LMFDB - Embedded newform 192.4.c.c.191.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("192.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, 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:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.4
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.4.c.c.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.89898 + 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 + 16.9706i) q^{9} +O(q^{10})$	\(q+(4.89898 + 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 + 16.9706i) q^{9} +29.3939 q^{11} +26.0000 q^{13} +(29.3939 - 83.1384i) q^{15} -67.8823i q^{17} -107.387i q^{19} +(-30.0000 + 84.8528i) q^{21} +176.363 q^{23} -163.000 q^{25} +(73.4847 + 119.512i) q^{27} +16.9706i q^{29} +31.1769i q^{31} +(144.000 + 50.9117i) q^{33} +293.939 q^{35} -206.000 q^{37} +(127.373 + 45.0333i) q^{39} +305.470i q^{41} -93.5307i q^{43} +(288.000 - 356.382i) q^{45} -117.576 q^{47} +43.0000 q^{49} +(117.576 - 332.554i) q^{51} +50.9117i q^{53} -498.831i q^{55} +(186.000 - 526.087i) q^{57} -558.484 q^{59} -278.000 q^{61} +(-293.939 + 363.731i) q^{63} -441.235i q^{65} +890.274i q^{67} +(864.000 + 305.470i) q^{69} +58.7878 q^{71} -422.000 q^{73} +(-798.534 - 282.324i) q^{75} +509.117i q^{77} +668.572i q^{79} +(153.000 + 712.764i) q^{81} -29.3939 q^{83} -1152.00 q^{85} +(-29.3939 + 83.1384i) q^{87} +373.352i q^{89} +450.333i q^{91} +(-54.0000 + 152.735i) q^{93} -1822.42 q^{95} -1070.00 q^{97} +(617.271 + 498.831i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 84 q^{9}+O(q^{10}) $ 4 * q + 84 * q^9	$ 4 q + 84 q^{9} + 104 q^{13} - 120 q^{21} - 652 q^{25} + 576 q^{33} - 824 q^{37} + 1152 q^{45} + 172 q^{49} + 744 q^{57} - 1112 q^{61} + 3456 q^{69} - 1688 q^{73} + 612 q^{81} - 4608 q^{85} - 216 q^{93} - 4280 q^{97}+O(q^{100}) $ 4 * q + 84 * q^9 + 104 * q^13 - 120 * q^21 - 652 * q^25 + 576 * q^33 - 824 * q^37 + 1152 * q^45 + 172 * q^49 + 744 * q^57 - 1112 * q^61 + 3456 * q^69 - 1688 * q^73 + 612 * q^81 - 4608 * q^85 - 216 * q^93 - 4280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	4.89898	+	1.73205i	0.942809	+	0.333333i
$3$	4.89898	+	1.73205i	0.942809	+	0.333333i
$4$		0			0
$5$		−	16.9706i		−	1.51789i	−0.651153	−	0.758947i	$-0.725714\pi$
$5$		−	16.9706i		−	1.51789i	0.651153	−	0.758947i	$-0.274286\pi$
$6$		0			0
$7$			17.3205i			0.935220i	0.883935	+	0.467610i	$0.154885\pi$
$7$			17.3205i			0.935220i	−0.883935	+	0.467610i	$0.845115\pi$
$8$		0			0
$9$	21.0000	+	16.9706i	0.777778	+	0.628539i
$10$		0			0
$11$	29.3939			0.805690			0.402845	−	0.915268i	$-0.368021\pi$
$11$	29.3939			0.805690			0.402845	+	0.915268i	$0.368021\pi$
$12$		0			0
$13$	26.0000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	29.3939	−	83.1384i	0.505964	−	1.43108i
$16$		0			0
$17$		−	67.8823i		−	0.968463i	−0.874940	−	0.484231i	$-0.839099\pi$
$17$		−	67.8823i		−	0.968463i	0.874940	−	0.484231i	$-0.160901\pi$
$18$		0			0
$19$		−	107.387i		−	1.29665i	−0.761365	−	0.648324i	$-0.775470\pi$
$19$		−	107.387i		−	1.29665i	0.761365	−	0.648324i	$-0.224530\pi$
$20$		0			0
$21$	−30.0000	+	84.8528i	−0.311740	+	0.881733i
$22$		0			0
$23$	176.363			1.59888			0.799441	−	0.600745i	$-0.205129\pi$
$23$	176.363			1.59888			0.799441	+	0.600745i	$0.205129\pi$
$24$		0			0
$25$	−163.000			−1.30400
$26$		0			0
$27$	73.4847	+	119.512i	0.523783	+	0.851852i
$28$		0			0
$29$			16.9706i			0.108667i	0.998523	+	0.0543337i	$0.0173035\pi$
$29$			16.9706i			0.108667i	−0.998523	+	0.0543337i	$0.982697\pi$
$30$		0			0
$31$			31.1769i			0.180630i	0.995913	+	0.0903151i	$0.0287874\pi$
$31$			31.1769i			0.180630i	−0.995913	+	0.0903151i	$0.971213\pi$
$32$		0			0
$33$	144.000	+	50.9117i	0.759612	+	0.268563i
$34$		0			0
$35$	293.939			1.41956
$36$		0			0
$37$	−206.000			−0.915302			−0.457651	−	0.889132i	$-0.651309\pi$
$37$	−206.000			−0.915302			−0.457651	+	0.889132i	$0.651309\pi$
$38$		0			0
$39$	127.373	+	45.0333i	0.522976	+	0.184900i
$40$		0			0
$41$			305.470i			1.16357i	0.813342	+	0.581786i	$0.197646\pi$
$41$			305.470i			1.16357i	−0.813342	+	0.581786i	$0.802354\pi$
$42$		0			0
$43$		−	93.5307i		−	0.331705i	−0.986151	−	0.165852i	$-0.946962\pi$
$43$		−	93.5307i		−	0.331705i	0.986151	−	0.165852i	$-0.0530375\pi$
$44$		0			0
$45$	288.000	−	356.382i	0.954056	−	1.18058i
$46$		0			0
$47$	−117.576			−0.364897			−0.182448	−	0.983215i	$-0.558402\pi$
$47$	−117.576			−0.364897			−0.182448	+	0.983215i	$0.558402\pi$
$48$		0			0
$49$	43.0000			0.125364
$50$		0			0
$51$	117.576	−	332.554i	0.322821	−	0.913075i
$52$		0			0
$53$			50.9117i			0.131948i	0.997821	+	0.0659741i	$0.0210155\pi$
