LMFDB - Embedded newform 1587.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1587,2,Mod(1,1587)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1587, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1587.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1587 = 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1587.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$12.6722588008$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.2803712.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 8x^{2} - 2 $ x^6 - 6x^4 + 8x^2 - 2
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			1.5
Root			$-1.20864$ of defining polynomial
Character	$\chi$	$=$	1587.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} -0.762528 q^{5} +2.70928 q^{6} -0.651685 q^{7} +9.04945 q^{8} +1.00000 q^{9} -2.06590 q^{10} +6.13793 q^{11} +5.34017 q^{12} -3.41855 q^{13} -1.76560 q^{14} -0.762528 q^{15} +13.8371 q^{16} -3.59096 q^{17} +2.70928 q^{18} -3.96119 q^{19} -4.07203 q^{20} -0.651685 q^{21} +16.6293 q^{22} +9.04945 q^{24} -4.41855 q^{25} -9.26180 q^{26} +1.00000 q^{27} -3.48011 q^{28} +0.921622 q^{29} -2.06590 q^{30} +3.07838 q^{31} +19.3896 q^{32} +6.13793 q^{33} -9.72889 q^{34} +0.496928 q^{35} +5.34017 q^{36} -1.89529 q^{37} -10.7320 q^{38} -3.41855 q^{39} -6.90046 q^{40} -6.68035 q^{41} -1.76560 q^{42} +3.48011 q^{43} +32.7776 q^{44} -0.762528 q^{45} -0.183417 q^{47} +13.8371 q^{48} -6.57531 q^{49} -11.9711 q^{50} -3.59096 q^{51} -18.2557 q^{52} +10.2100 q^{53} +2.70928 q^{54} -4.68035 q^{55} -5.89739 q^{56} -3.96119 q^{57} +2.49693 q^{58} +5.65983 q^{59} -4.07203 q^{60} -2.93927 q^{61} +8.34017 q^{62} -0.651685 q^{63} +24.8576 q^{64} +2.60674 q^{65} +16.6293 q^{66} -7.01130 q^{67} -19.1763 q^{68} +1.34632 q^{70} -4.00000 q^{71} +9.04945 q^{72} -0.680346 q^{73} -5.13486 q^{74} -4.41855 q^{75} -21.1534 q^{76} -4.00000 q^{77} -9.26180 q^{78} +11.1431 q^{79} -10.5512 q^{80} +1.00000 q^{81} -18.0989 q^{82} +12.4975 q^{83} -3.48011 q^{84} +2.73820 q^{85} +9.42858 q^{86} +0.921622 q^{87} +55.5449 q^{88} -3.36927 q^{89} -2.06590 q^{90} +2.22782 q^{91} +3.07838 q^{93} -0.496928 q^{94} +3.02052 q^{95} +19.3896 q^{96} -5.54601 q^{97} -17.8143 q^{98} +6.13793 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 6 q^{3} + 10 q^{4} + 2 q^{6} + 18 q^{8} + 6 q^{9} + 10 q^{12} + 8 q^{13} + 26 q^{16} + 2 q^{18} + 18 q^{24} + 2 q^{25} - 40 q^{26} + 6 q^{27} + 12 q^{29} + 12 q^{31} + 58 q^{32} - 32 q^{35}+ \cdots - 90 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 6 * q^3 + 10 * q^4 + 2 * q^6 + 18 * q^8 + 6 * q^9 + 10 * q^12 + 8 * q^13 + 26 * q^16 + 2 * q^18 + 18 * q^24 + 2 * q^25 - 40 * q^26 + 6 * q^27 + 12 * q^29 + 12 * q^31 + 58 * q^32 - 32 * q^35 + 10 * q^36 + 8 * q^39 + 4 * q^41 + 8 * q^47 + 26 * q^48 + 2 * q^49 - 42 * q^50 - 24 * q^52 + 2 * q^54 + 16 * q^55 - 20 * q^58 + 56 * q^59 + 28 * q^62 + 26 * q^64 + 56 * q^70 - 24 * q^71 + 18 * q^72 + 40 * q^73 + 2 * q^75 - 24 * q^77 - 40 * q^78 + 6 * q^81 - 36 * q^82 + 32 * q^85 + 12 * q^87 + 12 * q^93 + 32 * q^94 - 48 * q^95 + 58 * q^96 - 90 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.70928	1.91575	0.957873	−	0.287190i	$-0.0927213\pi$
$2$	2.70928	1.91575	0.957873	+	0.287190i	$0.0927213\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.34017	2.67009
$5$	−0.762528	−0.341013	−0.170506	−	0.985357i	$-0.554540\pi$
$5$	−0.762528	−0.341013	−0.170506	+	0.985357i	$0.554540\pi$
$6$	2.70928	1.10606
$7$	−0.651685	−0.246314	−0.123157	−	0.992387i	$-0.539302\pi$
$7$	−0.651685	−0.246314	−0.123157	+	0.992387i	$0.539302\pi$
$8$	9.04945	3.19946
$9$	1.00000	0.333333
$10$	−2.06590	−0.653295
$11$	6.13793	1.85066	0.925328	−	0.379168i	$-0.123790\pi$
$11$	6.13793	1.85066	0.925328	+	0.379168i	$0.123790\pi$
$12$	5.34017	1.54158
$13$	−3.41855	−0.948135	−0.474068	−	0.880488i	$-0.657215\pi$
$13$	−3.41855	−0.948135	−0.474068	+	0.880488i	$0.657215\pi$
$14$	−1.76560	−0.471875
$15$	−0.762528	−0.196884
$16$	13.8371	3.45928
$17$	−3.59096	−0.870935	−0.435467	−	0.900205i	$-0.643417\pi$
$17$	−3.59096	−0.870935	−0.435467	+	0.900205i	$0.643417\pi$
$18$	2.70928	0.638582
$19$	−3.96119	−0.908759	−0.454380	−	0.890808i	$-0.650139\pi$
$19$	−3.96119	−0.908759	−0.454380	+	0.890808i	$0.650139\pi$
$20$	−4.07203	−0.910534
$21$	−0.651685	−0.142209
$22$	16.6293	3.54539
$23$	0	0
$23$	0	0
$24$	9.04945	1.84721
$25$	−4.41855	−0.883710
$26$	−9.26180	−1.81639
$27$	1.00000	0.192450
$28$	−3.48011	−0.657679
$29$	0.921622	0.171141	0.0855705	−	0.996332i	$-0.472729\pi$
$29$	0.921622	0.171141	0.0855705	+	0.996332i	$0.472729\pi$
$30$	−2.06590	−0.377180
$31$	3.07838	0.552893	0.276446	−	0.961029i	$-0.410843\pi$
$31$	3.07838	0.552893	0.276446	+	0.961029i	$0.410843\pi$
$32$	19.3896	3.42763
$33$	6.13793	1.06848
$34$	−9.72889	−1.66849
$35$	0.496928	0.0839962
$36$	5.34017	0.890029
$37$	−1.89529	−0.311584	−0.155792	−	0.987790i	$-0.549793\pi$
$37$	−1.89529	−0.311584	−0.155792	+	0.987790i	$0.549793\pi$
$38$	−10.7320	−1.74095
$39$	−3.41855	−0.547406
$40$	−6.90046	−1.09106
$41$	−6.68035	−1.04329	−0.521647	−	0.853161i	$-0.674682\pi$
$41$	−6.68035	−1.04329	−0.521647	+	0.853161i	$0.674682\pi$
$42$	−1.76560	−0.272437
$43$	3.48011	0.530712	0.265356	−	0.964150i	$-0.414510\pi$
$43$	3.48011	0.530712	0.265356	+	0.964150i	$0.414510\pi$
$44$	32.7776	4.94141
$45$	−0.762528	−0.113671
$46$	0	0
$47$	−0.183417	−0.0267542	−0.0133771	−	0.999911i	$-0.504258\pi$
$47$	−0.183417	−0.0267542	−0.0133771	+	0.999911i	$0.504258\pi$
$48$	13.8371	1.99721
$49$	−6.57531	−0.939329
$50$	−11.9711	−1.69297
$51$	−3.59096	−0.502834
$52$	−18.2557	−2.53160
$53$	10.2100	1.40245	0.701223	−	0.712942i	$-0.252638\pi$
$53$	10.2100	1.40245	0.701223	+	0.712942i	$0.252638\pi$
