LMFDB - Embedded newform 1476.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1476,2,Mod(1,1476)] 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("1476.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1476, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1476 = 2^{2} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1476.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.7859193383$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25808.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 6x + 9 $ x^4 - 10x^2 - 6x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.55466$ of defining polynomial
Character	$\chi$	$=$	1476.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.33307 q^{5} +2.77840 q^{7} -2.02837 q^{11} -3.10932 q^{13} +7.91609 q^{17} +2.17964 q^{19} +4.75004 q^{23} +6.10932 q^{25} -3.85936 q^{29} -10.6661 q^{31} +9.26060 q^{35} +6.58303 q^{37} +1.00000 q^{41} +4.80677 q^{43} -4.03900 q^{47} +0.719526 q^{49} +1.94327 q^{53} -6.76068 q^{55} -4.75004 q^{59} +2.89068 q^{61} -10.3636 q^{65} +5.97163 q^{67} -14.4026 q^{71} -9.24916 q^{73} -5.63562 q^{77} -0.320282 q^{79} +3.69745 q^{83} +26.3849 q^{85} -8.96868 q^{89} -8.63895 q^{91} +7.26489 q^{95} +17.8322 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{11} + 4 q^{17} + 6 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{29} - 8 q^{31} + 26 q^{35} + 16 q^{37} + 4 q^{41} + 4 q^{43} + 6 q^{47} + 16 q^{49} + 16 q^{53} - 2 q^{55} - 12 q^{59} + 24 q^{61}+ \cdots + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^11 + 4 * q^17 + 6 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^29 - 8 * q^31 + 26 * q^35 + 16 * q^37 + 4 * q^41 + 4 * q^43 + 6 * q^47 + 16 * q^49 + 16 * q^53 - 2 * q^55 - 12 * q^59 + 24 * q^61 - 4 * q^65 + 28 * q^67 + 2 * q^71 + 8 * q^73 - 8 * q^77 - 18 * q^79 + 12 * q^83 + 32 * q^85 - 4 * q^89 + 36 * q^91 - 14 * q^95 + 16 * q^97

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
$5$	3.33307	1.49059	0.745296	−	0.666734i	$-0.232308\pi$
$5$	3.33307	1.49059	0.745296	+	0.666734i	$0.232308\pi$
$6$	0	0
$7$	2.77840	1.05014	0.525069	−	0.851060i	$-0.324040\pi$
$7$	2.77840	1.05014	0.525069	+	0.851060i	$0.324040\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.02837	−0.611575	−0.305788	−	0.952100i	$-0.598920\pi$
$11$	−2.02837	−0.611575	−0.305788	+	0.952100i	$0.598920\pi$
$12$	0	0
$13$	−3.10932	−0.862371	−0.431186	−	0.902263i	$-0.641905\pi$
$13$	−3.10932	−0.862371	−0.431186	+	0.902263i	$0.641905\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.91609	1.91993	0.959967	−	0.280112i	$-0.0903717\pi$
$17$	7.91609	1.91993	0.959967	+	0.280112i	$0.0903717\pi$
$18$	0	0
$19$	2.17964	0.500044	0.250022	−	0.968240i	$-0.419562\pi$
$19$	2.17964	0.500044	0.250022	+	0.968240i	$0.419562\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.75004	0.990451	0.495226	−	0.868764i	$-0.335085\pi$
$23$	4.75004	0.990451	0.495226	+	0.868764i	$0.335085\pi$
$24$	0	0
$25$	6.10932	1.22186
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.85936	−0.716665	−0.358333	−	0.933594i	$-0.616655\pi$
$29$	−3.85936	−0.716665	−0.358333	+	0.933594i	$0.616655\pi$
$30$	0	0
$31$	−10.6661	−1.91569	−0.957847	−	0.287280i	$-0.907249\pi$
$31$	−10.6661	−1.91569	−0.957847	+	0.287280i	$0.907249\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.26060	1.56533
$36$	0	0
$37$	6.58303	1.08224	0.541122	−	0.840944i	$-0.318000\pi$
$37$	6.58303	1.08224	0.541122	+	0.840944i	$0.318000\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.80677	0.733025	0.366513	−	0.930413i	$-0.380552\pi$
$43$	4.80677	0.733025	0.366513	+	0.930413i	$0.380552\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.03900	−0.589149	−0.294575	−	0.955628i	$-0.595178\pi$
$47$	−4.03900	−0.589149	−0.294575	+	0.955628i	$0.595178\pi$
$48$	0	0
$49$	0.719526	0.102789
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.94327	0.266928	0.133464	−	0.991054i	$-0.457390\pi$
$53$	1.94327	0.266928	0.133464	+	0.991054i	$0.457390\pi$
$54$	0	0
$55$	−6.76068	−0.911609
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.75004	−0.618402	−0.309201	−	0.950997i	$-0.600062\pi$
$59$	−4.75004	−0.618402	−0.309201	+	0.950997i	$0.600062\pi$
$60$	0	0
$61$	2.89068	0.370113	0.185057	−	0.982728i	$-0.440753\pi$
$61$	2.89068	0.370113	0.185057	+	0.982728i	$0.440753\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.3636	−1.28544
$66$	0	0
$67$	5.97163	0.729551	0.364776	−	0.931095i	$-0.381146\pi$
$67$	5.97163	0.729551	0.364776	+	0.931095i	$0.381146\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.4026	−1.70927	−0.854636	−	0.519228i	$-0.826220\pi$
