LMFDB - Embedded newform 729.2.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7459857.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 $ x^6 - 3x^5 - 6x^4 + 13x^3 + 12x^2 - 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.77773$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.12503 q^{2} +2.51575 q^{4} -2.07094 q^{5} +4.84867 q^{7} -1.09598 q^{8} +4.40081 q^{10} +4.14949 q^{11} -1.21602 q^{13} -10.3036 q^{14} -2.70251 q^{16} +2.36364 q^{17} -1.83801 q^{19} -5.20996 q^{20} -8.81778 q^{22} -4.30357 q^{23} -0.711206 q^{25} +2.58407 q^{26} +12.1980 q^{28} +2.98182 q^{29} +1.47359 q^{31} +7.93487 q^{32} -5.02280 q^{34} -10.0413 q^{35} +8.97108 q^{37} +3.90582 q^{38} +2.26970 q^{40} +2.26052 q^{41} -5.48486 q^{43} +10.4391 q^{44} +9.14521 q^{46} +7.18313 q^{47} +16.5096 q^{49} +1.51133 q^{50} -3.05919 q^{52} -6.32803 q^{53} -8.59334 q^{55} -5.31404 q^{56} -6.33645 q^{58} +0.262257 q^{59} -4.45561 q^{61} -3.13141 q^{62} -11.4568 q^{64} +2.51830 q^{65} +4.13110 q^{67} +5.94632 q^{68} +21.3381 q^{70} +3.08551 q^{71} +12.7601 q^{73} -19.0638 q^{74} -4.62396 q^{76} +20.1195 q^{77} +4.55241 q^{79} +5.59674 q^{80} -4.80368 q^{82} -8.45306 q^{83} -4.89495 q^{85} +11.6555 q^{86} -4.54775 q^{88} +16.9632 q^{89} -5.89606 q^{91} -10.8267 q^{92} -15.2644 q^{94} +3.80640 q^{95} -5.10977 q^{97} -35.0834 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

Coefficient data

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

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

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

$2$	−2.12503	−1.50262	−0.751311	−	0.659948i	$-0.770578\pi$
$2$	−2.12503	−1.50262	−0.751311	+	0.659948i	$0.770578\pi$
$3$	0	0
$3$	0	0
$4$	2.51575	1.25787
$5$	−2.07094	−0.926153	−0.463076	−	0.886318i	$-0.653254\pi$
$5$	−2.07094	−0.926153	−0.463076	+	0.886318i	$0.653254\pi$
$6$	0	0
$7$	4.84867	1.83263	0.916313	−	0.400463i	$-0.131151\pi$
$7$	4.84867	1.83263	0.916313	+	0.400463i	$0.131151\pi$
$8$	−1.09598	−0.387487
$9$	0	0
$10$	4.40081	1.39166
$11$	4.14949	1.25112	0.625559	−	0.780177i	$-0.284871\pi$
$11$	4.14949	1.25112	0.625559	+	0.780177i	$0.284871\pi$
$12$	0	0
$13$	−1.21602	−0.337262	−0.168631	−	0.985679i	$-0.553935\pi$
$13$	−1.21602	−0.337262	−0.168631	+	0.985679i	$0.553935\pi$
$14$	−10.3036	−2.75374
$15$	0	0
$16$	−2.70251	−0.675628
$17$	2.36364	0.573266	0.286633	−	0.958040i	$-0.407464\pi$
$17$	2.36364	0.573266	0.286633	+	0.958040i	$0.407464\pi$
$18$	0	0
$19$	−1.83801	−0.421668	−0.210834	−	0.977522i	$-0.567618\pi$
$19$	−1.83801	−0.421668	−0.210834	+	0.977522i	$0.567618\pi$
$20$	−5.20996	−1.16498
$21$	0	0
$22$	−8.81778	−1.87996
$23$	−4.30357	−0.897356	−0.448678	−	0.893693i	$-0.648105\pi$
$23$	−4.30357	−0.897356	−0.448678	+	0.893693i	$0.648105\pi$
$24$	0	0
$25$	−0.711206	−0.142241
$26$	2.58407	0.506777
$27$	0	0
$28$	12.1980	2.30521
$29$	2.98182	0.553710	0.276855	−	0.960912i	$-0.410708\pi$
$29$	2.98182	0.553710	0.276855	+	0.960912i	$0.410708\pi$
$30$	0	0
$31$	1.47359	0.264664	0.132332	−	0.991205i	$-0.457754\pi$
$31$	1.47359	0.264664	0.132332	+	0.991205i	$0.457754\pi$
$32$	7.93487	1.40270
$33$	0	0
$34$	−5.02280	−0.861403
$35$	−10.0413	−1.69729
$36$	0	0
$37$	8.97108	1.47484	0.737418	−	0.675436i	$-0.236045\pi$
$37$	8.97108	1.47484	0.737418	+	0.675436i	$0.236045\pi$
$38$	3.90582	0.633607
$39$	0	0
$40$	2.26970	0.358872
$41$	2.26052	0.353035	0.176517	−	0.984298i	$-0.443517\pi$
$41$	2.26052	0.353035	0.176517	+	0.984298i	$0.443517\pi$
$42$	0	0
$43$	−5.48486	−0.836433	−0.418216	−	0.908347i	$-0.637345\pi$
$43$	−5.48486	−0.836433	−0.418216	+	0.908347i	$0.637345\pi$
$44$	10.4391	1.57375
$45$	0	0
$46$	9.14521	1.34839
$47$	7.18313	1.04777	0.523884	−	0.851790i	$-0.324483\pi$
$47$	7.18313	1.04777	0.523884	+	0.851790i	$0.324483\pi$
$48$	0	0
$49$	16.5096	2.35852
$50$	1.51133	0.213735
$51$	0	0
$52$	−3.05919	−0.424233
$53$	−6.32803	−0.869222	−0.434611	−	0.900618i	$-0.643114\pi$
$53$	−6.32803	−0.869222	−0.434611	+	0.900618i	$0.643114\pi$
$54$	0	0
$55$	−8.59334	−1.15873
$56$	−5.31404	−0.710118
$57$	0	0
$58$	−6.33645	−0.832017
$59$	0.262257	0.0341429	0.0170715	−	0.999854i	$-0.494566\pi$
$59$	0.262257	0.0341429	0.0170715	+	0.999854i	$0.494566\pi$
$60$	0	0
$61$	−4.45561	−0.570482	−0.285241	−	0.958456i	$-0.592074\pi$
$61$	−4.45561	−0.570482	−0.285241	+	0.958456i	$0.592074\pi$
$62$	−3.13141	−0.397690
$63$	0	0
$64$	−11.4568	−1.43210
$65$	2.51830	0.312356
$66$	0	0
$67$	4.13110	0.504695	0.252347	−	0.967637i	$-0.418797\pi$
$67$	4.13110	0.504695	0.252347	+	0.967637i	$0.418797\pi$
$68$	5.94632	0.721097
$69$	0	0
$70$	21.3381	2.55039
$71$	3.08551	0.366183	0.183091	−	0.983096i	$-0.441390\pi$
$71$	3.08551	0.366183	0.183091	+	0.983096i	$0.441390\pi$
$72$	0	0
$73$	12.7601	1.49345	0.746726	−	0.665132i	$-0.231625\pi$
$73$	12.7601	1.49345	0.746726	+	0.665132i	$0.231625\pi$
$74$	−19.0638	−2.21612
$75$	0	0
$76$	−4.62396	−0.530405
$77$	20.1195	2.29283
$78$	0	0
$79$	4.55241	0.512186	0.256093	−	0.966652i	$-0.417565\pi$
