LMFDB - Embedded newform 2601.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, 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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 867)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} +1.22668 q^{7} +2.83750 q^{8} +O(q^{10})$	$q-0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} +1.22668 q^{7} +2.83750 q^{8} -1.18479 q^{10} +4.29086 q^{11} -5.34730 q^{13} -1.07873 q^{14} -0.0418891 q^{16} -5.59627 q^{19} -1.65270 q^{20} -3.77332 q^{22} -5.94356 q^{23} -3.18479 q^{25} +4.70233 q^{26} -1.50475 q^{28} +4.02229 q^{29} -2.81521 q^{31} -5.63816 q^{32} +1.65270 q^{35} +8.66044 q^{37} +4.92127 q^{38} +3.82295 q^{40} -8.22668 q^{41} -7.30541 q^{43} -5.26352 q^{44} +5.22668 q^{46} -5.78106 q^{47} -5.49525 q^{49} +2.80066 q^{50} +6.55943 q^{52} +9.87939 q^{53} +5.78106 q^{55} +3.48070 q^{56} -3.53714 q^{58} +9.68004 q^{59} -9.47565 q^{61} +2.47565 q^{62} +5.04189 q^{64} -7.20439 q^{65} +0.128356 q^{67} -1.45336 q^{70} +12.5175 q^{71} +3.81521 q^{73} -7.61587 q^{74} +6.86484 q^{76} +5.26352 q^{77} +1.36959 q^{79} -0.0564370 q^{80} +7.23442 q^{82} -12.3182 q^{83} +6.42427 q^{86} +12.1753 q^{88} -10.0915 q^{89} -6.55943 q^{91} +7.29086 q^{92} +5.08378 q^{94} -7.53983 q^{95} -1.93582 q^{97} +4.83244 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 3 q^{11} - 15 q^{13} - 12 q^{14} + 3 q^{16} - 3 q^{19} - 6 q^{20} - 18 q^{22} - 3 q^{23} - 6 q^{25} - 12 q^{26} - 21 q^{28} + 6 q^{29} - 12 q^{31} + 6 q^{35} + 3 q^{37} + 6 q^{38} - 9 q^{40} - 18 q^{41} - 24 q^{43} - 21 q^{44} + 9 q^{46} - 6 q^{50} - 6 q^{52} + 24 q^{53} - 24 q^{56} + 9 q^{58} + 9 q^{59} - 9 q^{61} - 12 q^{62} + 12 q^{64} - 21 q^{65} - 18 q^{67} + 9 q^{70} + 15 q^{71} + 15 q^{73} - 12 q^{74} - 3 q^{76} + 21 q^{77} - 3 q^{79} - 15 q^{80} - 9 q^{82} - 30 q^{86} + 6 q^{91} + 6 q^{92} + 9 q^{94} + 6 q^{95} - 15 q^{97} + 27 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 3 * q^11 - 15 * q^13 - 12 * q^14 + 3 * q^16 - 3 * q^19 - 6 * q^20 - 18 * q^22 - 3 * q^23 - 6 * q^25 - 12 * q^26 - 21 * q^28 + 6 * q^29 - 12 * q^31 + 6 * q^35 + 3 * q^37 + 6 * q^38 - 9 * q^40 - 18 * q^41 - 24 * q^43 - 21 * q^44 + 9 * q^46 - 6 * q^50 - 6 * q^52 + 24 * q^53 - 24 * q^56 + 9 * q^58 + 9 * q^59 - 9 * q^61 - 12 * q^62 + 12 * q^64 - 21 * q^65 - 18 * q^67 + 9 * q^70 + 15 * q^71 + 15 * q^73 - 12 * q^74 - 3 * q^76 + 21 * q^77 - 3 * q^79 - 15 * q^80 - 9 * q^82 - 30 * q^86 + 6 * q^91 + 6 * q^92 + 9 * q^94 + 6 * q^95 - 15 * q^97 + 27 * q^98

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 $

$2$	−0.879385	−0.621819	−0.310910	−	0.950439i	$-0.600634\pi$
$2$	−0.879385	−0.621819	−0.310910	+	0.950439i	$0.600634\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	1.22668	0.463642	0.231821	−	0.972758i	$-0.425532\pi$
$7$	1.22668	0.463642	0.231821	+	0.972758i	$0.425532\pi$
$8$	2.83750	1.00321
$9$	0	0
$10$	−1.18479	−0.374664
$11$	4.29086	1.29374	0.646871	−	0.762599i	$-0.276077\pi$
$11$	4.29086	1.29374	0.646871	+	0.762599i	$0.276077\pi$
$12$	0	0
$13$	−5.34730	−1.48307	−0.741537	−	0.670912i	$-0.765903\pi$
$13$	−5.34730	−1.48307	−0.741537	+	0.670912i	$0.765903\pi$
$14$	−1.07873	−0.288302
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.59627	−1.28387	−0.641936	−	0.766758i	$-0.721868\pi$
$19$	−5.59627	−1.28387	−0.641936	+	0.766758i	$0.721868\pi$
$20$	−1.65270	−0.369556
$21$	0	0
$22$	−3.77332	−0.804474
$23$	−5.94356	−1.23932	−0.619659	−	0.784871i	$-0.712729\pi$
$23$	−5.94356	−1.23932	−0.619659	+	0.784871i	$0.712729\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	4.70233	0.922203
$27$	0	0
$28$	−1.50475	−0.284371
$29$	4.02229	0.746920	0.373460	−	0.927646i	$-0.378171\pi$
$29$	4.02229	0.746920	0.373460	+	0.927646i	$0.378171\pi$
$30$	0	0
$31$	−2.81521	−0.505626	−0.252813	−	0.967515i	$-0.581356\pi$
$31$	−2.81521	−0.505626	−0.252813	+	0.967515i	$0.581356\pi$
$32$	−5.63816	−0.996695
$33$	0	0
$34$	0	0
$35$	1.65270	0.279358
$36$	0	0
$37$	8.66044	1.42377	0.711884	−	0.702297i	$-0.247842\pi$
$37$	8.66044	1.42377	0.711884	+	0.702297i	$0.247842\pi$
$38$	4.92127	0.798336
$39$	0	0
$40$	3.82295	0.604461
$41$	−8.22668	−1.28479	−0.642396	−	0.766373i	$-0.722059\pi$
$41$	−8.22668	−1.28479	−0.642396	+	0.766373i	$0.722059\pi$
$42$	0	0
$43$	−7.30541	−1.11406	−0.557032	−	0.830491i	$-0.688060\pi$
$43$	−7.30541	−1.11406	−0.557032	+	0.830491i	$0.688060\pi$
$44$	−5.26352	−0.793505
$45$	0	0
$46$	5.22668	0.770632
$47$	−5.78106	−0.843254	−0.421627	−	0.906769i	$-0.638541\pi$
$47$	−5.78106	−0.843254	−0.421627	+	0.906769i	$0.638541\pi$
$48$	0	0
$49$	−5.49525	−0.785036
$50$	2.80066	0.396073
$51$	0	0
$52$	6.55943	0.909629
$53$	9.87939	1.35704	0.678519	−	0.734583i	$-0.262622\pi$
$53$	9.87939	1.35704	0.678519	+	0.734583i	$0.262622\pi$
$54$	0	0
$55$	5.78106	0.779518
$56$	3.48070	0.465129
$57$	0	0
$58$	−3.53714	−0.464449
$59$	9.68004	1.26023	0.630117	−	0.776500i	$-0.283007\pi$
$59$	9.68004	1.26023	0.630117	+	0.776500i	$0.283007\pi$
$60$	0	0
$61$	−9.47565	−1.21323	−0.606616	−	0.794995i	$-0.707474\pi$
$61$	−9.47565	−1.21323	−0.606616	+	0.794995i	$0.707474\pi$
$62$	2.47565	0.314408
$63$	0	0
$64$	5.04189	0.630236
$65$	−7.20439	−0.893595
