{61} - 2 q^{63}+ \cdots + 24 q^{95}+O(q^{100}) $ 12 * q + 6 * q^3 - 2 * q^7 - 6 * q^9 + 10 * q^19 - 4 * q^21 + 12 * q^23 + 6 * q^25 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 4 * q^35 - 10 * q^37 + 6 * q^39 + 2 * q^49 + 16 * q^53 + 20 * q^57 + 24 * q^61 - 2 * q^63 + 4 * q^65 - 6 * q^67 + 30 * q^73 - 6 * q^75 + 36 * q^77 + 42 * q^79 - 6 * q^81 - 64 * q^83 - 8 * q^85 + 4 * q^87 + 14 * q^91 + 2 * q^93 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i
$6$		0			0
$7$	−2.52722	+	0.783034i	−0.955201	+	0.295959i
$7$	−2.52722	+	0.783034i	−0.955201	+	0.295959i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−3.57422	−	2.06357i	−1.07767	−	0.622191i	−0.147401	−	0.989077i	$-0.547091\pi$
$11$	−3.57422	−	2.06357i	−1.07767	−	0.622191i	−0.930266	+	0.366886i	$0.880424\pi$
$12$		0			0
$13$			6.61553i			1.83482i	0.397946	+	0.917409i	$0.369723\pi$
$13$			6.61553i			1.83482i	−0.397946	+	0.917409i	$0.630277\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	0.894746	+	0.516582i	0.217008	+	0.125289i	0.604564	−	0.796557i	$-0.293347\pi$
$17$	0.894746	+	0.516582i	0.217008	+	0.125289i	−0.387556	+	0.921846i	$0.626681\pi$
$18$		0			0
$19$	2.61021	+	4.52102i	0.598824	+	1.03719i	0.992995	+	0.118157i	$0.0376986\pi$
$19$	2.61021	+	4.52102i	0.598824	+	1.03719i	−0.394171	+	0.919037i	$0.628968\pi$
$20$		0			0
$21$	−0.585484	+	2.58016i	−0.127763	+	0.563036i
$22$		0			0
$23$	2.85625	−	1.64906i	0.595570	−	0.343853i	−0.171727	−	0.985145i	$-0.554935\pi$
$23$	2.85625	−	1.64906i	0.595570	−	0.343853i	0.767297	+	0.641292i	$0.221601\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.30501			0.428030			0.214015	−	0.976830i	$-0.431346\pi$
$29$	2.30501			0.428030			0.214015	+	0.976830i	$0.431346\pi$
$30$		0			0
$31$	−1.56951	+	2.71848i	−0.281893	+	0.488253i	−0.971851	−	0.235596i	$-0.924296\pi$
$31$	−1.56951	+	2.71848i	−0.281893	+	0.488253i	0.689958	+	0.723850i	$0.257629\pi$
$32$		0			0
$33$	−3.57422	+	2.06357i	−0.622191	+	0.359222i
$34$		0			0
$35$	−1.79712	+	1.94174i	−0.303769	+	0.328214i
$36$		0			0
$37$	4.14373	+	7.17715i	0.681225	+	1.17992i	0.974607	+	0.223921i	$0.0718859\pi$
$37$	4.14373	+	7.17715i	0.681225	+	1.17992i	−0.293382	+	0.955995i	$0.594781\pi$
$38$		0			0
$39$	5.72922	+	3.30776i	0.917409	+	0.529666i
$40$		0			0
$41$			7.90476i			1.23452i	0.786761	+	0.617258i	$0.211757\pi$
$41$			7.90476i			1.23452i	−0.786761	+	0.617258i	$0.788243\pi$
$42$		0			0
$43$		−	0.130144i		−	0.0198467i	−0.999951	−	0.00992337i	$-0.996841\pi$
$43$		−	0.130144i		−	0.0198467i	0.999951	−	0.00992337i	$-0.00315876\pi$
$44$		0			0
$45$	−0.866025	−	0.500000i	−0.129099	−	0.0745356i
$46$		0			0
$47$	5.85920	+	10.1484i	0.854652	+	1.48030i	0.876967	+	0.480550i	$0.159563\pi$
$47$	5.85920	+	10.1484i	0.854652	+	1.48030i	−0.0223151	+	0.999751i	$0.507104\pi$
$48$		0			0
$49$	5.77372	−	3.95780i	0.824817	−	0.565400i
$50$		0			0
$51$	0.894746	−	0.516582i	0.125289	−	0.0723359i
$52$		0			0
$53$	0.829031	−	1.43592i	0.113876	−	0.197239i	−0.803454	−	0.595367i	$-0.797007\pi$
$53$	0.829031	−	1.43592i	0.113876	−	0.197239i	0.917330	+	0.398128i	$0.130340\pi$
$54$		0			0
$55$	−4.12715			−0.556505
$56$		0			0
$57$	5.22043			0.691463
$58$		0			0
$59$	2.28498	−	3.95771i	0.297480	−	0.515250i	−0.678079	−	0.734989i	$-0.737187\pi$
$59$	2.28498	−	3.95771i	0.297480	−	0.515250i	0.975559	+	0.219739i	$0.0705206\pi$
$60$		0			0
$61$	8.22919	−	4.75113i	1.05364	−	0.608319i	0.129974	−	0.991517i	$-0.458511\pi$
$61$	8.22919	−	4.75113i	1.05364	−	0.608319i	0.923666	+	0.383198i	$0.125177\pi$
$62$		0			0
$63$	1.94174	+	1.79712i	0.244636	+	0.226416i
$64$		0			0
$65$	3.30776	+	5.72922i	0.410278	+	0.710622i
$66$		0			0
$67$	−5.66625	−	3.27141i	−0.692243	−	0.399667i	0.112209	−	0.993685i	$-0.464207\pi$
$67$	−5.66625	−	3.27141i	−0.692243	−	0.399667i	−0.804452	+	0.594018i	$0.797541\pi$
$68$		0			0
$69$		−	3.29812i		−	0.397047i
$70$		0			0
$71$		−	5.00499i		−	0.593983i	−0.954880	−	0.296991i	$-0.904017\pi$
$71$		−	5.00499i		−	0.593983i	0.954880	−	0.296991i	$-0.0959832\pi$
$72$		0			0
$73$	−0.175670	−	0.101423i	−0.0205607	−	0.0118707i	0.489684	−	0.871900i	$-0.337112\pi$
$73$	−0.175670	−	0.101423i	−0.0205607	−	0.0118707i	−0.510245	+	0.860029i	$0.670445\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$		0			0
$77$	10.6487	+	2.41638i	1.21353	+	0.275372i
$78$		0			0
$79$	−7.06366	+	4.07821i	−0.794724	+	0.458834i	−0.841623	−	0.540066i	$-0.818399\pi$
$79$	−7.06366	+	4.07821i	−0.794724	+	0.458834i	0.0468990	+	0.998900i	$0.485066\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−10.0544			−1.10362			−0.551809	−	0.833970i	$-0.686062\pi$
$83$	−10.0544			−1.10362			−0.551809	+	0.833970i	$0.686062\pi$
$84$		0			0
$85$	1.03316			0.112062
$86$		0			0
$87$	1.15251	−	1.99620i	0.123562	−	0.214015i
$88$		0			0
$89$	−13.0711	+	7.54663i	−1.38554	+	0.799941i	−0.992809	−	0.119713i	$-0.961802\pi$
$89$	−13.0711	+	7.54663i	−1.38554	+	0.799941i	−0.392730	+	0.919654i	$0.628469\pi$
$90$		0			0
$91$	−5.18018	−	16.7189i	−0.543031	−	1.75262i
$92$		0			0
$93$	1.56951	+	2.71848i	0.162751	+	0.281893i
$94$		0			0
$95$	4.52102	+	2.61021i	0.463847	+	0.267802i
$96$		0			0
$97$			8.29245i			0.841971i	0.907067	+	0.420985i	$0.138316\pi$
$97$			8.29245i			0.841971i	−0.907067	+	0.420985i	$0.861684\pi$
$98$		0			0
$99$			4.12715i			0.414794i
$100$		0			0
$101$	−8.49886	−	4.90682i	−0.845668	−	0.488247i	0.0135186	−	0.999909i	$-0.495697\pi$
$101$	−8.49886	−	4.90682i	−0.845668	−	0.488247i	−0.859187	+	0.511662i	$0.829030\pi$
$102$		0			0
$103$	−2.77576	−	4.80775i	−0.273503	−	0.473722i	0.696253	−	0.717796i	$-0.254849\pi$
$103$	−2.77576	−	4.80775i	−0.273503	−	0.473722i	−0.969756	+	0.244075i	$0.921516\pi$
$104$		0			0
$105$	0.783034	+	2.52722i	0.0764163	+	0.246632i
$106$		0			0
$107$	9.11853	−	5.26459i	0.881522	−	0.508947i	0.0103620	−	0.999946i	$-0.496702\pi$
$107$	9.11853	−	5.26459i	0.881522	−	0.508947i	0.871160	+	0.490999i	$0.163368\pi$
$108$		0			0
$109$	−7.11565	+	12.3247i	−0.681556	+	1.18049i	0.292950	+	0.956128i	$0.405363\pi$
$109$	−7.11565	+	12.3247i	−0.681556	+	1.18049i	−0.974506	+	0.224362i	$0.927970\pi$
$110$		0			0
$111$	8.28746			0.786611
$112$		0			0
$113$	8.62883			0.811732			0.405866	−	0.913933i	$-0.366970\pi$