$79$	4.55241	0.512186	0.256093	+	0.966652i	$0.417565\pi$
$80$	5.59674	0.625734
$81$	0	0
$82$	−4.80368	−0.530478
$83$	−8.45306	−0.927844	−0.463922	−	0.885876i	$-0.653558\pi$
$83$	−8.45306	−0.927844	−0.463922	+	0.885876i	$0.653558\pi$
$84$	0	0
$85$	−4.89495	−0.530932
$86$	11.6555	1.25684
$87$	0	0
$88$	−4.54775	−0.484791
$89$	16.9632	1.79809	0.899046	−	0.437854i	$-0.144261\pi$
$89$	16.9632	1.79809	0.899046	+	0.437854i	$0.144261\pi$
$90$	0	0
$91$	−5.89606	−0.618075
$92$	−10.8267	−1.12876
$93$	0	0
$94$	−15.2644	−1.57440
$95$	3.80640	0.390529
$96$	0	0
$97$	−5.10977	−0.518819	−0.259409	−	0.965767i	$-0.583528\pi$
$97$	−5.10977	−0.518819	−0.259409	+	0.965767i	$0.583528\pi$
$98$	−35.0834	−3.54396
$99$	0	0
$100$	−1.78921	−0.178921
$101$	18.5799	1.84877	0.924384	−	0.381464i	$-0.124580\pi$
$101$	18.5799	1.84877	0.924384	+	0.381464i	$0.124580\pi$
$102$	0	0
$103$	9.00577	0.887365	0.443682	−	0.896184i	$-0.353672\pi$
$103$	9.00577	0.887365	0.443682	+	0.896184i	$0.353672\pi$
$104$	1.33273	0.130684
$105$	0	0
$106$	13.4472	1.30611
$107$	7.42680	0.717976	0.358988	−	0.933342i	$-0.383122\pi$
$107$	7.42680	0.717976	0.358988	+	0.933342i	$0.383122\pi$
$108$	0	0
$109$	−5.62396	−0.538678	−0.269339	−	0.963045i	$-0.586805\pi$
$109$	−5.62396	−0.538678	−0.269339	+	0.963045i	$0.586805\pi$
$110$	18.2611	1.74113
$111$	0	0
$112$	−13.1036	−1.23817
$113$	−2.40267	−0.226024	−0.113012	−	0.993594i	$-0.536050\pi$
$113$	−2.40267	−0.226024	−0.113012	+	0.993594i	$0.536050\pi$
$114$	0	0
$115$	8.91243	0.831089
$116$	7.50150	0.696497
$117$	0	0
$118$	−0.557303	−0.0513039
$119$	11.4605	1.05058
$120$	0	0
$121$	6.21826	0.565296
$122$	9.46829	0.857219
$123$	0	0
$124$	3.70717	0.332914
$125$	11.8276	1.05789
$126$	0	0
$127$	−9.23469	−0.819447	−0.409723	−	0.912210i	$-0.634375\pi$
$127$	−9.23469	−0.819447	−0.409723	+	0.912210i	$0.634375\pi$
$128$	8.47629	0.749206
$129$	0	0
$130$	−5.35145	−0.469353
$131$	−15.3390	−1.34018	−0.670088	−	0.742282i	$-0.733743\pi$
$131$	−15.3390	−1.34018	−0.670088	+	0.742282i	$0.733743\pi$
$132$	0	0
$133$	−8.91189	−0.772759
$134$	−8.77871	−0.758365
$135$	0	0
$136$	−2.59049	−0.222133
$137$	−3.64397	−0.311325	−0.155663	−	0.987810i	$-0.549751\pi$
$137$	−3.64397	−0.311325	−0.155663	+	0.987810i	$0.549751\pi$
$138$	0	0
$139$	13.2755	1.12602	0.563008	−	0.826451i	$-0.309644\pi$
$139$	13.2755	1.12602	0.563008	+	0.826451i	$0.309644\pi$
$140$	−25.2614	−2.13498
$141$	0	0
$142$	−6.55680	−0.550234
$143$	−5.04584	−0.421954
$144$	0	0
$145$	−6.17517	−0.512820
$146$	−27.1155	−2.24409
$147$	0	0
$148$	22.5690	1.85516
$149$	8.91098	0.730016	0.365008	−	0.931004i	$-0.381066\pi$
$149$	8.91098	0.730016	0.365008	+	0.931004i	$0.381066\pi$
$150$	0	0
$151$	−0.712404	−0.0579746	−0.0289873	−	0.999580i	$-0.509228\pi$
$151$	−0.712404	−0.0579746	−0.0289873	+	0.999580i	$0.509228\pi$
$152$	2.01441	0.163391
$153$	0	0
$154$	−42.7545	−3.44526
$155$	−3.05171	−0.245119
$156$	0	0
$157$	−13.8181	−1.10280	−0.551400	−	0.834241i	$-0.685906\pi$
$157$	−13.8181	−1.10280	−0.551400	+	0.834241i	$0.685906\pi$
$158$	−9.67399	−0.769622
$159$	0	0
$160$	−16.4326	−1.29911
$161$	−20.8666	−1.64452
$162$	0	0
$163$	−1.19321	−0.0934597	−0.0467298	−	0.998908i	$-0.514880\pi$
$163$	−1.19321	−0.0934597	−0.0467298	+	0.998908i	$0.514880\pi$
$164$	5.68691	0.444073
$165$	0	0
$166$	17.9630	1.39420
$167$	−23.9363	−1.85225	−0.926124	−	0.377219i	$-0.876880\pi$
$167$	−23.9363	−1.85225	−0.926124	+	0.377219i	$0.876880\pi$
$168$	0	0
$169$	−11.5213	−0.886254
$170$	10.4019	0.797791
$171$	0	0
$172$	−13.7985	−1.05213
$173$	−9.14000	−0.694902	−0.347451	−	0.937698i	$-0.612953\pi$
$173$	−9.14000	−0.694902	−0.347451	+	0.937698i	$0.612953\pi$
$174$	0	0
$175$	−3.44841	−0.260675
$176$	−11.2140	−0.845290
$177$	0	0
$178$	−36.0472	−2.70185
$179$	10.6008	0.792337	0.396169	−	0.918178i	$-0.370340\pi$
$179$	10.6008	0.792337	0.396169	+	0.918178i	$0.370340\pi$
$180$	0	0
$181$	−1.46292	−0.108738	−0.0543690	−	0.998521i	$-0.517315\pi$
$181$	−1.46292	−0.108738	−0.0543690	+	0.998521i	$0.517315\pi$
$182$	12.5293	0.928733
$183$	0	0
$184$	4.71661	0.347713
$185$	−18.5786	−1.36592
$186$	0	0
$187$	9.80789	0.717224
$188$	18.0709	1.31796
$189$	0	0
$190$	−8.08871	−0.586817
$191$	12.4223	0.898844	0.449422	−	0.893320i	$-0.351630\pi$
$191$	12.4223	0.898844	0.449422	+	0.893320i	$0.351630\pi$
$192$	0	0
$193$	20.7733	1.49529	0.747647	−	0.664096i	$-0.231183\pi$
$193$	20.7733	1.49529	0.747647	+	0.664096i	$0.231183\pi$
$194$	10.8584	0.779588
$195$	0	0
$196$	41.5340	2.96672
$197$	−14.1887	−1.01090	−0.505450	−	0.862856i	$-0.668674\pi$
$197$	−14.1887	−1.01090	−0.505450	+	0.862856i	$0.668674\pi$
$198$	0	0
$199$	20.3286	1.44106	0.720529	−	0.693424i	$-0.243899\pi$
$199$	20.3286	1.44106	0.720529	+	0.693424i	$0.243899\pi$
$200$	0.779466	0.0551165
$201$	0	0
$202$	−39.4828	−2.77800
$203$	14.4579	1.01474
$204$	0	0
$205$	−4.68141	−0.326964
$206$	−19.1375	−1.33337
$207$	0	0
$208$	3.28629	0.227864
$209$	−7.62679	−0.527556
$210$	0	0
