LMFDB - Embedded newform 448.4.p.i.383.7

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:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			383.7
Character	$\chi$	$=$	448.383
Dual form			448.4.p.i.255.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.69336 - 4.66504i) q^{3} +(-12.5403 - 7.24016i) q^{5} +(18.3374 - 2.59641i) q^{7} +(-1.00837 + 1.74655i) q^{9} +O(q^{10})$	\(q+(-2.69336 - 4.66504i) q^{3} +(-12.5403 - 7.24016i) q^{5} +(18.3374 - 2.59641i) q^{7} +(-1.00837 + 1.74655i) q^{9} +(-50.5990 + 29.2133i) q^{11} +35.9239i q^{13} +78.0015i q^{15} +(-47.6229 + 27.4951i) q^{17} +(-2.40717 + 4.16933i) q^{19} +(-61.5014 - 78.5514i) q^{21} +(-22.4952 - 12.9876i) q^{23} +(42.3399 + 73.3349i) q^{25} -134.578 q^{27} +260.809 q^{29} +(-61.1481 - 105.912i) q^{31} +(272.563 + 157.364i) q^{33} +(-248.755 - 100.206i) q^{35} +(170.068 - 294.567i) q^{37} +(167.586 - 96.7559i) q^{39} +500.847i q^{41} +205.069i q^{43} +(25.2906 - 14.6015i) q^{45} +(-96.7271 + 167.536i) q^{47} +(329.517 - 95.2224i) q^{49} +(256.531 + 148.108i) q^{51} +(277.068 + 479.897i) q^{53} +846.038 q^{55} +25.9334 q^{57} +(-99.3874 - 172.144i) q^{59} +(309.030 + 178.419i) q^{61} +(-13.9561 + 34.6452i) q^{63} +(260.095 - 450.497i) q^{65} +(779.971 - 450.316i) q^{67} +139.921i q^{69} +451.482i q^{71} +(65.0285 - 37.5442i) q^{73} +(228.073 - 395.035i) q^{75} +(-852.002 + 667.071i) q^{77} +(-405.952 - 234.377i) q^{79} +(389.692 + 674.967i) q^{81} -624.057 q^{83} +796.277 q^{85} +(-702.452 - 1216.68i) q^{87} +(-525.349 - 303.311i) q^{89} +(93.2729 + 658.749i) q^{91} +(-329.387 + 570.516i) q^{93} +(60.3733 - 34.8565i) q^{95} -873.069i q^{97} -117.831i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 216 q^{9}+O(q^{10}) $ 48 * q - 216 * q^9	$ 48 q - 216 q^{9} + 104 q^{21} + 432 q^{25} - 112 q^{29} + 72 q^{33} + 504 q^{37} + 1320 q^{45} + 160 q^{49} - 392 q^{53} + 1360 q^{57} - 600 q^{61} - 744 q^{65} - 648 q^{73} + 2880 q^{77} - 400 q^{81} + 240 q^{85} + 3816 q^{89} - 2872 q^{93}+O(q^{100}) $ 48 * q - 216 * q^9 + 104 * q^21 + 432 * q^25 - 112 * q^29 + 72 * q^33 + 504 * q^37 + 1320 * q^45 + 160 * q^49 - 392 * q^53 + 1360 * q^57 - 600 * q^61 - 744 * q^65 - 648 * q^73 + 2880 * q^77 - 400 * q^81 + 240 * q^85 + 3816 * q^89 - 2872 * q^93

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
$3$	−2.69336	−	4.66504i	−0.518337	−	0.897786i	−0.999773	−	0.0213051i	$-0.993218\pi$
$3$	−2.69336	−	4.66504i	−0.518337	−	0.897786i	0.481436	−	0.876481i	$-0.340115\pi$
$4$		0			0
$5$	−12.5403	−	7.24016i	−1.12164	−	0.647580i	−0.179822	−	0.983699i	$-0.557552\pi$
$5$	−12.5403	−	7.24016i	−1.12164	−	0.647580i	−0.941819	+	0.336119i	$0.890885\pi$
$6$		0			0
$7$	18.3374	−	2.59641i	0.990124	−	0.140193i
$7$	18.3374	−	2.59641i	0.990124	−	0.140193i
$8$		0			0
$9$	−1.00837	+	1.74655i	−0.0373470	+	0.0646869i
$10$		0			0
$11$	−50.5990	+	29.2133i	−1.38692	+	0.800741i	−0.992967	−	0.118387i	$-0.962228\pi$
$11$	−50.5990	+	29.2133i	−1.38692	+	0.800741i	−0.393957	+	0.919129i	$0.628894\pi$
$12$		0			0
$13$			35.9239i			0.766422i	0.923661	+	0.383211i	$0.125182\pi$
$13$			35.9239i			0.766422i	−0.923661	+	0.383211i	$0.874818\pi$
$14$		0			0
$15$			78.0015i			1.34266i
$16$		0			0
$17$	−47.6229	+	27.4951i	−0.679427	+	0.392267i	−0.799639	−	0.600481i	$-0.794976\pi$
$17$	−47.6229	+	27.4951i	−0.679427	+	0.392267i	0.120212	+	0.992748i	$0.461643\pi$
$18$		0			0
$19$	−2.40717	+	4.16933i	−0.0290653	+	0.0503427i	−0.880192	−	0.474617i	$-0.842586\pi$
$19$	−2.40717	+	4.16933i	−0.0290653	+	0.0503427i	0.851127	+	0.524960i	$0.175920\pi$
$20$		0			0
$21$	−61.5014	−	78.5514i	−0.639081	−	0.816253i
$22$		0			0
$23$	−22.4952	−	12.9876i	−0.203938	−	0.117743i	0.394553	−	0.918873i	$-0.370899\pi$
$23$	−22.4952	−	12.9876i	−0.203938	−	0.117743i	−0.598491	+	0.801130i	$0.704233\pi$
$24$		0			0
$25$	42.3399	+	73.3349i	0.338720	+	0.586680i
$26$		0			0
$27$	−134.578			−0.959241
$28$		0			0
$29$	260.809			1.67003			0.835017	−	0.550223i	$-0.185457\pi$
$29$	260.809			1.67003			0.835017	+	0.550223i	$0.185457\pi$
$30$		0			0
$31$	−61.1481	−	105.912i	−0.354275	−	0.613622i	0.632719	−	0.774382i	$-0.281939\pi$
$31$	−61.1481	−	105.912i	−0.354275	−	0.613622i	−0.986994	+	0.160760i	$0.948606\pi$
$32$		0			0
$33$	272.563	+	157.364i	1.43779	+	0.830108i
$34$		0			0
$35$	−248.755	−	100.206i	−1.20135	−	0.483939i
$36$		0			0
$37$	170.068	−	294.567i	0.755650	−	1.30882i	−0.189401	−	0.981900i	$-0.560655\pi$
$37$	170.068	−	294.567i	0.755650	−	1.30882i	0.945051	−	0.326924i	$-0.106012\pi$
$38$		0			0
$39$	167.586	−	96.7559i	0.688083	−	0.397265i
$40$		0			0
$41$			500.847i			1.90778i	0.300152	+	0.953892i	$0.402963\pi$
$41$			500.847i			1.90778i	−0.300152	+	0.953892i	$0.597037\pi$
$42$		0			0
$43$			205.069i			0.727273i	0.931541	+	0.363636i	$0.118465\pi$
$43$			205.069i			0.727273i	−0.931541	+	0.363636i	$0.881535\pi$
$44$		0			0
$45$	25.2906	−	14.6015i	0.0837799	−	0.0483704i
$46$		0			0
$47$	−96.7271	+	167.536i	−0.300194	+	0.519951i	−0.976180	−	0.216964i	$-0.930385\pi$
$47$	−96.7271	+	167.536i	−0.300194	+	0.519951i	0.675986	+	0.736914i	$0.263718\pi$
$48$		0			0
$49$	329.517	−	95.2224i	0.960692	−	0.277616i
$50$		0			0
$51$	256.531	+	148.108i	0.704345	+	0.406654i
$52$		0			0
$53$	277.068	+	479.897i	0.718081	+	1.24375i	0.961759	+	0.273896i	$0.0883124\pi$
$53$	277.068	+	479.897i	0.718081	+	1.24375i	−0.243679	+	0.969856i	$0.578354\pi$
$54$		0			0
$55$	846.038			2.07418
$56$		0			0
$57$	25.9334			0.0602626
$58$		0			0
$59$	−99.3874	−	172.144i	−0.219307	−	0.379852i	0.735289	−	0.677754i	$-0.237046\pi$
$59$	−99.3874	−	172.144i	−0.219307	−	0.379852i	−0.954596	+	0.297902i	$0.903713\pi$
$60$		0			0
$61$	309.030	+	178.419i	0.648643	+	0.374494i	0.787936	−	0.615757i	$-0.211150\pi$
$61$	309.030	+	178.419i	0.648643	+	0.374494i	−0.139293	+	0.990251i	$0.544483\pi$
$62$		0			0
$63$	−13.9561	+	34.6452i	−0.0279095	+	0.0692839i
$64$		0			0
$65$	260.095	−	450.497i	0.496319	−	0.859651i
$66$		0			0
$67$	779.971	−	450.316i	1.42222	−	0.821118i	0.425730	−	0.904850i	$-0.360017\pi$
$67$	779.971	−	450.316i	1.42222	−	0.821118i	0.996488	+	0.0837325i	$0.0266841\pi$
$68$		0			0
$69$			139.921i			0.244123i
$70$		0			0
$71$			451.482i			0.754663i	0.926078	+	0.377331i	$0.123158\pi$
$71$			451.482i			0.754663i	−0.926078	+	0.377331i	$0.876842\pi$
$72$		0			0
$73$	65.0285	−	37.5442i	0.104260	−	0.0601948i	−0.446963	−	0.894552i	$-0.647494\pi$
$73$	65.0285	−	37.5442i	0.104260	−	0.0601948i	0.551224	+	0.834358i	$0.314161\pi$
$74$		0			0
