LMFDB - Embedded newform 688.2.i.d.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(49,688)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("688.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			337.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	688.337
Dual form			688.2.i.d.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +(2.50000 - 4.33013i) q^{13} +(-0.500000 + 0.866025i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} +3.00000 q^{21} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} +(2.50000 + 4.33013i) q^{31} +3.00000 q^{35} +(4.50000 + 7.79423i) q^{37} +5.00000 q^{39} -10.0000 q^{41} +(4.00000 + 5.19615i) q^{43} +2.00000 q^{45} +8.00000 q^{47} +(-1.00000 - 1.73205i) q^{49} -3.00000 q^{51} +(2.50000 + 4.33013i) q^{53} +(-0.500000 + 0.866025i) q^{57} -12.0000 q^{59} +(6.50000 - 11.2583i) q^{61} +(-3.00000 - 5.19615i) q^{63} +5.00000 q^{65} +(-1.50000 - 2.59808i) q^{67} +(3.50000 - 6.06218i) q^{69} +(-0.500000 + 0.866025i) q^{71} +(-5.50000 + 9.52628i) q^{73} +4.00000 q^{75} +(-2.50000 + 4.33013i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.50000 + 7.79423i) q^{83} -3.00000 q^{85} -3.00000 q^{87} +(0.500000 + 0.866025i) q^{89} +(-7.50000 - 12.9904i) q^{91} +(-2.50000 + 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} -2.00000 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + 3 * q^7 + 2 * q^9	$ 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + 5 q^{13} - q^{15} - 3 q^{17} + q^{19} + 6 q^{21} - 7 q^{23} + 4 q^{25} + 10 q^{27} - 3 q^{29} + 5 q^{31} + 6 q^{35} + 9 q^{37} + 10 q^{39} - 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 2 q^{49} - 6 q^{51} + 5 q^{53} - q^{57} - 24 q^{59} + 13 q^{61} - 6 q^{63} + 10 q^{65} - 3 q^{67} + 7 q^{69} - q^{71} - 11 q^{73} + 8 q^{75} - 5 q^{79} - q^{81} + 9 q^{83} - 6 q^{85} - 6 q^{87} + q^{89} - 15 q^{91} - 5 q^{93} - q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + q^3 + q^5 + 3 * q^7 + 2 * q^9 + 5 * q^13 - q^15 - 3 * q^17 + q^19 + 6 * q^21 - 7 * q^23 + 4 * q^25 + 10 * q^27 - 3 * q^29 + 5 * q^31 + 6 * q^35 + 9 * q^37 + 10 * q^39 - 20 * q^41 + 8 * q^43 + 4 * q^45 + 16 * q^47 - 2 * q^49 - 6 * q^51 + 5 * q^53 - q^57 - 24 * q^59 + 13 * q^61 - 6 * q^63 + 10 * q^65 - 3 * q^67 + 7 * q^69 - q^71 - 11 * q^73 + 8 * q^75 - 5 * q^79 - q^81 + 9 * q^83 - 6 * q^85 - 6 * q^87 + q^89 - 15 * q^91 - 5 * q^93 - q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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	0.973494	−	0.228714i	$-0.0734519\pi$
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i	−0.684819	+	0.728714i	$0.740119\pi$
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	−0.732294	+	0.680989i	$0.761550\pi$
$6$		0			0
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	−0.429919	−	0.902867i	$-0.641458\pi$
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	0.996866	−	0.0791130i	$-0.0252088\pi$
$8$		0			0
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$	−0.500000	+	0.866025i	−0.129099	+	0.223607i
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$	3.00000			0.654654
$22$		0			0
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	−0.956967	−	0.290196i	$-0.906280\pi$
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	0.227167	−	0.973856i	$-0.427054\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	−0.971023	−	0.238987i	$-0.923185\pi$
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	0.692480	+	0.721437i	$0.256518\pi$
$30$		0			0
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	0.998322	−	0.0579057i	$-0.0184423\pi$
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	−0.549309	+	0.835619i	$0.685109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	0.952587	+	0.304266i	$0.0984111\pi$
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	−0.212792	+	0.977098i	$0.568256\pi$
$38$		0			0
$39$	5.00000			0.800641
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	4.00000	+	5.19615i	0.609994	+	0.792406i
$43$	4.00000	+	5.19615i	0.609994	+	0.792406i
$44$		0			0
$45$	2.00000			0.298142
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	2.50000	+	4.33013i	0.343401	+	0.594789i	0.985062	−	0.172200i	$-0.0550875\pi$
$53$	2.50000	+	4.33013i	0.343401	+	0.594789i	−0.641661	+	0.766989i	$0.721754\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.500000	+	0.866025i	−0.0662266	+	0.114708i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$	−3.00000	−	5.19615i	−0.377964	−	0.654654i
$64$		0			0
$65$	5.00000			0.620174
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$		0			0
$69$	3.50000	−	6.06218i	0.421350	−	0.729800i
$70$		0			0
$71$	−0.500000	+	0.866025i	−0.0593391	+	0.102778i	−0.894169	−	0.447730i	$-0.852233\pi$
$71$	−0.500000	+	0.866025i	−0.0593391	+	0.102778i	0.834830	+	0.550508i	$0.185566\pi$
$72$		0			0
$73$	−5.50000	+	9.52628i	−0.643726	+	1.11497i	0.340868	+	0.940111i	$0.389279\pi$
$73$	−5.50000	+	9.52628i	−0.643726	+	1.11497i	−0.984594	+	0.174855i	$0.944054\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	−0.971698	−	0.236225i	$-0.924090\pi$
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	0.690426	+	0.723403i	$0.257423\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	$-0.00222321\pi$
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	−0.506036	+	0.862512i	$0.668890\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$	−3.00000			−0.321634
$88$		0			0
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	0.891308	−	0.453398i	$-0.149788\pi$
