LMFDB - Embedded newform 4020.2.q.e.841.1

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			841.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4020.841
Dual form			4020.2.q.e.3781.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +(-1.50000 + 2.59808i) q^{11} +(0.500000 + 0.866025i) q^{13} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} +(-0.500000 + 0.866025i) q^{21} +(-0.500000 - 0.866025i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{29} +(3.50000 - 6.06218i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(0.500000 - 0.866025i) q^{35} +(-3.50000 - 6.06218i) q^{37} +(0.500000 + 0.866025i) q^{39} +(-2.50000 + 4.33013i) q^{41} -8.00000 q^{43} -1.00000 q^{45} +(0.500000 - 0.866025i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-0.500000 - 0.866025i) q^{51} +6.00000 q^{53} +(1.50000 - 2.59808i) q^{55} +(2.50000 + 4.33013i) q^{57} -12.0000 q^{59} +(-1.50000 - 2.59808i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(-8.00000 - 1.73205i) q^{67} +(-0.500000 - 0.866025i) q^{69} +(-7.50000 + 12.9904i) q^{71} +(-5.50000 - 9.52628i) q^{73} +1.00000 q^{75} +(-1.50000 - 2.59808i) q^{77} +(1.50000 - 2.59808i) q^{79} +1.00000 q^{81} +(3.50000 + 6.06218i) q^{83} +(0.500000 + 0.866025i) q^{85} +(-2.50000 + 4.33013i) q^{87} -14.0000 q^{89} -1.00000 q^{91} +(3.50000 - 6.06218i) q^{93} +(-2.50000 - 4.33013i) q^{95} +(8.50000 + 14.7224i) q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 3 q^{11} + q^{13} - 2 q^{15} - q^{17} + 5 q^{19} - q^{21} - q^{23} + 2 q^{25} + 2 q^{27} - 5 q^{29} + 7 q^{31} - 3 q^{33} + q^{35} - 7 q^{37} + q^{39} - 5 q^{41} - 16 q^{43} - 2 q^{45} + q^{47} + 6 q^{49} - q^{51} + 12 q^{53} + 3 q^{55} + 5 q^{57} - 24 q^{59} - 3 q^{61} - q^{63} - q^{65} - 16 q^{67} - q^{69} - 15 q^{71} - 11 q^{73} + 2 q^{75} - 3 q^{77} + 3 q^{79} + 2 q^{81} + 7 q^{83} + q^{85} - 5 q^{87} - 28 q^{89} - 2 q^{91} + 7 q^{93} - 5 q^{95} + 17 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 3 * q^11 + q^13 - 2 * q^15 - q^17 + 5 * q^19 - q^21 - q^23 + 2 * q^25 + 2 * q^27 - 5 * q^29 + 7 * q^31 - 3 * q^33 + q^35 - 7 * q^37 + q^39 - 5 * q^41 - 16 * q^43 - 2 * q^45 + q^47 + 6 * q^49 - q^51 + 12 * q^53 + 3 * q^55 + 5 * q^57 - 24 * q^59 - 3 * q^61 - q^63 - q^65 - 16 * q^67 - q^69 - 15 * q^71 - 11 * q^73 + 2 * q^75 - 3 * q^77 + 3 * q^79 + 2 * q^81 + 7 * q^83 + q^85 - 5 * q^87 - 28 * q^89 - 2 * q^91 + 7 * q^93 - 5 * q^95 + 17 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	0.799000	−	0.601331i	$-0.205363\pi$
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	−0.920268	+	0.391289i	$0.872029\pi$
$18$		0			0
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	0.996199	+	0.0871106i	$0.0277634\pi$
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	−0.422659	+	0.906289i	$0.638903\pi$
$20$		0			0
$21$	−0.500000	+	0.866025i	−0.109109	+	0.188982i
$22$		0			0
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	0.809177	−	0.587565i	$-0.199913\pi$
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	−0.913434	+	0.406986i	$0.866580\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	−0.999167	−	0.0408130i	$-0.987005\pi$
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	0.534928	+	0.844897i	$0.320339\pi$
$30$		0			0
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$	−1.50000	+	2.59808i	−0.261116	+	0.452267i
$34$		0			0
$35$	0.500000	−	0.866025i	0.0845154	−	0.146385i
$36$		0			0
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	$-0.971514\pi$
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		0			0
$39$	0.500000	+	0.866025i	0.0800641	+	0.138675i
$40$		0			0
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	−0.992507	−	0.122189i	$-0.961009\pi$
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	0.602072	+	0.798441i	$0.294342\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	−0.827253	−	0.561830i	$-0.810098\pi$
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	0.900185	+	0.435507i	$0.143431\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$	−0.500000	−	0.866025i	−0.0700140	−	0.121268i
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	1.50000	−	2.59808i	0.202260	−	0.350325i
$56$		0			0
$57$	2.50000	+	4.33013i	0.331133	+	0.573539i
$58$		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
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	$-0.228182\pi$
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	−0.945931	+	0.324367i	$0.894849\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.0629941	+	0.109109i
$64$		0			0
$65$	−0.500000	−	0.866025i	−0.0620174	−	0.107417i
$66$		0			0
$67$	−8.00000	−	1.73205i	−0.977356	−	0.211604i
$67$	−8.00000	−	1.73205i	−0.977356	−	0.211604i
$68$		0			0
$69$	−0.500000	−	0.866025i	−0.0601929	−	0.104257i
$70$		0			0
$71$	−7.50000	+	12.9904i	−0.890086	+	1.54167i	−0.0503155	+	0.998733i	$0.516023\pi$
$71$	−7.50000	+	12.9904i	−0.890086	+	1.54167i	−0.839771	+	0.542941i	$0.817311\pi$
$72$		0			0
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	−0.984594	−	0.174855i	$-0.944054\pi$
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	0.340868	−	0.940111i	$-0.389279\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−1.50000	−	2.59808i	−0.170941	−	0.296078i
$78$		0			0
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	−0.769222	−	0.638982i	$-0.779356\pi$
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	0.937985	+	0.346675i	$0.112689\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	3.50000	+	6.06218i	0.384175	+	0.665410i	0.991654	−	0.128925i	$-0.0411526\pi$
