LMFDB - Embedded newform 76.5.j.a.21.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,5,Mod(13,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("76.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	76.j (of order $18$, degree $6$, 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:	$7.85611719437$
Analytic rank:	$0$
Dimension:	$42$
Relative dimension:	$7$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.1
Character	$\chi$	$=$	76.21
Dual form			76.5.j.a.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.26669 + 9.85185i) q^{3} +(-19.2135 + 6.99315i) q^{5} +(1.56085 + 2.70347i) q^{7} +(-14.6554 - 83.1149i) q^{9} +O(q^{10})$	\(q+(-8.26669 + 9.85185i) q^{3} +(-19.2135 + 6.99315i) q^{5} +(1.56085 + 2.70347i) q^{7} +(-14.6554 - 83.1149i) q^{9} +(57.2296 - 99.1246i) q^{11} +(-73.1984 - 87.2344i) q^{13} +(89.9367 - 247.099i) q^{15} +(11.6409 - 66.0190i) q^{17} +(113.685 - 342.632i) q^{19} +(-39.5372 - 6.97148i) q^{21} +(-229.764 - 83.6274i) q^{23} +(-158.522 + 133.016i) q^{25} +(37.8358 + 21.8445i) q^{27} +(-1363.50 + 240.421i) q^{29} +(708.058 - 408.797i) q^{31} +(503.462 + 1383.25i) q^{33} +(-48.8952 - 41.0279i) q^{35} +1099.85i q^{37} +1464.53 q^{39} +(-51.0022 + 60.7821i) q^{41} +(-317.379 + 115.516i) q^{43} +(862.817 + 1494.44i) q^{45} +(-275.570 - 1562.84i) q^{47} +(1195.63 - 2070.89i) q^{49} +(554.178 + 660.443i) q^{51} +(-876.707 + 2408.73i) q^{53} +(-406.389 + 2304.75i) q^{55} +(2435.76 + 3952.44i) q^{57} +(-4803.44 - 846.977i) q^{59} +(-1504.98 - 547.766i) q^{61} +(201.824 - 169.350i) q^{63} +(2016.44 + 1164.19i) q^{65} +(-4753.54 + 838.177i) q^{67} +(2723.28 - 1572.28i) q^{69} +(183.202 + 503.344i) q^{71} +(-5428.81 - 4555.31i) q^{73} -2661.34i q^{75} +357.307 q^{77} +(7499.16 - 8937.15i) q^{79} +(5895.91 - 2145.94i) q^{81} +(-3230.90 - 5596.08i) q^{83} +(238.018 + 1349.86i) q^{85} +(8903.00 - 15420.5i) q^{87} +(-6885.48 - 8205.80i) q^{89} +(121.584 - 334.049i) q^{91} +(-1825.88 + 10355.1i) q^{93} +(211.792 + 7378.19i) q^{95} +(-4876.88 - 859.926i) q^{97} +(-9077.46 - 3303.92i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{18}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−8.26669	+	9.85185i	−0.918521	+	1.09465i	0.0767054	+	0.997054i	$0.475560\pi$
$3$	−8.26669	+	9.85185i	−0.918521	+	1.09465i	−0.995226	+	0.0975965i	$0.968885\pi$
$4$		0			0
$5$	−19.2135	+	6.99315i	−0.768541	+	0.279726i	−0.696386	−	0.717667i	$-0.745210\pi$
$5$	−19.2135	+	6.99315i	−0.768541	+	0.279726i	−0.0721550	+	0.997393i	$0.522988\pi$
$6$		0			0
$7$	1.56085	+	2.70347i	0.0318540	+	0.0551728i	0.881513	−	0.472160i	$-0.156525\pi$
$7$	1.56085	+	2.70347i	0.0318540	+	0.0551728i	−0.849659	+	0.527333i	$0.823192\pi$
$8$		0			0
$9$	−14.6554	−	83.1149i	−0.180931	−	1.02611i
$10$		0			0
$11$	57.2296	−	99.1246i	0.472972	−	0.819212i	−0.526549	−	0.850145i	$-0.676514\pi$
$11$	57.2296	−	99.1246i	0.472972	−	0.819212i	0.999521	+	0.0309329i	$0.00984782\pi$
$12$		0			0
$13$	−73.1984	−	87.2344i	−0.433127	−	0.516180i	0.504695	−	0.863298i	$-0.331605\pi$
$13$	−73.1984	−	87.2344i	−0.433127	−	0.516180i	−0.937822	+	0.347118i	$0.887160\pi$
$14$		0			0
$15$	89.9367	−	247.099i	0.399719	−	1.09822i
$16$		0			0
$17$	11.6409	−	66.0190i	0.0402800	−	0.228439i	−0.958022	−	0.286696i	$-0.907443\pi$
$17$	11.6409	−	66.0190i	0.0402800	−	0.228439i	0.998302	+	0.0582566i	$0.0185541\pi$
$18$		0			0
$19$	113.685	−	342.632i	0.314916	−	0.949119i
$19$	113.685	−	342.632i	0.314916	−	0.949119i
$20$		0			0
$21$	−39.5372	−	6.97148i	−0.0896536	−	0.0158083i
$22$		0			0
$23$	−229.764	−	83.6274i	−0.434337	−	0.158086i	0.115593	−	0.993297i	$-0.463123\pi$
$23$	−229.764	−	83.6274i	−0.434337	−	0.158086i	−0.549930	+	0.835211i	$0.685346\pi$
$24$		0			0
$25$	−158.522	+	133.016i	−0.253636	+	0.212826i
$26$		0			0
$27$	37.8358	+	21.8445i	0.0519010	+	0.0299651i
$28$		0			0
$29$	−1363.50	+	240.421i	−1.62128	+	0.285875i	−0.909243	−	0.416266i	$-0.863338\pi$
$29$	−1363.50	+	240.421i	−1.62128	+	0.285875i	−0.712037	+	0.702142i	$0.752227\pi$
$30$		0			0
$31$	708.058	−	408.797i	0.736793	−	0.425387i	−0.0841094	−	0.996457i	$-0.526805\pi$
$31$	708.058	−	408.797i	0.736793	−	0.425387i	0.820902	+	0.571069i	$0.193471\pi$
$32$		0			0
$33$	503.462	+	1383.25i	0.462316	+	1.27020i
$34$		0			0
$35$	−48.8952	−	41.0279i	−0.0399144	−	0.0334922i
$36$		0			0
$37$			1099.85i			0.803397i	0.915772	+	0.401699i	$0.131580\pi$
$37$			1099.85i			0.803397i	−0.915772	+	0.401699i	$0.868420\pi$
$38$		0			0
$39$	1464.53			0.962872
$40$		0			0
$41$	−51.0022	+	60.7821i	−0.0303404	+	0.0361583i	−0.781001	−	0.624529i	$-0.785291\pi$
$41$	−51.0022	+	60.7821i	−0.0303404	+	0.0361583i	0.750661	+	0.660688i	$0.229735\pi$
$42$		0			0
$43$	−317.379	+	115.516i	−0.171649	+	0.0624751i	−0.426415	−	0.904528i	$-0.640224\pi$
$43$	−317.379	+	115.516i	−0.171649	+	0.0624751i	0.254766	+	0.967003i	$0.418001\pi$
$44$		0			0
$45$	862.817	+	1494.44i	0.426083	+	0.737997i
$46$		0			0
$47$	−275.570	−	1562.84i	−0.124749	−	0.707486i	−0.981457	−	0.191684i	$-0.938605\pi$
$47$	−275.570	−	1562.84i	−0.124749	−	0.707486i	0.856708	−	0.515802i	$-0.172506\pi$
$48$		0			0
$49$	1195.63	−	2070.89i	0.497971	−	0.862510i
$50$		0			0
$51$	554.178	+	660.443i	0.213063	+	0.253919i
$52$		0			0
$53$	−876.707	+	2408.73i	−0.312106	+	0.857505i	0.680125	+	0.733097i	$0.261926\pi$
$53$	−876.707	+	2408.73i	−0.312106	+	0.857505i	−0.992231	+	0.124409i	$0.960297\pi$
$54$		0			0
$55$	−406.389	+	2304.75i	−0.134344	+	0.761900i
$56$		0			0
$57$	2435.76	+	3952.44i	0.749697	+	1.21651i
$58$		0			0
$59$	−4803.44	−	846.977i	−1.37990	−	0.243314i	−0.566044	−	0.824375i	$-0.691527\pi$
$59$	−4803.44	−	846.977i	−1.37990	−	0.243314i	−0.813860	+	0.581061i	$0.802638\pi$
$60$		0			0
$61$	−1504.98	−	547.766i	−0.404455	−	0.147209i	0.131779	−	0.991279i	$-0.457931\pi$
$61$	−1504.98	−	547.766i	−0.404455	−	0.147209i	−0.536234	+	0.844070i	$0.680153\pi$
$62$		0			0
$63$	201.824	−	169.350i	0.0508500	−	0.0426682i
$64$		0			0
$65$	2016.44	+	1164.19i	0.477264	+	0.275549i
$66$		0			0
$67$	−4753.54	+	838.177i	−1.05893	+	0.186718i	−0.675881	−	0.737010i	$-0.736237\pi$
$67$	−4753.54	+	838.177i	−1.05893	+	0.186718i	−0.383048	+	0.923728i	$0.625126\pi$
$68$		0			0
$69$	2723.28	−	1572.28i	0.571997	−	0.330242i
$70$		0			0
$71$	183.202	+	503.344i	0.0363425	+	0.0998501i	0.956537	−	0.291612i	$-0.0941916\pi$
$71$	183.202	+	503.344i	0.0363425	+	0.0998501i	−0.920194	+	0.391462i	$0.871969\pi$
$72$		0			0
$73$	−5428.81	−	4555.31i	−1.01873	−	0.854816i	−0.0292626	−	0.999572i	$-0.509316\pi$
$73$	−5428.81	−	4555.31i	−1.01873	−	0.854816i	−0.989467	+	0.144756i	$0.953760\pi$
$74$		0			0
$75$		−	2661.34i		−	0.473127i
$76$		0			0
$77$	357.307			0.0602643
$78$		0			0
