LMFDB - Embedded newform 720.4.f.i.289.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(289,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.289"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-22,0,0,0,0,0,148,0,0,0,0,0,0,0,-160] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$5.17891i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.i.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.1789 + 0.178908i) q^{5} -33.0735i q^{7} +48.3578 q^{11} -60.3578i q^{13} -17.7891i q^{17} -130.863 q^{19} +70.8625i q^{23} +(124.936 - 4.00000i) q^{25} +104.505 q^{29} +210.441 q^{31} +(5.91712 + 369.725i) q^{35} -300.945i q^{37} -240.147 q^{41} +108.000i q^{43} +278.991i q^{47} -750.853 q^{49} -328.358i q^{53} +(-540.588 + 8.65162i) q^{55} -889.533 q^{59} -241.450 q^{61} +(10.7985 + 674.735i) q^{65} +103.834i q^{67} -277.597 q^{71} +274.403i q^{73} -1599.36i q^{77} +366.991 q^{79} +57.7251i q^{83} +(3.18262 + 198.863i) q^{85} -203.175 q^{89} -1996.24 q^{91} +(1462.90 - 23.4124i) q^{95} -1283.45i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{5} + 148 q^{11} - 160 q^{19} + 100 q^{29} + 24 q^{31} + 796 q^{35} - 688 q^{41} - 3276 q^{49} - 1072 q^{55} - 1332 q^{59} + 488 q^{61} - 820 q^{65} + 616 q^{71} + 2104 q^{79} + 2148 q^{85} + 1368 q^{89}+ \cdots + 2944 q^{95}+O(q^{100}) $ 4 * q - 22 * q^5 + 148 * q^11 - 160 * q^19 + 100 * q^29 + 24 * q^31 + 796 * q^35 - 688 * q^41 - 3276 * q^49 - 1072 * q^55 - 1332 * q^59 + 488 * q^61 - 820 * q^65 + 616 * q^71 + 2104 * q^79 + 2148 * q^85 + 1368 * q^89 - 1352 * q^91 + 2944 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−11.1789	+	0.178908i	−0.999872	+	0.0160020i
$5$	−11.1789	+	0.178908i	−0.999872	+	0.0160020i
$6$		0			0
$7$		−	33.0735i		−	1.78580i	−0.450257	−	0.892899i	$-0.648667\pi$
$7$		−	33.0735i		−	1.78580i	0.450257	−	0.892899i	$-0.351333\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	48.3578			1.32549			0.662747	−	0.748844i	$-0.269391\pi$
$11$	48.3578			1.32549			0.662747	+	0.748844i	$0.269391\pi$
$12$		0			0
$13$		−	60.3578i		−	1.28771i	−0.765147	−	0.643856i	$-0.777334\pi$
$13$		−	60.3578i		−	1.28771i	0.765147	−	0.643856i	$-0.222666\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	17.7891i		−	0.253793i	−0.991916	−	0.126897i	$-0.959498\pi$
$17$		−	17.7891i		−	0.253793i	0.991916	−	0.126897i	$-0.0405017\pi$
$18$		0			0
$19$	−130.863			−1.58010			−0.790051	−	0.613042i	$-0.789946\pi$
$19$	−130.863			−1.58010			−0.790051	+	0.613042i	$0.789946\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			70.8625i			0.642429i	0.947007	+	0.321214i	$0.104091\pi$
$23$			70.8625i			0.642429i	−0.947007	+	0.321214i	$0.895909\pi$
$24$		0			0
$25$	124.936	−	4.00000i	0.999488	−	0.0320000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	104.505			0.669174			0.334587	−	0.942365i	$-0.391403\pi$
$29$	104.505			0.669174			0.334587	+	0.942365i	$0.391403\pi$
$30$		0			0
$31$	210.441			1.21923			0.609617	−	0.792696i	$-0.291323\pi$
$31$	210.441			1.21923			0.609617	+	0.792696i	$0.291323\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.91712	+	369.725i	0.0285764	+	1.78557i
$36$		0			0
$37$		−	300.945i		−	1.33717i	−0.743638	−	0.668583i	$-0.766901\pi$
$37$		−	300.945i		−	1.33717i	0.743638	−	0.668583i	$-0.233099\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−240.147			−0.914747			−0.457374	−	0.889275i	$-0.651210\pi$
$41$	−240.147			−0.914747			−0.457374	+	0.889275i	$0.651210\pi$
$42$		0			0
$43$			108.000i			0.383020i	0.981491	+	0.191510i	$0.0613384\pi$
$43$			108.000i			0.383020i	−0.981491	+	0.191510i	$0.938662\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			278.991i			0.865850i	0.901430	+	0.432925i	$0.142519\pi$
$47$			278.991i			0.865850i	−0.901430	+	0.432925i	$0.857481\pi$
$48$		0			0
$49$	−750.853			−2.18908
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	328.358i		−	0.851008i	−0.904957	−	0.425504i	$-0.860097\pi$
$53$		−	328.358i		−	0.851008i	0.904957	−	0.425504i	$-0.139903\pi$
$54$		0			0
$55$	−540.588	+	8.65162i	−1.32532	+	0.0212106i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−889.533			−1.96284			−0.981418	−	0.191882i	$-0.938541\pi$
$59$	−889.533			−1.96284			−0.981418	+	0.191882i	$0.938541\pi$
$60$		0			0
$61$	−241.450			−0.506795			−0.253398	−	0.967362i	$-0.581548\pi$
$61$	−241.450			−0.506795			−0.253398	+	0.967362i	$0.581548\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.7985	+	674.735i	0.0206060	+	1.28755i
$66$		0			0
$67$			103.834i			0.189334i	0.995509	+	0.0946669i	$0.0301786\pi$
$67$			103.834i			0.189334i	−0.995509	+	0.0946669i	$0.969821\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−277.597			−0.464010			−0.232005	−	0.972715i	$-0.574529\pi$
$71$	−277.597			−0.464010			−0.232005	+	0.972715i	$0.574529\pi$
$72$		0			0
$73$			274.403i			0.439951i	0.975505	+	0.219976i	$0.0705978\pi$
$73$			274.403i			0.439951i	−0.975505	+	0.219976i	$0.929402\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1599.36i		−	2.36706i
$78$		0			0
$79$	366.991			0.522654			0.261327	−	0.965250i	$-0.415840\pi$
$79$	366.991			0.522654			0.261327	+	0.965250i	$0.415840\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			57.7251i			0.0763391i	0.999271	+	0.0381696i	$0.0121527\pi$
