LMFDB - Embedded newform 208.8.w.a.17.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(17,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 $ x^14 + 1279x^12 + 629380x^10 + 148562016x^8 + 16872573312x^6 + 790180980480x^4 + 10484997239808x^2 + 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.7
Root			$-0.679146i$ of defining polynomial
Character	$\chi$	$=$	208.17
Dual form			208.8.w.a.49.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(31.2267 - 54.0862i) q^{3} -439.155i q^{5} +(1128.14 - 651.329i) q^{7} +(-856.712 - 1483.87i) q^{9} +(-497.807 - 287.409i) q^{11} +(-7892.42 + 676.938i) q^{13} +(-23752.2 - 13713.4i) q^{15} +(-5634.27 - 9758.85i) q^{17} +(-35549.9 + 20524.7i) q^{19} -81355.4i q^{21} +(22132.8 - 38335.2i) q^{23} -114732. q^{25} +29576.4 q^{27} +(72756.1 - 126017. i) q^{29} +122045. i q^{31} +(-31089.7 + 17949.6i) q^{33} +(-286035. - 495427. i) q^{35} +(36374.8 + 21001.0i) q^{37} +(-209841. + 448010. i) q^{39} +(-75942.3 - 43845.3i) q^{41} +(374876. + 649305. i) q^{43} +(-651649. + 376230. i) q^{45} -940126. i q^{47} +(436688. - 756366. i) q^{49} -703759. q^{51} -924214. q^{53} +(-126217. + 218614. i) q^{55} +2.56368e6i q^{57} +(-547626. + 316172. i) q^{59} +(32203.2 + 55777.7i) q^{61} +(-1.93297e6 - 1.11600e6i) q^{63} +(297281. + 3.46600e6i) q^{65} +(1.68358e6 + 972013. i) q^{67} +(-1.38227e6 - 2.39416e6i) q^{69} +(4.09113e6 - 2.36202e6i) q^{71} -1.92731e6i q^{73} +(-3.58271e6 + 6.20544e6i) q^{75} -748791. q^{77} +1.50287e6 q^{79} +(2.79720e6 - 4.84490e6i) q^{81} -1.87974e6i q^{83} +(-4.28565e6 + 2.47432e6i) q^{85} +(-4.54386e6 - 7.87020e6i) q^{87} +(3.78860e6 + 2.18735e6i) q^{89} +(-8.46281e6 + 5.90424e6i) q^{91} +(6.60097e6 + 3.81107e6i) q^{93} +(9.01354e6 + 1.56119e7i) q^{95} +(-1.53602e6 + 886819. i) q^{97} +984906. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 26 q^{3} + 2772 q^{7} - 3491 q^{9} - 6516 q^{11} + 5109 q^{13} + 19266 q^{15} - 38403 q^{17} - 43254 q^{19} + 68550 q^{23} + 39380 q^{25} + 432400 q^{27} + 221583 q^{29} + 219756 q^{33} - 659616 q^{35}+ \cdots - 68556288 q^{97}+O(q^{100}) $ 14 * q - 26 * q^3 + 2772 * q^7 - 3491 * q^9 - 6516 * q^11 + 5109 * q^13 + 19266 * q^15 - 38403 * q^17 - 43254 * q^19 + 68550 * q^23 + 39380 * q^25 + 432400 * q^27 + 221583 * q^29 + 219756 * q^33 - 659616 * q^35 - 565347 * q^37 - 30082 * q^39 + 1716057 * q^41 + 882692 * q^43 - 4616853 * q^45 + 2653979 * q^49 + 38964 * q^51 - 1791210 * q^53 - 2453536 * q^55 + 1189032 * q^59 + 3858525 * q^61 - 7718520 * q^63 + 10424661 * q^65 - 4674330 * q^67 - 13743840 * q^69 + 25203126 * q^71 - 10316152 * q^75 + 19599336 * q^77 + 30688440 * q^79 + 13886221 * q^81 - 22436313 * q^85 - 12769686 * q^87 + 2123160 * q^89 - 37205532 * q^91 + 36638964 * q^93 + 43420386 * q^95 - 68556288 * 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
$3$	31.2267	−	54.0862i	0.667731	−	1.15654i	−0.310806	−	0.950473i	$-0.600599\pi$
$3$	31.2267	−	54.0862i	0.667731	−	1.15654i	0.978537	−	0.206071i	$-0.0660677\pi$
$4$		0			0
$5$		−	439.155i		−	1.57117i	−0.618754	−	0.785585i	$-0.712362\pi$
$5$		−	439.155i		−	1.57117i	0.618754	−	0.785585i	$-0.287638\pi$
$6$		0			0
$7$	1128.14	−	651.329i	1.24313	−	0.717724i	0.273403	−	0.961899i	$-0.411851\pi$
$7$	1128.14	−	651.329i	1.24313	−	0.717724i	0.969731	+	0.244175i	$0.0785173\pi$
$8$		0			0
$9$	−856.712	−	1483.87i	−0.391729	−	0.678495i
$10$		0			0
$11$	−497.807	−	287.409i	−0.112768	−	0.0651067i	0.442555	−	0.896741i	$-0.354072\pi$
$11$	−497.807	−	287.409i	−0.112768	−	0.0651067i	−0.555323	+	0.831635i	$0.687405\pi$
$12$		0			0
$13$	−7892.42	+	676.938i	−0.996342	+	0.0854568i
$13$	−7892.42	+	676.938i	−0.996342	+	0.0854568i
$14$		0			0
$15$	−23752.2	−	13713.4i	−1.81713	−	1.04912i
$16$		0			0
$17$	−5634.27	−	9758.85i	−0.278142	−	0.481756i	0.692781	−	0.721148i	$-0.256385\pi$
$17$	−5634.27	−	9758.85i	−0.278142	−	0.481756i	−0.970923	+	0.239392i	$0.923052\pi$
$18$		0			0
$19$	−35549.9	+	20524.7i	−1.18905	+	0.686499i	−0.958091	−	0.286464i	$-0.907520\pi$
$19$	−35549.9	+	20524.7i	−1.18905	+	0.686499i	−0.230960	+	0.972963i	$0.574187\pi$
$20$		0			0
$21$		−	81355.4i		−	1.91699i
$22$		0			0
$23$	22132.8	−	38335.2i	0.379306	−	0.656977i	−0.611655	−	0.791124i	$-0.709496\pi$
$23$	22132.8	−	38335.2i	0.379306	−	0.656977i	0.990961	+	0.134147i	$0.0428294\pi$
$24$		0			0
$25$	−114732.			−1.46857
$26$		0			0
$27$	29576.4			0.289183
$28$		0			0
$29$	72756.1	−	126017.i	0.553957	−	0.959482i	−0.444027	−	0.896014i	$-0.646450\pi$
$29$	72756.1	−	126017.i	0.553957	−	0.959482i	0.997984	−	0.0634685i	$-0.0202162\pi$
$30$		0			0
$31$			122045.i			0.735793i	0.929867	+	0.367896i	$0.119922\pi$
$31$			122045.i			0.735793i	−0.929867	+	0.367896i	$0.880078\pi$
$32$		0			0
$33$	−31089.7	+	17949.6i	−0.150597	+	0.0869475i
$34$		0			0
$35$	−286035.	−	495427.i	−1.12767	−	1.95318i
$36$		0			0
$37$	36374.8	+	21001.0i	0.118058	+	0.0681608i	0.557866	−	0.829931i	$-0.311620\pi$
$37$	36374.8	+	21001.0i	0.118058	+	0.0681608i	−0.439808	+	0.898092i	$0.644954\pi$
$38$		0			0
$39$	−209841.	+	448010.i	−0.566454	+	1.20938i
$40$		0			0
$41$	−75942.3	−	43845.3i	−0.172084	−	0.0993526i	0.411484	−	0.911417i	$-0.365011\pi$
$41$	−75942.3	−	43845.3i	−0.172084	−	0.0993526i	−0.583568	+	0.812064i	$0.698344\pi$
$42$		0			0
$43$	374876.	+	649305.i	0.719032	+	1.24540i	0.961384	+	0.275212i	$0.0887479\pi$
$43$	374876.	+	649305.i	0.719032	+	1.24540i	−0.242352	+	0.970188i	$0.577919\pi$
$44$		0			0
$45$	−651649.	+	376230.i	−1.06603	+	0.615473i
$46$		0			0
$47$		−	940126.i		−	1.32082i	−0.750905	−	0.660410i	$-0.770383\pi$
$47$		−	940126.i		−	1.32082i	0.750905	−	0.660410i	$-0.229617\pi$
$48$		0			0
$49$	436688.	−	756366.i	0.530255	−	0.918429i
$50$		0			0
$51$	−703759.			−0.742896
$52$		0			0
$53$	−924214.			−0.852721			−0.426360	−	0.904553i	$-0.640204\pi$
$53$	−924214.			−0.852721			−0.426360	+	0.904553i	$0.640204\pi$
$54$		0			0
$55$	−126217.	+	218614.i	−0.102294	+	0.177178i
$56$		0			0
$57$			2.56368e6i			1.83359i
$58$		0			0
$59$	−547626.	+	316172.i	−0.347138	+	0.200420i	−0.663424	−	0.748244i	$-0.730897\pi$
$59$	−547626.	+	316172.i	−0.347138	+	0.200420i	0.316286	+	0.948664i	$0.397564\pi$
$60$		0			0
$61$	32203.2	+	55777.7i	0.0181654	+	0.0314634i	0.874965	−	0.484186i	$-0.160884\pi$
$61$	32203.2	+	55777.7i	0.0181654	+	0.0314634i	−0.856800	+	0.515649i	$0.827551\pi$
$62$		0			0
$63$	−1.93297e6	−	1.11600e6i	−0.973944	−	0.562307i
$64$		0			0
$65$	297281.	+	3.46600e6i	0.134267	+	1.56542i
$66$		0			0
$67$	1.68358e6	+	972013.i	0.683866	+	0.394830i	0.801310	−	0.598249i	$-0.204137\pi$
$67$	1.68358e6	+	972013.i	0.683866	+	0.394830i	−0.117444	+	0.993079i	$0.537470\pi$
$68$		0			0
$69$	−1.38227e6	−	2.39416e6i	−0.506549	−	0.877368i
$70$		0			0
$71$	4.09113e6	−	2.36202e6i	1.35656	−	0.783211i	0.367402	−	0.930062i	$-0.380247\pi$
$71$	4.09113e6	−	2.36202e6i	1.35656	−	0.783211i	0.989159	+	0.146851i	$0.0469139\pi$
