LMFDB - Embedded newform 4761.2.a.bp.1.2

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			1.2
Root			$-0.830830$ 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)

$f(q)$	$=$	$q-1.59435 q^{2} +0.541956 q^{4} -2.53843 q^{5} -1.11325 q^{7} +2.32463 q^{8} +O(q^{10})$	$q-1.59435 q^{2} +0.541956 q^{4} -2.53843 q^{5} -1.11325 q^{7} +2.32463 q^{8} +4.04715 q^{10} +1.42094 q^{11} -4.22871 q^{13} +1.77491 q^{14} -4.79020 q^{16} +4.91899 q^{17} -5.79020 q^{19} -1.37572 q^{20} -2.26547 q^{22} +1.44362 q^{25} +6.74204 q^{26} -0.603331 q^{28} -6.42297 q^{29} +7.44168 q^{31} +2.98798 q^{32} -7.84259 q^{34} +2.82590 q^{35} -8.27686 q^{37} +9.23160 q^{38} -5.90092 q^{40} +4.60187 q^{41} +2.02446 q^{43} +0.770084 q^{44} -2.61810 q^{47} -5.76068 q^{49} -2.30164 q^{50} -2.29177 q^{52} +10.4294 q^{53} -3.60695 q^{55} -2.58789 q^{56} +10.2405 q^{58} -11.9787 q^{59} +11.1062 q^{61} -11.8646 q^{62} +4.81650 q^{64} +10.7343 q^{65} -11.3900 q^{67} +2.66587 q^{68} -4.50548 q^{70} -3.87666 q^{71} -12.3341 q^{73} +13.1962 q^{74} -3.13803 q^{76} -1.58185 q^{77} +5.45051 q^{79} +12.1596 q^{80} -7.33699 q^{82} +1.56378 q^{83} -12.4865 q^{85} -3.22770 q^{86} +3.30316 q^{88} +2.28331 q^{89} +4.70760 q^{91} +4.17416 q^{94} +14.6980 q^{95} -8.75140 q^{97} +9.18455 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.59435	−1.12738	−0.563688	−	0.825988i	$-0.690618\pi$
$2$	−1.59435	−1.12738	−0.563688	+	0.825988i	$0.690618\pi$
$3$	0	0
$3$	0	0
$4$	0.541956	0.270978
$5$	−2.53843	−1.13522	−0.567610	−	0.823298i	$-0.692132\pi$
$5$	−2.53843	−1.13522	−0.567610	+	0.823298i	$0.692132\pi$
$6$	0	0
$7$	−1.11325	−0.420768	−0.210384	−	0.977619i	$-0.567471\pi$
$7$	−1.11325	−0.420768	−0.210384	+	0.977619i	$0.567471\pi$
$8$	2.32463	0.821883
$9$	0	0
$10$	4.04715	1.27982
$11$	1.42094	0.428428	0.214214	−	0.976787i	$-0.431281\pi$
$11$	1.42094	0.428428	0.214214	+	0.976787i	$0.431281\pi$
$12$	0	0
$13$	−4.22871	−1.17283	−0.586416	−	0.810010i	$-0.699462\pi$
$13$	−4.22871	−1.17283	−0.586416	+	0.810010i	$0.699462\pi$
$14$	1.77491	0.474364
$15$	0	0
$16$	−4.79020	−1.19755
$17$	4.91899	1.19303	0.596515	−	0.802602i	$-0.296552\pi$
$17$	4.91899	1.19303	0.596515	+	0.802602i	$0.296552\pi$
$18$	0	0
$19$	−5.79020	−1.32836	−0.664181	−	0.747572i	$-0.731220\pi$
$19$	−5.79020	−1.32836	−0.664181	+	0.747572i	$0.731220\pi$
$20$	−1.37572	−0.307619
$21$	0	0
$22$	−2.26547	−0.483000
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.44362	0.288724
$26$	6.74204	1.32222
$27$	0	0
$28$	−0.603331	−0.114019
$29$	−6.42297	−1.19272	−0.596358	−	0.802719i	$-0.703386\pi$
$29$	−6.42297	−1.19272	−0.596358	+	0.802719i	$0.703386\pi$
$30$	0	0
$31$	7.44168	1.33656	0.668282	−	0.743908i	$-0.267030\pi$
$31$	7.44168	1.33656	0.668282	+	0.743908i	$0.267030\pi$
$32$	2.98798	0.528206
$33$	0	0
$34$	−7.84259	−1.34499
$35$	2.82590	0.477664
$36$	0	0
$37$	−8.27686	−1.36071	−0.680354	−	0.732884i	$-0.738174\pi$
$37$	−8.27686	−1.36071	−0.680354	+	0.732884i	$0.738174\pi$
$38$	9.23160	1.49756
$39$	0	0
$40$	−5.90092	−0.933017
$41$	4.60187	0.718691	0.359345	−	0.933205i	$-0.383000\pi$
$41$	4.60187	0.718691	0.359345	+	0.933205i	$0.383000\pi$
$42$	0	0
$43$	2.02446	0.308728	0.154364	−	0.988014i	$-0.450667\pi$
$43$	2.02446	0.308728	0.154364	+	0.988014i	$0.450667\pi$
$44$	0.770084	0.116095
$45$	0	0
$46$	0	0
$47$	−2.61810	−0.381888	−0.190944	−	0.981601i	$-0.561155\pi$
$47$	−2.61810	−0.381888	−0.190944	+	0.981601i	$0.561155\pi$
$48$	0	0
$49$	−5.76068	−0.822954
$50$	−2.30164	−0.325501
$51$	0	0
$52$	−2.29177	−0.317812
$53$	10.4294	1.43259	0.716296	−	0.697797i	$-0.245836\pi$
$53$	10.4294	1.43259	0.716296	+	0.697797i	$0.245836\pi$
$54$	0	0
$55$	−3.60695	−0.486360
$56$	−2.58789	−0.345822
$57$	0	0
$58$	10.2405	1.34464
$59$	−11.9787	−1.55950	−0.779748	−	0.626093i	$-0.784653\pi$
$59$	−11.9787	−1.55950	−0.779748	+	0.626093i	$0.784653\pi$
$60$	0	0
$61$	11.1062	1.42201	0.711004	−	0.703188i	$-0.248241\pi$
$61$	11.1062	1.42201	0.711004	+	0.703188i	$0.248241\pi$
$62$	−11.8646	−1.50681
$63$	0	0
$64$	4.81650	0.602062
$65$	10.7343	1.33142
$66$	0	0
$67$	−11.3900	−1.39151	−0.695755	−	0.718279i	$-0.744930\pi$
$67$	−11.3900	−1.39151	−0.695755	+	0.718279i	$0.744930\pi$
$68$	2.66587	0.323284
$69$	0	0
$70$	−4.50548	−0.538508
$71$	−3.87666	−0.460075	−0.230038	−	0.973182i	$-0.573885\pi$
$71$	−3.87666	−0.460075	−0.230038	+	0.973182i	$0.573885\pi$
$72$	0	0
$73$	−12.3341	−1.44359	−0.721797	−	0.692105i	$-0.756684\pi$
$73$	−12.3341	−1.44359	−0.721797	+	0.692105i	$0.756684\pi$
$74$	13.1962	1.53403
$75$	0	0
$76$	−3.13803	−0.359957
$77$	−1.58185	−0.180269
$78$	0	0
$79$	5.45051	0.613230	0.306615	−	0.951834i	$-0.400804\pi$
$79$	5.45051	0.613230	0.306615	+	0.951834i	$0.400804\pi$
$80$	12.1596	1.35948
$81$	0	0
$82$	−7.33699	−0.810235
$83$	1.56378	0.171647	0.0858234	−	0.996310i	$-0.472648\pi$
