LMFDB - Embedded newform 1035.2.a.p.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1035,2,Mod(1,1035)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("1035.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,-6,0,6,0,0,0,4] 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:	$8.26451660920$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.52269$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.605771 q^{2} -1.63304 q^{4} -1.00000 q^{5} +3.25359 q^{7} +2.20079 q^{8} +0.605771 q^{10} +0.759390 q^{11} -0.806561 q^{13} -1.97093 q^{14} +1.93291 q^{16} -5.43960 q^{17} +0.0290683 q^{19} +1.63304 q^{20} -0.460016 q^{22} +1.00000 q^{23} +1.00000 q^{25} +0.488591 q^{26} -5.31325 q^{28} -1.55297 q^{29} +4.63304 q^{31} -5.57248 q^{32} +3.29515 q^{34} -3.25359 q^{35} +5.52390 q^{37} -0.0176087 q^{38} -2.20079 q^{40} +5.76029 q^{41} +10.8320 q^{43} -1.24012 q^{44} -0.605771 q^{46} +0.299374 q^{47} +3.58587 q^{49} -0.605771 q^{50} +1.31715 q^{52} -0.633536 q^{53} -0.759390 q^{55} +7.16048 q^{56} +0.940742 q^{58} +5.27170 q^{59} +6.50669 q^{61} -2.80656 q^{62} -0.490173 q^{64} +0.806561 q^{65} +9.24263 q^{67} +8.88310 q^{68} +1.97093 q^{70} +10.9978 q^{71} +4.80656 q^{73} -3.34622 q^{74} -0.0474698 q^{76} +2.47075 q^{77} +11.6677 q^{79} -1.93291 q^{80} -3.48941 q^{82} +4.30770 q^{83} +5.43960 q^{85} -6.56173 q^{86} +1.67126 q^{88} -16.6163 q^{89} -2.62422 q^{91} -1.63304 q^{92} -0.181352 q^{94} -0.0290683 q^{95} +2.40449 q^{97} -2.17222 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 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^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.25359	1.22974	0.614871	−	0.788627i	$-0.289208\pi$
$7$	3.25359	1.22974	0.614871	+	0.788627i	$0.289208\pi$
$8$	2.20079	0.778097
$9$	0	0
$10$	0.605771	0.191562
$11$	0.759390	0.228965	0.114482	−	0.993425i	$-0.463479\pi$
$11$	0.759390	0.228965	0.114482	+	0.993425i	$0.463479\pi$
$12$	0	0
$13$	−0.806561	−0.223700	−0.111850	−	0.993725i	$-0.535678\pi$
$13$	−0.806561	−0.223700	−0.111850	+	0.993725i	$0.535678\pi$
$14$	−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	$0.498939\pi$
$20$	1.63304	0.365159
$21$	0	0
$22$	−0.460016	−0.0980758
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0.488591	0.0958206
$27$	0	0
$28$	−5.31325	−1.00411
$29$	−1.55297	−0.288379	−0.144189	−	0.989550i	$-0.546057\pi$
$29$	−1.55297	−0.288379	−0.144189	+	0.989550i	$0.546057\pi$
$30$	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
$35$	−3.25359	−0.549958
$36$	0	0
$37$	5.52390	0.908123	0.454062	−	0.890970i	$-0.349975\pi$
$37$	5.52390	0.908123	0.454062	+	0.890970i	$0.349975\pi$
$38$	−0.0176087	−0.00285652
$39$	0	0
$40$	−2.20079	−0.347976
$41$	5.76029	0.899606	0.449803	−	0.893128i	$-0.351494\pi$
$41$	5.76029	0.899606	0.449803	+	0.893128i	$0.351494\pi$
$42$	0	0
$43$	10.8320	1.65187	0.825935	−	0.563765i	$-0.190648\pi$
$43$	10.8320	1.65187	0.825935	+	0.563765i	$0.190648\pi$
$44$	−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
$50$	−0.605771	−0.0856689
$51$	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
$55$	−0.759390	−0.102396
$56$	7.16048	0.956859
$57$	0	0
$58$	0.940742	0.123526
$59$	5.27170	0.686316	0.343158	−	0.939278i	$-0.388503\pi$
$59$	5.27170	0.686316	0.343158	+	0.939278i	$0.388503\pi$
$60$	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
$65$	0.806561	0.100042
