LMFDB - Embedded newform 1188.3.e.c.485.2

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			485.2
Root			$4.98184i$ of defining polynomial
Character	$\chi$	$=$	1188.485
Dual form			1188.3.e.c.485.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.30468i q^{5} -0.564603 q^{7} -3.31662i q^{11} +19.8624 q^{13} +9.16530i q^{17} -2.84379 q^{19} -28.1995i q^{23} -3.13959 q^{25} +13.6044i q^{29} +31.1872 q^{31} +2.99504i q^{35} +11.9564 q^{37} +15.0484i q^{41} +0.558376 q^{43} -70.4155i q^{47} -48.6812 q^{49} -43.3730i q^{53} -17.5936 q^{55} -35.2353i q^{59} -56.4788 q^{61} -105.364i q^{65} +37.3372 q^{67} -100.568i q^{71} -101.202 q^{73} +1.87258i q^{77} +31.2293 q^{79} -14.3919i q^{83} +48.6190 q^{85} +50.1036i q^{89} -11.2144 q^{91} +15.0854i q^{95} +45.8208 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 5.30468i	− 1.06094i	−0.847705	−	0.530468i	$-0.822016\pi$
$5$	− 5.30468i	− 1.06094i	0.847705	−	0.530468i	$-0.177984\pi$
$6$	0	0
$7$	−0.564603	−0.0806575	−0.0403288	−	0.999186i	$-0.512841\pi$
$7$	−0.564603	−0.0806575	−0.0403288	+	0.999186i	$0.512841\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	0	0
$13$	19.8624	1.52788	0.763939	−	0.645288i	$-0.223263\pi$
$13$	19.8624	1.52788	0.763939	+	0.645288i	$0.223263\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.16530i	0.539136i	0.962981	+	0.269568i	$0.0868808\pi$
$17$	9.16530i	0.539136i	−0.962981	+	0.269568i	$0.913119\pi$
$18$	0	0
$19$	−2.84379	−0.149673	−0.0748366	−	0.997196i	$-0.523844\pi$
$19$	−2.84379	−0.149673	−0.0748366	+	0.997196i	$0.523844\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 28.1995i	− 1.22606i	−0.790058	−	0.613032i	$-0.789950\pi$
$23$	− 28.1995i	− 1.22606i	0.790058	−	0.613032i	$-0.210050\pi$
$24$	0	0
$25$	−3.13959	−0.125584
$26$	0	0
$27$	0	0
$28$	0	0
$29$	13.6044i	0.469117i	0.972102	+	0.234558i	$0.0753644\pi$
$29$	13.6044i	0.469117i	−0.972102	+	0.234558i	$0.924636\pi$
$30$	0	0
$31$	31.1872	1.00604	0.503020	−	0.864275i	$-0.332222\pi$
$31$	31.1872	1.00604	0.503020	+	0.864275i	$0.332222\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.99504i	0.0855724i
$36$	0	0
$37$	11.9564	0.323145	0.161573	−	0.986861i	$-0.448343\pi$
$37$	11.9564	0.323145	0.161573	+	0.986861i	$0.448343\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	15.0484i	0.367035i	0.983016	+	0.183518i	$0.0587484\pi$
$41$	15.0484i	0.367035i	−0.983016	+	0.183518i	$0.941252\pi$
$42$	0	0
$43$	0.558376	0.0129855	0.00649274	−	0.999979i	$-0.497933\pi$
$43$	0.558376	0.0129855	0.00649274	+	0.999979i	$0.497933\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 70.4155i	− 1.49820i	−0.662455	−	0.749101i	$-0.730486\pi$
$47$	− 70.4155i	− 1.49820i	0.662455	−	0.749101i	$-0.269514\pi$
$48$	0	0
$49$	−48.6812	−0.993494
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 43.3730i	− 0.818358i	−0.912454	−	0.409179i	$-0.865815\pi$
$53$	− 43.3730i	− 0.818358i	0.912454	−	0.409179i	$-0.134185\pi$
$54$	0	0
$55$	−17.5936	−0.319884
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 35.2353i	− 0.597209i	−0.954377	−	0.298604i	$-0.903479\pi$
$59$	− 35.2353i	− 0.597209i	0.954377	−	0.298604i	$-0.0965211\pi$
$60$	0	0
$61$	−56.4788	−0.925883	−0.462941	−	0.886389i	$-0.653206\pi$
$61$	−56.4788	−0.925883	−0.462941	+	0.886389i	$0.653206\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 105.364i	− 1.62098i
$66$	0	0
$67$	37.3372	0.557272	0.278636	−	0.960397i	$-0.410118\pi$
$67$	37.3372	0.557272	0.278636	+	0.960397i	$0.410118\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 100.568i	− 1.41644i	−0.705990	−	0.708222i	$-0.749498\pi$
$71$	− 100.568i	− 1.41644i	0.705990	−	0.708222i	$-0.250502\pi$
$72$	0	0
$73$	−101.202	−1.38632	−0.693161	−	0.720783i	$-0.743783\pi$
$73$	−101.202	−1.38632	−0.693161	+	0.720783i	$0.743783\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.87258i	0.0243192i
$78$	0	0
$79$	31.2293	0.395307	0.197654	−	0.980272i	$-0.436668\pi$
$79$	31.2293	0.395307	0.197654	+	0.980272i	$0.436668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 14.3919i	− 0.173396i	−0.996235	−	0.0866981i	$-0.972368\pi$
$83$	− 14.3919i	− 0.173396i	0.996235	−	0.0866981i	$-0.0276316\pi$
$84$	0	0
$85$	48.6190	0.571988
$86$	0	0
$87$	0	0
$88$	0	0
$89$	50.1036i	0.562961i	0.959567	+	0.281481i	$0.0908255\pi$
$89$	50.1036i	0.562961i	−0.959567	+	0.281481i	$0.909174\pi$
$90$	0	0
$91$	−11.2144	−0.123235
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.0854i	0.158794i
