LMFDB - Embedded newform 117.4.g.a.100.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			100.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.100
Dual form			117.4.g.a.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} -7.00000 q^{5} +(5.00000 + 8.66025i) q^{7} -15.0000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} -7.00000 q^{5} +(5.00000 + 8.66025i) q^{7} -15.0000 q^{8} +(3.50000 - 6.06218i) q^{10} +(-11.0000 + 19.0526i) q^{11} +(-45.5000 - 11.2583i) q^{13} -10.0000 q^{14} +(-20.5000 + 35.5070i) q^{16} +(18.5000 + 32.0429i) q^{17} +(-15.0000 - 25.9808i) q^{19} +(-24.5000 - 42.4352i) q^{20} +(-11.0000 - 19.0526i) q^{22} +(-81.0000 + 140.296i) q^{23} -76.0000 q^{25} +(32.5000 - 33.7750i) q^{26} +(-35.0000 + 60.6218i) q^{28} +(-56.5000 + 97.8609i) q^{29} +196.000 q^{31} +(-80.5000 - 139.430i) q^{32} -37.0000 q^{34} +(-35.0000 - 60.6218i) q^{35} +(-6.50000 + 11.2583i) q^{37} +30.0000 q^{38} +105.000 q^{40} +(142.500 - 246.817i) q^{41} +(123.000 + 213.042i) q^{43} -154.000 q^{44} +(-81.0000 - 140.296i) q^{46} +462.000 q^{47} +(121.500 - 210.444i) q^{49} +(38.0000 - 65.8179i) q^{50} +(-91.0000 - 315.233i) q^{52} +537.000 q^{53} +(77.0000 - 133.368i) q^{55} +(-75.0000 - 129.904i) q^{56} +(-56.5000 - 97.8609i) q^{58} +(288.000 + 498.831i) q^{59} +(317.500 + 549.926i) q^{61} +(-98.0000 + 169.741i) q^{62} -167.000 q^{64} +(318.500 + 78.8083i) q^{65} +(-101.000 + 174.937i) q^{67} +(-129.500 + 224.301i) q^{68} +70.0000 q^{70} +(-543.000 - 940.504i) q^{71} -805.000 q^{73} +(-6.50000 - 11.2583i) q^{74} +(105.000 - 181.865i) q^{76} -220.000 q^{77} +884.000 q^{79} +(143.500 - 248.549i) q^{80} +(142.500 + 246.817i) q^{82} -518.000 q^{83} +(-129.500 - 224.301i) q^{85} -246.000 q^{86} +(165.000 - 285.788i) q^{88} +(97.0000 - 168.009i) q^{89} +(-130.000 - 450.333i) q^{91} -1134.00 q^{92} +(-231.000 + 400.104i) q^{94} +(105.000 + 181.865i) q^{95} +(601.000 + 1040.96i) q^{97} +(121.500 + 210.444i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8}+O(q^{10}) $ 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8	$ 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8} + 7 q^{10} - 22 q^{11} - 91 q^{13} - 20 q^{14} - 41 q^{16} + 37 q^{17} - 30 q^{19} - 49 q^{20} - 22 q^{22} - 162 q^{23} - 152 q^{25} + 65 q^{26} - 70 q^{28} - 113 q^{29} + 392 q^{31} - 161 q^{32} - 74 q^{34} - 70 q^{35} - 13 q^{37} + 60 q^{38} + 210 q^{40} + 285 q^{41} + 246 q^{43} - 308 q^{44} - 162 q^{46} + 924 q^{47} + 243 q^{49} + 76 q^{50} - 182 q^{52} + 1074 q^{53} + 154 q^{55} - 150 q^{56} - 113 q^{58} + 576 q^{59} + 635 q^{61} - 196 q^{62} - 334 q^{64} + 637 q^{65} - 202 q^{67} - 259 q^{68} + 140 q^{70} - 1086 q^{71} - 1610 q^{73} - 13 q^{74} + 210 q^{76} - 440 q^{77} + 1768 q^{79} + 287 q^{80} + 285 q^{82} - 1036 q^{83} - 259 q^{85} - 492 q^{86} + 330 q^{88} + 194 q^{89} - 260 q^{91} - 2268 q^{92} - 462 q^{94} + 210 q^{95} + 1202 q^{97} + 243 q^{98}+O(q^{100}) $ 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8 + 7 * q^10 - 22 * q^11 - 91 * q^13 - 20 * q^14 - 41 * q^16 + 37 * q^17 - 30 * q^19 - 49 * q^20 - 22 * q^22 - 162 * q^23 - 152 * q^25 + 65 * q^26 - 70 * q^28 - 113 * q^29 + 392 * q^31 - 161 * q^32 - 74 * q^34 - 70 * q^35 - 13 * q^37 + 60 * q^38 + 210 * q^40 + 285 * q^41 + 246 * q^43 - 308 * q^44 - 162 * q^46 + 924 * q^47 + 243 * q^49 + 76 * q^50 - 182 * q^52 + 1074 * q^53 + 154 * q^55 - 150 * q^56 - 113 * q^58 + 576 * q^59 + 635 * q^61 - 196 * q^62 - 334 * q^64 + 637 * q^65 - 202 * q^67 - 259 * q^68 + 140 * q^70 - 1086 * q^71 - 1610 * q^73 - 13 * q^74 + 210 * q^76 - 440 * q^77 + 1768 * q^79 + 287 * q^80 + 285 * q^82 - 1036 * q^83 - 259 * q^85 - 492 * q^86 + 330 * q^88 + 194 * q^89 - 260 * q^91 - 2268 * q^92 - 462 * q^94 + 210 * q^95 + 1202 * q^97 + 243 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−0.500000	+	0.866025i	−0.176777	+	0.306186i	−0.940775	−	0.339032i	$-0.889900\pi$
$2$	−0.500000	+	0.866025i	−0.176777	+	0.306186i	0.763998	+	0.645219i	$0.223234\pi$
$3$		0			0
$3$		0			0
$4$	3.50000	+	6.06218i	0.437500	+	0.757772i
$5$	−7.00000			−0.626099			−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000			−0.626099			−0.313050	+	0.949737i	$0.601351\pi$
$6$		0			0
$7$	5.00000	+	8.66025i	0.269975	+	0.467610i	0.968855	−	0.247629i	$-0.0796514\pi$
$7$	5.00000	+	8.66025i	0.269975	+	0.467610i	−0.698880	+	0.715239i	$0.746318\pi$
$8$	−15.0000			−0.662913
$9$		0			0
$10$	3.50000	−	6.06218i	0.110680	−	0.191703i
$11$	−11.0000	+	19.0526i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−11.0000	+	19.0526i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−45.5000	−	11.2583i	−0.970725	−	0.240192i
$13$	−45.5000	−	11.2583i	−0.970725	−	0.240192i
$14$	−10.0000			−0.190901
$15$		0			0
$16$	−20.5000	+	35.5070i	−0.320312	+	0.554798i
$17$	18.5000	+	32.0429i	0.263936	+	0.457150i	0.967284	−	0.253695i	$-0.0816459\pi$
$17$	18.5000	+	32.0429i	0.263936	+	0.457150i	−0.703348	+	0.710845i	$0.748313\pi$
$18$		0			0
$19$	−15.0000	−	25.9808i	−0.181118	−	0.313705i	0.761144	−	0.648583i	$-0.224638\pi$
$19$	−15.0000	−	25.9808i	−0.181118	−	0.313705i	−0.942261	+	0.334878i	$0.891305\pi$
$20$	−24.5000	−	42.4352i	−0.273918	−	0.474440i
$21$		0			0
$22$	−11.0000	−	19.0526i	−0.106600	−	0.184637i
$23$	−81.0000	+	140.296i	−0.734333	+	1.27190i	0.220682	+	0.975346i	$0.429172\pi$
$23$	−81.0000	+	140.296i	−0.734333	+	1.27190i	−0.955015	+	0.296557i	$0.904162\pi$
$24$		0			0
$25$	−76.0000			−0.608000
$26$	32.5000	−	33.7750i	0.245145	−	0.254762i
$27$		0			0
$28$	−35.0000	+	60.6218i	−0.236228	+	0.409159i
$29$	−56.5000	+	97.8609i	−0.361786	+	0.626631i	−0.988255	−	0.152815i	$-0.951166\pi$
$29$	−56.5000	+	97.8609i	−0.361786	+	0.626631i	0.626469	+	0.779446i	$0.284499\pi$
$30$		0			0
$31$	196.000			1.13557			0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000			1.13557			0.567785	+	0.823177i	$0.307801\pi$
$32$	−80.5000	−	139.430i	−0.444704	−	0.770250i
$33$		0			0
$34$	−37.0000			−0.186631
$35$	−35.0000	−	60.6218i	−0.169031	−	0.292770i
$36$		0			0
$37$	−6.50000	+	11.2583i	−0.0288809	+	0.0500232i	−0.880105	−	0.474780i	$-0.842528\pi$
$37$	−6.50000	+	11.2583i	−0.0288809	+	0.0500232i	0.851224	+	0.524803i	$0.175861\pi$
$38$	30.0000			0.128070
$39$		0			0
$40$	105.000			0.415049
$41$	142.500	−	246.817i	0.542799	−	0.940156i	−0.455943	−	0.890009i	$-0.650698\pi$
$41$	142.500	−	246.817i	0.542799	−	0.940156i	0.998742	−	0.0501465i	$-0.0159688\pi$
$42$		0			0
$43$	123.000	+	213.042i	0.436217	+	0.755550i	0.997394	−	0.0721459i	$-0.0229847\pi$
$43$	123.000	+	213.042i	0.436217	+	0.755550i	−0.561177	+	0.827696i	$0.689651\pi$
$44$	−154.000			−0.527645
$45$		0			0
$46$	−81.0000	−	140.296i	−0.259626	−	0.449686i
$47$	462.000			1.43382			0.716911	−	0.697165i	$-0.245555\pi$
$47$	462.000			1.43382			0.716911	+	0.697165i	$0.245555\pi$
$48$		0			0
$49$	121.500	−	210.444i	0.354227	−	0.613540i
$50$	38.0000	−	65.8179i	0.107480	−	0.186161i
$51$		0			0
$52$	−91.0000	−	315.233i	−0.242681	−	0.840673i
$53$	537.000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$53$	537.000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$54$		0			0
$55$	77.0000	−	133.368i	0.188776	−	0.326970i
$56$	−75.0000	−	129.904i	−0.178970	−	0.309984i
$57$		0			0
$58$	−56.5000	−	97.8609i	−0.127911	−	0.221548i
$59$	288.000	+	498.831i	0.635498	+	1.10072i	0.986409	+	0.164307i	$0.0525387\pi$
$59$	288.000	+	498.831i	0.635498	+	1.10072i	−0.350911	+	0.936409i	$0.614128\pi$
