LMFDB - Embedded newform 5148.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18953 q^{5} -4.42723 q^{7} +O(q^{10})$	$q+3.18953 q^{5} -4.42723 q^{7} -1.00000 q^{11} -1.00000 q^{13} +1.49180 q^{17} -1.76231 q^{19} -1.61676 q^{23} +5.17313 q^{25} +9.36266 q^{29} -5.31867 q^{31} -14.1208 q^{35} -0.810466 q^{37} +1.91903 q^{41} +5.23353 q^{43} -8.37907 q^{47} +12.6004 q^{49} -6.61676 q^{53} -3.18953 q^{55} -11.4067 q^{59} +12.2499 q^{61} -3.18953 q^{65} -12.0768 q^{67} +10.4231 q^{71} -13.8503 q^{73} +4.42723 q^{77} -2.34625 q^{79} +0.935432 q^{83} +4.75814 q^{85} -12.2059 q^{89} +4.42723 q^{91} -5.62093 q^{95} -3.18953 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} - 7 q^{7}+O(q^{10}) $ 3 * q + q^5 - 7 * q^7	$ 3 q + q^{5} - 7 q^{7} - 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 10 q^{23} + 10 q^{25} + 14 q^{29} - 23 q^{31} + 6 q^{35} - 11 q^{37} + q^{41} - 14 q^{43} - 8 q^{47} + 26 q^{49} - 5 q^{53} - q^{55} + q^{59} + 4 q^{61} - q^{65} - 9 q^{67} - 7 q^{71} + 3 q^{73} + 7 q^{77} + 4 q^{79} - 5 q^{83} - 20 q^{85} - 25 q^{89} + 7 q^{91} - 34 q^{95} - q^{97}+O(q^{100}) $ 3 * q + q^5 - 7 * q^7 - 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 10 * q^23 + 10 * q^25 + 14 * q^29 - 23 * q^31 + 6 * q^35 - 11 * q^37 + q^41 - 14 * q^43 - 8 * q^47 + 26 * q^49 - 5 * q^53 - q^55 + q^59 + 4 * q^61 - q^65 - 9 * q^67 - 7 * q^71 + 3 * q^73 + 7 * q^77 + 4 * q^79 - 5 * q^83 - 20 * q^85 - 25 * q^89 + 7 * q^91 - 34 * q^95 - 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
$5$	3.18953	1.42640	0.713201	−	0.700959i	$-0.247244\pi$
$5$	3.18953	1.42640	0.713201	+	0.700959i	$0.247244\pi$
$6$	0	0
$7$	−4.42723	−1.67334	−0.836668	−	0.547711i	$-0.815499\pi$
$7$	−4.42723	−1.67334	−0.836668	+	0.547711i	$0.815499\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.49180	0.361814	0.180907	−	0.983500i	$-0.442097\pi$
$17$	1.49180	0.361814	0.180907	+	0.983500i	$0.442097\pi$
$18$	0	0
$19$	−1.76231	−0.404301	−0.202150	−	0.979355i	$-0.564793\pi$
$19$	−1.76231	−0.404301	−0.202150	+	0.979355i	$0.564793\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.61676	−0.337118	−0.168559	−	0.985692i	$-0.553911\pi$
$23$	−1.61676	−0.337118	−0.168559	+	0.985692i	$0.553911\pi$
$24$	0	0
$25$	5.17313	1.03463
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.36266	1.73860	0.869301	−	0.494283i	$-0.164569\pi$
$29$	9.36266	1.73860	0.869301	+	0.494283i	$0.164569\pi$
$30$	0	0
$31$	−5.31867	−0.955261	−0.477631	−	0.878561i	$-0.658504\pi$
$31$	−5.31867	−0.955261	−0.477631	+	0.878561i	$0.658504\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.1208	−2.38685
$36$	0	0
$37$	−0.810466	−0.133240	−0.0666199	−	0.997778i	$-0.521221\pi$
$37$	−0.810466	−0.133240	−0.0666199	+	0.997778i	$0.521221\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.91903	0.299701	0.149851	−	0.988709i	$-0.452121\pi$
$41$	1.91903	0.299701	0.149851	+	0.988709i	$0.452121\pi$
$42$	0	0
$43$	5.23353	0.798105	0.399053	−	0.916928i	$-0.369339\pi$
$43$	5.23353	0.798105	0.399053	+	0.916928i	$0.369339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.37907	−1.22221	−0.611106	−	0.791549i	$-0.709275\pi$
$47$	−8.37907	−1.22221	−0.611106	+	0.791549i	$0.709275\pi$
$48$	0	0
$49$	12.6004	1.80005
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.61676	−0.908882	−0.454441	−	0.890777i	$-0.650161\pi$
$53$	−6.61676	−0.908882	−0.454441	+	0.890777i	$0.650161\pi$
$54$	0	0
$55$	−3.18953	−0.430077
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.4067	−1.48502	−0.742510	−	0.669835i	$-0.766365\pi$
$59$	−11.4067	−1.48502	−0.742510	+	0.669835i	$0.766365\pi$
$60$	0	0
$61$	12.2499	1.56844	0.784222	−	0.620481i	$-0.213063\pi$
$61$	12.2499	1.56844	0.784222	+	0.620481i	$0.213063\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.18953	−0.395613
$66$	0	0
$67$	−12.0768	−1.47542	−0.737708	−	0.675120i	$-0.764092\pi$
$67$	−12.0768	−1.47542	−0.737708	+	0.675120i	$0.764092\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4231	1.23699	0.618495	−	0.785789i	$-0.287743\pi$
$71$	10.4231	1.23699	0.618495	+	0.785789i	$0.287743\pi$
$72$	0	0
$73$	−13.8503	−1.62105	−0.810527	−	0.585701i	$-0.800819\pi$
$73$	−13.8503	−1.62105	−0.810527	+	0.585701i	$0.800819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.42723	0.504530
$78$	0	0
$79$	−2.34625	−0.263974	−0.131987	−	0.991251i	$-0.542136\pi$
$79$	−2.34625	−0.263974	−0.131987	+	0.991251i	$0.542136\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.935432	0.102677	0.0513385	−	0.998681i	$-0.483651\pi$