$83$	3.50000	+	6.06218i	0.384175	+	0.665410i	−0.607479	+	0.794335i	$0.707819\pi$
$84$		0			0
$85$	0.500000	+	0.866025i	0.0542326	+	0.0939336i
$86$		0			0
$87$	−2.50000	+	4.33013i	−0.268028	+	0.464238i
$88$		0			0
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$	3.50000	−	6.06218i	0.362933	−	0.628619i
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	8.50000	+	14.7224i	0.863044	+	1.49484i	0.868976	+	0.494854i	$0.164778\pi$
$97$	8.50000	+	14.7224i	0.863044	+	1.49484i	−0.00593185	+	0.999982i	$0.501888\pi$
$98$		0			0
$99$	−1.50000	+	2.59808i	−0.150756	+	0.261116i
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	−6.50000	+	11.2583i	−0.640464	+	1.10932i	0.344865	+	0.938652i	$0.387925\pi$
$103$	−6.50000	+	11.2583i	−0.640464	+	1.10932i	−0.985329	+	0.170664i	$0.945409\pi$
$104$		0			0
$105$	0.500000	−	0.866025i	0.0487950	−	0.0845154i
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	−3.50000	−	6.06218i	−0.332205	−	0.575396i
$112$		0			0
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	−0.996262	−	0.0863794i	$-0.972470\pi$
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	0.572938	+	0.819599i	$0.305804\pi$
$114$		0			0
$115$	0.500000	+	0.866025i	0.0466252	+	0.0807573i
$116$		0			0
$117$	0.500000	+	0.866025i	0.0462250	+	0.0800641i
$118$		0			0
$119$	1.00000			0.0916698
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	−2.50000	+	4.33013i	−0.225417	+	0.390434i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	0.419064	+	0.907957i	$0.362358\pi$
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	−0.995846	+	0.0910585i	$0.970975\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−5.00000			−0.433555
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$	0.500000	−	0.866025i	0.0421076	−	0.0729325i
$142$		0			0
$143$	−3.00000			−0.250873
$144$		0			0
$145$	2.50000	−	4.33013i	0.207614	−	0.359597i
$146$		0			0
$147$	3.00000	+	5.19615i	0.247436	+	0.428571i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	8.50000	+	14.7224i	0.691720	+	1.19809i	0.971274	+	0.237964i	$0.0764802\pi$
$151$	8.50000	+	14.7224i	0.691720	+	1.19809i	−0.279554	+	0.960130i	$0.590186\pi$
$152$		0			0
$153$	−0.500000	−	0.866025i	−0.0404226	−	0.0700140i
$154$		0			0
$155$	−3.50000	+	6.06218i	−0.281127	+	0.486926i
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$	11.5000	−	19.9186i	0.900750	−	1.56014i	0.0742262	−	0.997241i	$-0.476351\pi$
$163$	11.5000	−	19.9186i	0.900750	−	1.56014i	0.826523	−	0.562902i	$-0.190315\pi$
$164$		0			0
$165$	1.50000	−	2.59808i	0.116775	−	0.202260i
$166$		0			0
$167$	−1.50000	+	2.59808i	−0.116073	+	0.201045i	−0.918208	−	0.396098i	$-0.870364\pi$
$167$	−1.50000	+	2.59808i	−0.116073	+	0.201045i	0.802135	+	0.597143i	$0.203697\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$	2.50000	+	4.33013i	0.191180	+	0.331133i
$172$		0			0
$173$	−6.50000	−	11.2583i	−0.494186	−	0.855955i	0.505792	−	0.862656i	$-0.331200\pi$
$173$	−6.50000	−	11.2583i	−0.494186	−	0.855955i	−0.999978	+	0.00670064i	$0.997867\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.0377964	+	0.0654654i
$176$		0			0
$177$	−12.0000			−0.901975
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		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
$183$	−1.50000	−	2.59808i	−0.110883	−	0.192055i
$184$		0			0
$185$	3.50000	+	6.06218i	0.257325	+	0.445700i
$186$		0			0
$187$	3.00000			0.219382
$188$		0			0
$189$	−0.500000	+	0.866025i	−0.0363696	+	0.0629941i
$190$		0			0
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	0.915177	−	0.403051i	$-0.132050\pi$
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	−0.806641	+	0.591041i	$0.798717\pi$
$192$		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
$195$	−0.500000	−	0.866025i	−0.0358057	−	0.0620174i
$196$		0			0
$197$	−10.5000	+	18.1865i	−0.748094	+	1.29574i	0.200641	+	0.979665i	$0.435697\pi$
$197$	−10.5000	+	18.1865i	−0.748094	+	1.29574i	−0.948735	+	0.316072i	$0.897636\pi$
$198$		0			0
$199$	6.50000	+	11.2583i	0.460773	+	0.798082i	0.999000	−	0.0447181i	$-0.0142390\pi$
$199$	6.50000	+	11.2583i	0.460773	+	0.798082i	−0.538227	+	0.842800i	$0.680906\pi$
$200$		0			0
$201$	−8.00000	−	1.73205i	−0.564276	−	0.122169i
$202$		0			0
$203$	−2.50000	−	4.33013i	−0.175466	−	0.303915i
$204$		0			0
$205$	2.50000	−	4.33013i	0.174608	−	0.302429i
$206$		0			0
$207$	−0.500000	−	0.866025i	−0.0347524	−	0.0601929i
$208$		0			0
$209$	−15.0000			−1.03757
$210$		0			0
$211$	2.50000	+	4.33013i	0.172107	+	0.298098i	0.939156	−	0.343490i	$-0.111609\pi$
$211$	2.50000	+	4.33013i	0.172107	+	0.298098i	−0.767049	+	0.641588i	$0.778276\pi$
$212$		0			0
$213$	−7.50000	+	12.9904i	−0.513892	+	0.890086i
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	3.50000	+	6.06218i	0.237595	+	0.411527i
$218$		0			0
$219$	−5.50000	−	9.52628i	−0.371656	−	0.643726i
$220$		0			0
$221$	0.500000	−	0.866025i	0.0336336	−	0.0582552i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	−0.677158	−	0.735838i	$-0.736789\pi$
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	0.975833	+	0.218517i	$0.0701218\pi$
$228$		0			0
$229$	4.50000	+	7.79423i	0.297368	+	0.515057i	0.975533	−	0.219853i	$-0.0705577\pi$
$229$	4.50000	+	7.79423i	0.297368	+	0.515057i	−0.678165	+	0.734910i	$0.737224\pi$
$230$		0			0
$231$	−1.50000	−	2.59808i	−0.0986928	−	0.170941i
$232$		0			0
$233$	−8.50000	+	14.7224i	−0.556854	+	0.964499i	0.440903	+	0.897555i	$0.354658\pi$
$233$	−8.50000	+	14.7224i	−0.556854	+	0.964499i	−0.997757	+	0.0669439i	$0.978675\pi$