$54$	2.70928	0.368686
$55$	−4.68035	−0.631098
$56$	−5.89739	−0.788072
$57$	−3.96119	−0.524672
$58$	2.49693	0.327863
$59$	5.65983	0.736847	0.368423	−	0.929658i	$-0.379898\pi$
$59$	5.65983	0.736847	0.368423	+	0.929658i	$0.379898\pi$
$60$	−4.07203	−0.525697
$61$	−2.93927	−0.376335	−0.188167	−	0.982137i	$-0.560255\pi$
$61$	−2.93927	−0.376335	−0.188167	+	0.982137i	$0.560255\pi$
$62$	8.34017	1.05920
$63$	−0.651685	−0.0821046
$64$	24.8576	3.10720
$65$	2.60674	0.323326
$66$	16.6293	2.04693
$67$	−7.01130	−0.856567	−0.428283	−	0.903644i	$-0.640882\pi$
$67$	−7.01130	−0.856567	−0.428283	+	0.903644i	$0.640882\pi$
$68$	−19.1763	−2.32547
$69$	0	0
$70$	1.34632	0.160916
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	9.04945	1.06649
$73$	−0.680346	−0.0796285	−0.0398142	−	0.999207i	$-0.512677\pi$
$73$	−0.680346	−0.0796285	−0.0398142	+	0.999207i	$0.512677\pi$
$74$	−5.13486	−0.596916
$75$	−4.41855	−0.510210
$76$	−21.1534	−2.42647
$77$	−4.00000	−0.455842
$78$	−9.26180	−1.04869
$79$	11.1431	1.25370	0.626848	−	0.779141i	$-0.284345\pi$
$79$	11.1431	1.25370	0.626848	+	0.779141i	$0.284345\pi$
$80$	−10.5512	−1.17966
$81$	1.00000	0.111111
$82$	−18.0989	−1.99869
$83$	12.4975	1.37178	0.685892	−	0.727703i	$-0.259412\pi$
$83$	12.4975	1.37178	0.685892	+	0.727703i	$0.259412\pi$
$84$	−3.48011	−0.379711
$85$	2.73820	0.297000
$86$	9.42858	1.01671
$87$	0.921622	0.0988083
$88$	55.5449	5.92111
$89$	−3.36927	−0.357142	−0.178571	−	0.983927i	$-0.557147\pi$
$89$	−3.36927	−0.357142	−0.178571	+	0.983927i	$0.557147\pi$
$90$	−2.06590	−0.217765
$91$	2.22782	0.233539
$92$	0	0
$93$	3.07838	0.319213
$94$	−0.496928	−0.0512543
$95$	3.02052	0.309899
$96$	19.3896	1.97894
$97$	−5.54601	−0.563112	−0.281556	−	0.959545i	$-0.590851\pi$
$97$	−5.54601	−0.563112	−0.281556	+	0.959545i	$0.590851\pi$
$98$	−17.8143	−1.79952
$99$	6.13793	0.616885
$100$	−23.5958	−2.35958
$101$	−9.51745	−0.947021	−0.473511	−	0.880788i	$-0.657013\pi$
$101$	−9.51745	−0.947021	−0.473511	+	0.880788i	$0.657013\pi$
$102$	−9.72889	−0.963303
$103$	15.4966	1.52692	0.763462	−	0.645853i	$-0.223498\pi$
$103$	15.4966	1.52692	0.763462	+	0.645853i	$0.223498\pi$
$104$	−30.9360	−3.03352
$105$	0.496928	0.0484953
$106$	27.6616	2.68673
$107$	−12.7569	−1.23326	−0.616630	−	0.787253i	$-0.711503\pi$
$107$	−12.7569	−1.23326	−0.616630	+	0.787253i	$0.711503\pi$
$108$	5.34017	0.513858
$109$	−16.7402	−1.60342	−0.801710	−	0.597714i	$-0.796076\pi$
$109$	−16.7402	−1.60342	−0.801710	+	0.597714i	$0.796076\pi$
$110$	−12.6803	−1.20902
$111$	−1.89529	−0.179893
$112$	−9.01744	−0.852068
$113$	−7.38154	−0.694397	−0.347198	−	0.937792i	$-0.612867\pi$
$113$	−7.38154	−0.694397	−0.347198	+	0.937792i	$0.612867\pi$
$114$	−10.7320	−1.00514
$115$	0	0
$116$	4.92162	0.456961
$117$	−3.41855	−0.316045
$118$	15.3340	1.41161
$119$	2.34017	0.214523
$120$	−6.90046	−0.629923
$121$	26.6742	2.42493
$122$	−7.96329	−0.720963
$123$	−6.68035	−0.602347
$124$	16.4391	1.47627
$125$	7.18191	0.642370
$126$	−1.76560	−0.157292
$127$	7.44521	0.660656	0.330328	−	0.943866i	$-0.392841\pi$
$127$	7.44521	0.660656	0.330328	+	0.943866i	$0.392841\pi$
$128$	28.5669	2.52498
$129$	3.48011	0.306407
$130$	7.06238	0.619412
$131$	−18.8371	−1.64581	−0.822903	−	0.568182i	$-0.807647\pi$
$131$	−18.8371	−1.64581	−0.822903	+	0.568182i	$0.807647\pi$
$132$	32.7776	2.85293
$133$	2.58145	0.223840
$134$	−18.9955	−1.64097
$135$	−0.762528	−0.0656280
$136$	−32.4962	−2.78652
$137$	−3.81264	−0.325736	−0.162868	−	0.986648i	$-0.552074\pi$
$137$	−3.81264	−0.325736	−0.162868	+	0.986648i	$0.552074\pi$
$138$	0	0
$139$	6.15676	0.522209	0.261105	−	0.965311i	$-0.415913\pi$
$139$	6.15676	0.522209	0.261105	+	0.965311i	$0.415913\pi$
$140$	2.65368	0.224277
$141$	−0.183417	−0.0154465
$142$	−10.8371	−0.909429
$143$	−20.9828	−1.75467
$144$	13.8371	1.15309
$145$	−0.702763	−0.0583613
$146$	−1.84324	−0.152548
$147$	−6.57531	−0.542322
$148$	−10.1212	−0.831956
$149$	−7.94444	−0.650834	−0.325417	−	0.945571i	$-0.605505\pi$
$149$	−7.94444	−0.650834	−0.325417	+	0.945571i	$0.605505\pi$
$150$	−11.9711	−0.977434
$151$	10.5236	0.856398	0.428199	−	0.903685i	$-0.359148\pi$
$151$	10.5236	0.856398	0.428199	+	0.903685i	$0.359148\pi$
$152$	−35.8466	−2.90754
$153$	−3.59096	−0.290312
$154$	−10.8371	−0.873279
$155$	−2.34735	−0.188544
$156$	−18.2557	−1.46162
$157$	−20.0497	−1.60014	−0.800070	−	0.599907i	$-0.795204\pi$
$157$	−20.0497	−1.60014	−0.800070	+	0.599907i	$0.795204\pi$
$158$	30.1897	2.40177
$159$	10.2100	0.809703
$160$	−14.7851	−1.16887
$161$	0	0
$162$	2.70928	0.212861
$163$	−14.5958	−1.14323	−0.571617	−	0.820521i	$-0.693684\pi$
$163$	−14.5958	−1.14323	−0.571617	+	0.820521i	$0.693684\pi$
$164$	−35.6742	−2.78569
$165$	−4.68035	−0.364364
$166$	33.8593	2.62799
$167$	14.8371	1.14813	0.574065	−	0.818810i	$-0.305366\pi$
$167$	14.8371	1.14813	0.574065	+	0.818810i	$0.305366\pi$
$168$	−5.89739	−0.454994
$169$	−1.31351	−0.101039
$170$	7.41855	0.568977
$171$	−3.96119	−0.302920
$172$	18.5844	1.41705
$173$	−9.23513	−0.702134	−0.351067	−	0.936350i	$-0.614181\pi$
$173$	−9.23513	−0.702134	−0.351067	+	0.936350i	$0.614181\pi$
$174$	2.49693	0.189292
$175$	2.87950	0.217670
$176$	84.9312	6.40193
$177$	5.65983	0.425419
$178$	−9.12828	−0.684193
$179$	2.02666	0.151480	0.0757399	−	0.997128i	$-0.475868\pi$
$179$	2.02666	0.151480	0.0757399	+	0.997128i	$0.475868\pi$
$180$	−4.07203	−0.303511
$181$	−22.9977	−1.70940	−0.854701	−	0.519121i	$-0.826260\pi$
$181$	−22.9977	−1.70940	−0.854701	+	0.519121i	$0.826260\pi$
$182$	6.03578	0.447402
$183$	−2.93927	−0.217277
$184$	0	0
$185$	1.44521	0.106254
$186$	8.34017	0.611531
$187$	−22.0410	−1.61180
$188$	−0.979481	−0.0714360
$189$	−0.651685	−0.0474031
$190$	8.18342	0.593688
$191$	−11.2319	−0.812711	−0.406355	−	0.913715i	$-0.633200\pi$
