LMFDB - Embedded newform 4752.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4752,2,Mod(1,4752)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4752, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4752 = 2^{4} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4752.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.9449110405$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{5} +3.73205 q^{7} +O(q^{10})$	$q+0.732051 q^{5} +3.73205 q^{7} +1.00000 q^{11} -0.267949 q^{13} +4.19615 q^{17} +5.19615 q^{19} +8.00000 q^{23} -4.46410 q^{25} +1.26795 q^{29} -0.535898 q^{31} +2.73205 q^{35} -6.46410 q^{37} -1.46410 q^{41} -3.46410 q^{43} +10.1962 q^{47} +6.92820 q^{49} -4.73205 q^{53} +0.732051 q^{55} +10.1962 q^{59} -4.26795 q^{61} -0.196152 q^{65} -5.92820 q^{67} -13.8564 q^{71} +3.19615 q^{73} +3.73205 q^{77} +5.73205 q^{79} +3.07180 q^{85} +0.339746 q^{89} -1.00000 q^{91} +3.80385 q^{95} -5.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7	$ 2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{17} + 16 q^{23} - 2 q^{25} + 6 q^{29} - 8 q^{31} + 2 q^{35} - 6 q^{37} + 4 q^{41} + 10 q^{47} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 12 q^{61} + 10 q^{65} + 2 q^{67} - 4 q^{73} + 4 q^{77} + 8 q^{79} + 20 q^{85} + 18 q^{89} - 2 q^{91} + 18 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^17 + 16 * q^23 - 2 * q^25 + 6 * q^29 - 8 * q^31 + 2 * q^35 - 6 * q^37 + 4 * q^41 + 10 * q^47 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 12 * q^61 + 10 * q^65 + 2 * q^67 - 4 * q^73 + 4 * q^77 + 8 * q^79 + 20 * q^85 + 18 * q^89 - 2 * q^91 + 18 * q^95 + 10 * q^97

Display 100 coefficients 10 coefficients

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
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	3.73205	1.41058	0.705291	−	0.708918i	$-0.250816\pi$
$7$	3.73205	1.41058	0.705291	+	0.708918i	$0.250816\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.267949	−0.0743157	−0.0371579	−	0.999309i	$-0.511830\pi$
$13$	−0.267949	−0.0743157	−0.0371579	+	0.999309i	$0.511830\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.19615	1.01772	0.508858	−	0.860850i	$-0.330068\pi$
$17$	4.19615	1.01772	0.508858	+	0.860850i	$0.330068\pi$
$18$	0	0
$19$	5.19615	1.19208	0.596040	−	0.802955i	$-0.296740\pi$
$19$	5.19615	1.19208	0.596040	+	0.802955i	$0.296740\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.26795	0.235452	0.117726	−	0.993046i	$-0.462440\pi$
$29$	1.26795	0.235452	0.117726	+	0.993046i	$0.462440\pi$
$30$	0	0
$31$	−0.535898	−0.0962502	−0.0481251	−	0.998841i	$-0.515325\pi$
$31$	−0.535898	−0.0962502	−0.0481251	+	0.998841i	$0.515325\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−6.46410	−1.06269	−0.531346	−	0.847155i	$-0.678314\pi$
$37$	−6.46410	−1.06269	−0.531346	+	0.847155i	$0.678314\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.46410	−0.228654	−0.114327	−	0.993443i	$-0.536471\pi$
$41$	−1.46410	−0.228654	−0.114327	+	0.993443i	$0.536471\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.1962	1.48726	0.743631	−	0.668590i	$-0.233102\pi$
$47$	10.1962	1.48726	0.743631	+	0.668590i	$0.233102\pi$
$48$	0	0
$49$	6.92820	0.989743
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.73205	−0.649997	−0.324999	−	0.945715i	$-0.605364\pi$
$53$	−4.73205	−0.649997	−0.324999	+	0.945715i	$0.605364\pi$
$54$	0	0
$55$	0.732051	0.0987097
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.1962	1.32743	0.663713	−	0.747987i	$-0.268980\pi$
$59$	10.1962	1.32743	0.663713	+	0.747987i	$0.268980\pi$
$60$	0	0
$61$	−4.26795	−0.546455	−0.273227	−	0.961949i	$-0.588091\pi$
$61$	−4.26795	−0.546455	−0.273227	+	0.961949i	$0.588091\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.196152	−0.0243297
$66$	0	0
$67$	−5.92820	−0.724245	−0.362123	−	0.932130i	$-0.617948\pi$
$67$	−5.92820	−0.724245	−0.362123	+	0.932130i	$0.617948\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	3.19615	0.374081	0.187041	−	0.982352i	$-0.440110\pi$
$73$	3.19615	0.374081	0.187041	+	0.982352i	$0.440110\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.73205	0.425307
$78$	0	0
$79$	5.73205	0.644906	0.322453	−	0.946585i	$-0.395493\pi$
$79$	5.73205	0.644906	0.322453	+	0.946585i	$0.395493\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.07180	0.333183
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.339746	0.0360130	0.0180065	−	0.999838i	$-0.494268\pi$
