LMFDB - Embedded newform 124.5.o.a.73.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,5,Mod(13,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 11]))

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

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

chi := DirichletCharacter("124.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	124.o (of order $30$, degree $8$, 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:	$12.8178754224$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$11$ over $\Q(\zeta_{30})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			73.5
Character	$\chi$	$=$	124.73
Dual form			124.5.o.a.17.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.07010 + 2.76433i) q^{3} +(-0.342480 - 0.593193i) q^{5} +(-3.29625 - 31.3617i) q^{7} +(-6.68282 + 63.5828i) q^{9} +O(q^{10})$	\(q+(-3.07010 + 2.76433i) q^{3} +(-0.342480 - 0.593193i) q^{5} +(-3.29625 - 31.3617i) q^{7} +(-6.68282 + 63.5828i) q^{9} +(21.2034 - 47.6236i) q^{11} +(-29.7989 - 140.193i) q^{13} +(2.69123 + 0.874433i) q^{15} +(-155.539 - 349.346i) q^{17} +(585.965 + 124.551i) q^{19} +(96.8139 + 87.1716i) q^{21} +(188.743 - 259.782i) q^{23} +(312.265 - 540.860i) q^{25} +(-351.937 - 484.399i) q^{27} +(1277.51 - 415.087i) q^{29} +(-824.331 - 493.963i) q^{31} +(66.5507 + 204.822i) q^{33} +(-17.4747 + 12.6961i) q^{35} +(-1140.98 - 658.744i) q^{37} +(479.024 + 348.031i) q^{39} +(1706.34 - 1895.09i) q^{41} +(-254.904 + 1199.23i) q^{43} +(40.0056 - 17.8116i) q^{45} +(-668.281 + 2056.76i) q^{47} +(1375.84 - 292.444i) q^{49} +(1443.23 + 642.566i) q^{51} +(-4164.47 - 437.703i) q^{53} +(-35.5117 + 3.73243i) q^{55} +(-2143.27 + 1237.42i) q^{57} +(-2620.01 - 2909.81i) q^{59} +2467.23i q^{61} +2016.09 q^{63} +(-72.9558 + 65.6897i) q^{65} +(1916.75 + 3319.91i) q^{67} +(138.664 + 1319.30i) q^{69} +(-959.539 + 9129.40i) q^{71} +(1740.13 - 3908.40i) q^{73} +(536.428 + 2523.70i) q^{75} +(-1563.45 - 507.995i) q^{77} +(518.236 + 1163.98i) q^{79} +(-2645.89 - 562.401i) q^{81} +(-760.880 - 685.100i) q^{83} +(-153.961 + 211.909i) q^{85} +(-2774.63 + 4805.80i) q^{87} +(-375.554 - 516.906i) q^{89} +(-4298.46 + 1396.65i) q^{91} +(3896.25 - 762.206i) q^{93} +(-126.799 - 390.246i) q^{95} +(-8845.42 + 6426.57i) q^{97} +(2886.34 + 1666.43i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9}+O(q^{10}) $ 88 * q - 9 * q^3 + 3 * q^5 - 215 * q^7 - 254 * q^9	$ 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9} - 42 q^{11} + 6 q^{13} + 665 q^{15} - 585 q^{17} - 153 q^{19} - 402 q^{21} - 1365 q^{23} - 5933 q^{25} - 9225 q^{27} - 1140 q^{29} + 117 q^{31} + 5151 q^{33} + 2898 q^{35} + 6594 q^{37} + 3173 q^{39} - 9393 q^{41} - 5322 q^{43} + 2010 q^{45} - 5112 q^{47} - 5210 q^{49} - 1829 q^{51} + 7395 q^{53} + 10585 q^{55} + 40485 q^{57} + 5625 q^{59} - 14954 q^{63} - 17094 q^{65} + 8909 q^{67} - 35370 q^{69} - 11811 q^{71} - 22105 q^{73} + 79377 q^{75} + 71490 q^{77} + 219 q^{79} - 5422 q^{81} + 10545 q^{83} - 53630 q^{85} + 13732 q^{87} - 40305 q^{89} + 42760 q^{91} - 1028 q^{93} + 62319 q^{95} + 35201 q^{97} + 16197 q^{99}+O(q^{100}) $ 88 * q - 9 * q^3 + 3 * q^5 - 215 * q^7 - 254 * q^9 - 42 * q^11 + 6 * q^13 + 665 * q^15 - 585 * q^17 - 153 * q^19 - 402 * q^21 - 1365 * q^23 - 5933 * q^25 - 9225 * q^27 - 1140 * q^29 + 117 * q^31 + 5151 * q^33 + 2898 * q^35 + 6594 * q^37 + 3173 * q^39 - 9393 * q^41 - 5322 * q^43 + 2010 * q^45 - 5112 * q^47 - 5210 * q^49 - 1829 * q^51 + 7395 * q^53 + 10585 * q^55 + 40485 * q^57 + 5625 * q^59 - 14954 * q^63 - 17094 * q^65 + 8909 * q^67 - 35370 * q^69 - 11811 * q^71 - 22105 * q^73 + 79377 * q^75 + 71490 * q^77 + 219 * q^79 - 5422 * q^81 + 10545 * q^83 - 53630 * q^85 + 13732 * q^87 - 40305 * q^89 + 42760 * q^91 - 1028 * q^93 + 62319 * q^95 + 35201 * q^97 + 16197 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$1$	$e\left(\frac{23}{30}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−3.07010	+	2.76433i	−0.341122	+	0.307148i	−0.821830	−	0.569733i	$-0.807046\pi$
$3$	−3.07010	+	2.76433i	−0.341122	+	0.307148i	0.480708	+	0.876881i	$0.340380\pi$
$4$		0			0
$5$	−0.342480	−	0.593193i	−0.0136992	−	0.0237277i	0.859095	−	0.511817i	$-0.171027\pi$
$5$	−0.342480	−	0.593193i	−0.0136992	−	0.0237277i	−0.872794	+	0.488089i	$0.837694\pi$
$6$		0			0
$7$	−3.29625	−	31.3617i	−0.0672704	−	0.640035i	−0.975263	−	0.221049i	$-0.929052\pi$
$7$	−3.29625	−	31.3617i	−0.0672704	−	0.640035i	0.907992	−	0.418987i	$-0.137615\pi$
$8$		0			0
$9$	−6.68282	+	63.5828i	−0.0825039	+	0.784972i
$10$		0			0
$11$	21.2034	−	47.6236i	0.175234	−	0.393583i	−0.804480	−	0.593979i	$-0.797556\pi$
$11$	21.2034	−	47.6236i	0.175234	−	0.393583i	0.979715	+	0.200396i	$0.0642229\pi$
$12$		0			0
$13$	−29.7989	−	140.193i	−0.176325	−	0.829543i	−0.974016	−	0.226477i	$-0.927279\pi$
$13$	−29.7989	−	140.193i	−0.176325	−	0.829543i	0.797692	−	0.603065i	$-0.206054\pi$
$14$		0			0
$15$	2.69123	+	0.874433i	0.0119610	+	0.00388637i
$16$		0			0
$17$	−155.539	−	349.346i	−0.538197	−	1.20881i	−0.954124	−	0.299413i	$-0.903209\pi$
$17$	−155.539	−	349.346i	−0.538197	−	1.20881i	0.415927	−	0.909398i	$-0.363457\pi$
$18$		0			0
$19$	585.965	+	124.551i	1.62317	+	0.345016i	0.927639	−	0.373479i	$-0.121835\pi$
$19$	585.965	+	124.551i	1.62317	+	0.345016i	0.695533	+	0.718494i	$0.255168\pi$
$20$		0			0
$21$	96.8139	+	87.1716i	0.219533	+	0.197668i
$22$		0			0
$23$	188.743	−	259.782i	0.356792	−	0.491082i	−0.592460	−	0.805600i	$-0.701843\pi$
$23$	188.743	−	259.782i	0.356792	−	0.491082i	0.949251	+	0.314518i	$0.101843\pi$
$24$		0			0
$25$	312.265	−	540.860i	0.499625	−	0.865375i
$26$		0			0
$27$	−351.937	−	484.399i	−0.482766	−	0.664471i
$28$		0			0
$29$	1277.51	−	415.087i	1.51903	−	0.493564i	0.573532	−	0.819183i	$-0.305573\pi$
$29$	1277.51	−	415.087i	1.51903	−	0.493564i	0.945501	+	0.325620i	$0.105573\pi$
$30$		0			0
$31$	−824.331	−	493.963i	−0.857784	−	0.514010i
$31$	−824.331	−	493.963i	−0.857784	−	0.514010i
$32$		0			0
$33$	66.5507	+	204.822i	0.0611118	+	0.188083i
$34$		0			0
$35$	−17.4747	+	12.6961i	−0.0142650	+	0.0103641i
$36$		0			0
$37$	−1140.98	−	658.744i	−0.833440	−	0.481187i	0.0215893	−	0.999767i	$-0.493127\pi$
$37$	−1140.98	−	658.744i	−0.833440	−	0.481187i	−0.855029	+	0.518580i	$0.826461\pi$
$38$		0			0
$39$	479.024	+	348.031i	0.314940	+	0.228817i
$40$		0			0
$41$	1706.34	−	1895.09i	1.01508	−	1.12736i	0.0232539	−	0.999730i	$-0.492597\pi$
$41$	1706.34	−	1895.09i	1.01508	−	1.12736i	0.991822	−	0.127627i	$-0.0407359\pi$
$42$		0			0
$43$	−254.904	+	1199.23i	−0.137861	+	0.648584i	0.853895	+	0.520445i	$0.174234\pi$
$43$	−254.904	+	1199.23i	−0.137861	+	0.648584i	−0.991756	+	0.128139i	$0.959100\pi$
$44$		0			0
$45$	40.0056	−	17.8116i	0.0197558	−	0.00879586i
$46$		0			0
$47$	−668.281	+	2056.76i	−0.302526	+	0.931081i	0.678062	+	0.735004i	$0.262820\pi$
$47$	−668.281	+	2056.76i	−0.302526	+	0.931081i	−0.980589	+	0.196076i	$0.937180\pi$
$48$		0			0
$49$	1375.84	−	292.444i	0.573028	−	0.121801i
$50$		0			0
$51$	1443.23	+	642.566i	0.554874	+	0.247046i
$52$		0			0
$53$	−4164.47	−	437.703i	−1.48255	−	0.155822i	−0.671587	−	0.740926i	$-0.734387\pi$
$53$	−4164.47	−	437.703i	−1.48255	−	0.155822i	−0.810958	+	0.585104i	$0.801054\pi$
$54$		0			0
