LMFDB - Embedded newform 15.5.c.a.11.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,5,Mod(11,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("15.11");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	15.c (of order $2$, degree $1$, 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:	$1.55054944626$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 73x^{4} + 1096x^{2} + 180 $ x^6 + 73x^4 + 1096x^2 + 180
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.2
Root			$-4.56632i$ of defining polynomial
Character	$\chi$	$=$	15.11
Dual form			15.5.c.a.11.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.56632i q^{2} +(-7.98405 - 4.15391i) q^{3} -4.85128 q^{4} -11.1803i q^{5} +(-18.9681 + 36.4577i) q^{6} +61.6068 q^{7} -50.9086i q^{8} +(46.4900 + 66.3301i) q^{9} +O(q^{10})$	\(q-4.56632i q^{2} +(-7.98405 - 4.15391i) q^{3} -4.85128 q^{4} -11.1803i q^{5} +(-18.9681 + 36.4577i) q^{6} +61.6068 q^{7} -50.9086i q^{8} +(46.4900 + 66.3301i) q^{9} -51.0530 q^{10} +108.827i q^{11} +(38.7328 + 20.1518i) q^{12} -63.7026 q^{13} -281.317i q^{14} +(-46.4422 + 89.2644i) q^{15} -310.086 q^{16} -175.143i q^{17} +(302.884 - 212.288i) q^{18} +301.723 q^{19} +54.2389i q^{20} +(-491.872 - 255.909i) q^{21} +496.937 q^{22} +1039.90i q^{23} +(-211.470 + 406.457i) q^{24} -125.000 q^{25} +290.886i q^{26} +(-95.6491 - 722.698i) q^{27} -298.872 q^{28} -179.873i q^{29} +(407.610 + 212.070i) q^{30} +1068.94 q^{31} +601.411i q^{32} +(452.056 - 868.876i) q^{33} -799.761 q^{34} -688.785i q^{35} +(-225.536 - 321.785i) q^{36} -1061.29 q^{37} -1377.76i q^{38} +(508.604 + 264.615i) q^{39} -569.176 q^{40} -173.768i q^{41} +(-1168.56 + 2246.04i) q^{42} -3032.96 q^{43} -527.948i q^{44} +(741.593 - 519.774i) q^{45} +4748.50 q^{46} +935.703i q^{47} +(2475.74 + 1288.07i) q^{48} +1394.40 q^{49} +570.790i q^{50} +(-727.531 + 1398.35i) q^{51} +309.039 q^{52} -423.304i q^{53} +(-3300.07 + 436.765i) q^{54} +1216.72 q^{55} -3136.32i q^{56} +(-2408.97 - 1253.33i) q^{57} -821.359 q^{58} +3188.72i q^{59} +(225.304 - 433.046i) q^{60} -2301.35 q^{61} -4881.11i q^{62} +(2864.10 + 4086.39i) q^{63} -2215.13 q^{64} +712.216i q^{65} +(-3967.57 - 2064.23i) q^{66} +4219.06 q^{67} +849.669i q^{68} +(4319.64 - 8302.58i) q^{69} -3145.21 q^{70} -475.261i q^{71} +(3376.77 - 2366.74i) q^{72} -2148.47 q^{73} +4846.18i q^{74} +(998.006 + 519.239i) q^{75} -1463.74 q^{76} +6704.46i q^{77} +(1208.32 - 2322.45i) q^{78} +3106.20 q^{79} +3466.86i q^{80} +(-2238.36 + 6167.37i) q^{81} -793.482 q^{82} -3769.87i q^{83} +(2386.21 + 1241.49i) q^{84} -1958.16 q^{85} +13849.5i q^{86} +(-747.178 + 1436.12i) q^{87} +5540.21 q^{88} -8260.68i q^{89} +(-2373.46 - 3386.35i) q^{90} -3924.51 q^{91} -5044.82i q^{92} +(-8534.44 - 4440.27i) q^{93} +4272.72 q^{94} -3373.37i q^{95} +(2498.21 - 4801.70i) q^{96} +9310.26 q^{97} -6367.29i q^{98} +(-7218.47 + 5059.35i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9}+O(q^{10}) $ 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9	$ 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27} - 3764 q^{28} + 2200 q^{30} + 3772 q^{31} + 4420 q^{33} + 3124 q^{34} - 7606 q^{36} + 1896 q^{37} - 1336 q^{39} - 4650 q^{40} - 1980 q^{42} - 7384 q^{43} + 1900 q^{45} + 8196 q^{46} + 14668 q^{48} - 1318 q^{49} - 8492 q^{51} + 8976 q^{52} - 278 q^{54} - 1300 q^{55} - 11584 q^{57} - 23740 q^{58} + 5050 q^{60} + 6452 q^{61} + 14796 q^{63} + 3174 q^{64} - 12760 q^{66} + 13816 q^{67} + 5472 q^{69} - 2100 q^{70} - 2040 q^{72} + 596 q^{73} - 1000 q^{75} + 21348 q^{76} - 1400 q^{78} - 16124 q^{79} + 5086 q^{81} - 31240 q^{82} - 14736 q^{84} - 3100 q^{85} - 4900 q^{87} + 15660 q^{88} + 7550 q^{90} - 11632 q^{91} - 8184 q^{93} + 34924 q^{94} + 14354 q^{96} + 9756 q^{97} + 9680 q^{99}+O(q^{100}) $ 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	4.56632i		−	1.14158i	−0.821096	−	0.570790i	$-0.806637\pi$
$2$		−	4.56632i		−	1.14158i	0.821096	−	0.570790i	$-0.193363\pi$
$3$	−7.98405	−	4.15391i	−0.887116	−	0.461546i
$3$	−7.98405	−	4.15391i	−0.887116	−	0.461546i
$4$	−4.85128			−0.303205
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$	−18.9681	+	36.4577i	−0.526892	+	1.01271i
$7$	61.6068			1.25728			0.628641	−	0.777695i	$-0.283611\pi$
$7$	61.6068			1.25728			0.628641	+	0.777695i	$0.283611\pi$
$8$		−	50.9086i		−	0.795447i
$9$	46.4900	+	66.3301i	0.573951	+	0.818890i
$10$	−51.0530			−0.510530
$11$			108.827i			0.899393i	0.893181	+	0.449696i	$0.148468\pi$
$11$			108.827i			0.899393i	−0.893181	+	0.449696i	$0.851532\pi$
$12$	38.7328	+	20.1518i	0.268978	+	0.139943i
$13$	−63.7026			−0.376938			−0.188469	−	0.982079i	$-0.560353\pi$
$13$	−63.7026			−0.376938			−0.188469	+	0.982079i	$0.560353\pi$
$14$		−	281.317i		−	1.43529i
$15$	−46.4422	+	89.2644i	−0.206410	+	0.396730i
$16$	−310.086			−1.21127
$17$		−	175.143i		−	0.606033i	−0.952985	−	0.303016i	$-0.902006\pi$
$17$		−	175.143i		−	0.606033i	0.952985	−	0.303016i	$-0.0979937\pi$
$18$	302.884	−	212.288i	0.934828	−	0.655211i
$19$	301.723			0.835798			0.417899	−	0.908493i	$-0.362767\pi$
$19$	301.723			0.835798			0.417899	+	0.908493i	$0.362767\pi$
$20$			54.2389i			0.135597i
$21$	−491.872	−	255.909i	−1.11536	−	0.580293i
$22$	496.937			1.02673
$23$			1039.90i			1.96578i	0.184203	+	0.982888i	$0.441030\pi$
$23$			1039.90i			1.96578i	−0.184203	+	0.982888i	$0.558970\pi$
$24$	−211.470	+	406.457i	−0.367136	+	0.705654i
$25$	−125.000			−0.200000
$26$			290.886i			0.430305i
$27$	−95.6491	−	722.698i	−0.131206	−	0.991355i
$28$	−298.872			−0.381214
$29$		−	179.873i		−	0.213880i	−0.994265	−	0.106940i	$-0.965895\pi$
$29$		−	179.873i		−	0.213880i	0.994265	−	0.106940i	$-0.0341053\pi$
$30$	407.610	+	212.070i	0.452900	+	0.235633i
$31$	1068.94			1.11232			0.556159	−	0.831076i	$-0.312275\pi$
$31$	1068.94			1.11232			0.556159	+	0.831076i	$0.312275\pi$
$32$			601.411i			0.587316i
$33$	452.056	−	868.876i	0.415111	−	0.797866i
$34$	−799.761			−0.691835
$35$		−	688.785i		−	0.562274i
$36$	−225.536	−	321.785i	−0.174025	−	0.248291i
$37$	−1061.29			−0.775229			−0.387615	−	0.921822i	$-0.626701\pi$
$37$	−1061.29			−0.775229			−0.387615	+	0.921822i	$0.626701\pi$
$38$		−	1377.76i		−	0.954130i
$39$	508.604	+	264.615i	0.334388	+	0.173974i
$40$	−569.176			−0.355735
$41$		−	173.768i		−	0.103372i	−0.998663	−	0.0516860i	$-0.983541\pi$
$41$		−	173.768i		−	0.103372i	0.998663	−	0.0516860i	$-0.0164595\pi$
$42$	−1168.56	+	2246.04i	−0.662451	+	1.27327i
$43$	−3032.96			−1.64032			−0.820162	−	0.572132i	$-0.806117\pi$
$43$	−3032.96			−1.64032			−0.820162	+	0.572132i	$0.806117\pi$
$44$		−	527.948i		−	0.272700i
$45$	741.593	−	519.774i	0.366219	−	0.256679i
$46$	4748.50			2.24409
$47$			935.703i			0.423587i	0.977314	+	0.211793i	$0.0679304\pi$
$47$			935.703i			0.423587i	−0.977314	+	0.211793i	$0.932070\pi$
$48$	2475.74	+	1288.07i	1.07454	+	0.559057i
$49$	1394.40			0.580759
$50$			570.790i			0.228316i
$51$	−727.531	+	1398.35i	−0.279712	+	0.537622i
$52$	309.039			0.114289
$53$		−	423.304i		−	0.150696i	−0.997157	−	0.0753478i	$-0.975993\pi$
$53$		−	423.304i		−	0.150696i	0.997157	−	0.0753478i	$-0.0240067\pi$
$54$	−3300.07	+	436.765i	−1.13171	+	0.149782i
$55$	1216.72			0.402221
$56$		−	3136.32i		−	1.00010i
$57$	−2408.97	−	1253.33i	−0.741450	−	0.385759i
$58$	−821.359			−0.244161
$59$			3188.72i			0.916034i	0.888943	+	0.458017i	$0.151440\pi$
$59$			3188.72i			0.916034i	−0.888943	+	0.458017i	$0.848560\pi$
$60$	225.304	−	433.046i	0.0625844	−	0.120291i
$61$	−2301.35			−0.618475			−0.309238	−	0.950985i	$-0.600074\pi$
