LMFDB - Embedded newform 208.8.w.a.49.7

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			49.7
Root			$0.679146i$ of defining polynomial
Character	$\chi$	$=$	208.49
Dual form			208.8.w.a.17.7

$q$-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 $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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	0.978537	+	0.206071i	$0.0660677\pi$
$3$	31.2267	+	54.0862i	0.667731	+	1.15654i	−0.310806	+	0.950473i	$0.600599\pi$
$4$		0			0
$5$			439.155i			1.57117i	0.618754	+	0.785585i	$0.287638\pi$
$5$			439.155i			1.57117i	−0.618754	+	0.785585i	$0.712362\pi$
$6$		0			0
$7$	1128.14	+	651.329i	1.24313	+	0.717724i	0.969731	−	0.244175i	$-0.0785173\pi$
$7$	1128.14	+	651.329i	1.24313	+	0.717724i	0.273403	+	0.961899i	$0.411851\pi$
$8$		0			0
$9$	−856.712	+	1483.87i	−0.391729	+	0.678495i
$10$		0			0
$11$	−497.807	+	287.409i	−0.112768	+	0.0651067i	−0.555323	−	0.831635i	$-0.687405\pi$
$11$	−497.807	+	287.409i	−0.112768	+	0.0651067i	0.442555	+	0.896741i	$0.354072\pi$
$12$		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	−0.970923	−	0.239392i	$-0.923052\pi$
$17$	−5634.27	+	9758.85i	−0.278142	+	0.481756i	0.692781	+	0.721148i	$0.256385\pi$
$18$		0			0
$19$	−35549.9	−	20524.7i	−1.18905	−	0.686499i	−0.230960	−	0.972963i	$-0.574187\pi$
$19$	−35549.9	−	20524.7i	−1.18905	−	0.686499i	−0.958091	+	0.286464i	$0.907520\pi$
$20$		0			0
$21$			81355.4i			1.91699i
$22$		0			0
$23$	22132.8	+	38335.2i	0.379306	+	0.656977i	0.990961	−	0.134147i	$-0.0428294\pi$
$23$	22132.8	+	38335.2i	0.379306	+	0.656977i	−0.611655	+	0.791124i	$0.709496\pi$
$24$		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	0.997984	+	0.0634685i	$0.0202162\pi$
$29$	72756.1	+	126017.i	0.553957	+	0.959482i	−0.444027	+	0.896014i	$0.646450\pi$
$30$		0			0
$31$		−	122045.i		−	0.735793i	−0.929867	−	0.367896i	$-0.880078\pi$
$31$		−	122045.i		−	0.735793i	0.929867	−	0.367896i	$-0.119922\pi$
$32$		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	−0.439808	−	0.898092i	$-0.644954\pi$
$37$	36374.8	−	21001.0i	0.118058	−	0.0681608i	0.557866	+	0.829931i	$0.311620\pi$
$38$		0			0
$39$	−209841.	−	448010.i	−0.566454	−	1.20938i
$40$		0			0
$41$	−75942.3	+	43845.3i	−0.172084	+	0.0993526i	−0.583568	−	0.812064i	$-0.698344\pi$
$41$	−75942.3	+	43845.3i	−0.172084	+	0.0993526i	0.411484	+	0.911417i	$0.365011\pi$
$42$		0			0
$43$	374876.	−	649305.i	0.719032	−	1.24540i	−0.242352	−	0.970188i	$-0.577919\pi$
$43$	374876.	−	649305.i	0.719032	−	1.24540i	0.961384	−	0.275212i	$-0.0887479\pi$
$44$		0			0
$45$	−651649.	−	376230.i	−1.06603	−	0.615473i
$46$		0			0
$47$			940126.i			1.32082i	0.750905	+	0.660410i	$0.229617\pi$
$47$			940126.i			1.32082i	−0.750905	+	0.660410i	$0.770383\pi$
$48$		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	0.316286	−	0.948664i	$-0.397564\pi$
$59$	−547626.	−	316172.i	−0.347138	−	0.200420i	−0.663424	+	0.748244i	$0.730897\pi$
$60$		0			0
$61$	32203.2	−	55777.7i	0.0181654	−	0.0314634i	−0.856800	−	0.515649i	$-0.827551\pi$
$61$	32203.2	−	55777.7i	0.0181654	−	0.0314634i	0.874965	+	0.484186i	$0.160884\pi$
$62$		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	−0.117444	−	0.993079i	$-0.537470\pi$
$67$	1.68358e6	−	972013.i	0.683866	−	0.394830i	0.801310	+	0.598249i	$0.204137\pi$
$68$		0			0
$69$	−1.38227e6	+	2.39416e6i	−0.506549	+	0.877368i
$70$		0			0
$71$	4.09113e6	+	2.36202e6i	1.35656	+	0.783211i	0.989159	−	0.146851i	$-0.0469139\pi$
