LMFDB - Embedded newform 3825.2.a.bq.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3825,2,Mod(1,3825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3825.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3825 = 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,11,0,0,1,9,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.5427787731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.60789$ of defining polynomial
Character	$\chi$	$=$	3825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80107 q^{2} +5.84602 q^{4} -1.33298 q^{7} +10.7730 q^{8} -1.09485 q^{11} +3.17083 q^{13} -3.73378 q^{14} +18.4839 q^{16} -1.00000 q^{17} +2.75613 q^{19} -3.06675 q^{22} -3.57111 q^{23} +8.88173 q^{26} -7.79262 q^{28} +0.180058 q^{29} +0.816039 q^{31} +30.2288 q^{32} -2.80107 q^{34} +8.44817 q^{37} +7.72013 q^{38} +7.97191 q^{41} +6.54798 q^{43} -6.40051 q^{44} -10.0029 q^{46} -0.576074 q^{47} -5.22317 q^{49} +18.5367 q^{52} -7.84602 q^{53} -14.3602 q^{56} +0.504355 q^{58} +9.76375 q^{59} -5.21577 q^{61} +2.28579 q^{62} +47.7053 q^{64} -2.64963 q^{67} -5.84602 q^{68} -13.4114 q^{71} -12.3299 q^{73} +23.6639 q^{74} +16.1124 q^{76} +1.45941 q^{77} +3.74745 q^{79} +22.3299 q^{82} +4.66596 q^{83} +18.3414 q^{86} -11.7948 q^{88} -3.00946 q^{89} -4.22665 q^{91} -20.8768 q^{92} -1.61363 q^{94} -12.2681 q^{97} -14.6305 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 11 q^{4} + q^{7} + 9 q^{8} - 4 q^{11} - 3 q^{13} + 7 q^{14} + 27 q^{16} - 5 q^{17} + 6 q^{19} + 18 q^{22} - 4 q^{23} + 5 q^{26} - 15 q^{28} - 2 q^{29} + 21 q^{31} + 9 q^{32} - q^{34} - 2 q^{37}+ \cdots - 38 q^{98}+O(q^{100}) $ 5 * q + q^2 + 11 * q^4 + q^7 + 9 * q^8 - 4 * q^11 - 3 * q^13 + 7 * q^14 + 27 * q^16 - 5 * q^17 + 6 * q^19 + 18 * q^22 - 4 * q^23 + 5 * q^26 - 15 * q^28 - 2 * q^29 + 21 * q^31 + 9 * q^32 - q^34 - 2 * q^37 + 34 * q^38 + 8 * q^41 - 4 * q^43 + 4 * q^44 - 6 * q^46 + 2 * q^47 + 10 * q^49 - 13 * q^52 - 21 * q^53 + 29 * q^56 + 22 * q^58 - 12 * q^59 - 2 * q^61 - 3 * q^62 + 43 * q^64 - 12 * q^67 - 11 * q^68 - 21 * q^71 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 + 41 * q^79 + 28 * q^82 + 8 * q^83 + 36 * q^86 + 14 * q^88 + 10 * q^89 + 13 * q^91 - 56 * q^92 - 10 * q^94 - 20 * q^97 - 38 * 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 $

$2$	2.80107	1.98066	0.990329	−	0.138737i	$-0.0443042\pi$
$2$	2.80107	1.98066	0.990329	+	0.138737i	$0.0443042\pi$
$3$	0	0
$3$	0	0
$4$	5.84602	2.92301
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.33298	−0.503819	−0.251909	−	0.967751i	$-0.581059\pi$
$7$	−1.33298	−0.503819	−0.251909	+	0.967751i	$0.581059\pi$
$8$	10.7730	3.80882
$9$	0	0
$10$	0	0
$11$	−1.09485	−0.330110	−0.165055	−	0.986284i	$-0.552780\pi$
$11$	−1.09485	−0.330110	−0.165055	+	0.986284i	$0.552780\pi$
$12$	0	0
$13$	3.17083	0.879430	0.439715	−	0.898137i	$-0.355079\pi$
$13$	3.17083	0.879430	0.439715	+	0.898137i	$0.355079\pi$
$14$	−3.73378	−0.997893
$15$	0	0
$16$	18.4839	4.62097
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.75613	0.632300	0.316150	−	0.948709i	$-0.397610\pi$
$19$	2.75613	0.632300	0.316150	+	0.948709i	$0.397610\pi$
$20$	0	0
$21$	0	0
$22$	−3.06675	−0.653834
$23$	−3.57111	−0.744628	−0.372314	−	0.928107i	$-0.621436\pi$
$23$	−3.57111	−0.744628	−0.372314	+	0.928107i	$0.621436\pi$
$24$	0	0
$25$	0	0
$26$	8.88173	1.74185
$27$	0	0
$28$	−7.79262	−1.47267
$29$	0.180058	0.0334359	0.0167179	−	0.999860i	$-0.494678\pi$
$29$	0.180058	0.0334359	0.0167179	+	0.999860i	$0.494678\pi$
$30$	0	0
$31$	0.816039	0.146565	0.0732825	−	0.997311i	$-0.476653\pi$
$31$	0.816039	0.146565	0.0732825	+	0.997311i	$0.476653\pi$
$32$	30.2288	5.34374
$33$	0	0
$34$	−2.80107	−0.480380
$35$	0	0
$36$	0	0
$37$	8.44817	1.38887	0.694435	−	0.719555i	$-0.255654\pi$
$37$	8.44817	1.38887	0.694435	+	0.719555i	$0.255654\pi$
$38$	7.72013	1.25237
$39$	0	0
$40$	0	0
$41$	7.97191	1.24500	0.622501	−	0.782619i	$-0.286117\pi$
$41$	7.97191	1.24500	0.622501	+	0.782619i	$0.286117\pi$
$42$	0	0
$43$	6.54798	0.998557	0.499279	−	0.866441i	$-0.333598\pi$
$43$	6.54798	0.998557	0.499279	+	0.866441i	$0.333598\pi$
$44$	−6.40051	−0.964913
$45$	0	0
$46$	−10.0029	−1.47485
$47$	−0.576074	−0.0840290	−0.0420145	−	0.999117i	$-0.513378\pi$
$47$	−0.576074	−0.0840290	−0.0420145	+	0.999117i	$0.513378\pi$
$48$	0	0
$49$	−5.22317	−0.746166
$50$	0	0
$51$	0	0
$52$	18.5367	2.57058
$53$	−7.84602	−1.07773	−0.538867	−	0.842391i	$-0.681147\pi$
$53$	−7.84602	−1.07773	−0.538867	+	0.842391i	$0.681147\pi$
$54$	0	0
$55$	0	0
$56$	−14.3602	−1.91896
$57$	0	0
$58$	0.504355	0.0662251
$59$	9.76375	1.27113	0.635566	−	0.772046i	$-0.280767\pi$
$59$	9.76375	1.27113	0.635566	+	0.772046i	$0.280767\pi$
$60$	0	0
$61$	−5.21577	−0.667811	−0.333906	−	0.942606i	$-0.608367\pi$
$61$	−5.21577	−0.667811	−0.333906	+	0.942606i	$0.608367\pi$
$62$	2.28579	0.290295