$191$	−11.2319	−0.812711	−0.406355	+	0.913715i	$0.633200\pi$
$192$	24.8576	1.79394
$193$	14.0989	1.01486	0.507430	−	0.861693i	$-0.330595\pi$
$193$	14.0989	1.01486	0.507430	+	0.861693i	$0.330595\pi$
$194$	−15.0257	−1.07878
$195$	2.60674	0.186673
$196$	−35.1133	−2.50809
$197$	2.39803	0.170853	0.0854263	−	0.996344i	$-0.472775\pi$
$197$	2.39803	0.170853	0.0854263	+	0.996344i	$0.472775\pi$
$198$	16.6293	1.18180
$199$	17.2810	1.22502	0.612510	−	0.790463i	$-0.290160\pi$
$199$	17.2810	1.22502	0.612510	+	0.790463i	$0.290160\pi$
$200$	−39.9854	−2.82740
$201$	−7.01130	−0.494539
$202$	−25.7854	−1.81425
$203$	−0.600608	−0.0421544
$204$	−19.1763	−1.34261
$205$	5.09395	0.355777
$206$	41.9845	2.92520
$207$	0	0
$208$	−47.3028	−3.27986
$209$	−24.3135	−1.68180
$210$	1.34632	0.0929046
$211$	13.6020	0.936398	0.468199	−	0.883623i	$-0.344903\pi$
$211$	13.6020	0.936398	0.468199	+	0.883623i	$0.344903\pi$
$212$	54.5230	3.74465
$213$	−4.00000	−0.274075
$214$	−34.5621	−2.36261
$215$	−2.65368	−0.180980
$216$	9.04945	0.615737
$217$	−2.00613	−0.136185
$218$	−45.3538	−3.07175
$219$	−0.680346	−0.0459735
$220$	−24.9939	−1.68509
$221$	12.2759	0.825764
$222$	−5.13486	−0.344630
$223$	15.2039	1.01813	0.509065	−	0.860728i	$-0.329991\pi$
$223$	15.2039	1.01813	0.509065	+	0.860728i	$0.329991\pi$
$224$	−12.6359	−0.844274
$225$	−4.41855	−0.294570
$226$	−19.9986	−1.33029
$227$	7.44130	0.493897	0.246948	−	0.969029i	$-0.420572\pi$
$227$	7.44130	0.493897	0.246948	+	0.969029i	$0.420572\pi$
$228$	−21.1534	−1.40092
$229$	22.3970	1.48004	0.740019	−	0.672586i	$-0.234816\pi$
$229$	22.3970	1.48004	0.740019	+	0.672586i	$0.234816\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	8.34017	0.547559
$233$	16.8371	1.10304	0.551518	−	0.834163i	$-0.314049\pi$
$233$	16.8371	1.10304	0.551518	+	0.834163i	$0.314049\pi$
$234$	−9.26180	−0.605462
$235$	0.139861	0.00912353
$236$	30.2245	1.96744
$237$	11.1431	0.723822
$238$	6.34017	0.410972
$239$	−9.84324	−0.636707	−0.318353	−	0.947972i	$-0.603130\pi$
$239$	−9.84324	−0.636707	−0.318353	+	0.947972i	$0.603130\pi$
$240$	−10.5512	−0.681076
$241$	13.4307	0.865146	0.432573	−	0.901599i	$-0.357606\pi$
$241$	13.4307	0.865146	0.432573	+	0.901599i	$0.357606\pi$
$242$	72.2678	4.64555
$243$	1.00000	0.0641500
$244$	−15.6962	−1.00485
$245$	5.01386	0.320324
$246$	−18.0989	−1.15394
$247$	13.5415	0.861627
$248$	27.8576	1.76896
$249$	12.4975	0.792000
$250$	19.4578	1.23062
$251$	−13.3575	−0.843121	−0.421560	−	0.906800i	$-0.638517\pi$
$251$	−13.3575	−0.843121	−0.421560	+	0.906800i	$0.638517\pi$
$252$	−3.48011	−0.219226
$253$	0	0
$254$	20.1711	1.26565
$255$	2.73820	0.171473
$256$	27.6803	1.73002
$257$	4.52359	0.282174	0.141087	−	0.989997i	$-0.454940\pi$
$257$	4.52359	0.282174	0.141087	+	0.989997i	$0.454940\pi$
$258$	9.42858	0.586998
$259$	1.23513	0.0767474
$260$	13.9204	0.863310
$261$	0.921622	0.0570470
$262$	−51.0349	−3.15295
$263$	25.5957	1.57830	0.789149	−	0.614201i	$-0.210522\pi$
$263$	25.5957	1.57830	0.789149	+	0.614201i	$0.210522\pi$
$264$	55.5449	3.41855
$265$	−7.78539	−0.478252
$266$	6.99386	0.428821
$267$	−3.36927	−0.206196
$268$	−37.4416	−2.28711
$269$	25.4329	1.55067	0.775336	−	0.631548i	$-0.217580\pi$
$269$	25.4329	1.55067	0.775336	+	0.631548i	$0.217580\pi$
$270$	−2.06590	−0.125727
$271$	−31.6430	−1.92218	−0.961088	−	0.276243i	$-0.910911\pi$
$271$	−31.6430	−1.92218	−0.961088	+	0.276243i	$0.910911\pi$
$272$	−49.6884	−3.01280
$273$	2.22782	0.134834
$274$	−10.3295	−0.624028
$275$	−27.1208	−1.63544
$276$	0	0
$277$	13.0472	0.783929	0.391965	−	0.919980i	$-0.371796\pi$
$277$	13.0472	0.783929	0.391965	+	0.919980i	$0.371796\pi$
$278$	16.6803	1.00042
$279$	3.07838	0.184298
$280$	4.49693	0.268743
$281$	3.93217	0.234574	0.117287	−	0.993098i	$-0.462580\pi$
$281$	3.93217	0.234574	0.117287	+	0.993098i	$0.462580\pi$
$282$	−0.496928	−0.0295917
$283$	20.9318	1.24426	0.622132	−	0.782913i	$-0.286267\pi$
$283$	20.9318	1.24426	0.622132	+	0.782913i	$0.286267\pi$
$284$	−21.3607	−1.26752
$285$	3.02052	0.178920
$286$	−56.8483	−3.36151
$287$	4.35348	0.256978
$288$	19.3896	1.14254
$289$	−4.10504	−0.241473
$290$	−1.90398	−0.111805
$291$	−5.54601	−0.325113
$292$	−3.63317	−0.212615
$293$	15.8668	0.926949	0.463475	−	0.886110i	$-0.346602\pi$
$293$	15.8668	0.926949	0.463475	+	0.886110i	$0.346602\pi$
$294$	−17.8143	−1.03895
$295$	−4.31578	−0.251274
$296$	−17.1513	−0.996901
$297$	6.13793	0.356159
$298$	−21.5237	−1.24683
$299$	0	0
$300$	−23.5958	−1.36231
$301$	−2.26794	−0.130722
$302$	28.5113	1.64064
$303$	−9.51745	−0.546763
$304$	−54.8114	−3.14365
$305$	2.24128	0.128335
$306$	−9.72889	−0.556163
$307$	32.7526	1.86929	0.934644	−	0.355584i	$-0.115718\pi$
$307$	32.7526	1.86929	0.934644	+	0.355584i	$0.115718\pi$
$308$	−21.3607	−1.21714
$309$	15.4966	0.881570
$310$	−6.35962	−0.361202
$311$	−14.0267	−0.795379	−0.397690	−	0.917520i	$-0.630188\pi$
$311$	−14.0267	−0.795379	−0.397690	+	0.917520i	$0.630188\pi$
$312$	−30.9360	−1.75141
$313$	15.4745	0.874672	0.437336	−	0.899298i	$-0.355922\pi$
$313$	15.4745	0.874672	0.437336	+	0.899298i	$0.355922\pi$
$314$	−54.3201	−3.06546
$315$	0.496928	0.0279987
$316$	59.5061	3.34748
$317$	19.4452	1.09215	0.546076	−	0.837736i	$-0.316121\pi$
$317$	19.4452	1.09215	0.546076	+	0.837736i	$0.316121\pi$
$318$	27.6616	1.55119
$319$	5.65685	0.316723
$320$	−18.9546	−1.05960
$321$	−12.7569	−0.712023
$322$	0	0
$323$	14.2245	0.791470
$324$	5.34017	0.296676
$325$	15.1050	0.837877
$326$	−39.5441	−2.19015
$327$	−16.7402	−0.925735
$328$	−60.4534	−3.33798
$329$	0.119530	0.00658993
$330$	−12.6803	−0.698030
$331$	17.1194	0.940968	0.470484	−	0.882408i	$-0.344079\pi$
$331$	17.1194	0.940968	0.470484	+	0.882408i	$0.344079\pi$
$332$	66.7391	3.66278