$66$	0	0
$67$	0.128356	0.0156811	0.00784056	−	0.999969i	$-0.497504\pi$
$67$	0.128356	0.0156811	0.00784056	+	0.999969i	$0.497504\pi$
$68$	0	0
$69$	0	0
$70$	−1.45336	−0.173710
$71$	12.5175	1.48556	0.742779	−	0.669536i	$-0.233507\pi$
$71$	12.5175	1.48556	0.742779	+	0.669536i	$0.233507\pi$
$72$	0	0
$73$	3.81521	0.446536	0.223268	−	0.974757i	$-0.428327\pi$
$73$	3.81521	0.446536	0.223268	+	0.974757i	$0.428327\pi$
$74$	−7.61587	−0.885327
$75$	0	0
$76$	6.86484	0.787451
$77$	5.26352	0.599834
$78$	0	0
$79$	1.36959	0.154090	0.0770452	−	0.997028i	$-0.475451\pi$
$79$	1.36959	0.154090	0.0770452	+	0.997028i	$0.475451\pi$
$80$	−0.0564370	−0.00630985
$81$	0	0
$82$	7.23442	0.798908
$83$	−12.3182	−1.35210	−0.676049	−	0.736857i	$-0.736309\pi$
$83$	−12.3182	−1.35210	−0.676049	+	0.736857i	$0.736309\pi$
$84$	0	0
$85$	0	0
$86$	6.42427	0.692747
$87$	0	0
$88$	12.1753	1.29789
$89$	−10.0915	−1.06970	−0.534849	−	0.844947i	$-0.679632\pi$
$89$	−10.0915	−1.06970	−0.534849	+	0.844947i	$0.679632\pi$
$90$	0	0
$91$	−6.55943	−0.687615
$92$	7.29086	0.760125
$93$	0	0
$94$	5.08378	0.524352
$95$	−7.53983	−0.773570
$96$	0	0
$97$	−1.93582	−0.196553	−0.0982765	−	0.995159i	$-0.531333\pi$
$97$	−1.93582	−0.196553	−0.0982765	+	0.995159i	$0.531333\pi$
$98$	4.83244	0.488151
$99$	0	0
$100$	3.90673	0.390673
$101$	−8.11381	−0.807354	−0.403677	−	0.914902i	$-0.632268\pi$
$101$	−8.11381	−0.807354	−0.403677	+	0.914902i	$0.632268\pi$
$102$	0	0
$103$	9.25402	0.911826	0.455913	−	0.890024i	$-0.349313\pi$
$103$	9.25402	0.911826	0.455913	+	0.890024i	$0.349313\pi$
$104$	−15.1729	−1.48783
$105$	0	0
$106$	−8.68779	−0.843832
$107$	−6.68004	−0.645784	−0.322892	−	0.946436i	$-0.604655\pi$
$107$	−6.68004	−0.645784	−0.322892	+	0.946436i	$0.604655\pi$
$108$	0	0
$109$	0.0418891	0.00401224	0.00200612	−	0.999998i	$-0.499361\pi$
$109$	0.0418891	0.00401224	0.00200612	+	0.999998i	$0.499361\pi$
$110$	−5.08378	−0.484719
$111$	0	0
$112$	−0.0513845	−0.00485538
$113$	−12.7169	−1.19630	−0.598152	−	0.801383i	$-0.704098\pi$
$113$	−12.7169	−1.19630	−0.598152	+	0.801383i	$0.704098\pi$
$114$	0	0
$115$	−8.00774	−0.746726
$116$	−4.93407	−0.458117
$117$	0	0
$118$	−8.51249	−0.783638
$119$	0	0
$120$	0	0
$121$	7.41147	0.673770
$122$	8.33275	0.754412
$123$	0	0
$124$	3.45336	0.310121
$125$	−11.0273	−0.986315
$126$	0	0
$127$	7.20708	0.639525	0.319763	−	0.947498i	$-0.396397\pi$
$127$	7.20708	0.639525	0.319763	+	0.947498i	$0.396397\pi$
$128$	6.84255	0.604802
$129$	0	0
$130$	6.33544	0.555655
$131$	−8.98040	−0.784621	−0.392311	−	0.919833i	$-0.628324\pi$
$131$	−8.98040	−0.784621	−0.392311	+	0.919833i	$0.628324\pi$
$132$	0	0
$133$	−6.86484	−0.595257
$134$	−0.112874	−0.00975083
$135$	0	0
$136$	0	0
$137$	16.1848	1.38276	0.691380	−	0.722491i	$-0.257003\pi$
$137$	16.1848	1.38276	0.691380	+	0.722491i	$0.257003\pi$
$138$	0	0
$139$	−21.6878	−1.83953	−0.919767	−	0.392465i	$-0.871622\pi$
$139$	−21.6878	−1.83953	−0.919767	+	0.392465i	$0.871622\pi$
$140$	−2.02734	−0.171342
$141$	0	0
$142$	−11.0077	−0.923749
$143$	−22.9445	−1.91872
$144$	0	0
$145$	5.41921	0.450041
$146$	−3.35504	−0.277665
$147$	0	0
$148$	−10.6236	−0.873255
$149$	−2.31315	−0.189500	−0.0947502	−	0.995501i	$-0.530205\pi$
$149$	−2.31315	−0.189500	−0.0947502	+	0.995501i	$0.530205\pi$
$150$	0	0
$151$	−7.51754	−0.611769	−0.305884	−	0.952069i	$-0.598952\pi$
$151$	−7.51754	−0.611769	−0.305884	+	0.952069i	$0.598952\pi$
$152$	−15.8794	−1.28799
$153$	0	0
$154$	−4.62866	−0.372988
$155$	−3.79292	−0.304655
$156$	0	0
$157$	−0.731429	−0.0583744	−0.0291872	−	0.999574i	$-0.509292\pi$
$157$	−0.731429	−0.0583744	−0.0291872	+	0.999574i	$0.509292\pi$
$158$	−1.20439	−0.0958164
$159$	0	0
$160$	−7.59627	−0.600538
$161$	−7.29086	−0.574600
$162$	0	0
$163$	−2.72193	−0.213198	−0.106599	−	0.994302i	$-0.533996\pi$
$163$	−2.72193	−0.213198	−0.106599	+	0.994302i	$0.533996\pi$
$164$	10.0915	0.788015
$165$	0	0
$166$	10.8324	0.840761
$167$	6.64084	0.513884	0.256942	−	0.966427i	$-0.417285\pi$
$167$	6.64084	0.513884	0.256942	+	0.966427i	$0.417285\pi$
$168$	0	0
$169$	15.5936	1.19951
$170$	0	0
$171$	0	0
$172$	8.96141	0.683301
$173$	10.7520	0.817457	0.408728	−	0.912656i	$-0.365972\pi$
$173$	10.7520	0.817457	0.408728	+	0.912656i	$0.365972\pi$
$174$	0	0
$175$	−3.90673	−0.295321
$176$	−0.179740	−0.0135484
$177$	0	0
$178$	8.87433	0.665159
$179$	−17.8007	−1.33048	−0.665242	−	0.746628i	$-0.731672\pi$
$179$	−17.8007	−1.33048	−0.665242	+	0.746628i	$0.731672\pi$
$180$	0	0
$181$	−3.40642	−0.253197	−0.126599	−	0.991954i	$-0.540406\pi$
$181$	−3.40642	−0.253197	−0.126599	+	0.991954i	$0.540406\pi$
$182$	5.76827	0.427572
$183$	0	0
$184$	−16.8648	−1.24329
$185$	11.6682	0.857862
$186$	0	0
$187$	0	0
$188$	7.09152	0.517202
$189$	0	0
$190$	6.63041	0.481021
$191$	16.0351	1.16026	0.580129	−	0.814525i	$-0.303002\pi$
$191$	16.0351	1.16026	0.580129	+	0.814525i	$0.303002\pi$
$192$	0	0
$193$	−20.7520	−1.49376	−0.746880	−	0.664959i	$-0.768449\pi$
$193$	−20.7520	−1.49376	−0.746880	+	0.664959i	$0.768449\pi$
$194$	1.70233	0.122220
$195$	0	0
$196$	6.74092	0.481495
$197$	−16.6655	−1.18737	−0.593684	−	0.804698i	$-0.702327\pi$
$197$	−16.6655	−1.18737	−0.593684	+	0.804698i	$0.702327\pi$
$198$	0	0
