LMFDB - Embedded newform 4761.2.a.bu.1.10

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			1.10
Root			$2.65972$ 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)

$f(q)$	$=$	$q+2.65972 q^{2} +5.07414 q^{4} -2.90602 q^{5} -2.29650 q^{7} +8.17636 q^{8} -7.72922 q^{10} +0.118967 q^{11} -4.31706 q^{13} -6.10806 q^{14} +11.5986 q^{16} +2.57465 q^{17} -1.72713 q^{19} -14.7456 q^{20} +0.316419 q^{22} +3.44497 q^{25} -11.4822 q^{26} -11.6528 q^{28} -1.42548 q^{29} -5.72727 q^{31} +14.4963 q^{32} +6.84785 q^{34} +6.67369 q^{35} -6.89385 q^{37} -4.59370 q^{38} -23.7607 q^{40} +6.51623 q^{41} -8.63357 q^{43} +0.603653 q^{44} -5.91880 q^{47} -1.72608 q^{49} +9.16268 q^{50} -21.9053 q^{52} +3.61800 q^{53} -0.345720 q^{55} -18.7770 q^{56} -3.79140 q^{58} +2.71366 q^{59} -12.5680 q^{61} -15.2329 q^{62} +15.3591 q^{64} +12.5455 q^{65} +0.0842144 q^{67} +13.0641 q^{68} +17.7502 q^{70} -15.4920 q^{71} -0.695461 q^{73} -18.3358 q^{74} -8.76371 q^{76} -0.273207 q^{77} +7.39276 q^{79} -33.7058 q^{80} +17.3314 q^{82} +2.78042 q^{83} -7.48198 q^{85} -22.9629 q^{86} +0.972714 q^{88} -13.2850 q^{89} +9.91413 q^{91} -15.7424 q^{94} +5.01909 q^{95} +8.52177 q^{97} -4.59090 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 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}+ \cdots - 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

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 $

$2$	2.65972	1.88071	0.940355	−	0.340195i	$-0.110493\pi$
$2$	2.65972	1.88071	0.940355	+	0.340195i	$0.110493\pi$
$3$	0	0
$3$	0	0
$4$	5.07414	2.53707
$5$	−2.90602	−1.29961	−0.649807	−	0.760100i	$-0.725150\pi$
$5$	−2.90602	−1.29961	−0.649807	+	0.760100i	$0.725150\pi$
$6$	0	0
$7$	−2.29650	−0.867996	−0.433998	−	0.900914i	$-0.642898\pi$
$7$	−2.29650	−0.867996	−0.433998	+	0.900914i	$0.642898\pi$
$8$	8.17636	2.89078
$9$	0	0
$10$	−7.72922	−2.44419
$11$	0.118967	0.0358698	0.0179349	−	0.999839i	$-0.494291\pi$
$11$	0.118967	0.0358698	0.0179349	+	0.999839i	$0.494291\pi$
$12$	0	0
$13$	−4.31706	−1.19734	−0.598668	−	0.800997i	$-0.704303\pi$
$13$	−4.31706	−1.19734	−0.598668	+	0.800997i	$0.704303\pi$
$14$	−6.10806	−1.63245
$15$	0	0
$16$	11.5986	2.89965
$17$	2.57465	0.624444	0.312222	−	0.950009i	$-0.398927\pi$
$17$	2.57465	0.624444	0.312222	+	0.950009i	$0.398927\pi$
$18$	0	0
$19$	−1.72713	−0.396231	−0.198116	−	0.980179i	$-0.563482\pi$
$19$	−1.72713	−0.396231	−0.198116	+	0.980179i	$0.563482\pi$
$20$	−14.7456	−3.29721
$21$	0	0
$22$	0.316419	0.0674607
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.44497	0.688994
$26$	−11.4822	−2.25184
$27$	0	0
$28$	−11.6528	−2.20217
$29$	−1.42548	−0.264706	−0.132353	−	0.991203i	$-0.542253\pi$
$29$	−1.42548	−0.264706	−0.132353	+	0.991203i	$0.542253\pi$
$30$	0	0
$31$	−5.72727	−1.02865	−0.514324	−	0.857596i	$-0.671957\pi$
$31$	−5.72727	−1.02865	−0.514324	+	0.857596i	$0.671957\pi$
$32$	14.4963	2.56261
$33$	0	0
$34$	6.84785	1.17440
