LMFDB - Embedded newform 1188.3.e.c.485.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1188,3,Mod(485,1188)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1188, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1188.485"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1188 = 2^{2} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1188.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3706554060$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 46x^{6} + 637x^{4} + 2880x^{2} + 2916 $ x^8 + 46x^6 + 637x^4 + 2880*x^2 + 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.8
Root			$1.18638i$ of defining polynomial
Character	$\chi$	$=$	1188.485
Dual form			1188.3.e.c.485.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.47276i q^{5} +2.36935 q^{7} -3.31662i q^{11} -11.3098 q^{13} -24.1204i q^{17} -27.4239 q^{19} -28.7633i q^{23} -16.8966 q^{25} +2.39074i q^{29} -46.9354 q^{31} +15.3363i q^{35} +69.0057 q^{37} +13.5656i q^{41} +55.5865 q^{43} -39.7402i q^{47} -43.3862 q^{49} -99.2946i q^{53} +21.4677 q^{55} +43.6773i q^{59} -8.08001 q^{61} -73.2056i q^{65} -7.40346 q^{67} +60.2234i q^{71} +104.544 q^{73} -7.85826i q^{77} -56.7966 q^{79} -153.111i q^{83} +156.126 q^{85} +40.0175i q^{89} -26.7969 q^{91} -177.508i q^{95} +54.2828 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 20 q^{13} + 16 q^{19} - 12 q^{25} - 32 q^{31} - 20 q^{37} - 48 q^{43} + 84 q^{49} + 36 q^{61} + 16 q^{67} + 200 q^{73} - 28 q^{79} - 60 q^{85} - 228 q^{91} - 120 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 20 * q^13 + 16 * q^19 - 12 * q^25 - 32 * q^31 - 20 * q^37 - 48 * q^43 + 84 * q^49 + 36 * q^61 + 16 * q^67 + 200 * q^73 - 28 * q^79 - 60 * q^85 - 228 * q^91 - 120 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1188\mathbb{Z}\right)^\times$.

$n$	$353$	$541$	$595$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	6.47276i	1.29455i	0.762256	+	0.647276i	$0.224092\pi$
$5$	6.47276i	1.29455i	−0.762256	+	0.647276i	$0.775908\pi$
$6$	0	0
$7$	2.36935	0.338479	0.169240	−	0.985575i	$-0.445869\pi$
$7$	2.36935	0.338479	0.169240	+	0.985575i	$0.445869\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	0	0
$13$	−11.3098	−0.869984	−0.434992	−	0.900434i	$-0.643249\pi$
$13$	−11.3098	−0.869984	−0.434992	+	0.900434i	$0.643249\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 24.1204i	− 1.41885i	−0.704782	−	0.709424i	$-0.748955\pi$
$17$	− 24.1204i	− 1.41885i	0.704782	−	0.709424i	$-0.251045\pi$
$18$	0	0
$19$	−27.4239	−1.44336	−0.721682	−	0.692225i	$-0.756631\pi$
$19$	−27.4239	−1.44336	−0.721682	+	0.692225i	$0.756631\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 28.7633i	− 1.25058i	−0.780394	−	0.625288i	$-0.784981\pi$
$23$	− 28.7633i	− 1.25058i	0.780394	−	0.625288i	$-0.215019\pi$
$24$	0	0
$25$	−16.8966	−0.675865
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.39074i	0.0824394i	0.999150	+	0.0412197i	$0.0131244\pi$
$29$	2.39074i	0.0824394i	−0.999150	+	0.0412197i	$0.986876\pi$
$30$	0	0
$31$	−46.9354	−1.51405	−0.757023	−	0.653388i	$-0.773347\pi$
$31$	−46.9354	−1.51405	−0.757023	+	0.653388i	$0.773347\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.3363i	0.438179i
$36$	0	0
$37$	69.0057	1.86502	0.932510	−	0.361144i	$-0.117614\pi$
$37$	69.0057	1.86502	0.932510	+	0.361144i	$0.117614\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	13.5656i	0.330869i	0.986221	+	0.165434i	$0.0529026\pi$
$41$	13.5656i	0.330869i	−0.986221	+	0.165434i	$0.947097\pi$
$42$	0	0
$43$	55.5865	1.29271	0.646355	−	0.763037i	$-0.276293\pi$
$43$	55.5865	1.29271	0.646355	+	0.763037i	$0.276293\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 39.7402i	− 0.845537i	−0.906238	−	0.422768i	$-0.861058\pi$
$47$	− 39.7402i	− 0.845537i	0.906238	−	0.422768i	$-0.138942\pi$
$48$	0	0
$49$	−43.3862	−0.885432
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 99.2946i	− 1.87348i	−0.350023	−	0.936741i	$-0.613826\pi$
$53$	− 99.2946i	− 1.87348i	0.350023	−	0.936741i	$-0.386174\pi$
$54$	0	0
$55$	21.4677	0.390322
$56$	0	0
$57$	0	0
$58$	0	0
$59$	43.6773i	0.740294i	0.928973	+	0.370147i	$0.120693\pi$
$59$	43.6773i	0.740294i	−0.928973	+	0.370147i	$0.879307\pi$
$60$	0	0
$61$	−8.08001	−0.132459	−0.0662296	−	0.997804i	$-0.521097\pi$
$61$	−8.08001	−0.132459	−0.0662296	+	0.997804i	$0.521097\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 73.2056i	− 1.12624i
$66$	0	0
$67$	−7.40346	−0.110499	−0.0552497	−	0.998473i	$-0.517595\pi$
$67$	−7.40346	−0.110499	−0.0552497	+	0.998473i	$0.517595\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	60.2234i	0.848217i	0.905611	+	0.424108i	$0.139412\pi$
