LMFDB - Embedded newform 160.3.e.c.79.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(79,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("160.79");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.e (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:	$4.35968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.53824000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{6} + 36x^{4} + 96x^{2} + 256 $ x^8 + 6x^6 + 36x^4 + 96*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			79.7
Root			$1.34500 - 1.48020i$ of defining polynomial
Character	$\chi$	$=$	160.79
Dual form			160.3.e.c.79.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.79002i q^{3} +(-4.35250 - 2.46084i) q^{5} -7.67752 q^{7} -13.9443 q^{9} +0.472136 q^{11} +4.10995 q^{13} +(11.7875 - 20.8486i) q^{15} -2.26154i q^{17} -26.3607 q^{19} -36.7754i q^{21} -9.73249 q^{23} +(12.8885 + 21.4216i) q^{25} -23.6832i q^{27} +41.6971i q^{29} +22.0104i q^{31} +2.26154i q^{33} +(33.4164 + 18.8931i) q^{35} -51.7449 q^{37} +19.6867i q^{39} +15.0557 q^{41} +9.31310i q^{43} +(60.6925 + 34.3146i) q^{45} -8.76226 q^{47} +9.94427 q^{49} +10.8328 q^{51} +39.9002 q^{53} +(-2.05497 - 1.16185i) q^{55} -126.268i q^{57} +77.1935 q^{59} +14.7650i q^{61} +107.057 q^{63} +(-17.8885 - 10.1139i) q^{65} -75.8395i q^{67} -46.6188i q^{69} +81.0705i q^{71} +83.4249i q^{73} +(-102.610 + 61.7364i) q^{75} -3.62483 q^{77} -100.757i q^{79} -12.0557 q^{81} -0.266939i q^{83} +(-5.56528 + 9.84336i) q^{85} -199.730 q^{87} +85.5542 q^{89} -31.5542 q^{91} -105.430 q^{93} +(114.735 + 64.8694i) q^{95} +99.1297i q^{97} -6.58359 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9} - 32 q^{11} - 32 q^{19} - 40 q^{25} + 160 q^{35} + 192 q^{41} + 8 q^{49} - 128 q^{51} + 224 q^{59} - 320 q^{75} - 168 q^{81} + 112 q^{89} + 320 q^{91} - 160 q^{99}+O(q^{100}) $ 8 * q - 40 * q^9 - 32 * q^11 - 32 * q^19 - 40 * q^25 + 160 * q^35 + 192 * q^41 + 8 * q^49 - 128 * q^51 + 224 * q^59 - 320 * q^75 - 168 * q^81 + 112 * q^89 + 320 * q^91 - 160 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			4.79002i			1.59667i	0.602212	+	0.798336i	$0.294286\pi$
$3$			4.79002i			1.59667i	−0.602212	+	0.798336i	$0.705714\pi$
$4$		0			0
$5$	−4.35250	−	2.46084i	−0.870500	−	0.492168i
$5$	−4.35250	−	2.46084i	−0.870500	−	0.492168i
$6$		0			0
$7$	−7.67752			−1.09679			−0.548394	−	0.836220i	$-0.684761\pi$
$7$	−7.67752			−1.09679			−0.548394	+	0.836220i	$0.684761\pi$
$8$		0			0
$9$	−13.9443			−1.54936
$10$		0			0
$11$	0.472136			0.0429215			0.0214607	−	0.999770i	$-0.493168\pi$
$11$	0.472136			0.0429215			0.0214607	+	0.999770i	$0.493168\pi$
$12$		0			0
$13$	4.10995			0.316150			0.158075	−	0.987427i	$-0.449471\pi$
$13$	4.10995			0.316150			0.158075	+	0.987427i	$0.449471\pi$
$14$		0			0
$15$	11.7875	−	20.8486i	0.785831	−	1.38990i
$16$		0			0
$17$		−	2.26154i		−	0.133032i	−0.997785	−	0.0665159i	$-0.978812\pi$
$17$		−	2.26154i		−	0.133032i	0.997785	−	0.0665159i	$-0.0211883\pi$
$18$		0			0
$19$	−26.3607			−1.38740			−0.693702	−	0.720262i	$-0.744022\pi$
$19$	−26.3607			−1.38740			−0.693702	+	0.720262i	$0.744022\pi$
$20$		0			0
$21$		−	36.7754i		−	1.75121i
$22$		0			0
$23$	−9.73249			−0.423152			−0.211576	−	0.977362i	$-0.567860\pi$
$23$	−9.73249			−0.423152			−0.211576	+	0.977362i	$0.567860\pi$
$24$		0			0
$25$	12.8885	+	21.4216i	0.515542	+	0.856864i
$26$		0			0
$27$		−	23.6832i		−	0.877154i
$28$		0			0
$29$			41.6971i			1.43783i	0.695097	+	0.718916i	$0.255361\pi$
$29$			41.6971i			1.43783i	−0.695097	+	0.718916i	$0.744639\pi$
$30$		0			0
$31$			22.0104i			0.710013i	0.934864	+	0.355007i	$0.115521\pi$
$31$			22.0104i			0.710013i	−0.934864	+	0.355007i	$0.884479\pi$
$32$		0			0
$33$			2.26154i			0.0685315i
$34$		0			0
$35$	33.4164	+	18.8931i	0.954755	+	0.539804i
$36$		0			0
$37$	−51.7449			−1.39851			−0.699256	−	0.714872i	$-0.746485\pi$
$37$	−51.7449			−1.39851			−0.699256	+	0.714872i	$0.746485\pi$
$38$		0			0
$39$			19.6867i			0.504787i
$40$		0			0
$41$	15.0557			0.367213			0.183606	−	0.983000i	$-0.441223\pi$
$41$	15.0557			0.367213			0.183606	+	0.983000i	$0.441223\pi$
$42$		0			0
$43$			9.31310i			0.216584i	0.994119	+	0.108292i	$0.0345381\pi$
$43$			9.31310i			0.216584i	−0.994119	+	0.108292i	$0.965462\pi$
$44$		0			0
$45$	60.6925	+	34.3146i	1.34872	+	0.762547i
$46$		0			0
$47$	−8.76226			−0.186431			−0.0932156	−	0.995646i	$-0.529715\pi$
$47$	−8.76226			−0.186431			−0.0932156	+	0.995646i	$0.529715\pi$
$48$		0			0
$49$	9.94427			0.202944
$50$		0			0
$51$	10.8328			0.212408
$52$		0			0
$53$	39.9002			0.752834			0.376417	−	0.926450i	$-0.377156\pi$
$53$	39.9002			0.752834			0.376417	+	0.926450i	$0.377156\pi$
$54$		0			0
$55$	−2.05497	−	1.16185i	−0.0373631	−	0.0211246i
$56$		0			0
$57$		−	126.268i		−	2.21523i
$58$		0			0
$59$	77.1935			1.30836			0.654182	−	0.756337i	$-0.273013\pi$
$59$	77.1935			1.30836			0.654182	+	0.756337i	$0.273013\pi$
$60$		0			0
$61$			14.7650i			0.242050i	0.992649	+	0.121025i	$0.0386181\pi$
$61$			14.7650i			0.242050i	−0.992649	+	0.121025i	$0.961382\pi$
$62$		0			0
$63$	107.057			1.69932
$64$		0			0
$65$	−17.8885	−	10.1139i	−0.275208	−	0.155599i
$66$		0			0
$67$		−	75.8395i		−	1.13193i	−0.824428	−	0.565966i	$-0.808503\pi$
$67$		−	75.8395i		−	1.13193i	0.824428	−	0.565966i	$-0.191497\pi$
$68$		0			0
$69$		−	46.6188i		−	0.675635i
$70$		0			0
$71$			81.0705i			1.14184i	0.821006	+	0.570919i	$0.193413\pi$
$71$			81.0705i			1.14184i	−0.821006	+	0.570919i	$0.806587\pi$
$72$		0			0
$73$			83.4249i			1.14281i	0.820669	+	0.571403i	$0.193601\pi$
$73$			83.4249i			1.14281i	−0.820669	+	0.571403i	$0.806399\pi$
$74$		0			0
$75$	−102.610	+	61.7364i	−1.36813	+	0.823151i
$76$		0			0
$77$	−3.62483			−0.0470757