$83$	0.935432	0.102677	0.0513385	+	0.998681i	$0.483651\pi$
$84$	0	0
$85$	4.75814	0.516092
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.2059	−1.29383	−0.646914	−	0.762563i	$-0.723941\pi$
$89$	−12.2059	−1.29383	−0.646914	+	0.762563i	$0.723941\pi$
$90$	0	0
$91$	4.42723	0.464100
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.62093	−0.576695
$96$	0	0
$97$	−3.18953	−0.323848	−0.161924	−	0.986803i	$-0.551770\pi$
$97$	−3.18953	−0.323848	−0.161924	+	0.986803i	$0.551770\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.2171	−1.41466	−0.707328	−	0.706885i	$-0.750100\pi$
$101$	−14.2171	−1.41466	−0.707328	+	0.706885i	$0.750100\pi$
$102$	0	0
$103$	−8.46004	−0.833593	−0.416796	−	0.909000i	$-0.636847\pi$
$103$	−8.46004	−0.833593	−0.416796	+	0.909000i	$0.636847\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.3791	1.19673	0.598365	−	0.801224i	$-0.295817\pi$
$107$	12.3791	1.19673	0.598365	+	0.801224i	$0.295817\pi$
$108$	0	0
$109$	−13.6332	−1.30582	−0.652910	−	0.757435i	$-0.726452\pi$
$109$	−13.6332	−1.30582	−0.652910	+	0.757435i	$0.726452\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.1854	−1.52259	−0.761296	−	0.648405i	$-0.775436\pi$
$113$	−16.1854	−1.52259	−0.761296	+	0.648405i	$0.775436\pi$
$114$	0	0
$115$	−5.15672	−0.480867
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.60453	−0.605436
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.552195	0.0493898
$126$	0	0
$127$	−14.0880	−1.25011	−0.625053	−	0.780582i	$-0.714923\pi$
$127$	−14.0880	−1.25011	−0.625053	+	0.780582i	$0.714923\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.08798	0.706650	0.353325	−	0.935501i	$-0.385051\pi$
$131$	8.08798	0.706650	0.353325	+	0.935501i	$0.385051\pi$
$132$	0	0
$133$	7.80213	0.676530
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.1567	−0.953183	−0.476591	−	0.879125i	$-0.658128\pi$
$137$	−11.1567	−0.953183	−0.476591	+	0.879125i	$0.658128\pi$
$138$	0	0
$139$	4.88727	0.414533	0.207266	−	0.978285i	$-0.433543\pi$
$139$	4.88727	0.414533	0.207266	+	0.978285i	$0.433543\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	29.8625	2.47995
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.06457	0.0872128	0.0436064	−	0.999049i	$-0.486115\pi$
$149$	1.06457	0.0872128	0.0436064	+	0.999049i	$0.486115\pi$
$150$	0	0
$151$	−15.7417	−1.28104	−0.640522	−	0.767940i	$-0.721282\pi$
$151$	−15.7417	−1.28104	−0.640522	+	0.767940i	$0.721282\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.9641	−1.36259
$156$	0	0
$157$	−22.0838	−1.76248	−0.881240	−	0.472669i	$-0.843291\pi$
$157$	−22.0838	−1.76248	−0.881240	+	0.472669i	$0.843291\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.15778	0.564112
$162$	0	0
$163$	−17.4918	−1.37006	−0.685032	−	0.728513i	$-0.740212\pi$
$163$	−17.4918	−1.37006	−0.685032	+	0.728513i	$0.740212\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.1578	1.01818	0.509090	−	0.860713i	$-0.329982\pi$
$167$	13.1578	1.01818	0.509090	+	0.860713i	$0.329982\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.34625	−0.178382	−0.0891912	−	0.996015i	$-0.528428\pi$
$173$	−2.34625	−0.178382	−0.0891912	+	0.996015i	$0.528428\pi$
$174$	0	0
$175$	−22.9026	−1.73128
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.8984	1.26305	0.631525	−	0.775356i	$-0.282429\pi$
$179$	16.8984	1.26305	0.631525	+	0.775356i	$0.282429\pi$
$180$	0	0
$181$	4.09738	0.304556	0.152278	−	0.988338i	$-0.451339\pi$
$181$	4.09738	0.304556	0.152278	+	0.988338i	$0.451339\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.58501	−0.190054
$186$	0	0
$187$	−1.49180	−0.109091
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.29809	−0.238642	−0.119321	−	0.992856i	$-0.538072\pi$
$191$	−3.29809	−0.238642	−0.119321	+	0.992856i	$0.538072\pi$
$192$	0	0
$193$	8.29809	0.597310	0.298655	−	0.954361i	$-0.403462\pi$
$193$	8.29809	0.597310	0.298655	+	0.954361i	$0.403462\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.528779	0.0376740	0.0188370	−	0.999823i	$-0.494004\pi$
$197$	0.528779	0.0376740	0.0188370	+	0.999823i	$0.494004\pi$
$198$	0	0
$199$	23.9794	1.69986	0.849928	−	0.526899i	$-0.176645\pi$
$199$	23.9794	1.69986	0.849928	+	0.526899i	$0.176645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−41.4506	−2.90926
$204$	0	0
$205$	6.12080	0.427495
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.76231	0.121901
$210$	0	0
$211$	−24.1208	−1.66054	−0.830272	−	0.557358i	$-0.811815\pi$
