LMFDB - Embedded newform 729.2.a.e.1.1

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			1.1
Root			$-0.578404$ 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)

$f(q)$	$=$	\(q-2.45779 q^{2} +4.04073 q^{4} -3.08026 q^{5} -2.65867 q^{7} -5.01568 q^{8} +7.57064 q^{10} -3.43434 q^{11} -3.34396 q^{13} +6.53444 q^{14} +4.24603 q^{16} +2.57282 q^{17} -2.09676 q^{19} -12.4465 q^{20} +8.44089 q^{22} +0.534444 q^{23} +4.48802 q^{25} +8.21874 q^{26} -10.7430 q^{28} +2.53089 q^{29} +7.71470 q^{31} -0.404491 q^{32} -6.32344 q^{34} +8.18939 q^{35} -10.2957 q^{37} +5.15340 q^{38} +15.4496 q^{40} -4.88501 q^{41} +2.74149 q^{43} -13.8772 q^{44} -1.31355 q^{46} -5.65800 q^{47} +0.0685109 q^{49} -11.0306 q^{50} -13.5120 q^{52} +6.42657 q^{53} +10.5787 q^{55} +13.3350 q^{56} -6.22040 q^{58} +1.65495 q^{59} +14.3722 q^{61} -18.9611 q^{62} -7.49791 q^{64} +10.3003 q^{65} -5.87898 q^{67} +10.3961 q^{68} -20.1278 q^{70} +14.8163 q^{71} +1.88140 q^{73} +25.3046 q^{74} -8.47245 q^{76} +9.13077 q^{77} +17.1935 q^{79} -13.0789 q^{80} +12.0063 q^{82} +3.96878 q^{83} -7.92496 q^{85} -6.73801 q^{86} +17.2256 q^{88} +5.09880 q^{89} +8.89047 q^{91} +2.15954 q^{92} +13.9062 q^{94} +6.45858 q^{95} -10.6319 q^{97} -0.168385 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.45779	−1.73792	−0.868960	−	0.494883i	$-0.835211\pi$
$2$	−2.45779	−1.73792	−0.868960	+	0.494883i	$0.835211\pi$
$3$	0	0
$3$	0	0
$4$	4.04073	2.02036
$5$	−3.08026	−1.37754	−0.688768	−	0.724982i	$-0.741848\pi$
$5$	−3.08026	−1.37754	−0.688768	+	0.724982i	$0.741848\pi$
$6$	0	0
$7$	−2.65867	−1.00488	−0.502441	−	0.864612i	$-0.667565\pi$
$7$	−2.65867	−1.00488	−0.502441	+	0.864612i	$0.667565\pi$
$8$	−5.01568	−1.77331
$9$	0	0
$10$	7.57064	2.39405
$11$	−3.43434	−1.03549	−0.517746	−	0.855534i	$-0.673229\pi$
$11$	−3.43434	−1.03549	−0.517746	+	0.855534i	$0.673229\pi$
$12$	0	0
$13$	−3.34396	−0.927447	−0.463723	−	0.885980i	$-0.653487\pi$
$13$	−3.34396	−0.927447	−0.463723	+	0.885980i	$0.653487\pi$
$14$	6.53444	1.74640
$15$	0	0
$16$	4.24603	1.06151
$17$	2.57282	0.624000	0.312000	−	0.950082i	$-0.399001\pi$
$17$	2.57282	0.624000	0.312000	+	0.950082i	$0.399001\pi$
$18$	0	0
$19$	−2.09676	−0.481030	−0.240515	−	0.970645i	$-0.577316\pi$
$19$	−2.09676	−0.481030	−0.240515	+	0.970645i	$0.577316\pi$
$20$	−12.4465	−2.78312
$21$	0	0
$22$	8.44089	1.79960
$23$	0.534444	0.111439	0.0557196	−	0.998446i	$-0.482255\pi$
$23$	0.534444	0.111439	0.0557196	+	0.998446i	$0.482255\pi$
$24$	0	0
$25$	4.48802	0.897604
$26$	8.21874	1.61183
$27$	0	0
$28$	−10.7430	−2.03023
$29$	2.53089	0.469975	0.234988	−	0.971998i	$-0.424495\pi$
$29$	2.53089	0.469975	0.234988	+	0.971998i	$0.424495\pi$
$30$	0	0
$31$	7.71470	1.38560	0.692801	−	0.721129i	$-0.256377\pi$
$31$	7.71470	1.38560	0.692801	+	0.721129i	$0.256377\pi$
$32$	−0.404491	−0.0715045
$33$	0	0
$34$	−6.32344	−1.08446
$35$	8.18939	1.38426
$36$	0	0
$37$	−10.2957	−1.69260	−0.846298	−	0.532709i	$-0.821174\pi$
$37$	−10.2957	−1.69260	−0.846298	+	0.532709i	$0.821174\pi$
$38$	5.15340	0.835992
$39$	0	0
$40$	15.4496	2.44280
$41$	−4.88501	−0.762910	−0.381455	−	0.924387i	$-0.624577\pi$
$41$	−4.88501	−0.762910	−0.381455	+	0.924387i	$0.624577\pi$
$42$	0	0
$43$	2.74149	0.418073	0.209037	−	0.977908i	$-0.432967\pi$
$43$	2.74149	0.418073	0.209037	+	0.977908i	$0.432967\pi$
$44$	−13.8772	−2.09207
$45$	0	0
$46$	−1.31355	−0.193673
$47$	−5.65800	−0.825304	−0.412652	−	0.910889i	$-0.635397\pi$
$47$	−5.65800	−0.825304	−0.412652	+	0.910889i	$0.635397\pi$
$48$	0	0
$49$	0.0685109	0.00978728
$50$	−11.0306	−1.55996
$51$	0	0
$52$	−13.5120	−1.87378
$53$	6.42657	0.882758	0.441379	−	0.897321i	$-0.354489\pi$
$53$	6.42657	0.882758	0.441379	+	0.897321i	$0.354489\pi$
$54$	0	0
$55$	10.5787	1.42643
$56$	13.3350	1.78197
$57$	0	0
$58$	−6.22040	−0.816779
$59$	1.65495	0.215456	0.107728	−	0.994180i	$-0.465642\pi$
$59$	1.65495	0.215456	0.107728	+	0.994180i	$0.465642\pi$
$60$	0	0
$61$	14.3722	1.84017	0.920086	−	0.391716i	$-0.128118\pi$
$61$	14.3722	1.84017	0.920086	+	0.391716i	$0.128118\pi$
$62$	−18.9611	−2.40806
$63$	0	0
$64$	−7.49791	−0.937239
$65$	10.3003	1.27759
$66$	0	0
$67$	−5.87898	−0.718232	−0.359116	−	0.933293i	$-0.616922\pi$
$67$	−5.87898	−0.718232	−0.359116	+	0.933293i	$0.616922\pi$
$68$	10.3961	1.26071
$69$	0	0
$70$	−20.1278	−2.40573
$71$	14.8163	1.75837	0.879184	−	0.476483i	$-0.158089\pi$
$71$	14.8163	1.75837	0.879184	+	0.476483i	$0.158089\pi$
$72$	0	0
$73$	1.88140	0.220201	0.110101	−	0.993920i	$-0.464883\pi$
$73$	1.88140	0.220201	0.110101	+	0.993920i	$0.464883\pi$
$74$	25.3046	2.94160
$75$	0	0
$76$	−8.47245	−0.971857
$77$	9.13077	1.04055
$78$	0	0
$79$	17.1935	1.93442	0.967209	−	0.253981i	$-0.0817400\pi$
$79$	17.1935	1.93442	0.967209	+	0.253981i	$0.0817400\pi$
$80$	−13.0789	−1.46227
$81$	0	0
$82$	12.0063	1.32588
$83$	3.96878	0.435631	0.217815	−	0.975990i	$-0.430107\pi$
$83$	3.96878	0.435631	0.217815	+	0.975990i	$0.430107\pi$
$84$	0	0