$234$		0			0
$235$	−0.500000	+	0.866025i	−0.0326164	+	0.0564933i
$236$		0			0
$237$	1.50000	−	2.59808i	0.0974355	−	0.168763i
$238$		0			0
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	−0.999854	−	0.0170808i	$-0.994563\pi$
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	0.514719	+	0.857359i	$0.327896\pi$
$240$		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
$247$	−2.50000	+	4.33013i	−0.159071	+	0.275519i
$248$		0			0
$249$	3.50000	+	6.06218i	0.221803	+	0.384175i
$250$		0			0
$251$	−10.5000	−	18.1865i	−0.662754	−	1.14792i	−0.979889	−	0.199543i	$-0.936054\pi$
$251$	−10.5000	−	18.1865i	−0.662754	−	1.14792i	0.317135	−	0.948380i	$-0.397279\pi$
$252$		0			0
$253$	3.00000			0.188608
$254$		0			0
$255$	0.500000	+	0.866025i	0.0313112	+	0.0542326i
$256$		0			0
$257$	−10.5000	+	18.1865i	−0.654972	+	1.13444i	0.326929	+	0.945049i	$0.393986\pi$
$257$	−10.5000	+	18.1865i	−0.654972	+	1.13444i	−0.981901	+	0.189396i	$0.939347\pi$
$258$		0			0
$259$	7.00000			0.434959
$260$		0			0
$261$	−2.50000	+	4.33013i	−0.154746	+	0.268028i
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$	−14.0000			−0.856786
$268$		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
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$	−1.00000			−0.0605228
$274$		0			0
$275$	−1.50000	+	2.59808i	−0.0904534	+	0.156670i
$276$		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
$279$	3.50000	−	6.06218i	0.209540	−	0.362933i
$280$		0			0
$281$	−2.50000	−	4.33013i	−0.149137	−	0.258314i	0.781771	−	0.623565i	$-0.214316\pi$
$281$	−2.50000	−	4.33013i	−0.149137	−	0.258314i	−0.930909	+	0.365251i	$0.880983\pi$
$282$		0			0
$283$	16.0000			0.951101			0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000			0.951101			0.475551	+	0.879688i	$0.342249\pi$
$284$		0			0
$285$	−2.50000	−	4.33013i	−0.148087	−	0.256495i
$286$		0			0
$287$	−2.50000	−	4.33013i	−0.147570	−	0.255599i
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$	8.50000	+	14.7224i	0.498279	+	0.863044i
$292$		0			0
$293$	26.0000			1.51894			0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000			1.51894			0.759468	+	0.650545i	$0.225459\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$	−1.50000	+	2.59808i	−0.0870388	+	0.150756i
$298$		0			0
$299$	0.500000	−	0.866025i	0.0289157	−	0.0500835i
$300$		0			0
$301$	4.00000	−	6.92820i	0.230556	−	0.399335i
$302$		0			0
$303$	1.50000	−	2.59808i	0.0861727	−	0.149256i
$304$		0			0
$305$	1.50000	+	2.59808i	0.0858898	+	0.148765i
$306$		0			0
$307$	−11.5000	−	19.9186i	−0.656340	−	1.13681i	−0.981556	−	0.191174i	$-0.938771\pi$
$307$	−11.5000	−	19.9186i	−0.656340	−	1.13681i	0.325216	−	0.945640i	$-0.394563\pi$
$308$		0			0
$309$	−6.50000	+	11.2583i	−0.369772	+	0.640464i
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$	0.500000	−	0.866025i	0.0281718	−	0.0487950i
$316$		0			0
$317$	9.50000	+	16.4545i	0.533573	+	0.924176i	0.999231	+	0.0392110i	$0.0124844\pi$
$317$	9.50000	+	16.4545i	0.533573	+	0.924176i	−0.465658	+	0.884965i	$0.654182\pi$
$318$		0			0
$319$	−7.50000	−	12.9904i	−0.419919	−	0.727322i
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$	2.50000	−	4.33013i	0.139104	−	0.240935i
$324$		0			0
$325$	0.500000	+	0.866025i	0.0277350	+	0.0480384i
$326$		0			0
$327$	2.00000			0.110600
$328$		0			0
$329$	0.500000	+	0.866025i	0.0275659	+	0.0477455i
$330$		0			0
$331$	13.5000	−	23.3827i	0.742027	−	1.28523i	−0.209544	−	0.977799i	$-0.567198\pi$
$331$	13.5000	−	23.3827i	0.742027	−	1.28523i	0.951571	−	0.307429i	$-0.0994688\pi$
$332$		0			0
$333$	−3.50000	−	6.06218i	−0.191799	−	0.332205i
$334$		0			0
$335$	8.00000	+	1.73205i	0.437087	+	0.0946320i
$336$		0			0
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	−0.886194	−	0.463314i	$-0.846660\pi$
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	0.0418554	−	0.999124i	$-0.486673\pi$
$338$		0			0
$339$	−4.50000	+	7.79423i	−0.244406	+	0.423324i
$340$		0			0
$341$	10.5000	+	18.1865i	0.568607	+	0.984856i
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$	0.500000	+	0.866025i	0.0269191	+	0.0466252i
$346$		0			0
$347$	−13.5000	+	23.3827i	−0.724718	+	1.25525i	0.234372	+	0.972147i	$0.424697\pi$
$347$	−13.5000	+	23.3827i	−0.724718	+	1.25525i	−0.959090	+	0.283101i	$0.908637\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$	0.500000	+	0.866025i	0.0266880	+	0.0462250i
$352$		0			0
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	0.721063	−	0.692869i	$-0.243654\pi$
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	−0.960574	+	0.278024i	$0.910320\pi$
$354$		0			0
$355$	7.50000	−	12.9904i	0.398059	−	0.689458i
$356$		0			0
$357$	1.00000			0.0529256
$358$		0			0
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$	1.00000	+	1.73205i	0.0524864	+	0.0909091i
$364$		0			0
$365$	5.50000	+	9.52628i	0.287883	+	0.498628i
$366$		0			0
$367$	1.50000	−	2.59808i	0.0782994	−	0.135618i	−0.824217	−	0.566274i	$-0.808384\pi$
$367$	1.50000	−	2.59808i	0.0782994	−	0.135618i	0.902516	+	0.430656i	$0.141718\pi$
$368$		0			0
$369$	−2.50000	+	4.33013i	−0.130145	+	0.225417i
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	8.50000	−	14.7224i	0.440113	−	0.762299i	−0.557584	−	0.830120i	$-0.688272\pi$
$373$	8.50000	−	14.7224i	0.440113	−	0.762299i	0.997697	+	0.0678218i	$0.0216049\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	−5.00000			−0.257513
$378$		0			0
$379$	12.5000	+	21.6506i	0.642082	+	1.11212i	0.984967	+	0.172741i	$0.0552624\pi$
$379$	12.5000	+	21.6506i	0.642082	+	1.11212i	−0.342885	+	0.939377i	$0.611404\pi$
