LMFDB - Embedded newform 76.5.j.a.41.4

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			41.4
Character	$\chi$	$=$	76.41
Dual form			76.5.j.a.13.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.180467 - 0.495829i) q^{3} +(-0.550159 + 3.12011i) q^{5} +(-25.9990 - 45.0316i) q^{7} +(61.8363 - 51.8868i) q^{9} +O(q^{10})$	\(q+(-0.180467 - 0.495829i) q^{3} +(-0.550159 + 3.12011i) q^{5} +(-25.9990 - 45.0316i) q^{7} +(61.8363 - 51.8868i) q^{9} +(-19.1428 + 33.1563i) q^{11} +(71.6974 - 196.987i) q^{13} +(1.64632 - 0.290291i) q^{15} +(-302.594 - 253.907i) q^{17} +(210.294 - 293.424i) q^{19} +(-17.6360 + 21.0178i) q^{21} +(-80.8917 - 458.759i) q^{23} +(577.875 + 210.329i) q^{25} +(-73.9000 - 42.6662i) q^{27} +(277.497 + 330.708i) q^{29} +(-1061.61 + 612.923i) q^{31} +(19.8945 + 3.50794i) q^{33} +(154.807 - 56.3452i) q^{35} -963.274i q^{37} -110.611 q^{39} +(866.860 + 2381.68i) q^{41} +(-372.808 + 2114.30i) q^{43} +(127.873 + 221.482i) q^{45} +(-1913.69 + 1605.78i) q^{47} +(-151.397 + 262.228i) q^{49} +(-71.2860 + 195.857i) q^{51} +(3624.29 - 639.061i) q^{53} +(-92.9197 - 77.9689i) q^{55} +(-183.439 - 51.3167i) q^{57} +(-897.018 + 1069.02i) q^{59} +(-83.2176 - 471.950i) q^{61} +(-3944.23 - 1435.58i) q^{63} +(575.175 + 332.078i) q^{65} +(-1813.84 - 2161.65i) q^{67} +(-212.868 + 122.899i) q^{69} +(5130.43 + 904.634i) q^{71} +(3346.39 - 1217.99i) q^{73} -324.485i q^{75} +1990.78 q^{77} +(-2475.89 - 6802.45i) q^{79} +(1127.57 - 6394.77i) q^{81} +(-3492.18 - 6048.63i) q^{83} +(958.691 - 804.437i) q^{85} +(113.895 - 197.273i) q^{87} +(-194.356 + 533.990i) q^{89} +(-10734.7 + 1892.82i) q^{91} +(495.491 + 415.766i) q^{93} +(799.817 + 817.570i) q^{95} +(6980.90 - 8319.52i) q^{97} +(536.656 + 3043.53i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

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
$3$	−0.180467	−	0.495829i	−0.0200519	−	0.0550921i	0.929263	−	0.369419i	$-0.120443\pi$
$3$	−0.180467	−	0.495829i	−0.0200519	−	0.0550921i	−0.949315	+	0.314327i	$0.898221\pi$
$4$		0			0
$5$	−0.550159	+	3.12011i	−0.0220064	+	0.124804i	−0.993832	−	0.110897i	$-0.964628\pi$
$5$	−0.550159	+	3.12011i	−0.0220064	+	0.124804i	0.971826	+	0.235701i	$0.0757387\pi$
$6$		0			0
$7$	−25.9990	−	45.0316i	−0.530592	−	0.919013i	−0.999363	−	0.0356926i	$-0.988636\pi$
$7$	−25.9990	−	45.0316i	−0.530592	−	0.919013i	0.468771	−	0.883320i	$-0.344697\pi$
$8$		0			0
$9$	61.8363	−	51.8868i	0.763411	−	0.640578i
$10$		0			0
$11$	−19.1428	+	33.1563i	−0.158205	+	0.274019i	−0.934221	−	0.356693i	$-0.883904\pi$
$11$	−19.1428	+	33.1563i	−0.158205	+	0.274019i	0.776016	+	0.630713i	$0.217237\pi$
$12$		0			0
$13$	71.6974	−	196.987i	0.424245	−	1.16560i	−0.525010	−	0.851096i	$-0.675939\pi$
$13$	71.6974	−	196.987i	0.424245	−	1.16560i	0.949255	−	0.314507i	$-0.101839\pi$
$14$		0			0
$15$	1.64632	−	0.290291i	0.00731700	−	0.00129018i
$16$		0			0
$17$	−302.594	−	253.907i	−1.04704	−	0.878570i	−0.0542595	−	0.998527i	$-0.517280\pi$
$17$	−302.594	−	253.907i	−1.04704	−	0.878570i	−0.992779	+	0.119957i	$0.961724\pi$
$18$		0			0
$19$	210.294	−	293.424i	0.582532	−	0.812808i
$19$	210.294	−	293.424i	0.582532	−	0.812808i
$20$		0			0
$21$	−17.6360	+	21.0178i	−0.0399910	+	0.0476594i
$22$		0			0
$23$	−80.8917	−	458.759i	−0.152914	−	0.867220i	−0.960668	−	0.277700i	$-0.910428\pi$
$23$	−80.8917	−	458.759i	−0.152914	−	0.867220i	0.807754	−	0.589520i	$-0.200683\pi$
$24$		0			0
$25$	577.875	+	210.329i	0.924601	+	0.336527i
$26$		0			0
$27$	−73.9000	−	42.6662i	−0.101372	−	0.0585270i
$28$		0			0
$29$	277.497	+	330.708i	0.329960	+	0.393231i	0.905363	−	0.424639i	$-0.139599\pi$
$29$	277.497	+	330.708i	0.329960	+	0.393231i	−0.575402	+	0.817870i	$0.695155\pi$
$30$		0			0
$31$	−1061.61	+	612.923i	−1.10470	+	0.637797i	−0.937451	−	0.348118i	$-0.886821\pi$
$31$	−1061.61	+	612.923i	−1.10470	+	0.637797i	−0.167246	+	0.985915i	$0.553487\pi$
$32$		0			0
$33$	19.8945	+	3.50794i	0.0182686	+	0.00322125i
$34$		0			0
$35$	154.807	−	56.3452i	0.126373	−	0.0459960i
$36$		0			0
$37$		−	963.274i		−	0.703634i	−0.936069	−	0.351817i	$-0.885564\pi$
$37$		−	963.274i		−	0.703634i	0.936069	−	0.351817i	$-0.114436\pi$
$38$		0			0
$39$	−110.611			−0.0727224
$40$		0			0
$41$	866.860	+	2381.68i	0.515681	+	1.41682i	0.875235	+	0.483698i	$0.160707\pi$
$41$	866.860	+	2381.68i	0.515681	+	1.41682i	−0.359554	+	0.933124i	$0.617071\pi$
$42$		0			0
$43$	−372.808	+	2114.30i	−0.201627	+	1.14348i	0.701034	+	0.713128i	$0.252722\pi$
$43$	−372.808	+	2114.30i	−0.201627	+	1.14348i	−0.902660	+	0.430354i	$0.858389\pi$
$44$		0			0
$45$	127.873	+	221.482i	0.0631470	+	0.109374i
$46$		0			0
$47$	−1913.69	+	1605.78i	−0.866316	+	0.726925i	−0.963319	−	0.268359i	$-0.913519\pi$
$47$	−1913.69	+	1605.78i	−0.866316	+	0.726925i	0.0970031	+	0.995284i	$0.469074\pi$
$48$		0			0
$49$	−151.397	+	262.228i	−0.0630560	+	0.109216i
$50$		0			0
$51$	−71.2860	+	195.857i	−0.0274072	+	0.0753005i
$52$		0			0
$53$	3624.29	−	639.061i	1.29024	−	0.227505i	0.513919	−	0.857839i	$-0.328193\pi$
$53$	3624.29	−	639.061i	1.29024	−	0.227505i	0.776324	+	0.630334i	$0.217082\pi$
$54$		0			0
$55$	−92.9197	−	77.9689i	−0.0307173	−	0.0257748i
$56$		0			0
$57$	−183.439	−	51.3167i	−0.0564601	−	0.0157946i
$58$		0			0
$59$	−897.018	+	1069.02i	−0.257690	+	0.307103i	−0.879342	−	0.476191i	$-0.842017\pi$
$59$	−897.018	+	1069.02i	−0.257690	+	0.307103i	0.621652	+	0.783293i	$0.286462\pi$
$60$		0			0
$61$	−83.2176	−	471.950i	−0.0223643	−	0.126834i	0.971582	−	0.236705i	$-0.0760675\pi$
$61$	−83.2176	−	471.950i	−0.0223643	−	0.126834i	−0.993946	+	0.109871i	$0.964956\pi$
$62$		0			0
$63$	−3944.23	−	1435.58i	−0.993759	−	0.361699i
$64$		0			0
$65$	575.175	+	332.078i	0.136136	+	0.0785983i
$66$		0			0
$67$	−1813.84	−	2161.65i	−0.404063	−	0.481544i	0.525191	−	0.850984i	$-0.323994\pi$
$67$	−1813.84	−	2161.65i	−0.404063	−	0.481544i	−0.929254	+	0.369441i	$0.879549\pi$
$68$		0			0
$69$	−212.868	+	122.899i	−0.0447107	+	0.0258138i
$70$		0			0
$71$	5130.43	+	904.634i	1.01774	+	0.179455i	0.657540	−	0.753419i	$-0.271597\pi$
$71$	5130.43	+	904.634i	1.01774	+	0.179455i	0.360201	+	0.932875i	$0.382708\pi$
$72$		0			0
$73$	3346.39	−	1217.99i	0.627959	−	0.228558i	−0.00838368	−	0.999965i	$-0.502669\pi$
$73$	3346.39	−	1217.99i	0.627959	−	0.228558i	0.636343	+	0.771406i	$0.280446\pi$
$74$		0			0
$75$		−	324.485i		−	0.0576862i
$76$		0			0
$77$	1990.78			0.335770
$78$		0			0
$79$	−2475.89	−	6802.45i	−0.396714	−	1.08996i	−0.963875	−	0.266355i	$-0.914181\pi$
$79$	−2475.89	−	6802.45i	−0.396714	−	1.08996i	0.567161	−	0.823607i	$-0.308042\pi$
$80$		0			0
$81$	1127.57	−	6394.77i	0.171860	−	0.974664i
$82$		0			0
$83$	−3492.18	−	6048.63i	−0.506921	−	0.878012i	−0.999968	−	0.00800975i	$-0.997450\pi$
