LMFDB - Embedded newform 867.2.d.f.577.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(577,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.6
Root			$2.06644i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.f.577.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.65222 q^{2} +1.00000i q^{3} +0.729840 q^{4} +4.06644i q^{5} +1.65222i q^{6} +0.922382i q^{7} -2.09859 q^{8} -1.00000 q^{9} +6.71866i q^{10} -2.27016i q^{11} +0.729840i q^{12} +3.57461 q^{13} +1.52398i q^{14} -4.06644 q^{15} -4.92701 q^{16} -1.65222 q^{18} -1.72984 q^{19} +2.96785i q^{20} -0.922382 q^{21} -3.75081i q^{22} +5.09859i q^{23} -2.09859i q^{24} -11.5359 q^{25} +5.90604 q^{26} -1.00000i q^{27} +0.673192i q^{28} +2.20372i q^{29} -6.71866 q^{30} -4.13287i q^{31} -3.94335 q^{32} +2.27016 q^{33} -3.75081 q^{35} -0.729840 q^{36} +5.95453i q^{37} -2.85808 q^{38} +3.57461i q^{39} -8.53377i q^{40} +5.05010i q^{41} -1.52398 q^{42} +5.23146 q^{43} -1.65685i q^{44} -4.06644i q^{45} +8.42400i q^{46} -6.96130 q^{47} -4.92701i q^{48} +6.14921 q^{49} -19.0599 q^{50} +2.60889 q^{52} +2.69555 q^{53} -1.65222i q^{54} +9.23146 q^{55} -1.93570i q^{56} -1.72984i q^{57} +3.64104i q^{58} +7.39840 q^{59} -2.96785 q^{60} -3.89023i q^{61} -6.82843i q^{62} -0.922382i q^{63} +3.33873 q^{64} +14.5359i q^{65} +3.75081 q^{66} +12.0419 q^{67} -5.09859 q^{69} -6.19717 q^{70} +1.52398i q^{71} +2.09859 q^{72} -1.16502i q^{73} +9.83822i q^{74} -11.5359i q^{75} -1.26251 q^{76} +2.09396 q^{77} +5.90604i q^{78} -11.8837i q^{79} -20.0354i q^{80} +1.00000 q^{81} +8.34389i q^{82} +6.38508 q^{83} -0.673192 q^{84} +8.64354 q^{86} -2.20372 q^{87} +4.76413i q^{88} +10.3597 q^{89} -6.71866i q^{90} +3.29715i q^{91} +3.72115i q^{92} +4.13287 q^{93} -11.5016 q^{94} -7.03429i q^{95} -3.94335i q^{96} +14.4695i q^{97} +10.1599 q^{98} +2.27016i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.65222	1.16830	0.584149	−	0.811646i	$-0.301428\pi$
$2$	1.65222	1.16830	0.584149	+	0.811646i	$0.301428\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0.729840	0.364920
$5$	4.06644i	1.81857i	0.416179	+	0.909283i	$0.363369\pi$
$5$	4.06644i	1.81857i	−0.416179	+	0.909283i	$0.636631\pi$
$6$	1.65222i	0.674517i
$7$	0.922382i	0.348628i	0.984690	+	0.174314i	$0.0557707\pi$
$7$	0.922382i	0.348628i	−0.984690	+	0.174314i	$0.944229\pi$
$8$	−2.09859	−0.741962
$9$	−1.00000	−0.333333
$10$	6.71866i	2.12463i
$11$	− 2.27016i	− 0.684479i	−0.939613	−	0.342239i	$-0.888815\pi$
$11$	− 2.27016i	− 0.684479i	0.939613	−	0.342239i	$-0.111185\pi$
$12$	0.729840i	0.210687i
$13$	3.57461	0.991417	0.495709	−	0.868489i	$-0.334908\pi$
$13$	3.57461	0.991417	0.495709	+	0.868489i	$0.334908\pi$
$14$	1.52398i	0.407301i
$15$	−4.06644	−1.04995
$16$	−4.92701	−1.23175
$17$	0	0
$17$	0	0
$18$	−1.65222	−0.389433
$19$	−1.72984	−0.396853	−0.198426	−	0.980116i	$-0.563583\pi$
$19$	−1.72984	−0.396853	−0.198426	+	0.980116i	$0.563583\pi$
$20$	2.96785i	0.663631i
$21$	−0.922382	−0.201280
$22$	− 3.75081i	− 0.799675i
$23$	5.09859i	1.06313i	0.847018	+	0.531564i	$0.178396\pi$
$23$	5.09859i	1.06313i	−0.847018	+	0.531564i	$0.821604\pi$
$24$	− 2.09859i	− 0.428372i
$25$	−11.5359	−2.30718
$26$	5.90604	1.15827
$27$	− 1.00000i	− 0.192450i
$28$	0.673192i	0.127221i
$29$	2.20372i	0.409221i	0.978843	+	0.204611i	$0.0655928\pi$
$29$	2.20372i	0.409221i	−0.978843	+	0.204611i	$0.934407\pi$
$30$	−6.71866	−1.22665
$31$	− 4.13287i	− 0.742286i	−0.928576	−	0.371143i	$-0.878966\pi$
$31$	− 4.13287i	− 0.742286i	0.928576	−	0.371143i	$-0.121034\pi$
$32$	−3.94335	−0.697093
$33$	2.27016	0.395184
$34$	0	0
$35$	−3.75081	−0.634003
$36$	−0.729840	−0.121640
$37$	5.95453i	0.978919i	0.872026	+	0.489460i	$0.162806\pi$
$37$	5.95453i	0.978919i	−0.872026	+	0.489460i	$0.837194\pi$
$38$	−2.85808	−0.463642
$39$	3.57461i	0.572395i
$40$	− 8.53377i	− 1.34931i
$41$	5.05010i	0.788693i	0.918962	+	0.394346i	$0.129029\pi$
$41$	5.05010i	0.788693i	−0.918962	+	0.394346i	$0.870971\pi$
$42$	−1.52398	−0.235155
$43$	5.23146	0.797790	0.398895	−	0.916997i	$-0.369394\pi$
$43$	5.23146	0.797790	0.398895	+	0.916997i	$0.369394\pi$
$44$	− 1.65685i	− 0.249780i
$45$	− 4.06644i	− 0.606189i
$46$	8.42400i	1.24205i
$47$	−6.96130	−1.01541	−0.507705	−	0.861531i	$-0.669506\pi$
$47$	−6.96130	−1.01541	−0.507705	+	0.861531i	$0.669506\pi$
$48$	− 4.92701i	− 0.711153i
$49$	6.14921	0.878459
$50$	−19.0599	−2.69548
$51$	0	0
$52$	2.60889	0.361788
$53$	2.69555	0.370263	0.185131	−	0.982714i	$-0.440729\pi$
$53$	2.69555	0.370263	0.185131	+	0.982714i	$0.440729\pi$
$54$	− 1.65222i	− 0.224839i
$55$	9.23146	1.24477
$56$	− 1.93570i	− 0.258669i
$57$	− 1.72984i	− 0.229123i
$58$	3.64104i	0.478092i
$59$	7.39840	0.963190	0.481595	−	0.876394i	$-0.340058\pi$
$59$	7.39840	0.963190	0.481595	+	0.876394i	$0.340058\pi$
$60$	−2.96785	−0.383148
$61$	− 3.89023i	− 0.498093i	−0.968492	−	0.249047i	$-0.919883\pi$
$61$	− 3.89023i	− 0.498093i	0.968492	−	0.249047i	$-0.0801172\pi$
$62$	− 6.82843i	− 0.867211i
$63$	− 0.922382i	− 0.116209i
$64$	3.33873	0.417341
$65$	14.5359i	1.80296i
$66$	3.75081	0.461693
$67$	12.0419	1.47116	0.735578	−	0.677440i	$-0.236910\pi$
$67$	12.0419	1.47116	0.735578	+	0.677440i	$0.236910\pi$
$68$	0	0
$69$	−5.09859	−0.613798
$70$	−6.19717	−0.740704
$71$	1.52398i	0.180863i	0.995903	+	0.0904317i	$0.0288247\pi$
$71$	1.52398i	0.180863i	−0.995903	+	0.0904317i	$0.971175\pi$
$72$	2.09859	0.247321
$73$	− 1.16502i	− 0.136356i	−0.997673	−	0.0681778i	$-0.978281\pi$
$73$	− 1.16502i	− 0.136356i	0.997673	−	0.0681778i	$-0.0217185\pi$
$74$	9.83822i	1.14367i
$75$	− 11.5359i	− 1.33205i
$76$	−1.26251	−0.144820
$77$	2.09396	0.238628
$78$	5.90604i	0.668728i
$79$	− 11.8837i	− 1.33702i	−0.743704	−	0.668509i	$-0.766933\pi$
$79$	− 11.8837i	− 1.33702i	0.743704	−	0.668509i	$-0.233067\pi$
$80$	− 20.0354i	− 2.24002i
$81$	1.00000	0.111111
$82$	8.34389i	0.921428i