$380$		0			0
$381$	−6.50000	+	11.2583i	−0.333005	+	0.576782i
$382$		0			0
$383$	−0.500000	−	0.866025i	−0.0255488	−	0.0442518i	0.852968	−	0.521963i	$-0.174800\pi$
$383$	−0.500000	−	0.866025i	−0.0255488	−	0.0442518i	−0.878517	+	0.477711i	$0.841467\pi$
$384$		0			0
$385$	1.50000	+	2.59808i	0.0764471	+	0.132410i
$386$		0			0
$387$	−8.00000			−0.406663
$388$		0			0
$389$	−16.5000	−	28.5788i	−0.836583	−	1.44900i	−0.892735	−	0.450582i	$-0.851216\pi$
$389$	−16.5000	−	28.5788i	−0.836583	−	1.44900i	0.0561516	−	0.998422i	$-0.482117\pi$
$390$		0			0
$391$	−0.500000	+	0.866025i	−0.0252861	+	0.0437968i
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$	−1.50000	+	2.59808i	−0.0754732	+	0.130723i
$396$		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
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$	7.00000			0.348695
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	21.0000			1.04093
$408$		0			0
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	−0.480560	−	0.876962i	$-0.659566\pi$
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	0.999751	−	0.0223042i	$-0.00710022\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$		0			0
$413$	6.00000	−	10.3923i	0.295241	−	0.511372i
$414$		0			0
$415$	−3.50000	−	6.06218i	−0.171808	−	0.297581i
$416$		0			0
$417$	−20.0000			−0.979404
$418$		0			0
$419$	13.5000	+	23.3827i	0.659518	+	1.14232i	0.980741	+	0.195315i	$0.0625730\pi$
$419$	13.5000	+	23.3827i	0.659518	+	1.14232i	−0.321222	+	0.947004i	$0.604094\pi$
$420$		0			0
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	0.827155	−	0.561973i	$-0.189958\pi$
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	−0.900261	+	0.435351i	$0.856624\pi$
$422$		0			0
$423$	0.500000	−	0.866025i	0.0243108	−	0.0421076i
$424$		0			0
$425$	−0.500000	−	0.866025i	−0.0242536	−	0.0420084i
$426$		0			0
$427$	3.00000			0.145180
$428$		0			0
$429$	−3.00000			−0.144841
$430$		0			0
$431$	−1.50000	+	2.59808i	−0.0722525	+	0.125145i	−0.899888	−	0.436121i	$-0.856352\pi$
$431$	−1.50000	+	2.59808i	−0.0722525	+	0.125145i	0.827636	+	0.561266i	$0.189685\pi$
$432$		0			0
$433$	14.5000	−	25.1147i	0.696826	−	1.20694i	−0.272736	−	0.962089i	$-0.587929\pi$
$433$	14.5000	−	25.1147i	0.696826	−	1.20694i	0.969561	−	0.244848i	$-0.0787382\pi$
$434$		0			0
$435$	2.50000	−	4.33013i	0.119866	−	0.207614i
$436$		0			0
$437$	2.50000	−	4.33013i	0.119591	−	0.207138i
$438$		0			0
$439$	8.50000	+	14.7224i	0.405683	+	0.702663i	0.994401	−	0.105675i	$-0.0337004\pi$
$439$	8.50000	+	14.7224i	0.405683	+	0.702663i	−0.588718	+	0.808339i	$0.700367\pi$
$440$		0			0
$441$	3.00000	+	5.19615i	0.142857	+	0.247436i
$442$		0			0
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	−0.899457	−	0.437009i	$-0.856038\pi$
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	0.828190	+	0.560448i	$0.189371\pi$
$444$		0			0
$445$	14.0000			0.663664
$446$		0			0
$447$	6.00000			0.283790
$448$		0			0
$449$	17.5000	−	30.3109i	0.825876	−	1.43046i	−0.0753719	−	0.997155i	$-0.524014\pi$
$449$	17.5000	−	30.3109i	0.825876	−	1.43046i	0.901248	−	0.433304i	$-0.142652\pi$
$450$		0			0
$451$	−7.50000	−	12.9904i	−0.353161	−	0.611693i
$452$		0			0
$453$	8.50000	+	14.7224i	0.399365	+	0.691720i
$454$		0			0
$455$	1.00000			0.0468807
$456$		0			0
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	0.461067	+	0.887365i	$0.347467\pi$
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	−0.999014	+	0.0443868i	$0.985867\pi$
$458$		0			0
$459$	−0.500000	−	0.866025i	−0.0233380	−	0.0404226i
$460$		0			0
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$	12.5000	+	21.6506i	0.580924	+	1.00619i	0.995370	+	0.0961164i	$0.0306421\pi$
$463$	12.5000	+	21.6506i	0.580924	+	1.00619i	−0.414446	+	0.910074i	$0.636025\pi$
$464$		0			0
$465$	−3.50000	+	6.06218i	−0.162309	+	0.281127i
$466$		0			0
$467$	−16.5000	−	28.5788i	−0.763529	−	1.32247i	−0.941021	−	0.338349i	$-0.890132\pi$
$467$	−16.5000	−	28.5788i	−0.763529	−	1.32247i	0.177492	−	0.984122i	$-0.443202\pi$
$468$		0			0
$469$	5.50000	−	6.06218i	0.253966	−	0.279925i
$470$		0			0
$471$	−3.50000	−	6.06218i	−0.161271	−	0.279330i
$472$		0			0
$473$	12.0000	−	20.7846i	0.551761	−	0.955677i
$474$		0			0
$475$	2.50000	+	4.33013i	0.114708	+	0.198680i
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	19.5000	+	33.7750i	0.890978	+	1.54322i	0.838705	+	0.544586i	$0.183313\pi$
$479$	19.5000	+	33.7750i	0.890978	+	1.54322i	0.0522726	+	0.998633i	$0.483354\pi$
$480$		0			0
$481$	3.50000	−	6.06218i	0.159586	−	0.276412i
$482$		0			0
$483$	1.00000			0.0455016
$484$		0			0
$485$	−8.50000	−	14.7224i	−0.385965	−	0.668511i
$486$		0			0
$487$	18.5000	+	32.0429i	0.838315	+	1.45200i	0.891303	+	0.453409i	$0.149792\pi$
$487$	18.5000	+	32.0429i	0.838315	+	1.45200i	−0.0529875	+	0.998595i	$0.516874\pi$
$488$		0			0
$489$	11.5000	−	19.9186i	0.520048	−	0.900750i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	5.00000			0.225189
$494$		0			0
$495$	1.50000	−	2.59808i	0.0674200	−	0.116775i
$496$		0			0
$497$	−7.50000	−	12.9904i	−0.336421	−	0.582698i
$498$		0			0
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	0.991136	−	0.132855i	$-0.0424144\pi$
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	−0.610623	+	0.791921i	$0.709081\pi$
$500$		0			0
$501$	−1.50000	+	2.59808i	−0.0670151	+	0.116073i
$502$		0			0
$503$	14.5000	−	25.1147i	0.646523	−	1.11981i	−0.337424	−	0.941353i	$-0.609556\pi$
$503$	14.5000	−	25.1147i	0.646523	−	1.11981i	0.983948	−	0.178458i	$-0.0571109\pi$
$504$		0			0
$505$	−1.50000	+	2.59808i	−0.0667491	+	0.115613i
