LMFDB - Embedded newform 5175.2.a.by.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5175,2,Mod(1,5175)] 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("5175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,0,0,-6,0,0,0,-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:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.98838128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 $ x^6 - x^5 - 10x^4 - x^3 + 16x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1035)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.52269$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.605771 q^{2} -1.63304 q^{4} -3.25359 q^{7} +2.20079 q^{8} -0.759390 q^{11} +0.806561 q^{13} +1.97093 q^{14} +1.93291 q^{16} -5.43960 q^{17} +0.0290683 q^{19} +0.460016 q^{22} +1.00000 q^{23} -0.488591 q^{26} +5.31325 q^{28} +1.55297 q^{29} +4.63304 q^{31} -5.57248 q^{32} +3.29515 q^{34} -5.52390 q^{37} -0.0176087 q^{38} -5.76029 q^{41} -10.8320 q^{43} +1.24012 q^{44} -0.605771 q^{46} +0.299374 q^{47} +3.58587 q^{49} -1.31715 q^{52} -0.633536 q^{53} -7.16048 q^{56} -0.940742 q^{58} -5.27170 q^{59} +6.50669 q^{61} -2.80656 q^{62} -0.490173 q^{64} -9.24263 q^{67} +8.88310 q^{68} -10.9978 q^{71} -4.80656 q^{73} +3.34622 q^{74} -0.0474698 q^{76} +2.47075 q^{77} +11.6677 q^{79} +3.48941 q^{82} +4.30770 q^{83} +6.56173 q^{86} -1.67126 q^{88} +16.6163 q^{89} -2.62422 q^{91} -1.63304 q^{92} -0.181352 q^{94} -2.40449 q^{97} -2.17222 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} - 4 q^{11} - 12 q^{13} + 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 8 q^{22} + 6 q^{23} + 12 q^{26} - 24 q^{28} + 6 q^{29} + 8 q^{31} + 20 q^{32} - 12 q^{34} - 22 q^{37} + 36 q^{38}+ \cdots - 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 - 4 * q^11 - 12 * q^13 + 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 8 * q^22 + 6 * q^23 + 12 * q^26 - 24 * q^28 + 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 22 * q^37 + 36 * q^38 + 18 * q^41 - 8 * q^43 - 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^56 + 16 * q^58 + 6 * q^59 + 14 * q^64 - 10 * q^67 + 28 * q^68 + 22 * q^71 - 12 * q^73 + 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^82 + 24 * q^83 - 20 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 12 * q^97 - 4 * 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 $

$2$	−0.605771	−0.428345	−0.214172	−	0.976796i	$-0.568705\pi$
$2$	−0.605771	−0.428345	−0.214172	+	0.976796i	$0.568705\pi$
$3$	0	0
$3$	0	0
$4$	−1.63304	−0.816521
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.25359	−1.22974	−0.614871	−	0.788627i	$-0.710792\pi$
$7$	−3.25359	−1.22974	−0.614871	+	0.788627i	$0.710792\pi$
$8$	2.20079	0.778097
$9$	0	0
$10$	0	0
$11$	−0.759390	−0.228965	−0.114482	−	0.993425i	$-0.536521\pi$
$11$	−0.759390	−0.228965	−0.114482	+	0.993425i	$0.536521\pi$
$12$	0	0
$13$	0.806561	0.223700	0.111850	−	0.993725i	$-0.464322\pi$
$13$	0.806561	0.223700	0.111850	+	0.993725i	$0.464322\pi$
$14$	1.97093	0.526754
$15$	0	0
$16$	1.93291	0.483227
$17$	−5.43960	−1.31930	−0.659649	−	0.751574i	$-0.729295\pi$
$17$	−5.43960	−1.31930	−0.659649	+	0.751574i	$0.729295\pi$
$18$	0	0
$19$	0.0290683	0.00666873	0.00333437	−	0.999994i	$-0.498939\pi$
$19$	0.0290683	0.00666873	0.00333437	+	0.999994i	$0.498939\pi$
$20$	0	0
$21$	0	0
$22$	0.460016	0.0980758
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	−0.488591	−0.0958206
$27$	0	0
$28$	5.31325	1.00411
$29$	1.55297	0.288379	0.144189	−	0.989550i	$-0.453943\pi$
$29$	1.55297	0.288379	0.144189	+	0.989550i	$0.453943\pi$
$30$	0	0
$31$	4.63304	0.832119	0.416059	−	0.909337i	$-0.363411\pi$
$31$	4.63304	0.832119	0.416059	+	0.909337i	$0.363411\pi$
$32$	−5.57248	−0.985085
$33$	0	0
$34$	3.29515	0.565114
$35$	0	0
$36$	0	0
$37$	−5.52390	−0.908123	−0.454062	−	0.890970i	$-0.650025\pi$
$37$	−5.52390	−0.908123	−0.454062	+	0.890970i	$0.650025\pi$
$38$	−0.0176087	−0.00285652
$39$	0	0
$40$	0	0
$41$	−5.76029	−0.899606	−0.449803	−	0.893128i	$-0.648506\pi$
$41$	−5.76029	−0.899606	−0.449803	+	0.893128i	$0.648506\pi$
$42$	0	0
$43$	−10.8320	−1.65187	−0.825935	−	0.563765i	$-0.809352\pi$
$43$	−10.8320	−1.65187	−0.825935	+	0.563765i	$0.809352\pi$
$44$	1.24012	0.186955
$45$	0	0
$46$	−0.605771	−0.0893160
$47$	0.299374	0.0436682	0.0218341	−	0.999762i	$-0.493049\pi$
$47$	0.299374	0.0436682	0.0218341	+	0.999762i	$0.493049\pi$
$48$	0	0
$49$	3.58587	0.512267
$50$	0	0
$51$	0	0
$52$	−1.31715	−0.182656
$53$	−0.633536	−0.0870228	−0.0435114	−	0.999053i	$-0.513854\pi$
$53$	−0.633536	−0.0870228	−0.0435114	+	0.999053i	$0.513854\pi$
$54$	0	0
$55$	0	0
$56$	−7.16048	−0.956859
$57$	0	0
$58$	−0.940742	−0.123526
$59$	−5.27170	−0.686316	−0.343158	−	0.939278i	$-0.611497\pi$
$59$	−5.27170	−0.686316	−0.343158	+	0.939278i	$0.611497\pi$
$60$	0	0
