LMFDB - Embedded newform 288.4.d.c.145.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,4,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	288.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$16.9925500817$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-10}, \sqrt{22})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 64 $ x^4 - 6*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.1
Root			$2.34521 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.4.d.c.145.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.32456i q^{5} -10.0000 q^{7} +37.9473i q^{11} -59.3296i q^{13} -75.0467 q^{17} +118.659i q^{19} -150.093 q^{23} +85.0000 q^{25} +246.658i q^{29} -62.0000 q^{31} +63.2456i q^{35} -59.3296i q^{37} -375.233 q^{41} -118.659i q^{43} -450.280 q^{47} -243.000 q^{49} +132.816i q^{53} +240.000 q^{55} +733.648i q^{59} +533.966i q^{61} -375.233 q^{65} -711.955i q^{67} +30.0000 q^{73} -379.473i q^{77} -94.0000 q^{79} -670.403i q^{83} +474.637i q^{85} +750.467 q^{89} +593.296i q^{91} +750.467 q^{95} +130.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 40 q^{7} + 340 q^{25} - 248 q^{31} - 972 q^{49} + 960 q^{55} + 120 q^{73} - 376 q^{79} + 520 q^{97}+O(q^{100}) $ 4 * q - 40 * q^7 + 340 * q^25 - 248 * q^31 - 972 * q^49 + 960 * q^55 + 120 * q^73 - 376 * q^79 + 520 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 6.32456i	− 0.565685i	−0.959166	−	0.282843i	$-0.908723\pi$
$5$	− 6.32456i	− 0.565685i	0.959166	−	0.282843i	$-0.0912774\pi$
$6$	0	0
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	37.9473i	1.04014i	0.854123	+	0.520071i	$0.174094\pi$
$11$	37.9473i	1.04014i	−0.854123	+	0.520071i	$0.825906\pi$
$12$	0	0
$13$	− 59.3296i	− 1.26577i	−0.774244	−	0.632887i	$-0.781870\pi$
$13$	− 59.3296i	− 1.26577i	0.774244	−	0.632887i	$-0.218130\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−75.0467	−1.07068	−0.535338	−	0.844638i	$-0.679816\pi$
$17$	−75.0467	−1.07068	−0.535338	+	0.844638i	$0.679816\pi$
$18$	0	0
$19$	118.659i	1.43275i	0.697715	+	0.716376i	$0.254200\pi$
$19$	118.659i	1.43275i	−0.697715	+	0.716376i	$0.745800\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−150.093	−1.36072	−0.680361	−	0.732877i	$-0.738177\pi$
$23$	−150.093	−1.36072	−0.680361	+	0.732877i	$0.738177\pi$
$24$	0	0
$25$	85.0000	0.680000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	246.658i	1.57942i	0.613480	+	0.789710i	$0.289769\pi$
$29$	246.658i	1.57942i	−0.613480	+	0.789710i	$0.710231\pi$
$30$	0	0
$31$	−62.0000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−62.0000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	63.2456i	0.305441i
$36$	0	0
$37$	− 59.3296i	− 0.263614i	−0.991275	−	0.131807i	$-0.957922\pi$
$37$	− 59.3296i	− 0.263614i	0.991275	−	0.131807i	$-0.0420779\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−375.233	−1.42931	−0.714654	−	0.699479i	$-0.753416\pi$
$41$	−375.233	−1.42931	−0.714654	+	0.699479i	$0.753416\pi$
$42$	0	0
$43$	− 118.659i	− 0.420822i	−0.977613	−	0.210411i	$-0.932520\pi$
$43$	− 118.659i	− 0.420822i	0.977613	−	0.210411i	$-0.0674802\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−450.280	−1.39745	−0.698724	−	0.715391i	$-0.746249\pi$
$47$	−450.280	−1.39745	−0.698724	+	0.715391i	$0.746249\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	0	0
$52$	0	0
$53$	132.816i	0.344220i	0.985078	+	0.172110i	$0.0550584\pi$
$53$	132.816i	0.344220i	−0.985078	+	0.172110i	$0.944942\pi$
$54$	0	0
$55$	240.000	0.588393
$56$	0	0
$57$	0	0
$58$	0	0
$59$	733.648i	1.61886i	0.587215	+	0.809431i	$0.300224\pi$
$59$	733.648i	1.61886i	−0.587215	+	0.809431i	$0.699776\pi$
$60$	0	0
$61$	533.966i	1.12078i	0.828230	+	0.560388i	$0.189348\pi$
$61$	533.966i	1.12078i	−0.828230	+	0.560388i	$0.810652\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−375.233	−0.716030
$66$	0	0
$67$	− 711.955i	− 1.29820i	−0.760705	−	0.649098i	$-0.775146\pi$
$67$	− 711.955i	− 1.29820i	0.760705	−	0.649098i	$-0.224854\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	30.0000	0.0480991	0.0240496	−	0.999711i	$-0.492344\pi$
$73$	30.0000	0.0480991	0.0240496	+	0.999711i	$0.492344\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 379.473i	− 0.561623i
$78$	0	0
$79$	−94.0000	−0.133871	−0.0669356	−	0.997757i	$-0.521322\pi$
$79$	−94.0000	−0.133871	−0.0669356	+	0.997757i	$0.521322\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 670.403i	− 0.886582i	−0.896378	−	0.443291i	$-0.853811\pi$
$83$	− 670.403i	− 0.886582i	0.896378	−	0.443291i	$-0.146189\pi$
$84$	0	0
$85$	474.637i	0.605666i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	750.467	0.893812	0.446906	−	0.894581i	$-0.352526\pi$