$71$	60.2234i	0.848217i	−0.905611	+	0.424108i	$0.860588\pi$
$72$	0	0
$73$	104.544	1.43211	0.716055	−	0.698044i	$-0.245946\pi$
$73$	104.544	1.43211	0.716055	+	0.698044i	$0.245946\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.85826i	− 0.102055i
$78$	0	0
$79$	−56.7966	−0.718944	−0.359472	−	0.933156i	$-0.617043\pi$
$79$	−56.7966	−0.718944	−0.359472	+	0.933156i	$0.617043\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 153.111i	− 1.84471i	−0.386346	−	0.922354i	$-0.626263\pi$
$83$	− 153.111i	− 1.84471i	0.386346	−	0.922354i	$-0.373737\pi$
$84$	0	0
$85$	156.126	1.83677
$86$	0	0
$87$	0	0
$88$	0	0
$89$	40.0175i	0.449635i	0.974401	+	0.224817i	$0.0721786\pi$
$89$	40.0175i	0.449635i	−0.974401	+	0.224817i	$0.927821\pi$
$90$	0	0
$91$	−26.7969	−0.294472
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 177.508i	− 1.86851i
$96$	0	0
$97$	54.2828	0.559616	0.279808	−	0.960056i	$-0.409729\pi$
$97$	54.2828	0.559616	0.279808	+	0.960056i	$0.409729\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 39.6119i	− 0.392197i	−0.980584	−	0.196098i	$-0.937173\pi$
$101$	− 39.6119i	− 0.392197i	0.980584	−	0.196098i	$-0.0628272\pi$
$102$	0	0
$103$	−153.093	−1.48634	−0.743169	−	0.669104i	$-0.766678\pi$
$103$	−153.093	−1.48634	−0.743169	+	0.669104i	$0.766678\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 176.770i	− 1.65205i	−0.563631	−	0.826027i	$-0.690596\pi$
$107$	− 176.770i	− 1.65205i	0.563631	−	0.826027i	$-0.309404\pi$
$108$	0	0
$109$	1.96578	0.0180346	0.00901732	−	0.999959i	$-0.497130\pi$
$109$	1.96578	0.0180346	0.00901732	+	0.999959i	$0.497130\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 29.4916i	− 0.260988i	−0.991449	−	0.130494i	$-0.958344\pi$
$113$	− 29.4916i	− 0.260988i	0.991449	−	0.130494i	$-0.0416563\pi$
$114$	0	0
$115$	186.178	1.61894
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 57.1498i	− 0.480250i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	52.4512i	0.419609i
$126$	0	0
$127$	−162.789	−1.28180	−0.640902	−	0.767622i	$-0.721440\pi$
$127$	−162.789	−1.28180	−0.640902	+	0.767622i	$0.721440\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15.7976i	− 0.120593i	−0.998181	−	0.0602964i	$-0.980795\pi$
$131$	− 15.7976i	− 0.120593i	0.998181	−	0.0602964i	$-0.0192046\pi$
$132$	0	0
$133$	−64.9770	−0.488549
$134$	0	0
$135$	0	0
$136$	0	0
$137$	123.988i	0.905022i	0.891759	+	0.452511i	$0.149472\pi$
$137$	123.988i	0.905022i	−0.891759	+	0.452511i	$0.850528\pi$
$138$	0	0
$139$	56.1505	0.403961	0.201980	−	0.979390i	$-0.435262\pi$
$139$	56.1505	0.403961	0.201980	+	0.979390i	$0.435262\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	37.5103i	0.262310i
$144$	0	0
$145$	−15.4747	−0.106722
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 200.205i	− 1.34365i	−0.740708	−	0.671827i	$-0.765510\pi$
$149$	− 200.205i	− 1.34365i	0.740708	−	0.671827i	$-0.234490\pi$
$150$	0	0
$151$	−116.060	−0.768609	−0.384305	−	0.923206i	$-0.625559\pi$
$151$	−116.060	−0.768609	−0.384305	+	0.923206i	$0.625559\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 303.802i	− 1.96001i
$156$	0	0
$157$	−209.506	−1.33443	−0.667215	−	0.744865i	$-0.732514\pi$
$157$	−209.506	−1.33443	−0.667215	+	0.744865i	$0.732514\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 68.1503i	− 0.423294i
$162$	0	0
$163$	11.4638	0.0703298	0.0351649	−	0.999382i	$-0.488804\pi$
$163$	11.4638	0.0703298	0.0351649	+	0.999382i	$0.488804\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	291.934i	1.74811i	0.485831	+	0.874053i	$0.338517\pi$
$167$	291.934i	1.74811i	−0.485831	+	0.874053i	$0.661483\pi$
$168$	0	0
$169$	−41.0886	−0.243128
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 270.241i	− 1.56208i	−0.624478	−	0.781042i	$-0.714688\pi$
$173$	− 270.241i	− 1.56208i	0.624478	−	0.781042i	$-0.285312\pi$
$174$	0	0
$175$	−40.0341	−0.228766
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 146.964i	− 0.821029i	−0.911854	−	0.410515i	$-0.865349\pi$
$179$	− 146.964i	− 0.821029i	0.911854	−	0.410515i	$-0.134651\pi$
$180$	0	0
$181$	−69.9537	−0.386484	−0.193242	−	0.981151i	$-0.561900\pi$
$181$	−69.9537	−0.386484	−0.193242	+	0.981151i	$0.561900\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	446.658i	2.41437i
$186$	0	0
$187$	−79.9983	−0.427799
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 306.674i	− 1.60562i	−0.596233	−	0.802812i	$-0.703336\pi$
$191$	− 306.674i	− 1.60562i	0.596233	−	0.802812i	$-0.296664\pi$
$192$	0	0
$193$	232.543	1.20489	0.602444	−	0.798161i	$-0.294194\pi$
