LMFDB - Embedded newform 13.4.c.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,4,Mod(3,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("13.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	13.c (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:	$0.767024830075$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	13.3
Dual form			13.4.c.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-4.00000 + 6.92820i) q^{4} +17.0000 q^{5} +(-4.00000 + 6.92820i) q^{6} +(-10.0000 + 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})$	\(q+(-2.00000 - 3.46410i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-4.00000 + 6.92820i) q^{4} +17.0000 q^{5} +(-4.00000 + 6.92820i) q^{6} +(-10.0000 + 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-34.0000 - 58.8897i) q^{10} +(16.0000 + 27.7128i) q^{11} +16.0000 q^{12} +(-45.5000 + 11.2583i) q^{13} +80.0000 q^{14} +(-17.0000 - 29.4449i) q^{15} +(32.0000 + 55.4256i) q^{16} +(6.50000 - 11.2583i) q^{17} -92.0000 q^{18} +(-15.0000 + 25.9808i) q^{19} +(-68.0000 + 117.779i) q^{20} +40.0000 q^{21} +(64.0000 - 110.851i) q^{22} +(-39.0000 - 67.5500i) q^{23} +164.000 q^{25} +(130.000 + 135.100i) q^{26} -100.000 q^{27} +(-80.0000 - 138.564i) q^{28} +(-98.5000 - 170.607i) q^{29} +(-68.0000 + 117.779i) q^{30} -74.0000 q^{31} +(128.000 - 221.703i) q^{32} +(32.0000 - 55.4256i) q^{33} -52.0000 q^{34} +(-170.000 + 294.449i) q^{35} +(92.0000 + 159.349i) q^{36} +(113.500 + 196.588i) q^{37} +120.000 q^{38} +(65.0000 + 67.5500i) q^{39} +(82.5000 + 142.894i) q^{41} +(-80.0000 - 138.564i) q^{42} +(78.0000 - 135.100i) q^{43} -256.000 q^{44} +(195.500 - 338.616i) q^{45} +(-156.000 + 270.200i) q^{46} -162.000 q^{47} +(64.0000 - 110.851i) q^{48} +(-28.5000 - 49.3634i) q^{49} +(-328.000 - 568.113i) q^{50} -26.0000 q^{51} +(104.000 - 360.267i) q^{52} +93.0000 q^{53} +(200.000 + 346.410i) q^{54} +(272.000 + 471.118i) q^{55} +60.0000 q^{57} +(-394.000 + 682.428i) q^{58} +(432.000 - 748.246i) q^{59} +272.000 q^{60} +(-72.5000 + 125.574i) q^{61} +(148.000 + 256.344i) q^{62} +(230.000 + 398.372i) q^{63} -512.000 q^{64} +(-773.500 + 191.392i) q^{65} -256.000 q^{66} +(-431.000 - 746.514i) q^{67} +(52.0000 + 90.0666i) q^{68} +(-78.0000 + 135.100i) q^{69} +1360.00 q^{70} +(-327.000 + 566.381i) q^{71} +215.000 q^{73} +(454.000 - 786.351i) q^{74} +(-164.000 - 284.056i) q^{75} +(-120.000 - 207.846i) q^{76} -640.000 q^{77} +(104.000 - 360.267i) q^{78} -76.0000 q^{79} +(544.000 + 942.236i) q^{80} +(-210.500 - 364.597i) q^{81} +(330.000 - 571.577i) q^{82} +628.000 q^{83} +(-160.000 + 277.128i) q^{84} +(110.500 - 191.392i) q^{85} -624.000 q^{86} +(-197.000 + 341.214i) q^{87} +(133.000 + 230.363i) q^{89} -1564.00 q^{90} +(260.000 - 900.666i) q^{91} +624.000 q^{92} +(74.0000 + 128.172i) q^{93} +(324.000 + 561.184i) q^{94} +(-255.000 + 441.673i) q^{95} -512.000 q^{96} +(-119.000 + 206.114i) q^{97} +(-114.000 + 197.454i) q^{98} +736.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9} - 68 q^{10} + 32 q^{11} + 32 q^{12} - 91 q^{13} + 160 q^{14} - 34 q^{15} + 64 q^{16} + 13 q^{17} - 184 q^{18} - 30 q^{19} - 136 q^{20} + 80 q^{21} + 128 q^{22} - 78 q^{23} + 328 q^{25} + 260 q^{26} - 200 q^{27} - 160 q^{28} - 197 q^{29} - 136 q^{30} - 148 q^{31} + 256 q^{32} + 64 q^{33} - 104 q^{34} - 340 q^{35} + 184 q^{36} + 227 q^{37} + 240 q^{38} + 130 q^{39} + 165 q^{41} - 160 q^{42} + 156 q^{43} - 512 q^{44} + 391 q^{45} - 312 q^{46} - 324 q^{47} + 128 q^{48} - 57 q^{49} - 656 q^{50} - 52 q^{51} + 208 q^{52} + 186 q^{53} + 400 q^{54} + 544 q^{55} + 120 q^{57} - 788 q^{58} + 864 q^{59} + 544 q^{60} - 145 q^{61} + 296 q^{62} + 460 q^{63} - 1024 q^{64} - 1547 q^{65} - 512 q^{66} - 862 q^{67} + 104 q^{68} - 156 q^{69} + 2720 q^{70} - 654 q^{71} + 430 q^{73} + 908 q^{74} - 328 q^{75} - 240 q^{76} - 1280 q^{77} + 208 q^{78} - 152 q^{79} + 1088 q^{80} - 421 q^{81} + 660 q^{82} + 1256 q^{83} - 320 q^{84} + 221 q^{85} - 1248 q^{86} - 394 q^{87} + 266 q^{89} - 3128 q^{90} + 520 q^{91} + 1248 q^{92} + 148 q^{93} + 648 q^{94} - 510 q^{95} - 1024 q^{96} - 238 q^{97} - 228 q^{98} + 1472 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9 - 68 * q^10 + 32 * q^11 + 32 * q^12 - 91 * q^13 + 160 * q^14 - 34 * q^15 + 64 * q^16 + 13 * q^17 - 184 * q^18 - 30 * q^19 - 136 * q^20 + 80 * q^21 + 128 * q^22 - 78 * q^23 + 328 * q^25 + 260 * q^26 - 200 * q^27 - 160 * q^28 - 197 * q^29 - 136 * q^30 - 148 * q^31 + 256 * q^32 + 64 * q^33 - 104 * q^34 - 340 * q^35 + 184 * q^36 + 227 * q^37 + 240 * q^38 + 130 * q^39 + 165 * q^41 - 160 * q^42 + 156 * q^43 - 512 * q^44 + 391 * q^45 - 312 * q^46 - 324 * q^47 + 128 * q^48 - 57 * q^49 - 656 * q^50 - 52 * q^51 + 208 * q^52 + 186 * q^53 + 400 * q^54 + 544 * q^55 + 120 * q^57 - 788 * q^58 + 864 * q^59 + 544 * q^60 - 145 * q^61 + 296 * q^62 + 460 * q^63 - 1024 * q^64 - 1547 * q^65 - 512 * q^66 - 862 * q^67 + 104 * q^68 - 156 * q^69 + 2720 * q^70 - 654 * q^71 + 430 * q^73 + 908 * q^74 - 328 * q^75 - 240 * q^76 - 1280 * q^77 + 208 * q^78 - 152 * q^79 + 1088 * q^80 - 421 * q^81 + 660 * q^82 + 1256 * q^83 - 320 * q^84 + 221 * q^85 - 1248 * q^86 - 394 * q^87 + 266 * q^89 - 3128 * q^90 + 520 * q^91 + 1248 * q^92 + 148 * q^93 + 648 * q^94 - 510 * q^95 - 1024 * q^96 - 238 * q^97 - 228 * q^98 + 1472 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$.

