LMFDB - Embedded newform 4356.3.f.h.1693.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4356,3,Mod(1693,4356)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4356, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4356.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4356.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-20,0,68,0,0,0, 0,0,96,0,0,0,0,0,-24,0,0,0,0,0,0,0,0,0,-248,0,-24,0,0,0,-76,0,0,0,0,0, -28,0,0,0,0,0,0,0,44,0,0,0,372,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-152, 0,320] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	no
Analytic conductor:	$118.692403155$
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} - 122x^{6} - 84x^{5} + 5617x^{4} + 5796x^{3} - 106140x^{2} - 56448x + 902232 $ x^8 - 122x^6 - 84x^5 + 5617x^4 + 5796x^3 - 106140x^2 - 56448x + 902232
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1452)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.6
Root			$-5.29611 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	4356.1693
Dual form			4356.3.f.h.1693.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.29611 q^{5} +11.2666i q^{7} -8.53702i q^{13} +18.8227i q^{17} -19.4817i q^{19} -34.9216 q^{23} -6.54344 q^{25} -9.50332i q^{29} +24.6664 q^{31} +48.4024i q^{35} -27.0065 q^{37} -57.3980i q^{41} -41.2949i q^{43} -53.4744 q^{47} -77.9357 q^{49} -25.0337 q^{53} +101.668 q^{59} +114.718i q^{61} -36.6760i q^{65} -73.4602 q^{67} -105.706 q^{71} -41.3560i q^{73} -76.6876i q^{79} -129.542i q^{83} +80.8645i q^{85} -154.763 q^{89} +96.1830 q^{91} -83.6956i q^{95} +106.580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 20 q^{23} + 68 q^{25} + 96 q^{31} - 24 q^{37} - 248 q^{47} - 24 q^{49} - 76 q^{53} - 28 q^{59} + 44 q^{67} + 372 q^{71} - 152 q^{89} + 320 q^{91} + 244 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 20 * q^23 + 68 * q^25 + 96 * q^31 - 24 * q^37 - 248 * q^47 - 24 * q^49 - 76 * q^53 - 28 * q^59 + 44 * q^67 + 372 * q^71 - 152 * q^89 + 320 * q^91 + 244 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4356\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$	$2179$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	4.29611	0.859222	0.429611	−	0.903014i	$-0.358651\pi$
$5$	4.29611	0.859222	0.429611	+	0.903014i	$0.358651\pi$
$6$	0	0
$7$	11.2666i	1.60951i	0.593607	+	0.804755i	$0.297703\pi$
$7$	11.2666i	1.60951i	−0.593607	+	0.804755i	$0.702297\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 8.53702i	− 0.656694i	−0.944557	−	0.328347i	$-0.893508\pi$
$13$	− 8.53702i	− 0.656694i	0.944557	−	0.328347i	$-0.106492\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.8227i	1.10722i	0.832776	+	0.553609i	$0.186750\pi$
$17$	18.8227i	1.10722i	−0.832776	+	0.553609i	$0.813250\pi$
$18$	0	0
$19$	− 19.4817i	− 1.02535i	−0.858582	−	0.512677i	$-0.828654\pi$
$19$	− 19.4817i	− 1.02535i	0.858582	−	0.512677i	$-0.171346\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−34.9216	−1.51833	−0.759166	−	0.650897i	$-0.774393\pi$
$23$	−34.9216	−1.51833	−0.759166	+	0.650897i	$0.774393\pi$
$24$	0	0
$25$	−6.54344	−0.261738
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.50332i	− 0.327701i	−0.986485	−	0.163850i	$-0.947609\pi$
$29$	− 9.50332i	− 0.327701i	0.986485	−	0.163850i	$-0.0523914\pi$
$30$	0	0
$31$	24.6664	0.795691	0.397846	−	0.917452i	$-0.369758\pi$
$31$	24.6664	0.795691	0.397846	+	0.917452i	$0.369758\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	48.4024i	1.38293i
$36$	0	0
$37$	−27.0065	−0.729906	−0.364953	−	0.931026i	$-0.618915\pi$
$37$	−27.0065	−0.729906	−0.364953	+	0.931026i	$0.618915\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 57.3980i	− 1.39995i	−0.714166	−	0.699976i	$-0.753194\pi$
$41$	− 57.3980i	− 1.39995i	0.714166	−	0.699976i	$-0.246806\pi$
$42$	0	0
$43$	− 41.2949i	− 0.960347i	−0.877174	−	0.480173i	$-0.840574\pi$
$43$	− 41.2949i	− 0.960347i	0.877174	−	0.480173i	$-0.159426\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−53.4744	−1.13775	−0.568876	−	0.822423i	$-0.692622\pi$
$47$	−53.4744	−1.13775	−0.568876	+	0.822423i	$0.692622\pi$
$48$	0	0
$49$	−77.9357	−1.59052
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−25.0337	−0.472335	−0.236167	−	0.971712i	$-0.575891\pi$
$53$	−25.0337	−0.472335	−0.236167	+	0.971712i	$0.575891\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	101.668	1.72318	0.861591	−	0.507603i	$-0.169468\pi$
$59$	101.668	1.72318	0.861591	+	0.507603i	$0.169468\pi$
$60$	0	0
$61$	114.718i	1.88063i	0.340306	+	0.940315i	$0.389469\pi$