$83$	6.38508	0.700854	0.350427	−	0.936590i	$-0.386036\pi$
$83$	6.38508	0.700854	0.350427	+	0.936590i	$0.386036\pi$
$84$	−0.673192	−0.0734513
$85$	0	0
$86$	8.64354	0.932057
$87$	−2.20372	−0.236264
$88$	4.76413i	0.507858i
$89$	10.3597	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$89$	10.3597	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$90$	− 6.71866i	− 0.708209i
$91$	3.29715i	0.345636i
$92$	3.72115i	0.387957i
$93$	4.13287	0.428559
$94$	−11.5016	−1.18630
$95$	− 7.03429i	− 0.721703i
$96$	− 3.94335i	− 0.402467i
$97$	14.4695i	1.46915i	0.678526	+	0.734576i	$0.262619\pi$
$97$	14.4695i	1.46915i	−0.678526	+	0.734576i	$0.737381\pi$
$98$	10.1599	1.02630
$99$	2.27016i	0.228160i
$100$	−8.41937	−0.841937
$101$	0.265746	0.0264427	0.0132213	−	0.999913i	$-0.495791\pi$
$101$	0.265746	0.0264427	0.0132213	+	0.999913i	$0.495791\pi$
$102$	0	0
$103$	−3.03429	−0.298977	−0.149489	−	0.988763i	$-0.547763\pi$
$103$	−3.03429	−0.298977	−0.149489	+	0.988763i	$0.547763\pi$
$104$	−7.50162	−0.735594
$105$	− 3.75081i	− 0.366042i
$106$	4.45366	0.432577
$107$	19.6524i	1.89987i	0.312443	+	0.949937i	$0.398853\pi$
$107$	19.6524i	1.89987i	−0.312443	+	0.949937i	$0.601147\pi$
$108$	− 0.729840i	− 0.0702289i
$109$	− 0.715639i	− 0.0685457i	−0.999413	−	0.0342729i	$-0.989088\pi$
$109$	− 0.715639i	− 0.0685457i	0.999413	−	0.0342729i	$-0.0109115\pi$
$110$	15.2524	1.45426
$111$	−5.95453	−0.565179
$112$	− 4.54459i	− 0.429423i
$113$	11.7876i	1.10888i	0.832223	+	0.554442i	$0.187068\pi$
$113$	11.7876i	1.10888i	−0.832223	+	0.554442i	$0.812932\pi$
$114$	− 2.85808i	− 0.267684i
$115$	−20.7331	−1.93337
$116$	1.60837i	0.149333i
$117$	−3.57461	−0.330472
$118$	12.2238	1.12529
$119$	0	0
$120$	8.53377	0.779023
$121$	5.84638	0.531489
$122$	− 6.42753i	− 0.581921i
$123$	−5.05010	−0.455352
$124$	− 3.01634i	− 0.270875i
$125$	− 26.5778i	− 2.37719i
$126$	− 1.52398i	− 0.135767i
$127$	−2.81048	−0.249390	−0.124695	−	0.992195i	$-0.539795\pi$
$127$	−2.81048	−0.249390	−0.124695	+	0.992195i	$0.539795\pi$
$128$	13.4030	1.18467
$129$	5.23146i	0.460604i
$130$	24.0166i	2.10639i
$131$	− 7.92701i	− 0.692586i	−0.938126	−	0.346293i	$-0.887440\pi$
$131$	− 7.92701i	− 0.692586i	0.938126	−	0.346293i	$-0.112560\pi$
$132$	1.65685	0.144211
$133$	− 1.59557i	− 0.138354i
$134$	19.8960	1.71875
$135$	4.06644	0.349983
$136$	0	0
$137$	−0.103218	−0.00881851	−0.00440926	−	0.999990i	$-0.501404\pi$
$137$	−0.103218	−0.00881851	−0.00440926	+	0.999990i	$0.501404\pi$
$138$	−8.42400	−0.717099
$139$	− 14.4464i	− 1.22532i	−0.790345	−	0.612662i	$-0.790099\pi$
$139$	− 14.4464i	− 1.22532i	0.790345	−	0.612662i	$-0.209901\pi$
$140$	−2.73749	−0.231360
$141$	− 6.96130i	− 0.586247i
$142$	2.51796i	0.211302i
$143$	− 8.11492i	− 0.678604i
$144$	4.92701	0.410584
$145$	−8.96130	−0.744195
$146$	− 1.92488i	− 0.159304i
$147$	6.14921i	0.507178i
$148$	4.34586i	0.357227i
$149$	3.59557	0.294561	0.147280	−	0.989095i	$-0.452948\pi$
$149$	3.59557	0.294561	0.147280	+	0.989095i	$0.452948\pi$
$150$	− 19.0599i	− 1.55623i
$151$	−7.58828	−0.617526	−0.308763	−	0.951139i	$-0.599915\pi$
$151$	−7.58828	−0.617526	−0.308763	+	0.951139i	$0.599915\pi$
$152$	3.63022	0.294450
$153$	0	0
$154$	3.45968	0.278789
$155$	16.8061	1.34990
$156$	2.60889i	0.208878i
$157$	−18.9569	−1.51292	−0.756462	−	0.654038i	$-0.773074\pi$
$157$	−18.9569	−1.51292	−0.756462	+	0.654038i	$0.773074\pi$
$158$	− 19.6345i	− 1.56204i
$159$	2.69555i	0.213771i
$160$	− 16.0354i	− 1.26771i
$161$	−4.70285	−0.370636
$162$	1.65222	0.129811
$163$	− 13.4330i	− 1.05216i	−0.850436	−	0.526079i	$-0.823662\pi$
$163$	− 13.4330i	− 1.05216i	0.850436	−	0.526079i	$-0.176338\pi$
$164$	3.68577i	0.287810i
$165$	9.23146i	0.718668i
$166$	10.5496	0.818806
$167$	− 2.13729i	− 0.165388i	−0.996575	−	0.0826941i	$-0.973648\pi$
$167$	− 2.13729i	− 0.165388i	0.996575	−	0.0826941i	$-0.0263524\pi$
$168$	1.93570	0.149342
$169$	−0.222197	−0.0170921
$170$	0	0
$171$	1.72984	0.132284
$172$	3.81813	0.291130
$173$	− 5.27995i	− 0.401427i	−0.979650	−	0.200713i	$-0.935674\pi$
$173$	− 5.27995i	− 0.401427i	0.979650	−	0.200713i	$-0.0643260\pi$
$174$	−3.64104	−0.276027
$175$	− 10.6405i	− 0.804347i
$176$	11.1851i	0.843109i
$177$	7.39840i	0.556098i
$178$	17.1165	1.28294
$179$	7.77619	0.581220	0.290610	−	0.956842i	$-0.406142\pi$
$179$	7.77619	0.581220	0.290610	+	0.956842i	$0.406142\pi$
$180$	− 2.96785i	− 0.221210i
$181$	− 0.915833i	− 0.0680733i	−0.999421	−	0.0340367i	$-0.989164\pi$
$181$	− 0.915833i	− 0.0680733i	0.999421	−	0.0340367i	$-0.0108363\pi$
$182$	5.44763i	0.403805i
$183$	3.89023	0.287574
$184$	− 10.6998i	− 0.788802i
$185$	−24.2137	−1.78023
$186$	6.82843	0.500685
$187$	0	0
$188$	−5.08064	−0.370544
$189$	0.922382	0.0670934
$190$	− 11.6222i	− 0.843164i
$191$	−19.1658	−1.38679	−0.693393	−	0.720560i	$-0.743885\pi$
$191$	−19.1658	−1.38679	−0.693393	+	0.720560i	$0.743885\pi$
$192$	3.33873i	0.240952i
$193$	− 4.71137i	− 0.339132i	−0.985519	−	0.169566i	$-0.945763\pi$
$193$	− 4.71137i	− 0.339132i	0.985519	−	0.169566i	$-0.0542365\pi$
$194$	23.9068i	1.71641i
$195$	−14.5359	−1.04094
$196$	4.48794	0.320567
$197$	− 14.8949i	− 1.06122i	−0.847618	−	0.530608i	$-0.821964\pi$
$197$	− 14.8949i	− 1.06122i	0.847618	−	0.530608i	$-0.178036\pi$
$198$	3.75081i	0.266558i
$199$	− 18.7837i	− 1.33154i	−0.746156	−	0.665771i	$-0.768103\pi$
$199$	− 18.7837i	− 1.33154i	0.746156	−	0.665771i	$-0.231897\pi$
$200$	24.2091	1.71184
$201$	12.0419i	0.849373i
$202$	0.439071	0.0308929
$203$	−2.03268	−0.142666
$204$	0	0
$205$	−20.5359	−1.43429
$206$	−5.01332	−0.349294
$207$	− 5.09859i	− 0.354376i
$208$	−17.6121	−1.22118
$209$	3.92701i	0.271637i
$210$	− 6.19717i	− 0.427646i
$211$	5.88066i	0.404841i	0.979299	+	0.202421i	$0.0648808\pi$
$211$	5.88066i	0.404841i	−0.979299	+	0.202421i	$0.935119\pi$
$212$	1.96732	0.135116
$213$	−1.52398	−0.104421
$214$	32.4702i	2.21962i
$215$	21.2734i	1.45083i