$35$	6.67369	1.12806
$36$	0	0
$37$	−6.89385	−1.13334	−0.566671	−	0.823944i	$-0.691769\pi$
$37$	−6.89385	−1.13334	−0.566671	+	0.823944i	$0.691769\pi$
$38$	−4.59370	−0.745196
$39$	0	0
$40$	−23.7607	−3.75689
$41$	6.51623	1.01766	0.508832	−	0.860866i	$-0.330078\pi$
$41$	6.51623	1.01766	0.508832	+	0.860866i	$0.330078\pi$
$42$	0	0
$43$	−8.63357	−1.31661	−0.658303	−	0.752753i	$-0.728726\pi$
$43$	−8.63357	−1.31661	−0.658303	+	0.752753i	$0.728726\pi$
$44$	0.603653	0.0910042
$45$	0	0
$46$	0	0
$47$	−5.91880	−0.863345	−0.431673	−	0.902030i	$-0.642076\pi$
$47$	−5.91880	−0.863345	−0.431673	+	0.902030i	$0.642076\pi$
$48$	0	0
$49$	−1.72608	−0.246583
$50$	9.16268	1.29580
$51$	0	0
$52$	−21.9053	−3.03772
$53$	3.61800	0.496970	0.248485	−	0.968636i	$-0.420067\pi$
$53$	3.61800	0.496970	0.248485	+	0.968636i	$0.420067\pi$
$54$	0	0
$55$	−0.345720	−0.0466169
$56$	−18.7770	−2.50918
$57$	0	0
$58$	−3.79140	−0.497835
$59$	2.71366	0.353288	0.176644	−	0.984275i	$-0.443476\pi$
$59$	2.71366	0.353288	0.176644	+	0.984275i	$0.443476\pi$
$60$	0	0
$61$	−12.5680	−1.60917	−0.804583	−	0.593841i	$-0.797611\pi$
$61$	−12.5680	−1.60917	−0.804583	+	0.593841i	$0.797611\pi$
$62$	−15.2329	−1.93459
$63$	0	0
$64$	15.3591	1.91989
$65$	12.5455	1.55607
$66$	0	0
$67$	0.0842144	0.0102884	0.00514421	−	0.999987i	$-0.498363\pi$
$67$	0.0842144	0.0102884	0.00514421	+	0.999987i	$0.498363\pi$
$68$	13.0641	1.58426
$69$	0	0
$70$	17.7502	2.12155
$71$	−15.4920	−1.83857	−0.919284	−	0.393595i	$-0.871231\pi$
$71$	−15.4920	−1.83857	−0.919284	+	0.393595i	$0.871231\pi$
$72$	0	0
$73$	−0.695461	−0.0813976	−0.0406988	−	0.999171i	$-0.512958\pi$
$73$	−0.695461	−0.0813976	−0.0406988	+	0.999171i	$0.512958\pi$
$74$	−18.3358	−2.13149
$75$	0	0
$76$	−8.76371	−1.00527
$77$	−0.273207	−0.0311349
$78$	0	0
$79$	7.39276	0.831751	0.415875	−	0.909422i	$-0.363475\pi$
$79$	7.39276	0.831751	0.415875	+	0.909422i	$0.363475\pi$
$80$	−33.7058	−3.76842
$81$	0	0
$82$	17.3314	1.91393
$83$	2.78042	0.305190	0.152595	−	0.988289i	$-0.451237\pi$
$83$	2.78042	0.305190	0.152595	+	0.988289i	$0.451237\pi$
$84$	0	0
$85$	−7.48198	−0.811535
$86$	−22.9629	−2.47615
$87$	0	0
$88$	0.972714	0.103692
$89$	−13.2850	−1.40820	−0.704101	−	0.710099i	$-0.748650\pi$
$89$	−13.2850	−1.40820	−0.704101	+	0.710099i	$0.748650\pi$
$90$	0	0
$91$	9.91413	1.03928
$92$	0	0
$93$	0	0
$94$	−15.7424	−1.62370
$95$	5.01909	0.514947
$96$	0	0
$97$	8.52177	0.865255	0.432627	−	0.901573i	$-0.357587\pi$
$97$	8.52177	0.865255	0.432627	+	0.901573i	$0.357587\pi$
$98$	−4.59090	−0.463751
$99$	0	0
$100$	17.4803	1.74803
$101$	−6.99251	−0.695781	−0.347891	−	0.937535i	$-0.613102\pi$
$101$	−6.99251	−0.695781	−0.347891	+	0.937535i	$0.613102\pi$
$102$	0	0
$103$	−13.8162	−1.36135	−0.680673	−	0.732587i	$-0.738312\pi$
$103$	−13.8162	−1.36135	−0.680673	+	0.732587i	$0.738312\pi$
$104$	−35.2978	−3.46123
$105$	0	0
$106$	9.62288	0.934657
$107$	8.22279	0.794927	0.397464	−	0.917618i	$-0.369890\pi$