$75$	228.073	−	395.035i	0.351142	−	0.608196i
$76$		0			0
$77$	−852.002	+	667.071i	−1.26097	+	0.987270i
$78$		0			0
$79$	−405.952	−	234.377i	−0.578142	−	0.333790i	0.182253	−	0.983252i	$-0.441661\pi$
$79$	−405.952	−	234.377i	−0.578142	−	0.333790i	−0.760395	+	0.649461i	$0.774994\pi$
$80$		0			0
$81$	389.692	+	674.967i	0.534557	+	0.925881i
$82$		0			0
$83$	−624.057			−0.825291			−0.412646	−	0.910892i	$-0.635395\pi$
$83$	−624.057			−0.825291			−0.412646	+	0.910892i	$0.635395\pi$
$84$		0			0
$85$	796.277			1.01610
$86$		0			0
$87$	−702.452	−	1216.68i	−0.865641	−	1.49933i
$88$		0			0
$89$	−525.349	−	303.311i	−0.625696	−	0.361246i	0.153388	−	0.988166i	$-0.450982\pi$
$89$	−525.349	−	303.311i	−0.625696	−	0.361246i	−0.779083	+	0.626921i	$0.784315\pi$
$90$		0			0
$91$	93.2729	+	658.749i	0.107447	+	0.758853i
$92$		0			0
$93$	−329.387	+	570.516i	−0.367268	+	0.636126i
$94$		0			0
$95$	60.3733	−	34.8565i	0.0652018	−	0.0376443i
$96$		0			0
$97$		−	873.069i		−	0.913884i	−0.889497	−	0.456942i	$-0.848945\pi$
$97$		−	873.069i		−	0.913884i	0.889497	−	0.456942i	$-0.151055\pi$
$98$		0			0
$99$		−	117.831i		−	0.119621i
$100$		0			0
$101$	1067.09	−	616.083i	1.05128	−	0.606956i	0.128271	−	0.991739i	$-0.459057\pi$
$101$	1067.09	−	616.083i	1.05128	−	0.606956i	0.923007	+	0.384784i	$0.125724\pi$
$102$		0			0
$103$	33.7274	−	58.4176i	0.0322646	−	0.0558840i	−0.849442	−	0.527682i	$-0.823061\pi$
$103$	33.7274	−	58.4176i	0.0322646	−	0.0558840i	0.881707	+	0.471798i	$0.156395\pi$
$104$		0			0
$105$	202.523	+	1430.34i	0.188231	+	1.32940i
$106$		0			0
$107$	843.403	+	486.939i	0.762008	+	0.439945i	0.830016	−	0.557739i	$-0.188331\pi$
$107$	843.403	+	486.939i	0.762008	+	0.439945i	−0.0680085	+	0.997685i	$0.521664\pi$
$108$		0			0
$109$	339.479	+	587.995i	0.298314	+	0.516695i	0.975750	−	0.218887i	$-0.0702425\pi$
$109$	339.479	+	587.995i	0.298314	+	0.516695i	−0.677437	+	0.735581i	$0.736909\pi$
$110$		0			0
$111$	−1832.22			−1.56673
$112$		0			0
$113$	−48.3836			−0.0402792			−0.0201396	−	0.999797i	$-0.506411\pi$
$113$	−48.3836			−0.0402792			−0.0201396	+	0.999797i	$0.506411\pi$
$114$		0			0
$115$	188.065	+	325.737i	0.152497	+	0.264132i
$116$		0			0
$117$	−62.7427	−	36.2245i	−0.0495775	−	0.0286236i
$118$		0			0
$119$	−801.890	+	627.836i	−0.617724	+	0.483644i
$120$		0			0
$121$	1041.34	−	1803.65i	0.782374	−	1.35511i
$122$		0			0
$123$	2336.47	−	1348.96i	1.71278	−	0.988875i
$124$		0			0
$125$			583.848i			0.417768i
$126$		0			0
$127$			2153.09i			1.50438i	0.658947	+	0.752189i	$0.271002\pi$
$127$			2153.09i			1.50438i	−0.658947	+	0.752189i	$0.728998\pi$
$128$		0			0
$129$	956.654	−	552.325i	0.652936	−	0.376973i
$130$		0			0
$131$	−1280.11	+	2217.21i	−0.853767	+	1.47877i	0.0240181	+	0.999712i	$0.492354\pi$
$131$	−1280.11	+	2217.21i	−0.853767	+	1.47877i	−0.877785	+	0.479056i	$0.840979\pi$
$132$		0			0
$133$	−33.3158	+	82.7045i	−0.0217206	+	0.0539202i
$134$		0			0
$135$	1687.65	+	974.365i	1.07592	+	0.621185i
$136$		0			0
$137$	829.902	+	1437.43i	0.517542	+	0.896409i	0.999792	+	0.0203758i	$0.00648626\pi$
$137$	829.902	+	1437.43i	0.517542	+	0.896409i	−0.482250	+	0.876034i	$0.660180\pi$
$138$		0			0
$139$	−888.312			−0.542055			−0.271027	−	0.962572i	$-0.587363\pi$
$139$	−888.312			−0.542055			−0.271027	+	0.962572i	$0.587363\pi$
$140$		0			0
$141$	1042.08			0.622406
$142$		0			0
$143$	−1049.46	−	1817.71i	−0.613706	−	1.06297i
$144$		0			0
$145$	−3270.63	−	1888.30i	−1.87318	−	1.08148i
$146$		0			0
$147$	−1331.72	−	1280.74i	−0.747203	−	0.718597i
$148$		0			0
$149$	−671.616	+	1163.27i	−0.369268	+	0.639590i	−0.989451	−	0.144866i	$-0.953725\pi$
$149$	−671.616	+	1163.27i	−0.369268	+	0.639590i	0.620184	+	0.784457i	$0.287058\pi$
$150$		0			0
$151$	−2161.38	+	1247.87i	−1.16484	+	0.672518i	−0.952458	−	0.304669i	$-0.901454\pi$
$151$	−2161.38	+	1247.87i	−1.16484	+	0.672518i	−0.212378	+	0.977188i	$0.568121\pi$
$152$		0			0
$153$		−	110.901i		−	0.0586000i
$154$		0			0
$155$			1770.89i			0.917685i
$156$		0			0
$157$	414.309	−	239.201i	0.210608	−	0.121594i	−0.390986	−	0.920397i	$-0.627866\pi$
$157$	414.309	−	239.201i	0.210608	−	0.121594i	0.601594	+	0.798802i	$0.294533\pi$
$158$		0			0
$159$	1492.49	−	2585.07i	0.744416	−	1.28937i
$160$		0			0
$161$	−446.223	−	179.752i	−0.218430	−	0.0879901i
$162$		0			0
$163$	−3250.17	−	1876.49i	−1.56180	−	0.901705i	−0.997075	−	0.0764242i	$-0.975650\pi$
$163$	−3250.17	−	1876.49i	−1.56180	−	0.901705i	−0.564723	−	0.825281i	$-0.691017\pi$
$164$		0			0
$165$	−2278.68	−	3946.80i	−1.07512	−	1.86217i
$166$		0			0
$167$	3742.97			1.73437			0.867185	−	0.497985i	$-0.165927\pi$
$167$	3742.97			1.73437			0.867185	+	0.497985i	$0.165927\pi$
$168$		0			0
$169$	906.476			0.412597
$170$		0			0
$171$	−4.85462	−	8.40845i	−0.00217101	−	0.00376030i
$172$		0			0
$173$	−3753.01	−	2166.80i	−1.64934	−	0.952248i	−0.977334	−	0.211703i	$-0.932099\pi$
$173$	−3753.01	−	2166.80i	−1.64934	−	0.952248i	−0.672007	−	0.740545i	$-0.734568\pi$
$174$		0			0
$175$	966.810	+	1234.84i	0.417623	+	0.533400i
$176$		0			0
$177$	−535.372	+	927.292i	−0.227350	+	0.393783i
$178$		0			0
$179$	−954.997	+	551.368i	−0.398770	+	0.230230i	−0.685953	−	0.727646i	$-0.740615\pi$
$179$	−954.997	+	551.368i	−0.398770	+	0.230230i	0.287183	+	0.957876i	$0.407281\pi$
$180$		0			0
$181$			1281.76i			0.526366i	0.964746	+	0.263183i	$0.0847724\pi$
$181$			1281.76i			0.526366i	−0.964746	+	0.263183i	$0.915228\pi$
$182$		0			0
$183$		−	1922.18i		−	0.776458i
$184$		0			0
$185$	−4265.42	+	2462.64i	−1.69514	+	0.978687i
$186$		0			0
$187$	1606.45	−	2782.45i	0.628209	−	1.08809i
$188$		0			0
$189$	−2467.80	+	349.419i	−0.949768	+	0.134479i
$190$		0			0
$191$	574.187	+	331.507i	0.217522	+	0.125586i	0.604802	−	0.796376i	$-0.293252\pi$
$191$	574.187	+	331.507i	0.217522	+	0.125586i	−0.387280	+	0.921962i	$0.626585\pi$
$192$		0			0
$193$	1278.35	+	2214.16i	0.476774	+	0.825797i	0.999646	−	0.0266145i	$-0.00847265\pi$
$193$	1278.35	+	2214.16i	0.476774	+	0.825797i	−0.522872	+	0.852411i	$0.675139\pi$
$194$		0			0
$195$	−2802.11			−1.02904
$196$		0			0
$197$	−3975.13			−1.43765			−0.718824	−	0.695192i	$-0.755319\pi$
$197$	−3975.13			−1.43765			−0.718824	+	0.695192i	$0.755319\pi$
$198$		0			0
$199$	890.677	+	1542.70i	0.317278	+	0.549542i	0.979919	−	0.199395i	$-0.0638977\pi$
$199$	890.677	+	1542.70i	0.317278	+	0.549542i	−0.662641	+	0.748937i	$0.730564\pi$
$200$		0			0
$201$	−4201.48	−	2425.73i	−1.47438	−	0.851232i
$202$		0			0
$203$	4782.55	−	677.166i	1.65354	−	0.234127i
$204$		0			0
