LMFDB - Embedded newform 4761.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.5791333887977.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 $ x^10 - 2x^9 - 12x^8 + 22x^7 + 49x^6 - 84x^5 - 73x^4 + 132x^3 + 17x^2 - 74*x + 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.41812$ 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)

$f(q)$	$=$	$q-1.41812 q^{2} +0.0110547 q^{4} -0.849430 q^{5} -5.06334 q^{7} +2.82056 q^{8} +O(q^{10})$	$q-1.41812 q^{2} +0.0110547 q^{4} -0.849430 q^{5} -5.06334 q^{7} +2.82056 q^{8} +1.20459 q^{10} -1.47830 q^{11} +1.01815 q^{13} +7.18040 q^{14} -4.02199 q^{16} +5.71934 q^{17} -4.56726 q^{19} -0.00939015 q^{20} +2.09640 q^{22} -4.27847 q^{25} -1.44385 q^{26} -0.0559734 q^{28} +2.87836 q^{29} +5.27820 q^{31} +0.0625339 q^{32} -8.11068 q^{34} +4.30095 q^{35} +0.462189 q^{37} +6.47690 q^{38} -2.39586 q^{40} +6.89693 q^{41} -4.85049 q^{43} -0.0163421 q^{44} -4.11922 q^{47} +18.6374 q^{49} +6.06737 q^{50} +0.0112552 q^{52} +2.82472 q^{53} +1.25571 q^{55} -14.2814 q^{56} -4.08185 q^{58} -2.11728 q^{59} -9.77162 q^{61} -7.48511 q^{62} +7.95529 q^{64} -0.864843 q^{65} +5.48351 q^{67} +0.0632253 q^{68} -6.09924 q^{70} +12.6544 q^{71} +1.75910 q^{73} -0.655438 q^{74} -0.0504895 q^{76} +7.48511 q^{77} +2.34094 q^{79} +3.41639 q^{80} -9.78065 q^{82} +16.4396 q^{83} -4.85817 q^{85} +6.87857 q^{86} -4.16962 q^{88} -10.1854 q^{89} -5.15521 q^{91} +5.84154 q^{94} +3.87956 q^{95} +11.2699 q^{97} -26.4300 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * 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 $

$2$	−1.41812	−1.00276	−0.501380	−	0.865227i	$-0.667174\pi$
$2$	−1.41812	−1.00276	−0.501380	+	0.865227i	$0.667174\pi$
$3$	0	0
$3$	0	0
$4$	0.0110547	0.00552733
$5$	−0.849430	−0.379876	−0.189938	−	0.981796i	$-0.560829\pi$
$5$	−0.849430	−0.379876	−0.189938	+	0.981796i	$0.560829\pi$
$6$	0	0
$7$	−5.06334	−1.91376	−0.956880	−	0.290482i	$-0.906184\pi$
$7$	−5.06334	−1.91376	−0.956880	+	0.290482i	$0.906184\pi$
$8$	2.82056	0.997217
$9$	0	0
$10$	1.20459	0.380925
$11$	−1.47830	−0.445723	−0.222862	−	0.974850i	$-0.571540\pi$
$11$	−1.47830	−0.445723	−0.222862	+	0.974850i	$0.571540\pi$
$12$	0	0
$13$	1.01815	0.282383	0.141191	−	0.989982i	$-0.454907\pi$
$13$	1.01815	0.282383	0.141191	+	0.989982i	$0.454907\pi$
$14$	7.18040	1.91904
$15$	0	0
$16$	−4.02199	−1.00550
$17$	5.71934	1.38714	0.693571	−	0.720388i	$-0.256036\pi$
$17$	5.71934	1.38714	0.693571	+	0.720388i	$0.256036\pi$
$18$	0	0
$19$	−4.56726	−1.04780	−0.523900	−	0.851780i	$-0.675524\pi$
$19$	−4.56726	−1.04780	−0.523900	+	0.851780i	$0.675524\pi$
$20$	−0.00939015	−0.00209970
$21$	0	0
$22$	2.09640	0.446953
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.27847	−0.855694
$26$	−1.44385	−0.283162
$27$	0	0
$28$	−0.0559734	−0.0105780
$29$	2.87836	0.534498	0.267249	−	0.963627i	$-0.413885\pi$
$29$	2.87836	0.534498	0.267249	+	0.963627i	$0.413885\pi$
$30$	0	0
$31$	5.27820	0.947993	0.473997	−	0.880527i	$-0.342811\pi$
$31$	5.27820	0.947993	0.473997	+	0.880527i	$0.342811\pi$
$32$	0.0625339	0.0110545
$33$	0	0
$34$	−8.11068	−1.39097
$35$	4.30095	0.726993
$36$	0	0
$37$	0.462189	0.0759835	0.0379917	−	0.999278i	$-0.487904\pi$
$37$	0.462189	0.0759835	0.0379917	+	0.999278i	$0.487904\pi$
$38$	6.47690	1.05069
$39$	0	0
$40$	−2.39586	−0.378819
$41$	6.89693	1.07712	0.538560	−	0.842587i	$-0.318969\pi$
$41$	6.89693	1.07712	0.538560	+	0.842587i	$0.318969\pi$
$42$	0	0
$43$	−4.85049	−0.739693	−0.369847	−	0.929093i	$-0.620590\pi$
$43$	−4.85049	−0.739693	−0.369847	+	0.929093i	$0.620590\pi$
$44$	−0.0163421	−0.00246366
$45$	0	0
$46$	0	0
$47$	−4.11922	−0.600850	−0.300425	−	0.953805i	$-0.597129\pi$
$47$	−4.11922	−0.600850	−0.300425	+	0.953805i	$0.597129\pi$
$48$	0	0
$49$	18.6374	2.66248
$50$	6.06737	0.858055
$51$	0	0
$52$	0.0112552	0.00156082
$53$	2.82472	0.388006	0.194003	−	0.981001i	$-0.437853\pi$
$53$	2.82472	0.388006	0.194003	+	0.981001i	$0.437853\pi$
$54$	0	0
$55$	1.25571	0.169320
$56$	−14.2814	−1.90844
$57$	0	0
$58$	−4.08185	−0.535973
$59$	−2.11728	−0.275646	−0.137823	−	0.990457i	$-0.544011\pi$
$59$	−2.11728	−0.275646	−0.137823	+	0.990457i	$0.544011\pi$
$60$	0	0
$61$	−9.77162	−1.25113	−0.625564	−	0.780173i	$-0.715131\pi$
$61$	−9.77162	−1.25113	−0.625564	+	0.780173i	$0.715131\pi$
$62$	−7.48511	−0.950609
$63$	0	0
$64$	7.95529	0.994412
$65$	−0.864843	−0.107271
$66$	0	0
$67$	5.48351	0.669918	0.334959	−	0.942233i	$-0.391278\pi$
$67$	5.48351	0.669918	0.334959	+	0.942233i	$0.391278\pi$
$68$	0.0632253	0.00766719
$69$	0	0
$70$	−6.09924	−0.728999
$71$	12.6544	1.50181	0.750903	−	0.660413i	$-0.229619\pi$
$71$	12.6544	1.50181	0.750903	+	0.660413i	$0.229619\pi$
$72$	0	0
$73$	1.75910	0.205887	0.102943	−	0.994687i	$-0.467174\pi$
$73$	1.75910	0.205887	0.102943	+	0.994687i	$0.467174\pi$
$74$	−0.655438	−0.0761932
$75$	0	0
$76$	−0.0504895	−0.00579154
$77$	7.48511	0.853008
$78$	0	0