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	−0.838308	+	0.545197i	$0.816455\pi$
$90$		0			0
$91$	−7.50000	−	12.9904i	−0.786214	−	1.36176i
$92$		0			0
$93$	−2.50000	+	4.33013i	−0.259238	+	0.449013i
$94$		0			0
$95$	−0.500000	+	0.866025i	−0.0512989	+	0.0888523i
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	−0.550474	−	0.834853i	$-0.685553\pi$
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	0.998240	+	0.0592978i	$0.0188862\pi$
$102$		0			0
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	0.985329	+	0.170664i	$0.0545913\pi$
$104$		0			0
$105$	1.50000	+	2.59808i	0.146385	+	0.253546i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	0.648292	−	0.761392i	$-0.275484\pi$
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	−0.983531	+	0.180741i	$0.942150\pi$
$110$		0			0
$111$	−4.50000	+	7.79423i	−0.427121	+	0.739795i
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	3.50000	−	6.06218i	0.326377	−	0.565301i
$116$		0			0
$117$	−5.00000	−	8.66025i	−0.462250	−	0.800641i
$118$		0			0
$119$	4.50000	+	7.79423i	0.412514	+	0.714496i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−5.00000	−	8.66025i	−0.450835	−	0.780869i
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	−2.50000	+	6.06218i	−0.220113	+	0.533745i
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$	3.00000			0.260133
$134$		0			0
$135$	2.50000	+	4.33013i	0.215166	+	0.372678i
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	0.998179	+	0.0603135i	$0.0192101\pi$
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	−0.446857	+	0.894606i	$0.647457\pi$
$140$		0			0
$141$	4.00000	+	6.92820i	0.336861	+	0.583460i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−3.00000			−0.249136
$146$		0			0
$147$	1.00000	−	1.73205i	0.0824786	−	0.142857i
$148$		0			0
$149$	10.5000	+	18.1865i	0.860194	+	1.48990i	0.871742	+	0.489966i	$0.162991\pi$
$149$	10.5000	+	18.1865i	0.860194	+	1.48990i	−0.0115483	+	0.999933i	$0.503676\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	3.00000	+	5.19615i	0.242536	+	0.420084i
$154$		0			0
$155$	−2.50000	+	4.33013i	−0.200805	+	0.347804i
$156$		0			0
$157$	0.500000	−	0.866025i	0.0399043	−	0.0691164i	−0.845383	−	0.534160i	$-0.820628\pi$
$157$	0.500000	−	0.866025i	0.0399043	−	0.0691164i	0.885288	+	0.465044i	$0.153961\pi$
$158$		0			0
$159$	−2.50000	+	4.33013i	−0.198263	+	0.343401i
$160$		0			0
$161$	−21.0000			−1.65503
$162$		0			0
$163$	−0.500000	+	0.866025i	−0.0391630	+	0.0678323i	−0.884943	−	0.465700i	$-0.845802\pi$
$163$	−0.500000	+	0.866025i	−0.0391630	+	0.0678323i	0.845780	+	0.533533i	$0.179136\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	0.802135	−	0.597143i	$-0.203697\pi$
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	−0.918208	+	0.396098i	$0.870364\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$	2.00000			0.152944
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$	−6.00000	−	10.3923i	−0.453557	−	0.785584i
$176$		0			0
$177$	−6.00000	−	10.3923i	−0.450988	−	0.781133i
$178$		0			0
$179$	−0.500000	+	0.866025i	−0.0373718	+	0.0647298i	−0.884106	−	0.467286i	$-0.845232\pi$
$179$	−0.500000	+	0.866025i	−0.0373718	+	0.0647298i	0.846735	+	0.532016i	$0.178565\pi$
$180$		0			0
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$	13.0000			0.960988
$184$		0			0
$185$	−4.50000	+	7.79423i	−0.330847	+	0.573043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	7.50000	−	12.9904i	0.545545	−	0.944911i
$190$		0			0
$191$	−9.50000	−	16.4545i	−0.687396	−	1.19060i	−0.972677	−	0.232161i	$-0.925420\pi$
$191$	−9.50000	−	16.4545i	−0.687396	−	1.19060i	0.285282	−	0.958444i	$-0.407913\pi$
$192$		0			0
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$		0			0
$195$	2.50000	+	4.33013i	0.179029	+	0.310087i
$196$		0			0
$197$	−5.50000	+	9.52628i	−0.391859	+	0.678719i	−0.992695	−	0.120653i	$-0.961501\pi$
$197$	−5.50000	+	9.52628i	−0.391859	+	0.678719i	0.600836	+	0.799372i	$0.294834\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	1.50000	−	2.59808i	0.105802	−	0.183254i
$202$		0			0
$203$	4.50000	+	7.79423i	0.315838	+	0.547048i
$204$		0			0
$205$	−5.00000	−	8.66025i	−0.349215	−	0.604858i
$206$		0			0
$207$	−14.0000			−0.973067
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$	−1.00000			−0.0685189
$214$		0			0
$215$	−2.50000	+	6.06218i	−0.170499	+	0.413437i
$216$		0			0
$217$	15.0000			1.01827
$218$		0			0
$219$	−11.0000			−0.743311
$220$		0			0
$221$	7.50000	+	12.9904i	0.504505	+	0.873828i
$222$		0			0
$223$	−20.0000			−1.33930			−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000			−1.33930			−0.669650	+	0.742677i	$0.733556\pi$
$224$		0			0
$225$	−4.00000	−	6.92820i	−0.266667	−	0.461880i
$226$		0			0
$227$	−3.50000	−	6.06218i	−0.232303	−	0.402361i	0.726182	−	0.687502i	$-0.241293\pi$
$227$	−3.50000	−	6.06218i	−0.232303	−	0.402361i	−0.958485	+	0.285141i	$0.907959\pi$
$228$		0			0
$229$	4.50000	−	7.79423i	0.297368	−	0.515057i	−0.678165	−	0.734910i	$-0.737224\pi$
$229$	4.50000	−	7.79423i	0.297368	−	0.515057i	0.975533	+	0.219853i	$0.0705577\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	−0.680135	−	0.733087i	$-0.738079\pi$
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	0.974939	+	0.222470i	$0.0714120\pi$
$234$		0			0
$235$	4.00000	+	6.92820i	0.260931	+	0.451946i
$236$		0			0
$237$	−5.00000			−0.324785
$238$		0			0
$239$	12.5000	+	21.6506i	0.808558	+	1.40046i	0.913863	+	0.406023i	$0.133085\pi$
$239$	12.5000	+	21.6506i	0.808558	+	1.40046i	−0.105305	+	0.994440i	$0.533582\pi$