$55$	−35.5117	+	3.73243i	−0.0117394	+	0.00123386i
$56$		0			0
$57$	−2143.27	+	1237.42i	−0.659670	+	0.380861i
$58$		0			0
$59$	−2620.01	−	2909.81i	−0.752659	−	0.835912i	0.238143	−	0.971230i	$-0.423461\pi$
$59$	−2620.01	−	2909.81i	−0.752659	−	0.835912i	−0.990802	+	0.135318i	$0.956795\pi$
$60$		0			0
$61$			2467.23i			0.663056i	0.943445	+	0.331528i	$0.107564\pi$
$61$			2467.23i			0.663056i	−0.943445	+	0.331528i	$0.892436\pi$
$62$		0			0
$63$	2016.09			0.507960
$64$		0			0
$65$	−72.9558	+	65.6897i	−0.0172676	+	0.0155479i
$66$		0			0
$67$	1916.75	+	3319.91i	0.426989	+	0.739566i	0.996604	−	0.0823455i	$-0.0262411\pi$
$67$	1916.75	+	3319.91i	0.426989	+	0.739566i	−0.569615	+	0.821911i	$0.692908\pi$
$68$		0			0
$69$	138.664	+	1319.30i	0.0291251	+	0.277106i
$70$		0			0
$71$	−959.539	+	9129.40i	−0.190347	+	1.81103i	0.316064	+	0.948738i	$0.397638\pi$
$71$	−959.539	+	9129.40i	−0.190347	+	1.81103i	−0.506411	+	0.862292i	$0.669028\pi$
$72$		0			0
$73$	1740.13	−	3908.40i	0.326540	−	0.733421i	−0.673443	−	0.739239i	$-0.735185\pi$
$73$	1740.13	−	3908.40i	0.326540	−	0.733421i	0.999983	+	0.00581831i	$0.00185204\pi$
$74$		0			0
$75$	536.428	+	2523.70i	0.0953650	+	0.448657i
$76$		0			0
$77$	−1563.45	−	507.995i	−0.263695	−	0.0856798i
$78$		0			0
$79$	518.236	+	1163.98i	0.0830373	+	0.186505i	0.950292	−	0.311360i	$-0.100784\pi$
$79$	518.236	+	1163.98i	0.0830373	+	0.186505i	−0.867255	+	0.497864i	$0.834118\pi$
$80$		0			0
$81$	−2645.89	−	562.401i	−0.403275	−	0.0857188i
$82$		0			0
$83$	−760.880	−	685.100i	−0.110449	−	0.0994483i	0.612056	−	0.790814i	$-0.290343\pi$
$83$	−760.880	−	685.100i	−0.110449	−	0.0994483i	−0.722505	+	0.691366i	$0.757009\pi$
$84$		0			0
$85$	−153.961	+	211.909i	−0.0213094	+	0.0293299i
$86$		0			0
$87$	−2774.63	+	4805.80i	−0.366578	+	0.634932i
$88$		0			0
$89$	−375.554	−	516.906i	−0.0474124	−	0.0652576i	0.784653	−	0.619936i	$-0.212841\pi$
$89$	−375.554	−	516.906i	−0.0474124	−	0.0652576i	−0.832065	+	0.554678i	$0.812841\pi$
$90$		0			0
$91$	−4298.46	+	1396.65i	−0.519075	+	0.168658i
$92$		0			0
$93$	3896.25	−	762.206i	0.450486	−	0.0881264i
$94$		0			0
$95$	−126.799	−	390.246i	−0.0140497	−	0.0432406i
$96$		0			0
$97$	−8845.42	+	6426.57i	−0.940102	+	0.683024i	−0.948445	−	0.316941i	$-0.897344\pi$
$97$	−8845.42	+	6426.57i	−0.940102	+	0.683024i	0.00834344	+	0.999965i	$0.497344\pi$
$98$		0			0
$99$	2886.34	+	1666.43i	0.294494	+	0.170026i
$100$		0			0
$101$	−7826.39	−	5686.20i	−0.767218	−	0.557416i	0.133898	−	0.990995i	$-0.457251\pi$
$101$	−7826.39	−	5686.20i	−0.767218	−	0.557416i	−0.901116	+	0.433579i	$0.857251\pi$
$102$		0			0
$103$	−4854.65	+	5391.64i	−0.457598	+	0.508214i	−0.927150	−	0.374691i	$-0.877749\pi$
$103$	−4854.65	+	5391.64i	−0.457598	+	0.508214i	0.469552	+	0.882905i	$0.344415\pi$
$104$		0			0
$105$	18.5528	−	87.2839i	0.00168279	−	0.00791690i
$106$		0			0
$107$	11085.0	−	4935.38i	0.968211	−	0.431075i	0.139173	−	0.990268i	$-0.455555\pi$
$107$	11085.0	−	4935.38i	0.968211	−	0.431075i	0.829038	+	0.559193i	$0.188889\pi$
$108$		0			0
$109$	7025.94	−	21623.6i	0.591360	−	1.82002i	0.0192877	−	0.999814i	$-0.493860\pi$
$109$	7025.94	−	21623.6i	0.591360	−	1.82002i	0.572072	−	0.820203i	$-0.306140\pi$
$110$		0			0
$111$	5323.90	−	1131.63i	0.432100	−	0.0918457i
$112$		0			0
$113$	1589.92	+	707.880i	0.124514	+	0.0554374i	0.468048	−	0.883703i	$-0.344957\pi$
$113$	1589.92	+	707.880i	0.124514	+	0.0554374i	−0.343534	+	0.939140i	$0.611624\pi$
$114$		0			0
$115$	−218.742	−	22.9907i	−0.0165400	−	0.00173842i
$116$		0			0
$117$	9112.98	−	957.813i	0.665715	−	0.0699695i
$118$		0			0
$119$	−10443.4	+	6029.50i	−0.737477	+	0.425782i
$120$		0			0
$121$	7978.32	+	8860.82i	0.544930	+	0.605206i
$122$		0			0
$123$			10535.0i			0.696344i
$124$		0			0
$125$	−855.879			−0.0547762
$126$		0			0
$127$	14706.6	−	13241.9i	0.911811	−	0.820999i	−0.0725128	−	0.997367i	$-0.523102\pi$
$127$	14706.6	−	13241.9i	0.911811	−	0.820999i	0.984324	+	0.176369i	$0.0564351\pi$
$128$		0			0
$129$	−2532.49	−	4386.40i	−0.152184	−	0.263590i
$130$		0			0
$131$	−2350.82	−	22366.6i	−0.136986	−	1.30334i	−0.819760	−	0.572707i	$-0.805893\pi$
$131$	−2350.82	−	22366.6i	−0.136986	−	1.30334i	0.682774	−	0.730630i	$-0.260774\pi$
$132$		0			0
$133$	1974.64	−	18787.4i	0.111631	−	1.06210i
$134$		0			0
$135$	−166.811	+	374.663i	−0.00915286	+	0.0205577i
$136$		0			0
$137$	−6326.11	−	29762.0i	−0.337051	−	1.58570i	−0.741380	−	0.671086i	$-0.765828\pi$
$137$	−6326.11	−	29762.0i	−0.337051	−	1.58570i	0.404329	−	0.914614i	$-0.367505\pi$
$138$		0			0
$139$	17686.0	+	5746.53i	0.915377	+	0.297424i	0.728569	−	0.684973i	$-0.240186\pi$
$139$	17686.0	+	5746.53i	0.915377	+	0.297424i	0.186808	+	0.982396i	$0.440186\pi$
$140$		0			0
$141$	−3633.86	−	8161.79i	−0.182781	−	0.410532i
$142$		0			0
$143$	−7308.31	−	1553.43i	−0.357392	−	0.0759660i
$144$		0			0
$145$	−683.747	−	615.648i	−0.0325207	−	0.0292817i
$146$		0			0
$147$	−3415.55	+	4701.10i	−0.158062	+	0.217553i
$148$		0			0
$149$	−1062.87	+	1840.94i	−0.0478748	+	0.0829216i	−0.888970	−	0.457966i	$-0.848578\pi$
$149$	−1062.87	+	1840.94i	−0.0478748	+	0.0829216i	0.841095	+	0.540887i	$0.181912\pi$
$150$		0			0
$151$	21913.5	+	30161.3i	0.961075	+	1.32281i	0.946429	+	0.322913i	$0.104662\pi$
$151$	21913.5	+	30161.3i	0.961075	+	1.32281i	0.0146464	+	0.999893i	$0.495338\pi$
$152$		0			0
$153$	23251.8	−	7554.98i	0.993286	−	0.322738i
$154$		0			0
$155$	−10.6986	+	658.160i	−0.000445311	+	0.0273948i
$156$		0			0
$157$	251.907	+	775.291i	0.0102198	+	0.0314533i	0.956036	−	0.293248i	$-0.0947362\pi$
$157$	251.907	+	775.291i	0.0102198	+	0.0314533i	−0.945817	+	0.324701i	$0.894736\pi$
$158$		0			0
$159$	13995.3	−	10168.2i	0.553589	−	0.402206i
$160$		0			0
$161$	−8769.36	−	5062.99i	−0.338311	−	0.195324i
$162$		0			0
$163$	−15881.8	−	11538.8i	−0.597758	−	0.434297i	0.247324	−	0.968933i	$-0.420449\pi$
$163$	−15881.8	−	11538.8i	−0.597758	−	0.434297i	−0.845083	+	0.534636i	$0.820449\pi$
$164$		0			0
$165$	98.7067	−	109.625i	0.00362559	−	0.00402662i
$166$		0			0
$167$	322.909	−	1519.17i	0.0115784	−	0.0544719i	−0.971979	−	0.235067i	$-0.924469\pi$
$167$	322.909	−	1519.17i	0.0115784	−	0.0544719i	0.983558	+	0.180595i	$0.0578024\pi$
$168$		0			0
$169$	7325.76	−	3261.64i	0.256495	−	0.114199i
$170$		0			0
$171$	−11835.2	+	36424.9i	−0.404746	+	1.24568i
$172$		0			0
$173$	−33892.8	+	7204.13i	−1.13244	+	0.240707i	−0.735755	−	0.677248i	$-0.763172\pi$
$173$	−33892.8	+	7204.13i	−1.13244	+	0.240707i	−0.396684	+	0.917955i	$0.629839\pi$
$174$		0			0
$175$	−17991.6	−	8010.37i	−0.587481	−	0.261563i
$176$		0			0
$177$	16087.3	+	1690.85i	0.513497	+	0.0539707i
$178$		0			0
$179$	−10987.1	+	1154.79i	−0.342909	+	0.0360412i	−0.274418	−	0.961611i	$-0.588485\pi$
$179$	−10987.1	+	1154.79i	−0.342909	+	0.0360412i	−0.0684910	+	0.997652i	$0.521818\pi$
$180$		0			0
$181$	8280.66	−	4780.84i	0.252760	−	0.145931i	−0.368267	−	0.929720i	$-0.620049\pi$
$181$	8280.66	−	4780.84i	0.252760	−	0.145931i	0.621027	+	0.783789i	$0.286716\pi$
$182$		0			0
$183$	−6820.23	−	7574.64i	−0.203656	−	0.226183i
$184$		0			0
$185$			902.427i			0.0263675i
$186$		0			0
$187$	−19935.1			−0.570078
$188$		0			0
$189$	−14031.5	+	12634.0i	−0.392809	+	0.353687i
$190$		0			0
