LMFDB - Embedded newform 2601.2.a.y.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:	$0$
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	$-0.597409\pi$
$5$	−1.34730	−0.602529	−0.301265	+	0.953541i	$0.597409\pi$
$6$	0	0
$7$	−1.22668	−0.463642	−0.231821	−	0.972758i	$-0.574468\pi$
$7$	−1.22668	−0.463642	−0.231821	+	0.972758i	$0.574468\pi$
$8$	2.83750	1.00321
$9$	0	0
$10$	1.18479	0.374664
$11$	−4.29086	−1.29374	−0.646871	−	0.762599i	$-0.723923\pi$
$11$	−4.29086	−1.29374	−0.646871	+	0.762599i	$0.723923\pi$
$12$	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	$-0.287271\pi$
$23$	5.94356	1.23932	0.619659	+	0.784871i	$0.287271\pi$
$24$	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	$-0.621829\pi$
$29$	−4.02229	−0.746920	−0.373460	+	0.927646i	$0.621829\pi$
$30$	0	0
$31$	2.81521	0.505626	0.252813	−	0.967515i	$-0.418644\pi$
$31$	2.81521	0.505626	0.252813	+	0.967515i	$0.418644\pi$
$32$	−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	$-0.752158\pi$
$37$	−8.66044	−1.42377	−0.711884	+	0.702297i	$0.752158\pi$
$38$	4.92127	0.798336
$39$	0	0
$40$	−3.82295	−0.604461
$41$	8.22668	1.28479	0.642396	−	0.766373i	$-0.277941\pi$
$41$	8.22668	1.28479	0.642396	+	0.766373i	$0.277941\pi$
$42$	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	$-0.292526\pi$
$61$	9.47565	1.21323	0.606616	+	0.794995i	$0.292526\pi$
$62$	−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	$-0.766493\pi$
$71$	−12.5175	−1.48556	−0.742779	+	0.669536i	$0.766493\pi$
$72$	0	0
$73$	−3.81521	−0.446536	−0.223268	−	0.974757i	$-0.571673\pi$
$73$	−3.81521	−0.446536	−0.223268	+	0.974757i	$0.571673\pi$
$74$	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	$-0.524549\pi$
$79$	−1.36959	−0.154090	−0.0770452	+	0.997028i	$0.524549\pi$
$80$	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	$-0.468667\pi$
$97$	1.93582	0.196553	0.0982765	+	0.995159i	$0.468667\pi$
$98$	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	$-0.395345\pi$
$107$	6.68004	0.645784	0.322892	+	0.946436i	$0.395345\pi$
$108$	0	0
$109$	−0.0418891	−0.00401224	−0.00200612	−	0.999998i	$-0.500639\pi$
$109$	−0.0418891	−0.00401224	−0.00200612	+	0.999998i	$0.500639\pi$
$110$	−5.08378	−0.484719
$111$	0	0
$112$	0.0513845	0.00485538
$113$	12.7169	1.19630	0.598152	−	0.801383i	$-0.295902\pi$
$113$	12.7169	1.19630	0.598152	+	0.801383i	$0.295902\pi$
$114$	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	$-0.371676\pi$
$131$	8.98040	0.784621	0.392311	+	0.919833i	$0.371676\pi$
$132$	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	$-0.128378\pi$
$139$	21.6878	1.83953	0.919767	+	0.392465i	$0.128378\pi$
$140$	−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	$-0.466004\pi$
$163$	2.72193	0.213198	0.106599	+	0.994302i	$0.466004\pi$
$164$	−10.0915	−0.788015
$165$	0	0
$166$	10.8324	0.840761
$167$	−6.64084	−0.513884	−0.256942	−	0.966427i	$-0.582715\pi$
$167$	−6.64084	−0.513884	−0.256942	+	0.966427i	$0.582715\pi$
$168$	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	$-0.634028\pi$
$173$	−10.7520	−0.817457	−0.408728	+	0.912656i	$0.634028\pi$
$174$	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	$-0.459594\pi$
$181$	3.40642	0.253197	0.126599	+	0.991954i	$0.459594\pi$
$182$	−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	$-0.231551\pi$
$193$	20.7520	1.49376	0.746880	+	0.664959i	$0.231551\pi$