$193$	232.543	1.20489	0.602444	+	0.798161i	$0.294194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	194.742i	0.988536i	0.869309	+	0.494268i	$0.164564\pi$
$197$	194.742i	0.988536i	−0.869309	+	0.494268i	$0.835436\pi$
$198$	0	0
$199$	−110.029	−0.552909	−0.276454	−	0.961027i	$-0.589159\pi$
$199$	−110.029	−0.552909	−0.276454	+	0.961027i	$0.589159\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.66451i	0.0279040i
$204$	0	0
$205$	−87.8070	−0.428327
$206$	0	0
$207$	0	0
$208$	0	0
$209$	90.9548i	0.435191i
$210$	0	0
$211$	175.701	0.832706	0.416353	−	0.909203i	$-0.363308\pi$
$211$	175.701	0.832706	0.416353	+	0.909203i	$0.363308\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	359.798i	1.67348i
$216$	0	0
$217$	−111.207	−0.512473
$218$	0	0
$219$	0	0
$220$	0	0
$221$	272.797i	1.23437i
$222$	0	0
$223$	192.105	0.861458	0.430729	−	0.902481i	$-0.358256\pi$
$223$	192.105	0.861458	0.430729	+	0.902481i	$0.358256\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 218.719i	− 0.963522i	−0.876303	−	0.481761i	$-0.839997\pi$
$227$	− 218.719i	− 0.963522i	0.876303	−	0.481761i	$-0.160003\pi$
$228$	0	0
$229$	−83.0039	−0.362462	−0.181231	−	0.983441i	$-0.558008\pi$
$229$	−83.0039	−0.362462	−0.181231	+	0.983441i	$0.558008\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 62.3899i	− 0.267768i	−0.990997	−	0.133884i	$-0.957255\pi$
$233$	− 62.3899i	− 0.267768i	0.990997	−	0.133884i	$-0.0427449\pi$
$234$	0	0
$235$	257.229	1.09459
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 248.939i	− 1.04159i	−0.853683	−	0.520793i	$-0.825636\pi$
$239$	− 248.939i	− 1.04159i	0.853683	−	0.520793i	$-0.174364\pi$
$240$	0	0
$241$	−391.340	−1.62382	−0.811909	−	0.583784i	$-0.801572\pi$
$241$	−391.340	−1.62382	−0.811909	+	0.583784i	$0.801572\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 280.828i	− 1.14624i
$246$	0	0
$247$	310.159	1.25570
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 300.168i	− 1.19589i	−0.801538	−	0.597944i	$-0.795984\pi$
$251$	− 300.168i	− 1.19589i	0.801538	−	0.597944i	$-0.204016\pi$
$252$	0	0
$253$	−95.3969	−0.377063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	264.747i	1.03015i	0.857147	+	0.515073i	$0.172235\pi$
$257$	264.747i	1.03015i	−0.857147	+	0.515073i	$0.827765\pi$
$258$	0	0
$259$	163.499	0.631271
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.9887i	0.0607937i	0.999538	+	0.0303968i	$0.00967711\pi$
$263$	15.9887i	0.0607937i	−0.999538	+	0.0303968i	$0.990323\pi$
$264$	0	0
$265$	642.710	2.42532
$266$	0	0
$267$	0	0
$268$	0	0
$269$	510.478i	1.89769i	0.315748	+	0.948843i	$0.397745\pi$
$269$	510.478i	1.89769i	−0.315748	+	0.948843i	$0.602255\pi$
$270$	0	0
$271$	192.719	0.711139	0.355569	−	0.934650i	$-0.384287\pi$
$271$	192.719	0.711139	0.355569	+	0.934650i	$0.384287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	56.0398i	0.203781i
$276$	0	0
$277$	75.1251	0.271210	0.135605	−	0.990763i	$-0.456702\pi$
$277$	75.1251	0.271210	0.135605	+	0.990763i	$0.456702\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 227.381i	− 0.809184i	−0.914497	−	0.404592i	$-0.867414\pi$
$281$	− 227.381i	− 0.809184i	0.914497	−	0.404592i	$-0.132586\pi$
$282$	0	0
$283$	6.80965	0.0240624	0.0120312	−	0.999928i	$-0.496170\pi$
$283$	6.80965	0.0240624	0.0120312	+	0.999928i	$0.496170\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	32.1418i	0.111992i
$288$	0	0
$289$	−292.794	−1.01313
$290$	0	0
$291$	0	0
$292$	0	0
$293$	234.435i	0.800120i	0.916489	+	0.400060i	$0.131011\pi$
$293$	234.435i	0.800120i	−0.916489	+	0.400060i	$0.868989\pi$
$294$	0	0
$295$	−282.713	−0.958349
$296$	0	0
$297$	0	0
$298$	0	0
$299$	325.306i	1.08798i
$300$	0	0
$301$	131.704	0.437555
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 52.3000i	− 0.171475i
$306$	0	0
$307$	−42.0857	−0.137087	−0.0685435	−	0.997648i	$-0.521835\pi$
$307$	−42.0857	−0.137087	−0.0685435	+	0.997648i	$0.521835\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 64.7703i	− 0.208265i	−0.994563	−	0.104132i	$-0.966793\pi$
$311$	− 64.7703i	− 0.208265i	0.994563	−	0.104132i	$-0.0332065\pi$
$312$	0	0
$313$	−212.875	−0.680112	−0.340056	−	0.940405i	$-0.610446\pi$
$313$	−212.875	−0.680112	−0.340056	+	0.940405i	$0.610446\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	336.207i	1.06059i	0.847813	+	0.530295i	$0.177919\pi$
$317$	336.207i	1.06059i	−0.847813	+	0.530295i	$0.822081\pi$
$318$	0	0
$319$	7.92919	0.0248564
$320$	0	0
$321$	0	0
$322$	0	0
$323$	661.476i	2.04791i
$324$	0	0
$325$	191.097	0.587992
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 94.1587i	− 0.286197i
$330$	0	0
$331$	248.741	0.751482	0.375741	−	0.926725i	$-0.377388\pi$