$113$	8.62883			0.811732			0.405866	+	0.913933i	$0.366970\pi$
$114$		0			0
$115$	1.64906	−	2.85625i	0.153776	−	0.266347i
$116$		0			0
$117$	5.72922	−	3.30776i	0.529666	−	0.305803i
$118$		0			0
$119$	−2.66572	−	0.604901i	−0.244366	−	0.0554512i
$120$		0			0
$121$	3.01668	+	5.22505i	0.274244	+	0.475004i
$122$		0			0
$123$	6.84572	+	3.95238i	0.617258	+	0.356374i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	1.15780i		−	0.102738i	−0.998680	−	0.0513692i	$-0.983641\pi$
$127$		−	1.15780i		−	0.102738i	0.998680	−	0.0513692i	$-0.0163585\pi$
$128$		0			0
$129$	−0.112708	−	0.0650719i	−0.00992337	−	0.00572926i
$130$		0			0
$131$	5.03017	+	8.71251i	0.439488	+	0.761215i	0.997650	−	0.0685165i	$-0.0218266\pi$
$131$	5.03017	+	8.71251i	0.439488	+	0.761215i	−0.558162	+	0.829732i	$0.688493\pi$
$132$		0			0
$133$	−10.1367	−	9.38175i	−0.878964	−	0.813501i
$134$		0			0
$135$	−0.866025	+	0.500000i	−0.0745356	+	0.0430331i
$136$		0			0
$137$	−4.58298	+	7.93795i	−0.391550	+	0.678185i	−0.992654	−	0.120986i	$-0.961394\pi$
$137$	−4.58298	+	7.93795i	−0.391550	+	0.678185i	0.601104	+	0.799171i	$0.294728\pi$
$138$		0			0
$139$	−0.0166158			−0.00140933			−0.000704667	−	1.00000i	$-0.500224\pi$
$139$	−0.0166158			−0.00140933			−0.000704667		1.00000i	$0.500224\pi$
$140$		0			0
$141$	11.7184			0.986867
$142$		0			0
$143$	13.6516	−	23.6453i	1.14161	−	1.97732i
$144$		0			0
$145$	1.99620	−	1.15251i	0.165775	−	0.0957105i
$146$		0			0
$147$	−0.540700	−	6.97909i	−0.0445962	−	0.575625i
$148$		0			0
$149$	3.80974	+	6.59866i	0.312106	+	0.540583i	0.978818	−	0.204732i	$-0.0656322\pi$
$149$	3.80974	+	6.59866i	0.312106	+	0.540583i	−0.666712	+	0.745315i	$0.732299\pi$
$150$		0			0
$151$	−5.01873	−	2.89757i	−0.408419	−	0.235801i	0.281691	−	0.959505i	$-0.409105\pi$
$151$	−5.01873	−	2.89757i	−0.408419	−	0.235801i	−0.690110	+	0.723704i	$0.742438\pi$
$152$		0			0
$153$		−	1.03316i		−	0.0835263i
$154$		0			0
$155$			3.13903i			0.252133i
$156$		0			0
$157$	17.8758	+	10.3206i	1.42665	+	0.823676i	0.996855	−	0.0792516i	$-0.0252530\pi$
$157$	17.8758	+	10.3206i	1.42665	+	0.823676i	0.429793	+	0.902927i	$0.358586\pi$
$158$		0			0
$159$	−0.829031	−	1.43592i	−0.0657464	−	0.113876i
$160$		0			0
$161$	−5.92712	+	6.40409i	−0.467123	+	0.504713i
$162$		0			0
$163$	−20.4148	+	11.7865i	−1.59901	+	0.923190i	−0.607334	+	0.794447i	$0.707761\pi$
$163$	−20.4148	+	11.7865i	−1.59901	+	0.923190i	−0.991678	+	0.128743i	$0.958906\pi$
$164$		0			0
$165$	−2.06357	+	3.57422i	−0.160649	+	0.278252i
$166$		0			0
$167$	19.3518			1.49749			0.748745	−	0.662858i	$-0.230657\pi$
$167$	19.3518			1.49749			0.748745	+	0.662858i	$0.230657\pi$
$168$		0			0
$169$	−30.7652			−2.36656
$170$		0			0
$171$	2.61021	−	4.52102i	0.199608	−	0.345731i
$172$		0			0
$173$	13.4292	−	7.75333i	1.02100	−	0.589475i	0.106607	−	0.994301i	$-0.466001\pi$
$173$	13.4292	−	7.75333i	1.02100	−	0.589475i	0.914394	+	0.404826i	$0.132668\pi$
$174$		0			0
$175$	−0.585484	+	2.58016i	−0.0442585	+	0.195042i
$176$		0			0
$177$	−2.28498	−	3.95771i	−0.171750	−	0.297480i
$178$		0			0
$179$	4.91542	+	2.83792i	0.367396	+	0.212116i	0.672320	−	0.740261i	$-0.265298\pi$
$179$	4.91542	+	2.83792i	0.367396	+	0.212116i	−0.304925	+	0.952377i	$0.598631\pi$
$180$		0			0
$181$		−	14.1447i		−	1.05137i	−0.850679	−	0.525685i	$-0.823809\pi$
$181$		−	14.1447i		−	1.05137i	0.850679	−	0.525685i	$-0.176191\pi$
$182$		0			0
$183$		−	9.50225i		−	0.702427i
$184$		0			0
$185$	7.17715	+	4.14373i	0.527675	+	0.304653i
$186$		0			0
$187$	−2.13201	−	3.69275i	−0.155908	−	0.270041i
$188$		0			0
$189$	2.52722	−	0.783034i	0.183828	−	0.0569573i
$190$		0			0
$191$	−17.0972	+	9.87108i	−1.23711	+	0.714246i	−0.968503	−	0.249003i	$-0.919897\pi$
$191$	−17.0972	+	9.87108i	−1.23711	+	0.714246i	−0.268608	+	0.963249i	$0.586564\pi$
$192$		0			0
$193$	−0.426237	+	0.738264i	−0.0306812	+	0.0531414i	−0.880958	−	0.473194i	$-0.843101\pi$
$193$	−0.426237	+	0.738264i	−0.0306812	+	0.0531414i	0.850277	+	0.526335i	$0.176434\pi$
$194$		0			0
$195$	6.61553			0.473748
$196$		0			0
$197$	25.9587			1.84948			0.924741	−	0.380596i	$-0.124281\pi$
$197$	25.9587			1.84948			0.924741	+	0.380596i	$0.124281\pi$
$198$		0			0
$199$	−12.7263	+	22.0425i	−0.902140	+	1.56255i	−0.0774298	+	0.996998i	$0.524671\pi$
$199$	−12.7263	+	22.0425i	−0.902140	+	1.56255i	−0.824711	+	0.565555i	$0.808662\pi$
$200$		0			0
$201$	−5.66625	+	3.27141i	−0.399667	+	0.230748i
$202$		0			0
$203$	−5.82528	+	1.80490i	−0.408855	+	0.126679i
$204$		0			0
$205$	3.95238	+	6.84572i	0.276046	+	0.478126i
$206$		0			0
$207$	−2.85625	−	1.64906i	−0.198523	−	0.114618i
$208$		0			0
$209$		−	21.5455i		−	1.49033i
$210$		0			0
$211$			17.4963i			1.20449i	0.798310	+	0.602246i	$0.205727\pi$
$211$			17.4963i			1.20449i	−0.798310	+	0.602246i	$0.794273\pi$
$212$		0			0
$213$	−4.33444	−	2.50249i	−0.296991	−	0.171468i
$214$		0			0
$215$	−0.0650719	−	0.112708i	−0.00443787	−	0.00768661i
$216$		0			0
$217$	1.83785	−	8.09919i	0.124762	−	0.549809i
$218$		0			0
$219$	−0.175670	+	0.101423i	−0.0118707	+	0.00685355i
$220$		0			0
$221$	−3.41746	+	5.91922i	−0.229883	+	0.398170i
$222$		0			0
$223$	−6.28803			−0.421078			−0.210539	−	0.977585i	$-0.567522\pi$
$223$	−6.28803			−0.421078			−0.210539	+	0.977585i	$0.567522\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	2.81939	−	4.88333i	0.187130	−	0.324118i	−0.757163	−	0.653227i	$-0.773415\pi$
$227$	2.81939	−	4.88333i	0.187130	−	0.324118i	0.944292	+	0.329109i	$0.106748\pi$
$228$		0			0
$229$	5.96603	−	3.44449i	0.394246	−	0.227618i	−0.289752	−	0.957102i	$-0.593573\pi$
$229$	5.96603	−	3.44449i	0.394246	−	0.227618i	0.683998	+	0.729483i	$0.260239\pi$
$230$		0			0
$231$	7.41699	−	8.01385i	0.488002	−	0.527272i
$232$		0			0
$233$	−9.74766	−	16.8834i	−0.638591	−	1.10607i	−0.985742	−	0.168262i	$-0.946184\pi$
$233$	−9.74766	−	16.8834i	−0.638591	−	1.10607i	0.347152	−	0.937809i	$-0.387149\pi$
$234$		0			0
$235$	10.1484	+	5.85920i	0.662011	+	0.382212i
$236$		0			0
$237$			8.15641i			0.529816i
$238$		0			0
$239$		−	6.88564i		−	0.445395i	−0.974888	−	0.222697i	$-0.928514\pi$
$239$		−	6.88564i		−	0.445395i	0.974888	−	0.222697i	$-0.0714862\pi$
$240$		0			0