$83$	−3492.18	−	6048.63i	−0.506921	−	0.878012i	0.493047	−	0.870003i	$-0.335883\pi$
$84$		0			0
$85$	958.691	−	804.437i	0.132691	−	0.111341i
$86$		0			0
$87$	113.895	−	197.273i	0.0150476	−	0.0260632i
$88$		0			0
$89$	−194.356	+	533.990i	−0.0245368	+	0.0674144i	−0.951356	−	0.308093i	$-0.900309\pi$
$89$	−194.356	+	533.990i	−0.0245368	+	0.0674144i	0.926819	+	0.375507i	$0.122531\pi$
$90$		0			0
$91$	−10734.7	+	1892.82i	−1.29630	+	0.228574i
$92$		0			0
$93$	495.491	+	415.766i	0.0572888	+	0.0480710i
$94$		0			0
$95$	799.817	+	817.570i	0.0886224	+	0.0905895i
$96$		0			0
$97$	6980.90	−	8319.52i	0.741939	−	0.884208i	−0.254625	−	0.967040i	$-0.581952\pi$
$97$	6980.90	−	8319.52i	0.741939	−	0.884208i	0.996564	+	0.0828316i	$0.0263964\pi$
$98$		0			0
$99$	536.656	+	3043.53i	0.0547552	+	0.310532i
$100$		0			0
$101$	6626.34	+	2411.79i	0.649578	+	0.236427i	0.645730	−	0.763566i	$-0.276553\pi$
$101$	6626.34	+	2411.79i	0.649578	+	0.236427i	0.00384764	+	0.999993i	$0.498775\pi$
$102$		0			0
$103$	4953.12	+	2859.68i	0.466879	+	0.269553i	0.714932	−	0.699194i	$-0.246458\pi$
$103$	4953.12	+	2859.68i	0.466879	+	0.269553i	−0.248053	+	0.968746i	$0.579791\pi$
$104$		0			0
$105$	−55.8751	−	66.5894i	−0.00506804	−	0.00603985i
$106$		0			0
$107$	1251.39	−	722.488i	0.109301	−	0.0631049i	−0.444353	−	0.895852i	$-0.646566\pi$
$107$	1251.39	−	722.488i	0.109301	−	0.0631049i	0.553654	+	0.832747i	$0.313233\pi$
$108$		0			0
$109$	−11571.3	−	2040.33i	−0.973931	−	0.171730i	−0.336032	−	0.941850i	$-0.609085\pi$
$109$	−11571.3	−	2040.33i	−0.973931	−	0.171730i	−0.637899	+	0.770120i	$0.720196\pi$
$110$		0			0
$111$	−477.619	+	173.839i	−0.0387646	+	0.0141092i
$112$		0			0
$113$			17403.4i			1.36294i	0.731845	+	0.681471i	$0.238659\pi$
$113$			17403.4i			1.36294i	−0.731845	+	0.681471i	$0.761341\pi$
$114$		0			0
$115$	1475.88			0.111598
$116$		0			0
$117$	−5787.53	−	15901.1i	−0.422787	−	1.16160i
$118$		0			0
$119$	−3566.68	+	20227.6i	−0.251866	+	1.42840i
$120$		0			0
$121$	6587.60	+	11410.1i	0.449942	+	0.779323i
$122$		0			0
$123$	1024.47	−	859.628i	0.0677153	−	0.0568199i
$124$		0			0
$125$	−1964.25	+	3402.18i	−0.125712	+	0.217739i
$126$		0			0
$127$	−525.524	+	1443.87i	−0.0325826	+	0.0895199i	−0.954917	−	0.296872i	$-0.904057\pi$
$127$	−525.524	+	1443.87i	−0.0325826	+	0.0895199i	0.922335	+	0.386392i	$0.126279\pi$
$128$		0			0
$129$	1115.61	−	196.712i	0.0670398	−	0.0118209i
$130$		0			0
$131$	1467.97	+	1231.77i	0.0855409	+	0.0717774i	0.684555	−	0.728961i	$-0.259996\pi$
$131$	1467.97	+	1231.77i	0.0855409	+	0.0717774i	−0.599014	+	0.800738i	$0.704441\pi$
$132$		0			0
$133$	−18680.8	−	1841.16i	−1.05607	−	0.104085i
$134$		0			0
$135$	173.780	−	207.103i	0.00953525	−	0.0113637i
$136$		0			0
$137$	5865.24	+	33263.4i	0.312496	+	1.77225i	0.585930	+	0.810362i	$0.300729\pi$
$137$	5865.24	+	33263.4i	0.312496	+	1.77225i	−0.273434	+	0.961891i	$0.588160\pi$
$138$		0			0
$139$	13242.3	+	4819.79i	0.685382	+	0.249459i	0.661157	−	0.750248i	$-0.270066\pi$
$139$	13242.3	+	4819.79i	0.685382	+	0.249459i	0.0242252	+	0.999707i	$0.492288\pi$
$140$		0			0
$141$	1141.55	+	659.074i	0.0574191	+	0.0331509i
$142$		0			0
$143$	5158.88	+	6148.11i	0.252280	+	0.300656i
$144$		0			0
$145$	−1184.51	+	683.877i	−0.0563382	+	0.0325269i
$146$		0			0
$147$	157.342	+	27.7437i	0.00728134	+	0.00128390i
$148$		0			0
$149$	17675.3	−	6433.30i	0.796151	−	0.289775i	0.0882604	−	0.996097i	$-0.471869\pi$
$149$	17675.3	−	6433.30i	0.796151	−	0.289775i	0.707890	+	0.706322i	$0.249647\pi$
$150$		0			0
$151$		−	24175.2i		−	1.06027i	−0.847913	−	0.530135i	$-0.822141\pi$
$151$		−	24175.2i		−	1.06027i	0.847913	−	0.530135i	$-0.177859\pi$
$152$		0			0
$153$	−31885.7			−1.36211
$154$		0			0
$155$	−1328.33	−	3649.55i	−0.0552894	−	0.151906i
$156$		0			0
$157$	1620.67	−	9191.29i	0.0657500	−	0.372887i	−0.934123	−	0.356951i	$-0.883816\pi$
$157$	1620.67	−	9191.29i	0.0657500	−	0.372887i	0.999873	−	0.0159357i	$-0.00507269\pi$
$158$		0			0
$159$	−970.930	−	1681.70i	−0.0384055	−	0.0665203i
$160$		0			0
$161$	−18555.6	+	15570.0i	−0.715851	+	0.600670i
$162$		0			0
$163$	−10154.6	+	17588.2i	−0.382196	+	0.661982i	−0.991376	−	0.131050i	$-0.958165\pi$
$163$	−10154.6	+	17588.2i	−0.382196	+	0.661982i	0.609180	+	0.793032i	$0.291499\pi$
$164$		0			0
$165$	−21.8903	+	60.1431i	−0.000804051	+	0.00220911i
$166$		0			0
$167$	37747.3	−	6655.86i	1.35348	−	0.238655i	0.550588	−	0.834777i	$-0.314403\pi$
$167$	37747.3	−	6655.86i	1.35348	−	0.238655i	0.802894	+	0.596121i	$0.203292\pi$
$168$		0			0
$169$	−11784.3	−	9888.23i	−0.412602	−	0.346215i
$170$		0			0
$171$	−2221.00	−	29055.7i	−0.0759549	−	0.993664i
$172$		0			0
$173$	16752.5	−	19964.8i	0.559741	−	0.667074i	−0.409751	−	0.912198i	$-0.634384\pi$
$173$	16752.5	−	19964.8i	0.559741	−	0.667074i	0.969492	+	0.245124i	$0.0788287\pi$
$174$		0			0
$175$	−5552.72	−	31491.0i	−0.181313	−	1.02828i
$176$		0			0
$177$	691.935	+	251.844i	0.0220861	+	0.00803868i
$178$		0			0
$179$	46834.0	+	27039.6i	1.46169	+	0.843907i	0.999090	−	0.0426605i	$-0.0135834\pi$
$179$	46834.0	+	27039.6i	1.46169	+	0.843907i	0.462600	+	0.886567i	$0.346917\pi$
$180$		0			0
$181$	−30865.0	−	36783.5i	−0.942127	−	1.12278i	−0.992277	−	0.124042i	$-0.960414\pi$
$181$	−30865.0	−	36783.5i	−0.942127	−	1.12278i	0.0501495	−	0.998742i	$-0.484030\pi$
$182$		0			0
$183$	−218.989	+	126.433i	−0.00653912	+	0.00377536i
$184$		0			0
$185$	3005.52	+	529.954i	0.0878165	+	0.0154844i
$186$		0			0
$187$	14211.1	−	5172.42i	0.406392	−	0.147915i
$188$		0			0
$189$			4437.12i			0.124216i
$190$		0			0
$191$	20431.6			0.560061			0.280030	−	0.959991i	$-0.409655\pi$
$191$	20431.6			0.560061			0.280030	+	0.959991i	$0.409655\pi$
$192$		0			0
$193$	−4955.27	−	13614.5i	−0.133031	−	0.365499i	0.855236	−	0.518240i	$-0.173412\pi$
$193$	−4955.27	−	13614.5i	−0.133031	−	0.365499i	−0.988266	+	0.152740i	$0.951190\pi$
$194$		0			0
$195$	60.8535	−	345.118i	0.00160036	−	0.00907607i
$196$		0			0
$197$	−27348.9	−	47369.8i	−0.704706	−	1.22059i	−0.966797	−	0.255544i	$-0.917745\pi$
$197$	−27348.9	−	47369.8i	−0.704706	−	1.22059i	0.262091	−	0.965043i	$-0.415588\pi$
$198$		0			0
$199$	−49582.7	+	41604.8i	−1.25206	+	1.05060i	−0.255574	+	0.966790i	$0.582264\pi$
$199$	−49582.7	+	41604.8i	−1.25206	+	1.05060i	−0.996482	+	0.0838097i	$0.973291\pi$
$200$		0			0
$201$	−744.470	+	1289.46i	−0.0184270	+	0.0319165i
$202$		0			0
$203$	7677.66	−	21094.2i	0.186310	−	0.511883i
$204$		0			0
$205$	−7908.00	+	1394.39i	−0.188174	+	0.0331801i
$206$		0			0
$207$	−28805.6	−	24170.8i	−0.672259	−	0.564092i
$208$		0			0
$209$	5703.23	+	12589.5i	0.130565	+	0.288215i
$210$		0			0
$211$	32105.5	−	38261.8i	0.721131	−	0.859411i	−0.273609	−	0.961841i	$-0.588217\pi$