$331$	248.741	0.751482	0.375741	+	0.926725i	$0.377388\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 47.9208i	− 0.143047i
$336$	0	0
$337$	−244.817	−0.726460	−0.363230	−	0.931699i	$-0.618326\pi$
$337$	−244.817	−0.726460	−0.363230	+	0.931699i	$0.618326\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	155.667i	0.456502i
$342$	0	0
$343$	−218.896	−0.638179
$344$	0	0
$345$	0	0
$346$	0	0
$347$	231.000i	0.665706i	0.942979	+	0.332853i	$0.108011\pi$
$347$	231.000i	0.665706i	−0.942979	+	0.332853i	$0.891989\pi$
$348$	0	0
$349$	431.359	1.23598	0.617992	−	0.786184i	$-0.287946\pi$
$349$	431.359	1.23598	0.617992	+	0.786184i	$0.287946\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.1441i	− 0.0853940i	−0.999088	−	0.0426970i	$-0.986405\pi$
$353$	− 30.1441i	− 0.0853940i	0.999088	−	0.0426970i	$-0.0135950\pi$
$354$	0	0
$355$	−389.812	−1.09806
$356$	0	0
$357$	0	0
$358$	0	0
$359$	166.652i	0.464211i	0.972691	+	0.232106i	$0.0745615\pi$
$359$	166.652i	0.464211i	−0.972691	+	0.232106i	$0.925439\pi$
$360$	0	0
$361$	391.071	1.08330
$362$	0	0
$363$	0	0
$364$	0	0
$365$	676.688i	1.85394i
$366$	0	0
$367$	349.845	0.953257	0.476629	−	0.879105i	$-0.341859\pi$
$367$	349.845	0.953257	0.476629	+	0.879105i	$0.341859\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 235.264i	− 0.634135i
$372$	0	0
$373$	−305.960	−0.820268	−0.410134	−	0.912025i	$-0.634518\pi$
$373$	−305.960	−0.820268	−0.410134	+	0.912025i	$0.634518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 27.0388i	− 0.0717209i
$378$	0	0
$379$	−452.238	−1.19324	−0.596620	−	0.802524i	$-0.703490\pi$
$379$	−452.238	−1.19324	−0.596620	+	0.802524i	$0.703490\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	67.6757i	0.176699i	0.996090	+	0.0883495i	$0.0281592\pi$
$383$	67.6757i	0.176699i	−0.996090	+	0.0883495i	$0.971841\pi$
$384$	0	0
$385$	50.8646	0.132116
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 199.299i	− 0.512338i	−0.966632	−	0.256169i	$-0.917540\pi$
$389$	− 199.299i	− 0.512338i	0.966632	−	0.256169i	$-0.0824603\pi$
$390$	0	0
$391$	−693.781	−1.77438
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 367.631i	− 0.930711i
$396$	0	0
$397$	−548.910	−1.38264	−0.691322	−	0.722546i	$-0.742972\pi$
$397$	−548.910	−1.38264	−0.691322	+	0.722546i	$0.742972\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	576.087i	1.43663i	0.695720	+	0.718313i	$0.255085\pi$
$401$	576.087i	1.43663i	−0.695720	+	0.718313i	$0.744915\pi$
$402$	0	0
$403$	530.830	1.31720
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 228.866i	− 0.562325i
$408$	0	0
$409$	462.589	1.13102	0.565512	−	0.824740i	$-0.308679\pi$
$409$	462.589	1.13102	0.565512	+	0.824740i	$0.308679\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	103.487i	0.250574i
$414$	0	0
$415$	991.049	2.38807
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.2312i	0.0530578i	0.999648	+	0.0265289i	$0.00844540\pi$
$419$	22.2312i	0.0530578i	−0.999648	+	0.0265289i	$0.991555\pi$
$420$	0	0
$421$	−323.082	−0.767416	−0.383708	−	0.923454i	$-0.625353\pi$
$421$	−323.082	−0.767416	−0.383708	+	0.923454i	$0.625353\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	407.554i	0.958950i
$426$	0	0
$427$	−19.1444	−0.0448347
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 119.463i	− 0.277177i	−0.990350	−	0.138588i	$-0.955743\pi$
$431$	− 119.463i	− 0.277177i	0.990350	−	0.138588i	$-0.0442565\pi$
$432$	0	0
$433$	−3.24221	−0.00748778	−0.00374389	−	0.999993i	$-0.501192\pi$
$433$	−3.24221	−0.00748778	−0.00374389	+	0.999993i	$0.501192\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	788.801i	1.80504i
$438$	0	0
$439$	−594.548	−1.35432	−0.677162	−	0.735834i	$-0.736790\pi$
$439$	−594.548	−1.35432	−0.677162	+	0.735834i	$0.736790\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17.6351i	− 0.0398083i	−0.999802	−	0.0199041i	$-0.993664\pi$
$443$	− 17.6351i	− 0.0398083i	0.999802	−	0.0199041i	$-0.00633610\pi$
$444$	0	0
$445$	−259.024	−0.582076
$446$	0	0
$447$	0	0
$448$	0	0
$449$	477.445i	1.06335i	0.846948	+	0.531676i	$0.178438\pi$
$449$	477.445i	1.06335i	−0.846948	+	0.531676i	$0.821562\pi$
$450$	0	0
$451$	44.9921	0.0997607
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 173.450i	− 0.381209i
$456$	0	0
$457$	−784.832	−1.71736	−0.858678	−	0.512516i	$-0.828714\pi$
$457$	−784.832	−1.71736	−0.858678	+	0.512516i	$0.828714\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	368.778i	0.799952i	0.916526	+	0.399976i	$0.130982\pi$
$461$	368.778i	0.799952i	−0.916526	+	0.399976i	$0.869018\pi$
$462$	0	0
$463$	−854.206	−1.84494	−0.922469	−	0.386072i	$-0.873832\pi$