$71$	−14.4026	−1.70927	−0.854636	+	0.519228i	$0.826220\pi$
$72$	0	0
$73$	−9.24916	−1.08253	−0.541266	−	0.840851i	$-0.682055\pi$
$73$	−9.24916	−1.08253	−0.541266	+	0.840851i	$0.682055\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.63562	−0.642238
$78$	0	0
$79$	−0.320282	−0.0360345	−0.0180173	−	0.999838i	$-0.505735\pi$
$79$	−0.320282	−0.0360345	−0.0180173	+	0.999838i	$0.505735\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.69745	0.405847	0.202924	−	0.979195i	$-0.434956\pi$
$83$	3.69745	0.405847	0.202924	+	0.979195i	$0.434956\pi$
$84$	0	0
$85$	26.3849	2.86184
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.96868	−0.950679	−0.475339	−	0.879803i	$-0.657675\pi$
$89$	−8.96868	−0.950679	−0.475339	+	0.879803i	$0.657675\pi$
$90$	0	0
$91$	−8.63895	−0.905608
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.26489	0.745362
$96$	0	0
$97$	17.8322	1.81058	0.905292	−	0.424790i	$-0.139652\pi$
$97$	17.8322	1.81058	0.905292	+	0.424790i	$0.139652\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.0254	1.29608	0.648039	−	0.761607i	$-0.275590\pi$
$101$	13.0254	1.29608	0.648039	+	0.761607i	$0.275590\pi$
$102$	0	0
$103$	−8.05673	−0.793853	−0.396927	−	0.917850i	$-0.629923\pi$
$103$	−8.05673	−0.793853	−0.396927	+	0.917850i	$0.629923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.41602	0.426912	0.213456	−	0.976953i	$-0.431528\pi$
$107$	4.41602	0.426912	0.213456	+	0.976953i	$0.431528\pi$
$108$	0	0
$109$	18.5255	1.77442	0.887210	−	0.461366i	$-0.152640\pi$
$109$	18.5255	1.77442	0.887210	+	0.461366i	$0.152640\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.08310	0.760394	0.380197	−	0.924905i	$-0.375856\pi$
$113$	8.08310	0.760394	0.380197	+	0.924905i	$0.375856\pi$
$114$	0	0
$115$	15.8322	1.47636
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.9941	2.01620
$120$	0	0
$121$	−6.88573	−0.625976
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.69745	0.330710
$126$	0	0
$127$	4.44748	0.394650	0.197325	−	0.980338i	$-0.436775\pi$
$127$	4.44748	0.394650	0.197325	+	0.980338i	$0.436775\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.85936	−0.861416	−0.430708	−	0.902491i	$-0.641736\pi$
$131$	−9.85936	−0.861416	−0.430708	+	0.902491i	$0.641736\pi$
$132$	0	0
$133$	6.05593	0.525115
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.30255	0.196720	0.0983602	−	0.995151i	$-0.468640\pi$
$137$	2.30255	0.196720	0.0983602	+	0.995151i	$0.468640\pi$
$138$	0	0
$139$	5.16605	0.438179	0.219090	−	0.975705i	$-0.429691\pi$
$139$	5.16605	0.438179	0.219090	+	0.975705i	$0.429691\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.30684	0.527405
$144$	0	0
$145$	−12.8635	−1.06826
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.85936	−0.316171	−0.158086	−	0.987425i	$-0.550532\pi$
$149$	−3.85936	−0.316171	−0.158086	+	0.987425i	$0.550532\pi$
$150$	0	0
$151$	−4.71103	−0.383379	−0.191689	−	0.981456i	$-0.561397\pi$
$151$	−4.71103	−0.383379	−0.191689	+	0.981456i	$0.561397\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−35.5509	−2.85552
$156$	0	0
$157$	8.30684	0.662958	0.331479	−	0.943463i	$-0.392452\pi$
$157$	8.30684	0.662958	0.331479	+	0.943463i	$0.392452\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.1975	1.04011
$162$	0	0
$163$	−13.6348	−1.06796	−0.533981	−	0.845497i	$-0.679304\pi$
$163$	−13.6348	−1.06796	−0.533981	+	0.845497i	$0.679304\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.97163	−0.771628	−0.385814	−	0.922577i	$-0.626079\pi$
$167$	−9.97163	−0.771628	−0.385814	+	0.922577i	$0.626079\pi$
$168$	0	0
$169$	−3.33211	−0.256316
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.11361	−0.692895	−0.346448	−	0.938069i	$-0.612612\pi$
$173$	−9.11361	−0.692895	−0.346448	+	0.938069i	$0.612612\pi$
$174$	0	0
$175$	16.9742	1.28313
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.9403	0.967205	0.483602	−	0.875288i	$-0.339328\pi$
$179$	12.9403	0.967205	0.483602	+	0.875288i	$0.339328\pi$
$180$	0	0
$181$	8.16191	0.606670	0.303335	−	0.952884i	$-0.401900\pi$
$181$	8.16191	0.606670	0.303335	+	0.952884i	$0.401900\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.9417	1.61318
$186$	0	0
$187$	−16.0567	−1.17418
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.52829	−0.327656	−0.163828	−	0.986489i	$-0.552384\pi$
$191$	−4.52829	−0.327656	−0.163828	+	0.986489i	$0.552384\pi$
$192$	0	0
$193$	−14.1662	−1.01971	−0.509853	−	0.860262i	$-0.670300\pi$
$193$	−14.1662	−1.01971	−0.509853	+	0.860262i	$0.670300\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.5042	0.890888	0.445444	−	0.895310i	$-0.353046\pi$
$197$	12.5042	0.890888	0.445444	+	0.895310i	$0.353046\pi$
