LMFDB - Embedded newform 867.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, 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("867.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.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:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.06644$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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	$-0.698572\pi$
$2$	−1.65222	−1.16830	−0.584149	+	0.811646i	$0.698572\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.729840	0.364920
$5$	4.06644	1.81857	0.909283	−	0.416179i	$-0.136631\pi$
$5$	4.06644	1.81857	0.909283	+	0.416179i	$0.136631\pi$
$6$	−1.65222	−0.674517
$7$	−0.922382	−0.348628	−0.174314	−	0.984690i	$-0.555771\pi$
$7$	−0.922382	−0.348628	−0.174314	+	0.984690i	$0.555771\pi$
$8$	2.09859	0.741962
$9$	1.00000	0.333333
$10$	−6.71866	−2.12463
$11$	2.27016	0.684479	0.342239	−	0.939613i	$-0.388815\pi$
$11$	2.27016	0.684479	0.342239	+	0.939613i	$0.388815\pi$
$12$	0.729840	0.210687
$13$	3.57461	0.991417	0.495709	−	0.868489i	$-0.334908\pi$
$13$	3.57461	0.991417	0.495709	+	0.868489i	$0.334908\pi$
$14$	1.52398	0.407301
$15$	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	$-0.436417\pi$
$19$	1.72984	0.396853	0.198426	+	0.980116i	$0.436417\pi$
$20$	2.96785	0.663631
$21$	−0.922382	−0.201280
$22$	−3.75081	−0.799675
$23$	−5.09859	−1.06313	−0.531564	−	0.847018i	$-0.678396\pi$
$23$	−5.09859	−1.06313	−0.531564	+	0.847018i	$0.678396\pi$
$24$	2.09859	0.428372
$25$	11.5359	2.30718
$26$	−5.90604	−1.15827
$27$	1.00000	0.192450
$28$	−0.673192	−0.127221
$29$	2.20372	0.409221	0.204611	−	0.978843i	$-0.434407\pi$
$29$	2.20372	0.409221	0.204611	+	0.978843i	$0.434407\pi$
$30$	−6.71866	−1.22665
$31$	−4.13287	−0.742286	−0.371143	−	0.928576i	$-0.621034\pi$
$31$	−4.13287	−0.742286	−0.371143	+	0.928576i	$0.621034\pi$
$32$	3.94335	0.697093
$33$	2.27016	0.395184
$34$	0	0
$35$	−3.75081	−0.634003
$36$	0.729840	0.121640
$37$	5.95453	0.978919	0.489460	−	0.872026i	$-0.337194\pi$
$37$	5.95453	0.978919	0.489460	+	0.872026i	$0.337194\pi$
$38$	−2.85808	−0.463642
$39$	3.57461	0.572395
$40$	8.53377	1.34931
$41$	−5.05010	−0.788693	−0.394346	−	0.918962i	$-0.629029\pi$
$41$	−5.05010	−0.788693	−0.394346	+	0.918962i	$0.629029\pi$
$42$	1.52398	0.235155
$43$	−5.23146	−0.797790	−0.398895	−	0.916997i	$-0.630606\pi$
$43$	−5.23146	−0.797790	−0.398895	+	0.916997i	$0.630606\pi$
$44$	1.65685	0.249780
$45$	4.06644	0.606189
$46$	8.42400	1.24205
$47$	−6.96130	−1.01541	−0.507705	−	0.861531i	$-0.669506\pi$
$47$	−6.96130	−1.01541	−0.507705	+	0.861531i	$0.669506\pi$
$48$	−4.92701	−0.711153
$49$	−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	$-0.559271\pi$
$53$	−2.69555	−0.370263	−0.185131	+	0.982714i	$0.559271\pi$
$54$	−1.65222	−0.224839
$55$	9.23146	1.24477
$56$	−1.93570	−0.258669
$57$	1.72984	0.229123
$58$	−3.64104	−0.478092
$59$	−7.39840	−0.963190	−0.481595	−	0.876394i	$-0.659942\pi$
$59$	−7.39840	−0.963190	−0.481595	+	0.876394i	$0.659942\pi$
$60$	2.96785	0.383148
$61$	3.89023	0.498093	0.249047	−	0.968492i	$-0.419883\pi$
$61$	3.89023	0.498093	0.249047	+	0.968492i	$0.419883\pi$
$62$	6.82843	0.867211
$63$	−0.922382	−0.116209
$64$	3.33873	0.417341
$65$	14.5359	1.80296
$66$	−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
$71$	1.52398	0.180863	0.0904317	−	0.995903i	$-0.471175\pi$
$71$	1.52398	0.180863	0.0904317	+	0.995903i	$0.471175\pi$
$72$	2.09859	0.247321
$73$	−1.16502	−0.136356	−0.0681778	−	0.997673i	$-0.521719\pi$
$73$	−1.16502	−0.136356	−0.0681778	+	0.997673i	$0.521719\pi$
$74$	−9.83822	−1.14367
$75$	11.5359	1.33205
$76$	1.26251	0.144820
$77$	−2.09396	−0.238628
$78$	−5.90604	−0.668728
$79$	11.8837	1.33702	0.668509	−	0.743704i	$-0.266933\pi$
$79$	11.8837	1.33702	0.668509	+	0.743704i	$0.266933\pi$
$80$	−20.0354	−2.24002
$81$	1.00000	0.111111
$82$	8.34389	0.921428
$83$	−6.38508	−0.700854	−0.350427	−	0.936590i	$-0.613964\pi$
$83$	−6.38508	−0.700854	−0.350427	+	0.936590i	$0.613964\pi$
$84$	−0.673192	−0.0734513
$85$	0	0
$86$	8.64354	0.932057
$87$	2.20372	0.236264
$88$	4.76413	0.507858
$89$	10.3597	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$89$	10.3597	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$90$	−6.71866	−0.708209
$91$	−3.29715	−0.345636
$92$	−3.72115	−0.387957
$93$	−4.13287	−0.428559
$94$	11.5016	1.18630
$95$	7.03429	0.721703
$96$	3.94335	0.402467
$97$	14.4695	1.46915	0.734576	−	0.678526i	$-0.237381\pi$
$97$	14.4695	1.46915	0.734576	+	0.678526i	$0.237381\pi$
$98$	10.1599	1.02630
$99$	2.27016	0.228160
$100$	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
$105$	−3.75081	−0.366042
$106$	4.45366	0.432577
$107$	19.6524	1.89987	0.949937	−	0.312443i	$-0.101147\pi$
$107$	19.6524	1.89987	0.949937	+	0.312443i	$0.101147\pi$
$108$	0.729840	0.0702289
$109$	0.715639	0.0685457	0.0342729	−	0.999413i	$-0.489088\pi$
