LMFDB - Embedded newform 4761.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, 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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.478891 q^{2} -1.77066 q^{4} +3.37703 q^{5} +4.53843 q^{7} +1.80574 q^{8} +O(q^{10})$	$q-0.478891 q^{2} -1.77066 q^{4} +3.37703 q^{5} +4.53843 q^{7} +1.80574 q^{8} -1.61723 q^{10} -0.982050 q^{11} -0.453800 q^{13} -2.17341 q^{14} +2.67657 q^{16} +3.28463 q^{17} +1.67657 q^{19} -5.97958 q^{20} +0.470295 q^{22} +6.40433 q^{25} +0.217321 q^{26} -8.03603 q^{28} +1.36926 q^{29} +6.46556 q^{31} -4.89326 q^{32} -1.57298 q^{34} +15.3264 q^{35} -3.55991 q^{37} -0.802897 q^{38} +6.09803 q^{40} -5.77408 q^{41} -11.0036 q^{43} +1.73888 q^{44} +10.0618 q^{47} +13.5973 q^{49} -3.06698 q^{50} +0.803526 q^{52} +12.3048 q^{53} -3.31641 q^{55} +8.19521 q^{56} -0.655726 q^{58} +10.6367 q^{59} +5.57388 q^{61} -3.09630 q^{62} -3.00980 q^{64} -1.53250 q^{65} -7.75309 q^{67} -5.81597 q^{68} -7.33969 q^{70} -2.98957 q^{71} -3.89717 q^{73} +1.70481 q^{74} -2.96865 q^{76} -4.45696 q^{77} -8.71170 q^{79} +9.03887 q^{80} +2.76516 q^{82} +7.48521 q^{83} +11.0923 q^{85} +5.26954 q^{86} -1.77333 q^{88} +7.23401 q^{89} -2.05954 q^{91} -4.81851 q^{94} +5.66184 q^{95} -5.63473 q^{97} -6.51165 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} + 7 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 4 * q^4 + 7 * q^5 + 3 * q^7 + 9 * q^8	$ 5 q - 2 q^{2} + 4 q^{4} + 7 q^{5} + 3 q^{7} + 9 q^{8} + 6 q^{10} + 2 q^{11} - 7 q^{13} - 10 q^{14} + 6 q^{16} + 16 q^{17} + q^{19} + 21 q^{20} - 14 q^{22} + 20 q^{25} + 5 q^{26} - 2 q^{28} - 18 q^{29} + 5 q^{31} + 6 q^{32} - 2 q^{34} + 24 q^{35} - 26 q^{37} + 26 q^{38} + 17 q^{40} - 9 q^{41} - 22 q^{43} + 28 q^{44} - 2 q^{47} + 2 q^{49} + 14 q^{50} + q^{52} + 35 q^{53} + 16 q^{55} - 10 q^{56} + 5 q^{58} - 6 q^{59} + 7 q^{61} - 13 q^{62} - 21 q^{64} + 10 q^{65} - 5 q^{67} + 15 q^{68} - 47 q^{70} + 18 q^{71} + 4 q^{73} + 28 q^{74} + 3 q^{76} + 21 q^{77} - 13 q^{79} + 37 q^{80} - 47 q^{82} + 24 q^{83} + 18 q^{85} + 11 q^{86} - 14 q^{88} - 7 q^{89} + 9 q^{91} - 8 q^{94} + 30 q^{95} - 6 q^{97} - 14 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 + 7 * q^5 + 3 * q^7 + 9 * q^8 + 6 * q^10 + 2 * q^11 - 7 * q^13 - 10 * q^14 + 6 * q^16 + 16 * q^17 + q^19 + 21 * q^20 - 14 * q^22 + 20 * q^25 + 5 * q^26 - 2 * q^28 - 18 * q^29 + 5 * q^31 + 6 * q^32 - 2 * q^34 + 24 * q^35 - 26 * q^37 + 26 * q^38 + 17 * q^40 - 9 * q^41 - 22 * q^43 + 28 * q^44 - 2 * q^47 + 2 * q^49 + 14 * q^50 + q^52 + 35 * q^53 + 16 * q^55 - 10 * q^56 + 5 * q^58 - 6 * q^59 + 7 * q^61 - 13 * q^62 - 21 * q^64 + 10 * q^65 - 5 * q^67 + 15 * q^68 - 47 * q^70 + 18 * q^71 + 4 * q^73 + 28 * q^74 + 3 * q^76 + 21 * q^77 - 13 * q^79 + 37 * q^80 - 47 * q^82 + 24 * q^83 + 18 * q^85 + 11 * q^86 - 14 * q^88 - 7 * q^89 + 9 * q^91 - 8 * q^94 + 30 * q^95 - 6 * q^97 - 14 * 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 $

$2$	−0.478891	−0.338627	−0.169314	−	0.985562i	$-0.554155\pi$
$2$	−0.478891	−0.338627	−0.169314	+	0.985562i	$0.554155\pi$
$3$	0	0
$3$	0	0
$4$	−1.77066	−0.885331
$5$	3.37703	1.51025	0.755127	−	0.655579i	$-0.227575\pi$
$5$	3.37703	1.51025	0.755127	+	0.655579i	$0.227575\pi$
$6$	0	0
$7$	4.53843	1.71536	0.857682	−	0.514180i	$-0.171904\pi$
$7$	4.53843	1.71536	0.857682	+	0.514180i	$0.171904\pi$
$8$	1.80574	0.638425
$9$	0	0
$10$	−1.61723	−0.511413
$11$	−0.982050	−0.296099	−0.148050	−	0.988980i	$-0.547300\pi$
$11$	−0.982050	−0.296099	−0.148050	+	0.988980i	$0.547300\pi$
$12$	0	0
$13$	−0.453800	−0.125861	−0.0629307	−	0.998018i	$-0.520045\pi$
$13$	−0.453800	−0.125861	−0.0629307	+	0.998018i	$0.520045\pi$
$14$	−2.17341	−0.580870
$15$	0	0
$16$	2.67657	0.669143
$17$	3.28463	0.796640	0.398320	−	0.917247i	$-0.369593\pi$
$17$	3.28463	0.796640	0.398320	+	0.917247i	$0.369593\pi$
$18$	0	0
$19$	1.67657	0.384632	0.192316	−	0.981333i	$-0.438400\pi$
$19$	1.67657	0.384632	0.192316	+	0.981333i	$0.438400\pi$
$20$	−5.97958	−1.33708
$21$	0	0
$22$	0.470295	0.100267
$23$	0	0
$23$	0	0
$24$	0	0
$25$	6.40433	1.28087
$26$	0.217321	0.0426201
$27$	0	0
$28$	−8.03603	−1.51867
$29$	1.36926	0.254265	0.127132	−	0.991886i	$-0.459423\pi$
$29$	1.36926	0.254265	0.127132	+	0.991886i	$0.459423\pi$
$30$	0	0
$31$	6.46556	1.16125	0.580624	−	0.814171i	$-0.302808\pi$
$31$	6.46556	1.16125	0.580624	+	0.814171i	$0.302808\pi$
$32$	−4.89326	−0.865015
$33$	0	0
$34$	−1.57298	−0.269764
$35$	15.3264	2.59064
$36$	0	0
$37$	−3.55991	−0.585245	−0.292622	−	0.956228i	$-0.594528\pi$
$37$	−3.55991	−0.585245	−0.292622	+	0.956228i	$0.594528\pi$
$38$	−0.802897	−0.130247
$39$	0	0
$40$	6.09803	0.964184
$41$	−5.77408	−0.901760	−0.450880	−	0.892585i	$-0.648890\pi$
$41$	−5.77408	−0.901760	−0.450880	+	0.892585i	$0.648890\pi$
$42$	0	0
$43$	−11.0036	−1.67804	−0.839018	−	0.544104i	$-0.816870\pi$
$43$	−11.0036	−1.67804	−0.839018	+	0.544104i	$0.816870\pi$
$44$	1.73888	0.262146
$45$	0	0
$46$	0	0
$47$	10.0618	1.46767	0.733833	−	0.679330i	$-0.237730\pi$
$47$	10.0618	1.46767	0.733833	+	0.679330i	$0.237730\pi$
$48$	0	0
$49$	13.5973	1.94248
$50$	−3.06698	−0.433737
$51$	0	0
$52$	0.803526	0.111429
$53$	12.3048	1.69020	0.845100	−	0.534608i	$-0.179541\pi$