$72$		0			0
$73$		−	1.92731e6i		−	0.579857i	−0.957048	−	0.289928i	$-0.906368\pi$
$73$		−	1.92731e6i		−	0.579857i	0.957048	−	0.289928i	$-0.0936315\pi$
$74$		0			0
$75$	−3.58271e6	+	6.20544e6i	−0.980612	+	1.69847i
$76$		0			0
$77$	−748791.			−0.186915
$78$		0			0
$79$	1.50287e6			0.342947			0.171474	−	0.985189i	$-0.445147\pi$
$79$	1.50287e6			0.342947			0.171474	+	0.985189i	$0.445147\pi$
$80$		0			0
$81$	2.79720e6	−	4.84490e6i	0.584826	−	1.01295i
$82$		0			0
$83$		−	1.87974e6i		−	0.360848i	−0.983589	−	0.180424i	$-0.942253\pi$
$83$		−	1.87974e6i		−	0.360848i	0.983589	−	0.180424i	$-0.0577470\pi$
$84$		0			0
$85$	−4.28565e6	+	2.47432e6i	−0.756921	+	0.437008i
$86$		0			0
$87$	−4.54386e6	−	7.87020e6i	−0.739789	−	1.28135i
$88$		0			0
$89$	3.78860e6	+	2.18735e6i	0.569657	+	0.328892i	0.757012	−	0.653400i	$-0.226658\pi$
$89$	3.78860e6	+	2.18735e6i	0.569657	+	0.328892i	−0.187355	+	0.982292i	$0.559992\pi$
$90$		0			0
$91$	−8.46281e6	+	5.90424e6i	−1.17725	+	0.821333i
$92$		0			0
$93$	6.60097e6	+	3.81107e6i	0.850977	+	0.491312i
$94$		0			0
$95$	9.01354e6	+	1.56119e7i	1.07861	+	1.86820i
$96$		0			0
$97$	−1.53602e6	+	886819.i	−0.170881	+	0.0986583i	−0.583001	−	0.812471i	$-0.698122\pi$
$97$	−1.53602e6	+	886819.i	−0.170881	+	0.0986583i	0.412120	+	0.911130i	$0.364788\pi$
$98$		0			0
$99$			984906.i			0.102017i
$100$		0			0
$101$	−4.81973e6	+	8.34802e6i	−0.465477	+	0.806229i	−0.999223	−	0.0394154i	$-0.987450\pi$
$101$	−4.81973e6	+	8.34802e6i	−0.465477	+	0.806229i	0.533746	+	0.845645i	$0.320784\pi$
$102$		0			0
$103$	−7.14668e6			−0.644427			−0.322214	−	0.946667i	$-0.604427\pi$
$103$	−7.14668e6			−0.644427			−0.322214	+	0.946667i	$0.604427\pi$
$104$		0			0
$105$	−3.57277e7			−3.01191
$106$		0			0
$107$	−1.03003e7	+	1.78406e7i	−0.812839	+	1.40788i	0.0980299	+	0.995183i	$0.468746\pi$
$107$	−1.03003e7	+	1.78406e7i	−0.812839	+	1.40788i	−0.910869	+	0.412695i	$0.864587\pi$
$108$		0			0
$109$			1.05713e7i			0.781872i	0.920418	+	0.390936i	$0.127849\pi$
$109$			1.05713e7i			0.781872i	−0.920418	+	0.390936i	$0.872151\pi$
$110$		0			0
$111$	2.27173e6	−	1.31158e6i	0.157662	−	0.0910261i
$112$		0			0
$113$	−1.34036e7	−	2.32157e7i	−0.873868	−	1.51358i	−0.857964	−	0.513711i	$-0.828270\pi$
$113$	−1.34036e7	−	2.32157e7i	−0.873868	−	1.51358i	−0.0159047	−	0.999874i	$-0.505063\pi$
$114$		0			0
$115$	−1.68351e7	−	9.71976e6i	−1.03222	−	0.595954i
$116$		0			0
$117$	7.76602e6	+	1.11314e7i	0.448278	+	0.642537i
$118$		0			0
$119$	−1.27124e7	−	7.33953e6i	−0.691536	−	0.399258i
$120$		0			0
$121$	−9.57838e6	−	1.65902e7i	−0.491522	−	0.851341i
$122$		0			0
$123$	−4.74285e6	+	2.73829e6i	−0.229811	+	0.132682i
$124$		0			0
$125$			1.60763e7i			0.736209i
$126$		0			0
$127$	4.03532e6	−	6.98938e6i	0.174810	−	0.302779i	−0.765286	−	0.643691i	$-0.777402\pi$
$127$	4.03532e6	−	6.98938e6i	0.174810	−	0.302779i	0.940095	+	0.340912i	$0.110736\pi$
$128$		0			0
$129$	4.68246e7			1.92048
$130$		0			0
$131$	−1.26390e7			−0.491206			−0.245603	−	0.969370i	$-0.578986\pi$
$131$	−1.26390e7			−0.491206			−0.245603	+	0.969370i	$0.578986\pi$
$132$		0			0
$133$	−2.67367e7	+	4.63094e7i	−0.985434	+	1.70682i
$134$		0			0
$135$		−	1.29886e7i		−	0.454355i
$136$		0			0
$137$	2.45282e7	−	1.41614e7i	0.814975	−	0.470526i	−0.0337058	−	0.999432i	$-0.510731\pi$
$137$	2.45282e7	−	1.41614e7i	0.814975	−	0.470526i	0.848680	+	0.528906i	$0.177398\pi$
$138$		0			0
$139$	−4.44074e6	−	7.69159e6i	−0.140250	−	0.242920i	0.787341	−	0.616518i	$-0.211457\pi$
$139$	−4.44074e6	−	7.69159e6i	−0.140250	−	0.242920i	−0.927591	+	0.373598i	$0.878124\pi$
$140$		0			0
$141$	−5.08478e7	−	2.93570e7i	−1.52759	−	0.881952i
$142$		0			0
$143$	4.12346e6	+	1.93137e6i	0.117919	+	0.0552317i
$144$		0			0
$145$	−5.53411e7	−	3.19512e7i	−1.50751	−	0.870361i
$146$		0			0
$147$	−2.72726e7	−	4.72376e7i	−0.708136	−	1.22653i
$148$		0			0
$149$	−4.40391e7	+	2.54260e7i	−1.09065	+	0.629689i	−0.933750	−	0.357926i	$-0.883484\pi$
$149$	−4.40391e7	+	2.54260e7i	−1.09065	+	0.629689i	−0.156902	+	0.987614i	$0.550151\pi$
$150$		0			0
$151$		−	9.76974e6i		−	0.230921i	−0.993312	−	0.115461i	$-0.963166\pi$
$151$		−	9.76974e6i		−	0.230921i	0.993312	−	0.115461i	$-0.0368344\pi$
$152$		0			0
$153$	−9.65390e6	+	1.67210e7i	−0.217913	+	0.377436i
$154$		0			0
$155$	5.35969e7			1.15606
$156$		0			0
$157$	6.53474e7			1.34766			0.673828	−	0.738888i	$-0.264649\pi$
$157$	6.53474e7			1.34766			0.673828	+	0.738888i	$0.264649\pi$
$158$		0			0
$159$	−2.88601e7	+	4.99872e7i	−0.569388	+	0.986209i
$160$		0			0
$161$		−	5.76631e7i		−	1.08895i
$162$		0			0
$163$	−1.74920e7	+	1.00990e7i	−0.316361	+	0.182651i	−0.649769	−	0.760132i	$-0.725134\pi$
$163$	−1.74920e7	+	1.00990e7i	−0.316361	+	0.182651i	0.333409	+	0.942782i	$0.391801\pi$
$164$		0			0
$165$	7.88268e6	+	1.36532e7i	0.136609	+	0.236614i
$166$		0			0
$167$	5.86371e6	+	3.38542e6i	0.0974238	+	0.0562477i	0.547920	−	0.836531i	$-0.315420\pi$
$167$	5.86371e6	+	3.38542e6i	0.0974238	+	0.0562477i	−0.450497	+	0.892778i	$0.648753\pi$
$168$		0			0
$169$	6.18320e7	−	1.06853e7i	0.985394	−	0.170288i
$170$		0			0
$171$	6.09120e7	+	3.51676e7i	0.931572	+	0.537843i
$172$		0			0
$173$	4.33522e7	+	7.50883e7i	0.636576	+	1.10258i	0.986179	+	0.165684i	$0.0529831\pi$
$173$	4.33522e7	+	7.50883e7i	0.636576	+	1.10258i	−0.349603	+	0.936898i	$0.613684\pi$
$174$		0			0
$175$	−1.29434e8	+	7.47285e7i	−1.82563	+	1.05403i
$176$		0			0
$177$			3.94920e7i			0.535307i
$178$		0			0
$179$	4.47221e7	−	7.74609e7i	0.582823	−	1.00948i	−0.412320	−	0.911039i	$-0.635281\pi$
$179$	4.47221e7	−	7.74609e7i	0.582823	−	1.00948i	0.995143	−	0.0984398i	$-0.0313852\pi$
$180$		0			0
$181$	−2.55748e7			−0.320581			−0.160290	−	0.987070i	$-0.551243\pi$
$181$	−2.55748e7			−0.320581			−0.160290	+	0.987070i	$0.551243\pi$
$182$		0			0
$183$	4.02240e6			0.0485184
$184$		0			0
$185$	9.22271e6	−	1.59742e7i	0.107092	−	0.185489i
$186$		0			0
$187$			6.47736e6i			0.0724356i
$188$		0			0
$189$	3.33662e7	−	1.92640e7i	0.359493	−	0.207553i
$190$		0			0
$191$	7.26412e7	+	1.25818e8i	0.754339	+	1.30655i	0.945702	+	0.325034i	$0.105376\pi$
$191$	7.26412e7	+	1.25818e8i	0.754339	+	1.30655i	−0.191364	+	0.981519i	$0.561291\pi$
$192$		0			0
$193$	−7.90017e7	−	4.56117e7i	−0.791017	−	0.456694i	0.0493033	−	0.998784i	$-0.484300\pi$
$193$	−7.90017e7	−	4.56117e7i	−0.791017	−	0.456694i	−0.840321	+	0.542090i	$0.817633\pi$
$194$		0			0
$195$	1.96746e8	+	9.21528e7i	1.90013	+	0.889995i
$196$		0			0
$197$	1.47447e8	+	8.51285e7i	1.37405	+	0.793311i	0.991436	−	0.130595i	$-0.0416889\pi$
$197$	1.47447e8	+	8.51285e7i	1.37405	+	0.793311i	0.382619	+	0.923906i	$0.375022\pi$
$198$		0			0
$199$	6.97613e7	+	1.20830e8i	0.627521	+	1.08690i	0.988047	+	0.154150i	$0.0492639\pi$
$199$	6.97613e7	+	1.20830e8i	0.627521	+	1.08690i	−0.360526	+	0.932749i	$0.617403\pi$
$200$		0			0
$201$	1.05145e8	−	6.07055e7i	0.913277	−	0.527281i
