LMFDB - Embedded newform 1587.2.a.s.1.3

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			1.3
Root			$0.569973$ 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)

$f(q)$	$=$	\(q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -2.36899 q^{5} +0.193937 q^{6} +0.954779 q^{7} -0.768452 q^{8} +1.00000 q^{9} -0.459434 q^{10} -4.18945 q^{11} -1.96239 q^{12} +1.61213 q^{13} +0.185167 q^{14} -2.36899 q^{15} +3.77575 q^{16} -5.19742 q^{17} +0.193937 q^{18} +7.97266 q^{19} +4.64888 q^{20} +0.954779 q^{21} -0.812488 q^{22} -0.768452 q^{24} +0.612127 q^{25} +0.312650 q^{26} +1.00000 q^{27} -1.87365 q^{28} -1.35026 q^{29} -0.459434 q^{30} +5.35026 q^{31} +2.26916 q^{32} -4.18945 q^{33} -1.00797 q^{34} -2.26187 q^{35} -1.96239 q^{36} +8.43209 q^{37} +1.54619 q^{38} +1.61213 q^{39} +1.82046 q^{40} +7.92478 q^{41} +0.185167 q^{42} +1.87365 q^{43} +8.22133 q^{44} -2.36899 q^{45} +11.6629 q^{47} +3.77575 q^{48} -6.08840 q^{49} +0.118714 q^{50} -5.19742 q^{51} -3.16362 q^{52} -8.83833 q^{53} +0.193937 q^{54} +9.92478 q^{55} -0.733702 q^{56} +7.97266 q^{57} -0.261865 q^{58} +12.9624 q^{59} +4.64888 q^{60} -6.15220 q^{61} +1.03761 q^{62} +0.954779 q^{63} -7.11142 q^{64} -3.81912 q^{65} -0.812488 q^{66} -1.50331 q^{67} +10.1994 q^{68} -0.438658 q^{70} -4.00000 q^{71} -0.768452 q^{72} +13.9248 q^{73} +1.63529 q^{74} +0.612127 q^{75} -15.6455 q^{76} -4.00000 q^{77} +0.312650 q^{78} +2.42218 q^{79} -8.94472 q^{80} +1.00000 q^{81} +1.53690 q^{82} -1.73136 q^{83} -1.87365 q^{84} +12.3127 q^{85} +0.363369 q^{86} -1.35026 q^{87} +3.21939 q^{88} +1.45012 q^{89} -0.459434 q^{90} +1.53923 q^{91} +5.35026 q^{93} +2.26187 q^{94} -18.8872 q^{95} +2.26916 q^{96} -2.33308 q^{97} -1.18076 q^{98} -4.18945 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.193937	0.137134	0.0685669	−	0.997647i	$-0.478157\pi$
$2$	0.193937	0.137134	0.0685669	+	0.997647i	$0.478157\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.96239	−0.981194
$5$	−2.36899	−1.05945	−0.529723	−	0.848171i	$-0.677704\pi$
$5$	−2.36899	−1.05945	−0.529723	+	0.848171i	$0.677704\pi$
$6$	0.193937	0.0791743
$7$	0.954779	0.360873	0.180436	−	0.983587i	$-0.442249\pi$
$7$	0.954779	0.360873	0.180436	+	0.983587i	$0.442249\pi$
$8$	−0.768452	−0.271689
$9$	1.00000	0.333333
$10$	−0.459434	−0.145286
$11$	−4.18945	−1.26317	−0.631583	−	0.775308i	$-0.717595\pi$
$11$	−4.18945	−1.26317	−0.631583	+	0.775308i	$0.717595\pi$
$12$	−1.96239	−0.566493
$13$	1.61213	0.447124	0.223562	−	0.974690i	$-0.428232\pi$
$13$	1.61213	0.447124	0.223562	+	0.974690i	$0.428232\pi$
$14$	0.185167	0.0494879
$15$	−2.36899	−0.611671
$16$	3.77575	0.943937
$17$	−5.19742	−1.26056	−0.630280	−	0.776368i	$-0.717060\pi$
$17$	−5.19742	−1.26056	−0.630280	+	0.776368i	$0.717060\pi$
$18$	0.193937	0.0457113
$19$	7.97266	1.82905	0.914526	−	0.404526i	$-0.132564\pi$
$19$	7.97266	1.82905	0.914526	+	0.404526i	$0.132564\pi$
$20$	4.64888	1.03952
$21$	0.954779	0.208350
$22$	−0.812488	−0.173223
$23$	0	0
$23$	0	0
$24$	−0.768452	−0.156860
$25$	0.612127	0.122425
$26$	0.312650	0.0613158
$27$	1.00000	0.192450
$28$	−1.87365	−0.354086
$29$	−1.35026	−0.250737	−0.125369	−	0.992110i	$-0.540011\pi$
$29$	−1.35026	−0.250737	−0.125369	+	0.992110i	$0.540011\pi$
$30$	−0.459434	−0.0838808
$31$	5.35026	0.960935	0.480468	−	0.877012i	$-0.340467\pi$
$31$	5.35026	0.960935	0.480468	+	0.877012i	$0.340467\pi$
$32$	2.26916	0.401134
$33$	−4.18945	−0.729290
$34$	−1.00797	−0.172865
$35$	−2.26187	−0.382325
$36$	−1.96239	−0.327065
$37$	8.43209	1.38623	0.693114	−	0.720828i	$-0.256238\pi$
$37$	8.43209	1.38623	0.693114	+	0.720828i	$0.256238\pi$
$38$	1.54619	0.250825
$39$	1.61213	0.258147
$40$	1.82046	0.287840
$41$	7.92478	1.23764	0.618821	−	0.785532i	$-0.287611\pi$
$41$	7.92478	1.23764	0.618821	+	0.785532i	$0.287611\pi$
$42$	0.185167	0.0285718
$43$	1.87365	0.285729	0.142864	−	0.989742i	$-0.454369\pi$
$43$	1.87365	0.285729	0.142864	+	0.989742i	$0.454369\pi$
$44$	8.22133	1.23941
$45$	−2.36899	−0.353149
$46$	0	0
$47$	11.6629	1.70121	0.850605	−	0.525805i	$-0.176236\pi$
$47$	11.6629	1.70121	0.850605	+	0.525805i	$0.176236\pi$
$48$	3.77575	0.544982
$49$	−6.08840	−0.869771
$50$	0.118714	0.0167887
$51$	−5.19742	−0.727784
$52$	−3.16362	−0.438715
$53$	−8.83833	−1.21404	−0.607019	−	0.794687i	$-0.707635\pi$
$53$	−8.83833	−1.21404	−0.607019	+	0.794687i	$0.707635\pi$
$54$	0.193937	0.0263914
$55$	9.92478	1.33826
$56$	−0.733702	−0.0980451
$57$	7.97266	1.05600
$58$	−0.261865	−0.0343846
$59$	12.9624	1.68756	0.843780	−	0.536690i	$-0.180325\pi$
$59$	12.9624	1.68756	0.843780	+	0.536690i	$0.180325\pi$
$60$	4.64888	0.600168
$61$	−6.15220	−0.787708	−0.393854	−	0.919173i	$-0.628859\pi$
$61$	−6.15220	−0.787708	−0.393854	+	0.919173i	$0.628859\pi$
$62$	1.03761	0.131777
$63$	0.954779	0.120291
$64$	−7.11142	−0.888927
$65$	−3.81912	−0.473703
$66$	−0.812488	−0.100010
$67$	−1.50331	−0.183659	−0.0918296	−	0.995775i	$-0.529271\pi$
$67$	−1.50331	−0.183659	−0.0918296	+	0.995775i	$0.529271\pi$
$68$	10.1994	1.23685
$69$	0	0
$70$	−0.438658	−0.0524297
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−0.768452	−0.0905629
$73$	13.9248	1.62977	0.814886	−	0.579621i	$-0.196800\pi$
$73$	13.9248	1.62977	0.814886	+	0.579621i	$0.196800\pi$
$74$	1.63529	0.190099
$75$	0.612127	0.0706823
$76$	−15.6455	−1.79466
$77$	−4.00000	−0.455842
$78$	0.312650	0.0354007
$79$	2.42218	0.272517	0.136258	−	0.990673i	$-0.456492\pi$
$79$	2.42218	0.272517	0.136258	+	0.990673i	$0.456492\pi$
$80$	−8.94472	−1.00005
$81$	1.00000	0.111111