$199$	9.14115	0.647999	0.323999	−	0.946057i	$-0.394972\pi$
$199$	9.14115	0.647999	0.323999	+	0.946057i	$0.394972\pi$
$200$	−9.03684	−0.639001
$201$	0	0
$202$	7.13516	0.502028
$203$	4.93407	0.346304
$204$	0	0
$205$	−11.0838	−0.774125
$206$	−8.13785	−0.566991
$207$	0	0
$208$	0.223993	0.0155311
$209$	−24.0128	−1.66100
$210$	0	0
$211$	−13.8229	−0.951611	−0.475806	−	0.879551i	$-0.657843\pi$
$211$	−13.8229	−0.951611	−0.475806	+	0.879551i	$0.657843\pi$
$212$	−12.1189	−0.832327
$213$	0	0
$214$	5.87433	0.401561
$215$	−9.84255	−0.671256
$216$	0	0
$217$	−3.45336	−0.234430
$218$	−0.0368366	−0.00249489
$219$	0	0
$220$	−7.09152	−0.478110
$221$	0	0
$222$	0	0
$223$	1.31996	0.0883907	0.0441954	−	0.999023i	$-0.485928\pi$
$223$	1.31996	0.0883907	0.0441954	+	0.999023i	$0.485928\pi$
$224$	−6.91622	−0.462110
$225$	0	0
$226$	11.1830	0.743885
$227$	9.17530	0.608986	0.304493	−	0.952515i	$-0.401513\pi$
$227$	9.17530	0.608986	0.304493	+	0.952515i	$0.401513\pi$
$228$	0	0
$229$	−26.1584	−1.72859	−0.864297	−	0.502981i	$-0.832237\pi$
$229$	−26.1584	−1.72859	−0.864297	+	0.502981i	$0.832237\pi$
$230$	7.04189	0.464328
$231$	0	0
$232$	11.4132	0.749315
$233$	−1.27126	−0.0832829	−0.0416415	−	0.999133i	$-0.513259\pi$
$233$	−1.27126	−0.0832829	−0.0416415	+	0.999133i	$0.513259\pi$
$234$	0	0
$235$	−7.78880	−0.508085
$236$	−11.8743	−0.772953
$237$	0	0
$238$	0	0
$239$	4.78880	0.309762	0.154881	−	0.987933i	$-0.450501\pi$
$239$	4.78880	0.309762	0.154881	+	0.987933i	$0.450501\pi$
$240$	0	0
$241$	−5.31046	−0.342077	−0.171038	−	0.985264i	$-0.554712\pi$
$241$	−5.31046	−0.342077	−0.171038	+	0.985264i	$0.554712\pi$
$242$	−6.51754	−0.418963
$243$	0	0
$244$	11.6236	0.744125
$245$	−7.40373	−0.473007
$246$	0	0
$247$	29.9249	1.90408
$248$	−7.98814	−0.507247
$249$	0	0
$250$	9.69728	0.613310
$251$	8.43882	0.532653	0.266327	−	0.963883i	$-0.414190\pi$
$251$	8.43882	0.532653	0.266327	+	0.963883i	$0.414190\pi$
$252$	0	0
$253$	−25.5030	−1.60336
$254$	−6.33780	−0.397669
$255$	0	0
$256$	−16.1010	−1.00631
$257$	3.81016	0.237671	0.118835	−	0.992914i	$-0.462084\pi$
$257$	3.81016	0.237671	0.118835	+	0.992914i	$0.462084\pi$
$258$	0	0
$259$	10.6236	0.660119
$260$	8.83750	0.548078
$261$	0	0
$262$	7.89723	0.487893
$263$	9.69728	0.597960	0.298980	−	0.954259i	$-0.403354\pi$
$263$	9.69728	0.597960	0.298980	+	0.954259i	$0.403354\pi$
$264$	0	0
$265$	13.3105	0.817655
$266$	6.03684	0.370142
$267$	0	0
$268$	−0.157451	−0.00961787
$269$	−6.90941	−0.421274	−0.210637	−	0.977564i	$-0.567554\pi$
$269$	−6.90941	−0.421274	−0.210637	+	0.977564i	$0.567554\pi$
$270$	0	0
$271$	−13.6604	−0.829813	−0.414906	−	0.909864i	$-0.636186\pi$
$271$	−13.6604	−0.829813	−0.414906	+	0.909864i	$0.636186\pi$
$272$	0	0
$273$	0	0
$274$	−14.2327	−0.859827
$275$	−13.6655	−0.824060
$276$	0	0
$277$	−0.514853	−0.0309345	−0.0154672	−	0.999880i	$-0.504924\pi$
$277$	−0.514853	−0.0309345	−0.0154672	+	0.999880i	$0.504924\pi$
$278$	19.0719	1.14386
$279$	0	0
$280$	4.68954	0.280254
$281$	−0.285807	−0.0170498	−0.00852491	−	0.999964i	$-0.502714\pi$
$281$	−0.285807	−0.0170498	−0.00852491	+	0.999964i	$0.502714\pi$
$282$	0	0
$283$	−13.9590	−0.829779	−0.414890	−	0.909872i	$-0.636180\pi$
$283$	−13.9590	−0.829779	−0.414890	+	0.909872i	$0.636180\pi$
$284$	−15.3550	−0.911154
$285$	0	0
$286$	20.1771	1.19309
$287$	−10.0915	−0.595684
$288$	0	0
$289$	0	0
$290$	−4.76558	−0.279844
$291$	0	0
$292$	−4.68004	−0.273879
$293$	17.4884	1.02169	0.510843	−	0.859674i	$-0.329333\pi$
$293$	17.4884	1.02169	0.510843	+	0.859674i	$0.329333\pi$
$294$	0	0
$295$	13.0419	0.759328
$296$	24.5740	1.42833
$297$	0	0
$298$	2.03415	0.117835
$299$	31.7820	1.83800
$300$	0	0
$301$	−8.96141	−0.516527
$302$	6.61081	0.380410
$303$	0	0
$304$	0.234422	0.0134450
$305$	−12.7665	−0.731008
$306$	0	0
$307$	6.32770	0.361141	0.180570	−	0.983562i	$-0.442206\pi$
$307$	6.32770	0.361141	0.180570	+	0.983562i	$0.442206\pi$
$308$	−6.45666	−0.367902
$309$	0	0
$310$	3.33544	0.189440
$311$	3.48070	0.197373	0.0986863	−	0.995119i	$-0.468536\pi$
$311$	3.48070	0.197373	0.0986863	+	0.995119i	$0.468536\pi$
$312$	0	0
$313$	17.3746	0.982073	0.491036	−	0.871139i	$-0.336618\pi$
$313$	17.3746	0.982073	0.491036	+	0.871139i	$0.336618\pi$
$314$	0.643208	0.0362983
$315$	0	0
$316$	−1.68004	−0.0945099
$317$	3.07604	0.172767	0.0863837	−	0.996262i	$-0.472469\pi$
$317$	3.07604	0.172767	0.0863837	+	0.996262i	$0.472469\pi$
$318$	0	0
$319$	17.2591	0.966323
$320$	6.79292	0.379736
$321$	0	0
$322$	6.41147	0.357297
$323$	0	0
$324$	0	0
$325$	17.0300	0.944656
$326$	2.39363	0.132571
$327$	0	0
$328$	−23.3432	−1.28891
$329$	−7.09152	−0.390968
$330$	0	0
$331$	11.7469	0.645669	0.322834	−	0.946455i	$-0.395364\pi$
$331$	11.7469	0.645669	0.322834	+	0.946455i	$0.395364\pi$
$332$	15.1105	0.829297
$333$	0	0
$334$	−5.83986	−0.319543
$335$	0.172933	0.00944834
$336$	0	0
$337$	26.6040	1.44921	0.724606	−	0.689163i	$-0.242022\pi$
$337$	26.6040	1.44921	0.724606	+	0.689163i	$0.242022\pi$
$338$	−13.7128	−0.745876
$339$	0	0
$340$	0	0
$341$	−12.0797	−0.654150
$342$	0	0
$343$	−15.3277	−0.827618
$344$	−20.7291	−1.11764
$345$	0	0
$346$	−9.45512	−0.508310
$347$	−2.00774	−0.107781	−0.0538906	−	0.998547i	$-0.517162\pi$