$202$		0			0
$203$		−	1.89553e8i		−	1.59035i
$204$		0			0
$205$	−1.92549e7	+	3.33504e7i	−0.156100	+	0.270373i
$206$		0			0
$207$	−7.58459e7			−0.594341
$208$		0			0
$209$	2.35960e7			0.178783
$210$		0			0
$211$	4.18539e7	−	7.24931e7i	0.306724	−	0.531261i	−0.670920	−	0.741530i	$-0.734101\pi$
$211$	4.18539e7	−	7.24931e7i	0.306724	−	0.531261i	0.977644	+	0.210269i	$0.0674338\pi$
$212$		0			0
$213$		−	2.95032e8i		−	2.09190i
$214$		0			0
$215$	2.85146e8	−	1.64629e8i	1.95673	−	1.12972i
$216$		0			0
$217$	7.94917e7	+	1.37684e8i	0.528096	+	0.914689i
$218$		0			0
$219$	−1.04241e8	−	6.01834e7i	−0.670630	−	0.387188i
$220$		0			0
$221$	5.10742e7	+	7.32069e7i	0.318294	+	0.456225i
$222$		0			0
$223$	8.23600e7	+	4.75506e7i	0.497335	+	0.287137i	0.727612	−	0.685988i	$-0.240630\pi$
$223$	8.23600e7	+	4.75506e7i	0.497335	+	0.287137i	−0.230277	+	0.973125i	$0.573963\pi$
$224$		0			0
$225$	9.82926e7	+	1.70248e8i	0.575283	+	0.996420i
$226$		0			0
$227$	−8.55988e7	+	4.94205e7i	−0.485710	+	0.280425i	−0.722793	−	0.691064i	$-0.757142\pi$
$227$	−8.55988e7	+	4.94205e7i	−0.485710	+	0.280425i	0.237083	+	0.971489i	$0.423809\pi$
$228$		0			0
$229$		−	8.72426e7i		−	0.480070i	−0.970764	−	0.240035i	$-0.922841\pi$
$229$		−	8.72426e7i		−	0.480070i	0.970764	−	0.240035i	$-0.0771589\pi$
$230$		0			0
$231$	−2.33823e7	+	4.04993e7i	−0.124809	+	0.216175i
$232$		0			0
$233$	−2.42490e8			−1.25588			−0.627940	−	0.778262i	$-0.716102\pi$
$233$	−2.42490e8			−1.25588			−0.627940	+	0.778262i	$0.716102\pi$
$234$		0			0
$235$	−4.12861e8			−2.07523
$236$		0			0
$237$	4.69298e7	−	8.12847e7i	0.228997	−	0.396634i
$238$		0			0
$239$		−	4.08512e8i		−	1.93558i	−0.251750	−	0.967792i	$-0.581006\pi$
$239$		−	4.08512e8i		−	1.93558i	0.251750	−	0.967792i	$-0.418994\pi$
$240$		0			0
$241$	1.89875e8	−	1.09624e8i	0.873792	−	0.504484i	0.00518518	−	0.999987i	$-0.498349\pi$
$241$	1.89875e8	−	1.09624e8i	0.873792	−	0.504484i	0.868606	+	0.495503i	$0.165016\pi$
$242$		0			0
$243$	−1.42353e8	−	2.46563e8i	−0.636421	−	1.10231i
$244$		0			0
$245$	−3.32162e8	−	1.91774e8i	−1.44301	−	0.833121i
$246$		0			0
$247$	2.66681e8	−	1.86055e8i	1.12604	−	0.785600i
$248$		0			0
$249$	−1.01668e8	−	5.86980e7i	−0.417337	−	0.240949i
$250$		0			0
$251$	−7.42822e6	−	1.28661e7i	−0.0296501	−	0.0513556i	0.850820	−	0.525458i	$-0.176106\pi$
$251$	−7.42822e6	−	1.28661e7i	−0.0296501	−	0.0513556i	−0.880470	+	0.474102i	$0.842773\pi$
$252$		0			0
$253$	−2.20358e7	+	1.27223e7i	−0.0855473	+	0.0493907i
$254$		0			0
$255$			3.09059e8i			1.16722i
$256$		0			0
$257$	2.49812e8	−	4.32688e8i	0.918011	−	1.59004i	0.115579	−	0.993298i	$-0.463128\pi$
$257$	2.49812e8	−	4.32688e8i	0.918011	−	1.59004i	0.802432	−	0.596744i	$-0.203539\pi$
$258$		0			0
$259$	5.47143e7			0.195682
$260$		0			0
$261$	−2.49324e8			−0.868005
$262$		0			0
$263$	1.51725e7	−	2.62796e7i	0.0514295	−	0.0890785i	−0.839165	−	0.543878i	$-0.816956\pi$
$263$	1.51725e7	−	2.62796e7i	0.0514295	−	0.0890785i	0.890594	+	0.454799i	$0.150289\pi$
$264$		0			0
$265$			4.05873e8i			1.33977i
$266$		0			0
$267$	2.36611e8	−	1.36607e8i	0.760756	−	0.439222i
$268$		0			0
$269$	−5.52630e7	−	9.57183e7i	−0.173102	−	0.299821i	0.766401	−	0.642363i	$-0.222046\pi$
$269$	−5.52630e7	−	9.57183e7i	−0.173102	−	0.299821i	−0.939503	+	0.342541i	$0.888712\pi$
$270$		0			0
$271$	−4.36530e7	−	2.52031e7i	−0.133236	−	0.0769238i	0.431901	−	0.901921i	$-0.357843\pi$
$271$	−4.36530e7	−	2.52031e7i	−0.133236	−	0.0769238i	−0.565136	+	0.824997i	$0.691176\pi$
$272$		0			0
$273$	5.50725e7	+	6.42091e8i	0.163820	+	1.90997i
$274$		0			0
$275$	5.71145e7	+	3.29751e7i	0.165608	+	0.0956140i
$276$		0			0
$277$	−2.69297e8	−	4.66436e8i	−0.761294	−	1.31860i	−0.942184	−	0.335097i	$-0.891231\pi$
$277$	−2.69297e8	−	4.66436e8i	−0.761294	−	1.31860i	0.180889	−	0.983503i	$-0.442102\pi$
$278$		0			0
$279$	1.81099e8	−	1.04558e8i	0.499232	−	0.288232i
$280$		0			0
$281$		−	1.43482e8i		−	0.385766i	−0.981222	−	0.192883i	$-0.938216\pi$
$281$		−	1.43482e8i		−	0.385766i	0.981222	−	0.192883i	$-0.0617839\pi$
$282$		0			0
$283$	−5.06236e7	+	8.76826e7i	−0.132770	+	0.229964i	−0.924743	−	0.380591i	$-0.875721\pi$
$283$	−5.06236e7	+	8.76826e7i	−0.132770	+	0.229964i	0.791973	+	0.610556i	$0.209054\pi$
$284$		0			0
$285$	1.12585e9			2.88088
$286$		0			0
$287$	−1.14231e8			−0.285231
$288$		0			0
$289$	1.41679e8	−	2.45396e8i	0.345274	−	0.598032i
$290$		0			0
$291$			1.10770e8i			0.263509i
$292$		0			0
$293$	−2.65267e8	+	1.53152e8i	−0.616094	+	0.355702i	−0.775347	−	0.631536i	$-0.782425\pi$
$293$	−2.65267e8	+	1.53152e8i	−0.616094	+	0.355702i	0.159252	+	0.987238i	$0.449092\pi$
$294$		0			0
$295$	1.38849e8	+	2.40493e8i	0.314894	+	0.545413i
$296$		0			0
$297$	−1.47233e7	−	8.50052e6i	−0.0326106	−	0.0188277i
$298$		0			0
$299$	−1.48731e8	+	3.17540e8i	−0.321775	+	0.686988i
$300$		0			0
$301$	8.45822e8	+	4.88336e8i	1.78771	+	1.03213i
$302$		0			0
$303$	3.01008e8	+	5.21362e8i	0.621626	+	1.07669i
$304$		0			0
$305$	2.44951e7	−	1.41422e7i	0.0494344	−	0.0285409i
$306$		0			0
$307$			8.86416e6i			0.0174845i	0.999962	+	0.00874225i	$0.00278278\pi$
$307$			8.86416e6i			0.0174845i	−0.999962	+	0.00874225i	$0.997217\pi$
$308$		0			0
$309$	−2.23167e8	+	3.86537e8i	−0.430304	+	0.745308i
$310$		0			0
$311$	3.95364e8			0.745309			0.372654	−	0.927970i	$-0.378448\pi$
$311$	3.95364e8			0.745309			0.372654	+	0.927970i	$0.378448\pi$
$312$		0			0
$313$	−8.88408e8			−1.63760			−0.818799	−	0.574080i	$-0.805360\pi$
$313$	−8.88408e8			−1.63760			−0.818799	+	0.574080i	$0.805360\pi$
$314$		0			0
$315$	−4.90099e8	+	8.48876e8i	−0.883480	+	1.53023i
$316$		0			0
$317$		−	3.40256e8i		−	0.599928i	−0.953951	−	0.299964i	$-0.903025\pi$
$317$		−	3.40256e8i		−	0.599928i	0.953951	−	0.299964i	$-0.0969746\pi$
$318$		0			0
$319$	−7.24369e7	+	4.18215e7i	−0.124937	+	0.0721326i
$320$		0			0
$321$	6.43286e8	+	1.11420e9i	1.08552	+	1.88017i
$322$		0			0
$323$	4.00595e8	+	2.31284e8i	0.661450	+	0.381888i
$324$		0			0
$325$	9.05516e8	−	7.76666e7i	1.46320	−	0.125500i
$326$		0			0
$327$	5.71762e8	+	3.30107e8i	0.904270	+	0.522080i
$328$		0			0
$329$	−6.12331e8	−	1.06059e9i	−0.947984	−	1.64196i
$330$		0			0
$331$	4.75802e8	−	2.74704e8i	0.721154	−	0.416359i	−0.0940230	−	0.995570i	$-0.529973\pi$
$331$	4.75802e8	−	2.74704e8i	0.721154	−	0.416359i	0.815177	+	0.579211i	$0.196639\pi$
$332$		0			0
$333$		−	7.19673e7i		−	0.106802i
$334$		0			0
$335$	4.26865e8	−	7.39351e8i	0.620345	−	1.07447i
$336$		0			0
$337$	−6.88538e8			−0.979993			−0.489997	−	0.871724i	$-0.663002\pi$
$337$	−6.88538e8			−0.979993			−0.489997	+	0.871724i	$0.663002\pi$
$338$		0			0
$339$	−1.67420e9			−2.33404
$340$		0			0
$341$	3.50769e7	−	6.07550e7i	0.0479050	−	0.0829739i
$342$		0			0
$343$		−	6.49158e7i		−	0.0868602i
$344$		0			0
$345$	−1.05141e9	+	6.07032e8i	−1.37849	+	0.795874i
$346$		0			0
$347$	−3.43577e8	−	5.95094e8i	−0.441440	−	0.764596i	0.556357	−	0.830943i	$-0.312199\pi$