$79$	2.34094	0.263377	0.131688	−	0.991291i	$-0.457960\pi$
$79$	2.34094	0.263377	0.131688	+	0.991291i	$0.457960\pi$
$80$	3.41639	0.381965
$81$	0	0
$82$	−9.78065	−1.08009
$83$	16.4396	1.80448	0.902240	−	0.431235i	$-0.141922\pi$
$83$	16.4396	1.80448	0.902240	+	0.431235i	$0.141922\pi$
$84$	0	0
$85$	−4.85817	−0.526943
$86$	6.87857	0.741735
$87$	0	0
$88$	−4.16962	−0.444483
$89$	−10.1854	−1.07965	−0.539823	−	0.841778i	$-0.681509\pi$
$89$	−10.1854	−1.07965	−0.539823	+	0.841778i	$0.681509\pi$
$90$	0	0
$91$	−5.15521	−0.540413
$92$	0	0
$93$	0	0
$94$	5.84154	0.602508
$95$	3.87956	0.398035
$96$	0	0
$97$	11.2699	1.14429	0.572145	−	0.820153i	$-0.306111\pi$
$97$	11.2699	1.14429	0.572145	+	0.820153i	$0.306111\pi$
$98$	−26.4300	−2.66983
$99$	0	0
$100$	−0.0472970	−0.00472970
$101$	15.1769	1.51016	0.755080	−	0.655633i	$-0.227598\pi$
$101$	15.1769	1.51016	0.755080	+	0.655633i	$0.227598\pi$
$102$	0	0
$103$	5.94292	0.585574	0.292787	−	0.956178i	$-0.405417\pi$
$103$	5.94292	0.585574	0.292787	+	0.956178i	$0.405417\pi$
$104$	2.87174	0.281597
$105$	0	0
$106$	−4.00579	−0.389077
$107$	−19.0726	−1.84382	−0.921908	−	0.387409i	$-0.873370\pi$
$107$	−19.0726	−1.84382	−0.921908	+	0.387409i	$0.873370\pi$
$108$	0	0
$109$	6.60928	0.633054	0.316527	−	0.948583i	$-0.397483\pi$
$109$	6.60928	0.633054	0.316527	+	0.948583i	$0.397483\pi$
$110$	−1.78074	−0.169787
$111$	0	0
$112$	20.3647	1.92428
$113$	−0.983338	−0.0925047	−0.0462523	−	0.998930i	$-0.514728\pi$
$113$	−0.983338	−0.0925047	−0.0462523	+	0.998930i	$0.514728\pi$
$114$	0	0
$115$	0	0
$116$	0.0318193	0.00295435
$117$	0	0
$118$	3.00255	0.276407
$119$	−28.9589	−2.65466
$120$	0	0
$121$	−8.81464	−0.801331
$122$	13.8573	1.25458
$123$	0	0
$124$	0.0583487	0.00523987
$125$	7.88141	0.704934
$126$	0	0
$127$	5.35349	0.475046	0.237523	−	0.971382i	$-0.423665\pi$
$127$	5.35349	0.475046	0.237523	+	0.971382i	$0.423665\pi$
$128$	−11.4066	−1.00821
$129$	0	0
$130$	1.22645	0.107567
$131$	−9.13374	−0.798018	−0.399009	−	0.916947i	$-0.630646\pi$
$131$	−9.13374	−0.798018	−0.399009	+	0.916947i	$0.630646\pi$
$132$	0	0
$133$	23.1256	2.00524
$134$	−7.77626	−0.671766
$135$	0	0
$136$	16.1317	1.38328
$137$	−5.65456	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$137$	−5.65456	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$138$	0	0
$139$	15.8561	1.34490	0.672448	−	0.740144i	$-0.265243\pi$
$139$	15.8561	1.34490	0.672448	+	0.740144i	$0.265243\pi$
$140$	0.0475455	0.00401833
$141$	0	0
$142$	−17.9455	−1.50595
$143$	−1.50512	−0.125865
$144$	0	0
$145$	−2.44496	−0.203043
$146$	−2.49461	−0.206455
$147$	0	0
$148$	0.00510935	0.000419986 0
$149$	0.421141	0.0345012	0.0172506	−	0.999851i	$-0.494509\pi$
$149$	0.421141	0.0345012	0.0172506	+	0.999851i	$0.494509\pi$
$150$	0	0
$151$	−21.6388	−1.76094	−0.880469	−	0.474103i	$-0.842772\pi$
$151$	−21.6388	−1.76094	−0.880469	+	0.474103i	$0.842772\pi$
$152$	−12.8822	−1.04488
$153$	0	0
$154$	−10.6148	−0.855362
$155$	−4.48346	−0.360120
$156$	0	0
$157$	−1.80298	−0.143894	−0.0719468	−	0.997408i	$-0.522921\pi$
$157$	−1.80298	−0.143894	−0.0719468	+	0.997408i	$0.522921\pi$
$158$	−3.31973	−0.264104
$159$	0	0
$160$	−0.0531181	−0.00419936
$161$	0	0
$162$	0	0
$163$	−12.3472	−0.967109	−0.483555	−	0.875314i	$-0.660655\pi$
$163$	−12.3472	−0.967109	−0.483555	+	0.875314i	$0.660655\pi$
$164$	0.0762432	0.00595359
$165$	0	0
$166$	−23.3133	−1.80946
$167$	−13.4779	−1.04295	−0.521474	−	0.853267i	$-0.674618\pi$
$167$	−13.4779	−1.04295	−0.521474	+	0.853267i	$0.674618\pi$
$168$	0	0
$169$	−11.9634	−0.920260
$170$	6.88945	0.528397
$171$	0	0
$172$	−0.0536206	−0.00408853
$173$	2.82027	0.214421	0.107211	−	0.994236i	$-0.465808\pi$
$173$	2.82027	0.214421	0.107211	+	0.994236i	$0.465808\pi$
$174$	0	0
$175$	21.6633	1.63759
$176$	5.94569	0.448173
$177$	0	0
$178$	14.4440	1.08263
$179$	−5.52509	−0.412965	−0.206482	−	0.978450i	$-0.566202\pi$
$179$	−5.52509	−0.412965	−0.206482	+	0.978450i	$0.566202\pi$
$180$	0	0
$181$	−11.8866	−0.883526	−0.441763	−	0.897132i	$-0.645647\pi$
$181$	−11.8866	−0.883526	−0.441763	+	0.897132i	$0.645647\pi$
$182$	7.31069	0.541904
$183$	0	0
$184$	0	0
$185$	−0.392597	−0.0288643
$186$	0	0
$187$	−8.45487	−0.618282
$188$	−0.0455366	−0.00332110
$189$	0	0
$190$	−5.50167	−0.399133
$191$	−17.3863	−1.25803	−0.629014	−	0.777394i	$-0.716541\pi$
$191$	−17.3863	−1.25803	−0.629014	+	0.777394i	$0.716541\pi$
$192$	0	0
$193$	−10.4207	−0.750095	−0.375048	−	0.927006i	$-0.622374\pi$
$193$	−10.4207	−0.750095	−0.375048	+	0.927006i	$0.622374\pi$
$194$	−15.9821	−1.14745
$195$	0	0
$196$	0.206030	0.0147164
$197$	10.4310	0.743179	0.371590	−	0.928397i	$-0.378813\pi$
$197$	10.4310	0.743179	0.371590	+	0.928397i	$0.378813\pi$
$198$	0	0
$199$	−4.60128	−0.326176	−0.163088	−	0.986612i	$-0.552145\pi$
$199$	−4.60128	−0.326176	−0.163088	+	0.986612i	$0.552145\pi$
$200$	−12.0677	−0.853313
$201$	0	0
$202$	−21.5226	−1.51433
$203$	−14.5741	−1.02290
$204$	0	0
$205$	−5.85845	−0.409172