$216$	2.09859i	0.142791i
$217$	3.81209	0.258781
$218$	− 1.18239i	− 0.0800819i
$219$	1.16502	0.0787250
$220$	6.73749	0.454242
$221$	0	0
$222$	−9.83822	−0.660298
$223$	5.58387	0.373923	0.186962	−	0.982367i	$-0.440136\pi$
$223$	5.58387	0.373923	0.186962	+	0.982367i	$0.440136\pi$
$224$	− 3.63728i	− 0.243026i
$225$	11.5359	0.769060
$226$	19.4757i	1.29551i
$227$	27.5299i	1.82722i	0.406589	+	0.913611i	$0.366718\pi$
$227$	27.5299i	1.82722i	−0.406589	+	0.913611i	$0.633282\pi$
$228$	− 1.26251i	− 0.0836116i
$229$	3.57019	0.235925	0.117962	−	0.993018i	$-0.462364\pi$
$229$	3.57019	0.235925	0.117962	+	0.993018i	$0.462364\pi$
$230$	−34.2557	−2.25875
$231$	2.09396i	0.137772i
$232$	− 4.62470i	− 0.303627i
$233$	− 1.37088i	− 0.0898095i	−0.998991	−	0.0449047i	$-0.985702\pi$
$233$	− 1.37088i	− 0.0898095i	0.998991	−	0.0449047i	$-0.0142984\pi$
$234$	−5.90604	−0.386090
$235$	− 28.3077i	− 1.84659i
$236$	5.39965	0.351487
$237$	11.8837	0.771928
$238$	0	0
$239$	21.6121	1.39797	0.698986	−	0.715135i	$-0.253635\pi$
$239$	21.6121	1.39797	0.698986	+	0.715135i	$0.253635\pi$
$240$	20.0354	1.29328
$241$	1.06181i	0.0683969i	0.999415	+	0.0341984i	$0.0108878\pi$
$241$	1.06181i	0.0683969i	−0.999415	+	0.0341984i	$0.989112\pi$
$242$	9.65952	0.620937
$243$	1.00000i	0.0641500i
$244$	− 2.83925i	− 0.181764i
$245$	25.0054i	1.59753i
$246$	−8.34389	−0.531987
$247$	−6.18350	−0.393446
$248$	8.67319i	0.550748i
$249$	6.38508i	0.404638i
$250$	− 43.9125i	− 2.77727i
$251$	−7.29715	−0.460592	−0.230296	−	0.973121i	$-0.573969\pi$
$251$	−7.29715	−0.460592	−0.230296	+	0.973121i	$0.573969\pi$
$252$	− 0.673192i	− 0.0424071i
$253$	11.5746	0.727689
$254$	−4.64354	−0.291361
$255$	0	0
$256$	15.4673	0.966708
$257$	22.5569	1.40706	0.703530	−	0.710666i	$-0.251606\pi$
$257$	22.5569	1.40706	0.703530	+	0.710666i	$0.251606\pi$
$258$	8.64354i	0.538123i
$259$	−5.49236	−0.341278
$260$	10.6089i	0.657936i
$261$	− 2.20372i	− 0.136407i
$262$	− 13.0972i	− 0.809147i
$263$	4.24919	0.262016	0.131008	−	0.991381i	$-0.458179\pi$
$263$	4.24919	0.262016	0.131008	+	0.991381i	$0.458179\pi$
$264$	−4.76413	−0.293212
$265$	10.9613i	0.673347i
$266$	− 2.63624i	− 0.161639i
$267$	10.3597i	0.634003i
$268$	8.78869	0.536855
$269$	− 11.8338i	− 0.721520i	−0.932659	−	0.360760i	$-0.882517\pi$
$269$	− 11.8338i	− 0.721520i	0.932659	−	0.360760i	$-0.117483\pi$
$270$	6.71866	0.408885
$271$	−11.8045	−0.717070	−0.358535	−	0.933516i	$-0.616724\pi$
$271$	−11.8045	−0.717070	−0.358535	+	0.933516i	$0.616724\pi$
$272$	0	0
$273$	−3.29715	−0.199553
$274$	−0.170539	−0.0103027
$275$	26.1883i	1.57922i
$276$	−3.72115	−0.223987
$277$	10.4275i	0.626530i	0.949666	+	0.313265i	$0.101423\pi$
$277$	10.4275i	0.626530i	−0.949666	+	0.313265i	$0.898577\pi$
$278$	− 23.8686i	− 1.43154i
$279$	4.13287i	0.247429i
$280$	7.87140	0.470406
$281$	5.46697	0.326132	0.163066	−	0.986615i	$-0.447862\pi$
$281$	5.46697	0.326132	0.163066	+	0.986615i	$0.447862\pi$
$282$	− 11.5016i	− 0.684911i
$283$	− 17.6375i	− 1.04844i	−0.851582	−	0.524221i	$-0.824357\pi$
$283$	− 17.6375i	− 1.04844i	0.851582	−	0.524221i	$-0.175643\pi$
$284$	1.11226i	0.0660007i
$285$	7.03429	0.416675
$286$	− 13.4077i	− 0.792812i
$287$	−4.65812	−0.274960
$288$	3.94335	0.232364
$289$	0	0
$290$	−14.8061	−0.869442
$291$	−14.4695	−0.848215
$292$	− 0.850281i	− 0.0497589i
$293$	−8.25845	−0.482464	−0.241232	−	0.970467i	$-0.577551\pi$
$293$	−8.25845	−0.482464	−0.241232	+	0.970467i	$0.577551\pi$
$294$	10.1599i	0.592535i
$295$	30.0851i	1.75162i
$296$	− 12.4961i	− 0.726321i
$297$	−2.27016	−0.131728
$298$	5.94069	0.344135
$299$	18.2254i	1.05400i
$300$	− 8.41937i	− 0.486093i
$301$	4.82541i	0.278132i
$302$	−12.5375	−0.721454
$303$	0.265746i	0.0152667i
$304$	8.52295	0.488825
$305$	15.8194	0.905815
$306$	0	0
$307$	−9.94601	−0.567649	−0.283824	−	0.958876i	$-0.591603\pi$
$307$	−9.94601	−0.567649	−0.283824	+	0.958876i	$0.591603\pi$
$308$	1.52825	0.0870803
$309$	− 3.03429i	− 0.172615i
$310$	27.7674	1.57708
$311$	− 16.5865i	− 0.940536i	−0.882524	−	0.470268i	$-0.844157\pi$
$311$	− 16.5865i	− 0.940536i	0.882524	−	0.470268i	$-0.155843\pi$
$312$	− 7.50162i	− 0.424696i
$313$	− 20.6116i	− 1.16504i	−0.812818	−	0.582518i	$-0.802067\pi$
$313$	− 20.6116i	− 1.16504i	0.812818	−	0.582518i	$-0.197933\pi$
$314$	−31.3210	−1.76755
$315$	3.75081	0.211334
$316$	− 8.67319i	− 0.487905i
$317$	− 4.26802i	− 0.239716i	−0.992791	−	0.119858i	$-0.961756\pi$
$317$	− 4.26802i	− 0.239716i	0.992791	−	0.119858i	$-0.0382439\pi$
$318$	4.45366i	0.249749i
$319$	5.00280	0.280103
$320$	13.5767i	0.758963i
$321$	−19.6524	−1.09689
$322$	−7.77015	−0.433014
$323$	0	0
$324$	0.729840	0.0405467
$325$	−41.2363	−2.28738
$326$	− 22.1944i	− 1.22923i
$327$	0.715639	0.0395749
$328$	− 10.5981i	− 0.585181i
$329$	− 6.42098i	− 0.354000i
$330$	15.2524i	0.839619i
$331$	28.6105	1.57258	0.786288	−	0.617860i	$-0.212000\pi$
$331$	28.6105	1.57258	0.786288	+	0.617860i	$0.212000\pi$
$332$	4.66009	0.255756
$333$	− 5.95453i	− 0.326306i
$334$	− 3.53127i	− 0.193223i
$335$	48.9678i	2.67540i
$336$	4.54459	0.247928
$337$	31.3368i	1.70703i	0.521072	+	0.853513i	$0.325532\pi$
$337$	31.3368i	1.70703i	−0.521072	+	0.853513i	$0.674468\pi$
$338$	−0.367119	−0.0199686
$339$	−11.7876	−0.640214
$340$	0	0
$341$	−9.38228	−0.508079
$342$	2.85808	0.154547
$343$	12.1286i	0.654883i
$344$	−10.9787	−0.591930
$345$	− 20.7331i	− 1.11623i
$346$	− 8.72365i	− 0.468986i
$347$	− 12.0359i	− 0.646121i	−0.946378	−	0.323060i	$-0.895288\pi$
$347$	− 12.0359i	− 0.646121i	0.946378	−	0.323060i	$-0.104712\pi$
$348$	−1.60837	−0.0862175
$349$	−33.3512	−1.78525	−0.892625	−	0.450799i	$-0.851139\pi$
$349$	−33.3512	−1.78525	−0.892625	+	0.450799i	$0.851139\pi$
$350$	− 17.5805i	− 0.939718i
$351$	− 3.57461i	− 0.190798i
$352$	8.95204i	0.477145i
$353$	−6.56570	−0.349457	−0.174729	−	0.984617i	$-0.555905\pi$
$353$	−6.56570	−0.349457	−0.174729	+	0.984617i	$0.555905\pi$