$85$	−7.92496	−0.859582
$86$	−6.73801	−0.726578
$87$	0	0
$88$	17.2256	1.83625
$89$	5.09880	0.540471	0.270236	−	0.962794i	$-0.412898\pi$
$89$	5.09880	0.540471	0.270236	+	0.962794i	$0.412898\pi$
$90$	0	0
$91$	8.89047	0.931974
$92$	2.15954	0.225148
$93$	0	0
$94$	13.9062	1.43431
$95$	6.45858	0.662637
$96$	0	0
$97$	−10.6319	−1.07950	−0.539752	−	0.841824i	$-0.681482\pi$
$97$	−10.6319	−1.07950	−0.539752	+	0.841824i	$0.681482\pi$
$98$	−0.168385	−0.0170095
$99$	0	0
$100$	18.1349	1.81349
$101$	5.65809	0.563001	0.281500	−	0.959561i	$-0.409168\pi$
$101$	5.65809	0.563001	0.281500	+	0.959561i	$0.409168\pi$
$102$	0	0
$103$	10.3705	1.02184	0.510920	−	0.859628i	$-0.329305\pi$
$103$	10.3705	1.02184	0.510920	+	0.859628i	$0.329305\pi$
$104$	16.7722	1.64465
$105$	0	0
$106$	−15.7952	−1.53416
$107$	−14.2457	−1.37719	−0.688594	−	0.725147i	$-0.741772\pi$
$107$	−14.2457	−1.37719	−0.688594	+	0.725147i	$0.741772\pi$
$108$	0	0
$109$	5.76064	0.551769	0.275884	−	0.961191i	$-0.411029\pi$
$109$	5.76064	0.551769	0.275884	+	0.961191i	$0.411029\pi$
$110$	−26.0001	−2.47902
$111$	0	0
$112$	−11.2888	−1.06669
$113$	−11.4167	−1.07399	−0.536996	−	0.843585i	$-0.680441\pi$
$113$	−11.4167	−1.07399	−0.536996	+	0.843585i	$0.680441\pi$
$114$	0	0
$115$	−1.64623	−0.153512
$116$	10.2267	0.949521
$117$	0	0
$118$	−4.06752	−0.374445
$119$	−6.84027	−0.627046
$120$	0	0
$121$	0.794696	0.0722451
$122$	−35.3239	−3.19807
$123$	0	0
$124$	31.1730	2.79942
$125$	1.57703	0.141054
$126$	0	0
$127$	2.59019	0.229843	0.114921	−	0.993375i	$-0.463338\pi$
$127$	2.59019	0.229843	0.114921	+	0.993375i	$0.463338\pi$
$128$	19.2373	1.70035
$129$	0	0
$130$	−25.3159	−2.22035
$131$	−4.41099	−0.385390	−0.192695	−	0.981259i	$-0.561723\pi$
$131$	−4.41099	−0.385390	−0.192695	+	0.981259i	$0.561723\pi$
$132$	0	0
$133$	5.57460	0.483379
$134$	14.4493	1.24823
$135$	0	0
$136$	−12.9044	−1.10655
$137$	18.1677	1.55217	0.776085	−	0.630628i	$-0.217203\pi$
$137$	18.1677	1.55217	0.776085	+	0.630628i	$0.217203\pi$
$138$	0	0
$139$	1.95794	0.166071	0.0830353	−	0.996547i	$-0.473539\pi$
$139$	1.95794	0.166071	0.0830353	+	0.996547i	$0.473539\pi$
$140$	33.0911	2.79671
$141$	0	0
$142$	−36.4153	−3.05590
$143$	11.4843	0.960364
$144$	0	0
$145$	−7.79582	−0.647407
$146$	−4.62408	−0.382692
$147$	0	0
$148$	−41.6020	−3.41966
$149$	7.31519	0.599284	0.299642	−	0.954052i	$-0.403133\pi$
$149$	7.31519	0.599284	0.299642	+	0.954052i	$0.403133\pi$
$150$	0	0
$151$	4.85099	0.394768	0.197384	−	0.980326i	$-0.436755\pi$
$151$	4.85099	0.394768	0.197384	+	0.980326i	$0.436755\pi$
$152$	10.5167	0.853017
$153$	0	0
$154$	−22.4415	−1.80839
$155$	−23.7633	−1.90871
$156$	0	0
$157$	1.47670	0.117854	0.0589269	−	0.998262i	$-0.481232\pi$
$157$	1.47670	0.117854	0.0589269	+	0.998262i	$0.481232\pi$
$158$	−42.2580	−3.36186
$159$	0	0
$160$	1.24594	0.0985000
$161$	−1.42091	−0.111983
$162$	0	0
$163$	−17.2536	−1.35141	−0.675703	−	0.737174i	$-0.736160\pi$
$163$	−17.2536	−1.35141	−0.675703	+	0.737174i	$0.736160\pi$
$164$	−19.7390	−1.54136
$165$	0	0
$166$	−9.75444	−0.757091
$167$	6.84036	0.529323	0.264662	−	0.964341i	$-0.414740\pi$
$167$	6.84036	0.529323	0.264662	+	0.964341i	$0.414740\pi$
$168$	0	0
$169$	−1.81796	−0.139843
$170$	19.4779	1.49388
$171$	0	0
$172$	11.0776	0.844661
$173$	−23.9674	−1.82221	−0.911103	−	0.412179i	$-0.864768\pi$
$173$	−23.9674	−1.82221	−0.911103	+	0.412179i	$0.864768\pi$
$174$	0	0
$175$	−11.9322	−0.901986
$176$	−14.5823	−1.09918
$177$	0	0
$178$	−12.5318	−0.939296
$179$	−20.5722	−1.53764	−0.768820	−	0.639466i	$-0.779156\pi$
$179$	−20.5722	−1.53764	−0.768820	+	0.639466i	$0.779156\pi$
$180$	0	0
$181$	−15.4701	−1.14989	−0.574943	−	0.818194i	$-0.694976\pi$
$181$	−15.4701	−1.14989	−0.574943	+	0.818194i	$0.694976\pi$
$182$	−21.8509	−1.61970
$183$	0	0
$184$	−2.68060	−0.197617
$185$	31.7133	2.33161
$186$	0	0
$187$	−8.83593	−0.646147
$188$	−22.8624	−1.66741
$189$	0	0
$190$	−15.8738	−1.15161
$191$	−10.9508	−0.792376	−0.396188	−	0.918170i	$-0.629667\pi$
$191$	−10.9508	−0.792376	−0.396188	+	0.918170i	$0.629667\pi$
$192$	0	0
$193$	1.14485	0.0824082	0.0412041	−	0.999151i	$-0.486881\pi$
$193$	1.14485	0.0824082	0.0412041	+	0.999151i	$0.486881\pi$
$194$	26.1309	1.87609
$195$	0	0
$196$	0.276834	0.0197739
$197$	−5.04195	−0.359224	−0.179612	−	0.983738i	$-0.557484\pi$
$197$	−5.04195	−0.359224	−0.179612	+	0.983738i	$0.557484\pi$
$198$	0	0
$199$	13.7258	0.972997	0.486499	−	0.873681i	$-0.338274\pi$
$199$	13.7258	0.972997	0.486499	+	0.873681i	$0.338274\pi$
$200$	−22.5105	−1.59173
$201$	0	0
$202$	−13.9064	−0.978450
$203$	−6.72880	−0.472269
$204$	0	0
$205$	15.0471	1.05094
$206$	−25.4886	−1.77588
$207$	0	0
$208$	−14.1985	−0.984492
$209$	7.20100	0.498104
$210$	0	0
$211$	−5.83513	−0.401707	−0.200854	−	0.979621i	$-0.564372\pi$
$211$	−5.83513	−0.401707	−0.200854	+	0.979621i	$0.564372\pi$
$212$	25.9680	1.78349
$213$	0	0
$214$	35.0130	2.39344
$215$	−8.44451	−0.575911
$216$	0	0
$217$	−20.5108	−1.39237
$218$	−14.1584	−0.958930
