LMFDB - Embedded newform 384.5.e.c.257.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(257,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.e (of order $2$, degree $1$, 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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 $ x^16 - 4x^15 - 32x^14 + 356x^13 + 1348x^12 - 8992x^11 + 22064x^10 + 391324x^9 + 724325x^8 - 2262056x^7 + 45109352x^6 + 210288484x^5 + 385003542x^4 + 866573964x^3 + 12462543468x^2 + 25672770504*x + 21479188203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{54}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.8
Root			$-3.40000 + 2.59434i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.c.257.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.622561 + 8.97844i) q^{3} +0.562050i q^{5} +55.8203 q^{7} +(-80.2248 + 11.1793i) q^{9} +O(q^{10})$	\(q+(0.622561 + 8.97844i) q^{3} +0.562050i q^{5} +55.8203 q^{7} +(-80.2248 + 11.1793i) q^{9} -87.7824i q^{11} +71.5994 q^{13} +(-5.04633 + 0.349910i) q^{15} -334.017i q^{17} +365.653 q^{19} +(34.7515 + 501.179i) q^{21} -789.593i q^{23} +624.684 q^{25} +(-150.317 - 713.334i) q^{27} +236.473i q^{29} -1280.52 q^{31} +(788.149 - 54.6499i) q^{33} +31.3738i q^{35} +513.856 q^{37} +(44.5750 + 642.851i) q^{39} -1189.01i q^{41} +204.172 q^{43} +(-6.28330 - 45.0903i) q^{45} -1316.07i q^{47} +714.903 q^{49} +(2998.95 - 207.946i) q^{51} +317.847i q^{53} +49.3381 q^{55} +(227.641 + 3283.00i) q^{57} +1274.33i q^{59} +5771.27 q^{61} +(-4478.17 + 624.029i) q^{63} +40.2424i q^{65} +7246.66 q^{67} +(7089.32 - 491.570i) q^{69} +4833.29i q^{71} +6984.08 q^{73} +(388.904 + 5608.69i) q^{75} -4900.04i q^{77} -9332.37 q^{79} +(6311.05 - 1793.71i) q^{81} +838.939i q^{83} +187.734 q^{85} +(-2123.16 + 147.219i) q^{87} +13943.2i q^{89} +3996.70 q^{91} +(-797.204 - 11497.1i) q^{93} +205.515i q^{95} +6578.49 q^{97} +(981.342 + 7042.33i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 80 q^{7}+O(q^{10}) $ 16 * q + 8 * q^3 - 80 * q^7	$ 16 q + 8 q^{3} - 80 q^{7} - 416 q^{15} - 816 q^{19} + 608 q^{21} - 2000 q^{25} - 280 q^{27} - 592 q^{31} - 496 q^{33} + 2240 q^{37} + 16 q^{39} - 368 q^{43} + 800 q^{45} + 3984 q^{49} + 352 q^{51} - 1920 q^{55} + 560 q^{57} - 3520 q^{61} + 816 q^{63} + 3536 q^{67} - 10784 q^{69} + 3680 q^{73} - 5112 q^{75} + 14448 q^{79} - 624 q^{81} - 11136 q^{85} + 14944 q^{87} + 22944 q^{91} + 13760 q^{93} + 3264 q^{97} + 26976 q^{99}+O(q^{100}) $ 16 * q + 8 * q^3 - 80 * q^7 - 416 * q^15 - 816 * q^19 + 608 * q^21 - 2000 * q^25 - 280 * q^27 - 592 * q^31 - 496 * q^33 + 2240 * q^37 + 16 * q^39 - 368 * q^43 + 800 * q^45 + 3984 * q^49 + 352 * q^51 - 1920 * q^55 + 560 * q^57 - 3520 * q^61 + 816 * q^63 + 3536 * q^67 - 10784 * q^69 + 3680 * q^73 - 5112 * q^75 + 14448 * q^79 - 624 * q^81 - 11136 * q^85 + 14944 * q^87 + 22944 * q^91 + 13760 * q^93 + 3264 * q^97 + 26976 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0.622561	+	8.97844i	0.0691735	+	0.997605i
$3$	0.622561	+	8.97844i	0.0691735	+	0.997605i
$4$		0			0
$5$			0.562050i			0.0224820i	0.999937	+	0.0112410i	$0.00357820\pi$
$5$			0.562050i			0.0224820i	−0.999937	+	0.0112410i	$0.996422\pi$
$6$		0			0
$7$	55.8203			1.13919			0.569595	−	0.821926i	$-0.307100\pi$
$7$	55.8203			1.13919			0.569595	+	0.821926i	$0.307100\pi$
$8$		0			0
$9$	−80.2248	+	11.1793i	−0.990430	+	0.138016i
$10$		0			0
$11$		−	87.7824i		−	0.725475i	−0.931891	−	0.362737i	$-0.881842\pi$
$11$		−	87.7824i		−	0.725475i	0.931891	−	0.362737i	$-0.118158\pi$
$12$		0			0
$13$	71.5994			0.423665			0.211832	−	0.977306i	$-0.432057\pi$
$13$	71.5994			0.423665			0.211832	+	0.977306i	$0.432057\pi$
$14$		0			0
$15$	−5.04633	+	0.349910i	−0.0224281	+	0.00155516i
$16$		0			0
$17$		−	334.017i		−	1.15577i	−0.816119	−	0.577884i	$-0.803879\pi$
$17$		−	334.017i		−	1.15577i	0.816119	−	0.577884i	$-0.196121\pi$
$18$		0			0
$19$	365.653			1.01289			0.506445	−	0.862272i	$-0.330959\pi$
$19$	365.653			1.01289			0.506445	+	0.862272i	$0.330959\pi$
$20$		0			0
$21$	34.7515	+	501.179i	0.0788017	+	1.13646i
$22$		0			0
$23$		−	789.593i		−	1.49262i	−0.665601	−	0.746308i	$-0.731825\pi$
$23$		−	789.593i		−	1.49262i	0.665601	−	0.746308i	$-0.268175\pi$
$24$		0			0
$25$	624.684			0.999495
$26$		0			0
$27$	−150.317	−	713.334i	−0.206196	−	0.978511i
$28$		0			0
$29$			236.473i			0.281181i	0.990068	+	0.140590i	$0.0449000\pi$
$29$			236.473i			0.281181i	−0.990068	+	0.140590i	$0.955100\pi$
$30$		0			0
$31$	−1280.52			−1.33249			−0.666245	−	0.745733i	$-0.732100\pi$
$31$	−1280.52			−1.33249			−0.666245	+	0.745733i	$0.732100\pi$
$32$		0			0
$33$	788.149	−	54.6499i	0.723737	−	0.0501836i
$34$		0			0
$35$			31.3738i			0.0256112i
$36$		0			0
$37$	513.856			0.375352			0.187676	−	0.982231i	$-0.439905\pi$
$37$	513.856			0.375352			0.187676	+	0.982231i	$0.439905\pi$
$38$		0			0
$39$	44.5750	+	642.851i	0.0293064	+	0.422650i
$40$		0			0
$41$		−	1189.01i		−	0.707323i	−0.935374	−	0.353661i	$-0.884937\pi$
$41$		−	1189.01i		−	0.707323i	0.935374	−	0.353661i	$-0.115063\pi$
$42$		0			0
$43$	204.172			0.110423			0.0552114	−	0.998475i	$-0.482417\pi$
$43$	204.172			0.110423			0.0552114	+	0.998475i	$0.482417\pi$
$44$		0			0
$45$	−6.28330	−	45.0903i	−0.00310286	−	0.0222668i
$46$		0			0
$47$		−	1316.07i		−	0.595779i	−0.954600	−	0.297889i	$-0.903717\pi$
$47$		−	1316.07i		−	0.595779i	0.954600	−	0.297889i	$-0.0962826\pi$
$48$		0			0
$49$	714.903			0.297752
$50$		0			0
$51$	2998.95	−	207.946i	1.15300	−	0.0799485i
$52$		0			0
$53$			317.847i			0.113153i	0.998398	+	0.0565766i	$0.0180185\pi$
$53$			317.847i			0.113153i	−0.998398	+	0.0565766i	$0.981981\pi$
$54$		0			0
$55$	49.3381			0.0163101
$56$		0			0
$57$	227.641	+	3283.00i	0.0700651	+	1.01046i
$58$		0			0
$59$			1274.33i			0.366081i	0.983105	+	0.183040i	$0.0585939\pi$
$59$			1274.33i			0.366081i	−0.983105	+	0.183040i	$0.941406\pi$
$60$		0			0
