LMFDB - Embedded newform 448.4.p.g.383.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(255,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.255");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			383.1
Root			$0.426664 + 0.739004i$ of defining polynomial
Character	$\chi$	$=$	448.383
Dual form			448.4.p.g.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.53129 - 6.11637i) q^{3} +(-1.05999 - 0.611983i) q^{5} +(16.5026 + 8.40621i) q^{7} +(-11.4400 + 19.8147i) q^{9} +O(q^{10})$	\(q+(-3.53129 - 6.11637i) q^{3} +(-1.05999 - 0.611983i) q^{5} +(16.5026 + 8.40621i) q^{7} +(-11.4400 + 19.8147i) q^{9} +(-3.96612 + 2.28984i) q^{11} -41.0611i q^{13} +8.64436i q^{15} +(-103.451 + 59.7275i) q^{17} +(-30.0913 + 52.1196i) q^{19} +(-6.85991 - 130.621i) q^{21} +(-7.23396 - 4.17653i) q^{23} +(-61.7510 - 106.956i) q^{25} -29.0976 q^{27} -164.382 q^{29} +(143.780 + 249.034i) q^{31} +(28.0110 + 16.1722i) q^{33} +(-12.3481 - 19.0098i) q^{35} +(-74.3709 + 128.814i) q^{37} +(-251.145 + 144.999i) q^{39} -358.778i q^{41} +360.388i q^{43} +(24.2525 - 14.0022i) q^{45} +(112.717 - 195.232i) q^{47} +(201.671 + 277.449i) q^{49} +(730.631 + 421.830i) q^{51} +(342.362 + 592.988i) q^{53} +5.60537 q^{55} +425.044 q^{57} +(-42.5261 - 73.6574i) q^{59} +(181.109 + 104.563i) q^{61} +(-355.356 + 230.827i) q^{63} +(-25.1287 + 43.5242i) q^{65} +(-361.327 + 208.612i) q^{67} +58.9941i q^{69} +982.974i q^{71} +(295.770 - 170.763i) q^{73} +(-436.121 + 755.384i) q^{75} +(-84.7001 + 4.44826i) q^{77} +(66.0721 + 38.1467i) q^{79} +(411.633 + 712.968i) q^{81} -523.643 q^{83} +146.209 q^{85} +(580.480 + 1005.42i) q^{87} +(-841.824 - 486.027i) q^{89} +(345.169 - 677.615i) q^{91} +(1015.46 - 1758.82i) q^{93} +(63.7926 - 36.8307i) q^{95} +676.730i q^{97} -104.783i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9}+O(q^{10}) $ 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9	$ 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} + 15 q^{21} - 15 q^{23} + 306 q^{25} - 362 q^{27} + 348 q^{29} + 205 q^{31} - 471 q^{33} + 1185 q^{35} + 249 q^{37} + 288 q^{39} - 2118 q^{45} + 75 q^{47} + 702 q^{49} + 2505 q^{51} + 645 q^{53} + 918 q^{55} - 6 q^{57} - 321 q^{59} + 1707 q^{61} - 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} - 555 q^{77} - 3447 q^{79} + 225 q^{81} - 24 q^{83} - 3786 q^{85} + 3642 q^{87} - 2607 q^{89} + 2448 q^{91} + 2991 q^{93} + 2085 q^{95}+O(q^{100}) $ 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 + 15 * q^21 - 15 * q^23 + 306 * q^25 - 362 * q^27 + 348 * q^29 + 205 * q^31 - 471 * q^33 + 1185 * q^35 + 249 * q^37 + 288 * q^39 - 2118 * q^45 + 75 * q^47 + 702 * q^49 + 2505 * q^51 + 645 * q^53 + 918 * q^55 - 6 * q^57 - 321 * q^59 + 1707 * q^61 - 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 - 555 * q^77 - 3447 * q^79 + 225 * q^81 - 24 * q^83 - 3786 * q^85 + 3642 * q^87 - 2607 * q^89 + 2448 * q^91 + 2991 * q^93 + 2085 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−3.53129	−	6.11637i	−0.679597	−	1.17710i	−0.975102	−	0.221755i	$-0.928821\pi$
$3$	−3.53129	−	6.11637i	−0.679597	−	1.17710i	0.295505	−	0.955341i	$-0.404512\pi$
$4$		0			0
$5$	−1.05999	−	0.611983i	−0.0948080	−	0.0547374i	0.451846	−	0.892096i	$-0.350766\pi$
$5$	−1.05999	−	0.611983i	−0.0948080	−	0.0547374i	−0.546654	+	0.837358i	$0.684099\pi$
$6$		0			0
$7$	16.5026	+	8.40621i	0.891056	+	0.453893i
$7$	16.5026	+	8.40621i	0.891056	+	0.453893i
$8$		0			0
$9$	−11.4400	+	19.8147i	−0.423704	+	0.733877i
$10$		0			0
$11$	−3.96612	+	2.28984i	−0.108712	+	0.0627647i	−0.553370	−	0.832936i	$-0.686658\pi$
$11$	−3.96612	+	2.28984i	−0.108712	+	0.0627647i	0.444658	+	0.895700i	$0.353325\pi$
$12$		0			0
$13$		−	41.0611i		−	0.876024i	−0.898969	−	0.438012i	$-0.855683\pi$
$13$		−	41.0611i		−	0.876024i	0.898969	−	0.438012i	$-0.144317\pi$
$14$		0			0
$15$			8.64436i			0.148798i
$16$		0			0
$17$	−103.451	+	59.7275i	−1.47592	+	0.852120i	−0.999631	−	0.0271712i	$-0.991350\pi$
$17$	−103.451	+	59.7275i	−1.47592	+	0.852120i	−0.476284	+	0.879291i	$0.658017\pi$
$18$		0			0
$19$	−30.0913	+	52.1196i	−0.363337	+	0.629319i	−0.988508	−	0.151169i	$-0.951696\pi$
$19$	−30.0913	+	52.1196i	−0.363337	+	0.629319i	0.625170	+	0.780488i	$0.285029\pi$
$20$		0			0
$21$	−6.85991	−	130.621i	−0.0712836	−	1.35732i
$22$		0			0
$23$	−7.23396	−	4.17653i	−0.0655819	−	0.0378637i	0.466850	−	0.884336i	$-0.345389\pi$
$23$	−7.23396	−	4.17653i	−0.0655819	−	0.0378637i	−0.532432	+	0.846473i	$0.678722\pi$
$24$		0			0
$25$	−61.7510	−	106.956i	−0.494008	−	0.855646i
$26$		0			0
$27$	−29.0976			−0.207401
$28$		0			0
$29$	−164.382			−1.05258			−0.526292	−	0.850304i	$-0.676418\pi$
$29$	−164.382			−1.05258			−0.526292	+	0.850304i	$0.676418\pi$
$30$		0			0
$31$	143.780	+	249.034i	0.833019	+	1.44283i	0.895633	+	0.444793i	$0.146723\pi$
$31$	143.780	+	249.034i	0.833019	+	1.44283i	−0.0626143	+	0.998038i	$0.519944\pi$
$32$		0			0
$33$	28.0110	+	16.1722i	0.147760	+	0.0853095i
$34$		0			0
$35$	−12.3481	−	19.0098i	−0.0596343	−	0.0918068i
$36$		0			0
$37$	−74.3709	+	128.814i	−0.330446	+	0.572349i	−0.982599	−	0.185738i	$-0.940533\pi$
$37$	−74.3709	+	128.814i	−0.330446	+	0.572349i	0.652153	+	0.758087i	$0.273866\pi$
$38$		0			0
$39$	−251.145	+	144.999i	−1.03116	+	0.595343i
$40$		0			0
$41$		−	358.778i		−	1.36663i	−0.730125	−	0.683314i	$-0.760538\pi$
$41$		−	358.778i		−	1.36663i	0.730125	−	0.683314i	$-0.239462\pi$
$42$		0			0
$43$			360.388i			1.27811i	0.769161	+	0.639055i	$0.220674\pi$
$43$			360.388i			1.27811i	−0.769161	+	0.639055i	$0.779326\pi$
$44$		0			0
$45$	24.2525	−	14.0022i	0.0803411	−	0.0463850i
$46$		0			0
$47$	112.717	−	195.232i	0.349819	−	0.605904i	−0.636398	−	0.771361i	$-0.719577\pi$
$47$	112.717	−	195.232i	0.349819	−	0.605904i	0.986217	+	0.165457i	$0.0529099\pi$
$48$		0			0
$49$	201.671	+	277.449i	0.587963	+	0.808888i
$50$		0			0
$51$	730.631	+	421.830i	2.00606	+	1.15820i
$52$		0			0
$53$	342.362	+	592.988i	0.887302	+	1.53685i	0.843052	+	0.537832i	$0.180756\pi$
$53$	342.362	+	592.988i	0.887302	+	1.53685i	0.0442500	+	0.999020i	$0.485910\pi$
$54$		0			0
$55$	5.60537			0.0137423
$56$		0			0
$57$	425.044			0.987692
$58$		0			0
$59$	−42.5261	−	73.6574i	−0.0938377	−	0.162532i	0.815285	−	0.579060i	$-0.196580\pi$
$59$	−42.5261	−	73.6574i	−0.0938377	−	0.162532i	−0.909123	+	0.416528i	$0.863247\pi$
$60$		0			0
$61$	181.109	+	104.563i	0.380141	+	0.219475i	0.677880	−	0.735173i	$-0.262899\pi$
$61$	181.109	+	104.563i	0.380141	+	0.219475i	−0.297738	+	0.954648i	$0.596232\pi$
$62$		0			0
$63$	−355.356	+	230.827i	−0.710646	+	0.461610i
$64$		0			0
$65$	−25.1287	+	43.5242i	−0.0479513	+	0.0830541i
$66$		0			0
$67$	−361.327	+	208.612i	−0.658853	+	0.380389i	−0.791840	−	0.610729i	$-0.790877\pi$