$347$	−3.43577e8	−	5.95094e8i	−0.441440	−	0.764596i	−0.997797	+	0.0663474i	$0.978865\pi$
$348$		0			0
$349$	−8.95711e8	−	5.17139e8i	−1.12792	−	0.651206i	−0.184510	−	0.982831i	$-0.559070\pi$
$349$	−8.95711e8	−	5.17139e8i	−1.12792	−	0.651206i	−0.943411	+	0.331625i	$0.892403\pi$
$350$		0			0
$351$	−2.33430e8	+	2.00214e7i	−0.288125	+	0.0247127i
$352$		0			0
$353$	9.29168e8	+	5.36455e8i	1.12430	+	0.649116i	0.942496	−	0.334219i	$-0.108472\pi$
$353$	9.29168e8	+	5.36455e8i	1.12430	+	0.649116i	0.181806	+	0.983334i	$0.441806\pi$
$354$		0			0
$355$	−1.03729e9	−	1.79664e9i	−1.23056	−	2.13139i
$356$		0			0
$357$	−7.93935e8	+	4.58379e8i	−0.923520	+	0.533194i
$358$		0			0
$359$		−	6.57582e8i		−	0.750100i	−0.927005	−	0.375050i	$-0.877626\pi$
$359$		−	6.57582e8i		−	0.750100i	0.927005	−	0.375050i	$-0.122374\pi$
$360$		0			0
$361$	3.95593e8	−	6.85188e8i	0.442562	−	0.766539i
$362$		0			0
$363$	−1.19640e9			−1.31282
$364$		0			0
$365$	−8.46387e8			−0.911054
$366$		0			0
$367$	−4.47343e8	+	7.74821e8i	−0.472400	+	0.818220i	−0.999501	−	0.0315822i	$-0.989945\pi$
$367$	−4.47343e8	+	7.74821e8i	−0.472400	+	0.818220i	0.527102	+	0.849802i	$0.323279\pi$
$368$		0			0
$369$			1.50251e8i			0.155677i
$370$		0			0
$371$	−1.04264e9	+	6.01967e8i	−1.06005	+	0.612018i
$372$		0			0
$373$	−3.42088e8	−	5.92515e8i	−0.341317	−	0.591178i	0.643361	−	0.765563i	$-0.277540\pi$
$373$	−3.42088e8	−	5.92515e8i	−0.341317	−	0.591178i	−0.984677	+	0.174385i	$0.944206\pi$
$374$		0			0
$375$	8.69507e8	+	5.02010e8i	0.851458	+	0.491590i
$376$		0			0
$377$	−4.88916e8	+	1.04383e9i	−0.469937	+	1.00331i
$378$		0			0
$379$	9.87024e8	+	5.69859e8i	0.931302	+	0.537687i	0.887223	−	0.461341i	$-0.152631\pi$
$379$	9.87024e8	+	5.69859e8i	0.931302	+	0.537687i	0.0440788	+	0.999028i	$0.485965\pi$
$380$		0			0
$381$	−2.52019e8	−	4.36510e8i	−0.233451	−	0.404350i
$382$		0			0
$383$	−1.16426e9	+	6.72186e8i	−1.05890	+	0.611355i	−0.925127	−	0.379657i	$-0.876042\pi$
$383$	−1.16426e9	+	6.72186e8i	−1.05890	+	0.611355i	−0.133771	+	0.991012i	$0.542709\pi$
$384$		0			0
$385$			3.28835e8i			0.293674i
$386$		0			0
$387$	6.42322e8	−	1.11253e9i	0.563332	−	0.975719i
$388$		0			0
$389$	4.06758e8			0.350359			0.175179	−	0.984537i	$-0.443949\pi$
$389$	4.06758e8			0.350359			0.175179	+	0.984537i	$0.443949\pi$
$390$		0			0
$391$	−4.98810e8			−0.422004
$392$		0			0
$393$	−3.94675e8	+	6.83597e8i	−0.327994	+	0.568102i
$394$		0			0
$395$		−	6.59995e8i		−	0.538829i
$396$		0			0
$397$	1.70506e8	−	9.84416e7i	0.136764	−	0.0789609i	−0.430057	−	0.902802i	$-0.641506\pi$
$397$	1.70506e8	−	9.84416e7i	0.136764	−	0.0789609i	0.566821	+	0.823841i	$0.308173\pi$
$398$		0			0
$399$	1.66980e9	+	2.89218e9i	1.31601	+	2.27939i
$400$		0			0
$401$	8.66177e8	+	5.00088e8i	0.670813	+	0.387294i	0.796385	−	0.604791i	$-0.206743\pi$
$401$	8.66177e8	+	5.00088e8i	0.670813	+	0.387294i	−0.125572	+	0.992085i	$0.540077\pi$
$402$		0			0
$403$	−8.26171e7	−	9.63233e8i	−0.0628785	−	0.733101i
$404$		0			0
$405$	−2.12766e9	−	1.22841e9i	−1.59151	−	0.918860i
$406$		0			0
$407$	−1.20718e7	−	2.09089e7i	−0.00887544	−	0.0153727i
$408$		0			0
$409$	1.40643e9	−	8.12004e8i	1.01645	−	0.586850i	0.103379	−	0.994642i	$-0.467035\pi$
$409$	1.40643e9	−	8.12004e8i	1.01645	−	0.586850i	0.913075	+	0.407793i	$0.133701\pi$
$410$		0			0
$411$		−	1.76885e9i		−	1.25674i
$412$		0			0
$413$	−4.11864e8	+	7.13370e8i	−0.287693	+	0.498299i
$414$		0			0
$415$	−8.25497e8			−0.566953
$416$		0			0
$417$	−5.54678e8			−0.374598
$418$		0			0
$419$	−9.59601e7	+	1.66208e8i	−0.0637297	+	0.110383i	−0.896130	−	0.443792i	$-0.853633\pi$
$419$	−9.59601e7	+	1.66208e8i	−0.0637297	+	0.110383i	0.832400	+	0.554175i	$0.186966\pi$
$420$		0			0
$421$			1.17016e9i			0.764292i	0.924102	+	0.382146i	$0.124815\pi$
$421$			1.17016e9i			0.764292i	−0.924102	+	0.382146i	$0.875185\pi$
$422$		0			0
$423$	−1.39502e9	+	8.05417e8i	−0.896169	+	0.517404i
$424$		0			0
$425$	6.46433e8	+	1.11966e9i	0.408472	+	0.707495i
$426$		0			0
$427$	7.26592e7	+	4.19498e7i	0.0451641	+	0.0260755i
$428$		0			0
$429$	2.33222e8	−	1.62712e8i	0.142616	−	0.0994990i
$430$		0			0
$431$	1.83241e8	+	1.05794e8i	0.110243	+	0.0636491i	0.554108	−	0.832445i	$-0.313059\pi$
$431$	1.83241e8	+	1.05794e8i	0.110243	+	0.0636491i	−0.443864	+	0.896094i	$0.646393\pi$
$432$		0			0
$433$	−5.87978e8	−	1.01841e9i	−0.348060	−	0.602857i	0.637845	−	0.770165i	$-0.279826\pi$
$433$	−5.87978e8	−	1.01841e9i	−0.348060	−	0.602857i	−0.985905	+	0.167308i	$0.946493\pi$
$434$		0			0
$435$	−3.45624e9	+	1.99546e9i	−2.01322	+	1.16233i
$436$		0			0
$437$			1.81708e9i			1.04157i
$438$		0			0
$439$	1.21215e9	−	2.09950e9i	0.683802	−	1.18438i	−0.290010	−	0.957024i	$-0.593659\pi$
$439$	1.21215e9	−	2.09950e9i	0.683802	−	1.18438i	0.973812	−	0.227356i	$-0.0730081\pi$
$440$		0			0
$441$	−1.49646e9			−0.830866
$442$		0			0
$443$	−8.11271e7			−0.0443356			−0.0221678	−	0.999754i	$-0.507057\pi$
$443$	−8.11271e7			−0.0443356			−0.0221678	+	0.999754i	$0.507057\pi$
$444$		0			0
$445$	9.60586e8	−	1.66378e9i	0.516745	−	0.895028i
$446$		0			0
$447$			3.17588e9i			1.68185i
$448$		0			0
$449$	−2.29405e7	+	1.32447e7i	−0.0119603	+	0.00690525i	−0.505968	−	0.862552i	$-0.668865\pi$
$449$	−2.29405e7	+	1.32447e7i	−0.0119603	+	0.00690525i	0.494008	+	0.869457i	$0.335531\pi$
$450$		0			0
$451$	2.52030e7	+	4.36529e7i	0.0129370	+	0.0224076i
$452$		0			0
$453$	−5.28408e8	−	3.05077e8i	−0.267070	−	0.154193i
$454$		0			0
$455$	2.59288e9	+	3.71649e9i	1.29045	+	1.84966i
$456$		0			0
$457$	−2.28711e9	−	1.32046e9i	−1.12094	−	0.647172i	−0.179296	−	0.983795i	$-0.557382\pi$
$457$	−2.28711e9	−	1.32046e9i	−1.12094	−	0.647172i	−0.941640	+	0.336623i	$0.890715\pi$
$458$		0			0
$459$	−1.66642e8	−	2.88632e8i	−0.0804339	−	0.139316i
$460$		0			0
$461$	−1.91120e8	+	1.10343e8i	−0.0908558	+	0.0524556i	−0.544740	−	0.838605i	$-0.683371\pi$
$461$	−1.91120e8	+	1.10343e8i	−0.0908558	+	0.0524556i	0.453884	+	0.891061i	$0.350038\pi$
$462$		0			0
$463$		−	1.40461e9i		−	0.657691i	−0.944384	−	0.328845i	$-0.893341\pi$
$463$		−	1.40461e9i		−	0.657691i	0.944384	−	0.328845i	$-0.106659\pi$
$464$		0			0
$465$	1.67365e9	−	2.89885e9i	0.771934	−	1.33703i
$466$		0			0
$467$	−1.70943e9			−0.776681			−0.388341	−	0.921516i	$-0.626952\pi$
$467$	−1.70943e9			−0.776681			−0.388341	+	0.921516i	$0.626952\pi$
$468$		0			0
$469$	2.53240e9			1.13352
$470$		0			0
$471$	2.04058e9	−	3.53439e9i	0.899872	−	1.55862i
$472$		0			0
$473$		−	4.30971e8i		−	0.187255i
$474$		0			0
$475$	4.07872e9	−	2.35485e9i	1.74621	−	1.00817i
$476$		0			0
$477$	7.91785e8	+	1.37141e9i	0.334036	+	0.578567i
$478$		0			0
$479$	1.87473e9	+	1.08237e9i	0.779405	+	0.449990i	0.836220	−	0.548395i	$-0.184761\pi$
$479$	1.87473e9	+	1.08237e9i	0.779405	+	0.449990i	−0.0568141	+	0.998385i	$0.518094\pi$
$480$		0			0
$481$	−3.01302e8	−	1.41125e8i	−0.123451	−	0.0578226i
$482$		0			0
$483$	−3.11878e9	−	1.80063e9i	−1.25942	−	0.727125i