$83$			57.7251i			0.0763391i	−0.999271	+	0.0381696i	$0.987847\pi$
$84$		0			0
$85$	3.18262	+	198.863i	0.00406121	+	0.253761i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−203.175			−0.241983			−0.120992	−	0.992654i	$-0.538607\pi$
$89$	−203.175			−0.241983			−0.120992	+	0.992654i	$0.538607\pi$
$90$		0			0
$91$	−1996.24			−2.29959
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1462.90	−	23.4124i	1.57990	−	0.0252849i
$96$		0			0
$97$		−	1283.45i		−	1.34345i	−0.740801	−	0.671725i	$-0.765554\pi$
$97$		−	1283.45i		−	1.34345i	0.740801	−	0.671725i	$-0.234446\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−886.908			−0.873768			−0.436884	−	0.899518i	$-0.643918\pi$
$101$	−886.908			−0.873768			−0.436884	+	0.899518i	$0.643918\pi$
$102$		0			0
$103$			783.055i			0.749094i	0.927208	+	0.374547i	$0.122202\pi$
$103$			783.055i			0.749094i	−0.927208	+	0.374547i	$0.877798\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1301.51i		−	1.17590i	−0.808897	−	0.587950i	$-0.799935\pi$
$107$		−	1301.51i		−	1.17590i	0.808897	−	0.587950i	$-0.200065\pi$
$108$		0			0
$109$	−1161.91			−1.02102			−0.510508	−	0.859873i	$-0.670543\pi$
$109$	−1161.91			−1.02102			−0.510508	+	0.859873i	$0.670543\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	507.808i		−	0.422748i	−0.977405	−	0.211374i	$-0.932206\pi$
$113$		−	507.808i		−	0.422748i	0.977405	−	0.211374i	$-0.0677938\pi$
$114$		0			0
$115$	−12.6779	−	792.166i	−0.0102802	−	0.642346i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−588.346			−0.453224
$120$		0			0
$121$	1007.48			0.756933
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1395.93	+	67.0677i	−0.998848	+	0.0479898i
$126$		0			0
$127$			796.064i			0.556215i	0.960550	+	0.278107i	$0.0897071\pi$
$127$			796.064i			0.556215i	−0.960550	+	0.278107i	$0.910293\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−91.4764			−0.0610102			−0.0305051	−	0.999535i	$-0.509712\pi$
$131$	−91.4764			−0.0610102			−0.0305051	+	0.999535i	$0.509712\pi$
$132$		0			0
$133$			4328.08i			2.82174i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2273.28i			1.41766i	0.705380	+	0.708829i	$0.250776\pi$
$137$			2273.28i			1.41766i	−0.705380	+	0.708829i	$0.749224\pi$
$138$		0			0
$139$	−738.735			−0.450782			−0.225391	−	0.974268i	$-0.572366\pi$
$139$	−738.735			−0.450782			−0.225391	+	0.974268i	$0.572366\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2918.77i		−	1.70685i
$144$		0			0
$145$	−1168.25	+	18.6968i	−0.669088	+	0.0107082i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1507.15			0.828661			0.414330	−	0.910127i	$-0.364016\pi$
$149$	1507.15			0.828661			0.414330	+	0.910127i	$0.364016\pi$
$150$		0			0
$151$	−154.365			−0.0831925			−0.0415962	−	0.999135i	$-0.513244\pi$
$151$	−154.365			−0.0831925			−0.0415962	+	0.999135i	$0.513244\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2352.50	+	37.6496i	−1.21908	+	0.0195102i
$156$		0			0
$157$			315.514i			0.160387i	0.996779	+	0.0801935i	$0.0255538\pi$
$157$			315.514i			0.160387i	−0.996779	+	0.0801935i	$0.974446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2343.67			1.14725
$162$		0			0
$163$			1457.85i			0.700539i	0.936649	+	0.350270i	$0.113910\pi$
$163$			1457.85i			0.700539i	−0.936649	+	0.350270i	$0.886090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2198.61i			1.01876i	0.860541	+	0.509381i	$0.170126\pi$
$167$			2198.61i			1.01876i	−0.860541	+	0.509381i	$0.829874\pi$
$168$		0			0
$169$	−1446.07			−0.658200
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2030.49i			0.892343i	0.894948	+	0.446171i	$0.147213\pi$
$173$			2030.49i			0.892343i	−0.894948	+	0.446171i	$0.852787\pi$
$174$		0			0
$175$	−132.294	−	4132.06i	−0.0571456	−	1.78488i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−427.220			−0.178391			−0.0891954	−	0.996014i	$-0.528430\pi$
$179$	−427.220			−0.178391			−0.0891954	+	0.996014i	$0.528430\pi$
$180$		0			0
$181$	−3779.97			−1.55228			−0.776140	−	0.630561i	$-0.782825\pi$
$181$	−3779.97			−1.55228			−0.776140	+	0.630561i	$0.782825\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	53.8416	+	3364.24i	0.0213974	+	1.33699i
$186$		0			0
$187$		−	860.241i		−	0.336401i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1565.60			0.593103			0.296551	−	0.955017i	$-0.404163\pi$
$191$	1565.60			0.593103			0.296551	+	0.955017i	$0.404163\pi$
$192$		0			0
$193$		−	2642.63i		−	0.985599i	−0.870143	−	0.492800i	$-0.835974\pi$
$193$		−	2642.63i		−	0.985599i	0.870143	−	0.492800i	$-0.164026\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			98.4522i			0.0356062i	0.999842	+	0.0178031i	$0.00566721\pi$
$197$			98.4522i			0.0356062i	−0.999842	+	0.0178031i	$0.994333\pi$
$198$		0			0
$199$	1131.62			0.403106			0.201553	−	0.979478i	$-0.435401\pi$
$199$	1131.62			0.403106			0.201553	+	0.979478i	$0.435401\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	3456.33i		−	1.19501i
$204$		0			0
$205$	2684.58	−	42.9643i	0.914630	−	0.0146378i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6328.23			−2.09441
$210$		0			0
$211$	1902.85			0.620841			0.310420	−	0.950599i	$-0.399530\pi$