$89$	750.467	0.893812	0.446906	+	0.894581i	$0.352526\pi$
$90$	0	0
$91$	593.296i	0.683454i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	750.467	0.810487
$96$	0	0
$97$	130.000	0.136077	0.0680387	−	0.997683i	$-0.478326\pi$
$97$	130.000	0.136077	0.0680387	+	0.997683i	$0.478326\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	562.885i	0.554546i	0.960791	+	0.277273i	$0.0894307\pi$
$101$	562.885i	0.554546i	−0.960791	+	0.277273i	$0.910569\pi$
$102$	0	0
$103$	570.000	0.545279	0.272640	−	0.962116i	$-0.412103\pi$
$103$	570.000	0.545279	0.272640	+	0.962116i	$0.412103\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1290.21i	− 1.16569i	−0.812582	−	0.582847i	$-0.801939\pi$
$107$	− 1290.21i	− 1.16569i	0.812582	−	0.582847i	$-0.198061\pi$
$108$	0	0
$109$	− 1720.56i	− 1.51192i	−0.654616	−	0.755961i	$-0.727170\pi$
$109$	− 1720.56i	− 1.51192i	0.654616	−	0.755961i	$-0.272830\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1350.84	1.12457	0.562285	−	0.826944i	$-0.309923\pi$
$113$	1350.84	1.12457	0.562285	+	0.826944i	$0.309923\pi$
$114$	0	0
$115$	949.273i	0.769741i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	750.467	0.578111
$120$	0	0
$121$	−109.000	−0.0818933
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1328.16i	− 0.950352i
$126$	0	0
$127$	−2530.00	−1.76773	−0.883863	−	0.467746i	$-0.845066\pi$
$127$	−2530.00	−1.76773	−0.883863	+	0.467746i	$0.845066\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 252.982i	− 0.168726i	−0.996435	−	0.0843632i	$-0.973114\pi$
$131$	− 252.982i	− 0.168726i	0.996435	−	0.0843632i	$-0.0268856\pi$
$132$	0	0
$133$	− 1186.59i	− 0.773613i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	675.420	0.421204	0.210602	−	0.977572i	$-0.432458\pi$
$137$	675.420	0.421204	0.210602	+	0.977572i	$0.432458\pi$
$138$	0	0
$139$	− 237.318i	− 0.144814i	−0.997375	−	0.0724068i	$-0.976932\pi$
$139$	− 237.318i	− 0.144814i	0.997375	−	0.0724068i	$-0.0230680\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2251.40	1.31658
$144$	0	0
$145$	1560.00	0.893455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2940.92i	− 1.61698i	−0.588513	−	0.808488i	$-0.700286\pi$
$149$	− 2940.92i	− 1.61698i	0.588513	−	0.808488i	$-0.299714\pi$
$150$	0	0
$151$	2262.00	1.21907	0.609533	−	0.792761i	$-0.291357\pi$
$151$	2262.00	1.21907	0.609533	+	0.792761i	$0.291357\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	392.122i	0.203200i
$156$	0	0
$157$	1720.56i	0.874621i	0.899311	+	0.437310i	$0.144069\pi$
$157$	1720.56i	0.874621i	−0.899311	+	0.437310i	$0.855931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1500.93	0.734721
$162$	0	0
$163$	2017.21i	0.969324i	0.874702	+	0.484662i	$0.161057\pi$
$163$	2017.21i	0.969324i	−0.874702	+	0.484662i	$0.838943\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1050.65	0.486838	0.243419	−	0.969921i	$-0.421731\pi$
$167$	1050.65	0.486838	0.243419	+	0.969921i	$0.421731\pi$
$168$	0	0
$169$	−1323.00	−0.602185
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1423.02i	0.625379i	0.949855	+	0.312690i	$0.101230\pi$
$173$	1423.02i	0.625379i	−0.949855	+	0.312690i	$0.898770\pi$
$174$	0	0
$175$	−850.000	−0.367165
$176$	0	0
$177$	0	0
$178$	0	0
$179$	126.491i	0.0528178i	0.999651	+	0.0264089i	$0.00840719\pi$
$179$	126.491i	0.0528178i	−0.999651	+	0.0264089i	$0.991593\pi$
$180$	0	0
$181$	59.3296i	0.0243643i	0.999926	+	0.0121821i	$0.00387779\pi$
$181$	59.3296i	0.0243643i	−0.999926	+	0.0121821i	$0.996122\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−375.233	−0.149123
$186$	0	0
$187$	− 2847.82i	− 1.11365i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3752.33	−1.42151	−0.710757	−	0.703437i	$-0.751648\pi$
$191$	−3752.33	−1.42151	−0.710757	+	0.703437i	$0.751648\pi$
$192$	0	0
$193$	−1350.00	−0.503498	−0.251749	−	0.967793i	$-0.581006\pi$
$193$	−1350.00	−0.503498	−0.251749	+	0.967793i	$0.581006\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1448.32i	0.523801i	0.965095	+	0.261900i	$0.0843492\pi$
$197$	1448.32i	0.523801i	−0.965095	+	0.261900i	$0.915651\pi$
$198$	0	0
$199$	3194.00	1.13777	0.568886	−	0.822416i	$-0.307375\pi$
$199$	3194.00	1.13777	0.568886	+	0.822416i	$0.307375\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2466.58i	− 0.852807i
$204$	0	0
$205$	2373.18i	0.808538i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4502.80	−1.49026
$210$	0	0
$211$	− 5458.32i	− 1.78088i	−0.455097	−	0.890442i	$-0.650396\pi$
$211$	− 5458.32i	− 1.78088i	0.455097	−	0.890442i	$-0.349604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−750.467	−0.238053
$216$	0	0