$66$	0	0
$67$	9.24263	1.12917	0.564583	−	0.825376i	$-0.309037\pi$
$67$	9.24263	1.12917	0.564583	+	0.825376i	$0.309037\pi$
$68$	8.88310	1.07723
$69$	0	0
$70$	1.97093	0.235571
$71$	10.9978	1.30520	0.652599	−	0.757703i	$-0.273679\pi$
$71$	10.9978	1.30520	0.652599	+	0.757703i	$0.273679\pi$
$72$	0	0
$73$	4.80656	0.562565	0.281283	−	0.959625i	$-0.409240\pi$
$73$	4.80656	0.562565	0.281283	+	0.959625i	$0.409240\pi$
$74$	−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	$0.272096\pi$
$80$	−1.93291	−0.216106
$81$	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
$85$	5.43960	0.590008
$86$	−6.56173	−0.707569
$87$	0	0
$88$	1.67126	0.178157
$89$	−16.6163	−1.76132	−0.880661	−	0.473748i	$-0.842901\pi$
$89$	−16.6163	−1.76132	−0.880661	+	0.473748i	$0.842901\pi$
$90$	0	0
$91$	−2.62422	−0.275093
$92$	−1.63304	−0.170256
$93$	0	0
$94$	−0.181352	−0.0187050
$95$	−0.0290683	−0.00298235
$96$	0	0
$97$	2.40449	0.244139	0.122069	−	0.992522i	$-0.461047\pi$
$97$	2.40449	0.244139	0.122069	+	0.992522i	$0.461047\pi$
$98$	−2.17222	−0.219427
$99$	0	0
$100$	−1.63304	−0.163304
$101$	−7.27907	−0.724294	−0.362147	−	0.932121i	$-0.617956\pi$
$101$	−7.27907	−0.724294	−0.362147	+	0.932121i	$0.617956\pi$
$102$	0	0
$103$	−7.73634	−0.762284	−0.381142	−	0.924517i	$-0.624469\pi$
$103$	−7.73634	−0.762284	−0.381142	+	0.924517i	$0.624469\pi$
$104$	−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	$0.773391\pi$
$110$	0.460016	0.0438608
$111$	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
$115$	−1.00000	−0.0932505
$116$	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
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.54363	−0.580653	−0.290327	−	0.956928i	$-0.593764\pi$
$127$	−6.54363	−0.580653	−0.290327	+	0.956928i	$0.593764\pi$
$128$	11.4419	1.01133
$129$	0	0
$130$	−0.488591	−0.0428523
$131$	17.2727	1.50913	0.754563	−	0.656228i	$-0.227849\pi$
$131$	17.2727	1.50913	0.754563	+	0.656228i	$0.227849\pi$
$132$	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	$0.561755\pi$
$140$	5.31325	0.449052
$141$	0	0
$142$	−6.66214	−0.559075
$143$	−0.612495	−0.0512194
$144$	0	0
$145$	1.55297	0.128967
$146$	−2.91167	−0.240972
$147$	0	0
$148$	−9.02076	−0.741502
$149$	−7.83190	−0.641614	−0.320807	−	0.947145i	$-0.603954\pi$
$149$	−7.83190	−0.641614	−0.320807	+	0.947145i	$0.603954\pi$
$150$	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
$155$	−4.63304	−0.372135
$156$	0	0
$157$	10.0783	0.804332	0.402166	−	0.915567i	$-0.368257\pi$
$157$	10.0783	0.804332	0.402166	+	0.915567i	$0.368257\pi$
$158$	−7.06793	−0.562294
$159$	0	0
$160$	5.57248	0.440543
$161$	3.25359	0.256419
$162$	0	0
$163$	−3.65296	−0.286122	−0.143061	−	0.989714i	$-0.545695\pi$
$163$	−3.65296	−0.286122	−0.143061	+	0.989714i	$0.545695\pi$
$164$	−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
$170$	−3.29515	−0.252727
$171$	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
$175$	3.25359	0.245949
$176$	1.46783	0.110642
$177$	0	0
$178$	10.0657	0.754452
$179$	−1.92327	−0.143752	−0.0718759	−	0.997414i	$-0.522899\pi$
$179$	−1.92327	−0.143752	−0.0718759	+	0.997414i	$0.522899\pi$
$180$	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
$185$	−5.52390	−0.406125
$186$	0	0
$187$	−4.13078	−0.302073
$188$	−0.488890	−0.0356560
$189$	0	0
$190$	0.0176087	0.00127747
$191$	−6.92978	−0.501421	−0.250711	−	0.968062i	$-0.580664\pi$