$198$	0	0
$199$	−14.5853	−1.03393	−0.516963	−	0.856008i	$-0.672938\pi$
$199$	−14.5853	−1.03393	−0.516963	+	0.856008i	$0.672938\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.7229	−0.752597
$204$	0	0
$205$	3.33307	0.232791
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.42111	−0.305815
$210$	0	0
$211$	−26.3860	−1.81649	−0.908245	−	0.418439i	$-0.862577\pi$
$211$	−26.3860	−1.81649	−0.908245	+	0.418439i	$0.862577\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0213	1.09264
$216$	0	0
$217$	−29.6348	−2.01174
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.6137	−1.65570
$222$	0	0
$223$	−1.89482	−0.126886	−0.0634432	−	0.997985i	$-0.520208\pi$
$223$	−1.89482	−0.126886	−0.0634432	+	0.997985i	$0.520208\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.9805	1.65801	0.829007	−	0.559238i	$-0.188906\pi$
$227$	24.9805	1.65801	0.829007	+	0.559238i	$0.188906\pi$
$228$	0	0
$229$	−2.52549	−0.166889	−0.0834446	−	0.996512i	$-0.526592\pi$
$229$	−2.52549	−0.166889	−0.0834446	+	0.996512i	$0.526592\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.24835	−0.605880	−0.302940	−	0.953010i	$-0.597968\pi$
$233$	−9.24835	−0.605880	−0.302940	+	0.953010i	$0.597968\pi$
$234$	0	0
$235$	−13.4623	−0.878181
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.34746	−0.151844	−0.0759222	−	0.997114i	$-0.524190\pi$
$239$	−2.34746	−0.151844	−0.0759222	+	0.997114i	$0.524190\pi$
$240$	0	0
$241$	2.89068	0.186205	0.0931024	−	0.995657i	$-0.470322\pi$
$241$	2.89068	0.186205	0.0931024	+	0.995657i	$0.470322\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.39823	0.153217
$246$	0	0
$247$	−6.77721	−0.431224
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9730	−1.57628	−0.788140	−	0.615496i	$-0.788956\pi$
$251$	−24.9730	−1.57628	−0.788140	+	0.615496i	$0.788956\pi$
$252$	0	0
$253$	−9.63481	−0.605736
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.7500	−0.670569	−0.335284	−	0.942117i	$-0.608832\pi$
$257$	−10.7500	−0.670569	−0.335284	+	0.942117i	$0.608832\pi$
$258$	0	0
$259$	18.2903	1.13650
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.90250	0.302301	0.151151	−	0.988511i	$-0.451702\pi$
$263$	4.90250	0.302301	0.151151	+	0.988511i	$0.451702\pi$
$264$	0	0
$265$	6.47704	0.397881
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.32797	−0.202910	−0.101455	−	0.994840i	$-0.532350\pi$
$269$	−3.32797	−0.202910	−0.101455	+	0.994840i	$0.532350\pi$
$270$	0	0
$271$	4.58222	0.278350	0.139175	−	0.990268i	$-0.455555\pi$
$271$	4.58222	0.278350	0.139175	+	0.990268i	$0.455555\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12.3919	−0.747262
$276$	0	0
$277$	−24.2178	−1.45511	−0.727555	−	0.686050i	$-0.759343\pi$
$277$	−24.2178	−1.45511	−0.727555	+	0.686050i	$0.759343\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.61369	−0.394540	−0.197270	−	0.980349i	$-0.563208\pi$
$281$	−6.61369	−0.394540	−0.197270	+	0.980349i	$0.563208\pi$
$282$	0	0
$283$	5.16605	0.307090	0.153545	−	0.988142i	$-0.450931\pi$
$283$	5.16605	0.307090	0.153545	+	0.988142i	$0.450931\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.77840	0.164004
$288$	0	0
$289$	45.6645	2.68615
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.2458	0.598567	0.299284	−	0.954164i	$-0.403252\pi$
$293$	10.2458	0.598567	0.299284	+	0.954164i	$0.403252\pi$
$294$	0	0
$295$	−15.8322	−0.921785
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.7694	−0.854137
$300$	0	0
$301$	13.3551	0.769778
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.63481	0.551688
$306$	0	0
$307$	−18.2754	−1.04303	−0.521515	−	0.853242i	$-0.674633\pi$
$307$	−18.2754	−1.04303	−0.521515	+	0.853242i	$0.674633\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.50303	−0.255343	−0.127672	−	0.991816i	$-0.540750\pi$
$311$	−4.50303	−0.255343	−0.127672	+	0.991816i	$0.540750\pi$
$312$	0	0
$313$	6.56095	0.370847	0.185423	−	0.982659i	$-0.440634\pi$
$313$	6.56095	0.370847	0.185423	+	0.982659i	$0.440634\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.8363	−0.720960	−0.360480	−	0.932767i	$-0.617387\pi$
$317$	−12.8363	−0.720960	−0.360480	+	0.932767i	$0.617387\pi$
$318$	0	0
$319$	7.82820	0.438295
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.2543	0.960052
$324$	0	0
$325$	−18.9959	−1.05370
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.2220	−0.618688
$330$	0	0
$331$	−8.99705	−0.494523	−0.247261	−	0.968949i	$-0.579531\pi$
$331$	−8.99705	−0.494523	−0.247261	+	0.968949i	$0.579531\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.9038	1.08746
$336$	0	0
$337$	−35.0502	−1.90930	−0.954652	−	0.297723i	$-0.903773\pi$