$217$	620.000	0.193955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4452.49i	1.35523i
$222$	0	0
$223$	−5330.00	−1.60055	−0.800276	−	0.599632i	$-0.795314\pi$
$223$	−5330.00	−1.60055	−0.800276	+	0.599632i	$0.795314\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4768.71i	1.39432i	0.716915	+	0.697160i	$0.245553\pi$
$227$	4768.71i	1.39432i	−0.716915	+	0.697160i	$0.754447\pi$
$228$	0	0
$229$	3619.10i	1.04435i	0.852837	+	0.522177i	$0.174880\pi$
$229$	3619.10i	1.04435i	−0.852837	+	0.522177i	$0.825120\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−150.093	−0.0422015	−0.0211007	−	0.999777i	$-0.506717\pi$
$233$	−150.093	−0.0422015	−0.0211007	+	0.999777i	$0.506717\pi$
$234$	0	0
$235$	2847.82i	0.790516i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2251.40	−0.609334	−0.304667	−	0.952459i	$-0.598545\pi$
$239$	−2251.40	−0.609334	−0.304667	+	0.952459i	$0.598545\pi$
$240$	0	0
$241$	−1162.00	−0.310585	−0.155293	−	0.987869i	$-0.549632\pi$
$241$	−1162.00	−0.310585	−0.155293	+	0.987869i	$0.549632\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1536.87i	0.400763i
$246$	0	0
$247$	7040.00	1.81354
$248$	0	0
$249$	0	0
$250$	0	0
$251$	645.105i	0.162226i	0.996705	+	0.0811128i	$0.0258474\pi$
$251$	645.105i	0.162226i	−0.996705	+	0.0811128i	$0.974153\pi$
$252$	0	0
$253$	− 5695.64i	− 1.41534i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1801.12	−0.437162	−0.218581	−	0.975819i	$-0.570143\pi$
$257$	−1801.12	−0.437162	−0.218581	+	0.975819i	$0.570143\pi$
$258$	0	0
$259$	593.296i	0.142338i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6604.11	−1.54839	−0.774195	−	0.632947i	$-0.781845\pi$
$263$	−6604.11	−1.54839	−0.774195	+	0.632947i	$0.781845\pi$
$264$	0	0
$265$	840.000	0.194720
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6052.60i	− 1.37187i	−0.727662	−	0.685936i	$-0.759393\pi$
$269$	− 6052.60i	− 1.37187i	0.727662	−	0.685936i	$-0.240607\pi$
$270$	0	0
$271$	−6402.00	−1.43503	−0.717516	−	0.696542i	$-0.754721\pi$
$271$	−6402.00	−1.43503	−0.717516	+	0.696542i	$0.754721\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3225.52i	0.707296i
$276$	0	0
$277$	− 2076.54i	− 0.450422i	−0.974310	−	0.225211i	$-0.927693\pi$
$277$	− 2076.54i	− 0.450422i	0.974310	−	0.225211i	$-0.0723072\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9005.60	−1.91185	−0.955923	−	0.293616i	$-0.905141\pi$
$281$	−9005.60	−1.91185	−0.955923	+	0.293616i	$0.905141\pi$
$282$	0	0
$283$	5221.00i	1.09667i	0.836260	+	0.548333i	$0.184737\pi$
$283$	5221.00i	1.09667i	−0.836260	+	0.548333i	$0.815263\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3752.33	0.771753
$288$	0	0
$289$	719.000	0.146346
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5293.65i	1.05549i	0.849403	+	0.527745i	$0.176962\pi$
$293$	5293.65i	1.05549i	−0.849403	+	0.527745i	$0.823038\pi$
$294$	0	0
$295$	4640.00	0.915767
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8904.97i	1.72237i
$300$	0	0
$301$	1186.59i	0.227223i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3377.10	0.634007
$306$	0	0
$307$	4983.69i	0.926495i	0.886229	+	0.463247i	$0.153316\pi$
$307$	4983.69i	0.926495i	−0.886229	+	0.463247i	$0.846684\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1500.93	−0.273666	−0.136833	−	0.990594i	$-0.543692\pi$
$311$	−1500.93	−0.273666	−0.136833	+	0.990594i	$0.543692\pi$
$312$	0	0
$313$	6350.00	1.14672	0.573360	−	0.819304i	$-0.305640\pi$
$313$	6350.00	1.14672	0.573360	+	0.819304i	$0.305640\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1739.25i	0.308158i	0.988059	+	0.154079i	$0.0492411\pi$
$317$	1739.25i	0.308158i	−0.988059	+	0.154079i	$0.950759\pi$
$318$	0	0
$319$	−9360.00	−1.64282
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8904.97i	− 1.53401i
$324$	0	0
$325$	− 5043.01i	− 0.860727i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4502.80	0.754551
$330$	0	0
$331$	− 3322.46i	− 0.551718i	−0.961198	−	0.275859i	$-0.911038\pi$
$331$	− 3322.46i	− 0.551718i	0.961198	−	0.275859i	$-0.0889623\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4502.80	−0.734371
$336$	0	0
$337$	−1430.00	−0.231149	−0.115574	−	0.993299i	$-0.536871\pi$
$337$	−1430.00	−0.231149	−0.115574	+	0.993299i	$0.536871\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2352.73i	− 0.373630i
$342$	0	0
$343$	5860.00	0.922479
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10106.6i	− 1.56355i	−0.623559	−	0.781776i	$-0.714314\pi$
$347$	− 10106.6i	− 1.56355i	0.623559	−	0.781776i	$-0.285686\pi$
$348$	0	0
$349$	− 1127.26i	− 0.172897i	−0.996256	−	0.0864484i	$-0.972448\pi$