$211$	−24.1208	−1.66054	−0.830272	+	0.557358i	$0.811815\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.6925	1.13842
$216$	0	0
$217$	23.5470	1.59847
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.49180	−0.100349
$222$	0	0
$223$	4.71414	0.315682	0.157841	−	0.987465i	$-0.449547\pi$
$223$	4.71414	0.315682	0.157841	+	0.987465i	$0.449547\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.8667	0.721248	0.360624	−	0.932711i	$-0.382564\pi$
$227$	10.8667	0.721248	0.360624	+	0.932711i	$0.382564\pi$
$228$	0	0
$229$	8.51938	0.562977	0.281488	−	0.959565i	$-0.409172\pi$
$229$	8.51938	0.562977	0.281488	+	0.959565i	$0.409172\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.5962	1.08725	0.543626	−	0.839327i	$-0.317051\pi$
$233$	16.5962	1.08725	0.543626	+	0.839327i	$0.317051\pi$
$234$	0	0
$235$	−26.7253	−1.74337
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.2294	−1.04979	−0.524895	−	0.851167i	$-0.675895\pi$
$239$	−16.2294	−1.04979	−0.524895	+	0.851167i	$0.675895\pi$
$240$	0	0
$241$	0.968246	0.0623702	0.0311851	−	0.999514i	$-0.490072\pi$
$241$	0.968246	0.0623702	0.0311851	+	0.999514i	$0.490072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	40.1893	2.56760
$246$	0	0
$247$	1.76231	0.112133
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.1801	1.52624	0.763118	−	0.646259i	$-0.223667\pi$
$251$	24.1801	1.52624	0.763118	+	0.646259i	$0.223667\pi$
$252$	0	0
$253$	1.61676	0.101645
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.3473	0.832583	0.416291	−	0.909231i	$-0.363330\pi$
$257$	13.3473	0.832583	0.416291	+	0.909231i	$0.363330\pi$
$258$	0	0
$259$	3.58812	0.222955
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.23353	−0.199388	−0.0996939	−	0.995018i	$-0.531786\pi$
$263$	−3.23353	−0.199388	−0.0996939	+	0.995018i	$0.531786\pi$
$264$	0	0
$265$	−21.1044	−1.29643
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.44364	−0.331904	−0.165952	−	0.986134i	$-0.553070\pi$
$269$	−5.44364	−0.331904	−0.165952	+	0.986134i	$0.553070\pi$
$270$	0	0
$271$	7.34209	0.446000	0.223000	−	0.974818i	$-0.428415\pi$
$271$	7.34209	0.446000	0.223000	+	0.974818i	$0.428415\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.17313	−0.311951
$276$	0	0
$277$	−26.4999	−1.59222	−0.796111	−	0.605150i	$-0.793113\pi$
$277$	−26.4999	−1.59222	−0.796111	+	0.605150i	$0.793113\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.0122	−1.43245	−0.716225	−	0.697869i	$-0.754132\pi$
$281$	−24.0122	−1.43245	−0.716225	+	0.697869i	$0.754132\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.49597	−0.501501
$288$	0	0
$289$	−14.7745	−0.869091
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.7417	1.27017	0.635083	−	0.772444i	$-0.280966\pi$
$293$	21.7417	1.27017	0.635083	+	0.772444i	$0.280966\pi$
$294$	0	0
$295$	−36.3819	−2.11824
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.61676	0.0934998
$300$	0	0
$301$	−23.1700	−1.33550
$302$	0	0
$303$	0	0
$304$	0	0
$305$	39.0716	2.23723
$306$	0	0
$307$	23.4835	1.34027	0.670136	−	0.742238i	$-0.266236\pi$
$307$	23.4835	1.34027	0.670136	+	0.742238i	$0.266236\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.9794	−0.679291	−0.339645	−	0.940554i	$-0.610307\pi$
$311$	−11.9794	−0.679291	−0.339645	+	0.940554i	$0.610307\pi$
$312$	0	0
$313$	−13.0757	−0.739085	−0.369542	−	0.929214i	$-0.620486\pi$
$313$	−13.0757	−0.739085	−0.369542	+	0.929214i	$0.620486\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.714144	−0.0401103	−0.0200552	−	0.999799i	$-0.506384\pi$
$317$	−0.714144	−0.0401103	−0.0200552	+	0.999799i	$0.506384\pi$
$318$	0	0
$319$	−9.36266	−0.524208
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.62900	−0.146282
$324$	0	0
$325$	−5.17313	−0.286953
$326$	0	0
$327$	0	0
$328$	0	0
$329$	37.0961	2.04517
$330$	0	0
$331$	9.78572	0.537872	0.268936	−	0.963158i	$-0.413328\pi$
$331$	9.78572	0.537872	0.268936	+	0.963158i	$0.413328\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−38.5194	−2.10454
$336$	0	0
$337$	18.2499	0.994137	0.497069	−	0.867711i	$-0.334410\pi$
$337$	18.2499	0.994137	0.497069	+	0.867711i	$0.334410\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.31867	0.288022
$342$	0	0
$343$	−24.7941	−1.33875
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.7170	1.43424	0.717121	−	0.696948i	$-0.245459\pi$
$347$	26.7170	1.43424	0.717121	+	0.696948i	$0.245459\pi$
$348$	0	0
$349$	11.4436	0.612564	0.306282	−	0.951941i	$-0.400915\pi$
$349$	11.4436	0.612564	0.306282	+	0.951941i	$0.400915\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.6813	−0.674959	−0.337480	−	0.941333i	$-0.609574\pi$