$109$	0.715639	0.0685457	0.0342729	+	0.999413i	$0.489088\pi$
$110$	−15.2524	−1.45426
$111$	5.95453	0.565179
$112$	4.54459	0.429423
$113$	−11.7876	−1.10888	−0.554442	−	0.832223i	$-0.687068\pi$
$113$	−11.7876	−1.10888	−0.554442	+	0.832223i	$0.687068\pi$
$114$	−2.85808	−0.267684
$115$	−20.7331	−1.93337
$116$	1.60837	0.149333
$117$	3.57461	0.330472
$118$	12.2238	1.12529
$119$	0	0
$120$	8.53377	0.779023
$121$	−5.84638	−0.531489
$122$	−6.42753	−0.581921
$123$	−5.05010	−0.455352
$124$	−3.01634	−0.270875
$125$	26.5778	2.37719
$126$	1.52398	0.135767
$127$	2.81048	0.249390	0.124695	−	0.992195i	$-0.460205\pi$
$127$	2.81048	0.249390	0.124695	+	0.992195i	$0.460205\pi$
$128$	−13.4030	−1.18467
$129$	−5.23146	−0.460604
$130$	−24.0166	−2.10639
$131$	−7.92701	−0.692586	−0.346293	−	0.938126i	$-0.612560\pi$
$131$	−7.92701	−0.692586	−0.346293	+	0.938126i	$0.612560\pi$
$132$	1.65685	0.144211
$133$	−1.59557	−0.138354
$134$	−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
$139$	−14.4464	−1.22532	−0.612662	−	0.790345i	$-0.709901\pi$
$139$	−14.4464	−1.22532	−0.612662	+	0.790345i	$0.709901\pi$
$140$	−2.73749	−0.231360
$141$	−6.96130	−0.586247
$142$	−2.51796	−0.211302
$143$	8.11492	0.678604
$144$	−4.92701	−0.410584
$145$	8.96130	0.744195
$146$	1.92488	0.159304
$147$	−6.14921	−0.507178
$148$	4.34586	0.357227
$149$	3.59557	0.294561	0.147280	−	0.989095i	$-0.452948\pi$
$149$	3.59557	0.294561	0.147280	+	0.989095i	$0.452948\pi$
$150$	−19.0599	−1.55623
$151$	7.58828	0.617526	0.308763	−	0.951139i	$-0.400085\pi$
$151$	7.58828	0.617526	0.308763	+	0.951139i	$0.400085\pi$
$152$	3.63022	0.294450
$153$	0	0
$154$	3.45968	0.278789
$155$	−16.8061	−1.34990
$156$	2.60889	0.208878
$157$	−18.9569	−1.51292	−0.756462	−	0.654038i	$-0.773074\pi$
$157$	−18.9569	−1.51292	−0.756462	+	0.654038i	$0.773074\pi$
$158$	−19.6345	−1.56204
$159$	−2.69555	−0.213771
$160$	16.0354	1.26771
$161$	4.70285	0.370636
$162$	−1.65222	−0.129811
$163$	13.4330	1.05216	0.526079	−	0.850436i	$-0.323662\pi$
$163$	13.4330	1.05216	0.526079	+	0.850436i	$0.323662\pi$
$164$	−3.68577	−0.287810
$165$	9.23146	0.718668
$166$	10.5496	0.818806
$167$	−2.13729	−0.165388	−0.0826941	−	0.996575i	$-0.526352\pi$
$167$	−2.13729	−0.165388	−0.0826941	+	0.996575i	$0.526352\pi$
$168$	−1.93570	−0.149342
$169$	−0.222197	−0.0170921
$170$	0	0
$171$	1.72984	0.132284
$172$	−3.81813	−0.291130
$173$	−5.27995	−0.401427	−0.200713	−	0.979650i	$-0.564326\pi$
$173$	−5.27995	−0.401427	−0.200713	+	0.979650i	$0.564326\pi$
$174$	−3.64104	−0.276027
$175$	−10.6405	−0.804347
$176$	−11.1851	−0.843109
$177$	−7.39840	−0.556098
$178$	−17.1165	−1.28294
$179$	−7.77619	−0.581220	−0.290610	−	0.956842i	$-0.593858\pi$
$179$	−7.77619	−0.581220	−0.290610	+	0.956842i	$0.593858\pi$
$180$	2.96785	0.221210
$181$	0.915833	0.0680733	0.0340367	−	0.999421i	$-0.489164\pi$
$181$	0.915833	0.0680733	0.0340367	+	0.999421i	$0.489164\pi$
$182$	5.44763	0.403805
$183$	3.89023	0.287574
$184$	−10.6998	−0.788802
$185$	24.2137	1.78023
$186$	6.82843	0.500685
$187$	0	0
$188$	−5.08064	−0.370544
$189$	−0.922382	−0.0670934
$190$	−11.6222	−0.843164
$191$	−19.1658	−1.38679	−0.693393	−	0.720560i	$-0.743885\pi$
$191$	−19.1658	−1.38679	−0.693393	+	0.720560i	$0.743885\pi$
$192$	3.33873	0.240952
$193$	4.71137	0.339132	0.169566	−	0.985519i	$-0.445763\pi$
$193$	4.71137	0.339132	0.169566	+	0.985519i	$0.445763\pi$
$194$	−23.9068	−1.71641
$195$	14.5359	1.04094
$196$	−4.48794	−0.320567
$197$	14.8949	1.06122	0.530608	−	0.847618i	$-0.321964\pi$
$197$	14.8949	1.06122	0.530608	+	0.847618i	$0.321964\pi$
$198$	−3.75081	−0.266558
$199$	−18.7837	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$199$	−18.7837	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$200$	24.2091	1.71184
$201$	12.0419	0.849373
$202$	−0.439071	−0.0308929
$203$	−2.03268	−0.142666
$204$	0	0
$205$	−20.5359	−1.43429
$206$	5.01332	0.349294
$207$	−5.09859	−0.354376
$208$	−17.6121	−1.22118
$209$	3.92701	0.271637
$210$	6.19717	0.427646
$211$	−5.88066	−0.404841	−0.202421	−	0.979299i	$-0.564881\pi$
$211$	−5.88066	−0.404841	−0.202421	+	0.979299i	$0.564881\pi$
$212$	−1.96732	−0.135116
$213$	1.52398	0.104421
$214$	−32.4702	−2.21962
$215$	−21.2734	−1.45083
$216$	2.09859	0.142791
$217$	3.81209	0.258781
$218$	−1.18239	−0.0800819
$219$	−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	$-0.559864\pi$
$223$	−5.58387	−0.373923	−0.186962	+	0.982367i	$0.559864\pi$
$224$	−3.63728	−0.243026
$225$	11.5359	0.769060
$226$	19.4757	1.29551
$227$	−27.5299	−1.82722	−0.913611	−	0.406589i	$-0.866718\pi$
$227$	−27.5299	−1.82722	−0.913611	+	0.406589i	$0.866718\pi$
$228$	1.26251	0.0836116
$229$	−3.57019	−0.235925	−0.117962	−	0.993018i	$-0.537636\pi$
$229$	−3.57019	−0.235925	−0.117962	+	0.993018i	$0.537636\pi$
$230$	34.2557	2.25875
$231$	−2.09396	−0.137772
$232$	4.62470	0.303627