$241$	−11.3287	−	6.54062i	−0.729745	−	0.421318i	0.0885842	−	0.996069i	$-0.471766\pi$
$241$	−11.3287	−	6.54062i	−0.729745	−	0.421318i	−0.818329	+	0.574750i	$0.805099\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	3.02128	−	6.31442i	0.193023	−	0.403413i
$246$		0			0
$247$	−29.9090	+	17.2680i	−1.90306	+	1.09873i
$248$		0			0
$249$	−5.02722	+	8.70741i	−0.318587	+	0.551809i
$250$		0			0
$251$	19.3042			1.21847			0.609235	−	0.792989i	$-0.291476\pi$
$251$	19.3042			1.21847			0.609235	+	0.792989i	$0.291476\pi$
$252$		0			0
$253$	−13.6118			−0.855768
$254$		0			0
$255$	0.516582	−	0.894746i	0.0323496	−	0.0560312i
$256$		0			0
$257$	−5.35258	+	3.09031i	−0.333885	+	0.192768i	−0.657564	−	0.753398i	$-0.728413\pi$
$257$	−5.35258	+	3.09031i	−0.333885	+	0.192768i	0.323680	+	0.946167i	$0.395080\pi$
$258$		0			0
$259$	−16.0921	−	14.8936i	−0.999914	−	0.925442i
$260$		0			0
$261$	−1.15251	−	1.99620i	−0.0713384	−	0.123562i
$262$		0			0
$263$	19.5925	+	11.3117i	1.20813	+	0.697512i	0.962350	−	0.271815i	$-0.0876238\pi$
$263$	19.5925	+	11.3117i	1.20813	+	0.697512i	0.245776	+	0.969327i	$0.420957\pi$
$264$		0			0
$265$		−	1.65806i		−	0.101854i
$266$		0			0
$267$			15.0933i			0.923692i
$268$		0			0
$269$	−21.9236	−	12.6576i	−1.33670	−	0.771746i	−0.350386	−	0.936605i	$-0.613950\pi$
$269$	−21.9236	−	12.6576i	−1.33670	−	0.771746i	−0.986317	+	0.164859i	$0.947283\pi$
$270$		0			0
$271$	3.77665	+	6.54135i	0.229415	+	0.397359i	0.957635	−	0.287985i	$-0.0929853\pi$
$271$	3.77665	+	6.54135i	0.229415	+	0.397359i	−0.728220	+	0.685344i	$0.759652\pi$
$272$		0			0
$273$	−17.0691	−	3.87329i	−1.03307	−	0.234422i
$274$		0			0
$275$	−3.57422	+	2.06357i	−0.215533	+	0.124438i
$276$		0			0
$277$	3.05473	−	5.29095i	0.183541	−	0.317902i	−0.759543	−	0.650457i	$-0.774577\pi$
$277$	3.05473	−	5.29095i	0.183541	−	0.317902i	0.943084	+	0.332555i	$0.107911\pi$
$278$		0			0
$279$	3.13903			0.187929
$280$		0			0
$281$	−10.6235			−0.633744			−0.316872	−	0.948468i	$-0.602633\pi$
$281$	−10.6235			−0.633744			−0.316872	+	0.948468i	$0.602633\pi$
$282$		0			0
$283$	14.9476	−	25.8901i	0.888544	−	1.53900i	0.0469479	−	0.998897i	$-0.485051\pi$
$283$	14.9476	−	25.8901i	0.888544	−	1.53900i	0.841597	−	0.540107i	$-0.181616\pi$
$284$		0			0
$285$	4.52102	−	2.61021i	0.267802	−	0.154616i
$286$		0			0
$287$	−6.18970	−	19.9771i	−0.365366	−	1.17921i
$288$		0			0
$289$	−7.96629	−	13.7980i	−0.468605	−	0.811648i
$290$		0			0
$291$	7.18147	+	4.14622i	0.420985	+	0.243056i
$292$		0			0
$293$			27.1903i			1.58847i	0.607608	+	0.794237i	$0.292129\pi$
$293$			27.1903i			1.58847i	−0.607608	+	0.794237i	$0.707871\pi$
$294$		0			0
$295$		−	4.56997i		−	0.266074i
$296$		0			0
$297$	3.57422	+	2.06357i	0.207397	+	0.119741i
$298$		0			0
$299$	10.9094	+	18.8956i	0.630907	+	1.09276i
$300$		0			0
$301$	0.101907	+	0.328902i	0.00587382	+	0.0189576i
$302$		0			0
$303$	−8.49886	+	4.90682i	−0.488247	+	0.281889i
$304$		0			0
$305$	4.75113	−	8.22919i	0.272049	−	0.471202i
$306$		0			0
$307$	−16.1475			−0.921586			−0.460793	−	0.887508i	$-0.652435\pi$
$307$	−16.1475			−0.921586			−0.460793	+	0.887508i	$0.652435\pi$
$308$		0			0
$309$	−5.55151			−0.315814
$310$		0			0
$311$	−8.65793	+	14.9960i	−0.490946	+	0.850343i	−0.999946	−	0.0104233i	$-0.996682\pi$
$311$	−8.65793	+	14.9960i	−0.490946	+	0.850343i	0.509000	+	0.860767i	$0.330015\pi$
$312$		0			0
$313$	−10.3161	+	5.95602i	−0.583102	+	0.336654i	−0.762365	−	0.647147i	$-0.775962\pi$
$313$	−10.3161	+	5.95602i	−0.583102	+	0.336654i	0.179263	+	0.983801i	$0.442629\pi$
$314$		0			0
$315$	2.58016	+	0.585484i	0.145375	+	0.0329883i
$316$		0			0
$317$	−3.96271	−	6.86362i	−0.222568	−	0.385499i	0.733019	−	0.680208i	$-0.238111\pi$
$317$	−3.96271	−	6.86362i	−0.222568	−	0.385499i	−0.955587	+	0.294709i	$0.904777\pi$
$318$		0			0
$319$	−8.23862	−	4.75657i	−0.461274	−	0.266317i
$320$		0			0
$321$		−	10.5292i		−	0.587681i
$322$		0			0
$323$			5.39356i			0.300106i
$324$		0			0
$325$	5.72922	+	3.30776i	0.317800	+	0.183482i
$326$		0			0
$327$	7.11565	+	12.3247i	0.393497	+	0.681556i
$328$		0			0
$329$	−22.7541	−	21.0594i	−1.25447	−	1.16104i
$330$		0			0
$331$	−0.704494	+	0.406740i	−0.0387225	+	0.0223564i	−0.519236	−	0.854631i	$-0.673784\pi$
$331$	−0.704494	+	0.406740i	−0.0387225	+	0.0223564i	0.480514	+	0.876987i	$0.340450\pi$
$332$		0			0
$333$	4.14373	−	7.17715i	0.227075	−	0.393306i
$334$		0			0
$335$	−6.54283			−0.357473
$336$		0			0
$337$	22.9573			1.25056			0.625282	−	0.780399i	$-0.284984\pi$
$337$	22.9573			1.25056			0.625282	+	0.780399i	$0.284984\pi$
$338$		0			0
$339$	4.31442	−	7.47279i	0.234327	−	0.405866i
$340$		0			0
$341$	11.2196	−	6.47762i	0.607574	−	0.350783i
$342$		0			0
$343$	−11.4924	+	14.5233i	−0.620530	+	0.784183i
$344$		0			0
$345$	−1.64906	−	2.85625i	−0.0887824	−	0.153776i
$346$		0			0
$347$	−8.32883	−	4.80865i	−0.447115	−	0.258142i	0.259496	−	0.965744i	$-0.416444\pi$
$347$	−8.32883	−	4.80865i	−0.447115	−	0.258142i	−0.706611	+	0.707602i	$0.749777\pi$
$348$		0			0
$349$		−	28.7303i		−	1.53790i	−0.639310	−	0.768949i	$-0.720780\pi$
$349$		−	28.7303i		−	1.53790i	0.639310	−	0.768949i	$-0.279220\pi$
$350$		0			0
$351$		−	6.61553i		−	0.353111i
$352$		0			0
$353$	−27.5718	−	15.9186i	−1.46750	−	0.847262i	−0.468163	−	0.883642i	$-0.655084\pi$
$353$	−27.5718	−	15.9186i	−1.46750	−	0.847262i	−0.999338	+	0.0363802i	$0.988417\pi$
$354$		0			0
$355$	−2.50249	−	4.33444i	−0.132819	−	0.230048i
$356$		0			0
$357$	−1.85672	+	2.00613i	−0.0982681	+	0.106176i
$358$		0			0
$359$	11.3821	−	6.57147i	0.600725	−	0.346829i	−0.168602	−	0.985684i	$-0.553925\pi$
$359$	11.3821	−	6.57147i	0.600725	−	0.346829i	0.769327	+	0.638855i	$0.220592\pi$
$360$		0			0
$361$	−4.12644	+	7.14721i	−0.217181	+	0.376169i
$362$		0			0
$363$	6.03336			0.316669
$364$		0			0
$365$	−0.202847			−0.0106175
$366$		0			0
$367$	7.22312	−	12.5108i	0.377044	−	0.653059i	−0.613587	−	0.789627i	$-0.710274\pi$
$367$	7.22312	−	12.5108i	0.377044	−	0.653059i	0.990631	+	0.136568i	$0.0436074\pi$
$368$		0			0
$369$	6.84572	−	3.95238i	0.356374	−	0.205753i
$370$		0			0
$371$	−0.970770	+	4.27806i	−0.0503998	+	0.222106i
$372$		0			0
$373$	−5.00580	−	8.67030i	−0.259191	−	0.448931i	0.706835	−	0.707379i	$-0.250122\pi$