$61$	−2301.35			−0.618475			−0.309238	+	0.950985i	$0.600074\pi$
$62$		−	4881.11i		−	1.26980i
$63$	2864.10	+	4086.39i	0.721618	+	1.02958i
$64$	−2215.13			−0.540804
$65$			712.216i			0.168572i
$66$	−3967.57	−	2064.23i	−0.910828	−	0.473883i
$67$	4219.06			0.939867			0.469933	−	0.882702i	$-0.344278\pi$
$67$	4219.06			0.939867			0.469933	+	0.882702i	$0.344278\pi$
$68$			849.669i			0.183752i
$69$	4319.64	−	8302.58i	0.907296	−	1.74387i
$70$	−3145.21			−0.641880
$71$		−	475.261i		−	0.0942791i	−0.998888	−	0.0471396i	$-0.984989\pi$
$71$		−	475.261i		−	0.0942791i	0.998888	−	0.0471396i	$-0.0150106\pi$
$72$	3376.77	−	2366.74i	0.651384	−	0.456548i
$73$	−2148.47			−0.403166			−0.201583	−	0.979471i	$-0.564609\pi$
$73$	−2148.47			−0.403166			−0.201583	+	0.979471i	$0.564609\pi$
$74$			4846.18i			0.884986i
$75$	998.006	+	519.239i	0.177423	+	0.0923092i
$76$	−1463.74			−0.253418
$77$			6704.46i			1.13079i
$78$	1208.32	−	2322.45i	0.198606	−	0.381731i
$79$	3106.20			0.497709			0.248855	−	0.968541i	$-0.419946\pi$
$79$	3106.20			0.497709			0.248855	+	0.968541i	$0.419946\pi$
$80$			3466.86i			0.541697i
$81$	−2238.36	+	6167.37i	−0.341161	+	0.940005i
$82$	−793.482			−0.118007
$83$		−	3769.87i		−	0.547231i	−0.961839	−	0.273615i	$-0.911780\pi$
$83$		−	3769.87i		−	0.547231i	0.961839	−	0.273615i	$-0.0882195\pi$
$84$	2386.21	+	1241.49i	0.338181	+	0.175948i
$85$	−1958.16			−0.271026
$86$			13849.5i			1.87256i
$87$	−747.178	+	1436.12i	−0.0987156	+	0.189737i
$88$	5540.21			0.715420
$89$		−	8260.68i		−	1.04288i	−0.853287	−	0.521442i	$-0.825394\pi$
$89$		−	8260.68i		−	1.04288i	0.853287	−	0.521442i	$-0.174606\pi$
$90$	−2373.46	−	3386.35i	−0.293019	−	0.418068i
$91$	−3924.51			−0.473918
$92$		−	5044.82i		−	0.596033i
$93$	−8534.44	−	4440.27i	−0.986755	−	0.513385i
$94$	4272.72			0.483558
$95$		−	3373.37i		−	0.373780i
$96$	2498.21	−	4801.70i	0.271073	−	0.521018i
$97$	9310.26			0.989505			0.494753	−	0.869034i	$-0.335259\pi$
$97$	9310.26			0.989505			0.494753	+	0.869034i	$0.335259\pi$
$98$		−	6367.29i		−	0.662983i
$99$	−7218.47	+	5059.35i	−0.736504	+	0.516207i
$100$	606.409			0.0606409
$101$		−	10239.8i		−	1.00380i	−0.864925	−	0.501902i	$-0.832634\pi$
$101$		−	10239.8i		−	1.00380i	0.864925	−	0.501902i	$-0.167366\pi$
$102$	6385.33	+	3322.14i	0.613738	+	0.319314i
$103$	−5400.15			−0.509016			−0.254508	−	0.967071i	$-0.581913\pi$
$103$	−5400.15			−0.509016			−0.254508	+	0.967071i	$0.581913\pi$
$104$			3243.01i			0.299835i
$105$	−2861.15	+	5499.29i	−0.259515	+	0.498802i
$106$	−1932.94			−0.172031
$107$		−	6786.14i		−	0.592728i	−0.955075	−	0.296364i	$-0.904226\pi$
$107$		−	6786.14i		−	0.592728i	0.955075	−	0.296364i	$-0.0957741\pi$
$108$	464.020	+	3506.01i	0.0397823	+	0.300584i
$109$	−4600.12			−0.387183			−0.193591	−	0.981082i	$-0.562014\pi$
$109$	−4600.12			−0.387183			−0.193591	+	0.981082i	$0.562014\pi$
$110$		−	5555.92i		−	0.459167i
$111$	8473.38	+	4408.50i	0.687719	+	0.357804i
$112$	−19103.4			−1.52291
$113$			14751.6i			1.15526i	0.816298	+	0.577632i	$0.196023\pi$
$113$			14751.6i			1.15526i	−0.816298	+	0.577632i	$0.803977\pi$
$114$	−5723.11	+	11000.1i	−0.440375	+	0.846425i
$115$	11626.4			0.879122
$116$			872.615i			0.0648495i
$117$	−2961.53	−	4225.39i	−0.216344	−	0.308671i
$118$	14560.7			1.04573
$119$		−	10790.0i		−	0.761954i
$120$	4544.33	+	2364.31i	0.315578	+	0.164188i
$121$	2797.78			0.191092
$122$			10508.7i			0.706039i
$123$	−721.818	+	1387.37i	−0.0477109	+	0.0917030i
$124$	−5185.71			−0.337260
$125$			1397.54i			0.0894427i
$126$	18659.7	−	13078.4i	1.17534	−	0.823785i
$127$	−15454.5			−0.958178			−0.479089	−	0.877766i	$-0.659033\pi$
$127$	−15454.5			−0.958178			−0.479089	+	0.877766i	$0.659033\pi$
$128$			19737.6i			1.20469i
$129$	24215.3	+	12598.6i	1.45516	+	0.757084i
$130$	3252.21			0.192438
$131$		−	21843.0i		−	1.27283i	−0.771347	−	0.636415i	$-0.780417\pi$
$131$		−	21843.0i		−	1.27283i	0.771347	−	0.636415i	$-0.219583\pi$
$132$	−2193.05	+	4215.16i	−0.125864	+	0.241917i
$133$	18588.2			1.05083
$134$		−	19265.6i		−	1.07293i
$135$	−8080.01	+	1069.39i	−0.443347	+	0.0586771i
$136$	−8916.32			−0.482067
$137$			18924.3i			1.00827i	0.863624	+	0.504137i	$0.168189\pi$
$137$			18924.3i			1.00827i	−0.863624	+	0.504137i	$0.831811\pi$
$138$	−37912.2	−	19724.8i	−1.99077	−	1.03575i
$139$	−21796.9			−1.12815			−0.564073	−	0.825725i	$-0.690766\pi$
$139$	−21796.9			−1.12815			−0.564073	+	0.825725i	$0.690766\pi$
$140$			3341.49i			0.170484i
$141$	3886.83	−	7470.70i	0.195505	−	0.375771i
$142$	−2170.19			−0.107627
$143$		−	6932.53i		−	0.339016i
$144$	−14415.9	−	20568.0i	−0.695210	−	0.991898i
$145$	−2011.04			−0.0956502
$146$			9810.60i			0.460246i
$147$	−11133.0	−	5792.22i	−0.515201	−	0.268047i
$148$	5148.60			0.235053
$149$			31031.9i			1.39777i	0.715235	+	0.698884i	$0.246320\pi$
$149$			31031.9i			1.39777i	−0.715235	+	0.698884i	$0.753680\pi$
$150$	2371.01	−	4557.21i	0.105378	−	0.202543i
$151$	13097.0			0.574405			0.287203	−	0.957870i	$-0.407275\pi$
$151$	13097.0			0.574405			0.287203	+	0.957870i	$0.407275\pi$
$152$		−	15360.3i		−	0.664834i
$153$	11617.3	−	8142.42i	0.496274	−	0.347833i
$154$	30614.7			1.29089
$155$		−	11951.1i		−	0.497443i
$156$	−2467.38	−	1283.72i	−0.101388	−	0.0527498i
$157$	12609.6			0.511567			0.255784	−	0.966734i	$-0.417667\pi$
$157$	12609.6			0.511567			0.255784	+	0.966734i	$0.417667\pi$
$158$		−	14183.9i		−	0.568175i
$159$	−1758.37	+	3379.68i	−0.0695529	+	0.133685i
$160$	6723.98			0.262656
$161$			64064.7i			2.47154i
$162$	28162.2	+	10221.1i	1.07309	+	0.389462i
$163$	1570.77			0.0591206			0.0295603	−	0.999563i	$-0.490589\pi$
$163$	1570.77			0.0591206			0.0295603	+	0.999563i	$0.490589\pi$
$164$			842.998i			0.0313429i
$165$	−9714.33	−	5054.14i	−0.356817	−	0.185643i
$166$	−17214.4			−0.624707
$167$		−	9867.19i		−	0.353802i	−0.984229	−	0.176901i	$-0.943393\pi$
$167$		−	9867.19i		−	0.353802i	0.984229	−	0.176901i	$-0.0566073\pi$
$168$	−13028.0	+	25040.5i	−0.461593	+	0.887207i
$169$	−24503.0			−0.857918
$170$			8941.60i			0.309398i
$171$	14027.1	+	20013.3i	0.479707	+	0.684427i
$172$	14713.7			0.497354
$173$		−	21586.3i		−	0.721250i	−0.932711	−	0.360625i	$-0.882563\pi$
$173$		−	21586.3i		−	0.721250i	0.932711	−	0.360625i	$-0.117437\pi$
$174$	6557.77	+	3411.85i	0.216600	+	0.112692i
$175$	−7700.85			−0.251456
$176$		−	33745.5i		−	1.08941i
$177$	13245.6	−	25458.9i	0.422792	−	0.812629i
$178$	−37720.9			−1.19054
$179$			22176.0i			0.692114i	0.938213	+	0.346057i	$0.112480\pi$
$179$			22176.0i			0.692114i	−0.938213	+	0.346057i	$0.887520\pi$
$180$	−3597.67	+	2521.57i	−0.111039	+	0.0778262i
$181$	−57041.1			−1.74113			−0.870565	−	0.492054i	$-0.836246\pi$
$181$	−57041.1			−1.74113			−0.870565	+	0.492054i	$0.836246\pi$
$182$			17920.6i			0.541015i
$183$	18374.1	+	9559.59i	0.548660	+	0.285455i
$184$	52939.7			1.56367
$185$			11865.6i			0.346693i
$186$	−20275.7	+	38971.0i	−0.586070	+	1.12646i
$187$	19060.3			0.545062
$188$		−	4539.35i		−	0.128434i
$189$	−5892.64	−	44523.1i	−0.164963	−	1.24641i
$190$	−15403.9			−0.426700
$191$		−	64290.5i		−	1.76230i	−0.472835	−	0.881151i	$-0.656769\pi$
$191$		−	64290.5i		−	1.76230i	0.472835	−	0.881151i	$-0.343231\pi$
$192$	17685.7	+	9201.46i	0.479756	+	0.249606i
$193$	36452.6			0.978621			0.489310	−	0.872110i	$-0.337249\pi$
$193$	36452.6			0.978621			0.489310	+	0.872110i	$0.337249\pi$
$194$		−	42513.6i		−	1.12960i
$195$	2958.48	−	5686.37i	0.0778037	−	0.149543i
$196$	−6764.63			−0.176089
$197$		−	63474.0i		−	1.63555i	−0.575540	−	0.817774i	$-0.695208\pi$