$233$	−1.37088	−0.0898095	−0.0449047	−	0.998991i	$-0.514298\pi$
$233$	−1.37088	−0.0898095	−0.0449047	+	0.998991i	$0.514298\pi$
$234$	−5.90604	−0.386090
$235$	−28.3077	−1.84659
$236$	−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
$241$	1.06181	0.0683969	0.0341984	−	0.999415i	$-0.489112\pi$
$241$	1.06181	0.0683969	0.0341984	+	0.999415i	$0.489112\pi$
$242$	9.65952	0.620937
$243$	1.00000	0.0641500
$244$	2.83925	0.181764
$245$	−25.0054	−1.59753
$246$	8.34389	0.531987
$247$	6.18350	0.393446
$248$	−8.67319	−0.550748
$249$	−6.38508	−0.404638
$250$	−43.9125	−2.77727
$251$	−7.29715	−0.460592	−0.230296	−	0.973121i	$-0.573969\pi$
$251$	−7.29715	−0.460592	−0.230296	+	0.973121i	$0.573969\pi$
$252$	−0.673192	−0.0424071
$253$	−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	$-0.748394\pi$
$257$	−22.5569	−1.40706	−0.703530	+	0.710666i	$0.748394\pi$
$258$	8.64354	0.538123
$259$	−5.49236	−0.341278
$260$	10.6089	0.657936
$261$	2.20372	0.136407
$262$	13.0972	0.809147
$263$	−4.24919	−0.262016	−0.131008	−	0.991381i	$-0.541821\pi$
$263$	−4.24919	−0.262016	−0.131008	+	0.991381i	$0.541821\pi$
$264$	4.76413	0.293212
$265$	−10.9613	−0.673347
$266$	2.63624	0.161639
$267$	10.3597	0.634003
$268$	8.78869	0.536855
$269$	−11.8338	−0.721520	−0.360760	−	0.932659i	$-0.617483\pi$
$269$	−11.8338	−0.721520	−0.360760	+	0.932659i	$0.617483\pi$
$270$	−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
$275$	26.1883	1.57922
$276$	−3.72115	−0.223987
$277$	10.4275	0.626530	0.313265	−	0.949666i	$-0.398577\pi$
$277$	10.4275	0.626530	0.313265	+	0.949666i	$0.398577\pi$
$278$	23.8686	1.43154
$279$	−4.13287	−0.247429
$280$	−7.87140	−0.470406
$281$	−5.46697	−0.326132	−0.163066	−	0.986615i	$-0.552138\pi$
$281$	−5.46697	−0.326132	−0.163066	+	0.986615i	$0.552138\pi$
$282$	11.5016	0.684911
$283$	17.6375	1.04844	0.524221	−	0.851582i	$-0.324357\pi$
$283$	17.6375	1.04844	0.524221	+	0.851582i	$0.324357\pi$
$284$	1.11226	0.0660007
$285$	7.03429	0.416675
$286$	−13.4077	−0.792812
$287$	4.65812	0.274960
$288$	3.94335	0.232364
$289$	0	0
$290$	−14.8061	−0.869442
$291$	14.4695	0.848215
$292$	−0.850281	−0.0497589
$293$	−8.25845	−0.482464	−0.241232	−	0.970467i	$-0.577551\pi$
$293$	−8.25845	−0.482464	−0.241232	+	0.970467i	$0.577551\pi$
$294$	10.1599	0.592535
$295$	−30.0851	−1.75162
$296$	12.4961	0.726321
$297$	2.27016	0.131728
$298$	−5.94069	−0.344135
$299$	−18.2254	−1.05400
$300$	8.41937	0.486093
$301$	4.82541	0.278132
$302$	−12.5375	−0.721454
$303$	0.265746	0.0152667
$304$	−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
$309$	−3.03429	−0.172615
$310$	27.7674	1.57708
$311$	−16.5865	−0.940536	−0.470268	−	0.882524i	$-0.655843\pi$
$311$	−16.5865	−0.940536	−0.470268	+	0.882524i	$0.655843\pi$
$312$	7.50162	0.424696
$313$	20.6116	1.16504	0.582518	−	0.812818i	$-0.302067\pi$
$313$	20.6116	1.16504	0.582518	+	0.812818i	$0.302067\pi$
$314$	31.3210	1.76755
$315$	−3.75081	−0.211334
$316$	8.67319	0.487905
$317$	4.26802	0.239716	0.119858	−	0.992791i	$-0.461756\pi$
$317$	4.26802	0.239716	0.119858	+	0.992791i	$0.461756\pi$
$318$	4.45366	0.249749
$319$	5.00280	0.280103
$320$	13.5767	0.758963
$321$	19.6524	1.09689
$322$	−7.77015	−0.433014
$323$	0	0
$324$	0.729840	0.0405467
$325$	41.2363	2.28738
$326$	−22.1944	−1.22923
$327$	0.715639	0.0395749
$328$	−10.5981	−0.585181
$329$	6.42098	0.354000
$330$	−15.2524	−0.839619
$331$	−28.6105	−1.57258	−0.786288	−	0.617860i	$-0.788000\pi$
$331$	−28.6105	−1.57258	−0.786288	+	0.617860i	$0.788000\pi$
$332$	−4.66009	−0.255756
$333$	5.95453	0.326306
$334$	3.53127	0.193223
$335$	48.9678	2.67540
$336$	4.54459	0.247928
$337$	31.3368	1.70703	0.853513	−	0.521072i	$-0.174468\pi$
$337$	31.3368	1.70703	0.853513	+	0.521072i	$0.174468\pi$
$338$	0.367119	0.0199686
$339$	−11.7876	−0.640214
$340$	0	0
$341$	−9.38228	−0.508079
$342$	−2.85808	−0.154547
$343$	12.1286	0.654883
$344$	−10.9787	−0.591930
$345$	−20.7331	−1.11623
$346$	8.72365	0.468986
$347$	12.0359	0.646121	0.323060	−	0.946378i	$-0.395288\pi$
$347$	12.0359	0.646121	0.323060	+	0.946378i	$0.395288\pi$
$348$	1.60837	0.0862175
$349$	33.3512	1.78525	0.892625	−	0.450799i	$-0.148861\pi$
$349$	33.3512	1.78525	0.892625	+	0.450799i	$0.148861\pi$
$350$	17.5805	0.939718
$351$	3.57461	0.190798
$352$	8.95204	0.477145
$353$	−6.56570	−0.349457	−0.174729	−	0.984617i	$-0.555905\pi$
$353$	−6.56570	−0.349457	−0.174729	+	0.984617i	$0.555905\pi$
$354$	12.2238	0.649688
$355$	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	$-0.307445\pi$
$359$	21.5508	1.13741	0.568705	+	0.822541i	$0.307445\pi$
$360$	8.53377	0.449769
$361$	−16.0077	−0.842508
$362$	−1.51316	−0.0795299
$363$	−5.84638	−0.306855
$364$	−2.40640	−0.126129
$365$	−4.73749	−0.247972
$366$	−6.42753	−0.335972
$367$	35.5644	1.85645	0.928223	−	0.372025i	$-0.121336\pi$
$367$	35.5644	1.85645	0.928223	+	0.372025i	$0.121336\pi$