$63$	0	0
$64$	47.7053	5.96316
$65$	0	0
$66$	0	0
$67$	−2.64963	−0.323704	−0.161852	−	0.986815i	$-0.551747\pi$
$67$	−2.64963	−0.323704	−0.161852	+	0.986815i	$0.551747\pi$
$68$	−5.84602	−0.708934
$69$	0	0
$70$	0	0
$71$	−13.4114	−1.59164	−0.795820	−	0.605534i	$-0.792960\pi$
$71$	−13.4114	−1.59164	−0.795820	+	0.605534i	$0.792960\pi$
$72$	0	0
$73$	−12.3299	−1.44311	−0.721553	−	0.692359i	$-0.756571\pi$
$73$	−12.3299	−1.44311	−0.721553	+	0.692359i	$0.756571\pi$
$74$	23.6639	2.75088
$75$	0	0
$76$	16.1124	1.84822
$77$	1.45941	0.166315
$78$	0	0
$79$	3.74745	0.421621	0.210810	−	0.977527i	$-0.432390\pi$
$79$	3.74745	0.421621	0.210810	+	0.977527i	$0.432390\pi$
$80$	0	0
$81$	0	0
$82$	22.3299	2.46592
$83$	4.66596	0.512156	0.256078	−	0.966656i	$-0.417570\pi$
$83$	4.66596	0.512156	0.256078	+	0.966656i	$0.417570\pi$
$84$	0	0
$85$	0	0
$86$	18.3414	1.97780
$87$	0	0
$88$	−11.7948	−1.25733
$89$	−3.00946	−0.319002	−0.159501	−	0.987198i	$-0.550988\pi$
$89$	−3.00946	−0.319002	−0.159501	+	0.987198i	$0.550988\pi$
$90$	0	0
$91$	−4.22665	−0.443074
$92$	−20.8768	−2.17655
$93$	0	0
$94$	−1.61363	−0.166433
$95$	0	0
$96$	0	0
$97$	−12.2681	−1.24564	−0.622819	−	0.782366i	$-0.714013\pi$
$97$	−12.2681	−1.24564	−0.622819	+	0.782366i	$0.714013\pi$
$98$	−14.6305	−1.47790
$99$	0	0
$100$	0	0
$101$	−2.98113	−0.296634	−0.148317	−	0.988940i	$-0.547386\pi$
$101$	−2.98113	−0.296634	−0.148317	+	0.988940i	$0.547386\pi$
$102$	0	0
$103$	−13.8440	−1.36409	−0.682045	−	0.731310i	$-0.738909\pi$
$103$	−13.8440	−1.36409	−0.682045	+	0.731310i	$0.738909\pi$
$104$	34.1593	3.34959
$105$	0	0
$106$	−21.9773	−2.13462
$107$	7.87103	0.760921	0.380461	−	0.924797i	$-0.375765\pi$
$107$	7.87103	0.760921	0.380461	+	0.924797i	$0.375765\pi$
$108$	0	0
$109$	13.6739	1.30972	0.654859	−	0.755751i	$-0.272728\pi$
$109$	13.6739	1.30972	0.654859	+	0.755751i	$0.272728\pi$
$110$	0	0
$111$	0	0
$112$	−24.6386	−2.32813
$113$	−19.5722	−1.84120	−0.920600	−	0.390508i	$-0.872300\pi$
$113$	−19.5722	−1.84120	−0.920600	+	0.390508i	$0.872300\pi$
$114$	0	0
$115$	0	0
$116$	1.05262	0.0977334
$117$	0	0
$118$	27.3490	2.51768
$119$	1.33298	0.122194
$120$	0	0
$121$	−9.80130	−0.891028
$122$	−14.6098	−1.32271
$123$	0	0
$124$	4.77058	0.428411
$125$	0	0
$126$	0	0
$127$	−5.00946	−0.444517	−0.222259	−	0.974988i	$-0.571343\pi$
$127$	−5.00946	−0.444517	−0.222259	+	0.974988i	$0.571343\pi$
$128$	73.1684	6.46724
$129$	0	0
$130$	0	0
$131$	−2.63893	−0.230564	−0.115282	−	0.993333i	$-0.536777\pi$
$131$	−2.63893	−0.230564	−0.115282	+	0.993333i	$0.536777\pi$
$132$	0	0
$133$	−3.67387	−0.318565
$134$	−7.42180	−0.641146
$135$	0	0
$136$	−10.7730	−0.923775
$137$	3.27064	0.279430	0.139715	−	0.990192i	$-0.455381\pi$
$137$	3.27064	0.279430	0.139715	+	0.990192i	$0.455381\pi$
$138$	0	0
$139$	−10.3901	−0.881276	−0.440638	−	0.897685i	$-0.645248\pi$
$139$	−10.3901	−0.881276	−0.440638	+	0.897685i	$0.645248\pi$
$140$	0	0
$141$	0	0
$142$	−37.5663	−3.15249
$143$	−3.47158	−0.290308
$144$	0	0
$145$	0	0
$146$	−34.5370	−2.85830
$147$	0	0
$148$	49.3881	4.05968
$149$	11.2566	0.922179	0.461090	−	0.887354i	$-0.347459\pi$
$149$	11.2566	0.922179	0.461090	+	0.887354i	$0.347459\pi$
$150$	0	0
$151$	7.13736	0.580831	0.290415	−	0.956901i	$-0.406207\pi$
$151$	7.13736	0.580831	0.290415	+	0.956901i	$0.406207\pi$
$152$	29.6917	2.40832
$153$	0	0
$154$	4.08792	0.329414
$155$	0	0
$156$	0	0
$157$	−8.30595	−0.662887	−0.331443	−	0.943475i	$-0.607536\pi$
$157$	−8.30595	−0.662887	−0.331443	+	0.943475i	$0.607536\pi$
$158$	10.4969	0.835087
$159$	0	0
$160$	0	0
$161$	4.76022	0.375158
$162$	0	0
$163$	11.6059	0.909042	0.454521	−	0.890736i	$-0.349811\pi$
$163$	11.6059	0.909042	0.454521	+	0.890736i	$0.349811\pi$
$164$	46.6039	3.63915
$165$	0	0
$166$	13.0697	1.01441
$167$	19.6832	1.52313	0.761566	−	0.648087i	$-0.224431\pi$
$167$	19.6832	1.52313	0.761566	+	0.648087i	$0.224431\pi$
$168$	0	0
$169$	−2.94583	−0.226602
$170$	0	0
$171$	0	0
$172$	38.2796	2.91879
$173$	−19.8536	−1.50944	−0.754722	−	0.656045i	$-0.772228\pi$
$173$	−19.8536	−1.50944	−0.754722	+	0.656045i	$0.772228\pi$
$174$	0	0
$175$	0	0
$176$	−20.2371	−1.52543
$177$	0	0
$178$	−8.42971	−0.631834
$179$	6.71142	0.501635	0.250817	−	0.968034i	$-0.419301\pi$
$179$	6.71142	0.501635	0.250817	+	0.968034i	$0.419301\pi$
$180$	0	0
$181$	1.49564	0.111170	0.0555852	−	0.998454i	$-0.482298\pi$
$181$	1.49564	0.111170	0.0555852	+	0.998454i	$0.482298\pi$
$182$	−11.8392	−0.877578
$183$	0	0
$184$	−38.4715	−2.83616
$185$	0	0
$186$	0	0
$187$	1.09485	0.0800633
$188$	−3.36774	−0.245617
$189$	0	0
$190$	0	0
$191$	15.9320	1.15280	0.576401	−	0.817167i	$-0.304457\pi$
$191$	15.9320	1.15280	0.576401	+	0.817167i	$0.304457\pi$
$192$	0	0
$193$	−17.4673	−1.25732	−0.628661	−	0.777680i	$-0.716397\pi$
$193$	−17.4673	−1.25732	−0.628661	+	0.777680i	$0.716397\pi$
$194$	−34.3639	−2.46718
$195$	0	0