$61$	114.718i	1.88063i	−0.340306	+	0.940315i	$0.610531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 36.6760i	− 0.564246i
$66$	0	0
$67$	−73.4602	−1.09642	−0.548210	−	0.836340i	$-0.684691\pi$
$67$	−73.4602	−1.09642	−0.548210	+	0.836340i	$0.684691\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−105.706	−1.48882	−0.744409	−	0.667724i	$-0.767269\pi$
$71$	−105.706	−1.48882	−0.744409	+	0.667724i	$0.767269\pi$
$72$	0	0
$73$	− 41.3560i	− 0.566521i	−0.959043	−	0.283260i	$-0.908584\pi$
$73$	− 41.3560i	− 0.566521i	0.959043	−	0.283260i	$-0.0914160\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 76.6876i	− 0.970729i	−0.874312	−	0.485365i	$-0.838687\pi$
$79$	− 76.6876i	− 0.970729i	0.874312	−	0.485365i	$-0.161313\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 129.542i	− 1.56075i	−0.625313	−	0.780374i	$-0.715029\pi$
$83$	− 129.542i	− 1.56075i	0.625313	−	0.780374i	$-0.284971\pi$
$84$	0	0
$85$	80.8645i	0.951347i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−154.763	−1.73891	−0.869457	−	0.494009i	$-0.835531\pi$
$89$	−154.763	−1.73891	−0.869457	+	0.494009i	$0.835531\pi$
$90$	0	0
$91$	96.1830	1.05696
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 83.6956i	− 0.881006i
$96$	0	0
$97$	106.580	1.09877	0.549384	−	0.835570i	$-0.314863\pi$
$97$	106.580	1.09877	0.549384	+	0.835570i	$0.314863\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 21.4654i	− 0.212529i	−0.994338	−	0.106264i	$-0.966111\pi$
$101$	− 21.4654i	− 0.212529i	0.994338	−	0.106264i	$-0.0338890\pi$
$102$	0	0
$103$	9.44447	0.0916939	0.0458470	−	0.998948i	$-0.485401\pi$
$103$	9.44447	0.0916939	0.0458470	+	0.998948i	$0.485401\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 77.4125i	− 0.723481i	−0.932279	−	0.361741i	$-0.882183\pi$
$107$	− 77.4125i	− 0.723481i	0.932279	−	0.361741i	$-0.117817\pi$
$108$	0	0
$109$	64.5647i	0.592337i	0.955136	+	0.296168i	$0.0957090\pi$
$109$	64.5647i	0.592337i	−0.955136	+	0.296168i	$0.904291\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−75.4226	−0.667457	−0.333728	−	0.942669i	$-0.608307\pi$
$113$	−75.4226	−0.667457	−0.333728	+	0.942669i	$0.608307\pi$
$114$	0	0
$115$	−150.027	−1.30458
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−212.068	−1.78208
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−135.514	−1.08411
$126$	0	0
$127$	109.910i	0.865436i	0.901529	+	0.432718i	$0.142445\pi$
$127$	109.910i	0.865436i	−0.901529	+	0.432718i	$0.857555\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 45.8062i	− 0.349666i	−0.984598	−	0.174833i	$-0.944061\pi$
$131$	− 45.8062i	− 0.349666i	0.984598	−	0.174833i	$-0.0559385\pi$
$132$	0	0
$133$	219.492	1.65032
$134$	0	0
$135$	0	0
$136$	0	0
$137$	177.446	1.29523	0.647613	−	0.761969i	$-0.275767\pi$
$137$	177.446	1.29523	0.647613	+	0.761969i	$0.275767\pi$
$138$	0	0
$139$	134.102i	0.964762i	0.875962	+	0.482381i	$0.160228\pi$
$139$	134.102i	0.964762i	−0.875962	+	0.482381i	$0.839772\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 40.8273i	− 0.281568i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 28.1457i	− 0.188897i	−0.995530	−	0.0944487i	$-0.969891\pi$
$149$	− 28.1457i	− 0.188897i	0.995530	−	0.0944487i	$-0.0301088\pi$
$150$	0	0
$151$	− 116.164i	− 0.769301i	−0.923062	−	0.384651i	$-0.874322\pi$
$151$	− 116.164i	− 0.769301i	0.923062	−	0.384651i	$-0.125678\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	105.970	0.683675
$156$	0	0
$157$	75.8469	0.483101	0.241551	−	0.970388i	$-0.422344\pi$
$157$	75.8469	0.483101	0.241551	+	0.970388i	$0.422344\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 393.447i	− 2.44377i
$162$	0	0
$163$	13.2014	0.0809901	0.0404951	−	0.999180i	$-0.487106\pi$
$163$	13.2014	0.0809901	0.0404951	+	0.999180i	$0.487106\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 196.988i	− 1.17957i	−0.807560	−	0.589785i	$-0.799212\pi$
$167$	− 196.988i	− 1.17957i	0.807560	−	0.589785i	$-0.200788\pi$
$168$	0	0
$169$	96.1192	0.568753
$170$	0	0
$171$	0	0
$172$	0	0
$173$	133.409i	0.771149i	0.922677	+	0.385574i	$0.125997\pi$
$173$	133.409i	0.771149i	−0.922677	+	0.385574i	$0.874003\pi$
$174$	0	0
$175$	− 73.7222i	− 0.421270i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−154.706	−0.864278	−0.432139	−	0.901807i	$-0.642241\pi$
$179$	−154.706	−0.864278	−0.432139	+	0.901807i	$0.642241\pi$
$180$	0	0
$181$	4.92735	0.0272229	0.0136115	−	0.999907i	$-0.495667\pi$
$181$	4.92735	0.0272229	0.0136115	+	0.999907i	$0.495667\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−116.023	−0.627151
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	374.082	1.95854	0.979271	−	0.202554i	$-0.0649243\pi$