$67$	−361.327	+	208.612i	−0.658853	+	0.380389i	0.132987	+	0.991118i	$0.457543\pi$
$68$		0			0
$69$			58.9941i			0.102928i
$70$		0			0
$71$			982.974i			1.64306i	0.570162	+	0.821532i	$0.306880\pi$
$71$			982.974i			1.64306i	−0.570162	+	0.821532i	$0.693120\pi$
$72$		0			0
$73$	295.770	−	170.763i	0.474210	−	0.273785i	−0.243791	−	0.969828i	$-0.578391\pi$
$73$	295.770	−	170.763i	0.474210	−	0.273785i	0.718000	+	0.696043i	$0.245058\pi$
$74$		0			0
$75$	−436.121	+	755.384i	−0.671452	+	1.16299i
$76$		0			0
$77$	−84.7001	+	4.44826i	−0.125357	+	0.00658346i
$78$		0			0
$79$	66.0721	+	38.1467i	0.0940973	+	0.0543271i	0.546310	−	0.837583i	$-0.316032\pi$
$79$	66.0721	+	38.1467i	0.0940973	+	0.0543271i	−0.452213	+	0.891910i	$0.649365\pi$
$80$		0			0
$81$	411.633	+	712.968i	0.564654	+	0.978009i
$82$		0			0
$83$	−523.643			−0.692498			−0.346249	−	0.938143i	$-0.612545\pi$
$83$	−523.643			−0.692498			−0.346249	+	0.938143i	$0.612545\pi$
$84$		0			0
$85$	146.209			0.186571
$86$		0			0
$87$	580.480	+	1005.42i	0.715334	+	1.23899i
$88$		0			0
$89$	−841.824	−	486.027i	−1.00262	−	0.578863i	−0.0935976	−	0.995610i	$-0.529837\pi$
$89$	−841.824	−	486.027i	−1.00262	−	0.578863i	−0.909022	+	0.416747i	$0.863170\pi$
$90$		0			0
$91$	345.169	−	677.615i	0.397621	−	0.780587i
$92$		0			0
$93$	1015.46	−	1758.82i	1.13223	−	1.96109i
$94$		0			0
$95$	63.7926	−	36.8307i	0.0688946	−	0.0397763i
$96$		0			0
$97$			676.730i			0.708367i	0.935176	+	0.354183i	$0.115241\pi$
$97$			676.730i			0.708367i	−0.935176	+	0.354183i	$0.884759\pi$
$98$		0			0
$99$		−	104.783i		−	0.106375i
$100$		0			0
$101$	−726.410	+	419.393i	−0.715649	+	0.413180i	−0.813149	−	0.582056i	$-0.802249\pi$
$101$	−726.410	+	419.393i	−0.715649	+	0.413180i	0.0975005	+	0.995235i	$0.468915\pi$
$102$		0			0
$103$	757.601	−	1312.20i	0.724744	−	1.25529i	−0.234335	−	0.972156i	$-0.575291\pi$
$103$	757.601	−	1312.20i	0.724744	−	1.25529i	0.959079	−	0.283138i	$-0.0913753\pi$
$104$		0			0
$105$	−72.6663	+	142.654i	−0.0675381	+	0.132587i
$106$		0			0
$107$	1461.57	+	843.836i	1.32051	+	0.762399i	0.983811	−	0.179211i	$-0.0573546\pi$
$107$	1461.57	+	843.836i	1.32051	+	0.762399i	0.336704	+	0.941611i	$0.390688\pi$
$108$		0			0
$109$	−788.117	−	1365.06i	−0.692550	−	1.19953i	−0.971000	−	0.239081i	$-0.923154\pi$
$109$	−788.117	−	1365.06i	−0.692550	−	1.19953i	0.278450	−	0.960451i	$-0.410179\pi$
$110$		0			0
$111$	1050.50			0.898281
$112$		0			0
$113$	−2192.01			−1.82484			−0.912422	−	0.409251i	$-0.865790\pi$
$113$	−2192.01			−1.82484			−0.912422	+	0.409251i	$0.865790\pi$
$114$		0			0
$115$	5.11193	+	8.85412i	0.00414513	+	0.00717957i
$116$		0			0
$117$	813.613	+	469.740i	0.642894	+	0.371175i
$118$		0			0
$119$	−2209.29	+	116.027i	−1.70189	+	0.0893797i
$120$		0			0
$121$	−655.013	+	1134.52i	−0.492121	+	0.852379i
$122$		0			0
$123$	−2194.42	+	1266.95i	−1.60865	+	0.928756i
$124$		0			0
$125$			304.158i			0.217638i
$126$		0			0
$127$			1085.60i			0.758515i	0.925291	+	0.379258i	$0.123821\pi$
$127$			1085.60i			0.758515i	−0.925291	+	0.379258i	$0.876179\pi$
$128$		0			0
$129$	2204.27	−	1272.64i	1.50446	−	0.868599i
$130$		0			0
$131$	766.676	−	1327.92i	0.511334	−	0.885657i	−0.488580	−	0.872519i	$-0.662485\pi$
$131$	766.676	−	1327.92i	0.511334	−	0.885657i	0.999914	−	0.0131373i	$-0.00418186\pi$
$132$		0			0
$133$	−934.713	+	607.155i	−0.609397	+	0.395842i
$134$		0			0
$135$	30.8431	+	17.8072i	0.0196633	+	0.0113526i
$136$		0			0
$137$	−460.087	−	796.895i	−0.286919	−	0.496958i	0.686154	−	0.727457i	$-0.259298\pi$
$137$	−460.087	−	796.895i	−0.286919	−	0.496958i	−0.973073	+	0.230498i	$0.925964\pi$
$138$		0			0
$139$	−1424.66			−0.869339			−0.434669	−	0.900590i	$-0.643135\pi$
$139$	−1424.66			−0.869339			−0.434669	+	0.900590i	$0.643135\pi$
$140$		0			0
$141$	−1592.15			−0.950943
$142$		0			0
$143$	94.0234	+	162.853i	0.0549834	+	0.0952341i
$144$		0			0
$145$	174.242	+	100.599i	0.0997934	+	0.0576158i
$146$		0			0
$147$	984.820	−	2213.25i	0.552562	−	1.24181i
$148$		0			0
$149$	−689.127	+	1193.60i	−0.378896	+	0.656267i	−0.990902	−	0.134586i	$-0.957029\pi$
$149$	−689.127	+	1193.60i	−0.378896	+	0.656267i	0.612006	+	0.790853i	$0.290363\pi$
$150$		0			0
$151$	−1155.09	+	666.890i	−0.622514	+	0.359409i	−0.777847	−	0.628453i	$-0.783688\pi$
$151$	−1155.09	+	666.890i	−0.622514	+	0.359409i	0.155333	+	0.987862i	$0.450355\pi$
$152$		0			0
$153$		−	2733.13i		−	1.44419i
$154$		0			0
$155$		−	351.963i		−	0.182389i
$156$		0			0
$157$	−1087.64	+	627.947i	−0.552884	+	0.319208i	−0.750285	−	0.661115i	$-0.770084\pi$
$157$	−1087.64	+	627.947i	−0.552884	+	0.319208i	0.197400	+	0.980323i	$0.436750\pi$
$158$		0			0
$159$	2417.96	−	4188.03i	1.20602	−	2.08888i
$160$		0			0
$161$	−84.2703	−	129.734i	−0.0412511	−	0.0635059i
$162$		0			0
$163$	2818.55	+	1627.29i	1.35439	+	0.781959i	0.988861	−	0.148840i	$-0.0475539\pi$
$163$	2818.55	+	1627.29i	1.35439	+	0.781959i	0.365532	+	0.930799i	$0.380887\pi$
$164$		0			0
$165$	−19.7942	−	34.2845i	−0.00933924	−	0.0161760i
$166$		0			0
$167$	−2736.24			−1.26788			−0.633941	−	0.773381i	$-0.718564\pi$
$167$	−2736.24			−1.26788			−0.633941	+	0.773381i	$0.718564\pi$
$168$		0			0
$169$	510.983			0.232582
$170$		0			0
$171$	−688.489	−	1192.50i	−0.307895	−	0.533290i
$172$		0			0
$173$	2927.28	+	1690.07i	1.28646	+	0.742737i	0.978021	−	0.208508i	$-0.0668608\pi$
$173$	2927.28	+	1690.07i	1.28646	+	0.742737i	0.308437	+	0.951245i	$0.400194\pi$
$174$		0			0
$175$	−119.958	−	2284.14i	−0.0518170	−	0.986656i
$176$		0			0
$177$	−300.344	+	520.211i	−0.127544	+	0.220912i
$178$		0			0
$179$	1054.81	−	608.992i	0.440446	−	0.254292i	−0.263341	−	0.964703i	$-0.584824\pi$
$179$	1054.81	−	608.992i	0.440446	−	0.254292i	0.703787	+	0.710411i	$0.251491\pi$
$180$		0			0
$181$		−	1757.61i		−	0.721781i	−0.932608	−	0.360890i	$-0.882473\pi$
$181$		−	1757.61i		−	0.721781i	0.932608	−	0.360890i	$-0.117527\pi$
$182$		0			0
$183$		−	1476.97i		−	0.596618i
$184$		0			0
$185$	157.664	−	91.0275i	0.0626579	−	0.0361755i
$186$		0			0
$187$	273.533	−	473.772i	0.106966	−	0.185271i
$188$		0			0
$189$	−480.186	−	244.601i	−0.184806	−	0.0941380i
$190$		0			0
$191$	−828.295	−	478.217i	−0.313787	−	0.181165i	0.334833	−	0.942278i	$-0.391320\pi$
$191$	−828.295	−	478.217i	−0.313787	−	0.181165i	−0.648620	+	0.761112i	$0.724654\pi$
$192$		0			0
$193$	−666.774	−	1154.89i	−0.248681	−	0.430729i	0.714479	−	0.699657i	$-0.246664\pi$
$193$	−666.774	−	1154.89i	−0.248681	−	0.430729i	−0.963160	+	0.268928i	$0.913330\pi$
$194$		0			0
$195$	354.947			0.130350
$196$		0			0
$197$	−1098.75			−0.397373			−0.198686	−	0.980063i	$-0.563668\pi$
$197$	−1098.75			−0.397373			−0.198686	+	0.980063i	$0.563668\pi$
$198$		0			0