$240$		0			0
$241$	−7.50000	+	12.9904i	−0.483117	+	0.836784i	−0.999812	−	0.0193858i	$-0.993829\pi$
$241$	−7.50000	+	12.9904i	−0.483117	+	0.836784i	0.516695	+	0.856170i	$0.327162\pi$
$242$		0			0
$243$	8.00000	−	13.8564i	0.513200	−	0.888889i
$244$		0			0
$245$	1.00000	−	1.73205i	0.0638877	−	0.110657i
$246$		0			0
$247$	5.00000			0.318142
$248$		0			0
$249$	−4.50000	+	7.79423i	−0.285176	+	0.493939i
$250$		0			0
$251$	3.50000	−	6.06218i	0.220918	−	0.382641i	−0.734169	−	0.678967i	$-0.762428\pi$
$251$	3.50000	−	6.06218i	0.220918	−	0.382641i	0.955087	+	0.296326i	$0.0957613\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.50000	−	2.59808i	−0.0939336	−	0.162698i
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	27.0000			1.67770
$260$		0			0
$261$	3.00000	+	5.19615i	0.185695	+	0.321634i
$262$		0			0
$263$	10.5000	+	18.1865i	0.647458	+	1.12143i	0.983728	+	0.179664i	$0.0575011\pi$
$263$	10.5000	+	18.1865i	0.647458	+	1.12143i	−0.336270	+	0.941766i	$0.609166\pi$
$264$		0			0
$265$	−2.50000	+	4.33013i	−0.153574	+	0.265998i
$266$		0			0
$267$	−0.500000	+	0.866025i	−0.0305995	+	0.0529999i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	11.5000	−	19.9186i	0.698575	−	1.20997i	−0.270385	−	0.962752i	$-0.587151\pi$
$271$	11.5000	−	19.9186i	0.698575	−	1.20997i	0.968960	−	0.247216i	$-0.0795156\pi$
$272$		0			0
$273$	7.50000	−	12.9904i	0.453921	−	0.786214i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.50000	−	6.06218i	−0.210295	−	0.364241i	0.741512	−	0.670940i	$-0.234109\pi$
$277$	−3.50000	−	6.06218i	−0.210295	−	0.364241i	−0.951807	+	0.306699i	$0.900776\pi$
$278$		0			0
$279$	10.0000			0.598684
$280$		0			0
$281$	8.50000	+	14.7224i	0.507067	+	0.878267i	0.999967	+	0.00818015i	$0.00260385\pi$
$281$	8.50000	+	14.7224i	0.507067	+	0.878267i	−0.492899	+	0.870087i	$0.664063\pi$
$282$		0			0
$283$	1.50000	−	2.59808i	0.0891657	−	0.154440i	−0.817993	−	0.575228i	$-0.804913\pi$
$283$	1.50000	−	2.59808i	0.0891657	−	0.154440i	0.907159	+	0.420789i	$0.138247\pi$
$284$		0			0
$285$	−1.00000			−0.0592349
$286$		0			0
$287$	−15.0000	+	25.9808i	−0.885422	+	1.53360i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	−1.00000	−	1.73205i	−0.0586210	−	0.101535i
$292$		0			0
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−35.0000			−2.02410
$300$		0			0
$301$	19.5000	−	2.59808i	1.12396	−	0.149751i
$302$		0			0
$303$	9.00000			0.517036
$304$		0			0
$305$	13.0000			0.744378
$306$		0			0
$307$	2.50000	+	4.33013i	0.142683	+	0.247133i	0.928506	−	0.371318i	$-0.121094\pi$
$307$	2.50000	+	4.33013i	0.142683	+	0.247133i	−0.785823	+	0.618451i	$0.787761\pi$
$308$		0			0
$309$	7.00000			0.398216
$310$		0			0
$311$	−1.50000	−	2.59808i	−0.0850572	−	0.147323i	0.820358	−	0.571850i	$-0.193774\pi$
$311$	−1.50000	−	2.59808i	−0.0850572	−	0.147323i	−0.905416	+	0.424526i	$0.860441\pi$
$312$		0			0
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	0.999748	−	0.0224310i	$-0.00714060\pi$
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	−0.519300	+	0.854592i	$0.673807\pi$
$314$		0			0
$315$	3.00000	−	5.19615i	0.169031	−	0.292770i
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−6.00000	−	10.3923i	−0.334887	−	0.580042i
$322$		0			0
$323$	−3.00000			−0.166924
$324$		0			0
$325$	−10.0000	−	17.3205i	−0.554700	−	0.960769i
$326$		0			0
$327$	3.50000	−	6.06218i	0.193550	−	0.335239i
$328$		0			0
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	−0.477500	−	0.878632i	$-0.658457\pi$
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	0.999667	−	0.0257885i	$-0.00820965\pi$
$332$		0			0
$333$	18.0000			0.986394
$334$		0			0
$335$	1.50000	−	2.59808i	0.0819538	−	0.141948i
$336$		0			0
$337$	−1.50000	+	2.59808i	−0.0817102	+	0.141526i	−0.903985	−	0.427565i	$-0.859372\pi$
$337$	−1.50000	+	2.59808i	−0.0817102	+	0.141526i	0.822274	+	0.569091i	$0.192705\pi$
$338$		0			0
$339$	−1.00000	−	1.73205i	−0.0543125	−	0.0940721i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$	7.00000			0.376867
$346$		0			0
$347$	18.5000	+	32.0429i	0.993132	+	1.72016i	0.597890	+	0.801578i	$0.296006\pi$
$347$	18.5000	+	32.0429i	0.993132	+	1.72016i	0.395242	+	0.918577i	$0.370661\pi$
$348$		0			0
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	0.879097	−	0.476642i	$-0.158146\pi$
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	−0.852333	+	0.523000i	$0.824813\pi$
$350$		0			0
$351$	12.5000	−	21.6506i	0.667201	−	1.15563i
$352$		0			0
$353$	12.5000	−	21.6506i	0.665308	−	1.15235i	−0.313894	−	0.949458i	$-0.601634\pi$
$353$	12.5000	−	21.6506i	0.665308	−	1.15235i	0.979202	−	0.202889i	$-0.0650330\pi$
$354$		0			0
$355$	−1.00000			−0.0530745
$356$		0			0
$357$	−4.50000	+	7.79423i	−0.238165	+	0.412514i
$358$		0			0
$359$	9.50000	−	16.4545i	0.501391	−	0.868434i	−0.498608	−	0.866828i	$-0.666155\pi$
$359$	9.50000	−	16.4545i	0.501391	−	0.868434i	0.999999	−	0.00160673i	$-0.000511438\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	−5.50000	−	9.52628i	−0.288675	−	0.500000i
$364$		0			0
$365$	−11.0000			−0.575766
$366$		0			0
$367$	6.50000	+	11.2583i	0.339297	+	0.587680i	0.984301	−	0.176500i	$-0.0564774\pi$
$367$	6.50000	+	11.2583i	0.339297	+	0.587680i	−0.645003	+	0.764180i	$0.723144\pi$
$368$		0			0
$369$	−10.0000	+	17.3205i	−0.520579	+	0.901670i
$370$		0			0
$371$	15.0000			0.778761
$372$		0			0
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	−0.972556	−	0.232671i	$-0.925254\pi$
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	0.687776	+	0.725923i	$0.258587\pi$
