LMFDB - Embedded newform 2601.2.a.bf.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:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.652223$ 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)

$f(q)$	$=$	$q-2.06644 q^{2} +2.27016 q^{4} +0.347777 q^{5} -4.33660 q^{7} -0.558268 q^{8} +O(q^{10})$	$q-2.06644 q^{2} +2.27016 q^{4} +0.347777 q^{5} -4.33660 q^{7} -0.558268 q^{8} -0.718659 q^{10} +0.729840 q^{11} -5.40303 q^{13} +8.96130 q^{14} -3.38669 q^{16} +3.27016 q^{19} +0.789510 q^{20} -1.50817 q^{22} -3.55827 q^{23} -4.87905 q^{25} +11.1650 q^{26} -9.84476 q^{28} +4.38206 q^{29} -3.30445 q^{31} +8.11492 q^{32} -1.50817 q^{35} -2.87389 q^{37} -6.75758 q^{38} -0.194153 q^{40} -11.8494 q^{41} +3.74618 q^{43} +1.65685 q^{44} +7.35293 q^{46} -0.476019 q^{47} +11.8061 q^{49} +10.0822 q^{50} -12.2657 q^{52} +10.1329 q^{53} +0.253822 q^{55} +2.42098 q^{56} -9.05526 q^{58} -5.29790 q^{59} +3.54709 q^{61} +6.82843 q^{62} -9.99559 q^{64} -1.87905 q^{65} -1.55666 q^{67} +3.11654 q^{70} +8.96130 q^{71} -4.09396 q^{73} +5.93872 q^{74} +7.42378 q^{76} -3.16502 q^{77} +0.812615 q^{79} -1.17781 q^{80} +24.4860 q^{82} -7.21351 q^{83} -7.74124 q^{86} -0.407446 q^{88} +9.77391 q^{89} +23.4308 q^{91} -8.07784 q^{92} +0.983662 q^{94} +1.13729 q^{95} -1.77317 q^{97} -24.3965 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8	$ 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8} + 12 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} + 6 q^{16} + 10 q^{19} + 16 q^{20} - 4 q^{22} - 6 q^{23} + 2 q^{25} + 20 q^{26} - 24 q^{28} + 16 q^{29} - 4 q^{31} + 14 q^{32} - 4 q^{35} - 12 q^{37} + 12 q^{38} + 8 q^{40} - 14 q^{41} + 14 q^{43} - 16 q^{44} + 12 q^{46} - 4 q^{47} + 30 q^{50} - 8 q^{52} + 20 q^{53} + 2 q^{55} - 16 q^{56} - 8 q^{58} + 24 q^{59} - 12 q^{61} + 16 q^{62} - 2 q^{64} + 14 q^{65} + 4 q^{67} - 4 q^{70} + 4 q^{71} - 20 q^{73} + 12 q^{74} + 40 q^{76} + 12 q^{77} - 8 q^{79} + 4 q^{82} + 4 q^{83} + 4 q^{86} - 16 q^{88} - 4 q^{89} + 28 q^{91} + 16 q^{92} + 8 q^{94} + 22 q^{95} - 24 q^{97} - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8 + 12 * q^10 + 6 * q^11 + 2 * q^13 + 4 * q^14 + 6 * q^16 + 10 * q^19 + 16 * q^20 - 4 * q^22 - 6 * q^23 + 2 * q^25 + 20 * q^26 - 24 * q^28 + 16 * q^29 - 4 * q^31 + 14 * q^32 - 4 * q^35 - 12 * q^37 + 12 * q^38 + 8 * q^40 - 14 * q^41 + 14 * q^43 - 16 * q^44 + 12 * q^46 - 4 * q^47 + 30 * q^50 - 8 * q^52 + 20 * q^53 + 2 * q^55 - 16 * q^56 - 8 * q^58 + 24 * q^59 - 12 * q^61 + 16 * q^62 - 2 * q^64 + 14 * q^65 + 4 * q^67 - 4 * q^70 + 4 * q^71 - 20 * q^73 + 12 * q^74 + 40 * q^76 + 12 * q^77 - 8 * q^79 + 4 * q^82 + 4 * q^83 + 4 * q^86 - 16 * q^88 - 4 * q^89 + 28 * q^91 + 16 * q^92 + 8 * q^94 + 22 * q^95 - 24 * q^97 - 38 * 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 $

$2$	−2.06644	−1.46119	−0.730596	−	0.682810i	$-0.760757\pi$
$2$	−2.06644	−1.46119	−0.730596	+	0.682810i	$0.760757\pi$
$3$	0	0
$3$	0	0
$4$	2.27016	1.13508
$5$	0.347777	0.155531	0.0777653	−	0.996972i	$-0.475222\pi$
$5$	0.347777	0.155531	0.0777653	+	0.996972i	$0.475222\pi$
$6$	0	0
$7$	−4.33660	−1.63908	−0.819540	−	0.573023i	$-0.805771\pi$
$7$	−4.33660	−1.63908	−0.819540	+	0.573023i	$0.805771\pi$
$8$	−0.558268	−0.197377
$9$	0	0
$10$	−0.718659	−0.227260
$11$	0.729840	0.220055	0.110028	−	0.993929i	$-0.464906\pi$
$11$	0.729840	0.220055	0.110028	+	0.993929i	$0.464906\pi$
$12$	0	0
$13$	−5.40303	−1.49853	−0.749266	−	0.662269i	$-0.769593\pi$
$13$	−5.40303	−1.49853	−0.749266	+	0.662269i	$0.769593\pi$
$14$	8.96130	2.39501
$15$	0	0
$16$	−3.38669	−0.846674
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.27016	0.750226	0.375113	−	0.926979i	$-0.377604\pi$
$19$	3.27016	0.750226	0.375113	+	0.926979i	$0.377604\pi$
$20$	0.789510	0.176540
$21$	0	0
$22$	−1.50817	−0.321543
$23$	−3.55827	−0.741950	−0.370975	−	0.928643i	$-0.620976\pi$
$23$	−3.55827	−0.741950	−0.370975	+	0.928643i	$0.620976\pi$
$24$	0	0
$25$	−4.87905	−0.975810
$26$	11.1650	2.18964
$27$	0	0
$28$	−9.84476	−1.86049
$29$	4.38206	0.813729	0.406864	−	0.913489i	$-0.366622\pi$
$29$	4.38206	0.813729	0.406864	+	0.913489i	$0.366622\pi$
$30$	0	0
$31$	−3.30445	−0.593496	−0.296748	−	0.954956i	$-0.595902\pi$
$31$	−3.30445	−0.593496	−0.296748	+	0.954956i	$0.595902\pi$
$32$	8.11492	1.43453
$33$	0	0
$34$	0	0
$35$	−1.50817	−0.254927
$36$	0	0
$37$	−2.87389	−0.472465	−0.236233	−	0.971697i	$-0.575913\pi$
$37$	−2.87389	−0.472465	−0.236233	+	0.971697i	$0.575913\pi$
$38$	−6.75758	−1.09622
$39$	0	0
$40$	−0.194153	−0.0306982
$41$	−11.8494	−1.85056	−0.925282	−	0.379279i	$-0.876172\pi$
$41$	−11.8494	−1.85056	−0.925282	+	0.379279i	$0.876172\pi$
$42$	0	0
$43$	3.74618	0.571287	0.285643	−	0.958336i	$-0.407793\pi$
$43$	3.74618	0.571287	0.285643	+	0.958336i	$0.407793\pi$
$44$	1.65685	0.249780
$45$	0	0
$46$	7.35293	1.08413
$47$	−0.476019	−0.0694345	−0.0347172	−	0.999397i	$-0.511053\pi$
$47$	−0.476019	−0.0694345	−0.0347172	+	0.999397i	$0.511053\pi$
$48$	0	0
$49$	11.8061	1.68658
$50$	10.0822	1.42585
$51$	0	0
$52$	−12.2657	−1.70095
$53$	10.1329	1.39186	0.695929	−	0.718111i	$-0.254993\pi$
$53$	10.1329	1.39186	0.695929	+	0.718111i	$0.254993\pi$
$54$	0	0
$55$	0.253822	0.0342253
$56$	2.42098	0.323517
$57$	0	0
$58$	−9.05526	−1.18901
$59$	−5.29790	−0.689727	−0.344864	−	0.938653i	$-0.612075\pi$
