LMFDB - Embedded newform 4761.2.a.bx.1.15

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			1.15
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	$-0.346504\pi$
$2$	1.31168	0.927499	0.463749	+	0.885966i	$0.346504\pi$
$3$	0	0
$3$	0	0
$4$	−0.279492	−0.139746
$5$	−3.00796	−1.34520	−0.672601	−	0.740006i	$-0.734823\pi$
$5$	−3.00796	−1.34520	−0.672601	+	0.740006i	$0.734823\pi$
$6$	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	$-0.458178\pi$
$11$	0.869027	0.262022	0.131011	+	0.991381i	$0.458178\pi$
$12$	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	$-0.681924\pi$
$17$	−4.46055	−1.08184	−0.540922	+	0.841073i	$0.681924\pi$
$18$	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	$-0.802992\pi$
$29$	−8.77250	−1.62901	−0.814507	+	0.580154i	$0.802992\pi$
$30$	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	$-0.578333\pi$
$41$	−3.11979	−0.487230	−0.243615	+	0.969872i	$0.578333\pi$
$42$	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	$-0.464022\pi$
$47$	1.54648	0.225578	0.112789	+	0.993619i	$0.464022\pi$
$48$	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	$-0.237920\pi$
$53$	10.6788	1.46685	0.733426	+	0.679769i	$0.237920\pi$
$54$	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	$-0.713743\pi$
$59$	−9.55774	−1.24431	−0.622156	+	0.782894i	$0.713743\pi$
$60$	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	$-0.344715\pi$
$71$	7.89904	0.937444	0.468722	+	0.883346i	$0.344715\pi$
$72$	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	$-0.301181\pi$
$83$	10.6552	1.16956	0.584780	+	0.811192i	$0.301181\pi$
$84$	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	$-0.318153\pi$
$89$	10.2022	1.08143	0.540716	+	0.841205i	$0.318153\pi$
$90$	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	$-0.338923\pi$
$101$	9.74267	0.969432	0.484716	+	0.874672i	$0.338923\pi$
$102$	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	$-0.402297\pi$
$107$	6.25085	0.604292	0.302146	+	0.953262i	$0.402297\pi$
$108$	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	$-0.263326\pi$
$113$	14.3909	1.35378	0.676892	+	0.736082i	$0.263326\pi$
$114$	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	$-0.495214\pi$
$131$	0.344199	0.0300728	0.0150364	+	0.999887i	$0.495214\pi$
$132$	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	$-0.191083\pi$
$137$	19.3166	1.65033	0.825163	+	0.564894i	$0.191083\pi$
$138$	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	$-0.380213\pi$
$149$	8.97184	0.735002	0.367501	+	0.930023i	$0.380213\pi$
$150$	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	$-0.497494\pi$
$167$	0.203490	0.0157466	0.00787328	+	0.999969i	$0.497494\pi$
$168$	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	$-0.138929\pi$
$173$	23.8398	1.81251	0.906255	+	0.422732i	$0.138929\pi$
$174$	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	$-0.151257\pi$
$179$	23.7936	1.77841	0.889207	+	0.457505i	$0.151257\pi$
$180$	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	$-0.532614\pi$
$191$	−2.82708	−0.204561	−0.102280	+	0.994756i	$0.532614\pi$
$192$	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	$-0.479245\pi$
$197$	1.82904	0.130314	0.0651570	+	0.997875i	$0.479245\pi$
$198$	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	$-0.525331\pi$
$227$	−2.39549	−0.158994	−0.0794972	+	0.996835i	$0.525331\pi$
$228$	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	$-0.616042\pi$
$233$	−10.8846	−0.713072	−0.356536	+	0.934282i	$0.616042\pi$
$234$	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	$-0.505773\pi$
$239$	−0.560729	−0.0362705	−0.0181353	+	0.999836i	$0.505773\pi$
$240$	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	$-0.699566\pi$
$251$	−18.5896	−1.17337	−0.586683	+	0.809817i	$0.699566\pi$
$252$	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	$-0.355831\pi$
$257$	14.0303	0.875184	0.437592	+	0.899174i	$0.355831\pi$
$258$	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	$-0.413760\pi$
$263$	8.68036	0.535254	0.267627	+	0.963523i	$0.413760\pi$
$264$	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	$-0.455561\pi$