$191$	374.082	1.95854	0.979271	+	0.202554i	$0.0649243\pi$
$192$	0	0
$193$	279.486i	1.44811i	0.689741	+	0.724056i	$0.257724\pi$
$193$	279.486i	1.44811i	−0.689741	+	0.724056i	$0.742276\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	102.226i	0.518914i	0.965755	+	0.259457i	$0.0835435\pi$
$197$	102.226i	0.518914i	−0.965755	+	0.259457i	$0.916456\pi$
$198$	0	0
$199$	356.042	1.78915	0.894577	−	0.446913i	$-0.147477\pi$
$199$	356.042	1.78915	0.894577	+	0.446913i	$0.147477\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	107.070	0.527438
$204$	0	0
$205$	− 246.588i	− 1.20287i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 233.984i	− 1.10893i	−0.832208	−	0.554463i	$-0.812924\pi$
$211$	− 233.984i	− 1.10893i	0.832208	−	0.554463i	$-0.187076\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 177.407i	− 0.825151i
$216$	0	0
$217$	277.906i	1.28067i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	160.690	0.727104
$222$	0	0
$223$	−230.249	−1.03250	−0.516252	−	0.856436i	$-0.672673\pi$
$223$	−230.249	−1.03250	−0.516252	+	0.856436i	$0.672673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 369.054i	− 1.62579i	−0.582412	−	0.812894i	$-0.697891\pi$
$227$	− 369.054i	− 1.62579i	0.582412	−	0.812894i	$-0.302109\pi$
$228$	0	0
$229$	−435.716	−1.90269	−0.951345	−	0.308129i	$-0.900297\pi$
$229$	−435.716	−1.90269	−0.951345	+	0.308129i	$0.900297\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 314.438i	− 1.34952i	−0.738038	−	0.674759i	$-0.764248\pi$
$233$	− 314.438i	− 1.34952i	0.738038	−	0.674759i	$-0.235752\pi$
$234$	0	0
$235$	−229.732	−0.977582
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 99.0828i	− 0.414572i	−0.978280	−	0.207286i	$-0.933537\pi$
$239$	− 99.0828i	− 0.414572i	0.978280	−	0.207286i	$-0.0664631\pi$
$240$	0	0
$241$	− 163.646i	− 0.679030i	−0.940601	−	0.339515i	$-0.889737\pi$
$241$	− 163.646i	− 0.679030i	0.940601	−	0.339515i	$-0.110263\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−334.820	−1.36661
$246$	0	0
$247$	−166.316	−0.673344
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.2866	0.0967594	0.0483797	−	0.998829i	$-0.484594\pi$
$251$	24.2866	0.0967594	0.0483797	+	0.998829i	$0.484594\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.21310	0.0163934	0.00819669	−	0.999966i	$-0.497391\pi$
$257$	4.21310	0.0163934	0.00819669	+	0.999966i	$0.497391\pi$
$258$	0	0
$259$	− 304.271i	− 1.17479i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 81.9742i	− 0.311689i	−0.987782	−	0.155845i	$-0.950190\pi$
$263$	− 81.9742i	− 0.311689i	0.987782	−	0.155845i	$-0.0498099\pi$
$264$	0	0
$265$	−107.548	−0.405840
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−475.097	−1.76616	−0.883079	−	0.469224i	$-0.844534\pi$
$269$	−475.097	−1.76616	−0.883079	+	0.469224i	$0.844534\pi$
$270$	0	0
$271$	− 356.841i	− 1.31676i	−0.752687	−	0.658379i	$-0.771242\pi$
$271$	− 356.841i	− 1.31676i	0.752687	−	0.658379i	$-0.228758\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	357.686i	1.29128i	0.763640	+	0.645642i	$0.223410\pi$
$277$	357.686i	1.29128i	−0.763640	+	0.645642i	$0.776590\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	101.725i	0.362012i	0.983482	+	0.181006i	$0.0579353\pi$
$281$	101.725i	0.362012i	−0.983482	+	0.181006i	$0.942065\pi$
$282$	0	0
$283$	478.049i	1.68922i	0.535382	+	0.844610i	$0.320168\pi$
$283$	478.049i	1.68922i	−0.535382	+	0.844610i	$0.679832\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	646.679	2.25324
$288$	0	0
$289$	−65.2949	−0.225934
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 281.330i	− 0.960169i	−0.877222	−	0.480085i	$-0.840606\pi$
$293$	− 281.330i	− 0.960169i	0.877222	−	0.480085i	$-0.159394\pi$
$294$	0	0
$295$	436.776	1.48060
$296$	0	0
$297$	0	0
$298$	0	0
$299$	298.127i	0.997079i
$300$	0	0
$301$	465.252	1.54569
$302$	0	0
$303$	0	0
$304$	0	0
$305$	492.843i	1.61588i
$306$	0	0
$307$	− 305.673i	− 0.995678i	−0.867269	−	0.497839i	$-0.834127\pi$
$307$	− 305.673i	− 0.995678i	0.867269	−	0.497839i	$-0.165873\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	146.798	0.472020	0.236010	−	0.971751i	$-0.424160\pi$
$311$	146.798	0.472020	0.236010	+	0.971751i	$0.424160\pi$
$312$	0	0
$313$	401.538	1.28287	0.641435	−	0.767178i	$-0.278339\pi$
$313$	401.538	1.28287	0.641435	+	0.767178i	$0.278339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−182.861	−0.576849	−0.288425	−	0.957503i	$-0.593131\pi$
$317$	−182.861	−0.576849	−0.288425	+	0.957503i	$0.593131\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	366.699	1.13529