$206$	−8.42776	−0.587190
$207$	0	0
$208$	−4.09497	−0.283935
$209$	6.75176	0.467029
$210$	0	0
$211$	−0.530765	−0.0365393	−0.0182697	−	0.999833i	$-0.505816\pi$
$211$	−0.530765	−0.0365393	−0.0182697	+	0.999833i	$0.505816\pi$
$212$	0.0312264	0.00214464
$213$	0	0
$214$	27.0471	1.84890
$215$	4.12015	0.280992
$216$	0	0
$217$	−26.7253	−1.81423
$218$	−9.37273	−0.634801
$219$	0	0
$220$	0.0138814	0.000935886 0
$221$	5.82311	0.391705
$222$	0	0
$223$	−3.29465	−0.220626	−0.110313	−	0.993897i	$-0.535185\pi$
$223$	−3.29465	−0.220626	−0.110313	+	0.993897i	$0.535185\pi$
$224$	−0.316630	−0.0211557
$225$	0	0
$226$	1.39449	0.0927600
$227$	13.0178	0.864018	0.432009	−	0.901869i	$-0.357805\pi$
$227$	13.0178	0.864018	0.432009	+	0.901869i	$0.357805\pi$
$228$	0	0
$229$	4.91857	0.325028	0.162514	−	0.986706i	$-0.448040\pi$
$229$	4.91857	0.325028	0.162514	+	0.986706i	$0.448040\pi$
$230$	0	0
$231$	0	0
$232$	8.11858	0.533011
$233$	16.6299	1.08946	0.544731	−	0.838611i	$-0.316632\pi$
$233$	16.6299	1.08946	0.544731	+	0.838611i	$0.316632\pi$
$234$	0	0
$235$	3.49899	0.228249
$236$	−0.0234058	−0.00152359
$237$	0	0
$238$	41.0671	2.66199
$239$	9.20520	0.595435	0.297718	−	0.954654i	$-0.403775\pi$
$239$	9.20520	0.595435	0.297718	+	0.954654i	$0.403775\pi$
$240$	0	0
$241$	−2.70054	−0.173957	−0.0869784	−	0.996210i	$-0.527721\pi$
$241$	−2.70054	−0.173957	−0.0869784	+	0.996210i	$0.527721\pi$
$242$	12.5002	0.803542
$243$	0	0
$244$	−0.108022	−0.00691539
$245$	−15.8311	−1.01141
$246$	0	0
$247$	−4.65013	−0.295881
$248$	14.8875	0.945355
$249$	0	0
$250$	−11.1768	−0.706880
$251$	−9.05985	−0.571853	−0.285926	−	0.958252i	$-0.592301\pi$
$251$	−9.05985	−0.571853	−0.285926	+	0.958252i	$0.592301\pi$
$252$	0	0
$253$	0	0
$254$	−7.59188	−0.476357
$255$	0	0
$256$	0.265304	0.0165815
$257$	20.4346	1.27468	0.637338	−	0.770585i	$-0.280036\pi$
$257$	20.4346	1.27468	0.637338	+	0.770585i	$0.280036\pi$
$258$	0	0
$259$	−2.34022	−0.145414
$260$	−0.00956054	−0.000592920 0
$261$	0	0
$262$	12.9527	0.800221
$263$	−3.10487	−0.191455	−0.0957273	−	0.995408i	$-0.530518\pi$
$263$	−3.10487	−0.191455	−0.0957273	+	0.995408i	$0.530518\pi$
$264$	0	0
$265$	−2.39940	−0.147394
$266$	−32.7947	−2.01077
$267$	0	0
$268$	0.0606183	0.00370285
$269$	−3.81666	−0.232706	−0.116353	−	0.993208i	$-0.537120\pi$
$269$	−3.81666	−0.232706	−0.116353	+	0.993208i	$0.537120\pi$
$270$	0	0
$271$	−1.98253	−0.120430	−0.0602150	−	0.998185i	$-0.519179\pi$
$271$	−1.98253	−0.120430	−0.0602150	+	0.998185i	$0.519179\pi$
$272$	−23.0031	−1.39477
$273$	0	0
$274$	8.01883	0.484435
$275$	6.32485	0.381403
$276$	0	0
$277$	−14.5994	−0.877195	−0.438598	−	0.898684i	$-0.644525\pi$
$277$	−14.5994	−0.877195	−0.438598	+	0.898684i	$0.644525\pi$
$278$	−22.4858	−1.34861
$279$	0	0
$280$	12.1311	0.724970
$281$	−14.4172	−0.860060	−0.430030	−	0.902815i	$-0.641497\pi$
$281$	−14.4172	−0.860060	−0.430030	+	0.902815i	$0.641497\pi$
$282$	0	0
$283$	−14.8202	−0.880970	−0.440485	−	0.897760i	$-0.645194\pi$
$283$	−14.8202	−0.880970	−0.440485	+	0.897760i	$0.645194\pi$
$284$	0.139891	0.00830097
$285$	0	0
$286$	2.13444	0.126212
$287$	−34.9215	−2.06135
$288$	0	0
$289$	15.7108	0.924164
$290$	3.46724	0.203604
$291$	0	0
$292$	0.0194462	0.00113801
$293$	−10.9526	−0.639857	−0.319929	−	0.947442i	$-0.603659\pi$
$293$	−10.9526	−0.639857	−0.319929	+	0.947442i	$0.603659\pi$
$294$	0	0
$295$	1.79848	0.104712
$296$	1.30363	0.0757720
$297$	0	0
$298$	−0.597228	−0.0345965
$299$	0	0
$300$	0	0
$301$	24.5597	1.41560
$302$	30.6863	1.76580
$303$	0	0
$304$	18.3695	1.05356
$305$	8.30030	0.475274
$306$	0	0
$307$	−12.6242	−0.720499	−0.360250	−	0.932856i	$-0.617308\pi$
$307$	−12.6242	−0.720499	−0.360250	+	0.932856i	$0.617308\pi$
$308$	0.0827453	0.00471485
$309$	0	0
$310$	6.35807	0.361114
$311$	−29.6291	−1.68011	−0.840055	−	0.542501i	$-0.817477\pi$
$311$	−29.6291	−1.68011	−0.840055	+	0.542501i	$0.817477\pi$
$312$	0	0
$313$	1.51000	0.0853501	0.0426751	−	0.999089i	$-0.486412\pi$
$313$	1.51000	0.0853501	0.0426751	+	0.999089i	$0.486412\pi$
$314$	2.55684	0.144291
$315$	0	0
$316$	0.0258783	0.00145577
$317$	13.2178	0.742386	0.371193	−	0.928556i	$-0.378949\pi$
$317$	13.2178	0.742386	0.371193	+	0.928556i	$0.378949\pi$
$318$	0	0
$319$	−4.25507	−0.238238
$320$	−6.75746	−0.377754
$321$	0	0
$322$	0	0
$323$	−26.1217	−1.45345
$324$	0	0
$325$	−4.35610	−0.241633
$326$	17.5098	0.969778
$327$	0	0
$328$	19.4532	1.07412
$329$	20.8570	1.14988
$330$	0	0
$331$	30.4156	1.67179	0.835897	−	0.548886i	$-0.184948\pi$
$331$	30.4156	1.67179	0.835897	+	0.548886i	$0.184948\pi$
$332$	0.181734	0.00997395
$333$	0	0
$334$	19.1132	1.04583
$335$	−4.65786	−0.254486
$336$	0	0
$337$	−0.939586	−0.0511825	−0.0255913	−	0.999672i	$-0.508147\pi$
$337$	−0.939586	−0.0511825	−0.0255913	+	0.999672i	$0.508147\pi$
$338$	16.9655	0.922800
$339$	0	0
$340$	−0.0537054	−0.00291259
$341$	−7.80275	−0.422543
$342$	0	0
$343$	−58.9239	−3.18159
$344$	−13.6811	−0.737635
