LMFDB - Embedded newform 288.3.b.c.271.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,3,Mod(271,288)] 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("288.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	288.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0] 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:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 16 $ x^4 - 2*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.2
Root			$-1.58114 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	288.271
Dual form			288.3.b.c.271.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89898i q^{5} +7.74597i q^{7} +12.6491 q^{11} -15.4919i q^{13} +25.2982 q^{17} -8.00000 q^{19} -39.1918i q^{23} +1.00000 q^{25} -24.4949i q^{29} -7.74597i q^{31} +37.9473 q^{35} +46.4758i q^{37} +25.2982 q^{41} +40.0000 q^{43} +39.1918i q^{47} -11.0000 q^{49} -14.6969i q^{53} -61.9677i q^{55} -25.2982 q^{59} +15.4919i q^{61} -75.8947 q^{65} -80.0000 q^{67} -10.0000 q^{73} +97.9796i q^{77} +54.2218i q^{79} -139.140 q^{83} -123.935i q^{85} -50.5964 q^{89} +120.000 q^{91} +39.1918i q^{95} +50.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{19} + 4 q^{25} + 160 q^{43} - 44 q^{49} - 320 q^{67} - 40 q^{73} + 480 q^{91} + 200 q^{97}+O(q^{100}) $ 4 * q - 32 * q^19 + 4 * q^25 + 160 * q^43 - 44 * q^49 - 320 * q^67 - 40 * q^73 + 480 * q^91 + 200 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	− 4.89898i	− 0.979796i	−0.871780	−	0.489898i	$-0.837034\pi$
$5$	− 4.89898i	− 0.979796i	0.871780	−	0.489898i	$-0.162966\pi$
$6$	0	0
$7$	7.74597i	1.10657i	0.832993	+	0.553283i	$0.186625\pi$
$7$	7.74597i	1.10657i	−0.832993	+	0.553283i	$0.813375\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	12.6491	1.14992	0.574960	−	0.818182i	$-0.305018\pi$
$11$	12.6491	1.14992	0.574960	+	0.818182i	$0.305018\pi$
$12$	0	0
$13$	− 15.4919i	− 1.19169i	−0.803101	−	0.595844i	$-0.796818\pi$
$13$	− 15.4919i	− 1.19169i	0.803101	−	0.595844i	$-0.203182\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25.2982	1.48813	0.744065	−	0.668107i	$-0.232895\pi$
$17$	25.2982	1.48813	0.744065	+	0.668107i	$0.232895\pi$
$18$	0	0
$19$	−8.00000	−0.421053	−0.210526	−	0.977588i	$-0.567518\pi$
$19$	−8.00000	−0.421053	−0.210526	+	0.977588i	$0.567518\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 39.1918i	− 1.70399i	−0.523548	−	0.851996i	$-0.675392\pi$
$23$	− 39.1918i	− 1.70399i	0.523548	−	0.851996i	$-0.324608\pi$
$24$	0	0
$25$	1.00000	0.0400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 24.4949i	− 0.844652i	−0.906444	−	0.422326i	$-0.861214\pi$
$29$	− 24.4949i	− 0.844652i	0.906444	−	0.422326i	$-0.138786\pi$
$30$	0	0
$31$	− 7.74597i	− 0.249870i	−0.992165	−	0.124935i	$-0.960128\pi$
$31$	− 7.74597i	− 0.249870i	0.992165	−	0.124935i	$-0.0398722\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	37.9473	1.08421
$36$	0	0
$37$	46.4758i	1.25610i	0.778172	+	0.628051i	$0.216147\pi$
$37$	46.4758i	1.25610i	−0.778172	+	0.628051i	$0.783853\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	25.2982	0.617030	0.308515	−	0.951220i	$-0.400168\pi$
$41$	25.2982	0.617030	0.308515	+	0.951220i	$0.400168\pi$
$42$	0	0
$43$	40.0000	0.930233	0.465116	−	0.885250i	$-0.346013\pi$
$43$	40.0000	0.930233	0.465116	+	0.885250i	$0.346013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	39.1918i	0.833869i	0.908937	+	0.416934i	$0.136896\pi$
$47$	39.1918i	0.833869i	−0.908937	+	0.416934i	$0.863104\pi$
$48$	0	0
$49$	−11.0000	−0.224490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 14.6969i	− 0.277301i	−0.990341	−	0.138650i	$-0.955724\pi$
$53$	− 14.6969i	− 0.277301i	0.990341	−	0.138650i	$-0.0442764\pi$
$54$	0	0
$55$	− 61.9677i	− 1.12669i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−25.2982	−0.428783	−0.214392	−	0.976748i	$-0.568777\pi$
$59$	−25.2982	−0.428783	−0.214392	+	0.976748i	$0.568777\pi$
$60$	0	0
$61$	15.4919i	0.253966i	0.991905	+	0.126983i	$0.0405294\pi$
$61$	15.4919i	0.253966i	−0.991905	+	0.126983i	$0.959471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−75.8947	−1.16761
$66$	0	0
$67$	−80.0000	−1.19403	−0.597015	−	0.802230i	$-0.703647\pi$
$67$	−80.0000	−1.19403	−0.597015	+	0.802230i	$0.703647\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−10.0000	−0.136986	−0.0684932	−	0.997652i	$-0.521819\pi$
$73$	−10.0000	−0.136986	−0.0684932	+	0.997652i	$0.521819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	97.9796i	1.27246i
$78$	0	0
$79$	54.2218i	0.686351i	0.939271	+	0.343176i	$0.111503\pi$
$79$	54.2218i	0.686351i	−0.939271	+	0.343176i	$0.888497\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−139.140	−1.67639	−0.838194	−	0.545372i	$-0.816388\pi$