$196$	−30.5347	−2.18105
$197$	14.0436	1.00057	0.500283	−	0.865862i	$-0.333229\pi$
$197$	14.0436	1.00057	0.500283	+	0.865862i	$0.333229\pi$
$198$	0	0
$199$	−4.58571	−0.325072	−0.162536	−	0.986703i	$-0.551967\pi$
$199$	−4.58571	−0.325072	−0.162536	+	0.986703i	$0.551967\pi$
$200$	0	0
$201$	0	0
$202$	−8.35037	−0.587530
$203$	−0.240013	−0.0168456
$204$	0	0
$205$	0	0
$206$	−38.7781	−2.70180
$207$	0	0
$208$	58.6093	4.06382
$209$	−3.01755	−0.208728
$210$	0	0
$211$	7.23345	0.497971	0.248986	−	0.968507i	$-0.419903\pi$
$211$	7.23345	0.497971	0.248986	+	0.968507i	$0.419903\pi$
$212$	−45.8680	−3.15022
$213$	0	0
$214$	22.0473	1.50713
$215$	0	0
$216$	0	0
$217$	−1.08776	−0.0738422
$218$	38.3015	2.59411
$219$	0	0
$220$	0	0
$221$	−3.17083	−0.213293
$222$	0	0
$223$	−22.2566	−1.49041	−0.745207	−	0.666833i	$-0.767649\pi$
$223$	−22.2566	−1.49041	−0.745207	+	0.666833i	$0.767649\pi$
$224$	−40.2943	−2.69228
$225$	0	0
$226$	−54.8232	−3.64679
$227$	−13.3541	−0.886342	−0.443171	−	0.896437i	$-0.646147\pi$
$227$	−13.3541	−0.886342	−0.443171	+	0.896437i	$0.646147\pi$
$228$	0	0
$229$	−20.4647	−1.35235	−0.676174	−	0.736742i	$-0.736363\pi$
$229$	−20.4647	−1.35235	−0.676174	+	0.736742i	$0.736363\pi$
$230$	0	0
$231$	0	0
$232$	1.93976	0.127351
$233$	16.1239	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$233$	16.1239	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$234$	0	0
$235$	0	0
$236$	57.0791	3.71553
$237$	0	0
$238$	3.73378	0.242025
$239$	4.38637	0.283731	0.141865	−	0.989886i	$-0.454690\pi$
$239$	4.38637	0.283731	0.141865	+	0.989886i	$0.454690\pi$
$240$	0	0
$241$	−14.8343	−0.955558	−0.477779	−	0.878480i	$-0.658558\pi$
$241$	−14.8343	−0.955558	−0.477779	+	0.878480i	$0.658558\pi$
$242$	−27.4542	−1.76482
$243$	0	0
$244$	−30.4915	−1.95202
$245$	0	0
$246$	0	0
$247$	8.73923	0.556064
$248$	8.79118	0.558240
$249$	0	0
$250$	0	0
$251$	−14.5998	−0.921534	−0.460767	−	0.887521i	$-0.652426\pi$
$251$	−14.5998	−0.921534	−0.460767	+	0.887521i	$0.652426\pi$
$252$	0	0
$253$	3.90983	0.245809
$254$	−14.0319	−0.880437
$255$	0	0
$256$	109.540	6.84623
$257$	−5.34722	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$257$	−5.34722	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$258$	0	0
$259$	−11.2612	−0.699739
$260$	0	0
$261$	0	0
$262$	−7.39183	−0.456669
$263$	−18.3754	−1.13307	−0.566537	−	0.824037i	$-0.691717\pi$
$263$	−18.3754	−1.13307	−0.566537	+	0.824037i	$0.691717\pi$
$264$	0	0
$265$	0	0
$266$	−10.2908	−0.630968
$267$	0	0
$268$	−15.4898	−0.946188
$269$	15.0812	0.919516	0.459758	−	0.888044i	$-0.347936\pi$
$269$	15.0812	0.919516	0.459758	+	0.888044i	$0.347936\pi$
$270$	0	0
$271$	20.6031	1.25155	0.625774	−	0.780004i	$-0.284783\pi$
$271$	20.6031	1.25155	0.625774	+	0.780004i	$0.284783\pi$
$272$	−18.4839	−1.12075
$273$	0	0
$274$	9.16132	0.553455
$275$	0	0
$276$	0	0
$277$	−3.93417	−0.236381	−0.118191	−	0.992991i	$-0.537709\pi$
$277$	−3.93417	−0.236381	−0.118191	+	0.992991i	$0.537709\pi$
$278$	−29.1034	−1.74551
$279$	0	0
$280$	0	0
$281$	−24.7878	−1.47872	−0.739358	−	0.673312i	$-0.764871\pi$
$281$	−24.7878	−1.47872	−0.739358	+	0.673312i	$0.764871\pi$
$282$	0	0
$283$	10.2892	0.611631	0.305815	−	0.952091i	$-0.401071\pi$
$283$	10.2892	0.611631	0.305815	+	0.952091i	$0.401071\pi$
$284$	−78.4032	−4.65237
$285$	0	0
$286$	−9.72416	−0.575002
$287$	−10.6264	−0.627256
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	−72.0808	−4.21821
$293$	4.20448	0.245628	0.122814	−	0.992430i	$-0.460808\pi$
$293$	4.20448	0.245628	0.122814	+	0.992430i	$0.460808\pi$
$294$	0	0
$295$	0	0
$296$	91.0119	5.28996
$297$	0	0
$298$	31.5307	1.82652
$299$	−11.3234	−0.654848
$300$	0	0
$301$	−8.72832	−0.503092
$302$	19.9923	1.15043
$303$	0	0
$304$	50.9440	2.92184
$305$	0	0
$306$	0	0
$307$	20.2969	1.15841	0.579204	−	0.815183i	$-0.303363\pi$
$307$	20.2969	1.15841	0.579204	+	0.815183i	$0.303363\pi$
$308$	8.53175	0.486141
$309$	0	0
$310$	0	0
$311$	−8.03850	−0.455822	−0.227911	−	0.973682i	$-0.573189\pi$
$311$	−8.03850	−0.455822	−0.227911	+	0.973682i	$0.573189\pi$
$312$	0	0
$313$	−29.6525	−1.67606	−0.838028	−	0.545627i	$-0.816292\pi$
$313$	−29.6525	−1.67606	−0.838028	+	0.545627i	$0.816292\pi$
$314$	−23.2656	−1.31295
$315$	0	0
$316$	21.9077	1.23240
$317$	−3.06410	−0.172097	−0.0860484	−	0.996291i	$-0.527424\pi$
$317$	−3.06410	−0.172097	−0.0860484	+	0.996291i	$0.527424\pi$
$318$	0	0
$319$	−0.197136	−0.0110375
$320$	0	0
$321$	0	0
$322$	13.3337	0.743059
$323$	−2.75613	−0.153355
$324$	0	0
$325$	0	0
$326$	32.5089	1.80050
$327$	0	0
$328$	85.8812	4.74199
$329$	0.767895	0.0423354
$330$	0	0
$331$	11.6185	0.638609	0.319305	−	0.947652i	$-0.396551\pi$
$331$	11.6185	0.638609	0.319305	+	0.947652i	$0.396551\pi$
$332$	27.2773	1.49704
$333$	0	0
$334$	55.1341	3.01681
$335$	0	0
$336$	0	0
$337$	−7.19732	−0.392063	−0.196032	−	0.980598i	$-0.562805\pi$