$324$	0	0
$325$	55.8615i	0.171882i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 602.473i	− 1.83123i
$330$	0	0
$331$	6.11805	0.0184835	0.00924177	−	0.999957i	$-0.497058\pi$
$331$	6.11805	0.0184835	0.00924177	+	0.999957i	$0.497058\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−315.593	−0.942069
$336$	0	0
$337$	330.000i	0.979227i	0.871939	+	0.489614i	$0.162862\pi$
$337$	330.000i	0.979227i	−0.871939	+	0.489614i	$0.837138\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 326.006i	− 0.950455i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	611.432i	1.76205i	0.473069	+	0.881025i	$0.343146\pi$
$347$	611.432i	1.76205i	−0.473069	+	0.881025i	$0.656854\pi$
$348$	0	0
$349$	− 292.437i	− 0.837929i	−0.908003	−	0.418965i	$-0.862393\pi$
$349$	− 292.437i	− 0.837929i	0.908003	−	0.418965i	$-0.137607\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−483.474	−1.36962	−0.684808	−	0.728724i	$-0.740114\pi$
$353$	−483.474	−1.36962	−0.684808	+	0.728724i	$0.740114\pi$
$354$	0	0
$355$	−454.125	−1.27923
$356$	0	0
$357$	0	0
$358$	0	0
$359$	226.676i	0.631409i	0.948858	+	0.315704i	$0.102241\pi$
$359$	226.676i	0.631409i	−0.948858	+	0.315704i	$0.897759\pi$
$360$	0	0
$361$	−18.5374	−0.0513502
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 177.670i	− 0.486767i
$366$	0	0
$367$	404.623	1.10252	0.551258	−	0.834335i	$-0.314148\pi$
$367$	404.623	1.10252	0.551258	+	0.834335i	$0.314148\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 282.044i	− 0.760227i
$372$	0	0
$373$	466.702i	1.25121i	0.780140	+	0.625605i	$0.215148\pi$
$373$	466.702i	1.25121i	−0.780140	+	0.625605i	$0.784852\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−81.1301	−0.215199
$378$	0	0
$379$	19.4870	0.0514168	0.0257084	−	0.999669i	$-0.491816\pi$
$379$	19.4870	0.0514168	0.0257084	+	0.999669i	$0.491816\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−575.275	−1.50202	−0.751012	−	0.660288i	$-0.770434\pi$
$383$	−575.275	−1.50202	−0.751012	+	0.660288i	$0.770434\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.3772	0.0292474	0.0146237	−	0.999893i	$-0.495345\pi$
$389$	11.3772	0.0292474	0.0146237	+	0.999893i	$0.495345\pi$
$390$	0	0
$391$	− 657.320i	− 1.68113i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 329.458i	− 0.834072i
$396$	0	0
$397$	−598.360	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$397$	−598.360	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−535.327	−1.33498	−0.667489	−	0.744619i	$-0.732631\pi$
$401$	−535.327	−1.33498	−0.667489	+	0.744619i	$0.732631\pi$
$402$	0	0
$403$	− 210.578i	− 0.522526i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 69.4847i	− 0.169889i	−0.996386	−	0.0849447i	$-0.972929\pi$
$409$	− 69.4847i	− 0.169889i	0.996386	−	0.0849447i	$-0.0270713\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1145.45i	2.77348i
$414$	0	0
$415$	− 556.527i	− 1.34103i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−166.637	−0.397701	−0.198851	−	0.980030i	$-0.563721\pi$
$419$	−166.637	−0.397701	−0.198851	+	0.980030i	$0.563721\pi$
$420$	0	0
$421$	−390.912	−0.928532	−0.464266	−	0.885696i	$-0.653682\pi$
$421$	−390.912	−0.928532	−0.464266	+	0.885696i	$0.653682\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 123.165i	− 0.289801i
$426$	0	0
$427$	−1292.48	−3.02689
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 23.6611i	− 0.0548981i	−0.999623	−	0.0274490i	$-0.991262\pi$
$431$	− 23.6611i	− 0.0548981i	0.999623	−	0.0274490i	$-0.00873840\pi$
$432$	0	0
$433$	252.280	0.582633	0.291317	−	0.956627i	$-0.405907\pi$
$433$	252.280	0.582633	0.291317	+	0.956627i	$0.405907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	680.333i	1.55683i
$438$	0	0
$439$	− 563.619i	− 1.28387i	−0.766759	−	0.641935i	$-0.778132\pi$
$439$	− 563.619i	− 1.28387i	0.766759	−	0.641935i	$-0.221868\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−444.572	−1.00355	−0.501775	−	0.864998i	$-0.667319\pi$
$443$	−444.572	−1.00355	−0.501775	+	0.864998i	$0.667319\pi$
$444$	0	0
$445$	−664.880	−1.49411
$446$	0	0
$447$	0	0
$448$	0	0
$449$	559.625	1.24638	0.623190	−	0.782070i	$-0.285836\pi$
$449$	559.625	1.24638	0.623190	+	0.782070i	$0.285836\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	413.213	0.908160
$456$	0	0
$457$	450.755i	0.986334i	0.869935	+	0.493167i	$0.164161\pi$
$457$	450.755i	0.986334i	−0.869935	+	0.493167i	$0.835839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 499.755i	− 1.08407i	−0.840357	−	0.542034i	$-0.817654\pi$
$461$	− 499.755i	− 1.08407i	0.840357	−	0.542034i	$-0.182346\pi$