$107$	8.22279	0.794927	0.397464	+	0.917618i	$0.369890\pi$
$108$	0	0
$109$	17.2715	1.65431	0.827154	−	0.561975i	$-0.189958\pi$
$109$	17.2715	1.65431	0.827154	+	0.561975i	$0.189958\pi$
$110$	−0.919520	−0.0876728
$111$	0	0
$112$	−26.6362	−2.51688
$113$	7.49774	0.705328	0.352664	−	0.935750i	$-0.385276\pi$
$113$	7.49774	0.705328	0.352664	+	0.935750i	$0.385276\pi$
$114$	0	0
$115$	0	0
$116$	−7.23310	−0.671577
$117$	0	0
$118$	7.21758	0.664432
$119$	−5.91268	−0.542015
$120$	0	0
$121$	−10.9858	−0.998713
$122$	−33.4274	−3.02637
$123$	0	0
$124$	−29.0609	−2.60975
$125$	4.51895	0.404187
$126$	0	0
$127$	−2.00858	−0.178233	−0.0891164	−	0.996021i	$-0.528404\pi$
$127$	−2.00858	−0.178233	−0.0891164	+	0.996021i	$0.528404\pi$
$128$	11.8583	1.04813
$129$	0	0
$130$	33.3675	2.92652
$131$	4.91198	0.429161	0.214581	−	0.976706i	$-0.431162\pi$
$131$	4.91198	0.429161	0.214581	+	0.976706i	$0.431162\pi$
$132$	0	0
$133$	3.96636	0.343927
$134$	0.223987	0.0193495
$135$	0	0
$136$	21.0512	1.80513
$137$	7.93152	0.677636	0.338818	−	0.940852i	$-0.389973\pi$
$137$	7.93152	0.677636	0.338818	+	0.940852i	$0.389973\pi$
$138$	0	0
$139$	10.7987	0.915936	0.457968	−	0.888969i	$-0.348577\pi$
$139$	10.7987	0.915936	0.457968	+	0.888969i	$0.348577\pi$
$140$	33.8632	2.86196
$141$	0	0
$142$	−41.2046	−3.45781
$143$	−0.513586	−0.0429482
$144$	0	0
$145$	4.14249	0.344015
$146$	−1.84974	−0.153085
$147$	0	0
$148$	−34.9804	−2.87537
$149$	16.5524	1.35603	0.678014	−	0.735049i	$-0.262841\pi$
$149$	16.5524	1.35603	0.678014	+	0.735049i	$0.262841\pi$
$150$	0	0
$151$	18.9005	1.53810	0.769051	−	0.639187i	$-0.220729\pi$
$151$	18.9005	1.53810	0.769051	+	0.639187i	$0.220729\pi$
$152$	−14.1217	−1.14542
$153$	0	0
$154$	−0.726656	−0.0585556
$155$	16.6436	1.33684
$156$	0	0
$157$	−7.56346	−0.603630	−0.301815	−	0.953367i	$-0.597592\pi$
$157$	−7.56346	−0.603630	−0.301815	+	0.953367i	$0.597592\pi$
$158$	19.6627	1.56428
$159$	0	0
$160$	−42.1267	−3.33041
$161$	0	0
$162$	0	0
$163$	1.66925	0.130745	0.0653727	−	0.997861i	$-0.479176\pi$
$163$	1.66925	0.130745	0.0653727	+	0.997861i	$0.479176\pi$
$164$	33.0642	2.58188
$165$	0	0
$166$	7.39514	0.573974
$167$	−21.4339	−1.65860	−0.829302	−	0.558800i	$-0.811262\pi$
$167$	−21.4339	−1.65860	−0.829302	+	0.558800i	$0.811262\pi$
$168$	0	0
$169$	5.63697	0.433613
$170$	−19.9000	−1.52626
$171$	0	0
$172$	−43.8079	−3.34032
$173$	6.33043	0.481294	0.240647	−	0.970613i	$-0.422640\pi$
$173$	6.33043	0.481294	0.240647	+	0.970613i	$0.422640\pi$
$174$	0	0
$175$	−7.91138	−0.598044
$176$	1.37985	0.104010
$177$	0	0
$178$	−35.3343	−2.64842
$179$	−8.19257	−0.612341	−0.306171	−	0.951977i	$-0.599048\pi$
$179$	−8.19257	−0.612341	−0.306171	+	0.951977i	$0.599048\pi$
$180$	0	0
$181$	−6.31237	−0.469195	−0.234597	−	0.972093i	$-0.575377\pi$
$181$	−6.31237	−0.469195	−0.234597	+	0.972093i	$0.575377\pi$
$182$	26.3688	1.95459
$183$	0	0
$184$	0	0
$185$	20.0337	1.47291
$186$	0	0
$187$	0.306297	0.0223987
