LMFDB - Embedded newform 160.3.h.b.159.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(159,160)]

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, 0, 1]))

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

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

chi := DirichletCharacter("160.159");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.h (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:	$4.35968422976$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			159.1
Root			$-2.65109i$ of defining polynomial
Character	$\chi$	$=$	160.159
Dual form			160.3.h.b.159.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} +O(q^{10})$	$q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} -6.19112i q^{11} -16.1132i q^{13} +(-11.8500 + 7.01762i) q^{15} -5.20875i q^{17} -36.2264i q^{19} +10.6044 q^{21} -22.0411 q^{23} +(12.0176 - 21.9221i) q^{25} +28.6823 q^{27} -20.0352 q^{29} +26.4175i q^{31} +17.0529i q^{33} +(-16.5633 + 9.80888i) q^{35} +69.3219i q^{37} +44.3822i q^{39} +11.6220 q^{41} +25.8542 q^{43} +(-6.08006 + 3.60065i) q^{45} +66.1853 q^{47} -34.1777 q^{49} +14.3470i q^{51} -39.5751i q^{53} +(-15.7736 - 26.6354i) q^{55} +99.7821i q^{57} -27.7736i q^{59} -54.1954 q^{61} +5.44096 q^{63} +(-41.0529 - 69.3219i) q^{65} +107.507 q^{67} +60.7101 q^{69} -70.7997i q^{71} +37.4351i q^{73} +(-33.1014 + 60.3822i) q^{75} +23.8356i q^{77} +97.6530i q^{79} -66.2835 q^{81} -126.163 q^{83} +(-13.2707 - 22.4090i) q^{85} +55.1852 q^{87} +133.635 q^{89} +62.0352i q^{91} -72.7645i q^{93} +(-92.2969 - 155.853i) q^{95} +6.40900i q^{97} +8.74960i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 2 q^{5} + 12 q^{7} + 18 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 2 * q^5 + 12 * q^7 + 18 * q^9	$ 6 q + 4 q^{3} - 2 q^{5} + 12 q^{7} + 18 q^{9} - 36 q^{15} + 8 q^{21} - 68 q^{23} - 10 q^{25} + 184 q^{27} + 44 q^{29} - 108 q^{35} - 68 q^{41} - 76 q^{43} - 6 q^{45} + 268 q^{47} - 62 q^{49} - 288 q^{55} - 100 q^{61} - 172 q^{63} + 308 q^{67} - 184 q^{69} - 284 q^{75} + 238 q^{81} - 204 q^{83} - 32 q^{85} + 584 q^{87} + 76 q^{89} - 32 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 - 2 * q^5 + 12 * q^7 + 18 * q^9 - 36 * q^15 + 8 * q^21 - 68 * q^23 - 10 * q^25 + 184 * q^27 + 44 * q^29 - 108 * q^35 - 68 * q^41 - 76 * q^43 - 6 * q^45 + 268 * q^47 - 62 * q^49 - 288 * q^55 - 100 * q^61 - 172 * q^63 + 308 * q^67 - 184 * q^69 - 284 * q^75 + 238 * q^81 - 204 * q^83 - 32 * q^85 + 584 * q^87 + 76 * q^89 - 32 * q^95

Display 100 coefficients 10 coefficients

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
$3$	−2.75441			−0.918135			−0.459068	−	0.888401i	$-0.651816\pi$
$3$	−2.75441			−0.918135			−0.459068	+	0.888401i	$0.651816\pi$
$4$		0			0
$5$	4.30219	−	2.54778i	0.860437	−	0.509556i
$5$	4.30219	−	2.54778i	0.860437	−	0.509556i
$6$		0			0
$7$	−3.84997			−0.549995			−0.274998	−	0.961445i	$-0.588677\pi$
$7$	−3.84997			−0.549995			−0.274998	+	0.961445i	$0.588677\pi$
$8$		0			0
$9$	−1.41325			−0.157028
$10$		0			0
$11$		−	6.19112i		−	0.562829i	−0.959586	−	0.281415i	$-0.909196\pi$
$11$		−	6.19112i		−	0.562829i	0.959586	−	0.281415i	$-0.0908037\pi$
$12$		0			0
$13$		−	16.1132i		−	1.23948i	−0.784809	−	0.619738i	$-0.787239\pi$
$13$		−	16.1132i		−	1.23948i	0.784809	−	0.619738i	$-0.212761\pi$
$14$		0			0
$15$	−11.8500	+	7.01762i	−0.789998	+	0.467842i
$16$		0			0
$17$		−	5.20875i		−	0.306397i	−0.988195	−	0.153198i	$-0.951043\pi$
$17$		−	5.20875i		−	0.306397i	0.988195	−	0.153198i	$-0.0489574\pi$
$18$		0			0
$19$		−	36.2264i		−	1.90665i	−0.301944	−	0.953326i	$-0.597636\pi$
$19$		−	36.2264i		−	1.90665i	0.301944	−	0.953326i	$-0.402364\pi$
$20$		0			0
$21$	10.6044			0.504970
$22$		0			0
$23$	−22.0411			−0.958308			−0.479154	−	0.877731i	$-0.659057\pi$
$23$	−22.0411			−0.958308			−0.479154	+	0.877731i	$0.659057\pi$
$24$		0			0
$25$	12.0176	−	21.9221i	0.480705	−	0.876882i
$26$		0			0
$27$	28.6823			1.06231
$28$		0			0
$29$	−20.0352			−0.690871			−0.345435	−	0.938443i	$-0.612269\pi$
$29$	−20.0352			−0.690871			−0.345435	+	0.938443i	$0.612269\pi$
$30$		0			0
$31$			26.4175i			0.852177i	0.904681	+	0.426089i	$0.140109\pi$
$31$			26.4175i			0.852177i	−0.904681	+	0.426089i	$0.859891\pi$
$32$		0			0
$33$			17.0529i			0.516754i
$34$		0			0
$35$	−16.5633	+	9.80888i	−0.473237	+	0.280254i
$36$		0			0
$37$			69.3219i			1.87357i	0.349911	+	0.936783i	$0.386212\pi$
$37$			69.3219i			1.87357i	−0.349911	+	0.936783i	$0.613788\pi$
$38$		0			0
$39$			44.3822i			1.13801i
$40$		0			0
$41$	11.6220			0.283463			0.141732	−	0.989905i	$-0.454733\pi$
$41$	11.6220			0.283463			0.141732	+	0.989905i	$0.454733\pi$
$42$		0			0
$43$	25.8542			0.601261			0.300630	−	0.953741i	$-0.402803\pi$
$43$	25.8542			0.601261			0.300630	+	0.953741i	$0.402803\pi$
$44$		0			0
$45$	−6.08006	+	3.60065i	−0.135112	+	0.0800144i
$46$		0			0
$47$	66.1853			1.40820			0.704099	−	0.710102i	$-0.251351\pi$
$47$	66.1853			1.40820			0.704099	+	0.710102i	$0.251351\pi$
$48$		0			0
$49$	−34.1777			−0.697505
$50$		0			0
$51$			14.3470i			0.281314i
$52$		0			0
$53$		−	39.5751i		−	0.746699i	−0.927691	−	0.373350i	$-0.878209\pi$
$53$		−	39.5751i		−	0.746699i	0.927691	−	0.373350i	$-0.121791\pi$
$54$		0			0
$55$	−15.7736	−	26.6354i	−0.286793	−	0.484280i
$56$		0			0
$57$			99.7821i			1.75056i
$58$		0			0
$59$		−	27.7736i		−	0.470739i	−0.971906	−	0.235370i	$-0.924370\pi$
$59$		−	27.7736i		−	0.470739i	0.971906	−	0.235370i	$-0.0756301\pi$
$60$		0			0
$61$	−54.1954			−0.888449			−0.444224	−	0.895916i	$-0.646521\pi$
$61$	−54.1954			−0.888449			−0.444224	+	0.895916i	$0.646521\pi$
$62$		0			0
$63$	5.44096			0.0863645
$64$		0			0
$65$	−41.0529	−	69.3219i	−0.631583	−	1.06649i
$66$		0			0
$67$	107.507			1.60459			0.802293	−	0.596931i	$-0.203613\pi$