$354$	12.2238i	0.649688i
$355$	−6.19717	−0.328912
$356$	7.56093	0.400728
$357$	0	0
$358$	12.8480	0.679038
$359$	−21.5508	−1.13741	−0.568705	−	0.822541i	$-0.692555\pi$
$359$	−21.5508	−1.13741	−0.568705	+	0.822541i	$0.692555\pi$
$360$	8.53377i	0.449769i
$361$	−16.0077	−0.842508
$362$	− 1.51316i	− 0.0795299i
$363$	5.84638i	0.306855i
$364$	2.40640i	0.126129i
$365$	4.73749	0.247972
$366$	6.42753	0.335972
$367$	− 35.5644i	− 1.85645i	−0.372025	−	0.928223i	$-0.621336\pi$
$367$	− 35.5644i	− 1.85645i	0.372025	−	0.928223i	$-0.378664\pi$
$368$	− 25.1208i	− 1.30951i
$369$	− 5.05010i	− 0.262898i
$370$	−40.0065	−2.07984
$371$	2.48633i	0.129084i
$372$	3.01634	0.156390
$373$	−35.8452	−1.85599	−0.927997	−	0.372588i	$-0.878471\pi$
$373$	−35.8452	−1.85599	−0.927997	+	0.372588i	$0.878471\pi$
$374$	0	0
$375$	26.5778	1.37247
$376$	14.6089	0.753396
$377$	7.87744i	0.405709i
$378$	1.52398	0.0783851
$379$	− 21.9206i	− 1.12599i	−0.826461	−	0.562994i	$-0.809650\pi$
$379$	− 21.9206i	− 1.12599i	0.826461	−	0.562994i	$-0.190350\pi$
$380$	− 5.13391i	− 0.263364i
$381$	− 2.81048i	− 0.143985i
$382$	−31.6661	−1.62018
$383$	34.3109	1.75321	0.876603	−	0.481215i	$-0.159804\pi$
$383$	34.3109	1.75321	0.876603	+	0.481215i	$0.159804\pi$
$384$	13.4030i	0.683971i
$385$	8.51494i	0.433961i
$386$	− 7.78423i	− 0.396207i
$387$	−5.23146	−0.265930
$388$	10.5604i	0.536123i
$389$	34.1956	1.73379	0.866894	−	0.498493i	$-0.166113\pi$
$389$	34.1956	1.73379	0.866894	+	0.498493i	$0.166113\pi$
$390$	−24.0166	−1.21613
$391$	0	0
$392$	−12.9047	−0.651783
$393$	7.92701	0.399865
$394$	− 24.6096i	− 1.23982i
$395$	48.3242	2.43146
$396$	1.65685i	0.0832601i
$397$	− 0.230357i	− 0.0115613i	−0.999983	−	0.00578065i	$-0.998160\pi$
$397$	− 0.230357i	− 0.0115613i	0.999983	−	0.00578065i	$-0.00184005\pi$
$398$	− 31.0349i	− 1.55564i
$399$	1.59557	0.0798786
$400$	56.8376	2.84188
$401$	− 0.606756i	− 0.0302999i	−0.999885	−	0.0151500i	$-0.995177\pi$
$401$	− 0.606756i	− 0.0302999i	0.999885	−	0.0151500i	$-0.00482257\pi$
$402$	19.8960i	0.992321i
$403$	− 14.7734i	− 0.735915i
$404$	0.193952	0.00964947
$405$	4.06644i	0.202063i
$406$	−3.35843	−0.166676
$407$	13.5177	0.670049
$408$	0	0
$409$	−1.50925	−0.0746278	−0.0373139	−	0.999304i	$-0.511880\pi$
$409$	−1.50925	−0.0746278	−0.0373139	+	0.999304i	$0.511880\pi$
$410$	−33.9299	−1.67568
$411$	− 0.103218i	− 0.00509137i
$412$	−2.21454	−0.109103
$413$	6.82416i	0.335795i
$414$	− 8.42400i	− 0.414017i
$415$	25.9645i	1.27455i
$416$	−14.0959	−0.691110
$417$	14.4464	0.707441
$418$	6.48830i	0.317353i
$419$	12.1198i	0.592090i	0.955174	+	0.296045i	$0.0956678\pi$
$419$	12.1198i	0.592090i	−0.955174	+	0.296045i	$0.904332\pi$
$420$	− 2.73749i	− 0.133576i
$421$	−17.8730	−0.871078	−0.435539	−	0.900170i	$-0.643442\pi$
$421$	−17.8730	−0.871078	−0.435539	+	0.900170i	$0.643442\pi$
$422$	9.71616i	0.472975i
$423$	6.96130	0.338470
$424$	−5.65685	−0.274721
$425$	0	0
$426$	−2.51796	−0.121995
$427$	3.58828	0.173649
$428$	14.3431i	0.693302i
$429$	8.11492	0.391792
$430$	35.1484i	1.69501i
$431$	− 11.1361i	− 0.536408i	−0.963362	−	0.268204i	$-0.913570\pi$
$431$	− 11.1361i	− 0.536408i	0.963362	−	0.268204i	$-0.0864301\pi$
$432$	4.92701i	0.237051i
$433$	29.1182	1.39933	0.699665	−	0.714471i	$-0.253333\pi$
$433$	29.1182	1.39933	0.699665	+	0.714471i	$0.253333\pi$
$434$	6.29842	0.302334
$435$	− 8.96130i	− 0.429661i
$436$	− 0.522302i	− 0.0250137i
$437$	− 8.81974i	− 0.421905i
$438$	1.92488	0.0919742
$439$	22.5958i	1.07844i	0.842165	+	0.539219i	$0.181281\pi$
$439$	22.5958i	1.07844i	−0.842165	+	0.539219i	$0.818719\pi$
$440$	−19.3730	−0.923572
$441$	−6.14921	−0.292820
$442$	0	0
$443$	32.1976	1.52975	0.764877	−	0.644176i	$-0.222800\pi$
$443$	32.1976	1.52975	0.764877	+	0.644176i	$0.222800\pi$
$444$	−4.34586	−0.206245
$445$	42.1271i	1.99701i
$446$	9.22579	0.436854
$447$	3.59557i	0.170065i
$448$	3.07959i	0.145497i
$449$	− 1.12030i	− 0.0528702i	−0.999651	−	0.0264351i	$-0.991584\pi$
$449$	− 1.12030i	− 0.0528702i	0.999651	−	0.0264351i	$-0.00841553\pi$
$450$	19.0599	0.898492
$451$	11.4645	0.539844
$452$	8.60306i	0.404654i
$453$	− 7.58828i	− 0.356529i
$454$	45.4855i	2.13474i
$455$	−13.4077	−0.628561
$456$	3.63022i	0.170001i
$457$	−40.0795	−1.87484	−0.937419	−	0.348203i	$-0.886792\pi$
$457$	−40.0795	−1.87484	−0.937419	+	0.348203i	$0.886792\pi$
$458$	5.89875	0.275631
$459$	0	0
$460$	−15.1318	−0.705526
$461$	−4.99595	−0.232684	−0.116342	−	0.993209i	$-0.537117\pi$
$461$	−4.99595	−0.232684	−0.116342	+	0.993209i	$0.537117\pi$
$462$	3.45968i	0.160959i
$463$	39.0452	1.81458	0.907292	−	0.420502i	$-0.138146\pi$
$463$	39.0452	1.81458	0.907292	+	0.420502i	$0.138146\pi$
$464$	− 10.8578i	− 0.504060i
$465$	16.8061i	0.779363i
$466$	− 2.26500i	− 0.104924i
$467$	−10.5242	−0.487002	−0.243501	−	0.969901i	$-0.578296\pi$
$467$	−10.5242	−0.487002	−0.243501	+	0.969901i	$0.578296\pi$
$468$	−2.60889	−0.120596
$469$	11.1073i	0.512886i
$470$	− 46.7706i	− 2.15737i
$471$	− 18.9569i	− 0.873487i
$472$	−15.5262	−0.714651
$473$	− 11.8762i	− 0.546070i
$474$	19.6345	0.901842
$475$	19.9553	0.915611
$476$	0	0
$477$	−2.69555	−0.123421
$478$	35.7081	1.63325
$479$	− 25.9640i	− 1.18632i	−0.805083	−	0.593162i	$-0.797879\pi$
$479$	− 25.9640i	− 1.18632i	0.805083	−	0.593162i	$-0.202121\pi$
$480$	16.0354	0.731912
$481$	21.2851i	0.970517i
$482$	1.75434i	0.0799079i
$483$	− 4.70285i	− 0.213987i
$484$	4.26692	0.193951
$485$	−58.8392	−2.67175
$486$	1.65222i	0.0749464i
$487$	− 25.5405i	− 1.15735i	−0.815557	−	0.578676i	$-0.803570\pi$
$487$	− 25.5405i	− 1.15735i	0.815557	−	0.578676i	$-0.196430\pi$
$488$	8.16399i	0.369566i
$489$	13.4330	0.607463
$490$	41.3145i	1.86640i
$491$	2.50764	0.113168	0.0565842	−	0.998398i	$-0.481979\pi$
$491$	2.50764	0.113168	0.0565842	+	0.998398i	$0.481979\pi$
$492$	−3.68577	−0.166167
$493$	0	0
$494$	−10.2165	−0.459663
