LMFDB - Embedded newform 576.4.f.b.287.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(287,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("576.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.f (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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.196571825135013064605696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 49x^{12} + 2145x^{8} - 12544x^{4} + 65536 $ x^16 - 49x^12 + 2145x^8 - 12544*x^4 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.14
Root			$1.50834 + 0.404160i$ of defining polynomial
Character	$\chi$	$=$	576.287
Dual form			576.4.f.b.287.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.9985 q^{5} -32.7386i q^{7} +O(q^{10})$	$q+14.9985 q^{5} -32.7386i q^{7} +45.2970i q^{11} +70.5614i q^{13} +116.640i q^{17} +34.6410 q^{19} +102.868 q^{23} +99.9545 q^{25} +11.9830 q^{29} +80.7386i q^{31} -491.030i q^{35} -53.6148i q^{37} -303.188i q^{41} +462.379 q^{43} +100.779 q^{47} -728.818 q^{49} +694.676 q^{53} +679.386i q^{55} -286.702i q^{59} +422.620i q^{61} +1058.31i q^{65} -683.766 q^{67} +192.156 q^{71} -134.091 q^{73} +1482.96 q^{77} -214.648i q^{79} -684.725i q^{83} +1749.43i q^{85} +755.126i q^{89} +2310.08 q^{91} +519.563 q^{95} +700.682 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} - 5328 q^{49} - 5312 q^{73} + 128 q^{97}+O(q^{100}) $ 16 * q + 16 * q^25 - 5328 * q^49 - 5312 * q^73 + 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	14.9985	1.34151	0.670753	−	0.741681i	$-0.265971\pi$
$5$	14.9985	1.34151	0.670753	+	0.741681i	$0.265971\pi$
$6$	0	0
$7$	− 32.7386i	− 1.76772i	−0.467751	−	0.883860i	$-0.654936\pi$
$7$	− 32.7386i	− 1.76772i	0.467751	−	0.883860i	$-0.345064\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	45.2970i	1.24160i	0.783970	+	0.620798i	$0.213191\pi$
$11$	45.2970i	1.24160i	−0.783970	+	0.620798i	$0.786809\pi$
$12$	0	0
$13$	70.5614i	1.50540i	0.658363	+	0.752700i	$0.271249\pi$
$13$	70.5614i	1.50540i	−0.658363	+	0.752700i	$0.728751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	116.640i	1.66409i	0.554711	+	0.832043i	$0.312829\pi$
$17$	116.640i	1.66409i	−0.554711	+	0.832043i	$0.687171\pi$
$18$	0	0
$19$	34.6410	0.418273	0.209137	−	0.977886i	$-0.432935\pi$
$19$	34.6410	0.418273	0.209137	+	0.977886i	$0.432935\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	102.868	0.932585	0.466292	−	0.884631i	$-0.345589\pi$
$23$	102.868	0.932585	0.466292	+	0.884631i	$0.345589\pi$
$24$	0	0
$25$	99.9545	0.799636
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11.9830	0.0767308	0.0383654	−	0.999264i	$-0.487785\pi$
$29$	11.9830	0.0767308	0.0383654	+	0.999264i	$0.487785\pi$
$30$	0	0
$31$	80.7386i	0.467777i	0.972263	+	0.233888i	$0.0751451\pi$
$31$	80.7386i	0.467777i	−0.972263	+	0.233888i	$0.924855\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 491.030i	− 2.37141i
$36$	0	0
$37$	− 53.6148i	− 0.238222i	−0.992881	−	0.119111i	$-0.961996\pi$
$37$	− 53.6148i	− 0.238222i	0.992881	−	0.119111i	$-0.0380045\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 303.188i	− 1.15488i	−0.816434	−	0.577439i	$-0.804052\pi$
$41$	− 303.188i	− 1.15488i	0.816434	−	0.577439i	$-0.195948\pi$
$42$	0	0
$43$	462.379	1.63982	0.819908	−	0.572495i	$-0.194024\pi$
$43$	462.379	1.63982	0.819908	+	0.572495i	$0.194024\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	100.779	0.312768	0.156384	−	0.987696i	$-0.450016\pi$
$47$	100.779	0.312768	0.156384	+	0.987696i	$0.450016\pi$
$48$	0	0
$49$	−728.818	−2.12483
$50$	0	0
$51$	0	0
$52$	0	0
$53$	694.676	1.80040	0.900199	−	0.435478i	$-0.143421\pi$
$53$	694.676	1.80040	0.900199	+	0.435478i	$0.143421\pi$
$54$	0	0
$55$	679.386i	1.66561i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 286.702i	− 0.632634i	−0.948654	−	0.316317i	$-0.897554\pi$
$59$	− 286.702i	− 0.632634i	0.948654	−	0.316317i	$-0.102446\pi$
$60$	0	0
$61$	422.620i	0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$	422.620i	0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1058.31i	2.01950i
$66$	0	0
$67$	−683.766	−1.24680	−0.623398	−	0.781905i	$-0.714248\pi$
$67$	−683.766	−1.24680	−0.623398	+	0.781905i	$0.714248\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	192.156	0.321194	0.160597	−	0.987020i	$-0.448658\pi$
$71$	192.156	0.321194	0.160597	+	0.987020i	$0.448658\pi$
$72$	0	0
$73$	−134.091	−0.214988	−0.107494	−	0.994206i	$-0.534283\pi$
$73$	−134.091	−0.214988	−0.107494	+	0.994206i	$0.534283\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1482.96	2.19479
$78$	0	0
$79$	− 214.648i	− 0.305693i	−0.988250	−	0.152847i	$-0.951156\pi$
$79$	− 214.648i	− 0.305693i	0.988250	−	0.152847i	$-0.0488440\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 684.725i	− 0.905522i	−0.891632	−	0.452761i	$-0.850439\pi$