$96$	0	0
$97$	45.8208	0.472380	0.236190	−	0.971707i	$-0.424101\pi$
$97$	45.8208	0.472380	0.236190	+	0.971707i	$0.424101\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 149.138i	− 1.47662i	−0.674464	−	0.738308i	$-0.735625\pi$
$101$	− 149.138i	− 1.47662i	0.674464	−	0.738308i	$-0.264375\pi$
$102$	0	0
$103$	77.8057	0.755395	0.377698	−	0.925929i	$-0.376716\pi$
$103$	77.8057	0.755395	0.377698	+	0.925929i	$0.376716\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 20.4889i	− 0.191485i	−0.995406	−	0.0957425i	$-0.969477\pi$
$107$	− 20.4889i	− 0.191485i	0.995406	−	0.0957425i	$-0.0305225\pi$
$108$	0	0
$109$	−159.379	−1.46219	−0.731096	−	0.682275i	$-0.760991\pi$
$109$	−159.379	−1.46219	−0.731096	+	0.682275i	$0.760991\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	75.9415i	0.672049i	0.941853	+	0.336025i	$0.109083\pi$
$113$	75.9415i	0.672049i	−0.941853	+	0.336025i	$0.890917\pi$
$114$	0	0
$115$	−149.589	−1.30077
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 5.17476i	− 0.0434853i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 115.962i	− 0.927699i
$126$	0	0
$127$	152.251	1.19883	0.599414	−	0.800439i	$-0.295400\pi$
$127$	152.251	1.19883	0.599414	+	0.800439i	$0.295400\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.2673i	0.0936433i	0.998903	+	0.0468217i	$0.0149092\pi$
$131$	12.2673i	0.0936433i	−0.998903	+	0.0468217i	$0.985091\pi$
$132$	0	0
$133$	1.60561	0.0120723
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 225.242i	− 1.64410i	−0.569415	−	0.822051i	$-0.692830\pi$
$137$	− 225.242i	− 1.64410i	0.569415	−	0.822051i	$-0.307170\pi$
$138$	0	0
$139$	−5.08546	−0.0365860	−0.0182930	−	0.999833i	$-0.505823\pi$
$139$	−5.08546	−0.0365860	−0.0182930	+	0.999833i	$0.505823\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 65.8762i	− 0.460673i
$144$	0	0
$145$	72.1669	0.497703
$146$	0	0
$147$	0	0
$148$	0	0
$149$	80.6159i	0.541047i	0.962713	+	0.270523i	$0.0871967\pi$
$149$	80.6159i	0.541047i	−0.962713	+	0.270523i	$0.912803\pi$
$150$	0	0
$151$	35.0789	0.232310	0.116155	−	0.993231i	$-0.462943\pi$
$151$	35.0789	0.232310	0.116155	+	0.993231i	$0.462943\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 165.438i	− 1.06734i
$156$	0	0
$157$	−62.4163	−0.397556	−0.198778	−	0.980045i	$-0.563697\pi$
$157$	−62.4163	−0.397556	−0.198778	+	0.980045i	$0.563697\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.9215i	0.0988913i
$162$	0	0
$163$	61.6952	0.378498	0.189249	−	0.981929i	$-0.439395\pi$
$163$	61.6952	0.378498	0.189249	+	0.981929i	$0.439395\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 294.745i	− 1.76494i	−0.470370	−	0.882469i	$-0.655879\pi$
$167$	− 294.745i	− 1.76494i	0.470370	−	0.882469i	$-0.344121\pi$
$168$	0	0
$169$	225.516	1.33441
$170$	0	0
$171$	0	0
$172$	0	0
$173$	55.4345i	0.320431i	0.987082	+	0.160215i	$0.0512189\pi$
$173$	55.4345i	0.320431i	−0.987082	+	0.160215i	$0.948781\pi$
$174$	0	0
$175$	1.77262	0.0101293
$176$	0	0
$177$	0	0
$178$	0	0
$179$	99.7982i	0.557532i	0.960359	+	0.278766i	$0.0899253\pi$
$179$	99.7982i	0.557532i	−0.960359	+	0.278766i	$0.910075\pi$
$180$	0	0
$181$	295.847	1.63451	0.817256	−	0.576275i	$-0.195494\pi$
$181$	295.847	1.63451	0.817256	+	0.576275i	$0.195494\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 63.4247i	− 0.342836i
$186$	0	0
$187$	30.3979	0.162555
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 214.856i	− 1.12490i	−0.826830	−	0.562451i	$-0.809858\pi$
$191$	− 214.856i	− 1.12490i	0.826830	−	0.562451i	$-0.190142\pi$
$192$	0	0
$193$	−118.089	−0.611859	−0.305930	−	0.952054i	$-0.598967\pi$
$193$	−118.089	−0.611859	−0.305930	+	0.952054i	$0.598967\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	281.424i	1.42855i	0.699867	+	0.714273i	$0.253243\pi$
$197$	281.424i	1.42855i	−0.699867	+	0.714273i	$0.746757\pi$
$198$	0	0
$199$	54.1057	0.271888	0.135944	−	0.990717i	$-0.456593\pi$
$199$	54.1057	0.271888	0.135944	+	0.990717i	$0.456593\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.68108i	− 0.0378378i
$204$	0	0
$205$	79.8271	0.389400
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.43179i	0.0451282i
$210$	0	0
$211$	26.0405	0.123415	0.0617075	−	0.998094i	$-0.480345\pi$
$211$	26.0405	0.123415	0.0617075	+	0.998094i	$0.480345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 2.96200i	− 0.0137768i
$216$	0	0
$217$	−17.6084	−0.0811447
$218$	0	0
$219$	0	0
$220$	0	0