$83$	−139.140	−1.67639	−0.838194	+	0.545372i	$0.816388\pi$
$84$	0	0
$85$	− 123.935i	− 1.45806i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−50.5964	−0.568499	−0.284250	−	0.958750i	$-0.591744\pi$
$89$	−50.5964	−0.568499	−0.284250	+	0.958750i	$0.591744\pi$
$90$	0	0
$91$	120.000	1.31868
$92$	0	0
$93$	0	0
$94$	0	0
$95$	39.1918i	0.412546i
$96$	0	0
$97$	50.0000	0.515464	0.257732	−	0.966216i	$-0.417025\pi$
$97$	50.0000	0.515464	0.257732	+	0.966216i	$0.417025\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	122.474i	1.21262i	0.795229	+	0.606309i	$0.207351\pi$
$101$	122.474i	1.21262i	−0.795229	+	0.606309i	$0.792649\pi$
$102$	0	0
$103$	178.157i	1.72968i	0.502046	+	0.864841i	$0.332581\pi$
$103$	178.157i	1.72968i	−0.502046	+	0.864841i	$0.667419\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−25.2982	−0.236432	−0.118216	−	0.992988i	$-0.537718\pi$
$107$	−25.2982	−0.236432	−0.118216	+	0.992988i	$0.537718\pi$
$108$	0	0
$109$	46.4758i	0.426383i	0.977010	+	0.213192i	$0.0683859\pi$
$109$	46.4758i	0.426383i	−0.977010	+	0.213192i	$0.931614\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−50.5964	−0.447756	−0.223878	−	0.974617i	$-0.571872\pi$
$113$	−50.5964	−0.447756	−0.223878	+	0.974617i	$0.571872\pi$
$114$	0	0
$115$	−192.000	−1.66957
$116$	0	0
$117$	0	0
$118$	0	0
$119$	195.959i	1.64672i
$120$	0	0
$121$	39.0000	0.322314
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 127.373i	− 1.01899i
$126$	0	0
$127$	− 116.190i	− 0.914878i	−0.889241	−	0.457439i	$-0.848767\pi$
$127$	− 116.190i	− 0.914878i	0.889241	−	0.457439i	$-0.151233\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	50.5964	0.386232	0.193116	−	0.981176i	$-0.438141\pi$
$131$	50.5964	0.386232	0.193116	+	0.981176i	$0.438141\pi$
$132$	0	0
$133$	− 61.9677i	− 0.465923i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−126.491	−0.923293	−0.461646	−	0.887064i	$-0.652741\pi$
$137$	−126.491	−0.923293	−0.461646	+	0.887064i	$0.652741\pi$
$138$	0	0
$139$	208.000	1.49640	0.748201	−	0.663472i	$-0.230918\pi$
$139$	208.000	1.49640	0.748201	+	0.663472i	$0.230918\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 195.959i	− 1.37034i
$144$	0	0
$145$	−120.000	−0.827586
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 122.474i	− 0.821976i	−0.911641	−	0.410988i	$-0.865184\pi$
$149$	− 122.474i	− 0.821976i	0.911641	−	0.410988i	$-0.134816\pi$
$150$	0	0
$151$	− 178.157i	− 1.17985i	−0.807458	−	0.589925i	$-0.799157\pi$
$151$	− 178.157i	− 1.17985i	0.807458	−	0.589925i	$-0.200843\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−37.9473	−0.244821
$156$	0	0
$157$	77.4597i	0.493374i	0.969095	+	0.246687i	$0.0793419\pi$
$157$	77.4597i	0.493374i	−0.969095	+	0.246687i	$0.920658\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	303.579	1.88558
$162$	0	0
$163$	−200.000	−1.22699	−0.613497	−	0.789697i	$-0.710238\pi$
$163$	−200.000	−1.22699	−0.613497	+	0.789697i	$0.710238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 39.1918i	− 0.234682i	−0.993092	−	0.117341i	$-0.962563\pi$
$167$	− 39.1918i	− 0.234682i	0.993092	−	0.117341i	$-0.0374370\pi$
$168$	0	0
$169$	−71.0000	−0.420118
$170$	0	0
$171$	0	0
$172$	0	0
$173$	181.262i	1.04776i	0.851793	+	0.523879i	$0.175516\pi$
$173$	181.262i	1.04776i	−0.851793	+	0.523879i	$0.824484\pi$
$174$	0	0
$175$	7.74597i	0.0442627i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	278.280	1.55464	0.777320	−	0.629106i	$-0.216579\pi$
$179$	278.280	1.55464	0.777320	+	0.629106i	$0.216579\pi$
$180$	0	0
$181$	325.331i	1.79741i	0.438557	+	0.898703i	$0.355490\pi$
$181$	325.331i	1.79741i	−0.438557	+	0.898703i	$0.644510\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	227.684	1.23072
$186$	0	0
$187$	320.000	1.71123
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 195.959i	− 1.02596i	−0.858399	−	0.512982i	$-0.828541\pi$
$191$	− 195.959i	− 1.02596i	0.858399	−	0.512982i	$-0.171459\pi$
$192$	0	0
$193$	170.000	0.880829	0.440415	−	0.897795i	$-0.354831\pi$
$193$	170.000	0.880829	0.440415	+	0.897795i	$0.354831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 132.272i	− 0.671434i	−0.941963	−	0.335717i	$-0.891021\pi$
$197$	− 132.272i	− 0.671434i	0.941963	−	0.335717i	$-0.108979\pi$
$198$	0	0
$199$	− 69.7137i	− 0.350320i	−0.984540	−	0.175160i	$-0.943956\pi$
$199$	− 69.7137i	− 0.350320i	0.984540	−	0.175160i	$-0.0560443\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	189.737	0.934663
$204$	0	0
$205$	− 123.935i	− 0.604563i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−101.193	−0.484176
$210$	0	0