$53$	12.3048	1.69020	0.845100	+	0.534608i	$0.179541\pi$
$54$	0	0
$55$	−3.31641	−0.447185
$56$	8.19521	1.09513
$57$	0	0
$58$	−0.655726	−0.0861011
$59$	10.6367	1.38477	0.692387	−	0.721526i	$-0.256559\pi$
$59$	10.6367	1.38477	0.692387	+	0.721526i	$0.256559\pi$
$60$	0	0
$61$	5.57388	0.713662	0.356831	−	0.934169i	$-0.383857\pi$
$61$	5.57388	0.713662	0.356831	+	0.934169i	$0.383857\pi$
$62$	−3.09630	−0.393231
$63$	0	0
$64$	−3.00980	−0.376226
$65$	−1.53250	−0.190083
$66$	0	0
$67$	−7.75309	−0.947190	−0.473595	−	0.880743i	$-0.657044\pi$
$67$	−7.75309	−0.947190	−0.473595	+	0.880743i	$0.657044\pi$
$68$	−5.81597	−0.705290
$69$	0	0
$70$	−7.33969	−0.877260
$71$	−2.98957	−0.354796	−0.177398	−	0.984139i	$-0.556768\pi$
$71$	−2.98957	−0.354796	−0.177398	+	0.984139i	$0.556768\pi$
$72$	0	0
$73$	−3.89717	−0.456129	−0.228064	−	0.973646i	$-0.573240\pi$
$73$	−3.89717	−0.456129	−0.228064	+	0.973646i	$0.573240\pi$
$74$	1.70481	0.198180
$75$	0	0
$76$	−2.96865	−0.340527
$77$	−4.45696	−0.507918
$78$	0	0
$79$	−8.71170	−0.980143	−0.490072	−	0.871682i	$-0.663029\pi$
$79$	−8.71170	−0.980143	−0.490072	+	0.871682i	$0.663029\pi$
$80$	9.03887	1.01058
$81$	0	0
$82$	2.76516	0.305361
$83$	7.48521	0.821608	0.410804	−	0.911724i	$-0.365248\pi$
$83$	7.48521	0.821608	0.410804	+	0.911724i	$0.365248\pi$
$84$	0	0
$85$	11.0923	1.20313
$86$	5.26954	0.568229
$87$	0	0
$88$	−1.77333	−0.189037
$89$	7.23401	0.766804	0.383402	−	0.923582i	$-0.374752\pi$
$89$	7.23401	0.766804	0.383402	+	0.923582i	$0.374752\pi$
$90$	0	0
$91$	−2.05954	−0.215898
$92$	0	0
$93$	0	0
$94$	−4.81851	−0.496992
$95$	5.66184	0.580892
$96$	0	0
$97$	−5.63473	−0.572120	−0.286060	−	0.958212i	$-0.592346\pi$
$97$	−5.63473	−0.572120	−0.286060	+	0.958212i	$0.592346\pi$
$98$	−6.51165	−0.657776
$99$	0	0
$100$	−11.3399	−1.13399
$101$	−7.89258	−0.785341	−0.392671	−	0.919679i	$-0.628449\pi$
$101$	−7.89258	−0.785341	−0.392671	+	0.919679i	$0.628449\pi$
$102$	0	0
$103$	−9.00385	−0.887176	−0.443588	−	0.896231i	$-0.646295\pi$
$103$	−9.00385	−0.887176	−0.443588	+	0.896231i	$0.646295\pi$
$104$	−0.819443	−0.0803530
$105$	0	0
$106$	−5.89269	−0.572348
$107$	3.61572	0.349545	0.174772	−	0.984609i	$-0.444081\pi$
$107$	3.61572	0.349545	0.174772	+	0.984609i	$0.444081\pi$
$108$	0	0
$109$	1.27585	0.122205	0.0611023	−	0.998132i	$-0.480538\pi$
$109$	1.27585	0.122205	0.0611023	+	0.998132i	$0.480538\pi$
$110$	1.58820	0.151429
$111$	0	0
$112$	12.1474	1.14782
$113$	1.18592	0.111562	0.0557810	−	0.998443i	$-0.482235\pi$
$113$	1.18592	0.111562	0.0557810	+	0.998443i	$0.482235\pi$
$114$	0	0
$115$	0	0
$116$	−2.42450	−0.225109
$117$	0	0
$118$	−5.09380	−0.468923
$119$	14.9071	1.36653
$120$	0	0
$121$	−10.0356	−0.912325
$122$	−2.66928	−0.241665
$123$	0	0
$124$	−11.4483	−1.02809
$125$	4.74248	0.424180
$126$	0	0
$127$	2.86103	0.253875	0.126938	−	0.991911i	$-0.459485\pi$
$127$	2.86103	0.253875	0.126938	+	0.991911i	$0.459485\pi$
$128$	11.2279	0.992415
$129$	0	0
$130$	0.733899	0.0643672
$131$	−0.385666	−0.0336958	−0.0168479	−	0.999858i	$-0.505363\pi$
$131$	−0.385666	−0.0336958	−0.0168479	+	0.999858i	$0.505363\pi$
$132$	0	0
$133$	7.60901	0.659785
$134$	3.71289	0.320745
$135$	0	0
$136$	5.93118	0.508595
$137$	−20.4664	−1.74856	−0.874282	−	0.485419i	$-0.838667\pi$
$137$	−20.4664	−1.74856	−0.874282	+	0.485419i	$0.838667\pi$
$138$	0	0
$139$	10.1030	0.856927	0.428463	−	0.903559i	$-0.359055\pi$
$139$	10.1030	0.856927	0.428463	+	0.903559i	$0.359055\pi$
$140$	−27.1379	−2.29357
$141$	0	0
$142$	1.43168	0.120144
$143$	0.445654	0.0372675
$144$	0	0
$145$	4.62403	0.384005
$146$	1.86632	0.154458
$147$	0	0
$148$	6.30339	0.518136
$149$	10.3047	0.844197	0.422099	−	0.906550i	$-0.361294\pi$
$149$	10.3047	0.844197	0.422099	+	0.906550i	$0.361294\pi$
$150$	0	0
$151$	0.500368	0.0407194	0.0203597	−	0.999793i	$-0.493519\pi$
$151$	0.500368	0.0407194	0.0203597	+	0.999793i	$0.493519\pi$
$152$	3.02745	0.245559
$153$	0	0
$154$	2.13440	0.171995
$155$	21.8344	1.75378
$156$	0	0
$157$	−15.5852	−1.24383	−0.621916	−	0.783084i	$-0.713645\pi$
$157$	−15.5852	−1.24383	−0.621916	+	0.783084i	$0.713645\pi$
$158$	4.17196	0.331903
$159$	0	0
$160$	−16.5247	−1.30639
$161$	0	0
$162$	0	0
$163$	−6.78939	−0.531786	−0.265893	−	0.964003i	$-0.585667\pi$
$163$	−6.78939	−0.531786	−0.265893	+	0.964003i	$0.585667\pi$
$164$	10.2240	0.798357
$165$	0	0
$166$	−3.58460	−0.278219
$167$	−20.0470	−1.55128	−0.775642	−	0.631173i	$-0.782574\pi$
$167$	−20.0470	−1.55128	−0.775642	+	0.631173i	$0.782574\pi$
$168$	0	0
$169$	−12.7941	−0.984159
$170$	−5.31200	−0.407412
$171$	0	0
$172$	19.4837	1.48562
$173$	−10.3866	−0.789677	−0.394838	−	0.918751i	$-0.629199\pi$
$173$	−10.3866	−0.789677	−0.394838	+	0.918751i	$0.629199\pi$
$174$	0	0
$175$	29.0656	2.19715
$176$	−2.62853	−0.198133
$177$	0	0
$178$	−3.46431	−0.259661
$179$	−4.19553	−0.313588	−0.156794	−	0.987631i	$-0.550116\pi$
$179$	−4.19553	−0.313588	−0.156794	+	0.987631i	$0.550116\pi$
$180$	0	0
$181$	−0.608963	−0.0452639	−0.0226319	−	0.999744i	$-0.507205\pi$
$181$	−0.608963	−0.0452639	−0.0226319	+	0.999744i	$0.507205\pi$
$182$	0.986295	0.0731090
$183$	0	0
$184$	0	0
$185$	−12.0219	−0.883868
$186$	0	0
$187$	−3.22567	−0.235884
$188$	−17.8161	−1.29937
$189$	0	0
$190$	−2.71141	−0.196706
