LMFDB - Embedded newform 208.8.w.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,8,Mod(17,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("208.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.w (of order $6$, degree $2$, not 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:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 22180 x^{14} + 184473654 x^{12} + 707524481236 x^{10} + \cdots + 18\!\cdots\!76 $ x^16 + 22180x^14 + 184473654x^12 + 707524481236x^10 + 1235305573179121x^8 + 834592841081820432x^6 + 142296444753813668448x^4 + 5907885179463590558976*x^2 + 18970385728293762437376
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{4}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.2
Root			$-50.2253i$ of defining polynomial
Character	$\chi$	$=$	208.17
Dual form			208.8.w.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-25.1127 + 43.4964i) q^{3} +469.361i q^{5} +(-64.3718 + 37.1651i) q^{7} +(-167.792 - 290.624i) q^{9} +(-2926.79 - 1689.78i) q^{11} +(5470.00 + 5729.54i) q^{13} +(-20415.5 - 11786.9i) q^{15} +(-4589.65 - 7949.50i) q^{17} +(-40877.8 + 23600.8i) q^{19} -3733.25i q^{21} +(21052.4 - 36463.9i) q^{23} -142175. q^{25} -92988.0 q^{27} +(-82803.7 + 143420. i) q^{29} -107861. i q^{31} +(146999. - 84869.8i) q^{33} +(-17443.8 - 30213.6i) q^{35} +(-228730. - 132058. i) q^{37} +(-386581. + 94041.1i) q^{39} +(384584. + 222040. i) q^{41} +(308598. + 534508. i) q^{43} +(136408. - 78754.9i) q^{45} -281242. i q^{47} +(-409009. + 708424. i) q^{49} +461033. q^{51} +1.57574e6 q^{53} +(793118. - 1.37372e6i) q^{55} -2.37072e6i q^{57} +(-1.46781e6 + 847438. i) q^{59} +(-1.64761e6 - 2.85375e6i) q^{61} +(21602.1 + 12472.0i) q^{63} +(-2.68922e6 + 2.56740e6i) q^{65} +(2.11527e6 + 1.22125e6i) q^{67} +(1.05737e6 + 1.83141e6i) q^{69} +(2.46302e6 - 1.42202e6i) q^{71} -3.94991e6i q^{73} +(3.57039e6 - 6.18410e6i) q^{75} +251203. q^{77} -3.64110e6 q^{79} +(2.70214e6 - 4.68024e6i) q^{81} +5.97315e6i q^{83} +(3.73119e6 - 2.15420e6i) q^{85} +(-4.15884e6 - 7.20333e6i) q^{87} +(-1.28990e6 - 744724. i) q^{89} +(-565052. - 165528. i) q^{91} +(4.69156e6 + 2.70868e6i) q^{93} +(-1.10773e7 - 1.91865e7i) q^{95} +(1.20731e7 - 6.97042e6i) q^{97} +1.13413e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2520 q^{7} - 4684 q^{9} + 8496 q^{11} - 3620 q^{13} - 51648 q^{15} + 41520 q^{17} + 54432 q^{19} + 7560 q^{23} - 273960 q^{25} - 133920 q^{27} - 346056 q^{29} + 486300 q^{33} + 283248 q^{35} - 68280 q^{37}+ \cdots + 83706300 q^{97}+O(q^{100}) $ 16 * q - 2520 * q^7 - 4684 * q^9 + 8496 * q^11 - 3620 * q^13 - 51648 * q^15 + 41520 * q^17 + 54432 * q^19 + 7560 * q^23 - 273960 * q^25 - 133920 * q^27 - 346056 * q^29 + 486300 * q^33 + 283248 * q^35 - 68280 * q^37 + 1122680 * q^39 - 1288008 * q^41 - 285360 * q^43 + 5743980 * q^45 - 2131660 * q^49 - 957280 * q^51 + 3666600 * q^53 + 1091472 * q^55 - 1799280 * q^59 - 4626000 * q^61 + 8609760 * q^63 - 11518020 * q^65 + 7094400 * q^67 + 18027316 * q^69 - 25958520 * q^71 + 10898032 * q^75 - 20577480 * q^77 - 34143488 * q^79 - 15745864 * q^81 + 7123932 * q^85 + 10460440 * q^87 + 10609452 * q^89 + 38683952 * q^91 - 52747080 * q^93 - 33837504 * q^95 + 83706300 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$	−25.1127	+	43.4964i	−0.536993	+	0.930099i	0.462071	+	0.886843i	$0.347106\pi$
$3$	−25.1127	+	43.4964i	−0.536993	+	0.930099i	−0.999064	+	0.0432559i	$0.986227\pi$
$4$		0			0
$5$			469.361i			1.67924i	0.543176	+	0.839619i	$0.317222\pi$
$5$			469.361i			1.67924i	−0.543176	+	0.839619i	$0.682778\pi$
$6$		0			0
$7$	−64.3718	+	37.1651i	−0.0709336	+	0.0409536i	−0.535047	−	0.844822i	$-0.679706\pi$
$7$	−64.3718	+	37.1651i	−0.0709336	+	0.0409536i	0.464114	+	0.885776i	$0.346373\pi$
$8$		0			0
$9$	−167.792	−	290.624i	−0.0767223	−	0.132887i
$10$		0			0
$11$	−2926.79	−	1689.78i	−0.663005	−	0.382786i	0.130416	−	0.991459i	$-0.458369\pi$
$11$	−2926.79	−	1689.78i	−0.663005	−	0.382786i	−0.793421	+	0.608673i	$0.791702\pi$
$12$		0			0
$13$	5470.00	+	5729.54i	0.690534	+	0.723300i
$13$	5470.00	+	5729.54i	0.690534	+	0.723300i
$14$		0			0
$15$	−20415.5	−	11786.9i	−1.56186	−	0.901738i
$16$		0			0
$17$	−4589.65	−	7949.50i	−0.226573	−	0.392436i	0.730217	−	0.683215i	$-0.239419\pi$
$17$	−4589.65	−	7949.50i	−0.226573	−	0.392436i	−0.956790	+	0.290779i	$0.906086\pi$
$18$		0			0
$19$	−40877.8	+	23600.8i	−1.36726	+	0.789387i	−0.990577	−	0.136957i	$-0.956268\pi$
$19$	−40877.8	+	23600.8i	−1.36726	+	0.789387i	−0.376681	+	0.926343i	$0.622935\pi$
$20$		0			0
$21$		−	3733.25i		−	0.0879671i
$22$		0			0
$23$	21052.4	−	36463.9i	0.360790	−	0.624907i	−0.627301	−	0.778777i	$-0.715840\pi$
$23$	21052.4	−	36463.9i	0.360790	−	0.624907i	0.988091	+	0.153870i	$0.0491737\pi$
$24$		0			0
$25$	−142175.			−1.81984
$26$		0			0
$27$	−92988.0			−0.909188
$28$		0			0
$29$	−82803.7	+	143420.i	−0.630459	+	1.09199i	0.356999	+	0.934105i	$0.383800\pi$
$29$	−82803.7	+	143420.i	−0.630459	+	1.09199i	−0.987458	+	0.157882i	$0.949533\pi$
$30$		0			0
$31$		−	107861.i		−	0.650277i	−0.945666	−	0.325138i	$-0.894589\pi$
$31$		−	107861.i		−	0.650277i	0.945666	−	0.325138i	$-0.105411\pi$
$32$		0			0
$33$	146999.	−	84869.8i	0.712058	−	0.411107i
$34$		0			0
$35$	−17443.8	−	30213.6i	−0.0687708	−	0.119114i
$36$		0			0
$37$	−228730.	−	132058.i	−0.742365	−	0.428605i	0.0805633	−	0.996749i	$-0.474328\pi$
$37$	−228730.	−	132058.i	−0.742365	−	0.428605i	−0.822929	+	0.568145i	$0.807661\pi$
$38$		0			0
$39$	−386581.	+	94041.1i	−1.04355	+	0.253858i
$40$		0			0
$41$	384584.	+	222040.i	0.871460	+	0.503138i	0.867833	−	0.496856i	$-0.165512\pi$
$41$	384584.	+	222040.i	0.871460	+	0.503138i	0.00362686	+	0.999993i	$0.498846\pi$
$42$		0			0
$43$	308598.	+	534508.i	0.591907	+	1.02521i	0.993975	+	0.109604i	$0.0349581\pi$
$43$	308598.	+	534508.i	0.591907	+	1.02521i	−0.402068	+	0.915610i	$0.631709\pi$
$44$		0			0
$45$	136408.	−	78754.9i	0.223149	−	0.128835i
$46$		0			0
$47$		−	281242.i		−	0.395128i	−0.980290	−	0.197564i	$-0.936697\pi$
$47$		−	281242.i		−	0.395128i	0.980290	−	0.197564i	$-0.0633030\pi$
$48$		0			0
$49$	−409009.	+	708424.i	−0.496646	+	0.860215i
$50$		0			0
$51$	461033.			0.486672
$52$		0			0
$53$	1.57574e6			1.45385			0.726924	−	0.686718i	$-0.240949\pi$
$53$	1.57574e6			1.45385			0.726924	+	0.686718i	$0.240949\pi$
$54$		0			0
$55$	793118.	−	1.37372e6i	0.642789	−	1.11334i
$56$		0			0
$57$		−	2.37072e6i		−	1.69558i
$58$		0			0
$59$	−1.46781e6	+	847438.i	−0.930436	+	0.537188i	−0.886950	−	0.461866i	$-0.847180\pi$
$59$	−1.46781e6	+	847438.i	−0.930436	+	0.537188i	−0.0434869	+	0.999054i	$0.513847\pi$
$60$		0			0
$61$	−1.64761e6	−	2.85375e6i	−0.929395	−	1.60976i	−0.784336	−	0.620336i	$-0.786996\pi$
$61$	−1.64761e6	−	2.85375e6i	−0.929395	−	1.60976i	−0.145059	−	0.989423i	$-0.546337\pi$
$62$		0			0
$63$	21602.1	+	12472.0i	0.0108844	+	0.00628410i
$64$		0			0
$65$	−2.68922e6	+	2.56740e6i	−1.21459	+	1.15957i
$66$		0			0