$347$	−2.00774	−0.107781	−0.0538906	+	0.998547i	$0.517162\pi$
$348$	0	0
$349$	−31.8462	−1.70469	−0.852343	−	0.522983i	$-0.824819\pi$
$349$	−31.8462	−1.70469	−0.852343	+	0.522983i	$0.824819\pi$
$350$	3.43552	0.183636
$351$	0	0
$352$	−24.1925	−1.28947
$353$	−12.8102	−0.681816	−0.340908	−	0.940097i	$-0.610734\pi$
$353$	−12.8102	−0.681816	−0.340908	+	0.940097i	$0.610734\pi$
$354$	0	0
$355$	16.8648	0.895093
$356$	12.3791	0.656090
$357$	0	0
$358$	15.6536	0.827320
$359$	13.9145	0.734377	0.367189	−	0.930146i	$-0.380320\pi$
$359$	13.9145	0.734377	0.367189	+	0.930146i	$0.380320\pi$
$360$	0	0
$361$	12.3182	0.648326
$362$	2.99556	0.157443
$363$	0	0
$364$	8.04633	0.421742
$365$	5.14022	0.269051
$366$	0	0
$367$	−7.82976	−0.408710	−0.204355	−	0.978897i	$-0.565510\pi$
$367$	−7.82976	−0.408710	−0.204355	+	0.978897i	$0.565510\pi$
$368$	0.248970	0.0129785
$369$	0	0
$370$	−10.2608	−0.533435
$371$	12.1189	0.629180
$372$	0	0
$373$	−34.9659	−1.81046	−0.905232	−	0.424919i	$-0.860303\pi$
$373$	−34.9659	−1.81046	−0.905232	+	0.424919i	$0.860303\pi$
$374$	0	0
$375$	0	0
$376$	−16.4037	−0.845958
$377$	−21.5084	−1.10774
$378$	0	0
$379$	−1.09564	−0.0562791	−0.0281396	−	0.999604i	$-0.508958\pi$
$379$	−1.09564	−0.0562791	−0.0281396	+	0.999604i	$0.508958\pi$
$380$	9.24897	0.474462
$381$	0	0
$382$	−14.1010	−0.721471
$383$	−8.95037	−0.457343	−0.228671	−	0.973504i	$-0.573438\pi$
$383$	−8.95037	−0.457343	−0.228671	+	0.973504i	$0.573438\pi$
$384$	0	0
$385$	7.09152	0.361417
$386$	18.2490	0.928848
$387$	0	0
$388$	2.37464	0.120554
$389$	28.4638	1.44317	0.721586	−	0.692325i	$-0.243414\pi$
$389$	28.4638	1.44317	0.721586	+	0.692325i	$0.243414\pi$
$390$	0	0
$391$	0	0
$392$	−15.5928	−0.787553
$393$	0	0
$394$	14.6554	0.738328
$395$	1.84524	0.0928439
$396$	0	0
$397$	23.5371	1.18130	0.590648	−	0.806930i	$-0.298872\pi$
$397$	23.5371	1.18130	0.590648	+	0.806930i	$0.298872\pi$
$398$	−8.03859	−0.402938
$399$	0	0
$400$	0.133408	0.00667040
$401$	35.1002	1.75282	0.876410	−	0.481566i	$-0.159932\pi$
$401$	35.1002	1.75282	0.876410	+	0.481566i	$0.159932\pi$
$402$	0	0
$403$	15.0537	0.749881
$404$	9.95306	0.495183
$405$	0	0
$406$	−4.33895	−0.215338
$407$	37.1607	1.84199
$408$	0	0
$409$	−1.95904	−0.0968685	−0.0484343	−	0.998826i	$-0.515423\pi$
$409$	−1.95904	−0.0968685	−0.0484343	+	0.998826i	$0.515423\pi$
$410$	9.74691	0.481366
$411$	0	0
$412$	−11.3517	−0.559260
$413$	11.8743	0.584298
$414$	0	0
$415$	−16.5963	−0.814679
$416$	30.1489	1.47817
$417$	0	0
$418$	21.1165	1.03284
$419$	−8.25671	−0.403367	−0.201683	−	0.979451i	$-0.564641\pi$
$419$	−8.25671	−0.403367	−0.201683	+	0.979451i	$0.564641\pi$
$420$	0	0
$421$	−21.6313	−1.05425	−0.527124	−	0.849789i	$-0.676730\pi$
$421$	−21.6313	−1.05425	−0.527124	+	0.849789i	$0.676730\pi$
$422$	12.1557	0.591730
$423$	0	0
$424$	28.0327	1.36139
$425$	0	0
$426$	0	0
$427$	−11.6236	−0.562506
$428$	8.19429	0.396086
$429$	0	0
$430$	8.65539	0.417400
$431$	−33.1857	−1.59850	−0.799250	−	0.600999i	$-0.794770\pi$
$431$	−33.1857	−1.59850	−0.799250	+	0.600999i	$0.794770\pi$
$432$	0	0
$433$	−3.83069	−0.184091	−0.0920456	−	0.995755i	$-0.529341\pi$
$433$	−3.83069	−0.184091	−0.0920456	+	0.995755i	$0.529341\pi$
$434$	3.03684	0.145773
$435$	0	0
$436$	−0.0513845	−0.00246087
$437$	33.2618	1.59113
$438$	0	0
$439$	−26.7023	−1.27443	−0.637216	−	0.770685i	$-0.719914\pi$
$439$	−26.7023	−1.27443	−0.637216	+	0.770685i	$0.719914\pi$
$440$	16.4037	0.782017
$441$	0	0
$442$	0	0
$443$	−35.3364	−1.67888	−0.839441	−	0.543451i	$-0.817117\pi$
$443$	−35.3364	−1.67888	−0.839441	+	0.543451i	$0.817117\pi$
$444$	0	0
$445$	−13.5963	−0.644525
$446$	−1.16075	−0.0549631
$447$	0	0
$448$	6.18479	0.292204
$449$	26.0205	1.22798	0.613992	−	0.789312i	$-0.289563\pi$
$449$	26.0205	1.22798	0.613992	+	0.789312i	$0.289563\pi$
$450$	0	0
$451$	−35.2995	−1.66219
$452$	15.5996	0.733742
$453$	0	0
$454$	−8.06862	−0.378679
$455$	−8.83750	−0.414308
$456$	0	0
$457$	30.1147	1.40871	0.704354	−	0.709849i	$-0.251237\pi$
$457$	30.1147	1.40871	0.704354	+	0.709849i	$0.251237\pi$
$458$	23.0033	1.07487
$459$	0	0
$460$	9.82295	0.457997
$461$	27.1753	1.26568	0.632840	−	0.774283i	$-0.281889\pi$
$461$	27.1753	1.26568	0.632840	+	0.774283i	$0.281889\pi$
$462$	0	0
$463$	−8.93407	−0.415201	−0.207601	−	0.978214i	$-0.566565\pi$
$463$	−8.93407	−0.415201	−0.207601	+	0.978214i	$0.566565\pi$
$464$	−0.168490	−0.00782195
$465$	0	0
$466$	1.11793	0.0517869
$467$	−39.8266	−1.84295	−0.921477	−	0.388433i	$-0.873016\pi$
$467$	−39.8266	−1.84295	−0.921477	+	0.388433i	$0.873016\pi$
$468$	0	0
$469$	0.157451	0.00727043
$470$	6.84936	0.315937
$471$	0	0
$472$	27.4671	1.26428
$473$	−31.3465	−1.44131
$474$	0	0
$475$	17.8229	0.817773
$476$	0	0
$477$	0	0
$478$	−4.21120	−0.192616
$479$	−20.3155	−0.928239	−0.464120	−	0.885772i	$-0.653629\pi$
$479$	−20.3155	−0.928239	−0.464120	+	0.885772i	$0.653629\pi$
$480$	0	0
$481$	−46.3100	−2.11155
$482$	4.66994	0.212710
$483$	0	0
$484$	−9.09152	−0.413251
$485$	−2.60813	−0.118429
$486$	0	0
$487$	−23.1952	−1.05108	−0.525538	−	0.850770i	$-0.676136\pi$
$487$	−23.1952	−1.05108	−0.525538	+	0.850770i	$0.676136\pi$
$488$	−26.8871	−1.21712
$489$	0	0
$490$	6.51073	0.294125