$191$	−6.92978	−0.501421	−0.250711	+	0.968062i	$0.580664\pi$
$192$	0	0
$193$	−3.26608	−0.235098	−0.117549	−	0.993067i	$-0.537504\pi$
$193$	−3.26608	−0.235098	−0.117549	+	0.993067i	$0.537504\pi$
$194$	−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	$0.649009\pi$
$200$	2.20079	0.155619
$201$	0	0
$202$	4.40945	0.310248
$203$	−5.05272	−0.354632
$204$	0	0
$205$	−5.76029	−0.402316
$206$	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
$215$	−10.8320	−0.738739
$216$	0	0
$217$	15.0740	1.02329
$218$	9.57662	0.648610
$219$	0	0
$220$	1.24012	0.0836086
$221$	4.38737	0.295127
$222$	0	0
$223$	−9.19320	−0.615622	−0.307811	−	0.951447i	$-0.599597\pi$
$223$	−9.19320	−0.615622	−0.307811	+	0.951447i	$0.599597\pi$
$224$	−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	$0.432758\pi$
$230$	0.605771	0.0399433
$231$	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
$235$	−0.299374	−0.0195290
$236$	−8.60890	−0.560392
$237$	0	0
$238$	10.7211	0.694945
$239$	−10.9137	−0.705948	−0.352974	−	0.935633i	$-0.614830\pi$
$239$	−10.9137	−0.705948	−0.352974	+	0.935633i	$0.614830\pi$
$240$	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
$245$	−3.58587	−0.229093
$246$	0	0
$247$	−0.0234454	−0.00149179
$248$	10.1964	0.647469
$249$	0	0
$250$	0.605771	0.0383123
$251$	−6.78486	−0.428257	−0.214128	−	0.976806i	$-0.568691\pi$
$251$	−6.78486	−0.428257	−0.214128	+	0.976806i	$0.568691\pi$
$252$	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
$260$	−1.31715	−0.0816860
$261$	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
$265$	0.633536	0.0389178
$266$	−0.0572917	−0.00351278
$267$	0	0
$268$	−15.0936	−0.921988
$269$	−3.97605	−0.242424	−0.121212	−	0.992627i	$-0.538678\pi$
$269$	−3.97605	−0.242424	−0.121212	+	0.992627i	$0.538678\pi$
$270$	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
$275$	0.759390	0.0457930
$276$	0	0
$277$	−0.495963	−0.0297995	−0.0148998	−	0.999889i	$-0.504743\pi$
$277$	−0.495963	−0.0297995	−0.0148998	+	0.999889i	$0.504743\pi$
$278$	2.75385	0.165165
$279$	0	0
$280$	−7.16048	−0.427920
$281$	−32.4709	−1.93705	−0.968525	−	0.248915i	$-0.919926\pi$
$281$	−32.4709	−1.93705	−0.968525	+	0.248915i	$0.919926\pi$
$282$	0	0
$283$	−3.26383	−0.194015	−0.0970073	−	0.995284i	$-0.530927\pi$
$283$	−3.26383	−0.194015	−0.0970073	+	0.995284i	$0.530927\pi$
$284$	−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
$290$	−0.940742	−0.0552423
$291$	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
$295$	−5.27170	−0.306930
$296$	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
$305$	−6.50669	−0.372572
$306$	0	0
$307$	−20.7502	−1.18428	−0.592139	−	0.805836i	$-0.701716\pi$
$307$	−20.7502	−1.18428	−0.592139	+	0.805836i	$0.701716\pi$
$308$	−4.03483	−0.229906
$309$	0	0
$310$	2.80656	0.159402
$311$	−18.2147	−1.03286	−0.516430	−	0.856329i	$-0.672739\pi$
$311$	−18.2147	−1.03286	−0.516430	+	0.856329i	$0.672739\pi$
$312$	0	0
$313$	−18.5051	−1.04597	−0.522986	−	0.852342i	$-0.675182\pi$
$313$	−18.5051	−1.04597	−0.522986	+	0.852342i	$0.675182\pi$
$314$	−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
$320$	0.490173	0.0274015
$321$	0	0
$322$	−1.97093	−0.109836
$323$	−0.158120	−0.00879804
$324$	0	0
$325$	−0.806561	−0.0447400
$326$	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
$335$	−9.24263	−0.504979
$336$	0	0
$337$	13.1864	0.718311	0.359156	−	0.933278i	$-0.383065\pi$