$463$	−854.206	−1.84494	−0.922469	+	0.386072i	$0.873832\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	69.8195i	0.149507i	0.997202	+	0.0747533i	$0.0238169\pi$
$467$	69.8195i	0.149507i	−0.997202	+	0.0747533i	$0.976183\pi$
$468$	0	0
$469$	−17.5414	−0.0374018
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 184.360i	− 0.389767i
$474$	0	0
$475$	463.372	0.975520
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 863.327i	− 1.80235i	−0.433452	−	0.901177i	$-0.642705\pi$
$479$	− 863.327i	− 1.80235i	0.433452	−	0.901177i	$-0.357295\pi$
$480$	0	0
$481$	−780.441	−1.62254
$482$	0	0
$483$	0	0
$484$	0	0
$485$	351.360i	0.724453i
$486$	0	0
$487$	−453.852	−0.931934	−0.465967	−	0.884802i	$-0.654293\pi$
$487$	−453.852	−0.931934	−0.465967	+	0.884802i	$0.654293\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 164.399i	− 0.334825i	−0.985887	−	0.167413i	$-0.946459\pi$
$491$	− 164.399i	− 0.334825i	0.985887	−	0.167413i	$-0.0535412\pi$
$492$	0	0
$493$	57.6657	0.116969
$494$	0	0
$495$	0	0
$496$	0	0
$497$	142.691i	0.287104i
$498$	0	0
$499$	426.028	0.853764	0.426882	−	0.904307i	$-0.359612\pi$
$499$	426.028	0.853764	0.426882	+	0.904307i	$0.359612\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	512.798i	1.01948i	0.860329	+	0.509740i	$0.170258\pi$
$503$	512.798i	1.01948i	−0.860329	+	0.509740i	$0.829742\pi$
$504$	0	0
$505$	256.398	0.507719
$506$	0	0
$507$	0	0
$508$	0	0
$509$	616.672i	1.21154i	0.795641	+	0.605769i	$0.207134\pi$
$509$	616.672i	1.21154i	−0.795641	+	0.605769i	$0.792866\pi$
$510$	0	0
$511$	247.702	0.484739
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 990.933i	− 1.92414i
$516$	0	0
$517$	−131.803	−0.254939
$518$	0	0
$519$	0	0
$520$	0	0
$521$	882.973i	1.69477i	0.530983	+	0.847383i	$0.321823\pi$
$521$	882.973i	1.69477i	−0.530983	+	0.847383i	$0.678177\pi$
$522$	0	0
$523$	173.680	0.332084	0.166042	−	0.986119i	$-0.446901\pi$
$523$	173.680	0.332084	0.166042	+	0.986119i	$0.446901\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1132.10i	2.14820i
$528$	0	0
$529$	−298.325	−0.563941
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 153.424i	− 0.287851i
$534$	0	0
$535$	1144.19	2.13867
$536$	0	0
$537$	0	0
$538$	0	0
$539$	143.896i	0.266968i
$540$	0	0
$541$	686.979	1.26983	0.634916	−	0.772581i	$-0.281035\pi$
$541$	686.979	1.26983	0.634916	+	0.772581i	$0.281035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.7240i	0.0233468i
$546$	0	0
$547$	226.292	0.413697	0.206848	−	0.978373i	$-0.433679\pi$
$547$	226.292	0.413697	0.206848	+	0.978373i	$0.433679\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 65.5635i	− 0.118990i
$552$	0	0
$553$	−134.571	−0.243348
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 972.179i	− 1.74538i	−0.488271	−	0.872692i	$-0.662372\pi$
$557$	− 972.179i	− 1.74538i	0.488271	−	0.872692i	$-0.337628\pi$
$558$	0	0
$559$	−628.672	−1.12464
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 280.795i	− 0.498748i	−0.968407	−	0.249374i	$-0.919775\pi$
$563$	− 280.795i	− 0.498748i	0.968407	−	0.249374i	$-0.0802248\pi$
$564$	0	0
$565$	190.892	0.337862
$566$	0	0
$567$	0	0
$568$	0	0
$569$	354.038i	0.622211i	0.950375	+	0.311105i	$0.100699\pi$
$569$	354.038i	0.622211i	−0.950375	+	0.311105i	$0.899301\pi$
$570$	0	0
$571$	287.332	0.503209	0.251605	−	0.967830i	$-0.419042\pi$
$571$	287.332	0.503209	0.251605	+	0.967830i	$0.419042\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	486.002i	0.845221i
$576$	0	0
$577$	822.994	1.42633	0.713167	−	0.700995i	$-0.247260\pi$
$577$	822.994	1.42633	0.713167	+	0.700995i	$0.247260\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 362.774i	− 0.624395i
$582$	0	0
$583$	−329.323	−0.564876
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 366.304i	− 0.624028i	−0.950078	−	0.312014i	$-0.898996\pi$
$587$	− 366.304i	− 0.624028i	0.950078	−	0.312014i	$-0.101004\pi$
$588$	0	0
$589$	1287.15	2.18532
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1018.50i	− 1.71754i	−0.512360	−	0.858771i	$-0.671228\pi$
$593$	− 1018.50i	− 1.71754i	0.512360	−	0.858771i	$-0.328772\pi$
$594$	0	0
$595$	369.917	0.621709
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 279.285i	− 0.466252i	−0.972447	−	0.233126i	$-0.925105\pi$
$599$	− 279.285i	− 0.466252i	0.972447	−	0.233126i	$-0.0748955\pi$
$600$	0	0
$601$	858.207	1.42797	0.713983	−	0.700163i	$-0.246889\pi$
$601$	858.207	1.42797	0.713983	+	0.700163i	$0.246889\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 71.2004i	− 0.117687i
$606$	0	0
$607$	−231.764	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$607$	−231.764	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	449.454i	0.735603i
$612$	0	0
