LMFDB - Embedded newform 1176.3.f.d.97.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(97,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 $ x^16 - 4x^15 - 66x^14 + 136x^13 + 3441x^12 - 4512x^11 - 100322x^10 + 24868x^9 + 2325060x^8 + 260012x^7 - 34273754x^6 - 26559312x^5 + 402103713x^4 + 286444088x^3 - 2633783418x^2 - 3060285788*x + 13841287201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$3.49578 + 1.44094i$ of defining polynomial
Character	$\chi$	$=$	1176.97
Dual form			1176.3.f.d.97.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.22784i q^{5} -3.00000 q^{9} -20.0696 q^{11} +9.54403i q^{13} -3.85873 q^{15} +3.05393i q^{17} +17.6370i q^{19} +24.8091 q^{23} +20.0367 q^{25} +5.19615i q^{27} +29.5215 q^{29} +21.7428i q^{31} +34.7616i q^{33} +32.7552 q^{37} +16.5307 q^{39} -25.9138i q^{41} +32.5577 q^{43} +6.68352i q^{45} +5.49107i q^{47} +5.28955 q^{51} -20.1438 q^{53} +44.7118i q^{55} +30.5482 q^{57} -87.8677i q^{59} +47.9302i q^{61} +21.2626 q^{65} -110.959 q^{67} -42.9705i q^{69} +56.6552 q^{71} -81.0874i q^{73} -34.7046i q^{75} +107.197 q^{79} +9.00000 q^{81} +122.200i q^{83} +6.80365 q^{85} -51.1327i q^{87} +133.565i q^{89} +37.6596 q^{93} +39.2924 q^{95} -12.3024i q^{97} +60.2088 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) $ 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	− 2.22784i	− 0.445568i	−0.974868	−	0.222784i	$-0.928486\pi$
$5$	− 2.22784i	− 0.445568i	0.974868	−	0.222784i	$-0.0715144\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−20.0696	−1.82451	−0.912255	−	0.409624i	$-0.865660\pi$
$11$	−20.0696	−1.82451	−0.912255	+	0.409624i	$0.865660\pi$
$12$	0	0
$13$	9.54403i	0.734156i	0.930190	+	0.367078i	$0.119642\pi$
$13$	9.54403i	0.734156i	−0.930190	+	0.367078i	$0.880358\pi$
$14$	0	0
$15$	−3.85873	−0.257249
$16$	0	0
$17$	3.05393i	0.179643i	0.995958	+	0.0898213i	$0.0286296\pi$
$17$	3.05393i	0.179643i	−0.995958	+	0.0898213i	$0.971370\pi$
$18$	0	0
$19$	17.6370i	0.928264i	0.885766	+	0.464132i	$0.153634\pi$
$19$	17.6370i	0.928264i	−0.885766	+	0.464132i	$0.846366\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	24.8091	1.07865	0.539327	−	0.842096i	$-0.318679\pi$
$23$	24.8091	1.07865	0.539327	+	0.842096i	$0.318679\pi$
$24$	0	0
$25$	20.0367	0.801469
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	29.5215	1.01798	0.508991	−	0.860772i	$-0.330019\pi$
$29$	29.5215	1.01798	0.508991	+	0.860772i	$0.330019\pi$
$30$	0	0
$31$	21.7428i	0.701379i	0.936492	+	0.350690i	$0.114053\pi$
$31$	21.7428i	0.701379i	−0.936492	+	0.350690i	$0.885947\pi$
$32$	0	0
$33$	34.7616i	1.05338i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	32.7552	0.885276	0.442638	−	0.896700i	$-0.354043\pi$
$37$	32.7552	0.885276	0.442638	+	0.896700i	$0.354043\pi$
$38$	0	0
$39$	16.5307	0.423865
$40$	0	0
$41$	− 25.9138i	− 0.632043i	−0.948752	−	0.316021i	$-0.897653\pi$
$41$	− 25.9138i	− 0.632043i	0.948752	−	0.316021i	$-0.102347\pi$
$42$	0	0
$43$	32.5577	0.757157	0.378578	−	0.925569i	$-0.376413\pi$
$43$	32.5577	0.757157	0.378578	+	0.925569i	$0.376413\pi$
$44$	0	0
$45$	6.68352i	0.148523i
$46$	0	0
$47$	5.49107i	0.116831i	0.998292	+	0.0584157i	$0.0186049\pi$
$47$	5.49107i	0.116831i	−0.998292	+	0.0584157i	$0.981395\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.28955	0.103717
$52$	0	0
$53$	−20.1438	−0.380071	−0.190036	−	0.981777i	$-0.560860\pi$
$53$	−20.1438	−0.380071	−0.190036	+	0.981777i	$0.560860\pi$
$54$	0	0
$55$	44.7118i	0.812942i
$56$	0	0
$57$	30.5482	0.535933
$58$	0	0
$59$	− 87.8677i	− 1.48928i	−0.667465	−	0.744641i	$-0.732621\pi$
$59$	− 87.8677i	− 1.48928i	0.667465	−	0.744641i	$-0.267379\pi$
$60$	0	0
$61$	47.9302i	0.785741i	0.919594	+	0.392870i	$0.128518\pi$
$61$	47.9302i	0.785741i	−0.919594	+	0.392870i	$0.871482\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.2626	0.327116
$66$	0	0
$67$	−110.959	−1.65610	−0.828051	−	0.560653i	$-0.810550\pi$
$67$	−110.959	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$68$	0	0
$69$	− 42.9705i	− 0.622762i
$70$	0	0
$71$	56.6552	0.797960	0.398980	−	0.916960i	$-0.369364\pi$
$71$	56.6552	0.797960	0.398980	+	0.916960i	$0.369364\pi$
$72$	0	0
$73$	− 81.0874i	− 1.11079i	−0.831588	−	0.555393i	$-0.812568\pi$
$73$	− 81.0874i	− 1.11079i	0.831588	−	0.555393i	$-0.187432\pi$
$74$	0	0
$75$	− 34.7046i	− 0.462729i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	107.197	1.35693	0.678464	−	0.734634i	$-0.262646\pi$
$79$	107.197	1.35693	0.678464	+	0.734634i	$0.262646\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	122.200i	1.47229i	0.676826	+	0.736143i	$0.263355\pi$
$83$	122.200i	1.47229i	−0.676826	+	0.736143i	$0.736645\pi$
$84$	0	0