$219$	0	0
$220$	42.7455	2.88190
$221$	−8.60339	−0.578727
$222$	0	0
$223$	8.72868	0.584516	0.292258	−	0.956340i	$-0.405593\pi$
$223$	8.72868	0.584516	0.292258	+	0.956340i	$0.405593\pi$
$224$	1.07541	0.0718536
$225$	0	0
$226$	28.0598	1.86651
$227$	24.2825	1.61169	0.805844	−	0.592128i	$-0.201712\pi$
$227$	24.2825	1.61169	0.805844	+	0.592128i	$0.201712\pi$
$228$	0	0
$229$	18.1514	1.19948	0.599740	−	0.800195i	$-0.295271\pi$
$229$	18.1514	1.19948	0.599740	+	0.800195i	$0.295271\pi$
$230$	4.04608	0.266791
$231$	0	0
$232$	−12.6942	−0.833412
$233$	10.5380	0.690367	0.345183	−	0.938535i	$-0.387817\pi$
$233$	10.5380	0.690367	0.345183	+	0.938535i	$0.387817\pi$
$234$	0	0
$235$	17.4281	1.13688
$236$	6.68720	0.435300
$237$	0	0
$238$	16.8119	1.08976
$239$	−9.53660	−0.616871	−0.308436	−	0.951245i	$-0.599805\pi$
$239$	−9.53660	−0.616871	−0.308436	+	0.951245i	$0.599805\pi$
$240$	0	0
$241$	−7.03728	−0.453311	−0.226656	−	0.973975i	$-0.572779\pi$
$241$	−7.03728	−0.453311	−0.226656	+	0.973975i	$0.572779\pi$
$242$	−1.95320	−0.125556
$243$	0	0
$244$	58.0742	3.71782
$245$	−0.211032	−0.0134823
$246$	0	0
$247$	7.01148	0.446130
$248$	−38.6945	−2.45710
$249$	0	0
$250$	−3.87602	−0.245141
$251$	−15.5870	−0.983843	−0.491921	−	0.870640i	$-0.663705\pi$
$251$	−15.5870	−0.983843	−0.491921	+	0.870640i	$0.663705\pi$
$252$	0	0
$253$	−1.83546	−0.115395
$254$	−6.36615	−0.399448
$255$	0	0
$256$	−32.2853	−2.01783
$257$	−12.1898	−0.760377	−0.380189	−	0.924909i	$-0.624141\pi$
$257$	−12.1898	−0.760377	−0.380189	+	0.924909i	$0.624141\pi$
$258$	0	0
$259$	27.3727	1.70086
$260$	41.6206	2.58120
$261$	0	0
$262$	10.8413	0.669776
$263$	6.88963	0.424833	0.212417	−	0.977179i	$-0.431867\pi$
$263$	6.88963	0.424833	0.212417	+	0.977179i	$0.431867\pi$
$264$	0	0
$265$	−19.7955	−1.21603
$266$	−13.7012	−0.840073
$267$	0	0
$268$	−23.7554	−1.45109
$269$	−7.05875	−0.430380	−0.215190	−	0.976572i	$-0.569037\pi$
$269$	−7.05875	−0.430380	−0.215190	+	0.976572i	$0.569037\pi$
$270$	0	0
$271$	23.7575	1.44316	0.721581	−	0.692330i	$-0.243416\pi$
$271$	23.7575	1.44316	0.721581	+	0.692330i	$0.243416\pi$
$272$	10.9243	0.662381
$273$	0	0
$274$	−44.6523	−2.69755
$275$	−15.4134	−0.929462
$276$	0	0
$277$	−0.104464	−0.00627662	−0.00313831	−	0.999995i	$-0.500999\pi$
$277$	−0.104464	−0.00627662	−0.00313831	+	0.999995i	$0.500999\pi$
$278$	−4.81221	−0.288617
$279$	0	0
$280$	−41.0754	−2.45472
$281$	−8.15220	−0.486319	−0.243160	−	0.969986i	$-0.578184\pi$
$281$	−8.15220	−0.486319	−0.243160	+	0.969986i	$0.578184\pi$
$282$	0	0
$283$	23.6160	1.40382	0.701912	−	0.712263i	$-0.252330\pi$
$283$	23.6160	1.40382	0.701912	+	0.712263i	$0.252330\pi$
$284$	59.8685	3.55254
$285$	0	0
$286$	−28.2260	−1.66904
$287$	12.9876	0.766634
$288$	0	0
$289$	−10.3806	−0.610624
$290$	19.1605	1.12514
$291$	0	0
$292$	7.60222	0.444886
$293$	−21.6258	−1.26339	−0.631695	−	0.775217i	$-0.717640\pi$
$293$	−21.6258	−1.26339	−0.631695	+	0.775217i	$0.717640\pi$
$294$	0	0
$295$	−5.09768	−0.296798
$296$	51.6398	3.00150
$297$	0	0
$298$	−17.9792	−1.04151
$299$	−1.78716	−0.103354
$300$	0	0
$301$	−7.28871	−0.420114
$302$	−11.9227	−0.686074
$303$	0	0
$304$	−8.90293	−0.510618
$305$	−44.2702	−2.53490
$306$	0	0
$307$	16.3599	0.933711	0.466855	−	0.884334i	$-0.345387\pi$
$307$	16.3599	0.933711	0.466855	+	0.884334i	$0.345387\pi$
$308$	36.8950	2.10229
$309$	0	0
$310$	58.4052	3.31719
$311$	−7.75486	−0.439738	−0.219869	−	0.975529i	$-0.570563\pi$
$311$	−7.75486	−0.439738	−0.219869	+	0.975529i	$0.570563\pi$
$312$	0	0
$313$	15.4181	0.871480	0.435740	−	0.900073i	$-0.356487\pi$
$313$	15.4181	0.871480	0.435740	+	0.900073i	$0.356487\pi$
$314$	−3.62942	−0.204820
$315$	0	0
$316$	69.4742	3.90823
$317$	8.63621	0.485058	0.242529	−	0.970144i	$-0.422023\pi$
$317$	8.63621	0.485058	0.242529	+	0.970144i	$0.422023\pi$
$318$	0	0
$319$	−8.69195	−0.486656
$320$	23.0955	1.29108
$321$	0	0
$322$	3.49229	0.194618
$323$	−5.39459	−0.300163
$324$	0	0
$325$	−15.0077	−0.832480
$326$	42.4057	2.34864
$327$	0	0
$328$	24.5016	1.35288
$329$	15.0427	0.829332
$330$	0	0
$331$	20.5445	1.12923	0.564614	−	0.825355i	$-0.309025\pi$
$331$	20.5445	1.12923	0.564614	+	0.825355i	$0.309025\pi$
$332$	16.0368	0.880133
$333$	0	0
$334$	−16.8122	−0.919921
$335$	18.1088	0.989390
$336$	0	0
$337$	15.9782	0.870388	0.435194	−	0.900337i	$-0.356680\pi$
$337$	15.9782	0.870388	0.435194	+	0.900337i	$0.356680\pi$
$338$	4.46816	0.243036
$339$	0	0
$340$	−32.0226	−1.73667
$341$	−26.4949	−1.43478
$342$	0	0
$343$	18.4285	0.995047
$344$	−13.7504	−0.741374
$345$	0	0
$346$	58.9068	3.16685
$347$	−9.80427	−0.526321	−0.263161	−	0.964752i	$-0.584765\pi$
$347$	−9.80427	−0.526321	−0.263161	+	0.964752i	$0.584765\pi$
$348$	0	0
$349$	−9.43497	−0.505043	−0.252521	−	0.967591i	$-0.581260\pi$
$349$	−9.43497	−0.505043	−0.252521	+	0.967591i	$0.581260\pi$
$350$	29.3267	1.56758
$351$	0	0
$352$	1.38916	0.0740424
$353$	−3.38516	−0.180174	−0.0900870	−	0.995934i	$-0.528715\pi$