$495$	−9.23146	−0.414923
$496$	20.3627i	0.914313i
$497$	−1.40569	−0.0630540
$498$	10.5496i	0.472738i
$499$	− 15.2270i	− 0.681656i	−0.940126	−	0.340828i	$-0.889293\pi$
$499$	− 15.2270i	− 0.681656i	0.940126	−	0.340828i	$-0.110707\pi$
$500$	− 19.3976i	− 0.867486i
$501$	2.13729	0.0954869
$502$	−12.0565	−0.538109
$503$	39.0990i	1.74334i	0.490094	+	0.871670i	$0.336962\pi$
$503$	39.0990i	1.74334i	−0.490094	+	0.871670i	$0.663038\pi$
$504$	1.93570i	0.0862229i
$505$	1.08064i	0.0480878i
$506$	19.1238	0.850158
$507$	− 0.222197i	− 0.00986811i
$508$	−2.05120	−0.0910073
$509$	−31.9138	−1.41455	−0.707277	−	0.706937i	$-0.750076\pi$
$509$	−31.9138	−1.41455	−0.707277	+	0.706937i	$0.750076\pi$
$510$	0	0
$511$	1.07460	0.0475374
$512$	−1.25058	−0.0552685
$513$	1.72984i	0.0763743i
$514$	37.2690	1.64386
$515$	− 12.3387i	− 0.543709i
$516$	3.81813i	0.168084i
$517$	15.8033i	0.695027i
$518$	−9.07460	−0.398715
$519$	5.27995	0.231764
$520$	− 30.5049i	− 1.33773i
$521$	4.83056i	0.211631i	0.994386	+	0.105815i	$0.0337453\pi$
$521$	4.83056i	0.211631i	−0.994386	+	0.105815i	$0.966255\pi$
$522$	− 3.64104i	− 0.159364i
$523$	36.5734	1.59924	0.799622	−	0.600503i	$-0.205033\pi$
$523$	36.5734	1.59924	0.799622	+	0.600503i	$0.205033\pi$
$524$	− 5.78546i	− 0.252739i
$525$	10.6405	0.464390
$526$	7.02061	0.306113
$527$	0	0
$528$	−11.1851	−0.486769
$529$	−2.99559	−0.130243
$530$	18.1105i	0.786670i
$531$	−7.39840	−0.321063
$532$	− 1.16451i	− 0.0504881i
$533$	18.0521i	0.781924i
$534$	17.1165i	0.740705i
$535$	−79.9154	−3.45504
$536$	−25.2711	−1.09154
$537$	7.77619i	0.335567i
$538$	− 19.5521i	− 0.842950i
$539$	− 13.9597i	− 0.601286i
$540$	2.96785	0.127716
$541$	11.0291i	0.474179i	0.971488	+	0.237090i	$0.0761935\pi$
$541$	11.0291i	0.474179i	−0.971488	+	0.237090i	$0.923806\pi$
$542$	−19.5036	−0.837751
$543$	0.915833	0.0393021
$544$	0	0
$545$	2.91010	0.124655
$546$	−5.44763	−0.233137
$547$	− 26.8766i	− 1.14916i	−0.818448	−	0.574580i	$-0.805165\pi$
$547$	− 26.8766i	− 1.14916i	0.818448	−	0.574580i	$-0.194835\pi$
$548$	−0.0753327	−0.00321805
$549$	3.89023i	0.166031i
$550$	43.2690i	1.84500i
$551$	− 3.81209i	− 0.162400i
$552$	10.6998	0.455415
$553$	10.9613	0.466122
$554$	17.2286i	0.731973i
$555$	− 24.2137i	− 1.02782i
$556$	− 10.5435i	− 0.447146i
$557$	−34.9146	−1.47938	−0.739690	−	0.672948i	$-0.765028\pi$
$557$	−34.9146	−1.47938	−0.739690	+	0.672948i	$0.765028\pi$
$558$	6.82843i	0.289070i
$559$	18.7004	0.790943
$560$	18.4803	0.780935
$561$	0	0
$562$	9.03266	0.381020
$563$	7.91334	0.333507	0.166754	−	0.985999i	$-0.446672\pi$
$563$	7.91334	0.333507	0.166754	+	0.985999i	$0.446672\pi$
$564$	− 5.08064i	− 0.213933i
$565$	−47.9335	−2.01658
$566$	− 29.1411i	− 1.22489i
$567$	0.922382i	0.0387364i
$568$	− 3.19821i	− 0.134194i
$569$	17.2956	0.725070	0.362535	−	0.931970i	$-0.381911\pi$
$569$	17.2956	0.725070	0.362535	+	0.931970i	$0.381911\pi$
$570$	11.6222	0.486801
$571$	19.3791i	0.810988i	0.914098	+	0.405494i	$0.132901\pi$
$571$	19.3791i	0.810988i	−0.914098	+	0.405494i	$0.867099\pi$
$572$	− 5.92260i	− 0.247636i
$573$	− 19.1658i	− 0.800661i
$574$	−7.69626	−0.321236
$575$	− 58.8168i	− 2.45283i
$576$	−3.33873	−0.139114
$577$	−2.89714	−0.120610	−0.0603048	−	0.998180i	$-0.519207\pi$
$577$	−2.89714	−0.120610	−0.0603048	+	0.998180i	$0.519207\pi$
$578$	0	0
$579$	4.71137	0.195798
$580$	−6.54032	−0.271572
$581$	5.88949i	0.244337i
$582$	−23.9068	−0.990968
$583$	− 6.11934i	− 0.253437i
$584$	2.44490i	0.101171i
$585$	− 14.5359i	− 0.600986i
$586$	−13.6448	−0.563662
$587$	−12.3165	−0.508357	−0.254178	−	0.967157i	$-0.581805\pi$
$587$	−12.3165	−0.508357	−0.254178	+	0.967157i	$0.581805\pi$
$588$	4.48794i	0.185080i
$589$	7.14921i	0.294578i
$590$	49.7073i	2.04642i
$591$	14.8949	0.612693
$592$	− 29.3381i	− 1.20579i
$593$	−43.1763	−1.77304	−0.886519	−	0.462693i	$-0.846883\pi$
$593$	−43.1763	−1.77304	−0.886519	+	0.462693i	$0.846883\pi$
$594$	−3.75081	−0.153898
$595$	0	0
$596$	2.62420	0.107491
$597$	18.7837	0.768766
$598$	30.1125i	1.23139i
$599$	6.70607	0.274003	0.137001	−	0.990571i	$-0.456254\pi$
$599$	6.70607	0.274003	0.137001	+	0.990571i	$0.456254\pi$
$600$	24.2091i	0.988332i
$601$	− 31.8393i	− 1.29875i	−0.760468	−	0.649375i	$-0.775030\pi$
$601$	− 31.8393i	− 1.29875i	0.760468	−	0.649375i	$-0.224970\pi$
$602$	7.97265i	0.324941i
$603$	−12.0419	−0.490386
$604$	−5.53823	−0.225348
$605$	23.7739i	0.966547i
$606$	0.439071i	0.0178360i
$607$	27.9168i	1.13311i	0.824025	+	0.566554i	$0.191724\pi$
$607$	27.9168i	1.13311i	−0.824025	+	0.566554i	$0.808276\pi$
$608$	6.82137	0.276643
$609$	− 2.03268i	− 0.0823682i
$610$	26.1371	1.05826
$611$	−24.8839	−1.00669
$612$	0	0
$613$	−5.84314	−0.236002	−0.118001	−	0.993013i	$-0.537649\pi$
$613$	−5.84314	−0.236002	−0.118001	+	0.993013i	$0.537649\pi$
$614$	−16.4330	−0.663183
$615$	− 20.5359i	− 0.828088i
$616$	−4.39435	−0.177053
$617$	37.0522i	1.49166i	0.666134	+	0.745832i	$0.267948\pi$
$617$	37.0522i	1.49166i	−0.666134	+	0.745832i	$0.732052\pi$
$618$	− 5.01332i	− 0.201665i
$619$	16.1686i	0.649870i	0.945736	+	0.324935i	$0.105342\pi$
$619$	16.1686i	0.649870i	−0.945736	+	0.324935i	$0.894658\pi$
$620$	12.2657	0.492604
$621$	5.09859	0.204599
$622$	− 27.4046i	− 1.09883i
$623$	9.55561i	0.382837i
$624$	− 17.6121i	− 0.705049i
$625$	50.3976	2.01590
$626$	− 34.0550i	− 1.36111i
$627$	−3.92701	−0.156830
$628$	−13.8355	−0.552097
$629$	0	0
$630$	6.19717	0.246901
$631$	31.6670	1.26064	0.630322	−	0.776334i	$-0.282923\pi$
$631$	31.6670	1.26064	0.630322	+	0.776334i	$0.282923\pi$
$632$	24.9389i	0.992018i
$633$	−5.88066	−0.233735
$634$	− 7.05173i	− 0.280060i
$635$	− 11.4286i	− 0.453531i
$636$	1.96732i	0.0780095i
$637$	21.9810	0.870919
$638$	8.26575	0.327244
$639$	− 1.52398i	− 0.0602878i
$640$	54.5026i	2.15440i
$641$	− 15.9293i	− 0.629169i	−0.949229	−	0.314585i	$-0.898135\pi$
$641$	− 15.9293i	− 0.629169i	0.949229	−	0.314585i	$-0.101865\pi$