$82$	1.53690	0.169723
$83$	−1.73136	−0.190041	−0.0950205	−	0.995475i	$-0.530292\pi$
$83$	−1.73136	−0.190041	−0.0950205	+	0.995475i	$0.530292\pi$
$84$	−1.87365	−0.204432
$85$	12.3127	1.33549
$86$	0.363369	0.0391831
$87$	−1.35026	−0.144763
$88$	3.21939	0.343188
$89$	1.45012	0.153713	0.0768564	−	0.997042i	$-0.475512\pi$
$89$	1.45012	0.153713	0.0768564	+	0.997042i	$0.475512\pi$
$90$	−0.459434	−0.0484286
$91$	1.53923	0.161355
$92$	0	0
$93$	5.35026	0.554796
$94$	2.26187	0.233294
$95$	−18.8872	−1.93778
$96$	2.26916	0.231595
$97$	−2.33308	−0.236889	−0.118444	−	0.992961i	$-0.537791\pi$
$97$	−2.33308	−0.236889	−0.118444	+	0.992961i	$0.537791\pi$
$98$	−1.18076	−0.119275
$99$	−4.18945	−0.421056
$100$	−1.20123	−0.120123
$101$	15.1490	1.50738	0.753692	−	0.657227i	$-0.228271\pi$
$101$	15.1490	1.50738	0.753692	+	0.657227i	$0.228271\pi$
$102$	−1.00797	−0.0998039
$103$	9.98860	0.984206	0.492103	−	0.870537i	$-0.336228\pi$
$103$	9.98860	0.984206	0.492103	+	0.870537i	$0.336228\pi$
$104$	−1.23884	−0.121478
$105$	−2.26187	−0.220735
$106$	−1.71408	−0.166486
$107$	18.2252	1.76190	0.880949	−	0.473212i	$-0.156906\pi$
$107$	18.2252	1.76190	0.880949	+	0.473212i	$0.156906\pi$
$108$	−1.96239	−0.188831
$109$	−2.51128	−0.240537	−0.120269	−	0.992741i	$-0.538376\pi$
$109$	−2.51128	−0.240537	−0.120269	+	0.992741i	$0.538376\pi$
$110$	1.92478	0.183520
$111$	8.43209	0.800339
$112$	3.60500	0.340641
$113$	11.6668	1.09752	0.548758	−	0.835981i	$-0.315101\pi$
$113$	11.6668	1.09752	0.548758	+	0.835981i	$0.315101\pi$
$114$	1.54619	0.144814
$115$	0	0
$116$	2.64974	0.246022
$117$	1.61213	0.149041
$118$	2.51388	0.231422
$119$	−4.96239	−0.454901
$120$	1.82046	0.166184
$121$	6.55149	0.595590
$122$	−1.19314	−0.108021
$123$	7.92478	0.714553
$124$	−10.4993	−0.942864
$125$	10.3948	0.929743
$126$	0.185167	0.0164960
$127$	−13.9756	−1.24013	−0.620065	−	0.784550i	$-0.712894\pi$
$127$	−13.9756	−1.24013	−0.620065	+	0.784550i	$0.712894\pi$
$128$	−5.91748	−0.523037
$129$	1.87365	0.164965
$130$	−0.740666	−0.0649607
$131$	−8.77575	−0.766741	−0.383370	−	0.923595i	$-0.625237\pi$
$131$	−8.77575	−0.766741	−0.383370	+	0.923595i	$0.625237\pi$
$132$	8.22133	0.715575
$133$	7.61213	0.660055
$134$	−0.291548	−0.0251859
$135$	−2.36899	−0.203890
$136$	3.99397	0.342480
$137$	−11.8450	−1.01198	−0.505992	−	0.862538i	$-0.668873\pi$
$137$	−11.8450	−1.01198	−0.505992	+	0.862538i	$0.668873\pi$
$138$	0	0
$139$	10.7005	0.907607	0.453803	−	0.891102i	$-0.350067\pi$
$139$	10.7005	0.907607	0.453803	+	0.891102i	$0.350067\pi$
$140$	4.43866	0.375135
$141$	11.6629	0.982194
$142$	−0.775746	−0.0650992
$143$	−6.75393	−0.564792
$144$	3.77575	0.314646
$145$	3.19876	0.265643
$146$	2.70052	0.223497
$147$	−6.08840	−0.502162
$148$	−16.5470	−1.36016
$149$	−12.7638	−1.04565	−0.522827	−	0.852439i	$-0.675123\pi$
$149$	−12.7638	−1.04565	−0.522827	+	0.852439i	$0.675123\pi$
$150$	0.118714	0.00969294
$151$	−8.62530	−0.701917	−0.350959	−	0.936391i	$-0.614144\pi$
$151$	−8.62530	−0.701917	−0.350959	+	0.936391i	$0.614144\pi$
$152$	−6.12660	−0.496933
$153$	−5.19742	−0.420186
$154$	−0.775746	−0.0625114
$155$	−12.6747	−1.01806
$156$	−3.16362	−0.253292
$157$	4.50659	0.359665	0.179833	−	0.983697i	$-0.442444\pi$
$157$	4.50659	0.359665	0.179833	+	0.983697i	$0.442444\pi$
$158$	0.469750	0.0373713
$159$	−8.83833	−0.700926
$160$	−5.37562	−0.424980
$161$	0	0
$162$	0.193937	0.0152371
$163$	7.79877	0.610847	0.305423	−	0.952217i	$-0.401202\pi$
$163$	7.79877	0.610847	0.305423	+	0.952217i	$0.401202\pi$
$164$	−15.5515	−1.21437
$165$	9.92478	0.772643
$166$	−0.335773	−0.0260611
$167$	4.77575	0.369558	0.184779	−	0.982780i	$-0.440843\pi$
$167$	4.77575	0.369558	0.184779	+	0.982780i	$0.440843\pi$
$168$	−0.733702	−0.0566063
$169$	−10.4010	−0.800081
$170$	2.38787	0.183142
$171$	7.97266	0.609684
$172$	−3.67683	−0.280355
$173$	−16.0508	−1.22032	−0.610159	−	0.792279i	$-0.708895\pi$
$173$	−16.0508	−1.22032	−0.610159	+	0.792279i	$0.708895\pi$
$174$	−0.261865	−0.0198519
$175$	0.584446	0.0441800
$176$	−15.8183	−1.19235
$177$	12.9624	0.974313
$178$	0.281232	0.0210792
$179$	−14.3634	−1.07357	−0.536787	−	0.843718i	$-0.680362\pi$
$179$	−14.3634	−1.07357	−0.536787	+	0.843718i	$0.680362\pi$
$180$	4.64888	0.346507
$181$	−9.45734	−0.702959	−0.351479	−	0.936196i	$-0.614321\pi$
$181$	−9.45734	−0.702959	−0.351479	+	0.936196i	$0.614321\pi$
$182$	0.298512	0.0221272
$183$	−6.15220	−0.454784
$184$	0	0
$185$	−19.9756	−1.46863
$186$	1.03761	0.0760814
$187$	21.7743	1.59230
$188$	−22.8872	−1.66922
$189$	0.954779	0.0694500
$190$	−3.66291	−0.265736
$191$	22.9632	1.66156	0.830779	−	0.556602i	$-0.187895\pi$
$191$	22.9632	1.66156	0.830779	+	0.556602i	$0.187895\pi$
$192$	−7.11142	−0.513222
$193$	−5.53690	−0.398555	−0.199278	−	0.979943i	$-0.563860\pi$
$193$	−5.53690	−0.398555	−0.199278	+	0.979943i	$0.563860\pi$
$194$	−0.452470	−0.0324854
$195$	−3.81912	−0.273493
$196$	11.9478	0.853414
$197$	19.2750	1.37329	0.686645	−	0.726993i	$-0.259083\pi$
$197$	19.2750	1.37329	0.686645	+	0.726993i	$0.259083\pi$
$198$	−0.812488	−0.0577410
$199$	−1.76727	−0.125278	−0.0626391	−	0.998036i	$-0.519952\pi$
$199$	−1.76727	−0.125278	−0.0626391	+	0.998036i	$0.519952\pi$
$200$	−0.470390	−0.0332616
$201$	−1.50331	−0.106036
$202$	2.93795	0.206714
$203$	−1.28920	−0.0904842
$204$	10.1994	0.714098
$205$	−18.7737	−1.31121
$206$	1.93715	0.134968
$207$	0	0
$208$	6.08698	0.422056
$209$	−33.4010	−2.31040
$210$	−0.438658	−0.0302703
$211$	−3.27504	−0.225463	−0.112731	−	0.993625i	$-0.535960\pi$