$211$	−9.86332	−0.679019	−0.339509	−	0.940603i	$-0.610261\pi$
$211$	−9.86332	−0.679019	−0.339509	+	0.940603i	$0.610261\pi$
$212$	−15.9197	−1.09337
$213$	0	0
$214$	−15.7822	−1.07885
$215$	11.3588	0.774665
$216$	0	0
$217$	7.14494	0.485030
$218$	11.9511	0.809429
$219$	0	0
$220$	−21.6187	−1.45753
$221$	−2.87422	−0.193341
$222$	0	0
$223$	15.0714	1.00925	0.504627	−	0.863337i	$-0.331630\pi$
$223$	15.0714	1.00925	0.504627	+	0.863337i	$0.331630\pi$
$224$	38.4736	2.57062
$225$	0	0
$226$	5.10574	0.339629
$227$	−22.6926	−1.50616	−0.753079	−	0.657930i	$-0.771432\pi$
$227$	−22.6926	−1.50616	−0.753079	+	0.657930i	$0.771432\pi$
$228$	0	0
$229$	8.71670	0.576016	0.288008	−	0.957628i	$-0.407007\pi$
$229$	8.71670	0.576016	0.288008	+	0.957628i	$0.407007\pi$
$230$	−18.9392	−1.24881
$231$	0	0
$232$	−3.26801	−0.214555
$233$	−23.5890	−1.54536	−0.772682	−	0.634793i	$-0.781085\pi$
$233$	−23.5890	−1.54536	−0.772682	+	0.634793i	$0.781085\pi$
$234$	0	0
$235$	−14.8758	−0.970393
$236$	0.659772	0.0429475
$237$	0	0
$238$	−24.3539	−1.57863
$239$	9.95452	0.643904	0.321952	−	0.946756i	$-0.395661\pi$
$239$	9.95452	0.643904	0.321952	+	0.946756i	$0.395661\pi$
$240$	0	0
$241$	5.71345	0.368036	0.184018	−	0.982923i	$-0.441090\pi$
$241$	5.71345	0.368036	0.184018	+	0.982923i	$0.441090\pi$
$242$	−13.2140	−0.849427
$243$	0	0
$244$	−11.2092	−0.717594
$245$	−34.1905	−2.18435
$246$	0	0
$247$	2.23504	0.142212
$248$	−1.61502	−0.102554
$249$	0	0
$250$	−25.1339	−1.58961
$251$	−7.28966	−0.460120	−0.230060	−	0.973176i	$-0.573892\pi$
$251$	−7.28966	−0.460120	−0.230060	+	0.973176i	$0.573892\pi$
$252$	0	0
$253$	−17.8576	−1.12270
$254$	19.6240	1.23132
$255$	0	0
$256$	4.90123	0.306327
$257$	−23.2431	−1.44986	−0.724932	−	0.688821i	$-0.758129\pi$
$257$	−23.2431	−1.44986	−0.724932	+	0.688821i	$0.758129\pi$
$258$	0	0
$259$	43.4978	2.70282
$260$	6.33539	0.392904
$261$	0	0
$262$	32.5958	2.01378
$263$	−27.3996	−1.68953	−0.844766	−	0.535136i	$-0.820260\pi$
$263$	−27.3996	−1.68953	−0.844766	+	0.535136i	$0.820260\pi$
$264$	0	0
$265$	13.1050	0.805032
$266$	18.9380	1.16116
$267$	0	0
$268$	10.3928	0.634842
$269$	9.41973	0.574331	0.287166	−	0.957881i	$-0.407287\pi$
$269$	9.41973	0.574331	0.287166	+	0.957881i	$0.407287\pi$
$270$	0	0
$271$	26.2797	1.59638	0.798189	−	0.602408i	$-0.205792\pi$
$271$	26.2797	1.59638	0.798189	+	0.602408i	$0.205792\pi$
$272$	−6.38776	−0.387315
$273$	0	0
$274$	7.74354	0.467804
$275$	−2.95114	−0.177961
$276$	0	0
$277$	0.374812	0.0225203	0.0112601	−	0.999937i	$-0.496416\pi$
$277$	0.374812	0.0225203	0.0112601	+	0.999937i	$0.496416\pi$
$278$	−28.2109	−1.69198
$279$	0	0
$280$	11.0051	0.657678
$281$	13.9816	0.834071	0.417036	−	0.908890i	$-0.363069\pi$
$281$	13.9816	0.834071	0.417036	+	0.908890i	$0.363069\pi$
$282$	0	0
$283$	15.5619	0.925058	0.462529	−	0.886604i	$-0.346942\pi$
$283$	15.5619	0.925058	0.462529	+	0.886604i	$0.346942\pi$
$284$	7.76236	0.460612
$285$	0	0
$286$	10.7226	0.634038
$287$	10.9605	0.646980
$288$	0	0
$289$	−11.4132	−0.671366
$290$	13.1224	0.770575
$291$	0	0
$292$	32.1011	1.87857
$293$	24.6242	1.43856	0.719281	−	0.694719i	$-0.244471\pi$
$293$	24.6242	1.43856	0.719281	+	0.694719i	$0.244471\pi$
$294$	0	0
$295$	−0.543118	−0.0316216
$296$	−9.83210	−0.571479
$297$	0	0
$298$	−18.9361	−1.09694
$299$	5.23321	0.302644
$300$	0	0
$301$	−26.5943	−1.53287
$302$	1.51388	0.0871139
$303$	0	0
$304$	4.96723	0.284890
$305$	9.22729	0.528353
$306$	0	0
$307$	−20.3912	−1.16379	−0.581893	−	0.813265i	$-0.697688\pi$
$307$	−20.3912	−1.16379	−0.581893	+	0.813265i	$0.697688\pi$
$308$	50.6156	2.88409
$309$	0	0
$310$	6.48497	0.368322
$311$	22.1790	1.25766	0.628828	−	0.777544i	$-0.283535\pi$
$311$	22.1790	1.25766	0.628828	+	0.777544i	$0.283535\pi$
$312$	0	0
$313$	−11.1720	−0.631481	−0.315740	−	0.948846i	$-0.602253\pi$
$313$	−11.1720	−0.631481	−0.315740	+	0.948846i	$0.602253\pi$
$314$	29.3638	1.65709
$315$	0	0
$316$	11.4527	0.644265
$317$	25.2167	1.41631	0.708154	−	0.706058i	$-0.249528\pi$
$317$	25.2167	1.41631	0.708154	+	0.706058i	$0.249528\pi$
$318$	0	0
$319$	12.3730	0.692757
$320$	23.7264	1.32634
$321$	0	0
$322$	44.3421	2.47109
$323$	−4.34438	−0.241728
$324$	0	0
$325$	0.864837	0.0479725
$326$	2.53561	0.140435
$327$	0	0
$328$	−2.47748	−0.136796
$329$	34.8287	1.92017
$330$	0	0
$331$	−26.1657	−1.43820	−0.719098	−	0.694908i	$-0.755445\pi$
$331$	−26.1657	−1.43820	−0.719098	+	0.694908i	$0.755445\pi$
$332$	−21.2658	−1.16711
$333$	0	0
$334$	50.8654	2.78323
$335$	−8.55527	−0.467424
$336$	0	0
$337$	10.9460	0.596269	0.298135	−	0.954524i	$-0.403636\pi$
$337$	10.9460	0.596269	0.298135	+	0.954524i	$0.403636\pi$
$338$	24.4831	1.33171
$339$	0	0
$340$	−12.3145	−0.667846
$341$	6.11463	0.331126
$342$	0	0
$343$	46.1091	2.48966
$344$	6.01128	0.324106
$345$	0	0
$346$	19.4228	1.04417
$347$	3.51338	0.188608	0.0943040	−	0.995543i	$-0.469937\pi$
$347$	3.51338	0.188608	0.0943040	+	0.995543i	$0.469937\pi$
$348$	0	0
$349$	−29.1968	−1.56287	−0.781436	−	0.623986i	$-0.785512\pi$