$333$	−1.89529	−0.103861
$334$	40.1978	2.19953
$335$	5.34632	0.292100
$336$	−9.01744	−0.491941
$337$	−11.2029	−0.610259	−0.305129	−	0.952311i	$-0.598700\pi$
$337$	−11.2029	−0.610259	−0.305129	+	0.952311i	$0.598700\pi$
$338$	−3.55866	−0.193566
$339$	−7.38154	−0.400910
$340$	14.6225	0.793016
$341$	18.8949	1.02321
$342$	−10.7320	−0.580318
$343$	8.84683	0.477684
$344$	31.4931	1.69799
$345$	0	0
$346$	−25.0205	−1.34511
$347$	9.17727	0.492662	0.246331	−	0.969186i	$-0.420775\pi$
$347$	9.17727	0.492662	0.246331	+	0.969186i	$0.420775\pi$
$348$	4.92162	0.263827
$349$	29.2495	1.56569	0.782845	−	0.622217i	$-0.213768\pi$
$349$	29.2495	1.56569	0.782845	+	0.622217i	$0.213768\pi$
$350$	7.80137	0.417001
$351$	−3.41855	−0.182469
$352$	119.012	6.34337
$353$	−30.2823	−1.61176	−0.805882	−	0.592076i	$-0.798309\pi$
$353$	−30.2823	−1.61176	−0.805882	+	0.592076i	$0.798309\pi$
$354$	15.3340	0.814994
$355$	3.05011	0.161883
$356$	−17.9925	−0.953600
$357$	2.34017	0.123855
$358$	5.49079	0.290197
$359$	−7.66299	−0.404437	−0.202219	−	0.979340i	$-0.564815\pi$
$359$	−7.66299	−0.404437	−0.202219	+	0.979340i	$0.564815\pi$
$360$	−6.90046	−0.363686
$361$	−3.30898	−0.174157
$362$	−62.3070	−3.27478
$363$	26.6742	1.40003
$364$	11.8969	0.623569
$365$	0.518783	0.0271543
$366$	−7.96329	−0.416248
$367$	25.1657	1.31364	0.656820	−	0.754048i	$-0.271901\pi$
$367$	25.1657	1.31364	0.656820	+	0.754048i	$0.271901\pi$
$368$	0	0
$369$	−6.68035	−0.347765
$370$	3.91548	0.203556
$371$	−6.65368	−0.345442
$372$	16.4391	0.852326
$373$	−3.07913	−0.159431	−0.0797157	−	0.996818i	$-0.525401\pi$
$373$	−3.07913	−0.159431	−0.0797157	+	0.996818i	$0.525401\pi$
$374$	−59.7152	−3.08780
$375$	7.18191	0.370872
$376$	−1.65983	−0.0855990
$377$	−3.15061	−0.162265
$378$	−1.76560	−0.0908124
$379$	−11.0613	−0.568180	−0.284090	−	0.958798i	$-0.591691\pi$
$379$	−11.0613	−0.568180	−0.284090	+	0.958798i	$0.591691\pi$
$380$	16.1301	0.827456
$381$	7.44521	0.381430
$382$	−30.4303	−1.55695
$383$	24.3742	1.24546	0.622731	−	0.782436i	$-0.286023\pi$
$383$	24.3742	1.24546	0.622731	+	0.782436i	$0.286023\pi$
$384$	28.5669	1.45780
$385$	3.05011	0.155448
$386$	38.1978	1.94422
$387$	3.48011	0.176904
$388$	−29.6167	−1.50356
$389$	−1.10374	−0.0559621	−0.0279810	−	0.999608i	$-0.508908\pi$
$389$	−1.10374	−0.0559621	−0.0279810	+	0.999608i	$0.508908\pi$
$390$	7.06238	0.357618
$391$	0	0
$392$	−59.5029	−3.00535
$393$	−18.8371	−0.950206
$394$	6.49693	0.327311
$395$	−8.49693	−0.427527
$396$	32.7776	1.64714
$397$	3.47641	0.174476	0.0872380	−	0.996187i	$-0.472196\pi$
$397$	3.47641	0.174476	0.0872380	+	0.996187i	$0.472196\pi$
$398$	46.8191	2.34683
$399$	2.58145	0.129234
$400$	−61.1399	−3.05700
$401$	−21.5237	−1.07484	−0.537420	−	0.843314i	$-0.680601\pi$
$401$	−21.5237	−1.07484	−0.537420	+	0.843314i	$0.680601\pi$
$402$	−18.9955	−0.947412
$403$	−10.5236	−0.524217
$404$	−50.8248	−2.52863
$405$	−0.762528	−0.0378903
$406$	−1.62721	−0.0807572
$407$	−11.6332	−0.576635
$408$	−32.4962	−1.60880
$409$	25.3028	1.25114	0.625572	−	0.780166i	$-0.284866\pi$
$409$	25.3028	1.25114	0.625572	+	0.780166i	$0.284866\pi$
$410$	13.8009	0.681579
$411$	−3.81264	−0.188064
$412$	82.7544	4.07702
$413$	−3.68843	−0.181496
$414$	0	0
$415$	−9.52973	−0.467796
$416$	−66.2844	−3.24986
$417$	6.15676	0.301498
$418$	−65.8720	−3.22190
$419$	−25.4558	−1.24360	−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558	−1.24360	−0.621800	+	0.783176i	$0.713598\pi$
$420$	2.65368	0.129487
$421$	−27.5287	−1.34167	−0.670833	−	0.741608i	$-0.734063\pi$
$421$	−27.5287	−1.34167	−0.670833	+	0.741608i	$0.734063\pi$
$422$	36.8515	1.79390
$423$	−0.183417	−0.00891806
$424$	92.3945	4.48708
$425$	15.8668	0.769654
$426$	−10.8371	−0.525059
$427$	1.91548	0.0926965
$428$	−68.1243	−3.29291
$429$	−20.9828	−1.01306
$430$	−7.18956	−0.346711
$431$	−28.3046	−1.36338	−0.681692	−	0.731639i	$-0.738756\pi$
$431$	−28.3046	−1.36338	−0.681692	+	0.731639i	$0.738756\pi$
$432$	13.8371	0.665738
$433$	−29.0161	−1.39442	−0.697211	−	0.716866i	$-0.745576\pi$
$433$	−29.0161	−1.39442	−0.697211	+	0.716866i	$0.745576\pi$
$434$	−5.43517	−0.260896
$435$	−0.702763	−0.0336949
$436$	−89.3955	−4.28127
$437$	0	0
$438$	−1.84324	−0.0880736
$439$	34.3545	1.63965	0.819827	−	0.572612i	$-0.194070\pi$
$439$	34.3545	1.63965	0.819827	+	0.572612i	$0.194070\pi$
$440$	−42.3545	−2.01917
$441$	−6.57531	−0.313110
$442$	33.2587	1.58195
$443$	−22.8371	−1.08502	−0.542512	−	0.840048i	$-0.682527\pi$
$443$	−22.8371	−1.08502	−0.542512	+	0.840048i	$0.682527\pi$
$444$	−10.1212	−0.480330
$445$	2.56916	0.121790
$446$	41.1917	1.95048
$447$	−7.94444	−0.375759
$448$	−16.1993	−0.765347
$449$	−16.4703	−0.777280	−0.388640	−	0.921390i	$-0.627055\pi$
$449$	−16.4703	−0.777280	−0.388640	+	0.921390i	$0.627055\pi$
$450$	−11.9711	−0.564322
$451$	−41.0035	−1.93078
$452$	−39.4187	−1.85410
$453$	10.5236	0.494441
$454$	20.1605	0.946181
$455$	−1.69878	−0.0796398
$456$	−35.8466	−1.67867
$457$	−15.0753	−0.705191	−0.352596	−	0.935776i	$-0.614701\pi$
$457$	−15.0753	−0.705191	−0.352596	+	0.935776i	$0.614701\pi$
$458$	60.6798	2.83538
$459$	−3.59096	−0.167611
$460$	0	0
$461$	−15.8120	−0.736440	−0.368220	−	0.929739i	$-0.620033\pi$
$461$	−15.8120	−0.736440	−0.368220	+	0.929739i	$0.620033\pi$
$462$	−10.8371	−0.504188
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	12.7526	0.592024
$465$	−2.34735	−0.108856
$466$	45.6163	2.11314
$467$	29.1269	1.34783	0.673916	−	0.738808i	$-0.264611\pi$
$467$	29.1269	1.34783	0.673916	+	0.738808i	$0.264611\pi$
$468$	−18.2557	−0.843868
$469$	4.56916	0.210984
$470$	0.378922	0.0174784
$471$	−20.0497	−0.923841
$472$	51.2183	2.35751
$473$	21.3607	0.982166
$474$	30.1897	1.38666
$475$	17.5027	0.803080
$476$	12.4969	0.572796