$211$	32105.5	−	38261.8i	0.721131	−	0.859411i	0.994740	+	0.102430i	$0.0326619\pi$
$212$		0			0
$213$	−477.330	−	2707.07i	−0.0105211	−	0.0596679i
$214$		0			0
$215$	−6391.73	−	2326.40i	−0.138274	−	0.0503278i
$216$		0			0
$217$	55201.8	+	31870.8i	1.17229	+	0.676820i
$218$		0			0
$219$	−1207.83	−	1439.43i	−0.0251835	−	0.0300126i
$220$		0			0
$221$	−71711.5	+	41402.6i	−1.46826	+	0.847703i
$222$		0			0
$223$	74551.7	+	13145.5i	1.49916	+	0.264342i	0.862205	−	0.506559i	$-0.169083\pi$
$223$	74551.7	+	13145.5i	1.49916	+	0.264342i	0.636955	+	0.770901i	$0.280194\pi$
$224$		0			0
$225$	46647.0	−	16978.1i	0.921423	−	0.335370i
$226$		0			0
$227$			13420.6i			0.260448i	0.991485	+	0.130224i	$0.0415697\pi$
$227$			13420.6i			0.260448i	−0.991485	+	0.130224i	$0.958430\pi$
$228$		0			0
$229$	−19611.7			−0.373977			−0.186988	−	0.982362i	$-0.559873\pi$
$229$	−19611.7			−0.373977			−0.186988	+	0.982362i	$0.559873\pi$
$230$		0			0
$231$	−359.270	−	987.085i	−0.00673281	−	0.0184983i
$232$		0			0
$233$	−15352.7	+	87069.6i	−0.282796	+	1.60382i	0.430256	+	0.902707i	$0.358423\pi$
$233$	−15352.7	+	87069.6i	−0.282796	+	1.60382i	−0.713053	+	0.701110i	$0.752688\pi$
$234$		0			0
$235$	−3957.36	−	6854.36i	−0.0716589	−	0.124117i
$236$		0			0
$237$	−2926.04	+	2455.24i	−0.0520934	+	0.0437116i
$238$		0			0
$239$	−24137.8	+	41807.8i	−0.422572	+	0.731917i	−0.996190	−	0.0872061i	$-0.972206\pi$
$239$	−24137.8	+	41807.8i	−0.422572	+	0.731917i	0.573618	+	0.819123i	$0.305539\pi$
$240$		0			0
$241$	−12652.2	+	34761.5i	−0.217836	+	0.598501i	−0.999688	−	0.0249635i	$-0.992053\pi$
$241$	−12652.2	+	34761.5i	−0.217836	+	0.598501i	0.781852	+	0.623464i	$0.214275\pi$
$242$		0			0
$243$	−10181.1	+	1795.21i	−0.172418	+	0.0304020i
$244$		0			0
$245$	−734.887	−	616.643i	−0.0122430	−	0.0102731i
$246$		0			0
$247$	−42723.1	−	62462.9i	−0.700275	−	1.02383i
$248$		0			0
$249$	−2368.86	+	2823.10i	−0.0382068	+	0.0455331i
$250$		0			0
$251$	17755.8	+	100698.i	0.281833	+	1.59835i	0.716384	+	0.697706i	$0.245796\pi$
$251$	17755.8	+	100698.i	0.281833	+	1.59835i	−0.434551	+	0.900647i	$0.643093\pi$
$252$		0			0
$253$	16759.3	+	6099.88i	0.261827	+	0.0952972i
$254$		0			0
$255$	−571.875	−	330.172i	−0.00879470	−	0.00507762i
$256$		0			0
$257$	−1331.43	−	1586.74i	−0.0201582	−	0.0240237i	0.755871	−	0.654720i	$-0.227214\pi$
$257$	−1331.43	−	1586.74i	−0.0201582	−	0.0240237i	−0.776029	+	0.630697i	$0.782769\pi$
$258$		0			0
$259$	−43377.8	+	25044.2i	−0.646648	+	0.373342i
$260$		0			0
$261$	34318.7	+	6051.32i	0.503791	+	0.0888319i
$262$		0			0
$263$	71950.3	−	26187.8i	1.04021	−	0.378606i	0.235251	−	0.971935i	$-0.424409\pi$
$263$	71950.3	−	26187.8i	1.04021	−	0.378606i	0.804960	+	0.593329i	$0.202187\pi$
$264$		0			0
$265$			11659.8i			0.166034i
$266$		0			0
$267$	299.842			0.00420601
$268$		0			0
$269$	−40959.5	−	112535.i	−0.566044	−	1.55519i	−0.810626	−	0.585565i	$-0.800873\pi$
$269$	−40959.5	−	112535.i	−0.566044	−	1.55519i	0.244582	−	0.969629i	$-0.421349\pi$
$270$		0			0
$271$	−23494.3	+	133243.i	−0.319908	+	1.81429i	0.223371	+	0.974733i	$0.428294\pi$
$271$	−23494.3	+	133243.i	−0.319908	+	1.81429i	−0.543279	+	0.839552i	$0.682817\pi$
$272$		0			0
$273$	2875.77	+	4980.98i	0.0385859	+	0.0668328i
$274$		0			0
$275$	−18035.9	+	15133.9i	−0.238492	+	0.200118i
$276$		0			0
$277$	18417.5	−	31900.1i	0.240033	−	0.415750i	−0.720690	−	0.693257i	$-0.756175\pi$
$277$	18417.5	−	31900.1i	0.240033	−	0.415750i	0.960724	+	0.277507i	$0.0895083\pi$
$278$		0			0
$279$	−33843.6	+	92984.6i	−0.434779	+	1.19455i
$280$		0			0
$281$	28595.8	−	5042.22i	0.362152	−	0.0638571i	0.0103881	−	0.999946i	$-0.496693\pi$
$281$	28595.8	−	5042.22i	0.362152	−	0.0638571i	0.351763	+	0.936089i	$0.385582\pi$
$282$		0			0
$283$	−29691.9	−	24914.4i	−0.370736	−	0.311084i	0.438317	−	0.898821i	$-0.355575\pi$
$283$	−29691.9	−	24914.4i	−0.370736	−	0.311084i	−0.809053	+	0.587736i	$0.800019\pi$
$284$		0			0
$285$	261.034	−	544.117i	0.00321372	−	0.00669889i
$286$		0			0
$287$	84713.3	−	100957.i	1.02846	−	1.22567i
$288$		0			0
$289$	12591.4	+	71409.2i	0.150757	+	0.854985i
$290$		0			0
$291$	−5384.88	−	1959.94i	−0.0635902	−	0.0231449i
$292$		0			0
$293$	26114.1	+	15077.0i	0.304187	+	0.175622i	0.644322	−	0.764754i	$-0.277140\pi$
$293$	26114.1	+	15077.0i	0.304187	+	0.175622i	−0.340136	+	0.940376i	$0.610473\pi$
$294$		0			0
$295$	−2841.97	−	3386.93i	−0.0326569	−	0.0389190i
$296$		0			0
$297$	2829.31	−	1633.50i	0.0320751	−	0.0185186i
$298$		0			0
$299$	−96169.3	−	16957.2i	−1.07571	−	0.189676i
$300$		0			0
$301$	104903.	−	38181.5i	1.15786	−	0.421425i
$302$		0			0
$303$		−	3720.78i		−	0.0405274i
$304$		0			0
$305$	1518.32			0.0163216
$306$		0			0
$307$	−60034.4	−	164943.i	−0.636977	−	1.75008i	−0.661020	−	0.750368i	$-0.729876\pi$
$307$	−60034.4	−	164943.i	−0.636977	−	1.75008i	0.0240437	−	0.999711i	$-0.492346\pi$
$308$		0			0
$309$	524.040	−	2971.98i	0.00548842	−	0.0311264i
$310$		0			0
$311$	−59834.2	−	103636.i	−0.618626	−	1.07149i	−0.989737	−	0.142904i	$-0.954356\pi$
$311$	−59834.2	−	103636.i	−0.618626	−	1.07149i	0.371110	−	0.928589i	$-0.378977\pi$
$312$		0			0
$313$	−59356.0	+	49805.6i	−0.605865	+	0.508381i	−0.893325	−	0.449412i	$-0.851634\pi$
$313$	−59356.0	+	49805.6i	−0.605865	+	0.508381i	0.287460	+	0.957793i	$0.407189\pi$
$314$		0			0
$315$	6649.13	−	11516.6i	0.0670106	−	0.116066i
$316$		0			0
$317$	50069.0	−	137564.i	0.498254	−	1.36894i	−0.394707	−	0.918807i	$-0.629154\pi$
$317$	50069.0	−	137564.i	0.498254	−	1.36894i	0.892961	−	0.450134i	$-0.148624\pi$
$318$		0			0
$319$	−16277.1	+	2870.10i	−0.159954	+	0.0282043i
$320$		0			0
$321$	−584.064	−	490.088i	−0.00566827	−	0.00475624i
$322$		0			0
$323$	−138136.	+	35393.1i	−1.32404	+	0.339246i
$324$		0			0
$325$	82864.3	−	98753.8i	0.784514	−	0.934948i
$326$		0			0
$327$	1076.58	+	6105.59i	0.0100682	+	0.0570994i
$328$		0			0
$329$	122065.	+	44428.0i	1.12771	+	0.410454i
$330$		0			0
$331$	171009.	+	98731.9i	1.56085	+	0.901160i	0.997171	+	0.0751687i	$0.0239495\pi$
$331$	171009.	+	98731.9i	1.56085	+	0.901160i	0.563683	+	0.825991i	$0.309384\pi$
$332$		0			0
$333$	−49981.3	−	59565.3i	−0.450732	−	0.537162i
$334$		0			0
$335$	7742.48	−	4470.12i	0.0689907	−	0.0398318i
$336$		0			0
$337$	47042.1	+	8294.80i	0.414216	+	0.0730375i	0.376873	−	0.926265i	$-0.376999\pi$
$337$	47042.1	+	8294.80i	0.414216	+	0.0730375i	0.0373435	+	0.999302i	$0.488110\pi$
$338$		0			0
$339$	8629.11	−	3140.74i	0.0750873	−	0.0273295i
$340$		0			0
$341$		−	46932.3i		−	0.403611i
$342$		0			0
$343$	−109103.			−0.927356
$344$		0			0
$345$	−266.348	−	731.785i	−0.00223775	−	0.00614816i
$346$		0			0
$347$	27447.3	−	155661.i	0.227950	−	1.29277i	−0.629013	−	0.777394i	$-0.716541\pi$