$89$	0.339746	0.0360130	0.0180065	+	0.999838i	$0.494268\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.80385	0.390267
$96$	0	0
$97$	−5.39230	−0.547506	−0.273753	−	0.961800i	$-0.588265\pi$
$97$	−5.39230	−0.547506	−0.273753	+	0.961800i	$0.588265\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.196152	0.0195179	0.00975895	−	0.999952i	$-0.496894\pi$
$101$	0.196152	0.0195179	0.00975895	+	0.999952i	$0.496894\pi$
$102$	0	0
$103$	−12.8564	−1.26678	−0.633390	−	0.773833i	$-0.718337\pi$
$103$	−12.8564	−1.26678	−0.633390	+	0.773833i	$0.718337\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.66025	0.547197	0.273599	−	0.961844i	$-0.411786\pi$
$107$	5.66025	0.547197	0.273599	+	0.961844i	$0.411786\pi$
$108$	0	0
$109$	11.8564	1.13564	0.567819	−	0.823154i	$-0.307787\pi$
$109$	11.8564	1.13564	0.567819	+	0.823154i	$0.307787\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.6603	1.09690	0.548452	−	0.836182i	$-0.315217\pi$
$113$	11.6603	1.09690	0.548452	+	0.836182i	$0.315217\pi$
$114$	0	0
$115$	5.85641	0.546113
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.6603	1.43557
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−5.46410	−0.484861	−0.242430	−	0.970169i	$-0.577945\pi$
$127$	−5.46410	−0.484861	−0.242430	+	0.970169i	$0.577945\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.1244	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$131$	−15.1244	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$132$	0	0
$133$	19.3923	1.68153
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.39230	0.717003	0.358501	−	0.933529i	$-0.383288\pi$
$137$	8.39230	0.717003	0.358501	+	0.933529i	$0.383288\pi$
$138$	0	0
$139$	7.19615	0.610370	0.305185	−	0.952293i	$-0.401282\pi$
$139$	7.19615	0.610370	0.305185	+	0.952293i	$0.401282\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.267949	−0.0224070
$144$	0	0
$145$	0.928203	0.0770831
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.85641	−0.807468	−0.403734	−	0.914876i	$-0.632288\pi$
$149$	−9.85641	−0.807468	−0.403734	+	0.914876i	$0.632288\pi$
$150$	0	0
$151$	4.80385	0.390932	0.195466	−	0.980711i	$-0.437378\pi$
$151$	4.80385	0.390932	0.195466	+	0.980711i	$0.437378\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.392305	−0.0315107
$156$	0	0
$157$	10.5359	0.840856	0.420428	−	0.907326i	$-0.361880\pi$
$157$	10.5359	0.840856	0.420428	+	0.907326i	$0.361880\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	29.8564	2.35301
$162$	0	0
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.7321	1.75906	0.879529	−	0.475844i	$-0.157857\pi$
$167$	22.7321	1.75906	0.879529	+	0.475844i	$0.157857\pi$
$168$	0	0
$169$	−12.9282	−0.994477
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.39230	0.333941	0.166970	−	0.985962i	$-0.446602\pi$
$173$	4.39230	0.333941	0.166970	+	0.985962i	$0.446602\pi$
$174$	0	0
$175$	−16.6603	−1.25940
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.1962	−1.06107	−0.530535	−	0.847663i	$-0.678009\pi$
$179$	−14.1962	−1.06107	−0.530535	+	0.847663i	$0.678009\pi$
$180$	0	0
$181$	−16.3205	−1.21309	−0.606547	−	0.795048i	$-0.707446\pi$
$181$	−16.3205	−1.21309	−0.606547	+	0.795048i	$0.707446\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.73205	−0.347907
$186$	0	0
$187$	4.19615	0.306853
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−2.80385	−0.201825	−0.100913	−	0.994895i	$-0.532176\pi$
$193$	−2.80385	−0.201825	−0.100913	+	0.994895i	$0.532176\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.66025	−0.688265	−0.344132	−	0.938921i	$-0.611827\pi$
$197$	−9.66025	−0.688265	−0.344132	+	0.938921i	$0.611827\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.73205	0.332125
$204$	0	0
$205$	−1.07180	−0.0748575
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.19615	0.359425
$210$	0	0
$211$	9.33975	0.642975	0.321487	−	0.946914i	$-0.395817\pi$
$211$	9.33975	0.642975	0.321487	+	0.946914i	$0.395817\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.53590	−0.172947
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.12436	−0.0756323
$222$	0	0
$223$	−17.8564	−1.19575	−0.597877	−	0.801588i	$-0.703989\pi$
$223$	−17.8564	−1.19575	−0.597877	+	0.801588i	$0.703989\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−29.6603	−1.96862	−0.984310	−	0.176447i	$-0.943540\pi$
$227$	−29.6603	−1.96862	−0.984310	+	0.176447i	$0.943540\pi$