$78$		0			0
$79$		−	100.757i		−	1.27541i	−0.770281	−	0.637704i	$-0.779884\pi$
$79$		−	100.757i		−	1.27541i	0.770281	−	0.637704i	$-0.220116\pi$
$80$		0			0
$81$	−12.0557			−0.148836
$82$		0			0
$83$		−	0.266939i		−	0.00321613i	−0.999999	−	0.00160806i	$-0.999488\pi$
$83$		−	0.266939i		−	0.00321613i	0.999999	−	0.00160806i	$-0.000511863\pi$
$84$		0			0
$85$	−5.56528	+	9.84336i	−0.0654739	+	0.115804i
$86$		0			0
$87$	−199.730			−2.29575
$88$		0			0
$89$	85.5542			0.961283			0.480641	−	0.876917i	$-0.340404\pi$
$89$	85.5542			0.961283			0.480641	+	0.876917i	$0.340404\pi$
$90$		0			0
$91$	−31.5542			−0.346749
$92$		0			0
$93$	−105.430			−1.13366
$94$		0			0
$95$	114.735	+	64.8694i	1.20774	+	0.682836i
$96$		0			0
$97$			99.1297i			1.02196i	0.859594	+	0.510978i	$0.170716\pi$
$97$			99.1297i			1.02196i	−0.859594	+	0.510978i	$0.829284\pi$
$98$		0			0
$99$	−6.58359			−0.0665009
$100$		0			0
$101$			105.405i			1.04361i	0.853065	+	0.521805i	$0.174741\pi$
$101$			105.405i			1.04361i	−0.853065	+	0.521805i	$0.825259\pi$
$102$		0			0
$103$	−101.863			−0.988958			−0.494479	−	0.869190i	$-0.664641\pi$
$103$	−101.863			−0.988958			−0.494479	+	0.869190i	$0.664641\pi$
$104$		0			0
$105$	−90.4984	+	160.065i	−0.861890	+	1.52443i
$106$		0			0
$107$		−	32.9962i		−	0.308376i	−0.988042	−	0.154188i	$-0.950724\pi$
$107$		−	32.9962i		−	0.308376i	0.988042	−	0.154188i	$-0.0492762\pi$
$108$		0			0
$109$		−	181.554i		−	1.66563i	−0.553552	−	0.832814i	$-0.686728\pi$
$109$		−	181.554i		−	1.66563i	0.553552	−	0.832814i	$-0.313272\pi$
$110$		0			0
$111$		−	247.859i		−	2.23296i
$112$		0			0
$113$		−	76.6403i		−	0.678233i	−0.940744	−	0.339116i	$-0.889872\pi$
$113$		−	76.6403i		−	0.678233i	0.940744	−	0.339116i	$-0.110128\pi$
$114$		0			0
$115$	42.3607	+	23.9501i	0.368354	+	0.208262i
$116$		0			0
$117$	−57.3102			−0.489831
$118$		0			0
$119$			17.3630i			0.145908i
$120$		0			0
$121$	−120.777			−0.998158
$122$		0			0
$123$			72.1172i			0.586319i
$124$		0			0
$125$	−3.38228	−	124.954i	−0.0270582	−	0.999634i
$126$		0			0
$127$	142.962			1.12569			0.562843	−	0.826564i	$-0.309708\pi$
$127$	142.962			1.12569			0.562843	+	0.826564i	$0.309708\pi$
$128$		0			0
$129$	−44.6099			−0.345813
$130$		0			0
$131$	−110.138			−0.840746			−0.420373	−	0.907351i	$-0.638101\pi$
$131$	−110.138			−0.840746			−0.420373	+	0.907351i	$0.638101\pi$
$132$		0			0
$133$	202.385			1.52169
$134$		0			0
$135$	−58.2804	+	103.081i	−0.431707	+	0.763563i
$136$		0			0
$137$		−	225.398i		−	1.64524i	−0.568592	−	0.822620i	$-0.692512\pi$
$137$		−	225.398i		−	1.64524i	0.568592	−	0.822620i	$-0.307488\pi$
$138$		0			0
$139$	−97.4164			−0.700837			−0.350419	−	0.936593i	$-0.613961\pi$
$139$	−97.4164			−0.700837			−0.350419	+	0.936593i	$0.613961\pi$
$140$		0			0
$141$		−	41.9714i		−	0.297669i
$142$		0			0
$143$	1.94045			0.0135696
$144$		0			0
$145$	102.610	−	181.487i	0.707655	−	1.25163i
$146$		0			0
$147$			47.6332i			0.324036i
$148$		0			0
$149$			191.397i			1.28454i	0.766477	+	0.642271i	$0.222008\pi$
$149$			191.397i			1.28454i	−0.766477	+	0.642271i	$0.777992\pi$
$150$		0			0
$151$		−	68.3549i		−	0.452682i	−0.974048	−	0.226341i	$-0.927324\pi$
$151$		−	68.3549i		−	0.452682i	0.974048	−	0.226341i	$-0.0726763\pi$
$152$		0			0
$153$			31.5355i			0.206115i
$154$		0			0
$155$	54.1641	−	95.8004i	0.349446	−	0.618067i
$156$		0			0
$157$	−155.235			−0.988756			−0.494378	−	0.869247i	$-0.664604\pi$
$157$	−155.235			−0.988756			−0.494378	+	0.869247i	$0.664604\pi$
$158$		0			0
$159$			191.123i			1.20203i
$160$		0			0
$161$	74.7214			0.464108
$162$		0			0
$163$			90.4765i			0.555070i	0.960715	+	0.277535i	$0.0895175\pi$
$163$			90.4765i			0.555070i	−0.960715	+	0.277535i	$0.910482\pi$
$164$		0			0
$165$	5.56528	−	9.84336i	0.0337290	−	0.0596567i
$166$		0			0
$167$	100.778			0.603461			0.301730	−	0.953393i	$-0.402436\pi$
$167$	100.778			0.603461			0.301730	+	0.953393i	$0.402436\pi$
$168$		0			0
$169$	−152.108			−0.900049
$170$		0			0
$171$	367.580			2.14959
$172$		0			0
$173$	54.6556			0.315928			0.157964	−	0.987445i	$-0.449507\pi$
$173$	54.6556			0.315928			0.157964	+	0.987445i	$0.449507\pi$
$174$		0			0
$175$	−98.9520	−	164.465i	−0.565440	−	0.939799i
$176$		0			0
$177$			369.758i			2.08903i
$178$		0			0
$179$	106.807			0.596684			0.298342	−	0.954459i	$-0.403566\pi$
$179$	106.807			0.596684			0.298342	+	0.954459i	$0.403566\pi$
$180$		0			0
$181$			144.778i			0.799879i	0.916542	+	0.399939i	$0.130969\pi$
$181$			144.778i			0.799879i	−0.916542	+	0.399939i	$0.869031\pi$
$182$		0			0
$183$	−70.7248			−0.386474
$184$		0			0
$185$	225.220	+	127.336i	1.21740	+	0.688302i
$186$		0			0
$187$		−	1.06775i		−	0.00570992i
$188$		0			0
$189$			181.828i			0.962052i
$190$		0			0
$191$			210.809i			1.10371i	0.833939	+	0.551857i	$0.186081\pi$
$191$			210.809i			1.10371i	−0.833939	+	0.551857i	$0.813919\pi$
$192$		0			0
$193$		−	108.176i		−	0.560496i	−0.959928	−	0.280248i	$-0.909583\pi$
$193$		−	108.176i		−	0.560496i	0.959928	−	0.280248i	$-0.0904168\pi$
$194$		0			0
$195$	48.4458	−	85.6864i	0.248440	−	0.439418i
$196$		0			0
$197$	308.300			1.56497			0.782487	−	0.622667i	$-0.213951\pi$
$197$	308.300			1.56497			0.782487	+	0.622667i	$0.213951\pi$
$198$		0			0
$199$			247.859i			1.24552i	0.782412	+	0.622761i	$0.213989\pi$
$199$			247.859i			1.24552i	−0.782412	+	0.622761i	$0.786011\pi$
$200$		0			0
$201$	363.272			1.80733
$202$		0			0
$203$		−	320.130i		−	1.57700i
$204$		0			0
$205$	−65.5301	−	37.0497i	−0.319659	−	0.180730i
$206$		0			0
$207$	135.712			0.655616
$208$		0			0
$209$	−12.4458			−0.0595494
$210$		0			0
$211$	−230.584			−1.09281			−0.546407	−	0.837520i	$-0.684005\pi$
$211$	−230.584			−1.09281			−0.546407	+	0.837520i	$0.684005\pi$