$61$	6.50669	0.833097	0.416548	−	0.909114i	$-0.363240\pi$
$61$	6.50669	0.833097	0.416548	+	0.909114i	$0.363240\pi$
$62$	−2.80656	−0.356434
$63$	0	0
$64$	−0.490173	−0.0612717
$65$	0	0
$66$	0	0
$67$	−9.24263	−1.12917	−0.564583	−	0.825376i	$-0.690963\pi$
$67$	−9.24263	−1.12917	−0.564583	+	0.825376i	$0.690963\pi$
$68$	8.88310	1.07723
$69$	0	0
$70$	0	0
$71$	−10.9978	−1.30520	−0.652599	−	0.757703i	$-0.726321\pi$
$71$	−10.9978	−1.30520	−0.652599	+	0.757703i	$0.726321\pi$
$72$	0	0
$73$	−4.80656	−0.562565	−0.281283	−	0.959625i	$-0.590760\pi$
$73$	−4.80656	−0.562565	−0.281283	+	0.959625i	$0.590760\pi$
$74$	3.34622	0.388990
$75$	0	0
$76$	−0.0474698	−0.00544516
$77$	2.47075	0.281568
$78$	0	0
$79$	11.6677	1.31271	0.656357	−	0.754450i	$-0.272096\pi$
$79$	11.6677	1.31271	0.656357	+	0.754450i	$0.272096\pi$
$80$	0	0
$81$	0	0
$82$	3.48941	0.385341
$83$	4.30770	0.472832	0.236416	−	0.971652i	$-0.424027\pi$
$83$	4.30770	0.472832	0.236416	+	0.971652i	$0.424027\pi$
$84$	0	0
$85$	0	0
$86$	6.56173	0.707569
$87$	0	0
$88$	−1.67126	−0.178157
$89$	16.6163	1.76132	0.880661	−	0.473748i	$-0.157099\pi$
$89$	16.6163	1.76132	0.880661	+	0.473748i	$0.157099\pi$
$90$	0	0
$91$	−2.62422	−0.275093
$92$	−1.63304	−0.170256
$93$	0	0
$94$	−0.181352	−0.0187050
$95$	0	0
$96$	0	0
$97$	−2.40449	−0.244139	−0.122069	−	0.992522i	$-0.538953\pi$
$97$	−2.40449	−0.244139	−0.122069	+	0.992522i	$0.538953\pi$
$98$	−2.17222	−0.219427
$99$	0	0
$100$	0	0
$101$	7.27907	0.724294	0.362147	−	0.932121i	$-0.382044\pi$
$101$	7.27907	0.724294	0.362147	+	0.932121i	$0.382044\pi$
$102$	0	0
$103$	7.73634	0.762284	0.381142	−	0.924517i	$-0.375531\pi$
$103$	7.73634	0.762284	0.381142	+	0.924517i	$0.375531\pi$
$104$	1.77507	0.174060
$105$	0	0
$106$	0.383777	0.0372757
$107$	5.11476	0.494462	0.247231	−	0.968957i	$-0.420479\pi$
$107$	5.11476	0.494462	0.247231	+	0.968957i	$0.420479\pi$
$108$	0	0
$109$	−15.8090	−1.51423	−0.757113	−	0.653284i	$-0.773391\pi$
$109$	−15.8090	−1.51423	−0.757113	+	0.653284i	$0.773391\pi$
$110$	0	0
$111$	0	0
$112$	−6.28890	−0.594245
$113$	8.92145	0.839259	0.419630	−	0.907695i	$-0.362160\pi$
$113$	8.92145	0.839259	0.419630	+	0.907695i	$0.362160\pi$
$114$	0	0
$115$	0	0
$116$	−2.53606	−0.235467
$117$	0	0
$118$	3.19344	0.293980
$119$	17.6983	1.62240
$120$	0	0
$121$	−10.4233	−0.947575
$122$	−3.94156	−0.356852
$123$	0	0
$124$	−7.56595	−0.679442
$125$	0	0
$126$	0	0
$127$	6.54363	0.580653	0.290327	−	0.956928i	$-0.406236\pi$
$127$	6.54363	0.580653	0.290327	+	0.956928i	$0.406236\pi$
$128$	11.4419	1.01133
$129$	0	0
$130$	0	0
$131$	−17.2727	−1.50913	−0.754563	−	0.656228i	$-0.772151\pi$
$131$	−17.2727	−1.50913	−0.754563	+	0.656228i	$0.772151\pi$
$132$	0	0
$133$	−0.0945766	−0.00820083
$134$	5.59891	0.483672
$135$	0	0
$136$	−11.9714	−1.02654
$137$	15.4734	1.32198	0.660991	−	0.750394i	$-0.270136\pi$
$137$	15.4734	1.32198	0.660991	+	0.750394i	$0.270136\pi$
$138$	0	0
$139$	−4.54603	−0.385589	−0.192795	−	0.981239i	$-0.561755\pi$
$139$	−4.54603	−0.385589	−0.192795	+	0.981239i	$0.561755\pi$
$140$	0	0
$141$	0	0
$142$	6.66214	0.559075
$143$	−0.612495	−0.0512194
$144$	0	0
$145$	0	0
$146$	2.91167	0.240972
$147$	0	0
$148$	9.02076	0.741502
$149$	7.83190	0.641614	0.320807	−	0.947145i	$-0.396046\pi$
$149$	7.83190	0.641614	0.320807	+	0.947145i	$0.396046\pi$
$150$	0	0
$151$	−8.29142	−0.674747	−0.337373	−	0.941371i	$-0.609538\pi$
$151$	−8.29142	−0.674747	−0.337373	+	0.941371i	$0.609538\pi$
$152$	0.0639733	0.00518892
$153$	0	0
$154$	−1.49671	−0.120608
$155$	0	0
$156$	0	0
$157$	−10.0783	−0.804332	−0.402166	−	0.915567i	$-0.631743\pi$
$157$	−10.0783	−0.804332	−0.402166	+	0.915567i	$0.631743\pi$
$158$	−7.06793	−0.562294
$159$	0	0
$160$	0	0
$161$	−3.25359	−0.256419
$162$	0	0
$163$	3.65296	0.286122	0.143061	−	0.989714i	$-0.454305\pi$
$163$	3.65296	0.286122	0.143061	+	0.989714i	$0.454305\pi$
$164$	9.40679	0.734547
$165$	0	0
$166$	−2.60948	−0.202535
$167$	17.6276	1.36407	0.682033	−	0.731321i	$-0.261096\pi$
$167$	17.6276	1.36407	0.682033	+	0.731321i	$0.261096\pi$
$168$	0	0
$169$	−12.3495	−0.949958
$170$	0	0
$171$	0	0
$172$	17.6892	1.34879
$173$	6.56173	0.498879	0.249440	−	0.968390i	$-0.419754\pi$
$173$	6.56173	0.498879	0.249440	+	0.968390i	$0.419754\pi$
$174$	0	0
$175$	0	0
$176$	−1.46783	−0.110642
$177$	0	0
$178$	−10.0657	−0.754452
$179$	1.92327	0.143752	0.0718759	−	0.997414i	$-0.477101\pi$
$179$	1.92327	0.143752	0.0718759	+	0.997414i	$0.477101\pi$
$180$	0	0
$181$	4.77012	0.354560	0.177280	−	0.984160i	$-0.443270\pi$
$181$	4.77012	0.354560	0.177280	+	0.984160i	$0.443270\pi$
$182$	1.58968	0.117835
$183$	0	0
$184$	2.20079	0.162244
$185$	0	0
$186$	0	0
$187$	4.13078	0.302073
$188$	−0.488890	−0.0356560
$189$	0	0
$190$	0	0
$191$	6.92978	0.501421	0.250711	−	0.968062i	$-0.419336\pi$
$191$	6.92978	0.501421	0.250711	+	0.968062i	$0.419336\pi$
$192$	0	0