$374$		0			0
$375$	4.50000	+	7.79423i	0.232379	+	0.402492i
$376$		0			0
$377$	7.50000	+	12.9904i	0.386270	+	0.669039i
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−8.00000	−	13.8564i	−0.409852	−	0.709885i
$382$		0			0
$383$	−8.00000			−0.408781			−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000			−0.408781			−0.204390	+	0.978889i	$0.565521\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.0000	−	1.73205i	0.660827	−	0.0880451i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	21.0000			1.06202
$392$		0			0
$393$	2.00000	+	3.46410i	0.100887	+	0.174741i
$394$		0			0
$395$	−5.00000			−0.251577
$396$		0			0
$397$	10.5000	+	18.1865i	0.526980	+	0.912756i	0.999506	+	0.0314391i	$0.0100090\pi$
$397$	10.5000	+	18.1865i	0.526980	+	0.912756i	−0.472526	+	0.881317i	$0.656658\pi$
$398$		0			0
$399$	1.50000	+	2.59808i	0.0750939	+	0.130066i
$400$		0			0
$401$	18.5000	−	32.0429i	0.923846	−	1.60015i	0.130439	−	0.991456i	$-0.458361\pi$
$401$	18.5000	−	32.0429i	0.923846	−	1.60015i	0.793407	−	0.608692i	$-0.208305\pi$
$402$		0			0
$403$	25.0000			1.24534
$404$		0			0
$405$	0.500000	−	0.866025i	0.0248452	−	0.0430331i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−18.0000			−0.890043			−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000			−0.890043			−0.445021	+	0.895520i	$0.646804\pi$
$410$		0			0
$411$	−9.00000	−	15.5885i	−0.443937	−	0.768922i
$412$		0			0
$413$	−18.0000	+	31.1769i	−0.885722	+	1.53412i
$414$		0			0
$415$	−4.50000	+	7.79423i	−0.220896	+	0.382604i
$416$		0			0
$417$	−6.50000	+	11.2583i	−0.318306	+	0.551323i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	18.5000	−	32.0429i	0.901635	−	1.56168i	0.0762630	−	0.997088i	$-0.475701\pi$
$421$	18.5000	−	32.0429i	0.901635	−	1.56168i	0.825372	−	0.564590i	$-0.190966\pi$
$422$		0			0
$423$	8.00000	−	13.8564i	0.388973	−	0.673722i
$424$		0			0
$425$	6.00000	+	10.3923i	0.291043	+	0.504101i
$426$		0			0
$427$	−19.5000	−	33.7750i	−0.943671	−	1.63449i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−13.5000	−	23.3827i	−0.648769	−	1.12370i	−0.983417	−	0.181357i	$-0.941951\pi$
$433$	−13.5000	−	23.3827i	−0.648769	−	1.12370i	0.334649	−	0.942343i	$-0.391382\pi$
$434$		0			0
$435$	−1.50000	−	2.59808i	−0.0719195	−	0.124568i
$436$		0			0
$437$	3.50000	−	6.06218i	0.167428	−	0.289993i
$438$		0			0
$439$	−6.50000	+	11.2583i	−0.310228	+	0.537331i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−6.50000	+	11.2583i	−0.310228	+	0.537331i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−4.00000			−0.190476
$442$		0			0
$443$	−18.5000	+	32.0429i	−0.878962	+	1.52241i	−0.0264796	+	0.999649i	$0.508430\pi$
$443$	−18.5000	+	32.0429i	−0.878962	+	1.52241i	−0.852482	+	0.522757i	$0.824904\pi$
$444$		0			0
$445$	−0.500000	+	0.866025i	−0.0237023	+	0.0410535i
$446$		0			0
$447$	−10.5000	+	18.1865i	−0.496633	+	0.860194i
$448$		0			0
$449$	10.5000	+	18.1865i	0.495526	+	0.858276i	0.999987	−	0.00515887i	$-0.00164213\pi$
$449$	10.5000	+	18.1865i	0.495526	+	0.858276i	−0.504461	+	0.863434i	$0.668309\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	4.00000	+	6.92820i	0.187936	+	0.325515i
$454$		0			0
$455$	7.50000	−	12.9904i	0.351605	−	0.608998i
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$		0			0
$459$	−7.50000	+	12.9904i	−0.350070	+	0.606339i
$460$		0			0
$461$	−1.50000	−	2.59808i	−0.0698620	−	0.121004i	0.828978	−	0.559281i	$-0.188923\pi$
$461$	−1.50000	−	2.59808i	−0.0698620	−	0.121004i	−0.898840	+	0.438276i	$0.855589\pi$
$462$		0			0
$463$	−11.5000	−	19.9186i	−0.534450	−	0.925695i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−11.5000	−	19.9186i	−0.534450	−	0.925695i	0.464739	−	0.885448i	$-0.346148\pi$
$464$		0			0
$465$	−5.00000			−0.231869
$466$		0			0
$467$	10.5000	+	18.1865i	0.485882	+	0.841572i	0.999868	−	0.0162260i	$-0.00516512\pi$
$467$	10.5000	+	18.1865i	0.485882	+	0.841572i	−0.513986	+	0.857798i	$0.671832\pi$
$468$		0			0
$469$	−9.00000			−0.415581
$470$		0			0
$471$	1.00000			0.0460776
$472$		0			0
$473$		0			0
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$	10.0000			0.457869
$478$		0			0
$479$	−7.50000	−	12.9904i	−0.342684	−	0.593546i	0.642246	−	0.766498i	$-0.278003\pi$
$479$	−7.50000	−	12.9904i	−0.342684	−	0.593546i	−0.984930	+	0.172953i	$0.944669\pi$
$480$		0			0
$481$	45.0000			2.05182
$482$		0			0
$483$	−10.5000	−	18.1865i	−0.477767	−	0.827516i
$484$		0			0
$485$	−1.00000	−	1.73205i	−0.0454077	−	0.0786484i
$486$		0			0
$487$	−4.50000	+	7.79423i	−0.203914	+	0.353190i	−0.949786	−	0.312899i	$-0.898700\pi$
$487$	−4.50000	+	7.79423i	−0.203914	+	0.353190i	0.745872	+	0.666089i	$0.232033\pi$
$488$		0			0
$489$	−1.00000			−0.0452216
$490$		0			0
$491$	−4.50000	+	7.79423i	−0.203082	+	0.351749i	−0.949520	−	0.313707i	$-0.898429\pi$
$491$	−4.50000	+	7.79423i	−0.203082	+	0.351749i	0.746438	+	0.665455i	$0.231763\pi$
$492$		0			0
$493$	−4.50000	−	7.79423i	−0.202670	−	0.351034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.50000	+	2.59808i	0.0672842	+	0.116540i
$498$		0			0
$499$	−8.50000	+	14.7224i	−0.380512	+	0.659067i	−0.991136	−	0.132855i	$-0.957586\pi$
$499$	−8.50000	+	14.7224i	−0.380512	+	0.659067i	0.610623	+	0.791921i	$0.290919\pi$
$500$		0			0
$501$	1.50000	−	2.59808i	0.0670151	−	0.116073i
$502$		0			0
$503$	−4.50000	+	7.79423i	−0.200645	+	0.347527i	−0.948736	−	0.316068i	$-0.897637\pi$
$503$	−4.50000	+	7.79423i	−0.200645	+	0.347527i	0.748091	+	0.663596i	$0.230970\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	−13.5000	+	23.3827i	−0.598377	+	1.03642i	0.394684	+	0.918817i	$0.370854\pi$