$79$	7499.16	−	8937.15i	1.20160	−	1.43201i	0.328469	−	0.944515i	$-0.393467\pi$
$79$	7499.16	−	8937.15i	1.20160	−	1.43201i	0.873127	−	0.487492i	$-0.162088\pi$
$80$		0			0
$81$	5895.91	−	2145.94i	0.898629	−	0.327074i
$82$		0			0
$83$	−3230.90	−	5596.08i	−0.468994	−	0.812321i	0.530378	−	0.847761i	$-0.322050\pi$
$83$	−3230.90	−	5596.08i	−0.468994	−	0.812321i	−0.999372	+	0.0354403i	$0.988717\pi$
$84$		0			0
$85$	238.018	+	1349.86i	0.0329436	+	0.186832i
$86$		0			0
$87$	8903.00	−	15420.5i	1.17625	−	2.03732i
$88$		0			0
$89$	−6885.48	−	8205.80i	−0.869269	−	1.03595i	−0.999014	−	0.0444070i	$-0.985860\pi$
$89$	−6885.48	−	8205.80i	−0.869269	−	1.03595i	0.129745	−	0.991547i	$-0.458584\pi$
$90$		0			0
$91$	121.584	−	334.049i	0.0146823	−	0.0403392i
$92$		0			0
$93$	−1825.88	+	10355.1i	−0.211109	+	1.19726i
$94$		0			0
$95$	211.792	+	7378.19i	0.0234672	+	0.817528i
$96$		0			0
$97$	−4876.88	−	859.926i	−0.518321	−	0.0913940i	−0.0916331	−	0.995793i	$-0.529209\pi$
$97$	−4876.88	−	859.926i	−0.518321	−	0.0913940i	−0.426688	+	0.904399i	$0.640320\pi$
$98$		0			0
$99$	−9077.46	−	3303.92i	−0.926177	−	0.337101i
$100$		0			0
$101$	−10204.2	+	8562.37i	−1.00032	+	0.839366i	−0.987028	−	0.160546i	$-0.948675\pi$
$101$	−10204.2	+	8562.37i	−1.00032	+	0.839366i	−0.0132891	+	0.999912i	$0.504230\pi$
$102$		0			0
$103$	17592.0	+	10156.8i	1.65822	+	0.957372i	0.973537	+	0.228529i	$0.0733914\pi$
$103$	17592.0	+	10156.8i	1.65822	+	0.957372i	0.684680	+	0.728844i	$0.259942\pi$
$104$		0			0
$105$	808.402	−	142.543i	0.0733244	−	0.0129291i
$106$		0			0
$107$	−446.474	+	257.772i	−0.0389968	+	0.0225148i	−0.519372	−	0.854548i	$-0.673834\pi$
$107$	−446.474	+	257.772i	−0.0389968	+	0.0225148i	0.480375	+	0.877063i	$0.340501\pi$
$108$		0			0
$109$	1442.56	+	3963.39i	0.121417	+	0.333591i	0.985480	−	0.169794i	$-0.0543101\pi$
$109$	1442.56	+	3963.39i	0.121417	+	0.333591i	−0.864062	+	0.503385i	$0.832088\pi$
$110$		0			0
$111$	−10835.6	−	9092.12i	−0.879439	−	0.737937i
$112$		0			0
$113$			21932.9i			1.71767i	0.512251	+	0.858836i	$0.328812\pi$
$113$			21932.9i			1.71767i	−0.512251	+	0.858836i	$0.671188\pi$
$114$		0			0
$115$	4999.40			0.378027
$116$		0			0
$117$	−6177.73	+	7362.33i	−0.451292	+	0.537828i
$118$		0			0
$119$	196.650	−	71.5748i	0.0138867	−	0.00505436i
$120$		0			0
$121$	770.041	+	1333.75i	0.0525949	+	0.0910970i
$122$		0			0
$123$	−177.197	−	1004.93i	−0.0117124	−	0.0664243i
$124$		0			0
$125$	8505.14	−	14731.3i	0.544329	−	0.942805i
$126$		0			0
$127$	8629.27	+	10284.0i	0.535016	+	0.637607i	0.964063	−	0.265675i	$-0.0855949\pi$
$127$	8629.27	+	10284.0i	0.535016	+	0.637607i	−0.429047	+	0.903282i	$0.641151\pi$
$128$		0			0
$129$	1485.62	−	4081.71i	0.0892747	−	0.245280i
$130$		0			0
$131$	−3972.35	+	22528.3i	−0.231476	+	1.31276i	0.618435	+	0.785836i	$0.287767\pi$
$131$	−3972.35	+	22528.3i	−0.231476	+	1.31276i	−0.849910	+	0.526927i	$0.823344\pi$
$132$		0			0
$133$	1103.74	−	227.453i	0.0623970	−	0.0128585i
$134$		0			0
$135$	−879.722	−	155.119i	−0.0482701	−	0.00851132i
$136$		0			0
$137$	−3938.12	−	1433.36i	−0.209820	−	0.0763684i	0.234972	−	0.972002i	$-0.424500\pi$
$137$	−3938.12	−	1433.36i	−0.209820	−	0.0763684i	−0.444792	+	0.895634i	$0.646722\pi$
$138$		0			0
$139$	17460.2	−	14650.9i	0.903691	−	0.758287i	−0.0672169	−	0.997738i	$-0.521412\pi$
$139$	17460.2	−	14650.9i	0.903691	−	0.758287i	0.970908	+	0.239451i	$0.0769675\pi$
$140$		0			0
$141$	17674.9	+	10204.6i	0.889034	+	0.513284i
$142$		0			0
$143$	−12836.2	+	2263.37i	−0.627717	+	0.110684i
$144$		0			0
$145$	24516.3	−	14154.5i	1.16605	−	0.673221i
$146$		0			0
$147$	10518.2	+	28898.5i	0.486751	+	1.33734i
$148$		0			0
$149$	−24075.2	−	20201.5i	−1.08442	−	0.909937i	−0.0881403	−	0.996108i	$-0.528092\pi$
$149$	−24075.2	−	20201.5i	−1.08442	−	0.909937i	−0.996280	+	0.0861711i	$0.972537\pi$
$150$		0			0
$151$			24250.1i			1.06355i	0.846885	+	0.531776i	$0.178475\pi$
$151$			24250.1i			1.06355i	−0.846885	+	0.531776i	$0.821525\pi$
$152$		0			0
$153$	−5657.77			−0.241692
$154$		0			0
$155$	−10745.5	+	12806.0i	−0.447263	+	0.533028i
$156$		0			0
$157$	4379.41	−	1593.98i	0.177671	−	0.0646669i	−0.251653	−	0.967818i	$-0.580974\pi$
$157$	4379.41	−	1593.98i	0.177671	−	0.0646669i	0.429324	+	0.903151i	$0.358752\pi$
$158$		0			0
$159$	−16483.0	−	28549.4i	−0.651992	−	1.12928i
$160$		0			0
$161$	−132.543	−	751.691i	−0.00511336	−	0.0289993i
$162$		0			0
$163$	1164.96	−	2017.78i	0.0438467	−	0.0759448i	−0.843269	−	0.537492i	$-0.819372\pi$
$163$	1164.96	−	2017.78i	0.0438467	−	0.0759448i	0.887116	+	0.461547i	$0.152705\pi$
$164$		0			0
$165$	−19346.6	−	23056.3i	−0.710617	−	0.846880i
$166$		0			0
$167$	−16240.9	+	44621.6i	−0.582341	+	1.59997i	0.201826	+	0.979421i	$0.435312\pi$
$167$	−16240.9	+	44621.6i	−0.582341	+	1.59997i	−0.784168	+	0.620549i	$0.786910\pi$
$168$		0			0
$169$	2707.72	−	15356.3i	0.0948049	−	0.537665i
$170$		0			0
$171$	−30143.9	−	4427.49i	−1.03088	−	0.151414i
$172$		0			0
$173$	7189.27	+	1267.66i	0.240211	+	0.0423557i	0.292457	−	0.956279i	$-0.405527\pi$
$173$	7189.27	+	1267.66i	0.240211	+	0.0423557i	−0.0522465	+	0.998634i	$0.516638\pi$
$174$		0			0
$175$	−607.034	−	220.942i	−0.0198215	−	0.00721445i
$176$		0			0
$177$	48052.9	−	40321.1i	1.53381	−	1.28702i
$178$		0			0
$179$	−14758.6	−	8520.88i	−0.460616	−	0.265937i	0.251687	−	0.967809i	$-0.419015\pi$
$179$	−14758.6	−	8520.88i	−0.460616	−	0.265937i	−0.712303	+	0.701872i	$0.752348\pi$
$180$		0			0
$181$	8504.11	−	1499.50i	0.259580	−	0.0457710i	−0.0423434	−	0.999103i	$-0.513482\pi$
$181$	8504.11	−	1499.50i	0.259580	−	0.0457710i	0.301924	+	0.953332i	$0.402371\pi$
$182$		0			0
$183$	17837.7	−	10298.6i	0.532643	−	0.307522i
$184$		0			0
$185$	−7691.42	−	21132.0i	−0.224731	−	0.617444i
$186$		0			0
$187$	−5877.90	−	4932.15i	−0.168089	−	0.141043i
$188$		0			0
$189$			136.384i			0.00381803i
$190$		0			0
$191$	−12103.5			−0.331776			−0.165888	−	0.986145i	$-0.553049\pi$
$191$	−12103.5			−0.331776			−0.165888	+	0.986145i	$0.553049\pi$
$192$		0			0
$193$	1445.25	−	1722.38i	0.0387996	−	0.0462396i	−0.746296	−	0.665615i	$-0.768169\pi$
$193$	1445.25	−	1722.38i	0.0387996	−	0.0462396i	0.785095	+	0.619375i	$0.212614\pi$
$194$		0			0
$195$	−28138.8	+	10241.7i	−0.740007	+	0.269340i
$196$		0			0
$197$	11434.8	+	19805.6i	0.294642	+	0.510335i	0.974902	−	0.222637i	$-0.0714663\pi$
$197$	11434.8	+	19805.6i	0.294642	+	0.510335i	−0.680260	+	0.732971i	$0.738133\pi$
$198$		0			0
$199$	−8358.98	−	47406.2i	−0.211080	−	1.19709i	−0.887580	−	0.460653i	$-0.847615\pi$
$199$	−8358.98	−	47406.2i	−0.211080	−	1.19709i	0.676500	−	0.736442i	$-0.263496\pi$
$200$		0			0
$201$	31038.4	−	53760.1i	0.768258	−	1.33066i
$202$		0			0
$203$	−2778.18	−	3310.91i	−0.0674169	−	0.0803443i
$204$		0			0