$462$	0	0
$463$	−513.790	−1.10970	−0.554849	−	0.831951i	$-0.687224\pi$
$463$	−513.790	−1.10970	−0.554849	+	0.831951i	$0.687224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−659.459	−1.41212	−0.706059	−	0.708153i	$-0.749529\pi$
$467$	−659.459	−1.41212	−0.706059	+	0.708153i	$0.749529\pi$
$468$	0	0
$469$	− 827.645i	− 1.76470i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	127.478i	0.268374i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	66.6672i	0.139180i	0.997576	+	0.0695900i	$0.0221691\pi$
$479$	66.6672i	0.139180i	−0.997576	+	0.0695900i	$0.977831\pi$
$480$	0	0
$481$	230.555i	0.479325i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	457.881	0.944085
$486$	0	0
$487$	−739.168	−1.51780	−0.758899	−	0.651208i	$-0.774262\pi$
$487$	−739.168	−1.51780	−0.758899	+	0.651208i	$0.774262\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 234.675i	− 0.477952i	−0.971025	−	0.238976i	$-0.923188\pi$
$491$	− 234.675i	− 0.477952i	0.971025	−	0.238976i	$-0.0768118\pi$
$492$	0	0
$493$	178.878	0.362836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1190.95i	− 2.39627i
$498$	0	0
$499$	−336.958	−0.675267	−0.337634	−	0.941278i	$-0.609626\pi$
$499$	−336.958	−0.675267	−0.337634	+	0.941278i	$0.609626\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 77.9744i	− 0.155019i	−0.996992	−	0.0775093i	$-0.975303\pi$
$503$	− 77.9744i	− 0.155019i	0.996992	−	0.0775093i	$-0.0246968\pi$
$504$	0	0
$505$	− 92.2177i	− 0.182609i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−679.368	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$509$	−679.368	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$510$	0	0
$511$	465.940	0.911821
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.5745	0.0787854
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	507.183	0.973480	0.486740	−	0.873547i	$-0.338186\pi$
$521$	507.183	0.973480	0.486740	+	0.873547i	$0.338186\pi$
$522$	0	0
$523$	− 114.146i	− 0.218253i	−0.994028	−	0.109127i	$-0.965195\pi$
$523$	− 114.146i	− 0.218253i	0.994028	−	0.109127i	$-0.0348054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	464.289i	0.881004i
$528$	0	0
$529$	690.520	1.30533
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−490.008	−0.919340
$534$	0	0
$535$	− 332.573i	− 0.621631i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	720.619i	1.33201i	0.745946	+	0.666006i	$0.231998\pi$
$541$	720.619i	1.33201i	−0.745946	+	0.666006i	$0.768002\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	277.377i	0.508949i
$546$	0	0
$547$	− 741.614i	− 1.35578i	−0.735161	−	0.677892i	$-0.762893\pi$
$547$	− 741.614i	− 1.35578i	0.735161	−	0.677892i	$-0.237107\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−185.141	−0.336009
$552$	0	0
$553$	864.007	1.56240
$554$	0	0
$555$	0	0
$556$	0	0
$557$	344.843i	0.619108i	0.950882	+	0.309554i	$0.100180\pi$
$557$	344.843i	0.619108i	−0.950882	+	0.309554i	$0.899820\pi$
$558$	0	0
$559$	−352.536	−0.630654
$560$	0	0
$561$	0	0
$562$	0	0
$563$	126.613i	0.224890i	0.993658	+	0.112445i	$0.0358683\pi$
$563$	126.613i	0.224890i	−0.993658	+	0.112445i	$0.964132\pi$
$564$	0	0
$565$	−324.024	−0.573493
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 559.627i	− 0.983526i	−0.870729	−	0.491763i	$-0.836353\pi$
$569$	− 559.627i	− 0.983526i	0.870729	−	0.491763i	$-0.163647\pi$
$570$	0	0
$571$	− 67.4557i	− 0.118136i	−0.998254	−	0.0590680i	$-0.981187\pi$
$571$	− 67.4557i	− 0.118136i	0.998254	−	0.0590680i	$-0.0188129\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	228.508	0.397405
$576$	0	0
$577$	−516.662	−0.895427	−0.447714	−	0.894177i	$-0.647762\pi$
$577$	−516.662	−0.895427	−0.447714	+	0.894177i	$0.647762\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1459.50	2.51204
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−355.032	−0.604825	−0.302413	−	0.953177i	$-0.597792\pi$
$587$	−355.032	−0.604825	−0.302413	+	0.953177i	$0.597792\pi$
$588$	0	0
$589$	− 480.544i	− 0.815865i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 691.664i	− 1.16638i	−0.812335	−	0.583191i	$-0.801804\pi$
$593$	− 691.664i	− 1.16638i	0.812335	−	0.583191i	$-0.198196\pi$
$594$	0	0
$595$	−911.066	−1.53120
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−649.689	−1.08462	−0.542312	−	0.840177i	$-0.682451\pi$
$599$	−649.689	−1.08462	−0.542312	+	0.840177i	$0.682451\pi$
$600$	0	0
$601$	− 801.856i	− 1.33420i	−0.744967	−	0.667101i	$-0.767535\pi$
$601$	− 801.856i	− 1.33420i	0.744967	−	0.667101i	$-0.232465\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 512.192i	− 0.843810i	−0.906640	−	0.421905i	$-0.861362\pi$