$477$	10.2100	0.467482
$478$	−26.6681	−1.21977
$479$	10.3719	0.473904	0.236952	−	0.971521i	$-0.423852\pi$
$479$	10.3719	0.473904	0.236952	+	0.971521i	$0.423852\pi$
$480$	−14.7851	−0.674846
$481$	6.47915	0.295424
$482$	36.3874	1.65740
$483$	0	0
$484$	142.445	6.47477
$485$	4.22899	0.192029
$486$	2.70928	0.122895
$487$	17.1629	0.777725	0.388863	−	0.921296i	$-0.372868\pi$
$487$	17.1629	0.777725	0.388863	+	0.921296i	$0.372868\pi$
$488$	−26.5988	−1.20407
$489$	−14.5958	−0.660046
$490$	13.5839	0.613659
$491$	−3.50307	−0.158091	−0.0790457	−	0.996871i	$-0.525187\pi$
$491$	−3.50307	−0.158091	−0.0790457	+	0.996871i	$0.525187\pi$
$492$	−35.6742	−1.60832
$493$	−3.30950	−0.149053
$494$	36.6877	1.65066
$495$	−4.68035	−0.210366
$496$	42.5958	1.91261
$497$	2.60674	0.116928
$498$	33.8593	1.51727
$499$	−8.92162	−0.399387	−0.199693	−	0.979858i	$-0.563995\pi$
$499$	−8.92162	−0.399387	−0.199693	+	0.979858i	$0.563995\pi$
$500$	38.3526	1.71518
$501$	14.8371	0.662873
$502$	−36.1893	−1.61521
$503$	11.0543	0.492888	0.246444	−	0.969157i	$-0.420738\pi$
$503$	11.0543	0.492888	0.246444	+	0.969157i	$0.420738\pi$
$504$	−5.89739	−0.262691
$505$	7.25732	0.322947
$506$	0	0
$507$	−1.31351	−0.0583351
$508$	39.7587	1.76401
$509$	6.71154	0.297484	0.148742	−	0.988876i	$-0.452478\pi$
$509$	6.71154	0.297484	0.148742	+	0.988876i	$0.452478\pi$
$510$	7.41855	0.328499
$511$	0.443371	0.0196136
$512$	17.8599	0.789303
$513$	−3.96119	−0.174891
$514$	12.2557	0.540574
$515$	−11.8166	−0.520701
$516$	18.5844	0.818133
$517$	−1.12580	−0.0495128
$518$	3.34632	0.147029
$519$	−9.23513	−0.405377
$520$	23.5896	1.03447
$521$	26.9588	1.18109	0.590544	−	0.807005i	$-0.298913\pi$
$521$	26.9588	1.18109	0.590544	+	0.807005i	$0.298913\pi$
$522$	2.49693	0.109288
$523$	−15.2749	−0.667925	−0.333962	−	0.942586i	$-0.608386\pi$
$523$	−15.2749	−0.667925	−0.333962	+	0.942586i	$0.608386\pi$
$524$	−100.593	−4.39444
$525$	2.87950	0.125672
$526$	69.3458	3.02362
$527$	−11.0543	−0.481534
$528$	84.9312	3.69616
$529$	0	0
$530$	−21.0928	−0.916211
$531$	5.65983	0.245616
$532$	13.7854	0.597672
$533$	22.8371	0.989185
$534$	−9.12828	−0.395019
$535$	9.72753	0.420558
$536$	−63.4484	−2.74055
$537$	2.02666	0.0874569
$538$	68.9048	2.97070
$539$	−40.3588	−1.73838
$540$	−4.07203	−0.175232
$541$	33.9299	1.45876	0.729379	−	0.684110i	$-0.239809\pi$
$541$	33.9299	1.45876	0.729379	+	0.684110i	$0.239809\pi$
$542$	−85.7296	−3.68240
$543$	−22.9977	−0.986924
$544$	−69.6273	−2.98524
$545$	12.7649	0.546787
$546$	6.03578	0.258307
$547$	2.45136	0.104812	0.0524062	−	0.998626i	$-0.483311\pi$
$547$	2.45136	0.104812	0.0524062	+	0.998626i	$0.483311\pi$
$548$	−20.3602	−0.869743
$549$	−2.93927	−0.125445
$550$	−73.4776	−3.13310
$551$	−3.65072	−0.155526
$552$	0	0
$553$	−7.26180	−0.308803
$554$	35.3484	1.50181
$555$	1.44521	0.0613459
$556$	32.8781	1.39434
$557$	27.6239	1.17046	0.585231	−	0.810867i	$-0.301004\pi$
$557$	27.6239	1.17046	0.585231	+	0.810867i	$0.301004\pi$
$558$	8.34017	0.353068
$559$	−11.8969	−0.503187
$560$	6.87605	0.290566
$561$	−22.0410	−0.930573
$562$	10.6533	0.449384
$563$	−2.22782	−0.0938914	−0.0469457	−	0.998897i	$-0.514949\pi$
$563$	−2.22782	−0.0938914	−0.0469457	+	0.998897i	$0.514949\pi$
$564$	−0.979481	−0.0412436
$565$	5.62863	0.236798
$566$	56.7099	2.38369
$567$	−0.651685	−0.0273682
$568$	−36.1978	−1.51883
$569$	40.9814	1.71803	0.859016	−	0.511949i	$-0.171076\pi$
$569$	40.9814	1.71803	0.859016	+	0.511949i	$0.171076\pi$
$570$	8.18342	0.342766
$571$	19.5907	0.819844	0.409922	−	0.912121i	$-0.365556\pi$
$571$	19.5907	0.819844	0.409922	+	0.912121i	$0.365556\pi$
$572$	−112.052	−4.68513
$573$	−11.2319	−0.469219
$574$	11.7948	0.492305
$575$	0	0
$576$	24.8576	1.03573
$577$	7.45959	0.310547	0.155273	−	0.987872i	$-0.450374\pi$
$577$	7.45959	0.310547	0.155273	+	0.987872i	$0.450374\pi$
$578$	−11.1217	−0.462601
$579$	14.0989	0.585930
$580$	−3.75288	−0.155830
$581$	−8.14447	−0.337890
$582$	−15.0257	−0.622834
$583$	62.6681	2.59545
$584$	−6.15676	−0.254768
$585$	2.60674	0.107775
$586$	42.9876	1.77580
$587$	−17.8432	−0.736470	−0.368235	−	0.929733i	$-0.620038\pi$
$587$	−17.8432	−0.736470	−0.368235	+	0.929733i	$0.620038\pi$
$588$	−35.1133	−1.44805
$589$	−12.1940	−0.502447
$590$	−11.6926	−0.481378
$591$	2.39803	0.0986418
$592$	−26.2253	−1.07785
$593$	−6.73367	−0.276519	−0.138259	−	0.990396i	$-0.544151\pi$
$593$	−6.73367	−0.276519	−0.138259	+	0.990396i	$0.544151\pi$
$594$	16.6293	0.682310
$595$	−1.78445	−0.0731552
$596$	−42.4247	−1.73778
$597$	17.2810	0.707266
$598$	0	0
$599$	−33.8720	−1.38397	−0.691986	−	0.721911i	$-0.743264\pi$
$599$	−33.8720	−1.38397	−0.691986	+	0.721911i	$0.743264\pi$
$600$	−39.9854	−1.63240
$601$	−19.2618	−0.785705	−0.392853	−	0.919601i	$-0.628512\pi$
$601$	−19.2618	−0.785705	−0.392853	+	0.919601i	$0.628512\pi$
$602$	−6.14447	−0.250430
$603$	−7.01130	−0.285522
$604$	56.1978	2.28666
$605$	−20.3398	−0.826932
$606$	−25.7854	−1.04746
$607$	11.6865	0.474340	0.237170	−	0.971468i	$-0.423780\pi$
$607$	11.6865	0.474340	0.237170	+	0.971468i	$0.423780\pi$
$608$	−76.8060	−3.11489
$609$	−0.600608	−0.0243379
$610$	6.07223	0.245858
$611$	0.627022	0.0253666
$612$	−19.1763	−0.775157
$613$	34.6352	1.39890	0.699451	−	0.714680i	$-0.253428\pi$
$613$	34.6352	1.39890	0.699451	+	0.714680i	$0.253428\pi$
$614$	88.7358	3.58108
$615$	5.09395	0.205408
$616$	−36.1978	−1.45845
$617$	23.2704	0.936832	0.468416	−	0.883508i	$-0.344825\pi$
$617$	23.2704	0.936832	0.468416	+	0.883508i	$0.344825\pi$
$618$	41.9845	1.68886
$619$	−16.7181	−0.671958	−0.335979	−	0.941869i	$-0.609067\pi$
$619$	−16.7181	−0.671958	−0.335979	+	0.941869i	$0.609067\pi$
$620$	−12.5353	−0.503428
$621$	0	0
$622$	−38.0021	−1.52374
$623$	2.19570	0.0879690
$624$	−47.3028	−1.89363
