LMFDB - Embedded newform 4020.2.q.f.3781.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(841,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("4020.841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.q (of order $3$, degree $2$, 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:	$32.0998616126$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3781.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4020.3781
Dual form			4020.2.q.f.841.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{9} +(0.500000 + 0.866025i) q^{11} +(-3.50000 + 6.06218i) q^{13} -1.00000 q^{15} +(1.50000 - 2.59808i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-0.500000 - 0.866025i) q^{21} +(1.50000 - 2.59808i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-0.500000 - 0.866025i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{35} +(0.500000 - 0.866025i) q^{37} +(-3.50000 + 6.06218i) q^{39} +(3.50000 + 6.06218i) q^{41} +8.00000 q^{43} -1.00000 q^{45} +(-1.50000 - 2.59808i) q^{47} +(3.00000 - 5.19615i) q^{49} +(1.50000 - 2.59808i) q^{51} -2.00000 q^{53} +(-0.500000 - 0.866025i) q^{55} +(2.50000 - 4.33013i) q^{57} +4.00000 q^{59} +(-5.50000 + 9.52628i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(3.50000 - 6.06218i) q^{65} +(8.00000 + 1.73205i) q^{67} +(1.50000 - 2.59808i) q^{69} +(6.50000 + 11.2583i) q^{71} +(-5.50000 + 9.52628i) q^{73} +1.00000 q^{75} +(0.500000 - 0.866025i) q^{77} +(5.50000 + 9.52628i) q^{79} +1.00000 q^{81} +(5.50000 - 9.52628i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(-0.500000 - 0.866025i) q^{87} +2.00000 q^{89} +7.00000 q^{91} +(-0.500000 - 0.866025i) q^{93} +(-2.50000 + 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} + q^{11} - 7 q^{13} - 2 q^{15} + 3 q^{17} + 5 q^{19} - q^{21} + 3 q^{23} + 2 q^{25} + 2 q^{27} - q^{29} - q^{31} + q^{33} + q^{35} + q^{37} - 7 q^{39} + 7 q^{41} + 16 q^{43} - 2 q^{45} - 3 q^{47} + 6 q^{49} + 3 q^{51} - 4 q^{53} - q^{55} + 5 q^{57} + 8 q^{59} - 11 q^{61} - q^{63} + 7 q^{65} + 16 q^{67} + 3 q^{69} + 13 q^{71} - 11 q^{73} + 2 q^{75} + q^{77} + 11 q^{79} + 2 q^{81} + 11 q^{83} - 3 q^{85} - q^{87} + 4 q^{89} + 14 q^{91} - q^{93} - 5 q^{95} - 7 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 + q^11 - 7 * q^13 - 2 * q^15 + 3 * q^17 + 5 * q^19 - q^21 + 3 * q^23 + 2 * q^25 + 2 * q^27 - q^29 - q^31 + q^33 + q^35 + q^37 - 7 * q^39 + 7 * q^41 + 16 * q^43 - 2 * q^45 - 3 * q^47 + 6 * q^49 + 3 * q^51 - 4 * q^53 - q^55 + 5 * q^57 + 8 * q^59 - 11 * q^61 - q^63 + 7 * q^65 + 16 * q^67 + 3 * q^69 + 13 * q^71 - 11 * q^73 + 2 * q^75 + q^77 + 11 * q^79 + 2 * q^81 + 11 * q^83 - 3 * q^85 - q^87 + 4 * q^89 + 14 * q^91 - q^93 - 5 * q^95 - 7 * q^97 + q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	−3.50000	+	6.06218i	−0.970725	+	1.68135i	−0.277350	+	0.960769i	$0.589456\pi$
$13$	−3.50000	+	6.06218i	−0.970725	+	1.68135i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$	−0.500000	−	0.866025i	−0.109109	−	0.188982i
$22$		0			0
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	$-0.732076\pi$
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	0.978961	+	0.204046i	$0.0654092\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	0.817625	−	0.575751i	$-0.195290\pi$
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	−0.907428	+	0.420208i	$0.861957\pi$
$32$		0			0
$33$	0.500000	+	0.866025i	0.0870388	+	0.150756i
$34$		0			0
$35$	0.500000	+	0.866025i	0.0845154	+	0.146385i
$36$		0			0
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$	−3.50000	+	6.06218i	−0.560449	+	0.970725i
$40$		0			0
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	0.998504	+	0.0546823i	$0.0174146\pi$
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	−0.451896	+	0.892071i	$0.649252\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$	−1.50000	−	2.59808i	−0.218797	−	0.378968i	0.735643	−	0.677369i	$-0.236880\pi$
$47$	−1.50000	−	2.59808i	−0.218797	−	0.378968i	−0.954441	+	0.298401i	$0.903547\pi$
$48$		0			0
$49$	3.00000	−	5.19615i	0.428571	−	0.742307i
$50$		0			0
$51$	1.50000	−	2.59808i	0.210042	−	0.363803i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−0.500000	−	0.866025i	−0.0674200	−	0.116775i
$56$		0			0
$57$	2.50000	−	4.33013i	0.331133	−	0.573539i
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$	−0.500000	−	0.866025i	−0.0629941	−	0.109109i
$64$		0			0
$65$	3.50000	−	6.06218i	0.434122	−	0.751921i
$66$		0			0
$67$	8.00000	+	1.73205i	0.977356	+	0.211604i
$67$	8.00000	+	1.73205i	0.977356	+	0.211604i
$68$		0			0
$69$	1.50000	−	2.59808i	0.180579	−	0.312772i
$70$		0			0
$71$	6.50000	+	11.2583i	0.771408	+	1.33612i	0.936791	+	0.349889i	$0.113781\pi$
$71$	6.50000	+	11.2583i	0.771408	+	1.33612i	−0.165383	+	0.986229i	$0.552886\pi$
$72$		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
$75$	1.00000			0.115470
$76$		0			0
$77$	0.500000	−	0.866025i	0.0569803	−	0.0986928i
$78$		0			0
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	0.989705	+	0.143120i	$0.0457135\pi$
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	−0.370907	+	0.928670i	$0.620953\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.50000	−	9.52628i	0.603703	−	1.04565i	−0.388552	−	0.921427i	$-0.627024\pi$
$83$	5.50000	−	9.52628i	0.603703	−	1.04565i	0.992255	−	0.124218i	$-0.0396422\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$	−0.500000	−	0.866025i	−0.0536056	−	0.0928477i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	7.00000			0.733799
$92$		0			0
$93$	−0.500000	−	0.866025i	−0.0518476	−	0.0898027i
$94$		0			0
$95$	−2.50000	+	4.33013i	−0.256495	+	0.444262i