$53$			50.9117i			0.131948i	−0.997821	+	0.0659741i	$0.978985\pi$
$54$		0			0
$55$		−	498.831i		−	1.22295i
$56$		0			0
$57$	186.000	−	526.087i	0.432216	−	1.22249i
$58$		0			0
$59$	−558.484			−1.23235			−0.616173	−	0.787611i	$-0.711318\pi$
$59$	−558.484			−1.23235			−0.616173	+	0.787611i	$0.711318\pi$
$60$		0			0
$61$	−278.000			−0.583512			−0.291756	−	0.956493i	$-0.594240\pi$
$61$	−278.000			−0.583512			−0.291756	+	0.956493i	$0.594240\pi$
$62$		0			0
$63$	−293.939	+	363.731i	−0.587822	+	0.727393i
$64$		0			0
$65$		−	441.235i		−	0.841976i
$66$		0			0
$67$			890.274i			1.62335i	0.584111	+	0.811674i	$0.301443\pi$
$67$			890.274i			1.62335i	−0.584111	+	0.811674i	$0.698557\pi$
$68$		0			0
$69$	864.000	+	305.470i	1.50744	+	0.532961i
$70$		0			0
$71$	58.7878			0.0982651			0.0491326	−	0.998792i	$-0.484354\pi$
$71$	58.7878			0.0982651			0.0491326	+	0.998792i	$0.484354\pi$
$72$		0			0
$73$	−422.000			−0.676594			−0.338297	−	0.941039i	$-0.609851\pi$
$73$	−422.000			−0.676594			−0.338297	+	0.941039i	$0.609851\pi$
$74$		0			0
$75$	−798.534	−	282.324i	−1.22942	−	0.434667i
$76$		0			0
$77$			509.117i			0.753497i
$78$		0			0
$79$			668.572i			0.952154i	0.879404	+	0.476077i	$0.157942\pi$
$79$			668.572i			0.952154i	−0.879404	+	0.476077i	$0.842058\pi$
$80$		0			0
$81$	153.000	+	712.764i	0.209877	+	0.977728i
$82$		0			0
$83$	−29.3939			−0.0388723			−0.0194361	−	0.999811i	$-0.506187\pi$
$83$	−29.3939			−0.0388723			−0.0194361	+	0.999811i	$0.506187\pi$
$84$		0			0
$85$	−1152.00			−1.47002
$86$		0			0
$87$	−29.3939	+	83.1384i	−0.0362225	+	0.102453i
$88$		0			0
$89$			373.352i			0.444666i	0.974971	+	0.222333i	$0.0713672\pi$
$89$			373.352i			0.444666i	−0.974971	+	0.222333i	$0.928633\pi$
$90$		0			0
$91$			450.333i			0.518766i
$92$		0			0
$93$	−54.0000	+	152.735i	−0.0602101	+	0.170300i
$94$		0			0
$95$	−1822.42			−1.96817
$96$		0			0
$97$	−1070.00			−1.12002			−0.560011	−	0.828486i	$-0.689203\pi$
$97$	−1070.00			−1.12002			−0.560011	+	0.828486i	$0.689203\pi$
$98$		0			0
$99$	617.271	+	498.831i	0.626648	+	0.506408i
$100$		0			0
$101$		−	1781.91i		−	1.75551i	−0.479109	−	0.877755i	$-0.659040\pi$
$101$		−	1781.91i		−	1.75551i	0.479109	−	0.877755i	$-0.340960\pi$
$102$		0			0
$103$			17.3205i			0.0165693i	0.999966	+	0.00828466i	$0.00263712\pi$
$103$			17.3205i			0.0165693i	−0.999966	+	0.00828466i	$0.997363\pi$
$104$		0			0
$105$	1440.00	+	509.117i	1.33838	+	0.473188i
$106$		0			0
$107$	−1381.51			−1.24819			−0.624093	−	0.781350i	$-0.714531\pi$
$107$	−1381.51			−1.24819			−0.624093	+	0.781350i	$0.714531\pi$
$108$		0			0
$109$	−358.000			−0.314589			−0.157294	−	0.987552i	$-0.550277\pi$
$109$	−358.000			−0.314589			−0.157294	+	0.987552i	$0.550277\pi$
$110$		0			0
$111$	−1009.19	−	356.802i	−0.862955	−	0.305101i
$112$		0			0
$113$		−	67.8823i		−	0.0565117i	−0.999601	−	0.0282559i	$-0.991005\pi$
$113$		−	67.8823i		−	0.0565117i	0.999601	−	0.0282559i	$-0.00899532\pi$
$114$		0			0
$115$		−	2992.98i		−	2.42693i
$116$		0			0
$117$	546.000	+	441.235i	0.431433	+	0.348651i
$118$		0			0
$119$	1175.76			0.905725
$120$		0			0
$121$	−467.000			−0.350864
$122$		0			0
$123$	−529.090	+	1496.49i	−0.387857	+	1.09703i
$124$		0			0
$125$			644.881i			0.461440i
$126$		0			0
$127$			31.1769i			0.0217835i	0.999941	+	0.0108917i	$0.00346702\pi$
$127$			31.1769i			0.0217835i	−0.999941	+	0.0108917i	$0.996533\pi$
$128$		0			0
$129$	162.000	−	458.205i	0.110568	−	0.312734i
$130$		0			0
$131$	1734.24			1.15665			0.578325	−	0.815806i	$-0.303707\pi$
$131$	1734.24			1.15665			0.578325	+	0.815806i	$0.303707\pi$
$132$		0			0
$133$	1860.00			1.21265
$134$		0			0
$135$	2028.18	−	1247.08i	1.29302	−	0.795046i
$136$		0			0
$137$		−	2274.06i		−	1.41814i	−0.705136	−	0.709072i	$-0.749114\pi$
$137$		−	2274.06i		−	1.41814i	0.705136	−	0.709072i	$-0.250886\pi$
$138$		0			0
$139$			1541.53i			0.940651i	0.882493	+	0.470325i	$0.155863\pi$
$139$			1541.53i			0.940651i	−0.882493	+	0.470325i	$0.844137\pi$
$140$		0			0
$141$	−576.000	−	203.647i	−0.344028	−	0.121632i
$142$		0			0
$143$	764.241			0.446916
$144$		0			0
$145$	288.000			0.164946
$146$		0			0
$147$	210.656	+	74.4782i	0.118195	+	0.0417881i
$148$		0			0
$149$			2358.91i			1.29698i	0.761225	+	0.648488i	$0.224598\pi$
$149$			2358.91i			1.29698i	−0.761225	+	0.648488i	$0.775402\pi$
$150$		0			0
$151$			2040.36i			1.09961i	0.835292	+	0.549807i	$0.185299\pi$
$151$			2040.36i			1.09961i	−0.835292	+	0.549807i	$0.814701\pi$
$152$		0			0
$153$	1152.00	−	1425.53i	0.608717	−	0.753249i
$154$		0			0
$155$	529.090			0.274178
$156$		0			0
$157$	106.000			0.0538836			0.0269418	−	0.999637i	$-0.491423\pi$