$60$		0			0
$61$	317.500	+	549.926i	0.666421	+	1.15428i	0.978898	+	0.204350i	$0.0655082\pi$
$61$	317.500	+	549.926i	0.666421	+	1.15428i	−0.312476	+	0.949926i	$0.601159\pi$
$62$	−98.0000	+	169.741i	−0.200742	+	0.347696i
$63$		0			0
$64$	−167.000			−0.326172
$65$	318.500	+	78.8083i	0.607770	+	0.150384i
$66$		0			0
$67$	−101.000	+	174.937i	−0.184166	+	0.318985i	−0.943295	−	0.331955i	$-0.892292\pi$
$67$	−101.000	+	174.937i	−0.184166	+	0.318985i	0.759129	+	0.650940i	$0.225625\pi$
$68$	−129.500	+	224.301i	−0.230944	+	0.400006i
$69$		0			0
$70$	70.0000			0.119523
$71$	−543.000	−	940.504i	−0.907637	−	1.57207i	−0.817338	−	0.576159i	$-0.804551\pi$
$71$	−543.000	−	940.504i	−0.907637	−	1.57207i	−0.0902997	−	0.995915i	$-0.528783\pi$
$72$		0			0
$73$	−805.000			−1.29066			−0.645330	−	0.763904i	$-0.723280\pi$
$73$	−805.000			−1.29066			−0.645330	+	0.763904i	$0.723280\pi$
$74$	−6.50000	−	11.2583i	−0.0102109	−	0.0176859i
$75$		0			0
$76$	105.000	−	181.865i	0.158478	−	0.274492i
$77$	−220.000			−0.325602
$78$		0			0
$79$	884.000			1.25896			0.629480	−	0.777017i	$-0.283268\pi$
$79$	884.000			1.25896			0.629480	+	0.777017i	$0.283268\pi$
$80$	143.500	−	248.549i	0.200547	−	0.347358i
$81$		0			0
$82$	142.500	+	246.817i	0.191908	+	0.332395i
$83$	−518.000			−0.685035			−0.342517	−	0.939511i	$-0.611280\pi$
$83$	−518.000			−0.685035			−0.342517	+	0.939511i	$0.611280\pi$
$84$		0			0
$85$	−129.500	−	224.301i	−0.165250	−	0.286221i
$86$	−246.000			−0.308452
$87$		0			0
$88$	165.000	−	285.788i	0.199876	−	0.346195i
$89$	97.0000	−	168.009i	0.115528	−	0.200100i	−0.802463	−	0.596702i	$-0.796477\pi$
$89$	97.0000	−	168.009i	0.115528	−	0.200100i	0.917991	+	0.396602i	$0.129811\pi$
$90$		0			0
$91$	−130.000	−	450.333i	−0.149755	−	0.518766i
$92$	−1134.00			−1.28508
$93$		0			0
$94$	−231.000	+	400.104i	−0.253466	+	0.439016i
$95$	105.000	+	181.865i	0.113398	+	0.196410i
$96$		0			0
$97$	601.000	+	1040.96i	0.629096	+	1.08963i	0.987733	+	0.156149i	$0.0499081\pi$
$97$	601.000	+	1040.96i	0.629096	+	1.08963i	−0.358638	+	0.933477i	$0.616759\pi$
$98$	121.500	+	210.444i	0.125238	+	0.216919i
$99$		0			0
$100$	−266.000	−	460.726i	−0.266000	−	0.460726i
$101$	−214.500	+	371.525i	−0.211322	+	0.366021i	−0.952129	−	0.305698i	$-0.901110\pi$
$101$	−214.500	+	371.525i	−0.211322	+	0.366021i	0.740806	+	0.671719i	$0.234444\pi$
$102$		0			0
$103$	−1302.00			−1.24553			−0.622766	−	0.782408i	$-0.713991\pi$
$103$	−1302.00			−1.24553			−0.622766	+	0.782408i	$0.713991\pi$
$104$	682.500	+	168.875i	0.643506	+	0.159226i
$105$		0			0
$106$	−268.500	+	465.056i	−0.246029	+	0.426134i
$107$	−669.000	+	1158.74i	−0.604436	+	1.04691i	0.387704	+	0.921784i	$0.373268\pi$
$107$	−669.000	+	1158.74i	−0.604436	+	1.04691i	−0.992140	+	0.125130i	$0.960065\pi$
$108$		0			0
$109$	−1034.00			−0.908617			−0.454308	−	0.890844i	$-0.650114\pi$
$109$	−1034.00			−0.908617			−0.454308	+	0.890844i	$0.650114\pi$
$110$	77.0000	+	133.368i	0.0667424	+	0.115601i
$111$		0			0
$112$	−410.000			−0.345905
$113$	538.500	+	932.709i	0.448299	+	0.776477i	0.998275	−	0.0587032i	$-0.0186966\pi$
$113$	538.500	+	932.709i	0.448299	+	0.776477i	−0.549976	+	0.835180i	$0.685363\pi$
$114$		0			0
$115$	567.000	−	982.073i	0.459765	−	0.796337i
$116$	−791.000			−0.633125
$117$		0			0
$118$	−576.000			−0.449365
$119$	−185.000	+	320.429i	−0.142512	+	0.246838i
$120$		0			0
$121$	423.500	+	733.524i	0.318182	+	0.551107i
$122$	−635.000			−0.471231
$123$		0			0
$124$	686.000	+	1188.19i	0.496811	+	0.860503i
$125$	1407.00			1.00677
$126$		0			0
$127$	494.000	−	855.633i	0.345161	−	0.597836i	−0.640222	−	0.768190i	$-0.721158\pi$
$127$	494.000	−	855.633i	0.345161	−	0.597836i	0.985383	+	0.170354i	$0.0544911\pi$
$128$	727.500	−	1260.07i	0.502363	−	0.870119i
$129$		0			0
$130$	−227.500	+	236.425i	−0.153485	+	0.159506i
$131$	−560.000			−0.373492			−0.186746	−	0.982408i	$-0.559794\pi$
$131$	−560.000			−0.373492			−0.186746	+	0.982408i	$0.559794\pi$
$132$		0			0
$133$	150.000	−	259.808i	0.0977944	−	0.169385i
$134$	−101.000	−	174.937i	−0.0651125	−	0.112778i
$135$		0			0
$136$	−277.500	−	480.644i	−0.174966	−	0.303051i
$137$	−259.500	−	449.467i	−0.161829	−	0.280296i	0.773696	−	0.633557i	$-0.218406\pi$
$137$	−259.500	−	449.467i	−0.161829	−	0.280296i	−0.935525	+	0.353261i	$0.885073\pi$
$138$		0			0
$139$	174.000	+	301.377i	0.106176	+	0.183903i	0.914218	−	0.405222i	$-0.132806\pi$
$139$	174.000	+	301.377i	0.106176	+	0.183903i	−0.808042	+	0.589125i	$0.799473\pi$
$140$	245.000	−	424.352i	0.147902	−	0.256174i
$141$		0			0
$142$	1086.00			0.641796
$143$	715.000	−	743.050i	0.418121	−	0.434524i
$144$		0			0
$145$	395.500	−	685.026i	0.226514	−	0.392333i
$146$	402.500	−	697.150i	0.228158	−	0.395182i
$147$		0			0
$148$	−91.0000			−0.0505416
$149$	−322.500	−	558.586i	−0.177317	−	0.307122i	0.763644	−	0.645638i	$-0.223408\pi$
$149$	−322.500	−	558.586i	−0.177317	−	0.307122i	−0.940961	+	0.338516i	$0.890075\pi$
$150$		0			0
$151$	2914.00			1.57045			0.785225	−	0.619211i	$-0.212547\pi$
$151$	2914.00			1.57045			0.785225	+	0.619211i	$0.212547\pi$
$152$	225.000	+	389.711i	0.120065	+	0.207959i
$153$		0			0
$154$	110.000	−	190.526i	0.0575588	−	0.0996947i
$155$	−1372.00			−0.710979
$156$		0			0
$157$	−2079.00			−1.05683			−0.528415	−	0.848986i	$-0.677213\pi$
$157$	−2079.00			−1.05683			−0.528415	+	0.848986i	$0.677213\pi$
$158$	−442.000	+	765.566i	−0.222555	+	0.385476i
$159$		0			0
$160$	563.500	+	976.011i	0.278429	+	0.482253i
$161$	−1620.00			−0.793006
$162$		0			0
$163$	−850.000	−	1472.24i	−0.408449	−	0.707454i	0.586267	−	0.810118i	$-0.300597\pi$
$163$	−850.000	−	1472.24i	−0.408449	−	0.707454i	−0.994716	+	0.102664i	$0.967263\pi$
$164$	1995.00			0.949898
$165$		0			0
$166$	259.000	−	448.601i	0.121098	−	0.209748i
$167$	1840.00	−	3186.97i	0.852596	−	1.47674i	−0.0262621	−	0.999655i	$-0.508360\pi$
$167$	1840.00	−	3186.97i	0.852596	−	1.47674i	0.878858	−	0.477084i	$-0.158306\pi$
$168$		0			0
$169$	1943.50	+	1024.51i	0.884615	+	0.466321i
$170$	259.000			0.116849
$171$		0			0
$172$	−861.000	+	1491.30i	−0.381690	+	0.661106i
$173$	2073.00	+	3590.54i	0.911025	+	1.57794i	0.812619	+	0.582795i	$0.198041\pi$
$173$	2073.00	+	3590.54i	0.911025	+	1.57794i	0.0984052	+	0.995146i	$0.468626\pi$
$174$		0			0
$175$	−380.000	−	658.179i	−0.164145	−	0.284307i
$176$	−451.000	−	781.155i	−0.193156	−	0.334555i
$177$		0			0
$178$	97.0000	+	168.009i	0.0408453	+	0.0707461i
$179$	1837.00	−	3181.78i	0.767060	−	1.32859i	−0.172090	−	0.985081i	$-0.555052\pi$
$179$	1837.00	−	3181.78i	0.767060	−	1.32859i	0.939150	−	0.343506i	$-0.111615\pi$
$180$		0			0
$181$	−3283.00			−1.34820			−0.674098	−	0.738642i	$-0.735467\pi$
$181$	−3283.00			−1.34820			−0.674098	+	0.738642i	$0.735467\pi$
$182$	455.000	+	112.583i	0.185312	+	0.0458529i
$183$		0			0
$184$	1215.00	−	2104.44i	0.486799	−	0.843160i
$185$	45.5000	−	78.8083i	0.0180823	−	0.0313195i
$186$		0			0
$187$	−814.000			−0.318319
$188$	1617.00	+	2800.73i	0.627297	+	1.08651i
$189$		0			0
$190$	−210.000			−0.0801842
$191$	−298.000	−	516.151i	−0.112893	−	0.195536i	0.804043	−	0.594572i	$-0.202678\pi$
$191$	−298.000	−	516.151i	−0.112893	−	0.195536i	−0.916935	+	0.399036i	$0.869345\pi$
$192$		0			0
$193$	196.500	−	340.348i	0.0732869	−	0.126937i	−0.827053	−	0.562124i	$-0.809984\pi$
$193$	196.500	−	340.348i	0.0732869	−	0.126937i	0.900340	+	0.435187i	$0.143318\pi$
$194$	−1202.00			−0.444838
$195$		0			0
$196$	1701.00			0.619898