$197$		−	63474.0i		−	1.63555i	0.575540	−	0.817774i	$-0.304792\pi$
$198$	23102.6	+	32961.9i	0.589292	+	0.840778i
$199$	−9634.07			−0.243279			−0.121639	−	0.992574i	$-0.538815\pi$
$199$	−9634.07			−0.243279			−0.121639	+	0.992574i	$0.538815\pi$
$200$			6363.58i			0.159089i
$201$	−33685.2	−	17525.6i	−0.833771	−	0.433792i
$202$	−46758.2			−1.14592
$203$		−	11081.4i		−	0.268908i
$204$	3529.45	−	6783.80i	0.0848100	−	0.163009i
$205$	−1942.79			−0.0462294
$206$			24658.8i			0.581082i
$207$	−68976.4	+	48344.8i	−1.60975	+	1.12826i
$208$	19753.2			0.456575
$209$			32835.5i			0.751711i
$210$	25111.5	+	13064.9i	0.569423	+	0.296257i
$211$	64309.8			1.44448			0.722241	−	0.691641i	$-0.243112\pi$
$211$	64309.8			1.44448			0.722241	+	0.691641i	$0.243112\pi$
$212$			2053.56i			0.0456916i
$213$	−1974.19	+	3794.51i	−0.0435141	+	0.0836365i
$214$	−30987.7			−0.676646
$215$			33909.5i			0.733575i
$216$	−36791.6	+	4869.37i	−0.788571	+	0.104367i
$217$	65853.8			1.39850
$218$			21005.6i			0.442000i
$219$	17153.5	+	8924.56i	0.357655	+	0.186079i
$220$	−5902.63			−0.121955
$221$			11157.1i			0.228437i
$222$	20130.6	−	38692.2i	0.408462	−	0.785086i
$223$	−51769.1			−1.04102			−0.520512	−	0.853854i	$-0.674259\pi$
$223$	−51769.1			−1.04102			−0.520512	+	0.853854i	$0.674259\pi$
$224$			37051.1i			0.738422i
$225$	−5811.25	−	8291.26i	−0.114790	−	0.163778i
$226$	67360.3			1.31883
$227$			69770.7i			1.35401i	0.735979	+	0.677004i	$0.236722\pi$
$227$			69770.7i			1.35401i	−0.735979	+	0.677004i	$0.763278\pi$
$228$	11686.6	+	6080.26i	0.224811	+	0.116964i
$229$	−2284.08			−0.0435553			−0.0217777	−	0.999763i	$-0.506933\pi$
$229$	−2284.08			−0.0435553			−0.0217777	+	0.999763i	$0.506933\pi$
$230$		−	53089.8i		−	1.00359i
$231$	27849.7	−	53528.7i	0.521912	−	1.00314i
$232$	−9157.11			−0.170131
$233$			22464.3i			0.413791i	0.978363	+	0.206895i	$0.0663360\pi$
$233$			22464.3i			0.413791i	−0.978363	+	0.206895i	$0.933664\pi$
$234$	−19294.5	+	13523.3i	−0.352372	+	0.246974i
$235$	10461.5			0.189434
$236$		−	15469.3i		−	0.277746i
$237$	−24800.1	−	12902.9i	−0.441526	−	0.229716i
$238$	−49270.7			−0.869832
$239$		−	42116.1i		−	0.737314i	−0.929565	−	0.368657i	$-0.879818\pi$
$239$		−	42116.1i		−	0.737314i	0.929565	−	0.368657i	$-0.120182\pi$
$240$	14401.0	−	27679.6i	0.250018	−	0.480548i
$241$	−975.921			−0.0168028			−0.00840138	−	0.999965i	$-0.502674\pi$
$241$	−975.921			−0.0168028			−0.00840138	+	0.999965i	$0.502674\pi$
$242$		−	12775.6i		−	0.218147i
$243$	43489.9	−	39942.7i	0.736505	−	0.676432i
$244$	11164.5			0.187525
$245$		−	15589.9i		−	0.259723i
$246$	6335.19	+	3296.05i	0.104686	+	0.0544658i
$247$	−19220.5			−0.315044
$248$		−	54418.1i		−	0.884790i
$249$	−15659.7	+	30098.8i	−0.252572	+	0.485457i
$250$	6381.63			0.102106
$251$			95434.5i			1.51481i	0.652945	+	0.757405i	$0.273533\pi$
$251$			95434.5i			1.51481i	−0.652945	+	0.757405i	$0.726467\pi$
$252$	−13894.6	−	19824.2i	−0.218798	−	0.312172i
$253$	−113168.			−1.76801
$254$			70570.0i			1.09384i
$255$	15634.1	+	8134.04i	0.240432	+	0.125091i
$256$	54686.0			0.834442
$257$			23114.1i			0.349953i	0.984573	+	0.174977i	$0.0559850\pi$
$257$			23114.1i			0.349953i	−0.984573	+	0.174977i	$0.944015\pi$
$258$	57529.4	−	110575.i	0.864272	−	1.66118i
$259$	−65382.6			−0.974682
$260$		−	3455.16i		−	0.0511118i
$261$	11931.0	−	8362.31i	0.175144	−	0.122757i
$262$	−99742.2			−1.45304
$263$			37165.6i			0.537316i	0.963236	+	0.268658i	$0.0865801\pi$
$263$			37165.6i			0.537316i	−0.963236	+	0.268658i	$0.913420\pi$
$264$	−44233.3	−	23013.6i	−0.634661	−	0.330199i
$265$	−4732.68			−0.0673931
$266$		−	84879.7i		−	1.19961i
$267$	−34314.1	+	65953.7i	−0.481339	+	0.925159i
$268$	−20467.8			−0.284972
$269$		−	134496.i		−	1.85869i	−0.369216	−	0.929344i	$-0.620374\pi$
$269$		−	134496.i		−	1.85869i	0.369216	−	0.929344i	$-0.379626\pi$
$270$	4883.18	+	36895.9i	0.0669846	+	0.506117i
$271$	111796.			1.52226			0.761128	−	0.648602i	$-0.224646\pi$
$271$	111796.			1.52226			0.761128	+	0.648602i	$0.224646\pi$
$272$			54309.5i			0.734070i
$273$	31333.5	+	16302.1i	0.420420	+	0.218735i
$274$	86414.3			1.15102
$275$		−	13603.3i		−	0.179879i
$276$	−20955.7	+	40278.1i	−0.275096	+	0.528750i
$277$	128402.			1.67345			0.836725	−	0.547624i	$-0.184467\pi$
$277$	128402.			1.67345			0.836725	+	0.547624i	$0.184467\pi$
$278$			99531.6i			1.28787i
$279$	49694.9	+	70902.7i	0.638415	+	0.910865i
$280$	−35065.1			−0.447259
$281$		−	29381.6i		−	0.372102i	−0.982540	−	0.186051i	$-0.940431\pi$
$281$		−	29381.6i		−	0.372102i	0.982540	−	0.186051i	$-0.0595690\pi$
$282$	−34113.6	−	17748.5i	−0.428972	−	0.223184i
$283$	60115.2			0.750605			0.375303	−	0.926902i	$-0.377539\pi$
$283$	60115.2			0.750605			0.375303	+	0.926902i	$0.377539\pi$
$284$			2305.62i			0.0285859i
$285$	−14012.7	+	26933.1i	−0.172517	+	0.331587i
$286$	−31656.1			−0.387013
$287$		−	10705.3i		−	0.129968i
$288$	−39891.7	+	27959.6i	−0.480947	+	0.337090i
$289$	52845.8			0.632724
$290$			9183.07i			0.109192i
$291$	−74333.5	−	38674.0i	−0.877806	−	0.456702i
$292$	10422.8			0.122242
$293$		−	3769.43i		−	0.0439076i	−0.999759	−	0.0219538i	$-0.993011\pi$
$293$		−	3769.43i		−	0.0439076i	0.999759	−	0.0219538i	$-0.00698868\pi$
$294$	−26449.1	+	50836.7i	−0.305997	+	0.588143i
$295$	35650.9			0.409663
$296$			54028.8i			0.616654i
$297$	78648.7	−	10409.2i	0.891618	−	0.118006i
$298$	141701.			1.59566
$299$		−	66244.0i		−	0.740976i
$300$	−4841.60	−	2518.97i	−0.0537956	−	0.0279886i
$301$	−186851.			−2.06235
$302$		−	59805.2i		−	0.655730i
$303$	−42535.2	+	81755.1i	−0.463302	+	0.890491i
$304$	−93560.0			−1.01238
$305$			25729.8i			0.276591i
$306$	−37180.9	−	53048.2i	−0.397079	−	0.566536i
$307$	−140911.			−1.49509			−0.747546	−	0.664210i	$-0.768768\pi$
$307$	−140911.			−1.49509			−0.747546	+	0.664210i	$0.768768\pi$
$308$		−	32525.2i		−	0.342861i
$309$	43115.0	+	22431.7i	0.451556	+	0.234934i
$310$	−54572.4			−0.567871
$311$		−	18018.5i		−	0.186294i	−0.995652	−	0.0931468i	$-0.970307\pi$
$311$		−	18018.5i		−	0.186294i	0.995652	−	0.0931468i	$-0.0296926\pi$
$312$	13471.2	−	25892.3i	0.138387	−	0.265988i
$313$	−44156.6			−0.450720			−0.225360	−	0.974276i	$-0.572356\pi$
$313$	−44156.6			−0.450720			−0.225360	+	0.974276i	$0.572356\pi$
$314$		−	57579.6i		−	0.583995i
$315$	45687.2	−	32021.6i	0.460440	−	0.322717i
$316$	−15069.0			−0.150908
$317$		−	193874.i		−	1.92931i	−0.263525	−	0.964653i	$-0.584885\pi$
$317$		−	193874.i		−	1.92931i	0.263525	−	0.964653i	$-0.415115\pi$
$318$	15432.7	+	8029.27i	0.152612	+	0.0794002i
$319$	19575.0			0.192362
$320$			24765.9i			0.241855i
$321$	−28189.0	+	54180.8i	−0.273571	+	0.525818i
$322$	292540.			2.82146
$323$		−	52844.8i		−	0.506521i
$324$	10858.9	−	29919.6i	0.103442	−	0.285014i
$325$	7962.82			0.0753876
$326$		−	7172.66i		−	0.0674909i
$327$	36727.5	+	19108.5i	0.343476	+	0.178703i
$328$	−8846.31			−0.0822270
$329$			57645.7i			0.532568i
$330$	−23078.8	+	44358.7i	−0.211927	+	0.407335i
$331$	−37018.2			−0.337878			−0.168939	−	0.985627i	$-0.554034\pi$
$331$	−37018.2			−0.337878			−0.168939	+	0.985627i	$0.554034\pi$
$332$			18288.7i			0.165923i
$333$	−49339.3	−	70395.4i	−0.444943	−	0.634827i
$334$	−45056.7			−0.403893
$335$		−	47170.6i		−	0.420321i
$336$	152522.	+	79353.8i	1.35100	+	0.702893i
$337$	−106760.			−0.940041			−0.470021	−	0.882655i	$-0.655754\pi$
$337$	−106760.			−0.940041			−0.470021	+	0.882655i	$0.655754\pi$
$338$			111888.i			0.979382i
$339$	61276.7	−	117777.i	0.533207	−	1.02485i
$340$	9499.59			0.0821764
$341$			116329.i			1.00041i