$157$	106.000			0.0538836			0.0269418	+	0.999637i	$0.491423\pi$
$158$		0			0
$159$	−88.1816	+	249.415i	−0.0439828	+	0.124402i
$160$		0			0
$161$			3054.70i			1.49531i
$162$		0			0
$163$		−	79.6743i		−	0.0382857i	−0.999817	−	0.0191429i	$-0.993906\pi$
$163$		−	79.6743i		−	0.0382857i	0.999817	−	0.0191429i	$-0.00609374\pi$
$164$		0			0
$165$	864.000	−	2443.76i	0.407650	−	1.15301i
$166$		0			0
$167$	3115.75			1.44374			0.721868	−	0.692030i	$-0.243284\pi$
$167$	3115.75			1.44374			0.721868	+	0.692030i	$0.243284\pi$
$168$		0			0
$169$	−1521.00			−0.692308
$170$		0			0
$171$	1822.42	−	2255.13i	0.814994	−	1.00850i
$172$		0			0
$173$		−	2223.14i		−	0.977009i	−0.872561	−	0.488504i	$-0.837543\pi$
$173$		−	2223.14i		−	0.977009i	0.872561	−	0.488504i	$-0.162457\pi$
$174$		0			0
$175$		−	2823.24i		−	1.21953i
$176$		0			0
$177$	−2736.00	−	967.322i	−1.16187	−	0.410782i
$178$		0			0
$179$	2792.42			1.16601			0.583003	−	0.812470i	$-0.301878\pi$
$179$	2792.42			1.16601			0.583003	+	0.812470i	$0.301878\pi$
$180$		0			0
$181$	−4510.00			−1.85208			−0.926038	−	0.377431i	$-0.876808\pi$
$181$	−4510.00			−1.85208			−0.926038	+	0.377431i	$0.876808\pi$
$182$		0			0
$183$	−1361.92	−	481.510i	−0.550141	−	0.194504i
$184$		0			0
$185$			3495.94i			1.38933i
$186$		0			0
$187$		−	1995.32i		−	0.780280i
$188$		0			0
$189$	−2070.00	+	1272.79i	−0.796668	+	0.489852i
$190$		0			0
$191$	−1881.21			−0.712667			−0.356334	−	0.934359i	$-0.615973\pi$
$191$	−1881.21			−0.712667			−0.356334	+	0.934359i	$0.615973\pi$
$192$		0			0
$193$	4994.00			1.86257			0.931285	−	0.364292i	$-0.118689\pi$
$193$	4994.00			1.86257			0.931285	+	0.364292i	$0.118689\pi$
$194$		0			0
$195$	764.241	−	2161.60i	0.280659	−	0.793822i
$196$		0			0
$197$			2155.26i			0.779472i	0.920927	+	0.389736i	$0.127434\pi$
$197$			2155.26i			0.779472i	−0.920927	+	0.389736i	$0.872566\pi$
$198$		0			0
$199$		−	3973.32i		−	1.41538i	−0.706521	−	0.707692i	$-0.749736\pi$
$199$		−	3973.32i		−	1.41538i	0.706521	−	0.707692i	$-0.250264\pi$
$200$		0			0
$201$	−1542.00	+	4361.43i	−0.541116	+	1.53051i
$202$		0			0
$203$	−293.939			−0.101628
$204$		0			0
$205$	5184.00			1.76618
$206$		0			0
$207$	3703.63	+	2992.98i	1.24357	+	1.00496i
$208$		0			0
$209$		−	3156.52i		−	1.04470i
$210$		0			0
$211$			4908.63i			1.60154i	0.598974	+	0.800768i	$0.295575\pi$
$211$			4908.63i			1.60154i	−0.598974	+	0.800768i	$0.704425\pi$
$212$		0			0
$213$	288.000	+	101.823i	0.0926452	+	0.0327550i
$214$		0			0
$215$	−1587.27			−0.503492
$216$		0			0
$217$	−540.000			−0.168929
$218$		0			0
$219$	−2067.37	−	730.925i	−0.637899	−	0.225531i
$220$		0			0
$221$		−	1764.94i		−	0.537206i
$222$		0			0
$223$			4021.82i			1.20772i	0.797091	+	0.603859i	$0.206371\pi$
$223$			4021.82i			1.20772i	−0.797091	+	0.603859i	$0.793629\pi$
$224$		0			0
$225$	−3423.00	−	2766.20i	−1.01422	−	0.819615i
$226$		0			0
$227$	−2968.78			−0.868039			−0.434020	−	0.900903i	$-0.642905\pi$
$227$	−2968.78			−0.868039			−0.434020	+	0.900903i	$0.642905\pi$
$228$		0			0
$229$	−430.000			−0.124084			−0.0620419	−	0.998074i	$-0.519761\pi$
$229$	−430.000			−0.124084			−0.0620419	+	0.998074i	$0.519761\pi$
$230$		0			0
$231$	−881.816	+	2494.15i	−0.251166	+	0.710404i
$232$		0			0
$233$		−	6346.99i		−	1.78457i	−0.451471	−	0.892286i	$-0.649101\pi$
$233$		−	6346.99i		−	1.78457i	0.451471	−	0.892286i	$-0.350899\pi$
$234$		0			0
$235$			1995.32i			0.553874i
$236$		0			0
$237$	−1158.00	+	3275.32i	−0.317385	+	0.897700i
$238$		0			0
$239$	4115.14			1.11375			0.556875	−	0.830596i	$-0.312000\pi$
$239$	4115.14			1.11375			0.556875	+	0.830596i	$0.312000\pi$
$240$		0			0
$241$	4690.00			1.25357			0.626783	−	0.779194i	$-0.284371\pi$
$241$	4690.00			1.25357			0.626783	+	0.779194i	$0.284371\pi$
$242$		0			0
$243$	−484.999	+	3756.82i	−0.128036	+	0.991770i
$244$		0			0
$245$		−	729.734i		−	0.190290i
$246$		0			0
$247$		−	2792.07i		−	0.719251i
$248$		0			0
$249$	−144.000	−	50.9117i	−0.0366491	−	0.0129574i
$250$		0			0
$251$	−1263.94			−0.317845			−0.158922	−	0.987291i	$-0.550802\pi$
$251$	−1263.94			−0.317845			−0.158922	+	0.987291i	$0.550802\pi$
$252$		0			0
$253$	5184.00			1.28820
$254$		0			0
$255$	−5643.62	−	1995.32i	−1.38595	−	0.490008i
$256$		0			0
$257$			271.529i			0.0659047i	0.999457	+	0.0329524i	$0.0104910\pi$
$257$			271.529i			0.0659047i	−0.999457	+	0.0329524i	$0.989509\pi$
$258$		0			0
$259$		−	3568.02i		−	0.856009i
$260$		0			0
$261$	−288.000	+	356.382i	−0.0683017	+	0.0845191i
$262$		0			0
$263$	−5114.53			−1.19915			−0.599574	−	0.800320i	$-0.704663\pi$
$263$	−5114.53			−1.19915			−0.599574	+	0.800320i	$0.704663\pi$