$96$		0			0
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	−0.987181	−	0.159602i	$-0.948979\pi$
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	0.631810	+	0.775123i	$0.282312\pi$
$98$		0			0
$99$	0.500000	+	0.866025i	0.0502519	+	0.0870388i
$100$		0			0
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	0.949595	+	0.313478i	$0.101494\pi$
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	−0.203317	+	0.979113i	$0.565172\pi$
$102$		0			0
$103$	−2.50000	−	4.33013i	−0.246332	−	0.426660i	0.716173	−	0.697923i	$-0.245892\pi$
$103$	−2.50000	−	4.33013i	−0.246332	−	0.426660i	−0.962505	+	0.271263i	$0.912559\pi$
$104$		0			0
$105$	0.500000	+	0.866025i	0.0487950	+	0.0845154i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	0.500000	−	0.866025i	0.0474579	−	0.0821995i
$112$		0			0
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	0.999796	+	0.0202056i	$0.00643208\pi$
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	−0.482399	+	0.875951i	$0.660235\pi$
$114$		0			0
$115$	−1.50000	+	2.59808i	−0.139876	+	0.242272i
$116$		0			0
$117$	−3.50000	+	6.06218i	−0.323575	+	0.560449i
$118$		0			0
$119$	−3.00000			−0.275010
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$	3.50000	+	6.06218i	0.315584	+	0.546608i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.50000	−	11.2583i	−0.576782	−	0.999015i	−0.995846	−	0.0910585i	$-0.970975\pi$
$127$	−6.50000	−	11.2583i	−0.576782	−	0.999015i	0.419064	−	0.907957i	$-0.362358\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		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
$133$	−5.00000			−0.433555
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	−1.50000	−	2.59808i	−0.126323	−	0.218797i
$142$		0			0
$143$	−7.00000			−0.585369
$144$		0			0
$145$	0.500000	+	0.866025i	0.0415227	+	0.0719195i
$146$		0			0
$147$	3.00000	−	5.19615i	0.247436	−	0.428571i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$		0			0
$153$	1.50000	−	2.59808i	0.121268	−	0.210042i
$154$		0			0
$155$	0.500000	+	0.866025i	0.0401610	+	0.0695608i
$156$		0			0
$157$	4.50000	−	7.79423i	0.359139	−	0.622047i	−0.628678	−	0.777666i	$-0.716404\pi$
$157$	4.50000	−	7.79423i	0.359139	−	0.622047i	0.987817	+	0.155618i	$0.0497370\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$	−3.00000			−0.236433
$162$		0			0
$163$	−0.500000	−	0.866025i	−0.0391630	−	0.0678323i	0.845780	−	0.533533i	$-0.179136\pi$
$163$	−0.500000	−	0.866025i	−0.0391630	−	0.0678323i	−0.884943	+	0.465700i	$0.845802\pi$
$164$		0			0
$165$	−0.500000	−	0.866025i	−0.0389249	−	0.0674200i
$166$		0			0
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	0.985933	−	0.167139i	$-0.0534527\pi$
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	−0.637713	+	0.770274i	$0.720119\pi$
$168$		0			0
$169$	−18.0000	−	31.1769i	−1.38462	−	2.39822i
$170$		0			0
$171$	2.50000	−	4.33013i	0.191180	−	0.331133i
$172$		0			0
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	−0.426329	−	0.904568i	$-0.640193\pi$
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	0.996544	−	0.0830722i	$-0.0264732\pi$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.0377964	−	0.0654654i
$176$		0			0
$177$	4.00000			0.300658
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−3.50000	−	6.06218i	−0.260153	−	0.450598i	0.706129	−	0.708083i	$-0.250440\pi$
$181$	−3.50000	−	6.06218i	−0.260153	−	0.450598i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	−5.50000	+	9.52628i	−0.406572	+	0.704203i
$184$		0			0
$185$	−0.500000	+	0.866025i	−0.0367607	+	0.0636715i
$186$		0			0
$187$	3.00000			0.219382
$188$		0			0
$189$	−0.500000	−	0.866025i	−0.0363696	−	0.0629941i
$190$		0			0
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	−0.981635	−	0.190767i	$-0.938902\pi$
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	0.656027	+	0.754738i	$0.272236\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	3.50000	−	6.06218i	0.250640	−	0.434122i
$196$		0			0
$197$	−8.50000	−	14.7224i	−0.605600	−	1.04893i	−0.991956	−	0.126580i	$-0.959600\pi$
$197$	−8.50000	−	14.7224i	−0.605600	−	1.04893i	0.386356	−	0.922350i	$-0.373733\pi$
$198$		0			0
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	−0.914282	−	0.405079i	$-0.867244\pi$
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	0.807950	+	0.589252i	$0.200577\pi$
$200$		0			0
$201$	8.00000	+	1.73205i	0.564276	+	0.122169i
$202$		0			0
$203$	−0.500000	+	0.866025i	−0.0350931	+	0.0607831i
$204$		0			0
$205$	−3.50000	−	6.06218i	−0.244451	−	0.423401i
$206$		0			0
$207$	1.50000	−	2.59808i	0.104257	−	0.180579i
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	−1.50000	+	2.59808i	−0.103264	+	0.178859i	−0.913028	−	0.407898i	$-0.866262\pi$
$211$	−1.50000	+	2.59808i	−0.103264	+	0.178859i	0.809763	+	0.586756i	$0.199595\pi$
$212$		0			0
$213$	6.50000	+	11.2583i	0.445373	+	0.771408i
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−0.500000	+	0.866025i	−0.0339422	+	0.0587896i
$218$		0			0
$219$	−5.50000	+	9.52628i	−0.371656	+	0.643726i
$220$		0			0
$221$	10.5000	+	18.1865i	0.706306	+	1.22336i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	6.50000	+	11.2583i	0.431420	+	0.747242i	0.996996	−	0.0774548i	$-0.0246793\pi$
$227$	6.50000	+	11.2583i	0.431420	+	0.747242i	−0.565576	+	0.824696i	$0.691346\pi$
$228$		0			0
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	−0.999987	−	0.00505719i	$-0.998390\pi$
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	0.504373	+	0.863486i	$0.331724\pi$
$230$		0			0
$231$	0.500000	−	0.866025i	0.0328976	−	0.0569803i
$232$		0			0