$607$	− 512.192i	− 0.843810i	0.906640	−	0.421905i	$-0.138638\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	456.512i	0.747155i
$612$	0	0
$613$	− 1015.96i	− 1.65736i	−0.559724	−	0.828679i	$-0.689093\pi$
$613$	− 1015.96i	− 1.65736i	0.559724	−	0.828679i	$-0.310907\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	790.584	1.28133	0.640667	−	0.767819i	$-0.278658\pi$
$617$	790.584	1.28133	0.640667	+	0.767819i	$0.278658\pi$
$618$	0	0
$619$	15.1503	0.0244754	0.0122377	−	0.999925i	$-0.496105\pi$
$619$	15.1503	0.0244754	0.0122377	+	0.999925i	$0.496105\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1743.65i	− 2.79880i
$624$	0	0
$625$	−418.597	−0.669756
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 508.336i	− 0.808166i
$630$	0	0
$631$	275.558	0.436701	0.218350	−	0.975870i	$-0.429932\pi$
$631$	275.558	0.436701	0.218350	+	0.975870i	$0.429932\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	472.187i	0.743602i
$636$	0	0
$637$	665.339i	1.04449i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	770.510	1.20204	0.601022	−	0.799232i	$-0.294760\pi$
$641$	770.510	1.20204	0.601022	+	0.799232i	$0.294760\pi$
$642$	0	0
$643$	428.201	0.665943	0.332971	−	0.942937i	$-0.391949\pi$
$643$	428.201	0.665943	0.332971	+	0.942937i	$0.391949\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−844.252	−1.30487	−0.652436	−	0.757844i	$-0.726253\pi$
$647$	−844.252	−1.30487	−0.652436	+	0.757844i	$0.726253\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−538.685	−0.824939	−0.412470	−	0.910971i	$-0.635334\pi$
$653$	−538.685	−0.824939	−0.412470	+	0.910971i	$0.635334\pi$
$654$	0	0
$655$	− 196.789i	− 0.300441i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	633.597i	0.961452i	0.876871	+	0.480726i	$0.159627\pi$
$659$	633.597i	0.961452i	−0.876871	+	0.480726i	$0.840373\pi$
$660$	0	0
$661$	336.321	0.508806	0.254403	−	0.967098i	$-0.418121\pi$
$661$	336.321	0.508806	0.254403	+	0.967098i	$0.418121\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	942.963	1.41799
$666$	0	0
$667$	331.871i	0.497558i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.5303i	0.0394210i	0.999806	+	0.0197105i	$0.00627446\pi$
$673$	26.5303i	0.0394210i	−0.999806	+	0.0197105i	$0.993726\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 949.002i	− 1.40178i	−0.713271	−	0.700888i	$-0.752787\pi$
$677$	− 949.002i	− 1.40178i	0.713271	−	0.700888i	$-0.247213\pi$
$678$	0	0
$679$	1200.80i	1.76848i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−212.285	−0.310813	−0.155406	−	0.987851i	$-0.549669\pi$
$683$	−212.285	−0.310813	−0.155406	+	0.987851i	$0.549669\pi$
$684$	0	0
$685$	762.327	1.11289
$686$	0	0
$687$	0	0
$688$	0	0
$689$	213.714i	0.310179i
$690$	0	0
$691$	1237.95	1.79153	0.895767	−	0.444523i	$-0.146627\pi$
$691$	1237.95	1.79153	0.895767	+	0.444523i	$0.146627\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	576.116i	0.828944i
$696$	0	0
$697$	1080.39	1.55005
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1131.57i	1.61423i	0.590397	+	0.807113i	$0.298971\pi$
$701$	1131.57i	1.61423i	−0.590397	+	0.807113i	$0.701029\pi$
$702$	0	0
$703$	526.133i	0.748412i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	241.842	0.342067
$708$	0	0
$709$	−513.897	−0.724820	−0.362410	−	0.932019i	$-0.618046\pi$
$709$	−513.897	−0.724820	−0.362410	+	0.932019i	$0.618046\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−861.392	−1.20812
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−860.203	−1.19639	−0.598194	−	0.801351i	$-0.704115\pi$
$719$	−860.203	−1.19639	−0.598194	+	0.801351i	$0.704115\pi$
$720$	0	0
$721$	106.407i	0.147582i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	62.1844i	0.0857717i
$726$	0	0
$727$	−42.6664	−0.0586883	−0.0293442	−	0.999569i	$-0.509342\pi$
$727$	−42.6664	−0.0586883	−0.0293442	+	0.999569i	$0.509342\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	777.283	1.06331
$732$	0	0
$733$	421.301i	0.574762i	0.957816	+	0.287381i	$0.0927846\pi$
$733$	421.301i	0.574762i	−0.957816	+	0.287381i	$0.907215\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 593.597i	− 0.803244i	−0.915806	−	0.401622i	$-0.868447\pi$
$739$	− 593.597i	− 0.803244i	0.915806	−	0.401622i	$-0.131553\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 344.394i	− 0.463518i	−0.972773	−	0.231759i	$-0.925552\pi$
$743$	− 344.394i	− 0.463518i	0.972773	−	0.231759i	$-0.0744481\pi$
$744$	0	0
$745$	− 120.917i	− 0.162305i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	872.174	1.16445
