LMFDB - Embedded newform 5625.2.a.s.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-5,0,11,0,0,10,-15,0,0,-1,0,10,8,0,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.499011$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.500989 q^{2} -1.74901 q^{4} +0.0237879 q^{7} +1.87821 q^{8} -3.58329 q^{11} +3.77587 q^{13} -0.0119175 q^{14} +2.55705 q^{16} +3.62303 q^{17} -2.43084 q^{19} +1.79519 q^{22} -1.71538 q^{23} -1.89167 q^{26} -0.0416052 q^{28} -3.85734 q^{29} -6.00979 q^{31} -5.03748 q^{32} -1.81510 q^{34} -0.369309 q^{37} +1.21782 q^{38} +7.80900 q^{41} +0.174574 q^{43} +6.26720 q^{44} +0.859388 q^{46} -7.81082 q^{47} -6.99943 q^{49} -6.60403 q^{52} +8.97184 q^{53} +0.0446787 q^{56} +1.93249 q^{58} +4.45536 q^{59} +9.21403 q^{61} +3.01084 q^{62} -2.59038 q^{64} -4.47385 q^{67} -6.33671 q^{68} +9.69458 q^{71} -3.95387 q^{73} +0.185020 q^{74} +4.25156 q^{76} -0.0852388 q^{77} -9.68349 q^{79} -3.91223 q^{82} +8.95717 q^{83} -0.0874600 q^{86} -6.73018 q^{88} +17.0151 q^{89} +0.0898200 q^{91} +3.00022 q^{92} +3.91314 q^{94} +2.76438 q^{97} +3.50664 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34}+ \cdots + 30 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * q^98

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 $

$2$	−0.500989	−0.354253	−0.177126	−	0.984188i	$-0.556680\pi$
$2$	−0.500989	−0.354253	−0.177126	+	0.984188i	$0.556680\pi$
$3$	0	0
$3$	0	0
$4$	−1.74901	−0.874505
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0237879	0.00899097	0.00449549	−	0.999990i	$-0.498569\pi$
$7$	0.0237879	0.00899097	0.00449549	+	0.999990i	$0.498569\pi$
$8$	1.87821	0.664049
$9$	0	0
$10$	0	0
$11$	−3.58329	−1.08040	−0.540201	−	0.841536i	$-0.681652\pi$
$11$	−3.58329	−1.08040	−0.540201	+	0.841536i	$0.681652\pi$
$12$	0	0
$13$	3.77587	1.04724	0.523619	−	0.851952i	$-0.324582\pi$
$13$	3.77587	1.04724	0.523619	+	0.851952i	$0.324582\pi$
$14$	−0.0119175	−0.00318508
$15$	0	0
$16$	2.55705	0.639264
$17$	3.62303	0.878713	0.439357	−	0.898313i	$-0.355206\pi$
$17$	3.62303	0.878713	0.439357	+	0.898313i	$0.355206\pi$
$18$	0	0
$19$	−2.43084	−0.557672	−0.278836	−	0.960339i	$-0.589949\pi$
$19$	−2.43084	−0.557672	−0.278836	+	0.960339i	$0.589949\pi$
$20$	0	0
$21$	0	0
$22$	1.79519	0.382736
$23$	−1.71538	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$23$	−1.71538	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$24$	0	0
$25$	0	0
$26$	−1.89167	−0.370987
$27$	0	0
$28$	−0.0416052	−0.00786265
$29$	−3.85734	−0.716290	−0.358145	−	0.933666i	$-0.616591\pi$
$29$	−3.85734	−0.716290	−0.358145	+	0.933666i	$0.616591\pi$
$30$	0	0
$31$	−6.00979	−1.07939	−0.539695	−	0.841861i	$-0.681460\pi$
$31$	−6.00979	−1.07939	−0.539695	+	0.841861i	$0.681460\pi$
$32$	−5.03748	−0.890510
$33$	0	0
$34$	−1.81510	−0.311287
$35$	0	0
$36$	0	0
$37$	−0.369309	−0.0607139	−0.0303570	−	0.999539i	$-0.509664\pi$
$37$	−0.369309	−0.0607139	−0.0303570	+	0.999539i	$0.509664\pi$
$38$	1.21782	0.197557
$39$	0	0
$40$	0	0
$41$	7.80900	1.21956	0.609780	−	0.792570i	$-0.291258\pi$
$41$	7.80900	1.21956	0.609780	+	0.792570i	$0.291258\pi$
$42$	0	0
$43$	0.174574	0.0266224	0.0133112	−	0.999911i	$-0.495763\pi$
$43$	0.174574	0.0266224	0.0133112	+	0.999911i	$0.495763\pi$
$44$	6.26720	0.944817
$45$	0	0
$46$	0.859388	0.126710
$47$	−7.81082	−1.13932	−0.569662	−	0.821879i	$-0.692926\pi$
$47$	−7.81082	−1.13932	−0.569662	+	0.821879i	$0.692926\pi$
$48$	0	0
$49$	−6.99943	−0.999919
$50$	0	0
$51$	0	0
$52$	−6.60403	−0.915815
$53$	8.97184	1.23238	0.616189	−	0.787599i	$-0.288676\pi$
$53$	8.97184	1.23238	0.616189	+	0.787599i	$0.288676\pi$
$54$	0	0
$55$	0	0
$56$	0.0446787	0.00597045
$57$	0	0
$58$	1.93249	0.253748
$59$	4.45536	0.580038	0.290019	−	0.957021i	$-0.406338\pi$
$59$	4.45536	0.580038	0.290019	+	0.957021i	$0.406338\pi$
$60$	0	0
$61$	9.21403	1.17974	0.589868	−	0.807500i	$-0.299180\pi$
$61$	9.21403	1.17974	0.589868	+	0.807500i	$0.299180\pi$
$62$	3.01084	0.382377
$63$	0	0
$64$	−2.59038	−0.323798
$65$	0	0
$66$	0	0
$67$	−4.47385	−0.546568	−0.273284	−	0.961933i	$-0.588110\pi$
$67$	−4.47385	−0.546568	−0.273284	+	0.961933i	$0.588110\pi$
$68$	−6.33671	−0.768439
$69$	0	0
$70$	0	0
$71$	9.69458	1.15054	0.575268	−	0.817965i	$-0.304898\pi$
$71$	9.69458	1.15054	0.575268	+	0.817965i	$0.304898\pi$
$72$	0	0
$73$	−3.95387	−0.462765	−0.231383	−	0.972863i	$-0.574325\pi$
$73$	−3.95387	−0.462765	−0.231383	+	0.972863i	$0.574325\pi$
$74$	0.185020	0.0215081
$75$	0	0
$76$	4.25156	0.487687
$77$	−0.0852388	−0.00971386
$78$	0	0
$79$	−9.68349	−1.08948	−0.544739	−	0.838606i	$-0.683371\pi$
$79$	−9.68349	−1.08948	−0.544739	+	0.838606i	$0.683371\pi$
$80$	0	0
$81$	0	0
$82$	−3.91223	−0.432033
$83$	8.95717	0.983177	0.491589	−	0.870828i	$-0.336416\pi$
$83$	8.95717	0.983177	0.491589	+	0.870828i	$0.336416\pi$
$84$	0	0