$194$	−1.70233	−0.122220
$195$	0	0
$196$	6.74092	0.481495
$197$	16.6655	1.18737	0.593684	−	0.804698i	$-0.297673\pi$
$197$	16.6655	1.18737	0.593684	+	0.804698i	$0.297673\pi$
$198$	0	0
$199$	−9.14115	−0.647999	−0.323999	−	0.946057i	$-0.605028\pi$
$199$	−9.14115	−0.647999	−0.323999	+	0.946057i	$0.605028\pi$
$200$	−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	$-0.342157\pi$
$211$	13.8229	0.951611	0.475806	+	0.879551i	$0.342157\pi$
$212$	−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	$-0.598487\pi$
$227$	−9.17530	−0.608986	−0.304493	+	0.952515i	$0.598487\pi$
$228$	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	$-0.486741\pi$
$233$	1.27126	0.0832829	0.0416415	+	0.999133i	$0.486741\pi$
$234$	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	$-0.445288\pi$
$241$	5.31046	0.342077	0.171038	+	0.985264i	$0.445288\pi$
$242$	−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	$-0.432446\pi$
$269$	6.90941	0.421274	0.210637	+	0.977564i	$0.432446\pi$
$270$	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	$-0.495076\pi$
$277$	0.514853	0.0309345	0.0154672	+	0.999880i	$0.495076\pi$
$278$	−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	$-0.363820\pi$
$283$	13.9590	0.829779	0.414890	+	0.909872i	$0.363820\pi$
$284$	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	$-0.531464\pi$
$311$	−3.48070	−0.197373	−0.0986863	+	0.995119i	$0.531464\pi$
$312$	0	0
$313$	−17.3746	−0.982073	−0.491036	−	0.871139i	$-0.663382\pi$
$313$	−17.3746	−0.982073	−0.491036	+	0.871139i	$0.663382\pi$
$314$	0.643208	0.0362983
$315$	0	0
$316$	1.68004	0.0945099
$317$	−3.07604	−0.172767	−0.0863837	−	0.996262i	$-0.527531\pi$
$317$	−3.07604	−0.172767	−0.0863837	+	0.996262i	$0.527531\pi$
$318$	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	$-0.757978\pi$
$337$	−26.6040	−1.44921	−0.724606	+	0.689163i	$0.757978\pi$
$338$	−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	$-0.482838\pi$
$347$	2.00774	0.107781	0.0538906	+	0.998547i	$0.482838\pi$
$348$	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	$-0.434490\pi$
$367$	7.82976	0.408710	0.204355	+	0.978897i	$0.434490\pi$
$368$	−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	$-0.491042\pi$
$379$	1.09564	0.0562791	0.0281396	+	0.999604i	$0.491042\pi$
$380$	−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	$-0.701128\pi$
$397$	−23.5371	−1.18130	−0.590648	+	0.806930i	$0.701128\pi$
$398$	8.03859	0.402938
$399$	0	0
$400$	0.133408	0.00667040
$401$	−35.1002	−1.75282	−0.876410	−	0.481566i	$-0.840068\pi$
$401$	−35.1002	−1.75282	−0.876410	+	0.481566i	$0.840068\pi$
$402$	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	$-0.435359\pi$
$419$	8.25671	0.403367	0.201683	+	0.979451i	$0.435359\pi$
$420$	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	$-0.205230\pi$
$431$	33.1857	1.59850	0.799250	+	0.600999i	$0.205230\pi$
$432$	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	$-0.280086\pi$
$439$	26.7023	1.27443	0.637216	+	0.770685i	$0.280086\pi$
$440$	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	$-0.710437\pi$
$449$	−26.0205	−1.22798	−0.613992	+	0.789312i	$0.710437\pi$
$450$	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	$-0.346371\pi$
$479$	20.3155	0.928239	0.464120	+	0.885772i	$0.346371\pi$
