LMFDB - Embedded newform 4761.2.a.bx.1.6

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 30 x^{18} + 376 x^{16} - 2566 x^{14} + 10441 x^{12} - 26158 x^{10} + 40383 x^{8} - 37458 x^{6} + \cdots + 529 $ x^20 - 30x^18 + 376x^16 - 2566x^14 + 10441x^12 - 26158x^10 + 40383x^8 - 37458x^6 + 19703x^4 - 5244*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.31168$ 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)

$f(q)$	$=$	$q-1.31168 q^{2} -0.279492 q^{4} +3.00796 q^{5} +3.44211 q^{7} +2.98997 q^{8} +O(q^{10})$	$q-1.31168 q^{2} -0.279492 q^{4} +3.00796 q^{5} +3.44211 q^{7} +2.98997 q^{8} -3.94549 q^{10} -0.869027 q^{11} +4.15129 q^{13} -4.51495 q^{14} -3.36290 q^{16} +4.46055 q^{17} +7.10356 q^{19} -0.840700 q^{20} +1.13989 q^{22} +4.04783 q^{25} -5.44517 q^{26} -0.962039 q^{28} +8.77250 q^{29} -1.53058 q^{31} -1.56888 q^{32} -5.85083 q^{34} +10.3537 q^{35} -1.21135 q^{37} -9.31761 q^{38} +8.99370 q^{40} +3.11979 q^{41} +0.568839 q^{43} +0.242886 q^{44} -1.54648 q^{47} +4.84809 q^{49} -5.30946 q^{50} -1.16025 q^{52} -10.6788 q^{53} -2.61400 q^{55} +10.2918 q^{56} -11.5067 q^{58} +9.55774 q^{59} -5.21393 q^{61} +2.00764 q^{62} +8.78367 q^{64} +12.4869 q^{65} +12.8509 q^{67} -1.24669 q^{68} -13.5808 q^{70} -7.89904 q^{71} +11.8667 q^{73} +1.58891 q^{74} -1.98539 q^{76} -2.99128 q^{77} +0.667910 q^{79} -10.1155 q^{80} -4.09218 q^{82} -10.6552 q^{83} +13.4172 q^{85} -0.746135 q^{86} -2.59836 q^{88} -10.2022 q^{89} +14.2892 q^{91} +2.02849 q^{94} +21.3672 q^{95} +6.61530 q^{97} -6.35915 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) $ 20 * q + 20 * q^4 + 18 * q^7	$ 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * q^97

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 $

$2$	−1.31168	−0.927499	−0.463749	−	0.885966i	$-0.653496\pi$
$2$	−1.31168	−0.927499	−0.463749	+	0.885966i	$0.653496\pi$
$3$	0	0
$3$	0	0
$4$	−0.279492	−0.139746
$5$	3.00796	1.34520	0.672601	−	0.740006i	$-0.265177\pi$
$5$	3.00796	1.34520	0.672601	+	0.740006i	$0.265177\pi$
$6$	0	0
$7$	3.44211	1.30099	0.650497	−	0.759509i	$-0.274561\pi$
$7$	3.44211	1.30099	0.650497	+	0.759509i	$0.274561\pi$
$8$	2.98997	1.05711
$9$	0	0
$10$	−3.94549	−1.24767
$11$	−0.869027	−0.262022	−0.131011	−	0.991381i	$-0.541822\pi$
$11$	−0.869027	−0.262022	−0.131011	+	0.991381i	$0.541822\pi$
$12$	0	0
$13$	4.15129	1.15136	0.575680	−	0.817675i	$-0.304737\pi$
$13$	4.15129	1.15136	0.575680	+	0.817675i	$0.304737\pi$
$14$	−4.51495	−1.20667
$15$	0	0
$16$	−3.36290	−0.840725
$17$	4.46055	1.08184	0.540922	−	0.841073i	$-0.318076\pi$
$17$	4.46055	1.08184	0.540922	+	0.841073i	$0.318076\pi$
$18$	0	0
$19$	7.10356	1.62967	0.814834	−	0.579694i	$-0.196828\pi$
$19$	7.10356	1.62967	0.814834	+	0.579694i	$0.196828\pi$
$20$	−0.840700	−0.187986
$21$	0	0
$22$	1.13989	0.243025
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.04783	0.809566
$26$	−5.44517	−1.06789
$27$	0	0
$28$	−0.962039	−0.181808
$29$	8.77250	1.62901	0.814507	−	0.580154i	$-0.197008\pi$
$29$	8.77250	1.62901	0.814507	+	0.580154i	$0.197008\pi$
$30$	0	0
$31$	−1.53058	−0.274901	−0.137450	−	0.990509i	$-0.543891\pi$
$31$	−1.53058	−0.274901	−0.137450	+	0.990509i	$0.543891\pi$
$32$	−1.56888	−0.277341
$33$	0	0
$34$	−5.85083	−1.00341
$35$	10.3537	1.75010
$36$	0	0
$37$	−1.21135	−0.199145	−0.0995724	−	0.995030i	$-0.531747\pi$
$37$	−1.21135	−0.199145	−0.0995724	+	0.995030i	$0.531747\pi$
$38$	−9.31761	−1.51152
$39$	0	0
$40$	8.99370	1.42203
$41$	3.11979	0.487230	0.243615	−	0.969872i	$-0.421667\pi$
$41$	3.11979	0.487230	0.243615	+	0.969872i	$0.421667\pi$
$42$	0	0
$43$	0.568839	0.0867471	0.0433736	−	0.999059i	$-0.486189\pi$
$43$	0.568839	0.0867471	0.0433736	+	0.999059i	$0.486189\pi$
$44$	0.242886	0.0366164
$45$	0	0
$46$	0	0
$47$	−1.54648	−0.225578	−0.112789	−	0.993619i	$-0.535978\pi$
$47$	−1.54648	−0.225578	−0.112789	+	0.993619i	$0.535978\pi$
$48$	0	0
$49$	4.84809	0.692584
$50$	−5.30946	−0.750872
$51$	0	0
$52$	−1.16025	−0.160898
$53$	−10.6788	−1.46685	−0.733426	−	0.679769i	$-0.762080\pi$
$53$	−10.6788	−1.46685	−0.733426	+	0.679769i	$0.762080\pi$
$54$	0	0
$55$	−2.61400	−0.352472
$56$	10.2918	1.37530
$57$	0	0
$58$	−11.5067	−1.51091
$59$	9.55774	1.24431	0.622156	−	0.782894i	$-0.286257\pi$
$59$	9.55774	1.24431	0.622156	+	0.782894i	$0.286257\pi$
$60$	0	0
$61$	−5.21393	−0.667575	−0.333788	−	0.942648i	$-0.608327\pi$
$61$	−5.21393	−0.667575	−0.333788	+	0.942648i	$0.608327\pi$
$62$	2.00764	0.254970
$63$	0	0
$64$	8.78367	1.09796
$65$	12.4869	1.54881
$66$	0	0
$67$	12.8509	1.56998	0.784992	−	0.619506i	$-0.212667\pi$
$67$	12.8509	1.56998	0.784992	+	0.619506i	$0.212667\pi$
$68$	−1.24669	−0.151183
$69$	0	0
$70$	−13.5808	−1.62321
$71$	−7.89904	−0.937444	−0.468722	−	0.883346i	$-0.655285\pi$
$71$	−7.89904	−0.937444	−0.468722	+	0.883346i	$0.655285\pi$
$72$	0	0
$73$	11.8667	1.38889	0.694445	−	0.719546i	$-0.255650\pi$
$73$	11.8667	1.38889	0.694445	+	0.719546i	$0.255650\pi$