$347$	27447.3	−	155661.i	0.227950	−	1.29277i	0.856964	−	0.515377i	$-0.172348\pi$
$348$		0			0
$349$	−19980.7	−	34607.6i	−0.164044	−	0.284133i	0.772271	−	0.635293i	$-0.219121\pi$
$349$	−19980.7	−	34607.6i	−0.164044	−	0.284133i	−0.936315	+	0.351160i	$0.885787\pi$
$350$		0			0
$351$	−13703.1	+	11498.3i	−0.111226	+	0.0933295i
$352$		0			0
$353$	−102732.	+	177938.i	−0.824438	+	1.42797i	0.0779095	+	0.996960i	$0.475175\pi$
$353$	−102732.	+	177938.i	−0.824438	+	1.42797i	−0.902348	+	0.431009i	$0.858158\pi$
$354$		0			0
$355$	−5645.11	+	15509.8i	−0.0447936	+	0.123069i
$356$		0			0
$357$	10673.1	−	1881.96i	0.0837441	−	0.0147664i
$358$		0			0
$359$	−46.7711	−	39.2456i	−0.000362901	−	0.000304510i	0.642606	−	0.766197i	$-0.277853\pi$
$359$	−46.7711	−	39.2456i	−0.000362901	−	0.000304510i	−0.642969	+	0.765892i	$0.722298\pi$
$360$		0			0
$361$	−41873.7	−	123411.i	−0.321312	−	0.946973i
$362$		0			0
$363$	4468.60	−	5325.46i	0.0339123	−	0.0404152i
$364$		0			0
$365$	1959.20	+	11111.2i	0.0147060	+	0.0834017i
$366$		0			0
$367$	42648.2	+	15522.7i	0.316642	+	0.115248i	0.495452	−	0.868635i	$-0.335002\pi$
$367$	42648.2	+	15522.7i	0.316642	+	0.115248i	−0.178810	+	0.983884i	$0.557225\pi$
$368$		0			0
$369$	177181.	+	102296.i	1.30126	+	0.751284i
$370$		0			0
$371$	−123006.	−	146593.i	−0.893673	−	1.06504i
$372$		0			0
$373$	−2182.83	+	1260.26i	−0.0156893	+	0.00905820i	−0.507824	−	0.861461i	$-0.669550\pi$
$373$	−2182.83	+	1260.26i	−0.0156893	+	0.00905820i	0.492135	+	0.870519i	$0.336217\pi$
$374$		0			0
$375$	2041.38	+	359.950i	0.0145165	+	0.00255965i
$376$		0			0
$377$	85040.9	−	30952.3i	0.598336	−	0.217776i
$378$		0			0
$379$			141908.i			0.987936i	0.869480	+	0.493968i	$0.164454\pi$
$379$			141908.i			0.987936i	−0.869480	+	0.493968i	$0.835546\pi$
$380$		0			0
$381$	810.750			0.00558518
$382$		0			0
$383$	−52054.1	−	143018.i	−0.354860	−	0.974971i	−0.980786	−	0.195088i	$-0.937501\pi$
$383$	−52054.1	−	143018.i	−0.354860	−	0.974971i	0.625925	−	0.779883i	$-0.284721\pi$
$384$		0			0
$385$	−1095.24	+	6211.44i	−0.00738907	+	0.0419055i
$386$		0			0
$387$	86651.2	+	150084.i	0.578566	+	1.00210i
$388$		0			0
$389$	−216517.	+	181680.i	−1.43085	+	1.20062i	−0.485638	+	0.874160i	$0.661413\pi$
$389$	−216517.	+	181680.i	−1.43085	+	1.20062i	−0.945210	+	0.326464i	$0.894143\pi$
$390$		0			0
$391$	−92004.7	+	159357.i	−0.601806	+	1.04236i
$392$		0			0
$393$	345.828	−	950.155i	0.00223911	−	0.00615190i
$394$		0			0
$395$	22586.5	−	3982.61i	0.144762	−	0.0255255i
$396$		0			0
$397$	−39373.6	−	33038.4i	−0.249818	−	0.209622i	0.509276	−	0.860603i	$-0.329913\pi$
$397$	−39373.6	−	33038.4i	−0.249818	−	0.209622i	−0.759094	+	0.650981i	$0.774358\pi$
$398$		0			0
$399$	2458.36	+	9594.74i	0.0154419	+	0.0602681i
$400$		0			0
$401$	11830.4	−	14098.9i	0.0735717	−	0.0876794i	−0.728003	−	0.685574i	$-0.759551\pi$
$401$	11830.4	−	14098.9i	0.0735717	−	0.0876794i	0.801575	+	0.597894i	$0.203996\pi$
$402$		0			0
$403$	44622.9	+	253069.i	0.274756	+	1.55822i
$404$		0			0
$405$	19332.0	+	7036.28i	0.117860	+	0.0428976i
$406$		0			0
$407$	31938.7	+	18439.8i	0.192809	+	0.111318i
$408$		0			0
$409$	121640.	+	144965.i	0.727158	+	0.866593i	0.995305	−	0.0967849i	$-0.0308559\pi$
$409$	121640.	+	144965.i	0.727158	+	0.866593i	−0.268147	+	0.963378i	$0.586411\pi$
$410$		0			0
$411$	15434.5	−	8911.10i	0.0913710	−	0.0527531i
$412$		0			0
$413$	71461.5	+	12600.6i	0.418959	+	0.0738738i
$414$		0			0
$415$	20793.6	−	7568.26i	0.120735	−	0.0439440i
$416$		0			0
$417$		−	7435.71i		−	0.0427612i
$418$		0			0
$419$	−133497.			−0.760400			−0.380200	−	0.924904i	$-0.624145\pi$
$419$	−133497.			−0.760400			−0.380200	+	0.924904i	$0.624145\pi$
$420$		0			0
$421$	24810.7	+	68166.9i	0.139983	+	0.384600i	0.989797	−	0.142482i	$-0.0455083\pi$
$421$	24810.7	+	68166.9i	0.139983	+	0.384600i	−0.849814	+	0.527082i	$0.823286\pi$
$422$		0			0
$423$	−35016.9	+	198591.i	−0.195703	+	1.10989i
$424$		0			0
$425$	−121458.	−	210371.i	−0.672430	−	1.16468i
$426$		0			0
$427$	−19089.1	+	16017.7i	−0.104696	+	0.0878503i
$428$		0			0
$429$	2117.40	−	3667.45i	0.0115051	−	0.0199274i
$430$		0			0
$431$	12011.1	−	33000.3i	0.0646590	−	0.177649i	−0.903156	−	0.429313i	$-0.858756\pi$
$431$	12011.1	−	33000.3i	0.0646590	−	0.177649i	0.967815	+	0.251664i	$0.0809778\pi$
$432$		0			0
$433$	241750.	−	42627.0i	1.28941	−	0.227357i	0.513437	−	0.858127i	$-0.328372\pi$
$433$	241750.	−	42627.0i	1.28941	−	0.227357i	0.775970	+	0.630770i	$0.217261\pi$
$434$		0			0
$435$	552.851	+	463.897i	0.00292166	+	0.00245156i
$436$		0			0
$437$	−151622.	−	72738.9i	−0.793960	−	0.380894i
$438$		0			0
$439$	66870.5	−	79693.2i	0.346981	−	0.413516i	−0.564124	−	0.825690i	$-0.690786\pi$
$439$	66870.5	−	79693.2i	0.346981	−	0.413516i	0.911105	+	0.412174i	$0.135230\pi$
$440$		0			0
$441$	4244.32	+	24070.8i	0.0218238	+	0.123769i
$442$		0			0
$443$	103744.	+	37759.7i	0.528634	+	0.192407i	0.592529	−	0.805549i	$-0.298130\pi$
$443$	103744.	+	37759.7i	0.528634	+	0.192407i	−0.0638940	+	0.997957i	$0.520352\pi$
$444$		0			0
$445$	−1559.18	−	900.192i	−0.00787364	−	0.00454585i
$446$		0			0
$447$	−6379.63	−	7602.95i	−0.0319286	−	0.0380511i
$448$		0			0
$449$	−286945.	+	165668.i	−1.42333	+	0.821760i	−0.996582	−	0.0826112i	$-0.973674\pi$
$449$	−286945.	+	165668.i	−1.42333	+	0.821760i	−0.426748	+	0.904371i	$0.640341\pi$
$450$		0			0
$451$	−95561.9	−	16850.1i	−0.469820	−	0.0828419i
$452$		0			0
$453$	−11986.8	+	4362.83i	−0.0584125	+	0.0212604i
$454$		0			0
$455$		−	34534.8i		−	0.166814i
$456$		0			0
$457$	5096.81			0.0244043			0.0122021	−	0.999926i	$-0.496116\pi$
$457$	5096.81			0.0244043			0.0122021	+	0.999926i	$0.496116\pi$
$458$		0			0
$459$	11528.5	+	31674.3i	0.0547201	+	0.150342i
$460$		0			0
$461$	−47130.4	+	267290.i	−0.221768	+	1.25771i	0.646999	+	0.762490i	$0.276024\pi$
$461$	−47130.4	+	267290.i	−0.221768	+	1.25771i	−0.868768	+	0.495220i	$0.835088\pi$
$462$		0			0
$463$	27622.4	+	47843.5i	0.128855	+	0.223183i	0.923233	−	0.384240i	$-0.125537\pi$
$463$	27622.4	+	47843.5i	0.128855	+	0.223183i	−0.794378	+	0.607423i	$0.792203\pi$
$464$		0			0
$465$	−1569.83	+	1317.25i	−0.00726019	+	0.00609202i
$466$		0			0
$467$	188119.	−	325832.i	0.862580	−	1.49403i	−0.00684959	−	0.999977i	$-0.502180\pi$
$467$	188119.	−	325832.i	0.862580	−	1.49403i	0.869430	−	0.494056i	$-0.164486\pi$
$468$		0			0
$469$	−50184.5	+	137881.i	−0.228152	+	0.626842i
$470$		0			0
$471$	−4849.79	+	855.148i	−0.0218615	+	0.00385478i
$472$		0			0
$473$	−62965.8	−	52834.6i	−0.281438	−	0.236154i
$474$		0			0
$475$	183239.	−	125331.i	0.812142	−	0.555484i
$476$		0			0
$477$	190954.	−	227570.i	0.839252	−	1.00018i
$478$		0			0
$479$	−37162.5	−	210759.i	−0.161970	−	0.918576i	−0.952134	−	0.305681i	$-0.901116\pi$