$353$	−12.6813	−0.674959	−0.337480	+	0.941333i	$0.609574\pi$
$354$	0	0
$355$	33.2447	1.76445
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.46004	−0.129836	−0.0649180	−	0.997891i	$-0.520679\pi$
$359$	−2.46004	−0.129836	−0.0649180	+	0.997891i	$0.520679\pi$
$360$	0	0
$361$	−15.8943	−0.836541
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−44.1760	−2.31228
$366$	0	0
$367$	−14.3473	−0.748924	−0.374462	−	0.927242i	$-0.622173\pi$
$367$	−14.3473	−0.748924	−0.374462	+	0.927242i	$0.622173\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	29.2939	1.52086
$372$	0	0
$373$	0.725323	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$373$	0.725323	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.36266	−0.482202
$378$	0	0
$379$	25.1484	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$379$	25.1484	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.3491	−1.85735	−0.928676	−	0.370892i	$-0.879052\pi$
$383$	−36.3491	−1.85735	−0.928676	+	0.370892i	$0.879052\pi$
$384$	0	0
$385$	14.1208	0.719662
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.51521	0.381036	0.190518	−	0.981684i	$-0.438983\pi$
$389$	7.51521	0.381036	0.190518	+	0.981684i	$0.438983\pi$
$390$	0	0
$391$	−2.41188	−0.121974
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.48346	−0.376534
$396$	0	0
$397$	37.0081	1.85738	0.928691	−	0.370855i	$-0.120935\pi$
$397$	37.0081	1.85738	0.928691	+	0.370855i	$0.120935\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.9424	−1.64507	−0.822534	−	0.568717i	$-0.807440\pi$
$401$	−32.9424	−1.64507	−0.822534	+	0.568717i	$0.807440\pi$
$402$	0	0
$403$	5.31867	0.264942
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.810466	0.0401733
$408$	0	0
$409$	−14.4999	−0.716972	−0.358486	−	0.933535i	$-0.616707\pi$
$409$	−14.4999	−0.716972	−0.358486	+	0.933535i	$0.616707\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	50.4999	2.48494
$414$	0	0
$415$	2.98359	0.146459
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.7816	0.575566	0.287783	−	0.957696i	$-0.407082\pi$
$419$	11.7816	0.575566	0.287783	+	0.957696i	$0.407082\pi$
$420$	0	0
$421$	30.3379	1.47858	0.739290	−	0.673387i	$-0.235161\pi$
$421$	30.3379	1.47858	0.739290	+	0.673387i	$0.235161\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.71725	0.374342
$426$	0	0
$427$	−54.2333	−2.62453
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.7735	0.807950	0.403975	−	0.914770i	$-0.367628\pi$
$431$	16.7735	0.807950	0.403975	+	0.914770i	$0.367628\pi$
$432$	0	0
$433$	13.7183	0.659260	0.329630	−	0.944110i	$-0.393076\pi$
$433$	13.7183	0.659260	0.329630	+	0.944110i	$0.393076\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.84923	0.136297
$438$	0	0
$439$	−0.0328135	−0.00156610	−0.000783052	−	1.00000i	$-0.500249\pi$
$439$	−0.0328135	−0.00156610	−0.000783052		1.00000i	$0.500249\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.1965	0.817032	0.408516	−	0.912751i	$-0.366046\pi$
$443$	17.1965	0.817032	0.408516	+	0.912751i	$0.366046\pi$
$444$	0	0
$445$	−38.9313	−1.84552
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.8185	0.557751	0.278876	−	0.960327i	$-0.410038\pi$
$449$	11.8185	0.557751	0.278876	+	0.960327i	$0.410038\pi$
$450$	0	0
$451$	−1.91903	−0.0903634
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.1208	0.661993
$456$	0	0
$457$	32.7927	1.53398	0.766990	−	0.641660i	$-0.221754\pi$
$457$	32.7927	1.53398	0.766990	+	0.641660i	$0.221754\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.6883	−0.730679	−0.365339	−	0.930874i	$-0.619047\pi$
$461$	−15.6883	−0.730679	−0.365339	+	0.930874i	$0.619047\pi$
$462$	0	0
$463$	−39.7529	−1.84747	−0.923737	−	0.383027i	$-0.874882\pi$
$463$	−39.7529	−1.84747	−0.923737	+	0.383027i	$0.874882\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.810466	−0.0375039	−0.0187519	−	0.999824i	$-0.505969\pi$
$467$	−0.810466	−0.0375039	−0.0187519	+	0.999824i	$0.505969\pi$
$468$	0	0
$469$	53.4668	2.46887
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.23353	−0.240638
$474$	0	0
$475$	−9.11663	−0.418300
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.7909	−0.675816	−0.337908	−	0.941179i	$-0.609719\pi$
$479$	−14.7909	−0.675816	−0.337908	+	0.941179i	$0.609719\pi$
$480$	0	0
$481$	0.810466	0.0369541
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.1731	−0.461938
$486$	0	0
$487$	22.9229	1.03874	0.519368	−	0.854550i	$-0.326167\pi$