$228$	0	0
$229$	−16.3923	−1.08323	−0.541617	−	0.840625i	$-0.682188\pi$
$229$	−16.3923	−1.08323	−0.541617	+	0.840625i	$0.682188\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.33975	0.415331	0.207665	−	0.978200i	$-0.433414\pi$
$233$	6.33975	0.415331	0.207665	+	0.978200i	$0.433414\pi$
$234$	0	0
$235$	7.46410	0.486904
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.3923	1.06033	0.530165	−	0.847894i	$-0.322130\pi$
$239$	16.3923	1.06033	0.530165	+	0.847894i	$0.322130\pi$
$240$	0	0
$241$	15.1962	0.978870	0.489435	−	0.872040i	$-0.337203\pi$
$241$	15.1962	0.978870	0.489435	+	0.872040i	$0.337203\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.07180	0.324025
$246$	0	0
$247$	−1.39230	−0.0885902
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.5885	0.668337	0.334169	−	0.942513i	$-0.391544\pi$
$251$	10.5885	0.668337	0.334169	+	0.942513i	$0.391544\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.7321	0.794204	0.397102	−	0.917775i	$-0.370016\pi$
$257$	12.7321	0.794204	0.397102	+	0.917775i	$0.370016\pi$
$258$	0	0
$259$	−24.1244	−1.49901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.87564	0.300645	0.150323	−	0.988637i	$-0.451969\pi$
$263$	4.87564	0.300645	0.150323	+	0.988637i	$0.451969\pi$
$264$	0	0
$265$	−3.46410	−0.212798
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.9282	1.64184	0.820921	−	0.571042i	$-0.193461\pi$
$269$	26.9282	1.64184	0.820921	+	0.571042i	$0.193461\pi$
$270$	0	0
$271$	2.26795	0.137768	0.0688841	−	0.997625i	$-0.478056\pi$
$271$	2.26795	0.137768	0.0688841	+	0.997625i	$0.478056\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.46410	−0.269195
$276$	0	0
$277$	−9.85641	−0.592214	−0.296107	−	0.955155i	$-0.595689\pi$
$277$	−9.85641	−0.592214	−0.296107	+	0.955155i	$0.595689\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	26.2487	1.56032	0.780162	−	0.625578i	$-0.215137\pi$
$283$	26.2487	1.56032	0.780162	+	0.625578i	$0.215137\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.46410	−0.322536
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.1244	1.58462	0.792311	−	0.610118i	$-0.208878\pi$
$293$	27.1244	1.58462	0.792311	+	0.610118i	$0.208878\pi$
$294$	0	0
$295$	7.46410	0.434577
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.14359	−0.123967
$300$	0	0
$301$	−12.9282	−0.745169
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.12436	−0.178900
$306$	0	0
$307$	27.3205	1.55926	0.779632	−	0.626238i	$-0.215406\pi$
$307$	27.3205	1.55926	0.779632	+	0.626238i	$0.215406\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.5885	1.73451	0.867256	−	0.497862i	$-0.165881\pi$
$311$	30.5885	1.73451	0.867256	+	0.497862i	$0.165881\pi$
$312$	0	0
$313$	−1.92820	−0.108988	−0.0544942	−	0.998514i	$-0.517355\pi$
$313$	−1.92820	−0.108988	−0.0544942	+	0.998514i	$0.517355\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.5885	−0.594707	−0.297354	−	0.954767i	$-0.596104\pi$
$317$	−10.5885	−0.594707	−0.297354	+	0.954767i	$0.596104\pi$
$318$	0	0
$319$	1.26795	0.0709915
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.8038	1.21320
$324$	0	0
$325$	1.19615	0.0663506
$326$	0	0
$327$	0	0
$328$	0	0
$329$	38.0526	2.09791
$330$	0	0
$331$	27.3923	1.50562	0.752809	−	0.658239i	$-0.228699\pi$
$331$	27.3923	1.50562	0.752809	+	0.658239i	$0.228699\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.33975	−0.237106
$336$	0	0
$337$	−29.9808	−1.63316	−0.816578	−	0.577235i	$-0.804132\pi$
$337$	−29.9808	−1.63316	−0.816578	+	0.577235i	$0.804132\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.535898	−0.0290205
$342$	0	0
$343$	−0.267949	−0.0144679
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.3397	0.555067	0.277533	−	0.960716i	$-0.410483\pi$
$347$	10.3397	0.555067	0.277533	+	0.960716i	$0.410483\pi$
$348$	0	0
$349$	−5.87564	−0.314516	−0.157258	−	0.987558i	$-0.550265\pi$
$349$	−5.87564	−0.314516	−0.157258	+	0.987558i	$0.550265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.1244	−1.12434	−0.562168	−	0.827023i	$-0.690033\pi$
$353$	−21.1244	−1.12434	−0.562168	+	0.827023i	$0.690033\pi$
$354$	0	0
$355$	−10.1436	−0.538366
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.53590	0.344952	0.172476	−	0.985014i	$-0.444823\pi$
$359$	6.53590	0.344952	0.172476	+	0.985014i	$0.444823\pi$
$360$	0	0
$361$	8.00000	0.421053
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.33975	0.122468