$480$	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	$-0.323864\pi$
$487$	23.1952	1.05108	0.525538	+	0.850770i	$0.323864\pi$
$488$	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	$-0.882099\pi$
$499$	−41.6468	−1.86437	−0.932184	+	0.361985i	$0.882099\pi$
$500$	−13.5270	−0.604947
$501$	0	0
$502$	−7.42097	−0.331214
$503$	22.4165	0.999504	0.499752	−	0.866169i	$-0.333425\pi$
$503$	22.4165	0.999504	0.499752	+	0.866169i	$0.333425\pi$
$504$	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	$-0.489381\pi$
$521$	1.52259	0.0667060	0.0333530	+	0.999444i	$0.489381\pi$
$522$	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	$-0.553839\pi$
$541$	−7.83069	−0.336668	−0.168334	+	0.985730i	$0.553839\pi$
$542$	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	$-0.781808\pi$
$547$	−36.2104	−1.54824	−0.774122	+	0.633036i	$0.781808\pi$
$548$	−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	$-0.645715\pi$
$571$	−21.1215	−0.883909	−0.441955	+	0.897037i	$0.645715\pi$
$572$	−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	$-0.330488\pi$
$601$	24.8939	1.01544	0.507722	+	0.861521i	$0.330488\pi$
$602$	−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	$-0.474454\pi$
$607$	3.95037	0.160341	0.0801703	+	0.996781i	$0.474454\pi$
$608$	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	$-0.756367\pi$
$617$	−35.8239	−1.44222	−0.721108	+	0.692823i	$0.756367\pi$
$618$	0	0
$619$	−31.1334	−1.25136	−0.625679	−	0.780081i	$-0.715178\pi$
$619$	−31.1334	−1.25136	−0.625679	+	0.780081i	$0.715178\pi$
$620$	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	$-0.801520\pi$
$641$	−41.1070	−1.62363	−0.811814	+	0.583915i	$0.801520\pi$
$642$	0	0
$643$	−12.1952	−0.480933	−0.240466	−	0.970657i	$-0.577300\pi$
$643$	−12.1952	−0.480933	−0.240466	+	0.970657i	$0.577300\pi$
$644$	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	$-0.845738\pi$
$653$	−45.2226	−1.76970	−0.884848	+	0.465880i	$0.845738\pi$
$654$	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	$-0.400325\pi$
$673$	15.9828	0.616090	0.308045	+	0.951372i	$0.400325\pi$
$674$	23.3952	0.901148
$675$	0	0
$676$	−19.1284	−0.735706
$677$	−30.5303	−1.17338	−0.586688	−	0.809813i	$-0.699569\pi$
$677$	−30.5303	−1.17338	−0.586688	+	0.809813i	$0.699569\pi$
$678$	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	$-0.443241\pi$
$683$	9.27093	0.354742	0.177371	+	0.984144i	$0.443241\pi$
$684$	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	$-0.170316\pi$
$691$	45.2259	1.72047	0.860236	+	0.509895i	$0.170316\pi$
$692$	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	$-0.572747\pi$
$709$	−12.0651	−0.453115	−0.226557	+	0.973998i	$0.572747\pi$
$710$	−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	$-0.528605\pi$
$719$	−4.81284	−0.179489	−0.0897444	+	0.995965i	$0.528605\pi$
$720$	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	$-0.523568\pi$
$743$	−4.03272	−0.147946	−0.0739730	+	0.997260i	$0.523568\pi$
$744$	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	$-0.321605\pi$
$751$	29.1343	1.06313	0.531564	+	0.847018i	$0.321605\pi$
$752$	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	$-0.503063\pi$
$787$	−0.539830	−0.0192428	−0.00962142	+	0.999954i	$0.503063\pi$
$788$	−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	$-0.531673\pi$
$809$	−5.65095	−0.198677	−0.0993384	+	0.995054i	$0.531673\pi$