$199$	−124.289	−	215.275i	−0.0442745	−	0.0766857i	0.843039	−	0.537853i	$-0.180764\pi$
$199$	−124.289	−	215.275i	−0.0442745	−	0.0766857i	−0.887313	+	0.461167i	$0.847431\pi$
$200$		0			0
$201$	2551.90	+	1473.34i	0.895509	+	0.517023i
$202$		0			0
$203$	−2712.73	−	1381.83i	−0.937912	−	0.477761i
$204$		0			0
$205$	−219.566	+	380.300i	−0.0748057	+	0.129567i
$206$		0			0
$207$	165.513	−	95.5590i	0.0555747	−	0.0320861i
$208$		0			0
$209$		−	275.617i		−	0.0912191i
$210$		0			0
$211$		−	2029.24i		−	0.662078i	−0.943617	−	0.331039i	$-0.892601\pi$
$211$		−	2029.24i		−	0.662078i	0.943617	−	0.331039i	$-0.107399\pi$
$212$		0			0
$213$	6012.24	−	3471.17i	1.93405	−	1.11662i
$214$		0			0
$215$	220.551	−	382.006i	0.0699604	−	0.121175i
$216$		0			0
$217$	279.308	+	5318.34i	0.0873762	+	1.66375i
$218$		0			0
$219$	−2088.90	−	1206.03i	−0.644543	−	0.372127i
$220$		0			0
$221$	2452.48	+	4247.82i	0.746477	+	1.29294i
$222$		0			0
$223$	3904.24			1.17241			0.586204	−	0.810164i	$-0.300622\pi$
$223$	3904.24			1.17241			0.586204	+	0.810164i	$0.300622\pi$
$224$		0			0
$225$	2825.73			0.837253
$226$		0			0
$227$	−2127.88	−	3685.59i	−0.622168	−	1.07763i	−0.989081	−	0.147371i	$-0.952919\pi$
$227$	−2127.88	−	3685.59i	−0.622168	−	1.07763i	0.366914	−	0.930255i	$-0.380414\pi$
$228$		0			0
$229$	−1500.18	−	866.127i	−0.432901	−	0.249936i	0.267681	−	0.963508i	$-0.413743\pi$
$229$	−1500.18	−	866.127i	−0.432901	−	0.249936i	−0.700582	+	0.713572i	$0.747076\pi$
$230$		0			0
$231$	326.308	+	502.349i	0.0929414	+	0.143083i
$232$		0			0
$233$	437.455	−	757.694i	0.122998	−	0.213039i	−0.797950	−	0.602723i	$-0.794082\pi$
$233$	437.455	−	757.694i	0.122998	−	0.213039i	0.920949	+	0.389684i	$0.127416\pi$
$234$		0			0
$235$	−238.957	+	137.962i	−0.0663312	+	0.0382963i
$236$		0			0
$237$		−	538.828i		−	0.147682i
$238$		0			0
$239$		−	1842.84i		−	0.498760i	−0.968406	−	0.249380i	$-0.919773\pi$
$239$		−	1842.84i		−	0.498760i	0.968406	−	0.249380i	$-0.0802269\pi$
$240$		0			0
$241$	−4056.11	+	2341.80i	−1.08414	+	0.625927i	−0.932010	−	0.362433i	$-0.881946\pi$
$241$	−4056.11	+	2341.80i	−1.08414	+	0.625927i	−0.152128	+	0.988361i	$0.548613\pi$
$242$		0			0
$243$	2514.37	−	4355.02i	0.663773	−	1.14969i
$244$		0			0
$245$	−43.9748	−	417.511i	−0.0114671	−	0.108873i
$246$		0			0
$247$	2140.09	+	1235.58i	0.551298	+	0.318292i
$248$		0			0
$249$	1849.14	+	3202.80i	0.470619	+	0.815137i
$250$		0			0
$251$	−1015.59			−0.255393			−0.127696	−	0.991813i	$-0.540758\pi$
$251$	−1015.59			−0.255393			−0.127696	+	0.991813i	$0.540758\pi$
$252$		0			0
$253$	38.2543			0.00950603
$254$		0			0
$255$	−516.306	−	894.268i	−0.126793	−	0.219613i
$256$		0			0
$257$	2612.42	+	1508.28i	0.634078	+	0.366085i	0.782330	−	0.622864i	$-0.214031\pi$
$257$	2612.42	+	1508.28i	0.634078	+	0.366085i	−0.148251	+	0.988950i	$0.547365\pi$
$258$		0			0
$259$	−2310.15	+	1500.59i	−0.554231	+	0.360008i
$260$		0			0
$261$	1880.53	−	3257.18i	0.445985	−	0.772468i
$262$		0			0
$263$	−2295.87	+	1325.52i	−0.538287	+	0.310780i	−0.744385	−	0.667751i	$-0.767257\pi$
$263$	−2295.87	+	1325.52i	−0.538287	+	0.310780i	0.206097	+	0.978532i	$0.433924\pi$
$264$		0			0
$265$		−	838.078i		−	0.194275i
$266$		0			0
$267$			6865.21i			1.57357i
$268$		0			0
$269$	−391.947	+	226.291i	−0.0888381	+	0.0512907i	−0.543761	−	0.839240i	$-0.683000\pi$
$269$	−391.947	+	226.291i	−0.0888381	+	0.0512907i	0.454923	+	0.890531i	$0.349667\pi$
$270$		0			0
$271$	−82.3630	+	142.657i	−0.0184620	+	0.0319771i	−0.875109	−	0.483926i	$-0.839210\pi$
$271$	−82.3630	+	142.657i	−0.0184620	+	0.0319771i	0.856647	+	0.515903i	$0.172544\pi$
$272$		0			0
$273$	−5363.44	+	281.676i	−1.18905	+	0.0624461i
$274$		0			0
$275$	489.823	+	282.799i	0.107409	+	0.0620125i
$276$		0			0
$277$	969.804	+	1679.75i	0.210360	+	0.364355i	0.951827	−	0.306634i	$-0.0992029\pi$
$277$	969.804	+	1679.75i	0.210360	+	0.364355i	−0.741467	+	0.670989i	$0.765870\pi$
$278$		0			0
$279$	−6579.37			−1.41181
$280$		0			0
$281$	−7152.79			−1.51850			−0.759252	−	0.650796i	$-0.774435\pi$
$281$	−7152.79			−1.51850			−0.759252	+	0.650796i	$0.774435\pi$
$282$		0			0
$283$	1647.77	+	2854.03i	0.346113	+	0.599485i	0.985555	−	0.169354i	$-0.0541681\pi$
$283$	1647.77	+	2854.03i	0.346113	+	0.599485i	−0.639442	+	0.768839i	$0.720835\pi$
$284$		0			0
$285$	−450.541	−	260.120i	−0.0936411	−	0.0540637i
$286$		0			0
$287$	3015.97	−	5920.77i	0.620303	−	1.21774i
$288$		0			0
$289$	4678.24	−	8102.95i	0.952217	−	1.64929i
$290$		0			0
$291$	4139.14	−	2389.73i	0.833816	−	0.481404i
$292$		0			0
$293$		−	7368.65i		−	1.46922i	−0.678491	−	0.734609i	$-0.737366\pi$
$293$		−	7368.65i		−	1.46922i	0.678491	−	0.734609i	$-0.262634\pi$
$294$		0			0
$295$			104.101i			0.0205457i
$296$		0			0
$297$	115.405	−	66.6289i	0.0225470	−	0.0130175i
$298$		0			0
$299$	−171.493	+	297.034i	−0.0331695	+	0.0574513i
$300$		0			0
$301$	−3029.50	+	5947.34i	−0.580124	+	1.13887i
$302$		0			0
$303$	5130.33	+	2962.00i	0.972705	+	0.561592i
$304$		0			0
$305$	−127.982	−	221.671i	−0.0240270	−	0.0416159i
$306$		0			0
$307$	2692.96			0.500636			0.250318	−	0.968164i	$-0.419465\pi$
$307$	2692.96			0.500636			0.250318	+	0.968164i	$0.419465\pi$
$308$		0			0
$309$	−10701.2			−1.97014
$310$		0			0
$311$	−1486.00	−	2573.82i	−0.270943	−	0.469286i	0.698161	−	0.715941i	$-0.254002\pi$
$311$	−1486.00	−	2573.82i	−0.270943	−	0.469286i	−0.969103	+	0.246655i	$0.920669\pi$
$312$		0			0
$313$	−987.269	−	570.000i	−0.178287	−	0.102934i	0.408201	−	0.912892i	$-0.366156\pi$
$313$	−987.269	−	570.000i	−0.178287	−	0.102934i	−0.586487	+	0.809958i	$0.699490\pi$
$314$		0			0
$315$	517.935	−	27.2008i	0.0926422	−	0.00486536i
$316$		0			0
$317$	−2822.56	+	4888.81i	−0.500097	+	0.866193i	0.499903	+	0.866081i	$0.333369\pi$
$317$	−2822.56	+	4888.81i	−0.500097	+	0.866193i	−1.00000		0.000111477i	$0.999965\pi$
$318$		0			0
$319$	651.958	−	376.408i	0.114428	−	0.0660652i
$320$		0			0
$321$		−	11919.3i		−	2.07250i
$322$		0			0
$323$		−	7189.10i		−	1.23843i
$324$		0			0
$325$	−4391.73	+	2535.56i	−0.749567	+	0.432762i
$326$		0			0
$327$	−5566.14	+	9640.83i	−0.941309	+	1.63040i
$328$		0			0
$329$	3501.28	−	2274.31i	0.586723	−	0.381114i
$330$		0			0
$331$	−6373.93	−	3679.99i	−1.05844	−	0.611089i	−0.133438	−	0.991057i	$-0.542602\pi$
$331$	−6373.93	−	3679.99i	−1.05844	−	0.611089i	−0.925000	+	0.379968i	$0.875935\pi$
$332$		0			0
$333$	−1701.61	−	2947.27i	−0.280023	−	0.485014i
$334$		0			0
$335$	510.669			0.0832861
$336$		0			0
$337$	−10163.9			−1.64292			−0.821459	−	0.570268i	$-0.806839\pi$
$337$	−10163.9			−1.64292			−0.821459	+	0.570268i	$0.806839\pi$
$338$		0			0
$339$	7740.64	+	13407.2i	1.24016	+	2.14802i
$340$		0			0
$341$	−1140.49	−	658.464i	−0.181118	−	0.104568i
$342$		0			0