$197$	−1761.00	+	3050.14i	−0.636884	+	1.10311i	0.349229	+	0.937037i	$0.386443\pi$
$197$	−1761.00	+	3050.14i	−0.636884	+	1.10311i	−0.986113	+	0.166077i	$0.946890\pi$
$198$		0			0
$199$	−1009.00	−	1747.64i	−0.359428	−	0.622547i	0.628438	−	0.777860i	$-0.283695\pi$
$199$	−1009.00	−	1747.64i	−0.359428	−	0.622547i	−0.987865	+	0.155313i	$0.950361\pi$
$200$	1140.00			0.403051
$201$		0			0
$202$	−214.500	−	371.525i	−0.0747137	−	0.129408i
$203$	−1130.00			−0.390692
$204$		0			0
$205$	−997.500	+	1727.72i	−0.339846	+	0.588630i
$206$	651.000	−	1127.57i	0.220181	−	0.381365i
$207$		0			0
$208$	1332.50	−	1384.77i	0.444194	−	0.461619i
$209$	660.000			0.218436
$210$		0			0
$211$	−80.0000	+	138.564i	−0.0261016	+	0.0452092i	−0.878781	−	0.477225i	$-0.841643\pi$
$211$	−80.0000	+	138.564i	−0.0261016	+	0.0452092i	0.852680	+	0.522434i	$0.174976\pi$
$212$	1879.50	+	3255.39i	0.608890	+	1.05463i
$213$		0			0
$214$	−669.000	−	1158.74i	−0.213700	−	0.370140i
$215$	−861.000	−	1491.30i	−0.273115	−	0.473049i
$216$		0			0
$217$	980.000	+	1697.41i	0.306575	+	0.531003i
$218$	517.000	−	895.470i	0.160622	−	0.278206i
$219$		0			0
$220$	1078.00			0.330358
$221$	−481.000	−	1666.23i	−0.146405	−	0.507163i
$222$		0			0
$223$	−2036.00	+	3526.46i	−0.611393	+	1.05896i	0.379613	+	0.925145i	$0.376057\pi$
$223$	−2036.00	+	3526.46i	−0.611393	+	1.05896i	−0.991006	+	0.133818i	$0.957276\pi$
$224$	805.000	−	1394.30i	0.240118	−	0.415896i
$225$		0			0
$226$	−1077.00			−0.316995
$227$	−2897.00	−	5017.75i	−0.847051	−	1.46714i	−0.883828	−	0.467812i	$-0.845042\pi$
$227$	−2897.00	−	5017.75i	−0.847051	−	1.46714i	0.0367765	−	0.999324i	$-0.488291\pi$
$228$		0			0
$229$	6482.00			1.87049			0.935246	−	0.353999i	$-0.115178\pi$
$229$	6482.00			1.87049			0.935246	+	0.353999i	$0.115178\pi$
$230$	567.000	+	982.073i	0.162552	+	0.281548i
$231$		0			0
$232$	847.500	−	1467.91i	0.239832	−	0.415402i
$233$	−6890.00			−1.93725			−0.968624	−	0.248530i	$-0.920053\pi$
$233$	−6890.00			−1.93725			−0.968624	+	0.248530i	$0.920053\pi$
$234$		0			0
$235$	−3234.00			−0.897714
$236$	−2016.00	+	3491.81i	−0.556061	+	0.963126i
$237$		0			0
$238$	−185.000	−	320.429i	−0.0503856	−	0.0872704i
$239$	−2466.00			−0.667415			−0.333708	−	0.942677i	$-0.608300\pi$
$239$	−2466.00			−0.667415			−0.333708	+	0.942677i	$0.608300\pi$
$240$		0			0
$241$	1808.50	+	3132.41i	0.483385	+	0.837247i	0.999818	−	0.0190805i	$-0.00607389\pi$
$241$	1808.50	+	3132.41i	0.483385	+	0.837247i	−0.516433	+	0.856327i	$0.672741\pi$
$242$	−847.000			−0.224989
$243$		0			0
$244$	−2222.50	+	3849.48i	−0.583119	+	1.00999i
$245$	−850.500	+	1473.11i	−0.221781	+	0.384137i
$246$		0			0
$247$	390.000	+	1351.00i	0.100466	+	0.348024i
$248$	−2940.00			−0.752783
$249$		0			0
$250$	−703.500	+	1218.50i	−0.177973	+	0.308258i
$251$	2430.00	+	4208.88i	0.611077	+	1.05842i	0.991059	+	0.133422i	$0.0425966\pi$
$251$	2430.00	+	4208.88i	0.611077	+	1.05842i	−0.379983	+	0.924994i	$0.624070\pi$
$252$		0			0
$253$	−1782.00	−	3086.51i	−0.442820	−	0.766986i
$254$	494.000	+	855.633i	0.122033	+	0.211367i
$255$		0			0
$256$	59.5000	+	103.057i	0.0145264	+	0.0251604i
$257$	282.500	−	489.304i	0.0685676	−	0.118763i	−0.829703	−	0.558204i	$-0.811490\pi$
$257$	282.500	−	489.304i	0.0685676	−	0.118763i	0.898271	+	0.439442i	$0.144824\pi$
$258$		0			0
$259$	−130.000			−0.0311884
$260$	637.000	+	2206.63i	0.151943	+	0.526344i
$261$		0			0
$262$	280.000	−	484.974i	0.0660246	−	0.114358i
$263$	−249.000	+	431.281i	−0.0583802	+	0.101118i	−0.893738	−	0.448588i	$-0.851927\pi$
$263$	−249.000	+	431.281i	−0.0583802	+	0.101118i	0.835358	+	0.549706i	$0.185260\pi$
$264$		0			0
$265$	−3759.00			−0.871372
$266$	150.000	+	259.808i	0.0345755	+	0.0598866i
$267$		0			0
$268$	−1414.00			−0.322290
$269$	2773.00	+	4802.98i	0.628523	+	1.08863i	0.987848	+	0.155422i	$0.0496737\pi$
$269$	2773.00	+	4802.98i	0.628523	+	1.08863i	−0.359325	+	0.933213i	$0.616993\pi$
$270$		0			0
$271$	1128.00	−	1953.75i	0.252845	−	0.437941i	−0.711463	−	0.702724i	$-0.751967\pi$
$271$	1128.00	−	1953.75i	0.252845	−	0.437941i	0.964308	+	0.264783i	$0.0853002\pi$
$272$	−1517.00			−0.338168
$273$		0			0
$274$	519.000			0.114430
$275$	836.000	−	1447.99i	0.183319	−	0.317518i
$276$		0			0
$277$	−1154.50	−	1999.65i	−0.250423	−	0.433745i	0.713219	−	0.700941i	$-0.247236\pi$
$277$	−1154.50	−	1999.65i	−0.250423	−	0.433745i	−0.963642	+	0.267196i	$0.913903\pi$
$278$	−348.000			−0.0750779
$279$		0			0
$280$	525.000	+	909.327i	0.112053	+	0.194081i
$281$	−5833.00			−1.23832			−0.619159	−	0.785265i	$-0.712527\pi$
$281$	−5833.00			−1.23832			−0.619159	+	0.785265i	$0.712527\pi$
$282$		0			0
$283$	−825.000	+	1428.94i	−0.173290	+	0.300148i	−0.939568	−	0.342362i	$-0.888773\pi$
$283$	−825.000	+	1428.94i	−0.173290	+	0.300148i	0.766278	+	0.642509i	$0.222107\pi$
$284$	3801.00	−	6583.53i	0.794183	−	1.37556i
$285$		0			0
$286$	286.000	+	990.733i	0.0591312	+	0.204837i
$287$	2850.00			0.586168
$288$		0			0
$289$	1772.00	−	3069.19i	0.360676	−	0.624709i
$290$	395.500	+	685.026i	0.0800847	+	0.138711i
$291$		0			0
$292$	−2817.50	−	4880.05i	−0.564663	−	0.978026i
$293$	1495.50	+	2590.28i	0.298184	+	0.516471i	0.975721	−	0.219019i	$-0.0702856\pi$
$293$	1495.50	+	2590.28i	0.298184	+	0.516471i	−0.677536	+	0.735489i	$0.736952\pi$
$294$		0			0
$295$	−2016.00	−	3491.81i	−0.397885	−	0.689157i
$296$	97.5000	−	168.875i	0.0191455	−	0.0331610i
$297$		0			0
$298$	645.000			0.125382
$299$	5265.00	−	5471.55i	1.01834	−	1.05829i
$300$		0			0
$301$	−1230.00	+	2130.42i	−0.235535	+	0.407959i
$302$	−1457.00	+	2523.60i	−0.277619	+	0.480850i
$303$		0			0
$304$	1230.00			0.232057
$305$	−2222.50	−	3849.48i	−0.417246	−	0.722691i
$306$		0			0
$307$	−2422.00			−0.450263			−0.225132	−	0.974328i	$-0.572281\pi$
$307$	−2422.00			−0.450263			−0.225132	+	0.974328i	$0.572281\pi$
$308$	−770.000	−	1333.68i	−0.142451	−	0.246732i
$309$		0			0
$310$	686.000	−	1188.19i	0.125684	−	0.217692i
$311$	3402.00			0.620288			0.310144	−	0.950690i	$-0.399623\pi$
$311$	3402.00			0.620288			0.310144	+	0.950690i	$0.399623\pi$
$312$		0			0
$313$	2310.00			0.417153			0.208577	−	0.978006i	$-0.433117\pi$
$313$	2310.00			0.417153			0.208577	+	0.978006i	$0.433117\pi$
$314$	1039.50	−	1800.47i	0.186823	−	0.323587i
$315$		0			0
$316$	3094.00	+	5358.97i	0.550795	+	0.954004i
$317$	257.000			0.0455349			0.0227674	−	0.999741i	$-0.492752\pi$
$317$	257.000			0.0455349			0.0227674	+	0.999741i	$0.492752\pi$
$318$		0			0
$319$	−1243.00	−	2152.94i	−0.218165	−	0.377873i
$320$	1169.00			0.204216
$321$		0			0
$322$	810.000	−	1402.96i	0.140185	−	0.242807i
$323$	555.000	−	961.288i	0.0956069	−	0.165596i
$324$		0			0
$325$	3458.00	+	855.633i	0.590201	+	0.146037i
$326$	1700.00			0.288817
$327$		0			0
$328$	−2137.50	+	3702.26i	−0.359828	+	0.623241i
$329$	2310.00	+	4001.04i	0.387096	+	0.670469i
$330$		0			0
$331$	−514.000	−	890.274i	−0.0853535	−	0.147837i	0.820188	−	0.572094i	$-0.193869\pi$
$331$	−514.000	−	890.274i	−0.0853535	−	0.147837i	−0.905542	+	0.424257i	$0.860535\pi$
$332$	−1813.00	−	3140.21i	−0.299703	−	0.519100i
$333$		0			0
$334$	1840.00	+	3186.97i	0.301438	+	0.522106i
$335$	707.000	−	1224.56i	0.115306	−	0.199716i
$336$		0			0
$337$	2487.00			0.402005			0.201002	−	0.979591i	$-0.435580\pi$
$337$	2487.00			0.402005			0.201002	+	0.979591i	$0.435580\pi$
$338$	−1859.00	+	1170.87i	−0.299161	+	0.188422i
$339$		0			0
$340$	906.500	−	1570.10i	0.144594	−	0.250444i
$341$	−2156.00	+	3734.30i	−0.342387	+	0.593032i
$342$		0			0
