LMFDB - Embedded newform 2592.3.e.i.161.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,3,Mod(161,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("2592.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2592.e (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:	$70.6268845222$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.12
Character	$\chi$	$=$	2592.161
Dual form			2592.3.e.i.161.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.832715i q^{5} -2.52101 q^{7} +O(q^{10})$	$q+0.832715i q^{5} -2.52101 q^{7} +10.9430i q^{11} -8.72906 q^{13} -20.8637i q^{17} +1.50150 q^{19} +1.15712i q^{23} +24.3066 q^{25} +18.1689i q^{29} +51.3818 q^{31} -2.09929i q^{35} -7.93951 q^{37} +25.2742i q^{41} -38.6837 q^{43} +68.8846i q^{47} -42.6445 q^{49} +46.5195i q^{53} -9.11241 q^{55} -102.919i q^{59} -88.3302 q^{61} -7.26882i q^{65} +22.6384 q^{67} -104.256i q^{71} -75.2115 q^{73} -27.5875i q^{77} -103.735 q^{79} +62.0650i q^{83} +17.3735 q^{85} -1.95722i q^{89} +22.0061 q^{91} +1.25032i q^{95} -118.434 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 120 q^{25} + 72 q^{49} + 192 q^{61} + 24 q^{73} - 192 q^{85} + 264 q^{97}+O(q^{100}) $ 24 * q - 120 * q^25 + 72 * q^49 + 192 * q^61 + 24 * q^73 - 192 * q^85 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	0.832715i	0.166543i	0.996527	+	0.0832715i	$0.0265369\pi$
$5$	0.832715i	0.166543i	−0.996527	+	0.0832715i	$0.973463\pi$
$6$	0	0
$7$	−2.52101	−0.360145	−0.180072	−	0.983653i	$-0.557633\pi$
$7$	−2.52101	−0.360145	−0.180072	+	0.983653i	$0.557633\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.9430i	0.994820i	0.867516	+	0.497410i	$0.165716\pi$
$11$	10.9430i	0.994820i	−0.867516	+	0.497410i	$0.834284\pi$
$12$	0	0
$13$	−8.72906	−0.671466	−0.335733	−	0.941957i	$-0.608984\pi$
$13$	−8.72906	−0.671466	−0.335733	+	0.941957i	$0.608984\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 20.8637i	− 1.22728i	−0.789586	−	0.613639i	$-0.789705\pi$
$17$	− 20.8637i	− 1.22728i	0.789586	−	0.613639i	$-0.210295\pi$
$18$	0	0
$19$	1.50150	0.0790265	0.0395132	−	0.999219i	$-0.487419\pi$
$19$	1.50150	0.0790265	0.0395132	+	0.999219i	$0.487419\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.15712i	0.0503094i	0.999684	+	0.0251547i	$0.00800784\pi$
$23$	1.15712i	0.0503094i	−0.999684	+	0.0251547i	$0.991992\pi$
$24$	0	0
$25$	24.3066	0.972263
$26$	0	0
$27$	0	0
$28$	0	0
$29$	18.1689i	0.626513i	0.949669	+	0.313257i	$0.101420\pi$
$29$	18.1689i	0.626513i	−0.949669	+	0.313257i	$0.898580\pi$
$30$	0	0
$31$	51.3818	1.65748	0.828739	−	0.559636i	$-0.189059\pi$
$31$	51.3818	1.65748	0.828739	+	0.559636i	$0.189059\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.09929i	− 0.0599796i
$36$	0	0
$37$	−7.93951	−0.214581	−0.107291	−	0.994228i	$-0.534218\pi$
$37$	−7.93951	−0.214581	−0.107291	+	0.994228i	$0.534218\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	25.2742i	0.616445i	0.951314	+	0.308222i	$0.0997341\pi$
$41$	25.2742i	0.616445i	−0.951314	+	0.308222i	$0.900266\pi$
$42$	0	0
$43$	−38.6837	−0.899620	−0.449810	−	0.893124i	$-0.648508\pi$
$43$	−38.6837	−0.899620	−0.449810	+	0.893124i	$0.648508\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	68.8846i	1.46563i	0.680427	+	0.732815i	$0.261794\pi$
$47$	68.8846i	1.46563i	−0.680427	+	0.732815i	$0.738206\pi$
$48$	0	0
$49$	−42.6445	−0.870296
$50$	0	0
$51$	0	0
$52$	0	0
$53$	46.5195i	0.877727i	0.898554	+	0.438864i	$0.144619\pi$
$53$	46.5195i	0.877727i	−0.898554	+	0.438864i	$0.855381\pi$
$54$	0	0
$55$	−9.11241	−0.165680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 102.919i	− 1.74439i	−0.489156	−	0.872196i	$-0.662695\pi$
$59$	− 102.919i	− 1.74439i	0.489156	−	0.872196i	$-0.337305\pi$
$60$	0	0
$61$	−88.3302	−1.44804	−0.724018	−	0.689781i	$-0.757707\pi$
$61$	−88.3302	−1.44804	−0.724018	+	0.689781i	$0.757707\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 7.26882i	− 0.111828i
$66$	0	0
$67$	22.6384	0.337886	0.168943	−	0.985626i	$-0.445965\pi$
$67$	22.6384	0.337886	0.168943	+	0.985626i	$0.445965\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 104.256i	− 1.46840i	−0.678935	−	0.734198i	$-0.737558\pi$
$71$	− 104.256i	− 1.46840i	0.678935	−	0.734198i	$-0.262442\pi$
$72$	0	0
$73$	−75.2115	−1.03030	−0.515148	−	0.857101i	$-0.672263\pi$
$73$	−75.2115	−1.03030	−0.515148	+	0.857101i	$0.672263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 27.5875i	− 0.358279i
$78$	0	0
$79$	−103.735	−1.31310	−0.656552	−	0.754281i	$-0.727986\pi$
$79$	−103.735	−1.31310	−0.656552	+	0.754281i	$0.727986\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	62.0650i	0.747772i	0.927475	+	0.373886i	$0.121975\pi$
$83$	62.0650i	0.747772i	−0.927475	+	0.373886i	$0.878025\pi$