$211$	1902.85			0.620841			0.310420	+	0.950599i	$0.399530\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−19.3221	−	1207.32i	−0.00612910	−	0.382971i
$216$		0			0
$217$		−	6960.00i		−	2.17731i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1073.71			−0.326813
$222$		0			0
$223$		−	4855.59i		−	1.45809i	−0.684465	−	0.729046i	$-0.739964\pi$
$223$		−	4855.59i		−	1.45809i	0.684465	−	0.729046i	$-0.260036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6536.19i			1.91111i	0.294812	+	0.955555i	$0.404743\pi$
$227$			6536.19i			1.91111i	−0.294812	+	0.955555i	$0.595257\pi$
$228$		0			0
$229$	5510.35			1.59011			0.795053	−	0.606539i	$-0.207443\pi$
$229$	5510.35			1.59011			0.795053	+	0.606539i	$0.207443\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5915.65i		−	1.66329i	−0.555306	−	0.831646i	$-0.687399\pi$
$233$		−	5915.65i		−	1.66329i	0.555306	−	0.831646i	$-0.312601\pi$
$234$		0			0
$235$	−49.9137	−	3118.81i	−0.0138554	−	0.865739i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2263.25			−0.612543			−0.306272	−	0.951944i	$-0.599082\pi$
$239$	−2263.25			−0.612543			−0.306272	+	0.951944i	$0.599082\pi$
$240$		0			0
$241$	772.493			0.206476			0.103238	−	0.994657i	$-0.467080\pi$
$241$	772.493			0.206476			0.103238	+	0.994657i	$0.467080\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8393.72	−	134.334i	2.18880	−	0.0350297i
$246$		0			0
$247$			7898.58i			2.03471i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	596.192			0.149926			0.0749628	−	0.997186i	$-0.476116\pi$
$251$	596.192			0.149926			0.0749628	+	0.997186i	$0.476116\pi$
$252$		0			0
$253$			3426.76i			0.851535i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4139.19i			1.00465i	0.864678	+	0.502326i	$0.167522\pi$
$257$			4139.19i			1.00465i	−0.864678	+	0.502326i	$0.832478\pi$
$258$		0			0
$259$	−9953.30			−2.38791
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1611.34i		−	0.377792i	−0.981997	−	0.188896i	$-0.939509\pi$
$263$		−	1611.34i		−	0.377792i	0.981997	−	0.188896i	$-0.0604909\pi$
$264$		0			0
$265$	58.7460	+	3670.68i	0.0136179	+	0.850899i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7031.33			−1.59371			−0.796855	−	0.604171i	$-0.793504\pi$
$269$	−7031.33			−1.59371			−0.796855	+	0.604171i	$0.793504\pi$
$270$		0			0
$271$	5441.27			1.21968			0.609840	−	0.792524i	$-0.291234\pi$
$271$	5441.27			1.21968			0.609840	+	0.792524i	$0.291234\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6041.63	−	193.431i	1.32481	−	0.0424158i
$276$		0			0
$277$		−	1080.43i		−	0.234355i	−0.993111	−	0.117178i	$-0.962615\pi$
$277$		−	1080.43i		−	0.234355i	0.993111	−	0.117178i	$-0.0373847\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1602.99			0.340308			0.170154	−	0.985417i	$-0.445573\pi$
$281$	1602.99			0.340308			0.170154	+	0.985417i	$0.445573\pi$
$282$		0			0
$283$		−	334.810i		−	0.0703265i	−0.999382	−	0.0351632i	$-0.988805\pi$
$283$		−	334.810i		−	0.0703265i	0.999382	−	0.0351632i	$-0.0111951\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			7942.49i			1.63355i
$288$		0			0
$289$	4596.55			0.935589
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	539.250i		−	0.107520i	−0.998554	−	0.0537599i	$-0.982879\pi$
$293$		−	539.250i		−	0.107520i	0.998554	−	0.0537599i	$-0.0171206\pi$
$294$		0			0
$295$	9944.01	−	159.145i	1.96258	−	0.0314094i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4277.11			0.827263
$300$		0			0
$301$	3571.93			0.683996
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2699.15	−	43.1974i	0.506731	−	0.00810977i
$306$		0			0
$307$			8477.58i			1.57603i	0.615656	+	0.788015i	$0.288891\pi$
$307$			8477.58i			1.57603i	−0.615656	+	0.788015i	$0.711109\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3646.92			0.664945			0.332473	−	0.943113i	$-0.392117\pi$
$311$	3646.92			0.664945			0.332473	+	0.943113i	$0.392117\pi$
$312$		0			0
$313$			7537.05i			1.36108i	0.732709	+	0.680542i	$0.238255\pi$
$313$			7537.05i			1.36108i	−0.732709	+	0.680542i	$0.761745\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10721.2i		−	1.89956i	−0.312916	−	0.949781i	$-0.601306\pi$
$317$		−	10721.2i		−	1.89956i	0.312916	−	0.949781i	$-0.398694\pi$
$318$		0			0
$319$	5053.62			0.886986
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2327.92i			0.401019i
$324$		0			0
$325$	−241.431	−	7540.86i	−0.0412068	−	1.28705i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9227.18			1.54623
$330$		0			0
$331$	4405.14			0.731505			0.365753	−	0.930712i	$-0.380812\pi$
$331$	4405.14			0.731505			0.365753	+	0.930712i	$0.380812\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.5768	−	1160.75i	−0.00302973	−	0.189310i
$336$		0			0
$337$			186.117i			0.0300844i	0.999887	+	0.0150422i	$0.00478827\pi$
$337$			186.117i			0.0300844i	−0.999887	+	0.0150422i	$0.995212\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10176.5			1.61609
$342$		0			0
$343$			13489.1i			2.12345i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5547.38i		−	0.858211i	−0.903254	−	0.429105i	$-0.858829\pi$
$347$		−	5547.38i		−	0.858211i	0.903254	−	0.429105i	$-0.141171\pi$
$348$		0			0
$349$	−9078.82			−1.39249			−0.696244	−	0.717805i	$-0.745147\pi$