$349$	−29.1968	−1.56287	−0.781436	+	0.623986i	$0.785512\pi$
$350$	7.32796	0.391696
$351$	0	0
$352$	32.9256	1.75494
$353$	−25.0768	−1.33470	−0.667352	−	0.744742i	$-0.732572\pi$
$353$	−25.0768	−1.33470	−0.667352	+	0.744742i	$0.732572\pi$
$354$	0	0
$355$	−6.38991	−0.339141
$356$	42.6750	2.26177
$357$	0	0
$358$	−22.5269	−1.19058
$359$	4.20724	0.222050	0.111025	−	0.993818i	$-0.464587\pi$
$359$	4.20724	0.222050	0.111025	+	0.993818i	$0.464587\pi$
$360$	0	0
$361$	−15.6217	−0.822196
$362$	3.10875	0.163392
$363$	0	0
$364$	−14.8330	−0.777460
$365$	−26.4253	−1.38316
$366$	0	0
$367$	17.4966	0.913317	0.456658	−	0.889642i	$-0.349046\pi$
$367$	17.4966	0.913317	0.456658	+	0.889642i	$0.349046\pi$
$368$	11.6304	0.606279
$369$	0	0
$370$	39.4800	2.05247
$371$	−30.6825	−1.59296
$372$	0	0
$373$	−29.6546	−1.53546	−0.767728	−	0.640776i	$-0.778613\pi$
$373$	−29.6546	−1.53546	−0.767728	+	0.640776i	$0.778613\pi$
$374$	−20.8420	−1.07772
$375$	0	0
$376$	−7.87255	−0.405996
$377$	−3.62594	−0.186745
$378$	0	0
$379$	20.9523	1.07625	0.538124	−	0.842865i	$-0.319133\pi$
$379$	20.9523	1.07625	0.538124	+	0.842865i	$0.319133\pi$
$380$	9.57594	0.491236
$381$	0	0
$382$	−26.3977	−1.35062
$383$	9.89678	0.505702	0.252851	−	0.967505i	$-0.418632\pi$
$383$	9.89678	0.505702	0.252851	+	0.967505i	$0.418632\pi$
$384$	0	0
$385$	−41.6663	−2.12351
$386$	−44.1439	−2.24686
$387$	0	0
$388$	−12.8549	−0.652608
$389$	−21.1148	−1.07056	−0.535281	−	0.844674i	$-0.679794\pi$
$389$	−21.1148	−1.07056	−0.535281	+	0.844674i	$0.679794\pi$
$390$	0	0
$391$	−10.1721	−0.514424
$392$	−18.0942	−0.913894
$393$	0	0
$394$	30.1513	1.51900
$395$	−9.42776	−0.474362
$396$	0	0
$397$	−9.77909	−0.490799	−0.245399	−	0.969422i	$-0.578919\pi$
$397$	−9.77909	−0.490799	−0.245399	+	0.969422i	$0.578919\pi$
$398$	−43.1989	−2.16537
$399$	0	0
$400$	1.92204	0.0961021
$401$	−19.1393	−0.955773	−0.477886	−	0.878422i	$-0.658597\pi$
$401$	−19.1393	−0.955773	−0.477886	+	0.878422i	$0.658597\pi$
$402$	0	0
$403$	−1.79190	−0.0892611
$404$	46.7423	2.32552
$405$	0	0
$406$	−30.7234	−1.52478
$407$	37.2254	1.84519
$408$	0	0
$409$	14.9956	0.741483	0.370741	−	0.928736i	$-0.379104\pi$
$409$	14.9956	0.741483	0.370741	+	0.928736i	$0.379104\pi$
$410$	9.94813	0.491303
$411$	0	0
$412$	22.6562	1.11619
$413$	1.27160	0.0625712
$414$	0	0
$415$	17.5058	0.859325
$416$	−9.64892	−0.473077
$417$	0	0
$418$	16.2071	0.792717
$419$	5.81180	0.283925	0.141963	−	0.989872i	$-0.454659\pi$
$419$	5.81180	0.283925	0.141963	+	0.989872i	$0.454659\pi$
$420$	0	0
$421$	−13.3135	−0.648861	−0.324431	−	0.945910i	$-0.605173\pi$
$421$	−13.3135	−0.648861	−0.324431	+	0.945910i	$0.605173\pi$
$422$	20.9598	1.02031
$423$	0	0
$424$	6.93538	0.336812
$425$	−1.68103	−0.0815421
$426$	0	0
$427$	−21.6038	−1.04548
$428$	18.6840	0.903123
$429$	0	0
$430$	−24.1378	−1.16403
$431$	−36.4166	−1.75413	−0.877064	−	0.480374i	$-0.840501\pi$
$431$	−36.4166	−1.75413	−0.877064	+	0.480374i	$0.840501\pi$
$432$	0	0
$433$	−10.8761	−0.522674	−0.261337	−	0.965248i	$-0.584163\pi$
$433$	−10.8761	−0.522674	−0.261337	+	0.965248i	$0.584163\pi$
$434$	−15.1832	−0.728817
$435$	0	0
$436$	−14.1485	−0.677589
$437$	7.90999	0.378386
$438$	0	0
$439$	9.45326	0.451180	0.225590	−	0.974222i	$-0.427569\pi$
$439$	9.45326	0.451180	0.225590	+	0.974222i	$0.427569\pi$
$440$	9.41811	0.448991
$441$	0	0
$442$	6.10780	0.290518
$443$	16.5239	0.785075	0.392537	−	0.919736i	$-0.371597\pi$
$443$	16.5239	0.785075	0.392537	+	0.919736i	$0.371597\pi$
$444$	0	0
$445$	−35.1297	−1.66531
$446$	−32.0271	−1.51653
$447$	0	0
$448$	−55.5503	−2.62450
$449$	−4.74362	−0.223865	−0.111933	−	0.993716i	$-0.535704\pi$
$449$	−4.74362	−0.223865	−0.111933	+	0.993716i	$0.535704\pi$
$450$	0	0
$451$	9.38002	0.441688
$452$	−6.04451	−0.284310
$453$	0	0
$454$	48.2223	2.26319
$455$	12.2104	0.572432
$456$	0	0
$457$	−22.4085	−1.04823	−0.524114	−	0.851648i	$-0.675603\pi$
$457$	−22.4085	−1.04823	−0.524114	+	0.851648i	$0.675603\pi$
$458$	−18.5232	−0.865534
$459$	0	0
$460$	22.4214	1.04540
$461$	−14.9223	−0.694999	−0.347499	−	0.937680i	$-0.612969\pi$
$461$	−14.9223	−0.694999	−0.347499	+	0.937680i	$0.612969\pi$
$462$	0	0
$463$	−14.8472	−0.690007	−0.345003	−	0.938601i	$-0.612122\pi$
$463$	−14.8472	−0.690007	−0.345003	+	0.938601i	$0.612122\pi$
$464$	−8.05840	−0.374102
$465$	0	0
$466$	50.1272	2.32210
$467$	−15.3514	−0.710379	−0.355190	−	0.934794i	$-0.615584\pi$
$467$	−15.3514	−0.710379	−0.355190	+	0.934794i	$0.615584\pi$
$468$	0	0
$469$	20.0304	0.924917
$470$	31.6116	1.45813
$471$	0	0
$472$	−0.287427	−0.0132299
$473$	−22.7594	−1.04648
$474$	0	0
$475$	1.30720	0.0599785
$476$	28.8317	1.32150
$477$	0	0
$478$	−21.1536	−0.967545
$479$	10.4486	0.477409	0.238705	−	0.971092i	$-0.423277\pi$
$479$	10.4486	0.477409	0.238705	+	0.971092i	$0.423277\pi$
$480$	0	0
$481$	−10.9090	−0.497406
$482$	−12.1412	−0.553018
$483$	0	0
$484$	15.6436	0.711071
$485$	10.5820	0.480505
$486$	0	0
$487$	−16.1649	−0.732500	−0.366250	−	0.930516i	$-0.619359\pi$