$343$	5860.00			0.922479
$344$	−1845.00	−	3195.63i	−0.289174	−	0.500863i
$345$		0			0
$346$	−4146.00			−0.644192
$347$	−1425.00	−	2468.17i	−0.220455	−	0.381840i	0.734491	−	0.678618i	$-0.237421\pi$
$347$	−1425.00	−	2468.17i	−0.220455	−	0.381840i	−0.954946	+	0.296779i	$0.904088\pi$
$348$		0			0
$349$	1009.00	−	1747.64i	0.154758	−	0.268049i	−0.778213	−	0.628001i	$-0.783874\pi$
$349$	1009.00	−	1747.64i	0.154758	−	0.268049i	0.932971	+	0.359952i	$0.117207\pi$
$350$	760.000			0.116068
$351$		0			0
$352$	3542.00			0.536333
$353$	−2643.50	+	4578.68i	−0.398582	+	0.690364i	−0.993551	−	0.113385i	$-0.963831\pi$
$353$	−2643.50	+	4578.68i	−0.398582	+	0.690364i	0.594970	+	0.803748i	$0.297164\pi$
$354$		0			0
$355$	3801.00	+	6583.53i	0.568271	+	0.984274i
$356$	1358.00			0.202174
$357$		0			0
$358$	1837.00	+	3181.78i	0.271197	+	0.469727i
$359$	7278.00			1.06997			0.534983	−	0.844863i	$-0.320318\pi$
$359$	7278.00			1.06997			0.534983	+	0.844863i	$0.320318\pi$
$360$		0			0
$361$	2979.50	−	5160.65i	0.434393	−	0.752390i
$362$	1641.50	−	2843.16i	0.238330	−	0.412799i
$363$		0			0
$364$	2275.00	−	2364.25i	0.327589	−	0.340440i
$365$	5635.00			0.808080
$366$		0			0
$367$	2101.00	−	3639.04i	0.298832	−	0.517592i	−0.677037	−	0.735949i	$-0.736736\pi$
$367$	2101.00	−	3639.04i	0.298832	−	0.517592i	0.975869	+	0.218357i	$0.0700697\pi$
$368$	−3321.00	−	5752.14i	−0.470432	−	0.814813i
$369$		0			0
$370$	45.5000	+	78.8083i	0.00639306	+	0.0110731i
$371$	2685.00	+	4650.56i	0.375737	+	0.650795i
$372$		0			0
$373$	791.500	+	1370.92i	0.109872	+	0.190304i	0.915718	−	0.401821i	$-0.131623\pi$
$373$	791.500	+	1370.92i	0.109872	+	0.190304i	−0.805846	+	0.592125i	$0.798289\pi$
$374$	407.000	−	704.945i	0.0562713	−	0.0974648i
$375$		0			0
$376$	−6930.00			−0.950499
$377$	3672.50	−	3816.57i	0.501707	−	0.521389i
$378$		0			0
$379$	1026.00	−	1777.08i	0.139056	−	0.240851i	−0.788084	−	0.615568i	$-0.788927\pi$
$379$	1026.00	−	1777.08i	0.139056	−	0.240851i	0.927139	+	0.374717i	$0.122260\pi$
$380$	−735.000	+	1273.06i	−0.0992229	+	0.171859i
$381$		0			0
$382$	596.000			0.0798273
$383$	−3436.00	−	5951.33i	−0.458411	−	0.793991i	0.540466	−	0.841366i	$-0.318248\pi$
$383$	−3436.00	−	5951.33i	−0.458411	−	0.793991i	−0.998877	+	0.0473746i	$0.984915\pi$
$384$		0			0
$385$	1540.00			0.203859
$386$	196.500	+	340.348i	0.0259108	+	0.0448789i
$387$		0			0
$388$	−4207.00	+	7286.74i	−0.550459	+	0.953423i
$389$	11653.0			1.51884			0.759422	−	0.650598i	$-0.225482\pi$
$389$	11653.0			1.51884			0.759422	+	0.650598i	$0.225482\pi$
$390$		0			0
$391$	−5994.00			−0.775268
$392$	−1822.50	+	3156.66i	−0.234822	+	0.406723i
$393$		0			0
$394$	−1761.00	−	3050.14i	−0.225172	−	0.390010i
$395$	−6188.00			−0.788233
$396$		0			0
$397$	−3067.00	−	5312.20i	−0.387729	−	0.671566i	0.604415	−	0.796670i	$-0.293407\pi$
$397$	−3067.00	−	5312.20i	−0.387729	−	0.671566i	−0.992144	+	0.125104i	$0.960074\pi$
$398$	2018.00			0.254154
$399$		0			0
$400$	1558.00	−	2698.54i	0.194750	−	0.337317i
$401$	−5397.50	+	9348.74i	−0.672165	+	1.16422i	0.305124	+	0.952313i	$0.401302\pi$
$401$	−5397.50	+	9348.74i	−0.672165	+	1.16422i	−0.977289	+	0.211912i	$0.932031\pi$
$402$		0			0
$403$	−8918.00	−	2206.63i	−1.10233	−	0.272755i
$404$	−3003.00			−0.369814
$405$		0			0
$406$	565.000	−	978.609i	0.0690652	−	0.119624i
$407$	−143.000	−	247.683i	−0.0174158	−	0.0301651i
$408$		0			0
$409$	4244.50	+	7351.69i	0.513147	+	0.888796i	0.999884	+	0.0152477i	$0.00485367\pi$
$409$	4244.50	+	7351.69i	0.513147	+	0.888796i	−0.486737	+	0.873549i	$0.661813\pi$
$410$	−997.500	−	1727.72i	−0.120154	−	0.208112i
$411$		0			0
$412$	−4557.00	−	7892.96i	−0.544921	−	0.943830i
$413$	−2880.00	+	4988.31i	−0.343137	+	0.594331i
$414$		0			0
$415$	3626.00			0.428900
$416$	2093.00	+	7250.36i	0.246677	+	0.854515i
$417$		0			0
$418$	−330.000	+	571.577i	−0.0386144	+	0.0668821i
$419$	748.000	−	1295.57i	0.0872129	−	0.151057i	−0.819119	−	0.573623i	$-0.805537\pi$
$419$	748.000	−	1295.57i	0.0872129	−	0.151057i	0.906332	+	0.422566i	$0.138871\pi$
$420$		0			0
$421$	−11695.0			−1.35387			−0.676935	−	0.736043i	$-0.736692\pi$
$421$	−11695.0			−1.35387			−0.676935	+	0.736043i	$0.736692\pi$
$422$	−80.0000	−	138.564i	−0.00922829	−	0.0159839i
$423$		0			0
$424$	−8055.00			−0.922607
$425$	−1406.00	−	2435.26i	−0.160473	−	0.277947i
$426$		0			0
$427$	−3175.00	+	5499.26i	−0.359834	+	0.623250i
$428$	−9366.00			−1.05776
$429$		0			0
$430$	1722.00			0.193121
$431$	5295.00	−	9171.21i	0.591766	−	1.02497i	−0.402228	−	0.915539i	$-0.631764\pi$
$431$	5295.00	−	9171.21i	0.591766	−	1.02497i	0.993995	−	0.109430i	$-0.0349024\pi$
$432$		0			0
$433$	6974.50	+	12080.2i	0.774072	+	1.34073i	0.935315	+	0.353817i	$0.115116\pi$
$433$	6974.50	+	12080.2i	0.774072	+	1.34073i	−0.161243	+	0.986915i	$0.551550\pi$
$434$	−1960.00			−0.216781
$435$		0			0
$436$	−3619.00	−	6268.29i	−0.397520	−	0.688525i
$437$	4860.00			0.532003
$438$		0			0
$439$	5363.00	−	9288.99i	0.583057	−	1.00988i	−0.412058	−	0.911158i	$-0.635190\pi$
$439$	5363.00	−	9288.99i	0.583057	−	1.00988i	0.995115	−	0.0987266i	$-0.0314769\pi$
$440$	−1155.00	+	2000.52i	−0.125142	+	0.216752i
$441$		0			0
$442$	1683.50	+	416.558i	0.181167	+	0.0448273i
$443$	−16228.0			−1.74044			−0.870221	−	0.492662i	$-0.836024\pi$
$443$	−16228.0			−1.74044			−0.870221	+	0.492662i	$0.836024\pi$
$444$		0			0
$445$	−679.000	+	1176.06i	−0.0723319	+	0.125282i
$446$	−2036.00	−	3526.46i	−0.216160	−	0.374400i
$447$		0			0
$448$	−835.000	−	1446.26i	−0.0880581	−	0.152521i
$449$	3769.00	+	6528.10i	0.396147	+	0.686147i	0.993247	−	0.116020i	$-0.0370136\pi$
$449$	3769.00	+	6528.10i	0.396147	+	0.686147i	−0.597100	+	0.802167i	$0.703680\pi$
$450$		0			0
$451$	3135.00	+	5429.98i	0.327320	+	0.566935i
$452$	−3769.50	+	6528.97i	−0.392262	+	0.679417i
$453$		0			0
$454$	5794.00			0.598956
$455$	910.000	+	3152.33i	0.0937614	+	0.324799i
$456$		0			0
$457$	−7769.50	+	13457.2i	−0.795278	+	1.37746i	0.127385	+	0.991853i	$0.459342\pi$
$457$	−7769.50	+	13457.2i	−0.795278	+	1.37746i	−0.922663	+	0.385608i	$0.873992\pi$
$458$	−3241.00	+	5613.58i	−0.330659	+	0.572719i
$459$		0			0
$460$	7938.00			0.804589
$461$	2405.50	+	4166.45i	0.243027	+	0.420935i	0.961575	−	0.274543i	$-0.0885264\pi$
$461$	2405.50	+	4166.45i	0.243027	+	0.420935i	−0.718548	+	0.695477i	$0.755193\pi$
$462$		0			0
$463$	562.000			0.0564111			0.0282056	−	0.999602i	$-0.491021\pi$
$463$	562.000			0.0564111			0.0282056	+	0.999602i	$0.491021\pi$
$464$	−2316.50	−	4012.30i	−0.231769	−	0.401436i
$465$		0			0
$466$	3445.00	−	5966.92i	0.342460	−	0.593159i
$467$	−4914.00			−0.486922			−0.243461	−	0.969911i	$-0.578283\pi$
$467$	−4914.00			−0.486922			−0.243461	+	0.969911i	$0.578283\pi$
$468$		0			0
$469$	−2020.00			−0.198880
$470$	1617.00	−	2800.73i	0.158695	−	0.274868i
$471$		0			0
$472$	−4320.00	−	7482.46i	−0.421280	−	0.729678i
$473$	−5412.00			−0.526097
$474$		0			0
$475$	1140.00	+	1974.54i	0.110120	+	0.190733i
$476$	−2590.00			−0.249396
$477$		0			0
$478$	1233.00	−	2135.62i	0.117983	−	0.204353i
$479$	−1800.00	+	3117.69i	−0.171700	+	0.297392i	−0.939014	−	0.343878i	$-0.888259\pi$
$479$	−1800.00	+	3117.69i	−0.171700	+	0.297392i	0.767315	+	0.641271i	$0.221593\pi$
$480$		0			0
$481$	422.500	−	439.075i	0.0400506	−	0.0416218i
$482$	−3617.00			−0.341805
$483$		0			0
$484$	−2964.50	+	5134.66i	−0.278409	+	0.482219i
$485$	−4207.00	−	7286.74i	−0.393876	−	0.682214i
$486$		0			0
$487$	8565.00	+	14835.0i	0.796955	+	1.38037i	0.921590	+	0.388164i	$0.126891\pi$