$71$	4.09113e6	+	2.36202e6i	1.35656	+	0.783211i	0.367402	+	0.930062i	$0.380247\pi$
$72$		0			0
$73$			1.92731e6i			0.579857i	0.957048	+	0.289928i	$0.0936315\pi$
$73$			1.92731e6i			0.579857i	−0.957048	+	0.289928i	$0.906368\pi$
$74$		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	$0.0577470\pi$
$83$			1.87974e6i			0.360848i	−0.983589	+	0.180424i	$0.942253\pi$
$84$		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	−0.187355	−	0.982292i	$-0.559992\pi$
$89$	3.78860e6	−	2.18735e6i	0.569657	−	0.328892i	0.757012	+	0.653400i	$0.226658\pi$
$90$		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	0.412120	−	0.911130i	$-0.364788\pi$
$97$	−1.53602e6	−	886819.i	−0.170881	−	0.0986583i	−0.583001	+	0.812471i	$0.698122\pi$
$98$		0			0
$99$		−	984906.i		−	0.102017i
$100$		0			0
$101$	−4.81973e6	−	8.34802e6i	−0.465477	−	0.806229i	0.533746	−	0.845645i	$-0.320784\pi$
$101$	−4.81973e6	−	8.34802e6i	−0.465477	−	0.806229i	−0.999223	+	0.0394154i	$0.987450\pi$
$102$		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	−0.910869	−	0.412695i	$-0.864587\pi$
$107$	−1.03003e7	−	1.78406e7i	−0.812839	−	1.40788i	0.0980299	−	0.995183i	$-0.468746\pi$
$108$		0			0
$109$		−	1.05713e7i		−	0.781872i	−0.920418	−	0.390936i	$-0.872151\pi$
$109$		−	1.05713e7i		−	0.781872i	0.920418	−	0.390936i	$-0.127849\pi$
$110$		0			0
$111$	2.27173e6	+	1.31158e6i	0.157662	+	0.0910261i
$112$		0			0
$113$	−1.34036e7	+	2.32157e7i	−0.873868	+	1.51358i	−0.0159047	+	0.999874i	$0.505063\pi$
$113$	−1.34036e7	+	2.32157e7i	−0.873868	+	1.51358i	−0.857964	+	0.513711i	$0.828270\pi$
$114$		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	0.940095	−	0.340912i	$-0.110736\pi$
$127$	4.03532e6	+	6.98938e6i	0.174810	+	0.302779i	−0.765286	+	0.643691i	$0.777402\pi$
$128$		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	0.848680	−	0.528906i	$-0.177398\pi$
$137$	2.45282e7	+	1.41614e7i	0.814975	+	0.470526i	−0.0337058	+	0.999432i	$0.510731\pi$
$138$		0			0
$139$	−4.44074e6	+	7.69159e6i	−0.140250	+	0.242920i	−0.927591	−	0.373598i	$-0.878124\pi$
$139$	−4.44074e6	+	7.69159e6i	−0.140250	+	0.242920i	0.787341	+	0.616518i	$0.211457\pi$
$140$		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	−0.156902	−	0.987614i	$-0.550151\pi$
$149$	−4.40391e7	−	2.54260e7i	−1.09065	−	0.629689i	−0.933750	+	0.357926i	$0.883484\pi$
$150$		0			0
$151$			9.76974e6i			0.230921i	0.993312	+	0.115461i	$0.0368344\pi$
$151$			9.76974e6i			0.230921i	−0.993312	+	0.115461i	$0.963166\pi$
$152$		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	0.333409	−	0.942782i	$-0.391801\pi$
$163$	−1.74920e7	−	1.00990e7i	−0.316361	−	0.182651i	−0.649769	+	0.760132i	$0.725134\pi$
$164$		0			0
$165$	7.88268e6	−	1.36532e7i	0.136609	−	0.236614i
$166$		0			0
$167$	5.86371e6	−	3.38542e6i	0.0974238	−	0.0562477i	−0.450497	−	0.892778i	$-0.648753\pi$
$167$	5.86371e6	−	3.38542e6i	0.0974238	−	0.0562477i	0.547920	+	0.836531i	$0.315420\pi$
$168$		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	−0.349603	−	0.936898i	$-0.613684\pi$
$173$	4.33522e7	−	7.50883e7i	0.636576	−	1.10258i	0.986179	−	0.165684i	$-0.0529831\pi$
$174$		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	0.995143	+	0.0984398i	$0.0313852\pi$
$179$	4.47221e7	+	7.74609e7i	0.582823	+	1.00948i	−0.412320	+	0.911039i	$0.635281\pi$
$180$		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	−0.191364	−	0.981519i	$-0.561291\pi$
$191$	7.26412e7	−	1.25818e8i	0.754339	−	1.30655i	0.945702	−	0.325034i	$-0.105376\pi$
$192$		0			0