$67$	107.507			1.60459			0.802293	+	0.596931i	$0.203613\pi$
$68$		0			0
$69$	60.7101			0.879857
$70$		0			0
$71$		−	70.7997i		−	0.997179i	−0.866838	−	0.498590i	$-0.833851\pi$
$71$		−	70.7997i		−	0.997179i	0.866838	−	0.498590i	$-0.166149\pi$
$72$		0			0
$73$			37.4351i			0.512810i	0.966569	+	0.256405i	$0.0825380\pi$
$73$			37.4351i			0.512810i	−0.966569	+	0.256405i	$0.917462\pi$
$74$		0			0
$75$	−33.1014	+	60.3822i	−0.441352	+	0.805097i
$76$		0			0
$77$			23.8356i			0.309554i
$78$		0			0
$79$			97.6530i			1.23611i	0.786133	+	0.618057i	$0.212080\pi$
$79$			97.6530i			1.23611i	−0.786133	+	0.618057i	$0.787920\pi$
$80$		0			0
$81$	−66.2835			−0.818315
$82$		0			0
$83$	−126.163			−1.52003			−0.760017	−	0.649904i	$-0.774809\pi$
$83$	−126.163			−1.52003			−0.760017	+	0.649904i	$0.774809\pi$
$84$		0			0
$85$	−13.2707	−	22.4090i	−0.156126	−	0.263635i
$86$		0			0
$87$	55.1852			0.634313
$88$		0			0
$89$	133.635			1.50151			0.750757	−	0.660579i	$-0.229689\pi$
$89$	133.635			1.50151			0.750757	+	0.660579i	$0.229689\pi$
$90$		0			0
$91$			62.0352i			0.681706i
$92$		0			0
$93$		−	72.7645i		−	0.782414i
$94$		0			0
$95$	−92.2969	−	155.853i	−0.971546	−	1.64055i
$96$		0			0
$97$			6.40900i			0.0660722i	0.999454	+	0.0330361i	$0.0105176\pi$
$97$			6.40900i			0.0660722i	−0.999454	+	0.0330361i	$0.989482\pi$
$98$		0			0
$99$			8.74960i			0.0883798i
$100$		0			0
$101$	121.564			1.20361			0.601803	−	0.798644i	$-0.294449\pi$
$101$	121.564			1.20361			0.601803	+	0.798644i	$0.294449\pi$
$102$		0			0
$103$	9.95891			0.0966884			0.0483442	−	0.998831i	$-0.484606\pi$
$103$	9.95891			0.0966884			0.0483442	+	0.998831i	$0.484606\pi$
$104$		0			0
$105$	45.6220	−	27.0176i	0.434495	−	0.257311i
$106$		0			0
$107$	−134.842			−1.26020			−0.630102	−	0.776512i	$-0.716987\pi$
$107$	−134.842			−1.26020			−0.630102	+	0.776512i	$0.716987\pi$
$108$		0			0
$109$	−28.2306			−0.258996			−0.129498	−	0.991580i	$-0.541337\pi$
$109$	−28.2306			−0.258996			−0.129498	+	0.991580i	$0.541337\pi$
$110$		0			0
$111$		−	190.941i		−	1.72019i
$112$		0			0
$113$		−	190.052i		−	1.68188i	−0.541130	−	0.840939i	$-0.682003\pi$
$113$		−	190.052i		−	1.68188i	0.541130	−	0.840939i	$-0.317997\pi$
$114$		0			0
$115$	−94.8249	+	56.1559i	−0.824564	+	0.488312i
$116$		0			0
$117$			22.7719i			0.194632i
$118$		0			0
$119$			20.0535i			0.168517i
$120$		0			0
$121$	82.6700			0.683223
$122$		0			0
$123$	−32.0117			−0.260258
$124$		0			0
$125$	−4.15055	−	124.931i	−0.0332044	−	0.999449i
$126$		0			0
$127$	60.0646			0.472950			0.236475	−	0.971638i	$-0.424008\pi$
$127$	60.0646			0.472950			0.236475	+	0.971638i	$0.424008\pi$
$128$		0			0
$129$	−71.2130			−0.552039
$130$		0			0
$131$			111.985i			0.854848i	0.904051	+	0.427424i	$0.140579\pi$
$131$			111.985i			0.854848i	−0.904051	+	0.427424i	$0.859421\pi$
$132$		0			0
$133$			139.470i			1.04865i
$134$		0			0
$135$	123.397	−	73.0763i	0.914049	−	0.541306i
$136$		0			0
$137$		−	42.6439i		−	0.311269i	−0.987815	−	0.155635i	$-0.950258\pi$
$137$		−	42.6439i		−	0.311269i	0.987815	−	0.155635i	$-0.0497422\pi$
$138$		0			0
$139$		−	222.332i		−	1.59951i	−0.600325	−	0.799756i	$-0.704962\pi$
$139$		−	222.332i		−	1.59951i	0.600325	−	0.799756i	$-0.295038\pi$
$140$		0			0
$141$	−182.301			−1.29292
$142$		0			0
$143$	−99.7587			−0.697614
$144$		0			0
$145$	−86.1954	+	51.0454i	−0.594451	+	0.352037i
$146$		0			0
$147$	94.1394			0.640404
$148$		0			0
$149$	20.0981			0.134887			0.0674434	−	0.997723i	$-0.478516\pi$
$149$	20.0981			0.134887			0.0674434	+	0.997723i	$0.478516\pi$
$150$		0			0
$151$			86.0522i			0.569882i	0.958545	+	0.284941i	$0.0919741\pi$
$151$			86.0522i			0.569882i	−0.958545	+	0.284941i	$0.908026\pi$
$152$		0			0
$153$			7.36126i			0.0481128i
$154$		0			0
$155$	67.3060	+	113.653i	0.434232	+	0.733245i
$156$		0			0
$157$		−	16.0342i		−	0.102129i	−0.998695	−	0.0510643i	$-0.983739\pi$
$157$		−	16.0342i		−	0.102129i	0.998695	−	0.0510643i	$-0.0162614\pi$
$158$		0			0
$159$			109.006i			0.685571i
$160$		0			0
$161$	84.8575			0.527065
$162$		0			0
$163$	179.157			1.09912			0.549561	−	0.835454i	$-0.314795\pi$
$163$	179.157			1.09912			0.549561	+	0.835454i	$0.314795\pi$
$164$		0			0
$165$	43.4470	+	73.3646i	0.263315	+	0.444634i
$166$		0			0
$167$	−137.800			−0.825149			−0.412574	−	0.910924i	$-0.635370\pi$
$167$	−137.800			−0.825149			−0.412574	+	0.910924i	$0.635370\pi$
$168$		0			0
$169$	−90.6347			−0.536300
$170$		0			0
$171$			51.1969i			0.299397i
$172$		0			0
$173$			62.8895i			0.363523i	0.983343	+	0.181762i	$0.0581799\pi$
$173$			62.8895i			0.363523i	−0.983343	+	0.181762i	$0.941820\pi$
$174$		0			0
$175$	−46.2675	+	84.3992i	−0.264385	+	0.482281i
$176$		0			0
$177$			76.4998i			0.432203i
$178$		0			0
$179$		−	238.020i		−	1.32972i	−0.746967	−	0.664861i	$-0.768491\pi$
$179$		−	238.020i		−	1.32972i	0.746967	−	0.664861i	$-0.231509\pi$
$180$		0			0
$181$	186.718			1.03159			0.515795	−	0.856712i	$-0.327497\pi$
$181$	186.718			1.03159			0.515795	+	0.856712i	$0.327497\pi$
$182$		0			0
$183$	149.276			0.815716
$184$		0			0
$185$	176.617	+	298.236i	0.954687	+	1.61209i
$186$		0			0
$187$	−32.2480			−0.172449
$188$		0			0
$189$	−110.426			−0.584264
$190$		0			0
$191$			123.447i			0.646319i	0.946344	+	0.323160i	$0.104745\pi$
$191$			123.447i			0.646319i	−0.946344	+	0.323160i	$0.895255\pi$
$192$		0			0
$193$			162.355i			0.841220i	0.907242	+	0.420610i	$0.138184\pi$
$193$			162.355i			0.841220i	−0.907242	+	0.420610i	$0.861816\pi$
$194$		0			0
$195$	113.076	+	190.941i	0.579878	+	0.979183i
$196$		0			0
$197$		−	113.540i		−	0.576344i	−0.957579	−	0.288172i	$-0.906952\pi$
$197$		−	113.540i		−	0.576344i	0.957579	−	0.288172i	$-0.0930475\pi$
$198$		0			0
$199$			325.928i			1.63783i	0.573915	+	0.818915i	$0.305424\pi$
$199$			325.928i			1.63783i	−0.573915	+	0.818915i	$0.694576\pi$
$200$		0			0