$85$	0	0
$86$	−0.0874600	−0.00943105
$87$	0	0
$88$	−6.73018	−0.717440
$89$	17.0151	1.80359	0.901797	−	0.432161i	$-0.142249\pi$
$89$	17.0151	1.80359	0.901797	+	0.432161i	$0.142249\pi$
$90$	0	0
$91$	0.0898200	0.00941569
$92$	3.00022	0.312795
$93$	0	0
$94$	3.91314	0.403609
$95$	0	0
$96$	0	0
$97$	2.76438	0.280680	0.140340	−	0.990103i	$-0.455180\pi$
$97$	2.76438	0.280680	0.140340	+	0.990103i	$0.455180\pi$
$98$	3.50664	0.354224
$99$	0	0
$100$	0	0
$101$	−10.3526	−1.03012	−0.515062	−	0.857153i	$-0.672231\pi$
$101$	−10.3526	−1.03012	−0.515062	+	0.857153i	$0.672231\pi$
$102$	0	0
$103$	18.2913	1.80230	0.901150	−	0.433508i	$-0.142724\pi$
$103$	18.2913	1.80230	0.901150	+	0.433508i	$0.142724\pi$
$104$	7.09189	0.695417
$105$	0	0
$106$	−4.49480	−0.436573
$107$	−15.8786	−1.53504	−0.767522	−	0.641023i	$-0.778510\pi$
$107$	−15.8786	−1.53504	−0.767522	+	0.641023i	$0.778510\pi$
$108$	0	0
$109$	5.96109	0.570969	0.285484	−	0.958383i	$-0.407845\pi$
$109$	5.96109	0.570969	0.285484	+	0.958383i	$0.407845\pi$
$110$	0	0
$111$	0	0
$112$	0.0608269	0.00574760
$113$	−2.42830	−0.228435	−0.114218	−	0.993456i	$-0.536436\pi$
$113$	−2.42830	−0.228435	−0.114218	+	0.993456i	$0.536436\pi$
$114$	0	0
$115$	0	0
$116$	6.74652	0.626399
$117$	0	0
$118$	−2.23209	−0.205480
$119$	0.0861842	0.00790049
$120$	0	0
$121$	1.83995	0.167268
$122$	−4.61613	−0.417925
$123$	0	0
$124$	10.5112	0.943932
$125$	0	0
$126$	0	0
$127$	−17.1073	−1.51803	−0.759013	−	0.651075i	$-0.774318\pi$
$127$	−17.1073	−1.51803	−0.759013	+	0.651075i	$0.774318\pi$
$128$	11.3727	1.00522
$129$	0	0
$130$	0	0
$131$	−4.78955	−0.418464	−0.209232	−	0.977866i	$-0.567096\pi$
$131$	−4.78955	−0.418464	−0.209232	+	0.977866i	$0.567096\pi$
$132$	0	0
$133$	−0.0578245	−0.00501402
$134$	2.24135	0.193623
$135$	0	0
$136$	6.80482	0.583509
$137$	−12.8104	−1.09446	−0.547231	−	0.836982i	$-0.684318\pi$
$137$	−12.8104	−1.09446	−0.547231	+	0.836982i	$0.684318\pi$
$138$	0	0
$139$	−7.94020	−0.673479	−0.336739	−	0.941598i	$-0.609324\pi$
$139$	−7.94020	−0.673479	−0.336739	+	0.941598i	$0.609324\pi$
$140$	0	0
$141$	0	0
$142$	−4.85688	−0.407581
$143$	−13.5300	−1.13144
$144$	0	0
$145$	0	0
$146$	1.98085	0.163936
$147$	0	0
$148$	0.645924	0.0530946
$149$	5.62724	0.461002	0.230501	−	0.973072i	$-0.425964\pi$
$149$	5.62724	0.461002	0.230501	+	0.973072i	$0.425964\pi$
$150$	0	0
$151$	−7.36960	−0.599730	−0.299865	−	0.953982i	$-0.596942\pi$
$151$	−7.36960	−0.599730	−0.299865	+	0.953982i	$0.596942\pi$
$152$	−4.56563	−0.370322
$153$	0	0
$154$	0.0427037	0.00344116
$155$	0	0
$156$	0	0
$157$	8.63091	0.688822	0.344411	−	0.938819i	$-0.388079\pi$
$157$	8.63091	0.688822	0.344411	+	0.938819i	$0.388079\pi$
$158$	4.85133	0.385951
$159$	0	0
$160$	0	0
$161$	−0.0408053	−0.00321591
$162$	0	0
$163$	4.74964	0.372020	0.186010	−	0.982548i	$-0.440444\pi$
$163$	4.74964	0.372020	0.186010	+	0.982548i	$0.440444\pi$
$164$	−13.6580	−1.06651
$165$	0	0
$166$	−4.48745	−0.348293
$167$	−18.5967	−1.43906	−0.719528	−	0.694463i	$-0.755642\pi$
$167$	−18.5967	−1.43906	−0.719528	+	0.694463i	$0.755642\pi$
$168$	0	0
$169$	1.25720	0.0967079
$170$	0	0
$171$	0	0
$172$	−0.305332	−0.0232814
$173$	−15.7957	−1.20092	−0.600462	−	0.799653i	$-0.705017\pi$
$173$	−15.7957	−1.20092	−0.600462	+	0.799653i	$0.705017\pi$
$174$	0	0
$175$	0	0
$176$	−9.16266	−0.690661
$177$	0	0
$178$	−8.52437	−0.638928
$179$	−9.36946	−0.700306	−0.350153	−	0.936692i	$-0.613870\pi$
$179$	−9.36946	−0.700306	−0.350153	+	0.936692i	$0.613870\pi$
$180$	0	0
$181$	−5.84630	−0.434552	−0.217276	−	0.976110i	$-0.569717\pi$
$181$	−5.84630	−0.434552	−0.217276	+	0.976110i	$0.569717\pi$
$182$	−0.0449988	−0.00333554
$183$	0	0
$184$	−3.22186	−0.237518
$185$	0	0
$186$	0	0
$187$	−12.9824	−0.949364
$188$	13.6612	0.996345
$189$	0	0
$190$	0	0
$191$	21.8947	1.58425	0.792123	−	0.610362i	$-0.208976\pi$
$191$	21.8947	1.58425	0.792123	+	0.610362i	$0.208976\pi$
$192$	0	0
$193$	−25.2541	−1.81783	−0.908916	−	0.416980i	$-0.863089\pi$
$193$	−25.2541	−1.81783	−0.908916	+	0.416980i	$0.863089\pi$
$194$	−1.38492	−0.0994317
$195$	0	0
$196$	12.2421	0.874434
$197$	−14.4538	−1.02979	−0.514897	−	0.857252i	$-0.672170\pi$
$197$	−14.4538	−1.02979	−0.514897	+	0.857252i	$0.672170\pi$
$198$	0	0
$199$	−3.77734	−0.267768	−0.133884	−	0.990997i	$-0.542745\pi$
$199$	−3.77734	−0.267768	−0.133884	+	0.990997i	$0.542745\pi$
$200$	0	0
$201$	0	0
$202$	5.18655	0.364924
$203$	−0.0917579	−0.00644014
$204$	0	0
$205$	0	0
$206$	−9.16377	−0.638470
$207$	0	0
$208$	9.65511	0.669461
$209$	8.71039	0.602510
$210$	0	0
$211$	−18.9006	−1.30117	−0.650586	−	0.759432i	$-0.725477\pi$
$211$	−18.9006	−1.30117	−0.650586	+	0.759432i	$0.725477\pi$
$212$	−15.6918	−1.07772
$213$	0	0
$214$	7.95502	0.543794
$215$	0	0
$216$	0	0
$217$	−0.142960	−0.00970477
$218$	−2.98644	−0.202267
$219$	0	0
$220$	0	0