$613$	567.997	0.926585	0.463292	−	0.886205i	$-0.346668\pi$
$613$	567.997	0.926585	0.463292	+	0.886205i	$0.346668\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 768.319i	− 1.24525i	−0.782521	−	0.622625i	$-0.786066\pi$
$617$	− 768.319i	− 1.24525i	0.782521	−	0.622625i	$-0.213934\pi$
$618$	0	0
$619$	−649.024	−1.04850	−0.524252	−	0.851563i	$-0.675655\pi$
$619$	−649.024	−1.04850	−0.524252	+	0.851563i	$0.675655\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	94.8156i	0.152192i
$624$	0	0
$625$	−761.920	−1.21907
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1664.45i	− 2.64618i
$630$	0	0
$631$	426.539	0.675973	0.337986	−	0.941151i	$-0.390254\pi$
$631$	426.539	0.675973	0.337986	+	0.941151i	$0.390254\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1053.70i	− 1.65936i
$636$	0	0
$637$	490.689	0.770312
$638$	0	0
$639$	0	0
$640$	0	0
$641$	565.451i	0.882139i	0.897473	+	0.441069i	$0.145401\pi$
$641$	565.451i	0.882139i	−0.897473	+	0.441069i	$0.854599\pi$
$642$	0	0
$643$	−1115.23	−1.73442	−0.867211	−	0.497940i	$-0.834090\pi$
$643$	−1115.23	−1.73442	−0.867211	+	0.497940i	$0.834090\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 562.467i	− 0.869346i	−0.900588	−	0.434673i	$-0.856864\pi$
$647$	− 562.467i	− 0.869346i	0.900588	−	0.434673i	$-0.143136\pi$
$648$	0	0
$649$	144.861	0.223207
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 738.483i	− 1.13091i	−0.824780	−	0.565454i	$-0.808701\pi$
$653$	− 738.483i	− 1.13091i	0.824780	−	0.565454i	$-0.191299\pi$
$654$	0	0
$655$	102.254	0.156114
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 16.6506i	− 0.0252665i	−0.999920	−	0.0126332i	$-0.995979\pi$
$659$	− 16.6506i	− 0.0252665i	0.999920	−	0.0126332i	$-0.00402139\pi$
$660$	0	0
$661$	−415.280	−0.628261	−0.314130	−	0.949380i	$-0.601713\pi$
$661$	−415.280	−0.628261	−0.314130	+	0.949380i	$0.601713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 420.580i	− 0.632452i
$666$	0	0
$667$	68.7655	0.103097
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.7984i	0.0399379i
$672$	0	0
$673$	−72.0072	−0.106994	−0.0534972	−	0.998568i	$-0.517037\pi$
$673$	−72.0072	−0.106994	−0.0534972	+	0.998568i	$0.517037\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	515.299i	0.761151i	0.924750	+	0.380576i	$0.124274\pi$
$677$	515.299i	0.761151i	−0.924750	+	0.380576i	$0.875726\pi$
$678$	0	0
$679$	128.615	0.189419
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1068.87i	1.56497i	0.622669	+	0.782485i	$0.286048\pi$
$683$	1068.87i	1.56497i	−0.622669	+	0.782485i	$0.713952\pi$
$684$	0	0
$685$	−802.545	−1.17160
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1123.00i	1.62990i
$690$	0	0
$691$	−844.368	−1.22195	−0.610975	−	0.791650i	$-0.709223\pi$
$691$	−844.368	−1.22195	−0.610975	+	0.791650i	$0.709223\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	363.449i	0.522948i
$696$	0	0
$697$	327.208	0.469452
$698$	0	0
$699$	0	0
$700$	0	0
$701$	726.985i	1.03707i	0.855057	+	0.518534i	$0.173522\pi$
$701$	726.985i	1.03707i	−0.855057	+	0.518534i	$0.826478\pi$
$702$	0	0
$703$	−1892.41	−2.69190
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 93.8546i	− 0.132751i
$708$	0	0
$709$	908.394	1.28123	0.640616	−	0.767861i	$-0.278679\pi$
$709$	908.394	1.28123	0.640616	+	0.767861i	$0.278679\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1350.02i	1.89343i
$714$	0	0
$715$	−242.795	−0.339574
$716$	0	0
$717$	0	0
$718$	0	0
$719$	231.093i	0.321409i	0.987003	+	0.160704i	$0.0513766\pi$
$719$	231.093i	0.321409i	−0.987003	+	0.160704i	$0.948623\pi$
$720$	0	0
$721$	−362.731	−0.503094
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 40.3955i	− 0.0557179i
$726$	0	0
$727$	118.093	0.162438	0.0812192	−	0.996696i	$-0.474119\pi$
$727$	118.093	0.162438	0.0812192	+	0.996696i	$0.474119\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1340.77i	− 1.83416i
$732$	0	0
$733$	831.429	1.13428	0.567141	−	0.823621i	$-0.308049\pi$
$733$	831.429	1.13428	0.567141	+	0.823621i	$0.308049\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.5545i	0.0333168i
$738$	0	0
$739$	538.823	0.729124	0.364562	−	0.931179i	$-0.381219\pi$
$739$	538.823	0.729124	0.364562	+	0.931179i	$0.381219\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 567.594i	− 0.763922i	−0.924178	−	0.381961i	$-0.875249\pi$
$743$	− 567.594i	− 0.763922i	0.924178	−	0.381961i	$-0.124751\pi$
$744$	0	0
$745$	1295.88	1.73943
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 418.830i	− 0.559186i
$750$	0	0
$751$	149.289	0.198787	0.0993933	−	0.995048i	$-0.468310\pi$