$233$	−14.5000	−	25.1147i	−0.949927	−	1.64532i	−0.745573	−	0.666424i	$-0.767824\pi$
$233$	−14.5000	−	25.1147i	−0.949927	−	1.64532i	−0.204354	−	0.978897i	$-0.565509\pi$
$234$		0			0
$235$	1.50000	+	2.59808i	0.0978492	+	0.169480i
$236$		0			0
$237$	5.50000	+	9.52628i	0.357263	+	0.618798i
$238$		0			0
$239$	2.50000	+	4.33013i	0.161712	+	0.280093i	0.935483	−	0.353373i	$-0.114965\pi$
$239$	2.50000	+	4.33013i	0.161712	+	0.280093i	−0.773771	+	0.633465i	$0.781632\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−3.00000	+	5.19615i	−0.191663	+	0.331970i
$246$		0			0
$247$	17.5000	+	30.3109i	1.11350	+	1.92864i
$248$		0			0
$249$	5.50000	−	9.52628i	0.348548	−	0.603703i
$250$		0			0
$251$	11.5000	−	19.9186i	0.725874	−	1.25725i	−0.232740	−	0.972539i	$-0.574769\pi$
$251$	11.5000	−	19.9186i	0.725874	−	1.25725i	0.958613	−	0.284711i	$-0.0918976\pi$
$252$		0			0
$253$	3.00000			0.188608
$254$		0			0
$255$	−1.50000	+	2.59808i	−0.0939336	+	0.162698i
$256$		0			0
$257$	15.5000	+	26.8468i	0.966863	+	1.67466i	0.704523	+	0.709681i	$0.251161\pi$
$257$	15.5000	+	26.8468i	0.966863	+	1.67466i	0.262341	+	0.964975i	$0.415506\pi$
$258$		0			0
$259$	−1.00000			−0.0621370
$260$		0			0
$261$	−0.500000	−	0.866025i	−0.0309492	−	0.0536056i
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$	2.00000			0.122859
$266$		0			0
$267$	2.00000			0.122398
$268$		0			0
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$	7.00000			0.423659
$274$		0			0
$275$	0.500000	+	0.866025i	0.0301511	+	0.0522233i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$	−0.500000	−	0.866025i	−0.0299342	−	0.0518476i
$280$		0			0
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	−0.550804	−	0.834634i	$-0.685679\pi$
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	0.998217	+	0.0596933i	$0.0190123\pi$
$282$		0			0
$283$	−16.0000			−0.951101			−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000			−0.951101			−0.475551	+	0.879688i	$0.657751\pi$
$284$		0			0
$285$	−2.50000	+	4.33013i	−0.148087	+	0.256495i
$286$		0			0
$287$	3.50000	−	6.06218i	0.206598	−	0.357839i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	−3.50000	+	6.06218i	−0.205174	+	0.355371i
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$		0			0
$297$	0.500000	+	0.866025i	0.0290129	+	0.0502519i
$298$		0			0
$299$	10.5000	+	18.1865i	0.607231	+	1.05175i
$300$		0			0
$301$	−4.00000	−	6.92820i	−0.230556	−	0.399335i
$302$		0			0
$303$	7.50000	+	12.9904i	0.430864	+	0.746278i
$304$		0			0
$305$	5.50000	−	9.52628i	0.314929	−	0.545473i
$306$		0			0
$307$	4.50000	−	7.79423i	0.256829	−	0.444840i	−0.708562	−	0.705649i	$-0.750656\pi$
$307$	4.50000	−	7.79423i	0.256829	−	0.444840i	0.965391	+	0.260808i	$0.0839891\pi$
$308$		0			0
$309$	−2.50000	−	4.33013i	−0.142220	−	0.246332i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$	0.500000	+	0.866025i	0.0281718	+	0.0487950i
$316$		0			0
$317$	−8.50000	+	14.7224i	−0.477408	+	0.826894i	−0.999665	−	0.0258939i	$-0.991757\pi$
$317$	−8.50000	+	14.7224i	−0.477408	+	0.826894i	0.522257	+	0.852788i	$0.325090\pi$
$318$		0			0
$319$	0.500000	−	0.866025i	0.0279946	−	0.0484881i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.50000	−	12.9904i	−0.417311	−	0.722804i
$324$		0			0
$325$	−3.50000	+	6.06218i	−0.194145	+	0.336269i
$326$		0			0
$327$	10.0000			0.553001
$328$		0			0
$329$	−1.50000	+	2.59808i	−0.0826977	+	0.143237i
$330$		0			0
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	0.789104	−	0.614260i	$-0.210545\pi$
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	−0.926516	+	0.376254i	$0.877212\pi$
$332$		0			0
$333$	0.500000	−	0.866025i	0.0273998	−	0.0474579i
$334$		0			0
$335$	−8.00000	−	1.73205i	−0.437087	−	0.0946320i
$336$		0			0
$337$	−11.5000	+	19.9186i	−0.626445	+	1.08503i	0.361815	+	0.932250i	$0.382157\pi$
$337$	−11.5000	+	19.9186i	−0.626445	+	1.08503i	−0.988260	+	0.152784i	$0.951176\pi$
$338$		0			0
$339$	5.50000	+	9.52628i	0.298719	+	0.517396i
$340$		0			0
$341$	0.500000	−	0.866025i	0.0270765	−	0.0468979i
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$	−1.50000	+	2.59808i	−0.0807573	+	0.139876i
$346$		0			0
$347$	16.5000	+	28.5788i	0.885766	+	1.53419i	0.844833	+	0.535031i	$0.179700\pi$
$347$	16.5000	+	28.5788i	0.885766	+	1.53419i	0.0409337	+	0.999162i	$0.486967\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	−3.50000	+	6.06218i	−0.186816	+	0.323575i
$352$		0			0
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	0.438733	+	0.898617i	$0.355427\pi$
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	−0.997592	+	0.0693543i	$0.977906\pi$
$354$		0			0
$355$	−6.50000	−	11.2583i	−0.344984	−	0.597530i
$356$		0			0
$357$	−3.00000			−0.158777
$358$		0			0
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$	5.00000	−	8.66025i	0.262432	−	0.454545i
$364$		0			0
$365$	5.50000	−	9.52628i	0.287883	−	0.498628i
$366$		0			0
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	0.793370	−	0.608740i	$-0.208325\pi$
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	−0.923869	+	0.382709i	$0.874991\pi$
$368$		0			0
$369$	3.50000	+	6.06218i	0.182203	+	0.315584i
$370$		0			0
$371$	1.00000	+	1.73205i	0.0519174	+	0.0899236i
$372$		0			0
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	0.761074	−	0.648665i	$-0.224672\pi$
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	−0.942297	+	0.334777i	$0.891339\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	7.00000			0.360518
$378$		0			0
$379$	−15.5000	+	26.8468i	−0.796182	+	1.37903i	0.125905	+	0.992042i	$0.459817\pi$
$379$	−15.5000	+	26.8468i	−0.796182	+	1.37903i	−0.922086	+	0.386985i	$0.873517\pi$