$221$	13.6801	0.920222
$222$	0	0
$223$	−11.3556	−0.760426	−0.380213	−	0.924899i	$-0.624149\pi$
$223$	−11.3556	−0.760426	−0.380213	+	0.924899i	$0.624149\pi$
$224$	−0.119831	−0.00800655
$225$	0	0
$226$	1.21655	0.0809239
$227$	−11.1977	−0.743216	−0.371608	−	0.928390i	$-0.621194\pi$
$227$	−11.1977	−0.743216	−0.371608	+	0.928390i	$0.621194\pi$
$228$	0	0
$229$	9.64969	0.637669	0.318835	−	0.947810i	$-0.396709\pi$
$229$	9.64969	0.637669	0.318835	+	0.947810i	$0.396709\pi$
$230$	0	0
$231$	0	0
$232$	−7.24491	−0.475651
$233$	15.5261	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$233$	15.5261	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$234$	0	0
$235$	0	0
$236$	−7.79246	−0.507246
$237$	0	0
$238$	−0.0431773	−0.00279877
$239$	−14.3342	−0.927203	−0.463601	−	0.886044i	$-0.653443\pi$
$239$	−14.3342	−0.927203	−0.463601	+	0.886044i	$0.653443\pi$
$240$	0	0
$241$	25.3114	1.63045	0.815224	−	0.579145i	$-0.196614\pi$
$241$	25.3114	1.63045	0.815224	+	0.579145i	$0.196614\pi$
$242$	−0.921794	−0.0592552
$243$	0	0
$244$	−16.1154	−1.03168
$245$	0	0
$246$	0	0
$247$	−9.17853	−0.584016
$248$	−11.2877	−0.716768
$249$	0	0
$250$	0	0
$251$	−4.23698	−0.267436	−0.133718	−	0.991019i	$-0.542692\pi$
$251$	−4.23698	−0.267436	−0.133718	+	0.991019i	$0.542692\pi$
$252$	0	0
$253$	6.14671	0.386440
$254$	8.57057	0.537766
$255$	0	0
$256$	−0.516849	−0.0323031
$257$	20.4007	1.27256	0.636281	−	0.771458i	$-0.280472\pi$
$257$	20.4007	1.27256	0.636281	+	0.771458i	$0.280472\pi$
$258$	0	0
$259$	−0.00878507	−0.000545877 0
$260$	0	0
$261$	0	0
$262$	2.39951	0.148242
$263$	−28.6678	−1.76773	−0.883865	−	0.467741i	$-0.845068\pi$
$263$	−28.6678	−1.76773	−0.883865	+	0.467741i	$0.845068\pi$
$264$	0	0
$265$	0	0
$266$	0.0289694	0.00177623
$267$	0	0
$268$	7.82481	0.477977
$269$	−7.94652	−0.484508	−0.242254	−	0.970213i	$-0.577887\pi$
$269$	−7.94652	−0.484508	−0.242254	+	0.970213i	$0.577887\pi$
$270$	0	0
$271$	9.55487	0.580417	0.290208	−	0.956963i	$-0.406275\pi$
$271$	9.55487	0.580417	0.290208	+	0.956963i	$0.406275\pi$
$272$	9.26428	0.561729
$273$	0	0
$274$	6.41785	0.387717
$275$	0	0
$276$	0	0
$277$	17.1904	1.03287	0.516436	−	0.856326i	$-0.327258\pi$
$277$	17.1904	1.03287	0.516436	+	0.856326i	$0.327258\pi$
$278$	3.97795	0.238582
$279$	0	0
$280$	0	0
$281$	−27.2182	−1.62370	−0.811851	−	0.583864i	$-0.801540\pi$
$281$	−27.2182	−1.62370	−0.811851	+	0.583864i	$0.801540\pi$
$282$	0	0
$283$	−3.47901	−0.206806	−0.103403	−	0.994640i	$-0.532973\pi$
$283$	−3.47901	−0.206806	−0.103403	+	0.994640i	$0.532973\pi$
$284$	−16.9559	−1.00615
$285$	0	0
$286$	6.77840	0.400815
$287$	0.185760	0.0109650
$288$	0	0
$289$	−3.87366	−0.227863
$290$	0	0
$291$	0	0
$292$	6.91535	0.404690
$293$	−23.4941	−1.37254	−0.686271	−	0.727346i	$-0.740754\pi$
$293$	−23.4941	−1.37254	−0.686271	+	0.727346i	$0.740754\pi$
$294$	0	0
$295$	0	0
$296$	−0.693640	−0.0403170
$297$	0	0
$298$	−2.81919	−0.163311
$299$	−6.47706	−0.374578
$300$	0	0
$301$	0.00415276	0.000239361 0
$302$	3.69209	0.212456
$303$	0	0
$304$	−6.21578	−0.356500
$305$	0	0
$306$	0	0
$307$	1.11253	0.0634952	0.0317476	−	0.999496i	$-0.489893\pi$
$307$	1.11253	0.0634952	0.0317476	+	0.999496i	$0.489893\pi$
$308$	0.149083	0.00849482
$309$	0	0
$310$	0	0
$311$	14.7529	0.836561	0.418280	−	0.908318i	$-0.362633\pi$
$311$	14.7529	0.836561	0.418280	+	0.908318i	$0.362633\pi$
$312$	0	0
$313$	4.94834	0.279697	0.139848	−	0.990173i	$-0.455339\pi$
$313$	4.94834	0.279697	0.139848	+	0.990173i	$0.455339\pi$
$314$	−4.32400	−0.244017
$315$	0	0
$316$	16.9365	0.952754
$317$	−22.6855	−1.27414	−0.637072	−	0.770804i	$-0.719855\pi$
$317$	−22.6855	−1.27414	−0.637072	+	0.770804i	$0.719855\pi$
$318$	0	0
$319$	13.8220	0.773881
$320$	0	0
$321$	0	0
$322$	0.0204430	0.00113925
$323$	−8.80699	−0.490034
$324$	0	0
$325$	0	0
$326$	−2.37952	−0.131789
$327$	0	0
$328$	14.6670	0.809848
$329$	−0.185803	−0.0102436
$330$	0	0
$331$	1.73258	0.0952312	0.0476156	−	0.998866i	$-0.484838\pi$
$331$	1.73258	0.0952312	0.0476156	+	0.998866i	$0.484838\pi$
$332$	−15.6662	−0.859793
$333$	0	0
$334$	9.31676	0.509790
$335$	0	0
$336$	0	0
$337$	13.9811	0.761596	0.380798	−	0.924658i	$-0.375649\pi$
$337$	13.9811	0.761596	0.380798	+	0.924658i	$0.375649\pi$
$338$	−0.629845	−0.0342590
$339$	0	0
$340$	0	0
$341$	21.5348	1.16617
$342$	0	0
$343$	−0.333017	−0.0179812
$344$	0.327888	0.0176785
$345$	0	0
$346$	7.91347	0.425431
$347$	−5.08173	−0.272802	−0.136401	−	0.990654i	$-0.543553\pi$
$347$	−5.08173	−0.272802	−0.136401	+	0.990654i	$0.543553\pi$
$348$	0	0
$349$	−8.88643	−0.475680	−0.237840	−	0.971304i	$-0.576439\pi$
$349$	−8.88643	−0.475680	−0.237840	+	0.971304i	$0.576439\pi$
$350$	0	0
$351$	0	0
$352$	18.0508	0.962108
$353$	−9.41440	−0.501078	−0.250539	−	0.968107i	$-0.580608\pi$
$353$	−9.41440	−0.501078	−0.250539	+	0.968107i	$0.580608\pi$