$212$		0			0
$213$	−388.329			−1.82314
$214$		0			0
$215$	22.9180	−	40.5353i	0.106595	−	0.188536i
$216$		0			0
$217$		−	168.985i		−	0.778734i
$218$		0			0
$219$	−399.607			−1.82469
$220$		0			0
$221$		−	9.29480i		−	0.0420579i
$222$		0			0
$223$	177.782			0.797229			0.398615	−	0.917118i	$-0.369491\pi$
$223$	177.782			0.797229			0.398615	+	0.917118i	$0.369491\pi$
$224$		0			0
$225$	−179.721	−	298.709i	−0.798762	−	1.32759i
$226$		0			0
$227$			203.301i			0.895601i	0.894134	+	0.447800i	$0.147792\pi$
$227$			203.301i			0.895601i	−0.894134	+	0.447800i	$0.852208\pi$
$228$		0			0
$229$		−	81.0705i		−	0.354020i	−0.984209	−	0.177010i	$-0.943358\pi$
$229$		−	81.0705i		−	0.354020i	0.984209	−	0.177010i	$-0.0566425\pi$
$230$		0			0
$231$		−	17.3630i		−	0.0751645i
$232$		0			0
$233$			56.2864i			0.241573i	0.992679	+	0.120786i	$0.0385416\pi$
$233$			56.2864i			0.241573i	−0.992679	+	0.120786i	$0.961458\pi$
$234$		0			0
$235$	38.1378	+	21.5625i	0.162288	+	0.0917554i
$236$		0			0
$237$	482.629			2.03641
$238$		0			0
$239$		−	360.235i		−	1.50726i	−0.657300	−	0.753629i	$-0.728301\pi$
$239$		−	360.235i		−	1.50726i	0.657300	−	0.753629i	$-0.271699\pi$
$240$		0			0
$241$	−197.495			−0.819483			−0.409741	−	0.912202i	$-0.634381\pi$
$241$	−197.495			−0.819483			−0.409741	+	0.912202i	$0.634381\pi$
$242$		0			0
$243$		−	270.896i		−	1.11480i
$244$		0			0
$245$	−43.2825	−	24.4713i	−0.176663	−	0.0998827i
$246$		0			0
$247$	−108.341			−0.438627
$248$		0			0
$249$	1.27864			0.00513510
$250$		0			0
$251$	178.361			0.710600			0.355300	−	0.934752i	$-0.384379\pi$
$251$	178.361			0.710600			0.355300	+	0.934752i	$0.384379\pi$
$252$		0			0
$253$	−4.59506			−0.0181623
$254$		0			0
$255$	−47.1498	−	26.6578i	−0.184901	−	0.104540i
$256$		0			0
$257$			453.183i			1.76336i	0.471849	+	0.881679i	$0.343587\pi$
$257$			453.183i			1.76336i	−0.471849	+	0.881679i	$0.656413\pi$
$258$		0			0
$259$	397.272			1.53387
$260$		0			0
$261$		−	581.436i		−	2.22772i
$262$		0			0
$263$	21.0921			0.0801981			0.0400990	−	0.999196i	$-0.487233\pi$
$263$	21.0921			0.0801981			0.0400990	+	0.999196i	$0.487233\pi$
$264$		0			0
$265$	−173.666	−	98.1879i	−0.655342	−	0.370520i
$266$		0			0
$267$			409.806i			1.53485i
$268$		0			0
$269$			92.4148i			0.343549i	0.985136	+	0.171775i	$0.0549501\pi$
$269$			92.4148i			0.343549i	−0.985136	+	0.171775i	$0.945050\pi$
$270$		0			0
$271$			188.799i			0.696675i	0.937369	+	0.348337i	$0.113254\pi$
$271$			188.799i			0.696675i	−0.937369	+	0.348337i	$0.886746\pi$
$272$		0			0
$273$		−	151.145i		−	0.553645i
$274$		0			0
$275$	6.08514	+	10.1139i	0.0221278	+	0.0367779i
$276$		0			0
$277$	−70.1251			−0.253159			−0.126580	−	0.991956i	$-0.540400\pi$
$277$	−70.1251			−0.253159			−0.126580	+	0.991956i	$0.540400\pi$
$278$		0			0
$279$		−	306.919i		−	1.10007i
$280$		0			0
$281$	−506.269			−1.80167			−0.900835	−	0.434161i	$-0.857045\pi$
$281$	−506.269			−1.80167			−0.900835	+	0.434161i	$0.857045\pi$
$282$		0			0
$283$			501.350i			1.77156i	0.464110	+	0.885778i	$0.346374\pi$
$283$			501.350i			1.77156i	−0.464110	+	0.885778i	$0.653626\pi$
$284$		0			0
$285$	−310.726	+	549.582i	−1.09026	+	1.92836i
$286$		0			0
$287$	−115.591			−0.402755
$288$		0			0
$289$	283.885			0.982303
$290$		0			0
$291$	−474.833			−1.63173
$292$		0			0
$293$	−324.026			−1.10589			−0.552945	−	0.833218i	$-0.686496\pi$
$293$	−324.026			−1.10589			−0.552945	+	0.833218i	$0.686496\pi$
$294$		0			0
$295$	−335.985	−	189.961i	−1.13893	−	0.643935i
$296$		0			0
$297$		−	11.1817i		−	0.0376487i
$298$		0			0
$299$	−40.0000			−0.133779
$300$		0			0
$301$		−	71.5015i		−	0.237546i
$302$		0			0
$303$	−504.890			−1.66630
$304$		0			0
$305$	36.3344	−	64.2648i	0.119129	−	0.210704i
$306$		0			0
$307$			9.84697i			0.0320748i	0.999871	+	0.0160374i	$0.00510509\pi$
$307$			9.84697i			0.0320748i	−0.999871	+	0.0160374i	$0.994895\pi$
$308$		0			0
$309$		−	487.924i		−	1.57904i
$310$		0			0
$311$			200.417i			0.644429i	0.946667	+	0.322214i	$0.104427\pi$
$311$			200.417i			0.644429i	−0.946667	+	0.322214i	$0.895573\pi$
$312$		0			0
$313$			261.582i			0.835727i	0.908510	+	0.417863i	$0.137221\pi$
$313$			261.582i			0.835727i	−0.908510	+	0.417863i	$0.862779\pi$
$314$		0			0
$315$	−465.967	−	263.451i	−1.47926	−	0.836352i
$316$		0			0
$317$	−322.341			−1.01685			−0.508425	−	0.861107i	$-0.669772\pi$
$317$	−322.341			−1.01685			−0.508425	+	0.861107i	$0.669772\pi$
$318$		0			0
$319$			19.6867i			0.0617138i
$320$		0			0
$321$	158.053			0.492376
$322$		0			0
$323$			59.6157i			0.184569i
$324$		0			0
$325$	52.9712	+	88.0416i	0.162988	+	0.270897i
$326$		0			0
$327$	869.645			2.65946
$328$		0			0
$329$	67.2724			0.204475
$330$		0			0
$331$	61.0883			0.184557			0.0922783	−	0.995733i	$-0.470585\pi$
$331$	61.0883			0.184557			0.0922783	+	0.995733i	$0.470585\pi$
$332$		0			0
$333$	721.545			2.16680
$334$		0			0
$335$	−186.629	+	330.091i	−0.557101	+	0.985348i
$336$		0			0
$337$			114.960i			0.341129i	0.985347	+	0.170564i	$0.0545591\pi$
$337$			114.960i			0.341129i	−0.985347	+	0.170564i	$0.945441\pi$
$338$		0			0
$339$	367.108			1.08292
$340$		0			0
$341$			10.3919i			0.0304748i
$342$		0			0
$343$	299.851			0.874201
$344$		0			0
$345$	−114.721	+	202.908i	−0.332526	+	0.588140i
$346$		0			0
$347$		−	143.967i		−	0.414892i	−0.978246	−	0.207446i	$-0.933485\pi$
$347$		−	143.967i		−	0.414892i	0.978246	−	0.207446i	$-0.0665151\pi$
$348$		0			0
$349$		−	35.9526i		−	0.103016i	−0.998673	−	0.0515080i	$-0.983597\pi$
$349$		−	35.9526i		−	0.103016i	0.998673	−	0.0515080i	$-0.0164028\pi$
$350$		0			0
$351$		−	97.3365i		−	0.277312i
$352$		0			0
$353$		−	252.536i		−	0.715400i	−0.933837	−	0.357700i	$-0.883561\pi$