$83$	1.56378	0.171647	0.0858234	+	0.996310i	$0.472648\pi$
$84$	0	0
$85$	−12.4865	−1.35435
$86$	−3.22770	−0.348052
$87$	0	0
$88$	3.30316	0.352118
$89$	2.28331	0.242031	0.121015	−	0.992651i	$-0.461385\pi$
$89$	2.28331	0.242031	0.121015	+	0.992651i	$0.461385\pi$
$90$	0	0
$91$	4.70760	0.493490
$92$	0	0
$93$	0	0
$94$	4.17416	0.430532
$95$	14.6980	1.50798
$96$	0	0
$97$	−8.75140	−0.888570	−0.444285	−	0.895886i	$-0.646542\pi$
$97$	−8.75140	−0.888570	−0.444285	+	0.895886i	$0.646542\pi$
$98$	9.18455	0.927779
$99$	0	0
$100$	0.782379	0.0782379
$101$	−2.59909	−0.258619	−0.129309	−	0.991604i	$-0.541276\pi$
$101$	−2.59909	−0.258619	−0.129309	+	0.991604i	$0.541276\pi$
$102$	0	0
$103$	0.233923	0.0230491	0.0115246	−	0.999934i	$-0.496332\pi$
$103$	0.233923	0.0230491	0.0115246	+	0.999934i	$0.496332\pi$
$104$	−9.83020	−0.963930
$105$	0	0
$106$	−16.6282	−1.61507
$107$	−1.43536	−0.138761	−0.0693807	−	0.997590i	$-0.522102\pi$
$107$	−1.43536	−0.138761	−0.0693807	+	0.997590i	$0.522102\pi$
$108$	0	0
$109$	−14.2583	−1.36570	−0.682848	−	0.730561i	$-0.739259\pi$
$109$	−14.2583	−1.36570	−0.682848	+	0.730561i	$0.739259\pi$
$110$	5.75074	0.548311
$111$	0	0
$112$	5.33267	0.503890
$113$	−4.59752	−0.432498	−0.216249	−	0.976338i	$-0.569382\pi$
$113$	−4.59752	−0.432498	−0.216249	+	0.976338i	$0.569382\pi$
$114$	0	0
$115$	0	0
$116$	−3.48096	−0.323199
$117$	0	0
$118$	19.0983	1.75814
$119$	−5.47605	−0.501989
$120$	0	0
$121$	−8.98094	−0.816449
$122$	−17.7072	−1.60314
$123$	0	0
$124$	4.03306	0.362179
$125$	9.02761	0.807454
$126$	0	0
$127$	2.70167	0.239734	0.119867	−	0.992790i	$-0.461753\pi$
$127$	2.70167	0.239734	0.119867	+	0.992790i	$0.461753\pi$
$128$	−13.6552	−1.20696
$129$	0	0
$130$	−17.1142	−1.50101
$131$	−7.07648	−0.618275	−0.309138	−	0.951017i	$-0.600040\pi$
$131$	−7.07648	−0.618275	−0.309138	+	0.951017i	$0.600040\pi$
$132$	0	0
$133$	6.44592	0.558932
$134$	18.1597	1.56876
$135$	0	0
$136$	11.4348	0.980530
$137$	4.85066	0.414420	0.207210	−	0.978296i	$-0.433562\pi$
$137$	4.85066	0.414420	0.207210	+	0.978296i	$0.433562\pi$
$138$	0	0
$139$	−20.0995	−1.70482	−0.852410	−	0.522875i	$-0.824860\pi$
$139$	−20.0995	−1.70482	−0.852410	+	0.522875i	$0.824860\pi$
$140$	1.53151	0.129436
$141$	0	0
$142$	6.18076	0.518678
$143$	−6.00872	−0.502475
$144$	0	0
$145$	16.3043	1.35399
$146$	19.6649	1.62747
$147$	0	0
$148$	−4.48569	−0.368721
$149$	−7.59966	−0.622588	−0.311294	−	0.950314i	$-0.600762\pi$
$149$	−7.59966	−0.622588	−0.311294	+	0.950314i	$0.600762\pi$
$150$	0	0
$151$	11.9845	0.975281	0.487641	−	0.873044i	$-0.337858\pi$
$151$	11.9845	0.975281	0.487641	+	0.873044i	$0.337858\pi$
$152$	−13.4601	−1.09176
$153$	0	0
$154$	2.52203	0.203231
$155$	−18.8902	−1.51729
$156$	0	0
$157$	6.06277	0.483862	0.241931	−	0.970294i	$-0.422219\pi$
$157$	6.06277	0.483862	0.241931	+	0.970294i	$0.422219\pi$
$158$	−8.69002	−0.691341
$159$	0	0
$160$	−7.58478	−0.599630
$161$	0	0
$162$	0	0
$163$	16.7017	1.30818	0.654090	−	0.756417i	$-0.273052\pi$
$163$	16.7017	1.30818	0.654090	+	0.756417i	$0.273052\pi$
$164$	2.49401	0.194749
$165$	0	0
$166$	−2.49321	−0.193510
$167$	−20.3186	−1.57230	−0.786151	−	0.618035i	$-0.787929\pi$
$167$	−20.3186	−1.57230	−0.786151	+	0.618035i	$0.787929\pi$
$168$	0	0
$169$	4.88197	0.375536
$170$	19.9079	1.52686
$171$	0	0
$172$	1.09717	0.0836583
$173$	1.94187	0.147637	0.0738187	−	0.997272i	$-0.476481\pi$
$173$	1.94187	0.147637	0.0738187	+	0.997272i	$0.476481\pi$
$174$	0	0
$175$	−1.60711	−0.121486
$176$	−6.80656	−0.513064
$177$	0	0
$178$	−3.64041	−0.272860
$179$	17.5279	1.31009	0.655047	−	0.755588i	$-0.272649\pi$
$179$	17.5279	1.31009	0.655047	+	0.755588i	$0.272649\pi$
$180$	0	0
$181$	13.0317	0.968636	0.484318	−	0.874892i	$-0.339068\pi$
$181$	13.0317	0.968636	0.484318	+	0.874892i	$0.339068\pi$
$182$	−7.50557	−0.556350
$183$	0	0
$184$	0	0
$185$	21.0102	1.54470
$186$	0	0
$187$	6.98957	0.511128
$188$	−1.41889	−0.103483
$189$	0	0
$190$	−23.4338	−1.70006
$191$	−14.1180	−1.02154	−0.510771	−	0.859717i	$-0.670640\pi$
$191$	−14.1180	−1.02154	−0.510771	+	0.859717i	$0.670640\pi$
$192$	0	0
$193$	12.1481	0.874442	0.437221	−	0.899354i	$-0.355963\pi$
$193$	12.1481	0.874442	0.437221	+	0.899354i	$0.355963\pi$
$194$	13.9528	1.00175
$195$	0	0
$196$	−3.12203	−0.223002
$197$	4.38380	0.312333	0.156166	−	0.987731i	$-0.450086\pi$
$197$	4.38380	0.312333	0.156166	+	0.987731i	$0.450086\pi$
$198$	0	0
$199$	16.9874	1.20420	0.602102	−	0.798419i	$-0.294330\pi$
$199$	16.9874	1.20420	0.602102	+	0.798419i	$0.294330\pi$
$200$	3.35589	0.237297
$201$	0	0
$202$	4.14386	0.291561
$203$	7.15036	0.501857
$204$	0	0
$205$	−11.6815	−0.815872
$206$	−0.372955	−0.0259850
$207$	0	0
$208$	20.2563	1.40452
$209$	−8.22750	−0.569108
$210$	0	0
$211$	−7.10490	−0.489121	−0.244561	−	0.969634i	$-0.578644\pi$
$211$	−7.10490	−0.489121	−0.244561	+	0.969634i	$0.578644\pi$