$221$	182.045i	0.823734i
$222$	0	0
$223$	−216.866	−0.972494	−0.486247	−	0.873821i	$-0.661635\pi$
$223$	−216.866	−0.972494	−0.486247	+	0.873821i	$0.661635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 310.745i	− 1.36892i	−0.729049	−	0.684461i	$-0.760038\pi$
$227$	− 310.745i	− 1.36892i	0.729049	−	0.684461i	$-0.239962\pi$
$228$	0	0
$229$	−327.571	−1.43044	−0.715220	−	0.698900i	$-0.753673\pi$
$229$	−327.571	−1.43044	−0.715220	+	0.698900i	$0.753673\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	194.165i	0.833326i	0.909061	+	0.416663i	$0.136800\pi$
$233$	194.165i	0.833326i	−0.909061	+	0.416663i	$0.863200\pi$
$234$	0	0
$235$	−373.532	−1.58950
$236$	0	0
$237$	0	0
$238$	0	0
$239$	378.856i	1.58517i	0.609760	+	0.792586i	$0.291266\pi$
$239$	378.856i	1.58517i	−0.609760	+	0.792586i	$0.708734\pi$
$240$	0	0
$241$	148.542	0.616357	0.308179	−	0.951328i	$-0.400281\pi$
$241$	148.542	0.616357	0.308179	+	0.951328i	$0.400281\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	258.238i	1.05403i
$246$	0	0
$247$	−56.4846	−0.228682
$248$	0	0
$249$	0	0
$250$	0	0
$251$	110.768i	0.441306i	0.975352	+	0.220653i	$0.0708189\pi$
$251$	110.768i	0.441306i	−0.975352	+	0.220653i	$0.929181\pi$
$252$	0	0
$253$	−93.5270	−0.369672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	186.962i	0.727480i	0.931501	+	0.363740i	$0.118500\pi$
$257$	186.962i	0.727480i	−0.931501	+	0.363740i	$0.881500\pi$
$258$	0	0
$259$	−6.75061	−0.0260641
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 235.994i	− 0.897317i	−0.893703	−	0.448659i	$-0.851902\pi$
$263$	− 235.994i	− 0.897317i	0.893703	−	0.448659i	$-0.148098\pi$
$264$	0	0
$265$	−230.080	−0.868225
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 190.384i	− 0.707747i	−0.935293	−	0.353874i	$-0.884864\pi$
$269$	− 190.384i	− 0.707747i	0.935293	−	0.353874i	$-0.115136\pi$
$270$	0	0
$271$	494.840	1.82598	0.912989	−	0.407983i	$-0.133768\pi$
$271$	494.840	1.82598	0.912989	+	0.407983i	$0.133768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.4129i	0.0378649i
$276$	0	0
$277$	−56.1559	−0.202729	−0.101364	−	0.994849i	$-0.532321\pi$
$277$	−56.1559	−0.202729	−0.101364	+	0.994849i	$0.532321\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	121.371i	0.431924i	0.976402	+	0.215962i	$0.0692888\pi$
$281$	121.371i	0.431924i	−0.976402	+	0.215962i	$0.930711\pi$
$282$	0	0
$283$	−6.76598	−0.0239081	−0.0119540	−	0.999929i	$-0.503805\pi$
$283$	−6.76598	−0.0239081	−0.0119540	+	0.999929i	$0.503805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.49639i	− 0.0296041i
$288$	0	0
$289$	204.997	0.709333
$290$	0	0
$291$	0	0
$292$	0	0
$293$	576.251i	1.96673i	0.181651	+	0.983363i	$0.441856\pi$
$293$	576.251i	1.96673i	−0.181651	+	0.983363i	$0.558144\pi$
$294$	0	0
$295$	−186.912	−0.633600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 560.110i	− 1.87328i
$300$	0	0
$301$	−0.315261	−0.00104738
$302$	0	0
$303$	0	0
$304$	0	0
$305$	299.602i	0.982302i
$306$	0	0
$307$	−488.407	−1.59090	−0.795451	−	0.606018i	$-0.792766\pi$
$307$	−488.407	−1.59090	−0.795451	+	0.606018i	$0.792766\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	249.127i	0.801052i	0.916285	+	0.400526i	$0.131173\pi$
$311$	249.127i	0.801052i	−0.916285	+	0.400526i	$0.868827\pi$
$312$	0	0
$313$	81.3321	0.259847	0.129923	−	0.991524i	$-0.458527\pi$
$313$	81.3321	0.259847	0.129923	+	0.991524i	$0.458527\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 327.067i	− 1.03176i	−0.856661	−	0.515879i	$-0.827465\pi$
$317$	− 327.067i	− 1.03176i	0.856661	−	0.515879i	$-0.172535\pi$
$318$	0	0
$319$	45.1207	0.141444
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 26.0642i	− 0.0806941i
$324$	0	0
$325$	−62.3599	−0.191877
$326$	0	0
$327$	0	0
$328$	0	0
$329$	39.7568i	0.120841i
$330$	0	0
$331$	−406.079	−1.22682	−0.613412	−	0.789763i	$-0.710204\pi$
$331$	−406.079	−1.22682	−0.613412	+	0.789763i	$0.710204\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 198.062i	− 0.591230i
$336$	0	0
$337$	451.688	1.34032	0.670160	−	0.742217i	$-0.266225\pi$
$337$	451.688	1.34032	0.670160	+	0.742217i	$0.266225\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 103.436i	− 0.303333i
$342$	0	0
$343$	55.1511	0.160790
$344$	0	0
$345$	0	0
$346$	0	0
$347$	345.330i	0.995187i	0.867410	+	0.497594i	$0.165783\pi$
$347$	345.330i	0.995187i	−0.867410	+	0.497594i	$0.834217\pi$
$348$	0	0
$349$	409.992	1.17476	0.587381	−	0.809310i	$-0.300159\pi$