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

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 $

$2$	−2.00000	−	3.46410i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−2.00000	−	3.46410i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$3$	−1.00000	−	1.73205i	−0.192450	−	0.333333i	0.753612	−	0.657320i	$-0.228310\pi$
$3$	−1.00000	−	1.73205i	−0.192450	−	0.333333i	−0.946062	+	0.323987i	$0.894977\pi$
$4$	−4.00000	+	6.92820i	−0.500000	+	0.866025i
$5$	17.0000			1.52053			0.760263	−	0.649615i	$-0.225070\pi$
$5$	17.0000			1.52053			0.760263	+	0.649615i	$0.225070\pi$
$6$	−4.00000	+	6.92820i	−0.272166	+	0.471405i
$7$	−10.0000	+	17.3205i	−0.539949	+	0.935220i	0.458957	+	0.888459i	$0.348223\pi$
$7$	−10.0000	+	17.3205i	−0.539949	+	0.935220i	−0.998906	+	0.0467610i	$0.985110\pi$
$8$		0			0
$9$	11.5000	−	19.9186i	0.425926	−	0.737725i
$10$	−34.0000	−	58.8897i	−1.07517	−	1.86226i
$11$	16.0000	+	27.7128i	0.438562	+	0.759612i	0.997579	−	0.0695447i	$-0.0221546\pi$
$11$	16.0000	+	27.7128i	0.438562	+	0.759612i	−0.559017	+	0.829156i	$0.688821\pi$
$12$	16.0000			0.384900
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$14$	80.0000			1.52721
$15$	−17.0000	−	29.4449i	−0.292625	−	0.506842i
$16$	32.0000	+	55.4256i	0.500000	+	0.866025i
$17$	6.50000	−	11.2583i	0.0927342	−	0.160620i	−0.815927	−	0.578156i	$-0.803773\pi$
$17$	6.50000	−	11.2583i	0.0927342	−	0.160620i	0.908661	+	0.417535i	$0.137106\pi$
$18$	−92.0000			−1.20470
$19$	−15.0000	+	25.9808i	−0.181118	+	0.313705i	−0.942261	−	0.334878i	$-0.891305\pi$
$19$	−15.0000	+	25.9808i	−0.181118	+	0.313705i	0.761144	+	0.648583i	$0.224638\pi$
$20$	−68.0000	+	117.779i	−0.760263	+	1.31681i
$21$	40.0000			0.415653
$22$	64.0000	−	110.851i	0.620220	−	1.07425i
$23$	−39.0000	−	67.5500i	−0.353568	−	0.612398i	0.633304	−	0.773903i	$-0.281698\pi$
$23$	−39.0000	−	67.5500i	−0.353568	−	0.612398i	−0.986872	+	0.161506i	$0.948365\pi$
$24$		0			0
$25$	164.000			1.31200
$26$	130.000	+	135.100i	0.980581	+	1.01905i
$27$	−100.000			−0.712778
$28$	−80.0000	−	138.564i	−0.539949	−	0.935220i
$29$	−98.5000	−	170.607i	−0.630724	−	1.09245i	−0.987404	−	0.158219i	$-0.949425\pi$
$29$	−98.5000	−	170.607i	−0.630724	−	1.09245i	0.356680	−	0.934227i	$-0.383909\pi$
$30$	−68.0000	+	117.779i	−0.413835	+	0.716783i
$31$	−74.0000			−0.428735			−0.214368	−	0.976753i	$-0.568769\pi$
$31$	−74.0000			−0.428735			−0.214368	+	0.976753i	$0.568769\pi$
$32$	128.000	−	221.703i	0.707107	−	1.22474i
$33$	32.0000	−	55.4256i	0.168803	−	0.292375i
$34$	−52.0000			−0.262292
$35$	−170.000	+	294.449i	−0.821007	+	1.42203i
$36$	92.0000	+	159.349i	0.425926	+	0.737725i
$37$	113.500	+	196.588i	0.504305	+	0.873482i	0.999988	+	0.00497814i	$0.00158460\pi$
$37$	113.500	+	196.588i	0.504305	+	0.873482i	−0.495683	+	0.868504i	$0.665082\pi$
$38$	120.000			0.512278
$39$	65.0000	+	67.5500i	0.266880	+	0.277350i
$40$		0			0
$41$	82.5000	+	142.894i	0.314252	+	0.544301i	0.979278	−	0.202520i	$-0.0649130\pi$
$41$	82.5000	+	142.894i	0.314252	+	0.544301i	−0.665026	+	0.746820i	$0.731580\pi$
$42$	−80.0000	−	138.564i	−0.293911	−	0.509069i
$43$	78.0000	−	135.100i	0.276625	−	0.479129i	−0.693919	−	0.720053i	$-0.744117\pi$
$43$	78.0000	−	135.100i	0.276625	−	0.479129i	0.970544	+	0.240924i	$0.0774506\pi$
$44$	−256.000			−0.877124
$45$	195.500	−	338.616i	0.647632	−	1.12173i
$46$	−156.000	+	270.200i	−0.500021	+	0.866061i
$47$	−162.000			−0.502769			−0.251384	−	0.967887i	$-0.580886\pi$
$47$	−162.000			−0.502769			−0.251384	+	0.967887i	$0.580886\pi$
$48$	64.0000	−	110.851i	0.192450	−	0.333333i
$49$	−28.5000	−	49.3634i	−0.0830904	−	0.143917i
$50$	−328.000	−	568.113i	−0.927724	−	1.60687i
$51$	−26.0000			−0.0713868
$52$	104.000	−	360.267i	0.277350	−	0.960769i
$53$	93.0000			0.241029			0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000			0.241029			0.120514	+	0.992712i	$0.461546\pi$
$54$	200.000	+	346.410i	0.504010	+	0.872971i
$55$	272.000	+	471.118i	0.666845	+	1.15501i
$56$		0			0
$57$	60.0000			0.139424
$58$	−394.000	+	682.428i	−0.891978	+	1.54495i
$59$	432.000	−	748.246i	0.953248	−	1.65107i	0.214919	−	0.976632i	$-0.431051\pi$
$59$	432.000	−	748.246i	0.953248	−	1.65107i	0.738328	−	0.674442i	$-0.235616\pi$
$60$	272.000			0.585251
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	−0.932027	−	0.362389i	$-0.881961\pi$
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	0.779852	+	0.625964i	$0.215294\pi$
$62$	148.000	+	256.344i	0.303162	+	0.525091i
$63$	230.000	+	398.372i	0.459957	+	0.796668i
$64$	−512.000			−1.00000
$65$	−773.500	+	191.392i	−1.47601	+	0.365219i
$66$	−256.000			−0.477446
$67$	−431.000	−	746.514i	−0.785896	−	1.36121i	−0.928462	−	0.371427i	$-0.878869\pi$
$67$	−431.000	−	746.514i	−0.785896	−	1.36121i	0.142566	−	0.989785i	$-0.454465\pi$
$68$	52.0000	+	90.0666i	0.0927342	+	0.160620i
$69$	−78.0000	+	135.100i	−0.136088	+	0.235712i
$70$	1360.00			2.32216
$71$	−327.000	+	566.381i	−0.546588	+	0.946718i	0.451917	+	0.892060i	$0.350740\pi$
$71$	−327.000	+	566.381i	−0.546588	+	0.946718i	−0.998505	+	0.0546585i	$0.982593\pi$
$72$		0			0
$73$	215.000			0.344710			0.172355	−	0.985035i	$-0.444862\pi$
$73$	215.000			0.344710			0.172355	+	0.985035i	$0.444862\pi$
$74$	454.000	−	786.351i	0.713195	−	1.23529i
$75$	−164.000	−	284.056i	−0.252495	−	0.437333i
$76$	−120.000	−	207.846i	−0.181118	−	0.313705i
$77$	−640.000			−0.947205
$78$	104.000	−	360.267i	0.150970	−	0.522976i
$79$	−76.0000			−0.108236			−0.0541182	−	0.998535i	$-0.517235\pi$
$79$	−76.0000			−0.108236			−0.0541182	+	0.998535i	$0.517235\pi$
$80$	544.000	+	942.236i	0.760263	+	1.31681i
$81$	−210.500	−	364.597i	−0.288752	−	0.500133i
$82$	330.000	−	571.577i	0.444420	−	0.769757i
$83$	628.000			0.830505			0.415253	−	0.909706i	$-0.363693\pi$
$83$	628.000			0.830505			0.415253	+	0.909706i	$0.363693\pi$
$84$	−160.000	+	277.128i	−0.207827	+	0.359966i
$85$	110.500	−	191.392i	0.141005	−	0.244227i
$86$	−624.000			−0.782415
$87$	−197.000	+	341.214i	−0.242766	+	0.420483i
$88$		0			0
$89$	133.000	+	230.363i	0.158404	+	0.274364i	0.934293	−	0.356505i	$-0.116032\pi$
$89$	133.000	+	230.363i	0.158404	+	0.274364i	−0.775889	+	0.630869i	$0.782698\pi$
$90$	−1564.00			−1.83178
$91$	260.000	−	900.666i	0.299510	−	1.03753i
$92$	624.000			0.707136
$93$	74.0000	+	128.172i	0.0825101	+	0.142912i
$94$	324.000	+	561.184i	0.355511	+	0.615763i
$95$	−255.000	+	441.673i	−0.275394	+	0.476997i
$96$	−512.000			−0.544331
$97$	−119.000	+	206.114i	−0.124563	+	0.215750i	−0.921562	−	0.388231i	$-0.873086\pi$
$97$	−119.000	+	206.114i	−0.124563	+	0.215750i	0.796999	+	0.603981i	$0.206420\pi$
$98$	−114.000	+	197.454i	−0.117508	+	0.203529i
$99$	736.000			0.747180
$100$	−656.000	+	1136.23i	−0.656000	+	1.13623i
$101$	409.500	+	709.275i	0.403433	+	0.698767i	0.994138	−	0.108121i	$-0.0344834\pi$
$101$	409.500	+	709.275i	0.403433	+	0.698767i	−0.590704	+	0.806888i	$0.701150\pi$
$102$	52.0000	+	90.0666i	0.0504781	+	0.0874307i
$103$	1638.00			1.56696			0.783480	−	0.621417i	$-0.213443\pi$
$103$	1638.00			1.56696			0.783480	+	0.621417i	$0.213443\pi$
$104$		0			0
$105$	680.000			0.632011
$106$	−186.000	−	322.161i	−0.170433	−	0.295199i
$107$	−261.000	−	452.065i	−0.235811	−	0.408437i	0.723697	−	0.690118i	$-0.242441\pi$