$193$	−7.90017e7	+	4.56117e7i	−0.791017	+	0.456694i	−0.840321	−	0.542090i	$-0.817633\pi$
$193$	−7.90017e7	+	4.56117e7i	−0.791017	+	0.456694i	0.0493033	+	0.998784i	$0.484300\pi$
$194$		0			0
$195$	1.96746e8	−	9.21528e7i	1.90013	−	0.889995i
$196$		0			0
$197$	1.47447e8	−	8.51285e7i	1.37405	−	0.793311i	0.382619	−	0.923906i	$-0.375022\pi$
$197$	1.47447e8	−	8.51285e7i	1.37405	−	0.793311i	0.991436	+	0.130595i	$0.0416889\pi$
$198$		0			0
$199$	6.97613e7	−	1.20830e8i	0.627521	−	1.08690i	−0.360526	−	0.932749i	$-0.617403\pi$
$199$	6.97613e7	−	1.20830e8i	0.627521	−	1.08690i	0.988047	−	0.154150i	$-0.0492639\pi$
$200$		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	0.977644	−	0.210269i	$-0.0674338\pi$
$211$	4.18539e7	+	7.24931e7i	0.306724	+	0.531261i	−0.670920	+	0.741530i	$0.734101\pi$
$212$		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	−0.230277	−	0.973125i	$-0.573963\pi$
$223$	8.23600e7	−	4.75506e7i	0.497335	−	0.287137i	0.727612	+	0.685988i	$0.240630\pi$
$224$		0			0
$225$	9.82926e7	−	1.70248e8i	0.575283	−	0.996420i
$226$		0			0
$227$	−8.55988e7	−	4.94205e7i	−0.485710	−	0.280425i	0.237083	−	0.971489i	$-0.423809\pi$
$227$	−8.55988e7	−	4.94205e7i	−0.485710	−	0.280425i	−0.722793	+	0.691064i	$0.757142\pi$
$228$		0			0
$229$			8.72426e7i			0.480070i	0.970764	+	0.240035i	$0.0771589\pi$
$229$			8.72426e7i			0.480070i	−0.970764	+	0.240035i	$0.922841\pi$
$230$		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	$0.418994\pi$
$239$			4.08512e8i			1.93558i	−0.251750	+	0.967792i	$0.581006\pi$
$240$		0			0
$241$	1.89875e8	+	1.09624e8i	0.873792	+	0.504484i	0.868606	−	0.495503i	$-0.165016\pi$
$241$	1.89875e8	+	1.09624e8i	0.873792	+	0.504484i	0.00518518	+	0.999987i	$0.498349\pi$
$242$		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	−0.880470	−	0.474102i	$-0.842773\pi$
$251$	−7.42822e6	+	1.28661e7i	−0.0296501	+	0.0513556i	0.850820	+	0.525458i	$0.176106\pi$
$252$		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	0.802432	+	0.596744i	$0.203539\pi$
$257$	2.49812e8	+	4.32688e8i	0.918011	+	1.59004i	0.115579	+	0.993298i	$0.463128\pi$
$258$		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	0.890594	−	0.454799i	$-0.150289\pi$
$263$	1.51725e7	+	2.62796e7i	0.0514295	+	0.0890785i	−0.839165	+	0.543878i	$0.816956\pi$
$264$		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	−0.939503	−	0.342541i	$-0.888712\pi$
$269$	−5.52630e7	+	9.57183e7i	−0.173102	+	0.299821i	0.766401	+	0.642363i	$0.222046\pi$
$270$		0			0
$271$	−4.36530e7	+	2.52031e7i	−0.133236	+	0.0769238i	−0.565136	−	0.824997i	$-0.691176\pi$
$271$	−4.36530e7	+	2.52031e7i	−0.133236	+	0.0769238i	0.431901	+	0.901921i	$0.357843\pi$
$272$		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	0.180889	+	0.983503i	$0.442102\pi$
$277$	−2.69297e8	+	4.66436e8i	−0.761294	+	1.31860i	−0.942184	+	0.335097i	$0.891231\pi$
$278$		0			0
$279$	1.81099e8	+	1.04558e8i	0.499232	+	0.288232i
$280$		0			0
$281$			1.43482e8i			0.385766i	0.981222	+	0.192883i	$0.0617839\pi$
$281$			1.43482e8i			0.385766i	−0.981222	+	0.192883i	$0.938216\pi$
$282$		0			0
$283$	−5.06236e7	−	8.76826e7i	−0.132770	−	0.229964i	0.791973	−	0.610556i	$-0.209054\pi$
$283$	−5.06236e7	−	8.76826e7i	−0.132770	−	0.229964i	−0.924743	+	0.380591i	$0.875721\pi$
$284$		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	0.159252	−	0.987238i	$-0.449092\pi$
$293$	−2.65267e8	−	1.53152e8i	−0.616094	−	0.355702i	−0.775347	+	0.631536i	$0.782425\pi$
$294$		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	$-0.997217\pi$
