LMFDB - Embedded newform 76.8.e.a.45.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, degree $2$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			45.3
Character	$\chi$	$=$	76.45
Dual form			76.8.e.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-22.8105 - 39.5090i) q^{3} +(216.474 + 374.944i) q^{5} -508.391 q^{7} +(52.8600 - 91.5562i) q^{9} +O(q^{10})$	\(q+(-22.8105 - 39.5090i) q^{3} +(216.474 + 374.944i) q^{5} -508.391 q^{7} +(52.8600 - 91.5562i) q^{9} -1596.64 q^{11} +(1340.66 - 2322.09i) q^{13} +(9875.77 - 17105.3i) q^{15} +(-3266.04 - 5656.95i) q^{17} +(-17595.4 - 24171.7i) q^{19} +(11596.7 + 20086.0i) q^{21} +(-10247.4 + 17749.1i) q^{23} +(-54659.5 + 94673.1i) q^{25} -104596. q^{27} +(-98589.9 + 170763. i) q^{29} -256272. q^{31} +(36420.2 + 63081.6i) q^{33} +(-110053. - 190618. i) q^{35} +157659. q^{37} -122324. q^{39} +(-85374.6 - 147873. i) q^{41} +(-473770. - 820594. i) q^{43} +45771.3 q^{45} +(-164204. + 284410. i) q^{47} -565082. q^{49} +(-149000. + 258076. i) q^{51} +(-558053. + 966577. i) q^{53} +(-345631. - 598651. i) q^{55} +(-553639. + 1.24655e6i) q^{57} +(333278. + 577255. i) q^{59} +(617751. - 1.06998e6i) q^{61} +(-26873.6 + 46546.4i) q^{63} +1.16087e6 q^{65} +(-94447.0 + 163587. i) q^{67} +934998. q^{69} +(677843. + 1.17406e6i) q^{71} +(-1.42377e6 - 2.46604e6i) q^{73} +4.98725e6 q^{75} +811717. q^{77} +(-2.34677e6 - 4.06472e6i) q^{79} +(2.27029e6 + 3.93226e6i) q^{81} +2.12538e6 q^{83} +(1.41403e6 - 2.44917e6i) q^{85} +8.99555e6 q^{87} +(-2.16036e6 + 3.74186e6i) q^{89} +(-681579. + 1.18053e6i) q^{91} +(5.84569e6 + 1.01250e7i) q^{93} +(5.25409e6 - 1.18299e7i) q^{95} +(-5.40154e6 - 9.35575e6i) q^{97} +(-84398.4 + 146182. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	−22.8105	−	39.5090i	−0.487765	−	0.844834i	0.512136	−	0.858905i	$-0.328854\pi$
$3$	−22.8105	−	39.5090i	−0.487765	−	0.844834i	−0.999901	+	0.0140703i	$0.995521\pi$
$4$		0			0
$5$	216.474	+	374.944i	0.774481	+	1.34144i	0.935086	+	0.354422i	$0.115322\pi$
$5$	216.474	+	374.944i	0.774481	+	1.34144i	−0.160604	+	0.987019i	$0.551344\pi$
$6$		0			0
$7$	−508.391			−0.560215			−0.280108	−	0.959969i	$-0.590370\pi$
$7$	−508.391			−0.560215			−0.280108	+	0.959969i	$0.590370\pi$
$8$		0			0
$9$	52.8600	−	91.5562i	0.0241701	−	0.0418638i
$10$		0			0
$11$	−1596.64			−0.361687			−0.180843	−	0.983512i	$-0.557883\pi$
$11$	−1596.64			−0.361687			−0.180843	+	0.983512i	$0.557883\pi$
$12$		0			0
$13$	1340.66	−	2322.09i	0.169245	−	0.293141i	−0.768909	−	0.639358i	$-0.779200\pi$
$13$	1340.66	−	2322.09i	0.169245	−	0.293141i	0.938155	+	0.346216i	$0.112534\pi$
$14$		0			0
$15$	9875.77	−	17105.3i	0.755530	−	1.30862i
$16$		0			0
$17$	−3266.04	−	5656.95i	−0.161232	−	0.279262i	0.774079	−	0.633089i	$-0.218213\pi$
$17$	−3266.04	−	5656.95i	−0.161232	−	0.279262i	−0.935311	+	0.353827i	$0.884880\pi$
$18$		0			0
$19$	−17595.4	−	24171.7i	−0.588522	−	0.808482i
$19$	−17595.4	−	24171.7i	−0.588522	−	0.808482i
$20$		0			0
$21$	11596.7	+	20086.0i	0.273253	+	0.473289i
$22$		0			0
$23$	−10247.4	+	17749.1i	−0.175618	+	0.304179i	−0.940375	−	0.340140i	$-0.889526\pi$
$23$	−10247.4	+	17749.1i	−0.175618	+	0.304179i	0.764757	+	0.644319i	$0.222859\pi$
$24$		0			0
$25$	−54659.5	+	94673.1i	−0.699642	+	1.21182i
$26$		0			0
$27$	−104596.			−1.02269
$28$		0			0
$29$	−98589.9	+	170763.i	−0.750653	+	1.30017i	0.196853	+	0.980433i	$0.436928\pi$
$29$	−98589.9	+	170763.i	−0.750653	+	1.30017i	−0.947506	+	0.319737i	$0.896406\pi$
$30$		0			0
$31$	−256272.			−1.54502			−0.772511	−	0.635002i	$-0.780999\pi$
$31$	−256272.			−1.54502			−0.772511	+	0.635002i	$0.780999\pi$
$32$		0			0
$33$	36420.2	+	63081.6i	0.176418	+	0.305565i
$34$		0			0
$35$	−110053.	−	190618.i	−0.433876	−	0.751495i
$36$		0			0
$37$	157659.			0.511698			0.255849	−	0.966717i	$-0.417645\pi$
$37$	157659.			0.511698			0.255849	+	0.966717i	$0.417645\pi$
$38$		0			0
$39$	−122324.			−0.330208
$40$		0			0
$41$	−85374.6	−	147873.i	−0.193457	−	0.335078i	0.752936	−	0.658093i	$-0.228637\pi$
$41$	−85374.6	−	147873.i	−0.193457	−	0.335078i	−0.946394	+	0.323015i	$0.895303\pi$
$42$		0			0
$43$	−473770.	−	820594.i	−0.908716	−	1.57394i	−0.815850	−	0.578264i	$-0.803730\pi$
$43$	−473770.	−	820594.i	−0.908716	−	1.57394i	−0.0928664	−	0.995679i	$-0.529603\pi$
$44$		0			0
$45$	45771.3			0.0748771
$46$		0			0
$47$	−164204.	+	284410.i	−0.230697	+	0.399579i	−0.958013	−	0.286724i	$-0.907434\pi$
$47$	−164204.	+	284410.i	−0.230697	+	0.399579i	0.727317	+	0.686302i	$0.240767\pi$
$48$		0			0
$49$	−565082.			−0.686159
$50$		0			0
$51$	−149000.	+	258076.i	−0.157287	+	0.272428i
$52$		0			0
$53$	−558053.	+	966577.i	−0.514885	+	0.891807i	0.484966	+	0.874533i	$0.338832\pi$
$53$	−558053.	+	966577.i	−0.514885	+	0.891807i	−0.999851	+	0.0172737i	$0.994501\pi$
$54$		0			0
$55$	−345631.	−	598651.i	−0.280120	−	0.485181i
$56$		0			0
$57$	−553639.	+	1.24655e6i	−0.395972	+	0.891552i
$58$		0			0
$59$	333278.	+	577255.i	0.211264	+	0.365920i	0.952110	−	0.305755i	$-0.0989088\pi$
$59$	333278.	+	577255.i	0.211264	+	0.365920i	−0.740847	+	0.671674i	$0.765575\pi$
$60$		0			0
$61$	617751.	−	1.06998e6i	0.348465	−	0.603558i	−0.637512	−	0.770440i	$-0.720036\pi$
$61$	617751.	−	1.06998e6i	0.348465	−	0.603558i	0.985977	+	0.166882i	$0.0533698\pi$
$62$		0			0
$63$	−26873.6	+	46546.4i	−0.0135405	+	0.0234528i
$64$		0			0
$65$	1.16087e6			0.524309
$66$		0			0
$67$	−94447.0	+	163587.i	−0.0383642	+	0.0664488i	−0.884570	−	0.466408i	$-0.845548\pi$
$67$	−94447.0	+	163587.i	−0.0383642	+	0.0664488i	0.846206	+	0.532856i	$0.178881\pi$
$68$		0			0
$69$	934998.			0.342641
$70$		0			0
$71$	677843.	+	1.17406e6i	0.224763	+	0.389301i	0.956248	−	0.292556i	$-0.0945058\pi$
$71$	677843.	+	1.17406e6i	0.224763	+	0.389301i	−0.731485	+	0.681857i	$0.761173\pi$
$72$		0			0
$73$	−1.42377e6	−	2.46604e6i	−0.428361	−	0.741942i	0.568367	−	0.822775i	$-0.307575\pi$
$73$	−1.42377e6	−	2.46604e6i	−0.428361	−	0.741942i	−0.996728	+	0.0808329i	$0.974242\pi$
$74$		0			0
$75$	4.98725e6			1.36504
$76$		0			0
$77$	811717.			0.202622
$78$		0			0
$79$	−2.34677e6	−	4.06472e6i	−0.535519	−	0.927546i	−0.999138	−	0.0415116i	$-0.986783\pi$
$79$	−2.34677e6	−	4.06472e6i	−0.535519	−	0.927546i	0.463619	−	0.886035i	$-0.346551\pi$
$80$		0			0
$81$	2.27029e6	+	3.93226e6i	0.474662	+	0.822138i
$82$		0			0
$83$	2.12538e6			0.408002			0.204001	−	0.978971i	$-0.434605\pi$
$83$	2.12538e6			0.408002			0.204001	+	0.978971i	$0.434605\pi$
$84$		0			0
$85$	1.41403e6	−	2.44917e6i	0.249742	−	0.432566i
$86$		0			0
$87$	8.99555e6			1.46457