$61$	5771.27			1.55100			0.775500	−	0.631347i	$-0.217498\pi$
$61$	5771.27			1.55100			0.775500	+	0.631347i	$0.217498\pi$
$62$		0			0
$63$	−4478.17	+	624.029i	−1.12829	+	0.157226i
$64$		0			0
$65$			40.2424i			0.00952483i
$66$		0			0
$67$	7246.66			1.61432			0.807158	−	0.590336i	$-0.201005\pi$
$67$	7246.66			1.61432			0.807158	+	0.590336i	$0.201005\pi$
$68$		0			0
$69$	7089.32	−	491.570i	1.48904	−	0.103249i
$70$		0			0
$71$			4833.29i			0.958796i	0.877598	+	0.479398i	$0.159145\pi$
$71$			4833.29i			0.958796i	−0.877598	+	0.479398i	$0.840855\pi$
$72$		0			0
$73$	6984.08			1.31058			0.655290	−	0.755377i	$-0.272546\pi$
$73$	6984.08			1.31058			0.655290	+	0.755377i	$0.272546\pi$
$74$		0			0
$75$	388.904	+	5608.69i	0.0691385	+	0.997100i
$76$		0			0
$77$		−	4900.04i		−	0.826453i
$78$		0			0
$79$	−9332.37			−1.49533			−0.747666	−	0.664075i	$-0.768826\pi$
$79$	−9332.37			−1.49533			−0.747666	+	0.664075i	$0.768826\pi$
$80$		0			0
$81$	6311.05	−	1793.71i	0.961903	−	0.273389i
$82$		0			0
$83$			838.939i			0.121780i	0.998144	+	0.0608898i	$0.0193938\pi$
$83$			838.939i			0.121780i	−0.998144	+	0.0608898i	$0.980606\pi$
$84$		0			0
$85$	187.734			0.0259840
$86$		0			0
$87$	−2123.16	+	147.219i	−0.280507	+	0.0194502i
$88$		0			0
$89$			13943.2i			1.76028i	0.474715	+	0.880140i	$0.342551\pi$
$89$			13943.2i			1.76028i	−0.474715	+	0.880140i	$0.657449\pi$
$90$		0			0
$91$	3996.70			0.482635
$92$		0			0
$93$	−797.204	−	11497.1i	−0.0921730	−	1.32930i
$94$		0			0
$95$			205.515i			0.0227718i
$96$		0			0
$97$	6578.49			0.699170			0.349585	−	0.936905i	$-0.386323\pi$
$97$	6578.49			0.699170			0.349585	+	0.936905i	$0.386323\pi$
$98$		0			0
$99$	981.342	+	7042.33i	0.100127	+	0.718532i
$100$		0			0
$101$			17934.3i			1.75810i	0.476732	+	0.879049i	$0.341821\pi$
$101$			17934.3i			1.75810i	−0.476732	+	0.879049i	$0.658179\pi$
$102$		0			0
$103$	−13246.5			−1.24861			−0.624303	−	0.781182i	$-0.714617\pi$
$103$	−13246.5			−1.24861			−0.624303	+	0.781182i	$0.714617\pi$
$104$		0			0
$105$	−281.688	+	19.5321i	−0.0255499	+	0.00177162i
$106$		0			0
$107$		−	11912.3i		−	1.04046i	−0.854025	−	0.520232i	$-0.825845\pi$
$107$		−	11912.3i		−	1.04046i	0.854025	−	0.520232i	$-0.174155\pi$
$108$		0			0
$109$	12868.3			1.08310			0.541548	−	0.840670i	$-0.317838\pi$
$109$	12868.3			1.08310			0.541548	+	0.840670i	$0.317838\pi$
$110$		0			0
$111$	319.907	+	4613.63i	0.0259644	+	0.374453i
$112$		0			0
$113$		−	22368.0i		−	1.75174i	−0.482544	−	0.875872i	$-0.660287\pi$
$113$		−	22368.0i		−	1.75174i	0.482544	−	0.875872i	$-0.339713\pi$
$114$		0			0
$115$	443.791			0.0335570
$116$		0			0
$117$	−5744.05	+	800.428i	−0.419610	+	0.0584723i
$118$		0			0
$119$		−	18644.9i		−	1.31664i
$120$		0			0
$121$	6935.25			0.473687
$122$		0			0
$123$	10675.5	−	740.231i	0.705628	−	0.0489280i
$124$		0			0
$125$			702.385i			0.0449526i
$126$		0			0
$127$	12221.9			0.757762			0.378881	−	0.925445i	$-0.376309\pi$
$127$	12221.9			0.757762			0.378881	+	0.925445i	$0.376309\pi$
$128$		0			0
$129$	127.109	+	1833.14i	0.00763832	+	0.110158i
$130$		0			0
$131$		−	15121.0i		−	0.881125i	−0.897722	−	0.440563i	$-0.854779\pi$
$131$		−	15121.0i		−	0.881125i	0.897722	−	0.440563i	$-0.145221\pi$
$132$		0			0
$133$	20410.9			1.15387
$134$		0			0
$135$	400.929	−	84.4857i	0.0219989	−	0.00463570i
$136$		0			0
$137$		−	2504.91i		−	0.133460i	−0.997771	−	0.0667300i	$-0.978743\pi$
$137$		−	2504.91i		−	0.133460i	0.997771	−	0.0667300i	$-0.0212566\pi$
$138$		0			0
$139$	28510.7			1.47563			0.737815	−	0.675003i	$-0.235858\pi$
$139$	28510.7			1.47563			0.737815	+	0.675003i	$0.235858\pi$
$140$		0			0
$141$	11816.3	−	819.337i	0.594351	−	0.0412121i
$142$		0			0
$143$		−	6285.17i		−	0.307358i
$144$		0			0
$145$	−132.910			−0.00632150
$146$		0			0
$147$	445.071	+	6418.71i	0.0205965	+	0.297039i
$148$		0			0
$149$			33444.1i			1.50642i	0.657778	+	0.753212i	$0.271496\pi$
$149$			33444.1i			1.50642i	−0.657778	+	0.753212i	$0.728504\pi$
$150$		0			0
$151$	−13142.6			−0.576404			−0.288202	−	0.957570i	$-0.593057\pi$
$151$	−13142.6			−0.576404			−0.288202	+	0.957570i	$0.593057\pi$
$152$		0			0
$153$	3734.06	+	26796.5i	0.159514	+	1.14471i
$154$		0			0
$155$		−	719.718i		−	0.0299570i
$156$		0			0
$157$	−31662.0			−1.28452			−0.642258	−	0.766488i	$-0.722002\pi$
$157$	−31662.0			−1.28452			−0.642258	+	0.766488i	$0.722002\pi$
$158$		0			0
$159$	−2853.77	+	197.879i	−0.112882	+	0.00782719i
$160$		0			0
$161$		−	44075.3i		−	1.70037i
$162$		0			0
$163$	−32147.7			−1.20997			−0.604985	−	0.796237i	$-0.706821\pi$
$163$	−32147.7			−1.20997			−0.604985	+	0.796237i	$0.706821\pi$
$164$		0			0
$165$	30.7160	+	442.979i	0.00112823	+	0.0162710i
$166$		0			0
$167$		−	13951.1i		−	0.500238i	−0.968215	−	0.250119i	$-0.919530\pi$
$167$		−	13951.1i		−	0.500238i	0.968215	−	0.250119i	$-0.0804698\pi$
$168$		0			0
$169$	−23434.5			−0.820508
$170$		0			0
$171$	−29334.5	+	4087.73i	−1.00320	+	0.139794i
$172$		0			0
$173$		−	32861.5i		−	1.09798i	−0.835828	−	0.548991i	$-0.815012\pi$
$173$		−	32861.5i		−	1.09798i	0.835828	−	0.548991i	$-0.184988\pi$
$174$		0			0
$175$	34870.0			1.13861
$176$		0			0
$177$	−11441.5	+	793.347i	−0.365204	+	0.0253231i
$178$		0			0
$179$		−	20985.9i		−	0.654971i	−0.944856	−	0.327486i	$-0.893799\pi$
$179$		−	20985.9i		−	0.654971i	0.944856	−	0.327486i	$-0.106201\pi$
$180$		0			0
$181$	−56067.7			−1.71142			−0.855708	−	0.517459i	$-0.826878\pi$
$181$	−56067.7			−1.71142			−0.855708	+	0.517459i	$0.826878\pi$
$182$		0			0
$183$	3592.97	+	51817.0i	0.107288	+	1.54728i
$184$		0			0
$185$			288.813i			0.00843865i
$186$		0			0
$187$	−29320.8			−0.838480
$188$		0			0
$189$	−8390.74	−	39818.5i	−0.234897	−	1.11471i
$190$		0			0
$191$		−	47798.6i		−	1.31023i	−0.755529	−	0.655116i	$-0.772620\pi$
$191$		−	47798.6i		−	1.31023i	0.755529	−	0.655116i	$-0.227380\pi$
$192$		0			0
$193$	43.1516			0.00115846			0.000579232	−	1.00000i	$-0.499816\pi$