$487$	22.9229	1.03874	0.519368	+	0.854550i	$0.326167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.79929	−0.216589	−0.108294	−	0.994119i	$-0.534539\pi$
$491$	−4.79929	−0.216589	−0.108294	+	0.994119i	$0.534539\pi$
$492$	0	0
$493$	13.9672	0.629050
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−46.1453	−2.06990
$498$	0	0
$499$	−40.1760	−1.79852	−0.899262	−	0.437411i	$-0.855895\pi$
$499$	−40.1760	−1.79852	−0.899262	+	0.437411i	$0.855895\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.3543	−0.862967	−0.431483	−	0.902121i	$-0.642010\pi$
$503$	−19.3543	−0.862967	−0.431483	+	0.902121i	$0.642010\pi$
$504$	0	0
$505$	−45.3460	−2.01787
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.0112	0.532386	0.266193	−	0.963920i	$-0.414234\pi$
$509$	12.0112	0.532386	0.266193	+	0.963920i	$0.414234\pi$
$510$	0	0
$511$	61.3184	2.71257
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−26.9836	−1.18904
$516$	0	0
$517$	8.37907	0.368511
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.60036	0.113924	0.0569618	−	0.998376i	$-0.481859\pi$
$521$	2.60036	0.113924	0.0569618	+	0.998376i	$0.481859\pi$
$522$	0	0
$523$	−31.4095	−1.37344	−0.686721	−	0.726921i	$-0.740950\pi$
$523$	−31.4095	−1.37344	−0.686721	+	0.726921i	$0.740950\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.93437	−0.345627
$528$	0	0
$529$	−20.3861	−0.886351
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.91903	−0.0831222
$534$	0	0
$535$	39.4835	1.70702
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.6004	−0.542736
$540$	0	0
$541$	−34.1606	−1.46868	−0.734340	−	0.678782i	$-0.762508\pi$
$541$	−34.1606	−1.46868	−0.734340	+	0.678782i	$0.762508\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−43.4835	−1.86263
$546$	0	0
$547$	4.97526	0.212727	0.106363	−	0.994327i	$-0.466079\pi$
$547$	4.97526	0.212727	0.106363	+	0.994327i	$0.466079\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.4999	−0.702918
$552$	0	0
$553$	10.3874	0.441717
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.6855	0.961215	0.480608	−	0.876936i	$-0.340416\pi$
$557$	22.6855	0.961215	0.480608	+	0.876936i	$0.340416\pi$
$558$	0	0
$559$	−5.23353	−0.221355
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.28275	0.0962063	0.0481031	−	0.998842i	$-0.484682\pi$
$563$	2.28275	0.0962063	0.0481031	+	0.998842i	$0.484682\pi$
$564$	0	0
$565$	−51.6238	−2.17183
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−44.0245	−1.84560	−0.922801	−	0.385277i	$-0.874106\pi$
$569$	−44.0245	−1.84560	−0.922801	+	0.385277i	$0.874106\pi$
$570$	0	0
$571$	26.9424	1.12751	0.563753	−	0.825943i	$-0.309357\pi$
$571$	26.9424	1.12751	0.563753	+	0.825943i	$0.309357\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.36372	−0.348791
$576$	0	0
$577$	−9.53579	−0.396980	−0.198490	−	0.980103i	$-0.563604\pi$
$577$	−9.53579	−0.396980	−0.198490	+	0.980103i	$0.563604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.14137	−0.171813
$582$	0	0
$583$	6.61676	0.274038
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.0796	1.81936	0.909681	−	0.415308i	$-0.136326\pi$
$587$	44.0796	1.81936	0.909681	+	0.415308i	$0.136326\pi$
$588$	0	0
$589$	9.37312	0.386213
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.5069	−0.595726	−0.297863	−	0.954609i	$-0.596274\pi$
$593$	−14.5069	−0.595726	−0.297863	+	0.954609i	$0.596274\pi$
$594$	0	0
$595$	−21.0654	−0.863595
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.4876	−0.918820	−0.459410	−	0.888224i	$-0.651939\pi$
$599$	−22.4876	−0.918820	−0.459410	+	0.888224i	$0.651939\pi$
$600$	0	0
$601$	−3.77454	−0.153967	−0.0769834	−	0.997032i	$-0.524529\pi$
$601$	−3.77454	−0.153967	−0.0769834	+	0.997032i	$0.524529\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.18953	0.129673
$606$	0	0
$607$	18.2171	0.739410	0.369705	−	0.929149i	$-0.379459\pi$
$607$	18.2171	0.739410	0.369705	+	0.929149i	$0.379459\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.37907	0.338981
$612$	0	0
$613$	2.73756	0.110569	0.0552845	−	0.998471i	$-0.482393\pi$
$613$	2.73756	0.110569	0.0552845	+	0.998471i	$0.482393\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.0716	−1.08986	−0.544930	−	0.838481i	$-0.683444\pi$
$617$	−27.0716	−1.08986	−0.544930	+	0.838481i	$0.683444\pi$
$618$	0	0
$619$	−5.03592	−0.202411	−0.101205	−	0.994866i	$-0.532270\pi$
$619$	−5.03592	−0.202411	−0.101205	+	0.994866i	$0.532270\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	54.0385	2.16501
$624$	0	0
$625$	−24.1044	−0.964176
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.20905	−0.0482080
$630$	0	0