$211$	−272.000	−1.28910	−0.644550	−	0.764562i	$-0.722955\pi$
$211$	−272.000	−1.28910	−0.644550	+	0.764562i	$0.722955\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 195.959i	− 0.911438i
$216$	0	0
$217$	60.0000	0.276498
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 391.918i	− 1.77339i
$222$	0	0
$223$	193.649i	0.868382i	0.900821	+	0.434191i	$0.142966\pi$
$223$	193.649i	0.868382i	−0.900821	+	0.434191i	$0.857034\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−215.035	−0.947290	−0.473645	−	0.880716i	$-0.657062\pi$
$227$	−215.035	−0.947290	−0.473645	+	0.880716i	$0.657062\pi$
$228$	0	0
$229$	− 108.444i	− 0.473553i	−0.971564	−	0.236776i	$-0.923909\pi$
$229$	− 108.444i	− 0.473553i	0.971564	−	0.236776i	$-0.0760908\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	252.982	1.08576	0.542880	−	0.839810i	$-0.317334\pi$
$233$	252.982	1.08576	0.542880	+	0.839810i	$0.317334\pi$
$234$	0	0
$235$	192.000	0.817021
$236$	0	0
$237$	0	0
$238$	0	0
$239$	195.959i	0.819913i	0.912105	+	0.409956i	$0.134456\pi$
$239$	195.959i	0.819913i	−0.912105	+	0.409956i	$0.865544\pi$
$240$	0	0
$241$	−202.000	−0.838174	−0.419087	−	0.907946i	$-0.637650\pi$
$241$	−202.000	−0.838174	−0.419087	+	0.907946i	$0.637650\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	53.8888i	0.219954i
$246$	0	0
$247$	123.935i	0.501763i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	316.228	1.25987	0.629936	−	0.776647i	$-0.283081\pi$
$251$	316.228	1.25987	0.629936	+	0.776647i	$0.283081\pi$
$252$	0	0
$253$	− 495.742i	− 1.95945i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−202.386	−0.787493	−0.393747	−	0.919219i	$-0.628821\pi$
$257$	−202.386	−0.787493	−0.393747	+	0.919219i	$0.628821\pi$
$258$	0	0
$259$	−360.000	−1.38996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	156.767i	0.596074i	0.954554	+	0.298037i	$0.0963318\pi$
$263$	156.767i	0.596074i	−0.954554	+	0.298037i	$0.903668\pi$
$264$	0	0
$265$	−72.0000	−0.271698
$266$	0	0
$267$	0	0
$268$	0	0
$269$	73.4847i	0.273177i	0.990628	+	0.136589i	$0.0436139\pi$
$269$	73.4847i	0.273177i	−0.990628	+	0.136589i	$0.956386\pi$
$270$	0	0
$271$	69.7137i	0.257246i	0.991694	+	0.128623i	$0.0410557\pi$
$271$	69.7137i	0.257246i	−0.991694	+	0.128623i	$0.958944\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.6491	0.0459968
$276$	0	0
$277$	15.4919i	0.0559276i	0.999609	+	0.0279638i	$0.00890231\pi$
$277$	15.4919i	0.0559276i	−0.999609	+	0.0279638i	$0.991098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−202.386	−0.720234	−0.360117	−	0.932907i	$-0.617263\pi$
$281$	−202.386	−0.720234	−0.360117	+	0.932907i	$0.617263\pi$
$282$	0	0
$283$	160.000	0.565371	0.282686	−	0.959213i	$-0.408775\pi$
$283$	160.000	0.565371	0.282686	+	0.959213i	$0.408775\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	195.959i	0.682785i
$288$	0	0
$289$	351.000	1.21453
$290$	0	0
$291$	0	0
$292$	0	0
$293$	259.646i	0.886164i	0.896481	+	0.443082i	$0.146115\pi$
$293$	259.646i	0.886164i	−0.896481	+	0.443082i	$0.853885\pi$
$294$	0	0
$295$	123.935i	0.420120i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−607.157	−2.03063
$300$	0	0
$301$	309.839i	1.02936i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	75.8947	0.248835
$306$	0	0
$307$	−80.0000	−0.260586	−0.130293	−	0.991476i	$-0.541592\pi$
$307$	−80.0000	−0.260586	−0.130293	+	0.991476i	$0.541592\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 391.918i	− 1.26019i	−0.776519	−	0.630094i	$-0.783016\pi$
$311$	− 391.918i	− 1.26019i	0.776519	−	0.630094i	$-0.216984\pi$
$312$	0	0
$313$	230.000	0.734824	0.367412	−	0.930058i	$-0.380244\pi$
$313$	230.000	0.734824	0.367412	+	0.930058i	$0.380244\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	504.595i	1.59178i	0.605440	+	0.795891i	$0.292997\pi$
$317$	504.595i	1.59178i	−0.605440	+	0.795891i	$0.707003\pi$
$318$	0	0
$319$	− 309.839i	− 0.971281i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−202.386	−0.626581
$324$	0	0
$325$	− 15.4919i	− 0.0476675i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−303.579	−0.922731
$330$	0	0
$331$	352.000	1.06344	0.531722	−	0.846919i	$-0.321545\pi$
$331$	352.000	1.06344	0.531722	+	0.846919i	$0.321545\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	391.918i	1.16991i
$336$	0	0
$337$	−550.000	−1.63205	−0.816024	−	0.578018i	$-0.803826\pi$
$337$	−550.000	−1.63205	−0.816024	+	0.578018i	$0.803826\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 97.9796i	− 0.287330i
$342$	0	0
$343$	294.347i	0.858154i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−366.824	−1.05713	−0.528565	−	0.848893i	$-0.677270\pi$
$347$	−366.824	−1.05713	−0.528565	+	0.848893i	$0.677270\pi$