$59$	−5.29790	−0.689727	−0.344864	+	0.938653i	$0.612075\pi$
$60$	0	0
$61$	3.54709	0.454158	0.227079	−	0.973876i	$-0.427082\pi$
$61$	3.54709	0.454158	0.227079	+	0.973876i	$0.427082\pi$
$62$	6.82843	0.867211
$63$	0	0
$64$	−9.99559	−1.24945
$65$	−1.87905	−0.233068
$66$	0	0
$67$	−1.55666	−0.190176	−0.0950880	−	0.995469i	$-0.530313\pi$
$67$	−1.55666	−0.190176	−0.0950880	+	0.995469i	$0.530313\pi$
$68$	0	0
$69$	0	0
$70$	3.11654	0.372497
$71$	8.96130	1.06351	0.531755	−	0.846898i	$-0.321533\pi$
$71$	8.96130	1.06351	0.531755	+	0.846898i	$0.321533\pi$
$72$	0	0
$73$	−4.09396	−0.479161	−0.239581	−	0.970876i	$-0.577010\pi$
$73$	−4.09396	−0.479161	−0.239581	+	0.970876i	$0.577010\pi$
$74$	5.93872	0.690362
$75$	0	0
$76$	7.42378	0.851566
$77$	−3.16502	−0.360688
$78$	0	0
$79$	0.812615	0.0914263	0.0457131	−	0.998955i	$-0.485444\pi$
$79$	0.812615	0.0914263	0.0457131	+	0.998955i	$0.485444\pi$
$80$	−1.17781	−0.131684
$81$	0	0
$82$	24.4860	2.70403
$83$	−7.21351	−0.791786	−0.395893	−	0.918297i	$-0.629565\pi$
$83$	−7.21351	−0.791786	−0.395893	+	0.918297i	$0.629565\pi$
$84$	0	0
$85$	0	0
$86$	−7.74124	−0.834759
$87$	0	0
$88$	−0.407446	−0.0434339
$89$	9.77391	1.03603	0.518016	−	0.855371i	$-0.326671\pi$
$89$	9.77391	1.03603	0.518016	+	0.855371i	$0.326671\pi$
$90$	0	0
$91$	23.4308	2.45621
$92$	−8.07784	−0.842173
$93$	0	0
$94$	0.983662	0.101457
$95$	1.13729	0.116683
$96$	0	0
$97$	−1.77317	−0.180038	−0.0900192	−	0.995940i	$-0.528693\pi$
$97$	−1.77317	−0.180038	−0.0900192	+	0.995940i	$0.528693\pi$
$98$	−24.3965	−2.46442
$99$	0	0
$100$	−11.0762	−1.10762
$101$	14.6089	1.45364	0.726820	−	0.686828i	$-0.240998\pi$
$101$	14.6089	1.45364	0.726820	+	0.686828i	$0.240998\pi$
$102$	0	0
$103$	2.86271	0.282072	0.141036	−	0.990005i	$-0.454957\pi$
$103$	2.86271	0.282072	0.141036	+	0.990005i	$0.454957\pi$
$104$	3.01634	0.295776
$105$	0	0
$106$	−20.9389	−2.03377
$107$	6.31812	0.610796	0.305398	−	0.952225i	$-0.401211\pi$
$107$	6.31812	0.610796	0.305398	+	0.952225i	$0.401211\pi$
$108$	0	0
$109$	−4.69857	−0.450042	−0.225021	−	0.974354i	$-0.572245\pi$
$109$	−4.69857	−0.450042	−0.225021	+	0.974354i	$0.572245\pi$
$110$	−0.524507	−0.0500097
$111$	0	0
$112$	14.6867	1.38777
$113$	−12.4256	−1.16890	−0.584452	−	0.811429i	$-0.698690\pi$
$113$	−12.4256	−1.16890	−0.584452	+	0.811429i	$0.698690\pi$
$114$	0	0
$115$	−1.23748	−0.115396
$116$	9.94798	0.923647
$117$	0	0
$118$	10.9478	1.00782
$119$	0	0
$120$	0	0
$121$	−10.4673	−0.951576
$122$	−7.32983	−0.663612
$123$	0	0
$124$	−7.50162	−0.673665
$125$	−3.43571	−0.307299
$126$	0	0
$127$	−1.81048	−0.160654	−0.0803270	−	0.996769i	$-0.525596\pi$
$127$	−1.81048	−0.160654	−0.0803270	+	0.996769i	$0.525596\pi$
$128$	4.42539	0.391153
$129$	0	0
$130$	3.88294	0.340556
$131$	−6.38669	−0.558008	−0.279004	−	0.960290i	$-0.590004\pi$
$131$	−6.38669	−0.558008	−0.279004	+	0.960290i	$0.590004\pi$
$132$	0	0
$133$	−14.1814	−1.22968
$134$	3.21673	0.277883
$135$	0	0
$136$	0	0
$137$	2.28156	0.194927	0.0974633	−	0.995239i	$-0.468927\pi$
$137$	2.28156	0.194927	0.0974633	+	0.995239i	$0.468927\pi$
$138$	0	0
$139$	16.6247	1.41009	0.705044	−	0.709163i	$-0.250927\pi$
$139$	16.6247	1.41009	0.705044	+	0.709163i	$0.250927\pi$
$140$	−3.42378	−0.289363
$141$	0	0
$142$	−18.5180	−1.55399
$143$	−3.94335	−0.329760
$144$	0	0
$145$	1.52398	0.126560
$146$	8.45990	0.700146
$147$	0	0
$148$	−6.52420	−0.536286
$149$	12.1814	0.997936	0.498968	−	0.866620i	$-0.333713\pi$
$149$	12.1814	0.997936	0.498968	+	0.866620i	$0.333713\pi$
$150$	0	0
$151$	19.3823	1.57731	0.788654	−	0.614837i	$-0.210778\pi$
$151$	19.3823	1.57731	0.788654	+	0.614837i	$0.210778\pi$
$152$	−1.82562	−0.148078
$153$	0	0
$154$	6.54032	0.527034
$155$	−1.14921	−0.0923068
$156$	0	0
$157$	1.81475	0.144833	0.0724164	−	0.997374i	$-0.476929\pi$
$157$	1.81475	0.144833	0.0724164	+	0.997374i	$0.476929\pi$
$158$	−1.67922	−0.133591
$159$	0	0
$160$	2.82219	0.223113
$161$	15.4308	1.21611
$162$	0	0
$163$	−14.7091	−1.15210	−0.576052	−	0.817413i	$-0.695408\pi$
$163$	−14.7091	−1.15210	−0.576052	+	0.817413i	$0.695408\pi$
$164$	−26.9000	−2.10054
$165$	0	0
$166$	14.9063	1.15695
$167$	−8.03429	−0.621712	−0.310856	−	0.950457i	$-0.600616\pi$
$167$	−8.03429	−0.621712	−0.310856	+	0.950457i	$0.600616\pi$
$168$	0	0
$169$	16.1928	1.24560
$170$	0	0
$171$	0	0
$172$	8.50442	0.648456
$173$	12.0373	0.915179	0.457590	−	0.889163i	$-0.348713\pi$
$173$	12.0373	0.915179	0.457590	+	0.889163i	$0.348713\pi$
$174$	0	0
$175$	21.1585	1.59943
$176$	−2.47175	−0.186315
$177$	0	0
$178$	−20.1972	−1.51384
$179$	9.05223	0.676596	0.338298	−	0.941039i	$-0.390149\pi$
$179$	9.05223	0.676596	0.338298	+	0.941039i	$0.390149\pi$
$180$	0	0
$181$	9.60212	0.713720	0.356860	−	0.934158i	$-0.383847\pi$
$181$	9.60212	0.713720	0.356860	+	0.934158i	$0.383847\pi$
$182$	−48.4182	−3.58900
$183$	0	0
$184$	1.98647	0.146444
$185$	−0.999475	−0.0734828
$186$	0	0
$187$	0	0
$188$	−1.08064	−0.0788136
$189$	0	0
$190$	−2.35013	−0.170496
$191$	−18.9231	−1.36923	−0.684615	−	0.728905i	$-0.740029\pi$
$191$	−18.9231	−1.36923	−0.684615	+	0.728905i	$0.740029\pi$
$192$	0	0
$193$	−24.8450	−1.78838	−0.894190	−	0.447687i	$-0.852248\pi$
$193$	−24.8450	−1.78838	−0.894190	+	0.447687i	$0.852248\pi$
$194$	3.66415	0.263070
$195$	0	0
$196$	26.8017	1.91440