$191$	−1.46435	−0.105957	−0.0529784	−	0.998596i	$-0.516871\pi$
$191$	−1.46435	−0.105957	−0.0529784	+	0.998596i	$0.516871\pi$
$192$	0	0
$193$	21.2482	1.52948	0.764739	−	0.644341i	$-0.222868\pi$
$193$	21.2482	1.52948	0.764739	+	0.644341i	$0.222868\pi$
$194$	2.69842	0.193736
$195$	0	0
$196$	−24.0763	−1.71974
$197$	6.08177	0.433308	0.216654	−	0.976248i	$-0.430486\pi$
$197$	6.08177	0.433308	0.216654	+	0.976248i	$0.430486\pi$
$198$	0	0
$199$	11.5681	0.820041	0.410021	−	0.912076i	$-0.365521\pi$
$199$	11.5681	0.820041	0.410021	+	0.912076i	$0.365521\pi$
$200$	11.5646	0.817737
$201$	0	0
$202$	3.77969	0.265938
$203$	6.21428	0.436157
$204$	0	0
$205$	−19.4993	−1.36189
$206$	4.31187	0.300422
$207$	0	0
$208$	−1.21463	−0.0842193
$209$	−1.64648	−0.113889
$210$	0	0
$211$	−2.76750	−0.190522	−0.0952612	−	0.995452i	$-0.530369\pi$
$211$	−2.76750	−0.190522	−0.0952612	+	0.995452i	$0.530369\pi$
$212$	−21.7877	−1.49639
$213$	0	0
$214$	−1.73154	−0.118365
$215$	−37.1595	−2.53426
$216$	0	0
$217$	29.3435	1.99197
$218$	−0.610996	−0.0413819
$219$	0	0
$220$	5.87225	0.395907
$221$	−1.49056	−0.100266
$222$	0	0
$223$	−18.4424	−1.23500	−0.617498	−	0.786572i	$-0.711854\pi$
$223$	−18.4424	−1.23500	−0.617498	+	0.786572i	$0.711854\pi$
$224$	−22.2077	−1.48382
$225$	0	0
$226$	−0.567927	−0.0377780
$227$	−7.25039	−0.481225	−0.240613	−	0.970621i	$-0.577348\pi$
$227$	−7.25039	−0.481225	−0.240613	+	0.970621i	$0.577348\pi$
$228$	0	0
$229$	9.95723	0.657992	0.328996	−	0.944331i	$-0.393290\pi$
$229$	9.95723	0.657992	0.328996	+	0.944331i	$0.393290\pi$
$230$	0	0
$231$	0	0
$232$	2.47252	0.162329
$233$	−5.40085	−0.353821	−0.176911	−	0.984227i	$-0.556610\pi$
$233$	−5.40085	−0.353821	−0.176911	+	0.984227i	$0.556610\pi$
$234$	0	0
$235$	33.9790	2.21655
$236$	−18.8339	−1.22598
$237$	0	0
$238$	−7.13886	−0.462744
$239$	14.8162	0.958381	0.479190	−	0.877711i	$-0.340930\pi$
$239$	14.8162	0.958381	0.479190	+	0.877711i	$0.340930\pi$
$240$	0	0
$241$	14.6527	0.943862	0.471931	−	0.881635i	$-0.343557\pi$
$241$	14.6527	0.943862	0.471931	+	0.881635i	$0.343557\pi$
$242$	4.80595	0.308938
$243$	0	0
$244$	−9.86946	−0.631827
$245$	45.9186	2.93363
$246$	0	0
$247$	−0.760828	−0.0484104
$248$	11.6751	0.741370
$249$	0	0
$250$	−2.27113	−0.143639
$251$	−0.963491	−0.0608150	−0.0304075	−	0.999538i	$-0.509681\pi$
$251$	−0.963491	−0.0608150	−0.0304075	+	0.999538i	$0.509681\pi$
$252$	0	0
$253$	0	0
$254$	−1.37012	−0.0859692
$255$	0	0
$256$	0.642664	0.0401665
$257$	0.223629	0.0139496	0.00697479	−	0.999976i	$-0.497780\pi$
$257$	0.223629	0.0139496	0.00697479	+	0.999976i	$0.497780\pi$
$258$	0	0
$259$	−16.1564	−1.00391
$260$	2.71353	0.168286
$261$	0	0
$262$	0.184692	0.0114103
$263$	22.3351	1.37724	0.688619	−	0.725123i	$-0.258217\pi$
$263$	22.3351	1.37724	0.688619	+	0.725123i	$0.258217\pi$
$264$	0	0
$265$	41.5538	2.55263
$266$	−3.64389	−0.223421
$267$	0	0
$268$	13.7281	0.838577
$269$	23.7208	1.44628	0.723142	−	0.690699i	$-0.242697\pi$
$269$	23.7208	1.44628	0.723142	+	0.690699i	$0.242697\pi$
$270$	0	0
$271$	−11.4998	−0.698565	−0.349282	−	0.937018i	$-0.613575\pi$
$271$	−11.4998	−0.698565	−0.349282	+	0.937018i	$0.613575\pi$
$272$	8.79155	0.533066
$273$	0	0
$274$	9.80119	0.592111
$275$	−6.28938	−0.379264
$276$	0	0
$277$	−6.25528	−0.375843	−0.187922	−	0.982184i	$-0.560175\pi$
$277$	−6.25528	−0.375843	−0.187922	+	0.982184i	$0.560175\pi$
$278$	−4.83825	−0.290179
$279$	0	0
$280$	27.6755	1.65393
$281$	21.4090	1.27715	0.638576	−	0.769559i	$-0.279524\pi$
$281$	21.4090	1.27715	0.638576	+	0.769559i	$0.279524\pi$
$282$	0	0
$283$	7.25638	0.431347	0.215673	−	0.976466i	$-0.430805\pi$
$283$	7.25638	0.431347	0.215673	+	0.976466i	$0.430805\pi$
$284$	5.29351	0.314112
$285$	0	0
$286$	−0.213420	−0.0126198
$287$	−26.2053	−1.54685
$288$	0	0
$289$	−6.21121	−0.365365
$290$	−2.21441	−0.130035
$291$	0	0
$292$	6.90057	0.403825
$293$	0.391405	0.0228661	0.0114331	−	0.999935i	$-0.496361\pi$
$293$	0.391405	0.0228661	0.0114331	+	0.999935i	$0.496361\pi$
$294$	0	0
$295$	35.9203	2.09136
$296$	−6.42826	−0.373635
$297$	0	0
$298$	−4.93485	−0.285868
$299$	0	0
$300$	0	0
$301$	−49.9391	−2.87844
$302$	−0.239622	−0.0137887
$303$	0	0
$304$	4.48747	0.257374
$305$	18.8232	1.07781
$306$	0	0
$307$	1.28817	0.0735195	0.0367598	−	0.999324i	$-0.488296\pi$
$307$	1.28817	0.0735195	0.0367598	+	0.999324i	$0.488296\pi$
$308$	7.89178	0.449676
$309$	0	0
$310$	−10.4563	−0.593878
$311$	1.25954	0.0714218	0.0357109	−	0.999362i	$-0.488630\pi$
$311$	1.25954	0.0714218	0.0357109	+	0.999362i	$0.488630\pi$
$312$	0	0
$313$	−10.4392	−0.590059	−0.295030	−	0.955488i	$-0.595330\pi$
$313$	−10.4392	−0.590059	−0.295030	+	0.955488i	$0.595330\pi$
$314$	7.46360	0.421195
$315$	0	0
$316$	15.4255	0.867752
$317$	27.1162	1.52300	0.761499	−	0.648166i	$-0.224464\pi$
$317$	27.1162	1.52300	0.761499	+	0.648166i	$0.224464\pi$
$318$	0	0
$319$	−1.34468	−0.0752877
$320$	−10.1642	−0.568196
$321$	0	0
$322$	0	0
$323$	5.50692	0.306413
$324$	0	0
$325$	−2.90628	−0.161212
$326$	3.25138	0.180077
$327$	0	0
$328$	−10.4265	−0.575706
$329$	45.6648	2.51758
$330$	0	0
$331$	−7.48281	−0.411292	−0.205646	−	0.978626i	$-0.565930\pi$
$331$	−7.48281	−0.411292	−0.205646	+	0.978626i	$0.565930\pi$
$332$	−13.2538	−0.727396
$333$	0	0
$334$	9.60034	0.525307