$337$	−35.0502	−1.90930	−0.954652	+	0.297723i	$0.903773\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.6348	1.17159
$342$	0	0
$343$	−17.4497	−0.942195
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.8982	−1.01451	−0.507255	−	0.861796i	$-0.669340\pi$
$347$	−18.8982	−1.01451	−0.507255	+	0.861796i	$0.669340\pi$
$348$	0	0
$349$	−23.1653	−1.24001	−0.620004	−	0.784599i	$-0.712869\pi$
$349$	−23.1653	−1.24001	−0.620004	+	0.784599i	$0.712869\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.55171	0.0825893	0.0412946	−	0.999147i	$-0.486852\pi$
$353$	1.55171	0.0825893	0.0412946	+	0.999147i	$0.486852\pi$
$354$	0	0
$355$	−48.0047	−2.54783
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.28128	0.225957	0.112979	−	0.993597i	$-0.463961\pi$
$359$	4.28128	0.225957	0.112979	+	0.993597i	$0.463961\pi$
$360$	0	0
$361$	−14.2492	−0.749956
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.8280	−1.61361
$366$	0	0
$367$	−15.3067	−0.799003	−0.399501	−	0.916733i	$-0.630817\pi$
$367$	−15.3067	−0.799003	−0.399501	+	0.916733i	$0.630817\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.39918	0.280312
$372$	0	0
$373$	−5.39489	−0.279337	−0.139668	−	0.990198i	$-0.544604\pi$
$373$	−5.39489	−0.279337	−0.139668	+	0.990198i	$0.544604\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	26.8009	1.37667	0.688334	−	0.725394i	$-0.258342\pi$
$379$	26.8009	1.37667	0.688334	+	0.725394i	$0.258342\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.6675	−1.10716	−0.553578	−	0.832797i	$-0.686738\pi$
$383$	−21.6675	−1.10716	−0.553578	+	0.832797i	$0.686738\pi$
$384$	0	0
$385$	−18.7839	−0.957315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.7187	−1.30399	−0.651995	−	0.758223i	$-0.726068\pi$
$389$	−25.7187	−1.30399	−0.651995	+	0.758223i	$0.726068\pi$
$390$	0	0
$391$	37.6017	1.90160
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.06752	−0.0537128
$396$	0	0
$397$	28.1603	1.41333	0.706663	−	0.707551i	$-0.250200\pi$
$397$	28.1603	1.41333	0.706663	+	0.707551i	$0.250200\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.7178	0.535220	0.267610	−	0.963527i	$-0.413766\pi$
$401$	10.7178	0.535220	0.267610	+	0.963527i	$0.413766\pi$
$402$	0	0
$403$	33.1644	1.65204
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.3528	−0.661873
$408$	0	0
$409$	22.7492	1.12488	0.562439	−	0.826839i	$-0.309863\pi$
$409$	22.7492	1.12488	0.562439	+	0.826839i	$0.309863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.1975	−0.649408
$414$	0	0
$415$	12.3238	0.604953
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.02542	0.343214	0.171607	−	0.985165i	$-0.445104\pi$
$419$	7.02542	0.343214	0.171607	+	0.985165i	$0.445104\pi$
$420$	0	0
$421$	16.5992	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$421$	16.5992	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	48.3620	2.34590
$426$	0	0
$427$	8.03147	0.388670
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.7229	0.516502	0.258251	−	0.966078i	$-0.416854\pi$
$431$	10.7229	0.516502	0.258251	+	0.966078i	$0.416854\pi$
$432$	0	0
$433$	31.1644	1.49767	0.748834	−	0.662758i	$-0.230614\pi$
$433$	31.1644	1.49767	0.748834	+	0.662758i	$0.230614\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.3534	0.495270
$438$	0	0
$439$	−34.7981	−1.66082	−0.830411	−	0.557152i	$-0.811894\pi$
$439$	−34.7981	−1.66082	−0.830411	+	0.557152i	$0.811894\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.8806	1.22963	0.614813	−	0.788673i	$-0.289231\pi$
$443$	25.8806	1.22963	0.614813	+	0.788673i	$0.289231\pi$
$444$	0	0
$445$	−29.8932	−1.41707
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.2231	−1.28474	−0.642368	−	0.766396i	$-0.722048\pi$
$449$	−27.2231	−1.28474	−0.642368	+	0.766396i	$0.722048\pi$
$450$	0	0
$451$	−2.02837	−0.0955120
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−28.7942	−1.34989
$456$	0	0
$457$	−6.19767	−0.289915	−0.144957	−	0.989438i	$-0.546305\pi$
$457$	−6.19767	−0.289915	−0.144957	+	0.989438i	$0.546305\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.9170	0.508458	0.254229	−	0.967144i	$-0.418178\pi$
$461$	10.9170	0.508458	0.254229	+	0.967144i	$0.418178\pi$
$462$	0	0
$463$	4.07047	0.189171	0.0945854	−	0.995517i	$-0.469847\pi$
$463$	4.07047	0.189171	0.0945854	+	0.995517i	$0.469847\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.63657	0.399653	0.199826	−	0.979831i	$-0.435962\pi$
$467$	8.63657	0.399653	0.199826	+	0.979831i	$0.435962\pi$
$468$	0	0
$469$	16.5916	0.766129
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.74989	−0.448300
$474$	0	0
$475$	13.3161	0.610986
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.5537	1.07620	0.538098	−	0.842882i	$-0.319143\pi$