$205$	3626.21	−	6280.78i	1.23544	−	2.13985i
$206$		0			0
$207$	45.3669	−	26.1926i	0.0152329	−	0.00879474i
$208$		0			0
$209$		−	281.285i		−	0.0930953i
$210$		0			0
$211$			3427.72i			1.11836i	0.829047	+	0.559179i	$0.188884\pi$
$211$			3427.72i			1.11836i	−0.829047	+	0.559179i	$0.811116\pi$
$212$		0			0
$213$	2106.18	−	1216.00i	0.677526	−	0.391170i
$214$		0			0
$215$	1484.73	−	2571.63i	0.470967	−	0.815739i
$216$		0			0
$217$	−1396.28	−	1783.37i	−0.436801	−	0.557895i
$218$		0			0
$219$	−350.290	−	202.240i	−0.108084	−	0.0624024i
$220$		0			0
$221$	−987.731	−	1710.80i	−0.300642	−	0.520728i
$222$		0			0
$223$	−4119.68			−1.23710			−0.618552	−	0.785744i	$-0.712280\pi$
$223$	−4119.68			−1.23710			−0.618552	+	0.785744i	$0.712280\pi$
$224$		0			0
$225$	−170.777			−0.0506007
$226$		0			0
$227$	866.444	+	1500.72i	0.253339	+	0.438796i	0.964443	−	0.264291i	$-0.0851379\pi$
$227$	866.444	+	1500.72i	0.253339	+	0.438796i	−0.711104	+	0.703087i	$0.751805\pi$
$228$		0			0
$229$	1575.12	+	909.397i	0.454528	+	0.262422i	0.709741	−	0.704463i	$-0.248812\pi$
$229$	1575.12	+	909.397i	0.454528	+	0.262422i	−0.255213	+	0.966885i	$0.582145\pi$
$230$		0			0
$231$	5406.66	+	2177.96i	1.53997	+	0.620343i
$232$		0			0
$233$	1465.77	−	2538.79i	0.412129	−	0.713828i	−0.582994	−	0.812477i	$-0.698119\pi$
$233$	1465.77	−	2538.79i	0.412129	−	0.713828i	0.995122	+	0.0986491i	$0.0314521\pi$
$234$		0			0
$235$	2425.98	−	1400.64i	0.673419	−	0.388799i
$236$		0			0
$237$			2525.04i			0.692064i
$238$		0			0
$239$		−	923.229i		−	0.249869i	−0.992165	−	0.124935i	$-0.960128\pi$
$239$		−	923.229i		−	0.249869i	0.992165	−	0.124935i	$-0.0398721\pi$
$240$		0			0
$241$	−6299.30	+	3636.90i	−1.68371	+	0.972089i	−0.724548	+	0.689224i	$0.757952\pi$
$241$	−6299.30	+	3636.90i	−1.68371	+	0.972089i	−0.959160	+	0.282865i	$0.908715\pi$
$242$		0			0
$243$	282.363	−	489.067i	0.0745415	−	0.129110i
$244$		0			0
$245$	−4821.68	−	1191.64i	−1.25733	−	0.310739i
$246$		0			0
$247$	−149.779	−	86.4747i	−0.0385837	−	0.0222763i
$248$		0			0
$249$	1680.81	+	2911.25i	0.427779	+	0.740936i
$250$		0			0
$251$	−3501.75			−0.880592			−0.440296	−	0.897853i	$-0.645127\pi$
$251$	−3501.75			−0.880592			−0.440296	+	0.897853i	$0.645127\pi$
$252$		0			0
$253$	1517.64			0.377128
$254$		0			0
$255$	−2144.66	−	3714.66i	−0.526681	−	0.912239i
$256$		0			0
$257$	6628.77	+	3827.12i	1.60891	+	0.928907i	0.989614	+	0.143753i	$0.0459172\pi$
$257$	6628.77	+	3827.12i	1.60891	+	0.928907i	0.619301	+	0.785154i	$0.287416\pi$
$258$		0			0
$259$	2353.79	−	5843.14i	0.564699	−	1.40183i
$260$		0			0
$261$	−262.992	+	455.515i	−0.0623708	+	0.108029i
$262$		0			0
$263$	2218.99	−	1281.14i	0.520262	−	0.300374i	−0.216780	−	0.976221i	$-0.569555\pi$
$263$	2218.99	−	1281.14i	0.520262	−	0.300374i	0.737042	+	0.675847i	$0.236222\pi$
$264$		0			0
$265$		−	8024.08i		−	1.86006i
$266$		0			0
$267$			3267.70i			0.748988i
$268$		0			0
$269$	1799.21	−	1038.78i	0.407807	−	0.235447i	−0.282040	−	0.959403i	$-0.591011\pi$
$269$	1799.21	−	1038.78i	0.407807	−	0.235447i	0.689847	+	0.723955i	$0.257678\pi$
$270$		0			0
$271$	−453.320	+	785.173i	−0.101613	+	0.176000i	−0.912350	−	0.409412i	$-0.865734\pi$
$271$	−453.320	+	785.173i	−0.101613	+	0.176000i	0.810736	+	0.585412i	$0.199067\pi$
$272$		0			0
$273$	2821.87	−	2209.37i	0.625594	−	0.489806i
$274$		0			0
$275$	−4284.72	−	2473.78i	−0.939557	−	0.542454i
$276$		0			0
$277$	837.572	+	1450.72i	0.181678	+	0.314676i	0.942452	−	0.334341i	$-0.108514\pi$
$277$	837.572	+	1450.72i	0.181678	+	0.314676i	−0.760774	+	0.649017i	$0.775180\pi$
$278$		0			0
$279$	246.639			0.0529244
$280$		0			0
$281$	−872.820			−0.185296			−0.0926479	−	0.995699i	$-0.529533\pi$
$281$	−872.820			−0.185296			−0.0926479	+	0.995699i	$0.529533\pi$
$282$		0			0
$283$	3118.93	+	5402.14i	0.655127	+	1.13471i	0.981862	+	0.189598i	$0.0607184\pi$
$283$	3118.93	+	5402.14i	0.655127	+	1.13471i	−0.326735	+	0.945116i	$0.605948\pi$
$284$		0			0
$285$	−325.214	−	187.762i	−0.0675930	−	0.0390249i
$286$		0			0
$287$	1300.40	+	9184.20i	0.267457	+	1.88894i
$288$		0			0
$289$	−944.537	+	1635.99i	−0.192253	+	0.332991i
$290$		0			0
$291$	−4072.90	+	2351.49i	−0.820472	+	0.473700i
$292$		0			0
$293$			6323.11i			1.26075i	0.776290	+	0.630376i	$0.217099\pi$
$293$			6323.11i			1.26075i	−0.776290	+	0.630376i	$0.782901\pi$
$294$		0			0
$295$			2878.33i			0.568076i
$296$		0			0
$297$	6809.50	−	3931.47i	1.33040	−	0.768104i
$298$		0			0
$299$	466.564	−	808.113i	0.0902412	−	0.156302i
$300$		0			0
$301$	532.442	+	3760.42i	0.101958	+	0.720091i
$302$		0			0
$303$	−5748.09	−	3318.66i	−1.08983	−	0.629215i
$304$		0			0
$305$	−2583.56	−	4474.86i	−0.485030	−	0.840097i
$306$		0			0
$307$	−1053.63			−0.195875			−0.0979376	−	0.995193i	$-0.531225\pi$
$307$	−1053.63			−0.195875			−0.0979376	+	0.995193i	$0.531225\pi$
$308$		0			0
$309$	−363.360			−0.0668959
$310$		0			0
$311$	−1489.99	−	2580.74i	−0.271671	−	0.470548i	0.697619	−	0.716469i	$-0.254243\pi$
$311$	−1489.99	−	2580.74i	−0.271671	−	0.470548i	−0.969290	+	0.245921i	$0.920910\pi$
$312$		0			0
$313$	1124.15	+	649.030i	0.203006	+	0.117206i	0.598057	−	0.801454i	$-0.295940\pi$
$313$	1124.15	+	649.030i	0.203006	+	0.117206i	−0.395051	+	0.918659i	$0.629273\pi$
$314$		0			0
$315$	425.851	−	333.418i	0.0761713	−	0.0596380i
$316$		0			0
$317$	−177.670	+	307.734i	−0.0314794	+	0.0545239i	−0.881336	−	0.472490i	$-0.843355\pi$
$317$	−177.670	+	307.734i	−0.0314794	+	0.0545239i	0.849857	+	0.527014i	$0.176689\pi$
$318$		0			0
$319$	−13196.7	+	7619.10i	−2.31621	+	1.33727i
$320$		0			0
$321$		−	5246.00i		−	0.912160i
$322$		0			0
$323$		−	264.741i		−	0.0456055i
$324$		0			0
$325$	−2634.47	+	1521.01i	−0.449644	+	0.259602i
$326$		0			0
$327$	1828.68	−	3167.36i	0.309254	−	0.535644i
$328$		0			0
$329$	−1338.73	+	3323.32i	−0.224336	+	0.556901i
$330$		0			0
$331$	5045.32	+	2912.92i	0.837813	+	0.483711i	0.856520	−	0.516114i	$-0.172622\pi$
$331$	5045.32	+	2912.92i	0.837813	+	0.483711i	−0.0187074	+	0.999825i	$0.505955\pi$
$332$		0			0
$333$	342.983	+	594.064i	0.0564425	+	0.0977613i
$334$		0			0
$335$	−13041.5			−2.12696
$336$		0			0
$337$	−8448.35			−1.36561			−0.682805	−	0.730600i	$-0.739240\pi$
$337$	−8448.35			−1.36561			−0.682805	+	0.730600i	$0.739240\pi$
$338$		0			0
$339$	130.315	+	225.711i	0.0208782	+	0.0361621i
$340$		0			0
$341$	6188.06	+	3572.68i	0.982705	+	0.567365i
$342$		0			0
$343$	5795.24	−	2601.69i	0.912285	−	0.409557i
$344$		0			0
$345$	1013.05	−	1754.66i	0.158089	−	0.273819i
$346$		0			0