$107$	−261.000	−	452.065i	−0.235811	−	0.408437i	−0.959508	+	0.281681i	$0.909108\pi$
$108$	400.000	−	692.820i	0.356389	−	0.617284i
$109$	−1634.00			−1.43586			−0.717930	−	0.696115i	$-0.754910\pi$
$109$	−1634.00			−1.43586			−0.717930	+	0.696115i	$0.754910\pi$
$110$	1088.00	−	1884.47i	0.943061	−	1.63343i
$111$	227.000	−	393.176i	0.194107	−	0.336203i
$112$	−1280.00			−1.07990
$113$	−163.500	+	283.190i	−0.136113	+	0.235755i	−0.926022	−	0.377469i	$-0.876794\pi$
$113$	−163.500	+	283.190i	−0.136113	+	0.235755i	0.789909	+	0.613224i	$0.210128\pi$
$114$	−120.000	−	207.846i	−0.0985880	−	0.170759i
$115$	−663.000	−	1148.35i	−0.537609	−	0.931167i
$116$	1576.00			1.26145
$117$	−299.000	+	1035.77i	−0.236261	+	0.818433i
$118$	−3456.00			−2.69619
$119$	130.000	+	225.167i	0.100144	+	0.173454i
$120$		0			0
$121$	153.500	−	265.870i	0.115327	−	0.199752i
$122$	580.000			0.430416
$123$	165.000	−	285.788i	0.120956	−	0.209501i
$124$	296.000	−	512.687i	0.214368	−	0.371296i
$125$	663.000			0.474404
$126$	920.000	−	1593.49i	0.650477	−	1.12666i
$127$	1079.00	+	1868.88i	0.753904	+	1.30580i	0.945918	+	0.324407i	$0.105165\pi$
$127$	1079.00	+	1868.88i	0.753904	+	1.30580i	−0.192014	+	0.981392i	$0.561502\pi$
$128$		0			0
$129$	−312.000			−0.212946
$130$	2210.00	+	2296.70i	1.49100	+	1.54949i
$131$	730.000			0.486873			0.243437	−	0.969917i	$-0.421725\pi$
$131$	730.000			0.486873			0.243437	+	0.969917i	$0.421725\pi$
$132$	256.000	+	443.405i	0.168803	+	0.292375i
$133$	−300.000	−	519.615i	−0.195589	−	0.338770i
$134$	−1724.00	+	2986.06i	−1.11142	+	1.92504i
$135$	−1700.00			−1.08380
$136$		0			0
$137$	−835.500	+	1447.13i	−0.521033	+	0.902456i	0.478667	+	0.877996i	$0.341120\pi$
$137$	−835.500	+	1447.13i	−0.521033	+	0.902456i	−0.999701	+	0.0244601i	$0.992213\pi$
$138$	624.000			0.384916
$139$	−456.000	+	789.815i	−0.278255	+	0.481951i	−0.970951	−	0.239278i	$-0.923089\pi$
$139$	−456.000	+	789.815i	−0.278255	+	0.481951i	0.692696	+	0.721229i	$0.256423\pi$
$140$	−1360.00	−	2355.59i	−0.821007	−	1.42203i
$141$	162.000	+	280.592i	0.0967579	+	0.167590i
$142$	2616.00			1.54598
$143$	−1040.00	−	1080.80i	−0.608176	−	0.632035i
$144$	1472.00			0.851852
$145$	−1674.50	−	2900.32i	−0.959032	−	1.66109i
$146$	−430.000	−	744.782i	−0.243747	−	0.422182i
$147$	−57.0000	+	98.7269i	−0.0319815	+	0.0553936i
$148$	−1816.00			−1.00861
$149$	1057.50	−	1831.64i	0.581435	−	1.00707i	−0.413875	−	0.910334i	$-0.635825\pi$
$149$	1057.50	−	1831.64i	0.581435	−	1.00707i	0.995310	−	0.0967407i	$-0.0308418\pi$
$150$	−656.000	+	1136.23i	−0.357081	+	0.618483i
$151$	514.000			0.277011			0.138506	−	0.990362i	$-0.455770\pi$
$151$	514.000			0.277011			0.138506	+	0.990362i	$0.455770\pi$
$152$		0			0
$153$	−149.500	−	258.942i	−0.0789958	−	0.136825i
$154$	1280.00	+	2217.03i	0.669775	+	1.16008i
$155$	−1258.00			−0.651903
$156$	−728.000	+	180.133i	−0.373632	+	0.0924500i
$157$	2901.00			1.47468			0.737341	−	0.675521i	$-0.236081\pi$
$157$	2901.00			1.47468			0.737341	+	0.675521i	$0.236081\pi$
$158$	152.000	+	263.272i	0.0765346	+	0.132562i
$159$	−93.0000	−	161.081i	−0.0463860	−	0.0803430i
$160$	2176.00	−	3768.94i	1.07517	−	1.86226i
$161$	1560.00			0.763635
$162$	−842.000	+	1458.39i	−0.408357	+	0.707294i
$163$	−1180.00	+	2043.82i	−0.567023	+	0.982112i	0.429835	+	0.902907i	$0.358572\pi$
$163$	−1180.00	+	2043.82i	−0.567023	+	0.982112i	−0.996858	+	0.0792052i	$0.974762\pi$
$164$	−1320.00			−0.628504
$165$	544.000	−	942.236i	0.256669	−	0.444563i
$166$	−1256.00	−	2175.46i	−0.587256	−	1.01716i
$167$	−140.000	−	242.487i	−0.0648714	−	0.112361i	0.831766	−	0.555127i	$-0.187330\pi$
$167$	−140.000	−	242.487i	−0.0648714	−	0.112361i	−0.896637	+	0.442767i	$0.853997\pi$
$168$		0			0
$169$	1943.50	−	1024.51i	0.884615	−	0.466321i
$170$	−884.000			−0.398822
$171$	345.000	+	597.558i	0.154285	+	0.267230i
$172$	624.000	+	1080.80i	0.276625	+	0.479129i
$173$	−663.000	+	1148.35i	−0.291370	+	0.504667i	−0.974134	−	0.225972i	$-0.927444\pi$
$173$	−663.000	+	1148.35i	−0.291370	+	0.504667i	0.682764	+	0.730639i	$0.260778\pi$
$174$	1576.00			0.686645
$175$	−1640.00	+	2840.56i	−0.708413	+	1.22701i
$176$	−1024.00	+	1773.62i	−0.438562	+	0.759612i
$177$	−1728.00			−0.733810
$178$	532.000	−	921.451i	0.224017	−	0.388009i
$179$	−2132.00	−	3692.73i	−0.890241	−	1.54194i	−0.839586	−	0.543227i	$-0.817202\pi$
$179$	−2132.00	−	3692.73i	−0.890241	−	1.54194i	−0.0506550	−	0.998716i	$-0.516131\pi$
$180$	1564.00	+	2708.93i	0.647632	+	1.12173i
$181$	−403.000			−0.165496			−0.0827479	−	0.996571i	$-0.526370\pi$
$181$	−403.000			−0.165496			−0.0827479	+	0.996571i	$0.526370\pi$
$182$	−3640.00	+	900.666i	−1.48250	+	0.366823i
$183$	290.000			0.117144
$184$		0			0
$185$	1929.50	+	3341.99i	0.766809	+	1.32815i
$186$	296.000	−	512.687i	0.116687	−	0.202108i
$187$	416.000			0.162679
$188$	648.000	−	1122.37i	0.251384	−	0.435410i
$189$	1000.00	−	1732.05i	0.384864	−	0.666604i
$190$	2040.00			0.778932
$191$	623.000	−	1079.07i	0.236014	−	0.408788i	−0.723553	−	0.690269i	$-0.757492\pi$
$191$	623.000	−	1079.07i	0.236014	−	0.408788i	0.959567	+	0.281481i	$0.0908255\pi$
$192$	512.000	+	886.810i	0.192450	+	0.333333i
$193$	−133.500	−	231.229i	−0.0497904	−	0.0862394i	0.840056	−	0.542500i	$-0.182522\pi$
$193$	−133.500	−	231.229i	−0.0497904	−	0.0862394i	−0.889846	+	0.456260i	$0.849189\pi$
$194$	952.000			0.352318
$195$	1105.00	+	1148.35i	0.405798	+	0.421718i
$196$	456.000			0.166181
$197$	−639.000	−	1106.78i	−0.231101	−	0.400278i	0.727032	−	0.686604i	$-0.240899\pi$
$197$	−639.000	−	1106.78i	−0.231101	−	0.400278i	−0.958132	+	0.286326i	$0.907566\pi$
$198$	−1472.00	−	2549.58i	−0.528336	−	0.915104i
$199$	−2119.00	+	3670.22i	−0.754834	+	1.30741i	0.190623	+	0.981663i	$0.438949\pi$
$199$	−2119.00	+	3670.22i	−0.754834	+	1.30741i	−0.945457	+	0.325747i	$0.894384\pi$
$200$		0			0
$201$	−862.000	+	1493.03i	−0.302492	+	0.523931i
$202$	1638.00	−	2837.10i	0.570541	−	0.988206i
$203$	3940.00			1.36224
$204$	104.000	−	180.133i	0.0356934	−	0.0618228i
$205$	1402.50	+	2429.20i	0.477829	+	0.827623i
$206$	−3276.00	−	5674.20i	−1.10801	−	1.91913i
$207$	−1794.00			−0.602375
$208$	−2080.00	−	2161.60i	−0.693375	−	0.720577i
$209$	−960.000			−0.317725
$210$	−1360.00	−	2355.59i	−0.446900	−	0.774053i
$211$	−1535.00	−	2658.70i	−0.500823	−	0.867452i	−1.00000		0.000951154i	$-0.999697\pi$
$211$	−1535.00	−	2658.70i	−0.500823	−	0.867452i	0.499176	−	0.866501i	$-0.333636\pi$
$212$	−372.000	+	644.323i	−0.120514	+	0.208737i
$213$	1308.00			0.420764
$214$	−1044.00	+	1808.26i	−0.333488	+	0.577618i
$215$	1326.00	−	2296.70i	0.420616	−	0.728528i
$216$		0			0
$217$	740.000	−	1281.72i	0.231495	−	0.400962i
$218$	3268.00	+	5660.34i	1.01531	+	1.75856i
$219$	−215.000	−	372.391i	−0.0663395	−	0.114903i
$220$	−4352.00			−1.33369
$221$	−169.000	+	585.433i	−0.0514397	+	0.178192i
$222$	−1816.00			−0.549018
$223$	2689.00	+	4657.48i	0.807483	+	1.39860i	0.914602	+	0.404356i	$0.132504\pi$
$223$	2689.00	+	4657.48i	0.807483	+	1.39860i	−0.107119	+	0.994246i	$0.534162\pi$
$224$	2560.00	+	4434.05i	0.763604	+	1.32260i
$225$	1886.00	−	3266.65i	0.558815	−	0.967896i
$226$	1308.00			0.384986
$227$	1987.00	−	3441.58i	0.580977	−	1.00628i	−0.414387	−	0.910101i	$-0.636004\pi$
$227$	1987.00	−	3441.58i	0.580977	−	1.00628i	0.995364	−	0.0961811i	$-0.0306628\pi$