$193$	43.1516			0.00115846			0.000579232		1.00000i	$0.499816\pi$
$194$		0			0
$195$	−361.314	+	25.0534i	−0.00950201	+	0.000658865i
$196$		0			0
$197$			40137.0i			1.03422i	0.855920	+	0.517109i	$0.172992\pi$
$197$			40137.0i			1.03422i	−0.855920	+	0.517109i	$0.827008\pi$
$198$		0			0
$199$	23378.7			0.590355			0.295178	−	0.955442i	$-0.404621\pi$
$199$	23378.7			0.590355			0.295178	+	0.955442i	$0.404621\pi$
$200$		0			0
$201$	4511.49	+	65063.7i	0.111668	+	1.61045i
$202$		0			0
$203$			13200.0i			0.320318i
$204$		0			0
$205$	668.282			0.0159020
$206$		0			0
$207$	8827.07	+	63345.0i	0.206004	+	1.47833i
$208$		0			0
$209$		−	32097.9i		−	0.734826i
$210$		0			0
$211$	−54577.0			−1.22587			−0.612935	−	0.790133i	$-0.710011\pi$
$211$	−54577.0			−1.22587			−0.612935	+	0.790133i	$0.710011\pi$
$212$		0			0
$213$	−43395.4	+	3009.02i	−0.956500	+	0.0663232i
$214$		0			0
$215$			114.755i			0.00248252i
$216$		0			0
$217$	−71479.2			−1.51796
$218$		0			0
$219$	4348.02	+	62706.2i	0.0906573	+	1.30744i
$220$		0			0
$221$		−	23915.4i		−	0.489658i
$222$		0			0
$223$	20007.6			0.402332			0.201166	−	0.979557i	$-0.435527\pi$
$223$	20007.6			0.402332			0.201166	+	0.979557i	$0.435527\pi$
$224$		0			0
$225$	−50115.2	+	6983.50i	−0.989929	+	0.137946i
$226$		0			0
$227$			18082.4i			0.350917i	0.984487	+	0.175458i	$0.0561407\pi$
$227$			18082.4i			0.350917i	−0.984487	+	0.175458i	$0.943859\pi$
$228$		0			0
$229$	56787.1			1.08288			0.541438	−	0.840741i	$-0.317880\pi$
$229$	56787.1			1.08288			0.541438	+	0.840741i	$0.317880\pi$
$230$		0			0
$231$	43994.7	−	3050.57i	0.824473	−	0.0571686i
$232$		0			0
$233$			57688.8i			1.06262i	0.847176	+	0.531312i	$0.178301\pi$
$233$			57688.8i			1.06262i	−0.847176	+	0.531312i	$0.821699\pi$
$234$		0			0
$235$	739.700			0.0133943
$236$		0			0
$237$	−5809.97	−	83790.1i	−0.103437	−	1.49175i
$238$		0			0
$239$			31777.3i			0.556315i	0.960535	+	0.278158i	$0.0897238\pi$
$239$			31777.3i			0.556315i	−0.960535	+	0.278158i	$0.910276\pi$
$240$		0			0
$241$	−28106.4			−0.483918			−0.241959	−	0.970286i	$-0.577790\pi$
$241$	−28106.4			−0.483918			−0.241959	+	0.970286i	$0.577790\pi$
$242$		0			0
$243$	20033.7	+	55546.7i	0.339273	+	0.940688i
$244$		0			0
$245$			401.811i			0.00669406i
$246$		0			0
$247$	26180.5			0.429126
$248$		0			0
$249$	−7532.37	+	522.291i	−0.121488	+	0.00842391i
$250$		0			0
$251$		−	114789.i		−	1.82202i	−0.412386	−	0.911009i	$-0.635304\pi$
$251$		−	114789.i		−	1.82202i	0.412386	−	0.911009i	$-0.364696\pi$
$252$		0			0
$253$	−69312.4			−1.08285
$254$		0			0
$255$	116.876	+	1685.56i	0.00179740	+	0.0259217i
$256$		0			0
$257$		−	50929.0i		−	0.771079i	−0.922691	−	0.385539i	$-0.874015\pi$
$257$		−	50929.0i		−	0.771079i	0.922691	−	0.385539i	$-0.125985\pi$
$258$		0			0
$259$	28683.6			0.427597
$260$		0			0
$261$	−2643.59	−	18971.0i	−0.0388073	−	0.278490i
$262$		0			0
$263$			2156.83i			0.0311820i	0.999878	+	0.0155910i	$0.00496297\pi$
$263$			2156.83i			0.0311820i	−0.999878	+	0.0155910i	$0.995037\pi$
$264$		0			0
$265$	−178.646			−0.00254391
$266$		0			0
$267$	−125188.	+	8680.48i	−1.75606	+	0.121765i
$268$		0			0
$269$			121224.i			1.67527i	0.546230	+	0.837635i	$0.316062\pi$
$269$			121224.i			1.67527i	−0.546230	+	0.837635i	$0.683938\pi$
$270$		0			0
$271$	21224.3			0.288998			0.144499	−	0.989505i	$-0.453843\pi$
$271$	21224.3			0.288998			0.144499	+	0.989505i	$0.453843\pi$
$272$		0			0
$273$	2488.19	+	35884.1i	0.0333855	+	0.481478i
$274$		0			0
$275$		−	54836.3i		−	0.725108i
$276$		0			0
$277$	−116163.			−1.51394			−0.756970	−	0.653450i	$-0.773321\pi$
$277$	−116163.			−1.51394			−0.756970	+	0.653450i	$0.773321\pi$
$278$		0			0
$279$	102730.	−	14315.3i	1.31974	−	0.183904i
$280$		0			0
$281$		−	117518.i		−	1.48830i	−0.668011	−	0.744151i	$-0.732854\pi$
$281$		−	117518.i		−	1.48830i	0.668011	−	0.744151i	$-0.267146\pi$
$282$		0			0
$283$	−73183.7			−0.913780			−0.456890	−	0.889523i	$-0.651037\pi$
$283$	−73183.7			−0.913780			−0.456890	+	0.889523i	$0.651037\pi$
$284$		0			0
$285$	−1845.21	+	127.946i	−0.0227172	+	0.00157520i
$286$		0			0
$287$		−	66370.8i		−	0.805774i
$288$		0			0
$289$	−28046.3			−0.335799
$290$		0			0
$291$	4095.51	+	59064.6i	0.0483640	+	0.697495i
$292$		0			0
$293$			90550.0i			1.05476i	0.849630	+	0.527380i	$0.176825\pi$
$293$			90550.0i			1.05476i	−0.849630	+	0.527380i	$0.823175\pi$
$294$		0			0
$295$	−716.235			−0.00823023
$296$		0			0
$297$	−62618.2	+	13195.2i	−0.709885	+	0.149590i
$298$		0			0
$299$		−	56534.4i		−	0.632369i
$300$		0			0
$301$	11396.9			0.125792
$302$		0			0
$303$	−161023.	+	11165.2i	−1.75389	+	0.121614i
$304$		0			0
$305$			3243.74i			0.0348696i
$306$		0			0
$307$	−47033.7			−0.499036			−0.249518	−	0.968370i	$-0.580272\pi$
$307$	−47033.7			−0.499036			−0.249518	+	0.968370i	$0.580272\pi$
$308$		0			0
$309$	−8246.73	−	118933.i	−0.0863704	−	1.24562i
$310$		0			0
$311$		−	166222.i		−	1.71857i	−0.511495	−	0.859286i	$-0.670908\pi$
$311$		−	166222.i		−	1.71857i	0.511495	−	0.859286i	$-0.329092\pi$
$312$		0			0
$313$	114083.			1.16448			0.582241	−	0.813016i	$-0.302176\pi$
$313$	114083.			1.16448			0.582241	+	0.813016i	$0.302176\pi$
$314$		0			0
$315$	−350.735	−	2516.96i	−0.00353475	−	0.0253661i
$316$		0			0
$317$			32769.6i			0.326101i	0.986618	+	0.163051i	$0.0521334\pi$
$317$			32769.6i			0.326101i	−0.986618	+	0.163051i	$0.947867\pi$
$318$		0			0
$319$	20758.2			0.203989
$320$		0			0
$321$	106954.	−	7416.12i	1.03797	−	0.0719725i
$322$		0			0
$323$		−	122134.i		−	1.17067i
$324$		0			0
$325$	44727.0			0.423451
$326$		0			0
$327$	8011.28	+	115537.i	0.0749215	+	1.08050i
$328$		0			0
$329$		−	73463.7i		−	0.678705i
$330$		0			0
$331$	137589.			1.25582			0.627912	−	0.778285i	$-0.283910\pi$
$331$	137589.			1.25582			0.627912	+	0.778285i	$0.283910\pi$
$332$		0			0
$333$	−41224.0	+	5744.53i	−0.371760	+	0.0518043i
$334$		0			0
$335$			4072.98i			0.0362930i
$336$		0			0