$354$	0	0
$355$	0	0
$356$	−29.7595	−1.57725
$357$	0	0
$358$	4.69400	0.248086
$359$	−9.11498	−0.481070	−0.240535	−	0.970640i	$-0.577323\pi$
$359$	−9.11498	−0.481070	−0.240535	+	0.970640i	$0.577323\pi$
$360$	0	0
$361$	−13.0910	−0.689002
$362$	2.92893	0.153941
$363$	0	0
$364$	−0.157096	−0.00823407
$365$	0	0
$366$	0	0
$367$	−17.8784	−0.933246	−0.466623	−	0.884456i	$-0.654529\pi$
$367$	−17.8784	−0.933246	−0.466623	+	0.884456i	$0.654529\pi$
$368$	−4.38633	−0.228653
$369$	0	0
$370$	0	0
$371$	0.213421	0.0110803
$372$	0	0
$373$	−3.07395	−0.159163	−0.0795814	−	0.996828i	$-0.525358\pi$
$373$	−3.07395	−0.159163	−0.0795814	+	0.996828i	$0.525358\pi$
$374$	6.50402	0.336315
$375$	0	0
$376$	−14.6704	−0.756567
$377$	−14.5648	−0.750126
$378$	0	0
$379$	25.5866	1.31430	0.657149	−	0.753761i	$-0.271762\pi$
$379$	25.5866	1.31430	0.657149	+	0.753761i	$0.271762\pi$
$380$	0	0
$381$	0	0
$382$	−10.9690	−0.561224
$383$	2.22529	0.113707	0.0568535	−	0.998383i	$-0.481893\pi$
$383$	2.22529	0.113707	0.0568535	+	0.998383i	$0.481893\pi$
$384$	0	0
$385$	0	0
$386$	12.6520	0.643972
$387$	0	0
$388$	−4.83492	−0.245456
$389$	−4.64413	−0.235467	−0.117733	−	0.993045i	$-0.537563\pi$
$389$	−4.64413	−0.235467	−0.117733	+	0.993045i	$0.537563\pi$
$390$	0	0
$391$	−6.21488	−0.314300
$392$	−13.1464	−0.663995
$393$	0	0
$394$	7.24122	0.364807
$395$	0	0
$396$	0	0
$397$	−37.9452	−1.90442	−0.952208	−	0.305452i	$-0.901193\pi$
$397$	−37.9452	−1.90442	−0.952208	+	0.305452i	$0.901193\pi$
$398$	1.89241	0.0948577
$399$	0	0
$400$	0	0
$401$	16.7187	0.834890	0.417445	−	0.908702i	$-0.362926\pi$
$401$	16.7187	0.834890	0.417445	+	0.908702i	$0.362926\pi$
$402$	0	0
$403$	−22.6922	−1.13038
$404$	18.1068	0.900848
$405$	0	0
$406$	0.0459697	0.00228144
$407$	1.32334	0.0655955
$408$	0	0
$409$	0.473132	0.0233949	0.0116974	−	0.999932i	$-0.496277\pi$
$409$	0.473132	0.0233949	0.0116974	+	0.999932i	$0.496277\pi$
$410$	0	0
$411$	0	0
$412$	−31.9917	−1.57612
$413$	0.105983	0.00521511
$414$	0	0
$415$	0	0
$416$	−19.0209	−0.932576
$417$	0	0
$418$	−4.36381	−0.213441
$419$	−24.2348	−1.18395	−0.591975	−	0.805957i	$-0.701651\pi$
$419$	−24.2348	−1.18395	−0.591975	+	0.805957i	$0.701651\pi$
$420$	0	0
$421$	−31.3482	−1.52782	−0.763908	−	0.645325i	$-0.776722\pi$
$421$	−31.3482	−1.52782	−0.763908	+	0.645325i	$0.776722\pi$
$422$	9.46901	0.460944
$423$	0	0
$424$	16.8510	0.818359
$425$	0	0
$426$	0	0
$427$	0.219182	0.0106070
$428$	27.7719	1.34240
$429$	0	0
$430$	0	0
$431$	2.00066	0.0963686	0.0481843	−	0.998838i	$-0.484657\pi$
$431$	2.00066	0.0963686	0.0481843	+	0.998838i	$0.484657\pi$
$432$	0	0
$433$	7.07253	0.339884	0.169942	−	0.985454i	$-0.445642\pi$
$433$	7.07253	0.339884	0.169942	+	0.985454i	$0.445642\pi$
$434$	0.0716215	0.00343794
$435$	0	0
$436$	−10.4260	−0.499315
$437$	4.16981	0.199469
$438$	0	0
$439$	12.9620	0.618644	0.309322	−	0.950957i	$-0.399898\pi$
$439$	12.9620	0.618644	0.309322	+	0.950957i	$0.399898\pi$
$440$	0	0
$441$	0	0
$442$	−6.85358	−0.325991
$443$	−21.8687	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$443$	−21.8687	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$444$	0	0
$445$	0	0
$446$	5.68902	0.269383
$447$	0	0
$448$	−0.0616197	−0.00291126
$449$	−0.399626	−0.0188595	−0.00942976	−	0.999956i	$-0.503002\pi$
$449$	−0.399626	−0.0188595	−0.00942976	+	0.999956i	$0.503002\pi$
$450$	0	0
$451$	−27.9819	−1.31762
$452$	4.24712	0.199768
$453$	0	0
$454$	5.60992	0.263287
$455$	0	0
$456$	0	0
$457$	−35.2247	−1.64774	−0.823872	−	0.566777i	$-0.808190\pi$
$457$	−35.2247	−1.64774	−0.823872	+	0.566777i	$0.808190\pi$
$458$	−4.83439	−0.225896
$459$	0	0
$460$	0	0
$461$	15.9615	0.743402	0.371701	−	0.928353i	$-0.378775\pi$
$461$	15.9615	0.743402	0.371701	+	0.928353i	$0.378775\pi$
$462$	0	0
$463$	−30.4831	−1.41667	−0.708336	−	0.705875i	$-0.750554\pi$
$463$	−30.4831	−1.41667	−0.708336	+	0.705875i	$0.750554\pi$
$464$	−9.86342	−0.457898
$465$	0	0
$466$	−7.77839	−0.360327
$467$	−6.64657	−0.307567	−0.153783	−	0.988105i	$-0.549146\pi$
$467$	−6.64657	−0.307567	−0.153783	+	0.988105i	$0.549146\pi$
$468$	0	0
$469$	−0.106424	−0.00491418
$470$	0	0
$471$	0	0
$472$	8.36811	0.385174
$473$	−0.625551	−0.0287628
$474$	0	0
$475$	0	0
$476$	−0.150737	−0.00690902
$477$	0	0
$478$	7.18129	0.328464
$479$	−22.2408	−1.01621	−0.508104	−	0.861296i	$-0.669653\pi$
$479$	−22.2408	−1.01621	−0.508104	+	0.861296i	$0.669653\pi$
$480$	0	0
$481$	−1.39446	−0.0635820
$482$	−12.6807	−0.577591
$483$	0	0
$484$	−3.21809	−0.146277
$485$	0	0
$486$	0	0
$487$	10.1851	0.461531	0.230765	−	0.973009i	$-0.425877\pi$
$487$	10.1851	0.461531	0.230765	+	0.973009i	$0.425877\pi$
$488$	17.3059	0.783402
$489$	0	0
$490$	0	0
$491$	−6.64004	−0.299661	−0.149830	−	0.988712i	$-0.547873\pi$
$491$	−6.64004	−0.299661	−0.149830	+	0.988712i	$0.547873\pi$
$492$	0	0