$335$	−26.1824	−1.43050
$336$	0	0
$337$	−25.7102	−1.40053	−0.700263	−	0.713885i	$-0.746934\pi$
$337$	−25.7102	−1.40053	−0.700263	+	0.713885i	$0.746934\pi$
$338$	6.12697	0.333263
$339$	0	0
$340$	−19.6407	−1.06517
$341$	−6.34950	−0.343845
$342$	0	0
$343$	29.9415	1.61669
$344$	−19.8696	−1.07130
$345$	0	0
$346$	4.97404	0.267406
$347$	24.6109	1.32118	0.660592	−	0.750745i	$-0.270306\pi$
$347$	24.6109	1.32118	0.660592	+	0.750745i	$0.270306\pi$
$348$	0	0
$349$	−19.3411	−1.03531	−0.517654	−	0.855590i	$-0.673194\pi$
$349$	−19.3411	−1.03531	−0.517654	+	0.855590i	$0.673194\pi$
$350$	−13.9193	−0.744017
$351$	0	0
$352$	4.80543	0.256130
$353$	23.5211	1.25190	0.625952	−	0.779862i	$-0.284711\pi$
$353$	23.5211	1.25190	0.625952	+	0.779862i	$0.284711\pi$
$354$	0	0
$355$	−10.0959	−0.535832
$356$	−12.8090	−0.678875
$357$	0	0
$358$	2.00920	0.106190
$359$	6.04520	0.319053	0.159527	−	0.987194i	$-0.449003\pi$
$359$	6.04520	0.319053	0.159527	+	0.987194i	$0.449003\pi$
$360$	0	0
$361$	−16.1891	−0.852058
$362$	0.291627	0.0153276
$363$	0	0
$364$	3.64675	0.191141
$365$	−13.1608	−0.688870
$366$	0	0
$367$	3.17802	0.165891	0.0829457	−	0.996554i	$-0.473567\pi$
$367$	3.17802	0.165891	0.0829457	+	0.996554i	$0.473567\pi$
$368$	0	0
$369$	0	0
$370$	5.75719	0.299302
$371$	55.8447	2.89931
$372$	0	0
$373$	−0.553883	−0.0286790	−0.0143395	−	0.999897i	$-0.504565\pi$
$373$	−0.553883	−0.0286790	−0.0143395	+	0.999897i	$0.504565\pi$
$374$	1.54475	0.0798769
$375$	0	0
$376$	18.1690	0.936994
$377$	−0.621369	−0.0320021
$378$	0	0
$379$	2.32749	0.119555	0.0597776	−	0.998212i	$-0.480961\pi$
$379$	2.32749	0.119555	0.0597776	+	0.998212i	$0.480961\pi$
$380$	−10.0252	−0.514282
$381$	0	0
$382$	0.701266	0.0358799
$383$	29.2792	1.49610	0.748049	−	0.663644i	$-0.230991\pi$
$383$	29.2792	1.49610	0.748049	+	0.663644i	$0.230991\pi$
$384$	0	0
$385$	−15.0513	−0.767085
$386$	−10.1756	−0.517923
$387$	0	0
$388$	9.97721	0.506516
$389$	−10.1901	−0.516660	−0.258330	−	0.966057i	$-0.583172\pi$
$389$	−10.1901	−0.516660	−0.258330	+	0.966057i	$0.583172\pi$
$390$	0	0
$391$	0	0
$392$	24.5532	1.24013
$393$	0	0
$394$	−2.91251	−0.146730
$395$	−29.4197	−1.48027
$396$	0	0
$397$	34.5240	1.73271	0.866356	−	0.499427i	$-0.166456\pi$
$397$	34.5240	1.73271	0.866356	+	0.499427i	$0.166456\pi$
$398$	−5.53987	−0.277688
$399$	0	0
$400$	17.1417	0.857084
$401$	−8.98946	−0.448912	−0.224456	−	0.974484i	$-0.572061\pi$
$401$	−8.98946	−0.448912	−0.224456	+	0.974484i	$0.572061\pi$
$402$	0	0
$403$	−2.93407	−0.146156
$404$	13.9751	0.695287
$405$	0	0
$406$	−2.97597	−0.147695
$407$	3.49601	0.173291
$408$	0	0
$409$	22.4088	1.10805	0.554023	−	0.832501i	$-0.313092\pi$
$409$	22.4088	1.10805	0.554023	+	0.832501i	$0.313092\pi$
$410$	9.33803	0.461172
$411$	0	0
$412$	15.9428	0.785445
$413$	48.2737	2.37539
$414$	0	0
$415$	25.2778	1.24084
$416$	2.22056	0.108872
$417$	0	0
$418$	0.788485	0.0385660
$419$	−10.2057	−0.498581	−0.249290	−	0.968429i	$-0.580197\pi$
$419$	−10.2057	−0.498581	−0.249290	+	0.968429i	$0.580197\pi$
$420$	0	0
$421$	16.0424	0.781860	0.390930	−	0.920420i	$-0.372153\pi$
$421$	16.0424	0.781860	0.390930	+	0.920420i	$0.372153\pi$
$422$	1.32533	0.0645161
$423$	0	0
$424$	22.2193	1.07907
$425$	21.0359	1.02039
$426$	0	0
$427$	25.2967	1.22419
$428$	−6.40222	−0.309463
$429$	0	0
$430$	17.7954	0.858170
$431$	38.4664	1.85286	0.926431	−	0.376464i	$-0.122860\pi$
$431$	38.4664	1.85286	0.926431	+	0.376464i	$0.122860\pi$
$432$	0	0
$433$	−7.77458	−0.373623	−0.186811	−	0.982396i	$-0.559815\pi$
$433$	−7.77458	−0.373623	−0.186811	+	0.982396i	$0.559815\pi$
$434$	−14.0523	−0.674534
$435$	0	0
$436$	−2.25911	−0.108192
$437$	0	0
$438$	0	0
$439$	34.9747	1.66925	0.834626	−	0.550816i	$-0.185684\pi$
$439$	34.9747	1.66925	0.834626	+	0.550816i	$0.185684\pi$
$440$	−5.98857	−0.285494
$441$	0	0
$442$	0.713818	0.0339529
$443$	10.0605	0.477987	0.238994	−	0.971021i	$-0.423182\pi$
$443$	10.0605	0.477987	0.238994	+	0.971021i	$0.423182\pi$
$444$	0	0
$445$	24.4295	1.15807
$446$	8.83192	0.418204
$447$	0	0
$448$	−13.6598	−0.645364
$449$	17.0613	0.805173	0.402586	−	0.915382i	$-0.368111\pi$
$449$	17.0613	0.805173	0.402586	+	0.915382i	$0.368111\pi$
$450$	0	0
$451$	5.67044	0.267011
$452$	−2.09987	−0.0987694
$453$	0	0
$454$	3.47215	0.162956
$455$	−6.95512	−0.326061
$456$	0	0
$457$	−24.3766	−1.14029	−0.570145	−	0.821544i	$-0.693113\pi$
$457$	−24.3766	−1.14029	−0.570145	+	0.821544i	$0.693113\pi$
$458$	−4.76843	−0.222814
$459$	0	0
$460$	0	0
$461$	−38.0631	−1.77278	−0.886388	−	0.462943i	$-0.846793\pi$
$461$	−38.0631	−1.77278	−0.886388	+	0.462943i	$0.846793\pi$
$462$	0	0
$463$	14.2735	0.663346	0.331673	−	0.943394i	$-0.392387\pi$
$463$	14.2735	0.663346	0.331673	+	0.943394i	$0.392387\pi$
$464$	3.66492	0.170140
$465$	0	0
$466$	2.58642	0.119814
$467$	3.90396	0.180654	0.0903269	−	0.995912i	$-0.471209\pi$
$467$	3.90396	0.180654	0.0903269	+	0.995912i	$0.471209\pi$
$468$	0	0
$469$	−35.1868	−1.62478
$470$	−16.2723	−0.750583
$471$	0	0
$472$	19.2070	0.884075
$473$	10.8061	0.496865
$474$	0	0
$475$	10.7373	0.492663
$476$	−26.3954	−1.20983
$477$	0	0
$478$	−7.09535	−0.324534
$479$	−18.4056	−0.840972	−0.420486	−	0.907299i	$-0.638140\pi$
$479$	−18.4056	−0.840972	−0.420486	+	0.907299i	$0.638140\pi$