$479$	−37162.5	−	210759.i	−0.161970	−	0.918576i	0.790164	−	0.612895i	$-0.209995\pi$
$480$		0			0
$481$	−189752.	−	69064.2i	−0.820157	−	0.298513i
$482$		0			0
$483$	11068.7	+	6390.52i	0.0474463	+	0.0273932i
$484$		0			0
$485$	22117.2	+	26358.2i	0.0940256	+	0.112055i
$486$		0			0
$487$	56468.3	−	32602.0i	0.238093	−	0.137463i	−0.376207	−	0.926536i	$-0.622772\pi$
$487$	56468.3	−	32602.0i	0.238093	−	0.137463i	0.614300	+	0.789073i	$0.289439\pi$
$488$		0			0
$489$	10553.3	+	1860.83i	0.0441337	+	0.00778197i
$490$		0			0
$491$	349139.	−	127076.i	1.44822	−	0.527110i	0.506128	−	0.862458i	$-0.331076\pi$
$491$	349139.	−	127076.i	1.44822	−	0.527110i	0.942094	+	0.335348i	$0.108854\pi$
$492$		0			0
$493$		−	170528.i		−	0.701622i
$494$		0			0
$495$	−9791.37			−0.0399607
$496$		0			0
$497$	−92649.1	−	254551.i	−0.375084	−	1.03053i
$498$		0			0
$499$	−24132.3	+	136861.i	−0.0969165	+	0.549641i	0.897227	+	0.441570i	$0.145578\pi$
$499$	−24132.3	+	136861.i	−0.0969165	+	0.549641i	−0.994143	+	0.108071i	$0.965533\pi$
$500$		0			0
$501$	−10112.3	−	17515.0i	−0.0402879	−	0.0697807i
$502$		0			0
$503$	55026.8	−	46173.0i	0.217490	−	0.182495i	−0.527533	−	0.849534i	$-0.676883\pi$
$503$	55026.8	−	46173.0i	0.217490	−	0.182495i	0.745023	+	0.667039i	$0.232439\pi$
$504$		0			0
$505$	−11170.6	+	19348.0i	−0.0438020	+	0.0758672i
$506$		0			0
$507$	−2776.19	+	7627.51i	−0.0108002	+	0.0296734i
$508$		0			0
$509$	−433988.	+	76523.8i	−1.67510	+	0.295366i	−0.928895	−	0.370344i	$-0.879240\pi$
$509$	−433988.	+	76523.8i	−1.67510	+	0.295366i	−0.746210	+	0.665710i	$0.768129\pi$
$510$		0			0
$511$	−141851.	−	119027.i	−0.543238	−	0.455831i
$512$		0			0
$513$	−28060.0	+	12711.6i	−0.106624	+	0.0483019i
$514$		0			0
$515$	−11647.5	+	13881.0i	−0.0439156	+	0.0523366i
$516$		0			0
$517$	−16608.3	−	94190.2i	−0.0621360	−	0.352391i
$518$		0			0
$519$	−12922.4	−	4703.38i	−0.0479743	−	0.0174612i
$520$		0			0
$521$	−261584.	−	151025.i	−0.963685	−	0.556384i	−0.0663798	−	0.997794i	$-0.521145\pi$
$521$	−261584.	−	151025.i	−0.963685	−	0.556384i	−0.897305	+	0.441411i	$0.854478\pi$
$522$		0			0
$523$	−222308.	−	264936.i	−0.812739	−	0.968584i	0.187167	−	0.982328i	$-0.440070\pi$
$523$	−222308.	−	264936.i	−0.812739	−	0.968584i	−0.999906	+	0.0137438i	$0.995625\pi$
$524$		0			0
$525$	−14612.1	+	8436.29i	−0.0530143	+	0.0306078i
$526$		0			0
$527$	476863.	+	84083.8i	1.71701	+	0.302755i
$528$		0			0
$529$	59047.8	−	21491.7i	0.211005	−	0.0767995i
$530$		0			0
$531$			112648.i			0.399516i
$532$		0			0
$533$	531311.			1.87023
$534$		0			0
$535$	1565.78	+	4301.94i	0.00547045	+	0.0150299i
$536$		0			0
$537$	4955.04	−	28101.4i	0.0171830	−	0.0974495i
$538$		0			0
$539$	−5796.35	−	10039.6i	−0.0199516	−	0.0345571i
$540$		0			0
$541$	85186.7	−	71480.1i	0.291057	−	0.244225i	−0.485554	−	0.874207i	$-0.661382\pi$
$541$	85186.7	−	71480.1i	0.291057	−	0.244225i	0.776610	+	0.629981i	$0.216938\pi$
$542$		0			0
$543$	−12668.2	+	21942.0i	−0.0429651	+	0.0744177i
$544$		0			0
$545$	12732.1	−	34981.1i	0.0428654	−	0.117772i
$546$		0			0
$547$	185387.	−	32688.7i	0.619591	−	0.109251i	0.144963	−	0.989437i	$-0.453694\pi$
$547$	185387.	−	32688.7i	0.619591	−	0.109251i	0.474628	+	0.880187i	$0.342583\pi$
$548$		0			0
$549$	−29633.9	−	24865.8i	−0.0983204	−	0.0825006i
$550$		0			0
$551$	155393.	−	11878.2i	0.511834	−	0.0391242i
$552$		0			0
$553$	−241955.	+	288350.i	−0.791195	+	0.942910i
$554$		0			0
$555$	−279.630	−	1585.86i	−0.000907817	−	0.00514849i
$556$		0			0
$557$	−100049.	−	36414.7i	−0.322478	−	0.117372i	0.175709	−	0.984442i	$-0.443778\pi$
$557$	−100049.	−	36414.7i	−0.322478	−	0.117372i	−0.498187	+	0.867070i	$0.666001\pi$
$558$		0			0
$559$	389760.	+	225028.i	1.24731	+	0.720133i
$560$		0			0
$561$	−5129.28	−	6112.83i	−0.0162978	−	0.0194230i
$562$		0			0
$563$	−234974.	+	135662.i	−0.741315	+	0.427998i	−0.822547	−	0.568697i	$-0.807448\pi$
$563$	−234974.	+	135662.i	−0.741315	+	0.427998i	0.0812322	+	0.996695i	$0.474114\pi$
$564$		0			0
$565$	−54300.5	−	9574.64i	−0.170101	−	0.0299934i
$566$		0			0
$567$	−317283.	+	115481.i	−0.986916	+	0.359208i
$568$		0			0
$569$			31356.3i			0.0968501i	0.998827	+	0.0484250i	$0.0154202\pi$
$569$			31356.3i			0.0968501i	−0.998827	+	0.0484250i	$0.984580\pi$
$570$		0			0
$571$	−208073.			−0.638180			−0.319090	−	0.947724i	$-0.603377\pi$
$571$	−208073.			−0.638180			−0.319090	+	0.947724i	$0.603377\pi$
$572$		0			0
$573$	−3687.22	−	10130.6i	−0.0112303	−	0.0308549i
$574$		0			0
$575$	49745.3	−	282120.i	0.150458	−	0.853292i
$576$		0			0
$577$	−113879.	−	197245.i	−0.342053	−	0.592454i	0.642761	−	0.766067i	$-0.277789\pi$
$577$	−113879.	−	197245.i	−0.342053	−	0.592454i	−0.984814	+	0.173613i	$0.944456\pi$
$578$		0			0
$579$	−5856.19	+	4913.93i	−0.0174686	+	0.0146579i
$580$		0			0
$581$	−181586.	+	314517.i	−0.537936	+	0.931733i
$582$		0			0
$583$	−48190.3	+	132402.i	−0.141782	+	0.389544i
$584$		0			0
$585$	52797.2	−	9309.57i	0.154276	−	0.0272031i
$586$		0			0
$587$	39581.8	+	33213.1i	0.114873	+	0.0963903i	0.698415	−	0.715693i	$-0.253889\pi$
$587$	39581.8	+	33213.1i	0.114873	+	0.0963903i	−0.583542	+	0.812083i	$0.698333\pi$
$588$		0			0
$589$	−43405.2	+	440396.i	−0.125115	+	1.26944i
$590$		0			0
$591$	−18551.7	+	22109.1i	−0.0531140	+	0.0632988i
$592$		0			0
$593$	38149.6	+	216357.i	0.108488	+	0.615264i	0.989770	+	0.142673i	$0.0455699\pi$
$593$	38149.6	+	216357.i	0.108488	+	0.615264i	−0.881282	+	0.472590i	$0.843319\pi$
$594$		0			0
$595$	−61150.1	−	22256.8i	−0.172728	−	0.0628679i
$596$		0			0
$597$	29576.9	+	17076.2i	0.0829858	+	0.0479119i
$598$		0			0
$599$	172539.	+	205624.i	0.480877	+	0.573086i	0.950873	−	0.309582i	$-0.100189\pi$
$599$	172539.	+	205624.i	0.480877	+	0.573086i	−0.469996	+	0.882668i	$0.655745\pi$
$600$		0			0
$601$	352411.	−	203465.i	0.975664	−	0.563300i	0.0747060	−	0.997206i	$-0.476198\pi$
$601$	352411.	−	203465.i	0.975664	−	0.563300i	0.900958	+	0.433906i	$0.142865\pi$
$602$		0			0
$603$	−224322.	−	39554.1i	−0.616933	−	0.108782i
$604$		0			0
$605$	−39224.9	+	14276.7i	−0.107164	+	0.0390047i
$606$		0			0
$607$			169315.i			0.459535i	0.973246	+	0.229768i	$0.0737966\pi$
$607$			169315.i			0.459535i	−0.973246	+	0.229768i	$0.926203\pi$
$608$		0			0
$609$	−11844.7			−0.0319366
$610$		0			0
$611$	179111.	+	492102.i	0.479776	+	1.31817i
$612$		0			0
$613$	21097.6	−	119651.i	0.0561452	−	0.318415i	−0.943781	−	0.330572i	$-0.892758\pi$
$613$	21097.6	−	119651.i	0.0561452	−	0.318415i	0.999926	+	0.0121563i	$0.00386957\pi$
$614$		0			0
$615$	2118.51	+	3669.37i	0.00560120	+	0.00970156i
$616$		0			0
$617$	−105120.	+	88206.3i	−0.276131	+	0.231702i	−0.770327	−	0.637649i	$-0.779907\pi$
$617$	−105120.	+	88206.3i	−0.276131	+	0.231702i	0.494196	+	0.869351i	$0.335463\pi$