$342$	91387.2	−	64052.3i	0.781328	−	0.547624i
$343$	−62013.3			−0.527104
$344$			154404.i			1.30479i
$345$	−92825.6	−	48295.0i	−0.779883	−	0.405755i
$346$	−98570.0			−0.823365
$347$			63787.6i			0.529758i	0.964282	+	0.264879i	$0.0853320\pi$
$347$			63787.6i			0.529758i	−0.964282	+	0.264879i	$0.914668\pi$
$348$	3624.77	−	6967.00i	0.0299310	−	0.0575291i
$349$	127165.			1.04404			0.522021	−	0.852932i	$-0.325178\pi$
$349$	127165.			1.04404			0.522021	+	0.852932i	$0.325178\pi$
$350$			35164.6i			0.287058i
$351$	6093.09	+	46037.7i	0.0494565	+	0.373680i
$352$	−65449.5			−0.528228
$353$		−	67251.4i		−	0.539699i	−0.962902	−	0.269850i	$-0.913026\pi$
$353$		−	67251.4i		−	0.539699i	0.962902	−	0.269850i	$-0.0869740\pi$
$354$	−116253.	−	60483.9i	−0.927681	−	0.482651i
$355$	−5313.58			−0.0421629
$356$			40074.8i			0.316207i
$357$	−44820.9	+	86148.1i	−0.351677	+	0.675942i
$358$	101263.			0.790104
$359$			143827.i			1.11597i	0.829851	+	0.557984i	$0.188425\pi$
$359$			143827.i			1.11597i	−0.829851	+	0.557984i	$0.811575\pi$
$360$	−26461.0	−	37753.5i	−0.204174	−	0.291308i
$361$	−39284.1			−0.301441
$362$			260468.i			1.98764i
$363$	−22337.6	−	11621.7i	−0.169521	−	0.0881979i
$364$	19038.9			0.143694
$365$			24020.6i			0.180301i
$366$	43652.2	−	83901.8i	0.325869	−	0.626339i
$367$	122411.			0.908838			0.454419	−	0.890788i	$-0.349847\pi$
$367$	122411.			0.908838			0.454419	+	0.890788i	$0.349847\pi$
$368$		−	322457.i		−	2.38109i
$369$	11526.1	−	8078.49i	0.0846502	−	0.0593304i
$370$	54182.0			0.395778
$371$		−	26078.4i		−	0.189467i
$372$	41402.9	+	21541.0i	0.299189	+	0.155661i
$373$	193520.			1.39094			0.695471	−	0.718554i	$-0.255196\pi$
$373$	193520.			1.39094			0.695471	+	0.718554i	$0.255196\pi$
$374$		−	87035.2i		−	0.622231i
$375$	5805.27	−	11158.0i	0.0412819	−	0.0793461i
$376$	47635.4			0.336941
$377$			11458.4i			0.0806196i
$378$	−203307.	+	26907.7i	−1.42288	+	0.188318i
$379$	50009.6			0.348157			0.174079	−	0.984732i	$-0.444305\pi$
$379$	50009.6			0.348157			0.174079	+	0.984732i	$0.444305\pi$
$380$			16365.1i			0.113332i
$381$	123389.	+	64196.5i	0.850016	+	0.442243i
$382$	−293571.			−2.01181
$383$			72111.0i			0.491591i	0.969322	+	0.245796i	$0.0790492\pi$
$383$			72111.0i			0.491591i	−0.969322	+	0.245796i	$0.920951\pi$
$384$	81988.2	−	157586.i	0.556018	−	1.06870i
$385$	74958.1			0.505705
$386$		−	166454.i		−	1.11717i
$387$	−141002.	−	201176.i	−0.941465	−	1.34324i
$388$	−45166.6			−0.300023
$389$		−	25283.5i		−	0.167085i	−0.996504	−	0.0835426i	$-0.973377\pi$
$389$		−	25283.5i		−	0.167085i	0.996504	−	0.0835426i	$-0.0266235\pi$
$390$	−25965.8	−	13509.4i	−0.170715	−	0.0888191i
$391$	182131.			1.19132
$392$		−	70987.1i		−	0.461963i
$393$	−90734.0	+	174396.i	−0.587469	+	1.12915i
$394$	−289842.			−1.86711
$395$		−	34728.4i		−	0.222582i
$396$	35018.8	−	24544.3i	0.223311	−	0.156517i
$397$	−28110.1			−0.178353			−0.0891766	−	0.996016i	$-0.528424\pi$
$397$	−28110.1			−0.178353			−0.0891766	+	0.996016i	$0.528424\pi$
$398$			43992.3i			0.277722i
$399$	−148409.	−	77213.8i	−0.932212	−	0.485008i
$400$	38760.7			0.242254
$401$		−	58113.8i		−	0.361402i	−0.983538	−	0.180701i	$-0.942163\pi$
$401$		−	58113.8i		−	0.361402i	0.983538	−	0.180701i	$-0.0578366\pi$
$402$	−80027.6	+	153817.i	−0.495208	+	0.951817i
$403$	−68094.0			−0.419275
$404$			49676.1i			0.304358i
$405$	68953.3	+	25025.6i	0.420383	+	0.152572i
$406$	−50601.3			−0.306980
$407$		−	115496.i		−	0.697236i
$408$	71188.3	+	37037.6i	0.427650	+	0.222496i
$409$	−8511.55			−0.0508818			−0.0254409	−	0.999676i	$-0.508099\pi$
$409$	−8511.55			−0.0508818			−0.0254409	+	0.999676i	$0.508099\pi$
$410$			8871.39i			0.0527745i
$411$	78609.8	−	151092.i	0.465364	−	0.894456i
$412$	26197.6			0.154336
$413$			196447.i			1.15171i
$414$	220758.	+	314968.i	1.28800	+	1.83766i
$415$	−42148.4			−0.244729
$416$		−	38311.4i		−	0.221382i
$417$	174028.	+	90542.4i	1.00080	+	0.520691i
$418$	149937.			0.858138
$419$			292724.i			1.66736i	0.552244	+	0.833682i	$0.313772\pi$
$419$			292724.i			1.66736i	−0.552244	+	0.833682i	$0.686228\pi$
$420$	13880.2	−	26678.6i	0.0786862	−	0.151239i
$421$	−152199.			−0.858713			−0.429357	−	0.903135i	$-0.641260\pi$
$421$	−152199.			−0.858713			−0.429357	+	0.903135i	$0.641260\pi$
$422$		−	293659.i		−	1.64899i
$423$	−62065.3	+	43500.9i	−0.346871	+	0.243118i
$424$	−21549.8			−0.119870
$425$			21892.9i			0.121207i
$426$	17326.9	+	9014.80i	0.0954778	+	0.0496749i
$427$	−141779.			−0.777598
$428$			32921.4i			0.179718i
$429$	−28797.1	+	55349.6i	−0.156471	+	0.300746i
$430$	154842.			0.837434
$431$		−	252810.i		−	1.36094i	−0.732775	−	0.680471i	$-0.761775\pi$
$431$		−	252810.i		−	1.36094i	0.732775	−	0.680471i	$-0.238225\pi$
$432$	29659.4	+	224098.i	0.158926	+	1.20080i
$433$	330313.			1.76178			0.880888	−	0.473326i	$-0.156947\pi$
$433$	330313.			1.76178			0.880888	+	0.473326i	$0.156947\pi$
$434$		−	300710.i		−	1.59650i
$435$	16056.3	+	8353.71i	0.0848528	+	0.0441469i
$436$	22316.4			0.117396
$437$			313761.i			1.64299i
$438$	40752.4	−	78328.3i	0.212425	−	0.408292i
$439$	−229425.			−1.19045			−0.595227	−	0.803558i	$-0.702938\pi$
$439$	−229425.			−1.19045			−0.595227	+	0.803558i	$0.702938\pi$
$440$		−	61941.4i		−	0.319945i
$441$	64825.8	+	92490.8i	0.333327	+	0.475578i
$442$	50946.8			0.260779
$443$			6149.63i			0.0313358i	0.999877	+	0.0156679i	$0.00498746\pi$
$443$			6149.63i			0.0313358i	−0.999877	+	0.0156679i	$0.995013\pi$
$444$	−41106.7	−	21386.9i	−0.208520	−	0.108488i
$445$	−92357.2			−0.466392
$446$			236394.i			1.18841i
$447$	128904.	−	247760.i	0.645134	−	1.23998i
$448$	−136467.			−0.679943
$449$		−	219554.i		−	1.08905i	−0.838744	−	0.544526i	$-0.816710\pi$
$449$		−	219554.i		−	1.08905i	0.838744	−	0.544526i	$-0.183290\pi$
$450$	−37860.5	+	26536.0i	−0.186966	+	0.131042i
$451$	18910.6			0.0929720
$452$		−	71563.9i		−	0.350281i
$453$	−104567.	−	54403.9i	−0.509564	−	0.265114i
$454$	318595.			1.54571
$455$			43877.4i			0.211942i
$456$	−63805.4	+	122637.i	−0.306851	+	0.589785i
$457$	−138678.			−0.664010			−0.332005	−	0.943278i	$-0.607725\pi$
$457$	−138678.			−0.664010			−0.332005	+	0.943278i	$0.607725\pi$
$458$			10429.9i			0.0497219i
$459$	−126576.	+	16752.3i	−0.600794	+	0.0795151i
$460$	−56402.8			−0.266554
$461$			3892.31i			0.0183149i	0.999958	+	0.00915746i	$0.00291495\pi$
$461$			3892.31i			0.0183149i	−0.999958	+	0.00915746i	$0.997085\pi$
$462$	−244429.	−	127171.i	−1.14517	−	0.595804i
$463$	−279654.			−1.30455			−0.652273	−	0.757984i	$-0.726184\pi$
$463$	−279654.			−1.30455			−0.652273	+	0.757984i	$0.726184\pi$
$464$			55776.1i			0.259067i
$465$	−49643.7	+	95418.0i	−0.229593	+	0.441290i
$466$	102579.			0.472375
$467$			202921.i			0.930452i	0.885192	+	0.465226i	$0.154027\pi$
$467$			202921.i			0.930452i	−0.885192	+	0.465226i	$0.845973\pi$
$468$	14367.2	+	20498.6i	0.0655965	+	0.0935905i
$469$	259923.			1.18168
$470$		−	47770.5i		−	0.216254i
$471$	−100676.	−	52379.3i	−0.453820	−	0.236112i
$472$	162333.			0.728657
$473$		−	330066.i		−	1.47530i
$474$	−58918.7	+	113245.i	−0.262239	+	0.504037i
$475$	−37715.4			−0.167160
$476$			52345.4i			0.231028i
$477$	28077.8	−	19679.4i	0.123403	−	0.0864918i
$478$	−192316.			−0.841703
$479$		−	308474.i		−	1.34446i	−0.740343	−	0.672230i	$-0.765337\pi$
$479$		−	308474.i		−	1.34446i	0.740343	−	0.672230i	$-0.234663\pi$
$480$	−53684.6	−	27930.8i	−0.233006	−	0.121228i
$481$	67606.8			0.292213
$482$			4456.37i			0.0191817i
$483$	266119.	−	511495.i	1.14073	−	2.19254i
$484$	−13572.8			−0.0579401