$67$	2.11527e6	+	1.22125e6i	0.859219	+	0.496071i	0.863751	−	0.503919i	$-0.168109\pi$
$67$	2.11527e6	+	1.22125e6i	0.859219	+	0.496071i	−0.00453136	+	0.999990i	$0.501442\pi$
$68$		0			0
$69$	1.05737e6	+	1.83141e6i	0.387483	+	0.671141i
$70$		0			0
$71$	2.46302e6	−	1.42202e6i	0.816701	−	0.471523i	−0.0325765	−	0.999469i	$-0.510371\pi$
$71$	2.46302e6	−	1.42202e6i	0.816701	−	0.471523i	0.849278	+	0.527947i	$0.177038\pi$
$72$		0			0
$73$		−	3.94991e6i		−	1.18839i	−0.804323	−	0.594193i	$-0.797472\pi$
$73$		−	3.94991e6i		−	1.18839i	0.804323	−	0.594193i	$-0.202528\pi$
$74$		0			0
$75$	3.57039e6	−	6.18410e6i	0.977240	−	1.69263i
$76$		0			0
$77$	251203.			0.0627058
$78$		0			0
$79$	−3.64110e6			−0.830880			−0.415440	−	0.909621i	$-0.636372\pi$
$79$	−3.64110e6			−0.830880			−0.415440	+	0.909621i	$0.636372\pi$
$80$		0			0
$81$	2.70214e6	−	4.68024e6i	0.564950	−	0.978522i
$82$		0			0
$83$			5.97315e6i			1.14665i	0.819329	+	0.573324i	$0.194346\pi$
$83$			5.97315e6i			1.14665i	−0.819329	+	0.573324i	$0.805654\pi$
$84$		0			0
$85$	3.73119e6	−	2.15420e6i	0.658993	−	0.380470i
$86$		0			0
$87$	−4.15884e6	−	7.20333e6i	−0.677104	−	1.17278i
$88$		0			0
$89$	−1.28990e6	−	744724.i	−0.193950	−	0.111977i	0.399880	−	0.916567i	$-0.369052\pi$
$89$	−1.28990e6	−	744724.i	−0.193950	−	0.111977i	−0.593831	+	0.804590i	$0.702385\pi$
$90$		0			0
$91$	−565052.	−	165528.i	−0.0786038	−	0.0230264i
$92$		0			0
$93$	4.69156e6	+	2.70868e6i	0.604822	+	0.349194i
$94$		0			0
$95$	−1.10773e7	−	1.91865e7i	−1.32557	−	2.29595i
$96$		0			0
$97$	1.20731e7	−	6.97042e6i	1.34313	−	0.775457i	0.355866	−	0.934537i	$-0.384186\pi$
$97$	1.20731e7	−	6.97042e6i	1.34313	−	0.775457i	0.987266	+	0.159080i	$0.0508526\pi$
$98$		0			0
$99$			1.13413e6i			0.117473i
$100$		0			0
$101$	−1.00902e6	+	1.74767e6i	−0.0974485	+	0.168786i	−0.910628	−	0.413227i	$-0.864401\pi$
$101$	−1.00902e6	+	1.74767e6i	−0.0974485	+	0.168786i	0.813179	+	0.582013i	$0.197735\pi$
$102$		0			0
$103$	1.67184e7			1.50753			0.753763	−	0.657147i	$-0.228237\pi$
$103$	1.67184e7			1.50753			0.753763	+	0.657147i	$0.228237\pi$
$104$		0			0
$105$	1.75224e6			0.147718
$106$		0			0
$107$	2.27240e6	−	3.93592e6i	0.179326	−	0.310601i	−0.762324	−	0.647195i	$-0.775942\pi$
$107$	2.27240e6	−	3.93592e6i	0.179326	−	0.310601i	0.941650	+	0.336594i	$0.109275\pi$
$108$		0			0
$109$			1.28345e7i			0.949265i	0.880184	+	0.474632i	$0.157419\pi$
$109$			1.28345e7i			0.949265i	−0.880184	+	0.474632i	$0.842581\pi$
$110$		0			0
$111$	1.14881e7	−	6.63263e6i	0.797290	−	0.460315i
$112$		0			0
$113$	−7.89884e6	−	1.36812e7i	−0.514978	−	0.891968i	−0.999849	−	0.0173821i	$-0.994467\pi$
$113$	−7.89884e6	−	1.36812e7i	−0.514978	−	0.891968i	0.484871	−	0.874586i	$-0.338867\pi$
$114$		0			0
$115$	1.71147e7	+	9.88119e6i	1.04937	+	0.605853i
$116$		0			0
$117$	747321.	−	2.55108e6i	0.0431377	−	0.147256i
$118$		0			0
$119$	590887.	+	341149.i	0.0321433	+	0.0185579i
$120$		0			0
$121$	−4.03286e6	−	6.98512e6i	−0.206949	−	0.358447i
$122$		0			0
$123$	−1.93158e7	+	1.11520e7i	−0.935935	+	0.540363i
$124$		0			0
$125$		−	3.00625e7i		−	1.37670i
$126$		0			0
$127$	−1.38226e7	+	2.39414e7i	−0.598793	+	1.03714i	0.394207	+	0.919022i	$0.371019\pi$
$127$	−1.38226e7	+	2.39414e7i	−0.598793	+	1.03714i	−0.993000	+	0.118118i	$0.962314\pi$
$128$		0			0
$129$	−3.09989e7			−1.27140
$130$		0			0
$131$	9.74162e6			0.378601			0.189300	−	0.981919i	$-0.439378\pi$
$131$	9.74162e6			0.378601			0.189300	+	0.981919i	$0.439378\pi$
$132$		0			0
$133$	1.75425e6	−	3.03845e6i	0.0646564	−	0.111988i
$134$		0			0
$135$		−	4.36450e7i		−	1.52674i
$136$		0			0
$137$	2.47332e6	−	1.42797e6i	0.0821785	−	0.0474458i	−0.458348	−	0.888773i	$-0.651559\pi$
$137$	2.47332e6	−	1.42797e6i	0.0821785	−	0.0474458i	0.540526	+	0.841327i	$0.318225\pi$
$138$		0			0
$139$	−2.24368e7	−	3.88616e7i	−0.708612	−	1.22735i	−0.965372	−	0.260877i	$-0.915988\pi$
$139$	−2.24368e7	−	3.88616e7i	−0.708612	−	1.22735i	0.256760	−	0.966475i	$-0.417345\pi$
$140$		0			0
$141$	1.22330e7	+	7.06273e6i	0.367508	+	0.212181i
$142$		0			0
$143$	−6.32784e6	−	2.60123e7i	−0.180959	−	0.743878i
$144$		0			0
$145$	−6.73159e7	−	3.88648e7i	−1.83371	−	1.05869i
$146$		0			0
$147$	−2.05426e7	−	3.55808e7i	−0.533390	−	0.923859i
$148$		0			0
$149$	2.11798e7	−	1.22282e7i	0.524531	−	0.302838i	−0.214256	−	0.976778i	$-0.568733\pi$
$149$	2.11798e7	−	1.22282e7i	0.524531	−	0.302838i	0.738786	+	0.673940i	$0.235399\pi$
$150$		0			0
$151$			5.94949e7i			1.40624i	0.711069	+	0.703122i	$0.248211\pi$
$151$			5.94949e7i			1.40624i	−0.711069	+	0.703122i	$0.751789\pi$
$152$		0			0
$153$	−1.54021e6	+	2.66772e6i	−0.0347664	+	0.0602172i
$154$		0			0
$155$	5.06257e7			1.09197
$156$		0			0
$157$	4.61680e7			0.952122			0.476061	−	0.879412i	$-0.342064\pi$
$157$	4.61680e7			0.952122			0.476061	+	0.879412i	$0.342064\pi$
$158$		0			0
$159$	−3.95710e7	+	6.85390e7i	−0.780706	+	1.35222i
$160$		0			0
$161$			3.12966e6i			0.0591026i
$162$		0			0
$163$	8.58326e6	−	4.95554e6i	0.155237	−	0.0896262i	−0.420369	−	0.907353i	$-0.638099\pi$
$163$	8.58326e6	−	4.95554e6i	0.155237	−	0.0896262i	0.575606	+	0.817727i	$0.304766\pi$
$164$		0			0
$165$	3.98346e7	+	6.89956e7i	0.690346	+	1.19571i
$166$		0			0
$167$	−5.08074e7	−	2.93337e7i	−0.844150	−	0.487370i	0.0145226	−	0.999895i	$-0.495377\pi$
$167$	−5.08074e7	−	2.93337e7i	−0.844150	−	0.487370i	−0.858673	+	0.512524i	$0.828710\pi$
$168$		0			0
$169$	−2.90681e6	+	6.26812e7i	−0.0463248	+	0.998926i
$170$		0			0
$171$	1.37179e7	+	7.92005e6i	0.209798	+	0.121127i
$172$		0			0
$173$	3.14153e6	+	5.44129e6i	0.0461296	+	0.0798989i	0.888168	−	0.459518i	$-0.151978\pi$
$173$	3.14153e6	+	5.44129e6i	0.0461296	+	0.0798989i	−0.842039	+	0.539417i	$0.818645\pi$
$174$		0			0
$175$	9.15205e6	−	5.28394e6i	0.129088	−	0.0745289i
$176$		0			0
$177$		−	8.51257e7i		−	1.15386i
$178$		0			0
$179$	2.66019e7	−	4.60759e7i	0.346679	−	0.600465i	−0.638979	−	0.769225i	$-0.720643\pi$
$179$	2.66019e7	−	4.60759e7i	0.346679	−	0.600465i	0.985657	+	0.168759i	$0.0539761\pi$
$180$		0			0
$181$	−3.34020e7			−0.418695			−0.209347	−	0.977841i	$-0.567134\pi$
$181$	−3.34020e7			−0.418695			−0.209347	+	0.977841i	$0.567134\pi$
$182$		0			0
$183$	1.65504e8			1.99631
$184$		0			0
$185$	6.19827e7	−	1.07357e8i	0.719729	−	1.24661i
$186$		0			0
$187$			3.10220e7i			0.346916i
$188$		0			0
$189$	5.98580e6	−	3.45590e6i	0.0644920	−	0.0372345i
$190$		0			0
$191$	1.59471e7	+	2.76212e7i	0.165602	+	0.286831i	0.936869	−	0.349681i	$-0.113710\pi$
$191$	1.59471e7	+	2.76212e7i	0.165602	+	0.286831i	−0.771267	+	0.636512i	$0.780377\pi$
$192$		0			0
$193$	1.33692e8	+	7.71874e7i	1.33862	+	0.772851i	0.986602	−	0.163143i	$-0.0521633\pi$
$193$	1.33692e8	+	7.71874e7i	1.33862	+	0.772851i	0.352015	+	0.935994i	$0.385497\pi$
$194$		0			0
$195$	−4.41392e7	−	1.81446e8i	−0.426288	−	1.75237i
$196$		0			0
$197$	1.48984e7	+	8.60161e6i	0.138838	+	0.0801582i	0.567810	−	0.823160i	$-0.307791\pi$
$197$	1.48984e7	+	8.60161e6i	0.138838	+	0.0801582i	−0.428972	+	0.903318i	$0.641124\pi$