$349$	− 1127.26i	− 0.172897i	0.996256	−	0.0864484i	$-0.0275518\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1350.84	−0.203677	−0.101838	−	0.994801i	$-0.532472\pi$
$353$	−1350.84	−0.203677	−0.101838	+	0.994801i	$0.532472\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12007.5	1.76526	0.882632	−	0.470065i	$-0.155769\pi$
$359$	12007.5	1.76526	0.882632	+	0.470065i	$0.155769\pi$
$360$	0	0
$361$	−7221.00	−1.05278
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 189.737i	− 0.0272090i
$366$	0	0
$367$	−5070.00	−0.721122	−0.360561	−	0.932736i	$-0.617415\pi$
$367$	−5070.00	−0.721122	−0.360561	+	0.932736i	$0.617415\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1328.16i	− 0.185861i
$372$	0	0
$373$	− 2432.51i	− 0.337670i	−0.985644	−	0.168835i	$-0.946000\pi$
$373$	− 2432.51i	− 0.337670i	0.985644	−	0.168835i	$-0.0540004\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14634.1	1.99919
$378$	0	0
$379$	− 7712.85i	− 1.04534i	−0.852536	−	0.522668i	$-0.824937\pi$
$379$	− 7712.85i	− 1.04534i	0.852536	−	0.522668i	$-0.175063\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5853.64	0.780958	0.390479	−	0.920612i	$-0.372309\pi$
$383$	5853.64	0.780958	0.390479	+	0.920612i	$0.372309\pi$
$384$	0	0
$385$	−2400.00	−0.317702
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2156.67i	− 0.281099i	−0.990074	−	0.140550i	$-0.955113\pi$
$389$	− 2156.67i	− 0.281099i	0.990074	−	0.140550i	$-0.0448870\pi$
$390$	0	0
$391$	11264.0	1.45689
$392$	0	0
$393$	0	0
$394$	0	0
$395$	594.508i	0.0757290i
$396$	0	0
$397$	12162.6i	1.53759i	0.639498	+	0.768793i	$0.279142\pi$
$397$	12162.6i	1.53759i	−0.639498	+	0.768793i	$0.720858\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3377.10	0.420559	0.210280	−	0.977641i	$-0.432563\pi$
$401$	3377.10	0.420559	0.210280	+	0.977641i	$0.432563\pi$
$402$	0	0
$403$	3678.43i	0.454680i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2251.40	0.274196
$408$	0	0
$409$	3526.00	0.426282	0.213141	−	0.977021i	$-0.431631\pi$
$409$	3526.00	0.426282	0.213141	+	0.977021i	$0.431631\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7336.48i	− 0.874104i
$414$	0	0
$415$	−4240.00	−0.501526
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2466.58i	− 0.287590i	−0.989608	−	0.143795i	$-0.954069\pi$
$419$	− 2466.58i	− 0.287590i	0.989608	−	0.143795i	$-0.0459306\pi$
$420$	0	0
$421$	13586.5i	1.57284i	0.617694	+	0.786418i	$0.288067\pi$
$421$	13586.5i	1.57284i	−0.617694	+	0.786418i	$0.711933\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6378.97	−0.728059
$426$	0	0
$427$	− 5339.66i	− 0.605163i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6754.20	0.754845	0.377423	−	0.926041i	$-0.376810\pi$
$431$	6754.20	0.754845	0.377423	+	0.926041i	$0.376810\pi$
$432$	0	0
$433$	7790.00	0.864581	0.432290	−	0.901734i	$-0.357706\pi$
$433$	7790.00	0.864581	0.432290	+	0.901734i	$0.357706\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17809.9i	− 1.94958i
$438$	0	0
$439$	9354.00	1.01695	0.508476	−	0.861076i	$-0.330209\pi$
$439$	9354.00	1.01695	0.508476	+	0.861076i	$0.330209\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6488.99i	0.695940i	0.937506	+	0.347970i	$0.113129\pi$
$443$	6488.99i	0.695940i	−0.937506	+	0.347970i	$0.886871\pi$
$444$	0	0
$445$	− 4746.37i	− 0.505617i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10131.3	1.06487	0.532434	−	0.846472i	$-0.321278\pi$
$449$	10131.3	1.06487	0.532434	+	0.846472i	$0.321278\pi$
$450$	0	0
$451$	− 14239.1i	− 1.48668i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3752.33	0.386620
$456$	0	0
$457$	−12010.0	−1.22933	−0.614665	−	0.788788i	$-0.710709\pi$
$457$	−12010.0	−1.22933	−0.614665	+	0.788788i	$0.710709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9316.07i	0.941199i	0.882347	+	0.470599i	$0.155962\pi$
$461$	9316.07i	0.941199i	−0.882347	+	0.470599i	$0.844038\pi$
$462$	0	0
$463$	−14770.0	−1.48255	−0.741274	−	0.671202i	$-0.765778\pi$
$463$	−14770.0	−1.48255	−0.741274	+	0.671202i	$0.765778\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8740.54i	− 0.866089i	−0.901372	−	0.433045i	$-0.857439\pi$
$467$	− 8740.54i	− 0.866089i	0.901372	−	0.433045i	$-0.142561\pi$
$468$	0	0
$469$	7119.55i	0.700960i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4502.80	0.437714
$474$	0	0
$475$	10086.0i	0.974271i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3001.87	−0.286344	−0.143172	−	0.989698i	$-0.545730\pi$
$479$	−3001.87	−0.286344	−0.143172	+	0.989698i	$0.545730\pi$
$480$	0	0
$481$	−3520.00	−0.333676
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 822.192i	− 0.0769770i
$486$	0	0
$487$	9910.00	0.922105	0.461052	−	0.887373i	$-0.347472\pi$