$750$	0	0
$751$	−42.5009	−0.0565924	−0.0282962	−	0.999600i	$-0.509008\pi$
$751$	−42.5009	−0.0565924	−0.0282962	+	0.999600i	$0.509008\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 499.055i	− 0.661000i
$756$	0	0
$757$	612.622	0.809276	0.404638	−	0.914477i	$-0.367398\pi$
$757$	612.622	0.809276	0.404638	+	0.914477i	$0.367398\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 971.925i	− 1.27717i	−0.769552	−	0.638584i	$-0.779521\pi$
$761$	− 971.925i	− 1.27717i	0.769552	−	0.638584i	$-0.220479\pi$
$762$	0	0
$763$	−727.423	−0.953372
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 867.940i	− 1.13160i
$768$	0	0
$769$	97.6695i	0.127008i	0.997982	+	0.0635042i	$0.0202276\pi$
$769$	97.6695i	0.127008i	−0.997982	+	0.0635042i	$0.979772\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−47.4526	−0.0613876	−0.0306938	−	0.999529i	$-0.509772\pi$
$773$	−47.4526	−0.0613876	−0.0306938	+	0.999529i	$0.509772\pi$
$774$	0	0
$775$	−161.403	−0.208262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1118.21	−1.43545
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	325.846	0.415091
$786$	0	0
$787$	− 19.1045i	− 0.0242751i	−0.999926	−	0.0121376i	$-0.996136\pi$
$787$	− 19.1045i	− 0.0242751i	0.999926	−	0.0121376i	$-0.00386360\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 849.754i	− 1.07428i
$792$	0	0
$793$	979.354	1.23500
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−402.913	−0.505537	−0.252769	−	0.967527i	$-0.581341\pi$
$797$	−402.913	−0.505537	−0.252769	+	0.967527i	$0.581341\pi$
$798$	0	0
$799$	− 1006.53i	− 1.25974i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 1690.29i	− 2.09974i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	460.834i	0.569635i	0.958582	+	0.284817i	$0.0919329\pi$
$809$	460.834i	0.569635i	−0.958582	+	0.284817i	$0.908067\pi$
$810$	0	0
$811$	− 138.120i	− 0.170308i	−0.996368	−	0.0851542i	$-0.972862\pi$
$811$	− 138.120i	− 0.170308i	0.996368	−	0.0851542i	$-0.0271383\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	56.7146	0.0695885
$816$	0	0
$817$	−804.496	−0.984695
$818$	0	0
$819$	0	0
$820$	0	0
$821$	253.642i	0.308943i	0.987997	+	0.154472i	$0.0493675\pi$
$821$	253.642i	0.308943i	−0.987997	+	0.154472i	$0.950632\pi$
$822$	0	0
$823$	−539.230	−0.655201	−0.327600	−	0.944816i	$-0.606240\pi$
$823$	−539.230	−0.655201	−0.327600	+	0.944816i	$0.606240\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 389.528i	− 0.471013i	−0.971873	−	0.235506i	$-0.924325\pi$
$827$	− 389.528i	− 0.471013i	0.971873	−	0.235506i	$-0.0756749\pi$
$828$	0	0
$829$	−535.197	−0.645594	−0.322797	−	0.946468i	$-0.604623\pi$
$829$	−535.197	−0.645594	−0.322797	+	0.946468i	$0.604623\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1466.96i	− 1.76106i
$834$	0	0
$835$	− 846.283i	− 1.01351i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−252.148	−0.300534	−0.150267	−	0.988645i	$-0.548013\pi$
$839$	−252.148	−0.300534	−0.150267	+	0.988645i	$0.548013\pi$
$840$	0	0
$841$	750.687	0.892612
$842$	0	0
$843$	0	0
$844$	0	0
$845$	412.939	0.488685
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	943.111	1.10824
$852$	0	0
$853$	1191.72i	1.39709i	0.715566	+	0.698545i	$0.246169\pi$
$853$	1191.72i	1.39709i	−0.715566	+	0.698545i	$0.753831\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1368.07i	1.59635i	0.602425	+	0.798176i	$0.294201\pi$
$857$	1368.07i	1.59635i	−0.602425	+	0.798176i	$0.705799\pi$
$858$	0	0
$859$	525.360	0.611595	0.305798	−	0.952097i	$-0.401077\pi$
$859$	525.360	0.611595	0.305798	+	0.952097i	$0.401077\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	773.424	0.896204	0.448102	−	0.893982i	$-0.352100\pi$
$863$	773.424	0.896204	0.448102	+	0.893982i	$0.352100\pi$
$864$	0	0
$865$	573.138i	0.662588i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	627.131i	0.720013i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1526.78i	− 1.74489i
$876$	0	0
$877$	789.916i	0.900702i	0.892852	+	0.450351i	$0.148701\pi$
$877$	789.916i	0.900702i	−0.892852	+	0.450351i	$0.851299\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	147.871	0.167844	0.0839221	−	0.996472i	$-0.473255\pi$
$881$	147.871	0.167844	0.0839221	+	0.996472i	$0.473255\pi$
$882$	0	0
$883$	307.458	0.348197	0.174098	−	0.984728i	$-0.444299\pi$
$883$	307.458	0.348197	0.174098	+	0.984728i	$0.444299\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 457.872i	− 0.516203i	−0.966118	−	0.258102i	$-0.916903\pi$
$887$	− 457.872i	− 0.516203i	0.966118	−	0.258102i	$-0.0830970\pi$
$888$	0	0
$889$	−1238.31	−1.39293