$197$	11.1762	0.796272	0.398136	−	0.917326i	$-0.369657\pi$
$197$	11.1762	0.796272	0.398136	+	0.917326i	$0.369657\pi$
$198$	0	0
$199$	−17.1268	−1.21409	−0.607045	−	0.794667i	$-0.707645\pi$
$199$	−17.1268	−1.21409	−0.607045	+	0.794667i	$0.707645\pi$
$200$	2.72382	0.192603
$201$	0	0
$202$	−30.1883	−2.12404
$203$	−19.0032	−1.33377
$204$	0	0
$205$	−4.12095	−0.287820
$206$	−5.91562	−0.412160
$207$	0	0
$208$	18.2984	1.26877
$209$	2.38669	0.165091
$210$	0	0
$211$	4.60462	0.316995	0.158498	−	0.987359i	$-0.449335\pi$
$211$	4.60462	0.316995	0.158498	+	0.987359i	$0.449335\pi$
$212$	23.0032	1.57987
$213$	0	0
$214$	−13.0560	−0.892490
$215$	1.30284	0.0888526
$216$	0	0
$217$	14.3300	0.972787
$218$	9.70931	0.657597
$219$	0	0
$220$	0.576216	0.0388485
$221$	0	0
$222$	0	0
$223$	−4.04355	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$223$	−4.04355	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$224$	−35.1911	−2.35131
$225$	0	0
$226$	25.6767	1.70799
$227$	17.3583	1.15211	0.576056	−	0.817410i	$-0.304591\pi$
$227$	17.3583	1.15211	0.576056	+	0.817410i	$0.304591\pi$
$228$	0	0
$229$	18.7418	1.23849	0.619245	−	0.785198i	$-0.287439\pi$
$229$	18.7418	1.23849	0.619245	+	0.785198i	$0.287439\pi$
$230$	2.55718	0.168616
$231$	0	0
$232$	−2.44636	−0.160612
$233$	9.78510	0.641043	0.320521	−	0.947241i	$-0.396142\pi$
$233$	9.78510	0.641043	0.320521	+	0.947241i	$0.396142\pi$
$234$	0	0
$235$	−0.165548	−0.0107992
$236$	−12.0271	−0.782896
$237$	0	0
$238$	0	0
$239$	14.2984	0.924888	0.462444	−	0.886648i	$-0.346973\pi$
$239$	14.2984	0.924888	0.462444	+	0.886648i	$0.346973\pi$
$240$	0	0
$241$	6.37551	0.410683	0.205341	−	0.978690i	$-0.434170\pi$
$241$	6.37551	0.410683	0.205341	+	0.978690i	$0.434170\pi$
$242$	21.6301	1.39043
$243$	0	0
$244$	8.05245	0.515505
$245$	4.10588	0.262315
$246$	0	0
$247$	−17.6688	−1.12424
$248$	1.84476	0.117143
$249$	0	0
$250$	7.09967	0.449023
$251$	27.4308	1.73141	0.865707	−	0.500550i	$-0.166869\pi$
$251$	27.4308	1.73141	0.865707	+	0.500550i	$0.166869\pi$
$252$	0	0
$253$	−2.59697	−0.163270
$254$	3.74124	0.234746
$255$	0	0
$256$	10.8464	0.677898
$257$	−0.657380	−0.0410062	−0.0205031	−	0.999790i	$-0.506527\pi$
$257$	−0.657380	−0.0410062	−0.0205031	+	0.999790i	$0.506527\pi$
$258$	0	0
$259$	12.4629	0.774408
$260$	−4.26575	−0.264550
$261$	0	0
$262$	13.1977	0.815357
$263$	9.50817	0.586299	0.293149	−	0.956067i	$-0.405297\pi$
$263$	9.50817	0.586299	0.293149	+	0.956067i	$0.405297\pi$
$264$	0	0
$265$	3.52398	0.216476
$266$	29.3049	1.79680
$267$	0	0
$268$	−3.53386	−0.215865
$269$	17.2775	1.05342	0.526712	−	0.850044i	$-0.323424\pi$
$269$	17.2775	1.05342	0.526712	+	0.850044i	$0.323424\pi$
$270$	0	0
$271$	21.2897	1.29326	0.646630	−	0.762804i	$-0.276178\pi$
$271$	21.2897	1.29326	0.646630	+	0.762804i	$0.276178\pi$
$272$	0	0
$273$	0	0
$274$	−4.71470	−0.284825
$275$	−3.56093	−0.214732
$276$	0	0
$277$	−11.3298	−0.680743	−0.340372	−	0.940291i	$-0.610553\pi$
$277$	−11.3298	−0.680743	−0.340372	+	0.940291i	$0.610553\pi$
$278$	−34.3539	−2.06041
$279$	0	0
$280$	0.841962	0.0503168
$281$	−19.0233	−1.13484	−0.567418	−	0.823430i	$-0.692058\pi$
$281$	−19.0233	−1.13484	−0.567418	+	0.823430i	$0.692058\pi$
$282$	0	0
$283$	11.7380	0.697753	0.348876	−	0.937169i	$-0.386563\pi$
$283$	11.7380	0.697753	0.348876	+	0.937169i	$0.386563\pi$
$284$	20.3436	1.20717
$285$	0	0
$286$	8.14869	0.481842
$287$	51.3860	3.03322
$288$	0	0
$289$	0	0
$290$	−3.14921	−0.184928
$291$	0	0
$292$	−9.29393	−0.543886
$293$	20.9548	1.22419	0.612095	−	0.790784i	$-0.290327\pi$
$293$	20.9548	1.22419	0.612095	+	0.790784i	$0.290327\pi$
$294$	0	0
$295$	−1.84249	−0.107274
$296$	1.60440	0.0932540
$297$	0	0
$298$	−25.1720	−1.45818
$299$	19.2254	1.11184
$300$	0	0
$301$	−16.2457	−0.936384
$302$	−40.0523	−2.30475
$303$	0	0
$304$	−11.0750	−0.635197
$305$	1.23360	0.0706355
$306$	0	0
$307$	33.4019	1.90634	0.953172	−	0.302428i	$-0.0977971\pi$
$307$	33.4019	1.90634	0.953172	+	0.302428i	$0.0977971\pi$
$308$	−7.18511	−0.409409
$309$	0	0
$310$	2.37477	0.134878
$311$	16.2434	0.921078	0.460539	−	0.887640i	$-0.347656\pi$
$311$	16.2434	0.921078	0.460539	+	0.887640i	$0.347656\pi$
$312$	0	0
$313$	−7.91531	−0.447400	−0.223700	−	0.974658i	$-0.571814\pi$
$313$	−7.91531	−0.447400	−0.223700	+	0.974658i	$0.571814\pi$
$314$	−3.75007	−0.211629
$315$	0	0
$316$	1.84476	0.103776
$317$	10.8030	0.606759	0.303380	−	0.952870i	$-0.401885\pi$
$317$	10.8030	0.606759	0.303380	+	0.952870i	$0.401885\pi$
$318$	0	0
$319$	3.19821	0.179065
$320$	−3.47624	−0.194328
$321$	0	0
$322$	−31.8867	−1.77698
$323$	0	0
$324$	0	0
$325$	26.3617	1.46228
$326$	30.3954	1.68345
$327$	0	0
$328$	6.61513	0.365260
$329$	2.06430	0.113809
$330$	0	0
$331$	22.4389	1.23336	0.616678	−	0.787215i	$-0.288478\pi$
$331$	22.4389	1.23336	0.616678	+	0.787215i	$0.288478\pi$
$332$	−16.3758	−0.898740
$333$	0	0
$334$	16.6023	0.908440
$335$	−0.541370	−0.0295782
$336$	0	0
$337$	−16.4622	−0.896752	−0.448376	−	0.893845i	$-0.647997\pi$
$337$	−16.4622	−0.896752	−0.448376	+	0.893845i	$0.647997\pi$
$338$	−33.4613	−1.82006
$339$	0	0
$340$	0	0
$341$	−2.41172	−0.130602
$342$	0	0
$343$	−20.8420	−1.12536
$344$	−2.09137	−0.112759
$345$	0	0
$346$	−24.8743	−1.33725
$347$	21.2778	1.14225	0.571126	−	0.820862i	$-0.306507\pi$
$347$	21.2778	1.14225	0.571126	+	0.820862i	$0.306507\pi$
$348$	0	0