$625$	16.6163	0.664654
$626$	41.9247	1.67565
$627$	−24.3135	−0.970988
$628$	−107.069	−4.27251
$629$	6.80590	0.271369
$630$	1.34632	0.0536385
$631$	−5.10732	−0.203319	−0.101660	−	0.994819i	$-0.532415\pi$
$631$	−5.10732	−0.203319	−0.101660	+	0.994819i	$0.532415\pi$
$632$	100.839	4.01116
$633$	13.6020	0.540630
$634$	52.6824	2.09229
$635$	−5.67718	−0.225292
$636$	54.5230	2.16198
$637$	22.4780	0.890611
$638$	15.3260	0.606761
$639$	−4.00000	−0.158238
$640$	−21.7831	−0.861051
$641$	−28.2622	−1.11629	−0.558145	−	0.829743i	$-0.688487\pi$
$641$	−28.2622	−1.11629	−0.558145	+	0.829743i	$0.688487\pi$
$642$	−34.5621	−1.36406
$643$	−4.88564	−0.192671	−0.0963354	−	0.995349i	$-0.530712\pi$
$643$	−4.88564	−0.192671	−0.0963354	+	0.995349i	$0.530712\pi$
$644$	0	0
$645$	−2.65368	−0.104489
$646$	38.5380	1.51626
$647$	1.49079	0.0586088	0.0293044	−	0.999571i	$-0.490671\pi$
$647$	1.49079	0.0586088	0.0293044	+	0.999571i	$0.490671\pi$
$648$	9.04945	0.355496
$649$	34.7396	1.36365
$650$	40.9237	1.60516
$651$	−2.00613	−0.0786266
$652$	−77.9442	−3.05253
$653$	40.6681	1.59146	0.795732	−	0.605649i	$-0.207086\pi$
$653$	40.6681	1.59146	0.795732	+	0.605649i	$0.207086\pi$
$654$	−45.3538	−1.77347
$655$	14.3638	0.561241
$656$	−92.4366	−3.60904
$657$	−0.680346	−0.0265428
$658$	0.323841	0.0126246
$659$	−3.61301	−0.140743	−0.0703715	−	0.997521i	$-0.522418\pi$
$659$	−3.61301	−0.140743	−0.0703715	+	0.997521i	$0.522418\pi$
$660$	−24.9939	−0.972885
$661$	13.2090	0.513771	0.256885	−	0.966442i	$-0.417304\pi$
$661$	13.2090	0.513771	0.256885	+	0.966442i	$0.417304\pi$
$662$	46.3812	1.80266
$663$	12.2759	0.476755
$664$	113.096	4.38897
$665$	−1.96843	−0.0763324
$666$	−5.13486	−0.198972
$667$	0	0
$668$	79.2327	3.06560
$669$	15.2039	0.587818
$670$	14.4846	0.559591
$671$	−18.0410	−0.696467
$672$	−12.6359	−0.487442
$673$	14.9360	0.575740	0.287870	−	0.957669i	$-0.407053\pi$
$673$	14.9360	0.575740	0.287870	+	0.957669i	$0.407053\pi$
$674$	−30.3516	−1.16910
$675$	−4.41855	−0.170070
$676$	−7.01438	−0.269784
$677$	−7.10181	−0.272945	−0.136472	−	0.990644i	$-0.543577\pi$
$677$	−7.10181	−0.272945	−0.136472	+	0.990644i	$0.543577\pi$
$678$	−19.9986	−0.768042
$679$	3.61425	0.138702
$680$	24.7792	0.950241
$681$	7.44130	0.285151
$682$	51.1914	1.96022
$683$	−41.7275	−1.59666	−0.798330	−	0.602221i	$-0.794283\pi$
$683$	−41.7275	−1.59666	−0.798330	+	0.602221i	$0.794283\pi$
$684$	−21.1534	−0.808822
$685$	2.90725	0.111080
$686$	23.9685	0.915121
$687$	22.3970	0.854501
$688$	48.1547	1.83588
$689$	−34.9033	−1.32971
$690$	0	0
$691$	11.7587	0.447323	0.223661	−	0.974667i	$-0.428199\pi$
$691$	11.7587	0.447323	0.223661	+	0.974667i	$0.428199\pi$
$692$	−49.3172	−1.87476
$693$	−4.00000	−0.151947
$694$	24.8638	0.943816
$695$	−4.69470	−0.178080
$696$	8.34017	0.316133
$697$	23.9888	0.908642
$698$	79.2450	2.99947
$699$	16.8371	0.636838
$700$	15.3771	0.581198
$701$	−52.0735	−1.96679	−0.983394	−	0.181484i	$-0.941910\pi$
$701$	−52.0735	−1.96679	−0.983394	+	0.181484i	$0.941910\pi$
$702$	−9.26180	−0.349564
$703$	7.50761	0.283155
$704$	152.574	5.75036
$705$	0.139861	0.00526747
$706$	−82.0431	−3.08773
$707$	6.20238	0.233265
$708$	30.2245	1.13590
$709$	−6.10891	−0.229425	−0.114713	−	0.993399i	$-0.536595\pi$
$709$	−6.10891	−0.229425	−0.114713	+	0.993399i	$0.536595\pi$
$710$	8.26360	0.310127
$711$	11.1431	0.417899
$712$	−30.4900	−1.14266
$713$	0	0
$714$	6.34017	0.237275
$715$	16.0000	0.598366
$716$	10.8227	0.404464
$717$	−9.84324	−0.367603
$718$	−20.7611	−0.774799
$719$	3.81658	0.142335	0.0711673	−	0.997464i	$-0.477328\pi$
$719$	3.81658	0.142335	0.0711673	+	0.997464i	$0.477328\pi$
$720$	−10.5512	−0.393219
$721$	−10.0989	−0.376103
$722$	−8.96493	−0.333640
$723$	13.4307	0.499493
$724$	−122.811	−4.56425
$725$	−4.07223	−0.151239
$726$	72.2678	2.68211
$727$	−44.8452	−1.66322	−0.831608	−	0.555364i	$-0.812579\pi$
$727$	−44.8452	−1.66322	−0.831608	+	0.555364i	$0.812579\pi$
$728$	20.1605	0.747199
$729$	1.00000	0.0370370
$730$	1.40553	0.0520208
$731$	−12.4969	−0.462216
$732$	−15.6962	−0.580149
$733$	−28.4154	−1.04955	−0.524774	−	0.851241i	$-0.675850\pi$
$733$	−28.4154	−1.04955	−0.524774	+	0.851241i	$0.675850\pi$
$734$	68.1808	2.51660
$735$	5.01386	0.184939
$736$	0	0
$737$	−43.0349	−1.58521
$738$	−18.0989	−0.666230
$739$	6.52359	0.239974	0.119987	−	0.992775i	$-0.461715\pi$
$739$	6.52359	0.239974	0.119987	+	0.992775i	$0.461715\pi$
$740$	7.71769	0.283708
$741$	13.5415	0.497460
$742$	−18.0267	−0.661780
$743$	3.22768	0.118412	0.0592060	−	0.998246i	$-0.481143\pi$
$743$	3.22768	0.118412	0.0592060	+	0.998246i	$0.481143\pi$
$744$	27.8576	1.02131
$745$	6.05786	0.221943
$746$	−8.34221	−0.305430
$747$	12.4975	0.457261
$748$	−117.703	−4.30365
$749$	8.31351	0.303769
$750$	19.4578	0.710497
$751$	9.87744	0.360433	0.180216	−	0.983627i	$-0.442320\pi$
$751$	9.87744	0.360433	0.180216	+	0.983627i	$0.442320\pi$
$752$	−2.53797	−0.0925501
$753$	−13.3575	−0.486776
$754$	−8.53588	−0.310858
$755$	−8.02453	−0.292043
$756$	−3.48011	−0.126570
$757$	11.1652	0.405805	0.202902	−	0.979199i	$-0.434963\pi$
$757$	11.1652	0.405805	0.202902	+	0.979199i	$0.434963\pi$
$758$	−29.9680	−1.08849
$759$	0	0
$760$	27.3340	0.991509
$761$	−4.12556	−0.149551	−0.0747757	−	0.997200i	$-0.523824\pi$
$761$	−4.12556	−0.149551	−0.0747757	+	0.997200i	$0.523824\pi$
$762$	20.1711	0.730723
$763$	10.9093	0.394944
$764$	−59.9802	−2.17001
$765$	2.73820	0.0990000
$766$	66.0363	2.38599
$767$	−19.3484	−0.698630
$768$	27.6803	0.998828
$769$	−23.8844	−0.861293	−0.430647	−	0.902521i	$-0.641714\pi$
$769$	−23.8844	−0.861293	−0.430647	+	0.902521i	$0.641714\pi$
$770$	8.26360	0.297799
$771$	4.52359	0.162913
$772$	75.2905	2.70977
$773$	8.38781	0.301689	0.150844	−	0.988558i	$-0.451801\pi$