$347$	6521.74	−	3765.33i	1.00895	−	0.582517i	0.0980649	−	0.995180i	$-0.468735\pi$
$347$	6521.74	−	3765.33i	1.00895	−	0.582517i	0.910884	+	0.412663i	$0.135401\pi$
$348$		0			0
$349$			2537.51i			0.389198i	0.980883	+	0.194599i	$0.0623405\pi$
$349$			2537.51i			0.389198i	−0.980883	+	0.194599i	$0.937660\pi$
$350$		0			0
$351$		−	4834.55i		−	0.735183i
$352$		0			0
$353$	−4433.19	+	2559.51i	−0.668428	+	0.385917i	−0.795481	−	0.605979i	$-0.792782\pi$
$353$	−4433.19	+	2559.51i	−0.668428	+	0.385917i	0.127053	+	0.991896i	$0.459448\pi$
$354$		0			0
$355$	3268.80	−	5661.74i	0.488705	−	0.846461i
$356$		0			0
$357$	5088.66	+	2049.86i	0.754399	+	0.303894i
$358$		0			0
$359$	−1033.72	−	596.817i	−0.151971	−	0.0877403i	0.422086	−	0.906556i	$-0.361298\pi$
$359$	−1033.72	−	596.817i	−0.151971	−	0.0877403i	−0.574057	+	0.818815i	$0.694631\pi$
$360$		0			0
$361$	3417.91	+	5920.00i	0.498310	+	0.863099i
$362$		0			0
$363$	−11218.8			−1.62213
$364$		0			0
$365$	−1087.31			−0.155924
$366$		0			0
$367$	1006.67	+	1743.60i	0.143182	+	0.247998i	0.928693	−	0.370849i	$-0.120933\pi$
$367$	1006.67	+	1743.60i	0.143182	+	0.247998i	−0.785512	+	0.618847i	$0.787600\pi$
$368$		0			0
$369$	−874.752	−	505.038i	−0.123409	−	0.0712500i
$370$		0			0
$371$	6326.71	+	8080.65i	0.885354	+	1.13080i
$372$		0			0
$373$	−5124.65	+	8876.15i	−0.711379	+	1.23214i	0.252961	+	0.967477i	$0.418596\pi$
$373$	−5124.65	+	8876.15i	−0.711379	+	1.23214i	−0.964340	+	0.264668i	$0.914738\pi$
$374$		0			0
$375$	2723.67	−	1572.51i	0.375066	−	0.216545i
$376$		0			0
$377$			9369.26i			1.27995i
$378$		0			0
$379$		−	318.732i		−	0.0431983i	−0.999767	−	0.0215992i	$-0.993124\pi$
$379$		−	318.732i		−	0.0431983i	0.999767	−	0.0215992i	$-0.00687576\pi$
$380$		0			0
$381$	10044.3	−	5799.05i	1.35061	−	0.779775i
$382$		0			0
$383$	4629.64	−	8018.77i	0.617659	−	1.06982i	−0.372253	−	0.928131i	$-0.621415\pi$
$383$	4629.64	−	8018.77i	0.617659	−	1.06982i	0.989912	−	0.141685i	$-0.0452521\pi$
$384$		0			0
$385$	15514.1	−	2196.66i	2.05369	−	0.290784i
$386$		0			0
$387$	−358.163	−	206.785i	−0.0470450	−	0.0271615i
$388$		0			0
$389$	−3638.33	−	6301.77i	−0.474217	−	0.821368i	0.525347	−	0.850888i	$-0.323936\pi$
$389$	−3638.33	−	6301.77i	−0.474217	−	0.821368i	−0.999564	+	0.0295198i	$0.990602\pi$
$390$		0			0
$391$	1428.38			0.184748
$392$		0			0
$393$	13791.1			1.77016
$394$		0			0
$395$	3393.85	+	5878.33i	0.432312	+	0.748786i
$396$		0			0
$397$	3325.00	+	1919.69i	0.420345	+	0.242687i	0.695225	−	0.718792i	$-0.255305\pi$
$397$	3325.00	+	1919.69i	0.420345	+	0.242687i	−0.274880	+	0.961479i	$0.588638\pi$
$398$		0			0
$399$	475.551	−	67.3337i	0.0596675	−	0.00844838i
$400$		0			0
$401$	−4025.18	+	6971.81i	−0.501266	+	0.868218i	0.498733	+	0.866756i	$0.333799\pi$
$401$	−4025.18	+	6971.81i	−0.501266	+	0.868218i	−0.999999	+	0.00146254i	$0.999534\pi$
$402$		0			0
$403$	3804.75	−	2196.67i	0.470293	−	0.271524i
$404$		0			0
$405$		−	11285.7i		−	1.38467i
$406$		0			0
$407$			19873.0i			2.42032i
$408$		0			0
$409$	2541.83	−	1467.53i	0.307300	−	0.177420i	−0.338418	−	0.940996i	$-0.609892\pi$
$409$	2541.83	−	1467.53i	0.307300	−	0.177420i	0.645718	+	0.763576i	$0.276558\pi$
$410$		0			0
$411$	4470.45	−	7743.04i	0.536523	−	0.929285i
$412$		0			0
$413$	−2269.46	−	2898.62i	−0.270394	−	0.345355i
$414$		0			0
$415$	7825.89	+	4518.28i	0.925681	+	0.534442i
$416$		0			0
$417$	2392.54	+	4144.01i	0.280967	+	0.486650i
$418$		0			0
$419$	−12068.8			−1.40716			−0.703580	−	0.710616i	$-0.748416\pi$
$419$	−12068.8			−1.40716			−0.703580	+	0.710616i	$0.748416\pi$
$420$		0			0
$421$	16472.4			1.90693			0.953463	−	0.301512i	$-0.0974912\pi$
$421$	16472.4			1.90693			0.953463	+	0.301512i	$0.0974912\pi$
$422$		0			0
$423$	−195.073	−	337.877i	−0.0224227	−	0.0388372i
$424$		0			0
$425$	−4032.71	−	2328.28i	−0.460270	−	0.265737i
$426$		0			0
$427$	6130.04	+	2469.36i	0.694739	+	0.279861i
$428$		0			0
$429$	−5653.12	+	9791.50i	−0.636213	+	1.10195i
$430$		0			0
$431$	11297.6	−	6522.69i	1.26262	−	0.728972i	0.289036	−	0.957318i	$-0.406665\pi$
$431$	11297.6	−	6522.69i	1.26262	−	0.728972i	0.973580	+	0.228347i	$0.0733319\pi$
$432$		0			0
$433$			2116.10i			0.234858i	0.993081	+	0.117429i	$0.0374652\pi$
$433$			2116.10i			0.234858i	−0.993081	+	0.117429i	$0.962535\pi$
$434$		0			0
$435$			20343.5i			2.24229i
$436$		0			0
$437$	108.299	−	62.5266i	0.0118550	−	0.00684451i
$438$		0			0
$439$	4575.14	−	7924.37i	0.497402	−	0.861525i	−0.502594	−	0.864523i	$-0.667621\pi$
$439$	4575.14	−	7924.37i	0.497402	−	0.861525i	0.999996	+	0.00299765i	$0.000954184\pi$
$440$		0			0
$441$	−165.965	+	671.537i	−0.0179208	+	0.0725124i
$442$		0			0
$443$	−5877.85	−	3393.58i	−0.630395	−	0.363959i	0.150510	−	0.988609i	$-0.451908\pi$
$443$	−5877.85	−	3393.58i	−0.630395	−	0.363959i	−0.780905	+	0.624650i	$0.785242\pi$
$444$		0			0
$445$	4392.04	+	7607.23i	0.467871	+	0.810376i
$446$		0			0
$447$	7235.61			0.765621
$448$		0			0
$449$	8586.18			0.902465			0.451233	−	0.892406i	$-0.350984\pi$
$449$	8586.18			0.902465			0.451233	+	0.892406i	$0.350984\pi$
$450$		0			0
$451$	−14631.4	−	25342.3i	−1.52764	−	2.64595i
$452$		0			0
$453$	11642.7	+	6721.93i	1.20756	+	0.697183i
$454$		0			0
$455$	3599.77	−	8936.24i	0.370901	−	0.920741i
$456$		0			0
$457$	−648.401	+	1123.06i	−0.0663696	+	0.114956i	−0.897301	−	0.441420i	$-0.854475\pi$
$457$	−648.401	+	1123.06i	−0.0663696	+	0.114956i	0.830931	+	0.556375i	$0.187808\pi$
$458$		0			0
$459$	6408.99	−	3700.23i	0.651734	−	0.376279i
$460$		0			0
$461$		−	4276.06i		−	0.432008i	−0.976392	−	0.216004i	$-0.930698\pi$
$461$		−	4276.06i		−	0.432008i	0.976392	−	0.216004i	$-0.0693025\pi$
$462$		0			0
$463$			8953.82i			0.898746i	0.893344	+	0.449373i	$0.148353\pi$
$463$			8953.82i			0.898746i	−0.893344	+	0.449373i	$0.851647\pi$
$464$		0			0
$465$	8261.25	−	4769.64i	0.823885	−	0.475670i
$466$		0			0
$467$	−789.125	+	1366.80i	−0.0781935	+	0.135435i	−0.902471	−	0.430752i	$-0.858249\pi$
$467$	−789.125	+	1366.80i	−0.0781935	+	0.135435i	0.824277	+	0.566187i	$0.191582\pi$
$468$		0			0
$469$	13133.4	−	10282.7i	1.29306	−	1.01239i
$470$		0			0
$471$	−2231.76	−	1288.51i	−0.218332	−	0.126054i
$472$		0			0
$473$	−5990.75	−	10376.3i	−0.582358	−	1.00867i
$474$		0			0
$475$	−407.677			−0.0393800
$476$		0			0
$477$	−1117.55			−0.107273
$478$		0			0
$479$	−6925.80	−	11995.8i	−0.660642	−	1.14427i	−0.980447	−	0.196783i	$-0.936951\pi$
$479$	−6925.80	−	11995.8i	−0.660642	−	1.14427i	0.319805	−	0.947484i	$-0.396383\pi$
$480$		0			0
$481$	10582.0	+	6109.50i	1.00311	+	0.579146i