$353$		−	252.536i		−	0.715400i	0.933837	−	0.357700i	$-0.116439\pi$
$354$		0			0
$355$	199.502	−	352.860i	0.561976	−	0.993971i
$356$		0			0
$357$	−83.1691			−0.232967
$358$		0			0
$359$		−	594.152i		−	1.65502i	−0.561452	−	0.827509i	$-0.689757\pi$
$359$		−	594.152i		−	1.65502i	0.561452	−	0.827509i	$-0.310243\pi$
$360$		0			0
$361$	333.885			0.924890
$362$		0			0
$363$		−	578.524i		−	1.59373i
$364$		0			0
$365$	205.295	−	363.107i	0.562453	−	0.994814i
$366$		0			0
$367$	−88.9062			−0.242251			−0.121126	−	0.992637i	$-0.538650\pi$
$367$	−88.9062			−0.242251			−0.121126	+	0.992637i	$0.538650\pi$
$368$		0			0
$369$	−209.941			−0.568946
$370$		0			0
$371$	−306.334			−0.825699
$372$		0			0
$373$	83.6542			0.224274			0.112137	−	0.993693i	$-0.464230\pi$
$373$	83.6542			0.224274			0.112137	+	0.993693i	$0.464230\pi$
$374$		0			0
$375$	598.533	−	16.2012i	1.59609	−	0.0432031i
$376$		0			0
$377$			171.373i			0.454570i
$378$		0			0
$379$	135.135			0.356556			0.178278	−	0.983980i	$-0.442947\pi$
$379$	135.135			0.356556			0.178278	+	0.983980i	$0.442947\pi$
$380$		0			0
$381$			684.791i			1.79735i
$382$		0			0
$383$	−498.526			−1.30164			−0.650818	−	0.759234i	$-0.725574\pi$
$383$	−498.526			−1.30164			−0.650818	+	0.759234i	$0.725574\pi$
$384$		0			0
$385$	15.7771	+	8.92013i	0.0409794	+	0.0231692i
$386$		0			0
$387$		−	129.864i		−	0.335567i
$388$		0			0
$389$			308.420i			0.792854i	0.918066	+	0.396427i	$0.129750\pi$
$389$			308.420i			0.792854i	−0.918066	+	0.396427i	$0.870250\pi$
$390$		0			0
$391$			22.0104i			0.0562926i
$392$		0			0
$393$		−	527.562i		−	1.34240i
$394$		0			0
$395$	−247.947	+	438.546i	−0.627715	+	1.11024i
$396$		0			0
$397$	297.682			0.749828			0.374914	−	0.927060i	$-0.377672\pi$
$397$	297.682			0.749828			0.374914	+	0.927060i	$0.377672\pi$
$398$		0			0
$399$			969.426i			2.42964i
$400$		0			0
$401$	148.663			0.370729			0.185365	−	0.982670i	$-0.440653\pi$
$401$	148.663			0.370729			0.185365	+	0.982670i	$0.440653\pi$
$402$		0			0
$403$			90.4616i			0.224470i
$404$		0			0
$405$	52.4726	+	29.6672i	0.129562	+	0.0732524i
$406$		0			0
$407$	−24.4306			−0.0600261
$408$		0			0
$409$	442.387			1.08163			0.540815	−	0.841141i	$-0.318116\pi$
$409$	442.387			1.08163			0.540815	+	0.841141i	$0.318116\pi$
$410$		0			0
$411$	1079.66			2.62691
$412$		0			0
$413$	−592.654			−1.43500
$414$		0			0
$415$	−0.656893	+	1.16185i	−0.00158287	+	0.00279964i
$416$		0			0
$417$		−	466.626i		−	1.11901i
$418$		0			0
$419$	−536.184			−1.27968			−0.639838	−	0.768510i	$-0.720999\pi$
$419$	−536.184			−1.27968			−0.639838	+	0.768510i	$0.720999\pi$
$420$		0			0
$421$			514.582i			1.22228i	0.791521	+	0.611142i	$0.209290\pi$
$421$			514.582i			1.22228i	−0.791521	+	0.611142i	$0.790710\pi$
$422$		0			0
$423$	122.183			0.288850
$424$		0			0
$425$	48.4458	−	29.1480i	0.113990	−	0.0685834i
$426$		0			0
$427$		−	113.359i		−	0.265477i
$428$		0			0
$429$			9.29480i			0.0216662i
$430$		0			0
$431$			86.8151i			0.201427i	0.994915	+	0.100714i	$0.0321126\pi$
$431$			86.8151i			0.201427i	−0.994915	+	0.100714i	$0.967887\pi$
$432$		0			0
$433$			494.017i			1.14092i	0.821327	+	0.570458i	$0.193234\pi$
$433$			494.017i			1.14092i	−0.821327	+	0.570458i	$0.806766\pi$
$434$		0			0
$435$	869.325	+	491.503i	1.99845	+	1.12989i
$436$		0			0
$437$	256.555			0.587082
$438$		0			0
$439$			524.700i			1.19522i	0.801789	+	0.597608i	$0.203882\pi$
$439$			524.700i			1.19522i	−0.801789	+	0.597608i	$0.796118\pi$
$440$		0			0
$441$	−138.666			−0.314435
$442$		0			0
$443$			496.575i			1.12094i	0.828176	+	0.560469i	$0.189379\pi$
$443$			496.575i			1.12094i	−0.828176	+	0.560469i	$0.810621\pi$
$444$		0			0
$445$	−372.375	−	210.535i	−0.836797	−	0.473112i
$446$		0			0
$447$	−916.795			−2.05099
$448$		0			0
$449$	−332.158			−0.739773			−0.369886	−	0.929077i	$-0.620603\pi$
$449$	−332.158			−0.739773			−0.369886	+	0.929077i	$0.620603\pi$
$450$		0			0
$451$	7.10835			0.0157613
$452$		0			0
$453$	327.421			0.722785
$454$		0			0
$455$	137.340	+	77.6497i	0.301845	+	0.170659i
$456$		0			0
$457$		−	252.788i		−	0.553147i	−0.960993	−	0.276574i	$-0.910801\pi$
$457$		−	252.788i		−	0.553147i	0.960993	−	0.276574i	$-0.0891990\pi$
$458$		0			0
$459$	−53.5604			−0.116689
$460$		0			0
$461$		−	832.716i		−	1.80633i	−0.429299	−	0.903163i	$-0.641239\pi$
$461$		−	832.716i		−	1.80633i	0.429299	−	0.903163i	$-0.358761\pi$
$462$		0			0
$463$	854.205			1.84494			0.922468	−	0.386074i	$-0.126169\pi$
$463$	854.205			1.84494			0.922468	+	0.386074i	$0.126169\pi$
$464$		0			0
$465$	458.885	+	259.447i	0.986850	+	0.557950i
$466$		0			0
$467$			102.726i			0.219970i	0.993933	+	0.109985i	$0.0350802\pi$
$467$			102.726i			0.219970i	−0.993933	+	0.109985i	$0.964920\pi$
$468$		0			0
$469$			582.259i			1.24149i
$470$		0			0
$471$		−	743.577i		−	1.57872i
$472$		0			0
$473$			4.39705i			0.00929608i
$474$		0			0
$475$	−339.751	−	564.688i	−0.715265	−	1.18882i
$476$		0			0
$477$	−556.379			−1.16641
$478$		0			0
$479$		−	268.772i		−	0.561111i	−0.959838	−	0.280556i	$-0.909481\pi$
$479$		−	268.772i		−	0.561111i	0.959838	−	0.280556i	$-0.0905187\pi$
$480$		0			0
$481$	−212.669			−0.442139
$482$		0			0
$483$			357.917i			0.741028i
$484$		0			0
$485$	243.942	−	431.462i	0.502973	−	0.889612i
$486$		0			0
$487$	−504.065			−1.03504			−0.517520	−	0.855671i	$-0.673145\pi$
$487$	−504.065			−1.03504			−0.517520	+	0.855671i	$0.673145\pi$
$488$		0			0
$489$	−433.384			−0.886266
$490$		0			0
$491$	−130.020			−0.264807			−0.132403	−	0.991196i	$-0.542269\pi$
$491$	−130.020			−0.264807			−0.132403	+	0.991196i	$0.542269\pi$
$492$		0			0
$493$	94.2997			0.191277
$494$		0			0
$495$	28.6551	+	16.2012i	0.0578891	+	0.0327296i
$496$		0			0
$497$		−	622.421i		−	1.25236i