$228$	−240.000	+	415.692i	−0.0697122	+	0.120745i
$229$	−6298.00			−1.81740			−0.908698	−	0.417455i	$-0.862922\pi$
$229$	−6298.00			−1.81740			−0.908698	+	0.417455i	$0.862922\pi$
$230$	−2652.00	+	4593.40i	−0.760294	+	1.31687i
$231$	640.000	+	1108.51i	0.182290	+	0.315735i
$232$		0			0
$233$	4030.00			1.13311			0.566554	−	0.824025i	$-0.308276\pi$
$233$	4030.00			1.13311			0.566554	+	0.824025i	$0.308276\pi$
$234$	4186.00	−	1035.77i	1.16943	−	0.289360i
$235$	−2754.00			−0.764473
$236$	3456.00	+	5985.97i	0.953248	+	1.65107i
$237$	76.0000	+	131.636i	0.0208301	+	0.0360788i
$238$	520.000	−	900.666i	0.141624	−	0.245301i
$239$	−984.000			−0.266317			−0.133158	−	0.991095i	$-0.542512\pi$
$239$	−984.000			−0.266317			−0.133158	+	0.991095i	$0.542512\pi$
$240$	1088.00	−	1884.47i	0.292625	−	0.506842i
$241$	−471.500	+	816.662i	−0.126025	+	0.218281i	−0.922133	−	0.386873i	$-0.873555\pi$
$241$	−471.500	+	816.662i	−0.126025	+	0.218281i	0.796108	+	0.605154i	$0.206889\pi$
$242$	−1228.00			−0.326194
$243$	−1771.00	+	3067.46i	−0.467530	+	0.809785i
$244$	−580.000	−	1004.59i	−0.152175	−	0.263575i
$245$	−484.500	−	839.179i	−0.126341	−	0.218829i
$246$	−1320.00			−0.342114
$247$	390.000	−	1351.00i	0.100466	−	0.348024i
$248$		0			0
$249$	−628.000	−	1087.73i	−0.159831	−	0.276835i
$250$	−1326.00	−	2296.70i	−0.335454	−	0.581024i
$251$	1365.00	−	2364.25i	0.343259	−	0.594542i	−0.641777	−	0.766891i	$-0.721802\pi$
$251$	1365.00	−	2364.25i	0.343259	−	0.594542i	0.985036	+	0.172349i	$0.0551358\pi$
$252$	−3680.00			−0.919914
$253$	1248.00	−	2161.60i	0.310123	−	0.537149i
$254$	4316.00	−	7475.53i	1.06618	−	1.84668i
$255$	−442.000			−0.108546
$256$	−2048.00	+	3547.24i	−0.500000	+	0.866025i
$257$	942.500	+	1632.46i	0.228761	+	0.396225i	0.957441	−	0.288629i	$-0.0931993\pi$
$257$	942.500	+	1632.46i	0.228761	+	0.396225i	−0.728680	+	0.684854i	$0.759866\pi$
$258$	624.000	+	1080.80i	0.150576	+	0.260805i
$259$	−4540.00			−1.08920
$260$	1768.00	−	6124.53i	0.421718	−	1.46087i
$261$	−4531.00			−1.07457
$262$	−1460.00	−	2528.79i	−0.344271	−	0.596296i
$263$	−2016.00	−	3491.81i	−0.472669	−	0.818686i	0.526842	−	0.849963i	$-0.323376\pi$
$263$	−2016.00	−	3491.81i	−0.472669	−	0.818686i	−0.999511	+	0.0312769i	$0.990043\pi$
$264$		0			0
$265$	1581.00			0.366491
$266$	−1200.00	+	2078.46i	−0.276604	+	0.479093i
$267$	266.000	−	460.726i	0.0609698	−	0.105603i
$268$	6896.00			1.57179
$269$	−2003.00	+	3469.30i	−0.453997	+	0.786345i	−0.998630	−	0.0523292i	$-0.983335\pi$
$269$	−2003.00	+	3469.30i	−0.453997	+	0.786345i	0.544633	+	0.838674i	$0.316669\pi$
$270$	3400.00	+	5888.97i	0.766361	+	1.32738i
$271$	2148.00	+	3720.45i	0.481482	+	0.833952i	0.999774	−	0.0212520i	$-0.00676523\pi$
$271$	2148.00	+	3720.45i	0.481482	+	0.833952i	−0.518292	+	0.855204i	$0.673432\pi$
$272$	832.000			0.185468
$273$	−1820.00	+	450.333i	−0.403485	+	0.0998367i
$274$	6684.00			1.47371
$275$	2624.00	+	4544.90i	0.575393	+	0.996610i
$276$	−624.000	−	1080.80i	−0.136088	−	0.235712i
$277$	2775.50	−	4807.31i	0.602035	−	1.04275i	−0.390478	−	0.920612i	$-0.627690\pi$
$277$	2775.50	−	4807.31i	0.602035	−	1.04275i	0.992513	−	0.122142i	$-0.0389765\pi$
$278$	3648.00			0.787023
$279$	−851.000	+	1473.98i	−0.182609	+	0.316289i
$280$		0			0
$281$	−5557.00			−1.17973			−0.589863	−	0.807504i	$-0.700818\pi$
$281$	−5557.00			−1.17973			−0.589863	+	0.807504i	$0.700818\pi$
$282$	648.000	−	1122.37i	0.136836	−	0.237007i
$283$	−1560.00	−	2702.00i	−0.327676	−	0.567552i	0.654374	−	0.756171i	$-0.272932\pi$
$283$	−1560.00	−	2702.00i	−0.327676	−	0.567552i	−0.982050	+	0.188619i	$0.939599\pi$
$284$	−2616.00	−	4531.04i	−0.546588	−	0.946718i
$285$	1020.00			0.211999
$286$	−1664.00	+	5764.27i	−0.344036	+	1.19178i
$287$	−3300.00			−0.678721
$288$	−2944.00	−	5099.16i	−0.602350	−	1.04330i
$289$	2372.00	+	4108.42i	0.482801	+	0.836235i
$290$	−6698.00	+	11601.3i	−1.35628	+	2.34914i
$291$	476.000			0.0958887
$292$	−860.000	+	1489.56i	−0.172355	+	0.298528i
$293$	−4150.50	+	7188.88i	−0.827559	+	1.43337i	0.0723887	+	0.997376i	$0.476938\pi$
$293$	−4150.50	+	7188.88i	−0.827559	+	1.43337i	−0.899948	+	0.435998i	$0.856396\pi$
$294$	456.000			0.0904573
$295$	7344.00	−	12720.2i	1.44944	−	2.51050i
$296$		0			0
$297$	−1600.00	−	2771.28i	−0.312597	−	0.541435i
$298$	−8460.00			−1.64455
$299$	2535.00	+	2634.45i	0.490310	+	0.509546i
$300$	2624.00			0.504989
$301$	1560.00	+	2702.00i	0.298727	+	0.517411i
$302$	−1028.00	−	1780.55i	−0.195877	−	0.339268i
$303$	819.000	−	1418.55i	0.155282	−	0.268956i
$304$	−1920.00			−0.362235
$305$	−1232.50	+	2134.75i	−0.231386	+	0.400772i
$306$	−598.000	+	1035.77i	−0.111717	+	0.193499i
$307$	8678.00			1.61329			0.806644	−	0.591037i	$-0.201281\pi$
$307$	8678.00			1.61329			0.806644	+	0.591037i	$0.201281\pi$
$308$	2560.00	−	4434.05i	0.473602	−	0.820303i
$309$	−1638.00	−	2837.10i	−0.301562	−	0.522320i
$310$	2516.00	+	4357.84i	0.460965	+	0.798415i
$311$	8658.00			1.57862			0.789309	−	0.613996i	$-0.210439\pi$
$311$	8658.00			1.57862			0.789309	+	0.613996i	$0.210439\pi$
$312$		0			0
$313$	−5250.00			−0.948075			−0.474038	−	0.880505i	$-0.657204\pi$
$313$	−5250.00			−0.948075			−0.474038	+	0.880505i	$0.657204\pi$
$314$	−5802.00	−	10049.4i	−1.04276	−	1.80611i
$315$	3910.00	+	6772.32i	0.699376	+	1.21136i
$316$	304.000	−	526.543i	0.0541182	−	0.0937354i
$317$	6413.00			1.13625			0.568123	−	0.822944i	$-0.307670\pi$
$317$	6413.00			1.13625			0.568123	+	0.822944i	$0.307670\pi$
$318$	−372.000	+	644.323i	−0.0655998	+	0.113622i
$319$	3152.00	−	5459.42i	0.553223	−	0.958210i
$320$	−8704.00			−1.52053
$321$	−522.000	+	904.131i	−0.0907639	+	0.157208i
$322$	−3120.00	−	5404.00i	−0.539971	−	0.935258i
$323$	195.000	+	337.750i	0.0335916	+	0.0581824i
$324$	3368.00			0.577503
$325$	−7462.00	+	1846.37i	−1.27359	+	0.315132i
$326$	9440.00			1.60378
$327$	1634.00	+	2830.17i	0.276332	+	0.478620i
$328$		0			0
$329$	1620.00	−	2805.92i	0.271470	−	0.470199i
$330$	−4352.00			−0.725969
$331$	−1744.00	+	3020.70i	−0.289604	+	0.501609i	−0.973715	−	0.227769i	$-0.926857\pi$
$331$	−1744.00	+	3020.70i	−0.289604	+	0.501609i	0.684111	+	0.729378i	$0.260190\pi$
$332$	−2512.00	+	4350.91i	−0.415253	+	0.719239i
$333$	5221.00			0.859186
$334$	−560.000	+	969.948i	−0.0917420	+	0.158902i
$335$	−7327.00	−	12690.7i	−1.19498	−	2.06976i
$336$	1280.00	+	2217.03i	0.207827	+	0.359966i
$337$	−1833.00			−0.296290			−0.148145	−	0.988966i	$-0.547330\pi$
$337$	−1833.00			−0.296290			−0.148145	+	0.988966i	$0.547330\pi$
$338$	−7436.00	−	4683.47i	−1.19664	−	0.753689i
$339$	654.000			0.104780
$340$	884.000	+	1531.13i	0.141005	+	0.244227i
$341$	−1184.00	−	2050.75i	−0.188027	−	0.325672i
$342$	1380.00	−	2390.23i	0.218193	−	0.377921i
$343$	−5720.00			−0.900440
$344$		0			0
$345$	−1326.00	+	2296.70i	−0.206926	+	0.358406i
$346$	5304.00			0.824118
$347$	−3615.00	+	6261.36i	−0.559260	+	0.968667i	0.438298	+	0.898830i	$0.355581\pi$
$347$	−3615.00	+	6261.36i	−0.559260	+	0.968667i	−0.997558	+	0.0698377i	$0.977752\pi$
$348$	−1576.00	−	2729.71i	−0.242766	−	0.420483i
$349$	2629.00	+	4553.56i	0.403230	+	0.698414i	0.994114	−	0.108342i	$-0.0345543\pi$
$349$	2629.00	+	4553.56i	0.403230	+	0.698414i	−0.590884	+	0.806757i	$0.701221\pi$
$350$	13120.0			2.00370
$351$	4550.00	−	1125.83i	0.691912	−	0.171204i