$366$	0	0
$367$	−13.9282	−0.727046	−0.363523	−	0.931585i	$-0.618426\pi$
$367$	−13.9282	−0.727046	−0.363523	+	0.931585i	$0.618426\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.6603	−0.916875
$372$	0	0
$373$	−20.1244	−1.04200	−0.521000	−	0.853557i	$-0.674441\pi$
$373$	−20.1244	−1.04200	−0.521000	+	0.853557i	$0.674441\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.339746	−0.0174978
$378$	0	0
$379$	−14.8564	−0.763122	−0.381561	−	0.924344i	$-0.624613\pi$
$379$	−14.8564	−0.763122	−0.381561	+	0.924344i	$0.624613\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.7321	1.26375	0.631874	−	0.775071i	$-0.282286\pi$
$383$	24.7321	1.26375	0.631874	+	0.775071i	$0.282286\pi$
$384$	0	0
$385$	2.73205	0.139238
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−21.4641	−1.08827	−0.544137	−	0.838997i	$-0.683143\pi$
$389$	−21.4641	−1.08827	−0.544137	+	0.838997i	$0.683143\pi$
$390$	0	0
$391$	33.5692	1.69767
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.19615	0.211131
$396$	0	0
$397$	20.3923	1.02346	0.511730	−	0.859146i	$-0.329005\pi$
$397$	20.3923	1.02346	0.511730	+	0.859146i	$0.329005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.53590	0.326387	0.163194	−	0.986594i	$-0.447820\pi$
$401$	6.53590	0.326387	0.163194	+	0.986594i	$0.447820\pi$
$402$	0	0
$403$	0.143594	0.00715290
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.46410	−0.320414
$408$	0	0
$409$	32.9090	1.62724	0.813622	−	0.581394i	$-0.197493\pi$
$409$	32.9090	1.62724	0.813622	+	0.581394i	$0.197493\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	38.0526	1.87244
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.7128	1.15845	0.579223	−	0.815169i	$-0.303356\pi$
$419$	23.7128	1.15845	0.579223	+	0.815169i	$0.303356\pi$
$420$	0	0
$421$	20.8564	1.01648	0.508240	−	0.861216i	$-0.330296\pi$
$421$	20.8564	1.01648	0.508240	+	0.861216i	$0.330296\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.7321	−0.908638
$426$	0	0
$427$	−15.9282	−0.770820
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.5359	0.700170	0.350085	−	0.936718i	$-0.386153\pi$
$431$	14.5359	0.700170	0.350085	+	0.936718i	$0.386153\pi$
$432$	0	0
$433$	−3.60770	−0.173375	−0.0866874	−	0.996236i	$-0.527628\pi$
$433$	−3.60770	−0.173375	−0.0866874	+	0.996236i	$0.527628\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.5692	1.98853
$438$	0	0
$439$	−30.5359	−1.45740	−0.728699	−	0.684834i	$-0.759875\pi$
$439$	−30.5359	−1.45740	−0.728699	+	0.684834i	$0.759875\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.26795	−0.155265	−0.0776325	−	0.996982i	$-0.524736\pi$
$443$	−3.26795	−0.155265	−0.0776325	+	0.996982i	$0.524736\pi$
$444$	0	0
$445$	0.248711	0.0117900
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.2487	0.483667	0.241833	−	0.970318i	$-0.422251\pi$
$449$	10.2487	0.483667	0.241833	+	0.970318i	$0.422251\pi$
$450$	0	0
$451$	−1.46410	−0.0689419
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.732051	−0.0343191
$456$	0	0
$457$	−11.0718	−0.517917	−0.258958	−	0.965888i	$-0.583379\pi$
$457$	−11.0718	−0.517917	−0.258958	+	0.965888i	$0.583379\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.6603	1.38142	0.690708	−	0.723134i	$-0.257299\pi$
$461$	29.6603	1.38142	0.690708	+	0.723134i	$0.257299\pi$
$462$	0	0
$463$	25.7846	1.19831	0.599156	−	0.800632i	$-0.295503\pi$
$463$	25.7846	1.19831	0.599156	+	0.800632i	$0.295503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.9282	−0.875893	−0.437946	−	0.899001i	$-0.644294\pi$
$467$	−18.9282	−0.875893	−0.437946	+	0.899001i	$0.644294\pi$
$468$	0	0
$469$	−22.1244	−1.02161
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.46410	−0.159280
$474$	0	0
$475$	−23.1962	−1.06431
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.5167	−0.526210	−0.263105	−	0.964767i	$-0.584747\pi$
$479$	−11.5167	−0.526210	−0.263105	+	0.964767i	$0.584747\pi$
$480$	0	0
$481$	1.73205	0.0789747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.94744	−0.179244
$486$	0	0
$487$	−18.8564	−0.854465	−0.427233	−	0.904142i	$-0.640511\pi$
$487$	−18.8564	−0.854465	−0.427233	+	0.904142i	$0.640511\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.0526	−1.35625	−0.678126	−	0.734945i	$-0.737208\pi$
$491$	−30.0526	−1.35625	−0.678126	+	0.734945i	$0.737208\pi$
$492$	0	0
$493$	5.32051	0.239624
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−51.7128	−2.31964