$337$	−7.19732	−0.392063	−0.196032	+	0.980598i	$0.562805\pi$
$338$	−8.25149	−0.448822
$339$	0	0
$340$	0	0
$341$	−0.893440	−0.0483825
$342$	0	0
$343$	16.2932	0.879752
$344$	70.5412	3.80333
$345$	0	0
$346$	−55.6115	−2.98969
$347$	−8.49365	−0.455963	−0.227982	−	0.973665i	$-0.573213\pi$
$347$	−8.49365	−0.455963	−0.227982	+	0.973665i	$0.573213\pi$
$348$	0	0
$349$	−0.281605	−0.0150739	−0.00753697	−	0.999972i	$-0.502399\pi$
$349$	−0.281605	−0.0150739	−0.00753697	+	0.999972i	$0.502399\pi$
$350$	0	0
$351$	0	0
$352$	−33.0960	−1.76402
$353$	−7.56460	−0.402623	−0.201311	−	0.979527i	$-0.564520\pi$
$353$	−7.56460	−0.402623	−0.201311	+	0.979527i	$0.564520\pi$
$354$	0	0
$355$	0	0
$356$	−17.5933	−0.932445
$357$	0	0
$358$	18.7992	0.993568
$359$	−4.24214	−0.223891	−0.111946	−	0.993714i	$-0.535708\pi$
$359$	−4.24214	−0.223891	−0.111946	+	0.993714i	$0.535708\pi$
$360$	0	0
$361$	−11.4037	−0.600197
$362$	4.18941	0.220191
$363$	0	0
$364$	−24.7091	−1.29511
$365$	0	0
$366$	0	0
$367$	−27.9623	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$367$	−27.9623	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$368$	−66.0080	−3.44090
$369$	0	0
$370$	0	0
$371$	10.4586	0.542982
$372$	0	0
$373$	−9.39715	−0.486566	−0.243283	−	0.969955i	$-0.578224\pi$
$373$	−9.39715	−0.486566	−0.243283	+	0.969955i	$0.578224\pi$
$374$	3.06675	0.158578
$375$	0	0
$376$	−6.20603	−0.320052
$377$	0.570933	0.0294045
$378$	0	0
$379$	23.9952	1.23255	0.616275	−	0.787531i	$-0.288641\pi$
$379$	23.9952	1.23255	0.616275	+	0.787531i	$0.288641\pi$
$380$	0	0
$381$	0	0
$382$	44.6268	2.28331
$383$	−1.97888	−0.101116	−0.0505581	−	0.998721i	$-0.516100\pi$
$383$	−1.97888	−0.101116	−0.0505581	+	0.998721i	$0.516100\pi$
$384$	0	0
$385$	0	0
$386$	−48.9271	−2.49032
$387$	0	0
$388$	−71.7196	−3.64101
$389$	26.5739	1.34735	0.673674	−	0.739028i	$-0.264715\pi$
$389$	26.5739	1.34735	0.673674	+	0.739028i	$0.264715\pi$
$390$	0	0
$391$	3.57111	0.180599
$392$	−56.2691	−2.84202
$393$	0	0
$394$	39.3372	1.98178
$395$	0	0
$396$	0	0
$397$	6.32026	0.317205	0.158602	−	0.987343i	$-0.449301\pi$
$397$	6.32026	0.317205	0.158602	+	0.987343i	$0.449301\pi$
$398$	−12.8449	−0.643857
$399$	0	0
$400$	0	0
$401$	14.9995	0.749041	0.374520	−	0.927219i	$-0.377807\pi$
$401$	14.9995	0.749041	0.374520	+	0.927219i	$0.377807\pi$
$402$	0	0
$403$	2.58752	0.128894
$404$	−17.4277	−0.867063
$405$	0	0
$406$	−0.672295	−0.0333654
$407$	−9.24947	−0.458479
$408$	0	0
$409$	18.0132	0.890697	0.445348	−	0.895357i	$-0.353080\pi$
$409$	18.0132	0.890697	0.445348	+	0.895357i	$0.353080\pi$
$410$	0	0
$411$	0	0
$412$	−80.9322	−3.98725
$413$	−13.0149	−0.640421
$414$	0	0
$415$	0	0
$416$	95.8503	4.69945
$417$	0	0
$418$	−8.45238	−0.413419
$419$	11.2294	0.548594	0.274297	−	0.961645i	$-0.411555\pi$
$419$	11.2294	0.548594	0.274297	+	0.961645i	$0.411555\pi$
$420$	0	0
$421$	37.8927	1.84678	0.923389	−	0.383866i	$-0.125408\pi$
$421$	37.8927	1.84678	0.923389	+	0.383866i	$0.125408\pi$
$422$	20.2614	0.986311
$423$	0	0
$424$	−84.5250	−4.10490
$425$	0	0
$426$	0	0
$427$	6.95252	0.336456
$428$	46.0142	2.22418
$429$	0	0
$430$	0	0
$431$	−14.7151	−0.708803	−0.354402	−	0.935093i	$-0.615315\pi$
$431$	−14.7151	−0.708803	−0.354402	+	0.935093i	$0.615315\pi$
$432$	0	0
$433$	−37.5179	−1.80299	−0.901497	−	0.432786i	$-0.857531\pi$
$433$	−37.5179	−1.80299	−0.901497	+	0.432786i	$0.857531\pi$
$434$	−3.04691	−0.146256
$435$	0	0
$436$	79.9377	3.82832
$437$	−9.84245	−0.470828
$438$	0	0
$439$	34.8342	1.66255	0.831273	−	0.555864i	$-0.187612\pi$
$439$	34.8342	1.66255	0.831273	+	0.555864i	$0.187612\pi$
$440$	0	0
$441$	0	0
$442$	−8.88173	−0.422461
$443$	−9.84325	−0.467667	−0.233833	−	0.972277i	$-0.575127\pi$
$443$	−9.84325	−0.467667	−0.233833	+	0.972277i	$0.575127\pi$
$444$	0	0
$445$	0	0
$446$	−62.3425	−2.95200
$447$	0	0
$448$	−63.5901	−3.00435
$449$	−27.7439	−1.30932	−0.654658	−	0.755925i	$-0.727187\pi$
$449$	−27.7439	−1.30932	−0.654658	+	0.755925i	$0.727187\pi$
$450$	0	0
$451$	−8.72804	−0.410987
$452$	−114.420	−5.38184
$453$	0	0
$454$	−37.4058	−1.75554
$455$	0	0
$456$	0	0
$457$	28.0953	1.31424	0.657120	−	0.753786i	$-0.271774\pi$
$457$	28.0953	1.31424	0.657120	+	0.753786i	$0.271774\pi$
$458$	−57.3232	−2.67854
$459$	0	0
$460$	0	0
$461$	−5.20430	−0.242388	−0.121194	−	0.992629i	$-0.538672\pi$
$461$	−5.20430	−0.242388	−0.121194	+	0.992629i	$0.538672\pi$
$462$	0	0
$463$	−1.28264	−0.0596092	−0.0298046	−	0.999556i	$-0.509489\pi$
$463$	−1.28264	−0.0596092	−0.0298046	+	0.999556i	$0.509489\pi$
$464$	3.32817	0.154506
$465$	0	0
$466$	45.1642	2.09219
$467$	−24.5006	−1.13375	−0.566876	−	0.823803i	$-0.691848\pi$
$467$	−24.5006	−1.13375	−0.566876	+	0.823803i	$0.691848\pi$
$468$	0	0
$469$	3.53190	0.163088
$470$	0	0
$471$	0	0
$472$	105.185	4.84152
$473$	−7.16905	−0.329633
$474$	0	0
$475$	0	0
$476$	7.79262	0.357174
$477$	0	0
$478$	12.2866	0.561974