$348$	0	0
$349$	573.202i	1.64241i	0.570632	+	0.821206i	$0.306698\pi$
$349$	573.202i	1.64241i	−0.570632	+	0.821206i	$0.693302\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−354.175	−1.00333	−0.501664	−	0.865062i	$-0.667279\pi$
$353$	−354.175	−1.00333	−0.501664	+	0.865062i	$0.667279\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−297.000	−0.822715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	48.9898i	0.134219i
$366$	0	0
$367$	− 503.488i	− 1.37190i	−0.727648	−	0.685951i	$-0.759387\pi$
$367$	− 503.488i	− 1.37190i	0.727648	−	0.685951i	$-0.240613\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	113.842	0.306852
$372$	0	0
$373$	− 201.395i	− 0.539933i	−0.962870	−	0.269967i	$-0.912987\pi$
$373$	− 201.395i	− 0.539933i	0.962870	−	0.269967i	$-0.0870127\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−379.473	−1.00656
$378$	0	0
$379$	−152.000	−0.401055	−0.200528	−	0.979688i	$-0.564266\pi$
$379$	−152.000	−0.401055	−0.200528	+	0.979688i	$0.564266\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	431.110i	1.12561i	0.826588	+	0.562807i	$0.190279\pi$
$383$	431.110i	1.12561i	−0.826588	+	0.562807i	$0.809721\pi$
$384$	0	0
$385$	480.000	1.24675
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 24.4949i	− 0.0629689i	−0.999504	−	0.0314844i	$-0.989977\pi$
$389$	− 24.4949i	− 0.0629689i	0.999504	−	0.0314844i	$-0.0100235\pi$
$390$	0	0
$391$	− 991.484i	− 2.53576i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	265.631	0.672484
$396$	0	0
$397$	− 46.4758i	− 0.117068i	−0.998285	−	0.0585338i	$-0.981357\pi$
$397$	− 46.4758i	− 0.117068i	0.998285	−	0.0585338i	$-0.0186425\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	480.666	1.19867	0.599334	−	0.800499i	$-0.295432\pi$
$401$	480.666	1.19867	0.599334	+	0.800499i	$0.295432\pi$
$402$	0	0
$403$	−120.000	−0.297767
$404$	0	0
$405$	0	0
$406$	0	0
$407$	587.878i	1.44442i
$408$	0	0
$409$	62.0000	0.151589	0.0757946	−	0.997123i	$-0.475851\pi$
$409$	62.0000	0.151589	0.0757946	+	0.997123i	$0.475851\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 195.959i	− 0.474477i
$414$	0	0
$415$	681.645i	1.64252i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−63.2456	−0.150944	−0.0754720	−	0.997148i	$-0.524046\pi$
$419$	−63.2456	−0.150944	−0.0754720	+	0.997148i	$0.524046\pi$
$420$	0	0
$421$	15.4919i	0.0367979i	0.999831	+	0.0183990i	$0.00585690\pi$
$421$	15.4919i	0.0367979i	−0.999831	+	0.0183990i	$0.994143\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.2982	0.0595252
$426$	0	0
$427$	−120.000	−0.281030
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 587.878i	− 1.36399i	−0.731359	−	0.681993i	$-0.761114\pi$
$431$	− 587.878i	− 1.36399i	0.731359	−	0.681993i	$-0.238886\pi$
$432$	0	0
$433$	−370.000	−0.854503	−0.427252	−	0.904133i	$-0.640518\pi$
$433$	−370.000	−0.854503	−0.427252	+	0.904133i	$0.640518\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	313.535i	0.717471i
$438$	0	0
$439$	426.028i	0.970451i	0.874389	+	0.485226i	$0.161263\pi$
$439$	426.028i	0.970451i	−0.874389	+	0.485226i	$0.838737\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	240.333	0.542513	0.271256	−	0.962507i	$-0.412561\pi$
$443$	240.333	0.542513	0.271256	+	0.962507i	$0.412561\pi$
$444$	0	0
$445$	247.871i	0.557013i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.2982	0.0563435	0.0281717	−	0.999603i	$-0.491031\pi$
$449$	25.2982	0.0563435	0.0281717	+	0.999603i	$0.491031\pi$
$450$	0	0
$451$	320.000	0.709534
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 587.878i	− 1.29204i
$456$	0	0
$457$	−130.000	−0.284464	−0.142232	−	0.989833i	$-0.545428\pi$
$457$	−130.000	−0.284464	−0.142232	+	0.989833i	$0.545428\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 759.342i	− 1.64716i	−0.567198	−	0.823581i	$-0.691973\pi$
$461$	− 759.342i	− 1.64716i	0.567198	−	0.823581i	$-0.308027\pi$
$462$	0	0
$463$	503.488i	1.08745i	0.839265	+	0.543723i	$0.182986\pi$
$463$	503.488i	1.08745i	−0.839265	+	0.543723i	$0.817014\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−366.824	−0.785491	−0.392745	−	0.919647i	$-0.628475\pi$
$467$	−366.824	−0.785491	−0.392745	+	0.919647i	$0.628475\pi$
$468$	0	0
$469$	− 619.677i	− 1.32127i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	505.964	1.06969
$474$	0	0
$475$	−8.00000	−0.0168421
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 783.837i	− 1.63640i	−0.574932	−	0.818201i	$-0.694972\pi$
$479$	− 783.837i	− 1.63640i	0.574932	−	0.818201i	$-0.305028\pi$
$480$	0	0
$481$	720.000	1.49688
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 244.949i	− 0.505049i
$486$	0	0