$493$	−13.9752	−0.629413
$494$	4.59834	0.206889
$495$	0	0
$496$	−15.3674	−0.690015
$497$	0.230614	0.0103444
$498$	0	0
$499$	1.08397	0.0485253	0.0242626	−	0.999706i	$-0.492276\pi$
$499$	1.08397	0.0485253	0.0242626	+	0.999706i	$0.492276\pi$
$500$	0	0
$501$	0	0
$502$	2.12268	0.0947399
$503$	1.98603	0.0885525	0.0442762	−	0.999019i	$-0.485902\pi$
$503$	1.98603	0.0885525	0.0442762	+	0.999019i	$0.485902\pi$
$504$	0	0
$505$	0	0
$506$	−3.07944	−0.136898
$507$	0	0
$508$	29.9208	1.32752
$509$	−2.30915	−0.102351	−0.0511757	−	0.998690i	$-0.516297\pi$
$509$	−2.30915	−0.102351	−0.0511757	+	0.998690i	$0.516297\pi$
$510$	0	0
$511$	−0.0940541	−0.00416071
$512$	−22.4865	−0.993773
$513$	0	0
$514$	−10.2205	−0.450809
$515$	0	0
$516$	0	0
$517$	27.9884	1.23093
$518$	0.00440122	0.000193379 0
$519$	0	0
$520$	0	0
$521$	11.0447	0.483879	0.241940	−	0.970291i	$-0.422216\pi$
$521$	11.0447	0.483879	0.241940	+	0.970291i	$0.422216\pi$
$522$	0	0
$523$	26.0025	1.13701	0.568505	−	0.822680i	$-0.307522\pi$
$523$	26.0025	1.13701	0.568505	+	0.822680i	$0.307522\pi$
$524$	8.37696	0.365949
$525$	0	0
$526$	14.3623	0.626224
$527$	−21.7736	−0.948475
$528$	0	0
$529$	−20.0575	−0.872064
$530$	0	0
$531$	0	0
$532$	0.101136	0.00438478
$533$	29.4858	1.27717
$534$	0	0
$535$	0	0
$536$	−8.40286	−0.362948
$537$	0	0
$538$	3.98112	0.171638
$539$	25.0810	1.08031
$540$	0	0
$541$	−0.0182391	−0.000784162 0	−0.000392081	−	1.00000i	$-0.500125\pi$
$541$	−0.0182391	−0.000784162 0	−0.000392081		1.00000i	$0.500125\pi$
$542$	−4.78689	−0.205614
$543$	0	0
$544$	−18.2510	−0.782503
$545$	0	0
$546$	0	0
$547$	−9.96048	−0.425879	−0.212940	−	0.977065i	$-0.568304\pi$
$547$	−9.96048	−0.425879	−0.212940	+	0.977065i	$0.568304\pi$
$548$	22.4054	0.957113
$549$	0	0
$550$	0	0
$551$	9.37656	0.399455
$552$	0	0
$553$	−0.230350	−0.00979547
$554$	−8.61222	−0.365898
$555$	0	0
$556$	13.8875	0.588960
$557$	34.9291	1.47999	0.739996	−	0.672611i	$-0.234827\pi$
$557$	34.9291	1.47999	0.739996	+	0.672611i	$0.234827\pi$
$558$	0	0
$559$	0.659171	0.0278800
$560$	0	0
$561$	0	0
$562$	13.6360	0.575201
$563$	−15.7939	−0.665634	−0.332817	−	0.942991i	$-0.607999\pi$
$563$	−15.7939	−0.665634	−0.332817	+	0.942991i	$0.607999\pi$
$564$	0	0
$565$	0	0
$566$	1.74295	0.0732616
$567$	0	0
$568$	18.2085	0.764012
$569$	14.5605	0.610408	0.305204	−	0.952287i	$-0.401275\pi$
$569$	14.5605	0.610408	0.305204	+	0.952287i	$0.401275\pi$
$570$	0	0
$571$	27.6138	1.15560	0.577802	−	0.816177i	$-0.303911\pi$
$571$	27.6138	1.15560	0.577802	+	0.816177i	$0.303911\pi$
$572$	23.6642	0.989448
$573$	0	0
$574$	−0.0930636	−0.00388440
$575$	0	0
$576$	0	0
$577$	−27.1520	−1.13035	−0.565176	−	0.824970i	$-0.691192\pi$
$577$	−27.1520	−1.13035	−0.565176	+	0.824970i	$0.691192\pi$
$578$	1.94066	0.0807210
$579$	0	0
$580$	0	0
$581$	0.213072	0.00883972
$582$	0	0
$583$	−32.1487	−1.33146
$584$	−7.42621	−0.307299
$585$	0	0
$586$	11.7703	0.486227
$587$	10.7037	0.441787	0.220894	−	0.975298i	$-0.429103\pi$
$587$	10.7037	0.441787	0.220894	+	0.975298i	$0.429103\pi$
$588$	0	0
$589$	14.6088	0.601946
$590$	0	0
$591$	0	0
$592$	−0.944342	−0.0388122
$593$	3.84629	0.157948	0.0789740	−	0.996877i	$-0.474836\pi$
$593$	3.84629	0.157948	0.0789740	+	0.996877i	$0.474836\pi$
$594$	0	0
$595$	0	0
$596$	−9.84210	−0.403148
$597$	0	0
$598$	3.24494	0.132695
$599$	−46.1423	−1.88532	−0.942662	−	0.333750i	$-0.891686\pi$
$599$	−46.1423	−1.88532	−0.942662	+	0.333750i	$0.891686\pi$
$600$	0	0
$601$	−13.3119	−0.543005	−0.271503	−	0.962438i	$-0.587521\pi$
$601$	−13.3119	−0.543005	−0.271503	+	0.962438i	$0.587521\pi$
$602$	−0.00208049	−8.47943e−5 0
$603$	0	0
$604$	12.8895	0.524466
$605$	0	0
$606$	0	0
$607$	34.0838	1.38342	0.691709	−	0.722176i	$-0.256858\pi$
$607$	34.0838	1.38342	0.691709	+	0.722176i	$0.256858\pi$
$608$	12.2453	0.496613
$609$	0	0
$610$	0	0
$611$	−29.4926	−1.19314
$612$	0	0
$613$	25.4809	1.02916	0.514582	−	0.857441i	$-0.327947\pi$
$613$	25.4809	1.02916	0.514582	+	0.857441i	$0.327947\pi$
$614$	−0.557363	−0.0224933
$615$	0	0
$616$	−0.160097	−0.00645048
$617$	26.9453	1.08478	0.542389	−	0.840127i	$-0.317520\pi$
$617$	26.9453	1.08478	0.542389	+	0.840127i	$0.317520\pi$
$618$	0	0
$619$	39.3289	1.58076	0.790381	−	0.612615i	$-0.209882\pi$
$619$	39.3289	1.58076	0.790381	+	0.612615i	$0.209882\pi$
$620$	0	0
$621$	0	0
$622$	−7.39105	−0.296354
$623$	0.404752	0.0162161
$624$	0	0
$625$	0	0
$626$	−2.47906	−0.0990833
$627$	0	0
$628$	−15.0956	−0.602378
$629$	−1.33802	−0.0533502
$630$	0	0
$631$	11.2443	0.447630	0.223815	−	0.974632i	$-0.428149\pi$
$631$	11.2443	0.447630	0.223815	+	0.974632i	$0.428149\pi$
$632$	−18.1877	−0.723467
$633$	0	0
$634$	11.3652	0.451369
$635$	0	0
$636$	0	0
$637$	−26.4290	−1.04715
$638$	−6.92465	−0.274150
$639$	0	0
$640$	0	0
$641$	−17.4773	−0.690311	−0.345155	−	0.938546i	$-0.612174\pi$