$479$	23.5537	1.07620	0.538098	+	0.842882i	$0.319143\pi$
$480$	0	0
$481$	−20.4688	−0.933295
$482$	0	0
$483$	0	0
$484$	0	0
$485$	59.4358	2.69884
$486$	0	0
$487$	19.9102	0.902217	0.451108	−	0.892469i	$-0.351029\pi$
$487$	19.9102	0.902217	0.451108	+	0.892469i	$0.351029\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.7585	−1.34298	−0.671490	−	0.741013i	$-0.734346\pi$
$491$	−29.7585	−1.34298	−0.671490	+	0.741013i	$0.734346\pi$
$492$	0	0
$493$	−30.5511	−1.37595
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−40.0162	−1.79497
$498$	0	0
$499$	18.9297	0.847409	0.423704	−	0.905801i	$-0.360730\pi$
$499$	18.9297	0.847409	0.423704	+	0.905801i	$0.360730\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.93382	0.0862248	0.0431124	−	0.999070i	$-0.486273\pi$
$503$	1.93382	0.0862248	0.0431124	+	0.999070i	$0.486273\pi$
$504$	0	0
$505$	43.4146	1.93192
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.6602	−1.53629	−0.768144	−	0.640277i	$-0.778819\pi$
$509$	−34.6602	−1.53629	−0.768144	+	0.640277i	$0.778819\pi$
$510$	0	0
$511$	−25.6979	−1.13681
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−26.8536	−1.18331
$516$	0	0
$517$	8.19258	0.360309
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.33402	0.277498	0.138749	−	0.990328i	$-0.455692\pi$
$521$	6.33402	0.277498	0.138749	+	0.990328i	$0.455692\pi$
$522$	0	0
$523$	−29.9460	−1.30944	−0.654722	−	0.755869i	$-0.727215\pi$
$523$	−29.9460	−1.30944	−0.654722	+	0.755869i	$0.727215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−84.4341	−3.67801
$528$	0	0
$529$	−0.437142	−0.0190062
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.10932	−0.134680
$534$	0	0
$535$	14.7189	0.636352
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.45946	−0.0628635
$540$	0	0
$541$	0.834750	0.0358887	0.0179443	−	0.999839i	$-0.494288\pi$
$541$	0.834750	0.0358887	0.0179443	+	0.999839i	$0.494288\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	61.7467	2.64494
$546$	0	0
$547$	−20.6527	−0.883045	−0.441523	−	0.897250i	$-0.645562\pi$
$547$	−20.6527	−0.883045	−0.441523	+	0.897250i	$0.645562\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.41203	−0.358364
$552$	0	0
$553$	−0.889872	−0.0378412
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.72447	0.369668	0.184834	−	0.982770i	$-0.440825\pi$
$557$	8.72447	0.369668	0.184834	+	0.982770i	$0.440825\pi$
$558$	0	0
$559$	−14.9458	−0.632140
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.80144	−0.413081	−0.206541	−	0.978438i	$-0.566221\pi$
$563$	−9.80144	−0.413081	−0.206541	+	0.978438i	$0.566221\pi$
$564$	0	0
$565$	26.9415	1.13344
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.1570	−1.05464	−0.527318	−	0.849668i	$-0.676802\pi$
$569$	−25.1570	−1.05464	−0.527318	+	0.849668i	$0.676802\pi$
$570$	0	0
$571$	27.1928	1.13798	0.568992	−	0.822343i	$-0.307334\pi$
$571$	27.1928	1.13798	0.568992	+	0.822343i	$0.307334\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	29.0195	1.21020
$576$	0	0
$577$	−0.365186	−0.0152029	−0.00760144	−	0.999971i	$-0.502420\pi$
$577$	−0.365186	−0.0152029	−0.00760144	+	0.999971i	$0.502420\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.2730	0.426196
$582$	0	0
$583$	−3.94166	−0.163247
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.7242	1.55704	0.778522	−	0.627617i	$-0.215970\pi$
$587$	37.7242	1.55704	0.778522	+	0.627617i	$0.215970\pi$
$588$	0	0
$589$	−23.2484	−0.957932
$590$	0	0
$591$	0	0
$592$	0	0
$593$	45.1333	1.85340	0.926701	−	0.375800i	$-0.122632\pi$
$593$	45.1333	1.85340	0.926701	+	0.375800i	$0.122632\pi$
$594$	0	0
$595$	73.3078	3.00533
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.4982	−0.633238	−0.316619	−	0.948553i	$-0.602548\pi$
$599$	−15.4982	−0.633238	−0.316619	+	0.948553i	$0.602548\pi$
$600$	0	0
$601$	−5.18924	−0.211674	−0.105837	−	0.994384i	$-0.533752\pi$
$601$	−5.18924	−0.211674	−0.105837	+	0.994384i	$0.533752\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.9506	−0.933074
$606$	0	0
$607$	−16.8280	−0.683029	−0.341515	−	0.939876i	$-0.610940\pi$
$607$	−16.8280	−0.683029	−0.341515	+	0.939876i	$0.610940\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.5586	0.508065
$612$	0	0
$613$	25.1042	1.01395	0.506975	−	0.861961i	$-0.330764\pi$
$613$	25.1042	1.01395	0.506975	+	0.861961i	$0.330764\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.3324	−0.577001	−0.288501	−	0.957480i	$-0.593157\pi$
$617$	−14.3324	−0.577001	−0.288501	+	0.957480i	$0.593157\pi$
$618$	0	0
$619$	19.2712	0.774576	0.387288	−	0.921959i	$-0.373412\pi$