$337$	13.1864	0.718311	0.359156	+	0.933278i	$0.383065\pi$
$338$	7.48094	0.406910
$339$	0	0
$340$	−8.88310	−0.481754
$341$	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	$0.304732\pi$
$350$	−1.97093	−0.105351
$351$	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
$355$	−10.9978	−0.583702
$356$	27.1351	1.43816
$357$	0	0
$358$	1.16506	0.0615753
$359$	17.8421	0.941672	0.470836	−	0.882221i	$-0.343952\pi$
$359$	17.8421	0.941672	0.470836	+	0.882221i	$0.343952\pi$
$360$	0	0
$361$	−18.9992	−0.999956
$362$	−2.88960	−0.151874
$363$	0	0
$364$	4.28546	0.224619
$365$	−4.80656	−0.251587
$366$	0	0
$367$	−24.0011	−1.25285	−0.626425	−	0.779482i	$-0.715482\pi$
$367$	−24.0011	−1.25285	−0.626425	+	0.779482i	$0.715482\pi$
$368$	1.93291	0.100760
$369$	0	0
$370$	3.34622	0.173961
$371$	−2.06127	−0.107016
$372$	0	0
$373$	−2.49263	−0.129063	−0.0645317	−	0.997916i	$-0.520555\pi$
$373$	−2.49263	−0.129063	−0.0645317	+	0.997916i	$0.520555\pi$
$374$	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	$0.536087\pi$
$380$	0.0474698	0.00243515
$381$	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
$385$	−2.47075	−0.125921
$386$	1.97850	0.100703
$387$	0	0
$388$	−3.92663	−0.199344
$389$	−19.0506	−0.965903	−0.482951	−	0.875647i	$-0.660435\pi$
$389$	−19.0506	−0.965903	−0.482951	+	0.875647i	$0.660435\pi$
$390$	0	0
$391$	−5.43960	−0.275093
$392$	7.89175	0.398594
$393$	0	0
$394$	−14.4342	−0.727184
$395$	−11.6677	−0.587064
$396$	0	0
$397$	20.0028	1.00391	0.501955	−	0.864894i	$-0.332614\pi$
$397$	20.0028	1.00391	0.501955	+	0.864894i	$0.332614\pi$
$398$	7.71166	0.386550
$399$	0	0
$400$	1.93291	0.0966455
$401$	35.5317	1.77437	0.887184	−	0.461416i	$-0.152659\pi$
$401$	35.5317	1.77437	0.887184	+	0.461416i	$0.152659\pi$
$402$	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	$0.358905\pi$
$410$	3.48941	0.172330
$411$	0	0
$412$	12.6338	0.622421
$413$	17.1520	0.843993
$414$	0	0
$415$	−4.30770	−0.211457
$416$	4.49455	0.220363
$417$	0	0
$418$	−0.0133719	−0.000654042 0
$419$	31.4740	1.53761	0.768804	−	0.639485i	$-0.220852\pi$
$419$	31.4740	1.53761	0.768804	+	0.639485i	$0.220852\pi$
$420$	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
$425$	−5.43960	−0.263859
$426$	0	0
$427$	21.1701	1.02449
$428$	−8.35261	−0.403739
$429$	0	0
$430$	6.56173	0.316435
$431$	−18.9261	−0.911639	−0.455820	−	0.890072i	$-0.650654\pi$
$431$	−18.9261	−0.911639	−0.455820	+	0.890072i	$0.650654\pi$
$432$	0	0
$433$	14.9973	0.720724	0.360362	−	0.932812i	$-0.382653\pi$
$433$	14.9973	0.720724	0.360362	+	0.932812i	$0.382653\pi$
$434$	−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	$0.632485\pi$
$440$	−1.67126	−0.0796741
$441$	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
$445$	16.6163	0.787687
$446$	5.56897	0.263699
$447$	0	0
$448$	−1.59482	−0.0753484
$449$	−37.8186	−1.78477	−0.892384	−	0.451276i	$-0.850969\pi$
$449$	−37.8186	−1.78477	−0.892384	+	0.451276i	$0.850969\pi$
$450$	0	0
$451$	4.37431	0.205978
$452$	−14.5691	−0.685273
$453$	0	0
$454$	15.2058	0.713642
$455$	2.62422	0.123025
$456$	0	0
$457$	8.19036	0.383129	0.191565	−	0.981480i	$-0.438644\pi$
$457$	8.19036	0.383129	0.191565	+	0.981480i	$0.438644\pi$
$458$	−3.84425	−0.179630
$459$	0	0
$460$	1.63304	0.0761410
$461$	−16.4728	−0.767214	−0.383607	−	0.923497i	$-0.625318\pi$
$461$	−16.4728	−0.767214	−0.383607	+	0.923497i	$0.625318\pi$
$462$	0	0