$84$	0	0
$85$	17.3735	0.204395
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.95722i	− 0.0219912i	−0.999940	−	0.0109956i	$-0.996500\pi$
$89$	− 1.95722i	− 0.0219912i	0.999940	−	0.0109956i	$-0.00350008\pi$
$90$	0	0
$91$	22.0061	0.241825
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.25032i	0.0131613i
$96$	0	0
$97$	−118.434	−1.22097	−0.610486	−	0.792027i	$-0.709026\pi$
$97$	−118.434	−1.22097	−0.610486	+	0.792027i	$0.709026\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 55.7039i	− 0.551524i	−0.961226	−	0.275762i	$-0.911070\pi$
$101$	− 55.7039i	− 0.551524i	0.961226	−	0.275762i	$-0.0889301\pi$
$102$	0	0
$103$	201.947	1.96066	0.980328	−	0.197377i	$-0.0632425\pi$
$103$	201.947	1.96066	0.980328	+	0.197377i	$0.0632425\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	55.1900i	0.515794i	0.966172	+	0.257897i	$0.0830295\pi$
$107$	55.1900i	0.515794i	−0.966172	+	0.257897i	$0.916970\pi$
$108$	0	0
$109$	−44.6887	−0.409988	−0.204994	−	0.978763i	$-0.565717\pi$
$109$	−44.6887	−0.409988	−0.204994	+	0.978763i	$0.565717\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 108.164i	− 0.957201i	−0.878033	−	0.478600i	$-0.841144\pi$
$113$	− 108.164i	− 0.957201i	0.878033	−	0.478600i	$-0.158856\pi$
$114$	0	0
$115$	−0.963548	−0.00837868
$116$	0	0
$117$	0	0
$118$	0	0
$119$	52.5978i	0.441998i
$120$	0	0
$121$	1.25027	0.0103328
$122$	0	0
$123$	0	0
$124$	0	0
$125$	41.0583i	0.328467i
$126$	0	0
$127$	−172.177	−1.35573	−0.677863	−	0.735188i	$-0.737094\pi$
$127$	−172.177	−1.35573	−0.677863	+	0.735188i	$0.737094\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 88.7027i	− 0.677120i	−0.940945	−	0.338560i	$-0.890060\pi$
$131$	− 88.7027i	− 0.677120i	0.940945	−	0.338560i	$-0.109940\pi$
$132$	0	0
$133$	−3.78531	−0.0284610
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 89.6228i	− 0.654181i	−0.944993	−	0.327090i	$-0.893932\pi$
$137$	− 89.6228i	− 0.654181i	0.944993	−	0.327090i	$-0.106068\pi$
$138$	0	0
$139$	−149.584	−1.07614	−0.538071	−	0.842900i	$-0.680847\pi$
$139$	−149.584	−1.07614	−0.538071	+	0.842900i	$0.680847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 95.5223i	− 0.667988i
$144$	0	0
$145$	−15.1295	−0.104341
$146$	0	0
$147$	0	0
$148$	0	0
$149$	140.079i	0.940130i	0.882632	+	0.470065i	$0.155770\pi$
$149$	140.079i	0.940130i	−0.882632	+	0.470065i	$0.844230\pi$
$150$	0	0
$151$	−145.132	−0.961140	−0.480570	−	0.876956i	$-0.659570\pi$
$151$	−145.132	−0.961140	−0.480570	+	0.876956i	$0.659570\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	42.7864i	0.276041i
$156$	0	0
$157$	−247.079	−1.57375	−0.786876	−	0.617111i	$-0.788303\pi$
$157$	−247.079	−1.57375	−0.786876	+	0.617111i	$0.788303\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.91711i	− 0.0181187i
$162$	0	0
$163$	189.342	1.16161	0.580805	−	0.814043i	$-0.302738\pi$
$163$	189.342	1.16161	0.580805	+	0.814043i	$0.302738\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	231.590i	1.38677i	0.720568	+	0.693384i	$0.243881\pi$
$167$	231.590i	1.38677i	−0.720568	+	0.693384i	$0.756119\pi$
$168$	0	0
$169$	−92.8035	−0.549133
$170$	0	0
$171$	0	0
$172$	0	0
$173$	259.699i	1.50115i	0.660786	+	0.750574i	$0.270223\pi$
$173$	259.699i	1.50115i	−0.660786	+	0.750574i	$0.729777\pi$
$174$	0	0
$175$	−61.2772	−0.350156
$176$	0	0
$177$	0	0
$178$	0	0
$179$	220.358i	1.23105i	0.788118	+	0.615524i	$0.211056\pi$
$179$	220.358i	1.23105i	−0.788118	+	0.615524i	$0.788944\pi$
$180$	0	0
$181$	−286.189	−1.58116	−0.790578	−	0.612362i	$-0.790220\pi$
$181$	−286.189	−1.58116	−0.790578	+	0.612362i	$0.790220\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 6.61135i	− 0.0357370i
$186$	0	0
$187$	228.312	1.22092
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.57954i	− 0.00826983i	−0.999991	−	0.00413491i	$-0.998684\pi$
$191$	− 1.57954i	− 0.00826983i	0.999991	−	0.00413491i	$-0.00131619\pi$
$192$	0	0
$193$	−232.133	−1.20276	−0.601381	−	0.798962i	$-0.705383\pi$
$193$	−232.133	−1.20276	−0.601381	+	0.798962i	$0.705383\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 214.012i	− 1.08636i	−0.839618	−	0.543178i	$-0.817221\pi$
$197$	− 214.012i	− 1.08636i	0.839618	−	0.543178i	$-0.182779\pi$
$198$	0	0
$199$	25.8912	0.130106	0.0650532	−	0.997882i	$-0.479278\pi$
$199$	25.8912	0.130106	0.0650532	+	0.997882i	$0.479278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 45.8040i	− 0.225635i
$204$	0	0
$205$	−21.0462	−0.102664
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.4310i	0.0786171i
$210$	0	0
$211$	−93.9504	−0.445263	−0.222631	−	0.974903i	$-0.571465\pi$
$211$	−93.9504	−0.445263	−0.222631	+	0.974903i	$0.571465\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 32.2124i	− 0.149825i