$487$	−16.1649	−0.732500	−0.366250	+	0.930516i	$0.619359\pi$
$488$	4.88324	0.221054
$489$	0	0
$490$	72.6557	3.28225
$491$	−10.7773	−0.486375	−0.243187	−	0.969979i	$-0.578193\pi$
$491$	−10.7773	−0.486375	−0.243187	+	0.969979i	$0.578193\pi$
$492$	0	0
$493$	7.04794	0.317423
$494$	−4.74953	−0.213692
$495$	0	0
$496$	−3.98238	−0.178814
$497$	14.9606	0.671076
$498$	0	0
$499$	−19.1872	−0.858936	−0.429468	−	0.903082i	$-0.641299\pi$
$499$	−19.1872	−0.858936	−0.429468	+	0.903082i	$0.641299\pi$
$500$	29.7552	1.33069
$501$	0	0
$502$	15.4907	0.691386
$503$	−12.0251	−0.536171	−0.268086	−	0.963395i	$-0.586391\pi$
$503$	−12.0251	−0.536171	−0.268086	+	0.963395i	$0.586391\pi$
$504$	0	0
$505$	−38.4778	−1.71224
$506$	37.9479	1.68699
$507$	0	0
$508$	−23.2322	−1.03076
$509$	−14.9530	−0.662781	−0.331391	−	0.943494i	$-0.607518\pi$
$509$	−14.9530	−0.662781	−0.331391	+	0.943494i	$0.607518\pi$
$510$	0	0
$511$	61.8694	2.73694
$512$	−27.3678	−1.20950
$513$	0	0
$514$	49.3922	2.17860
$515$	−18.6504	−0.821835
$516$	0	0
$517$	29.8063	1.31088
$518$	−92.4342	−4.06132
$519$	0	0
$520$	−2.75999	−0.121034
$521$	37.4188	1.63935	0.819673	−	0.572832i	$-0.194155\pi$
$521$	37.4188	1.63935	0.819673	+	0.572832i	$0.194155\pi$
$522$	0	0
$523$	−8.44979	−0.369483	−0.184742	−	0.982787i	$-0.559145\pi$
$523$	−8.44979	−0.369483	−0.184742	+	0.982787i	$0.559145\pi$
$524$	−38.5891	−1.68577
$525$	0	0
$526$	58.2250	2.53873
$527$	3.48303	0.151723
$528$	0	0
$529$	−4.47929	−0.194752
$530$	−27.8484	−1.20966
$531$	0	0
$532$	−22.4201	−0.972033
$533$	−2.74883	−0.119065
$534$	0	0
$535$	−15.3805	−0.664956
$536$	−4.52760	−0.195562
$537$	0	0
$538$	−20.0172	−0.863003
$539$	68.5065	2.95078
$540$	0	0
$541$	12.6259	0.542828	0.271414	−	0.962463i	$-0.412509\pi$
$541$	12.6259	0.542828	0.271414	+	0.962463i	$0.412509\pi$
$542$	−55.8451	−2.39875
$543$	0	0
$544$	18.7552	0.804121
$545$	11.6469	0.498898
$546$	0	0
$547$	31.3951	1.34236	0.671178	−	0.741296i	$-0.265789\pi$
$547$	31.3951	1.34236	0.671178	+	0.741296i	$0.265789\pi$
$548$	−9.16731	−0.391608
$549$	0	0
$550$	6.27126	0.267407
$551$	−5.48060	−0.233482
$552$	0	0
$553$	22.0731	0.938645
$554$	−0.796486	−0.0338394
$555$	0	0
$556$	33.3979	1.41639
$557$	−15.9303	−0.674988	−0.337494	−	0.941328i	$-0.609579\pi$
$557$	−15.9303	−0.674988	−0.337494	+	0.941328i	$0.609579\pi$
$558$	0	0
$559$	6.66967	0.282097
$560$	27.1368	1.14674
$561$	0	0
$562$	−29.7113	−1.25329
$563$	−24.8609	−1.04776	−0.523880	−	0.851792i	$-0.675516\pi$
$563$	−24.8609	−1.04776	−0.523880	+	0.851792i	$0.675516\pi$
$564$	0	0
$565$	4.97578	0.209333
$566$	−33.0695	−1.39001
$567$	0	0
$568$	−3.38165	−0.141891
$569$	19.1900	0.804487	0.402243	−	0.915533i	$-0.368231\pi$
$569$	19.1900	0.804487	0.402243	+	0.915533i	$0.368231\pi$
$570$	0	0
$571$	−20.3792	−0.852844	−0.426422	−	0.904524i	$-0.640226\pi$
$571$	−20.3792	−0.852844	−0.426422	+	0.904524i	$0.640226\pi$
$572$	−12.6941	−0.530765
$573$	0	0
$574$	−23.2915	−0.972167
$575$	3.06072	0.127641
$576$	0	0
$577$	23.2991	0.969953	0.484976	−	0.874527i	$-0.338828\pi$
$577$	23.2991	0.969953	0.484976	+	0.874527i	$0.338828\pi$
$578$	24.2534	1.00881
$579$	0	0
$580$	−15.5352	−0.645063
$581$	−40.9861	−1.70039
$582$	0	0
$583$	−26.2581	−1.08750
$584$	−13.9847	−0.578693
$585$	0	0
$586$	−52.3272	−2.16162
$587$	−36.9093	−1.52341	−0.761704	−	0.647925i	$-0.775637\pi$
$587$	−36.9093	−1.52341	−0.761704	+	0.647925i	$0.775637\pi$
$588$	0	0
$589$	−2.70846	−0.111600
$590$	1.15414	0.0475153
$591$	0	0
$592$	−24.2444	−0.996440
$593$	4.36830	0.179385	0.0896923	−	0.995970i	$-0.471412\pi$
$593$	4.36830	0.179385	0.0896923	+	0.995970i	$0.471412\pi$
$594$	0	0
$595$	−23.7340	−0.973000
$596$	22.4178	0.918268
$597$	0	0
$598$	−11.1207	−0.454760
$599$	−31.2056	−1.27503	−0.637513	−	0.770439i	$-0.720037\pi$
$599$	−31.2056	−1.27503	−0.637513	+	0.770439i	$0.720037\pi$
$600$	0	0
$601$	43.9371	1.79223	0.896116	−	0.443820i	$-0.146377\pi$
$601$	43.9371	1.79223	0.896116	+	0.443820i	$0.146377\pi$
$602$	56.5136	2.30332
$603$	0	0
$604$	−1.79223	−0.0729247
$605$	−12.8776	−0.523551
$606$	0	0
$607$	17.3587	0.704566	0.352283	−	0.935894i	$-0.385405\pi$
$607$	17.3587	0.704566	0.352283	+	0.935894i	$0.385405\pi$
$608$	−14.5843	−0.591473
$609$	0	0
$610$	−19.6083	−0.793915
$611$	−8.73480	−0.353372
$612$	0	0
$613$	−1.19805	−0.0483887	−0.0241944	−	0.999707i	$-0.507702\pi$
$613$	−1.19805	−0.0483887	−0.0241944	+	0.999707i	$0.507702\pi$
$614$	43.3318	1.74873
$615$	0	0
$616$	−22.0505	−0.888441
$617$	26.0137	1.04727	0.523636	−	0.851942i	$-0.324575\pi$
$617$	26.0137	1.04727	0.523636	+	0.851942i	$0.324575\pi$
$618$	0	0
$619$	9.63135	0.387117	0.193558	−	0.981089i	$-0.437997\pi$
$619$	9.63135	0.387117	0.193558	+	0.981089i	$0.437997\pi$
$620$	−7.67733	−0.308329
$621$	0	0
$622$	−47.1311	−1.88978
$623$	82.2488	3.29523
$624$	0	0
$625$	−20.9382	−0.837526
$626$	23.7409	0.948877
$627$	0	0
$628$	−34.7627	−1.38718
$629$	21.2044	0.845474
$630$	0	0
$631$	−14.1673	−0.563992	−0.281996	−	0.959416i	$-0.590997\pi$