$337$	−8517.04			−0.0749944			−0.0374972	−	0.999297i	$-0.511939\pi$
$337$	−8517.04			−0.0749944			−0.0374972	+	0.999297i	$0.511939\pi$
$338$		0			0
$339$	200830.	−	13925.5i	1.74755	−	0.121174i
$340$		0			0
$341$			112407.i			0.966688i
$342$		0			0
$343$	−94118.4			−0.799993
$344$		0			0
$345$	276.287	+	3984.55i	0.00232125	+	0.0334766i
$346$		0			0
$347$			83527.7i			0.693699i	0.937921	+	0.346850i	$0.112749\pi$
$347$			83527.7i			0.693699i	−0.937921	+	0.346850i	$0.887251\pi$
$348$		0			0
$349$	133537.			1.09635			0.548177	−	0.836363i	$-0.315322\pi$
$349$	133537.			1.09635			0.548177	+	0.836363i	$0.315322\pi$
$350$		0			0
$351$	−10762.6	−	51074.3i	−0.0873582	−	0.414561i
$352$		0			0
$353$		−	4971.02i		−	0.0398929i	−0.999801	−	0.0199465i	$-0.993650\pi$
$353$		−	4971.02i		−	0.0398929i	0.999801	−	0.0199465i	$-0.00634958\pi$
$354$		0			0
$355$	−2716.55			−0.0215556
$356$		0			0
$357$	167402.	−	11607.6i	1.31348	−	0.0910764i
$358$		0			0
$359$			101188.i			0.785125i	0.919725	+	0.392562i	$0.128411\pi$
$359$			101188.i			0.785125i	−0.919725	+	0.392562i	$0.871589\pi$
$360$		0			0
$361$	3381.22			0.0259453
$362$		0			0
$363$	4317.61	+	62267.7i	0.0327665	+	0.472552i
$364$		0			0
$365$			3925.40i			0.0294644i
$366$		0			0
$367$	72491.9			0.538217			0.269108	−	0.963110i	$-0.413271\pi$
$367$	72491.9			0.538217			0.269108	+	0.963110i	$0.413271\pi$
$368$		0			0
$369$	13292.2	+	95388.1i	0.0976215	+	0.700554i
$370$		0			0
$371$			17742.3i			0.128903i
$372$		0			0
$373$	91510.6			0.657739			0.328870	−	0.944375i	$-0.393332\pi$
$373$	91510.6			0.657739			0.328870	+	0.944375i	$0.393332\pi$
$374$		0			0
$375$	−6306.32	+	437.277i	−0.0448449	+	0.00310953i
$376$		0			0
$377$			16931.3i			0.119126i
$378$		0			0
$379$	150530.			1.04796			0.523981	−	0.851730i	$-0.324446\pi$
$379$	150530.			1.04796			0.523981	+	0.851730i	$0.324446\pi$
$380$		0			0
$381$	7608.91	+	109734.i	0.0524170	+	0.755947i
$382$		0			0
$383$		−	228289.i		−	1.55628i	−0.628091	−	0.778140i	$-0.716163\pi$
$383$		−	228289.i		−	1.55628i	0.628091	−	0.778140i	$-0.283837\pi$
$384$		0			0
$385$	2754.07			0.0185803
$386$		0			0
$387$	−16379.6	+	2282.49i	−0.109366	+	0.0152400i
$388$		0			0
$389$		−	280509.i		−	1.85374i	−0.375385	−	0.926869i	$-0.622490\pi$
$389$		−	280509.i		−	1.85374i	0.375385	−	0.926869i	$-0.377510\pi$
$390$		0			0
$391$	−263738.			−1.72512
$392$		0			0
$393$	135763.	−	9413.74i	0.879015	−	0.0609505i
$394$		0			0
$395$		−	5245.25i		−	0.0336180i
$396$		0			0
$397$	119440.			0.757827			0.378913	−	0.925432i	$-0.376298\pi$
$397$	119440.			0.757827			0.378913	+	0.925432i	$0.376298\pi$
$398$		0			0
$399$	12707.0	+	183258.i	0.0798174	+	1.15111i
$400$		0			0
$401$		−	1318.85i		−	0.00820176i	−0.999992	−	0.00410088i	$-0.998695\pi$
$401$		−	1318.85i		−	0.00820176i	0.999992	−	0.00410088i	$-0.00130535\pi$
$402$		0			0
$403$	−91684.7			−0.564530
$404$		0			0
$405$	1008.15	+	3547.12i	0.00614634	+	0.0216255i
$406$		0			0
$407$		−	45107.6i		−	0.272308i
$408$		0			0
$409$	38558.7			0.230503			0.115251	−	0.993336i	$-0.463233\pi$
$409$	38558.7			0.230503			0.115251	+	0.993336i	$0.463233\pi$
$410$		0			0
$411$	22490.2	−	1559.46i	0.133140	−	0.00923188i
$412$		0			0
$413$			71133.3i			0.417035i
$414$		0			0
$415$	−471.526			−0.00273785
$416$		0			0
$417$	17749.6	+	255981.i	0.102074	+	1.47210i
$418$		0			0
$419$			99777.3i			0.568334i	0.958775	+	0.284167i	$0.0917170\pi$
$419$			99777.3i			0.568334i	−0.958775	+	0.284167i	$0.908283\pi$
$420$		0			0
$421$	−109123.			−0.615674			−0.307837	−	0.951439i	$-0.599605\pi$
$421$	−109123.			−0.615674			−0.307837	+	0.951439i	$0.599605\pi$
$422$		0			0
$423$	14712.7	+	105582.i	0.0822267	+	0.590077i
$424$		0			0
$425$		−	208655.i		−	1.15518i
$426$		0			0
$427$	322154.			1.76688
$428$		0			0
$429$	56431.0	−	3912.90i	0.306622	−	0.0212610i
$430$		0			0
$431$			340800.i			1.83462i	0.398178	+	0.917308i	$0.369643\pi$
$431$			340800.i			1.83462i	−0.398178	+	0.917308i	$0.630357\pi$
$432$		0			0
$433$	255345.			1.36192			0.680959	−	0.732321i	$-0.261563\pi$
$433$	255345.			1.36192			0.680959	+	0.732321i	$0.261563\pi$
$434$		0			0
$435$	−82.7443	−	1193.32i	−0.000437280	−	0.00630636i
$436$		0			0
$437$		−	288717.i		−	1.51185i
$438$		0			0
$439$	−257441.			−1.33582			−0.667912	−	0.744240i	$-0.732812\pi$
$439$	−257441.			−1.33582			−0.667912	+	0.744240i	$0.732812\pi$
$440$		0			0
$441$	−57353.0	+	7992.08i	−0.294903	+	0.0410944i
$442$		0			0
$443$			118677.i			0.604729i	0.953192	+	0.302364i	$0.0977759\pi$
$443$			118677.i			0.604729i	−0.953192	+	0.302364i	$0.902224\pi$
$444$		0			0
$445$	−7836.76			−0.0395746
$446$		0			0
$447$	−300276.	+	20821.0i	−1.50282	+	0.104205i
$448$		0			0
$449$			351950.i			1.74578i	0.487919	+	0.872889i	$0.337756\pi$
$449$			351950.i			1.74578i	−0.487919	+	0.872889i	$0.662244\pi$
$450$		0			0
$451$	−104374.			−0.513145
$452$		0			0
$453$	−8182.06	−	118000.i	−0.0398718	−	0.575023i
$454$		0			0
$455$			2246.34i			0.0108506i
$456$		0			0
$457$	279904.			1.34022			0.670110	−	0.742261i	$-0.266247\pi$
$457$	279904.			1.34022			0.670110	+	0.742261i	$0.266247\pi$
$458$		0			0
$459$	−238266.	+	50208.5i	−1.13093	+	0.238315i
$460$		0			0
$461$			239756.i			1.12815i	0.825723	+	0.564076i	$0.190767\pi$
$461$			239756.i			1.12815i	−0.825723	+	0.564076i	$0.809233\pi$
$462$		0			0
$463$	7111.21			0.0331728			0.0165864	−	0.999862i	$-0.494720\pi$
$463$	7111.21			0.0331728			0.0165864	+	0.999862i	$0.494720\pi$
$464$		0			0
$465$	6461.95	−	448.068i	0.0298853	−	0.00207223i
$466$		0			0
$467$			174901.i			0.801969i	0.916085	+	0.400984i	$0.131332\pi$
$467$			174901.i			0.801969i	−0.916085	+	0.400984i	$0.868668\pi$
$468$		0			0
$469$	404511.			1.83901
$470$		0			0
$471$	−19711.6	−	284276.i	−0.0888544	−	1.28144i
$472$		0			0
$473$		−	17922.7i		−	0.0801089i
$474$		0			0
$475$	228418.			1.01238
$476$		0			0
$477$	−3553.30	−	25499.2i	−0.0156169	−	0.112070i
$478$		0			0