$373$	−5.00580	−	8.67030i	−0.259191	−	0.448931i	−0.966025	+	0.258448i	$0.916789\pi$
$374$		0			0
$375$	−0.866025	−	0.500000i	−0.0447214	−	0.0258199i
$376$		0			0
$377$			15.2489i			0.785357i
$378$		0			0
$379$		−	30.6798i		−	1.57592i	−0.615729	−	0.787958i	$-0.711139\pi$
$379$		−	30.6798i		−	1.57592i	0.615729	−	0.787958i	$-0.288861\pi$
$380$		0			0
$381$	−1.00269	−	0.578901i	−0.0513692	−	0.0296580i
$382$		0			0
$383$	15.2758	+	26.4584i	0.780556	+	1.35196i	0.931618	+	0.363438i	$0.118397\pi$
$383$	15.2758	+	26.4584i	0.780556	+	1.35196i	−0.151062	+	0.988524i	$0.548269\pi$
$384$		0			0
$385$	10.4302	−	3.23170i	0.531574	−	0.164703i
$386$		0			0
$387$	−0.112708	+	0.0650719i	−0.00572926	+	0.00330779i
$388$		0			0
$389$	14.1030	−	24.4271i	0.715051	−	1.23851i	−0.247888	−	0.968789i	$-0.579737\pi$
$389$	14.1030	−	24.4271i	0.715051	−	1.23851i	0.962940	−	0.269717i	$-0.0869300\pi$
$390$		0			0
$391$	3.40750			0.172324
$392$		0			0
$393$	10.0603			0.507477
$394$		0			0
$395$	−4.07821	+	7.06366i	−0.205197	+	0.355411i
$396$		0			0
$397$	11.4781	−	6.62687i	0.576068	−	0.332593i	−0.183501	−	0.983019i	$-0.558743\pi$
$397$	11.4781	−	6.62687i	0.576068	−	0.332593i	0.759569	+	0.650427i	$0.225410\pi$
$398$		0			0
$399$	−13.1932	+	4.08777i	−0.660486	+	0.204645i
$400$		0			0
$401$	−13.5079	−	23.3964i	−0.674552	−	1.16836i	−0.976600	−	0.215065i	$-0.931004\pi$
$401$	−13.5079	−	23.3964i	−0.674552	−	1.16836i	0.302048	−	0.953293i	$-0.402330\pi$
$402$		0			0
$403$	−17.9842	−	10.3832i	−0.895856	−	0.517223i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	34.2036i		−	1.69541i
$408$		0			0
$409$	−18.6089	−	10.7438i	−0.920149	−	0.531248i	−0.0364665	−	0.999335i	$-0.511610\pi$
$409$	−18.6089	−	10.7438i	−0.920149	−	0.531248i	−0.883683	+	0.468087i	$0.844944\pi$
$410$		0			0
$411$	4.58298	+	7.93795i	0.226062	+	0.391550i
$412$		0			0
$413$	−2.67565	+	11.7912i	−0.131660	+	0.580209i
$414$		0			0
$415$	−8.70741	+	5.02722i	−0.427430	+	0.246777i
$416$		0			0
$417$	−0.00830791	+	0.0143897i	−0.000406840	+	0.000704667i
$418$		0			0
$419$	33.3478			1.62914			0.814572	−	0.580062i	$-0.196972\pi$
$419$	33.3478			1.62914			0.814572	+	0.580062i	$0.196972\pi$
$420$		0			0
$421$	14.9806			0.730110			0.365055	−	0.930986i	$-0.381050\pi$
$421$	14.9806			0.730110			0.365055	+	0.930986i	$0.381050\pi$
$422$		0			0
$423$	5.85920	−	10.1484i	0.284884	−	0.493434i
$424$		0			0
$425$	0.894746	−	0.516582i	0.0434016	−	0.0250579i
$426$		0			0
$427$	−17.0767	+	18.4509i	−0.826400	+	0.892901i
$428$		0			0
$429$	−13.6516	−	23.6453i	−0.659107	−	1.14161i
$430$		0			0
$431$	−23.9972	−	13.8548i	−1.15591	−	0.667363i	−0.205587	−	0.978639i	$-0.565910\pi$
$431$	−23.9972	−	13.8548i	−1.15591	−	0.667363i	−0.950320	+	0.311276i	$0.899244\pi$
$432$		0			0
$433$		−	20.0565i		−	0.963854i	−0.876211	−	0.481927i	$-0.839937\pi$
$433$		−	20.0565i		−	0.963854i	0.876211	−	0.481927i	$-0.160063\pi$
$434$		0			0
$435$		−	2.30501i		−	0.110517i
$436$		0			0
$437$	14.9109	+	8.60880i	0.713284	+	0.411815i
$438$		0			0
$439$	8.92508	+	15.4587i	0.425971	+	0.737803i	0.996511	−	0.0834667i	$-0.0265992\pi$
$439$	8.92508	+	15.4587i	0.425971	+	0.737803i	−0.570540	+	0.821270i	$0.693266\pi$
$440$		0			0
$441$	−6.31442	−	3.02128i	−0.300686	−	0.143871i
$442$		0			0
$443$	−13.0756	+	7.54920i	−0.621241	+	0.358673i	−0.777352	−	0.629066i	$-0.783437\pi$
$443$	−13.0756	+	7.54920i	−0.621241	+	0.358673i	0.156111	+	0.987739i	$0.450104\pi$
$444$		0			0
$445$	−7.54663	+	13.0711i	−0.357744	+	0.619631i
$446$		0			0
$447$	7.61948			0.360389
$448$		0			0
$449$	−19.3459			−0.912991			−0.456496	−	0.889726i	$-0.650896\pi$
$449$	−19.3459			−0.912991			−0.456496	+	0.889726i	$0.650896\pi$
$450$		0			0
$451$	16.3121	−	28.2533i	0.768105	−	1.33040i
$452$		0			0
$453$	−5.01873	+	2.89757i	−0.235801	+	0.136140i
$454$		0			0
$455$	−12.8456	−	11.8889i	−0.602212	−	0.557361i
$456$		0			0
$457$	16.4300	+	28.4577i	0.768565	+	1.33119i	0.938341	+	0.345711i	$0.112362\pi$
$457$	16.4300	+	28.4577i	0.768565	+	1.33119i	−0.169776	+	0.985483i	$0.554304\pi$
$458$		0			0
$459$	−0.894746	−	0.516582i	−0.0417632	−	0.0241120i
$460$		0			0
$461$		−	1.76716i		−	0.0823047i	−0.999153	−	0.0411523i	$-0.986897\pi$
$461$		−	1.76716i		−	0.0823047i	0.999153	−	0.0411523i	$-0.0131029\pi$
$462$		0			0
$463$		−	0.373594i		−	0.0173624i	−0.999962	−	0.00868118i	$-0.997237\pi$
$463$		−	0.373594i		−	0.0173624i	0.999962	−	0.00868118i	$-0.00276334\pi$
$464$		0			0
$465$	2.71848	+	1.56951i	0.126066	+	0.0727845i
$466$		0			0
$467$	−4.14402	−	7.17765i	−0.191762	−	0.332142i	0.754072	−	0.656792i	$-0.228087\pi$
$467$	−4.14402	−	7.17765i	−0.191762	−	0.332142i	−0.945834	+	0.324650i	$0.894754\pi$
$468$		0			0
$469$	16.8815	+	3.83072i	0.779516	+	0.176886i
$470$		0			0
$471$	17.8758	−	10.3206i	0.823676	−	0.475549i
$472$		0			0
$473$	−0.268561	+	0.465162i	−0.0123485	+	0.0213882i
$474$		0			0
$475$	5.22043			0.239530
$476$		0			0
$477$	−1.65806			−0.0759174
$478$		0			0
$479$	10.0176	−	17.3509i	0.457714	−	0.792783i	−0.541126	−	0.840942i	$-0.682002\pi$
$479$	10.0176	−	17.3509i	0.457714	−	0.792783i	0.998840	+	0.0481581i	$0.0153351\pi$
$480$		0			0
$481$	−47.4807	+	27.4130i	−2.16493	+	1.24992i
$482$		0			0
$483$	2.58254	+	8.33508i	0.117510	+	0.379259i
$484$		0			0
$485$	4.14622	+	7.18147i	0.188270	+	0.326094i
$486$		0			0
$487$	23.3329	+	13.4713i	1.05732	+	0.610441i	0.924688	−	0.380725i	$-0.124325\pi$
$487$	23.3329	+	13.4713i	1.05732	+	0.610441i	0.132627	+	0.991166i	$0.457659\pi$
$488$		0			0
$489$			23.5730i			1.06601i
$490$		0			0
$491$			12.8001i			0.577659i	0.957381	+	0.288830i	$0.0932661\pi$
$491$			12.8001i			0.577659i	−0.957381	+	0.288830i	$0.906734\pi$
$492$		0			0
$493$	2.06240	+	1.19073i	0.0928859	+	0.0536277i
$494$		0			0
$495$	2.06357	+	3.57422i	0.0927508	+	0.160649i
$496$		0			0
$497$	3.91907	+	12.6487i	0.175794	+	0.567373i
$498$		0			0
$499$	−9.07074	+	5.23700i	−0.406062	+	0.234440i	−0.689096	−	0.724670i	$-0.741992\pi$
$499$	−9.07074	+	5.23700i	−0.406062	+	0.234440i	0.283034	+	0.959110i	$0.408659\pi$
$500$		0			0
$501$	9.67592	−	16.7592i	0.432288	−	0.748745i
$502$		0			0
$503$	−14.1978			−0.633051			−0.316525	−	0.948584i	$-0.602516\pi$