$631$	−14.1673	−0.563992	−0.281996	+	0.959416i	$0.590997\pi$
$632$	−4.98933	−0.198465
$633$	0	0
$634$	−53.5861	−2.12818
$635$	19.1245	0.758933
$636$	0	0
$637$	−20.0760	−0.795438
$638$	−26.2930	−1.04095
$639$	0	0
$640$	−17.5539	−0.693879
$641$	22.2009	0.876883	0.438442	−	0.898760i	$-0.355531\pi$
$641$	22.2009	0.876883	0.438442	+	0.898760i	$0.355531\pi$
$642$	0	0
$643$	21.5368	0.849328	0.424664	−	0.905351i	$-0.360392\pi$
$643$	21.5368	0.849328	0.424664	+	0.905351i	$0.360392\pi$
$644$	−52.4951	−2.06860
$645$	0	0
$646$	9.23194	0.363226
$647$	13.4037	0.526952	0.263476	−	0.964666i	$-0.415131\pi$
$647$	13.4037	0.526952	0.263476	+	0.964666i	$0.415131\pi$
$648$	0	0
$649$	1.08823	0.0427168
$650$	−1.83780	−0.0720846
$651$	0	0
$652$	−3.00182	−0.117560
$653$	−18.9109	−0.740039	−0.370020	−	0.929024i	$-0.620649\pi$
$653$	−18.9109	−0.740039	−0.370020	+	0.929024i	$0.620649\pi$
$654$	0	0
$655$	31.7662	1.24121
$656$	−6.10909	−0.238520
$657$	0	0
$658$	−74.0119	−2.88528
$659$	−0.0281814	−0.00109779	−0.000548897	−	1.00000i	$-0.500175\pi$
$659$	−0.0281814	−0.00109779	−0.000548897		1.00000i	$0.500175\pi$
$660$	0	0
$661$	−45.0417	−1.75192	−0.875960	−	0.482383i	$-0.839771\pi$
$661$	−45.0417	−1.75192	−0.875960	+	0.482383i	$0.839771\pi$
$662$	55.6028	2.16107
$663$	0	0
$664$	9.26436	0.359527
$665$	18.4560	0.715693
$666$	0	0
$667$	−12.8325	−0.496875
$668$	−60.2177	−2.32989
$669$	0	0
$670$	18.1802	0.702362
$671$	−18.4885	−0.713740
$672$	0	0
$673$	−9.75969	−0.376208	−0.188104	−	0.982149i	$-0.560234\pi$
$673$	−9.75969	−0.376208	−0.188104	+	0.982149i	$0.560234\pi$
$674$	−23.2607	−0.895968
$675$	0	0
$676$	−28.9847	−1.11480
$677$	40.8808	1.57118	0.785588	−	0.618750i	$-0.212361\pi$
$677$	40.8808	1.57118	0.785588	+	0.618750i	$0.212361\pi$
$678$	0	0
$679$	−24.7756	−0.950801
$680$	5.36476	0.205729
$681$	0	0
$682$	−12.9938	−0.497557
$683$	−44.0251	−1.68457	−0.842287	−	0.539029i	$-0.818791\pi$
$683$	−44.0251	−1.68457	−0.842287	+	0.539029i	$0.818791\pi$
$684$	0	0
$685$	7.54644	0.288335
$686$	−97.9831	−3.74101
$687$	0	0
$688$	14.8229	0.565117
$689$	7.69498	0.293155
$690$	0	0
$691$	21.5516	0.819862	0.409931	−	0.912117i	$-0.365553\pi$
$691$	21.5516	0.819862	0.409931	+	0.912117i	$0.365553\pi$
$692$	−22.9939	−0.874098
$693$	0	0
$694$	−7.46603	−0.283407
$695$	−27.4928	−1.04286
$696$	0	0
$697$	5.34306	0.202383
$698$	62.0441	2.34841
$699$	0	0
$700$	−8.67532	−0.327896
$701$	−12.8521	−0.485419	−0.242709	−	0.970099i	$-0.578036\pi$
$701$	−12.8521	−0.485419	−0.242709	+	0.970099i	$0.578036\pi$
$702$	0	0
$703$	−16.4889	−0.621891
$704$	−47.5399	−1.79173
$705$	0	0
$706$	53.2889	2.00556
$707$	90.0878	3.38810
$708$	0	0
$709$	−49.7117	−1.86696	−0.933481	−	0.358627i	$-0.883245\pi$
$709$	−49.7117	−1.86696	−0.933481	+	0.358627i	$0.883245\pi$
$710$	13.5787	0.509601
$711$	0	0
$712$	−18.5912	−0.696737
$713$	−6.34168	−0.237498
$714$	0	0
$715$	10.4496	0.390794
$716$	26.6688	0.996660
$717$	0	0
$718$	−8.94051	−0.333657
$719$	−5.63745	−0.210242	−0.105121	−	0.994459i	$-0.533523\pi$
$719$	−5.63745	−0.210242	−0.105121	+	0.994459i	$0.533523\pi$
$720$	0	0
$721$	43.6660	1.62621
$722$	33.1966	1.23545
$723$	0	0
$724$	−3.68034	−0.136779
$725$	−2.12069	−0.0787604
$726$	0	0
$727$	−45.4143	−1.68432	−0.842161	−	0.539226i	$-0.818717\pi$
$727$	−45.4143	−1.68432	−0.842161	+	0.539226i	$0.818717\pi$
$728$	6.46195	0.239496
$729$	0	0
$730$	56.1546	2.07837
$731$	−12.9642	−0.479499
$732$	0	0
$733$	−25.6669	−0.948027	−0.474013	−	0.880518i	$-0.657195\pi$
$733$	−25.6669	−0.948027	−0.474013	+	0.880518i	$0.657195\pi$
$734$	−37.1808	−1.37237
$735$	0	0
$736$	−34.1483	−1.25872
$737$	17.1420	0.631433
$738$	0	0
$739$	−14.4553	−0.531745	−0.265873	−	0.964008i	$-0.585660\pi$
$739$	−14.4553	−0.531745	−0.265873	+	0.964008i	$0.585660\pi$
$740$	−46.7390	−1.71816
$741$	0	0
$742$	65.2013	2.39361
$743$	34.7937	1.27646	0.638229	−	0.769846i	$-0.279667\pi$
$743$	34.7937	1.27646	0.638229	+	0.769846i	$0.279667\pi$
$744$	0	0
$745$	−18.4541	−0.676107
$746$	63.0168	2.30721
$747$	0	0
$748$	24.6742	0.902177
$749$	36.0101	1.31578
$750$	0	0
$751$	18.1184	0.661152	0.330576	−	0.943779i	$-0.392757\pi$
$751$	18.1184	0.661152	0.330576	+	0.943779i	$0.392757\pi$
$752$	−19.4125	−0.707901
$753$	0	0
$754$	7.70522	0.280608
$755$	1.47535	0.0536933
$756$	0	0
$757$	−37.0045	−1.34495	−0.672475	−	0.740120i	$-0.734769\pi$
$757$	−37.0045	−1.34495	−0.672475	+	0.740120i	$0.734769\pi$
$758$	−44.5243	−1.61720
$759$	0	0
$760$	−4.17173	−0.151325
$761$	−12.4444	−0.451110	−0.225555	−	0.974230i	$-0.572420\pi$
$761$	−12.4444	−0.451110	−0.225555	+	0.974230i	$0.572420\pi$
$762$	0	0
$763$	−27.2688	−0.987195
$764$	31.2513	1.13063
$765$	0	0
$766$	−21.0309	−0.759878
$767$	−0.318908	−0.0115151
$768$	0	0
$769$	−40.8856	−1.47437	−0.737186	−	0.675690i	$-0.763846\pi$
$769$	−40.8856	−1.47437	−0.737186	+	0.675690i	$0.763846\pi$
$770$	88.5421	3.19084
$771$	0	0
$772$	52.2604	1.88089
$773$	−36.4162	−1.30980	−0.654900	−	0.755716i	$-0.727289\pi$