$83$	− 684.725i	− 0.905522i	0.891632	−	0.452761i	$-0.149561\pi$
$84$	0	0
$85$	1749.43i	2.23238i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	755.126i	0.899361i	0.893189	+	0.449681i	$0.148462\pi$
$89$	755.126i	0.899361i	−0.893189	+	0.449681i	$0.851538\pi$
$90$	0	0
$91$	2310.08	2.66113
$92$	0	0
$93$	0	0
$94$	0	0
$95$	519.563	0.561116
$96$	0	0
$97$	700.682	0.733438	0.366719	−	0.930332i	$-0.380481\pi$
$97$	700.682	0.733438	0.366719	+	0.930332i	$0.380481\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−247.292	−0.243628	−0.121814	−	0.992553i	$-0.538871\pi$
$101$	−247.292	−0.243628	−0.121814	+	0.992553i	$0.538871\pi$
$102$	0	0
$103$	− 524.807i	− 0.502046i	−0.967981	−	0.251023i	$-0.919233\pi$
$103$	− 524.807i	− 0.502046i	0.967981	−	0.251023i	$-0.0807670\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1121.79i	− 1.01353i	−0.862084	−	0.506765i	$-0.830841\pi$
$107$	− 1121.79i	− 1.01353i	0.862084	−	0.506765i	$-0.169159\pi$
$108$	0	0
$109$	− 234.712i	− 0.206251i	−0.994668	−	0.103126i	$-0.967116\pi$
$109$	− 234.712i	− 0.206251i	0.994668	−	0.103126i	$-0.0328844\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 678.372i	− 0.564743i	−0.959305	−	0.282371i	$-0.908879\pi$
$113$	− 678.372i	− 0.564743i	0.959305	−	0.282371i	$-0.0911210\pi$
$114$	0	0
$115$	1542.86	1.25107
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3818.65	2.94164
$120$	0	0
$121$	−720.818	−0.541561
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−375.644	−0.268789
$126$	0	0
$127$	661.420i	0.462138i	0.972937	+	0.231069i	$0.0742224\pi$
$127$	661.420i	0.462138i	−0.972937	+	0.231069i	$0.925778\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2000.75i	− 1.33440i	−0.744879	−	0.667200i	$-0.767493\pi$
$131$	− 2000.75i	− 1.33440i	0.744879	−	0.667200i	$-0.232507\pi$
$132$	0	0
$133$	− 1134.10i	− 0.739390i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2479.08i	1.54600i	0.634404	+	0.773002i	$0.281246\pi$
$137$	2479.08i	1.54600i	−0.634404	+	0.773002i	$0.718754\pi$
$138$	0	0
$139$	−2034.45	−1.24144	−0.620719	−	0.784033i	$-0.713159\pi$
$139$	−2034.45	−1.24144	−0.620719	+	0.784033i	$0.713159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3196.22	−1.86910
$144$	0	0
$145$	179.727	0.102935
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1726.03	0.949007	0.474503	−	0.880254i	$-0.342628\pi$
$149$	1726.03	0.949007	0.474503	+	0.880254i	$0.342628\pi$
$150$	0	0
$151$	3332.44i	1.79596i	0.440034	+	0.897981i	$0.354967\pi$
$151$	3332.44i	1.79596i	−0.440034	+	0.897981i	$0.645033\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1210.96i	0.627525i
$156$	0	0
$157$	2160.65i	1.09834i	0.835711	+	0.549169i	$0.185056\pi$
$157$	2160.65i	1.09834i	−0.835711	+	0.549169i	$0.814944\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3367.76i	− 1.64855i
$162$	0	0
$163$	272.011	0.130709	0.0653544	−	0.997862i	$-0.479182\pi$
$163$	272.011	0.130709	0.0653544	+	0.997862i	$0.479182\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3676.63	−1.70363	−0.851816	−	0.523842i	$-0.824498\pi$
$167$	−3676.63	−1.70363	−0.851816	+	0.523842i	$0.824498\pi$
$168$	0	0
$169$	−2781.91	−1.26623
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1027.82	0.451696	0.225848	−	0.974163i	$-0.427485\pi$
$173$	1027.82	0.451696	0.225848	+	0.974163i	$0.427485\pi$
$174$	0	0
$175$	− 3272.37i	− 1.41353i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3492.16i	1.45819i	0.684413	+	0.729095i	$0.260059\pi$
$179$	3492.16i	1.45819i	−0.684413	+	0.729095i	$0.739941\pi$
$180$	0	0
$181$	− 415.909i	− 0.170797i	−0.996347	−	0.0853985i	$-0.972784\pi$
$181$	− 415.909i	− 0.170797i	0.996347	−	0.0853985i	$-0.0272163\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 804.141i	− 0.319576i
$186$	0	0
$187$	−5283.46	−2.06612
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−853.735	−0.323425	−0.161712	−	0.986838i	$-0.551702\pi$
$191$	−853.735	−0.323425	−0.161712	+	0.986838i	$0.551702\pi$
$192$	0	0
$193$	350.364	0.130672	0.0653361	−	0.997863i	$-0.479188\pi$
$193$	350.364	0.130672	0.0653361	+	0.997863i	$0.479188\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	186.249	0.0673589	0.0336795	−	0.999433i	$-0.489277\pi$
$197$	186.249	0.0673589	0.0336795	+	0.999433i	$0.489277\pi$
$198$	0	0
$199$	276.421i	0.0984670i	0.998787	+	0.0492335i	$0.0156779\pi$
$199$	276.421i	0.0984670i	−0.998787	+	0.0492335i	$0.984322\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 392.308i	− 0.135638i
$204$	0	0
$205$	− 4547.36i	− 1.54928i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1569.13i	0.519327i
$210$	0	0
$211$	2168.76	0.707601	0.353801	−	0.935321i	$-0.384889\pi$