$380$		0			0
$381$	−6.50000	−	11.2583i	−0.333005	−	0.576782i
$382$		0			0
$383$	9.50000	−	16.4545i	0.485427	−	0.840785i	−0.514432	−	0.857531i	$-0.671997\pi$
$383$	9.50000	−	16.4545i	0.485427	−	0.840785i	0.999860	+	0.0167461i	$0.00533070\pi$
$384$		0			0
$385$	−0.500000	+	0.866025i	−0.0254824	+	0.0441367i
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	−0.825491	−	0.564416i	$-0.809102\pi$
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	0.901544	+	0.432688i	$0.142435\pi$
$390$		0			0
$391$	−4.50000	−	7.79423i	−0.227575	−	0.394171i
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$	−5.50000	−	9.52628i	−0.276735	−	0.479319i
$396$		0			0
$397$	−30.0000			−1.50566			−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000			−1.50566			−0.752828	+	0.658217i	$0.771311\pi$
$398$		0			0
$399$	−5.00000			−0.250313
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$	7.00000			0.348695
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	1.00000			0.0495682
$408$		0			0
$409$	−13.5000	−	23.3827i	−0.667532	−	1.15620i	−0.978592	−	0.205809i	$-0.934017\pi$
$409$	−13.5000	−	23.3827i	−0.667532	−	1.15620i	0.311060	−	0.950390i	$-0.399316\pi$
$410$		0			0
$411$	18.0000			0.887875
$412$		0			0
$413$	−2.00000	−	3.46410i	−0.0984136	−	0.170457i
$414$		0			0
$415$	−5.50000	+	9.52628i	−0.269984	+	0.467627i
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$	−8.50000	+	14.7224i	−0.415252	+	0.719238i	−0.995455	−	0.0952342i	$-0.969640\pi$
$419$	−8.50000	+	14.7224i	−0.415252	+	0.719238i	0.580203	+	0.814472i	$0.302973\pi$
$420$		0			0
$421$	2.50000	−	4.33013i	0.121843	−	0.211037i	−0.798652	−	0.601793i	$-0.794453\pi$
$421$	2.50000	−	4.33013i	0.121843	−	0.211037i	0.920494	+	0.390756i	$0.127786\pi$
$422$		0			0
$423$	−1.50000	−	2.59808i	−0.0729325	−	0.126323i
$424$		0			0
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$	11.0000			0.532327
$428$		0			0
$429$	−7.00000			−0.337963
$430$		0			0
$431$	−7.50000	−	12.9904i	−0.361262	−	0.625725i	0.626907	−	0.779094i	$-0.284321\pi$
$431$	−7.50000	−	12.9904i	−0.361262	−	0.625725i	−0.988169	+	0.153370i	$0.950987\pi$
$432$		0			0
$433$	−1.50000	−	2.59808i	−0.0720854	−	0.124856i	0.827730	−	0.561127i	$-0.189632\pi$
$433$	−1.50000	−	2.59808i	−0.0720854	−	0.124856i	−0.899815	+	0.436271i	$0.856299\pi$
$434$		0			0
$435$	0.500000	+	0.866025i	0.0239732	+	0.0415227i
$436$		0			0
$437$	−7.50000	−	12.9904i	−0.358774	−	0.621414i
$438$		0			0
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	−0.853847	−	0.520524i	$-0.825737\pi$
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	0.877711	+	0.479191i	$0.159070\pi$
$440$		0			0
$441$	3.00000	−	5.19615i	0.142857	−	0.247436i
$442$		0			0
$443$	−11.5000	−	19.9186i	−0.546381	−	0.946360i	−0.998519	−	0.0544120i	$-0.982672\pi$
$443$	−11.5000	−	19.9186i	−0.546381	−	0.946360i	0.452137	−	0.891948i	$-0.350662\pi$
$444$		0			0
$445$	−2.00000			−0.0948091
$446$		0			0
$447$	14.0000			0.662177
$448$		0			0
$449$	−4.50000	−	7.79423i	−0.212368	−	0.367832i	0.740087	−	0.672511i	$-0.234784\pi$
$449$	−4.50000	−	7.79423i	−0.212368	−	0.367832i	−0.952455	+	0.304679i	$0.901451\pi$
$450$		0			0
$451$	−3.50000	+	6.06218i	−0.164809	+	0.285457i
$452$		0			0
$453$	8.50000	−	14.7224i	0.399365	−	0.691720i
$454$		0			0
$455$	−7.00000			−0.328165
$456$		0			0
$457$	−7.50000	−	12.9904i	−0.350835	−	0.607664i	0.635561	−	0.772051i	$-0.280769\pi$
$457$	−7.50000	−	12.9904i	−0.350835	−	0.607664i	−0.986396	+	0.164386i	$0.947436\pi$
$458$		0			0
$459$	1.50000	−	2.59808i	0.0700140	−	0.121268i
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	−15.5000	+	26.8468i	−0.720346	+	1.24768i	0.240515	+	0.970645i	$0.422684\pi$
$463$	−15.5000	+	26.8468i	−0.720346	+	1.24768i	−0.960861	+	0.277031i	$0.910650\pi$
$464$		0			0
$465$	0.500000	+	0.866025i	0.0231869	+	0.0401610i
$466$		0			0
$467$	5.50000	−	9.52628i	0.254510	−	0.440824i	−0.710253	−	0.703947i	$-0.751419\pi$
$467$	5.50000	−	9.52628i	0.254510	−	0.440824i	0.964762	+	0.263123i	$0.0847526\pi$
$468$		0			0
$469$	−2.50000	−	7.79423i	−0.115439	−	0.359904i
$470$		0			0
$471$	4.50000	−	7.79423i	0.207349	−	0.359139i
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	−0.373230	−	0.927739i	$-0.621750\pi$
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	0.990060	−	0.140643i	$-0.0449170\pi$
$480$		0			0
$481$	3.50000	+	6.06218i	0.159586	+	0.276412i
$482$		0			0
$483$	−3.00000			−0.136505
$484$		0			0
$485$	3.50000	−	6.06218i	0.158927	−	0.275269i
$486$		0			0
$487$	−1.50000	+	2.59808i	−0.0679715	+	0.117730i	−0.898008	−	0.439979i	$-0.854986\pi$
$487$	−1.50000	+	2.59808i	−0.0679715	+	0.117730i	0.830037	+	0.557709i	$0.188319\pi$
$488$		0			0
$489$	−0.500000	−	0.866025i	−0.0226108	−	0.0391630i
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−3.00000			−0.135113
$494$		0			0
$495$	−0.500000	−	0.866025i	−0.0224733	−	0.0389249i
$496$		0			0
$497$	6.50000	−	11.2583i	0.291565	−	0.505005i
$498$		0			0
$499$	8.50000	−	14.7224i	0.380512	−	0.659067i	−0.610623	−	0.791921i	$-0.709081\pi$
$499$	8.50000	−	14.7224i	0.380512	−	0.659067i	0.991136	+	0.132855i	$0.0424144\pi$
$500$		0			0
$501$	4.50000	+	7.79423i	0.201045	+	0.348220i
$502$		0			0
$503$	4.50000	+	7.79423i	0.200645	+	0.347527i	0.948736	−	0.316068i	$-0.102363\pi$
$503$	4.50000	+	7.79423i	0.200645	+	0.347527i	−0.748091	+	0.663596i	$0.769030\pi$
$504$		0			0
$505$	−7.50000	−	12.9904i	−0.333746	−	0.578064i
$506$		0			0
$507$	−18.0000	−	31.1769i	−0.799408	−	1.38462i
$508$		0			0
$509$	14.0000			0.620539			0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000			0.620539			0.310270	+	0.950649i	$0.399581\pi$