$485$		−	104092.i		−	0.442520i
$486$	−182391.	−	198589.i	−0.772202	−	0.840779i
$487$	210519.			0.887633			0.443816	−	0.896118i	$-0.353624\pi$
$487$	210519.			0.887633			0.443816	+	0.896118i	$0.353624\pi$
$488$			117158.i			0.491965i
$489$	−12541.1	−	6524.86i	−0.0524468	−	0.0272869i
$490$	−71188.4			−0.296495
$491$			442555.i			1.83571i	0.396913	+	0.917856i	$0.370082\pi$
$491$			442555.i			1.83571i	−0.396913	+	0.917856i	$0.629918\pi$
$492$	3501.74	−	6730.53i	0.0144662	−	0.0278048i
$493$	−31503.6			−0.129618
$494$			87767.1i			0.359648i
$495$	56565.2	+	80705.0i	0.230855	+	0.329374i
$496$	−331462.			−1.34732
$497$		−	29279.3i		−	0.118535i
$498$	137441.	+	71507.3i	0.554188	+	0.288331i
$499$	170123.			0.683224			0.341612	−	0.939841i	$-0.389027\pi$
$499$	170123.			0.683224			0.341612	+	0.939841i	$0.389027\pi$
$500$		−	6779.86i		−	0.0271195i
$501$	−40987.4	+	78780.1i	−0.163296	+	0.313864i
$502$	435785.			1.72928
$503$			232933.i			0.920653i	0.887750	+	0.460327i	$0.152268\pi$
$503$			232933.i			0.920653i	−0.887750	+	0.460327i	$0.847732\pi$
$504$	208032.	−	145808.i	0.818973	−	0.574009i
$505$	−114484.			−0.448915
$506$			516762.i			2.01832i
$507$	195633.	+	101783.i	0.761073	+	0.395968i
$508$	74973.8			0.290524
$509$			97567.9i			0.376592i	0.982112	+	0.188296i	$0.0602965\pi$
$509$			97567.9i			0.376592i	−0.982112	+	0.188296i	$0.939704\pi$
$510$	37142.6	−	71390.2i	0.142801	−	0.274472i
$511$	−132360.			−0.506893
$512$			66087.5i			0.252104i
$513$	−28859.6	−	218055.i	−0.109662	−	0.828573i
$514$	105546.			0.399500
$515$			60375.5i			0.227639i
$516$	−117475.	−	61119.5i	−0.441211	−	0.229552i
$517$	−101829.			−0.380971
$518$			298558.i			1.11268i
$519$	−89667.6	+	172346.i	−0.332890	+	0.639833i
$520$	36258.0			0.134090
$521$		−	436403.i		−	1.60773i	−0.594815	−	0.803863i	$-0.702775\pi$
$521$		−	436403.i		−	1.60773i	0.594815	−	0.803863i	$-0.297225\pi$
$522$	−38185.0	−	54480.8i	−0.140137	−	0.199941i
$523$	143160.			0.523381			0.261691	−	0.965152i	$-0.415720\pi$
$523$	143160.			0.523381			0.261691	+	0.965152i	$0.415720\pi$
$524$			105967.i			0.385928i
$525$	61484.0	+	31988.7i	0.223071	+	0.116059i
$526$	169710.			0.613389
$527$		−	187217.i		−	0.674101i
$528$	−140176.	+	269426.i	−0.502812	+	0.966433i
$529$	−801542.			−2.86428
$530$			21610.9i			0.0769346i
$531$	−211508.	+	148243.i	−0.750131	+	0.525759i
$532$	−90176.5			−0.318618
$533$			11069.5i			0.0389648i
$534$	301166.	+	156689.i	1.05614	+	0.549487i
$535$	−75871.3			−0.265076
$536$		−	214787.i		−	0.747615i
$537$	92117.3	−	177054.i	0.319442	−	0.613986i
$538$	−614154.			−2.12184
$539$			151748.i			0.522330i
$540$	39198.3	−	5187.91i	0.134425	−	0.0177912i
$541$	264951.			0.905254			0.452627	−	0.891700i	$-0.350487\pi$
$541$	264951.			0.905254			0.452627	+	0.891700i	$0.350487\pi$
$542$		−	510496.i		−	1.73778i
$543$	455419.	+	236944.i	1.54458	+	0.803611i
$544$	105333.			0.355933
$545$			51430.9i			0.173153i
$546$	74440.5	−	143079.i	0.249703	−	0.479943i
$547$	241732.			0.807903			0.403952	−	0.914780i	$-0.367636\pi$
$547$	241732.			0.807903			0.403952	+	0.914780i	$0.367636\pi$
$548$		−	91806.9i		−	0.305713i
$549$	−106990.	−	152649.i	−0.354974	−	0.506463i
$550$	−62117.1			−0.205346
$551$		−	54271.9i		−	0.178761i
$552$	−422673.	−	219907.i	−1.38716	−	0.721706i
$553$	191363.			0.625761
$554$		−	586325.i		−	1.91038i
$555$	49288.5	−	94735.3i	0.160015	−	0.307557i
$556$	105743.			0.342059
$557$			246479.i			0.794455i	0.917720	+	0.397228i	$0.130028\pi$
$557$			246479.i			0.794455i	−0.917720	+	0.397228i	$0.869972\pi$
$558$	323764.	−	226923.i	1.03983	−	0.728802i
$559$	193207.			0.618300
$560$			213582.i			0.681066i
$561$	−152178.	−	79174.7i	−0.483533	−	0.251571i
$562$	−134166.			−0.424784
$563$		−	44133.4i		−	0.139236i	−0.997574	−	0.0696179i	$-0.977822\pi$
$563$		−	44133.4i		−	0.139236i	0.997574	−	0.0696179i	$-0.0221780\pi$
$564$	−18856.1	+	36242.4i	−0.0592780	+	0.113935i
$565$	164927.			0.516649
$566$		−	274505.i		−	0.856876i
$567$	−137898.	+	379952.i	−0.428936	+	1.18185i
$568$	−24194.9			−0.0749941
$569$			196815.i			0.607901i	0.952688	+	0.303951i	$0.0983058\pi$
$569$			196815.i			0.607901i	−0.952688	+	0.303951i	$0.901694\pi$
$570$	122985.	+	63986.3i	0.378533	+	0.196942i
$571$	200156.			0.613897			0.306949	−	0.951726i	$-0.400692\pi$
$571$	200156.			0.613897			0.306949	+	0.951726i	$0.400692\pi$
$572$			33631.6i			0.102791i
$573$	−267057.	+	513299.i	−0.813383	+	1.56337i
$574$	−48883.9			−0.148369
$575$		−	129987.i		−	0.393155i
$576$	−102981.	−	146930.i	−0.310395	−	0.442859i
$577$	−426161.			−1.28004			−0.640018	−	0.768360i	$-0.721073\pi$
$577$	−426161.			−1.28004			−0.640018	+	0.768360i	$0.721073\pi$
$578$		−	241311.i		−	0.722305i
$579$	−291040.	−	151421.i	−0.868150	−	0.451678i
$580$	9756.13			0.0290016
$581$		−	232250.i		−	0.688023i
$582$	−176598.	+	339431.i	−0.521362	+	1.00209i
$583$	46066.7			0.135535
$584$			109376.i			0.320697i
$585$	−47241.4	+	33110.9i	−0.138042	+	0.0967520i
$586$	−17212.4			−0.0501241
$587$			175893.i			0.510472i	0.966879	+	0.255236i	$0.0821532\pi$
$587$			175893.i			0.510472i	−0.966879	+	0.255236i	$0.917847\pi$
$588$	54009.1	+	28099.7i	0.156211	+	0.0812731i
$589$	322523.			0.929673
$590$		−	162793.i		−	0.467663i
$591$	−263665.	+	506779.i	−0.754880	+	1.45092i
$592$	329090.			0.939013
$593$		−	338113.i		−	0.961506i	−0.876856	−	0.480753i	$-0.840363\pi$
$593$		−	338113.i		−	0.961506i	0.876856	−	0.480753i	$-0.159637\pi$
$594$	−47531.6	−	359135.i	−0.134713	−	1.01785i
$595$	−120636.			−0.340756
$596$		−	150544.i		−	0.423810i
$597$	76918.9	+	40019.1i	0.215816	+	0.112284i
$598$	−302491.			−0.845883
$599$			184189.i			0.513345i	0.966498	+	0.256673i	$0.0826262\pi$
$599$			184189.i			0.513345i	−0.966498	+	0.256673i	$0.917374\pi$
$600$	26433.8	−	50807.1i	0.0734271	−	0.141131i
$601$	459733.			1.27279			0.636394	−	0.771364i	$-0.280425\pi$
$601$	459733.			1.27279			0.636394	+	0.771364i	$0.280425\pi$
$602$			853221.i			2.35434i
$603$	196144.	+	279851.i	0.539437	+	0.769647i
$604$	−63537.2			−0.174162
$605$		−	31280.2i		−	0.0854591i
$606$	373320.	+	194230.i	1.01657	+	0.528896i
$607$	−396870.			−1.07714			−0.538568	−	0.842582i	$-0.681035\pi$
$607$	−396870.			−1.07714			−0.538568	+	0.842582i	$0.681035\pi$
$608$			181460.i			0.490878i
$609$	−46031.3	+	88474.6i	−0.124113	+	0.238553i
$610$	117491.			0.315750
$611$		−	59606.7i		−	0.159666i
$612$	−56358.6	+	39501.1i	−0.150473	+	0.105465i
$613$	−112592.			−0.299631			−0.149815	−	0.988714i	$-0.547868\pi$
$613$	−112592.			−0.299631			−0.149815	+	0.988714i	$0.547868\pi$
$614$			643445.i			1.70677i
$615$	15511.3	+	8070.17i	0.0410108	+	0.0213370i
$616$	341315.			0.899485
$617$		−	705193.i		−	1.85241i	−0.377019	−	0.926206i	$-0.623051\pi$
$617$		−	705193.i		−	1.85241i	0.377019	−	0.926206i	$-0.376949\pi$
$618$	102431.	−	196877.i	0.268196	−	0.515487i
$619$	600978.			1.56847			0.784237	−	0.620461i	$-0.213055\pi$
$619$	600978.			1.56847			0.784237	+	0.620461i	$0.213055\pi$
$620$			57978.0i			0.150827i
$621$	751530.	−	99465.1i	1.94878	−	0.257922i
$622$	−82278.2			−0.212669
$623$		−	508914.i		−	1.31120i
$624$	−157711.	−	82053.2i	−0.405035	−	0.210730i
$625$	15625.0			0.0400000
$626$			201633.i			0.514533i
$627$	136396.	−	262160.i	0.346949	−	0.666855i
$628$	−61172.8			−0.155110
$629$			185878.i			0.469814i
$630$	−146221.	−	208622.i	−0.368408	−	0.525629i
$631$	173024.			0.434558			0.217279	−	0.976110i	$-0.430282\pi$
$631$	173024.			0.434558			0.217279	+	0.976110i	$0.430282\pi$