$509$	−13.5000	+	23.3827i	−0.598377	+	1.03642i	−0.993061	+	0.117602i	$0.962479\pi$
$510$		0			0
$511$	16.5000	+	28.5788i	0.729917	+	1.26425i
$512$		0			0
$513$	2.50000	+	4.33013i	0.110378	+	0.191180i
$514$		0			0
$515$	7.00000			0.308457
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−3.00000	−	5.19615i	−0.131685	−	0.228086i
$520$		0			0
$521$	−17.5000	−	30.3109i	−0.766689	−	1.32794i	−0.939349	−	0.342963i	$-0.888570\pi$
$521$	−17.5000	−	30.3109i	−0.766689	−	1.32794i	0.172660	−	0.984981i	$-0.444764\pi$
$522$		0			0
$523$	−10.5000	+	18.1865i	−0.459133	+	0.795242i	−0.998915	−	0.0465630i	$-0.985173\pi$
$523$	−10.5000	+	18.1865i	−0.459133	+	0.795242i	0.539782	+	0.841805i	$0.318507\pi$
$524$		0			0
$525$	6.00000	−	10.3923i	0.261861	−	0.453557i
$526$		0			0
$527$	−15.0000			−0.653410
$528$		0			0
$529$	−13.0000	+	22.5167i	−0.565217	+	0.978985i
$530$		0			0
$531$	−12.0000	+	20.7846i	−0.520756	+	0.901975i
$532$		0			0
$533$	−25.0000	+	43.3013i	−1.08287	+	1.87559i
$534$		0			0
$535$	−6.00000	−	10.3923i	−0.259403	−	0.449299i
$536$		0			0
$537$	−1.00000			−0.0431532
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.5000	+	26.8468i	−0.666397	+	1.15423i	0.312507	+	0.949915i	$0.398831\pi$
$541$	−15.5000	+	26.8468i	−0.666397	+	1.15423i	−0.978905	+	0.204318i	$0.934502\pi$
$542$		0			0
$543$	−7.00000			−0.300399
$544$		0			0
$545$	3.50000	−	6.06218i	0.149924	−	0.259675i
$546$		0			0
$547$	0.500000	+	0.866025i	0.0213785	+	0.0370286i	0.876517	−	0.481371i	$-0.159861\pi$
$547$	0.500000	+	0.866025i	0.0213785	+	0.0370286i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	−13.0000	−	22.5167i	−0.554826	−	0.960988i
$550$		0			0
$551$	−3.00000			−0.127804
$552$		0			0
$553$	7.50000	+	12.9904i	0.318932	+	0.552407i
$554$		0			0
$555$	−9.00000			−0.382029
$556$		0			0
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$	32.5000	−	4.33013i	1.37460	−	0.183145i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$	−1.00000	−	1.73205i	−0.0420703	−	0.0728679i
$566$		0			0
$567$	−3.00000			−0.125988
$568$		0			0
$569$	8.50000	+	14.7224i	0.356339	+	0.617196i	0.987346	−	0.158580i	$-0.0506917\pi$
$569$	8.50000	+	14.7224i	0.356339	+	0.617196i	−0.631008	+	0.775777i	$0.717358\pi$
$570$		0			0
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	0.969378	−	0.245573i	$-0.0789761\pi$
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	−0.697362	+	0.716720i	$0.745643\pi$
$572$		0			0
$573$	9.50000	−	16.4545i	0.396868	−	0.687396i
$574$		0			0
$575$	−28.0000			−1.16768
$576$		0			0
$577$	2.50000	−	4.33013i	0.104076	−	0.180266i	−0.809284	−	0.587417i	$-0.800145\pi$
$577$	2.50000	−	4.33013i	0.104076	−	0.180266i	0.913360	+	0.407152i	$0.133478\pi$
$578$		0			0
$579$	3.00000	+	5.19615i	0.124676	+	0.215945i
$580$		0			0
$581$	27.0000			1.12015
$582$		0			0
$583$		0			0
$584$		0			0
$585$	5.00000	−	8.66025i	0.206725	−	0.358057i
$586$		0			0
$587$	−6.50000	+	11.2583i	−0.268284	+	0.464681i	−0.968419	−	0.249329i	$-0.919790\pi$
$587$	−6.50000	+	11.2583i	−0.268284	+	0.464681i	0.700135	+	0.714010i	$0.253123\pi$
$588$		0			0
$589$	−2.50000	+	4.33013i	−0.103011	+	0.178420i
$590$		0			0
$591$	−11.0000			−0.452480
$592$		0			0
$593$	0.500000	−	0.866025i	0.0205325	−	0.0355634i	−0.855577	−	0.517676i	$-0.826797\pi$
$593$	0.500000	−	0.866025i	0.0205325	−	0.0355634i	0.876109	+	0.482113i	$0.160130\pi$
$594$		0			0
$595$	−4.50000	+	7.79423i	−0.184482	+	0.319532i
$596$		0			0
$597$	−4.00000	−	6.92820i	−0.163709	−	0.283552i
$598$		0			0
$599$	−15.5000	−	26.8468i	−0.633313	−	1.09693i	−0.986870	−	0.161517i	$-0.948361\pi$
$599$	−15.5000	−	26.8468i	−0.633313	−	1.09693i	0.353557	−	0.935413i	$-0.384972\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$	−6.00000			−0.244339
$604$		0			0
$605$	−5.50000	−	9.52628i	−0.223607	−	0.387298i
$606$		0			0
$607$	−21.5000	−	37.2391i	−0.872658	−	1.51149i	−0.859237	−	0.511578i	$-0.829061\pi$
$607$	−21.5000	−	37.2391i	−0.872658	−	1.51149i	−0.0134214	−	0.999910i	$-0.504272\pi$
$608$		0			0
$609$	−4.50000	+	7.79423i	−0.182349	+	0.315838i
$610$		0			0
$611$	20.0000	−	34.6410i	0.809113	−	1.40143i
$612$		0			0
$613$	−30.0000			−1.21169			−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000			−1.21169			−0.605844	+	0.795583i	$0.707165\pi$
$614$		0			0
$615$	5.00000	−	8.66025i	0.201619	−	0.349215i
$616$		0			0
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		0			0
$619$	−10.5000	+	18.1865i	−0.422031	+	0.730978i	−0.996138	−	0.0878015i	$-0.972016\pi$
$619$	−10.5000	+	18.1865i	−0.422031	+	0.730978i	0.574107	+	0.818780i	$0.305349\pi$
$620$		0			0
$621$	−17.5000	−	30.3109i	−0.702251	−	1.21633i
$622$		0			0
$623$	3.00000			0.120192
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.0000			−1.07656
$630$		0			0
$631$	−4.50000	+	7.79423i	−0.179142	+	0.310283i	−0.941587	−	0.336770i	$-0.890666\pi$
$631$	−4.50000	+	7.79423i	−0.179142	+	0.310283i	0.762445	+	0.647053i	$0.223999\pi$
$632$		0			0
$633$	−4.00000	−	6.92820i	−0.158986	−	0.275371i
$634$		0			0
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$	−10.0000			−0.396214
$638$		0			0
$639$	1.00000	+	1.73205i	0.0395594	+	0.0685189i
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$	−12.0000			−0.473234			−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000			−0.473234			−0.236617	+	0.971603i	$0.576039\pi$
$644$		0			0
$645$	−6.50000	+	0.866025i	−0.255937	+	0.0340997i
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	7.50000	+	12.9904i	0.293948	+	0.509133i