$345$	0	0
$346$	−3.99948	−0.215013
$347$	−27.8946	−1.49746	−0.748731	−	0.662874i	$-0.769337\pi$
$347$	−27.8946	−1.49746	−0.748731	+	0.662874i	$0.769337\pi$
$348$	0	0
$349$	21.9082	1.17272	0.586360	−	0.810050i	$-0.300560\pi$
$349$	21.9082	1.17272	0.586360	+	0.810050i	$0.300560\pi$
$350$	−30.7211	−1.64211
$351$	0	0
$352$	−0.0924436	−0.00492726
$353$	17.4523	0.928894	0.464447	−	0.885601i	$-0.346253\pi$
$353$	17.4523	0.928894	0.464447	+	0.885601i	$0.346253\pi$
$354$	0	0
$355$	−10.7491	−0.570501
$356$	−0.112596	−0.00596756
$357$	0	0
$358$	7.83522	0.414104
$359$	16.0538	0.847290	0.423645	−	0.905828i	$-0.360750\pi$
$359$	16.0538	0.847290	0.423645	+	0.905828i	$0.360750\pi$
$360$	0	0
$361$	1.85984	0.0978863
$362$	16.8566	0.885964
$363$	0	0
$364$	−0.0569891	−0.00298704
$365$	−1.49423	−0.0782116
$366$	0	0
$367$	−12.0449	−0.628736	−0.314368	−	0.949301i	$-0.601793\pi$
$367$	−12.0449	−0.628736	−0.314368	+	0.949301i	$0.601793\pi$
$368$	0	0
$369$	0	0
$370$	0.556749	0.0289440
$371$	−14.3025	−0.742550
$372$	0	0
$373$	−16.9618	−0.878246	−0.439123	−	0.898427i	$-0.644711\pi$
$373$	−16.9618	−0.878246	−0.439123	+	0.898427i	$0.644711\pi$
$374$	11.9900	0.619988
$375$	0	0
$376$	−11.6185	−0.599178
$377$	2.93059	0.150933
$378$	0	0
$379$	6.60600	0.339327	0.169664	−	0.985502i	$-0.445732\pi$
$379$	6.60600	0.339327	0.169664	+	0.985502i	$0.445732\pi$
$380$	0.0428872	0.00220007
$381$	0	0
$382$	24.6558	1.26150
$383$	11.6289	0.594211	0.297106	−	0.954845i	$-0.403979\pi$
$383$	11.6289	0.594211	0.297106	+	0.954845i	$0.403979\pi$
$384$	0	0
$385$	−6.35808	−0.324038
$386$	14.7777	0.752165
$387$	0	0
$388$	0.124585	0.00632487
$389$	−16.6432	−0.843846	−0.421923	−	0.906632i	$-0.638645\pi$
$389$	−16.6432	−0.843846	−0.421923	+	0.906632i	$0.638645\pi$
$390$	0	0
$391$	0	0
$392$	52.5677	2.65507
$393$	0	0
$394$	−14.7924	−0.745230
$395$	−1.98847	−0.100051
$396$	0	0
$397$	−31.0645	−1.55908	−0.779541	−	0.626352i	$-0.784547\pi$
$397$	−31.0645	−1.55908	−0.779541	+	0.626352i	$0.784547\pi$
$398$	6.52515	0.327076
$399$	0	0
$400$	17.2079	0.860397
$401$	16.2430	0.811139	0.405569	−	0.914064i	$-0.367073\pi$
$401$	16.2430	0.811139	0.405569	+	0.914064i	$0.367073\pi$
$402$	0	0
$403$	5.37398	0.267697
$404$	0.167776	0.00834715
$405$	0	0
$406$	20.6678	1.02572
$407$	−0.683253	−0.0338676
$408$	0	0
$409$	−5.66031	−0.279884	−0.139942	−	0.990160i	$-0.544692\pi$
$409$	−5.66031	−0.279884	−0.139942	+	0.990160i	$0.544692\pi$
$410$	8.30797	0.410301
$411$	0	0
$412$	0.0656970	0.00323666
$413$	10.7205	0.527521
$414$	0	0
$415$	−13.9643	−0.685479
$416$	0.0636686	0.00312161
$417$	0	0
$418$	−9.57478	−0.468318
$419$	16.4074	0.801552	0.400776	−	0.916176i	$-0.368741\pi$
$419$	16.4074	0.801552	0.400776	+	0.916176i	$0.368741\pi$
$420$	0	0
$421$	−3.17259	−0.154623	−0.0773113	−	0.997007i	$-0.524634\pi$
$421$	−3.17259	−0.154623	−0.0773113	+	0.997007i	$0.524634\pi$
$422$	0.752686	0.0366402
$423$	0	0
$424$	7.96729	0.386926
$425$	−24.4700	−1.18697
$426$	0	0
$427$	49.4770	2.39436
$428$	−0.210841	−0.0101914
$429$	0	0
$430$	−5.84286	−0.281768
$431$	−24.8521	−1.19708	−0.598542	−	0.801092i	$-0.704253\pi$
$431$	−24.8521	−1.19708	−0.598542	+	0.801092i	$0.704253\pi$
$432$	0	0
$433$	−15.3426	−0.737319	−0.368659	−	0.929565i	$-0.620183\pi$
$433$	−15.3426	−0.737319	−0.368659	+	0.929565i	$0.620183\pi$
$434$	37.8996	1.81924
$435$	0	0
$436$	0.0730633	0.00349910
$437$	0	0
$438$	0	0
$439$	−25.5666	−1.22023	−0.610114	−	0.792314i	$-0.708876\pi$
$439$	−25.5666	−1.22023	−0.610114	+	0.792314i	$0.708876\pi$
$440$	3.54180	0.168849
$441$	0	0
$442$	−8.25786	−0.392786
$443$	23.9302	1.13696	0.568479	−	0.822698i	$-0.307532\pi$
$443$	23.9302	1.13696	0.568479	+	0.822698i	$0.307532\pi$
$444$	0	0
$445$	8.65175	0.410132
$446$	4.67220	0.221235
$447$	0	0
$448$	−40.2803	−1.90307
$449$	−8.08322	−0.381471	−0.190735	−	0.981642i	$-0.561087\pi$
$449$	−8.08322	−0.381471	−0.190735	+	0.981642i	$0.561087\pi$
$450$	0	0
$451$	−10.1957	−0.480097
$452$	−0.0108705	−0.000511304 0
$453$	0	0
$454$	−18.4607	−0.866403
$455$	4.37899	0.205290
$456$	0	0
$457$	−22.7288	−1.06321	−0.531603	−	0.846994i	$-0.678410\pi$
$457$	−22.7288	−1.06321	−0.531603	+	0.846994i	$0.678410\pi$
$458$	−6.97510	−0.325925
$459$	0	0
$460$	0	0
$461$	7.53588	0.350981	0.175490	−	0.984481i	$-0.443849\pi$
$461$	7.53588	0.350981	0.175490	+	0.984481i	$0.443849\pi$
$462$	0	0
$463$	−22.9762	−1.06780	−0.533898	−	0.845549i	$-0.679273\pi$
$463$	−22.9762	−1.06780	−0.533898	+	0.845549i	$0.679273\pi$
$464$	−11.5767	−0.537436
$465$	0	0
$466$	−23.5831	−1.09247
$467$	−11.2062	−0.518560	−0.259280	−	0.965802i	$-0.583485\pi$
$467$	−11.2062	−0.518560	−0.259280	+	0.965802i	$0.583485\pi$
$468$	0	0
$469$	−27.7649	−1.28206
$470$	−4.96197	−0.228879
$471$	0	0
$472$	−5.97191	−0.274879
$473$	7.17047	0.329699
$474$	0	0
$475$	19.5409	0.896597
$476$	−0.320131	−0.0146732
$477$	0	0
$478$	−13.0541	−0.597078