$307$		−	8.86416e6i		−	0.0174845i	0.999962	−	0.00874225i	$-0.00278278\pi$
$308$		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	$0.0969746\pi$
$317$			3.40256e8i			0.599928i	−0.953951	+	0.299964i	$0.903025\pi$
$318$		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	0.815177	−	0.579211i	$-0.196639\pi$
$331$	4.75802e8	+	2.74704e8i	0.721154	+	0.416359i	−0.0940230	+	0.995570i	$0.529973\pi$
$332$		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	−0.997797	−	0.0663474i	$-0.978865\pi$
$347$	−3.43577e8	+	5.95094e8i	−0.441440	+	0.764596i	0.556357	+	0.830943i	$0.312199\pi$
$348$		0			0
$349$	−8.95711e8	+	5.17139e8i	−1.12792	+	0.651206i	−0.943411	−	0.331625i	$-0.892403\pi$
$349$	−8.95711e8	+	5.17139e8i	−1.12792	+	0.651206i	−0.184510	+	0.982831i	$0.559070\pi$
$350$		0			0
$351$	−2.33430e8	−	2.00214e7i	−0.288125	−	0.0247127i
$352$		0			0
$353$	9.29168e8	−	5.36455e8i	1.12430	−	0.649116i	0.181806	−	0.983334i	$-0.441806\pi$
$353$	9.29168e8	−	5.36455e8i	1.12430	−	0.649116i	0.942496	+	0.334219i	$0.108472\pi$
$354$		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	$0.122374\pi$
$359$			6.57582e8i			0.750100i	−0.927005	+	0.375050i	$0.877626\pi$
$360$		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	0.527102	−	0.849802i	$-0.323279\pi$
$367$	−4.47343e8	−	7.74821e8i	−0.472400	−	0.818220i	−0.999501	+	0.0315822i	$0.989945\pi$
$368$		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	−0.984677	−	0.174385i	$-0.944206\pi$
$373$	−3.42088e8	+	5.92515e8i	−0.341317	+	0.591178i	0.643361	+	0.765563i	$0.277540\pi$
$374$		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	0.0440788	−	0.999028i	$-0.485965\pi$
$379$	9.87024e8	−	5.69859e8i	0.931302	−	0.537687i	0.887223	+	0.461341i	$0.152631\pi$
$380$		0			0
$381$	−2.52019e8	+	4.36510e8i	−0.233451	+	0.404350i
$382$		0			0
$383$	−1.16426e9	−	6.72186e8i	−1.05890	−	0.611355i	−0.133771	−	0.991012i	$-0.542709\pi$
$383$	−1.16426e9	−	6.72186e8i	−1.05890	−	0.611355i	−0.925127	+	0.379657i	$0.876042\pi$
$384$		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	0.566821	−	0.823841i	$-0.308173\pi$
$397$	1.70506e8	+	9.84416e7i	0.136764	+	0.0789609i	−0.430057	+	0.902802i	$0.641506\pi$
$398$		0			0
$399$	1.66980e9	−	2.89218e9i	1.31601	−	2.27939i
$400$		0			0
$401$	8.66177e8	−	5.00088e8i	0.670813	−	0.387294i	−0.125572	−	0.992085i	$-0.540077\pi$
$401$	8.66177e8	−	5.00088e8i	0.670813	−	0.387294i	0.796385	+	0.604791i	$0.206743\pi$
$402$		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	0.913075	−	0.407793i	$-0.133701\pi$
$409$	1.40643e9	+	8.12004e8i	1.01645	+	0.586850i	0.103379	+	0.994642i	$0.467035\pi$
$410$		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	0.832400	−	0.554175i	$-0.186966\pi$
$419$	−9.59601e7	−	1.66208e8i	−0.0637297	−	0.110383i	−0.896130	+	0.443792i	$0.853633\pi$
$420$		0			0
$421$		−	1.17016e9i		−	0.764292i	−0.924102	−	0.382146i	$-0.875185\pi$
$421$		−	1.17016e9i		−	0.764292i	0.924102	−	0.382146i	$-0.124815\pi$
$422$		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	−0.443864	−	0.896094i	$-0.646393\pi$
$431$	1.83241e8	−	1.05794e8i	0.110243	−	0.0636491i	0.554108	+	0.832445i	$0.313059\pi$
$432$		0			0
$433$	−5.87978e8	+	1.01841e9i	−0.348060	+	0.602857i	−0.985905	−	0.167308i	$-0.946493\pi$
$433$	−5.87978e8	+	1.01841e9i	−0.348060	+	0.602857i	0.637845	+	0.770165i	$0.279826\pi$
$434$		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	0.973812	+	0.227356i	$0.0730081\pi$