$353$	−3.38516	−0.180174	−0.0900870	+	0.995934i	$0.528715\pi$
$354$	0	0
$355$	−45.6380	−2.42221
$356$	20.6029	1.09195
$357$	0	0
$358$	50.5622	2.67229
$359$	35.2273	1.85923	0.929614	−	0.368535i	$-0.120141\pi$
$359$	35.2273	1.85923	0.929614	+	0.368535i	$0.120141\pi$
$360$	0	0
$361$	−14.6036	−0.768610
$362$	38.0223	1.99841
$363$	0	0
$364$	35.9240	1.88293
$365$	−5.79520	−0.303335
$366$	0	0
$367$	−31.5454	−1.64666	−0.823329	−	0.567564i	$-0.807886\pi$
$367$	−31.5454	−1.64666	−0.823329	+	0.567564i	$0.807886\pi$
$368$	2.26927	0.118294
$369$	0	0
$370$	−77.9447	−4.05215
$371$	−17.0861	−0.887067
$372$	0	0
$373$	14.4423	0.747795	0.373897	−	0.927470i	$-0.378021\pi$
$373$	14.4423	0.747795	0.373897	+	0.927470i	$0.378021\pi$
$374$	21.7169	1.12295
$375$	0	0
$376$	28.3787	1.46352
$377$	−8.46320	−0.435877
$378$	0	0
$379$	1.00099	0.0514176	0.0257088	−	0.999669i	$-0.491816\pi$
$379$	1.00099	0.0514176	0.0257088	+	0.999669i	$0.491816\pi$
$380$	26.0974	1.33877
$381$	0	0
$382$	26.9149	1.37709
$383$	4.11734	0.210386	0.105193	−	0.994452i	$-0.466454\pi$
$383$	4.11734	0.210386	0.105193	+	0.994452i	$0.466454\pi$
$384$	0	0
$385$	−28.1252	−1.43339
$386$	−2.81380	−0.143219
$387$	0	0
$388$	−42.9605	−2.18099
$389$	−15.5199	−0.786891	−0.393445	−	0.919348i	$-0.628717\pi$
$389$	−15.5199	−0.786891	−0.393445	+	0.919348i	$0.628717\pi$
$390$	0	0
$391$	1.37503	0.0695381
$392$	−0.343629	−0.0173559
$393$	0	0
$394$	12.3920	0.624302
$395$	−52.9605	−2.66473
$396$	0	0
$397$	−1.54893	−0.0777384	−0.0388692	−	0.999244i	$-0.512376\pi$
$397$	−1.54893	−0.0777384	−0.0388692	+	0.999244i	$0.512376\pi$
$398$	−33.7352	−1.69099
$399$	0	0
$400$	19.0563	0.952814
$401$	7.94738	0.396873	0.198437	−	0.980114i	$-0.436414\pi$
$401$	7.94738	0.396873	0.198437	+	0.980114i	$0.436414\pi$
$402$	0	0
$403$	−25.7976	−1.28507
$404$	22.8628	1.13747
$405$	0	0
$406$	16.5380	0.820766
$407$	35.3588	1.75267
$408$	0	0
$409$	26.4554	1.30814	0.654068	−	0.756436i	$-0.273061\pi$
$409$	26.4554	1.30814	0.654068	+	0.756436i	$0.273061\pi$
$410$	−36.9826	−1.82644
$411$	0	0
$412$	41.9045	2.06449
$413$	−4.39996	−0.216508
$414$	0	0
$415$	−12.2249	−0.600097
$416$	1.35260	0.0663166
$417$	0	0
$418$	−17.6985	−0.865664
$419$	38.3400	1.87303	0.936516	−	0.350626i	$-0.114031\pi$
$419$	38.3400	1.87303	0.936516	+	0.350626i	$0.114031\pi$
$420$	0	0
$421$	7.34450	0.357949	0.178974	−	0.983854i	$-0.442722\pi$
$421$	7.34450	0.357949	0.178974	+	0.983854i	$0.442722\pi$
$422$	14.3415	0.698134
$423$	0	0
$424$	−32.2337	−1.56540
$425$	11.5469	0.560105
$426$	0	0
$427$	−38.2109	−1.84916
$428$	−57.5632	−2.78242
$429$	0	0
$430$	20.7548	1.00089
$431$	−15.8463	−0.763289	−0.381644	−	0.924309i	$-0.624642\pi$
$431$	−15.8463	−0.763289	−0.381644	+	0.924309i	$0.624642\pi$
$432$	0	0
$433$	−23.8507	−1.14619	−0.573097	−	0.819488i	$-0.694258\pi$
$433$	−23.8507	−1.14619	−0.573097	+	0.819488i	$0.694258\pi$
$434$	50.4113	2.41982
$435$	0	0
$436$	23.2772	1.11477
$437$	−1.12060	−0.0536057
$438$	0	0
$439$	−21.4699	−1.02470	−0.512352	−	0.858776i	$-0.671226\pi$
$439$	−21.4699	−1.02470	−0.512352	+	0.858776i	$0.671226\pi$
$440$	−53.0593	−2.52950
$441$	0	0
$442$	21.1453	1.00578
$443$	31.5479	1.49889	0.749443	−	0.662068i	$-0.230321\pi$
$443$	31.5479	1.49889	0.749443	+	0.662068i	$0.230321\pi$
$444$	0	0
$445$	−15.7056	−0.744518
$446$	−21.4533	−1.01584
$447$	0	0
$448$	19.9345	0.941815
$449$	20.7898	0.981130	0.490565	−	0.871405i	$-0.336790\pi$
$449$	20.7898	0.981130	0.490565	+	0.871405i	$0.336790\pi$
$450$	0	0
$451$	16.7768	0.789988
$452$	−46.1318	−2.16986
$453$	0	0
$454$	−59.6813	−2.80098
$455$	−27.3850	−1.28383
$456$	0	0
$457$	−17.4277	−0.815235	−0.407618	−	0.913153i	$-0.633640\pi$
$457$	−17.4277	−0.815235	−0.407618	+	0.913153i	$0.633640\pi$
$458$	−44.6124	−2.08460
$459$	0	0
$460$	−6.65196	−0.310149
$461$	−31.0045	−1.44402	−0.722011	−	0.691882i	$-0.756782\pi$
$461$	−31.0045	−1.44402	−0.722011	+	0.691882i	$0.756782\pi$
$462$	0	0
$463$	6.48795	0.301521	0.150760	−	0.988570i	$-0.451828\pi$
$463$	6.48795	0.301521	0.150760	+	0.988570i	$0.451828\pi$
$464$	10.7463	0.498883
$465$	0	0
$466$	−25.9002	−1.19980
$467$	1.94390	0.0899530	0.0449765	−	0.998988i	$-0.485679\pi$
$467$	1.94390	0.0899530	0.0449765	+	0.998988i	$0.485679\pi$
$468$	0	0
$469$	15.6303	0.721738
$470$	−42.8346	−1.97581
$471$	0	0
$472$	−8.30070	−0.382071
$473$	−9.41521	−0.432912
$474$	0	0
$475$	−9.41031	−0.431775
$476$	−27.6397	−1.26686
$477$	0	0
$478$	23.4390	1.07207
$479$	1.32939	0.0607414	0.0303707	−	0.999539i	$-0.490331\pi$
$479$	1.32939	0.0607414	0.0303707	+	0.999539i	$0.490331\pi$
$480$	0	0
$481$	34.4282	1.56979
$482$	17.2962	0.787818
$483$	0	0
$484$	3.21115	0.145961
$485$	32.7490	1.48705
$486$	0	0
$487$	21.2040	0.960844	0.480422	−	0.877037i	$-0.340484\pi$
$487$	21.2040	0.960844	0.480422	+	0.877037i	$0.340484\pi$
$488$	−72.0864	−3.26320
$489$	0	0
$490$	0.518671	0.0234312
$491$	27.0101	1.21895	0.609475	−	0.792805i	$-0.291380\pi$