$498$	0	0
$499$	19.4641	0.871333	0.435666	−	0.900108i	$-0.356513\pi$
$499$	19.4641	0.871333	0.435666	+	0.900108i	$0.356513\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.9282	1.02232	0.511159	−	0.859486i	$-0.329216\pi$
$503$	22.9282	1.02232	0.511159	+	0.859486i	$0.329216\pi$
$504$	0	0
$505$	0.143594	0.00638983
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.0526	−0.711517	−0.355759	−	0.934578i	$-0.615777\pi$
$509$	−16.0526	−0.711517	−0.355759	+	0.934578i	$0.615777\pi$
$510$	0	0
$511$	11.9282	0.527673
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.41154	−0.414722
$516$	0	0
$517$	10.1962	0.448426
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	−39.9808	−1.74824	−0.874118	−	0.485713i	$-0.838560\pi$
$523$	−39.9808	−1.74824	−0.874118	+	0.485713i	$0.838560\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.24871	−0.0979554
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.392305	0.0169926
$534$	0	0
$535$	4.14359	0.179143
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.92820	0.298419
$540$	0	0
$541$	−23.5885	−1.01415	−0.507073	−	0.861903i	$-0.669273\pi$
$541$	−23.5885	−1.01415	−0.507073	+	0.861903i	$0.669273\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.67949	0.371789
$546$	0	0
$547$	16.8038	0.718481	0.359240	−	0.933245i	$-0.383036\pi$
$547$	16.8038	0.718481	0.359240	+	0.933245i	$0.383036\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.58846	0.280678
$552$	0	0
$553$	21.3923	0.909693
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.2679	−1.24012	−0.620061	−	0.784553i	$-0.712892\pi$
$557$	−29.2679	−1.24012	−0.620061	+	0.784553i	$0.712892\pi$
$558$	0	0
$559$	0.928203	0.0392588
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.7128	−1.50512	−0.752558	−	0.658526i	$-0.771180\pi$
$563$	−35.7128	−1.50512	−0.752558	+	0.658526i	$0.771180\pi$
$564$	0	0
$565$	8.53590	0.359108
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.3205	−1.48071	−0.740356	−	0.672215i	$-0.765343\pi$
$569$	−35.3205	−1.48071	−0.740356	+	0.672215i	$0.765343\pi$
$570$	0	0
$571$	−32.1244	−1.34436	−0.672181	−	0.740387i	$-0.734642\pi$
$571$	−32.1244	−1.34436	−0.672181	+	0.740387i	$0.734642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−35.7128	−1.48933
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.73205	−0.195982
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.41154	−0.0582606	−0.0291303	−	0.999576i	$-0.509274\pi$
$587$	−1.41154	−0.0582606	−0.0291303	+	0.999576i	$0.509274\pi$
$588$	0	0
$589$	−2.78461	−0.114738
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−47.7128	−1.95933	−0.979665	−	0.200639i	$-0.935698\pi$
$593$	−47.7128	−1.95933	−0.979665	+	0.200639i	$0.935698\pi$
$594$	0	0
$595$	11.4641	0.469982
$596$	0	0
$597$	0	0
$598$	0	0
$599$	46.2487	1.88967	0.944836	−	0.327545i	$-0.106221\pi$
$599$	46.2487	1.88967	0.944836	+	0.327545i	$0.106221\pi$
$600$	0	0
$601$	3.85641	0.157306	0.0786531	−	0.996902i	$-0.474938\pi$
$601$	3.85641	0.157306	0.0786531	+	0.996902i	$0.474938\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.732051	0.0297621
$606$	0	0
$607$	37.5885	1.52567	0.762834	−	0.646594i	$-0.223807\pi$
$607$	37.5885	1.52567	0.762834	+	0.646594i	$0.223807\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.73205	−0.110527
$612$	0	0
$613$	22.9090	0.925284	0.462642	−	0.886545i	$-0.346901\pi$
$613$	22.9090	0.925284	0.462642	+	0.886545i	$0.346901\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−37.8564	−1.52404	−0.762021	−	0.647553i	$-0.775793\pi$
$617$	−37.8564	−1.52404	−0.762021	+	0.647553i	$0.775793\pi$
$618$	0	0
$619$	−20.4641	−0.822522	−0.411261	−	0.911518i	$-0.634911\pi$
$619$	−20.4641	−0.822522	−0.411261	+	0.911518i	$0.634911\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.26795	0.0507993
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.1244	−1.08152
$630$	0	0
$631$	3.39230	0.135046	0.0675228	−	0.997718i	$-0.478490\pi$
$631$	3.39230	0.135046	0.0675228	+	0.997718i	$0.478490\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−1.85641	−0.0735535
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.87564	−0.113581	−0.0567906	−	0.998386i	$-0.518087\pi$
$641$	−2.87564	−0.113581	−0.0567906	+	0.998386i	$0.518087\pi$
$642$	0	0
$643$	−30.3923	−1.19856	−0.599278	−	0.800541i	$-0.704545\pi$