$503$	−14.1978			−0.633051			−0.316525	+	0.948584i	$0.602516\pi$
$504$		0			0
$505$	−9.81364			−0.436701
$506$		0			0
$507$	−15.3826	+	26.6435i	−0.683166	+	1.18328i
$508$		0			0
$509$	−36.2907	+	20.9525i	−1.60856	+	0.928701i	−0.618864	+	0.785498i	$0.712407\pi$
$509$	−36.2907	+	20.9525i	−1.60856	+	0.928701i	−0.989693	+	0.143203i	$0.954260\pi$
$510$		0			0
$511$	0.523376	+	0.118764i	0.0231528	+	0.00525379i
$512$		0			0
$513$	−2.61021	−	4.52102i	−0.115244	−	0.199608i
$514$		0			0
$515$	−4.80775	−	2.77576i	−0.211855	−	0.122314i
$516$		0			0
$517$		−	48.3636i		−	2.12703i
$518$		0			0
$519$		−	15.5067i		−	0.680667i
$520$		0			0
$521$	−0.361218	−	0.208549i	−0.0158253	−	0.00913672i	0.492066	−	0.870558i	$-0.336242\pi$
$521$	−0.361218	−	0.208549i	−0.0158253	−	0.00913672i	−0.507892	+	0.861421i	$0.669575\pi$
$522$		0			0
$523$	15.4485	+	26.7576i	0.675517	+	1.17003i	0.976317	+	0.216343i	$0.0694129\pi$
$523$	15.4485	+	26.7576i	0.675517	+	1.17003i	−0.300800	+	0.953687i	$0.597254\pi$
$524$		0			0
$525$	1.94174	+	1.79712i	0.0847444	+	0.0784329i
$526$		0			0
$527$	−2.80863	+	1.62157i	−0.122346	+	0.0706365i
$528$		0			0
$529$	−6.06121	+	10.4983i	−0.263531	+	0.456449i
$530$		0			0
$531$	−4.56997			−0.198320
$532$		0			0
$533$	−52.2942			−2.26511
$534$		0			0
$535$	5.26459	−	9.11853i	0.227608	−	0.394229i
$536$		0			0
$537$	4.91542	−	2.83792i	0.212116	−	0.122465i
$538$		0			0
$539$	−28.8037	+	2.23155i	−1.24066	+	0.0961196i
$540$		0			0
$541$	2.56205	+	4.43760i	0.110151	+	0.190787i	0.915831	−	0.401564i	$-0.131533\pi$
$541$	2.56205	+	4.43760i	0.110151	+	0.190787i	−0.805680	+	0.592351i	$0.798200\pi$
$542$		0			0
$543$	−12.2497	−	7.07237i	−0.525685	−	0.303504i
$544$		0			0
$545$			14.2313i			0.609602i
$546$		0			0
$547$			6.78023i			0.289902i	0.989439	+	0.144951i	$0.0463024\pi$
$547$			6.78023i			0.289902i	−0.989439	+	0.144951i	$0.953698\pi$
$548$		0			0
$549$	−8.22919	−	4.75113i	−0.351213	−	0.202773i
$550$		0			0
$551$	6.01658	+	10.4210i	0.256315	+	0.443950i
$552$		0			0
$553$	14.6581	−	15.8376i	0.623325	−	0.673484i
$554$		0			0
$555$	7.17715	−	4.14373i	0.304653	−	0.175892i
$556$		0			0
$557$	11.5156	−	19.9455i	0.487930	−	0.845119i	−0.511974	−	0.859001i	$-0.671086\pi$
$557$	11.5156	−	19.9455i	0.487930	−	0.845119i	0.999904	+	0.0138820i	$0.00441891\pi$
$558$		0			0
$559$	0.860970			0.0364152
$560$		0			0
$561$	−4.26402			−0.180027
$562$		0			0
$563$	10.7679	−	18.6505i	0.453812	−	0.786026i	−0.544807	−	0.838562i	$-0.683397\pi$
$563$	10.7679	−	18.6505i	0.453812	−	0.786026i	0.998619	+	0.0525355i	$0.0167303\pi$
$564$		0			0
$565$	7.47279	−	4.31442i	0.314383	−	0.181509i
$566$		0			0
$567$	0.585484	−	2.58016i	0.0245880	−	0.108356i
$568$		0			0
$569$	19.8062	+	34.3053i	0.830318	+	1.43815i	0.897786	+	0.440432i	$0.145175\pi$
$569$	19.8062	+	34.3053i	0.830318	+	1.43815i	−0.0674674	+	0.997721i	$0.521492\pi$
$570$		0			0
$571$	−2.11083	−	1.21869i	−0.0883354	−	0.0510005i	0.455182	−	0.890399i	$-0.349574\pi$
$571$	−2.11083	−	1.21869i	−0.0883354	−	0.0510005i	−0.543517	+	0.839398i	$0.682908\pi$
$572$		0			0
$573$			19.7422i			0.824741i
$574$		0			0
$575$		−	3.29812i		−	0.137541i
$576$		0			0
$577$	−14.7858	−	8.53656i	−0.615539	−	0.355382i	0.159591	−	0.987183i	$-0.448982\pi$
$577$	−14.7858	−	8.53656i	−0.615539	−	0.355382i	−0.775130	+	0.631802i	$0.782316\pi$
$578$		0			0
$579$	0.426237	+	0.738264i	0.0177138	+	0.0306812i
$580$		0			0
$581$	25.4098	−	7.87297i	1.05418	−	0.326626i
$582$		0			0
$583$	−5.92627	+	3.42154i	−0.245441	+	0.141706i
$584$		0			0
$585$	3.30776	−	5.72922i	0.136759	−	0.236874i
$586$		0			0
$587$	40.6886			1.67940			0.839699	−	0.543053i	$-0.182732\pi$
$587$	40.6886			1.67940			0.839699	+	0.543053i	$0.182732\pi$
$588$		0			0
$589$	−16.3871			−0.675218
$590$		0			0
$591$	12.9794	−	22.4809i	0.533900	−	0.924741i
$592$		0			0
$593$	38.1175	−	22.0071i	1.56530	−	0.903724i	0.568591	−	0.822621i	$-0.307489\pi$
$593$	38.1175	−	22.0071i	1.56530	−	0.903724i	0.996706	−	0.0811037i	$-0.0258445\pi$
$594$		0			0
$595$	−2.61104	+	0.809002i	−0.107042	+	0.0331659i
$596$		0			0
$597$	12.7263	+	22.0425i	0.520851	+	0.902140i
$598$		0			0
$599$	41.4461	+	23.9289i	1.69344	+	0.977710i	0.951702	+	0.307025i	$0.0993334\pi$
$599$	41.4461	+	23.9289i	1.69344	+	0.977710i	0.741742	+	0.670685i	$0.234000\pi$
$600$		0			0
$601$		−	16.5232i		−	0.673995i	−0.941506	−	0.336997i	$-0.890589\pi$
$601$		−	16.5232i		−	0.673995i	0.941506	−	0.336997i	$-0.109411\pi$
$602$		0			0
$603$			6.54283i			0.266444i
$604$		0			0
$605$	5.22505	+	3.01668i	0.212428	+	0.122646i
$606$		0			0
$607$	23.3334	+	40.4147i	0.947075	+	1.64038i	0.751543	+	0.659685i	$0.229310\pi$
$607$	23.3334	+	40.4147i	0.947075	+	1.64038i	0.195532	+	0.980697i	$0.437357\pi$
$608$		0			0
$609$	−1.34955	+	5.94729i	−0.0546865	+	0.240997i
$610$		0			0
$611$	−67.1372	+	38.7617i	−2.71608	+	1.56813i
$612$		0			0
$613$	3.59308	−	6.22341i	0.145123	−	0.251361i	−0.784296	−	0.620387i	$-0.786975\pi$
$613$	3.59308	−	6.22341i	0.145123	−	0.251361i	0.929419	+	0.369026i	$0.120309\pi$
$614$		0			0
$615$	7.90476			0.318751
$616$		0			0
$617$	16.0989			0.648118			0.324059	−	0.946037i	$-0.394952\pi$
$617$	16.0989			0.648118			0.324059	+	0.946037i	$0.394952\pi$
$618$		0			0
$619$	−2.73407	+	4.73555i	−0.109892	+	0.190338i	−0.915726	−	0.401803i	$-0.868384\pi$
$619$	−2.73407	+	4.73555i	−0.109892	+	0.190338i	0.805835	+	0.592141i	$0.201717\pi$
$620$		0			0
$621$	−2.85625	+	1.64906i	−0.114618	+	0.0661745i
$622$		0			0
$623$	27.1244	−	29.3072i	1.08672	−	1.17417i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−18.6589	−	10.7727i	−0.745167	−	0.430222i
$628$		0			0
$629$			8.56230i			0.341401i
$630$		0			0
$631$		−	33.6308i		−	1.33882i	−0.742893	−	0.669410i	$-0.766547\pi$
$631$		−	33.6308i		−	1.33882i	0.742893	−	0.669410i	$-0.233453\pi$
$632$		0			0
$633$	15.1522	+	8.74813i	0.602246	+	0.347707i
$634$		0			0
$635$	−0.578901	−	1.00269i	−0.0229730	−	0.0397904i
$636$		0			0
$637$	26.1830	+	38.1962i	1.03741	+	1.51339i
$638$		0			0
$639$	−4.33444	+	2.50249i	−0.171468	+	0.0989971i
$640$		0			0
$641$	11.3261	−	19.6174i	0.447355	−	0.774842i	−0.550858	−	0.834599i	$-0.685699\pi$
$641$	11.3261	−	19.6174i	0.447355	−	0.774842i	0.998213	+	0.0597575i	$0.0190327\pi$