$349$	6.41832	0.343565	0.171782	−	0.985135i	$-0.445047\pi$
$349$	6.41832	0.343565	0.171782	+	0.985135i	$0.445047\pi$
$350$	−43.7226	−2.33707
$351$	0	0
$352$	5.92260	0.315676
$353$	10.0201	0.533315	0.266658	−	0.963791i	$-0.414081\pi$
$353$	10.0201	0.533315	0.266658	+	0.963791i	$0.414081\pi$
$354$	0	0
$355$	3.11654	0.165409
$356$	22.1883	1.17598
$357$	0	0
$358$	−18.7059	−0.988636
$359$	30.1366	1.59055	0.795275	−	0.606248i	$-0.207326\pi$
$359$	30.1366	1.59055	0.795275	+	0.606248i	$0.207326\pi$
$360$	0	0
$361$	−8.30606	−0.437161
$362$	−19.8422	−1.04288
$363$	0	0
$364$	53.1916	2.78800
$365$	−1.42378	−0.0745243
$366$	0	0
$367$	24.8365	1.29645	0.648226	−	0.761448i	$-0.275511\pi$
$367$	24.8365	1.29645	0.648226	+	0.761448i	$0.275511\pi$
$368$	12.0508	0.628190
$369$	0	0
$370$	2.06535	0.107372
$371$	−43.9422	−2.28136
$372$	0	0
$373$	−6.09593	−0.315635	−0.157818	−	0.987468i	$-0.550446\pi$
$373$	−6.09593	−0.315635	−0.157818	+	0.987468i	$0.550446\pi$
$374$	0	0
$375$	0	0
$376$	0.265746	0.0137048
$377$	−23.6764	−1.21940
$378$	0	0
$379$	−27.9623	−1.43632	−0.718162	−	0.695876i	$-0.755017\pi$
$379$	−27.9623	−1.43632	−0.718162	+	0.695876i	$0.755017\pi$
$380$	2.58182	0.132445
$381$	0	0
$382$	39.1034	2.00071
$383$	36.1155	1.84542	0.922708	−	0.385500i	$-0.125971\pi$
$383$	36.1155	1.84542	0.922708	+	0.385500i	$0.125971\pi$
$384$	0	0
$385$	−1.10072	−0.0560980
$386$	51.3406	2.61317
$387$	0	0
$388$	−4.02538	−0.204358
$389$	−23.1246	−1.17246	−0.586231	−	0.810144i	$-0.699389\pi$
$389$	−23.1246	−1.17246	−0.586231	+	0.810144i	$0.699389\pi$
$390$	0	0
$391$	0	0
$392$	−6.59094	−0.332893
$393$	0	0
$394$	−23.0949	−1.16351
$395$	0.282609	0.0142196
$396$	0	0
$397$	−4.21329	−0.211459	−0.105730	−	0.994395i	$-0.533718\pi$
$397$	−4.21329	−0.211459	−0.105730	+	0.994395i	$0.533718\pi$
$398$	35.3915	1.77402
$399$	0	0
$400$	16.5239	0.826193
$401$	−6.19254	−0.309241	−0.154620	−	0.987974i	$-0.549415\pi$
$401$	−6.19254	−0.309241	−0.154620	+	0.987974i	$0.549415\pi$
$402$	0	0
$403$	17.8540	0.889372
$404$	33.1645	1.65000
$405$	0	0
$406$	39.2690	1.94889
$407$	−2.09748	−0.103968
$408$	0	0
$409$	−34.6034	−1.71103	−0.855515	−	0.517778i	$-0.826759\pi$
$409$	−34.6034	−1.71103	−0.855515	+	0.517778i	$0.826759\pi$
$410$	8.51568	0.420559
$411$	0	0
$412$	6.49882	0.320174
$413$	22.9748	1.13052
$414$	0	0
$415$	−2.50869	−0.123147
$416$	−43.8452	−2.14969
$417$	0	0
$418$	−4.93195	−0.241230
$419$	5.83550	0.285083	0.142541	−	0.989789i	$-0.454473\pi$
$419$	5.83550	0.285083	0.142541	+	0.989789i	$0.454473\pi$
$420$	0	0
$421$	27.0152	1.31664	0.658319	−	0.752739i	$-0.271268\pi$
$421$	27.0152	1.31664	0.658319	+	0.752739i	$0.271268\pi$
$422$	−9.51515	−0.463190
$423$	0	0
$424$	−5.65685	−0.274721
$425$	0	0
$426$	0	0
$427$	−15.3823	−0.744401
$428$	14.3431	0.693302
$429$	0	0
$430$	−2.69223	−0.129831
$431$	−17.3371	−0.835100	−0.417550	−	0.908654i	$-0.637111\pi$
$431$	−17.3371	−0.835100	−0.417550	+	0.908654i	$0.637111\pi$
$432$	0	0
$433$	3.97603	0.191076	0.0955378	−	0.995426i	$-0.469543\pi$
$433$	3.97603	0.191076	0.0955378	+	0.995426i	$0.469543\pi$
$434$	−29.6121	−1.42143
$435$	0	0
$436$	−10.6665	−0.510833
$437$	−11.6361	−0.556630
$438$	0	0
$439$	2.79680	0.133484	0.0667420	−	0.997770i	$-0.478740\pi$
$439$	2.79680	0.133484	0.0667420	+	0.997770i	$0.478740\pi$
$440$	−0.141700	−0.00675531
$441$	0	0
$442$	0	0
$443$	−9.88565	−0.469682	−0.234841	−	0.972034i	$-0.575457\pi$
$443$	−9.88565	−0.469682	−0.234841	+	0.972034i	$0.575457\pi$
$444$	0	0
$445$	3.39914	0.161135
$446$	8.35574	0.395656
$447$	0	0
$448$	43.3468	2.04794
$449$	−40.0492	−1.89004	−0.945020	−	0.327012i	$-0.893958\pi$
$449$	−40.0492	−1.89004	−0.945020	+	0.327012i	$0.893958\pi$
$450$	0	0
$451$	−8.64817	−0.407226
$452$	−28.2081	−1.32680
$453$	0	0
$454$	−35.8698	−1.68346
$455$	8.14869	0.382016
$456$	0	0
$457$	−0.452046	−0.0211458	−0.0105729	−	0.999944i	$-0.503366\pi$
$457$	−0.452046	−0.0211458	−0.0105729	+	0.999944i	$0.503366\pi$
$458$	−38.7287	−1.80967
$459$	0	0
$460$	−2.80929	−0.130984
$461$	−11.5310	−0.537051	−0.268525	−	0.963273i	$-0.586536\pi$
$461$	−11.5310	−0.537051	−0.268525	+	0.963273i	$0.586536\pi$
$462$	0	0
$463$	4.41067	0.204981	0.102491	−	0.994734i	$-0.467319\pi$
$463$	4.41067	0.204981	0.102491	+	0.994734i	$0.467319\pi$
$464$	−14.8407	−0.688963
$465$	0	0
$466$	−20.2203	−0.936686
$467$	−8.34586	−0.386200	−0.193100	−	0.981179i	$-0.561854\pi$
$467$	−8.34586	−0.386200	−0.193100	+	0.981179i	$0.561854\pi$
$468$	0	0
$469$	6.75059	0.311713
$470$	0.342095	0.0157797
$471$	0	0
$472$	2.95764	0.136137
$473$	2.73411	0.125715
$474$	0	0
$475$	−15.9553	−0.732078
$476$	0	0
$477$	0	0
$478$	−29.5468	−1.35144
$479$	12.7630	0.583154	0.291577	−	0.956547i	$-0.405820\pi$
$479$	12.7630	0.583154	0.291577	+	0.956547i	$0.405820\pi$
$480$	0	0
$481$	15.5277	0.708004
$482$	−13.1746	−0.600086
$483$	0	0
$484$	−23.7625	−1.08011
$485$	−0.616669	−0.0280015
$486$	0	0
$487$	−12.8442	−0.582028	−0.291014	−	0.956719i	$-0.593993\pi$
$487$	−12.8442	−0.582028	−0.291014	+	0.956719i	$0.593993\pi$
$488$	−1.98022	−0.0896405
$489$	0	0
$490$	−8.48454	−0.383292
$491$	20.4629	0.923479	0.461739	−	0.887016i	$-0.347226\pi$
$491$	20.4629	0.923479	0.461739	+	0.887016i	$0.347226\pi$
$492$	0	0
$493$	0	0
$494$	36.5114	1.64273
$495$	0	0
$496$	11.1911	0.502497