$439$	1.21215e9	+	2.09950e9i	0.683802	+	1.18438i	−0.290010	+	0.957024i	$0.593659\pi$
$440$		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	0.494008	−	0.869457i	$-0.335531\pi$
$449$	−2.29405e7	−	1.32447e7i	−0.0119603	−	0.00690525i	−0.505968	+	0.862552i	$0.668865\pi$
$450$		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	−0.941640	−	0.336623i	$-0.890715\pi$
$457$	−2.28711e9	+	1.32046e9i	−1.12094	+	0.647172i	−0.179296	+	0.983795i	$0.557382\pi$
$458$		0			0
$459$	−1.66642e8	+	2.88632e8i	−0.0804339	+	0.139316i
$460$		0			0
$461$	−1.91120e8	−	1.10343e8i	−0.0908558	−	0.0524556i	0.453884	−	0.891061i	$-0.350038\pi$
$461$	−1.91120e8	−	1.10343e8i	−0.0908558	−	0.0524556i	−0.544740	+	0.838605i	$0.683371\pi$
$462$		0			0
$463$			1.40461e9i			0.657691i	0.944384	+	0.328845i	$0.106659\pi$
$463$			1.40461e9i			0.657691i	−0.944384	+	0.328845i	$0.893341\pi$
$464$		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	−0.0568141	−	0.998385i	$-0.518094\pi$
$479$	1.87473e9	−	1.08237e9i	0.779405	−	0.449990i	0.836220	+	0.548395i	$0.184761\pi$
$480$		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	−0.262474	−	0.964939i	$-0.584538\pi$
$487$	−3.13354e9	−	1.80915e9i	−1.22937	−	0.709779i	−0.966899	+	0.255160i	$0.917872\pi$
$488$		0			0
$489$		−	1.26143e9i		−	0.487847i
$490$		0			0
$491$	8.96441e8	+	1.55268e9i	0.341772	+	0.591966i	0.984762	−	0.173908i	$-0.0556397\pi$
$491$	8.96441e8	+	1.55268e9i	0.341772	+	0.591966i	−0.642990	+	0.765875i	$0.722306\pi$
$492$		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	$-0.890149\pi$
$499$		−	1.87794e9i		−	0.676596i	0.941039	−	0.338298i	$-0.109851\pi$
$500$		0			0
$501$	3.66209e8	+	2.11431e8i	0.130106	+	0.0751166i
$502$		0			0
$503$	−5.65977e7	+	9.80300e7i	−0.0198294	+	0.0343456i	−0.875770	−	0.482729i	$-0.839646\pi$
$503$	−5.65977e7	+	9.80300e7i	−0.0198294	+	0.0343456i	0.855940	+	0.517074i	$0.172979\pi$
$504$		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	0.584517	−	0.811381i	$-0.301284\pi$
$509$	4.69915e9	−	2.71305e9i	1.57945	−	0.911898i	0.994935	−	0.100516i	$-0.0320495\pi$
$510$		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	0.989828	+	0.142268i	$0.0454395\pi$
$523$	2.02223e9	+	3.50260e9i	0.618122	+	1.07062i	−0.371706	+	0.928350i	$0.621227\pi$
$524$		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	$0.396390\pi$
$541$			6.97904e9i			1.89498i	−0.319782	+	0.947491i	$0.603610\pi$
$542$		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	−0.476778	−	0.879024i	$-0.658195\pi$
$557$	1.87977e8	−	1.08528e8i	0.0460905	−	0.0266103i	0.522868	+	0.852414i	$0.324862\pi$
$558$		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	0.430151	+	0.902757i	$0.358460\pi$
$563$	−2.39971e9	+	4.15643e9i	−0.566735	+	0.981614i	−0.996886	+	0.0788572i	$0.974873\pi$
$564$		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	0.993629	+	0.112699i	$0.0359497\pi$
$569$	2.61206e9	+	4.52422e9i	0.594415	+	1.02956i	−0.399214	+	0.916858i	$0.630717\pi$
$570$		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	$0.000255829\pi$
$577$			7.41729e6i			0.00160742i	−1.00000		0.000803711i	$0.999744\pi$
$578$		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	−0.926007	−	0.377505i	$-0.876782\pi$
$587$	−5.20465e9	+	3.00490e9i	−1.06208	+	0.613193i	−0.136074	+	0.990699i	$0.543449\pi$
$588$		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	$0.0315381\pi$
$593$			1.00461e9i			0.197836i	−0.995096	+	0.0989178i	$0.968462\pi$
$594$		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	−0.996697	−	0.0812066i	$-0.974123\pi$