$463$	13.6276	0.633330	0.316665	−	0.948537i	$-0.397437\pi$
$463$	13.6276	0.633330	0.316665	+	0.948537i	$0.397437\pi$
$464$	−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
$470$	0.181352	0.00836514
$471$	0	0
$472$	11.6019	0.534021
$473$	8.22574	0.378220
$474$	0	0
$475$	0.0290683	0.00133375
$476$	28.9020	1.32472
$477$	0	0
$478$	6.61119	0.302389
$479$	23.4710	1.07242	0.536208	−	0.844086i	$-0.319856\pi$
$479$	23.4710	1.07242	0.536208	+	0.844086i	$0.319856\pi$
$480$	0	0
$481$	−4.45536	−0.203147
$482$	−0.203725	−0.00927943
$483$	0	0
$484$	17.0217	0.773715
$485$	−2.40449	−0.109182
$486$	0	0
$487$	−3.73378	−0.169194	−0.0845969	−	0.996415i	$-0.526960\pi$
$487$	−3.73378	−0.169194	−0.0845969	+	0.996415i	$0.526960\pi$
$488$	14.3199	0.648230
$489$	0	0
$490$	2.17222	0.0981307
$491$	−26.9239	−1.21506	−0.607530	−	0.794297i	$-0.707840\pi$
$491$	−26.9239	−1.21506	−0.607530	+	0.794297i	$0.707840\pi$
$492$	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	$0.638402\pi$
$500$	1.63304	0.0730319
$501$	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
$505$	7.27907	0.323914
$506$	−0.460016	−0.0204502
$507$	0	0
$508$	10.6860	0.474115
$509$	−7.00770	−0.310611	−0.155305	−	0.987867i	$-0.549636\pi$
$509$	−7.00770	−0.310611	−0.155305	+	0.987867i	$0.549636\pi$
$510$	0	0
$511$	15.6386	0.691811
$512$	−19.2790	−0.852018
$513$	0	0
$514$	3.23622	0.142744
$515$	7.73634	0.340904
$516$	0	0
$517$	0.227342	0.00999847
$518$	−10.8872	−0.478357
$519$	0	0
$520$	1.77507	0.0778420
$521$	−5.60958	−0.245760	−0.122880	−	0.992422i	$-0.539213\pi$
$521$	−5.60958	−0.245760	−0.122880	+	0.992422i	$0.539213\pi$
$522$	0	0
$523$	−11.9919	−0.524371	−0.262185	−	0.965018i	$-0.584443\pi$
$523$	−11.9919	−0.524371	−0.262185	+	0.965018i	$0.584443\pi$
$524$	−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
$530$	−0.383777	−0.0166702
$531$	0	0
$532$	−0.154447	−0.00669615
$533$	−4.64602	−0.201242
$534$	0	0
$535$	−5.11476	−0.221130
$536$	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
$545$	15.8090	0.677182
$546$	0	0
$547$	−19.3357	−0.826735	−0.413368	−	0.910564i	$-0.635648\pi$
$547$	−19.3357	−0.826735	−0.413368	+	0.910564i	$0.635648\pi$
$548$	−25.2687	−1.07943
$549$	0	0
$550$	−0.460016	−0.0196152
$551$	−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
$560$	−6.28890	−0.265755
$561$	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
$565$	−8.92145	−0.375328
$566$	1.97713	0.0831051
$567$	0	0
$568$	24.2038	1.01557
$569$	−6.48889	−0.272028	−0.136014	−	0.990707i	$-0.543429\pi$
$569$	−6.48889	−0.272028	−0.136014	+	0.990707i	$0.543429\pi$
$570$	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
$575$	1.00000	0.0417029
$576$	0	0
$577$	−46.4119	−1.93215	−0.966076	−	0.258259i	$-0.916851\pi$
$577$	−46.4119	−1.93215	−0.966076	+	0.258259i	$0.916851\pi$
$578$	−7.62622	−0.317209
$579$	0	0
$580$	−2.53606	−0.105304
$581$	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
$590$	3.19344	0.131472
$591$	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
$595$	17.6983	0.725558
$596$	12.7898	0.523892
$597$	0	0
$598$	0.488591	0.0199800
$599$	3.41317	0.139459	0.0697293	−	0.997566i	$-0.477786\pi$
$599$	3.41317	0.139459	0.0697293	+	0.997566i	$0.477786\pi$
$600$	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
$605$	10.4233	0.423768
$606$	0	0
$607$	−34.2006	−1.38816	−0.694080	−	0.719898i	$-0.744189\pi$
$607$	−34.2006	−1.38816	−0.694080	+	0.719898i	$0.744189\pi$