$88$		0			0
$89$	−2.16036e6	+	3.74186e6i	−0.324834	+	0.562629i	−0.981479	−	0.191571i	$-0.938642\pi$
$89$	−2.16036e6	+	3.74186e6i	−0.324834	+	0.562629i	0.656645	+	0.754200i	$0.271975\pi$
$90$		0			0
$91$	−681579.	+	1.18053e6i	−0.0948137	+	0.164222i
$92$		0			0
$93$	5.84569e6	+	1.01250e7i	0.753608	+	1.30529i
$94$		0			0
$95$	5.25409e6	−	1.18299e7i	0.628731	−	1.41562i
$96$		0			0
$97$	−5.40154e6	−	9.35575e6i	−0.600920	−	1.04082i	−0.992682	−	0.120757i	$-0.961468\pi$
$97$	−5.40154e6	−	9.35575e6i	−0.600920	−	1.04082i	0.391762	−	0.920067i	$-0.371866\pi$
$98$		0			0
$99$	−84398.4	+	146182.i	−0.00874200	+	0.0151416i
$100$		0			0
$101$	6.81849e6	−	1.18100e7i	0.658512	−	1.14058i	−0.322489	−	0.946573i	$-0.604520\pi$
$101$	6.81849e6	−	1.18100e7i	0.658512	−	1.14058i	0.981001	−	0.194002i	$-0.0621470\pi$
$102$		0			0
$103$	7.04217e6			0.635004			0.317502	−	0.948258i	$-0.397156\pi$
$103$	7.04217e6			0.635004			0.317502	+	0.948258i	$0.397156\pi$
$104$		0			0
$105$	−5.02075e6	+	8.69620e6i	−0.423259	+	0.733107i
$106$		0			0
$107$	−6.52313e6			−0.514770			−0.257385	−	0.966309i	$-0.582861\pi$
$107$	−6.52313e6			−0.514770			−0.257385	+	0.966309i	$0.582861\pi$
$108$		0			0
$109$	7.68667e6	+	1.33137e7i	0.568519	+	0.984704i	0.996713	+	0.0810173i	$0.0258169\pi$
$109$	7.68667e6	+	1.33137e7i	0.568519	+	0.984704i	−0.428193	+	0.903687i	$0.640850\pi$
$110$		0			0
$111$	−3.59629e6	−	6.22896e6i	−0.249589	−	0.432300i
$112$		0			0
$113$	−4.27249e6			−0.278552			−0.139276	−	0.990254i	$-0.544478\pi$
$113$	−4.27249e6			−0.278552			−0.139276	+	0.990254i	$0.544478\pi$
$114$		0			0
$115$	−8.87322e6			−0.544050
$116$		0			0
$117$	−141734.	−	245491.i	−0.00818135	−	0.0141705i
$118$		0			0
$119$	1.66043e6	+	2.87594e6i	0.0903245	+	0.156447i
$120$		0			0
$121$	−1.69379e7			−0.869183
$122$		0			0
$123$	−3.89488e6	+	6.74613e6i	−0.188724	+	0.326879i
$124$		0			0
$125$	−1.35054e7			−0.618476
$126$		0			0
$127$	−9.41761e6	+	1.63118e7i	−0.407970	+	0.706624i	−0.994662	−	0.103186i	$-0.967096\pi$
$127$	−9.41761e6	+	1.63118e7i	−0.407970	+	0.706624i	0.586692	+	0.809810i	$0.300430\pi$
$128$		0			0
$129$	−2.16139e7	+	3.74364e7i	−0.886480	+	1.53543i
$130$		0			0
$131$	2.59599e6	+	4.49639e6i	0.100891	+	0.174749i	0.912052	−	0.410074i	$-0.134497\pi$
$131$	2.59599e6	+	4.49639e6i	0.100891	+	0.174749i	−0.811161	+	0.584823i	$0.801164\pi$
$132$		0			0
$133$	8.94536e6	+	1.22887e7i	0.329699	+	0.452924i
$134$		0			0
$135$	−2.26424e7	−	3.92178e7i	−0.792052	−	1.37188i
$136$		0			0
$137$	−884376.	+	1.53178e6i	−0.0293843	+	0.0508951i	−0.880344	−	0.474337i	$-0.842688\pi$
$137$	−884376.	+	1.53178e6i	−0.0293843	+	0.0508951i	0.850959	+	0.525232i	$0.176021\pi$
$138$		0			0
$139$	1.85331e7	−	3.21003e7i	0.585325	−	1.01381i	−0.409510	−	0.912306i	$-0.634300\pi$
$139$	1.85331e7	−	3.21003e7i	0.585325	−	1.01381i	0.994835	−	0.101507i	$-0.0323663\pi$
$140$		0			0
$141$	1.49823e7			0.450104
$142$		0			0
$143$	−2.14055e6	+	3.70754e6i	−0.0612137	+	0.106025i
$144$		0			0
$145$	−8.53686e7			−2.32547
$146$		0			0
$147$	1.28898e7	+	2.23258e7i	0.334685	+	0.579691i
$148$		0			0
$149$	−3.95922e7	−	6.85756e7i	−0.980522	−	1.69831i	−0.660357	−	0.750952i	$-0.729595\pi$
$149$	−3.95922e7	−	6.85756e7i	−0.980522	−	1.69831i	−0.320165	−	0.947362i	$-0.603739\pi$
$150$		0			0
$151$	5.67510e7			1.34139			0.670693	−	0.741735i	$-0.265997\pi$
$151$	5.67510e7			1.34139			0.670693	+	0.741735i	$0.265997\pi$
$152$		0			0
$153$	−690572.			−0.0155880
$154$		0			0
$155$	−5.54761e7	−	9.60875e7i	−1.19659	−	2.07255i
$156$		0			0
$157$	2.71518e7	+	4.70283e7i	0.559951	+	0.969863i	0.997500	+	0.0706688i	$0.0225134\pi$
$157$	2.71518e7	+	4.70283e7i	0.559951	+	0.969863i	−0.437549	+	0.899195i	$0.644153\pi$
$158$		0			0
$159$	5.09179e7			1.00457
$160$		0			0
$161$	5.20971e6	−	9.02348e6i	0.0983836	−	0.170405i
$162$		0			0
$163$	1.15429e7			0.208765			0.104383	−	0.994537i	$-0.466713\pi$
$163$	1.15429e7			0.208765			0.104383	+	0.994537i	$0.466713\pi$
$164$		0			0
$165$	−1.57681e7	+	2.73111e7i	−0.273265	+	0.473309i
$166$		0			0
$167$	−2.94834e7	+	5.10668e7i	−0.489858	+	0.848459i	−0.999932	−	0.0116716i	$-0.996285\pi$
$167$	−2.94834e7	+	5.10668e7i	−0.489858	+	0.848459i	0.510074	+	0.860131i	$0.329618\pi$
$168$		0			0
$169$	2.77795e7	+	4.81156e7i	0.442712	+	0.766800i
$170$		0			0
$171$	−3.14317e6	+	333254.i	−0.0480708	+	0.00509669i
$172$		0			0
$173$	2.77863e7	+	4.81272e7i	0.408008	+	0.706691i	0.994666	−	0.103144i	$-0.0328902\pi$
$173$	2.77863e7	+	4.81272e7i	0.408008	+	0.706691i	−0.586658	+	0.809835i	$0.699557\pi$
$174$		0			0
$175$	2.77884e7	−	4.81310e7i	0.391950	−	0.678877i
$176$		0			0
$177$	1.52045e7	−	2.63350e7i	0.206094	−	0.356966i
$178$		0			0
$179$	−4.88613e7			−0.636766			−0.318383	−	0.947962i	$-0.603140\pi$
$179$	−4.88613e7			−0.636766			−0.318383	+	0.947962i	$0.603140\pi$
$180$		0			0
$181$	−6.92377e7	+	1.19923e8i	−0.867896	+	1.50324i	−0.00375381	+	0.999993i	$0.501195\pi$
$181$	−6.92377e7	+	1.19923e8i	−0.867896	+	1.50324i	−0.864142	+	0.503247i	$0.832138\pi$
$182$		0			0
$183$	−5.63649e7			−0.679876
$184$		0			0
$185$	3.41292e7	+	5.91135e7i	0.396301	+	0.686413i
$186$		0			0
$187$	5.21469e6	+	9.03211e6i	0.0583154	+	0.101005i
$188$		0			0
$189$	5.31758e7			0.572925
$190$		0			0
$191$	1.68538e8			1.75017			0.875085	−	0.483970i	$-0.160806\pi$
$191$	1.68538e8			1.75017			0.875085	+	0.483970i	$0.160806\pi$
$192$		0			0
$193$	−4.68747e6	−	8.11893e6i	−0.0469340	−	0.0812921i	0.841604	−	0.540095i	$-0.181612\pi$
$193$	−4.68747e6	−	8.11893e6i	−0.0469340	−	0.0812921i	−0.888538	+	0.458803i	$0.848278\pi$
$194$		0			0
$195$	−2.64801e7	−	4.58648e7i	−0.255740	−	0.442954i
$196$		0			0
$197$	9.60473e7			0.895062			0.447531	−	0.894268i	$-0.352303\pi$
$197$	9.60473e7			0.895062			0.447531	+	0.894268i	$0.352303\pi$
$198$		0			0
$199$	5.88987e7	−	1.02015e8i	0.529809	−	0.917657i	−0.469586	−	0.882887i	$-0.655597\pi$
$199$	5.88987e7	−	1.02015e8i	0.529809	−	0.917657i	0.999395	−	0.0347699i	$-0.0110698\pi$
$200$		0			0
$201$	8.61754e6			0.0748510
$202$		0			0
$203$	5.01222e7	−	8.68142e7i	0.420527	−	0.728375i
$204$		0			0
$205$	3.69628e7	−	6.40214e7i	0.299658	−	0.519023i
$206$		0			0
$207$	1.08336e6	+	1.87643e6i	0.00848939	+	0.0147041i
$208$		0			0
$209$	2.80936e7	+	3.85935e7i	0.212860	+	0.292417i
$210$		0			0
$211$	−3.02590e7	−	5.24102e7i	−0.221751	−	0.384085i	0.733589	−	0.679594i	$-0.237844\pi$
$211$	−3.02590e7	−	5.24102e7i	−0.221751	−	0.384085i	−0.955340	+	0.295509i	$0.904511\pi$
$212$		0			0
$213$	3.09239e7	−	5.35618e7i	0.219263	−	0.379775i