$205$	554.874	−	1524.50i	0.0132034	−	0.0362761i
$206$		0			0
$207$	−3583.40	+	20322.4i	−0.0836285	+	0.474281i
$208$		0			0
$209$	−27457.1	−	30877.7i	−0.628583	−	0.706890i
$210$		0			0
$211$	48912.2	+	8624.53i	1.09863	+	0.193718i	0.693440	−	0.720514i	$-0.256094\pi$
$211$	48912.2	+	8624.53i	1.09863	+	0.193718i	0.405191	+	0.914232i	$0.367205\pi$
$212$		0			0
$213$	−6473.35	−	2356.11i	−0.142682	−	0.0519321i
$214$		0			0
$215$	5290.14	−	4438.96i	0.114443	−	0.0960293i
$216$		0			0
$217$	2210.34	+	1276.14i	0.0469397	+	0.0271006i
$218$		0			0
$219$	89756.6	−	15826.5i	1.87145	−	0.329987i
$220$		0			0
$221$	−6611.23	+	3816.99i	−0.135362	+	0.0781514i
$222$		0			0
$223$	22601.8	+	62097.9i	0.454499	+	1.24873i	0.929527	+	0.368755i	$0.120216\pi$
$223$	22601.8	+	62097.9i	0.454499	+	1.24873i	−0.475027	+	0.879971i	$0.657562\pi$
$224$		0			0
$225$	13378.8	+	11226.2i	0.264273	+	0.221752i
$226$		0			0
$227$		−	66333.9i		−	1.28731i	−0.765315	−	0.643656i	$-0.777417\pi$
$227$		−	66333.9i		−	1.28731i	0.765315	−	0.643656i	$-0.222583\pi$
$228$		0			0
$229$	−92938.4			−1.77225			−0.886124	−	0.463449i	$-0.846612\pi$
$229$	−92938.4			−1.77225			−0.886124	+	0.463449i	$0.846612\pi$
$230$		0			0
$231$	−2953.75	+	3520.14i	−0.0553540	+	0.0659683i
$232$		0			0
$233$	−84238.9	+	30660.5i	−1.55168	+	0.564764i	−0.968810	−	0.247805i	$-0.920291\pi$
$233$	−84238.9	+	30660.5i	−1.55168	+	0.564764i	−0.582866	+	0.812569i	$0.698069\pi$
$234$		0			0
$235$	16223.8	+	28100.5i	0.293777	+	0.508836i
$236$		0			0
$237$	26054.3	+	147761.i	0.463856	+	2.63066i
$238$		0			0
$239$	−24727.5	+	42829.3i	−0.432897	+	0.749800i	−0.997121	−	0.0758216i	$-0.975842\pi$
$239$	−24727.5	+	42829.3i	−0.432897	+	0.749800i	0.564224	+	0.825622i	$0.309175\pi$
$240$		0			0
$241$	−58976.1	−	70284.9i	−1.01541	−	1.21012i	−0.977521	−	0.210837i	$-0.932381\pi$
$241$	−58976.1	−	70284.9i	−1.01541	−	1.21012i	−0.0378895	−	0.999282i	$-0.512063\pi$
$242$		0			0
$243$	−28808.5	+	79150.8i	−0.487875	+	1.34043i
$244$		0			0
$245$	−8490.19	+	48150.3i	−0.141444	+	0.802170i
$246$		0			0
$247$	−38210.9	+	15162.9i	−0.626315	+	0.248535i
$248$		0			0
$249$	81840.6	+	14430.7i	1.31999	+	0.232749i
$250$		0			0
$251$	42840.7	+	15592.7i	0.680000	+	0.247500i	0.658847	−	0.752277i	$-0.271044\pi$
$251$	42840.7	+	15592.7i	0.680000	+	0.247500i	0.0211523	+	0.999776i	$0.493267\pi$
$252$		0			0
$253$	−21438.9	+	17989.3i	−0.334935	+	0.281044i
$254$		0			0
$255$	−15266.3	−	8813.99i	−0.234776	−	0.135548i
$256$		0			0
$257$	101228.	−	17849.2i	1.53262	−	0.270242i	0.657240	−	0.753681i	$-0.271724\pi$
$257$	101228.	−	17849.2i	1.53262	−	0.270242i	0.875378	+	0.483439i	$0.160612\pi$
$258$		0			0
$259$	−2973.41	+	1716.70i	−0.0443257	+	0.0255914i
$260$		0			0
$261$	39965.2	+	109803.i	0.586679	+	1.61189i
$262$		0			0
$263$	−55109.3	−	46242.2i	−0.796735	−	0.668540i	0.150668	−	0.988584i	$-0.451858\pi$
$263$	−55109.3	−	46242.2i	−0.796735	−	0.668540i	−0.947402	+	0.320045i	$0.896302\pi$
$264$		0			0
$265$		−	52411.2i		−	0.746332i
$266$		0			0
$267$	137762.			1.93245
$268$		0			0
$269$	−3420.43	+	4076.32i	−0.0472690	+	0.0563330i	−0.789161	−	0.614186i	$-0.789484\pi$
$269$	−3420.43	+	4076.32i	−0.0472690	+	0.0563330i	0.741892	+	0.670519i	$0.233929\pi$
$270$		0			0
$271$	−85921.4	+	31272.8i	−1.16994	+	0.425822i	−0.852638	−	0.522503i	$-0.824998\pi$
$271$	−85921.4	+	31272.8i	−1.16994	+	0.425822i	−0.317300	+	0.948325i	$0.602776\pi$
$272$		0			0
$273$	2285.91	+	3959.31i	0.0306714	+	0.0531244i
$274$		0			0
$275$	4112.99	+	23325.9i	0.0543867	+	0.308442i
$276$		0			0
$277$	−4412.53	+	7642.73i	−0.0575080	+	0.0996068i	−0.893346	−	0.449369i	$-0.851649\pi$
$277$	−4412.53	+	7642.73i	−0.0575080	+	0.0996068i	0.835838	+	0.548976i	$0.184982\pi$
$278$		0			0
$279$	−44354.0	−	52859.1i	−0.569803	−	0.679065i
$280$		0			0
$281$	12846.7	−	35295.9i	0.162696	−	0.447005i	−0.831378	−	0.555707i	$-0.812447\pi$
$281$	12846.7	−	35295.9i	0.162696	−	0.447005i	0.994074	+	0.108703i	$0.0346697\pi$
$282$		0			0
$283$	21836.2	−	123839.i	0.272649	−	1.54627i	−0.473683	−	0.880696i	$-0.657076\pi$
$283$	21836.2	−	123839.i	0.272649	−	1.54627i	0.746332	−	0.665574i	$-0.231813\pi$
$284$		0			0
$285$	−74439.6	−	58906.6i	−0.916462	−	0.725228i
$286$		0			0
$287$	−243.929	−	43.0113i	−0.00296142	−	0.000522178i
$288$		0			0
$289$	74261.1	+	27028.8i	0.889131	+	0.323617i
$290$		0			0
$291$	48787.5	−	40937.6i	0.576133	−	0.483433i
$292$		0			0
$293$	−9067.24	−	5234.97i	−0.105618	−	0.0609788i	0.446260	−	0.894903i	$-0.352756\pi$
$293$	−9067.24	−	5234.97i	−0.105618	−	0.0609788i	−0.551879	+	0.833924i	$0.686089\pi$
$294$		0			0
$295$	98214.1	−	17317.8i	1.12857	−	0.198998i
$296$		0			0
$297$	4330.66	−	2500.31i	0.0490955	−	0.0283453i
$298$		0			0
$299$	9523.19	+	26164.8i	0.106522	+	0.292667i
$300$		0			0
$301$	−807.675	−	677.720i	−0.00891464	−	0.00748027i
$302$		0			0
$303$		−	171313.i		−	1.86597i
$304$		0			0
$305$	32746.5			0.352018
$306$		0			0
$307$	−2931.65	+	3493.80i	−0.0311053	+	0.0370699i	−0.781373	−	0.624064i	$-0.785480\pi$
$307$	−2931.65	+	3493.80i	−0.0311053	+	0.0370699i	0.750268	+	0.661134i	$0.229925\pi$
$308$		0			0
$309$	−245491.	+	89351.3i	−2.57109	+	0.935802i
$310$		0			0
$311$	10594.0	+	18349.3i	0.109531	+	0.189714i	0.915580	−	0.402135i	$-0.131732\pi$
$311$	10594.0	+	18349.3i	0.109531	+	0.189714i	−0.806049	+	0.591848i	$0.798398\pi$
$312$		0			0
$313$	−4983.07	−	28260.4i	−0.0508637	−	0.288463i	0.948757	−	0.316007i	$-0.102342\pi$
$313$	−4983.07	−	28260.4i	−0.0508637	−	0.288463i	−0.999621	+	0.0275445i	$0.991231\pi$
$314$		0			0
$315$	−2693.45	+	4665.20i	−0.0271449	+	0.0470164i
$316$		0			0
$317$	−8523.70	−	10158.2i	−0.0848222	−	0.101087i	0.721964	−	0.691931i	$-0.243240\pi$
$317$	−8523.70	−	10158.2i	−0.0848222	−	0.101087i	−0.806786	+	0.590843i	$0.798795\pi$
$318$		0			0
$319$	−54200.7	+	148915.i	−0.532628	+	1.46338i
$320$		0			0
$321$	1151.33	−	6529.52i	0.0111735	−	0.0633681i
$322$		0			0
$323$	−21296.8	−	11493.9i	−0.204131	−	0.110170i
$324$		0			0
$325$	23207.2	+	4092.05i	0.219713	+	0.0387413i
$326$		0			0
$327$	−50971.9	−	18552.3i	−0.476689	−	0.173501i
$328$		0			0
$329$	3794.95	−	3184.34i	0.0350602	−	0.0294190i
$330$		0			0
$331$	−99006.8	−	57161.6i	−0.903669	−	0.521733i	−0.0252800	−	0.999680i	$-0.508048\pi$
$331$	−99006.8	−	57161.6i	−0.903669	−	0.521733i	−0.878389	+	0.477947i	$0.841381\pi$
$332$		0			0
$333$	91414.0	−	16118.8i	0.824374	−	0.145359i
$334$		0			0
$335$	85470.7	−	49346.5i	0.761601	−	0.439711i
$336$		0			0
$337$	49309.4	+	135477.i	0.434180	+	1.19290i	0.943223	+	0.332160i	$0.107777\pi$
$337$	49309.4	+	135477.i	0.434180	+	1.19290i	−0.509043	+	0.860741i	$0.670001\pi$
$338$		0			0
$339$	−216080.	−	181313.i	−1.88025	−	1.57772i
$340$		0			0