$773$	8.38781	0.301689	0.150844	+	0.988558i	$0.451801\pi$
$774$	9.42858	0.338903
$775$	−13.6020	−0.488597
$776$	−50.1883	−1.80166
$777$	1.23513	0.0443102
$778$	−2.99035	−0.107209
$779$	26.4621	0.948104
$780$	13.9204	0.498432
$781$	−24.5517	−0.878530
$782$	0	0
$783$	0.921622	0.0329361
$784$	−90.9832	−3.24940
$785$	15.2885	0.545668
$786$	−51.0349	−1.82035
$787$	5.67023	0.202122	0.101061	−	0.994880i	$-0.467776\pi$
$787$	5.67023	0.202122	0.101061	+	0.994880i	$0.467776\pi$
$788$	12.8059	0.456191
$789$	25.5957	0.911231
$790$	−23.0205	−0.819033
$791$	4.81044	0.171040
$792$	55.5449	1.97370
$793$	10.0480	0.356816
$794$	9.41855	0.334252
$795$	−7.78539	−0.276119
$796$	92.2837	3.27091
$797$	14.2664	0.505340	0.252670	−	0.967553i	$-0.418691\pi$
$797$	14.2664	0.505340	0.252670	+	0.967553i	$0.418691\pi$
$798$	6.99386	0.247580
$799$	0.658644	0.0233011
$800$	−85.6740	−3.02903
$801$	−3.36927	−0.119047
$802$	−58.3136	−2.05912
$803$	−4.17592	−0.147365
$804$	−37.4416	−1.32046
$805$	0	0
$806$	−28.5113	−1.00427
$807$	25.4329	0.895281
$808$	−86.1276	−3.02996
$809$	−48.6369	−1.70998	−0.854991	−	0.518644i	$-0.826437\pi$
$809$	−48.6369	−1.70998	−0.854991	+	0.518644i	$0.826437\pi$
$810$	−2.06590	−0.0725883
$811$	−40.1256	−1.40900	−0.704499	−	0.709705i	$-0.748828\pi$
$811$	−40.1256	−1.40900	−0.704499	+	0.709705i	$0.748828\pi$
$812$	−3.20735	−0.112556
$813$	−31.6430	−1.10977
$814$	−31.5174	−1.10469
$815$	11.1297	0.389857
$816$	−49.6884	−1.73944
$817$	−13.7854	−0.482290
$818$	68.5523	2.39688
$819$	2.22782	0.0778463
$820$	27.2026	0.949956
$821$	−46.7091	−1.63016	−0.815079	−	0.579349i	$-0.803307\pi$
$821$	−46.7091	−1.63016	−0.815079	+	0.579349i	$0.803307\pi$
$822$	−10.3295	−0.360282
$823$	23.2762	0.811356	0.405678	−	0.914016i	$-0.367035\pi$
$823$	23.2762	0.811356	0.405678	+	0.914016i	$0.367035\pi$
$824$	140.236	4.88534
$825$	−27.1208	−0.944224
$826$	−9.99296	−0.347700
$827$	−8.58744	−0.298614	−0.149307	−	0.988791i	$-0.547704\pi$
$827$	−8.58744	−0.298614	−0.149307	+	0.988791i	$0.547704\pi$
$828$	0	0
$829$	21.2208	0.737027	0.368514	−	0.929622i	$-0.379867\pi$
$829$	21.2208	0.737027	0.368514	+	0.929622i	$0.379867\pi$
$830$	−25.8187	−0.896179
$831$	13.0472	0.452602
$832$	−84.9770	−2.94605
$833$	23.6116	0.818095
$834$	16.6803	0.577593
$835$	−11.3137	−0.391527
$836$	−129.838	−4.49055
$837$	3.07838	0.106404
$838$	−68.9669	−2.38242
$839$	18.9767	0.655148	0.327574	−	0.944825i	$-0.393769\pi$
$839$	18.9767	0.655148	0.327574	+	0.944825i	$0.393769\pi$
$840$	4.49693	0.155159
$841$	−28.1506	−0.970711
$842$	−74.5828	−2.57029
$843$	3.93217	0.135431
$844$	72.6369	2.50026
$845$	1.00159	0.0344557
$846$	−0.496928	−0.0170848
$847$	−17.3832	−0.597293
$848$	141.276	4.85145
$849$	20.9318	0.718376
$850$	42.9876	1.47446
$851$	0	0
$852$	−21.3607	−0.731805
$853$	−17.8843	−0.612346	−0.306173	−	0.951976i	$-0.599049\pi$
$853$	−17.8843	−0.612346	−0.306173	+	0.951976i	$0.599049\pi$
$854$	5.18956	0.177583
$855$	3.02052	0.103300
$856$	−115.443	−3.94577
$857$	−3.72753	−0.127330	−0.0636649	−	0.997971i	$-0.520279\pi$
$857$	−3.72753	−0.127330	−0.0636649	+	0.997971i	$0.520279\pi$
$858$	−56.8483	−1.94077
$859$	−20.9939	−0.716301	−0.358151	−	0.933664i	$-0.616593\pi$
$859$	−20.9939	−0.716301	−0.358151	+	0.933664i	$0.616593\pi$
$860$	−14.1711	−0.483232
$861$	4.35348	0.148366
$862$	−76.6850	−2.61190
$863$	−21.1773	−0.720883	−0.360441	−	0.932782i	$-0.617374\pi$
$863$	−21.1773	−0.720883	−0.360441	+	0.932782i	$0.617374\pi$
$864$	19.3896	0.659648
$865$	7.04205	0.239437
$866$	−78.6125	−2.67136
$867$	−4.10504	−0.139414
$868$	−10.7131	−0.363626
$869$	68.3956	2.32016
$870$	−1.90398	−0.0645509
$871$	23.9685	0.812141
$872$	−151.489	−5.13008
$873$	−5.54601	−0.187704
$874$	0	0
$875$	−4.68035	−0.158225
$876$	−3.63317	−0.122753
$877$	38.3545	1.29514	0.647571	−	0.762006i	$-0.275785\pi$
$877$	38.3545	1.29514	0.647571	+	0.762006i	$0.275785\pi$
$878$	93.0759	3.14116
$879$	15.8668	0.535174
$880$	−64.7624	−2.18314
$881$	−29.0468	−0.978612	−0.489306	−	0.872112i	$-0.662750\pi$
$881$	−29.0468	−0.978612	−0.489306	+	0.872112i	$0.662750\pi$
$882$	−17.8143	−0.599839
$883$	−19.6430	−0.661040	−0.330520	−	0.943799i	$-0.607224\pi$
$883$	−19.6430	−0.661040	−0.330520	+	0.943799i	$0.607224\pi$
$884$	65.5552	2.20486
$885$	−4.31578	−0.145073
$886$	−61.8720	−2.07863
$887$	−22.9672	−0.771163	−0.385581	−	0.922674i	$-0.625999\pi$
$887$	−22.9672	−0.771163	−0.385581	+	0.922674i	$0.625999\pi$
$888$	−17.1513	−0.575561
$889$	−4.85194	−0.162729
$890$	6.96057	0.233319
$891$	6.13793	0.205628
$892$	81.1917	2.71850
$893$	0.726551	0.0243131
$894$	−21.5237	−0.719859
$895$	−1.54539	−0.0516566
$896$	−18.6166	−0.621938
$897$	0	0
$898$	−44.6225	−1.48907
$899$	2.83710	0.0946226
$900$	−23.5958	−0.786528
$901$	−36.6635	−1.22144
$902$	−111.090	−3.69889
$903$	−2.26794	−0.0754723
$904$	−66.7988	−2.22170
$905$	17.5364	0.582928
$906$	28.5113	0.947225
$907$	−23.9819	−0.796305	−0.398152	−	0.917319i	$-0.630348\pi$
$907$	−23.9819	−0.796305	−0.398152	+	0.917319i	$0.630348\pi$
$908$	39.7378	1.31875
$909$	−9.51745	−0.315674
$910$	−4.60245	−0.152570
$911$	8.96636	0.297069	0.148534	−	0.988907i	$-0.452544\pi$
$911$	8.96636	0.297069	0.148534	+	0.988907i	$0.452544\pi$
$912$	−54.8114	−1.81499
$913$	76.7091	2.53870
$914$	−40.8431	−1.35097
$915$	2.24128	0.0740943
$916$	119.604	3.95183
$917$	12.2759	0.405385
$918$	−9.72889	−0.321101
$919$	16.6601	0.549566	0.274783	−	0.961506i	$-0.411394\pi$
$919$	16.6601	0.549566	0.274783	+	0.961506i	$0.411394\pi$
$920$	0	0
$921$	32.7526	1.07923
$922$	−42.8392	−1.41083
$923$	13.6742	0.450092
$924$	−21.3607	−0.702715
$925$	8.37444	0.275350
$926$	−21.6742	−0.712259
$927$	15.4966	0.508975
$928$	17.8699	0.586608