$343$	995.807	+	6273.91i	0.156759	+	0.987637i
$344$		0			0
$345$	36.1034	−	62.5329i	0.00563403	−	0.00975843i
$346$		0			0
$347$	−9132.03	+	5272.38i	−1.41278	+	0.815666i	−0.995649	−	0.0931819i	$-0.970296\pi$
$347$	−9132.03	+	5272.38i	−1.41278	+	0.815666i	−0.417127	+	0.908848i	$0.636963\pi$
$348$		0			0
$349$		−	6048.02i		−	0.927631i	−0.885932	−	0.463816i	$-0.846480\pi$
$349$		−	6048.02i		−	0.927631i	0.885932	−	0.463816i	$-0.153520\pi$
$350$		0			0
$351$			1194.78i			0.181689i
$352$		0			0
$353$	5446.54	−	3144.56i	0.821218	−	0.474131i	−0.0296180	−	0.999561i	$-0.509429\pi$
$353$	5446.54	−	3144.56i	0.821218	−	0.474131i	0.850836	+	0.525431i	$0.176096\pi$
$354$		0			0
$355$	601.563	−	1041.94i	0.0899371	−	0.155776i
$356$		0			0
$357$	8511.32	+	13103.1i	1.26181	+	1.94255i
$358$		0			0
$359$	−133.941	−	77.3311i	−0.0196912	−	0.0113687i	0.490122	−	0.871654i	$-0.336952\pi$
$359$	−133.941	−	77.3311i	−0.0196912	−	0.0113687i	−0.509813	+	0.860285i	$0.670286\pi$
$360$		0			0
$361$	1618.53	+	2803.38i	0.235972	+	0.408715i
$362$		0			0
$363$	9252.17			1.33778
$364$		0			0
$365$	−418.016			−0.0599451
$366$		0			0
$367$	−2945.21	−	5101.26i	−0.418907	−	0.725568i	0.576923	−	0.816799i	$-0.304253\pi$
$367$	−2945.21	−	5101.26i	−0.418907	−	0.725568i	−0.995830	+	0.0912308i	$0.970920\pi$
$368$		0			0
$369$	7109.08	+	4104.43i	1.00294	+	0.579046i
$370$		0			0
$371$	665.075	+	12663.8i	0.0930700	+	1.77216i
$372$		0			0
$373$	175.150	−	303.369i	0.0243135	−	0.0421122i	−0.853613	−	0.520908i	$-0.825593\pi$
$373$	175.150	−	303.369i	0.0243135	−	0.0421122i	0.877926	+	0.478796i	$0.158927\pi$
$374$		0			0
$375$	1860.34	−	1074.07i	0.256181	−	0.147906i
$376$		0			0
$377$			6749.71i			0.922089i
$378$		0			0
$379$			8247.17i			1.11775i	0.829251	+	0.558877i	$0.188767\pi$
$379$			8247.17i			1.11775i	−0.829251	+	0.558877i	$0.811233\pi$
$380$		0			0
$381$	6639.93	−	3833.57i	0.892845	−	0.515485i
$382$		0			0
$383$	−6340.08	+	10981.3i	−0.845856	+	1.46507i	0.0390189	+	0.999238i	$0.487577\pi$
$383$	−6340.08	+	10981.3i	−0.845856	+	1.46507i	−0.884875	+	0.465828i	$0.845757\pi$
$384$		0			0
$385$	92.5031	+	47.1199i	0.0122452	+	0.00623754i
$386$		0			0
$387$	−7140.98	−	4122.85i	−0.937975	−	0.541540i
$388$		0			0
$389$	−3863.76	−	6692.24i	−0.503601	−	0.872262i	−0.999991	−	0.00416264i	$-0.998675\pi$
$389$	−3863.76	−	6692.24i	−0.503601	−	0.872262i	0.496391	−	0.868099i	$-0.334658\pi$
$390$		0			0
$391$	997.814			0.129058
$392$		0			0
$393$	−10829.4			−1.39000
$394$		0			0
$395$	−46.6903	−	80.8699i	−0.00594745	−	0.0103013i
$396$		0			0
$397$	8743.56	+	5048.10i	1.10536	+	0.638178i	0.937623	−	0.347654i	$-0.113022\pi$
$397$	8743.56	+	5048.10i	1.10536	+	0.638178i	0.167734	+	0.985832i	$0.446355\pi$
$398$		0			0
$399$	7014.33	+	3573.01i	0.880089	+	0.448306i
$400$		0			0
$401$	−3839.25	+	6649.77i	−0.478112	+	0.828114i	−0.999685	−	0.0250922i	$-0.992012\pi$
$401$	−3839.25	+	6649.77i	−0.478112	+	0.828114i	0.521573	+	0.853207i	$0.325345\pi$
$402$		0			0
$403$	10225.6	−	5903.76i	1.26395	−	0.729744i
$404$		0			0
$405$		−	1007.65i		−	0.123631i
$406$		0			0
$407$		−	681.190i		−	0.0829615i
$408$		0			0
$409$	3214.30	−	1855.78i	0.388599	−	0.224358i	−0.292954	−	0.956127i	$-0.594638\pi$
$409$	3214.30	−	1855.78i	0.388599	−	0.224358i	0.681553	+	0.731769i	$0.261305\pi$
$410$		0			0
$411$	−3249.40	+	5628.13i	−0.389979	+	0.675463i
$412$		0			0
$413$	−82.6116	−	1573.02i	−0.00984273	−	0.187417i
$414$		0			0
$415$	555.054	+	320.461i	0.0656543	+	0.0379055i
$416$		0			0
$417$	5030.89	+	8713.75i	0.590800	+	1.02330i
$418$		0			0
$419$	−8868.09			−1.03397			−0.516986	−	0.855994i	$-0.672946\pi$
$419$	−8868.09			−1.03397			−0.516986	+	0.855994i	$0.672946\pi$
$420$		0			0
$421$	9571.99			1.10810			0.554050	−	0.832483i	$-0.313081\pi$
$421$	9571.99			1.10810			0.554050	+	0.832483i	$0.313081\pi$
$422$		0			0
$423$	2578.97	+	4466.91i	0.296439	+	0.513448i
$424$		0			0
$425$	12776.4	+	7376.46i	1.45823	+	0.841908i
$426$		0			0
$427$	2109.79	+	3248.01i	0.239109	+	0.368108i
$428$		0			0
$429$	664.047	−	1150.16i	0.0747331	−	0.129442i
$430$		0			0
$431$	6678.90	−	3856.06i	0.746430	−	0.430952i	−0.0779726	−	0.996956i	$-0.524845\pi$
$431$	6678.90	−	3856.06i	0.746430	−	0.430952i	0.824403	+	0.566004i	$0.191511\pi$
$432$		0			0
$433$			2222.71i			0.246690i	0.992364	+	0.123345i	$0.0393622\pi$
$433$			2222.71i			0.246690i	−0.992364	+	0.123345i	$0.960638\pi$
$434$		0			0
$435$		−	1420.98i		−	0.156622i
$436$		0			0
$437$	435.358	−	251.354i	0.0476567	−	0.0275146i
$438$		0			0
$439$	−5306.28	+	9190.74i	−0.576890	+	0.999203i	0.418943	+	0.908012i	$0.362401\pi$
$439$	−5306.28	+	9190.74i	−0.576890	+	0.999203i	−0.995833	+	0.0911907i	$0.970933\pi$
$440$		0			0
$441$	−7804.68	+	822.036i	−0.842747	+	0.0887632i
$442$		0			0
$443$	−13502.3	−	7795.53i	−1.44811	−	0.836065i	−0.449738	−	0.893160i	$-0.648483\pi$
$443$	−13502.3	−	7795.53i	−1.44811	−	0.836065i	−0.998369	+	0.0570953i	$0.981816\pi$
$444$		0			0
$445$	594.881	+	1030.36i	0.0633709	+	0.109762i
$446$		0			0
$447$	9734.03			1.02999
$448$		0			0
$449$	2759.74			0.290067			0.145033	−	0.989427i	$-0.453671\pi$
$449$	2759.74			0.290067			0.145033	+	0.989427i	$0.453671\pi$
$450$		0			0
$451$	821.544	+	1422.96i	0.0857761	+	0.148569i
$452$		0			0
$453$	8157.89	+	4709.96i	0.846117	+	0.488506i
$454$		0			0
$455$	−780.563	+	507.025i	−0.0804249	+	0.0522411i
$456$		0			0
$457$	3836.15	−	6644.40i	0.392664	−	0.680113i	−0.600136	−	0.799898i	$-0.704887\pi$
$457$	3836.15	−	6644.40i	0.392664	−	0.680113i	0.992800	+	0.119784i	$0.0382203\pi$
$458$		0			0
$459$	3010.18	−	1737.93i	0.306107	−	0.176731i
$460$		0			0
$461$		−	13864.8i		−	1.40076i	−0.713771	−	0.700379i	$-0.753014\pi$
$461$		−	13864.8i		−	1.40076i	0.713771	−	0.700379i	$-0.246986\pi$
$462$		0			0
$463$			11518.3i			1.15616i	0.815981	+	0.578079i	$0.196197\pi$
$463$			11518.3i			1.15616i	−0.815981	+	0.578079i	$0.803803\pi$
$464$		0			0
$465$	−2152.74	+	1242.88i	−0.214690	+	0.123951i
$466$		0			0
$467$	−1407.96	+	2438.66i	−0.139513	+	0.241644i	−0.927312	−	0.374288i	$-0.877887\pi$
$467$	−1407.96	+	2438.66i	−0.139513	+	0.241644i	0.787799	+	0.615932i	$0.211220\pi$
$468$		0			0
$469$	−7716.48	+	405.252i	−0.759731	+	0.0398994i
$470$		0			0
$471$	7681.52	+	4434.93i	0.751477	+	0.433866i
$472$		0			0
$473$	−825.231	−	1429.34i	−0.0802202	−	0.138945i
$474$		0			0
$475$	7432.66			0.717966
$476$		0			0
$477$	−15666.5			−1.50381
$478$		0			0
$479$	−1560.08	−	2702.13i	−0.148813	−	0.257753i	0.781976	−	0.623309i	$-0.214212\pi$
$479$	−1560.08	−	2702.13i	−0.148813	−	0.257753i	−0.930789	+	0.365556i	$0.880879\pi$
$480$		0			0
$481$	5289.26	+	3053.75i	0.501392	+	0.289479i
$482$		0			0
$483$	−495.917	+	973.556i	−0.0467184	+	0.0917149i