$642$	−32.4702	−1.28150
$643$	− 16.6863i	− 0.658043i	−0.944322	−	0.329022i	$-0.893281\pi$
$643$	− 16.6863i	− 0.658043i	0.944322	−	0.329022i	$-0.106719\pi$
$644$	−3.43233	−0.135253
$645$	−21.2734	−0.837639
$646$	0	0
$647$	−8.35044	−0.328290	−0.164145	−	0.986436i	$-0.552486\pi$
$647$	−8.35044	−0.328290	−0.164145	+	0.986436i	$0.552486\pi$
$648$	−2.09859	−0.0824403
$649$	− 16.7956i	− 0.659283i
$650$	−68.1316	−2.67234
$651$	3.81209i	0.149408i
$652$	− 9.80398i	− 0.383954i
$653$	− 28.6977i	− 1.12303i	−0.827467	−	0.561514i	$-0.810219\pi$
$653$	− 28.6977i	− 1.12303i	0.827467	−	0.561514i	$-0.189781\pi$
$654$	1.18239	0.0462353
$655$	32.2347	1.25951
$656$	− 24.8819i	− 0.971475i
$657$	1.16502i	0.0454519i
$658$	− 10.6089i	− 0.413578i
$659$	−40.3690	−1.57255	−0.786276	−	0.617876i	$-0.787993\pi$
$659$	−40.3690	−1.57255	−0.786276	+	0.617876i	$0.787993\pi$
$660$	6.73749i	0.262257i
$661$	39.4706	1.53523	0.767614	−	0.640913i	$-0.221444\pi$
$661$	39.4706	1.53523	0.767614	+	0.640913i	$0.221444\pi$
$662$	47.2710	1.83724
$663$	0	0
$664$	−13.3997	−0.520007
$665$	6.48830	0.251606
$666$	− 9.83822i	− 0.381223i
$667$	−11.2359	−0.435055
$668$	− 1.55988i	− 0.0603535i
$669$	5.58387i	0.215885i
$670$	80.9057i	3.12566i
$671$	−8.83145	−0.340934
$672$	3.63728	0.140311
$673$	21.6056i	0.832834i	0.909174	+	0.416417i	$0.136714\pi$
$673$	21.6056i	0.832834i	−0.909174	+	0.416417i	$0.863286\pi$
$674$	51.7754i	1.99431i
$675$	11.5359i	0.444017i
$676$	−0.162168	−0.00623724
$677$	2.51083i	0.0964990i	0.998835	+	0.0482495i	$0.0153643\pi$
$677$	2.51083i	0.0964990i	−0.998835	+	0.0482495i	$0.984636\pi$
$678$	−19.4757	−0.747961
$679$	−13.3464	−0.512187
$680$	0	0
$681$	−27.5299	−1.05495
$682$	−15.5016	−0.593588
$683$	18.7423i	0.717156i	0.933500	+	0.358578i	$0.116738\pi$
$683$	18.7423i	0.717156i	−0.933500	+	0.358578i	$0.883262\pi$
$684$	1.26251	0.0482732
$685$	− 0.419730i	− 0.0160370i
$686$	20.0392i	0.765098i
$687$	3.57019i	0.136211i
$688$	−25.7755	−0.982681
$689$	9.63554	0.367085
$690$	− 34.2557i	− 1.30409i
$691$	38.5702i	1.46728i	0.679538	+	0.733640i	$0.262180\pi$
$691$	38.5702i	1.46728i	−0.679538	+	0.733640i	$0.737820\pi$
$692$	− 3.85352i	− 0.146489i
$693$	−2.09396	−0.0795428
$694$	− 19.8860i	− 0.754862i
$695$	58.7452	2.22833
$696$	4.62470	0.175299
$697$	0	0
$698$	−55.1037	−2.08570
$699$	1.37088	0.0518515
$700$	− 7.76588i	− 0.293523i
$701$	−28.7827	−1.08711	−0.543553	−	0.839375i	$-0.682921\pi$
$701$	−28.7827	−1.08711	−0.543553	+	0.839375i	$0.682921\pi$
$702$	− 5.90604i	− 0.222909i
$703$	− 10.3004i	− 0.388487i
$704$	− 7.57945i	− 0.285661i
$705$	28.3077	1.06613
$706$	−10.8480	−0.408270
$707$	0.245119i	0.00921866i
$708$	5.39965i	0.202931i
$709$	− 5.14588i	− 0.193258i	−0.995320	−	0.0966288i	$-0.969194\pi$
$709$	− 5.14588i	− 0.193258i	0.995320	−	0.0966288i	$-0.0308060\pi$
$710$	−10.2391	−0.384267
$711$	11.8837i	0.445673i
$712$	−21.7407	−0.814768
$713$	21.0718	0.789146
$714$	0	0
$715$	32.9988	1.23409
$716$	5.67538	0.212099
$717$	21.6121i	0.807120i
$718$	−35.6068	−1.32883
$719$	− 31.5022i	− 1.17483i	−0.809285	−	0.587417i	$-0.800145\pi$
$719$	− 31.5022i	− 1.17483i	0.809285	−	0.587417i	$-0.199855\pi$
$720$	20.0354i	0.746675i
$721$	− 2.79877i	− 0.104232i
$722$	−26.4482	−0.984300
$723$	−1.06181	−0.0394890
$724$	− 0.668412i	− 0.0248413i
$725$	− 25.4219i	− 0.944147i
$726$	9.65952i	0.358498i
$727$	31.2775	1.16002	0.580010	−	0.814610i	$-0.303049\pi$
$727$	31.2775	1.16002	0.580010	+	0.814610i	$0.303049\pi$
$728$	− 6.91936i	− 0.256449i
$729$	−1.00000	−0.0370370
$730$	7.82739	0.289705
$731$	0	0
$732$	2.83925	0.104942
$733$	−18.8153	−0.694960	−0.347480	−	0.937687i	$-0.612963\pi$
$733$	−18.8153	−0.694960	−0.347480	+	0.937687i	$0.612963\pi$
$734$	− 58.7603i	− 2.16888i
$735$	−25.0054	−0.922337
$736$	− 20.1055i	− 0.741099i
$737$	− 27.3371i	− 1.00698i
$738$	− 8.34389i	− 0.307143i
$739$	−6.94762	−0.255572	−0.127786	−	0.991802i	$-0.540787\pi$
$739$	−6.94762	−0.255572	−0.127786	+	0.991802i	$0.540787\pi$
$740$	−17.6722	−0.649641
$741$	− 6.18350i	− 0.227156i
$742$	4.10797i	0.150808i
$743$	− 33.9563i	− 1.24574i	−0.782327	−	0.622868i	$-0.785967\pi$
$743$	− 33.9563i	− 1.24574i	0.782327	−	0.622868i	$-0.214033\pi$
$744$	−8.67319	−0.317975
$745$	14.6212i	0.535678i
$746$	−59.2243	−2.16835
$747$	−6.38508	−0.233618
$748$	0	0
$749$	−18.1271	−0.662349
$750$	43.9125	1.60346
$751$	− 6.91509i	− 0.252335i	−0.992009	−	0.126168i	$-0.959732\pi$
$751$	− 6.91509i	− 0.252335i	0.992009	−	0.126168i	$-0.0402677\pi$
$752$	34.2984	1.25073
$753$	− 7.29715i	− 0.265923i
$754$	13.0153i	0.473989i
$755$	− 30.8573i	− 1.12301i
$756$	0.673192	0.0244838
$757$	−1.86166	−0.0676633	−0.0338316	−	0.999428i	$-0.510771\pi$
$757$	−1.86166	−0.0676633	−0.0338316	+	0.999428i	$0.510771\pi$
$758$	− 36.2178i	− 1.31549i
$759$	11.5746i	0.420132i
$760$	14.7621i	0.535476i
$761$	49.2335	1.78471	0.892357	−	0.451330i	$-0.149050\pi$
$761$	49.2335	1.78471	0.892357	+	0.451330i	$0.149050\pi$
$762$	− 4.64354i	− 0.168218i
$763$	0.660093	0.0238970
$764$	−13.9880	−0.506066
$765$	0	0
$766$	56.6893	2.04827
$767$	26.4464	0.954923
$768$	15.4673i	0.558129i
$769$	50.9303	1.83659	0.918296	−	0.395895i	$-0.129565\pi$
$769$	50.9303	1.83659	0.918296	+	0.395895i	$0.129565\pi$
$770$	14.0686i	0.506996i
$771$	22.5569i	0.812366i
$772$	− 3.43855i	− 0.123756i
$773$	−2.91614	−0.104886	−0.0524431	−	0.998624i	$-0.516701\pi$
$773$	−2.91614	−0.104886	−0.0524431	+	0.998624i	$0.516701\pi$
$774$	−8.64354	−0.310686
$775$	47.6764i	1.71259i
$776$	− 30.3654i	− 1.09006i
$777$	− 5.49236i	− 0.197037i
$778$	56.4988	2.02558
$779$	− 8.73586i	− 0.312995i
$780$	−10.6089	−0.379859
$781$	3.45968	0.123797
$782$	0	0
$783$	2.20372	0.0787546
$784$	−30.2972	−1.08204
$785$	− 77.0870i	− 2.75135i
$786$	13.0972	0.467161
$787$	− 50.8932i	− 1.81415i	−0.420974	−	0.907073i	$-0.638312\pi$