$216$	0	0
$217$	−129.534	−0.596932
$218$	0	0
$219$	0	0
$220$	0	0
$221$	182.121i	0.824076i
$222$	0	0
$223$	299.372	1.34247	0.671237	−	0.741243i	$-0.265763\pi$
$223$	299.372	1.34247	0.671237	+	0.741243i	$0.265763\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 364.774i	− 1.60693i	−0.595349	−	0.803467i	$-0.702986\pi$
$227$	− 364.774i	− 1.60693i	0.595349	−	0.803467i	$-0.297014\pi$
$228$	0	0
$229$	−281.958	−1.23126	−0.615629	−	0.788036i	$-0.711098\pi$
$229$	−281.958	−1.23126	−0.615629	+	0.788036i	$0.711098\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 208.568i	− 0.895140i	−0.894249	−	0.447570i	$-0.852290\pi$
$233$	− 208.568i	− 0.895140i	0.894249	−	0.447570i	$-0.147710\pi$
$234$	0	0
$235$	−57.3613	−0.244090
$236$	0	0
$237$	0	0
$238$	0	0
$239$	29.0098i	0.121380i	0.998157	+	0.0606899i	$0.0193301\pi$
$239$	29.0098i	0.121380i	−0.998157	+	0.0606899i	$0.980670\pi$
$240$	0	0
$241$	−232.911	−0.966437	−0.483218	−	0.875500i	$-0.660532\pi$
$241$	−232.911	−0.966437	−0.483218	+	0.875500i	$0.660532\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 35.5107i	− 0.144942i
$246$	0	0
$247$	−13.1067	−0.0530636
$248$	0	0
$249$	0	0
$250$	0	0
$251$	112.869i	0.449676i	0.974396	+	0.224838i	$0.0721853\pi$
$251$	112.869i	0.449676i	−0.974396	+	0.224838i	$0.927815\pi$
$252$	0	0
$253$	−12.6624	−0.0500488
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 337.414i	− 1.31290i	−0.754371	−	0.656448i	$-0.772058\pi$
$257$	− 337.414i	− 1.31290i	0.754371	−	0.656448i	$-0.227942\pi$
$258$	0	0
$259$	20.0156	0.0772804
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 505.730i	− 1.92293i	−0.274930	−	0.961464i	$-0.588655\pi$
$263$	− 505.730i	− 1.92293i	0.274930	−	0.961464i	$-0.411345\pi$
$264$	0	0
$265$	−38.7375	−0.146179
$266$	0	0
$267$	0	0
$268$	0	0
$269$	337.414i	1.25433i	0.778887	+	0.627164i	$0.215784\pi$
$269$	337.414i	1.25433i	−0.778887	+	0.627164i	$0.784216\pi$
$270$	0	0
$271$	−12.0273	−0.0443813	−0.0221907	−	0.999754i	$-0.507064\pi$
$271$	−12.0273	−0.0443813	−0.0221907	+	0.999754i	$0.507064\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	265.988i	0.967227i
$276$	0	0
$277$	−199.886	−0.721611	−0.360805	−	0.932641i	$-0.617498\pi$
$277$	−199.886	−0.721611	−0.360805	+	0.932641i	$0.617498\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 334.906i	− 1.19184i	−0.803046	−	0.595918i	$-0.796788\pi$
$281$	− 334.906i	− 1.19184i	0.803046	−	0.595918i	$-0.203212\pi$
$282$	0	0
$283$	−470.153	−1.66132	−0.830659	−	0.556782i	$-0.812036\pi$
$283$	−470.153	−1.66132	−0.830659	+	0.556782i	$0.812036\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 63.7167i	− 0.222009i
$288$	0	0
$289$	−146.296	−0.506213
$290$	0	0
$291$	0	0
$292$	0	0
$293$	151.557i	0.517259i	0.965977	+	0.258630i	$0.0832709\pi$
$293$	151.557i	0.517259i	−0.965977	+	0.258630i	$0.916729\pi$
$294$	0	0
$295$	85.7023	0.290516
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 10.1005i	− 0.0337811i
$300$	0	0
$301$	97.5220	0.323994
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 73.5538i	− 0.241160i
$306$	0	0
$307$	123.852	0.403426	0.201713	−	0.979445i	$-0.435349\pi$
$307$	123.852	0.403426	0.201713	+	0.979445i	$0.435349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	96.8945i	0.311558i	0.987792	+	0.155779i	$0.0497887\pi$
$311$	96.8945i	0.311558i	−0.987792	+	0.155779i	$0.950211\pi$
$312$	0	0
$313$	−96.4999	−0.308306	−0.154153	−	0.988047i	$-0.549265\pi$
$313$	−96.4999	−0.308306	−0.154153	+	0.988047i	$0.549265\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	138.376i	0.436516i	0.975891	+	0.218258i	$0.0700375\pi$
$317$	138.376i	0.436516i	−0.975891	+	0.218258i	$0.929963\pi$
$318$	0	0
$319$	−198.822	−0.623268
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 31.3270i	− 0.0969875i
$324$	0	0
$325$	−212.174	−0.652842
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 173.659i	− 0.527839i
$330$	0	0
$331$	298.986	0.903281	0.451641	−	0.892200i	$-0.350839\pi$
$331$	298.986	0.903281	0.451641	+	0.892200i	$0.350839\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.8513i	0.0562725i
$336$	0	0
$337$	−47.7082	−0.141567	−0.0707837	−	0.997492i	$-0.522550\pi$
$337$	−47.7082	−0.141567	−0.0707837	+	0.997492i	$0.522550\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	562.272i	1.64889i
$342$	0	0
$343$	231.037	0.673577
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 120.983i	− 0.348654i	−0.984688	−	0.174327i	$-0.944225\pi$
$347$	− 120.983i	− 0.348654i	0.984688	−	0.174327i	$-0.0557749\pi$
$348$	0	0
$349$	−387.512	−1.11035	−0.555175	−	0.831733i	$-0.687349\pi$