$211$	−3.27504	−0.225463	−0.112731	+	0.993625i	$0.535960\pi$
$212$	17.3442	1.19121
$213$	−4.00000	−0.274075
$214$	3.53453	0.241616
$215$	−4.43866	−0.302714
$216$	−0.768452	−0.0522865
$217$	5.10832	0.346775
$218$	−0.487030	−0.0329858
$219$	13.9248	0.940949
$220$	−19.4763	−1.31309
$221$	−8.37890	−0.563626
$222$	1.63529	0.109754
$223$	−18.5501	−1.24221	−0.621103	−	0.783729i	$-0.713315\pi$
$223$	−18.5501	−1.24221	−0.621103	+	0.783729i	$0.713315\pi$
$224$	2.16655	0.144758
$225$	0.612127	0.0408085
$226$	2.26261	0.150507
$227$	−6.09901	−0.404805	−0.202403	−	0.979302i	$-0.564875\pi$
$227$	−6.09901	−0.404805	−0.202403	+	0.979302i	$0.564875\pi$
$228$	−15.6455	−1.03615
$229$	8.16814	0.539766	0.269883	−	0.962893i	$-0.413015\pi$
$229$	8.16814	0.539766	0.269883	+	0.962893i	$0.413015\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	1.03761	0.0681225
$233$	6.77575	0.443894	0.221947	−	0.975059i	$-0.428759\pi$
$233$	6.77575	0.443894	0.221947	+	0.975059i	$0.428759\pi$
$234$	0.312650	0.0204386
$235$	−27.6294	−1.80234
$236$	−25.4372	−1.65582
$237$	2.42218	0.157338
$238$	−0.962389	−0.0623824
$239$	−5.29948	−0.342795	−0.171397	−	0.985202i	$-0.554828\pi$
$239$	−5.29948	−0.342795	−0.171397	+	0.985202i	$0.554828\pi$
$240$	−8.94472	−0.577379
$241$	9.52916	0.613827	0.306914	−	0.951737i	$-0.400704\pi$
$241$	9.52916	0.613827	0.306914	+	0.951737i	$0.400704\pi$
$242$	1.27057	0.0816756
$243$	1.00000	0.0641500
$244$	12.0730	0.772895
$245$	14.4234	0.921475
$246$	1.53690	0.0979894
$247$	12.8529	0.817813
$248$	−4.11142	−0.261075
$249$	−1.73136	−0.109720
$250$	2.01594	0.127499
$251$	16.9360	1.06899	0.534496	−	0.845171i	$-0.320502\pi$
$251$	16.9360	1.06899	0.534496	+	0.845171i	$0.320502\pi$
$252$	−1.87365	−0.118029
$253$	0	0
$254$	−2.71037	−0.170064
$255$	12.3127	0.771048
$256$	13.0752	0.817201
$257$	−14.6253	−0.912301	−0.456151	−	0.889903i	$-0.650772\pi$
$257$	−14.6253	−0.912301	−0.456151	+	0.889903i	$0.650772\pi$
$258$	0.363369	0.0226224
$259$	8.05079	0.500251
$260$	7.49459	0.464795
$261$	−1.35026	−0.0835791
$262$	−1.70194	−0.105146
$263$	−2.17351	−0.134024	−0.0670122	−	0.997752i	$-0.521347\pi$
$263$	−2.17351	−0.134024	−0.0670122	+	0.997752i	$0.521347\pi$
$264$	3.21939	0.198140
$265$	20.9380	1.28621
$266$	1.47627	0.0905159
$267$	1.45012	0.0887462
$268$	2.95009	0.180205
$269$	−7.02302	−0.428201	−0.214101	−	0.976812i	$-0.568682\pi$
$269$	−7.02302	−0.428201	−0.214101	+	0.976812i	$0.568682\pi$
$270$	−0.459434	−0.0279603
$271$	29.0494	1.76462	0.882312	−	0.470665i	$-0.155986\pi$
$271$	29.0494	1.76462	0.882312	+	0.470665i	$0.155986\pi$
$272$	−19.6241	−1.18989
$273$	1.53923	0.0931582
$274$	−2.29717	−0.138777
$275$	−2.56447	−0.154644
$276$	0	0
$277$	−25.2506	−1.51716	−0.758581	−	0.651579i	$-0.774107\pi$
$277$	−25.2506	−1.51716	−0.758581	+	0.651579i	$0.774107\pi$
$278$	2.07522	0.124464
$279$	5.35026	0.320312
$280$	1.73813	0.103873
$281$	22.9805	1.37090	0.685450	−	0.728120i	$-0.259606\pi$
$281$	22.9805	1.37090	0.685450	+	0.728120i	$0.259606\pi$
$282$	2.26187	0.134692
$283$	8.99791	0.534870	0.267435	−	0.963576i	$-0.413824\pi$
$283$	8.99791	0.534870	0.267435	+	0.963576i	$0.413824\pi$
$284$	7.84955	0.465785
$285$	−18.8872	−1.11878
$286$	−1.30983	−0.0774521
$287$	7.56641	0.446631
$288$	2.26916	0.133711
$289$	10.0132	0.589010
$290$	0.620357	0.0364286
$291$	−2.33308	−0.136768
$292$	−27.3258	−1.59912
$293$	−3.18148	−0.185864	−0.0929320	−	0.995672i	$-0.529624\pi$
$293$	−3.18148	−0.185864	−0.0929320	+	0.995672i	$0.529624\pi$
$294$	−1.18076	−0.0688635
$295$	−30.7078	−1.78788
$296$	−6.47966	−0.376622
$297$	−4.18945	−0.243097
$298$	−2.47537	−0.143395
$299$	0	0
$300$	−1.20123	−0.0693531
$301$	1.78892	0.103112
$302$	−1.67276	−0.0962566
$303$	15.1490	0.870289
$304$	30.1027	1.72651
$305$	14.5745	0.834534
$306$	−1.00797	−0.0576218
$307$	14.9018	0.850488	0.425244	−	0.905079i	$-0.360188\pi$
$307$	14.9018	0.850488	0.425244	+	0.905079i	$0.360188\pi$
$308$	7.84955	0.447270
$309$	9.98860	0.568231
$310$	−2.45809	−0.139610
$311$	2.36344	0.134018	0.0670091	−	0.997752i	$-0.478654\pi$
$311$	2.36344	0.134018	0.0670091	+	0.997752i	$0.478654\pi$
$312$	−1.23884	−0.0701356
$313$	−18.7205	−1.05815	−0.529074	−	0.848576i	$-0.677461\pi$
$313$	−18.7205	−1.05815	−0.529074	+	0.848576i	$0.677461\pi$
$314$	0.873993	0.0493223
$315$	−2.26187	−0.127442
$316$	−4.75326	−0.267392
$317$	−1.97556	−0.110959	−0.0554793	−	0.998460i	$-0.517669\pi$
$317$	−1.97556	−0.110959	−0.0554793	+	0.998460i	$0.517669\pi$
$318$	−1.71408	−0.0961206
$319$	5.65685	0.316723
$320$	16.8469	0.941770
$321$	18.2252	1.01723
$322$	0	0
$323$	−41.4372	−2.30563
$324$	−1.96239	−0.109022
$325$	0.986826	0.0547393
$326$	1.51247	0.0837678
$327$	−2.51128	−0.138874
$328$	−6.08981	−0.336254
$329$	11.1355	0.613920
$330$	1.92478	0.105955
$331$	−24.4241	−1.34247	−0.671234	−	0.741245i	$-0.734235\pi$
$331$	−24.4241	−1.34247	−0.671234	+	0.741245i	$0.734235\pi$
$332$	3.39759	0.186467
$333$	8.43209	0.462076
$334$	0.926192	0.0506790
$335$	3.56134	0.194577
$336$	3.60500	0.196669
$337$	−7.98994	−0.435240	−0.217620	−	0.976034i	$-0.569829\pi$
$337$	−7.98994	−0.435240	−0.217620	+	0.976034i	$0.569829\pi$
$338$	−2.01714	−0.109718
$339$	11.6668	0.633652
$340$	−24.1622	−1.31038
$341$	−22.4147	−1.21382
$342$	1.54619	0.0836084
$343$	−12.4965	−0.674749
$344$	−1.43981	−0.0776293
$345$	0	0
$346$	−3.11283	−0.167347
$347$	−8.18664	−0.439482	−0.219741	−	0.975558i	$-0.570521\pi$
$347$	−8.18664	−0.439482	−0.219741	+	0.975558i	$0.570521\pi$
$348$	2.64974	0.142041
$349$	8.63989	0.462483	0.231241	−	0.972896i	$-0.425721\pi$