$193$	3.26608	0.235098	0.117549	−	0.993067i	$-0.462496\pi$
$193$	3.26608	0.235098	0.117549	+	0.993067i	$0.462496\pi$
$194$	1.45657	0.104575
$195$	0	0
$196$	−5.85588	−0.418277
$197$	23.8278	1.69766	0.848831	−	0.528665i	$-0.177307\pi$
$197$	23.8278	1.69766	0.848831	+	0.528665i	$0.177307\pi$
$198$	0	0
$199$	−12.7303	−0.902429	−0.451214	−	0.892416i	$-0.649009\pi$
$199$	−12.7303	−0.902429	−0.451214	+	0.892416i	$0.649009\pi$
$200$	0	0
$201$	0	0
$202$	−4.40945	−0.310248
$203$	−5.05272	−0.354632
$204$	0	0
$205$	0	0
$206$	−4.68645	−0.326520
$207$	0	0
$208$	1.55901	0.108098
$209$	−0.0220742	−0.00152691
$210$	0	0
$211$	22.4213	1.54355	0.771773	−	0.635898i	$-0.219370\pi$
$211$	22.4213	1.54355	0.771773	+	0.635898i	$0.219370\pi$
$212$	1.03459	0.0710559
$213$	0	0
$214$	−3.09837	−0.211800
$215$	0	0
$216$	0	0
$217$	−15.0740	−1.02329
$218$	9.57662	0.648610
$219$	0	0
$220$	0	0
$221$	−4.38737	−0.295127
$222$	0	0
$223$	9.19320	0.615622	0.307811	−	0.951447i	$-0.400403\pi$
$223$	9.19320	0.615622	0.307811	+	0.951447i	$0.400403\pi$
$224$	18.1306	1.21140
$225$	0	0
$226$	−5.40435	−0.359492
$227$	−25.1015	−1.66605	−0.833023	−	0.553238i	$-0.813392\pi$
$227$	−25.1015	−1.66605	−0.833023	+	0.553238i	$0.813392\pi$
$228$	0	0
$229$	6.34605	0.419359	0.209679	−	0.977770i	$-0.432758\pi$
$229$	6.34605	0.419359	0.209679	+	0.977770i	$0.432758\pi$
$230$	0	0
$231$	0	0
$232$	3.41776	0.224387
$233$	26.0075	1.70381	0.851904	−	0.523697i	$-0.175448\pi$
$233$	26.0075	1.70381	0.851904	+	0.523697i	$0.175448\pi$
$234$	0	0
$235$	0	0
$236$	8.60890	0.560392
$237$	0	0
$238$	−10.7211	−0.694945
$239$	10.9137	0.705948	0.352974	−	0.935633i	$-0.385170\pi$
$239$	10.9137	0.705948	0.352974	+	0.935633i	$0.385170\pi$
$240$	0	0
$241$	0.336307	0.0216635	0.0108317	−	0.999941i	$-0.496552\pi$
$241$	0.336307	0.0216635	0.0108317	+	0.999941i	$0.496552\pi$
$242$	6.31415	0.405889
$243$	0	0
$244$	−10.6257	−0.680241
$245$	0	0
$246$	0	0
$247$	0.0234454	0.00149179
$248$	10.1964	0.647469
$249$	0	0
$250$	0	0
$251$	6.78486	0.428257	0.214128	−	0.976806i	$-0.431309\pi$
$251$	6.78486	0.428257	0.214128	+	0.976806i	$0.431309\pi$
$252$	0	0
$253$	−0.759390	−0.0477425
$254$	−3.96394	−0.248720
$255$	0	0
$256$	−5.95082	−0.371926
$257$	−5.34232	−0.333245	−0.166622	−	0.986021i	$-0.553286\pi$
$257$	−5.34232	−0.333245	−0.166622	+	0.986021i	$0.553286\pi$
$258$	0	0
$259$	17.9725	1.11676
$260$	0	0
$261$	0	0
$262$	10.4633	0.646426
$263$	25.2748	1.55851	0.779255	−	0.626707i	$-0.215598\pi$
$263$	25.2748	1.55851	0.779255	+	0.626707i	$0.215598\pi$
$264$	0	0
$265$	0	0
$266$	0.0572917	0.00351278
$267$	0	0
$268$	15.0936	0.921988
$269$	3.97605	0.242424	0.121212	−	0.992627i	$-0.461322\pi$
$269$	3.97605	0.242424	0.121212	+	0.992627i	$0.461322\pi$
$270$	0	0
$271$	−22.1177	−1.34355	−0.671776	−	0.740754i	$-0.734468\pi$
$271$	−22.1177	−1.34355	−0.671776	+	0.740754i	$0.734468\pi$
$272$	−10.5143	−0.637521
$273$	0	0
$274$	−9.37333	−0.566264
$275$	0	0
$276$	0	0
$277$	0.495963	0.0297995	0.0148998	−	0.999889i	$-0.495257\pi$
$277$	0.495963	0.0297995	0.0148998	+	0.999889i	$0.495257\pi$
$278$	2.75385	0.165165
$279$	0	0
$280$	0	0
$281$	32.4709	1.93705	0.968525	−	0.248915i	$-0.0800738\pi$
$281$	32.4709	1.93705	0.968525	+	0.248915i	$0.0800738\pi$
$282$	0	0
$283$	3.26383	0.194015	0.0970073	−	0.995284i	$-0.469073\pi$
$283$	3.26383	0.194015	0.0970073	+	0.995284i	$0.469073\pi$
$284$	17.9599	1.06572
$285$	0	0
$286$	0.371031	0.0219395
$287$	18.7416	1.10628
$288$	0	0
$289$	12.5893	0.740546
$290$	0	0
$291$	0	0
$292$	7.84932	0.459346
$293$	−28.3698	−1.65738	−0.828689	−	0.559709i	$-0.810913\pi$
$293$	−28.3698	−1.65738	−0.828689	+	0.559709i	$0.810913\pi$
$294$	0	0
$295$	0	0
$296$	−12.1569	−0.706608
$297$	0	0
$298$	−4.74434	−0.274832
$299$	0.806561	0.0466446
$300$	0	0
$301$	35.2430	2.03138
$302$	5.02270	0.289024
$303$	0	0
$304$	0.0561865	0.00322252
$305$	0	0
$306$	0	0
$307$	20.7502	1.18428	0.592139	−	0.805836i	$-0.298284\pi$
$307$	20.7502	1.18428	0.592139	+	0.805836i	$0.298284\pi$
$308$	−4.03483	−0.229906
$309$	0	0
$310$	0	0
$311$	18.2147	1.03286	0.516430	−	0.856329i	$-0.327261\pi$
$311$	18.2147	1.03286	0.516430	+	0.856329i	$0.327261\pi$
$312$	0	0
$313$	18.5051	1.04597	0.522986	−	0.852342i	$-0.324818\pi$
$313$	18.5051	1.04597	0.522986	+	0.852342i	$0.324818\pi$
$314$	6.10511	0.344531
$315$	0	0
$316$	−19.0538	−1.07186
$317$	2.12539	0.119374	0.0596870	−	0.998217i	$-0.480990\pi$
$317$	2.12539	0.119374	0.0596870	+	0.998217i	$0.480990\pi$
$318$	0	0
$319$	−1.17931	−0.0660286
$320$	0	0
$321$	0	0
$322$	1.97093	0.109836
$323$	−0.158120	−0.00879804
$324$	0	0
$325$	0	0
$326$	−2.21286	−0.122559
$327$	0	0
$328$	−12.6772	−0.699980
$329$	−0.974041	−0.0537006
$330$	0	0
$331$	18.8396	1.03552	0.517760	−	0.855526i	$-0.326766\pi$
$331$	18.8396	1.03552	0.517760	+	0.855526i	$0.326766\pi$
$332$	−7.03465	−0.386077