$264$		0			0
$265$	864.000			0.200283
$266$		0			0
$267$	−646.665	+	1829.05i	−0.148222	+	0.419235i
$268$		0			0
$269$		−	3173.50i		−	0.719299i	−0.933087	−	0.359649i	$-0.882896\pi$
$269$		−	3173.50i		−	0.719299i	0.933087	−	0.359649i	$-0.117104\pi$
$270$		0			0
$271$			668.572i			0.149863i	0.997189	+	0.0749314i	$0.0238738\pi$
$271$			668.572i			0.149863i	−0.997189	+	0.0749314i	$0.976126\pi$
$272$		0			0
$273$	−780.000	+	2206.17i	−0.172922	+	0.489098i
$274$		0			0
$275$	−4791.20			−1.05062
$276$		0			0
$277$	2018.00			0.437725			0.218863	−	0.975756i	$-0.429765\pi$
$277$	2018.00			0.437725			0.218863	+	0.975756i	$0.429765\pi$
$278$		0			0
$279$	−529.090	+	654.715i	−0.113533	+	0.140490i
$280$		0			0
$281$			4717.82i			1.00157i	0.865572	+	0.500785i	$0.166955\pi$
$281$			4717.82i			1.00157i	−0.865572	+	0.500785i	$0.833045\pi$
$282$		0			0
$283$		−	5081.84i		−	1.06743i	−0.845663	−	0.533717i	$-0.820795\pi$
$283$		−	5081.84i		−	1.06743i	0.845663	−	0.533717i	$-0.179205\pi$
$284$		0			0
$285$	−8928.00	−	3156.52i	−1.85561	−	0.656057i
$286$		0			0
$287$	−5290.90			−1.08819
$288$		0			0
$289$	305.000			0.0620802
$290$		0			0
$291$	−5241.91	−	1853.29i	−1.05597	−	0.373340i
$292$		0			0
$293$		−	424.264i		−	0.0845931i	−0.999105	−	0.0422965i	$-0.986533\pi$
$293$		−	424.264i		−	0.0845931i	0.999105	−	0.0422965i	$-0.0134674\pi$
$294$		0			0
$295$			9477.78i			1.87057i
$296$		0			0
$297$	2160.00	+	3512.91i	0.422006	+	0.686328i
$298$		0			0
$299$	4585.44			0.886900
$300$		0			0
$301$	1620.00			0.310217
$302$		0			0
$303$	3086.36	−	8729.54i	0.585170	−	1.65511i
$304$		0			0
$305$			4717.82i			0.885709i
$306$		0			0
$307$			3522.99i			0.654944i	0.944861	+	0.327472i	$0.106197\pi$
$307$			3522.99i			0.654944i	−0.944861	+	0.327472i	$0.893803\pi$
$308$		0			0
$309$	−30.0000	+	84.8528i	−0.00552311	+	0.0156217i
$310$		0			0
$311$	−9464.83			−1.72573			−0.862864	−	0.505437i	$-0.831331\pi$
$311$	−9464.83			−1.72573			−0.862864	+	0.505437i	$0.831331\pi$
$312$		0			0
$313$	−3718.00			−0.671418			−0.335709	−	0.941966i	$-0.608976\pi$
$313$	−3718.00			−0.671418			−0.335709	+	0.941966i	$0.608976\pi$
$314$		0			0
$315$	6172.71	+	4988.31i	1.10410	+	0.892251i
$316$		0			0
$317$			7212.49i			1.27790i	0.769249	+	0.638949i	$0.220631\pi$
$317$			7212.49i			1.27790i	−0.769249	+	0.638949i	$0.779369\pi$
$318$		0			0
$319$			498.831i			0.0875522i
$320$		0			0
$321$	−6768.00	−	2392.85i	−1.17680	−	0.416062i
$322$		0			0
$323$	−7289.68			−1.25575
$324$		0			0
$325$	−4238.00			−0.723329
$326$		0			0
$327$	−1753.83	−	620.074i	−0.296597	−	0.104863i
$328$		0			0
$329$		−	2036.47i		−	0.341259i
$330$		0			0
$331$			1541.53i			0.255982i	0.991775	+	0.127991i	$0.0408528\pi$
$331$			1541.53i			0.255982i	−0.991775	+	0.127991i	$0.959147\pi$
$332$		0			0
$333$	−4326.00	−	3495.94i	−0.711902	−	0.575304i
$334$		0			0
$335$	15108.5			2.46407
$336$		0			0
$337$	2530.00			0.408955			0.204478	−	0.978871i	$-0.434450\pi$
$337$	2530.00			0.408955			0.204478	+	0.978871i	$0.434450\pi$
$338$		0			0
$339$	117.576	−	332.554i	0.0188372	−	0.0532798i
$340$		0			0
$341$			916.410i			0.145532i
$342$		0			0
$343$			6685.72i			1.05246i
$344$		0			0
$345$	5184.00	−	14662.6i	0.808977	−	2.28813i
$346$		0			0
$347$	5555.44			0.859458			0.429729	−	0.902958i	$-0.358609\pi$
$347$	5555.44			0.859458			0.429729	+	0.902958i	$0.358609\pi$
$348$		0			0
$349$	7786.00			1.19420			0.597099	−	0.802168i	$-0.296320\pi$
$349$	7786.00			1.19420			0.597099	+	0.802168i	$0.296320\pi$
$350$		0			0
$351$	1910.60	+	3107.30i	0.290542	+	0.472522i
$352$		0			0
$353$			407.294i			0.0614109i	0.999528	+	0.0307054i	$0.00977538\pi$
$353$			407.294i			0.0614109i	−0.999528	+	0.0307054i	$0.990225\pi$
$354$		0			0
$355$		−	997.661i		−	0.149156i
$356$		0			0
$357$	5760.00	+	2036.47i	0.853926	+	0.301908i
$358$		0			0
$359$	8759.38			1.28775			0.643875	−	0.765131i	$-0.277326\pi$
$359$	8759.38			1.28775			0.643875	+	0.765131i	$0.277326\pi$
$360$		0			0
$361$	−4673.00			−0.681295
$362$		0			0
$363$	−2287.82	−	808.868i	−0.330798	−	0.116955i
$364$		0			0
$365$			7161.58i			1.02700i
$366$		0			0
$367$		−	12578.2i		−	1.78903i	−0.447037	−	0.894515i	$-0.647521\pi$
$367$		−	12578.2i		−	1.78903i	0.447037	−	0.894515i	$-0.352479\pi$
$368$		0			0
$369$	−5184.00	+	6414.87i	−0.731350	+	0.905000i
$370$		0			0
$371$	−881.816			−0.123401
$372$		0			0
$373$	4258.00			0.591075			0.295537	−	0.955331i	$-0.404501\pi$
$373$	4258.00			0.591075			0.295537	+	0.955331i	$0.404501\pi$
$374$		0			0
$375$	−1116.97	+	3159.26i	−0.153813	+	0.435049i
$376$		0			0