$341$		−	93581.3i		−	0.804785i
$342$		0			0
$343$	14960.0			0.127158
$344$		0			0
$345$	−41328.5	+	49253.4i	−0.347225	+	0.413807i
$346$		0			0
$347$	−44410.2	+	16164.0i	−0.368828	+	0.134243i	−0.519783	−	0.854298i	$-0.673987\pi$
$347$	−44410.2	+	16164.0i	−0.368828	+	0.134243i	0.150955	+	0.988541i	$0.451765\pi$
$348$		0			0
$349$	55218.0	+	95640.3i	0.453346	+	0.785218i	0.998591	−	0.0530586i	$-0.0168970\pi$
$349$	55218.0	+	95640.3i	0.453346	+	0.785218i	−0.545246	+	0.838276i	$0.683564\pi$
$350$		0			0
$351$	−863.927	−	4899.57i	−0.00701234	−	0.0397689i
$352$		0			0
$353$	102767.	−	177998.i	0.824717	−	1.42845i	−0.0774187	−	0.996999i	$-0.524668\pi$
$353$	102767.	−	177998.i	0.824717	−	1.42845i	0.902136	−	0.431453i	$-0.141999\pi$
$354$		0			0
$355$	−7039.93	−	8389.86i	−0.0558613	−	0.0665730i
$356$		0			0
$357$	−920.500	+	2529.05i	−0.00722250	+	0.0198436i
$358$		0			0
$359$	22847.0	−	129572.i	0.177272	−	1.00536i	−0.758216	−	0.652003i	$-0.773929\pi$
$359$	22847.0	−	129572.i	0.177272	−	1.00536i	0.935488	−	0.353357i	$-0.114960\pi$
$360$		0			0
$361$	−104473.	−	77904.2i	−0.801655	−	0.597787i
$362$		0			0
$363$	−19505.6	−	3439.36i	−0.148029	−	0.0261015i
$364$		0			0
$365$	136163.	+	49559.1i	1.02205	+	0.371996i
$366$		0			0
$367$	190217.	−	159611.i	1.41227	−	1.18504i	0.456940	−	0.889497i	$-0.348945\pi$
$367$	190217.	−	159611.i	1.41227	−	1.18504i	0.955331	−	0.295539i	$-0.0954991\pi$
$368$		0			0
$369$	5799.36	+	3348.26i	0.0425919	+	0.0245905i
$370$		0			0
$371$	−7880.34	+	1389.52i	−0.0572529	+	0.0100952i
$372$		0			0
$373$	65920.1	−	38059.0i	0.473805	−	0.273552i	−0.244026	−	0.969769i	$-0.578468\pi$
$373$	65920.1	−	38059.0i	0.473805	−	0.273552i	0.717831	+	0.696217i	$0.245135\pi$
$374$		0			0
$375$	74821.6	+	205571.i	0.532065	+	1.46184i
$376$		0			0
$377$	120779.	+	101345.i	0.849783	+	0.713052i
$378$		0			0
$379$			45055.4i			0.313667i	0.987625	+	0.156834i	$0.0501286\pi$
$379$			45055.4i			0.313667i	−0.987625	+	0.156834i	$0.949871\pi$
$380$		0			0
$381$	−172651.			−1.18938
$382$		0			0
$383$	34187.5	−	40743.1i	0.233061	−	0.277752i	−0.636820	−	0.771012i	$-0.719751\pi$
$383$	34187.5	−	40743.1i	0.233061	−	0.277752i	0.869882	+	0.493261i	$0.164195\pi$
$384$		0			0
$385$	−6865.13	+	2498.70i	−0.0463156	+	0.0168575i
$386$		0			0
$387$	14252.5	+	24686.0i	0.0951629	+	0.164827i
$388$		0			0
$389$	19761.0	+	112070.i	0.130590	+	0.740611i	0.977830	+	0.209401i	$0.0671514\pi$
$389$	19761.0	+	112070.i	0.130590	+	0.740611i	−0.847240	+	0.531210i	$0.821738\pi$
$390$		0			0
$391$	−8195.67	+	14195.3i	−0.0536082	+	0.0928521i
$392$		0			0
$393$	−189108.	−	225370.i	−1.22440	−	1.45919i
$394$		0			0
$395$	−81586.5	+	224157.i	−0.522906	+	1.43667i
$396$		0			0
$397$	−39544.9	+	224270.i	−0.250905	+	1.42295i	0.555464	+	0.831541i	$0.312541\pi$
$397$	−39544.9	+	224270.i	−0.250905	+	1.42295i	−0.806369	+	0.591413i	$0.798570\pi$
$398$		0			0
$399$	−6883.43	+	12754.2i	−0.0432374	+	0.0801136i
$400$		0			0
$401$	302245.	+	53294.0i	1.87962	+	0.331428i	0.991701	−	0.128565i	$-0.0410370\pi$
$401$	302245.	+	53294.0i	1.87962	+	0.331428i	0.887922	+	0.459993i	$0.152148\pi$
$402$		0			0
$403$	−87489.9	−	31843.7i	−0.538701	−	0.196071i
$404$		0			0
$405$	−98274.3	+	82462.0i	−0.599142	+	0.502740i
$406$		0			0
$407$	109022.	+	62944.0i	0.658152	+	0.379984i
$408$		0			0
$409$	−262236.	+	46239.3i	−1.56764	+	0.276417i	−0.888948	−	0.458008i	$-0.848563\pi$
$409$	−262236.	+	46239.3i	−1.56764	+	0.276417i	−0.678690	+	0.734425i	$0.737452\pi$
$410$		0			0
$411$	46676.4	−	26948.6i	0.276321	−	0.159534i
$412$		0			0
$413$	−5207.67	−	14308.0i	−0.0305312	−	0.0838837i
$414$		0			0
$415$	101211.	+	84926.2i	0.587668	+	0.493112i
$416$		0			0
$417$			293130.i			1.68573i
$418$		0			0
$419$	115258.			0.656514			0.328257	−	0.944588i	$-0.393539\pi$
$419$	115258.			0.656514			0.328257	+	0.944588i	$0.393539\pi$
$420$		0			0
$421$	−22344.9	+	26629.6i	−0.126071	+	0.150245i	−0.825387	−	0.564567i	$-0.809043\pi$
$421$	−22344.9	+	26629.6i	−0.126071	+	0.150245i	0.699317	+	0.714812i	$0.253488\pi$
$422$		0			0
$423$	−125856.	+	45808.0i	−0.703387	+	0.256012i
$424$		0			0
$425$	6936.24	+	12013.9i	0.0384013	+	0.0665131i
$426$		0			0
$427$	−868.170	−	4923.64i	−0.00476156	−	0.0270041i
$428$		0			0
$429$	83814.4	−	145171.i	0.455412	−	0.788796i
$430$		0			0
$431$	−9788.89	−	11665.9i	−0.0526962	−	0.0628008i	0.739053	−	0.673648i	$-0.235274\pi$
$431$	−9788.89	−	11665.9i	−0.0526962	−	0.0628008i	−0.791749	+	0.610847i	$0.790829\pi$
$432$		0			0
$433$	1615.17	−	4437.64i	0.00861474	−	0.0236688i	−0.935311	−	0.353827i	$-0.884880\pi$
$433$	1615.17	−	4437.64i	0.00861474	−	0.0236688i	0.943925	+	0.330159i	$0.107102\pi$
$434$		0			0
$435$	−63220.5	+	358541.i	−0.334102	+	1.89479i
$436$		0			0
$437$	−54774.2	+	69217.5i	−0.286822	+	0.362454i
$438$		0			0
$439$	−256793.	−	45279.5i	−1.33246	−	0.234949i	−0.538349	−	0.842722i	$-0.680952\pi$
$439$	−256793.	−	45279.5i	−1.33246	−	0.234949i	−0.794111	+	0.607773i	$0.792063\pi$
$440$		0			0
$441$	−189644.	−	69024.8i	−0.975129	−	0.354918i
$442$		0			0
$443$	128893.	−	108154.i	0.656783	−	0.551106i	−0.252338	−	0.967639i	$-0.581199\pi$
$443$	128893.	−	108154.i	0.656783	−	0.551106i	0.909121	+	0.416533i	$0.136755\pi$
$444$		0			0
$445$	189679.	+	109511.i	0.957852	+	0.553016i
$446$		0			0
$447$	398045.	−	70186.0i	1.99213	−	0.351265i
$448$		0			0
$449$	44967.9	−	25962.2i	0.223054	−	0.128780i	−0.384310	−	0.923204i	$-0.625560\pi$
$449$	44967.9	−	25962.2i	0.223054	−	0.128780i	0.607364	+	0.794424i	$0.292227\pi$
$450$		0			0
$451$	3106.16	+	8534.11i	0.0152711	+	0.0419571i
$452$		0			0
$453$	−238908.	−	200468.i	−1.16422	−	0.976895i
$454$		0			0
$455$			7268.52i			0.0351094i
$456$		0			0
$457$	−313833.			−1.50268			−0.751341	−	0.659915i	$-0.770592\pi$
$457$	−313833.			−1.50268			−0.751341	+	0.659915i	$0.770592\pi$
$458$		0			0
$459$	1882.60	−	2243.59i	0.00893578	−	0.0106492i
$460$		0			0
$461$	285937.	−	104073.i	1.34545	−	0.489705i	0.433928	−	0.900947i	$-0.357127\pi$
$461$	285937.	−	104073.i	1.34545	−	0.489705i	0.911526	+	0.411242i	$0.134905\pi$
$462$		0			0
$463$	−38084.9	−	65964.9i	−0.177660	−	0.307717i	0.763418	−	0.645904i	$-0.223520\pi$
$463$	−38084.9	−	65964.9i	−0.177660	−	0.307717i	−0.941079	+	0.338188i	$0.890186\pi$
$464$		0			0
$465$	−37333.0	−	211726.i	−0.172658	−	0.979194i
$466$		0			0
$467$	107590.	−	186351.i	0.493330	−	0.854473i	−0.506640	−	0.862158i	$-0.669113\pi$
$467$	107590.	−	186351.i	0.493330	−	0.854473i	0.999970	+	0.00768464i	$0.00244612\pi$
$468$		0			0
$469$	−9685.53	−	11542.8i	−0.0440330	−	0.0524764i
$470$		0			0
$471$	−20499.6	+	56322.2i	−0.0924068	+	0.253886i
$472$		0			0
$473$	−6712.95	+	38071.0i	−0.0300048	+	0.170166i
$474$		0			0