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	−42.0000			−1.86162			−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000			−1.86162			−0.930809	+	0.365507i	$0.880896\pi$
$510$		0			0
$511$	11.0000			0.486611
$512$		0			0
$513$	2.50000	+	4.33013i	0.110378	+	0.191180i
$514$		0			0
$515$	6.50000	−	11.2583i	0.286424	−	0.496101i
$516$		0			0
$517$	1.50000	+	2.59808i	0.0659699	+	0.114263i
$518$		0			0
$519$	−6.50000	−	11.2583i	−0.285318	−	0.494186i
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	2.50000	+	4.33013i	0.109317	+	0.189343i	0.915494	−	0.402332i	$-0.131800\pi$
$523$	2.50000	+	4.33013i	0.109317	+	0.189343i	−0.806177	+	0.591675i	$0.798467\pi$
$524$		0			0
$525$	−0.500000	+	0.866025i	−0.0218218	+	0.0377964i
$526$		0			0
$527$	−7.00000			−0.304925
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	−5.00000			−0.216574
$534$		0			0
$535$	8.00000			0.345870
$536$		0			0
$537$	−16.0000			−0.690451
$538$		0			0
$539$	−18.0000			−0.775315
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$	−3.50000	+	6.06218i	−0.150199	+	0.260153i
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	−6.50000	+	11.2583i	−0.277920	+	0.481371i	−0.970868	−	0.239616i	$-0.922978\pi$
$547$	−6.50000	+	11.2583i	−0.277920	+	0.481371i	0.692948	+	0.720988i	$0.256312\pi$
$548$		0			0
$549$	−1.50000	−	2.59808i	−0.0640184	−	0.110883i
$550$		0			0
$551$	−25.0000			−1.06504
$552$		0			0
$553$	1.50000	+	2.59808i	0.0637865	+	0.110481i
$554$		0			0
$555$	3.50000	+	6.06218i	0.148567	+	0.257325i
$556$		0			0
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	−0.945473	−	0.325701i	$-0.894400\pi$
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	0.754802	+	0.655953i	$0.227733\pi$
$558$		0			0
$559$	−4.00000	−	6.92820i	−0.169182	−	0.293032i
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	4.50000	−	7.79423i	0.189316	−	0.327906i
$566$		0			0
$567$	−0.500000	+	0.866025i	−0.0209980	+	0.0363696i
$568$		0			0
$569$	−22.5000	+	38.9711i	−0.943249	+	1.63376i	−0.184030	+	0.982921i	$0.558914\pi$
$569$	−22.5000	+	38.9711i	−0.943249	+	1.63376i	−0.759220	+	0.650835i	$0.774419\pi$
$570$		0			0
$571$	−10.5000	+	18.1865i	−0.439411	+	0.761083i	−0.997644	−	0.0686016i	$-0.978146\pi$
$571$	−10.5000	+	18.1865i	−0.439411	+	0.761083i	0.558233	+	0.829684i	$0.311480\pi$
$572$		0			0
$573$	1.50000	+	2.59808i	0.0626634	+	0.108536i
$574$		0			0
$575$	−0.500000	−	0.866025i	−0.0208514	−	0.0361158i
$576$		0			0
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	−0.633062	−	0.774101i	$-0.718202\pi$
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	0.986922	+	0.161198i	$0.0515357\pi$
$578$		0			0
$579$	−2.00000			−0.0831172
$580$		0			0
$581$	−7.00000			−0.290409
$582$		0			0
$583$	−9.00000	+	15.5885i	−0.372742	+	0.645608i
$584$		0			0
$585$	−0.500000	−	0.866025i	−0.0206725	−	0.0358057i
$586$		0			0
$587$	−6.50000	−	11.2583i	−0.268284	−	0.464681i	0.700135	−	0.714010i	$-0.253123\pi$
$587$	−6.50000	−	11.2583i	−0.268284	−	0.464681i	−0.968419	+	0.249329i	$0.919790\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$	−10.5000	+	18.1865i	−0.431912	+	0.748094i
$592$		0			0
$593$	19.5000	+	33.7750i	0.800769	+	1.38697i	0.919111	+	0.394000i	$0.128909\pi$
$593$	19.5000	+	33.7750i	0.800769	+	1.38697i	−0.118342	+	0.992973i	$0.537758\pi$
$594$		0			0
$595$	−1.00000			−0.0409960
$596$		0			0
$597$	6.50000	+	11.2583i	0.266027	+	0.460773i
$598$		0			0
$599$	−7.50000	+	12.9904i	−0.306442	+	0.530773i	−0.977581	−	0.210558i	$-0.932472\pi$
$599$	−7.50000	+	12.9904i	−0.306442	+	0.530773i	0.671140	+	0.741331i	$0.265805\pi$
$600$		0			0
$601$	−3.50000	−	6.06218i	−0.142768	−	0.247281i	0.785770	−	0.618519i	$-0.212267\pi$
$601$	−3.50000	−	6.06218i	−0.142768	−	0.247281i	−0.928538	+	0.371237i	$0.878934\pi$
$602$		0			0
$603$	−8.00000	−	1.73205i	−0.325785	−	0.0705346i
$604$		0			0
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.0134214	−	0.999910i	$-0.495728\pi$
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.859237	−	0.511578i	$-0.170939\pi$
$608$		0			0
$609$	−2.50000	−	4.33013i	−0.101305	−	0.175466i
$610$		0			0
$611$	1.00000			0.0404557
$612$		0			0
$613$	−13.5000	−	23.3827i	−0.545260	−	0.944418i	−0.998591	−	0.0530754i	$-0.983098\pi$
$613$	−13.5000	−	23.3827i	−0.545260	−	0.944418i	0.453331	−	0.891342i	$-0.350236\pi$
$614$		0			0
$615$	2.50000	−	4.33013i	0.100810	−	0.174608i
$616$		0			0
$617$	−10.0000			−0.402585			−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000			−0.402585			−0.201292	+	0.979531i	$0.564514\pi$
$618$		0			0
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	0.996138	−	0.0878015i	$-0.0279841\pi$
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	−0.574107	+	0.818780i	$0.694651\pi$
$620$		0			0
$621$	−0.500000	−	0.866025i	−0.0200643	−	0.0347524i
$622$		0			0
$623$	7.00000	−	12.1244i	0.280449	−	0.485752i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−15.0000			−0.599042
$628$		0			0
$629$	−3.50000	+	6.06218i	−0.139554	+	0.241715i
$630$		0			0
$631$	10.5000	+	18.1865i	0.417998	+	0.723994i	0.995738	−	0.0922266i	$-0.0293984\pi$
$631$	10.5000	+	18.1865i	0.417998	+	0.723994i	−0.577740	+	0.816221i	$0.696065\pi$
$632$		0			0
$633$	2.50000	+	4.33013i	0.0993661	+	0.172107i
$634$		0			0
$635$	6.50000	−	11.2583i	0.257945	−	0.446773i
$636$		0			0
$637$	−3.00000	+	5.19615i	−0.118864	+	0.205879i
$638$		0			0
$639$	−7.50000	+	12.9904i	−0.296695	+	0.513892i
$640$		0			0
$641$	17.5000	−	30.3109i	0.691208	−	1.19721i	−0.280234	−	0.959932i	$-0.590412\pi$