$377$			441.235i			0.0602778i
$378$		0			0
$379$		−	2726.25i		−	0.369493i	−0.982786	−	0.184747i	$-0.940854\pi$
$379$		−	2726.25i		−	0.369493i	0.982786	−	0.184747i	$-0.0591465\pi$
$380$		0			0
$381$	−54.0000	+	152.735i	−0.00726116	+	0.0205377i
$382$		0			0
$383$	−4232.72			−0.564704			−0.282352	−	0.959311i	$-0.591115\pi$
$383$	−4232.72			−0.564704			−0.282352	+	0.959311i	$0.591115\pi$
$384$		0			0
$385$	8640.00			1.14373
$386$		0			0
$387$	1587.27	−	1964.15i	0.208489	−	0.257993i
$388$		0			0
$389$		−	5447.55i		−	0.710030i	−0.934861	−	0.355015i	$-0.884476\pi$
$389$		−	5447.55i		−	0.710030i	0.934861	−	0.355015i	$-0.115524\pi$
$390$		0			0
$391$		−	11971.9i		−	1.54846i
$392$		0			0
$393$	8496.00	+	3003.79i	1.09050	+	0.385550i
$394$		0			0
$395$	11346.0			1.44527
$396$		0			0
$397$	−13574.0			−1.71602			−0.858009	−	0.513634i	$-0.828299\pi$
$397$	−13574.0			−1.71602			−0.858009	+	0.513634i	$0.828299\pi$
$398$		0			0
$399$	9112.10	+	3221.61i	1.14330	+	0.404217i
$400$		0			0
$401$		−	11743.6i		−	1.46247i	−0.682128	−	0.731233i	$-0.738945\pi$
$401$		−	11743.6i		−	1.46247i	0.682128	−	0.731233i	$-0.261055\pi$
$402$		0			0
$403$			810.600i			0.100196i
$404$		0			0
$405$	12096.0	−	2596.50i	1.48409	−	0.318570i
$406$		0			0
$407$	−6055.14			−0.737450
$408$		0			0
$409$	2890.00			0.349392			0.174696	−	0.984622i	$-0.444106\pi$
$409$	2890.00			0.349392			0.174696	+	0.984622i	$0.444106\pi$
$410$		0			0
$411$	3938.78	−	11140.6i	0.472715	−	1.33704i
$412$		0			0
$413$		−	9673.22i		−	1.15251i
$414$		0			0
$415$			498.831i			0.0590039i
$416$		0			0
$417$	−2670.00	+	7551.90i	−0.313550	+	0.886854i
$418$		0			0
$419$	8318.47			0.969890			0.484945	−	0.874545i	$-0.338840\pi$
$419$	8318.47			0.969890			0.484945	+	0.874545i	$0.338840\pi$
$420$		0			0
$421$	−10414.0			−1.20558			−0.602788	−	0.797902i	$-0.705943\pi$
$421$	−10414.0			−1.20558			−0.602788	+	0.797902i	$0.705943\pi$
$422$		0			0
$423$	−2469.09	−	1995.32i	−0.283809	−	0.229352i
$424$		0			0
$425$			11064.8i			1.26288i
$426$		0			0
$427$		−	4815.10i		−	0.545712i
$428$		0			0
$429$	3744.00	+	1323.70i	0.421357	+	0.148972i
$430$		0			0
$431$	13050.9			1.45856			0.729279	−	0.684216i	$-0.239855\pi$
$431$	13050.9			1.45856			0.729279	+	0.684216i	$0.239855\pi$
$432$		0			0
$433$	−9214.00			−1.02262			−0.511312	−	0.859395i	$-0.670841\pi$
$433$	−9214.00			−1.02262			−0.511312	+	0.859395i	$0.670841\pi$
$434$		0			0
$435$	1410.91	+	498.831i	0.155512	+	0.0549818i
$436$		0			0
$437$		−	18939.1i		−	2.07319i
$438$		0			0
$439$		−	7326.57i		−	0.796534i	−0.917270	−	0.398267i	$-0.869612\pi$
$439$		−	7326.57i		−	0.796534i	0.917270	−	0.398267i	$-0.130388\pi$
$440$		0			0
$441$	903.000	+	729.734i	0.0975057	+	0.0787965i
$442$		0			0
$443$	−2204.54			−0.236435			−0.118218	−	0.992988i	$-0.537718\pi$
$443$	−2204.54			−0.236435			−0.118218	+	0.992988i	$0.537718\pi$
$444$		0			0
$445$	6336.00			0.674956
$446$		0			0
$447$	−4085.75	+	11556.2i	−0.432325	+	1.22280i
$448$		0			0
$449$		−	2851.05i		−	0.299665i	−0.988711	−	0.149832i	$-0.952127\pi$
$449$		−	2851.05i		−	0.299665i	0.988711	−	0.149832i	$-0.0478734\pi$
$450$		0			0
$451$			8978.95i			0.937477i
$452$		0			0
$453$	−3534.00	+	9995.66i	−0.366538	+	1.03673i
$454$		0			0
$455$	7642.41			0.787432
$456$		0			0
$457$	−38.0000			−0.00388964			−0.00194482	−	0.999998i	$-0.500619\pi$
$457$	−38.0000			−0.00388964			−0.00194482	+	0.999998i	$0.500619\pi$
$458$		0			0
$459$	8112.71	−	4988.31i	0.824987	−	0.507264i
$460$		0			0
$461$			15833.5i			1.59966i	0.600230	+	0.799828i	$0.295076\pi$
$461$			15833.5i			1.59966i	−0.600230	+	0.799828i	$0.704924\pi$
$462$		0			0
$463$			11365.7i			1.14084i	0.821353	+	0.570421i	$0.193220\pi$
$463$			11365.7i			1.14084i	−0.821353	+	0.570421i	$0.806780\pi$
$464$		0			0
$465$	2592.00	+	916.410i	0.258497	+	0.0913925i
$466$		0			0
$467$	−499.696			−0.0495143			−0.0247571	−	0.999693i	$-0.507881\pi$
$467$	−499.696			−0.0495143			−0.0247571	+	0.999693i	$0.507881\pi$
$468$		0			0
$469$	−15420.0			−1.51819
$470$		0			0
$471$	519.292	+	183.597i	0.0508019	+	0.0179612i
$472$		0			0
$473$		−	2749.23i		−	0.267251i
$474$		0			0
$475$			17504.1i			1.69083i
$476$		0			0
$477$	−864.000	+	1069.15i	−0.0829347	+	0.102626i
$478$		0			0
$479$	−8700.59			−0.829937			−0.414969	−	0.909836i	$-0.636207\pi$
$479$	−8700.59			−0.829937			−0.414969	+	0.909836i	$0.636207\pi$
$480$		0			0
$481$	−5356.00			−0.507718
$482$		0			0
$483$	−5290.90	+	14964.9i	−0.498435	+	1.40979i
$484$		0			0
$485$			18158.5i			1.70007i
$486$		0			0
$487$			4007.97i			0.372933i	0.982461	+	0.186466i	$0.0597035\pi$
