LMFDB - Embedded newform 1525.2.a.m.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1525,2,Mod(1,1525)] 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("1525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1525 = 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,5,7,11,0,4,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.1771863082$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5x^{9} - 3x^{8} + 39x^{7} - 9x^{6} - 106x^{5} + 37x^{4} + 118x^{3} - 39x^{2} - 45x + 13 $ x^10 - 5x^9 - 3x^8 + 39x^7 - 9x^6 - 106x^5 + 37x^4 + 118x^3 - 39x^2 - 45*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.279450$ of defining polynomial
Character	$\chi$	$=$	1525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.720550 q^{2} +0.439334 q^{3} -1.48081 q^{4} +0.316562 q^{6} -1.55598 q^{7} -2.50810 q^{8} -2.80699 q^{9} +1.48709 q^{11} -0.650569 q^{12} +5.11309 q^{13} -1.12117 q^{14} +1.15441 q^{16} -4.91648 q^{17} -2.02257 q^{18} +2.65659 q^{19} -0.683597 q^{21} +1.07152 q^{22} +7.51428 q^{23} -1.10189 q^{24} +3.68424 q^{26} -2.55121 q^{27} +2.30411 q^{28} +7.73136 q^{29} +1.77957 q^{31} +5.84800 q^{32} +0.653329 q^{33} -3.54257 q^{34} +4.15660 q^{36} +10.2069 q^{37} +1.91421 q^{38} +2.24635 q^{39} -2.46759 q^{41} -0.492566 q^{42} -5.18165 q^{43} -2.20209 q^{44} +5.41442 q^{46} +0.817989 q^{47} +0.507170 q^{48} -4.57891 q^{49} -2.15998 q^{51} -7.57150 q^{52} +7.10407 q^{53} -1.83827 q^{54} +3.90256 q^{56} +1.16713 q^{57} +5.57083 q^{58} +3.46647 q^{59} +1.00000 q^{61} +1.28227 q^{62} +4.36763 q^{63} +1.90497 q^{64} +0.470756 q^{66} +4.18821 q^{67} +7.28036 q^{68} +3.30128 q^{69} +10.8582 q^{71} +7.04019 q^{72} +0.305565 q^{73} +7.35458 q^{74} -3.93390 q^{76} -2.31389 q^{77} +1.61861 q^{78} -16.3167 q^{79} +7.30012 q^{81} -1.77802 q^{82} +4.51834 q^{83} +1.01228 q^{84} -3.73364 q^{86} +3.39665 q^{87} -3.72976 q^{88} -9.13297 q^{89} -7.95588 q^{91} -11.1272 q^{92} +0.781826 q^{93} +0.589402 q^{94} +2.56923 q^{96} +16.8035 q^{97} -3.29934 q^{98} -4.17424 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{2} + 7 q^{3} + 11 q^{4} + 4 q^{6} + 6 q^{7} + 15 q^{8} + 11 q^{9} + 19 q^{12} + 3 q^{13} - 5 q^{14} + 17 q^{16} + 11 q^{17} + 5 q^{18} - 2 q^{19} + 9 q^{21} - 6 q^{22} + 33 q^{23} - 7 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 5 * q^2 + 7 * q^3 + 11 * q^4 + 4 * q^6 + 6 * q^7 + 15 * q^8 + 11 * q^9 + 19 * q^12 + 3 * q^13 - 5 * q^14 + 17 * q^16 + 11 * q^17 + 5 * q^18 - 2 * q^19 + 9 * q^21 - 6 * q^22 + 33 * q^23 - 7 * q^24 + 12 * q^26 + 19 * q^27 + 21 * q^28 - 3 * q^29 + 20 * q^32 + 7 * q^33 - 3 * q^34 + 45 * q^36 - q^37 + 11 * q^38 - 22 * q^39 + 4 * q^41 - 16 * q^42 + 30 * q^43 - 22 * q^44 + 23 * q^46 + 5 * q^47 + 20 * q^48 - 10 * q^49 - q^51 - 35 * q^52 + 35 * q^53 - 52 * q^54 + 13 * q^56 + 4 * q^57 + 26 * q^58 - 4 * q^59 + 10 * q^61 + 22 * q^62 + 47 * q^63 + q^64 + 25 * q^66 + 28 * q^68 + 26 * q^69 - 11 * q^71 - 27 * q^72 - 9 * q^73 - 41 * q^74 + 44 * q^76 - 4 * q^77 + 51 * q^78 + 16 * q^79 + 14 * q^81 - 22 * q^82 + 18 * q^83 + 2 * q^84 + 7 * q^86 - q^87 - 32 * q^88 - 7 * q^89 - 12 * q^91 + 80 * q^92 - 3 * q^93 - 73 * q^94 + 53 * q^96 + 8 * q^97 - 8 * q^98 - 12 * q^99

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 $

$2$	0.720550	0.509506	0.254753	−	0.967006i	$-0.418006\pi$
$2$	0.720550	0.509506	0.254753	+	0.967006i	$0.418006\pi$
$3$	0.439334	0.253650	0.126825	−	0.991925i	$-0.459521\pi$
$3$	0.439334	0.253650	0.126825	+	0.991925i	$0.459521\pi$
$4$	−1.48081	−0.740404
$5$	0	0
$5$	0	0
$6$	0.316562	0.129236
$7$	−1.55598	−0.588107	−0.294053	−	0.955789i	$-0.595004\pi$
$7$	−1.55598	−0.588107	−0.294053	+	0.955789i	$0.595004\pi$
$8$	−2.50810	−0.886746
$9$	−2.80699	−0.935662
$10$	0	0
$11$	1.48709	0.448374	0.224187	−	0.974546i	$-0.428027\pi$
$11$	1.48709	0.448374	0.224187	+	0.974546i	$0.428027\pi$
$12$	−0.650569	−0.187803
$13$	5.11309	1.41812	0.709058	−	0.705151i	$-0.249121\pi$
$13$	5.11309	1.41812	0.709058	+	0.705151i	$0.249121\pi$
$14$	−1.12117	−0.299644
$15$	0	0
$16$	1.15441	0.288601
$17$	−4.91648	−1.19242	−0.596211	−	0.802828i	$-0.703328\pi$
$17$	−4.91648	−1.19242	−0.596211	+	0.802828i	$0.703328\pi$
$18$	−2.02257	−0.476725
$19$	2.65659	0.609463	0.304732	−	0.952438i	$-0.401433\pi$
$19$	2.65659	0.609463	0.304732	+	0.952438i	$0.401433\pi$
$20$	0	0
$21$	−0.683597	−0.149173
$22$	1.07152	0.228449
$23$	7.51428	1.56684	0.783418	−	0.621495i	$-0.213474\pi$
$23$	7.51428	1.56684	0.783418	+	0.621495i	$0.213474\pi$
$24$	−1.10189	−0.224923
$25$	0	0
$26$	3.68424	0.722538
$27$	−2.55121	−0.490980
$28$	2.30411	0.435437
$29$	7.73136	1.43568	0.717838	−	0.696210i	$-0.245132\pi$
$29$	7.73136	1.43568	0.717838	+	0.696210i	$0.245132\pi$
$30$	0	0
$31$	1.77957	0.319620	0.159810	−	0.987148i	$-0.448912\pi$
$31$	1.77957	0.319620	0.159810	+	0.987148i	$0.448912\pi$
$32$	5.84800	1.03379
$33$	0.653329	0.113730
$34$	−3.54257	−0.607546
$35$	0	0
$36$	4.15660	0.692767
$37$	10.2069	1.67800	0.839001	−	0.544130i	$-0.183140\pi$
$37$	10.2069	1.67800	0.839001	+	0.544130i	$0.183140\pi$
$38$	1.91421	0.310525
$39$	2.24635	0.359704
$40$	0	0
$41$	−2.46759	−0.385372	−0.192686	−	0.981260i	$-0.561720\pi$
$41$	−2.46759	−0.385372	−0.192686	+	0.981260i	$0.561720\pi$
$42$	−0.492566	−0.0760046
$43$	−5.18165	−0.790194	−0.395097	−	0.918639i	$-0.629289\pi$
$43$	−5.18165	−0.790194	−0.395097	+	0.918639i	$0.629289\pi$
$44$	−2.20209	−0.331978
$45$	0	0
$46$	5.41442	0.798312
$47$	0.817989	0.119316	0.0596580	−	0.998219i	$-0.480999\pi$