$510$		0			0
$511$	11.0000			0.486611
$512$		0			0
$513$	2.50000	−	4.33013i	0.110378	−	0.191180i
$514$		0			0
$515$	2.50000	+	4.33013i	0.110163	+	0.190808i
$516$		0			0
$517$	1.50000	−	2.59808i	0.0659699	−	0.114263i
$518$		0			0
$519$	7.50000	−	12.9904i	0.329213	−	0.570214i
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$	−1.50000	+	2.59808i	−0.0655904	+	0.113606i	−0.896956	−	0.442120i	$-0.854226\pi$
$523$	−1.50000	+	2.59808i	−0.0655904	+	0.113606i	0.831365	+	0.555726i	$0.187560\pi$
$524$		0			0
$525$	−0.500000	−	0.866025i	−0.0218218	−	0.0377964i
$526$		0			0
$527$	−3.00000			−0.130682
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$	−49.0000			−2.12243
$534$		0			0
$535$		0			0
$536$		0			0
$537$	24.0000			1.03568
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	14.0000			0.601907			0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000			0.601907			0.300954	+	0.953639i	$0.402695\pi$
$542$		0			0
$543$	−3.50000	−	6.06218i	−0.150199	−	0.260153i
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	−0.989490	−	0.144604i	$-0.953809\pi$
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	0.369514	−	0.929225i	$-0.379524\pi$
$548$		0			0
$549$	−5.50000	+	9.52628i	−0.234734	+	0.406572i
$550$		0			0
$551$	−5.00000			−0.213007
$552$		0			0
$553$	5.50000	−	9.52628i	0.233884	−	0.405099i
$554$		0			0
$555$	−0.500000	+	0.866025i	−0.0212238	+	0.0367607i
$556$		0			0
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	0.896053	−	0.443947i	$-0.146422\pi$
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	−0.832496	+	0.554031i	$0.813089\pi$
$558$		0			0
$559$	−28.0000	+	48.4974i	−1.18427	+	2.05122i
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	24.0000			1.01148			0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000			1.01148			0.505740	+	0.862686i	$0.331220\pi$
$564$		0			0
$565$	−5.50000	−	9.52628i	−0.231387	−	0.400774i
$566$		0			0
$567$	−0.500000	−	0.866025i	−0.0209980	−	0.0363696i
$568$		0			0
$569$	−4.50000	−	7.79423i	−0.188650	−	0.326751i	0.756151	−	0.654398i	$-0.227078\pi$
$569$	−4.50000	−	7.79423i	−0.188650	−	0.326751i	−0.944800	+	0.327647i	$0.893744\pi$
$570$		0			0
$571$	21.5000	+	37.2391i	0.899747	+	1.55841i	0.827817	+	0.560998i	$0.189582\pi$
$571$	21.5000	+	37.2391i	0.899747	+	1.55841i	0.0719297	+	0.997410i	$0.477084\pi$
$572$		0			0
$573$	−4.50000	+	7.79423i	−0.187990	+	0.325609i
$574$		0			0
$575$	1.50000	−	2.59808i	0.0625543	−	0.108347i
$576$		0			0
$577$	−15.5000	−	26.8468i	−0.645273	−	1.11765i	−0.984238	−	0.176847i	$-0.943410\pi$
$577$	−15.5000	−	26.8468i	−0.645273	−	1.11765i	0.338965	−	0.940799i	$-0.389923\pi$
$578$		0			0
$579$	−2.00000			−0.0831172
$580$		0			0
$581$	−11.0000			−0.456357
$582$		0			0
$583$	−1.00000	−	1.73205i	−0.0414158	−	0.0717342i
$584$		0			0
$585$	3.50000	−	6.06218i	0.144707	−	0.250640i
$586$		0			0
$587$	−0.500000	+	0.866025i	−0.0206372	+	0.0357447i	−0.876160	−	0.482021i	$-0.839903\pi$
$587$	−0.500000	+	0.866025i	−0.0206372	+	0.0357447i	0.855522	+	0.517766i	$0.173236\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$	−8.50000	−	14.7224i	−0.349643	−	0.605600i
$592$		0			0
$593$	−2.50000	+	4.33013i	−0.102663	+	0.177817i	−0.912781	−	0.408450i	$-0.866070\pi$
$593$	−2.50000	+	4.33013i	−0.102663	+	0.177817i	0.810118	+	0.586267i	$0.199403\pi$
$594$		0			0
$595$	3.00000			0.122988
$596$		0			0
$597$	−1.50000	+	2.59808i	−0.0613909	+	0.106332i
$598$		0			0
$599$	22.5000	+	38.9711i	0.919325	+	1.59232i	0.800443	+	0.599409i	$0.204598\pi$
$599$	22.5000	+	38.9711i	0.919325	+	1.59232i	0.118882	+	0.992908i	$0.462069\pi$
$600$		0			0
$601$	8.50000	−	14.7224i	0.346722	−	0.600541i	−0.638943	−	0.769254i	$-0.720628\pi$
$601$	8.50000	−	14.7224i	0.346722	−	0.600541i	0.985665	+	0.168714i	$0.0539613\pi$
$602$		0			0
$603$	8.00000	+	1.73205i	0.325785	+	0.0705346i
$604$		0			0
$605$	−5.00000	+	8.66025i	−0.203279	+	0.352089i
$606$		0			0
$607$	13.5000	+	23.3827i	0.547948	+	0.949074i	0.998415	+	0.0562808i	$0.0179242\pi$
$607$	13.5000	+	23.3827i	0.547948	+	0.949074i	−0.450467	+	0.892793i	$0.648742\pi$
$608$		0			0
$609$	−0.500000	+	0.866025i	−0.0202610	+	0.0350931i
$610$		0			0
$611$	21.0000			0.849569
$612$		0			0
$613$	10.5000	−	18.1865i	0.424091	−	0.734547i	−0.572244	−	0.820083i	$-0.693927\pi$
$613$	10.5000	−	18.1865i	0.424091	−	0.734547i	0.996335	+	0.0855362i	$0.0272603\pi$
$614$		0			0
$615$	−3.50000	−	6.06218i	−0.141134	−	0.244451i
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	0.263887	+	0.964554i	$0.414995\pi$
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	−0.967271	+	0.253744i	$0.918338\pi$
$620$		0			0
$621$	1.50000	−	2.59808i	0.0601929	−	0.104257i
$622$		0			0
$623$	−1.00000	−	1.73205i	−0.0400642	−	0.0693932i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.00000			0.199681
$628$		0			0
$629$	−1.50000	−	2.59808i	−0.0598089	−	0.103592i
$630$		0			0
$631$	−13.5000	+	23.3827i	−0.537427	+	0.930850i	0.461615	+	0.887080i	$0.347270\pi$
$631$	−13.5000	+	23.3827i	−0.537427	+	0.930850i	−0.999042	+	0.0437697i	$0.986063\pi$
$632$		0			0
$633$	−1.50000	+	2.59808i	−0.0596196	+	0.103264i
$634$		0			0
$635$	6.50000	+	11.2583i	0.257945	+	0.446773i
$636$		0			0
$637$	21.0000	+	36.3731i	0.832050	+	1.44115i
$638$		0			0
$639$	6.50000	+	11.2583i	0.257136	+	0.445373i
$640$		0			0
$641$	7.50000	+	12.9904i	0.296232	+	0.513089i	0.975271	−	0.221013i	$-0.0709364\pi$
$641$	7.50000	+	12.9904i	0.296232	+	0.513089i	−0.679039	+	0.734103i	$0.737603\pi$
$642$		0			0