$201$	−296.118			−1.47323
$202$		0			0
$203$	77.1351			0.379976
$204$		0			0
$205$	50.0000	−	29.6103i	0.243902	−	0.144441i
$206$		0			0
$207$	31.1496			0.150481
$208$		0			0
$209$	−224.282			−1.07312
$210$		0			0
$211$		−	130.731i		−	0.619580i	−0.950805	−	0.309790i	$-0.899741\pi$
$211$		−	130.731i		−	0.619580i	0.950805	−	0.309790i	$-0.100259\pi$
$212$		0			0
$213$			195.011i			0.915546i
$214$		0			0
$215$	111.230	−	65.8709i	0.517347	−	0.306376i
$216$		0			0
$217$		−	101.707i		−	0.468694i
$218$		0			0
$219$		−	103.112i		−	0.470829i
$220$		0			0
$221$	−83.9295			−0.379772
$222$		0			0
$223$	93.3889			0.418784			0.209392	−	0.977832i	$-0.432851\pi$
$223$	93.3889			0.418784			0.209392	+	0.977832i	$0.432851\pi$
$224$		0			0
$225$	−16.9839	+	30.9813i	−0.0754840	+	0.137695i
$226$		0			0
$227$	−14.9957			−0.0660602			−0.0330301	−	0.999454i	$-0.510516\pi$
$227$	−14.9957			−0.0660602			−0.0330301	+	0.999454i	$0.510516\pi$
$228$		0			0
$229$	−144.106			−0.629283			−0.314641	−	0.949211i	$-0.601884\pi$
$229$	−144.106			−0.629283			−0.314641	+	0.949211i	$0.601884\pi$
$230$		0			0
$231$		−	65.6530i		−	0.284212i
$232$		0			0
$233$		−	126.528i		−	0.543040i	−0.962433	−	0.271520i	$-0.912474\pi$
$233$		−	126.528i		−	0.543040i	0.962433	−	0.271520i	$-0.0875263\pi$
$234$		0			0
$235$	284.741	−	168.626i	1.21167	−	0.717556i
$236$		0			0
$237$		−	268.976i		−	1.13492i
$238$		0			0
$239$			1.65300i			0.00691630i	0.999994	+	0.00345815i	$0.00110077\pi$
$239$			1.65300i			0.00691630i	−0.999994	+	0.00345815i	$0.998899\pi$
$240$		0			0
$241$	206.928			0.858622			0.429311	−	0.903157i	$-0.358756\pi$
$241$	206.928			0.858622			0.429311	+	0.903157i	$0.358756\pi$
$242$		0			0
$243$	−75.5692			−0.310984
$244$		0			0
$245$	−147.039	+	87.0774i	−0.600159	+	0.355418i
$246$		0			0
$247$	−583.722			−2.36325
$248$		0			0
$249$	347.503			1.39560
$250$		0			0
$251$		−	74.1206i		−	0.295301i	−0.989040	−	0.147651i	$-0.952829\pi$
$251$		−	74.1206i		−	0.295301i	0.989040	−	0.147651i	$-0.0471711\pi$
$252$		0			0
$253$			136.459i			0.539364i
$254$		0			0
$255$	36.5530	+	61.7235i	0.143345	+	0.242053i
$256$		0			0
$257$		−	274.682i		−	1.06880i	−0.845231	−	0.534402i	$-0.820537\pi$
$257$		−	274.682i		−	1.06880i	0.845231	−	0.534402i	$-0.179463\pi$
$258$		0			0
$259$		−	266.887i		−	1.03045i
$260$		0			0
$261$	28.3148			0.108486
$262$		0			0
$263$	−75.5382			−0.287218			−0.143609	−	0.989635i	$-0.545871\pi$
$263$	−75.5382			−0.287218			−0.143609	+	0.989635i	$0.545871\pi$
$264$		0			0
$265$	−100.829	−	170.259i	−0.380485	−	0.642488i
$266$		0			0
$267$	−368.084			−1.37859
$268$		0			0
$269$	314.087			1.16761			0.583804	−	0.811895i	$-0.301564\pi$
$269$	314.087			1.16761			0.583804	+	0.811895i	$0.301564\pi$
$270$		0			0
$271$			128.158i			0.472908i	0.971643	+	0.236454i	$0.0759852\pi$
$271$			128.158i			0.472908i	−0.971643	+	0.236454i	$0.924015\pi$
$272$		0			0
$273$		−	170.870i		−	0.625898i
$274$		0			0
$275$	−135.722	−	74.4026i	−0.493535	−	0.270555i
$276$		0			0
$277$		−	242.118i		−	0.874073i	−0.899444	−	0.437037i	$-0.856028\pi$
$277$		−	242.118i		−	0.874073i	0.899444	−	0.437037i	$-0.143972\pi$
$278$		0			0
$279$		−	37.3345i		−	0.133815i
$280$		0			0
$281$	28.8562			0.102691			0.0513456	−	0.998681i	$-0.483649\pi$
$281$	28.8562			0.102691			0.0513456	+	0.998681i	$0.483649\pi$
$282$		0			0
$283$	269.993			0.954039			0.477020	−	0.878893i	$-0.341717\pi$
$283$	269.993			0.954039			0.477020	+	0.878893i	$0.341717\pi$
$284$		0			0
$285$	254.223	+	429.281i	0.892011	+	1.50625i
$286$		0			0
$287$	−44.7443			−0.155904
$288$		0			0
$289$	261.869			0.906121
$290$		0			0
$291$		−	17.6530i		−	0.0606632i
$292$		0			0
$293$		−	353.448i		−	1.20631i	−0.797625	−	0.603154i	$-0.793911\pi$
$293$		−	353.448i		−	1.20631i	0.797625	−	0.603154i	$-0.206089\pi$
$294$		0			0
$295$	−70.7611	−	119.487i	−0.239868	−	0.405042i
$296$		0			0
$297$		−	177.576i		−	0.597898i
$298$		0			0
$299$			355.152i			1.18780i
$300$		0			0
$301$	−99.5379			−0.330691
$302$		0			0
$303$	−334.837			−1.10507
$304$		0			0
$305$	−233.159	+	138.078i	−0.764454	+	0.452715i
$306$		0			0
$307$	−260.946			−0.849985			−0.424993	−	0.905197i	$-0.639723\pi$
$307$	−260.946			−0.849985			−0.424993	+	0.905197i	$0.639723\pi$
$308$		0			0
$309$	−27.4309			−0.0887730
$310$		0			0
$311$		−	141.570i		−	0.455208i	−0.973754	−	0.227604i	$-0.926911\pi$
$311$		−	141.570i		−	0.455208i	0.973754	−	0.227604i	$-0.0730892\pi$
$312$		0			0
$313$			365.950i			1.16917i	0.811333	+	0.584584i	$0.198742\pi$
$313$			365.950i			1.16917i	−0.811333	+	0.584584i	$0.801258\pi$
$314$		0			0
$315$	23.4080	−	13.8624i	0.0743113	−	0.0440076i
$316$		0			0
$317$			9.22805i			0.0291106i	0.999894	+	0.0145553i	$0.00463326\pi$
$317$			9.22805i			0.0291106i	−0.999894	+	0.0145553i	$0.995367\pi$
$318$		0			0
$319$			124.041i			0.388842i
$320$		0			0
$321$	371.409			1.15704
$322$		0			0
$323$	−188.694			−0.584192
$324$		0			0
$325$	−353.234	−	193.642i	−1.08687	−	0.595822i
$326$		0			0
$327$	77.7586			0.237794
$328$		0			0
$329$	−254.811			−0.774502
$330$		0			0
$331$			53.3799i			0.161268i	0.996744	+	0.0806342i	$0.0256946\pi$
$331$			53.3799i			0.161268i	−0.996744	+	0.0806342i	$0.974305\pi$
$332$		0			0
$333$		−	97.9692i		−	0.294202i
$334$		0			0
$335$	462.516	−	273.905i	1.38065	−	0.817626i
$336$		0			0
$337$		−	350.458i		−	1.03994i	−0.854186	−	0.519968i	$-0.825944\pi$
$337$		−	350.458i		−	1.03994i	0.854186	−	0.519968i	$-0.174056\pi$
$338$		0			0
$339$			523.481i			1.54419i
$340$		0			0
$341$	163.554			0.479630
$342$		0			0
$343$	320.232			0.933620
$344$		0			0
$345$	261.186	−	154.676i	0.757062	−	0.448336i
$346$		0			0
$347$	70.8302			0.204122			0.102061	−	0.994778i	$-0.467456\pi$
$347$	70.8302			0.204122			0.102061	+	0.994778i	$0.467456\pi$
$348$		0			0