$47$	0.817989	0.119316	0.0596580	+	0.998219i	$0.480999\pi$
$48$	0.507170	0.0732036
$49$	−4.57891	−0.654130
$50$	0	0
$51$	−2.15998	−0.302458
$52$	−7.57150	−1.04998
$53$	7.10407	0.975819	0.487909	−	0.872894i	$-0.337760\pi$
$53$	7.10407	0.975819	0.487909	+	0.872894i	$0.337760\pi$
$54$	−1.83827	−0.250157
$55$	0	0
$56$	3.90256	0.521501
$57$	1.16713	0.154590
$58$	5.57083	0.731486
$59$	3.46647	0.451296	0.225648	−	0.974209i	$-0.427550\pi$
$59$	3.46647	0.451296	0.225648	+	0.974209i	$0.427550\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.28227	0.162848
$63$	4.36763	0.550269
$64$	1.90497	0.238121
$65$	0	0
$66$	0.470756	0.0579461
$67$	4.18821	0.511671	0.255836	−	0.966720i	$-0.417649\pi$
$67$	4.18821	0.511671	0.255836	+	0.966720i	$0.417649\pi$
$68$	7.28036	0.882874
$69$	3.30128	0.397428
$70$	0	0
$71$	10.8582	1.28863	0.644317	−	0.764759i	$-0.277142\pi$
$71$	10.8582	1.28863	0.644317	+	0.764759i	$0.277142\pi$
$72$	7.04019	0.829694
$73$	0.305565	0.0357637	0.0178818	−	0.999840i	$-0.494308\pi$
$73$	0.305565	0.0357637	0.0178818	+	0.999840i	$0.494308\pi$
$74$	7.35458	0.854952
$75$	0	0
$76$	−3.93390	−0.451249
$77$	−2.31389	−0.263692
$78$	1.61861	0.183272
$79$	−16.3167	−1.83577	−0.917887	−	0.396843i	$-0.870106\pi$
$79$	−16.3167	−1.83577	−0.917887	+	0.396843i	$0.870106\pi$
$80$	0	0
$81$	7.30012	0.811125
$82$	−1.77802	−0.196350
$83$	4.51834	0.495952	0.247976	−	0.968766i	$-0.420235\pi$
$83$	4.51834	0.495952	0.247976	+	0.968766i	$0.420235\pi$
$84$	1.01228	0.110448
$85$	0	0
$86$	−3.73364	−0.402608
$87$	3.39665	0.364159
$88$	−3.72976	−0.397594
$89$	−9.13297	−0.968093	−0.484046	−	0.875042i	$-0.660833\pi$
$89$	−9.13297	−0.968093	−0.484046	+	0.875042i	$0.660833\pi$
$90$	0	0
$91$	−7.95588	−0.834003
$92$	−11.1272	−1.16009
$93$	0.781826	0.0810716
$94$	0.589402	0.0607922
$95$	0	0
$96$	2.56923	0.262221
$97$	16.8035	1.70614	0.853071	−	0.521796i	$-0.174738\pi$
$97$	16.8035	1.70614	0.853071	+	0.521796i	$0.174738\pi$
$98$	−3.29934	−0.333283
$99$	−4.17424	−0.419526
$100$	0	0
$101$	−12.9907	−1.29262	−0.646310	−	0.763075i	$-0.723689\pi$
$101$	−12.9907	−1.29262	−0.646310	+	0.763075i	$0.723689\pi$
$102$	−1.55637	−0.154104
$103$	5.55458	0.547309	0.273655	−	0.961828i	$-0.411767\pi$
$103$	5.55458	0.547309	0.273655	+	0.961828i	$0.411767\pi$
$104$	−12.8241	−1.25751
$105$	0	0
$106$	5.11884	0.497186
$107$	−1.41509	−0.136802	−0.0684011	−	0.997658i	$-0.521790\pi$
$107$	−1.41509	−0.136802	−0.0684011	+	0.997658i	$0.521790\pi$
$108$	3.77785	0.363523
$109$	4.18011	0.400382	0.200191	−	0.979757i	$-0.435844\pi$
$109$	4.18011	0.400382	0.200191	+	0.979757i	$0.435844\pi$
$110$	0	0
$111$	4.48423	0.425625
$112$	−1.79624	−0.169728
$113$	13.1683	1.23877	0.619387	−	0.785086i	$-0.287381\pi$
$113$	13.1683	1.23877	0.619387	+	0.785086i	$0.287381\pi$
$114$	0.840976	0.0787646
$115$	0	0
$116$	−11.4486	−1.06298
$117$	−14.3524	−1.32688
$118$	2.49776	0.229938
$119$	7.64997	0.701272
$120$	0	0
$121$	−8.78857	−0.798961
$122$	0.720550	0.0652356
$123$	−1.08410	−0.0977496
$124$	−2.63520	−0.236648
$125$	0	0
$126$	3.14709	0.280365
$127$	3.68461	0.326957	0.163478	−	0.986547i	$-0.447729\pi$
$127$	3.68461	0.326957	0.163478	+	0.986547i	$0.447729\pi$
$128$	−10.3234	−0.912466
$129$	−2.27647	−0.200432
$130$	0	0
$131$	−22.6546	−1.97934	−0.989670	−	0.143361i	$-0.954209\pi$
$131$	−22.6546	−1.97934	−0.989670	+	0.143361i	$0.954209\pi$
$132$	−0.967454	−0.0842061
$133$	−4.13361	−0.358430
$134$	3.01782	0.260700
$135$	0	0
$136$	12.3310	1.05738
$137$	18.5621	1.58586	0.792932	−	0.609310i	$-0.208553\pi$
$137$	18.5621	1.58586	0.792932	+	0.609310i	$0.208553\pi$
$138$	2.37874	0.202492
$139$	−21.5135	−1.82475	−0.912375	−	0.409356i	$-0.865753\pi$
$139$	−21.5135	−1.82475	−0.912375	+	0.409356i	$0.865753\pi$
$140$	0	0
$141$	0.359371	0.0302645
$142$	7.82389	0.656566
$143$	7.60361	0.635846
$144$	−3.24040	−0.270033
$145$	0	0
$146$	0.220175	0.0182218
$147$	−2.01167	−0.165920
$148$	−15.1144	−1.24240
$149$	−2.99790	−0.245598	−0.122799	−	0.992432i	$-0.539187\pi$
$149$	−2.99790	−0.245598	−0.122799	+	0.992432i	$0.539187\pi$
$150$	0	0
$151$	21.5771	1.75592	0.877959	−	0.478737i	$-0.158905\pi$
$151$	21.5771	1.75592	0.877959	+	0.478737i	$0.158905\pi$
$152$	−6.66298	−0.540439
$153$	13.8005	1.11570
$154$	−1.66727	−0.134353
$155$	0	0
$156$	−3.32642	−0.266327
$157$	−8.92442	−0.712246	−0.356123	−	0.934439i	$-0.615902\pi$
$157$	−8.92442	−0.712246	−0.356123	+	0.934439i	$0.615902\pi$
$158$	−11.7570	−0.935337
$159$	3.12106	0.247516
$160$	0	0
$161$	−11.6921	−0.921467
$162$	5.26011	0.413273
$163$	−0.757606	−0.0593403	−0.0296701	−	0.999560i	$-0.509446\pi$
$163$	−0.757606	−0.0593403	−0.0296701	+	0.999560i	$0.509446\pi$
$164$	3.65402	0.285331
$165$	0	0
$166$	3.25569	0.252690
$167$	17.9496	1.38898	0.694490	−	0.719502i	$-0.255630\pi$
$167$	17.9496	1.38898	0.694490	+	0.719502i	$0.255630\pi$
$168$	1.71453	0.132279
$169$	13.1437	1.01105
$170$	0	0
$171$	−7.45701	−0.570251
$172$	7.67302	0.585062
$173$	−11.1374	−0.846763	−0.423382	−	0.905951i	$-0.639157\pi$
$173$	−11.1374	−0.846763	−0.423382	+	0.905951i	$0.639157\pi$
$174$	2.44746	0.185541
$175$	0	0
$176$	1.71670	0.129401
$177$	1.52294	0.114471
$178$	−6.58076	−0.493249
$179$	−5.24241	−0.391836	−0.195918	−	0.980620i	$-0.562769\pi$
$179$	−5.24241	−0.391836	−0.195918	+	0.980620i	$0.562769\pi$
$180$	0	0
$181$	20.5187	1.52515	0.762573	−	0.646903i	$-0.223936\pi$
$181$	20.5187	1.52515	0.762573	+	0.646903i	$0.223936\pi$
$182$	−5.73261	−0.424930
$183$	0.439334	0.0324765
$184$	−18.8465	−1.38939
$185$	0	0
$186$	0.563345	0.0413065