$368$	25.1208	1.30951
$369$	−5.05010	−0.262898
$370$	−40.0065	−2.07984
$371$	2.48633	0.129084
$372$	−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
$377$	7.87744	0.405709
$378$	1.52398	0.0783851
$379$	−21.9206	−1.12599	−0.562994	−	0.826461i	$-0.690350\pi$
$379$	−21.9206	−1.12599	−0.562994	+	0.826461i	$0.690350\pi$
$380$	5.13391	0.263364
$381$	2.81048	0.143985
$382$	31.6661	1.62018
$383$	−34.3109	−1.75321	−0.876603	−	0.481215i	$-0.840196\pi$
$383$	−34.3109	−1.75321	−0.876603	+	0.481215i	$0.840196\pi$
$384$	−13.4030	−0.683971
$385$	−8.51494	−0.433961
$386$	−7.78423	−0.396207
$387$	−5.23146	−0.265930
$388$	10.5604	0.536123
$389$	−34.1956	−1.73379	−0.866894	−	0.498493i	$-0.833887\pi$
$389$	−34.1956	−1.73379	−0.866894	+	0.498493i	$0.833887\pi$
$390$	−24.0166	−1.21613
$391$	0	0
$392$	−12.9047	−0.651783
$393$	−7.92701	−0.399865
$394$	−24.6096	−1.23982
$395$	48.3242	2.43146
$396$	1.65685	0.0832601
$397$	0.230357	0.0115613	0.00578065	−	0.999983i	$-0.498160\pi$
$397$	0.230357	0.0115613	0.00578065	+	0.999983i	$0.498160\pi$
$398$	31.0349	1.55564
$399$	−1.59557	−0.0798786
$400$	−56.8376	−2.84188
$401$	0.606756	0.0302999	0.0151500	−	0.999885i	$-0.495177\pi$
$401$	0.606756	0.0302999	0.0151500	+	0.999885i	$0.495177\pi$
$402$	−19.8960	−0.992321
$403$	−14.7734	−0.735915
$404$	0.193952	0.00964947
$405$	4.06644	0.202063
$406$	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
$411$	−0.103218	−0.00509137
$412$	−2.21454	−0.109103
$413$	6.82416	0.335795
$414$	8.42400	0.414017
$415$	−25.9645	−1.27455
$416$	14.0959	0.691110
$417$	−14.4464	−0.707441
$418$	−6.48830	−0.317353
$419$	−12.1198	−0.592090	−0.296045	−	0.955174i	$-0.595668\pi$
$419$	−12.1198	−0.592090	−0.296045	+	0.955174i	$0.595668\pi$
$420$	−2.73749	−0.133576
$421$	−17.8730	−0.871078	−0.435539	−	0.900170i	$-0.643442\pi$
$421$	−17.8730	−0.871078	−0.435539	+	0.900170i	$0.643442\pi$
$422$	9.71616	0.472975
$423$	−6.96130	−0.338470
$424$	−5.65685	−0.274721
$425$	0	0
$426$	−2.51796	−0.121995
$427$	−3.58828	−0.173649
$428$	14.3431	0.693302
$429$	8.11492	0.391792
$430$	35.1484	1.69501
$431$	11.1361	0.536408	0.268204	−	0.963362i	$-0.413570\pi$
$431$	11.1361	0.536408	0.268204	+	0.963362i	$0.413570\pi$
$432$	−4.92701	−0.237051
$433$	−29.1182	−1.39933	−0.699665	−	0.714471i	$-0.746667\pi$
$433$	−29.1182	−1.39933	−0.699665	+	0.714471i	$0.746667\pi$
$434$	−6.29842	−0.302334
$435$	8.96130	0.429661
$436$	0.522302	0.0250137
$437$	−8.81974	−0.421905
$438$	1.92488	0.0919742
$439$	22.5958	1.07844	0.539219	−	0.842165i	$-0.318719\pi$
$439$	22.5958	1.07844	0.539219	+	0.842165i	$0.318719\pi$
$440$	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
$445$	42.1271	1.99701
$446$	9.22579	0.436854
$447$	3.59557	0.170065
$448$	−3.07959	−0.145497
$449$	1.12030	0.0528702	0.0264351	−	0.999651i	$-0.491584\pi$
$449$	1.12030	0.0528702	0.0264351	+	0.999651i	$0.491584\pi$
$450$	−19.0599	−0.898492
$451$	−11.4645	−0.539844
$452$	−8.60306	−0.404654
$453$	7.58828	0.356529
$454$	45.4855	2.13474
$455$	−13.4077	−0.628561
$456$	3.63022	0.170001
$457$	40.0795	1.87484	0.937419	−	0.348203i	$-0.113208\pi$
$457$	40.0795	1.87484	0.937419	+	0.348203i	$0.113208\pi$
$458$	5.89875	0.275631
$459$	0	0
$460$	−15.1318	−0.705526
$461$	4.99595	0.232684	0.116342	−	0.993209i	$-0.462883\pi$
$461$	4.99595	0.232684	0.116342	+	0.993209i	$0.462883\pi$
$462$	3.45968	0.160959
$463$	39.0452	1.81458	0.907292	−	0.420502i	$-0.138146\pi$
$463$	39.0452	1.81458	0.907292	+	0.420502i	$0.138146\pi$
$464$	−10.8578	−0.504060
$465$	−16.8061	−0.779363
$466$	2.26500	0.104924
$467$	10.5242	0.487002	0.243501	−	0.969901i	$-0.421704\pi$
$467$	10.5242	0.487002	0.243501	+	0.969901i	$0.421704\pi$
$468$	2.60889	0.120596
$469$	−11.1073	−0.512886
$470$	46.7706	2.15737
$471$	−18.9569	−0.873487
$472$	−15.5262	−0.714651
$473$	−11.8762	−0.546070
$474$	−19.6345	−0.901842
$475$	19.9553	0.915611
$476$	0	0
$477$	−2.69555	−0.123421
$478$	−35.7081	−1.63325
$479$	−25.9640	−1.18632	−0.593162	−	0.805083i	$-0.702121\pi$
$479$	−25.9640	−1.18632	−0.593162	+	0.805083i	$0.702121\pi$
$480$	16.0354	0.731912
$481$	21.2851	0.970517
$482$	−1.75434	−0.0799079
$483$	4.70285	0.213987
$484$	−4.26692	−0.193951
$485$	58.8392	2.67175
$486$	−1.65222	−0.0749464
$487$	25.5405	1.15735	0.578676	−	0.815557i	$-0.303570\pi$
$487$	25.5405	1.15735	0.578676	+	0.815557i	$0.303570\pi$
$488$	8.16399	0.369566
$489$	13.4330	0.607463
$490$	41.3145	1.86640
$491$	−2.50764	−0.113168	−0.0565842	−	0.998398i	$-0.518021\pi$
$491$	−2.50764	−0.113168	−0.0565842	+	0.998398i	$0.518021\pi$
$492$	−3.68577	−0.166167
$493$	0	0
$494$	−10.2165	−0.459663
$495$	9.23146	0.414923
$496$	20.3627	0.914313
$497$	−1.40569	−0.0630540
$498$	10.5496	0.472738
$499$	15.2270	0.681656	0.340828	−	0.940126i	$-0.389293\pi$
$499$	15.2270	0.681656	0.340828	+	0.940126i	$0.389293\pi$
$500$	19.3976	0.867486