$479$	−20.7310	−0.947224	−0.473612	−	0.880734i	$-0.657050\pi$
$479$	−20.7310	−0.947224	−0.473612	+	0.880734i	$0.657050\pi$
$480$	0	0
$481$	0.470576	0.0214564
$482$	3.82968	0.174437
$483$	0	0
$484$	−0.0974428	−0.00442922
$485$	−9.57303	−0.434689
$486$	0	0
$487$	−25.1231	−1.13844	−0.569219	−	0.822186i	$-0.692754\pi$
$487$	−25.1231	−1.13844	−0.569219	+	0.822186i	$0.692754\pi$
$488$	−27.5614	−1.24765
$489$	0	0
$490$	22.4504	1.01421
$491$	−17.3156	−0.781440	−0.390720	−	0.920509i	$-0.627774\pi$
$491$	−17.3156	−0.781440	−0.390720	+	0.920509i	$0.627774\pi$
$492$	0	0
$493$	16.4623	0.741425
$494$	6.59443	0.296697
$495$	0	0
$496$	−21.2289	−0.953204
$497$	−64.0737	−2.87410
$498$	0	0
$499$	38.7837	1.73620	0.868098	−	0.496393i	$-0.165342\pi$
$499$	38.7837	1.73620	0.868098	+	0.496393i	$0.165342\pi$
$500$	0.0871262	0.00389640
$501$	0	0
$502$	12.8479	0.573431
$503$	−16.8432	−0.751001	−0.375501	−	0.926822i	$-0.622529\pi$
$503$	−16.8432	−0.751001	−0.375501	+	0.926822i	$0.622529\pi$
$504$	0	0
$505$	−12.8917	−0.573674
$506$	0	0
$507$	0	0
$508$	0.0591810	0.00262573
$509$	−24.3526	−1.07941	−0.539705	−	0.841854i	$-0.681464\pi$
$509$	−24.3526	−1.07941	−0.539705	+	0.841854i	$0.681464\pi$
$510$	0	0
$511$	−8.90691	−0.394018
$512$	22.4370	0.991583
$513$	0	0
$514$	−28.9786	−1.27819
$515$	−5.04809	−0.222446
$516$	0	0
$517$	6.08943	0.267813
$518$	3.31870	0.145816
$519$	0	0
$520$	−2.43934	−0.106972
$521$	27.5718	1.20794	0.603971	−	0.797006i	$-0.293584\pi$
$521$	27.5718	1.20794	0.603971	+	0.797006i	$0.293584\pi$
$522$	0	0
$523$	21.2198	0.927876	0.463938	−	0.885868i	$-0.346436\pi$
$523$	21.2198	0.927876	0.463938	+	0.885868i	$0.346436\pi$
$524$	−0.100970	−0.00441091
$525$	0	0
$526$	4.40307	0.191983
$527$	30.1878	1.31500
$528$	0	0
$529$	0	0
$530$	3.40264	0.147801
$531$	0	0
$532$	0.255645	0.0110836
$533$	7.02207	0.304160
$534$	0	0
$535$	16.2008	0.700422
$536$	15.4666	0.668053
$537$	0	0
$538$	5.41247	0.233348
$539$	−27.5516	−1.18673
$540$	0	0
$541$	12.7195	0.546854	0.273427	−	0.961893i	$-0.411843\pi$
$541$	12.7195	0.546854	0.273427	+	0.961893i	$0.411843\pi$
$542$	2.81146	0.120762
$543$	0	0
$544$	0.357652	0.0153342
$545$	−5.61412	−0.240482
$546$	0	0
$547$	−25.6219	−1.09551	−0.547756	−	0.836638i	$-0.684518\pi$
$547$	−25.6219	−1.09551	−0.547756	+	0.836638i	$0.684518\pi$
$548$	−0.0625093	−0.00267026
$549$	0	0
$550$	−8.96937	−0.382455
$551$	−13.1462	−0.560047
$552$	0	0
$553$	−11.8530	−0.504040
$554$	20.7037	0.879616
$555$	0	0
$556$	0.175284	0.00743369
$557$	32.5069	1.37736	0.688680	−	0.725065i	$-0.258190\pi$
$557$	32.5069	1.37736	0.688680	+	0.725065i	$0.258190\pi$
$558$	0	0
$559$	−4.93851	−0.208877
$560$	−17.2984	−0.730989
$561$	0	0
$562$	20.4453	0.862433
$563$	30.8728	1.30113	0.650567	−	0.759449i	$-0.274531\pi$
$563$	30.8728	1.30113	0.650567	+	0.759449i	$0.274531\pi$
$564$	0	0
$565$	0.835276	0.0351403
$566$	21.0168	0.883401
$567$	0	0
$568$	35.6926	1.49763
$569$	22.9336	0.961428	0.480714	−	0.876877i	$-0.340377\pi$
$569$	22.9336	0.961428	0.480714	+	0.876877i	$0.340377\pi$
$570$	0	0
$571$	−12.7017	−0.531548	−0.265774	−	0.964035i	$-0.585627\pi$
$571$	−12.7017	−0.531548	−0.265774	+	0.964035i	$0.585627\pi$
$572$	−0.0166386	−0.000695695 0
$573$	0	0
$574$	49.5227	2.06704
$575$	0	0
$576$	0	0
$577$	10.9907	0.457548	0.228774	−	0.973480i	$-0.426528\pi$
$577$	10.9907	0.457548	0.228774	+	0.973480i	$0.426528\pi$
$578$	−22.2797	−0.926715
$579$	0	0
$580$	−0.0270282	−0.00112229
$581$	−83.2392	−3.45334
$582$	0	0
$583$	−4.17578	−0.172943
$584$	4.96164	0.205314
$585$	0	0
$586$	15.5320	0.641623
$587$	20.7804	0.857701	0.428850	−	0.903376i	$-0.358919\pi$
$587$	20.7804	0.857701	0.428850	+	0.903376i	$0.358919\pi$
$588$	0	0
$589$	−24.1069	−0.993308
$590$	−2.55045	−0.105001
$591$	0	0
$592$	−1.85892	−0.0764011
$593$	−31.5977	−1.29756	−0.648780	−	0.760976i	$-0.724721\pi$
$593$	−31.5977	−1.29756	−0.648780	+	0.760976i	$0.724721\pi$
$594$	0	0
$595$	24.5986	1.00844
$596$	0.00465557	0.000190700 0
$597$	0	0
$598$	0	0
$599$	−21.9650	−0.897466	−0.448733	−	0.893666i	$-0.648125\pi$
$599$	−21.9650	−0.897466	−0.448733	+	0.893666i	$0.648125\pi$
$600$	0	0
$601$	−21.3616	−0.871356	−0.435678	−	0.900103i	$-0.643491\pi$
$601$	−21.3616	−0.871356	−0.435678	+	0.900103i	$0.643491\pi$
$602$	−34.8285	−1.41950
$603$	0	0
$604$	−0.239209	−0.00973329
$605$	7.48742	0.304407
$606$	0	0
$607$	−5.88927	−0.239038	−0.119519	−	0.992832i	$-0.538135\pi$
$607$	−5.88927	−0.239038	−0.119519	+	0.992832i	$0.538135\pi$
$608$	−0.285608	−0.0115829
$609$	0	0
$610$	−11.7708	−0.476585
$611$	−4.19397	−0.169670
$612$	0	0
$613$	−5.06195	−0.204450	−0.102225	−	0.994761i	$-0.532596\pi$
$613$	−5.06195	−0.204450	−0.102225	+	0.994761i	$0.532596\pi$
$614$	17.9025	0.722488
$615$	0	0
$616$	21.1122	0.850634
$617$	−8.04564	−0.323905	−0.161953	−	0.986799i	$-0.551779\pi$
$617$	−8.04564	−0.323905	−0.161953	+	0.986799i	$0.551779\pi$
$618$	0	0
$619$	7.14502	0.287183	0.143591	−	0.989637i	$-0.454135\pi$