$187$	−7.31124	−0.534651
$188$	−1.21128	−0.0883420
$189$	3.96964	0.288749
$190$	0	0
$191$	24.4453	1.76880	0.884402	−	0.466727i	$-0.154567\pi$
$191$	24.4453	1.76880	0.884402	+	0.466727i	$0.154567\pi$
$192$	0.836917	0.0603993
$193$	−15.6775	−1.12849	−0.564246	−	0.825607i	$-0.690833\pi$
$193$	−15.6775	−1.12849	−0.564246	+	0.825607i	$0.690833\pi$
$194$	12.1078	0.869289
$195$	0	0
$196$	6.78049	0.484320
$197$	10.1638	0.724141	0.362071	−	0.932151i	$-0.382070\pi$
$197$	10.1638	0.724141	0.362071	+	0.932151i	$0.382070\pi$
$198$	−3.00775	−0.213751
$199$	19.9611	1.41500	0.707502	−	0.706711i	$-0.249822\pi$
$199$	19.9611	1.41500	0.707502	+	0.706711i	$0.249822\pi$
$200$	0	0
$201$	1.84002	0.129785
$202$	−9.36043	−0.658598
$203$	−12.0299	−0.844331
$204$	3.19851	0.223941
$205$	0	0
$206$	4.00236	0.278857
$207$	−21.0925	−1.46603
$208$	5.90257	0.409270
$209$	3.95058	0.273267
$210$	0	0
$211$	−2.93714	−0.202201	−0.101100	−	0.994876i	$-0.532236\pi$
$211$	−2.93714	−0.202201	−0.101100	+	0.994876i	$0.532236\pi$
$212$	−10.5198	−0.722500
$213$	4.77039	0.326862
$214$	−1.01965	−0.0697016
$215$	0	0
$216$	6.39867	0.435375
$217$	−2.76898	−0.187971
$218$	3.01198	0.203997
$219$	0.134245	0.00907145
$220$	0	0
$221$	−25.1384	−1.69099
$222$	3.23112	0.216858
$223$	−3.52494	−0.236047	−0.118024	−	0.993011i	$-0.537656\pi$
$223$	−3.52494	−0.236047	−0.118024	+	0.993011i	$0.537656\pi$
$224$	−9.09940	−0.607979
$225$	0	0
$226$	9.48845	0.631162
$227$	11.8501	0.786522	0.393261	−	0.919427i	$-0.371347\pi$
$227$	11.8501	0.786522	0.393261	+	0.919427i	$0.371347\pi$
$228$	−1.72829	−0.114459
$229$	7.67875	0.507426	0.253713	−	0.967280i	$-0.418348\pi$
$229$	7.67875	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$230$	0	0
$231$	−1.01657	−0.0668854
$232$	−19.3910	−1.27308
$233$	−21.4561	−1.40563	−0.702816	−	0.711371i	$-0.748074\pi$
$233$	−21.4561	−1.40563	−0.702816	+	0.711371i	$0.748074\pi$
$234$	−10.3416	−0.676051
$235$	0	0
$236$	−5.13317	−0.334141
$237$	−7.16849	−0.465643
$238$	5.51219	0.357302
$239$	7.26667	0.470042	0.235021	−	0.971990i	$-0.424484\pi$
$239$	7.26667	0.470042	0.235021	+	0.971990i	$0.424484\pi$
$240$	0	0
$241$	−13.5399	−0.872180	−0.436090	−	0.899903i	$-0.643637\pi$
$241$	−13.5399	−0.872180	−0.436090	+	0.899903i	$0.643637\pi$
$242$	−6.33261	−0.407075
$243$	10.8608	0.696722
$244$	−1.48081	−0.0947990
$245$	0	0
$246$	−0.781145	−0.0498040
$247$	13.5834	0.864289
$248$	−4.46334	−0.283422
$249$	1.98506	0.125798
$250$	0	0
$251$	−18.4018	−1.16151	−0.580755	−	0.814079i	$-0.697242\pi$
$251$	−18.4018	−1.16151	−0.580755	+	0.814079i	$0.697242\pi$
$252$	−6.46761	−0.407421
$253$	11.1744	0.702529
$254$	2.65495	0.166586
$255$	0	0
$256$	−11.2484	−0.703028
$257$	−0.511671	−0.0319172	−0.0159586	−	0.999873i	$-0.505080\pi$
$257$	−0.511671	−0.0319172	−0.0159586	+	0.999873i	$0.505080\pi$
$258$	−1.64031	−0.102122
$259$	−15.8818	−0.986845
$260$	0	0
$261$	−21.7018	−1.34331
$262$	−16.3238	−1.00849
$263$	−7.12243	−0.439188	−0.219594	−	0.975591i	$-0.570473\pi$
$263$	−7.12243	−0.439188	−0.219594	+	0.975591i	$0.570473\pi$
$264$	−1.63861	−0.100850
$265$	0	0
$266$	−2.97847	−0.182622
$267$	−4.01242	−0.245556
$268$	−6.20193	−0.378843
$269$	−9.90914	−0.604171	−0.302085	−	0.953281i	$-0.597683\pi$
$269$	−9.90914	−0.604171	−0.302085	+	0.953281i	$0.597683\pi$
$270$	0	0
$271$	−3.24162	−0.196914	−0.0984572	−	0.995141i	$-0.531391\pi$
$271$	−3.24162	−0.196914	−0.0984572	+	0.995141i	$0.531391\pi$
$272$	−5.67561	−0.344135
$273$	−3.49529	−0.211545
$274$	13.3749	0.808007
$275$	0	0
$276$	−4.88856	−0.294257
$277$	16.8362	1.01159	0.505793	−	0.862655i	$-0.331200\pi$
$277$	16.8362	1.01159	0.505793	+	0.862655i	$0.331200\pi$
$278$	−15.5015	−0.929721
$279$	−4.99523	−0.299057
$280$	0	0
$281$	−13.6382	−0.813589	−0.406795	−	0.913520i	$-0.633354\pi$
$281$	−13.6382	−0.813589	−0.406795	+	0.913520i	$0.633354\pi$
$282$	0.258945	0.0154199
$283$	15.6794	0.932043	0.466021	−	0.884773i	$-0.345687\pi$
$283$	15.6794	0.932043	0.466021	+	0.884773i	$0.345687\pi$
$284$	−16.0789	−0.954109
$285$	0	0
$286$	5.47878	0.323967
$287$	3.83953	0.226640
$288$	−16.4153	−0.967278
$289$	7.17180	0.421871
$290$	0	0
$291$	7.38237	0.432762
$292$	−0.452483	−0.0264796
$293$	1.82255	0.106475	0.0532374	−	0.998582i	$-0.483046\pi$
$293$	1.82255	0.106475	0.0532374	+	0.998582i	$0.483046\pi$
$294$	−1.44951	−0.0845372
$295$	0	0
$296$	−25.5999	−1.48796
$297$	−3.79387	−0.220143
$298$	−2.16014	−0.125133
$299$	38.4212	2.22195
$300$	0	0
$301$	8.06256	0.464718
$302$	15.5474	0.894650
$303$	−5.70724	−0.327873
$304$	3.06678	0.175892
$305$	0	0
$306$	9.94395	0.568458
$307$	−10.2173	−0.583130	−0.291565	−	0.956551i	$-0.594176\pi$
$307$	−10.2173	−0.583130	−0.291565	+	0.956551i	$0.594176\pi$
$308$	3.42642	0.195238
$309$	2.44032	0.138825
$310$	0	0
$311$	10.3527	0.587050	0.293525	−	0.955951i	$-0.405172\pi$
$311$	10.3527	0.587050	0.293525	+	0.955951i	$0.405172\pi$
$312$	−5.63407	−0.318966
$313$	−24.2900	−1.37295	−0.686477	−	0.727152i	$-0.740844\pi$
$313$	−24.2900	−1.37295	−0.686477	+	0.727152i	$0.740844\pi$
$314$	−6.43049	−0.362894
$315$	0	0
$316$	24.1619	1.35921
$317$	0.342674	0.0192465	0.00962323	−	0.999954i	$-0.496937\pi$
$317$	0.342674	0.0192465	0.00962323	+	0.999954i	$0.496937\pi$
$318$	2.24888	0.126111
$319$	11.4972	0.643720
$320$	0	0
$321$	−0.621699	−0.0346999
$322$	−8.42475	−0.469493
$323$	−13.0611	−0.726737
$324$	−10.8101	−0.600560
$325$	0	0
$326$	−0.545893	−0.0302342
$327$	1.83646	0.101557
$328$	6.18895	0.341727
$329$	−1.27278	−0.0701706
$330$	0	0