$74$	1.58891	0.184707
$75$	0	0
$76$	−1.98539	−0.227739
$77$	−2.99128	−0.340888
$78$	0	0
$79$	0.667910	0.0751458	0.0375729	−	0.999294i	$-0.488037\pi$
$79$	0.667910	0.0751458	0.0375729	+	0.999294i	$0.488037\pi$
$80$	−10.1155	−1.13094
$81$	0	0
$82$	−4.09218	−0.451905
$83$	−10.6552	−1.16956	−0.584780	−	0.811192i	$-0.698819\pi$
$83$	−10.6552	−1.16956	−0.584780	+	0.811192i	$0.698819\pi$
$84$	0	0
$85$	13.4172	1.45530
$86$	−0.746135	−0.0804578
$87$	0	0
$88$	−2.59836	−0.276986
$89$	−10.2022	−1.08143	−0.540716	−	0.841205i	$-0.681847\pi$
$89$	−10.2022	−1.08143	−0.540716	+	0.841205i	$0.681847\pi$
$90$	0	0
$91$	14.2892	1.49791
$92$	0	0
$93$	0	0
$94$	2.02849	0.209223
$95$	21.3672	2.19223
$96$	0	0
$97$	6.61530	0.671682	0.335841	−	0.941919i	$-0.390980\pi$
$97$	6.61530	0.671682	0.335841	+	0.941919i	$0.390980\pi$
$98$	−6.35915	−0.642371
$99$	0	0
$100$	−1.13133	−0.113133
$101$	−9.74267	−0.969432	−0.484716	−	0.874672i	$-0.661077\pi$
$101$	−9.74267	−0.969432	−0.484716	+	0.874672i	$0.661077\pi$
$102$	0	0
$103$	6.29792	0.620553	0.310276	−	0.950646i	$-0.399578\pi$
$103$	6.29792	0.620553	0.310276	+	0.950646i	$0.399578\pi$
$104$	12.4122	1.21712
$105$	0	0
$106$	14.0072	1.36050
$107$	−6.25085	−0.604292	−0.302146	−	0.953262i	$-0.597703\pi$
$107$	−6.25085	−0.604292	−0.302146	+	0.953262i	$0.597703\pi$
$108$	0	0
$109$	−10.4297	−0.998985	−0.499493	−	0.866318i	$-0.666480\pi$
$109$	−10.4297	−0.998985	−0.499493	+	0.866318i	$0.666480\pi$
$110$	3.42874	0.326917
$111$	0	0
$112$	−11.5755	−1.09378
$113$	−14.3909	−1.35378	−0.676892	−	0.736082i	$-0.736674\pi$
$113$	−14.3909	−1.35378	−0.676892	+	0.736082i	$0.736674\pi$
$114$	0	0
$115$	0	0
$116$	−2.45184	−0.227648
$117$	0	0
$118$	−12.5367	−1.15410
$119$	15.3537	1.40747
$120$	0	0
$121$	−10.2448	−0.931345
$122$	6.83902	0.619175
$123$	0	0
$124$	0.427785	0.0384162
$125$	−2.86409	−0.256172
$126$	0	0
$127$	15.2928	1.35701	0.678507	−	0.734594i	$-0.262627\pi$
$127$	15.2928	1.35701	0.678507	+	0.734594i	$0.262627\pi$
$128$	−8.38362	−0.741015
$129$	0	0
$130$	−16.3789	−1.43652
$131$	−0.344199	−0.0300728	−0.0150364	−	0.999887i	$-0.504786\pi$
$131$	−0.344199	−0.0300728	−0.0150364	+	0.999887i	$0.504786\pi$
$132$	0	0
$133$	24.4512	2.12019
$134$	−16.8563	−1.45616
$135$	0	0
$136$	13.3369	1.14363
$137$	−19.3166	−1.65033	−0.825163	−	0.564894i	$-0.808917\pi$
$137$	−19.3166	−1.65033	−0.825163	+	0.564894i	$0.808917\pi$
$138$	0	0
$139$	−22.0403	−1.86943	−0.934717	−	0.355393i	$-0.884347\pi$
$139$	−22.0403	−1.86943	−0.934717	+	0.355393i	$0.884347\pi$
$140$	−2.89378	−0.244569
$141$	0	0
$142$	10.3610	0.869478
$143$	−3.60759	−0.301681
$144$	0	0
$145$	26.3874	2.19135
$146$	−15.5653	−1.28819
$147$	0	0
$148$	0.338562	0.0278296
$149$	−8.97184	−0.735002	−0.367501	−	0.930023i	$-0.619787\pi$
$149$	−8.97184	−0.735002	−0.367501	+	0.930023i	$0.619787\pi$
$150$	0	0
$151$	4.52230	0.368020	0.184010	−	0.982924i	$-0.441092\pi$
$151$	4.52230	0.368020	0.184010	+	0.982924i	$0.441092\pi$
$152$	21.2394	1.72274
$153$	0	0
$154$	3.92361	0.316174
$155$	−4.60393	−0.369797
$156$	0	0
$157$	5.96135	0.475768	0.237884	−	0.971294i	$-0.423546\pi$
$157$	5.96135	0.475768	0.237884	+	0.971294i	$0.423546\pi$
$158$	−0.876086	−0.0696976
$159$	0	0
$160$	−4.71912	−0.373080
$161$	0	0
$162$	0	0
$163$	−7.19093	−0.563237	−0.281619	−	0.959526i	$-0.590871\pi$
$163$	−7.19093	−0.563237	−0.281619	+	0.959526i	$0.590871\pi$
$164$	−0.871956	−0.0680883
$165$	0	0
$166$	13.9762	1.08477
$167$	−0.203490	−0.0157466	−0.00787328	−	0.999969i	$-0.502506\pi$
$167$	−0.203490	−0.0157466	−0.00787328	+	0.999969i	$0.502506\pi$
$168$	0	0
$169$	4.23322	0.325632
$170$	−17.5991	−1.34979
$171$	0	0
$172$	−0.158986	−0.0121225
$173$	−23.8398	−1.81251	−0.906255	−	0.422732i	$-0.861071\pi$
$173$	−23.8398	−1.81251	−0.906255	+	0.422732i	$0.861071\pi$
$174$	0	0
$175$	13.9331	1.05324
$176$	2.92245	0.220288
$177$	0	0
$178$	13.3821	1.00303
$179$	−23.7936	−1.77841	−0.889207	−	0.457505i	$-0.848743\pi$
$179$	−23.7936	−1.77841	−0.889207	+	0.457505i	$0.848743\pi$
$180$	0	0
$181$	−5.81574	−0.432281	−0.216140	−	0.976362i	$-0.569347\pi$
$181$	−5.81574	−0.432281	−0.216140	+	0.976362i	$0.569347\pi$
$182$	−18.7429	−1.38931
$183$	0	0
$184$	0	0
$185$	−3.64370	−0.267890
$186$	0	0
$187$	−3.87634	−0.283466
$188$	0.432229	0.0315235
$189$	0	0
$190$	−28.0270	−2.03329
$191$	2.82708	0.204561	0.102280	−	0.994756i	$-0.467386\pi$
$191$	2.82708	0.204561	0.102280	+	0.994756i	$0.467386\pi$
$192$	0	0
$193$	−20.4283	−1.47046	−0.735232	−	0.677815i	$-0.762927\pi$
$193$	−20.4283	−1.47046	−0.735232	+	0.677815i	$0.762927\pi$
$194$	−8.67716	−0.622984
$195$	0	0
$196$	−1.35500	−0.0967857
$197$	−1.82904	−0.130314	−0.0651570	−	0.997875i	$-0.520755\pi$
$197$	−1.82904	−0.130314	−0.0651570	+	0.997875i	$0.520755\pi$
$198$	0	0
$199$	−6.50793	−0.461335	−0.230668	−	0.973033i	$-0.574091\pi$
$199$	−6.50793	−0.461335	−0.230668	+	0.973033i	$0.574091\pi$