$501$	−2.13729	−0.0954869
$502$	12.0565	0.538109
$503$	−39.0990	−1.74334	−0.871670	−	0.490094i	$-0.836962\pi$
$503$	−39.0990	−1.74334	−0.871670	+	0.490094i	$0.836962\pi$
$504$	−1.93570	−0.0862229
$505$	1.08064	0.0480878
$506$	19.1238	0.850158
$507$	−0.222197	−0.00986811
$508$	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
$513$	1.72984	0.0763743
$514$	37.2690	1.64386
$515$	−12.3387	−0.543709
$516$	−3.81813	−0.168084
$517$	−15.8033	−0.695027
$518$	9.07460	0.398715
$519$	−5.27995	−0.231764
$520$	30.5049	1.33773
$521$	−4.83056	−0.211631	−0.105815	−	0.994386i	$-0.533745\pi$
$521$	−4.83056	−0.211631	−0.105815	+	0.994386i	$0.533745\pi$
$522$	−3.64104	−0.159364
$523$	36.5734	1.59924	0.799622	−	0.600503i	$-0.205033\pi$
$523$	36.5734	1.59924	0.799622	+	0.600503i	$0.205033\pi$
$524$	−5.78546	−0.252739
$525$	−10.6405	−0.464390
$526$	7.02061	0.306113
$527$	0	0
$528$	−11.1851	−0.486769
$529$	2.99559	0.130243
$530$	18.1105	0.786670
$531$	−7.39840	−0.321063
$532$	−1.16451	−0.0504881
$533$	−18.0521	−0.781924
$534$	−17.1165	−0.740705
$535$	79.9154	3.45504
$536$	25.2711	1.09154
$537$	−7.77619	−0.335567
$538$	19.5521	0.842950
$539$	−13.9597	−0.601286
$540$	2.96785	0.127716
$541$	11.0291	0.474179	0.237090	−	0.971488i	$-0.423806\pi$
$541$	11.0291	0.474179	0.237090	+	0.971488i	$0.423806\pi$
$542$	19.5036	0.837751
$543$	0.915833	0.0393021
$544$	0	0
$545$	2.91010	0.124655
$546$	5.44763	0.233137
$547$	−26.8766	−1.14916	−0.574580	−	0.818448i	$-0.694835\pi$
$547$	−26.8766	−1.14916	−0.574580	+	0.818448i	$0.694835\pi$
$548$	−0.0753327	−0.00321805
$549$	3.89023	0.166031
$550$	−43.2690	−1.84500
$551$	3.81209	0.162400
$552$	−10.6998	−0.455415
$553$	−10.9613	−0.466122
$554$	−17.2286	−0.731973
$555$	24.2137	1.02782
$556$	−10.5435	−0.447146
$557$	−34.9146	−1.47938	−0.739690	−	0.672948i	$-0.765028\pi$
$557$	−34.9146	−1.47938	−0.739690	+	0.672948i	$0.765028\pi$
$558$	6.82843	0.289070
$559$	−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	$-0.553328\pi$
$563$	−7.91334	−0.333507	−0.166754	+	0.985999i	$0.553328\pi$
$564$	−5.08064	−0.213933
$565$	−47.9335	−2.01658
$566$	−29.1411	−1.22489
$567$	−0.922382	−0.0387364
$568$	3.19821	0.134194
$569$	−17.2956	−0.725070	−0.362535	−	0.931970i	$-0.618089\pi$
$569$	−17.2956	−0.725070	−0.362535	+	0.931970i	$0.618089\pi$
$570$	−11.6222	−0.486801
$571$	−19.3791	−0.810988	−0.405494	−	0.914098i	$-0.632901\pi$
$571$	−19.3791	−0.810988	−0.405494	+	0.914098i	$0.632901\pi$
$572$	5.92260	0.247636
$573$	−19.1658	−0.800661
$574$	−7.69626	−0.321236
$575$	−58.8168	−2.45283
$576$	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
$581$	5.88949	0.244337
$582$	−23.9068	−0.990968
$583$	−6.11934	−0.253437
$584$	−2.44490	−0.101171
$585$	14.5359	0.600986
$586$	13.6448	0.563662
$587$	12.3165	0.508357	0.254178	−	0.967157i	$-0.418195\pi$
$587$	12.3165	0.508357	0.254178	+	0.967157i	$0.418195\pi$
$588$	−4.48794	−0.185080
$589$	−7.14921	−0.294578
$590$	49.7073	2.04642
$591$	14.8949	0.612693
$592$	−29.3381	−1.20579
$593$	43.1763	1.77304	0.886519	−	0.462693i	$-0.153117\pi$
$593$	43.1763	1.77304	0.886519	+	0.462693i	$0.153117\pi$
$594$	−3.75081	−0.153898
$595$	0	0
$596$	2.62420	0.107491
$597$	−18.7837	−0.768766
$598$	30.1125	1.23139
$599$	6.70607	0.274003	0.137001	−	0.990571i	$-0.456254\pi$
$599$	6.70607	0.274003	0.137001	+	0.990571i	$0.456254\pi$
$600$	24.2091	0.988332
$601$	31.8393	1.29875	0.649375	−	0.760468i	$-0.275030\pi$
$601$	31.8393	1.29875	0.649375	+	0.760468i	$0.275030\pi$
$602$	−7.97265	−0.324941
$603$	12.0419	0.490386
$604$	5.53823	0.225348
$605$	−23.7739	−0.966547
$606$	−0.439071	−0.0178360
$607$	27.9168	1.13311	0.566554	−	0.824025i	$-0.308276\pi$
$607$	27.9168	1.13311	0.566554	+	0.824025i	$0.308276\pi$
$608$	6.82137	0.276643
$609$	−2.03268	−0.0823682
$610$	−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
$615$	−20.5359	−0.828088
$616$	−4.39435	−0.177053
$617$	37.0522	1.49166	0.745832	−	0.666134i	$-0.232052\pi$
$617$	37.0522	1.49166	0.745832	+	0.666134i	$0.232052\pi$
$618$	5.01332	0.201665
$619$	−16.1686	−0.649870	−0.324935	−	0.945736i	$-0.605342\pi$
$619$	−16.1686	−0.649870	−0.324935	+	0.945736i	$0.605342\pi$
$620$	−12.2657	−0.492604
$621$	−5.09859	−0.204599
$622$	27.4046	1.09883
$623$	−9.55561	−0.382837
$624$	−17.6121	−0.705049
$625$	50.3976	2.01590
$626$	−34.0550	−1.36111
$627$	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	$-0.717077\pi$
$631$	−31.6670	−1.26064	−0.630322	+	0.776334i	$0.717077\pi$
$632$	24.9389	0.992018
$633$	−5.88066	−0.233735
$634$	−7.05173	−0.280060
$635$	11.4286	0.453531
$636$	−1.96732	−0.0780095
$637$	−21.9810	−0.870919
$638$	−8.26575	−0.327244
$639$	1.52398	0.0602878
$640$	−54.5026	−2.15440
$641$	−15.9293	−0.629169	−0.314585	−	0.949229i	$-0.601865\pi$
$641$	−15.9293	−0.629169	−0.314585	+	0.949229i	$0.601865\pi$
$642$	−32.4702	−1.28150