$619$	19.2712	0.774576	0.387288	+	0.921959i	$0.373412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.9186	−0.998344
$624$	0	0
$625$	−18.2228	−0.728911
$626$	0	0
$627$	0	0
$628$	0	0
$629$	52.1119	2.07784
$630$	0	0
$631$	−30.1856	−1.20167	−0.600834	−	0.799374i	$-0.705165\pi$
$631$	−30.1856	−1.20167	−0.600834	+	0.799374i	$0.705165\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.8238	0.588263
$636$	0	0
$637$	−2.23724	−0.0886427
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.9984	1.02687	0.513437	−	0.858127i	$-0.328372\pi$
$641$	25.9984	1.02687	0.513437	+	0.858127i	$0.328372\pi$
$642$	0	0
$643$	11.4402	0.451159	0.225580	−	0.974225i	$-0.427572\pi$
$643$	11.4402	0.451159	0.225580	+	0.974225i	$0.427572\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.6915	−1.48181	−0.740904	−	0.671611i	$-0.765603\pi$
$647$	−37.6915	−1.48181	−0.740904	+	0.671611i	$0.765603\pi$
$648$	0	0
$649$	9.63481	0.378200
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.6814	1.70938	0.854692	−	0.519136i	$-0.173746\pi$
$653$	43.6814	1.70938	0.854692	+	0.519136i	$0.173746\pi$
$654$	0	0
$655$	−32.8619	−1.28402
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.0900	−0.977367	−0.488684	−	0.872461i	$-0.662523\pi$
$659$	−25.0900	−0.977367	−0.488684	+	0.872461i	$0.662523\pi$
$660$	0	0
$661$	−37.7392	−1.46788	−0.733942	−	0.679212i	$-0.762322\pi$
$661$	−37.7392	−1.46788	−0.733942	+	0.679212i	$0.762322\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.1848	0.782733
$666$	0	0
$667$	−18.3321	−0.709822
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.86335	−0.226352
$672$	0	0
$673$	3.63642	0.140174	0.0700869	−	0.997541i	$-0.477672\pi$
$673$	3.63642	0.140174	0.0700869	+	0.997541i	$0.477672\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.1948	−1.12205	−0.561024	−	0.827800i	$-0.689592\pi$
$677$	−29.1948	−1.12205	−0.561024	+	0.827800i	$0.689592\pi$
$678$	0	0
$679$	49.5450	1.90136
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.6527	1.09636	0.548182	−	0.836359i	$-0.315320\pi$
$683$	28.6527	1.09636	0.548182	+	0.836359i	$0.315320\pi$
$684$	0	0
$685$	7.67456	0.293230
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.04225	−0.230191
$690$	0	0
$691$	17.6880	0.672883	0.336442	−	0.941704i	$-0.390777\pi$
$691$	17.6880	0.672883	0.336442	+	0.941704i	$0.390777\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.2188	0.653146
$696$	0	0
$697$	7.91609	0.299843
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.79339	−0.105505	−0.0527525	−	0.998608i	$-0.516799\pi$
$701$	−2.79339	−0.105505	−0.0527525	+	0.998608i	$0.516799\pi$
$702$	0	0
$703$	14.3486	0.541169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.1899	1.36106
$708$	0	0
$709$	20.4117	0.766578	0.383289	−	0.923628i	$-0.374791\pi$
$709$	20.4117	0.766578	0.383289	+	0.923628i	$0.374791\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−50.6645	−1.89740
$714$	0	0
$715$	21.0211	0.786145
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.9848	−1.30471	−0.652356	−	0.757912i	$-0.726219\pi$
$719$	−34.9848	−1.30471	−0.652356	+	0.757912i	$0.726219\pi$
$720$	0	0
$721$	−22.3849	−0.833655
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−23.5781	−0.875668
$726$	0	0
$727$	−24.8796	−0.922734	−0.461367	−	0.887209i	$-0.652641\pi$
$727$	−24.8796	−0.922734	−0.461367	+	0.887209i	$0.652641\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.0508	1.40736
$732$	0	0
$733$	−28.7737	−1.06278	−0.531390	−	0.847127i	$-0.678330\pi$
$733$	−28.7737	−1.06278	−0.531390	+	0.847127i	$0.678330\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.1127	−0.446176
$738$	0	0
$739$	−30.1856	−1.11039	−0.555197	−	0.831719i	$-0.687357\pi$
$739$	−30.1856	−1.11039	−0.555197	+	0.831719i	$0.687357\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.7127	0.833247	0.416624	−	0.909079i	$-0.363213\pi$
$743$	22.7127	0.833247	0.416624	+	0.909079i	$0.363213\pi$
$744$	0	0
$745$	−12.8635	−0.471282
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.2695	0.448317
$750$	0	0
$751$	−0.0200854	−0.000732927 0	−0.000366463	−	1.00000i	$-0.500117\pi$
$751$	−0.0200854	−0.000732927 0	−0.000366463		1.00000i	$0.500117\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.7022	−0.571461
$756$	0	0
$757$	44.9002	1.63192	0.815962	−	0.578106i	$-0.196208\pi$
$757$	44.9002	1.63192	0.815962	+	0.578106i	$0.196208\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.0203	0.689486	0.344743	−	0.938697i	$-0.387966\pi$
$761$	19.0203	0.689486	0.344743	+	0.938697i	$0.387966\pi$
$762$	0	0
$763$	51.4713	1.86339
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.7694	0.533292
$768$	0	0