$601$	−3.02639e9	−	5.24187e9i	−0.568676	−	0.984975i	0.428022	−	0.903768i	$-0.359211\pi$
$602$		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	0.337152	+	0.941450i	$0.390536\pi$
$607$	−3.56363e9	+	6.17239e9i	−0.646744	+	1.12019i	−0.983896	+	0.178743i	$0.942797\pi$
$608$		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	−0.574045	−	0.818824i	$-0.694627\pi$
$613$	−8.66562e8	+	5.00310e8i	−0.151946	+	0.0877258i	0.422100	+	0.906549i	$0.361293\pi$
$614$		0			0
$615$	1.20253e9	−	2.08285e9i	0.208465	−	0.361073i
$616$		0			0
$617$	−3.21958e9	−	1.85882e9i	−0.551825	−	0.318596i	0.198033	−	0.980195i	$-0.436545\pi$
$617$	−3.21958e9	−	1.85882e9i	−0.551825	−	0.318596i	−0.749858	+	0.661599i	$0.769878\pi$
$618$		0			0
$619$		−	8.30664e9i		−	1.40769i	−0.710352	−	0.703847i	$-0.751464\pi$
$619$		−	8.30664e9i		−	1.40769i	0.710352	−	0.703847i	$-0.248536\pi$
$620$		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	−0.145963	−	0.989290i	$-0.546628\pi$
$631$	−6.78879e9	−	3.91951e9i	−1.07570	−	0.621053i	−0.929732	+	0.368237i	$0.879962\pi$
$632$		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	−0.814382	−	0.580330i	$-0.802924\pi$
$641$	6.36068e8	−	1.10170e9i	0.0953895	−	0.165220i	0.909771	+	0.415110i	$0.136257\pi$
$642$		0			0
$643$	−4.91826e9	−	2.83956e9i	−0.729581	−	0.421224i	0.0886881	−	0.996059i	$-0.471733\pi$
$643$	−4.91826e9	−	2.83956e9i	−0.729581	−	0.421224i	−0.818269	+	0.574836i	$0.805066\pi$
$644$		0			0
$645$			2.05633e10i			3.01740i
$646$		0			0
$647$	4.91513e9	+	8.51325e9i	0.713460	+	1.23575i	0.963551	+	0.267526i	$0.0862063\pi$
$647$	4.91513e9	+	8.51325e9i	0.713460	+	1.23575i	−0.250091	+	0.968222i	$0.580460\pi$
$648$		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	0.969075	−	0.246765i	$-0.0793677\pi$
$653$	1.92707e9	+	3.33778e9i	0.270832	+	0.469096i	−0.698243	+	0.715861i	$0.746034\pi$
$654$		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	−0.983388	−	0.181515i	$-0.941900\pi$
$659$	−2.45749e9	+	4.25650e9i	−0.334498	+	0.579367i	0.648891	+	0.760882i	$0.275233\pi$
$660$		0			0
$661$	1.10777e10	−	6.39573e9i	1.49192	−	0.861360i	0.491962	−	0.870616i	$-0.336280\pi$
$661$	1.10777e10	−	6.39573e9i	1.49192	−	0.861360i	0.999957	+	0.00925625i	$0.00294640\pi$
$662$		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	−0.970322	−	0.241818i	$-0.922256\pi$
$673$	−5.49257e9	−	9.51341e9i	−0.694581	−	1.20305i	0.275741	−	0.961232i	$-0.411077\pi$
$674$		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	0.989561	+	0.144113i	$0.0460328\pi$
$683$	1.33989e10	+	7.73584e9i	1.60915	+	0.929042i	0.619586	+	0.784929i	$0.287301\pi$
$684$		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	−0.251110	−	0.967959i	$-0.580796\pi$
$691$	4.00359e9	−	2.31148e9i	0.461612	−	0.266512i	0.712722	+	0.701447i	$0.247462\pi$
$692$		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	0.824124	−	0.566410i	$-0.191668\pi$
$709$	7.07624e9	+	4.08547e9i	0.745660	+	0.430507i	−0.0784635	+	0.996917i	$0.525001\pi$
$710$		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	−0.161752	−	0.986831i	$-0.551714\pi$
$719$	7.71168e9	−	1.33570e10i	0.773745	−	1.34017i	0.935497	−	0.353334i	$-0.114952\pi$
$720$		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	$-0.714841\pi$
$733$		−	1.66495e10i		−	1.56148i	0.624854	−	0.780742i	$-0.285159\pi$
$734$		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	−0.292759	−	0.956186i	$-0.594573\pi$