$484$		0			0
$485$	3.89451e8	+	6.74549e8i	0.155009	+	0.268483i
$486$		0			0
$487$	−3.13354e9	+	1.80915e9i	−1.22937	+	0.709779i	−0.966899	−	0.255160i	$-0.917872\pi$
$487$	−3.13354e9	+	1.80915e9i	−1.22937	+	0.709779i	−0.262474	+	0.964939i	$0.584538\pi$
$488$		0			0
$489$			1.26143e9i			0.487847i
$490$		0			0
$491$	8.96441e8	−	1.55268e9i	0.341772	−	0.591966i	−0.642990	−	0.765875i	$-0.722306\pi$
$491$	8.96441e8	−	1.55268e9i	0.341772	−	0.591966i	0.984762	+	0.173908i	$0.0556397\pi$
$492$		0			0
$493$	−1.63971e9			−0.616315
$494$		0			0
$495$	4.32527e8			0.160286
$496$		0			0
$497$	3.07690e9	−	5.32935e9i	1.12426	−	1.94727i
$498$		0			0
$499$			1.87794e9i			0.676596i	0.941039	+	0.338298i	$0.109851\pi$
$499$			1.87794e9i			0.676596i	−0.941039	+	0.338298i	$0.890149\pi$
$500$		0			0
$501$	3.66209e8	−	2.11431e8i	0.130106	−	0.0751166i
$502$		0			0
$503$	−5.65977e7	−	9.80300e7i	−0.0198294	−	0.0343456i	0.855940	−	0.517074i	$-0.172979\pi$
$503$	−5.65977e7	−	9.80300e7i	−0.0198294	−	0.0343456i	−0.875770	+	0.482729i	$0.839646\pi$
$504$		0			0
$505$	3.66608e9	+	2.11661e9i	1.26672	+	0.731343i
$506$		0			0
$507$	1.35288e9	−	3.67793e9i	0.461032	−	1.25336i
$508$		0			0
$509$	4.69915e9	+	2.71305e9i	1.57945	+	0.911898i	0.994935	+	0.100516i	$0.0320495\pi$
$509$	4.69915e9	+	2.71305e9i	1.57945	+	0.911898i	0.584517	+	0.811381i	$0.301284\pi$
$510$		0			0
$511$	−1.25531e9	−	2.17426e9i	−0.416177	−	0.720840i
$512$		0			0
$513$	−1.05144e9	+	6.07048e8i	−0.343853	+	0.198524i
$514$		0			0
$515$			3.13850e9i			1.01250i
$516$		0			0
$517$	−2.70200e8	+	4.68001e8i	−0.0859942	+	0.148946i
$518$		0			0
$519$	5.41499e9			1.70025
$520$		0			0
$521$	−2.65903e9			−0.823741			−0.411871	−	0.911242i	$-0.635124\pi$
$521$	−2.65903e9			−0.823741			−0.411871	+	0.911242i	$0.635124\pi$
$522$		0			0
$523$	2.02223e9	−	3.50260e9i	0.618122	−	1.07062i	−0.371706	−	0.928350i	$-0.621227\pi$
$523$	2.02223e9	−	3.50260e9i	0.618122	−	1.07062i	0.989828	−	0.142268i	$-0.0454395\pi$
$524$		0			0
$525$			9.33410e9i			2.81524i
$526$		0			0
$527$	1.19102e9	−	6.87637e8i	0.354473	−	0.204655i
$528$		0			0
$529$	7.22687e8	+	1.25173e9i	0.212254	+	0.367634i
$530$		0			0
$531$	9.38316e8	+	5.41737e8i	0.271968	+	0.157021i
$532$		0			0
$533$	6.29049e8	+	2.94637e8i	0.179945	+	0.0842835i
$534$		0			0
$535$	7.83478e9	+	4.52341e9i	2.21202	+	1.27711i
$536$		0			0
$537$	−2.79305e9	−	4.83770e9i	−0.778338	−	1.34812i
$538$		0			0
$539$	−4.34772e8	+	2.51016e8i	−0.119592	+	0.0690463i
$540$		0			0
$541$		−	6.97904e9i		−	1.89498i	−0.319782	−	0.947491i	$-0.603610\pi$
$541$		−	6.97904e9i		−	1.89498i	0.319782	−	0.947491i	$-0.396390\pi$
$542$		0			0
$543$	−7.98617e8	+	1.38325e9i	−0.214062	+	0.370766i
$544$		0			0
$545$	4.64244e9			1.22845
$546$		0			0
$547$	−5.47434e7			−0.0143013			−0.00715065	−	0.999974i	$-0.502276\pi$
$547$	−5.47434e7			−0.0143013			−0.00715065	+	0.999974i	$0.502276\pi$
$548$		0			0
$549$	5.51778e7	−	9.55708e7i	0.0142318	−	0.0246503i
$550$		0			0
$551$			5.97320e9i			1.52116i
$552$		0			0
$553$	1.69544e9	−	9.78866e8i	0.426330	−	0.246142i
$554$		0			0
$555$	−5.75989e8	−	9.97643e8i	−0.143017	−	0.247713i
$556$		0			0
$557$	1.87977e8	+	1.08528e8i	0.0460905	+	0.0266103i	0.522868	−	0.852414i	$-0.324862\pi$
$557$	1.87977e8	+	1.08528e8i	0.0460905	+	0.0266103i	−0.476778	+	0.879024i	$0.658195\pi$
$558$		0			0
$559$	−3.39822e9	−	4.87082e9i	−0.822830	−	1.17940i
$560$		0			0
$561$	3.50336e8	+	2.02266e8i	0.0837750	+	0.0483675i
$562$		0			0
$563$	−2.39971e9	−	4.15643e9i	−0.566735	−	0.981614i	−0.996886	−	0.0788572i	$-0.974873\pi$
$563$	−2.39971e9	−	4.15643e9i	−0.566735	−	0.981614i	0.430151	−	0.902757i	$-0.358460\pi$
$564$		0			0
$565$	−1.01953e10	+	5.88625e9i	−2.37810	+	1.37300i
$566$		0			0
$567$		−	7.28760e9i		−	1.67897i
$568$		0			0
$569$	2.61206e9	−	4.52422e9i	0.594415	−	1.02956i	−0.399214	−	0.916858i	$-0.630717\pi$
$569$	2.61206e9	−	4.52422e9i	0.594415	−	1.02956i	0.993629	−	0.112699i	$-0.0359497\pi$
$570$		0			0
$571$	9.53468e8			0.214328			0.107164	−	0.994241i	$-0.465823\pi$
$571$	9.53468e8			0.214328			0.107164	+	0.994241i	$0.465823\pi$
$572$		0			0
$573$	9.07338e9			2.01478
$574$		0			0
$575$	−2.53935e9	+	4.39829e9i	−0.557039	+	0.964820i
$576$		0			0
$577$		−	7.41729e6i		−	0.00160742i	−1.00000		0.000803711i	$-0.999744\pi$
$577$		−	7.41729e6i		−	0.00160742i	1.00000		0.000803711i	$-0.000255829\pi$
$578$		0			0
$579$	−4.93392e9	+	2.84860e9i	−1.05637	+	0.609898i
$580$		0			0
$581$	−1.22433e9	−	2.12060e9i	−0.258989	−	0.448583i
$582$		0			0
$583$	4.60080e8	+	2.65627e8i	0.0961597	+	0.0555178i
$584$		0			0
$585$	4.88840e9	−	3.41049e9i	1.00953	−	0.704321i
$586$		0			0
$587$	−5.20465e9	−	3.00490e9i	−1.06208	−	0.613193i	−0.136074	−	0.990699i	$-0.543449\pi$
$587$	−5.20465e9	−	3.00490e9i	−1.06208	−	0.613193i	−0.926007	+	0.377505i	$0.876782\pi$
$588$		0			0
$589$	−2.50495e9	−	4.33870e9i	−0.505121	−	0.874895i
$590$		0			0
$591$	9.20856e9	−	5.31656e9i	1.83500	−	1.05944i
$592$		0			0
$593$		−	1.00461e9i		−	0.197836i	−0.995096	−	0.0989178i	$-0.968462\pi$
$593$		−	1.00461e9i		−	0.197836i	0.995096	−	0.0989178i	$-0.0315381\pi$
$594$		0			0
$595$	−3.22320e9	+	5.58274e9i	−0.627303	+	1.08652i
$596$		0			0
$597$	8.71365e9			1.67606
$598$		0			0
$599$	4.70768e9			0.894979			0.447489	−	0.894289i	$-0.352318\pi$
$599$	4.70768e9			0.894979			0.447489	+	0.894289i	$0.352318\pi$
$600$		0			0
$601$	−3.02639e9	+	5.24187e9i	−0.568676	+	0.984975i	0.428022	+	0.903768i	$0.359211\pi$
$601$	−3.02639e9	+	5.24187e9i	−0.568676	+	0.984975i	−0.996697	+	0.0812066i	$0.974123\pi$
$602$		0			0
$603$		−	3.33094e9i		−	0.618666i
$604$		0			0
$605$	−7.28569e9	+	4.20640e9i	−1.33760	+	0.772265i
$606$		0			0
$607$	−3.56363e9	−	6.17239e9i	−0.646744	−	1.12019i	−0.983896	−	0.178743i	$-0.942797\pi$
$607$	−3.56363e9	−	6.17239e9i	−0.646744	−	1.12019i	0.337152	−	0.941450i	$-0.390536\pi$
$608$		0			0
$609$	−1.02522e10	−	5.91910e9i	−1.83931	−	1.06193i
$610$		0			0
$611$	6.36406e8	+	7.41987e9i	0.112873	+	1.31599i
$612$		0			0
$613$	−8.66562e8	−	5.00310e8i	−0.151946	−	0.0877258i	0.422100	−	0.906549i	$-0.361293\pi$
$613$	−8.66562e8	−	5.00310e8i	−0.151946	−	0.0877258i	−0.574045	+	0.818824i	$0.694627\pi$
$614$		0			0
$615$	1.20253e9	+	2.08285e9i	0.208465	+	0.361073i
$616$		0			0
$617$	−3.21958e9	+	1.85882e9i	−0.551825	+	0.318596i	−0.749858	−	0.661599i	$-0.769878\pi$
$617$	−3.21958e9	+	1.85882e9i	−0.551825	+	0.318596i	0.198033	+	0.980195i	$0.436545\pi$
$618$		0			0
$619$			8.30664e9i			1.40769i	0.710352	+	0.703847i	$0.248536\pi$
$619$			8.30664e9i			1.40769i	−0.710352	+	0.703847i	$0.751464\pi$
$620$		0			0
$621$	6.54611e8	−	1.13382e9i	0.109689	−	0.189987i
$622$		0			0
$623$	5.69874e9			0.944214
$624$		0			0
$625$	−1.90347e9			−0.311864
$626$		0			0
$627$	7.36823e8	−	1.27622e9i	0.119379	−	0.206770i
$628$		0			0
$629$		−	4.73302e8i		−	0.0758335i
$630$		0			0