$352$	8192.00			1.24044
$353$	−1581.50	−	2739.24i	−0.238455	−	0.413017i	0.721816	−	0.692085i	$-0.243308\pi$
$353$	−1581.50	−	2739.24i	−0.238455	−	0.413017i	−0.960271	+	0.279068i	$0.909974\pi$
$354$	3456.00	+	5985.97i	0.518882	+	0.898730i
$355$	−5559.00	+	9628.47i	−0.831102	+	1.43951i
$356$	−2128.00			−0.316808
$357$	260.000	−	450.333i	0.0385453	−	0.0667624i
$358$	−8528.00	+	14770.9i	−1.25899	+	2.18064i
$359$	−10068.0			−1.48014			−0.740068	−	0.672532i	$-0.765207\pi$
$359$	−10068.0			−1.48014			−0.740068	+	0.672532i	$0.765207\pi$
$360$		0			0
$361$	2979.50	+	5160.65i	0.434393	+	0.752390i
$362$	806.000	+	1396.03i	0.117023	+	0.202690i
$363$	−614.000			−0.0887786
$364$	5200.00	+	5404.00i	0.748775	+	0.778150i
$365$	3655.00			0.524141
$366$	−580.000	−	1004.59i	−0.0828336	−	0.143472i
$367$	−3719.00	−	6441.50i	−0.528965	−	0.916195i	−0.999429	−	0.0337755i	$-0.989247\pi$
$367$	−3719.00	−	6441.50i	−0.528965	−	0.916195i	0.470464	−	0.882419i	$-0.344086\pi$
$368$	2496.00	−	4323.20i	0.353568	−	0.612398i
$369$	3795.00			0.535392
$370$	7718.00	−	13368.0i	1.08443	−	1.87829i
$371$	−930.000	+	1610.81i	−0.130143	+	0.225415i
$372$	−1184.00			−0.165020
$373$	4841.50	−	8385.72i	0.672073	−	1.16407i	−0.305242	−	0.952275i	$-0.598737\pi$
$373$	4841.50	−	8385.72i	0.672073	−	1.16407i	0.977315	−	0.211790i	$-0.0679294\pi$
$374$	−832.000	−	1441.07i	−0.115031	−	0.199240i
$375$	−663.000	−	1148.35i	−0.0912991	−	0.158135i
$376$		0			0
$377$	6402.50	+	6653.67i	0.874657	+	0.908970i
$378$	−8000.00			−1.08856
$379$	531.000	+	919.719i	0.0719674	+	0.124651i	0.899763	−	0.436378i	$-0.143739\pi$
$379$	531.000	+	919.719i	0.0719674	+	0.124651i	−0.827796	+	0.561029i	$0.810406\pi$
$380$	−2040.00	−	3533.38i	−0.275394	−	0.476997i
$381$	2158.00	−	3737.77i	0.290178	−	0.502602i
$382$	−4984.00			−0.667549
$383$	1766.00	−	3058.80i	0.235609	−	0.408087i	−0.723840	−	0.689968i	$-0.757625\pi$
$383$	1766.00	−	3058.80i	0.235609	−	0.408087i	0.959450	+	0.281880i	$0.0909581\pi$
$384$		0			0
$385$	−10880.0			−1.44025
$386$	−534.000	+	924.915i	−0.0704142	+	0.121961i
$387$	−1794.00	−	3107.30i	−0.235644	−	0.408147i
$388$	−952.000	−	1648.91i	−0.124563	−	0.215750i
$389$	−11063.0			−1.44194			−0.720972	−	0.692964i	$-0.756304\pi$
$389$	−11063.0			−1.44194			−0.720972	+	0.692964i	$0.756304\pi$
$390$	1768.00	−	6124.53i	0.229554	−	0.795199i
$391$	−1014.00			−0.131151
$392$		0			0
$393$	−730.000	−	1264.40i	−0.0936988	−	0.162291i
$394$	−2556.00	+	4427.12i	−0.326826	+	0.566079i
$395$	−1292.00			−0.164576
$396$	−2944.00	+	5099.16i	−0.373590	+	0.647077i
$397$	2993.00	−	5184.03i	0.378374	−	0.655362i	−0.612452	−	0.790508i	$-0.709817\pi$
$397$	2993.00	−	5184.03i	0.378374	−	0.655362i	0.990826	+	0.135145i	$0.0431501\pi$
$398$	16952.0			2.13499
$399$	−600.000	+	1039.23i	−0.0752821	+	0.130392i
$400$	5248.00	+	9089.80i	0.656000	+	1.13623i
$401$	−2967.50	−	5139.86i	−0.369551	−	0.640081i	0.619945	−	0.784646i	$-0.287155\pi$
$401$	−2967.50	−	5139.86i	−0.369551	−	0.640081i	−0.989495	+	0.144565i	$0.953822\pi$
$402$	6896.00			0.855575
$403$	3367.00	−	833.116i	0.416184	−	0.102979i
$404$	−6552.00			−0.806867
$405$	−3578.50	−	6198.14i	−0.439055	−	0.760465i
$406$	−7880.00	−	13648.6i	−0.963246	−	1.66839i
$407$	−3632.00	+	6290.81i	−0.442338	+	0.766152i
$408$		0			0
$409$	7544.50	−	13067.5i	0.912106	−	1.57981i	0.101023	−	0.994884i	$-0.467788\pi$
$409$	7544.50	−	13067.5i	0.912106	−	1.57981i	0.811083	−	0.584931i	$-0.198878\pi$
$410$	5610.00	−	9716.81i	0.675752	−	1.17044i
$411$	3342.00			0.401092
$412$	−6552.00	+	11348.4i	−0.783480	+	1.35703i
$413$	8640.00	+	14964.9i	1.02941	+	1.78299i
$414$	3588.00	+	6214.60i	0.425943	+	0.737756i
$415$	10676.0			1.26281
$416$	−3328.00	+	11528.5i	−0.392232	+	1.35873i
$417$	1824.00			0.214201
$418$	1920.00	+	3325.54i	0.224666	+	0.389132i
$419$	5407.00	+	9365.20i	0.630428	+	1.09193i	0.987464	+	0.157843i	$0.0504538\pi$
$419$	5407.00	+	9365.20i	0.630428	+	1.09193i	−0.357037	+	0.934090i	$0.616213\pi$
$420$	−2720.00	+	4711.18i	−0.316006	+	0.547338i
$421$	−6535.00			−0.756524			−0.378262	−	0.925699i	$-0.623478\pi$
$421$	−6535.00			−0.756524			−0.378262	+	0.925699i	$0.623478\pi$
$422$	−6140.00	+	10634.8i	−0.708271	+	1.22676i
$423$	−1863.00	+	3226.81i	−0.214142	+	0.370905i
$424$		0			0
$425$	1066.00	−	1846.37i	0.121667	−	0.210734i
$426$	−2616.00	−	4531.04i	−0.297525	−	0.515328i
$427$	−1450.00	−	2511.47i	−0.164334	−	0.284634i
$428$	4176.00			0.471623
$429$	−832.000	+	2882.13i	−0.0936348	+	0.324361i
$430$	−10608.0			−1.18968
$431$	−990.000	−	1714.73i	−0.110642	−	0.191637i	0.805387	−	0.592749i	$-0.201957\pi$
$431$	−990.000	−	1714.73i	−0.110642	−	0.191637i	−0.916029	+	0.401112i	$0.868624\pi$
$432$	−3200.00	−	5542.56i	−0.356389	−	0.617284i
$433$	3464.50	−	6000.69i	0.384511	−	0.665993i	−0.607190	−	0.794556i	$-0.707703\pi$
$433$	3464.50	−	6000.69i	0.384511	−	0.665993i	0.991701	+	0.128564i	$0.0410368\pi$
$434$	−5920.00			−0.654767
$435$	−3349.00	+	5800.64i	−0.369132	+	0.639355i
$436$	6536.00	−	11320.7i	0.717930	−	1.24349i
$437$	2340.00			0.256150
$438$	−860.000	+	1489.56i	−0.0938182	+	0.162498i
$439$	2288.00	+	3962.93i	0.248748	+	0.430844i	0.963179	−	0.268862i	$-0.0866476\pi$
$439$	2288.00	+	3962.93i	0.248748	+	0.430844i	−0.714431	+	0.699706i	$0.753314\pi$
$440$		0			0
$441$	−1311.00			−0.141561
$442$	2366.00	−	585.433i	0.254613	−	0.0630005i
$443$	−8812.00			−0.945081			−0.472540	−	0.881309i	$-0.656663\pi$
$443$	−8812.00			−0.945081			−0.472540	+	0.881309i	$0.656663\pi$
$444$	1816.00	+	3145.40i	0.194107	+	0.336203i
$445$	2261.00	+	3916.17i	0.240858	+	0.417178i
$446$	10756.0	−	18629.9i	1.14195	−	1.97792i
$447$	−4230.00			−0.447589
$448$	5120.00	−	8868.10i	0.539949	−	0.935220i
$449$	−959.000	+	1661.04i	−0.100797	+	0.174586i	−0.912013	−	0.410160i	$-0.865473\pi$
$449$	−959.000	+	1661.04i	−0.100797	+	0.174586i	0.811216	+	0.584747i	$0.198806\pi$
$450$	−15088.0			−1.58057
$451$	−2640.00	+	4572.61i	−0.275638	+	0.477419i
$452$	−1308.00	−	2265.52i	−0.136113	−	0.235755i
$453$	−514.000	−	890.274i	−0.0533109	−	0.0923371i
$454$	−15896.0			−1.64325
$455$	4420.00	−	15311.3i	0.455413	−	1.57760i
$456$		0			0
$457$	5880.50	+	10185.3i	0.601922	+	1.04256i	0.992530	+	0.122002i	$0.0389314\pi$
$457$	5880.50	+	10185.3i	0.601922	+	1.04256i	−0.390608	+	0.920557i	$0.627735\pi$
$458$	12596.0	+	21816.9i	1.28509	+	2.22585i
$459$	−650.000	+	1125.83i	−0.0660989	+	0.114487i
$460$	10608.0			1.07522
$461$	−450.500	+	780.289i	−0.0455138	+	0.0788323i	−0.887885	−	0.460066i	$-0.847826\pi$
$461$	−450.500	+	780.289i	−0.0455138	+	0.0788323i	0.842371	+	0.538898i	$0.181159\pi$
$462$	2560.00	−	4434.05i	0.257796	−	0.446517i
$463$	1372.00			0.137715			0.0688577	−	0.997626i	$-0.478065\pi$
$463$	1372.00			0.137715			0.0688577	+	0.997626i	$0.478065\pi$
$464$	6304.00	−	10918.8i	0.630724	−	1.09245i
$465$	1258.00	+	2178.92i	0.125459	+	0.217301i
$466$	−8060.00	−	13960.3i	−0.801228	−	1.38777i
$467$	−6396.00			−0.633772			−0.316886	−	0.948464i	$-0.602637\pi$
$467$	−6396.00			−0.633772			−0.316886	+	0.948464i	$0.602637\pi$
$468$	−5980.00	−	6214.60i	−0.590653	−	0.613825i
$469$	17240.0			1.69738
$470$	5508.00	+	9540.14i	0.540564	+	0.936284i
$471$	−2901.00	−	5024.68i	−0.283803	−	0.491561i
$472$		0			0
$473$	4992.00			0.485269