$200$	12.1029	0.855803
$201$	0	0
$202$	12.7793	0.899147
$203$	30.1959	2.11934
$204$	0	0
$205$	9.38422	0.655422
$206$	−8.26087	−0.575562
$207$	0	0
$208$	−13.9604	−0.967978
$209$	−6.17319	−0.427008
$210$	0	0
$211$	−17.8994	−1.23224	−0.616122	−	0.787651i	$-0.711297\pi$
$211$	−17.8994	−1.23224	−0.616122	+	0.787651i	$0.711297\pi$
$212$	2.98465	0.204986
$213$	0	0
$214$	8.19912	0.560480
$215$	1.71104	0.116692
$216$	0	0
$217$	−5.26842	−0.357644
$218$	13.6805	0.926558
$219$	0	0
$220$	0.730591	0.0492564
$221$	18.5171	1.24559
$222$	0	0
$223$	−7.77423	−0.520601	−0.260300	−	0.965528i	$-0.583822\pi$
$223$	−7.77423	−0.520601	−0.260300	+	0.965528i	$0.583822\pi$
$224$	−5.40024	−0.360819
$225$	0	0
$226$	18.8763	1.25563
$227$	2.39549	0.158994	0.0794972	−	0.996835i	$-0.474669\pi$
$227$	2.39549	0.158994	0.0794972	+	0.996835i	$0.474669\pi$
$228$	0	0
$229$	14.7075	0.971897	0.485949	−	0.873987i	$-0.338474\pi$
$229$	14.7075	0.971897	0.485949	+	0.873987i	$0.338474\pi$
$230$	0	0
$231$	0	0
$232$	26.2295	1.72205
$233$	10.8846	0.713072	0.356536	−	0.934282i	$-0.383958\pi$
$233$	10.8846	0.713072	0.356536	+	0.934282i	$0.383958\pi$
$234$	0	0
$235$	−4.65176	−0.303447
$236$	−2.67131	−0.173887
$237$	0	0
$238$	−20.1392	−1.30543
$239$	0.560729	0.0362705	0.0181353	−	0.999836i	$-0.494227\pi$
$239$	0.560729	0.0362705	0.0181353	+	0.999836i	$0.494227\pi$
$240$	0	0
$241$	−28.3856	−1.82847	−0.914237	−	0.405179i	$-0.867209\pi$
$241$	−28.3856	−1.82847	−0.914237	+	0.405179i	$0.867209\pi$
$242$	13.4379	0.863821
$243$	0	0
$244$	1.45725	0.0932908
$245$	14.5829	0.931665
$246$	0	0
$247$	29.4889	1.87634
$248$	−4.57639	−0.290601
$249$	0	0
$250$	3.75678	0.237599
$251$	18.5896	1.17337	0.586683	−	0.809817i	$-0.300434\pi$
$251$	18.5896	1.17337	0.586683	+	0.809817i	$0.300434\pi$
$252$	0	0
$253$	0	0
$254$	−20.0592	−1.25863
$255$	0	0
$256$	−6.57070	−0.410669
$257$	−14.0303	−0.875184	−0.437592	−	0.899174i	$-0.644169\pi$
$257$	−14.0303	−0.875184	−0.437592	+	0.899174i	$0.644169\pi$
$258$	0	0
$259$	−4.16960	−0.259086
$260$	−3.48999	−0.216440
$261$	0	0
$262$	0.451479	0.0278925
$263$	−8.68036	−0.535254	−0.267627	−	0.963523i	$-0.586240\pi$
$263$	−8.68036	−0.535254	−0.267627	+	0.963523i	$0.586240\pi$
$264$	0	0
$265$	−32.1215	−1.97321
$266$	−32.0722	−1.96647
$267$	0	0
$268$	−3.59171	−0.219399
$269$	−4.56468	−0.278314	−0.139157	−	0.990270i	$-0.544439\pi$
$269$	−4.56468	−0.278314	−0.139157	+	0.990270i	$0.544439\pi$
$270$	0	0
$271$	2.47870	0.150571	0.0752853	−	0.997162i	$-0.476013\pi$
$271$	2.47870	0.150571	0.0752853	+	0.997162i	$0.476013\pi$
$272$	−15.0004	−0.909533
$273$	0	0
$274$	25.3372	1.53068
$275$	−3.51767	−0.212124
$276$	0	0
$277$	−11.7486	−0.705905	−0.352953	−	0.935641i	$-0.614822\pi$
$277$	−11.7486	−0.705905	−0.352953	+	0.935641i	$0.614822\pi$
$278$	28.9099	1.73390
$279$	0	0
$280$	30.9573	1.85005
$281$	−0.852554	−0.0508591	−0.0254296	−	0.999677i	$-0.508095\pi$
$281$	−0.852554	−0.0508591	−0.0254296	+	0.999677i	$0.508095\pi$
$282$	0	0
$283$	14.9843	0.890725	0.445363	−	0.895350i	$-0.353075\pi$
$283$	14.9843	0.890725	0.445363	+	0.895350i	$0.353075\pi$
$284$	2.20771	0.131004
$285$	0	0
$286$	4.73200	0.279809
$287$	10.7387	0.633883
$288$	0	0
$289$	2.89654	0.170385
$290$	−34.6118	−2.03248
$291$	0	0
$292$	−3.31664	−0.194091
$293$	15.9494	0.931774	0.465887	−	0.884844i	$-0.345735\pi$
$293$	15.9494	0.931774	0.465887	+	0.884844i	$0.345735\pi$
$294$	0	0
$295$	28.7493	1.67385
$296$	−3.62190	−0.210519
$297$	0	0
$298$	11.7682	0.681714
$299$	0	0
$300$	0	0
$301$	1.95800	0.112857
$302$	−5.93182	−0.341338
$303$	0	0
$304$	−23.8886	−1.37010
$305$	−15.6833	−0.898023
$306$	0	0
$307$	20.0111	1.14209	0.571047	−	0.820917i	$-0.306537\pi$
$307$	20.0111	1.14209	0.571047	+	0.820917i	$0.306537\pi$
$308$	0.836038	0.0476377
$309$	0	0
$310$	6.03889	0.342986
$311$	6.60327	0.374437	0.187219	−	0.982318i	$-0.440053\pi$
$311$	6.60327	0.374437	0.187219	+	0.982318i	$0.440053\pi$
$312$	0	0
$313$	−13.0292	−0.736451	−0.368226	−	0.929736i	$-0.620035\pi$
$313$	−13.0292	−0.736451	−0.368226	+	0.929736i	$0.620035\pi$
$314$	−7.81939	−0.441274
$315$	0	0
$316$	−0.186675	−0.0105013
$317$	−9.81512	−0.551272	−0.275636	−	0.961262i	$-0.588888\pi$
$317$	−9.81512	−0.551272	−0.275636	+	0.961262i	$0.588888\pi$
$318$	0	0
$319$	−7.62355	−0.426837
$320$	26.4209	1.47698
$321$	0	0
$322$	0	0
$323$	31.6858	1.76305
$324$	0	0
$325$	16.8037	0.932103
$326$	9.43221	0.522402
$327$	0	0
$328$	9.32808	0.515057
$329$	−5.32316	−0.293475
$330$	0	0
$331$	6.37949	0.350649	0.175324	−	0.984511i	$-0.443903\pi$
$331$	6.37949	0.350649	0.175324	+	0.984511i	$0.443903\pi$
$332$	2.97804	0.163441
$333$	0	0
$334$	0.266915	0.0146049
$335$	38.6549	2.11194
$336$	0	0
$337$	13.0202	0.709258	0.354629	−	0.935007i	$-0.384607\pi$
$337$	13.0202	0.709258	0.354629	+	0.935007i	$0.384607\pi$
$338$	−5.55263	−0.302023
$339$	0	0
$340$	−3.74999	−0.203372
$341$	1.33012	0.0720299
$342$	0	0