$498$		0			0
$499$	−664.184			−1.33103			−0.665515	−	0.746384i	$-0.731788\pi$
$499$	−664.184			−1.33103			−0.665515	+	0.746384i	$0.731788\pi$
$500$		0			0
$501$			482.728i			0.963529i
$502$		0			0
$503$	550.015			1.09347			0.546735	−	0.837306i	$-0.315871\pi$
$503$	550.015			1.09347			0.546735	+	0.837306i	$0.315871\pi$
$504$		0			0
$505$	259.384	−	458.774i	0.513631	−	0.908463i
$506$		0			0
$507$		−	728.602i		−	1.43708i
$508$		0			0
$509$		−	349.843i		−	0.687314i	−0.939095	−	0.343657i	$-0.888334\pi$
$509$		−	349.843i		−	0.687314i	0.939095	−	0.343657i	$-0.111666\pi$
$510$		0			0
$511$		−	640.496i		−	1.25342i
$512$		0			0
$513$			624.304i			1.21697i
$514$		0			0
$515$	443.358	+	250.668i	0.860888	+	0.486733i
$516$		0			0
$517$	−4.13698			−0.00800189
$518$		0			0
$519$			261.801i			0.504434i
$520$		0			0
$521$	29.7771			0.0571537			0.0285769	−	0.999592i	$-0.490902\pi$
$521$	29.7771			0.0571537			0.0285769	+	0.999592i	$0.490902\pi$
$522$		0			0
$523$		−	603.023i		−	1.15301i	−0.817094	−	0.576504i	$-0.804416\pi$
$523$		−	603.023i		−	1.15301i	0.817094	−	0.576504i	$-0.195584\pi$
$524$		0			0
$525$	787.789	−	473.982i	1.50055	−	0.902823i
$526$		0			0
$527$	49.7774			0.0944543
$528$		0			0
$529$	−434.279			−0.820943
$530$		0			0
$531$	−1076.41			−2.02713
$532$		0			0
$533$	61.8782			0.116094
$534$		0			0
$535$	−81.1985	+	143.616i	−0.151773	+	0.268442i
$536$		0			0
$537$			511.605i			0.952710i
$538$		0			0
$539$	4.69505			0.00871066
$540$		0			0
$541$		−	163.368i		−	0.301974i	−0.988536	−	0.150987i	$-0.951755\pi$
$541$		−	163.368i		−	0.301974i	0.988536	−	0.150987i	$-0.0482451\pi$
$542$		0			0
$543$	−693.490			−1.27714
$544$		0			0
$545$	−446.774	+	790.212i	−0.819769	+	1.44993i
$546$		0			0
$547$		−	524.218i		−	0.958350i	−0.877719	−	0.479175i	$-0.840936\pi$
$547$		−	524.218i		−	0.958350i	0.877719	−	0.479175i	$-0.159064\pi$
$548$		0			0
$549$		−	205.888i		−	0.375023i
$550$		0			0
$551$		−	1099.16i		−	1.99485i
$552$		0			0
$553$			773.566i			1.39885i
$554$		0			0
$555$	−609.941	+	1078.81i	−1.09899	+	1.94380i
$556$		0			0
$557$	−589.515			−1.05837			−0.529187	−	0.848505i	$-0.677503\pi$
$557$	−589.515			−1.05837			−0.529187	+	0.848505i	$0.677503\pi$
$558$		0			0
$559$			38.2763i			0.0684728i
$560$		0			0
$561$	5.11456			0.00911687
$562$		0			0
$563$		−	557.763i		−	0.990698i	−0.868694	−	0.495349i	$-0.835040\pi$
$563$		−	557.763i		−	0.990698i	0.868694	−	0.495349i	$-0.164960\pi$
$564$		0			0
$565$	−188.599	+	333.577i	−0.333804	+	0.590402i
$566$		0			0
$567$	92.5581			0.163242
$568$		0			0
$569$	−542.715			−0.953805			−0.476903	−	0.878956i	$-0.658241\pi$
$569$	−542.715			−0.953805			−0.476903	+	0.878956i	$0.658241\pi$
$570$		0			0
$571$	476.695			0.834842			0.417421	−	0.908713i	$-0.362934\pi$
$571$	476.695			0.834842			0.417421	+	0.908713i	$0.362934\pi$
$572$		0			0
$573$	−1009.78			−1.76227
$574$		0			0
$575$	−125.438	−	208.486i	−0.218152	−	0.362584i
$576$		0			0
$577$		−	865.659i		−	1.50027i	−0.661282	−	0.750137i	$-0.729987\pi$
$577$		−	865.659i		−	1.50027i	0.661282	−	0.750137i	$-0.270013\pi$
$578$		0			0
$579$	518.164			0.894929
$580$		0			0
$581$			2.04943i			0.00352741i
$582$		0			0
$583$	18.8383			0.0323127
$584$		0			0
$585$	249.443	+	141.031i	0.426398	+	0.241079i
$586$		0			0
$587$			567.091i			0.966083i	0.875597	+	0.483041i	$0.160468\pi$
$587$			567.091i			0.966083i	−0.875597	+	0.483041i	$0.839532\pi$
$588$		0			0
$589$		−	580.209i		−	0.985075i
$590$		0			0
$591$			1476.76i			2.49875i
$592$		0			0
$593$		−	421.774i		−	0.711254i	−0.934628	−	0.355627i	$-0.884267\pi$
$593$		−	421.774i		−	0.711254i	0.934628	−	0.355627i	$-0.115733\pi$
$594$		0			0
$595$	42.7276	−	75.5725i	0.0718110	−	0.127013i
$596$		0			0
$597$	−1187.25			−1.98869
$598$		0			0
$599$		−	120.444i		−	0.201075i	−0.994933	−	0.100538i	$-0.967944\pi$
$599$		−	120.444i		−	0.201075i	0.994933	−	0.100538i	$-0.0320563\pi$
$600$		0			0
$601$	340.158			0.565986			0.282993	−	0.959122i	$-0.408673\pi$
$601$	340.158			0.565986			0.282993	+	0.959122i	$0.408673\pi$
$602$		0			0
$603$			1057.53i			1.75377i
$604$		0			0
$605$	525.682	+	297.213i	0.868897	+	0.491261i
$606$		0			0
$607$	−115.621			−0.190479			−0.0952396	−	0.995454i	$-0.530362\pi$
$607$	−115.621			−0.190479			−0.0952396	+	0.995454i	$0.530362\pi$
$608$		0			0
$609$	1533.43			2.51795
$610$		0			0
$611$	−36.0124			−0.0589401
$612$		0			0
$613$	1095.63			1.78733			0.893663	−	0.448738i	$-0.148126\pi$
$613$	1095.63			1.78733			0.893663	+	0.448738i	$0.148126\pi$
$614$		0			0
$615$	177.469	−	313.890i	0.288567	−	0.510391i
$616$		0			0
$617$			716.775i			1.16171i	0.814007	+	0.580855i	$0.197282\pi$
$617$			716.775i			1.16171i	−0.814007	+	0.580855i	$0.802718\pi$
$618$		0			0
$619$	−118.584			−0.191573			−0.0957864	−	0.995402i	$-0.530537\pi$
$619$	−118.584			−0.191573			−0.0957864	+	0.995402i	$0.530537\pi$
$620$		0			0
$621$			230.496i			0.371169i
$622$		0			0
$623$	−656.844			−1.05432
$624$		0			0
$625$	−292.771	+	552.187i	−0.468433	+	0.883499i
$626$		0			0
$627$		−	59.6157i		−	0.0950809i
$628$		0			0
$629$			117.023i			0.186046i
$630$		0			0
$631$			1168.62i			1.85201i	0.377515	+	0.926004i	$0.376779\pi$
$631$			1168.62i			1.85201i	−0.377515	+	0.926004i	$0.623221\pi$
$632$		0			0
$633$		−	1104.50i		−	1.74486i
$634$		0			0
$635$	−622.243	−	351.807i	−0.979910	−	0.554026i
$636$		0			0
$637$	40.8704			0.0641608
$638$		0			0
$639$		−	1130.47i		−	1.76912i
$640$		0			0
$641$	−244.158			−0.380902			−0.190451	−	0.981697i	$-0.560995\pi$
$641$	−244.158			−0.380902			−0.190451	+	0.981697i	$0.560995\pi$
$642$		0			0
$643$			868.847i			1.35124i	0.737250	+	0.675620i	$0.236124\pi$
$643$			868.847i			1.35124i	−0.737250	+	0.675620i	$0.763876\pi$