$652$		0			0
$653$	−50.0000			−1.95665			−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000			−1.95665			−0.978326	+	0.207072i	$0.933606\pi$
$654$		0			0
$655$	2.00000	+	3.46410i	0.0781465	+	0.135354i
$656$		0			0
$657$	11.0000	+	19.0526i	0.429151	+	0.743311i
$658$		0			0
$659$	11.5000	−	19.9186i	0.447976	−	0.775918i	−0.550278	−	0.834982i	$-0.685478\pi$
$659$	11.5000	−	19.9186i	0.447976	−	0.775918i	0.998254	+	0.0590638i	$0.0188115\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$	−7.50000	+	12.9904i	−0.291276	+	0.504505i
$664$		0			0
$665$	1.50000	+	2.59808i	0.0581675	+	0.100749i
$666$		0			0
$667$	21.0000			0.813123
$668$		0			0
$669$	−10.0000	−	17.3205i	−0.386622	−	0.669650i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−23.5000	+	40.7032i	−0.905858	+	1.56899i	−0.0860977	+	0.996287i	$0.527440\pi$
$673$	−23.5000	+	40.7032i	−0.905858	+	1.56899i	−0.819761	+	0.572706i	$0.805894\pi$
$674$		0			0
$675$	10.0000	−	17.3205i	0.384900	−	0.666667i
$676$		0			0
$677$	−14.0000			−0.538064			−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000			−0.538064			−0.269032	+	0.963131i	$0.586704\pi$
$678$		0			0
$679$	−3.00000	+	5.19615i	−0.115129	+	0.199410i
$680$		0			0
$681$	3.50000	−	6.06218i	0.134120	−	0.232303i
$682$		0			0
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	0.939184	−	0.343413i	$-0.111583\pi$
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	−0.766997	+	0.641651i	$0.778250\pi$
$684$		0			0
$685$	−9.00000	−	15.5885i	−0.343872	−	0.595604i
$686$		0			0
$687$	9.00000			0.343371
$688$		0			0
$689$	25.0000			0.952424
$690$		0			0
$691$	2.50000	+	4.33013i	0.0951045	+	0.164726i	0.909652	−	0.415371i	$-0.136348\pi$
$691$	2.50000	+	4.33013i	0.0951045	+	0.164726i	−0.814548	+	0.580097i	$0.803015\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.50000	+	11.2583i	−0.246559	+	0.427053i
$696$		0			0
$697$	15.0000	−	25.9808i	0.568166	−	0.984092i
$698$		0			0
$699$	9.00000			0.340411
$700$		0			0
$701$	−3.50000	+	6.06218i	−0.132193	+	0.228965i	−0.924522	−	0.381129i	$-0.875535\pi$
$701$	−3.50000	+	6.06218i	−0.132193	+	0.228965i	0.792329	+	0.610095i	$0.208869\pi$
$702$		0			0
$703$	−4.50000	+	7.79423i	−0.169721	+	0.293965i
$704$		0			0
$705$	−4.00000	+	6.92820i	−0.150649	+	0.260931i
$706$		0			0
$707$	−13.5000	−	23.3827i	−0.507720	−	0.879396i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$	5.00000	+	8.66025i	0.187515	+	0.324785i
$712$		0			0
$713$	17.5000	−	30.3109i	0.655380	−	1.13515i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.5000	+	21.6506i	−0.466821	+	0.808558i
$718$		0			0
$719$	−15.5000	−	26.8468i	−0.578052	−	1.00122i	−0.995703	−	0.0926083i	$-0.970480\pi$
$719$	−15.5000	−	26.8468i	−0.578052	−	1.00122i	0.417650	−	0.908608i	$-0.362854\pi$
$720$		0			0
$721$	−10.5000	−	18.1865i	−0.391040	−	0.677302i
$722$		0			0
$723$	−15.0000			−0.557856
$724$		0			0
$725$	6.00000	+	10.3923i	0.222834	+	0.385961i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−19.5000	+	2.59808i	−0.721234	+	0.0960933i
$732$		0			0
$733$	26.0000			0.960332			0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000			0.960332			0.480166	+	0.877178i	$0.340576\pi$
$734$		0			0
$735$	2.00000			0.0737711
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$	2.50000	+	4.33013i	0.0918398	+	0.159071i
$742$		0			0
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	0.991798	−	0.127815i	$-0.0407965\pi$
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	−0.606590	+	0.795015i	$0.707463\pi$
$744$		0			0
$745$	−10.5000	+	18.1865i	−0.384690	+	0.666303i
$746$		0			0
$747$	18.0000			0.658586
$748$		0			0
$749$	−18.0000	+	31.1769i	−0.657706	+	1.13918i
$750$		0			0
$751$	−13.5000	−	23.3827i	−0.492622	−	0.853246i	0.507342	−	0.861745i	$-0.330628\pi$
$751$	−13.5000	−	23.3827i	−0.492622	−	0.853246i	−0.999964	+	0.00849853i	$0.997295\pi$
$752$		0			0
$753$	7.00000			0.255094
$754$		0			0
$755$	4.00000	+	6.92820i	0.145575	+	0.252143i
$756$		0			0
$757$	2.50000	−	4.33013i	0.0908640	−	0.157381i	−0.817011	−	0.576622i	$-0.804370\pi$
$757$	2.50000	−	4.33013i	0.0908640	−	0.157381i	0.907875	+	0.419241i	$0.137704\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.5000	−	32.0429i	0.670624	−	1.16156i	−0.307103	−	0.951676i	$-0.599360\pi$
$761$	18.5000	−	32.0429i	0.670624	−	1.16156i	0.977727	−	0.209879i	$-0.0673071\pi$
$762$		0			0
$763$	−21.0000			−0.760251
$764$		0			0
$765$	−3.00000	+	5.19615i	−0.108465	+	0.187867i
$766$		0			0
$767$	−30.0000	+	51.9615i	−1.08324	+	1.87622i
$768$		0			0
$769$	16.5000	+	28.5788i	0.595005	+	1.03058i	0.993546	+	0.113429i	$0.0361834\pi$
$769$	16.5000	+	28.5788i	0.595005	+	1.03058i	−0.398541	+	0.917151i	$0.630483\pi$
$770$		0			0
$771$	3.00000	+	5.19615i	0.108042	+	0.187135i
$772$		0			0
$773$	26.0000			0.935155			0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000			0.935155			0.467578	+	0.883952i	$0.345127\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$		0			0
$777$	13.5000	+	23.3827i	0.484310	+	0.838849i
$778$		0			0
$779$	−5.00000	−	8.66025i	−0.179144	−	0.310286i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−7.50000	+	12.9904i	−0.268028	+	0.464238i
$784$		0			0
$785$	1.00000			0.0356915
$786$		0			0
$787$	−2.50000	+	4.33013i	−0.0891154	+	0.154352i	−0.907137	−	0.420834i	$-0.861737\pi$
$787$	−2.50000	+	4.33013i	−0.0891154	+	0.154352i	0.818022	+	0.575187i	$0.195071\pi$
$788$		0			0
$789$	−10.5000	+	18.1865i	−0.373810	+	0.647458i
$790$		0			0