$212$	5.65229	0.388201
$213$	0	0
$214$	2.28847	0.156436
$215$	−5.13895	−0.350474
$216$	0	0
$217$	−8.28443	−0.562384
$218$	22.7327	1.53965
$219$	0	0
$220$	−1.95480	−0.131793
$221$	−20.8010	−1.39922
$222$	0	0
$223$	−5.10101	−0.341589	−0.170794	−	0.985307i	$-0.554633\pi$
$223$	−5.10101	−0.341589	−0.170794	+	0.985307i	$0.554633\pi$
$224$	−3.32637	−0.222252
$225$	0	0
$226$	7.33005	0.487588
$227$	13.9768	0.927674	0.463837	−	0.885921i	$-0.346472\pi$
$227$	13.9768	0.927674	0.463837	+	0.885921i	$0.346472\pi$
$228$	0	0
$229$	2.53723	0.167665	0.0838324	−	0.996480i	$-0.473284\pi$
$229$	2.53723	0.167665	0.0838324	+	0.996480i	$0.473284\pi$
$230$	0	0
$231$	0	0
$232$	−14.9311	−0.980272
$233$	−1.22954	−0.0805499	−0.0402750	−	0.999189i	$-0.512823\pi$
$233$	−1.22954	−0.0805499	−0.0402750	+	0.999189i	$0.512823\pi$
$234$	0	0
$235$	6.64585	0.433527
$236$	−6.49193	−0.422589
$237$	0	0
$238$	8.73075	0.565930
$239$	10.1247	0.654910	0.327455	−	0.944867i	$-0.393809\pi$
$239$	10.1247	0.654910	0.327455	+	0.944867i	$0.393809\pi$
$240$	0	0
$241$	20.1723	1.29941	0.649707	−	0.760185i	$-0.274892\pi$
$241$	20.1723	1.29941	0.649707	+	0.760185i	$0.274892\pi$
$242$	14.3188	0.920445
$243$	0	0
$244$	6.01909	0.385332
$245$	14.6231	0.934234
$246$	0	0
$247$	24.4850	1.55795
$248$	17.2992	1.09850
$249$	0	0
$250$	−14.3932	−0.910305
$251$	15.9853	1.00898	0.504492	−	0.863417i	$-0.331680\pi$
$251$	15.9853	1.00898	0.504492	+	0.863417i	$0.331680\pi$
$252$	0	0
$253$	0	0
$254$	−4.30740	−0.270270
$255$	0	0
$256$	12.1381	0.758632
$257$	21.7160	1.35460	0.677302	−	0.735705i	$-0.263149\pi$
$257$	21.7160	1.35460	0.677302	+	0.735705i	$0.263149\pi$
$258$	0	0
$259$	9.21419	0.572542
$260$	5.81750	0.360786
$261$	0	0
$262$	11.2824	0.697029
$263$	16.1777	0.997559	0.498779	−	0.866729i	$-0.333782\pi$
$263$	16.1777	0.997559	0.498779	+	0.866729i	$0.333782\pi$
$264$	0	0
$265$	−26.4744	−1.62631
$266$	−10.2771	−0.630127
$267$	0	0
$268$	−6.17287	−0.377068
$269$	−1.94306	−0.118471	−0.0592354	−	0.998244i	$-0.518866\pi$
$269$	−1.94306	−0.118471	−0.0592354	+	0.998244i	$0.518866\pi$
$270$	0	0
$271$	4.94901	0.300631	0.150315	−	0.988638i	$-0.451971\pi$
$271$	4.94901	0.300631	0.150315	+	0.988638i	$0.451971\pi$
$272$	−23.5629	−1.42871
$273$	0	0
$274$	−7.73366	−0.467208
$275$	2.05129	0.123698
$276$	0	0
$277$	20.3847	1.22480	0.612400	−	0.790548i	$-0.290204\pi$
$277$	20.3847	1.22480	0.612400	+	0.790548i	$0.290204\pi$
$278$	32.0457	1.92197
$279$	0	0
$280$	6.56919	0.392584
$281$	20.6517	1.23198	0.615989	−	0.787755i	$-0.288757\pi$
$281$	20.6517	1.23198	0.615989	+	0.787755i	$0.288757\pi$
$282$	0	0
$283$	−16.8827	−1.00357	−0.501786	−	0.864992i	$-0.667323\pi$
$283$	−16.8827	−1.00357	−0.501786	+	0.864992i	$0.667323\pi$
$284$	−2.10098	−0.124670
$285$	0	0
$286$	9.58002	0.566478
$287$	−5.12302	−0.302402
$288$	0	0
$289$	7.19642	0.423319
$290$	−25.9947	−1.52646
$291$	0	0
$292$	−6.68452	−0.391182
$293$	18.6621	1.09025	0.545125	−	0.838355i	$-0.316482\pi$
$293$	18.6621	1.09025	0.545125	+	0.838355i	$0.316482\pi$
$294$	0	0
$295$	30.4071	1.77037
$296$	−19.2407	−1.11834
$297$	0	0
$298$	12.1165	0.701891
$299$	0	0
$300$	0	0
$301$	−2.25373	−0.129903
$302$	−19.1074	−1.09951
$303$	0	0
$304$	27.7362	1.59078
$305$	−28.1924	−1.61429
$306$	0	0
$307$	−11.1163	−0.634443	−0.317221	−	0.948352i	$-0.602750\pi$
$307$	−11.1163	−0.634443	−0.317221	+	0.948352i	$0.602750\pi$
$308$	−0.857295	−0.0488489
$309$	0	0
$310$	30.1176	1.71056
$311$	5.55334	0.314901	0.157451	−	0.987527i	$-0.449673\pi$
$311$	5.55334	0.314901	0.157451	+	0.987527i	$0.449673\pi$
$312$	0	0
$313$	11.8225	0.668247	0.334123	−	0.942529i	$-0.391560\pi$
$313$	11.8225	0.668247	0.334123	+	0.942529i	$0.391560\pi$
$314$	−9.66618	−0.545494
$315$	0	0
$316$	2.95393	0.166172
$317$	−33.9941	−1.90930	−0.954651	−	0.297728i	$-0.903771\pi$
$317$	−33.9941	−1.90930	−0.954651	+	0.297728i	$0.903771\pi$
$318$	0	0
$319$	−9.12663	−0.510993
$320$	−12.2263	−0.683473
$321$	0	0
$322$	0	0
$323$	−28.4819	−1.58477
$324$	0	0
$325$	−6.10465	−0.338625
$326$	−26.6284	−1.47481
$327$	0	0
$328$	10.6977	0.590679
$329$	2.91459	0.160686
$330$	0	0
$331$	−6.63442	−0.364661	−0.182330	−	0.983237i	$-0.558364\pi$
$331$	−6.63442	−0.364661	−0.182330	+	0.983237i	$0.558364\pi$
$332$	0.847497	0.0465124
$333$	0	0
$334$	32.3950	1.77258
$335$	28.9127	1.57967
$336$	0	0
$337$	17.5874	0.958049	0.479024	−	0.877802i	$-0.340991\pi$
$337$	17.5874	0.958049	0.479024	+	0.877802i	$0.340991\pi$
$338$	−7.78357	−0.423370
$339$	0	0
$340$	−6.76713	−0.366999
$341$	10.5741	0.572622
$342$	0	0
$343$	14.2058	0.767041
$344$	4.70614	0.253738
$345$	0	0
$346$	−3.09602	−0.166443
$347$	2.91940	0.156722	0.0783609	−	0.996925i	$-0.475031\pi$
$347$	2.91940	0.156722	0.0783609	+	0.996925i	$0.475031\pi$
$348$	0	0
$349$	−2.35657	−0.126144	−0.0630721	−	0.998009i	$-0.520090\pi$