$349$	−9078.82			−1.39249			−0.696244	+	0.717805i	$0.745147\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10678.1i		−	1.61002i	−0.593262	−	0.805009i	$-0.702160\pi$
$353$		−	10678.1i		−	1.61002i	0.593262	−	0.805009i	$-0.297840\pi$
$354$		0			0
$355$	3103.23	−	49.6644i	0.463951	−	0.00742511i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	265.733			0.0390665			0.0195332	−	0.999809i	$-0.493782\pi$
$359$	265.733			0.0390665			0.0195332	+	0.999809i	$0.493782\pi$
$360$		0			0
$361$	10266.0			1.49672
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−49.0930	−	3067.53i	−0.00704012	−	0.439895i
$366$		0			0
$367$		−	5854.21i		−	0.832663i	−0.909213	−	0.416332i	$-0.863316\pi$
$367$		−	5854.21i		−	0.832663i	0.909213	−	0.416332i	$-0.136684\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10859.9			−1.51973
$372$		0			0
$373$		−	10134.5i		−	1.40682i	−0.710787	−	0.703408i	$-0.751661\pi$
$373$		−	10134.5i		−	1.40682i	0.710787	−	0.703408i	$-0.248339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6307.68i		−	0.861703i
$378$		0			0
$379$	−4235.70			−0.574072			−0.287036	−	0.957920i	$-0.592670\pi$
$379$	−4235.70			−0.574072			−0.287036	+	0.957920i	$0.592670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			8100.84i			1.08077i	0.841419	+	0.540383i	$0.181721\pi$
$383$			8100.84i			1.08077i	−0.841419	+	0.540383i	$0.818279\pi$
$384$		0			0
$385$	286.139	+	17879.1i	0.0378779	+	2.36676i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13820.2			−1.80131			−0.900656	−	0.434532i	$-0.856914\pi$
$389$	−13820.2			−1.80131			−0.900656	+	0.434532i	$0.856914\pi$
$390$		0			0
$391$	1260.58			0.163044
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4102.55	+	65.6577i	−0.522587	+	0.00836353i
$396$		0			0
$397$		−	11523.9i		−	1.45685i	−0.685125	−	0.728425i	$-0.740252\pi$
$397$		−	11523.9i		−	1.45685i	0.685125	−	0.728425i	$-0.259748\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−700.915			−0.0872869			−0.0436434	−	0.999047i	$-0.513897\pi$
$401$	−700.915			−0.0872869			−0.0436434	+	0.999047i	$0.513897\pi$
$402$		0			0
$403$		−	12701.7i		−	1.57002i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14553.1i		−	1.77240i
$408$		0			0
$409$	−6650.69			−0.804047			−0.402024	−	0.915629i	$-0.631693\pi$
$409$	−6650.69			−0.804047			−0.402024	+	0.915629i	$0.631693\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			29419.9i			3.50523i
$414$		0			0
$415$	−10.3275	−	645.303i	−0.00122158	−	0.0763294i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13844.4			−1.61418			−0.807091	−	0.590426i	$-0.798960\pi$
$419$	−13844.4			−1.61418			−0.807091	+	0.590426i	$0.798960\pi$
$420$		0			0
$421$	12576.3			1.45590			0.727949	−	0.685631i	$-0.240474\pi$
$421$	12576.3			1.45590			0.727949	+	0.685631i	$0.240474\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−71.1563	−	2222.50i	−0.00812139	−	0.253663i
$426$		0			0
$427$			7985.59i			0.905035i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13440.6			1.50212			0.751059	−	0.660236i	$-0.229544\pi$
$431$	13440.6			1.50212			0.751059	+	0.660236i	$0.229544\pi$
$432$		0			0
$433$			3012.87i			0.334387i	0.985924	+	0.167193i	$0.0534704\pi$
$433$			3012.87i			0.334387i	−0.985924	+	0.167193i	$0.946530\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	9273.25i		−	1.01510i
$438$		0			0
$439$	10675.0			1.16057			0.580283	−	0.814415i	$-0.302942\pi$
$439$	10675.0			1.16057			0.580283	+	0.814415i	$0.302942\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			125.868i			0.0134992i	0.999977	+	0.00674962i	$0.00214849\pi$
$443$			125.868i			0.0134992i	−0.999977	+	0.00674962i	$0.997852\pi$
$444$		0			0
$445$	2271.28	−	36.3497i	0.241952	−	0.00387223i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9707.21			−1.02029			−0.510146	−	0.860088i	$-0.670409\pi$
$449$	−9707.21			−1.02029			−0.510146	+	0.860088i	$0.670409\pi$
$450$		0			0
$451$	−11613.0			−1.21249
$452$		0			0
$453$		0			0
$454$		0			0
$455$	22315.8	−	357.144i	2.29930	−	0.0367982i
$456$		0			0
$457$			1279.97i			0.131016i	0.997852	+	0.0655082i	$0.0208668\pi$
$457$			1279.97i			0.131016i	−0.997852	+	0.0655082i	$0.979133\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3080.48			−0.311220			−0.155610	−	0.987819i	$-0.549734\pi$
$461$	−3080.48			−0.311220			−0.155610	+	0.987819i	$0.549734\pi$
$462$		0			0
$463$		−	18017.3i		−	1.80850i	−0.427008	−	0.904248i	$-0.640432\pi$
$463$		−	18017.3i		−	1.80850i	0.427008	−	0.904248i	$-0.359568\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			7236.77i			0.717083i	0.933514	+	0.358541i	$0.116726\pi$
$467$			7236.77i			0.717083i	−0.933514	+	0.358541i	$0.883274\pi$
$468$		0			0
$469$	3434.16			0.338112
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5222.64i			0.507690i
$474$		0			0
$475$	−16349.4	+	523.450i	−1.57929	+	0.0505632i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4932.78			−0.470532			−0.235266	−	0.971931i	$-0.575596\pi$
$479$	−4932.78			−0.470532			−0.235266	+	0.971931i	$0.575596\pi$
$480$		0			0
$481$	−18164.4			−1.72188
$482$		0			0
$483$		0			0
$484$		0			0
$485$	229.620	+	14347.6i	0.0214979	+	1.34328i
$486$		0			0
$487$		−	5937.91i		−	0.552510i	−0.961084	−	0.276255i	$-0.910907\pi$