$85$	6.80365	0.0800430
$86$	0	0
$87$	− 51.1327i	− 0.587732i
$88$	0	0
$89$	133.565i	1.50073i	0.661024	+	0.750365i	$0.270122\pi$
$89$	133.565i	1.50073i	−0.661024	+	0.750365i	$0.729878\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	37.6596	0.404941
$94$	0	0
$95$	39.2924	0.413604
$96$	0	0
$97$	− 12.3024i	− 0.126829i	−0.997987	−	0.0634143i	$-0.979801\pi$
$97$	− 12.3024i	− 0.126829i	0.997987	−	0.0634143i	$-0.0201990\pi$
$98$	0	0
$99$	60.2088	0.608170
$100$	0	0
$101$	− 71.0412i	− 0.703379i	−0.936117	−	0.351689i	$-0.885607\pi$
$101$	− 71.0412i	− 0.703379i	0.936117	−	0.351689i	$-0.114393\pi$
$102$	0	0
$103$	43.0893i	0.418342i	0.977879	+	0.209171i	$0.0670766\pi$
$103$	43.0893i	0.418342i	−0.977879	+	0.209171i	$0.932923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−51.8168	−0.484269	−0.242134	−	0.970243i	$-0.577847\pi$
$107$	−51.8168	−0.484269	−0.242134	+	0.970243i	$0.577847\pi$
$108$	0	0
$109$	149.268	1.36943	0.684714	−	0.728812i	$-0.259927\pi$
$109$	149.268	1.36943	0.684714	+	0.728812i	$0.259927\pi$
$110$	0	0
$111$	− 56.7337i	− 0.511114i
$112$	0	0
$113$	81.8648	0.724467	0.362234	−	0.932087i	$-0.382014\pi$
$113$	81.8648	0.724467	0.362234	+	0.932087i	$0.382014\pi$
$114$	0	0
$115$	− 55.2706i	− 0.480614i
$116$	0	0
$117$	− 28.6321i	− 0.244719i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	281.789	2.32883
$122$	0	0
$123$	−44.8839	−0.364910
$124$	0	0
$125$	− 100.335i	− 0.802677i
$126$	0	0
$127$	−137.263	−1.08081	−0.540404	−	0.841406i	$-0.681729\pi$
$127$	−137.263	−1.08081	−0.540404	+	0.841406i	$0.681729\pi$
$128$	0	0
$129$	− 56.3917i	− 0.437145i
$130$	0	0
$131$	145.922i	1.11391i	0.830543	+	0.556955i	$0.188030\pi$
$131$	145.922i	1.11391i	−0.830543	+	0.556955i	$0.811970\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.5762	0.0857495
$136$	0	0
$137$	156.287	1.14078	0.570392	−	0.821373i	$-0.306791\pi$
$137$	156.287	1.14078	0.570392	+	0.821373i	$0.306791\pi$
$138$	0	0
$139$	177.430i	1.27648i	0.769838	+	0.638239i	$0.220337\pi$
$139$	177.430i	1.27648i	−0.769838	+	0.638239i	$0.779663\pi$
$140$	0	0
$141$	9.51082	0.0674526
$142$	0	0
$143$	− 191.545i	− 1.33947i
$144$	0	0
$145$	− 65.7691i	− 0.453580i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	232.905	1.56312	0.781559	−	0.623831i	$-0.214425\pi$
$149$	232.905	1.56312	0.781559	+	0.623831i	$0.214425\pi$
$150$	0	0
$151$	−136.542	−0.904254	−0.452127	−	0.891954i	$-0.649335\pi$
$151$	−136.542	−0.904254	−0.452127	+	0.891954i	$0.649335\pi$
$152$	0	0
$153$	− 9.16178i	− 0.0598809i
$154$	0	0
$155$	48.4393	0.312512
$156$	0	0
$157$	307.425i	1.95812i	0.203565	+	0.979061i	$0.434747\pi$
$157$	307.425i	1.95812i	−0.203565	+	0.979061i	$0.565253\pi$
$158$	0	0
$159$	34.8900i	0.219434i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	137.704	0.844808	0.422404	−	0.906408i	$-0.361186\pi$
$163$	137.704	0.844808	0.422404	+	0.906408i	$0.361186\pi$
$164$	0	0
$165$	77.4432	0.469352
$166$	0	0
$167$	48.8954i	0.292787i	0.989226	+	0.146393i	$0.0467666\pi$
$167$	48.8954i	0.292787i	−0.989226	+	0.146393i	$0.953233\pi$
$168$	0	0
$169$	77.9115	0.461015
$170$	0	0
$171$	− 52.9110i	− 0.309421i
$172$	0	0
$173$	− 175.401i	− 1.01388i	−0.861981	−	0.506940i	$-0.830777\pi$
$173$	− 175.401i	− 1.01388i	0.861981	−	0.506940i	$-0.169223\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−152.191	−0.859837
$178$	0	0
$179$	−159.718	−0.892279	−0.446140	−	0.894963i	$-0.647201\pi$
$179$	−159.718	−0.892279	−0.446140	+	0.894963i	$0.647201\pi$
$180$	0	0
$181$	311.027i	1.71838i	0.511658	+	0.859189i	$0.329032\pi$
$181$	311.027i	1.71838i	−0.511658	+	0.859189i	$0.670968\pi$
$182$	0	0
$183$	83.0175	0.453648
$184$	0	0
$185$	− 72.9733i	− 0.394450i
$186$	0	0
$187$	− 61.2911i	− 0.327760i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−182.712	−0.956606	−0.478303	−	0.878195i	$-0.658748\pi$
$191$	−182.712	−0.956606	−0.478303	+	0.878195i	$0.658748\pi$
$192$	0	0
$193$	−134.304	−0.695875	−0.347937	−	0.937518i	$-0.613118\pi$
$193$	−134.304	−0.695875	−0.347937	+	0.937518i	$0.613118\pi$
$194$	0	0
$195$	− 36.8278i	− 0.188861i
$196$	0	0
$197$	−40.8435	−0.207327	−0.103664	−	0.994612i	$-0.533057\pi$
$197$	−40.8435	−0.207327	−0.103664	+	0.994612i	$0.533057\pi$
$198$	0	0
$199$	− 332.159i	− 1.66914i	−0.550900	−	0.834571i	$-0.685715\pi$
$199$	− 332.159i	− 1.66914i	0.550900	−	0.834571i	$-0.314285\pi$
$200$	0	0
$201$	192.186i	0.956150i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−57.7317	−0.281618
$206$	0	0
$207$	−74.4272	−0.359552
$208$	0	0
$209$	− 353.968i	− 1.69363i
$210$	0	0
$211$	−20.9652	−0.0993612	−0.0496806	−	0.998765i	$-0.515820\pi$
$211$	−20.9652	−0.0993612	−0.0496806	+	0.998765i	$0.515820\pi$