$214$		0			0
$215$	2.05118e8	−	3.55275e8i	1.40757	−	2.43798i
$216$		0			0
$217$	1.30286e8			0.865544
$218$		0			0
$219$	−6.49538e7	+	1.12503e8i	−0.417879	+	0.723787i
$220$		0			0
$221$	−1.75146e7			−0.109151
$222$		0			0
$223$	1.12008e8	+	1.94004e8i	0.676368	+	1.17150i	0.976067	+	0.217471i	$0.0697806\pi$
$223$	1.12008e8	+	1.94004e8i	0.676368	+	1.17150i	−0.299698	+	0.954034i	$0.596886\pi$
$224$		0			0
$225$	5.77861e6	+	1.00088e7i	0.0338208	+	0.0585794i
$226$		0			0
$227$	−1.57027e8			−0.891011			−0.445506	−	0.895279i	$-0.646976\pi$
$227$	−1.57027e8			−0.891011			−0.445506	+	0.895279i	$0.646976\pi$
$228$		0			0
$229$	−1.45255e8			−0.799295			−0.399648	−	0.916669i	$-0.630867\pi$
$229$	−1.45255e8			−0.799295			−0.399648	+	0.916669i	$0.630867\pi$
$230$		0			0
$231$	−1.85157e7	−	3.20701e7i	−0.0988321	−	0.171182i
$232$		0			0
$233$	1.48486e8	+	2.57185e8i	0.769022	+	1.33199i	0.938094	+	0.346381i	$0.112590\pi$
$233$	1.48486e8	+	2.57185e8i	0.769022	+	1.33199i	−0.169072	+	0.985604i	$0.554077\pi$
$234$		0			0
$235$	−1.42184e8			−0.714681
$236$		0			0
$237$	−1.07062e8	+	1.85437e8i	−0.522415	+	0.904850i
$238$		0			0
$239$	−9.66836e7			−0.458100			−0.229050	−	0.973415i	$-0.573562\pi$
$239$	−9.66836e7			−0.458100			−0.229050	+	0.973415i	$0.573562\pi$
$240$		0			0
$241$	−6.16188e7	+	1.06727e8i	−0.283566	+	0.491150i	−0.972260	−	0.233901i	$-0.924851\pi$
$241$	−6.16188e7	+	1.06727e8i	−0.283566	+	0.491150i	0.688695	+	0.725051i	$0.258184\pi$
$242$		0			0
$243$	−1.08030e7	+	1.87113e7i	−0.0482971	+	0.0836531i
$244$		0			0
$245$	−1.22325e8	−	2.11874e8i	−0.531417	−	0.920442i
$246$		0			0
$247$	−7.97184e7	+	8.45212e6i	−0.336604	+	0.0356883i
$248$		0			0
$249$	−4.84810e7	−	8.39715e7i	−0.199009	−	0.344694i
$250$		0			0
$251$	−9.05608e7	+	1.56856e8i	−0.361478	+	0.626098i	−0.988204	−	0.153141i	$-0.951061\pi$
$251$	−9.05608e7	+	1.56856e8i	−0.361478	+	0.626098i	0.626726	+	0.779240i	$0.284394\pi$
$252$		0			0
$253$	1.63615e7	−	2.83389e7i	0.0635185	−	0.110017i
$254$		0			0
$255$	−1.29019e8			−0.487262
$256$		0			0
$257$	5.70493e7	−	9.88122e7i	0.209645	−	0.363115i	−0.741958	−	0.670447i	$-0.766103\pi$
$257$	5.70493e7	−	9.88122e7i	0.209645	−	0.363115i	0.951603	+	0.307331i	$0.0994359\pi$
$258$		0			0
$259$	−8.01526e7			−0.286661
$260$		0			0
$261$	1.04229e7	+	1.80530e7i	0.0362867	+	0.0628505i
$262$		0			0
$263$	2.80357e8	+	4.85593e8i	0.950313	+	1.64599i	0.744748	+	0.667346i	$0.232570\pi$
$263$	2.80357e8	+	4.85593e8i	0.950313	+	1.64599i	0.205565	+	0.978643i	$0.434097\pi$
$264$		0			0
$265$	−4.83216e8			−1.59507
$266$		0			0
$267$	1.97116e8			0.633771
$268$		0			0
$269$	−2.68775e8	−	4.65532e8i	−0.841891	−	1.45820i	−0.888294	−	0.459276i	$-0.848109\pi$
$269$	−2.68775e8	−	4.65532e8i	−0.841891	−	1.45820i	0.0464022	−	0.998923i	$-0.485224\pi$
$270$		0			0
$271$	1.01239e8	+	1.75351e8i	0.308998	+	0.535200i	0.978143	−	0.207932i	$-0.0666732\pi$
$271$	1.01239e8	+	1.75351e8i	0.308998	+	0.535200i	−0.669146	+	0.743131i	$0.733340\pi$
$272$		0			0
$273$	6.21887e7			0.184987
$274$		0			0
$275$	8.72716e7	−	1.51159e8i	0.253051	−	0.438298i
$276$		0			0
$277$	−5.94618e8			−1.68097			−0.840483	−	0.541837i	$-0.817729\pi$
$277$	−5.94618e8			−1.68097			−0.840483	+	0.541837i	$0.817729\pi$
$278$		0			0
$279$	−1.35465e7	+	2.34633e7i	−0.0373433	+	0.0646805i
$280$		0			0
$281$	3.19191e8	−	5.52855e8i	0.858180	−	1.48641i	−0.0154832	−	0.999880i	$-0.504929\pi$
$281$	3.19191e8	−	5.52855e8i	0.858180	−	1.48641i	0.873663	−	0.486531i	$-0.161738\pi$
$282$		0			0
$283$	−3.17486e7	−	5.49902e7i	−0.0832669	−	0.144222i	0.821384	−	0.570375i	$-0.193202\pi$
$283$	−3.17486e7	−	5.49902e7i	−0.0832669	−	0.144222i	−0.904651	+	0.426152i	$0.859869\pi$
$284$		0			0
$285$	−5.87234e8	+	6.22614e7i	−1.50264	+	0.159317i
$286$		0			0
$287$	4.34037e7	+	7.51774e7i	0.108378	+	0.187716i
$288$		0			0
$289$	1.83835e8	−	3.18412e8i	0.448009	−	0.775974i
$290$		0			0
$291$	−2.46424e8	+	4.26819e8i	−0.586216	+	1.01536i
$292$		0			0
$293$	−2.09497e6			−0.00486565			−0.00243282	−	0.999997i	$-0.500774\pi$
$293$	−2.09497e6			−0.00486565			−0.00243282	+	0.999997i	$0.500774\pi$
$294$		0			0
$295$	−1.44292e8	+	2.49921e8i	−0.327240	+	0.566796i
$296$		0			0
$297$	1.67003e8			0.369893
$298$		0			0
$299$	2.74766e7	+	4.75909e7i	0.0594449	+	0.102962i
$300$		0			0
$301$	2.40861e8	+	4.17183e8i	0.509076	+	0.881746i
$302$		0			0
$303$	−6.22133e8			−1.28480
$304$		0			0
$305$	5.34908e8			1.07952
$306$		0			0
$307$	2.62337e8	+	4.54382e8i	0.517459	+	0.896265i	0.999794	+	0.0202788i	$0.00645538\pi$
$307$	2.62337e8	+	4.54382e8i	0.517459	+	0.896265i	−0.482335	+	0.875987i	$0.660211\pi$
$308$		0			0
$309$	−1.60636e8	−	2.78229e8i	−0.309733	−	0.536473i
$310$		0			0
$311$	−4.39369e8			−0.828262			−0.414131	−	0.910217i	$-0.635914\pi$
$311$	−4.39369e8			−0.828262			−0.414131	+	0.910217i	$0.635914\pi$
$312$		0			0
$313$	1.84522e8	−	3.19602e8i	0.340129	−	0.589121i	−0.644328	−	0.764750i	$-0.722863\pi$
$313$	1.84522e8	−	3.19602e8i	0.340129	−	0.589121i	0.984456	+	0.175629i	$0.0561960\pi$
$314$		0			0
$315$	−2.32697e7			−0.0419473
$316$		0			0
$317$	5.22049e8	−	9.04216e8i	0.920458	−	1.59428i	0.121750	−	0.992561i	$-0.461149\pi$
$317$	5.22049e8	−	9.04216e8i	0.920458	−	1.59428i	0.798708	−	0.601719i	$-0.205517\pi$
$318$		0			0
$319$	1.57413e8	−	2.72646e8i	0.271501	−	0.470254i
$320$		0			0
$321$	1.48796e8	+	2.57722e8i	0.251087	+	0.434895i
$322$		0			0
$323$	−7.92709e7	+	1.78482e8i	−0.130889	+	0.294704i
$324$		0			0
$325$	1.46560e8	+	2.53849e8i	0.236822	+	0.410188i
$326$		0			0
$327$	3.50674e8	−	6.07385e8i	0.554608	−	0.960609i
$328$		0			0
$329$	8.34799e7	−	1.44591e8i	0.129240	−	0.223850i
$330$		0			0
$331$	3.40389e8			0.515915			0.257957	−	0.966156i	$-0.416951\pi$
$331$	3.40389e8			0.515915			0.257957	+	0.966156i	$0.416951\pi$
$332$		0			0
$333$	8.33388e6	−	1.44347e7i	0.0123678	−	0.0214217i
$334$		0			0
$335$	−8.17813e7			−0.118849
$336$		0			0
$337$	4.65079e8	+	8.05540e8i	0.661945	+	1.14652i	0.980104	+	0.198485i	$0.0636021\pi$
$337$	4.65079e8	+	8.05540e8i	0.661945	+	1.14652i	−0.318159	+	0.948037i	$0.603065\pi$
$338$		0			0
$339$	9.74577e7	+	1.68802e8i	0.135868	+	0.235330i
$340$		0			0
$341$	4.09173e8			0.558814
$342$		0			0
$343$	7.05964e8			0.944612
$344$		0			0
$345$	2.02403e8	+	3.50572e8i	0.265369	+	0.459632i
$346$		0			0
$347$	−3.71576e8	−	6.43589e8i	−0.477414	−	0.826905i	0.522251	−	0.852792i	$-0.325092\pi$
$347$	−3.71576e8	−	6.43589e8i	−0.477414	−	0.826905i	−0.999665	+	0.0258867i	$0.991759\pi$
$348$		0			0