$739$	4.26718e9	−	2.46366e9i	0.388943	−	0.224556i	0.681702	+	0.731630i	$0.261240\pi$
$740$		0			0
$741$	−1.73545e9	+	2.02336e10i	−0.156693	+	1.82688i
$742$		0			0
$743$	8.58753e9	−	4.95801e9i	0.768082	−	0.443452i	−0.0641082	−	0.997943i	$-0.520420\pi$
$743$	8.58753e9	−	4.95801e9i	0.768082	−	0.443452i	0.832190	+	0.554491i	$0.187087\pi$
$744$		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	−0.694180	−	0.719801i	$-0.744233\pi$
$751$	−1.12646e10	−	1.95109e10i	−0.970456	−	1.68088i	−0.276276	−	0.961078i	$-0.589100\pi$
$752$		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	0.785846	+	0.618423i	$0.212228\pi$
$757$	1.10819e10	+	1.91944e10i	0.928493	+	1.60820i	0.142647	+	0.989774i	$0.454439\pi$
$758$		0			0
$759$		−	1.58911e9i		−	0.131919i
$760$		0			0
$761$	6.64485e9	+	3.83641e9i	0.546561	+	0.315557i	0.747734	−	0.663999i	$-0.231142\pi$
$761$	6.64485e9	+	3.83641e9i	0.546561	+	0.315557i	−0.201173	+	0.979556i	$0.564475\pi$
$762$		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	−0.775528	−	0.631314i	$-0.782516\pi$
$769$	−7.77528e9	+	4.48906e9i	−0.616558	+	0.355970i	0.158970	+	0.987283i	$0.449183\pi$
$770$		0			0
$771$	−1.56016e10	+	2.70228e10i	−1.22597	+	2.12344i
$772$		0			0
$773$	9.70027e9	+	5.60045e9i	0.755363	+	0.436109i	0.827628	−	0.561276i	$-0.189690\pi$
$773$	9.70027e9	+	5.60045e9i	0.755363	+	0.436109i	−0.0722655	+	0.997385i	$0.523023\pi$
$774$		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	0.534594	−	0.845109i	$-0.320464\pi$
$787$	9.57297e8	+	5.52696e8i	0.0700060	+	0.0404180i	−0.464588	+	0.885527i	$0.653798\pi$
$788$		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	0.475052	+	0.879958i	$0.342429\pi$
$797$	−7.49691e9	+	1.29850e10i	−0.524540	+	0.908529i	−0.999592	+	0.0285717i	$0.990904\pi$
$798$		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	−0.999949	−	0.0100514i	$-0.996800\pi$
$809$	−7.66062e9	−	1.32686e10i	−0.508679	−	0.881059i	0.491270	−	0.871007i	$-0.336533\pi$
$810$		0			0
$811$			1.14391e10i			0.753041i	0.926408	+	0.376521i	$0.122880\pi$
$811$			1.14391e10i			0.753041i	−0.926408	+	0.376521i	$0.877120\pi$
$812$		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	0.231339	−	0.972873i	$-0.425689\pi$
$821$	1.88617e10	−	1.08898e10i	1.18954	−	0.686782i	0.958202	+	0.286091i	$0.0923561\pi$
$822$		0			0
$823$	1.08157e9	−	1.87333e9i	0.0676324	−	0.117143i	−0.830226	−	0.557426i	$-0.811789\pi$
$823$	1.08157e9	−	1.87333e9i	0.0676324	−	0.117143i	0.897859	+	0.440284i	$0.145122\pi$
$824$		0			0
$825$	3.56699e9	+	2.05941e9i	0.221164	+	0.127689i
$826$		0			0
$827$		−	2.27847e10i		−	1.40079i	−0.713755	−	0.700395i	$-0.753007\pi$
$827$		−	2.27847e10i		−	1.40079i	0.713755	−	0.700395i	$-0.246993\pi$
$828$		0			0
$829$	4.91919e9	+	8.52030e9i	0.299884	+	0.519414i	0.976109	−	0.217281i	$-0.0697187\pi$
$829$	4.91919e9	+	8.52030e9i	0.299884	+	0.519414i	−0.676225	+	0.736695i	$0.736385\pi$
$830$		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	0.925276	−	0.379294i	$-0.123833\pi$
$839$	1.81235e10	+	1.04636e10i	1.05944	+	0.611666i	0.134160	+	0.990960i	$0.457166\pi$
$840$		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	$-0.754452\pi$
$853$		−	2.52741e10i		−	1.39429i	0.716928	−	0.697147i	$-0.245548\pi$
$854$		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	$-0.949677\pi$
$863$		−	5.94525e9i		−	0.314871i	0.987529	−	0.157435i	$-0.0503226\pi$
$864$		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	0.712697	−	0.701472i	$-0.247474\pi$
$877$	9.21976e9	+	5.32303e9i	0.461552	+	0.266477i	−0.251144	+	0.967950i	$0.580807\pi$