$212$	0	0
$213$	− 98.1297i	− 0.460703i
$214$	0	0
$215$	− 72.5334i	− 0.337365i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−140.448	−0.641313
$220$	0	0
$221$	−29.1468	−0.131886
$222$	0	0
$223$	343.267i	1.53931i	0.638457	+	0.769657i	$0.279573\pi$
$223$	343.267i	1.53931i	−0.638457	+	0.769657i	$0.720427\pi$
$224$	0	0
$225$	−60.1102	−0.267156
$226$	0	0
$227$	159.320i	0.701851i	0.936403	+	0.350926i	$0.114133\pi$
$227$	159.320i	0.701851i	−0.936403	+	0.350926i	$0.885867\pi$
$228$	0	0
$229$	417.319i	1.82235i	0.412015	+	0.911177i	$0.364825\pi$
$229$	417.319i	1.82235i	−0.412015	+	0.911177i	$0.635175\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	294.589	1.26433	0.632165	−	0.774834i	$-0.282167\pi$
$233$	294.589	1.26433	0.632165	+	0.774834i	$0.282167\pi$
$234$	0	0
$235$	12.2332	0.0520563
$236$	0	0
$237$	− 185.671i	− 0.783423i
$238$	0	0
$239$	190.059	0.795225	0.397613	−	0.917553i	$-0.369839\pi$
$239$	190.059	0.795225	0.397613	+	0.917553i	$0.369839\pi$
$240$	0	0
$241$	426.607i	1.77015i	0.465446	+	0.885076i	$0.345894\pi$
$241$	426.607i	1.77015i	−0.465446	+	0.885076i	$0.654106\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−168.328	−0.681491
$248$	0	0
$249$	211.656	0.850024
$250$	0	0
$251$	− 294.663i	− 1.17396i	−0.809602	−	0.586979i	$-0.800317\pi$
$251$	− 294.663i	− 1.17396i	0.809602	−	0.586979i	$-0.199683\pi$
$252$	0	0
$253$	−497.908	−1.96802
$254$	0	0
$255$	− 11.7843i	− 0.0462128i
$256$	0	0
$257$	− 73.7314i	− 0.286893i	−0.989658	−	0.143446i	$-0.954182\pi$
$257$	− 73.7314i	− 0.286893i	0.989658	−	0.143446i	$-0.0458184\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−88.5644	−0.339327
$262$	0	0
$263$	75.3342	0.286442	0.143221	−	0.989691i	$-0.454254\pi$
$263$	75.3342	0.286442	0.143221	+	0.989691i	$0.454254\pi$
$264$	0	0
$265$	44.8771i	0.169347i
$266$	0	0
$267$	231.341	0.866447
$268$	0	0
$269$	418.118i	1.55434i	0.629290	+	0.777171i	$0.283346\pi$
$269$	418.118i	1.55434i	−0.629290	+	0.777171i	$0.716654\pi$
$270$	0	0
$271$	− 436.300i	− 1.60996i	−0.593300	−	0.804982i	$-0.702175\pi$
$271$	− 436.300i	− 1.60996i	0.593300	−	0.804982i	$-0.297825\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−402.129	−1.46229
$276$	0	0
$277$	199.932	0.721777	0.360889	−	0.932609i	$-0.382473\pi$
$277$	199.932	0.721777	0.360889	+	0.932609i	$0.382473\pi$
$278$	0	0
$279$	− 65.2283i	− 0.233793i
$280$	0	0
$281$	216.492	0.770435	0.385218	−	0.922826i	$-0.374126\pi$
$281$	216.492	0.770435	0.385218	+	0.922826i	$0.374126\pi$
$282$	0	0
$283$	474.868i	1.67798i	0.544149	+	0.838989i	$0.316853\pi$
$283$	474.868i	1.67798i	−0.544149	+	0.838989i	$0.683147\pi$
$284$	0	0
$285$	− 68.0565i	− 0.238795i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	279.674	0.967729
$290$	0	0
$291$	−21.3083	−0.0732245
$292$	0	0
$293$	− 380.161i	− 1.29748i	−0.761011	−	0.648739i	$-0.775297\pi$
$293$	− 380.161i	− 1.29748i	0.761011	−	0.648739i	$-0.224703\pi$
$294$	0	0
$295$	−195.755	−0.663576
$296$	0	0
$297$	− 104.285i	− 0.351127i
$298$	0	0
$299$	236.778i	0.791901i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−123.047	−0.406096
$304$	0	0
$305$	106.781	0.350101
$306$	0	0
$307$	− 70.8402i	− 0.230750i	−0.993322	−	0.115375i	$-0.963193\pi$
$307$	− 70.8402i	− 0.230750i	0.993322	−	0.115375i	$-0.0368069\pi$
$308$	0	0
$309$	74.6328	0.241530
$310$	0	0
$311$	− 278.391i	− 0.895146i	−0.894247	−	0.447573i	$-0.852288\pi$
$311$	− 278.391i	− 0.895146i	0.894247	−	0.447573i	$-0.147712\pi$
$312$	0	0
$313$	− 35.7406i	− 0.114187i	−0.998369	−	0.0570936i	$-0.981817\pi$
$313$	− 35.7406i	− 0.114187i	0.998369	−	0.0570936i	$-0.0181833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−541.358	−1.70775	−0.853877	−	0.520474i	$-0.825755\pi$
$317$	−541.358	−1.70775	−0.853877	+	0.520474i	$0.825755\pi$
$318$	0	0
$319$	−592.484	−1.85732
$320$	0	0
$321$	89.7493i	0.279593i
$322$	0	0
$323$	−53.8621	−0.166756
$324$	0	0
$325$	191.231i	0.588404i
$326$	0	0
$327$	− 258.539i	− 0.790639i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	96.3508	0.291090	0.145545	−	0.989352i	$-0.453506\pi$
$331$	96.3508	0.291090	0.145545	+	0.989352i	$0.453506\pi$
$332$	0	0
$333$	−98.2656	−0.295092
$334$	0	0
$335$	247.198i	0.737905i
$336$	0	0
$337$	−181.063	−0.537280	−0.268640	−	0.963241i	$-0.586574\pi$
$337$	−181.063	−0.537280	−0.268640	+	0.963241i	$0.586574\pi$
$338$	0	0
$339$	− 141.794i	− 0.418271i
$340$	0	0
$341$	− 436.368i	− 1.27967i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−95.7314	−0.277482
$346$	0	0
$347$	119.923	0.345600	0.172800	−	0.984957i	$-0.444719\pi$
$347$	119.923	0.345600	0.172800	+	0.984957i	$0.444719\pi$
$348$	0	0