$349$	373.045			1.06890			0.534449	−	0.845201i	$-0.320519\pi$
$349$	373.045			1.06890			0.534449	+	0.845201i	$0.320519\pi$
$350$		0			0
$351$		−	462.163i		−	1.31670i
$352$		0			0
$353$			543.568i			1.53985i	0.638132	+	0.769927i	$0.279707\pi$
$353$			543.568i			1.53985i	−0.638132	+	0.769927i	$0.720293\pi$
$354$		0			0
$355$	−180.382	−	304.594i	−0.508119	−	0.858010i
$356$		0			0
$357$		−	55.2355i		−	0.154721i
$358$		0			0
$359$		−	500.805i		−	1.39500i	−0.716585	−	0.697500i	$-0.754296\pi$
$359$		−	500.805i		−	1.39500i	0.716585	−	0.697500i	$-0.245704\pi$
$360$		0			0
$361$	−951.350			−2.63532
$362$		0			0
$363$	−227.707			−0.627291
$364$		0			0
$365$	95.3765	+	161.053i	0.261305	+	0.441241i
$366$		0			0
$367$	142.499			0.388281			0.194140	−	0.980974i	$-0.437808\pi$
$367$	142.499			0.388281			0.194140	+	0.980974i	$0.437808\pi$
$368$		0			0
$369$	−16.4248			−0.0445116
$370$		0			0
$371$			152.363i			0.410681i
$372$		0			0
$373$		−	160.000i		−	0.428954i	−0.976729	−	0.214477i	$-0.931195\pi$
$373$		−	160.000i		−	0.428954i	0.976729	−	0.214477i	$-0.0688047\pi$
$374$		0			0
$375$	11.4323	+	344.111i	0.0304862	+	0.917629i
$376$		0			0
$377$			322.832i			0.856317i
$378$		0			0
$379$			192.796i			0.508698i	0.967113	+	0.254349i	$0.0818611\pi$
$379$			192.796i			0.508698i	−0.967113	+	0.254349i	$0.918139\pi$
$380$		0			0
$381$	−165.442			−0.434232
$382$		0			0
$383$	605.286			1.58038			0.790191	−	0.612861i	$-0.209981\pi$
$383$	605.286			1.58038			0.790191	+	0.612861i	$0.209981\pi$
$384$		0			0
$385$	60.7280	+	102.545i	0.157735	+	0.266352i
$386$		0			0
$387$	−36.5384			−0.0944146
$388$		0			0
$389$	−522.159			−1.34231			−0.671155	−	0.741317i	$-0.734202\pi$
$389$	−522.159			−1.34231			−0.671155	+	0.741317i	$0.734202\pi$
$390$		0			0
$391$			114.806i			0.293623i
$392$		0			0
$393$		−	308.452i		−	0.784866i
$394$		0			0
$395$	248.798	+	420.121i	0.629870	+	1.06360i
$396$		0			0
$397$			357.537i			0.900598i	0.892878	+	0.450299i	$0.148683\pi$
$397$			357.537i			0.900598i	−0.892878	+	0.450299i	$0.851317\pi$
$398$		0			0
$399$		−	384.158i		−	0.962802i
$400$		0			0
$401$	262.506			0.654629			0.327315	−	0.944915i	$-0.393856\pi$
$401$	262.506			0.654629			0.327315	+	0.944915i	$0.393856\pi$
$402$		0			0
$403$	425.670			1.05625
$404$		0			0
$405$	−285.164	+	168.876i	−0.704108	+	0.416977i
$406$		0			0
$407$	429.181			1.05450
$408$		0			0
$409$	−63.2015			−0.154527			−0.0772634	−	0.997011i	$-0.524618\pi$
$409$	−63.2015			−0.154527			−0.0772634	+	0.997011i	$0.524618\pi$
$410$		0			0
$411$			117.459i			0.285787i
$412$		0			0
$413$			106.928i			0.258905i
$414$		0			0
$415$	−542.776	+	321.435i	−1.30789	+	0.774542i
$416$		0			0
$417$			612.393i			1.46857i
$418$		0			0
$419$			673.390i			1.60714i	0.595213	+	0.803568i	$0.297068\pi$
$419$			673.390i			1.60714i	−0.595213	+	0.803568i	$0.702932\pi$
$420$		0			0
$421$	84.6877			0.201158			0.100579	−	0.994929i	$-0.467930\pi$
$421$	84.6877			0.201158			0.100579	+	0.994929i	$0.467930\pi$
$422$		0			0
$423$	−93.5363			−0.221126
$424$		0			0
$425$	−114.186	−	62.5968i	−0.268674	−	0.147286i
$426$		0			0
$427$	208.650			0.488643
$428$		0			0
$429$	274.776			0.640504
$430$		0			0
$431$			672.158i			1.55953i	0.626072	+	0.779766i	$0.284662\pi$
$431$			672.158i			1.55953i	−0.626072	+	0.779766i	$0.715338\pi$
$432$		0			0
$433$			562.185i			1.29835i	0.760640	+	0.649174i	$0.224885\pi$
$433$			562.185i			1.29835i	−0.760640	+	0.649174i	$0.775115\pi$
$434$		0			0
$435$	237.417	−	140.600i	0.545786	−	0.323218i
$436$		0			0
$437$			798.469i			1.82716i
$438$		0			0
$439$		−	384.842i		−	0.876633i	−0.898821	−	0.438316i	$-0.855575\pi$
$439$		−	384.842i		−	0.876633i	0.898821	−	0.438316i	$-0.144425\pi$
$440$		0			0
$441$	48.3017			0.109528
$442$		0			0
$443$	−461.625			−1.04204			−0.521021	−	0.853544i	$-0.674449\pi$
$443$	−461.625			−1.04204			−0.521021	+	0.853544i	$0.674449\pi$
$444$		0			0
$445$	574.922	−	340.472i	1.29196	−	0.765106i
$446$		0			0
$447$	−55.3584			−0.123844
$448$		0			0
$449$	48.7390			0.108550			0.0542750	−	0.998526i	$-0.482715\pi$
$449$	48.7390			0.108550			0.0542750	+	0.998526i	$0.482715\pi$
$450$		0			0
$451$		−	71.9532i		−	0.159542i
$452$		0			0
$453$		−	237.023i		−	0.523229i
$454$		0			0
$455$	158.052	+	266.887i	0.347368	+	0.586565i
$456$		0			0
$457$		−	762.588i		−	1.66868i	−0.551248	−	0.834341i	$-0.685848\pi$
$457$		−	762.588i		−	1.66868i	0.551248	−	0.834341i	$-0.314152\pi$
$458$		0			0
$459$		−	149.399i		−	0.325488i
$460$		0			0
$461$	−406.436			−0.881639			−0.440820	−	0.897596i	$-0.645312\pi$
$461$	−406.436			−0.881639			−0.440820	+	0.897596i	$0.645312\pi$
$462$		0			0
$463$	−260.743			−0.563159			−0.281580	−	0.959538i	$-0.590858\pi$
$463$	−260.743			−0.563159			−0.281580	+	0.959538i	$0.590858\pi$
$464$		0			0
$465$	−185.388	−	313.046i	−0.398684	−	0.673218i
$466$		0			0
$467$	594.738			1.27353			0.636765	−	0.771058i	$-0.280272\pi$
$467$	594.738			1.27353			0.636765	+	0.771058i	$0.280272\pi$
$468$		0			0
$469$	−413.899			−0.882515
$470$		0			0
$471$			44.1647i			0.0937679i
$472$		0			0
$473$		−	160.067i		−	0.338407i
$474$		0			0
$475$	−794.157	−	435.355i	−1.67191	−	0.916537i
$476$		0			0
$477$			55.9294i			0.117252i
$478$		0			0
$479$		−	534.894i		−	1.11669i	−0.829609	−	0.558344i	$-0.811437\pi$
$479$		−	534.894i		−	1.11669i	0.829609	−	0.558344i	$-0.188563\pi$
$480$		0			0
$481$	1117.00			2.32224
$482$		0			0
$483$	−233.732			−0.483917
$484$		0			0
$485$	16.3287	+	27.5727i	0.0336675	+	0.0568510i
$486$		0			0
$487$	264.298			0.542706			0.271353	−	0.962480i	$-0.412529\pi$
$487$	264.298			0.542706			0.271353	+	0.962480i	$0.412529\pi$
$488$		0			0
$489$	−493.471			−1.00914
$490$		0			0
$491$		−	539.150i		−	1.09807i	−0.835801	−	0.549033i	$-0.814996\pi$
$491$		−	539.150i		−	1.09807i	0.835801	−	0.549033i	$-0.185004\pi$