$487$	9910.00	0.922105	0.461052	+	0.887373i	$0.347472\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3389.96i	0.311582i	0.987790	+	0.155791i	$0.0497927\pi$
$491$	3389.96i	0.311582i	−0.987790	+	0.155791i	$0.950207\pi$
$492$	0	0
$493$	− 18510.8i	− 1.69105i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4271.73i	0.383224i	0.981471	+	0.191612i	$0.0613716\pi$
$499$	4271.73i	0.383224i	−0.981471	+	0.191612i	$0.938628\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	900.560	0.0798290	0.0399145	−	0.999203i	$-0.487291\pi$
$503$	900.560	0.0798290	0.0399145	+	0.999203i	$0.487291\pi$
$504$	0	0
$505$	3560.00	0.313699
$506$	0	0
$507$	0	0
$508$	0	0
$509$	853.815i	0.0743510i	0.999309	+	0.0371755i	$0.0118361\pi$
$509$	853.815i	0.0743510i	−0.999309	+	0.0371755i	$0.988164\pi$
$510$	0	0
$511$	−300.000	−0.0259711
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3605.00i	− 0.308457i
$516$	0	0
$517$	− 17086.9i	− 1.45354i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7879.90	−0.662619	−0.331310	−	0.943522i	$-0.607490\pi$
$521$	−7879.90	−0.662619	−0.331310	+	0.943522i	$0.607490\pi$
$522$	0	0
$523$	10323.3i	0.863114i	0.902086	+	0.431557i	$0.142036\pi$
$523$	10323.3i	0.863114i	−0.902086	+	0.431557i	$0.857964\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4652.89	0.384598
$528$	0	0
$529$	10361.0	0.851566
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22262.4i	1.80918i
$534$	0	0
$535$	−8160.00	−0.659416
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 9221.20i	− 0.736893i
$540$	0	0
$541$	15603.7i	1.24003i	0.784591	+	0.620014i	$0.212873\pi$
$541$	15603.7i	1.24003i	−0.784591	+	0.620014i	$0.787127\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10881.8	−0.855273
$546$	0	0
$547$	2254.52i	0.176228i	0.996110	+	0.0881138i	$0.0280839\pi$
$547$	2254.52i	0.176228i	−0.996110	+	0.0881138i	$0.971916\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29268.2	−2.26292
$552$	0	0
$553$	940.000	0.0722837
$554$	0	0
$555$	0	0
$556$	0	0
$557$	284.605i	0.0216501i	0.999941	+	0.0108250i	$0.00344579\pi$
$557$	284.605i	0.0216501i	−0.999941	+	0.0108250i	$0.996554\pi$
$558$	0	0
$559$	−7040.00	−0.532666
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18328.6i	1.37204i	0.727584	+	0.686018i	$0.240643\pi$
$563$	18328.6i	1.37204i	−0.727584	+	0.686018i	$0.759357\pi$
$564$	0	0
$565$	− 8543.46i	− 0.636152i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7879.90	−0.580567	−0.290283	−	0.956941i	$-0.593750\pi$
$569$	−7879.90	−0.580567	−0.290283	+	0.956941i	$0.593750\pi$
$570$	0	0
$571$	8306.14i	0.608759i	0.952551	+	0.304379i	$0.0984491\pi$
$571$	8306.14i	0.608759i	−0.952551	+	0.304379i	$0.901551\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12757.9	−0.925291
$576$	0	0
$577$	−6330.00	−0.456709	−0.228355	−	0.973578i	$-0.573335\pi$
$577$	−6330.00	−0.456709	−0.228355	+	0.973578i	$0.573335\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6704.03i	0.478709i
$582$	0	0
$583$	−5040.00	−0.358037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1922.66i	− 0.135191i	−0.997713	−	0.0675953i	$-0.978467\pi$
$587$	− 1922.66i	− 0.135191i	0.997713	−	0.0675953i	$-0.0215327\pi$
$588$	0	0
$589$	− 7356.87i	− 0.514660i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10356.4	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$593$	−10356.4	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$594$	0	0
$595$	− 4746.37i	− 0.327029i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	750.467	0.0511907	0.0255954	−	0.999672i	$-0.491852\pi$
$599$	750.467	0.0511907	0.0255954	+	0.999672i	$0.491852\pi$
$600$	0	0
$601$	18578.0	1.26092	0.630460	−	0.776222i	$-0.282866\pi$
$601$	18578.0	1.26092	0.630460	+	0.776222i	$0.282866\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	689.377i	0.0463259i
$606$	0	0
$607$	8030.00	0.536948	0.268474	−	0.963287i	$-0.413481\pi$
$607$	8030.00	0.536948	0.268474	+	0.963287i	$0.413481\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26714.9i	1.76885i
$612$	0	0
$613$	− 3856.42i	− 0.254094i	−0.991897	−	0.127047i	$-0.959450\pi$
$613$	− 3856.42i	− 0.254094i	0.991897	−	0.127047i	$-0.0405499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−300.187	−0.0195868	−0.00979340	−	0.999952i	$-0.503117\pi$
$617$	−300.187	−0.0195868	−0.00979340	+	0.999952i	$0.503117\pi$
$618$	0	0
$619$	− 1423.91i	− 0.0924584i	−0.998931	−	0.0462292i	$-0.985280\pi$
$619$	− 1423.91i	− 0.0924584i	0.998931	−	0.0462292i	$-0.0147205\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7504.67	−0.482613
$624$	0	0
$625$	2225.00	0.142400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4452.49i	0.282245i