$331$	10.5422	0.579449	0.289725	−	0.957110i	$-0.406436\pi$
$331$	10.5422	0.579449	0.289725	+	0.957110i	$0.406436\pi$
$332$	−6.69079	−0.367205
$333$	−28.6506	−1.57004
$334$	12.9336	0.707694
$335$	0	0
$336$	−0.789148	−0.0430516
$337$	−0.563439	−0.0306925	−0.0153462	−	0.999882i	$-0.504885\pi$
$337$	−0.563439	−0.0306925	−0.0153462	+	0.999882i	$0.504885\pi$
$338$	9.47066	0.515136
$339$	5.78530	0.314214
$340$	0	0
$341$	2.64638	0.143309
$342$	−5.37315	−0.290547
$343$	18.0166	0.972805
$344$	12.9961	0.700701
$345$	0	0
$346$	−8.02508	−0.431431
$347$	−1.76141	−0.0945572	−0.0472786	−	0.998882i	$-0.515055\pi$
$347$	−1.76141	−0.0945572	−0.0472786	+	0.998882i	$0.515055\pi$
$348$	−5.02978	−0.269625
$349$	24.1482	1.29262	0.646311	−	0.763074i	$-0.276311\pi$
$349$	24.1482	1.29262	0.646311	+	0.763074i	$0.276311\pi$
$350$	0	0
$351$	−13.0445	−0.696266
$352$	8.69649	0.463525
$353$	−3.42979	−0.182550	−0.0912748	−	0.995826i	$-0.529094\pi$
$353$	−3.42979	−0.182550	−0.0912748	+	0.995826i	$0.529094\pi$
$354$	1.09735	0.0583236
$355$	0	0
$356$	13.5242	0.716779
$357$	3.36089	0.177877
$358$	−3.77742	−0.199643
$359$	25.7065	1.35674	0.678368	−	0.734722i	$-0.262687\pi$
$359$	25.7065	1.35674	0.678368	+	0.734722i	$0.262687\pi$
$360$	0	0
$361$	−11.9425	−0.628555
$362$	14.7848	0.777071
$363$	−3.86112	−0.202656
$364$	11.7811	0.617499
$365$	0	0
$366$	0.316562	0.0165470
$367$	−18.5776	−0.969743	−0.484872	−	0.874585i	$-0.661134\pi$
$367$	−18.5776	−0.969743	−0.484872	+	0.874585i	$0.661134\pi$
$368$	8.67453	0.452191
$369$	6.92648	0.360578
$370$	0	0
$371$	−11.0538	−0.573886
$372$	−1.15773	−0.0600257
$373$	−19.7060	−1.02034	−0.510168	−	0.860075i	$-0.670417\pi$
$373$	−19.7060	−1.02034	−0.510168	+	0.860075i	$0.670417\pi$
$374$	−5.26812	−0.272408
$375$	0	0
$376$	−2.05160	−0.105803
$377$	39.5311	2.03595
$378$	2.86032	0.147119
$379$	9.63249	0.494788	0.247394	−	0.968915i	$-0.420426\pi$
$379$	9.63249	0.494788	0.247394	+	0.968915i	$0.420426\pi$
$380$	0	0
$381$	1.61878	0.0829324
$382$	17.6141	0.901216
$383$	32.2998	1.65044	0.825222	−	0.564808i	$-0.191050\pi$
$383$	32.2998	1.65044	0.825222	+	0.564808i	$0.191050\pi$
$384$	−4.53541	−0.231447
$385$	0	0
$386$	−11.2964	−0.574973
$387$	14.5448	0.739354
$388$	−24.8828	−1.26323
$389$	−12.4875	−0.633142	−0.316571	−	0.948569i	$-0.602532\pi$
$389$	−12.4875	−0.633142	−0.316571	+	0.948569i	$0.602532\pi$
$390$	0	0
$391$	−36.9438	−1.86833
$392$	11.4844	0.580047
$393$	−9.95294	−0.502059
$394$	7.32353	0.368954
$395$	0	0
$396$	6.18124	0.310619
$397$	−26.4161	−1.32578	−0.662892	−	0.748715i	$-0.730671\pi$
$397$	−26.4161	−1.32578	−0.662892	+	0.748715i	$0.730671\pi$
$398$	14.3830	0.720953
$399$	−1.81604	−0.0909155
$400$	0	0
$401$	−16.2783	−0.812900	−0.406450	−	0.913673i	$-0.633233\pi$
$401$	−16.2783	−0.812900	−0.406450	+	0.913673i	$0.633233\pi$
$402$	1.32583	0.0661264
$403$	9.09910	0.453258
$404$	19.2367	0.957061
$405$	0	0
$406$	−8.66813	−0.430192
$407$	15.1785	0.752372
$408$	5.41744	0.268203
$409$	22.6122	1.11810	0.559051	−	0.829133i	$-0.311166\pi$
$409$	22.6122	1.11810	0.559051	+	0.829133i	$0.311166\pi$
$410$	0	0
$411$	8.15495	0.402254
$412$	−8.22527	−0.405230
$413$	−5.39377	−0.265410
$414$	−15.1982	−0.746950
$415$	0	0
$416$	29.9013	1.46603
$417$	−9.45161	−0.462847
$418$	2.84659	0.139231
$419$	23.9325	1.16918	0.584590	−	0.811329i	$-0.301256\pi$
$419$	23.9325	1.16918	0.584590	+	0.811329i	$0.301256\pi$
$420$	0	0
$421$	−2.68092	−0.130660	−0.0653299	−	0.997864i	$-0.520810\pi$
$421$	−2.68092	−0.130660	−0.0653299	+	0.997864i	$0.520810\pi$
$422$	−2.11636	−0.103023
$423$	−2.29608	−0.111639
$424$	−17.8177	−0.865304
$425$	0	0
$426$	3.43730	0.166538
$427$	−1.55598	−0.0752994
$428$	2.09548	0.101289
$429$	3.34053	0.161282
$430$	0	0
$431$	−20.8236	−1.00304	−0.501520	−	0.865146i	$-0.667225\pi$
$431$	−20.8236	−1.00304	−0.501520	+	0.865146i	$0.667225\pi$
$432$	−2.94513	−0.141697
$433$	−22.3913	−1.07606	−0.538029	−	0.842926i	$-0.680831\pi$
$433$	−22.3913	−1.07606	−0.538029	+	0.842926i	$0.680831\pi$
$434$	−1.99519	−0.0957723
$435$	0	0
$436$	−6.18993	−0.296444
$437$	19.9624	0.954929
$438$	0.0967304	0.00462196
$439$	−41.8296	−1.99642	−0.998208	−	0.0598369i	$-0.980942\pi$
$439$	−41.8296	−1.99642	−0.998208	+	0.0598369i	$0.980942\pi$
$440$	0	0
$441$	12.8529	0.612045
$442$	−18.1135	−0.861570
$443$	15.2626	0.725149	0.362575	−	0.931955i	$-0.381898\pi$
$443$	15.2626	0.725149	0.362575	+	0.931955i	$0.381898\pi$
$444$	−6.64029	−0.315134
$445$	0	0
$446$	−2.53990	−0.120268
$447$	−1.31708	−0.0622957
$448$	−2.96410	−0.140041
$449$	−14.7033	−0.693893	−0.346947	−	0.937885i	$-0.612782\pi$
$449$	−14.7033	−0.693893	−0.346947	+	0.937885i	$0.612782\pi$
$450$	0	0
$451$	−3.66952	−0.172791
$452$	−19.4998	−0.917192
$453$	9.47954	0.445388
$454$	8.53862	0.400737
$455$	0	0
$456$	−2.92727	−0.137082
$457$	−34.3838	−1.60841	−0.804204	−	0.594354i	$-0.797408\pi$
$457$	−34.3838	−1.60841	−0.804204	+	0.594354i	$0.797408\pi$
$458$	5.53292	0.258536
$459$	12.5430	0.585455
$460$	0	0
$461$	27.0268	1.25876	0.629382	−	0.777096i	$-0.283308\pi$
$461$	27.0268	1.25876	0.629382	+	0.777096i	$0.283308\pi$
$462$	−0.732489	−0.0340785
$463$	17.7573	0.825252	0.412626	−	0.910900i	$-0.364612\pi$
$463$	17.7573	0.825252	0.412626	+	0.910900i	$0.364612\pi$
$464$	8.92512	0.414338
$465$	0	0
$466$	−15.4602	−0.716178
$467$	24.3805	1.12819	0.564097	−	0.825708i	$-0.309224\pi$
$467$	24.3805	1.12819	0.564097	+	0.825708i	$0.309224\pi$
$468$	21.2531	0.982424
$469$	−6.51679	−0.300917
$470$	0	0
$471$	−3.92080	−0.180661
$472$	−8.69423	−0.400184