$479$	31.1171	1.42178	0.710888	−	0.703306i	$-0.248293\pi$
$479$	31.1171	1.42178	0.710888	+	0.703306i	$0.248293\pi$
$480$	0	0
$481$	26.7877	1.22141
$482$	−41.5518	−1.89263
$483$	0	0
$484$	−57.2986	−2.60448
$485$	0	0
$486$	0	0
$487$	19.8828	0.900978	0.450489	−	0.892782i	$-0.351250\pi$
$487$	19.8828	0.900978	0.450489	+	0.892782i	$0.351250\pi$
$488$	−56.1894	−2.54358
$489$	0	0
$490$	0	0
$491$	−19.8895	−0.897602	−0.448801	−	0.893632i	$-0.648149\pi$
$491$	−19.8895	−0.897602	−0.448801	+	0.893632i	$0.648149\pi$
$492$	0	0
$493$	−0.180058	−0.00810939
$494$	24.4792	1.10137
$495$	0	0
$496$	15.0836	0.677272
$497$	17.8771	0.801898
$498$	0	0
$499$	4.68299	0.209639	0.104820	−	0.994491i	$-0.466573\pi$
$499$	4.68299	0.209639	0.104820	+	0.994491i	$0.466573\pi$
$500$	0	0
$501$	0	0
$502$	−40.8952	−1.82524
$503$	−39.2233	−1.74888	−0.874440	−	0.485134i	$-0.838771\pi$
$503$	−39.2233	−1.74888	−0.874440	+	0.485134i	$0.838771\pi$
$504$	0	0
$505$	0	0
$506$	10.9517	0.486863
$507$	0	0
$508$	−29.2854	−1.29933
$509$	−16.9770	−0.752494	−0.376247	−	0.926519i	$-0.622786\pi$
$509$	−16.9770	−0.752494	−0.376247	+	0.926519i	$0.622786\pi$
$510$	0	0
$511$	16.4355	0.727064
$512$	160.492	7.09281
$513$	0	0
$514$	−14.9780	−0.660649
$515$	0	0
$516$	0	0
$517$	0.630714	0.0277388
$518$	−31.5436	−1.38594
$519$	0	0
$520$	0	0
$521$	24.6927	1.08181	0.540903	−	0.841085i	$-0.318083\pi$
$521$	24.6927	1.08181	0.540903	+	0.841085i	$0.318083\pi$
$522$	0	0
$523$	−41.2117	−1.80206	−0.901032	−	0.433753i	$-0.857189\pi$
$523$	−41.2117	−1.80206	−0.901032	+	0.433753i	$0.857189\pi$
$524$	−15.4272	−0.673941
$525$	0	0
$526$	−51.4707	−2.24423
$527$	−0.816039	−0.0355472
$528$	0	0
$529$	−10.2472	−0.445529
$530$	0	0
$531$	0	0
$532$	−21.4775	−0.931167
$533$	25.2776	1.09489
$534$	0	0
$535$	0	0
$536$	−28.5444	−1.23293
$537$	0	0
$538$	42.2435	1.82125
$539$	5.71858	0.246317
$540$	0	0
$541$	−3.70168	−0.159147	−0.0795737	−	0.996829i	$-0.525356\pi$
$541$	−3.70168	−0.159147	−0.0795737	+	0.996829i	$0.525356\pi$
$542$	57.7108	2.47889
$543$	0	0
$544$	−30.2288	−1.29605
$545$	0	0
$546$	0	0
$547$	31.0162	1.32616	0.663079	−	0.748550i	$-0.269250\pi$
$547$	31.0162	1.32616	0.663079	+	0.748550i	$0.269250\pi$
$548$	19.1202	0.816776
$549$	0	0
$550$	0	0
$551$	0.496263	0.0211415
$552$	0	0
$553$	−4.99527	−0.212421
$554$	−11.0199	−0.468191
$555$	0	0
$556$	−60.7407	−2.57598
$557$	40.6170	1.72100	0.860498	−	0.509453i	$-0.170152\pi$
$557$	40.6170	1.72100	0.860498	+	0.509453i	$0.170152\pi$
$558$	0	0
$559$	20.7625	0.878162
$560$	0	0
$561$	0	0
$562$	−69.4325	−2.92883
$563$	6.63255	0.279529	0.139764	−	0.990185i	$-0.455366\pi$
$563$	6.63255	0.279529	0.139764	+	0.990185i	$0.455366\pi$
$564$	0	0
$565$	0	0
$566$	28.8209	1.21143
$567$	0	0
$568$	−144.481	−6.06227
$569$	20.9921	0.880035	0.440017	−	0.897989i	$-0.354972\pi$
$569$	20.9921	0.880035	0.440017	+	0.897989i	$0.354972\pi$
$570$	0	0
$571$	9.22867	0.386208	0.193104	−	0.981178i	$-0.438145\pi$
$571$	9.22867	0.386208	0.193104	+	0.981178i	$0.438145\pi$
$572$	−20.2949	−0.848574
$573$	0	0
$574$	−29.7653	−1.24238
$575$	0	0
$576$	0	0
$577$	−1.17031	−0.0487208	−0.0243604	−	0.999703i	$-0.507755\pi$
$577$	−1.17031	−0.0487208	−0.0243604	+	0.999703i	$0.507755\pi$
$578$	2.80107	0.116509
$579$	0	0
$580$	0	0
$581$	−6.21963	−0.258034
$582$	0	0
$583$	8.59021	0.355770
$584$	−132.830	−5.49653
$585$	0	0
$586$	11.7771	0.486506
$587$	−32.1314	−1.32620	−0.663102	−	0.748529i	$-0.730761\pi$
$587$	−32.1314	−1.32620	−0.663102	+	0.748529i	$0.730761\pi$
$588$	0	0
$589$	2.24911	0.0926730
$590$	0	0
$591$	0	0
$592$	156.155	6.41793
$593$	24.9096	1.02291	0.511456	−	0.859309i	$-0.329106\pi$
$593$	24.9096	1.02291	0.511456	+	0.859309i	$0.329106\pi$
$594$	0	0
$595$	0	0
$596$	65.8065	2.69554
$597$	0	0
$598$	−31.7176	−1.29703
$599$	30.4971	1.24608	0.623039	−	0.782191i	$-0.285898\pi$
$599$	30.4971	1.24608	0.623039	+	0.782191i	$0.285898\pi$
$600$	0	0
$601$	−4.43000	−0.180703	−0.0903517	−	0.995910i	$-0.528799\pi$
$601$	−4.43000	−0.180703	−0.0903517	+	0.995910i	$0.528799\pi$
$602$	−24.4487	−0.996454
$603$	0	0
$604$	41.7252	1.69777
$605$	0	0
$606$	0	0
$607$	28.4929	1.15649	0.578246	−	0.815862i	$-0.303737\pi$
$607$	28.4929	1.15649	0.578246	+	0.815862i	$0.303737\pi$
$608$	83.3145	3.37885
$609$	0	0
$610$	0	0
$611$	−1.82663	−0.0738976
$612$	0	0
$613$	−5.63970	−0.227785	−0.113893	−	0.993493i	$-0.536332\pi$
$613$	−5.63970	−0.227785	−0.113893	+	0.993493i	$0.536332\pi$
$614$	56.8533	2.29441
$615$	0	0
$616$	15.7222	0.633466
$617$	−5.49860	−0.221365	−0.110683	−	0.993856i	$-0.535304\pi$
$617$	−5.49860	−0.221365	−0.110683	+	0.993856i	$0.535304\pi$
$618$	0	0
$619$	−20.7770	−0.835096	−0.417548	−	0.908655i	$-0.637111\pi$
$619$	−20.7770	−0.835096	−0.417548	+	0.908655i	$0.637111\pi$
$620$	0	0
$621$	0	0
$622$	−22.5164	−0.902827
$623$	4.01154	0.160719
$624$	0	0
$625$	0	0