$487$			4007.97i			0.372933i	−0.982461	+	0.186466i	$0.940296\pi$
$488$		0			0
$489$	138.000	−	390.323i	0.0127619	−	0.0360961i
$490$		0			0
$491$	−16901.5			−1.55347			−0.776734	−	0.629828i	$-0.783125\pi$
$491$	−16901.5			−1.55347			−0.776734	+	0.629828i	$0.783125\pi$
$492$		0			0
$493$	1152.00			0.105240
$494$		0			0
$495$	8465.44	−	10475.4i	0.768673	−	0.951184i
$496$		0			0
$497$			1018.23i			0.0918994i
$498$		0			0
$499$		−	12439.6i		−	1.11598i	−0.829849	−	0.557988i	$-0.811573\pi$
$499$		−	12439.6i		−	1.11598i	0.829849	−	0.557988i	$-0.188427\pi$
$500$		0			0
$501$	15264.0	+	5396.64i	1.36117	+	0.481246i
$502$		0			0
$503$	−12756.9			−1.13082			−0.565411	−	0.824809i	$-0.691283\pi$
$503$	−12756.9			−1.13082			−0.565411	+	0.824809i	$0.691283\pi$
$504$		0			0
$505$	−30240.0			−2.66468
$506$		0			0
$507$	−7451.35	−	2634.45i	−0.652714	−	0.230769i
$508$		0			0
$509$		−	5549.37i		−	0.483245i	−0.970370	−	0.241622i	$-0.922320\pi$
$509$		−	5549.37i		−	0.483245i	0.970370	−	0.241622i	$-0.0776796\pi$
$510$		0			0
$511$		−	7309.25i		−	0.632764i
$512$		0			0
$513$	12834.0	−	7891.31i	1.10455	−	0.679162i
$514$		0			0
$515$	293.939			0.0251505
$516$		0			0
$517$	−3456.00			−0.293994
$518$		0			0
$519$	3850.60	−	10891.1i	0.325670	−	0.921133i
$520$		0			0
$521$		−	11506.0i		−	0.967541i	−0.875195	−	0.483770i	$-0.839267\pi$
$521$		−	11506.0i		−	0.967541i	0.875195	−	0.483770i	$-0.160733\pi$
$522$		0			0
$523$		−	17441.8i		−	1.45827i	−0.684371	−	0.729134i	$-0.739923\pi$
$523$		−	17441.8i		−	1.45827i	0.684371	−	0.729134i	$-0.260077\pi$
$524$		0			0
$525$	4890.00	−	13831.0i	0.406509	−	1.14978i
$526$		0			0
$527$	2116.36			0.174934
$528$		0			0
$529$	18937.0			1.55642
$530$		0			0
$531$	−11728.2	−	9477.78i	−0.958491	−	0.774578i
$532$		0			0
$533$			7942.22i			0.645433i
$534$		0			0
$535$			23445.0i			1.89461i
$536$		0			0
$537$	13680.0	+	4836.61i	1.09932	+	0.388669i
$538$		0			0
$539$	1263.94			0.101005
$540$		0			0
$541$	6794.00			0.539920			0.269960	−	0.962871i	$-0.412989\pi$
$541$	6794.00			0.539920			0.269960	+	0.962871i	$0.412989\pi$
$542$		0			0
$543$	−22094.4	−	7811.55i	−1.74615	−	0.617358i
$544$		0			0
$545$			6075.46i			0.477512i
$546$		0			0
$547$			3910.97i			0.305706i	0.988249	+	0.152853i	$0.0488461\pi$
$547$			3910.97i			0.305706i	−0.988249	+	0.152853i	$0.951154\pi$
$548$		0			0
$549$	−5838.00	−	4717.82i	−0.453843	−	0.366760i
$550$		0			0
$551$	1822.42			0.140903
$552$		0			0
$553$	−11580.0			−0.890473
$554$		0			0
$555$	−6055.14	+	17126.5i	−0.463110	+	1.30987i
$556$		0			0
$557$		−	22587.8i		−	1.71827i	−0.511749	−	0.859135i	$-0.671002\pi$
$557$		−	22587.8i		−	1.71827i	0.511749	−	0.859135i	$-0.328998\pi$
$558$		0			0
$559$		−	2431.80i		−	0.183997i
$560$		0			0
$561$	3456.00	−	9775.04i	0.260093	−	0.735655i
$562$		0			0
$563$	−8729.98			−0.653508			−0.326754	−	0.945109i	$-0.605955\pi$
$563$	−8729.98			−0.653508			−0.326754	+	0.945109i	$0.605955\pi$
$564$		0			0
$565$	−1152.00			−0.0857788
$566$		0			0
$567$	−12345.4	+	2650.04i	−0.914390	+	0.196281i
$568$		0			0
$569$			16122.0i			1.18782i	0.804531	+	0.593911i	$0.202417\pi$
$569$			16122.0i			1.18782i	−0.804531	+	0.593911i	$0.797583\pi$
$570$		0			0
$571$			9495.10i			0.695898i	0.937513	+	0.347949i	$0.113122\pi$
$571$			9495.10i			0.695898i	−0.937513	+	0.347949i	$0.886878\pi$
$572$		0			0
$573$	−9216.00	−	3258.35i	−0.671909	−	0.237556i
$574$		0			0
$575$	−28747.2			−2.08494
$576$		0			0
$577$	−8974.00			−0.647474			−0.323737	−	0.946147i	$-0.604939\pi$
$577$	−8974.00			−0.647474			−0.323737	+	0.946147i	$0.604939\pi$
$578$		0			0
$579$	24465.5	+	8649.86i	1.75605	+	0.620857i
$580$		0			0
$581$		−	509.117i		−	0.0363541i
$582$		0			0
$583$			1496.49i			0.106309i
$584$		0			0
$585$	7488.00	−	9265.93i	0.529215	−	0.654870i
$586$		0			0
$587$	−4908.78			−0.345157			−0.172578	−	0.984996i	$-0.555210\pi$
$587$	−4908.78			−0.345157			−0.172578	+	0.984996i	$0.555210\pi$
$588$		0			0
$589$	3348.00			0.234214
$590$		0			0
$591$	−3733.02	+	10558.6i	−0.259824	+	0.734893i
$592$		0			0
$593$			14730.4i			1.02008i	0.860151	+	0.510040i	$0.170369\pi$
$593$			14730.4i			1.02008i	−0.860151	+	0.510040i	$0.829631\pi$
$594$		0			0
$595$		−	19953.2i		−	1.37479i
$596$		0			0
$597$	6882.00	−	19465.2i	0.471795	−	1.33444i
$598$		0			0
$599$	10052.7			0.685714			0.342857	−	0.939388i	$-0.388605\pi$
$599$	10052.7			0.685714			0.342857	+	0.939388i	$0.388605\pi$
$600$		0			0
$601$	23690.0			1.60788			0.803939	−	0.594711i	$-0.202734\pi$
$601$	23690.0			1.60788			0.803939	+	0.594711i	$0.202734\pi$
$602$		0			0
$603$	−15108.5	+	18695.8i	−1.02034	+	1.26260i