$484$		0			0
$485$	414.147	−	717.324i	0.0387742	−	0.0671588i
$486$		0			0
$487$	1721.99	−	994.191i	0.160227	−	0.0925074i	−0.417742	−	0.908566i	$-0.637178\pi$
$487$	1721.99	−	994.191i	0.160227	−	0.0925074i	0.577970	+	0.816058i	$0.303845\pi$
$488$		0			0
$489$		−	22985.8i		−	2.12567i
$490$		0			0
$491$			3035.51i			0.279003i	0.990222	+	0.139502i	$0.0445501\pi$
$491$			3035.51i			0.279003i	−0.990222	+	0.139502i	$0.955450\pi$
$492$		0			0
$493$	17005.5	−	9818.12i	1.55353	−	0.896929i
$494$		0			0
$495$	−64.1255	+	111.069i	−0.00582268	+	0.0100852i
$496$		0			0
$497$	−8263.09	+	16221.6i	−0.745775	+	1.46406i
$498$		0			0
$499$	14306.5	+	8259.87i	1.28346	+	0.741007i	0.977479	−	0.211031i	$-0.0676820\pi$
$499$	14306.5	+	8259.87i	1.28346	+	0.741007i	0.305982	+	0.952037i	$0.401015\pi$
$500$		0			0
$501$	9662.45	+	16735.8i	0.861649	+	1.49242i
$502$		0			0
$503$	−16446.0			−1.45783			−0.728917	−	0.684602i	$-0.759976\pi$
$503$	−16446.0			−1.45783			−0.728917	+	0.684602i	$0.759976\pi$
$504$		0			0
$505$	1026.65			0.0904656
$506$		0			0
$507$	−1804.43	−	3125.36i	−0.158062	−	0.273772i
$508$		0			0
$509$	−4057.43	−	2342.56i	−0.353325	−	0.203992i	0.312824	−	0.949811i	$-0.398725\pi$
$509$	−4057.43	−	2342.56i	−0.353325	−	0.203992i	−0.666149	+	0.745819i	$0.732058\pi$
$510$		0			0
$511$	6316.45	−	331.726i	0.546817	−	0.0287176i
$512$		0			0
$513$	875.585	−	1516.56i	0.0753567	−	0.130522i
$514$		0			0
$515$	−1606.09	+	927.278i	−0.137423	+	0.0793412i
$516$		0			0
$517$			1032.42i			0.0878251i
$518$		0			0
$519$		−	23872.5i		−	2.01905i
$520$		0			0
$521$	1904.98	−	1099.84i	0.160189	−	0.0924853i	−0.417762	−	0.908556i	$-0.637186\pi$
$521$	1904.98	−	1099.84i	0.160189	−	0.0924853i	0.577952	+	0.816071i	$0.303852\pi$
$522$		0			0
$523$	−5232.69	+	9063.28i	−0.437494	+	0.757762i	−0.997496	−	0.0707295i	$-0.977467\pi$
$523$	−5232.69	+	9063.28i	−0.437494	+	0.757762i	0.560001	+	0.828492i	$0.310801\pi$
$524$		0			0
$525$	−13547.0	+	8799.67i	−1.12617	+	0.731522i
$526$		0			0
$527$	−29748.3	−	17175.2i	−2.45893	−	1.41966i
$528$		0			0
$529$	−6048.61	−	10476.5i	−0.497133	−	0.861059i
$530$		0			0
$531$	1946.00			0.159038
$532$		0			0
$533$	−14731.8			−1.19720
$534$		0			0
$535$	−1032.83	−	1788.91i	−0.0834635	−	0.144563i
$536$		0			0
$537$	−7449.65	−	4301.06i	−0.598652	−	0.345632i
$538$		0			0
$539$	−1435.16	−	638.599i	−0.114688	−	0.0510323i
$540$		0			0
$541$	−1467.47	+	2541.74i	−0.116620	+	0.201992i	−0.918426	−	0.395592i	$-0.870539\pi$
$541$	−1467.47	+	2541.74i	−0.116620	+	0.201992i	0.801806	+	0.597584i	$0.203873\pi$
$542$		0			0
$543$	−10750.2	+	6206.64i	−0.849606	+	0.490520i
$544$		0			0
$545$			1929.26i			0.151634i
$546$		0			0
$547$		−	9168.13i		−	0.716638i	−0.933599	−	0.358319i	$-0.883350\pi$
$547$		−	9168.13i		−	0.716638i	0.933599	−	0.358319i	$-0.116650\pi$
$548$		0			0
$549$	−4143.78	+	2392.41i	−0.322135	+	0.185985i
$550$		0			0
$551$	4946.46	−	8567.52i	0.382444	−	0.662412i
$552$		0			0
$553$	769.691	+	1184.94i	0.0591873	+	0.0911186i
$554$		0			0
$555$	−1113.52	−	642.889i	−0.0851642	−	0.0491696i
$556$		0			0
$557$	2309.07	+	3999.42i	0.175652	+	0.304239i	0.940387	−	0.340107i	$-0.110463\pi$
$557$	2309.07	+	3999.42i	0.175652	+	0.304239i	−0.764734	+	0.644346i	$0.777130\pi$
$558$		0			0
$559$	14797.9			1.11965
$560$		0			0
$561$	−3863.69			−0.290776
$562$		0			0
$563$	−11894.8	−	20602.4i	−0.890418	−	1.54225i	−0.839375	−	0.543552i	$-0.817079\pi$
$563$	−11894.8	−	20602.4i	−0.890418	−	1.54225i	−0.0510424	−	0.998696i	$-0.516254\pi$
$564$		0			0
$565$	2323.50	+	1341.48i	0.173010	+	0.0998873i
$566$		0			0
$567$	799.641	+	15226.1i	0.0592271	+	1.12775i
$568$		0			0
$569$	7386.30	−	12793.4i	0.544200	−	0.942582i	−0.454457	−	0.890769i	$-0.650167\pi$
$569$	7386.30	−	12793.4i	0.544200	−	0.942582i	0.998657	−	0.0518134i	$-0.0165001\pi$
$570$		0			0
$571$	21040.7	−	12147.8i	1.54208	−	0.890318i	0.543369	−	0.839494i	$-0.317149\pi$
$571$	21040.7	−	12147.8i	1.54208	−	0.890318i	0.998708	−	0.0508238i	$-0.0161847\pi$
$572$		0			0
$573$			6754.88i			0.492477i
$574$		0			0
$575$			1031.62i			0.0748199i
$576$		0			0
$577$	8681.56	−	5012.30i	0.626375	−	0.361638i	−0.152972	−	0.988231i	$-0.548884\pi$
$577$	8681.56	−	5012.30i	0.626375	−	0.361638i	0.779347	+	0.626593i	$0.215551\pi$
$578$		0			0
$579$	−4709.15	+	8156.48i	−0.338006	+	0.585444i
$580$		0			0
$581$	−8641.47	−	4401.86i	−0.617054	−	0.314320i
$582$		0			0
$583$	−2715.69	−	1567.91i	−0.192920	−	0.111383i
$584$		0			0
$585$	−574.946	−	995.835i	−0.0406343	−	0.0703807i
$586$		0			0
$587$	10134.7			0.712613			0.356306	−	0.934369i	$-0.384036\pi$
$587$	10134.7			0.712613			0.356306	+	0.934369i	$0.384036\pi$
$588$		0			0
$589$	−17306.1			−1.21067
$590$		0			0
$591$	3879.99	+	6720.34i	0.270053	+	0.467746i
$592$		0			0
$593$	−1903.10	−	1098.76i	−0.131789	−	0.0760885i	0.432656	−	0.901559i	$-0.357576\pi$
$593$	−1903.10	−	1098.76i	−0.131789	−	0.0760885i	−0.564445	+	0.825471i	$0.690910\pi$
$594$		0			0
$595$	2412.82	+	1229.06i	0.166246	+	0.0846834i
$596$		0			0
$597$	−877.803	+	1520.40i	−0.0601777	+	0.104231i
$598$		0			0
$599$	21359.3	−	12331.8i	1.45696	−	0.841174i	0.458095	−	0.888903i	$-0.348532\pi$
$599$	21359.3	−	12331.8i	1.45696	−	0.841174i	0.998860	+	0.0477293i	$0.0151985\pi$
$600$		0			0
$601$		−	21532.0i		−	1.46142i	−0.682691	−	0.730708i	$-0.739190\pi$
$601$		−	21532.0i		−	1.46142i	0.682691	−	0.730708i	$-0.260810\pi$
$602$		0			0
$603$		−	9546.12i		−	0.644690i
$604$		0			0
$605$	1388.61	−	801.714i	0.0933140	−	0.0538749i
$606$		0			0
$607$	493.715	−	855.140i	0.0330136	−	0.0571813i	−0.849047	−	0.528318i	$-0.822823\pi$
$607$	493.715	−	855.140i	0.0330136	−	0.0571813i	0.882060	+	0.471137i	$0.156156\pi$
$608$		0			0
$609$	1127.65	+	21471.7i	0.0750320	+	1.42870i
$610$		0			0
$611$	−8016.43	−	4628.29i	−0.530786	−	0.306449i
$612$		0			0
$613$	3639.26	+	6303.38i	0.239785	+	0.415320i	0.960653	−	0.277753i	$-0.0895897\pi$
$613$	3639.26	+	6303.38i	0.239785	+	0.415320i	−0.720867	+	0.693073i	$0.756256\pi$
$614$		0			0
$615$	3101.41			0.203351
$616$		0			0
$617$	−2543.97			−0.165991			−0.0829953	−	0.996550i	$-0.526449\pi$
$617$	−2543.97			−0.165991			−0.0829953	+	0.996550i	$0.526449\pi$
$618$		0			0
$619$	−3926.57	−	6801.02i	−0.254963	−	0.441609i	0.709922	−	0.704280i	$-0.248730\pi$
$619$	−3926.57	−	6801.02i	−0.254963	−	0.441609i	−0.964886	+	0.262671i	$0.915397\pi$
$620$		0			0
$621$	210.491	+	121.527i	0.0136018	+	0.00785300i
$622$		0			0
$623$	−9806.63	−	15097.3i	−0.630649	−	0.970882i
$624$		0			0
$625$	−7532.73	+	13047.1i	−0.482095	+	0.835013i
$626$		0			0
$627$	−1685.77	+	973.282i	−0.107374	+	0.0619923i
$628$		0			0
$629$		−	17768.0i		−	1.12632i
$630$		0			0