$773$	−36.4162	−1.30980	−0.654900	+	0.755716i	$0.727289\pi$
$774$	0	0
$775$	−1.04802	−0.0376461
$776$	5.60019	0.201035
$777$	0	0
$778$	44.8695	1.60865
$779$	−4.15486	−0.148863
$780$	0	0
$781$	12.8033	0.458138
$782$	21.6160	0.772985
$783$	0	0
$784$	−44.6174	−1.59348
$785$	28.6164	1.02136
$786$	0	0
$787$	18.3472	0.654005	0.327003	−	0.945023i	$-0.393961\pi$
$787$	18.3472	0.654005	0.327003	+	0.945023i	$0.393961\pi$
$788$	−35.6951	−1.27158
$789$	0	0
$790$	20.0343	0.712787
$791$	−11.6498	−0.414218
$792$	0	0
$793$	5.41808	0.192402
$794$	20.7808	0.737485
$795$	0	0
$796$	51.1417	1.81267
$797$	3.46492	0.122734	0.0613670	−	0.998115i	$-0.480454\pi$
$797$	3.46492	0.122734	0.0613670	+	0.998115i	$0.480454\pi$
$798$	0	0
$799$	16.9783	0.600650
$800$	−5.64333	−0.199522
$801$	0	0
$802$	40.6716	1.43617
$803$	52.9477	1.86849
$804$	0	0
$805$	43.2135	1.52307
$806$	3.80785	0.134126
$807$	0	0
$808$	−20.3631	−0.716372
$809$	−24.8406	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$809$	−24.8406	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$810$	0	0
$811$	−40.3286	−1.41613	−0.708063	−	0.706149i	$-0.750431\pi$
$811$	−40.3286	−1.41613	−0.708063	+	0.706149i	$0.750431\pi$
$812$	36.3723	1.27642
$813$	0	0
$814$	−79.1050	−2.77263
$815$	2.47107	0.0865579
$816$	0	0
$817$	10.0812	0.352697
$818$	−31.8660	−1.11417
$819$	0	0
$820$	−11.7772	−0.411279
$821$	40.1816	1.40235	0.701174	−	0.712990i	$-0.252659\pi$
$821$	40.1816	1.40235	0.701174	+	0.712990i	$0.252659\pi$
$822$	0	0
$823$	−47.9554	−1.67162	−0.835810	−	0.549019i	$-0.815001\pi$
$823$	−47.9554	−1.67162	−0.835810	+	0.549019i	$0.815001\pi$
$824$	−9.87012	−0.343842
$825$	0	0
$826$	−2.70218	−0.0940209
$827$	−5.00048	−0.173884	−0.0869419	−	0.996213i	$-0.527709\pi$
$827$	−5.00048	−0.173884	−0.0869419	+	0.996213i	$0.527709\pi$
$828$	0	0
$829$	29.7037	1.03165	0.515826	−	0.856693i	$-0.327485\pi$
$829$	29.7037	1.03165	0.515826	+	0.856693i	$0.327485\pi$
$830$	−37.2003	−1.29124
$831$	0	0
$832$	13.9316	0.482993
$833$	39.0228	1.35206
$834$	0	0
$835$	49.5707	1.71546
$836$	−19.1871	−0.663599
$837$	0	0
$838$	−12.3502	−0.426632
$839$	−25.9045	−0.894323	−0.447162	−	0.894453i	$-0.647565\pi$
$839$	−25.9045	−0.894323	−0.447162	+	0.894453i	$0.647565\pi$
$840$	0	0
$841$	−20.1088	−0.693405
$842$	28.2916	0.974993
$843$	0	0
$844$	−24.8136	−0.854120
$845$	23.8599	0.820807
$846$	0	0
$847$	30.1503	1.03598
$848$	17.1016	0.587270
$849$	0	0
$850$	3.57224	0.122527
$851$	−38.6077	−1.32345
$852$	0	0
$853$	−4.99166	−0.170911	−0.0854556	−	0.996342i	$-0.527235\pi$
$853$	−4.99166	−0.170911	−0.0854556	+	0.996342i	$0.527235\pi$
$854$	45.9086	1.57096
$855$	0	0
$856$	−8.13961	−0.278206
$857$	−14.6830	−0.501562	−0.250781	−	0.968044i	$-0.580687\pi$
$857$	−14.6830	−0.501562	−0.250781	+	0.968044i	$0.580687\pi$
$858$	0	0
$859$	−17.4849	−0.596576	−0.298288	−	0.954476i	$-0.596416\pi$
$859$	−17.4849	−0.596576	−0.298288	+	0.954476i	$0.596416\pi$
$860$	28.5759	0.974430
$861$	0	0
$862$	77.3864	2.63579
$863$	6.33263	0.215565	0.107783	−	0.994174i	$-0.465625\pi$
$863$	6.33263	0.215565	0.107783	+	0.994174i	$0.465625\pi$
$864$	0	0
$865$	18.9284	0.643585
$866$	23.1121	0.785381
$867$	0	0
$868$	17.9749	0.610106
$869$	18.8902	0.640805
$870$	0	0
$871$	−5.02349	−0.170214
$872$	6.16374	0.208730
$873$	0	0
$874$	−16.8090	−0.568571
$875$	57.3480	1.93872
$876$	0	0
$877$	32.5310	1.09849	0.549247	−	0.835660i	$-0.314915\pi$
$877$	32.5310	1.09849	0.549247	+	0.835660i	$0.314915\pi$
$878$	−20.0885	−0.677953
$879$	0	0
$880$	23.2236	0.782868
$881$	−33.3599	−1.12393	−0.561963	−	0.827163i	$-0.689954\pi$
$881$	−33.3599	−1.12393	−0.561963	+	0.827163i	$0.689954\pi$
$882$	0	0
$883$	−54.8511	−1.84589	−0.922944	−	0.384934i	$-0.874224\pi$
$883$	−54.8511	−1.84589	−0.922944	+	0.384934i	$0.874224\pi$
$884$	−7.23081	−0.243198
$885$	0	0
$886$	−35.1138	−1.17967
$887$	21.0477	0.706714	0.353357	−	0.935489i	$-0.385040\pi$
$887$	21.0477	0.706714	0.353357	+	0.935489i	$0.385040\pi$
$888$	0	0
$889$	−44.7760	−1.50174
$890$	74.6516	2.50233
$891$	0	0
$892$	37.9158	1.26951
$893$	−13.2026	−0.441810
$894$	0	0
$895$	−21.9535	−0.733825
$896$	41.0988	1.37301
$897$	0	0
$898$	10.0803	0.336385
$899$	4.39397	0.146547
$900$	0	0
$901$	−14.9572	−0.498296
$902$	−19.9328	−0.663690
$903$	0	0
$904$	2.63327	0.0875813
$905$	3.02962	0.100708
$906$	0	0
$907$	14.8380	0.492689	0.246344	−	0.969182i	$-0.420771\pi$
$907$	14.8380	0.492689	0.246344	+	0.969182i	$0.420771\pi$
$908$	−57.0887	−1.89456
$909$	0	0
$910$	−25.9474	−0.860149
$911$	18.0295	0.597345	0.298673	−	0.954356i	$-0.403456\pi$
$911$	18.0295	0.597345	0.298673	+	0.954356i	$0.403456\pi$
$912$	0	0
$913$	−35.0759	−1.16084
$914$	47.6188	1.57509
$915$	0	0
$916$	21.9290	0.724555
$917$	−74.3738	−2.45604
$918$	0	0
$919$	13.4881	0.444932	0.222466	−	0.974940i	$-0.428589\pi$
$919$	13.4881	0.444932	0.222466	+	0.974940i	$0.428589\pi$
$920$	−9.76783	−0.322036
$921$	0	0
$922$	31.7102	1.04432
$923$	−3.75203	−0.123499
$924$	0	0
$925$	−6.38029	−0.209783
$926$	31.5507	1.03682