$497$	−38.8615	−1.74318
$498$	0	0
$499$	7.08491	0.317164	0.158582	−	0.987346i	$-0.449308\pi$
$499$	7.08491	0.317164	0.158582	+	0.987346i	$0.449308\pi$
$500$	−7.79961	−0.348809
$501$	0	0
$502$	−56.6839	−2.52993
$503$	−18.3274	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$503$	−18.3274	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$504$	0	0
$505$	5.08064	0.226085
$506$	5.36647	0.238569
$507$	0	0
$508$	−4.11008	−0.182355
$509$	−9.62950	−0.426820	−0.213410	−	0.976963i	$-0.568457\pi$
$509$	−9.62950	−0.426820	−0.213410	+	0.976963i	$0.568457\pi$
$510$	0	0
$511$	17.7538	0.785383
$512$	−31.2641	−1.38169
$513$	0	0
$514$	1.35843	0.0599179
$515$	0.995586	0.0438708
$516$	0	0
$517$	−0.347418	−0.0152794
$518$	−25.7538	−1.13156
$519$	0	0
$520$	1.04901	0.0460023
$521$	3.24478	0.142156	0.0710781	−	0.997471i	$-0.477356\pi$
$521$	3.24478	0.142156	0.0710781	+	0.997471i	$0.477356\pi$
$522$	0	0
$523$	−6.77444	−0.296226	−0.148113	−	0.988970i	$-0.547320\pi$
$523$	−6.77444	−0.296226	−0.148113	+	0.988970i	$0.547320\pi$
$524$	−14.4988	−0.633384
$525$	0	0
$526$	−19.6480	−0.856695
$527$	0	0
$528$	0	0
$529$	−10.3387	−0.449510
$530$	−7.28208	−0.316313
$531$	0	0
$532$	−32.1940	−1.39578
$533$	64.0227	2.77313
$534$	0	0
$535$	2.19730	0.0949975
$536$	0.869031	0.0375364
$537$	0	0
$538$	−35.7028	−1.53926
$539$	8.61654	0.371141
$540$	0	0
$541$	−24.6277	−1.05883	−0.529414	−	0.848363i	$-0.677588\pi$
$541$	−24.6277	−1.05883	−0.529414	+	0.848363i	$0.677588\pi$
$542$	−43.9939	−1.88970
$543$	0	0
$544$	0	0
$545$	−1.63406	−0.0699953
$546$	0	0
$547$	32.1356	1.37402	0.687009	−	0.726649i	$-0.258923\pi$
$547$	32.1356	1.37402	0.687009	+	0.726649i	$0.258923\pi$
$548$	5.17950	0.221257
$549$	0	0
$550$	7.35843	0.313765
$551$	14.3300	0.610480
$552$	0	0
$553$	−3.52398	−0.149855
$554$	23.4124	0.994696
$555$	0	0
$556$	37.7407	1.60056
$557$	−43.4415	−1.84068	−0.920338	−	0.391124i	$-0.872086\pi$
$557$	−43.4415	−1.84068	−0.920338	+	0.391124i	$0.872086\pi$
$558$	0	0
$559$	−20.2407	−0.856091
$560$	5.10771	0.215840
$561$	0	0
$562$	39.3105	1.65821
$563$	−14.3986	−0.606829	−0.303415	−	0.952859i	$-0.598127\pi$
$563$	−14.3986	−0.606829	−0.303415	+	0.952859i	$0.598127\pi$
$564$	0	0
$565$	−4.32134	−0.181800
$566$	−24.2559	−1.01955
$567$	0	0
$568$	−5.00280	−0.209913
$569$	−16.8103	−0.704726	−0.352363	−	0.935863i	$-0.614622\pi$
$569$	−16.8103	−0.704726	−0.352363	+	0.935863i	$0.614622\pi$
$570$	0	0
$571$	−22.6928	−0.949663	−0.474831	−	0.880077i	$-0.657491\pi$
$571$	−22.6928	−0.949663	−0.474831	+	0.880077i	$0.657491\pi$
$572$	−8.95204	−0.374303
$573$	0	0
$574$	−106.186	−4.43212
$575$	17.3610	0.724002
$576$	0	0
$577$	−20.5881	−0.857095	−0.428548	−	0.903519i	$-0.640975\pi$
$577$	−20.5881	−0.857095	−0.428548	+	0.903519i	$0.640975\pi$
$578$	0	0
$579$	0	0
$580$	3.45968	0.143655
$581$	31.2821	1.29780
$582$	0	0
$583$	7.39538	0.306285
$584$	2.28552	0.0945756
$585$	0	0
$586$	−43.3017	−1.78878
$587$	−10.5119	−0.433873	−0.216937	−	0.976186i	$-0.569606\pi$
$587$	−10.5119	−0.433873	−0.216937	+	0.976186i	$0.569606\pi$
$588$	0	0
$589$	−10.8061	−0.445256
$590$	3.80738	0.156747
$591$	0	0
$592$	9.73300	0.400024
$593$	−7.79428	−0.320073	−0.160036	−	0.987111i	$-0.551161\pi$
$593$	−7.79428	−0.320073	−0.160036	+	0.987111i	$0.551161\pi$
$594$	0	0
$595$	0	0
$596$	27.6536	1.13274
$597$	0	0
$598$	−39.7281	−1.62460
$599$	−16.8503	−0.688484	−0.344242	−	0.938881i	$-0.611864\pi$
$599$	−16.8503	−0.688484	−0.344242	+	0.938881i	$0.611864\pi$
$600$	0	0
$601$	30.3662	1.23866	0.619331	−	0.785130i	$-0.287404\pi$
$601$	30.3662	1.23866	0.619331	+	0.785130i	$0.287404\pi$
$602$	33.5706	1.36824
$603$	0	0
$604$	44.0009	1.79037
$605$	−3.64030	−0.147999
$606$	0	0
$607$	25.0467	1.01662	0.508308	−	0.861175i	$-0.330271\pi$
$607$	25.0467	1.01662	0.508308	+	0.861175i	$0.330271\pi$
$608$	26.5371	1.07622
$609$	0	0
$610$	−2.54915	−0.103212
$611$	2.57194	0.104050
$612$	0	0
$613$	−31.5000	−1.27227	−0.636137	−	0.771576i	$-0.719469\pi$
$613$	−31.5000	−1.27227	−0.636137	+	0.771576i	$0.719469\pi$
$614$	−69.0228	−2.78553
$615$	0	0
$616$	1.76693	0.0711916
$617$	−11.5547	−0.465174	−0.232587	−	0.972576i	$-0.574719\pi$
$617$	−11.5547	−0.465174	−0.232587	+	0.972576i	$0.574719\pi$
$618$	0	0
$619$	−23.7249	−0.953585	−0.476792	−	0.879016i	$-0.658201\pi$
$619$	−23.7249	−0.953585	−0.476792	+	0.879016i	$0.658201\pi$
$620$	−2.60889	−0.104776
$621$	0	0
$622$	−33.5659	−1.34587
$623$	−42.3855	−1.69814
$624$	0	0
$625$	23.2004	0.928016
$626$	16.3565	0.653736
$627$	0	0
$628$	4.11977	0.164397
$629$	0	0
$630$	0	0
$631$	36.8092	1.46535	0.732675	−	0.680579i	$-0.238272\pi$
$631$	36.8092	1.46535	0.732675	+	0.680579i	$0.238272\pi$
$632$	−0.453656	−0.0180455
$633$	0	0
$634$	−22.3238	−0.886592
$635$	−0.629643	−0.0249866
$636$	0	0
$637$	−63.7886	−2.52739
$638$	−6.60889	−0.261649
$639$	0	0
$640$	1.53905	0.0608363
$641$	−35.7986	−1.41396	−0.706980	−	0.707233i	$-0.749943\pi$
$641$	−35.7986	−1.41396	−0.706980	+	0.707233i	$0.749943\pi$
$642$	0	0
$643$	16.6863	0.658043	0.329022	−	0.944322i	$-0.393281\pi$
$643$	16.6863	0.658043	0.329022	+	0.944322i	$0.393281\pi$
$644$	35.0303	1.38039
$645$	0	0
$646$	0	0
$647$	−19.2205	−0.755636	−0.377818	−	0.925880i	$-0.623325\pi$
$647$	−19.2205	−0.755636	−0.377818	+	0.925880i	$0.623325\pi$
$648$	0	0
$649$	−3.86662	−0.151778