$349$	208.343i	0.596971i	0.954414	+	0.298486i	$0.0964815\pi$
$349$	208.343i	0.596971i	−0.954414	+	0.298486i	$0.903518\pi$
$350$	0	0
$351$	−49.5922	−0.141288
$352$	0	0
$353$	− 43.2517i	− 0.122526i	−0.998122	−	0.0612630i	$-0.980487\pi$
$353$	− 43.2517i	− 0.122526i	0.998122	−	0.0612630i	$-0.0195128\pi$
$354$	0	0
$355$	− 126.219i	− 0.355545i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−675.239	−1.88089	−0.940444	−	0.339948i	$-0.889591\pi$
$359$	−675.239	−1.88089	−0.940444	+	0.339948i	$0.889591\pi$
$360$	0	0
$361$	49.9358	0.138326
$362$	0	0
$363$	− 488.073i	− 1.34455i
$364$	0	0
$365$	−180.650	−0.494931
$366$	0	0
$367$	483.729i	1.31806i	0.752116	+	0.659031i	$0.229033\pi$
$367$	483.729i	1.31806i	−0.752116	+	0.659031i	$0.770967\pi$
$368$	0	0
$369$	77.7413i	0.210681i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−294.723	−0.790142	−0.395071	−	0.918651i	$-0.629280\pi$
$373$	−294.723	−0.790142	−0.395071	+	0.918651i	$0.629280\pi$
$374$	0	0
$375$	−173.785	−0.463426
$376$	0	0
$377$	281.754i	0.747358i
$378$	0	0
$379$	−302.851	−0.799080	−0.399540	−	0.916716i	$-0.630830\pi$
$379$	−302.851	−0.799080	−0.399540	+	0.916716i	$0.630830\pi$
$380$	0	0
$381$	237.746i	0.624005i
$382$	0	0
$383$	215.575i	0.562858i	0.959582	+	0.281429i	$0.0908085\pi$
$383$	215.575i	0.562858i	−0.959582	+	0.281429i	$0.909192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−97.6732	−0.252386
$388$	0	0
$389$	127.583	0.327977	0.163989	−	0.986462i	$-0.447564\pi$
$389$	127.583	0.327977	0.163989	+	0.986462i	$0.447564\pi$
$390$	0	0
$391$	75.7650i	0.193772i
$392$	0	0
$393$	252.745	0.643116
$394$	0	0
$395$	− 238.818i	− 0.604603i
$396$	0	0
$397$	− 632.191i	− 1.59242i	−0.605020	−	0.796211i	$-0.706835\pi$
$397$	− 632.191i	− 1.59242i	0.605020	−	0.796211i	$-0.293165\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−130.220	−0.324739	−0.162369	−	0.986730i	$-0.551914\pi$
$401$	−130.220	−0.324739	−0.162369	+	0.986730i	$0.551914\pi$
$402$	0	0
$403$	−207.514	−0.514922
$404$	0	0
$405$	− 20.0505i	− 0.0495075i
$406$	0	0
$407$	−657.384	−1.61519
$408$	0	0
$409$	− 95.5551i	− 0.233631i	−0.993154	−	0.116816i	$-0.962731\pi$
$409$	− 95.5551i	− 0.233631i	0.993154	−	0.116816i	$-0.0372686\pi$
$410$	0	0
$411$	− 270.698i	− 0.658632i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	272.241	0.656003
$416$	0	0
$417$	307.319	0.736975
$418$	0	0
$419$	385.093i	0.919076i	0.888158	+	0.459538i	$0.151985\pi$
$419$	385.093i	0.919076i	−0.888158	+	0.459538i	$0.848015\pi$
$420$	0	0
$421$	−621.622	−1.47654	−0.738269	−	0.674507i	$-0.764356\pi$
$421$	−621.622	−1.47654	−0.738269	+	0.674507i	$0.764356\pi$
$422$	0	0
$423$	− 16.4732i	− 0.0389438i
$424$	0	0
$425$	61.1907i	0.143978i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−331.765	−0.773346
$430$	0	0
$431$	608.595	1.41205	0.706027	−	0.708185i	$-0.250486\pi$
$431$	608.595	1.41205	0.706027	+	0.708185i	$0.250486\pi$
$432$	0	0
$433$	313.524i	0.724073i	0.932164	+	0.362037i	$0.117918\pi$
$433$	313.524i	0.724073i	−0.932164	+	0.362037i	$0.882082\pi$
$434$	0	0
$435$	−113.915	−0.261875
$436$	0	0
$437$	437.558i	1.00128i
$438$	0	0
$439$	− 322.662i	− 0.734993i	−0.930025	−	0.367497i	$-0.880215\pi$
$439$	− 322.662i	− 0.734993i	0.930025	−	0.367497i	$-0.119785\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−112.627	−0.254237	−0.127118	−	0.991888i	$-0.540573\pi$
$443$	−112.627	−0.254237	−0.127118	+	0.991888i	$0.540573\pi$
$444$	0	0
$445$	297.561	0.668677
$446$	0	0
$447$	− 403.403i	− 0.902467i
$448$	0	0
$449$	5.71585	0.0127302	0.00636509	−	0.999980i	$-0.497974\pi$
$449$	5.71585	0.0127302	0.00636509	+	0.999980i	$0.497974\pi$
$450$	0	0
$451$	520.079i	1.15317i
$452$	0	0
$453$	236.498i	0.522071i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	229.967	0.503211	0.251606	−	0.967830i	$-0.419041\pi$
$457$	229.967	0.503211	0.251606	+	0.967830i	$0.419041\pi$
$458$	0	0
$459$	−15.8687	−0.0345723
$460$	0	0
$461$	− 898.515i	− 1.94906i	−0.224264	−	0.974529i	$-0.571998\pi$
$461$	− 898.515i	− 1.94906i	0.224264	−	0.974529i	$-0.428002\pi$
$462$	0	0
$463$	−180.373	−0.389574	−0.194787	−	0.980846i	$-0.562402\pi$
$463$	−180.373	−0.389574	−0.194787	+	0.980846i	$0.562402\pi$
$464$	0	0
$465$	− 83.8994i	− 0.180429i
$466$	0	0
$467$	− 198.234i	− 0.424483i	−0.977217	−	0.212242i	$-0.931924\pi$
$467$	− 198.234i	− 0.424483i	0.977217	−	0.212242i	$-0.0680764\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	532.476	1.13052
$472$	0	0
$473$	−653.421	−1.38144
$474$	0	0
$475$	353.388i	0.743975i
$476$	0	0
$477$	60.4313	0.126690
$478$	0	0
$479$	70.9833i	0.148191i	0.997251	+	0.0740953i	$0.0236069\pi$
$479$	70.9833i	0.148191i	−0.997251	+	0.0740953i	$0.976393\pi$
$480$	0	0