$343$	−7.40711	−0.399946
$344$	1.70081	0.0917015
$345$	0	0
$346$	31.2703	1.68110
$347$	−7.75767	−0.416453	−0.208227	−	0.978081i	$-0.566769\pi$
$347$	−7.75767	−0.416453	−0.208227	+	0.978081i	$0.566769\pi$
$348$	0	0
$349$	20.2596	1.08447	0.542237	−	0.840226i	$-0.317578\pi$
$349$	20.2596	1.08447	0.542237	+	0.840226i	$0.317578\pi$
$350$	−18.2757	−0.976879
$351$	0	0
$352$	1.36340	0.0726694
$353$	8.29023	0.441245	0.220622	−	0.975359i	$-0.429191\pi$
$353$	8.29023	0.441245	0.220622	+	0.975359i	$0.429191\pi$
$354$	0	0
$355$	−23.7600	−1.26105
$356$	2.85143	0.151126
$357$	0	0
$358$	31.2096	1.64948
$359$	17.0498	0.899855	0.449928	−	0.893065i	$-0.351450\pi$
$359$	17.0498	0.899855	0.449928	+	0.893065i	$0.351450\pi$
$360$	0	0
$361$	31.4606	1.65582
$362$	7.62840	0.400940
$363$	0	0
$364$	−3.99370	−0.209327
$365$	35.6945	1.86834
$366$	0	0
$367$	17.2878	0.902418	0.451209	−	0.892418i	$-0.350993\pi$
$367$	17.2878	0.902418	0.451209	+	0.892418i	$0.350993\pi$
$368$	0	0
$369$	0	0
$370$	4.77937	0.248468
$371$	−36.7577	−1.90836
$372$	0	0
$373$	30.4013	1.57412	0.787060	−	0.616876i	$-0.211602\pi$
$373$	30.4013	1.57412	0.787060	+	0.616876i	$0.211602\pi$
$374$	5.08453	0.262915
$375$	0	0
$376$	−4.62393	−0.238461
$377$	36.4172	1.87558
$378$	0	0
$379$	−3.82270	−0.196359	−0.0981795	−	0.995169i	$-0.531302\pi$
$379$	−3.82270	−0.196359	−0.0981795	+	0.995169i	$0.531302\pi$
$380$	−5.97196	−0.306355
$381$	0	0
$382$	−3.70823	−0.189730
$383$	16.7643	0.856616	0.428308	−	0.903633i	$-0.359110\pi$
$383$	16.7643	0.856616	0.428308	+	0.903633i	$0.359110\pi$
$384$	0	0
$385$	−8.99766	−0.458563
$386$	26.7955	1.36385
$387$	0	0
$388$	−1.84892	−0.0938647
$389$	13.8918	0.704344	0.352172	−	0.935935i	$-0.385443\pi$
$389$	13.8918	0.704344	0.352172	+	0.935935i	$0.385443\pi$
$390$	0	0
$391$	0	0
$392$	14.4956	0.732139
$393$	0	0
$394$	2.39912	0.120866
$395$	2.00905	0.101086
$396$	0	0
$397$	24.7907	1.24421	0.622104	−	0.782934i	$-0.286278\pi$
$397$	24.7907	1.24421	0.622104	+	0.782934i	$0.286278\pi$
$398$	8.53633	0.427888
$399$	0	0
$400$	−13.6125	−0.680623
$401$	7.43263	0.371168	0.185584	−	0.982628i	$-0.440582\pi$
$401$	7.43263	0.371168	0.185584	+	0.982628i	$0.440582\pi$
$402$	0	0
$403$	−6.35389	−0.316510
$404$	2.72299	0.135474
$405$	0	0
$406$	−39.6074	−1.96568
$407$	1.05270	0.0521803
$408$	0	0
$409$	15.7172	0.777166	0.388583	−	0.921414i	$-0.372965\pi$
$409$	15.7172	0.777166	0.388583	+	0.921414i	$0.372965\pi$
$410$	−12.3091	−0.607903
$411$	0	0
$412$	−1.76022	−0.0867196
$413$	32.8987	1.61884
$414$	0	0
$415$	−32.0504	−1.57329
$416$	−6.51287	−0.319320
$417$	0	0
$418$	8.09726	0.396050
$419$	26.2299	1.28142	0.640708	−	0.767785i	$-0.278641\pi$
$419$	26.2299	1.28142	0.640708	+	0.767785i	$0.278641\pi$
$420$	0	0
$421$	−7.65828	−0.373242	−0.186621	−	0.982432i	$-0.559754\pi$
$421$	−7.65828	−0.373242	−0.186621	+	0.982432i	$0.559754\pi$
$422$	23.4783	1.14290
$423$	0	0
$424$	−31.9294	−1.55063
$425$	18.0556	0.875823
$426$	0	0
$427$	−17.9469	−0.868511
$428$	1.74706	0.0844473
$429$	0	0
$430$	−2.24435	−0.108232
$431$	−22.7211	−1.09444	−0.547218	−	0.836990i	$-0.684313\pi$
$431$	−22.7211	−1.09444	−0.547218	+	0.836990i	$0.684313\pi$
$432$	0	0
$433$	11.2995	0.543020	0.271510	−	0.962436i	$-0.412477\pi$
$433$	11.2995	0.543020	0.271510	+	0.962436i	$0.412477\pi$
$434$	6.91050	0.331714
$435$	0	0
$436$	2.91502	0.139604
$437$	0	0
$438$	0	0
$439$	−11.3755	−0.542925	−0.271463	−	0.962449i	$-0.587507\pi$
$439$	−11.3755	−0.542925	−0.271463	+	0.962449i	$0.587507\pi$
$440$	−7.81578	−0.372602
$441$	0	0
$442$	−24.2885	−1.15529
$443$	21.2761	1.01086	0.505429	−	0.862868i	$-0.331334\pi$
$443$	21.2761	1.01086	0.505429	+	0.862868i	$0.331334\pi$
$444$	0	0
$445$	−30.6879	−1.45474
$446$	10.1973	0.482857
$447$	0	0
$448$	30.2343	1.42844
$449$	−39.9613	−1.88589	−0.942944	−	0.332951i	$-0.891956\pi$
$449$	−39.9613	−1.88589	−0.942944	+	0.332951i	$0.891956\pi$
$450$	0	0
$451$	−2.71119	−0.127665
$452$	4.02214	0.189186
$453$	0	0
$454$	−3.14212	−0.147467
$455$	42.9813	2.01499
$456$	0	0
$457$	−7.41656	−0.346932	−0.173466	−	0.984840i	$-0.555497\pi$
$457$	−7.41656	−0.346932	−0.173466	+	0.984840i	$0.555497\pi$
$458$	−19.2915	−0.901434
$459$	0	0
$460$	0	0
$461$	−33.8156	−1.57495	−0.787474	−	0.616348i	$-0.788611\pi$
$461$	−33.8156	−1.57495	−0.787474	+	0.616348i	$0.788611\pi$
$462$	0	0
$463$	21.2302	0.986652	0.493326	−	0.869845i	$-0.335781\pi$
$463$	21.2302	0.986652	0.493326	+	0.869845i	$0.335781\pi$
$464$	−29.5011	−1.36955
$465$	0	0
$466$	−14.2771	−0.661374
$467$	3.12039	0.144394	0.0721972	−	0.997390i	$-0.476999\pi$
$467$	3.12039	0.144394	0.0721972	+	0.997390i	$0.476999\pi$
$468$	0	0
$469$	44.2341	2.04254
$470$	6.10163	0.281447
$471$	0	0
$472$	28.5773	1.31538
$473$	−0.494337	−0.0227296
$474$	0	0
$475$	28.7540	1.31932
$476$	−4.29123	−0.196688
$477$	0	0
$478$	−0.735497	−0.0336409
$479$	−29.2097	−1.33462	−0.667312	−	0.744778i	$-0.732555\pi$