$191$	7970.92	+	13806.0i	0.218495	+	0.378445i	0.954348	−	0.298697i	$-0.0965519\pi$
$191$	7970.92	+	13806.0i	0.218495	+	0.378445i	−0.735853	+	0.677141i	$0.763219\pi$
$192$		0			0
$193$	7212.23	+	68619.8i	0.193622	+	1.84219i	0.471853	+	0.881677i	$0.343585\pi$
$193$	7212.23	+	68619.8i	0.193622	+	1.84219i	−0.278231	+	0.960514i	$0.589748\pi$
$194$		0			0
$195$	42.3935	−	403.347i	0.00111489	−	0.0106074i
$196$		0			0
$197$	3956.70	−	8886.89i	0.101953	−	0.228990i	−0.855366	−	0.518025i	$-0.826668\pi$
$197$	3956.70	−	8886.89i	0.101953	−	0.228990i	0.957319	+	0.289034i	$0.0933342\pi$
$198$		0			0
$199$	9984.58	+	46973.8i	0.252130	+	1.18618i	0.903895	+	0.427755i	$0.140696\pi$
$199$	9984.58	+	46973.8i	0.252130	+	1.18618i	−0.651765	+	0.758421i	$0.725971\pi$
$200$		0			0
$201$	−15061.9	−	4893.92i	−0.372811	−	0.121134i
$202$		0			0
$203$	−17228.8	−	38696.6i	−0.418084	−	0.939032i
$204$		0			0
$205$	−1708.54	−	363.161i	−0.0406553	−	0.00864155i
$206$		0			0
$207$	15256.3	+	13736.9i	0.356049	+	0.320588i
$208$		0			0
$209$	18356.0	−	25264.8i	0.420228	−	0.578394i
$210$		0			0
$211$	−18584.5	+	32189.3i	−0.417432	+	0.723014i	−0.995680	−	0.0928470i	$-0.970403\pi$
$211$	−18584.5	+	32189.3i	−0.417432	+	0.723014i	0.578248	+	0.815861i	$0.303737\pi$
$212$		0			0
$213$	−22290.8	−	30680.6i	−0.491322	−	0.676247i
$214$		0			0
$215$	798.675	−	259.505i	0.0172780	−	0.00561396i
$216$		0			0
$217$	−12774.3	+	27480.7i	−0.271281	+	0.583590i
$218$		0			0
$219$	5461.72	+	16809.5i	0.113878	+	0.350482i
$220$		0			0
$221$	−44340.9	+	32215.6i	−0.907862	+	0.659601i
$222$		0			0
$223$	20796.5	+	12006.9i	0.418197	+	0.241446i	0.694305	−	0.719680i	$-0.255712\pi$
$223$	20796.5	+	12006.9i	0.418197	+	0.241446i	−0.276109	+	0.961126i	$0.589045\pi$
$224$		0			0
$225$	32302.5	+	23469.2i	0.638075	+	0.463588i
$226$		0			0
$227$	17305.2	−	19219.3i	0.335834	−	0.372981i	−0.551448	−	0.834209i	$-0.685924\pi$
$227$	17305.2	−	19219.3i	0.335834	−	0.372981i	0.887282	+	0.461228i	$0.152591\pi$
$228$		0			0
$229$	−3331.57	+	15673.8i	−0.0635298	+	0.298884i	−0.998429	−	0.0560287i	$-0.982156\pi$
$229$	−3331.57	+	15673.8i	−0.0635298	+	0.298884i	0.934899	+	0.354913i	$0.115489\pi$
$230$		0			0
$231$	6204.21	−	2762.29i	0.116269	−	0.0517661i
$232$		0			0
$233$	−20386.0	+	62741.7i	−0.375509	+	1.15570i	0.567625	+	0.823287i	$0.307862\pi$
$233$	−20386.0	+	62741.7i	−0.375509	+	1.15570i	−0.943134	+	0.332412i	$0.892138\pi$
$234$		0			0
$235$	1448.93	−	307.979i	0.0262368	−	0.00557680i
$236$		0			0
$237$	−4808.65	−	2140.95i	−0.0856103	−	0.0381162i
$238$		0			0
$239$	−91589.8	−	9626.48i	−1.60344	−	0.168528i	−0.739865	−	0.672756i	$-0.765111\pi$
$239$	−91589.8	−	9626.48i	−1.60344	−	0.168528i	−0.863571	+	0.504228i	$0.831777\pi$
$240$		0			0
$241$	76246.0	−	8013.77i	1.31275	−	0.137976i	0.577847	−	0.816145i	$-0.303893\pi$
$241$	76246.0	−	8013.77i	1.31275	−	0.137976i	0.734905	+	0.678170i	$0.237227\pi$
$242$		0			0
$243$	51679.0	−	29836.9i	0.875188	−	0.505290i
$244$		0			0
$245$	−644.673	−	715.982i	−0.0107401	−	0.0119281i
$246$		0			0
$247$		−	85859.5i		−	1.40732i
$248$		0			0
$249$	4229.82			0.0682217
$250$		0			0
$251$	7570.10	−	6816.14i	0.120158	−	0.108191i	−0.606861	−	0.794808i	$-0.707572\pi$
$251$	7570.10	−	6816.14i	0.120158	−	0.108191i	0.727020	+	0.686617i	$0.240905\pi$
$252$		0			0
$253$	−8369.77	−	14496.9i	−0.130759	−	0.226482i
$254$		0			0
$255$	−113.111	−	1076.18i	−0.00173950	−	0.0165502i
$256$		0			0
$257$	−3782.88	+	35991.7i	−0.0572738	+	0.544924i	0.927835	+	0.372991i	$0.121668\pi$
$257$	−3782.88	+	35991.7i	−0.0572738	+	0.544924i	−0.985109	+	0.171933i	$0.944999\pi$
$258$		0			0
$259$	−16898.4	+	37954.5i	−0.251911	+	0.565800i
$260$		0			0
$261$	17855.0	+	84001.3i	0.262108	+	1.23312i
$262$		0			0
$263$	−87145.1	−	28315.1i	−1.25989	−	0.409362i	−0.398432	−	0.917198i	$-0.630446\pi$
$263$	−87145.1	−	28315.1i	−1.25989	−	0.409362i	−0.861454	+	0.507836i	$0.830446\pi$
$264$		0			0
$265$	1166.61	+	2620.24i	0.0166124	+	0.0373120i
$266$		0			0
$267$	2581.88	+	548.796i	0.0362171	+	0.00769819i
$268$		0			0
$269$	−42811.7	−	38547.9i	−0.591641	−	0.532716i	0.318011	−	0.948087i	$-0.396985\pi$
$269$	−42811.7	−	38547.9i	−0.591641	−	0.532716i	−0.909652	+	0.415371i	$0.863652\pi$
$270$		0			0
$271$	7437.60	−	10237.0i	0.101273	−	0.139391i	−0.755373	−	0.655295i	$-0.772544\pi$
$271$	7437.60	−	10237.0i	0.101273	−	0.139391i	0.856646	+	0.515905i	$0.172544\pi$
$272$		0			0
$273$	9335.88	−	16170.2i	0.125265	−	0.216965i
$274$		0			0
$275$	−19136.6	−	26339.2i	−0.253046	−	0.348287i
$276$		0			0
$277$	70509.6	−	22909.9i	0.918943	−	0.298583i	0.188910	−	0.981994i	$-0.439505\pi$
$277$	70509.6	−	22909.9i	0.918943	−	0.298583i	0.730033	+	0.683412i	$0.239505\pi$
$278$		0			0
$279$	36916.4	−	49112.2i	0.474254	−	0.630929i
$280$		0			0
$281$	−3169.58	−	9754.96i	−0.0401410	−	0.123541i	0.928978	−	0.370135i	$-0.120689\pi$
$281$	−3169.58	−	9754.96i	−0.0401410	−	0.123541i	−0.969119	+	0.246594i	$0.920689\pi$
$282$		0			0
$283$	119596.	−	86891.6i	1.49329	−	1.08494i	0.520328	−	0.853966i	$-0.325810\pi$
$283$	119596.	−	86891.6i	1.49329	−	1.08494i	0.972961	−	0.230971i	$-0.0741904\pi$
$284$		0			0
$285$	1468.05	+	847.581i	0.0180739	+	0.0104350i
$286$		0			0
$287$	−65057.7	−	47267.2i	−0.789832	−	0.573847i
$288$		0			0
$289$	−41964.0	+	46605.8i	−0.502437	+	0.558012i
$290$		0			0
$291$	9391.14	−	44181.8i	0.110900	−	0.521744i
$292$		0			0
$293$	−94.2132	+	41.9464i	−0.00109743	+	0.000488607i	−0.407285	−	0.913301i	$-0.633525\pi$
$293$	−94.2132	+	41.9464i	−0.00109743	+	0.000488607i	0.406188	+	0.913790i	$0.366858\pi$
$294$		0			0
$295$	−828.779	+	2550.72i	−0.00952346	+	0.0293102i
$296$		0			0
$297$	−30531.1	+	6489.58i	−0.346122	+	0.0735705i
$298$		0			0
$299$	−42043.9	−	18719.1i	−0.470284	−	0.209384i
$300$		0			0
$301$	38450.2	+	4041.28i	0.424390	+	0.0446052i
$302$		0			0
$303$	39746.3	−	4177.50i	0.432924	−	0.0455021i
$304$		0			0
$305$	1463.54	−	844.977i	0.0157328	−	0.00908333i
$306$		0			0
$307$	74791.2	+	83064.0i	0.793549	+	0.881326i	0.995173	−	0.0981375i	$-0.0312885\pi$
$307$	74791.2	+	83064.0i	0.793549	+	0.881326i	−0.201624	+	0.979463i	$0.564622\pi$
$308$		0			0
$309$		−	29972.7i		−	0.313913i
$310$		0			0
$311$	163082.			1.68610			0.843052	−	0.537833i	$-0.180757\pi$
$311$	163082.			1.68610			0.843052	+	0.537833i	$0.180757\pi$
$312$		0			0
$313$	25596.9	−	23047.6i	0.261276	−	0.235254i	−0.528073	−	0.849199i	$-0.677085\pi$
$313$	25596.9	−	23047.6i	0.261276	−	0.235254i	0.789348	+	0.613945i	$0.210419\pi$
$314$		0			0
$315$	−690.472	−	1195.93i	−0.00695865	−	0.0120527i
$316$		0			0
$317$	−4339.13	−	41284.0i	−0.0431801	−	0.410831i	−0.994667	−	0.103141i	$-0.967111\pi$
$317$	−4339.13	−	41284.0i	−0.0431801	−	0.410831i	0.951487	−	0.307690i	$-0.0995560\pi$
$318$		0			0
$319$	7319.53	−	69640.6i	0.0719286	−	0.684355i
$320$		0			0
$321$	−20389.2	+	45794.8i	−0.197874	+	0.444433i
$322$		0			0
$323$	−47629.1	−	224077.i	−0.456528	−	2.14779i
$324$		0			0
$325$	−85129.7	−	27660.3i	−0.805962	−	0.261873i
$326$		0			0
$327$	38204.5	+	85808.7i	0.357288	+	0.802482i
$328$		0			0
$329$	66706.3	+	14178.9i	0.616276	+	0.130993i
$330$		0			0
$331$	−82314.3	−	74116.1i	−0.751310	−	0.676483i	0.201691	−	0.979449i	$-0.435356\pi$