$349$	−2.35657	−0.126144	−0.0630721	+	0.998009i	$0.520090\pi$
$350$	2.56229	0.136960
$351$	0	0
$352$	4.24573	0.226298
$353$	−2.13948	−0.113873	−0.0569364	−	0.998378i	$-0.518133\pi$
$353$	−2.13948	−0.113873	−0.0569364	+	0.998378i	$0.518133\pi$
$354$	0	0
$355$	9.84063	0.522286
$356$	1.23746	0.0655850
$357$	0	0
$358$	−27.9456	−1.47697
$359$	18.9299	0.999081	0.499541	−	0.866290i	$-0.333502\pi$
$359$	18.9299	0.999081	0.499541	+	0.866290i	$0.333502\pi$
$360$	0	0
$361$	14.5264	0.764545
$362$	−20.7771	−1.09202
$363$	0	0
$364$	2.55131	0.133725
$365$	31.3092	1.63880
$366$	0	0
$367$	7.75649	0.404885	0.202443	−	0.979294i	$-0.435112\pi$
$367$	7.75649	0.404885	0.202443	+	0.979294i	$0.435112\pi$
$368$	0	0
$369$	0	0
$370$	−33.4977	−1.74146
$371$	−11.6105	−0.602789
$372$	0	0
$373$	−19.6495	−1.01741	−0.508705	−	0.860941i	$-0.669876\pi$
$373$	−19.6495	−1.01741	−0.508705	+	0.860941i	$0.669876\pi$
$374$	−11.1438	−0.576233
$375$	0	0
$376$	−6.08612	−0.313867
$377$	27.1609	1.39886
$378$	0	0
$379$	−9.61188	−0.493729	−0.246864	−	0.969050i	$-0.579400\pi$
$379$	−9.61188	−0.493729	−0.246864	+	0.969050i	$0.579400\pi$
$380$	7.96566	0.408630
$381$	0	0
$382$	22.5090	1.15166
$383$	2.13032	0.108854	0.0544271	−	0.998518i	$-0.482667\pi$
$383$	2.13032	0.108854	0.0544271	+	0.998518i	$0.482667\pi$
$384$	0	0
$385$	4.01542	0.204645
$386$	−19.3684	−0.985826
$387$	0	0
$388$	−4.74287	−0.240783
$389$	33.5013	1.69858	0.849291	−	0.527926i	$-0.177030\pi$
$389$	33.5013	1.69858	0.849291	+	0.527926i	$0.177030\pi$
$390$	0	0
$391$	0	0
$392$	−13.3915	−0.676372
$393$	0	0
$394$	−6.98931	−0.352117
$395$	−13.8357	−0.696151
$396$	0	0
$397$	21.9087	1.09957	0.549784	−	0.835307i	$-0.314710\pi$
$397$	21.9087	1.09957	0.549784	+	0.835307i	$0.314710\pi$
$398$	−27.0839	−1.35759
$399$	0	0
$400$	−6.91523	−0.345761
$401$	6.15242	0.307237	0.153619	−	0.988130i	$-0.450907\pi$
$401$	6.15242	0.307237	0.153619	+	0.988130i	$0.450907\pi$
$402$	0	0
$403$	−31.4687	−1.56757
$404$	−1.40859	−0.0700799
$405$	0	0
$406$	−11.4002	−0.565781
$407$	−11.7609	−0.582966
$408$	0	0
$409$	3.68280	0.182103	0.0910514	−	0.995846i	$-0.470977\pi$
$409$	3.68280	0.182103	0.0910514	+	0.995846i	$0.470977\pi$
$410$	18.6244	0.919795
$411$	0	0
$412$	0.126776	0.00624580
$413$	13.3353	0.656186
$414$	0	0
$415$	−3.96953	−0.194857
$416$	−12.6353	−0.619497
$417$	0	0
$418$	13.1175	0.641599
$419$	−36.1103	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$419$	−36.1103	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$420$	0	0
$421$	−19.4209	−0.946516	−0.473258	−	0.880924i	$-0.656922\pi$
$421$	−19.4209	−0.946516	−0.473258	+	0.880924i	$0.656922\pi$
$422$	11.3277	0.551424
$423$	0	0
$424$	24.2446	1.17742
$425$	7.10115	0.344456
$426$	0	0
$427$	−12.3640	−0.598335
$428$	−0.777901	−0.0376013
$429$	0	0
$430$	8.19330	0.395116
$431$	−5.13266	−0.247231	−0.123616	−	0.992330i	$-0.539449\pi$
$431$	−5.13266	−0.247231	−0.123616	+	0.992330i	$0.539449\pi$
$432$	0	0
$433$	−16.5765	−0.796616	−0.398308	−	0.917252i	$-0.630402\pi$
$433$	−16.5765	−0.796616	−0.398308	+	0.917252i	$0.630402\pi$
$434$	13.2083	0.634018
$435$	0	0
$436$	−7.72735	−0.370073
$437$	0	0
$438$	0	0
$439$	1.25289	0.0597971	0.0298986	−	0.999553i	$-0.490482\pi$
$439$	1.25289	0.0597971	0.0298986	+	0.999553i	$0.490482\pi$
$440$	−8.38483	−0.399731
$441$	0	0
$442$	33.1640	1.57745
$443$	21.2968	1.01184	0.505920	−	0.862580i	$-0.331153\pi$
$443$	21.2968	1.01184	0.505920	+	0.862580i	$0.331153\pi$
$444$	0	0
$445$	−5.79603	−0.274758
$446$	8.13279	0.385099
$447$	0	0
$448$	−5.36195	−0.253328
$449$	8.26756	0.390170	0.195085	−	0.980786i	$-0.437502\pi$
$449$	8.26756	0.390170	0.195085	+	0.980786i	$0.437502\pi$
$450$	0	0
$451$	6.53896	0.307908
$452$	−2.49165	−0.117197
$453$	0	0
$454$	−22.2839	−1.04584
$455$	−11.9499	−0.560220
$456$	0	0
$457$	−15.3123	−0.716278	−0.358139	−	0.933668i	$-0.616589\pi$
$457$	−15.3123	−0.716278	−0.358139	+	0.933668i	$0.616589\pi$
$458$	−4.04524	−0.189021
$459$	0	0
$460$	0	0
$461$	−28.6751	−1.33553	−0.667767	−	0.744370i	$-0.732750\pi$
$461$	−28.6751	−1.33553	−0.667767	+	0.744370i	$0.732750\pi$
$462$	0	0
$463$	−29.6589	−1.37836	−0.689182	−	0.724588i	$-0.742030\pi$
$463$	−29.6589	−1.37836	−0.689182	+	0.724588i	$0.742030\pi$
$464$	30.7673	1.42833
$465$	0	0
$466$	1.96032	0.0908101
$467$	29.6855	1.37368	0.686840	−	0.726809i	$-0.258997\pi$
$467$	29.6855	1.37368	0.686840	+	0.726809i	$0.258997\pi$
$468$	0	0
$469$	12.6799	0.585503
$470$	−10.5958	−0.488749
$471$	0	0
$472$	−27.8461	−1.28172
$473$	2.87663	0.132268
$474$	0	0
$475$	−8.35885	−0.383530
$476$	−2.96778	−0.136028
$477$	0	0
$478$	−16.1423	−0.738330
$479$	2.55600	0.116787	0.0583934	−	0.998294i	$-0.481402\pi$
$479$	2.55600	0.116787	0.0583934	+	0.998294i	$0.481402\pi$
$480$	0	0
$481$	35.0004	1.59588
$482$	−32.1618	−1.46493
$483$	0	0
$484$	−4.86727	−0.221240