$769$	42.1220	1.51896	0.759480	−	0.650531i	$-0.225454\pi$
$769$	42.1220	1.51896	0.759480	+	0.650531i	$0.225454\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.7753	−0.387561	−0.193780	−	0.981045i	$-0.562075\pi$
$773$	−10.7753	−0.387561	−0.193780	+	0.981045i	$0.562075\pi$
$774$	0	0
$775$	−65.1628	−2.34072
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.17964	0.0780938
$780$	0	0
$781$	29.2137	1.04535
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.6873	0.988201
$786$	0	0
$787$	22.8592	0.814843	0.407421	−	0.913240i	$-0.366428\pi$
$787$	22.8592	0.814843	0.407421	+	0.913240i	$0.366428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.4581	0.798519
$792$	0	0
$793$	−8.98805	−0.319175
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.00444	0.177266	0.0886332	−	0.996064i	$-0.471750\pi$
$797$	5.00444	0.177266	0.0886332	+	0.996064i	$0.471750\pi$
$798$	0	0
$799$	−31.9731	−1.13113
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.7607	0.662050
$804$	0	0
$805$	43.9882	1.55038
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.52135	−0.264437	−0.132218	−	0.991221i	$-0.542210\pi$
$809$	−7.52135	−0.264437	−0.132218	+	0.991221i	$0.542210\pi$
$810$	0	0
$811$	29.4103	1.03273	0.516367	−	0.856367i	$-0.327284\pi$
$811$	29.4103	1.03273	0.516367	+	0.856367i	$0.327284\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−45.4457	−1.59189
$816$	0	0
$817$	10.4770	0.366545
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.0205	−1.32692	−0.663462	−	0.748210i	$-0.730913\pi$
$821$	−38.0205	−1.32692	−0.663462	+	0.748210i	$0.730913\pi$
$822$	0	0
$823$	25.7849	0.898805	0.449403	−	0.893329i	$-0.351637\pi$
$823$	25.7849	0.898805	0.449403	+	0.893329i	$0.351637\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.00295	−0.243516	−0.121758	−	0.992560i	$-0.538853\pi$
$827$	−7.00295	−0.243516	−0.121758	+	0.992560i	$0.538853\pi$
$828$	0	0
$829$	0.166709	0.00579003	0.00289501	−	0.999996i	$-0.499078\pi$
$829$	0.166709	0.00579003	0.00289501	+	0.999996i	$0.499078\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.69584	0.197349
$834$	0	0
$835$	−33.2361	−1.15018
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.1105	0.452625	0.226313	−	0.974055i	$-0.427333\pi$
$839$	13.1105	0.452625	0.226313	+	0.974055i	$0.427333\pi$
$840$	0	0
$841$	−14.1053	−0.486391
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.1061	−0.382063
$846$	0	0
$847$	−19.1313	−0.657361
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.2696	1.07191
$852$	0	0
$853$	−0.445573	−0.0152561	−0.00762806	−	0.999971i	$-0.502428\pi$
$853$	−0.445573	−0.0152561	−0.00762806	+	0.999971i	$0.502428\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.5519	0.702038	0.351019	−	0.936368i	$-0.385835\pi$
$857$	20.5519	0.702038	0.351019	+	0.936368i	$0.385835\pi$
$858$	0	0
$859$	−18.8891	−0.644487	−0.322243	−	0.946657i	$-0.604437\pi$
$859$	−18.8891	−0.644487	−0.322243	+	0.946657i	$0.604437\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.9204	1.63123	0.815614	−	0.578596i	$-0.196399\pi$
$863$	47.9204	1.63123	0.815614	+	0.578596i	$0.196399\pi$
$864$	0	0
$865$	−30.3763	−1.03282
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.649649	0.0220378
$870$	0	0
$871$	−18.5677	−0.629144
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.2730	0.347291
$876$	0	0
$877$	−50.2738	−1.69762	−0.848812	−	0.528694i	$-0.822682\pi$
$877$	−50.2738	−1.69762	−0.848812	+	0.528694i	$0.822682\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.6019	1.30053	0.650265	−	0.759707i	$-0.274658\pi$
$881$	38.6019	1.30053	0.650265	+	0.759707i	$0.274658\pi$
$882$	0	0
$883$	3.07860	0.103603	0.0518016	−	0.998657i	$-0.483504\pi$
$883$	3.07860	0.103603	0.0518016	+	0.998657i	$0.483504\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.0770	1.04346	0.521731	−	0.853110i	$-0.325286\pi$
$887$	31.0770	1.04346	0.521731	+	0.853110i	$0.325286\pi$
$888$	0	0
$889$	12.3569	0.414437
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.80358	−0.294601
$894$	0	0
$895$	43.1309	1.44171
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.1644	1.37291
$900$	0	0
$901$	15.3831	0.512485
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.2042	0.904298
$906$	0	0
$907$	42.5534	1.41296	0.706482	−	0.707731i	$-0.250281\pi$
$907$	42.5534	1.41296	0.706482	+	0.707731i	$0.250281\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.8694	0.492645	0.246323	−	0.969188i	$-0.420778\pi$
$911$	14.8694	0.492645	0.246323	+	0.969188i	$0.420778\pi$
$912$	0	0
$913$	−7.49977	−0.248206
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.3933	−0.904606
$918$	0	0