$491$	22.4730	1.01419	0.507095	−	0.861890i	$-0.330719\pi$
$491$	22.4730	1.01419	0.507095	+	0.861890i	$0.330719\pi$
$492$	0	0
$493$	0	0
$494$	−26.3155	−1.18399
$495$	0	0
$496$	0.117926	0.00529505
$497$	15.3550	0.688768
$498$	0	0
$499$	41.6468	1.86437	0.932184	−	0.361985i	$-0.117901\pi$
$499$	41.6468	1.86437	0.932184	+	0.361985i	$0.117901\pi$
$500$	13.5270	0.604947
$501$	0	0
$502$	−7.42097	−0.331214
$503$	−22.4165	−0.999504	−0.499752	−	0.866169i	$-0.666575\pi$
$503$	−22.4165	−0.999504	−0.499752	+	0.866169i	$0.666575\pi$
$504$	0	0
$505$	−10.9317	−0.486454
$506$	22.4270	0.997000
$507$	0	0
$508$	−8.84079	−0.392247
$509$	−21.6732	−0.960649	−0.480325	−	0.877091i	$-0.659481\pi$
$509$	−21.6732	−0.960649	−0.480325	+	0.877091i	$0.659481\pi$
$510$	0	0
$511$	4.68004	0.207033
$512$	0.473897	0.0209435
$513$	0	0
$514$	−3.35059	−0.147788
$515$	12.4679	0.549402
$516$	0	0
$517$	−24.8057	−1.09095
$518$	−9.34224	−0.410475
$519$	0	0
$520$	−20.4424	−0.896460
$521$	−1.52259	−0.0667060	−0.0333530	−	0.999444i	$-0.510619\pi$
$521$	−1.52259	−0.0667060	−0.0333530	+	0.999444i	$0.510619\pi$
$522$	0	0
$523$	−6.41921	−0.280693	−0.140346	−	0.990102i	$-0.544822\pi$
$523$	−6.41921	−0.280693	−0.140346	+	0.990102i	$0.544822\pi$
$524$	11.0161	0.481240
$525$	0	0
$526$	−8.52765	−0.371823
$527$	0	0
$528$	0	0
$529$	12.3259	0.535910
$530$	−11.7050	−0.508434
$531$	0	0
$532$	8.42097	0.365095
$533$	43.9905	1.90544
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0.364208	0.0157314
$537$	0	0
$538$	6.07604	0.261957
$539$	−23.5794	−1.01563
$540$	0	0
$541$	7.83069	0.336668	0.168334	−	0.985730i	$-0.446161\pi$
$541$	7.83069	0.336668	0.168334	+	0.985730i	$0.446161\pi$
$542$	12.0128	0.515994
$543$	0	0
$544$	0	0
$545$	0.0564370	0.00241749
$546$	0	0
$547$	36.2104	1.54824	0.774122	−	0.633036i	$-0.218192\pi$
$547$	36.2104	1.54824	0.774122	+	0.633036i	$0.218192\pi$
$548$	−19.8536	−0.848103
$549$	0	0
$550$	12.0172	0.512417
$551$	−22.5098	−0.958950
$552$	0	0
$553$	1.68004	0.0714428
$554$	0.452754	0.0192357
$555$	0	0
$556$	26.6040	1.12826
$557$	−1.84524	−0.0781852	−0.0390926	−	0.999236i	$-0.512447\pi$
$557$	−1.84524	−0.0781852	−0.0390926	+	0.999236i	$0.512447\pi$
$558$	0	0
$559$	39.0642	1.65224
$560$	−0.0692302	−0.00292551
$561$	0	0
$562$	0.251334	0.0106019
$563$	5.10700	0.215234	0.107617	−	0.994192i	$-0.465678\pi$
$563$	5.10700	0.215234	0.107617	+	0.994192i	$0.465678\pi$
$564$	0	0
$565$	−17.1334	−0.720808
$566$	12.2754	0.515973
$567$	0	0
$568$	35.5185	1.49032
$569$	34.2080	1.43407	0.717037	−	0.697035i	$-0.245498\pi$
$569$	34.2080	1.43407	0.717037	+	0.697035i	$0.245498\pi$
$570$	0	0
$571$	21.1215	0.883909	0.441955	−	0.897037i	$-0.354285\pi$
$571$	21.1215	0.883909	0.441955	+	0.897037i	$0.354285\pi$
$572$	28.1456	1.17683
$573$	0	0
$574$	8.87433	0.370407
$575$	18.9290	0.789394
$576$	0	0
$577$	19.7246	0.821147	0.410573	−	0.911828i	$-0.365329\pi$
$577$	19.7246	0.821147	0.410573	+	0.911828i	$0.365329\pi$
$578$	0	0
$579$	0	0
$580$	−6.64765	−0.276029
$581$	−15.1105	−0.626890
$582$	0	0
$583$	42.3911	1.75566
$584$	10.8256	0.447968
$585$	0	0
$586$	−15.3791	−0.635304
$587$	−15.7597	−0.650473	−0.325236	−	0.945633i	$-0.605444\pi$
$587$	−15.7597	−0.650473	−0.325236	+	0.945633i	$0.605444\pi$
$588$	0	0
$589$	15.7547	0.649159
$590$	−11.4688	−0.472165
$591$	0	0
$592$	−0.362778	−0.0149101
$593$	19.2608	0.790947	0.395474	−	0.918477i	$-0.370580\pi$
$593$	19.2608	0.790947	0.395474	+	0.918477i	$0.370580\pi$
$594$	0	0
$595$	0	0
$596$	2.83750	0.116228
$597$	0	0
$598$	−27.9486	−1.14290
$599$	−22.7196	−0.928296	−0.464148	−	0.885758i	$-0.653640\pi$
$599$	−22.7196	−0.928296	−0.464148	+	0.885758i	$0.653640\pi$
$600$	0	0
$601$	−24.8939	−1.01544	−0.507722	−	0.861521i	$-0.669512\pi$
$601$	−24.8939	−1.01544	−0.507722	+	0.861521i	$0.669512\pi$
$602$	7.88053	0.321186
$603$	0	0
$604$	9.22163	0.375223
$605$	9.98545	0.405966
$606$	0	0
$607$	−3.95037	−0.160341	−0.0801703	−	0.996781i	$-0.525546\pi$
$607$	−3.95037	−0.160341	−0.0801703	+	0.996781i	$0.525546\pi$
$608$	31.5526	1.27963
$609$	0	0
$610$	11.2267	0.454555
$611$	30.9130	1.25061
$612$	0	0
$613$	−22.8357	−0.922327	−0.461163	−	0.887315i	$-0.652568\pi$
$613$	−22.8357	−0.922327	−0.461163	+	0.887315i	$0.652568\pi$
$614$	−5.56448	−0.224564
$615$	0	0
$616$	14.9352	0.601757
$617$	35.8239	1.44222	0.721108	−	0.692823i	$-0.243633\pi$
$617$	35.8239	1.44222	0.721108	+	0.692823i	$0.243633\pi$
$618$	0	0
$619$	31.1334	1.25136	0.625679	−	0.780081i	$-0.284822\pi$
$619$	31.1334	1.25136	0.625679	+	0.780081i	$0.284822\pi$
$620$	4.65270	0.186857
$621$	0	0
$622$	−3.06088	−0.122730
$623$	−12.3791	−0.495957
$624$	0	0
$625$	1.06687	0.0426746
$626$	−15.2790	−0.610672
$627$	0	0
$628$	0.897231	0.0358034
$629$	0	0
$630$	0	0
$631$	−6.20027	−0.246829	−0.123415	−	0.992355i	$-0.539384\pi$
$631$	−6.20027	−0.246829	−0.123415	+	0.992355i	$0.539384\pi$
$632$	3.88619	0.154584
$633$	0	0
$634$	−2.70502	−0.107430
$635$	9.71007	0.385333
$636$	0	0
$637$	29.3847	1.16427
$638$	−15.1774	−0.600878
$639$	0	0
$640$	9.21894	0.364411
$641$	41.1070	1.62363	0.811814	−	0.583915i	$-0.198480\pi$
$641$	41.1070	1.62363	0.811814	+	0.583915i	$0.198480\pi$
$642$	0	0
$643$	12.1952	0.480933	0.240466	−	0.970657i	$-0.422700\pi$