$487$		−	5937.91i		−	0.552510i	0.961084	−	0.276255i	$-0.0890934\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15703.5			1.44336			0.721678	−	0.692229i	$-0.243371\pi$
$491$	15703.5			1.44336			0.721678	+	0.692229i	$0.243371\pi$
$492$		0			0
$493$		−	1859.04i		−	0.169832i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9181.09i			0.828628i
$498$		0			0
$499$	−5656.77			−0.507478			−0.253739	−	0.967273i	$-0.581660\pi$
$499$	−5656.77			−0.507478			−0.253739	+	0.967273i	$0.581660\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			290.441i			0.0257457i	0.999917	+	0.0128729i	$0.00409768\pi$
$503$			290.441i			0.0257457i	−0.999917	+	0.0128729i	$0.995902\pi$
$504$		0			0
$505$	9914.66	−	158.675i	0.873657	−	0.0139821i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17330.6			1.50916			0.754582	−	0.656206i	$-0.227840\pi$
$509$	17330.6			1.50916			0.754582	+	0.656206i	$0.227840\pi$
$510$		0			0
$511$	9075.45			0.785664
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−140.095	−	8753.70i	−0.0119870	−	0.748998i
$516$		0			0
$517$			13491.4i			1.14768i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6174.36			−0.519201			−0.259601	−	0.965716i	$-0.583591\pi$
$521$	−6174.36			−0.519201			−0.259601	+	0.965716i	$0.583591\pi$
$522$		0			0
$523$			13389.4i			1.11946i	0.828675	+	0.559730i	$0.189095\pi$
$523$			13389.4i			1.11946i	−0.828675	+	0.559730i	$0.810905\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3743.55i		−	0.309434i
$528$		0			0
$529$	7145.50			0.587285
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14494.7i			1.17793i
$534$		0			0
$535$	232.850	+	14549.4i	0.0188168	+	1.17575i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−36309.6			−2.90161
$540$		0			0
$541$	−14355.5			−1.14084			−0.570418	−	0.821354i	$-0.693219\pi$
$541$	−14355.5			−1.14084			−0.570418	+	0.821354i	$0.693219\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12988.9	−	207.875i	1.02089	−	0.0163384i
$546$		0			0
$547$		−	21133.9i		−	1.65195i	−0.563704	−	0.825977i	$-0.690624\pi$
$547$		−	21133.9i		−	1.65195i	0.563704	−	0.825977i	$-0.309376\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13675.8			−1.05736
$552$		0			0
$553$		−	12137.6i		−	0.933355i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3098.60i			0.235713i	0.993031	+	0.117856i	$0.0376023\pi$
$557$			3098.60i			0.235713i	−0.993031	+	0.117856i	$0.962398\pi$
$558$		0			0
$559$	6518.64			0.493219
$560$		0			0
$561$		0			0
$562$		0			0
$563$			7908.77i			0.592033i	0.955183	+	0.296017i	$0.0956584\pi$
$563$			7908.77i			0.592033i	−0.955183	+	0.296017i	$0.904342\pi$
$564$		0			0
$565$	90.8511	+	5676.74i	0.00676484	+	0.422694i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10740.7			−0.791345			−0.395673	−	0.918392i	$-0.629488\pi$
$569$	−10740.7			−0.791345			−0.395673	+	0.918392i	$0.629488\pi$
$570$		0			0
$571$	−14701.2			−1.07745			−0.538725	−	0.842482i	$-0.681094\pi$
$571$	−14701.2			−1.07745			−0.538725	+	0.842482i	$0.681094\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	283.450	+	8853.28i	0.0205577	+	0.642100i
$576$		0			0
$577$		−	15788.1i		−	1.13911i	−0.821954	−	0.569554i	$-0.807116\pi$
$577$		−	15788.1i		−	1.13911i	0.821954	−	0.569554i	$-0.192884\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1909.17			0.136326
$582$		0			0
$583$		−	15878.7i		−	1.12801i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	14778.5i		−	1.03913i	−0.854430	−	0.519567i	$-0.826093\pi$
$587$		−	14778.5i		−	1.03913i	0.854430	−	0.519567i	$-0.173907\pi$
$588$		0			0
$589$	−27538.8			−1.92651
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11143.9i		−	0.771713i	−0.922559	−	0.385857i	$-0.873906\pi$
$593$		−	11143.9i		−	0.771713i	0.922559	−	0.385857i	$-0.126094\pi$
$594$		0			0
$595$	6577.07	−	105.260i	0.453166	−	0.00725251i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7707.87			0.525768			0.262884	−	0.964827i	$-0.415326\pi$
$599$	7707.87			0.525768			0.262884	+	0.964827i	$0.415326\pi$
$600$		0			0
$601$	−13681.4			−0.928580			−0.464290	−	0.885683i	$-0.653691\pi$
$601$	−13681.4			−0.928580			−0.464290	+	0.885683i	$0.653691\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11262.5	+	180.246i	−0.756837	+	0.0121125i
$606$		0			0
$607$			11552.4i			0.772481i	0.922398	+	0.386241i	$0.126227\pi$
$607$			11552.4i			0.772481i	−0.922398	+	0.386241i	$0.873773\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16839.3			1.11496
$612$		0			0
$613$		−	9904.66i		−	0.652603i	−0.945266	−	0.326301i	$-0.894198\pi$
$613$		−	9904.66i		−	0.652603i	0.945266	−	0.326301i	$-0.105802\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			20323.5i			1.32608i	0.748582	+	0.663042i	$0.230735\pi$
$617$			20323.5i			1.32608i	−0.748582	+	0.663042i	$0.769265\pi$
$618$		0			0
$619$	−9223.99			−0.598940			−0.299470	−	0.954106i	$-0.596810\pi$
$619$	−9223.99			−0.598940			−0.299470	+	0.954106i	$0.596810\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6719.70i			0.432134i
$624$		0			0
$625$	15593.0	−	999.488i	0.997952	−	0.0639672i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5353.54			−0.339364
$630$		0			0
$631$	16916.0			1.06722			0.533610	−	0.845730i	$-0.320835\pi$