$604$		0			0
$605$			7925.25i			0.532574i
$606$		0			0
$607$			22589.4i			1.51050i	0.655435	+	0.755252i	$0.272485\pi$
$607$			22589.4i			1.51050i	−0.655435	+	0.755252i	$0.727515\pi$
$608$		0			0
$609$	−1440.00	−	509.117i	−0.0958157	−	0.0338760i
$610$		0			0
$611$	−3056.96			−0.202408
$612$		0			0
$613$	−9422.00			−0.620801			−0.310400	−	0.950606i	$-0.600463\pi$
$613$	−9422.00			−0.620801			−0.310400	+	0.950606i	$0.600463\pi$
$614$		0			0
$615$	25396.3	+	8978.95i	1.66517	+	0.588726i
$616$		0			0
$617$			984.293i			0.0642239i	0.999484	+	0.0321119i	$0.0102233\pi$
$617$			984.293i			0.0642239i	−0.999484	+	0.0321119i	$0.989777\pi$
$618$		0			0
$619$			12238.7i			0.794691i	0.917669	+	0.397345i	$0.130069\pi$
$619$			12238.7i			0.794691i	−0.917669	+	0.397345i	$0.869931\pi$
$620$		0			0
$621$	12960.0	+	21077.4i	0.837467	+	1.36201i
$622$		0			0
$623$	−6466.65			−0.415860
$624$		0			0
$625$	−9431.00			−0.603584
$626$		0			0
$627$	5467.26	−	15463.7i	0.348232	−	0.984948i
$628$		0			0
$629$			13983.7i			0.886436i
$630$		0			0
$631$		−	13922.2i		−	0.878344i	−0.898403	−	0.439172i	$-0.855272\pi$
$631$		−	13922.2i		−	0.878344i	0.898403	−	0.439172i	$-0.144728\pi$
$632$		0			0
$633$	−8502.00	+	24047.3i	−0.533845	+	1.50994i
$634$		0			0
$635$	529.090			0.0330650
$636$		0			0
$637$	1118.00			0.0695397
$638$		0			0
$639$	1234.54	+	997.661i	0.0764284	+	0.0617635i
$640$		0			0
$641$		−	16427.5i		−	1.01224i	−0.862462	−	0.506121i	$-0.831079\pi$
$641$		−	16427.5i		−	1.01224i	0.862462	−	0.506121i	$-0.168921\pi$
$642$		0			0
$643$		−	1465.31i		−	0.0898700i	−0.998990	−	0.0449350i	$-0.985692\pi$
$643$		−	1465.31i		−	0.0898700i	0.998990	−	0.0449350i	$-0.0143081\pi$
$644$		0			0
$645$	−7776.00	−	2749.23i	−0.474697	−	0.167831i
$646$		0			0
$647$	12286.6			0.746581			0.373290	−	0.927715i	$-0.378230\pi$
$647$	12286.6			0.746581			0.373290	+	0.927715i	$0.378230\pi$
$648$		0			0
$649$	−16416.0			−0.992888
$650$		0			0
$651$	−2645.45	−	935.307i	−0.159268	−	0.0563097i
$652$		0			0
$653$			8502.25i			0.509523i	0.967004	+	0.254762i	$0.0819970\pi$
$653$			8502.25i			0.509523i	−0.967004	+	0.254762i	$0.918003\pi$
$654$		0			0
$655$		−	29431.0i		−	1.75567i
$656$		0			0
$657$	−8862.00	−	7161.58i	−0.526240	−	0.425266i
$658$		0			0
$659$	17136.6			1.01297			0.506486	−	0.862248i	$-0.330944\pi$
$659$	17136.6			1.01297			0.506486	+	0.862248i	$0.330944\pi$
$660$		0			0
$661$	10018.0			0.589493			0.294747	−	0.955575i	$-0.404765\pi$
$661$	10018.0			0.589493			0.294747	+	0.955575i	$0.404765\pi$
$662$		0			0
$663$	3056.96	−	8646.40i	0.179069	−	0.506483i
$664$		0			0
$665$		−	31565.2i		−	1.84067i
$666$		0			0
$667$			2992.98i			0.173746i
$668$		0			0
$669$	−6966.00	+	19702.8i	−0.402573	+	1.13865i
$670$		0			0
$671$	−8171.50			−0.470130
$672$		0			0
$673$	1682.00			0.0963393			0.0481696	−	0.998839i	$-0.484661\pi$
$673$	1682.00			0.0963393			0.0481696	+	0.998839i	$0.484661\pi$
$674$		0			0
$675$	−11978.0	−	19480.4i	−0.683013	−	1.11081i
$676$		0			0
$677$		−	9113.19i		−	0.517354i	−0.965964	−	0.258677i	$-0.916714\pi$
$677$		−	9113.19i		−	0.517354i	0.965964	−	0.258677i	$-0.0832865\pi$
$678$		0			0
$679$		−	18532.9i		−	1.04747i
$680$		0			0
$681$	−14544.0	−	5142.08i	−0.818395	−	0.289346i
$682$		0			0
$683$	−23250.6			−1.30257			−0.651287	−	0.758832i	$-0.725771\pi$
$683$	−23250.6			−1.30257			−0.651287	+	0.758832i	$0.725771\pi$
$684$		0			0
$685$	−38592.0			−2.15259
$686$		0			0
$687$	−2106.56	−	744.782i	−0.116987	−	0.0413613i
$688$		0			0
$689$			1323.70i			0.0731917i
$690$		0			0
$691$			25222.1i			1.38856i	0.719705	+	0.694280i	$0.244277\pi$
$691$			25222.1i			1.38856i	−0.719705	+	0.694280i	$0.755723\pi$
$692$		0			0
$693$	−8640.00	+	10691.5i	−0.473602	+	0.586053i
$694$		0			0
$695$	26160.6			1.42781
$696$		0			0
$697$	20736.0			1.12688
$698$		0			0
$699$	10993.3	−	31093.8i	0.594857	−	1.68251i
$700$		0			0
$701$		−	16410.5i		−	0.884190i	−0.896968	−	0.442095i	$-0.854235\pi$
$701$		−	16410.5i		−	0.884190i	0.896968	−	0.442095i	$-0.145765\pi$
$702$		0			0
$703$			22121.8i			1.18682i
$704$		0			0
$705$	−3456.00	+	9775.04i	−0.184625	+	0.522198i
$706$		0			0
$707$	30863.6			1.64179
$708$		0			0
$709$	27602.0			1.46208			0.731040	−	0.682334i	$-0.239035\pi$
$709$	27602.0			1.46208			0.731040	+	0.682334i	$0.239035\pi$
$710$		0			0
$711$	−11346.0	+	14040.0i	−0.598466	+	0.740564i
$712$		0			0
$713$			5498.46i			0.288806i
$714$		0			0
$715$		−	12969.6i		−	0.678371i
$716$		0			0
$717$	20160.0	+	7127.64i	1.05005	+	0.371250i
$718$		0			0
$719$	31627.8			1.64050			0.820249	−	0.572006i	$-0.193835\pi$