$608$	−0.161983	−0.00656927
$609$	0	0
$610$	3.94156	0.159589
$611$	−0.241463	−0.00976856
$612$	0	0
$613$	−20.6739	−0.835009	−0.417505	−	0.908675i	$-0.637095\pi$
$613$	−20.6739	−0.835009	−0.417505	+	0.908675i	$0.637095\pi$
$614$	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	$0.348721\pi$
$620$	7.56595	0.303856
$621$	0	0
$622$	11.0339	0.442420
$623$	−54.0626	−2.16597
$624$	0	0
$625$	1.00000	0.0400000
$626$	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
$635$	6.54363	0.259676
$636$	0	0
$637$	−2.89222	−0.114594
$638$	0.714390	0.0282830
$639$	0	0
$640$	−11.4419	−0.452281
$641$	−34.7903	−1.37413	−0.687067	−	0.726594i	$-0.741102\pi$
$641$	−34.7903	−1.37413	−0.687067	+	0.726594i	$0.741102\pi$
$642$	0	0
$643$	42.5726	1.67890	0.839449	−	0.543439i	$-0.182878\pi$
$643$	42.5726	1.67890	0.839449	+	0.543439i	$0.182878\pi$
$644$	−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
$650$	0.488591	0.0191641
$651$	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
$655$	−17.2727	−0.674901
$656$	11.1341	0.434714
$657$	0	0
$658$	−0.590045	−0.0230024
$659$	10.6150	0.413501	0.206750	−	0.978394i	$-0.433711\pi$
$659$	10.6150	0.413501	0.206750	+	0.978394i	$0.433711\pi$
$660$	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
$665$	−0.0945766	−0.00366752
$666$	0	0
$667$	−1.55297	−0.0601311
$668$	−28.7867	−1.11379
$669$	0	0
$670$	5.59891	0.216305
$671$	4.94112	0.190750
$672$	0	0
$673$	41.7219	1.60826	0.804130	−	0.594454i	$-0.202632\pi$
$673$	41.7219	1.60826	0.804130	+	0.594454i	$0.202632\pi$
$674$	−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
$680$	11.9714	0.459083
$681$	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
$685$	−15.4734	−0.591208
$686$	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
$695$	4.54603	0.172441
$696$	0	0
$697$	−31.3337	−1.18685
$698$	−13.0300	−0.493191
$699$	0	0
$700$	−5.31325	−0.200822
$701$	32.2539	1.21821	0.609106	−	0.793089i	$-0.291528\pi$
$701$	32.2539	1.21821	0.609106	+	0.793089i	$0.291528\pi$
$702$	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	$0.275748\pi$
$710$	6.66214	0.250026
$711$	0	0
$712$	−36.5689	−1.37048
$713$	4.63304	0.173509
$714$	0	0
$715$	0.612495	0.0229060
$716$	3.14078	0.117376
$717$	0	0
$718$	−10.8082	−0.403360
$719$	23.1282	0.862536	0.431268	−	0.902224i	$-0.358066\pi$
$719$	23.1282	0.862536	0.431268	+	0.902224i	$0.358066\pi$
$720$	0	0
$721$	−25.1709	−0.937413
$722$	11.5091	0.428326
$723$	0	0
$724$	−7.78981	−0.289506
$725$	−1.55297	−0.0576758
$726$	0	0
$727$	41.9820	1.55703	0.778513	−	0.627628i	$-0.215974\pi$
$727$	41.9820	1.55703	0.778513	+	0.627628i	$0.215974\pi$
$728$	−5.77536	−0.214049
$729$	0	0
$730$	2.91167	0.107766
$731$	−58.9220	−2.17931
$732$	0	0
$733$	6.33790	0.234096	0.117048	−	0.993126i	$-0.462657\pi$
$733$	6.33790	0.234096	0.117048	+	0.993126i	$0.462657\pi$
$734$	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	$0.747187\pi$
$740$	9.02076	0.331610
$741$	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
$745$	7.83190	0.286939
$746$	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
$755$	8.29142	0.301756
$756$	0	0
$757$	45.9553	1.67028	0.835138	−	0.550041i	$-0.185388\pi$
$757$	45.9553	1.67028	0.835138	+	0.550041i	$0.185388\pi$
$758$	2.66827	0.0969158
$759$	0	0
$760$	−0.0639733	−0.00232056
$761$	−43.6112	−1.58090	−0.790452	−	0.612523i	$-0.790155\pi$