$619$	7.14502	0.287183	0.143591	+	0.989637i	$0.454135\pi$
$620$	−0.0495631	−0.00199050
$621$	0	0
$622$	42.0175	1.68475
$623$	51.5719	2.06618
$624$	0	0
$625$	14.6976	0.587906
$626$	−2.14135	−0.0855857
$627$	0	0
$628$	−0.0199313	−0.000795347 0
$629$	2.64342	0.105400
$630$	0	0
$631$	−38.2370	−1.52219	−0.761096	−	0.648640i	$-0.775338\pi$
$631$	−38.2370	−1.52219	−0.761096	+	0.648640i	$0.775338\pi$
$632$	6.60276	0.262644
$633$	0	0
$634$	−18.7444	−0.744435
$635$	−4.54741	−0.180459
$636$	0	0
$637$	18.9755	0.751839
$638$	6.03419	0.238896
$639$	0	0
$640$	9.68910	0.382996
$641$	−32.3783	−1.27887	−0.639434	−	0.768846i	$-0.720831\pi$
$641$	−32.3783	−1.27887	−0.639434	+	0.768846i	$0.720831\pi$
$642$	0	0
$643$	−35.7909	−1.41146	−0.705728	−	0.708483i	$-0.749380\pi$
$643$	−35.7909	−1.41146	−0.705728	+	0.708483i	$0.749380\pi$
$644$	0	0
$645$	0	0
$646$	37.0436	1.45746
$647$	4.46706	0.175618	0.0878092	−	0.996137i	$-0.472013\pi$
$647$	4.46706	0.175618	0.0878092	+	0.996137i	$0.472013\pi$
$648$	0	0
$649$	3.12997	0.122862
$650$	6.17746	0.242300
$651$	0	0
$652$	−0.136494	−0.00534553
$653$	−21.2846	−0.832931	−0.416465	−	0.909152i	$-0.636731\pi$
$653$	−21.2846	−0.832931	−0.416465	+	0.909152i	$0.636731\pi$
$654$	0	0
$655$	7.75847	0.303148
$656$	−27.7394	−1.08304
$657$	0	0
$658$	−29.5777	−1.15306
$659$	−46.0318	−1.79314	−0.896572	−	0.442898i	$-0.853950\pi$
$659$	−46.0318	−1.79314	−0.896572	+	0.442898i	$0.853950\pi$
$660$	0	0
$661$	−33.0817	−1.28673	−0.643365	−	0.765560i	$-0.722462\pi$
$661$	−33.0817	−1.28673	−0.643365	+	0.765560i	$0.722462\pi$
$662$	−43.1329	−1.67641
$663$	0	0
$664$	46.3688	1.79946
$665$	−19.6435	−0.761743
$666$	0	0
$667$	0	0
$668$	−0.148993	−0.00576472
$669$	0	0
$670$	6.60538	0.255188
$671$	14.4453	0.557656
$672$	0	0
$673$	41.3111	1.59243	0.796213	−	0.605016i	$-0.206833\pi$
$673$	41.3111	1.59243	0.796213	+	0.605016i	$0.206833\pi$
$674$	1.33244	0.0513238
$675$	0	0
$676$	−0.132251	−0.00508658
$677$	30.0261	1.15400	0.576998	−	0.816745i	$-0.304224\pi$
$677$	30.0261	1.15400	0.576998	+	0.816745i	$0.304224\pi$
$678$	0	0
$679$	−57.0635	−2.18990
$680$	−13.7027	−0.525476
$681$	0	0
$682$	11.0652	0.423709
$683$	40.4613	1.54821	0.774104	−	0.633058i	$-0.218201\pi$
$683$	40.4613	1.54821	0.774104	+	0.633058i	$0.218201\pi$
$684$	0	0
$685$	4.80315	0.183519
$686$	83.5610	3.19037
$687$	0	0
$688$	19.5086	0.743759
$689$	2.87598	0.109566
$690$	0	0
$691$	3.24119	0.123301	0.0616503	−	0.998098i	$-0.480364\pi$
$691$	3.24119	0.123301	0.0616503	+	0.998098i	$0.480364\pi$
$692$	0.0311772	0.00118518
$693$	0	0
$694$	39.5579	1.50160
$695$	−13.4686	−0.510895
$696$	0	0
$697$	39.4458	1.49412
$698$	−31.0684	−1.17596
$699$	0	0
$700$	0.239481	0.00905152
$701$	29.7702	1.12440	0.562202	−	0.827000i	$-0.309954\pi$
$701$	29.7702	1.12440	0.562202	+	0.827000i	$0.309954\pi$
$702$	0	0
$703$	−2.11094	−0.0796155
$704$	−11.7603	−0.443232
$705$	0	0
$706$	−24.7495	−0.931458
$707$	−76.8458	−2.89009
$708$	0	0
$709$	−31.9434	−1.19966	−0.599830	−	0.800127i	$-0.704765\pi$
$709$	−31.9434	−1.19966	−0.599830	+	0.800127i	$0.704765\pi$
$710$	15.2434	0.572075
$711$	0	0
$712$	−28.7284	−1.07664
$713$	0	0
$714$	0	0
$715$	1.27849	0.0478130
$716$	−0.0610780	−0.00228259
$717$	0	0
$718$	−22.7662	−0.849628
$719$	11.6380	0.434023	0.217012	−	0.976169i	$-0.430369\pi$
$719$	11.6380	0.434023	0.217012	+	0.976169i	$0.430369\pi$
$720$	0	0
$721$	−30.0910	−1.12065
$722$	−2.63747	−0.0981565
$723$	0	0
$724$	−0.131403	−0.00488354
$725$	−12.3150	−0.457367
$726$	0	0
$727$	−36.0566	−1.33726	−0.668632	−	0.743594i	$-0.733120\pi$
$727$	−36.0566	−1.33726	−0.668632	+	0.743594i	$0.733120\pi$
$728$	−14.5406	−0.538909
$729$	0	0
$730$	2.11899	0.0784275
$731$	−27.7416	−1.02606
$732$	0	0
$733$	−1.63846	−0.0605179	−0.0302589	−	0.999542i	$-0.509633\pi$
$733$	−1.63846	−0.0605179	−0.0302589	+	0.999542i	$0.509633\pi$
$734$	17.0810	0.630472
$735$	0	0
$736$	0	0
$737$	−8.10626	−0.298598
$738$	0	0
$739$	36.0410	1.32579	0.662894	−	0.748713i	$-0.269328\pi$
$739$	36.0410	1.32579	0.662894	+	0.748713i	$0.269328\pi$
$740$	−0.00434003	−0.000159543 0
$741$	0	0
$742$	20.2827	0.744600
$743$	−26.5705	−0.974777	−0.487389	−	0.873185i	$-0.662050\pi$
$743$	−26.5705	−0.974777	−0.487389	+	0.873185i	$0.662050\pi$
$744$	0	0
$745$	−0.357730	−0.0131062
$746$	24.0537	0.880670
$747$	0	0
$748$	−0.0934657	−0.00341745
$749$	96.5709	3.52862
$750$	0	0
$751$	−31.0283	−1.13224	−0.566119	−	0.824323i	$-0.691556\pi$
$751$	−31.0283	−1.13224	−0.566119	+	0.824323i	$0.691556\pi$
$752$	16.5675	0.604153
$753$	0	0
$754$	−4.15592	−0.151350
$755$	18.3806	0.668939
$756$	0	0
$757$	10.8695	0.395058	0.197529	−	0.980297i	$-0.436708\pi$
$757$	10.8695	0.395058	0.197529	+	0.980297i	$0.436708\pi$
$758$	−9.36808	−0.340264
$759$	0	0
$760$	10.9425	0.396927
$761$	−17.1144	−0.620397	−0.310198	−	0.950672i	$-0.600395\pi$
$761$	−17.1144	−0.620397	−0.310198	+	0.950672i	$0.600395\pi$