$349$	8.63989	0.462483	0.231241	+	0.972896i	$0.425721\pi$
$350$	0.113345	0.00605857
$351$	1.61213	0.0860490
$352$	−9.50653	−0.506700
$353$	1.19982	0.0638598	0.0319299	−	0.999490i	$-0.489835\pi$
$353$	1.19982	0.0638598	0.0319299	+	0.999490i	$0.489835\pi$
$354$	2.51388	0.133611
$355$	9.47597	0.502932
$356$	−2.84571	−0.150822
$357$	−4.96239	−0.262637
$358$	−2.78560	−0.147223
$359$	−0.548535	−0.0289506	−0.0144753	−	0.999895i	$-0.504608\pi$
$359$	−0.548535	−0.0289506	−0.0144753	+	0.999895i	$0.504608\pi$
$360$	1.82046	0.0959465
$361$	44.5633	2.34543
$362$	−1.83412	−0.0963994
$363$	6.55149	0.343864
$364$	−3.02056	−0.158320
$365$	−32.9877	−1.72665
$366$	−1.19314	−0.0623662
$367$	5.42881	0.283382	0.141691	−	0.989911i	$-0.454746\pi$
$367$	5.42881	0.283382	0.141691	+	0.989911i	$0.454746\pi$
$368$	0	0
$369$	7.92478	0.412547
$370$	−3.87399	−0.201399
$371$	−8.43866	−0.438113
$372$	−10.4993	−0.544363
$373$	21.4772	1.11204	0.556022	−	0.831167i	$-0.312327\pi$
$373$	21.4772	1.11204	0.556022	+	0.831167i	$0.312327\pi$
$374$	4.22284	0.218358
$375$	10.3948	0.536787
$376$	−8.96239	−0.462200
$377$	−2.17679	−0.112111
$378$	0.185167	0.00952394
$379$	31.8547	1.63627	0.818133	−	0.575028i	$-0.195009\pi$
$379$	31.8547	1.63627	0.818133	+	0.575028i	$0.195009\pi$
$380$	37.0640	1.90134
$381$	−13.9756	−0.715990
$382$	4.45340	0.227856
$383$	34.0129	1.73798	0.868990	−	0.494829i	$-0.164769\pi$
$383$	34.0129	1.73798	0.868990	+	0.494829i	$0.164769\pi$
$384$	−5.91748	−0.301975
$385$	9.47597	0.482940
$386$	−1.07381	−0.0546554
$387$	1.87365	0.0952429
$388$	4.57841	0.232434
$389$	−20.1520	−1.02175	−0.510875	−	0.859655i	$-0.670678\pi$
$389$	−20.1520	−1.02175	−0.510875	+	0.859655i	$0.670678\pi$
$390$	−0.740666	−0.0375051
$391$	0	0
$392$	4.67864	0.236307
$393$	−8.77575	−0.442678
$394$	3.73813	0.188325
$395$	−5.73813	−0.288717
$396$	8.22133	0.413137
$397$	22.6253	1.13553	0.567766	−	0.823190i	$-0.307808\pi$
$397$	22.6253	1.13553	0.567766	+	0.823190i	$0.307808\pi$
$398$	−0.342738	−0.0171799
$399$	7.61213	0.381083
$400$	2.31124	0.115562
$401$	−2.47537	−0.123614	−0.0618071	−	0.998088i	$-0.519686\pi$
$401$	−2.47537	−0.123614	−0.0618071	+	0.998088i	$0.519686\pi$
$402$	−0.291548	−0.0145411
$403$	8.62530	0.429657
$404$	−29.7283	−1.47904
$405$	−2.36899	−0.117716
$406$	−0.250023	−0.0124085
$407$	−35.3258	−1.75104
$408$	3.99397	0.197731
$409$	−28.0870	−1.38881	−0.694406	−	0.719583i	$-0.744333\pi$
$409$	−28.0870	−1.38881	−0.694406	+	0.719583i	$0.744333\pi$
$410$	−3.64091	−0.179812
$411$	−11.8450	−0.584269
$412$	−19.6015	−0.965697
$413$	12.3762	0.608994
$414$	0	0
$415$	4.10157	0.201338
$416$	3.65817	0.179357
$417$	10.7005	0.524007
$418$	−6.47768	−0.316834
$419$	−25.4558	−1.24360	−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558	−1.24360	−0.621800	+	0.783176i	$0.713598\pi$
$420$	4.43866	0.216584
$421$	33.7470	1.64473	0.822364	−	0.568962i	$-0.192655\pi$
$421$	33.7470	1.64473	0.822364	+	0.568962i	$0.192655\pi$
$422$	−0.635150	−0.0309186
$423$	11.6629	0.567070
$424$	6.79184	0.329841
$425$	−3.18148	−0.154324
$426$	−0.775746	−0.0375850
$427$	−5.87399	−0.284262
$428$	−35.7649	−1.72876
$429$	−6.75393	−0.326083
$430$	−0.860818	−0.0415123
$431$	10.4806	0.504832	0.252416	−	0.967619i	$-0.418775\pi$
$431$	10.4806	0.504832	0.252416	+	0.967619i	$0.418775\pi$
$432$	3.77575	0.181661
$433$	5.86762	0.281980	0.140990	−	0.990011i	$-0.454972\pi$
$433$	5.86762	0.281980	0.140990	+	0.990011i	$0.454972\pi$
$434$	0.990690	0.0475546
$435$	3.19876	0.153369
$436$	4.92812	0.236014
$437$	0	0
$438$	2.70052	0.129036
$439$	−0.373285	−0.0178159	−0.00890795	−	0.999960i	$-0.502836\pi$
$439$	−0.373285	−0.0178159	−0.00890795	+	0.999960i	$0.502836\pi$
$440$	−7.62672	−0.363589
$441$	−6.08840	−0.289924
$442$	−1.62498	−0.0772922
$443$	−12.7757	−0.606994	−0.303497	−	0.952832i	$-0.598154\pi$
$443$	−12.7757	−0.606994	−0.303497	+	0.952832i	$0.598154\pi$
$444$	−16.5470	−0.785288
$445$	−3.43533	−0.162850
$446$	−3.59754	−0.170348
$447$	−12.7638	−0.603709
$448$	−6.78984	−0.320790
$449$	−30.1016	−1.42058	−0.710290	−	0.703909i	$-0.751436\pi$
$449$	−30.1016	−1.42058	−0.710290	+	0.703909i	$0.751436\pi$
$450$	0.118714	0.00559622
$451$	−33.2005	−1.56335
$452$	−22.8947	−1.07688
$453$	−8.62530	−0.405252
$454$	−1.18282	−0.0555125
$455$	−3.64641	−0.170947
$456$	−6.12660	−0.286905
$457$	−25.4027	−1.18829	−0.594143	−	0.804359i	$-0.702509\pi$
$457$	−25.4027	−1.18829	−0.594143	+	0.804359i	$0.702509\pi$
$458$	1.58410	0.0740202
$459$	−5.19742	−0.242595
$460$	0	0
$461$	29.3014	1.36470	0.682351	−	0.731025i	$-0.260958\pi$
$461$	29.3014	1.36470	0.682351	+	0.731025i	$0.260958\pi$
$462$	−0.775746	−0.0360910
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−5.09825	−0.236680
$465$	−12.6747	−0.587777
$466$	1.31406	0.0608729
$467$	−2.54384	−0.117715	−0.0588575	−	0.998266i	$-0.518746\pi$
$467$	−2.54384	−0.117715	−0.0588575	+	0.998266i	$0.518746\pi$
$468$	−3.16362	−0.146238
$469$	−1.43533	−0.0662775
$470$	−5.35834	−0.247162
$471$	4.50659	0.207653
$472$	−9.96097	−0.458491
$473$	−7.84955	−0.360923
$474$	0.469750	0.0215763
$475$	4.88028	0.223922
$476$	9.73813	0.446347
$477$	−8.83833	−0.404680
$478$	−1.02776	−0.0470088
$479$	−7.75854	−0.354497	−0.177248	−	0.984166i	$-0.556720\pi$
$479$	−7.75854	−0.354497	−0.177248	+	0.984166i	$0.556720\pi$
$480$	−5.37562	−0.245362
$481$	13.5936	0.619815
$482$	1.84805	0.0841765
$483$	0	0
$484$	−12.8566	−0.584390
$485$	5.52705	0.250971
$486$	0.193937	0.00879714