$642$		0			0
$643$	50.0029			1.97192			0.985962	−	0.166972i	$-0.0533989\pi$
$643$	50.0029			1.97192			0.985962	+	0.166972i	$0.0533989\pi$
$644$		0			0
$645$	−0.130144			−0.00512441
$646$		0			0
$647$	13.3724	−	23.1616i	0.525722	−	0.910578i	−0.473829	−	0.880617i	$-0.657128\pi$
$647$	13.3724	−	23.1616i	0.525722	−	0.910578i	0.999551	−	0.0299609i	$-0.00953827\pi$
$648$		0			0
$649$	−16.3341	+	9.43047i	−0.641168	+	0.370178i
$650$		0			0
$651$	−6.09518	−	5.64122i	−0.238889	−	0.221097i
$652$		0			0
$653$	−16.9276	−	29.3194i	−0.662427	−	1.14736i	−0.979976	−	0.199115i	$-0.936193\pi$
$653$	−16.9276	−	29.3194i	−0.662427	−	1.14736i	0.317549	−	0.948242i	$-0.397140\pi$
$654$		0			0
$655$	8.71251	+	5.03017i	0.340426	+	0.196545i
$656$		0			0
$657$			0.202847i			0.00791380i
$658$		0			0
$659$		−	33.0953i		−	1.28921i	−0.764516	−	0.644604i	$-0.777022\pi$
$659$		−	33.0953i		−	1.28921i	0.764516	−	0.644604i	$-0.222978\pi$
$660$		0			0
$661$	10.6420	+	6.14416i	0.413926	+	0.238980i	0.692475	−	0.721442i	$-0.256520\pi$
$661$	10.6420	+	6.14416i	0.413926	+	0.238980i	−0.278549	+	0.960422i	$0.589854\pi$
$662$		0			0
$663$	3.41746	+	5.91922i	0.132723	+	0.229883i
$664$		0			0
$665$	−13.4695	−	3.05648i	−0.522326	−	0.118525i
$666$		0			0
$667$	6.58370	−	3.80110i	0.254922	−	0.147179i
$668$		0			0
$669$	−3.14402	+	5.44560i	−0.121555	+	0.210539i
$670$		0			0
$671$	−39.2172			−1.51396
$672$		0			0
$673$	−10.7864			−0.415785			−0.207893	−	0.978152i	$-0.566660\pi$
$673$	−10.7864			−0.415785			−0.207893	+	0.978152i	$0.566660\pi$
$674$		0			0
$675$	−0.500000	+	0.866025i	−0.0192450	+	0.0333333i
$676$		0			0
$677$	26.1193	−	15.0800i	1.00385	−	0.579570i	0.0944617	−	0.995528i	$-0.469887\pi$
$677$	26.1193	−	15.0800i	1.00385	−	0.579570i	0.909384	+	0.415958i	$0.136554\pi$
$678$		0			0
$679$	−6.49327	−	20.9569i	−0.249189	−	0.804251i
$680$		0			0
$681$	−2.81939	−	4.88333i	−0.108039	−	0.187130i
$682$		0			0
$683$	−24.0260	−	13.8714i	−0.919330	−	0.530776i	−0.0359089	−	0.999355i	$-0.511433\pi$
$683$	−24.0260	−	13.8714i	−0.919330	−	0.530776i	−0.883421	+	0.468580i	$0.844766\pi$
$684$		0			0
$685$			9.16595i			0.350213i
$686$		0			0
$687$		−	6.88898i		−	0.262831i
$688$		0			0
$689$	9.49940	+	5.48448i	0.361898	+	0.208942i
$690$		0			0
$691$	4.71795	+	8.17173i	0.179479	+	0.310867i	0.941702	−	0.336447i	$-0.109225\pi$
$691$	4.71795	+	8.17173i	0.179479	+	0.310867i	−0.762223	+	0.647314i	$0.775892\pi$
$692$		0			0
$693$	−3.23170	−	10.4302i	−0.122762	−	0.396212i
$694$		0			0
$695$	−0.0143897	+	0.00830791i	−0.000545833	+	0.000315137i
$696$		0			0
$697$	−4.08346	+	7.07275i	−0.154672	+	0.267900i
$698$		0			0
$699$	−19.4953			−0.737381
$700$		0			0
$701$	46.0773			1.74031			0.870157	−	0.492774i	$-0.164017\pi$
$701$	46.0773			1.74031			0.870157	+	0.492774i	$0.164017\pi$
$702$		0			0
$703$	−21.6321	+	37.4678i	−0.815869	+	1.41313i
$704$		0			0
$705$	10.1484	−	5.85920i	0.382212	−	0.220670i
$706$		0			0
$707$	25.3207	+	5.74573i	0.952284	+	0.216091i
$708$		0			0
$709$	−10.6812	−	18.5004i	−0.401141	−	0.694797i	0.592723	−	0.805407i	$-0.298053\pi$
$709$	−10.6812	−	18.5004i	−0.401141	−	0.694797i	−0.993864	+	0.110610i	$0.964720\pi$
$710$		0			0
$711$	7.06366	+	4.07821i	0.264908	+	0.152945i
$712$		0			0
$713$			10.3529i			0.387719i
$714$		0			0
$715$		−	27.3033i		−	1.02108i
$716$		0			0
$717$	−5.96314	−	3.44282i	−0.222697	−	0.128574i
$718$		0			0
$719$	−26.1733	−	45.3334i	−0.976098	−	1.69065i	−0.676265	−	0.736659i	$-0.736402\pi$
$719$	−26.1733	−	45.3334i	−0.976098	−	1.69065i	−0.299833	−	0.953992i	$-0.596931\pi$
$720$		0			0
$721$	10.7796	+	9.97675i	0.401453	+	0.371553i
$722$		0			0
$723$	−11.3287	+	6.54062i	−0.421318	+	0.243248i
$724$		0			0
$725$	1.15251	−	1.99620i	0.0428030	−	0.0741370i
$726$		0			0
$727$	−33.2600			−1.23355			−0.616773	−	0.787141i	$-0.711560\pi$
$727$	−33.2600			−1.23355			−0.616773	+	0.787141i	$0.711560\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	0.0672299	−	0.116446i	0.00248659	−	0.00430690i
$732$		0			0
$733$	−10.8156	+	6.24440i	−0.399484	+	0.230642i	−0.686261	−	0.727355i	$-0.740749\pi$
$733$	−10.8156	+	6.24440i	−0.399484	+	0.230642i	0.286777	+	0.957997i	$0.407416\pi$
$734$		0			0
$735$	−3.95780	−	5.77372i	−0.145986	−	0.212967i
$736$		0			0
$737$	13.5016	+	23.3855i	0.497338	+	0.861415i
$738$		0			0
$739$	11.9229	+	6.88368i	0.438590	+	0.253220i	0.702999	−	0.711190i	$-0.251844\pi$
$739$	11.9229	+	6.88368i	0.438590	+	0.253220i	−0.264409	+	0.964411i	$0.585177\pi$
$740$		0			0
$741$			34.5359i			1.26871i
$742$		0			0
$743$			41.5627i			1.52479i	0.647113	+	0.762394i	$0.275976\pi$
$743$			41.5627i			1.52479i	−0.647113	+	0.762394i	$0.724024\pi$
$744$		0			0
$745$	6.59866	+	3.80974i	0.241756	+	0.139578i
$746$		0			0
$747$	5.02722	+	8.70741i	0.183936	+	0.318587i
$748$		0			0
$749$	−18.9222	+	20.4449i	−0.691403	+	0.747041i
$750$		0			0
$751$	−17.1062	+	9.87628i	−0.624215	+	0.360391i	−0.778508	−	0.627634i	$-0.784023\pi$
$751$	−17.1062	+	9.87628i	−0.624215	+	0.360391i	0.154293	+	0.988025i	$0.450690\pi$
$752$		0			0
$753$	9.65211	−	16.7179i	0.351742	−	0.609235i
$754$		0			0
$755$	−5.79513			−0.210906
$756$		0			0
$757$	−36.2853			−1.31881			−0.659406	−	0.751787i	$-0.729192\pi$
$757$	−36.2853			−1.31881			−0.659406	+	0.751787i	$0.729192\pi$
$758$		0			0
$759$	−6.80591	+	11.7882i	−0.247039	+	0.427884i
$760$		0			0
$761$	−19.4517	+	11.2304i	−0.705123	+	0.407103i	−0.809253	−	0.587461i	$-0.800128\pi$
$761$	−19.4517	+	11.2304i	−0.705123	+	0.407103i	0.104129	+	0.994564i	$0.466794\pi$
$762$		0			0
$763$	8.33221	−	36.7190i	0.301646	−	1.32932i
$764$		0			0
$765$	−0.516582	−	0.894746i	−0.0186771	−	0.0323496i
$766$		0			0
$767$	26.1823	+	15.1164i	0.945389	+	0.545821i
$768$		0			0
$769$		−	9.04388i		−	0.326131i	−0.986615	−	0.163065i	$-0.947862\pi$
$769$		−	9.04388i		−	0.326131i	0.986615	−	0.163065i	$-0.0521381\pi$
$770$		0			0
$771$			6.18062i			0.222590i
$772$		0			0
$773$	15.9583	+	9.21351i	0.573979	+	0.331387i	0.758737	−	0.651397i	$-0.225817\pi$
$773$	15.9583	+	9.21351i	0.573979	+	0.331387i	−0.184758	+	0.982784i	$0.559150\pi$
$774$		0			0
$775$	1.56951	+	2.71848i	0.0563786	+	0.0976507i