$626$	−83.0588	−3.31970
$627$	0	0
$628$	−48.5567	−1.93762
$629$	−8.44817	−0.336850
$630$	0	0
$631$	−9.55265	−0.380285	−0.190142	−	0.981757i	$-0.560895\pi$
$631$	−9.55265	−0.380285	−0.190142	+	0.981757i	$0.560895\pi$
$632$	40.3712	1.60588
$633$	0	0
$634$	−8.58276	−0.340865
$635$	0	0
$636$	0	0
$637$	−16.5618	−0.656201
$638$	−0.552193	−0.0218615
$639$	0	0
$640$	0	0
$641$	38.2638	1.51133	0.755664	−	0.654959i	$-0.227314\pi$
$641$	38.2638	1.51133	0.755664	+	0.654959i	$0.227314\pi$
$642$	0	0
$643$	2.28036	0.0899286	0.0449643	−	0.998989i	$-0.485683\pi$
$643$	2.28036	0.0899286	0.0449643	+	0.998989i	$0.485683\pi$
$644$	27.8283	1.09659
$645$	0	0
$646$	−7.72013	−0.303744
$647$	−15.3848	−0.604840	−0.302420	−	0.953175i	$-0.597794\pi$
$647$	−15.3848	−0.604840	−0.302420	+	0.953175i	$0.597794\pi$
$648$	0	0
$649$	−10.6898	−0.419613
$650$	0	0
$651$	0	0
$652$	67.8481	2.65714
$653$	−24.7373	−0.968046	−0.484023	−	0.875055i	$-0.660825\pi$
$653$	−24.7373	−0.968046	−0.484023	+	0.875055i	$0.660825\pi$
$654$	0	0
$655$	0	0
$656$	147.352	5.75312
$657$	0	0
$658$	2.15093	0.0838520
$659$	17.3744	0.676812	0.338406	−	0.941000i	$-0.390112\pi$
$659$	17.3744	0.676812	0.338406	+	0.941000i	$0.390112\pi$
$660$	0	0
$661$	4.18538	0.162793	0.0813963	−	0.996682i	$-0.474062\pi$
$661$	4.18538	0.162793	0.0813963	+	0.996682i	$0.474062\pi$
$662$	32.5442	1.26487
$663$	0	0
$664$	50.2663	1.95071
$665$	0	0
$666$	0	0
$667$	−0.643006	−0.0248973
$668$	115.068	4.45213
$669$	0	0
$670$	0	0
$671$	5.71049	0.220451
$672$	0	0
$673$	16.8948	0.651246	0.325623	−	0.945500i	$-0.394426\pi$
$673$	16.8948	0.651246	0.325623	+	0.945500i	$0.394426\pi$
$674$	−20.1602	−0.776543
$675$	0	0
$676$	−17.2214	−0.662361
$677$	−6.13351	−0.235730	−0.117865	−	0.993030i	$-0.537605\pi$
$677$	−6.13351	−0.235730	−0.117865	+	0.993030i	$0.537605\pi$
$678$	0	0
$679$	16.3531	0.627576
$680$	0	0
$681$	0	0
$682$	−2.50259	−0.0958292
$683$	26.7588	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$683$	26.7588	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$684$	0	0
$685$	0	0
$686$	45.6386	1.74249
$687$	0	0
$688$	121.032	4.61430
$689$	−24.8784	−0.947791
$690$	0	0
$691$	23.1367	0.880160	0.440080	−	0.897959i	$-0.354950\pi$
$691$	23.1367	0.880160	0.440080	+	0.897959i	$0.354950\pi$
$692$	−116.065	−4.41212
$693$	0	0
$694$	−23.7914	−0.903107
$695$	0	0
$696$	0	0
$697$	−7.97191	−0.301957
$698$	−0.788795	−0.0298563
$699$	0	0
$700$	0	0
$701$	−41.0017	−1.54861	−0.774307	−	0.632810i	$-0.781902\pi$
$701$	−41.0017	−1.54861	−0.774307	+	0.632810i	$0.781902\pi$
$702$	0	0
$703$	23.2843	0.878182
$704$	−52.2301	−1.96850
$705$	0	0
$706$	−21.1890	−0.797458
$707$	3.97379	0.149450
$708$	0	0
$709$	−10.7096	−0.402207	−0.201103	−	0.979570i	$-0.564453\pi$
$709$	−10.7096	−0.402207	−0.201103	+	0.979570i	$0.564453\pi$
$710$	0	0
$711$	0	0
$712$	−32.4208	−1.21502
$713$	−2.91417	−0.109136
$714$	0	0
$715$	0	0
$716$	39.2351	1.46628
$717$	0	0
$718$	−11.8825	−0.443452
$719$	6.17046	0.230119	0.115060	−	0.993359i	$-0.463294\pi$
$719$	6.17046	0.230119	0.115060	+	0.993359i	$0.463294\pi$
$720$	0	0
$721$	18.4538	0.687254
$722$	−31.9427	−1.18878
$723$	0	0
$724$	8.74357	0.324952
$725$	0	0
$726$	0	0
$727$	−38.9277	−1.44375	−0.721875	−	0.692024i	$-0.756719\pi$
$727$	−38.9277	−1.44375	−0.721875	+	0.692024i	$0.756719\pi$
$728$	−45.5337	−1.68759
$729$	0	0
$730$	0	0
$731$	−6.54798	−0.242186
$732$	0	0
$733$	31.8680	1.17707	0.588535	−	0.808472i	$-0.299705\pi$
$733$	31.8680	1.17707	0.588535	+	0.808472i	$0.299705\pi$
$734$	−78.3245	−2.89101
$735$	0	0
$736$	−107.950	−3.97910
$737$	2.90094	0.106858
$738$	0	0
$739$	29.7808	1.09551	0.547753	−	0.836640i	$-0.315483\pi$
$739$	29.7808	1.09551	0.547753	+	0.836640i	$0.315483\pi$
$740$	0	0
$741$	0	0
$742$	29.2953	1.07546
$743$	−33.4684	−1.22784	−0.613918	−	0.789370i	$-0.710408\pi$
$743$	−33.4684	−1.22784	−0.613918	+	0.789370i	$0.710408\pi$
$744$	0	0
$745$	0	0
$746$	−26.3221	−0.963721
$747$	0	0
$748$	6.40051	0.234026
$749$	−10.4919	−0.383367
$750$	0	0
$751$	15.9989	0.583808	0.291904	−	0.956448i	$-0.405711\pi$
$751$	15.9989	0.583808	0.291904	+	0.956448i	$0.405711\pi$
$752$	−10.6481	−0.388295
$753$	0	0
$754$	1.59922	0.0582403
$755$	0	0
$756$	0	0
$757$	−20.6978	−0.752275	−0.376138	−	0.926564i	$-0.622748\pi$
$757$	−20.6978	−0.752275	−0.376138	+	0.926564i	$0.622748\pi$
$758$	67.2123	2.44126
$759$	0	0
$760$	0	0
$761$	22.4458	0.813660	0.406830	−	0.913504i	$-0.366634\pi$
$761$	22.4458	0.813660	0.406830	+	0.913504i	$0.366634\pi$
$762$	0	0
$763$	−18.2270	−0.659861
$764$	93.1390	3.36965
$765$	0	0
$766$	−5.54300	−0.200277
$767$	30.9592	1.11787
$768$	0	0
$769$	30.6260	1.10440	0.552202	−	0.833711i	$-0.313788\pi$
$769$	30.6260	1.10440	0.552202	+	0.833711i	$0.313788\pi$
$770$	0	0
$771$	0	0
$772$	−102.114	−3.67516
$773$	10.9578	0.394126	0.197063	−	0.980391i	$-0.436860\pi$