$487$	27.2243	1.23365	0.616824	−	0.787101i	$-0.288419\pi$
$487$	27.2243	1.23365	0.616824	+	0.787101i	$0.288419\pi$
$488$	4.72767	0.214012
$489$	7.79877	0.352673
$490$	2.79722	0.126365
$491$	−6.26187	−0.282594	−0.141297	−	0.989967i	$-0.545127\pi$
$491$	−6.26187	−0.282594	−0.141297	+	0.989967i	$0.545127\pi$
$492$	−15.5515	−0.701115
$493$	7.01788	0.316069
$494$	2.49265	0.112150
$495$	9.92478	0.446086
$496$	20.2012	0.907062
$497$	−3.81912	−0.171311
$498$	−0.335773	−0.0150464
$499$	−6.64974	−0.297683	−0.148842	−	0.988861i	$-0.547554\pi$
$499$	−6.64974	−0.297683	−0.148842	+	0.988861i	$0.547554\pi$
$500$	−20.3987	−0.912258
$501$	4.77575	0.213365
$502$	3.28451	0.146595
$503$	27.8076	1.23988	0.619939	−	0.784650i	$-0.287157\pi$
$503$	27.8076	1.23988	0.619939	+	0.784650i	$0.287157\pi$
$504$	−0.733702	−0.0326817
$505$	−35.8879	−1.59699
$506$	0	0
$507$	−10.4010	−0.461927
$508$	27.4255	1.21681
$509$	32.6761	1.44834	0.724171	−	0.689620i	$-0.242223\pi$
$509$	32.6761	1.44834	0.724171	+	0.689620i	$0.242223\pi$
$510$	2.38787	0.105737
$511$	13.2951	0.588140
$512$	14.3707	0.635103
$513$	7.97266	0.352001
$514$	−2.83638	−0.125107
$515$	−23.6629	−1.04271
$516$	−3.67683	−0.161863
$517$	−48.8612	−2.14891
$518$	1.56134	0.0686014
$519$	−16.0508	−0.704551
$520$	2.93481	0.128700
$521$	1.48468	0.0650452	0.0325226	−	0.999471i	$-0.489646\pi$
$521$	1.48468	0.0650452	0.0325226	+	0.999471i	$0.489646\pi$
$522$	−0.261865	−0.0114615
$523$	−3.34105	−0.146094	−0.0730470	−	0.997328i	$-0.523272\pi$
$523$	−3.34105	−0.146094	−0.0730470	+	0.997328i	$0.523272\pi$
$524$	17.2214	0.752321
$525$	0.584446	0.0255073
$526$	−0.421523	−0.0183793
$527$	−27.8076	−1.21132
$528$	−15.8183	−0.688403
$529$	0	0
$530$	4.06063	0.176383
$531$	12.9624	0.562520
$532$	−14.9380	−0.647642
$533$	12.7757	0.553379
$534$	0.281232	0.0121701
$535$	−43.1754	−1.86663
$536$	1.15523	0.0498981
$537$	−14.3634	−0.619828
$538$	−1.36202	−0.0587209
$539$	25.5070	1.09867
$540$	4.64888	0.200056
$541$	−1.28489	−0.0552417	−0.0276208	−	0.999618i	$-0.508793\pi$
$541$	−1.28489	−0.0552417	−0.0276208	+	0.999618i	$0.508793\pi$
$542$	5.63374	0.241990
$543$	−9.45734	−0.405853
$544$	−11.7938	−0.505654
$545$	5.94921	0.254836
$546$	0.298512	0.0127751
$547$	−13.4518	−0.575159	−0.287579	−	0.957757i	$-0.592851\pi$
$547$	−13.4518	−0.575159	−0.287579	+	0.957757i	$0.592851\pi$
$548$	23.2444	0.992953
$549$	−6.15220	−0.262569
$550$	−0.497345	−0.0212069
$551$	−10.7652	−0.458612
$552$	0	0
$553$	2.31265	0.0983439
$554$	−4.89701	−0.208054
$555$	−19.9756	−0.847915
$556$	−20.9986	−0.890538
$557$	21.4273	0.907905	0.453952	−	0.891026i	$-0.350014\pi$
$557$	21.4273	0.907905	0.453952	+	0.891026i	$0.350014\pi$
$558$	1.03761	0.0439256
$559$	3.02056	0.127756
$560$	−8.54023	−0.360891
$561$	21.7743	0.919313
$562$	4.45675	0.187997
$563$	−1.53923	−0.0648706	−0.0324353	−	0.999474i	$-0.510326\pi$
$563$	−1.53923	−0.0648706	−0.0324353	+	0.999474i	$0.510326\pi$
$564$	−22.8872	−0.963724
$565$	−27.6385	−1.16276
$566$	1.74502	0.0733488
$567$	0.954779	0.0400970
$568$	3.07381	0.128974
$569$	4.49131	0.188286	0.0941428	−	0.995559i	$-0.469989\pi$
$569$	4.49131	0.188286	0.0941428	+	0.995559i	$0.469989\pi$
$570$	−3.66291	−0.153423
$571$	34.0489	1.42490	0.712450	−	0.701723i	$-0.247585\pi$
$571$	34.0489	1.42490	0.712450	+	0.701723i	$0.247585\pi$
$572$	13.2538	0.554170
$573$	22.9632	0.959301
$574$	1.46740	0.0612483
$575$	0	0
$576$	−7.11142	−0.296309
$577$	−41.3865	−1.72294	−0.861470	−	0.507808i	$-0.830456\pi$
$577$	−41.3865	−1.72294	−0.861470	+	0.507808i	$0.830456\pi$
$578$	1.94192	0.0807732
$579$	−5.53690	−0.230106
$580$	−6.27721	−0.260647
$581$	−1.65306	−0.0685806
$582$	−0.452470	−0.0187555
$583$	37.0278	1.53353
$584$	−10.7005	−0.442791
$585$	−3.81912	−0.157901
$586$	−0.617005	−0.0254883
$587$	−13.2995	−0.548928	−0.274464	−	0.961597i	$-0.588500\pi$
$587$	−13.2995	−0.548928	−0.274464	+	0.961597i	$0.588500\pi$
$588$	11.9478	0.492719
$589$	42.6558	1.75760
$590$	−5.95537	−0.245179
$591$	19.2750	0.792869
$592$	31.8374	1.30851
$593$	40.6516	1.66936	0.834682	−	0.550733i	$-0.185652\pi$
$593$	40.6516	1.66936	0.834682	+	0.550733i	$0.185652\pi$
$594$	−0.812488	−0.0333368
$595$	11.7559	0.481943
$596$	25.0476	1.02599
$597$	−1.76727	−0.0723294
$598$	0	0
$599$	25.5223	1.04281	0.521407	−	0.853308i	$-0.325407\pi$
$599$	25.5223	1.04281	0.521407	+	0.853308i	$0.325407\pi$
$600$	−0.470390	−0.0192036
$601$	−9.68735	−0.395155	−0.197578	−	0.980287i	$-0.563307\pi$
$601$	−9.68735	−0.395155	−0.197578	+	0.980287i	$0.563307\pi$
$602$	0.346937	0.0141401
$603$	−1.50331	−0.0612197
$604$	16.9262	0.688717
$605$	−15.5204	−0.630996
$606$	2.93795	0.119346
$607$	2.59895	0.105488	0.0527441	−	0.998608i	$-0.483203\pi$
$607$	2.59895	0.105488	0.0527441	+	0.998608i	$0.483203\pi$
$608$	18.0912	0.733696
$609$	−1.28920	−0.0522411
$610$	2.82653	0.114443
$611$	18.8021	0.760651
$612$	10.1994	0.412285
$613$	22.9306	0.926159	0.463080	−	0.886317i	$-0.346744\pi$
$613$	22.9306	0.926159	0.463080	+	0.886317i	$0.346744\pi$
$614$	2.89000	0.116631
$615$	−18.7737	−0.757030
$616$	3.07381	0.123847
$617$	13.8609	0.558019	0.279009	−	0.960288i	$-0.409994\pi$
$617$	13.8609	0.558019	0.279009	+	0.960288i	$0.409994\pi$
$618$	1.93715	0.0779238
$619$	26.1979	1.05298	0.526490	−	0.850181i	$-0.323508\pi$
$619$	26.1979	1.05298	0.526490	+	0.850181i	$0.323508\pi$
$620$	24.8727	0.998914
$621$	0	0
$622$	0.458357	0.0183784
$623$	1.38455	0.0554708
$624$	6.08698	0.243674
$625$	−27.6859	−1.10744
$626$	−3.63060	−0.145108
$627$	−33.4010	−1.33391
$628$	−8.84369	−0.352902