$474$	304.000	−	526.543i	0.0294582	−	0.0510231i
$475$	−2460.00	+	4260.84i	−0.237626	+	0.411581i
$476$	−2080.00			−0.200287
$477$	1069.50	−	1852.43i	0.102660	−	0.177813i
$478$	1968.00	+	3408.68i	0.188314	+	0.326170i
$479$	−1635.00	−	2831.90i	−0.155960	−	0.270131i	0.777448	−	0.628947i	$-0.216514\pi$
$479$	−1635.00	−	2831.90i	−0.155960	−	0.270131i	−0.933408	+	0.358816i	$0.883181\pi$
$480$	−8704.00			−0.827670
$481$	−7377.50	−	7666.92i	−0.699345	−	0.726781i
$482$	3772.00			0.356452
$483$	−1560.00	−	2702.00i	−0.146962	−	0.254545i
$484$	1228.00	+	2126.96i	0.115327	+	0.199752i
$485$	−2023.00	+	3503.94i	−0.189401	+	0.328053i
$486$	14168.0			1.32237
$487$	−9960.00	+	17251.2i	−0.926757	+	1.60519i	−0.138046	+	0.990426i	$0.544082\pi$
$487$	−9960.00	+	17251.2i	−0.926757	+	1.60519i	−0.788711	+	0.614765i	$0.789251\pi$
$488$		0			0
$489$	4720.00			0.436494
$490$	−1938.00	+	3356.71i	−0.178673	+	0.309471i
$491$	−3276.00	−	5674.20i	−0.301108	−	0.521534i	0.675280	−	0.737562i	$-0.264023\pi$
$491$	−3276.00	−	5674.20i	−0.301108	−	0.521534i	−0.976387	+	0.216028i	$0.930690\pi$
$492$	1320.00	+	2286.31i	0.120956	+	0.209501i
$493$	−2561.00			−0.233959
$494$	−5460.00	+	1351.00i	−0.497281	+	0.123045i
$495$	12512.0			1.13611
$496$	−2368.00	−	4101.50i	−0.214368	−	0.371296i
$497$	−6540.00	−	11327.6i	−0.590260	−	1.02236i
$498$	−2512.00	+	4350.91i	−0.226035	+	0.391504i
$499$	1746.00			0.156637			0.0783183	−	0.996928i	$-0.475045\pi$
$499$	1746.00			0.156637			0.0783183	+	0.996928i	$0.475045\pi$
$500$	−2652.00	+	4593.40i	−0.237202	+	0.410846i
$501$	−280.000	+	484.974i	−0.0249690	+	0.0432476i
$502$	−10920.0			−0.970883
$503$	−7346.00	+	12723.6i	−0.651177	+	1.12787i	0.331661	+	0.943399i	$0.392391\pi$
$503$	−7346.00	+	12723.6i	−0.651177	+	1.12787i	−0.982838	+	0.184473i	$0.940942\pi$
$504$		0			0
$505$	6961.50	+	12057.7i	0.613431	+	1.06249i
$506$	−9984.00			−0.877160
$507$	−3718.00	−	2341.73i	−0.325685	−	0.205128i
$508$	−17264.0			−1.50781
$509$	−4038.50	−	6994.89i	−0.351677	−	0.609122i	0.634867	−	0.772622i	$-0.281055\pi$
$509$	−4038.50	−	6994.89i	−0.351677	−	0.609122i	−0.986543	+	0.163500i	$0.947722\pi$
$510$	884.000	+	1531.13i	0.0767533	+	0.132941i
$511$	−2150.00	+	3723.91i	−0.186126	+	0.322380i
$512$	16384.0			1.41421
$513$	1500.00	−	2598.08i	0.129097	−	0.223602i
$514$	3770.00	−	6529.83i	0.323517	−	0.560347i
$515$	27846.0			2.38260
$516$	1248.00	−	2161.60i	0.106473	−	0.184417i
$517$	−2592.00	−	4489.48i	−0.220495	−	0.381909i
$518$	9080.00	+	15727.0i	0.770178	+	1.33399i
$519$	2652.00			0.224296
$520$		0			0
$521$	11247.0			0.945758			0.472879	−	0.881127i	$-0.343215\pi$
$521$	11247.0			0.945758			0.472879	+	0.881127i	$0.343215\pi$
$522$	9062.00	+	15695.8i	0.759833	+	1.31607i
$523$	−1366.00	−	2365.98i	−0.114208	−	0.197815i	0.803255	−	0.595636i	$-0.203100\pi$
$523$	−1366.00	−	2365.98i	−0.114208	−	0.197815i	−0.917463	+	0.397821i	$0.869767\pi$
$524$	−2920.00	+	5057.59i	−0.243437	+	0.421645i
$525$	6560.00			0.545337
$526$	−8064.00	+	13967.3i	−0.668455	+	1.15780i
$527$	−481.000	+	833.116i	−0.0397584	+	0.0688636i
$528$	4096.00			0.337605
$529$	3041.50	−	5268.03i	0.249979	−	0.432977i
$530$	−3162.00	−	5476.74i	−0.259148	−	0.448858i
$531$	−9936.00	−	17209.7i	−0.812026	−	1.40647i
$532$	4800.00			0.391177
$533$	−5362.50	−	5572.87i	−0.435789	−	0.452885i
$534$	−2128.00			−0.172449
$535$	−4437.00	−	7685.11i	−0.358557	−	0.621040i
$536$		0			0
$537$	−4264.00	+	7385.46i	−0.342654	+	0.593494i
$538$	16024.0			1.28410
$539$	912.000	−	1579.63i	0.0728806	−	0.126233i
$540$	6800.00	−	11777.9i	0.541899	−	0.938596i
$541$	−18375.0			−1.46026			−0.730132	−	0.683306i	$-0.760542\pi$
$541$	−18375.0			−1.46026			−0.730132	+	0.683306i	$0.760542\pi$
$542$	8592.00	−	14881.8i	0.680919	−	1.17939i
$543$	403.000	+	698.016i	0.0318497	+	0.0551653i
$544$	−1664.00	−	2882.13i	−0.131146	−	0.227151i
$545$	−27778.0			−2.18326
$546$	5200.00	+	5404.00i	0.407581	+	0.423571i
$547$	−10346.0			−0.808708			−0.404354	−	0.914603i	$-0.632504\pi$
$547$	−10346.0			−0.808708			−0.404354	+	0.914603i	$0.632504\pi$
$548$	−6684.00	−	11577.0i	−0.521033	−	0.902456i
$549$	1667.50	+	2888.19i	0.129631	+	0.224527i
$550$	10496.0	−	18179.6i	0.813729	−	1.40942i
$551$	5910.00			0.456941
$552$		0			0
$553$	760.000	−	1316.36i	0.0584421	−	0.101225i
$554$	−22204.0			−1.70281
$555$	3859.00	−	6683.98i	0.295145	−	0.511206i
$556$	−3648.00	−	6318.52i	−0.278255	−	0.481951i
$557$	−172.500	−	298.779i	−0.0131222	−	0.0227283i	0.859390	−	0.511321i	$-0.170844\pi$
$557$	−172.500	−	298.779i	−0.0131222	−	0.0227283i	−0.872512	+	0.488593i	$0.837510\pi$
$558$	6808.00			0.516498
$559$	−2028.00	+	7025.20i	−0.153444	+	0.531546i
$560$	−21760.0			−1.64201
$561$	−416.000	−	720.533i	−0.0313075	−	0.0542263i
$562$	11114.0	+	19250.0i	0.834192	+	1.44486i
$563$	4290.00	−	7430.50i	0.321140	−	0.556231i	−0.659583	−	0.751631i	$-0.729267\pi$
$563$	4290.00	−	7430.50i	0.321140	−	0.556231i	0.980724	+	0.195400i	$0.0626006\pi$
$564$	−2592.00			−0.193516
$565$	−2779.50	+	4814.24i	−0.206964	+	0.358472i
$566$	−6240.00	+	10808.0i	−0.463404	+	0.802640i
$567$	8420.00			0.623645
$568$		0			0
$569$	9841.00	+	17045.1i	0.725055	+	1.25583i	0.958951	+	0.283570i	$0.0915189\pi$
$569$	9841.00	+	17045.1i	0.725055	+	1.25583i	−0.233897	+	0.972261i	$0.575148\pi$
$570$	−2040.00	−	3533.38i	−0.149906	−	0.259644i
$571$	26624.0			1.95128			0.975639	−	0.219382i	$-0.0704042\pi$
$571$	26624.0			1.95128			0.975639	+	0.219382i	$0.0704042\pi$
$572$	11648.0	−	2882.13i	0.851446	−	0.210678i
$573$	−2492.00			−0.181684
$574$	6600.00	+	11431.5i	0.479928	+	0.831260i
$575$	−6396.00	−	11078.2i	−0.463881	−	0.803466i
$576$	−5888.00	+	10198.3i	−0.425926	+	0.737725i
$577$	−14101.0			−1.01739			−0.508694	−	0.860948i	$-0.669871\pi$
$577$	−14101.0			−1.01739			−0.508694	+	0.860948i	$0.669871\pi$
$578$	9488.00	−	16433.7i	0.682783	−	1.18262i
$579$	−267.000	+	462.458i	−0.0191643	+	0.0331936i
$580$	26792.0			1.91806
$581$	−6280.00	+	10877.3i	−0.448431	+	0.776705i
$582$	−952.000	−	1648.91i	−0.0678036	−	0.117439i
$583$	1488.00	+	2577.29i	0.105706	+	0.183088i
$584$		0			0
$585$	−5083.00	+	17608.0i	−0.359241	+	1.24445i
$586$	33204.0			2.34069
$587$	−704.000	−	1219.36i	−0.0495012	−	0.0857386i	0.840213	−	0.542256i	$-0.182430\pi$
$587$	−704.000	−	1219.36i	−0.0495012	−	0.0857386i	−0.889714	+	0.456518i	$0.849096\pi$
$588$	−456.000	−	789.815i	−0.0319815	−	0.0553936i
$589$	1110.00	−	1922.58i	0.0776515	−	0.134496i
$590$	−58752.0			−4.09963
$591$	−1278.00	+	2213.56i	−0.0889508	+	0.154067i
$592$	−7264.00	+	12581.6i	−0.504305	+	0.873482i
$593$	−1241.00			−0.0859389			−0.0429694	−	0.999076i	$-0.513682\pi$
$593$	−1241.00			−0.0859389			−0.0429694	+	0.999076i	$0.513682\pi$
$594$	−6400.00	+	11085.1i	−0.442079	+	0.765704i
$595$	2210.00	+	3827.83i	0.152271	+	0.263741i
$596$	8460.00	+	14653.1i	0.581435	+	1.00707i
$597$	8476.00			0.581071
$598$	4056.00	−	14050.4i	0.277361	−	0.960808i
$599$	11078.0			0.755651			0.377825	−	0.925877i	$-0.376672\pi$
$599$	11078.0			0.755651			0.377825	+	0.925877i	$0.376672\pi$
$600$		0			0
$601$	6908.50	+	11965.9i	0.468891	+	0.812143i	0.999368	−	0.0355563i	$-0.0113203\pi$
$601$	6908.50	+	11965.9i	0.468891	+	0.812143i	−0.530477	+	0.847700i	$0.677987\pi$
$602$	6240.00	−	10808.0i	0.422464	−	0.731729i