$643$	12.1952	0.480933	0.240466	+	0.970657i	$0.422700\pi$
$644$	8.94356	0.352426
$645$	0	0
$646$	0	0
$647$	−2.00774	−0.0789324	−0.0394662	−	0.999221i	$-0.512566\pi$
$647$	−2.00774	−0.0789324	−0.0394662	+	0.999221i	$0.512566\pi$
$648$	0	0
$649$	41.5357	1.63042
$650$	−14.9760	−0.587405
$651$	0	0
$652$	3.33895	0.130763
$653$	45.2226	1.76970	0.884848	−	0.465880i	$-0.154262\pi$
$653$	45.2226	1.76970	0.884848	+	0.465880i	$0.154262\pi$
$654$	0	0
$655$	−12.0993	−0.472757
$656$	0.344608	0.0134547
$657$	0	0
$658$	6.23618	0.243111
$659$	−13.0443	−0.508132	−0.254066	−	0.967187i	$-0.581768\pi$
$659$	−13.0443	−0.508132	−0.254066	+	0.967187i	$0.581768\pi$
$660$	0	0
$661$	−5.40736	−0.210322	−0.105161	−	0.994455i	$-0.533536\pi$
$661$	−5.40736	−0.210322	−0.105161	+	0.994455i	$0.533536\pi$
$662$	−10.3301	−0.401489
$663$	0	0
$664$	−34.9528	−1.35643
$665$	−9.24897	−0.358660
$666$	0	0
$667$	−23.9067	−0.925672
$668$	−8.14620	−0.315186
$669$	0	0
$670$	−0.152075	−0.00587516
$671$	−40.6587	−1.56961
$672$	0	0
$673$	−15.9828	−0.616090	−0.308045	−	0.951372i	$-0.599675\pi$
$673$	−15.9828	−0.616090	−0.308045	+	0.951372i	$0.599675\pi$
$674$	−23.3952	−0.901148
$675$	0	0
$676$	−19.1284	−0.735706
$677$	30.5303	1.17338	0.586688	−	0.809813i	$-0.300431\pi$
$677$	30.5303	1.17338	0.586688	+	0.809813i	$0.300431\pi$
$678$	0	0
$679$	−2.37464	−0.0911302
$680$	0	0
$681$	0	0
$682$	10.6227	0.406763
$683$	−9.27093	−0.354742	−0.177371	−	0.984144i	$-0.556759\pi$
$683$	−9.27093	−0.354742	−0.177371	+	0.984144i	$0.556759\pi$
$684$	0	0
$685$	21.8057	0.833153
$686$	13.4789	0.514629
$687$	0	0
$688$	0.306017	0.0116668
$689$	−52.8280	−2.01259
$690$	0	0
$691$	−45.2259	−1.72047	−0.860236	−	0.509895i	$-0.829684\pi$
$691$	−45.2259	−1.72047	−0.860236	+	0.509895i	$0.829684\pi$
$692$	−13.1892	−0.501380
$693$	0	0
$694$	1.76558	0.0670204
$695$	−29.2199	−1.10837
$696$	0	0
$697$	0	0
$698$	28.0051	1.06001
$699$	0	0
$700$	4.79231	0.181132
$701$	35.5553	1.34291	0.671453	−	0.741047i	$-0.265671\pi$
$701$	35.5553	1.34291	0.671453	+	0.741047i	$0.265671\pi$
$702$	0	0
$703$	−48.4662	−1.82794
$704$	21.6340	0.815363
$705$	0	0
$706$	11.2651	0.423966
$707$	−9.95306	−0.374323
$708$	0	0
$709$	12.0651	0.453115	0.226557	−	0.973998i	$-0.427253\pi$
$709$	12.0651	0.453115	0.226557	+	0.973998i	$0.427253\pi$
$710$	−14.8307	−0.556586
$711$	0	0
$712$	−28.6346	−1.07313
$713$	16.7324	0.626632
$714$	0	0
$715$	−30.9130	−1.15608
$716$	21.8357	0.816040
$717$	0	0
$718$	−12.2362	−0.456650
$719$	4.81284	0.179489	0.0897444	−	0.995965i	$-0.471395\pi$
$719$	4.81284	0.179489	0.0897444	+	0.995965i	$0.471395\pi$
$720$	0	0
$721$	11.3517	0.422761
$722$	−10.8324	−0.403142
$723$	0	0
$724$	4.17859	0.155296
$725$	−12.8102	−0.475757
$726$	0	0
$727$	−21.3824	−0.793029	−0.396514	−	0.918029i	$-0.629780\pi$
$727$	−21.3824	−0.793029	−0.396514	+	0.918029i	$0.629780\pi$
$728$	−18.6124	−0.689820
$729$	0	0
$730$	−4.52023	−0.167301
$731$	0	0
$732$	0	0
$733$	26.7793	0.989116	0.494558	−	0.869145i	$-0.335330\pi$
$733$	26.7793	0.989116	0.494558	+	0.869145i	$0.335330\pi$
$734$	6.88537	0.254144
$735$	0	0
$736$	33.5107	1.23522
$737$	0.550756	0.0202873
$738$	0	0
$739$	27.4270	1.00892	0.504458	−	0.863436i	$-0.331692\pi$
$739$	27.4270	1.00892	0.504458	+	0.863436i	$0.331692\pi$
$740$	−14.3131	−0.526162
$741$	0	0
$742$	−10.6571	−0.391236
$743$	4.03272	0.147946	0.0739730	−	0.997260i	$-0.476432\pi$
$743$	4.03272	0.147946	0.0739730	+	0.997260i	$0.476432\pi$
$744$	0	0
$745$	−3.11650	−0.114180
$746$	30.7485	1.12578
$747$	0	0
$748$	0	0
$749$	−8.19429	−0.299413
$750$	0	0
$751$	−29.1343	−1.06313	−0.531564	−	0.847018i	$-0.678395\pi$
$751$	−29.1343	−1.06313	−0.531564	+	0.847018i	$0.678395\pi$
$752$	0.242163	0.00883078
$753$	0	0
$754$	18.9141	0.688812
$755$	−10.1284	−0.368609
$756$	0	0
$757$	−18.2659	−0.663885	−0.331942	−	0.943300i	$-0.607704\pi$
$757$	−18.2659	−0.663885	−0.331942	+	0.943300i	$0.607704\pi$
$758$	0.963488	0.0349954
$759$	0	0
$760$	−21.3942	−0.776051
$761$	−10.5422	−0.382154	−0.191077	−	0.981575i	$-0.561198\pi$
$761$	−10.5422	−0.382154	−0.191077	+	0.981575i	$0.561198\pi$
$762$	0	0
$763$	0.0513845	0.00186025
$764$	−19.6699	−0.711633
$765$	0	0
$766$	7.87082	0.284384
$767$	−51.7621	−1.86902
$768$	0	0
$769$	44.8025	1.61562	0.807810	−	0.589443i	$-0.200653\pi$
$769$	44.8025	1.61562	0.807810	+	0.589443i	$0.200653\pi$
$770$	−6.23618	−0.224736
$771$	0	0
$772$	25.4561	0.916183
$773$	−26.8780	−0.966733	−0.483366	−	0.875418i	$-0.660586\pi$
$773$	−26.8780	−0.966733	−0.483366	+	0.875418i	$0.660586\pi$
$774$	0	0
$775$	8.96585	0.322063
$776$	−5.49289	−0.197183
$777$	0	0
$778$	−25.0306	−0.897392
$779$	46.0387	1.64951
$780$	0	0
$781$	53.7110	1.92193
$782$	0	0
$783$	0	0
$784$	0.230191	0.00822111
$785$	−0.985452	−0.0351723
$786$	0	0
$787$	0.539830	0.0192428	0.00962142	−	0.999954i	$-0.496937\pi$
$787$	0.539830	0.0192428	0.00962142	+	0.999954i	$0.496937\pi$
$788$	20.4433	0.728261
$789$	0	0
$790$	−1.62267	−0.0577322
$791$	−15.5996	−0.554657
$792$	0	0
$793$	50.6691	1.79931
$794$	−20.6982	−0.734552
$795$	0	0
$796$	−11.2133	−0.397444
$797$	12.6709	0.448825	0.224413	−	0.974494i	$-0.427954\pi$