$480$	0	0
$481$	1.61548	0.0736597
$482$	−7.01704	−0.319618
$483$	0	0
$484$	17.7696	0.807710
$485$	−19.0287	−0.864047
$486$	0	0
$487$	13.0454	0.591142	0.295571	−	0.955321i	$-0.404490\pi$
$487$	13.0454	0.591142	0.295571	+	0.955321i	$0.404490\pi$
$488$	10.0650	0.455620
$489$	0	0
$490$	−21.9900	−0.993408
$491$	−23.9726	−1.08187	−0.540934	−	0.841065i	$-0.681929\pi$
$491$	−23.9726	−1.08187	−0.540934	+	0.841065i	$0.681929\pi$
$492$	0	0
$493$	4.49751	0.202558
$494$	0.364354	0.0163931
$495$	0	0
$496$	17.3055	0.777042
$497$	−13.5679	−0.608605
$498$	0	0
$499$	−8.80105	−0.393989	−0.196994	−	0.980405i	$-0.563118\pi$
$499$	−8.80105	−0.393989	−0.196994	+	0.980405i	$0.563118\pi$
$500$	−8.39733	−0.375540
$501$	0	0
$502$	0.461408	0.0205936
$503$	36.0877	1.60907	0.804535	−	0.593905i	$-0.202415\pi$
$503$	36.0877	1.60907	0.804535	+	0.593905i	$0.202415\pi$
$504$	0	0
$505$	−26.6535	−1.18606
$506$	0	0
$507$	0	0
$508$	−5.06592	−0.224764
$509$	−34.3598	−1.52297	−0.761486	−	0.648181i	$-0.775530\pi$
$509$	−34.3598	−1.52297	−0.761486	+	0.648181i	$0.775530\pi$
$510$	0	0
$511$	−17.6870	−0.782427
$512$	−22.7636	−1.00602
$513$	0	0
$514$	−0.107094	−0.00472371
$515$	−30.4063	−1.33986
$516$	0	0
$517$	−9.88120	−0.434575
$518$	7.73715	0.339951
$519$	0	0
$520$	−2.76729	−0.121353
$521$	−29.4985	−1.29235	−0.646176	−	0.763189i	$-0.723633\pi$
$521$	−29.4985	−1.29235	−0.646176	+	0.763189i	$0.723633\pi$
$522$	0	0
$523$	20.4143	0.892655	0.446327	−	0.894870i	$-0.352732\pi$
$523$	20.4143	0.892655	0.446327	+	0.894870i	$0.352732\pi$
$524$	0.682885	0.0298320
$525$	0	0
$526$	−10.6961	−0.466371
$527$	21.2370	0.925097
$528$	0	0
$529$	0	0
$530$	−19.8998	−0.864391
$531$	0	0
$532$	−13.4730	−0.584128
$533$	2.62028	0.113497
$534$	0	0
$535$	12.2104	0.527902
$536$	−14.0000	−0.604710
$537$	0	0
$538$	−11.3597	−0.489752
$539$	−13.3533	−0.575166
$540$	0	0
$541$	−19.5835	−0.841959	−0.420980	−	0.907070i	$-0.638314\pi$
$541$	−19.5835	−0.841959	−0.420980	+	0.907070i	$0.638314\pi$
$542$	5.50717	0.236553
$543$	0	0
$544$	−16.0726	−0.689105
$545$	4.30860	0.184560
$546$	0	0
$547$	43.3154	1.85203	0.926017	−	0.377481i	$-0.123210\pi$
$547$	43.3154	1.85203	0.926017	+	0.377481i	$0.123210\pi$
$548$	36.2391	1.54806
$549$	0	0
$550$	3.01193	0.128429
$551$	2.29566	0.0977985
$552$	0	0
$553$	−39.5374	−1.68130
$554$	2.99560	0.127271
$555$	0	0
$556$	−17.8890	−0.758664
$557$	−35.5099	−1.50460	−0.752302	−	0.658818i	$-0.771057\pi$
$557$	−35.5099	−1.50460	−0.752302	+	0.658818i	$0.771057\pi$
$558$	0	0
$559$	4.99344	0.211200
$560$	41.0223	1.73351
$561$	0	0
$562$	−10.2526	−0.432479
$563$	−13.6324	−0.574537	−0.287268	−	0.957850i	$-0.592747\pi$
$563$	−13.6324	−0.574537	−0.287268	+	0.957850i	$0.592747\pi$
$564$	0	0
$565$	4.00489	0.168487
$566$	−3.47502	−0.146066
$567$	0	0
$568$	−5.39837	−0.226511
$569$	15.2541	0.639485	0.319742	−	0.947505i	$-0.396404\pi$
$569$	15.2541	0.639485	0.319742	+	0.947505i	$0.396404\pi$
$570$	0	0
$571$	−2.81949	−0.117992	−0.0589959	−	0.998258i	$-0.518790\pi$
$571$	−2.81949	−0.117992	−0.0589959	+	0.998258i	$0.518790\pi$
$572$	−0.789103	−0.0329941
$573$	0	0
$574$	12.5495	0.523805
$575$	0	0
$576$	0	0
$577$	5.37437	0.223738	0.111869	−	0.993723i	$-0.464316\pi$
$577$	5.37437	0.223738	0.111869	+	0.993723i	$0.464316\pi$
$578$	2.97449	0.123723
$579$	0	0
$580$	−8.18760	−0.339971
$581$	33.9711	1.40936
$582$	0	0
$583$	−12.0840	−0.500467
$584$	−7.03726	−0.291204
$585$	0	0
$586$	−0.187440	−0.00774309
$587$	−7.22107	−0.298046	−0.149023	−	0.988834i	$-0.547613\pi$
$587$	−7.22107	−0.298046	−0.149023	+	0.988834i	$0.547613\pi$
$588$	0	0
$589$	10.8400	0.446654
$590$	−17.2019	−0.708192
$591$	0	0
$592$	−9.52835	−0.391613
$593$	1.60336	0.0658421	0.0329211	−	0.999458i	$-0.489519\pi$
$593$	1.60336	0.0658421	0.0329211	+	0.999458i	$0.489519\pi$
$594$	0	0
$595$	50.3416	2.06380
$596$	−18.2462	−0.747394
$597$	0	0
$598$	0	0
$599$	−7.66948	−0.313366	−0.156683	−	0.987649i	$-0.550080\pi$
$599$	−7.66948	−0.313366	−0.156683	+	0.987649i	$0.550080\pi$
$600$	0	0
$601$	27.4846	1.12112	0.560560	−	0.828114i	$-0.310586\pi$
$601$	27.4846	1.12112	0.560560	+	0.828114i	$0.310586\pi$
$602$	23.9154	0.974720
$603$	0	0
$604$	−0.885983	−0.0360501
$605$	−33.8905	−1.37784
$606$	0	0
$607$	−35.2790	−1.43193	−0.715964	−	0.698137i	$-0.754013\pi$
$607$	−35.2790	−1.43193	−0.715964	+	0.698137i	$0.754013\pi$
$608$	−8.20392	−0.332713
$609$	0	0
$610$	−9.01425	−0.364976
$611$	−4.56604	−0.184722
$612$	0	0
$613$	−6.23774	−0.251940	−0.125970	−	0.992034i	$-0.540204\pi$
$613$	−6.23774	−0.251940	−0.125970	+	0.992034i	$0.540204\pi$
$614$	−0.616892	−0.0248957
$615$	0	0
$616$	−8.04811	−0.324268
$617$	−32.5436	−1.31016	−0.655079	−	0.755560i	$-0.727365\pi$
$617$	−32.5436	−1.31016	−0.655079	+	0.755560i	$0.727365\pi$
$618$	0	0
$619$	−36.7609	−1.47754	−0.738772	−	0.673955i	$-0.764594\pi$
$619$	−36.7609	−1.47754	−0.738772	+	0.673955i	$0.764594\pi$
$620$	−38.6614	−1.55268
$621$	0	0
$622$	−0.603182	−0.0241854
$623$	32.8310	1.31535
$624$	0	0
$625$	−16.0062	−0.640247
$626$	4.99925	0.199810
$627$	0	0
$628$	27.5961	1.10120
$629$	−11.6930	−0.466229
$630$	0	0
$631$	−31.9574	−1.27220	−0.636102	−	0.771605i	$-0.719454\pi$
$631$	−31.9574	−1.27220	−0.636102	+	0.771605i	$0.719454\pi$