$333$	0	0
$334$	−10.6783	−0.584291
$335$	0	0
$336$	0	0
$337$	−13.1864	−0.718311	−0.359156	−	0.933278i	$-0.616935\pi$
$337$	−13.1864	−0.718311	−0.359156	+	0.933278i	$0.616935\pi$
$338$	7.48094	0.406910
$339$	0	0
$340$	0	0
$341$	−3.51829	−0.190526
$342$	0	0
$343$	11.1082	0.599786
$344$	−23.8390	−1.28531
$345$	0	0
$346$	−3.97490	−0.213692
$347$	−19.1319	−1.02705	−0.513527	−	0.858073i	$-0.671661\pi$
$347$	−19.1319	−1.02705	−0.513527	+	0.858073i	$0.671661\pi$
$348$	0	0
$349$	21.5097	1.15139	0.575695	−	0.817665i	$-0.304732\pi$
$349$	21.5097	1.15139	0.575695	+	0.817665i	$0.304732\pi$
$350$	0	0
$351$	0	0
$352$	4.23169	0.225550
$353$	0.160343	0.00853422	0.00426711	−	0.999991i	$-0.498642\pi$
$353$	0.160343	0.00853422	0.00426711	+	0.999991i	$0.498642\pi$
$354$	0	0
$355$	0	0
$356$	−27.1351	−1.43816
$357$	0	0
$358$	−1.16506	−0.0615753
$359$	−17.8421	−0.941672	−0.470836	−	0.882221i	$-0.656048\pi$
$359$	−17.8421	−0.941672	−0.470836	+	0.882221i	$0.656048\pi$
$360$	0	0
$361$	−18.9992	−0.999956
$362$	−2.88960	−0.151874
$363$	0	0
$364$	4.28546	0.224619
$365$	0	0
$366$	0	0
$367$	24.0011	1.25285	0.626425	−	0.779482i	$-0.284518\pi$
$367$	24.0011	1.25285	0.626425	+	0.779482i	$0.284518\pi$
$368$	1.93291	0.100760
$369$	0	0
$370$	0	0
$371$	2.06127	0.107016
$372$	0	0
$373$	2.49263	0.129063	0.0645317	−	0.997916i	$-0.479445\pi$
$373$	2.49263	0.129063	0.0645317	+	0.997916i	$0.479445\pi$
$374$	−2.50231	−0.129391
$375$	0	0
$376$	0.658859	0.0339781
$377$	1.25256	0.0645103
$378$	0	0
$379$	−4.40475	−0.226257	−0.113128	−	0.993580i	$-0.536087\pi$
$379$	−4.40475	−0.226257	−0.113128	+	0.993580i	$0.536087\pi$
$380$	0	0
$381$	0	0
$382$	−4.19786	−0.214781
$383$	−3.80138	−0.194242	−0.0971208	−	0.995273i	$-0.530963\pi$
$383$	−3.80138	−0.194242	−0.0971208	+	0.995273i	$0.530963\pi$
$384$	0	0
$385$	0	0
$386$	−1.97850	−0.100703
$387$	0	0
$388$	3.92663	0.199344
$389$	19.0506	0.965903	0.482951	−	0.875647i	$-0.339565\pi$
$389$	19.0506	0.965903	0.482951	+	0.875647i	$0.339565\pi$
$390$	0	0
$391$	−5.43960	−0.275093
$392$	7.89175	0.398594
$393$	0	0
$394$	−14.4342	−0.727184
$395$	0	0
$396$	0	0
$397$	−20.0028	−1.00391	−0.501955	−	0.864894i	$-0.667386\pi$
$397$	−20.0028	−1.00391	−0.501955	+	0.864894i	$0.667386\pi$
$398$	7.71166	0.386550
$399$	0	0
$400$	0	0
$401$	−35.5317	−1.77437	−0.887184	−	0.461416i	$-0.847341\pi$
$401$	−35.5317	−1.77437	−0.887184	+	0.461416i	$0.847341\pi$
$402$	0	0
$403$	3.73683	0.186145
$404$	−11.8870	−0.591401
$405$	0	0
$406$	3.06079	0.151905
$407$	4.19480	0.207928
$408$	0	0
$409$	17.3475	0.857781	0.428891	−	0.903356i	$-0.358905\pi$
$409$	17.3475	0.857781	0.428891	+	0.903356i	$0.358905\pi$
$410$	0	0
$411$	0	0
$412$	−12.6338	−0.622421
$413$	17.1520	0.843993
$414$	0	0
$415$	0	0
$416$	−4.49455	−0.220363
$417$	0	0
$418$	0.0133719	0.000654042 0
$419$	−31.4740	−1.53761	−0.768804	−	0.639485i	$-0.779148\pi$
$419$	−31.4740	−1.53761	−0.768804	+	0.639485i	$0.779148\pi$
$420$	0	0
$421$	9.78764	0.477020	0.238510	−	0.971140i	$-0.423341\pi$
$421$	9.78764	0.477020	0.238510	+	0.971140i	$0.423341\pi$
$422$	−13.5822	−0.661170
$423$	0	0
$424$	−1.39428	−0.0677122
$425$	0	0
$426$	0	0
$427$	−21.1701	−1.02449
$428$	−8.35261	−0.403739
$429$	0	0
$430$	0	0
$431$	18.9261	0.911639	0.455820	−	0.890072i	$-0.349346\pi$
$431$	18.9261	0.911639	0.455820	+	0.890072i	$0.349346\pi$
$432$	0	0
$433$	−14.9973	−0.720724	−0.360362	−	0.932812i	$-0.617347\pi$
$433$	−14.9973	−0.720724	−0.360362	+	0.932812i	$0.617347\pi$
$434$	9.13141	0.438322
$435$	0	0
$436$	25.8167	1.23640
$437$	0.0290683	0.00139053
$438$	0	0
$439$	−16.9421	−0.808603	−0.404301	−	0.914626i	$-0.632485\pi$
$439$	−16.9421	−0.808603	−0.404301	+	0.914626i	$0.632485\pi$
$440$	0	0
$441$	0	0
$442$	2.65774	0.126416
$443$	29.4600	1.39969	0.699844	−	0.714296i	$-0.253253\pi$
$443$	29.4600	1.39969	0.699844	+	0.714296i	$0.253253\pi$
$444$	0	0
$445$	0	0
$446$	−5.56897	−0.263699
$447$	0	0
$448$	1.59482	0.0753484
$449$	37.8186	1.78477	0.892384	−	0.451276i	$-0.149031\pi$
$449$	37.8186	1.78477	0.892384	+	0.451276i	$0.149031\pi$
$450$	0	0
$451$	4.37431	0.205978
$452$	−14.5691	−0.685273
$453$	0	0
$454$	15.2058	0.713642
$455$	0	0
$456$	0	0
$457$	−8.19036	−0.383129	−0.191565	−	0.981480i	$-0.561356\pi$
$457$	−8.19036	−0.383129	−0.191565	+	0.981480i	$0.561356\pi$
$458$	−3.84425	−0.179630
$459$	0	0
$460$	0	0
$461$	16.4728	0.767214	0.383607	−	0.923497i	$-0.374682\pi$
$461$	16.4728	0.767214	0.383607	+	0.923497i	$0.374682\pi$
$462$	0	0
$463$	−13.6276	−0.633330	−0.316665	−	0.948537i	$-0.602563\pi$
$463$	−13.6276	−0.633330	−0.316665	+	0.948537i	$0.602563\pi$
$464$	3.00175	0.139353
$465$	0	0
$466$	−15.7546	−0.729817
$467$	4.42450	0.204741	0.102371	−	0.994746i	$-0.467357\pi$
$467$	4.42450	0.204741	0.102371	+	0.994746i	$0.467357\pi$
$468$	0	0
$469$	30.0718	1.38858
$470$	0	0