$762$	0	0
$763$	−33.4650	−1.21151
$764$	−0.192199	−0.00695353
$765$	0	0
$766$	−16.4912	−0.595851
$767$	−2.15570	−0.0778378
$768$	0	0
$769$	27.9833	1.00910	0.504552	−	0.863381i	$-0.331658\pi$
$769$	27.9833	1.00910	0.504552	+	0.863381i	$0.331658\pi$
$770$	9.01649	0.324932
$771$	0	0
$772$	−0.115197	−0.00414602
$773$	10.7778	0.387649	0.193825	−	0.981036i	$-0.437911\pi$
$773$	10.7778	0.387649	0.193825	+	0.981036i	$0.437911\pi$
$774$	0	0
$775$	−22.5826	−0.811192
$776$	31.7875	1.14111
$777$	0	0
$778$	23.6021	0.846175
$779$	−31.5000	−1.12861
$780$	0	0
$781$	−18.7070	−0.669390
$782$	0	0
$783$	0	0
$784$	−74.9593	−2.67712
$785$	1.53150	0.0546618
$786$	0	0
$787$	33.0236	1.17716	0.588582	−	0.808437i	$-0.299686\pi$
$787$	33.0236	1.17716	0.588582	+	0.808437i	$0.299686\pi$
$788$	0.115311	0.00410779
$789$	0	0
$790$	2.81988	0.100327
$791$	4.97897	0.177032
$792$	0	0
$793$	−9.94892	−0.353297
$794$	44.0531	1.56338
$795$	0	0
$796$	−0.0508656	−0.00180288
$797$	51.5655	1.82654	0.913272	−	0.407351i	$-0.133548\pi$
$797$	51.5655	1.82654	0.913272	+	0.407351i	$0.133548\pi$
$798$	0	0
$799$	−23.5592	−0.833465
$800$	−0.267549	−0.00945929
$801$	0	0
$802$	−23.0345	−0.813377
$803$	−2.60047	−0.0917686
$804$	0	0
$805$	0	0
$806$	−7.62093	−0.268436
$807$	0	0
$808$	42.8074	1.50596
$809$	6.57465	0.231152	0.115576	−	0.993299i	$-0.463129\pi$
$809$	6.57465	0.231152	0.115576	+	0.993299i	$0.463129\pi$
$810$	0	0
$811$	8.09852	0.284378	0.142189	−	0.989840i	$-0.454586\pi$
$811$	8.09852	0.284378	0.142189	+	0.989840i	$0.454586\pi$
$812$	−0.161112	−0.00565391
$813$	0	0
$814$	0.968932	0.0339611
$815$	10.4881	0.367382
$816$	0	0
$817$	22.1535	0.775051
$818$	8.02698	0.280657
$819$	0	0
$820$	−0.0647632	−0.00226163
$821$	−52.4466	−1.83040	−0.915199	−	0.403003i	$-0.867967\pi$
$821$	−52.4466	−1.83040	−0.915199	+	0.403003i	$0.867967\pi$
$822$	0	0
$823$	7.64434	0.266465	0.133232	−	0.991085i	$-0.457464\pi$
$823$	7.64434	0.266465	0.133232	+	0.991085i	$0.457464\pi$
$824$	16.7623	0.583944
$825$	0	0
$826$	−15.2029	−0.528977
$827$	−10.4045	−0.361800	−0.180900	−	0.983501i	$-0.557901\pi$
$827$	−10.4045	−0.361800	−0.180900	+	0.983501i	$0.557901\pi$
$828$	0	0
$829$	35.0711	1.21807	0.609035	−	0.793144i	$-0.291557\pi$
$829$	35.0711	1.21807	0.609035	+	0.793144i	$0.291557\pi$
$830$	19.8030	0.687371
$831$	0	0
$832$	8.09965	0.280805
$833$	106.593	3.69324
$834$	0	0
$835$	11.4485	0.396192
$836$	0.0746384	0.00258142
$837$	0	0
$838$	−23.2675	−0.803764
$839$	16.5869	0.572644	0.286322	−	0.958133i	$-0.407567\pi$
$839$	16.5869	0.572644	0.286322	+	0.958133i	$0.407567\pi$
$840$	0	0
$841$	−20.7150	−0.714312
$842$	4.49911	0.155049
$843$	0	0
$844$	−0.00586742	−0.000201965 0
$845$	10.1620	0.349585
$846$	0	0
$847$	44.6315	1.53356
$848$	−11.3610	−0.390139
$849$	0	0
$850$	34.7013	1.19025
$851$	0	0
$852$	0	0
$853$	−27.1767	−0.930513	−0.465257	−	0.885176i	$-0.654038\pi$
$853$	−27.1767	−0.930513	−0.465257	+	0.885176i	$0.654038\pi$
$854$	−70.1641	−2.40097
$855$	0	0
$856$	−53.7953	−1.83868
$857$	20.4008	0.696878	0.348439	−	0.937331i	$-0.386712\pi$
$857$	20.4008	0.696878	0.348439	+	0.937331i	$0.386712\pi$
$858$	0	0
$859$	9.58902	0.327173	0.163587	−	0.986529i	$-0.447694\pi$
$859$	9.58902	0.327173	0.163587	+	0.986529i	$0.447694\pi$
$860$	0.0455469	0.00155314
$861$	0	0
$862$	35.2432	1.20039
$863$	20.6679	0.703543	0.351772	−	0.936086i	$-0.385579\pi$
$863$	20.6679	0.703543	0.351772	+	0.936086i	$0.385579\pi$
$864$	0	0
$865$	−2.39562	−0.0814537
$866$	21.7576	0.739353
$867$	0	0
$868$	−0.295439	−0.0100279
$869$	−3.46061	−0.117393
$870$	0	0
$871$	5.58301	0.189173
$872$	18.6418	0.631292
$873$	0	0
$874$	0	0
$875$	−39.9062	−1.34908
$876$	0	0
$877$	−49.3397	−1.66608	−0.833041	−	0.553211i	$-0.813402\pi$
$877$	−49.3397	−1.66608	−0.833041	+	0.553211i	$0.813402\pi$
$878$	36.2564	1.22360
$879$	0	0
$880$	−5.05044	−0.170250
$881$	−27.9208	−0.940676	−0.470338	−	0.882486i	$-0.655868\pi$
$881$	−27.9208	−0.940676	−0.470338	+	0.882486i	$0.655868\pi$
$882$	0	0
$883$	−5.63664	−0.189688	−0.0948440	−	0.995492i	$-0.530235\pi$
$883$	−5.63664	−0.189688	−0.0948440	+	0.995492i	$0.530235\pi$
$884$	0.0643725	0.00216508
$885$	0	0
$886$	−33.9358	−1.14009
$887$	17.4634	0.586362	0.293181	−	0.956057i	$-0.405286\pi$
$887$	17.4634	0.586362	0.293181	+	0.956057i	$0.405286\pi$
$888$	0	0
$889$	−27.1065	−0.909124
$890$	−12.2692	−0.411264
$891$	0	0
$892$	−0.0364212	−0.00121947
$893$	18.8135	0.629571
$894$	0	0
$895$	4.69318	0.156876
$896$	57.7555	1.92947
$897$	0	0
$898$	11.4629	0.382523
$899$	15.1926	0.506701
$900$	0	0
$901$	16.1555	0.538219
$902$	14.4587	0.481422
$903$	0	0
$904$	−2.77356	−0.0922473
$905$	10.0968	0.335631
$906$	0	0
$907$	36.7709	1.22096	0.610479	−	0.792032i	$-0.290977\pi$
$907$	36.7709	1.22096	0.610479	+	0.792032i	$0.290977\pi$
$908$	0.143907	0.00477571
$909$	0	0
$910$	−6.20992	−0.205857
$911$	13.7142	0.454370	0.227185	−	0.973852i	$-0.427048\pi$