$618$		0			0
$619$	26097.4	−	45202.1i	0.0681108	−	0.117971i	−0.829959	−	0.557825i	$-0.811636\pi$
$619$	26097.4	−	45202.1i	0.0681108	−	0.117971i	0.898070	+	0.439853i	$0.144970\pi$
$620$		0			0
$621$	−13595.6	+	37353.7i	−0.0352546	+	0.0968613i
$622$		0			0
$623$	29099.5	−	5131.03i	0.0749738	−	0.0132199i
$624$		0			0
$625$	284896.	+	239056.i	0.729333	+	0.611983i
$626$		0			0
$627$	5213.01	−	5099.82i	0.0132603	−	0.0129724i
$628$		0			0
$629$	−244582.	+	291481.i	−0.618191	+	0.736731i
$630$		0			0
$631$	−23375.9	−	132571.i	−0.0587097	−	0.332960i	0.941279	−	0.337629i	$-0.109625\pi$
$631$	−23375.9	−	132571.i	−0.0587097	−	0.332960i	−0.999989	+	0.00466924i	$0.998514\pi$
$632$		0			0
$633$	−24765.3	−	9013.83i	−0.0618068	−	0.0224958i
$634$		0			0
$635$	−4215.90	−	2434.05i	−0.0104554	−	0.00603645i
$636$		0			0
$637$	40800.7	+	48624.4i	0.100552	+	0.119833i
$638$		0			0
$639$	364186.	−	210263.i	0.891911	−	0.514945i
$640$		0			0
$641$	−204371.	−	36036.2i	−0.497398	−	0.0877046i	−0.0806789	−	0.996740i	$-0.525709\pi$
$641$	−204371.	−	36036.2i	−0.497398	−	0.0877046i	−0.416719	+	0.909036i	$0.636820\pi$
$642$		0			0
$643$	−198736.	+	72334.0i	−0.480679	+	0.174953i	−0.570984	−	0.820961i	$-0.693438\pi$
$643$	−198736.	+	72334.0i	−0.480679	+	0.174953i	0.0903050	+	0.995914i	$0.471216\pi$
$644$		0			0
$645$			3589.04i			0.00862699i
$646$		0			0
$647$	−212208.			−0.506936			−0.253468	−	0.967344i	$-0.581571\pi$
$647$	−212208.			−0.506936			−0.253468	+	0.967344i	$0.581571\pi$
$648$		0			0
$649$	−18273.5	−	50206.0i	−0.0433842	−	0.119197i
$650$		0			0
$651$	5840.35	−	33122.3i	0.0137809	−	0.0781552i
$652$		0			0
$653$	327740.	+	567662.i	0.768604	+	1.33126i	0.938320	+	0.345769i	$0.112382\pi$
$653$	327740.	+	567662.i	0.768604	+	1.33126i	−0.169716	+	0.985493i	$0.554285\pi$
$654$		0			0
$655$	−4650.87	+	3902.55i	−0.0108406	+	0.00909631i
$656$		0			0
$657$	143731.	−	248950.i	0.332982	−	0.576741i
$658$		0			0
$659$	−60523.7	+	166288.i	−0.139365	+	0.382903i	−0.989666	−	0.143395i	$-0.954198\pi$
$659$	−60523.7	+	166288.i	−0.139365	+	0.382903i	0.850300	+	0.526298i	$0.176420\pi$
$660$		0			0
$661$	−29843.7	+	5262.25i	−0.0683045	+	0.0120439i	−0.207696	−	0.978193i	$-0.566596\pi$
$661$	−29843.7	+	5262.25i	−0.0683045	+	0.0120439i	0.139391	+	0.990237i	$0.455485\pi$
$662$		0			0
$663$	33470.2	+	28084.8i	0.0761432	+	0.0638917i
$664$		0			0
$665$	16022.0	−	57273.1i	0.0362305	−	0.129511i
$666$		0			0
$667$	129268.	−	154056.i	0.290563	−	0.346279i
$668$		0			0
$669$	−6936.22	−	39337.2i	−0.0154978	−	0.0878925i
$670$		0			0
$671$	17241.2	+	6275.27i	0.0382932	+	0.0139376i
$672$		0			0
$673$	−763517.	−	440817.i	−1.68573	−	0.973258i	−0.957723	−	0.287692i	$-0.907112\pi$
$673$	−763517.	−	440817.i	−1.68573	−	0.973258i	−0.728010	−	0.685566i	$-0.759555\pi$
$674$		0			0
$675$	−33731.1	−	40199.1i	−0.0740325	−	0.0882285i
$676$		0			0
$677$	93382.7	−	53914.5i	0.203746	−	0.117633i	−0.394656	−	0.918829i	$-0.629136\pi$
$677$	93382.7	−	53914.5i	0.203746	−	0.117633i	0.598402	+	0.801196i	$0.295803\pi$
$678$		0			0
$679$	−556138.	−	98062.1i	−1.20627	−	0.212697i
$680$		0			0
$681$	6654.34	−	2421.98i	0.0143486	−	0.00522248i
$682$		0			0
$683$		−	275330.i		−	0.590217i	−0.955464	−	0.295109i	$-0.904644\pi$
$683$		−	275330.i		−	0.590217i	0.955464	−	0.295109i	$-0.0953559\pi$
$684$		0			0
$685$	−107012.			−0.228062
$686$		0			0
$687$	3539.27	+	9724.06i	0.00749894	+	0.0206032i
$688$		0			0
$689$	133966.	−	759757.i	0.282199	−	1.60043i
$690$		0			0
$691$	248965.	+	431220.i	0.521413	+	0.903113i	0.999690	+	0.0249044i	$0.00792815\pi$
$691$	248965.	+	431220.i	0.521413	+	0.903113i	−0.478277	+	0.878209i	$0.658739\pi$
$692$		0			0
$693$	123102.	−	103295.i	0.256330	−	0.215087i
$694$		0			0
$695$	−22323.6	+	38665.6i	−0.0462162	+	0.0800489i
$696$		0			0
$697$	342417.	−	940783.i	0.704839	−	1.93653i
$698$		0			0
$699$	45942.3	−	8100.87i	0.0940283	−	0.0165797i
$700$		0			0
$701$	707525.	+	593684.i	1.43981	+	1.20815i	0.939615	+	0.342234i	$0.111184\pi$
$701$	707525.	+	593684.i	1.43981	+	1.20815i	0.500197	+	0.865912i	$0.333261\pi$
$702$		0			0
$703$	−282647.	−	202571.i	−0.571919	−	0.409889i
$704$		0			0
$705$	−2684.41	+	3199.16i	−0.00540097	+	0.00643662i
$706$		0			0
$707$	−63671.5	−	361099.i	−0.127382	−	0.722417i
$708$		0			0
$709$	844331.	+	307311.i	1.67966	+	0.611345i	0.993263	−	0.115885i	$-0.0369705\pi$
$709$	844331.	+	307311.i	1.67966	+	0.611345i	0.686394	+	0.727230i	$0.259193\pi$
$710$		0			0
$711$	−506058.	−	292172.i	−1.00106	−	0.577963i
$712$		0			0
$713$	367060.	+	437445.i	0.722034	+	0.860487i
$714$		0			0
$715$	−22021.0	+	12713.8i	−0.0430749	+	0.0248693i
$716$		0			0
$717$	25085.6	+	4423.27i	0.0487962	+	0.00860409i
$718$		0			0
$719$	−697825.	+	253987.i	−1.34986	+	0.491309i	−0.912904	−	0.408175i	$-0.866165\pi$
$719$	−697825.	+	253987.i	−1.34986	+	0.491309i	−0.436955	+	0.899483i	$0.643943\pi$
$720$		0			0
$721$		−	297396.i		−	0.572090i
$722$		0			0
$723$	19519.1			0.0373407
$724$		0			0
$725$	90801.0	+	249474.i	0.172749	+	0.474623i
$726$		0			0
$727$	10629.1	−	60280.8i	0.0201108	−	0.114054i	−0.973100	−	0.230384i	$-0.926002\pi$
$727$	10629.1	−	60280.8i	0.0201108	−	0.114054i	0.993211	+	0.116330i	$0.0371130\pi$
$728$		0			0
$729$	−260256.	−	450777.i	−0.489718	−	0.848216i
$730$		0			0
$731$	649644.	−	545116.i	1.21574	−	1.02013i
$732$		0			0
$733$	313840.	−	543587.i	0.584118	−	1.01172i	−0.410866	−	0.911696i	$-0.634774\pi$
$733$	313840.	−	543587.i	0.584118	−	1.01172i	0.994985	−	0.100027i	$-0.0318929\pi$
$734$		0			0
$735$	−173.127	+	475.662i	−0.000320472	+	0.000880488i
$736$		0			0
$737$	106394.	−	18760.2i	0.195877	−	0.0345384i
$738$		0			0
$739$	−681044.	−	571463.i	−1.24706	−	1.04640i	−0.996939	−	0.0781890i	$-0.975086\pi$
$739$	−681044.	−	571463.i	−1.24706	−	1.04640i	−0.250118	−	0.968215i	$-0.580469\pi$
$740$		0			0
$741$	−23260.8	+	32455.8i	−0.0423632	+	0.0591093i
$742$		0			0
$743$	−202266.	+	241051.i	−0.366392	+	0.436649i	−0.917470	−	0.397805i	$-0.869772\pi$
$743$	−202266.	+	241051.i	−0.366392	+	0.436649i	0.551078	+	0.834453i	$0.314217\pi$
$744$		0			0
$745$	10348.3	+	58688.3i	0.0186448	+	0.105740i
$746$		0			0
$747$	−529787.	−	192827.i	−0.949424	−	0.345562i
$748$		0			0
$749$	−65069.6	−	37568.0i	−0.115988	−	0.0669659i
$750$		0			0
$751$	9165.30	+	10922.8i	0.0162505	+	0.0193666i	0.774108	−	0.633053i	$-0.218198\pi$
$751$	9165.30	+	10922.8i	0.0162505	+	0.0193666i	−0.757858	+	0.652420i	$0.773754\pi$
$752$		0			0
$753$	46724.6	−	26976.5i	0.0824054	−	0.0475768i
$754$		0			0
$755$	75429.3	+	13300.2i	0.132326	+	0.0233327i
$756$		0			0
$757$	−644273.	+	234496.i	−1.12429	+	0.409208i	−0.836216	−	0.548401i	$-0.815237\pi$