$331$	−82314.3	−	74116.1i	−0.751310	−	0.676483i	−0.953001	+	0.302966i	$0.902023\pi$
$332$		0			0
$333$	49509.7	−	68144.3i	0.446480	−	0.614527i
$334$		0			0
$335$	1312.90	−	2274.01i	0.0116988	−	0.0202629i
$336$		0			0
$337$	121498.	+	167228.i	1.06982	+	1.47248i	0.870259	+	0.492594i	$0.163951\pi$
$337$	121498.	+	167228.i	1.06982	+	1.47248i	0.199560	+	0.979886i	$0.436049\pi$
$338$		0			0
$339$	−6838.03	+	2221.81i	−0.0595020	+	0.0193334i
$340$		0			0
$341$	−41002.9	+	28783.9i	−0.352619	+	0.247537i
$342$		0			0
$343$	−37103.7	−	114193.i	−0.315376	−	0.970627i
$344$		0			0
$345$	735.112	−	534.090i	0.00617611	−	0.00448721i
$346$		0			0
$347$	7247.70	+	4184.46i	0.0601923	+	0.0347521i	0.529794	−	0.848126i	$-0.322269\pi$
$347$	7247.70	+	4184.46i	0.0601923	+	0.0347521i	−0.469602	+	0.882878i	$0.655603\pi$
$348$		0			0
$349$	−111740.	−	81183.5i	−0.917394	−	0.666526i	0.0254797	−	0.999675i	$-0.491889\pi$
$349$	−111740.	−	81183.5i	−0.917394	−	0.666526i	−0.942874	+	0.333149i	$0.891889\pi$
$350$		0			0
$351$	−57421.9	+	63773.5i	−0.466083	+	0.517638i
$352$		0			0
$353$	−17359.5	+	81670.2i	−0.139312	+	0.655412i	0.851963	+	0.523602i	$0.175412\pi$
$353$	−17359.5	+	81670.2i	−0.139312	+	0.655412i	−0.991275	+	0.131810i	$0.957921\pi$
$354$		0			0
$355$	5744.12	−	2557.45i	0.0455792	−	0.0202932i
$356$		0			0
$357$	15394.8	−	47380.2i	0.120791	−	0.371758i
$358$		0			0
$359$	96626.0	−	20538.5i	0.749730	−	0.159360i	0.182828	−	0.983145i	$-0.441475\pi$
$359$	96626.0	−	20538.5i	0.749730	−	0.159360i	0.566903	+	0.823785i	$0.308142\pi$
$360$		0			0
$361$	208788.	+	92958.4i	1.60210	+	0.713303i
$362$		0			0
$363$	−48988.4	−	5148.89i	−0.371775	−	0.0390751i
$364$		0			0
$365$	−2914.39	+	306.315i	−0.0218757	+	0.00229923i
$366$		0			0
$367$	−131551.	+	75951.1i	−0.976703	+	0.563900i	−0.901273	−	0.433251i	$-0.857366\pi$
$367$	−131551.	+	75951.1i	−0.976703	+	0.563900i	−0.0754298	+	0.997151i	$0.524033\pi$
$368$		0			0
$369$	109092.	+	121159.i	0.801196	+	0.889818i
$370$		0			0
$371$			132048.i			0.959363i
$372$		0			0
$373$	−135986.			−0.977409			−0.488705	−	0.872449i	$-0.662530\pi$
$373$	−135986.			−0.977409			−0.488705	+	0.872449i	$0.662530\pi$
$374$		0			0
$375$	2627.63	−	2365.93i	0.0186854	−	0.0168244i
$376$		0			0
$377$	−96260.4	−	166728.i	−0.677275	−	1.17307i
$378$		0			0
$379$	14028.3	+	133470.i	0.0976622	+	0.929194i	0.928162	+	0.372176i	$0.121388\pi$
$379$	14028.3	+	133470.i	0.0976622	+	0.929194i	−0.830500	+	0.557018i	$0.811945\pi$
$380$		0			0
$381$	−8545.79	+	81307.8i	−0.0588711	+	0.560121i
$382$		0			0
$383$	−792.879	+	1780.83i	−0.00540517	+	0.0121402i	−0.916228	−	0.400658i	$-0.868782\pi$
$383$	−792.879	+	1780.83i	−0.00540517	+	0.0121402i	0.910823	+	0.412798i	$0.135448\pi$
$384$		0			0
$385$	234.111	+	1101.40i	0.00157943	+	0.00743063i
$386$		0			0
$387$	−74546.9	−	24221.8i	−0.497746	−	0.161728i
$388$		0			0
$389$	−83915.4	−	188477.i	−0.554552	−	1.24555i	−0.945647	−	0.325195i	$-0.894570\pi$
$389$	−83915.4	−	188477.i	−0.554552	−	1.24555i	0.391095	−	0.920350i	$-0.372097\pi$
$390$		0			0
$391$	−120111.	−	25530.3i	−0.785649	−	0.166995i
$392$		0			0
$393$	69045.8	+	62169.1i	0.447046	+	0.402522i
$394$		0			0
$395$	512.977	−	706.052i	0.00328779	−	0.00452525i
$396$		0			0
$397$	−76391.5	+	132314.i	−0.484690	+	0.839507i	−0.999845	−	0.0175895i	$-0.994401\pi$
$397$	−76391.5	+	132314.i	−0.484690	+	0.839507i	0.515156	+	0.857097i	$0.327734\pi$
$398$		0			0
$399$	45872.3	+	63137.8i	0.288141	+	0.396592i
$400$		0			0
$401$	−76390.3	+	24820.7i	−0.475061	+	0.154357i	−0.536755	−	0.843738i	$-0.680350\pi$
$401$	−76390.3	+	24820.7i	−0.475061	+	0.154357i	0.0616933	+	0.998095i	$0.480350\pi$
$402$		0			0
$403$	−44685.9	+	130285.i	−0.275144	+	0.802201i
$404$		0			0
$405$	572.552	+	1762.13i	0.00349064	+	0.0107431i
$406$		0			0
$407$	−55564.4	+	40369.9i	−0.335434	+	0.243707i
$408$		0			0
$409$	249216.	+	143885.i	1.48981	+	0.860141i	0.999932	−	0.0116519i	$-0.00370899\pi$
$409$	249216.	+	143885.i	1.48981	+	0.860141i	0.489875	+	0.871793i	$0.337042\pi$
$410$		0			0
$411$	101694.	+	73884.8i	0.602019	+	0.437392i
$412$		0			0
$413$	−82620.5	+	91759.4i	−0.484382	+	0.537960i
$414$		0			0
$415$	−145.810	+	685.982i	−0.000846624	+	0.00398305i
$416$		0			0
$417$	−70183.0	+	31247.5i	−0.403608	+	0.179698i
$418$		0			0
$419$	−14107.4	+	43418.0i	−0.0803559	+	0.247310i	−0.983161	−	0.182739i	$-0.941504\pi$
$419$	−14107.4	+	43418.0i	−0.0803559	+	0.247310i	0.902806	+	0.430049i	$0.141504\pi$
$420$		0			0
$421$	−54591.6	+	11603.8i	−0.308008	+	0.0654691i	−0.359322	−	0.933213i	$-0.616992\pi$
$421$	−54591.6	+	11603.8i	−0.308008	+	0.0654691i	0.0513147	+	0.998683i	$0.483659\pi$
$422$		0			0
$423$	−126308.	−	56236.1i	−0.705913	−	0.314293i
$424$		0			0
$425$	−237517.	−	24964.0i	−1.31497	−	0.138209i
$426$		0			0
$427$	77376.6	−	8132.61i	0.424379	−	0.0446040i
$428$		0			0
$429$	26731.4	−	15433.4i	0.145247	−	0.0838584i
$430$		0			0
$431$	17698.3	+	19655.9i	0.0952744	+	0.105813i	0.788896	−	0.614526i	$-0.210653\pi$
$431$	17698.3	+	19655.9i	0.0952744	+	0.105813i	−0.693622	+	0.720339i	$0.743986\pi$
$432$		0			0
$433$		−	235426.i		−	1.25568i	−0.778343	−	0.627840i	$-0.783939\pi$
$433$		−	235426.i		−	1.25568i	0.778343	−	0.627840i	$-0.216061\pi$
$434$		0			0
$435$	3801.02			0.0200873
$436$		0			0
$437$	142953.	−	128715.i	0.748565	−	0.674011i
$438$		0			0
$439$	−23280.5	−	40323.1i	−0.120799	−	0.209230i	0.799284	−	0.600954i	$-0.205212\pi$
$439$	−23280.5	−	40323.1i	−0.120799	−	0.209230i	−0.920083	+	0.391723i	$0.871879\pi$
$440$		0			0
$441$	9399.90	+	89434.0i	0.0483332	+	0.459860i
$442$		0			0
$443$	−27991.6	+	266323.i	−0.142633	+	1.35706i	0.655779	+	0.754953i	$0.272340\pi$
$443$	−27991.6	+	266323.i	−0.142633	+	1.35706i	−0.798412	+	0.602111i	$0.794326\pi$
$444$		0			0
$445$	−178.005	+	399.806i	−0.000898902	+	0.00201897i
$446$		0			0
$447$	−1825.86	−	8589.99i	−0.00913802	−	0.0429910i
$448$		0			0
$449$	81201.5	+	26384.0i	0.402783	+	0.130872i	0.503401	−	0.864053i	$-0.332082\pi$
$449$	81201.5	+	26384.0i	0.402783	+	0.130872i	−0.100618	+	0.994925i	$0.532082\pi$
$450$		0			0
$451$	−54070.5	−	121444.i	−0.265832	−	0.597069i
$452$		0			0
$453$	−150652.	−	32022.1i	−0.734140	−	0.156046i
$454$		0			0
$455$	2300.62	+	2071.49i	0.0111128	+	0.0100060i
$456$		0			0
$457$	2314.10	−	3185.08i	0.0110802	−	0.0152507i	−0.803441	−	0.595384i	$-0.797000\pi$
$457$	2314.10	−	3185.08i	0.0110802	−	0.0152507i	0.814521	+	0.580134i	$0.197000\pi$
$458$		0			0
$459$	−114483.	+	198291.i	−0.543396	+	0.941190i
$460$		0			0
$461$	76338.6	+	105071.i	0.359205	+	0.494403i	0.949927	−	0.312472i	$-0.101157\pi$
$461$	76338.6	+	105071.i	0.359205	+	0.494403i	−0.590722	+	0.806875i	$0.701157\pi$
$462$		0			0
$463$	127841.	−	41538.1i	0.596361	−	0.193769i	0.00474408	−	0.999989i	$-0.498490\pi$
$463$	127841.	−	41538.1i	0.596361	−	0.193769i	0.591617	+	0.806219i	$0.298490\pi$
$464$		0			0
$465$	−1786.52	−	2050.19i	−0.00826234	−	0.00948174i
$466$		0			0
$467$	80965.7	+	249187.i	0.371251	+	1.14259i	0.945973	+	0.324244i	$0.105110\pi$
$467$	80965.7	+	249187.i	0.371251	+	1.14259i	−0.574723	+	0.818348i	$0.694890\pi$
$468$		0			0
$469$	97800.1	−	71055.9i	0.444625	−	0.323039i
$470$		0			0