$482$		0			0
$483$	363.292	+	2565.78i	0.0342243	+	0.241712i
$484$		0			0
$485$	−6321.16	+	10948.6i	−0.591813	+	1.02505i
$486$		0			0
$487$	1360.19	−	785.308i	0.126563	−	0.0730712i	−0.435382	−	0.900246i	$-0.643387\pi$
$487$	1360.19	−	785.308i	0.126563	−	0.0730712i	0.561945	+	0.827175i	$0.310053\pi$
$488$		0			0
$489$			20216.2i			1.86955i
$490$		0			0
$491$		−	15339.7i		−	1.40992i	−0.709248	−	0.704959i	$-0.750965\pi$
$491$		−	15339.7i		−	1.40992i	0.709248	−	0.704959i	$-0.249035\pi$
$492$		0			0
$493$	−12420.5	+	7170.97i	−1.13467	+	0.655100i
$494$		0			0
$495$	−853.118	+	1477.64i	−0.0774643	+	0.134172i
$496$		0			0
$497$	1172.23	+	8278.99i	0.105798	+	0.747210i
$498$		0			0
$499$	−7906.61	−	4564.88i	−0.709316	−	0.409524i	0.101492	−	0.994836i	$-0.467638\pi$
$499$	−7906.61	−	4564.88i	−0.709316	−	0.409524i	−0.810808	+	0.585313i	$0.800972\pi$
$500$		0			0
$501$	−10081.2	−	17461.1i	−0.898989	−	1.55709i
$502$		0			0
$503$	−2887.52			−0.255961			−0.127980	−	0.991777i	$-0.540849\pi$
$503$	−2887.52			−0.255961			−0.127980	+	0.991777i	$0.540849\pi$
$504$		0			0
$505$	−17842.2			−1.57221
$506$		0			0
$507$	−2441.47	−	4228.74i	−0.213865	−	0.370424i
$508$		0			0
$509$	−7925.41	−	4575.74i	−0.690153	−	0.398460i	0.113516	−	0.993536i	$-0.463789\pi$
$509$	−7925.41	−	4575.74i	−0.690153	−	0.398460i	−0.803669	+	0.595076i	$0.797122\pi$
$510$		0			0
$511$	1094.97	−	857.303i	0.0947920	−	0.0742169i
$512$		0			0
$513$	323.951	−	561.100i	0.0278807	−	0.0482907i
$514$		0			0
$515$	−845.905	+	488.384i	−0.0723787	+	0.0417879i
$516$		0			0
$517$		−	11302.9i		−	0.961510i
$518$		0			0
$519$			23343.9i			1.97434i
$520$		0			0
$521$	−7771.71	+	4487.00i	−0.653521	+	0.377311i	−0.789804	−	0.613359i	$-0.789818\pi$
$521$	−7771.71	+	4487.00i	−0.653521	+	0.377311i	0.136283	+	0.990670i	$0.456484\pi$
$522$		0			0
$523$	−7788.74	+	13490.5i	−0.651200	+	1.12791i	0.331632	+	0.943409i	$0.392401\pi$
$523$	−7788.74	+	13490.5i	−0.651200	+	1.12791i	−0.982832	+	0.184503i	$0.940932\pi$
$524$		0			0
$525$	3156.59	−	7836.06i	0.262410	−	0.651417i
$526$		0			0
$527$	5824.10	+	3362.55i	0.481408	+	0.277941i
$528$		0			0
$529$	−5746.14	−	9952.61i	−0.472273	−	0.818001i
$530$		0			0
$531$	400.877			0.0327619
$532$		0			0
$533$	−17992.3			−1.46217
$534$		0			0
$535$	−7051.03	−	12212.7i	−0.569800	−	0.986922i
$536$		0			0
$537$	5144.30	+	2970.06i	0.413395	+	0.238674i
$538$		0			0
$539$	−13891.5	+	14444.5i	−1.11011	+	1.15430i
$540$		0			0
$541$	8017.44	−	13886.6i	0.637148	−	1.10357i	−0.348908	−	0.937157i	$-0.613448\pi$
$541$	8017.44	−	13886.6i	0.637148	−	1.10357i	0.986056	−	0.166415i	$-0.0532191\pi$
$542$		0			0
$543$	5979.45	−	3452.23i	0.472565	−	0.272835i
$544$		0			0
$545$		−	9831.54i		−	0.772728i
$546$		0			0
$547$			20672.1i			1.61586i	0.589277	+	0.807931i	$0.299413\pi$
$547$			20672.1i			1.61586i	−0.589277	+	0.807931i	$0.700587\pi$
$548$		0			0
$549$	−623.233	+	359.824i	−0.0484498	+	0.0279725i
$550$		0			0
$551$	−627.810	+	1087.40i	−0.0485401	+	0.0840740i
$552$		0			0
$553$	−8052.63	−	3243.83i	−0.619227	−	0.249443i
$554$		0			0
$555$	22976.6	+	13265.6i	1.75730	+	1.01458i
$556$		0			0
$557$	6966.07	+	12065.6i	0.529914	+	0.917838i	0.999391	+	0.0348932i	$0.0111091\pi$
$557$	6966.07	+	12065.6i	0.529914	+	0.917838i	−0.469477	+	0.882945i	$0.655558\pi$
$558$		0			0
$559$	−7366.87			−0.557398
$560$		0			0
$561$	−17307.0			−1.30250
$562$		0			0
$563$	3880.02	+	6720.39i	0.290450	+	0.503074i	0.973916	−	0.226908i	$-0.0728618\pi$
$563$	3880.02	+	6720.39i	0.290450	+	0.503074i	−0.683466	+	0.729982i	$0.739528\pi$
$564$		0			0
$565$	606.747	+	350.305i	0.0451788	+	0.0260840i
$566$		0			0
$567$	8898.42	+	11365.3i	0.659080	+	0.841796i
$568$		0			0
$569$	−6388.05	+	11064.4i	−0.470652	+	0.815194i	−0.999437	−	0.0335626i	$-0.989315\pi$
$569$	−6388.05	+	11064.4i	−0.470652	+	0.815194i	0.528784	+	0.848756i	$0.322648\pi$
$570$		0			0
$571$	−1507.45	+	870.328i	−0.110481	+	0.0637865i	−0.554223	−	0.832369i	$-0.686984\pi$
$571$	−1507.45	+	870.328i	−0.110481	+	0.0637865i	0.443741	+	0.896155i	$0.353651\pi$
$572$		0			0
$573$		−	3571.47i		−	0.260385i
$574$		0			0
$575$		−	2199.58i		−	0.159528i
$576$		0			0
$577$	270.498	−	156.172i	0.0195164	−	0.0112678i	−0.490210	−	0.871604i	$-0.663080\pi$
$577$	270.498	−	156.172i	0.0195164	−	0.0112678i	0.509726	+	0.860337i	$0.329747\pi$
$578$		0			0
$579$	6886.09	−	11927.1i	0.494260	−	0.856083i
$580$		0			0
$581$	−11443.6	+	1620.31i	−0.817141	+	0.115700i
$582$		0			0
$583$	−28038.8	−	16188.2i	−1.99185	−	1.14999i
$584$		0			0
$585$	524.543	+	908.535i	0.0370721	+	0.0642108i
$586$		0			0
$587$	−3731.70			−0.262391			−0.131196	−	0.991357i	$-0.541882\pi$
$587$	−3731.70			−0.262391			−0.131196	+	0.991357i	$0.541882\pi$
$588$		0			0
$589$	588.774			0.0411885
$590$		0			0
$591$	10706.5	+	18544.1i	0.745186	+	1.29070i
$592$		0			0
$593$	5905.30	+	3409.43i	0.408940	+	0.236102i	0.690334	−	0.723490i	$-0.257463\pi$
$593$	5905.30	+	3409.43i	0.408940	+	0.236102i	−0.281394	+	0.959592i	$0.590797\pi$
$594$		0			0
$595$	14601.6	−	2067.46i	1.00606	−	0.142450i
$596$		0			0
$597$	4797.82	−	8310.07i	0.328914	−	0.569696i
$598$		0			0
$599$	6269.69	−	3619.81i	0.427667	−	0.246914i	−0.270685	−	0.962668i	$-0.587250\pi$
$599$	6269.69	−	3619.81i	0.427667	−	0.246914i	0.698352	+	0.715754i	$0.253917\pi$
$600$		0			0
$601$		−	15554.1i		−	1.05568i	−0.849343	−	0.527841i	$-0.823002\pi$
$601$		−	15554.1i		−	1.05568i	0.849343	−	0.527841i	$-0.176998\pi$
$602$		0			0
$603$			1816.34i			0.122665i
$604$		0			0
$605$	−26117.5	+	15078.9i	−1.75509	+	1.01330i
$606$		0			0
$607$	−2133.04	+	3694.54i	−0.142632	+	0.247046i	−0.928487	−	0.371365i	$-0.878890\pi$
$607$	−2133.04	+	3694.54i	−0.142632	+	0.247046i	0.785855	+	0.618411i	$0.212223\pi$
$608$		0			0
$609$	−16040.1	−	20486.9i	−1.06729	−	1.36317i
$610$		0			0
$611$	−6018.55	−	3474.81i	−0.398502	−	0.230075i
$612$		0			0
$613$	−902.274	−	1562.78i	−0.0594494	−	0.102969i	0.834769	−	0.550600i	$-0.185601\pi$
$613$	−902.274	−	1562.78i	−0.0594494	−	0.102969i	−0.894218	+	0.447631i	$0.852268\pi$
$614$		0			0
$615$	−39066.8			−2.56150
$616$		0			0
$617$	−14258.9			−0.930376			−0.465188	−	0.885212i	$-0.654013\pi$
$617$	−14258.9			−0.930376			−0.465188	+	0.885212i	$0.654013\pi$
$618$		0			0
$619$	−14936.7	−	25871.2i	−0.969884	−	1.67989i	−0.695879	−	0.718159i	$-0.744985\pi$
$619$	−14936.7	−	25871.2i	−0.969884	−	1.67989i	−0.274005	−	0.961728i	$-0.588348\pi$
$620$		0			0
$621$	3027.35	+	1747.84i	0.195625	+	0.112944i