$349$	−2.17401e8			−0.273762			−0.136881	−	0.990588i	$-0.543708\pi$
$349$	−2.17401e8			−0.273762			−0.136881	+	0.990588i	$0.543708\pi$
$350$		0			0
$351$	−1.40228e8	+	2.42882e8i	−0.173085	+	0.299792i
$352$		0			0
$353$	4.44846e8			0.538267			0.269134	−	0.963103i	$-0.413263\pi$
$353$	4.44846e8			0.538267			0.269134	+	0.963103i	$0.413263\pi$
$354$		0			0
$355$	−2.93471e8	+	5.08307e8i	−0.348150	+	0.603013i
$356$		0			0
$357$	7.57504e7	−	1.31204e8i	0.0881143	−	0.152618i
$358$		0			0
$359$	−5.51367e8	−	9.54995e8i	−0.628941	−	1.08936i	−0.987765	−	0.155952i	$-0.950155\pi$
$359$	−5.51367e8	−	9.54995e8i	−0.628941	−	1.08936i	0.358824	−	0.933405i	$-0.383178\pi$
$360$		0			0
$361$	−2.74673e8	+	8.50624e8i	−0.307285	+	0.951618i
$362$		0			0
$363$	3.86363e8	+	6.69200e8i	0.423957	+	0.734315i
$364$		0			0
$365$	6.16418e8	−	1.06767e9i	0.663514	−	1.14924i
$366$		0			0
$367$	−5.26617e8	+	9.12127e8i	−0.556113	+	0.963217i	0.441703	+	0.897162i	$0.354375\pi$
$367$	−5.26617e8	+	9.12127e8i	−0.556113	+	0.963217i	−0.997816	+	0.0660551i	$0.978959\pi$
$368$		0			0
$369$	−1.80516e7			−0.0187035
$370$		0			0
$371$	2.83709e8	−	4.91399e8i	0.288446	−	0.499604i
$372$		0			0
$373$	6.77179e8			0.675652			0.337826	−	0.941209i	$-0.390309\pi$
$373$	6.77179e8			0.675652			0.337826	+	0.941209i	$0.390309\pi$
$374$		0			0
$375$	3.08066e8	+	5.33585e8i	0.301671	+	0.522510i
$376$		0			0
$377$	2.64351e8	+	4.57869e8i	0.254089	+	0.440095i
$378$		0			0
$379$	−7.35547e8			−0.694022			−0.347011	−	0.937861i	$-0.612803\pi$
$379$	−7.35547e8			−0.694022			−0.347011	+	0.937861i	$0.612803\pi$
$380$		0			0
$381$	8.59283e8			0.795974
$382$		0			0
$383$	−1.35670e8	−	2.34988e8i	−0.123393	−	0.213722i	0.797711	−	0.603040i	$-0.206044\pi$
$383$	−1.35670e8	−	2.34988e8i	−0.123393	−	0.213722i	−0.921104	+	0.389318i	$0.872711\pi$
$384$		0			0
$385$	1.75716e8	+	3.04349e8i	0.156927	+	0.271806i
$386$		0			0
$387$	−1.00174e8			−0.0878550
$388$		0			0
$389$	1.60630e8	−	2.78220e8i	0.138358	−	0.239643i	−0.788517	−	0.615012i	$-0.789151\pi$
$389$	1.60630e8	−	2.78220e8i	0.138358	−	0.239643i	0.926875	+	0.375370i	$0.122484\pi$
$390$		0			0
$391$	1.33874e8			0.113261
$392$		0			0
$393$	1.18432e8	−	2.05130e8i	0.0984225	−	0.170473i
$394$		0			0
$395$	1.01603e9	−	1.75981e9i	0.829499	−	1.43673i
$396$		0			0
$397$	−2.95681e8	−	5.12135e8i	−0.237169	−	0.410788i	0.722732	−	0.691128i	$-0.242886\pi$
$397$	−2.95681e8	−	5.12135e8i	−0.237169	−	0.410788i	−0.959901	+	0.280340i	$0.909553\pi$
$398$		0			0
$399$	2.81465e8	−	6.33734e8i	0.221830	−	0.499461i
$400$		0			0
$401$	−8.43066e8	−	1.46023e9i	−0.652915	−	1.13088i	−0.982412	−	0.186725i	$-0.940213\pi$
$401$	−8.43066e8	−	1.46023e9i	−0.652915	−	1.13088i	0.329498	−	0.944156i	$-0.393121\pi$
$402$		0			0
$403$	−3.43573e8	+	5.95085e8i	−0.261487	+	0.452910i
$404$		0			0
$405$	−9.82918e8	+	1.70246e9i	−0.735233	+	1.27346i
$406$		0			0
$407$	−2.51725e8			−0.185074
$408$		0			0
$409$	1.10054e9	−	1.90619e9i	0.795380	−	1.37764i	−0.127218	−	0.991875i	$-0.540605\pi$
$409$	1.10054e9	−	1.90619e9i	0.795380	−	1.37764i	0.922598	−	0.385763i	$-0.126062\pi$
$410$		0			0
$411$	8.06923e7			0.0573305
$412$		0			0
$413$	−1.69436e8	−	2.93471e8i	−0.118353	−	0.204994i
$414$		0			0
$415$	4.60089e8	+	7.96897e8i	0.315990	+	0.547311i
$416$		0			0
$417$	−1.69100e9			−1.14200
$418$		0			0
$419$	−2.51296e9			−1.66892			−0.834462	−	0.551065i	$-0.814222\pi$
$419$	−2.51296e9			−1.66892			−0.834462	+	0.551065i	$0.814222\pi$
$420$		0			0
$421$	−1.33314e9	−	2.30906e9i	−0.870737	−	1.50816i	−0.861236	−	0.508205i	$-0.830309\pi$
$421$	−1.33314e9	−	2.30906e9i	−0.870737	−	1.50816i	−0.00950083	−	0.999955i	$-0.503024\pi$
$422$		0			0
$423$	1.73597e7	+	3.00678e7i	0.0111519	+	0.0193157i
$424$		0			0
$425$	7.14081e8			0.451218
$426$		0			0
$427$	−3.14059e8	+	5.43966e8i	−0.195215	+	0.338123i
$428$		0			0
$429$	1.95308e8			0.119432
$430$		0			0
$431$	−1.35965e9	+	2.35499e9i	−0.818009	+	1.41683i	0.0891387	+	0.996019i	$0.471589\pi$
$431$	−1.35965e9	+	2.35499e9i	−0.818009	+	1.41683i	−0.907147	+	0.420813i	$0.861745\pi$
$432$		0			0
$433$	−1.39158e9	+	2.41028e9i	−0.823759	+	1.42679i	0.0791059	+	0.996866i	$0.474793\pi$
$433$	−1.39158e9	+	2.41028e9i	−0.823759	+	1.42679i	−0.902864	+	0.429925i	$0.858540\pi$
$434$		0			0
$435$	1.94730e9	+	3.37283e9i	1.13428	+	1.96463i
$436$		0			0
$437$	6.09334e8	−	6.46045e7i	0.349278	−	0.0370321i
$438$		0			0
$439$	−2.87000e8	−	4.97098e8i	−0.161903	−	0.280425i	0.773648	−	0.633616i	$-0.218430\pi$
$439$	−2.87000e8	−	4.97098e8i	−0.161903	−	0.280425i	−0.935551	+	0.353191i	$0.885097\pi$
$440$		0			0
$441$	−2.98702e7	+	5.17367e7i	−0.0165845	+	0.0287253i
$442$		0			0
$443$	1.15965e9	−	2.00857e9i	0.633742	−	1.09767i	−0.353039	−	0.935609i	$-0.614852\pi$
$443$	1.15965e9	−	2.00857e9i	0.633742	−	1.09767i	0.986780	−	0.162064i	$-0.0518151\pi$
$444$		0			0
$445$	−1.87065e9			−1.00631
$446$		0			0
$447$	−1.80624e9	+	3.12849e9i	−0.956529	+	1.65676i
$448$		0			0
$449$	−3.57267e8			−0.186265			−0.0931325	−	0.995654i	$-0.529688\pi$
$449$	−3.57267e8			−0.186265			−0.0931325	+	0.995654i	$0.529688\pi$
$450$		0			0
$451$	1.36313e8	+	2.36100e8i	0.0699710	+	0.121193i
$452$		0			0
$453$	−1.29452e9	−	2.24217e9i	−0.654282	−	1.13325i
$454$		0			0
$455$	−5.90176e8			−0.293726
$456$		0			0
$457$	−2.95976e9			−1.45061			−0.725303	−	0.688430i	$-0.758300\pi$
$457$	−2.95976e9			−1.45061			−0.725303	+	0.688430i	$0.758300\pi$
$458$		0			0
$459$	3.41616e8	+	5.91696e8i	0.164890	+	0.285597i
$460$		0			0
$461$	−4.24003e6	−	7.34394e6i	−0.00201565	−	0.00349121i	0.865016	−	0.501745i	$-0.167308\pi$
$461$	−4.24003e6	−	7.34394e6i	−0.00201565	−	0.00349121i	−0.867031	+	0.498253i	$0.833975\pi$
$462$		0			0
$463$	−2.78532e9			−1.30419			−0.652096	−	0.758136i	$-0.726110\pi$
$463$	−2.78532e9			−1.30419			−0.652096	+	0.758136i	$0.726110\pi$
$464$		0			0
$465$	−2.53088e9	+	4.38361e9i	−1.16731	+	2.02184i
$466$		0			0
$467$	3.44386e9			1.56472			0.782359	−	0.622828i	$-0.214016\pi$
$467$	3.44386e9			1.56472			0.782359	+	0.622828i	$0.214016\pi$
$468$		0			0
$469$	4.80160e7	−	8.31662e7i	0.0214922	−	0.0372256i
$470$		0			0
$471$	1.23869e9	−	2.14548e9i	0.546249	−	0.946131i
$472$		0			0
$473$	7.56440e8	+	1.31019e9i	0.328671	+	0.569274i
$474$		0			0
$475$	3.25017e9	−	3.44599e8i	1.39149	−	0.147532i
$476$		0			0
$477$	5.89974e7	+	1.02186e8i	0.0248896	+	0.0431101i
$478$		0			0
$479$	9.10040e8	−	1.57624e9i	0.378344	−	0.655310i	−0.612478	−	0.790488i	$-0.709827\pi$
$479$	9.10040e8	−	1.57624e9i	0.378344	−	0.655310i	0.990821	+	0.135177i	$0.0431605\pi$