$650$	−54.4747	−2.13667
$651$	0	0
$652$	−33.3920	−1.30773
$653$	28.0597	1.09806	0.549030	−	0.835803i	$-0.314997\pi$
$653$	28.0597	1.09806	0.549030	+	0.835803i	$0.314997\pi$
$654$	0	0
$655$	−2.22115	−0.0867874
$656$	40.1303	1.56682
$657$	0	0
$658$	−4.26575	−0.166296
$659$	27.6727	1.07797	0.538987	−	0.842314i	$-0.318807\pi$
$659$	27.6727	1.07797	0.538987	+	0.842314i	$0.318807\pi$
$660$	0	0
$661$	−13.8137	−0.537291	−0.268645	−	0.963239i	$-0.586576\pi$
$661$	−13.8137	−0.537291	−0.268645	+	0.963239i	$0.586576\pi$
$662$	−46.3687	−1.80217
$663$	0	0
$664$	4.02707	0.156281
$665$	−4.93195	−0.191253
$666$	0	0
$667$	−15.5926	−0.603746
$668$	−18.2391	−0.705692
$669$	0	0
$670$	1.11871	0.0432194
$671$	2.58881	0.0999398
$672$	0	0
$673$	−19.5639	−0.754135	−0.377067	−	0.926186i	$-0.623067\pi$
$673$	−19.5639	−0.754135	−0.377067	+	0.926186i	$0.623067\pi$
$674$	34.0180	1.31033
$675$	0	0
$676$	36.7601	1.41385
$677$	−34.0377	−1.30818	−0.654088	−	0.756418i	$-0.726948\pi$
$677$	−34.0377	−1.30818	−0.654088	+	0.756418i	$0.726948\pi$
$678$	0	0
$679$	7.68953	0.295097
$680$	0	0
$681$	0	0
$682$	4.98366	0.190834
$683$	6.68407	0.255759	0.127879	−	0.991790i	$-0.459183\pi$
$683$	6.68407	0.255759	0.127879	+	0.991790i	$0.459183\pi$
$684$	0	0
$685$	0.793474	0.0303171
$686$	43.0686	1.64437
$687$	0	0
$688$	−12.6872	−0.483694
$689$	−54.7482	−2.08574
$690$	0	0
$691$	−35.0555	−1.33357	−0.666787	−	0.745248i	$-0.732331\pi$
$691$	−35.0555	−1.33357	−0.666787	+	0.745248i	$0.732331\pi$
$692$	27.3266	1.03880
$693$	0	0
$694$	−43.9692	−1.66905
$695$	5.78169	0.219312
$696$	0	0
$697$	0	0
$698$	−13.2630	−0.502014
$699$	0	0
$700$	48.0331	1.81548
$701$	39.3006	1.48436	0.742182	−	0.670199i	$-0.233791\pi$
$701$	39.3006	1.48436	0.742182	+	0.670199i	$0.233791\pi$
$702$	0	0
$703$	−9.39809	−0.354456
$704$	−7.29518	−0.274948
$705$	0	0
$706$	−20.7059	−0.779276
$707$	−63.3529	−2.38263
$708$	0	0
$709$	12.2095	0.458536	0.229268	−	0.973363i	$-0.426367\pi$
$709$	12.2095	0.458536	0.229268	+	0.973363i	$0.426367\pi$
$710$	−6.44012	−0.241693
$711$	0	0
$712$	−5.45646	−0.204489
$713$	11.7581	0.440344
$714$	0	0
$715$	−1.37141	−0.0512877
$716$	20.5500	0.767990
$717$	0	0
$718$	−62.2754	−2.32410
$719$	−31.2379	−1.16498	−0.582489	−	0.812839i	$-0.697921\pi$
$719$	−31.2379	−1.16498	−0.582489	+	0.812839i	$0.697921\pi$
$720$	0	0
$721$	−12.4144	−0.462338
$722$	17.1639	0.638776
$723$	0	0
$724$	21.7984	0.810129
$725$	−21.3803	−0.794045
$726$	0	0
$727$	−36.9344	−1.36982	−0.684910	−	0.728627i	$-0.740159\pi$
$727$	−36.9344	−1.36982	−0.684910	+	0.728627i	$0.740159\pi$
$728$	−13.0806	−0.484801
$729$	0	0
$730$	2.94216	0.108894
$731$	0	0
$732$	0	0
$733$	8.29737	0.306470	0.153235	−	0.988190i	$-0.451031\pi$
$733$	8.29737	0.306470	0.153235	+	0.988190i	$0.451031\pi$
$734$	−51.3230	−1.89437
$735$	0	0
$736$	−28.8751	−1.06435
$737$	−1.13611	−0.0418492
$738$	0	0
$739$	−21.2613	−0.782111	−0.391055	−	0.920367i	$-0.627890\pi$
$739$	−21.2613	−0.782111	−0.391055	+	0.920367i	$0.627890\pi$
$740$	−2.26897	−0.0834089
$741$	0	0
$742$	90.8037	3.33351
$743$	48.3827	1.77499	0.887495	−	0.460817i	$-0.152444\pi$
$743$	48.3827	1.77499	0.887495	+	0.460817i	$0.152444\pi$
$744$	0	0
$745$	4.23640	0.155210
$746$	12.5968	0.461203
$747$	0	0
$748$	0	0
$749$	−27.3991	−1.00114
$750$	0	0
$751$	29.2270	1.06651	0.533255	−	0.845955i	$-0.320969\pi$
$751$	29.2270	1.06651	0.533255	+	0.845955i	$0.320969\pi$
$752$	1.61213	0.0587883
$753$	0	0
$754$	48.9258	1.78177
$755$	6.74071	0.245320
$756$	0	0
$757$	42.3932	1.54081	0.770403	−	0.637557i	$-0.220055\pi$
$757$	42.3932	1.54081	0.770403	+	0.637557i	$0.220055\pi$
$758$	57.7822	2.09875
$759$	0	0
$760$	−0.634910	−0.0230306
$761$	15.1503	0.549197	0.274598	−	0.961559i	$-0.411455\pi$
$761$	15.1503	0.549197	0.274598	+	0.961559i	$0.411455\pi$
$762$	0	0
$763$	20.3758	0.737654
$764$	−42.9585	−1.55418
$765$	0	0
$766$	−74.6304	−2.69651
$767$	28.6247	1.03358
$768$	0	0
$769$	28.3540	1.02247	0.511236	−	0.859440i	$-0.329188\pi$
$769$	28.3540	1.02247	0.511236	+	0.859440i	$0.329188\pi$
$770$	2.27457	0.0819699
$771$	0	0
$772$	−56.4021	−2.02996
$773$	21.2004	0.762526	0.381263	−	0.924467i	$-0.375489\pi$
$773$	21.2004	0.762526	0.381263	+	0.924467i	$0.375489\pi$
$774$	0	0
$775$	16.1226	0.579139
$776$	0.989904	0.0355355
$777$	0	0
$778$	47.7855	1.71319
$779$	−38.7494	−1.38834
$780$	0	0
$781$	6.54032	0.234031
$782$	0	0
$783$	0	0
$784$	−39.9835	−1.42798
$785$	0.631129	0.0225260
$786$	0	0
$787$	36.0185	1.28392	0.641961	−	0.766737i	$-0.278121\pi$
$787$	36.0185	1.28392	0.641961	+	0.766737i	$0.278121\pi$
$788$	25.3718	0.903832
$789$	0	0
$790$	−0.583993	−0.0207775
$791$	53.8849	1.91593
$792$	0	0
$793$	−19.1650	−0.680570
$794$	8.70650	0.308982
$795$	0	0
$796$	−38.8807	−1.37809
$797$	0.536047	0.0189878	0.00949388	−	0.999955i	$-0.496978\pi$
$797$	0.536047	0.0189878	0.00949388	+	0.999955i	$0.496978\pi$
$798$	0	0
$799$	0	0
$800$	−39.5931	−1.39983
$801$	0	0
$802$	12.7965	0.451860
$803$	−2.98793	−0.105442
$804$	0	0
$805$	5.36647	0.189143
$806$	−36.8942	−1.29954
$807$	0	0
$808$	−8.15567	−0.286915
$809$	4.96166	0.174443	0.0872213	−	0.996189i	$-0.472201\pi$
$809$	4.96166	0.174443	0.0872213	+	0.996189i	$0.472201\pi$
$810$	0	0
$811$	46.5272	1.63379	0.816896	−	0.576785i	$-0.195693\pi$
$811$	46.5272	1.63379	0.816896	+	0.576785i	$0.195693\pi$