$481$	312.617i	0.649931i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−27.4077	−0.0565107
$486$	0	0
$487$	419.126	0.860628	0.430314	−	0.902679i	$-0.358403\pi$
$487$	419.126	0.860628	0.430314	+	0.902679i	$0.358403\pi$
$488$	0	0
$489$	− 238.510i	− 0.487750i
$490$	0	0
$491$	2.83664	0.00577727	0.00288864	−	0.999996i	$-0.499081\pi$
$491$	2.83664	0.00577727	0.00288864	+	0.999996i	$0.499081\pi$
$492$	0	0
$493$	90.1564i	0.182873i
$494$	0	0
$495$	− 134.135i	− 0.270981i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−135.351	−0.271244	−0.135622	−	0.990761i	$-0.543303\pi$
$499$	−135.351	−0.271244	−0.135622	+	0.990761i	$0.543303\pi$
$500$	0	0
$501$	84.6893	0.169041
$502$	0	0
$503$	− 604.629i	− 1.20205i	−0.799232	−	0.601023i	$-0.794760\pi$
$503$	− 604.629i	− 1.20205i	0.799232	−	0.601023i	$-0.205240\pi$
$504$	0	0
$505$	−158.268	−0.313403
$506$	0	0
$507$	− 134.947i	− 0.266167i
$508$	0	0
$509$	− 872.707i	− 1.71455i	−0.514857	−	0.857276i	$-0.672155\pi$
$509$	− 872.707i	− 1.71455i	0.514857	−	0.857276i	$-0.327845\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−91.6446	−0.178644
$514$	0	0
$515$	95.9959	0.186400
$516$	0	0
$517$	− 110.204i	− 0.213160i
$518$	0	0
$519$	−303.804	−0.585364
$520$	0	0
$521$	− 664.285i	− 1.27502i	−0.770442	−	0.637510i	$-0.779964\pi$
$521$	− 664.285i	− 1.27502i	0.770442	−	0.637510i	$-0.220036\pi$
$522$	0	0
$523$	− 71.8993i	− 0.137475i	−0.997635	−	0.0687374i	$-0.978103\pi$
$523$	− 71.8993i	− 0.137475i	0.997635	−	0.0687374i	$-0.0218970\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−66.4008	−0.125998
$528$	0	0
$529$	86.4892	0.163496
$530$	0	0
$531$	263.603i	0.496427i
$532$	0	0
$533$	247.322	0.464018
$534$	0	0
$535$	115.439i	0.215775i
$536$	0	0
$537$	276.640i	0.515158i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−650.281	−1.20200	−0.600999	−	0.799250i	$-0.705230\pi$
$541$	−650.281	−1.20200	−0.600999	+	0.799250i	$0.705230\pi$
$542$	0	0
$543$	538.714	0.992106
$544$	0	0
$545$	− 332.544i	− 0.610173i
$546$	0	0
$547$	475.388	0.869082	0.434541	−	0.900652i	$-0.356911\pi$
$547$	475.388	0.869082	0.434541	+	0.900652i	$0.356911\pi$
$548$	0	0
$549$	− 143.791i	− 0.261914i
$550$	0	0
$551$	520.671i	0.944956i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−126.393	−0.227736
$556$	0	0
$557$	−885.427	−1.58963	−0.794817	−	0.606849i	$-0.792433\pi$
$557$	−885.427	−1.58963	−0.794817	+	0.606849i	$0.792433\pi$
$558$	0	0
$559$	310.732i	0.555871i
$560$	0	0
$561$	−106.159	−0.189232
$562$	0	0
$563$	340.621i	0.605011i	0.953148	+	0.302505i	$0.0978230\pi$
$563$	340.621i	0.605011i	−0.953148	+	0.302505i	$0.902177\pi$
$564$	0	0
$565$	− 182.382i	− 0.322799i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	291.899	0.513003	0.256501	−	0.966544i	$-0.417430\pi$
$569$	291.899	0.513003	0.256501	+	0.966544i	$0.417430\pi$
$570$	0	0
$571$	79.4220	0.139093	0.0695464	−	0.997579i	$-0.477845\pi$
$571$	79.4220	0.139093	0.0695464	+	0.997579i	$0.477845\pi$
$572$	0	0
$573$	316.466i	0.552297i
$574$	0	0
$575$	497.092	0.864509
$576$	0	0
$577$	− 527.869i	− 0.914850i	−0.889248	−	0.457425i	$-0.848772\pi$
$577$	− 527.869i	− 0.914850i	0.889248	−	0.457425i	$-0.151228\pi$
$578$	0	0
$579$	232.621i	0.401764i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	404.278	0.693444
$584$	0	0
$585$	−63.7877	−0.109039
$586$	0	0
$587$	1123.75i	1.91440i	0.289428	+	0.957200i	$0.406535\pi$
$587$	1123.75i	1.91440i	−0.289428	+	0.957200i	$0.593465\pi$
$588$	0	0
$589$	−383.477	−0.651065
$590$	0	0
$591$	70.7430i	0.119700i
$592$	0	0
$593$	− 816.159i	− 1.37632i	−0.725558	−	0.688161i	$-0.758418\pi$
$593$	− 816.159i	− 1.37632i	0.725558	−	0.688161i	$-0.241582\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−575.317	−0.963680
$598$	0	0
$599$	242.867	0.405455	0.202727	−	0.979235i	$-0.435019\pi$
$599$	242.867	0.405455	0.202727	+	0.979235i	$0.435019\pi$
$600$	0	0
$601$	687.163i	1.14337i	0.820475	+	0.571683i	$0.193709\pi$
$601$	687.163i	1.14337i	−0.820475	+	0.571683i	$0.806291\pi$
$602$	0	0
$603$	332.876	0.552034
$604$	0	0
$605$	− 627.780i	− 1.03765i
$606$	0	0
$607$	107.603i	0.177270i	0.996064	+	0.0886350i	$0.0282505\pi$
$607$	107.603i	0.177270i	−0.996064	+	0.0886350i	$0.971750\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−52.4070	−0.0857725
$612$	0	0
$613$	−648.141	−1.05733	−0.528663	−	0.848832i	$-0.677307\pi$
$613$	−648.141	−1.05733	−0.528663	+	0.848832i	$0.677307\pi$
$614$	0	0
$615$	99.9942i	0.162592i
$616$	0	0
$617$	795.796	1.28978	0.644891	−	0.764274i	$-0.276903\pi$
$617$	795.796	1.28978	0.644891	+	0.764274i	$0.276903\pi$
$618$	0	0
$619$	− 256.766i	− 0.414808i	−0.978255	−	0.207404i	$-0.933499\pi$