$761$	−43.6112	−1.58090	−0.790452	+	0.612523i	$0.790155\pi$
$762$	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	$0.706594\pi$
$770$	1.49671	0.0539376
$771$	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
$775$	4.63304	0.166424
$776$	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
$785$	−10.0783	−0.359708
$786$	0	0
$787$	−18.9710	−0.676244	−0.338122	−	0.941102i	$-0.609792\pi$
$787$	−18.9710	−0.676244	−0.338122	+	0.941102i	$0.609792\pi$
$788$	−38.9118	−1.38618
$789$	0	0
$790$	7.06793	0.251466
$791$	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
$800$	−5.57248	−0.197017
$801$	0	0
$802$	−21.5241	−0.760041
$803$	3.65006	0.128808
$804$	0	0
$805$	−3.25359	−0.114674
$806$	2.26366	0.0797341
$807$	0	0
$808$	−16.0197	−0.563571
$809$	−42.5832	−1.49715	−0.748573	−	0.663052i	$-0.769261\pi$
$809$	−42.5832	−1.49715	−0.748573	+	0.663052i	$0.769261\pi$
$810$	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
$815$	3.65296	0.127958
$816$	0	0
$817$	0.314869	0.0110159
$818$	−10.5086	−0.367426
$819$	0	0
$820$	9.40679	0.328499
$821$	13.8087	0.481929	0.240964	−	0.970534i	$-0.422536\pi$
$821$	13.8087	0.481929	0.240964	+	0.970534i	$0.422536\pi$
$822$	0	0
$823$	−44.7579	−1.56016	−0.780082	−	0.625678i	$-0.784822\pi$
$823$	−44.7579	−1.56016	−0.780082	+	0.625678i	$0.784822\pi$
$824$	−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	$0.381013\pi$
$830$	2.60948	0.0905763
$831$	0	0
$832$	0.395355	0.0137065
$833$	−19.5057	−0.675833
$834$	0	0
$835$	−17.6276	−0.610029
$836$	−0.0360481	−0.00124675
$837$	0	0
$838$	−19.0661	−0.658626
$839$	57.6216	1.98932	0.994659	−	0.103213i	$-0.0329124\pi$
$839$	57.6216	1.98932	0.994659	+	0.103213i	$0.0329124\pi$
$840$	0	0
$841$	−26.5883	−0.916838
$842$	−5.92906	−0.204329
$843$	0	0
$844$	−36.6149	−1.26034
$845$	12.3495	0.424834
$846$	0	0
$847$	−33.9133	−1.16527
$848$	−1.22457	−0.0420518
$849$	0	0
$850$	3.29515	0.113023
$851$	5.52390	0.189357
$852$	0	0
$853$	−21.6926	−0.742741	−0.371370	−	0.928485i	$-0.621112\pi$
$853$	−21.6926	−0.742741	−0.371370	+	0.928485i	$0.621112\pi$
$854$	−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	$0.577812\pi$
$860$	17.6892	0.603196
$861$	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
$865$	−6.56173	−0.223105
$866$	−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
$875$	−3.25359	−0.109992
$876$	0	0
$877$	−3.21780	−0.108657	−0.0543287	−	0.998523i	$-0.517302\pi$
$877$	−3.21780	−0.108657	−0.0543287	+	0.998523i	$0.517302\pi$
$878$	10.2630	0.346361
$879$	0	0
$880$	−1.46783	−0.0494806
$881$	−52.6836	−1.77495	−0.887477	−	0.460852i	$-0.847544\pi$
$881$	−52.6836	−1.77495	−0.887477	+	0.460852i	$0.847544\pi$
$882$	0	0
$883$	−6.07047	−0.204287	−0.102144	−	0.994770i	$-0.532570\pi$
$883$	−6.07047	−0.204287	−0.102144	+	0.994770i	$0.532570\pi$
$884$	−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
$890$	−10.0657	−0.337401
$891$	0	0
$892$	15.0129	0.502669
$893$	0.00870230	0.000291211 0
$894$	0	0
$895$	1.92327	0.0642877
$896$	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
$905$	−4.77012	−0.158564
$906$	0	0
$907$	44.5820	1.48032	0.740160	−	0.672430i	$-0.234750\pi$
$907$	44.5820	1.48032	0.740160	+	0.672430i	$0.234750\pi$
$908$	40.9918	1.36036
$909$	0	0
$910$	−1.58968	−0.0526973
$911$	−4.99864	−0.165612	−0.0828062	−	0.996566i	$-0.526388\pi$
$911$	−4.99864	−0.165612	−0.0828062	+	0.996566i	$0.526388\pi$