$641$	17.5000	−	30.3109i	0.691208	−	1.19721i	0.971442	−	0.237276i	$-0.0762547\pi$
$642$		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
$645$	8.00000			0.315000
$646$		0			0
$647$	13.5000	+	23.3827i	0.530740	+	0.919268i	0.999357	+	0.0358667i	$0.0114192\pi$
$647$	13.5000	+	23.3827i	0.530740	+	0.919268i	−0.468617	+	0.883402i	$0.655247\pi$
$648$		0			0
$649$	18.0000	−	31.1769i	0.706562	−	1.22380i
$650$		0			0
$651$	3.50000	+	6.06218i	0.137176	+	0.237595i
$652$		0			0
$653$	−10.5000	−	18.1865i	−0.410897	−	0.711694i	0.584091	−	0.811688i	$-0.301451\pi$
$653$	−10.5000	−	18.1865i	−0.410897	−	0.711694i	−0.994988	+	0.0999939i	$0.968118\pi$
$654$		0			0
$655$	−12.0000			−0.468879
$656$		0			0
$657$	−5.50000	−	9.52628i	−0.214575	−	0.371656i
$658$		0			0
$659$	10.5000	−	18.1865i	0.409022	−	0.708447i	−0.585758	−	0.810486i	$-0.699203\pi$
$659$	10.5000	−	18.1865i	0.409022	−	0.708447i	0.994780	+	0.102039i	$0.0325366\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$		0			0
$663$	0.500000	−	0.866025i	0.0194184	−	0.0336336i
$664$		0			0
$665$	5.00000			0.193892
$666$		0			0
$667$	5.00000			0.193601
$668$		0			0
$669$	24.0000			0.927894
$670$		0			0
$671$	9.00000			0.347441
$672$		0			0
$673$	46.0000			1.77317			0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000			1.77317			0.886585	+	0.462566i	$0.153071\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	−18.5000	+	32.0429i	−0.711013	+	1.23151i	0.253465	+	0.967345i	$0.418430\pi$
$677$	−18.5000	+	32.0429i	−0.711013	+	1.23151i	−0.964477	+	0.264166i	$0.914903\pi$
$678$		0			0
$679$	−17.0000			−0.652400
$680$		0			0
$681$	4.50000	−	7.79423i	0.172440	−	0.298675i
$682$		0			0
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	0.592168	−	0.805814i	$-0.298272\pi$
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	−0.993940	+	0.109926i	$0.964939\pi$
$684$		0			0
$685$	−2.00000			−0.0764161
$686$		0			0
$687$	4.50000	+	7.79423i	0.171686	+	0.297368i
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	−16.5000	+	28.5788i	−0.627690	+	1.08719i	0.360325	+	0.932827i	$0.382666\pi$
$691$	−16.5000	+	28.5788i	−0.627690	+	1.08719i	−0.988014	+	0.154363i	$0.950667\pi$
$692$		0			0
$693$	−1.50000	−	2.59808i	−0.0569803	−	0.0986928i
$694$		0			0
$695$	20.0000			0.758643
$696$		0			0
$697$	5.00000			0.189389
$698$		0			0
$699$	−8.50000	+	14.7224i	−0.321500	+	0.556854i
$700$		0			0
$701$	−18.5000	+	32.0429i	−0.698735	+	1.21025i	0.270170	+	0.962813i	$0.412920\pi$
$701$	−18.5000	+	32.0429i	−0.698735	+	1.21025i	−0.968905	+	0.247432i	$0.920413\pi$
$702$		0			0
$703$	17.5000	−	30.3109i	0.660025	−	1.14320i
$704$		0			0
$705$	−0.500000	+	0.866025i	−0.0188311	+	0.0326164i
$706$		0			0
$707$	1.50000	+	2.59808i	0.0564133	+	0.0977107i
$708$		0			0
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	0.413114	−	0.910679i	$-0.364441\pi$
$710$		0			0
$711$	1.50000	−	2.59808i	0.0562544	−	0.0974355i
$712$		0			0
$713$	−7.00000			−0.262152
$714$		0			0
$715$	3.00000			0.112194
$716$		0			0
$717$	−7.50000	+	12.9904i	−0.280093	+	0.485135i
$718$		0			0
$719$	19.5000	+	33.7750i	0.727227	+	1.25959i	0.958051	+	0.286599i	$0.0925247\pi$
$719$	19.5000	+	33.7750i	0.727227	+	1.25959i	−0.230823	+	0.972996i	$0.574142\pi$
$720$		0			0
$721$	−6.50000	−	11.2583i	−0.242073	−	0.419282i
$722$		0			0
$723$	−26.0000			−0.966950
$724$		0			0
$725$	−2.50000	+	4.33013i	−0.0928477	+	0.160817i
$726$		0			0
$727$	0.500000	+	0.866025i	0.0185440	+	0.0321191i	0.875148	−	0.483854i	$-0.160764\pi$
$727$	0.500000	+	0.866025i	0.0185440	+	0.0321191i	−0.856605	+	0.515974i	$0.827430\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	4.00000	+	6.92820i	0.147945	+	0.256249i
$732$		0			0
$733$	2.50000	−	4.33013i	0.0923396	−	0.159937i	−0.816156	−	0.577832i	$-0.803899\pi$
$733$	2.50000	−	4.33013i	0.0923396	−	0.159937i	0.908495	+	0.417895i	$0.137232\pi$
$734$		0			0
$735$	−3.00000	−	5.19615i	−0.110657	−	0.191663i
$736$		0			0
$737$	16.5000	−	18.1865i	0.607785	−	0.669910i
$738$		0			0
$739$	8.50000	+	14.7224i	0.312678	+	0.541573i	0.978941	−	0.204143i	$-0.0654407\pi$
$739$	8.50000	+	14.7224i	0.312678	+	0.541573i	−0.666264	+	0.745716i	$0.732107\pi$
$740$		0			0
$741$	−2.50000	+	4.33013i	−0.0918398	+	0.159071i
$742$		0			0
$743$	−8.50000	−	14.7224i	−0.311835	−	0.540114i	0.666925	−	0.745125i	$-0.267610\pi$
$743$	−8.50000	−	14.7224i	−0.311835	−	0.540114i	−0.978760	+	0.205011i	$0.934277\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$		0			0
$747$	3.50000	+	6.06218i	0.128058	+	0.221803i
$748$		0			0
$749$	4.00000	−	6.92820i	0.146157	−	0.253151i
$750$		0			0
$751$	52.0000			1.89751			0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000			1.89751			0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$	−10.5000	−	18.1865i	−0.382641	−	0.662754i
$754$		0			0
$755$	−8.50000	−	14.7224i	−0.309347	−	0.535804i
$756$		0			0
$757$	10.5000	−	18.1865i	0.381629	−	0.661001i	−0.609666	−	0.792658i	$-0.708697\pi$
$757$	10.5000	−	18.1865i	0.381629	−	0.661001i	0.991295	+	0.131657i	$0.0420299\pi$
$758$		0			0
$759$	3.00000			0.108893
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	−1.00000	+	1.73205i	−0.0362024	+	0.0627044i
$764$		0			0
$765$	0.500000	+	0.866025i	0.0180775	+	0.0313112i
$766$		0			0
$767$	−6.00000	−	10.3923i	−0.216647	−	0.375244i
$768$		0			0
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	−0.817423	−	0.576038i	$-0.804598\pi$
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	0.907575	+	0.419890i	$0.137931\pi$
$770$		0			0
$771$	−10.5000	+	18.1865i	−0.378148	+	0.654972i