$198$		0			0
$199$	1.05808e8	+	1.83264e8i	0.951769	+	1.64851i	0.741594	+	0.670849i	$0.234070\pi$
$199$	1.05808e8	+	1.83264e8i	0.951769	+	1.64851i	0.210175	+	0.977664i	$0.432597\pi$
$200$		0			0
$201$	−1.06240e8	+	6.13378e7i	−0.922789	+	0.532773i
$202$		0			0
$203$		−	1.23096e7i		−	0.103278i
$204$		0			0
$205$	−1.04217e8	+	1.80509e8i	−0.844888	+	1.46339i
$206$		0			0
$207$	−1.41297e7			−0.110723
$208$		0			0
$209$	1.59521e8			1.20867
$210$		0			0
$211$	−2.35499e7	+	4.07897e7i	−0.172584	+	0.298925i	−0.939323	−	0.343035i	$-0.888545\pi$
$211$	−2.35499e7	+	4.07897e7i	−0.172584	+	0.298925i	0.766738	+	0.641960i	$0.221878\pi$
$212$		0			0
$213$			1.42843e8i			1.01282i
$214$		0			0
$215$	−2.50877e8	+	1.44844e8i	−1.72158	+	0.993953i
$216$		0			0
$217$	4.00866e6	+	6.94320e6i	0.0266312	+	0.0461265i
$218$		0			0
$219$	1.71807e8	+	9.91928e7i	1.10532	+	0.638154i
$220$		0			0
$221$	2.04417e7	−	6.97803e7i	0.127392	−	0.434871i
$222$		0			0
$223$	−1.46451e8	−	8.45534e7i	−0.884351	−	0.510580i	−0.0122607	−	0.999925i	$-0.503903\pi$
$223$	−1.46451e8	−	8.45534e7i	−0.884351	−	0.510580i	−0.872091	+	0.489344i	$0.837236\pi$
$224$		0			0
$225$	2.38558e7	+	4.13194e7i	0.139622	+	0.241833i
$226$		0			0
$227$	−2.75344e8	+	1.58970e8i	−1.56237	+	0.902036i	−0.565355	+	0.824848i	$0.691261\pi$
$227$	−2.75344e8	+	1.58970e8i	−1.56237	+	0.902036i	−0.997017	+	0.0771881i	$0.975406\pi$
$228$		0			0
$229$			1.16369e8i			0.640345i	0.947359	+	0.320173i	$0.103741\pi$
$229$			1.16369e8i			0.640345i	−0.947359	+	0.320173i	$0.896259\pi$
$230$		0			0
$231$	−6.30839e6	+	1.09264e7i	−0.0336726	+	0.0583226i
$232$		0			0
$233$	−3.60170e8			−1.86536			−0.932679	−	0.360707i	$-0.882535\pi$
$233$	−3.60170e8			−1.86536			−0.932679	+	0.360707i	$0.882535\pi$
$234$		0			0
$235$	1.32004e8			0.663513
$236$		0			0
$237$	9.14378e7	−	1.58375e8i	0.446177	−	0.772800i
$238$		0			0
$239$			1.23549e8i			0.585394i	0.956205	+	0.292697i	$0.0945527\pi$
$239$			1.23549e8i			0.585394i	−0.956205	+	0.292697i	$0.905447\pi$
$240$		0			0
$241$	−2.01715e7	+	1.16460e7i	−0.0928280	+	0.0535943i	−0.545695	−	0.837984i	$-0.683734\pi$
$241$	−2.01715e7	+	1.16460e7i	−0.0928280	+	0.0535943i	0.452867	+	0.891578i	$0.350401\pi$
$242$		0			0
$243$	3.40333e7	+	5.89475e7i	0.152154	+	0.263538i
$244$		0			0
$245$	−3.32507e8	−	1.91973e8i	−1.44451	−	0.833986i
$246$		0			0
$247$	−3.58824e8	−	1.05115e8i	−1.51510	−	0.443839i
$248$		0			0
$249$	−2.59810e8	−	1.50002e8i	−1.06649	−	0.615741i
$250$		0			0
$251$	−1.65291e7	−	2.86293e7i	−0.0659769	−	0.114275i	0.831150	−	0.556048i	$-0.187683\pi$
$251$	−1.65291e7	−	2.86293e7i	−0.0659769	−	0.114275i	−0.897127	+	0.441773i	$0.854350\pi$
$252$		0			0
$253$	−1.23232e8	+	7.11480e7i	−0.478412	+	0.276211i
$254$		0			0
$255$			2.16391e8i			0.817238i
$256$		0			0
$257$	−1.32117e8	+	2.28834e8i	−0.485505	+	0.840919i	−0.999861	−	0.0166574i	$-0.994698\pi$
$257$	−1.32117e8	+	2.28834e8i	−0.485505	+	0.840919i	0.514356	+	0.857577i	$0.328031\pi$
$258$		0			0
$259$	1.96317e7			0.0702116
$260$		0			0
$261$	5.55751e7			0.193481
$262$		0			0
$263$	4.04021e7	−	6.99785e7i	0.136949	−	0.237202i	−0.789391	−	0.613890i	$-0.789604\pi$
$263$	4.04021e7	−	6.99785e7i	0.136949	−	0.237202i	0.926340	+	0.376688i	$0.122937\pi$
$264$		0			0
$265$			7.39591e8i			2.44136i
$266$		0			0
$267$	6.47856e7	−	3.74040e7i	0.208300	−	0.120262i
$268$		0			0
$269$	−2.58417e8	−	4.47591e8i	−0.809445	−	1.40200i	−0.913249	−	0.407403i	$-0.866435\pi$
$269$	−2.58417e8	−	4.47591e8i	−0.809445	−	1.40200i	0.103803	−	0.994598i	$-0.466899\pi$
$270$		0			0
$271$	1.37813e8	+	7.95666e7i	0.420629	+	0.242850i	0.695346	−	0.718675i	$-0.255251\pi$
$271$	1.37813e8	+	7.95666e7i	0.420629	+	0.242850i	−0.274718	+	0.961525i	$0.588584\pi$
$272$		0			0
$273$	2.13898e7	−	2.04209e7i	0.0636265	−	0.0607443i
$274$		0			0
$275$	4.16116e8	+	2.40245e8i	1.20656	+	0.696609i
$276$		0			0
$277$	1.07125e8	+	1.85545e8i	0.302838	+	0.524530i	0.976778	−	0.214256i	$-0.0687327\pi$
$277$	1.07125e8	+	1.85545e8i	0.302838	+	0.524530i	−0.673940	+	0.738786i	$0.735399\pi$
$278$		0			0
$279$	−3.13470e7	+	1.80982e7i	−0.0864133	+	0.0498908i
$280$		0			0
$281$			3.43238e8i			0.922833i	0.887184	+	0.461416i	$0.152659\pi$
$281$			3.43238e8i			0.922833i	−0.887184	+	0.461416i	$0.847341\pi$
$282$		0			0
$283$	−4.02846e7	+	6.97750e7i	−0.105654	+	0.182998i	−0.914005	−	0.405702i	$-0.867027\pi$
$283$	−4.02846e7	+	6.97750e7i	−0.105654	+	0.182998i	0.808351	+	0.588701i	$0.200360\pi$
$284$		0			0
$285$	1.11272e9			2.84728
$286$		0			0
$287$	−3.30084e7			−0.0824211
$288$		0			0
$289$	1.63040e8	−	2.82393e8i	0.397329	−	0.688195i
$290$		0			0
$291$			7.00183e8i			1.66566i
$292$		0			0
$293$	−3.64956e8	+	2.10707e8i	−0.847625	+	0.489377i	−0.859849	−	0.510549i	$-0.829442\pi$
$293$	−3.64956e8	+	2.10707e8i	−0.847625	+	0.489377i	0.0122236	+	0.999925i	$0.496109\pi$
$294$		0			0
$295$	−3.97755e8	−	6.88931e8i	−0.902066	−	1.56242i
$296$		0			0
$297$	2.72156e8	+	1.57129e8i	0.602796	+	0.348025i
$298$		0			0
$299$	3.24078e8	−	7.88365e7i	0.701133	−	0.170560i
$300$		0			0
$301$	−3.97300e7	−	2.29381e7i	−0.0839723	−	0.0484814i
$302$		0			0
$303$	−5.06784e7	−	8.77775e7i	−0.104658	−	0.181273i
$304$		0			0
$305$	1.33944e9	−	7.73325e8i	2.70317	−	1.56067i
$306$		0			0
$307$		−	8.05591e8i		−	1.58902i	−0.607248	−	0.794512i	$-0.707727\pi$
$307$		−	8.05591e8i		−	1.58902i	0.607248	−	0.794512i	$-0.292273\pi$
$308$		0			0
$309$	−4.19844e8	+	7.27191e8i	−0.809530	+	1.40215i
$310$		0			0
$311$	−4.44406e8			−0.837758			−0.418879	−	0.908042i	$-0.637577\pi$
$311$	−4.44406e8			−0.837758			−0.418879	+	0.908042i	$0.637577\pi$
$312$		0			0
$313$	4.53945e8			0.836755			0.418377	−	0.908273i	$-0.362599\pi$
$313$	4.53945e8			0.836755			0.418377	+	0.908273i	$0.362599\pi$
$314$		0			0
$315$	−5.85386e6	+	1.01392e7i	−0.0105525	+	0.0182775i
$316$		0			0
$317$		−	6.82323e8i		−	1.20305i	−0.798855	−	0.601524i	$-0.794561\pi$
$317$		−	6.82323e8i		−	1.20305i	0.798855	−	0.601524i	$-0.205439\pi$
$318$		0			0
$319$	4.84698e8	−	2.79840e8i	0.835995	−	0.482662i
$320$		0			0
$321$	1.14132e8	+	1.97683e8i	0.192593	+	0.333581i
$322$		0			0
$323$	3.75230e8	+	2.16639e8i	0.619567	+	0.357707i
$324$		0			0
$325$	−7.77696e8	−	8.14597e8i	−1.25666	−	1.31629i
$326$		0			0
$327$	−5.58256e8	−	3.22309e8i	−0.882910	−	0.509748i
$328$		0			0
$329$	1.04524e7	+	1.81040e7i	0.0161819	+	0.0280278i
$330$		0			0
$331$	8.80479e8	−	5.08345e8i	1.33451	−	0.770478i	0.348520	−	0.937301i	$-0.386684\pi$
$331$	8.80479e8	−	5.08345e8i	1.33451	−	0.770478i	0.985987	+	0.166823i	$0.0533509\pi$
$332$		0			0
$333$			8.86326e7i			0.131534i
$334$		0			0
$335$	−5.73208e8	+	9.92826e8i	−0.833020	+	1.44283i
$336$		0			0
$337$	3.52031e8			0.501044			0.250522	−	0.968111i	$-0.419398\pi$
$337$	3.52031e8			0.501044			0.250522	+	0.968111i	$0.419398\pi$
$338$		0			0
$339$	7.93443e8			1.10616
$340$		0			0
$341$	−1.82261e8	+	3.15686e8i	−0.248917	+	0.431137i