$188$	−30.0328	−2.19037
$189$	0	0
$190$	13.3494	0.968467
$191$	12.9462	0.936756	0.468378	−	0.883528i	$-0.344839\pi$
$191$	12.9462	0.936756	0.468378	+	0.883528i	$0.344839\pi$
$192$	0	0
$193$	12.6113	0.907781	0.453891	−	0.891057i	$-0.350036\pi$
$193$	12.6113	0.907781	0.453891	+	0.891057i	$0.350036\pi$
$194$	22.6656	1.62729
$195$	0	0
$196$	−8.75836	−0.625597
$197$	14.5326	1.03540	0.517701	−	0.855561i	$-0.326788\pi$
$197$	14.5326	1.03540	0.517701	+	0.855561i	$0.326788\pi$
$198$	0	0
$199$	−20.1733	−1.43005	−0.715023	−	0.699101i	$-0.753584\pi$
$199$	−20.1733	−1.43005	−0.715023	+	0.699101i	$0.753584\pi$
$200$	28.1673	1.99173
$201$	0	0
$202$	−18.5982	−1.30856
$203$	3.27363	0.229764
$204$	0	0
$205$	−18.9363	−1.32257
$206$	−36.7472	−2.56030
$207$	0	0
$208$	−50.0717	−3.47185
$209$	−0.205471	−0.0142127
$210$	0	0
$211$	7.45381	0.513141	0.256571	−	0.966525i	$-0.417407\pi$
$211$	7.45381	0.513141	0.256571	+	0.966525i	$0.417407\pi$
$212$	18.3582	1.26085
$213$	0	0
$214$	21.8704	1.49503
$215$	25.0893	1.71108
$216$	0	0
$217$	13.1527	0.892862
$218$	45.9374	3.11127
$219$	0	0
$220$	−1.75423	−0.118270
$221$	−11.1149	−0.747669
$222$	0	0
$223$	1.68197	0.112633	0.0563164	−	0.998413i	$-0.482064\pi$
$223$	1.68197	0.112633	0.0563164	+	0.998413i	$0.482064\pi$
$224$	−33.2909	−2.22434
$225$	0	0
$226$	19.9419	1.32652
$227$	10.7814	0.715587	0.357794	−	0.933801i	$-0.383529\pi$
$227$	10.7814	0.715587	0.357794	+	0.933801i	$0.383529\pi$
$228$	0	0
$229$	−21.9077	−1.44770	−0.723852	−	0.689955i	$-0.757630\pi$
$229$	−21.9077	−1.44770	−0.723852	+	0.689955i	$0.757630\pi$
$230$	0	0
$231$	0	0
$232$	−11.6553	−0.765206
$233$	−24.8712	−1.62937	−0.814683	−	0.579907i	$-0.803089\pi$
$233$	−24.8712	−1.62937	−0.814683	+	0.579907i	$0.803089\pi$
$234$	0	0
$235$	17.2002	1.12201
$236$	13.7695	0.896316
$237$	0	0
$238$	−15.7261	−1.01937
$239$	−4.19683	−0.271470	−0.135735	−	0.990745i	$-0.543340\pi$
$239$	−4.19683	−0.271470	−0.135735	+	0.990745i	$0.543340\pi$
$240$	0	0
$241$	27.4295	1.76689	0.883444	−	0.468538i	$-0.155219\pi$
$241$	27.4295	1.76689	0.883444	+	0.468538i	$0.155219\pi$
$242$	−29.2193	−1.87829
$243$	0	0
$244$	−63.7717	−4.08256
$245$	5.01603	0.320462
$246$	0	0
$247$	7.45613	0.474422
$248$	−46.8282	−2.97359
$249$	0	0
$250$	12.0192	0.760159
$251$	−12.9237	−0.815737	−0.407868	−	0.913041i	$-0.633728\pi$
$251$	−12.9237	−0.815737	−0.407868	+	0.913041i	$0.633728\pi$
$252$	0	0
$253$	0	0
$254$	−5.34227	−0.335204
$255$	0	0
$256$	0.821585	0.0513491
$257$	1.43556	0.0895479	0.0447739	−	0.998997i	$-0.485743\pi$
$257$	1.43556	0.0895479	0.0447739	+	0.998997i	$0.485743\pi$
$258$	0	0
$259$	15.8317	0.983737
$260$	63.6574	3.94786
$261$	0	0
$262$	13.0645	0.807128
$263$	14.3651	0.885790	0.442895	−	0.896574i	$-0.353951\pi$
$263$	14.3651	0.885790	0.442895	+	0.896574i	$0.353951\pi$
$264$	0	0
$265$	−10.5140	−0.645869
$266$	10.5494	0.646827
$267$	0	0
$268$	0.427315	0.0261024