$787$	− 50.8932i	− 1.81415i	0.420974	−	0.907073i	$-0.361688\pi$
$788$	− 10.8709i	− 0.387259i
$789$	4.24919i	0.151275i
$790$	79.8424	2.84067
$791$	−10.8727	−0.386588
$792$	− 4.76413i	− 0.169286i
$793$	− 13.9060i	− 0.493818i
$794$	− 0.380602i	− 0.0135070i
$795$	−10.9613	−0.388757
$796$	− 13.7091i	− 0.485906i
$797$	13.6061	0.481952	0.240976	−	0.970531i	$-0.422532\pi$
$797$	13.6061	0.481952	0.240976	+	0.970531i	$0.422532\pi$
$798$	2.63624	0.0933220
$799$	0	0
$800$	45.4901	1.60832
$801$	−10.3597	−0.366042
$802$	− 1.00250i	− 0.0353993i
$803$	−2.64479	−0.0933326
$804$	8.78869i	0.309953i
$805$	− 19.1238i	− 0.674026i
$806$	− 24.4089i	− 0.859768i
$807$	11.8338	0.416570
$808$	−0.557691	−0.0196195
$809$	− 17.3937i	− 0.611529i	−0.952107	−	0.305765i	$-0.901088\pi$
$809$	− 17.3937i	− 0.611529i	0.952107	−	0.305765i	$-0.0989121\pi$
$810$	6.71866i	0.236070i
$811$	− 32.9286i	− 1.15628i	−0.815937	−	0.578140i	$-0.803779\pi$
$811$	− 32.9286i	− 1.15628i	0.815937	−	0.578140i	$-0.196221\pi$
$812$	−1.48353	−0.0520617
$813$	− 11.8045i	− 0.414000i
$814$	22.3343	0.782817
$815$	54.6246	1.91342
$816$	0	0
$817$	−9.04959	−0.316605
$818$	−2.49362	−0.0871875
$819$	− 3.29715i	− 0.115212i
$820$	−14.9879	−0.523401
$821$	43.7368i	1.52643i	0.646147	+	0.763213i	$0.276379\pi$
$821$	43.7368i	1.52643i	−0.646147	+	0.763213i	$0.723621\pi$
$822$	− 0.170539i	− 0.00594824i
$823$	− 1.67402i	− 0.0583528i	−0.999574	−	0.0291764i	$-0.990712\pi$
$823$	− 1.67402i	− 0.0583528i	0.999574	−	0.0291764i	$-0.00928846\pi$
$824$	6.36771	0.221830
$825$	−26.1883	−0.911761
$826$	11.2750i	0.392308i
$827$	2.27016i	0.0789412i	0.999221	+	0.0394706i	$0.0125671\pi$
$827$	2.27016i	0.0789412i	−0.999221	+	0.0394706i	$0.987433\pi$
$828$	− 3.72115i	− 0.129319i
$829$	19.2597	0.668918	0.334459	−	0.942410i	$-0.391446\pi$
$829$	19.2597	0.668918	0.334459	+	0.942410i	$0.391446\pi$
$830$	42.8992i	1.48905i
$831$	−10.4275	−0.361727
$832$	11.9346	0.413760
$833$	0	0
$834$	23.8686	0.826502
$835$	8.69114	0.300769
$836$	2.86609i	0.0991259i
$837$	−4.13287	−0.142853
$838$	20.0246i	0.691737i
$839$	− 14.8059i	− 0.511157i	−0.966788	−	0.255579i	$-0.917734\pi$
$839$	− 14.8059i	− 0.511157i	0.966788	−	0.255579i	$-0.0822660\pi$
$840$	7.87140i	0.271589i
$841$	24.1436	0.832538
$842$	−29.5302	−1.01768
$843$	5.46697i	0.188293i
$844$	4.29194i	0.147735i
$845$	− 0.903549i	− 0.0310830i
$846$	11.5016	0.395434
$847$	5.39259i	0.185292i
$848$	−13.2810	−0.456073
$849$	17.6375	0.605318
$850$	0	0
$851$	−30.3597	−1.04072
$852$	−1.11226	−0.0381055
$853$	6.35771i	0.217684i	0.994059	+	0.108842i	$0.0347142\pi$
$853$	6.35771i	0.217684i	−0.994059	+	0.108842i	$0.965286\pi$
$854$	5.92864	0.202874
$855$	7.03429i	0.240568i
$856$	− 41.2423i	− 1.40963i
$857$	40.8311i	1.39477i	0.716699	+	0.697383i	$0.245652\pi$
$857$	40.8311i	1.39477i	−0.716699	+	0.697383i	$0.754348\pi$
$858$	13.4077	0.457730
$859$	−22.1464	−0.755626	−0.377813	−	0.925882i	$-0.623324\pi$
$859$	−22.1464	−0.755626	−0.377813	+	0.925882i	$0.623324\pi$
$860$	15.5262i	0.529439i
$861$	− 4.65812i	− 0.158748i
$862$	− 18.3993i	− 0.626684i
$863$	17.3423	0.590339	0.295170	−	0.955445i	$-0.404624\pi$
$863$	17.3423	0.590339	0.295170	+	0.955445i	$0.404624\pi$
$864$	3.94335i	0.134156i
$865$	21.4706	0.730021
$866$	48.1097	1.63483
$867$	0	0
$868$	2.78222	0.0944346
$869$	−26.9779	−0.915161
$870$	− 14.8061i	− 0.501973i
$871$	43.0452	1.45853
$872$	1.50183i	0.0508584i
$873$	− 14.4695i	− 0.489717i
$874$	− 14.5722i	− 0.492911i
$875$	24.5149	0.828756
$876$	0.850281	0.0287283
$877$	− 30.2306i	− 1.02081i	−0.859933	−	0.510407i	$-0.829495\pi$
$877$	− 30.2306i	− 1.02081i	0.859933	−	0.510407i	$-0.170505\pi$
$878$	37.3333i	1.25994i
$879$	− 8.25845i	− 0.278551i
$880$	−45.4835	−1.53325
$881$	10.5647i	0.355933i	0.984037	+	0.177966i	$0.0569519\pi$
$881$	10.5647i	0.355933i	−0.984037	+	0.177966i	$0.943048\pi$
$882$	−10.1599	−0.342100
$883$	13.9150	0.468276	0.234138	−	0.972203i	$-0.424773\pi$
$883$	13.9150	0.468276	0.234138	+	0.972203i	$0.424773\pi$
$884$	0	0
$885$	−30.0851	−1.01130
$886$	53.1976	1.78721
$887$	35.9644i	1.20757i	0.797149	+	0.603783i	$0.206341\pi$
$887$	35.9644i	1.20757i	−0.797149	+	0.603783i	$0.793659\pi$
$888$	12.4961	0.419342
$889$	− 2.59234i	− 0.0869442i
$890$	69.6033i	2.33311i
$891$	− 2.27016i	− 0.0760532i
$892$	4.07533	0.136452
$893$	12.0419	0.402968
$894$	5.94069i	0.198686i
$895$	31.6214i	1.05699i
$896$	12.3627i	0.413010i
$897$	−18.2254	−0.608530
$898$	− 1.85098i	− 0.0617681i
$899$	9.10771	0.303759
$900$	8.41937	0.280646
$901$	0	0
$902$	18.9420	0.630698
$903$	−4.82541	−0.160579
$904$	− 24.7373i	− 0.822750i
$905$	3.72418	0.123796
$906$	− 12.5375i	− 0.416532i
$907$	39.1843i	1.30109i	0.759466	+	0.650547i	$0.225460\pi$
$907$	39.1843i	1.30109i	−0.759466	+	0.650547i	$0.774540\pi$
$908$	20.0924i	0.666790i
$909$	−0.265746	−0.00881423
$910$	−22.1524	−0.734347
$911$	− 9.92806i	− 0.328931i	−0.986383	−	0.164466i	$-0.947410\pi$
$911$	− 9.92806i	− 0.328931i	0.986383	−	0.164466i	$-0.0525900\pi$
$912$	8.52295i	0.282223i
$913$	− 14.4952i	− 0.479720i
$914$	−66.2202	−2.19037
$915$	15.8194i	0.522973i
$916$	2.60567	0.0860938
$917$	7.31174	0.241455
$918$	0	0
$919$	42.8795	1.41446	0.707232	−	0.706982i	$-0.249944\pi$
$919$	42.8795	1.41446	0.707232	+	0.706982i	$0.249944\pi$
$920$	43.5102	1.43449
$921$	− 9.94601i	− 0.327732i
$922$	−8.25442	−0.271845
$923$	5.44763i	0.179311i
$924$	1.52825i	0.0502758i
$925$	− 68.6909i	− 2.25854i
$926$	64.5113	2.11997
$927$	3.03429	0.0996590
$928$	− 8.69006i	− 0.285265i
$929$	18.1624i	0.595888i	0.954583	+	0.297944i	$0.0963008\pi$
$929$	18.1624i	0.595888i	−0.954583	+	0.297944i	$0.903699\pi$
$930$	27.7674i	0.910528i
$931$	−10.6372	−0.348619
$932$	− 1.00053i	− 0.0327733i
$933$	16.5865	0.543019
$934$	−17.3883	−0.568963
$935$	0	0
$936$	7.50162	0.245198
$937$	−1.60243	−0.0523492	−0.0261746	−	0.999657i	$-0.508333\pi$