$719$	31627.8			1.64050			0.820249	+	0.572006i	$0.193835\pi$
$720$		0			0
$721$	−300.000			−0.0154960
$722$		0			0
$723$	22976.2	+	8123.32i	1.18187	+	0.417855i
$724$		0			0
$725$		−	2766.20i		−	0.141702i
$726$		0			0
$727$		−	27279.8i		−	1.39168i	−0.718197	−	0.695840i	$-0.755032\pi$
$727$		−	27279.8i		−	1.39168i	0.718197	−	0.695840i	$-0.244968\pi$
$728$		0			0
$729$	−8883.00	+	17564.5i	−0.451303	+	0.892371i
$730$		0			0
$731$	−6349.08			−0.321244
$732$		0			0
$733$	15914.0			0.801906			0.400953	−	0.916099i	$-0.368679\pi$
$733$	15914.0			0.801906			0.400953	+	0.916099i	$0.368679\pi$
$734$		0			0
$735$	1263.94	−	3574.95i	0.0634299	−	0.179407i
$736$		0			0
$737$			26168.6i			1.30791i
$738$		0			0
$739$			2275.91i			0.113289i	0.998394	+	0.0566447i	$0.0180402\pi$
$739$			2275.91i			0.113289i	−0.998394	+	0.0566447i	$0.981960\pi$
$740$		0			0
$741$	4836.00	−	13678.3i	0.239750	−	0.678116i
$742$		0			0
$743$	−8641.80			−0.426698			−0.213349	−	0.976976i	$-0.568437\pi$
$743$	−8641.80			−0.426698			−0.213349	+	0.976976i	$0.568437\pi$
$744$		0			0
$745$	40032.0			1.96867
$746$		0			0
$747$	−617.271	−	498.831i	−0.0302340	−	0.0244327i
$748$		0			0
$749$		−	23928.5i		−	1.16733i
$750$		0			0
$751$			16631.2i			0.808095i	0.914738	+	0.404048i	$0.132397\pi$
$751$			16631.2i			0.808095i	−0.914738	+	0.404048i	$0.867603\pi$
$752$		0			0
$753$	−6192.00	−	2189.20i	−0.299667	−	0.105948i
$754$		0			0
$755$	34626.0			1.66910
$756$		0			0
$757$	−11422.0			−0.548401			−0.274201	−	0.961673i	$-0.588413\pi$
$757$	−11422.0			−0.548401			−0.274201	+	0.961673i	$0.588413\pi$
$758$		0			0
$759$	25396.3	+	8978.95i	1.21453	+	0.429401i
$760$		0			0
$761$		−	12660.0i		−	0.603057i	−0.953457	−	0.301528i	$-0.902503\pi$
$761$		−	12660.0i		−	0.603057i	0.953457	−	0.301528i	$-0.0974968\pi$
$762$		0			0
$763$		−	6200.74i		−	0.294210i
$764$		0			0
$765$	−24192.0	−	19550.1i	−1.14335	−	0.923967i
$766$		0			0
$767$	−14520.6			−0.683582
$768$		0			0
$769$	18818.0			0.882437			0.441219	−	0.897400i	$-0.354546\pi$
$769$	18818.0			0.882437			0.441219	+	0.897400i	$0.354546\pi$
$770$		0			0
$771$	−470.302	+	1330.22i	−0.0219682	+	0.0621356i
$772$		0			0
$773$		−	1917.67i		−	0.0892289i	−0.999004	−	0.0446144i	$-0.985794\pi$
$773$		−	1917.67i		−	0.0892289i	0.999004	−	0.0446144i	$-0.0142059\pi$
$774$		0			0
$775$		−	5081.84i		−	0.235542i
$776$		0			0
$777$	6180.00	−	17479.7i	0.285336	−	0.807053i
$778$		0			0
$779$	32803.6			1.50874
$780$		0			0
$781$	1728.00			0.0791712
$782$		0			0
$783$	−2028.18	+	1247.08i	−0.0925685	+	0.0569181i
$784$		0			0
$785$		−	1798.88i		−	0.0817895i
$786$		0			0
$787$			19845.8i			0.898892i	0.893308	+	0.449446i	$0.148379\pi$
$787$			19845.8i			0.898892i	−0.893308	+	0.449446i	$0.851621\pi$
$788$		0			0
$789$	−25056.0	−	8858.63i	−1.13057	−	0.399716i
$790$		0			0

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.89898 + 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 + 16.9706i) q^{9} +O(q^{10})\)	\(q+(4.89898 + 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 + 16.9706i) q^{9} +29.3939 q^{11} +26.0000 q^{13} +(29.3939 - 83.1384i) q^{15} -67.8823i q^{17} -107.387i q^{19} +(-30.0000 + 84.8528i) q^{21} +176.363 q^{23} -163.000 q^{25} +(73.4847 + 119.512i) q^{27} +16.9706i q^{29} +31.1769i q^{31} +(144.000 + 50.9117i) q^{33} +293.939 q^{35} -206.000 q^{37} +(127.373 + 45.0333i) q^{39} +305.470i q^{41} -93.5307i q^{43} +(288.000 - 356.382i) q^{45} -117.576 q^{47} +43.0000 q^{49} +(117.576 - 332.554i) q^{51} +50.9117i q^{53} -498.831i q^{55} +(186.000 - 526.087i) q^{57} -558.484 q^{59} -278.000 q^{61} +(-293.939 + 363.731i) q^{63} -441.235i q^{65} +890.274i q^{67} +(864.000 + 305.470i) q^{69} +58.7878 q^{71} -422.000 q^{73} +(-798.534 - 282.324i) q^{75} +509.117i q^{77} +668.572i q^{79} +(153.000 + 712.764i) q^{81} -29.3939 q^{83} -1152.00 q^{85} +(-29.3939 + 83.1384i) q^{87} +373.352i q^{89} +450.333i q^{91} +(-54.0000 + 152.735i) q^{93} -1822.42 q^{95} -1070.00 q^{97} +(617.271 + 498.831i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 84 q^{9}+O(q^{10}) \) 4 * q + 84 * q^9	\( 4 q + 84 q^{9} + 104 q^{13} - 120 q^{21} - 652 q^{25} + 576 q^{33} - 824 q^{37} + 1152 q^{45} + 172 q^{49} + 744 q^{57} - 1112 q^{61} + 3456 q^{69} - 1688 q^{73} + 612 q^{81} - 4608 q^{85} - 216 q^{93} - 4280 q^{97}+O(q^{100}) \) 4 * q + 84 * q^9 + 104 * q^13 - 120 * q^21 - 652 * q^25 + 576 * q^33 - 824 * q^37 + 1152 * q^45 + 172 * q^49 + 744 * q^57 - 1112 * q^61 + 3456 * q^69 - 1688 * q^73 + 612 * q^81 - 4608 * q^85 - 216 * q^93 - 4280 * q^97

Introduction

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