$644$		0			0
$645$	194.165	+	109.778i	0.301031	+	0.170198i
$646$		0			0
$647$	35.1333			0.0543019			0.0271510	−	0.999631i	$-0.491357\pi$
$647$	35.1333			0.0543019			0.0271510	+	0.999631i	$0.491357\pi$
$648$		0			0
$649$	36.4458			0.0561569
$650$		0			0
$651$	809.443			1.24338
$652$		0			0
$653$	580.068			0.888313			0.444157	−	0.895949i	$-0.353503\pi$
$653$	580.068			0.888313			0.444157	+	0.895949i	$0.353503\pi$
$654$		0			0
$655$	479.375	+	271.031i	0.731870	+	0.413788i
$656$		0			0
$657$		−	1163.30i		−	1.77062i
$658$		0			0
$659$	992.788			1.50651			0.753253	−	0.657731i	$-0.228483\pi$
$659$	992.788			1.50651			0.753253	+	0.657731i	$0.228483\pi$
$660$		0			0
$661$		−	719.647i		−	1.08872i	−0.838850	−	0.544362i	$-0.816772\pi$
$661$		−	719.647i		−	1.08872i	0.838850	−	0.544362i	$-0.183228\pi$
$662$		0			0
$663$	44.5223			0.0671528
$664$		0			0
$665$	−880.879	−	498.036i	−1.32463	−	0.748926i
$666$		0			0
$667$		−	405.817i		−	0.608421i
$668$		0			0
$669$			851.580i			1.27291i
$670$		0			0
$671$			6.97110i			0.0103891i
$672$		0			0
$673$			189.591i			0.281711i	0.990030	+	0.140855i	$0.0449852\pi$
$673$			189.591i			0.281711i	−0.990030	+	0.140855i	$0.955015\pi$
$674$		0			0
$675$	507.331	−	305.241i	0.751602	−	0.452209i
$676$		0			0
$677$	327.192			0.483297			0.241649	−	0.970364i	$-0.422312\pi$
$677$	327.192			0.483297			0.241649	+	0.970364i	$0.422312\pi$
$678$		0			0
$679$		−	761.070i		−	1.12087i
$680$		0			0
$681$	−973.817			−1.42998
$682$		0			0
$683$		−	673.539i		−	0.986148i	−0.869987	−	0.493074i	$-0.835873\pi$
$683$		−	673.539i		−	0.986148i	0.869987	−	0.493074i	$-0.164127\pi$
$684$		0			0
$685$	−554.668	+	981.044i	−0.809734	+	1.43218i
$686$		0			0
$687$	388.329			0.565254
$688$		0			0
$689$	163.988			0.238008
$690$		0			0
$691$	−237.036			−0.343033			−0.171516	−	0.985181i	$-0.554867\pi$
$691$	−237.036			−0.343033			−0.171516	+	0.985181i	$0.554867\pi$
$692$		0			0
$693$	50.5456			0.0729374
$694$		0			0
$695$	424.005	+	239.726i	0.610079	+	0.344930i
$696$		0			0
$697$		−	34.0491i		−	0.0488510i
$698$		0			0
$699$	−269.613			−0.385712
$700$		0			0
$701$			476.306i			0.679466i	0.940522	+	0.339733i	$0.110337\pi$
$701$			476.306i			0.679466i	−0.940522	+	0.339733i	$0.889663\pi$
$702$		0			0
$703$	1364.03			1.94030
$704$		0			0
$705$	−103.285	+	182.681i	−0.146503	+	0.259121i
$706$		0			0
$707$		−	809.246i		−	1.14462i
$708$		0			0
$709$		−	964.778i		−	1.36076i	−0.732860	−	0.680380i	$-0.761815\pi$
$709$		−	964.778i		−	1.36076i	0.732860	−	0.680380i	$-0.238185\pi$
$710$		0			0
$711$			1404.99i			1.97607i
$712$		0			0
$713$		−	214.216i		−	0.300443i
$714$		0			0
$715$	−8.44582	−	4.77514i	−0.0118123	−	0.00667852i
$716$		0			0
$717$	1725.53			2.40660
$718$		0			0
$719$			899.974i			1.25170i	0.779943	+	0.625851i	$0.215248\pi$
$719$			899.974i			1.25170i	−0.779943	+	0.625851i	$0.784752\pi$
$720$		0			0
$721$	782.053			1.08468
$722$		0			0
$723$		−	946.006i		−	1.30845i
$724$		0			0
$725$	−893.220	+	537.415i	−1.23203	+	0.741262i
$726$		0			0
$727$	−908.888			−1.25019			−0.625095	−	0.780549i	$-0.714940\pi$
$727$	−908.888			−1.25019			−0.625095	+	0.780549i	$0.714940\pi$
$728$		0			0
$729$	1189.09			1.63113
$730$		0			0
$731$	21.0619			0.0288125
$732$		0			0
$733$	433.283			0.591109			0.295554	−	0.955326i	$-0.404496\pi$
$733$	433.283			0.591109			0.295554	+	0.955326i	$0.404496\pi$
$734$		0			0
$735$	117.218	−	207.324i	0.159480	−	0.282073i
$736$		0			0
$737$		−	35.8065i		−	0.0485842i
$738$		0			0
$739$	886.138			1.19910			0.599552	−	0.800336i	$-0.295345\pi$
$739$	886.138			1.19910			0.599552	+	0.800336i	$0.295345\pi$
$740$		0			0
$741$		−	518.955i		−	0.700344i
$742$		0			0
$743$	−895.305			−1.20499			−0.602493	−	0.798124i	$-0.705826\pi$
$743$	−895.305			−1.20499			−0.602493	+	0.798124i	$0.705826\pi$
$744$		0			0
$745$	470.997	−	833.055i	0.632211	−	1.11820i
$746$		0			0
$747$			3.72226i			0.00498295i
$748$		0			0
$749$			253.329i			0.338223i
$750$		0			0
$751$		−	1297.13i		−	1.72720i	−0.504176	−	0.863601i	$-0.668204\pi$
$751$		−	1297.13i		−	1.72720i	0.504176	−	0.863601i	$-0.331796\pi$
$752$		0			0
$753$			854.351i			1.13460i
$754$		0			0
$755$	−168.210	+	297.515i	−0.222795	+	0.394060i
$756$		0			0
$757$	−421.694			−0.557060			−0.278530	−	0.960428i	$-0.589847\pi$
$757$	−421.694			−0.557060			−0.278530	+	0.960428i	$0.589847\pi$
$758$		0			0
$759$		−	22.0104i		−	0.0289992i
$760$		0			0
$761$	1415.76			1.86039			0.930196	−	0.367063i	$-0.119637\pi$
$761$	1415.76			1.86039			0.930196	+	0.367063i	$0.119637\pi$
$762$		0			0
$763$			1393.88i			1.82684i
$764$		0			0
$765$	77.6038	−	137.258i	0.101443	−	0.179423i
$766$		0			0
$767$	317.261			0.413639
$768$		0			0
$769$	−414.210			−0.538635			−0.269318	−	0.963051i	$-0.586798\pi$
$769$	−414.210			−0.538635			−0.269318	+	0.963051i	$0.586798\pi$
$770$		0			0
$771$	−2170.76			−2.81551
$772$		0			0
$773$	−727.056			−0.940564			−0.470282	−	0.882516i	$-0.655848\pi$
$773$	−727.056			−0.940564			−0.470282	+	0.882516i	$0.655848\pi$
$774$		0			0
$775$	−471.498	+	283.682i	−0.608385	+	0.366042i
$776$		0			0
$777$			1902.94i			2.44909i
$778$		0			0
$779$	−396.879			−0.509473
$780$		0			0
$781$			38.2763i			0.0490094i
$782$		0			0
$783$	987.519			1.26120
$784$		0			0
$785$	675.659	+	382.008i	0.860713	+	0.486634i
$786$		0			0
$787$			1030.11i			1.30890i	0.756104	+	0.654451i	$0.227100\pi$
$787$			1030.11i			1.30890i	−0.756104	+	0.654451i	$0.772900\pi$
$788$		0			0
$789$			101.032i			0.128050i
$790$		0			0
$791$			588.407i			0.743878i
$792$		0			0
$793$			60.6835i			0.0765239i
$794$		0			0
$795$	470.322	−	831.861i	0.591600	−	1.04637i
$796$		0			0
$797$	−371.202			−0.465750			−0.232875	−	0.972507i	$-0.574813\pi$