$485$	22.2148	1.00872
$486$	0	0
$487$	32.4321	1.46964	0.734819	−	0.678264i	$-0.237267\pi$
$487$	32.4321	1.46964	0.734819	+	0.678264i	$0.237267\pi$
$488$	25.8179	1.16872
$489$	0	0
$490$	−23.3143	−1.05323
$491$	6.08953	0.274816	0.137408	−	0.990515i	$-0.456123\pi$
$491$	6.08953	0.274816	0.137408	+	0.990515i	$0.456123\pi$
$492$	0	0
$493$	−31.5945	−1.42294
$494$	−39.0378	−1.75639
$495$	0	0
$496$	−35.6471	−1.60060
$497$	4.31569	0.193585
$498$	0	0
$499$	20.4394	0.914995	0.457497	−	0.889211i	$-0.348746\pi$
$499$	20.4394	0.914995	0.457497	+	0.889211i	$0.348746\pi$
$500$	4.89257	0.218802
$501$	0	0
$502$	−25.4862	−1.13750
$503$	−2.83315	−0.126324	−0.0631620	−	0.998003i	$-0.520118\pi$
$503$	−2.83315	−0.126324	−0.0631620	+	0.998003i	$0.520118\pi$
$504$	0	0
$505$	6.59760	0.293589
$506$	0	0
$507$	0	0
$508$	1.46418	0.0649626
$509$	34.5628	1.53197	0.765984	−	0.642859i	$-0.222252\pi$
$509$	34.5628	1.53197	0.765984	+	0.642859i	$0.222252\pi$
$510$	0	0
$511$	13.7309	0.607419
$512$	7.95788	0.351692
$513$	0	0
$514$	−34.6229	−1.52715
$515$	−0.593797	−0.0261658
$516$	0	0
$517$	−3.72015	−0.163612
$518$	−14.6907	−0.645471
$519$	0	0
$520$	24.9533	1.09427
$521$	16.8578	0.738552	0.369276	−	0.929320i	$-0.379606\pi$
$521$	16.8578	0.738552	0.369276	+	0.929320i	$0.379606\pi$
$522$	0	0
$523$	26.1482	1.14338	0.571690	−	0.820470i	$-0.306288\pi$
$523$	26.1482	1.14338	0.571690	+	0.820470i	$0.306288\pi$
$524$	−3.83514	−0.167539
$525$	0	0
$526$	−25.7929	−1.12462
$527$	36.6055	1.59456
$528$	0	0
$529$	0	0
$530$	42.2094	1.83346
$531$	0	0
$532$	3.49340	0.151458
$533$	−19.4600	−0.842904
$534$	0	0
$535$	3.64356	0.157525
$536$	−26.4776	−1.14366
$537$	0	0
$538$	3.09793	0.133561
$539$	−8.18556	−0.352577
$540$	0	0
$541$	14.7684	0.634944	0.317472	−	0.948268i	$-0.397166\pi$
$541$	14.7684	0.634944	0.317472	+	0.948268i	$0.397166\pi$
$542$	−7.89046	−0.338924
$543$	0	0
$544$	14.6979	0.630165
$545$	36.1936	1.55036
$546$	0	0
$547$	3.76910	0.161155	0.0805775	−	0.996748i	$-0.474324\pi$
$547$	3.76910	0.161155	0.0805775	+	0.996748i	$0.474324\pi$
$548$	2.62884	0.112299
$549$	0	0
$550$	−3.27048	−0.139454
$551$	37.1902	1.58436
$552$	0	0
$553$	−6.06777	−0.258028
$554$	−32.5004	−1.38081
$555$	0	0
$556$	−10.8930	−0.461968
$557$	0.764998	0.0324140	0.0162070	−	0.999869i	$-0.494841\pi$
$557$	0.764998	0.0324140	0.0162070	+	0.999869i	$0.494841\pi$
$558$	0	0
$559$	−8.56086	−0.362086
$560$	−13.5366	−0.572026
$561$	0	0
$562$	−32.9261	−1.38890
$563$	−15.3879	−0.648523	−0.324261	−	0.945968i	$-0.605116\pi$
$563$	−15.3879	−0.648523	−0.324261	+	0.945968i	$0.605116\pi$
$564$	0	0
$565$	11.6705	0.490980
$566$	26.9169	1.13140
$567$	0	0
$568$	−9.01182	−0.378128
$569$	41.0368	1.72035	0.860176	−	0.509998i	$-0.170354\pi$
$569$	41.0368	1.72035	0.860176	+	0.509998i	$0.170354\pi$
$570$	0	0
$571$	21.6410	0.905649	0.452824	−	0.891600i	$-0.350416\pi$
$571$	21.6410	0.905649	0.452824	+	0.891600i	$0.350416\pi$
$572$	−3.25646	−0.136159
$573$	0	0
$574$	8.16789	0.340921
$575$	0	0
$576$	0	0
$577$	−32.8601	−1.36798	−0.683992	−	0.729489i	$-0.739758\pi$
$577$	−32.8601	−1.36798	−0.683992	+	0.729489i	$0.739758\pi$
$578$	−11.4736	−0.477240
$579$	0	0
$580$	8.83618	0.366902
$581$	−1.74087	−0.0722235
$582$	0	0
$583$	14.8196	0.613763
$584$	−28.6722	−1.18647
$585$	0	0
$586$	−29.7539	−1.22912
$587$	10.4614	0.431789	0.215895	−	0.976417i	$-0.430733\pi$
$587$	10.4614	0.431789	0.215895	+	0.976417i	$0.430733\pi$
$588$	0	0
$589$	−43.0888	−1.77544
$590$	−48.4796	−1.99587
$591$	0	0
$592$	39.6478	1.62951
$593$	23.1714	0.951536	0.475768	−	0.879571i	$-0.342170\pi$
$593$	23.1714	0.951536	0.475768	+	0.879571i	$0.342170\pi$
$594$	0	0
$595$	13.9006	0.569868
$596$	−4.11868	−0.168708
$597$	0	0
$598$	0	0
$599$	24.4836	1.00037	0.500187	−	0.865917i	$-0.333265\pi$
$599$	24.4836	1.00037	0.500187	+	0.865917i	$0.333265\pi$
$600$	0	0
$601$	26.6428	1.08678	0.543391	−	0.839480i	$-0.317140\pi$
$601$	26.6428	1.08678	0.543391	+	0.839480i	$0.317140\pi$
$602$	3.59323	0.146449
$603$	0	0
$604$	6.49504	0.264280
$605$	22.7975	0.926849
$606$	0	0
$607$	−22.6218	−0.918189	−0.459095	−	0.888387i	$-0.651826\pi$
$607$	−22.6218	−0.918189	−0.459095	+	0.888387i	$0.651826\pi$
$608$	−17.3010	−0.701649
$609$	0	0
$610$	44.9486	1.81991
$611$	11.0712	0.447891
$612$	0	0
$613$	−33.1077	−1.33721	−0.668603	−	0.743619i	$-0.733108\pi$
$613$	−33.1077	−1.33721	−0.668603	+	0.743619i	$0.733108\pi$
$614$	17.7233	0.715256
$615$	0	0
$616$	−3.67723	−0.148160
$617$	−12.4787	−0.502373	−0.251187	−	0.967939i	$-0.580821\pi$
$617$	−12.4787	−0.502373	−0.251187	+	0.967939i	$0.580821\pi$
$618$	0	0
$619$	35.0347	1.40816	0.704081	−	0.710120i	$-0.251359\pi$
$619$	35.0347	1.40816	0.704081	+	0.710120i	$0.251359\pi$
$620$	−10.2376	−0.411153
$621$	0	0
$622$	−8.85398	−0.355012
$623$	−2.54190	−0.101839
$624$	0	0
$625$	−30.1341	−1.20536