$342$		0			0
$343$		−	1.22017e8i		−	0.163265i
$344$		0			0
$345$	−8.59593e8	+	4.96286e8i	−1.12701	+	0.650677i
$346$		0			0
$347$	−3.04896e8	−	5.28095e8i	−0.391740	−	0.678514i	0.600939	−	0.799295i	$-0.294793\pi$
$347$	−3.04896e8	−	5.28095e8i	−0.391740	−	0.678514i	−0.992679	+	0.120781i	$0.961460\pi$
$348$		0			0
$349$	−6.42715e8	−	3.71072e8i	−0.809337	−	0.467271i	0.0373883	−	0.999301i	$-0.488096\pi$
$349$	−6.42715e8	−	3.71072e8i	−0.809337	−	0.467271i	−0.846726	+	0.532030i	$0.821429\pi$
$350$		0			0
$351$	−5.08644e8	−	5.32779e8i	−0.627826	−	0.657615i
$352$		0			0
$353$	−9.41104e8	−	5.43346e8i	−1.13874	−	0.657454i	−0.192624	−	0.981273i	$-0.561700\pi$
$353$	−9.41104e8	−	5.43346e8i	−1.13874	−	0.657454i	−0.946119	+	0.323819i	$0.895033\pi$
$354$		0			0
$355$	6.67442e8	+	1.15604e9i	0.791798	+	1.37143i
$356$		0			0
$357$	−2.96775e7	+	1.71343e7i	−0.0345214	+	0.0199310i
$358$		0			0
$359$		−	1.56901e9i		−	1.78976i	−0.446302	−	0.894882i	$-0.647259\pi$
$359$		−	1.56901e9i		−	1.78976i	0.446302	−	0.894882i	$-0.352741\pi$
$360$		0			0
$361$	6.67063e8	−	1.15539e9i	0.746262	−	1.29256i
$362$		0			0
$363$	4.05103e8			0.444521
$364$		0			0
$365$	1.85393e9			1.99558
$366$		0			0
$367$	−2.69422e8	+	4.66653e8i	−0.284513	+	0.492791i	−0.972491	−	0.232940i	$-0.925165\pi$
$367$	−2.69422e8	+	4.66653e8i	−0.284513	+	0.492791i	0.687978	+	0.725732i	$0.258499\pi$
$368$		0			0
$369$		−	1.49026e8i		−	0.154408i
$370$		0			0
$371$	−1.01433e8	+	5.85624e7i	−0.103127	+	0.0595402i
$372$		0			0
$373$	−7.04483e8	−	1.22020e9i	−0.702894	−	1.21745i	−0.967446	−	0.253077i	$-0.918558\pi$
$373$	−7.04483e8	−	1.22020e9i	−0.702894	−	1.21745i	0.264552	−	0.964371i	$-0.414776\pi$
$374$		0			0
$375$	1.30761e9	+	7.54950e8i	1.28047	+	0.739280i
$376$		0			0
$377$	−1.27467e9	+	3.10081e8i	−1.22519	+	0.298044i
$378$		0			0
$379$	−1.37512e9	−	7.93925e8i	−1.29749	−	0.749104i	−0.317517	−	0.948253i	$-0.602849\pi$
$379$	−1.37512e9	−	7.93925e8i	−1.29749	−	0.749104i	−0.979969	+	0.199149i	$0.936182\pi$
$380$		0			0
$381$	−6.94244e8	−	1.20247e9i	−0.643095	−	1.11387i
$382$		0			0
$383$	−8.74416e8	+	5.04845e8i	−0.795285	+	0.459158i	−0.841820	−	0.539759i	$-0.818515\pi$
$383$	−8.74416e8	+	5.04845e8i	−0.795285	+	0.459158i	0.0465351	+	0.998917i	$0.485182\pi$
$384$		0			0
$385$			1.17905e8i			0.105298i
$386$		0			0
$387$	1.03560e8	−	1.79372e8i	0.0908250	−	0.157313i
$388$		0			0
$389$	−6.14149e8			−0.528993			−0.264497	−	0.964387i	$-0.585206\pi$
$389$	−6.14149e8			−0.528993			−0.264497	+	0.964387i	$0.585206\pi$
$390$		0			0
$391$	−3.86493e8			−0.326981
$392$		0			0
$393$	−2.44638e8	+	4.23726e8i	−0.203306	+	0.352136i
$394$		0			0
$395$		−	1.70899e9i		−	1.39524i
$396$		0			0
$397$	2.26743e8	−	1.30910e8i	0.181873	−	0.105004i	−0.406300	−	0.913740i	$-0.633181\pi$
$397$	2.26743e8	−	1.30910e8i	0.181873	−	0.105004i	0.588172	+	0.808736i	$0.299848\pi$
$398$		0			0
$399$	8.81079e7	+	1.52607e8i	0.0694400	+	0.120274i
$400$		0			0
$401$	7.19925e6	+	4.15649e6i	0.00557547	+	0.00321900i	0.502785	−	0.864411i	$-0.332309\pi$
$401$	7.19925e6	+	4.15649e6i	0.00557547	+	0.00321900i	−0.497210	+	0.867630i	$0.665642\pi$
$402$		0			0
$403$	6.17994e8	−	5.89999e8i	0.470345	−	0.449039i
$404$		0			0
$405$	2.19672e9	+	1.26828e9i	1.64317	+	0.948685i
$406$		0			0
$407$	4.46297e8	+	7.73009e8i	0.328128	+	0.568334i
$408$		0			0
$409$	−1.36216e8	+	7.86441e7i	−0.0984454	+	0.0568375i	−0.548414	−	0.836207i	$-0.684768\pi$
$409$	−1.36216e8	+	7.86441e7i	−0.0984454	+	0.0568375i	0.449969	+	0.893044i	$0.351435\pi$
$410$		0			0
$411$			1.43441e8i			0.101912i
$412$		0			0
$413$	6.29902e7	−	1.09102e8i	0.0439995	−	0.0762094i
$414$		0			0
$415$	−2.80356e9			−1.92549
$416$		0			0
$417$	2.25379e9			1.52208
$418$		0			0
$419$	4.85746e8	−	8.41336e8i	0.322597	−	0.558754i	−0.658426	−	0.752645i	$-0.728778\pi$
$419$	4.85746e8	−	8.41336e8i	0.322597	−	0.558754i	0.981023	+	0.193891i	$0.0621109\pi$
$420$		0			0
$421$		−	4.85973e8i		−	0.317413i	−0.987326	−	0.158707i	$-0.949268\pi$
$421$		−	4.85973e8i		−	0.317413i	0.987326	−	0.158707i	$-0.0507324\pi$
$422$		0			0
$423$	−8.17356e7	+	4.71901e7i	−0.0525073	+	0.0303151i
$424$		0			0
$425$	6.52533e8	+	1.13022e9i	0.412326	+	0.714170i
$426$		0			0
$427$	2.12119e8	+	1.22467e8i	0.131851	+	0.0761241i
$428$		0			0
$429$	1.29035e9	+	3.77999e8i	0.789054	+	0.231148i
$430$		0			0
$431$	−2.39621e8	−	1.38345e8i	−0.144163	−	0.0832328i	0.426183	−	0.904637i	$-0.359858\pi$
$431$	−2.39621e8	−	1.38345e8i	−0.144163	−	0.0832328i	−0.570347	+	0.821404i	$0.693191\pi$
$432$		0			0
$433$	−1.17994e9	−	2.04371e9i	−0.698476	−	1.20980i	−0.968995	−	0.247081i	$-0.920529\pi$
$433$	−1.17994e9	−	2.04371e9i	−0.698476	−	1.20980i	0.270519	−	0.962715i	$-0.412805\pi$
$434$		0			0
$435$	3.38096e9	−	1.95200e9i	1.96937	−	1.13702i
$436$		0			0
$437$			1.98742e9i			1.13921i
$438$		0			0
$439$	−5.02423e8	+	8.70222e8i	−0.283428	+	0.490912i	−0.972227	−	0.234041i	$-0.924805\pi$
$439$	−5.02423e8	+	8.70222e8i	−0.283428	+	0.490912i	0.688798	+	0.724953i	$0.258138\pi$
$440$		0			0
$441$	2.74513e8			0.152415
$442$		0			0
$443$	9.97546e8			0.545155			0.272578	−	0.962134i	$-0.412124\pi$
$443$	9.97546e8			0.545155			0.272578	+	0.962134i	$0.412124\pi$
$444$		0			0
$445$	3.49544e8	−	6.05429e8i	0.188037	−	0.325689i
$446$		0			0
$447$			1.22833e9i			0.650487i
$448$		0			0
$449$	1.25644e7	−	7.25408e6i	0.00655060	−	0.00378199i	−0.496721	−	0.867910i	$-0.665463\pi$
$449$	1.25644e7	−	7.25408e6i	0.00655060	−	0.00378199i	0.503272	+	0.864128i	$0.332129\pi$
$450$		0			0
$451$	−7.50397e8	−	1.29973e9i	−0.385188	−	0.667166i
$452$		0			0
$453$	−2.58782e9	−	1.49408e9i	−1.30795	−	0.755143i
$454$		0			0
$455$	7.76925e7	−	2.65213e8i	0.0386669	−	0.131994i
$456$		0			0
$457$	−6.28752e8	−	3.63010e8i	−0.308157	−	0.177915i	0.337944	−	0.941166i	$-0.390268\pi$
$457$	−6.28752e8	−	3.63010e8i	−0.308157	−	0.177915i	−0.646102	+	0.763251i	$0.723602\pi$
$458$		0			0
$459$	4.26782e8	+	7.39208e8i	0.205997	+	0.356798i
$460$		0			0
$461$	−1.36362e9	+	7.87284e8i	−0.648244	+	0.374264i	−0.787783	−	0.615953i	$-0.788771\pi$
$461$	−1.36362e9	+	7.87284e8i	−0.648244	+	0.374264i	0.139539	+	0.990217i	$0.455438\pi$
$462$		0			0
$463$			2.40145e9i			1.12445i	0.826984	+	0.562225i	$0.190054\pi$
$463$			2.40145e9i			1.12445i	−0.826984	+	0.562225i	$0.809946\pi$
$464$		0			0
$465$	−1.27135e9	+	2.20204e9i	−0.586380	+	1.01564i
$466$		0			0
$467$	−2.88669e9			−1.31157			−0.655784	−	0.754948i	$-0.727662\pi$
$467$	−2.88669e9			−1.31157			−0.655784	+	0.754948i	$0.727662\pi$
$468$		0			0
$469$	−1.81552e8			−0.0812634
$470$		0			0
$471$	−1.15940e9	+	2.00814e9i	−0.511283	+	0.885567i
$472$		0			0
$473$		−	2.08585e9i		−	0.906296i
$474$		0			0
$475$	5.81180e9	−	3.35545e9i	2.48819	−	1.43656i
$476$		0			0
$477$	−2.64396e8	−	4.57947e8i	−0.111543	−	0.193197i
$478$		0			0
$479$	−1.36402e9	−	7.87515e8i	−0.567081	−	0.327404i	0.188902	−	0.981996i	$-0.439507\pi$
$479$	−1.36402e9	−	7.87515e8i	−0.567081	−	0.327404i	−0.755983	+	0.654592i	$0.772841\pi$