$479$		−	227873.i		−	0.993166i	−0.867989	−	0.496583i	$-0.834588\pi$
$479$		−	227873.i		−	0.993166i	0.867989	−	0.496583i	$-0.165412\pi$
$480$		0			0
$481$	36791.8			0.159023
$482$		0			0
$483$	395728.	−	27439.6i	1.69630	−	0.117621i
$484$		0			0
$485$			3697.44i			0.0157187i
$486$		0			0
$487$	−151441.			−0.638537			−0.319268	−	0.947664i	$-0.603437\pi$
$487$	−151441.			−0.638537			−0.319268	+	0.947664i	$0.603437\pi$
$488$		0			0
$489$	−20013.9	−	288636.i	−0.0836978	−	1.20707i
$490$		0			0
$491$		−	187262.i		−	0.776758i	−0.921500	−	0.388379i	$-0.873035\pi$
$491$		−	187262.i		−	0.776758i	0.921500	−	0.388379i	$-0.126965\pi$
$492$		0			0
$493$	78985.9			0.324979
$494$		0			0
$495$	−3958.14	+	551.563i	−0.0161540	+	0.00225105i
$496$		0			0
$497$			269796.i			1.09225i
$498$		0			0
$499$	−67671.2			−0.271771			−0.135885	−	0.990725i	$-0.543388\pi$
$499$	−67671.2			−0.271771			−0.135885	+	0.990725i	$0.543388\pi$
$500$		0			0
$501$	125260.	−	8685.44i	0.499040	−	0.0346032i
$502$		0			0
$503$			20555.7i			0.0812450i	0.999175	+	0.0406225i	$0.0129341\pi$
$503$			20555.7i			0.0812450i	−0.999175	+	0.0406225i	$0.987066\pi$
$504$		0			0
$505$	−10080.0			−0.0395255
$506$		0			0
$507$	−14589.4	−	210406.i	−0.0567574	−	0.818543i
$508$		0			0
$509$			246028.i			0.949620i	0.880088	+	0.474810i	$0.157483\pi$
$509$			246028.i			0.949620i	−0.880088	+	0.474810i	$0.842517\pi$
$510$		0			0
$511$	389853.			1.49300
$512$		0			0
$513$	−54963.9	−	260833.i	−0.208854	−	0.991123i
$514$		0			0
$515$		−	7445.17i		−	0.0280712i
$516$		0			0
$517$	−115528.			−0.432222
$518$		0			0
$519$	295045.	−	20458.3i	1.09535	−	0.0759512i
$520$		0			0
$521$		−	310680.i		−	1.14456i	−0.820059	−	0.572278i	$-0.806060\pi$
$521$		−	310680.i		−	1.14456i	0.820059	−	0.572278i	$-0.193940\pi$
$522$		0			0
$523$	87436.7			0.319662			0.159831	−	0.987144i	$-0.448905\pi$
$523$	87436.7			0.319662			0.159831	+	0.987144i	$0.448905\pi$
$524$		0			0
$525$	21708.7	+	313079.i	0.0787618	+	1.13589i
$526$		0			0
$527$			427717.i			1.54005i
$528$		0			0
$529$	−343617.			−1.22790
$530$		0			0
$531$	−14246.0	−	102233.i	−0.0505248	−	0.362577i
$532$		0			0
$533$		−	85132.3i		−	0.299668i
$534$		0			0
$535$	6695.29			0.0233917
$536$		0			0
$537$	188421.	−	13065.0i	0.653402	−	0.0453066i
$538$		0			0
$539$		−	62755.9i		−	0.216012i
$540$		0			0
$541$	−259352.			−0.886125			−0.443063	−	0.896491i	$-0.646108\pi$
$541$	−259352.			−0.886125			−0.443063	+	0.896491i	$0.646108\pi$
$542$		0			0
$543$	−34905.6	−	503401.i	−0.118385	−	1.70732i
$544$		0			0
$545$			7232.61i			0.0243502i
$546$		0			0
$547$	−301313.			−1.00703			−0.503515	−	0.863986i	$-0.667960\pi$
$547$	−301313.			−1.00703			−0.503515	+	0.863986i	$0.667960\pi$
$548$		0			0
$549$	−462999.	+	64518.5i	−1.53616	+	0.214062i
$550$		0			0
$551$			86467.0i			0.284805i
$552$		0			0
$553$	−520935.			−1.70347
$554$		0			0
$555$	−2593.09	+	179.804i	−0.00841844	+	0.000583731i
$556$		0			0
$557$		−	213353.i		−	0.687682i	−0.939028	−	0.343841i	$-0.888272\pi$
$557$		−	213353.i		−	0.687682i	0.939028	−	0.343841i	$-0.111728\pi$
$558$		0			0
$559$	14618.6			0.0467822
$560$		0			0
$561$	−18254.0	−	263255.i	−0.0580006	−	0.836472i
$562$		0			0
$563$		−	372694.i		−	1.17581i	−0.808931	−	0.587904i	$-0.799953\pi$
$563$		−	372694.i		−	1.17581i	0.808931	−	0.587904i	$-0.200047\pi$
$564$		0			0
$565$	12571.9			0.0393827
$566$		0			0
$567$	352284.	−	100125.i	1.09579	−	0.311442i
$568$		0			0
$569$			181871.i			0.561744i	0.959745	+	0.280872i	$0.0906236\pi$
$569$			181871.i			0.561744i	−0.959745	+	0.280872i	$0.909376\pi$
$570$		0			0
$571$	−166399.			−0.510361			−0.255180	−	0.966893i	$-0.582135\pi$
$571$	−166399.			−0.510361			−0.255180	+	0.966893i	$0.582135\pi$
$572$		0			0
$573$	429157.	−	29757.5i	1.30709	−	0.0906332i
$574$		0			0
$575$		−	493246.i		−	1.49186i
$576$		0			0
$577$	124612.			0.374289			0.187145	−	0.982332i	$-0.440077\pi$
$577$	124612.			0.374289			0.187145	+	0.982332i	$0.440077\pi$
$578$		0			0
$579$	26.8645	+	387.434i	8.01350e−5	+	0.00115569i
$580$		0			0
$581$			46829.8i			0.138730i
$582$		0			0
$583$	27901.4			0.0820897
$584$		0			0
$585$	−449.880	−	3228.44i	−0.00131457	−	0.00943368i
$586$		0			0
$587$			412727.i			1.19781i	0.800821	+	0.598904i	$0.204397\pi$
$587$			412727.i			1.19781i	−0.800821	+	0.598904i	$0.795603\pi$
$588$		0			0
$589$	−468228.			−1.34967
$590$		0			0
$591$	−360367.	+	24987.7i	−1.03174	+	0.0715404i
$592$		0			0
$593$			26644.9i			0.0757714i	0.999282	+	0.0378857i	$0.0120623\pi$
$593$			26644.9i			0.0757714i	−0.999282	+	0.0378857i	$0.987938\pi$
$594$		0			0
$595$	10479.4			0.0296006
$596$		0			0
$597$	14554.6	+	209904.i	0.0408369	+	0.588941i
$598$		0			0
$599$			227404.i			0.633789i	0.948461	+	0.316895i	$0.102640\pi$
$599$			227404.i			0.633789i	−0.948461	+	0.316895i	$0.897360\pi$
$600$		0			0
$601$	−152020.			−0.420873			−0.210437	−	0.977607i	$-0.567489\pi$
$601$	−152020.			−0.420873			−0.210437	+	0.977607i	$0.567489\pi$
$602$		0			0
$603$	−581362.	+	81012.3i	−1.59887	+	0.222801i
$604$		0			0
$605$			3897.95i			0.0106494i
$606$		0			0
$607$	249269.			0.676535			0.338267	−	0.941050i	$-0.390159\pi$
$607$	249269.			0.676535			0.338267	+	0.941050i	$0.390159\pi$
$608$		0			0
$609$	−118515.	+	8217.79i	−0.319551	+	0.0221575i
$610$		0			0
$611$		−	94230.1i		−	0.252410i
$612$		0			0
$613$	−351135.			−0.934443			−0.467221	−	0.884140i	$-0.654745\pi$
$613$	−351135.			−0.934443			−0.467221	+	0.884140i	$0.654745\pi$
$614$		0			0
$615$	416.047	+	6000.14i	0.00110000	+	0.0158639i
$616$		0			0
$617$			362003.i			0.950915i	0.879739	+	0.475458i	$0.157718\pi$
$617$			362003.i			0.950915i	−0.879739	+	0.475458i	$0.842282\pi$
$618$		0			0
$619$	−599882.			−1.56561			−0.782807	−	0.622265i	$-0.786213\pi$
$619$	−599882.			−1.56561			−0.782807	+	0.622265i	$0.786213\pi$
$620$		0			0
$621$	−563244.	+	118689.i	−1.46054	+	0.307772i
$622$		0			0