$622$		0			0
$623$	−10421.0	−	4197.89i	−0.670160	−	0.269960i
$624$		0			0
$625$	9519.65	−	16488.5i	0.609258	−	1.05527i
$626$		0			0
$627$	−1312.21	+	757.603i	−0.0835797	+	0.0482548i
$628$		0			0
$629$			18704.2i			1.18567i
$630$		0			0
$631$			17023.6i			1.07401i	0.843579	+	0.537005i	$0.180444\pi$
$631$			17023.6i			1.07401i	−0.843579	+	0.537005i	$0.819556\pi$
$632$		0			0
$633$	15990.4	−	9232.07i	1.00405	−	0.579687i
$634$		0			0
$635$	15588.7	−	27000.5i	0.974205	−	1.68737i
$636$		0			0
$637$	3420.76	+	11837.5i	0.212771	+	0.736295i
$638$		0			0
$639$	−788.535	−	455.261i	−0.0488168	−	0.0281844i
$640$		0			0
$641$	−13415.6	−	23236.5i	−0.826652	−	1.43180i	−0.900650	−	0.434545i	$-0.856909\pi$
$641$	−13415.6	−	23236.5i	−0.826652	−	1.43180i	0.0739983	−	0.997258i	$-0.476424\pi$
$642$		0			0
$643$	13857.5			0.849902			0.424951	−	0.905216i	$-0.360291\pi$
$643$	13857.5			0.849902			0.424951	+	0.905216i	$0.360291\pi$
$644$		0			0
$645$	−15995.7			−0.976480
$646$		0			0
$647$	−1488.14	−	2577.53i	−0.0904245	−	0.156620i	0.817265	−	0.576262i	$-0.195489\pi$
$647$	−1488.14	−	2577.53i	−0.0904245	−	0.156620i	−0.907690	+	0.419642i	$0.862156\pi$
$648$		0			0
$649$	10057.8	+	5806.88i	0.608326	+	0.351217i
$650$		0			0
$651$	−4558.80	+	11317.0i	−0.274460	+	0.681332i
$652$		0			0
$653$	1550.88	−	2686.20i	0.0929412	−	0.160979i	−0.815806	−	0.578325i	$-0.803706\pi$
$653$	1550.88	−	2686.20i	0.0929412	−	0.160979i	0.908747	+	0.417346i	$0.137040\pi$
$654$		0			0
$655$	32105.9	−	18536.4i	1.91524	−	1.10576i
$656$		0			0
$657$			151.434i			0.00899239i
$658$		0			0
$659$			24304.3i			1.43666i	0.695700	+	0.718332i	$0.255094\pi$
$659$			24304.3i			1.43666i	−0.695700	+	0.718332i	$0.744906\pi$
$660$		0			0
$661$	6642.15	−	3834.85i	0.390847	−	0.225655i	−0.291680	−	0.956516i	$-0.594214\pi$
$661$	6642.15	−	3834.85i	0.390847	−	0.225655i	0.682527	+	0.730860i	$0.260881\pi$
$662$		0			0
$663$	−5320.63	+	9215.60i	−0.311668	+	0.539825i
$664$		0			0
$665$	1016.59	−	795.930i	0.0592804	−	0.0464133i
$666$		0			0
$667$	−5866.94	−	3387.28i	−0.340583	−	0.196636i
$668$		0			0
$669$	11095.8	+	19218.5i	0.641237	+	1.11066i
$670$		0			0
$671$	−20848.8			−1.19949
$672$		0			0
$673$	24315.9			1.39273			0.696367	−	0.717686i	$-0.254798\pi$
$673$	24315.9			1.39273			0.696367	+	0.717686i	$0.254798\pi$
$674$		0			0
$675$	−5698.02	−	9869.26i	−0.324914	−	0.562767i
$676$		0			0
$677$	−7377.97	−	4259.67i	−0.418846	−	0.241821i	0.275738	−	0.961233i	$-0.411078\pi$
$677$	−7377.97	−	4259.67i	−0.418846	−	0.241821i	−0.694583	+	0.719412i	$0.744411\pi$
$678$		0			0
$679$	−2266.84	−	16009.8i	−0.128120	−	0.904858i
$680$		0			0
$681$	4667.29	−	8083.98i	0.262630	−	0.454888i
$682$		0			0
$683$	−10577.6	+	6106.97i	−0.592591	+	0.342133i	−0.766121	−	0.642696i	$-0.777816\pi$
$683$	−10577.6	+	6106.97i	−0.592591	+	0.342133i	0.173530	+	0.984829i	$0.444483\pi$
$684$		0			0
$685$		−	24034.5i		−	1.34060i
$686$		0			0
$687$		−	9797.33i		−	0.544092i
$688$		0			0
$689$	−17239.7	+	9953.36i	−0.953239	+	0.550353i
$690$		0			0
$691$	9636.86	−	16691.5i	0.530541	−	0.918923i	−0.468824	−	0.883291i	$-0.655322\pi$
$691$	9636.86	−	16691.5i	0.530541	−	0.918923i	0.999365	−	0.0356318i	$-0.0113443\pi$
$692$		0			0
$693$	−305.938	−	2160.72i	−0.0167700	−	0.118440i
$694$		0			0
$695$	11139.7	+	6431.52i	0.607991	+	0.351024i
$696$		0			0
$697$	−13770.8	−	23851.8i	−0.748361	−	1.29620i
$698$		0			0
$699$	−15791.4			−0.854486
$700$		0			0
$701$	−12603.8			−0.679083			−0.339542	−	0.940591i	$-0.610272\pi$
$701$	−12603.8			−0.679083			−0.339542	+	0.940591i	$0.610272\pi$
$702$		0			0
$703$	818.764	+	1418.14i	0.0439264	+	0.0760828i
$704$		0			0
$705$	−13068.1	−	7544.86i	−0.698117	−	0.403058i
$706$		0			0
$707$	17967.9	−	14067.9i	0.955805	−	0.748343i
$708$		0			0
$709$	6712.60	−	11626.6i	0.355567	−	0.615860i	−0.631648	−	0.775256i	$-0.717621\pi$
$709$	6712.60	−	11626.6i	0.355567	−	0.615860i	0.987215	+	0.159395i	$0.0509544\pi$
$710$		0			0
$711$	818.700	−	472.677i	0.0431838	−	0.0249322i
$712$		0			0
$713$			3176.66i			0.166854i
$714$		0			0
$715$			30392.9i			1.58969i
$716$		0			0
$717$	−4306.90	+	2486.59i	−0.224329	+	0.129516i
$718$		0			0
$719$	2237.85	−	3876.08i	0.116075	−	0.201048i	−0.802134	−	0.597144i	$-0.796302\pi$
$719$	2237.85	−	3876.08i	0.116075	−	0.201048i	0.918209	+	0.396096i	$0.129635\pi$
$720$		0			0
$721$	466.796	−	1158.79i	0.0241115	−	0.0598554i
$722$		0			0
$723$	33932.6	+	19591.0i	1.74546	+	1.00774i
$724$		0			0
$725$	11042.6	+	19126.4i	0.565674	+	0.979775i
$726$		0			0
$727$	27241.1			1.38971			0.694853	−	0.719152i	$-0.255469\pi$
$727$	27241.1			1.38971			0.694853	+	0.719152i	$0.255469\pi$
$728$		0			0
$729$	18001.4			0.914564
$730$		0			0
$731$	−5638.40	−	9765.99i	−0.285285	−	0.494129i
$732$		0			0
$733$	−30644.6	−	17692.7i	−1.54418	−	0.891534i	−0.998568	−	0.0534982i	$-0.982963\pi$
$733$	−30644.6	−	17692.7i	−1.54418	−	0.891534i	−0.545615	−	0.838036i	$-0.683704\pi$
$734$		0			0
$735$	7427.49	+	25702.8i	0.372744	+	1.28988i
$736$		0			0
$737$	−26310.5	+	45571.1i	−1.31501	+	2.27766i
$738$		0			0
$739$	6850.58	−	3955.18i	0.341005	−	0.196879i	−0.319711	−	0.947515i	$-0.603586\pi$
$739$	6850.58	−	3955.18i	0.341005	−	0.196879i	0.660716	+	0.750636i	$0.270253\pi$
$740$		0			0
$741$			931.629i			0.0461866i
$742$		0			0
$743$		−	22494.1i		−	1.11067i	−0.831626	−	0.555336i	$-0.812590\pi$
$743$		−	22494.1i		−	1.11067i	0.831626	−	0.555336i	$-0.187410\pi$
$744$		0			0
$745$	16844.6	−	9725.21i	0.828372	−	0.478261i
$746$		0			0
$747$	629.280	−	1089.95i	0.0308222	−	0.0533856i
$748$		0			0
$749$	16730.1	+	6739.35i	0.816159	+	0.328773i
$750$		0			0
$751$	16198.4	+	9352.16i	0.787069	+	0.454414i	0.838930	−	0.544240i	$-0.183182\pi$
$751$	16198.4	+	9352.16i	0.787069	+	0.454414i	−0.0518607	+	0.998654i	$0.516515\pi$
$752$		0			0
$753$	9431.48	+	16335.8i	0.456444	+	0.790584i
$754$		0			0
$755$	36139.2			1.74204
$756$		0			0
$757$	3190.02			0.153161			0.0765807	−	0.997063i	$-0.475600\pi$
$757$	3190.02			0.153161			0.0765807	+	0.997063i	$0.475600\pi$
$758$		0			0
$759$	−4087.56	−	7079.86i	−0.195480	−	0.338581i
$760$		0			0
$761$	−14520.5	−	8383.40i	−0.691678	−	0.399341i	0.112562	−	0.993645i	$-0.464094\pi$
$761$	−14520.5	−	8383.40i	−0.691678	−	0.399341i	−0.804240	+	0.594304i	$0.797428\pi$
$762$		0			0
$763$	7751.82	+	9900.85i	0.367805	+	0.469770i
$764$		0			0
$765$	−802.941	+	1390.73i	−0.0379482	+	0.0657282i
$766$		0			0
$767$	6184.08	−	3570.38i	0.291127	−	0.168082i
$768$		0			0