$473$	−7.70557	−0.354302
$474$	−5.16526	−0.237248
$475$	0	0
$476$	−11.3281	−0.519224
$477$	−19.9410	−0.913037
$478$	5.23600	0.239489
$479$	−10.6712	−0.487581	−0.243791	−	0.969828i	$-0.578391\pi$
$479$	−10.6712	−0.487581	−0.243791	+	0.969828i	$0.578391\pi$
$480$	0	0
$481$	52.1887	2.37960
$482$	−9.75616	−0.444381
$483$	−5.13674	−0.233730
$484$	13.0142	0.591553
$485$	0	0
$486$	7.82576	0.354984
$487$	−25.8899	−1.17319	−0.586593	−	0.809882i	$-0.699531\pi$
$487$	−25.8899	−1.17319	−0.586593	+	0.809882i	$0.699531\pi$
$488$	−2.50810	−0.113536
$489$	−0.332842	−0.0150516
$490$	0	0
$491$	−18.4203	−0.831298	−0.415649	−	0.909525i	$-0.636446\pi$
$491$	−18.4203	−0.831298	−0.415649	+	0.909525i	$0.636446\pi$
$492$	1.60534	0.0723742
$493$	−38.0111	−1.71193
$494$	9.78750	0.440360
$495$	0	0
$496$	2.05435	0.0922428
$497$	−16.8952	−0.757854
$498$	1.43033	0.0640948
$499$	−16.5577	−0.741223	−0.370611	−	0.928788i	$-0.620852\pi$
$499$	−16.5577	−0.741223	−0.370611	+	0.928788i	$0.620852\pi$
$500$	0	0
$501$	7.88587	0.352315
$502$	−13.2594	−0.591796
$503$	2.93512	0.130871	0.0654354	−	0.997857i	$-0.479156\pi$
$503$	2.93512	0.130871	0.0654354	+	0.997857i	$0.479156\pi$
$504$	−10.9544	−0.487949
$505$	0	0
$506$	8.05172	0.357942
$507$	5.77446	0.256453
$508$	−5.45620	−0.242080
$509$	3.99813	0.177214	0.0886069	−	0.996067i	$-0.471759\pi$
$509$	3.99813	0.177214	0.0886069	+	0.996067i	$0.471759\pi$
$510$	0	0
$511$	−0.475455	−0.0210329
$512$	12.5417	0.554269
$513$	−6.77751	−0.299234
$514$	−0.368685	−0.0162620
$515$	0	0
$516$	3.37102	0.148401
$517$	1.21642	0.0534982
$518$	−11.4436	−0.502803
$519$	−4.89305	−0.214781
$520$	0	0
$521$	11.7960	0.516794	0.258397	−	0.966039i	$-0.416806\pi$
$521$	11.7960	0.516794	0.258397	+	0.966039i	$0.416806\pi$
$522$	−15.6372	−0.684423
$523$	15.9467	0.697300	0.348650	−	0.937253i	$-0.386640\pi$
$523$	15.9467	0.697300	0.348650	+	0.937253i	$0.386640\pi$
$524$	33.5471	1.46551
$525$	0	0
$526$	−5.13207	−0.223769
$527$	−8.74923	−0.381122
$528$	0.754206	0.0328226
$529$	33.4644	1.45498
$530$	0	0
$531$	−9.73032	−0.422260
$532$	6.12108	0.265383
$533$	−12.6170	−0.546502
$534$	−2.89115	−0.125112
$535$	0	0
$536$	−10.5044	−0.453722
$537$	−2.30317	−0.0993891
$538$	−7.14003	−0.307829
$539$	−6.80925	−0.293295
$540$	0	0
$541$	−31.5537	−1.35660	−0.678299	−	0.734786i	$-0.737283\pi$
$541$	−31.5537	−1.35660	−0.678299	+	0.734786i	$0.737283\pi$
$542$	−2.33575	−0.100329
$543$	9.01458	0.386853
$544$	−28.7516	−1.23271
$545$	0	0
$546$	−2.51853	−0.107783
$547$	35.5484	1.51994	0.759969	−	0.649959i	$-0.225214\pi$
$547$	35.5484	1.51994	0.759969	+	0.649959i	$0.225214\pi$
$548$	−27.4868	−1.17418
$549$	−2.80699	−0.119799
$550$	0	0
$551$	20.5390	0.874992
$552$	−8.27993	−0.352417
$553$	25.3886	1.07963
$554$	12.1313	0.515410
$555$	0	0
$556$	31.8573	1.35105
$557$	−13.8404	−0.586438	−0.293219	−	0.956045i	$-0.594727\pi$
$557$	−13.8404	−0.586438	−0.293219	+	0.956045i	$0.594727\pi$
$558$	−3.59931	−0.152371
$559$	−26.4942	−1.12059
$560$	0	0
$561$	−3.21208	−0.135614
$562$	−9.82704	−0.414529
$563$	21.1352	0.890742	0.445371	−	0.895346i	$-0.353072\pi$
$563$	21.1352	0.890742	0.445371	+	0.895346i	$0.353072\pi$
$564$	−0.532159	−0.0224079
$565$	0	0
$566$	11.2978	0.474881
$567$	−11.3589	−0.477028
$568$	−27.2335	−1.14269
$569$	21.0044	0.880550	0.440275	−	0.897863i	$-0.354881\pi$
$569$	21.0044	0.880550	0.440275	+	0.897863i	$0.354881\pi$
$570$	0	0
$571$	−28.2745	−1.18325	−0.591626	−	0.806213i	$-0.701514\pi$
$571$	−28.2745	−1.18325	−0.591626	+	0.806213i	$0.701514\pi$
$572$	−11.2595	−0.470783
$573$	10.7397	0.448656
$574$	2.76657	0.115475
$575$	0	0
$576$	−5.34722	−0.222801
$577$	19.5259	0.812876	0.406438	−	0.913678i	$-0.366771\pi$
$577$	19.5259	0.812876	0.406438	+	0.913678i	$0.366771\pi$
$578$	5.16764	0.214946
$579$	−6.88766	−0.286241
$580$	0	0
$581$	−7.03046	−0.291673
$582$	5.31937	0.220495
$583$	10.5644	0.437532
$584$	−0.766387	−0.0317133
$585$	0	0
$586$	1.31324	0.0542495
$587$	10.9396	0.451525	0.225763	−	0.974182i	$-0.427513\pi$
$587$	10.9396	0.451525	0.225763	+	0.974182i	$0.427513\pi$
$588$	2.97890	0.122848
$589$	4.72759	0.194797
$590$	0	0
$591$	4.46531	0.183678
$592$	11.7829	0.484274
$593$	39.8088	1.63475	0.817375	−	0.576106i	$-0.195428\pi$
$593$	39.8088	1.63475	0.817375	+	0.576106i	$0.195428\pi$
$594$	−2.73367	−0.112164
$595$	0	0
$596$	4.43931	0.181841
$597$	8.76959	0.358915
$598$	27.6844	1.13210
$599$	3.73150	0.152465	0.0762325	−	0.997090i	$-0.475711\pi$
$599$	3.73150	0.152465	0.0762325	+	0.997090i	$0.475711\pi$
$600$	0	0
$601$	26.6864	1.08856	0.544281	−	0.838903i	$-0.316803\pi$
$601$	26.6864	1.08856	0.544281	+	0.838903i	$0.316803\pi$
$602$	5.80948	0.236777
$603$	−11.7562	−0.478751
$604$	−31.9515	−1.30009
$605$	0	0
$606$	−4.11236	−0.167053
$607$	−2.79869	−0.113595	−0.0567977	−	0.998386i	$-0.518089\pi$
$607$	−2.79869	−0.113595	−0.0567977	+	0.998386i	$0.518089\pi$
$608$	15.5357	0.630057
$609$	−5.28513	−0.214164
$610$	0	0
$611$	4.18245	0.169204
$612$	−20.4359	−0.826071
$613$	−45.1022	−1.82166	−0.910830	−	0.412782i	$-0.864557\pi$
$613$	−45.1022	−1.82166	−0.910830	+	0.412782i	$0.864557\pi$
$614$	−7.36205	−0.297108
$615$	0	0
$616$	5.80345	0.233828
$617$	15.9680	0.642848	0.321424	−	0.946935i	$-0.395839\pi$
$617$	15.9680	0.642848	0.321424	+	0.946935i	$0.395839\pi$
$618$	1.75837	0.0707321
$619$	−42.2993	−1.70015	−0.850076	−	0.526659i	$-0.823444\pi$
$619$	−42.2993	−1.70015	−0.850076	+	0.526659i	$0.823444\pi$
$620$	0	0
$621$	−19.1705	−0.769285
$622$	7.45967	0.299105
$623$	14.2108	0.569342