$773$	10.9578	0.394126	0.197063	+	0.980391i	$0.436860\pi$
$774$	0	0
$775$	0	0
$776$	−132.164	−4.74441
$777$	0	0
$778$	74.4354	2.66864
$779$	21.9716	0.787215
$780$	0	0
$781$	14.6835	0.525415
$782$	10.0029	0.357705
$783$	0	0
$784$	−96.5444	−3.44801
$785$	0	0
$786$	0	0
$787$	31.1539	1.11052	0.555259	−	0.831677i	$-0.312619\pi$
$787$	31.1539	1.11052	0.555259	+	0.831677i	$0.312619\pi$
$788$	82.0993	2.92467
$789$	0	0
$790$	0	0
$791$	26.0894	0.927631
$792$	0	0
$793$	−16.5383	−0.587294
$794$	17.7035	0.628274
$795$	0	0
$796$	−26.8081	−0.950189
$797$	−15.2518	−0.540246	−0.270123	−	0.962826i	$-0.587064\pi$
$797$	−15.2518	−0.540246	−0.270123	+	0.962826i	$0.587064\pi$
$798$	0	0
$799$	0.576074	0.0203800
$800$	0	0
$801$	0	0
$802$	42.0148	1.48359
$803$	13.4994	0.476383
$804$	0	0
$805$	0	0
$806$	7.24784	0.255294
$807$	0	0
$808$	−32.1157	−1.12983
$809$	2.66539	0.0937100	0.0468550	−	0.998902i	$-0.485080\pi$
$809$	2.66539	0.0937100	0.0468550	+	0.998902i	$0.485080\pi$
$810$	0	0
$811$	5.54482	0.194705	0.0973525	−	0.995250i	$-0.468963\pi$
$811$	5.54482	0.194705	0.0973525	+	0.995250i	$0.468963\pi$
$812$	−1.40312	−0.0492399
$813$	0	0
$814$	−25.9085	−0.908091
$815$	0	0
$816$	0	0
$817$	18.0471	0.631388
$818$	50.4564	1.76417
$819$	0	0
$820$	0	0
$821$	−31.2022	−1.08896	−0.544482	−	0.838773i	$-0.683274\pi$
$821$	−31.2022	−1.08896	−0.544482	+	0.838773i	$0.683274\pi$
$822$	0	0
$823$	50.4682	1.75921	0.879606	−	0.475703i	$-0.157806\pi$
$823$	50.4682	1.75921	0.879606	+	0.475703i	$0.157806\pi$
$824$	−149.141	−5.19558
$825$	0	0
$826$	−36.4557	−1.26845
$827$	25.7704	0.896123	0.448062	−	0.894003i	$-0.352114\pi$
$827$	25.7704	0.896123	0.448062	+	0.894003i	$0.352114\pi$
$828$	0	0
$829$	4.34228	0.150814	0.0754068	−	0.997153i	$-0.475974\pi$
$829$	4.34228	0.150814	0.0754068	+	0.997153i	$0.475974\pi$
$830$	0	0
$831$	0	0
$832$	151.265	5.24418
$833$	5.22317	0.180972
$834$	0	0
$835$	0	0
$836$	−17.6406	−0.610114
$837$	0	0
$838$	31.4545	1.08658
$839$	−44.9482	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$839$	−44.9482	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$840$	0	0
$841$	−28.9676	−0.998882
$842$	106.140	3.65784
$843$	0	0
$844$	42.2869	1.45557
$845$	0	0
$846$	0	0
$847$	13.0649	0.448917
$848$	−145.025	−4.98017
$849$	0	0
$850$	0	0
$851$	−30.1693	−1.03419
$852$	0	0
$853$	−15.7009	−0.537590	−0.268795	−	0.963197i	$-0.586625\pi$
$853$	−15.7009	−0.537590	−0.268795	+	0.963197i	$0.586625\pi$
$854$	19.4745	0.666405
$855$	0	0
$856$	84.7945	2.89821
$857$	−6.77401	−0.231396	−0.115698	−	0.993284i	$-0.536910\pi$
$857$	−6.77401	−0.231396	−0.115698	+	0.993284i	$0.536910\pi$
$858$	0	0
$859$	36.0071	1.22855	0.614273	−	0.789094i	$-0.289449\pi$
$859$	36.0071	1.22855	0.614273	+	0.789094i	$0.289449\pi$
$860$	0	0
$861$	0	0
$862$	−41.2182	−1.40390
$863$	35.5716	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$863$	35.5716	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$864$	0	0
$865$	0	0
$866$	−105.090	−3.57111
$867$	0	0
$868$	−6.35909	−0.215841
$869$	−4.10289	−0.139181
$870$	0	0
$871$	−8.40152	−0.284675
$872$	147.308	4.98849
$873$	0	0
$874$	−27.5694	−0.932550
$875$	0	0
$876$	0	0
$877$	−24.1451	−0.815321	−0.407660	−	0.913134i	$-0.633655\pi$
$877$	−24.1451	−0.815321	−0.407660	+	0.913134i	$0.633655\pi$
$878$	97.5733	3.29294
$879$	0	0
$880$	0	0
$881$	−32.9124	−1.10885	−0.554423	−	0.832235i	$-0.687061\pi$
$881$	−32.9124	−1.10885	−0.554423	+	0.832235i	$0.687061\pi$
$882$	0	0
$883$	15.5885	0.524594	0.262297	−	0.964987i	$-0.415520\pi$
$883$	15.5885	0.524594	0.262297	+	0.964987i	$0.415520\pi$
$884$	−18.5367	−0.623458
$885$	0	0
$886$	−27.5717	−0.926289
$887$	−8.59951	−0.288743	−0.144372	−	0.989524i	$-0.546116\pi$
$887$	−8.59951	−0.288743	−0.144372	+	0.989524i	$0.546116\pi$
$888$	0	0
$889$	6.67750	0.223956
$890$	0	0
$891$	0	0
$892$	−130.113	−4.35649
$893$	−1.58773	−0.0531315
$894$	0	0
$895$	0	0
$896$	−97.5320	−3.25832
$897$	0	0
$898$	−77.7127	−2.59331
$899$	0.146934	0.00490053
$900$	0	0
$901$	7.84602	0.261389
$902$	−24.4479	−0.814025
$903$	0	0
$904$	−210.851	−7.01280
$905$	0	0
$906$	0	0
$907$	−44.9819	−1.49360	−0.746800	−	0.665049i	$-0.768411\pi$
$907$	−44.9819	−1.49360	−0.746800	+	0.665049i	$0.768411\pi$
$908$	−78.0682	−2.59079
$909$	0	0
$910$	0	0
$911$	−50.9800	−1.68904	−0.844522	−	0.535521i	$-0.820115\pi$
$911$	−50.9800	−1.68904	−0.844522	+	0.535521i	$0.820115\pi$
$912$	0	0
$913$	−5.10852	−0.169067
$914$	78.6969	2.60306
$915$	0	0
$916$	−119.637	−3.95292
$917$	3.51763	0.116163
$918$	0	0
$919$	−13.0346	−0.429973	−0.214986	−	0.976617i	$-0.568971\pi$
$919$	−13.0346	−0.429973	−0.214986	+	0.976617i	$0.568971\pi$
$920$	0	0
$921$	0	0
$922$	−14.5776	−0.480088
$923$	−42.5252	−1.39974
$924$	0	0
$925$	0	0
$926$	−3.59276	−0.118066
$927$	0	0
$928$	5.44292	0.178673
$929$	−37.8748	−1.24263	−0.621316	−	0.783560i	$-0.713402\pi$