$211$	2168.76	0.707601	0.353801	+	0.935321i	$0.384889\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6934.98	2.19982
$216$	0	0
$217$	2643.27	0.826899
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8230.31	−2.50512
$222$	0	0
$223$	− 4738.22i	− 1.42284i	−0.702765	−	0.711422i	$-0.748051\pi$
$223$	− 4738.22i	− 1.42284i	0.702765	−	0.711422i	$-0.251949\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5969.91i	1.74554i	0.488135	+	0.872768i	$0.337677\pi$
$227$	5969.91i	1.74554i	−0.488135	+	0.872768i	$0.662323\pi$
$228$	0	0
$229$	− 4401.87i	− 1.27023i	−0.772416	−	0.635117i	$-0.780952\pi$
$229$	− 4401.87i	− 1.27023i	0.772416	−	0.635117i	$-0.219048\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3393.41i	− 0.954117i	−0.878871	−	0.477059i	$-0.841703\pi$
$233$	− 3393.41i	− 0.954117i	0.878871	−	0.477059i	$-0.158297\pi$
$234$	0	0
$235$	1511.53	0.419580
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6221.67	−1.68388	−0.841938	−	0.539574i	$-0.818585\pi$
$239$	−6221.67	−1.68388	−0.841938	+	0.539574i	$0.818585\pi$
$240$	0	0
$241$	5291.77	1.41441	0.707205	−	0.707008i	$-0.249956\pi$
$241$	5291.77	1.41441	0.707205	+	0.707008i	$0.249956\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10931.2	−2.85048
$246$	0	0
$247$	2444.32i	0.629669i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2463.46i	− 0.619492i	−0.950819	−	0.309746i	$-0.899756\pi$
$251$	− 2463.46i	− 0.619492i	0.950819	−	0.309746i	$-0.100244\pi$
$252$	0	0
$253$	4659.61i	1.15789i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 30.9844i	− 0.00752045i	−0.999993	−	0.00376022i	$-0.998803\pi$
$257$	− 30.9844i	− 0.00752045i	0.999993	−	0.00376022i	$-0.00119692\pi$
$258$	0	0
$259$	−1755.28	−0.421110
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2013.49	0.472079	0.236040	−	0.971743i	$-0.424150\pi$
$263$	2013.49	0.472079	0.236040	+	0.971743i	$0.424150\pi$
$264$	0	0
$265$	10419.1	2.41524
$266$	0	0
$267$	0	0
$268$	0	0
$269$	894.272	0.202694	0.101347	−	0.994851i	$-0.467685\pi$
$269$	894.272	0.202694	0.101347	+	0.994851i	$0.467685\pi$
$270$	0	0
$271$	− 4374.90i	− 0.980650i	−0.871540	−	0.490325i	$-0.836878\pi$
$271$	− 4374.90i	− 0.980650i	0.871540	−	0.490325i	$-0.163122\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4527.64i	0.992825i
$276$	0	0
$277$	326.826i	0.0708919i	0.999372	+	0.0354460i	$0.0112852\pi$
$277$	326.826i	0.0708919i	−0.999372	+	0.0354460i	$0.988715\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6935.17i	− 1.47231i	−0.676815	−	0.736153i	$-0.736641\pi$
$281$	− 6935.17i	− 1.47231i	0.676815	−	0.736153i	$-0.263359\pi$
$282$	0	0
$283$	−3065.89	−0.643986	−0.321993	−	0.946742i	$-0.604353\pi$
$283$	−3065.89	−0.643986	−0.321993	+	0.946742i	$0.604353\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9925.96	−2.04150
$288$	0	0
$289$	−8692.00	−1.76918
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3164.39	0.630941	0.315471	−	0.948935i	$-0.397838\pi$
$293$	3164.39	0.630941	0.315471	+	0.948935i	$0.397838\pi$
$294$	0	0
$295$	− 4300.09i	− 0.848681i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7258.51i	1.40391i
$300$	0	0
$301$	− 15137.7i	− 2.89874i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6338.67i	1.19000i
$306$	0	0
$307$	−8990.84	−1.67145	−0.835724	−	0.549150i	$-0.814951\pi$
$307$	−8990.84	−1.67145	−0.835724	+	0.549150i	$0.814951\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5317.80	0.969597	0.484798	−	0.874626i	$-0.338893\pi$
$311$	5317.80	0.969597	0.484798	+	0.874626i	$0.338893\pi$
$312$	0	0
$313$	−3947.64	−0.712887	−0.356443	−	0.934317i	$-0.616011\pi$
$313$	−3947.64	−0.712887	−0.356443	+	0.934317i	$0.616011\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3967.23	0.702908	0.351454	−	0.936205i	$-0.385687\pi$
$317$	3967.23	0.702908	0.351454	+	0.936205i	$0.385687\pi$
$318$	0	0
$319$	542.795i	0.0952686i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4040.54i	0.696043i
$324$	0	0
$325$	7052.93i	1.20377i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3299.36i	− 0.552886i
$330$	0	0
$331$	−5505.80	−0.914278	−0.457139	−	0.889395i	$-0.651126\pi$
$331$	−5505.80	−0.914278	−0.457139	+	0.889395i	$0.651126\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10255.5	−1.67258
$336$	0	0
$337$	−8701.27	−1.40649	−0.703247	−	0.710946i	$-0.748267\pi$
$337$	−8701.27	−1.40649	−0.703247	+	0.710946i	$0.748267\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3657.22	−0.580790
$342$	0	0
$343$	12631.2i	1.98839i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8278.18i	1.28068i	0.768092	+	0.640340i	$0.221207\pi$
$347$	8278.18i	1.28068i	−0.768092	+	0.640340i	$0.778793\pi$
$348$	0	0