$471$	0	0
$472$	−11.6019	−0.534021
$473$	8.22574	0.378220
$474$	0	0
$475$	0	0
$476$	−28.9020	−1.32472
$477$	0	0
$478$	−6.61119	−0.302389
$479$	−23.4710	−1.07242	−0.536208	−	0.844086i	$-0.680144\pi$
$479$	−23.4710	−1.07242	−0.536208	+	0.844086i	$0.680144\pi$
$480$	0	0
$481$	−4.45536	−0.203147
$482$	−0.203725	−0.00927943
$483$	0	0
$484$	17.0217	0.773715
$485$	0	0
$486$	0	0
$487$	3.73378	0.169194	0.0845969	−	0.996415i	$-0.473040\pi$
$487$	3.73378	0.169194	0.0845969	+	0.996415i	$0.473040\pi$
$488$	14.3199	0.648230
$489$	0	0
$490$	0	0
$491$	26.9239	1.21506	0.607530	−	0.794297i	$-0.292160\pi$
$491$	26.9239	1.21506	0.607530	+	0.794297i	$0.292160\pi$
$492$	0	0
$493$	−8.44753	−0.380457
$494$	−0.0142025	−0.000639002 0
$495$	0	0
$496$	8.95525	0.402103
$497$	35.7824	1.60506
$498$	0	0
$499$	−18.8192	−0.842461	−0.421231	−	0.906954i	$-0.638402\pi$
$499$	−18.8192	−0.842461	−0.421231	+	0.906954i	$0.638402\pi$
$500$	0	0
$501$	0	0
$502$	−4.11007	−0.183441
$503$	18.3550	0.818410	0.409205	−	0.912443i	$-0.365806\pi$
$503$	18.3550	0.818410	0.409205	+	0.912443i	$0.365806\pi$
$504$	0	0
$505$	0	0
$506$	0.460016	0.0204502
$507$	0	0
$508$	−10.6860	−0.474115
$509$	7.00770	0.310611	0.155305	−	0.987867i	$-0.450364\pi$
$509$	7.00770	0.310611	0.155305	+	0.987867i	$0.450364\pi$
$510$	0	0
$511$	15.6386	0.691811
$512$	−19.2790	−0.852018
$513$	0	0
$514$	3.23622	0.142744
$515$	0	0
$516$	0	0
$517$	−0.227342	−0.00999847
$518$	−10.8872	−0.478357
$519$	0	0
$520$	0	0
$521$	5.60958	0.245760	0.122880	−	0.992422i	$-0.460787\pi$
$521$	5.60958	0.245760	0.122880	+	0.992422i	$0.460787\pi$
$522$	0	0
$523$	11.9919	0.524371	0.262185	−	0.965018i	$-0.415557\pi$
$523$	11.9919	0.524371	0.262185	+	0.965018i	$0.415557\pi$
$524$	28.2071	1.23223
$525$	0	0
$526$	−15.3107	−0.667579
$527$	−25.2019	−1.09781
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0.154447	0.00669615
$533$	−4.64602	−0.201242
$534$	0	0
$535$	0	0
$536$	−20.3411	−0.878601
$537$	0	0
$538$	−2.40857	−0.103841
$539$	−2.72308	−0.117291
$540$	0	0
$541$	37.3061	1.60391	0.801957	−	0.597382i	$-0.203793\pi$
$541$	37.3061	1.60391	0.801957	+	0.597382i	$0.203793\pi$
$542$	13.3982	0.575504
$543$	0	0
$544$	30.3121	1.29962
$545$	0	0
$546$	0	0
$547$	19.3357	0.826735	0.413368	−	0.910564i	$-0.364352\pi$
$547$	19.3357	0.826735	0.413368	+	0.910564i	$0.364352\pi$
$548$	−25.2687	−1.07943
$549$	0	0
$550$	0	0
$551$	0.0451422	0.00192312
$552$	0	0
$553$	−37.9618	−1.61430
$554$	−0.300440	−0.0127645
$555$	0	0
$556$	7.42386	0.314842
$557$	17.6322	0.747099	0.373550	−	0.927610i	$-0.378141\pi$
$557$	17.6322	0.747099	0.373550	+	0.927610i	$0.378141\pi$
$558$	0	0
$559$	−8.73670	−0.369523
$560$	0	0
$561$	0	0
$562$	−19.6699	−0.829725
$563$	25.7172	1.08385	0.541924	−	0.840427i	$-0.317696\pi$
$563$	25.7172	1.08385	0.541924	+	0.840427i	$0.317696\pi$
$564$	0	0
$565$	0	0
$566$	−1.97713	−0.0831051
$567$	0	0
$568$	−24.2038	−1.01557
$569$	6.48889	0.272028	0.136014	−	0.990707i	$-0.456571\pi$
$569$	6.48889	0.272028	0.136014	+	0.990707i	$0.456571\pi$
$570$	0	0
$571$	−40.4991	−1.69483	−0.847417	−	0.530928i	$-0.821843\pi$
$571$	−40.4991	−1.69483	−0.847417	+	0.530928i	$0.821843\pi$
$572$	1.00023	0.0418217
$573$	0	0
$574$	−11.3531	−0.473871
$575$	0	0
$576$	0	0
$577$	46.4119	1.93215	0.966076	−	0.258259i	$-0.0831487\pi$
$577$	46.4119	1.93215	0.966076	+	0.258259i	$0.0831487\pi$
$578$	−7.62622	−0.317209
$579$	0	0
$580$	0	0
$581$	−14.0155	−0.581461
$582$	0	0
$583$	0.481101	0.0199252
$584$	−10.5782	−0.437730
$585$	0	0
$586$	17.1856	0.709929
$587$	−23.0323	−0.950644	−0.475322	−	0.879812i	$-0.657668\pi$
$587$	−23.0323	−0.950644	−0.475322	+	0.879812i	$0.657668\pi$
$588$	0	0
$589$	0.134675	0.00554918
$590$	0	0
$591$	0	0
$592$	−10.6772	−0.438830
$593$	−23.0064	−0.944759	−0.472380	−	0.881395i	$-0.656605\pi$
$593$	−23.0064	−0.944759	−0.472380	+	0.881395i	$0.656605\pi$
$594$	0	0
$595$	0	0
$596$	−12.7898	−0.523892
$597$	0	0
$598$	−0.488591	−0.0199800
$599$	−3.41317	−0.139459	−0.0697293	−	0.997566i	$-0.522214\pi$
$599$	−3.41317	−0.139459	−0.0697293	+	0.997566i	$0.522214\pi$
$600$	0	0
$601$	44.6751	1.82233	0.911167	−	0.412037i	$-0.135183\pi$
$601$	44.6751	1.82233	0.911167	+	0.412037i	$0.135183\pi$
$602$	−21.3492	−0.870128
$603$	0	0
$604$	13.5402	0.550945
$605$	0	0
$606$	0	0
$607$	34.2006	1.38816	0.694080	−	0.719898i	$-0.255811\pi$
$607$	34.2006	1.38816	0.694080	+	0.719898i	$0.255811\pi$
$608$	−0.161983	−0.00656927
$609$	0	0
$610$	0	0
$611$	0.241463	0.00976856
$612$	0	0
$613$	20.6739	0.835009	0.417505	−	0.908675i	$-0.362905\pi$
$613$	20.6739	0.835009	0.417505	+	0.908675i	$0.362905\pi$
$614$	−12.5699	−0.507279
$615$	0	0
$616$	5.43760	0.219087
$617$	−10.9355	−0.440245	−0.220123	−	0.975472i	$-0.570646\pi$
$617$	−10.9355	−0.440245	−0.220123	+	0.975472i	$0.570646\pi$
$618$	0	0