$631$	−25.0192	−0.996001	−0.498000	−	0.867177i	$-0.665932\pi$
$631$	−25.0192	−0.996001	−0.498000	+	0.867177i	$0.665932\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−44.9341	−1.78316
$636$	0	0
$637$	−12.6004	−0.499244
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.2541	0.878984	0.439492	−	0.898246i	$-0.355158\pi$
$641$	22.2541	0.878984	0.439492	+	0.898246i	$0.355158\pi$
$642$	0	0
$643$	−26.1648	−1.03184	−0.515919	−	0.856637i	$-0.672550\pi$
$643$	−26.1648	−1.03184	−0.515919	+	0.856637i	$0.672550\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.0974	−0.790110	−0.395055	−	0.918658i	$-0.629274\pi$
$647$	−20.0974	−0.790110	−0.395055	+	0.918658i	$0.629274\pi$
$648$	0	0
$649$	11.4067	0.447750
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.5030	0.606678	0.303339	−	0.952883i	$-0.401898\pi$
$653$	15.5030	0.606678	0.303339	+	0.952883i	$0.401898\pi$
$654$	0	0
$655$	25.7969	1.00797
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.0164	−1.59777	−0.798886	−	0.601482i	$-0.794577\pi$
$659$	−41.0164	−1.59777	−0.798886	+	0.601482i	$0.794577\pi$
$660$	0	0
$661$	−25.6014	−0.995780	−0.497890	−	0.867240i	$-0.665892\pi$
$661$	−25.6014	−0.995780	−0.497890	+	0.867240i	$0.665892\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.8852	0.965005
$666$	0	0
$667$	−15.1372	−0.586115
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.2499	−0.472903
$672$	0	0
$673$	40.1453	1.54749	0.773743	−	0.633499i	$-0.218382\pi$
$673$	40.1453	1.54749	0.773743	+	0.633499i	$0.218382\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.5222	−1.71113	−0.855564	−	0.517697i	$-0.826789\pi$
$677$	−44.5222	−1.71113	−0.855564	+	0.517697i	$0.826789\pi$
$678$	0	0
$679$	14.1208	0.541906
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.5163	1.12941	0.564704	−	0.825293i	$-0.308990\pi$
$683$	29.5163	1.12941	0.564704	+	0.825293i	$0.308990\pi$
$684$	0	0
$685$	−35.5847	−1.35962
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.61676	0.252079
$690$	0	0
$691$	44.8573	1.70645	0.853226	−	0.521541i	$-0.174643\pi$
$691$	44.8573	1.70645	0.853226	+	0.521541i	$0.174643\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.5881	0.591291
$696$	0	0
$697$	2.86280	0.108436
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.6454	1.27077	0.635385	−	0.772196i	$-0.280842\pi$
$701$	33.6454	1.27077	0.635385	+	0.772196i	$0.280842\pi$
$702$	0	0
$703$	1.42829	0.0538689
$704$	0	0
$705$	0	0
$706$	0	0
$707$	62.9424	2.36719
$708$	0	0
$709$	39.2692	1.47479	0.737393	−	0.675465i	$-0.236057\pi$
$709$	39.2692	1.47479	0.737393	+	0.675465i	$0.236057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.59903	0.322036
$714$	0	0
$715$	3.18953	0.119282
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.8656	−0.703570	−0.351785	−	0.936081i	$-0.614425\pi$
$719$	−18.8656	−0.703570	−0.351785	+	0.936081i	$0.614425\pi$
$720$	0	0
$721$	37.4545	1.39488
$722$	0	0
$723$	0	0
$724$	0	0
$725$	48.4342	1.79880
$726$	0	0
$727$	−43.7376	−1.62214	−0.811068	−	0.584952i	$-0.801113\pi$
$727$	−43.7376	−1.62214	−0.811068	+	0.584952i	$0.801113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.80736	0.288766
$732$	0	0
$733$	−22.6115	−0.835176	−0.417588	−	0.908636i	$-0.637124\pi$
$733$	−22.6115	−0.835176	−0.417588	+	0.908636i	$0.637124\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0768	0.444855
$738$	0	0
$739$	26.3777	0.970321	0.485160	−	0.874425i	$-0.338761\pi$
$739$	26.3777	0.970321	0.485160	+	0.874425i	$0.338761\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.9282	1.83169	0.915843	−	0.401536i	$-0.131524\pi$
$743$	49.9282	1.83169	0.915843	+	0.401536i	$0.131524\pi$
$744$	0	0
$745$	3.39547	0.124401
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−54.8050	−2.00253
$750$	0	0
$751$	46.0890	1.68181	0.840907	−	0.541180i	$-0.182022\pi$
$751$	46.0890	1.68181	0.840907	+	0.541180i	$0.182022\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−50.2088	−1.82728
$756$	0	0
$757$	3.91069	0.142136	0.0710682	−	0.997471i	$-0.477359\pi$
$757$	3.91069	0.142136	0.0710682	+	0.997471i	$0.477359\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.40248	−0.0508400	−0.0254200	−	0.999677i	$-0.508092\pi$
$761$	−1.40248	−0.0508400	−0.0254200	+	0.999677i	$0.508092\pi$
$762$	0	0
$763$	60.3572	2.18508
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.4067	0.411870
$768$	0	0
$769$	−24.1361	−0.870372	−0.435186	−	0.900341i	$-0.643317\pi$
$769$	−24.1361	−0.870372	−0.435186	+	0.900341i	$0.643317\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.9201	0.680508	0.340254	−	0.940334i	$-0.389487\pi$