$791$	−3.00000	+	5.19615i	−0.106668	+	0.184754i
$792$		0			0
$793$	−32.5000	−	56.2917i	−1.15411	−	1.99898i
$794$		0			0
$795$	−5.00000			−0.177332
$796$		0			0
$797$	10.5000	+	18.1865i	0.371929	+	0.644200i	0.989862	−	0.142031i	$-0.0453631\pi$
$797$	10.5000	+	18.1865i	0.371929	+	0.644200i	−0.617933	+	0.786231i	$0.712030\pi$
$798$		0			0
$799$	−12.0000	+	20.7846i	−0.424529	+	0.735307i
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−10.5000	−	18.1865i	−0.370076	−	0.640991i
$806$		0			0
$807$	7.00000	+	12.1244i	0.246412	+	0.426798i
$808$		0			0
$809$	−34.0000			−1.19538			−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000			−1.19538			−0.597688	+	0.801729i	$0.703914\pi$
$810$		0			0
$811$	−23.5000	−	40.7032i	−0.825197	−	1.42928i	−0.901769	−	0.432218i	$-0.857731\pi$
$811$	−23.5000	−	40.7032i	−0.825197	−	1.42928i	0.0765723	−	0.997064i	$-0.475602\pi$
$812$		0			0
$813$	23.0000			0.806645
$814$		0			0
$815$	−1.00000			−0.0350285
$816$		0			0
$817$	−2.50000	+	6.06218i	−0.0874639	+	0.212089i
$818$		0			0
$819$	−30.0000			−1.04828
$820$		0			0
$821$	10.0000			0.349002			0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000			0.349002			0.174501	+	0.984657i	$0.444169\pi$
$822$		0			0
$823$	−15.5000	−	26.8468i	−0.540296	−	0.935820i	−0.998887	−	0.0471726i	$-0.984979\pi$
$823$	−15.5000	−	26.8468i	−0.540296	−	0.935820i	0.458591	−	0.888648i	$-0.348354\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.5000	+	28.5788i	0.573761	+	0.993784i	0.996175	+	0.0873805i	$0.0278496\pi$
$827$	16.5000	+	28.5788i	0.573761	+	0.993784i	−0.422414	+	0.906403i	$0.638817\pi$
$828$		0			0
$829$	−5.50000	−	9.52628i	−0.191023	−	0.330861i	0.754567	−	0.656223i	$-0.227847\pi$
$829$	−5.50000	−	9.52628i	−0.191023	−	0.330861i	−0.945589	+	0.325362i	$0.894514\pi$
$830$		0			0
$831$	3.50000	−	6.06218i	0.121414	−	0.210295i
$832$		0			0
$833$	6.00000			0.207888
$834$		0			0
$835$	1.50000	−	2.59808i	0.0519096	−	0.0899101i
$836$		0			0
$837$	12.5000	+	21.6506i	0.432063	+	0.748355i
$838$		0			0
$839$	52.0000			1.79524			0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000			1.79524			0.897620	+	0.440771i	$0.145295\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$		0			0
$843$	−8.50000	+	14.7224i	−0.292756	+	0.507067i
$844$		0			0
$845$	6.00000	−	10.3923i	0.206406	−	0.357506i
$846$		0			0
$847$	−16.5000	+	28.5788i	−0.566947	+	0.981981i
$848$		0			0
$849$	3.00000			0.102960
$850$		0			0
$851$	31.5000	−	54.5596i	1.07981	−	1.87028i
$852$		0			0
$853$	2.50000	−	4.33013i	0.0855984	−	0.148261i	−0.820048	−	0.572295i	$-0.806053\pi$
$853$	2.50000	−	4.33013i	0.0855984	−	0.148261i	0.905646	+	0.424034i	$0.139386\pi$
$854$		0			0
$855$	1.00000	+	1.73205i	0.0341993	+	0.0592349i
$856$		0			0
$857$	−17.5000	−	30.3109i	−0.597789	−	1.03540i	−0.993147	−	0.116873i	$-0.962713\pi$
$857$	−17.5000	−	30.3109i	−0.597789	−	1.03540i	0.395358	−	0.918527i	$-0.370620\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	−30.0000			−1.02240
$862$		0			0
$863$	−13.5000	−	23.3827i	−0.459545	−	0.795956i	0.539392	−	0.842055i	$-0.318654\pi$
$863$	−13.5000	−	23.3827i	−0.459545	−	0.795956i	−0.998937	+	0.0460992i	$0.985321\pi$
$864$		0			0
$865$	−3.00000	−	5.19615i	−0.102003	−	0.176674i
$866$		0			0
$867$	−4.00000	+	6.92820i	−0.135847	+	0.235294i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−15.0000			−0.508256
$872$		0			0
$873$	−2.00000	+	3.46410i	−0.0676897	+	0.117242i
$874$		0			0
$875$	13.5000	−	23.3827i	0.456383	−	0.790479i
$876$		0			0
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	−0.992225	−	0.124459i	$-0.960280\pi$
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	0.603897	+	0.797062i	$0.293614\pi$
$878$		0			0
$879$	−7.00000	−	12.1244i	−0.236104	−	0.408944i
$880$		0			0
$881$	46.0000			1.54978			0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000			1.54978			0.774890	+	0.632096i	$0.217805\pi$
$882$		0			0
$883$	6.50000	+	11.2583i	0.218742	+	0.378873i	0.954424	−	0.298455i	$-0.0964712\pi$
$883$	6.50000	+	11.2583i	0.218742	+	0.378873i	−0.735681	+	0.677328i	$0.763138\pi$
$884$		0			0
$885$	6.00000	−	10.3923i	0.201688	−	0.349334i
$886$		0			0
$887$	16.0000			0.537227			0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000			0.537227			0.268614	+	0.963248i	$0.413434\pi$
$888$		0			0
$889$	−24.0000	+	41.5692i	−0.804934	+	1.39419i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	+	6.92820i	0.133855	+	0.231843i
$894$		0			0
$895$	−1.00000			−0.0334263
$896$		0			0
$897$	−17.5000	−	30.3109i	−0.584308	−	1.01205i
$898$		0			0
$899$	−15.0000			−0.500278
$900$		0			0
$901$	−15.0000			−0.499722
$902$		0			0
$903$	12.0000	+	15.5885i	0.399335	+	0.518751i
$904$		0			0
$905$	−7.00000			−0.232688
$906$		0			0
$907$	−20.0000			−0.664089			−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000			−0.664089			−0.332045	+	0.943264i	$0.607738\pi$
$908$		0			0
$909$	−9.00000	−	15.5885i	−0.298511	−	0.517036i
$910$		0			0
$911$	52.0000			1.72284			0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000			1.72284			0.861418	+	0.507896i	$0.169577\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	6.50000	+	11.2583i	0.214883	+	0.372189i
$916$		0			0
$917$	6.00000	−	10.3923i	0.198137	−	0.343184i
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$		0			0
$921$	−2.50000	+	4.33013i	−0.0823778	+	0.142683i
$922$		0			0
$923$	2.50000	+	4.33013i	0.0822885	+	0.142528i
$924$		0			0
$925$	36.0000			1.18367
$926$		0			0
$927$	−7.00000	−	12.1244i	−0.229910	−	0.398216i