$631$	−6.78879e9	+	3.91951e9i	−1.07570	+	0.621053i	−0.929732	−	0.368237i	$-0.879962\pi$
$631$	−6.78879e9	+	3.91951e9i	−1.07570	+	0.621053i	−0.145963	+	0.989290i	$0.546628\pi$
$632$		0			0
$633$	−2.61392e9	−	4.52744e9i	−0.409618	−	0.709479i
$634$		0			0
$635$	−3.06942e9	−	1.77213e9i	−0.475717	−	0.274655i
$636$		0			0
$637$	−2.93451e9	+	6.26517e9i	−0.449830	+	0.960384i
$638$		0			0
$639$	−7.00984e9	−	4.04713e9i	−1.06281	−	0.613613i
$640$		0			0
$641$	6.36068e8	+	1.10170e9i	0.0953895	+	0.165220i	0.909771	−	0.415110i	$-0.136257\pi$
$641$	6.36068e8	+	1.10170e9i	0.0953895	+	0.165220i	−0.814382	+	0.580330i	$0.802924\pi$
$642$		0			0
$643$	−4.91826e9	+	2.83956e9i	−0.729581	+	0.421224i	−0.818269	−	0.574836i	$-0.805066\pi$
$643$	−4.91826e9	+	2.83956e9i	−0.729581	+	0.421224i	0.0886881	+	0.996059i	$0.471733\pi$
$644$		0			0
$645$		−	2.05633e10i		−	3.01740i
$646$		0			0
$647$	4.91513e9	−	8.51325e9i	0.713460	−	1.23575i	−0.250091	−	0.968222i	$-0.580460\pi$
$647$	4.91513e9	−	8.51325e9i	0.713460	−	1.23575i	0.963551	−	0.267526i	$-0.0862063\pi$
$648$		0			0
$649$	3.63483e8			0.0521948
$650$		0			0
$651$	9.92905e9			1.41050
$652$		0			0
$653$	1.92707e9	−	3.33778e9i	0.270832	−	0.469096i	−0.698243	−	0.715861i	$-0.746034\pi$
$653$	1.92707e9	−	3.33778e9i	0.270832	−	0.469096i	0.969075	+	0.246765i	$0.0793677\pi$
$654$		0			0
$655$			5.55049e9i			0.771769i
$656$		0			0
$657$	−2.85987e9	+	1.65115e9i	−0.393430	+	0.227147i
$658$		0			0
$659$	−2.45749e9	−	4.25650e9i	−0.334498	−	0.579367i	0.648891	−	0.760882i	$-0.275233\pi$
$659$	−2.45749e9	−	4.25650e9i	−0.334498	−	0.579367i	−0.983388	+	0.181515i	$0.941900\pi$
$660$		0			0
$661$	1.10777e10	+	6.39573e9i	1.49192	+	0.861360i	0.999957	−	0.00925625i	$-0.00294640\pi$
$661$	1.10777e10	+	6.39573e9i	1.49192	+	0.861360i	0.491962	+	0.870616i	$0.336280\pi$
$662$		0			0
$663$	5.55436e9	−	4.76401e8i	0.740179	−	0.0634856i
$664$		0			0
$665$	2.03370e10	+	1.17416e10i	2.68171	+	1.54828i
$666$		0			0
$667$	−3.22060e9	−	5.57824e9i	−0.420239	−	0.727875i
$668$		0			0
$669$	5.14366e9	−	2.96969e9i	0.664172	−	0.383460i
$670$		0			0
$671$		−	3.70220e7i		−	0.00473076i
$672$		0			0
$673$	−5.49257e9	+	9.51341e9i	−0.694581	+	1.20305i	0.275741	+	0.961232i	$0.411077\pi$
$673$	−5.49257e9	+	9.51341e9i	−0.694581	+	1.20305i	−0.970322	+	0.241818i	$0.922256\pi$
$674$		0			0
$675$	−3.39337e9			−0.424686
$676$		0			0
$677$	3.14306e9			0.389307			0.194653	−	0.980872i	$-0.437642\pi$
$677$	3.14306e9			0.389307			0.194653	+	0.980872i	$0.437642\pi$
$678$		0			0
$679$	−1.15522e9	+	2.00090e9i	−0.141619	+	0.245291i
$680$		0			0
$681$			6.17296e9i			0.748994i
$682$		0			0
$683$	1.33989e10	−	7.73584e9i	1.60915	−	0.929042i	0.619586	−	0.784929i	$-0.287301\pi$
$683$	1.33989e10	−	7.73584e9i	1.60915	−	0.929042i	0.989561	−	0.144113i	$-0.0460328\pi$
$684$		0			0
$685$	−6.21904e9	−	1.07717e10i	−0.739276	−	1.28046i
$686$		0			0
$687$	−4.71862e9	−	2.72430e9i	−0.555222	−	0.320557i
$688$		0			0
$689$	7.29428e9	−	6.25635e8i	0.849601	−	0.0728708i
$690$		0			0
$691$	4.00359e9	+	2.31148e9i	0.461612	+	0.266512i	0.712722	−	0.701447i	$-0.247462\pi$
$691$	4.00359e9	+	2.31148e9i	0.461612	+	0.266512i	−0.251110	+	0.967959i	$0.580796\pi$
$692$		0			0
$693$	6.41498e8	+	1.11111e9i	0.0732199	+	0.126821i
$694$		0			0
$695$	−3.37780e9	+	1.95017e9i	−0.381669	+	0.220357i
$696$		0			0
$697$			9.88145e8i			0.110537i
$698$		0			0
$699$	−7.57216e9	+	1.31154e10i	−0.838590	+	1.45248i
$700$		0			0
$701$	−1.41767e10			−1.55440			−0.777200	−	0.629253i	$-0.783361\pi$
$701$	−1.41767e10			−1.55440			−0.777200	+	0.629253i	$0.783361\pi$
$702$		0			0
$703$	−1.72416e9			−0.187169
$704$		0			0
$705$	−1.28923e10	+	2.23301e10i	−1.38570	+	2.40010i
$706$		0			0
$707$			1.25569e10i			1.33634i
$708$		0			0
$709$	7.07624e9	−	4.08547e9i	0.745660	−	0.430507i	−0.0784635	−	0.996917i	$-0.525001\pi$
$709$	7.07624e9	−	4.08547e9i	0.745660	−	0.430507i	0.824124	+	0.566410i	$0.191668\pi$
$710$		0			0
$711$	−1.28753e9	−	2.23007e9i	−0.134343	−	0.232688i
$712$		0			0
$713$	4.67864e9	+	2.70121e9i	0.483399	+	0.279091i
$714$		0			0
$715$	8.48170e8	−	1.81084e9i	0.0867784	−	0.185271i
$716$		0			0
$717$	−2.20949e10	−	1.27565e10i	−2.23859	−	1.29245i
$718$		0			0
$719$	7.71168e9	+	1.33570e10i	0.773745	+	1.34017i	0.935497	+	0.353334i	$0.114952\pi$
$719$	7.71168e9	+	1.33570e10i	0.773745	+	1.34017i	−0.161752	+	0.986831i	$0.551714\pi$
$720$		0			0
$721$	−8.06242e9	+	4.65484e9i	−0.801110	+	0.462521i
$722$		0			0
$723$		−	1.36928e10i		−	1.34744i
$724$		0			0
$725$	−8.34748e9	+	1.44583e10i	−0.813527	+	1.40907i
$726$		0			0
$727$	−5.70391e9			−0.550557			−0.275279	−	0.961365i	$-0.588770\pi$
$727$	−5.70391e9			−0.550557			−0.275279	+	0.961365i	$0.588770\pi$
$728$		0			0
$729$	−5.54588e9			−0.530181
$730$		0			0
$731$	4.22431e9	−	7.31672e9i	0.399986	−	0.692796i
$732$		0			0
$733$			1.66495e10i			1.56148i	0.624854	+	0.780742i	$0.285159\pi$
$733$			1.66495e10i			1.56148i	−0.624854	+	0.780742i	$0.714841\pi$
$734$		0			0
$735$	−2.07446e10	+	1.19769e10i	−1.92708	+	1.11260i
$736$		0			0
$737$	−5.58730e8	−	9.67749e8i	−0.0514122	−	0.0890485i
$738$		0			0
$739$	4.26718e9	+	2.46366e9i	0.388943	+	0.224556i	0.681702	−	0.731630i	$-0.261240\pi$
$739$	4.26718e9	+	2.46366e9i	0.388943	+	0.224556i	−0.292759	+	0.956186i	$0.594573\pi$
$740$		0			0
$741$	−1.73545e9	−	2.02336e10i	−0.156693	−	1.82688i
$742$		0			0
$743$	8.58753e9	+	4.95801e9i	0.768082	+	0.443452i	0.832190	−	0.554491i	$-0.187087\pi$
$743$	8.58753e9	+	4.95801e9i	0.768082	+	0.443452i	−0.0641082	+	0.997943i	$0.520420\pi$
$744$		0			0
$745$	1.11660e10	+	1.93400e10i	0.989348	+	1.71360i
$746$		0			0
$747$	−2.78929e9	+	1.61040e9i	−0.244834	+	0.141355i
$748$		0			0
$749$			2.68354e10i			2.33358i
$750$		0			0
$751$	−1.12646e10	+	1.95109e10i	−0.970456	+	1.68088i	−0.276276	+	0.961078i	$0.589100\pi$
$751$	−1.12646e10	+	1.95109e10i	−0.970456	+	1.68088i	−0.694180	+	0.719801i	$0.744233\pi$
$752$		0			0
$753$	−9.27835e8			−0.0791933
$754$		0			0
$755$	−4.29043e9			−0.362816
$756$		0			0
$757$	1.10819e10	−	1.91944e10i	0.928493	−	1.60820i	0.142647	−	0.989774i	$-0.454439\pi$
$757$	1.10819e10	−	1.91944e10i	0.928493	−	1.60820i	0.785846	−	0.618423i	$-0.212228\pi$
$758$		0			0
$759$			1.58911e9i			0.131919i
$760$		0			0
$761$	6.64485e9	−	3.83641e9i	0.546561	−	0.315557i	−0.201173	−	0.979556i	$-0.564475\pi$
$761$	6.64485e9	−	3.83641e9i	0.546561	−	0.315557i	0.747734	+	0.663999i	$0.231142\pi$
$762$		0			0
$763$	6.88540e9	+	1.19259e10i	0.561168	+	0.971972i
$764$		0			0
$765$	7.34313e9	+	4.23956e9i	0.593016	+	0.342378i
$766$		0			0
$767$	4.10807e9	−	2.86607e9i	0.328741	−	0.229352i
$768$		0			0
$769$	−7.77528e9	−	4.48906e9i	−0.616558	−	0.355970i	0.158970	−	0.987283i	$-0.449183\pi$
$769$	−7.77528e9	−	4.48906e9i	−0.616558	−	0.355970i	−0.775528	+	0.631314i	$0.782516\pi$
$770$		0			0
$771$	−1.56016e10	−	2.70228e10i	−1.22597	−	2.12344i