$643$	−36.0000			−1.41970			−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000			−1.41970			−0.709851	+	0.704352i	$0.751238\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	−4.50000	+	7.79423i	−0.176913	+	0.306423i	−0.940822	−	0.338902i	$-0.889945\pi$
$647$	−4.50000	+	7.79423i	−0.176913	+	0.306423i	0.763908	+	0.645325i	$0.223278\pi$
$648$		0			0
$649$	2.00000	+	3.46410i	0.0785069	+	0.135978i
$650$		0			0
$651$	−0.500000	+	0.866025i	−0.0195965	+	0.0339422i
$652$		0			0
$653$	−16.5000	+	28.5788i	−0.645695	+	1.11838i	0.338446	+	0.940986i	$0.390099\pi$
$653$	−16.5000	+	28.5788i	−0.645695	+	1.11838i	−0.984141	+	0.177390i	$0.943234\pi$
$654$		0			0
$655$	−4.00000			−0.156293
$656$		0			0
$657$	−5.50000	+	9.52628i	−0.214575	+	0.371656i
$658$		0			0
$659$	−7.50000	−	12.9904i	−0.292159	−	0.506033i	0.682161	−	0.731202i	$-0.261040\pi$
$659$	−7.50000	−	12.9904i	−0.292159	−	0.506033i	−0.974320	+	0.225168i	$0.927707\pi$
$660$		0			0
$661$	50.0000			1.94477			0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000			1.94477			0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$	10.5000	+	18.1865i	0.407786	+	0.706306i
$664$		0			0
$665$	5.00000			0.193892
$666$		0			0
$667$	−3.00000			−0.116160
$668$		0			0
$669$	8.00000			0.309298
$670$		0			0
$671$	−11.0000			−0.424650
$672$		0			0
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	3.50000	+	6.06218i	0.134516	+	0.232988i	0.925412	−	0.378962i	$-0.123719\pi$
$677$	3.50000	+	6.06218i	0.134516	+	0.232988i	−0.790897	+	0.611950i	$0.790385\pi$
$678$		0			0
$679$	7.00000			0.268635
$680$		0			0
$681$	6.50000	+	11.2583i	0.249081	+	0.431420i
$682$		0			0
$683$	7.50000	−	12.9904i	0.286980	−	0.497063i	−0.686108	−	0.727500i	$-0.740682\pi$
$683$	7.50000	−	12.9904i	0.286980	−	0.497063i	0.973087	+	0.230437i	$0.0740155\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$		0			0
$687$	−7.50000	+	12.9904i	−0.286143	+	0.495614i
$688$		0			0
$689$	7.00000	−	12.1244i	0.266679	−	0.461901i
$690$		0			0
$691$	15.5000	+	26.8468i	0.589648	+	1.02130i	0.994278	+	0.106820i	$0.0340668\pi$
$691$	15.5000	+	26.8468i	0.589648	+	1.02130i	−0.404631	+	0.914480i	$0.632600\pi$
$692$		0			0
$693$	0.500000	−	0.866025i	0.0189934	−	0.0328976i
$694$		0			0
$695$	4.00000			0.151729
$696$		0			0
$697$	21.0000			0.795432
$698$		0			0
$699$	−14.5000	−	25.1147i	−0.548440	−	0.949927i
$700$		0			0
$701$	−0.500000	−	0.866025i	−0.0188847	−	0.0327093i	0.856429	−	0.516265i	$-0.172678\pi$
$701$	−0.500000	−	0.866025i	−0.0188847	−	0.0327093i	−0.875313	+	0.483556i	$0.839345\pi$
$702$		0			0
$703$	−2.50000	−	4.33013i	−0.0942893	−	0.163314i
$704$		0			0
$705$	1.50000	+	2.59808i	0.0564933	+	0.0978492i
$706$		0			0
$707$	7.50000	−	12.9904i	0.282067	−	0.488554i
$708$		0			0
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	−0.167727	−	0.985834i	$-0.553643\pi$
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	0.937620	−	0.347661i	$-0.113024\pi$
$710$		0			0
$711$	5.50000	+	9.52628i	0.206266	+	0.357263i
$712$		0			0
$713$	−3.00000			−0.112351
$714$		0			0
$715$	7.00000			0.261785
$716$		0			0
$717$	2.50000	+	4.33013i	0.0933642	+	0.161712i
$718$		0			0
$719$	17.5000	−	30.3109i	0.652640	−	1.13041i	−0.329840	−	0.944037i	$-0.606995\pi$
$719$	17.5000	−	30.3109i	0.652640	−	1.13041i	0.982480	−	0.186369i	$-0.0596719\pi$
$720$		0			0
$721$	−2.50000	+	4.33013i	−0.0931049	+	0.161262i
$722$		0			0
$723$	−26.0000			−0.966950
$724$		0			0
$725$	−0.500000	−	0.866025i	−0.0185695	−	0.0321634i
$726$		0			0
$727$	4.50000	−	7.79423i	0.166896	−	0.289072i	−0.770431	−	0.637523i	$-0.779959\pi$
$727$	4.50000	−	7.79423i	0.166896	−	0.289072i	0.937327	+	0.348451i	$0.113292\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	12.0000	−	20.7846i	0.443836	−	0.768747i
$732$		0			0
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	0.635515	−	0.772088i	$-0.280788\pi$
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	−0.986406	+	0.164328i	$0.947454\pi$
$734$		0			0
$735$	−3.00000	+	5.19615i	−0.110657	+	0.191663i
$736$		0			0
$737$	2.50000	+	7.79423i	0.0920887	+	0.287104i
$738$		0			0
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	−0.771310	−	0.636460i	$-0.780398\pi$
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	0.936845	+	0.349744i	$0.113732\pi$
$740$		0			0
$741$	17.5000	+	30.3109i	0.642879	+	1.11350i
$742$		0			0
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	0.467351	+	0.884072i	$0.345209\pi$
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	−0.999304	+	0.0372984i	$0.988125\pi$
$744$		0			0
$745$	−14.0000			−0.512920
$746$		0			0
$747$	5.50000	−	9.52628i	0.201234	−	0.348548i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−44.0000			−1.60558			−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000			−1.60558			−0.802791	+	0.596260i	$0.796653\pi$
$752$		0			0
$753$	11.5000	−	19.9186i	0.419083	−	0.725874i
$754$		0			0
$755$	−8.50000	+	14.7224i	−0.309347	+	0.535804i
$756$		0			0
$757$	−21.5000	−	37.2391i	−0.781431	−	1.35348i	−0.931108	−	0.364743i	$-0.881157\pi$
$757$	−21.5000	−	37.2391i	−0.781431	−	1.35348i	0.149677	−	0.988735i	$-0.452176\pi$
$758$		0			0
$759$	3.00000			0.108893
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$	−5.00000	−	8.66025i	−0.181012	−	0.313522i
$764$		0			0
$765$	−1.50000	+	2.59808i	−0.0542326	+	0.0939336i
$766$		0			0
$767$	−14.0000	+	24.2487i	−0.505511	+	0.875570i
$768$		0			0
$769$	−25.5000	−	44.1673i	−0.919554	−	1.59271i	−0.800094	−	0.599874i	$-0.795217\pi$
$769$	−25.5000	−	44.1673i	−0.919554	−	1.59271i	−0.119459	−	0.992839i	$-0.538116\pi$
$770$		0			0
$771$	15.5000	+	26.8468i	0.558219	+	0.966863i
$772$		0			0