$349$	409.992	1.17476	0.587381	+	0.809310i	$0.300159\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	314.931i	0.892156i	0.894994	+	0.446078i	$0.147180\pi$
$353$	314.931i	0.892156i	−0.894994	+	0.446078i	$0.852820\pi$
$354$	0	0
$355$	−533.478	−1.50276
$356$	0	0
$357$	0	0
$358$	0	0
$359$	660.887i	1.84091i	0.390847	+	0.920455i	$0.372182\pi$
$359$	660.887i	1.84091i	−0.390847	+	0.920455i	$0.627818\pi$
$360$	0	0
$361$	−352.913	−0.977598
$362$	0	0
$363$	0	0
$364$	0	0
$365$	536.841i	1.47080i
$366$	0	0
$367$	15.9186	0.0433748	0.0216874	−	0.999765i	$-0.493096\pi$
$367$	15.9186	0.0433748	0.0216874	+	0.999765i	$0.493096\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.4885i	0.0660068i
$372$	0	0
$373$	253.307	0.679108	0.339554	−	0.940587i	$-0.389724\pi$
$373$	253.307	0.679108	0.339554	+	0.940587i	$0.389724\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	270.216i	0.716754i
$378$	0	0
$379$	315.972	0.833699	0.416850	−	0.908975i	$-0.363134\pi$
$379$	315.972	0.833699	0.416850	+	0.908975i	$0.363134\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	131.220i	0.342611i	0.985218	+	0.171305i	$0.0547985\pi$
$383$	131.220i	0.342611i	−0.985218	+	0.171305i	$0.945202\pi$
$384$	0	0
$385$	9.93341	0.0258011
$386$	0	0
$387$	0	0
$388$	0	0
$389$	452.761i	1.16391i	0.813221	+	0.581955i	$0.197712\pi$
$389$	452.761i	1.16391i	−0.813221	+	0.581955i	$0.802288\pi$
$390$	0	0
$391$	258.457	0.661014
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 165.661i	− 0.419395i
$396$	0	0
$397$	227.897	0.574048	0.287024	−	0.957923i	$-0.407334\pi$
$397$	227.897	0.574048	0.287024	+	0.957923i	$0.407334\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	192.027i	0.478870i	0.970912	+	0.239435i	$0.0769622\pi$
$401$	192.027i	0.478870i	−0.970912	+	0.239435i	$0.923038\pi$
$402$	0	0
$403$	619.454	1.53711
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 39.6548i	− 0.0974320i
$408$	0	0
$409$	100.235	0.245074	0.122537	−	0.992464i	$-0.460897\pi$
$409$	100.235	0.245074	0.122537	+	0.992464i	$0.460897\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.8940i	0.0481694i
$414$	0	0
$415$	−76.3443	−0.183962
$416$	0	0
$417$	0	0
$418$	0	0
$419$	64.1202i	0.153031i	0.997068	+	0.0765157i	$0.0243795\pi$
$419$	64.1202i	0.153031i	−0.997068	+	0.0765157i	$0.975620\pi$
$420$	0	0
$421$	−410.137	−0.974196	−0.487098	−	0.873347i	$-0.661945\pi$
$421$	−410.137	−0.974196	−0.487098	+	0.873347i	$0.661945\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 28.7753i	− 0.0677067i
$426$	0	0
$427$	31.8881	0.0746794
$428$	0	0
$429$	0	0
$430$	0	0
$431$	409.896i	0.951035i	0.879706	+	0.475517i	$0.157739\pi$
$431$	409.896i	0.951035i	−0.879706	+	0.475517i	$0.842261\pi$
$432$	0	0
$433$	254.402	0.587533	0.293766	−	0.955877i	$-0.405091\pi$
$433$	254.402	0.587533	0.293766	+	0.955877i	$0.405091\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	80.1934i	0.183509i
$438$	0	0
$439$	−90.6491	−0.206490	−0.103245	−	0.994656i	$-0.532923\pi$
$439$	−90.6491	−0.206490	−0.103245	+	0.994656i	$0.532923\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 735.324i	− 1.65987i	−0.557857	−	0.829937i	$-0.688376\pi$
$443$	− 735.324i	− 1.65987i	0.557857	−	0.829937i	$-0.311624\pi$
$444$	0	0
$445$	265.783	0.597266
$446$	0	0
$447$	0	0
$448$	0	0
$449$	747.700i	1.66526i	0.553832	+	0.832628i	$0.313165\pi$
$449$	747.700i	1.66526i	−0.553832	+	0.832628i	$0.686835\pi$
$450$	0	0
$451$	49.9100	0.110665
$452$	0	0
$453$	0	0
$454$	0	0
$455$	59.4887i	0.130744i
$456$	0	0
$457$	−96.9797	−0.212209	−0.106105	−	0.994355i	$-0.533838\pi$
$457$	−96.9797	−0.212209	−0.106105	+	0.994355i	$0.533838\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	242.338i	0.525680i	0.964839	+	0.262840i	$0.0846591\pi$
$461$	242.338i	0.525680i	−0.964839	+	0.262840i	$0.915341\pi$
$462$	0	0
$463$	241.667	0.521960	0.260980	−	0.965344i	$-0.415954\pi$
$463$	241.667	0.521960	0.260980	+	0.965344i	$0.415954\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	284.779i	0.609805i	0.952384	+	0.304903i	$0.0986239\pi$
$467$	284.779i	0.609805i	−0.952384	+	0.304903i	$0.901376\pi$
$468$	0	0
$469$	−21.0807	−0.0449482
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.85192i	− 0.00391527i
$474$	0	0
$475$	8.92835	0.0187965
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 17.7155i	− 0.0369843i	−0.999829	−	0.0184921i	$-0.994113\pi$
$479$	− 17.7155i	− 0.0369843i	0.999829	−	0.0184921i	$-0.00588656\pi$