$349$	−387.512	−1.11035	−0.555175	+	0.831733i	$0.687349\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.92090i	− 0.0224388i	−0.999937	−	0.0112194i	$-0.996429\pi$
$353$	− 7.92090i	− 0.0224388i	0.999937	−	0.0112194i	$-0.00357132\pi$
$354$	0	0
$355$	86.8156	0.244551
$356$	0	0
$357$	0	0
$358$	0	0
$359$	210.527i	0.586425i	0.956047	+	0.293212i	$0.0947243\pi$
$359$	210.527i	0.586425i	−0.956047	+	0.293212i	$0.905276\pi$
$360$	0	0
$361$	−358.745	−0.993755
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 62.6298i	− 0.171588i
$366$	0	0
$367$	343.880	0.937004	0.468502	−	0.883462i	$-0.344794\pi$
$367$	343.880	0.937004	0.468502	+	0.883462i	$0.344794\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 117.276i	− 0.316109i
$372$	0	0
$373$	354.708	0.950960	0.475480	−	0.879726i	$-0.342274\pi$
$373$	354.708	0.950960	0.475480	+	0.879726i	$0.342274\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 158.597i	− 0.420682i
$378$	0	0
$379$	−269.497	−0.711073	−0.355536	−	0.934662i	$-0.615702\pi$
$379$	−269.497	−0.711073	−0.355536	+	0.934662i	$0.615702\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 83.9367i	− 0.219156i	−0.993978	−	0.109578i	$-0.965050\pi$
$383$	− 83.9367i	− 0.219156i	0.993978	−	0.109578i	$-0.0349499\pi$
$384$	0	0
$385$	22.9725	0.0596689
$386$	0	0
$387$	0	0
$388$	0	0
$389$	310.634i	0.798544i	0.916833	+	0.399272i	$0.130737\pi$
$389$	310.634i	0.798544i	−0.916833	+	0.399272i	$0.869263\pi$
$390$	0	0
$391$	24.1418	0.0617437
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 86.3819i	− 0.218688i
$396$	0	0
$397$	436.104	1.09850	0.549249	−	0.835658i	$-0.314914\pi$
$397$	436.104	1.09850	0.549249	+	0.835658i	$0.314914\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	409.658i	1.02159i	0.859703	+	0.510795i	$0.170649\pi$
$401$	409.658i	1.02159i	−0.859703	+	0.510795i	$0.829351\pi$
$402$	0	0
$403$	−448.515	−1.11294
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 86.8823i	− 0.213470i
$408$	0	0
$409$	−422.163	−1.03218	−0.516092	−	0.856533i	$-0.672614\pi$
$409$	−422.163	−1.03218	−0.516092	+	0.856533i	$0.672614\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	259.461i	0.628234i
$414$	0	0
$415$	−51.6825	−0.124536
$416$	0	0
$417$	0	0
$418$	0	0
$419$	635.062i	1.51566i	0.652452	+	0.757830i	$0.273741\pi$
$419$	635.062i	1.51566i	−0.652452	+	0.757830i	$0.726259\pi$
$420$	0	0
$421$	−47.5562	−0.112960	−0.0564800	−	0.998404i	$-0.517988\pi$
$421$	−47.5562	−0.112960	−0.0564800	+	0.998404i	$0.517988\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 507.126i	− 1.19324i
$426$	0	0
$427$	222.682	0.521503
$428$	0	0
$429$	0	0
$430$	0	0
$431$	614.503i	1.42576i	0.701286	+	0.712880i	$0.252610\pi$
$431$	614.503i	1.42576i	−0.701286	+	0.712880i	$0.747390\pi$
$432$	0	0
$433$	118.672	0.274070	0.137035	−	0.990566i	$-0.456243\pi$
$433$	118.672	0.274070	0.137035	+	0.990566i	$0.456243\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.73741i	0.00397578i
$438$	0	0
$439$	−490.147	−1.11651	−0.558254	−	0.829670i	$-0.688528\pi$
$439$	−490.147	−1.11651	−0.558254	+	0.829670i	$0.688528\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	193.088i	0.435864i	0.975964	+	0.217932i	$0.0699312\pi$
$443$	193.088i	0.435864i	−0.975964	+	0.217932i	$0.930069\pi$
$444$	0	0
$445$	1.62981	0.00366248
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 449.191i	− 1.00043i	−0.865902	−	0.500213i	$-0.833255\pi$
$449$	− 449.191i	− 1.00043i	0.865902	−	0.500213i	$-0.166745\pi$
$450$	0	0
$451$	−276.576	−0.613252
$452$	0	0
$453$	0	0
$454$	0	0
$455$	18.3248i	0.0402743i
$456$	0	0
$457$	94.0564	0.205813	0.102906	−	0.994691i	$-0.467186\pi$
$457$	94.0564	0.205813	0.102906	+	0.994691i	$0.467186\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	340.966i	0.739623i	0.929107	+	0.369811i	$0.120578\pi$
$461$	340.966i	0.739623i	−0.929107	+	0.369811i	$0.879422\pi$
$462$	0	0
$463$	−571.299	−1.23391	−0.616953	−	0.787000i	$-0.711633\pi$
$463$	−571.299	−1.23391	−0.616953	+	0.787000i	$0.711633\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 421.650i	− 0.902891i	−0.892299	−	0.451446i	$-0.850909\pi$
$467$	− 421.650i	− 0.902891i	0.892299	−	0.451446i	$-0.149091\pi$
$468$	0	0
$469$	−57.0716	−0.121688
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 423.316i	− 0.894960i
$474$	0	0
$475$	36.4964	0.0768345
$476$	0	0
$477$	0	0
$478$	0	0
$479$	808.679i	1.68827i	0.536134	+	0.844133i	$0.319884\pi$
$479$	808.679i	1.68827i	−0.536134	+	0.844133i	$0.680116\pi$
$480$	0	0
$481$	69.3045	0.144084
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 98.6219i	− 0.203344i
$486$	0	0
$487$	155.199	0.318684	0.159342	−	0.987223i	$-0.449063\pi$
$487$	155.199	0.318684	0.159342	+	0.987223i	$0.449063\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 631.410i	− 1.28597i	−0.765880	−	0.642984i	$-0.777696\pi$