$631$	16916.0			1.06722			0.533610	+	0.845730i	$0.320835\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−142.422	−	8899.13i	−0.00890057	−	0.556143i
$636$		0			0
$637$			45319.9i			2.81890i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5811.35			0.358088			0.179044	−	0.983841i	$-0.442700\pi$
$641$	5811.35			0.358088			0.179044	+	0.983841i	$0.442700\pi$
$642$		0			0
$643$		−	27931.7i		−	1.71309i	−0.516069	−	0.856547i	$-0.672605\pi$
$643$		−	27931.7i		−	1.71309i	0.516069	−	0.856547i	$-0.327395\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25437.8i		−	1.54569i	−0.634595	−	0.772845i	$-0.718833\pi$
$647$		−	25437.8i		−	1.54569i	0.634595	−	0.772845i	$-0.281167\pi$
$648$		0			0
$649$	−43015.9			−2.60173
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	22647.7i		−	1.35723i	−0.734493	−	0.678617i	$-0.762580\pi$
$653$		−	22647.7i		−	1.35723i	0.734493	−	0.678617i	$-0.237420\pi$
$654$		0			0
$655$	1022.61	−	16.3659i	0.0610024	−	0.000976288i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−25158.5			−1.48716			−0.743579	−	0.668649i	$-0.766873\pi$
$659$	−25158.5			−1.48716			−0.743579	+	0.668649i	$0.766873\pi$
$660$		0			0
$661$	23441.0			1.37935			0.689675	−	0.724119i	$-0.257753\pi$
$661$	23441.0			1.37935			0.689675	+	0.724119i	$0.257753\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−774.329	−	48383.2i	−0.0451537	−	2.82138i
$666$		0			0
$667$			7405.47i			0.429896i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11676.0			−0.671754
$672$		0			0
$673$		−	3145.33i		−	0.180154i	−0.995935	−	0.0900770i	$-0.971289\pi$
$673$		−	3145.33i		−	0.180154i	0.995935	−	0.0900770i	$-0.0287113\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	10606.2i		−	0.602109i	−0.953607	−	0.301054i	$-0.902661\pi$
$677$		−	10606.2i		−	0.602109i	0.953607	−	0.301054i	$-0.0973386\pi$
$678$		0			0
$679$	−42448.1			−2.39913
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7825.06i		−	0.438386i	−0.975682	−	0.219193i	$-0.929658\pi$
$683$		−	7825.06i		−	0.438386i	0.975682	−	0.219193i	$-0.0703424\pi$
$684$		0			0
$685$	−406.708	−	25412.8i	−0.0226854	−	1.41748i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19819.0			−1.09585
$690$		0			0
$691$	−22750.9			−1.25251			−0.626256	−	0.779618i	$-0.715413\pi$
$691$	−22750.9			−1.25251			−0.626256	+	0.779618i	$0.715413\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8258.25	−	132.166i	0.450724	−	0.00721343i
$696$		0			0
$697$			4271.99i			0.232157i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	133.598			0.00719817			0.00359908	−	0.999994i	$-0.498854\pi$
$701$	133.598			0.00719817			0.00359908	+	0.999994i	$0.498854\pi$
$702$		0			0
$703$			39382.5i			2.11286i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			29333.1i			1.56037i
$708$		0			0
$709$	−6886.26			−0.364766			−0.182383	−	0.983228i	$-0.558381\pi$
$709$	−6886.26			−0.364766			−0.182383	+	0.983228i	$0.558381\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			14912.4i			0.783271i
$714$		0			0
$715$	522.193	+	32628.7i	0.0273131	+	1.70663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1570.18			0.0814436			0.0407218	−	0.999171i	$-0.487034\pi$
$719$	1570.18			0.0814436			0.0407218	+	0.999171i	$0.487034\pi$
$720$		0			0
$721$	25898.3			1.33773
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13056.4	−	418.019i	0.668831	−	0.0214136i
$726$		0			0
$727$		−	3399.18i		−	0.173409i	−0.996234	−	0.0867047i	$-0.972366\pi$
$727$		−	3399.18i		−	0.173409i	0.996234	−	0.0867047i	$-0.0276337\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1921.22			0.0972078
$732$		0			0
$733$		−	14152.1i		−	0.713125i	−0.934272	−	0.356562i	$-0.883949\pi$
$733$		−	14152.1i		−	0.713125i	0.934272	−	0.356562i	$-0.116051\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5021.20i			0.250961i
$738$		0			0
$739$	−14919.5			−0.742655			−0.371328	−	0.928502i	$-0.621097\pi$
$739$	−14919.5			−0.742655			−0.371328	+	0.928502i	$0.621097\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			7287.29i			0.359818i	0.983683	+	0.179909i	$0.0575803\pi$
$743$			7287.29i			0.359818i	−0.983683	+	0.179909i	$0.942420\pi$
$744$		0			0
$745$	−16848.3	+	269.642i	−0.828555	+	0.0132603i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−43045.3			−2.09992
$750$		0			0
$751$	−18783.4			−0.912670			−0.456335	−	0.889808i	$-0.650838\pi$
$751$	−18783.4			−0.912670			−0.456335	+	0.889808i	$0.650838\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1725.63	−	27.6172i	0.0831818	−	0.00133125i
$756$		0			0
$757$			30614.3i			1.46988i	0.678134	+	0.734939i	$0.262789\pi$
$757$			30614.3i			1.46988i	−0.678134	+	0.734939i	$0.737211\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15277.2			0.727723			0.363861	−	0.931453i	$-0.381458\pi$
$761$	15277.2			0.727723			0.363861	+	0.931453i	$0.381458\pi$
$762$		0			0
$763$			38428.4i			1.82333i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			53690.3i			2.52757i
$768$		0			0
$769$	17700.1			0.830016			0.415008	−	0.909818i	$-0.363779\pi$
$769$	17700.1			0.830016			0.415008	+	0.909818i	$0.363779\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	29362.5i		−	1.36623i	−0.730310	−	0.683116i	$-0.760625\pi$
$773$		−	29362.5i		−	1.36623i	0.730310	−	0.683116i	$-0.239375\pi$