$269$	19.3890	1.18217	0.591083	−	0.806611i	$-0.298701\pi$
$269$	19.3890	1.18217	0.591083	+	0.806611i	$0.298701\pi$
$270$	0	0
$271$	16.3398	0.992571	0.496285	−	0.868159i	$-0.334697\pi$
$271$	16.3398	0.992571	0.496285	+	0.868159i	$0.334697\pi$
$272$	29.8623	1.81067
$273$	0	0
$274$	21.0957	1.27444
$275$	0.409837	0.0247141
$276$	0	0
$277$	20.1827	1.21266	0.606332	−	0.795212i	$-0.292640\pi$
$277$	20.1827	1.21266	0.606332	+	0.795212i	$0.292640\pi$
$278$	28.7216	1.72261
$279$	0	0
$280$	54.5664	3.26097
$281$	8.51231	0.507802	0.253901	−	0.967230i	$-0.418286\pi$
$281$	8.51231	0.507802	0.253901	+	0.967230i	$0.418286\pi$
$282$	0	0
$283$	3.37844	0.200828	0.100414	−	0.994946i	$-0.467983\pi$
$283$	3.37844	0.200828	0.100414	+	0.994946i	$0.467983\pi$
$284$	−78.6088	−4.66457
$285$	0	0
$286$	−1.36600	−0.0807731
$287$	−14.9645	−0.883328
$288$	0	0
$289$	−10.3712	−0.610070
$290$	11.0179	0.646992
$291$	0	0
$292$	−3.52887	−0.206511
$293$	−4.12667	−0.241083	−0.120541	−	0.992708i	$-0.538463\pi$
$293$	−4.12667	−0.241083	−0.120541	+	0.992708i	$0.538463\pi$
$294$	0	0
$295$	−7.88595	−0.459138
$296$	−56.3666	−3.27624
$297$	0	0
$298$	44.0249	2.55030
$299$	0	0
$300$	0	0
$301$	19.8270	1.14281
$302$	50.2702	2.89272
$303$	0	0
$304$	−20.0323	−1.14893
$305$	36.5229	2.09129
$306$	0	0
$307$	10.4058	0.593893	0.296946	−	0.954894i	$-0.404032\pi$
$307$	10.4058	0.593893	0.296946	+	0.954894i	$0.404032\pi$
$308$	−1.38629	−0.0789912
$309$	0	0
$310$	44.2673	2.51421
$311$	−13.1922	−0.748061	−0.374030	−	0.927416i	$-0.622024\pi$
$311$	−13.1922	−0.748061	−0.374030	+	0.927416i	$0.622024\pi$
$312$	0	0
$313$	13.7604	0.777783	0.388892	−	0.921283i	$-0.372858\pi$
$313$	13.7604	0.777783	0.388892	+	0.921283i	$0.372858\pi$
$314$	−20.1167	−1.13525
$315$	0	0
$316$	37.5119	2.11021
$317$	14.3418	0.805518	0.402759	−	0.915306i	$-0.368051\pi$
$317$	14.3418	0.805518	0.402759	+	0.915306i	$0.368051\pi$
$318$	0	0
$319$	−0.169585	−0.00949495
$320$	−44.6339	−2.49511
$321$	0	0
$322$	0	0
$323$	−4.44676	−0.247424
$324$	0	0
$325$	−14.8721	−0.824957
$326$	4.43974	0.245894
$327$	0	0
$328$	53.2790	2.94184
$329$	13.5925	0.749380
$330$	0	0
$331$	−11.8344	−0.650480	−0.325240	−	0.945632i	$-0.605445\pi$
$331$	−11.8344	−0.650480	−0.325240	+	0.945632i	$0.605445\pi$
$332$	14.1082	0.774288
$333$	0	0
$334$	−57.0083	−3.11935
$335$	−0.244729	−0.0133710
$336$	0	0
$337$	−26.7632	−1.45788	−0.728941	−	0.684576i	$-0.759987\pi$
$337$	−26.7632	−1.45788	−0.728941	+	0.684576i	$0.759987\pi$
$338$	14.9928	0.815500
$339$	0	0
$340$	−37.9646	−2.05892
$341$	−0.681354	−0.0368974
$342$	0	0
$343$	20.0395	1.08203
$344$	−70.5911	−3.80602
$345$	0	0
$346$	16.8372	0.905174
$347$	8.84567	0.474861	0.237430	−	0.971405i	$-0.423695\pi$
$347$	8.84567	0.474861	0.237430	+	0.971405i	$0.423695\pi$
$348$	0	0
$349$	−29.7789	−1.59403	−0.797015	−	0.603960i	$-0.793589\pi$
$349$	−29.7789	−1.59403	−0.797015	+	0.603960i	$0.793589\pi$
$350$	−21.0421	−1.12475
$351$	0	0
$352$	1.72458	0.0919205