$776$		0			0
$777$	−20.9443	+	6.48936i	−0.751371	+	0.232805i
$778$		0			0
$779$	−35.7376	+	20.6331i	−1.28043	+	0.739258i
$780$		0			0
$781$	−10.3282	+	17.8889i	−0.369571	+	0.640115i
$782$		0			0
$783$	−2.30501			−0.0823744
$784$		0			0
$785$	20.6413			0.736718
$786$		0			0
$787$	23.2608	−	40.2890i	0.829160	−	1.43615i	−0.0695387	−	0.997579i	$-0.522153\pi$
$787$	23.2608	−	40.2890i	0.829160	−	1.43615i	0.898698	−	0.438567i	$-0.144514\pi$
$788$		0			0
$789$	19.5925	−	11.3117i	0.697512	−	0.402709i
$790$		0			0
$791$	−21.8070	+	6.75667i	−0.775367	+	0.240239i
$792$		0			0
$793$	31.4312	+	54.4404i	1.11615	+	1.93324i
$794$		0			0
$795$	−1.43592	−	0.829031i	−0.0509270	−	0.0294027i
$796$		0			0
$797$			33.2995i			1.17953i	0.807575	+	0.589764i	$0.200779\pi$
$797$			33.2995i			1.17953i	−0.807575	+	0.589764i	$0.799221\pi$
$798$		0			0
$799$			12.1070i			0.428316i
$800$		0			0
$801$	13.0711	+	7.54663i	0.461846	+	0.266647i
$802$		0			0
$803$	0.418589	+	0.725018i	0.0147717	+	0.0255853i
$804$		0			0
$805$	−1.93100	+	8.50966i	−0.0680587	+	0.299926i
$806$		0			0
$807$	−21.9236	+	12.6576i	−0.771746	+	0.445568i
$808$		0			0
$809$	7.79307	−	13.4980i	0.273990	−	0.474564i	−0.695890	−	0.718148i	$-0.744990\pi$
$809$	7.79307	−	13.4980i	0.273990	−	0.474564i	0.969880	+	0.243584i	$0.0783232\pi$
$810$		0			0
$811$	44.5340			1.56380			0.781900	−	0.623404i	$-0.214251\pi$
$811$	44.5340			1.56380			0.781900	+	0.623404i	$0.214251\pi$
$812$		0			0
$813$	7.55330			0.264906
$814$		0			0
$815$	−11.7865	+	20.4148i	−0.412863	+	0.715100i
$816$		0			0
$817$	0.588383	−	0.339703i	0.0205849	−	0.0118847i
$818$		0			0
$819$	−11.8889	+	12.8456i	−0.415432	+	0.448863i
$820$		0			0
$821$	−5.80712	−	10.0582i	−0.202670	−	0.351034i	0.746718	−	0.665141i	$-0.231628\pi$
$821$	−5.80712	−	10.0582i	−0.202670	−	0.351034i	−0.949388	+	0.314106i	$0.898295\pi$
$822$		0			0
$823$	−31.4145	−	18.1372i	−1.09504	−	0.632223i	−0.160128	−	0.987096i	$-0.551191\pi$
$823$	−31.4145	−	18.1372i	−1.09504	−	0.632223i	−0.934914	+	0.354874i	$0.884524\pi$
$824$		0			0
$825$			4.12715i			0.143689i
$826$		0			0
$827$			2.95704i			0.102826i	0.998677	+	0.0514131i	$0.0163725\pi$
$827$			2.95704i			0.102826i	−0.998677	+	0.0514131i	$0.983627\pi$
$828$		0			0
$829$	27.3000	+	15.7617i	0.948169	+	0.547426i	0.892512	−	0.451024i	$-0.148941\pi$
$829$	27.3000	+	15.7617i	0.948169	+	0.547426i	0.0556576	+	0.998450i	$0.482274\pi$
$830$		0			0
$831$	−3.05473	−	5.29095i	−0.105967	−	0.183541i
$832$		0			0
$833$	7.21054	−	0.558631i	0.249830	−	0.0193554i
$834$		0			0
$835$	16.7592	−	9.67592i	0.579975	−	0.334849i
$836$		0			0
$837$	1.56951	−	2.71848i	0.0542504	−	0.0939644i
$838$		0			0
$839$	−18.8148			−0.649559			−0.324779	−	0.945790i	$-0.605290\pi$
$839$	−18.8148			−0.649559			−0.324779	+	0.945790i	$0.605290\pi$
$840$		0			0
$841$	−23.6869			−0.816790
$842$		0			0
$843$	−5.31174	+	9.20020i	−0.182946	+	0.316872i
$844$		0			0
$845$	−26.6435	+	15.3826i	−0.916563	+	0.529178i
$846$		0			0
$847$	−11.7152	−	10.8427i	−0.402540	−	0.372559i
$848$		0			0
$849$	−14.9476	−	25.8901i	−0.513001	−	0.888544i
$850$		0			0
$851$	23.6711	+	13.6665i	0.811435	+	0.468482i
$852$		0			0
$853$		−	2.71110i		−	0.0928263i	−0.998922	−	0.0464132i	$-0.985221\pi$
$853$		−	2.71110i		−	0.0928263i	0.998922	−	0.0464132i	$-0.0147791\pi$
$854$		0			0
$855$		−	5.22043i		−	0.178535i
$856$		0			0
$857$	11.9290	+	6.88721i	0.407487	+	0.235263i	0.689709	−	0.724086i	$-0.257738\pi$
$857$	11.9290	+	6.88721i	0.407487	+	0.235263i	−0.282222	+	0.959349i	$0.591072\pi$
$858$		0			0
$859$	−4.44373	−	7.69676i	−0.151618	−	0.262610i	0.780204	−	0.625525i	$-0.215115\pi$
$859$	−4.44373	−	7.69676i	−0.151618	−	0.262610i	−0.931822	+	0.362914i	$0.881782\pi$
$860$		0			0
$861$	−20.3955	−	4.62811i	−0.695077	−	0.157726i
$862$		0			0
$863$	−26.4367	+	15.2632i	−0.899916	+	0.519567i	−0.877173	−	0.480174i	$-0.840573\pi$
$863$	−26.4367	+	15.2632i	−0.899916	+	0.519567i	−0.0227433	+	0.999741i	$0.507240\pi$
$864$		0			0
$865$	7.75333	−	13.4292i	0.263621	−	0.456605i
$866$		0			0
$867$	−15.9326			−0.541099
$868$		0			0
$869$	33.6627			1.14193
$870$		0			0
$871$	21.6421	−	37.4853i	0.733315	−	1.27014i
$872$		0			0
$873$	7.18147	−	4.14622i	0.243056	−	0.140328i
$874$		0			0
$875$	0.783034	+	2.52722i	0.0264714	+	0.0854357i
$876$		0			0
$877$	9.22043	+	15.9703i	0.311352	+	0.539277i	0.978655	−	0.205509i	$-0.0658849\pi$
$877$	9.22043	+	15.9703i	0.311352	+	0.539277i	−0.667303	+	0.744786i	$0.732552\pi$
$878$		0			0
$879$	23.5475	+	13.5952i	0.794237	+	0.458553i
$880$		0			0
$881$			1.25682i			0.0423434i	0.999776	+	0.0211717i	$0.00673967\pi$
$881$			1.25682i			0.0423434i	−0.999776	+	0.0211717i	$0.993260\pi$
$882$		0			0
$883$			38.3476i			1.29050i	0.763972	+	0.645249i	$0.223246\pi$
$883$			38.3476i			1.29050i	−0.763972	+	0.645249i	$0.776754\pi$
$884$		0			0
$885$	−3.95771	−	2.28498i	−0.133037	−	0.0768089i
$886$		0			0
$887$	18.5284	+	32.0921i	0.622121	+	1.07755i	0.989090	+	0.147313i	$0.0470623\pi$
$887$	18.5284	+	32.0921i	0.622121	+	1.07755i	−0.366969	+	0.930233i	$0.619604\pi$
$888$		0			0
$889$	0.906599	+	2.92603i	0.0304063	+	0.0981358i
$890$		0			0
$891$	3.57422	−	2.06357i	0.119741	−	0.0691324i
$892$		0			0
$893$	−30.5875	+	52.9792i	−1.02357	+	1.77288i
$894$		0			0
$895$	5.67583			0.189722
$896$		0			0
$897$	21.8188			0.728508
$898$		0			0
$899$	−3.61775	+	6.26613i	−0.120659	+	0.208987i
$900$		0			0
$901$	1.48354	−	0.856525i	0.0494240	−	0.0285350i
$902$		0			0
$903$	0.335791	+	0.0761972i	0.0111744	+	0.00253568i
$904$		0			0
$905$	−7.07237	−	12.2497i	−0.235093	−	0.407194i
$906$		0			0
$907$	17.1370	+	9.89407i	0.569026	+	0.328527i	0.756760	−	0.653693i	$-0.226781\pi$
$907$	17.1370	+	9.89407i	0.569026	+	0.328527i	−0.187734	+	0.982220i	$0.560114\pi$
$908$		0			0
$909$			9.81364i			0.325498i
$910$		0			0
$911$			35.6168i			1.18004i	0.807390	+	0.590018i	$0.200879\pi$
$911$			35.6168i			1.18004i	−0.807390	+	0.590018i	$0.799121\pi$
$912$		0			0
$913$	35.9368	+	20.7481i	1.18933	+	0.686662i
$914$		0			0
$915$	−4.75113	−	8.22919i	−0.157067	−	0.272049i
$916$		0			0
$917$	−19.5346	−	18.0797i	−0.645088	−	0.597043i
$918$		0			0
$919$	33.5619	−	19.3770i	1.10710	−	0.639187i	0.169027	−	0.985611i	$-0.445938\pi$