$475$	27554.0	+	69436.8i	0.122123	+	0.307753i
$476$		0			0
$477$	213050.	+	37566.5i	0.936365	+	0.165106i
$478$		0			0
$479$	−115062.	−	41879.2i	−0.501490	−	0.182527i	0.0788742	−	0.996885i	$-0.474867\pi$
$479$	−115062.	−	41879.2i	−0.501490	−	0.182527i	−0.580364	+	0.814357i	$0.697090\pi$
$480$		0			0
$481$	95944.8	−	80507.3i	0.414698	−	0.347973i
$482$		0			0
$483$	8501.24	+	4908.19i	0.0364408	+	0.0210391i
$484$		0			0
$485$	99715.7	−	17582.6i	0.423916	−	0.0747479i
$486$		0			0
$487$	143586.	−	82899.5i	0.605417	−	0.349538i	−0.165752	−	0.986167i	$-0.553005\pi$
$487$	143586.	−	82899.5i	0.605417	−	0.349538i	0.771170	+	0.636630i	$0.219672\pi$
$488$		0			0
$489$	10248.4	+	28157.4i	0.0428588	+	0.117754i
$490$		0			0
$491$	−341110.	−	286226.i	−1.41492	−	1.18726i	−0.953996	−	0.299819i	$-0.903074\pi$
$491$	−341110.	−	286226.i	−1.41492	−	1.18726i	−0.460924	−	0.887440i	$-0.652482\pi$
$492$		0			0
$493$			92815.4i			0.381879i
$494$		0			0
$495$	197515.			0.806101
$496$		0			0
$497$	−1074.82	+	1280.93i	−0.00435136	+	0.00518575i
$498$		0			0
$499$	30922.7	−	11254.9i	0.124187	−	0.0452003i	−0.279179	−	0.960239i	$-0.590062\pi$
$499$	30922.7	−	11254.9i	0.124187	−	0.0452003i	0.403366	+	0.915039i	$0.367840\pi$
$500$		0			0
$501$	−305346.	−	528876.i	−1.21651	−	2.10707i
$502$		0			0
$503$	9171.59	+	52014.6i	0.0362500	+	0.205584i	0.997553	−	0.0699076i	$-0.0222704\pi$
$503$	9171.59	+	52014.6i	0.0362500	+	0.205584i	−0.961303	+	0.275492i	$0.911159\pi$
$504$		0			0
$505$	136181.	−	235873.i	0.533992	−	0.924902i
$506$		0			0
$507$	128904.	+	153621.i	0.501475	+	0.597635i
$508$		0			0
$509$	97679.4	−	268372.i	0.377023	−	1.03586i	−0.595561	−	0.803310i	$-0.703070\pi$
$509$	97679.4	−	268372.i	0.377023	−	1.03586i	0.972584	−	0.232552i	$-0.0747074\pi$
$510$		0			0
$511$	3841.60	−	21786.8i	0.0147119	−	0.0834356i
$512$		0			0
$513$	11786.0	−	10480.4i	0.0447849	−	0.0398238i
$514$		0			0
$515$	−409033.	−	72123.5i	−1.54221	−	0.271933i
$516$		0			0
$517$	−170686.	−	62124.7i	−0.638583	−	0.232425i
$518$		0			0
$519$	−71920.3	+	60348.3i	−0.267003	+	0.224042i
$520$		0			0
$521$	319762.	+	184615.i	1.17802	+	0.680128i	0.955555	−	0.294814i	$-0.0952577\pi$
$521$	319762.	+	184615.i	1.17802	+	0.680128i	0.222461	+	0.974942i	$0.428591\pi$
$522$		0			0
$523$	−450051.	+	79356.1i	−1.64535	+	0.290119i	−0.918128	−	0.396284i	$-0.870300\pi$
$523$	−450051.	+	79356.1i	−1.64535	+	0.290119i	−0.727221	+	0.686403i	$0.759188\pi$
$524$		0			0
$525$	7194.85	−	4153.95i	0.0261038	−	0.0150710i
$526$		0			0
$527$	−18745.9	−	51504.0i	−0.0674972	−	0.185447i
$528$		0			0
$529$	−168572.	−	141449.i	−0.602387	−	0.505462i
$530$		0			0
$531$			411651.i			1.45996i
$532$		0			0
$533$	9035.57			0.0318054
$534$		0			0
$535$	6775.70	−	8074.97i	0.0236726	−	0.0282120i
$536$		0			0
$537$	205951.	−	74960.1i	0.714193	−	0.259945i
$538$		0			0
$539$	−136851.	−	237032.i	−0.471052	−	0.815887i
$540$		0			0
$541$	8700.04	+	49340.4i	0.0297253	+	0.168581i	0.996057	−	0.0887206i	$-0.0282778\pi$
$541$	8700.04	+	49340.4i	0.0297253	+	0.168581i	−0.966331	+	0.257301i	$0.917167\pi$
$542$		0			0
$543$	−55527.9	+	96177.1i	−0.188327	+	0.326191i
$544$		0			0
$545$	−55433.2	−	66062.7i	−0.186628	−	0.222415i
$546$		0			0
$547$	−66561.1	+	182875.i	−0.222457	+	0.611195i	−0.999841	−	0.0178308i	$-0.994324\pi$
$547$	−66561.1	+	182875.i	−0.222457	+	0.611195i	0.777384	+	0.629026i	$0.216546\pi$
$548$		0			0
$549$	−23471.5	+	133114.i	−0.0778748	+	0.441650i
$550$		0			0
$551$	−72632.9	+	494510.i	−0.239238	+	1.62882i
$552$		0			0
$553$	35866.4	+	6324.21i	0.117284	+	0.0206803i
$554$		0			0
$555$	271772.	+	98916.9i	0.882305	+	0.321133i
$556$		0			0
$557$	−10023.2	+	8410.44i	−0.0323069	+	0.0271087i	−0.658798	−	0.752320i	$-0.728935\pi$
$557$	−10023.2	+	8410.44i	−0.0323069	+	0.0271087i	0.626491	+	0.779428i	$0.284490\pi$
$558$		0			0
$559$	33308.6	+	19230.7i	0.106594	+	0.0615421i
$560$		0			0
$561$	97181.5	−	17135.7i	0.308786	−	0.0544474i
$562$		0			0
$563$	−133471.	+	77059.4i	−0.421085	+	0.243113i	−0.695541	−	0.718486i	$-0.744835\pi$
$563$	−133471.	+	77059.4i	−0.421085	+	0.243113i	0.274456	+	0.961600i	$0.411502\pi$
$564$		0			0
$565$	−153380.	−	421409.i	−0.480477	−	1.32010i
$566$		0			0
$567$	15004.1	+	12589.9i	0.0466706	+	0.0391613i
$568$		0			0
$569$			80777.1i			0.249496i	0.992188	+	0.124748i	$0.0398122\pi$
$569$			80777.1i			0.249496i	−0.992188	+	0.124748i	$0.960188\pi$
$570$		0			0
$571$	296393.			0.909065			0.454533	−	0.890730i	$-0.349806\pi$
$571$	296393.			0.909065			0.454533	+	0.890730i	$0.349806\pi$
$572$		0			0
$573$	100056.	−	119242.i	0.304743	−	0.363179i
$574$		0			0
$575$	47546.6	−	17305.5i	0.143808	−	0.0523419i
$576$		0			0
$577$	185757.	+	321741.i	0.557949	+	0.966396i	0.997668	+	0.0682605i	$0.0217449\pi$
$577$	185757.	+	321741.i	0.557949	+	0.966396i	−0.439718	+	0.898136i	$0.644922\pi$
$578$		0			0
$579$	5021.21	+	28476.7i	0.0149779	+	0.0849440i
$580$		0			0
$581$	10085.9	−	17469.3i	0.0298787	−	0.0517514i
$582$		0			0
$583$	188591.	+	224754.i	0.554861	+	0.661257i
$584$		0			0
$585$	67210.1	−	184658.i	0.196392	−	0.539581i
$586$		0			0
$587$	4885.12	−	27704.9i	0.0141775	−	0.0804045i	−0.976898	−	0.213706i	$-0.931447\pi$
$587$	4885.12	−	27704.9i	0.0141775	−	0.0804045i	0.991076	+	0.133301i	$0.0425578\pi$
$588$		0			0
$589$	−59571.7	−	289077.i	−0.171715	−	0.833266i
$590$		0			0
$591$	−289649.	−	51073.0i	−0.829273	−	0.146223i
$592$		0			0
$593$	−16303.7	−	5934.06i	−0.0463635	−	0.0168749i	0.318734	−	0.947844i	$-0.396742\pi$
$593$	−16303.7	−	5934.06i	−0.0463635	−	0.0168749i	−0.365098	+	0.930969i	$0.618964\pi$
$594$		0			0
$595$	−3277.81	+	2750.41i	−0.00925869	+	0.00776896i
$596$		0			0
$597$	536140.	+	309540.i	1.50428	+	0.868498i
$598$		0			0
$599$	−464365.	+	81880.1i	−1.29421	+	0.228205i	−0.778005	−	0.628258i	$-0.783768\pi$
$599$	−464365.	+	81880.1i	−1.29421	+	0.228205i	−0.516209	+	0.856463i	$0.672657\pi$
$600$		0			0
$601$	29997.0	−	17318.8i	0.0830478	−	0.0479477i	−0.457901	−	0.889003i	$-0.651399\pi$
$601$	29997.0	−	17318.8i	0.0830478	−	0.0479477i	0.540949	+	0.841055i	$0.318065\pi$
$602$		0			0
$603$	139330.	+	382806.i	0.383186	+	1.05280i
$604$		0			0
$605$	−24122.3	−	20241.0i	−0.0659035	−	0.0552996i
$606$		0			0
$607$			65.3741i			0.000177431i	1.00000		8.87153e-5i	$2.82389e-5\pi$
$607$			65.3741i			0.000177431i	−1.00000		8.87153e-5i	$0.999972\pi$
$608$		0			0
$609$	55585.0			0.149873
$610$		0			0
$611$	−116162.	+	138436.i	−0.311158	+	0.370824i
$612$		0			0
$613$	189683.	−	69038.8i	0.504785	−	0.183727i	−0.0770598	−	0.997026i	$-0.524553\pi$
$613$	189683.	−	69038.8i	0.504785	−	0.183727i	0.581845	+	0.813300i	$0.302331\pi$
$614$		0			0
$615$	10432.2	+	18069.1i	0.0275821	+	0.0477735i