$492$		0			0
$493$			104.359i			0.211681i
$494$		0			0
$495$	22.2921	+	37.6424i	0.0450345	+	0.0760453i
$496$		0			0
$497$			272.577i			0.548444i
$498$		0			0
$499$		−	138.218i		−	0.276991i	−0.990363	−	0.138495i	$-0.955773\pi$
$499$		−	138.218i		−	0.276991i	0.990363	−	0.138495i	$-0.0442266\pi$
$500$		0			0
$501$	379.557			0.757598
$502$		0			0
$503$	−389.170			−0.773697			−0.386848	−	0.922143i	$-0.626436\pi$
$503$	−389.170			−0.773697			−0.386848	+	0.922143i	$0.626436\pi$
$504$		0			0
$505$	522.992	−	309.719i	1.03563	−	0.613305i
$506$		0			0
$507$	249.645			0.492396
$508$		0			0
$509$	−468.599			−0.920627			−0.460314	−	0.887756i	$-0.652263\pi$
$509$	−468.599			−0.920627			−0.460314	+	0.887756i	$0.652263\pi$
$510$		0			0
$511$		−	144.124i		−	0.282043i
$512$		0			0
$513$		−	1039.06i		−	2.02545i
$514$		0			0
$515$	42.8451	−	25.3731i	0.0831943	−	0.0492682i
$516$		0			0
$517$		−	409.761i		−	0.792575i
$518$		0			0
$519$		−	173.223i		−	0.333764i
$520$		0			0
$521$	−931.151			−1.78724			−0.893619	−	0.448826i	$-0.851842\pi$
$521$	−931.151			−1.78724			−0.893619	+	0.448826i	$0.851842\pi$
$522$		0			0
$523$	227.656			0.435289			0.217645	−	0.976028i	$-0.430163\pi$
$523$	227.656			0.435289			0.217645	+	0.976028i	$0.430163\pi$
$524$		0			0
$525$	127.439	−	232.470i	0.242742	−	0.442799i
$526$		0			0
$527$	137.602			0.261104
$528$		0			0
$529$	−43.1902			−0.0816451
$530$		0			0
$531$			39.2511i			0.0739191i
$532$		0			0
$533$		−	187.267i		−	0.351346i
$534$		0			0
$535$	−580.115	+	343.548i	−1.08433	+	0.642145i
$536$		0			0
$537$			655.605i			1.22087i
$538$		0			0
$539$			211.599i			0.392576i
$540$		0			0
$541$	388.174			0.717511			0.358756	−	0.933431i	$-0.383201\pi$
$541$	388.174			0.717511			0.358756	+	0.933431i	$0.383201\pi$
$542$		0			0
$543$	−514.296			−0.947139
$544$		0			0
$545$	−121.453	+	71.9254i	−0.222850	+	0.131973i
$546$		0			0
$547$	−473.059			−0.864824			−0.432412	−	0.901676i	$-0.642337\pi$
$547$	−473.059			−0.864824			−0.432412	+	0.901676i	$0.642337\pi$
$548$		0			0
$549$	76.5916			0.139511
$550$		0			0
$551$			725.804i			1.31725i
$552$		0			0
$553$		−	375.961i		−	0.679857i
$554$		0			0
$555$	−486.475	−	821.463i	−0.876532	−	1.48011i
$556$		0			0
$557$			419.101i			0.752426i	0.926533	+	0.376213i	$0.122774\pi$
$557$			419.101i			0.752426i	−0.926533	+	0.376213i	$0.877226\pi$
$558$		0			0
$559$		−	416.594i		−	0.745248i
$560$		0			0
$561$	88.8241			0.158332
$562$		0			0
$563$	145.910			0.259165			0.129582	−	0.991569i	$-0.458636\pi$
$563$	145.910			0.259165			0.129582	+	0.991569i	$0.458636\pi$
$564$		0			0
$565$	−484.211	−	817.640i	−0.857011	−	1.44715i
$566$		0			0
$567$	255.189			0.450069
$568$		0			0
$569$	950.513			1.67050			0.835249	−	0.549872i	$-0.185323\pi$
$569$	950.513			1.67050			0.835249	+	0.549872i	$0.185323\pi$
$570$		0			0
$571$		−	404.107i		−	0.707717i	−0.935299	−	0.353859i	$-0.884869\pi$
$571$		−	404.107i		−	0.707717i	0.935299	−	0.353859i	$-0.115131\pi$
$572$		0			0
$573$		−	340.023i		−	0.593408i
$574$		0			0
$575$	−264.882	+	483.186i	−0.460664	+	0.840324i
$576$		0			0
$577$			847.944i			1.46957i	0.678298	+	0.734787i	$0.262718\pi$
$577$			847.944i			1.46957i	−0.678298	+	0.734787i	$0.737282\pi$
$578$		0			0
$579$		−	447.193i		−	0.772354i
$580$		0			0
$581$	485.723			0.836011
$582$		0			0
$583$	−245.014			−0.420264
$584$		0			0
$585$	58.0179	+	97.9692i	0.0991760	+	0.167469i
$586$		0			0
$587$	658.243			1.12137			0.560684	−	0.828030i	$-0.310538\pi$
$587$	658.243			1.12137			0.560684	+	0.828030i	$0.310538\pi$
$588$		0			0
$589$	957.010			1.62480
$590$		0			0
$591$			312.735i			0.529162i
$592$		0			0
$593$		−	282.430i		−	0.476274i	−0.971232	−	0.238137i	$-0.923463\pi$
$593$		−	282.430i		−	0.476274i	0.971232	−	0.238137i	$-0.0765367\pi$
$594$		0			0
$595$	51.0920	+	86.2739i	0.0858688	+	0.144998i
$596$		0			0
$597$		−	897.739i		−	1.50375i
$598$		0			0
$599$			498.597i			0.832382i	0.909277	+	0.416191i	$0.136635\pi$
$599$			498.597i			0.832382i	−0.909277	+	0.416191i	$0.863365\pi$
$600$		0			0
$601$	−287.496			−0.478363			−0.239182	−	0.970975i	$-0.576879\pi$
$601$	−287.496			−0.478363			−0.239182	+	0.970975i	$0.576879\pi$
$602$		0			0
$603$	−151.934			−0.251964
$604$		0			0
$605$	355.662	−	210.625i	0.587871	−	0.348141i
$606$		0			0
$607$	−844.260			−1.39087			−0.695437	−	0.718587i	$-0.744789\pi$
$607$	−844.260			−1.39087			−0.695437	+	0.718587i	$0.744789\pi$
$608$		0			0
$609$	−212.461			−0.348869
$610$		0			0
$611$		−	1066.46i		−	1.74543i
$612$		0			0
$613$			975.340i			1.59109i	0.605892	+	0.795547i	$0.292816\pi$
$613$			975.340i			1.59109i	−0.605892	+	0.795547i	$0.707184\pi$
$614$		0			0
$615$	−137.720	+	81.5588i	−0.223935	+	0.132616i
$616$		0			0
$617$			319.229i			0.517389i	0.965959	+	0.258695i	$0.0832923\pi$
$617$			319.229i			0.517389i	−0.965959	+	0.258695i	$0.916708\pi$
$618$		0			0
$619$			845.837i			1.36646i	0.730205	+	0.683228i	$0.239425\pi$
$619$			845.837i			1.36646i	−0.730205	+	0.683228i	$0.760575\pi$
$620$		0			0
$621$	−632.190			−1.01802
$622$		0			0
$623$	−514.489			−0.825826
$624$		0			0
$625$	−336.153	−	526.902i	−0.537846	−	0.843043i
$626$		0			0
$627$	617.764			0.985269
$628$		0			0
$629$	361.080			0.574055
$630$		0			0
$631$		−	322.653i		−	0.511335i	−0.966765	−	0.255668i	$-0.917705\pi$
$631$		−	322.653i		−	0.511335i	0.966765	−	0.255668i	$-0.0822953\pi$
$632$		0			0
$633$			360.087i			0.568858i
$634$		0			0
$635$	258.409	−	153.032i	0.406944	−	0.240995i
$636$		0			0
$637$			550.712i			0.864541i
$638$		0			0
$639$			100.058i			0.156585i
$640$		0			0
$641$	167.659			0.261558			0.130779	−	0.991412i	$-0.458252\pi$
$641$	167.659			0.261558			0.130779	+	0.991412i	$0.458252\pi$
$642$		0			0
$643$	−118.227			−0.183867			−0.0919335	−	0.995765i	$-0.529305\pi$