$491$	− 631.410i	− 1.28597i	0.765880	−	0.642984i	$-0.222304\pi$
$492$	0	0
$493$	379.071	0.768906
$494$	0	0
$495$	0	0
$496$	0	0
$497$	262.831i	0.528836i
$498$	0	0
$499$	891.640	1.78685	0.893427	−	0.449208i	$-0.148294\pi$
$499$	891.640	1.78685	0.893427	+	0.449208i	$0.148294\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 281.433i	− 0.559509i	−0.960072	−	0.279754i	$-0.909747\pi$
$503$	− 281.433i	− 0.559509i	0.960072	−	0.279754i	$-0.0902531\pi$
$504$	0	0
$505$	46.3854	0.0918523
$506$	0	0
$507$	0	0
$508$	0	0
$509$	492.144i	0.966884i	0.875376	+	0.483442i	$0.160614\pi$
$509$	492.144i	0.966884i	−0.875376	+	0.483442i	$0.839386\pi$
$510$	0	0
$511$	189.609	0.371056
$512$	0	0
$513$	0	0
$514$	0	0
$515$	168.165i	0.326533i
$516$	0	0
$517$	−753.806	−1.45804
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 633.968i	− 1.21683i	−0.793619	−	0.608415i	$-0.791806\pi$
$521$	− 633.968i	− 1.21683i	0.793619	−	0.608415i	$-0.208194\pi$
$522$	0	0
$523$	−42.6893	−0.0816240	−0.0408120	−	0.999167i	$-0.512994\pi$
$523$	−42.6893	−0.0816240	−0.0408120	+	0.999167i	$0.512994\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1072.02i	− 2.03419i
$528$	0	0
$529$	527.661	0.997469
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 220.620i	− 0.413922i
$534$	0	0
$535$	−45.9575	−0.0859019
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 466.660i	− 0.865788i
$540$	0	0
$541$	−585.520	−1.08229	−0.541146	−	0.840929i	$-0.682009\pi$
$541$	−585.520	−1.08229	−0.541146	+	0.840929i	$0.682009\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 37.2129i	− 0.0682806i
$546$	0	0
$547$	−859.622	−1.57152	−0.785761	−	0.618531i	$-0.787728\pi$
$547$	−859.622	−1.57152	−0.785761	+	0.618531i	$0.787728\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	27.2806i	0.0495111i
$552$	0	0
$553$	261.518	0.472908
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 394.616i	− 0.708467i	−0.935157	−	0.354234i	$-0.884742\pi$
$557$	− 394.616i	− 0.708467i	0.935157	−	0.354234i	$-0.115258\pi$
$558$	0	0
$559$	337.672	0.604065
$560$	0	0
$561$	0	0
$562$	0	0
$563$	433.763i	0.770449i	0.922823	+	0.385225i	$0.125876\pi$
$563$	433.763i	0.770449i	−0.922823	+	0.385225i	$0.874124\pi$
$564$	0	0
$565$	90.0695	0.159415
$566$	0	0
$567$	0	0
$568$	0	0
$569$	216.219i	0.379999i	0.981784	+	0.189999i	$0.0608486\pi$
$569$	216.219i	0.379999i	−0.981784	+	0.189999i	$0.939151\pi$
$570$	0	0
$571$	1050.22	1.83927	0.919633	−	0.392778i	$-0.128486\pi$
$571$	1050.22	1.83927	0.919633	+	0.392778i	$0.128486\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.1256i	0.0489140i
$576$	0	0
$577$	235.000	0.407279	0.203640	−	0.979046i	$-0.434723\pi$
$577$	235.000	0.407279	0.203640	+	0.979046i	$0.434723\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 156.467i	− 0.269306i
$582$	0	0
$583$	−509.064	−0.873181
$584$	0	0
$585$	0	0
$586$	0	0
$587$	773.013i	1.31689i	0.752630	+	0.658443i	$0.228785\pi$
$587$	773.013i	1.31689i	−0.752630	+	0.658443i	$0.771215\pi$
$588$	0	0
$589$	77.1499	0.130985
$590$	0	0
$591$	0	0
$592$	0	0
$593$	504.381i	0.850557i	0.905062	+	0.425279i	$0.139824\pi$
$593$	504.381i	0.850557i	−0.905062	+	0.425279i	$0.860176\pi$
$594$	0	0
$595$	−43.7989	−0.0736117
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 43.7974i	− 0.0731176i	−0.999332	−	0.0365588i	$-0.988360\pi$
$599$	− 43.7974i	− 0.0731176i	0.999332	−	0.0365588i	$-0.0116396\pi$
$600$	0	0
$601$	−9.79174	−0.0162924	−0.00814621	−	0.999967i	$-0.502593\pi$
$601$	−9.79174	−0.0162924	−0.00814621	+	0.999967i	$0.502593\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.04112i	0.00172086i
$606$	0	0
$607$	351.448	0.578991	0.289496	−	0.957179i	$-0.406512\pi$
$607$	351.448	0.578991	0.289496	+	0.957179i	$0.406512\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 601.298i	− 0.984122i
$612$	0	0
$613$	733.118	1.19595	0.597976	−	0.801514i	$-0.295972\pi$
$613$	733.118	1.19595	0.597976	+	0.801514i	$0.295972\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 643.528i	− 1.04299i	−0.853253	−	0.521497i	$-0.825374\pi$
$617$	− 643.528i	− 1.04299i	0.853253	−	0.521497i	$-0.174626\pi$
$618$	0	0
$619$	248.639	0.401678	0.200839	−	0.979624i	$-0.435633\pi$
$619$	248.639	0.401678	0.200839	+	0.979624i	$0.435633\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.93418i	0.00792003i
$624$	0	0
$625$	573.475	0.917560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	165.648i	0.263351i
$630$	0	0
$631$	−479.055	−0.759200	−0.379600	−	0.925151i	$-0.623938\pi$
$631$	−479.055	−0.759200	−0.379600	+	0.925151i	$0.623938\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 143.375i	− 0.225787i