$603$	−19826.0			−1.33893
$604$	−2056.00	+	3561.10i	−0.138506	+	0.239899i
$605$	2609.50	−	4519.79i	0.175357	−	0.303728i
$606$	−6552.00			−0.439203
$607$	−4135.00	+	7162.03i	−0.276498	+	0.478909i	−0.970512	−	0.241053i	$-0.922507\pi$
$607$	−4135.00	+	7162.03i	−0.276498	+	0.478909i	0.694014	+	0.719962i	$0.255841\pi$
$608$	3840.00	+	6651.08i	0.256139	+	0.443646i
$609$	−3940.00	−	6824.28i	−0.262162	−	0.454078i
$610$	9860.00			0.654459
$611$	7371.00	−	1823.85i	0.488050	−	0.120761i
$612$	2392.00			0.157992
$613$	−11136.5	−	19289.0i	−0.733767	−	1.27092i	−0.955262	−	0.295760i	$-0.904427\pi$
$613$	−11136.5	−	19289.0i	−0.733767	−	1.27092i	0.221496	−	0.975161i	$-0.428906\pi$
$614$	−17356.0	−	30061.5i	−1.14077	−	1.97587i
$615$	2805.00	−	4858.40i	0.183916	−	0.318552i
$616$		0			0
$617$	9494.50	−	16445.0i	0.619504	−	1.07301i	−0.370072	−	0.929003i	$-0.620667\pi$
$617$	9494.50	−	16445.0i	0.619504	−	1.07301i	0.989576	−	0.144010i	$-0.0459997\pi$
$618$	−6552.00	+	11348.4i	−0.426473	+	0.738672i
$619$	72.0000			0.00467516			0.00233758	−	0.999997i	$-0.499256\pi$
$619$	72.0000			0.00467516			0.00233758	+	0.999997i	$0.499256\pi$
$620$	5032.00	−	8715.68i	0.325952	−	0.564565i
$621$	3900.00	+	6755.00i	0.252015	+	0.436504i
$622$	−17316.0	−	29992.2i	−1.11625	−	1.93340i
$623$	−5320.00			−0.342121
$624$	−1664.00	+	5764.27i	−0.106752	+	0.369800i
$625$	−9229.00			−0.590656
$626$	10500.0	+	18186.5i	0.670390	+	1.16115i
$627$	960.000	+	1662.77i	0.0611463	+	0.105908i
$628$	−11604.0	+	20098.7i	−0.737341	+	1.27711i
$629$	2951.00			0.187065
$630$	15640.0	−	27089.3i	0.989067	−	1.71312i
$631$	11690.0	−	20247.7i	0.737514	−	1.27741i	−0.216097	−	0.976372i	$-0.569333\pi$
$631$	11690.0	−	20247.7i	0.737514	−	1.27741i	0.953611	−	0.301040i	$-0.0973339\pi$
$632$		0			0
$633$	−3070.00	+	5317.40i	−0.192767	+	0.333882i
$634$	−12826.0	−	22215.3i	−0.803447	−	1.39161i
$635$	18343.0	+	31771.0i	1.14633	+	1.98550i
$636$	1488.00			0.0927721
$637$	1852.50	+	1925.17i	0.115226	+	0.119746i
$638$	−25216.0			−1.56475
$639$	7521.00	+	13026.8i	0.465612	+	0.806464i
$640$		0			0
$641$	−3191.50	+	5527.84i	−0.196656	+	0.340619i	−0.947442	−	0.319927i	$-0.896342\pi$
$641$	−3191.50	+	5527.84i	−0.196656	+	0.340619i	0.750786	+	0.660546i	$0.229675\pi$
$642$	4176.00			0.256719
$643$	8552.00	−	14812.5i	0.524507	−	0.908473i	−0.475086	−	0.879939i	$-0.657583\pi$
$643$	8552.00	−	14812.5i	0.524507	−	0.908473i	0.999593	−	0.0285332i	$-0.00908365\pi$
$644$	−6240.00	+	10808.0i	−0.381817	+	0.661327i
$645$	−5304.00			−0.323790
$646$	780.000	−	1351.00i	0.0475057	−	0.0822823i
$647$	−3497.00	−	6056.98i	−0.212490	−	0.368044i	0.740003	−	0.672604i	$-0.234824\pi$
$647$	−3497.00	−	6056.98i	−0.212490	−	0.368044i	−0.952493	+	0.304560i	$0.901491\pi$
$648$		0			0
$649$	27648.0			1.67223
$650$	21320.0	+	22156.4i	1.28652	+	1.33699i
$651$	−2960.00			−0.178205
$652$	−9440.00	−	16350.6i	−0.567023	−	0.982112i
$653$	2625.00	+	4546.63i	0.157311	+	0.272471i	0.933898	−	0.357539i	$-0.116384\pi$
$653$	2625.00	+	4546.63i	0.157311	+	0.272471i	−0.776587	+	0.630010i	$0.783051\pi$
$654$	6536.00	−	11320.7i	0.390792	−	0.676871i
$655$	12410.0			0.740304
$656$	−5280.00	+	9145.23i	−0.314252	+	0.544301i
$657$	2472.50	−	4282.50i	0.146821	−	0.254301i
$658$	−12960.0			−0.767832
$659$	2170.00	−	3758.55i	0.128272	−	0.222173i	−0.794735	−	0.606956i	$-0.792390\pi$
$659$	2170.00	−	3758.55i	0.128272	−	0.222173i	0.923007	+	0.384783i	$0.125724\pi$
$660$	4352.00	+	7537.89i	0.256669	+	0.444563i
$661$	2089.50	+	3619.12i	0.122953	+	0.212961i	0.920931	−	0.389726i	$-0.127430\pi$
$661$	2089.50	+	3619.12i	0.122953	+	0.212961i	−0.797978	+	0.602687i	$0.794097\pi$
$662$	13952.0			0.819124
$663$	1183.00	−	292.717i	0.0692970	−	0.0171466i
$664$		0			0
$665$	−5100.00	−	8833.46i	−0.297398	−	0.515108i
$666$	−10442.0	−	18086.1i	−0.607536	−	1.05228i
$667$	−7683.00	+	13307.3i	−0.446007	+	0.772508i
$668$	2240.00			0.129743
$669$	5378.00	−	9314.97i	0.310800	−	0.538322i
$670$	−29308.0	+	50762.9i	−1.68995	+	2.92708i
$671$	−4640.00			−0.266953
$672$	5120.00	−	8868.10i	0.293911	−	0.509069i
$673$	−11433.5	−	19803.4i	−0.654872	−	1.13427i	−0.981926	−	0.189266i	$-0.939389\pi$
$673$	−11433.5	−	19803.4i	−0.654872	−	1.13427i	0.327054	−	0.945006i	$-0.393944\pi$
$674$	3666.00	+	6349.70i	0.209509	+	0.362880i
$675$	−16400.0			−0.935165
$676$	−676.000	+	17563.0i	−0.0384615	+	0.999260i
$677$	5410.00			0.307124			0.153562	−	0.988139i	$-0.450925\pi$
$677$	5410.00			0.307124			0.153562	+	0.988139i	$0.450925\pi$
$678$	−1308.00	−	2265.52i	−0.0740906	−	0.128329i
$679$	−2380.00	−	4122.28i	−0.134515	−	0.232988i
$680$		0			0
$681$	−7948.00			−0.447236
$682$	−4736.00	+	8202.99i	−0.265910	+	0.460570i
$683$	6789.00	−	11758.9i	0.380342	−	0.658772i	−0.610769	−	0.791809i	$-0.709139\pi$
$683$	6789.00	−	11758.9i	0.380342	−	0.658772i	0.991111	+	0.133037i	$0.0424728\pi$
$684$	−5520.00			−0.308571
$685$	−14203.5	+	24601.2i	−0.792245	+	1.37221i
$686$	11440.0	+	19814.7i	0.636707	+	1.10281i
$687$	6298.00	+	10908.5i	0.349758	+	0.605798i
$688$	9984.00			0.553251
$689$	−4231.50	+	1047.02i	−0.233973	+	0.0578933i
$690$	10608.0			0.585275
$691$	−6372.00	−	11036.6i	−0.350799	−	0.607602i	0.635590	−	0.772026i	$-0.280757\pi$
$691$	−6372.00	−	11036.6i	−0.350799	−	0.607602i	−0.986390	+	0.164424i	$0.947423\pi$
$692$	−5304.00	−	9186.80i	−0.291370	−	0.504667i
$693$	−7360.00	+	12747.9i	−0.403439	+	0.698777i
$694$	28920.0			1.58183
$695$	−7752.00	+	13426.9i	−0.423094	+	0.732820i
$696$		0			0
$697$	2145.00			0.116568
$698$	10516.0	−	18214.2i	0.570253	−	0.987707i
$699$	−4030.00	−	6980.16i	−0.218067	−	0.377703i
$700$	−13120.0	−	22724.5i	−0.708413	−	1.22701i
$701$	16406.0			0.883946			0.441973	−	0.897028i	$-0.354279\pi$
$701$	16406.0			0.883946			0.441973	+	0.897028i	$0.354279\pi$
$702$	−13000.0	−	13510.0i	−0.698936	−	0.726356i
$703$	−6810.00			−0.365354
$704$	−8192.00	−	14189.0i	−0.438562	−	0.759612i
$705$	2754.00	+	4770.07i	0.147123	+	0.254824i
$706$	−6326.00	+	10957.0i	−0.337227	+	0.584094i
$707$	−16380.0			−0.871334
$708$	6912.00	−	11971.9i	0.366905	−	0.635498i
$709$	−354.500	+	614.012i	−0.0187779	+	0.0325243i	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−354.500	+	614.012i	−0.0187779	+	0.0325243i	0.856484	+	0.516174i	$0.172644\pi$
$710$	44472.0			2.35071
$711$	−874.000	+	1513.81i	−0.0461006	+	0.0798487i
$712$		0			0
$713$	2886.00	+	4998.70i	0.151587	+	0.262556i
$714$	−2080.00			−0.109022
$715$	−17680.0	−	18373.6i	−0.924748	−	0.961026i
$716$	34112.0			1.78048
$717$	984.000	+	1704.34i	0.0512527	+	0.0887722i
$718$	20136.0	+	34876.6i	1.04661	+	1.81279i
$719$	3822.00	−	6619.90i	0.198243	−	0.343367i	−0.749716	−	0.661760i	$-0.769810\pi$
$719$	3822.00	−	6619.90i	0.198243	−	0.343367i	0.947959	+	0.318393i	$0.103143\pi$
$720$	25024.0			1.29526
$721$	−16380.0	+	28371.0i	−0.846079	+	1.46545i
$722$	11918.0	−	20642.6i	0.614324	−	1.06404i
$723$	1886.00			0.0970140
$724$	1612.00	−	2792.07i	0.0827479	−	0.143324i
$725$	−16154.0	−	27979.5i	−0.827510	−	1.43329i
$726$	1228.00	+	2126.96i	0.0627760	+	0.108731i
$727$	−15808.0			−0.806446			−0.403223	−	0.915102i	$-0.632110\pi$
$727$	−15808.0			−0.806446			−0.403223	+	0.915102i	$0.632110\pi$
$728$		0			0
$729$	−4283.00			−0.217599