$643$	−118.227			−0.183867			−0.0919335	+	0.995765i	$0.529305\pi$
$644$		0			0
$645$	−306.372	+	181.435i	−0.474995	+	0.281295i
$646$		0			0
$647$	783.464			1.21092			0.605459	−	0.795876i	$-0.292989\pi$
$647$	783.464			1.21092			0.605459	+	0.795876i	$0.292989\pi$
$648$		0			0
$649$	−171.950			−0.264946
$650$		0			0
$651$			280.141i			0.430324i
$652$		0			0
$653$		−	28.4352i		−	0.0435455i	−0.999763	−	0.0217727i	$-0.993069\pi$
$653$		−	28.4352i		−	0.0435455i	0.999763	−	0.0217727i	$-0.00693102\pi$
$654$		0			0
$655$	285.314	+	481.781i	0.435593	+	0.735543i
$656$		0			0
$657$		−	52.9051i		−	0.0805253i
$658$		0			0
$659$			594.296i			0.901814i	0.892571	+	0.450907i	$0.148899\pi$
$659$			594.296i			0.901814i	−0.892571	+	0.450907i	$0.851101\pi$
$660$		0			0
$661$	−495.511			−0.749638			−0.374819	−	0.927098i	$-0.622295\pi$
$661$	−495.511			−0.749638			−0.374819	+	0.927098i	$0.622295\pi$
$662$		0			0
$663$	231.176			0.348682
$664$		0			0
$665$	355.340	+	600.028i	0.534346	+	0.902297i
$666$		0			0
$667$	441.599			0.662067
$668$		0			0
$669$	−257.231			−0.384501
$670$		0			0
$671$			335.530i			0.500045i
$672$		0			0
$673$		−	168.874i		−	0.250927i	−0.992098	−	0.125464i	$-0.959958\pi$
$673$		−	168.874i		−	0.250927i	0.992098	−	0.125464i	$-0.0400418\pi$
$674$		0			0
$675$	344.693	−	628.775i	0.510657	−	0.931519i
$676$		0			0
$677$			895.899i			1.32334i	0.749797	+	0.661668i	$0.230151\pi$
$677$			895.899i			1.32334i	−0.749797	+	0.661668i	$0.769849\pi$
$678$		0			0
$679$		−	24.6745i		−	0.0363394i
$680$		0			0
$681$	41.3042			0.0606522
$682$		0			0
$683$	359.410			0.526223			0.263112	−	0.964765i	$-0.415251\pi$
$683$	359.410			0.526223			0.263112	+	0.964765i	$0.415251\pi$
$684$		0			0
$685$	−108.647	−	183.462i	−0.158609	−	0.267828i
$686$		0			0
$687$	396.926			0.577767
$688$		0			0
$689$	−637.680			−0.925516
$690$		0			0
$691$			515.701i			0.746311i	0.927769	+	0.373155i	$0.121724\pi$
$691$			515.701i			0.746311i	−0.927769	+	0.373155i	$0.878276\pi$
$692$		0			0
$693$		−	33.6857i		−	0.0486085i
$694$		0			0
$695$	−566.454	−	956.514i	−0.815041	−	1.37628i
$696$		0			0
$697$		−	60.5360i		−	0.0868523i
$698$		0			0
$699$			348.510i			0.498584i
$700$		0			0
$701$	−1370.37			−1.95488			−0.977438	−	0.211221i	$-0.932256\pi$
$701$	−1370.37			−1.95488			−0.977438	+	0.211221i	$0.932256\pi$
$702$		0			0
$703$	2511.28			3.57224
$704$		0			0
$705$	−784.293	+	464.463i	−1.11247	+	0.658813i
$706$		0			0
$707$	−468.018			−0.661978
$708$		0			0
$709$	−662.128			−0.933890			−0.466945	−	0.884286i	$-0.654645\pi$
$709$	−662.128			−0.933890			−0.466945	+	0.884286i	$0.654645\pi$
$710$		0			0
$711$		−	138.008i		−	0.194104i
$712$		0			0
$713$		−	582.270i		−	0.816649i
$714$		0			0
$715$	−429.181	+	254.163i	−0.600253	+	0.355473i
$716$		0			0
$717$		−	4.55302i		−	0.00635010i
$718$		0			0
$719$			370.003i			0.514608i	0.966331	+	0.257304i	$0.0828341\pi$
$719$			370.003i			0.514608i	−0.966331	+	0.257304i	$0.917166\pi$
$720$		0			0
$721$	−38.3415			−0.0531782
$722$		0			0
$723$	−569.964			−0.788331
$724$		0			0
$725$	−240.776	+	439.214i	−0.332105	+	0.605812i
$726$		0			0
$727$	607.695			0.835894			0.417947	−	0.908471i	$-0.362750\pi$
$727$	607.695			0.835894			0.417947	+	0.908471i	$0.362750\pi$
$728$		0			0
$729$	804.700			1.10384
$730$		0			0
$731$		−	134.668i		−	0.184224i
$732$		0			0
$733$			1066.76i			1.45533i	0.685931	+	0.727666i	$0.259395\pi$
$733$			1066.76i			1.45533i	−0.685931	+	0.727666i	$0.740605\pi$
$734$		0			0
$735$	405.005	−	239.847i	0.551027	−	0.326322i
$736$		0			0
$737$		−	665.591i		−	0.903108i
$738$		0			0
$739$			558.366i			0.755570i	0.925893	+	0.377785i	$0.123314\pi$
$739$			558.366i			0.755570i	−0.925893	+	0.377785i	$0.876686\pi$
$740$		0			0
$741$	1607.81			2.16978
$742$		0			0
$743$	−1112.00			−1.49664			−0.748319	−	0.663339i	$-0.769139\pi$
$743$	−1112.00			−1.49664			−0.748319	+	0.663339i	$0.769139\pi$
$744$		0			0
$745$	86.4659	−	51.2056i	0.116062	−	0.0687324i
$746$		0			0
$747$	178.299			0.238687
$748$		0			0
$749$	519.137			0.693107
$750$		0			0
$751$		−	1207.34i		−	1.60764i	−0.594873	−	0.803820i	$-0.702798\pi$
$751$		−	1207.34i		−	1.60764i	0.594873	−	0.803820i	$-0.297202\pi$
$752$		0			0
$753$			204.158i			0.271127i
$754$		0			0
$755$	219.242	+	370.213i	0.290387	+	0.490348i
$756$		0			0
$757$			87.7776i			0.115955i	0.998318	+	0.0579773i	$0.0184651\pi$
$757$			87.7776i			0.115955i	−0.998318	+	0.0579773i	$0.981535\pi$
$758$		0			0
$759$		−	375.864i		−	0.495209i
$760$		0			0
$761$	67.0202			0.0880686			0.0440343	−	0.999030i	$-0.485979\pi$
$761$	67.0202			0.0880686			0.0440343	+	0.999030i	$0.485979\pi$
$762$		0			0
$763$	108.687			0.142447
$764$		0			0
$765$	18.7549	+	31.6695i	0.0245162	+	0.0413980i
$766$		0			0
$767$	−447.522			−0.583470
$768$		0			0
$769$	342.869			0.445863			0.222932	−	0.974834i	$-0.428437\pi$
$769$	342.869			0.445863			0.222932	+	0.974834i	$0.428437\pi$
$770$		0			0
$771$			756.587i			0.981306i
$772$		0			0
$773$		−	244.756i		−	0.316631i	−0.987389	−	0.158316i	$-0.949394\pi$
$773$		−	244.756i		−	0.316631i	0.987389	−	0.158316i	$-0.0506063\pi$
$774$		0			0
$775$	579.126	+	317.475i	0.747259	+	0.409646i
$776$		0			0
$777$			735.116i			0.946095i
$778$		0			0
$779$		−	421.023i		−	0.540466i
$780$		0			0
$781$	−438.330			−0.561242
$782$		0			0
$783$	−574.657			−0.733917
$784$		0			0
$785$	−40.8516	−	68.9821i	−0.0520403	−	0.0878753i
$786$		0			0
$787$	−1120.38			−1.42361			−0.711806	−	0.702376i	$-0.752123\pi$
$787$	−1120.38			−1.42361			−0.711806	+	0.702376i	$0.752123\pi$
$788$		0			0
$789$	208.063			0.263705
$790$		0			0
$791$			731.695i			0.925025i
$792$		0			0
$793$			873.260i			1.10121i
$794$		0			0
$795$	277.723	+	468.963i	0.349337	+	0.589891i
$796$		0			0
$797$		−	1344.51i		−	1.68696i	−0.537161	−	0.843480i	$-0.680503\pi$