$626$	−18.8492	−0.753366
$627$	0	0
$628$	3.28575	0.131116
$629$	−40.7137	−1.62336
$630$	0	0
$631$	30.5284	1.21532	0.607659	−	0.794198i	$-0.292109\pi$
$631$	30.5284	1.21532	0.607659	+	0.794198i	$0.292109\pi$
$632$	12.6704	0.504003
$633$	0	0
$634$	54.1986	2.15250
$635$	−6.85798	−0.272151
$636$	0	0
$637$	24.3602	0.965187
$638$	14.5511	0.576082
$639$	0	0
$640$	34.6626	1.37016
$641$	−15.8076	−0.624364	−0.312182	−	0.950022i	$-0.601060\pi$
$641$	−15.8076	−0.624364	−0.312182	+	0.950022i	$0.601060\pi$
$642$	0	0
$643$	−21.2193	−0.836809	−0.418404	−	0.908261i	$-0.637411\pi$
$643$	−21.2193	−0.836809	−0.418404	+	0.908261i	$0.637411\pi$
$644$	0	0
$645$	0	0
$646$	45.4101	1.78664
$647$	−34.8220	−1.36900	−0.684498	−	0.729015i	$-0.739978\pi$
$647$	−34.8220	−1.36900	−0.684498	+	0.729015i	$0.739978\pi$
$648$	0	0
$649$	−17.0210	−0.668133
$650$	9.73296	0.381758
$651$	0	0
$652$	9.05159	0.354488
$653$	−28.3949	−1.11118	−0.555590	−	0.831456i	$-0.687508\pi$
$653$	−28.3949	−1.11118	−0.555590	+	0.831456i	$0.687508\pi$
$654$	0	0
$655$	17.9632	0.701878
$656$	−22.0438	−0.860667
$657$	0	0
$658$	−4.64688	−0.181154
$659$	19.6903	0.767027	0.383513	−	0.923535i	$-0.374714\pi$
$659$	19.6903	0.767027	0.383513	+	0.923535i	$0.374714\pi$
$660$	0	0
$661$	29.6153	1.15190	0.575951	−	0.817484i	$-0.304632\pi$
$661$	29.6153	1.15190	0.575951	+	0.817484i	$0.304632\pi$
$662$	10.5776	0.411110
$663$	0	0
$664$	3.63521	0.141073
$665$	−16.3625	−0.634511
$666$	0	0
$667$	0	0
$668$	−11.0118	−0.426059
$669$	0	0
$670$	−46.0970	−1.78088
$671$	15.7813	0.609228
$672$	0	0
$673$	−27.2425	−1.05012	−0.525060	−	0.851065i	$-0.675957\pi$
$673$	−27.2425	−1.05012	−0.525060	+	0.851065i	$0.675957\pi$
$674$	−28.0406	−1.08008
$675$	0	0
$676$	2.64581	0.101762
$677$	2.16744	0.0833016	0.0416508	−	0.999132i	$-0.486738\pi$
$677$	2.16744	0.0833016	0.0416508	+	0.999132i	$0.486738\pi$
$678$	0	0
$679$	9.74247	0.373882
$680$	−29.0265	−1.11312
$681$	0	0
$682$	−16.8589	−0.645561
$683$	−10.0823	−0.385789	−0.192895	−	0.981219i	$-0.561788\pi$
$683$	−10.0823	−0.385789	−0.192895	+	0.981219i	$0.561788\pi$
$684$	0	0
$685$	−12.3131	−0.470458
$686$	−22.6490	−0.864744
$687$	0	0
$688$	−9.69757	−0.369716
$689$	−44.1030	−1.68019
$690$	0	0
$691$	−31.1667	−1.18564	−0.592819	−	0.805336i	$-0.701985\pi$
$691$	−31.1667	−1.18564	−0.592819	+	0.805336i	$0.701985\pi$
$692$	1.05240	0.0400064
$693$	0	0
$694$	−4.65455	−0.176684
$695$	51.0212	1.93534
$696$	0	0
$697$	22.6365	0.857419
$698$	3.75720	0.142212
$699$	0	0
$700$	−0.870981	−0.0329200
$701$	46.2776	1.74788	0.873941	−	0.486033i	$-0.161556\pi$
$701$	46.2776	1.74788	0.873941	+	0.486033i	$0.161556\pi$
$702$	0	0
$703$	47.9246	1.80751
$704$	6.84393	0.257940
$705$	0	0
$706$	3.41107	0.128377
$707$	2.89343	0.108819
$708$	0	0
$709$	−29.0665	−1.09162	−0.545808	−	0.837910i	$-0.683777\pi$
$709$	−29.0665	−1.09162	−0.545808	+	0.837910i	$0.683777\pi$
$710$	−15.6894	−0.588814
$711$	0	0
$712$	5.30787	0.198921
$713$	0	0
$714$	0	0
$715$	15.2527	0.570419
$716$	9.49932	0.355006
$717$	0	0
$718$	−30.1809	−1.12634
$719$	36.7130	1.36916	0.684582	−	0.728936i	$-0.259985\pi$
$719$	36.7130	1.36916	0.684582	+	0.728936i	$0.259985\pi$
$720$	0	0
$721$	−0.260414	−0.00969833
$722$	−23.1601	−0.861930
$723$	0	0
$724$	7.06258	0.262479
$725$	−9.27233	−0.344366
$726$	0	0
$727$	33.7769	1.25271	0.626357	−	0.779536i	$-0.284545\pi$
$727$	33.7769	1.25271	0.626357	+	0.779536i	$0.284545\pi$
$728$	10.9434	0.405591
$729$	0	0
$730$	−49.9178	−1.84754
$731$	9.95830	0.368321
$732$	0	0
$733$	48.2444	1.78195	0.890974	−	0.454054i	$-0.150023\pi$
$733$	48.2444	1.78195	0.890974	+	0.454054i	$0.150023\pi$
$734$	−12.3666	−0.456458
$735$	0	0
$736$	0	0
$737$	−16.1845	−0.596162
$738$	0	0
$739$	−42.8366	−1.57577	−0.787886	−	0.615822i	$-0.788824\pi$
$739$	−42.8366	−1.57577	−0.787886	+	0.615822i	$0.788824\pi$
$740$	11.3866	0.418580
$741$	0	0
$742$	18.5113	0.679570
$743$	30.3769	1.11442	0.557211	−	0.830371i	$-0.311871\pi$
$743$	30.3769	1.11442	0.557211	+	0.830371i	$0.311871\pi$
$744$	0	0
$745$	19.2912	0.706774
$746$	31.3281	1.14700
$747$	0	0
$748$	3.78803	0.138504
$749$	1.59791	0.0583864
$750$	0	0
$751$	17.2766	0.630431	0.315215	−	0.949020i	$-0.397923\pi$
$751$	17.2766	0.630431	0.315215	+	0.949020i	$0.397923\pi$
$752$	12.5412	0.457330
$753$	0	0
$754$	−43.3039	−1.57704
$755$	−30.4217	−1.10716
$756$	0	0
$757$	29.6047	1.07600	0.538001	−	0.842944i	$-0.319180\pi$
$757$	29.6047	1.07600	0.538001	+	0.842944i	$0.319180\pi$
$758$	15.3247	0.556618
$759$	0	0
$760$	34.1675	1.23938
$761$	−11.3935	−0.413015	−0.206507	−	0.978445i	$-0.566210\pi$
$761$	−11.3935	−0.413015	−0.206507	+	0.978445i	$0.566210\pi$
$762$	0	0
$763$	15.8730	0.574641
$764$	−7.65133	−0.276815
$765$	0	0
$766$	−3.39647	−0.122720
$767$	50.6545	1.82903
$768$	0	0
$769$	−14.3770	−0.518449	−0.259224	−	0.965817i	$-0.583467\pi$