$480$		0			0
$481$	−4.94525e8	−	2.03287e9i	−0.202619	−	0.832919i
$482$		0			0
$483$	−1.36129e8	−	7.85941e7i	−0.0549712	−	0.0317377i
$484$		0			0
$485$	3.27165e9	+	5.66666e9i	1.30218	+	2.25544i
$486$		0			0
$487$	−1.70387e9	+	9.83730e8i	−0.668475	+	0.385944i	−0.795499	−	0.605955i	$-0.792791\pi$
$487$	−1.70387e9	+	9.83730e8i	−0.668475	+	0.385944i	0.127023	+	0.991900i	$0.459458\pi$
$488$		0			0
$489$			4.97788e8i			0.192514i
$490$		0			0
$491$	3.80549e8	−	6.59129e8i	0.145086	−	0.251296i	−0.784319	−	0.620358i	$-0.786988\pi$
$491$	3.80549e8	−	6.59129e8i	0.145086	−	0.251296i	0.929405	+	0.369062i	$0.120321\pi$
$492$		0			0
$493$	1.52016e9			0.571380
$494$		0			0
$495$	−5.32314e8			−0.197265
$496$		0			0
$497$	−1.05699e8	+	1.83076e8i	−0.0386211	+	0.0668936i
$498$		0			0
$499$			5.96355e8i			0.214859i	0.994213	+	0.107429i	$0.0342620\pi$
$499$			5.96355e8i			0.214859i	−0.994213	+	0.107429i	$0.965738\pi$
$500$		0			0
$501$	2.55182e9	−	1.47329e9i	0.906605	−	0.523429i
$502$		0			0
$503$	−2.42561e8	−	4.20128e8i	−0.0849832	−	0.147195i	0.820401	−	0.571789i	$-0.193750\pi$
$503$	−2.42561e8	−	4.20128e8i	−0.0849832	−	0.147195i	−0.905384	+	0.424594i	$0.860417\pi$
$504$		0			0
$505$	−8.20290e8	−	4.73595e8i	−0.283431	−	0.163639i
$506$		0			0
$507$	−2.65341e9	−	1.70053e9i	−0.904224	−	0.579503i
$508$		0			0
$509$	−1.92665e9	−	1.11235e9i	−0.647576	−	0.373878i	0.139951	−	0.990158i	$-0.455305\pi$
$509$	−1.92665e9	−	1.11235e9i	−0.647576	−	0.373878i	−0.787527	+	0.616280i	$0.788639\pi$
$510$		0			0
$511$	1.46799e8	+	2.54263e8i	0.0486686	+	0.0842965i
$512$		0			0
$513$	3.80115e9	−	2.19459e9i	1.24309	−	0.717701i
$514$		0			0
$515$			7.84697e9i			2.53149i
$516$		0			0
$517$	−4.75237e8	+	8.23135e8i	−0.151249	+	0.261972i
$518$		0			0
$519$	−3.15569e8			−0.0990851
$520$		0			0
$521$	−5.35828e9			−1.65994			−0.829972	−	0.557805i	$-0.811644\pi$
$521$	−5.35828e9			−1.65994			−0.829972	+	0.557805i	$0.811644\pi$
$522$		0			0
$523$	−3.99109e8	+	6.91277e8i	−0.121993	+	0.211299i	−0.920554	−	0.390616i	$-0.872262\pi$
$523$	−3.99109e8	+	6.91277e8i	−0.121993	+	0.211299i	0.798560	+	0.601915i	$0.205595\pi$
$524$		0			0
$525$			5.30775e8i			0.160086i
$526$		0			0
$527$	−8.57441e8	+	4.95044e8i	−0.255192	+	0.147335i
$528$		0			0
$529$	8.16003e8	+	1.41336e9i	0.239661	+	0.415105i
$530$		0			0
$531$	4.92572e8	+	2.84386e8i	0.142770	+	0.0824286i
$532$		0			0
$533$	8.31486e8	+	3.41804e9i	0.237854	+	0.977761i
$534$		0			0
$535$	1.84737e9	+	1.06658e9i	0.521573	+	0.301130i
$536$		0			0
$537$	1.33609e9	+	2.31417e9i	0.372328	+	0.644891i
$538$		0			0
$539$	2.39417e9	−	1.38227e9i	0.658557	−	0.380218i
$540$		0			0
$541$			3.89614e9i			1.05790i	0.848654	+	0.528949i	$0.177414\pi$
$541$			3.89614e9i			1.05790i	−0.848654	+	0.528949i	$0.822586\pi$
$542$		0			0
$543$	8.38814e8	−	1.45287e9i	0.224836	−	0.389428i
$544$		0			0
$545$	−6.02403e9			−1.59404
$546$		0			0
$547$	2.21748e8			0.0579300			0.0289650	−	0.999580i	$-0.490779\pi$
$547$	2.21748e8			0.0579300			0.0289650	+	0.999580i	$0.490779\pi$
$548$		0			0
$549$	−5.52911e8	+	9.57670e8i	−0.142611	+	0.247009i
$550$		0			0
$551$		−	7.81695e9i		−	1.99070i
$552$		0			0
$553$	2.34384e8	−	1.35322e8i	0.0589374	−	0.0340275i
$554$		0			0
$555$	3.11310e9	+	5.39205e9i	0.772979	+	1.33884i
$556$		0			0
$557$	−1.24412e9	−	7.18292e8i	−0.305048	−	0.176120i	0.339660	−	0.940548i	$-0.389688\pi$
$557$	−1.24412e9	−	7.18292e8i	−0.305048	−	0.176120i	−0.644708	+	0.764429i	$0.723021\pi$
$558$		0			0
$559$	−1.37445e9	+	4.69188e9i	−0.332804	+	1.13607i
$560$		0			0
$561$	−1.34935e9	−	7.79045e8i	−0.322666	−	0.186291i
$562$		0			0
$563$	2.93914e9	+	5.09074e9i	0.694130	+	1.20227i	0.970473	+	0.241209i	$0.0775439\pi$
$563$	2.93914e9	+	5.09074e9i	0.694130	+	1.20227i	−0.276344	+	0.961059i	$0.589123\pi$
$564$		0			0
$565$	6.42142e9	−	3.70741e9i	1.49783	−	0.864770i
$566$		0			0
$567$			4.01700e8i			0.0925468i
$568$		0			0
$569$	1.49786e9	−	2.59437e9i	0.340862	−	0.590390i	−0.643731	−	0.765252i	$-0.722614\pi$
$569$	1.49786e9	−	2.59437e9i	0.340862	−	0.590390i	0.984593	+	0.174862i	$0.0559478\pi$
$570$		0			0
$571$	5.32432e9			1.19685			0.598423	−	0.801181i	$-0.295794\pi$
$571$	5.32432e9			1.19685			0.598423	+	0.801181i	$0.295794\pi$
$572$		0			0
$573$	−1.60190e9			−0.355708
$574$		0			0
$575$	−2.99313e9	+	5.18425e9i	−0.656580	+	1.13723i
$576$		0			0
$577$			2.16008e9i			0.468118i	0.972222	+	0.234059i	$0.0752009\pi$
$577$			2.16008e9i			0.468118i	−0.972222	+	0.234059i	$0.924799\pi$
$578$		0			0
$579$	−6.71475e9	+	3.87676e9i	−1.43766	+	0.830031i
$580$		0			0
$581$	−2.21992e8	−	3.84502e8i	−0.0469593	−	0.0813359i
$582$		0			0
$583$	−4.61186e9	−	2.66266e9i	−0.963908	−	0.556513i
$584$		0			0
$585$	1.19738e9	+	3.50764e8i	0.247278	+	0.0724384i
$586$		0			0
$587$	4.55850e9	+	2.63185e9i	0.930227	+	0.537067i	0.886883	−	0.461994i	$-0.152866\pi$
$587$	4.55850e9	+	2.63185e9i	0.930227	+	0.537067i	0.0433434	+	0.999060i	$0.486199\pi$
$588$		0			0
$589$	2.54561e9	+	4.40912e9i	0.513320	+	0.889096i
$590$		0			0
$591$	−7.48278e8	+	4.32019e8i	−0.149110	+	0.0860888i
$592$		0			0
$593$			6.23434e9i			1.22772i	0.789415	+	0.613859i	$0.210384\pi$
$593$			6.23434e9i			1.22772i	−0.789415	+	0.613859i	$0.789616\pi$
$594$		0			0
$595$	−1.60122e8	+	2.77340e8i	−0.0311632	+	0.0539762i
$596$		0			0
$597$	−1.06285e10			−2.04437
$598$		0			0
$599$	−3.87748e9			−0.737149			−0.368575	−	0.929598i	$-0.620154\pi$
$599$	−3.87748e9			−0.737149			−0.368575	+	0.929598i	$0.620154\pi$
$600$		0			0
$601$	−3.87101e9	+	6.70479e9i	−0.727385	+	1.25987i	0.230600	+	0.973049i	$0.425931\pi$
$601$	−3.87101e9	+	6.70479e9i	−0.727385	+	1.25987i	−0.957985	+	0.286819i	$0.907402\pi$
$602$		0			0
$603$		−	8.19664e8i		−	0.152239i
$604$		0			0
$605$	3.27854e9	−	1.89287e9i	0.601918	−	0.347517i
$606$		0			0
$607$	−7.39619e8	−	1.28106e9i	−0.134229	−	0.232492i	0.791073	−	0.611721i	$-0.209523\pi$
$607$	−7.39619e8	−	1.28106e9i	−0.134229	−	0.232492i	−0.925303	+	0.379229i	$0.876189\pi$
$608$		0			0
$609$	5.35424e8	+	3.09127e8i	0.0960589	+	0.0554596i
$610$		0			0
$611$	1.61139e9	−	1.53839e9i	0.285796	−	0.272849i
$612$		0			0
$613$	−2.65542e8	−	1.53311e8i	−0.0465609	−	0.0268820i	0.476539	−	0.879153i	$-0.341891\pi$
$613$	−2.65542e8	−	1.53311e8i	−0.0465609	−	0.0268820i	−0.523100	+	0.852271i	$0.675224\pi$
$614$		0			0
$615$	−5.23432e9	−	9.06611e9i	−0.907397	−	1.57166i
$616$		0			0
$617$	3.53172e9	−	2.03904e9i	0.605325	−	0.349484i	−0.165809	−	0.986158i	$-0.553023\pi$
$617$	3.53172e9	−	2.03904e9i	0.605325	−	0.349484i	0.771133	+	0.636674i	$0.219690\pi$
$618$		0			0
$619$		−	8.66187e9i		−	1.46789i	−0.679208	−	0.733946i	$-0.737676\pi$
$619$		−	8.66187e9i		−	1.46789i	0.679208	−	0.733946i	$-0.262324\pi$
$620$		0			0
$621$	−1.95762e9	+	3.39070e9i	−0.328026	+	0.568158i
$622$		0			0
$623$	1.10711e8			0.0183435
$624$		0			0
$625$	3.00277e9			0.491973
$626$		0			0
$627$	−4.00600e9	+	6.93859e9i	−0.649044	+	1.12418i