$797$	−371.202			−0.465750			−0.232875	+	0.972507i	$0.574813\pi$
$798$		0			0
$799$			19.8162i			0.0248013i
$800$		0			0
$801$	−1192.99			−1.48938
$802$		0			0
$803$			39.3879i			0.0490509i
$804$		0			0
$805$	−325.225	−	183.877i	−0.404006	−	0.228419i
$806$		0			0
$807$	−442.668			−0.548536
$808$		0			0
$809$	77.7771			0.0961398			0.0480699	−	0.998844i	$-0.484693\pi$
$809$	77.7771			0.0961398			0.0480699	+	0.998844i	$0.484693\pi$
$810$		0			0
$811$	1407.90			1.73600			0.868000	−	0.496564i	$-0.165405\pi$
$811$	1407.90			1.73600			0.868000	+	0.496564i	$0.165405\pi$
$812$		0			0
$813$	−904.350			−1.11236
$814$		0			0
$815$	222.648	−	393.799i	0.273188	−	0.483189i
$816$		0			0
$817$		−	245.500i		−	0.300489i
$818$		0			0
$819$	440.000			0.537241
$820$		0			0
$821$		−	820.275i		−	0.999116i	−0.866280	−	0.499558i	$-0.833496\pi$
$821$		−	820.275i		−	0.999116i	0.866280	−	0.499558i	$-0.166504\pi$
$822$		0			0
$823$	−1385.41			−1.68336			−0.841681	−	0.539976i	$-0.818433\pi$
$823$	−1385.41			−1.68336			−0.841681	+	0.539976i	$0.818433\pi$
$824$		0			0
$825$	−48.4458	+	29.1480i	−0.0587222	+	0.0353309i
$826$		0			0
$827$			258.898i			0.313057i	0.987673	+	0.156529i	$0.0500303\pi$
$827$			258.898i			0.313057i	−0.987673	+	0.156529i	$0.949970\pi$
$828$		0			0
$829$			1299.31i			1.56732i	0.621190	+	0.783660i	$0.286649\pi$
$829$			1299.31i			1.56732i	−0.621190	+	0.783660i	$0.713351\pi$
$830$		0			0
$831$		−	335.901i		−	0.404213i
$832$		0			0
$833$		−	22.4894i		−	0.0269980i
$834$		0			0
$835$	−438.636	−	247.998i	−0.525313	−	0.297004i
$836$		0			0
$837$	521.276			0.622791
$838$		0			0
$839$			804.961i			0.959429i	0.877425	+	0.479715i	$0.159260\pi$
$839$			804.961i			0.959429i	−0.877425	+	0.479715i	$0.840740\pi$
$840$		0			0
$841$	−897.650			−1.06736
$842$		0			0
$843$		−	2425.04i		−	2.87668i
$844$		0			0
$845$	662.052	+	374.314i	0.783493	+	0.442975i
$846$		0			0
$847$	927.268			1.09477
$848$		0			0
$849$	−2401.48			−2.82859
$850$		0			0
$851$	503.607			0.591782
$852$		0			0
$853$	−470.811			−0.551948			−0.275974	−	0.961165i	$-0.589000\pi$
$853$	−470.811			−0.551948			−0.275974	+	0.961165i	$0.589000\pi$
$854$		0			0
$855$	−1599.89	−	904.556i	−1.87122	−	1.05796i
$856$		0			0
$857$			266.105i			0.310508i	0.987875	+	0.155254i	$0.0496196\pi$
$857$			266.105i			0.310508i	−0.987875	+	0.155254i	$0.950380\pi$
$858$		0			0
$859$	−1442.02			−1.67872			−0.839360	−	0.543576i	$-0.817070\pi$
$859$	−1442.02			−1.67872			−0.839360	+	0.543576i	$0.817070\pi$
$860$		0			0
$861$		−	553.681i		−	0.643067i
$862$		0			0
$863$	−364.495			−0.422358			−0.211179	−	0.977447i	$-0.567730\pi$
$863$	−364.495			−0.422358			−0.211179	+	0.977447i	$0.567730\pi$
$864$		0			0
$865$	−237.889	−	134.499i	−0.275016	−	0.155490i
$866$		0			0
$867$			1359.82i			1.56842i
$868$		0			0
$869$		−	47.5711i		−	0.0547424i
$870$		0			0
$871$		−	311.696i		−	0.357860i
$872$		0			0
$873$		−	1382.29i		−	1.58338i
$874$		0			0
$875$	25.9675	+	959.338i	0.0296771	+	1.09639i
$876$		0			0
$877$	−77.4289			−0.0882883			−0.0441442	−	0.999025i	$-0.514056\pi$
$877$	−77.4289			−0.0882883			−0.0441442	+	0.999025i	$0.514056\pi$
$878$		0			0
$879$		−	1552.09i		−	1.76574i
$880$		0			0
$881$	724.932			0.822851			0.411426	−	0.911443i	$-0.365031\pi$
$881$	724.932			0.822851			0.411426	+	0.911443i	$0.365031\pi$
$882$		0			0
$883$			493.342i			0.558711i	0.960188	+	0.279356i	$0.0901208\pi$
$883$			493.342i			0.558711i	−0.960188	+	0.279356i	$0.909879\pi$
$884$		0			0
$885$	909.915	−	1609.37i	1.02815	−	1.81850i
$886$		0			0
$887$	1514.47			1.70741			0.853705	−	0.520758i	$-0.174351\pi$
$887$	1514.47			1.70741			0.853705	+	0.520758i	$0.174351\pi$
$888$		0			0
$889$	−1097.59			−1.23464
$890$		0			0
$891$	−5.69194			−0.00638826
$892$		0			0
$893$	230.979			0.258655
$894$		0			0
$895$	−464.876	−	262.834i	−0.519414	−	0.293669i
$896$		0			0
$897$		−	191.601i		−	0.213602i
$898$		0			0
$899$	−917.771			−1.02088
$900$		0			0
$901$		−	90.2359i		−	0.100151i
$902$		0			0
$903$	342.493			0.379284
$904$		0			0
$905$	356.276	−	630.147i	0.393675	−	0.696295i
$906$		0			0
$907$			851.570i			0.938887i	0.882963	+	0.469443i	$0.155545\pi$
$907$			851.570i			0.938887i	−0.882963	+	0.469443i	$0.844455\pi$
$908$		0			0
$909$		−	1469.79i		−	1.61693i
$910$		0			0
$911$		−	517.728i		−	0.568308i	−0.958779	−	0.284154i	$-0.908287\pi$
$911$		−	517.728i		−	0.568308i	0.958779	−	0.284154i	$-0.0917127\pi$
$912$		0			0
$913$		−	0.126031i		−	0.000138041i
$914$		0			0
$915$	307.830	+	174.042i	0.336426	+	0.190210i
$916$		0			0
$917$	845.585			0.922121
$918$		0			0
$919$			657.730i			0.715701i	0.933779	+	0.357851i	$0.116490\pi$
$919$			657.730i			0.715701i	−0.933779	+	0.357851i	$0.883510\pi$
$920$		0			0
$921$	−47.1672			−0.0512130
$922$		0			0
$923$			333.195i			0.360992i
$924$		0			0
$925$	−666.917	−	1108.46i	−0.720991	−	1.19833i
$926$		0			0
$927$	1420.40			1.53226
$928$		0			0
$929$	1206.28			1.29847			0.649237	−	0.760586i	$-0.275088\pi$
$929$	1206.28			1.29847			0.649237	+	0.760586i	$0.275088\pi$
$930$		0			0
$931$	−262.138			−0.281566
$932$		0			0
$933$	−960.003			−1.02894
$934$		0			0
$935$	−2.62757	+	4.64740i	−0.00281024	+	0.00497048i
$936$		0			0
$937$			216.730i			0.231302i	0.993290	+	0.115651i	$0.0368954\pi$
$937$			216.730i			0.231302i	−0.993290	+	0.115651i	$0.963105\pi$
$938$		0			0
$939$	−1252.98			−1.33438
$940$		0			0
$941$			749.322i			0.796304i	0.917320	+	0.398152i	$0.130348\pi$
$941$			749.322i			0.796304i	−0.917320	+	0.398152i	$0.869652\pi$
$942$		0			0
$943$	−146.530			−0.155387
$944$		0			0
$945$	447.449	−	791.406i	0.473491	−	0.837466i
$946$		0			0
$947$		−	114.160i		−	0.120549i	−0.998182	−	0.0602743i	$-0.980802\pi$