$643$	−30.3923	−1.19856	−0.599278	+	0.800541i	$0.704545\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.2487	1.03194	0.515972	−	0.856606i	$-0.327431\pi$
$647$	26.2487	1.03194	0.515972	+	0.856606i	$0.327431\pi$
$648$	0	0
$649$	10.1962	0.400234
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.4641	1.15302	0.576510	−	0.817090i	$-0.304414\pi$
$653$	29.4641	1.15302	0.576510	+	0.817090i	$0.304414\pi$
$654$	0	0
$655$	−11.0718	−0.432611
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−37.6603	−1.46704	−0.733518	−	0.679670i	$-0.762123\pi$
$659$	−37.6603	−1.46704	−0.733518	+	0.679670i	$0.762123\pi$
$660$	0	0
$661$	2.60770	0.101428	0.0507138	−	0.998713i	$-0.483850\pi$
$661$	2.60770	0.101428	0.0507138	+	0.998713i	$0.483850\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14.1962	0.550503
$666$	0	0
$667$	10.1436	0.392762
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.26795	−0.164762
$672$	0	0
$673$	−9.44486	−0.364073	−0.182036	−	0.983292i	$-0.558269\pi$
$673$	−9.44486	−0.364073	−0.182036	+	0.983292i	$0.558269\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.2487	−1.16255	−0.581276	−	0.813706i	$-0.697446\pi$
$677$	−30.2487	−1.16255	−0.581276	+	0.813706i	$0.697446\pi$
$678$	0	0
$679$	−20.1244	−0.772302
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.5885	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$683$	−22.5885	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$684$	0	0
$685$	6.14359	0.234735
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.26795	0.0483050
$690$	0	0
$691$	7.21539	0.274486	0.137243	−	0.990537i	$-0.456176\pi$
$691$	7.21539	0.274486	0.137243	+	0.990537i	$0.456176\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.26795	0.199825
$696$	0	0
$697$	−6.14359	−0.232705
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.94744	−0.375710	−0.187855	−	0.982197i	$-0.560153\pi$
$701$	−9.94744	−0.375710	−0.187855	+	0.982197i	$0.560153\pi$
$702$	0	0
$703$	−33.5885	−1.26681
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.732051	0.0275316
$708$	0	0
$709$	−35.3923	−1.32919	−0.664593	−	0.747206i	$-0.731395\pi$
$709$	−35.3923	−1.32919	−0.664593	+	0.747206i	$0.731395\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.28719	−0.160556
$714$	0	0
$715$	−0.196152	−0.00733568
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.9808	0.857038	0.428519	−	0.903533i	$-0.359036\pi$
$719$	22.9808	0.857038	0.428519	+	0.903533i	$0.359036\pi$
$720$	0	0
$721$	−47.9808	−1.78690
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.66025	−0.210217
$726$	0	0
$727$	29.3205	1.08744	0.543719	−	0.839268i	$-0.317016\pi$
$727$	29.3205	1.08744	0.543719	+	0.839268i	$0.317016\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.5359	−0.537630
$732$	0	0
$733$	24.9282	0.920744	0.460372	−	0.887726i	$-0.347716\pi$
$733$	24.9282	0.920744	0.460372	+	0.887726i	$0.347716\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.92820	−0.218368
$738$	0	0
$739$	34.2487	1.25986	0.629930	−	0.776652i	$-0.283084\pi$
$739$	34.2487	1.25986	0.629930	+	0.776652i	$0.283084\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.5885	1.04881	0.524404	−	0.851469i	$-0.324288\pi$
$743$	28.5885	1.04881	0.524404	+	0.851469i	$0.324288\pi$
$744$	0	0
$745$	−7.21539	−0.264351
$746$	0	0
$747$	0	0
$748$	0	0
$749$	21.1244	0.771867
$750$	0	0
$751$	22.3205	0.814487	0.407243	−	0.913320i	$-0.366490\pi$
$751$	22.3205	0.814487	0.407243	+	0.913320i	$0.366490\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.51666	0.127984
$756$	0	0
$757$	28.3205	1.02933	0.514663	−	0.857392i	$-0.327917\pi$
$757$	28.3205	1.02933	0.514663	+	0.857392i	$0.327917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.679492	0.0246316	0.0123158	−	0.999924i	$-0.496080\pi$
$761$	0.679492	0.0246316	0.0123158	+	0.999924i	$0.496080\pi$
$762$	0	0
$763$	44.2487	1.60191
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.73205	−0.0986486
$768$	0	0
$769$	−25.5885	−0.922743	−0.461372	−	0.887207i	$-0.652643\pi$
$769$	−25.5885	−0.922743	−0.461372	+	0.887207i	$0.652643\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.1244	−0.759790	−0.379895	−	0.925030i	$-0.624040\pi$
$773$	−21.1244	−0.759790	−0.379895	+	0.925030i	$0.624040\pi$
$774$	0	0
$775$	2.39230	0.0859341
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.60770	−0.272574
$780$	0	0
$781$	−13.8564	−0.495821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.71281	0.275282