$624$	2.59320	0.103811
$625$	0	0
$626$	−17.5022	−0.699528
$627$	1.73563	0.0693142
$628$	13.2153	0.527350
$629$	−50.1820	−2.00089
$630$	0	0
$631$	29.6232	1.17928	0.589640	−	0.807666i	$-0.299270\pi$
$631$	29.6232	1.17928	0.589640	+	0.807666i	$0.299270\pi$
$632$	40.9239	1.62786
$633$	−1.29039	−0.0512882
$634$	0.246914	0.00980619
$635$	0	0
$636$	−4.62169	−0.183262
$637$	−23.4124	−0.927632
$638$	8.28432	0.327979
$639$	−30.4789	−1.20573
$640$	0	0
$641$	17.0693	0.674196	0.337098	−	0.941470i	$-0.390555\pi$
$641$	17.0693	0.674196	0.337098	+	0.941470i	$0.390555\pi$
$642$	−0.447965	−0.0176798
$643$	−35.6791	−1.40705	−0.703523	−	0.710672i	$-0.748391\pi$
$643$	−35.6791	−1.40705	−0.703523	+	0.710672i	$0.748391\pi$
$644$	17.3138	0.682258
$645$	0	0
$646$	−9.41116	−0.370277
$647$	−27.3740	−1.07618	−0.538091	−	0.842887i	$-0.680854\pi$
$647$	−27.3740	−1.07618	−0.538091	+	0.842887i	$0.680854\pi$
$648$	−18.3094	−0.719262
$649$	5.15494	0.202349
$650$	0	0
$651$	−1.21651	−0.0476788
$652$	1.12187	0.0439357
$653$	16.5800	0.648827	0.324413	−	0.945915i	$-0.394833\pi$
$653$	16.5800	0.648827	0.324413	+	0.945915i	$0.394833\pi$
$654$	1.32326	0.0517437
$655$	0	0
$656$	−2.84860	−0.111219
$657$	−0.857717	−0.0334627
$658$	−0.917101	−0.0357523
$659$	−24.9575	−0.972205	−0.486103	−	0.873902i	$-0.661582\pi$
$659$	−24.9575	−0.972205	−0.486103	+	0.873902i	$0.661582\pi$
$660$	0	0
$661$	−2.21189	−0.0860325	−0.0430162	−	0.999074i	$-0.513697\pi$
$661$	−2.21189	−0.0860325	−0.0430162	+	0.999074i	$0.513697\pi$
$662$	7.59615	0.295233
$663$	−11.0442	−0.428920
$664$	−11.3324	−0.439783
$665$	0	0
$666$	−20.6442	−0.799946
$667$	58.0956	2.24947
$668$	−26.5799	−1.02841
$669$	−1.54863	−0.0598734
$670$	0	0
$671$	1.48709	0.0574084
$672$	−3.99768	−0.154214
$673$	−11.7315	−0.452217	−0.226108	−	0.974102i	$-0.572600\pi$
$673$	−11.7315	−0.452217	−0.226108	+	0.974102i	$0.572600\pi$
$674$	−0.405986	−0.0156380
$675$	0	0
$676$	−19.4632	−0.748585
$677$	15.4890	0.595291	0.297645	−	0.954677i	$-0.403799\pi$
$677$	15.4890	0.595291	0.297645	+	0.954677i	$0.403799\pi$
$678$	4.16860	0.160094
$679$	−26.1461	−1.00339
$680$	0	0
$681$	5.20617	0.199501
$682$	1.90685	0.0730170
$683$	33.6069	1.28593	0.642966	−	0.765895i	$-0.277704\pi$
$683$	33.6069	1.28593	0.642966	+	0.765895i	$0.277704\pi$
$684$	11.0424	0.422216
$685$	0	0
$686$	12.9819	0.495650
$687$	3.37354	0.128708
$688$	−5.98172	−0.228051
$689$	36.3237	1.38382
$690$	0	0
$691$	14.1226	0.537250	0.268625	−	0.963245i	$-0.413431\pi$
$691$	14.1226	0.537250	0.268625	+	0.963245i	$0.413431\pi$
$692$	16.4924	0.626947
$693$	6.49505	0.246726
$694$	−1.26918	−0.0481775
$695$	0	0
$696$	−8.51912	−0.322917
$697$	12.1319	0.459527
$698$	17.4000	0.658599
$699$	−9.42638	−0.356538
$700$	0	0
$701$	−7.49228	−0.282980	−0.141490	−	0.989940i	$-0.545189\pi$
$701$	−7.49228	−0.282980	−0.141490	+	0.989940i	$0.545189\pi$
$702$	−9.39925	−0.354752
$703$	27.1155	1.02268
$704$	2.83285	0.106767
$705$	0	0
$706$	−2.47134	−0.0930101
$707$	20.2133	0.760199
$708$	−2.25518	−0.0847547
$709$	49.5744	1.86181	0.930903	−	0.365268i	$-0.119023\pi$
$709$	49.5744	1.86181	0.930903	+	0.365268i	$0.119023\pi$
$710$	0	0
$711$	45.8008	1.71766
$712$	22.9064	0.858452
$713$	13.3722	0.500793
$714$	2.42169	0.0906296
$715$	0	0
$716$	7.76300	0.290117
$717$	3.19250	0.119226
$718$	18.5228	0.691265
$719$	19.0834	0.711691	0.355846	−	0.934545i	$-0.384193\pi$
$719$	19.0834	0.711691	0.355846	+	0.934545i	$0.384193\pi$
$720$	0	0
$721$	−8.64285	−0.321876
$722$	−8.60520	−0.320252
$723$	−5.94853	−0.221228
$724$	−30.3843	−1.12922
$725$	0	0
$726$	−2.78213	−0.103255
$727$	30.5643	1.13357	0.566783	−	0.823867i	$-0.308188\pi$
$727$	30.5643	1.13357	0.566783	+	0.823867i	$0.308188\pi$
$728$	19.9541	0.739549
$729$	−17.1288	−0.634402
$730$	0	0
$731$	25.4755	0.942245
$732$	−0.650569	−0.0240457
$733$	28.3333	1.04652	0.523258	−	0.852174i	$-0.324716\pi$
$733$	28.3333	1.04652	0.523258	+	0.852174i	$0.324716\pi$
$734$	−13.3861	−0.494090
$735$	0	0
$736$	43.9435	1.61978
$737$	6.22824	0.229420
$738$	4.99088	0.183717
$739$	−25.7427	−0.946960	−0.473480	−	0.880805i	$-0.657002\pi$
$739$	−25.7427	−0.946960	−0.473480	+	0.880805i	$0.657002\pi$
$740$	0	0
$741$	5.96764	0.219227
$742$	−7.96483	−0.292398
$743$	0.355184	0.0130304	0.00651522	−	0.999979i	$-0.497926\pi$
$743$	0.355184	0.0130304	0.00651522	+	0.999979i	$0.497926\pi$
$744$	−1.96090	−0.0718899
$745$	0	0
$746$	−14.1991	−0.519867
$747$	−12.6829	−0.464043
$748$	10.8265	0.395858
$749$	2.20186	0.0804544
$750$	0	0
$751$	12.9908	0.474040	0.237020	−	0.971505i	$-0.423829\pi$
$751$	12.9908	0.474040	0.237020	+	0.971505i	$0.423829\pi$
$752$	0.944291	0.0344348
$753$	−8.08452	−0.294616
$754$	28.4841	1.03733
$755$	0	0
$756$	−5.87827	−0.213791
$757$	9.95760	0.361915	0.180958	−	0.983491i	$-0.442080\pi$
$757$	9.95760	0.361915	0.180958	+	0.983491i	$0.442080\pi$
$758$	6.94069	0.252097
$759$	4.90930	0.178196
$760$	0	0
$761$	0.893562	0.0323916	0.0161958	−	0.999869i	$-0.494844\pi$
$761$	0.893562	0.0323916	0.0161958	+	0.999869i	$0.494844\pi$
$762$	1.16641	0.0422546
$763$	−6.50418	−0.235467
$764$	−36.1988	−1.30963
$765$	0	0
$766$	23.2736	0.840911
$767$	17.7243	0.639989
$768$	−4.94183	−0.178323
$769$	−12.4312	−0.448280	−0.224140	−	0.974557i	$-0.571957\pi$
$769$	−12.4312	−0.448280	−0.224140	+	0.974557i	$0.571957\pi$
$770$	0	0
$771$	−0.224795	−0.00809578
$772$	23.2154	0.835539
$773$	6.83312	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$773$	6.83312	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$774$	10.4803	0.376705