$751$	149.289	0.198787	0.0993933	+	0.995048i	$0.468310\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 751.229i	− 0.995005i
$756$	0	0
$757$	−845.341	−1.11670	−0.558349	−	0.829606i	$-0.688565\pi$
$757$	−845.341	−1.11670	−0.558349	+	0.829606i	$0.688565\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1413.64i	− 1.85761i	−0.370568	−	0.928805i	$-0.620837\pi$
$761$	− 1413.64i	− 1.85761i	0.370568	−	0.928805i	$-0.379163\pi$
$762$	0	0
$763$	4.65762	0.00610435
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 493.982i	− 0.644044i
$768$	0	0
$769$	−985.743	−1.28185	−0.640925	−	0.767604i	$-0.721449\pi$
$769$	−985.743	−1.28185	−0.640925	+	0.767604i	$0.721449\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.1448i	0.0558148i	0.999611	+	0.0279074i	$0.00888435\pi$
$773$	43.1448i	0.0558148i	−0.999611	+	0.0279074i	$0.991116\pi$
$774$	0	0
$775$	793.051	1.02329
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 372.023i	− 0.477564i
$780$	0	0
$781$	199.738	0.255747
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1356.08i	− 1.72749i
$786$	0	0
$787$	1137.96	1.44594	0.722971	−	0.690879i	$-0.242776\pi$
$787$	1137.96	1.44594	0.722971	+	0.690879i	$0.242776\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 69.8760i	− 0.0883389i
$792$	0	0
$793$	91.3832	0.115237
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 719.024i	− 0.902164i	−0.892483	−	0.451082i	$-0.851038\pi$
$797$	− 719.024i	− 0.902164i	0.892483	−	0.451082i	$-0.148962\pi$
$798$	0	0
$799$	−958.550	−1.19969
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 346.733i	− 0.431797i
$804$	0	0
$805$	441.121	0.547976
$806$	0	0
$807$	0	0
$808$	0	0
$809$	785.000i	0.970334i	0.874422	+	0.485167i	$0.161241\pi$
$809$	785.000i	0.970334i	−0.874422	+	0.485167i	$0.838759\pi$
$810$	0	0
$811$	−1113.43	−1.37291	−0.686454	−	0.727173i	$-0.740834\pi$
$811$	−1113.43	−1.37291	−0.686454	+	0.727173i	$0.740834\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	74.2022i	0.0910456i
$816$	0	0
$817$	−1524.40	−1.86585
$818$	0	0
$819$	0	0
$820$	0	0
$821$	82.6636i	0.100686i	0.998732	+	0.0503432i	$0.0160315\pi$
$821$	82.6636i	0.100686i	−0.998732	+	0.0503432i	$0.983968\pi$
$822$	0	0
$823$	−452.004	−0.549215	−0.274608	−	0.961556i	$-0.588548\pi$
$823$	−452.004	−0.549215	−0.274608	+	0.961556i	$0.588548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 897.131i	− 1.08480i	−0.840120	−	0.542401i	$-0.817515\pi$
$827$	− 897.131i	− 1.08480i	0.840120	−	0.542401i	$-0.182485\pi$
$828$	0	0
$829$	−1269.92	−1.53187	−0.765934	−	0.642919i	$-0.777723\pi$
$829$	−1269.92	−1.53187	−0.765934	+	0.642919i	$0.777723\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1046.49i	1.25629i
$834$	0	0
$835$	−1889.62	−2.26301
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.94159i	0.00469796i	0.999997	+	0.00234898i	$0.000747704\pi$
$839$	3.94159i	0.00469796i	−0.999997	+	0.00234898i	$0.999252\pi$
$840$	0	0
$841$	835.284	0.993204
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 265.956i	− 0.314741i
$846$	0	0
$847$	−26.0629	−0.0307708
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1984.83i	− 2.33235i
$852$	0	0
$853$	164.757	0.193150	0.0965749	−	0.995326i	$-0.469211\pi$
$853$	164.757	0.193150	0.0965749	+	0.995326i	$0.469211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	73.4834i	0.0857449i	0.999081	+	0.0428725i	$0.0136509\pi$
$857$	73.4834i	0.0857449i	−0.999081	+	0.0428725i	$0.986349\pi$
$858$	0	0
$859$	757.689	0.882059	0.441029	−	0.897493i	$-0.354613\pi$
$859$	757.689	0.882059	0.441029	+	0.897493i	$0.354613\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1561.72i	− 1.80964i	−0.425795	−	0.904820i	$-0.640006\pi$
$863$	− 1561.72i	− 1.80964i	0.425795	−	0.904820i	$-0.359994\pi$
$864$	0	0
$865$	1749.20	2.02220
$866$	0	0
$867$	0	0
$868$	0	0
$869$	188.373i	0.216770i
$870$	0	0
$871$	83.7316	0.0961328
$872$	0	0
$873$	0	0
$874$	0	0
$875$	124.275i	0.142029i
$876$	0	0
$877$	750.204	0.855421	0.427710	−	0.903916i	$-0.359320\pi$
$877$	750.204	0.855421	0.427710	+	0.903916i	$0.359320\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1473.45i	− 1.67247i	−0.548371	−	0.836235i	$-0.684752\pi$
$881$	− 1473.45i	− 1.67247i	0.548371	−	0.836235i	$-0.315248\pi$
$882$	0	0
$883$	77.1642	0.0873886	0.0436943	−	0.999045i	$-0.486087\pi$
$883$	77.1642	0.0873886	0.0436943	+	0.999045i	$0.486087\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1599.71i	1.80350i	0.432257	+	0.901751i	$0.357717\pi$
$887$	1599.71i	1.80350i	−0.432257	+	0.901751i	$0.642283\pi$
$888$	0	0
$889$	−385.705	−0.433864
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1089.83i	1.22042i
$894$	0	0
$895$	951.264	1.06287