$786$	0	0
$787$	−23.0526	−0.821735	−0.410867	−	0.911695i	$-0.634774\pi$
$787$	−23.0526	−0.821735	−0.410867	+	0.911695i	$0.634774\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	43.5167	1.54727
$792$	0	0
$793$	1.14359	0.0406102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	42.7846	1.51361
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.19615	0.112790
$804$	0	0
$805$	21.8564	0.770337
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.4115	−0.823106	−0.411553	−	0.911386i	$-0.635013\pi$
$809$	−23.4115	−0.823106	−0.411553	+	0.911386i	$0.635013\pi$
$810$	0	0
$811$	−20.3923	−0.716071	−0.358035	−	0.933708i	$-0.616553\pi$
$811$	−20.3923	−0.716071	−0.358035	+	0.933708i	$0.616553\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.9090	−0.487210
$816$	0	0
$817$	−18.0000	−0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.3923	0.432494	0.216247	−	0.976339i	$-0.430618\pi$
$821$	12.3923	0.432494	0.216247	+	0.976339i	$0.430618\pi$
$822$	0	0
$823$	−48.7128	−1.69802	−0.849011	−	0.528375i	$-0.822801\pi$
$823$	−48.7128	−1.69802	−0.849011	+	0.528375i	$0.822801\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.9808	−1.56413	−0.782067	−	0.623194i	$-0.785835\pi$
$827$	−44.9808	−1.56413	−0.782067	+	0.623194i	$0.785835\pi$
$828$	0	0
$829$	−0.607695	−0.0211061	−0.0105531	−	0.999944i	$-0.503359\pi$
$829$	−0.607695	−0.0211061	−0.0105531	+	0.999944i	$0.503359\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29.0718	1.00728
$834$	0	0
$835$	16.6410	0.575886
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.0333	−1.20948	−0.604742	−	0.796421i	$-0.706724\pi$
$839$	−35.0333	−1.20948	−0.604742	+	0.796421i	$0.706724\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.46410	−0.325575
$846$	0	0
$847$	3.73205	0.128235
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−51.7128	−1.77269
$852$	0	0
$853$	12.2679	0.420047	0.210023	−	0.977696i	$-0.432646\pi$
$853$	12.2679	0.420047	0.210023	+	0.977696i	$0.432646\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.8038	−0.813124	−0.406562	−	0.913623i	$-0.633272\pi$
$857$	−23.8038	−0.813124	−0.406562	+	0.913623i	$0.633272\pi$
$858$	0	0
$859$	−28.8564	−0.984568	−0.492284	−	0.870435i	$-0.663838\pi$
$859$	−28.8564	−0.984568	−0.492284	+	0.870435i	$0.663838\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.9090	−0.609628	−0.304814	−	0.952412i	$-0.598594\pi$
$863$	−17.9090	−0.609628	−0.304814	+	0.952412i	$0.598594\pi$
$864$	0	0
$865$	3.21539	0.109327
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.73205	0.194447
$870$	0	0
$871$	1.58846	0.0538228
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−25.8564	−0.874106
$876$	0	0
$877$	−11.1962	−0.378067	−0.189034	−	0.981971i	$-0.560536\pi$
$877$	−11.1962	−0.378067	−0.189034	+	0.981971i	$0.560536\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.3205	1.72903	0.864516	−	0.502605i	$-0.167625\pi$
$881$	51.3205	1.72903	0.864516	+	0.502605i	$0.167625\pi$
$882$	0	0
$883$	5.67949	0.191130	0.0955651	−	0.995423i	$-0.469534\pi$
$883$	5.67949	0.191130	0.0955651	+	0.995423i	$0.469534\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.73205	0.226040	0.113020	−	0.993593i	$-0.463948\pi$
$887$	6.73205	0.226040	0.113020	+	0.993593i	$0.463948\pi$
$888$	0	0
$889$	−20.3923	−0.683936
$890$	0	0
$891$	0	0
$892$	0	0
$893$	52.9808	1.77293
$894$	0	0
$895$	−10.3923	−0.347376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.679492	−0.0226623
$900$	0	0
$901$	−19.8564	−0.661513
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.9474	−0.397146
$906$	0	0
$907$	−12.4641	−0.413864	−0.206932	−	0.978355i	$-0.566348\pi$
$907$	−12.4641	−0.413864	−0.206932	+	0.978355i	$0.566348\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.46410	−0.313560	−0.156780	−	0.987634i	$-0.550111\pi$
$911$	−9.46410	−0.313560	−0.156780	+	0.987634i	$0.550111\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−56.4449	−1.86397
$918$	0	0
$919$	43.3205	1.42901	0.714506	−	0.699629i	$-0.246652\pi$
$919$	43.3205	1.42901	0.714506	+	0.699629i	$0.246652\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.71281	0.122209
$924$	0	0
$925$	28.8564	0.948793
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.2679	−1.41958	−0.709788	−	0.704416i	$-0.751209\pi$
$929$	−43.2679	−1.41958	−0.709788	+	0.704416i	$0.751209\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.07180	0.100458