$619$	− 256.766i	− 0.414808i	0.978255	−	0.207404i	$-0.0665014\pi$
$620$	0	0
$621$	128.912i	0.207587i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	277.389	0.443823
$626$	0	0
$627$	−613.090	−0.977815
$628$	0	0
$629$	100.032i	0.159033i
$630$	0	0
$631$	254.265	0.402956	0.201478	−	0.979493i	$-0.435426\pi$
$631$	254.265	0.402956	0.201478	+	0.979493i	$0.435426\pi$
$632$	0	0
$633$	36.3128i	0.0573662i
$634$	0	0
$635$	305.799i	0.481573i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−169.966	−0.265987
$640$	0	0
$641$	−597.368	−0.931932	−0.465966	−	0.884803i	$-0.654293\pi$
$641$	−597.368	−0.931932	−0.465966	+	0.884803i	$0.654293\pi$
$642$	0	0
$643$	− 1246.06i	− 1.93789i	−0.247285	−	0.968943i	$-0.579538\pi$
$643$	− 1246.06i	− 1.93789i	0.247285	−	0.968943i	$-0.420462\pi$
$644$	0	0
$645$	−125.631	−0.194778
$646$	0	0
$647$	339.682i	0.525011i	0.964931	+	0.262505i	$0.0845488\pi$
$647$	339.682i	0.525011i	−0.964931	+	0.262505i	$0.915451\pi$
$648$	0	0
$649$	1763.47i	2.71721i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−60.6679	−0.0929064	−0.0464532	−	0.998920i	$-0.514792\pi$
$653$	−60.6679	−0.0929064	−0.0464532	+	0.998920i	$0.514792\pi$
$654$	0	0
$655$	325.091	0.496322
$656$	0	0
$657$	243.262i	0.370262i
$658$	0	0
$659$	−940.346	−1.42693	−0.713465	−	0.700691i	$-0.752875\pi$
$659$	−940.346	−1.42693	−0.713465	+	0.700691i	$0.752875\pi$
$660$	0	0
$661$	55.3497i	0.0837363i	0.999123	+	0.0418681i	$0.0133309\pi$
$661$	55.3497i	0.0837363i	−0.999123	+	0.0418681i	$0.986669\pi$
$662$	0	0
$663$	50.4837i	0.0761443i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	732.400	1.09805
$668$	0	0
$669$	594.556	0.888724
$670$	0	0
$671$	− 961.940i	− 1.43359i
$672$	0	0
$673$	852.361	1.26651	0.633255	−	0.773944i	$-0.281719\pi$
$673$	852.361	1.26651	0.633255	+	0.773944i	$0.281719\pi$
$674$	0	0
$675$	104.114i	0.154243i
$676$	0	0
$677$	187.396i	0.276803i	0.990376	+	0.138402i	$0.0441964\pi$
$677$	187.396i	0.276803i	−0.990376	+	0.138402i	$0.955804\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	275.951	0.405214
$682$	0	0
$683$	340.416	0.498414	0.249207	−	0.968450i	$-0.419830\pi$
$683$	340.416	0.498414	0.249207	+	0.968450i	$0.419830\pi$
$684$	0	0
$685$	− 348.183i	− 0.508296i
$686$	0	0
$687$	722.818	1.05214
$688$	0	0
$689$	− 192.253i	− 0.279032i
$690$	0	0
$691$	54.9928i	0.0795844i	0.999208	+	0.0397922i	$0.0126696\pi$
$691$	54.9928i	0.0795844i	−0.999208	+	0.0397922i	$0.987330\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	395.286	0.568758
$696$	0	0
$697$	79.1387	0.113542
$698$	0	0
$699$	− 510.243i	− 0.729961i
$700$	0	0
$701$	404.818	0.577486	0.288743	−	0.957407i	$-0.406763\pi$
$701$	404.818	0.577486	0.288743	+	0.957407i	$0.406763\pi$
$702$	0	0
$703$	577.704i	0.821769i
$704$	0	0
$705$	− 21.1886i	− 0.0300547i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−773.202	−1.09055	−0.545277	−	0.838256i	$-0.683575\pi$
$709$	−773.202	−1.09055	−0.545277	+	0.838256i	$0.683575\pi$
$710$	0	0
$711$	−321.592	−0.452309
$712$	0	0
$713$	539.417i	0.756546i
$714$	0	0
$715$	−426.731	−0.596827
$716$	0	0
$717$	− 329.192i	− 0.459123i
$718$	0	0
$719$	1153.25i	1.60397i	0.597345	+	0.801985i	$0.296222\pi$
$719$	1153.25i	1.60397i	−0.597345	+	0.801985i	$0.703778\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	738.904	1.02200
$724$	0	0
$725$	591.514	0.815882
$726$	0	0
$727$	− 432.079i	− 0.594331i	−0.954826	−	0.297166i	$-0.903959\pi$
$727$	− 432.079i	− 0.594331i	0.954826	−	0.297166i	$-0.0960413\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	99.4289i	0.136018i
$732$	0	0
$733$	− 626.422i	− 0.854601i	−0.904110	−	0.427300i	$-0.859465\pi$
$733$	− 626.422i	− 0.854601i	0.904110	−	0.427300i	$-0.140535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2226.90	3.02157
$738$	0	0
$739$	26.8808	0.0363746	0.0181873	−	0.999835i	$-0.494210\pi$
$739$	26.8808	0.0363746	0.0181873	+	0.999835i	$0.494210\pi$
$740$	0	0
$741$	291.553i	0.393459i
$742$	0	0
$743$	43.0835	0.0579858	0.0289929	−	0.999580i	$-0.490770\pi$
$743$	43.0835	0.0579858	0.0289929	+	0.999580i	$0.490770\pi$
$744$	0	0
$745$	− 518.874i	− 0.696475i
$746$	0	0
$747$	− 366.599i	− 0.490762i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1253.57	1.66920	0.834599	−	0.550859i	$-0.185700\pi$
$751$	1253.57	1.66920	0.834599	+	0.550859i	$0.185700\pi$
$752$	0	0
$753$	−510.372	−0.677785
$754$	0	0
$755$	304.194i	0.402906i
$756$	0	0
$757$	1417.09	1.87199	0.935994	−	0.352016i	$-0.114504\pi$
$757$	1417.09	1.87199	0.935994	+	0.352016i	$0.114504\pi$
$758$	0	0
$759$	862.402i	1.13623i
$760$	0	0
$761$	263.209i	0.345872i	0.984933	+	0.172936i	$0.0553254\pi$