$730$	−7310.00	−	12661.3i	−0.370624	−	0.641939i
$731$	−1014.00	−	1756.30i	−0.0513053	−	0.0888633i
$732$	−1160.00	+	2009.18i	−0.0585722	+	0.101450i
$733$	−2583.00			−0.130157			−0.0650786	−	0.997880i	$-0.520730\pi$
$733$	−2583.00			−0.130157			−0.0650786	+	0.997880i	$0.520730\pi$
$734$	−14876.0	+	25766.0i	−0.748070	+	1.29569i
$735$	−969.000	+	1678.36i	−0.0486287	+	0.0842274i
$736$	−19968.0			−1.00004
$737$	13792.0	−	23888.4i	0.689328	−	1.19395i
$738$	−7590.00	−	13146.3i	−0.378580	−	0.655719i
$739$	−2038.00	−	3529.92i	−0.101447	−	0.175711i	0.810834	−	0.585276i	$-0.199014\pi$
$739$	−2038.00	−	3529.92i	−0.101447	−	0.175711i	−0.912281	+	0.409565i	$0.865680\pi$
$740$	−30872.0			−1.53362
$741$	−2730.00	+	675.500i	−0.135343	+	0.0334887i
$742$	7440.00			0.368101
$743$	17028.0	+	29493.4i	0.840776	+	1.45627i	0.889239	+	0.457442i	$0.151234\pi$
$743$	17028.0	+	29493.4i	0.840776	+	1.45627i	−0.0484632	+	0.998825i	$0.515432\pi$
$744$		0			0
$745$	17977.5	−	31137.9i	0.884087	−	1.53128i
$746$	−38732.0			−1.90091
$747$	7222.00	−	12508.9i	0.353734	−	0.612685i
$748$	−1664.00	+	2882.13i	−0.0813394	+	0.140884i
$749$	10440.0			0.509305
$750$	−2652.00	+	4593.40i	−0.129116	+	0.223636i
$751$	182.000	+	315.233i	0.00884324	+	0.0153169i	0.870413	−	0.492322i	$-0.163852\pi$
$751$	182.000	+	315.233i	0.00884324	+	0.0153169i	−0.861570	+	0.507639i	$0.830518\pi$
$752$	−5184.00	−	8978.95i	−0.251384	−	0.435410i
$753$	−5460.00			−0.264241
$754$	10244.0	−	35486.3i	0.494780	−	1.71397i
$755$	8738.00			0.421203
$756$	8000.00	+	13856.4i	0.384864	+	0.666604i
$757$	3457.00	+	5987.70i	0.165980	+	0.287486i	0.937003	−	0.349322i	$-0.113588\pi$
$757$	3457.00	+	5987.70i	0.165980	+	0.287486i	−0.771023	+	0.636807i	$0.780255\pi$
$758$	2124.00	−	3678.88i	0.101777	−	0.176283i
$759$	−4992.00			−0.238733
$760$		0			0
$761$	−6991.00	+	12108.8i	−0.333014	+	0.576797i	−0.983101	−	0.183062i	$-0.941399\pi$
$761$	−6991.00	+	12108.8i	−0.333014	+	0.576797i	0.650087	+	0.759859i	$0.274732\pi$
$762$	−17264.0			−0.820746
$763$	16340.0	−	28301.7i	0.775292	−	1.34284i
$764$	4984.00	+	8632.54i	0.236014	+	0.408788i
$765$	−2541.50	−	4402.01i	−0.120115	−	0.208046i
$766$	−14128.0			−0.666404
$767$	−11232.0	+	38908.8i	−0.528767	+	1.83170i
$768$	8192.00			0.384900

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:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-4.00000 + 6.92820i) q^{4} +17.0000 q^{5} +(-4.00000 + 6.92820i) q^{6} +(-10.0000 + 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-4.00000 + 6.92820i) q^{4} +17.0000 q^{5} +(-4.00000 + 6.92820i) q^{6} +(-10.0000 + 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-34.0000 - 58.8897i) q^{10} +(16.0000 + 27.7128i) q^{11} +16.0000 q^{12} +(-45.5000 + 11.2583i) q^{13} +80.0000 q^{14} +(-17.0000 - 29.4449i) q^{15} +(32.0000 + 55.4256i) q^{16} +(6.50000 - 11.2583i) q^{17} -92.0000 q^{18} +(-15.0000 + 25.9808i) q^{19} +(-68.0000 + 117.779i) q^{20} +40.0000 q^{21} +(64.0000 - 110.851i) q^{22} +(-39.0000 - 67.5500i) q^{23} +164.000 q^{25} +(130.000 + 135.100i) q^{26} -100.000 q^{27} +(-80.0000 - 138.564i) q^{28} +(-98.5000 - 170.607i) q^{29} +(-68.0000 + 117.779i) q^{30} -74.0000 q^{31} +(128.000 - 221.703i) q^{32} +(32.0000 - 55.4256i) q^{33} -52.0000 q^{34} +(-170.000 + 294.449i) q^{35} +(92.0000 + 159.349i) q^{36} +(113.500 + 196.588i) q^{37} +120.000 q^{38} +(65.0000 + 67.5500i) q^{39} +(82.5000 + 142.894i) q^{41} +(-80.0000 - 138.564i) q^{42} +(78.0000 - 135.100i) q^{43} -256.000 q^{44} +(195.500 - 338.616i) q^{45} +(-156.000 + 270.200i) q^{46} -162.000 q^{47} +(64.0000 - 110.851i) q^{48} +(-28.5000 - 49.3634i) q^{49} +(-328.000 - 568.113i) q^{50} -26.0000 q^{51} +(104.000 - 360.267i) q^{52} +93.0000 q^{53} +(200.000 + 346.410i) q^{54} +(272.000 + 471.118i) q^{55} +60.0000 q^{57} +(-394.000 + 682.428i) q^{58} +(432.000 - 748.246i) q^{59} +272.000 q^{60} +(-72.5000 + 125.574i) q^{61} +(148.000 + 256.344i) q^{62} +(230.000 + 398.372i) q^{63} -512.000 q^{64} +(-773.500 + 191.392i) q^{65} -256.000 q^{66} +(-431.000 - 746.514i) q^{67} +(52.0000 + 90.0666i) q^{68} +(-78.0000 + 135.100i) q^{69} +1360.00 q^{70} +(-327.000 + 566.381i) q^{71} +215.000 q^{73} +(454.000 - 786.351i) q^{74} +(-164.000 - 284.056i) q^{75} +(-120.000 - 207.846i) q^{76} -640.000 q^{77} +(104.000 - 360.267i) q^{78} -76.0000 q^{79} +(544.000 + 942.236i) q^{80} +(-210.500 - 364.597i) q^{81} +(330.000 - 571.577i) q^{82} +628.000 q^{83} +(-160.000 + 277.128i) q^{84} +(110.500 - 191.392i) q^{85} -624.000 q^{86} +(-197.000 + 341.214i) q^{87} +(133.000 + 230.363i) q^{89} -1564.00 q^{90} +(260.000 - 900.666i) q^{91} +624.000 q^{92} +(74.0000 + 128.172i) q^{93} +(324.000 + 561.184i) q^{94} +(-255.000 + 441.673i) q^{95} -512.000 q^{96} +(-119.000 + 206.114i) q^{97} +(-114.000 + 197.454i) q^{98} +736.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9} - 68 q^{10} + 32 q^{11} + 32 q^{12} - 91 q^{13} + 160 q^{14} - 34 q^{15} + 64 q^{16} + 13 q^{17} - 184 q^{18} - 30 q^{19} - 136 q^{20} + 80 q^{21} + 128 q^{22} - 78 q^{23} + 328 q^{25} + 260 q^{26} - 200 q^{27} - 160 q^{28} - 197 q^{29} - 136 q^{30} - 148 q^{31} + 256 q^{32} + 64 q^{33} - 104 q^{34} - 340 q^{35} + 184 q^{36} + 227 q^{37} + 240 q^{38} + 130 q^{39} + 165 q^{41} - 160 q^{42} + 156 q^{43} - 512 q^{44} + 391 q^{45} - 312 q^{46} - 324 q^{47} + 128 q^{48} - 57 q^{49} - 656 q^{50} - 52 q^{51} + 208 q^{52} + 186 q^{53} + 400 q^{54} + 544 q^{55} + 120 q^{57} - 788 q^{58} + 864 q^{59} + 544 q^{60} - 145 q^{61} + 296 q^{62} + 460 q^{63} - 1024 q^{64} - 1547 q^{65} - 512 q^{66} - 862 q^{67} + 104 q^{68} - 156 q^{69} + 2720 q^{70} - 654 q^{71} + 430 q^{73} + 908 q^{74} - 328 q^{75} - 240 q^{76} - 1280 q^{77} + 208 q^{78} - 152 q^{79} + 1088 q^{80} - 421 q^{81} + 660 q^{82} + 1256 q^{83} - 320 q^{84} + 221 q^{85} - 1248 q^{86} - 394 q^{87} + 266 q^{89} - 3128 q^{90} + 520 q^{91} + 1248 q^{92} + 148 q^{93} + 648 q^{94} - 510 q^{95} - 1024 q^{96} - 238 q^{97} - 228 q^{98} + 1472 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9 - 68 * q^10 + 32 * q^11 + 32 * q^12 - 91 * q^13 + 160 * q^14 - 34 * q^15 + 64 * q^16 + 13 * q^17 - 184 * q^18 - 30 * q^19 - 136 * q^20 + 80 * q^21 + 128 * q^22 - 78 * q^23 + 328 * q^25 + 260 * q^26 - 200 * q^27 - 160 * q^28 - 197 * q^29 - 136 * q^30 - 148 * q^31 + 256 * q^32 + 64 * q^33 - 104 * q^34 - 340 * q^35 + 184 * q^36 + 227 * q^37 + 240 * q^38 + 130 * q^39 + 165 * q^41 - 160 * q^42 + 156 * q^43 - 512 * q^44 + 391 * q^45 - 312 * q^46 - 324 * q^47 + 128 * q^48 - 57 * q^49 - 656 * q^50 - 52 * q^51 + 208 * q^52 + 186 * q^53 + 400 * q^54 + 544 * q^55 + 120 * q^57 - 788 * q^58 + 864 * q^59 + 544 * q^60 - 145 * q^61 + 296 * q^62 + 460 * q^63 - 1024 * q^64 - 1547 * q^65 - 512 * q^66 - 862 * q^67 + 104 * q^68 - 156 * q^69 + 2720 * q^70 - 654 * q^71 + 430 * q^73 + 908 * q^74 - 328 * q^75 - 240 * q^76 - 1280 * q^77 + 208 * q^78 - 152 * q^79 + 1088 * q^80 - 421 * q^81 + 660 * q^82 + 1256 * q^83 - 320 * q^84 + 221 * q^85 - 1248 * q^86 - 394 * q^87 + 266 * q^89 - 3128 * q^90 + 520 * q^91 + 1248 * q^92 + 148 * q^93 + 648 * q^94 - 510 * q^95 - 1024 * q^96 - 238 * q^97 - 228 * q^98 + 1472 * q^99

Introduction

L-functions

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