$797$	12.6709	0.448825	0.224413	+	0.974494i	$0.427954\pi$
$798$	0	0
$799$	0	0
$800$	17.9564	0.634853
$801$	0	0
$802$	−30.8666	−1.08994
$803$	16.3705	0.577703
$804$	0	0
$805$	−9.82295	−0.346213
$806$	−13.2380	−0.466290
$807$	0	0
$808$	−23.0229	−0.809943
$809$	5.65095	0.198677	0.0993384	−	0.995054i	$-0.468327\pi$
$809$	5.65095	0.198677	0.0993384	+	0.995054i	$0.468327\pi$
$810$	0	0
$811$	46.0634	1.61750	0.808752	−	0.588150i	$-0.200144\pi$
$811$	46.0634	1.61750	0.808752	+	0.588150i	$0.200144\pi$
$812$	−6.05253	−0.212402
$813$	0	0
$814$	−32.6786	−1.14538
$815$	−3.66725	−0.128458
$816$	0	0
$817$	40.8830	1.43032
$818$	1.72275	0.0602347
$819$	0	0
$820$	13.5963	0.474802
$821$	−56.9172	−1.98642	−0.993211	−	0.116325i	$-0.962889\pi$
$821$	−56.9172	−1.98642	−0.993211	+	0.116325i	$0.962889\pi$
$822$	0	0
$823$	−10.1889	−0.355163	−0.177582	−	0.984106i	$-0.556827\pi$
$823$	−10.1889	−0.355163	−0.177582	+	0.984106i	$0.556827\pi$
$824$	26.2583	0.914750
$825$	0	0
$826$	−10.4421	−0.363328
$827$	−33.0036	−1.14765	−0.573824	−	0.818979i	$-0.694541\pi$
$827$	−33.0036	−1.14765	−0.573824	+	0.818979i	$0.694541\pi$
$828$	0	0
$829$	57.2532	1.98849	0.994243	−	0.107149i	$-0.0341723\pi$
$829$	57.2532	1.98849	0.994243	+	0.107149i	$0.0341723\pi$
$830$	14.5945	0.506583
$831$	0	0
$832$	−26.9605	−0.934686
$833$	0	0
$834$	0	0
$835$	8.94719	0.309630
$836$	29.4561	1.01876
$837$	0	0
$838$	7.26083	0.250821
$839$	−20.3396	−0.702199	−0.351100	−	0.936338i	$-0.614192\pi$
$839$	−20.3396	−0.702199	−0.351100	+	0.936338i	$0.614192\pi$
$840$	0	0
$841$	−12.8212	−0.442110
$842$	19.0223	0.655551
$843$	0	0
$844$	16.9564	0.583662
$845$	21.0092	0.722737
$846$	0	0
$847$	9.09152	0.312388
$848$	−0.413838	−0.0142113
$849$	0	0
$850$	0	0
$851$	−51.4739	−1.76450
$852$	0	0
$853$	−10.7445	−0.367886	−0.183943	−	0.982937i	$-0.558886\pi$
$853$	−10.7445	−0.367886	−0.183943	+	0.982937i	$0.558886\pi$
$854$	10.2216	0.349777
$855$	0	0
$856$	−18.9546	−0.647855
$857$	6.67324	0.227953	0.113977	−	0.993483i	$-0.463641\pi$
$857$	6.67324	0.227953	0.113977	+	0.993483i	$0.463641\pi$
$858$	0	0
$859$	58.0310	1.97999	0.989995	−	0.141099i	$-0.0450637\pi$
$859$	58.0310	1.97999	0.989995	+	0.141099i	$0.0450637\pi$
$860$	12.0737	0.411709
$861$	0	0
$862$	29.1830	0.993978
$863$	29.0009	0.987203	0.493602	−	0.869688i	$-0.335680\pi$
$863$	29.0009	0.987203	0.493602	+	0.869688i	$0.335680\pi$
$864$	0	0
$865$	14.4861	0.492542
$866$	3.36865	0.114471
$867$	0	0
$868$	4.23618	0.143785
$869$	5.87670	0.199353
$870$	0	0
$871$	−0.686355	−0.0232563
$872$	0.118860	0.00402511
$873$	0	0
$874$	−29.2499	−0.989393
$875$	−13.5270	−0.457297
$876$	0	0
$877$	46.3319	1.56452	0.782259	−	0.622953i	$-0.214067\pi$
$877$	46.3319	1.56452	0.782259	+	0.622953i	$0.214067\pi$
$878$	23.4816	0.792467
$879$	0	0
$880$	−0.242163	−0.00816332
$881$	−32.9299	−1.10944	−0.554719	−	0.832038i	$-0.687174\pi$
$881$	−32.9299	−1.10944	−0.554719	+	0.832038i	$0.687174\pi$
$882$	0	0
$883$	19.7314	0.664015	0.332008	−	0.943277i	$-0.392274\pi$
$883$	19.7314	0.664015	0.332008	+	0.943277i	$0.392274\pi$
$884$	0	0
$885$	0	0
$886$	31.0743	1.04396
$887$	11.2199	0.376727	0.188363	−	0.982099i	$-0.439682\pi$
$887$	11.2199	0.376727	0.188363	+	0.982099i	$0.439682\pi$
$888$	0	0
$889$	8.84079	0.296511
$890$	11.9564	0.400778
$891$	0	0
$892$	−1.61916	−0.0542136
$893$	32.3523	1.08263
$894$	0	0
$895$	−23.9828	−0.801655
$896$	8.39363	0.280411
$897$	0	0
$898$	−22.8821	−0.763585
$899$	−11.3236	−0.377662
$900$	0	0
$901$	0	0
$902$	31.0419	1.03358
$903$	0	0
$904$	−36.0841	−1.20014
$905$	−4.58946	−0.152559
$906$	0	0
$907$	22.4088	0.744072	0.372036	−	0.928218i	$-0.378660\pi$
$907$	22.4088	0.744072	0.372036	+	0.928218i	$0.378660\pi$
$908$	−11.2552	−0.373516
$909$	0	0
$910$	7.77156	0.257625
$911$	−32.3732	−1.07257	−0.536286	−	0.844036i	$-0.680173\pi$
$911$	−32.3732	−1.07257	−0.536286	+	0.844036i	$0.680173\pi$
$912$	0	0
$913$	−52.8557	−1.74927
$914$	−26.4825	−0.875962
$915$	0	0
$916$	32.0880	1.06022
$917$	−11.0161	−0.363783
$918$	0	0
$919$	−26.8060	−0.884250	−0.442125	−	0.896954i	$-0.645775\pi$
$919$	−26.8060	−0.884250	−0.442125	+	0.896954i	$0.645775\pi$
$920$	−22.7219	−0.749120
$921$	0	0
$922$	−23.8976	−0.787024
$923$	−66.9350	−2.20319
$924$	0	0
$925$	−27.5817	−0.906881
$926$	7.85649	0.258180
$927$	0	0
$928$	−22.6783	−0.744451
$929$	17.5193	0.574789	0.287395	−	0.957812i	$-0.407211\pi$
$929$	17.5193	0.574789	0.287395	+	0.957812i	$0.407211\pi$
$930$	0	0
$931$	30.7529	1.00789
$932$	1.55943	0.0510808
$933$	0	0
$934$	35.0229	1.14598
$935$	0	0
$936$	0	0
$937$	−20.0615	−0.655380	−0.327690	−	0.944785i	$-0.606270\pi$
$937$	−20.0615	−0.655380	−0.327690	+	0.944785i	$0.606270\pi$
$938$	−0.138460	−0.00452089
$939$	0	0
$940$	9.55438	0.311629
$941$	13.9121	0.453522	0.226761	−	0.973950i	$-0.427186\pi$
$941$	13.9121	0.453522	0.226761	+	0.973950i	$0.427186\pi$
$942$	0	0
$943$	48.8958	1.59227
$944$	−0.405488	−0.0131975
$945$	0	0
$946$	27.5656	0.896236
$947$	26.4793	0.860461	0.430230	−	0.902719i	$-0.358432\pi$
$947$	26.4793	0.860461	0.430230	+	0.902719i	$0.358432\pi$
$948$	0	0
$949$	−20.4010	−0.662246
$950$	−15.6732	−0.508507
$951$	0	0
$952$	0	0
$953$	39.7377	1.28723	0.643616	−	0.765349i	$-0.277433\pi$