$774$		0			0
$775$	26291.6	−	841.763i	1.21861	−	0.0390155i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31426.2			1.44539
$780$		0			0
$781$	−13424.0			−0.615042
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−56.4481	−	3527.10i	−0.00256652	−	0.160367i
$786$		0			0
$787$		−	2816.92i		−	0.127589i	−0.997963	−	0.0637943i	$-0.979680\pi$
$787$		−	2816.92i		−	0.127589i	0.997963	−	0.0637943i	$-0.0203202\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−16795.0			−0.754943
$792$		0			0
$793$			14573.4i			0.652606i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	18433.9i		−	0.819276i	−0.912248	−	0.409638i	$-0.865655\pi$
$797$		−	18433.9i		−	0.819276i	0.912248	−	0.409638i	$-0.134345\pi$
$798$		0			0
$799$	4962.99			0.219747
$800$		0			0
$801$		0			0
$802$		0			0
$803$			13269.5i			0.583153i
$804$		0			0
$805$	−26199.7	+	419.302i	−1.14710	+	0.0183583i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13276.6			−0.576983			−0.288492	−	0.957482i	$-0.593154\pi$
$809$	−13276.6			−0.576983			−0.288492	+	0.957482i	$0.593154\pi$
$810$		0			0
$811$	18842.5			0.815845			0.407923	−	0.913016i	$-0.366253\pi$
$811$	18842.5			0.815845			0.407923	+	0.913016i	$0.366253\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−260.822	−	16297.2i	−0.0112101	−	0.700449i
$816$		0			0
$817$		−	14133.2i		−	0.605210i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5863.62			−0.249259			−0.124629	−	0.992203i	$-0.539774\pi$
$821$	−5863.62			−0.249259			−0.124629	+	0.992203i	$0.539774\pi$
$822$		0			0
$823$		−	5018.31i		−	0.212548i	−0.994337	−	0.106274i	$-0.966108\pi$
$823$		−	5018.31i		−	0.212548i	0.994337	−	0.106274i	$-0.0338921\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	20303.9i		−	0.853730i	−0.904315	−	0.426865i	$-0.859618\pi$
$827$		−	20303.9i		−	0.853730i	0.904315	−	0.426865i	$-0.140382\pi$
$828$		0			0
$829$	38156.2			1.59857			0.799287	−	0.600949i	$-0.205211\pi$
$829$	38156.2			1.59857			0.799287	+	0.600949i	$0.205211\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			13357.0i			0.555573i
$834$		0			0
$835$	−393.349	−	24578.0i	−0.0163023	−	1.01863i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−41004.6			−1.68729			−0.843645	−	0.536901i	$-0.819595\pi$
$839$	−41004.6			−1.68729			−0.843645	+	0.536901i	$0.819595\pi$
$840$		0			0
$841$	−13467.8			−0.552206
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16165.4	−	258.713i	0.658116	−	0.0105326i
$846$		0			0
$847$		−	33320.8i		−	1.35173i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	21325.8			0.859033
$852$		0			0
$853$		−	22451.8i		−	0.901214i	−0.892722	−	0.450607i	$-0.851208\pi$
$853$		−	22451.8i		−	0.901214i	0.892722	−	0.450607i	$-0.148792\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			39858.0i			1.58871i	0.607455	+	0.794354i	$0.292191\pi$
$857$			39858.0i			1.58871i	−0.607455	+	0.794354i	$0.707809\pi$
$858$		0			0
$859$	32585.0			1.29428			0.647139	−	0.762372i	$-0.275965\pi$
$859$	32585.0			1.29428			0.647139	+	0.762372i	$0.275965\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	15593.6i		−	0.615078i	−0.951535	−	0.307539i	$-0.900495\pi$
$863$		−	15593.6i		−	0.615078i	0.951535	−	0.307539i	$-0.0995055\pi$
$864$		0			0
$865$	−363.272	−	22698.7i	−0.0142793	−	0.892228i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17746.9			0.692775
$870$		0			0
$871$	6267.21			0.243807
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2218.16	+	46168.3i	0.0857000	+	1.78374i
$876$		0			0
$877$			1804.75i			0.0694892i	0.999396	+	0.0347446i	$0.0110618\pi$
$877$			1804.75i			0.0694892i	−0.999396	+	0.0347446i	$0.988938\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14788.0			0.565515			0.282758	−	0.959191i	$-0.408751\pi$
$881$	14788.0			0.565515			0.282758	+	0.959191i	$0.408751\pi$
$882$		0			0
$883$		−	1999.24i		−	0.0761947i	−0.999274	−	0.0380973i	$-0.987870\pi$
$883$		−	1999.24i		−	0.0761947i	0.999274	−	0.0380973i	$-0.0121297\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	32474.4i		−	1.22929i	−0.788802	−	0.614647i	$-0.789299\pi$
$887$		−	32474.4i		−	1.22929i	0.788802	−	0.614647i	$-0.210701\pi$
$888$		0			0
$889$	26328.6			0.993287
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36509.4i		−	1.36813i
$894$		0			0
$895$	4775.86	−	76.4333i	0.178368	−	0.00285462i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21992.0			0.815880
$900$		0			0
$901$	−5841.18			−0.215980
$902$		0			0
$903$		0			0
$904$		0			0
$905$	42255.9	−	676.268i	1.55208	−	0.0248397i
$906$		0			0
$907$		−	27597.9i		−	1.01033i	−0.863022	−	0.505167i	$-0.831431\pi$
$907$		−	27597.9i		−	1.01033i	0.863022	−	0.505167i	$-0.168569\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19861.4			−0.722325			−0.361163	−	0.932503i	$-0.617620\pi$
$911$	−19861.4			−0.722325			−0.361163	+	0.932503i	$0.617620\pi$
$912$		0			0
$913$			2791.46i			0.101187i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			3025.44i			0.108952i
$918$		0			0
$919$	−20886.0			−0.749691			−0.374845	−	0.927087i	$-0.622304\pi$
$919$	−20886.0			−0.749691			−0.374845	+	0.927087i	$0.622304\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			16755.2i			0.597511i