$772$		0			0
$773$	25.5000	−	44.1673i	0.917171	−	1.58859i	0.113480	−	0.993540i	$-0.463800\pi$
$773$	25.5000	−	44.1673i	0.917171	−	1.58859i	0.803691	−	0.595047i	$-0.202867\pi$
$774$		0			0
$775$	3.50000	−	6.06218i	0.125724	−	0.217760i
$776$		0			0
$777$	7.00000			0.251124
$778$		0			0
$779$	−25.0000			−0.895718
$780$		0			0
$781$	−22.5000	−	38.9711i	−0.805113	−	1.39450i
$782$		0			0
$783$	−2.50000	+	4.33013i	−0.0893427	+	0.154746i
$784$		0			0
$785$	3.50000	+	6.06218i	0.124920	+	0.216368i
$786$		0			0
$787$	20.5000	+	35.5070i	0.730746	+	1.26569i	0.956565	+	0.291520i	$0.0941610\pi$
$787$	20.5000	+	35.5070i	0.730746	+	1.26569i	−0.225819	+	0.974169i	$0.572506\pi$
$788$		0			0
$789$	16.0000			0.569615
$790$		0			0
$791$	−4.50000	−	7.79423i	−0.160002	−	0.277131i
$792$		0			0
$793$	1.50000	−	2.59808i	0.0532666	−	0.0922604i
$794$		0			0
$795$	−6.00000			−0.212798
$796$		0			0
$797$	15.5000	−	26.8468i	0.549038	−	0.950962i	−0.449303	−	0.893380i	$-0.648327\pi$
$797$	15.5000	−	26.8468i	0.549038	−	0.950962i	0.998341	−	0.0575824i	$-0.0183392\pi$
$798$		0			0
$799$	−1.00000			−0.0353775
$800$		0			0
$801$	−14.0000			−0.494666
$802$		0			0
$803$	33.0000			1.16454
$804$		0			0
$805$	−1.00000			−0.0352454
$806$		0			0
$807$	14.0000			0.492823
$808$		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
$811$	−18.5000	+	32.0429i	−0.649623	+	1.12518i	0.333590	+	0.942718i	$0.391740\pi$
$811$	−18.5000	+	32.0429i	−0.649623	+	1.12518i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	−11.5000	+	19.9186i	−0.402827	+	0.697718i
$816$		0			0
$817$	−20.0000	−	34.6410i	−0.699711	−	1.21194i
$818$		0			0
$819$	−1.00000			−0.0349428
$820$		0			0
$821$	−0.500000	−	0.866025i	−0.0174501	−	0.0302245i	0.857168	−	0.515036i	$-0.172221\pi$
$821$	−0.500000	−	0.866025i	−0.0174501	−	0.0302245i	−0.874619	+	0.484812i	$0.838888\pi$
$822$		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
$825$	−1.50000	+	2.59808i	−0.0522233	+	0.0904534i
$826$		0			0
$827$	7.50000	+	12.9904i	0.260801	+	0.451720i	0.966455	−	0.256836i	$-0.0826802\pi$
$827$	7.50000	+	12.9904i	0.260801	+	0.451720i	−0.705654	+	0.708556i	$0.749347\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	3.00000	−	5.19615i	0.103944	−	0.180036i
$834$		0			0
$835$	1.50000	−	2.59808i	0.0519096	−	0.0899101i
$836$		0			0
$837$	3.50000	−	6.06218i	0.120978	−	0.209540i
$838$		0			0
$839$	8.50000	−	14.7224i	0.293453	−	0.508275i	−0.681171	−	0.732124i	$-0.738529\pi$
$839$	8.50000	−	14.7224i	0.293453	−	0.508275i	0.974624	+	0.223849i	$0.0718624\pi$
$840$		0			0
$841$	2.00000	+	3.46410i	0.0689655	+	0.119452i
$842$		0			0
$843$	−2.50000	−	4.33013i	−0.0861046	−	0.149137i
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$	−2.00000			−0.0687208
$848$		0			0
$849$	16.0000			0.549119
$850$		0			0
$851$	−3.50000	+	6.06218i	−0.119978	+	0.207809i
$852$		0			0
$853$	−11.5000	−	19.9186i	−0.393753	−	0.681999i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	−11.5000	−	19.9186i	−0.393753	−	0.681999i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$	−2.50000	−	4.33013i	−0.0854982	−	0.148087i
$856$		0			0
$857$	10.0000			0.341593			0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000			0.341593			0.170797	+	0.985306i	$0.445366\pi$
$858$		0			0
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	−0.999982	−	0.00607046i	$-0.998068\pi$
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	0.505248	+	0.862974i	$0.331401\pi$
$860$		0			0
$861$	−2.50000	−	4.33013i	−0.0851998	−	0.147570i
$862$		0			0
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$	6.50000	+	11.2583i	0.221007	+	0.382795i
$866$		0			0
$867$	8.00000	−	13.8564i	0.271694	−	0.470588i
$868$		0			0
$869$	4.50000	+	7.79423i	0.152652	+	0.264401i
$870$		0			0
$871$	−2.50000	−	7.79423i	−0.0847093	−	0.264097i
$872$		0			0
$873$	8.50000	+	14.7224i	0.287681	+	0.498279i
$874$		0			0
$875$	0.500000	−	0.866025i	0.0169031	−	0.0292770i
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$		0			0
$879$	26.0000			0.876958
$880$		0			0
$881$	−10.5000	−	18.1865i	−0.353754	−	0.612720i	0.633150	−	0.774029i	$-0.281762\pi$
$881$	−10.5000	−	18.1865i	−0.353754	−	0.612720i	−0.986904	+	0.161309i	$0.948428\pi$
$882$		0			0
$883$	15.5000	−	26.8468i	0.521617	−	0.903466i	−0.478067	−	0.878323i	$-0.658663\pi$
$883$	15.5000	−	26.8468i	0.521617	−	0.903466i	0.999684	−	0.0251431i	$-0.00800415\pi$
$884$		0			0
$885$	12.0000			0.403376
$886$		0			0
$887$	9.50000	+	16.4545i	0.318979	+	0.552487i	0.980275	−	0.197637i	$-0.0633268\pi$
$887$	9.50000	+	16.4545i	0.318979	+	0.552487i	−0.661296	+	0.750125i	$0.729993\pi$
$888$		0			0
$889$	−6.50000	−	11.2583i	−0.218003	−	0.377592i
$890$		0			0
$891$	−1.50000	+	2.59808i	−0.0502519	+	0.0870388i
$892$		0			0
$893$	5.00000			0.167319
$894$		0			0
$895$	16.0000			0.534821
$896$		0			0
$897$	0.500000	−	0.866025i	0.0166945	−	0.0289157i
$898$		0			0
$899$	17.5000	+	30.3109i	0.583658	+	1.01092i
$900$		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
$907$	5.50000	−	9.52628i	0.182625	−	0.316315i	−0.760149	−	0.649749i	$-0.774874\pi$
$907$	5.50000	−	9.52628i	0.182625	−	0.316315i	0.942773	+	0.333434i	$0.108207\pi$
$908$		0			0
$909$	1.50000	−	2.59808i	0.0497519	−	0.0861727i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−21.0000			−0.694999
$914$		0			0
$915$	1.50000	+	2.59808i	0.0495885	+	0.0858898i
$916$		0			0
$917$	−6.00000	+	10.3923i	−0.198137	+	0.343184i
$918$		0			0
$919$	−5.50000	−	9.52628i	−0.181428	−	0.314243i	0.760939	−	0.648824i	$-0.224739\pi$