$480$	0	0
$481$	237.483	0.493727
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 243.065i	− 0.501164i
$486$	0	0
$487$	−475.591	−0.976572	−0.488286	−	0.872684i	$-0.662378\pi$
$487$	−475.591	−0.976572	−0.488286	+	0.872684i	$0.662378\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 551.069i	− 1.12234i	−0.827700	−	0.561170i	$-0.810351\pi$
$491$	− 551.069i	− 1.12234i	0.827700	−	0.561170i	$-0.189649\pi$
$492$	0	0
$493$	−124.688	−0.252918
$494$	0	0
$495$	0	0
$496$	0	0
$497$	56.7807i	0.114247i
$498$	0	0
$499$	749.533	1.50207	0.751035	−	0.660262i	$-0.229555\pi$
$499$	749.533	1.50207	0.751035	+	0.660262i	$0.229555\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	439.461i	0.873681i	0.899539	+	0.436840i	$0.143903\pi$
$503$	439.461i	0.873681i	−0.899539	+	0.436840i	$0.856097\pi$
$504$	0	0
$505$	−791.130	−1.56659
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 172.832i	− 0.339553i	−0.985483	−	0.169776i	$-0.945695\pi$
$509$	− 172.832i	− 0.339553i	0.985483	−	0.169776i	$-0.0543046\pi$
$510$	0	0
$511$	57.1387	0.111817
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 412.734i	− 0.801426i
$516$	0	0
$517$	−233.542	−0.451725
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 843.421i	− 1.61885i	−0.587223	−	0.809425i	$-0.699779\pi$
$521$	− 843.421i	− 1.61885i	0.587223	−	0.809425i	$-0.300221\pi$
$522$	0	0
$523$	−117.956	−0.225538	−0.112769	−	0.993621i	$-0.535972\pi$
$523$	−117.956	−0.225538	−0.112769	+	0.993621i	$0.535972\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	285.841i	0.542392i
$528$	0	0
$529$	−266.210	−0.503232
$530$	0	0
$531$	0	0
$532$	0	0
$533$	298.898i	0.560785i
$534$	0	0
$535$	−108.687	−0.203153
$536$	0	0
$537$	0	0
$538$	0	0
$539$	161.457i	0.299550i
$540$	0	0
$541$	−83.6365	−0.154596	−0.0772980	−	0.997008i	$-0.524629\pi$
$541$	−83.6365	−0.154596	−0.0772980	+	0.997008i	$0.524629\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	845.454i	1.55129i
$546$	0	0
$547$	90.6732	0.165765	0.0828823	−	0.996559i	$-0.473587\pi$
$547$	90.6732	0.165765	0.0828823	+	0.996559i	$0.473587\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 38.6880i	− 0.0702142i
$552$	0	0
$553$	−17.6321	−0.0318845
$554$	0	0
$555$	0	0
$556$	0	0
$557$	126.065i	0.226328i	0.993576	+	0.113164i	$0.0360985\pi$
$557$	126.065i	0.226328i	−0.993576	+	0.113164i	$0.963901\pi$
$558$	0	0
$559$	11.0907	0.0198402
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 574.544i	− 1.02050i	−0.860025	−	0.510252i	$-0.829552\pi$
$563$	− 574.544i	− 1.02050i	0.860025	−	0.510252i	$-0.170448\pi$
$564$	0	0
$565$	402.845	0.713001
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 411.936i	− 0.723964i	−0.932185	−	0.361982i	$-0.882100\pi$
$569$	− 411.936i	− 0.723964i	0.932185	−	0.361982i	$-0.117900\pi$
$570$	0	0
$571$	−421.300	−0.737828	−0.368914	−	0.929464i	$-0.620270\pi$
$571$	−421.300	−0.737828	−0.368914	+	0.929464i	$0.620270\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	88.5349i	0.153974i
$576$	0	0
$577$	−377.236	−0.653788	−0.326894	−	0.945061i	$-0.606002\pi$
$577$	−377.236	−0.653788	−0.326894	+	0.945061i	$0.606002\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.12570i	0.0139857i
$582$	0	0
$583$	−143.852	−0.246744
$584$	0	0
$585$	0	0
$586$	0	0
$587$	417.587i	0.711391i	0.934602	+	0.355696i	$0.115756\pi$
$587$	417.587i	0.711391i	−0.934602	+	0.355696i	$0.884244\pi$
$588$	0	0
$589$	−88.6900	−0.150577
$590$	0	0
$591$	0	0
$592$	0	0
$593$	575.043i	0.969718i	0.874592	+	0.484859i	$0.161129\pi$
$593$	575.043i	0.969718i	−0.874592	+	0.484859i	$0.838871\pi$
$594$	0	0
$595$	−27.4504	−0.0461351
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 170.090i	− 0.283957i	−0.989870	−	0.141978i	$-0.954654\pi$
$599$	− 170.090i	− 0.283957i	0.989870	−	0.141978i	$-0.0453464\pi$
$600$	0	0
$601$	−226.671	−0.377156	−0.188578	−	0.982058i	$-0.560388\pi$
$601$	−226.671	−0.377156	−0.188578	+	0.982058i	$0.560388\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	58.3514i	0.0964487i
$606$	0	0
$607$	961.338	1.58375	0.791876	−	0.610681i	$-0.209104\pi$
$607$	961.338	1.58375	0.791876	+	0.610681i	$0.209104\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1398.62i	− 2.28907i
$612$	0	0
$613$	854.731	1.39434	0.697171	−	0.716905i	$-0.254442\pi$
$613$	854.731	1.39434	0.697171	+	0.716905i	$0.254442\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	497.960i	0.807066i	0.914965	+	0.403533i	$0.132218\pi$