$479$	−29.2097	−1.33462	−0.667312	+	0.744778i	$0.732555\pi$
$480$	0	0
$481$	−5.02867	−0.229288
$482$	37.2328	1.69591
$483$	0	0
$484$	2.86333	0.130151
$485$	19.8986	0.903547
$486$	0	0
$487$	−37.5944	−1.70356	−0.851782	−	0.523897i	$-0.824478\pi$
$487$	−37.5944	−1.70356	−0.851782	+	0.523897i	$0.824478\pi$
$488$	−15.5895	−0.705703
$489$	0	0
$490$	−19.1281	−0.864118
$491$	25.3413	1.14364	0.571818	−	0.820381i	$-0.306238\pi$
$491$	25.3413	1.14364	0.571818	+	0.820381i	$0.306238\pi$
$492$	0	0
$493$	39.1302	1.76234
$494$	−38.6801	−1.74030
$495$	0	0
$496$	5.14720	0.231116
$497$	−27.1893	−1.21961
$498$	0	0
$499$	−16.3957	−0.733973	−0.366986	−	0.930226i	$-0.619610\pi$
$499$	−16.3957	−0.733973	−0.366986	+	0.930226i	$0.619610\pi$
$500$	0.800489	0.0357990
$501$	0	0
$502$	−24.3836	−1.08830
$503$	−28.8117	−1.28465	−0.642325	−	0.766432i	$-0.722030\pi$
$503$	−28.8117	−1.28465	−0.642325	+	0.766432i	$0.722030\pi$
$504$	0	0
$505$	−29.3056	−1.30408
$506$	0	0
$507$	0	0
$508$	−4.27420	−0.189637
$509$	−6.93755	−0.307501	−0.153751	−	0.988110i	$-0.549135\pi$
$509$	−6.93755	−0.307501	−0.153751	+	0.988110i	$0.549135\pi$
$510$	0	0
$511$	40.8464	1.80694
$512$	25.3859	1.12191
$513$	0	0
$514$	18.4032	0.811732
$515$	18.9439	0.834768
$516$	0	0
$517$	1.34394	0.0591063
$518$	5.46918	0.240302
$519$	0	0
$520$	37.3355	1.63727
$521$	9.57574	0.419521	0.209760	−	0.977753i	$-0.432732\pi$
$521$	9.57574	0.419521	0.209760	+	0.977753i	$0.432732\pi$
$522$	0	0
$523$	6.79947	0.297320	0.148660	−	0.988888i	$-0.452504\pi$
$523$	6.79947	0.297320	0.148660	+	0.988888i	$0.452504\pi$
$524$	0.0962007	0.00420255
$525$	0	0
$526$	11.3859	0.496448
$527$	−6.82724	−0.297399
$528$	0	0
$529$	0	0
$530$	42.1332	1.83015
$531$	0	0
$532$	−6.83390	−0.296287
$533$	12.9512	0.560977
$534$	0	0
$535$	−18.8023	−0.812894
$536$	38.4237	1.65965
$537$	0	0
$538$	5.98741	0.258136
$539$	−4.21312	−0.181472
$540$	0	0
$541$	−20.1102	−0.864606	−0.432303	−	0.901728i	$-0.642299\pi$
$541$	−20.1102	−0.864606	−0.432303	+	0.901728i	$0.642299\pi$
$542$	−3.25127	−0.139654
$543$	0	0
$544$	−6.99807	−0.300040
$545$	−31.3722	−1.34384
$546$	0	0
$547$	21.5894	0.923094	0.461547	−	0.887116i	$-0.347295\pi$
$547$	21.5894	0.923094	0.461547	+	0.887116i	$0.347295\pi$
$548$	5.39882	0.230626
$549$	0	0
$550$	4.61407	0.196745
$551$	62.3160	2.65475
$552$	0	0
$553$	2.29902	0.0977642
$554$	15.4104	0.654726
$555$	0	0
$556$	6.16008	0.261246
$557$	9.89328	0.419192	0.209596	−	0.977788i	$-0.432785\pi$
$557$	9.89328	0.419192	0.209596	+	0.977788i	$0.432785\pi$
$558$	0	0
$559$	2.36142	0.0998772
$560$	−34.8185	−1.47135
$561$	0	0
$562$	1.11828	0.0471718
$563$	21.6575	0.912757	0.456378	−	0.889786i	$-0.349146\pi$
$563$	21.6575	0.912757	0.456378	+	0.889786i	$0.349146\pi$
$564$	0	0
$565$	−43.2873	−1.82111
$566$	−19.6547	−0.826147
$567$	0	0
$568$	−23.6179	−0.990984
$569$	−11.0246	−0.462177	−0.231088	−	0.972933i	$-0.574229\pi$
$569$	−11.0246	−0.462177	−0.231088	+	0.972933i	$0.574229\pi$
$570$	0	0
$571$	−20.9227	−0.875586	−0.437793	−	0.899076i	$-0.644240\pi$
$571$	−20.9227	−0.875586	−0.437793	+	0.899076i	$0.644240\pi$
$572$	1.00829	0.0421587
$573$	0	0
$574$	−14.0857	−0.587926
$575$	0	0
$576$	0	0
$577$	24.3897	1.01536	0.507678	−	0.861547i	$-0.330504\pi$
$577$	24.3897	1.01536	0.507678	+	0.861547i	$0.330504\pi$
$578$	−3.79933	−0.158031
$579$	0	0
$580$	−7.37504	−0.306232
$581$	−36.6763	−1.52159
$582$	0	0
$583$	9.28021	0.384347
$584$	35.4810	1.46821
$585$	0	0
$586$	−20.9205	−0.864219
$587$	−28.0277	−1.15683	−0.578413	−	0.815744i	$-0.696328\pi$
$587$	−28.0277	−1.15683	−0.578413	+	0.815744i	$0.696328\pi$
$588$	0	0
$589$	−10.8726	−0.447997
$590$	−37.7099	−1.55249
$591$	0	0
$592$	4.07365	0.167426
$593$	−0.787202	−0.0323265	−0.0161633	−	0.999869i	$-0.505145\pi$
$593$	−0.787202	−0.0323265	−0.0161633	+	0.999869i	$0.505145\pi$
$594$	0	0
$595$	46.1833	1.89333
$596$	2.50755	0.102713
$597$	0	0
$598$	0	0
$599$	3.65036	0.149150	0.0745748	−	0.997215i	$-0.476240\pi$
$599$	3.65036	0.149150	0.0745748	+	0.997215i	$0.476240\pi$
$600$	0	0
$601$	−18.4283	−0.751705	−0.375852	−	0.926680i	$-0.622650\pi$
$601$	−18.4283	−0.751705	−0.375852	+	0.926680i	$0.622650\pi$
$602$	−2.56828	−0.104675
$603$	0	0
$604$	−1.26394	−0.0514292
$605$	−30.8159	−1.25285
$606$	0	0
$607$	−45.5013	−1.84684	−0.923421	−	0.383788i	$-0.874619\pi$
$607$	−45.5013	−1.84684	−0.923421	+	0.383788i	$0.874619\pi$
$608$	−11.1446	−0.451974
$609$	0	0
$610$	20.5715	0.832915
$611$	−6.41990	−0.259721
$612$	0	0
$613$	8.86139	0.357908	0.178954	−	0.983857i	$-0.442729\pi$
$613$	8.86139	0.357908	0.178954	+	0.983857i	$0.442729\pi$
$614$	−26.2482	−1.05929
$615$	0	0
$616$	−8.94384	−0.360358
$617$	−29.0336	−1.16885	−0.584423	−	0.811449i	$-0.698679\pi$
$617$	−29.0336	−1.16885	−0.584423	+	0.811449i	$0.698679\pi$
$618$	0	0
$619$	−14.3749	−0.577774	−0.288887	−	0.957363i	$-0.593285\pi$
$619$	−14.3749	−0.577774	−0.288887	+	0.957363i	$0.593285\pi$
$620$	1.28676	0.0516775