$632$	−15.7311	−0.625748
$633$	0	0
$634$	−12.9857	−0.515729
$635$	9.66179	0.383416
$636$	0	0
$637$	−6.17047	−0.244483
$638$	0.643956	0.0254945
$639$	0	0
$640$	37.9170	1.49880
$641$	4.56788	0.180420	0.0902102	−	0.995923i	$-0.471246\pi$
$641$	4.56788	0.180420	0.0902102	+	0.995923i	$0.471246\pi$
$642$	0	0
$643$	11.6990	0.461362	0.230681	−	0.973029i	$-0.425905\pi$
$643$	11.6990	0.461362	0.230681	+	0.973029i	$0.425905\pi$
$644$	0	0
$645$	0	0
$646$	−2.63722	−0.103760
$647$	−26.6323	−1.04702	−0.523512	−	0.852018i	$-0.675379\pi$
$647$	−26.6323	−1.04702	−0.523512	+	0.852018i	$0.675379\pi$
$648$	0	0
$649$	−10.4457	−0.410031
$650$	1.39179	0.0545907
$651$	0	0
$652$	12.0217	0.470807
$653$	−40.2675	−1.57579	−0.787895	−	0.615810i	$-0.788829\pi$
$653$	−40.2675	−1.57579	−0.787895	+	0.615810i	$0.788829\pi$
$654$	0	0
$655$	−1.30241	−0.0508892
$656$	−15.4548	−0.603407
$657$	0	0
$658$	−21.8685	−0.852522
$659$	−48.9366	−1.90630	−0.953150	−	0.302498i	$-0.902180\pi$
$659$	−48.9366	−1.90630	−0.953150	+	0.302498i	$0.902180\pi$
$660$	0	0
$661$	−6.54139	−0.254431	−0.127215	−	0.991875i	$-0.540604\pi$
$661$	−6.54139	−0.254431	−0.127215	+	0.991875i	$0.540604\pi$
$662$	3.58345	0.139275
$663$	0	0
$664$	13.5163	0.524535
$665$	25.6959	0.996443
$666$	0	0
$667$	0	0
$668$	35.4965	1.37340
$669$	0	0
$670$	12.5385	0.484406
$671$	−5.47383	−0.211315
$672$	0	0
$673$	39.2059	1.51128	0.755639	−	0.654989i	$-0.227327\pi$
$673$	39.2059	1.51128	0.755639	+	0.654989i	$0.227327\pi$
$674$	12.3124	0.474257
$675$	0	0
$676$	22.6540	0.871307
$677$	38.6263	1.48453	0.742265	−	0.670107i	$-0.233752\pi$
$677$	38.6263	1.48453	0.742265	+	0.670107i	$0.233752\pi$
$678$	0	0
$679$	−25.5728	−0.981395
$680$	20.0298	0.768107
$681$	0	0
$682$	3.04072	0.116435
$683$	−25.3556	−0.970206	−0.485103	−	0.874457i	$-0.661218\pi$
$683$	−25.3556	−0.970206	−0.485103	+	0.874457i	$0.661218\pi$
$684$	0	0
$685$	−69.1157	−2.64077
$686$	−14.3387	−0.547456
$687$	0	0
$688$	−29.4520	−1.12285
$689$	−5.58394	−0.212731
$690$	0	0
$691$	−46.4896	−1.76855	−0.884273	−	0.466970i	$-0.845346\pi$
$691$	−46.4896	−1.76855	−0.884273	+	0.466970i	$0.845346\pi$
$692$	18.3911	0.699126
$693$	0	0
$694$	−11.7860	−0.447389
$695$	34.1182	1.29418
$696$	0	0
$697$	−18.9657	−0.718378
$698$	9.26231	0.350584
$699$	0	0
$700$	−51.4654	−1.94521
$701$	37.8283	1.42875	0.714377	−	0.699761i	$-0.246710\pi$
$701$	37.8283	1.42875	0.714377	+	0.699761i	$0.246710\pi$
$702$	0	0
$703$	−5.96844	−0.225104
$704$	2.95578	0.111400
$705$	0	0
$706$	−11.2641	−0.423929
$707$	−35.8199	−1.34715
$708$	0	0
$709$	−38.9942	−1.46446	−0.732229	−	0.681059i	$-0.761520\pi$
$709$	−38.9942	−1.46446	−0.732229	+	0.681059i	$0.761520\pi$
$710$	4.83482	0.181448
$711$	0	0
$712$	13.0627	0.489547
$713$	0	0
$714$	0	0
$715$	1.50499	0.0562833
$716$	7.42886	0.277630
$717$	0	0
$718$	−2.89499	−0.108040
$719$	−7.76292	−0.289508	−0.144754	−	0.989468i	$-0.546239\pi$
$719$	−7.76292	−0.289508	−0.144754	+	0.989468i	$0.546239\pi$
$720$	0	0
$721$	−40.8633	−1.52183
$722$	7.75282	0.288530
$723$	0	0
$724$	1.07827	0.0400735
$725$	8.76919	0.325680
$726$	0	0
$727$	33.9017	1.25734	0.628671	−	0.777671i	$-0.283599\pi$
$727$	33.9017	1.25734	0.628671	+	0.777671i	$0.283599\pi$
$728$	−3.71899	−0.137835
$729$	0	0
$730$	6.30262	0.233270
$731$	−36.1428	−1.33679
$732$	0	0
$733$	−28.4087	−1.04930	−0.524650	−	0.851318i	$-0.675804\pi$
$733$	−28.4087	−1.04930	−0.524650	+	0.851318i	$0.675804\pi$
$734$	−1.52193	−0.0561754
$735$	0	0
$736$	0	0
$737$	7.61392	0.280462
$738$	0	0
$739$	−38.8992	−1.43093	−0.715465	−	0.698649i	$-0.753785\pi$
$739$	−38.8992	−1.43093	−0.715465	+	0.698649i	$0.753785\pi$
$740$	21.2867	0.782516
$741$	0	0
$742$	−26.7435	−0.981786
$743$	−22.7410	−0.834288	−0.417144	−	0.908840i	$-0.636969\pi$
$743$	−22.7410	−0.834288	−0.417144	+	0.908840i	$0.636969\pi$
$744$	0	0
$745$	34.7994	1.27495
$746$	0.265250	0.00971149
$747$	0	0
$748$	5.71158	0.208836
$749$	16.4097	0.599597
$750$	0	0
$751$	−3.22011	−0.117504	−0.0587518	−	0.998273i	$-0.518712\pi$
$751$	−3.22011	−0.117504	−0.0587518	+	0.998273i	$0.518712\pi$
$752$	26.9312	0.982078
$753$	0	0
$754$	0.297568	0.0108368
$755$	1.68976	0.0614966
$756$	0	0
$757$	34.7559	1.26323	0.631613	−	0.775284i	$-0.282393\pi$
$757$	34.7559	1.26323	0.631613	+	0.775284i	$0.282393\pi$
$758$	−1.11462	−0.0404847
$759$	0	0
$760$	10.2238	0.370856
$761$	−28.9240	−1.04849	−0.524247	−	0.851566i	$-0.675653\pi$
$761$	−28.9240	−1.04849	−0.524247	+	0.851566i	$0.675653\pi$
$762$	0	0
$763$	5.79037	0.209626
$764$	2.59287	0.0938069
$765$	0	0
$766$	−14.0216	−0.506620
$767$	−4.82691	−0.174290
$768$	0	0
$769$	20.5145	0.739773	0.369887	−	0.929077i	$-0.379397\pi$
$769$	20.5145	0.739773	0.369887	+	0.929077i	$0.379397\pi$
$770$	7.20794	0.259756
$771$	0	0
$772$	−37.6234	−1.35409
$773$	30.0464	1.08069	0.540346	−	0.841443i	$-0.318293\pi$
$773$	30.0464	1.08069	0.540346	+	0.841443i	$0.318293\pi$
$774$	0	0
$775$	41.4076	1.48741
$776$	−10.1748	−0.365256
$777$	0	0
$778$	4.87996	0.174955
$779$	−9.68067	−0.346846
$780$	0	0
$781$	2.93590	0.105055
$782$	0	0
$783$	0	0
$784$	36.3943	1.29980
$785$	−52.6316	−1.87850
$786$	0	0
$787$	17.7376	0.632278	0.316139	−	0.948713i	$-0.397613\pi$
$787$	17.7376	0.632278	0.316139	+	0.948713i	$0.397613\pi$