$471$	−2916.54	−	1683.87i	−0.0131470	−	0.00759042i
$472$		0			0
$473$	51706.8	+	37567.2i	0.231114	+	0.167914i
$474$		0			0
$475$	250341.	−	278032.i	1.10954	−	1.23227i
$476$		0			0
$477$	55660.8	−	261863.i	0.244632	−	1.15090i
$478$		0			0
$479$	359841.	−	160212.i	1.56834	−	0.698270i	0.575509	−	0.817796i	$-0.304804\pi$
$479$	359841.	−	160212.i	1.56834	−	0.698270i	0.992831	+	0.119526i	$0.0381375\pi$
$480$		0			0
$481$	−58351.3	+	179587.i	−0.252209	+	0.776219i
$482$		0			0
$483$	40918.6	−	8697.51i	0.175399	−	0.0372821i
$484$		0			0
$485$	6841.57	+	3046.07i	0.0290852	+	0.0129496i
$486$		0			0
$487$	−105726.	−	11112.3i	−0.445785	−	0.0468539i	−0.121023	−	0.992650i	$-0.538618\pi$
$487$	−105726.	−	11112.3i	−0.445785	−	0.0468539i	−0.324762	+	0.945796i	$0.605284\pi$
$488$		0			0
$489$	80655.9	−	8477.28i	0.337302	−	0.0354518i
$490$		0			0
$491$	−127359.	+	73530.8i	−0.528283	+	0.305004i	−0.740317	−	0.672258i	$-0.765325\pi$
$491$	−127359.	+	73530.8i	−0.528283	+	0.305004i	0.212034	+	0.977262i	$0.431991\pi$
$492$		0			0
$493$	−343711.	−	381730.i	−1.41416	−	1.57059i
$494$		0			0
$495$		−	2282.87i		−	0.00931690i
$496$		0			0
$497$	289477.			1.17193
$498$		0			0
$499$	−208857.	+	188056.i	−0.838780	+	0.755241i	−0.971783	−	0.235875i	$-0.924204\pi$
$499$	−208857.	+	188056.i	−0.838780	+	0.755241i	0.133004	+	0.991116i	$0.457538\pi$
$500$		0			0
$501$	3208.12	+	5556.62i	0.0127813	+	0.0221378i
$502$		0			0
$503$	2406.59	+	22897.2i	0.00951189	+	0.0904996i	0.998248	−	0.0591704i	$-0.0188455\pi$
$503$	2406.59	+	22897.2i	0.00951189	+	0.0904996i	−0.988736	+	0.149670i	$0.952179\pi$
$504$		0			0
$505$	−692.633	+	6589.97i	−0.00271594	+	0.0258405i
$506$		0			0
$507$	−13474.5	+	30264.3i	−0.0524202	+	0.117738i
$508$		0			0
$509$	−92657.6	−	435920.i	−0.357640	−	1.68256i	−0.677838	−	0.735211i	$-0.737083\pi$
$509$	−92657.6	−	435920.i	−0.357640	−	1.68256i	0.320198	−	0.947351i	$-0.396251\pi$
$510$		0			0
$511$	−128310.	−	41690.5i	−0.491382	−	0.159660i
$512$		0			0
$513$	−145890.	−	327675.i	−0.554360	−	1.24511i
$514$		0			0
$515$	4860.90	+	1033.22i	0.0183275	+	0.00389562i
$516$		0			0
$517$	83780.3	+	75436.1i	0.313445	+	0.282227i
$518$		0			0
$519$	84139.5	−	115808.i	0.312367	−	0.429936i
$520$		0			0
$521$	−3344.11	+	5792.17i	−0.0123198	+	0.0213386i	−0.872120	−	0.489293i	$-0.837255\pi$
$521$	−3344.11	+	5792.17i	−0.0123198	+	0.0213386i	0.859800	+	0.510631i	$0.170588\pi$
$522$		0			0
$523$	−134347.	−	184913.i	−0.491163	−	0.676028i	0.489439	−	0.872038i	$-0.337202\pi$
$523$	−134347.	−	184913.i	−0.491163	−	0.676028i	−0.980602	+	0.196009i	$0.937202\pi$
$524$		0			0
$525$	77379.3	−	25142.0i	0.280741	−	0.0912183i
$526$		0			0
$527$	−44348.6	+	364807.i	−0.159683	+	1.31354i
$528$		0			0
$529$	54612.7	+	168081.i	0.195156	+	0.600629i
$530$		0			0
$531$	202523.	−	147141.i	0.718265	−	0.521850i
$532$		0			0
$533$	−316524.	−	182745.i	−1.11417	−	0.643268i
$534$		0			0
$535$	−6724.04	−	4885.30i	−0.0234922	−	0.0170680i
$536$		0			0
$537$	30539.4	−	33917.4i	0.105904	−	0.117618i
$538$		0			0
$539$	15245.2	−	71723.2i	0.0524755	−	0.246878i
$540$		0			0
$541$	318331.	−	141730.i	1.08764	−	0.484247i	0.216999	−	0.976172i	$-0.430373\pi$
$541$	318331.	−	141730.i	1.08764	−	0.484247i	0.870638	+	0.491925i	$0.163707\pi$
$542$		0			0
$543$	−12206.6	+	37568.1i	−0.0413996	+	0.127415i
$544$		0			0
$545$	−15233.2	+	3237.92i	−0.0512860	+	0.0109012i
$546$		0			0
$547$	125578.	+	55910.8i	0.419699	+	0.186862i	0.605710	−	0.795686i	$-0.292889\pi$
$547$	125578.	+	55910.8i	0.419699	+	0.186862i	−0.186011	+	0.982548i	$0.559556\pi$
$548$		0			0
$549$	−156873.	−	16488.0i	−0.520480	−	0.0547047i
$550$		0			0
$551$	800273.	−	84112.1i	2.63594	−	0.277048i
$552$		0			0
$553$	34796.1	−	20089.5i	0.113784	−	0.0656930i
$554$		0			0
$555$	−2494.61	−	2770.54i	−0.00809871	−	0.00899453i
$556$		0			0
$557$			276000.i			0.889607i	0.895628	+	0.444803i	$0.146726\pi$
$557$			276000.i			0.889607i	−0.895628	+	0.444803i	$0.853274\pi$
$558$		0			0
$559$	175719.			0.562336
$560$		0			0
$561$	61202.6	−	55107.1i	0.194466	−	0.175098i
$562$		0			0
$563$	−237762.	−	411816.i	−0.750112	−	1.29923i	−0.947768	−	0.318960i	$-0.896666\pi$
$563$	−237762.	−	411816.i	−0.750112	−	1.29923i	0.197656	−	0.980271i	$-0.436667\pi$
$564$		0			0
$565$	−124.608	−	1185.57i	−0.000390346	−	0.00371389i
$566$		0			0
$567$	−8916.36	+	84833.5i	−0.0277346	+	0.263877i
$568$		0			0
$569$	−75559.9	+	169710.i	−0.233382	+	0.524184i	−0.991833	−	0.127546i	$-0.959290\pi$
$569$	−75559.9	+	169710.i	−0.233382	+	0.524184i	0.758451	+	0.651730i	$0.225957\pi$
$570$		0			0
$571$	63514.6	+	298813.i	0.194805	+	0.916488i	0.961570	+	0.274559i	$0.0885320\pi$
$571$	63514.6	+	298813.i	0.194805	+	0.916488i	−0.766765	+	0.641928i	$0.778135\pi$
$572$		0			0
$573$	−62635.9	−	20351.6i	−0.190772	−	0.0619855i
$574$		0			0
$575$	−81567.8	−	183204.i	−0.246708	−	0.554115i
$576$		0			0
$577$	15706.9	+	3338.60i	0.0471779	+	0.0100280i	0.231440	−	0.972849i	$-0.425656\pi$
$577$	15706.9	+	3338.60i	0.0471779	+	0.0100280i	−0.184262	+	0.982877i	$0.558990\pi$
$578$		0			0
$579$	−211830.	−	190732.i	−0.631873	−	0.568941i
$580$		0			0
$581$	−18977.9	+	26120.8i	−0.0562205	+	0.0773809i
$582$		0			0
$583$	−109146.	+	189046.i	−0.321122	+	0.556199i
$584$		0			0
$585$	−3689.18	−	5077.72i	−0.0107800	−	0.0148374i
$586$		0			0
$587$	431994.	−	140363.i	1.25372	−	0.407359i	0.394469	−	0.918909i	$-0.370928\pi$
$587$	431994.	−	140363.i	1.25372	−	0.407359i	0.859254	+	0.511550i	$0.170928\pi$
$588$		0			0
$589$	−421506.	−	392116.i	−1.21499	−	1.13027i
$590$		0			0
$591$	12418.8	+	38221.2i	0.0355554	+	0.109428i
$592$		0			0
$593$	265678.	−	193026.i	0.755520	−	0.548917i	−0.142013	−	0.989865i	$-0.545358\pi$
$593$	265678.	−	193026.i	0.755520	−	0.548917i	0.897533	+	0.440948i	$0.145358\pi$
$594$		0			0
$595$	7153.32	+	4129.97i	0.0202057	+	0.0116658i
$596$		0			0
$597$	−160505.	−	116613.i	−0.450338	−	0.327190i
$598$		0			0
$599$	−330120.	+	366636.i	−0.920065	+	1.02184i	0.0796220	+	0.996825i	$0.474629\pi$
$599$	−330120.	+	366636.i	−0.920065	+	1.02184i	−0.999687	+	0.0250108i	$0.992038\pi$
$600$		0			0
$601$	86993.9	−	409274.i	0.240846	−	1.13309i	−0.676946	−	0.736032i	$-0.736697\pi$
$601$	86993.9	−	409274.i	0.240846	−	1.13309i	0.917793	−	0.397060i	$-0.129970\pi$
$602$		0			0
$603$	−223898.	+	99686.0i	−0.615767	+	0.274157i
$604$		0			0
$605$	2523.76	−	7767.34i	0.00689505	−	0.0212208i
$606$		0			0
$607$	−480771.	+	102191.i	−1.30485	+	0.277354i	−0.807331	−	0.590099i	$-0.799089\pi$
$607$	−480771.	+	102191.i	−1.30485	+	0.277354i	−0.497519	+	0.867453i	$0.665755\pi$
$608$		0			0
$609$	159864.	+	71176.1i	0.431039	+	0.191911i
$610$		0			0
$611$	308256.	+	32399.0i	0.825714	+	0.0867860i
$612$		0			0
$613$	551819.	−	57998.5i	1.46851	−	0.154346i	0.663751	−	0.747953i	$-0.268963\pi$
$613$	551819.	−	57998.5i	1.46851	−	0.154346i	0.804755	+	0.593607i	$0.202297\pi$
$614$		0			0
$615$	6249.28	−	3608.02i	0.0165227	−	0.00953936i
$616$		0			0
$617$	362428.	+	402518.i	0.952033	+	1.05734i	0.998293	+	0.0584102i	$0.0186031\pi$
$617$	362428.	+	402518.i	0.952033	+	1.05734i	−0.0462597	+	0.998929i	$0.514730\pi$
$618$		0			0
$619$			260416.i			0.679653i	0.940488	+	0.339826i	$0.110368\pi$