$349$	3155.56i	0.483992i	0.970277	+	0.241996i	$0.0778021\pi$
$349$	3155.56i	0.483992i	−0.970277	+	0.241996i	$0.922198\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6524.09i	− 0.983689i	−0.870683	−	0.491844i	$-0.836323\pi$
$353$	− 6524.09i	− 0.983689i	0.870683	−	0.491844i	$-0.163677\pi$
$354$	0	0
$355$	2882.05	0.430883
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2329.90	0.342528	0.171264	−	0.985225i	$-0.445215\pi$
$359$	2329.90	0.342528	0.171264	+	0.985225i	$0.445215\pi$
$360$	0	0
$361$	−5659.00	−0.825047
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2011.16	−0.288408
$366$	0	0
$367$	− 6861.53i	− 0.975938i	−0.872861	−	0.487969i	$-0.837738\pi$
$367$	− 6861.53i	− 0.975938i	0.872861	−	0.487969i	$-0.162262\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 22742.7i	− 3.18260i
$372$	0	0
$373$	− 8068.84i	− 1.12008i	−0.828467	−	0.560038i	$-0.810786\pi$
$373$	− 8068.84i	− 1.12008i	0.828467	−	0.560038i	$-0.189214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	845.539i	0.115511i
$378$	0	0
$379$	11760.2	1.59387	0.796937	−	0.604062i	$-0.206452\pi$
$379$	11760.2	1.59387	0.796937	+	0.604062i	$0.206452\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1667.33	0.222445	0.111222	−	0.993796i	$-0.464523\pi$
$383$	1667.33	0.222445	0.111222	+	0.993796i	$0.464523\pi$
$384$	0	0
$385$	22242.2	2.94433
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5992.21	0.781021	0.390510	−	0.920599i	$-0.372299\pi$
$389$	5992.21	0.781021	0.390510	+	0.920599i	$0.372299\pi$
$390$	0	0
$391$	11998.6i	1.55190i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 3219.39i	− 0.410089i
$396$	0	0
$397$	8973.52i	1.13443i	0.823570	+	0.567214i	$0.191979\pi$
$397$	8973.52i	1.13443i	−0.823570	+	0.567214i	$0.808021\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	988.632i	0.123117i	0.998103	+	0.0615585i	$0.0196071\pi$
$401$	988.632i	0.123117i	−0.998103	+	0.0615585i	$0.980393\pi$
$402$	0	0
$403$	−5697.03	−0.704192
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2428.59	0.295776
$408$	0	0
$409$	8638.32	1.04435	0.522173	−	0.852840i	$-0.325122\pi$
$409$	8638.32	1.04435	0.522173	+	0.852840i	$0.325122\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9386.22	−1.11832
$414$	0	0
$415$	− 10269.8i	− 1.21476i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9365.12i	− 1.09192i	−0.837810	−	0.545962i	$-0.816164\pi$
$419$	− 9365.12i	− 1.09192i	0.837810	−	0.545962i	$-0.183836\pi$
$420$	0	0
$421$	− 7897.97i	− 0.914308i	−0.889388	−	0.457154i	$-0.848869\pi$
$421$	− 7897.97i	− 0.914308i	0.889388	−	0.457154i	$-0.151131\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11658.7i	1.33066i
$426$	0	0
$427$	13836.0	1.56808
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8680.10	0.970083	0.485042	−	0.874491i	$-0.338804\pi$
$431$	8680.10	0.970083	0.485042	+	0.874491i	$0.338804\pi$
$432$	0	0
$433$	−9146.91	−1.01518	−0.507589	−	0.861599i	$-0.669463\pi$
$433$	−9146.91	−1.01518	−0.507589	+	0.861599i	$0.669463\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3563.45	0.390075
$438$	0	0
$439$	− 4879.56i	− 0.530498i	−0.964180	−	0.265249i	$-0.914546\pi$
$439$	− 4879.56i	− 0.530498i	0.964180	−	0.265249i	$-0.0854541\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2320.06i	− 0.248824i	−0.992231	−	0.124412i	$-0.960295\pi$
$443$	− 2320.06i	− 0.248824i	0.992231	−	0.124412i	$-0.0397045\pi$
$444$	0	0
$445$	11325.7i	1.20650i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	87.0063i	0.00914495i	0.999990	+	0.00457247i	$0.00145547\pi$
$449$	87.0063i	0.00914495i	−0.999990	+	0.00457247i	$0.998545\pi$
$450$	0	0
$451$	13733.5	1.43389
$452$	0	0
$453$	0	0
$454$	0	0
$455$	34647.7	3.56992
$456$	0	0
$457$	−17554.7	−1.79688	−0.898439	−	0.439098i	$-0.855298\pi$
$457$	−17554.7	−1.79688	−0.898439	+	0.439098i	$0.855298\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14247.1	1.43938	0.719689	−	0.694297i	$-0.244284\pi$
$461$	14247.1	1.43938	0.719689	+	0.694297i	$0.244284\pi$
$462$	0	0
$463$	− 3262.78i	− 0.327504i	−0.986502	−	0.163752i	$-0.947640\pi$
$463$	− 3262.78i	− 0.327504i	0.986502	−	0.163752i	$-0.0523597\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12691.4i	1.25757i	0.777578	+	0.628787i	$0.216448\pi$
$467$	12691.4i	1.25757i	−0.777578	+	0.628787i	$0.783552\pi$
$468$	0	0
$469$	22385.6i	2.20399i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20944.4i	2.03599i
$474$	0	0
$475$	3462.53	0.334467
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17697.5	−1.68814	−0.844071	−	0.536231i	$-0.819847\pi$
$479$	−17697.5	−1.68814	−0.844071	+	0.536231i	$0.819847\pi$
$480$	0	0
$481$	3783.14	0.358620
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10509.2	0.983910
$486$	0	0