$616$		0			0
$617$	13154.6	+	74603.7i	0.0345548	+	0.195970i	0.997198	−	0.0748014i	$-0.0238323\pi$
$617$	13154.6	+	74603.7i	0.0345548	+	0.195970i	−0.962644	+	0.270772i	$0.912721\pi$
$618$		0			0
$619$	291209.	−	504389.i	0.760017	−	1.31639i	−0.182824	−	0.983146i	$-0.558524\pi$
$619$	291209.	−	504389.i	0.760017	−	1.31639i	0.942841	−	0.333243i	$-0.108143\pi$
$620$		0			0
$621$	−6866.53	−	8183.21i	−0.0178055	−	0.0212198i
$622$		0			0
$623$	11436.9	−	31422.7i	0.0294668	−	0.0809594i
$624$		0			0
$625$	−37936.4	+	215148.i	−0.0971173	+	0.550779i
$626$		0			0
$627$	531182.	−	15247.6i	1.35116	−	0.0387853i
$628$		0			0
$629$	72611.0	+	12803.3i	0.183528	+	0.0323609i
$630$		0			0
$631$	322731.	+	117464.i	0.810553	+	0.295017i	0.713852	−	0.700296i	$-0.246949\pi$
$631$	322731.	+	117464.i	0.810553	+	0.295017i	0.0967008	+	0.995313i	$0.469171\pi$
$632$		0			0
$633$	−489309.	+	410579.i	−1.22117	+	1.02468i
$634$		0			0
$635$	−237716.	−	137245.i	−0.589537	−	0.340369i
$636$		0			0
$637$	−268171.	+	47285.7i	−0.660895	+	0.116534i
$638$		0			0
$639$	39150.5	−	22603.6i	0.0958817	−	0.0553573i
$640$		0			0
$641$	−22622.2	−	62153.9i	−0.0550577	−	0.151270i	0.909115	−	0.416545i	$-0.136759\pi$
$641$	−22622.2	−	62153.9i	−0.0550577	−	0.151270i	−0.964173	+	0.265276i	$0.914537\pi$
$642$		0			0
$643$	323935.	+	271814.i	0.783496	+	0.657431i	0.944126	−	0.329584i	$-0.106908\pi$
$643$	323935.	+	271814.i	0.783496	+	0.657431i	−0.160631	+	0.987015i	$0.551353\pi$
$644$		0			0
$645$			88813.1i			0.213480i
$646$		0			0
$647$	−748603.			−1.78831			−0.894156	−	0.447757i	$-0.852223\pi$
$647$	−748603.			−1.78831			−0.894156	+	0.447757i	$0.852223\pi$
$648$		0			0
$649$	−358856.	+	427667.i	−0.851982	+	1.01535i
$650$		0			0
$651$	−30844.6	+	11226.5i	−0.0727807	+	0.0264900i
$652$		0			0
$653$	395000.	+	684159.i	0.926340	+	1.60447i	0.789392	+	0.613889i	$0.210396\pi$
$653$	395000.	+	684159.i	0.926340	+	1.60447i	0.136947	+	0.990578i	$0.456271\pi$
$654$		0			0
$655$	−81221.1	−	460628.i	−0.189316	−	1.07366i
$656$		0			0
$657$	−299053.	+	517975.i	−0.692816	+	1.19999i
$658$		0			0
$659$	75605.7	+	90103.4i	0.174094	+	0.207477i	0.846035	−	0.533128i	$-0.178983\pi$
$659$	75605.7	+	90103.4i	0.174094	+	0.207477i	−0.671941	+	0.740605i	$0.734539\pi$
$660$		0			0
$661$	−107073.	+	294181.i	−0.245063	+	0.673304i	0.754787	+	0.655970i	$0.227740\pi$
$661$	−107073.	+	294181.i	−0.245063	+	0.673304i	−0.999850	+	0.0173341i	$0.994482\pi$
$662$		0			0
$663$	17048.5	−	96686.7i	0.0387845	−	0.219958i
$664$		0			0
$665$	−19616.1	+	12088.8i	−0.0443578	+	0.0273363i
$666$		0			0
$667$	333389.	+	58785.5i	0.749375	+	0.132135i
$668$		0			0
$669$	−798621.	−	290674.i	−1.78439	−	0.649463i
$670$		0			0
$671$	−140426.	+	117832.i	−0.311891	+	0.261708i
$672$		0			0
$673$	−596200.	−	344216.i	−1.31632	−	0.759978i	−0.333186	−	0.942861i	$-0.608124\pi$
$673$	−596200.	−	344216.i	−1.31632	−	0.759978i	−0.983135	+	0.182883i	$0.941457\pi$
$674$		0			0
$675$	−8903.50	+	1569.93i	−0.0195413	+	0.00344566i
$676$		0			0
$677$	−272892.	+	157554.i	−0.595406	+	0.343758i	−0.767232	−	0.641369i	$-0.778367\pi$
$677$	−272892.	+	157554.i	−0.595406	+	0.343758i	0.171826	+	0.985127i	$0.445033\pi$
$678$		0			0
$679$	−5287.29	−	14526.7i	−0.0114682	−	0.0315085i
$680$		0			0
$681$	653512.	+	548361.i	1.40916	+	1.18242i
$682$		0			0
$683$		−	732776.i		−	1.57083i	−0.618968	−	0.785416i	$-0.712449\pi$
$683$		−	732776.i		−	1.57083i	0.618968	−	0.785416i	$-0.287551\pi$
$684$		0			0
$685$	85688.8			0.182618
$686$		0			0
$687$	768293.	−	915616.i	1.62785	−	1.93999i
$688$		0			0
$689$	274298.	−	99836.3i	0.577809	−	0.210305i
$690$		0			0
$691$	10550.9	+	18274.7i	0.0220970	+	0.0382732i	0.876862	−	0.480741i	$-0.159632\pi$
$691$	10550.9	+	18274.7i	0.0220970	+	0.0382732i	−0.854765	+	0.519014i	$0.826299\pi$
$692$		0			0
$693$	−5236.48	−	29697.5i	−0.0109037	−	0.0618378i
$694$		0			0
$695$	−233017.	+	403597.i	−0.482411	+	0.835561i
$696$		0			0
$697$	3419.06	+	4074.68i	0.00703787	+	0.00838740i
$698$		0			0
$699$	394314.	−	1.08337e6i	0.807027	−	2.21729i
$700$		0			0
$701$	93054.4	−	527738.i	0.189366	−	1.07395i	−0.730851	−	0.682537i	$-0.760877\pi$
$701$	93054.4	−	527738.i	0.189366	−	1.07395i	0.920217	−	0.391409i	$-0.128012\pi$
$702$		0			0
$703$	376844.	+	125036.i	0.762520	+	0.253003i
$704$		0			0
$705$	−410959.	−	72463.2i	−0.826838	−	0.145794i
$706$		0			0
$707$	−39075.4	−	14222.3i	−0.0781744	−	0.0284531i
$708$		0			0
$709$	483631.	−	405815.i	0.962103	−	0.807300i	−0.0191907	−	0.999816i	$-0.506109\pi$
$709$	483631.	−	405815.i	0.962103	−	0.807300i	0.981294	+	0.192515i	$0.0616645\pi$
$710$		0			0
$711$	−852714.	−	492315.i	−1.68680	−	0.973876i
$712$		0			0
$713$	−196873.	+	34714.0i	−0.387264	+	0.0682851i
$714$		0			0
$715$	230800.	−	133253.i	0.451466	−	0.260654i
$716$		0			0
$717$	−217534.	−	597669.i	−0.423144	−	1.16258i
$718$		0			0
$719$	198461.	+	166529.i	0.383899	+	0.322130i	0.814231	−	0.580541i	$-0.197159\pi$
$719$	198461.	+	166529.i	0.383899	+	0.322130i	−0.430332	+	0.902671i	$0.641603\pi$
$720$		0			0
$721$			63412.7i			0.121985i
$722$		0			0
$723$	1.17997e6			2.25733
$724$		0			0
$725$	184165.	−	219479.i	0.350373	−	0.417558i
$726$		0			0
$727$	−182622.	+	66469.1i	−0.345529	+	0.125762i	−0.508955	−	0.860793i	$-0.669968\pi$
$727$	−182622.	+	66469.1i	−0.345529	+	0.125762i	0.163425	+	0.986556i	$0.447746\pi$
$728$		0			0
$729$	−287522.	−	498003.i	−0.541023	−	0.937080i
$730$		0			0
$731$	3931.69	+	22297.7i	0.00735775	+	0.0417279i
$732$		0			0
$733$	−96413.3	+	166993.i	−0.179444	+	0.310806i	−0.941690	−	0.336481i	$-0.890763\pi$
$733$	−96413.3	+	166993.i	−0.179444	+	0.310806i	0.762246	+	0.647287i	$0.224097\pi$
$734$		0			0
$735$	−404183.	−	481687.i	−0.748176	−	0.891642i
$736$		0			0
$737$	−188959.	+	519161.i	−0.347883	+	0.955800i
$738$		0			0
$739$	139019.	−	788416.i	0.254557	−	1.44367i	−0.542650	−	0.839959i	$-0.682579\pi$
$739$	139019.	−	788416.i	0.254557	−	1.44367i	0.797207	−	0.603706i	$-0.206310\pi$
$740$		0			0
$741$	166495.	−	501795.i	0.303224	−	0.913881i
$742$		0			0
$743$	−329829.	−	58157.8i	−0.597463	−	0.105349i	−0.133265	−	0.991080i	$-0.542546\pi$
$743$	−329829.	−	58157.8i	−0.597463	−	0.105349i	−0.464198	+	0.885731i	$0.653657\pi$
$744$		0			0
$745$	603842.	+	219781.i	1.08795	+	0.395983i
$746$		0			0
$747$	−417768.	+	350549.i	−0.748675	+	0.628213i
$748$		0			0
$749$	−1393.76	−	804.686i	−0.00248441	−	0.00143438i
$750$		0			0
$751$	−16726.8	+	2949.38i	−0.0296574	+	0.00522939i	−0.188457	−	0.982081i	$-0.560349\pi$
$751$	−16726.8	+	2949.38i	−0.0296574	+	0.00522939i	0.158800	+	0.987311i	$0.449238\pi$
$752$		0			0
$753$	−507768.	+	293160.i	−0.895519	+	0.517028i
$754$		0			0
$755$	−169584.	−	465929.i	−0.297503	−	0.817383i