$641$	−17.4773	−0.690311	−0.345155	+	0.938546i	$0.612174\pi$
$642$	0	0
$643$	−8.72320	−0.344009	−0.172005	−	0.985096i	$-0.555024\pi$
$643$	−8.72320	−0.344009	−0.172005	+	0.985096i	$0.555024\pi$
$644$	0.0713689	0.00281233
$645$	0	0
$646$	4.41221	0.173596
$647$	−25.0654	−0.985421	−0.492710	−	0.870193i	$-0.663994\pi$
$647$	−25.0654	−0.985421	−0.492710	+	0.870193i	$0.663994\pi$
$648$	0	0
$649$	−15.9648	−0.626674
$650$	0	0
$651$	0	0
$652$	−8.30716	−0.325334
$653$	−29.9777	−1.17312	−0.586559	−	0.809906i	$-0.699518\pi$
$653$	−29.9777	−1.17312	−0.586559	+	0.809906i	$0.699518\pi$
$654$	0	0
$655$	0	0
$656$	19.9680	0.779621
$657$	0	0
$658$	0.0930852	0.00362884
$659$	4.36601	0.170076	0.0850379	−	0.996378i	$-0.472899\pi$
$659$	4.36601	0.170076	0.0850379	+	0.996378i	$0.472899\pi$
$660$	0	0
$661$	−19.4529	−0.756628	−0.378314	−	0.925677i	$-0.623496\pi$
$661$	−19.4529	−0.756628	−0.378314	+	0.925677i	$0.623496\pi$
$662$	−0.868004	−0.0337359
$663$	0	0
$664$	16.8235	0.652878
$665$	0	0
$666$	0	0
$667$	6.61681	0.256204
$668$	32.5258	1.25846
$669$	0	0
$670$	0	0
$671$	−33.0165	−1.27459
$672$	0	0
$673$	−36.2275	−1.39647	−0.698234	−	0.715870i	$-0.746030\pi$
$673$	−36.2275	−1.39647	−0.698234	+	0.715870i	$0.746030\pi$
$674$	−7.00436	−0.269798
$675$	0	0
$676$	−2.19886	−0.0845715
$677$	−2.04046	−0.0784214	−0.0392107	−	0.999231i	$-0.512484\pi$
$677$	−2.04046	−0.0784214	−0.0392107	+	0.999231i	$0.512484\pi$
$678$	0	0
$679$	0.0657586	0.00252359
$680$	0	0
$681$	0	0
$682$	−10.7887	−0.413121
$683$	−15.9105	−0.608799	−0.304400	−	0.952544i	$-0.598456\pi$
$683$	−15.9105	−0.608799	−0.304400	+	0.952544i	$0.598456\pi$
$684$	0	0
$685$	0	0
$686$	0.166838	0.00636990
$687$	0	0
$688$	0.446396	0.0170187
$689$	33.8765	1.29059
$690$	0	0
$691$	−3.90166	−0.148426	−0.0742130	−	0.997242i	$-0.523644\pi$
$691$	−3.90166	−0.148426	−0.0742130	+	0.997242i	$0.523644\pi$
$692$	27.6268	1.05021
$693$	0	0
$694$	2.54589	0.0966408
$695$	0	0
$696$	0	0
$697$	28.2922	1.07164
$698$	4.45201	0.168511
$699$	0	0
$700$	0	0
$701$	−30.3587	−1.14663	−0.573316	−	0.819335i	$-0.694343\pi$
$701$	−30.3587	−1.14663	−0.573316	+	0.819335i	$0.694343\pi$
$702$	0	0
$703$	0.897729	0.0338585
$704$	9.28208	0.349832
$705$	0	0
$706$	4.71651	0.177508
$707$	−0.246267	−0.00926181
$708$	0	0
$709$	−48.5677	−1.82400	−0.911999	−	0.410193i	$-0.865461\pi$
$709$	−48.5677	−1.82400	−0.911999	+	0.410193i	$0.865461\pi$
$710$	0	0
$711$	0	0
$712$	31.9579	1.19767
$713$	10.3091	0.386078
$714$	0	0
$715$	0	0
$716$	16.3873	0.612421
$717$	0	0
$718$	4.56651	0.170421
$719$	−4.76238	−0.177607	−0.0888033	−	0.996049i	$-0.528304\pi$
$719$	−4.76238	−0.177607	−0.0888033	+	0.996049i	$0.528304\pi$
$720$	0	0
$721$	0.435112	0.0162044
$722$	6.55847	0.244081
$723$	0	0
$724$	10.2252	0.380018
$725$	0	0
$726$	0	0
$727$	−3.38691	−0.125614	−0.0628068	−	0.998026i	$-0.520005\pi$
$727$	−3.38691	−0.125614	−0.0628068	+	0.998026i	$0.520005\pi$
$728$	0.168701	0.00625248
$729$	0	0
$730$	0	0
$731$	0.632488	0.0233934
$732$	0	0
$733$	15.4009	0.568847	0.284423	−	0.958699i	$-0.408198\pi$
$733$	15.4009	0.568847	0.284423	+	0.958699i	$0.408198\pi$
$734$	8.95690	0.330605
$735$	0	0
$736$	8.64121	0.318519
$737$	16.0311	0.590513
$738$	0	0
$739$	9.98465	0.367291	0.183646	−	0.982993i	$-0.441210\pi$
$739$	9.98465	0.367291	0.183646	+	0.982993i	$0.441210\pi$
$740$	0	0
$741$	0	0
$742$	−0.106922	−0.00392522
$743$	41.4419	1.52036	0.760178	−	0.649715i	$-0.225112\pi$
$743$	41.4419	1.52036	0.760178	+	0.649715i	$0.225112\pi$
$744$	0	0
$745$	0	0
$746$	1.54001	0.0563839
$747$	0	0
$748$	22.7063	0.830223
$749$	−0.377719	−0.0138015
$750$	0	0
$751$	21.1036	0.770082	0.385041	−	0.922900i	$-0.374187\pi$
$751$	21.1036	0.770082	0.385041	+	0.922900i	$0.374187\pi$
$752$	−19.9727	−0.728329
$753$	0	0
$754$	7.29682	0.265734
$755$	0	0
$756$	0	0
$757$	40.7168	1.47988	0.739938	−	0.672675i	$-0.234855\pi$
$757$	40.7168	1.47988	0.739938	+	0.672675i	$0.234855\pi$
$758$	−12.8186	−0.465594
$759$	0	0
$760$	0	0
$761$	5.07664	0.184028	0.0920140	−	0.995758i	$-0.470670\pi$
$761$	5.07664	0.184028	0.0920140	+	0.995758i	$0.470670\pi$
$762$	0	0
$763$	0.141802	0.00513357
$764$	−38.2941	−1.38543
$765$	0	0
$766$	−1.11485	−0.0402810
$767$	16.8229	0.607438
$768$	0	0
$769$	−46.7687	−1.68652	−0.843261	−	0.537505i	$-0.819367\pi$
$769$	−46.7687	−1.68652	−0.843261	+	0.537505i	$0.819367\pi$
$770$	0	0
$771$	0	0
$772$	44.1697	1.58970
$773$	−19.0938	−0.686756	−0.343378	−	0.939197i	$-0.611571\pi$
$773$	−19.0938	−0.686756	−0.343378	+	0.939197i	$0.611571\pi$
$774$	0	0
$775$	0	0
$776$	5.19209	0.186385
$777$	0	0
$778$	2.32666	0.0834147
$779$	−18.9824	−0.680115
$780$	0	0
$781$	−34.7385	−1.24304
$782$	3.11359	0.111342
$783$	0	0
$784$	−17.8979	−0.639212
$785$	0	0
$786$	0	0
$787$	−4.02513	−0.143481	−0.0717403	−	0.997423i	$-0.522855\pi$