$788$	−10.7688	−0.383621
$789$	0	0
$790$	14.0888	0.501258
$791$	5.38222	0.191370
$792$	0	0
$793$	−2.52942	−0.0898225
$794$	−16.5333	−0.586744
$795$	0	0
$796$	−20.4832	−0.726008
$797$	−28.1813	−0.998233	−0.499117	−	0.866535i	$-0.666342\pi$
$797$	−28.1813	−0.998233	−0.499117	+	0.866535i	$0.666342\pi$
$798$	0	0
$799$	33.0493	1.16920
$800$	−31.3381	−1.10797
$801$	0	0
$802$	4.30498	0.152014
$803$	3.82721	0.135059
$804$	0	0
$805$	0	0
$806$	1.40510	0.0494926
$807$	0	0
$808$	−14.2519	−0.501381
$809$	−13.8665	−0.487521	−0.243760	−	0.969835i	$-0.578381\pi$
$809$	−13.8665	−0.487521	−0.243760	+	0.969835i	$0.578381\pi$
$810$	0	0
$811$	−34.1914	−1.20062	−0.600311	−	0.799767i	$-0.704956\pi$
$811$	−34.1914	−1.20062	−0.600311	+	0.799767i	$0.704956\pi$
$812$	−11.0034	−0.386144
$813$	0	0
$814$	−1.67421	−0.0586809
$815$	−22.9280	−0.803132
$816$	0	0
$817$	−18.4484	−0.645427
$818$	−10.7314	−0.375215
$819$	0	0
$820$	34.5266	1.20572
$821$	−21.9060	−0.764524	−0.382262	−	0.924054i	$-0.624855\pi$
$821$	−21.9060	−0.764524	−0.382262	+	0.924054i	$0.624855\pi$
$822$	0	0
$823$	55.8315	1.94616	0.973082	−	0.230461i	$-0.0740233\pi$
$823$	55.8315	1.94616	0.973082	+	0.230461i	$0.0740233\pi$
$824$	−16.2586	−0.566395
$825$	0	0
$826$	−23.1179	−0.804373
$827$	14.1505	0.492060	0.246030	−	0.969262i	$-0.420874\pi$
$827$	14.1505	0.492060	0.246030	+	0.969262i	$0.420874\pi$
$828$	0	0
$829$	−14.1731	−0.492253	−0.246126	−	0.969238i	$-0.579158\pi$
$829$	−14.1731	−0.492253	−0.246126	+	0.969238i	$0.579158\pi$
$830$	−12.1053	−0.420181
$831$	0	0
$832$	1.36585	0.0473523
$833$	44.6622	1.54745
$834$	0	0
$835$	−67.6994	−2.34283
$836$	2.91536	0.100830
$837$	0	0
$838$	4.88742	0.168833
$839$	−4.91915	−0.169828	−0.0849139	−	0.996388i	$-0.527062\pi$
$839$	−4.91915	−0.169828	−0.0849139	+	0.996388i	$0.527062\pi$
$840$	0	0
$841$	−27.1251	−0.935349
$842$	−7.68258	−0.264759
$843$	0	0
$844$	4.90031	0.168676
$845$	−43.2059	−1.48633
$846$	0	0
$847$	−45.5458	−1.56497
$848$	32.9348	1.13099
$849$	0	0
$850$	−10.0739	−0.345532
$851$	0	0
$852$	0	0
$853$	3.97343	0.136048	0.0680238	−	0.997684i	$-0.478331\pi$
$853$	3.97343	0.136048	0.0680238	+	0.997684i	$0.478331\pi$
$854$	−12.1143	−0.414545
$855$	0	0
$856$	6.52905	0.223158
$857$	34.0277	1.16236	0.581182	−	0.813774i	$-0.302590\pi$
$857$	34.0277	1.16236	0.581182	+	0.813774i	$0.302590\pi$
$858$	0	0
$859$	35.8429	1.22294	0.611472	−	0.791266i	$-0.290578\pi$
$859$	35.8429	1.22294	0.611472	+	0.791266i	$0.290578\pi$
$860$	65.7970	2.24366
$861$	0	0
$862$	−18.4212	−0.627430
$863$	8.91432	0.303447	0.151724	−	0.988423i	$-0.451518\pi$
$863$	8.91432	0.303447	0.151724	+	0.988423i	$0.451518\pi$
$864$	0	0
$865$	−35.0758	−1.19261
$866$	3.72318	0.126519
$867$	0	0
$868$	−51.9574	−1.76355
$869$	8.55533	0.290220
$870$	0	0
$871$	3.51835	0.119215
$872$	2.30386	0.0780185
$873$	0	0
$874$	0	0
$875$	21.5234	0.727624
$876$	0	0
$877$	−41.8207	−1.41219	−0.706093	−	0.708119i	$-0.749544\pi$
$877$	−41.8207	−1.41219	−0.706093	+	0.708119i	$0.749544\pi$
$878$	−16.7491	−0.565255
$879$	0	0
$880$	−8.87662	−0.299231
$881$	−39.9441	−1.34575	−0.672875	−	0.739756i	$-0.734941\pi$
$881$	−39.9441	−1.34575	−0.672875	+	0.739756i	$0.734941\pi$
$882$	0	0
$883$	54.3174	1.82793	0.913963	−	0.405798i	$-0.133006\pi$
$883$	54.3174	1.82793	0.913963	+	0.405798i	$0.133006\pi$
$884$	2.63929	0.0887688
$885$	0	0
$886$	−4.81787	−0.161860
$887$	−42.0408	−1.41159	−0.705796	−	0.708415i	$-0.749411\pi$
$887$	−42.0408	−1.41159	−0.705796	+	0.708415i	$0.749411\pi$
$888$	0	0
$889$	12.9846	0.435489
$890$	−11.6991	−0.392154
$891$	0	0
$892$	32.6553	1.09338
$893$	16.8694	0.564511
$894$	0	0
$895$	−14.1684	−0.473598
$896$	50.9570	1.70235
$897$	0	0
$898$	−8.17051	−0.272654
$899$	8.85303	0.295265
$900$	0	0
$901$	40.4169	1.34648
$902$	−2.71552	−0.0904171
$903$	0	0
$904$	2.14146	0.0712240
$905$	−2.05649	−0.0683599
$906$	0	0
$907$	−10.7171	−0.355856	−0.177928	−	0.984044i	$-0.556939\pi$
$907$	−10.7171	−0.355856	−0.177928	+	0.984044i	$0.556939\pi$
$908$	12.8380	0.426044
$909$	0	0
$910$	3.33075	0.110413
$911$	−4.65134	−0.154106	−0.0770530	−	0.997027i	$-0.524551\pi$
$911$	−4.65134	−0.154106	−0.0770530	+	0.997027i	$0.524551\pi$
$912$	0	0
$913$	−7.35085	−0.243278
$914$	11.6738	0.386134
$915$	0	0
$916$	−17.6309	−0.582541
$917$	−1.75032	−0.0578006
$918$	0	0
$919$	45.2701	1.49332	0.746661	−	0.665205i	$-0.231656\pi$
$919$	45.2701	1.49332	0.746661	+	0.665205i	$0.231656\pi$
$920$	0	0
$921$	0	0
$922$	18.2281	0.600310
$923$	1.35666	0.0446551
$924$	0	0
$925$	−22.7988	−0.749621
$926$	−6.83546	−0.224627
$927$	0	0
$928$	−6.70015	−0.219943
$929$	33.5458	1.10060	0.550301	−	0.834966i	$-0.314513\pi$
$929$	33.5458	1.10060	0.550301	+	0.834966i	$0.314513\pi$
$930$	0	0
$931$	22.7969	0.747139
$932$	9.56308	0.313249
$933$	0	0
$934$	−1.86957	−0.0611744
$935$	−10.8932	−0.356245
$936$	0	0
$937$	4.43887	0.145011	0.0725057	−	0.997368i	$-0.476900\pi$
$937$	4.43887	0.145011	0.0725057	+	0.997368i	$0.476900\pi$
$938$	16.8507	0.550194
$939$	0	0
$940$	−60.1654	−1.96238
$941$	−17.9751	−0.585972	−0.292986	−	0.956117i	$-0.594649\pi$
$941$	−17.9751	−0.585972	−0.292986	+	0.956117i	$0.594649\pi$
$942$	0	0
$943$	0	0
$944$	28.4698	0.926613
$945$	0	0
$946$	−5.17495	−0.168252