$756$		0			0
$757$	483204.	+	405457.i	0.843216	+	0.707543i	0.958285	−	0.285815i	$-0.0922643\pi$
$757$	483204.	+	405457.i	0.843216	+	0.707543i	−0.115068	+	0.993358i	$0.536709\pi$
$758$		0			0
$759$		−	359925.i		−	0.624782i
$760$		0			0
$761$	−411928.			−0.711299			−0.355650	−	0.934619i	$-0.615740\pi$
$761$	−411928.			−0.711299			−0.355650	+	0.934619i	$0.615740\pi$
$762$		0			0
$763$	−8463.30	+	10086.2i	−0.0145375	+	0.0173251i
$764$		0			0
$765$	108706.	−	39565.6i	0.185750	−	0.0676075i
$766$		0			0
$767$	277719.	+	481023.i	0.472079	+	0.817665i
$768$		0			0
$769$	−14069.9	−	79794.5i	−0.0237924	−	0.134934i	0.970598	−	0.240707i	$-0.0773793\pi$
$769$	−14069.9	−	79794.5i	−0.0237924	−	0.134934i	−0.994390	+	0.105773i	$0.966268\pi$
$770$		0			0
$771$	−660972.	+	1.14484e6i	−1.11192	+	1.92590i
$772$		0			0
$773$	533118.	+	635345.i	0.892204	+	1.06329i	0.997626	+	0.0688597i	$0.0219361\pi$
$773$	533118.	+	635345.i	0.892204	+	1.06329i	−0.105422	+	0.994428i	$0.533619\pi$
$774$		0			0
$775$	−57866.4	+	158987.i	−0.0963436	+	0.264702i
$776$		0			0
$777$	7667.58	−	43485.0i	0.0127004	−	0.0720274i
$778$		0			0
$779$	15027.7	+	24385.0i	0.0247638	+	0.0401835i
$780$		0			0
$781$	60378.4	+	10646.3i	0.0989873	+	0.0174541i
$782$		0			0
$783$	−56840.9	−	20688.4i	−0.0927124	−	0.0337445i
$784$		0			0
$785$	−72997.0	+	61251.8i	−0.118458	+	0.0993984i
$786$		0			0
$787$	−743260.	−	429121.i	−1.20003	−	0.692836i	−0.239466	−	0.970905i	$-0.576972\pi$
$787$	−743260.	−	429121.i	−1.20003	−	0.692836i	−0.960561	+	0.278069i	$0.910306\pi$
$788$		0			0
$789$	911143.	−	160659.i	1.46363	−	0.258078i
$790$		0			0
$791$	−59295.0	+	34234.0i	−0.0947688	+	0.0547148i
$792$		0			0
$793$	62377.7	+	171381.i	0.0991935	+	0.272532i
$794$		0			0
$795$	516347.	+	433267.i	0.816973	+	0.685522i
$796$		0			0
$797$		−	275787.i		−	0.434167i	−0.976153	−	0.217083i	$-0.930346\pi$
$797$		−	275787.i		−	0.434167i	0.976153	−	0.217083i	$-0.0696544\pi$
$798$		0			0
$799$	−106385.			−0.166643
$800$		0			0
$801$	−581115.	+	692545.i	−0.905726	+	1.07940i
$802$		0			0
$803$	−762233.	+	277430.i	−1.18211	+	0.430251i
$804$		0			0
$805$	7803.31	+	13515.7i	0.0120417	+	0.0208568i
$806$		0			0
$807$	−11883.6	−	67395.2i	−0.0182474	−	0.103486i
$808$		0			0
$809$	−564969.	+	978555.i	−0.863233	+	1.49516i	0.00555901	+	0.999985i	$0.498231\pi$
$809$	−564969.	+	978555.i	−0.863233	+	1.49516i	−0.868792	+	0.495178i	$0.835103\pi$
$810$		0			0
$811$	432761.	+	515745.i	0.657971	+	0.784139i	0.987093	−	0.160148i	$-0.0511972\pi$
$811$	432761.	+	515745.i	0.657971	+	0.784139i	−0.329122	+	0.944287i	$0.606753\pi$
$812$		0			0
$813$	402190.	−	1.10501e6i	0.608485	−	1.67180i
$814$		0			0
$815$	−8272.44	+	46915.4i	−0.0124543	+	0.0706317i
$816$		0			0
$817$	3498.48	+	121877.i	0.00524126	+	0.182590i
$818$		0			0
$819$	−29546.3	−	5209.82i	−0.0440490	−	0.00776702i
$820$		0			0
$821$	644340.	+	234521.i	0.955937	+	0.347932i	0.772440	−	0.635088i	$-0.219036\pi$
$821$	644340.	+	234521.i	0.955937	+	0.347932i	0.183497	+	0.983020i	$0.441258\pi$
$822$		0			0
$823$	−522167.	+	438150.i	−0.770921	+	0.646879i	−0.940945	−	0.338561i	$-0.890060\pi$
$823$	−522167.	+	438150.i	−0.770921	+	0.646879i	0.170024	+	0.985440i	$0.445615\pi$
$824$		0			0
$825$	−263804.	−	152308.i	−0.387591	−	0.223776i
$826$		0			0
$827$	−177534.	+	31304.0i	−0.259579	+	0.0457708i	−0.301923	−	0.953332i	$-0.597629\pi$
$827$	−177534.	+	31304.0i	−0.259579	+	0.0457708i	0.0423440	+	0.999103i	$0.486517\pi$
$828$		0			0
$829$	−928800.	+	536243.i	−1.35149	+	0.780283i	−0.988458	−	0.151493i	$-0.951592\pi$
$829$	−928800.	+	536243.i	−1.35149	+	0.780283i	−0.363032	+	0.931777i	$0.618258\pi$
$830$		0			0
$831$	−38818.0	−	106652.i	−0.0562123	−	0.154442i
$832$		0			0
$833$	−122800.	−	103041.i	−0.176973	−	0.148498i
$834$		0			0
$835$		−	970913.i		−	1.39254i
$836$		0			0
$837$	35719.9			0.0509870
$838$		0			0
$839$	−146454.	+	174538.i	−0.208055	+	0.247950i	−0.859973	−	0.510339i	$-0.829520\pi$
$839$	−146454.	+	174538.i	−0.208055	+	0.247950i	0.651918	+	0.758289i	$0.273964\pi$
$840$		0			0
$841$	1.13669e6	−	413723.i	1.60713	−	0.584948i
$842$		0			0
$843$	241531.	+	418344.i	0.339874	+	0.588679i
$844$		0			0
$845$	55363.7	+	313983.i	0.0775375	+	0.439737i
$846$		0			0
$847$	−2403.84	+	4163.56i	−0.00335072	+	0.00580361i
$848$		0			0
$849$	1.03953e6	+	1.23887e6i	1.44219	+	1.71874i
$850$		0			0
$851$	91977.7	−	252707.i	0.127006	−	0.348945i
$852$		0			0
$853$	160381.	−	909569.i	0.220423	−	1.25008i	−0.650822	−	0.759230i	$-0.725576\pi$
$853$	160381.	−	909569.i	0.220423	−	1.25008i	0.871245	−	0.490849i	$-0.163313\pi$
$854$		0			0
$855$	610133.	−	125733.i	0.834627	−	0.171996i
$856$		0			0
$857$	727876.	+	128344.i	0.991051	+	0.174749i	0.645590	−	0.763684i	$-0.276612\pi$
$857$	727876.	+	128344.i	0.991051	+	0.174749i	0.345461	+	0.938433i	$0.387723\pi$
$858$		0			0
$859$	990958.	+	360679.i	1.34298	+	0.488804i	0.910749	−	0.412960i	$-0.135505\pi$
$859$	990958.	+	360679.i	1.34298	+	0.488804i	0.432229	+	0.901764i	$0.357727\pi$
$860$		0			0
$861$	2440.23	−	2047.59i	0.00329173	−	0.00276209i
$862$		0			0
$863$	529157.	+	305509.i	0.710498	+	0.410206i	0.811245	−	0.584706i	$-0.198790\pi$
$863$	529157.	+	305509.i	0.710498	+	0.410206i	−0.100747	+	0.994912i	$0.532123\pi$
$864$		0			0
$865$	−146996.	+	25919.4i	−0.196460	+	0.0346412i
$866$		0			0
$867$	−880177.	+	508170.i	−1.17093	+	0.676038i
$868$		0			0
$869$	−456718.	−	1.25482e6i	−0.604795	−	1.66166i
$870$		0			0
$871$	421069.	+	353319.i	0.555031	+	0.465726i
$872$		0			0
$873$			417944.i			0.548391i
$874$		0			0
$875$	53100.9			0.0693563
$876$		0			0
$877$	141645.	−	168806.i	0.184163	−	0.219477i	−0.666062	−	0.745896i	$-0.732021\pi$
$877$	141645.	−	168806.i	0.184163	−	0.219477i	0.850225	+	0.526420i	$0.176466\pi$
$878$		0			0
$879$	126530.	−	46053.2i	0.163763	−	0.0596049i
$880$		0			0
$881$	−445430.	−	771508.i	−0.573889	−	0.994005i	−0.996161	−	0.0875361i	$-0.972101\pi$
$881$	−445430.	−	771508.i	−0.573889	−	0.994005i	0.422272	−	0.906469i	$-0.361233\pi$
$882$		0			0
$883$	157356.	+	892408.i	0.201818	+	1.14457i	0.902368	+	0.430966i	$0.141827\pi$
$883$	157356.	+	892408.i	0.201818	+	1.14457i	−0.700550	+	0.713603i	$0.747062\pi$
$884$		0			0
$885$	−641293.	+	1.11075e6i	−0.818785	+	1.41818i
$886$		0			0
$887$	−236211.	−	281505.i	−0.300229	−	0.357799i	0.594747	−	0.803913i	$-0.297252\pi$
$887$	−236211.	−	281505.i	−0.300229	−	0.357799i	−0.894976	+	0.446114i	$0.852808\pi$
$888$		0			0
$889$	−14333.4	+	39380.7i	−0.0181362	+	0.0498287i
$890$		0			0
$891$	124706.	−	707241.i	0.157083	−	0.890865i
$892$		0			0
$893$	−566806.	−	83251.6i	−0.710774	−	0.104397i
$894$		0			0
$895$	343152.	+	60507.0i	0.428392	+	0.0755370i
$896$		0			0