$912$	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	$0.744082\pi$
$920$	−2.20079	−0.0725579
$921$	0	0
$922$	9.97872	0.328632
$923$	−8.87039	−0.291973
$924$	0	0
$925$	5.52390	0.181625
$926$	−8.25522	−0.271283
$927$	0	0
$928$	8.65388	0.284078
$929$	55.0370	1.80570	0.902852	−	0.429951i	$-0.141469\pi$
$929$	55.0370	1.80570	0.902852	+	0.429951i	$0.141469\pi$
$930$	0	0
$931$	0.104235	0.00341617
$932$	−42.4714	−1.39120
$933$	0	0
$934$	−2.68023	−0.0876999
$935$	4.13078	0.135091
$936$	0	0
$937$	31.2596	1.02121	0.510603	−	0.859817i	$-0.329422\pi$
$937$	31.2596	1.02121	0.510603	+	0.859817i	$0.329422\pi$
$938$	−18.2166	−0.594793
$939$	0	0
$940$	0.488890	0.0159458
$941$	4.14617	0.135161	0.0675806	−	0.997714i	$-0.478472\pi$
$941$	4.14617	0.135161	0.0675806	+	0.997714i	$0.478472\pi$
$942$	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
$950$	−0.0176087	−0.000571303 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
$955$	6.92978	0.224242
$956$	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
$965$	3.26608	0.105139
$966$	0	0
$967$	−40.1338	−1.29062	−0.645308	−	0.763922i	$-0.723271\pi$
$967$	−40.1338	−1.29062	−0.645308	+	0.763922i	$0.723271\pi$
$968$	−22.9396	−0.737305
$969$	0	0
$970$	1.45657	0.0467676
$971$	−2.29254	−0.0735710	−0.0367855	−	0.999323i	$-0.511712\pi$
$971$	−2.29254	−0.0735710	−0.0367855	+	0.999323i	$0.511712\pi$
$972$	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
$980$	5.85588	0.187059
$981$	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
$985$	−23.8278	−0.759217
$986$	−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
$995$	12.7303	0.403578
$996$	0	0
$997$	−26.1503	−0.828189	−0.414094	−	0.910234i	$-0.635902\pi$
$997$	−26.1503	−0.828189	−0.414094	+	0.910234i	$0.635902\pi$
$998$	11.4001	0.360864
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.3	✓	6	1.1	even	1	trivial
1035.2.a.q.1.4	yes	6	3.2	odd	2
5175.2.a.by.1.3		6	15.14	odd	2
5175.2.a.bz.1.4		6	5.4	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q-0.605771 q^{2} -1.63304 q^{4} -1.00000 q^{5} +3.25359 q^{7} +2.20079 q^{8} +0.605771 q^{10} +0.759390 q^{11} -0.806561 q^{13} -1.97093 q^{14} +1.93291 q^{16} -5.43960 q^{17} +0.0290683 q^{19} +1.63304 q^{20} -0.460016 q^{22} +1.00000 q^{23} +1.00000 q^{25} +0.488591 q^{26} -5.31325 q^{28} -1.55297 q^{29} +4.63304 q^{31} -5.57248 q^{32} +3.29515 q^{34} -3.25359 q^{35} +5.52390 q^{37} -0.0176087 q^{38} -2.20079 q^{40} +5.76029 q^{41} +10.8320 q^{43} -1.24012 q^{44} -0.605771 q^{46} +0.299374 q^{47} +3.58587 q^{49} -0.605771 q^{50} +1.31715 q^{52} -0.633536 q^{53} -0.759390 q^{55} +7.16048 q^{56} +0.940742 q^{58} +5.27170 q^{59} +6.50669 q^{61} -2.80656 q^{62} -0.490173 q^{64} +0.806561 q^{65} +9.24263 q^{67} +8.88310 q^{68} +1.97093 q^{70} +10.9978 q^{71} +4.80656 q^{73} -3.34622 q^{74} -0.0474698 q^{76} +2.47075 q^{77} +11.6677 q^{79} -1.93291 q^{80} -3.48941 q^{82} +4.30770 q^{83} +5.43960 q^{85} -6.56173 q^{86} +1.67126 q^{88} -16.6163 q^{89} -2.62422 q^{91} -1.63304 q^{92} -0.181352 q^{94} -0.0290683 q^{95} +2.40449 q^{97} -2.17222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 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^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 12 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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