$617$	497.960i	0.807066i	−0.914965	+	0.403533i	$0.867782\pi$
$618$	0	0
$619$	−833.541	−1.34659	−0.673297	−	0.739373i	$-0.735122\pi$
$619$	−833.541	−1.34659	−0.673297	+	0.739373i	$0.735122\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 28.2886i	− 0.0454071i
$624$	0	0
$625$	−693.633	−1.10981
$626$	0	0
$627$	0	0
$628$	0	0
$629$	109.584i	0.174219i
$630$	0	0
$631$	504.769	0.799950	0.399975	−	0.916526i	$-0.369019\pi$
$631$	504.769	0.799950	0.399975	+	0.916526i	$0.369019\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 807.643i	− 1.27188i
$636$	0	0
$637$	−966.927	−1.51794
$638$	0	0
$639$	0	0
$640$	0	0
$641$	168.339i	0.262619i	0.991341	+	0.131309i	$0.0419181\pi$
$641$	168.339i	0.262619i	−0.991341	+	0.131309i	$0.958082\pi$
$642$	0	0
$643$	−760.587	−1.18287	−0.591436	−	0.806352i	$-0.701439\pi$
$643$	−760.587	−1.18287	−0.591436	+	0.806352i	$0.701439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	886.660i	1.37042i	0.728347	+	0.685209i	$0.240289\pi$
$647$	886.660i	1.37042i	−0.728347	+	0.685209i	$0.759711\pi$
$648$	0	0
$649$	−116.862	−0.180065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	882.562i	1.35155i	0.737108	+	0.675775i	$0.236191\pi$
$653$	882.562i	1.35155i	−0.737108	+	0.675775i	$0.763809\pi$
$654$	0	0
$655$	65.0739	0.0993495
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1082.97i	− 1.64335i	−0.569956	−	0.821675i	$-0.693040\pi$
$659$	− 1082.97i	− 1.64335i	0.569956	−	0.821675i	$-0.306960\pi$
$660$	0	0
$661$	−0.939382	−0.00142115	−0.000710576	−	1.00000i	$-0.500226\pi$
$661$	−0.939382	−0.00142115	−0.000710576		1.00000i	$0.500226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 8.51725i	− 0.0128079i
$666$	0	0
$667$	383.636	0.575167
$668$	0	0
$669$	0	0
$670$	0	0
$671$	187.319i	0.279164i
$672$	0	0
$673$	801.952	1.19161	0.595804	−	0.803130i	$-0.296834\pi$
$673$	801.952	1.19161	0.595804	+	0.803130i	$0.296834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	679.226i	1.00329i	0.865074	+	0.501644i	$0.167271\pi$
$677$	679.226i	1.00329i	−0.865074	+	0.501644i	$0.832729\pi$
$678$	0	0
$679$	−25.8706	−0.0381010
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 633.610i	− 0.927686i	−0.885917	−	0.463843i	$-0.846470\pi$
$683$	− 633.610i	− 0.927686i	0.885917	−	0.463843i	$-0.153530\pi$
$684$	0	0
$685$	−1194.84	−1.74428
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 861.493i	− 1.25035i
$690$	0	0
$691$	613.990	0.888552	0.444276	−	0.895890i	$-0.353461\pi$
$691$	613.990	0.888552	0.444276	+	0.895890i	$0.353461\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26.9767i	0.0388154i
$696$	0	0
$697$	−137.924	−0.197882
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1243.75i	− 1.77425i	−0.461533	−	0.887123i	$-0.652701\pi$
$701$	− 1243.75i	− 1.77425i	0.461533	−	0.887123i	$-0.347299\pi$
$702$	0	0
$703$	−34.0014	−0.0483662
$704$	0	0
$705$	0	0
$706$	0	0
$707$	84.2039i	0.119100i
$708$	0	0
$709$	−319.728	−0.450956	−0.225478	−	0.974248i	$-0.572394\pi$
$709$	−319.728	−0.450956	−0.225478	+	0.974248i	$0.572394\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 879.463i	− 1.23347i
$714$	0	0
$715$	−349.452	−0.488744
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1172.26i	− 1.63040i	−0.579178	−	0.815201i	$-0.696626\pi$
$719$	− 1172.26i	− 1.63040i	0.579178	−	0.815201i	$-0.303374\pi$
$720$	0	0
$721$	−43.9293	−0.0609283
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 42.7123i	− 0.0589135i
$726$	0	0
$727$	1013.77	1.39446	0.697229	−	0.716848i	$-0.254416\pi$
$727$	1013.77	1.39446	0.697229	+	0.716848i	$0.254416\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.11768i	0.00700094i
$732$	0	0
$733$	817.660	1.11550	0.557749	−	0.830010i	$-0.311665\pi$
$733$	817.660	1.11550	0.557749	+	0.830010i	$0.311665\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 123.834i	− 0.168024i
$738$	0	0
$739$	388.311	0.525454	0.262727	−	0.964870i	$-0.415378\pi$
$739$	388.311	0.525454	0.262727	+	0.964870i	$0.415378\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1399.55i	− 1.88365i	−0.336105	−	0.941824i	$-0.609110\pi$
$743$	− 1399.55i	− 1.88365i	0.336105	−	0.941824i	$-0.390890\pi$
$744$	0	0
$745$	427.641	0.574015
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.5681i	0.0154447i
$750$	0	0
$751$	−540.976	−0.720341	−0.360170	−	0.932887i	$-0.617281\pi$
$751$	−540.976	−0.720341	−0.360170	+	0.932887i	$0.617281\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 186.082i	− 0.246466i