$487$	11978.3i	1.11456i	0.830325	+	0.557279i	$0.188155\pi$
$487$	11978.3i	1.11456i	−0.830325	+	0.557279i	$0.811845\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 306.706i	− 0.0281904i	−0.999901	−	0.0140952i	$-0.995513\pi$
$491$	− 306.706i	− 0.0281904i	0.999901	−	0.0140952i	$-0.00448678\pi$
$492$	0	0
$493$	1397.71i	0.127687i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6290.94i	− 0.567781i
$498$	0	0
$499$	−13338.0	−1.19658	−0.598290	−	0.801280i	$-0.704153\pi$
$499$	−13338.0	−1.19658	−0.598290	+	0.801280i	$0.704153\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6066.67	−0.537772	−0.268886	−	0.963172i	$-0.586656\pi$
$503$	−6066.67	−0.537772	−0.268886	+	0.963172i	$0.586656\pi$
$504$	0	0
$505$	−3709.00	−0.326828
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18826.2	1.63941	0.819703	−	0.572789i	$-0.194139\pi$
$509$	18826.2	1.63941	0.819703	+	0.572789i	$0.194139\pi$
$510$	0	0
$511$	4389.95i	0.380039i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 7871.31i	− 0.673497i
$516$	0	0
$517$	4564.98i	0.388332i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13427.7i	1.12914i	0.825387	+	0.564568i	$0.190957\pi$
$521$	13427.7i	1.12914i	−0.825387	+	0.564568i	$0.809043\pi$
$522$	0	0
$523$	11680.2	0.976554	0.488277	−	0.872689i	$-0.337626\pi$
$523$	11680.2	0.976554	0.488277	+	0.872689i	$0.337626\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9417.39	−0.778421
$528$	0	0
$529$	−1585.18	−0.130285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21393.4	1.73855
$534$	0	0
$535$	− 16825.2i	− 1.35966i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 33013.3i	− 2.63819i
$540$	0	0
$541$	4847.76i	0.385252i	0.981272	+	0.192626i	$0.0617004\pi$
$541$	4847.76i	0.385252i	−0.981272	+	0.192626i	$0.938300\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3520.33i	− 0.276687i
$546$	0	0
$547$	−11470.3	−0.896594	−0.448297	−	0.893885i	$-0.647969\pi$
$547$	−11470.3	−0.896594	−0.448297	+	0.893885i	$0.647969\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	415.104	0.0320944
$552$	0	0
$553$	−7027.27	−0.540380
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4316.47	0.328357	0.164178	−	0.986431i	$-0.447503\pi$
$557$	4316.47	0.328357	0.164178	+	0.986431i	$0.447503\pi$
$558$	0	0
$559$	32626.1i	2.46858i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9366.64i	− 0.701167i	−0.936531	−	0.350583i	$-0.885983\pi$
$563$	− 9366.64i	− 0.701167i	0.936531	−	0.350583i	$-0.114017\pi$
$564$	0	0
$565$	− 10174.6i	− 0.757605i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 21420.8i	− 1.57822i	−0.614252	−	0.789110i	$-0.710542\pi$
$569$	− 21420.8i	− 1.57822i	0.614252	−	0.789110i	$-0.289458\pi$
$570$	0	0
$571$	−15247.9	−1.11752	−0.558760	−	0.829330i	$-0.688723\pi$
$571$	−15247.9	−1.11752	−0.558760	+	0.829330i	$0.688723\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10282.1	0.745729
$576$	0	0
$577$	−6503.82	−0.469250	−0.234625	−	0.972086i	$-0.575386\pi$
$577$	−6503.82	−0.469250	−0.234625	+	0.972086i	$0.575386\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22417.0	−1.60071
$582$	0	0
$583$	31466.7i	2.23537i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9899.31i	− 0.696062i	−0.937483	−	0.348031i	$-0.886850\pi$
$587$	− 9899.31i	− 0.696062i	0.937483	−	0.348031i	$-0.113150\pi$
$588$	0	0
$589$	2796.87i	0.195659i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14773.3i	1.02305i	0.859269	+	0.511524i	$0.170919\pi$
$593$	14773.3i	1.02305i	−0.859269	+	0.511524i	$0.829081\pi$
$594$	0	0
$595$	57274.0	3.94622
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2720.03	−0.185538	−0.0927691	−	0.995688i	$-0.529572\pi$
$599$	−2720.03	−0.185538	−0.0927691	+	0.995688i	$0.529572\pi$
$600$	0	0
$601$	−3918.36	−0.265946	−0.132973	−	0.991120i	$-0.542452\pi$
$601$	−3918.36	−0.265946	−0.132973	+	0.991120i	$0.542452\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10811.2	−0.726507
$606$	0	0
$607$	− 15988.8i	− 1.06913i	−0.845126	−	0.534567i	$-0.820475\pi$
$607$	− 15988.8i	− 1.06913i	0.845126	−	0.534567i	$-0.179525\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7111.09i	0.470841i
$612$	0	0
$613$	6085.56i	0.400968i	0.979697	+	0.200484i	$0.0642515\pi$
$613$	6085.56i	0.400968i	−0.979697	+	0.200484i	$0.935749\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9147.13i	− 0.596839i	−0.954435	−	0.298420i	$-0.903541\pi$
$617$	− 9147.13i	− 0.596839i	0.954435	−	0.298420i	$-0.0964594\pi$
$618$	0	0
$619$	−10897.7	−0.707621	−0.353811	−	0.935317i	$-0.615114\pi$
$619$	−10897.7	−0.707621	−0.353811	+	0.935317i	$0.615114\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24721.8	1.58982
$624$	0	0
$625$	−18128.4	−1.16022