$947$	3.24237	0.105363	0.0526815	−	0.998611i	$-0.483223\pi$
$947$	3.24237	0.105363	0.0526815	+	0.998611i	$0.483223\pi$
$948$	0	0
$949$	1.76853	0.0574090
$950$	−5.14202	−0.166829
$951$	0	0
$952$	26.9182	0.872425
$953$	−13.2453	−0.429058	−0.214529	−	0.976718i	$-0.568822\pi$
$953$	−13.2453	−0.429058	−0.214529	+	0.976718i	$0.568822\pi$
$954$	0	0
$955$	−4.94516	−0.160022
$956$	−26.2345	−0.848485
$957$	0	0
$958$	8.81427	0.284776
$959$	−92.8853	−2.99942
$960$	0	0
$961$	10.8035	0.348499
$962$	−0.773641	−0.0249432
$963$	0	0
$964$	−25.9450	−0.835631
$965$	71.7557	2.30990
$966$	0	0
$967$	31.9173	1.02639	0.513196	−	0.858272i	$-0.328461\pi$
$967$	31.9173	1.02639	0.513196	+	0.858272i	$0.328461\pi$
$968$	−18.1216	−0.582451
$969$	0	0
$970$	9.11266	0.292590
$971$	55.7941	1.79052	0.895259	−	0.445547i	$-0.146991\pi$
$971$	55.7941	1.79052	0.895259	+	0.445547i	$0.146991\pi$
$972$	0	0
$973$	45.8518	1.46994
$974$	−6.24731	−0.200177
$975$	0	0
$976$	14.9189	0.477542
$977$	20.9426	0.670014	0.335007	−	0.942216i	$-0.391261\pi$
$977$	20.9426	0.670014	0.335007	+	0.942216i	$0.391261\pi$
$978$	0	0
$979$	−7.10416	−0.227050
$980$	−81.3064	−2.59724
$981$	0	0
$982$	11.4803	0.366350
$983$	−22.6442	−0.722238	−0.361119	−	0.932520i	$-0.617605\pi$
$983$	−22.6442	−0.722238	−0.361119	+	0.932520i	$0.617605\pi$
$984$	0	0
$985$	20.5383	0.654406
$986$	−2.15382	−0.0685915
$987$	0	0
$988$	1.34717	0.0428592
$989$	0	0
$990$	0	0
$991$	38.3804	1.21919	0.609596	−	0.792712i	$-0.291332\pi$
$991$	38.3804	1.21919	0.609596	+	0.792712i	$0.291332\pi$
$992$	−31.6377	−1.00450
$993$	0	0
$994$	6.49757	0.206090
$995$	39.0658	1.23847
$996$	0	0
$997$	−47.7368	−1.51184	−0.755919	−	0.654665i	$-0.772810\pi$
$997$	−47.7368	−1.51184	−0.755919	+	0.654665i	$0.772810\pi$
$998$	4.21475	0.133415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bp.1.3		5
3.2	odd	2		1587.2.a.q.1.3		5
23.3	even	11		207.2.i.a.55.1		10
23.8	even	11		207.2.i.a.64.1		10
23.22	odd	2		4761.2.a.bm.1.3		5
69.8	odd	22		69.2.e.b.64.1	yes	10
69.26	odd	22		69.2.e.b.55.1	✓	10
69.68	even	2		1587.2.a.r.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.b.55.1	✓	10	69.26	odd	22
69.2.e.b.64.1	yes	10	69.8	odd	22
207.2.i.a.55.1		10	23.3	even	11
207.2.i.a.64.1		10	23.8	even	11
1587.2.a.q.1.3		5	3.2	odd	2
1587.2.a.r.1.3		5	69.68	even	2
4761.2.a.bm.1.3		5	23.22	odd	2
4761.2.a.bp.1.3		5	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-0.478891 q^{2} -1.77066 q^{4} +3.37703 q^{5} +4.53843 q^{7} +1.80574 q^{8} +O(q^{10})\)	\(q-0.478891 q^{2} -1.77066 q^{4} +3.37703 q^{5} +4.53843 q^{7} +1.80574 q^{8} -1.61723 q^{10} -0.982050 q^{11} -0.453800 q^{13} -2.17341 q^{14} +2.67657 q^{16} +3.28463 q^{17} +1.67657 q^{19} -5.97958 q^{20} +0.470295 q^{22} +6.40433 q^{25} +0.217321 q^{26} -8.03603 q^{28} +1.36926 q^{29} +6.46556 q^{31} -4.89326 q^{32} -1.57298 q^{34} +15.3264 q^{35} -3.55991 q^{37} -0.802897 q^{38} +6.09803 q^{40} -5.77408 q^{41} -11.0036 q^{43} +1.73888 q^{44} +10.0618 q^{47} +13.5973 q^{49} -3.06698 q^{50} +0.803526 q^{52} +12.3048 q^{53} -3.31641 q^{55} +8.19521 q^{56} -0.655726 q^{58} +10.6367 q^{59} +5.57388 q^{61} -3.09630 q^{62} -3.00980 q^{64} -1.53250 q^{65} -7.75309 q^{67} -5.81597 q^{68} -7.33969 q^{70} -2.98957 q^{71} -3.89717 q^{73} +1.70481 q^{74} -2.96865 q^{76} -4.45696 q^{77} -8.71170 q^{79} +9.03887 q^{80} +2.76516 q^{82} +7.48521 q^{83} +11.0923 q^{85} +5.26954 q^{86} -1.77333 q^{88} +7.23401 q^{89} -2.05954 q^{91} -4.81851 q^{94} +5.66184 q^{95} -5.63473 q^{97} -6.51165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} + 7 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 4 * q^4 + 7 * q^5 + 3 * q^7 + 9 * q^8	\( 5 q - 2 q^{2} + 4 q^{4} + 7 q^{5} + 3 q^{7} + 9 q^{8} + 6 q^{10} + 2 q^{11} - 7 q^{13} - 10 q^{14} + 6 q^{16} + 16 q^{17} + q^{19} + 21 q^{20} - 14 q^{22} + 20 q^{25} + 5 q^{26} - 2 q^{28} - 18 q^{29} + 5 q^{31} + 6 q^{32} - 2 q^{34} + 24 q^{35} - 26 q^{37} + 26 q^{38} + 17 q^{40} - 9 q^{41} - 22 q^{43} + 28 q^{44} - 2 q^{47} + 2 q^{49} + 14 q^{50} + q^{52} + 35 q^{53} + 16 q^{55} - 10 q^{56} + 5 q^{58} - 6 q^{59} + 7 q^{61} - 13 q^{62} - 21 q^{64} + 10 q^{65} - 5 q^{67} + 15 q^{68} - 47 q^{70} + 18 q^{71} + 4 q^{73} + 28 q^{74} + 3 q^{76} + 21 q^{77} - 13 q^{79} + 37 q^{80} - 47 q^{82} + 24 q^{83} + 18 q^{85} + 11 q^{86} - 14 q^{88} - 7 q^{89} + 9 q^{91} - 8 q^{94} + 30 q^{95} - 6 q^{97} - 14 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 + 7 * q^5 + 3 * q^7 + 9 * q^8 + 6 * q^10 + 2 * q^11 - 7 * q^13 - 10 * q^14 + 6 * q^16 + 16 * q^17 + q^19 + 21 * q^20 - 14 * q^22 + 20 * q^25 + 5 * q^26 - 2 * q^28 - 18 * q^29 + 5 * q^31 + 6 * q^32 - 2 * q^34 + 24 * q^35 - 26 * q^37 + 26 * q^38 + 17 * q^40 - 9 * q^41 - 22 * q^43 + 28 * q^44 - 2 * q^47 + 2 * q^49 + 14 * q^50 + q^52 + 35 * q^53 + 16 * q^55 - 10 * q^56 + 5 * q^58 - 6 * q^59 + 7 * q^61 - 13 * q^62 - 21 * q^64 + 10 * q^65 - 5 * q^67 + 15 * q^68 - 47 * q^70 + 18 * q^71 + 4 * q^73 + 28 * q^74 + 3 * q^76 + 21 * q^77 - 13 * q^79 + 37 * q^80 - 47 * q^82 + 24 * q^83 + 18 * q^85 + 11 * q^86 - 14 * q^88 - 7 * q^89 + 9 * q^91 - 8 * q^94 + 30 * q^95 - 6 * q^97 - 14 * q^98

Introduction

L-functions

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