$632$		−	158133.i		−	0.395901i
$633$	−513453.	−	267137.i	−1.28142	−	0.666695i
$634$	−885290.			−2.20246
$635$			172786.i			0.428510i
$636$	8530.32	−	16395.7i	0.0210888	−	0.0405338i
$637$	−88827.0			−0.218910
$638$		−	89385.7i		−	0.219597i
$639$	31524.1	−	22094.9i	0.0772042	−	0.0541116i
$640$	220673.			0.538752
$641$			398714.i			0.970388i	0.874407	+	0.485194i	$0.161251\pi$
$641$			398714.i			0.970388i	−0.874407	+	0.485194i	$0.838749\pi$
$642$	247407.	+	128720.i	0.600264	+	0.312303i
$643$	−476538.			−1.15259			−0.576296	−	0.817241i	$-0.695502\pi$
$643$	−476538.			−1.15259			−0.576296	+	0.817241i	$0.695502\pi$
$644$		−	310795.i		−	0.749381i
$645$	140857.	−	270735.i	0.338578	−	0.650766i
$646$	−241306.			−0.578234
$647$			115282.i			0.275392i	0.990475	+	0.137696i	$0.0439697\pi$
$647$			115282.i			0.275392i	−0.990475	+	0.137696i	$0.956030\pi$
$648$	313973.	+	113952.i	0.747725	+	0.271376i
$649$	−347017.			−0.823875
$650$		−	36360.8i		−	0.0860610i
$651$	−525780.	−	273551.i	−1.24063	−	0.645470i
$652$	−7620.26			−0.0179256
$653$			128829.i			0.302124i	0.988524	+	0.151062i	$0.0482694\pi$
$653$			128829.i			0.302124i	−0.988524	+	0.151062i	$0.951731\pi$
$654$	87255.4	−	167710.i	0.204003	−	0.392105i
$655$	−244212.			−0.569227
$656$			53883.0i			0.125212i
$657$	−99882.4	−	142508.i	−0.231397	−	0.330148i
$658$	263229.			0.607969
$659$		−	273000.i		−	0.628626i	−0.949319	−	0.314313i	$-0.898226\pi$
$659$		−	273000.i		−	0.628626i	0.949319	−	0.314313i	$-0.101774\pi$
$660$	47126.9	+	24519.0i	0.108188	+	0.0562879i
$661$	−727370.			−1.66476			−0.832382	−	0.554202i	$-0.813023\pi$
$661$	−727370.			−1.66476			−0.832382	+	0.554202i	$0.813023\pi$
$662$			169037.i			0.385714i
$663$	46345.6	−	89078.7i	0.105434	−	0.202650i
$664$	−191919.			−0.435293
$665$		−	207822.i		−	0.469947i
$666$	−321448.	+	225299.i	−0.724706	+	0.507939i
$667$	187049.			0.420441
$668$			47868.5i			0.107274i
$669$	413327.	+	215044.i	0.923509	+	0.480480i
$670$	−215396.			−0.479830
$671$		−	250448.i		−	0.556252i
$672$	153907.	−	295817.i	0.340816	−	0.655066i
$673$	589602.			1.30175			0.650877	−	0.759183i	$-0.274401\pi$
$673$	589602.			1.30175			0.650877	+	0.759183i	$0.274401\pi$
$674$			487498.i			1.07313i
$675$	11956.1	+	90337.2i	0.0262412	+	0.198271i
$676$	118871.			0.260125
$677$		−	132672.i		−	0.289470i	−0.989470	−	0.144735i	$-0.953767\pi$
$677$		−	132672.i		−	0.289470i	0.989470	−	0.144735i	$-0.0462329\pi$
$678$	−537808.	−	279809.i	−1.16995	−	0.608698i
$679$	573575.			1.24409
$680$			99687.4i			0.215587i
$681$	289822.	−	557053.i	0.624937	−	1.20116i
$682$	531194.			1.14205
$683$			804075.i			1.72367i	0.507185	+	0.861837i	$0.330686\pi$
$683$			804075.i			1.72367i	−0.507185	+	0.861837i	$0.669314\pi$
$684$	−68049.4	−	97090.1i	−0.145449	−	0.207521i
$685$	211580.			0.450914
$686$			283173.i			0.601732i
$687$	18236.2	+	9487.88i	0.0386386	+	0.0201028i
$688$	940476.			1.98688
$689$			26965.5i			0.0568029i
$690$	−220530.	+	423871.i	−0.463202	+	0.890299i
$691$	−489997.			−1.02621			−0.513106	−	0.858325i	$-0.671505\pi$
$691$	−489997.			−1.02621			−0.513106	+	0.858325i	$0.671505\pi$
$692$			104721.i			0.218687i
$693$	−444707.	+	311690.i	−0.925993	+	0.649018i
$694$	291275.			0.604761
$695$			243697.i			0.504522i
$696$	73110.8	+	38037.8i	0.150926	+	0.0785230i
$697$	−30434.4			−0.0626468
$698$		−	580678.i		−	1.19186i
$699$	93314.7	−	179356.i	0.190983	−	0.367081i
$700$	37359.0			0.0762428
$701$			68039.2i			0.138460i	0.997601	+	0.0692298i	$0.0220542\pi$
$701$			68039.2i			0.138460i	−0.997601	+	0.0692298i	$0.977946\pi$
$702$	210223.	−	27823.0i	0.426585	−	0.0564586i
$703$	−320215.			−0.647935
$704$		−	241065.i		−	0.486395i
$705$	−83524.9	−	43456.1i	−0.168050	−	0.0874324i
$706$	−307091.			−0.616110
$707$		−	630842.i		−	1.26206i
$708$	−64258.3	+	123508.i	−0.128192	+	0.246393i
$709$	49483.1			0.0984384			0.0492192	−	0.998788i	$-0.484327\pi$
$709$	49483.1			0.0984384			0.0492192	+	0.998788i	$0.484327\pi$
$710$			24263.5i			0.0481323i
$711$	144407.	+	206035.i	0.285661	+	0.407569i
$712$	−420540.			−0.829559
$713$			1.11158e6i			2.18657i
$714$	393380.	+	204666.i	0.771642	+	0.401467i
$715$	−77508.0			−0.151612
$716$		−	107582.i		−	0.209852i
$717$	−174947.	+	336257.i	−0.340304	+	0.654084i
$718$	656761.			1.27397
$719$		−	772948.i		−	1.49518i	−0.664163	−	0.747588i	$-0.731212\pi$
$719$		−	772948.i		−	1.49518i	0.664163	−	0.747588i	$-0.268788\pi$
$720$	−229957.	+	161174.i	−0.443590	+	0.310908i
$721$	−332686.			−0.639976
$722$			179384.i			0.344119i
$723$	7791.80	+	4053.89i	0.0149060	+	0.00775525i
$724$	276722.			0.527919
$725$			22484.2i			0.0427761i
$726$	−53068.6	+	102001.i	−0.100685	+	0.193522i
$727$	475534.			0.899730			0.449865	−	0.893096i	$-0.351472\pi$
$727$	475534.			0.899730			0.449865	+	0.893096i	$0.351472\pi$
$728$			199792.i			0.376977i
$729$	−513143.	+	138251.i	−0.965570	+	0.260143i
$730$	109686.			0.205828
$731$			531203.i			0.994090i
$732$	−89137.6	−	46376.2i	−0.166356	−	0.0865512i
$733$	78220.7			0.145584			0.0727920	−	0.997347i	$-0.476809\pi$
$733$	78220.7			0.145584			0.0727920	+	0.997347i	$0.476809\pi$
$734$		−	558966.i		−	1.03751i
$735$	−64759.0	+	124470.i	−0.119874	+	0.230405i
$736$	−625405.			−1.15453
$737$			459146.i			0.845310i
$738$	−36889.0	−	52631.7i	−0.0677304	−	0.0966350i
$739$	−493941.			−0.904454			−0.452227	−	0.891903i	$-0.649370\pi$
$739$	−493941.			−0.904454			−0.452227	+	0.891903i	$0.649370\pi$
$740$		−	57563.2i		−	0.105119i
$741$	153458.	+	79840.4i	0.279481	+	0.145407i
$742$	−119082.			−0.216292
$743$		−	881286.i		−	1.59639i	−0.602398	−	0.798196i	$-0.705788\pi$
$743$		−	881286.i		−	1.59639i	0.602398	−	0.798196i	$-0.294212\pi$
$744$	−226048.	+	434477.i	−0.408371	+	0.784912i
$745$	346947.			0.625101
$746$		−	883676.i		−	1.58787i
$747$	250056.	−	175261.i	0.448122	−	0.314083i
$748$	−92466.6			−0.165265
$749$		−	418072.i		−	0.745226i
$750$	−50951.2	−	26508.7i	−0.0905799	−	0.0471266i
$751$	−455517.			−0.807653			−0.403827	−	0.914836i	$-0.632320\pi$
$751$	−455517.			−0.807653			−0.403827	+	0.914836i	$0.632320\pi$
$752$		−	290148.i		−	0.513079i
$753$	396427.	−	761954.i	0.699154	−	1.34381i
$754$	52322.7			0.0920338
$755$		−	146429.i		−	0.256882i
$756$	28586.8	+	215994.i	0.0500175	+	0.377918i
$757$	−210339.			−0.367053			−0.183526	−	0.983015i	$-0.558751\pi$
$757$	−210339.			−0.367053			−0.183526	+	0.983015i	$0.558751\pi$
$758$		−	228360.i		−	0.397449i
$759$	903541.	+	470091.i	1.56843	+	0.816016i
$760$	−171734.			−0.297323
$761$			299076.i			0.516431i	0.966087	+	0.258216i	$0.0831345\pi$
$761$			299076.i			0.516431i	−0.966087	+	0.258216i	$0.916866\pi$
$762$	293142.	−	563434.i	0.504856	−	0.970361i
$763$	−283399.			−0.486798
$764$			311891.i			0.534338i
$765$	−91035.0	−	129885.i	−0.155556	−	0.221940i
$766$	329282.			0.561191
$767$		−	203129.i		−	0.345288i
$768$	−436616.	−	227161.i	−0.740247	−	0.385133i
$769$	215663.			0.364689			0.182345	−	0.983235i	$-0.441631\pi$
$769$	215663.			0.364689			0.182345	+	0.983235i	$0.441631\pi$
$770$		−	342283.i		−	0.577303i
$771$	96013.8	−	184544.i	0.161519	−	0.310449i
$772$	−176842.			−0.296722
$773$			233916.i			0.391473i	0.980657	+	0.195736i	$0.0627097\pi$
$773$			233916.i			0.391473i	−0.980657	+	0.195736i	$0.937290\pi$
$774$	−918635.	+	643861.i	−1.53342	+	1.07476i
$775$	−133617.			−0.222463
$776$		−	473972.i		−	0.787100i
$777$	522018.	+	271594.i	0.864656	+	0.449860i
$778$	−115453.			−0.190741
$779$		−	52429.9i		−	0.0863981i
$780$	−14352.4	+	27586.1i	−0.0235904	+	0.0453421i