$491$	27.0101	1.21895	0.609475	+	0.792805i	$0.291380\pi$
$492$	0	0
$493$	6.51153	0.293265
$494$	−17.2328	−0.775338
$495$	0	0
$496$	32.7569	1.47083
$497$	−39.3915	−1.76695
$498$	0	0
$499$	15.0389	0.673234	0.336617	−	0.941642i	$-0.390717\pi$
$499$	15.0389	0.673234	0.336617	+	0.941642i	$0.390717\pi$
$500$	6.37237	0.284981
$501$	0	0
$502$	38.3096	1.70984
$503$	4.60650	0.205394	0.102697	−	0.994713i	$-0.467253\pi$
$503$	4.60650	0.205394	0.102697	+	0.994713i	$0.467253\pi$
$504$	0	0
$505$	−17.4284	−0.775554
$506$	4.51118	0.200546
$507$	0	0
$508$	10.4663	0.464366
$509$	29.7967	1.32071	0.660357	−	0.750952i	$-0.270405\pi$
$509$	29.7967	1.32071	0.660357	+	0.750952i	$0.270405\pi$
$510$	0	0
$511$	−5.00201	−0.221276
$512$	40.8760	1.80648
$513$	0	0
$514$	29.9599	1.32147
$515$	−31.9440	−1.40762
$516$	0	0
$517$	19.4315	0.854596
$518$	−67.2764	−2.95596
$519$	0	0
$520$	−51.6629	−2.26557
$521$	−11.7621	−0.515306	−0.257653	−	0.966237i	$-0.582949\pi$
$521$	−11.7621	−0.515306	−0.257653	+	0.966237i	$0.582949\pi$
$522$	0	0
$523$	29.3853	1.28493	0.642464	−	0.766316i	$-0.277912\pi$
$523$	29.3853	1.28493	0.642464	+	0.766316i	$0.277912\pi$
$524$	−17.8236	−0.778627
$525$	0	0
$526$	−16.9333	−0.738326
$527$	19.8485	0.864615
$528$	0	0
$529$	−22.7144	−0.987581
$530$	48.6533	2.11336
$531$	0	0
$532$	22.5254	0.976601
$533$	16.3352	0.707558
$534$	0	0
$535$	43.8806	1.89712
$536$	29.4871	1.27365
$537$	0	0
$538$	17.3489	0.747965
$539$	−0.235290	−0.0101347
$540$	0	0
$541$	−22.9116	−0.985046	−0.492523	−	0.870300i	$-0.663925\pi$
$541$	−22.9116	−0.985046	−0.492523	+	0.870300i	$0.663925\pi$
$542$	−58.3908	−2.50810
$543$	0	0
$544$	−1.04068	−0.0446188
$545$	−17.7443	−0.760081
$546$	0	0
$547$	−13.7577	−0.588237	−0.294118	−	0.955769i	$-0.595026\pi$
$547$	−13.7577	−0.588237	−0.294118	+	0.955769i	$0.595026\pi$
$548$	73.4107	3.13595
$549$	0	0
$550$	37.8829	1.61533
$551$	−5.30668	−0.226072
$552$	0	0
$553$	−45.7118	−1.94386
$554$	0.256750	0.0109083
$555$	0	0
$556$	7.91152	0.335523
$557$	21.9041	0.928105	0.464053	−	0.885808i	$-0.346395\pi$
$557$	21.9041	0.928105	0.464053	+	0.885808i	$0.346395\pi$
$558$	0	0
$559$	−9.16742	−0.387741
$560$	34.7724	1.46940
$561$	0	0
$562$	20.0364	0.845184
$563$	14.5684	0.613986	0.306993	−	0.951712i	$-0.400677\pi$
$563$	14.5684	0.613986	0.306993	+	0.951712i	$0.400677\pi$
$564$	0	0
$565$	35.1664	1.47946
$566$	−58.0431	−2.43973
$567$	0	0
$568$	−74.3137	−3.11813
$569$	−22.3570	−0.937255	−0.468628	−	0.883396i	$-0.655251\pi$
$569$	−22.3570	−0.937255	−0.468628	+	0.883396i	$0.655251\pi$
$570$	0	0
$571$	−15.4261	−0.645564	−0.322782	−	0.946473i	$-0.604618\pi$
$571$	−15.4261	−0.645564	−0.322782	+	0.946473i	$0.604618\pi$
$572$	46.4049	1.94029
$573$	0	0
$574$	−31.9208	−1.33235
$575$	2.39860	0.100028
$576$	0	0
$577$	32.8081	1.36582	0.682909	−	0.730503i	$-0.260714\pi$
$577$	32.8081	1.36582	0.682909	+	0.730503i	$0.260714\pi$
$578$	25.5133	1.06122
$579$	0	0
$580$	−31.5008	−1.30800
$581$	−10.5517	−0.437757
$582$	0	0
$583$	−22.0710	−0.914089
$584$	−9.43650	−0.390485
$585$	0	0
$586$	53.1515	2.19567
$587$	30.2343	1.24790	0.623951	−	0.781464i	$-0.285527\pi$
$587$	30.2343	1.24790	0.623951	+	0.781464i	$0.285527\pi$
$588$	0	0
$589$	−16.1759	−0.666516
$590$	12.5290	0.515812
$591$	0	0
$592$	−43.7157	−1.79671
$593$	41.1023	1.68787	0.843935	−	0.536446i	$-0.180234\pi$
$593$	41.1023	1.68787	0.843935	+	0.536446i	$0.180234\pi$
$594$	0	0
$595$	21.0698	0.863778
$596$	29.5587	1.21077
$597$	0	0
$598$	4.39246	0.179621
$599$	36.5505	1.49341	0.746707	−	0.665153i	$-0.231634\pi$
$599$	36.5505	1.49341	0.746707	+	0.665153i	$0.231634\pi$
$600$	0	0
$601$	3.94580	0.160953	0.0804764	−	0.996757i	$-0.474356\pi$
$601$	3.94580	0.160953	0.0804764	+	0.996757i	$0.474356\pi$
$602$	17.9141	0.730125
$603$	0	0
$604$	19.6015	0.797575
$605$	−2.44787	−0.0995202
$606$	0	0
$607$	10.0115	0.406355	0.203178	−	0.979142i	$-0.434873\pi$
$607$	10.0115	0.406355	0.203178	+	0.979142i	$0.434873\pi$
$608$	0.848121	0.0343958
$609$	0	0
$610$	108.807	4.40546
$611$	18.9201	0.765425
$612$	0	0
$613$	−18.7568	−0.757578	−0.378789	−	0.925483i	$-0.623659\pi$
$613$	−18.7568	−0.757578	−0.378789	+	0.925483i	$0.623659\pi$
$614$	−40.2093	−1.62271
$615$	0	0
$616$	−45.7970	−1.84522
$617$	−23.0101	−0.926350	−0.463175	−	0.886267i	$-0.653290\pi$
$617$	−23.0101	−0.926350	−0.463175	+	0.886267i	$0.653290\pi$
$618$	0	0
$619$	7.41402	0.297995	0.148997	−	0.988838i	$-0.452395\pi$
$619$	7.41402	0.297995	0.148997	+	0.988838i	$0.452395\pi$
$620$	−96.0211	−3.85630
$621$	0	0
$622$	19.0598	0.764229
$623$	−13.5560	−0.543110
$624$	0	0
$625$	−27.2978	−1.09191
$626$	−37.8943	−1.51456
$627$	0	0
$628$	5.96696	0.238107
$629$	−26.4889	−1.05618
$630$	0	0
$631$	−30.9924	−1.23379	−0.616894	−	0.787046i	$-0.711609\pi$
$631$	−30.9924	−1.23379	−0.616894	+	0.787046i	$0.711609\pi$
$632$	−86.2371	−3.43033
$633$	0	0
$634$	−21.2260	−0.842991
$635$	−7.97848	−0.316616
$636$	0	0
$637$	−0.229098	−0.00907717