$619$	22.7683	0.915136	0.457568	−	0.889175i	$-0.348721\pi$
$619$	22.7683	0.915136	0.457568	+	0.889175i	$0.348721\pi$
$620$	0	0
$621$	0	0
$622$	−11.0339	−0.442420
$623$	−54.0626	−2.16597
$624$	0	0
$625$	0	0
$626$	−11.2099	−0.448036
$627$	0	0
$628$	16.4582	0.656754
$629$	30.0478	1.19808
$630$	0	0
$631$	−27.7031	−1.10284	−0.551422	−	0.834227i	$-0.685914\pi$
$631$	−27.7031	−1.10284	−0.551422	+	0.834227i	$0.685914\pi$
$632$	25.6781	1.02142
$633$	0	0
$634$	−1.28750	−0.0511332
$635$	0	0
$636$	0	0
$637$	2.89222	0.114594
$638$	0.714390	0.0282830
$639$	0	0
$640$	0	0
$641$	34.7903	1.37413	0.687067	−	0.726594i	$-0.258898\pi$
$641$	34.7903	1.37413	0.687067	+	0.726594i	$0.258898\pi$
$642$	0	0
$643$	−42.5726	−1.67890	−0.839449	−	0.543439i	$-0.817122\pi$
$643$	−42.5726	−1.67890	−0.839449	+	0.543439i	$0.817122\pi$
$644$	5.31325	0.209372
$645$	0	0
$646$	0.0957846	0.00376859
$647$	10.0030	0.393259	0.196630	−	0.980478i	$-0.437000\pi$
$647$	10.0030	0.393259	0.196630	+	0.980478i	$0.437000\pi$
$648$	0	0
$649$	4.00327	0.157142
$650$	0	0
$651$	0	0
$652$	−5.96544	−0.233625
$653$	28.6000	1.11920	0.559602	−	0.828762i	$-0.310954\pi$
$653$	28.6000	1.11920	0.559602	+	0.828762i	$0.310954\pi$
$654$	0	0
$655$	0	0
$656$	−11.1341	−0.434714
$657$	0	0
$658$	0.590045	0.0230024
$659$	−10.6150	−0.413501	−0.206750	−	0.978394i	$-0.566289\pi$
$659$	−10.6150	−0.413501	−0.206750	+	0.978394i	$0.566289\pi$
$660$	0	0
$661$	−29.6190	−1.15204	−0.576022	−	0.817434i	$-0.695396\pi$
$661$	−29.6190	−1.15204	−0.576022	+	0.817434i	$0.695396\pi$
$662$	−11.4125	−0.443559
$663$	0	0
$664$	9.48035	0.367909
$665$	0	0
$666$	0	0
$667$	1.55297	0.0601311
$668$	−28.7867	−1.11379
$669$	0	0
$670$	0	0
$671$	−4.94112	−0.190750
$672$	0	0
$673$	−41.7219	−1.60826	−0.804130	−	0.594454i	$-0.797368\pi$
$673$	−41.7219	−1.60826	−0.804130	+	0.594454i	$0.797368\pi$
$674$	7.98796	0.307685
$675$	0	0
$676$	20.1672	0.775661
$677$	40.2597	1.54731	0.773653	−	0.633610i	$-0.218428\pi$
$677$	40.2597	1.54731	0.773653	+	0.633610i	$0.218428\pi$
$678$	0	0
$679$	7.82322	0.300228
$680$	0	0
$681$	0	0
$682$	2.13128	0.0816107
$683$	−4.98626	−0.190794	−0.0953970	−	0.995439i	$-0.530412\pi$
$683$	−4.98626	−0.190794	−0.0953970	+	0.995439i	$0.530412\pi$
$684$	0	0
$685$	0	0
$686$	−6.72901	−0.256915
$687$	0	0
$688$	−20.9373	−0.798229
$689$	−0.510985	−0.0194670
$690$	0	0
$691$	−8.04362	−0.305994	−0.152997	−	0.988227i	$-0.548892\pi$
$691$	−8.04362	−0.305994	−0.152997	+	0.988227i	$0.548892\pi$
$692$	−10.7156	−0.407345
$693$	0	0
$694$	11.5895	0.439933
$695$	0	0
$696$	0	0
$697$	31.3337	1.18685
$698$	−13.0300	−0.493191
$699$	0	0
$700$	0	0
$701$	−32.2539	−1.21821	−0.609106	−	0.793089i	$-0.708472\pi$
$701$	−32.2539	−1.21821	−0.609106	+	0.793089i	$0.708472\pi$
$702$	0	0
$703$	−0.160571	−0.00605603
$704$	0.372233	0.0140291
$705$	0	0
$706$	−0.0971314	−0.00365559
$707$	−23.6831	−0.890696
$708$	0	0
$709$	34.4906	1.29532	0.647660	−	0.761929i	$-0.275748\pi$
$709$	34.4906	1.29532	0.647660	+	0.761929i	$0.275748\pi$
$710$	0	0
$711$	0	0
$712$	36.5689	1.37048
$713$	4.63304	0.173509
$714$	0	0
$715$	0	0
$716$	−3.14078	−0.117376
$717$	0	0
$718$	10.8082	0.403360
$719$	−23.1282	−0.862536	−0.431268	−	0.902224i	$-0.641934\pi$
$719$	−23.1282	−0.862536	−0.431268	+	0.902224i	$0.641934\pi$
$720$	0	0
$721$	−25.1709	−0.937413
$722$	11.5091	0.428326
$723$	0	0
$724$	−7.78981	−0.289506
$725$	0	0
$726$	0	0
$727$	−41.9820	−1.55703	−0.778513	−	0.627628i	$-0.784026\pi$
$727$	−41.9820	−1.55703	−0.778513	+	0.627628i	$0.784026\pi$
$728$	−5.77536	−0.214049
$729$	0	0
$730$	0	0
$731$	58.9220	2.17931
$732$	0	0
$733$	−6.33790	−0.234096	−0.117048	−	0.993126i	$-0.537343\pi$
$733$	−6.33790	−0.234096	−0.117048	+	0.993126i	$0.537343\pi$
$734$	−14.5392	−0.536651
$735$	0	0
$736$	−5.57248	−0.205404
$737$	7.01876	0.258539
$738$	0	0
$739$	−38.1035	−1.40166	−0.700831	−	0.713328i	$-0.747187\pi$
$739$	−38.1035	−1.40166	−0.700831	+	0.713328i	$0.747187\pi$
$740$	0	0
$741$	0	0
$742$	−1.24866	−0.0458396
$743$	−47.4077	−1.73922	−0.869611	−	0.493738i	$-0.835630\pi$
$743$	−47.4077	−1.73922	−0.869611	+	0.493738i	$0.835630\pi$
$744$	0	0
$745$	0	0
$746$	−1.50996	−0.0552836
$747$	0	0
$748$	−6.74574	−0.246649
$749$	−16.6413	−0.608061
$750$	0	0
$751$	24.2044	0.883230	0.441615	−	0.897205i	$-0.354406\pi$
$751$	24.2044	0.883230	0.441615	+	0.897205i	$0.354406\pi$
$752$	0.578663	0.0211017
$753$	0	0
$754$	−0.758766	−0.0276326
$755$	0	0
$756$	0	0
$757$	−45.9553	−1.67028	−0.835138	−	0.550041i	$-0.814612\pi$
$757$	−45.9553	−1.67028	−0.835138	+	0.550041i	$0.814612\pi$
$758$	2.66827	0.0969158
$759$	0	0
$760$	0	0
$761$	43.6112	1.58090	0.790452	−	0.612523i	$-0.209845\pi$
$761$	43.6112	1.58090	0.790452	+	0.612523i	$0.209845\pi$
$762$	0	0
$763$	51.4360	1.86211
$764$	−11.3166	−0.409421
$765$	0	0
$766$	2.30277	0.0832024
$767$	−4.25194	−0.153529
$768$	0	0