$621$	0	0
$622$	−8.66139	−0.347290
$623$	−35.1171	−1.40694
$624$	0	0
$625$	−28.8542	−1.15417
$626$	17.0901	0.683058
$627$	0	0
$628$	−1.66615	−0.0664865
$629$	−5.40329	−0.215443
$630$	0	0
$631$	27.3801	1.08999	0.544993	−	0.838441i	$-0.316532\pi$
$631$	27.3801	1.08999	0.544993	+	0.838441i	$0.316532\pi$
$632$	1.99703	0.0794376
$633$	0	0
$634$	12.8743	0.511304
$635$	46.0001	1.82546
$636$	0	0
$637$	20.1258	0.797414
$638$	9.99967	0.395891
$639$	0	0
$640$	−25.2176	−0.996814
$641$	−19.6305	−0.775356	−0.387678	−	0.921795i	$-0.626723\pi$
$641$	−19.6305	−0.775356	−0.387678	+	0.921795i	$0.626723\pi$
$642$	0	0
$643$	9.24128	0.364441	0.182220	−	0.983258i	$-0.441672\pi$
$643$	9.24128	0.364441	0.182220	+	0.983258i	$0.441672\pi$
$644$	0	0
$645$	0	0
$646$	−41.5617	−1.63522
$647$	−4.19396	−0.164881	−0.0824407	−	0.996596i	$-0.526272\pi$
$647$	−4.19396	−0.164881	−0.0824407	+	0.996596i	$0.526272\pi$
$648$	0	0
$649$	−8.30594	−0.326036
$650$	−22.0411	−0.864524
$651$	0	0
$652$	2.00980	0.0787100
$653$	32.2900	1.26360	0.631802	−	0.775130i	$-0.282316\pi$
$653$	32.2900	1.26360	0.631802	+	0.775130i	$0.282316\pi$
$654$	0	0
$655$	−1.03534	−0.0404540
$656$	−10.4916	−0.409627
$657$	0	0
$658$	6.98229	0.272198
$659$	9.77371	0.380730	0.190365	−	0.981713i	$-0.439033\pi$
$659$	9.77371	0.380730	0.190365	+	0.981713i	$0.439033\pi$
$660$	0	0
$661$	−23.0720	−0.897396	−0.448698	−	0.893683i	$-0.648112\pi$
$661$	−23.0720	−0.897396	−0.448698	+	0.893683i	$0.648112\pi$
$662$	−8.36786	−0.325226
$663$	0	0
$664$	−31.8587	−1.23636
$665$	73.5483	2.85208
$666$	0	0
$667$	0	0
$668$	0.0568738	0.00220052
$669$	0	0
$670$	−50.7030	−1.95883
$671$	4.53105	0.174919
$672$	0	0
$673$	8.09755	0.312138	0.156069	−	0.987746i	$-0.450118\pi$
$673$	8.09755	0.312138	0.156069	+	0.987746i	$0.450118\pi$
$674$	−17.0784	−0.657836
$675$	0	0
$676$	−1.18315	−0.0455057
$677$	37.0305	1.42320	0.711599	−	0.702586i	$-0.247971\pi$
$677$	37.0305	1.42320	0.711599	+	0.702586i	$0.247971\pi$
$678$	0	0
$679$	22.7706	0.873854
$680$	40.1169	1.53841
$681$	0	0
$682$	−1.74469	−0.0668077
$683$	31.9874	1.22396	0.611982	−	0.790871i	$-0.290372\pi$
$683$	31.9874	1.22396	0.611982	+	0.790871i	$0.290372\pi$
$684$	0	0
$685$	−58.1035	−2.22002
$686$	9.71577	0.370950
$687$	0	0
$688$	−1.91295	−0.0729305
$689$	−44.3310	−1.68888
$690$	0	0
$691$	20.8081	0.791579	0.395789	−	0.918341i	$-0.370471\pi$
$691$	20.8081	0.791579	0.395789	+	0.918341i	$0.370471\pi$
$692$	6.66303	0.253291
$693$	0	0
$694$	10.1756	0.386260
$695$	−66.2964	−2.51477
$696$	0	0
$697$	13.9160	0.527106
$698$	−26.5742	−1.00585
$699$	0	0
$700$	−3.89417	−0.147186
$701$	7.33144	0.276905	0.138452	−	0.990369i	$-0.455787\pi$
$701$	7.33144	0.276905	0.138452	+	0.990369i	$0.455787\pi$
$702$	0	0
$703$	−8.60490	−0.324540
$704$	−7.63325	−0.287689
$705$	0	0
$706$	−10.8741	−0.409254
$707$	−33.5353	−1.26122
$708$	0	0
$709$	13.2194	0.496465	0.248233	−	0.968700i	$-0.420150\pi$
$709$	13.2194	0.496465	0.248233	+	0.968700i	$0.420150\pi$
$710$	31.1656	1.16962
$711$	0	0
$712$	−30.5043	−1.14320
$713$	0	0
$714$	0	0
$715$	−10.8515	−0.405822
$716$	6.65010	0.248526
$717$	0	0
$718$	−22.3639	−0.834615
$719$	21.4899	0.801438	0.400719	−	0.916201i	$-0.368760\pi$
$719$	21.4899	0.801438	0.400719	+	0.916201i	$0.368760\pi$
$720$	0	0
$721$	21.6781	0.807335
$722$	−41.2663	−1.53577
$723$	0	0
$724$	1.62545	0.0604094
$725$	35.5096	1.31879
$726$	0	0
$727$	−28.0773	−1.04133	−0.520665	−	0.853761i	$-0.674316\pi$
$727$	−28.0773	−1.04133	−0.520665	+	0.853761i	$0.674316\pi$
$728$	42.7242	1.58346
$729$	0	0
$730$	−46.8198	−1.73288
$731$	2.53734	0.0938468
$732$	0	0
$733$	−31.1406	−1.15021	−0.575103	−	0.818081i	$-0.695038\pi$
$733$	−31.1406	−1.15021	−0.575103	+	0.818081i	$0.695038\pi$
$734$	−22.6761	−0.836992
$735$	0	0
$736$	0	0
$737$	−11.1678	−0.411370
$738$	0	0
$739$	36.1208	1.32872	0.664362	−	0.747411i	$-0.268703\pi$
$739$	36.1208	1.32872	0.664362	+	0.747411i	$0.268703\pi$
$740$	1.01838	0.0374365
$741$	0	0
$742$	48.2144	1.77001
$743$	41.7020	1.52990	0.764950	−	0.644090i	$-0.222764\pi$
$743$	41.7020	1.52990	0.764950	+	0.644090i	$0.222764\pi$
$744$	0	0
$745$	−26.9870	−0.988726
$746$	−39.8768	−1.46000
$747$	0	0
$748$	1.08341	0.0396132
$749$	−21.5161	−0.786180
$750$	0	0
$751$	21.6303	0.789300	0.394650	−	0.918832i	$-0.370866\pi$
$751$	21.6303	0.789300	0.394650	+	0.918832i	$0.370866\pi$
$752$	5.20067	0.189649
$753$	0	0
$754$	−47.7678	−1.73960
$755$	13.6029	0.495060
$756$	0	0
$757$	2.42970	0.0883089	0.0441544	−	0.999025i	$-0.485941\pi$
$757$	2.42970	0.0883089	0.0441544	+	0.999025i	$0.485941\pi$
$758$	5.01417	0.182123
$759$	0	0
$760$	63.8873	2.31744
$761$	20.3244	0.736760	0.368380	−	0.929675i	$-0.379913\pi$
$761$	20.3244	0.736760	0.368380	+	0.929675i	$0.379913\pi$
$762$	0	0
$763$	−35.9002	−1.29967
$764$	−0.790146	−0.0285865
$765$	0	0
$766$	−21.9894	−0.794510
$767$	39.6769	1.43265
$768$	0	0