$636$	0	0
$637$	372.246	0.584374
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1177.76i	− 1.83738i	−0.394981	−	0.918689i	$-0.629249\pi$
$641$	− 1177.76i	− 1.83738i	0.394981	−	0.918689i	$-0.370751\pi$
$642$	0	0
$643$	176.171	0.273983	0.136992	−	0.990572i	$-0.456257\pi$
$643$	176.171	0.273983	0.136992	+	0.990572i	$0.456257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	317.529i	0.490771i	0.969426	+	0.245386i	$0.0789146\pi$
$647$	317.529i	0.490771i	−0.969426	+	0.245386i	$0.921085\pi$
$648$	0	0
$649$	1126.25	1.73536
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1020.62i	1.56298i	0.623921	+	0.781488i	$0.285539\pi$
$653$	1020.62i	1.56298i	−0.623921	+	0.781488i	$0.714461\pi$
$654$	0	0
$655$	73.8641	0.112770
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 147.334i	− 0.223572i	−0.993732	−	0.111786i	$-0.964343\pi$
$659$	− 147.334i	− 0.223572i	0.993732	−	0.111786i	$-0.0356571\pi$
$660$	0	0
$661$	773.957	1.17089	0.585444	−	0.810713i	$-0.300920\pi$
$661$	773.957	1.17089	0.585444	+	0.810713i	$0.300920\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.15208i	− 0.00473997i
$666$	0	0
$667$	−21.0235	−0.0315195
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 966.599i	− 1.44054i
$672$	0	0
$673$	−125.473	−0.186438	−0.0932189	−	0.995646i	$-0.529716\pi$
$673$	−125.473	−0.186438	−0.0932189	+	0.995646i	$0.529716\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1101.31i	1.62675i	0.581742	+	0.813374i	$0.302371\pi$
$677$	1101.31i	1.62675i	−0.581742	+	0.813374i	$0.697629\pi$
$678$	0	0
$679$	298.574	0.439726
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 469.992i	− 0.688128i	−0.938946	−	0.344064i	$-0.888196\pi$
$683$	− 469.992i	− 0.688128i	0.938946	−	0.344064i	$-0.111804\pi$
$684$	0	0
$685$	74.6302	0.108949
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 406.072i	− 0.589364i
$690$	0	0
$691$	−527.739	−0.763733	−0.381866	−	0.924218i	$-0.624719\pi$
$691$	−527.739	−0.763733	−0.381866	+	0.924218i	$0.624719\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 124.561i	− 0.179224i
$696$	0	0
$697$	527.315	0.756549
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1162.38i	1.65818i	0.559116	+	0.829089i	$0.311141\pi$
$701$	1162.38i	1.65818i	−0.559116	+	0.829089i	$0.688859\pi$
$702$	0	0
$703$	−11.9212	−0.0169576
$704$	0	0
$705$	0	0
$706$	0	0
$707$	140.430i	0.198628i
$708$	0	0
$709$	200.058	0.282169	0.141085	−	0.989998i	$-0.454941\pi$
$709$	200.058	0.282169	0.141085	+	0.989998i	$0.454941\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	59.4547i	0.0833867i
$714$	0	0
$715$	79.5428	0.111249
$716$	0	0
$717$	0	0
$718$	0	0
$719$	447.907i	0.622958i	0.950253	+	0.311479i	$0.100824\pi$
$719$	447.907i	0.622958i	−0.950253	+	0.311479i	$0.899176\pi$
$720$	0	0
$721$	−509.112	−0.706120
$722$	0	0
$723$	0	0
$724$	0	0
$725$	441.623i	0.609136i
$726$	0	0
$727$	−507.475	−0.698039	−0.349020	−	0.937115i	$-0.613485\pi$
$727$	−507.475	−0.698039	−0.349020	+	0.937115i	$0.613485\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	807.086i	1.10408i
$732$	0	0
$733$	1091.08	1.48852	0.744260	−	0.667890i	$-0.232802\pi$
$733$	1091.08	1.48852	0.744260	+	0.667890i	$0.232802\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	247.732i	0.336136i
$738$	0	0
$739$	−114.753	−0.155281	−0.0776406	−	0.996981i	$-0.524739\pi$
$739$	−114.753	−0.155281	−0.0776406	+	0.996981i	$0.524739\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	613.178i	0.825273i	0.910896	+	0.412637i	$0.135392\pi$
$743$	613.178i	0.825273i	−0.910896	+	0.412637i	$0.864608\pi$
$744$	0	0
$745$	−116.646	−0.156572
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 139.135i	− 0.185761i
$750$	0	0
$751$	−387.889	−0.516497	−0.258248	−	0.966079i	$-0.583145\pi$
$751$	−387.889	−0.516497	−0.258248	+	0.966079i	$0.583145\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 120.854i	− 0.160071i
$756$	0	0
$757$	732.340	0.967424	0.483712	−	0.875227i	$-0.339288\pi$
$757$	732.340	0.967424	0.483712	+	0.875227i	$0.339288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 890.685i	− 1.17041i	−0.810884	−	0.585207i	$-0.801013\pi$
$761$	− 890.685i	− 1.17041i	0.810884	−	0.585207i	$-0.198987\pi$
$762$	0	0
$763$	112.661	0.147655
$764$	0	0
$765$	0	0
$766$	0	0
$767$	898.388i	1.17130i
$768$	0	0
$769$	488.712	0.635516	0.317758	−	0.948172i	$-0.397070\pi$
$769$	488.712	0.635516	0.317758	+	0.948172i	$0.397070\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 579.450i	− 0.749612i	−0.927103	−	0.374806i	$-0.877709\pi$
$773$	− 579.450i	− 0.749612i	0.927103	−	0.374806i	$-0.122291\pi$
$774$	0	0
$775$	1248.92	1.61150
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.9493i	0.0487154i