$797$		−	1344.51i		−	1.68696i	0.537161	−	0.843480i	$-0.319497\pi$
$798$		0			0
$799$		−	344.742i		−	0.431467i
$800$		0			0
$801$	−188.859			−0.235779
$802$		0			0
$803$	231.765			0.288624
$804$		0			0
$805$	365.073	−	216.198i	0.453507	−	0.268569i
$806$		0			0
$807$	−865.122			−1.07202
$808$		0			0
$809$	345.363			0.426901			0.213451	−	0.976954i	$-0.431530\pi$
$809$	345.363			0.426901			0.213451	+	0.976954i	$0.431530\pi$
$810$		0			0
$811$			1275.44i			1.57268i	0.617797	+	0.786338i	$0.288025\pi$
$811$			1275.44i			1.57268i	−0.617797	+	0.786338i	$0.711975\pi$
$812$		0			0
$813$		−	352.999i		−	0.434193i
$814$		0			0
$815$	770.766	−	456.452i	0.945725	−	0.560064i
$816$		0			0
$817$		−	936.604i		−	1.14639i
$818$		0			0
$819$		−	87.6713i		−	0.107047i
$820$		0			0
$821$	1257.99			1.53226			0.766131	−	0.642684i	$-0.222179\pi$
$821$	1257.99			1.53226			0.766131	+	0.642684i	$0.222179\pi$
$822$		0			0
$823$	206.093			0.250417			0.125208	−	0.992130i	$-0.460040\pi$
$823$	206.093			0.250417			0.125208	+	0.992130i	$0.460040\pi$
$824$		0			0
$825$	373.834	+	204.935i	0.453132	+	0.248406i
$826$		0			0
$827$	−615.606			−0.744384			−0.372192	−	0.928156i	$-0.621394\pi$
$827$	−615.606			−0.744384			−0.372192	+	0.928156i	$0.621394\pi$
$828$		0			0
$829$	−1214.12			−1.46456			−0.732281	−	0.681002i	$-0.761544\pi$
$829$	−1214.12			−1.46456			−0.732281	+	0.681002i	$0.761544\pi$
$830$		0			0
$831$			666.892i			0.802518i
$832$		0			0
$833$			178.023i			0.213713i
$834$		0			0
$835$	−592.841	+	351.084i	−0.709989	+	0.420460i
$836$		0			0
$837$			757.715i			0.905275i
$838$		0			0
$839$		−	1338.99i		−	1.59593i	−0.602701	−	0.797967i	$-0.705909\pi$
$839$		−	1338.99i		−	1.59593i	0.602701	−	0.797967i	$-0.294091\pi$
$840$		0			0
$841$	−439.589			−0.522698
$842$		0			0
$843$	−79.4817			−0.0942844
$844$		0			0
$845$	−389.928	+	230.917i	−0.461453	+	0.273275i
$846$		0			0
$847$	−318.277			−0.375769
$848$		0			0
$849$	−743.671			−0.875937
$850$		0			0
$851$		−	1527.93i		−	1.79545i
$852$		0			0
$853$		−	112.066i		−	0.131379i	−0.997840	−	0.0656893i	$-0.979075\pi$
$853$		−	112.066i		−	0.131379i	0.997840	−	0.0656893i	$-0.0209246\pi$
$854$		0			0
$855$	130.438	+	220.259i	0.152560	+	0.257612i
$856$		0			0
$857$		−	1089.15i		−	1.27089i	−0.772148	−	0.635443i	$-0.780817\pi$
$857$		−	1089.15i		−	1.27089i	0.772148	−	0.635443i	$-0.219183\pi$
$858$		0			0
$859$		−	719.610i		−	0.837729i	−0.908049	−	0.418865i	$-0.862428\pi$
$859$		−	719.610i		−	0.837729i	0.908049	−	0.418865i	$-0.137572\pi$
$860$		0			0
$861$	123.244			0.143141
$862$		0			0
$863$	114.135			0.132254			0.0661269	−	0.997811i	$-0.478936\pi$
$863$	114.135			0.132254			0.0661269	+	0.997811i	$0.478936\pi$
$864$		0			0
$865$	160.229	+	270.563i	0.185236	+	0.312789i
$866$		0			0
$867$	−721.293			−0.831942
$868$		0			0
$869$	604.582			0.695721
$870$		0			0
$871$		−	1732.28i		−	1.98884i
$872$		0			0
$873$		−	9.05752i		−	0.0103752i
$874$		0			0
$875$	15.9795	+	480.981i	0.0182623	+	0.549692i
$876$		0			0
$877$		−	1427.99i		−	1.62827i	−0.580675	−	0.814135i	$-0.697211\pi$
$877$		−	1427.99i		−	1.62827i	0.580675	−	0.814135i	$-0.302789\pi$
$878$		0			0
$879$			973.539i			1.10755i
$880$		0			0
$881$	364.165			0.413354			0.206677	−	0.978409i	$-0.433735\pi$
$881$	364.165			0.413354			0.206677	+	0.978409i	$0.433735\pi$
$882$		0			0
$883$	800.458			0.906521			0.453260	−	0.891378i	$-0.350261\pi$
$883$	800.458			0.906521			0.453260	+	0.891378i	$0.350261\pi$
$884$		0			0
$885$	194.905	+	329.117i	0.220231	+	0.371883i
$886$		0			0
$887$	−591.672			−0.667049			−0.333524	−	0.942741i	$-0.608238\pi$
$887$	−591.672			−0.667049			−0.333524	+	0.942741i	$0.608238\pi$
$888$		0			0
$889$	−231.247			−0.260120
$890$		0			0
$891$			410.369i			0.460572i
$892$		0			0
$893$		−	2397.65i		−	2.68494i
$894$		0			0
$895$	−606.424	−	1024.01i	−0.677568	−	1.14414i
$896$		0			0
$897$		−	978.233i		−	1.09056i
$898$		0			0
$899$		−	529.281i		−	0.588744i
$900$		0			0
$901$	−206.136			−0.228786
$902$		0			0
$903$	274.168			0.303619
$904$		0			0
$905$	803.295	−	475.716i	0.887618	−	0.525653i
$906$		0			0
$907$	−545.657			−0.601606			−0.300803	−	0.953686i	$-0.597255\pi$
$907$	−545.657			−0.601606			−0.300803	+	0.953686i	$0.597255\pi$
$908$		0			0
$909$	−171.801			−0.189000
$910$		0			0
$911$		−	929.286i		−	1.02007i	−0.860153	−	0.510036i	$-0.829632\pi$
$911$		−	929.286i		−	1.02007i	0.860153	−	0.510036i	$-0.170368\pi$
$912$		0			0
$913$			781.089i			0.855520i
$914$		0			0
$915$	642.213	−	380.323i	0.701873	−	0.415653i
$916$		0			0
$917$		−	431.139i		−	0.470163i
$918$		0			0
$919$			171.353i			0.186456i	0.995645	+	0.0932278i	$0.0297185\pi$
$919$			171.353i			0.186456i	−0.995645	+	0.0932278i	$0.970282\pi$
$920$		0			0
$921$	718.750			0.780402
$922$		0			0
$923$	−1140.81			−1.23598
$924$		0			0
$925$	1519.68	+	833.085i	1.64290	+	0.900632i
$926$		0			0
$927$	−14.0744			−0.0151828
$928$		0			0
$929$	−1178.62			−1.26870			−0.634350	−	0.773046i	$-0.718732\pi$
$929$	−1178.62			−1.26870			−0.634350	+	0.773046i	$0.718732\pi$
$930$		0			0
$931$			1238.14i			1.32990i
$932$		0			0
$933$			389.940i			0.417943i
$934$		0			0
$935$	−138.737	+	82.1609i	−0.148382	+	0.0878726i
$936$		0			0
$937$		−	412.624i		−	0.440367i	−0.975458	−	0.220183i	$-0.929334\pi$
$937$		−	412.624i		−	0.440367i	0.975458	−	0.220183i	$-0.0706656\pi$
$938$		0			0
$939$		−	1007.97i		−	1.07346i
$940$		0			0
$941$	−1121.33			−1.19163			−0.595817	−	0.803120i	$-0.703172\pi$
$941$	−1121.33			−1.19163			−0.595817	+	0.803120i	$0.703172\pi$
$942$		0			0
$943$	−256.161			−0.271645
$944$		0			0
$945$	−475.073	+	281.341i	−0.502723	+	0.297716i
$946$		0			0
$947$	−1395.22			−1.47331			−0.736653	−	0.676271i	$-0.763595\pi$
$947$	−1395.22			−1.47331			−0.736653	+	0.676271i	$0.763595\pi$
$948$		0			0