$897$	−336497.	−	122475.i	−0.418211	−	0.152216i
$898$		0			0
$899$	−867151.	+	727626.i	−1.07294	+	0.900303i
$900$		0			0
$901$	148816.	+	85919.2i	0.183316	+	0.105838i
$902$		0			0
$903$	13353.6	−	2354.60i	0.0163766	−	0.00288763i
$904$		0			0
$905$	−152908.	+	88281.2i	−0.186695	+	0.107788i
$906$		0			0
$907$	−459883.	−	1.26352e6i	−0.559027	−	1.53592i	−0.821052	−	0.570854i	$-0.806612\pi$
$907$	−459883.	−	1.26352e6i	−0.559027	−	1.53592i	0.262024	−	0.965061i	$-0.415610\pi$
$908$		0			0
$909$	861208.	+	722639.i	1.04227	+	0.874569i
$910$		0			0
$911$		−	33412.4i		−	0.0402597i	−0.999797	−	0.0201299i	$-0.993592\pi$
$911$		−	33412.4i		−	0.0402597i	0.999797	−	0.0201299i	$-0.00640797\pi$
$912$		0			0
$913$	−739612.			−0.887284
$914$		0			0
$915$	−270705.	+	322614.i	−0.323336	+	0.385337i
$916$		0			0
$917$	−67104.9	+	24424.2i	−0.0798023	+	0.0290457i
$918$		0			0
$919$	642264.	+	1.11243e6i	0.760471	+	1.31717i	0.942608	+	0.333902i	$0.108365\pi$
$919$	642264.	+	1.11243e6i	0.760471	+	1.31717i	−0.182137	+	0.983273i	$0.558301\pi$
$920$		0			0
$921$	−10185.4	−	57764.3i	−0.0120077	−	0.0680989i
$922$		0			0
$923$	30498.8	−	52825.5i	0.0357997	−	0.0620070i
$924$		0			0
$925$	−146298.	−	174351.i	−0.170984	−	0.203770i
$926$		0			0
$927$	586360.	−	1.61101e6i	0.682347	−	1.87473i
$928$		0			0
$929$	144556.	−	819819.i	0.167496	−	0.949919i	−0.778957	−	0.627078i	$-0.784251\pi$
$929$	144556.	−	819819.i	0.167496	−	0.949919i	0.946453	−	0.322842i	$-0.104638\pi$
$930$		0			0
$931$	−573628.	−	645089.i	−0.661806	−	0.744252i
$932$		0			0
$933$	−268351.	−	47317.6i	−0.308277	−	0.0543575i
$934$		0			0
$935$	147426.	+	53658.8i	0.168637	+	0.0613788i
$936$		0			0
$937$	243893.	−	204651.i	0.277792	−	0.233096i	−0.493237	−	0.869895i	$-0.664186\pi$
$937$	243893.	−	204651.i	0.277792	−	0.233096i	0.771030	+	0.636799i	$0.219742\pi$
$938$		0			0
$939$	319611.	+	184527.i	0.362485	+	0.209281i
$940$		0			0
$941$	−341216.	+	60165.7i	−0.385346	+	0.0679469i	−0.362965	−	0.931803i	$-0.618236\pi$
$941$	−341216.	+	60165.7i	−0.385346	+	0.0679469i	−0.0223805	+	0.999750i	$0.507125\pi$
$942$		0			0
$943$	16801.5	−	9700.38i	0.0188941	−	0.0109085i
$944$		0			0
$945$	−953.754	−	2620.42i	−0.00106800	−	0.00293432i
$946$		0			0
$947$	14070.7	+	11806.7i	0.0156898	+	0.0131653i	0.650599	−	0.759422i	$-0.274518\pi$
$947$	14070.7	+	11806.7i	0.0156898	+	0.0131653i	−0.634909	+	0.772587i	$0.718962\pi$
$948$		0			0
$949$			807021.i			0.896092i
$950$		0			0
$951$	170539.			0.188566
$952$		0			0
$953$	−335588.	+	399938.i	−0.369505	+	0.440359i	−0.918473	−	0.395484i	$-0.870577\pi$
$953$	−335588.	+	399938.i	−0.369505	+	0.440359i	0.548967	+	0.835844i	$0.315021\pi$
$954$		0			0
$955$	232551.	−	84641.8i	0.254984	−	0.0928064i
$956$		0			0
$957$	−1.01903e6	−	1.76501e6i	−1.11266	−	1.92719i
$958$		0			0
$959$	−2271.77	−	12883.8i	−0.00247017	−	0.0140090i
$960$		0			0
$961$	−127530.	+	220888.i	−0.138091	+	0.239181i
$962$		0			0
$963$	27968.0	+	33330.9i	0.0301584	+	0.0359414i
$964$		0			0
$965$	−15723.4	+	43199.8i	−0.0168847	+	0.0463903i
$966$		0			0
$967$	−90193.0	+	511510.i	−0.0964539	+	0.547017i	0.897838	+	0.440325i	$0.145137\pi$
$967$	−90193.0	+	511510.i	−0.0964539	+	0.547017i	−0.994292	+	0.106692i	$0.965974\pi$
$968$		0			0
$969$	289291.	−	114797.i	0.308097	−	0.122259i
$970$		0			0
$971$	183235.	+	32309.3i	0.194344	+	0.0342680i	0.269973	−	0.962868i	$-0.412985\pi$
$971$	183235.	+	32309.3i	0.194344	+	0.0342680i	−0.0756290	+	0.997136i	$0.524096\pi$
$972$		0			0
$973$	66860.9	+	24335.4i	0.0706231	+	0.0257047i
$974$		0			0
$975$	−232161.	+	194806.i	−0.244219	+	0.204924i
$976$		0			0
$977$	−1.31897e6	−	761510.i	−1.38181	−	0.797786i	−0.389433	−	0.921055i	$-0.627329\pi$
$977$	−1.31897e6	−	761510.i	−1.38181	−	0.797786i	−0.992373	+	0.123269i	$0.960662\pi$
$978$		0			0
$979$	−1.20745e6	+	212906.i	−1.25981	+	0.222138i
$980$		0			0
$981$	308276.	−	177983.i	0.320333	−	0.184944i
$982$		0			0
$983$	−309658.	−	850779.i	−0.320461	−	0.880460i	−0.990423	−	0.138064i	$-0.955912\pi$
$983$	−309658.	−	850779.i	−0.320461	−	0.880460i	0.669962	−	0.742395i	$-0.266310\pi$
$984$		0			0
$985$	−358205.	−	300570.i	−0.369198	−	0.309794i
$986$		0			0
$987$			63711.3i			0.0654007i
$988$		0			0
$989$	82582.7			0.0844299
$990$		0			0
$991$	862721.	−	1.02815e6i	0.878463	−	1.04691i	−0.120070	−	0.992765i	$-0.538312\pi$
$991$	862721.	−	1.02815e6i	0.878463	−	1.04691i	0.998533	−	0.0541456i	$-0.0172435\pi$
$992$		0			0
$993$	1.38161e6	−	502864.i	1.40115	−	0.509978i
$994$		0			0
$995$	492124.	+	852384.i	0.497082	+	0.860972i
$996$		0			0
$997$	−270146.	−	1.53207e6i	−0.271774	−	1.54131i	−0.749025	−	0.662542i	$-0.769478\pi$
$997$	−270146.	−	1.53207e6i	−0.271774	−	1.54131i	0.477251	−	0.878767i	$-0.341633\pi$
$998$		0			0
$999$	−24025.7	+	41613.8i	−0.0240738	+	0.0416971i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.21.1	✓	42
19.10	odd	18	inner	76.5.j.a.29.1	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.21.1	✓	42	1.1	even	1	trivial
76.5.j.a.29.1	yes	42	19.10	odd	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.26669 + 9.85185i) q^{3} +(-19.2135 + 6.99315i) q^{5} +(1.56085 + 2.70347i) q^{7} +(-14.6554 - 83.1149i) q^{9} +O(q^{10})\)	\(q+(-8.26669 + 9.85185i) q^{3} +(-19.2135 + 6.99315i) q^{5} +(1.56085 + 2.70347i) q^{7} +(-14.6554 - 83.1149i) q^{9} +(57.2296 - 99.1246i) q^{11} +(-73.1984 - 87.2344i) q^{13} +(89.9367 - 247.099i) q^{15} +(11.6409 - 66.0190i) q^{17} +(113.685 - 342.632i) q^{19} +(-39.5372 - 6.97148i) q^{21} +(-229.764 - 83.6274i) q^{23} +(-158.522 + 133.016i) q^{25} +(37.8358 + 21.8445i) q^{27} +(-1363.50 + 240.421i) q^{29} +(708.058 - 408.797i) q^{31} +(503.462 + 1383.25i) q^{33} +(-48.8952 - 41.0279i) q^{35} +1099.85i q^{37} +1464.53 q^{39} +(-51.0022 + 60.7821i) q^{41} +(-317.379 + 115.516i) q^{43} +(862.817 + 1494.44i) q^{45} +(-275.570 - 1562.84i) q^{47} +(1195.63 - 2070.89i) q^{49} +(554.178 + 660.443i) q^{51} +(-876.707 + 2408.73i) q^{53} +(-406.389 + 2304.75i) q^{55} +(2435.76 + 3952.44i) q^{57} +(-4803.44 - 846.977i) q^{59} +(-1504.98 - 547.766i) q^{61} +(201.824 - 169.350i) q^{63} +(2016.44 + 1164.19i) q^{65} +(-4753.54 + 838.177i) q^{67} +(2723.28 - 1572.28i) q^{69} +(183.202 + 503.344i) q^{71} +(-5428.81 - 4555.31i) q^{73} -2661.34i q^{75} +357.307 q^{77} +(7499.16 - 8937.15i) q^{79} +(5895.91 - 2145.94i) q^{81} +(-3230.90 - 5596.08i) q^{83} +(238.018 + 1349.86i) q^{85} +(8903.00 - 15420.5i) q^{87} +(-6885.48 - 8205.80i) q^{89} +(121.584 - 334.049i) q^{91} +(-1825.88 + 10355.1i) q^{93} +(211.792 + 7378.19i) q^{95} +(-4876.88 - 859.926i) q^{97} +(-9077.46 - 3303.92i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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