$757$	−644273.	+	234496.i	−1.12429	+	0.409208i	−0.288073	+	0.957608i	$0.593015\pi$
$758$		0			0
$759$		−	9410.56i		−	0.0163355i
$760$		0			0
$761$	−184923.			−0.319317			−0.159658	−	0.987172i	$-0.551039\pi$
$761$	−184923.			−0.319317			−0.159658	+	0.987172i	$0.551039\pi$
$762$		0			0
$763$	208963.	+	574120.i	0.358938	+	0.986174i
$764$		0			0
$765$	17542.2	−	99486.9i	0.0299752	−	0.169998i
$766$		0			0
$767$	146270.	+	253347.i	0.248636	+	0.430651i
$768$		0			0
$769$	585144.	−	490994.i	0.989486	−	0.830277i	0.00399290	−	0.999992i	$-0.498729\pi$
$769$	585144.	−	490994.i	0.989486	−	0.830277i	0.985493	+	0.169715i	$0.0542846\pi$
$770$		0			0
$771$	−546.471	+	946.516i	−0.000919303	+	0.00159228i
$772$		0			0
$773$	178470.	−	490342.i	0.298680	−	0.820616i	−0.696041	−	0.718002i	$-0.745057\pi$
$773$	178470.	−	490342.i	0.298680	−	0.820616i	0.994721	−	0.102615i	$-0.0327209\pi$
$774$		0			0
$775$	−742396.	+	130904.i	−1.23604	+	0.217947i
$776$		0			0
$777$	20245.9	+	16988.3i	0.0335347	+	0.0281390i
$778$		0			0
$779$	881136.	+	246496.i	1.45200	+	0.406195i
$780$		0			0
$781$	−128205.	+	152789.i	−0.210186	+	0.250490i
$782$		0			0
$783$	−6396.97	−	36279.0i	−0.0104340	−	0.0591742i
$784$		0			0
$785$	27786.2	+	10113.3i	0.0450910	+	0.0164118i
$786$		0			0
$787$	−519955.	−	300196.i	−0.839491	−	0.484680i	0.0176001	−	0.999845i	$-0.494397\pi$
$787$	−519955.	−	300196.i	−0.839491	−	0.484680i	−0.857091	+	0.515165i	$0.827731\pi$
$788$		0			0
$789$	−25969.3	−	30949.0i	−0.0417164	−	0.0497156i
$790$		0			0
$791$	783703.	−	452471.i	1.25256	−	0.723166i
$792$		0			0
$793$	−98934.5	−	17444.8i	−0.157326	−	0.0277409i
$794$		0			0
$795$	5781.25	−	2104.20i	0.00914719	−	0.00332930i
$796$		0			0
$797$			917854.i			1.44496i	0.691390	+	0.722482i	$0.256999\pi$
$797$			917854.i			1.44496i	−0.691390	+	0.722482i	$0.743001\pi$
$798$		0			0
$799$	986790.			1.54572
$800$		0			0
$801$	15688.8	+	43104.5i	0.0244525	+	0.0671827i
$802$		0			0
$803$	−23675.4	+	134270.i	−0.0367169	+	0.208232i
$804$		0			0
$805$	−38371.5	−	66461.3i	−0.0592129	−	0.102560i
$806$		0			0
$807$	−48406.4	+	40617.8i	−0.0743286	+	0.0623691i
$808$		0			0
$809$	−269448.	+	466698.i	−0.411697	+	0.713081i	−0.995076	−	0.0991198i	$-0.968397\pi$
$809$	−269448.	+	466698.i	−0.411697	+	0.713081i	0.583378	+	0.812201i	$0.301731\pi$
$810$		0			0
$811$	−373554.	+	1.02633e6i	−0.567952	+	1.56044i	0.239743	+	0.970836i	$0.422937\pi$
$811$	−373554.	+	1.02633e6i	−0.567952	+	1.56044i	−0.807696	+	0.589600i	$0.799285\pi$
$812$		0			0
$813$	70305.7	−	12396.8i	0.106368	−	0.0187555i
$814$		0			0
$815$	−49290.5	−	41359.6i	−0.0742075	−	0.0622675i
$816$		0			0
$817$	541986.	+	554015.i	0.811977	+	0.829999i
$818$		0			0
$819$	−565582.	+	674034.i	−0.843195	+	1.00488i
$820$		0			0
$821$	−35168.1	−	199448.i	−0.0521750	−	0.295899i	0.947543	−	0.319627i	$-0.103558\pi$
$821$	−35168.1	−	199448.i	−0.0521750	−	0.295899i	−0.999718	+	0.0237279i	$0.992446\pi$
$822$		0			0
$823$	−925765.	−	336951.i	−1.36679	−	0.497470i	−0.448642	−	0.893712i	$-0.648092\pi$
$823$	−925765.	−	336951.i	−1.36679	−	0.497470i	−0.918146	+	0.396241i	$0.870314\pi$
$824$		0			0
$825$	10758.7	+	6211.56i	0.0158071	+	0.00912625i
$826$		0			0
$827$	399650.	+	476284.i	0.584344	+	0.696394i	0.974508	−	0.224351i	$-0.0720262\pi$
$827$	399650.	+	476284.i	0.584344	+	0.696394i	−0.390164	+	0.920745i	$0.627582\pi$
$828$		0			0
$829$	−795518.	+	459293.i	−1.15755	+	0.668314i	−0.950716	−	0.310062i	$-0.899650\pi$
$829$	−795518.	+	459293.i	−1.15755	+	0.668314i	−0.206837	+	0.978375i	$0.566317\pi$
$830$		0			0
$831$	−19140.7	−	3375.03i	−0.0277177	−	0.00488737i
$832$		0			0
$833$	112393.	−	40907.9i	0.161976	−	0.0589545i
$834$		0			0
$835$			121437.i			0.174172i
$836$		0			0
$837$	104604.			0.149313
$838$		0			0
$839$	151725.	+	416860.i	0.215542	+	0.592197i	0.999594	−	0.0284982i	$-0.00907250\pi$
$839$	151725.	+	416860.i	0.215542	+	0.592197i	−0.784052	+	0.620696i	$0.786850\pi$
$840$		0			0
$841$	90454.9	−	512995.i	0.127891	−	0.725306i
$842$		0			0
$843$	−7660.68	−	13268.7i	−0.0107798	−	0.0186712i
$844$		0			0
$845$	37335.6	−	31328.3i	0.0522889	−	0.0438756i
$846$		0			0
$847$	342542.	−	593301.i	0.477472	−	0.827005i
$848$		0			0
$849$	−6994.89	+	19218.3i	−0.00970433	+	0.0266624i
$850$		0			0
$851$	−441911.	+	77920.9i	−0.610205	+	0.107596i
$852$		0			0
$853$	802327.	+	673232.i	1.10269	+	0.925267i	0.997603	−	0.0691966i	$-0.0220436\pi$
$853$	802327.	+	673232.i	1.10269	+	0.925267i	0.105087	+	0.994463i	$0.466488\pi$
$854$		0			0
$855$	91878.9	+	9055.52i	0.125685	+	0.0123874i
$856$		0			0
$857$	−782317.	+	932329.i	−1.06517	+	1.26943i	−0.103678	+	0.994611i	$0.533061\pi$
$857$	−782317.	+	932329.i	−1.06517	+	1.26943i	−0.961497	+	0.274815i	$0.911383\pi$
$858$		0			0
$859$	−245004.	−	1.38949e6i	−0.332038	−	1.88308i	−0.454722	−	0.890633i	$-0.650261\pi$
$859$	−245004.	−	1.38949e6i	−0.332038	−	1.88308i	0.122684	−	0.992446i	$-0.460850\pi$
$860$		0			0
$861$	−65345.5	−	23783.8i	−0.0881474	−	0.0320830i
$862$		0			0
$863$	272525.	+	157343.i	0.365919	+	0.211264i	0.671674	−	0.740847i	$-0.265576\pi$
$863$	272525.	+	157343.i	0.365919	+	0.211264i	−0.305755	+	0.952110i	$0.598909\pi$
$864$		0			0
$865$	53075.9	+	63253.4i	0.0709358	+	0.0845379i
$866$		0			0
$867$	33134.4	−	19130.2i	0.0440799	−	0.0254496i
$868$		0			0
$869$	272940.	+	48126.7i	0.361433	+	0.0637303i
$870$		0			0
$871$	−555864.	+	202318.i	−0.732711	+	0.266685i
$872$		0			0
$873$		−	876665.i		−	1.15028i
$874$		0			0
$875$	204274.			0.266807
$876$		0			0
$877$	453424.	+	1.24577e6i	0.589530	+	1.61972i	0.771367	+	0.636391i	$0.219573\pi$
$877$	453424.	+	1.24577e6i	0.589530	+	1.61972i	−0.181837	+	0.983329i	$0.558204\pi$
$878$		0			0
$879$	2762.87	−	15669.0i	0.00357588	−	0.0202798i
$880$		0			0
$881$	−71718.1	−	124219.i	−0.0924011	−	0.160043i	0.816120	−	0.577883i	$-0.196121\pi$
$881$	−71718.1	−	124219.i	−0.0924011	−	0.160043i	−0.908521	+	0.417839i	$0.862788\pi$
$882$		0			0
$883$	49737.0	−	41734.3i	0.0637908	−	0.0535268i	−0.610335	−	0.792144i	$-0.708965\pi$
$883$	49737.0	−	41734.3i	0.0637908	−	0.0535268i	0.674126	+	0.738617i	$0.264521\pi$
$884$		0			0
$885$	−1166.45	+	2020.36i	−0.00148930	+	0.00257954i
$886$		0			0
$887$	−410854.	+	1.12881e6i	−0.522204	+	1.43474i	0.345857	+	0.938287i	$0.387588\pi$
$887$	−410854.	+	1.12881e6i	−0.522204	+	1.43474i	−0.868061	+	0.496457i	$0.834634\pi$
$888$		0			0
$889$	78682.7	−	13873.9i	0.0995580	−	0.0175548i
$890$		0			0
$891$	190442.	+	159800.i	0.239888	+	0.201290i
$892$		0			0
$893$	68734.8	+	899208.i	0.0861933	+	1.12761i
$894$		0			0
$895$	−110133.	+	131251.i	−0.137490	+	0.163854i
$896$		0			0
$897$	8947.49	+	50743.7i	0.0111203	+	0.0630663i
$898$		0			0