$780$	0	0
$781$	1140.88	1.46079
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 205.746i	− 0.262097i
$786$	0	0
$787$	175.881	0.223483	0.111741	−	0.993737i	$-0.464357\pi$
$787$	175.881	0.223483	0.111741	+	0.993737i	$0.464357\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	272.682i	0.344731i
$792$	0	0
$793$	771.040	0.972308
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 459.082i	− 0.576012i	−0.957629	−	0.288006i	$-0.907008\pi$
$797$	− 459.082i	− 0.576012i	0.957629	−	0.288006i	$-0.0929924\pi$
$798$	0	0
$799$	1437.19	1.79874
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 823.042i	− 1.02496i
$804$	0	0
$805$	2.42912	0.00301754
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 89.1339i	− 0.110178i	−0.998481	−	0.0550890i	$-0.982456\pi$
$809$	− 89.1339i	− 0.110178i	0.998481	−	0.0550890i	$-0.0175442\pi$
$810$	0	0
$811$	−625.256	−0.770969	−0.385484	−	0.922714i	$-0.625966\pi$
$811$	−625.256	−0.770969	−0.385484	+	0.922714i	$0.625966\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	157.668i	0.193458i
$816$	0	0
$817$	−58.0836	−0.0710938
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 722.140i	− 0.879586i	−0.898099	−	0.439793i	$-0.855052\pi$
$821$	− 722.140i	− 0.879586i	0.898099	−	0.439793i	$-0.144948\pi$
$822$	0	0
$823$	−299.067	−0.363386	−0.181693	−	0.983355i	$-0.558158\pi$
$823$	−299.067	−0.363386	−0.181693	+	0.983355i	$0.558158\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1178.17i	1.42463i	0.701858	+	0.712317i	$0.252354\pi$
$827$	1178.17i	1.42463i	−0.701858	+	0.712317i	$0.747646\pi$
$828$	0	0
$829$	−451.760	−0.544946	−0.272473	−	0.962163i	$-0.587841\pi$
$829$	−451.760	−0.544946	−0.272473	+	0.962163i	$0.587841\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	889.723i	1.06810i
$834$	0	0
$835$	−192.849	−0.230956
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 317.246i	− 0.378124i	−0.981965	−	0.189062i	$-0.939455\pi$
$839$	− 317.246i	− 0.378124i	0.981965	−	0.189062i	$-0.0605448\pi$
$840$	0	0
$841$	510.892	0.607481
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 77.2788i	− 0.0914542i
$846$	0	0
$847$	−3.15196	−0.00372132
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 9.18694i	− 0.0107955i
$852$	0	0
$853$	−111.847	−0.131122	−0.0655609	−	0.997849i	$-0.520884\pi$
$853$	−111.847	−0.131122	−0.0655609	+	0.997849i	$0.520884\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 602.799i	− 0.703383i	−0.936116	−	0.351692i	$-0.885607\pi$
$857$	− 602.799i	− 0.703383i	0.936116	−	0.351692i	$-0.114393\pi$
$858$	0	0
$859$	−1298.35	−1.51147	−0.755733	−	0.654880i	$-0.772719\pi$
$859$	−1298.35	−1.51147	−0.755733	+	0.654880i	$0.772719\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1332.38i	− 1.54390i	−0.635685	−	0.771949i	$-0.719282\pi$
$863$	− 1332.38i	− 1.54390i	0.635685	−	0.771949i	$-0.280718\pi$
$864$	0	0
$865$	−216.255	−0.250006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1135.18i	− 1.30630i
$870$	0	0
$871$	−197.612	−0.226879
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 103.509i	− 0.118296i
$876$	0	0
$877$	−814.823	−0.929103	−0.464551	−	0.885546i	$-0.653784\pi$
$877$	−814.823	−0.929103	−0.464551	+	0.885546i	$0.653784\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	86.4877i	0.0981700i	0.998795	+	0.0490850i	$0.0156305\pi$
$881$	86.4877i	0.0981700i	−0.998795	+	0.0490850i	$0.984369\pi$
$882$	0	0
$883$	367.217	0.415875	0.207937	−	0.978142i	$-0.433325\pi$
$883$	367.217	0.415875	0.207937	+	0.978142i	$0.433325\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	678.335i	0.764752i	0.924007	+	0.382376i	$0.124894\pi$
$887$	678.335i	0.764752i	−0.924007	+	0.382376i	$0.875106\pi$
$888$	0	0
$889$	434.061	0.488258
$890$	0	0
$891$	0	0
$892$	0	0
$893$	103.430i	0.115824i
$894$	0	0
$895$	−183.495	−0.205022
$896$	0	0
$897$	0	0
$898$	0	0
$899$	933.550i	1.03843i
$900$	0	0
$901$	970.571	1.07722
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 238.314i	− 0.263330i
$906$	0	0
$907$	862.742	0.951204	0.475602	−	0.879661i	$-0.342230\pi$
$907$	862.742	0.951204	0.475602	+	0.879661i	$0.342230\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1163.44i	− 1.27710i	−0.769581	−	0.638549i	$-0.779535\pi$
$911$	− 1163.44i	− 1.27710i	0.769581	−	0.638549i	$-0.220465\pi$
$912$	0	0
$913$	−679.179	−0.743898
$914$	0	0
$915$	0	0
$916$	0	0
$917$	223.621i	0.243861i
$918$	0	0
$919$	−832.360	−0.905724	−0.452862	−	0.891581i	$-0.649597\pi$
$919$	−832.360	−0.905724	−0.452862	+	0.891581i	$0.649597\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	910.059i	0.985979i
$924$	0	0
$925$	−192.982	−0.208630
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1778.36i	1.91427i	0.289643	+	0.957135i	$0.406464\pi$