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 112.211i	− 0.124817i
$900$	0	0
$901$	−2395.02	−2.65819
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 452.793i	− 0.500324i
$906$	0	0
$907$	58.8324	0.0648648	0.0324324	−	0.999474i	$-0.489675\pi$
$907$	58.8324	0.0648648	0.0324324	+	0.999474i	$0.489675\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	446.671i	0.490308i	0.969484	+	0.245154i	$0.0788386\pi$
$911$	446.671i	0.490308i	−0.969484	+	0.245154i	$0.921161\pi$
$912$	0	0
$913$	−507.811	−0.556200
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 37.4302i	− 0.0408181i
$918$	0	0
$919$	−1130.10	−1.22971	−0.614854	−	0.788641i	$-0.710785\pi$
$919$	−1130.10	−1.22971	−0.614854	+	0.788641i	$0.710785\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 681.114i	− 0.737935i
$924$	0	0
$925$	−1165.96	−1.26050
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 369.630i	− 0.397879i	−0.980012	−	0.198939i	$-0.936250\pi$
$929$	− 369.630i	− 0.397879i	0.980012	−	0.198939i	$-0.0637497\pi$
$930$	0	0
$931$	1189.82	1.27800
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 517.810i	− 0.553808i
$936$	0	0
$937$	−445.723	−0.475692	−0.237846	−	0.971303i	$-0.576441\pi$
$937$	−445.723	−0.475692	−0.237846	+	0.971303i	$0.576441\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 42.8294i	− 0.0455148i	−0.999741	−	0.0227574i	$-0.992755\pi$
$941$	− 42.8294i	− 0.0455148i	0.999741	−	0.0227574i	$-0.00724453\pi$
$942$	0	0
$943$	390.191	0.413777
$944$	0	0
$945$	0	0
$946$	0	0
$947$	735.865i	0.777049i	0.921438	+	0.388525i	$0.127015\pi$
$947$	735.865i	0.777049i	−0.921438	+	0.388525i	$0.872985\pi$
$948$	0	0
$949$	−1182.37	−1.24591
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 739.927i	− 0.776419i	−0.921571	−	0.388210i	$-0.873094\pi$
$953$	− 739.927i	− 0.776419i	0.921571	−	0.388210i	$-0.126906\pi$
$954$	0	0
$955$	1985.03	2.07856
$956$	0	0
$957$	0	0
$958$	0	0
$959$	293.772i	0.306331i
$960$	0	0
$961$	1241.94	1.29234
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1505.20i	1.55979i
$966$	0	0
$967$	935.751	0.967684	0.483842	−	0.875155i	$-0.339241\pi$
$967$	935.751	0.967684	0.483842	+	0.875155i	$0.339241\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1731.44i	− 1.78315i	−0.452869	−	0.891577i	$-0.649599\pi$
$971$	− 1731.44i	− 1.78315i	0.452869	−	0.891577i	$-0.350401\pi$
$972$	0	0
$973$	133.040	0.136732
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1135.16i	− 1.16189i	−0.813944	−	0.580944i	$-0.802684\pi$
$977$	− 1135.16i	− 1.16189i	0.813944	−	0.580944i	$-0.197316\pi$
$978$	0	0
$979$	132.723	0.135570
$980$	0	0
$981$	0	0
$982$	0	0
$983$	689.115i	0.701032i	0.936557	+	0.350516i	$0.113994\pi$
$983$	689.115i	0.701032i	−0.936557	+	0.350516i	$0.886006\pi$
$984$	0	0
$985$	−1260.52	−1.27971
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1598.85i	− 1.61663i
$990$	0	0
$991$	665.099	0.671139	0.335569	−	0.942015i	$-0.391071\pi$
$991$	665.099	0.671139	0.335569	+	0.942015i	$0.391071\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 712.190i	− 0.715769i
$996$	0	0
$997$	1024.30	1.02738	0.513691	−	0.857975i	$-0.328278\pi$
$997$	1024.30	1.02738	0.513691	+	0.857975i	$0.328278\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1188.3.e.c.485.8	yes	8
3.2	odd	2	inner	1188.3.e.c.485.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1188.3.e.c.485.1	✓	8	3.2	odd	2	inner
1188.3.e.c.485.8	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1188 = 2^{2} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1188.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.47276i q^{5} +2.36935 q^{7} -3.31662i q^{11} -11.3098 q^{13} -24.1204i q^{17} -27.4239 q^{19} -28.7633i q^{23} -16.8966 q^{25} +2.39074i q^{29} -46.9354 q^{31} +15.3363i q^{35} +69.0057 q^{37} +13.5656i q^{41} +55.5865 q^{43} -39.7402i q^{47} -43.3862 q^{49} -99.2946i q^{53} +21.4677 q^{55} +43.6773i q^{59} -8.08001 q^{61} -73.2056i q^{65} -7.40346 q^{67} +60.2234i q^{71} +104.544 q^{73} -7.85826i q^{77} -56.7966 q^{79} -153.111i q^{83} +156.126 q^{85} +40.0175i q^{89} -26.7969 q^{91} -177.508i q^{95} +54.2828 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 20 q^{13} + 16 q^{19} - 12 q^{25} - 32 q^{31} - 20 q^{37} - 48 q^{43} + 84 q^{49} + 36 q^{61} + 16 q^{67} + 200 q^{73} - 28 q^{79} - 60 q^{85} - 228 q^{91} - 120 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 20 * q^13 + 16 * q^19 - 12 * q^25 - 32 * q^31 - 20 * q^37 - 48 * q^43 + 84 * q^49 + 36 * q^61 + 16 * q^67 + 200 * q^73 - 28 * q^79 - 60 * q^85 - 228 * q^91 - 120 * q^97

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