$480$		0			0
$481$	2.11367e8	−	3.66099e8i	0.0866025	−	0.150000i
$482$		0			0
$483$	−4.75345e8			−0.191952
$484$		0			0
$485$	2.33859e9	−	4.05055e9i	0.930803	−	1.61220i
$486$		0			0
$487$	−2.44141e9			−0.957832			−0.478916	−	0.877861i	$-0.658970\pi$
$487$	−2.44141e9			−0.957832			−0.478916	+	0.877861i	$0.658970\pi$
$488$		0			0
$489$	−2.63299e8	−	4.56048e8i	−0.101828	−	0.176372i
$490$		0			0
$491$	−1.42574e9	−	2.46945e9i	−0.543568	−	0.941487i	−0.998696	−	0.0510611i	$-0.983740\pi$
$491$	−1.42574e9	−	2.46945e9i	−0.543568	−	0.941487i	0.455128	−	0.890426i	$-0.349594\pi$
$492$		0			0
$493$	1.28800e9			0.484117
$494$		0			0
$495$	−7.30802e7			−0.0270821
$496$		0			0
$497$	−3.44609e8	−	5.96881e8i	−0.125916	−	0.218092i
$498$		0			0
$499$	−2.46068e9	−	4.26202e9i	−0.886548	−	1.53555i	−0.843929	−	0.536456i	$-0.819763\pi$
$499$	−2.46068e9	−	4.26202e9i	−0.886548	−	1.53555i	−0.0426198	−	0.999091i	$-0.513570\pi$
$500$		0			0
$501$	2.69013e9			0.955743
$502$		0			0
$503$	5.28043e8	−	9.14597e8i	0.185004	−	0.320436i	−0.758574	−	0.651587i	$-0.774104\pi$
$503$	5.28043e8	−	9.14597e8i	0.185004	−	0.320436i	0.943578	+	0.331151i	$0.107437\pi$
$504$		0			0
$505$	5.90410e9			2.04002
$506$		0			0
$507$	1.26733e9	−	2.19508e9i	0.431879	−	0.748037i
$508$		0			0
$509$	−1.23055e9	+	2.13137e9i	−0.413605	+	0.716384i	−0.995281	−	0.0970361i	$-0.969064\pi$
$509$	−1.23055e9	+	2.13137e9i	−0.413605	+	0.716384i	0.581676	+	0.813420i	$0.302397\pi$
$510$		0			0
$511$	7.23831e8	+	1.25371e9i	0.239974	+	0.415647i
$512$		0			0
$513$	1.84042e9	+	2.52827e9i	0.601874	+	0.826824i
$514$		0			0
$515$	1.52445e9	+	2.64042e9i	0.491798	+	0.851820i
$516$		0			0
$517$	2.62175e8	−	4.54100e8i	0.0834400	−	0.144522i
$518$		0			0
$519$	1.26764e9	−	2.19561e9i	0.398024	−	0.689398i
$520$		0			0
$521$	−7.34701e8			−0.227603			−0.113802	−	0.993503i	$-0.536303\pi$
$521$	−7.34701e8			−0.227603			−0.113802	+	0.993503i	$0.536303\pi$
$522$		0			0
$523$	9.36595e8	−	1.62223e9i	0.286283	−	0.495857i	−0.686636	−	0.727001i	$-0.740913\pi$
$523$	9.36595e8	−	1.62223e9i	0.286283	−	0.495857i	0.972920	+	0.231144i	$0.0742468\pi$
$524$		0			0
$525$	−2.53547e9			−0.764718
$526$		0			0
$527$	8.36994e8	+	1.44972e9i	0.249106	+	0.431465i
$528$		0			0
$529$	1.49239e9	+	2.58490e9i	0.438317	+	0.759187i
$530$		0			0
$531$	7.04684e7			0.0204251
$532$		0			0
$533$	−4.57833e8			−0.130967
$534$		0			0
$535$	−1.41209e9	−	2.44581e9i	−0.398680	−	0.690533i
$536$		0			0
$537$	1.11455e9	+	1.93046e9i	0.310592	+	0.537962i
$538$		0			0
$539$	9.02232e8			0.248175
$540$		0			0
$541$	−1.89905e9	+	3.28924e9i	−0.515638	+	0.893111i	0.484197	+	0.874959i	$0.339112\pi$
$541$	−1.89905e9	+	3.28924e9i	−0.515638	+	0.893111i	−0.999835	+	0.0181523i	$0.994222\pi$
$542$		0			0
$543$	6.31740e9			1.69332
$544$		0			0
$545$	−3.32793e9	+	5.76414e9i	−0.880615	+	1.52527i
$546$		0			0
$547$	8.74777e7	−	1.51516e8i	0.0228529	−	0.0395824i	−0.854373	−	0.519661i	$-0.826058\pi$
$547$	8.74777e7	−	1.51516e8i	0.0228529	−	0.0395824i	0.877226	+	0.480078i	$0.159392\pi$
$548$		0			0
$549$	−6.53086e7	−	1.13118e8i	−0.0168449	−	0.0291761i
$550$		0			0
$551$	5.86236e9	−	6.21556e8i	1.49294	−	0.158289i
$552$		0			0
$553$	1.19307e9	+	2.06647e9i	0.300006	+	0.519625i
$554$		0			0
$555$	1.55701e9	−	2.69682e9i	0.386603	−	0.669617i
$556$		0			0
$557$	−3.29497e9	+	5.70705e9i	−0.807901	+	1.39932i	0.106415	+	0.994322i	$0.466063\pi$
$557$	−3.29497e9	+	5.70705e9i	−0.807901	+	1.39932i	−0.914315	+	0.405003i	$0.867270\pi$
$558$		0			0
$559$	−2.54066e9			−0.615183
$560$		0			0
$561$	2.37900e8	−	4.12054e8i	0.0568884	−	0.0985337i
$562$		0			0
$563$	−4.78616e9			−1.13034			−0.565169	−	0.824975i	$-0.691189\pi$
$563$	−4.78616e9			−1.13034			−0.565169	+	0.824975i	$0.691189\pi$
$564$		0			0
$565$	−9.24883e8	−	1.60194e9i	−0.215733	−	0.373661i
$566$		0			0
$567$	−1.15420e9	−	1.99913e9i	−0.265913	−	0.460574i
$568$		0			0
$569$	6.39881e9			1.45615			0.728076	−	0.685497i	$-0.240415\pi$
$569$	6.39881e9			1.45615			0.728076	+	0.685497i	$0.240415\pi$
$570$		0			0
$571$	−1.98904e9			−0.447113			−0.223557	−	0.974691i	$-0.571767\pi$
$571$	−1.98904e9			−0.447113			−0.223557	+	0.974691i	$0.571767\pi$
$572$		0			0
$573$	−3.84443e9	−	6.65875e9i	−0.853672	−	1.47860i
$574$		0			0
$575$	−1.12024e9	−	1.94031e9i	−0.245739	−	0.425632i
$576$		0			0
$577$	8.46742e9			1.83500			0.917500	−	0.397736i	$-0.130204\pi$
$577$	8.46742e9			1.83500			0.917500	+	0.397736i	$0.130204\pi$
$578$		0			0
$579$	−2.13847e8	+	3.70394e8i	−0.0457856	+	0.0793029i
$580$		0			0
$581$	−1.08052e9			−0.228569
$582$		0			0
$583$	8.91010e8	−	1.54327e9i	0.186227	−	0.322555i
$584$		0			0
$585$	6.13637e7	−	1.06285e8i	0.0126726	−	0.0219496i
$586$		0			0
$587$	3.39477e9	+	5.87991e9i	0.692750	+	1.19988i	0.970933	+	0.239351i	$0.0769346\pi$
$587$	3.39477e9	+	5.87991e9i	0.692750	+	1.19988i	−0.278183	+	0.960528i	$0.589732\pi$
$588$		0			0
$589$	4.50921e9	+	6.19453e9i	0.909278	+	1.24912i
$590$		0			0
$591$	−2.19089e9	−	3.79473e9i	−0.436580	−	0.756179i
$592$		0			0
$593$	4.01807e9	−	6.95951e9i	0.791273	−	1.37053i	−0.133906	−	0.990994i	$-0.542752\pi$
$593$	4.01807e9	−	6.95951e9i	0.791273	−	1.37053i	0.925179	−	0.379531i	$-0.123915\pi$
$594$		0			0
$595$	−7.18879e8	+	1.24513e9i	−0.139909	+	0.242330i
$596$		0			0
$597$	−5.37404e9			−1.03369
$598$		0			0
$599$	1.84947e9	−	3.20338e9i	0.351604	−	0.608996i	−0.634927	−	0.772572i	$-0.718970\pi$
$599$	1.84947e9	−	3.20338e9i	0.351604	−	0.608996i	0.986531	+	0.163576i	$0.0523030\pi$
$600$		0			0
$601$	1.34787e9			0.253272			0.126636	−	0.991949i	$-0.459582\pi$
$601$	1.34787e9			0.253272			0.126636	+	0.991949i	$0.459582\pi$
$602$		0			0
$603$	9.98494e6	+	1.72944e7i	0.00185453	+	0.00321215i
$604$		0			0
$605$	−3.66662e9	−	6.35077e9i	−0.673166	−	1.16596i
$606$		0			0
$607$	−5.39320e9			−0.978783			−0.489391	−	0.872064i	$-0.662781\pi$
$607$	−5.39320e9			−0.978783			−0.489391	+	0.872064i	$0.662781\pi$
$608$		0			0
$609$	−4.57326e9			−0.820474
$610$		0			0
$611$	4.40283e8	+	7.62593e8i	0.0780887	+	0.135254i
$612$		0			0
$613$	−1.84849e9	−	3.20167e9i	−0.324119	−	0.561391i	0.657214	−	0.753704i	$-0.271735\pi$
$613$	−1.84849e9	−	3.20167e9i	−0.324119	−	0.561391i	−0.981334	+	0.192313i	$0.938401\pi$
$614$		0			0
$615$	−3.37256e9			−0.584652
$616$		0			0
$617$	−4.53667e9	+	7.85774e9i	−0.777569	+	1.34679i	0.155770	+	0.987793i	$0.450214\pi$
$617$	−4.53667e9	+	7.85774e9i	−0.777569	+	1.34679i	−0.933339	+	0.358996i	$0.883119\pi$
$618$		0			0
$619$	8.32148e9			1.41021			0.705104	−	0.709104i	$-0.250900\pi$