$787$	−4.02513	−0.143481	−0.0717403	+	0.997423i	$0.522855\pi$
$788$	25.2799	0.900559
$789$	0	0
$790$	0	0
$791$	−0.0577642	−0.00205386
$792$	0	0
$793$	34.7910	1.23546
$794$	19.0101	0.674645
$795$	0	0
$796$	6.60660	0.234165
$797$	15.9847	0.566205	0.283103	−	0.959090i	$-0.408636\pi$
$797$	15.9847	0.566205	0.283103	+	0.959090i	$0.408636\pi$
$798$	0	0
$799$	−28.2988	−1.00114
$800$	0	0
$801$	0	0
$802$	−8.37587	−0.295762
$803$	14.1678	0.499972
$804$	0	0
$805$	0	0
$806$	11.3685	0.400440
$807$	0	0
$808$	−19.4444	−0.684052
$809$	29.2015	1.02667	0.513335	−	0.858188i	$-0.328410\pi$
$809$	29.2015	1.02667	0.513335	+	0.858188i	$0.328410\pi$
$810$	0	0
$811$	7.11719	0.249918	0.124959	−	0.992162i	$-0.460120\pi$
$811$	7.11719	0.249918	0.124959	+	0.992162i	$0.460120\pi$
$812$	0.160485	0.00563194
$813$	0	0
$814$	−0.662978	−0.0232374
$815$	0	0
$816$	0	0
$817$	−0.424362	−0.0148466
$818$	−0.237034	−0.00828770
$819$	0	0
$820$	0	0
$821$	32.6032	1.13786	0.568929	−	0.822387i	$-0.307358\pi$
$821$	32.6032	1.13786	0.568929	+	0.822387i	$0.307358\pi$
$822$	0	0
$823$	27.1361	0.945905	0.472953	−	0.881088i	$-0.343188\pi$
$823$	27.1361	0.945905	0.472953	+	0.881088i	$0.343188\pi$
$824$	34.3551	1.19681
$825$	0	0
$826$	−0.0530966	−0.00184747
$827$	54.8232	1.90639	0.953195	−	0.302357i	$-0.0977736\pi$
$827$	54.8232	1.90639	0.953195	+	0.302357i	$0.0977736\pi$
$828$	0	0
$829$	0.471969	0.0163922	0.00819608	−	0.999966i	$-0.497391\pi$
$829$	0.471969	0.0163922	0.00819608	+	0.999966i	$0.497391\pi$
$830$	0	0
$831$	0	0
$832$	−9.78095	−0.339093
$833$	−25.3591	−0.878642
$834$	0	0
$835$	0	0
$836$	−15.2346	−0.526898
$837$	0	0
$838$	12.1414	0.419418
$839$	38.4737	1.32826	0.664129	−	0.747618i	$-0.268802\pi$
$839$	38.4737	1.32826	0.664129	+	0.747618i	$0.268802\pi$
$840$	0	0
$841$	−14.1209	−0.486929
$842$	15.7051	0.541233
$843$	0	0
$844$	33.0574	1.13788
$845$	0	0
$846$	0	0
$847$	0.0437684	0.00150390
$848$	22.9415	0.787814
$849$	0	0
$850$	0	0
$851$	0.633505	0.0217163
$852$	0	0
$853$	−35.8541	−1.22762	−0.613811	−	0.789453i	$-0.710364\pi$
$853$	−35.8541	−1.22762	−0.613811	+	0.789453i	$0.710364\pi$
$854$	−0.109808	−0.00375755
$855$	0	0
$856$	−29.8234	−1.01934
$857$	−0.386299	−0.0131957	−0.00659786	−	0.999978i	$-0.502100\pi$
$857$	−0.386299	−0.0131957	−0.00659786	+	0.999978i	$0.502100\pi$
$858$	0	0
$859$	22.2889	0.760487	0.380244	−	0.924886i	$-0.375840\pi$
$859$	22.2889	0.760487	0.380244	+	0.924886i	$0.375840\pi$
$860$	0	0
$861$	0	0
$862$	−1.00231	−0.0341389
$863$	−9.29808	−0.316510	−0.158255	−	0.987398i	$-0.550587\pi$
$863$	−9.29808	−0.316510	−0.158255	+	0.987398i	$0.550587\pi$
$864$	0	0
$865$	0	0
$866$	−3.54326	−0.120405
$867$	0	0
$868$	0.250039	0.00848686
$869$	34.6987	1.17707
$870$	0	0
$871$	−16.8927	−0.572387
$872$	11.1962	0.379151
$873$	0	0
$874$	−2.08903	−0.0706626
$875$	0	0
$876$	0	0
$877$	29.1786	0.985292	0.492646	−	0.870230i	$-0.336030\pi$
$877$	29.1786	0.985292	0.492646	+	0.870230i	$0.336030\pi$
$878$	−6.49384	−0.219156
$879$	0	0
$880$	0	0
$881$	40.0737	1.35012	0.675059	−	0.737764i	$-0.264118\pi$
$881$	40.0737	1.35012	0.675059	+	0.737764i	$0.264118\pi$
$882$	0	0
$883$	30.8339	1.03764	0.518822	−	0.854882i	$-0.326371\pi$
$883$	30.8339	1.03764	0.518822	+	0.854882i	$0.326371\pi$
$884$	−23.9266	−0.804739
$885$	0	0
$886$	10.9560	0.368073
$887$	38.5528	1.29448	0.647239	−	0.762287i	$-0.275924\pi$
$887$	38.5528	1.29448	0.647239	+	0.762287i	$0.275924\pi$
$888$	0	0
$889$	−0.406946	−0.0136485
$890$	0	0
$891$	0	0
$892$	19.8610	0.664996
$893$	18.9868	0.635370
$894$	0	0
$895$	0	0
$896$	0.270533	0.00903787
$897$	0	0
$898$	0.200208	0.00668104
$899$	23.1818	0.773156
$900$	0	0
$901$	32.5052	1.08291
$902$	14.0186	0.466769
$903$	0	0
$904$	−4.56087	−0.151692
$905$	0	0
$906$	0	0
$907$	−4.77670	−0.158608	−0.0793038	−	0.996850i	$-0.525270\pi$
$907$	−4.77670	−0.158608	−0.0793038	+	0.996850i	$0.525270\pi$
$908$	19.5849	0.649946
$909$	0	0
$910$	0	0
$911$	11.5332	0.382111	0.191056	−	0.981579i	$-0.438809\pi$
$911$	11.5332	0.382111	0.191056	+	0.981579i	$0.438809\pi$
$912$	0	0
$913$	−32.0961	−1.06223
$914$	17.6472	0.583718
$915$	0	0
$916$	−16.8774	−0.557645
$917$	−0.113933	−0.00376240
$918$	0	0
$919$	−41.3268	−1.36325	−0.681623	−	0.731704i	$-0.738726\pi$
$919$	−41.3268	−1.36325	−0.681623	+	0.731704i	$0.738726\pi$
$920$	0	0
$921$	0	0
$922$	−7.99655	−0.263352
$923$	36.6055	1.20488
$924$	0	0
$925$	0	0
$926$	15.2717	0.501860
$927$	0	0
$928$	19.4313	0.637863
$929$	13.4174	0.440212	0.220106	−	0.975476i	$-0.429360\pi$
$929$	13.4174	0.440212	0.220106	+	0.975476i	$0.429360\pi$
$930$	0	0
$931$	17.0145	0.557627
$932$	−27.1552	−0.889499
$933$	0	0
$934$	3.32986	0.108956
$935$	0	0
$936$	0	0
$937$	41.9929	1.37185	0.685923	−	0.727674i	$-0.259398\pi$
$937$	41.9929	1.37185	0.685923	+	0.727674i	$0.259398\pi$
$938$	0.0533170	0.00174086