$487$	8565.00	+	14835.0i	0.796955	+	1.38037i	−0.124635	+	0.992203i	$0.539776\pi$
$488$	−4762.50	−	8248.89i	−0.441779	−	0.765184i
$489$		0			0
$490$	−850.500	−	1473.11i	−0.0784116	−	0.135813i
$491$	−5919.00	+	10252.0i	−0.544034	+	0.942295i	0.454633	+	0.890679i	$0.349770\pi$
$491$	−5919.00	+	10252.0i	−0.544034	+	0.942295i	−0.998667	+	0.0516158i	$0.983563\pi$
$492$		0			0
$493$	−4181.00			−0.381953
$494$	−1365.00	−	337.750i	−0.124320	−	0.0307613i
$495$		0			0
$496$	−4018.00	+	6959.38i	−0.363737	+	0.630011i
$497$	5430.00	−	9405.04i	0.490078	−	0.848840i
$498$		0			0
$499$	8976.00			0.805252			0.402626	−	0.915364i	$-0.368097\pi$
$499$	8976.00			0.805252			0.402626	+	0.915364i	$0.368097\pi$
$500$	4924.50	+	8529.48i	0.440461	+	0.762900i
$501$		0			0
$502$	−4860.00			−0.432096
$503$	841.000	+	1456.65i	0.0745494	+	0.129123i	0.900890	−	0.434047i	$-0.142915\pi$
$503$	841.000	+	1456.65i	0.0745494	+	0.129123i	−0.826341	+	0.563170i	$0.809581\pi$
$504$		0			0
$505$	1501.50	−	2600.67i	0.132309	−	0.229165i
$506$	3564.00			0.313121
$507$		0			0
$508$	6916.00			0.604031
$509$	7583.50	−	13135.0i	0.660379	−	1.14381i	−0.320138	−	0.947371i	$-0.603729\pi$
$509$	7583.50	−	13135.0i	0.660379	−	1.14381i	0.980516	−	0.196438i	$-0.0629375\pi$
$510$		0			0
$511$	−4025.00	−	6971.50i	−0.348445	−	0.603525i
$512$	11521.0			0.994455
$513$		0			0
$514$	282.500	+	489.304i	0.0242423	+	0.0419889i
$515$	9114.00			0.779827
$516$		0			0
$517$	−5082.00	+	8802.28i	−0.432314	+	0.748789i
$518$	65.0000	−	112.583i	0.00551339	−	0.00954947i
$519$		0			0
$520$	−4777.50	−	1182.12i	−0.402899	−	0.0996915i
$521$	6783.00			0.570381			0.285191	−	0.958471i	$-0.407943\pi$
$521$	6783.00			0.570381			0.285191	+	0.958471i	$0.407943\pi$
$522$		0			0
$523$	6959.00	−	12053.3i	0.581828	−	1.00775i	−0.413435	−	0.910534i	$-0.635671\pi$
$523$	6959.00	−	12053.3i	0.581828	−	1.00775i	0.995263	−	0.0972214i	$-0.0309955\pi$
$524$	−1960.00	−	3394.82i	−0.163403	−	0.283022i
$525$		0			0
$526$	−249.000	−	431.281i	−0.0206405	−	0.0357504i
$527$	3626.00	+	6280.42i	0.299717	+	0.519126i
$528$		0			0
$529$	−7038.50	−	12191.0i	−0.578491	−	1.00198i
$530$	1879.50	−	3255.39i	0.154038	−	0.266802i
$531$		0			0
$532$	2100.00			0.171140
$533$	−9262.50	+	9625.87i	−0.752727	+	0.782257i
$534$		0			0
$535$	4683.00	−	8111.19i	0.378437	−	0.655472i
$536$	1515.00	−	2624.06i	0.122086	−	0.211459i
$537$		0			0
$538$	−5546.00			−0.444433
$539$	2673.00	+	4629.77i	0.213607	+	0.369978i
$540$		0			0
$541$	−1335.00			−0.106093			−0.0530463	−	0.998592i	$-0.516893\pi$
$541$	−1335.00			−0.106093			−0.0530463	+	0.998592i	$0.516893\pi$
$542$	1128.00	+	1953.75i	0.0893944	+	0.154836i
$543$		0			0
$544$	2978.50	−	5158.91i	0.234747	−	0.406593i
$545$	7238.00			0.568884
$546$		0			0
$547$	−3806.00			−0.297501			−0.148750	−	0.988875i	$-0.547525\pi$
$547$	−3806.00			−0.297501			−0.148750	+	0.988875i	$0.547525\pi$
$548$	1816.50	−	3146.27i	0.141600	−	0.245259i
$549$		0			0
$550$	836.000	+	1447.99i	0.0648130	+	0.112259i
$551$	3390.00			0.262103
$552$		0			0
$553$	4420.00	+	7655.66i	0.339887	+	0.588702i
$554$	2309.00			0.177076
$555$		0			0
$556$	−1218.00	+	2109.64i	−0.0929041	+	0.160915i
$557$	−952.500	+	1649.78i	−0.0724573	+	0.125500i	−0.899978	−	0.435936i	$-0.856417\pi$
$557$	−952.500	+	1649.78i	−0.0724573	+	0.125500i	0.827520	+	0.561436i	$0.189751\pi$
$558$		0			0
$559$	−3198.00	−	11078.2i	−0.241970	−	0.838207i
$560$	2870.00			0.216571
$561$		0			0
$562$	2916.50	−	5051.53i	0.218906	−	0.379156i
$563$	−2400.00	−	4156.92i	−0.179659	−	0.311178i	0.762105	−	0.647454i	$-0.224166\pi$
$563$	−2400.00	−	4156.92i	−0.179659	−	0.311178i	−0.941764	+	0.336275i	$0.890833\pi$
$564$		0			0
$565$	−3769.50	−	6528.97i	−0.280680	−	0.486152i
$566$	−825.000	−	1428.94i	−0.0612674	−	0.106118i
$567$		0			0
$568$	8145.00	+	14107.6i	0.601684	+	1.04215i
$569$	7339.00	−	12711.5i	0.540715	−	0.936546i	−0.458148	−	0.888876i	$-0.651487\pi$
$569$	7339.00	−	12711.5i	0.540715	−	0.936546i	0.998863	−	0.0476701i	$-0.0151796\pi$
$570$		0			0
$571$	−586.000			−0.0429481			−0.0214740	−	0.999769i	$-0.506836\pi$
$571$	−586.000			−0.0429481			−0.0214740	+	0.999769i	$0.506836\pi$
$572$	7007.00	+	1733.78i	0.512198	+	0.126736i
$573$		0			0
$574$	−1425.00	+	2468.17i	−0.103621	+	0.179477i
$575$	6156.00	−	10662.5i	0.446475	−	0.773317i
$576$		0			0
$577$	8939.00			0.644949			0.322474	−	0.946578i	$-0.395485\pi$
$577$	8939.00			0.644949			0.322474	+	0.946578i	$0.395485\pi$
$578$	1772.00	+	3069.19i	0.127518	+	0.220868i
$579$		0			0
$580$	5537.00			0.396399
$581$	−2590.00	−	4486.01i	−0.184942	−	0.320329i
$582$		0			0
$583$	−5907.00	+	10231.2i	−0.419628	+	0.726816i
$584$	12075.0			0.855594
$585$		0			0
$586$	−2991.00			−0.210848
$587$	−6896.00	+	11944.2i	−0.484887	+	0.839848i	−0.999849	−	0.0173645i	$-0.994472\pi$
$587$	−6896.00	+	11944.2i	−0.484887	+	0.839848i	0.514963	+	0.857213i	$0.327806\pi$
$588$		0			0
$589$	−2940.00	−	5092.23i	−0.205672	−	0.356234i
$590$	4032.00			0.281347
$591$		0			0
$592$	−266.500	−	461.592i	−0.0185018	−	0.0320461i
$593$	−9569.00			−0.662650			−0.331325	−	0.943517i	$-0.607496\pi$
$593$	−9569.00			−0.662650			−0.331325	+	0.943517i	$0.607496\pi$
$594$		0			0
$595$	1295.00	−	2243.01i	0.0892266	−	0.154545i
$596$	2257.50	−	3910.10i	0.155152	−	0.268732i
$597$		0			0
$598$	2106.00	+	7295.40i	0.144015	+	0.498881i
$599$	5192.00			0.354156			0.177078	−	0.984197i	$-0.443336\pi$
$599$	5192.00			0.354156			0.177078	+	0.984197i	$0.443336\pi$
$600$		0			0
$601$	1838.50	−	3184.38i	0.124782	−	0.216129i	−0.796866	−	0.604156i	$-0.793510\pi$
$601$	1838.50	−	3184.38i	0.124782	−	0.216129i	0.921648	+	0.388028i	$0.126844\pi$
$602$	−1230.00	−	2130.42i	−0.0832742	−	0.144235i
$603$		0			0
$604$	10199.0	+	17665.2i	0.687072	+	1.19004i
$605$	−2964.50	−	5134.66i	−0.199213	−	0.345048i
$606$		0			0
$607$	5480.00	+	9491.64i	0.366435	+	0.634685i	0.989005	−	0.147879i	$-0.0472447\pi$
$607$	5480.00	+	9491.64i	0.366435	+	0.634685i	−0.622570	+	0.782564i	$0.713911\pi$
$608$	−2415.00	+	4182.90i	−0.161087	+	0.279012i
$609$		0			0
$610$	4445.00			0.295037
$611$	−21021.0	−	5201.35i	−1.39185	−	0.344393i
$612$		0			0
$613$	13013.5	−	22540.0i	0.857439	−	1.48513i	−0.0169241	−	0.999857i	$-0.505387\pi$
$613$	13013.5	−	22540.0i	0.857439	−	1.48513i	0.874363	−	0.485272i	$-0.161279\pi$
$614$	1211.00	−	2097.51i	0.0795961	−	0.137864i
$615$		0			0
$616$	3300.00			0.215845
$617$	8840.50	+	15312.2i	0.576832	+	0.999102i	0.995840	+	0.0911193i	$0.0290445\pi$
$617$	8840.50	+	15312.2i	0.576832	+	0.999102i	−0.419008	+	0.907982i	$0.637622\pi$
$618$		0			0
$619$	3192.00			0.207265			0.103633	−	0.994616i	$-0.466953\pi$
$619$	3192.00			0.207265			0.103633	+	0.994616i	$0.466953\pi$
$620$	−4802.00	−	8317.31i	−0.311053	−	0.538760i
$621$		0			0
$622$	−1701.00	+	2946.22i	−0.109653	+	0.189924i
$623$	1940.00			0.124758
$624$		0			0
$625$	−349.000			−0.0223360
$626$	−1155.00	+	2000.52i	−0.0737429	+	0.127727i
$627$		0			0
$628$	−7276.50	−	12603.3i	−0.462363	−	0.800836i
$629$	−481.000			−0.0304908
$630$		0			0
$631$	−3790.00	−	6564.47i	−0.239109	−	0.414148i	0.721350	−	0.692571i	$-0.243522\pi$
$631$	−3790.00	−	6564.47i	−0.239109	−	0.414148i	−0.960459	+	0.278422i	$0.910189\pi$
$632$	−13260.0			−0.834580
$633$		0			0
$634$	−128.500	+	222.569i	−0.00804951	+	0.0139422i
$635$	−3458.00	+	5989.43i	−0.216105	+	0.374304i
$636$		0			0
$637$	−7897.50	+	8207.32i	−0.491225	+	0.510496i