$812$	−43.1404	−1.51393
$813$	0	0
$814$	4.33432	0.151918
$815$	−5.11548	−0.179188
$816$	0	0
$817$	12.2506	0.428594
$818$	71.5058	2.50014
$819$	0	0
$820$	−9.35521	−0.326698
$821$	31.3048	1.09255	0.546273	−	0.837607i	$-0.316046\pi$
$821$	31.3048	1.09255	0.546273	+	0.837607i	$0.316046\pi$
$822$	0	0
$823$	45.6568	1.59150	0.795748	−	0.605628i	$-0.207078\pi$
$823$	45.6568	1.59150	0.795748	+	0.605628i	$0.207078\pi$
$824$	−1.59816	−0.0556745
$825$	0	0
$826$	−47.4760	−1.65190
$827$	−0.729840	−0.0253790	−0.0126895	−	0.999919i	$-0.504039\pi$
$827$	−0.729840	−0.0253790	−0.0126895	+	0.999919i	$0.504039\pi$
$828$	0	0
$829$	−24.0881	−0.836616	−0.418308	−	0.908305i	$-0.637377\pi$
$829$	−24.0881	−0.836616	−0.418308	+	0.908305i	$0.637377\pi$
$830$	5.18406	0.179941
$831$	0	0
$832$	54.0065	1.87234
$833$	0	0
$834$	0	0
$835$	−2.79414	−0.0966952
$836$	5.41818	0.187392
$837$	0	0
$838$	−12.0587	−0.416561
$839$	−32.6344	−1.12666	−0.563331	−	0.826231i	$-0.690481\pi$
$839$	−32.6344	−1.12666	−0.563331	+	0.826231i	$0.690481\pi$
$840$	0	0
$841$	−9.79752	−0.337846
$842$	−55.8251	−1.92386
$843$	0	0
$844$	10.4532	0.359815
$845$	5.63147	0.193729
$846$	0	0
$847$	45.3926	1.55971
$848$	−34.3169	−1.17845
$849$	0	0
$850$	0	0
$851$	10.2261	0.350546
$852$	0	0
$853$	−23.7844	−0.814363	−0.407182	−	0.913347i	$-0.633488\pi$
$853$	−23.7844	−0.814363	−0.407182	+	0.913347i	$0.633488\pi$
$854$	31.7865	1.08771
$855$	0	0
$856$	−3.52720	−0.120557
$857$	−43.0095	−1.46918	−0.734588	−	0.678514i	$-0.762624\pi$
$857$	−43.0095	−1.46918	−0.734588	+	0.678514i	$0.762624\pi$
$858$	0	0
$859$	5.99573	0.204572	0.102286	−	0.994755i	$-0.467384\pi$
$859$	5.99573	0.204572	0.102286	+	0.994755i	$0.467384\pi$
$860$	2.95764	0.100855
$861$	0	0
$862$	35.8261	1.22024
$863$	−54.1552	−1.84346	−0.921732	−	0.387828i	$-0.873225\pi$
$863$	−54.1552	−1.84346	−0.921732	+	0.387828i	$0.873225\pi$
$864$	0	0
$865$	4.18630	0.142338
$866$	−8.21621	−0.279198
$867$	0	0
$868$	32.5315	1.10419
$869$	0.593079	0.0201188
$870$	0	0
$871$	8.41067	0.284985
$872$	2.62306	0.0888281
$873$	0	0
$874$	24.0453	0.813343
$875$	14.8993	0.503688
$876$	0	0
$877$	26.7159	0.902131	0.451065	−	0.892491i	$-0.351044\pi$
$877$	26.7159	0.902131	0.451065	+	0.892491i	$0.351044\pi$
$878$	−5.77941	−0.195046
$879$	0	0
$880$	−0.859617	−0.0289777
$881$	−12.1210	−0.408368	−0.204184	−	0.978933i	$-0.565454\pi$
$881$	−12.1210	−0.408368	−0.204184	+	0.978933i	$0.565454\pi$
$882$	0	0
$883$	−44.5718	−1.49996	−0.749981	−	0.661460i	$-0.769937\pi$
$883$	−44.5718	−1.49996	−0.749981	+	0.661460i	$0.769937\pi$
$884$	0	0
$885$	0	0
$886$	20.4281	0.686295
$887$	−21.9938	−0.738481	−0.369240	−	0.929334i	$-0.620382\pi$
$887$	−21.9938	−0.738481	−0.369240	+	0.929334i	$0.620382\pi$
$888$	0	0
$889$	7.85131	0.263325
$890$	−7.02411	−0.235449
$891$	0	0
$892$	−9.17950	−0.307352
$893$	−1.55666	−0.0520915
$894$	0	0
$895$	3.14816	0.105231
$896$	−19.1911	−0.641131
$897$	0	0
$898$	82.7592	2.76171
$899$	−14.4803	−0.482945
$900$	0	0
$901$	0	0
$902$	17.8709	0.595036
$903$	0	0
$904$	6.93682	0.230715
$905$	3.33940	0.111005
$906$	0	0
$907$	15.9700	0.530276	0.265138	−	0.964210i	$-0.414582\pi$
$907$	15.9700	0.530276	0.265138	+	0.964210i	$0.414582\pi$
$908$	39.4061	1.30774
$909$	0	0
$910$	−16.8387	−0.558199
$911$	−38.0408	−1.26035	−0.630173	−	0.776454i	$-0.717016\pi$
$911$	−38.0408	−1.26035	−0.630173	+	0.776454i	$0.717016\pi$
$912$	0	0
$913$	−5.26471	−0.174237
$914$	0.934124	0.0308981
$915$	0	0
$916$	42.5468	1.40579
$917$	27.6965	0.914619
$918$	0	0
$919$	7.23321	0.238602	0.119301	−	0.992858i	$-0.461935\pi$
$919$	7.23321	0.238602	0.119301	+	0.992858i	$0.461935\pi$
$920$	0.690847	0.0227766
$921$	0	0
$922$	23.8280	0.784734
$923$	−48.4182	−1.59370
$924$	0	0
$925$	14.0219	0.461037
$926$	−9.11436	−0.299517
$927$	0	0
$928$	35.5601	1.16732
$929$	−44.1930	−1.44992	−0.724962	−	0.688789i	$-0.758143\pi$
$929$	−44.1930	−1.44992	−0.724962	+	0.688789i	$0.758143\pi$
$930$	0	0
$931$	38.6077	1.26532
$932$	22.2137	0.727635
$933$	0	0
$934$	17.2462	0.564312
$935$	0	0
$936$	0	0
$937$	−22.5141	−0.735504	−0.367752	−	0.929924i	$-0.619873\pi$
$937$	−22.5141	−0.735504	−0.367752	+	0.929924i	$0.619873\pi$
$938$	−13.9497	−0.455473
$939$	0	0
$940$	−0.375821	−0.0122579
$941$	2.23138	0.0727410	0.0363705	−	0.999338i	$-0.488420\pi$
$941$	2.23138	0.0727410	0.0363705	+	0.999338i	$0.488420\pi$
$942$	0	0
$943$	42.1633	1.37303
$944$	17.9424	0.583974
$945$	0	0
$946$	−5.64987	−0.183693
$947$	−13.8093	−0.448742	−0.224371	−	0.974504i	$-0.572033\pi$
$947$	−13.8093	−0.448742	−0.224371	+	0.974504i	$0.572033\pi$
$948$	0	0
$949$	22.1198	0.718038
$950$	32.9706	1.06971
$951$	0	0
$952$	0	0
$953$	7.23736	0.234441	0.117221	−	0.993106i	$-0.462602\pi$
$953$	7.23736	0.234441	0.117221	+	0.993106i	$0.462602\pi$
$954$	0	0
$955$	−6.58103	−0.212957
$956$	32.4597	1.04982
$957$	0	0
$958$	−26.3738	−0.852100
$959$	−9.89420	−0.319500
$960$	0	0
$961$	−20.0806	−0.647763
$962$	−32.0871	−1.03453
$963$	0	0
$964$	14.4734	0.466158
$965$	−8.64052	−0.278148
$966$	0	0
$967$	−10.6240	−0.341646	−0.170823	−	0.985302i	$-0.554643\pi$
$967$	−10.6240	−0.341646	−0.170823	+	0.985302i	$0.554643\pi$
$968$	5.84357	0.187820
$969$	0	0
$970$	1.27431	0.0409155
$971$	25.6491	0.823118	0.411559	−	0.911383i	$-0.364984\pi$