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6253.66	0.396422
$630$	0	0
$631$	5869.01i	0.370272i	0.982713	+	0.185136i	$0.0592726\pi$
$631$	5869.01i	0.370272i	−0.982713	+	0.185136i	$0.940727\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9920.30i	0.619961i
$636$	0	0
$637$	− 51426.4i	− 3.19873i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3789.93i	− 0.233531i	−0.993160	−	0.116765i	$-0.962747\pi$
$641$	− 3789.93i	− 0.233531i	0.993160	−	0.116765i	$-0.0372526\pi$
$642$	0	0
$643$	−6468.19	−0.396704	−0.198352	−	0.980131i	$-0.563559\pi$
$643$	−6468.19	−0.396704	−0.198352	+	0.980131i	$0.563559\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19330.4	1.17459	0.587293	−	0.809374i	$-0.300194\pi$
$647$	19330.4	1.17459	0.587293	+	0.809374i	$0.300194\pi$
$648$	0	0
$649$	12986.7	0.785475
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4048.75	−0.242634	−0.121317	−	0.992614i	$-0.538712\pi$
$653$	−4048.75	−0.242634	−0.121317	+	0.992614i	$0.538712\pi$
$654$	0	0
$655$	− 30008.2i	− 1.79010i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1132.07i	0.0669185i	0.999440	+	0.0334592i	$0.0106524\pi$
$659$	1132.07i	0.0669185i	−0.999440	+	0.0334592i	$0.989348\pi$
$660$	0	0
$661$	19461.5i	1.14518i	0.819842	+	0.572590i	$0.194061\pi$
$661$	19461.5i	1.14518i	−0.819842	+	0.572590i	$0.805939\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 17009.8i	− 0.991896i
$666$	0	0
$667$	1232.67	0.0715579
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19143.4	−1.10138
$672$	0	0
$673$	24992.0	1.43146	0.715729	−	0.698378i	$-0.246095\pi$
$673$	24992.0	1.43146	0.715729	+	0.698378i	$0.246095\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16473.0	−0.935168	−0.467584	−	0.883949i	$-0.654875\pi$
$677$	−16473.0	−0.935168	−0.467584	+	0.883949i	$0.654875\pi$
$678$	0	0
$679$	− 22939.4i	− 1.29651i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2919.74i	− 0.163573i	−0.996650	−	0.0817867i	$-0.973937\pi$
$683$	− 2919.74i	− 0.163573i	0.996650	−	0.0817867i	$-0.0260626\pi$
$684$	0	0
$685$	37182.5i	2.07397i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49017.3i	2.71032i
$690$	0	0
$691$	−1257.23	−0.0692148	−0.0346074	−	0.999401i	$-0.511018\pi$
$691$	−1257.23	−0.0692148	−0.0346074	+	0.999401i	$0.511018\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−30513.7	−1.66540
$696$	0	0
$697$	35364.0	1.92182
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25509.1	1.37442	0.687209	−	0.726460i	$-0.258836\pi$
$701$	25509.1	1.37442	0.687209	+	0.726460i	$0.258836\pi$
$702$	0	0
$703$	− 1857.27i	− 0.0996420i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8095.99i	0.430666i
$708$	0	0
$709$	166.571i	0.00882327i	0.999990	+	0.00441164i	$0.00140427\pi$
$709$	166.571i	0.00882327i	−0.999990	+	0.00441164i	$0.998596\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8305.42i	0.436242i
$714$	0	0
$715$	−47938.4	−2.50741
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1087.13	0.0563881	0.0281941	−	0.999602i	$-0.491024\pi$
$719$	1087.13	0.0563881	0.0281941	+	0.999602i	$0.491024\pi$
$720$	0	0
$721$	−17181.5	−0.887477
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1197.76	0.0613567
$726$	0	0
$727$	− 28180.7i	− 1.43764i	−0.695196	−	0.718820i	$-0.744682\pi$
$727$	− 28180.7i	− 1.43764i	0.695196	−	0.718820i	$-0.255318\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	53932.1i	2.72880i
$732$	0	0
$733$	8135.54i	0.409950i	0.978767	+	0.204975i	$0.0657112\pi$
$733$	8135.54i	0.409950i	−0.978767	+	0.204975i	$0.934289\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 30972.6i	− 1.54802i
$738$	0	0
$739$	22561.9	1.12307	0.561537	−	0.827452i	$-0.310210\pi$
$739$	22561.9	1.12307	0.561537	+	0.827452i	$0.310210\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26274.2	−1.29732	−0.648658	−	0.761080i	$-0.724669\pi$
$743$	−26274.2	−1.29732	−0.648658	+	0.761080i	$0.724669\pi$
$744$	0	0
$745$	25887.9	1.27310
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36725.9	−1.79164
$750$	0	0
$751$	19450.6i	0.945092i	0.881306	+	0.472546i	$0.156665\pi$
$751$	19450.6i	0.945092i	−0.881306	+	0.472546i	$0.843335\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	49981.6i	2.40929i
$756$	0	0
$757$	− 24848.1i	− 1.19302i	−0.802604	−	0.596512i	$-0.796553\pi$
$757$	− 24848.1i	− 1.19302i	0.802604	−	0.596512i	$-0.203447\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17940.3i	− 0.854580i	−0.904115	−	0.427290i	$-0.859468\pi$
$761$	− 17940.3i	− 0.854580i	0.904115	−	0.427290i	$-0.140532\pi$
$762$	0	0
$763$	−7684.17	−0.364594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20230.1	0.952367
$768$	0	0
$769$	−14399.3	−0.675229	−0.337614	−	0.941284i	$-0.609620\pi$