$619$	8.32148e9			1.41021			0.705104	+	0.709104i	$0.250900\pi$
$620$		0			0
$621$	1.07184e9	−	1.85649e9i	0.179602	−	0.311080i
$622$		0			0
$623$	1.09831e9	−	1.90233e9i	0.181977	−	0.315193i
$624$		0			0
$625$	1.34670e9	+	2.33256e9i	0.220644	+	0.382167i
$626$		0			0
$627$	8.83963e8	−	1.99029e9i	0.143218	−	0.322463i
$628$		0			0
$629$	−5.14922e8	−	8.91872e8i	−0.0825020	−	0.142898i
$630$		0			0
$631$	−1.94586e9	+	3.37032e9i	−0.308324	+	0.534034i	−0.977996	−	0.208624i	$-0.933102\pi$
$631$	−1.94586e9	+	3.37032e9i	−0.308324	+	0.534034i	0.669671	+	0.742657i	$0.266435\pi$
$632$		0			0
$633$	−1.38045e9	+	2.39101e9i	−0.216325	+	0.374686i
$634$		0			0
$635$	−8.15468e9			−1.26386
$636$		0			0
$637$	−7.57581e8	+	1.31217e9i	−0.116129	+	0.201142i
$638$		0			0
$639$	1.43323e8			0.0217302
$640$		0			0
$641$	3.47499e9	+	6.01886e9i	0.521136	+	0.902633i	0.999698	+	0.0245797i	$0.00782475\pi$
$641$	3.47499e9	+	6.01886e9i	0.521136	+	0.902633i	−0.478562	+	0.878054i	$0.658842\pi$
$642$		0			0
$643$	4.16974e9	+	7.22220e9i	0.618544	+	1.07135i	0.989752	+	0.142800i	$0.0456105\pi$
$643$	4.16974e9	+	7.22220e9i	0.618544	+	1.07135i	−0.371208	+	0.928550i	$0.621056\pi$
$644$		0			0
$645$	−1.87154e10			−2.74625
$646$		0			0
$647$	−1.04436e9			−0.151595			−0.0757973	−	0.997123i	$-0.524150\pi$
$647$	−1.04436e9			−0.151595			−0.0757973	+	0.997123i	$0.524150\pi$
$648$		0			0
$649$	−5.32125e8	−	9.21668e8i	−0.0764113	−	0.132348i
$650$		0			0
$651$	−2.97190e9	−	5.14747e9i	−0.422182	−	0.731241i
$652$		0			0
$653$	−1.16658e10			−1.63952			−0.819761	−	0.572706i	$-0.805894\pi$
$653$	−1.16658e10			−1.63952			−0.819761	+	0.572706i	$0.805894\pi$
$654$		0			0
$655$	−1.12393e9	+	1.94670e9i	−0.156277	+	0.270679i
$656$		0			0
$657$	−3.01042e8			−0.0414141
$658$		0			0
$659$	−2.99336e9	+	5.18465e9i	−0.407436	+	0.705700i	−0.994602	−	0.103767i	$-0.966910\pi$
$659$	−2.99336e9	+	5.18465e9i	−0.407436	+	0.705700i	0.587166	+	0.809467i	$0.300244\pi$
$660$		0			0
$661$	4.92978e9	−	8.53862e9i	0.663930	−	1.14996i	−0.315645	−	0.948877i	$-0.602221\pi$
$661$	4.92978e9	−	8.53862e9i	0.663930	−	1.14996i	0.979574	−	0.201082i	$-0.0644459\pi$
$662$		0			0
$663$	3.99517e8	+	6.91984e8i	0.0532400	+	0.0922143i
$664$		0			0
$665$	−2.67113e9	+	6.01419e9i	−0.352225	+	0.793052i
$666$		0			0
$667$	−2.02059e9	−	3.49976e9i	−0.263656	−	0.456665i
$668$		0			0
$669$	5.10994e9	−	8.85067e9i	0.659818	−	1.14284i
$670$		0			0
$671$	−9.86325e8	+	1.70837e9i	−0.126035	+	0.218299i
$672$		0			0
$673$	4.87344e9			0.616287			0.308144	−	0.951340i	$-0.400292\pi$
$673$	4.87344e9			0.616287			0.308144	+	0.951340i	$0.400292\pi$
$674$		0			0
$675$	5.71718e9	−	9.90245e9i	0.715515	−	1.23931i
$676$		0			0
$677$	2.31850e9			0.287175			0.143587	−	0.989638i	$-0.454136\pi$
$677$	2.31850e9			0.287175			0.143587	+	0.989638i	$0.454136\pi$
$678$		0			0
$679$	2.74610e9	+	4.75638e9i	0.336644	+	0.583085i
$680$		0			0
$681$	3.58186e9	+	6.20397e9i	0.434604	+	0.752757i
$682$		0			0
$683$	−1.49916e10			−1.80043			−0.900216	−	0.435444i	$-0.856591\pi$
$683$	−1.49916e10			−1.80043			−0.900216	+	0.435444i	$0.856591\pi$
$684$		0			0
$685$	−7.65778e8			−0.0910303
$686$		0			0
$687$	3.31334e9	+	5.73888e9i	0.389868	+	0.675272i
$688$		0			0
$689$	1.49632e9	+	2.59170e9i	0.174284	+	0.301868i
$690$		0			0
$691$	−1.43178e10			−1.65083			−0.825416	−	0.564524i	$-0.809060\pi$
$691$	−1.43178e10			−1.65083			−0.825416	+	0.564524i	$0.809060\pi$
$692$		0			0
$693$	4.29074e7	−	7.43178e7i	0.00489740	−	0.00848255i
$694$		0			0
$695$	1.60478e10			1.81329
$696$		0			0
$697$	−5.57674e8	+	9.65920e8i	−0.0623830	+	0.108050i
$698$		0			0
$699$	6.77407e9	−	1.17330e10i	0.750204	−	1.29939i
$700$		0			0
$701$	−9.38902e8	−	1.62623e9i	−0.102945	−	0.178307i	0.809952	−	0.586497i	$-0.199493\pi$
$701$	−9.38902e8	−	1.62623e9i	−0.102945	−	0.178307i	−0.912897	+	0.408190i	$0.866160\pi$
$702$		0			0
$703$	−2.77409e9	−	3.81090e9i	−0.301145	−	0.413699i
$704$		0			0
$705$	3.24329e9	+	5.61754e9i	0.348597	+	0.603787i
$706$		0			0
$707$	−3.46646e9	+	6.00408e9i	−0.368908	+	0.638968i
$708$		0			0
$709$	−3.26084e9	+	5.64793e9i	−0.343611	+	0.595152i	−0.985100	−	0.171980i	$-0.944984\pi$
$709$	−3.26084e9	+	5.64793e9i	−0.343611	+	0.595152i	0.641489	+	0.767132i	$0.278317\pi$
$710$		0			0
$711$	−4.96200e8			−0.0517742
$712$		0			0
$713$	2.62613e9	−	4.54859e9i	0.271333	−	0.469962i
$714$		0			0
$715$	−1.85349e9			−0.189636
$716$		0			0
$717$	2.20540e9	+	3.81987e9i	0.223445	+	0.387018i
$718$		0			0
$719$	6.79683e9	+	1.17724e10i	0.681954	+	1.18118i	0.974384	+	0.224892i	$0.0722029\pi$
$719$	6.79683e9	+	1.17724e10i	0.681954	+	1.18118i	−0.292430	+	0.956287i	$0.594464\pi$
$720$		0			0
$721$	−3.58018e9			−0.355739
$722$		0			0
$723$	5.62223e9			0.553254
$724$		0			0
$725$	−1.07778e10	−	1.86676e10i	−1.05038	−	1.81931i
$726$		0			0
$727$	−3.13772e9	−	5.43469e9i	−0.302861	−	0.524571i	0.673922	−	0.738803i	$-0.264609\pi$
$727$	−3.13772e9	−	5.43469e9i	−0.302861	−	0.524571i	−0.976783	+	0.214232i	$0.931275\pi$
$728$		0			0
$729$	1.09159e10			1.04355
$730$		0			0
$731$	−3.09471e9	+	5.36019e9i	−0.293028	+	0.507539i
$732$		0			0
$733$	−7.78524e8			−0.0730143			−0.0365072	−	0.999333i	$-0.511623\pi$
$733$	−7.78524e8			−0.0730143			−0.0365072	+	0.999333i	$0.511623\pi$
$734$		0			0
$735$	−5.58062e9	+	9.66591e9i	−0.518414	+	0.897919i
$736$		0			0
$737$	1.50798e8	−	2.61190e8i	0.0138758	−	0.0240336i
$738$		0			0
$739$	1.75699e8	+	3.04320e8i	0.0160145	+	0.0277379i	0.873922	−	0.486067i	$-0.161569\pi$
$739$	1.75699e8	+	3.04320e8i	0.0160145	+	0.0277379i	−0.857907	+	0.513805i	$0.828236\pi$
$740$		0			0
$741$	2.15235e9	+	2.95679e9i	0.194334	+	0.266967i
$742$		0			0
$743$	3.72945e9	+	6.45959e9i	0.333567	+	0.577756i	0.983209	−	0.182485i	$-0.0584142\pi$
$743$	3.72945e9	+	6.45959e9i	0.333567	+	0.577756i	−0.649641	+	0.760241i	$0.725081\pi$
$744$		0			0
$745$	1.71414e10	−	2.96897e10i	1.51879	−	2.63062i
$746$		0			0
$747$	1.12347e8	−	1.94591e8i	0.00986146	−	0.0170805i
$748$		0			0
$749$	3.31630e9			0.288382
$750$		0			0
$751$	−9.94993e8	+	1.72338e9i	−0.0857196	+	0.148471i	−0.905698	−	0.423924i	$-0.860652\pi$
$751$	−9.94993e8	+	1.72338e9i	−0.0857196	+	0.148471i	0.819978	+	0.572395i	$0.193986\pi$
$752$		0			0
$753$	8.26295e9			0.705266
$754$		0			0
$755$	1.22851e10	+	2.12784e10i	1.03888	+	1.79939i
$756$		0			0
$757$	3.98090e9	+	6.89512e9i	0.333538	+	0.577705i	0.983203	−	0.182516i	$-0.0584240\pi$
$757$	3.98090e9	+	6.89512e9i	0.333538	+	0.577705i	−0.649665	+	0.760221i	$0.725091\pi$
$758$		0			0