$487$	255.617i	0.524881i	0.964948	+	0.262440i	$0.0845273\pi$
$487$	255.617i	0.524881i	−0.964948	+	0.262440i	$0.915473\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	657.754	1.33962	0.669810	−	0.742532i	$-0.266375\pi$
$491$	657.754	1.33962	0.669810	+	0.742532i	$0.266375\pi$
$492$	0	0
$493$	− 619.677i	− 1.25695i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	352.000	0.705411	0.352705	−	0.935734i	$-0.385262\pi$
$499$	352.000	0.705411	0.352705	+	0.935734i	$0.385262\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	235.151i	0.467497i	0.972297	+	0.233749i	$0.0750992\pi$
$503$	235.151i	0.467497i	−0.972297	+	0.233749i	$0.924901\pi$
$504$	0	0
$505$	600.000	1.18812
$506$	0	0
$507$	0	0
$508$	0	0
$509$	759.342i	1.49183i	0.666041	+	0.745915i	$0.267988\pi$
$509$	759.342i	1.49183i	−0.666041	+	0.745915i	$0.732012\pi$
$510$	0	0
$511$	− 77.4597i	− 0.151584i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	872.789	1.69474
$516$	0	0
$517$	495.742i	0.958882i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	936.034	1.79661	0.898305	−	0.439372i	$-0.144799\pi$
$521$	936.034	1.79661	0.898305	+	0.439372i	$0.144799\pi$
$522$	0	0
$523$	−920.000	−1.75908	−0.879541	−	0.475823i	$-0.842150\pi$
$523$	−920.000	−1.75908	−0.879541	+	0.475823i	$0.842150\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 195.959i	− 0.371839i
$528$	0	0
$529$	−1007.00	−1.90359
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 391.918i	− 0.735306i
$534$	0	0
$535$	123.935i	0.231655i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−139.140	−0.258145
$540$	0	0
$541$	− 511.234i	− 0.944979i	−0.881336	−	0.472490i	$-0.843355\pi$
$541$	− 511.234i	− 0.944979i	0.881336	−	0.472490i	$-0.156645\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	227.684	0.417769
$546$	0	0
$547$	40.0000	0.0731261	0.0365631	−	0.999331i	$-0.488359\pi$
$547$	40.0000	0.0731261	0.0365631	+	0.999331i	$0.488359\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	195.959i	0.355643i
$552$	0	0
$553$	−420.000	−0.759494
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 308.636i	− 0.554104i	−0.960855	−	0.277052i	$-0.910643\pi$
$557$	− 308.636i	− 0.554104i	0.960855	−	0.277052i	$-0.0893574\pi$
$558$	0	0
$559$	− 619.677i	− 1.10855i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	88.5438	0.157271	0.0786357	−	0.996903i	$-0.474944\pi$
$563$	88.5438	0.157271	0.0786357	+	0.996903i	$0.474944\pi$
$564$	0	0
$565$	247.871i	0.438710i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	328.877	0.577991	0.288995	−	0.957330i	$-0.406679\pi$
$569$	328.877	0.577991	0.288995	+	0.957330i	$0.406679\pi$
$570$	0	0
$571$	−368.000	−0.644483	−0.322242	−	0.946657i	$-0.604436\pi$
$571$	−368.000	−0.644483	−0.322242	+	0.946657i	$0.604436\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 39.1918i	− 0.0681597i
$576$	0	0
$577$	710.000	1.23050	0.615251	−	0.788331i	$-0.289055\pi$
$577$	710.000	1.23050	0.615251	+	0.788331i	$0.289055\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1077.78i	− 1.85504i
$582$	0	0
$583$	− 185.903i	− 0.318873i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−101.193	−0.172390	−0.0861950	−	0.996278i	$-0.527471\pi$
$587$	−101.193	−0.172390	−0.0861950	+	0.996278i	$0.527471\pi$
$588$	0	0
$589$	61.9677i	0.105208i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−961.332	−1.62113	−0.810567	−	0.585646i	$-0.800841\pi$
$593$	−961.332	−1.62113	−0.810567	+	0.585646i	$0.800841\pi$
$594$	0	0
$595$	960.000	1.61345
$596$	0	0
$597$	0	0
$598$	0	0
$599$	195.959i	0.327144i	0.986531	+	0.163572i	$0.0523016\pi$
$599$	195.959i	0.327144i	−0.986531	+	0.163572i	$0.947698\pi$
$600$	0	0
$601$	−22.0000	−0.0366057	−0.0183028	−	0.999832i	$-0.505826\pi$
$601$	−22.0000	−0.0366057	−0.0183028	+	0.999832i	$0.505826\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 191.060i	− 0.315802i
$606$	0	0
$607$	− 240.125i	− 0.395593i	−0.980243	−	0.197797i	$-0.936621\pi$
$607$	− 240.125i	− 0.395593i	0.980243	−	0.197797i	$-0.0633785\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	607.157	0.993711
$612$	0	0
$613$	790.089i	1.28889i	0.764651	+	0.644444i	$0.222911\pi$
$613$	790.089i	1.28889i	−0.764651	+	0.644444i	$0.777089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1113.12	−1.80409	−0.902044	−	0.431645i	$-0.857933\pi$
$617$	−1113.12	−1.80409	−0.902044	+	0.431645i	$0.857933\pi$
$618$	0	0
$619$	−32.0000	−0.0516963	−0.0258481	−	0.999666i	$-0.508229\pi$
$619$	−32.0000	−0.0516963	−0.0258481	+	0.999666i	$0.508229\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 391.918i	− 0.629082i
$624$	0	0
$625$	−599.000	−0.958400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1175.76i	1.86924i