$630$	0	0
$631$	12902.0	0.813979	0.406989	−	0.913433i	$-0.366579\pi$
$631$	12902.0	0.813979	0.406989	+	0.913433i	$0.366579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16001.1i	0.999977i
$636$	0	0
$637$	14417.1i	0.896744i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19887.4	−1.22543	−0.612717	−	0.790302i	$-0.709924\pi$
$641$	−19887.4	−1.22543	−0.612717	+	0.790302i	$0.709924\pi$
$642$	0	0
$643$	29783.5i	1.82666i	0.407216	+	0.913332i	$0.366500\pi$
$643$	29783.5i	1.82666i	−0.407216	+	0.913332i	$0.633500\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13958.7	0.848180	0.424090	−	0.905620i	$-0.360594\pi$
$647$	13958.7	0.848180	0.424090	+	0.905620i	$0.360594\pi$
$648$	0	0
$649$	−27840.0	−1.68385
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18461.4i	− 1.10635i	−0.833064	−	0.553177i	$-0.813415\pi$
$653$	− 18461.4i	− 1.10635i	0.833064	−	0.553177i	$-0.186585\pi$
$654$	0	0
$655$	−1600.00	−0.0954461
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5160.84i	− 0.305065i	−0.988298	−	0.152532i	$-0.951257\pi$
$659$	− 5160.84i	− 0.305065i	0.988298	−	0.152532i	$-0.0487428\pi$
$660$	0	0
$661$	13467.8i	0.792492i	0.918144	+	0.396246i	$0.129687\pi$
$661$	13467.8i	0.792492i	−0.918144	+	0.396246i	$0.870313\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7504.67	−0.437622
$666$	0	0
$667$	− 37021.7i	− 2.14915i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20262.6	−1.16577
$672$	0	0
$673$	−15010.0	−0.859722	−0.429861	−	0.902895i	$-0.641437\pi$
$673$	−15010.0	−0.859722	−0.429861	+	0.902895i	$0.641437\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6280.28i	− 0.356530i	−0.983983	−	0.178265i	$-0.942952\pi$
$677$	− 6280.28i	− 0.356530i	0.983983	−	0.178265i	$-0.0570485\pi$
$678$	0	0
$679$	−1300.00	−0.0734748
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12510.0i	− 0.700850i	−0.936591	−	0.350425i	$-0.886037\pi$
$683$	− 12510.0i	− 0.700850i	0.936591	−	0.350425i	$-0.113963\pi$
$684$	0	0
$685$	− 4271.73i	− 0.238269i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7879.90	0.435704
$690$	0	0
$691$	− 28359.5i	− 1.56128i	−0.624978	−	0.780642i	$-0.714892\pi$
$691$	− 28359.5i	− 1.56128i	0.624978	−	0.780642i	$-0.285108\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1500.93	−0.0819189
$696$	0	0
$697$	28160.0	1.53032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19802.2i	1.06693i	0.845822	+	0.533465i	$0.179110\pi$
$701$	19802.2i	1.06693i	−0.845822	+	0.533465i	$0.820890\pi$
$702$	0	0
$703$	7040.00	0.377694
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5628.85i	− 0.299427i
$708$	0	0
$709$	12874.5i	0.681964i	0.940070	+	0.340982i	$0.110760\pi$
$709$	12874.5i	0.681964i	−0.940070	+	0.340982i	$0.889240\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9305.78	0.488786
$714$	0	0
$715$	− 14239.1i	− 0.744772i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1500.93	−0.0778517	−0.0389258	−	0.999242i	$-0.512394\pi$
$719$	−1500.93	−0.0778517	−0.0389258	+	0.999242i	$0.512394\pi$
$720$	0	0
$721$	−5700.00	−0.294423
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20965.9i	1.07401i
$726$	0	0
$727$	−23850.0	−1.21671	−0.608355	−	0.793665i	$-0.708170\pi$
$727$	−23850.0	−1.21671	−0.608355	+	0.793665i	$0.708170\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8904.97i	0.450564i
$732$	0	0
$733$	− 2195.19i	− 0.110616i	−0.998469	−	0.0553079i	$-0.982386\pi$
$733$	− 2195.19i	− 0.110616i	0.998469	−	0.0553079i	$-0.0176140\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27016.8	1.35031
$738$	0	0
$739$	− 2610.50i	− 0.129944i	−0.997887	−	0.0649722i	$-0.979304\pi$
$739$	− 2610.50i	− 0.129944i	0.997887	−	0.0649722i	$-0.0206959\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2851.77	0.140809	0.0704047	−	0.997519i	$-0.477571\pi$
$743$	2851.77	0.140809	0.0704047	+	0.997519i	$0.477571\pi$
$744$	0	0
$745$	−18600.0	−0.914700
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12902.1i	0.629416i
$750$	0	0
$751$	20578.0	0.999869	0.499935	−	0.866063i	$-0.333357\pi$
$751$	20578.0	0.999869	0.499935	+	0.866063i	$0.333357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 14306.1i	− 0.689608i
$756$	0	0
$757$	20468.7i	0.982758i	0.870946	+	0.491379i	$0.163507\pi$
$757$	20468.7i	0.982758i	−0.870946	+	0.491379i	$0.836493\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22138.8	−1.05457	−0.527286	−	0.849688i	$-0.676790\pi$
$761$	−22138.8	−1.05457	−0.527286	+	0.849688i	$0.676790\pi$
$762$	0	0
$763$	17205.6i	0.816362i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	43527.1	2.04911
$768$	0	0
$769$	−6854.00	−0.321406	−0.160703	−	0.987003i	$-0.551376\pi$