$638$	21.3630	0.845769
$639$	0	0
$640$	−59.2559	−2.34229
$641$	−35.3461	−1.39609	−0.698044	−	0.716055i	$-0.745946\pi$
$641$	−35.3461	−1.39609	−0.698044	+	0.716055i	$0.745946\pi$
$642$	0	0
$643$	−19.7009	−0.776929	−0.388465	−	0.921464i	$-0.626994\pi$
$643$	−19.7009	−0.776929	−0.388465	+	0.921464i	$0.626994\pi$
$644$	−5.74151	−0.226247
$645$	0	0
$646$	13.2588	0.521659
$647$	46.8317	1.84114	0.920572	−	0.390572i	$-0.127723\pi$
$647$	46.8317	1.84114	0.920572	+	0.390572i	$0.127723\pi$
$648$	0	0
$649$	−5.68366	−0.223103
$650$	36.8859	1.44678
$651$	0	0
$652$	−69.7171	−2.73033
$653$	17.3267	0.678047	0.339023	−	0.940778i	$-0.389903\pi$
$653$	17.3267	0.678047	0.339023	+	0.940778i	$0.389903\pi$
$654$	0	0
$655$	13.5870	0.530888
$656$	−20.7419	−0.809835
$657$	0	0
$658$	−36.9719	−1.44131
$659$	43.7342	1.70364	0.851821	−	0.523833i	$-0.175498\pi$
$659$	43.7342	1.70364	0.851821	+	0.523833i	$0.175498\pi$
$660$	0	0
$661$	−9.08686	−0.353438	−0.176719	−	0.984261i	$-0.556548\pi$
$661$	−9.08686	−0.353438	−0.176719	+	0.984261i	$0.556548\pi$
$662$	−50.4941	−1.96251
$663$	0	0
$664$	−19.9062	−0.772509
$665$	−17.1712	−0.665871
$666$	0	0
$667$	1.35262	0.0523737
$668$	27.6401	1.06943
$669$	0	0
$670$	−44.5076	−1.71948
$671$	−49.3591	−1.90549
$672$	0	0
$673$	7.49668	0.288976	0.144488	−	0.989507i	$-0.453846\pi$
$673$	7.49668	0.288976	0.144488	+	0.989507i	$0.453846\pi$
$674$	−39.2711	−1.51266
$675$	0	0
$676$	−7.34587	−0.282534
$677$	15.3626	0.590431	0.295215	−	0.955431i	$-0.404609\pi$
$677$	15.3626	0.590431	0.295215	+	0.955431i	$0.404609\pi$
$678$	0	0
$679$	28.2666	1.08477
$680$	39.7491	1.52431
$681$	0	0
$682$	65.1189	2.49353
$683$	6.62157	0.253367	0.126684	−	0.991943i	$-0.459567\pi$
$683$	6.62157	0.253367	0.126684	+	0.991943i	$0.459567\pi$
$684$	0	0
$685$	−55.9612	−2.13817
$686$	−45.2934	−1.72931
$687$	0	0
$688$	11.6405	0.443788
$689$	−21.4902	−0.818711
$690$	0	0
$691$	−18.6569	−0.709740	−0.354870	−	0.934916i	$-0.615475\pi$
$691$	−18.6569	−0.709740	−0.354870	+	0.934916i	$0.615475\pi$
$692$	−96.8457	−3.68152
$693$	0	0
$694$	24.0968	0.914704
$695$	−6.03098	−0.228768
$696$	0	0
$697$	−12.5682	−0.476056
$698$	23.1892	0.877723
$699$	0	0
$700$	−48.2146	−1.82234
$701$	24.8903	0.940092	0.470046	−	0.882642i	$-0.344237\pi$
$701$	24.8903	0.940092	0.470046	+	0.882642i	$0.344237\pi$
$702$	0	0
$703$	21.5876	0.814190
$704$	25.7504	0.970504
$705$	0	0
$706$	8.32002	0.313128
$707$	−15.0430	−0.565749
$708$	0	0
$709$	−26.1787	−0.983163	−0.491582	−	0.870831i	$-0.663581\pi$
$709$	−26.1787	−0.983163	−0.491582	+	0.870831i	$0.663581\pi$
$710$	112.169	4.20961
$711$	0	0
$712$	−25.5739	−0.958424
$713$	4.12308	0.154410
$714$	0	0
$715$	−35.3746	−1.32294
$716$	−83.1267	−3.10659
$717$	0	0
$718$	−86.5814	−3.23119
$719$	21.0290	0.784251	0.392125	−	0.919912i	$-0.371740\pi$
$719$	21.0290	0.784251	0.392125	+	0.919912i	$0.371740\pi$
$720$	0	0
$721$	−27.5718	−1.02683
$722$	35.8925	1.33578
$723$	0	0
$724$	−62.5106	−2.32319
$725$	11.3587	0.421852
$726$	0	0
$727$	41.3957	1.53528	0.767640	−	0.640881i	$-0.221431\pi$
$727$	41.3957	1.53528	0.767640	+	0.640881i	$0.221431\pi$
$728$	−44.5918	−1.65268
$729$	0	0
$730$	14.2434	0.527171
$731$	7.05336	0.260878
$732$	0	0
$733$	33.6267	1.24203	0.621016	−	0.783798i	$-0.286720\pi$
$733$	33.6267	1.24203	0.621016	+	0.783798i	$0.286720\pi$
$734$	77.5320	2.86176
$735$	0	0
$736$	−0.216178	−0.00796841
$737$	20.1904	0.743724
$738$	0	0
$739$	−12.9454	−0.476202	−0.238101	−	0.971240i	$-0.576525\pi$
$739$	−12.9454	−0.476202	−0.238101	+	0.971240i	$0.576525\pi$
$740$	128.145	4.71071
$741$	0	0
$742$	41.9941	1.54165
$743$	−0.0879292	−0.00322581	−0.00161290	−	0.999999i	$-0.500513\pi$
$743$	−0.0879292	−0.00322581	−0.00161290	+	0.999999i	$0.500513\pi$
$744$	0	0
$745$	−22.5327	−0.825534
$746$	−35.4962	−1.29961
$747$	0	0
$748$	−35.7036	−1.30545
$749$	37.8747	1.38391
$750$	0	0
$751$	49.8960	1.82073	0.910366	−	0.413804i	$-0.135800\pi$
$751$	49.8960	1.82073	0.910366	+	0.413804i	$0.135800\pi$
$752$	−24.0240	−0.876067
$753$	0	0
$754$	20.8008	0.757519
$755$	−14.9423	−0.543806
$756$	0	0
$757$	−11.8679	−0.431348	−0.215674	−	0.976465i	$-0.569195\pi$
$757$	−11.8679	−0.431348	−0.215674	+	0.976465i	$0.569195\pi$
$758$	−2.46023	−0.0893596
$759$	0	0
$760$	−32.3942	−1.17506
$761$	3.79030	0.137398	0.0686992	−	0.997637i	$-0.478115\pi$
$761$	3.79030	0.137398	0.0686992	+	0.997637i	$0.478115\pi$
$762$	0	0
$763$	−15.3156	−0.554462
$764$	−44.2494	−1.60089
$765$	0	0
$766$	−10.1196	−0.365634
$767$	−5.53408	−0.199824
$768$	0	0
$769$	6.26950	0.226084	0.113042	−	0.993590i	$-0.463941\pi$
$769$	6.26950	0.226084	0.113042	+	0.993590i	$0.463941\pi$
$770$	69.1257	2.49112
$771$	0	0
$772$	4.62603	0.166495
$773$	−1.29536	−0.0465907	−0.0232954	−	0.999729i	$-0.507416\pi$
$773$	−1.29536	−0.0465907	−0.0232954	+	0.999729i	$0.507416\pi$
$774$	0	0
$775$	34.6237	1.24372
$776$	53.3261	1.91430
$777$	0	0
$778$	38.1447	1.36755
$779$	10.2427	0.366983
$780$	0	0