$629$	−43.8251	−1.74742
$630$	−0.438658	−0.0174766
$631$	−2.12367	−0.0845420	−0.0422710	−	0.999106i	$-0.513459\pi$
$631$	−2.12367	−0.0845420	−0.0422710	+	0.999106i	$0.513459\pi$
$632$	−1.86133	−0.0740398
$633$	−3.27504	−0.130171
$634$	−0.383134	−0.0152162
$635$	33.1080	1.31385
$636$	17.3442	0.687744
$637$	−9.81527	−0.388895
$638$	1.09707	0.0434335
$639$	−4.00000	−0.158238
$640$	14.0185	0.554129
$641$	0.424874	0.0167815	0.00839076	−	0.999965i	$-0.497329\pi$
$641$	0.424874	0.0167815	0.00839076	+	0.999965i	$0.497329\pi$
$642$	3.53453	0.139497
$643$	4.52387	0.178404	0.0892021	−	0.996014i	$-0.471568\pi$
$643$	4.52387	0.178404	0.0892021	+	0.996014i	$0.471568\pi$
$644$	0	0
$645$	−4.43866	−0.174772
$646$	−8.03620	−0.316180
$647$	−6.78560	−0.266769	−0.133385	−	0.991064i	$-0.542585\pi$
$647$	−6.78560	−0.266769	−0.133385	+	0.991064i	$0.542585\pi$
$648$	−0.768452	−0.0301876
$649$	−54.3053	−2.13167
$650$	0.191382	0.00750661
$651$	5.10832	0.200211
$652$	−15.3042	−0.599359
$653$	15.0278	0.588082	0.294041	−	0.955793i	$-0.405000\pi$
$653$	15.0278	0.588082	0.294041	+	0.955793i	$0.405000\pi$
$654$	−0.487030	−0.0190444
$655$	20.7897	0.812320
$656$	29.9219	1.16826
$657$	13.9248	0.543257
$658$	2.15958	0.0841893
$659$	−33.9066	−1.32081	−0.660406	−	0.750909i	$-0.729616\pi$
$659$	−33.9066	−1.32081	−0.660406	+	0.750909i	$0.729616\pi$
$660$	−19.4763	−0.758113
$661$	2.88162	0.112082	0.0560410	−	0.998428i	$-0.482152\pi$
$661$	2.88162	0.112082	0.0560410	+	0.998428i	$0.482152\pi$
$662$	−4.73672	−0.184098
$663$	−8.37890	−0.325410
$664$	1.33046	0.0516320
$665$	−18.0331	−0.699293
$666$	1.63529	0.0633662
$667$	0	0
$668$	−9.37187	−0.362609
$669$	−18.5501	−0.717187
$670$	0.690674	0.0266831
$671$	25.7743	0.995007
$672$	2.16655	0.0835763
$673$	−14.7612	−0.569001	−0.284500	−	0.958676i	$-0.591828\pi$
$673$	−14.7612	−0.569001	−0.284500	+	0.958676i	$0.591828\pi$
$674$	−1.54954	−0.0596861
$675$	0.612127	0.0235608
$676$	20.4109	0.785034
$677$	−43.5919	−1.67537	−0.837687	−	0.546150i	$-0.816093\pi$
$677$	−43.5919	−1.67537	−0.837687	+	0.546150i	$0.816093\pi$
$678$	2.26261	0.0868951
$679$	−2.22758	−0.0854866
$680$	−9.46168	−0.362839
$681$	−6.09901	−0.233715
$682$	−4.34702	−0.166456
$683$	11.1754	0.427614	0.213807	−	0.976876i	$-0.431414\pi$
$683$	11.1754	0.427614	0.213807	+	0.976876i	$0.431414\pi$
$684$	−15.6455	−0.598219
$685$	28.0606	1.07214
$686$	−2.42353	−0.0925310
$687$	8.16814	0.311634
$688$	7.07442	0.269710
$689$	−14.2485	−0.542825
$690$	0	0
$691$	−0.574515	−0.0218556	−0.0109278	−	0.999940i	$-0.503478\pi$
$691$	−0.574515	−0.0218556	−0.0109278	+	0.999940i	$0.503478\pi$
$692$	31.4979	1.19737
$693$	−4.00000	−0.151947
$694$	−1.58769	−0.0602679
$695$	−25.3495	−0.961560
$696$	1.03761	0.0393306
$697$	−41.1884	−1.56012
$698$	1.67559	0.0634220
$699$	6.77575	0.256282
$700$	−1.14691	−0.0433491
$701$	−9.15748	−0.345873	−0.172937	−	0.984933i	$-0.555326\pi$
$701$	−9.15748	−0.345873	−0.172937	+	0.984933i	$0.555326\pi$
$702$	0.312650	0.0118002
$703$	67.2262	2.53548
$704$	29.7929	1.12286
$705$	−27.6294	−1.04058
$706$	0.232688	0.00875734
$707$	14.4640	0.543974
$708$	−25.4372	−0.955990
$709$	−26.7637	−1.00513	−0.502565	−	0.864539i	$-0.667610\pi$
$709$	−26.7637	−1.00513	−0.502565	+	0.864539i	$0.667610\pi$
$710$	1.83774	0.0689691
$711$	2.42218	0.0908390
$712$	−1.11435	−0.0417621
$713$	0	0
$714$	−0.962389	−0.0360165
$715$	16.0000	0.598366
$716$	28.1866	1.05338
$717$	−5.29948	−0.197913
$718$	−0.106381	−0.00397011
$719$	15.6629	0.584128	0.292064	−	0.956399i	$-0.405658\pi$
$719$	15.6629	0.584128	0.292064	+	0.956399i	$0.405658\pi$
$720$	−8.94472	−0.333350
$721$	9.53690	0.355173
$722$	8.64244	0.321638
$723$	9.52916	0.354393
$724$	18.5590	0.689739
$725$	−0.826531	−0.0306966
$726$	1.27057	0.0471554
$727$	−14.0923	−0.522654	−0.261327	−	0.965250i	$-0.584160\pi$
$727$	−14.0923	−0.522654	−0.261327	+	0.965250i	$0.584160\pi$
$728$	−1.18282	−0.0438383
$729$	1.00000	0.0370370
$730$	−6.39752	−0.236783
$731$	−9.73813	−0.360178
$732$	12.0730	0.446231
$733$	7.15682	0.264343	0.132172	−	0.991227i	$-0.457805\pi$
$733$	7.15682	0.264343	0.132172	+	0.991227i	$0.457805\pi$
$734$	1.05285	0.0388612
$735$	14.4234	0.532014
$736$	0	0
$737$	6.29806	0.231992
$738$	1.53690	0.0565742
$739$	−12.6253	−0.464429	−0.232215	−	0.972665i	$-0.574597\pi$
$739$	−12.6253	−0.464429	−0.232215	+	0.972665i	$0.574597\pi$
$740$	39.1998	1.44101
$741$	12.8529	0.472164
$742$	−1.63656	−0.0600802
$743$	−41.2948	−1.51496	−0.757479	−	0.652859i	$-0.773569\pi$
$743$	−41.2948	−1.51496	−0.757479	+	0.652859i	$0.773569\pi$
$744$	−4.11142	−0.150732
$745$	30.2374	1.10781
$746$	4.16521	0.152499
$747$	−1.73136	−0.0633470
$748$	−42.7297	−1.56235
$749$	17.4010	0.635820
$750$	2.01594	0.0736117
$751$	−18.8097	−0.686374	−0.343187	−	0.939267i	$-0.611506\pi$
$751$	−18.8097	−0.686374	−0.343187	+	0.939267i	$0.611506\pi$
$752$	44.0362	1.60583
$753$	16.9360	0.617182
$754$	−0.422160	−0.0153742
$755$	20.4333	0.743643
$756$	−1.87365	−0.0681439
$757$	31.1313	1.13149	0.565744	−	0.824581i	$-0.308589\pi$
$757$	31.1313	1.13149	0.565744	+	0.824581i	$0.308589\pi$
$758$	6.17779	0.224388
$759$	0	0
$760$	14.5139	0.526474
$761$	31.9003	1.15639	0.578193	−	0.815900i	$-0.303758\pi$
$761$	31.9003	1.15639	0.578193	+	0.815900i	$0.303758\pi$
$762$	−2.71037	−0.0981864
$763$	−2.39772	−0.0868034
$764$	−45.0627	−1.63031
$765$	12.3127	0.445165
$766$	6.59635	0.238336
$767$	20.8970	0.754547
$768$	13.0752	0.471811
$769$	−36.0475	−1.29991	−0.649953	−	0.759974i	$-0.725212\pi$