$756$	0	0
$757$	1242.72	1.64164	0.820820	−	0.571187i	$-0.193517\pi$
$757$	1242.72	1.64164	0.820820	+	0.571187i	$0.193517\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	61.0013i	0.0801594i	0.999196	+	0.0400797i	$0.0127612\pi$
$761$	61.0013i	0.0801594i	−0.999196	+	0.0400797i	$0.987239\pi$
$762$	0	0
$763$	89.9858	0.117937
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 699.859i	− 0.912463i
$768$	0	0
$769$	−275.424	−0.358159	−0.179080	−	0.983835i	$-0.557312\pi$
$769$	−275.424	−0.358159	−0.179080	+	0.983835i	$0.557312\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 508.827i	− 0.658250i	−0.944286	−	0.329125i	$-0.893246\pi$
$773$	− 508.827i	− 0.658250i	0.944286	−	0.329125i	$-0.106754\pi$
$774$	0	0
$775$	−97.9153	−0.126342
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 42.7946i	− 0.0549353i
$780$	0	0
$781$	−333.545	−0.427074
$782$	0	0
$783$	0	0
$784$	0	0
$785$	331.098i	0.421781i
$786$	0	0
$787$	−72.1748	−0.0917088	−0.0458544	−	0.998948i	$-0.514601\pi$
$787$	−72.1748	−0.0917088	−0.0458544	+	0.998948i	$0.514601\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 42.8768i	− 0.0542058i
$792$	0	0
$793$	−1121.81	−1.41464
$794$	0	0
$795$	0	0
$796$	0	0
$797$	302.879i	0.380024i	0.981782	+	0.190012i	$0.0608527\pi$
$797$	302.879i	0.380024i	−0.981782	+	0.190012i	$0.939147\pi$
$798$	0	0
$799$	645.380	0.807734
$800$	0	0
$801$	0	0
$802$	0	0
$803$	335.648i	0.417992i
$804$	0	0
$805$	84.4584	0.104917
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 870.791i	− 1.07638i	−0.842824	−	0.538190i	$-0.819108\pi$
$809$	− 870.791i	− 1.07638i	0.842824	−	0.538190i	$-0.180892\pi$
$810$	0	0
$811$	1169.49	1.44204	0.721018	−	0.692917i	$-0.243675\pi$
$811$	1169.49	1.44204	0.721018	+	0.692917i	$0.243675\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 327.273i	− 0.401562i
$816$	0	0
$817$	−1.58790	−0.00194358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1086.31i	− 1.32315i	−0.749879	−	0.661575i	$-0.769888\pi$
$821$	− 1086.31i	− 1.32315i	0.749879	−	0.661575i	$-0.230112\pi$
$822$	0	0
$823$	−729.935	−0.886920	−0.443460	−	0.896294i	$-0.646249\pi$
$823$	−729.935	−0.886920	−0.443460	+	0.896294i	$0.646249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1365.26i	1.65086i	0.564503	+	0.825431i	$0.309068\pi$
$827$	1365.26i	1.65086i	−0.564503	+	0.825431i	$0.690932\pi$
$828$	0	0
$829$	1261.78	1.52205	0.761024	−	0.648724i	$-0.224697\pi$
$829$	1261.78	1.52205	0.761024	+	0.648724i	$0.224697\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 446.178i	− 0.535628i
$834$	0	0
$835$	−1563.53	−1.87249
$836$	0	0
$837$	0	0
$838$	0	0
$839$	144.807i	0.172595i	0.996269	+	0.0862973i	$0.0275035\pi$
$839$	144.807i	0.172595i	−0.996269	+	0.0862973i	$0.972497\pi$
$840$	0	0
$841$	655.921	0.779929
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1196.29i	− 1.41573i
$846$	0	0
$847$	6.21063	0.00733250
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 337.163i	− 0.396197i
$852$	0	0
$853$	−64.1119	−0.0751605	−0.0375802	−	0.999294i	$-0.511965\pi$
$853$	−64.1119	−0.0751605	−0.0375802	+	0.999294i	$0.511965\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1104.09i	1.28832i	0.764891	+	0.644160i	$0.222793\pi$
$857$	1104.09i	1.28832i	−0.764891	+	0.644160i	$0.777207\pi$
$858$	0	0
$859$	920.695	1.07182	0.535911	−	0.844274i	$-0.319968\pi$
$859$	920.695	1.07182	0.535911	+	0.844274i	$0.319968\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1289.42i	1.49411i	0.664763	+	0.747055i	$0.268533\pi$
$863$	1289.42i	1.49411i	−0.664763	+	0.747055i	$0.731467\pi$
$864$	0	0
$865$	294.062	0.339956
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 103.576i	− 0.119190i
$870$	0	0
$871$	741.608	0.851444
$872$	0	0
$873$	0	0
$874$	0	0
$875$	65.4727i	0.0748259i
$876$	0	0
$877$	−715.659	−0.816030	−0.408015	−	0.912975i	$-0.633779\pi$
$877$	−715.659	−0.816030	−0.408015	+	0.912975i	$0.633779\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 723.537i	− 0.821267i	−0.911800	−	0.410634i	$-0.865307\pi$
$881$	− 723.537i	− 0.821267i	0.911800	−	0.410634i	$-0.134693\pi$
$882$	0	0
$883$	−474.053	−0.536866	−0.268433	−	0.963298i	$-0.586506\pi$
$883$	−474.053	−0.536866	−0.268433	+	0.963298i	$0.586506\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 445.907i	− 0.502713i	−0.967895	−	0.251357i	$-0.919123\pi$
$887$	− 445.907i	− 0.502713i	0.967895	−	0.251357i	$-0.0808767\pi$
$888$	0	0