$619$			260416.i			0.679653i	−0.940488	+	0.339826i	$0.889632\pi$
$620$		0			0
$621$	−192264.			−0.498557
$622$		0			0
$623$	−14973.1	+	13481.9i	−0.0385777	+	0.0347355i
$624$		0			0
$625$	−194873.	−	337530.i	−0.498874	−	0.864076i
$626$		0			0
$627$	13485.7	+	128307.i	0.0343034	+	0.326375i
$628$		0			0
$629$	−52663.2	+	501057.i	−0.133109	+	1.26644i
$630$		0			0
$631$	−163314.	+	366810.i	−0.410171	+	0.921260i	0.583829	+	0.811877i	$0.301554\pi$
$631$	−163314.	+	366810.i	−0.410171	+	0.921260i	−0.994000	+	0.109383i	$0.965113\pi$
$632$		0			0
$633$	−31925.6	−	150198.i	−0.0796767	−	0.374849i
$634$		0			0
$635$	−12891.7	−	4188.77i	−0.0319715	−	0.0103882i
$636$		0			0
$637$	−81997.0	−	184168.i	−0.202078	−	0.453874i
$638$		0			0
$639$	−574060.	−	122020.i	−1.40590	−	0.298834i
$640$		0			0
$641$	292218.	+	263114.i	0.711199	+	0.640367i	0.943153	−	0.332360i	$-0.107845\pi$
$641$	292218.	+	263114.i	0.711199	+	0.640367i	−0.231953	+	0.972727i	$0.574512\pi$
$642$		0			0
$643$	−13684.1	+	18834.6i	−0.0330975	+	0.0455548i	−0.825245	−	0.564775i	$-0.808963\pi$
$643$	−13684.1	+	18834.6i	−0.0330975	+	0.0455548i	0.792148	+	0.610330i	$0.208963\pi$
$644$		0			0
$645$	−1734.65	+	3004.51i	−0.00416959	+	0.00722194i
$646$		0			0
$647$	368154.	+	506721.i	0.879470	+	1.21049i	0.976567	+	0.215212i	$0.0690442\pi$
$647$	368154.	+	506721.i	0.879470	+	1.21049i	−0.0970974	+	0.995275i	$0.530956\pi$
$648$		0			0
$649$	−194129.	+	63076.2i	−0.460893	+	0.149753i
$650$		0			0
$651$	−36747.1	−	119681.i	−0.0867084	−	0.282399i
$652$		0			0
$653$	51619.5	+	158868.i	0.121056	+	0.372573i	0.993162	−	0.116745i	$-0.0372459\pi$
$653$	51619.5	+	158868.i	0.121056	+	0.372573i	−0.872106	+	0.489317i	$0.837246\pi$
$654$		0			0
$655$	−12462.6	+	9054.60i	−0.0290486	+	0.0211051i
$656$		0			0
$657$	236878.	+	136761.i	0.548774	+	0.316835i
$658$		0			0
$659$	149676.	+	108746.i	0.344652	+	0.250404i	0.746622	−	0.665249i	$-0.231674\pi$
$659$	149676.	+	108746.i	0.344652	+	0.250404i	−0.401970	+	0.915653i	$0.631674\pi$
$660$		0			0
$661$	−260702.	+	289539.i	−0.596679	+	0.662680i	−0.963530	−	0.267600i	$-0.913769\pi$
$661$	−260702.	+	289539.i	−0.596679	+	0.662680i	0.366851	+	0.930280i	$0.380436\pi$
$662$		0			0
$663$	47076.5	−	221478.i	0.107097	−	0.503852i
$664$		0			0
$665$	−11820.8	+	5262.98i	−0.0267304	+	0.0119011i
$666$		0			0
$667$	133288.	−	410218.i	0.299598	−	0.922068i
$668$		0			0
$669$	−97038.2	+	20626.1i	−0.216816	+	0.0460856i
$670$		0			0
$671$	117498.	+	52313.6i	0.260967	+	0.116190i
$672$		0			0
$673$	−82860.0	−	8708.93i	−0.182943	−	0.0192280i	0.0126141	−	0.999920i	$-0.495985\pi$
$673$	−82860.0	−	8708.93i	−0.182943	−	0.0192280i	−0.195557	+	0.980692i	$0.562651\pi$
$674$		0			0
$675$	−371890.	+	39087.2i	−0.816219	+	0.0857881i
$676$		0			0
$677$	−567336.	+	327552.i	−1.23784	+	0.714665i	−0.968652	−	0.248423i	$-0.920088\pi$
$677$	−567336.	+	327552.i	−1.23784	+	0.714665i	−0.269185	+	0.963089i	$0.586754\pi$
$678$		0			0
$679$	230705.	+	256224.i	0.500400	+	0.555751i
$680$		0			0
$681$			106842.i			0.230383i
$682$		0			0
$683$	167265.			0.358561			0.179281	−	0.983798i	$-0.442623\pi$
$683$	167265.			0.358561			0.179281	+	0.983798i	$0.442623\pi$
$684$		0			0
$685$	−15488.0	+	13945.5i	−0.0330077	+	0.0297203i
$686$		0			0
$687$	−33099.3	−	57329.6i	−0.0701302	−	0.121469i
$688$		0			0
$689$	62733.7	+	596871.i	0.132149	+	1.25731i
$690$		0			0
$691$	−74062.3	+	704656.i	−0.155110	+	1.47578i	0.589223	+	0.807970i	$0.299434\pi$
$691$	−74062.3	+	704656.i	−0.155110	+	1.47578i	−0.744334	+	0.667808i	$0.767233\pi$
$692$		0			0
$693$	42748.0	−	96013.5i	0.0890121	−	0.199925i
$694$		0			0
$695$	−2648.30	−	12459.3i	−0.00548274	−	0.0257943i
$696$		0			0
$697$	−927444.	−	301345.i	−1.90907	−	0.620295i
$698$		0			0
$699$	−110852.	−	248977.i	−0.226876	−	0.509571i
$700$		0			0
$701$	−620741.	−	131943.i	−1.26321	−	0.268503i	−0.472846	−	0.881145i	$-0.656773\pi$
$701$	−620741.	−	131943.i	−1.26321	−	0.268503i	−0.790361	+	0.612642i	$0.790107\pi$
$702$		0			0
$703$	−586527.	−	528111.i	−1.18680	−	1.06860i
$704$		0			0
$705$	−3596.99	+	4950.83i	−0.00723704	+	0.00996093i
$706$		0			0
$707$	−152531.	+	264192.i	−0.305155	+	0.528544i
$708$		0			0
$709$	325661.	+	448233.i	0.647848	+	0.891686i	0.999004	−	0.0446313i	$-0.0142113\pi$
$709$	325661.	+	448233.i	0.647848	+	0.891686i	−0.351156	+	0.936317i	$0.614211\pi$
$710$		0			0
$711$	−77472.1	+	25172.2i	−0.153252	+	0.0497946i
$712$		0			0
$713$	−283909.	+	120914.i	−0.558471	+	0.237848i
$714$		0			0
$715$	1581.47	+	4867.26i	0.00309349	+	0.00952077i
$716$		0			0
$717$	307800.	−	223630.i	0.598730	−	0.435003i
$718$		0			0
$719$	481265.	+	277858.i	0.930950	+	0.537484i	0.887112	−	0.461554i	$-0.152708\pi$
$719$	481265.	+	277858.i	0.930950	+	0.537484i	0.0438383	+	0.999039i	$0.486041\pi$
$720$		0			0
$721$	185093.	+	134478.i	0.356057	+	0.258691i
$722$		0			0
$723$	−211930.	+	235372.i	−0.405430	+	0.450275i
$724$		0			0
$725$	174417.	−	820569.i	0.331828	−	1.56113i
$726$		0			0
$727$	285346.	−	127044.i	0.539887	−	0.240373i	−0.118631	−	0.992938i	$-0.537850\pi$
$727$	285346.	−	127044.i	0.539887	−	0.240373i	0.658518	+	0.752565i	$0.271184\pi$
$728$		0			0
$729$	−8473.44	+	26078.6i	−0.0159443	+	0.0490714i
$730$		0			0
$731$	458595.	−	97477.3i	0.858211	−	0.182418i
$732$		0			0
$733$	258651.	+	115159.i	0.481401	+	0.214334i	0.633065	−	0.774099i	$-0.281797\pi$
$733$	258651.	+	115159.i	0.481401	+	0.214334i	−0.151664	+	0.988432i	$0.548463\pi$
$734$		0			0
$735$	3958.42	+	416.047i	0.00732735	+	0.000770136i
$736$		0			0
$737$	198748.	−	20889.2i	0.365904	−	0.0384580i
$738$		0			0
$739$	−54229.6	+	31309.5i	−0.0992996	+	0.0573306i	−0.548827	−	0.835936i	$-0.684926\pi$
$739$	−54229.6	+	31309.5i	−0.0992996	+	0.0573306i	0.449528	+	0.893266i	$0.351592\pi$
$740$		0			0
$741$	237344.	+	263597.i	0.432256	+	0.480069i
$742$		0			0
$743$		−	884141.i		−	1.60156i	−0.598956	−	0.800782i	$-0.704418\pi$
$743$		−	884141.i		−	1.60156i	0.598956	−	0.800782i	$-0.295582\pi$
$744$		0			0
$745$	1456.04			0.00262339
$746$		0			0
$747$	48645.3	−	43800.5i	0.0871766	−	0.0784942i
$748$		0			0
$749$	−191321.	−	331378.i	−0.341035	−	0.590691i
$750$		0			0
$751$	58592.2	+	557467.i	0.103887	+	0.988415i	0.914980	+	0.403499i	$0.132206\pi$
$751$	58592.2	+	557467.i	0.103887	+	0.988415i	−0.811094	+	0.584916i	$0.801127\pi$
$752$		0			0
$753$	−4398.87	+	41852.5i	−0.00775803	+	0.0738127i
$754$		0			0
$755$	10386.5	−	23328.6i	0.0182212	−	0.0409255i
$756$		0			0
$757$	3456.58	+	16261.9i	0.00603191	+	0.0283779i	0.981061	−	0.193701i	$-0.0620491\pi$
$757$	3456.58	+	16261.9i	0.00603191	+	0.0283779i	−0.975029	+	0.222079i	$0.928716\pi$
$758$		0			0
$759$	65770.1	+	21370.0i	0.114168	+	0.0370955i
$760$		0			0
$761$	285964.	+	642286.i	0.493790	+	1.10907i	0.972881	+	0.231305i	$0.0742995\pi$
$761$	285964.	+	642286.i	0.493790	+	1.10907i	−0.479092	+	0.877765i	$0.659034\pi$
$762$		0			0
$763$	−701314.	−	149069.i	−1.20466	−	0.256058i
$764$		0			0
$765$	−12444.9	−	11205.4i	−0.0212651	−	0.0191472i
$766$		0			0
$767$	−329861.	+	454015.i	−0.560713	+	0.771755i
$768$		0			0
$769$	25123.3	−	43514.8i	0.0424839	−	0.0735842i	−0.844002	−	0.536341i	$-0.819806\pi$
$769$	25123.3	−	43514.8i	0.0424839	−	0.0735842i	0.886485	+	0.462756i	$0.153140\pi$