$638$	2486.00			0.154266
$639$		0			0
$640$	−5092.50	+	8820.47i	−0.314529	+	0.544781i
$641$	−13853.5	−	23995.0i	−0.853635	−	1.47854i	−0.877905	−	0.478835i	$-0.841059\pi$
$641$	−13853.5	−	23995.0i	−0.853635	−	1.47854i	0.0242696	−	0.999705i	$-0.492274\pi$
$642$		0			0
$643$	−5608.00	−	9713.34i	−0.343947	−	0.595734i	0.641215	−	0.767361i	$-0.278431\pi$
$643$	−5608.00	−	9713.34i	−0.343947	−	0.595734i	−0.985162	+	0.171628i	$0.945097\pi$
$644$	−5670.00	−	9820.73i	−0.346940	−	0.600918i
$645$		0			0
$646$	555.000	+	961.288i	0.0338021	+	0.0585470i
$647$	−1268.00	+	2196.24i	−0.0770483	+	0.133452i	−0.901975	−	0.431788i	$-0.857883\pi$
$647$	−1268.00	+	2196.24i	−0.0770483	+	0.133452i	0.824927	+	0.565239i	$0.191216\pi$
$648$		0			0
$649$	−12672.0			−0.766440
$650$	−2470.00	+	2566.90i	−0.149048	+	0.154895i
$651$		0			0
$652$	5950.00	−	10305.7i	0.357393	−	0.619022i
$653$	8865.00	−	15354.6i	0.531262	−	0.920173i	−0.468072	−	0.883690i	$-0.655051\pi$
$653$	8865.00	−	15354.6i	0.531262	−	0.920173i	0.999334	−	0.0364829i	$-0.0116154\pi$
$654$		0			0
$655$	3920.00			0.233843
$656$	5842.50	+	10119.5i	0.347731	+	0.602287i
$657$		0			0
$658$	−4620.00			−0.273718
$659$	9460.00	+	16385.2i	0.559195	+	0.968554i	0.997564	+	0.0697586i	$0.0222229\pi$
$659$	9460.00	+	16385.2i	0.559195	+	0.968554i	−0.438369	+	0.898795i	$0.644444\pi$
$660$		0			0
$661$	−2620.50	+	4538.84i	−0.154199	+	0.267081i	−0.932767	−	0.360480i	$-0.882613\pi$
$661$	−2620.50	+	4538.84i	−0.154199	+	0.267081i	0.778568	+	0.627560i	$0.215946\pi$
$662$	1028.00			0.0603540
$663$		0			0
$664$	7770.00			0.454118
$665$	−1050.00	+	1818.65i	−0.0612290	+	0.106052i
$666$		0			0
$667$	−9153.00	−	15853.5i	−0.531343	−	0.920313i
$668$	25760.0			1.49204
$669$		0			0
$670$	707.000	+	1224.56i	0.0407669	+	0.0706103i
$671$	−13970.0			−0.803735
$672$		0			0
$673$	−10233.5	+	17724.9i	−0.586140	+	1.01522i	0.408592	+	0.912717i	$0.366020\pi$
$673$	−10233.5	+	17724.9i	−0.586140	+	1.01522i	−0.994732	+	0.102508i	$0.967313\pi$
$674$	−1243.50	+	2153.81i	−0.0710650	+	0.123088i
$675$		0			0
$676$	591.500	+	15367.6i	0.0336538	+	0.874353i
$677$	70.0000			0.00397388			0.00198694	−	0.999998i	$-0.499368\pi$
$677$	70.0000			0.00397388			0.00198694	+	0.999998i	$0.499368\pi$
$678$		0			0
$679$	−6010.00	+	10409.6i	−0.339680	+	0.588343i
$680$	1942.50	+	3364.51i	0.109546	+	0.189740i
$681$		0			0
$682$	−2156.00	−	3734.30i	−0.121052	−	0.209668i
$683$	3216.00	+	5570.28i	0.180171	+	0.312065i	0.941939	−	0.335785i	$-0.109002\pi$
$683$	3216.00	+	5570.28i	0.180171	+	0.312065i	−0.761768	+	0.647850i	$0.775668\pi$
$684$		0			0
$685$	1816.50	+	3146.27i	0.101321	+	0.175493i
$686$	−2930.00	+	5074.91i	−0.163073	+	0.282450i
$687$		0			0
$688$	−10086.0			−0.558903
$689$	−24433.5	−	6045.72i	−1.35100	−	0.334287i
$690$		0			0
$691$	3333.00	−	5772.93i	0.183492	−	0.317818i	−0.759575	−	0.650420i	$-0.774593\pi$
$691$	3333.00	−	5772.93i	0.183492	−	0.317818i	0.943067	+	0.332601i	$0.107926\pi$
$692$	−14511.0	+	25133.8i	−0.797147	+	1.38070i
$693$		0			0
$694$	2850.00			0.155885
$695$	−1218.00	−	2109.64i	−0.0664768	−	0.115141i
$696$		0			0
$697$	10545.0			0.573056
$698$	1009.00	+	1747.64i	0.0547152	+	0.0947695i
$699$		0			0
$700$	2660.00	−	4607.26i	0.143626	−	0.248768i
$701$	14054.0			0.757221			0.378611	−	0.925556i	$-0.376402\pi$
$701$	14054.0			0.757221			0.378611	+	0.925556i	$0.376402\pi$
$702$		0			0
$703$	390.000			0.0209234
$704$	1837.00	−	3181.78i	0.0983445	−	0.170338i
$705$		0			0
$706$	−2643.50	−	4578.68i	−0.140920	−	0.244080i
$707$	−4290.00			−0.228207
$708$		0			0
$709$	35.5000	+	61.4878i	0.00188044	+	0.00325701i	0.866964	−	0.498371i	$-0.166068\pi$
$709$	35.5000	+	61.4878i	0.00188044	+	0.00325701i	−0.865084	+	0.501628i	$0.832735\pi$
$710$	−7602.00			−0.401828
$711$		0			0
$712$	−1455.00	+	2520.13i	−0.0765849	+	0.132649i
$713$	−15876.0	+	27498.0i	−0.833886	+	1.44433i
$714$		0			0
$715$	−5005.00	+	5201.35i	−0.261785	+	0.272055i
$716$	25718.0			1.34236
$717$		0			0
$718$	−3639.00	+	6302.93i	−0.189145	+	0.327609i
$719$	1968.00	+	3408.68i	0.102078	+	0.176804i	0.912541	−	0.408986i	$-0.134118\pi$
$719$	1968.00	+	3408.68i	0.102078	+	0.176804i	−0.810463	+	0.585790i	$0.800784\pi$
$720$		0			0
$721$	−6510.00	−	11275.7i	−0.336262	−	0.582423i
$722$	2979.50	+	5160.65i	0.153581	+	0.266010i
$723$		0			0
$724$	−11490.5	−	19902.1i	−0.589836	−	1.02163i
$725$	4294.00	−	7437.43i	0.219966	−	0.380992i
$726$		0			0
$727$	34202.0			1.74482			0.872409	−	0.488777i	$-0.162557\pi$
$727$	34202.0			1.74482			0.872409	+	0.488777i	$0.162557\pi$
$728$	1950.00	+	6755.00i	0.0992745	+	0.343897i
$729$		0			0
$730$	−2817.50	+	4880.05i	−0.142850	+	0.247423i
$731$	−4551.00	+	7882.56i	−0.230267	+	0.398833i
$732$		0			0
$733$	−27363.0			−1.37882			−0.689410	−	0.724371i	$-0.742130\pi$
$733$	−27363.0			−1.37882			−0.689410	+	0.724371i	$0.742130\pi$
$734$	2101.00	+	3639.04i	0.105653	+	0.182996i
$735$		0			0
$736$	26082.0			1.30624
$737$	−2222.00	−	3848.62i	−0.111056	−	0.192355i
$738$		0			0
$739$	−10888.0	+	18858.6i	−0.541978	+	0.938733i	0.456813	+	0.889563i	$0.348991\pi$
$739$	−10888.0	+	18858.6i	−0.541978	+	0.938733i	−0.998790	+	0.0491701i	$0.984342\pi$
$740$	637.000			0.0316440
$741$		0			0
$742$	−5370.00			−0.265686
$743$	1242.00	−	2151.21i	0.0613251	−	0.106218i	−0.833733	−	0.552168i	$-0.813801\pi$
$743$	1242.00	−	2151.21i	0.0613251	−	0.106218i	0.895058	+	0.445950i	$0.147134\pi$
$744$		0			0
$745$	2257.50	+	3910.10i	0.111018	+	0.192289i
$746$	−1583.00			−0.0776914
$747$		0			0
$748$	−2849.00	−	4934.61i	−0.139264	−	0.241213i
$749$	−13380.0			−0.652730
$750$		0			0
$751$	−16453.0	+	28497.4i	−0.799439	+	1.38467i	0.120543	+	0.992708i	$0.461536\pi$
$751$	−16453.0	+	28497.4i	−0.799439	+	1.38467i	−0.919982	+	0.391960i	$0.871797\pi$
$752$	−9471.00	+	16404.3i	−0.459271	+	0.795481i
$753$		0			0
$754$	1469.00	+	5088.77i	0.0709520	+	0.245785i
$755$	−20398.0			−0.983257
$756$		0			0
$757$	1957.00	−	3389.62i	0.0939609	−	0.162745i	−0.815214	−	0.579160i	$-0.803380\pi$
$757$	1957.00	−	3389.62i	0.0939609	−	0.162745i	0.909174	+	0.416415i	$0.136714\pi$
$758$	1026.00	+	1777.08i	0.0491636	+	0.0851538i
$759$		0			0
$760$	−1575.00	−	2727.98i	−0.0751727	−	0.130203i
$761$	−16519.0	−	28611.7i	−0.786877	−	1.36291i	−0.927871	−	0.372900i	$-0.878363\pi$
$761$	−16519.0	−	28611.7i	−0.786877	−	1.36291i	0.140995	−	0.990010i	$-0.454970\pi$
$762$		0			0
$763$	−5170.00	−	8954.70i	−0.245303	−	0.424878i
$764$	2086.00	−	3613.06i	0.0987812	−	0.171094i
$765$		0			0
$766$	6872.00			0.324145
$767$	−7488.00	−	25939.2i	−0.352511	−	1.22113i
$768$		0			0
$769$	8793.00	−	15229.9i	0.412332	−	0.714181i	−0.582812	−	0.812607i	$-0.698048\pi$
$769$	8793.00	−	15229.9i	0.412332	−	0.714181i	0.995144	+	0.0984263i	$0.0313809\pi$
$770$	−770.000	+	1333.68i	−0.0360375	+	0.0624188i
$771$		0			0
$772$	2751.00			0.128252
$773$	9157.00	+	15860.4i	0.426073	+	0.737980i	0.996520	−	0.0833544i	$-0.0265633\pi$
$773$	9157.00	+	15860.4i	0.426073	+	0.737980i	−0.570447	+	0.821334i	$0.693230\pi$
$774$		0			0
$775$	−14896.0			−0.690426
$776$	−9015.00	−	15614.4i	−0.417036	−	0.722327i
$777$		0			0
$778$	−5826.50	+	10091.8i	−0.268496	+	0.465049i
$779$	−8550.00			−0.393242
$780$		0			0
$781$	23892.0			1.09465
$782$	2997.00	−	5190.96i	0.137049	−	0.237376i
$783$		0			0
$784$	4981.50	+	8628.21i	0.226927	+	0.393049i
$785$	14553.0			0.661680
$786$		0			0
$787$	21034.0	+	36432.0i	0.952708	+	1.65014i	0.739528	+	0.673125i	$0.235049\pi$