$947$		−	114.160i		−	0.120549i	0.998182	−	0.0602743i	$-0.0191976\pi$
$948$		0			0
$949$			342.872i			0.361298i
$950$		0			0
$951$		−	1544.02i		−	1.62358i
$952$		0			0
$953$			820.680i			0.861154i	0.902554	+	0.430577i	$0.141690\pi$
$953$			820.680i			0.861154i	−0.902554	+	0.430577i	$0.858310\pi$
$954$		0			0
$955$	518.768	−	917.548i	0.543212	−	0.960783i
$956$		0			0
$957$	−94.2997			−0.0985368
$958$		0			0
$959$			1730.50i			1.80448i
$960$		0			0
$961$	476.542			0.495881
$962$		0			0
$963$			460.109i			0.477787i
$964$		0			0
$965$	−266.203	+	470.835i	−0.275858	+	0.487912i
$966$		0			0
$967$	−1108.56			−1.14639			−0.573194	−	0.819420i	$-0.694296\pi$
$967$	−1108.56			−1.14639			−0.573194	+	0.819420i	$0.694296\pi$
$968$		0			0
$969$	−285.560			−0.294696
$970$		0			0
$971$	1372.41			1.41340			0.706698	−	0.707516i	$-0.250184\pi$
$971$	1372.41			1.41340			0.706698	+	0.707516i	$0.250184\pi$
$972$		0			0
$973$	747.916			0.768670
$974$		0			0
$975$	−421.721	+	253.733i	−0.432534	+	0.260239i
$976$		0			0
$977$			927.224i			0.949052i	0.880241	+	0.474526i	$0.157381\pi$
$977$			927.224i			0.949052i	−0.880241	+	0.474526i	$0.842619\pi$
$978$		0			0
$979$	40.3932			0.0412597
$980$		0			0
$981$			2531.63i			2.58066i
$982$		0			0
$983$	354.106			0.360230			0.180115	−	0.983646i	$-0.442353\pi$
$983$	354.106			0.360230			0.180115	+	0.983646i	$0.442353\pi$
$984$		0			0
$985$	−1341.88	−	758.677i	−1.36231	−	0.770230i
$986$		0			0
$987$			322.236i			0.326480i
$988$		0			0
$989$		−	90.6396i		−	0.0916478i
$990$		0			0
$991$			537.545i			0.542427i	0.962519	+	0.271213i	$0.0874249\pi$
$991$			537.545i			0.542427i	−0.962519	+	0.271213i	$0.912575\pi$
$992$		0			0
$993$			292.614i			0.294677i
$994$		0			0
$995$	609.941	−	1078.81i	0.613006	−	1.08423i
$996$		0			0
$997$	−441.759			−0.443088			−0.221544	−	0.975150i	$-0.571110\pi$
$997$	−441.759			−0.443088			−0.221544	+	0.975150i	$0.571110\pi$
$998$		0			0
$999$			1225.48i			1.22671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.e.c.79.7		8
3.2	odd	2		1440.3.p.g.559.8		8
4.3	odd	2		40.3.e.c.19.2	yes	8
5.2	odd	4		800.3.g.h.751.7		8
5.3	odd	4		800.3.g.h.751.2		8
5.4	even	2	inner	160.3.e.c.79.2		8
8.3	odd	2	inner	160.3.e.c.79.8		8
8.5	even	2		40.3.e.c.19.8	yes	8
12.11	even	2		360.3.p.g.19.7		8
15.14	odd	2		1440.3.p.g.559.2		8
16.3	odd	4		1280.3.h.m.1279.14		16
16.5	even	4		1280.3.h.m.1279.15		16
16.11	odd	4		1280.3.h.m.1279.3		16
16.13	even	4		1280.3.h.m.1279.2		16
20.3	even	4		200.3.g.h.51.6		8
20.7	even	4		200.3.g.h.51.3		8
20.19	odd	2		40.3.e.c.19.7	yes	8
24.5	odd	2		360.3.p.g.19.1		8
24.11	even	2		1440.3.p.g.559.1		8
40.3	even	4		800.3.g.h.751.1		8
40.13	odd	4		200.3.g.h.51.5		8
40.19	odd	2	inner	160.3.e.c.79.1		8
40.27	even	4		800.3.g.h.751.8		8
40.29	even	2		40.3.e.c.19.1	✓	8
40.37	odd	4		200.3.g.h.51.4		8
60.59	even	2		360.3.p.g.19.2		8
80.19	odd	4		1280.3.h.m.1279.1		16
80.29	even	4		1280.3.h.m.1279.13		16
80.59	odd	4		1280.3.h.m.1279.16		16
80.69	even	4		1280.3.h.m.1279.4		16
120.29	odd	2		360.3.p.g.19.8		8
120.59	even	2		1440.3.p.g.559.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.e.c.19.1	✓	8	40.29	even	2
40.3.e.c.19.2	yes	8	4.3	odd	2
40.3.e.c.19.7	yes	8	20.19	odd	2
40.3.e.c.19.8	yes	8	8.5	even	2
160.3.e.c.79.1		8	40.19	odd	2	inner
160.3.e.c.79.2		8	5.4	even	2	inner
160.3.e.c.79.7		8	1.1	even	1	trivial
160.3.e.c.79.8		8	8.3	odd	2	inner
200.3.g.h.51.3		8	20.7	even	4
200.3.g.h.51.4		8	40.37	odd	4
200.3.g.h.51.5		8	40.13	odd	4
200.3.g.h.51.6		8	20.3	even	4
360.3.p.g.19.1		8	24.5	odd	2
360.3.p.g.19.2		8	60.59	even	2
360.3.p.g.19.7		8	12.11	even	2
360.3.p.g.19.8		8	120.29	odd	2
800.3.g.h.751.1		8	40.3	even	4
800.3.g.h.751.2		8	5.3	odd	4
800.3.g.h.751.7		8	5.2	odd	4
800.3.g.h.751.8		8	40.27	even	4
1280.3.h.m.1279.1		16	80.19	odd	4
1280.3.h.m.1279.2		16	16.13	even	4
1280.3.h.m.1279.3		16	16.11	odd	4
1280.3.h.m.1279.4		16	80.69	even	4
1280.3.h.m.1279.13		16	80.29	even	4
1280.3.h.m.1279.14		16	16.3	odd	4
1280.3.h.m.1279.15		16	16.5	even	4
1280.3.h.m.1279.16		16	80.59	odd	4
1440.3.p.g.559.1		8	24.11	even	2
1440.3.p.g.559.2		8	15.14	odd	2
1440.3.p.g.559.7		8	120.59	even	2
1440.3.p.g.559.8		8	3.2	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.79002i q^{3} +(-4.35250 - 2.46084i) q^{5} -7.67752 q^{7} -13.9443 q^{9} +0.472136 q^{11} +4.10995 q^{13} +(11.7875 - 20.8486i) q^{15} -2.26154i q^{17} -26.3607 q^{19} -36.7754i q^{21} -9.73249 q^{23} +(12.8885 + 21.4216i) q^{25} -23.6832i q^{27} +41.6971i q^{29} +22.0104i q^{31} +2.26154i q^{33} +(33.4164 + 18.8931i) q^{35} -51.7449 q^{37} +19.6867i q^{39} +15.0557 q^{41} +9.31310i q^{43} +(60.6925 + 34.3146i) q^{45} -8.76226 q^{47} +9.94427 q^{49} +10.8328 q^{51} +39.9002 q^{53} +(-2.05497 - 1.16185i) q^{55} -126.268i q^{57} +77.1935 q^{59} +14.7650i q^{61} +107.057 q^{63} +(-17.8885 - 10.1139i) q^{65} -75.8395i q^{67} -46.6188i q^{69} +81.0705i q^{71} +83.4249i q^{73} +(-102.610 + 61.7364i) q^{75} -3.62483 q^{77} -100.757i q^{79} -12.0557 q^{81} -0.266939i q^{83} +(-5.56528 + 9.84336i) q^{85} -199.730 q^{87} +85.5542 q^{89} -31.5542 q^{91} -105.430 q^{93} +(114.735 + 64.8694i) q^{95} +99.1297i q^{97} -6.58359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9} - 32 q^{11} - 32 q^{19} - 40 q^{25} + 160 q^{35} + 192 q^{41} + 8 q^{49} - 128 q^{51} + 224 q^{59} - 320 q^{75} - 168 q^{81} + 112 q^{89} + 320 q^{91} - 160 q^{99}+O(q^{100}) \) 8 * q - 40 * q^9 - 32 * q^11 - 32 * q^19 - 40 * q^25 + 160 * q^35 + 192 * q^41 + 8 * q^49 - 128 * q^51 + 224 * q^59 - 320 * q^75 - 168 * q^81 + 112 * q^89 + 320 * q^91 - 160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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