$628$		0			0
$629$			2.42439e9i			0.388441i
$630$		0			0
$631$	−5.94670e8	+	3.43333e8i	−0.0942266	+	0.0544017i	−0.546373	−	0.837542i	$-0.683992\pi$
$631$	−5.94670e8	+	3.43333e8i	−0.0942266	+	0.0544017i	0.452146	+	0.891944i	$0.350658\pi$
$632$		0			0
$633$	−1.18280e9	−	2.04867e9i	−0.185353	−	0.321041i
$634$		0			0
$635$	−1.12372e10	−	6.48779e9i	−1.74160	−	1.00552i
$636$		0			0
$637$	−6.29623e9	+	1.53164e9i	−0.965144	+	0.234785i
$638$		0			0
$639$	−8.26548e8	−	4.77207e8i	−0.125318	−	0.0723526i
$640$		0			0
$641$	−2.49684e9	−	4.32466e9i	−0.374445	−	0.648558i	0.615799	−	0.787903i	$-0.288833\pi$
$641$	−2.49684e9	−	4.32466e9i	−0.374445	−	0.648558i	−0.990244	+	0.139346i	$0.955500\pi$
$642$		0			0
$643$	1.76995e9	−	1.02188e9i	0.262557	−	0.151587i	−0.362944	−	0.931811i	$-0.618228\pi$
$643$	1.76995e9	−	1.02188e9i	0.262557	−	0.151587i	0.625500	+	0.780224i	$0.284895\pi$
$644$		0			0
$645$		−	1.45497e10i		−	2.13498i
$646$		0			0
$647$	2.19517e9	−	3.80215e9i	0.318642	−	0.551904i	−0.661563	−	0.749890i	$-0.730107\pi$
$647$	2.19517e9	−	3.80215e9i	0.318642	−	0.551904i	0.980205	+	0.197985i	$0.0634399\pi$
$648$		0			0
$649$	5.72794e9			0.822512
$650$		0			0
$651$	−4.02672e8			−0.0572029
$652$		0			0
$653$	7.64503e8	−	1.32416e9i	0.107444	−	0.186099i	−0.807290	−	0.590155i	$-0.799067\pi$
$653$	7.64503e8	−	1.32416e9i	0.107444	−	0.186099i	0.914734	+	0.404056i	$0.132400\pi$
$654$		0			0
$655$			4.57234e9i			0.635761i
$656$		0			0
$657$	−1.14794e9	+	6.62762e8i	−0.157921	+	0.0911757i
$658$		0			0
$659$	−3.30987e9	−	5.73287e9i	−0.450518	−	0.780320i	0.547900	−	0.836544i	$-0.315427\pi$
$659$	−3.30987e9	−	5.73287e9i	−0.450518	−	0.780320i	−0.998418	+	0.0562235i	$0.982094\pi$
$660$		0			0
$661$	1.02805e10	+	5.93546e9i	1.38455	+	0.799372i	0.992695	−	0.120653i	$-0.0384988\pi$
$661$	1.02805e10	+	5.93546e9i	1.38455	+	0.799372i	0.391859	+	0.920025i	$0.371832\pi$
$662$		0			0
$663$	2.52185e9	+	2.64151e9i	0.336064	+	0.352010i
$664$		0			0
$665$	1.42613e9	+	8.23378e8i	0.188055	+	0.108573i
$666$		0			0
$667$	3.48644e9	+	6.03869e9i	0.454927	+	0.787956i
$668$		0			0
$669$	7.35554e9	−	4.24672e9i	0.949780	−	0.548356i
$670$		0			0
$671$			1.11364e10i			1.42304i
$672$		0			0
$673$	−5.07802e8	+	8.79539e8i	−0.0642158	+	0.111225i	−0.896346	−	0.443356i	$-0.853788\pi$
$673$	−5.07802e8	+	8.79539e8i	−0.0642158	+	0.111225i	0.832130	+	0.554581i	$0.187121\pi$
$674$		0			0
$675$	1.32206e10			1.65458
$676$		0			0
$677$	−1.52605e10			−1.89020			−0.945099	−	0.326785i	$-0.894035\pi$
$677$	−1.52605e10			−1.89020			−0.945099	+	0.326785i	$0.894035\pi$
$678$		0			0
$679$	−5.18112e8	+	8.97397e8i	−0.0635155	+	0.110012i
$680$		0			0
$681$		−	1.59686e10i		−	1.93755i
$682$		0			0
$683$	−1.02242e10	+	5.90295e9i	−1.22788	+	0.708918i	−0.966587	−	0.256340i	$-0.917483\pi$
$683$	−1.02242e10	+	5.90295e9i	−1.22788	+	0.708918i	−0.261296	+	0.965259i	$0.584150\pi$
$684$		0			0
$685$	6.70235e8	+	1.16088e9i	0.0796728	+	0.137997i
$686$		0			0
$687$	−5.06164e9	−	2.92234e9i	−0.595584	−	0.343861i
$688$		0			0
$689$	8.61929e9	+	9.02827e9i	1.00393	+	1.05157i
$690$		0			0
$691$	1.95302e9	+	1.12758e9i	0.225182	+	0.130009i	0.608348	−	0.793671i	$-0.291833\pi$
$691$	1.95302e9	+	1.12758e9i	0.225182	+	0.130009i	−0.383165	+	0.923680i	$0.625166\pi$
$692$		0			0
$693$	−4.21498e7	−	7.30057e7i	−0.00481094	−	0.00833279i
$694$		0			0
$695$	1.82401e10	−	1.05310e10i	2.06102	−	1.18993i
$696$		0			0
$697$		−	4.07633e9i		−	0.455990i
$698$		0			0
$699$	9.04484e9	−	1.56661e10i	1.00168	−	1.73497i
$700$		0			0
$701$	−1.12762e10			−1.23637			−0.618187	−	0.786031i	$-0.712132\pi$
$701$	−1.12762e10			−1.23637			−0.618187	+	0.786031i	$0.712132\pi$
$702$		0			0
$703$	1.24667e10			1.35334
$704$		0			0
$705$	−3.31497e9	+	5.74170e9i	−0.356302	+	0.617133i
$706$		0			0
$707$		−	1.50001e8i		−	0.0159634i
$708$		0			0
$709$	6.54595e9	−	3.77931e9i	0.689780	−	0.398245i	−0.113749	−	0.993509i	$-0.536286\pi$
$709$	6.54595e9	−	3.77931e9i	0.689780	−	0.398245i	0.803530	+	0.595265i	$0.202953\pi$
$710$		0			0
$711$	6.10947e8	+	1.05819e9i	0.0637470	+	0.110413i
$712$		0			0
$713$	−3.93303e9	−	2.27074e9i	−0.406363	−	0.234614i
$714$		0			0
$715$	1.22091e10	−	2.97004e9i	1.24915	−	0.303873i
$716$		0			0
$717$	−5.37396e9	−	3.10265e9i	−0.544474	−	0.314352i
$718$		0			0
$719$	−1.23044e9	−	2.13118e9i	−0.123455	−	0.213831i	0.797673	−	0.603090i	$-0.206064\pi$
$719$	−1.23044e9	−	2.13118e9i	−0.123455	−	0.213831i	−0.921128	+	0.389260i	$0.872731\pi$
$720$		0			0
$721$	−1.07619e9	+	6.21340e8i	−0.106934	+	0.0617385i
$722$		0			0
$723$		−	1.16985e9i		−	0.115119i
$724$		0			0
$725$	1.17726e10	−	2.03908e10i	1.14733	−	1.98724i
$726$		0			0
$727$	−1.00899e10			−0.973902			−0.486951	−	0.873429i	$-0.661891\pi$
$727$	−1.00899e10			−0.973902			−0.486951	+	0.873429i	$0.661891\pi$
$728$		0			0
$729$	8.40048e9			0.803078
$730$		0			0
$731$	2.83271e9	−	4.90640e9i	0.268220	−	0.464571i
$732$		0			0
$733$			8.44259e8i			0.0791793i	0.999216	+	0.0395897i	$0.0126051\pi$
$733$			8.44259e8i			0.0791793i	−0.999216	+	0.0395897i	$0.987395\pi$
$734$		0			0
$735$	1.67003e10	−	9.64190e9i	1.55138	−	0.895689i
$736$		0			0
$737$	−4.12730e9	−	7.14869e9i	−0.379778	−	0.657795i
$738$		0			0
$739$	6.39606e9	+	3.69277e9i	0.582984	+	0.336586i	0.762318	−	0.647202i	$-0.224061\pi$
$739$	6.39606e9	+	3.69277e9i	0.582984	+	0.336586i	−0.179334	+	0.983788i	$0.557394\pi$
$740$		0			0
$741$	1.35831e10	−	1.29678e10i	1.22641	−	1.17086i
$742$		0			0
$743$	1.15376e10	+	6.66123e9i	1.03194	+	0.595791i	0.917540	−	0.397644i	$-0.130172\pi$
$743$	1.15376e10	+	6.66123e9i	1.03194	+	0.595791i	0.114400	+	0.993435i	$0.463505\pi$
$744$		0			0
$745$	5.73944e9	+	9.94099e9i	0.508537	+	0.880811i
$746$		0			0
$747$	1.73594e9	−	1.00224e9i	0.152374	−	0.0879734i
$748$		0			0
$749$			3.37816e8i			0.0293761i
$750$		0			0
$751$	−6.71361e9	+	1.16283e10i	−0.578384	+	1.00179i	0.417281	+	0.908778i	$0.362983\pi$
$751$	−6.71361e9	+	1.16283e10i	−0.578384	+	1.00179i	−0.995665	+	0.0930132i	$0.970350\pi$
$752$		0			0
$753$	1.66036e9			0.141716
$754$		0			0
$755$	−2.79246e10			−2.36142
$756$		0			0
$757$	−2.20189e9	+	3.81378e9i	−0.184484	+	0.319536i	−0.943403	−	0.331650i	$-0.892395\pi$
$757$	−2.20189e9	+	3.81378e9i	−0.184484	+	0.319536i	0.758918	+	0.651186i	$0.225728\pi$
$758$		0			0
$759$		−	7.14687e9i		−	0.593293i
$760$		0			0
$761$	6.51209e9	−	3.75976e9i	0.535642	−	0.309253i	−0.207669	−	0.978199i	$-0.566588\pi$
$761$	6.51209e9	−	3.75976e9i	0.535642	−	0.309253i	0.743311	+	0.668946i	$0.233254\pi$
$762$		0			0
$763$	−4.76996e8	−	8.26182e8i	−0.0388758	−	0.0673348i
$764$		0			0
$765$	−1.25212e9	−	7.22915e8i	−0.101119	−	0.0583811i
$766$		0			0
$767$	−1.28843e10	−	3.77438e9i	−1.03105	−	0.302038i
$768$		0			0
$769$	1.97283e10	+	1.13902e10i	1.56440	+	0.903208i	0.996803	+	0.0799025i	$0.0254609\pi$
$769$	1.97283e10	+	1.13902e10i	1.56440	+	0.903208i	0.567599	+	0.823305i	$0.307872\pi$
$770$		0			0