$927$	0	0
$928$	23.6603	0.776689
$929$	−37.6211	−1.23431	−0.617155	−	0.786842i	$-0.711715\pi$
$929$	−37.6211	−1.23431	−0.617155	+	0.786842i	$0.711715\pi$
$930$	0	0
$931$	−30.3448	−0.994511
$932$	−59.3439	−1.94387
$933$	0	0
$934$	32.6222	1.06743
$935$	−20.3116	−0.664259
$936$	0	0
$937$	4.14458	0.135398	0.0676988	−	0.997706i	$-0.478434\pi$
$937$	4.14458	0.135398	0.0676988	+	0.997706i	$0.478434\pi$
$938$	−42.5651	−1.38980
$939$	0	0
$940$	−37.4238	−1.22063
$941$	−3.53033	−0.115085	−0.0575427	−	0.998343i	$-0.518327\pi$
$941$	−3.53033	−0.115085	−0.0575427	+	0.998343i	$0.518327\pi$
$942$	0	0
$943$	−9.72832	−0.316798
$944$	−0.708752	−0.0230679
$945$	0	0
$946$	48.3643	1.57246
$947$	−14.2551	−0.463228	−0.231614	−	0.972808i	$-0.574401\pi$
$947$	−14.2551	−0.463228	−0.231614	+	0.972808i	$0.574401\pi$
$948$	0	0
$949$	−15.5164	−0.503685
$950$	−2.77784	−0.0901250
$951$	0	0
$952$	−12.5605	−0.407087
$953$	11.6426	0.377141	0.188570	−	0.982060i	$-0.439615\pi$
$953$	11.6426	0.377141	0.188570	+	0.982060i	$0.439615\pi$
$954$	0	0
$955$	−25.7258	−0.832467
$956$	25.0430	0.809950
$957$	0	0
$958$	−22.2036	−0.717365
$959$	−17.6684	−0.570543
$960$	0	0
$961$	−28.8285	−0.929953
$962$	23.1819	0.747414
$963$	0	0
$964$	14.3736	0.462942
$965$	−43.0203	−1.38487
$966$	0	0
$967$	29.0681	0.934768	0.467384	−	0.884054i	$-0.345197\pi$
$967$	29.0681	0.934768	0.467384	+	0.884054i	$0.345197\pi$
$968$	−6.81507	−0.219045
$969$	0	0
$970$	−22.4871	−0.722018
$971$	47.5792	1.52689	0.763444	−	0.645874i	$-0.223507\pi$
$971$	47.5792	1.52689	0.763444	+	0.645874i	$0.223507\pi$
$972$	0	0
$973$	64.3687	2.06357
$974$	34.3508	1.10067
$975$	0	0
$976$	12.0413	0.385433
$977$	6.11630	0.195678	0.0978389	−	0.995202i	$-0.468807\pi$
$977$	6.11630	0.195678	0.0978389	+	0.995202i	$0.468807\pi$
$978$	0	0
$979$	70.3885	2.24963
$980$	−86.0145	−2.74763
$981$	0	0
$982$	22.9022	0.730837
$983$	10.6518	0.339740	0.169870	−	0.985466i	$-0.445665\pi$
$983$	10.6518	0.339740	0.169870	+	0.985466i	$0.445665\pi$
$984$	0	0
$985$	29.3839	0.936248
$986$	−14.9771	−0.476967
$987$	0	0
$988$	5.62281	0.178885
$989$	23.6045	0.750578
$990$	0	0
$991$	23.9856	0.761928	0.380964	−	0.924590i	$-0.375592\pi$
$991$	23.9856	0.761928	0.380964	+	0.924590i	$0.375592\pi$
$992$	11.6927	0.371244
$993$	0	0
$994$	−31.7918	−1.00837
$995$	−42.0994	−1.33464
$996$	0	0
$997$	4.31418	0.136631	0.0683157	−	0.997664i	$-0.478237\pi$
$997$	4.31418	0.136631	0.0683157	+	0.997664i	$0.478237\pi$
$998$	40.7733	1.29066
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.b.1.2	✓	6
3.2	odd	2		729.2.a.e.1.5	yes	6
9.2	odd	6		729.2.c.a.244.2		12
9.4	even	3		729.2.c.d.487.5		12
9.5	odd	6		729.2.c.a.487.2		12
9.7	even	3		729.2.c.d.244.5		12
27.2	odd	18		729.2.e.l.568.1		12
27.4	even	9		729.2.e.t.406.1		12
27.5	odd	18		729.2.e.u.649.2		12
27.7	even	9		729.2.e.t.325.1		12
27.11	odd	18		729.2.e.u.82.2		12
27.13	even	9		729.2.e.s.163.2		12
27.14	odd	18		729.2.e.l.163.1		12
27.16	even	9		729.2.e.j.82.1		12
27.20	odd	18		729.2.e.k.325.2		12
27.22	even	9		729.2.e.j.649.1		12
27.23	odd	18		729.2.e.k.406.2		12
27.25	even	9		729.2.e.s.568.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.b.1.2	✓	6	1.1	even	1	trivial
729.2.a.e.1.5	yes	6	3.2	odd	2
729.2.c.a.244.2		12	9.2	odd	6
729.2.c.a.487.2		12	9.5	odd	6
729.2.c.d.244.5		12	9.7	even	3
729.2.c.d.487.5		12	9.4	even	3
729.2.e.j.82.1		12	27.16	even	9
729.2.e.j.649.1		12	27.22	even	9
729.2.e.k.325.2		12	27.20	odd	18
729.2.e.k.406.2		12	27.23	odd	18
729.2.e.l.163.1		12	27.14	odd	18
729.2.e.l.568.1		12	27.2	odd	18
729.2.e.s.163.2		12	27.13	even	9
729.2.e.s.568.2		12	27.25	even	9
729.2.e.t.325.1		12	27.7	even	9
729.2.e.t.406.1		12	27.4	even	9
729.2.e.u.82.2		12	27.11	odd	18
729.2.e.u.649.2		12	27.5	odd	18

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12503 q^{2} +2.51575 q^{4} -2.07094 q^{5} +4.84867 q^{7} -1.09598 q^{8} +4.40081 q^{10} +4.14949 q^{11} -1.21602 q^{13} -10.3036 q^{14} -2.70251 q^{16} +2.36364 q^{17} -1.83801 q^{19} -5.20996 q^{20} -8.81778 q^{22} -4.30357 q^{23} -0.711206 q^{25} +2.58407 q^{26} +12.1980 q^{28} +2.98182 q^{29} +1.47359 q^{31} +7.93487 q^{32} -5.02280 q^{34} -10.0413 q^{35} +8.97108 q^{37} +3.90582 q^{38} +2.26970 q^{40} +2.26052 q^{41} -5.48486 q^{43} +10.4391 q^{44} +9.14521 q^{46} +7.18313 q^{47} +16.5096 q^{49} +1.51133 q^{50} -3.05919 q^{52} -6.32803 q^{53} -8.59334 q^{55} -5.31404 q^{56} -6.33645 q^{58} +0.262257 q^{59} -4.45561 q^{61} -3.13141 q^{62} -11.4568 q^{64} +2.51830 q^{65} +4.13110 q^{67} +5.94632 q^{68} +21.3381 q^{70} +3.08551 q^{71} +12.7601 q^{73} -19.0638 q^{74} -4.62396 q^{76} +20.1195 q^{77} +4.55241 q^{79} +5.59674 q^{80} -4.80368 q^{82} -8.45306 q^{83} -4.89495 q^{85} +11.6555 q^{86} -4.54775 q^{88} +16.9632 q^{89} -5.89606 q^{91} -10.8267 q^{92} -15.2644 q^{94} +3.80640 q^{95} -5.10977 q^{97} -35.0834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

Introduction

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