$889$	−85.9614	−0.0966945
$890$	0	0
$891$	0	0
$892$	0	0
$893$	200.247i	0.224241i
$894$	0	0
$895$	529.397	0.591505
$896$	0	0
$897$	0	0
$898$	0	0
$899$	424.283i	0.471950i
$900$	0	0
$901$	397.527	0.441206
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1569.37i	− 1.73411i
$906$	0	0
$907$	1273.61	1.40420	0.702100	−	0.712078i	$-0.252246\pi$
$907$	1273.61	1.40420	0.702100	+	0.712078i	$0.252246\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	968.407i	1.06302i	0.847054	+	0.531508i	$0.178374\pi$
$911$	968.407i	1.06302i	−0.847054	+	0.531508i	$0.821626\pi$
$912$	0	0
$913$	−47.7325	−0.0522809
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.92614i	− 0.00755304i
$918$	0	0
$919$	−1445.48	−1.57288	−0.786442	−	0.617664i	$-0.788079\pi$
$919$	−1445.48	−1.57288	−0.786442	+	0.617664i	$0.788079\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1997.51i	− 2.16415i
$924$	0	0
$925$	−37.5382	−0.0405818
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1029.02i	1.10766i	0.832629	+	0.553832i	$0.186835\pi$
$929$	1029.02i	1.10766i	−0.832629	+	0.553832i	$0.813165\pi$
$930$	0	0
$931$	138.439	0.148699
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 161.251i	− 0.172461i
$936$	0	0
$937$	960.013	1.02456	0.512280	−	0.858818i	$-0.328801\pi$
$937$	960.013	1.02456	0.512280	+	0.858818i	$0.328801\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 698.272i	− 0.742053i	−0.928622	−	0.371027i	$-0.879006\pi$
$941$	− 698.272i	− 0.742053i	0.928622	−	0.371027i	$-0.120994\pi$
$942$	0	0
$943$	424.358	0.450008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 50.9442i	− 0.0537954i	−0.999638	−	0.0268977i	$-0.991437\pi$
$947$	− 50.9442i	− 0.0537954i	0.999638	−	0.0268977i	$-0.00856283\pi$
$948$	0	0
$949$	−2010.11	−2.11813
$950$	0	0
$951$	0	0
$952$	0	0
$953$	120.929i	0.126893i	0.997985	+	0.0634467i	$0.0202093\pi$
$953$	120.929i	0.126893i	−0.997985	+	0.0634467i	$0.979791\pi$
$954$	0	0
$955$	−1139.74	−1.19345
$956$	0	0
$957$	0	0
$958$	0	0
$959$	127.172i	0.132609i
$960$	0	0
$961$	11.6442	0.0121167
$962$	0	0
$963$	0	0
$964$	0	0
$965$	626.423i	0.649143i
$966$	0	0
$967$	−1331.55	−1.37699	−0.688496	−	0.725240i	$-0.741729\pi$
$967$	−1331.55	−1.37699	−0.688496	+	0.725240i	$0.741729\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 755.114i	− 0.777666i	−0.921308	−	0.388833i	$-0.872878\pi$
$971$	− 755.114i	− 0.777666i	0.921308	−	0.388833i	$-0.127122\pi$
$972$	0	0
$973$	2.87126	0.00295094
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1177.46i	− 1.20517i	−0.798053	−	0.602587i	$-0.794137\pi$
$977$	− 1177.46i	− 1.20517i	0.798053	−	0.602587i	$-0.205863\pi$
$978$	0	0
$979$	166.175	0.169739
$980$	0	0
$981$	0	0
$982$	0	0
$983$	830.282i	0.844641i	0.906446	+	0.422321i	$0.138784\pi$
$983$	830.282i	0.844641i	−0.906446	+	0.422321i	$0.861216\pi$
$984$	0	0
$985$	1492.86	1.51560
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 15.7459i	− 0.0159210i
$990$	0	0
$991$	−1659.37	−1.67444	−0.837220	−	0.546866i	$-0.815821\pi$
$991$	−1659.37	−1.67444	−0.837220	+	0.546866i	$0.815821\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 287.013i	− 0.288455i
$996$	0	0
$997$	878.060	0.880702	0.440351	−	0.897826i	$-0.354854\pi$
$997$	878.060	0.880702	0.440351	+	0.897826i	$0.354854\pi$
$998$	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.485.2	✓	8
3.2	odd	2	inner	1188.3.e.c.485.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1188.3.e.c.485.2	✓	8	1.1	even	1	trivial
1188.3.e.c.485.7	yes	8	3.2	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-5.30468i q^{5} -0.564603 q^{7} -3.31662i q^{11} +19.8624 q^{13} +9.16530i q^{17} -2.84379 q^{19} -28.1995i q^{23} -3.13959 q^{25} +13.6044i q^{29} +31.1872 q^{31} +2.99504i q^{35} +11.9564 q^{37} +15.0484i q^{41} +0.558376 q^{43} -70.4155i q^{47} -48.6812 q^{49} -43.3730i q^{53} -17.5936 q^{55} -35.2353i q^{59} -56.4788 q^{61} -105.364i q^{65} +37.3372 q^{67} -100.568i q^{71} -101.202 q^{73} +1.87258i q^{77} +31.2293 q^{79} -14.3919i q^{83} +48.6190 q^{85} +50.1036i q^{89} -11.2144 q^{91} +15.0854i q^{95} +45.8208 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 20 q^{13} + 16 q^{19} - 12 q^{25} - 32 q^{31} - 20 q^{37} - 48 q^{43} + 84 q^{49} + 36 q^{61} + 16 q^{67} + 200 q^{73} - 28 q^{79} - 60 q^{85} - 228 q^{91} - 120 q^{97}+O(q^{100}) \) 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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