$353$	17.2117	0.916086	0.458043	−	0.888930i	$-0.348551\pi$
$353$	17.2117	0.916086	0.458043	+	0.888930i	$0.348551\pi$
$354$	0	0
$355$	45.0203	2.38943
$356$	−67.4097	−3.57271
$357$	0	0
$358$	−21.7900	−1.15164
$359$	−7.12729	−0.376164	−0.188082	−	0.982153i	$-0.560227\pi$
$359$	−7.12729	−0.376164	−0.188082	+	0.982153i	$0.560227\pi$
$360$	0	0
$361$	−16.0170	−0.843001
$362$	−16.7892	−0.882419
$363$	0	0
$364$	50.3056	2.63673
$365$	2.02103	0.105785
$366$	0	0
$367$	−13.5836	−0.709060	−0.354530	−	0.935045i	$-0.615359\pi$
$367$	−13.5836	−0.709060	−0.354530	+	0.935045i	$0.615359\pi$
$368$	0	0
$369$	0	0
$370$	53.2841	2.77011
$371$	−8.30874	−0.431368
$372$	0	0
$373$	−18.7273	−0.969664	−0.484832	−	0.874607i	$-0.661119\pi$
$373$	−18.7273	−0.969664	−0.484832	+	0.874607i	$0.661119\pi$
$374$	0.814666	0.0421254
$375$	0	0
$376$	−48.3942	−2.49574
$377$	6.15389	0.316942
$378$	0	0
$379$	10.0172	0.514549	0.257274	−	0.966338i	$-0.417176\pi$
$379$	10.0172	0.514549	0.257274	+	0.966338i	$0.417176\pi$
$380$	25.4675	1.30646
$381$	0	0
$382$	34.4334	1.76177
$383$	−29.1229	−1.48811	−0.744056	−	0.668117i	$-0.767101\pi$
$383$	−29.1229	−1.48811	−0.744056	+	0.668117i	$0.767101\pi$
$384$	0	0
$385$	0.793947	0.0404633
$386$	33.5426	1.70727
$387$	0	0
$388$	43.2406	2.19521
$389$	4.38322	0.222238	0.111119	−	0.993807i	$-0.464556\pi$
$389$	4.38322	0.222238	0.111119	+	0.993807i	$0.464556\pi$
$390$	0	0
$391$	0	0
$392$	−14.1130	−0.712816
$393$	0	0
$394$	38.6526	1.94729
$395$	−21.4835	−1.08095
$396$	0	0
$397$	0.145723	0.00731363	0.00365682	−	0.999993i	$-0.498836\pi$
$397$	0.145723	0.00731363	0.00365682	+	0.999993i	$0.498836\pi$
$398$	−53.6554	−2.68950
$399$	0	0
$400$	39.9568	1.99784
$401$	−33.6620	−1.68100	−0.840500	−	0.541812i	$-0.817739\pi$
$401$	−33.6620	−1.68100	−0.840500	+	0.541812i	$0.817739\pi$
$402$	0	0
$403$	24.7249	1.23164
$404$	−35.4810	−1.76524
$405$	0	0
$406$	8.70695	0.432119
$407$	−0.820139	−0.0406528
$408$	0	0
$409$	−30.2682	−1.49667	−0.748334	−	0.663322i	$-0.769146\pi$
$409$	−30.2682	−1.49667	−0.748334	+	0.663322i	$0.769146\pi$
$410$	−50.3654	−2.48737
$411$	0	0
$412$	−70.1050	−3.45383
$413$	−6.23192	−0.306653
$414$	0	0
$415$	−8.07995	−0.396629
$416$	−62.5815	−3.06831
$417$	0	0
$418$	−0.546497	−0.0267300
$419$	18.7368	0.915353	0.457676	−	0.889119i	$-0.348682\pi$
$419$	18.7368	0.915353	0.457676	+	0.889119i	$0.348682\pi$
$420$	0	0
$421$	−39.1961	−1.91030	−0.955151	−	0.296120i	$-0.904307\pi$
$421$	−39.1961	−1.91030	−0.955151	+	0.296120i	$0.904307\pi$
$422$	19.8251	0.965070
$423$	0	0
$424$	29.5820	1.43663
$425$	8.86958	0.430238
$426$	0	0
$427$	28.8624	1.39675
$428$	41.7236	2.01678
$429$	0	0
$430$	66.7308	3.21804
$431$	21.4693	1.03414	0.517069	−	0.855944i	$-0.327023\pi$
$431$	21.4693	1.03414	0.517069	+	0.855944i	$0.327023\pi$
$432$	0	0
$433$	0.758349	0.0364439	0.0182220	−	0.999834i	$-0.494199\pi$
$433$	0.758349	0.0364439	0.0182220	+	0.999834i	$0.494199\pi$