$759$	−1.49285e9			−0.123929
$760$		0			0
$761$	5.78040e9			0.475458			0.237729	−	0.971332i	$-0.423597\pi$
$761$	5.78040e9			0.475458			0.237729	+	0.971332i	$0.423597\pi$
$762$		0			0
$763$	−3.90783e9	−	6.76856e9i	−0.318493	−	0.551646i
$764$		0			0
$765$	−1.49491e8	−	2.58926e8i	−0.0120726	−	0.0209103i
$766$		0			0
$767$	1.78725e9			0.143021
$768$		0			0
$769$	7.57347e9	−	1.31176e10i	0.600555	−	1.04019i	−0.392182	−	0.919888i	$-0.628280\pi$
$769$	7.57347e9	−	1.31176e10i	0.600555	−	1.04019i	0.992737	−	0.120304i	$-0.0383870\pi$
$770$		0			0
$771$	−5.20529e9			−0.409030
$772$		0			0
$773$	8.91740e9	−	1.54454e10i	0.694400	−	1.20274i	−0.275982	−	0.961163i	$-0.589003\pi$
$773$	8.91740e9	−	1.54454e10i	0.694400	−	1.20274i	0.970382	−	0.241574i	$-0.0776636\pi$
$774$		0			0
$775$	1.40077e10	−	2.42620e10i	1.08096	−	1.87228i
$776$		0			0
$777$	1.82832e9	+	3.16675e9i	0.139823	+	0.242181i
$778$		0			0
$779$	−2.07215e9	+	4.66555e9i	−0.157051	+	0.353607i
$780$		0			0
$781$	−1.08227e9	−	1.87455e9i	−0.0812938	−	0.140805i
$782$		0			0
$783$	1.03121e10	−	1.78611e10i	0.767684	−	1.32967i
$784$		0			0
$785$	−1.17553e10	+	2.03608e10i	−0.867343	+	1.50228i
$786$		0			0
$787$	6.24456e8			0.0456657			0.0228329	−	0.999739i	$-0.492731\pi$
$787$	6.24456e8			0.0456657			0.0228329	+	0.999739i	$0.492731\pi$
$788$		0			0
$789$	1.27902e10	−	2.21533e10i	0.927059	−	1.60571i
$790$		0			0
$791$	2.17210e9			0.156049
$792$		0			0
$793$	−1.65638e9	−	2.86894e9i	−0.117952	−	0.204299i
$794$		0			0
$795$	1.10224e10	+	1.90914e10i	0.778022	+	1.34757i
$796$		0			0
$797$	−1.65367e10			−1.15703			−0.578516	−	0.815671i	$-0.696368\pi$
$797$	−1.65367e10			−1.15703			−0.578516	+	0.815671i	$0.696368\pi$
$798$		0			0
$799$	2.14519e9			0.148783
$800$		0			0
$801$	2.28393e8	+	3.95589e8i	0.0157025	+	0.0271976i
$802$		0			0
$803$	2.27325e9	+	3.93738e9i	0.154932	+	0.268351i
$804$		0			0
$805$	4.51107e9			0.304785
$806$		0			0
$807$	−1.22618e10	+	2.12381e10i	−0.821291	+	1.42252i
$808$		0			0
$809$	−8.33903e9			−0.553727			−0.276863	−	0.960909i	$-0.589295\pi$
$809$	−8.33903e9			−0.553727			−0.276863	+	0.960909i	$0.589295\pi$
$810$		0			0
$811$	6.50340e9	−	1.12642e10i	0.428122	−	0.741529i	−0.568585	−	0.822625i	$-0.692509\pi$
$811$	6.50340e9	−	1.12642e10i	0.428122	−	0.741529i	0.996706	+	0.0810962i	$0.0258421\pi$
$812$		0			0
$813$	4.61863e9	−	7.99970e9i	0.301437	−	0.522103i
$814$		0			0
$815$	2.49874e9	+	4.32794e9i	0.161685	+	0.280046i
$816$		0			0
$817$	−1.14990e10	+	2.58906e10i	−0.737704	+	1.66098i
$818$		0			0
$819$	7.20565e7	+	1.24806e8i	0.00458331	+	0.00793853i
$820$		0			0
$821$	8.08874e9	−	1.40101e10i	0.510129	−	0.883569i	−0.489802	−	0.871833i	$-0.662931\pi$
$821$	8.08874e9	−	1.40101e10i	0.510129	−	0.883569i	0.999931	−	0.0117353i	$-0.00373556\pi$
$822$		0			0
$823$	−7.06225e8	+	1.22322e9i	−0.0441615	+	0.0764899i	−0.887261	−	0.461267i	$-0.847395\pi$
$823$	−7.06225e8	+	1.22322e9i	−0.0441615	+	0.0764899i	0.843100	+	0.537757i	$0.180728\pi$
$824$		0			0
$825$	−7.96284e9			−0.493718
$826$		0			0
$827$	−1.49921e10	+	2.59671e10i	−0.921707	+	1.59644i	−0.124934	+	0.992165i	$0.539872\pi$
$827$	−1.49921e10	+	2.59671e10i	−0.921707	+	1.59644i	−0.796773	+	0.604279i	$0.793461\pi$
$828$		0			0
$829$	−5.64833e9			−0.344333			−0.172167	−	0.985068i	$-0.555077\pi$
$829$	−5.64833e9			−0.344333			−0.172167	+	0.985068i	$0.555077\pi$
$830$		0			0
$831$	1.35636e10	+	2.34928e10i	0.819917	+	1.42014i
$832$		0			0
$833$	1.84558e9	+	3.19664e9i	0.110631	+	0.191618i
$834$		0			0
$835$	−2.55296e10			−1.51754
$836$		0			0
$837$	2.68051e10			1.58007
$838$		0			0
$839$	−3.83395e9	−	6.64059e9i	−0.224119	−	0.388186i	0.731936	−	0.681374i	$-0.238617\pi$
$839$	−3.83395e9	−	6.64059e9i	−0.224119	−	0.388186i	−0.956055	+	0.293188i	$0.905284\pi$
$840$		0			0
$841$	−1.08150e10	−	1.87321e10i	−0.626961	−	1.08593i
$842$		0			0
$843$	−2.91236e10			−1.67436
$844$		0			0
$845$	−1.20271e10	+	2.08315e10i	−0.685744	+	1.18774i
$846$		0			0
$847$	8.61108e9			0.486929
$848$		0			0
$849$	−1.44841e9	+	2.50871e9i	−0.0812294	+	0.140693i
$850$		0			0
$851$	−1.61561e9	+	2.79831e9i	−0.0898632	+	0.155648i
$852$		0			0
$853$	−6.00149e9	−	1.03949e10i	−0.331083	−	0.573453i	0.651641	−	0.758527i	$-0.274081\pi$
$853$	−6.00149e9	−	1.03949e10i	−0.331083	−	0.573453i	−0.982725	+	0.185074i	$0.940747\pi$
$854$		0			0
$855$	−8.05366e8	−	1.10637e9i	−0.0440668	−	0.0605368i
$856$		0			0
$857$	3.27140e9	+	5.66624e9i	0.177542	+	0.307512i	0.941038	−	0.338301i	$-0.109852\pi$
$857$	3.27140e9	+	5.66624e9i	0.177542	+	0.307512i	−0.763496	+	0.645813i	$0.776519\pi$
$858$		0			0
$859$	7.30042e9	−	1.26447e10i	0.392981	−	0.680663i	−0.599860	−	0.800105i	$-0.704777\pi$
$859$	7.30042e9	−	1.26447e10i	0.392981	−	0.680663i	0.992841	+	0.119442i	$0.0381105\pi$
$860$		0			0
$861$	1.98012e9	−	3.42967e9i	0.105726	−	0.183122i
$862$		0			0
$863$	−1.73730e10			−0.920102			−0.460051	−	0.887893i	$-0.652169\pi$
$863$	−1.73730e10			−0.920102			−0.460051	+	0.887893i	$0.652169\pi$
$864$		0			0
$865$	−1.20300e10	+	2.08366e10i	−0.631989	+	1.09464i
$866$		0			0
$867$	−1.67735e10			−0.874092
$868$		0			0
$869$	3.74694e9	+	6.48989e9i	0.193690	+	0.335481i
$870$		0			0
$871$	2.53242e8	+	4.38629e8i	0.0129859	+	0.0224923i
$872$		0			0
$873$	−1.14210e9			−0.0580972
$874$		0			0
$875$	6.86603e9			0.346480
$876$		0			0
$877$	−1.26637e10	−	2.19342e10i	−0.633962	−	1.09805i	−0.986734	−	0.162345i	$-0.948094\pi$
$877$	−1.26637e10	−	2.19342e10i	−0.633962	−	1.09805i	0.352772	−	0.935709i	$-0.385239\pi$
$878$		0			0
$879$	4.77873e7	+	8.27700e7i	0.00237329	+	0.00411066i
$880$		0			0
$881$	1.85122e10			0.912099			0.456049	−	0.889955i	$-0.349264\pi$
$881$	1.85122e10			0.912099			0.456049	+	0.889955i	$0.349264\pi$
$882$		0			0
$883$	5.63509e9	−	9.76026e9i	0.275447	−	0.477088i	−0.694801	−	0.719202i	$-0.744507\pi$
$883$	5.63509e9	−	9.76026e9i	0.275447	−	0.477088i	0.970248	+	0.242114i	$0.0778408\pi$
$884$		0			0
$885$	1.31655e10			0.638464
$886$		0			0
$887$	−1.20525e10	+	2.08756e10i	−0.579890	+	1.00440i	0.415602	+	0.909547i	$0.363571\pi$
$887$	−1.20525e10	+	2.08756e10i	−0.579890	+	1.00440i	−0.995491	+	0.0948518i	$0.969762\pi$
$888$		0			0
$889$	4.78783e9	−	8.29277e9i	0.228551	−	0.395862i
$890$		0			0
$891$	−3.62484e9	−	6.27840e9i	−0.171679	−	0.297356i
$892$		0			0
$893$	9.76392e9	−	1.03522e9i	0.458822	−	0.0486465i
$894$		0			0
$895$	−1.05772e10	−	1.83203e10i	−0.493163	−	0.854184i
$896$		0			0
$897$	1.25351e9	−	2.17115e9i	0.0579903	−	0.100442i
$898$		0			0