$623$			778312.i			2.00529i
$624$		0			0
$625$	390033.			0.998484
$626$		0			0
$627$	288189.	−	19982.9i	0.733065	−	0.0508304i
$628$		0			0
$629$		−	171637.i		−	0.433819i
$630$		0			0
$631$	332404.			0.834847			0.417424	−	0.908712i	$-0.362933\pi$
$631$	332404.			0.834847			0.417424	+	0.908712i	$0.362933\pi$
$632$		0			0
$633$	−33977.5	−	490016.i	−0.0847977	−	1.22293i
$634$		0			0
$635$			6869.34i			0.0170360i
$636$		0			0
$637$	51186.6			0.126147
$638$		0			0
$639$	−54032.6	−	387750.i	−0.132329	−	0.949621i
$640$		0			0
$641$			196470.i			0.478166i	0.970999	+	0.239083i	$0.0768469\pi$
$641$			196470.i			0.478166i	−0.970999	+	0.239083i	$0.923153\pi$
$642$		0			0
$643$	−79225.2			−0.191620			−0.0958102	−	0.995400i	$-0.530544\pi$
$643$	−79225.2			−0.191620			−0.0958102	+	0.995400i	$0.530544\pi$
$644$		0			0
$645$	−1030.32	+	71.4417i	−0.00247658	+	0.000171725i
$646$		0			0
$647$			222500.i			0.531523i	0.964039	+	0.265762i	$0.0856234\pi$
$647$			222500.i			0.531523i	−0.964039	+	0.265762i	$0.914377\pi$
$648$		0			0
$649$	111864.			0.265582
$650$		0			0
$651$	−44500.2	−	641772.i	−0.105002	−	1.51432i
$652$		0			0
$653$		−	456863.i		−	1.07142i	−0.844402	−	0.535710i	$-0.820044\pi$
$653$		−	456863.i		−	1.07142i	0.844402	−	0.535710i	$-0.179956\pi$
$654$		0			0
$655$	8498.75			0.0198094
$656$		0			0
$657$	−560297.	+	78076.8i	−1.29804	+	0.180880i
$658$		0			0
$659$			681304.i			1.56881i	0.620249	+	0.784405i	$0.287031\pi$
$659$			681304.i			1.56881i	−0.620249	+	0.784405i	$0.712969\pi$
$660$		0			0
$661$	107338.			0.245670			0.122835	−	0.992427i	$-0.460801\pi$
$661$	107338.			0.245670			0.122835	+	0.992427i	$0.460801\pi$
$662$		0			0
$663$	214723.	−	14888.8i	0.488485	−	0.0338714i
$664$		0			0
$665$			11471.9i			0.0259414i
$666$		0			0
$667$	186717.			0.419694
$668$		0			0
$669$	12455.9	+	179637.i	0.0278307	+	0.401368i
$670$		0			0
$671$		−	506616.i		−	1.12521i
$672$		0			0
$673$	−650619.			−1.43647			−0.718235	−	0.695800i	$-0.755050\pi$
$673$	−650619.			−1.43647			−0.718235	+	0.695800i	$0.755050\pi$
$674$		0			0
$675$	−93900.7	−	445609.i	−0.206092	−	0.978016i
$676$		0			0
$677$		−	549716.i		−	1.19939i	−0.800228	−	0.599695i	$-0.795288\pi$
$677$		−	549716.i		−	1.19939i	0.800228	−	0.599695i	$-0.204712\pi$
$678$		0			0
$679$	367213.			0.796487
$680$		0			0
$681$	−162352.	+	11257.4i	−0.350076	+	0.0242741i
$682$		0			0
$683$			525635.i			1.12679i	0.826188	+	0.563395i	$0.190505\pi$
$683$			525635.i			1.12679i	−0.826188	+	0.563395i	$0.809495\pi$
$684$		0			0
$685$	1407.88			0.00300044
$686$		0			0
$687$	35353.5	+	509860.i	0.0749063	+	1.08028i
$688$		0			0
$689$			22757.7i			0.0479390i
$690$		0			0
$691$	−269536.			−0.564495			−0.282247	−	0.959342i	$-0.591080\pi$
$691$	−269536.			−0.564495			−0.282247	+	0.959342i	$0.591080\pi$
$692$		0			0
$693$	54778.8	+	393105.i	0.114063	+	0.818544i
$694$		0			0
$695$			16024.4i			0.0331751i
$696$		0			0
$697$	−397149.			−0.817501
$698$		0			0
$699$	−517955.	+	35914.8i	−1.06008	+	0.0735054i
$700$		0			0
$701$		−	535305.i		−	1.08935i	−0.838649	−	0.544673i	$-0.816654\pi$
$701$		−	535305.i		−	1.08935i	0.838649	−	0.544673i	$-0.183346\pi$
$702$		0			0
$703$	187893.			0.380190
$704$		0			0
$705$	460.508	+	6641.35i	0.000926529	+	0.0133622i
$706$		0			0
$707$			1.00110e6i			2.00281i
$708$		0			0
$709$	−601177.			−1.19594			−0.597971	−	0.801518i	$-0.704026\pi$
$709$	−601177.			−1.19594			−0.597971	+	0.801518i	$0.704026\pi$
$710$		0			0
$711$	748688.	−	104329.i	1.48102	−	0.206379i
$712$		0			0
$713$			1.01109e6i			1.98890i
$714$		0			0
$715$	3532.58			0.00691002
$716$		0			0
$717$	−285311.	+	19783.3i	−0.554983	+	0.0384823i
$718$		0			0
$719$			794011.i			1.53592i	0.640497	+	0.767960i	$0.278728\pi$
$719$			794011.i			1.53592i	−0.640497	+	0.767960i	$0.721272\pi$
$720$		0			0
$721$	−739421.			−1.42240
$722$		0			0
$723$	−17498.0	−	252352.i	−0.0334743	−	0.482759i
$724$		0			0
$725$			147721.i			0.281038i
$726$		0			0
$727$	−257472.			−0.487149			−0.243574	−	0.969882i	$-0.578320\pi$
$727$	−257472.			−0.487149			−0.243574	+	0.969882i	$0.578320\pi$
$728$		0			0
$729$	−486251.	+	214453.i	−0.914966	+	0.403531i
$730$		0			0
$731$		−	68196.8i		−	0.127623i
$732$		0			0
$733$	113268.			0.210813			0.105407	−	0.994429i	$-0.466386\pi$
$733$	113268.			0.210813			0.105407	+	0.994429i	$0.466386\pi$
$734$		0			0
$735$	−3607.64	+	250.152i	−0.00667802	+	0.000463051i
$736$		0			0
$737$		−	636129.i		−	1.17114i
$738$		0			0
$739$	−309173.			−0.566125			−0.283062	−	0.959102i	$-0.591350\pi$
$739$	−309173.			−0.566125			−0.283062	+	0.959102i	$0.591350\pi$
$740$		0			0
$741$	16299.0	+	235060.i	0.0296841	+	0.428098i
$742$		0			0
$743$			693611.i			1.25643i	0.778040	+	0.628215i	$0.216214\pi$
$743$			693611.i			1.25643i	−0.778040	+	0.628215i	$0.783786\pi$
$744$		0			0
$745$	−18797.3			−0.0338674
$746$		0			0
$747$	−9378.72	−	67303.8i	−0.0168075	−	0.120614i
$748$		0			0
$749$		−	664947.i		−	1.18529i
$750$		0			0
$751$	642817.			1.13974			0.569872	−	0.821733i	$-0.306993\pi$
$751$	642817.			1.13974			0.569872	+	0.821733i	$0.306993\pi$
$752$		0			0
$753$	1.03063e6	−	71463.2i	1.81765	−	0.126035i
$754$		0			0
$755$		−	7386.79i		−	0.0129587i
$756$		0			0
$757$	127284.			0.222117			0.111058	−	0.993814i	$-0.464576\pi$
$757$	127284.			0.222117			0.111058	+	0.993814i	$0.464576\pi$
$758$		0			0
$759$	−43151.2	−	622318.i	−0.0749048	−	1.08026i
$760$		0			0
$761$		−	125775.i		−	0.217183i	−0.994086	−	0.108592i	$-0.965366\pi$
$761$		−	125775.i		−	0.217183i	0.994086	−	0.108592i	$-0.0346341\pi$
$762$		0			0
$763$	718310.			1.23385
$764$		0			0
$765$	−15060.9	+	2098.73i	−0.0257353	+	0.00358619i
$766$		0			0
$767$			91241.0i			0.155096i
$768$		0			0
$769$	636165.			1.07576			0.537882	−	0.843020i	$-0.319225\pi$
$769$	636165.			1.07576			0.537882	+	0.843020i	$0.319225\pi$
$770$		0			0
$771$	457263.	−	31706.4i	0.769231	−	0.0533382i