$769$	−33.5219	−1.20883	−0.604416	−	0.796669i	$-0.706594\pi$
$769$	−33.5219	−1.20883	−0.604416	+	0.796669i	$0.706594\pi$
$770$	0	0
$771$	0	0
$772$	−5.33365	−0.191962
$773$	28.2390	1.01569	0.507843	−	0.861450i	$-0.330443\pi$
$773$	28.2390	1.01569	0.507843	+	0.861450i	$0.330443\pi$
$774$	0	0
$775$	0	0
$776$	−5.29177	−0.189964
$777$	0	0
$778$	−11.5403	−0.413739
$779$	−0.167442	−0.00599923
$780$	0	0
$781$	8.35162	0.298844
$782$	3.29515	0.117834
$783$	0	0
$784$	6.93116	0.247542
$785$	0	0
$786$	0	0
$787$	18.9710	0.676244	0.338122	−	0.941102i	$-0.390208\pi$
$787$	18.9710	0.676244	0.338122	+	0.941102i	$0.390208\pi$
$788$	−38.9118	−1.38618
$789$	0	0
$790$	0	0
$791$	−29.0268	−1.03207
$792$	0	0
$793$	5.24805	0.186364
$794$	12.1171	0.430020
$795$	0	0
$796$	20.7891	0.736852
$797$	−29.0624	−1.02944	−0.514722	−	0.857357i	$-0.672105\pi$
$797$	−29.0624	−1.02944	−0.514722	+	0.857357i	$0.672105\pi$
$798$	0	0
$799$	−1.62848	−0.0576113
$800$	0	0
$801$	0	0
$802$	21.5241	0.760041
$803$	3.65006	0.128808
$804$	0	0
$805$	0	0
$806$	−2.26366	−0.0797341
$807$	0	0
$808$	16.0197	0.563571
$809$	42.5832	1.49715	0.748573	−	0.663052i	$-0.230739\pi$
$809$	42.5832	1.49715	0.748573	+	0.663052i	$0.230739\pi$
$810$	0	0
$811$	−46.5666	−1.63517	−0.817587	−	0.575805i	$-0.804689\pi$
$811$	−46.5666	−1.63517	−0.817587	+	0.575805i	$0.804689\pi$
$812$	8.25131	0.289564
$813$	0	0
$814$	−2.54108	−0.0890649
$815$	0	0
$816$	0	0
$817$	−0.314869	−0.0110159
$818$	−10.5086	−0.367426
$819$	0	0
$820$	0	0
$821$	−13.8087	−0.481929	−0.240964	−	0.970534i	$-0.577464\pi$
$821$	−13.8087	−0.481929	−0.240964	+	0.970534i	$0.577464\pi$
$822$	0	0
$823$	44.7579	1.56016	0.780082	−	0.625678i	$-0.215178\pi$
$823$	44.7579	1.56016	0.780082	+	0.625678i	$0.215178\pi$
$824$	17.0261	0.593131
$825$	0	0
$826$	−10.3902	−0.361520
$827$	−7.56243	−0.262972	−0.131486	−	0.991318i	$-0.541975\pi$
$827$	−7.56243	−0.262972	−0.131486	+	0.991318i	$0.541975\pi$
$828$	0	0
$829$	21.0279	0.730328	0.365164	−	0.930943i	$-0.381013\pi$
$829$	21.0279	0.730328	0.365164	+	0.930943i	$0.381013\pi$
$830$	0	0
$831$	0	0
$832$	−0.395355	−0.0137065
$833$	−19.5057	−0.675833
$834$	0	0
$835$	0	0
$836$	0.0360481	0.00124675
$837$	0	0
$838$	19.0661	0.658626
$839$	−57.6216	−1.98932	−0.994659	−	0.103213i	$-0.967088\pi$
$839$	−57.6216	−1.98932	−0.994659	+	0.103213i	$0.967088\pi$
$840$	0	0
$841$	−26.5883	−0.916838
$842$	−5.92906	−0.204329
$843$	0	0
$844$	−36.6149	−1.26034
$845$	0	0
$846$	0	0
$847$	33.9133	1.16527
$848$	−1.22457	−0.0420518
$849$	0	0
$850$	0	0
$851$	−5.52390	−0.189357
$852$	0	0
$853$	21.6926	0.742741	0.371370	−	0.928485i	$-0.378888\pi$
$853$	21.6926	0.742741	0.371370	+	0.928485i	$0.378888\pi$
$854$	12.8242	0.438837
$855$	0	0
$856$	11.2565	0.384739
$857$	29.3518	1.00264	0.501319	−	0.865262i	$-0.332848\pi$
$857$	29.3518	1.00264	0.501319	+	0.865262i	$0.332848\pi$
$858$	0	0
$859$	−14.1869	−0.484051	−0.242025	−	0.970270i	$-0.577812\pi$
$859$	−14.1869	−0.484051	−0.242025	+	0.970270i	$0.577812\pi$
$860$	0	0
$861$	0	0
$862$	−11.4649	−0.390496
$863$	−36.9884	−1.25910	−0.629550	−	0.776960i	$-0.716761\pi$
$863$	−36.9884	−1.25910	−0.629550	+	0.776960i	$0.716761\pi$
$864$	0	0
$865$	0	0
$866$	9.08493	0.308718
$867$	0	0
$868$	24.6165	0.835539
$869$	−8.86031	−0.300565
$870$	0	0
$871$	−7.45474	−0.252594
$872$	−34.7923	−1.17821
$873$	0	0
$874$	−0.0176087	−0.000595625 0
$875$	0	0
$876$	0	0
$877$	3.21780	0.108657	0.0543287	−	0.998523i	$-0.482698\pi$
$877$	3.21780	0.108657	0.0543287	+	0.998523i	$0.482698\pi$
$878$	10.2630	0.346361
$879$	0	0
$880$	0	0
$881$	52.6836	1.77495	0.887477	−	0.460852i	$-0.152456\pi$
$881$	52.6836	1.77495	0.887477	+	0.460852i	$0.152456\pi$
$882$	0	0
$883$	6.07047	0.204287	0.102144	−	0.994770i	$-0.467430\pi$
$883$	6.07047	0.204287	0.102144	+	0.994770i	$0.467430\pi$
$884$	7.16476	0.240977
$885$	0	0
$886$	−17.8460	−0.599549
$887$	7.90948	0.265574	0.132787	−	0.991145i	$-0.457607\pi$
$887$	7.90948	0.265574	0.132787	+	0.991145i	$0.457607\pi$
$888$	0	0
$889$	−21.2903	−0.714054
$890$	0	0
$891$	0	0
$892$	−15.0129	−0.502669
$893$	0.00870230	0.000291211 0
$894$	0	0
$895$	0	0
$896$	−37.2273	−1.24368
$897$	0	0
$898$	−22.9094	−0.764496
$899$	7.19496	0.239965
$900$	0	0
$901$	3.44618	0.114809
$902$	−2.64983	−0.0882296
$903$	0	0
$904$	19.6342	0.653025
$905$	0	0
$906$	0	0
$907$	−44.5820	−1.48032	−0.740160	−	0.672430i	$-0.765250\pi$
$907$	−44.5820	−1.48032	−0.740160	+	0.672430i	$0.765250\pi$
$908$	40.9918	1.36036
$909$	0	0
$910$	0	0
$911$	4.99864	0.165612	0.0828062	−	0.996566i	$-0.473612\pi$
$911$	4.99864	0.165612	0.0828062	+	0.996566i	$0.473612\pi$
$912$	0	0
$913$	−3.27123	−0.108262
$914$	4.96148	0.164111
$915$	0	0
$916$	−10.3634	−0.342415
$917$	56.1984	1.85584
$918$	0	0
$919$	−42.0674	−1.38768	−0.693838	−	0.720131i	$-0.744082\pi$