$919$	26.5539	0.875931	0.437965	−	0.898992i	$-0.355699\pi$
$919$	26.5539	0.875931	0.437965	+	0.898992i	$0.355699\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	44.7823	1.47403
$924$	0	0
$925$	40.2178	1.32235
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.70749	−0.0888298	−0.0444149	−	0.999013i	$-0.514142\pi$
$929$	−2.70749	−0.0888298	−0.0444149	+	0.999013i	$0.514142\pi$
$930$	0	0
$931$	1.56831	0.0513993
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−53.5181	−1.75023
$936$	0	0
$937$	−36.0067	−1.17629	−0.588143	−	0.808757i	$-0.700141\pi$
$937$	−36.0067	−1.17629	−0.588143	+	0.808757i	$0.700141\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.6135	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$941$	−11.6135	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$942$	0	0
$943$	4.75004	0.154683
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.9061	0.679355	0.339678	−	0.940542i	$-0.389682\pi$
$947$	20.9061	0.679355	0.339678	+	0.940542i	$0.389682\pi$
$948$	0	0
$949$	28.7586	0.933544
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.7510	0.445438	0.222719	−	0.974883i	$-0.428507\pi$
$953$	13.7510	0.445438	0.222719	+	0.974883i	$0.428507\pi$
$954$	0	0
$955$	−15.0931	−0.488401
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.39742	0.206584
$960$	0	0
$961$	82.7663	2.66988
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−47.2169	−1.51997
$966$	0	0
$967$	7.07935	0.227656	0.113828	−	0.993500i	$-0.463689\pi$
$967$	7.07935	0.227656	0.113828	+	0.993500i	$0.463689\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.7382	0.601338	0.300669	−	0.953729i	$-0.402790\pi$
$971$	18.7382	0.601338	0.300669	+	0.953729i	$0.402790\pi$
$972$	0	0
$973$	14.3534	0.460148
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−60.6756	−1.94118	−0.970592	−	0.240729i	$-0.922613\pi$
$977$	−60.6756	−1.94118	−0.970592	+	0.240729i	$0.922613\pi$
$978$	0	0
$979$	18.1918	0.581412
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.3720	−1.51093	−0.755466	−	0.655188i	$-0.772590\pi$
$983$	−47.3720	−1.51093	−0.755466	+	0.655188i	$0.772590\pi$
$984$	0	0
$985$	41.6774	1.32795
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.8323	0.726026
$990$	0	0
$991$	−19.2110	−0.610256	−0.305128	−	0.952311i	$-0.598699\pi$
$991$	−19.2110	−0.610256	−0.305128	+	0.952311i	$0.598699\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−48.6138	−1.54116
$996$	0	0
$997$	−59.1077	−1.87196	−0.935980	−	0.352053i	$-0.885484\pi$
$997$	−59.1077	−1.87196	−0.935980	+	0.352053i	$0.885484\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1476.2.a.g.1.4		4
3.2	odd	2		164.2.a.a.1.4	✓	4
4.3	odd	2		5904.2.a.bp.1.4		4
12.11	even	2		656.2.a.i.1.1		4
15.2	even	4		4100.2.d.c.1149.2		8
15.8	even	4		4100.2.d.c.1149.7		8
15.14	odd	2		4100.2.a.c.1.1		4
21.20	even	2		8036.2.a.i.1.1		4
24.5	odd	2		2624.2.a.v.1.1		4
24.11	even	2		2624.2.a.y.1.4		4
123.122	odd	2		6724.2.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.4	✓	4	3.2	odd	2
656.2.a.i.1.1		4	12.11	even	2
1476.2.a.g.1.4		4	1.1	even	1	trivial
2624.2.a.v.1.1		4	24.5	odd	2
2624.2.a.y.1.4		4	24.11	even	2
4100.2.a.c.1.1		4	15.14	odd	2
4100.2.d.c.1149.2		8	15.2	even	4
4100.2.d.c.1149.7		8	15.8	even	4
5904.2.a.bp.1.4		4	4.3	odd	2
6724.2.a.c.1.1		4	123.122	odd	2
8036.2.a.i.1.1		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1476.a (trivial)

\(f(q)\)	\(=\)	\(q+3.33307 q^{5} +2.77840 q^{7} -2.02837 q^{11} -3.10932 q^{13} +7.91609 q^{17} +2.17964 q^{19} +4.75004 q^{23} +6.10932 q^{25} -3.85936 q^{29} -10.6661 q^{31} +9.26060 q^{35} +6.58303 q^{37} +1.00000 q^{41} +4.80677 q^{43} -4.03900 q^{47} +0.719526 q^{49} +1.94327 q^{53} -6.76068 q^{55} -4.75004 q^{59} +2.89068 q^{61} -10.3636 q^{65} +5.97163 q^{67} -14.4026 q^{71} -9.24916 q^{73} -5.63562 q^{77} -0.320282 q^{79} +3.69745 q^{83} +26.3849 q^{85} -8.96868 q^{89} -8.63895 q^{91} +7.26489 q^{95} +17.8322 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{11} + 4 q^{17} + 6 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{29} - 8 q^{31} + 26 q^{35} + 16 q^{37} + 4 q^{41} + 4 q^{43} + 6 q^{47} + 16 q^{49} + 16 q^{53} - 2 q^{55} - 12 q^{59} + 24 q^{61}+ \cdots + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^11 + 4 * q^17 + 6 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^29 - 8 * q^31 + 26 * q^35 + 16 * q^37 + 4 * q^41 + 4 * q^43 + 6 * q^47 + 16 * q^49 + 16 * q^53 - 2 * q^55 - 12 * q^59 + 24 * q^61 - 4 * q^65 + 28 * q^67 + 2 * q^71 + 8 * q^73 - 8 * q^77 - 18 * q^79 + 12 * q^83 + 32 * q^85 - 4 * q^89 + 36 * q^91 - 14 * q^95 + 16 * q^97

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