$643$	−16.6863	−0.658043	−0.329022	−	0.944322i	$-0.606719\pi$
$643$	−16.6863	−0.658043	−0.329022	+	0.944322i	$0.606719\pi$
$644$	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
$649$	−16.7956	−0.659283
$650$	−68.1316	−2.67234
$651$	3.81209	0.149408
$652$	9.80398	0.383954
$653$	28.6977	1.12303	0.561514	−	0.827467i	$-0.310219\pi$
$653$	28.6977	1.12303	0.561514	+	0.827467i	$0.310219\pi$
$654$	−1.18239	−0.0462353
$655$	−32.2347	−1.25951
$656$	24.8819	0.971475
$657$	−1.16502	−0.0454519
$658$	−10.6089	−0.413578
$659$	−40.3690	−1.57255	−0.786276	−	0.617876i	$-0.787993\pi$
$659$	−40.3690	−1.57255	−0.786276	+	0.617876i	$0.787993\pi$
$660$	6.73749	0.262257
$661$	−39.4706	−1.53523	−0.767614	−	0.640913i	$-0.778556\pi$
$661$	−39.4706	−1.53523	−0.767614	+	0.640913i	$0.778556\pi$
$662$	47.2710	1.83724
$663$	0	0
$664$	−13.3997	−0.520007
$665$	−6.48830	−0.251606
$666$	−9.83822	−0.381223
$667$	−11.2359	−0.435055
$668$	−1.55988	−0.0603535
$669$	−5.58387	−0.215885
$670$	−80.9057	−3.12566
$671$	8.83145	0.340934
$672$	−3.63728	−0.140311
$673$	−21.6056	−0.832834	−0.416417	−	0.909174i	$-0.636714\pi$
$673$	−21.6056	−0.832834	−0.416417	+	0.909174i	$0.636714\pi$
$674$	−51.7754	−1.99431
$675$	11.5359	0.444017
$676$	−0.162168	−0.00623724
$677$	2.51083	0.0964990	0.0482495	−	0.998835i	$-0.484636\pi$
$677$	2.51083	0.0964990	0.0482495	+	0.998835i	$0.484636\pi$
$678$	19.4757	0.747961
$679$	−13.3464	−0.512187
$680$	0	0
$681$	−27.5299	−1.05495
$682$	15.5016	0.593588
$683$	18.7423	0.717156	0.358578	−	0.933500i	$-0.383262\pi$
$683$	18.7423	0.717156	0.358578	+	0.933500i	$0.383262\pi$
$684$	1.26251	0.0482732
$685$	−0.419730	−0.0160370
$686$	−20.0392	−0.765098
$687$	−3.57019	−0.136211
$688$	25.7755	0.982681
$689$	−9.63554	−0.367085
$690$	34.2557	1.30409
$691$	−38.5702	−1.46728	−0.733640	−	0.679538i	$-0.762180\pi$
$691$	−38.5702	−1.46728	−0.733640	+	0.679538i	$0.762180\pi$
$692$	−3.85352	−0.146489
$693$	−2.09396	−0.0795428
$694$	−19.8860	−0.754862
$695$	−58.7452	−2.22833
$696$	4.62470	0.175299
$697$	0	0
$698$	−55.1037	−2.08570
$699$	−1.37088	−0.0518515
$700$	−7.76588	−0.293523
$701$	−28.7827	−1.08711	−0.543553	−	0.839375i	$-0.682921\pi$
$701$	−28.7827	−1.08711	−0.543553	+	0.839375i	$0.682921\pi$
$702$	−5.90604	−0.222909
$703$	10.3004	0.388487
$704$	7.57945	0.285661
$705$	−28.3077	−1.06613
$706$	10.8480	0.408270
$707$	−0.245119	−0.00921866
$708$	−5.39965	−0.202931
$709$	−5.14588	−0.193258	−0.0966288	−	0.995320i	$-0.530806\pi$
$709$	−5.14588	−0.193258	−0.0966288	+	0.995320i	$0.530806\pi$
$710$	−10.2391	−0.384267
$711$	11.8837	0.445673
$712$	21.7407	0.814768
$713$	21.0718	0.789146
$714$	0	0
$715$	32.9988	1.23409
$716$	−5.67538	−0.212099
$717$	21.6121	0.807120
$718$	−35.6068	−1.32883
$719$	−31.5022	−1.17483	−0.587417	−	0.809285i	$-0.699855\pi$
$719$	−31.5022	−1.17483	−0.587417	+	0.809285i	$0.699855\pi$
$720$	−20.0354	−0.746675
$721$	2.79877	0.104232
$722$	26.4482	0.984300
$723$	1.06181	0.0394890
$724$	0.668412	0.0248413
$725$	25.4219	0.944147
$726$	9.65952	0.358498
$727$	31.2775	1.16002	0.580010	−	0.814610i	$-0.303049\pi$
$727$	31.2775	1.16002	0.580010	+	0.814610i	$0.303049\pi$
$728$	−6.91936	−0.256449
$729$	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	$-0.387037\pi$
$733$	18.8153	0.694960	0.347480	+	0.937687i	$0.387037\pi$
$734$	−58.7603	−2.16888
$735$	−25.0054	−0.922337
$736$	−20.1055	−0.741099
$737$	27.3371	1.00698
$738$	8.34389	0.307143
$739$	6.94762	0.255572	0.127786	−	0.991802i	$-0.459213\pi$
$739$	6.94762	0.255572	0.127786	+	0.991802i	$0.459213\pi$
$740$	17.6722	0.649641
$741$	6.18350	0.227156
$742$	−4.10797	−0.150808
$743$	−33.9563	−1.24574	−0.622868	−	0.782327i	$-0.714033\pi$
$743$	−33.9563	−1.24574	−0.622868	+	0.782327i	$0.714033\pi$
$744$	−8.67319	−0.317975
$745$	14.6212	0.535678
$746$	59.2243	2.16835
$747$	−6.38508	−0.233618
$748$	0	0
$749$	−18.1271	−0.662349
$750$	−43.9125	−1.60346
$751$	−6.91509	−0.252335	−0.126168	−	0.992009i	$-0.540268\pi$
$751$	−6.91509	−0.252335	−0.126168	+	0.992009i	$0.540268\pi$
$752$	34.2984	1.25073
$753$	−7.29715	−0.265923
$754$	−13.0153	−0.473989
$755$	30.8573	1.12301
$756$	−0.673192	−0.0244838
$757$	1.86166	0.0676633	0.0338316	−	0.999428i	$-0.489229\pi$
$757$	1.86166	0.0676633	0.0338316	+	0.999428i	$0.489229\pi$
$758$	36.2178	1.31549
$759$	−11.5746	−0.420132
$760$	14.7621	0.535476
$761$	49.2335	1.78471	0.892357	−	0.451330i	$-0.149050\pi$
$761$	49.2335	1.78471	0.892357	+	0.451330i	$0.149050\pi$
$762$	−4.64354	−0.168218
$763$	−0.660093	−0.0238970
$764$	−13.9880	−0.506066
$765$	0	0
$766$	56.6893	2.04827
$767$	−26.4464	−0.954923
$768$	15.4673	0.558129
$769$	50.9303	1.83659	0.918296	−	0.395895i	$-0.129565\pi$
$769$	50.9303	1.83659	0.918296	+	0.395895i	$0.129565\pi$
$770$	14.0686	0.506996
$771$	−22.5569	−0.812366
$772$	3.43855	0.123756
$773$	2.91614	0.104886	0.0524431	−	0.998624i	$-0.483299\pi$