$769$	−14399.3	−0.675229	−0.337614	+	0.941284i	$0.609620\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18771.6	−0.873439	−0.436719	−	0.899598i	$-0.643860\pi$
$773$	−18771.6	−0.873439	−0.436719	+	0.899598i	$0.643860\pi$
$774$	0	0
$775$	8070.19i	0.374051i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 10502.7i	− 0.483055i
$780$	0	0
$781$	8704.10i	0.398793i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	32406.5i	1.47343i
$786$	0	0
$787$	−28982.6	−1.31273	−0.656364	−	0.754444i	$-0.727906\pi$
$787$	−28982.6	−1.31273	−0.656364	+	0.754444i	$0.727906\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22209.0	−0.998307
$792$	0	0
$793$	−29820.7	−1.33539
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24283.5	−1.07925	−0.539627	−	0.841904i	$-0.681435\pi$
$797$	−24283.5	−1.07925	−0.539627	+	0.841904i	$0.681435\pi$
$798$	0	0
$799$	11754.9i	0.520473i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6073.92i	− 0.266929i
$804$	0	0
$805$	− 50511.2i	− 2.21154i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 9507.08i	− 0.413166i	−0.978429	−	0.206583i	$-0.933766\pi$
$809$	− 9507.08i	− 0.413166i	0.978429	−	0.206583i	$-0.0662343\pi$
$810$	0	0
$811$	10922.8	0.472936	0.236468	−	0.971639i	$-0.424010\pi$
$811$	10922.8	0.472936	0.236468	+	0.971639i	$0.424010\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4079.75	0.175346
$816$	0	0
$817$	16017.3	0.685892
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8248.72	0.350649	0.175324	−	0.984511i	$-0.443903\pi$
$821$	8248.72	0.350649	0.175324	+	0.984511i	$0.443903\pi$
$822$	0	0
$823$	− 24647.9i	− 1.04395i	−0.852961	−	0.521975i	$-0.825195\pi$
$823$	− 24647.9i	− 1.04395i	0.852961	−	0.521975i	$-0.174805\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23080.8i	− 0.970492i	−0.874378	−	0.485246i	$-0.838730\pi$
$827$	− 23080.8i	− 0.970492i	0.874378	−	0.485246i	$-0.161270\pi$
$828$	0	0
$829$	− 32214.7i	− 1.34965i	−0.737977	−	0.674826i	$-0.764219\pi$
$829$	− 32214.7i	− 1.34965i	0.737977	−	0.674826i	$-0.235781\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 85009.7i	− 3.53591i
$834$	0	0
$835$	−55143.9	−2.28543
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19935.6	0.820327	0.410164	−	0.912012i	$-0.365472\pi$
$839$	19935.6	0.820327	0.410164	+	0.912012i	$0.365472\pi$
$840$	0	0
$841$	−24245.4	−0.994112
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41724.4	−1.69866
$846$	0	0
$847$	23598.6i	0.957329i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5515.25i	− 0.222162i
$852$	0	0
$853$	− 47052.6i	− 1.88869i	−0.328962	−	0.944343i	$-0.606699\pi$
$853$	− 47052.6i	− 1.88869i	0.328962	−	0.944343i	$-0.393301\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 17591.2i	− 0.701171i	−0.936531	−	0.350585i	$-0.885983\pi$
$857$	− 17591.2i	− 0.701171i	0.936531	−	0.350585i	$-0.114017\pi$
$858$	0	0
$859$	32388.2	1.28646	0.643230	−	0.765673i	$-0.277594\pi$
$859$	32388.2	1.28646	0.643230	+	0.765673i	$0.277594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23425.3	−0.923993	−0.461997	−	0.886882i	$-0.652867\pi$
$863$	−23425.3	−0.923993	−0.461997	+	0.886882i	$0.652867\pi$
$864$	0	0
$865$	15415.7	0.605952
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9722.90	0.379547
$870$	0	0
$871$	− 48247.5i	− 1.87693i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12298.1i	0.475144i
$876$	0	0
$877$	− 39520.1i	− 1.52166i	−0.648949	−	0.760832i	$-0.724791\pi$
$877$	− 39520.1i	− 1.52166i	0.648949	−	0.760832i	$-0.275209\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 14020.1i	− 0.536152i	−0.963398	−	0.268076i	$-0.913612\pi$
$881$	− 14020.1i	− 0.536152i	0.963398	−	0.268076i	$-0.0863879\pi$
$882$	0	0
$883$	579.530	0.0220869	0.0110435	−	0.999939i	$-0.496485\pi$
$883$	579.530	0.0220869	0.0110435	+	0.999939i	$0.496485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42843.1	−1.62179	−0.810897	−	0.585189i	$-0.801020\pi$
$887$	−42843.1	−1.62179	−0.810897	+	0.585189i	$0.801020\pi$
$888$	0	0
$889$	21654.0	0.816931
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3491.08	0.130823
$894$	0	0
$895$	52377.0i	1.95617i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	967.493i	0.0358929i
$900$	0	0
$901$	81027.4i	2.99602i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 6238.00i	− 0.229125i
$906$	0	0
$907$	7234.70	0.264856	0.132428	−	0.991193i	$-0.457723\pi$
$907$	7234.70	0.264856	0.132428	+	0.991193i	$0.457723\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3307.85	0.120301	0.0601503	−	0.998189i	$-0.480842\pi$
$911$	3307.85	0.120301	0.0601503	+	0.998189i	$0.480842\pi$
$912$	0	0
$913$	31016.0	1.12429
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−65501.8	−2.35885
$918$	0	0