$787$	21034.0	+	36432.0i	0.952708	+	1.65014i	0.213180	+	0.977013i	$0.431618\pi$
$788$	−24654.0			−1.11455
$789$		0			0
$790$	3094.00	−	5358.97i	0.139341	−	0.241346i
$791$	−5385.00	+	9327.09i	−0.242059	+	0.419258i
$792$		0			0
$793$	−8255.00	−	28596.2i	−0.369664	−	1.28055i
$794$	6134.00			0.274166
$795$		0			0
$796$	7063.00	−	12233.5i	0.314499	−	0.544729i
$797$	−2141.00	−	3708.32i	−0.0951545	−	0.164812i	0.814519	−	0.580137i	$-0.197001\pi$
$797$	−2141.00	−	3708.32i	−0.0951545	−	0.164812i	−0.909673	+	0.415325i	$0.863668\pi$
$798$		0			0
$799$	8547.00	+	14803.8i	0.378437	+	0.655472i
$800$	6118.00	+	10596.7i	0.270380	+	0.468312i
$801$		0			0
$802$	−5397.50	−	9348.74i	−0.237646	−	0.411616i
$803$	8855.00	−	15337.3i	0.389148	−	0.674025i
$804$		0			0
$805$	11340.0			0.496500
$806$	6370.00	−	6619.90i	0.278379	−	0.289300i
$807$		0			0
$808$	3217.50	−	5572.87i	0.140088	−	0.242640i
$809$	20110.5	−	34832.4i	0.873977	−	1.51377i	0.0161288	−	0.999870i	$-0.494866\pi$
$809$	20110.5	−	34832.4i	0.873977	−	1.51377i	0.857848	−	0.513903i	$-0.171801\pi$
$810$		0			0
$811$	−7084.00			−0.306724			−0.153362	−	0.988170i	$-0.549010\pi$
$811$	−7084.00			−0.306724			−0.153362	+	0.988170i	$0.549010\pi$
$812$	−3955.00	−	6850.26i	−0.170928	−	0.296055i
$813$		0			0
$814$	286.000			0.0123149
$815$	5950.00	+	10305.7i	0.255729	+	0.442936i
$816$		0			0
$817$	3690.00	−	6391.27i	0.158013	−	0.273687i
$818$	−8489.00			−0.362850
$819$		0			0
$820$	−13965.0			−0.594730
$821$	8669.00	−	15015.1i	0.368514	−	0.638285i	−0.620819	−	0.783954i	$-0.713200\pi$
$821$	8669.00	−	15015.1i	0.368514	−	0.638285i	0.989333	+	0.145668i	$0.0465332\pi$
$822$		0			0
$823$	−17748.0	−	30740.4i	−0.751709	−	1.30200i	−0.946994	−	0.321251i	$-0.895897\pi$
$823$	−17748.0	−	30740.4i	−0.751709	−	1.30200i	0.195285	−	0.980747i	$-0.437437\pi$
$824$	19530.0			0.825679
$825$		0			0
$826$	−2880.00	−	4988.31i	−0.121317	−	0.210128i
$827$	14992.0			0.630378			0.315189	−	0.949029i	$-0.397932\pi$
$827$	14992.0			0.630378			0.315189	+	0.949029i	$0.397932\pi$
$828$		0			0
$829$	10329.5	−	17891.2i	0.432760	−	0.749563i	−0.564349	−	0.825536i	$-0.690873\pi$
$829$	10329.5	−	17891.2i	0.432760	−	0.749563i	0.997110	+	0.0759730i	$0.0242063\pi$
$830$	−1813.00	+	3140.21i	−0.0758195	+	0.131323i
$831$		0			0
$832$	7598.50	+	1880.14i	0.316623	+	0.0783440i
$833$	8991.00			0.373973
$834$		0			0
$835$	−12880.0	+	22308.8i	−0.533809	+	0.924585i
$836$	2310.00	+	4001.04i	0.0955658	+	0.165525i
$837$		0			0
$838$	748.000	+	1295.57i	0.0308344	+	0.0534068i
$839$	14358.0	+	24868.8i	0.590814	+	1.02332i	0.994123	+	0.108256i	$0.0345268\pi$
$839$	14358.0	+	24868.8i	0.590814	+	1.02332i	−0.403309	+	0.915064i	$0.632140\pi$
$840$		0			0
$841$	5810.00	+	10063.2i	0.238222	+	0.412613i
$842$	5847.50	−	10128.2i	0.239333	−	0.414536i
$843$		0			0
$844$	−1120.00			−0.0456777
$845$	−13604.5	−	7171.56i	−0.553857	−	0.291963i
$846$		0			0
$847$	−4235.00	+	7335.24i	−0.171802	+	0.297570i
$848$	−11008.5	+	19067.3i	−0.445794	+	0.772138i
$849$		0			0
$850$	2812.00			0.113472
$851$	−1053.00	−	1823.85i	−0.0424164	−	0.0734674i
$852$		0			0
$853$	13377.0			0.536952			0.268476	−	0.963286i	$-0.413480\pi$
$853$	13377.0			0.536952			0.268476	+	0.963286i	$0.413480\pi$
$854$	−3175.00	−	5499.26i	−0.127220	−	0.220352i
$855$		0			0
$856$	10035.0	−	17381.1i	0.400688	−	0.694013i
$857$	27419.0			1.09290			0.546450	−	0.837492i	$-0.315979\pi$
$857$	27419.0			1.09290			0.546450	+	0.837492i	$0.315979\pi$
$858$		0			0
$859$	2422.00			0.0962021			0.0481010	−	0.998842i	$-0.484683\pi$
$859$	2422.00			0.0962021			0.0481010	+	0.998842i	$0.484683\pi$
$860$	6027.00	−	10439.1i	0.238976	−	0.413918i
$861$		0			0
$862$	5295.00	+	9171.21i	0.209221	+	0.362381i
$863$	34522.0			1.36169			0.680847	−	0.732425i	$-0.261612\pi$
$863$	34522.0			1.36169			0.680847	+	0.732425i	$0.261612\pi$
$864$		0			0
$865$	−14511.0	−	25133.8i	−0.570392	−	0.987947i
$866$	−13949.0			−0.547351
$867$		0			0
$868$	−6860.00	+	11881.9i	−0.268253	+	0.464628i
$869$	−9724.00	+	16842.5i	−0.379590	+	0.657470i
$870$		0			0
$871$	6565.00	−	6822.55i	0.255392	−	0.265411i
$872$	15510.0			0.602334
$873$		0			0
$874$	−2430.00	+	4208.88i	−0.0940457	+	0.162892i
$875$	7035.00	+	12185.0i	0.271802	+	0.470774i
$876$		0			0
$877$	−6866.50	−	11893.1i	−0.264385	−	0.457927i	0.703018	−	0.711172i	$-0.251835\pi$
$877$	−6866.50	−	11893.1i	−0.264385	−	0.457927i	−0.967402	+	0.253245i	$0.918502\pi$
$878$	5363.00	+	9288.99i	0.206142	+	0.357048i
$879$		0			0
$880$	3157.00	+	5468.08i	0.120935	+	0.209465i
$881$	−11379.5	+	19709.9i	−0.435170	+	0.753737i	−0.997310	−	0.0733055i	$-0.976645\pi$
$881$	−11379.5	+	19709.9i	−0.435170	+	0.753737i	0.562139	+	0.827043i	$0.309979\pi$
$882$		0			0
$883$	−2168.00			−0.0826263			−0.0413131	−	0.999146i	$-0.513154\pi$
$883$	−2168.00			−0.0826263			−0.0413131	+	0.999146i	$0.513154\pi$
$884$	8417.50	−	8747.72i	0.320261	−	0.332826i
$885$		0			0
$886$	8114.00	−	14053.9i	0.307669	−	0.532899i
$887$	7944.00	−	13759.4i	0.300714	−	0.520852i	−0.675584	−	0.737283i	$-0.736108\pi$
$887$	7944.00	−	13759.4i	0.300714	−	0.520852i	0.976298	+	0.216431i	$0.0694417\pi$
$888$		0			0
$889$	9880.00			0.372739
$890$	−679.000	−	1176.06i	−0.0255732	−	0.0442941i
$891$		0			0
$892$	−28504.0			−1.06994
$893$	−6930.00	−	12003.1i	−0.259690	−	0.449797i
$894$		0			0
$895$	−12859.0	+	22272.4i	−0.480256	+	0.831827i
$896$	14550.0			0.542502
$897$		0			0
$898$	−7538.00			−0.280118
$899$	−11074.0	+	19180.7i	−0.410833	+	0.711583i
$900$		0			0
$901$	9934.50	+	17207.1i	0.367332	+	0.636238i
$902$	−6270.00			−0.231450
$903$		0			0
$904$	−8077.50	−	13990.6i	−0.297183	−	0.514736i
$905$	22981.0			0.844104
$906$		0			0
$907$	5814.00	−	10070.1i	0.212845	−	0.368659i	−0.739759	−	0.672872i	$-0.765060\pi$
$907$	5814.00	−	10070.1i	0.212845	−	0.368659i	0.952604	+	0.304214i	$0.0983936\pi$
$908$	20279.0	−	35124.3i	0.741170	−	1.28374i
$909$		0			0
$910$	−3185.00	−	788.083i	−0.116024	−	0.0287085i
$911$	12584.0			0.457658			0.228829	−	0.973467i	$-0.426510\pi$
$911$	12584.0			0.457658			0.228829	+	0.973467i	$0.426510\pi$
$912$		0			0
$913$	5698.00	−	9869.23i	0.206546	−	0.357748i
$914$	−7769.50	−	13457.2i	−0.281173	−	0.487006i
$915$		0			0
$916$	22687.0	+	39295.0i	0.818340	+	1.41741i
$917$	−2800.00	−	4849.74i	−0.100833	−	0.174648i
$918$		0			0
$919$	−8592.00	−	14881.8i	−0.308405	−	0.534173i	0.669609	−	0.742714i	$-0.266462\pi$
$919$	−8592.00	−	14881.8i	−0.308405	−	0.534173i	−0.978014	+	0.208541i	$0.933128\pi$
$920$	−8505.00	+	14731.1i	−0.304784	+	0.527902i
$921$		0			0
$922$	−4811.00			−0.171846
$923$	14118.0	+	48906.2i	0.503467	+	1.74406i
$924$		0			0
$925$	494.000	−	855.633i	0.0175596	−	0.0304141i
$926$	−281.000	+	486.706i	−0.00997217	+	0.0172723i
$927$		0			0
$928$	18193.0			0.643550
$929$	6388.50	+	11065.2i	0.225619	+	0.390783i	0.956505	−	0.291716i	$-0.0942263\pi$
$929$	6388.50	+	11065.2i	0.225619	+	0.390783i	−0.730886	+	0.682499i	$0.760893\pi$
$930$		0			0
$931$	−7290.00			−0.256627
$932$	−24115.0	−	41768.4i	−0.847546	−	1.46799i
$933$		0			0
$934$	2457.00	−	4255.65i	0.0860765	−	0.149089i
$935$	5698.00			0.199299
$936$		0			0
$937$	9191.00			0.320445			0.160222	−	0.987081i	$-0.448779\pi$
$937$	9191.00			0.320445			0.160222	+	0.987081i	$0.448779\pi$
$938$	1010.00	−	1749.37i	0.0351574	−	0.0608945i
$939$		0			0
$940$	−11319.0	−	19605.1i	−0.392750	−	0.680263i
$941$	−50498.0			−1.74940			−0.874701	−	0.484662i	$-0.838942\pi$