$631$		−	16369.0i		−	1.03271i	−0.856375	−	0.516355i	$-0.827289\pi$
$631$		−	16369.0i		−	1.03271i	0.856375	−	0.516355i	$-0.172711\pi$
$632$		0			0
$633$	−12411.6	+	7165.83i	−0.779330	+	0.449946i
$634$		0			0
$635$	664.369	−	1150.72i	0.0415192	−	0.0719133i
$636$		0			0
$637$	11392.4	−	8280.85i	0.708605	−	0.515069i
$638$		0			0
$639$	−19477.3	−	11245.2i	−1.20581	−	0.696173i
$640$		0			0
$641$	−7248.62	−	12555.0i	−0.446651	−	0.773622i	0.551515	−	0.834165i	$-0.314050\pi$
$641$	−7248.62	−	12555.0i	−0.446651	−	0.773622i	−0.998166	+	0.0605431i	$0.980717\pi$
$642$		0			0
$643$	−29771.6			−1.82594			−0.912969	−	0.408029i	$-0.866216\pi$
$643$	−29771.6			−1.82594			−0.912969	+	0.408029i	$0.866216\pi$
$644$		0			0
$645$	−3115.32			−0.190179
$646$		0			0
$647$	5607.76	+	9712.93i	0.340748	+	0.590193i	0.984572	−	0.174981i	$-0.0559864\pi$
$647$	5607.76	+	9712.93i	0.340748	+	0.590193i	−0.643824	+	0.765174i	$0.722653\pi$
$648$		0			0
$649$	337.327	+	194.756i	0.0204025	+	0.0117794i
$650$		0			0
$651$	31542.7	−	20489.0i	1.89901	−	1.23353i
$652$		0			0
$653$	10200.5	−	17667.8i	0.611295	−	1.05879i	−0.379727	−	0.925099i	$-0.623982\pi$
$653$	10200.5	−	17667.8i	0.611295	−	1.05879i	0.991022	−	0.133696i	$-0.0426846\pi$
$654$		0			0
$655$	−1625.33	+	938.385i	−0.0969571	+	0.0559782i
$656$		0			0
$657$			7814.13i			0.464016i
$658$		0			0
$659$		−	13427.1i		−	0.793693i	−0.917885	−	0.396846i	$-0.870105\pi$
$659$		−	13427.1i		−	0.793693i	0.917885	−	0.396846i	$-0.129895\pi$
$660$		0			0
$661$	2003.84	−	1156.92i	0.117913	−	0.0680771i	−0.439884	−	0.898055i	$-0.644980\pi$
$661$	2003.84	−	1156.92i	0.117913	−	0.0680771i	0.557797	+	0.829978i	$0.311647\pi$
$662$		0			0
$663$	17320.8	−	30000.5i	1.01461	−	1.75735i
$664$		0			0
$665$	1362.35	−	71.5476i	0.0794431	−	0.00417218i
$666$		0			0
$667$	1189.13	+	686.546i	0.0690305	+	0.0398548i
$668$		0			0
$669$	−13787.0	−	23879.8i	−0.796765	−	1.38004i
$670$		0			0
$671$	−957.732			−0.0551011
$672$		0			0
$673$	−16395.5			−0.939081			−0.469540	−	0.882911i	$-0.655580\pi$
$673$	−16395.5			−0.939081			−0.469540	+	0.882911i	$0.655580\pi$
$674$		0			0
$675$	1796.81	+	3112.16i	0.102458	+	0.177462i
$676$		0			0
$677$	21618.4	+	12481.4i	1.22727	+	0.708566i	0.966458	−	0.256823i	$-0.0826758\pi$
$677$	21618.4	+	12481.4i	1.22727	+	0.708566i	0.260814	+	0.965389i	$0.416009\pi$
$678$		0			0
$679$	−5688.74	+	11167.8i	−0.321522	+	0.631195i
$680$		0			0
$681$	−15028.3	+	26029.8i	−0.845647	+	1.46470i
$682$		0			0
$683$	−17644.8	+	10187.3i	−0.988523	+	0.570724i	−0.904832	−	0.425768i	$-0.860004\pi$
$683$	−17644.8	+	10187.3i	−0.988523	+	0.570724i	−0.0836904	+	0.996492i	$0.526671\pi$
$684$		0			0
$685$			1126.26i			0.0628208i
$686$		0			0
$687$			12234.2i			0.679422i
$688$		0			0
$689$	24348.8	−	14057.8i	1.34632	−	0.777298i
$690$		0			0
$691$	−3196.66	+	5536.78i	−0.175986	+	0.304817i	−0.940502	−	0.339788i	$-0.889645\pi$
$691$	−3196.66	+	5536.78i	−0.175986	+	0.304817i	0.764516	+	0.644605i	$0.222978\pi$
$692$		0			0
$693$	880.829	−	1729.19i	0.0482827	−	0.0947859i
$694$		0			0
$695$	1510.12	+	871.868i	0.0824203	+	0.0475854i
$696$		0			0
$697$	21428.9	+	37116.0i	1.16453	+	2.01703i
$698$		0			0
$699$	−6179.12			−0.334357
$700$		0			0
$701$	19401.6			1.04535			0.522674	−	0.852533i	$-0.324934\pi$
$701$	19401.6			1.04535			0.522674	+	0.852533i	$0.324934\pi$
$702$		0			0
$703$	−4475.83	−	7752.37i	−0.240127	−	0.415912i
$704$		0			0
$705$	1687.65	+	974.367i	0.0901570	+	0.0520521i
$706$		0			0
$707$	−15513.2	+	814.716i	−0.825222	+	0.0433388i
$708$		0			0
$709$	7388.85	−	12797.9i	0.391388	−	0.677904i	−0.601245	−	0.799065i	$-0.705328\pi$
$709$	7388.85	−	12797.9i	0.391388	−	0.677904i	0.992633	+	0.121161i	$0.0386616\pi$
$710$		0			0
$711$	−1511.73	+	872.798i	−0.0797389	+	0.0460372i
$712$		0			0
$713$		−	2402.00i		−	0.126165i
$714$		0			0
$715$		−	230.163i		−	0.0120386i
$716$		0			0
$717$	−11271.5	+	6507.62i	−0.587089	+	0.338956i
$718$		0			0
$719$	−3218.52	+	5574.64i	−0.166941	+	0.289150i	−0.937343	−	0.348408i	$-0.886722\pi$
$719$	−3218.52	+	5574.64i	−0.166941	+	0.289150i	0.770402	+	0.637559i	$0.220056\pi$
$720$		0			0
$721$	23533.0	−	15286.2i	1.21556	−	0.789581i
$722$		0			0
$723$	28646.6	+	16539.1i	1.47355	+	0.850757i
$724$		0			0
$725$	10150.7	+	17581.6i	0.519985	+	0.900640i
$726$		0			0
$727$	13490.8			0.688235			0.344118	−	0.938927i	$-0.388178\pi$
$727$	13490.8			0.688235			0.344118	+	0.938927i	$0.388178\pi$
$728$		0			0
$729$	−13287.7			−0.675086
$730$		0			0
$731$	−21525.1	−	37282.5i	−1.08910	−	1.88638i
$732$		0			0
$733$	18019.0	+	10403.3i	0.907978	+	0.524221i	0.879780	−	0.475381i	$-0.157690\pi$
$733$	18019.0	+	10403.3i	0.907978	+	0.524221i	0.0281980	+	0.999602i	$0.491023\pi$
$734$		0			0
$735$	−2398.36	+	1743.32i	−0.120361	+	0.0874874i
$736$		0			0
$737$	955.377	−	1654.76i	0.0477500	−	0.0827055i
$738$		0			0
$739$	−14968.2	+	8641.87i	−0.745078	+	0.430171i	−0.823913	−	0.566717i	$-0.808213\pi$
$739$	−14968.2	+	8641.87i	−0.745078	+	0.430171i	0.0788348	+	0.996888i	$0.474880\pi$
$740$		0			0
$741$		−	17452.8i		−	0.865242i
$742$		0			0
$743$			11763.6i			0.580839i	0.956899	+	0.290420i	$0.0937948\pi$
$743$			11763.6i			0.580839i	−0.956899	+	0.290420i	$0.906205\pi$
$744$		0			0
$745$	1460.93	−	843.468i	0.0718447	−	0.0414795i
$746$		0			0
$747$	5990.49	−	10375.8i	0.293414	−	0.508208i
$748$		0			0
$749$	17026.2	+	26211.7i	0.830605	+	1.27871i
$750$		0			0
$751$	10719.5	+	6188.89i	0.520851	+	0.300713i	0.737283	−	0.675584i	$-0.236109\pi$
$751$	10719.5	+	6188.89i	0.520851	+	0.300713i	−0.216432	+	0.976298i	$0.569442\pi$
$752$		0			0
$753$	3586.35	+	6211.74i	0.173564	+	0.300622i
$754$		0			0
$755$	1632.50			0.0786924
$756$		0			0
$757$	−18350.8			−0.881071			−0.440536	−	0.897735i	$-0.645211\pi$
$757$	−18350.8			−0.881071			−0.440536	+	0.897735i	$0.645211\pi$
$758$		0			0
$759$	−135.087	−	233.977i	−0.00646027	−	0.0111895i
$760$		0			0
$761$	14340.0	+	8279.21i	0.683081	+	0.394377i	0.801015	−	0.598644i	$-0.204294\pi$
$761$	14340.0	+	8279.21i	0.683081	+	0.394377i	−0.117934	+	0.993021i	$0.537627\pi$
$762$		0			0
$763$	−1531.00	−	29152.1i	−0.0726422	−	1.38319i
$764$		0			0
$765$	−1672.63	+	2897.08i	−0.0790511	+	0.136921i
$766$		0			0
$767$	−3024.46	+	1746.17i	−0.142382	+	0.0822041i
$768$		0			0
$769$		−	3663.72i		−	0.171804i	−0.996304	−	0.0859019i	$-0.972623\pi$
$769$		−	3663.72i		−	0.171804i	0.996304	−	0.0859019i	$-0.0273772\pi$
$770$		0			0
$771$		−	21304.7i		−	0.995162i
$772$		0			0
$773$	−34341.3	+	19827.0i	−1.59789	+	0.922544i	−0.606000	+	0.795465i	$0.707227\pi$
$773$	−34341.3	+	19827.0i	−1.59789	+	0.922544i	−0.991893	+	0.127079i	$0.959440\pi$
$774$		0			0
$775$	17757.1	−	30756.1i	0.823035	−	1.42554i