$775$	0	0
$776$	−42.1449	−1.51291
$777$	−6.97740	−0.250313
$778$	−8.99788	−0.322590
$779$	−6.55536	−0.234870
$780$	0	0
$781$	16.1471	0.577790
$782$	−26.6199	−0.951925
$783$	−19.7243	−0.704889
$784$	−5.28592	−0.188783
$785$	0	0
$786$	−7.17159	−0.255802
$787$	−8.58979	−0.306193	−0.153096	−	0.988211i	$-0.548925\pi$
$787$	−8.58979	−0.306193	−0.153096	+	0.988211i	$0.548925\pi$
$788$	−15.0506	−0.536157
$789$	−3.12913	−0.111400
$790$	0	0
$791$	−20.4897	−0.728531
$792$	10.4694	0.372013
$793$	5.11309	0.181571
$794$	−19.0341	−0.675495
$795$	0	0
$796$	−29.5585	−1.04767
$797$	−31.8453	−1.12802	−0.564008	−	0.825769i	$-0.690741\pi$
$797$	−31.8453	−1.12802	−0.564008	+	0.825769i	$0.690741\pi$
$798$	−1.30855	−0.0463220
$799$	−4.02163	−0.142275
$800$	0	0
$801$	25.6361	0.905807
$802$	−11.7293	−0.414177
$803$	0.454402	0.0160355
$804$	−2.72472	−0.0960935
$805$	0	0
$806$	6.55636	0.230938
$807$	−4.35342	−0.153248
$808$	32.5819	1.14623
$809$	47.3911	1.66618	0.833091	−	0.553136i	$-0.186569\pi$
$809$	47.3911	1.66618	0.833091	+	0.553136i	$0.186569\pi$
$810$	0	0
$811$	39.7124	1.39449	0.697245	−	0.716833i	$-0.254409\pi$
$811$	39.7124	1.39449	0.697245	+	0.716833i	$0.254409\pi$
$812$	17.8139	0.625146
$813$	−1.42415	−0.0499473
$814$	10.9369	0.383338
$815$	0	0
$816$	−2.49349	−0.0872896
$817$	−13.7655	−0.481594
$818$	16.2932	0.569680
$819$	22.3321	0.780345
$820$	0	0
$821$	−36.5634	−1.27607	−0.638036	−	0.770006i	$-0.720253\pi$
$821$	−36.5634	−1.27607	−0.638036	+	0.770006i	$0.720253\pi$
$822$	5.87605	0.204951
$823$	−24.7315	−0.862085	−0.431042	−	0.902332i	$-0.641854\pi$
$823$	−24.7315	−0.862085	−0.431042	+	0.902332i	$0.641854\pi$
$824$	−13.9314	−0.485324
$825$	0	0
$826$	−3.88648	−0.135228
$827$	16.7282	0.581696	0.290848	−	0.956769i	$-0.406063\pi$
$827$	16.7282	0.581696	0.290848	+	0.956769i	$0.406063\pi$
$828$	31.2339	1.08545
$829$	−10.2609	−0.356377	−0.178188	−	0.983996i	$-0.557024\pi$
$829$	−10.2609	−0.356377	−0.178188	+	0.983996i	$0.557024\pi$
$830$	0	0
$831$	7.39670	0.256589
$832$	9.74026	0.337683
$833$	22.5121	0.779999
$834$	−6.81036	−0.235823
$835$	0	0
$836$	−5.85005	−0.202328
$837$	−4.54005	−0.156927
$838$	17.2446	0.595704
$839$	−37.5973	−1.29800	−0.649001	−	0.760787i	$-0.724813\pi$
$839$	−37.5973	−1.29800	−0.649001	+	0.760787i	$0.724813\pi$
$840$	0	0
$841$	30.7739	1.06117
$842$	−1.93173	−0.0665720
$843$	−5.99175	−0.206367
$844$	4.34934	0.149710
$845$	0	0
$846$	−1.65444	−0.0568810
$847$	13.6749	0.469874
$848$	8.20097	0.281623
$849$	6.88849	0.236412
$850$	0	0
$851$	76.6974	2.62915
$852$	−7.06402	−0.242009
$853$	−15.1524	−0.518807	−0.259403	−	0.965769i	$-0.583526\pi$
$853$	−15.1524	−0.518807	−0.259403	+	0.965769i	$0.583526\pi$
$854$	−1.12117	−0.0383655
$855$	0	0
$856$	3.54919	0.121309
$857$	14.3812	0.491252	0.245626	−	0.969365i	$-0.421006\pi$
$857$	14.3812	0.491252	0.245626	+	0.969365i	$0.421006\pi$
$858$	2.40702	0.0821742
$859$	18.1643	0.619758	0.309879	−	0.950776i	$-0.399711\pi$
$859$	18.1643	0.619758	0.309879	+	0.950776i	$0.399711\pi$
$860$	0	0
$861$	1.68684	0.0574872
$862$	−15.0045	−0.511054
$863$	39.6611	1.35008	0.675039	−	0.737782i	$-0.264127\pi$
$863$	39.6611	1.35008	0.675039	+	0.737782i	$0.264127\pi$
$864$	−14.9195	−0.507570
$865$	0	0
$866$	−16.1341	−0.548258
$867$	3.15082	0.107007
$868$	4.10033	0.139174
$869$	−24.2644	−0.823113
$870$	0	0
$871$	21.4147	0.725609
$872$	−10.4841	−0.355037
$873$	−47.1673	−1.59637
$874$	14.3839	0.486542
$875$	0	0
$876$	−0.198791	−0.00671653
$877$	−0.333555	−0.0112634	−0.00563168	−	0.999984i	$-0.501793\pi$
$877$	−0.333555	−0.0112634	−0.00563168	+	0.999984i	$0.501793\pi$
$878$	−30.1403	−1.01719
$879$	0.800710	0.0270073
$880$	0	0
$881$	−54.2989	−1.82938	−0.914689	−	0.404159i	$-0.867564\pi$
$881$	−54.2989	−1.82938	−0.914689	+	0.404159i	$0.867564\pi$
$882$	9.26119	0.311840
$883$	11.8126	0.397526	0.198763	−	0.980048i	$-0.436308\pi$
$883$	11.8126	0.397526	0.198763	+	0.980048i	$0.436308\pi$
$884$	37.2251	1.25202
$885$	0	0
$886$	10.9975	0.369468
$887$	−10.9451	−0.367500	−0.183750	−	0.982973i	$-0.558824\pi$
$887$	−10.9451	−0.367500	−0.183750	+	0.982973i	$0.558824\pi$
$888$	−11.2469	−0.377421
$889$	−5.73320	−0.192285
$890$	0	0
$891$	10.8559	0.363687
$892$	5.21976	0.174770
$893$	2.17306	0.0727187
$894$	−0.949022	−0.0317401
$895$	0	0
$896$	16.0630	0.536628
$897$	16.8797	0.563598
$898$	−10.5945	−0.353543
$899$	13.7585	0.458871
$900$	0	0
$901$	−34.9270	−1.16359
$902$	−2.64407	−0.0880380
$903$	3.54216	0.117876
$904$	−33.0275	−1.09848
$905$	0	0
$906$	6.83049	0.226928
$907$	−27.6292	−0.917413	−0.458707	−	0.888588i	$-0.651687\pi$
$907$	−27.6292	−0.917413	−0.458707	+	0.888588i	$0.651687\pi$
$908$	−17.5478	−0.582344
$909$	36.4646	1.20946
$910$	0	0
$911$	6.55505	0.217178	0.108589	−	0.994087i	$-0.465367\pi$
$911$	6.55505	0.217178	0.108589	+	0.994087i	$0.465367\pi$
$912$	1.34734	0.0446149
$913$	6.71916	0.222372
$914$	−24.7753	−0.819493
$915$	0	0
$916$	−11.3707	−0.375700
$917$	35.2502	1.16406
$918$	9.03784	0.298293
$919$	44.7098	1.47484	0.737420	−	0.675435i	$-0.236044\pi$
$919$	44.7098	1.47484	0.737420	+	0.675435i	$0.236044\pi$
$920$	0	0
$921$	−4.48879	−0.147911
$922$	19.4742	0.641348
$923$	55.5190	1.82743
$924$	1.50534	0.0495222
$925$	0	0
$926$	12.7950	0.420471
$927$	−15.5916	−0.512096
$928$	45.2130	1.48419
$929$	−16.0043	−0.525083	−0.262541	−	0.964921i	$-0.584561\pi$
$929$	−16.0043	−0.525083	−0.262541	+	0.964921i	$0.584561\pi$
$930$	0	0
$931$	−12.1643	−0.398668
$932$	31.7723	1.04074
$933$	4.54831	0.148905