$937$	−1.60243	−0.0523492	−0.0261746	+	0.999657i	$0.508333\pi$
$938$	18.3517i	0.599204i
$939$	20.6116	0.672634
$940$	− 20.6601i	− 0.673858i
$941$	− 32.3544i	− 1.05472i	−0.849641	−	0.527362i	$-0.823181\pi$
$941$	− 32.3544i	− 1.05472i	0.849641	−	0.527362i	$-0.176819\pi$
$942$	− 31.3210i	− 1.02049i
$943$	−25.7484	−0.838482
$944$	−36.4520	−1.18641
$945$	3.75081i	0.122014i
$946$	− 19.6222i	− 0.637973i
$947$	25.1819i	0.818301i	0.912467	+	0.409151i	$0.134175\pi$
$947$	25.1819i	0.818301i	−0.912467	+	0.409151i	$0.865825\pi$
$948$	8.67319	0.281692
$949$	− 4.16450i	− 0.135185i
$950$	32.9706	1.06971
$951$	4.26802	0.138400
$952$	0	0
$953$	54.0658	1.75136	0.875681	−	0.482889i	$-0.160413\pi$
$953$	54.0658	1.75136	0.875681	+	0.482889i	$0.160413\pi$
$954$	−4.45366	−0.144192
$955$	− 77.9364i	− 2.52196i
$956$	15.7734	0.510148
$957$	5.00280i	0.161718i
$958$	− 42.8983i	− 1.38598i
$959$	− 0.0952065i	− 0.00307438i
$960$	−13.5767	−0.438187
$961$	13.9194	0.449012
$962$	35.1677i	1.13385i
$963$	− 19.6524i	− 0.633291i
$964$	0.774948i	0.0249594i
$965$	19.1585	0.616733
$966$	− 7.77015i	− 0.250001i
$967$	2.17495	0.0699418	0.0349709	−	0.999388i	$-0.488866\pi$
$967$	2.17495	0.0699418	0.0349709	+	0.999388i	$0.488866\pi$
$968$	−12.2691	−0.394345
$969$	0	0
$970$	−97.2154	−3.12140
$971$	−47.4481	−1.52268	−0.761340	−	0.648352i	$-0.775458\pi$
$971$	−47.4481	−1.52268	−0.761340	+	0.648352i	$0.775458\pi$
$972$	0.729840i	0.0234096i
$973$	13.3251	0.427182
$974$	− 42.1987i	− 1.35213i
$975$	− 41.2363i	− 1.32062i
$976$	19.1672i	0.613528i
$977$	20.8226	0.666175	0.333087	−	0.942896i	$-0.391910\pi$
$977$	20.8226	0.666175	0.333087	+	0.942896i	$0.391910\pi$
$978$	22.1944	0.709698
$979$	− 23.5182i	− 0.751644i
$980$	18.2499i	0.582973i
$981$	0.715639i	0.0228486i
$982$	4.14319	0.132214
$983$	2.43039i	0.0775173i	0.999249	+	0.0387586i	$0.0123403\pi$
$983$	2.43039i	0.0775173i	−0.999249	+	0.0387586i	$0.987660\pi$
$984$	10.5981	0.337854
$985$	60.5690	1.92989
$986$	0	0
$987$	6.42098	0.204382
$988$	−4.51297	−0.143577
$989$	26.6731i	0.848154i
$990$	−15.2524	−0.484754
$991$	36.5777i	1.16193i	0.813929	+	0.580964i	$0.197324\pi$
$991$	36.5777i	1.16193i	−0.813929	+	0.580964i	$0.802676\pi$
$992$	16.2974i	0.517442i
$993$	28.6105i	0.907927i
$994$	−2.32252	−0.0736658
$995$	76.3827	2.42150
$996$	4.66009i	0.147661i
$997$	− 20.7109i	− 0.655920i	−0.944692	−	0.327960i	$-0.893639\pi$
$997$	− 20.7109i	− 0.655920i	0.944692	−	0.327960i	$-0.106361\pi$
$998$	− 25.1585i	− 0.796378i
$999$	5.95453	0.188393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.f.577.6		8
17.2	even	8		51.2.e.a.13.3	yes	8
17.3	odd	16		867.2.h.i.688.3		16
17.4	even	4		867.2.a.l.1.2		4
17.5	odd	16		867.2.h.k.757.1		16
17.6	odd	16		867.2.h.i.712.4		16
17.7	odd	16		867.2.h.k.733.2		16
17.8	even	8		51.2.e.a.4.2	✓	8
17.9	even	8		867.2.e.g.616.2		8
17.10	odd	16		867.2.h.k.733.1		16
17.11	odd	16		867.2.h.i.712.3		16
17.12	odd	16		867.2.h.k.757.2		16
17.13	even	4		867.2.a.k.1.2		4
17.14	odd	16		867.2.h.i.688.4		16
17.15	even	8		867.2.e.g.829.3		8
17.16	even	2	inner	867.2.d.f.577.5		8
51.2	odd	8		153.2.f.b.64.2		8
51.8	odd	8		153.2.f.b.55.3		8
51.38	odd	4		2601.2.a.be.1.3		4
51.47	odd	4		2601.2.a.bf.1.3		4
68.19	odd	8		816.2.bd.e.625.1		8
68.59	odd	8		816.2.bd.e.769.1		8
204.59	even	8		2448.2.be.x.1585.4		8
204.155	even	8		2448.2.be.x.1441.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.2	✓	8	17.8	even	8
51.2.e.a.13.3	yes	8	17.2	even	8
153.2.f.b.55.3		8	51.8	odd	8
153.2.f.b.64.2		8	51.2	odd	8
816.2.bd.e.625.1		8	68.19	odd	8
816.2.bd.e.769.1		8	68.59	odd	8
867.2.a.k.1.2		4	17.13	even	4
867.2.a.l.1.2		4	17.4	even	4
867.2.d.f.577.5		8	17.16	even	2	inner
867.2.d.f.577.6		8	1.1	even	1	trivial
867.2.e.g.616.2		8	17.9	even	8
867.2.e.g.829.3		8	17.15	even	8
867.2.h.i.688.3		16	17.3	odd	16
867.2.h.i.688.4		16	17.14	odd	16
867.2.h.i.712.3		16	17.11	odd	16
867.2.h.i.712.4		16	17.6	odd	16
867.2.h.k.733.1		16	17.10	odd	16
867.2.h.k.733.2		16	17.7	odd	16
867.2.h.k.757.1		16	17.5	odd	16
867.2.h.k.757.2		16	17.12	odd	16
2448.2.be.x.1441.4		8	204.155	even	8
2448.2.be.x.1585.4		8	204.59	even	8
2601.2.a.be.1.3		4	51.38	odd	4
2601.2.a.bf.1.3		4	51.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.65222 q^{2} +1.00000i q^{3} +0.729840 q^{4} +4.06644i q^{5} +1.65222i q^{6} +0.922382i q^{7} -2.09859 q^{8} -1.00000 q^{9} +6.71866i q^{10} -2.27016i q^{11} +0.729840i q^{12} +3.57461 q^{13} +1.52398i q^{14} -4.06644 q^{15} -4.92701 q^{16} -1.65222 q^{18} -1.72984 q^{19} +2.96785i q^{20} -0.922382 q^{21} -3.75081i q^{22} +5.09859i q^{23} -2.09859i q^{24} -11.5359 q^{25} +5.90604 q^{26} -1.00000i q^{27} +0.673192i q^{28} +2.20372i q^{29} -6.71866 q^{30} -4.13287i q^{31} -3.94335 q^{32} +2.27016 q^{33} -3.75081 q^{35} -0.729840 q^{36} +5.95453i q^{37} -2.85808 q^{38} +3.57461i q^{39} -8.53377i q^{40} +5.05010i q^{41} -1.52398 q^{42} +5.23146 q^{43} -1.65685i q^{44} -4.06644i q^{45} +8.42400i q^{46} -6.96130 q^{47} -4.92701i q^{48} +6.14921 q^{49} -19.0599 q^{50} +2.60889 q^{52} +2.69555 q^{53} -1.65222i q^{54} +9.23146 q^{55} -1.93570i q^{56} -1.72984i q^{57} +3.64104i q^{58} +7.39840 q^{59} -2.96785 q^{60} -3.89023i q^{61} -6.82843i q^{62} -0.922382i q^{63} +3.33873 q^{64} +14.5359i q^{65} +3.75081 q^{66} +12.0419 q^{67} -5.09859 q^{69} -6.19717 q^{70} +1.52398i q^{71} +2.09859 q^{72} -1.16502i q^{73} +9.83822i q^{74} -11.5359i q^{75} -1.26251 q^{76} +2.09396 q^{77} +5.90604i q^{78} -11.8837i q^{79} -20.0354i q^{80} +1.00000 q^{81} +8.34389i q^{82} +6.38508 q^{83} -0.673192 q^{84} +8.64354 q^{86} -2.20372 q^{87} +4.76413i q^{88} +10.3597 q^{89} -6.71866i q^{90} +3.29715i q^{91} +3.72115i q^{92} +4.13287 q^{93} -11.5016 q^{94} -7.03429i q^{95} -3.94335i q^{96} +14.4695i q^{97} +10.1599 q^{98} +2.27016i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more