$769$		−	1993.29i		−	0.0934720i	−0.998907	−	0.0467360i	$-0.985118\pi$
$769$		−	1993.29i		−	0.0934720i	0.998907	−	0.0467360i	$-0.0148819\pi$
$770$		0			0
$771$		−	41231.2i		−	1.92595i
$772$		0			0
$773$	−4045.15	+	2335.47i	−0.188220	+	0.108669i	−0.591149	−	0.806562i	$-0.701325\pi$
$773$	−4045.15	+	2335.47i	−0.188220	+	0.108669i	0.402929	+	0.915231i	$0.367992\pi$
$774$		0			0
$775$	5178.01	−	8968.58i	0.240000	−	0.415691i
$776$		0			0
$777$	−33598.1	+	4757.18i	−1.55125	+	0.219644i
$778$		0			0
$779$	−2088.20	−	1205.62i	−0.0960429	−	0.0554504i
$780$		0			0
$781$	−13189.3	−	22844.5i	−0.604290	−	1.04666i
$782$		0			0
$783$	−35099.1			−1.60197
$784$		0			0
$785$	−6927.42			−0.314969
$786$		0			0
$787$	14545.2	+	25193.1i	0.658807	+	1.14109i	0.980925	+	0.194387i	$0.0622719\pi$
$787$	14545.2	+	25193.1i	0.658807	+	1.14109i	−0.322118	+	0.946700i	$0.604395\pi$
$788$		0			0
$789$	−11953.1	−	6901.12i	−0.539343	−	0.311390i
$790$		0			0
$791$	−887.228	+	125.624i	−0.0398814	+	0.00564685i
$792$		0			0
$793$	−6409.48	+	11101.6i	−0.287021	+	0.497135i
$794$		0			0
$795$	−37432.6	+	21611.7i	−1.66994	+	0.964137i
$796$		0			0
$797$		−	7294.99i		−	0.324218i	−0.986773	−	0.162109i	$-0.948170\pi$
$797$		−	7294.99i		−	0.324218i	0.986773	−	0.162109i	$-0.0518296\pi$
$798$		0			0
$799$		−	10638.1i		−	0.471025i
$800$		0			0
$801$	1059.49	−	611.698i	0.0467357	−	0.0269829i
$802$		0			0
$803$	−2193.59	+	3799.40i	−0.0964010	+	0.166971i
$804$		0			0
$805$	4294.35	+	5484.87i	0.188020	+	0.240145i
$806$		0			0
$807$	−9691.86	−	5595.60i	−0.422763	−	0.244082i
$808$		0			0
$809$	−8307.35	−	14388.8i	−0.361027	−	0.625317i	0.627103	−	0.778936i	$-0.284240\pi$
$809$	−8307.35	−	14388.8i	−0.361027	−	0.625317i	−0.988130	+	0.153619i	$0.950907\pi$
$810$		0			0
$811$	−12794.0			−0.553956			−0.276978	−	0.960876i	$-0.589333\pi$
$811$	−12794.0			−0.553956			−0.276978	+	0.960876i	$0.589333\pi$
$812$		0			0
$813$	4883.81			0.210680
$814$		0			0
$815$	27172.2	+	47063.6i	1.16785	+	2.02278i
$816$		0			0
$817$	−855.001	−	493.635i	−0.0366128	−	0.0211384i
$818$		0			0
$819$	−1244.59	−	501.356i	−0.0531007	−	0.0213905i
$820$		0			0
$821$	6472.34	−	11210.4i	0.275136	−	0.476549i	−0.695034	−	0.718977i	$-0.744611\pi$
$821$	6472.34	−	11210.4i	0.275136	−	0.476549i	0.970169	+	0.242428i	$0.0779439\pi$
$822$		0			0
$823$	−22317.4	+	12885.0i	−0.945244	+	0.545737i	−0.891600	−	0.452823i	$-0.850417\pi$
$823$	−22317.4	+	12885.0i	−0.945244	+	0.545737i	−0.0536439	+	0.998560i	$0.517084\pi$
$824$		0			0
$825$			26651.1i			1.12470i
$826$		0			0
$827$		−	41940.5i		−	1.76350i	−0.471718	−	0.881749i	$-0.656366\pi$
$827$		−	41940.5i		−	1.76350i	0.471718	−	0.881749i	$-0.343634\pi$
$828$		0			0
$829$	256.068	−	147.841i	0.0107281	−	0.00619387i	−0.494626	−	0.869106i	$-0.664695\pi$
$829$	256.068	−	147.841i	0.0107281	−	0.00619387i	0.505354	+	0.862912i	$0.331362\pi$
$830$		0			0
$831$	4511.77	−	7814.61i	0.188341	−	0.326216i
$832$		0			0
$833$	−13074.4	+	13594.9i	−0.543820	+	0.565468i
$834$		0			0
$835$	−46938.1	−	27099.7i	−1.94534	−	1.12314i
$836$		0			0
$837$	8229.17	+	14253.3i	0.339835	+	0.588611i
$838$		0			0
$839$	−12790.0			−0.526295			−0.263147	−	0.964756i	$-0.584761\pi$
$839$	−12790.0			−0.526295			−0.263147	+	0.964756i	$0.584761\pi$
$840$		0			0
$841$	43632.3			1.78902
$842$		0			0
$843$	2350.82	+	4071.74i	0.0960457	+	0.166356i
$844$		0			0
$845$	−11367.5	−	6563.04i	−0.462786	−	0.267190i
$846$		0			0
$847$	14412.4	−	35778.0i	0.584670	−	1.45141i
$848$		0			0
$849$	16800.8	−	29099.8i	0.679154	−	1.17633i
$850$		0			0
$851$	−7651.43	+	4417.55i	−0.308211	+	0.177946i
$852$		0			0
$853$			36133.3i			1.45039i	0.688546	+	0.725193i	$0.258249\pi$
$853$			36133.3i			1.45039i	−0.688546	+	0.725193i	$0.741751\pi$
$854$		0			0
$855$			140.593i			0.00562360i
$856$		0			0
$857$	20654.6	−	11924.9i	0.823277	−	0.475319i	−0.0282685	−	0.999600i	$-0.508999\pi$
$857$	20654.6	−	11924.9i	0.823277	−	0.475319i	0.851545	+	0.524281i	$0.175666\pi$
$858$		0			0
$859$	−15005.5	+	25990.3i	−0.596020	+	1.03234i	0.397382	+	0.917653i	$0.369919\pi$
$859$	−15005.5	+	25990.3i	−0.596020	+	1.03234i	−0.993402	+	0.114684i	$0.963415\pi$
$860$		0			0
$861$	39342.2	−	30802.8i	1.55723	−	1.21923i
$862$		0			0
$863$	568.130	+	328.010i	0.0224095	+	0.0129381i	0.511163	−	0.859484i	$-0.329215\pi$
$863$	568.130	+	328.010i	0.0224095	+	0.0129381i	−0.488753	+	0.872422i	$0.662548\pi$
$864$		0			0
$865$	31376.0	+	54344.8i	1.23331	+	2.13616i
$866$		0			0
$867$	10175.9			0.398607
$868$		0			0
$869$	27387.7			1.06912
$870$		0			0
$871$	16177.1	+	28019.6i	0.629323	+	1.09002i
$872$		0			0
$873$	1524.86	+	880.376i	0.0591163	+	0.0341308i
$874$		0			0
$875$	1515.91	+	10706.2i	0.0585680	+	0.413642i
$876$		0			0
$877$	7112.50	−	12319.2i	0.273857	−	0.474334i	−0.695989	−	0.718052i	$-0.745034\pi$
$877$	7112.50	−	12319.2i	0.273857	−	0.474334i	0.969846	+	0.243719i	$0.0783673\pi$
$878$		0			0
$879$	29497.5	−	17030.4i	1.13189	−	0.653495i
$880$		0			0
$881$			30208.2i			1.15521i	0.816317	+	0.577605i	$0.196012\pi$
$881$			30208.2i			1.15521i	−0.816317	+	0.577605i	$0.803988\pi$
$882$		0			0
$883$			36178.1i			1.37881i	0.724376	+	0.689405i	$0.242128\pi$
$883$			36178.1i			1.37881i	−0.724376	+	0.689405i	$0.757872\pi$
$884$		0			0
$885$	13427.5	−	7752.36i	0.510011	−	0.294455i
$886$		0			0
$887$	712.309	−	1233.76i	0.0269639	−	0.0467029i	−0.852229	−	0.523170i	$-0.824749\pi$
$887$	712.309	−	1233.76i	0.0269639	−	0.0467029i	0.879192	+	0.476467i	$0.158083\pi$
$888$		0			0
$889$	5590.30	+	39482.0i	0.210903	+	1.48952i
$890$		0			0
$891$	−39436.1	−	22768.4i	−1.48278	−	0.856085i
$892$		0			0
$893$	−465.676	−	806.575i	−0.0174505	−	0.0302251i
$894$		0			0
$895$	15968.0			0.596369
$896$		0			0
$897$	−5026.50			−0.187102
$898$		0			0
$899$	−15948.0	−	27622.7i	−0.591651	−	1.02477i
$900$		0			0
$901$	−26389.6	−	15236.1i	−0.975767	−	0.563359i
$902$		0			0
$903$	16108.5	−	12612.0i	0.593639	−	0.464787i
$904$		0			0
$905$	9280.14	−	16073.7i	0.340864	−	0.590394i
$906$		0			0
$907$	−26578.5	+	15345.1i	−0.973014	+	0.561770i	−0.900154	−	0.435572i	$-0.856546\pi$
$907$	−26578.5	+	15345.1i	−0.973014	+	0.561770i	−0.0728605	+	0.997342i	$0.523213\pi$
$908$		0			0
$909$			2484.96i			0.0906719i
$910$		0			0
$911$			31797.5i			1.15642i	0.815889	+	0.578209i	$0.196248\pi$
$911$			31797.5i			1.15642i	−0.815889	+	0.578209i	$0.803752\pi$
$912$		0			0
$913$	31576.7	−	18230.8i	1.14462	−	0.660845i
$914$		0			0
$915$	−13916.9	+	24104.8i	−0.502818	+	0.870907i
$916$		0			0