$781$	−50.8841	−1.82078
$782$	−3.37953	−0.120852
$783$	0	0
$784$	0.290900	0.0103893
$785$	−4.54863	−0.162348
$786$	0	0
$787$	−12.3922	−0.441735	−0.220867	−	0.975304i	$-0.570889\pi$
$787$	−12.3922	−0.441735	−0.220867	+	0.975304i	$0.570889\pi$
$788$	−20.3731	−0.725763
$789$	0	0
$790$	130.166	4.63109
$791$	30.3532	1.07924
$792$	0	0
$793$	−48.0600	−1.70666
$794$	3.80693	0.135103
$795$	0	0
$796$	55.4623	1.96581
$797$	24.3827	0.863680	0.431840	−	0.901950i	$-0.357865\pi$
$797$	24.3827	0.863680	0.431840	+	0.901950i	$0.357865\pi$
$798$	0	0
$799$	−14.5570	−0.514989
$800$	−1.81536	−0.0641827
$801$	0	0
$802$	−19.5330	−0.689734
$803$	−6.46136	−0.228017
$804$	0	0
$805$	4.37677	0.154261
$806$	63.4051	2.23335
$807$	0	0
$808$	−28.3792	−0.998376
$809$	24.1156	0.847861	0.423930	−	0.905695i	$-0.360650\pi$
$809$	24.1156	0.847861	0.423930	+	0.905695i	$0.360650\pi$
$810$	0	0
$811$	48.1121	1.68944	0.844721	−	0.535206i	$-0.179766\pi$
$811$	48.1121	1.68944	0.844721	+	0.535206i	$0.179766\pi$
$812$	−27.1893	−0.954156
$813$	0	0
$814$	−86.9045	−3.04600
$815$	53.1456	1.86161
$816$	0	0
$817$	−5.74826	−0.201106
$818$	−65.0218	−2.27343
$819$	0	0
$820$	60.8013	2.12327
$821$	−16.9388	−0.591169	−0.295585	−	0.955317i	$-0.595514\pi$
$821$	−16.9388	−0.591169	−0.295585	+	0.955317i	$0.595514\pi$
$822$	0	0
$823$	11.5559	0.402815	0.201407	−	0.979508i	$-0.435449\pi$
$823$	11.5559	0.402815	0.201407	+	0.979508i	$0.435449\pi$
$824$	−52.0153	−1.81204
$825$	0	0
$826$	10.8142	0.376273
$827$	−10.2374	−0.355988	−0.177994	−	0.984032i	$-0.556961\pi$
$827$	−10.2374	−0.355988	−0.177994	+	0.984032i	$0.556961\pi$
$828$	0	0
$829$	9.35244	0.324824	0.162412	−	0.986723i	$-0.448073\pi$
$829$	9.35244	0.324824	0.162412	+	0.986723i	$0.448073\pi$
$830$	30.0462	1.04292
$831$	0	0
$832$	25.0727	0.869239
$833$	0.176266	0.00610726
$834$	0	0
$835$	−21.0701	−0.729161
$836$	29.0973	1.00635
$837$	0	0
$838$	−94.2316	−3.25518
$839$	11.8451	0.408939	0.204469	−	0.978873i	$-0.434453\pi$
$839$	11.8451	0.408939	0.204469	+	0.978873i	$0.434453\pi$
$840$	0	0
$841$	−22.5946	−0.779123
$842$	−18.0512	−0.622086
$843$	0	0
$844$	−23.5782	−0.811595
$845$	5.59979	0.192639
$846$	0	0
$847$	−2.11283	−0.0725978
$848$	27.2874	0.937055
$849$	0	0
$850$	−28.3797	−0.973417
$851$	−5.50246	−0.188622
$852$	0	0
$853$	−37.2713	−1.27614	−0.638072	−	0.769976i	$-0.720268\pi$
$853$	−37.2713	−1.27614	−0.638072	+	0.769976i	$0.720268\pi$
$854$	93.9144	3.21368
$855$	0	0
$856$	71.4521	2.44218
$857$	41.4318	1.41528	0.707642	−	0.706571i	$-0.249759\pi$
$857$	41.4318	1.41528	0.707642	+	0.706571i	$0.249759\pi$
$858$	0	0
$859$	−9.25851	−0.315896	−0.157948	−	0.987447i	$-0.550488\pi$
$859$	−9.25851	−0.315896	−0.157948	+	0.987447i	$0.550488\pi$
$860$	−34.1220	−1.16355
$861$	0	0
$862$	38.9468	1.32653
$863$	−51.4748	−1.75222	−0.876110	−	0.482110i	$-0.839870\pi$
$863$	−51.4748	−1.75222	−0.876110	+	0.482110i	$0.839870\pi$
$864$	0	0
$865$	73.8258	2.51015
$866$	58.6201	1.99199
$867$	0	0
$868$	−82.8787	−2.81309
$869$	−59.0483	−2.00308
$870$	0	0
$871$	19.6591	0.666122
$872$	−28.8935	−0.978458
$873$	0	0
$874$	2.75421	0.0931624
$875$	−4.19281	−0.141743
$876$	0	0
$877$	−24.8990	−0.840781	−0.420390	−	0.907343i	$-0.638107\pi$
$877$	−24.8990	−0.840781	−0.420390	+	0.907343i	$0.638107\pi$
$878$	52.7686	1.78085
$879$	0	0
$880$	44.9174	1.51416
$881$	1.43361	0.0482997	0.0241498	−	0.999708i	$-0.492312\pi$
$881$	1.43361	0.0482997	0.0241498	+	0.999708i	$0.492312\pi$
$882$	0	0
$883$	26.2046	0.881855	0.440928	−	0.897543i	$-0.354650\pi$
$883$	26.2046	0.881855	0.440928	+	0.897543i	$0.354650\pi$
$884$	−34.7640	−1.16924
$885$	0	0
$886$	−77.5381	−2.60494
$887$	−43.3726	−1.45631	−0.728154	−	0.685414i	$-0.759621\pi$
$887$	−43.3726	−1.45631	−0.728154	+	0.685414i	$0.759621\pi$
$888$	0	0
$889$	−6.88646	−0.230965
$890$	38.6011	1.29391
$891$	0	0
$892$	35.2702	1.18093
$893$	11.8635	0.396996
$894$	0	0
$895$	63.3678	2.11815
$896$	−51.1455	−1.70865
$897$	0	0
$898$	−51.0969	−1.70513
$899$	19.5251	0.651198
$900$	0	0
$901$	16.5344	0.550841
$902$	−41.2338	−1.37293
$903$	0	0
$904$	57.2625	1.90452
$905$	47.6521	1.58401
$906$	0	0
$907$	23.8718	0.792649	0.396325	−	0.918110i	$-0.370286\pi$
$907$	23.8718	0.792649	0.396325	+	0.918110i	$0.370286\pi$
$908$	98.1191	3.25620
$909$	0	0
$910$	67.3065	2.23119
$911$	−4.32052	−0.143145	−0.0715726	−	0.997435i	$-0.522802\pi$
$911$	−4.32052	−0.143145	−0.0715726	+	0.997435i	$0.522802\pi$
$912$	0	0
$913$	−13.6302	−0.451092
$914$	42.8337	1.41681
$915$	0	0
$916$	73.3450	2.42339
$917$	11.7273	0.387271
$918$	0	0
$919$	3.56151	0.117483	0.0587417	−	0.998273i	$-0.481291\pi$
$919$	3.56151	0.117483	0.0587417	+	0.998273i	$0.481291\pi$
$920$	8.25696	0.272224
$921$	0	0
$922$	76.2025	2.50959
$923$	−49.5449	−1.63079
$924$	0	0
$925$	−46.2071	−1.51928
$926$	−15.9460	−0.524019
$927$	0	0
$928$	−1.02372	−0.0336053
$929$	33.3258	1.09338	0.546692	−	0.837334i	$-0.315887\pi$
$929$	33.3258	1.09338	0.546692	+	0.837334i	$0.315887\pi$