$772$		0			0
$773$			346515.i			0.579913i	0.957040	+	0.289957i	$0.0936409\pi$
$773$			346515.i			0.579913i	−0.957040	+	0.289957i	$0.906359\pi$
$774$		0			0
$775$	−799923.			−1.33182
$776$		0			0
$777$	17857.3	+	257534.i	0.0295783	+	0.426572i
$778$		0			0
$779$		−	434765.i		−	0.716440i
$780$		0			0
$781$	424278.			0.695582
$782$		0			0
$783$	168684.	−	35545.9i	0.275138	−	0.0579784i
$784$		0			0
$785$		−	17795.6i		−	0.0288785i
$786$		0			0
$787$	−666749.			−1.07650			−0.538249	−	0.842786i	$-0.680914\pi$
$787$	−666749.			−1.07650			−0.538249	+	0.842786i	$0.680914\pi$
$788$		0			0
$789$	−19364.9	+	1342.76i	−0.0311073	+	0.00215697i
$790$		0			0
$791$		−	1.24859e6i		−	1.99557i
$792$		0			0
$793$	413219.			0.657104
$794$		0			0
$795$	−111.218	−	1603.96i	−0.000175971	−	0.00253781i
$796$		0			0
$797$		−	512411.i		−	0.806681i	−0.915050	−	0.403341i	$-0.867849\pi$
$797$		−	512411.i		−	0.806681i	0.915050	−	0.403341i	$-0.132151\pi$
$798$		0			0
$799$	−439591.			−0.688582
$800$		0			0
$801$	−155874.	−	1.11859e6i	−0.242946	−	1.74343i
$802$		0			0
$803$		−	613080.i		−	0.950792i
$804$		0			0
$805$	24772.5			0.0382277
$806$		0			0
$807$	−1.08840e6	+	75469.5i	−1.67126	+	0.115884i
$808$		0			0
$809$			450334.i			0.688079i	0.938955	+	0.344039i	$0.111795\pi$
$809$			450334.i			0.688079i	−0.938955	+	0.344039i	$0.888205\pi$
$810$		0			0
$811$	−128497.			−0.195367			−0.0976833	−	0.995218i	$-0.531143\pi$
$811$	−128497.			−0.195367			−0.0976833	+	0.995218i	$0.531143\pi$
$812$		0			0
$813$	13213.4	+	190561.i	0.0199910	+	0.288305i
$814$		0			0
$815$		−	18068.6i		−	0.0272025i
$816$		0			0
$817$	74656.0			0.111846
$818$		0			0
$819$	−320634.	+	44680.1i	−0.478016	+	0.0666110i
$820$		0			0
$821$			1.03116e6i			1.52982i	0.644140	+	0.764908i	$0.277215\pi$
$821$			1.03116e6i			1.52982i	−0.644140	+	0.764908i	$0.722785\pi$
$822$		0			0
$823$	−456962.			−0.674652			−0.337326	−	0.941388i	$-0.609523\pi$
$823$	−456962.			−0.674652			−0.337326	+	0.941388i	$0.609523\pi$
$824$		0			0
$825$	492344.	−	34138.9i	0.723371	−	0.0501582i
$826$		0			0
$827$			476279.i			0.696386i	0.937423	+	0.348193i	$0.113205\pi$
$827$			476279.i			0.696386i	−0.937423	+	0.348193i	$0.886795\pi$
$828$		0			0
$829$	−527555.			−0.767641			−0.383821	−	0.923408i	$-0.625392\pi$
$829$	−527555.			−0.767641			−0.383821	+	0.923408i	$0.625392\pi$
$830$		0			0
$831$	−72318.6	−	1.04296e6i	−0.104724	−	1.51031i
$832$		0			0
$833$		−	238790.i		−	0.344132i
$834$		0			0
$835$	7841.24			0.0112464
$836$		0			0
$837$	192485.	+	913441.i	0.274755	+	1.30386i
$838$		0			0
$839$			1.17995e6i			1.67626i	0.545472	+	0.838129i	$0.316350\pi$
$839$			1.17995e6i			1.67626i	−0.545472	+	0.838129i	$0.683650\pi$
$840$		0			0
$841$	651362.			0.920937
$842$		0			0
$843$	1.05513e6	−	73162.0i	1.48474	−	0.102951i
$844$		0			0
$845$		−	13171.4i		−	0.0184467i
$846$		0			0
$847$	387127.			0.539619
$848$		0			0
$849$	−45561.3	−	657076.i	−0.0632093	−	0.911591i
$850$		0			0
$851$		−	405738.i		−	0.560256i
$852$		0			0
$853$	105304.			0.144727			0.0723633	−	0.997378i	$-0.476946\pi$
$853$	105304.			0.144727			0.0723633	+	0.997378i	$0.476946\pi$
$854$		0			0
$855$	−2297.51	−	16487.4i	−0.00314286	−	0.0225538i
$856$		0			0
$857$			610982.i			0.831891i	0.909389	+	0.415946i	$0.136549\pi$
$857$			610982.i			0.831891i	−0.909389	+	0.415946i	$0.863451\pi$
$858$		0			0
$859$	732588.			0.992827			0.496413	−	0.868086i	$-0.334650\pi$
$859$	732588.			0.992827			0.496413	+	0.868086i	$0.334650\pi$
$860$		0			0
$861$	595907.	−	41319.9i	0.803844	−	0.0557382i
$862$		0			0
$863$		−	956613.i		−	1.28444i	−0.766519	−	0.642222i	$-0.778013\pi$
$863$		−	956613.i		−	1.28444i	0.766519	−	0.642222i	$-0.221987\pi$
$864$		0			0
$865$	18469.8			0.0246848
$866$		0			0
$867$	−17460.5	−	251812.i	−0.0232284	−	0.334995i
$868$		0			0
$869$			819218.i			1.08483i
$870$		0			0
$871$	518856.			0.683929
$872$		0			0
$873$	−527758.	+	73542.6i	−0.692479	+	0.0964963i
$874$		0			0
$875$			39207.3i			0.0512095i
$876$		0			0
$877$	778581.			1.01229			0.506144	−	0.862449i	$-0.331070\pi$
$877$	778581.			1.01229			0.506144	+	0.862449i	$0.331070\pi$
$878$		0			0
$879$	−812998.	+	56372.9i	−1.05223	+	0.0729613i
$880$		0			0
$881$		−	431230.i		−	0.555594i	−0.960640	−	0.277797i	$-0.910396\pi$
$881$		−	431230.i		−	0.555594i	0.960640	−	0.277797i	$-0.0896042\pi$
$882$		0			0
$883$	1.05031e6			1.34708			0.673541	−	0.739150i	$-0.264772\pi$
$883$	1.05031e6			1.34708			0.673541	+	0.739150i	$0.264772\pi$
$884$		0			0
$885$	−445.900	−	6430.68i	−0.000569313	−	0.00821051i
$886$		0			0
$887$		−	422335.i		−	0.536797i	−0.963308	−	0.268398i	$-0.913506\pi$
$887$		−	422335.i		−	0.536797i	0.963308	−	0.268398i	$-0.0864944\pi$
$888$		0			0
$889$	682232.			0.863234
$890$		0			0
$891$	−157456.	−	553999.i	−0.198337	−	0.697837i
$892$		0			0
$893$		−	481227.i		−	0.603458i
$894$		0			0
$895$	11795.1			0.0147251
$896$		0			0
$897$	507591.	−	35196.1i	0.630854	−	0.0437431i
$898$		0			0
$899$		−	302809.i		−	0.374671i
$900$		0			0
$901$	106166.			0.130779
$902$		0			0
$903$	7095.28	+	102327.i	0.00870149	+	0.125491i
$904$		0			0
$905$		−	31512.8i		−	0.0384760i
$906$		0			0
$907$	268248.			0.326078			0.163039	−	0.986620i	$-0.447870\pi$
$907$	268248.			0.326078			0.163039	+	0.986620i	$0.447870\pi$
$908$		0			0
$909$	−200493.	−	1.43878e6i	−0.242645	−	1.74127i
$910$		0			0
$911$			1.04830e6i			1.26313i	0.775324	+	0.631564i	$0.217587\pi$
$911$			1.04830e6i			1.26313i	−0.775324	+	0.631564i	$0.782413\pi$
$912$		0			0
$913$	73644.1			0.0883480
$914$		0			0
$915$	−29123.7	+	2019.43i	−0.0347860	+	0.00241205i
$916$		0			0
$917$		−	844058.i		−	1.00377i
$918$		0			0
$919$	−220409.			−0.260975			−0.130487	−	0.991450i	$-0.541654\pi$
$919$	−220409.			−0.260975			−0.130487	+	0.991450i	$0.541654\pi$
$920$		0			0
$921$	−29281.3	−	422289.i	−0.0345201	−	0.497841i
$922$		0			0
$923$			346061.i			0.406208i
$924$		0			0