$939$	0	0
$940$	0	0
$941$	−40.0565	−1.30580	−0.652902	−	0.757443i	$-0.726449\pi$
$941$	−40.0565	−1.30580	−0.652902	+	0.757443i	$0.726449\pi$
$942$	0	0
$943$	−13.3954	−0.436215
$944$	11.3926	0.370797
$945$	0	0
$946$	0.313394	0.0101893
$947$	−43.9751	−1.42900	−0.714499	−	0.699637i	$-0.753345\pi$
$947$	−43.9751	−1.42900	−0.714499	+	0.699637i	$0.753345\pi$
$948$	0	0
$949$	−14.9293	−0.484625
$950$	0	0
$951$	0	0
$952$	0.161872	0.00524631
$953$	−40.6171	−1.31572	−0.657859	−	0.753141i	$-0.728538\pi$
$953$	−40.6171	−1.31572	−0.657859	+	0.753141i	$0.728538\pi$
$954$	0	0
$955$	0	0
$956$	25.0707	0.810843
$957$	0	0
$958$	11.1424	0.359994
$959$	−0.304731	−0.00984028
$960$	0	0
$961$	5.11756	0.165083
$962$	0.698610	0.0225241
$963$	0	0
$964$	−44.2698	−1.42584
$965$	0	0
$966$	0	0
$967$	−16.0002	−0.514531	−0.257265	−	0.966341i	$-0.582821\pi$
$967$	−16.0002	−0.514531	−0.257265	+	0.966341i	$0.582821\pi$
$968$	3.45581	0.111074
$969$	0	0
$970$	0	0
$971$	−55.5201	−1.78173	−0.890863	−	0.454272i	$-0.849899\pi$
$971$	−55.5201	−1.78173	−0.890863	+	0.454272i	$0.849899\pi$
$972$	0	0
$973$	−0.188880	−0.00605523
$974$	−5.10263	−0.163499
$975$	0	0
$976$	23.5608	0.754162
$977$	−6.32539	−0.202367	−0.101184	−	0.994868i	$-0.532263\pi$
$977$	−6.32539	−0.202367	−0.101184	+	0.994868i	$0.532263\pi$
$978$	0	0
$979$	−60.9699	−1.94861
$980$	0	0
$981$	0	0
$982$	3.32659	0.106156
$983$	−48.6422	−1.55144	−0.775722	−	0.631074i	$-0.782614\pi$
$983$	−48.6422	−1.55144	−0.775722	+	0.631074i	$0.782614\pi$
$984$	0	0
$985$	0	0
$986$	7.00145	0.222972
$987$	0	0
$988$	16.0533	0.510725
$989$	−0.299462	−0.00952234
$990$	0	0
$991$	−44.1763	−1.40331	−0.701653	−	0.712519i	$-0.747554\pi$
$991$	−44.1763	−1.40331	−0.701653	+	0.712519i	$0.747554\pi$
$992$	30.2742	0.961207
$993$	0	0
$994$	−0.115535	−0.00366455
$995$	0	0
$996$	0	0
$997$	40.2262	1.27398	0.636988	−	0.770873i	$-0.280180\pi$
$997$	40.2262	1.27398	0.636988	+	0.770873i	$0.280180\pi$
$998$	−0.543059	−0.0171902
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.s.1.5		8
3.2	odd	2		625.2.a.g.1.4	yes	8
5.4	even	2		5625.2.a.be.1.4		8
12.11	even	2		10000.2.a.be.1.1		8
15.2	even	4		625.2.b.d.624.10		16
15.8	even	4		625.2.b.d.624.7		16
15.14	odd	2		625.2.a.e.1.5	✓	8
60.59	even	2		10000.2.a.bn.1.8		8
75.2	even	20		625.2.e.k.124.5		32
75.8	even	20		625.2.e.j.374.5		32
75.11	odd	10		625.2.d.n.501.2		16
75.14	odd	10		625.2.d.p.501.3		16
75.17	even	20		625.2.e.j.374.4		32
75.23	even	20		625.2.e.k.124.4		32
75.29	odd	10		625.2.d.q.376.2		16
75.38	even	20		625.2.e.k.499.5		32
75.41	odd	10		625.2.d.n.126.2		16
75.44	odd	10		625.2.d.q.251.2		16
75.47	even	20		625.2.e.j.249.5		32
75.53	even	20		625.2.e.j.249.4		32
75.56	odd	10		625.2.d.m.251.3		16
75.59	odd	10		625.2.d.p.126.3		16
75.62	even	20		625.2.e.k.499.4		32
75.71	odd	10		625.2.d.m.376.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.5	✓	8	15.14	odd	2
625.2.a.g.1.4	yes	8	3.2	odd	2
625.2.b.d.624.7		16	15.8	even	4
625.2.b.d.624.10		16	15.2	even	4
625.2.d.m.251.3		16	75.56	odd	10
625.2.d.m.376.3		16	75.71	odd	10
625.2.d.n.126.2		16	75.41	odd	10
625.2.d.n.501.2		16	75.11	odd	10
625.2.d.p.126.3		16	75.59	odd	10
625.2.d.p.501.3		16	75.14	odd	10
625.2.d.q.251.2		16	75.44	odd	10
625.2.d.q.376.2		16	75.29	odd	10
625.2.e.j.249.4		32	75.53	even	20
625.2.e.j.249.5		32	75.47	even	20
625.2.e.j.374.4		32	75.17	even	20
625.2.e.j.374.5		32	75.8	even	20
625.2.e.k.124.4		32	75.23	even	20
625.2.e.k.124.5		32	75.2	even	20
625.2.e.k.499.4		32	75.62	even	20
625.2.e.k.499.5		32	75.38	even	20
5625.2.a.s.1.5		8	1.1	even	1	trivial
5625.2.a.be.1.4		8	5.4	even	2
10000.2.a.be.1.1		8	12.11	even	2
10000.2.a.bn.1.8		8	60.59	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.500989 q^{2} -1.74901 q^{4} +0.0237879 q^{7} +1.87821 q^{8} -3.58329 q^{11} +3.77587 q^{13} -0.0119175 q^{14} +2.55705 q^{16} +3.62303 q^{17} -2.43084 q^{19} +1.79519 q^{22} -1.71538 q^{23} -1.89167 q^{26} -0.0416052 q^{28} -3.85734 q^{29} -6.00979 q^{31} -5.03748 q^{32} -1.81510 q^{34} -0.369309 q^{37} +1.21782 q^{38} +7.80900 q^{41} +0.174574 q^{43} +6.26720 q^{44} +0.859388 q^{46} -7.81082 q^{47} -6.99943 q^{49} -6.60403 q^{52} +8.97184 q^{53} +0.0446787 q^{56} +1.93249 q^{58} +4.45536 q^{59} +9.21403 q^{61} +3.01084 q^{62} -2.59038 q^{64} -4.47385 q^{67} -6.33671 q^{68} +9.69458 q^{71} -3.95387 q^{73} +0.185020 q^{74} +4.25156 q^{76} -0.0852388 q^{77} -9.68349 q^{79} -3.91223 q^{82} +8.95717 q^{83} -0.0874600 q^{86} -6.73018 q^{88} +17.0151 q^{89} +0.0898200 q^{91} +3.00022 q^{92} +3.91314 q^{94} +2.76438 q^{97} +3.50664 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34}+ \cdots + 30 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * q^98

Introduction

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