$929$	17.8843	0.586764	0.293382	−	0.955995i	$-0.405219\pi$
$929$	17.8843	0.586764	0.293382	+	0.955995i	$0.405219\pi$
$930$	−6.35962	−0.208540
$931$	26.0460	0.853624
$932$	89.9130	2.94520
$933$	−14.0267	−0.459212
$934$	78.9128	2.58211
$935$	16.8069	0.549645
$936$	−30.9360	−1.01117
$937$	44.6391	1.45830	0.729148	−	0.684356i	$-0.239916\pi$
$937$	44.6391	1.45830	0.729148	+	0.684356i	$0.239916\pi$
$938$	12.3791	0.404193
$939$	15.4745	0.504992
$940$	0.746882	0.0243606
$941$	−8.58275	−0.279790	−0.139895	−	0.990166i	$-0.544676\pi$
$941$	−8.58275	−0.279790	−0.139895	+	0.990166i	$0.544676\pi$
$942$	−54.3201	−1.76985
$943$	0	0
$944$	78.3156	2.54896
$945$	0.496928	0.0161651
$946$	57.8720	1.88158
$947$	2.65368	0.0862331	0.0431166	−	0.999070i	$-0.486271\pi$
$947$	2.65368	0.0862331	0.0431166	+	0.999070i	$0.486271\pi$
$948$	59.5061	1.93267
$949$	2.32580	0.0754986
$950$	47.4197	1.53850
$951$	19.4452	0.630554
$952$	21.1773	0.686359
$953$	−29.3880	−0.951971	−0.475986	−	0.879453i	$-0.657909\pi$
$953$	−29.3880	−0.951971	−0.475986	+	0.879453i	$0.657909\pi$
$954$	27.6616	0.895577
$955$	8.56463	0.277145
$956$	−52.5646	−1.70006
$957$	5.65685	0.182860
$958$	28.1003	0.907879
$959$	2.48464	0.0802333
$960$	−18.9546	−0.611758
$961$	−21.5236	−0.694309
$962$	17.5538	0.565957
$963$	−12.7569	−0.411087
$964$	71.7222	2.31002
$965$	−10.7508	−0.346081
$966$	0	0
$967$	−17.9155	−0.576123	−0.288061	−	0.957612i	$-0.593011\pi$
$967$	−17.9155	−0.576123	−0.288061	+	0.957612i	$0.593011\pi$
$968$	241.387	7.75847
$969$	14.2245	0.456955
$970$	11.4575	0.367878
$971$	21.1227	0.677859	0.338930	−	0.940812i	$-0.389935\pi$
$971$	21.1227	0.677859	0.338930	+	0.940812i	$0.389935\pi$
$972$	5.34017	0.171286
$973$	−4.01227	−0.128627
$974$	46.4990	1.48992
$975$	15.1050	0.483748
$976$	−40.6710	−1.30185
$977$	−42.5645	−1.36176	−0.680880	−	0.732395i	$-0.738403\pi$
$977$	−42.5645	−1.36176	−0.680880	+	0.732395i	$0.738403\pi$
$978$	−39.5441	−1.26448
$979$	−20.6803	−0.660947
$980$	26.7749	0.855292
$981$	−16.7402	−0.534473
$982$	−9.49079	−0.302863
$983$	18.1921	0.580238	0.290119	−	0.956991i	$-0.406305\pi$
$983$	18.1921	0.580238	0.290119	+	0.956991i	$0.406305\pi$
$984$	−60.4534	−1.92719
$985$	−1.82857	−0.0582630
$986$	−8.96636	−0.285547
$987$	0.119530	0.00380470
$988$	72.3141	2.30062
$989$	0	0
$990$	−12.6803	−0.403008
$991$	29.9155	0.950297	0.475148	−	0.879906i	$-0.342394\pi$
$991$	29.9155	0.950297	0.475148	+	0.879906i	$0.342394\pi$
$992$	59.6886	1.89511
$993$	17.1194	0.543268
$994$	7.06238	0.224005
$995$	−13.1773	−0.417748
$996$	66.7391	2.11471
$997$	−2.52813	−0.0800665	−0.0400333	−	0.999198i	$-0.512746\pi$
$997$	−2.52813	−0.0800665	−0.0400333	+	0.999198i	$0.512746\pi$
$998$	−24.1711	−0.765124
$999$	−1.89529	−0.0599643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1587.2.a.s.1.5	✓	6
3.2	odd	2		4761.2.a.bs.1.2		6
23.22	odd	2	inner	1587.2.a.s.1.6	yes	6
69.68	even	2		4761.2.a.bs.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1587.2.a.s.1.5	✓	6	1.1	even	1	trivial
1587.2.a.s.1.6	yes	6	23.22	odd	2	inner
4761.2.a.bs.1.1		6	69.68	even	2
4761.2.a.bs.1.2		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1587 = 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1587.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} -0.762528 q^{5} +2.70928 q^{6} -0.651685 q^{7} +9.04945 q^{8} +1.00000 q^{9} -2.06590 q^{10} +6.13793 q^{11} +5.34017 q^{12} -3.41855 q^{13} -1.76560 q^{14} -0.762528 q^{15} +13.8371 q^{16} -3.59096 q^{17} +2.70928 q^{18} -3.96119 q^{19} -4.07203 q^{20} -0.651685 q^{21} +16.6293 q^{22} +9.04945 q^{24} -4.41855 q^{25} -9.26180 q^{26} +1.00000 q^{27} -3.48011 q^{28} +0.921622 q^{29} -2.06590 q^{30} +3.07838 q^{31} +19.3896 q^{32} +6.13793 q^{33} -9.72889 q^{34} +0.496928 q^{35} +5.34017 q^{36} -1.89529 q^{37} -10.7320 q^{38} -3.41855 q^{39} -6.90046 q^{40} -6.68035 q^{41} -1.76560 q^{42} +3.48011 q^{43} +32.7776 q^{44} -0.762528 q^{45} -0.183417 q^{47} +13.8371 q^{48} -6.57531 q^{49} -11.9711 q^{50} -3.59096 q^{51} -18.2557 q^{52} +10.2100 q^{53} +2.70928 q^{54} -4.68035 q^{55} -5.89739 q^{56} -3.96119 q^{57} +2.49693 q^{58} +5.65983 q^{59} -4.07203 q^{60} -2.93927 q^{61} +8.34017 q^{62} -0.651685 q^{63} +24.8576 q^{64} +2.60674 q^{65} +16.6293 q^{66} -7.01130 q^{67} -19.1763 q^{68} +1.34632 q^{70} -4.00000 q^{71} +9.04945 q^{72} -0.680346 q^{73} -5.13486 q^{74} -4.41855 q^{75} -21.1534 q^{76} -4.00000 q^{77} -9.26180 q^{78} +11.1431 q^{79} -10.5512 q^{80} +1.00000 q^{81} -18.0989 q^{82} +12.4975 q^{83} -3.48011 q^{84} +2.73820 q^{85} +9.42858 q^{86} +0.921622 q^{87} +55.5449 q^{88} -3.36927 q^{89} -2.06590 q^{90} +2.22782 q^{91} +3.07838 q^{93} -0.496928 q^{94} +3.02052 q^{95} +19.3896 q^{96} -5.54601 q^{97} -17.8143 q^{98} +6.13793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 6 q^{3} + 10 q^{4} + 2 q^{6} + 18 q^{8} + 6 q^{9} + 10 q^{12} + 8 q^{13} + 26 q^{16} + 2 q^{18} + 18 q^{24} + 2 q^{25} - 40 q^{26} + 6 q^{27} + 12 q^{29} + 12 q^{31} + 58 q^{32} - 32 q^{35}+ \cdots - 90 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 6 * q^3 + 10 * q^4 + 2 * q^6 + 18 * q^8 + 6 * q^9 + 10 * q^12 + 8 * q^13 + 26 * q^16 + 2 * q^18 + 18 * q^24 + 2 * q^25 - 40 * q^26 + 6 * q^27 + 12 * q^29 + 12 * q^31 + 58 * q^32 - 32 * q^35 + 10 * q^36 + 8 * q^39 + 4 * q^41 + 8 * q^47 + 26 * q^48 + 2 * q^49 - 42 * q^50 - 24 * q^52 + 2 * q^54 + 16 * q^55 - 20 * q^58 + 56 * q^59 + 28 * q^62 + 26 * q^64 + 56 * q^70 - 24 * q^71 + 18 * q^72 + 40 * q^73 + 2 * q^75 - 24 * q^77 - 40 * q^78 + 6 * q^81 - 36 * q^82 + 32 * q^85 + 12 * q^87 + 12 * q^93 + 32 * q^94 - 48 * q^95 + 58 * q^96 - 90 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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