$773$	23.5000	+	40.7032i	0.845236	+	1.46399i	0.885416	+	0.464800i	$0.153874\pi$
$773$	23.5000	+	40.7032i	0.845236	+	1.46399i	−0.0401796	+	0.999192i	$0.512793\pi$
$774$		0			0
$775$	−0.500000	−	0.866025i	−0.0179605	−	0.0311086i
$776$		0			0
$777$	−1.00000			−0.0358748
$778$		0			0
$779$	35.0000			1.25401
$780$		0			0
$781$	−6.50000	+	11.2583i	−0.232588	+	0.402855i
$782$		0			0
$783$	−0.500000	−	0.866025i	−0.0178685	−	0.0309492i
$784$		0			0
$785$	−4.50000	+	7.79423i	−0.160612	+	0.278188i
$786$		0			0
$787$	−7.50000	+	12.9904i	−0.267346	+	0.463057i	−0.968176	−	0.250272i	$-0.919480\pi$
$787$	−7.50000	+	12.9904i	−0.267346	+	0.463057i	0.700830	+	0.713329i	$0.252813\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$	5.50000	−	9.52628i	0.195557	−	0.338716i
$792$		0			0
$793$	−38.5000	−	66.6840i	−1.36718	−	2.36802i
$794$		0			0
$795$	2.00000			0.0709327
$796$		0			0
$797$	−26.5000	−	45.8993i	−0.938678	−	1.62584i	−0.767940	−	0.640522i	$-0.778718\pi$
$797$	−26.5000	−	45.8993i	−0.938678	−	1.62584i	−0.170738	−	0.985316i	$-0.554615\pi$
$798$		0			0
$799$	−9.00000			−0.318397
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$	−11.0000			−0.388182
$804$		0			0
$805$	3.00000			0.105736
$806$		0			0
$807$	30.0000			1.05605
$808$		0			0
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	17.5000	+	30.3109i	0.614508	+	1.06436i	0.990471	+	0.137724i	$0.0439788\pi$
$811$	17.5000	+	30.3109i	0.614508	+	1.06436i	−0.375962	+	0.926635i	$0.622688\pi$
$812$		0			0
$813$	−8.00000			−0.280572
$814$		0			0
$815$	0.500000	+	0.866025i	0.0175142	+	0.0303355i
$816$		0			0
$817$	20.0000	−	34.6410i	0.699711	−	1.21194i
$818$		0			0
$819$	7.00000			0.244600
$820$		0			0
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	−0.838663	−	0.544651i	$-0.816662\pi$
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	0.891013	+	0.453978i	$0.149995\pi$
$822$		0			0
$823$	−7.50000	+	12.9904i	−0.261434	+	0.452816i	−0.966623	−	0.256203i	$-0.917529\pi$
$823$	−7.50000	+	12.9904i	−0.261434	+	0.452816i	0.705190	+	0.709019i	$0.250862\pi$
$824$		0			0
$825$	0.500000	+	0.866025i	0.0174078	+	0.0301511i
$826$		0			0
$827$	−6.50000	+	11.2583i	−0.226027	+	0.391491i	−0.956627	−	0.291315i	$-0.905907\pi$
$827$	−6.50000	+	11.2583i	−0.226027	+	0.391491i	0.730600	+	0.682806i	$0.239240\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	−9.00000	−	15.5885i	−0.311832	−	0.540108i
$834$		0			0
$835$	−4.50000	−	7.79423i	−0.155729	−	0.269730i
$836$		0			0
$837$	−0.500000	−	0.866025i	−0.0172825	−	0.0299342i
$838$		0			0
$839$	26.5000	+	45.8993i	0.914882	+	1.58462i	0.807075	+	0.590450i	$0.201050\pi$
$839$	26.5000	+	45.8993i	0.914882	+	1.58462i	0.107807	+	0.994172i	$0.465617\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$		0			0
$843$	7.50000	−	12.9904i	0.258314	−	0.447412i
$844$		0			0
$845$	18.0000	+	31.1769i	0.619219	+	1.07252i
$846$		0			0
$847$	−10.0000			−0.343604
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$	−1.50000	−	2.59808i	−0.0514193	−	0.0890609i
$852$		0			0
$853$	−7.50000	+	12.9904i	−0.256795	+	0.444782i	−0.965382	−	0.260842i	$-0.916000\pi$
$853$	−7.50000	+	12.9904i	−0.256795	+	0.444782i	0.708586	+	0.705624i	$0.249333\pi$
$854$		0			0
$855$	−2.50000	+	4.33013i	−0.0854982	+	0.148087i
$856$		0			0
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$	−6.50000	−	11.2583i	−0.221777	−	0.384129i	0.733571	−	0.679613i	$-0.237852\pi$
$859$	−6.50000	−	11.2583i	−0.221777	−	0.384129i	−0.955348	+	0.295484i	$0.904519\pi$
$860$		0			0
$861$	3.50000	−	6.06218i	0.119280	−	0.206598i
$862$		0			0
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$		0			0
$865$	−7.50000	+	12.9904i	−0.255008	+	0.441686i
$866$		0			0
$867$	4.00000	+	6.92820i	0.135847	+	0.235294i
$868$		0			0
$869$	−5.50000	+	9.52628i	−0.186575	+	0.323157i
$870$		0			0
$871$	−38.5000	+	42.4352i	−1.30452	+	1.43786i
$872$		0			0
$873$	−3.50000	+	6.06218i	−0.118457	+	0.205174i
$874$		0			0
$875$	0.500000	+	0.866025i	0.0169031	+	0.0292770i
$876$		0			0
$877$	2.50000	−	4.33013i	0.0844190	−	0.146218i	−0.820724	−	0.571324i	$-0.806430\pi$
$877$	2.50000	−	4.33013i	0.0844190	−	0.146218i	0.905143	+	0.425106i	$0.139763\pi$
$878$		0			0
$879$	−6.00000			−0.202375
$880$		0			0
$881$	11.5000	−	19.9186i	0.387445	−	0.671074i	−0.604660	−	0.796484i	$-0.706691\pi$
$881$	11.5000	−	19.9186i	0.387445	−	0.671074i	0.992105	+	0.125409i	$0.0400244\pi$
$882$		0			0
$883$	−12.5000	−	21.6506i	−0.420658	−	0.728602i	0.575346	−	0.817910i	$-0.304868\pi$
$883$	−12.5000	−	21.6506i	−0.420658	−	0.728602i	−0.996004	+	0.0893086i	$0.971534\pi$
$884$		0			0
$885$	−4.00000			−0.134459
$886$		0			0
$887$	11.5000	−	19.9186i	0.386132	−	0.668801i	−0.605793	−	0.795622i	$-0.707144\pi$
$887$	11.5000	−	19.9186i	0.386132	−	0.668801i	0.991926	+	0.126821i	$0.0404775\pi$
$888$		0			0
$889$	−6.50000	+	11.2583i	−0.218003	+	0.377592i
$890$		0			0
$891$	0.500000	+	0.866025i	0.0167506	+	0.0290129i
$892$		0			0
$893$	−15.0000			−0.501956
$894$		0			0
$895$	−24.0000			−0.802232
$896$		0			0
$897$	10.5000	+	18.1865i	0.350585	+	0.607231i
$898$		0			0
$899$	−0.500000	+	0.866025i	−0.0166759	+	0.0288836i
$900$		0			0
$901$	−3.00000	+	5.19615i	−0.0999445	+	0.173109i
$902$		0			0
$903$	−4.00000	−	6.92820i	−0.133112	−	0.230556i
$904$		0			0
$905$	3.50000	+	6.06218i	0.116344	+	0.201514i
$906$		0			0
$907$	−18.5000	−	32.0429i	−0.614282	−	1.06397i	−0.990510	−	0.137441i	$-0.956112\pi$
$907$	−18.5000	−	32.0429i	−0.614282	−	1.06397i	0.376228	−	0.926527i	$-0.377221\pi$
$908$		0			0
$909$	7.50000	+	12.9904i	0.248759	+	0.430864i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	11.0000			0.364047