$919$	38132.6i	1.36875i	0.729132	+	0.684373i	$0.239924\pi$
$919$	38132.6i	1.36875i	−0.729132	+	0.684373i	$0.760076\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13558.8i	0.483525i
$924$	0	0
$925$	− 5359.05i	− 0.190491i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7060.62i	0.249356i	0.992197	+	0.124678i	$0.0397897\pi$
$929$	7060.62i	0.249356i	−0.992197	+	0.124678i	$0.960210\pi$
$930$	0	0
$931$	−25247.0	−0.888762
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−79243.9	−2.77172
$936$	0	0
$937$	−29576.9	−1.03120	−0.515601	−	0.856829i	$-0.672431\pi$
$937$	−29576.9	−1.03120	−0.515601	+	0.856829i	$0.672431\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23649.7	0.819296	0.409648	−	0.912244i	$-0.365652\pi$
$941$	23649.7	0.819296	0.409648	+	0.912244i	$0.365652\pi$
$942$	0	0
$943$	− 31188.3i	− 1.07702i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14659.5i	0.503030i	0.967853	+	0.251515i	$0.0809288\pi$
$947$	14659.5i	0.503030i	−0.967853	+	0.251515i	$0.919071\pi$
$948$	0	0
$949$	− 9461.64i	− 0.323644i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42151.6i	1.43276i	0.697709	+	0.716381i	$0.254203\pi$
$953$	42151.6i	1.43276i	−0.697709	+	0.716381i	$0.745797\pi$
$954$	0	0
$955$	−12804.7	−0.433876
$956$	0	0
$957$	0	0
$958$	0	0
$959$	81161.8	2.73290
$960$	0	0
$961$	23272.3	0.781185
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5254.92	0.175297
$966$	0	0
$967$	− 29626.8i	− 0.985246i	−0.870243	−	0.492623i	$-0.836038\pi$
$967$	− 29626.8i	− 0.985246i	0.870243	−	0.492623i	$-0.163962\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53017.0i	1.75221i	0.482121	+	0.876105i	$0.339867\pi$
$971$	53017.0i	1.75221i	−0.482121	+	0.876105i	$0.660133\pi$
$972$	0	0
$973$	66605.1i	2.19451i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9593.54i	0.314150i	0.987587	+	0.157075i	$0.0502065\pi$
$977$	9593.54i	0.314150i	−0.987587	+	0.157075i	$0.949794\pi$
$978$	0	0
$979$	−34204.9	−1.11664
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35754.4	1.16011	0.580055	−	0.814578i	$-0.303031\pi$
$983$	35754.4	1.16011	0.580055	+	0.814578i	$0.303031\pi$
$984$	0	0
$985$	2793.46	0.0903623
$986$	0	0
$987$	0	0
$988$	0	0
$989$	47564.0	1.52927
$990$	0	0
$991$	− 46510.6i	− 1.49088i	−0.666575	−	0.745438i	$-0.732240\pi$
$991$	− 46510.6i	− 1.49088i	0.666575	−	0.745438i	$-0.267760\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4145.89i	0.132094i
$996$	0	0
$997$	4881.39i	0.155060i	0.996990	+	0.0775302i	$0.0247034\pi$
$997$	4881.39i	0.155060i	−0.996990	+	0.0775302i	$0.975297\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.f.b.287.14	yes	16
3.2	odd	2	inner	576.4.f.b.287.2	yes	16
4.3	odd	2	inner	576.4.f.b.287.16	yes	16
8.3	odd	2	inner	576.4.f.b.287.3	yes	16
8.5	even	2	inner	576.4.f.b.287.1	✓	16
12.11	even	2	inner	576.4.f.b.287.4	yes	16
16.3	odd	4		2304.4.c.m.2303.1		16
16.5	even	4		2304.4.c.m.2303.16		16
16.11	odd	4		2304.4.c.m.2303.14		16
16.13	even	4		2304.4.c.m.2303.3		16
24.5	odd	2	inner	576.4.f.b.287.13	yes	16
24.11	even	2	inner	576.4.f.b.287.15	yes	16
48.5	odd	4		2304.4.c.m.2303.4		16
48.11	even	4		2304.4.c.m.2303.2		16
48.29	odd	4		2304.4.c.m.2303.15		16
48.35	even	4		2304.4.c.m.2303.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.4.f.b.287.1	✓	16	8.5	even	2	inner
576.4.f.b.287.2	yes	16	3.2	odd	2	inner
576.4.f.b.287.3	yes	16	8.3	odd	2	inner
576.4.f.b.287.4	yes	16	12.11	even	2	inner
576.4.f.b.287.13	yes	16	24.5	odd	2	inner
576.4.f.b.287.14	yes	16	1.1	even	1	trivial
576.4.f.b.287.15	yes	16	24.11	even	2	inner
576.4.f.b.287.16	yes	16	4.3	odd	2	inner
2304.4.c.m.2303.1		16	16.3	odd	4
2304.4.c.m.2303.2		16	48.11	even	4
2304.4.c.m.2303.3		16	16.13	even	4
2304.4.c.m.2303.4		16	48.5	odd	4
2304.4.c.m.2303.13		16	48.35	even	4
2304.4.c.m.2303.14		16	16.11	odd	4
2304.4.c.m.2303.15		16	48.29	odd	4
2304.4.c.m.2303.16		16	16.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+14.9985 q^{5} -32.7386i q^{7} +O(q^{10})\)	\(q+14.9985 q^{5} -32.7386i q^{7} +45.2970i q^{11} +70.5614i q^{13} +116.640i q^{17} +34.6410 q^{19} +102.868 q^{23} +99.9545 q^{25} +11.9830 q^{29} +80.7386i q^{31} -491.030i q^{35} -53.6148i q^{37} -303.188i q^{41} +462.379 q^{43} +100.779 q^{47} -728.818 q^{49} +694.676 q^{53} +679.386i q^{55} -286.702i q^{59} +422.620i q^{61} +1058.31i q^{65} -683.766 q^{67} +192.156 q^{71} -134.091 q^{73} +1482.96 q^{77} -214.648i q^{79} -684.725i q^{83} +1749.43i q^{85} +755.126i q^{89} +2310.08 q^{91} +519.563 q^{95} +700.682 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} - 5328 q^{49} - 5312 q^{73} + 128 q^{97}+O(q^{100}) \) 16 * q + 16 * q^25 - 5328 * q^49 - 5312 * q^73 + 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more