$761$	263.209i	0.345872i	−0.984933	+	0.172936i	$0.944675\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−20.4110	−0.0266810
$766$	0	0
$767$	838.612	1.09337
$768$	0	0
$769$	802.451i	1.04350i	0.853099	+	0.521750i	$0.174721\pi$
$769$	802.451i	1.04350i	−0.853099	+	0.521750i	$0.825279\pi$
$770$	0	0
$771$	−127.707	−0.165638
$772$	0	0
$773$	436.597i	0.564808i	0.959296	+	0.282404i	$0.0911319\pi$
$773$	436.597i	0.564808i	−0.959296	+	0.282404i	$0.908868\pi$
$774$	0	0
$775$	435.654i	0.562134i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	457.041	0.586702
$780$	0	0
$781$	−1137.05	−1.45589
$782$	0	0
$783$	153.398i	0.195911i
$784$	0	0
$785$	684.894	0.872476
$786$	0	0
$787$	− 1506.37i	− 1.91407i	−0.289974	−	0.957035i	$-0.593646\pi$
$787$	− 1506.37i	− 1.91407i	0.289974	−	0.957035i	$-0.406354\pi$
$788$	0	0
$789$	− 130.483i	− 0.165377i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−457.447	−0.576856
$794$	0	0
$795$	77.7294	0.0977728
$796$	0	0
$797$	1134.30i	1.42321i	0.702581	+	0.711603i	$0.252031\pi$
$797$	1134.30i	1.42321i	−0.702581	+	0.711603i	$0.747969\pi$
$798$	0	0
$799$	−16.7693	−0.0209879
$800$	0	0
$801$	− 400.695i	− 0.500243i
$802$	0	0
$803$	1627.39i	2.02664i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	724.202	0.897400
$808$	0	0
$809$	−975.020	−1.20522	−0.602608	−	0.798037i	$-0.705872\pi$
$809$	−975.020	−1.20522	−0.602608	+	0.798037i	$0.705872\pi$
$810$	0	0
$811$	− 356.666i	− 0.439785i	−0.975524	−	0.219893i	$-0.929429\pi$
$811$	− 356.666i	− 0.439785i	0.975524	−	0.219893i	$-0.0705707\pi$
$812$	0	0
$813$	−755.694	−0.929513
$814$	0	0
$815$	− 306.782i	− 0.376419i
$816$	0	0
$817$	574.221i	0.702841i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1303.78	−1.58804	−0.794019	−	0.607892i	$-0.792015\pi$
$821$	−1303.78	−1.58804	−0.794019	+	0.607892i	$0.792015\pi$
$822$	0	0
$823$	−434.258	−0.527653	−0.263826	−	0.964570i	$-0.584985\pi$
$823$	−434.258	−0.527653	−0.263826	+	0.964570i	$0.584985\pi$
$824$	0	0
$825$	696.508i	0.844253i
$826$	0	0
$827$	809.328	0.978631	0.489315	−	0.872107i	$-0.337247\pi$
$827$	809.328	0.978631	0.489315	+	0.872107i	$0.337247\pi$
$828$	0	0
$829$	380.094i	0.458497i	0.973368	+	0.229248i	$0.0736268\pi$
$829$	380.094i	0.458497i	−0.973368	+	0.229248i	$0.926373\pi$
$830$	0	0
$831$	− 346.293i	− 0.416718i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	108.931	0.130456
$836$	0	0
$837$	−112.979	−0.134980
$838$	0	0
$839$	589.288i	0.702370i	0.936306	+	0.351185i	$0.114221\pi$
$839$	589.288i	0.702370i	−0.936306	+	0.351185i	$0.885779\pi$
$840$	0	0
$841$	30.5180	0.0362877
$842$	0	0
$843$	− 374.976i	− 0.444811i
$844$	0	0
$845$	− 173.574i	− 0.205413i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	822.495	0.968781
$850$	0	0
$851$	812.626	0.954907
$852$	0	0
$853$	− 115.367i	− 0.135249i	−0.997711	−	0.0676244i	$-0.978458\pi$
$853$	− 115.367i	− 0.135249i	0.997711	−	0.0676244i	$-0.0215420\pi$
$854$	0	0
$855$	−117.877	−0.137868
$856$	0	0
$857$	− 593.647i	− 0.692703i	−0.938105	−	0.346352i	$-0.887420\pi$
$857$	− 593.647i	− 0.692703i	0.938105	−	0.346352i	$-0.112580\pi$
$858$	0	0
$859$	− 534.129i	− 0.621803i	−0.950442	−	0.310902i	$-0.899369\pi$
$859$	− 534.129i	− 0.621803i	0.950442	−	0.310902i	$-0.100631\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−281.026	−0.325639	−0.162819	−	0.986656i	$-0.552059\pi$
$863$	−281.026	−0.325639	−0.162819	+	0.986656i	$0.552059\pi$
$864$	0	0
$865$	−390.766	−0.451752
$866$	0	0
$867$	− 484.409i	− 0.558718i
$868$	0	0
$869$	−2151.41	−2.47573
$870$	0	0
$871$	− 1058.99i	− 1.21584i
$872$	0	0
$873$	36.9071i	0.0422762i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	158.299	0.180500	0.0902501	−	0.995919i	$-0.471233\pi$
$877$	158.299	0.180500	0.0902501	+	0.995919i	$0.471233\pi$
$878$	0	0
$879$	−658.459	−0.749100
$880$	0	0
$881$	− 371.670i	− 0.421873i	−0.977500	−	0.210937i	$-0.932349\pi$
$881$	− 371.670i	− 0.421873i	0.977500	−	0.210937i	$-0.0676513\pi$
$882$	0	0
$883$	1192.21	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$883$	1192.21	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$884$	0	0
$885$	339.057i	0.383116i
$886$	0	0
$887$	194.236i	0.218981i	0.993988	+	0.109491i	$0.0349219\pi$
$887$	194.236i	0.218981i	−0.993988	+	0.109491i	$0.965078\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−180.626	−0.202723
$892$	0	0
$893$	−96.8461	−0.108450
$894$	0	0
$895$	355.826i	0.397571i
$896$	0	0
$897$	410.112	0.457204
$898$	0	0
$899$	641.878i	0.713992i
$900$	0	0
$901$	− 61.5176i	− 0.0682770i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	692.917	0.765654
$906$	0	0
$907$	1642.44	1.81084	0.905422	−	0.424512i	$-0.139554\pi$
$907$	1642.44	1.81084	0.905422	+	0.424512i	$0.139554\pi$