$769$	−6854.00	−0.321406	−0.160703	+	0.987003i	$0.551376\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38902.3i	− 1.81012i	−0.425288	−	0.905058i	$-0.639827\pi$
$773$	− 38902.3i	− 1.81012i	0.425288	−	0.905058i	$-0.360173\pi$
$774$	0	0
$775$	−5270.00	−0.244263
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 44524.9i	− 2.04784i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10881.8	0.494760
$786$	0	0
$787$	13171.2i	0.596571i	0.954477	+	0.298286i	$0.0964148\pi$
$787$	13171.2i	0.596571i	−0.954477	+	0.298286i	$0.903585\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13508.4	−0.607210
$792$	0	0
$793$	31680.0	1.41865
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13730.6i	− 0.610242i	−0.952314	−	0.305121i	$-0.901303\pi$
$797$	− 13730.6i	− 0.610242i	0.952314	−	0.305121i	$-0.0986970\pi$
$798$	0	0
$799$	33792.0	1.49621
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1138.42i	0.0500298i
$804$	0	0
$805$	− 9492.73i	− 0.415621i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30393.9	1.32088	0.660440	−	0.750879i	$-0.270370\pi$
$809$	30393.9	1.32088	0.660440	+	0.750879i	$0.270370\pi$
$810$	0	0
$811$	27647.6i	1.19709i	0.801090	+	0.598544i	$0.204254\pi$
$811$	27647.6i	1.19709i	−0.801090	+	0.598544i	$0.795746\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12757.9	0.548332
$816$	0	0
$817$	14080.0	0.602934
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 26303.8i	− 1.11816i	−0.829114	−	0.559080i	$-0.811154\pi$
$821$	− 26303.8i	− 1.11816i	0.829114	−	0.559080i	$-0.188846\pi$
$822$	0	0
$823$	−22970.0	−0.972884	−0.486442	−	0.873713i	$-0.661706\pi$
$823$	−22970.0	−0.972884	−0.486442	+	0.873713i	$0.661706\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7665.36i	0.322310i	0.986929	+	0.161155i	$0.0515220\pi$
$827$	7665.36i	0.322310i	−0.986929	+	0.161155i	$0.948478\pi$
$828$	0	0
$829$	− 17383.6i	− 0.728295i	−0.931341	−	0.364147i	$-0.881360\pi$
$829$	− 17383.6i	− 0.728295i	0.931341	−	0.364147i	$-0.118640\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18236.3	0.758525
$834$	0	0
$835$	− 6644.91i	− 0.275397i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33020.5	1.35875	0.679377	−	0.733789i	$-0.262250\pi$
$839$	33020.5	1.35875	0.679377	+	0.733789i	$0.262250\pi$
$840$	0	0
$841$	−36451.0	−1.49457
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8367.39i	0.340647i
$846$	0	0
$847$	1090.00	0.0442182
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8904.97i	0.358706i
$852$	0	0
$853$	− 46099.1i	− 1.85041i	−0.379463	−	0.925207i	$-0.623891\pi$
$853$	− 46099.1i	− 1.85041i	0.379463	−	0.925207i	$-0.376109\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44352.6	1.76786	0.883929	−	0.467620i	$-0.154889\pi$
$857$	44352.6	1.76786	0.883929	+	0.467620i	$0.154889\pi$
$858$	0	0
$859$	7712.85i	0.306355i	0.988199	+	0.153177i	$0.0489506\pi$
$859$	7712.85i	0.306355i	−0.988199	+	0.153177i	$0.951049\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21163.2	0.834765	0.417383	−	0.908731i	$-0.362948\pi$
$863$	21163.2	0.834765	0.417383	+	0.908731i	$0.362948\pi$
$864$	0	0
$865$	9000.00	0.353768
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3567.05i	− 0.139245i
$870$	0	0
$871$	−42240.0	−1.64322
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13281.6i	0.513142i
$876$	0	0
$877$	17146.3i	0.660191i	0.943947	+	0.330096i	$0.107081\pi$
$877$	17146.3i	0.660191i	−0.943947	+	0.330096i	$0.892919\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42026.1	−1.60715	−0.803573	−	0.595206i	$-0.797071\pi$
$881$	−42026.1	−1.60715	−0.803573	+	0.595206i	$0.797071\pi$
$882$	0	0
$883$	4865.03i	0.185415i	0.995693	+	0.0927073i	$0.0295521\pi$
$883$	4865.03i	0.185415i	−0.995693	+	0.0927073i	$0.970448\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11707.3	0.443170	0.221585	−	0.975141i	$-0.428877\pi$
$887$	11707.3	0.443170	0.221585	+	0.975141i	$0.428877\pi$
$888$	0	0
$889$	25300.0	0.954482
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 53429.8i	− 2.00220i
$894$	0	0
$895$	800.000	0.0298783
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 15292.8i	− 0.567344i
$900$	0	0
$901$	− 9967.37i	− 0.368547i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	375.233	0.0137825
$906$	0	0
$907$	12933.9i	0.473497i	0.971571	+	0.236748i	$0.0760817\pi$
$907$	12933.9i	0.473497i	−0.971571	+	0.236748i	$0.923918\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20262.6	−0.736915	−0.368458	−	0.929645i	$-0.620114\pi$
$911$	−20262.6	−0.736915	−0.368458	+	0.929645i	$0.620114\pi$
$912$	0	0
$913$	25440.0	0.922170