$434$	34.9825	1.67921
$435$	0	0
$436$	87.6378	4.19709
$437$	0	0
$438$	0	0
$439$	13.1094	0.625677	0.312838	−	0.949806i	$-0.398720\pi$
$439$	13.1094	0.625677	0.312838	+	0.949806i	$0.398720\pi$
$440$	−2.82673	−0.134759
$441$	0	0
$442$	−29.5626	−1.40615
$443$	3.05970	0.145371	0.0726854	−	0.997355i	$-0.476843\pi$
$443$	3.05970	0.145371	0.0726854	+	0.997355i	$0.476843\pi$
$444$	0	0
$445$	38.6064	1.83012
$446$	4.47357	0.211829
$447$	0	0
$448$	−35.2722	−1.66645
$449$	7.66937	0.361940	0.180970	−	0.983489i	$-0.442076\pi$
$449$	7.66937	0.361940	0.180970	+	0.983489i	$0.442076\pi$
$450$	0	0
$451$	0.775214	0.0365034
$452$	38.0446	1.78947
$453$	0	0
$454$	28.6756	1.34581
$455$	−28.8107	−1.35067
$456$	0	0
$457$	−8.96378	−0.419308	−0.209654	−	0.977776i	$-0.567234\pi$
$457$	−8.96378	−0.419308	−0.209654	+	0.977776i	$0.567234\pi$
$458$	−58.2686	−2.72271
$459$	0	0
$460$	0	0
$461$	7.53704	0.351035	0.175517	−	0.984476i	$-0.443840\pi$
$461$	7.53704	0.351035	0.175517	+	0.984476i	$0.443840\pi$
$462$	0	0
$463$	−32.9644	−1.53199	−0.765993	−	0.642849i	$-0.777752\pi$
$463$	−32.9644	−1.53199	−0.765993	+	0.642849i	$0.777752\pi$
$464$	−16.5336	−0.767553
$465$	0	0
$466$	−66.1505	−3.06436
$467$	−13.4842	−0.623976	−0.311988	−	0.950086i	$-0.600995\pi$
$467$	−13.4842	−0.623976	−0.311988	+	0.950086i	$0.600995\pi$
$468$	0	0
$469$	−0.193398	−0.00893031
$470$	45.7477	2.11018
$471$	0	0
$472$	22.1878	1.02128
$473$	−1.02711	−0.0472264
$474$	0	0
$475$	−5.94992	−0.273001
$476$	−30.0017	−1.37513
$477$	0	0
$478$	−11.1624	−0.510557
$479$	−20.8722	−0.953675	−0.476838	−	0.878991i	$-0.658217\pi$
$479$	−20.8722	−0.953675	−0.476838	+	0.878991i	$0.658217\pi$
$480$	0	0
$481$	29.7612	1.35699
$482$	72.9548	3.32300
$483$	0	0
$484$	−55.7437	−2.53380
$485$	−24.7645	−1.12450
$486$	0	0
$487$	−8.26550	−0.374545	−0.187273	−	0.982308i	$-0.559965\pi$
$487$	−8.26550	−0.374545	−0.187273	+	0.982308i	$0.559965\pi$
$488$	−102.760	−4.65174
$489$	0	0
$490$	13.3413	0.602697
$491$	11.1071	0.501255	0.250628	−	0.968084i	$-0.419363\pi$
$491$	11.1071	0.501255	0.250628	+	0.968084i	$0.419363\pi$
$492$	0	0
$493$	−3.67012	−0.165294
$494$	19.8312	0.892250
$495$	0	0
$496$	−66.4282	−2.98271
$497$	35.5775	1.59587
$498$	0	0
$499$	−6.08342	−0.272331	−0.136166	−	0.990686i	$-0.543478\pi$
$499$	−6.08342	−0.272331	−0.136166	+	0.990686i	$0.543478\pi$
$500$	22.9298	1.02545
$501$	0	0
$502$	−34.3735	−1.53416
$503$	−7.86791	−0.350813	−0.175406	−	0.984496i	$-0.556124\pi$
$503$	−7.86791	−0.350813	−0.175406	+	0.984496i	$0.556124\pi$
$504$	0	0
$505$	20.3204	0.904246
$506$	0	0
$507$	0	0
$508$	−10.1918	−0.452189
$509$	−29.7373	−1.31808	−0.659041	−	0.752107i	$-0.729038\pi$
$509$	−29.7373	−1.31808	−0.659041	+	0.752107i	$0.729038\pi$
$510$	0	0
$511$	1.59713	0.0706528
$512$	−21.5314	−0.951561
$513$	0	0
$514$	3.81820	0.168414
$515$	40.1501	1.76922
$516$	0	0
$517$	−0.704140	−0.0309680
$518$	42.1081	1.85012
$519$	0	0
$520$	102.576	4.49826