$914$		0			0
$915$	5.50000	−	9.52628i	0.181824	−	0.314929i
$916$		0			0
$917$	−2.00000	−	3.46410i	−0.0660458	−	0.114395i
$918$		0			0
$919$	14.5000	−	25.1147i	0.478311	−	0.828459i	−0.521380	−	0.853325i	$-0.674583\pi$
$919$	14.5000	−	25.1147i	0.478311	−	0.828459i	0.999691	+	0.0248659i	$0.00791589\pi$
$920$		0			0
$921$	4.50000	−	7.79423i	0.148280	−	0.256829i
$922$		0			0
$923$	−91.0000			−2.99530
$924$		0			0
$925$	0.500000	−	0.866025i	0.0164399	−	0.0284747i
$926$		0			0
$927$	−2.50000	−	4.33013i	−0.0821108	−	0.142220i
$928$		0			0
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	−15.0000	−	25.9808i	−0.491605	−	0.851485i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.00000			−0.0981105
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$	21.0000			0.683854
$944$		0			0
$945$	0.500000	+	0.866025i	0.0162650	+	0.0281718i
$946$		0			0
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$		0			0
$949$	−38.5000	−	66.6840i	−1.24976	−	2.16465i
$950$		0			0
$951$	−8.50000	+	14.7224i	−0.275631	+	0.477408i
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	4.50000	−	7.79423i	0.145617	−	0.252215i
$956$		0			0
$957$	0.500000	−	0.866025i	0.0161627	−	0.0279946i
$958$		0			0
$959$	−9.00000	−	15.5885i	−0.290625	−	0.503378i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.00000			0.0643823
$966$		0			0
$967$	1.50000	+	2.59808i	0.0482367	+	0.0835485i	0.889136	−	0.457644i	$-0.151306\pi$
$967$	1.50000	+	2.59808i	0.0482367	+	0.0835485i	−0.840899	+	0.541192i	$0.817973\pi$
$968$		0			0
$969$	−7.50000	−	12.9904i	−0.240935	−	0.417311i
$970$		0			0
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	0.720224	−	0.693742i	$-0.244039\pi$
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	−0.960910	+	0.276861i	$0.910706\pi$
$972$		0			0
$973$	2.00000	+	3.46410i	0.0641171	+	0.111054i
$974$		0			0
$975$	−3.50000	+	6.06218i	−0.112090	+	0.194145i
$976$		0			0
$977$	25.5000	−	44.1673i	0.815817	−	1.41304i	−0.0929223	−	0.995673i	$-0.529621\pi$
$977$	25.5000	−	44.1673i	0.815817	−	1.41304i	0.908740	−	0.417364i	$-0.137046\pi$
$978$		0			0
$979$	1.00000	+	1.73205i	0.0319601	+	0.0553566i
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−8.00000			−0.255160			−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000			−0.255160			−0.127580	+	0.991828i	$0.540721\pi$
$984$		0			0
$985$	8.50000	+	14.7224i	0.270833	+	0.469096i
$986$		0			0
$987$	−1.50000	+	2.59808i	−0.0477455	+	0.0826977i
$988$		0			0
$989$	12.0000	−	20.7846i	0.381578	−	0.660912i
$990$		0			0
$991$	48.0000			1.52477			0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000			1.52477			0.762385	+	0.647124i	$0.224028\pi$
$992$		0			0
$993$	−2.50000	−	4.33013i	−0.0793351	−	0.137412i
$994$		0			0
$995$	1.50000	−	2.59808i	0.0475532	−	0.0823646i
$996$		0			0
$997$	−30.0000			−0.950110			−0.475055	−	0.879956i	$-0.657572\pi$
$997$	−30.0000			−0.950110			−0.475055	+	0.879956i	$0.657572\pi$
$998$		0			0
$999$	0.500000	−	0.866025i	0.0158193	−	0.0273998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.q.f.3781.1	yes	2
67.37	even	3	inner	4020.2.q.f.841.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.q.f.841.1	✓	2	67.37	even	3	inner
4020.2.q.f.3781.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{9} +(0.500000 + 0.866025i) q^{11} +(-3.50000 + 6.06218i) q^{13} -1.00000 q^{15} +(1.50000 - 2.59808i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-0.500000 - 0.866025i) q^{21} +(1.50000 - 2.59808i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-0.500000 - 0.866025i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{35} +(0.500000 - 0.866025i) q^{37} +(-3.50000 + 6.06218i) q^{39} +(3.50000 + 6.06218i) q^{41} +8.00000 q^{43} -1.00000 q^{45} +(-1.50000 - 2.59808i) q^{47} +(3.00000 - 5.19615i) q^{49} +(1.50000 - 2.59808i) q^{51} -2.00000 q^{53} +(-0.500000 - 0.866025i) q^{55} +(2.50000 - 4.33013i) q^{57} +4.00000 q^{59} +(-5.50000 + 9.52628i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(3.50000 - 6.06218i) q^{65} +(8.00000 + 1.73205i) q^{67} +(1.50000 - 2.59808i) q^{69} +(6.50000 + 11.2583i) q^{71} +(-5.50000 + 9.52628i) q^{73} +1.00000 q^{75} +(0.500000 - 0.866025i) q^{77} +(5.50000 + 9.52628i) q^{79} +1.00000 q^{81} +(5.50000 - 9.52628i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(-0.500000 - 0.866025i) q^{87} +2.00000 q^{89} +7.00000 q^{91} +(-0.500000 - 0.866025i) q^{93} +(-2.50000 + 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} + q^{11} - 7 q^{13} - 2 q^{15} + 3 q^{17} + 5 q^{19} - q^{21} + 3 q^{23} + 2 q^{25} + 2 q^{27} - q^{29} - q^{31} + q^{33} + q^{35} + q^{37} - 7 q^{39} + 7 q^{41} + 16 q^{43} - 2 q^{45} - 3 q^{47} + 6 q^{49} + 3 q^{51} - 4 q^{53} - q^{55} + 5 q^{57} + 8 q^{59} - 11 q^{61} - q^{63} + 7 q^{65} + 16 q^{67} + 3 q^{69} + 13 q^{71} - 11 q^{73} + 2 q^{75} + q^{77} + 11 q^{79} + 2 q^{81} + 11 q^{83} - 3 q^{85} - q^{87} + 4 q^{89} + 14 q^{91} - q^{93} - 5 q^{95} - 7 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 + q^11 - 7 * q^13 - 2 * q^15 + 3 * q^17 + 5 * q^19 - q^21 + 3 * q^23 + 2 * q^25 + 2 * q^27 - q^29 - q^31 + q^33 + q^35 + q^37 - 7 * q^39 + 7 * q^41 + 16 * q^43 - 2 * q^45 - 3 * q^47 + 6 * q^49 + 3 * q^51 - 4 * q^53 - q^55 + 5 * q^57 + 8 * q^59 - 11 * q^61 - q^63 + 7 * q^65 + 16 * q^67 + 3 * q^69 + 13 * q^71 - 11 * q^73 + 2 * q^75 + q^77 + 11 * q^79 + 2 * q^81 + 11 * q^83 - 3 * q^85 - q^87 + 4 * q^89 + 14 * q^91 - q^93 - 5 * q^95 - 7 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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