$936$	0	0
$937$	−45.1962	−1.47649	−0.738247	−	0.674531i	$-0.764346\pi$
$937$	−45.1962	−1.47649	−0.738247	+	0.674531i	$0.764346\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.5359	−0.865046	−0.432523	−	0.901623i	$-0.642376\pi$
$941$	−26.5359	−0.865046	−0.432523	+	0.901623i	$0.642376\pi$
$942$	0	0
$943$	−11.7128	−0.381422
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.6603	−0.768855	−0.384427	−	0.923155i	$-0.625601\pi$
$947$	−23.6603	−0.768855	−0.384427	+	0.923155i	$0.625601\pi$
$948$	0	0
$949$	−0.856406	−0.0278001
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.14359	0.199011	0.0995053	−	0.995037i	$-0.468274\pi$
$953$	6.14359	0.199011	0.0995053	+	0.995037i	$0.468274\pi$
$954$	0	0
$955$	−2.92820	−0.0947544
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.3205	1.01139
$960$	0	0
$961$	−30.7128	−0.990736
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.05256	−0.0660742
$966$	0	0
$967$	−30.4115	−0.977969	−0.488985	−	0.872292i	$-0.662633\pi$
$967$	−30.4115	−0.977969	−0.488985	+	0.872292i	$0.662633\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.58846	0.0830675	0.0415338	−	0.999137i	$-0.486776\pi$
$971$	2.58846	0.0830675	0.0415338	+	0.999137i	$0.486776\pi$
$972$	0	0
$973$	26.8564	0.860977
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.6603	1.14087	0.570436	−	0.821342i	$-0.306774\pi$
$977$	35.6603	1.14087	0.570436	+	0.821342i	$0.306774\pi$
$978$	0	0
$979$	0.339746	0.0108583
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.0526	−0.511997	−0.255999	−	0.966677i	$-0.582404\pi$
$983$	−16.0526	−0.511997	−0.255999	+	0.966677i	$0.582404\pi$
$984$	0	0
$985$	−7.07180	−0.225326
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27.7128	−0.881216
$990$	0	0
$991$	26.4641	0.840660	0.420330	−	0.907371i	$-0.361914\pi$
$991$	26.4641	0.840660	0.420330	+	0.907371i	$0.361914\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.9808	0.348114
$996$	0	0
$997$	0.143594	0.00454765	0.00227383	−	0.999997i	$-0.499276\pi$
$997$	0.143594	0.00454765	0.00227383	+	0.999997i	$0.499276\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4752.2.a.w.1.2		2
3.2	odd	2		4752.2.a.bf.1.1		2
4.3	odd	2		297.2.a.e.1.1	✓	2
12.11	even	2		297.2.a.f.1.2	yes	2
20.19	odd	2		7425.2.a.bl.1.2		2
36.7	odd	6		891.2.e.p.595.2		4
36.11	even	6		891.2.e.m.595.1		4
36.23	even	6		891.2.e.m.298.1		4
36.31	odd	6		891.2.e.p.298.2		4
44.43	even	2		3267.2.a.q.1.2		2
60.59	even	2		7425.2.a.z.1.1		2
132.131	odd	2		3267.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.a.e.1.1	✓	2	4.3	odd	2
297.2.a.f.1.2	yes	2	12.11	even	2
891.2.e.m.298.1		4	36.23	even	6
891.2.e.m.595.1		4	36.11	even	6
891.2.e.p.298.2		4	36.31	odd	6
891.2.e.p.595.2		4	36.7	odd	6
3267.2.a.l.1.1		2	132.131	odd	2
3267.2.a.q.1.2		2	44.43	even	2
4752.2.a.w.1.2		2	1.1	even	1	trivial
4752.2.a.bf.1.1		2	3.2	odd	2
7425.2.a.z.1.1		2	60.59	even	2
7425.2.a.bl.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4752 = 2^{4} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4752.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{5} +3.73205 q^{7} +O(q^{10})\)	\(q+0.732051 q^{5} +3.73205 q^{7} +1.00000 q^{11} -0.267949 q^{13} +4.19615 q^{17} +5.19615 q^{19} +8.00000 q^{23} -4.46410 q^{25} +1.26795 q^{29} -0.535898 q^{31} +2.73205 q^{35} -6.46410 q^{37} -1.46410 q^{41} -3.46410 q^{43} +10.1962 q^{47} +6.92820 q^{49} -4.73205 q^{53} +0.732051 q^{55} +10.1962 q^{59} -4.26795 q^{61} -0.196152 q^{65} -5.92820 q^{67} -13.8564 q^{71} +3.19615 q^{73} +3.73205 q^{77} +5.73205 q^{79} +3.07180 q^{85} +0.339746 q^{89} -1.00000 q^{91} +3.80385 q^{95} -5.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7	\( 2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{17} + 16 q^{23} - 2 q^{25} + 6 q^{29} - 8 q^{31} + 2 q^{35} - 6 q^{37} + 4 q^{41} + 10 q^{47} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 12 q^{61} + 10 q^{65} + 2 q^{67} - 4 q^{73} + 4 q^{77} + 8 q^{79} + 20 q^{85} + 18 q^{89} - 2 q^{91} + 18 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^17 + 16 * q^23 - 2 * q^25 + 6 * q^29 - 8 * q^31 + 2 * q^35 - 6 * q^37 + 4 * q^41 + 10 * q^47 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 12 * q^61 + 10 * q^65 + 2 * q^67 - 4 * q^73 + 4 * q^77 + 8 * q^79 + 20 * q^85 + 18 * q^89 - 2 * q^91 + 18 * q^95 + 10 * q^97

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