$776$		0			0
$777$	17336.0	+	8830.73i	0.800419	+	0.407723i
$778$		0			0
$779$	18699.4	+	10796.1i	0.860045	+	0.496547i
$780$		0			0
$781$	−2250.85	−	3898.59i	−0.103126	−	0.178620i
$782$		0			0
$783$	4783.12			0.218308
$784$		0			0
$785$	1537.17			0.0698905
$786$		0			0
$787$	17734.3	+	30716.8i	0.803254	+	1.39128i	0.917464	+	0.397820i	$0.130233\pi$
$787$	17734.3	+	30716.8i	0.803254	+	1.39128i	−0.114210	+	0.993457i	$0.536434\pi$
$788$		0			0
$789$	16214.8	+	9361.61i	0.731637	+	0.422411i
$790$		0			0
$791$	−36173.9	−	18426.5i	−1.62604	−	0.828284i
$792$		0			0
$793$	4293.49	−	7436.54i	0.192265	−	0.333013i
$794$		0			0
$795$	−5126.00	+	2959.50i	−0.228680	+	0.132028i
$796$		0			0
$797$			32026.0i			1.42336i	0.702503	+	0.711681i	$0.252066\pi$
$797$			32026.0i			1.42336i	−0.702503	+	0.711681i	$0.747934\pi$
$798$		0			0
$799$			26929.2i			1.19235i
$800$		0			0
$801$	19261.0	−	11120.3i	0.849629	−	0.490533i
$802$		0			0
$803$	−782.040	+	1354.53i	−0.0343681	+	0.0595273i
$804$		0			0
$805$	9.93047	+	189.088i	0.000434786	+	0.00827884i
$806$		0			0
$807$	2768.16	+	1598.20i	0.120748	+	0.0697140i
$808$		0			0
$809$	−1708.60	−	2959.39i	−0.0742537	−	0.128611i	0.826508	−	0.562925i	$-0.190324\pi$
$809$	−1708.60	−	2959.39i	−0.0742537	−	0.128611i	−0.900761	+	0.434314i	$0.856991\pi$
$810$		0			0
$811$	27667.8			1.19796			0.598982	−	0.800763i	$-0.295572\pi$
$811$	27667.8			1.19796			0.598982	+	0.800763i	$0.295572\pi$
$812$		0			0
$813$	1163.39			0.0501868
$814$		0			0
$815$	−1991.75	−	3449.81i	−0.0856048	−	0.148272i
$816$		0			0
$817$	−18783.3	−	10844.5i	−0.804338	−	0.464385i
$818$		0			0
$819$	9478.00	+	14591.3i	0.404381	+	0.622543i
$820$		0			0
$821$	14342.5	−	24841.9i	0.609690	−	1.05601i	−0.381601	−	0.924327i	$-0.624627\pi$
$821$	14342.5	−	24841.9i	0.609690	−	1.05601i	0.991291	−	0.131687i	$-0.0420395\pi$
$822$		0			0
$823$	18633.9	−	10758.3i	0.789231	−	0.455663i	−0.0504606	−	0.998726i	$-0.516069\pi$
$823$	18633.9	−	10758.3i	0.789231	−	0.455663i	0.839692	+	0.543063i	$0.182736\pi$
$824$		0			0
$825$		−	3994.59i		−	0.168574i
$826$		0			0
$827$			28539.8i			1.20003i	0.799988	+	0.600016i	$0.204839\pi$
$827$			28539.8i			1.20003i	−0.799988	+	0.600016i	$0.795161\pi$
$828$		0			0
$829$	3778.09	−	2181.28i	0.158285	−	0.0913862i	−0.418765	−	0.908095i	$-0.637537\pi$
$829$	3778.09	−	2181.28i	0.158285	−	0.0913862i	0.577051	+	0.816708i	$0.304204\pi$
$830$		0			0
$831$	6849.32	−	11863.4i	0.285921	−	0.495229i
$832$		0			0
$833$	−37434.4	−	16657.0i	−1.55705	−	0.692835i
$834$		0			0
$835$	2900.37	+	1674.53i	0.120205	+	0.0694006i
$836$		0			0
$837$	−4183.65	−	7246.29i	−0.172769	−	0.299245i
$838$		0			0
$839$	−34373.2			−1.41442			−0.707208	−	0.707006i	$-0.750045\pi$
$839$	−34373.2			−1.41442			−0.707208	+	0.707006i	$0.750045\pi$
$840$		0			0
$841$	2632.42			0.107935
$842$		0			0
$843$	25258.6	+	43749.2i	1.03197	+	1.78743i
$844$		0			0
$845$	−541.635	−	312.713i	−0.0220507	−	0.0127310i
$846$		0			0
$847$	−20346.4	+	13216.3i	−0.825396	+	0.536147i
$848$		0			0
$849$	11637.5	−	20156.8i	0.470435	−	0.814816i
$850$		0			0
$851$	1075.99	−	621.224i	0.0433426	−	0.0250238i
$852$		0			0
$853$		−	9468.45i		−	0.380063i	−0.981778	−	0.190031i	$-0.939141\pi$
$853$		−	9468.45i		−	0.380063i	0.981778	−	0.190031i	$-0.0608590\pi$
$854$		0			0
$855$			1685.37i			0.0674136i
$856$		0			0
$857$	−9863.17	+	5694.50i	−0.393138	+	0.226978i	−0.683519	−	0.729933i	$-0.739551\pi$
$857$	−9863.17	+	5694.50i	−0.393138	+	0.226978i	0.290381	+	0.956911i	$0.406218\pi$
$858$		0			0
$859$	4227.95	−	7323.02i	0.167935	−	0.290871i	−0.769759	−	0.638335i	$-0.779624\pi$
$859$	4227.95	−	7323.02i	0.167935	−	0.290871i	0.937694	+	0.347464i	$0.112957\pi$
$860$		0			0
$861$	−46863.9	+	2461.19i	−1.85496	+	0.0974182i
$862$		0			0
$863$	−6473.10	−	3737.24i	−0.255326	−	0.147413i	0.366874	−	0.930271i	$-0.380428\pi$
$863$	−6473.10	−	3737.24i	−0.255326	−	0.147413i	−0.622201	+	0.782858i	$0.713761\pi$
$864$		0			0
$865$	−2068.58	−	3582.89i	−0.0813110	−	0.140835i
$866$		0			0
$867$	−66080.9			−2.58850
$868$		0			0
$869$	−349.399			−0.0136393
$870$		0			0
$871$	8565.86	+	14836.5i	0.333230	+	0.577171i
$872$		0			0
$873$	−13409.2	−	7741.81i	−0.519854	−	0.300138i
$874$		0			0
$875$	−2556.82	+	5019.39i	−0.0987842	+	0.193927i
$876$		0			0
$877$	−16028.3	+	27761.9i	−0.617147	+	1.06893i	0.372857	+	0.927889i	$0.378378\pi$
$877$	−16028.3	+	27761.9i	−0.617147	+	1.06893i	−0.990004	+	0.141041i	$0.954955\pi$
$878$		0			0
$879$	−45069.4	+	26020.8i	−1.72941	+	0.998476i
$880$		0			0
$881$		−	27139.7i		−	1.03787i	−0.854815	−	0.518933i	$-0.826329\pi$
$881$		−	27139.7i		−	1.03787i	0.854815	−	0.518933i	$-0.173671\pi$
$882$		0			0
$883$			43490.8i			1.65751i	0.559611	+	0.828755i	$0.310951\pi$
$883$			43490.8i			1.65751i	−0.559611	+	0.828755i	$0.689049\pi$
$884$		0			0
$885$	636.721	−	367.611i	0.0241843	−	0.0139628i
$886$		0			0
$887$	−16755.6	+	29021.6i	−0.634271	+	1.09859i	0.352398	+	0.935850i	$0.385366\pi$
$887$	−16755.6	+	29021.6i	−0.634271	+	1.09859i	−0.986669	+	0.162740i	$0.947967\pi$
$888$		0			0
$889$	−9125.78	+	17915.2i	−0.344284	+	0.675880i
$890$		0			0
$891$	−3265.16	−	1885.14i	−0.122769	−	0.0708807i
$892$		0			0
$893$	6783.60	+	11749.5i	0.254204	+	0.440295i
$894$		0			0
$895$	−1490.77			−0.0556771
$896$		0			0
$897$	2422.36			0.0901677
$898$		0			0
$899$	−23634.8	−	40936.6i	−0.876823	−	1.51870i
$900$		0			0
$901$	−70835.4	−	40896.8i	−2.61917	−	1.51218i
$902$		0			0
$903$	47074.2	−	2472.23i	1.73481	−	0.0911082i
$904$		0			0
$905$	−1075.63	+	1863.04i	−0.0395084	+	0.0684306i
$906$		0			0
$907$	6521.56	−	3765.22i	0.238748	−	0.137841i	−0.375853	−	0.926679i	$-0.622650\pi$
$907$	6521.56	−	3765.22i	0.238748	−	0.137841i	0.614601	+	0.788838i	$0.289317\pi$
$908$		0			0
$909$		−	19191.4i		−	0.700264i
$910$		0			0
$911$			19971.6i			0.726332i	0.931724	+	0.363166i	$0.118304\pi$
$911$			19971.6i			0.726332i	−0.931724	+	0.363166i	$0.881696\pi$
$912$		0			0
$913$	2076.83	−	1199.06i	0.0752826	−	0.0434644i
$914$		0			0
$915$	−903.882	+	1565.57i	−0.0326573	+	0.0565641i
$916$		0			0
$917$	23814.9	−	15469.3i	0.857621	−	0.557079i
$918$		0			0
$919$	40488.0	+	23375.8i	1.45329	+	0.839059i	0.998667	−	0.0516235i	$-0.0164396\pi$
$919$	40488.0	+	23375.8i	1.45329	+	0.839059i	0.454626	+	0.890682i	$0.349773\pi$
$920$		0			0
$921$	−9509.61	−	16471.1i	−0.340230	−	0.589296i
$922$		0			0
$923$	40362.0			1.43936
$924$		0			0
$925$	18369.9			0.652972
$926$		0			0
$927$	17333.9	+	30023.2i	0.614154	+	1.06375i
$928$		0			0
$929$	−16257.2	−	9386.12i	−0.574147	−	0.331484i	0.184657	−	0.982803i	$-0.440883\pi$
$929$	−16257.2	−	9386.12i	−0.574147	−	0.331484i	−0.758804	+	0.651319i	$0.774216\pi$
$930$		0			0