$919$	−42.0674	−1.38768	−0.693838	+	0.720131i	$0.744082\pi$
$920$	0	0
$921$	0	0
$922$	−9.97872	−0.328632
$923$	−8.87039	−0.291973
$924$	0	0
$925$	0	0
$926$	8.25522	0.271283
$927$	0	0
$928$	−8.65388	−0.284078
$929$	−55.0370	−1.80570	−0.902852	−	0.429951i	$-0.858531\pi$
$929$	−55.0370	−1.80570	−0.902852	+	0.429951i	$0.858531\pi$
$930$	0	0
$931$	0.104235	0.00341617
$932$	−42.4714	−1.39120
$933$	0	0
$934$	−2.68023	−0.0876999
$935$	0	0
$936$	0	0
$937$	−31.2596	−1.02121	−0.510603	−	0.859817i	$-0.670578\pi$
$937$	−31.2596	−1.02121	−0.510603	+	0.859817i	$0.670578\pi$
$938$	−18.2166	−0.594793
$939$	0	0
$940$	0	0
$941$	−4.14617	−0.135161	−0.0675806	−	0.997714i	$-0.521528\pi$
$941$	−4.14617	−0.135161	−0.0675806	+	0.997714i	$0.521528\pi$
$942$	0	0
$943$	−5.76029	−0.187581
$944$	−10.1897	−0.331647
$945$	0	0
$946$	−4.98291	−0.162008
$947$	34.8241	1.13163	0.565816	−	0.824532i	$-0.308561\pi$
$947$	34.8241	1.13163	0.565816	+	0.824532i	$0.308561\pi$
$948$	0	0
$949$	−3.87678	−0.125846
$950$	0	0
$951$	0	0
$952$	38.9502	1.26238
$953$	22.7088	0.735611	0.367805	−	0.929903i	$-0.380109\pi$
$953$	22.7088	0.735611	0.367805	+	0.929903i	$0.380109\pi$
$954$	0	0
$955$	0	0
$956$	−17.8225	−0.576421
$957$	0	0
$958$	14.2180	0.459364
$959$	−50.3442	−1.62570
$960$	0	0
$961$	−9.53492	−0.307578
$962$	2.69893	0.0870169
$963$	0	0
$964$	−0.549204	−0.0176887
$965$	0	0
$966$	0	0
$967$	40.1338	1.29062	0.645308	−	0.763922i	$-0.276729\pi$
$967$	40.1338	1.29062	0.645308	+	0.763922i	$0.276729\pi$
$968$	−22.9396	−0.737305
$969$	0	0
$970$	0	0
$971$	2.29254	0.0735710	0.0367855	−	0.999323i	$-0.488288\pi$
$971$	2.29254	0.0735710	0.0367855	+	0.999323i	$0.488288\pi$
$972$	0	0
$973$	14.7909	0.474176
$974$	−2.26182	−0.0724733
$975$	0	0
$976$	12.5768	0.402575
$977$	−21.7795	−0.696786	−0.348393	−	0.937348i	$-0.613273\pi$
$977$	−21.7795	−0.696786	−0.348393	+	0.937348i	$0.613273\pi$
$978$	0	0
$979$	−12.6182	−0.403281
$980$	0	0
$981$	0	0
$982$	−16.3097	−0.520464
$983$	−29.9728	−0.955983	−0.477991	−	0.878365i	$-0.658635\pi$
$983$	−29.9728	−0.955983	−0.477991	+	0.878365i	$0.658635\pi$
$984$	0	0
$985$	0	0
$986$	5.11726	0.162967
$987$	0	0
$988$	−0.0382873	−0.00121808
$989$	−10.8320	−0.344439
$990$	0	0
$991$	28.6163	0.909027	0.454513	−	0.890740i	$-0.349813\pi$
$991$	28.6163	0.909027	0.454513	+	0.890740i	$0.349813\pi$
$992$	−25.8175	−0.819708
$993$	0	0
$994$	−21.6759	−0.687518
$995$	0	0
$996$	0	0
$997$	26.1503	0.828189	0.414094	−	0.910234i	$-0.364098\pi$
$997$	26.1503	0.828189	0.414094	+	0.910234i	$0.364098\pi$
$998$	11.4001	0.360864
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.by.1.3		6
3.2	odd	2		5175.2.a.bz.1.4		6
5.4	even	2		1035.2.a.q.1.4	yes	6
15.14	odd	2		1035.2.a.p.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.3	✓	6	15.14	odd	2
1035.2.a.q.1.4	yes	6	5.4	even	2
5175.2.a.by.1.3		6	1.1	even	1	trivial
5175.2.a.bz.1.4		6	3.2	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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q-0.605771 q^{2} -1.63304 q^{4} -3.25359 q^{7} +2.20079 q^{8} -0.759390 q^{11} +0.806561 q^{13} +1.97093 q^{14} +1.93291 q^{16} -5.43960 q^{17} +0.0290683 q^{19} +0.460016 q^{22} +1.00000 q^{23} -0.488591 q^{26} +5.31325 q^{28} +1.55297 q^{29} +4.63304 q^{31} -5.57248 q^{32} +3.29515 q^{34} -5.52390 q^{37} -0.0176087 q^{38} -5.76029 q^{41} -10.8320 q^{43} +1.24012 q^{44} -0.605771 q^{46} +0.299374 q^{47} +3.58587 q^{49} -1.31715 q^{52} -0.633536 q^{53} -7.16048 q^{56} -0.940742 q^{58} -5.27170 q^{59} +6.50669 q^{61} -2.80656 q^{62} -0.490173 q^{64} -9.24263 q^{67} +8.88310 q^{68} -10.9978 q^{71} -4.80656 q^{73} +3.34622 q^{74} -0.0474698 q^{76} +2.47075 q^{77} +11.6677 q^{79} +3.48941 q^{82} +4.30770 q^{83} +6.56173 q^{86} -1.67126 q^{88} +16.6163 q^{89} -2.62422 q^{91} -1.63304 q^{92} -0.181352 q^{94} -2.40449 q^{97} -2.17222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} - 4 q^{11} - 12 q^{13} + 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 8 q^{22} + 6 q^{23} + 12 q^{26} - 24 q^{28} + 6 q^{29} + 8 q^{31} + 20 q^{32} - 12 q^{34} - 22 q^{37} + 36 q^{38}+ \cdots - 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 - 4 * q^11 - 12 * q^13 + 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 8 * q^22 + 6 * q^23 + 12 * q^26 - 24 * q^28 + 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 22 * q^37 + 36 * q^38 + 18 * q^41 - 8 * q^43 - 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^56 + 16 * q^58 + 6 * q^59 + 14 * q^64 - 10 * q^67 + 28 * q^68 + 22 * q^71 - 12 * q^73 + 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^82 + 24 * q^83 - 20 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 12 * q^97 - 4 * q^98

Introduction

L-functions

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Database

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