$769$	−36.0475	−1.29991	−0.649953	+	0.759974i	$0.725212\pi$
$770$	1.83774	0.0662275
$771$	−14.6253	−0.526717
$772$	10.8656	0.391060
$773$	26.0589	0.937274	0.468637	−	0.883391i	$-0.344745\pi$
$773$	26.0589	0.937274	0.468637	+	0.883391i	$0.344745\pi$
$774$	0.363369	0.0130610
$775$	3.27504	0.117643
$776$	1.79286	0.0643600
$777$	8.05079	0.288820
$778$	−3.90822	−0.140116
$779$	63.1815	2.26371
$780$	7.49459	0.268349
$781$	16.7578	0.599641
$782$	0	0
$783$	−1.35026	−0.0482544
$784$	−22.9882	−0.821009
$785$	−10.6761	−0.381046
$786$	−1.70194	−0.0607061
$787$	26.5543	0.946557	0.473279	−	0.880913i	$-0.343070\pi$
$787$	26.5543	0.946557	0.473279	+	0.880913i	$0.343070\pi$
$788$	−37.8251	−1.34746
$789$	−2.17351	−0.0773790
$790$	−1.11283	−0.0395929
$791$	11.1392	0.396064
$792$	3.21939	0.114396
$793$	−9.91813	−0.352203
$794$	4.38787	0.155720
$795$	20.9380	0.742593
$796$	3.46806	0.122922
$797$	38.3633	1.35890	0.679449	−	0.733723i	$-0.262219\pi$
$797$	38.3633	1.35890	0.679449	+	0.733723i	$0.262219\pi$
$798$	1.47627	0.0522594
$799$	−60.6171	−2.14448
$800$	1.38901	0.0491090
$801$	1.45012	0.0512376
$802$	−0.480066	−0.0169517
$803$	−58.3372	−2.05867
$804$	2.95009	0.104042
$805$	0	0
$806$	1.67276	0.0589205
$807$	−7.02302	−0.247222
$808$	−11.6413	−0.409540
$809$	17.5731	0.617837	0.308919	−	0.951088i	$-0.400033\pi$
$809$	17.5731	0.617837	0.308919	+	0.951088i	$0.400033\pi$
$810$	−0.459434	−0.0161429
$811$	−4.09966	−0.143959	−0.0719793	−	0.997406i	$-0.522932\pi$
$811$	−4.09966	−0.143959	−0.0719793	+	0.997406i	$0.522932\pi$
$812$	2.52992	0.0887826
$813$	29.0494	1.01881
$814$	−6.85097	−0.240126
$815$	−18.4752	−0.647159
$816$	−19.6241	−0.686982
$817$	14.9380	0.522613
$818$	−5.44709	−0.190453
$819$	1.53923	0.0537849
$820$	36.8414	1.28656
$821$	22.7466	0.793861	0.396930	−	0.917849i	$-0.370075\pi$
$821$	22.7466	0.793861	0.396930	+	0.917849i	$0.370075\pi$
$822$	−2.29717	−0.0801231
$823$	−13.7235	−0.478373	−0.239186	−	0.970974i	$-0.576881\pi$
$823$	−13.7235	−0.478373	−0.239186	+	0.970974i	$0.576881\pi$
$824$	−7.67576	−0.267398
$825$	−2.56447	−0.0892836
$826$	2.40020	0.0835137
$827$	−3.99732	−0.139000	−0.0695002	−	0.997582i	$-0.522140\pi$
$827$	−3.99732	−0.139000	−0.0695002	+	0.997582i	$0.522140\pi$
$828$	0	0
$829$	55.4617	1.92626	0.963132	−	0.269030i	$-0.0867030\pi$
$829$	55.4617	1.92626	0.963132	+	0.269030i	$0.0867030\pi$
$830$	0.795444	0.0276103
$831$	−25.2506	−0.875934
$832$	−11.4645	−0.397460
$833$	31.6440	1.09640
$834$	2.07522	0.0718591
$835$	−11.3137	−0.391527
$836$	65.5458	2.26695
$837$	5.35026	0.184932
$838$	−4.93682	−0.170540
$839$	11.8622	0.409530	0.204765	−	0.978811i	$-0.434357\pi$
$839$	11.8622	0.409530	0.204765	+	0.978811i	$0.434357\pi$
$840$	1.73813	0.0599714
$841$	−27.1768	−0.937131
$842$	6.54478	0.225548
$843$	22.9805	0.791489
$844$	6.42690	0.221223
$845$	24.6400	0.847642
$846$	2.26187	0.0777645
$847$	6.25523	0.214932
$848$	−33.3713	−1.14598
$849$	8.99791	0.308807
$850$	−0.617005	−0.0211631
$851$	0	0
$852$	7.84955	0.268921
$853$	30.4749	1.04344	0.521720	−	0.853117i	$-0.325291\pi$
$853$	30.4749	1.04344	0.521720	+	0.853117i	$0.325291\pi$
$854$	−1.13918	−0.0389820
$855$	−18.8872	−0.645927
$856$	−14.0052	−0.478688
$857$	49.1754	1.67980	0.839899	−	0.542742i	$-0.182614\pi$
$857$	49.1754	1.67980	0.839899	+	0.542742i	$0.182614\pi$
$858$	−1.30983	−0.0447170
$859$	−15.4763	−0.528044	−0.264022	−	0.964517i	$-0.585049\pi$
$859$	−15.4763	−0.528044	−0.264022	+	0.964517i	$0.585049\pi$
$860$	8.71037	0.297021
$861$	7.56641	0.257863
$862$	2.03257	0.0692296
$863$	−3.81336	−0.129808	−0.0649041	−	0.997892i	$-0.520674\pi$
$863$	−3.81336	−0.129808	−0.0649041	+	0.997892i	$0.520674\pi$
$864$	2.26916	0.0771984
$865$	38.0242	1.29286
$866$	1.13795	0.0386690
$867$	10.0132	0.340065
$868$	−10.0245	−0.340254
$869$	−10.1476	−0.344234
$870$	0.620357	0.0210321
$871$	−2.42353	−0.0821183
$872$	1.92980	0.0653513
$873$	−2.33308	−0.0789629
$874$	0	0
$875$	9.92478	0.335519
$876$	−27.3258	−0.923254
$877$	3.62672	0.122465	0.0612327	−	0.998124i	$-0.480497\pi$
$877$	3.62672	0.122465	0.0612327	+	0.998124i	$0.480497\pi$
$878$	−0.0723936	−0.00244316
$879$	−3.18148	−0.107309
$880$	37.4734	1.26323
$881$	−30.6533	−1.03273	−0.516367	−	0.856367i	$-0.672716\pi$
$881$	−30.6533	−1.03273	−0.516367	+	0.856367i	$0.672716\pi$
$882$	−1.18076	−0.0397583
$883$	41.0494	1.38142	0.690711	−	0.723131i	$-0.257298\pi$
$883$	41.0494	1.38142	0.690711	+	0.723131i	$0.257298\pi$
$884$	16.4427	0.553026
$885$	−30.7078	−1.03223
$886$	−2.47768	−0.0832394
$887$	−33.8397	−1.13623	−0.568113	−	0.822951i	$-0.692326\pi$
$887$	−33.8397	−1.13623	−0.568113	+	0.822951i	$0.692326\pi$
$888$	−6.47966	−0.217443
$889$	−13.3436	−0.447529
$890$	−0.666237	−0.0223323
$891$	−4.18945	−0.140352
$892$	36.4025	1.21884
$893$	92.9844	3.11160
$894$	−2.47537	−0.0827889
$895$	34.0269	1.13739
$896$	−5.64989	−0.188750
$897$	0	0
$898$	−5.83780	−0.194810
$899$	−7.22425	−0.240942
$900$	−1.20123	−0.0400410
$901$	45.9365	1.53037
$902$	−6.43878	−0.214388
$903$	1.78892	0.0595315
$904$	−8.96535	−0.298183
$905$	22.4044	0.744747
$906$	−1.67276	−0.0555738
$907$	−18.4739	−0.613415	−0.306708	−	0.951804i	$-0.599227\pi$
$907$	−18.4739	−0.613415	−0.306708	+	0.951804i	$0.599227\pi$
$908$	11.9686	0.397193
$909$	15.1490	0.502462
$910$	−0.707173	−0.0234426
$911$	−1.36102	−0.0450927	−0.0225464	−	0.999746i	$-0.507177\pi$
$911$	−1.36102	−0.0450927	−0.0225464	+	0.999746i	$0.507177\pi$
$912$	30.1027	0.996801
$913$	7.25343	0.240054
$914$	−4.92650	−0.162954