$919$	33.5619	−	19.3770i	1.10710	−	0.639187i	0.938078	+	0.346424i	$0.112604\pi$
$920$		0			0
$921$	−8.07375	+	13.9841i	−0.266039	+	0.460793i
$922$		0			0
$923$	33.1106			1.08985
$924$		0			0
$925$	8.28746			0.272490
$926$		0			0
$927$	−2.77576	+	4.80775i	−0.0911678	+	0.157907i
$928$		0			0
$929$	41.2685	−	23.8264i	1.35397	−	0.781717i	0.365171	−	0.930941i	$-0.381011\pi$
$929$	41.2685	−	23.8264i	1.35397	−	0.781717i	0.988804	+	0.149223i	$0.0476772\pi$
$930$		0			0
$931$	32.9640	+	15.7724i	1.08035	+	0.516919i
$932$		0			0
$933$	8.65793	+	14.9960i	0.283448	+	0.490946i
$934$		0			0
$935$	−3.69275	−	2.13201i	−0.120766	−	0.0697242i
$936$		0			0
$937$		−	14.8030i		−	0.483593i	−0.970327	−	0.241796i	$-0.922263\pi$
$937$		−	14.8030i		−	0.483593i	0.970327	−	0.241796i	$-0.0777366\pi$
$938$		0			0
$939$			11.9120i			0.388735i
$940$		0			0
$941$	11.2774	+	6.51104i	0.367634	+	0.212254i	0.672424	−	0.740166i	$-0.265253\pi$
$941$	11.2774	+	6.51104i	0.367634	+	0.212254i	−0.304790	+	0.952420i	$0.598586\pi$
$942$		0			0
$943$	13.0354	+	22.5780i	0.424492	+	0.735241i
$944$		0			0
$945$	1.79712	−	1.94174i	0.0584604	−	0.0631648i
$946$		0			0
$947$	−3.47305	+	2.00516i	−0.112859	+	0.0651591i	−0.555367	−	0.831605i	$-0.687422\pi$
$947$	−3.47305	+	2.00516i	−0.112859	+	0.0651591i	0.442508	+	0.896765i	$0.354089\pi$
$948$		0			0
$949$	0.670969	−	1.16215i	0.0217806	−	0.0377251i
$950$		0			0
$951$	−7.92543			−0.257000
$952$		0			0
$953$	−23.0850			−0.747796			−0.373898	−	0.927470i	$-0.621979\pi$
$953$	−23.0850			−0.747796			−0.373898	+	0.927470i	$0.621979\pi$
$954$		0			0
$955$	−9.87108	+	17.0972i	−0.319421	+	0.553253i
$956$		0			0
$957$	−8.23862	+	4.75657i	−0.266317	+	0.153758i
$958$		0			0
$959$	5.36652	−	23.6496i	0.173294	−	0.763685i
$960$		0			0
$961$	10.5732	+	18.3134i	0.341072	+	0.590755i
$962$		0			0
$963$	−9.11853	−	5.26459i	−0.293841	−	0.169649i
$964$		0			0
$965$			0.852473i			0.0274421i
$966$		0			0
$967$			9.68718i			0.311519i	0.987795	+	0.155759i	$0.0497824\pi$
$967$			9.68718i			0.311519i	−0.987795	+	0.155759i	$0.950218\pi$
$968$		0			0
$969$	4.67096	+	2.69678i	0.150053	+	0.0866330i
$970$		0			0
$971$	−11.8548	−	20.5330i	−0.380437	−	0.658937i	0.610688	−	0.791872i	$-0.290893\pi$
$971$	−11.8548	−	20.5330i	−0.380437	−	0.658937i	−0.991125	+	0.132935i	$0.957560\pi$
$972$		0			0
$973$	0.0419919	−	0.0130107i	0.00134620	−	0.000417105i
$974$		0			0
$975$	5.72922	−	3.30776i	0.183482	−	0.105933i
$976$		0			0
$977$	−4.00221	+	6.93204i	−0.128042	+	0.221776i	−0.922918	−	0.384997i	$-0.874203\pi$
$977$	−4.00221	+	6.93204i	−0.128042	+	0.221776i	0.794876	+	0.606772i	$0.207536\pi$
$978$		0			0
$979$	62.2921			1.99086
$980$		0			0
$981$	14.2313			0.454371
$982$		0			0
$983$	−0.859471	+	1.48865i	−0.0274129	+	0.0474805i	−0.879406	−	0.476072i	$-0.842060\pi$
$983$	−0.859471	+	1.48865i	−0.0274129	+	0.0474805i	0.851994	+	0.523552i	$0.175394\pi$
$984$		0			0
$985$	22.4809	−	12.9794i	0.716301	−	0.413557i
$986$		0			0
$987$	−29.6150	+	9.17590i	−0.942656	+	0.292072i
$988$		0			0
$989$	−0.214615	−	0.371724i	−0.00682436	−	0.0118201i
$990$		0			0
$991$	29.4356	+	16.9947i	0.935054	+	0.539854i	0.888407	−	0.459058i	$-0.151813\pi$
$991$	29.4356	+	16.9947i	0.935054	+	0.539854i	0.0466477	+	0.998911i	$0.485146\pi$
$992$		0			0
$993$			0.813479i			0.0258150i
$994$		0			0
$995$			25.4525i			0.806899i
$996$		0			0
$997$	45.2103	+	26.1022i	1.43182	+	0.826664i	0.997260	−	0.0739773i	$-0.0235692\pi$
$997$	45.2103	+	26.1022i	1.43182	+	0.826664i	0.434564	+	0.900641i	$0.356903\pi$
$998$		0			0
$999$	−4.14373	−	7.17715i	−0.131102	−	0.227075i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.dx.h.31.4	yes	12
4.3	odd	2		1680.2.dx.f.31.6	✓	12
7.5	odd	6		1680.2.dx.f.271.6	yes	12
28.19	even	6	inner	1680.2.dx.h.271.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.dx.f.31.6	✓	12	4.3	odd	2
1680.2.dx.f.271.6	yes	12	7.5	odd	6
1680.2.dx.h.31.4	yes	12	1.1	even	1	trivial
1680.2.dx.h.271.4	yes	12	28.19	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.dx (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(0.866025 - 0.500000i) q^{5} +(-2.52722 + 0.783034i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.57422 - 2.06357i) q^{11} +6.61553i q^{13} -1.00000i q^{15} +(0.894746 + 0.516582i) q^{17} +(2.61021 + 4.52102i) q^{19} +(-0.585484 + 2.58016i) q^{21} +(2.85625 - 1.64906i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +2.30501 q^{29} +(-1.56951 + 2.71848i) q^{31} +(-3.57422 + 2.06357i) q^{33} +(-1.79712 + 1.94174i) q^{35} +(4.14373 + 7.17715i) q^{37} +(5.72922 + 3.30776i) q^{39} +7.90476i q^{41} -0.130144i q^{43} +(-0.866025 - 0.500000i) q^{45} +(5.85920 + 10.1484i) q^{47} +(5.77372 - 3.95780i) q^{49} +(0.894746 - 0.516582i) q^{51} +(0.829031 - 1.43592i) q^{53} -4.12715 q^{55} +5.22043 q^{57} +(2.28498 - 3.95771i) q^{59} +(8.22919 - 4.75113i) q^{61} +(1.94174 + 1.79712i) q^{63} +(3.30776 + 5.72922i) q^{65} +(-5.66625 - 3.27141i) q^{67} -3.29812i q^{69} -5.00499i q^{71} +(-0.175670 - 0.101423i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(10.6487 + 2.41638i) q^{77} +(-7.06366 + 4.07821i) q^{79} +(-0.500000 + 0.866025i) q^{81} -10.0544 q^{83} +1.03316 q^{85} +(1.15251 - 1.99620i) q^{87} +(-13.0711 + 7.54663i) q^{89} +(-5.18018 - 16.7189i) q^{91} +(1.56951 + 2.71848i) q^{93} +(4.52102 + 2.61021i) q^{95} +8.29245i q^{97} +4.12715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} - 2 q^{7} - 6 q^{9} + 10 q^{19} - 4 q^{21} + 12 q^{23} + 6 q^{25} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 4 q^{35} - 10 q^{37} + 6 q^{39} + 2 q^{49} + 16 q^{53} + 20 q^{57} + 24 q^{61} - 2 q^{63}+ \cdots + 24 q^{95}+O(q^{100}) \) 12 * q + 6 * q^3 - 2 * q^7 - 6 * q^9 + 10 * q^19 - 4 * q^21 + 12 * q^23 + 6 * q^25 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 4 * q^35 - 10 * q^37 + 6 * q^39 + 2 * q^49 + 16 * q^53 + 20 * q^57 + 24 * q^61 - 2 * q^63 + 4 * q^65 - 6 * q^67 + 30 * q^73 - 6 * q^75 + 36 * q^77 + 42 * q^79 - 6 * q^81 - 64 * q^83 - 8 * q^85 + 4 * q^87 + 14 * q^91 + 2 * q^93 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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