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2529.82i	0.0911037i
$918$	0	0
$919$	−22746.0	−0.816454	−0.408227	−	0.912880i	$-0.633853\pi$
$919$	−22746.0	−0.816454	−0.408227	+	0.912880i	$0.633853\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 5043.01i	− 0.179258i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13883.6	0.490320	0.245160	−	0.969483i	$-0.421160\pi$
$929$	13883.6	0.490320	0.245160	+	0.969483i	$0.421160\pi$
$930$	0	0
$931$	− 28834.2i	− 1.01504i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18011.2	−0.629978
$936$	0	0
$937$	−27850.0	−0.970992	−0.485496	−	0.874239i	$-0.661361\pi$
$937$	−27850.0	−0.970992	−0.485496	+	0.874239i	$0.661361\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36789.9i	1.27451i	0.770651	+	0.637257i	$0.219931\pi$
$941$	36789.9i	1.27451i	−0.770651	+	0.637257i	$0.780069\pi$
$942$	0	0
$943$	56320.0	1.94489
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42728.7i	− 1.46620i	−0.680118	−	0.733102i	$-0.738072\pi$
$947$	− 42728.7i	− 1.46620i	0.680118	−	0.733102i	$-0.261928\pi$
$948$	0	0
$949$	− 1779.89i	− 0.0608826i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24990.5	−0.849447	−0.424723	−	0.905323i	$-0.639629\pi$
$953$	−24990.5	−0.849447	−0.424723	+	0.905323i	$0.639629\pi$
$954$	0	0
$955$	23731.8i	0.804130i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6754.20	−0.227429
$960$	0	0
$961$	−25947.0	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8538.15i	0.284822i
$966$	0	0
$967$	−10550.0	−0.350843	−0.175421	−	0.984493i	$-0.556129\pi$
$967$	−10550.0	−0.350843	−0.175421	+	0.984493i	$0.556129\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53012.4i	1.75206i	0.482257	+	0.876030i	$0.339817\pi$
$971$	53012.4i	1.75206i	−0.482257	+	0.876030i	$0.660183\pi$
$972$	0	0
$973$	2373.18i	0.0781920i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34596.5	1.13290	0.566448	−	0.824097i	$-0.308317\pi$
$977$	34596.5	1.13290	0.566448	+	0.824097i	$0.308317\pi$
$978$	0	0
$979$	28478.2i	0.929691i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33921.1	1.10063	0.550313	−	0.834959i	$-0.314509\pi$
$983$	33921.1	1.10063	0.550313	+	0.834959i	$0.314509\pi$
$984$	0	0
$985$	9160.00	0.296306
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17809.9i	0.572622i
$990$	0	0
$991$	8818.00	0.282657	0.141328	−	0.989963i	$-0.454863\pi$
$991$	8818.00	0.282657	0.141328	+	0.989963i	$0.454863\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 20200.6i	− 0.643621i
$996$	0	0
$997$	25096.4i	0.797203i	0.917124	+	0.398602i	$0.130504\pi$
$997$	25096.4i	0.797203i	−0.917124	+	0.398602i	$0.869496\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.4.d.c.145.1		4
3.2	odd	2	inner	288.4.d.c.145.3		4
4.3	odd	2		72.4.d.c.37.1	✓	4
8.3	odd	2		72.4.d.c.37.2	yes	4
8.5	even	2	inner	288.4.d.c.145.4		4
12.11	even	2		72.4.d.c.37.4	yes	4
16.3	odd	4		2304.4.a.bx.1.2		4
16.5	even	4		2304.4.a.cc.1.3		4
16.11	odd	4		2304.4.a.bx.1.3		4
16.13	even	4		2304.4.a.cc.1.2		4
24.5	odd	2	inner	288.4.d.c.145.2		4
24.11	even	2		72.4.d.c.37.3	yes	4
48.5	odd	4		2304.4.a.cc.1.1		4
48.11	even	4		2304.4.a.bx.1.1		4
48.29	odd	4		2304.4.a.cc.1.4		4
48.35	even	4		2304.4.a.bx.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.d.c.37.1	✓	4	4.3	odd	2
72.4.d.c.37.2	yes	4	8.3	odd	2
72.4.d.c.37.3	yes	4	24.11	even	2
72.4.d.c.37.4	yes	4	12.11	even	2
288.4.d.c.145.1		4	1.1	even	1	trivial
288.4.d.c.145.2		4	24.5	odd	2	inner
288.4.d.c.145.3		4	3.2	odd	2	inner
288.4.d.c.145.4		4	8.5	even	2	inner
2304.4.a.bx.1.1		4	48.11	even	4
2304.4.a.bx.1.2		4	16.3	odd	4
2304.4.a.bx.1.3		4	16.11	odd	4
2304.4.a.bx.1.4		4	48.35	even	4
2304.4.a.cc.1.1		4	48.5	odd	4
2304.4.a.cc.1.2		4	16.13	even	4
2304.4.a.cc.1.3		4	16.5	even	4
2304.4.a.cc.1.4		4	48.29	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.32456i q^{5} -10.0000 q^{7} +37.9473i q^{11} -59.3296i q^{13} -75.0467 q^{17} +118.659i q^{19} -150.093 q^{23} +85.0000 q^{25} +246.658i q^{29} -62.0000 q^{31} +63.2456i q^{35} -59.3296i q^{37} -375.233 q^{41} -118.659i q^{43} -450.280 q^{47} -243.000 q^{49} +132.816i q^{53} +240.000 q^{55} +733.648i q^{59} +533.966i q^{61} -375.233 q^{65} -711.955i q^{67} +30.0000 q^{73} -379.473i q^{77} -94.0000 q^{79} -670.403i q^{83} +474.637i q^{85} +750.467 q^{89} +593.296i q^{91} +750.467 q^{95} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 40 q^{7} + 340 q^{25} - 248 q^{31} - 972 q^{49} + 960 q^{55} + 120 q^{73} - 376 q^{79} + 520 q^{97}+O(q^{100}) \) 4 * q - 40 * q^7 + 340 * q^25 - 248 * q^31 - 972 * q^49 + 960 * q^55 + 120 * q^73 - 376 * q^79 + 520 * q^97

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