$931$	−20529.1	+	2162.24i	−0.722677	+	0.0761168i
$932$		0			0
$933$	−10495.0	+	18177.8i	−0.368264	+	0.637851i
$934$		0			0
$935$	−579.881	+	334.794i	−0.0202825	+	0.0117101i
$936$		0			0
$937$			13717.6i			0.478263i	0.970987	+	0.239132i	$0.0768628\pi$
$937$			13717.6i			0.478263i	−0.970987	+	0.239132i	$0.923137\pi$
$938$		0			0
$939$			8051.34i			0.279814i
$940$		0			0
$941$	15972.8	−	9221.91i	0.553346	−	0.319475i	−0.197124	−	0.980379i	$-0.563160\pi$
$941$	15972.8	−	9221.91i	0.553346	−	0.319475i	0.750471	+	0.660904i	$0.229827\pi$
$942$		0			0
$943$	−1498.45	+	2595.39i	−0.0517456	+	0.0896261i
$944$		0			0
$945$	359.299	+	553.139i	0.0123683	+	0.0190409i
$946$		0			0
$947$	28785.9	+	16619.6i	0.987769	+	0.570289i	0.904607	−	0.426247i	$-0.140165\pi$
$947$	28785.9	+	16619.6i	0.987769	+	0.570289i	0.0831621	+	0.996536i	$0.473498\pi$
$948$		0			0
$949$	−7011.73	−	12144.7i	−0.239842	−	0.415419i
$950$		0			0
$951$	39869.1			1.35946
$952$		0			0
$953$	32914.7			1.11880			0.559398	−	0.828899i	$-0.311033\pi$
$953$	32914.7			1.11880			0.559398	+	0.828899i	$0.311033\pi$
$954$		0			0
$955$	585.321	+	1013.81i	0.0198330	+	0.0343518i
$956$		0			0
$957$	−4604.50	−	2658.41i	−0.155530	−	0.0897955i
$958$		0			0
$959$	−893.769	−	17018.4i	−0.0300952	−	0.573048i
$960$		0			0
$961$	−26449.7	+	45812.2i	−0.887841	+	1.53779i
$962$		0			0
$963$	−33440.7	+	19307.0i	−1.11902	+	0.646064i
$964$		0			0
$965$			1632.22i			0.0544487i
$966$		0			0
$967$		−	29368.7i		−	0.976663i	−0.872658	−	0.488332i	$-0.837606\pi$
$967$		−	29368.7i		−	0.976663i	0.872658	−	0.488332i	$-0.162394\pi$
$968$		0			0
$969$	−43971.2	+	25386.8i	−1.45775	+	0.841632i
$970$		0			0
$971$	15783.6	−	27338.0i	0.521647	−	0.903520i	−0.478035	−	0.878341i	$-0.658651\pi$
$971$	15783.6	−	27338.0i	0.521647	−	0.903520i	0.999683	−	0.0251794i	$-0.00801569\pi$
$972$		0			0
$973$	−23510.6	−	11976.0i	−0.774630	−	0.394587i
$974$		0			0
$975$	31016.9	+	17907.6i	1.01881	+	0.588208i
$976$		0			0
$977$	−11625.0	−	20135.1i	−0.380672	−	0.659343i	0.610486	−	0.792027i	$-0.290974\pi$
$977$	−11625.0	−	20135.1i	−0.380672	−	0.659343i	−0.991158	+	0.132683i	$0.957641\pi$
$978$		0			0
$979$	4451.70			0.145329
$980$		0			0
$981$	36064.3			1.17374
$982$		0			0
$983$	2028.77	+	3513.93i	0.0658268	+	0.114015i	0.897060	−	0.441908i	$-0.145698\pi$
$983$	2028.77	+	3513.93i	0.0658268	+	0.114015i	−0.831234	+	0.555923i	$0.812365\pi$
$984$		0			0
$985$	1164.66	+	672.414i	0.0376741	+	0.0217512i
$986$		0			0
$987$	−26274.5	−	13383.9i	−0.847344	−	0.431626i
$988$		0			0
$989$	1505.17	−	2607.03i	0.0483940	−	0.0838208i
$990$		0			0
$991$	−36725.1	+	21203.3i	−1.17721	+	0.679661i	−0.955367	−	0.295422i	$-0.904540\pi$
$991$	−36725.1	+	21203.3i	−1.17721	+	0.679661i	−0.221841	+	0.975083i	$0.571206\pi$
$992$		0			0
$993$			51980.4i			1.66118i
$994$		0			0
$995$			304.252i			0.00969389i
$996$		0			0
$997$	49960.1	−	28844.5i	1.58701	−	0.916263i	0.593219	−	0.805041i	$-0.297857\pi$
$997$	49960.1	−	28844.5i	1.58701	−	0.916263i	0.993796	−	0.111222i	$-0.0354766\pi$
$998$		0			0
$999$	2164.02	−	3748.19i	0.0685350	−	0.118706i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.p.g.383.1		6
4.3	odd	2		448.4.p.f.383.3		6
7.3	odd	6		448.4.p.f.255.3		6
8.3	odd	2		112.4.p.g.47.1	yes	6
8.5	even	2		112.4.p.f.47.3	yes	6
28.3	even	6	inner	448.4.p.g.255.1		6
56.3	even	6		112.4.p.f.31.3	✓	6
56.5	odd	6		784.4.f.g.783.6		6
56.19	even	6		784.4.f.h.783.2		6
56.37	even	6		784.4.f.h.783.1		6
56.45	odd	6		112.4.p.g.31.1	yes	6
56.51	odd	6		784.4.f.g.783.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.3	✓	6	56.3	even	6
112.4.p.f.47.3	yes	6	8.5	even	2
112.4.p.g.31.1	yes	6	56.45	odd	6
112.4.p.g.47.1	yes	6	8.3	odd	2
448.4.p.f.255.3		6	7.3	odd	6
448.4.p.f.383.3		6	4.3	odd	2
448.4.p.g.255.1		6	28.3	even	6	inner
448.4.p.g.383.1		6	1.1	even	1	trivial
784.4.f.g.783.5		6	56.51	odd	6
784.4.f.g.783.6		6	56.5	odd	6
784.4.f.h.783.1		6	56.37	even	6
784.4.f.h.783.2		6	56.19	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.53129 - 6.11637i) q^{3} +(-1.05999 - 0.611983i) q^{5} +(16.5026 + 8.40621i) q^{7} +(-11.4400 + 19.8147i) q^{9} +O(q^{10})\)	\(q+(-3.53129 - 6.11637i) q^{3} +(-1.05999 - 0.611983i) q^{5} +(16.5026 + 8.40621i) q^{7} +(-11.4400 + 19.8147i) q^{9} +(-3.96612 + 2.28984i) q^{11} -41.0611i q^{13} +8.64436i q^{15} +(-103.451 + 59.7275i) q^{17} +(-30.0913 + 52.1196i) q^{19} +(-6.85991 - 130.621i) q^{21} +(-7.23396 - 4.17653i) q^{23} +(-61.7510 - 106.956i) q^{25} -29.0976 q^{27} -164.382 q^{29} +(143.780 + 249.034i) q^{31} +(28.0110 + 16.1722i) q^{33} +(-12.3481 - 19.0098i) q^{35} +(-74.3709 + 128.814i) q^{37} +(-251.145 + 144.999i) q^{39} -358.778i q^{41} +360.388i q^{43} +(24.2525 - 14.0022i) q^{45} +(112.717 - 195.232i) q^{47} +(201.671 + 277.449i) q^{49} +(730.631 + 421.830i) q^{51} +(342.362 + 592.988i) q^{53} +5.60537 q^{55} +425.044 q^{57} +(-42.5261 - 73.6574i) q^{59} +(181.109 + 104.563i) q^{61} +(-355.356 + 230.827i) q^{63} +(-25.1287 + 43.5242i) q^{65} +(-361.327 + 208.612i) q^{67} +58.9941i q^{69} +982.974i q^{71} +(295.770 - 170.763i) q^{73} +(-436.121 + 755.384i) q^{75} +(-84.7001 + 4.44826i) q^{77} +(66.0721 + 38.1467i) q^{79} +(411.633 + 712.968i) q^{81} -523.643 q^{83} +146.209 q^{85} +(580.480 + 1005.42i) q^{87} +(-841.824 - 486.027i) q^{89} +(345.169 - 677.615i) q^{91} +(1015.46 - 1758.82i) q^{93} +(63.7926 - 36.8307i) q^{95} +676.730i q^{97} -104.783i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9}+O(q^{10}) \) 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9	\( 6 q + 7 q^{3} + 3 q^{5} + 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} + 15 q^{21} - 15 q^{23} + 306 q^{25} - 362 q^{27} + 348 q^{29} + 205 q^{31} - 471 q^{33} + 1185 q^{35} + 249 q^{37} + 288 q^{39} - 2118 q^{45} + 75 q^{47} + 702 q^{49} + 2505 q^{51} + 645 q^{53} + 918 q^{55} - 6 q^{57} - 321 q^{59} + 1707 q^{61} - 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} - 555 q^{77} - 3447 q^{79} + 225 q^{81} - 24 q^{83} - 3786 q^{85} + 3642 q^{87} - 2607 q^{89} + 2448 q^{91} + 2991 q^{93} + 2085 q^{95}+O(q^{100}) \) 6 * q + 7 * q^3 + 3 * q^5 + 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 + 15 * q^21 - 15 * q^23 + 306 * q^25 - 362 * q^27 + 348 * q^29 + 205 * q^31 - 471 * q^33 + 1185 * q^35 + 249 * q^37 + 288 * q^39 - 2118 * q^45 + 75 * q^47 + 702 * q^49 + 2505 * q^51 + 645 * q^53 + 918 * q^55 - 6 * q^57 - 321 * q^59 + 1707 * q^61 - 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 - 555 * q^77 - 3447 * q^79 + 225 * q^81 - 24 * q^83 - 3786 * q^85 + 3642 * q^87 - 2607 * q^89 + 2448 * q^91 + 2991 * q^93 + 2085 * q^95

Introduction

L-functions

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