$771$	−6.63563e9	−	1.14933e10i	−0.521425	−	0.903135i
$772$		0			0
$773$	−1.25609e10	+	7.25204e9i	−0.978122	+	0.564719i	−0.901703	−	0.432357i	$-0.857682\pi$
$773$	−1.25609e10	+	7.25204e9i	−0.978122	+	0.564719i	−0.0764191	+	0.997076i	$0.524349\pi$
$774$		0			0
$775$			1.53351e10i			1.18340i
$776$		0			0
$777$	−4.93004e8	+	8.53908e8i	−0.0377031	+	0.0653037i
$778$		0			0
$779$	−2.09613e10			−1.58868
$780$		0			0
$781$	−9.61164e9			−0.721969
$782$		0			0
$783$	7.69975e9	−	1.33364e10i	0.573206	−	0.992821i
$784$		0			0
$785$			2.16695e10i			1.59884i
$786$		0			0
$787$	7.41960e9	−	4.28371e9i	0.542586	−	0.313262i	−0.203540	−	0.979067i	$-0.565245\pi$
$787$	7.41960e9	−	4.28371e9i	0.542586	−	0.313262i	0.746126	+	0.665804i	$0.231911\pi$
$788$		0			0
$789$	2.02921e9	+	3.51469e9i	0.147081	+	0.254752i
$790$		0			0
$791$	1.01692e9	+	5.87121e8i	0.0730585	+	0.0421804i
$792$		0			0
$793$	7.33824e9	−	2.50500e10i	0.522559	−	1.78383i
$794$		0			0
$795$	−3.21695e10	−	1.85731e10i	−2.27070	−	1.31099i
$796$		0			0
$797$	−8.46996e9	−	1.46704e10i	−0.592621	−	1.02645i	−0.993878	−	0.110484i	$-0.964760\pi$
$797$	−8.46996e9	−	1.46704e10i	−0.592621	−	1.02645i	0.401257	−	0.915966i	$-0.368573\pi$
$798$		0			0
$799$	−2.23573e9	+	1.29080e9i	−0.155062	+	0.0895253i
$800$		0			0
$801$			4.99834e8i			0.0343646i
$802$		0			0
$803$	−6.67449e9	+	1.15606e10i	−0.454897	+	0.787906i
$804$		0			0
$805$	−1.46894e9			−0.0992473
$806$		0			0
$807$	2.59581e10			1.73866
$808$		0			0
$809$	−4.69798e8	+	8.13714e8i	−0.0311955	+	0.0540321i	−0.881202	−	0.472741i	$-0.843265\pi$
$809$	−4.69798e8	+	8.13714e8i	−0.0311955	+	0.0540321i	0.850006	+	0.526773i	$0.176598\pi$
$810$		0			0
$811$			8.49475e9i			0.559213i	0.960115	+	0.279607i	$0.0902041\pi$
$811$			8.49475e9i			0.559213i	−0.960115	+	0.279607i	$0.909796\pi$
$812$		0			0
$813$	−6.92172e9	+	3.99626e9i	−0.451749	+	0.260817i
$814$		0			0
$815$	2.32594e9	+	4.02865e9i	0.150504	+	0.260680i
$816$		0			0
$817$	−2.52296e10	−	1.45663e10i	−1.61858	−	0.934487i
$818$		0			0
$819$	4.67047e7	+	1.91992e8i	0.00297075	+	0.0122121i
$820$		0			0
$821$	9.50583e9	+	5.48819e9i	0.599499	+	0.346121i	0.768845	−	0.639436i	$-0.220832\pi$
$821$	9.50583e9	+	5.48819e9i	0.599499	+	0.346121i	−0.169345	+	0.985557i	$0.554165\pi$
$822$		0			0
$823$	−7.58620e9	−	1.31397e10i	−0.474379	−	0.821648i	0.525191	−	0.850984i	$-0.323994\pi$
$823$	−7.58620e9	−	1.31397e10i	−0.474379	−	0.821648i	−0.999570	+	0.0293366i	$0.990661\pi$
$824$		0			0
$825$	−2.08995e10	+	1.20664e10i	−1.29583	+	0.748148i
$826$		0			0
$827$			1.08176e10i			0.665062i	0.943092	+	0.332531i	$0.107903\pi$
$827$			1.08176e10i			0.665062i	−0.943092	+	0.332531i	$0.892097\pi$
$828$		0			0
$829$	2.83648e9	−	4.91293e9i	0.172918	−	0.299502i	−0.766521	−	0.642219i	$-0.778014\pi$
$829$	2.83648e9	−	4.91293e9i	0.172918	−	0.299502i	0.939439	+	0.342717i	$0.111347\pi$
$830$		0			0
$831$	−1.07607e10			−0.650486
$832$		0			0
$833$	7.50883e9			0.450106
$834$		0			0
$835$	1.37681e10	−	2.38470e10i	0.818410	−	1.41753i
$836$		0			0
$837$			1.00298e10i			0.591224i
$838$		0			0
$839$	8.48728e9	−	4.90013e9i	0.496137	−	0.286445i	−0.230980	−	0.972959i	$-0.574193\pi$
$839$	8.48728e9	−	4.90013e9i	0.496137	−	0.286445i	0.727117	+	0.686514i	$0.240860\pi$
$840$		0			0
$841$	−5.08797e9	−	8.81262e9i	−0.294957	−	0.510880i
$842$		0			0
$843$	−1.49296e10	−	8.61961e9i	−0.858325	−	0.495554i
$844$		0			0
$845$	−2.94201e10	−	1.36434e9i	−1.67743	−	0.0777903i
$846$		0			0
$847$	5.19205e8	+	2.99763e8i	0.0293594	+	0.0169506i
$848$		0			0
$849$	−2.02331e9	−	3.50447e9i	−0.113471	−	0.196538i
$850$		0			0
$851$	−9.63066e9	+	5.56026e9i	−0.535676	+	0.309273i
$852$		0			0
$853$			1.09563e10i			0.604424i	0.953241	+	0.302212i	$0.0977251\pi$
$853$			1.09563e10i			0.604424i	−0.953241	+	0.302212i	$0.902275\pi$
$854$		0			0
$855$	−3.71736e9	+	6.43866e9i	−0.203401	+	0.352301i
$856$		0			0
$857$	4.83453e9			0.262374			0.131187	−	0.991358i	$-0.458121\pi$
$857$	4.83453e9			0.262374			0.131187	+	0.991358i	$0.458121\pi$
$858$		0			0
$859$	4.19822e9			0.225990			0.112995	−	0.993596i	$-0.463956\pi$
$859$	4.19822e9			0.225990			0.112995	+	0.993596i	$0.463956\pi$
$860$		0			0
$861$	8.28930e8	−	1.43575e9i	0.0442595	−	0.0766598i
$862$		0			0
$863$			1.58890e10i			0.841510i	0.907174	+	0.420755i	$0.138235\pi$
$863$			1.58890e10i			0.841510i	−0.907174	+	0.420755i	$0.861765\pi$
$864$		0			0
$865$	−2.55393e9	+	1.47451e9i	−0.134169	+	0.0774626i
$866$		0			0
$867$	8.18872e9	+	1.41833e10i	0.426726	+	0.739111i
$868$		0			0
$869$	1.06567e10	+	6.15267e9i	0.550878	+	0.318049i
$870$		0			0
$871$	4.57330e9	+	1.87998e10i	0.234513	+	0.964027i
$872$		0			0
$873$	−4.05154e9	−	2.33916e9i	−0.206096	−	0.118990i
$874$		0			0
$875$	1.11728e9	+	1.93518e9i	0.0563809	+	0.0976546i
$876$		0			0
$877$	−7.52789e9	+	4.34623e9i	−0.376855	+	0.217578i	−0.676449	−	0.736489i	$-0.736482\pi$
$877$	−7.52789e9	+	4.34623e9i	−0.376855	+	0.217578i	0.299594	+	0.954067i	$0.403149\pi$
$878$		0			0
$879$		−	2.11657e10i		−	1.05117i
$880$		0			0
$881$	−3.97881e9	+	6.89150e9i	−0.196037	+	0.339546i	−0.947240	−	0.320525i	$-0.896141\pi$
$881$	−3.97881e9	+	6.89150e9i	−0.196037	+	0.339546i	0.751203	+	0.660071i	$0.229474\pi$
$882$		0			0
$883$	−2.25480e8			−0.0110216			−0.00551080	−	0.999985i	$-0.501754\pi$
$883$	−2.25480e8			−0.0110216			−0.00551080	+	0.999985i	$0.501754\pi$
$884$		0			0
$885$	3.99547e10			1.93761
$886$		0			0
$887$	−1.14024e9	+	1.97496e9i	−0.0548611	+	0.0950222i	−0.892152	−	0.451736i	$-0.850805\pi$
$887$	−1.14024e9	+	1.97496e9i	−0.0548611	+	0.0950222i	0.837291	+	0.546758i	$0.184138\pi$
$888$		0			0
$889$		−	2.05487e9i		−	0.0980908i
$890$		0			0
$891$	−1.58172e10	+	9.13204e9i	−0.749129	+	0.432510i
$892$		0			0
$893$	6.63754e9	+	1.14966e10i	0.311909	+	0.540241i
$894$		0			0
$895$	2.16262e10	+	1.24859e10i	1.00832	+	0.582156i
$896$		0			0
$897$	−4.70936e9	+	1.60760e10i	−0.217865	+	0.743713i
$898$		0			0
$899$	1.54694e10	+	8.93129e9i	0.710094	+	0.409973i
$900$		0			0
$901$	−7.23209e9	−	1.25263e10i	−0.329403	−	0.570542i
$902$		0			0
$903$	1.99545e9	−	1.15208e9i	0.0901850	−	0.0520683i
$904$		0			0
$905$		−	1.56776e10i		−	0.703088i
$906$		0			0
$907$	1.00273e9	−	1.73678e9i	0.0446231	−	0.0772894i	−0.842851	−	0.538147i	$-0.819125\pi$
$907$	1.00273e9	−	1.73678e9i	0.0446231	−	0.0772894i	0.887474	+	0.460857i	$0.152458\pi$
$908$		0			0
$909$	6.77221e8			0.0299059
$910$		0			0
$911$	2.82977e10			1.24004			0.620022	−	0.784584i	$-0.287124\pi$
$911$	2.82977e10			1.24004			0.620022	+	0.784584i	$0.287124\pi$
$912$		0			0
$913$	1.00933e10	−	1.74821e10i	0.438921	−	0.760233i
$914$		0			0
$915$			7.76810e10i			3.35228i
$916$		0			0
$917$	−6.27085e8	+	3.62048e8i	−0.0268555	+	0.0155051i
$918$		0			0
$919$	−1.59245e10	−	2.75820e10i	−0.676802	−	1.17226i	−0.975939	−	0.218045i	$-0.930032\pi$
$919$	−1.59245e10	−	2.75820e10i	−0.676802	−	1.17226i	0.299137	−	0.954210i	$-0.403301\pi$
$920$		0			0