$781$	51721.0			0.0847940
$782$		−	831668.i		−	1.35999i
$783$	−129994.	+	17204.7i	−0.212031	+	0.0280624i
$784$	−432384.			−0.703457
$785$		−	140980.i		−	0.228780i
$786$	796347.	+	414320.i	1.28901	+	0.670643i
$787$	381652.			0.616194			0.308097	−	0.951355i	$-0.400308\pi$
$787$	381652.			0.616194			0.308097	+	0.951355i	$0.400308\pi$
$788$			307930.i			0.495906i
$789$	154383.	−	296732.i	0.247996	−	0.476662i
$790$	−158581.			−0.254095
$791$			908797.i			1.45249i
$792$	257565.	+	367483.i	0.410616	+	0.585850i
$793$	146602.			0.233127
$794$			128360.i			0.203604i
$795$	37785.9	+	19659.1i	0.0597855	+	0.0311050i
$796$	46737.5			0.0737632
$797$			1.02716e6i			1.61705i	0.588464	+	0.808523i	$0.299733\pi$
$797$			1.02716e6i			1.61705i	−0.588464	+	0.808523i	$0.700267\pi$
$798$	−352583.	+	677684.i	−0.553676	+	1.06419i
$799$	163882.			0.256707
$800$		−	75176.4i		−	0.117463i
$801$	547932.	−	384039.i	0.854007	−	0.598564i
$802$	−265366.			−0.412569
$803$		−	233811.i		−	0.362604i
$804$	163416.	+	85021.6i	0.252803	+	0.131528i
$805$	716265.			1.10530
$806$			310939.i			0.478636i
$807$	−558687.	+	1.07383e6i	−0.857869	+	1.64887i
$808$	−521294.			−0.798473
$809$			674223.i			1.03016i	0.857141	+	0.515082i	$0.172239\pi$
$809$			674223.i			1.03016i	−0.857141	+	0.515082i	$0.827761\pi$
$810$	114275.	−	314863.i	0.174173	−	0.479901i
$811$	591215.			0.898884			0.449442	−	0.893310i	$-0.351623\pi$
$811$	591215.			0.898884			0.449442	+	0.893310i	$0.351623\pi$
$812$			53759.1i			0.0815342i
$813$	−892584.	−	464391.i	−1.35042	−	0.702591i
$814$	−527393.			−0.795950
$815$		−	17561.8i		−	0.0264395i
$816$	225597.	−	433609.i	0.338807	−	0.651206i
$817$	−915114.			−1.37098
$818$			38866.5i			0.0580856i
$819$	−182451.	−	260313.i	−0.272005	−	0.388086i
$820$	9425.00			0.0140170
$821$			731133.i			1.08470i	0.840152	+	0.542351i	$0.182466\pi$
$821$			731133.i			1.08470i	−0.840152	+	0.542351i	$0.817534\pi$
$822$	−689936.	−	358958.i	−1.02109	−	0.531251i
$823$	490743.			0.724527			0.362264	−	0.932076i	$-0.382004\pi$
$823$	490743.			0.724527			0.362264	+	0.932076i	$0.382004\pi$
$824$			274914.i			0.404895i
$825$	−56507.0	+	108610.i	−0.0830222	+	0.159573i
$826$	897038.			1.31477
$827$			48569.2i			0.0710150i	0.999369	+	0.0355075i	$0.0113048\pi$
$827$			48569.2i			0.0710150i	−0.999369	+	0.0355075i	$0.988695\pi$
$828$	334623.	−	234534.i	0.488085	−	0.342093i
$829$	481266.			0.700287			0.350144	−	0.936696i	$-0.386133\pi$
$829$	481266.			0.700287			0.350144	+	0.936696i	$0.386133\pi$
$830$			192463.i			0.279378i
$831$	−1.02517e6	−	533371.i	−1.48454	−	0.772374i
$832$	141110.			0.203850
$833$		−	244220.i		−	0.351959i
$834$	413446.	−	794665.i	0.594410	−	1.14249i
$835$	−110319.			−0.158225
$836$		−	159294.i		−	0.227922i
$837$	−102243.	−	772518.i	−0.145943	−	1.10270i
$838$	1.33667e6			1.90343
$839$		−	654006.i		−	0.929091i	−0.885549	−	0.464545i	$-0.846218\pi$
$839$		−	654006.i		−	0.929091i	0.885549	−	0.464545i	$-0.153782\pi$
$840$	279962.	+	145657.i	0.396771	+	0.206431i
$841$	674927.			0.954255
$842$			694990.i			0.980290i
$843$	−122048.	+	234584.i	−0.171742	+	0.330098i
$844$	−311985.			−0.437974
$845$			273952.i			0.383672i
$846$	198639.	+	283410.i	0.277539	+	0.395981i
$847$	172363.			0.240257
$848$			131260.i			0.182533i
$849$	−479963.	−	249713.i	−0.665874	−	0.346439i
$850$	99970.1			0.138367
$851$		−	1.10363e6i		−	1.52393i
$852$	9577.35	−	18408.2i	0.0131937	−	0.0253590i
$853$	39930.2			0.0548786			0.0274393	−	0.999623i	$-0.491265\pi$
$853$	39930.2			0.0548786			0.0274393	+	0.999623i	$0.491265\pi$
$854$			647407.i			0.887690i
$855$	223756.	−	156828.i	0.306085	−	0.214532i
$856$	−345473.			−0.471484
$857$			558770.i			0.760801i	0.924822	+	0.380401i	$0.124214\pi$
$857$			558770.i			0.760801i	−0.924822	+	0.380401i	$0.875786\pi$
$858$	252744.	+	131497.i	0.343326	+	0.178624i
$859$	−816282.			−1.10625			−0.553126	−	0.833098i	$-0.686565\pi$
$859$	−816282.			−1.10625			−0.553126	+	0.833098i	$0.686565\pi$
$860$		−	164504.i		−	0.222423i
$861$	−44468.9	+	85471.7i	−0.0599861	+	0.115297i
$862$	−1.15441e6			−1.55362
$863$		−	525248.i		−	0.705250i	−0.935765	−	0.352625i	$-0.885289\pi$
$863$		−	525248.i		−	0.705250i	0.935765	−	0.352625i	$-0.114711\pi$
$864$	434639.	−	57524.5i	0.582239	−	0.0770593i
$865$	−241342.			−0.322553
$866$		−	1.50832e6i		−	2.01121i
$867$	−421923.	−	219517.i	−0.561300	−	0.292031i
$868$	−319475.			−0.424031
$869$			338037.i			0.447636i
$870$	38145.7	−	73318.1i	0.0503973	−	0.0968663i
$871$	−268765.			−0.354272
$872$			234186.i			0.307983i
$873$	432834.	+	617550.i	0.567927	+	0.810296i
$874$	1.43273e6			1.87561
$875$			86098.2i			0.112455i
$876$	−83216.3	−	43295.5i	−0.108443	−	0.0564202i
$877$	−53533.8			−0.0696031			−0.0348015	−	0.999394i	$-0.511080\pi$
$877$	−53533.8			−0.0696031			−0.0348015	+	0.999394i	$0.511080\pi$
$878$			1.04763e6i			1.35900i
$879$	−15657.9	+	30095.3i	−0.0202654	+	0.0389512i
$880$	−377287.			−0.487199
$881$		−	1.15608e6i		−	1.48949i	−0.667349	−	0.744746i	$-0.732571\pi$
$881$		−	1.15608e6i		−	1.48949i	0.667349	−	0.744746i	$-0.267429\pi$
$882$	422343.	−	296015.i	0.542910	−	0.380519i
$883$	−1.24115e6			−1.59186			−0.795929	−	0.605390i	$-0.793017\pi$
$883$	−1.24115e6			−1.59186			−0.795929	+	0.605390i	$0.793017\pi$
$884$		−	54126.1i		−	0.0692631i
$885$	−284639.	−	148091.i	−0.363419	−	0.189078i
$886$	28081.2			0.0357724
$887$		−	747316.i		−	0.949855i	−0.880025	−	0.474927i	$-0.842474\pi$
$887$		−	747316.i		−	0.949855i	0.880025	−	0.474927i	$-0.157526\pi$
$888$	224431.	−	431368.i	0.284614	−	0.547044i
$889$	−952100.			−1.20470
$890$			421733.i			0.532423i
$891$	−671174.	−	243593.i	−0.845434	−	0.306838i
$892$	251146.			0.315643
$893$			282323.i			0.354033i
$894$	−1.13135e6	−	588615.i	−1.41554	−	0.736472i
$895$	247936.			0.309523
$896$			1.21597e6i			1.51463i
$897$	−275172.	+	528895.i	−0.341994	+	0.657332i
$898$	−1.00255e6			−1.24324
$899$		−	192273.i		−	0.237903i
$900$	28192.0	+	40223.2i	0.0348049	+	0.0496583i
$901$	−74138.9			−0.0913264
$902$		−	86351.9i		−	0.106135i
$903$	1.49183e6	+	776162.i	1.82954	+	0.951869i
$904$	750982.			0.918951
$905$			637739.i			0.778657i
$906$	−248425.	+	477487.i	−0.302649	+	0.581708i
$907$	952017.			1.15726			0.578629	−	0.815591i	$-0.303588\pi$
$907$	952017.			1.15726			0.578629	+	0.815591i	$0.303588\pi$
$908$		−	338477.i		−	0.410542i
$909$	679207.	−	476049.i	0.822005	−	0.576134i
$910$	200358.			0.241949
$911$			1824.09i			0.00219790i	0.999999	+	0.00109895i	$0.000349808\pi$
$911$			1824.09i			0.00219790i	−0.999999	+	0.00109895i	$0.999650\pi$
$912$	746987.	+	388640.i	0.898098	+	0.467259i
$913$	410262.			0.492175
$914$			633248.i			0.758021i
$915$	106880.	−	205428.i	0.127659	−	0.245368i
$916$	11080.7			0.0132062
$917$		−	1.34568e6i		−	1.60031i
$918$	76496.5	+	577986.i	0.0907728	+	0.685854i
$919$	−852396.			−1.00928			−0.504638	−	0.863331i	$-0.668374\pi$
$919$	−852396.			−1.00928			−0.504638	+	0.863331i	$0.668374\pi$
$920$		−	591884.i		−	0.699295i
$921$	1.12504e6	+	585332.i	1.32632	+	0.690054i
$922$	17773.5			0.0209079
$923$			30275.3i			0.0355374i
$924$	−135107.	+	259683.i	−0.158246	+	0.304158i
$925$	132661.			0.155046
$926$			1.27699e6i			1.48924i
$927$	−251053.	−	358192.i	−0.292150	−	0.416828i
$928$	108178.			0.125615
$929$			1.31428e6i			1.52284i	0.648258	+	0.761421i	$0.275498\pi$
$929$			1.31428e6i			1.52284i	−0.648258	+	0.761421i	$0.724502\pi$
$930$	435709.	+	226689.i	0.503768	+	0.262099i
$931$	420723.			0.485397
$932$		−	108980.i		−	0.125463i