$810$	0	0
$811$	−46.0634	−1.61750	−0.808752	−	0.588150i	$-0.799856\pi$
$811$	−46.0634	−1.61750	−0.808752	+	0.588150i	$0.799856\pi$
$812$	−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	$-0.0371114\pi$
$821$	56.9172	1.98642	0.993211	+	0.116325i	$0.0371114\pi$
$822$	0	0
$823$	10.1889	0.355163	0.177582	−	0.984106i	$-0.443173\pi$
$823$	10.1889	0.355163	0.177582	+	0.984106i	$0.443173\pi$
$824$	26.2583	0.914750
$825$	0	0
$826$	10.4421	0.363328
$827$	33.0036	1.14765	0.573824	−	0.818979i	$-0.305459\pi$
$827$	33.0036	1.14765	0.573824	+	0.818979i	$0.305459\pi$
$828$	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	$-0.385808\pi$
$839$	20.3396	0.702199	0.351100	+	0.936338i	$0.385808\pi$
$840$	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	$-0.441114\pi$
$853$	10.7445	0.367886	0.183943	+	0.982937i	$0.441114\pi$
$854$	10.2216	0.349777
$855$	0	0
$856$	18.9546	0.647855
$857$	−6.67324	−0.227953	−0.113977	−	0.993483i	$-0.536359\pi$
$857$	−6.67324	−0.227953	−0.113977	+	0.993483i	$0.536359\pi$
$858$	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	$-0.785933\pi$
$877$	−46.3319	−1.56452	−0.782259	+	0.622953i	$0.785933\pi$
$878$	−23.4816	−0.792467
$879$	0	0
$880$	−0.242163	−0.00816332
$881$	32.9299	1.10944	0.554719	−	0.832038i	$-0.312826\pi$
$881$	32.9299	1.10944	0.554719	+	0.832038i	$0.312826\pi$
$882$	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	$-0.560318\pi$
$887$	−11.2199	−0.376727	−0.188363	+	0.982099i	$0.560318\pi$
$888$	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	$-0.621340\pi$
$907$	−22.4088	−0.744072	−0.372036	+	0.928218i	$0.621340\pi$
$908$	11.2552	0.373516
$909$	0	0
$910$	7.77156	0.257625
$911$	32.3732	1.07257	0.536286	−	0.844036i	$-0.319827\pi$
$911$	32.3732	1.07257	0.536286	+	0.844036i	$0.319827\pi$
$912$	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	$-0.592789\pi$
$929$	−17.5193	−0.574789	−0.287395	+	0.957812i	$0.592789\pi$
$930$	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	$-0.572814\pi$
$941$	−13.9121	−0.453522	−0.226761	+	0.973950i	$0.572814\pi$
$942$	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	$-0.641568\pi$
$947$	−26.4793	−0.860461	−0.430230	+	0.902719i	$0.641568\pi$
$948$	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	$-0.181346\pi$
$983$	52.8016	1.68411	0.842055	+	0.539392i	$0.181346\pi$
$984$	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	$-0.279339\pi$
$991$	40.2330	1.27804	0.639022	+	0.769189i	$0.279339\pi$
$992$	−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	$-0.833320\pi$
$997$	−54.6887	−1.73201	−0.866005	+	0.500036i	$0.833320\pi$
$998$	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.y.1.1		3
3.2	odd	2		867.2.a.i.1.3	✓	3
17.16	even	2		2601.2.a.z.1.1		3
51.2	odd	8		867.2.e.j.616.5		12
51.5	even	16		867.2.h.l.688.2		24
51.8	odd	8		867.2.e.j.829.2		12
51.11	even	16		867.2.h.l.733.6		24
51.14	even	16		867.2.h.l.757.6		24
51.20	even	16		867.2.h.l.757.5		24
51.23	even	16		867.2.h.l.733.5		24
51.26	odd	8		867.2.e.j.829.1		12
51.29	even	16		867.2.h.l.688.1		24
51.32	odd	8		867.2.e.j.616.6		12
51.38	odd	4		867.2.d.d.577.2		6
51.41	even	16		867.2.h.l.712.2		24
51.44	even	16		867.2.h.l.712.1		24
51.47	odd	4		867.2.d.d.577.1		6
51.50	odd	2		867.2.a.j.1.3	yes	3

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more