$773$	2.91614	0.104886	0.0524431	+	0.998624i	$0.483299\pi$
$774$	8.64354	0.310686
$775$	−47.6764	−1.71259
$776$	30.3654	1.09006
$777$	−5.49236	−0.197037
$778$	56.4988	2.02558
$779$	−8.73586	−0.312995
$780$	10.6089	0.379859
$781$	3.45968	0.123797
$782$	0	0
$783$	2.20372	0.0787546
$784$	30.2972	1.08204
$785$	−77.0870	−2.75135
$786$	13.0972	0.467161
$787$	−50.8932	−1.81415	−0.907073	−	0.420974i	$-0.861688\pi$
$787$	−50.8932	−1.81415	−0.907073	+	0.420974i	$0.861688\pi$
$788$	10.8709	0.387259
$789$	−4.24919	−0.151275
$790$	−79.8424	−2.84067
$791$	10.8727	0.386588
$792$	4.76413	0.169286
$793$	13.9060	0.493818
$794$	−0.380602	−0.0135070
$795$	−10.9613	−0.388757
$796$	−13.7091	−0.485906
$797$	−13.6061	−0.481952	−0.240976	−	0.970531i	$-0.577468\pi$
$797$	−13.6061	−0.481952	−0.240976	+	0.970531i	$0.577468\pi$
$798$	2.63624	0.0933220
$799$	0	0
$800$	45.4901	1.60832
$801$	10.3597	0.366042
$802$	−1.00250	−0.0353993
$803$	−2.64479	−0.0933326
$804$	8.78869	0.309953
$805$	19.1238	0.674026
$806$	24.4089	0.859768
$807$	−11.8338	−0.416570
$808$	0.557691	0.0196195
$809$	17.3937	0.611529	0.305765	−	0.952107i	$-0.401088\pi$
$809$	17.3937	0.611529	0.305765	+	0.952107i	$0.401088\pi$
$810$	−6.71866	−0.236070
$811$	−32.9286	−1.15628	−0.578140	−	0.815937i	$-0.696221\pi$
$811$	−32.9286	−1.15628	−0.578140	+	0.815937i	$0.696221\pi$
$812$	−1.48353	−0.0520617
$813$	−11.8045	−0.414000
$814$	−22.3343	−0.782817
$815$	54.6246	1.91342
$816$	0	0
$817$	−9.04959	−0.316605
$818$	2.49362	0.0871875
$819$	−3.29715	−0.115212
$820$	−14.9879	−0.523401
$821$	43.7368	1.52643	0.763213	−	0.646147i	$-0.223621\pi$
$821$	43.7368	1.52643	0.763213	+	0.646147i	$0.223621\pi$
$822$	0.170539	0.00594824
$823$	1.67402	0.0583528	0.0291764	−	0.999574i	$-0.490712\pi$
$823$	1.67402	0.0583528	0.0291764	+	0.999574i	$0.490712\pi$
$824$	−6.36771	−0.221830
$825$	26.1883	0.911761
$826$	−11.2750	−0.392308
$827$	−2.27016	−0.0789412	−0.0394706	−	0.999221i	$-0.512567\pi$
$827$	−2.27016	−0.0789412	−0.0394706	+	0.999221i	$0.512567\pi$
$828$	−3.72115	−0.129319
$829$	19.2597	0.668918	0.334459	−	0.942410i	$-0.391446\pi$
$829$	19.2597	0.668918	0.334459	+	0.942410i	$0.391446\pi$
$830$	42.8992	1.48905
$831$	10.4275	0.361727
$832$	11.9346	0.413760
$833$	0	0
$834$	23.8686	0.826502
$835$	−8.69114	−0.300769
$836$	2.86609	0.0991259
$837$	−4.13287	−0.142853
$838$	20.0246	0.691737
$839$	14.8059	0.511157	0.255579	−	0.966788i	$-0.417734\pi$
$839$	14.8059	0.511157	0.255579	+	0.966788i	$0.417734\pi$
$840$	−7.87140	−0.271589
$841$	−24.1436	−0.832538
$842$	29.5302	1.01768
$843$	−5.46697	−0.188293
$844$	−4.29194	−0.147735
$845$	−0.903549	−0.0310830
$846$	11.5016	0.395434
$847$	5.39259	0.185292
$848$	13.2810	0.456073
$849$	17.6375	0.605318
$850$	0	0
$851$	−30.3597	−1.04072
$852$	1.11226	0.0381055
$853$	6.35771	0.217684	0.108842	−	0.994059i	$-0.465286\pi$
$853$	6.35771	0.217684	0.108842	+	0.994059i	$0.465286\pi$
$854$	5.92864	0.202874
$855$	7.03429	0.240568
$856$	41.2423	1.40963
$857$	−40.8311	−1.39477	−0.697383	−	0.716699i	$-0.745652\pi$
$857$	−40.8311	−1.39477	−0.697383	+	0.716699i	$0.745652\pi$
$858$	−13.4077	−0.457730
$859$	22.1464	0.755626	0.377813	−	0.925882i	$-0.376676\pi$
$859$	22.1464	0.755626	0.377813	+	0.925882i	$0.376676\pi$
$860$	−15.5262	−0.529439
$861$	4.65812	0.158748
$862$	−18.3993	−0.626684
$863$	17.3423	0.590339	0.295170	−	0.955445i	$-0.404624\pi$
$863$	17.3423	0.590339	0.295170	+	0.955445i	$0.404624\pi$
$864$	3.94335	0.134156
$865$	−21.4706	−0.730021
$866$	48.1097	1.63483
$867$	0	0
$868$	2.78222	0.0944346
$869$	26.9779	0.915161
$870$	−14.8061	−0.501973
$871$	43.0452	1.45853
$872$	1.50183	0.0508584
$873$	14.4695	0.489717
$874$	14.5722	0.492911
$875$	−24.5149	−0.828756
$876$	−0.850281	−0.0287283
$877$	30.2306	1.02081	0.510407	−	0.859933i	$-0.329495\pi$
$877$	30.2306	1.02081	0.510407	+	0.859933i	$0.329495\pi$
$878$	−37.3333	−1.25994
$879$	−8.25845	−0.278551
$880$	−45.4835	−1.53325
$881$	10.5647	0.355933	0.177966	−	0.984037i	$-0.443048\pi$
$881$	10.5647	0.355933	0.177966	+	0.984037i	$0.443048\pi$
$882$	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
$887$	35.9644	1.20757	0.603783	−	0.797149i	$-0.293659\pi$
$887$	35.9644	1.20757	0.603783	+	0.797149i	$0.293659\pi$
$888$	12.4961	0.419342
$889$	−2.59234	−0.0869442
$890$	−69.6033	−2.33311
$891$	2.27016	0.0760532
$892$	−4.07533	−0.136452
$893$	−12.0419	−0.402968
$894$	−5.94069	−0.198686
$895$	−31.6214	−1.05699
$896$	12.3627	0.413010
$897$	−18.2254	−0.608530
$898$	−1.85098	−0.0617681
$899$	−9.10771	−0.303759
$900$	8.41937	0.280646
$901$	0	0
$902$	18.9420	0.630698
$903$	4.82541	0.160579
$904$	−24.7373	−0.822750
$905$	3.72418	0.123796
$906$	−12.5375	−0.416532
$907$	−39.1843	−1.30109	−0.650547	−	0.759466i	$-0.725460\pi$
$907$	−39.1843	−1.30109	−0.650547	+	0.759466i	$0.725460\pi$
$908$	−20.0924	−0.666790