$773$	18.9201	0.680508	0.340254	+	0.940334i	$0.389487\pi$
$774$	0	0
$775$	−27.5142	−0.988338
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.38191	−0.121169
$780$	0	0
$781$	−10.4231	−0.372966
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−70.4371	−2.51401
$786$	0	0
$787$	17.0318	0.607116	0.303558	−	0.952813i	$-0.401825\pi$
$787$	17.0318	0.607116	0.303558	+	0.952813i	$0.401825\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	71.6563	2.54781
$792$	0	0
$793$	−12.2499	−0.435008
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.9313	1.30817	0.654086	−	0.756420i	$-0.273053\pi$
$797$	36.9313	1.30817	0.654086	+	0.756420i	$0.273053\pi$
$798$	0	0
$799$	−12.4999	−0.442213
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.8503	0.488766
$804$	0	0
$805$	22.8300	0.804651
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13.4611	0.473267	0.236634	−	0.971599i	$-0.423956\pi$
$809$	13.4611	0.473267	0.236634	+	0.971599i	$0.423956\pi$
$810$	0	0
$811$	35.6301	1.25114	0.625570	−	0.780168i	$-0.284866\pi$
$811$	35.6301	1.25114	0.625570	+	0.780168i	$0.284866\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.7907	−1.95426
$816$	0	0
$817$	−9.22307	−0.322674
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.5222	−1.90284	−0.951419	−	0.307898i	$-0.900374\pi$
$821$	−54.5222	−1.90284	−0.951419	+	0.307898i	$0.900374\pi$
$822$	0	0
$823$	−34.1906	−1.19181	−0.595905	−	0.803055i	$-0.703206\pi$
$823$	−34.1906	−1.19181	−0.595905	+	0.803055i	$0.703206\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0122	1.11317	0.556587	−	0.830789i	$-0.312110\pi$
$827$	32.0122	1.11317	0.556587	+	0.830789i	$0.312110\pi$
$828$	0	0
$829$	−35.8584	−1.24541	−0.622706	−	0.782456i	$-0.713967\pi$
$829$	−35.8584	−1.24541	−0.622706	+	0.782456i	$0.713967\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.7972	0.651283
$834$	0	0
$835$	41.9672	1.45233
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.38191	−0.220328	−0.110164	−	0.993913i	$-0.535138\pi$
$839$	−6.38191	−0.220328	−0.110164	+	0.993913i	$0.535138\pi$
$840$	0	0
$841$	58.6594	2.02274
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.18953	0.109723
$846$	0	0
$847$	−4.42723	−0.152121
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.31033	0.0449176
$852$	0	0
$853$	9.22129	0.315731	0.157865	−	0.987461i	$-0.449539\pi$
$853$	9.22129	0.315731	0.157865	+	0.987461i	$0.449539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.7501	−0.606331	−0.303165	−	0.952938i	$-0.598043\pi$
$857$	−17.7501	−0.606331	−0.303165	+	0.952938i	$0.598043\pi$
$858$	0	0
$859$	2.02342	0.0690381	0.0345190	−	0.999404i	$-0.489010\pi$
$859$	2.02342	0.0690381	0.0345190	+	0.999404i	$0.489010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.2007	−0.789761	−0.394881	−	0.918732i	$-0.629214\pi$
$863$	−23.2007	−0.789761	−0.394881	+	0.918732i	$0.629214\pi$
$864$	0	0
$865$	−7.48346	−0.254445
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.34625	0.0795912
$870$	0	0
$871$	12.0768	0.409207
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.44470	−0.0826458
$876$	0	0
$877$	−31.6883	−1.07004	−0.535020	−	0.844840i	$-0.679696\pi$
$877$	−31.6883	−1.07004	−0.535020	+	0.844840i	$0.679696\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.34758	0.180165	0.0900823	−	0.995934i	$-0.471287\pi$
$881$	5.34758	0.180165	0.0900823	+	0.995934i	$0.471287\pi$
$882$	0	0
$883$	−21.1885	−0.713049	−0.356524	−	0.934286i	$-0.616038\pi$
$883$	−21.1885	−0.713049	−0.356524	+	0.934286i	$0.616038\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.93437	−0.266410	−0.133205	−	0.991088i	$-0.542527\pi$
$887$	−7.93437	−0.266410	−0.133205	+	0.991088i	$0.542527\pi$
$888$	0	0
$889$	62.3707	2.09185
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.7665	0.494141
$894$	0	0
$895$	53.8982	1.80162
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−49.7969	−1.66082
$900$	0	0
$901$	−9.87086	−0.328846
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.0687	0.434420
$906$	0	0
$907$	−44.7323	−1.48531	−0.742656	−	0.669673i	$-0.766434\pi$
$907$	−44.7323	−1.48531	−0.742656	+	0.669673i	$0.766434\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.38013	0.310777	0.155389	−	0.987853i	$-0.450337\pi$
$911$	9.38013	0.310777	0.155389	+	0.987853i	$0.450337\pi$
$912$	0	0
$913$	−0.935432	−0.0309583
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−35.8074	−1.18246
$918$	0	0