$953$	39.7377	1.28723	0.643616	+	0.765349i	$0.277433\pi$
$954$	0	0
$955$	21.6040	0.699089
$956$	−5.87433	−0.189990
$957$	0	0
$958$	17.8652	0.577197
$959$	19.8536	0.641106
$960$	0	0
$961$	−23.0746	−0.744342
$962$	40.7243	1.31300
$963$	0	0
$964$	6.51424	0.209810
$965$	−27.9590	−0.900033
$966$	0	0
$967$	−20.9753	−0.674522	−0.337261	−	0.941411i	$-0.609500\pi$
$967$	−20.9753	−0.674522	−0.337261	+	0.941411i	$0.609500\pi$
$968$	21.0300	0.675931
$969$	0	0
$970$	2.29355	0.0736414
$971$	−30.3310	−0.973368	−0.486684	−	0.873578i	$-0.661794\pi$
$971$	−30.3310	−0.973368	−0.486684	+	0.873578i	$0.661794\pi$
$972$	0	0
$973$	−26.6040	−0.852885
$974$	20.3975	0.653579
$975$	0	0
$976$	0.396926	0.0127053
$977$	−16.9469	−0.542178	−0.271089	−	0.962554i	$-0.587384\pi$
$977$	−16.9469	−0.542178	−0.271089	+	0.962554i	$0.587384\pi$
$978$	0	0
$979$	−43.3013	−1.38392
$980$	9.08202	0.290115
$981$	0	0
$982$	−19.7624	−0.630643
$983$	−52.8016	−1.68411	−0.842055	−	0.539392i	$-0.818654\pi$
$983$	−52.8016	−1.68411	−0.842055	+	0.539392i	$0.818654\pi$
$984$	0	0
$985$	−22.4534	−0.715424
$986$	0	0
$987$	0	0
$988$	−36.7083	−1.16785
$989$	43.4201	1.38068
$990$	0	0
$991$	−40.2330	−1.27804	−0.639022	−	0.769189i	$-0.720661\pi$
$991$	−40.2330	−1.27804	−0.639022	+	0.769189i	$0.720661\pi$
$992$	15.8726	0.503955
$993$	0	0
$994$	−13.5030	−0.428289
$995$	12.3158	0.390438
$996$	0	0
$997$	54.6887	1.73201	0.866005	−	0.500036i	$-0.166680\pi$
$997$	54.6887	1.73201	0.866005	+	0.500036i	$0.166680\pi$
$998$	−36.6236	−1.15930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.z.1.1		3
3.2	odd	2		867.2.a.j.1.3	yes	3
17.16	even	2		2601.2.a.y.1.1		3
51.2	odd	8		867.2.e.j.616.6		12
51.5	even	16		867.2.h.l.688.1		24
51.8	odd	8		867.2.e.j.829.1		12
51.11	even	16		867.2.h.l.733.5		24
51.14	even	16		867.2.h.l.757.5		24
51.20	even	16		867.2.h.l.757.6		24
51.23	even	16		867.2.h.l.733.6		24
51.26	odd	8		867.2.e.j.829.2		12
51.29	even	16		867.2.h.l.688.2		24
51.32	odd	8		867.2.e.j.616.5		12
51.38	odd	4		867.2.d.d.577.1		6
51.41	even	16		867.2.h.l.712.1		24
51.44	even	16		867.2.h.l.712.2		24
51.47	odd	4		867.2.d.d.577.2		6
51.50	odd	2		867.2.a.i.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
867.2.a.i.1.3	✓	3	51.50	odd	2
867.2.a.j.1.3	yes	3	3.2	odd	2
867.2.d.d.577.1		6	51.38	odd	4
867.2.d.d.577.2		6	51.47	odd	4
867.2.e.j.616.5		12	51.32	odd	8
867.2.e.j.616.6		12	51.2	odd	8
867.2.e.j.829.1		12	51.8	odd	8
867.2.e.j.829.2		12	51.26	odd	8
867.2.h.l.688.1		24	51.5	even	16
867.2.h.l.688.2		24	51.29	even	16
867.2.h.l.712.1		24	51.41	even	16
867.2.h.l.712.2		24	51.44	even	16
867.2.h.l.733.5		24	51.11	even	16
867.2.h.l.733.6		24	51.23	even	16
867.2.h.l.757.5		24	51.14	even	16
867.2.h.l.757.6		24	51.20	even	16
2601.2.a.y.1.1		3	17.16	even	2
2601.2.a.z.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} +1.22668 q^{7} +2.83750 q^{8} +O(q^{10})\)	\(q-0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} +1.22668 q^{7} +2.83750 q^{8} -1.18479 q^{10} +4.29086 q^{11} -5.34730 q^{13} -1.07873 q^{14} -0.0418891 q^{16} -5.59627 q^{19} -1.65270 q^{20} -3.77332 q^{22} -5.94356 q^{23} -3.18479 q^{25} +4.70233 q^{26} -1.50475 q^{28} +4.02229 q^{29} -2.81521 q^{31} -5.63816 q^{32} +1.65270 q^{35} +8.66044 q^{37} +4.92127 q^{38} +3.82295 q^{40} -8.22668 q^{41} -7.30541 q^{43} -5.26352 q^{44} +5.22668 q^{46} -5.78106 q^{47} -5.49525 q^{49} +2.80066 q^{50} +6.55943 q^{52} +9.87939 q^{53} +5.78106 q^{55} +3.48070 q^{56} -3.53714 q^{58} +9.68004 q^{59} -9.47565 q^{61} +2.47565 q^{62} +5.04189 q^{64} -7.20439 q^{65} +0.128356 q^{67} -1.45336 q^{70} +12.5175 q^{71} +3.81521 q^{73} -7.61587 q^{74} +6.86484 q^{76} +5.26352 q^{77} +1.36959 q^{79} -0.0564370 q^{80} +7.23442 q^{82} -12.3182 q^{83} +6.42427 q^{86} +12.1753 q^{88} -10.0915 q^{89} -6.55943 q^{91} +7.29086 q^{92} +5.08378 q^{94} -7.53983 q^{95} -1.93582 q^{97} +4.83244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 3 q^{11} - 15 q^{13} - 12 q^{14} + 3 q^{16} - 3 q^{19} - 6 q^{20} - 18 q^{22} - 3 q^{23} - 6 q^{25} - 12 q^{26} - 21 q^{28} + 6 q^{29} - 12 q^{31} + 6 q^{35} + 3 q^{37} + 6 q^{38} - 9 q^{40} - 18 q^{41} - 24 q^{43} - 21 q^{44} + 9 q^{46} - 6 q^{50} - 6 q^{52} + 24 q^{53} - 24 q^{56} + 9 q^{58} + 9 q^{59} - 9 q^{61} - 12 q^{62} + 12 q^{64} - 21 q^{65} - 18 q^{67} + 9 q^{70} + 15 q^{71} + 15 q^{73} - 12 q^{74} - 3 q^{76} + 21 q^{77} - 3 q^{79} - 15 q^{80} - 9 q^{82} - 30 q^{86} + 6 q^{91} + 6 q^{92} + 9 q^{94} + 6 q^{95} - 15 q^{97} + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 3 * q^11 - 15 * q^13 - 12 * q^14 + 3 * q^16 - 3 * q^19 - 6 * q^20 - 18 * q^22 - 3 * q^23 - 6 * q^25 - 12 * q^26 - 21 * q^28 + 6 * q^29 - 12 * q^31 + 6 * q^35 + 3 * q^37 + 6 * q^38 - 9 * q^40 - 18 * q^41 - 24 * q^43 - 21 * q^44 + 9 * q^46 - 6 * q^50 - 6 * q^52 + 24 * q^53 - 24 * q^56 + 9 * q^58 + 9 * q^59 - 9 * q^61 - 12 * q^62 + 12 * q^64 - 21 * q^65 - 18 * q^67 + 9 * q^70 + 15 * q^71 + 15 * q^73 - 12 * q^74 - 3 * q^76 + 21 * q^77 - 3 * q^79 - 15 * q^80 - 9 * q^82 - 30 * q^86 + 6 * q^91 + 6 * q^92 + 9 * q^94 + 6 * q^95 - 15 * q^97 + 27 * q^98

Introduction

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