$769$	−14.3770	−0.518449	−0.259224	+	0.965817i	$0.583467\pi$
$770$	−6.40200	−0.230712
$771$	0	0
$772$	6.58375	0.236954
$773$	19.6563	0.706987	0.353494	−	0.935437i	$-0.384994\pi$
$773$	19.6563	0.706987	0.353494	+	0.935437i	$0.384994\pi$
$774$	0	0
$775$	10.7430	0.385899
$776$	−20.3438	−0.730300
$777$	0	0
$778$	−53.4128	−1.91494
$779$	−26.6457	−0.954682
$780$	0	0
$781$	−5.50849	−0.197109
$782$	0	0
$783$	0	0
$784$	27.5948	0.985528
$785$	−15.3899	−0.549289
$786$	0	0
$787$	−3.37271	−0.120224	−0.0601121	−	0.998192i	$-0.519146\pi$
$787$	−3.37271	−0.120224	−0.0601121	+	0.998192i	$0.519146\pi$
$788$	2.37582	0.0846352
$789$	0	0
$790$	22.0590	0.784824
$791$	5.11817	0.181981
$792$	0	0
$793$	−46.9650	−1.66778
$794$	−34.9302	−1.23963
$795$	0	0
$796$	9.20641	0.326313
$797$	−39.5981	−1.40264	−0.701318	−	0.712849i	$-0.747405\pi$
$797$	−39.5981	−1.40264	−0.701318	+	0.712849i	$0.747405\pi$
$798$	0	0
$799$	−12.8784	−0.455604
$800$	4.31352	0.152506
$801$	0	0
$802$	−9.80912	−0.346372
$803$	−17.5259	−0.618477
$804$	0	0
$805$	0	0
$806$	50.1721	1.76724
$807$	0	0
$808$	−6.04193	−0.212554
$809$	27.5449	0.968426	0.484213	−	0.874950i	$-0.339106\pi$
$809$	27.5449	0.968426	0.484213	+	0.874950i	$0.339106\pi$
$810$	0	0
$811$	−51.8083	−1.81924	−0.909618	−	0.415446i	$-0.863626\pi$
$811$	−51.8083	−1.81924	−0.909618	+	0.415446i	$0.863626\pi$
$812$	3.87518	0.135992
$813$	0	0
$814$	18.7510	0.657222
$815$	−42.3961	−1.48507
$816$	0	0
$817$	−11.7220	−0.410102
$818$	−5.87168	−0.205298
$819$	0	0
$820$	−6.33086	−0.221083
$821$	27.8580	0.972252	0.486126	−	0.873889i	$-0.338410\pi$
$821$	27.8580	0.972252	0.486126	+	0.873889i	$0.338410\pi$
$822$	0	0
$823$	−2.72040	−0.0948271	−0.0474136	−	0.998875i	$-0.515098\pi$
$823$	−2.72040	−0.0948271	−0.0474136	+	0.998875i	$0.515098\pi$
$824$	0.543786	0.0189437
$825$	0	0
$826$	−21.2611	−0.739769
$827$	24.9432	0.867361	0.433680	−	0.901067i	$-0.357215\pi$
$827$	24.9432	0.867361	0.433680	+	0.901067i	$0.357215\pi$
$828$	0	0
$829$	12.8583	0.446587	0.223294	−	0.974751i	$-0.428319\pi$
$829$	12.8583	0.446587	0.223294	+	0.974751i	$0.428319\pi$
$830$	6.32883	0.219677
$831$	0	0
$832$	−20.3675	−0.706118
$833$	−28.3367	−0.981809
$834$	0	0
$835$	51.5773	1.78491
$836$	−4.45894	−0.154216
$837$	0	0
$838$	57.5725	1.98881
$839$	22.3878	0.772912	0.386456	−	0.922308i	$-0.373699\pi$
$839$	22.3878	0.772912	0.386456	+	0.922308i	$0.373699\pi$
$840$	0	0
$841$	12.2545	0.422570
$842$	30.9637	1.06708
$843$	0	0
$844$	−3.85054	−0.132541
$845$	−12.3925	−0.426316
$846$	0	0
$847$	9.99801	0.343536
$848$	−49.9590	−1.71560
$849$	0	0
$850$	−11.3217	−0.388332
$851$	0	0
$852$	0	0
$853$	−17.2990	−0.592306	−0.296153	−	0.955141i	$-0.595704\pi$
$853$	−17.2990	−0.592306	−0.296153	+	0.955141i	$0.595704\pi$
$854$	19.7125	0.674549
$855$	0	0
$856$	−3.33669	−0.114046
$857$	40.7943	1.39351	0.696754	−	0.717311i	$-0.254627\pi$
$857$	40.7943	1.39351	0.696754	+	0.717311i	$0.254627\pi$
$858$	0	0
$859$	−6.52166	−0.222516	−0.111258	−	0.993792i	$-0.535488\pi$
$859$	−6.52166	−0.222516	−0.111258	+	0.993792i	$0.535488\pi$
$860$	−2.78508	−0.0949706
$861$	0	0
$862$	8.18326	0.278723
$863$	6.31300	0.214897	0.107449	−	0.994211i	$-0.465732\pi$
$863$	6.31300	0.214897	0.107449	+	0.994211i	$0.465732\pi$
$864$	0	0
$865$	−4.92929	−0.167601
$866$	26.4288	0.898086
$867$	0	0
$868$	−4.48979	−0.152394
$869$	7.74482	0.262725
$870$	0	0
$871$	48.1650	1.63201
$872$	−33.1453	−1.12244
$873$	0	0
$874$	0	0
$875$	−10.0500	−0.339751
$876$	0	0
$877$	40.8670	1.37998	0.689989	−	0.723819i	$-0.257615\pi$
$877$	40.8670	1.37998	0.689989	+	0.723819i	$0.257615\pi$
$878$	−1.99754	−0.0674139
$879$	0	0
$880$	17.2780	0.582440
$881$	−2.32275	−0.0782554	−0.0391277	−	0.999234i	$-0.512458\pi$
$881$	−2.32275	−0.0782554	−0.0391277	+	0.999234i	$0.512458\pi$
$882$	0	0
$883$	−31.0257	−1.04410	−0.522049	−	0.852916i	$-0.674832\pi$
$883$	−31.0257	−1.04410	−0.522049	+	0.852916i	$0.674832\pi$
$884$	−11.2732	−0.379158
$885$	0	0
$886$	−33.9545	−1.14072
$887$	40.1986	1.34974	0.674868	−	0.737939i	$-0.264201\pi$
$887$	40.1986	1.34974	0.674868	+	0.737939i	$0.264201\pi$
$888$	0	0
$889$	−3.00762	−0.100872
$890$	9.24091	0.309756
$891$	0	0
$892$	−2.76452	−0.0925629
$893$	15.1593	0.507286
$894$	0	0
$895$	−44.4932	−1.48724
$896$	15.2016	0.507849
$897$	0	0
$898$	−13.1814	−0.439869
$899$	−47.7977	−1.59414
$900$	0	0
$901$	51.3022	1.70912
$902$	−10.4254	−0.347128
$903$	0	0
$904$	−10.6875	−0.355462
$905$	−33.0800	−1.09962
$906$	0	0
$907$	−34.2643	−1.13773	−0.568863	−	0.822432i	$-0.692617\pi$
$907$	−34.2643	−1.13773	−0.568863	+	0.822432i	$0.692617\pi$
$908$	7.57481	0.251379
$909$	0	0
$910$	19.0523	0.631579
$911$	46.9445	1.55534	0.777670	−	0.628672i	$-0.216401\pi$
$911$	46.9445	1.55534	0.777670	+	0.628672i	$0.216401\pi$
$912$	0	0
$913$	2.22203	0.0735383