$919$	−5.50000	−	9.52628i	−0.181428	−	0.314243i	−0.942367	+	0.334581i	$0.891405\pi$
$920$		0			0
$921$	−11.5000	−	19.9186i	−0.378938	−	0.656340i
$922$		0			0
$923$	−15.0000			−0.493731
$924$		0			0
$925$	−3.50000	−	6.06218i	−0.115079	−	0.199323i
$926$		0			0
$927$	−6.50000	+	11.2583i	−0.213488	+	0.369772i
$928$		0			0
$929$	−2.00000			−0.0656179			−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000			−0.0656179			−0.0328089	+	0.999462i	$0.510445\pi$
$930$		0			0
$931$	−15.0000	+	25.9808i	−0.491605	+	0.851485i
$932$		0			0
$933$	−24.0000			−0.785725
$934$		0			0
$935$	−3.00000			−0.0981105
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	6.00000			0.195803
$940$		0			0
$941$	14.0000			0.456387			0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000			0.456387			0.228193	+	0.973616i	$0.426718\pi$
$942$		0			0
$943$	5.00000			0.162822
$944$		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
$949$	5.50000	−	9.52628i	0.178538	−	0.309236i
$950$		0			0
$951$	9.50000	+	16.4545i	0.308059	+	0.533573i
$952$		0			0
$953$	22.0000			0.712650			0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000			0.712650			0.356325	+	0.934362i	$0.384030\pi$
$954$		0			0
$955$	−1.50000	−	2.59808i	−0.0485389	−	0.0840718i
$956$		0			0
$957$	−7.50000	−	12.9904i	−0.242441	−	0.419919i
$958$		0			0
$959$	−1.00000	+	1.73205i	−0.0322917	+	0.0559308i
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$		0			0
$963$	−8.00000			−0.257796
$964$		0			0
$965$	2.00000			0.0643823
$966$		0			0
$967$	−10.5000	+	18.1865i	−0.337657	+	0.584839i	−0.983992	−	0.178215i	$-0.942968\pi$
$967$	−10.5000	+	18.1865i	−0.337657	+	0.584839i	0.646334	+	0.763054i	$0.276301\pi$
$968$		0			0
$969$	2.50000	−	4.33013i	0.0803116	−	0.139104i
$970$		0			0
$971$	22.5000	−	38.9711i	0.722059	−	1.25064i	−0.238114	−	0.971237i	$-0.576529\pi$
$971$	22.5000	−	38.9711i	0.722059	−	1.25064i	0.960173	−	0.279406i	$-0.0901376\pi$
$972$		0			0
$973$	10.0000	−	17.3205i	0.320585	−	0.555270i
$974$		0			0
$975$	0.500000	+	0.866025i	0.0160128	+	0.0277350i
$976$		0			0
$977$	15.5000	+	26.8468i	0.495889	+	0.858905i	0.999989	−	0.00474056i	$-0.00150897\pi$
$977$	15.5000	+	26.8468i	0.495889	+	0.858905i	−0.504100	+	0.863645i	$0.668176\pi$
$978$		0			0
$979$	21.0000	−	36.3731i	0.671163	−	1.16249i
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$	40.0000			1.27580			0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000			1.27580			0.637901	+	0.770118i	$0.279803\pi$
$984$		0			0
$985$	10.5000	−	18.1865i	0.334558	−	0.579471i
$986$		0			0
$987$	0.500000	+	0.866025i	0.0159152	+	0.0275659i
$988$		0			0
$989$	4.00000	+	6.92820i	0.127193	+	0.220304i
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$	13.5000	−	23.3827i	0.428410	−	0.742027i
$994$		0			0
$995$	−6.50000	−	11.2583i	−0.206064	−	0.356913i
$996$		0			0
$997$	42.0000			1.33015			0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000			1.33015			0.665077	+	0.746775i	$0.268399\pi$
$998$		0			0
$999$	−3.50000	−	6.06218i	−0.110735	−	0.191799i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.q.e.841.1	✓	2	1.1	even	1	trivial
4020.2.q.e.3781.1	yes	2	67.29	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +(-1.50000 + 2.59808i) q^{11} +(0.500000 + 0.866025i) q^{13} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{17} +(2.50000 + 4.33013i) q^{19} +(-0.500000 + 0.866025i) q^{21} +(-0.500000 - 0.866025i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{29} +(3.50000 - 6.06218i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(0.500000 - 0.866025i) q^{35} +(-3.50000 - 6.06218i) q^{37} +(0.500000 + 0.866025i) q^{39} +(-2.50000 + 4.33013i) q^{41} -8.00000 q^{43} -1.00000 q^{45} +(0.500000 - 0.866025i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-0.500000 - 0.866025i) q^{51} +6.00000 q^{53} +(1.50000 - 2.59808i) q^{55} +(2.50000 + 4.33013i) q^{57} -12.0000 q^{59} +(-1.50000 - 2.59808i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(-8.00000 - 1.73205i) q^{67} +(-0.500000 - 0.866025i) q^{69} +(-7.50000 + 12.9904i) q^{71} +(-5.50000 - 9.52628i) q^{73} +1.00000 q^{75} +(-1.50000 - 2.59808i) q^{77} +(1.50000 - 2.59808i) q^{79} +1.00000 q^{81} +(3.50000 + 6.06218i) q^{83} +(0.500000 + 0.866025i) q^{85} +(-2.50000 + 4.33013i) q^{87} -14.0000 q^{89} -1.00000 q^{91} +(3.50000 - 6.06218i) q^{93} +(-2.50000 - 4.33013i) q^{95} +(8.50000 + 14.7224i) q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} + q^{13} - 2 q^{15} - q^{17} + 5 q^{19} - q^{21} - q^{23} + 2 q^{25} + 2 q^{27} - 5 q^{29} + 7 q^{31} - 3 q^{33} + q^{35} - 7 q^{37} + q^{39} - 5 q^{41} - 16 q^{43} - 2 q^{45} + q^{47} + 6 q^{49} - q^{51} + 12 q^{53} + 3 q^{55} + 5 q^{57} - 24 q^{59} - 3 q^{61} - q^{63} - q^{65} - 16 q^{67} - q^{69} - 15 q^{71} - 11 q^{73} + 2 q^{75} - 3 q^{77} + 3 q^{79} + 2 q^{81} + 7 q^{83} + q^{85} - 5 q^{87} - 28 q^{89} - 2 q^{91} + 7 q^{93} - 5 q^{95} + 17 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 3 * q^11 + q^13 - 2 * q^15 - q^17 + 5 * q^19 - q^21 - q^23 + 2 * q^25 + 2 * q^27 - 5 * q^29 + 7 * q^31 - 3 * q^33 + q^35 - 7 * q^37 + q^39 - 5 * q^41 - 16 * q^43 - 2 * q^45 + q^47 + 6 * q^49 - q^51 + 12 * q^53 + 3 * q^55 + 5 * q^57 - 24 * q^59 - 3 * q^61 - q^63 - q^65 - 16 * q^67 - q^69 - 15 * q^71 - 11 * q^73 + 2 * q^75 - 3 * q^77 + 3 * q^79 + 2 * q^81 + 7 * q^83 + q^85 - 5 * q^87 - 28 * q^89 - 2 * q^91 + 7 * q^93 - 5 * q^95 + 17 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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