$770$		0			0
$771$	−87879.0	−	120955.i	−0.147835	−	0.203477i
$772$		0			0
$773$	894997.	−	290802.i	1.49783	−	0.486674i	0.558446	−	0.829541i	$-0.311398\pi$
$773$	894997.	−	290802.i	1.49783	−	0.486674i	0.939384	+	0.342867i	$0.111398\pi$
$774$		0			0
$775$	−524575.	+	291600.i	−0.873381	+	0.485494i
$776$		0			0
$777$	−53038.8	−	163237.i	−0.0878520	−	0.270381i
$778$		0			0
$779$	1.23589e6	−	897928.i	2.03660	−	1.47968i
$780$		0			0
$781$	414429.	+	239271.i	0.679435	+	0.392272i
$782$		0			0
$783$	−650669.	−	472739.i	−1.06130	−	0.771077i
$784$		0			0
$785$	373.624	−	414.952i	0.000606311	−	0.000673377i
$786$		0			0
$787$	213697.	−	1.00536e6i	0.345023	−	1.62321i	−0.373449	−	0.927651i	$-0.621825\pi$
$787$	213697.	−	1.00536e6i	0.345023	−	1.62321i	0.718472	−	0.695556i	$-0.244842\pi$
$788$		0			0
$789$	345816.	−	153967.i	0.555509	−	0.247329i
$790$		0			0
$791$	16959.5	−	52196.1i	0.0271057	−	0.0834229i
$792$		0			0
$793$	345888.	−	73520.7i	0.550033	−	0.116913i
$794$		0			0
$795$	−10824.8	−	4819.51i	−0.0171272	−	0.00762550i
$796$		0			0
$797$	−165842.	−	17430.7i	−0.261083	−	0.0274409i	−0.0269174	−	0.999638i	$-0.508569\pi$
$797$	−165842.	−	17430.7i	−0.261083	−	0.0274409i	−0.234166	+	0.972197i	$0.575236\pi$
$798$		0			0
$799$	822464.	−	86444.5i	1.28832	−	0.135408i
$800$		0			0
$801$	35376.0	−	20424.4i	0.0551371	−	0.0318334i
$802$		0			0
$803$	−149235.	−	165742.i	−0.231441	−	0.257041i
$804$		0			0
$805$			6935.90i			0.0107031i
$806$		0			0
$807$	237995.			0.365444
$808$		0			0
$809$	−698839.	+	629237.i	−1.06778	+	0.961429i	−0.999344	−	0.0362267i	$-0.988466\pi$
$809$	−698839.	+	629237.i	−1.06778	+	0.961429i	−0.0684317	+	0.997656i	$0.521800\pi$
$810$		0			0
$811$	−611347.	−	1.05888e6i	−0.929494	−	1.60993i	−0.784170	−	0.620546i	$-0.786911\pi$
$811$	−611347.	−	1.05888e6i	−0.929494	−	1.60993i	−0.145323	−	0.989384i	$-0.546422\pi$
$812$		0			0
$813$	5464.21	+	51988.5i	0.00826697	+	0.0786550i
$814$		0			0
$815$	−1405.54	+	13372.8i	−0.00211606	+	0.0201330i
$816$		0			0
$817$	−298730.	+	670959.i	−0.447543	+	1.00520i
$818$		0			0
$819$	−60077.3	−	282642.i	−0.0895659	−	0.421374i
$820$		0			0
$821$	−204502.	−	66446.7i	−0.303397	−	0.0985796i	0.153362	−	0.988170i	$-0.450990\pi$
$821$	−204502.	−	66446.7i	−0.303397	−	0.0985796i	−0.456759	+	0.889590i	$0.650990\pi$
$822$		0			0
$823$	−492597.	−	1.10639e6i	−0.727264	−	1.63346i	−0.772919	−	0.634505i	$-0.781204\pi$
$823$	−492597.	−	1.10639e6i	−0.727264	−	1.63346i	0.0456547	−	0.998957i	$-0.485463\pi$
$824$		0			0
$825$	131561.	+	27964.2i	0.193295	+	0.0410861i
$826$		0			0
$827$	915404.	+	824234.i	1.33845	+	1.20515i	0.960158	+	0.279458i	$0.0901549\pi$
$827$	915404.	+	824234.i	1.33845	+	1.20515i	0.378291	+	0.925687i	$0.376512\pi$
$828$		0			0
$829$	711502.	−	979298.i	1.03530	−	1.42497i	0.134410	−	0.990926i	$-0.457086\pi$
$829$	711502.	−	979298.i	1.03530	−	1.42497i	0.900892	−	0.434044i	$-0.142914\pi$
$830$		0			0
$831$	−153141.	+	265247.i	−0.221763	+	0.384104i
$832$		0			0
$833$	−316161.	−	435158.i	−0.455636	−	0.627129i
$834$		0			0
$835$	−1011.75	+	328.737i	−0.00145111	+	0.000471494i
$836$		0			0
$837$	50836.8	+	573149.i	0.0725651	+	0.818120i
$838$		0			0
$839$	−31354.4	−	96498.8i	−0.0445424	−	0.137088i	0.926312	−	0.376757i	$-0.122961\pi$
$839$	−31354.4	−	96498.8i	−0.0445424	−	0.137088i	−0.970855	+	0.239670i	$0.922961\pi$
$840$		0			0
$841$	887523.	−	644823.i	1.25484	−	0.911693i
$842$		0			0
$843$	36696.8	+	21186.9i	0.0516384	+	0.0298135i
$844$		0			0
$845$	−4443.70	−	3228.54i	−0.00622346	−	0.00452161i
$846$		0			0
$847$	251592.	−	279421.i	0.350696	−	0.389487i
$848$		0			0
$849$	−126975.	+	597368.i	−0.176158	+	0.828756i
$850$		0			0
$851$	−386482.	+	172073.i	−0.533666	+	0.237604i
$852$		0			0
$853$	−39418.9	+	121319.i	−0.0541759	+	0.166736i	−0.974483	−	0.224459i	$-0.927938\pi$
$853$	−39418.9	+	121319.i	−0.0541759	+	0.166736i	0.920308	+	0.391196i	$0.127938\pi$
$854$		0			0
$855$	25660.3	−	5454.27i	0.0351018	−	0.00746112i
$856$		0			0
$857$	−18507.2	−	8239.93i	−0.0251987	−	0.0112192i	0.394099	−	0.919068i	$-0.371057\pi$
$857$	−18507.2	−	8239.93i	−0.0251987	−	0.0112192i	−0.419297	+	0.907849i	$0.637724\pi$
$858$		0			0
$859$	−1.00341e6	−	105463.i	−1.35986	−	0.142927i	−0.603646	−	0.797252i	$-0.706286\pi$
$859$	−1.00341e6	−	105463.i	−1.35986	−	0.142927i	−0.756213	+	0.654325i	$0.772953\pi$
$860$		0			0
$861$	330395.	−	34726.0i	0.445685	−	0.0468434i
$862$		0			0
$863$	589167.	−	340156.i	0.791073	−	0.456726i	−0.0492668	−	0.998786i	$-0.515688\pi$
$863$	589167.	−	340156.i	0.791073	−	0.456726i	0.840340	+	0.542059i	$0.182355\pi$
$864$		0			0
$865$	15881.0	+	17637.7i	0.0212249	+	0.0235727i
$866$		0			0
$867$		−	259087.i		−	0.344672i
$868$		0			0
$869$	66421.0			0.0879561
$870$		0			0
$871$	408310.	−	367644.i	0.538213	−	0.484609i
$872$		0			0
$873$	−349507.	−	605364.i	−0.458593	−	0.794306i
$874$		0			0
$875$	2821.19	+	26841.8i	0.00368482	+	0.0350587i
$876$		0			0
$877$	−128274.	+	1.22045e6i	−0.166778	+	1.58679i	0.516275	+	0.856423i	$0.327318\pi$
$877$	−128274.	+	1.22045e6i	−0.166778	+	1.58679i	−0.683054	+	0.730368i	$0.739349\pi$
$878$		0			0
$879$	173.290	−	389.216i	0.000224283	−	0.000503747i
$880$		0			0
$881$	−240697.	−	1.13239e6i	−0.310113	−	1.45897i	−0.806691	−	0.590974i	$-0.798744\pi$
$881$	−240697.	−	1.13239e6i	−0.310113	−	1.45897i	0.496578	−	0.867992i	$-0.334590\pi$
$882$		0			0
$883$	−400736.	−	130207.i	−0.513969	−	0.166999i	0.0405378	−	0.999178i	$-0.487093\pi$
$883$	−400736.	−	130207.i	−0.513969	−	0.166999i	−0.554506	+	0.832179i	$0.687093\pi$
$884$		0			0
$885$	−4506.60	−	10122.0i	−0.00575390	−	0.0129235i
$886$		0			0
$887$	1.34356e6	+	285583.i	1.70770	+	0.362982i	0.955280	−	0.295703i	$-0.0955540\pi$
$887$	1.34356e6	+	285583.i	1.70770	+	0.362982i	0.752418	+	0.658686i	$0.228887\pi$
$888$		0			0
$889$	−463765.	−	417576.i	−0.586806	−	0.528363i
$890$		0			0
$891$	−82885.3	+	114082.i	−0.104405	+	0.143701i
$892$		0			0
$893$	−647760.	+	1.12195e6i	−0.812290	+	1.40693i
$894$		0			0
$895$	4447.89	+	6122.00i	0.00555275	+	0.00764271i
$896$		0			0
$897$	180825.	−	58753.5i	0.224736	−	0.0730212i
$898$		0			0
$899$	−1.25813e6	−	288872.i	−1.55670	−	0.357426i
$900$		0			0
$901$	494827.	+	1.52292e6i	0.609543	+	1.87598i
$902$		0			0
$903$	−129217.	+	93881.8i	−0.158469	+	0.115135i
$904$		0			0
$905$	−5671.92	−	3274.69i	−0.00692521	−	0.00399827i
$906$		0			0
$907$	836562.	+	607798.i	1.01691	+	0.738831i	0.965648	−	0.259855i	$-0.0836746\pi$
$907$	836562.	+	607798.i	1.01691	+	0.738831i	0.0512652	+	0.998685i	$0.483675\pi$
$908$		0			0
$909$	413847.	−	459623.i	0.500855	−	0.556256i
$910$		0			0
$911$	22157.2	−	104242.i	0.0266980	−	0.125604i	−0.962780	−	0.270285i	$-0.912882\pi$
$911$	22157.2	−	104242.i	0.0266980	−	0.125604i	0.989478	+	0.144681i	$0.0462155\pi$
$912$		0			0
$913$	−48760.1	+	21709.4i	−0.0584956	+	0.0260439i
$914$		0			0
$915$	−2157.43	+	6639.87i	−0.00257688	+	0.00793081i
$916$		0			0
$917$	−693706.	+	147452.i	−0.824967	+	0.175352i
$918$		0			0
$919$	−41321.9	−	18397.7i	−0.0489271	−	0.0217837i	0.382127	−	0.924110i	$-0.375192\pi$
$919$	−41321.9	−	18397.7i	−0.0489271	−	0.0217837i	−0.431054	+	0.902326i	$0.641858\pi$
$920$		0			0