$769$	55.1569	1.98901	0.994504	−	0.104700i	$-0.0333884\pi$
$769$	55.1569	1.98901	0.994504	+	0.104700i	$0.0333884\pi$
$770$	11.8021	0.425317
$771$	0	0
$772$	5.70955	0.205491
$773$	5.23547	0.188307	0.0941534	−	0.995558i	$-0.469986\pi$
$773$	5.23547	0.188307	0.0941534	+	0.995558i	$0.469986\pi$
$774$	0	0
$775$	−6.19554	−0.222550
$776$	19.7795	0.710043
$777$	0	0
$778$	−18.2217	−0.653279
$779$	22.1616	0.794023
$780$	0	0
$781$	6.86448	0.245630
$782$	0	0
$783$	0	0
$784$	−16.3036	−0.582273
$785$	17.9315	0.640003
$786$	0	0
$787$	−13.2953	−0.473928	−0.236964	−	0.971518i	$-0.576152\pi$
$787$	−13.2953	−0.473928	−0.236964	+	0.971518i	$0.576152\pi$
$788$	0.511202	0.0182108
$789$	0	0
$790$	−2.63523	−0.0937573
$791$	−49.5351	−1.76126
$792$	0	0
$793$	−21.6445	−0.768620
$794$	−32.5175	−1.15400
$795$	0	0
$796$	1.81891	0.0644696
$797$	25.1354	0.890342	0.445171	−	0.895446i	$-0.353143\pi$
$797$	25.1354	0.890342	0.445171	+	0.895446i	$0.353143\pi$
$798$	0	0
$799$	−6.89817	−0.244040
$800$	−6.35055	−0.224526
$801$	0	0
$802$	−9.74924	−0.344258
$803$	−10.3125	−0.363919
$804$	0	0
$805$	0	0
$806$	8.33428	0.293563
$807$	0	0
$808$	−29.1303	−1.02480
$809$	−28.5499	−1.00376	−0.501880	−	0.864937i	$-0.667358\pi$
$809$	−28.5499	−1.00376	−0.501880	+	0.864937i	$0.667358\pi$
$810$	0	0
$811$	−21.2485	−0.746137	−0.373069	−	0.927804i	$-0.621694\pi$
$811$	−21.2485	−0.746137	−0.373069	+	0.927804i	$0.621694\pi$
$812$	−8.43949	−0.296168
$813$	0	0
$814$	−1.38080	−0.0483971
$815$	−21.6300	−0.757667
$816$	0	0
$817$	4.04078	0.141369
$818$	−20.6160	−0.720821
$819$	0	0
$820$	−2.62281	−0.0915925
$821$	34.4139	1.20105	0.600526	−	0.799605i	$-0.294958\pi$
$821$	34.4139	1.20105	0.600526	+	0.799605i	$0.294958\pi$
$822$	0	0
$823$	−6.90364	−0.240646	−0.120323	−	0.992735i	$-0.538393\pi$
$823$	−6.90364	−0.240646	−0.120323	+	0.992735i	$0.538393\pi$
$824$	18.8306	0.655994
$825$	0	0
$826$	−43.1527	−1.50147
$827$	−39.7475	−1.38216	−0.691079	−	0.722779i	$-0.742864\pi$
$827$	−39.7475	−1.38216	−0.691079	+	0.722779i	$0.742864\pi$
$828$	0	0
$829$	−32.5685	−1.13115	−0.565575	−	0.824697i	$-0.691346\pi$
$829$	−32.5685	−1.13115	−0.565575	+	0.824697i	$0.691346\pi$
$830$	42.0399	1.45923
$831$	0	0
$832$	36.4636	1.26415
$833$	21.6252	0.749267
$834$	0	0
$835$	−0.612091	−0.0211823
$836$	1.72535	0.0596726
$837$	0	0
$838$	−34.4053	−1.18851
$839$	18.7928	0.648800	0.324400	−	0.945920i	$-0.394838\pi$
$839$	18.7928	0.648800	0.324400	+	0.945920i	$0.394838\pi$
$840$	0	0
$841$	47.9568	1.65368
$842$	10.0452	0.346181
$843$	0	0
$844$	5.00272	0.172201
$845$	12.7334	0.438041
$846$	0	0
$847$	−35.2636	−1.21167
$848$	35.9119	1.23322
$849$	0	0
$850$	−23.6831	−0.812325
$851$	0	0
$852$	0	0
$853$	5.62752	0.192683	0.0963413	−	0.995348i	$-0.469286\pi$
$853$	5.62752	0.192683	0.0963413	+	0.995348i	$0.469286\pi$
$854$	23.5406	0.805543
$855$	0	0
$856$	−18.6898	−0.638805
$857$	−21.4197	−0.731684	−0.365842	−	0.930677i	$-0.619219\pi$
$857$	−21.4197	−0.731684	−0.365842	+	0.930677i	$0.619219\pi$
$858$	0	0
$859$	22.2648	0.759664	0.379832	−	0.925055i	$-0.375982\pi$
$859$	22.2648	0.759664	0.379832	+	0.925055i	$0.375982\pi$
$860$	−0.478223	−0.0163073
$861$	0	0
$862$	29.8028	1.01509
$863$	28.9122	0.984184	0.492092	−	0.870543i	$-0.336232\pi$
$863$	28.9122	0.984184	0.492092	+	0.870543i	$0.336232\pi$
$864$	0	0
$865$	−71.7093	−2.43819
$866$	−14.8214	−0.503651
$867$	0	0
$868$	1.47248	0.0499792
$869$	−0.580432	−0.0196898
$870$	0	0
$871$	53.3477	1.80762
$872$	−31.1845	−1.05604
$873$	0	0
$874$	0	0
$875$	−9.85850	−0.333278
$876$	0	0
$877$	29.2722	0.988452	0.494226	−	0.869333i	$-0.335451\pi$
$877$	29.2722	0.988452	0.494226	+	0.869333i	$0.335451\pi$
$878$	14.9211	0.503563
$879$	0	0
$880$	8.79063	0.296332
$881$	8.29984	0.279629	0.139814	−	0.990178i	$-0.455349\pi$
$881$	8.29984	0.279629	0.139814	+	0.990178i	$0.455349\pi$
$882$	0	0
$883$	52.9343	1.78138	0.890691	−	0.454609i	$-0.150221\pi$
$883$	52.9343	1.78138	0.890691	+	0.454609i	$0.150221\pi$
$884$	−5.17536	−0.174066
$885$	0	0
$886$	−27.9075	−0.937570
$887$	−36.6267	−1.22980	−0.614901	−	0.788604i	$-0.710804\pi$
$887$	−36.6267	−1.22980	−0.614901	+	0.788604i	$0.710804\pi$
$888$	0	0
$889$	52.6393	1.76547
$890$	40.2527	1.34927
$891$	0	0
$892$	2.17283	0.0727518
$893$	−10.9855	−0.367617
$894$	0	0
$895$	−71.5701	−2.39232
$896$	−28.8573	−0.964055
$897$	0	0
$898$	52.4165	1.74916
$899$	−13.4270	−0.447817
$900$	0	0
$901$	−47.6335	−1.58690
$902$	3.55621	0.118409
$903$	0	0
$904$	−43.0284	−1.43110
$905$	−17.4935	−0.581505
$906$	0	0
$907$	−27.4655	−0.911978	−0.455989	−	0.889985i	$-0.650714\pi$
$907$	−27.4655	−0.911978	−0.455989	+	0.889985i	$0.650714\pi$
$908$	−0.669519	−0.0222188
$909$	0	0
$910$	−56.3778	−1.86891
$911$	23.1347	0.766487	0.383243	−	0.923647i	$-0.374807\pi$
$911$	23.1347	0.766487	0.383243	+	0.923647i	$0.374807\pi$
$912$	0	0
$913$	9.25966	0.306450
$914$	9.72816	0.321779
$915$	0	0
$916$	−4.11061	−0.135819
$917$	−1.18477	−0.0391245