$921$	3.50403e10	+	2.02305e10i	1.47795	+	0.853294i
$922$		0			0
$923$	2.16202e10	+	6.33350e9i	0.905012	+	0.265117i
$924$		0			0
$925$	3.25197e10	+	1.87753e10i	1.35099	+	0.779992i
$926$		0			0
$927$	−2.80521e9	−	4.85877e9i	−0.115661	−	0.200330i
$928$		0			0
$929$	−2.32073e10	+	1.33988e10i	−0.949665	+	0.548289i	−0.892977	−	0.450102i	$-0.851387\pi$
$929$	−2.32073e10	+	1.33988e10i	−0.949665	+	0.548289i	−0.0566883	+	0.998392i	$0.518054\pi$
$930$		0			0
$931$		−	3.86118e10i		−	1.56818i
$932$		0			0
$933$	1.11602e10	−	1.93301e10i	0.449870	−	0.779197i
$934$		0			0
$935$	−1.45605e10			−0.582554
$936$		0			0
$937$	−6.16082e9			−0.244653			−0.122326	−	0.992490i	$-0.539035\pi$
$937$	−6.16082e9			−0.244653			−0.122326	+	0.992490i	$0.539035\pi$
$938$		0			0
$939$	−1.13998e10	+	1.97450e10i	−0.449331	+	0.778264i
$940$		0			0
$941$			7.68185e9i			0.300540i	0.988645	+	0.150270i	$0.0480143\pi$
$941$			7.68185e9i			0.300540i	−0.988645	+	0.150270i	$0.951986\pi$
$942$		0			0
$943$	1.61928e10	−	9.34894e9i	0.628829	−	0.363054i
$944$		0			0
$945$	1.62207e9	+	2.80950e9i	0.0625255	+	0.108297i
$946$		0			0
$947$	−2.89463e10	−	1.67121e10i	−1.10756	−	0.639451i	−0.169365	−	0.985553i	$-0.554172\pi$
$947$	−2.89463e10	−	1.67121e10i	−1.10756	−	0.639451i	−0.938197	+	0.346102i	$0.887505\pi$
$948$		0			0
$949$	2.26312e10	−	2.16060e10i	0.859559	−	0.820621i
$950$		0			0
$951$	2.96786e10	+	1.71350e10i	1.11895	+	0.646028i
$952$		0			0
$953$	−1.24555e10	−	2.15736e10i	−0.466162	−	0.807416i	0.533091	−	0.846058i	$-0.321030\pi$
$953$	−1.24555e10	−	2.15736e10i	−0.466162	−	0.807416i	−0.999253	+	0.0386419i	$0.987697\pi$
$954$		0			0
$955$	−1.29643e10	+	7.48496e9i	−0.481657	+	0.278085i
$956$		0			0
$957$			2.81101e10i			1.03674i
$958$		0			0
$959$	−1.06141e8	+	1.83842e8i	−0.00388615	+	0.00673101i
$960$		0			0
$961$	1.58786e10			0.577140
$962$		0			0
$963$	−1.52516e9			−0.0550331
$964$		0			0
$965$	−3.62288e10	+	6.27501e10i	−1.29780	+	2.24786i
$966$		0			0
$967$			7.88858e9i			0.280547i	0.990113	+	0.140274i	$0.0447982\pi$
$967$			7.88858e9i			0.280547i	−0.990113	+	0.140274i	$0.955202\pi$
$968$		0			0
$969$	−1.88460e10	+	1.08808e10i	−0.665406	+	0.384172i
$970$		0			0
$971$	1.61821e10	+	2.80282e10i	0.567241	+	0.982491i	0.996837	+	0.0794697i	$0.0253227\pi$
$971$	1.61821e10	+	2.80282e10i	0.567241	+	0.982491i	−0.429596	+	0.903021i	$0.641344\pi$
$972$		0			0
$973$	2.88859e9	+	1.66773e9i	0.100529	+	0.0580404i
$974$		0			0
$975$	5.49621e10	−	1.33703e10i	1.89910	−	0.461981i
$976$		0			0
$977$	−1.42381e10	−	8.22037e9i	−0.488451	−	0.282007i	0.235481	−	0.971879i	$-0.424334\pi$
$977$	−1.42381e10	−	8.22037e9i	−0.488451	−	0.282007i	−0.723932	+	0.689872i	$0.757667\pi$
$978$		0			0
$979$	2.51684e9	+	4.35930e9i	0.0857268	+	0.148483i
$980$		0			0
$981$	3.73002e9	−	2.15353e9i	0.126145	−	0.0728298i
$982$		0			0
$983$		−	5.12228e10i		−	1.71999i	−0.510301	−	0.859996i	$-0.670466\pi$
$983$		−	5.12228e10i		−	1.71999i	0.510301	−	0.859996i	$-0.329534\pi$
$984$		0			0
$985$	−4.03726e9	+	6.99274e9i	−0.134605	+	0.233142i
$986$		0			0
$987$	−1.04995e9			−0.0347582
$988$		0			0
$989$	2.59870e10			0.854217
$990$		0			0
$991$	1.27794e10	−	2.21346e10i	0.417113	−	0.722460i	−0.578535	−	0.815657i	$-0.696375\pi$
$991$	1.27794e10	−	2.21346e10i	0.417113	−	0.722460i	0.995648	+	0.0931974i	$0.0297088\pi$
$992$		0			0
$993$			5.10635e10i			1.65496i
$994$		0			0
$995$	−8.60172e10	+	4.96620e10i	−2.76824	+	1.59825i
$996$		0			0
$997$	−1.70676e10	−	2.95620e10i	−0.545432	−	0.944715i	−0.998580	−	0.0532799i	$-0.983032\pi$
$997$	−1.70676e10	−	2.95620e10i	−0.545432	−	0.944715i	0.453148	−	0.891435i	$-0.350301\pi$
$998$		0			0
$999$	2.12692e10	+	1.22798e10i	0.674950	+	0.389682i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.w.b.17.2		16
4.3	odd	2		26.8.e.a.17.4	✓	16
12.11	even	2		234.8.l.c.199.5		16
13.10	even	6	inner	208.8.w.b.49.2		16
52.7	even	12		338.8.a.m.1.2		8
52.19	even	12		338.8.a.n.1.2		8
52.23	odd	6		26.8.e.a.23.4	yes	16
52.35	odd	6		338.8.b.i.337.10		16
52.43	odd	6		338.8.b.i.337.2		16
156.23	even	6		234.8.l.c.127.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.e.a.17.4	✓	16	4.3	odd	2
26.8.e.a.23.4	yes	16	52.23	odd	6
208.8.w.b.17.2		16	1.1	even	1	trivial
208.8.w.b.49.2		16	13.10	even	6	inner
234.8.l.c.127.8		16	156.23	even	6
234.8.l.c.199.5		16	12.11	even	2
338.8.a.m.1.2		8	52.7	even	12
338.8.a.n.1.2		8	52.19	even	12
338.8.b.i.337.2		16	52.43	odd	6
338.8.b.i.337.10		16	52.35	odd	6

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-25.1127 + 43.4964i) q^{3} +469.361i q^{5} +(-64.3718 + 37.1651i) q^{7} +(-167.792 - 290.624i) q^{9} +(-2926.79 - 1689.78i) q^{11} +(5470.00 + 5729.54i) q^{13} +(-20415.5 - 11786.9i) q^{15} +(-4589.65 - 7949.50i) q^{17} +(-40877.8 + 23600.8i) q^{19} -3733.25i q^{21} +(21052.4 - 36463.9i) q^{23} -142175. q^{25} -92988.0 q^{27} +(-82803.7 + 143420. i) q^{29} -107861. i q^{31} +(146999. - 84869.8i) q^{33} +(-17443.8 - 30213.6i) q^{35} +(-228730. - 132058. i) q^{37} +(-386581. + 94041.1i) q^{39} +(384584. + 222040. i) q^{41} +(308598. + 534508. i) q^{43} +(136408. - 78754.9i) q^{45} -281242. i q^{47} +(-409009. + 708424. i) q^{49} +461033. q^{51} +1.57574e6 q^{53} +(793118. - 1.37372e6i) q^{55} -2.37072e6i q^{57} +(-1.46781e6 + 847438. i) q^{59} +(-1.64761e6 - 2.85375e6i) q^{61} +(21602.1 + 12472.0i) q^{63} +(-2.68922e6 + 2.56740e6i) q^{65} +(2.11527e6 + 1.22125e6i) q^{67} +(1.05737e6 + 1.83141e6i) q^{69} +(2.46302e6 - 1.42202e6i) q^{71} -3.94991e6i q^{73} +(3.57039e6 - 6.18410e6i) q^{75} +251203. q^{77} -3.64110e6 q^{79} +(2.70214e6 - 4.68024e6i) q^{81} +5.97315e6i q^{83} +(3.73119e6 - 2.15420e6i) q^{85} +(-4.15884e6 - 7.20333e6i) q^{87} +(-1.28990e6 - 744724. i) q^{89} +(-565052. - 165528. i) q^{91} +(4.69156e6 + 2.70868e6i) q^{93} +(-1.10773e7 - 1.91865e7i) q^{95} +(1.20731e7 - 6.97042e6i) q^{97} +1.13413e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2520 q^{7} - 4684 q^{9} + 8496 q^{11} - 3620 q^{13} - 51648 q^{15} + 41520 q^{17} + 54432 q^{19} + 7560 q^{23} - 273960 q^{25} - 133920 q^{27} - 346056 q^{29} + 486300 q^{33} + 283248 q^{35} - 68280 q^{37}+ \cdots + 83706300 q^{97}+O(q^{100}) \) 16 * q - 2520 * q^7 - 4684 * q^9 + 8496 * q^11 - 3620 * q^13 - 51648 * q^15 + 41520 * q^17 + 54432 * q^19 + 7560 * q^23 - 273960 * q^25 - 133920 * q^27 - 346056 * q^29 + 486300 * q^33 + 283248 * q^35 - 68280 * q^37 + 1122680 * q^39 - 1288008 * q^41 - 285360 * q^43 + 5743980 * q^45 - 2131660 * q^49 - 957280 * q^51 + 3666600 * q^53 + 1091472 * q^55 - 1799280 * q^59 - 4626000 * q^61 + 8609760 * q^63 - 11518020 * q^65 + 7094400 * q^67 + 18027316 * q^69 - 25958520 * q^71 + 10898032 * q^75 - 20577480 * q^77 - 34143488 * q^79 - 15745864 * q^81 + 7123932 * q^85 + 10460440 * q^87 + 10609452 * q^89 + 38683952 * q^91 - 52747080 * q^93 - 33837504 * q^95 + 83706300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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