$924$		0			0
$925$	−1203.78	−	37598.9i	−0.0427893	−	1.33648i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	37461.2			1.32299			0.661497	−	0.749948i	$-0.269922\pi$
$929$	37461.2			1.32299			0.661497	+	0.749948i	$0.269922\pi$
$930$		0			0
$931$	98258.5			3.45896
$932$		0			0
$933$		0			0
$934$		0			0
$935$	153.904	+	9616.56i	0.00538311	+	0.336358i
$936$		0			0
$937$		−	27149.0i		−	0.946552i	−0.880914	−	0.473276i	$-0.843071\pi$
$937$		−	27149.0i		−	0.946552i	0.880914	−	0.473276i	$-0.156929\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	43068.3			1.49202			0.746008	−	0.665937i	$-0.231968\pi$
$941$	43068.3			1.49202			0.746008	+	0.665937i	$0.231968\pi$
$942$		0			0
$943$		−	17017.4i		−	0.587660i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	30021.1i		−	1.03015i	−0.857145	−	0.515076i	$-0.827764\pi$
$947$		−	30021.1i		−	1.03015i	0.857145	−	0.515076i	$-0.172236\pi$
$948$		0			0
$949$	16562.4			0.566530
$950$		0			0
$951$		0			0
$952$		0			0
$953$			53820.4i			1.82939i	0.404140	+	0.914697i	$0.367571\pi$
$953$			53820.4i			1.82939i	−0.404140	+	0.914697i	$0.632429\pi$
$954$		0			0
$955$	−17501.7	+	280.098i	−0.593027	+	0.00949086i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	75185.1			2.53165
$960$		0			0
$961$	14494.3			0.486532
$962$		0			0
$963$		0			0
$964$		0			0
$965$	472.788	+	29541.7i	0.0157716	+	0.985473i
$966$		0			0
$967$		−	32210.2i		−	1.07116i	−0.844485	−	0.535580i	$-0.820093\pi$
$967$		−	32210.2i		−	1.07116i	0.844485	−	0.535580i	$-0.179907\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21616.1			−0.714411			−0.357206	−	0.934026i	$-0.616270\pi$
$971$	−21616.1			−0.714411			−0.357206	+	0.934026i	$0.616270\pi$
$972$		0			0
$973$			24432.5i			0.805005i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			46002.8i			1.50641i	0.657787	+	0.753204i	$0.271493\pi$
$977$			46002.8i			1.50641i	−0.657787	+	0.753204i	$0.728507\pi$
$978$		0			0
$979$	−9825.11			−0.320748
$980$		0			0
$981$		0			0
$982$		0			0
$983$			3926.43i			0.127400i	0.997969	+	0.0636998i	$0.0202900\pi$
$983$			3926.43i			0.127400i	−0.997969	+	0.0636998i	$0.979710\pi$
$984$		0			0
$985$	−17.6139	−	1100.59i	−0.000569772	−	0.0356017i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7653.15			−0.246063
$990$		0			0
$991$	−55802.3			−1.78872			−0.894359	−	0.447351i	$-0.852368\pi$
$991$	−55802.3			−1.78872			−0.894359	+	0.447351i	$0.852368\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−12650.2	+	202.456i	−0.403055	+	0.00645052i
$996$		0			0
$997$			43880.8i			1.39390i	0.717119	+	0.696951i	$0.245460\pi$
$997$			43880.8i			1.39390i	−0.717119	+	0.696951i	$0.754540\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.i.289.2		4
3.2	odd	2		240.4.f.g.49.4		4
4.3	odd	2		360.4.f.d.289.2		4
5.4	even	2	inner	720.4.f.i.289.1		4
12.11	even	2		120.4.f.d.49.2	✓	4
15.2	even	4		1200.4.a.bq.1.2		2
15.8	even	4		1200.4.a.bo.1.1		2
15.14	odd	2		240.4.f.g.49.2		4
20.3	even	4		1800.4.a.bl.1.2		2
20.7	even	4		1800.4.a.bn.1.1		2
20.19	odd	2		360.4.f.d.289.1		4
24.5	odd	2		960.4.f.o.769.1		4
24.11	even	2		960.4.f.n.769.3		4
60.23	odd	4		600.4.a.v.1.2		2
60.47	odd	4		600.4.a.t.1.1		2
60.59	even	2		120.4.f.d.49.4	yes	4
120.29	odd	2		960.4.f.o.769.3		4
120.59	even	2		960.4.f.n.769.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.d.49.2	✓	4	12.11	even	2
120.4.f.d.49.4	yes	4	60.59	even	2
240.4.f.g.49.2		4	15.14	odd	2
240.4.f.g.49.4		4	3.2	odd	2
360.4.f.d.289.1		4	20.19	odd	2
360.4.f.d.289.2		4	4.3	odd	2
600.4.a.t.1.1		2	60.47	odd	4
600.4.a.v.1.2		2	60.23	odd	4
720.4.f.i.289.1		4	5.4	even	2	inner
720.4.f.i.289.2		4	1.1	even	1	trivial
960.4.f.n.769.1		4	120.59	even	2
960.4.f.n.769.3		4	24.11	even	2
960.4.f.o.769.1		4	24.5	odd	2
960.4.f.o.769.3		4	120.29	odd	2
1200.4.a.bo.1.1		2	15.8	even	4
1200.4.a.bq.1.2		2	15.2	even	4
1800.4.a.bl.1.2		2	20.3	even	4
1800.4.a.bn.1.1		2	20.7	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-11.1789 + 0.178908i) q^{5} -33.0735i q^{7} +48.3578 q^{11} -60.3578i q^{13} -17.7891i q^{17} -130.863 q^{19} +70.8625i q^{23} +(124.936 - 4.00000i) q^{25} +104.505 q^{29} +210.441 q^{31} +(5.91712 + 369.725i) q^{35} -300.945i q^{37} -240.147 q^{41} +108.000i q^{43} +278.991i q^{47} -750.853 q^{49} -328.358i q^{53} +(-540.588 + 8.65162i) q^{55} -889.533 q^{59} -241.450 q^{61} +(10.7985 + 674.735i) q^{65} +103.834i q^{67} -277.597 q^{71} +274.403i q^{73} -1599.36i q^{77} +366.991 q^{79} +57.7251i q^{83} +(3.18262 + 198.863i) q^{85} -203.175 q^{89} -1996.24 q^{91} +(1462.90 - 23.4124i) q^{95} -1283.45i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{5} + 148 q^{11} - 160 q^{19} + 100 q^{29} + 24 q^{31} + 796 q^{35} - 688 q^{41} - 3276 q^{49} - 1072 q^{55} - 1332 q^{59} + 488 q^{61} - 820 q^{65} + 616 q^{71} + 2104 q^{79} + 2148 q^{85} + 1368 q^{89}+ \cdots + 2944 q^{95}+O(q^{100}) \) 4 * q - 22 * q^5 + 148 * q^11 - 160 * q^19 + 100 * q^29 + 24 * q^31 + 796 * q^35 - 688 * q^41 - 3276 * q^49 - 1072 * q^55 - 1332 * q^59 + 488 * q^61 - 820 * q^65 + 616 * q^71 + 2104 * q^79 + 2148 * q^85 + 1368 * q^89 - 1352 * q^91 + 2944 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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