$971$	25.6491	0.823118	0.411559	+	0.911383i	$0.364984\pi$
$972$	0	0
$973$	−72.0946	−2.31125
$974$	26.5418	0.850454
$975$	0	0
$976$	−12.0129	−0.384524
$977$	−17.2663	−0.552397	−0.276198	−	0.961101i	$-0.589075\pi$
$977$	−17.2663	−0.552397	−0.276198	+	0.961101i	$0.589075\pi$
$978$	0	0
$979$	7.13340	0.227984
$980$	9.32100	0.297748
$981$	0	0
$982$	−42.2853	−1.34938
$983$	34.9960	1.11620	0.558100	−	0.829774i	$-0.311531\pi$
$983$	34.9960	1.11620	0.558100	+	0.829774i	$0.311531\pi$
$984$	0	0
$985$	3.88683	0.123845
$986$	0	0
$987$	0	0
$988$	−40.1109	−1.27610
$989$	−13.3299	−0.423866
$990$	0	0
$991$	22.9208	0.728105	0.364052	−	0.931378i	$-0.381393\pi$
$991$	22.9208	0.728105	0.364052	+	0.931378i	$0.381393\pi$
$992$	−26.8153	−0.851388
$993$	0	0
$994$	80.3049	2.54712
$995$	−5.95633	−0.188828
$996$	0	0
$997$	59.8236	1.89463	0.947316	−	0.320301i	$-0.103784\pi$
$997$	59.8236	1.89463	0.947316	+	0.320301i	$0.103784\pi$
$998$	−14.6405	−0.463437
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bf.1.1		4
3.2	odd	2		867.2.a.k.1.4		4
17.8	even	8		153.2.f.b.64.4		8
17.15	even	8		153.2.f.b.55.1		8
17.16	even	2		2601.2.a.be.1.1		4
51.2	odd	8		867.2.e.g.616.4		8
51.5	even	16		867.2.h.i.688.2		16
51.8	odd	8		51.2.e.a.13.1	yes	8
51.11	even	16		867.2.h.k.733.4		16
51.14	even	16		867.2.h.k.757.4		16
51.20	even	16		867.2.h.k.757.3		16
51.23	even	16		867.2.h.k.733.3		16
51.26	odd	8		867.2.e.g.829.1		8
51.29	even	16		867.2.h.i.688.1		16
51.32	odd	8		51.2.e.a.4.4	✓	8
51.38	odd	4		867.2.d.f.577.2		8
51.41	even	16		867.2.h.i.712.2		16
51.44	even	16		867.2.h.i.712.1		16
51.47	odd	4		867.2.d.f.577.1		8
51.50	odd	2		867.2.a.l.1.4		4
68.15	odd	8		2448.2.be.x.1585.2		8
68.59	odd	8		2448.2.be.x.1441.2		8
204.59	even	8		816.2.bd.e.625.2		8
204.83	even	8		816.2.bd.e.769.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.4	✓	8	51.32	odd	8
51.2.e.a.13.1	yes	8	51.8	odd	8
153.2.f.b.55.1		8	17.15	even	8
153.2.f.b.64.4		8	17.8	even	8
816.2.bd.e.625.2		8	204.59	even	8
816.2.bd.e.769.2		8	204.83	even	8
867.2.a.k.1.4		4	3.2	odd	2
867.2.a.l.1.4		4	51.50	odd	2
867.2.d.f.577.1		8	51.47	odd	4
867.2.d.f.577.2		8	51.38	odd	4
867.2.e.g.616.4		8	51.2	odd	8
867.2.e.g.829.1		8	51.26	odd	8
867.2.h.i.688.1		16	51.29	even	16
867.2.h.i.688.2		16	51.5	even	16
867.2.h.i.712.1		16	51.44	even	16
867.2.h.i.712.2		16	51.41	even	16
867.2.h.k.733.3		16	51.23	even	16
867.2.h.k.733.4		16	51.11	even	16
867.2.h.k.757.3		16	51.20	even	16
867.2.h.k.757.4		16	51.14	even	16
2448.2.be.x.1441.2		8	68.59	odd	8
2448.2.be.x.1585.2		8	68.15	odd	8
2601.2.a.be.1.1		4	17.16	even	2
2601.2.a.bf.1.1		4	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)

\(f(q)\)	\(=\)	\(q-2.06644 q^{2} +2.27016 q^{4} +0.347777 q^{5} -4.33660 q^{7} -0.558268 q^{8} +O(q^{10})\)	\(q-2.06644 q^{2} +2.27016 q^{4} +0.347777 q^{5} -4.33660 q^{7} -0.558268 q^{8} -0.718659 q^{10} +0.729840 q^{11} -5.40303 q^{13} +8.96130 q^{14} -3.38669 q^{16} +3.27016 q^{19} +0.789510 q^{20} -1.50817 q^{22} -3.55827 q^{23} -4.87905 q^{25} +11.1650 q^{26} -9.84476 q^{28} +4.38206 q^{29} -3.30445 q^{31} +8.11492 q^{32} -1.50817 q^{35} -2.87389 q^{37} -6.75758 q^{38} -0.194153 q^{40} -11.8494 q^{41} +3.74618 q^{43} +1.65685 q^{44} +7.35293 q^{46} -0.476019 q^{47} +11.8061 q^{49} +10.0822 q^{50} -12.2657 q^{52} +10.1329 q^{53} +0.253822 q^{55} +2.42098 q^{56} -9.05526 q^{58} -5.29790 q^{59} +3.54709 q^{61} +6.82843 q^{62} -9.99559 q^{64} -1.87905 q^{65} -1.55666 q^{67} +3.11654 q^{70} +8.96130 q^{71} -4.09396 q^{73} +5.93872 q^{74} +7.42378 q^{76} -3.16502 q^{77} +0.812615 q^{79} -1.17781 q^{80} +24.4860 q^{82} -7.21351 q^{83} -7.74124 q^{86} -0.407446 q^{88} +9.77391 q^{89} +23.4308 q^{91} -8.07784 q^{92} +0.983662 q^{94} +1.13729 q^{95} -1.77317 q^{97} -24.3965 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8	\( 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8} + 12 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} + 6 q^{16} + 10 q^{19} + 16 q^{20} - 4 q^{22} - 6 q^{23} + 2 q^{25} + 20 q^{26} - 24 q^{28} + 16 q^{29} - 4 q^{31} + 14 q^{32} - 4 q^{35} - 12 q^{37} + 12 q^{38} + 8 q^{40} - 14 q^{41} + 14 q^{43} - 16 q^{44} + 12 q^{46} - 4 q^{47} + 30 q^{50} - 8 q^{52} + 20 q^{53} + 2 q^{55} - 16 q^{56} - 8 q^{58} + 24 q^{59} - 12 q^{61} + 16 q^{62} - 2 q^{64} + 14 q^{65} + 4 q^{67} - 4 q^{70} + 4 q^{71} - 20 q^{73} + 12 q^{74} + 40 q^{76} + 12 q^{77} - 8 q^{79} + 4 q^{82} + 4 q^{83} + 4 q^{86} - 16 q^{88} - 4 q^{89} + 28 q^{91} + 16 q^{92} + 8 q^{94} + 22 q^{95} - 24 q^{97} - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8 + 12 * q^10 + 6 * q^11 + 2 * q^13 + 4 * q^14 + 6 * q^16 + 10 * q^19 + 16 * q^20 - 4 * q^22 - 6 * q^23 + 2 * q^25 + 20 * q^26 - 24 * q^28 + 16 * q^29 - 4 * q^31 + 14 * q^32 - 4 * q^35 - 12 * q^37 + 12 * q^38 + 8 * q^40 - 14 * q^41 + 14 * q^43 - 16 * q^44 + 12 * q^46 - 4 * q^47 + 30 * q^50 - 8 * q^52 + 20 * q^53 + 2 * q^55 - 16 * q^56 - 8 * q^58 + 24 * q^59 - 12 * q^61 + 16 * q^62 - 2 * q^64 + 14 * q^65 + 4 * q^67 - 4 * q^70 + 4 * q^71 - 20 * q^73 + 12 * q^74 + 40 * q^76 + 12 * q^77 - 8 * q^79 + 4 * q^82 + 4 * q^83 + 4 * q^86 - 16 * q^88 - 4 * q^89 + 28 * q^91 + 16 * q^92 + 8 * q^94 + 22 * q^95 - 24 * q^97 - 38 * q^98

Introduction

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