$929$	−37.8748	−1.24263	−0.621316	+	0.783560i	$0.713402\pi$
$930$	0	0
$931$	−14.3957	−0.471801
$932$	94.2604	3.08760
$933$	0	0
$934$	−68.6280	−2.24558
$935$	0	0
$936$	0	0
$937$	22.9509	0.749772	0.374886	−	0.927071i	$-0.377682\pi$
$937$	22.9509	0.749772	0.374886	+	0.927071i	$0.377682\pi$
$938$	9.89311	0.323022
$939$	0	0
$940$	0	0
$941$	33.8175	1.10242	0.551209	−	0.834367i	$-0.314167\pi$
$941$	33.8175	1.10242	0.551209	+	0.834367i	$0.314167\pi$
$942$	0	0
$943$	−28.4685	−0.927064
$944$	180.472	5.87387
$945$	0	0
$946$	−20.0810	−0.652891
$947$	−10.0614	−0.326953	−0.163476	−	0.986547i	$-0.552271\pi$
$947$	−10.0614	−0.326953	−0.163476	+	0.986547i	$0.552271\pi$
$948$	0	0
$949$	−39.0960	−1.26911
$950$	0	0
$951$	0	0
$952$	14.3602	0.465416
$953$	−38.8829	−1.25954	−0.629770	−	0.776781i	$-0.716851\pi$
$953$	−38.8829	−1.25954	−0.629770	+	0.776781i	$0.716851\pi$
$954$	0	0
$955$	0	0
$956$	25.6428	0.829348
$957$	0	0
$958$	87.1612	2.81605
$959$	−4.35970	−0.140782
$960$	0	0
$961$	−30.3341	−0.978519
$962$	75.0344	2.41920
$963$	0	0
$964$	−86.7213	−2.79311
$965$	0	0
$966$	0	0
$967$	−5.40033	−0.173663	−0.0868315	−	0.996223i	$-0.527674\pi$
$967$	−5.40033	−0.173663	−0.0868315	+	0.996223i	$0.527674\pi$
$968$	−105.589	−3.39377
$969$	0	0
$970$	0	0
$971$	−10.1469	−0.325629	−0.162815	−	0.986657i	$-0.552057\pi$
$971$	−10.1469	−0.325629	−0.162815	+	0.986657i	$0.552057\pi$
$972$	0	0
$973$	13.8498	0.444004
$974$	55.6933	1.78453
$975$	0	0
$976$	−96.4077	−3.08594
$977$	9.09855	0.291088	0.145544	−	0.989352i	$-0.453507\pi$
$977$	9.09855	0.291088	0.145544	+	0.989352i	$0.453507\pi$
$978$	0	0
$979$	3.29490	0.105306
$980$	0	0
$981$	0	0
$982$	−55.7121	−1.77784
$983$	−49.1437	−1.56744	−0.783721	−	0.621113i	$-0.786681\pi$
$983$	−49.1437	−1.56744	−0.783721	+	0.621113i	$0.786681\pi$
$984$	0	0
$985$	0	0
$986$	−0.504355	−0.0160619
$987$	0	0
$988$	51.0897	1.62538
$989$	−23.3836	−0.743554
$990$	0	0
$991$	43.2065	1.37250	0.686249	−	0.727366i	$-0.259256\pi$
$991$	43.2065	1.37250	0.686249	+	0.727366i	$0.259256\pi$
$992$	24.6679	0.783205
$993$	0	0
$994$	50.0751	1.58829
$995$	0	0
$996$	0	0
$997$	48.5452	1.53744	0.768721	−	0.639584i	$-0.220893\pi$
$997$	48.5452	1.53744	0.768721	+	0.639584i	$0.220893\pi$
$998$	13.1174	0.415224
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.bq.1.5		5
3.2	odd	2		425.2.a.i.1.1	✓	5
5.4	even	2		3825.2.a.bl.1.1		5
12.11	even	2		6800.2.a.bz.1.1		5
15.2	even	4		425.2.b.f.324.1		10
15.8	even	4		425.2.b.f.324.10		10
15.14	odd	2		425.2.a.j.1.5	yes	5
51.50	odd	2		7225.2.a.x.1.1		5
60.59	even	2		6800.2.a.cd.1.5		5
255.254	odd	2		7225.2.a.y.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.1	✓	5	3.2	odd	2
425.2.a.j.1.5	yes	5	15.14	odd	2
425.2.b.f.324.1		10	15.2	even	4
425.2.b.f.324.10		10	15.8	even	4
3825.2.a.bl.1.1		5	5.4	even	2
3825.2.a.bq.1.5		5	1.1	even	1	trivial
6800.2.a.bz.1.1		5	12.11	even	2
6800.2.a.cd.1.5		5	60.59	even	2
7225.2.a.x.1.1		5	51.50	odd	2
7225.2.a.y.1.5		5	255.254	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80107 q^{2} +5.84602 q^{4} -1.33298 q^{7} +10.7730 q^{8} -1.09485 q^{11} +3.17083 q^{13} -3.73378 q^{14} +18.4839 q^{16} -1.00000 q^{17} +2.75613 q^{19} -3.06675 q^{22} -3.57111 q^{23} +8.88173 q^{26} -7.79262 q^{28} +0.180058 q^{29} +0.816039 q^{31} +30.2288 q^{32} -2.80107 q^{34} +8.44817 q^{37} +7.72013 q^{38} +7.97191 q^{41} +6.54798 q^{43} -6.40051 q^{44} -10.0029 q^{46} -0.576074 q^{47} -5.22317 q^{49} +18.5367 q^{52} -7.84602 q^{53} -14.3602 q^{56} +0.504355 q^{58} +9.76375 q^{59} -5.21577 q^{61} +2.28579 q^{62} +47.7053 q^{64} -2.64963 q^{67} -5.84602 q^{68} -13.4114 q^{71} -12.3299 q^{73} +23.6639 q^{74} +16.1124 q^{76} +1.45941 q^{77} +3.74745 q^{79} +22.3299 q^{82} +4.66596 q^{83} +18.3414 q^{86} -11.7948 q^{88} -3.00946 q^{89} -4.22665 q^{91} -20.8768 q^{92} -1.61363 q^{94} -12.2681 q^{97} -14.6305 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 11 q^{4} + q^{7} + 9 q^{8} - 4 q^{11} - 3 q^{13} + 7 q^{14} + 27 q^{16} - 5 q^{17} + 6 q^{19} + 18 q^{22} - 4 q^{23} + 5 q^{26} - 15 q^{28} - 2 q^{29} + 21 q^{31} + 9 q^{32} - q^{34} - 2 q^{37}+ \cdots - 38 q^{98}+O(q^{100}) \) 5 * q + q^2 + 11 * q^4 + q^7 + 9 * q^8 - 4 * q^11 - 3 * q^13 + 7 * q^14 + 27 * q^16 - 5 * q^17 + 6 * q^19 + 18 * q^22 - 4 * q^23 + 5 * q^26 - 15 * q^28 - 2 * q^29 + 21 * q^31 + 9 * q^32 - q^34 - 2 * q^37 + 34 * q^38 + 8 * q^41 - 4 * q^43 + 4 * q^44 - 6 * q^46 + 2 * q^47 + 10 * q^49 - 13 * q^52 - 21 * q^53 + 29 * q^56 + 22 * q^58 - 12 * q^59 - 2 * q^61 - 3 * q^62 + 43 * q^64 - 12 * q^67 - 11 * q^68 - 21 * q^71 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 + 41 * q^79 + 28 * q^82 + 8 * q^83 + 36 * q^86 + 14 * q^88 + 10 * q^89 + 13 * q^91 - 56 * q^92 - 10 * q^94 - 20 * q^97 - 38 * q^98

Introduction

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