$878$		0			0
$879$		−	1.91297e10i		−	0.950054i
$880$		0			0
$881$	−1.91599e9	−	3.31860e9i	−0.0944014	−	0.163508i	0.814957	−	0.579521i	$-0.196760\pi$
$881$	−1.91599e9	−	3.31860e9i	−0.0944014	−	0.163508i	−0.909359	+	0.416013i	$0.863427\pi$
$882$		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	0.707333	−	0.706880i	$-0.249898\pi$
$887$	−5.37291e9	−	9.30616e9i	−0.258510	−	0.447752i	−0.965843	+	0.259128i	$0.916565\pi$
$888$		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	0.999013	−	0.0444245i	$-0.0141454\pi$
$907$	1.03600e10	+	1.79440e10i	0.461034	+	0.798534i	−0.537979	+	0.842958i	$0.680812\pi$
$908$		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	−0.651046	−	0.759038i	$-0.725670\pi$
$919$	7.80748e9	−	1.35229e10i	0.331823	−	0.574734i	0.982869	+	0.184304i	$0.0590030\pi$
$920$		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	0.901648	−	0.432471i	$-0.142358\pi$
$929$	2.38983e10	+	1.37977e10i	0.977941	+	0.564615i	0.0762934	+	0.997085i	$0.475691\pi$
$930$		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	$-0.735940\pi$
$941$		−	3.77084e10i		−	1.47528i	0.675193	−	0.737641i	$-0.264060\pi$
$942$		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	0.518340	−	0.855174i	$-0.326550\pi$
$947$	3.96761e10	−	2.29070e10i	1.51811	−	0.876483i	0.999773	−	0.0213086i	$-0.00678325\pi$
$948$		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	−0.642391	−	0.766377i	$-0.722057\pi$
$953$	9.15154e9	−	1.58509e10i	0.342507	−	0.593239i	0.984898	+	0.173138i	$0.0553907\pi$
$954$		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	$0.154272\pi$
$967$			2.62013e10i			0.931816i	−0.884833	+	0.465908i	$0.845728\pi$
$968$		0			0
$969$	2.50185e10	+	1.44445e10i	0.883342	+	0.509998i
$970$		0			0
$971$	1.52480e9	−	2.64103e9i	0.0534497	−	0.0925777i	−0.838063	−	0.545574i	$-0.816312\pi$
$971$	1.52480e9	−	2.64103e9i	0.0534497	−	0.0925777i	0.891512	+	0.452996i	$0.149645\pi$
$972$		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	0.327074	−	0.944999i	$-0.393937\pi$
$977$	3.81568e10	−	2.20298e10i	1.30900	−	0.755754i	0.981930	+	0.189245i	$0.0606040\pi$
$978$		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	$-0.819889\pi$
$983$		−	3.19323e10i		−	1.07224i	0.844141	−	0.536122i	$-0.180111\pi$
$984$		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	0.993075	+	0.117478i	$0.0374811\pi$
$991$	1.83299e10	+	3.17483e10i	0.598277	+	1.03625i	−0.394798	+	0.918768i	$0.629186\pi$
$992$		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	−0.164495	+	0.986378i	$0.552600\pi$
$997$	−2.93042e10	+	5.07563e10i	−0.936476	+	1.62202i	−0.771981	+	0.635646i	$0.780734\pi$
$998$		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.49.7		14
4.3	odd	2		13.8.e.a.10.4	yes	14
12.11	even	2		117.8.q.b.10.4		14
13.4	even	6	inner	208.8.w.a.17.7		14
52.3	odd	6		169.8.b.d.168.8		14
52.11	even	12		169.8.a.g.1.7		14
52.15	even	12		169.8.a.g.1.8		14
52.23	odd	6		169.8.b.d.168.7		14
52.43	odd	6		13.8.e.a.4.4	✓	14
156.95	even	6		117.8.q.b.82.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.4	✓	14	52.43	odd	6
13.8.e.a.10.4	yes	14	4.3	odd	2
117.8.q.b.10.4		14	12.11	even	2
117.8.q.b.82.4		14	156.95	even	6
169.8.a.g.1.7		14	52.11	even	12
169.8.a.g.1.8		14	52.15	even	12
169.8.b.d.168.7		14	52.23	odd	6
169.8.b.d.168.8		14	52.3	odd	6
208.8.w.a.17.7		14	13.4	even	6	inner
208.8.w.a.49.7		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Related objects

Downloads

Learn more