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1041.77i	1.16660i
$894$	0	0
$895$	−664.633	−0.742606
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 234.413i	− 0.260749i
$900$	0	0
$901$	− 471.203i	− 0.522978i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.1684	0.0233905
$906$	0	0
$907$	613.197	0.676071	0.338036	−	0.941133i	$-0.390238\pi$
$907$	613.197	0.676071	0.338036	+	0.941133i	$0.390238\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−604.244	−0.663275	−0.331638	−	0.943407i	$-0.607601\pi$
$911$	−604.244	−0.663275	−0.331638	+	0.943407i	$0.607601\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	516.079	0.562791
$918$	0	0
$919$	1179.83i	1.28382i	0.766781	+	0.641909i	$0.221857\pi$
$919$	1179.83i	1.28382i	−0.766781	+	0.641909i	$0.778143\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	902.416i	0.977698i
$924$	0	0
$925$	176.716	0.191044
$926$	0	0
$927$	0	0
$928$	0	0
$929$	129.409	0.139299	0.0696496	−	0.997572i	$-0.477812\pi$
$929$	129.409	0.139299	0.0696496	+	0.997572i	$0.477812\pi$
$930$	0	0
$931$	1518.32i	1.63085i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	124.330i	0.132690i	0.997797	+	0.0663448i	$0.0211337\pi$
$937$	124.330i	0.132690i	−0.997797	+	0.0663448i	$0.978866\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1462.43i	− 1.55412i	−0.629425	−	0.777061i	$-0.716709\pi$
$941$	− 1462.43i	− 1.55412i	0.629425	−	0.777061i	$-0.283291\pi$
$942$	0	0
$943$	2004.43i	2.12559i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1866.17	−1.97061	−0.985304	−	0.170809i	$-0.945362\pi$
$947$	−1866.17	−1.97061	−0.985304	+	0.170809i	$0.945362\pi$
$948$	0	0
$949$	−353.057	−0.372031
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1125.12i	1.18061i	0.807181	+	0.590304i	$0.200992\pi$
$953$	1125.12i	1.18061i	−0.807181	+	0.590304i	$0.799008\pi$
$954$	0	0
$955$	1607.10	1.68282
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1999.21i	2.08468i
$960$	0	0
$961$	−352.568	−0.366876
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1200.70i	1.24425i
$966$	0	0
$967$	577.349i	0.597052i	0.954402	+	0.298526i	$0.0964950\pi$
$967$	577.349i	0.597052i	−0.954402	+	0.298526i	$0.903505\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	629.291	0.648085	0.324043	−	0.946042i	$-0.394958\pi$
$971$	629.291	0.648085	0.324043	+	0.946042i	$0.394958\pi$
$972$	0	0
$973$	−1510.87	−1.55279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	567.655	0.581018	0.290509	−	0.956872i	$-0.406175\pi$
$977$	567.655	0.581018	0.290509	+	0.956872i	$0.406175\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1108.33	1.12749	0.563747	−	0.825947i	$-0.309359\pi$
$983$	1108.33	1.12749	0.563747	+	0.825947i	$0.309359\pi$
$984$	0	0
$985$	439.174i	0.445862i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1442.09i	1.45812i
$990$	0	0
$991$	−534.101	−0.538952	−0.269476	−	0.963007i	$-0.586850\pi$
$991$	−534.101	−0.538952	−0.269476	+	0.963007i	$0.586850\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1529.59	1.53728
$996$	0	0
$997$	1577.87i	1.58262i	0.611416	+	0.791309i	$0.290600\pi$
$997$	1577.87i	1.58262i	−0.611416	+	0.791309i	$0.709400\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.3.f.h.1693.6		8
3.2	odd	2		1452.3.f.c.241.2	yes	8
11.10	odd	2	inner	4356.3.f.h.1693.5		8
33.32	even	2		1452.3.f.c.241.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.3.f.c.241.1	✓	8	33.32	even	2
1452.3.f.c.241.2	yes	8	3.2	odd	2
4356.3.f.h.1693.5		8	11.10	odd	2	inner
4356.3.f.h.1693.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4356.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.29611 q^{5} +11.2666i q^{7} -8.53702i q^{13} +18.8227i q^{17} -19.4817i q^{19} -34.9216 q^{23} -6.54344 q^{25} -9.50332i q^{29} +24.6664 q^{31} +48.4024i q^{35} -27.0065 q^{37} -57.3980i q^{41} -41.2949i q^{43} -53.4744 q^{47} -77.9357 q^{49} -25.0337 q^{53} +101.668 q^{59} +114.718i q^{61} -36.6760i q^{65} -73.4602 q^{67} -105.706 q^{71} -41.3560i q^{73} -76.6876i q^{79} -129.542i q^{83} +80.8645i q^{85} -154.763 q^{89} +96.1830 q^{91} -83.6956i q^{95} +106.580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 20 q^{23} + 68 q^{25} + 96 q^{31} - 24 q^{37} - 248 q^{47} - 24 q^{49} - 76 q^{53} - 28 q^{59} + 44 q^{67} + 372 q^{71} - 152 q^{89} + 320 q^{91} + 244 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 20 * q^23 + 68 * q^25 + 96 * q^31 - 24 * q^37 - 248 * q^47 - 24 * q^49 - 76 * q^53 - 28 * q^59 + 44 * q^67 + 372 * q^71 - 152 * q^89 + 320 * q^91 + 244 * q^97

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