$772$		0			0
$773$	9.70027e9	−	5.60045e9i	0.755363	−	0.436109i	−0.0722655	−	0.997385i	$-0.523023\pi$
$773$	9.70027e9	−	5.60045e9i	0.755363	−	0.436109i	0.827628	+	0.561276i	$0.189690\pi$
$774$		0			0
$775$		−	1.40026e10i		−	1.08057i
$776$		0			0
$777$	1.70855e9	−	2.95929e9i	0.130663	−	0.226315i
$778$		0			0
$779$	3.59965e9			0.272822
$780$		0			0
$781$	−2.71546e9			−0.203969
$782$		0			0
$783$	2.15186e9	−	3.72714e9i	0.160195	−	0.277466i
$784$		0			0
$785$		−	2.86976e10i		−	2.11740i
$786$		0			0
$787$	9.57297e8	−	5.52696e8i	0.0700060	−	0.0404180i	−0.464588	−	0.885527i	$-0.653798\pi$
$787$	9.57297e8	−	5.52696e8i	0.0700060	−	0.0404180i	0.534594	+	0.845109i	$0.320464\pi$
$788$		0			0
$789$	−9.47575e8	−	1.64125e9i	−0.0686822	−	0.118961i
$790$		0			0
$791$	−3.02421e10	−	1.74603e10i	−2.17267	−	1.25439i
$792$		0			0
$793$	−2.91920e8	−	4.18421e8i	−0.0207877	−	0.0297960i
$794$		0			0
$795$	2.19521e10	+	1.26741e10i	1.54950	+	0.894605i
$796$		0			0
$797$	−7.49691e9	−	1.29850e10i	−0.524540	−	0.908529i	−0.999592	−	0.0285717i	$-0.990904\pi$
$797$	−7.49691e9	−	1.29850e10i	−0.524540	−	0.908529i	0.475052	−	0.879958i	$-0.342429\pi$
$798$		0			0
$799$	−9.17454e9	+	5.29693e9i	−0.636313	+	0.367375i
$800$		0			0
$801$		−	7.49571e9i		−	0.515346i
$802$		0			0
$803$	−5.53925e8	+	9.59426e8i	−0.0377526	+	0.0653894i
$804$		0			0
$805$	−2.53230e10			−1.71092
$806$		0			0
$807$	−6.90272e9			−0.462342
$808$		0			0
$809$	−7.66062e9	+	1.32686e10i	−0.508679	+	0.881059i	0.491270	+	0.871007i	$0.336533\pi$
$809$	−7.66062e9	+	1.32686e10i	−0.508679	+	0.881059i	−0.999949	+	0.0100514i	$0.996800\pi$
$810$		0			0
$811$		−	1.14391e10i		−	0.753041i	−0.926408	−	0.376521i	$-0.877120\pi$
$811$		−	1.14391e10i		−	0.753041i	0.926408	−	0.376521i	$-0.122880\pi$
$812$		0			0
$813$	−2.72628e9	+	1.57402e9i	−0.177931	+	0.102729i
$814$		0			0
$815$	4.43503e9	+	7.68170e9i	0.286976	+	0.497056i
$816$		0			0
$817$	−2.66536e10	−	1.53885e10i	−1.70993	−	0.987229i
$818$		0			0
$819$	1.60113e10	+	7.49946e9i	1.01843	+	0.477020i
$820$		0			0
$821$	1.88617e10	+	1.08898e10i	1.18954	+	0.686782i	0.958202	−	0.286091i	$-0.0923561\pi$
$821$	1.88617e10	+	1.08898e10i	1.18954	+	0.686782i	0.231339	+	0.972873i	$0.425689\pi$
$822$		0			0
$823$	1.08157e9	+	1.87333e9i	0.0676324	+	0.117143i	0.897859	−	0.440284i	$-0.145122\pi$
$823$	1.08157e9	+	1.87333e9i	0.0676324	+	0.117143i	−0.830226	+	0.557426i	$0.811789\pi$
$824$		0			0
$825$	3.56699e9	−	2.05941e9i	0.221164	−	0.127689i
$826$		0			0
$827$			2.27847e10i			1.40079i	0.713755	+	0.700395i	$0.246993\pi$
$827$			2.27847e10i			1.40079i	−0.713755	+	0.700395i	$0.753007\pi$
$828$		0			0
$829$	4.91919e9	−	8.52030e9i	0.299884	−	0.519414i	−0.676225	−	0.736695i	$-0.736385\pi$
$829$	4.91919e9	−	8.52030e9i	0.299884	−	0.519414i	0.976109	+	0.217281i	$0.0697187\pi$
$830$		0			0
$831$	−3.36370e10			−2.03336
$832$		0			0
$833$	−9.84168e9			−0.589945
$834$		0			0
$835$	1.48672e9	−	2.57508e9i	0.0883746	−	0.153069i
$836$		0			0
$837$			3.60967e9i			0.212779i
$838$		0			0
$839$	1.81235e10	−	1.04636e10i	1.05944	−	0.611666i	0.134160	−	0.990960i	$-0.457166\pi$
$839$	1.81235e10	−	1.04636e10i	1.05944	−	0.611666i	0.925276	+	0.379294i	$0.123833\pi$
$840$		0			0
$841$	−1.96195e9	−	3.39821e9i	−0.113737	−	0.196999i
$842$		0			0
$843$	−7.76038e9	−	4.48046e9i	−0.446156	−	0.257588i
$844$		0			0
$845$	−4.69253e9	−	2.71539e10i	−0.267552	−	1.54822i
$846$		0			0
$847$	−2.16114e10	−	1.24774e10i	−1.22206	−	0.705555i
$848$		0			0
$849$	3.16161e9	+	5.47607e9i	0.177309	+	0.307109i
$850$		0			0
$851$	1.61016e9	−	9.29625e8i	0.0895602	−	0.0517076i
$852$		0			0
$853$			2.52741e10i			1.39429i	0.716928	+	0.697147i	$0.245548\pi$
$853$			2.52741e10i			1.39429i	−0.716928	+	0.697147i	$0.754452\pi$
$854$		0			0
$855$	1.54440e10	−	2.67498e10i	0.845043	−	1.46366i
$856$		0			0
$857$	−2.67167e10			−1.44994			−0.724969	−	0.688782i	$-0.758146\pi$
$857$	−2.67167e10			−1.44994			−0.724969	+	0.688782i	$0.758146\pi$
$858$		0			0
$859$	8.51182e9			0.458191			0.229095	−	0.973404i	$-0.426423\pi$
$859$	8.51182e9			0.458191			0.229095	+	0.973404i	$0.426423\pi$
$860$		0			0
$861$	−3.56705e9	+	6.17832e9i	−0.190458	+	0.329882i
$862$		0			0
$863$			5.94525e9i			0.314871i	0.987529	+	0.157435i	$0.0503226\pi$
$863$			5.94525e9i			0.314871i	−0.987529	+	0.157435i	$0.949677\pi$
$864$		0			0
$865$	3.29754e10	−	1.90384e10i	1.73234	−	1.00017i
$866$		0			0
$867$	−8.84835e9	−	1.53258e10i	−0.461100	−	0.798649i
$868$		0			0
$869$	−7.48140e8	−	4.31939e8i	−0.0386735	−	0.0223282i
$870$		0			0
$871$	−1.39455e10	−	6.53186e9i	−0.715105	−	0.334945i
$872$		0			0
$873$	2.63184e9	+	1.51950e9i	0.133878	+	0.0772947i
$874$		0			0
$875$	1.04710e10	+	1.81363e10i	0.528395	+	0.915207i
$876$		0			0
$877$	9.21976e9	−	5.32303e9i	0.461552	−	0.266477i	−0.251144	−	0.967950i	$-0.580807\pi$
$877$	9.21976e9	−	5.32303e9i	0.461552	−	0.266477i	0.712697	+	0.701472i	$0.247474\pi$
$878$		0			0
$879$			1.91297e10i			0.950054i
$880$		0			0
$881$	−1.91599e9	+	3.31860e9i	−0.0944014	+	0.163508i	−0.909359	−	0.416013i	$-0.863427\pi$
$881$	−1.91599e9	+	3.31860e9i	−0.0944014	+	0.163508i	0.814957	+	0.579521i	$0.196760\pi$
$882$		0			0
$883$	1.75487e10			0.857792			0.428896	−	0.903354i	$-0.358903\pi$
$883$	1.75487e10			0.857792			0.428896	+	0.903354i	$0.358903\pi$
$884$		0			0
$885$	1.73431e10			0.841058
$886$		0			0
$887$	−5.37291e9	+	9.30616e9i	−0.258510	+	0.447752i	−0.965843	−	0.259128i	$-0.916565\pi$
$887$	−5.37291e9	+	9.30616e9i	−0.258510	+	0.447752i	0.707333	+	0.706880i	$0.249898\pi$
$888$		0			0
$889$		−	1.05133e10i		−	0.501860i
$890$		0			0
$891$	−2.78493e9	+	1.60788e9i	−0.131899	+	0.0761521i
$892$		0			0
$893$	1.92958e10	+	3.34214e10i	0.906741	+	1.57052i
$894$		0			0
$895$	−3.40174e10	−	1.96399e10i	−1.58606	−	0.915714i
$896$		0			0
$897$	1.25302e10	+	1.79600e10i	0.579673	+	0.830871i
$898$		0			0
$899$	1.53798e10	+	8.87954e9i	0.705980	+	0.407598i
$900$		0			0
$901$	5.20727e9	+	9.01926e9i	0.237178	+	0.410803i
$902$		0			0
$903$	5.28244e10	−	3.04982e10i	2.38741	−	1.37837i
$904$		0			0
$905$			1.12313e10i			0.503687i
$906$		0			0
$907$	1.03600e10	−	1.79440e10i	0.461034	−	0.798534i	−0.537979	−	0.842958i	$-0.680812\pi$
$907$	1.03600e10	−	1.79440e10i	0.461034	−	0.798534i	0.999013	+	0.0444245i	$0.0141454\pi$
$908$		0			0
$909$	1.65165e10			0.729363
$910$		0			0
$911$	3.76677e10			1.65065			0.825325	−	0.564658i	$-0.190992\pi$
$911$	3.76677e10			1.65065			0.825325	+	0.564658i	$0.190992\pi$
$912$		0			0
$913$	−5.40254e8	+	9.35747e8i	−0.0234936	+	0.0406921i
$914$		0			0
$915$		−	1.76646e9i		−	0.0762307i
$916$		0			0
$917$	−1.42585e10	+	8.23217e9i	−0.610636	+	0.352551i
$918$		0			0
$919$	7.80748e9	+	1.35229e10i	0.331823	+	0.574734i	0.982869	−	0.184304i	$-0.0590030\pi$
$919$	7.80748e9	+	1.35229e10i	0.331823	+	0.574734i	−0.651046	+	0.759038i	$0.725670\pi$
$920$		0			0