$899$	−497292.	−	181000.i	−0.615308	−	0.223954i
$900$		0			0
$901$	−1.25895e6	−	726856.i	−1.55081	−	0.895362i
$902$		0			0
$903$	−37863.0	−	45123.4i	−0.0464344	−	0.0553383i
$904$		0			0
$905$	131749.	−	76065.4i	0.160861	−	0.0928732i
$906$		0			0
$907$	1.01442e6	+	178870.i	1.23312	+	0.217432i	0.751963	−	0.659205i	$-0.229107\pi$
$907$	1.01442e6	+	178870.i	1.23312	+	0.217432i	0.481153	+	0.876637i	$0.340218\pi$
$908$		0			0
$909$	534889.	−	194684.i	0.647345	−	0.235614i
$910$		0			0
$911$		−	522407.i		−	0.629466i	−0.949180	−	0.314733i	$-0.898085\pi$
$911$		−	522407.i		−	0.629466i	0.949180	−	0.314733i	$-0.101915\pi$
$912$		0			0
$913$	267400.			0.320790
$914$		0			0
$915$	−274.006	−	752.826i	−0.000327279	−	0.000899192i
$916$		0			0
$917$	17302.9	−	98129.8i	0.0205769	−	0.116698i
$918$		0			0
$919$	515181.	+	892320.i	0.609999	+	1.05655i	0.991240	+	0.132074i	$0.0421635\pi$
$919$	515181.	+	892320.i	0.609999	+	1.05655i	−0.381241	+	0.924476i	$0.624503\pi$
$920$		0			0
$921$	−70949.4	+	59533.6i	−0.0836429	+	0.0701848i
$922$		0			0
$923$	546040.	−	945769.i	0.640945	−	1.11015i
$924$		0			0
$925$	202605.	−	556653.i	0.236792	−	0.650580i
$926$		0			0
$927$	454662.	−	80169.2i	0.529090	−	0.0932928i
$928$		0			0
$929$	−980501.	−	822738.i	−1.13610	−	0.953302i	−0.136796	−	0.990599i	$-0.543681\pi$
$929$	−980501.	−	822738.i	−1.13610	−	0.953302i	−0.999304	+	0.0372977i	$0.988125\pi$
$930$		0			0
$931$	45105.9	+	99568.6i	0.0520396	+	0.114874i
$932$		0			0
$933$	−40587.5	+	48370.3i	−0.0466261	+	0.0555669i
$934$		0			0
$935$	8320.14	+	47185.9i	0.00951716	+	0.0539745i
$936$		0			0
$937$	−864430.	−	314627.i	−0.984579	−	0.358358i	−0.200960	−	0.979599i	$-0.564406\pi$
$937$	−864430.	−	314627.i	−0.984579	−	0.358358i	−0.783619	+	0.621242i	$0.786628\pi$
$938$		0			0
$939$	35406.8	+	20442.1i	0.0401565	+	0.0231844i
$940$		0			0
$941$	−671279.	−	799999.i	−0.758095	−	0.903463i	0.239631	−	0.970864i	$-0.422974\pi$
$941$	−671279.	−	799999.i	−0.758095	−	0.903463i	−0.997726	+	0.0674016i	$0.978529\pi$
$942$		0			0
$943$	1.02250e6	−	590338.i	1.14984	−	0.663861i
$944$		0			0
$945$	−13844.3	−	2441.12i	−0.0155027	−	0.00273354i
$946$		0			0
$947$	−1.59628e6	+	580997.i	−1.77995	+	0.647849i	−0.780201	+	0.625529i	$0.784883\pi$
$947$	−1.59628e6	+	580997.i	−1.77995	+	0.647849i	−0.999751	+	0.0223200i	$0.992895\pi$
$948$		0			0
$949$		−	746522.i		−	0.828916i
$950$		0			0
$951$	−77243.8			−0.0854088
$952$		0			0
$953$	122646.	+	336967.i	0.135042	+	0.371024i	0.988720	−	0.149777i	$-0.0478556\pi$
$953$	122646.	+	336967.i	0.135042	+	0.371024i	−0.853678	+	0.520801i	$0.825633\pi$
$954$		0			0
$955$	−11240.6	+	63748.7i	−0.0123249	+	0.0698980i
$956$		0			0
$957$	4360.56	+	7552.71i	0.00476122	+	0.00824668i
$958$		0			0
$959$	1.34541e6	−	1.12894e6i	1.46291	−	1.22753i
$960$		0			0
$961$	289588.	−	501581.i	0.313570	−	0.543119i
$962$		0			0
$963$	39893.5	−	109606.i	0.0430179	−	0.118191i
$964$		0			0
$965$	45204.8	−	7970.83i	0.0485434	−	0.00855951i
$966$		0			0
$967$	890932.	+	747581.i	0.952778	+	0.799476i	0.979763	−	0.200160i	$-0.0641462\pi$
$967$	890932.	+	747581.i	0.952778	+	0.799476i	−0.0269850	+	0.999636i	$0.508591\pi$
$968$		0			0
$969$	42477.9	+	62104.5i	0.0452393	+	0.0661417i
$970$		0			0
$971$	493231.	−	587810.i	0.523133	−	0.623445i	−0.438186	−	0.898884i	$-0.644379\pi$
$971$	493231.	−	587810.i	0.523133	−	0.623445i	0.961318	+	0.275439i	$0.0888233\pi$
$972$		0			0
$973$	−127243.	−	721630.i	−0.134403	−	0.762235i
$974$		0			0
$975$	−63919.3	−	23264.7i	−0.0672392	−	0.0244731i
$976$		0			0
$977$	−1.10521e6	−	638093.i	−1.15786	−	0.668490i	−0.207068	−	0.978326i	$-0.566392\pi$
$977$	−1.10521e6	−	638093.i	−1.15786	−	0.668490i	−0.950790	+	0.309837i	$0.899726\pi$
$978$		0			0
$979$	−13984.6	−	16666.2i	−0.0145910	−	0.0173889i
$980$		0			0
$981$	−821391.	+	474231.i	−0.853517	+	0.492778i
$982$		0			0
$983$	−1.25475e6	−	221246.i	−1.29852	−	0.228965i	−0.518696	−	0.854959i	$-0.673582\pi$
$983$	−1.25475e6	−	221246.i	−1.29852	−	0.228965i	−0.779828	+	0.625994i	$0.784693\pi$
$984$		0			0
$985$	162845.	−	59270.7i	0.167843	−	0.0610897i
$986$		0			0
$987$		−	68541.1i		−	0.0703585i
$988$		0			0
$989$	1.00011e6			1.02248
$990$		0			0
$991$	109943.	+	302067.i	0.111949	+	0.307578i	0.982997	−	0.183619i	$-0.0587814\pi$
$991$	109943.	+	302067.i	0.111949	+	0.307578i	−0.871048	+	0.491198i	$0.836559\pi$
$992$		0			0
$993$	18092.7	−	102609.i	0.0183487	−	0.104061i
$994$		0			0
$995$	−102533.	−	177592.i	−0.103566	−	0.179382i
$996$		0			0
$997$	−213777.	+	179381.i	−0.215066	+	0.180462i	−0.743956	−	0.668229i	$-0.767053\pi$
$997$	−213777.	+	179381.i	−0.215066	+	0.180462i	0.528890	+	0.848690i	$0.322608\pi$
$998$		0			0
$999$	−41099.3	+	71186.0i	−0.0411816	+	0.0713286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.41.4	yes	42
19.13	odd	18	inner	76.5.j.a.13.4	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.13.4	✓	42	19.13	odd	18	inner
76.5.j.a.41.4	yes	42	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-0.180467 - 0.495829i) q^{3} +(-0.550159 + 3.12011i) q^{5} +(-25.9990 - 45.0316i) q^{7} +(61.8363 - 51.8868i) q^{9} +O(q^{10})\)	\(q+(-0.180467 - 0.495829i) q^{3} +(-0.550159 + 3.12011i) q^{5} +(-25.9990 - 45.0316i) q^{7} +(61.8363 - 51.8868i) q^{9} +(-19.1428 + 33.1563i) q^{11} +(71.6974 - 196.987i) q^{13} +(1.64632 - 0.290291i) q^{15} +(-302.594 - 253.907i) q^{17} +(210.294 - 293.424i) q^{19} +(-17.6360 + 21.0178i) q^{21} +(-80.8917 - 458.759i) q^{23} +(577.875 + 210.329i) q^{25} +(-73.9000 - 42.6662i) q^{27} +(277.497 + 330.708i) q^{29} +(-1061.61 + 612.923i) q^{31} +(19.8945 + 3.50794i) q^{33} +(154.807 - 56.3452i) q^{35} -963.274i q^{37} -110.611 q^{39} +(866.860 + 2381.68i) q^{41} +(-372.808 + 2114.30i) q^{43} +(127.873 + 221.482i) q^{45} +(-1913.69 + 1605.78i) q^{47} +(-151.397 + 262.228i) q^{49} +(-71.2860 + 195.857i) q^{51} +(3624.29 - 639.061i) q^{53} +(-92.9197 - 77.9689i) q^{55} +(-183.439 - 51.3167i) q^{57} +(-897.018 + 1069.02i) q^{59} +(-83.2176 - 471.950i) q^{61} +(-3944.23 - 1435.58i) q^{63} +(575.175 + 332.078i) q^{65} +(-1813.84 - 2161.65i) q^{67} +(-212.868 + 122.899i) q^{69} +(5130.43 + 904.634i) q^{71} +(3346.39 - 1217.99i) q^{73} -324.485i q^{75} +1990.78 q^{77} +(-2475.89 - 6802.45i) q^{79} +(1127.57 - 6394.77i) q^{81} +(-3492.18 - 6048.63i) q^{83} +(958.691 - 804.437i) q^{85} +(113.895 - 197.273i) q^{87} +(-194.356 + 533.990i) q^{89} +(-10734.7 + 1892.82i) q^{91} +(495.491 + 415.766i) q^{93} +(799.817 + 817.570i) q^{95} +(6980.90 - 8319.52i) q^{97} +(536.656 + 3043.53i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 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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