$929$	1778.36i	1.91427i	−0.289643	+	0.957135i	$0.593536\pi$
$930$	0	0
$931$	−64.0308	−0.0687764
$932$	0	0
$933$	0	0
$934$	0	0
$935$	190.119i	0.203336i
$936$	0	0
$937$	432.361	0.461431	0.230715	−	0.973021i	$-0.425893\pi$
$937$	432.361	0.461431	0.230715	+	0.973021i	$0.425893\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1756.76i	1.86690i	0.358703	+	0.933452i	$0.383219\pi$
$941$	1756.76i	1.86690i	−0.358703	+	0.933452i	$0.616781\pi$
$942$	0	0
$943$	−29.2452	−0.0310130
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1132.48i	− 1.19586i	−0.801549	−	0.597929i	$-0.795990\pi$
$947$	− 1132.48i	− 1.19586i	0.801549	−	0.597929i	$-0.204010\pi$
$948$	0	0
$949$	656.526	0.691809
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1509.90i	− 1.58436i	−0.610286	−	0.792181i	$-0.708945\pi$
$953$	− 1509.90i	− 1.58436i	0.610286	−	0.792181i	$-0.291055\pi$
$954$	0	0
$955$	1.31530	0.00137728
$956$	0	0
$957$	0	0
$958$	0	0
$959$	225.940i	0.235600i
$960$	0	0
$961$	1679.09	1.74723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 193.301i	− 0.200312i
$966$	0	0
$967$	489.980	0.506701	0.253350	−	0.967375i	$-0.418467\pi$
$967$	489.980	0.506701	0.253350	+	0.967375i	$0.418467\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	253.051i	0.260608i	0.991474	+	0.130304i	$0.0415954\pi$
$971$	253.051i	0.260608i	−0.991474	+	0.130304i	$0.958405\pi$
$972$	0	0
$973$	377.103	0.387567
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1625.27i	1.66353i	0.555127	+	0.831765i	$0.312670\pi$
$977$	1625.27i	1.66353i	−0.555127	+	0.831765i	$0.687330\pi$
$978$	0	0
$979$	21.4179	0.0218773
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1162.97i	− 1.18308i	−0.806276	−	0.591540i	$-0.798520\pi$
$983$	− 1162.97i	− 1.18308i	0.806276	−	0.591540i	$-0.201480\pi$
$984$	0	0
$985$	178.211	0.180925
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 44.7615i	− 0.0452594i
$990$	0	0
$991$	−1411.24	−1.42405	−0.712027	−	0.702152i	$-0.752223\pi$
$991$	−1411.24	−1.42405	−0.712027	+	0.702152i	$0.752223\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.5599i	0.0216683i
$996$	0	0
$997$	1123.23	1.12661	0.563304	−	0.826250i	$-0.309530\pi$
$997$	1123.23	1.12661	0.563304	+	0.826250i	$0.309530\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.3.e.i.161.12		24
3.2	odd	2	inner	2592.3.e.i.161.11		24
4.3	odd	2	inner	2592.3.e.i.161.14		24
9.2	odd	6		864.3.q.a.449.5		24
9.4	even	3		864.3.q.a.737.5		24
9.5	odd	6		288.3.q.b.65.8	yes	24
9.7	even	3		288.3.q.b.257.8	yes	24
12.11	even	2	inner	2592.3.e.i.161.13		24
36.7	odd	6		288.3.q.b.257.5	yes	24
36.11	even	6		864.3.q.a.449.6		24
36.23	even	6		288.3.q.b.65.5	✓	24
36.31	odd	6		864.3.q.a.737.6		24
72.5	odd	6		576.3.q.l.65.5		24
72.11	even	6		1728.3.q.k.449.8		24
72.13	even	6		1728.3.q.k.1601.7		24
72.29	odd	6		1728.3.q.k.449.7		24
72.43	odd	6		576.3.q.l.257.8		24
72.59	even	6		576.3.q.l.65.8		24
72.61	even	6		576.3.q.l.257.5		24
72.67	odd	6		1728.3.q.k.1601.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.5	✓	24	36.23	even	6
288.3.q.b.65.8	yes	24	9.5	odd	6
288.3.q.b.257.5	yes	24	36.7	odd	6
288.3.q.b.257.8	yes	24	9.7	even	3
576.3.q.l.65.5		24	72.5	odd	6
576.3.q.l.65.8		24	72.59	even	6
576.3.q.l.257.5		24	72.61	even	6
576.3.q.l.257.8		24	72.43	odd	6
864.3.q.a.449.5		24	9.2	odd	6
864.3.q.a.449.6		24	36.11	even	6
864.3.q.a.737.5		24	9.4	even	3
864.3.q.a.737.6		24	36.31	odd	6
1728.3.q.k.449.7		24	72.29	odd	6
1728.3.q.k.449.8		24	72.11	even	6
1728.3.q.k.1601.7		24	72.13	even	6
1728.3.q.k.1601.8		24	72.67	odd	6
2592.3.e.i.161.11		24	3.2	odd	2	inner
2592.3.e.i.161.12		24	1.1	even	1	trivial
2592.3.e.i.161.13		24	12.11	even	2	inner
2592.3.e.i.161.14		24	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2592.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.832715i q^{5} -2.52101 q^{7} +O(q^{10})\)	\(q+0.832715i q^{5} -2.52101 q^{7} +10.9430i q^{11} -8.72906 q^{13} -20.8637i q^{17} +1.50150 q^{19} +1.15712i q^{23} +24.3066 q^{25} +18.1689i q^{29} +51.3818 q^{31} -2.09929i q^{35} -7.93951 q^{37} +25.2742i q^{41} -38.6837 q^{43} +68.8846i q^{47} -42.6445 q^{49} +46.5195i q^{53} -9.11241 q^{55} -102.919i q^{59} -88.3302 q^{61} -7.26882i q^{65} +22.6384 q^{67} -104.256i q^{71} -75.2115 q^{73} -27.5875i q^{77} -103.735 q^{79} +62.0650i q^{83} +17.3735 q^{85} -1.95722i q^{89} +22.0061 q^{91} +1.25032i q^{95} -118.434 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 120 q^{25} + 72 q^{49} + 192 q^{61} + 24 q^{73} - 192 q^{85} + 264 q^{97}+O(q^{100}) \) 24 * q - 120 * q^25 + 72 * q^49 + 192 * q^61 + 24 * q^73 - 192 * q^85 + 264 * q^97

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