$909$	0.265746	0.00881423
$910$	22.1524	0.734347
$911$	9.92806	0.328931	0.164466	−	0.986383i	$-0.447410\pi$
$911$	9.92806	0.328931	0.164466	+	0.986383i	$0.447410\pi$
$912$	−8.52295	−0.282223
$913$	−14.4952	−0.479720
$914$	−66.2202	−2.19037
$915$	15.8194	0.522973
$916$	−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
$921$	−9.94601	−0.327732
$922$	−8.25442	−0.271845
$923$	5.44763	0.179311
$924$	−1.52825	−0.0502758
$925$	68.6909	2.25854
$926$	−64.5113	−2.11997
$927$	−3.03429	−0.0996590
$928$	8.69006	0.285265
$929$	−18.1624	−0.595888	−0.297944	−	0.954583i	$-0.596301\pi$
$929$	−18.1624	−0.595888	−0.297944	+	0.954583i	$0.596301\pi$
$930$	27.7674	0.910528
$931$	−10.6372	−0.348619
$932$	−1.00053	−0.0327733
$933$	−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	$-0.491667\pi$
$937$	1.60243	0.0523492	0.0261746	+	0.999657i	$0.491667\pi$
$938$	18.3517	0.599204
$939$	20.6116	0.672634
$940$	−20.6601	−0.673858
$941$	32.3544	1.05472	0.527362	−	0.849641i	$-0.323181\pi$
$941$	32.3544	1.05472	0.527362	+	0.849641i	$0.323181\pi$
$942$	31.3210	1.02049
$943$	25.7484	0.838482
$944$	36.4520	1.18641
$945$	−3.75081	−0.122014
$946$	19.6222	0.637973
$947$	25.1819	0.818301	0.409151	−	0.912467i	$-0.365825\pi$
$947$	25.1819	0.818301	0.409151	+	0.912467i	$0.365825\pi$
$948$	8.67319	0.281692
$949$	−4.16450	−0.135185
$950$	−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
$955$	−77.9364	−2.52196
$956$	15.7734	0.510148
$957$	5.00280	0.161718
$958$	42.8983	1.38598
$959$	0.0952065	0.00307438
$960$	13.5767	0.438187
$961$	−13.9194	−0.449012
$962$	−35.1677	−1.13385
$963$	19.6524	0.633291
$964$	0.774948	0.0249594
$965$	19.1585	0.616733
$966$	−7.77015	−0.250001
$967$	−2.17495	−0.0699418	−0.0349709	−	0.999388i	$-0.511134\pi$
$967$	−2.17495	−0.0699418	−0.0349709	+	0.999388i	$0.511134\pi$
$968$	−12.2691	−0.394345
$969$	0	0
$970$	−97.2154	−3.12140
$971$	47.4481	1.52268	0.761340	−	0.648352i	$-0.224542\pi$
$971$	47.4481	1.52268	0.761340	+	0.648352i	$0.224542\pi$
$972$	0.729840	0.0234096
$973$	13.3251	0.427182
$974$	−42.1987	−1.35213
$975$	41.2363	1.32062
$976$	−19.1672	−0.613528
$977$	−20.8226	−0.666175	−0.333087	−	0.942896i	$-0.608090\pi$
$977$	−20.8226	−0.666175	−0.333087	+	0.942896i	$0.608090\pi$
$978$	−22.1944	−0.709698
$979$	23.5182	0.751644
$980$	−18.2499	−0.582973
$981$	0.715639	0.0228486
$982$	4.14319	0.132214
$983$	2.43039	0.0775173	0.0387586	−	0.999249i	$-0.487660\pi$
$983$	2.43039	0.0775173	0.0387586	+	0.999249i	$0.487660\pi$
$984$	−10.5981	−0.337854
$985$	60.5690	1.92989
$986$	0	0
$987$	6.42098	0.204382
$988$	4.51297	0.143577
$989$	26.6731	0.848154
$990$	−15.2524	−0.484754
$991$	36.5777	1.16193	0.580964	−	0.813929i	$-0.302676\pi$
$991$	36.5777	1.16193	0.580964	+	0.813929i	$0.302676\pi$
$992$	−16.2974	−0.517442
$993$	−28.6105	−0.907927
$994$	2.32252	0.0736658
$995$	−76.3827	−2.42150
$996$	−4.66009	−0.147661
$997$	20.7109	0.655920	0.327960	−	0.944692i	$-0.393639\pi$
$997$	20.7109	0.655920	0.327960	+	0.944692i	$0.393639\pi$
$998$	−25.1585	−0.796378
$999$	5.95453	0.188393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.a.l.1.2		4
3.2	odd	2		2601.2.a.be.1.3		4
17.2	even	8		51.2.e.a.4.2	✓	8
17.3	odd	16		867.2.h.k.757.2		16
17.4	even	4		867.2.d.f.577.5		8
17.5	odd	16		867.2.h.i.688.3		16
17.6	odd	16		867.2.h.k.733.2		16
17.7	odd	16		867.2.h.i.712.3		16
17.8	even	8		867.2.e.g.829.3		8
17.9	even	8		51.2.e.a.13.3	yes	8
17.10	odd	16		867.2.h.i.712.4		16
17.11	odd	16		867.2.h.k.733.1		16
17.12	odd	16		867.2.h.i.688.4		16
17.13	even	4		867.2.d.f.577.6		8
17.14	odd	16		867.2.h.k.757.1		16
17.15	even	8		867.2.e.g.616.2		8
17.16	even	2		867.2.a.k.1.2		4
51.2	odd	8		153.2.f.b.55.3		8
51.26	odd	8		153.2.f.b.64.2		8
51.50	odd	2		2601.2.a.bf.1.3		4
68.19	odd	8		816.2.bd.e.769.1		8
68.43	odd	8		816.2.bd.e.625.1		8
204.155	even	8		2448.2.be.x.1585.4		8
204.179	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	17.2	even	8
51.2.e.a.13.3	yes	8	17.9	even	8
153.2.f.b.55.3		8	51.2	odd	8
153.2.f.b.64.2		8	51.26	odd	8
816.2.bd.e.625.1		8	68.43	odd	8
816.2.bd.e.769.1		8	68.19	odd	8
867.2.a.k.1.2		4	17.16	even	2
867.2.a.l.1.2		4	1.1	even	1	trivial
867.2.d.f.577.5		8	17.4	even	4
867.2.d.f.577.6		8	17.13	even	4
867.2.e.g.616.2		8	17.15	even	8
867.2.e.g.829.3		8	17.8	even	8
867.2.h.i.688.3		16	17.5	odd	16
867.2.h.i.688.4		16	17.12	odd	16
867.2.h.i.712.3		16	17.7	odd	16
867.2.h.i.712.4		16	17.10	odd	16
867.2.h.k.733.1		16	17.11	odd	16
867.2.h.k.733.2		16	17.6	odd	16
867.2.h.k.757.1		16	17.14	odd	16
867.2.h.k.757.2		16	17.3	odd	16
2448.2.be.x.1441.4		8	204.179	even	8
2448.2.be.x.1585.4		8	204.155	even	8
2601.2.a.be.1.3		4	3.2	odd	2
2601.2.a.bf.1.3		4	51.50	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

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

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