$630$	0	0
$631$	− 116.190i	− 0.184135i	−0.995753	−	0.0920677i	$-0.970652\pi$
$631$	− 116.190i	− 0.184135i	0.995753	−	0.0920677i	$-0.0293476\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−569.210	−0.896394
$636$	0	0
$637$	170.411i	0.267522i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1239.61	1.93387	0.966937	−	0.255017i	$-0.0820809\pi$
$641$	1239.61	1.93387	0.966937	+	0.255017i	$0.0820809\pi$
$642$	0	0
$643$	1000.00	1.55521	0.777605	−	0.628753i	$-0.216434\pi$
$643$	1000.00	1.55521	0.777605	+	0.628753i	$0.216434\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	823.029i	1.27207i	0.771661	+	0.636034i	$0.219426\pi$
$647$	823.029i	1.27207i	−0.771661	+	0.636034i	$0.780574\pi$
$648$	0	0
$649$	−320.000	−0.493066
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 328.232i	− 0.502652i	−0.967903	−	0.251326i	$-0.919133\pi$
$653$	− 328.232i	− 0.502652i	0.967903	−	0.251326i	$-0.0808666\pi$
$654$	0	0
$655$	− 247.871i	− 0.378429i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−708.350	−1.07489	−0.537443	−	0.843300i	$-0.680610\pi$
$659$	−708.350	−1.07489	−0.537443	+	0.843300i	$0.680610\pi$
$660$	0	0
$661$	728.121i	1.10154i	0.834656	+	0.550772i	$0.185667\pi$
$661$	728.121i	1.10154i	−0.834656	+	0.550772i	$0.814333\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−303.579	−0.456509
$666$	0	0
$667$	−960.000	−1.43928
$668$	0	0
$669$	0	0
$670$	0	0
$671$	195.959i	0.292041i
$672$	0	0
$673$	1070.00	1.58990	0.794948	−	0.606678i	$-0.207498\pi$
$673$	1070.00	1.58990	0.794948	+	0.606678i	$0.207498\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1092.47i	1.61370i	0.590759	+	0.806848i	$0.298828\pi$
$677$	1092.47i	1.61370i	−0.590759	+	0.806848i	$0.701172\pi$
$678$	0	0
$679$	387.298i	0.570395i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−594.508	−0.870437	−0.435218	−	0.900325i	$-0.643329\pi$
$683$	−594.508	−0.870437	−0.435218	+	0.900325i	$0.643329\pi$
$684$	0	0
$685$	619.677i	0.904638i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−227.684	−0.330456
$690$	0	0
$691$	−8.00000	−0.0115774	−0.00578871	−	0.999983i	$-0.501843\pi$
$691$	−8.00000	−0.0115774	−0.00578871	+	0.999983i	$0.501843\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1018.99i	− 1.46617i
$696$	0	0
$697$	640.000	0.918221
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1249.24i	1.78208i	0.453922	+	0.891041i	$0.350024\pi$
$701$	1249.24i	1.78208i	−0.453922	+	0.891041i	$0.649976\pi$
$702$	0	0
$703$	− 371.806i	− 0.528885i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−948.683	−1.34184
$708$	0	0
$709$	− 914.024i	− 1.28917i	−0.764531	−	0.644587i	$-0.777029\pi$
$709$	− 914.024i	− 1.28917i	0.764531	−	0.644587i	$-0.222971\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−303.579	−0.425777
$714$	0	0
$715$	−960.000	−1.34266
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1175.76i	1.63526i	0.575741	+	0.817632i	$0.304714\pi$
$719$	1175.76i	1.63526i	−0.575741	+	0.817632i	$0.695286\pi$
$720$	0	0
$721$	−1380.00	−1.91401
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.4949i	− 0.0337861i
$726$	0	0
$727$	− 549.964i	− 0.756484i	−0.925707	−	0.378242i	$-0.876529\pi$
$727$	− 549.964i	− 0.756484i	0.925707	−	0.378242i	$-0.123471\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1011.93	1.38431
$732$	0	0
$733$	− 945.008i	− 1.28923i	−0.764506	−	0.644617i	$-0.777017\pi$
$733$	− 945.008i	− 1.28923i	0.764506	−	0.644617i	$-0.222983\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1011.93	−1.37304
$738$	0	0
$739$	1072.00	1.45061	0.725304	−	0.688428i	$-0.241699\pi$
$739$	1072.00	1.45061	0.725304	+	0.688428i	$0.241699\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	431.110i	0.580229i	0.956992	+	0.290115i	$0.0936934\pi$
$743$	431.110i	0.580229i	−0.956992	+	0.290115i	$0.906307\pi$
$744$	0	0
$745$	−600.000	−0.805369
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 195.959i	− 0.261628i
$750$	0	0
$751$	1293.58i	1.72247i	0.508205	+	0.861236i	$0.330309\pi$
$751$	1293.58i	1.72247i	−0.508205	+	0.861236i	$0.669691\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−872.789	−1.15601
$756$	0	0
$757$	− 790.089i	− 1.04371i	−0.853034	−	0.521855i	$-0.825240\pi$
$757$	− 790.089i	− 1.04371i	0.853034	−	0.521855i	$-0.174760\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−733.648	−0.964058	−0.482029	−	0.876155i	$-0.660100\pi$
$761$	−733.648	−0.964058	−0.482029	+	0.876155i	$0.660100\pi$
$762$	0	0
$763$	−360.000	−0.471822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	391.918i	0.510976i
$768$	0	0
$769$	938.000	1.21977	0.609883	−	0.792491i	$-0.291216\pi$