$917$	−17717.0	+	43981.4i	−0.638023	+	1.58385i
$918$		0			0
$919$	−24610.4	−	14208.8i	−0.883376	−	0.510018i	−0.0116062	−	0.999933i	$-0.503694\pi$
$919$	−24610.4	−	14208.8i	−0.883376	−	0.510018i	−0.871770	+	0.489915i	$0.837028\pi$
$920$		0			0
$921$	2837.80	+	4915.21i	0.101529	+	0.175854i
$922$		0			0
$923$	−16219.0			−0.578390
$924$		0			0
$925$	28802.7			1.02381
$926$		0			0
$927$	68.0193	+	117.813i	0.00240998	+	0.00417420i
$928$		0			0
$929$	−33144.8	−	19136.2i	−1.17055	−	0.675820i	−0.216744	−	0.976228i	$-0.569544\pi$
$929$	−33144.8	−	19136.2i	−1.17055	−	0.675820i	−0.953811	+	0.300409i	$0.902877\pi$
$930$		0			0
$931$	−396.189	+	1603.08i	−0.0139469	+	0.0564328i
$932$		0			0
$933$	−8026.17	+	13901.7i	−0.281634	+	0.487805i
$934$		0			0
$935$	−40290.8	+	23261.9i	−1.40925	+	0.813632i
$936$		0			0
$937$		−	388.650i		−	0.0135503i	−0.999977	−	0.00677516i	$-0.997843\pi$
$937$		−	388.650i		−	0.0135503i	0.999977	−	0.00677516i	$-0.00215662\pi$
$938$		0			0
$939$		−	6992.28i		−	0.243008i
$940$		0			0
$941$	9588.10	−	5535.69i	0.332161	−	0.191773i	−0.324639	−	0.945838i	$-0.605243\pi$
$941$	9588.10	−	5535.69i	0.332161	−	0.191773i	0.656800	+	0.754065i	$0.271910\pi$
$942$		0			0
$943$	6504.79	−	11266.6i	0.224629	−	0.389069i
$944$		0			0
$945$	33476.9	+	13485.5i	1.15238	+	0.464214i
$946$		0			0
$947$	17601.9	+	10162.5i	0.603996	+	0.348717i	0.770612	−	0.637304i	$-0.219951\pi$
$947$	17601.9	+	10162.5i	0.603996	+	0.348717i	−0.166616	+	0.986022i	$0.553284\pi$
$948$		0			0
$949$	1348.73	+	2336.08i	0.0461346	+	0.0799075i
$950$		0			0
$951$	1914.12			0.0652677
$952$		0			0
$953$	14683.7			0.499110			0.249555	−	0.968361i	$-0.419716\pi$
$953$	14683.7			0.499110			0.249555	+	0.968361i	$0.419716\pi$
$954$		0			0
$955$	−4800.33	−	8314.42i	−0.162655	−	0.281726i
$956$		0			0
$957$	71086.8	+	41042.0i	2.40116	+	1.38631i
$958$		0			0
$959$	18950.4	+	24203.9i	0.638101	+	0.815001i
$960$		0			0
$961$	7417.33	−	12847.2i	0.248979	−	0.431244i
$962$		0			0
$963$	−1700.92	+	982.028i	−0.0569174	+	0.0328613i
$964$		0			0
$965$		−	37021.8i		−	1.23500i
$966$		0			0
$967$			2737.32i			0.0910304i	0.998964	+	0.0455152i	$0.0144929\pi$
$967$			2737.32i			0.0910304i	−0.998964	+	0.0455152i	$0.985507\pi$
$968$		0			0
$969$	−1235.03	+	713.043i	−0.0409440	+	0.0236391i
$970$		0			0
$971$	6671.74	−	11555.8i	0.220501	−	0.381919i	−0.734459	−	0.678653i	$-0.762564\pi$
$971$	6671.74	−	11555.8i	0.220501	−	0.381919i	0.954960	+	0.296734i	$0.0958974\pi$
$972$		0			0
$973$	−16289.3	+	2306.42i	−0.536702	+	0.0759922i
$974$		0			0
$975$	14191.2	+	8193.28i	0.466135	+	0.269123i
$976$		0			0
$977$	29635.0	+	51329.3i	0.970428	+	1.68083i	0.694265	+	0.719719i	$0.255730\pi$
$977$	29635.0	+	51329.3i	0.970428	+	1.68083i	0.276163	+	0.961111i	$0.410937\pi$
$978$		0			0
$979$	35442.9			1.15706
$980$		0			0
$981$	−1369.28			−0.0445645
$982$		0			0
$983$	−4500.81	−	7795.64i	−0.146036	−	0.252942i	0.783723	−	0.621111i	$-0.213318\pi$
$983$	−4500.81	−	7795.64i	−0.146036	−	0.252942i	−0.929759	+	0.368168i	$0.879985\pi$
$984$		0			0
$985$	49849.5	+	28780.6i	1.61253	+	0.930992i
$986$		0			0
$987$	19109.1	−	2705.67i	0.616259	−	0.0872568i
$988$		0			0
$989$	2663.35	−	4613.06i	0.0856317	−	0.148318i
$990$		0			0
$991$	−25889.6	+	14947.4i	−0.829881	+	0.479132i	−0.853812	−	0.520582i	$-0.825715\pi$
$991$	−25889.6	+	14947.4i	−0.829881	+	0.479132i	0.0239312	+	0.999714i	$0.492382\pi$
$992$		0			0
$993$		−	31382.1i		−	1.00290i
$994$		0			0
$995$		−	25794.6i		−	0.821852i
$996$		0			0
$997$	13119.9	−	7574.77i	0.416761	−	0.240617i	−0.276929	−	0.960890i	$-0.589317\pi$
$997$	13119.9	−	7574.77i	0.416761	−	0.240617i	0.693691	+	0.720273i	$0.255984\pi$
$998$		0			0
$999$	−22887.4	+	39642.1i	−0.724850	+	1.25548i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.p.i.383.7		48
4.3	odd	2	inner	448.4.p.i.383.18		48
7.3	odd	6	inner	448.4.p.i.255.18		48
8.3	odd	2		224.4.p.a.159.7	yes	48
8.5	even	2		224.4.p.a.159.18	yes	48
28.3	even	6	inner	448.4.p.i.255.7		48
56.3	even	6		224.4.p.a.31.18	yes	48
56.45	odd	6		224.4.p.a.31.7	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.p.a.31.7	✓	48	56.45	odd	6
224.4.p.a.31.18	yes	48	56.3	even	6
224.4.p.a.159.7	yes	48	8.3	odd	2
224.4.p.a.159.18	yes	48	8.5	even	2
448.4.p.i.255.7		48	28.3	even	6	inner
448.4.p.i.255.18		48	7.3	odd	6	inner
448.4.p.i.383.7		48	1.1	even	1	trivial
448.4.p.i.383.18		48	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-2.69336 - 4.66504i) q^{3} +(-12.5403 - 7.24016i) q^{5} +(18.3374 - 2.59641i) q^{7} +(-1.00837 + 1.74655i) q^{9} +O(q^{10})\)	\(q+(-2.69336 - 4.66504i) q^{3} +(-12.5403 - 7.24016i) q^{5} +(18.3374 - 2.59641i) q^{7} +(-1.00837 + 1.74655i) q^{9} +(-50.5990 + 29.2133i) q^{11} +35.9239i q^{13} +78.0015i q^{15} +(-47.6229 + 27.4951i) q^{17} +(-2.40717 + 4.16933i) q^{19} +(-61.5014 - 78.5514i) q^{21} +(-22.4952 - 12.9876i) q^{23} +(42.3399 + 73.3349i) q^{25} -134.578 q^{27} +260.809 q^{29} +(-61.1481 - 105.912i) q^{31} +(272.563 + 157.364i) q^{33} +(-248.755 - 100.206i) q^{35} +(170.068 - 294.567i) q^{37} +(167.586 - 96.7559i) q^{39} +500.847i q^{41} +205.069i q^{43} +(25.2906 - 14.6015i) q^{45} +(-96.7271 + 167.536i) q^{47} +(329.517 - 95.2224i) q^{49} +(256.531 + 148.108i) q^{51} +(277.068 + 479.897i) q^{53} +846.038 q^{55} +25.9334 q^{57} +(-99.3874 - 172.144i) q^{59} +(309.030 + 178.419i) q^{61} +(-13.9561 + 34.6452i) q^{63} +(260.095 - 450.497i) q^{65} +(779.971 - 450.316i) q^{67} +139.921i q^{69} +451.482i q^{71} +(65.0285 - 37.5442i) q^{73} +(228.073 - 395.035i) q^{75} +(-852.002 + 667.071i) q^{77} +(-405.952 - 234.377i) q^{79} +(389.692 + 674.967i) q^{81} -624.057 q^{83} +796.277 q^{85} +(-702.452 - 1216.68i) q^{87} +(-525.349 - 303.311i) q^{89} +(93.2729 + 658.749i) q^{91} +(-329.387 + 570.516i) q^{93} +(60.3733 - 34.8565i) q^{95} -873.069i q^{97} -117.831i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 216 q^{9}+O(q^{10}) \) 48 * q - 216 * q^9	\( 48 q - 216 q^{9} + 104 q^{21} + 432 q^{25} - 112 q^{29} + 72 q^{33} + 504 q^{37} + 1320 q^{45} + 160 q^{49} - 392 q^{53} + 1360 q^{57} - 600 q^{61} - 744 q^{65} - 648 q^{73} + 2880 q^{77} - 400 q^{81} + 240 q^{85} + 3816 q^{89} - 2872 q^{93}+O(q^{100}) \) 48 * q - 216 * q^9 + 104 * q^21 + 432 * q^25 - 112 * q^29 + 72 * q^33 + 504 * q^37 + 1320 * q^45 + 160 * q^49 - 392 * q^53 + 1360 * q^57 - 600 * q^61 - 744 * q^65 - 648 * q^73 + 2880 * q^77 - 400 * q^81 + 240 * q^85 + 3816 * q^89 - 2872 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more