$925$	320998.			0.375162
$926$		0			0
$927$	1.06270e6	−	148086.i	1.23666	−	0.172327i
$928$		0			0
$929$			510188.i			0.591152i	0.955319	+	0.295576i	$0.0955115\pi$
$929$			510188.i			0.591152i	−0.955319	+	0.295576i	$0.904489\pi$
$930$		0			0
$931$	261406.			0.301590
$932$		0			0
$933$	1.49242e6	−	103483.i	1.71446	−	0.118880i
$934$		0			0
$935$		−	16479.8i		−	0.0188507i
$936$		0			0
$937$	−899577.			−1.02461			−0.512306	−	0.858803i	$-0.671208\pi$
$937$	−899577.			−1.02461			−0.512306	+	0.858803i	$0.671208\pi$
$938$		0			0
$939$	71023.7	+	1.02429e6i	0.0805512	+	1.16169i
$940$		0			0
$941$			1.71356e6i			1.93517i	0.252546	+	0.967585i	$0.418732\pi$
$941$			1.71356e6i			1.93517i	−0.252546	+	0.967585i	$0.581268\pi$
$942$		0			0
$943$	−938834.			−1.05576
$944$		0			0
$945$	22380.0	−	4716.02i	0.0250609	−	0.00528095i
$946$		0			0
$947$		−	149981.i		−	0.167239i	−0.996498	−	0.0836193i	$-0.973352\pi$
$947$		−	149981.i		−	0.167239i	0.996498	−	0.0836193i	$-0.0266480\pi$
$948$		0			0
$949$	500056.			0.555247
$950$		0			0
$951$	−294220.	+	20401.1i	−0.325320	+	0.0225575i
$952$		0			0
$953$			530581.i			0.584206i	0.956387	+	0.292103i	$0.0943549\pi$
$953$			530581.i			0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$954$		0			0
$955$	26865.2			0.0294566
$956$		0			0
$957$	12923.2	+	186376.i	0.0141106	+	0.203501i
$958$		0			0
$959$		−	139825.i		−	0.152036i
$960$		0			0
$961$	716220.			0.775532
$962$		0			0
$963$	133170.	+	955661.i	0.143600	+	1.03051i
$964$		0			0
$965$			24.2534i			2.60446e-5i
$966$		0			0
$967$	−1.45353e6			−1.55443			−0.777216	−	0.629233i	$-0.783369\pi$
$967$	−1.45353e6			−1.55443			−0.777216	+	0.629233i	$0.783369\pi$
$968$		0			0
$969$	1.09658e6	−	76036.1i	1.16786	−	0.0809790i
$970$		0			0
$971$		−	177413.i		−	0.188169i	−0.995564	−	0.0940845i	$-0.970008\pi$
$971$		−	177413.i		−	0.188169i	0.995564	−	0.0940845i	$-0.0299924\pi$
$972$		0			0
$973$	1.59147e6			1.68102
$974$		0			0
$975$	27845.3	+	401579.i	0.0292916	+	0.422436i
$976$		0			0
$977$		−	301481.i		−	0.315842i	−0.987452	−	0.157921i	$-0.949521\pi$
$977$		−	301481.i		−	0.315842i	0.987452	−	0.157921i	$-0.0504792\pi$
$978$		0			0
$979$	1.22397e6			1.27704
$980$		0			0
$981$	−1.03235e6	+	143858.i	−1.07273	+	0.149484i
$982$		0			0
$983$			178724.i			0.184959i	0.995715	+	0.0924795i	$0.0294793\pi$
$983$			178724.i			0.184959i	−0.995715	+	0.0924795i	$0.970521\pi$
$984$		0			0
$985$	−22559.0			−0.0232513
$986$		0			0
$987$	659589.	−	45735.6i	0.677079	−	0.0469483i
$988$		0			0
$989$		−	161213.i		−	0.164819i
$990$		0			0
$991$	−630243.			−0.641743			−0.320871	−	0.947123i	$-0.603976\pi$
$991$	−630243.			−0.641743			−0.320871	+	0.947123i	$0.603976\pi$
$992$		0			0
$993$	85657.7	+	1.23534e6i	0.0868696	+	1.25282i
$994$		0			0
$995$			13140.0i			0.0132724i
$996$		0			0
$997$	544850.			0.548134			0.274067	−	0.961711i	$-0.411631\pi$
$997$	544850.			0.548134			0.274067	+	0.961711i	$0.411631\pi$
$998$		0			0
$999$	−77241.4	−	366551.i	−0.0773961	−	0.367286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.c.257.8	yes	16
3.2	odd	2	inner	384.5.e.c.257.7	yes	16
4.3	odd	2		384.5.e.b.257.9	yes	16
8.3	odd	2		384.5.e.d.257.8	yes	16
8.5	even	2		384.5.e.a.257.9	✓	16
12.11	even	2		384.5.e.b.257.10	yes	16
24.5	odd	2		384.5.e.a.257.10	yes	16
24.11	even	2		384.5.e.d.257.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.9	✓	16	8.5	even	2
384.5.e.a.257.10	yes	16	24.5	odd	2
384.5.e.b.257.9	yes	16	4.3	odd	2
384.5.e.b.257.10	yes	16	12.11	even	2
384.5.e.c.257.7	yes	16	3.2	odd	2	inner
384.5.e.c.257.8	yes	16	1.1	even	1	trivial
384.5.e.d.257.7	yes	16	24.11	even	2
384.5.e.d.257.8	yes	16	8.3	odd	2

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.622561 + 8.97844i) q^{3} +0.562050i q^{5} +55.8203 q^{7} +(-80.2248 + 11.1793i) q^{9} +O(q^{10})\)	\(q+(0.622561 + 8.97844i) q^{3} +0.562050i q^{5} +55.8203 q^{7} +(-80.2248 + 11.1793i) q^{9} -87.7824i q^{11} +71.5994 q^{13} +(-5.04633 + 0.349910i) q^{15} -334.017i q^{17} +365.653 q^{19} +(34.7515 + 501.179i) q^{21} -789.593i q^{23} +624.684 q^{25} +(-150.317 - 713.334i) q^{27} +236.473i q^{29} -1280.52 q^{31} +(788.149 - 54.6499i) q^{33} +31.3738i q^{35} +513.856 q^{37} +(44.5750 + 642.851i) q^{39} -1189.01i q^{41} +204.172 q^{43} +(-6.28330 - 45.0903i) q^{45} -1316.07i q^{47} +714.903 q^{49} +(2998.95 - 207.946i) q^{51} +317.847i q^{53} +49.3381 q^{55} +(227.641 + 3283.00i) q^{57} +1274.33i q^{59} +5771.27 q^{61} +(-4478.17 + 624.029i) q^{63} +40.2424i q^{65} +7246.66 q^{67} +(7089.32 - 491.570i) q^{69} +4833.29i q^{71} +6984.08 q^{73} +(388.904 + 5608.69i) q^{75} -4900.04i q^{77} -9332.37 q^{79} +(6311.05 - 1793.71i) q^{81} +838.939i q^{83} +187.734 q^{85} +(-2123.16 + 147.219i) q^{87} +13943.2i q^{89} +3996.70 q^{91} +(-797.204 - 11497.1i) q^{93} +205.515i q^{95} +6578.49 q^{97} +(981.342 + 7042.33i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 80 q^{7}+O(q^{10}) \) 16 * q + 8 * q^3 - 80 * q^7	\( 16 q + 8 q^{3} - 80 q^{7} - 416 q^{15} - 816 q^{19} + 608 q^{21} - 2000 q^{25} - 280 q^{27} - 592 q^{31} - 496 q^{33} + 2240 q^{37} + 16 q^{39} - 368 q^{43} + 800 q^{45} + 3984 q^{49} + 352 q^{51} - 1920 q^{55} + 560 q^{57} - 3520 q^{61} + 816 q^{63} + 3536 q^{67} - 10784 q^{69} + 3680 q^{73} - 5112 q^{75} + 14448 q^{79} - 624 q^{81} - 11136 q^{85} + 14944 q^{87} + 22944 q^{91} + 13760 q^{93} + 3264 q^{97} + 26976 q^{99}+O(q^{100}) \) 16 * q + 8 * q^3 - 80 * q^7 - 416 * q^15 - 816 * q^19 + 608 * q^21 - 2000 * q^25 - 280 * q^27 - 592 * q^31 - 496 * q^33 + 2240 * q^37 + 16 * q^39 - 368 * q^43 + 800 * q^45 + 3984 * q^49 + 352 * q^51 - 1920 * q^55 + 560 * q^57 - 3520 * q^61 + 816 * q^63 + 3536 * q^67 - 10784 * q^69 + 3680 * q^73 - 5112 * q^75 + 14448 * q^79 - 624 * q^81 - 11136 * q^85 + 14944 * q^87 + 22944 * q^91 + 13760 * q^93 + 3264 * q^97 + 26976 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more