$941$	−50498.0			−1.74940			−0.874701	+	0.484662i	$0.838942\pi$
$942$		0			0
$943$	23085.0	+	39984.4i	0.797191	+	1.38078i
$944$	−23616.0			−0.814232
$945$		0			0
$946$	2706.00	−	4686.93i	0.0930017	−	0.161084i
$947$	780.000	−	1351.00i	0.0267651	−	0.0463586i	−0.852333	−	0.523000i	$-0.824813\pi$
$947$	780.000	−	1351.00i	0.0267651	−	0.0463586i	0.879098	+	0.476642i	$0.158146\pi$
$948$		0			0
$949$	36627.5	+	9062.96i	1.25288	+	0.310006i
$950$	−2280.00			−0.0778663
$951$		0			0
$952$	2775.00	−	4806.44i	0.0944730	−	0.163632i
$953$	−10749.0	−	18617.8i	−0.365366	−	0.632833i	0.623468	−	0.781849i	$-0.285723\pi$
$953$	−10749.0	−	18617.8i	−0.365366	−	0.632833i	−0.988835	+	0.149015i	$0.952390\pi$
$954$		0			0
$955$	2086.00	+	3613.06i	0.0706821	+	0.122425i
$956$	−8631.00	−	14949.3i	−0.291994	−	0.505749i
$957$		0			0
$958$	−1800.00	−	3117.69i	−0.0607050	−	0.105144i
$959$	2595.00	−	4494.67i	0.0873795	−	0.151346i
$960$		0			0
$961$	8625.00			0.289517
$962$	169.000	+	585.433i	0.00566401	+	0.0196207i
$963$		0			0
$964$	−12659.5	+	21926.9i	−0.422962	+	0.732591i
$965$	−1375.50	+	2382.44i	−0.0458849	+	0.0794749i
$966$		0			0
$967$	−418.000			−0.0139007			−0.00695035	−	0.999976i	$-0.502212\pi$
$967$	−418.000			−0.0139007			−0.00695035	+	0.999976i	$0.502212\pi$
$968$	−6352.50	−	11002.9i	−0.210927	−	0.365336i
$969$		0			0
$970$	8414.00			0.278513
$971$	9066.00	+	15702.8i	0.299631	+	0.518976i	0.976052	−	0.217539i	$-0.0698031\pi$
$971$	9066.00	+	15702.8i	0.299631	+	0.518976i	−0.676420	+	0.736516i	$0.736470\pi$
$972$		0			0
$973$	−1740.00	+	3013.77i	−0.0573297	+	0.0992980i
$974$	−17130.0			−0.563532
$975$		0			0
$976$	−26035.0			−0.853853
$977$	6250.50	−	10826.2i	0.204679	−	0.354514i	−0.745352	−	0.666672i	$-0.767718\pi$
$977$	6250.50	−	10826.2i	0.204679	−	0.354514i	0.950030	+	0.312158i	$0.101052\pi$
$978$		0			0
$979$	2134.00	+	3696.20i	0.0696659	+	0.120665i
$980$	−11907.0			−0.388118
$981$		0			0
$982$	−5919.00	−	10252.0i	−0.192345	−	0.333151i
$983$	43708.0			1.41818			0.709089	−	0.705119i	$-0.249106\pi$
$983$	43708.0			1.41818			0.709089	+	0.705119i	$0.249106\pi$
$984$		0			0
$985$	12327.0	−	21351.0i	0.398752	−	0.690659i
$986$	2090.50	−	3620.85i	0.0675204	−	0.116949i
$987$		0			0
$988$	−6825.00	+	7092.75i	−0.219769	+	0.228391i
$989$	−39852.0			−1.28131
$990$		0			0
$991$	19807.0	−	34306.7i	0.634904	−	1.09969i	−0.351631	−	0.936139i	$-0.614373\pi$
$991$	19807.0	−	34306.7i	0.634904	−	1.09969i	0.986535	−	0.163548i	$-0.0522938\pi$
$992$	−15778.0	−	27328.3i	−0.504992	−	0.874672i
$993$		0			0
$994$	5430.00	+	9405.04i	0.173269	+	0.300110i
$995$	7063.00	+	12233.5i	0.225037	+	0.389776i
$996$		0			0
$997$	18251.5	+	31612.5i	0.579770	+	1.00419i	0.995505	+	0.0947056i	$0.0301910\pi$
$997$	18251.5	+	31612.5i	0.579770	+	1.00419i	−0.415735	+	0.909486i	$0.636476\pi$
$998$	−4488.00	+	7773.44i	−0.142350	+	0.246557i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.g.a.100.1		2
3.2	odd	2		39.4.e.b.22.1	yes	2
12.11	even	2		624.4.q.c.529.1		2
13.3	even	3	inner	117.4.g.a.55.1		2
13.4	even	6		1521.4.a.e.1.1		1
13.9	even	3		1521.4.a.h.1.1		1
39.17	odd	6		507.4.a.d.1.1		1
39.20	even	12		507.4.b.d.337.1		2
39.29	odd	6		39.4.e.b.16.1	✓	2
39.32	even	12		507.4.b.d.337.2		2
39.35	odd	6		507.4.a.b.1.1		1
156.107	even	6		624.4.q.c.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.b.16.1	✓	2	39.29	odd	6
39.4.e.b.22.1	yes	2	3.2	odd	2
117.4.g.a.55.1		2	13.3	even	3	inner
117.4.g.a.100.1		2	1.1	even	1	trivial
507.4.a.b.1.1		1	39.35	odd	6
507.4.a.d.1.1		1	39.17	odd	6
507.4.b.d.337.1		2	39.20	even	12
507.4.b.d.337.2		2	39.32	even	12
624.4.q.c.289.1		2	156.107	even	6
624.4.q.c.529.1		2	12.11	even	2
1521.4.a.e.1.1		1	13.4	even	6
1521.4.a.h.1.1		1	13.9	even	3

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} -7.00000 q^{5} +(5.00000 + 8.66025i) q^{7} -15.0000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} -7.00000 q^{5} +(5.00000 + 8.66025i) q^{7} -15.0000 q^{8} +(3.50000 - 6.06218i) q^{10} +(-11.0000 + 19.0526i) q^{11} +(-45.5000 - 11.2583i) q^{13} -10.0000 q^{14} +(-20.5000 + 35.5070i) q^{16} +(18.5000 + 32.0429i) q^{17} +(-15.0000 - 25.9808i) q^{19} +(-24.5000 - 42.4352i) q^{20} +(-11.0000 - 19.0526i) q^{22} +(-81.0000 + 140.296i) q^{23} -76.0000 q^{25} +(32.5000 - 33.7750i) q^{26} +(-35.0000 + 60.6218i) q^{28} +(-56.5000 + 97.8609i) q^{29} +196.000 q^{31} +(-80.5000 - 139.430i) q^{32} -37.0000 q^{34} +(-35.0000 - 60.6218i) q^{35} +(-6.50000 + 11.2583i) q^{37} +30.0000 q^{38} +105.000 q^{40} +(142.500 - 246.817i) q^{41} +(123.000 + 213.042i) q^{43} -154.000 q^{44} +(-81.0000 - 140.296i) q^{46} +462.000 q^{47} +(121.500 - 210.444i) q^{49} +(38.0000 - 65.8179i) q^{50} +(-91.0000 - 315.233i) q^{52} +537.000 q^{53} +(77.0000 - 133.368i) q^{55} +(-75.0000 - 129.904i) q^{56} +(-56.5000 - 97.8609i) q^{58} +(288.000 + 498.831i) q^{59} +(317.500 + 549.926i) q^{61} +(-98.0000 + 169.741i) q^{62} -167.000 q^{64} +(318.500 + 78.8083i) q^{65} +(-101.000 + 174.937i) q^{67} +(-129.500 + 224.301i) q^{68} +70.0000 q^{70} +(-543.000 - 940.504i) q^{71} -805.000 q^{73} +(-6.50000 - 11.2583i) q^{74} +(105.000 - 181.865i) q^{76} -220.000 q^{77} +884.000 q^{79} +(143.500 - 248.549i) q^{80} +(142.500 + 246.817i) q^{82} -518.000 q^{83} +(-129.500 - 224.301i) q^{85} -246.000 q^{86} +(165.000 - 285.788i) q^{88} +(97.0000 - 168.009i) q^{89} +(-130.000 - 450.333i) q^{91} -1134.00 q^{92} +(-231.000 + 400.104i) q^{94} +(105.000 + 181.865i) q^{95} +(601.000 + 1040.96i) q^{97} +(121.500 + 210.444i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8}+O(q^{10}) \) 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8	\( 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8} + 7 q^{10} - 22 q^{11} - 91 q^{13} - 20 q^{14} - 41 q^{16} + 37 q^{17} - 30 q^{19} - 49 q^{20} - 22 q^{22} - 162 q^{23} - 152 q^{25} + 65 q^{26} - 70 q^{28} - 113 q^{29} + 392 q^{31} - 161 q^{32} - 74 q^{34} - 70 q^{35} - 13 q^{37} + 60 q^{38} + 210 q^{40} + 285 q^{41} + 246 q^{43} - 308 q^{44} - 162 q^{46} + 924 q^{47} + 243 q^{49} + 76 q^{50} - 182 q^{52} + 1074 q^{53} + 154 q^{55} - 150 q^{56} - 113 q^{58} + 576 q^{59} + 635 q^{61} - 196 q^{62} - 334 q^{64} + 637 q^{65} - 202 q^{67} - 259 q^{68} + 140 q^{70} - 1086 q^{71} - 1610 q^{73} - 13 q^{74} + 210 q^{76} - 440 q^{77} + 1768 q^{79} + 287 q^{80} + 285 q^{82} - 1036 q^{83} - 259 q^{85} - 492 q^{86} + 330 q^{88} + 194 q^{89} - 260 q^{91} - 2268 q^{92} - 462 q^{94} + 210 q^{95} + 1202 q^{97} + 243 q^{98}+O(q^{100}) \) 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8 + 7 * q^10 - 22 * q^11 - 91 * q^13 - 20 * q^14 - 41 * q^16 + 37 * q^17 - 30 * q^19 - 49 * q^20 - 22 * q^22 - 162 * q^23 - 152 * q^25 + 65 * q^26 - 70 * q^28 - 113 * q^29 + 392 * q^31 - 161 * q^32 - 74 * q^34 - 70 * q^35 - 13 * q^37 + 60 * q^38 + 210 * q^40 + 285 * q^41 + 246 * q^43 - 308 * q^44 - 162 * q^46 + 924 * q^47 + 243 * q^49 + 76 * q^50 - 182 * q^52 + 1074 * q^53 + 154 * q^55 - 150 * q^56 - 113 * q^58 + 576 * q^59 + 635 * q^61 - 196 * q^62 - 334 * q^64 + 637 * q^65 - 202 * q^67 - 259 * q^68 + 140 * q^70 - 1086 * q^71 - 1610 * q^73 - 13 * q^74 + 210 * q^76 - 440 * q^77 + 1768 * q^79 + 287 * q^80 + 285 * q^82 - 1036 * q^83 - 259 * q^85 - 492 * q^86 + 330 * q^88 + 194 * q^89 - 260 * q^91 - 2268 * q^92 - 462 * q^94 + 210 * q^95 + 1202 * q^97 + 243 * q^98

Introduction

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