$919$	−23.3627	−0.770663	−0.385332	−	0.922778i	$-0.625913\pi$
$919$	−23.3627	−0.770663	−0.385332	+	0.922778i	$0.625913\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.4231	−0.343079
$924$	0	0
$925$	−4.19264	−0.137853
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.1840	1.48244	0.741220	−	0.671262i	$-0.234247\pi$
$929$	45.1840	1.48244	0.741220	+	0.671262i	$0.234247\pi$
$930$	0	0
$931$	−22.2057	−0.727761
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.75814	−0.155608
$936$	0	0
$937$	27.1596	0.887264	0.443632	−	0.896209i	$-0.353690\pi$
$937$	27.1596	0.887264	0.443632	+	0.896209i	$0.353690\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	59.0427	1.92474	0.962368	−	0.271750i	$-0.0876023\pi$
$941$	59.0427	1.92474	0.962368	+	0.271750i	$0.0876023\pi$
$942$	0	0
$943$	−3.10261	−0.101035
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.5194	0.601799	0.300900	−	0.953656i	$-0.402713\pi$
$947$	18.5194	0.601799	0.300900	+	0.953656i	$0.402713\pi$
$948$	0	0
$949$	13.8503	0.449599
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.8297	−0.383202	−0.191601	−	0.981473i	$-0.561368\pi$
$953$	−11.8297	−0.383202	−0.191601	+	0.981473i	$0.561368\pi$
$954$	0	0
$955$	−10.5194	−0.340399
$956$	0	0
$957$	0	0
$958$	0	0
$959$	49.3934	1.59499
$960$	0	0
$961$	−2.71176	−0.0874760
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.4671	0.852005
$966$	0	0
$967$	3.84195	0.123549	0.0617744	−	0.998090i	$-0.480324\pi$
$967$	3.84195	0.123549	0.0617744	+	0.998090i	$0.480324\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.1219	1.15921	0.579603	−	0.814899i	$-0.303208\pi$
$971$	36.1219	1.15921	0.579603	+	0.814899i	$0.303208\pi$
$972$	0	0
$973$	−21.6371	−0.693653
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.6178	−0.531651	−0.265826	−	0.964021i	$-0.585645\pi$
$977$	−16.6178	−0.531651	−0.265826	+	0.964021i	$0.585645\pi$
$978$	0	0
$979$	12.2059	0.390104
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.48657	0.206889	0.103445	−	0.994635i	$-0.467014\pi$
$983$	6.48657	0.206889	0.103445	+	0.994635i	$0.467014\pi$
$984$	0	0
$985$	1.68656	0.0537382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.46137	−0.269056
$990$	0	0
$991$	51.7456	1.64375	0.821877	−	0.569665i	$-0.192927\pi$
$991$	51.7456	1.64375	0.821877	+	0.569665i	$0.192927\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	76.4832	2.42468
$996$	0	0
$997$	25.2663	0.800193	0.400096	−	0.916473i	$-0.368977\pi$
$997$	25.2663	0.800193	0.400096	+	0.916473i	$0.368977\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.o.1.3		3
3.2	odd	2		572.2.a.e.1.3	✓	3
12.11	even	2		2288.2.a.w.1.1		3
24.5	odd	2		9152.2.a.cc.1.1		3
24.11	even	2		9152.2.a.bv.1.3		3
33.32	even	2		6292.2.a.p.1.3		3
39.38	odd	2		7436.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.e.1.3	✓	3	3.2	odd	2
2288.2.a.w.1.1		3	12.11	even	2
5148.2.a.o.1.3		3	1.1	even	1	trivial
6292.2.a.p.1.3		3	33.32	even	2
7436.2.a.l.1.3		3	39.38	odd	2
9152.2.a.bv.1.3		3	24.11	even	2
9152.2.a.cc.1.1		3	24.5	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+3.18953 q^{5} -4.42723 q^{7} +O(q^{10})\)	\(q+3.18953 q^{5} -4.42723 q^{7} -1.00000 q^{11} -1.00000 q^{13} +1.49180 q^{17} -1.76231 q^{19} -1.61676 q^{23} +5.17313 q^{25} +9.36266 q^{29} -5.31867 q^{31} -14.1208 q^{35} -0.810466 q^{37} +1.91903 q^{41} +5.23353 q^{43} -8.37907 q^{47} +12.6004 q^{49} -6.61676 q^{53} -3.18953 q^{55} -11.4067 q^{59} +12.2499 q^{61} -3.18953 q^{65} -12.0768 q^{67} +10.4231 q^{71} -13.8503 q^{73} +4.42723 q^{77} -2.34625 q^{79} +0.935432 q^{83} +4.75814 q^{85} -12.2059 q^{89} +4.42723 q^{91} -5.62093 q^{95} -3.18953 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} - 7 q^{7}+O(q^{10}) \) 3 * q + q^5 - 7 * q^7	\( 3 q + q^{5} - 7 q^{7} - 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 10 q^{23} + 10 q^{25} + 14 q^{29} - 23 q^{31} + 6 q^{35} - 11 q^{37} + q^{41} - 14 q^{43} - 8 q^{47} + 26 q^{49} - 5 q^{53} - q^{55} + q^{59} + 4 q^{61} - q^{65} - 9 q^{67} - 7 q^{71} + 3 q^{73} + 7 q^{77} + 4 q^{79} - 5 q^{83} - 20 q^{85} - 25 q^{89} + 7 q^{91} - 34 q^{95} - q^{97}+O(q^{100}) \) 3 * q + q^5 - 7 * q^7 - 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 10 * q^23 + 10 * q^25 + 14 * q^29 - 23 * q^31 + 6 * q^35 - 11 * q^37 + q^41 - 14 * q^43 - 8 * q^47 + 26 * q^49 - 5 * q^53 - q^55 + q^59 + 4 * q^61 - q^65 - 9 * q^67 - 7 * q^71 + 3 * q^73 + 7 * q^77 + 4 * q^79 - 5 * q^83 - 20 * q^85 - 25 * q^89 + 7 * q^91 - 34 * q^95 - q^97

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