$269$	4.56468	0.278314	0.139157	+	0.990270i	$0.455561\pi$
$270$	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	$-0.491905\pi$
$281$	0.852554	0.0508591	0.0254296	+	0.999677i	$0.491905\pi$
$282$	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	$-0.654265\pi$
$293$	−15.9494	−0.931774	−0.465887	+	0.884844i	$0.654265\pi$
$294$	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	$-0.559947\pi$
$311$	−6.60327	−0.374437	−0.187219	+	0.982318i	$0.559947\pi$
$312$	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	$-0.411112\pi$
$317$	9.81512	0.551272	0.275636	+	0.961262i	$0.411112\pi$
$318$	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	$-0.433231\pi$
$347$	7.75767	0.416453	0.208227	+	0.978081i	$0.433231\pi$
$348$	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	$-0.570809\pi$
$353$	−8.29023	−0.441245	−0.220622	+	0.975359i	$0.570809\pi$
$354$	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	$-0.648550\pi$
$359$	−17.0498	−0.899855	−0.449928	+	0.893065i	$0.648550\pi$
$360$	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	$-0.640890\pi$
$383$	−16.7643	−0.856616	−0.428308	+	0.903633i	$0.640890\pi$
$384$	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	$-0.614557\pi$
$389$	−13.8918	−0.704344	−0.352172	+	0.935935i	$0.614557\pi$
$390$	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	$-0.559418\pi$
$401$	−7.43263	−0.371168	−0.185584	+	0.982628i	$0.559418\pi$
$402$	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	$-0.721359\pi$
$419$	−26.2299	−1.28142	−0.640708	+	0.767785i	$0.721359\pi$
$420$	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	$-0.315687\pi$
$431$	22.7211	1.09444	0.547218	+	0.836990i	$0.315687\pi$
$432$	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	$-0.668666\pi$
$443$	−21.2761	−1.01086	−0.505429	+	0.862868i	$0.668666\pi$
$444$	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	$-0.108044\pi$
$449$	39.9613	1.88589	0.942944	+	0.332951i	$0.108044\pi$
$450$	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	$-0.211389\pi$
$461$	33.8156	1.57495	0.787474	+	0.616348i	$0.211389\pi$
$462$	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	$-0.523001\pi$
$467$	−3.12039	−0.144394	−0.0721972	+	0.997390i	$0.523001\pi$
$468$	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	$-0.267445\pi$
$479$	29.2097	1.33462	0.667312	+	0.744778i	$0.267445\pi$
$480$	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	$-0.693762\pi$
$491$	−25.3413	−1.14364	−0.571818	+	0.820381i	$0.693762\pi$
$492$	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	$-0.277970\pi$
$503$	28.8117	1.28465	0.642325	+	0.766432i	$0.277970\pi$
$504$	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	$-0.450865\pi$
$509$	6.93755	0.307501	0.153751	+	0.988110i	$0.450865\pi$
$510$	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	$-0.567268\pi$
$521$	−9.57574	−0.419521	−0.209760	+	0.977753i	$0.567268\pi$
$522$	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	$-0.567215\pi$
$557$	−9.89328	−0.419192	−0.209596	+	0.977788i	$0.567215\pi$
$558$	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	$-0.650854\pi$
$563$	−21.6575	−0.912757	−0.456378	+	0.889786i	$0.650854\pi$
$564$	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	$-0.425771\pi$
$569$	11.0246	0.462177	0.231088	+	0.972933i	$0.425771\pi$
$570$	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	$-0.303672\pi$
$587$	28.0277	1.15683	0.578413	+	0.815744i	$0.303672\pi$
$588$	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	$-0.494855\pi$
$593$	0.787202	0.0323265	0.0161633	+	0.999869i	$0.494855\pi$
$594$	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	$-0.523760\pi$
$599$	−3.65036	−0.149150	−0.0745748	+	0.997215i	$0.523760\pi$
$600$	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	$-0.301321\pi$
$617$	29.0336	1.16885	0.584423	+	0.811449i	$0.301321\pi$
$618$	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	$-0.373277\pi$
$641$	19.6305	0.775356	0.387678	+	0.921795i	$0.373277\pi$
$642$	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	$-0.473728\pi$
$647$	4.19396	0.164881	0.0824407	+	0.996596i	$0.473728\pi$