$769$	938.000	1.21977	0.609883	+	0.792491i	$0.291216\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 426.211i	− 0.551373i	−0.961248	−	0.275686i	$-0.911095\pi$
$773$	− 426.211i	− 0.551373i	0.961248	−	0.275686i	$-0.0889051\pi$
$774$	0	0
$775$	− 7.74597i	− 0.00999480i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−202.386	−0.259802
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	379.473	0.483406
$786$	0	0
$787$	520.000	0.660737	0.330368	−	0.943852i	$-0.392827\pi$
$787$	520.000	0.660737	0.330368	+	0.943852i	$0.392827\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 391.918i	− 0.495472i
$792$	0	0
$793$	240.000	0.302648
$794$	0	0
$795$	0	0
$796$	0	0
$797$	83.2827i	0.104495i	0.998634	+	0.0522476i	$0.0166385\pi$
$797$	83.2827i	0.104495i	−0.998634	+	0.0522476i	$0.983362\pi$
$798$	0	0
$799$	991.484i	1.24091i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−126.491	−0.157523
$804$	0	0
$805$	− 1487.23i	− 1.84749i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−430.070	−0.531607	−0.265803	−	0.964027i	$-0.585637\pi$
$809$	−430.070	−0.531607	−0.265803	+	0.964027i	$0.585637\pi$
$810$	0	0
$811$	−872.000	−1.07522	−0.537608	−	0.843195i	$-0.680672\pi$
$811$	−872.000	−1.07522	−0.537608	+	0.843195i	$0.680672\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	979.796i	1.20220i
$816$	0	0
$817$	−320.000	−0.391677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1053.28i	− 1.28292i	−0.767155	−	0.641462i	$-0.778328\pi$
$821$	− 1053.28i	− 1.28292i	0.767155	−	0.641462i	$-0.221672\pi$
$822$	0	0
$823$	− 240.125i	− 0.291768i	−0.989302	−	0.145884i	$-0.953397\pi$
$823$	− 240.125i	− 0.291768i	0.989302	−	0.145884i	$-0.0466026\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1087.82	−1.31539	−0.657693	−	0.753286i	$-0.728467\pi$
$827$	−1087.82	−1.31539	−0.657693	+	0.753286i	$0.728467\pi$
$828$	0	0
$829$	1533.70i	1.85006i	0.379892	+	0.925031i	$0.375961\pi$
$829$	1533.70i	1.85006i	−0.379892	+	0.925031i	$0.624039\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−278.280	−0.334070
$834$	0	0
$835$	−192.000	−0.229940
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 783.837i	− 0.934251i	−0.884191	−	0.467126i	$-0.845290\pi$
$839$	− 783.837i	− 0.934251i	0.884191	−	0.467126i	$-0.154710\pi$
$840$	0	0
$841$	241.000	0.286564
$842$	0	0
$843$	0	0
$844$	0	0
$845$	347.828i	0.411630i
$846$	0	0
$847$	302.093i	0.356662i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1821.47	2.14039
$852$	0	0
$853$	1037.96i	1.21683i	0.793617	+	0.608417i	$0.208195\pi$
$853$	1037.96i	1.21683i	−0.793617	+	0.608417i	$0.791805\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	936.034	1.09222	0.546111	−	0.837713i	$-0.316108\pi$
$857$	936.034	1.09222	0.546111	+	0.837713i	$0.316108\pi$
$858$	0	0
$859$	−872.000	−1.01513	−0.507567	−	0.861612i	$-0.669455\pi$
$859$	−872.000	−1.01513	−0.507567	+	0.861612i	$0.669455\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	352.727i	0.408721i	0.978896	+	0.204361i	$0.0655115\pi$
$863$	352.727i	0.408721i	−0.978896	+	0.204361i	$0.934488\pi$
$864$	0	0
$865$	888.000	1.02659
$866$	0	0
$867$	0	0
$868$	0	0
$869$	685.857i	0.789249i
$870$	0	0
$871$	1239.35i	1.42291i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	986.631	1.12758
$876$	0	0
$877$	− 480.250i	− 0.547605i	−0.961786	−	0.273803i	$-0.911718\pi$
$877$	− 480.250i	− 0.547605i	0.961786	−	0.273803i	$-0.0882815\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	404.772	0.459446	0.229723	−	0.973256i	$-0.426218\pi$
$881$	404.772	0.459446	0.229723	+	0.973256i	$0.426218\pi$
$882$	0	0
$883$	1720.00	1.94790	0.973952	−	0.226752i	$-0.0728108\pi$
$883$	1720.00	1.94790	0.973952	+	0.226752i	$0.0728108\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	548.686i	0.618586i	0.950967	+	0.309293i	$0.100092\pi$
$887$	548.686i	0.618586i	−0.950967	+	0.309293i	$0.899908\pi$
$888$	0	0
$889$	900.000	1.01237
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 313.535i	− 0.351103i
$894$	0	0
$895$	− 1363.29i	− 1.52323i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−189.737	−0.211053
$900$	0	0
$901$	− 371.806i	− 0.412660i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1593.79	1.76109
$906$	0	0
$907$	760.000	0.837927	0.418964	−	0.908003i	$-0.362393\pi$
$907$	760.000	0.837927	0.418964	+	0.908003i	$0.362393\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1371.71i	− 1.50572i	−0.658179	−	0.752862i	$-0.728673\pi$
$911$	− 1371.71i	− 1.50572i	0.658179	−	0.752862i	$-0.271327\pi$
$912$	0	0
$913$	−1760.00	−1.92771
$914$	0	0
$915$	0	0
$916$	0	0
$917$	391.918i	0.427392i
$918$	0	0