$921$	4.79429e8	+	2.76798e8i	0.0202216	+	0.0116749i
$922$		0			0
$923$	−3.06900e10	+	2.14115e10i	−1.28467	+	0.896273i
$924$		0			0
$925$	−4.17337e9	−	2.40950e9i	−0.173377	−	0.100099i
$926$		0			0
$927$	6.12264e9	+	1.06047e10i	0.252441	+	0.437241i
$928$		0			0
$929$	2.38983e10	−	1.37977e10i	0.977941	−	0.564615i	0.0762934	−	0.997085i	$-0.475691\pi$
$929$	2.38983e10	−	1.37977e10i	0.977941	−	0.564615i	0.901648	+	0.432471i	$0.142358\pi$
$930$		0			0
$931$			3.58516e10i			1.45608i
$932$		0			0
$933$	1.23459e10	−	2.13838e10i	0.497666	−	0.861982i
$934$		0			0
$935$	2.84457e9			0.113809
$936$		0			0
$937$	−3.75571e9			−0.149143			−0.0745715	−	0.997216i	$-0.523759\pi$
$937$	−3.75571e9			−0.149143			−0.0745715	+	0.997216i	$0.523759\pi$
$938$		0			0
$939$	−2.77421e10	+	4.80506e10i	−1.09348	+	1.89395i
$940$		0			0
$941$			3.77084e10i			1.47528i	0.675193	+	0.737641i	$0.264060\pi$
$941$			3.77084e10i			1.47528i	−0.675193	+	0.737641i	$0.735940\pi$
$942$		0			0
$943$	−3.36164e9	+	1.94084e9i	−0.130545	+	0.0753701i
$944$		0			0
$945$	−8.45988e9	−	1.46529e10i	−0.326102	−	0.564825i
$946$		0			0
$947$	3.96761e10	+	2.29070e10i	1.51811	+	0.876483i	0.999773	+	0.0213086i	$0.00678325\pi$
$947$	3.96761e10	+	2.29070e10i	1.51811	+	0.876483i	0.518340	+	0.855174i	$0.326550\pi$
$948$		0			0
$949$	1.30467e9	+	1.52111e10i	0.0495527	+	0.577736i
$950$		0			0
$951$	−1.84032e10	−	1.06251e10i	−0.693843	−	0.400590i
$952$		0			0
$953$	9.15154e9	+	1.58509e10i	0.342507	+	0.593239i	0.984898	−	0.173138i	$-0.0553907\pi$
$953$	9.15154e9	+	1.58509e10i	0.342507	+	0.593239i	−0.642391	+	0.766377i	$0.722057\pi$
$954$		0			0
$955$	5.52538e10	−	3.19008e10i	2.05282	−	1.18519i
$956$		0			0
$957$			5.22378e9i			0.192661i
$958$		0			0
$959$	1.84474e10	−	3.19519e10i	0.675415	−	1.16985i
$960$		0			0
$961$	1.26175e10			0.458609
$962$		0			0
$963$	3.52974e10			1.27365
$964$		0			0
$965$	−2.00306e10	+	3.46940e10i	−0.717544	+	1.24282i
$966$		0			0
$967$		−	2.62013e10i		−	0.931816i	−0.884833	−	0.465908i	$-0.845728\pi$
$967$		−	2.62013e10i		−	0.931816i	0.884833	−	0.465908i	$-0.154272\pi$
$968$		0			0
$969$	2.50185e10	−	1.44445e10i	0.883342	−	0.509998i
$970$		0			0
$971$	1.52480e9	+	2.64103e9i	0.0534497	+	0.0925777i	0.891512	−	0.452996i	$-0.149645\pi$
$971$	1.52480e9	+	2.64103e9i	0.0534497	+	0.0925777i	−0.838063	+	0.545574i	$0.816312\pi$
$972$		0			0
$973$	−1.00195e10	−	5.78477e9i	−0.348700	−	0.201322i
$974$		0			0
$975$	2.40756e10	−	5.14012e10i	0.831879	−	1.77606i
$976$		0			0
$977$	3.81568e10	+	2.20298e10i	1.30900	+	0.755754i	0.981930	−	0.189245i	$-0.0606040\pi$
$977$	3.81568e10	+	2.20298e10i	1.30900	+	0.755754i	0.327074	+	0.944999i	$0.393937\pi$
$978$		0			0
$979$	−1.25733e9	−	2.17775e9i	−0.0428261	−	0.0741770i
$980$		0			0
$981$	1.56864e10	−	9.05656e9i	0.530496	−	0.306282i
$982$		0			0
$983$			3.19323e10i			1.07224i	0.844141	+	0.536122i	$0.180111\pi$
$983$			3.19323e10i			1.07224i	−0.844141	+	0.536122i	$0.819889\pi$
$984$		0			0
$985$	3.73846e10	−	6.47521e10i	1.24643	−	2.15887i
$986$		0			0
$987$	−7.64843e10			−2.53199
$988$		0			0
$989$	3.31883e10			1.09093
$990$		0			0
$991$	1.83299e10	−	3.17483e10i	0.598277	−	1.03625i	−0.394798	−	0.918768i	$-0.629186\pi$
$991$	1.83299e10	−	3.17483e10i	0.598277	−	1.03625i	0.993075	−	0.117478i	$-0.0374811\pi$
$992$		0			0
$993$		−	3.43124e10i		−	1.11206i
$994$		0			0
$995$	5.30632e10	−	3.06360e10i	1.70770	−	0.985943i
$996$		0			0
$997$	−2.93042e10	−	5.07563e10i	−0.936476	−	1.62202i	−0.771981	−	0.635646i	$-0.780734\pi$
$997$	−2.93042e10	−	5.07563e10i	−0.936476	−	1.62202i	−0.164495	−	0.986378i	$-0.552600\pi$
$998$		0			0
$999$	1.07584e9	+	6.21135e8i	0.0341403	+	0.0197109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.w.a.17.7		14
4.3	odd	2		13.8.e.a.4.4	✓	14
12.11	even	2		117.8.q.b.82.4		14
13.10	even	6	inner	208.8.w.a.49.7		14
52.7	even	12		169.8.a.g.1.8		14
52.19	even	12		169.8.a.g.1.7		14
52.23	odd	6		13.8.e.a.10.4	yes	14
52.35	odd	6		169.8.b.d.168.7		14
52.43	odd	6		169.8.b.d.168.8		14
156.23	even	6		117.8.q.b.10.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.4	✓	14	4.3	odd	2
13.8.e.a.10.4	yes	14	52.23	odd	6
117.8.q.b.10.4		14	156.23	even	6
117.8.q.b.82.4		14	12.11	even	2
169.8.a.g.1.7		14	52.19	even	12
169.8.a.g.1.8		14	52.7	even	12
169.8.b.d.168.7		14	52.35	odd	6
169.8.b.d.168.8		14	52.43	odd	6
208.8.w.a.17.7		14	1.1	even	1	trivial
208.8.w.a.49.7		14	13.10	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(31.2267 - 54.0862i) q^{3} -439.155i q^{5} +(1128.14 - 651.329i) q^{7} +(-856.712 - 1483.87i) q^{9} +(-497.807 - 287.409i) q^{11} +(-7892.42 + 676.938i) q^{13} +(-23752.2 - 13713.4i) q^{15} +(-5634.27 - 9758.85i) q^{17} +(-35549.9 + 20524.7i) q^{19} -81355.4i q^{21} +(22132.8 - 38335.2i) q^{23} -114732. q^{25} +29576.4 q^{27} +(72756.1 - 126017. i) q^{29} +122045. i q^{31} +(-31089.7 + 17949.6i) q^{33} +(-286035. - 495427. i) q^{35} +(36374.8 + 21001.0i) q^{37} +(-209841. + 448010. i) q^{39} +(-75942.3 - 43845.3i) q^{41} +(374876. + 649305. i) q^{43} +(-651649. + 376230. i) q^{45} -940126. i q^{47} +(436688. - 756366. i) q^{49} -703759. q^{51} -924214. q^{53} +(-126217. + 218614. i) q^{55} +2.56368e6i q^{57} +(-547626. + 316172. i) q^{59} +(32203.2 + 55777.7i) q^{61} +(-1.93297e6 - 1.11600e6i) q^{63} +(297281. + 3.46600e6i) q^{65} +(1.68358e6 + 972013. i) q^{67} +(-1.38227e6 - 2.39416e6i) q^{69} +(4.09113e6 - 2.36202e6i) q^{71} -1.92731e6i q^{73} +(-3.58271e6 + 6.20544e6i) q^{75} -748791. q^{77} +1.50287e6 q^{79} +(2.79720e6 - 4.84490e6i) q^{81} -1.87974e6i q^{83} +(-4.28565e6 + 2.47432e6i) q^{85} +(-4.54386e6 - 7.87020e6i) q^{87} +(3.78860e6 + 2.18735e6i) q^{89} +(-8.46281e6 + 5.90424e6i) q^{91} +(6.60097e6 + 3.81107e6i) q^{93} +(9.01354e6 + 1.56119e7i) q^{95} +(-1.53602e6 + 886819. i) q^{97} +984906. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 26 q^{3} + 2772 q^{7} - 3491 q^{9} - 6516 q^{11} + 5109 q^{13} + 19266 q^{15} - 38403 q^{17} - 43254 q^{19} + 68550 q^{23} + 39380 q^{25} + 432400 q^{27} + 221583 q^{29} + 219756 q^{33} - 659616 q^{35}+ \cdots - 68556288 q^{97}+O(q^{100}) \) 14 * q - 26 * q^3 + 2772 * q^7 - 3491 * q^9 - 6516 * q^11 + 5109 * q^13 + 19266 * q^15 - 38403 * q^17 - 43254 * q^19 + 68550 * q^23 + 39380 * q^25 + 432400 * q^27 + 221583 * q^29 + 219756 * q^33 - 659616 * q^35 - 565347 * q^37 - 30082 * q^39 + 1716057 * q^41 + 882692 * q^43 - 4616853 * q^45 + 2653979 * q^49 + 38964 * q^51 - 1791210 * q^53 - 2453536 * q^55 + 1189032 * q^59 + 3858525 * q^61 - 7718520 * q^63 + 10424661 * q^65 - 4674330 * q^67 - 13743840 * q^69 + 25203126 * q^71 - 10316152 * q^75 + 19599336 * q^77 + 30688440 * q^79 + 13886221 * q^81 - 22436313 * q^85 - 12769686 * q^87 + 2123160 * q^89 - 37205532 * q^91 + 36638964 * q^93 + 43420386 * q^95 - 68556288 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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