$914$	24.4131	0.807515
$915$	0	0
$916$	1.37507	0.0454335
$917$	7.87788	0.260150
$918$	0	0
$919$	4.61902	0.152367	0.0761837	−	0.997094i	$-0.475726\pi$
$919$	4.61902	0.152367	0.0761837	+	0.997094i	$0.475726\pi$
$920$	0	0
$921$	0	0
$922$	45.7182	1.50565
$923$	16.3933	0.539591
$924$	0	0
$925$	−11.9486	−0.392869
$926$	47.2866	1.55394
$927$	0	0
$928$	−19.1917	−0.629999
$929$	−10.4007	−0.341236	−0.170618	−	0.985337i	$-0.554576\pi$
$929$	−10.4007	−0.341236	−0.170618	+	0.985337i	$0.554576\pi$
$930$	0	0
$931$	33.3555	1.09318
$932$	−0.666357	−0.0218272
$933$	0	0
$934$	−47.3291	−1.54865
$935$	−17.7425	−0.580242
$936$	0	0
$937$	−41.3646	−1.35132	−0.675662	−	0.737212i	$-0.736142\pi$
$937$	−41.3646	−1.35132	−0.675662	+	0.737212i	$0.736142\pi$
$938$	−20.2162	−0.660082
$939$	0	0
$940$	3.60175	0.117476
$941$	−17.6096	−0.574058	−0.287029	−	0.957922i	$-0.592668\pi$
$941$	−17.6096	−0.574058	−0.287029	+	0.957922i	$0.592668\pi$
$942$	0	0
$943$	0	0
$944$	57.3804	1.86757
$945$	0	0
$946$	−4.58636	−0.149115
$947$	−52.4716	−1.70510	−0.852550	−	0.522646i	$-0.824945\pi$
$947$	−52.4716	−1.70510	−0.852550	+	0.522646i	$0.824945\pi$
$948$	0	0
$949$	52.1572	1.69309
$950$	13.3269	0.432383
$951$	0	0
$952$	−12.7298	−0.412576
$953$	−46.9568	−1.52108	−0.760540	−	0.649291i	$-0.775066\pi$
$953$	−46.9568	−1.52108	−0.760540	+	0.649291i	$0.775066\pi$
$954$	0	0
$955$	35.8375	1.15968
$956$	5.48712	0.177466
$957$	0	0
$958$	−4.07516	−0.131663
$959$	−5.39999	−0.174375
$960$	0	0
$961$	24.3786	0.786405
$962$	−55.8029	−1.79916
$963$	0	0
$964$	10.9325	0.352112
$965$	−30.8372	−0.992684
$966$	0	0
$967$	−15.7485	−0.506439	−0.253220	−	0.967409i	$-0.581490\pi$
$967$	−15.7485	−0.506439	−0.253220	+	0.967409i	$0.581490\pi$
$968$	−20.8774	−0.671025
$969$	0	0
$970$	−35.4182	−1.13721
$971$	−13.2136	−0.424045	−0.212023	−	0.977265i	$-0.568005\pi$
$971$	−13.2136	−0.424045	−0.212023	+	0.977265i	$0.568005\pi$
$972$	0	0
$973$	22.3757	0.717334
$974$	−51.7081	−1.65683
$975$	0	0
$976$	−53.2010	−1.70292
$977$	−59.5223	−1.90429	−0.952144	−	0.305650i	$-0.901126\pi$
$977$	−59.5223	−1.90429	−0.952144	+	0.305650i	$0.901126\pi$
$978$	0	0
$979$	3.24445	0.103693
$980$	7.92506	0.253157
$981$	0	0
$982$	−9.70884	−0.309822
$983$	−40.3416	−1.28670	−0.643348	−	0.765573i	$-0.722455\pi$
$983$	−40.3416	−1.28670	−0.643348	+	0.765573i	$0.722455\pi$
$984$	0	0
$985$	−11.1280	−0.354566
$986$	50.3727	1.60419
$987$	0	0
$988$	13.2698	0.422169
$989$	0	0
$990$	0	0
$991$	−9.90941	−0.314783	−0.157391	−	0.987536i	$-0.550308\pi$
$991$	−9.90941	−0.314783	−0.157391	+	0.987536i	$0.550308\pi$
$992$	22.2356	0.705981
$993$	0	0
$994$	−6.88072	−0.218243
$995$	−43.1213	−1.36704
$996$	0	0
$997$	−15.8969	−0.503460	−0.251730	−	0.967797i	$-0.581000\pi$
$997$	−15.8969	−0.503460	−0.251730	+	0.967797i	$0.581000\pi$
$998$	−32.5876	−1.03154
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bp.1.2		5
3.2	odd	2		1587.2.a.q.1.4		5
23.9	even	11		207.2.i.a.127.1		10
23.18	even	11		207.2.i.a.163.1		10
23.22	odd	2		4761.2.a.bm.1.2		5
69.32	odd	22		69.2.e.b.58.1	yes	10
69.41	odd	22		69.2.e.b.25.1	✓	10
69.68	even	2		1587.2.a.r.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.b.25.1	✓	10	69.41	odd	22
69.2.e.b.58.1	yes	10	69.32	odd	22
207.2.i.a.127.1		10	23.9	even	11
207.2.i.a.163.1		10	23.18	even	11
1587.2.a.q.1.4		5	3.2	odd	2
1587.2.a.r.1.4		5	69.68	even	2
4761.2.a.bm.1.2		5	23.22	odd	2
4761.2.a.bp.1.2		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)

\(f(q)\)	\(=\)	\(q-1.59435 q^{2} +0.541956 q^{4} -2.53843 q^{5} -1.11325 q^{7} +2.32463 q^{8} +O(q^{10})\)	\(q-1.59435 q^{2} +0.541956 q^{4} -2.53843 q^{5} -1.11325 q^{7} +2.32463 q^{8} +4.04715 q^{10} +1.42094 q^{11} -4.22871 q^{13} +1.77491 q^{14} -4.79020 q^{16} +4.91899 q^{17} -5.79020 q^{19} -1.37572 q^{20} -2.26547 q^{22} +1.44362 q^{25} +6.74204 q^{26} -0.603331 q^{28} -6.42297 q^{29} +7.44168 q^{31} +2.98798 q^{32} -7.84259 q^{34} +2.82590 q^{35} -8.27686 q^{37} +9.23160 q^{38} -5.90092 q^{40} +4.60187 q^{41} +2.02446 q^{43} +0.770084 q^{44} -2.61810 q^{47} -5.76068 q^{49} -2.30164 q^{50} -2.29177 q^{52} +10.4294 q^{53} -3.60695 q^{55} -2.58789 q^{56} +10.2405 q^{58} -11.9787 q^{59} +11.1062 q^{61} -11.8646 q^{62} +4.81650 q^{64} +10.7343 q^{65} -11.3900 q^{67} +2.66587 q^{68} -4.50548 q^{70} -3.87666 q^{71} -12.3341 q^{73} +13.1962 q^{74} -3.13803 q^{76} -1.58185 q^{77} +5.45051 q^{79} +12.1596 q^{80} -7.33699 q^{82} +1.56378 q^{83} -12.4865 q^{85} -3.22770 q^{86} +3.30316 q^{88} +2.28331 q^{89} +4.70760 q^{91} +4.17416 q^{94} +14.6980 q^{95} -8.75140 q^{97} +9.18455 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} + 7 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Representations

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Database

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