$928$		0			0
$929$	28.5000	−	49.3634i	0.935055	−	1.61956i	0.160518	−	0.987033i	$-0.448683\pi$
$929$	28.5000	−	49.3634i	0.935055	−	1.61956i	0.774536	−	0.632529i	$-0.217983\pi$
$930$		0			0
$931$	1.00000	−	1.73205i	0.0327737	−	0.0567657i
$932$		0			0
$933$	1.50000	−	2.59808i	0.0491078	−	0.0850572i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−17.5000	+	30.3109i	−0.571700	+	0.990214i	0.424691	+	0.905338i	$0.360383\pi$
$937$	−17.5000	+	30.3109i	−0.571700	+	0.990214i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$	−8.50000	+	14.7224i	−0.277387	+	0.480448i
$940$		0			0
$941$	4.50000	+	7.79423i	0.146696	+	0.254085i	0.930004	−	0.367549i	$-0.119803\pi$
$941$	4.50000	+	7.79423i	0.146696	+	0.254085i	−0.783309	+	0.621633i	$0.786469\pi$
$942$		0			0
$943$	35.0000	+	60.6218i	1.13976	+	1.97412i
$944$		0			0
$945$	15.0000			0.487950
$946$		0			0
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$	27.5000	+	47.6314i	0.892688	+	1.54618i
$950$		0			0
$951$	−9.00000	−	15.5885i	−0.291845	−	0.505490i
$952$		0			0
$953$	20.5000	−	35.5070i	0.664060	−	1.15019i	−0.315479	−	0.948933i	$-0.602165\pi$
$953$	20.5000	−	35.5070i	0.664060	−	1.15019i	0.979539	−	0.201253i	$-0.0645015\pi$
$954$		0			0
$955$	9.50000	−	16.4545i	0.307413	−	0.532455i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−27.0000	+	46.7654i	−0.871875	+	1.51013i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$	−12.0000	+	20.7846i	−0.386695	+	0.669775i
$964$		0			0
$965$	3.00000	+	5.19615i	0.0965734	+	0.167270i
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$		0			0
$969$	−1.50000	−	2.59808i	−0.0481869	−	0.0834622i
$970$		0			0
$971$	17.5000	−	30.3109i	0.561602	−	0.972723i	−0.435755	−	0.900065i	$-0.643519\pi$
$971$	17.5000	−	30.3109i	0.561602	−	0.972723i	0.997357	−	0.0726575i	$-0.0231480\pi$
$972$		0			0
$973$	39.0000			1.25028
$974$		0			0
$975$	10.0000	−	17.3205i	0.320256	−	0.554700i
$976$		0			0
$977$	20.5000	+	35.5070i	0.655853	+	1.13597i	0.981679	+	0.190541i	$0.0610243\pi$
$977$	20.5000	+	35.5070i	0.655853	+	1.13597i	−0.325826	+	0.945430i	$0.605642\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$	−25.5000	−	44.1673i	−0.813324	−	1.40872i	−0.910525	−	0.413453i	$-0.864323\pi$
$983$	−25.5000	−	44.1673i	−0.813324	−	1.40872i	0.0972017	−	0.995265i	$-0.469011\pi$
$984$		0			0
$985$	−11.0000			−0.350489
$986$		0			0
$987$	24.0000			0.763928
$988$		0			0
$989$	17.5000	−	42.4352i	0.556468	−	1.34936i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$	19.0000			0.602947
$994$		0			0
$995$	−4.00000	−	6.92820i	−0.126809	−	0.219639i
$996$		0			0
$997$	−14.0000			−0.443384			−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000			−0.443384			−0.221692	+	0.975117i	$0.571158\pi$
$998$		0			0
$999$	22.5000	+	38.9711i	0.711868	+	1.23299i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.i.d.337.1		2
4.3	odd	2		43.2.c.a.36.1	yes	2
12.11	even	2		387.2.h.a.208.1		2
43.6	even	3	inner	688.2.i.d.49.1		2
172.7	even	6		1849.2.a.a.1.1		1
172.79	odd	6		1849.2.a.c.1.1		1
172.135	odd	6		43.2.c.a.6.1	✓	2
516.479	even	6		387.2.h.a.307.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.c.a.6.1	✓	2	172.135	odd	6
43.2.c.a.36.1	yes	2	4.3	odd	2
387.2.h.a.208.1		2	12.11	even	2
387.2.h.a.307.1		2	516.479	even	6
688.2.i.d.49.1		2	43.6	even	3	inner
688.2.i.d.337.1		2	1.1	even	1	trivial
1849.2.a.a.1.1		1	172.7	even	6
1849.2.a.c.1.1		1	172.79	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +(2.50000 - 4.33013i) q^{13} +(-0.500000 + 0.866025i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} +3.00000 q^{21} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} +(2.50000 + 4.33013i) q^{31} +3.00000 q^{35} +(4.50000 + 7.79423i) q^{37} +5.00000 q^{39} -10.0000 q^{41} +(4.00000 + 5.19615i) q^{43} +2.00000 q^{45} +8.00000 q^{47} +(-1.00000 - 1.73205i) q^{49} -3.00000 q^{51} +(2.50000 + 4.33013i) q^{53} +(-0.500000 + 0.866025i) q^{57} -12.0000 q^{59} +(6.50000 - 11.2583i) q^{61} +(-3.00000 - 5.19615i) q^{63} +5.00000 q^{65} +(-1.50000 - 2.59808i) q^{67} +(3.50000 - 6.06218i) q^{69} +(-0.500000 + 0.866025i) q^{71} +(-5.50000 + 9.52628i) q^{73} +4.00000 q^{75} +(-2.50000 + 4.33013i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.50000 + 7.79423i) q^{83} -3.00000 q^{85} -3.00000 q^{87} +(0.500000 + 0.866025i) q^{89} +(-7.50000 - 12.9904i) q^{91} +(-2.50000 + 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + 3 * q^7 + 2 * q^9	\( 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + 5 q^{13} - q^{15} - 3 q^{17} + q^{19} + 6 q^{21} - 7 q^{23} + 4 q^{25} + 10 q^{27} - 3 q^{29} + 5 q^{31} + 6 q^{35} + 9 q^{37} + 10 q^{39} - 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 2 q^{49} - 6 q^{51} + 5 q^{53} - q^{57} - 24 q^{59} + 13 q^{61} - 6 q^{63} + 10 q^{65} - 3 q^{67} + 7 q^{69} - q^{71} - 11 q^{73} + 8 q^{75} - 5 q^{79} - q^{81} + 9 q^{83} - 6 q^{85} - 6 q^{87} + q^{89} - 15 q^{91} - 5 q^{93} - q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + q^3 + q^5 + 3 * q^7 + 2 * q^9 + 5 * q^13 - q^15 - 3 * q^17 + q^19 + 6 * q^21 - 7 * q^23 + 4 * q^25 + 10 * q^27 - 3 * q^29 + 5 * q^31 + 6 * q^35 + 9 * q^37 + 10 * q^39 - 20 * q^41 + 8 * q^43 + 4 * q^45 + 16 * q^47 - 2 * q^49 - 6 * q^51 + 5 * q^53 - q^57 - 24 * q^59 + 13 * q^61 - 6 * q^63 + 10 * q^65 - 3 * q^67 + 7 * q^69 - q^71 - 11 * q^73 + 8 * q^75 - 5 * q^79 - q^81 + 9 * q^83 - 6 * q^85 - 6 * q^87 + q^89 - 15 * q^91 - 5 * q^93 - q^95 - 4 * q^97

Introduction

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