$911$	13.7142	0.454370	0.227185	+	0.973852i	$0.427048\pi$
$912$	0	0
$913$	−24.3026	−0.804298
$914$	32.2320	1.06614
$915$	0	0
$916$	0.0543731	0.00179654
$917$	46.2472	1.52722
$918$	0	0
$919$	14.1731	0.467528	0.233764	−	0.972293i	$-0.424896\pi$
$919$	14.1731	0.467528	0.233764	+	0.972293i	$0.424896\pi$
$920$	0	0
$921$	0	0
$922$	−10.6868	−0.351950
$923$	12.8841	0.424084
$924$	0	0
$925$	−1.97746	−0.0650186
$926$	32.5830	1.07074
$927$	0	0
$928$	0.179995	0.00590863
$929$	−13.7665	−0.451665	−0.225833	−	0.974166i	$-0.572510\pi$
$929$	−13.7665	−0.451665	−0.225833	+	0.974166i	$0.572510\pi$
$930$	0	0
$931$	−85.1217	−2.78975
$932$	0.183838	0.00602181
$933$	0	0
$934$	15.8917	0.519991
$935$	7.18182	0.234871
$936$	0	0
$937$	2.34666	0.0766619	0.0383310	−	0.999265i	$-0.487796\pi$
$937$	2.34666	0.0766619	0.0383310	+	0.999265i	$0.487796\pi$
$938$	39.3738	1.28560
$939$	0	0
$940$	0.0386801	0.00126161
$941$	−51.4505	−1.67724	−0.838619	−	0.544719i	$-0.816636\pi$
$941$	−51.4505	−1.67724	−0.838619	+	0.544719i	$0.816636\pi$
$942$	0	0
$943$	0	0
$944$	8.51567	0.277161
$945$	0	0
$946$	−10.1686	−0.330608
$947$	−35.3492	−1.14869	−0.574347	−	0.818612i	$-0.694744\pi$
$947$	−35.3492	−1.14869	−0.574347	+	0.818612i	$0.694744\pi$
$948$	0	0
$949$	1.79102	0.0581389
$950$	−27.7112	−0.899071
$951$	0	0
$952$	−81.6802	−2.64727
$953$	−10.7136	−0.347048	−0.173524	−	0.984830i	$-0.555515\pi$
$953$	−10.7136	−0.347048	−0.173524	+	0.984830i	$0.555515\pi$
$954$	0	0
$955$	14.7684	0.477895
$956$	0.101760	0.00329117
$957$	0	0
$958$	29.3990	0.949838
$959$	28.6309	0.924542
$960$	0	0
$961$	−3.14058	−0.101309
$962$	−0.667332	−0.0215156
$963$	0	0
$964$	−0.0298535	−0.000961517 0
$965$	8.85161	0.284943
$966$	0	0
$967$	−44.9115	−1.44426	−0.722128	−	0.691759i	$-0.756836\pi$
$967$	−44.9115	−1.44426	−0.722128	+	0.691759i	$0.756836\pi$
$968$	−24.8622	−0.799101
$969$	0	0
$970$	13.5757	0.435888
$971$	−28.7318	−0.922048	−0.461024	−	0.887388i	$-0.652518\pi$
$971$	−28.7318	−0.922048	−0.461024	+	0.887388i	$0.652518\pi$
$972$	0	0
$973$	−80.2847	−2.57381
$974$	35.6275	1.14158
$975$	0	0
$976$	39.3013	1.25800
$977$	42.4434	1.35788	0.678942	−	0.734192i	$-0.262439\pi$
$977$	42.4434	1.35788	0.678942	+	0.734192i	$0.262439\pi$
$978$	0	0
$979$	15.0570	0.481223
$980$	−0.175008	−0.00559042
$981$	0	0
$982$	24.5555	0.783597
$983$	0.617034	0.0196803	0.00984016	−	0.999952i	$-0.496868\pi$
$983$	0.617034	0.0196803	0.00984016	+	0.999952i	$0.496868\pi$
$984$	0	0
$985$	−8.86041	−0.282316
$986$	−23.3455	−0.743471
$987$	0	0
$988$	−0.0514056	−0.00163543
$989$	0	0
$990$	0	0
$991$	−14.1083	−0.448165	−0.224082	−	0.974570i	$-0.571938\pi$
$991$	−14.1083	−0.448165	−0.224082	+	0.974570i	$0.571938\pi$
$992$	0.330066	0.0104796
$993$	0	0
$994$	90.8639	2.88203
$995$	3.90846	0.123907
$996$	0	0
$997$	54.7604	1.73428	0.867139	−	0.498066i	$-0.165956\pi$
$997$	54.7604	1.73428	0.867139	+	0.498066i	$0.165956\pi$
$998$	−54.9998	−1.74099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bu.1.3		10
3.2	odd	2		1587.2.a.t.1.8		10
23.5	odd	22		207.2.i.d.163.1		20
23.14	odd	22		207.2.i.d.127.1		20
23.22	odd	2		4761.2.a.bt.1.3		10
69.5	even	22		69.2.e.c.25.2	✓	20
69.14	even	22		69.2.e.c.58.2	yes	20
69.68	even	2		1587.2.a.u.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.25.2	✓	20	69.5	even	22
69.2.e.c.58.2	yes	20	69.14	even	22
207.2.i.d.127.1		20	23.14	odd	22
207.2.i.d.163.1		20	23.5	odd	22
1587.2.a.t.1.8		10	3.2	odd	2
1587.2.a.u.1.8		10	69.68	even	2
4761.2.a.bt.1.3		10	23.22	odd	2
4761.2.a.bu.1.3		10	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)

\(f(q)\)	\(=\)	\(q-1.41812 q^{2} +0.0110547 q^{4} -0.849430 q^{5} -5.06334 q^{7} +2.82056 q^{8} +O(q^{10})\)	\(q-1.41812 q^{2} +0.0110547 q^{4} -0.849430 q^{5} -5.06334 q^{7} +2.82056 q^{8} +1.20459 q^{10} -1.47830 q^{11} +1.01815 q^{13} +7.18040 q^{14} -4.02199 q^{16} +5.71934 q^{17} -4.56726 q^{19} -0.00939015 q^{20} +2.09640 q^{22} -4.27847 q^{25} -1.44385 q^{26} -0.0559734 q^{28} +2.87836 q^{29} +5.27820 q^{31} +0.0625339 q^{32} -8.11068 q^{34} +4.30095 q^{35} +0.462189 q^{37} +6.47690 q^{38} -2.39586 q^{40} +6.89693 q^{41} -4.85049 q^{43} -0.0163421 q^{44} -4.11922 q^{47} +18.6374 q^{49} +6.06737 q^{50} +0.0112552 q^{52} +2.82472 q^{53} +1.25571 q^{55} -14.2814 q^{56} -4.08185 q^{58} -2.11728 q^{59} -9.77162 q^{61} -7.48511 q^{62} +7.95529 q^{64} -0.864843 q^{65} +5.48351 q^{67} +0.0632253 q^{68} -6.09924 q^{70} +12.6544 q^{71} +1.75910 q^{73} -0.655438 q^{74} -0.0504895 q^{76} +7.48511 q^{77} +2.34094 q^{79} +3.41639 q^{80} -9.78065 q^{82} +16.4396 q^{83} -4.85817 q^{85} +6.87857 q^{86} -4.16962 q^{88} -10.1854 q^{89} -5.15521 q^{91} +5.84154 q^{94} +3.87956 q^{95} +11.2699 q^{97} -26.4300 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * q^98

Introduction

L-functions

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Database

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