$648$	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	$-0.717684\pi$
$653$	−32.2900	−1.26360	−0.631802	+	0.775130i	$0.717684\pi$
$654$	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	$-0.560967\pi$
$659$	−9.77371	−0.380730	−0.190365	+	0.981713i	$0.560967\pi$
$660$	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	$-0.752029\pi$
$677$	−37.0305	−1.42320	−0.711599	+	0.702586i	$0.752029\pi$
$678$	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	$-0.709628\pi$
$683$	−31.9874	−1.22396	−0.611982	+	0.790871i	$0.709628\pi$
$684$	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	$-0.544213\pi$
$701$	−7.33144	−0.276905	−0.138452	+	0.990369i	$0.544213\pi$
$702$	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	$-0.631240\pi$
$719$	−21.4899	−0.801438	−0.400719	+	0.916201i	$0.631240\pi$
$720$	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	$-0.777236\pi$
$743$	−41.7020	−1.52990	−0.764950	+	0.644090i	$0.777236\pi$
$744$	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	$-0.620087\pi$
$761$	−20.3244	−0.736760	−0.368380	+	0.929675i	$0.620087\pi$
$762$	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	$-0.530014\pi$
$773$	−5.23547	−0.188307	−0.0941534	+	0.995558i	$0.530014\pi$
$774$	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	$-0.646857\pi$
$797$	−25.1354	−0.890342	−0.445171	+	0.895446i	$0.646857\pi$
$798$	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	$-0.332642\pi$
$809$	28.5499	1.00376	0.501880	+	0.864937i	$0.332642\pi$
$810$	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	$-0.705042\pi$
$821$	−34.4139	−1.20105	−0.600526	+	0.799605i	$0.705042\pi$
$822$	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	$-0.257136\pi$
$827$	39.7475	1.38216	0.691079	+	0.722779i	$0.257136\pi$
$828$	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	$-0.605162\pi$
$839$	−18.7928	−0.648800	−0.324400	+	0.945920i	$0.605162\pi$
$840$	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	$-0.380781\pi$
$857$	21.4197	0.731684	0.365842	+	0.930677i	$0.380781\pi$
$858$	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	$-0.663768\pi$
$863$	−28.9122	−0.984184	−0.492092	+	0.870543i	$0.663768\pi$
$864$	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	$-0.544651\pi$
$881$	−8.29984	−0.279629	−0.139814	+	0.990178i	$0.544651\pi$
$882$	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	$-0.289196\pi$
$887$	36.6267	1.22980	0.614901	+	0.788604i	$0.289196\pi$
$888$	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	$-0.625193\pi$
$911$	−23.1347	−0.766487	−0.383243	+	0.923647i	$0.625193\pi$
$912$	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	$-0.239273\pi$
$929$	44.5324	1.46106	0.730531	+	0.682880i	$0.239273\pi$
$930$	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	$-0.391787\pi$
$941$	20.4576	0.666898	0.333449	+	0.942768i	$0.391787\pi$
$942$	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	$-0.672191\pi$
$947$	−31.6938	−1.02991	−0.514955	+	0.857217i	$0.672191\pi$
$948$	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	$-0.640755\pi$
$953$	−26.4207	−0.855851	−0.427926	+	0.903814i	$0.640755\pi$
$954$	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	$-0.906499\pi$
$971$	−59.6523	−1.91433	−0.957167	+	0.289537i	$0.906499\pi$
$972$	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	$-0.422981\pi$
$977$	14.9789	0.479218	0.239609	+	0.970869i	$0.422981\pi$
$978$	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	$-0.472763\pi$
$983$	5.35910	0.170929	0.0854644	+	0.996341i	$0.472763\pi$
$984$	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.1.15		20
3.2	odd	2	inner	4761.2.a.bx.1.6		20
23.17	odd	22		207.2.i.e.82.2	✓	40
23.19	odd	22		207.2.i.e.154.2	yes	40
23.22	odd	2		4761.2.a.bw.1.15		20
69.17	even	22		207.2.i.e.82.3	yes	40
69.65	even	22		207.2.i.e.154.3	yes	40
69.68	even	2		4761.2.a.bw.1.6		20

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

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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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