$918$	0	0
$919$	−21.7546	−0.717618	−0.358809	−	0.933411i	$-0.616817\pi$
$919$	−21.7546	−0.717618	−0.358809	+	0.933411i	$0.616817\pi$
$920$	0	0
$921$	0	0
$922$	44.3552	1.46076
$923$	−32.7912	−1.07934
$924$	0	0
$925$	−4.90334	−0.161221
$926$	−27.8473	−0.915118
$927$	0	0
$928$	−13.7630	−0.451792
$929$	−44.5324	−1.46106	−0.730531	−	0.682880i	$-0.760727\pi$
$929$	−44.5324	−1.46106	−0.730531	+	0.682880i	$0.760727\pi$
$930$	0	0
$931$	34.4387	1.12868
$932$	−3.04215	−0.0996488
$933$	0	0
$934$	−4.09295	−0.133926
$935$	−11.6599	−0.381319
$936$	0	0
$937$	−13.1920	−0.430963	−0.215482	−	0.976508i	$-0.569132\pi$
$937$	−13.1920	−0.430963	−0.215482	+	0.976508i	$0.569132\pi$
$938$	−58.0210	−1.89445
$939$	0	0
$940$	1.30013	0.0424055
$941$	−20.4576	−0.666898	−0.333449	−	0.942768i	$-0.608213\pi$
$941$	−20.4576	−0.666898	−0.333449	+	0.942768i	$0.608213\pi$
$942$	0	0
$943$	0	0
$944$	−32.1417	−1.04612
$945$	0	0
$946$	0.648412	0.0210817
$947$	31.6938	1.02991	0.514955	−	0.857217i	$-0.327809\pi$
$947$	31.6938	1.02991	0.514955	+	0.857217i	$0.327809\pi$
$948$	0	0
$949$	49.2620	1.59911
$950$	−37.7161	−1.22367
$951$	0	0
$952$	45.9070	1.48786
$953$	26.4207	0.855851	0.427926	−	0.903814i	$-0.359245\pi$
$953$	26.4207	0.855851	0.427926	+	0.903814i	$0.359245\pi$
$954$	0	0
$955$	8.50376	0.275175
$956$	−0.156719	−0.00506865
$957$	0	0
$958$	38.3138	1.23786
$959$	−66.4897	−2.14706
$960$	0	0
$961$	−28.6573	−0.924430
$962$	6.59601	0.212664
$963$	0	0
$964$	7.93352	0.255522
$965$	−61.4477	−1.97807
$966$	0	0
$967$	41.2669	1.32705	0.663527	−	0.748153i	$-0.269059\pi$
$967$	41.2669	1.32705	0.663527	+	0.748153i	$0.269059\pi$
$968$	−30.6316	−0.984537
$969$	0	0
$970$	−26.1006	−0.838039
$971$	59.6523	1.91433	0.957167	−	0.289537i	$-0.0935014\pi$
$971$	59.6523	1.91433	0.957167	+	0.289537i	$0.0935014\pi$
$972$	0	0
$973$	−75.8650	−2.43212
$974$	49.3118	1.58005
$975$	0	0
$976$	17.5339	0.561247
$977$	−14.9789	−0.479218	−0.239609	−	0.970869i	$-0.577019\pi$
$977$	−14.9789	−0.479218	−0.239609	+	0.970869i	$0.577019\pi$
$978$	0	0
$979$	8.86600	0.283359
$980$	−4.07579	−0.130196
$981$	0	0
$982$	−33.2397	−1.06072
$983$	−5.35910	−0.170929	−0.0854644	−	0.996341i	$-0.527237\pi$
$983$	−5.35910	−0.170929	−0.0854644	+	0.996341i	$0.527237\pi$
$984$	0	0
$985$	−5.50169	−0.175298
$986$	−51.3264	−1.63457
$987$	0	0
$988$	−8.24191	−0.262210
$989$	0	0
$990$	0	0
$991$	−43.6942	−1.38799	−0.693997	−	0.719978i	$-0.744152\pi$
$991$	−43.6942	−1.38799	−0.693997	+	0.719978i	$0.744152\pi$
$992$	2.40130	0.0762413
$993$	0	0
$994$	35.6637	1.13119
$995$	−19.5756	−0.620588
$996$	0	0
$997$	−35.7589	−1.13250	−0.566248	−	0.824235i	$-0.691606\pi$
$997$	−35.7589	−1.13250	−0.566248	+	0.824235i	$0.691606\pi$
$998$	21.5059	0.680759
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bx.1.6		20
3.2	odd	2	inner	4761.2.a.bx.1.15		20
23.17	odd	22		207.2.i.e.82.3	yes	40
23.19	odd	22		207.2.i.e.154.3	yes	40
23.22	odd	2		4761.2.a.bw.1.6		20
69.17	even	22		207.2.i.e.82.2	✓	40
69.65	even	22		207.2.i.e.154.2	yes	40
69.68	even	2		4761.2.a.bw.1.15		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.i.e.82.2	✓	40	69.17	even	22
207.2.i.e.82.3	yes	40	23.17	odd	22
207.2.i.e.154.2	yes	40	69.65	even	22
207.2.i.e.154.3	yes	40	23.19	odd	22
4761.2.a.bw.1.6		20	23.22	odd	2
4761.2.a.bw.1.15		20	69.68	even	2
4761.2.a.bx.1.6		20	1.1	even	1	trivial
4761.2.a.bx.1.15		20	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.31168 q^{2} -0.279492 q^{4} +3.00796 q^{5} +3.44211 q^{7} +2.98997 q^{8} +O(q^{10})\)	\(q-1.31168 q^{2} -0.279492 q^{4} +3.00796 q^{5} +3.44211 q^{7} +2.98997 q^{8} -3.94549 q^{10} -0.869027 q^{11} +4.15129 q^{13} -4.51495 q^{14} -3.36290 q^{16} +4.46055 q^{17} +7.10356 q^{19} -0.840700 q^{20} +1.13989 q^{22} +4.04783 q^{25} -5.44517 q^{26} -0.962039 q^{28} +8.77250 q^{29} -1.53058 q^{31} -1.56888 q^{32} -5.85083 q^{34} +10.3537 q^{35} -1.21135 q^{37} -9.31761 q^{38} +8.99370 q^{40} +3.11979 q^{41} +0.568839 q^{43} +0.242886 q^{44} -1.54648 q^{47} +4.84809 q^{49} -5.30946 q^{50} -1.16025 q^{52} -10.6788 q^{53} -2.61400 q^{55} +10.2918 q^{56} -11.5067 q^{58} +9.55774 q^{59} -5.21393 q^{61} +2.00764 q^{62} +8.78367 q^{64} +12.4869 q^{65} +12.8509 q^{67} -1.24669 q^{68} -13.5808 q^{70} -7.89904 q^{71} +11.8667 q^{73} +1.58891 q^{74} -1.98539 q^{76} -2.99128 q^{77} +0.667910 q^{79} -10.1155 q^{80} -4.09218 q^{82} -10.6552 q^{83} +13.4172 q^{85} -0.746135 q^{86} -2.59836 q^{88} -10.2022 q^{89} +14.2892 q^{91} +2.02849 q^{94} +21.3672 q^{95} +6.61530 q^{97} -6.35915 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) \) 20 * q + 20 * q^4 + 18 * q^7	\( 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * q^97

Introduction

L-functions

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Varieties

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Groups

Database

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