$908$	0	0
$909$	213.124i	0.234460i
$910$	0	0
$911$	1176.96	1.29194	0.645972	−	0.763361i	$-0.276452\pi$
$911$	1176.96	1.29194	0.645972	+	0.763361i	$0.276452\pi$
$912$	0	0
$913$	− 2452.50i	− 2.68620i
$914$	0	0
$915$	− 184.950i	− 0.202131i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−674.311	−0.733744	−0.366872	−	0.930271i	$-0.619571\pi$
$919$	−674.311	−0.733744	−0.366872	+	0.930271i	$0.619571\pi$
$920$	0	0
$921$	−122.699	−0.133223
$922$	0	0
$923$	540.719i	0.585828i
$924$	0	0
$925$	656.307	0.709521
$926$	0	0
$927$	− 129.268i	− 0.139447i
$928$	0	0
$929$	1066.17i	1.14765i	0.818978	+	0.573825i	$0.194541\pi$
$929$	1066.17i	1.14765i	−0.818978	+	0.573825i	$0.805459\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−482.187	−0.516813
$934$	0	0
$935$	−136.547	−0.146039
$936$	0	0
$937$	439.829i	0.469401i	0.972068	+	0.234700i	$0.0754109\pi$
$937$	439.829i	0.469401i	−0.972068	+	0.234700i	$0.924589\pi$
$938$	0	0
$939$	−61.9045	−0.0659260
$940$	0	0
$941$	− 134.763i	− 0.143213i	−0.997433	−	0.0716063i	$-0.977187\pi$
$941$	− 134.763i	− 0.143213i	0.997433	−	0.0716063i	$-0.0228125\pi$
$942$	0	0
$943$	− 642.896i	− 0.681756i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1323.24	1.39729	0.698646	−	0.715468i	$-0.253786\pi$
$947$	1323.24	1.39729	0.698646	+	0.715468i	$0.253786\pi$
$948$	0	0
$949$	773.901	0.815491
$950$	0	0
$951$	937.660i	0.985973i
$952$	0	0
$953$	−507.028	−0.532033	−0.266017	−	0.963968i	$-0.585708\pi$
$953$	−507.028	−0.532033	−0.266017	+	0.963968i	$0.585708\pi$
$954$	0	0
$955$	407.052i	0.426233i
$956$	0	0
$957$	1026.21i	1.07232i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	488.253	0.508067
$962$	0	0
$963$	155.450	0.161423
$964$	0	0
$965$	299.207i	0.310059i
$966$	0	0
$967$	965.582	0.998533	0.499267	−	0.866448i	$-0.333603\pi$
$967$	965.582	0.998533	0.499267	+	0.866448i	$0.333603\pi$
$968$	0	0
$969$	93.2919i	0.0962765i
$970$	0	0
$971$	− 964.565i	− 0.993373i	−0.867930	−	0.496687i	$-0.834550\pi$
$971$	− 964.565i	− 0.993373i	0.867930	−	0.496687i	$-0.165450\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	331.222	0.339715
$976$	0	0
$977$	−786.333	−0.804845	−0.402422	−	0.915454i	$-0.631832\pi$
$977$	−786.333	−0.804845	−0.402422	+	0.915454i	$0.631832\pi$
$978$	0	0
$979$	− 2680.59i	− 2.73809i
$980$	0	0
$981$	−447.803	−0.456476
$982$	0	0
$983$	− 430.391i	− 0.437834i	−0.975743	−	0.218917i	$-0.929748\pi$
$983$	− 430.391i	− 0.437834i	0.975743	−	0.218917i	$-0.0702524\pi$
$984$	0	0
$985$	90.9927i	0.0923784i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	807.727	0.816710
$990$	0	0
$991$	−721.038	−0.727586	−0.363793	−	0.931480i	$-0.618518\pi$
$991$	−721.038	−0.727586	−0.363793	+	0.931480i	$0.618518\pi$
$992$	0	0
$993$	− 166.885i	− 0.168061i
$994$	0	0
$995$	−739.997	−0.743716
$996$	0	0
$997$	− 6.33734i	− 0.00635641i	−0.999995	−	0.00317820i	$-0.998988\pi$
$997$	− 6.33734i	− 0.00635641i	0.999995	−	0.00317820i	$-0.00101166\pi$
$998$	0	0
$999$	170.201i	0.170371i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.f.d.97.4	✓	16
3.2	odd	2		3528.3.f.j.2449.9		16
4.3	odd	2		2352.3.f.m.97.12		16
7.2	even	3		1176.3.z.h.913.4		16
7.3	odd	6		1176.3.z.h.313.4		16
7.4	even	3		1176.3.z.g.313.5		16
7.5	odd	6		1176.3.z.g.913.5		16
7.6	odd	2	inner	1176.3.f.d.97.13	yes	16
21.20	even	2		3528.3.f.j.2449.8		16
28.27	even	2		2352.3.f.m.97.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.f.d.97.4	✓	16	1.1	even	1	trivial
1176.3.f.d.97.13	yes	16	7.6	odd	2	inner
1176.3.z.g.313.5		16	7.4	even	3
1176.3.z.g.913.5		16	7.5	odd	6
1176.3.z.h.313.4		16	7.3	odd	6
1176.3.z.h.913.4		16	7.2	even	3
2352.3.f.m.97.5		16	28.27	even	2
2352.3.f.m.97.12		16	4.3	odd	2
3528.3.f.j.2449.8		16	21.20	even	2
3528.3.f.j.2449.9		16	3.2	odd	2

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.22784i q^{5} -3.00000 q^{9} -20.0696 q^{11} +9.54403i q^{13} -3.85873 q^{15} +3.05393i q^{17} +17.6370i q^{19} +24.8091 q^{23} +20.0367 q^{25} +5.19615i q^{27} +29.5215 q^{29} +21.7428i q^{31} +34.7616i q^{33} +32.7552 q^{37} +16.5307 q^{39} -25.9138i q^{41} +32.5577 q^{43} +6.68352i q^{45} +5.49107i q^{47} +5.28955 q^{51} -20.1438 q^{53} +44.7118i q^{55} +30.5482 q^{57} -87.8677i q^{59} +47.9302i q^{61} +21.2626 q^{65} -110.959 q^{67} -42.9705i q^{69} +56.6552 q^{71} -81.0874i q^{73} -34.7046i q^{75} +107.197 q^{79} +9.00000 q^{81} +122.200i q^{83} +6.80365 q^{85} -51.1327i q^{87} +133.565i q^{89} +37.6596 q^{93} +39.2924 q^{95} -12.3024i q^{97} +60.2088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) \) 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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