$899$	2.52658e10	−	4.37616e10i	1.15978	−	2.00879i
$900$		0			0
$901$	7.29050e9			0.332063
$902$		0			0
$903$	1.09883e10	−	1.90323e10i	0.496620	−	0.860170i
$904$		0			0
$905$	−5.99527e10			−2.68868
$906$		0			0
$907$	7.17982e9	+	1.24358e10i	0.319513	+	0.553412i	0.980386	−	0.197085i	$-0.0631474\pi$
$907$	7.17982e9	+	1.24358e10i	0.319513	+	0.553412i	−0.660874	+	0.750497i	$0.729814\pi$
$908$		0			0
$909$	−7.20851e8	−	1.24855e9i	−0.0318326	−	0.0551356i
$910$		0			0
$911$	−3.38894e10			−1.48508			−0.742538	−	0.669803i	$-0.766378\pi$
$911$	−3.38894e10			−1.48508			−0.742538	+	0.669803i	$0.766378\pi$
$912$		0			0
$913$	−3.39346e9			−0.147569
$914$		0			0
$915$	−1.22015e10	−	2.11337e10i	−0.526551	−	0.912013i
$916$		0			0
$917$	−1.31978e9	−	2.28592e9i	−0.0565208	−	0.0978969i
$918$		0			0
$919$	−1.15820e10			−0.492243			−0.246122	−	0.969239i	$-0.579156\pi$
$919$	−1.15820e10			−0.492243			−0.246122	+	0.969239i	$0.579156\pi$
$920$		0			0
$921$	1.19681e10	−	2.07294e10i	0.504797	−	0.874334i
$922$		0			0
$923$	3.63502e9			0.152160
$924$		0			0
$925$	−8.61759e9	+	1.49261e10i	−0.358006	+	0.620084i
$926$		0			0
$927$	3.72249e8	−	6.44755e8i	0.0153481	−	0.0265837i
$928$		0			0
$929$	8.22688e9	+	1.42494e10i	0.336651	+	0.583097i	0.983801	−	0.179266i	$-0.0573723\pi$
$929$	8.22688e9	+	1.42494e10i	0.336651	+	0.583097i	−0.647149	+	0.762363i	$0.724039\pi$
$930$		0			0
$931$	9.94285e9	+	1.36590e10i	0.403819	+	0.554747i
$932$		0			0
$933$	1.00222e10	+	1.73590e10i	0.403998	+	0.699744i
$934$		0			0
$935$	−2.25769e9	+	3.91044e9i	−0.0903283	+	0.156453i
$936$		0			0
$937$	−3.41425e9	+	5.91366e9i	−0.135584	+	0.234838i	−0.925820	−	0.377964i	$-0.876624\pi$
$937$	−3.41425e9	+	5.91366e9i	−0.135584	+	0.234838i	0.790237	+	0.612802i	$0.209958\pi$
$938$		0			0
$939$	−1.68362e10			−0.663612
$940$		0			0
$941$	−4.51212e9	+	7.81522e9i	−0.176529	+	0.305758i	−0.940689	−	0.339269i	$-0.889820\pi$
$941$	−4.51212e9	+	7.81522e9i	−0.176529	+	0.305758i	0.764160	+	0.645027i	$0.223154\pi$
$942$		0			0
$943$	3.49949e9			0.135898
$944$		0			0
$945$	1.15112e10	+	1.99380e10i	0.443720	+	0.768545i
$946$		0			0
$947$	−8.80572e9	−	1.52519e10i	−0.336930	−	0.583580i	0.646924	−	0.762555i	$-0.276055\pi$
$947$	−8.80572e9	−	1.52519e10i	−0.336930	−	0.583580i	−0.983854	+	0.178975i	$0.942722\pi$
$948$		0			0
$949$	−7.63515e9			−0.289992
$950$		0			0
$951$	−4.76329e10			−1.79587
$952$		0			0
$953$	−4.31832e9	−	7.47955e9i	−0.161618	−	0.279930i	0.773831	−	0.633392i	$-0.218338\pi$
$953$	−4.31832e9	−	7.47955e9i	−0.161618	−	0.279930i	−0.935449	+	0.353462i	$0.885005\pi$
$954$		0			0
$955$	3.64840e10	+	6.31922e10i	1.35547	+	2.34775i
$956$		0			0
$957$	−1.43626e10			−0.529716
$958$		0			0
$959$	4.49609e8	−	7.78746e8i	0.0164615	−	0.0285122i
$960$		0			0
$961$	3.81625e10			1.38709
$962$		0			0
$963$	−3.44813e8	+	5.97234e8i	−0.0124420	+	0.0215502i
$964$		0			0
$965$	2.02943e9	−	3.51508e9i	0.0726990	−	0.125918i
$966$		0			0
$967$	2.37440e10	+	4.11258e10i	0.844424	+	1.46259i	0.886121	+	0.463455i	$0.153390\pi$
$967$	2.37440e10	+	4.11258e10i	0.844424	+	1.46259i	−0.0416966	+	0.999130i	$0.513276\pi$
$968$		0			0
$969$	8.85987e9	−	9.39366e8i	0.312820	−	0.0331666i
$970$		0			0
$971$	−1.02801e10	−	1.78057e10i	−0.360355	−	0.624154i	0.627664	−	0.778485i	$-0.284011\pi$
$971$	−1.02801e10	−	1.78057e10i	−0.360355	−	0.624154i	−0.988019	+	0.154331i	$0.950678\pi$
$972$		0			0
$973$	−9.42208e9	+	1.63195e10i	−0.327908	+	0.567953i
$974$		0			0
$975$	6.68620e9	−	1.15808e10i	0.231027	−	0.400151i
$976$		0			0
$977$	−3.53009e10			−1.21103			−0.605514	−	0.795835i	$-0.707032\pi$
$977$	−3.53009e10			−1.21103			−0.605514	+	0.795835i	$0.707032\pi$
$978$		0			0
$979$	3.44932e9	−	5.97440e9i	0.117488	−	0.203495i
$980$		0			0
$981$	1.62527e9			0.0549647
$982$		0			0
$983$	7.84352e9	+	1.35854e10i	0.263374	+	0.456178i	0.967136	−	0.254258i	$-0.0818312\pi$
$983$	7.84352e9	+	1.35854e10i	0.263374	+	0.456178i	−0.703762	+	0.710436i	$0.748498\pi$
$984$		0			0
$985$	2.07917e10	+	3.60124e10i	0.693209	+	1.20067i
$986$		0			0
$987$	−7.61688e9			−0.252155
$988$		0			0
$989$	1.94197e10			0.638346
$990$		0			0
$991$	−1.82705e10	−	3.16455e10i	−0.596339	−	1.03289i	−0.993356	−	0.115078i	$-0.963288\pi$
$991$	−1.82705e10	−	3.16455e10i	−0.596339	−	1.03289i	0.397018	−	0.917811i	$-0.370045\pi$
$992$		0			0
$993$	−7.76446e9	−	1.34484e10i	−0.251645	−	0.435863i
$994$		0			0
$995$	5.10001e10			1.64131
$996$		0			0
$997$	−1.75026e10	+	3.03154e10i	−0.559331	+	0.968790i	0.438221	+	0.898867i	$0.355609\pi$
$997$	−1.75026e10	+	3.03154e10i	−0.559331	+	0.968790i	−0.997552	+	0.0699232i	$0.977725\pi$
$998$		0			0
$999$	−1.64906e10			−0.523308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.45.3	✓	22
19.11	even	3	inner	76.8.e.a.49.3	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.3	✓	22	1.1	even	1	trivial
76.8.e.a.49.3	yes	22	19.11	even	3	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-22.8105 - 39.5090i) q^{3} +(216.474 + 374.944i) q^{5} -508.391 q^{7} +(52.8600 - 91.5562i) q^{9} +O(q^{10})\)	\(q+(-22.8105 - 39.5090i) q^{3} +(216.474 + 374.944i) q^{5} -508.391 q^{7} +(52.8600 - 91.5562i) q^{9} -1596.64 q^{11} +(1340.66 - 2322.09i) q^{13} +(9875.77 - 17105.3i) q^{15} +(-3266.04 - 5656.95i) q^{17} +(-17595.4 - 24171.7i) q^{19} +(11596.7 + 20086.0i) q^{21} +(-10247.4 + 17749.1i) q^{23} +(-54659.5 + 94673.1i) q^{25} -104596. q^{27} +(-98589.9 + 170763. i) q^{29} -256272. q^{31} +(36420.2 + 63081.6i) q^{33} +(-110053. - 190618. i) q^{35} +157659. q^{37} -122324. q^{39} +(-85374.6 - 147873. i) q^{41} +(-473770. - 820594. i) q^{43} +45771.3 q^{45} +(-164204. + 284410. i) q^{47} -565082. q^{49} +(-149000. + 258076. i) q^{51} +(-558053. + 966577. i) q^{53} +(-345631. - 598651. i) q^{55} +(-553639. + 1.24655e6i) q^{57} +(333278. + 577255. i) q^{59} +(617751. - 1.06998e6i) q^{61} +(-26873.6 + 46546.4i) q^{63} +1.16087e6 q^{65} +(-94447.0 + 163587. i) q^{67} +934998. q^{69} +(677843. + 1.17406e6i) q^{71} +(-1.42377e6 - 2.46604e6i) q^{73} +4.98725e6 q^{75} +811717. q^{77} +(-2.34677e6 - 4.06472e6i) q^{79} +(2.27029e6 + 3.93226e6i) q^{81} +2.12538e6 q^{83} +(1.41403e6 - 2.44917e6i) q^{85} +8.99555e6 q^{87} +(-2.16036e6 + 3.74186e6i) q^{89} +(-681579. + 1.18053e6i) q^{91} +(5.84569e6 + 1.01250e7i) q^{93} +(5.25409e6 - 1.18299e7i) q^{95} +(-5.40154e6 - 9.35575e6i) q^{97} +(-84398.4 + 146182. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

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Database

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