$919$	− 1231.61i	− 1.34016i	−0.742288	−	0.670081i	$-0.766259\pi$
$919$	− 1231.61i	− 1.34016i	0.742288	−	0.670081i	$-0.233741\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	46.4758i	0.0502441i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−430.070	−0.462938	−0.231469	−	0.972842i	$-0.574353\pi$
$929$	−430.070	−0.462938	−0.231469	+	0.972842i	$0.574353\pi$
$930$	0	0
$931$	88.0000	0.0945220
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1567.67i	− 1.67666i
$936$	0	0
$937$	−1330.00	−1.41942	−0.709712	−	0.704492i	$-0.751175\pi$
$937$	−1330.00	−1.41942	−0.709712	+	0.704492i	$0.751175\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 563.383i	− 0.598706i	−0.954142	−	0.299353i	$-0.903229\pi$
$941$	− 563.383i	− 0.598706i	0.954142	−	0.299353i	$-0.0967709\pi$
$942$	0	0
$943$	− 991.484i	− 1.05141i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1037.23	1.09528	0.547638	−	0.836715i	$-0.315527\pi$
$947$	1037.23	1.09528	0.547638	+	0.836715i	$0.315527\pi$
$948$	0	0
$949$	154.919i	0.163245i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.2982	0.0265459	0.0132729	−	0.999912i	$-0.495775\pi$
$953$	25.2982	0.0265459	0.0132729	+	0.999912i	$0.495775\pi$
$954$	0	0
$955$	−960.000	−1.00524
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 979.796i	− 1.02168i
$960$	0	0
$961$	901.000	0.937565
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 832.827i	− 0.863033i
$966$	0	0
$967$	1107.67i	1.14547i	0.819739	+	0.572737i	$0.194118\pi$
$967$	1107.67i	1.14547i	−0.819739	+	0.572737i	$0.805882\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−746.298	−0.768587	−0.384293	−	0.923211i	$-0.625555\pi$
$971$	−746.298	−0.768587	−0.384293	+	0.923211i	$0.625555\pi$
$972$	0	0
$973$	1611.16i	1.65587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	480.666	0.491982	0.245991	−	0.969272i	$-0.420887\pi$
$977$	480.666	0.491982	0.245991	+	0.969272i	$0.420887\pi$
$978$	0	0
$979$	−640.000	−0.653728
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 548.686i	− 0.558175i	−0.960266	−	0.279087i	$-0.909968\pi$
$983$	− 548.686i	− 0.558175i	0.960266	−	0.279087i	$-0.0900319\pi$
$984$	0	0
$985$	−648.000	−0.657868
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1567.67i	− 1.58511i
$990$	0	0
$991$	− 1185.13i	− 1.19590i	−0.801535	−	0.597948i	$-0.795983\pi$
$991$	− 1185.13i	− 1.19590i	0.801535	−	0.597948i	$-0.204017\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−341.526	−0.343242
$996$	0	0
$997$	666.153i	0.668158i	0.942545	+	0.334079i	$0.108425\pi$
$997$	666.153i	0.668158i	−0.942545	+	0.334079i	$0.891575\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.3.b.c.271.2		4
3.2	odd	2	inner	288.3.b.c.271.4		4
4.3	odd	2		72.3.b.c.19.1	✓	4
8.3	odd	2	inner	288.3.b.c.271.3		4
8.5	even	2		72.3.b.c.19.2	yes	4
12.11	even	2		72.3.b.c.19.4	yes	4
16.3	odd	4		2304.3.g.y.1279.4		8
16.5	even	4		2304.3.g.y.1279.5		8
16.11	odd	4		2304.3.g.y.1279.7		8
16.13	even	4		2304.3.g.y.1279.2		8
24.5	odd	2		72.3.b.c.19.3	yes	4
24.11	even	2	inner	288.3.b.c.271.1		4
48.5	odd	4		2304.3.g.y.1279.1		8
48.11	even	4		2304.3.g.y.1279.3		8
48.29	odd	4		2304.3.g.y.1279.6		8
48.35	even	4		2304.3.g.y.1279.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.b.c.19.1	✓	4	4.3	odd	2
72.3.b.c.19.2	yes	4	8.5	even	2
72.3.b.c.19.3	yes	4	24.5	odd	2
72.3.b.c.19.4	yes	4	12.11	even	2
288.3.b.c.271.1		4	24.11	even	2	inner
288.3.b.c.271.2		4	1.1	even	1	trivial
288.3.b.c.271.3		4	8.3	odd	2	inner
288.3.b.c.271.4		4	3.2	odd	2	inner
2304.3.g.y.1279.1		8	48.5	odd	4
2304.3.g.y.1279.2		8	16.13	even	4
2304.3.g.y.1279.3		8	48.11	even	4
2304.3.g.y.1279.4		8	16.3	odd	4
2304.3.g.y.1279.5		8	16.5	even	4
2304.3.g.y.1279.6		8	48.29	odd	4
2304.3.g.y.1279.7		8	16.11	odd	4
2304.3.g.y.1279.8		8	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.89898i q^{5} +7.74597i q^{7} +12.6491 q^{11} -15.4919i q^{13} +25.2982 q^{17} -8.00000 q^{19} -39.1918i q^{23} +1.00000 q^{25} -24.4949i q^{29} -7.74597i q^{31} +37.9473 q^{35} +46.4758i q^{37} +25.2982 q^{41} +40.0000 q^{43} +39.1918i q^{47} -11.0000 q^{49} -14.6969i q^{53} -61.9677i q^{55} -25.2982 q^{59} +15.4919i q^{61} -75.8947 q^{65} -80.0000 q^{67} -10.0000 q^{73} +97.9796i q^{77} +54.2218i q^{79} -139.140 q^{83} -123.935i q^{85} -50.5964 q^{89} +120.000 q^{91} +39.1918i q^{95} +50.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{19} + 4 q^{25} + 160 q^{43} - 44 q^{49} - 320 q^{67} - 40 q^{73} + 480 q^{91} + 200 q^{97}+O(q^{100}) \) 4 * q - 32 * q^19 + 4 * q^25 + 160 * q^43 - 44 * q^49 - 320 * q^67 - 40 * q^73 + 480 * q^91 + 200 * q^97

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