$930$	0	0
$931$	−0.143651	−0.00470798
$932$	42.5812	1.39479
$933$	0	0
$934$	−4.77770	−0.156331
$935$	27.2170	0.890091
$936$	0	0
$937$	26.0594	0.851322	0.425661	−	0.904883i	$-0.360042\pi$
$937$	26.0594	0.851322	0.425661	+	0.904883i	$0.360042\pi$
$938$	−38.4159	−1.25432
$939$	0	0
$940$	70.4223	2.29692
$941$	23.9936	0.782167	0.391084	−	0.920355i	$-0.372100\pi$
$941$	23.9936	0.782167	0.391084	+	0.920355i	$0.372100\pi$
$942$	0	0
$943$	−2.61076	−0.0850181
$944$	7.02697	0.228708
$945$	0	0
$946$	23.1406	0.752366
$947$	−29.8847	−0.971121	−0.485561	−	0.874203i	$-0.661385\pi$
$947$	−29.8847	−0.971121	−0.485561	+	0.874203i	$0.661385\pi$
$948$	0	0
$949$	−6.29131	−0.204225
$950$	23.1286	0.750390
$951$	0	0
$952$	34.3086	1.11195
$953$	10.1934	0.330196	0.165098	−	0.986277i	$-0.447206\pi$
$953$	10.1934	0.330196	0.165098	+	0.986277i	$0.447206\pi$
$954$	0	0
$955$	33.7315	1.09153
$956$	−38.5348	−1.24631
$957$	0	0
$958$	−3.26736	−0.105564
$959$	−48.3018	−1.55975
$960$	0	0
$961$	28.5166	0.919891
$962$	−84.6174	−2.72817
$963$	0	0
$964$	−28.4357	−0.915854
$965$	−3.52644	−0.113520
$966$	0	0
$967$	−18.7666	−0.603493	−0.301747	−	0.953388i	$-0.597570\pi$
$967$	−18.7666	−0.603493	−0.301747	+	0.953388i	$0.597570\pi$
$968$	−3.98594	−0.128113
$969$	0	0
$970$	−80.4901	−2.58438
$971$	51.9535	1.66727	0.833633	−	0.552319i	$-0.186257\pi$
$971$	51.9535	1.66727	0.833633	+	0.552319i	$0.186257\pi$
$972$	0	0
$973$	−5.20552	−0.166881
$974$	−52.1149	−1.66987
$975$	0	0
$976$	61.0249	1.95336
$977$	−32.5841	−1.04246	−0.521229	−	0.853417i	$-0.674526\pi$
$977$	−32.5841	−1.04246	−0.521229	+	0.853417i	$0.674526\pi$
$978$	0	0
$979$	−17.5110	−0.559654
$980$	−0.852722	−0.0272392
$981$	0	0
$982$	−66.3852	−2.11844
$983$	−35.0795	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$983$	−35.0795	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$984$	0	0
$985$	15.5305	0.494844
$986$	−16.0040	−0.509670
$987$	0	0
$988$	28.3315	0.901345
$989$	1.46517	0.0465898
$990$	0	0
$991$	−54.7635	−1.73962	−0.869810	−	0.493386i	$-0.835759\pi$
$991$	−54.7635	−1.73962	−0.869810	+	0.493386i	$0.835759\pi$
$992$	−3.12052	−0.0990767
$993$	0	0
$994$	96.8161	3.07082
$995$	−42.2791	−1.34034
$996$	0	0
$997$	50.3301	1.59397	0.796984	−	0.604000i	$-0.206427\pi$
$997$	50.3301	1.59397	0.796984	+	0.604000i	$0.206427\pi$
$998$	−36.9625	−1.17003
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.e.1.1	yes	6
3.2	odd	2		729.2.a.b.1.6	✓	6
9.2	odd	6		729.2.c.d.244.1		12
9.4	even	3		729.2.c.a.487.6		12
9.5	odd	6		729.2.c.d.487.1		12
9.7	even	3		729.2.c.a.244.6		12
27.2	odd	18		729.2.e.j.568.1		12
27.4	even	9		729.2.e.l.406.1		12
27.5	odd	18		729.2.e.t.649.2		12
27.7	even	9		729.2.e.l.325.1		12
27.11	odd	18		729.2.e.t.82.2		12
27.13	even	9		729.2.e.u.163.2		12
27.14	odd	18		729.2.e.j.163.1		12
27.16	even	9		729.2.e.k.82.1		12
27.20	odd	18		729.2.e.s.325.2		12
27.22	even	9		729.2.e.k.649.1		12
27.23	odd	18		729.2.e.s.406.2		12
27.25	even	9		729.2.e.u.568.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.b.1.6	✓	6	3.2	odd	2
729.2.a.e.1.1	yes	6	1.1	even	1	trivial
729.2.c.a.244.6		12	9.7	even	3
729.2.c.a.487.6		12	9.4	even	3
729.2.c.d.244.1		12	9.2	odd	6
729.2.c.d.487.1		12	9.5	odd	6
729.2.e.j.163.1		12	27.14	odd	18
729.2.e.j.568.1		12	27.2	odd	18
729.2.e.k.82.1		12	27.16	even	9
729.2.e.k.649.1		12	27.22	even	9
729.2.e.l.325.1		12	27.7	even	9
729.2.e.l.406.1		12	27.4	even	9
729.2.e.s.325.2		12	27.20	odd	18
729.2.e.s.406.2		12	27.23	odd	18
729.2.e.t.82.2		12	27.11	odd	18
729.2.e.t.649.2		12	27.5	odd	18
729.2.e.u.163.2		12	27.13	even	9
729.2.e.u.568.2		12	27.25	even	9

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)

\(f(q)\)	\(=\)	\(q-2.45779 q^{2} +4.04073 q^{4} -3.08026 q^{5} -2.65867 q^{7} -5.01568 q^{8} +7.57064 q^{10} -3.43434 q^{11} -3.34396 q^{13} +6.53444 q^{14} +4.24603 q^{16} +2.57282 q^{17} -2.09676 q^{19} -12.4465 q^{20} +8.44089 q^{22} +0.534444 q^{23} +4.48802 q^{25} +8.21874 q^{26} -10.7430 q^{28} +2.53089 q^{29} +7.71470 q^{31} -0.404491 q^{32} -6.32344 q^{34} +8.18939 q^{35} -10.2957 q^{37} +5.15340 q^{38} +15.4496 q^{40} -4.88501 q^{41} +2.74149 q^{43} -13.8772 q^{44} -1.31355 q^{46} -5.65800 q^{47} +0.0685109 q^{49} -11.0306 q^{50} -13.5120 q^{52} +6.42657 q^{53} +10.5787 q^{55} +13.3350 q^{56} -6.22040 q^{58} +1.65495 q^{59} +14.3722 q^{61} -18.9611 q^{62} -7.49791 q^{64} +10.3003 q^{65} -5.87898 q^{67} +10.3961 q^{68} -20.1278 q^{70} +14.8163 q^{71} +1.88140 q^{73} +25.3046 q^{74} -8.47245 q^{76} +9.13077 q^{77} +17.1935 q^{79} -13.0789 q^{80} +12.0063 q^{82} +3.96878 q^{83} -7.92496 q^{85} -6.73801 q^{86} +17.2256 q^{88} +5.09880 q^{89} +8.89047 q^{91} +2.15954 q^{92} +13.9062 q^{94} +6.45858 q^{95} -10.6319 q^{97} -0.168385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 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

L-functions

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