$921$	−459233.	−	48267.3i	−0.541394	−	0.0569028i
$922$		0			0
$923$	1.30847e6	−	137526.i	1.53589	−	0.161428i
$924$		0			0
$925$	−712576.	+	411406.i	−0.832814	+	0.480825i
$926$		0			0
$927$	−310373.	−	344704.i	−0.361180	−	0.401131i
$928$		0			0
$929$		−	334839.i		−	0.387976i	−0.981004	−	0.193988i	$-0.937858\pi$
$929$		−	334839.i		−	0.387976i	0.981004	−	0.193988i	$-0.0621423\pi$
$930$		0			0
$931$	842618.			0.972146
$932$		0			0
$933$	−500676.	+	450811.i	−0.575167	+	0.517882i
$934$		0			0
$935$	6827.36	+	11825.3i	0.00780962	+	0.0135267i
$936$		0			0
$937$	58.3104	+	554.786i	6.64151e−5	+	0.000631897i	0.994555	−	0.104213i	$-0.0332322\pi$
$937$	58.3104	+	554.786i	6.64151e−5	+	0.000631897i	−0.994489	+	0.104844i	$0.966566\pi$
$938$		0			0
$939$	−14874.0	+	141517.i	−0.0168693	+	0.160500i
$940$		0			0
$941$	−427661.	+	960543.i	−0.482971	+	1.08477i	0.493623	+	0.869676i	$0.335672\pi$
$941$	−427661.	+	960543.i	−0.482971	+	1.08477i	−0.976593	+	0.215094i	$0.930994\pi$
$942$		0			0
$943$	−170250.	−	800961.i	−0.191453	−	0.900717i
$944$		0			0
$945$	12299.9	+	3996.49i	0.0137733	+	0.00447523i
$946$		0			0
$947$	−552870.	−	1.24177e6i	−0.616486	−	1.38465i	−0.904273	−	0.426955i	$-0.859586\pi$
$947$	−552870.	−	1.24177e6i	−0.616486	−	1.38465i	0.287787	−	0.957694i	$-0.407080\pi$
$948$		0			0
$949$	−599783.	−	127488.i	−0.665981	−	0.141559i
$950$		0			0
$951$	127444.	+	114751.i	0.140916	+	0.126881i
$952$		0			0
$953$	439932.	−	605515.i	0.484395	−	0.666713i	−0.494947	−	0.868923i	$-0.664812\pi$
$953$	439932.	−	605515.i	0.484395	−	0.666713i	0.979342	+	0.202210i	$0.0648124\pi$
$954$		0			0
$955$	5459.76	−	9456.58i	0.00598642	−	0.0103688i
$956$		0			0
$957$	170038.	+	234037.i	0.185661	+	0.255541i
$958$		0			0
$959$	−912535.	+	296501.i	−0.992230	+	0.322395i
$960$		0			0
$961$	435522.	+	814378.i	0.471588	+	0.881819i
$962$		0			0
$963$	239726.	+	737800.i	0.258501	+	0.795584i
$964$		0			0
$965$	38234.7	−	27779.1i	0.0410585	−	0.0298308i
$966$		0			0
$967$	−190597.	−	110041.i	−0.203828	−	0.117680i	0.394612	−	0.918848i	$-0.370879\pi$
$967$	−190597.	−	110041.i	−0.203828	−	0.117680i	−0.598440	+	0.801168i	$0.704212\pi$
$968$		0			0
$969$	765649.	+	556276.i	0.815421	+	0.592438i
$970$		0			0
$971$	−505475.	+	561387.i	−0.536119	+	0.595420i	−0.948965	−	0.315380i	$-0.897868\pi$
$971$	−505475.	+	561387.i	−0.536119	+	0.595420i	0.412847	+	0.910801i	$0.364535\pi$
$972$		0			0
$973$	121924.	−	573605.i	0.128784	−	0.605881i
$974$		0			0
$975$	337819.	−	150407.i	0.355365	−	0.158219i
$976$		0			0
$977$	−523232.	+	1.61034e6i	−0.548157	+	1.68705i	0.165207	+	0.986259i	$0.447171\pi$
$977$	−523232.	+	1.61034e6i	−0.548157	+	1.68705i	−0.713364	+	0.700794i	$0.752829\pi$
$978$		0			0
$979$	−32579.9	+	6925.07i	−0.0339926	+	0.00722535i
$980$		0			0
$981$	1.32794e6	+	591236.i	1.37987	+	0.614359i
$982$		0			0
$983$	559956.	+	58853.7i	0.579491	+	0.0609070i	0.389739	−	0.920925i	$-0.372565\pi$
$983$	559956.	+	58853.7i	0.579491	+	0.0609070i	0.189752	+	0.981832i	$0.439232\pi$
$984$		0			0
$985$	−6626.73	+	696.497i	−0.00683009	+	0.000717872i
$986$		0			0
$987$	−243990.	+	140868.i	−0.250459	+	0.144603i
$988$		0			0
$989$	263428.	+	292566.i	0.269320	+	0.299110i
$990$		0			0
$991$			715895.i			0.728957i	0.931212	+	0.364479i	$0.118753\pi$
$991$			715895.i			0.728957i	−0.931212	+	0.364479i	$0.881247\pi$
$992$		0			0
$993$	457594.			0.464069
$994$		0			0
$995$	24445.0	−	22010.4i	0.0246913	−	0.0222321i
$996$		0			0
$997$	−412156.	−	713876.i	−0.414640	−	0.718178i	0.580750	−	0.814082i	$-0.302759\pi$
$997$	−412156.	−	713876.i	−0.414640	−	0.718178i	−0.995391	+	0.0959035i	$0.969426\pi$
$998$		0			0
$999$	82457.0	+	784526.i	0.0826222	+	0.786097i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.o.a.73.5	yes	88
31.17	odd	30	inner	124.5.o.a.17.5	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.o.a.17.5	✓	88	31.17	odd	30	inner
124.5.o.a.73.5	yes	88	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.o (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.07010 + 2.76433i) q^{3} +(-0.342480 - 0.593193i) q^{5} +(-3.29625 - 31.3617i) q^{7} +(-6.68282 + 63.5828i) q^{9} +O(q^{10})\)	\(q+(-3.07010 + 2.76433i) q^{3} +(-0.342480 - 0.593193i) q^{5} +(-3.29625 - 31.3617i) q^{7} +(-6.68282 + 63.5828i) q^{9} +(21.2034 - 47.6236i) q^{11} +(-29.7989 - 140.193i) q^{13} +(2.69123 + 0.874433i) q^{15} +(-155.539 - 349.346i) q^{17} +(585.965 + 124.551i) q^{19} +(96.8139 + 87.1716i) q^{21} +(188.743 - 259.782i) q^{23} +(312.265 - 540.860i) q^{25} +(-351.937 - 484.399i) q^{27} +(1277.51 - 415.087i) q^{29} +(-824.331 - 493.963i) q^{31} +(66.5507 + 204.822i) q^{33} +(-17.4747 + 12.6961i) q^{35} +(-1140.98 - 658.744i) q^{37} +(479.024 + 348.031i) q^{39} +(1706.34 - 1895.09i) q^{41} +(-254.904 + 1199.23i) q^{43} +(40.0056 - 17.8116i) q^{45} +(-668.281 + 2056.76i) q^{47} +(1375.84 - 292.444i) q^{49} +(1443.23 + 642.566i) q^{51} +(-4164.47 - 437.703i) q^{53} +(-35.5117 + 3.73243i) q^{55} +(-2143.27 + 1237.42i) q^{57} +(-2620.01 - 2909.81i) q^{59} +2467.23i q^{61} +2016.09 q^{63} +(-72.9558 + 65.6897i) q^{65} +(1916.75 + 3319.91i) q^{67} +(138.664 + 1319.30i) q^{69} +(-959.539 + 9129.40i) q^{71} +(1740.13 - 3908.40i) q^{73} +(536.428 + 2523.70i) q^{75} +(-1563.45 - 507.995i) q^{77} +(518.236 + 1163.98i) q^{79} +(-2645.89 - 562.401i) q^{81} +(-760.880 - 685.100i) q^{83} +(-153.961 + 211.909i) q^{85} +(-2774.63 + 4805.80i) q^{87} +(-375.554 - 516.906i) q^{89} +(-4298.46 + 1396.65i) q^{91} +(3896.25 - 762.206i) q^{93} +(-126.799 - 390.246i) q^{95} +(-8845.42 + 6426.57i) q^{97} +(2886.34 + 1666.43i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9}+O(q^{10}) \) 88 * q - 9 * q^3 + 3 * q^5 - 215 * q^7 - 254 * q^9	\( 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9} - 42 q^{11} + 6 q^{13} + 665 q^{15} - 585 q^{17} - 153 q^{19} - 402 q^{21} - 1365 q^{23} - 5933 q^{25} - 9225 q^{27} - 1140 q^{29} + 117 q^{31} + 5151 q^{33} + 2898 q^{35} + 6594 q^{37} + 3173 q^{39} - 9393 q^{41} - 5322 q^{43} + 2010 q^{45} - 5112 q^{47} - 5210 q^{49} - 1829 q^{51} + 7395 q^{53} + 10585 q^{55} + 40485 q^{57} + 5625 q^{59} - 14954 q^{63} - 17094 q^{65} + 8909 q^{67} - 35370 q^{69} - 11811 q^{71} - 22105 q^{73} + 79377 q^{75} + 71490 q^{77} + 219 q^{79} - 5422 q^{81} + 10545 q^{83} - 53630 q^{85} + 13732 q^{87} - 40305 q^{89} + 42760 q^{91} - 1028 q^{93} + 62319 q^{95} + 35201 q^{97} + 16197 q^{99}+O(q^{100}) \) 88 * q - 9 * q^3 + 3 * q^5 - 215 * q^7 - 254 * q^9 - 42 * q^11 + 6 * q^13 + 665 * q^15 - 585 * q^17 - 153 * q^19 - 402 * q^21 - 1365 * q^23 - 5933 * q^25 - 9225 * q^27 - 1140 * q^29 + 117 * q^31 + 5151 * q^33 + 2898 * q^35 + 6594 * q^37 + 3173 * q^39 - 9393 * q^41 - 5322 * q^43 + 2010 * q^45 - 5112 * q^47 - 5210 * q^49 - 1829 * q^51 + 7395 * q^53 + 10585 * q^55 + 40485 * q^57 + 5625 * q^59 - 14954 * q^63 - 17094 * q^65 + 8909 * q^67 - 35370 * q^69 - 11811 * q^71 - 22105 * q^73 + 79377 * q^75 + 71490 * q^77 + 219 * q^79 - 5422 * q^81 + 10545 * q^83 - 53630 * q^85 + 13732 * q^87 - 40305 * q^89 + 42760 * q^91 - 1028 * q^93 + 62319 * q^95 + 35201 * q^97 + 16197 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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