$933$	−74847.3	+	143861.i	−0.0859830	+	0.165264i
$934$	926604.			1.06219
$935$		−	213100.i		−	0.243759i
$936$	−215109.	+	150768.i	−0.245531	+	0.172090i
$937$	1.57670e6			1.79585			0.897926	−	0.440146i	$-0.145073\pi$
$937$	1.57670e6			1.79585			0.897926	+	0.440146i	$0.145073\pi$
$938$		−	1.18689e6i		−	1.34898i
$939$	352548.	+	183423.i	0.399841	+	0.208028i
$940$	−50751.5			−0.0574372
$941$		−	603453.i		−	0.681498i	−0.940154	−	0.340749i	$-0.889319\pi$
$941$		−	603453.i		−	0.681498i	0.940154	−	0.340749i	$-0.110681\pi$
$942$	−239181.	+	459718.i	−0.269540	+	0.518072i
$943$	180701.			0.203206
$944$		−	988774.i		−	1.10957i
$945$	−497784.	+	65881.7i	−0.557413	+	0.0737737i
$946$	−1.50719e6			−1.68417
$947$		−	267321.i		−	0.298080i	−0.988831	−	0.149040i	$-0.952382\pi$
$947$		−	267321.i		−	0.298080i	0.988831	−	0.149040i	$-0.0476183\pi$
$948$	120312.	+	62595.5i	0.133873	+	0.0696509i
$949$	136863.			0.151969
$950$			172221.i			0.190826i
$951$	−805335.	+	1.54790e6i	−0.890463	+	1.71152i
$952$	−549306.			−0.606095
$953$		−	1.44335e6i		−	1.58923i	−0.607117	−	0.794613i	$-0.707674\pi$
$953$		−	1.44335e6i		−	1.58923i	0.607117	−	0.794613i	$-0.292326\pi$
$954$	−89862.4	−	128212.i	−0.0987374	−	0.140874i
$955$	−718790.			−0.788125
$956$			204317.i			0.223557i
$957$	−156288.	−	81312.8i	−0.170648	−	0.0887841i
$958$	−1.40859e6			−1.53481
$959$			1.16587e6i			1.26768i
$960$	102875.	−	197732.i	0.111627	−	0.214553i
$961$	219105.			0.237250
$962$		−	308714.i		−	0.333585i
$963$	450125.	−	315488.i	0.485379	−	0.340196i
$964$	4734.46			0.00509468
$965$		−	407553.i		−	0.437652i
$966$	−2.33565e6	−	1.21518e6i	−2.50296	−	1.30223i
$967$	890704.			0.952534			0.476267	−	0.879301i	$-0.341990\pi$
$967$	890704.			0.952534			0.476267	+	0.879301i	$0.341990\pi$
$968$		−	142431.i		−	0.152004i
$969$	−219513.	+	421916.i	−0.233783	+	0.449343i
$970$	−475317.			−0.505172
$971$			1.14658e6i			1.21609i	0.793904	+	0.608044i	$0.208045\pi$
$971$			1.14658e6i			1.21609i	−0.793904	+	0.608044i	$0.791955\pi$
$972$	−210981.	+	193773.i	−0.223312	+	0.205097i
$973$	−1.34284e6			−1.41840
$974$		−	961297.i		−	1.01330i
$975$	−63575.5	−	33076.9i	−0.0668776	−	0.0347949i
$976$	713614.			0.749142
$977$			352274.i			0.369055i	0.982827	+	0.184527i	$0.0590755\pi$
$977$			352274.i			0.369055i	−0.982827	+	0.184527i	$0.940925\pi$
$978$	−29794.6	+	57266.8i	−0.0311501	+	0.0598722i
$979$	898981.			0.937962
$980$			75630.9i			0.0787493i
$981$	−213860.	−	305126.i	−0.222224	−	0.317060i
$982$	2.02085e6			2.09561
$983$			794769.i			0.822496i	0.911524	+	0.411248i	$0.134907\pi$
$983$			794769.i			0.822496i	−0.911524	+	0.411248i	$0.865093\pi$
$984$	70629.3	+	36746.8i	0.0729449	+	0.0379515i
$985$	−709660.			−0.731439
$986$			143856.i			0.147970i
$987$	239455.	−	460246.i	0.245805	−	0.472450i
$988$	93244.1			0.0955229
$989$		−	3.15396e6i		−	3.22451i
$990$	368525.	−	258295.i	0.376007	−	0.263539i
$991$	903807.			0.920298			0.460149	−	0.887842i	$-0.347796\pi$
$991$	903807.			0.920298			0.460149	+	0.887842i	$0.347796\pi$
$992$			642871.i			0.653282i
$993$	295555.	+	153770.i	0.299737	+	0.155946i
$994$	−133699.			−0.135318
$995$			107712.i			0.108797i
$996$	75969.6	−	146018.i	0.0765810	−	0.147193i
$997$	−1.64889e6			−1.65883			−0.829413	−	0.558636i	$-0.811325\pi$
$997$	−1.64889e6			−1.65883			−0.829413	+	0.558636i	$0.811325\pi$
$998$		−	776838.i		−	0.779955i
$999$	101511.	+	766991.i	0.101715	+	0.768527i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.5.c.a.11.2	✓	6
3.2	odd	2	inner	15.5.c.a.11.5	yes	6
4.3	odd	2		240.5.l.d.161.6		6
5.2	odd	4		75.5.d.d.74.10		12
5.3	odd	4		75.5.d.d.74.3		12
5.4	even	2		75.5.c.i.26.5		6
12.11	even	2		240.5.l.d.161.5		6
15.2	even	4		75.5.d.d.74.4		12
15.8	even	4		75.5.d.d.74.9		12
15.14	odd	2		75.5.c.i.26.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.c.a.11.2	✓	6	1.1	even	1	trivial
15.5.c.a.11.5	yes	6	3.2	odd	2	inner
75.5.c.i.26.2		6	15.14	odd	2
75.5.c.i.26.5		6	5.4	even	2
75.5.d.d.74.3		12	5.3	odd	4
75.5.d.d.74.4		12	15.2	even	4
75.5.d.d.74.9		12	15.8	even	4
75.5.d.d.74.10		12	5.2	odd	4
240.5.l.d.161.5		6	12.11	even	2
240.5.l.d.161.6		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	15.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.56632i q^{2} +(-7.98405 - 4.15391i) q^{3} -4.85128 q^{4} -11.1803i q^{5} +(-18.9681 + 36.4577i) q^{6} +61.6068 q^{7} -50.9086i q^{8} +(46.4900 + 66.3301i) q^{9} +O(q^{10})\)	\(q-4.56632i q^{2} +(-7.98405 - 4.15391i) q^{3} -4.85128 q^{4} -11.1803i q^{5} +(-18.9681 + 36.4577i) q^{6} +61.6068 q^{7} -50.9086i q^{8} +(46.4900 + 66.3301i) q^{9} -51.0530 q^{10} +108.827i q^{11} +(38.7328 + 20.1518i) q^{12} -63.7026 q^{13} -281.317i q^{14} +(-46.4422 + 89.2644i) q^{15} -310.086 q^{16} -175.143i q^{17} +(302.884 - 212.288i) q^{18} +301.723 q^{19} +54.2389i q^{20} +(-491.872 - 255.909i) q^{21} +496.937 q^{22} +1039.90i q^{23} +(-211.470 + 406.457i) q^{24} -125.000 q^{25} +290.886i q^{26} +(-95.6491 - 722.698i) q^{27} -298.872 q^{28} -179.873i q^{29} +(407.610 + 212.070i) q^{30} +1068.94 q^{31} +601.411i q^{32} +(452.056 - 868.876i) q^{33} -799.761 q^{34} -688.785i q^{35} +(-225.536 - 321.785i) q^{36} -1061.29 q^{37} -1377.76i q^{38} +(508.604 + 264.615i) q^{39} -569.176 q^{40} -173.768i q^{41} +(-1168.56 + 2246.04i) q^{42} -3032.96 q^{43} -527.948i q^{44} +(741.593 - 519.774i) q^{45} +4748.50 q^{46} +935.703i q^{47} +(2475.74 + 1288.07i) q^{48} +1394.40 q^{49} +570.790i q^{50} +(-727.531 + 1398.35i) q^{51} +309.039 q^{52} -423.304i q^{53} +(-3300.07 + 436.765i) q^{54} +1216.72 q^{55} -3136.32i q^{56} +(-2408.97 - 1253.33i) q^{57} -821.359 q^{58} +3188.72i q^{59} +(225.304 - 433.046i) q^{60} -2301.35 q^{61} -4881.11i q^{62} +(2864.10 + 4086.39i) q^{63} -2215.13 q^{64} +712.216i q^{65} +(-3967.57 - 2064.23i) q^{66} +4219.06 q^{67} +849.669i q^{68} +(4319.64 - 8302.58i) q^{69} -3145.21 q^{70} -475.261i q^{71} +(3376.77 - 2366.74i) q^{72} -2148.47 q^{73} +4846.18i q^{74} +(998.006 + 519.239i) q^{75} -1463.74 q^{76} +6704.46i q^{77} +(1208.32 - 2322.45i) q^{78} +3106.20 q^{79} +3466.86i q^{80} +(-2238.36 + 6167.37i) q^{81} -793.482 q^{82} -3769.87i q^{83} +(2386.21 + 1241.49i) q^{84} -1958.16 q^{85} +13849.5i q^{86} +(-747.178 + 1436.12i) q^{87} +5540.21 q^{88} -8260.68i q^{89} +(-2373.46 - 3386.35i) q^{90} -3924.51 q^{91} -5044.82i q^{92} +(-8534.44 - 4440.27i) q^{93} +4272.72 q^{94} -3373.37i q^{95} +(2498.21 - 4801.70i) q^{96} +9310.26 q^{97} -6367.29i q^{98} +(-7218.47 + 5059.35i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9}+O(q^{10}) \) 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9	\( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27} - 3764 q^{28} + 2200 q^{30} + 3772 q^{31} + 4420 q^{33} + 3124 q^{34} - 7606 q^{36} + 1896 q^{37} - 1336 q^{39} - 4650 q^{40} - 1980 q^{42} - 7384 q^{43} + 1900 q^{45} + 8196 q^{46} + 14668 q^{48} - 1318 q^{49} - 8492 q^{51} + 8976 q^{52} - 278 q^{54} - 1300 q^{55} - 11584 q^{57} - 23740 q^{58} + 5050 q^{60} + 6452 q^{61} + 14796 q^{63} + 3174 q^{64} - 12760 q^{66} + 13816 q^{67} + 5472 q^{69} - 2100 q^{70} - 2040 q^{72} + 596 q^{73} - 1000 q^{75} + 21348 q^{76} - 1400 q^{78} - 16124 q^{79} + 5086 q^{81} - 31240 q^{82} - 14736 q^{84} - 3100 q^{85} - 4900 q^{87} + 15660 q^{88} + 7550 q^{90} - 11632 q^{91} - 8184 q^{93} + 34924 q^{94} + 14354 q^{96} + 9756 q^{97} + 9680 q^{99}+O(q^{100}) \) 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * q^99

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