$949$	603.199			0.635615
$950$		0			0
$951$		−	25.4178i		−	0.0267275i
$952$		0			0
$953$			476.331i			0.499822i	0.968269	+	0.249911i	$0.0804014\pi$
$953$			476.331i			0.499822i	−0.968269	+	0.249911i	$0.919599\pi$
$954$		0			0
$955$	314.516	+	531.092i	0.329336	+	0.556117i
$956$		0			0
$957$		−	341.658i		−	0.357010i
$958$		0			0
$959$			164.178i			0.171197i
$960$		0			0
$961$	263.116			0.273794
$962$		0			0
$963$	190.565			0.197887
$964$		0			0
$965$	413.646	+	698.484i	0.428649	+	0.723817i
$966$		0			0
$967$	140.846			0.145653			0.0728263	−	0.997345i	$-0.476798\pi$
$967$	140.846			0.145653			0.0728263	+	0.997345i	$0.476798\pi$
$968$		0			0
$969$	519.740			0.536367
$970$		0			0
$971$		−	1144.98i		−	1.17918i	−0.807704	−	0.589588i	$-0.799290\pi$
$971$		−	1144.98i		−	1.17918i	0.807704	−	0.589588i	$-0.200710\pi$
$972$		0			0
$973$			855.971i			0.879724i
$974$		0			0
$975$	972.950	+	533.369i	0.997898	+	0.547045i
$976$		0			0
$977$			55.8912i			0.0572070i	0.999591	+	0.0286035i	$0.00910602\pi$
$977$			55.8912i			0.0572070i	−0.999591	+	0.0286035i	$0.990894\pi$
$978$		0			0
$979$		−	827.349i		−	0.845096i
$980$		0			0
$981$	39.8969			0.0406696
$982$		0			0
$983$	1028.74			1.04653			0.523267	−	0.852169i	$-0.324713\pi$
$983$	1028.74			1.04653			0.523267	+	0.852169i	$0.324713\pi$
$984$		0			0
$985$	−289.275	−	488.469i	−0.293680	−	0.495908i
$986$		0			0
$987$	701.853			0.711098
$988$		0			0
$989$	−569.855			−0.576193
$990$		0			0
$991$			666.207i			0.672257i	0.941816	+	0.336129i	$0.109118\pi$
$991$			666.207i			0.672257i	−0.941816	+	0.336129i	$0.890882\pi$
$992$		0			0
$993$		−	147.030i		−	0.148066i
$994$		0			0
$995$	830.394	+	1402.20i	0.834567	+	1.40925i
$996$		0			0
$997$		−	737.006i		−	0.739224i	−0.929186	−	0.369612i	$-0.879491\pi$
$997$		−	737.006i		−	0.739224i	0.929186	−	0.369612i	$-0.120509\pi$
$998$		0			0
$999$			1988.31i			1.99030i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.h.b.159.1	yes	6
3.2	odd	2		1440.3.j.b.1279.2		6
4.3	odd	2		160.3.h.a.159.5	✓	6
5.2	odd	4		800.3.b.h.351.5		6
5.3	odd	4		800.3.b.i.351.2		6
5.4	even	2		160.3.h.a.159.6	yes	6
8.3	odd	2		320.3.h.g.319.2		6
8.5	even	2		320.3.h.f.319.6		6
12.11	even	2		1440.3.j.a.1279.2		6
15.14	odd	2		1440.3.j.a.1279.1		6
16.3	odd	4		1280.3.e.g.639.5		6
16.5	even	4		1280.3.e.h.639.5		6
16.11	odd	4		1280.3.e.i.639.2		6
16.13	even	4		1280.3.e.f.639.2		6
20.3	even	4		800.3.b.i.351.5		6
20.7	even	4		800.3.b.h.351.2		6
20.19	odd	2	inner	160.3.h.b.159.2	yes	6
40.3	even	4		1600.3.b.w.1151.2		6
40.13	odd	4		1600.3.b.w.1151.5		6
40.19	odd	2		320.3.h.f.319.5		6
40.27	even	4		1600.3.b.v.1151.5		6
40.29	even	2		320.3.h.g.319.1		6
40.37	odd	4		1600.3.b.v.1151.2		6
60.59	even	2		1440.3.j.b.1279.1		6
80.19	odd	4		1280.3.e.h.639.2		6
80.29	even	4		1280.3.e.i.639.5		6
80.59	odd	4		1280.3.e.f.639.5		6
80.69	even	4		1280.3.e.g.639.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.h.a.159.5	✓	6	4.3	odd	2
160.3.h.a.159.6	yes	6	5.4	even	2
160.3.h.b.159.1	yes	6	1.1	even	1	trivial
160.3.h.b.159.2	yes	6	20.19	odd	2	inner
320.3.h.f.319.5		6	40.19	odd	2
320.3.h.f.319.6		6	8.5	even	2
320.3.h.g.319.1		6	40.29	even	2
320.3.h.g.319.2		6	8.3	odd	2
800.3.b.h.351.2		6	20.7	even	4
800.3.b.h.351.5		6	5.2	odd	4
800.3.b.i.351.2		6	5.3	odd	4
800.3.b.i.351.5		6	20.3	even	4
1280.3.e.f.639.2		6	16.13	even	4
1280.3.e.f.639.5		6	80.59	odd	4
1280.3.e.g.639.2		6	80.69	even	4
1280.3.e.g.639.5		6	16.3	odd	4
1280.3.e.h.639.2		6	80.19	odd	4
1280.3.e.h.639.5		6	16.5	even	4
1280.3.e.i.639.2		6	16.11	odd	4
1280.3.e.i.639.5		6	80.29	even	4
1440.3.j.a.1279.1		6	15.14	odd	2
1440.3.j.a.1279.2		6	12.11	even	2
1440.3.j.b.1279.1		6	60.59	even	2
1440.3.j.b.1279.2		6	3.2	odd	2
1600.3.b.v.1151.2		6	40.37	odd	4
1600.3.b.v.1151.5		6	40.27	even	4
1600.3.b.w.1151.2		6	40.3	even	4
1600.3.b.w.1151.5		6	40.13	odd	4

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} +O(q^{10})\)	\(q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} -6.19112i q^{11} -16.1132i q^{13} +(-11.8500 + 7.01762i) q^{15} -5.20875i q^{17} -36.2264i q^{19} +10.6044 q^{21} -22.0411 q^{23} +(12.0176 - 21.9221i) q^{25} +28.6823 q^{27} -20.0352 q^{29} +26.4175i q^{31} +17.0529i q^{33} +(-16.5633 + 9.80888i) q^{35} +69.3219i q^{37} +44.3822i q^{39} +11.6220 q^{41} +25.8542 q^{43} +(-6.08006 + 3.60065i) q^{45} +66.1853 q^{47} -34.1777 q^{49} +14.3470i q^{51} -39.5751i q^{53} +(-15.7736 - 26.6354i) q^{55} +99.7821i q^{57} -27.7736i q^{59} -54.1954 q^{61} +5.44096 q^{63} +(-41.0529 - 69.3219i) q^{65} +107.507 q^{67} +60.7101 q^{69} -70.7997i q^{71} +37.4351i q^{73} +(-33.1014 + 60.3822i) q^{75} +23.8356i q^{77} +97.6530i q^{79} -66.2835 q^{81} -126.163 q^{83} +(-13.2707 - 22.4090i) q^{85} +55.1852 q^{87} +133.635 q^{89} +62.0352i q^{91} -72.7645i q^{93} +(-92.2969 - 155.853i) q^{95} +6.40900i q^{97} +8.74960i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 2 q^{5} + 12 q^{7} + 18 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 2 * q^5 + 12 * q^7 + 18 * q^9	\( 6 q + 4 q^{3} - 2 q^{5} + 12 q^{7} + 18 q^{9} - 36 q^{15} + 8 q^{21} - 68 q^{23} - 10 q^{25} + 184 q^{27} + 44 q^{29} - 108 q^{35} - 68 q^{41} - 76 q^{43} - 6 q^{45} + 268 q^{47} - 62 q^{49} - 288 q^{55} - 100 q^{61} - 172 q^{63} + 308 q^{67} - 184 q^{69} - 284 q^{75} + 238 q^{81} - 204 q^{83} - 32 q^{85} + 584 q^{87} + 76 q^{89} - 32 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 - 2 * q^5 + 12 * q^7 + 18 * q^9 - 36 * q^15 + 8 * q^21 - 68 * q^23 - 10 * q^25 + 184 * q^27 + 44 * q^29 - 108 * q^35 - 68 * q^41 - 76 * q^43 - 6 * q^45 + 268 * q^47 - 62 * q^49 - 288 * q^55 - 100 * q^61 - 172 * q^63 + 308 * q^67 - 184 * q^69 - 284 * q^75 + 238 * q^81 - 204 * q^83 - 32 * q^85 + 584 * q^87 + 76 * q^89 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more