$915$	14.5745	0.481819
$916$	−16.0291	−0.529615
$917$	−8.37890	−0.276696
$918$	−1.00797	−0.0332680
$919$	35.7084	1.17791	0.588956	−	0.808165i	$-0.299539\pi$
$919$	35.7084	1.17791	0.588956	+	0.808165i	$0.299539\pi$
$920$	0	0
$921$	14.9018	0.491029
$922$	5.68261	0.187147
$923$	−6.44851	−0.212255
$924$	7.84955	0.258231
$925$	5.16151	0.169709
$926$	−1.55149	−0.0509852
$927$	9.98860	0.328069
$928$	−3.06396	−0.100579
$929$	−30.4749	−0.999848	−0.499924	−	0.866069i	$-0.666639\pi$
$929$	−30.4749	−0.999848	−0.499924	+	0.866069i	$0.666639\pi$
$930$	−2.45809	−0.0806041
$931$	−48.5407	−1.59086
$932$	−13.2966	−0.435546
$933$	2.36344	0.0773754
$934$	−0.493344	−0.0161427
$935$	−51.5832	−1.68695
$936$	−1.23884	−0.0404928
$937$	−44.4058	−1.45067	−0.725337	−	0.688394i	$-0.758316\pi$
$937$	−44.4058	−1.45067	−0.725337	+	0.688394i	$0.758316\pi$
$938$	−0.278364	−0.00908890
$939$	−18.7205	−0.610922
$940$	54.2195	1.76845
$941$	9.08836	0.296272	0.148136	−	0.988967i	$-0.452673\pi$
$941$	9.08836	0.296272	0.148136	+	0.988967i	$0.452673\pi$
$942$	0.873993	0.0284762
$943$	0	0
$944$	48.9427	1.59295
$945$	−2.26187	−0.0735785
$946$	−1.52232	−0.0494948
$947$	4.43866	0.144237	0.0721185	−	0.997396i	$-0.477024\pi$
$947$	4.43866	0.144237	0.0721185	+	0.997396i	$0.477024\pi$
$948$	−4.75326	−0.154379
$949$	22.4485	0.728709
$950$	0.946464	0.0307074
$951$	−1.97556	−0.0640620
$952$	3.81336	0.123592
$953$	−48.4363	−1.56901	−0.784503	−	0.620125i	$-0.787082\pi$
$953$	−48.4363	−1.56901	−0.784503	+	0.620125i	$0.787082\pi$
$954$	−1.71408	−0.0554953
$955$	−54.3996	−1.76033
$956$	10.3996	0.336348
$957$	5.65685	0.182860
$958$	−1.50467	−0.0486135
$959$	−11.3093	−0.365197
$960$	16.8469	0.543731
$961$	−2.37470	−0.0766032
$962$	2.63630	0.0849976
$963$	18.2252	0.587299
$964$	−18.6999	−0.602284
$965$	13.1169	0.422247
$966$	0	0
$967$	−10.1260	−0.325630	−0.162815	−	0.986657i	$-0.552057\pi$
$967$	−10.1260	−0.325630	−0.162815	+	0.986657i	$0.552057\pi$
$968$	−5.03451	−0.161815
$969$	−41.4372	−1.33116
$970$	1.07190	0.0344166
$971$	−20.8754	−0.669924	−0.334962	−	0.942232i	$-0.608724\pi$
$971$	−20.8754	−0.669924	−0.334962	+	0.942232i	$0.608724\pi$
$972$	−1.96239	−0.0629436
$973$	10.2166	0.327530
$974$	5.27978	0.169175
$975$	0.986826	0.0316037
$976$	−23.2291	−0.743547
$977$	52.6770	1.68529	0.842643	−	0.538473i	$-0.180999\pi$
$977$	52.6770	1.68529	0.842643	+	0.538473i	$0.180999\pi$
$978$	1.51247	0.0483633
$979$	−6.07522	−0.194165
$980$	−28.3043	−0.904146
$981$	−2.51128	−0.0801791
$982$	−1.21440	−0.0387532
$983$	−19.2159	−0.612892	−0.306446	−	0.951888i	$-0.599140\pi$
$983$	−19.2159	−0.612892	−0.306446	+	0.951888i	$0.599140\pi$
$984$	−6.08981	−0.194136
$985$	−45.6624	−1.45493
$986$	1.36102	0.0433438
$987$	11.1355	0.354447
$988$	−25.2225	−0.802433
$989$	0	0
$990$	1.92478	0.0611734
$991$	22.1260	0.702856	0.351428	−	0.936215i	$-0.385696\pi$
$991$	22.1260	0.702856	0.351428	+	0.936215i	$0.385696\pi$
$992$	12.1406	0.385464
$993$	−24.4241	−0.775074
$994$	−0.740666	−0.0234925
$995$	4.18664	0.132725
$996$	3.39759	0.107657
$997$	−40.3390	−1.27755	−0.638774	−	0.769394i	$-0.720558\pi$
$997$	−40.3390	−1.27755	−0.638774	+	0.769394i	$0.720558\pi$
$998$	−1.28963	−0.0408224
$999$	8.43209	0.266780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1587.2.a.s.1.3	✓	6
3.2	odd	2		4761.2.a.bs.1.4		6
23.22	odd	2	inner	1587.2.a.s.1.4	yes	6
69.68	even	2		4761.2.a.bs.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1587.2.a.s.1.3	✓	6	1.1	even	1	trivial
1587.2.a.s.1.4	yes	6	23.22	odd	2	inner
4761.2.a.bs.1.3		6	69.68	even	2
4761.2.a.bs.1.4		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -2.36899 q^{5} +0.193937 q^{6} +0.954779 q^{7} -0.768452 q^{8} +1.00000 q^{9} -0.459434 q^{10} -4.18945 q^{11} -1.96239 q^{12} +1.61213 q^{13} +0.185167 q^{14} -2.36899 q^{15} +3.77575 q^{16} -5.19742 q^{17} +0.193937 q^{18} +7.97266 q^{19} +4.64888 q^{20} +0.954779 q^{21} -0.812488 q^{22} -0.768452 q^{24} +0.612127 q^{25} +0.312650 q^{26} +1.00000 q^{27} -1.87365 q^{28} -1.35026 q^{29} -0.459434 q^{30} +5.35026 q^{31} +2.26916 q^{32} -4.18945 q^{33} -1.00797 q^{34} -2.26187 q^{35} -1.96239 q^{36} +8.43209 q^{37} +1.54619 q^{38} +1.61213 q^{39} +1.82046 q^{40} +7.92478 q^{41} +0.185167 q^{42} +1.87365 q^{43} +8.22133 q^{44} -2.36899 q^{45} +11.6629 q^{47} +3.77575 q^{48} -6.08840 q^{49} +0.118714 q^{50} -5.19742 q^{51} -3.16362 q^{52} -8.83833 q^{53} +0.193937 q^{54} +9.92478 q^{55} -0.733702 q^{56} +7.97266 q^{57} -0.261865 q^{58} +12.9624 q^{59} +4.64888 q^{60} -6.15220 q^{61} +1.03761 q^{62} +0.954779 q^{63} -7.11142 q^{64} -3.81912 q^{65} -0.812488 q^{66} -1.50331 q^{67} +10.1994 q^{68} -0.438658 q^{70} -4.00000 q^{71} -0.768452 q^{72} +13.9248 q^{73} +1.63529 q^{74} +0.612127 q^{75} -15.6455 q^{76} -4.00000 q^{77} +0.312650 q^{78} +2.42218 q^{79} -8.94472 q^{80} +1.00000 q^{81} +1.53690 q^{82} -1.73136 q^{83} -1.87365 q^{84} +12.3127 q^{85} +0.363369 q^{86} -1.35026 q^{87} +3.21939 q^{88} +1.45012 q^{89} -0.459434 q^{90} +1.53923 q^{91} +5.35026 q^{93} +2.26187 q^{94} -18.8872 q^{95} +2.26916 q^{96} -2.33308 q^{97} -1.18076 q^{98} -4.18945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 6 q^{3} + 10 q^{4} + 2 q^{6} + 18 q^{8} + 6 q^{9} + 10 q^{12} + 8 q^{13} + 26 q^{16} + 2 q^{18} + 18 q^{24} + 2 q^{25} - 40 q^{26} + 6 q^{27} + 12 q^{29} + 12 q^{31} + 58 q^{32} - 32 q^{35}+ \cdots - 90 q^{98}+O(q^{100}) \) 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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