$934$	17.5674	0.574822
$935$	0	0
$936$	35.9971	1.17660
$937$	−47.5058	−1.55195	−0.775973	−	0.630766i	$-0.782741\pi$
$937$	−47.5058	−1.55195	−0.775973	+	0.630766i	$0.782741\pi$
$938$	−4.69567	−0.153319
$939$	−10.6714	−0.348249
$940$	0	0
$941$	23.8312	0.776875	0.388438	−	0.921475i	$-0.373015\pi$
$941$	23.8312	0.776875	0.388438	+	0.921475i	$0.373015\pi$
$942$	−2.82513	−0.0920479
$943$	−18.5421	−0.603815
$944$	4.00171	0.130244
$945$	0	0
$946$	−5.55225	−0.180519
$947$	55.3675	1.79920	0.899601	−	0.436712i	$-0.143857\pi$
$947$	55.3675	1.79920	0.899601	+	0.436712i	$0.143857\pi$
$948$	10.6152	0.344764
$949$	1.56238	0.0507170
$950$	0	0
$951$	0.150548	0.00488186
$952$	−19.1869	−0.621850
$953$	−8.18421	−0.265113	−0.132556	−	0.991175i	$-0.542319\pi$
$953$	−8.18421	−0.265113	−0.132556	+	0.991175i	$0.542319\pi$
$954$	−14.3685	−0.465198
$955$	0	0
$956$	−10.7605	−0.348021
$957$	5.05112	0.163279
$958$	−7.68916	−0.248426
$959$	−28.8823	−0.932658
$960$	0	0
$961$	−27.8331	−0.897843
$962$	37.6046	1.21242
$963$	3.97215	0.128001
$964$	20.0500	0.645766
$965$	0	0
$966$	−3.70128	−0.119087
$967$	61.6534	1.98264	0.991320	−	0.131474i	$-0.0419709\pi$
$967$	61.6534	1.98264	0.991320	+	0.131474i	$0.0419709\pi$
$968$	22.0426	0.708475
$969$	−5.73817	−0.184337
$970$	0	0
$971$	−15.9922	−0.513213	−0.256607	−	0.966516i	$-0.582604\pi$
$971$	−15.9922	−0.513213	−0.256607	+	0.966516i	$0.582604\pi$
$972$	−16.0828	−0.515855
$973$	33.4746	1.07315
$974$	−18.6550	−0.597745
$975$	0	0
$976$	1.15441	0.0369516
$977$	−49.5344	−1.58475	−0.792373	−	0.610037i	$-0.791155\pi$
$977$	−49.5344	−1.58475	−0.792373	+	0.610037i	$0.791155\pi$
$978$	−0.239829	−0.00766890
$979$	−13.5815	−0.434068
$980$	0	0
$981$	−11.7335	−0.374622
$982$	−13.2728	−0.423551
$983$	−13.9480	−0.444871	−0.222436	−	0.974947i	$-0.571401\pi$
$983$	−13.9480	−0.444871	−0.222436	+	0.974947i	$0.571401\pi$
$984$	2.71902	0.0866791
$985$	0	0
$986$	−27.3889	−0.872240
$987$	−0.559175	−0.0177987
$988$	−20.1143	−0.639923
$989$	−38.9364	−1.23810
$990$	0	0
$991$	−13.9405	−0.442834	−0.221417	−	0.975179i	$-0.571068\pi$
$991$	−13.9405	−0.442834	−0.221417	+	0.975179i	$0.571068\pi$
$992$	10.4069	0.330420
$993$	4.63153	0.146977
$994$	−12.1739	−0.386131
$995$	0	0
$996$	−2.93949	−0.0931413
$997$	−23.4773	−0.743533	−0.371766	−	0.928326i	$-0.621248\pi$
$997$	−23.4773	−0.743533	−0.371766	+	0.928326i	$0.621248\pi$
$998$	−11.9306	−0.377657
$999$	−26.0399	−0.823865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1525.2.a.m.1.6	yes	10
5.2	odd	4		1525.2.b.h.1099.13		20
5.3	odd	4		1525.2.b.h.1099.8		20
5.4	even	2		1525.2.a.l.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1525.2.a.l.1.5	✓	10	5.4	even	2
1525.2.a.m.1.6	yes	10	1.1	even	1	trivial
1525.2.b.h.1099.8		20	5.3	odd	4
1525.2.b.h.1099.13		20	5.2	odd	4

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:	\( N \)	\(=\)	\( 1525 = 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1525.a (trivial)

\(f(q)\)	\(=\)	\(q+0.720550 q^{2} +0.439334 q^{3} -1.48081 q^{4} +0.316562 q^{6} -1.55598 q^{7} -2.50810 q^{8} -2.80699 q^{9} +1.48709 q^{11} -0.650569 q^{12} +5.11309 q^{13} -1.12117 q^{14} +1.15441 q^{16} -4.91648 q^{17} -2.02257 q^{18} +2.65659 q^{19} -0.683597 q^{21} +1.07152 q^{22} +7.51428 q^{23} -1.10189 q^{24} +3.68424 q^{26} -2.55121 q^{27} +2.30411 q^{28} +7.73136 q^{29} +1.77957 q^{31} +5.84800 q^{32} +0.653329 q^{33} -3.54257 q^{34} +4.15660 q^{36} +10.2069 q^{37} +1.91421 q^{38} +2.24635 q^{39} -2.46759 q^{41} -0.492566 q^{42} -5.18165 q^{43} -2.20209 q^{44} +5.41442 q^{46} +0.817989 q^{47} +0.507170 q^{48} -4.57891 q^{49} -2.15998 q^{51} -7.57150 q^{52} +7.10407 q^{53} -1.83827 q^{54} +3.90256 q^{56} +1.16713 q^{57} +5.57083 q^{58} +3.46647 q^{59} +1.00000 q^{61} +1.28227 q^{62} +4.36763 q^{63} +1.90497 q^{64} +0.470756 q^{66} +4.18821 q^{67} +7.28036 q^{68} +3.30128 q^{69} +10.8582 q^{71} +7.04019 q^{72} +0.305565 q^{73} +7.35458 q^{74} -3.93390 q^{76} -2.31389 q^{77} +1.61861 q^{78} -16.3167 q^{79} +7.30012 q^{81} -1.77802 q^{82} +4.51834 q^{83} +1.01228 q^{84} -3.73364 q^{86} +3.39665 q^{87} -3.72976 q^{88} -9.13297 q^{89} -7.95588 q^{91} -11.1272 q^{92} +0.781826 q^{93} +0.589402 q^{94} +2.56923 q^{96} +16.8035 q^{97} -3.29934 q^{98} -4.17424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{2} + 7 q^{3} + 11 q^{4} + 4 q^{6} + 6 q^{7} + 15 q^{8} + 11 q^{9} + 19 q^{12} + 3 q^{13} - 5 q^{14} + 17 q^{16} + 11 q^{17} + 5 q^{18} - 2 q^{19} + 9 q^{21} - 6 q^{22} + 33 q^{23} - 7 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 5 * q^2 + 7 * q^3 + 11 * q^4 + 4 * q^6 + 6 * q^7 + 15 * q^8 + 11 * q^9 + 19 * q^12 + 3 * q^13 - 5 * q^14 + 17 * q^16 + 11 * q^17 + 5 * q^18 - 2 * q^19 + 9 * q^21 - 6 * q^22 + 33 * q^23 - 7 * q^24 + 12 * q^26 + 19 * q^27 + 21 * q^28 - 3 * q^29 + 20 * q^32 + 7 * q^33 - 3 * q^34 + 45 * q^36 - q^37 + 11 * q^38 - 22 * q^39 + 4 * q^41 - 16 * q^42 + 30 * q^43 - 22 * q^44 + 23 * q^46 + 5 * q^47 + 20 * q^48 - 10 * q^49 - q^51 - 35 * q^52 + 35 * q^53 - 52 * q^54 + 13 * q^56 + 4 * q^57 + 26 * q^58 - 4 * q^59 + 10 * q^61 + 22 * q^62 + 47 * q^63 + q^64 + 25 * q^66 + 28 * q^68 + 26 * q^69 - 11 * q^71 - 27 * q^72 - 9 * q^73 - 41 * q^74 + 44 * q^76 - 4 * q^77 + 51 * q^78 + 16 * q^79 + 14 * q^81 - 22 * q^82 + 18 * q^83 + 2 * q^84 + 7 * q^86 - q^87 - 32 * q^88 - 7 * q^89 - 12 * q^91 + 80 * q^92 - 3 * q^93 - 73 * q^94 + 53 * q^96 + 8 * q^97 - 8 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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