$521$	−18.9754	−0.831325	−0.415663	−	0.909519i	$-0.636450\pi$
$521$	−18.9754	−0.831325	−0.415663	+	0.909519i	$0.636450\pi$
$522$	0	0
$523$	26.4941	1.15851	0.579254	−	0.815147i	$-0.303344\pi$
$523$	26.4941	1.15851	0.579254	+	0.815147i	$0.303344\pi$
$524$	24.9240	1.08881
$525$	0	0
$526$	38.2072	1.66591
$527$	−14.7457	−0.642332
$528$	0	0
$529$	0	0
$530$	−27.9643	−1.21469
$531$	0	0
$532$	20.1259	0.872567
$533$	−28.1309	−1.21848
$534$	0	0
$535$	−23.8956	−1.03310
$536$	0.688567	0.0297416
$537$	0	0
$538$	51.5693	2.22331
$539$	−0.205346	−0.00884488
$540$	0	0
$541$	32.6681	1.40451	0.702255	−	0.711926i	$-0.252177\pi$
$541$	32.6681	1.40451	0.702255	+	0.711926i	$0.252177\pi$
$542$	43.4593	1.86674
$543$	0	0
$544$	37.3229	1.60021
$545$	−50.1913	−2.14996
$546$	0	0
$547$	45.4508	1.94334	0.971668	−	0.236349i	$-0.0759509\pi$
$547$	45.4508	1.94334	0.971668	+	0.236349i	$0.0759509\pi$
$548$	40.2456	1.71921
$549$	0	0
$550$	1.09005	0.0464800
$551$	2.46200	0.104885
$552$	0	0
$553$	−16.9775	−0.721957
$554$	53.6805	2.28067
$555$	0	0
$556$	54.7942	2.32379
$557$	15.0491	0.637650	0.318825	−	0.947814i	$-0.396712\pi$
$557$	15.0491	0.637650	0.318825	+	0.947814i	$0.396712\pi$
$558$	0	0
$559$	37.2716	1.57642
$560$	77.4053	3.27097
$561$	0	0
$562$	22.6404	0.955027
$563$	−34.1730	−1.44022	−0.720111	−	0.693859i	$-0.755909\pi$
$563$	−34.1730	−1.44022	−0.720111	+	0.693859i	$0.755909\pi$
$564$	0	0
$565$	−21.7886	−0.916654
$566$	8.98573	0.377699
$567$	0	0
$568$	−126.669	−5.31489
$569$	−18.4407	−0.773076	−0.386538	−	0.922274i	$-0.626329\pi$
$569$	−18.4407	−0.773076	−0.386538	+	0.922274i	$0.626329\pi$
$570$	0	0
$571$	−6.63353	−0.277605	−0.138802	−	0.990320i	$-0.544325\pi$
$571$	−6.63353	−0.277605	−0.138802	+	0.990320i	$0.544325\pi$
$572$	−2.60600	−0.108963
$573$	0	0
$574$	−39.8015	−1.66128
$575$	0	0
$576$	0	0
$577$	−12.5728	−0.523412	−0.261706	−	0.965148i	$-0.584285\pi$
$577$	−12.5728	−0.523412	−0.261706	+	0.965148i	$0.584285\pi$
$578$	−27.5845	−1.14736
$579$	0	0
$580$	21.0196	0.872790
$581$	−6.38523	−0.264904
$582$	0	0
$583$	0.430421	0.0178262
$584$	−5.68634	−0.235302
$585$	0	0
$586$	−10.9758	−0.453406
$587$	−1.09521	−0.0452041	−0.0226021	−	0.999745i	$-0.507195\pi$
$587$	−1.09521	−0.0452041	−0.0226021	+	0.999745i	$0.507195\pi$
$588$	0	0
$589$	9.89175	0.407582
$590$	−20.9745	−0.863505
$591$	0	0
$592$	−79.9590	−3.28629
$593$	−2.32202	−0.0953538	−0.0476769	−	0.998863i	$-0.515182\pi$
$593$	−2.32202	−0.0953538	−0.0476769	+	0.998863i	$0.515182\pi$
$594$	0	0
$595$	17.1824	0.704409
$596$	83.9893	3.44034
$597$	0	0
$598$	0	0
$599$	43.0508	1.75901	0.879504	−	0.475892i	$-0.157875\pi$
$599$	43.0508	1.75901	0.879504	+	0.475892i	$0.157875\pi$
$600$	0	0
$601$	13.2724	0.541391	0.270695	−	0.962665i	$-0.412746\pi$
$601$	13.2724	0.541391	0.270695	+	0.962665i	$0.412746\pi$
$602$	52.7344	2.14929
$603$	0	0
$604$	95.9038	3.90227
$605$	31.9251	1.29794
$606$	0	0
$607$	−32.9113	−1.33583	−0.667914	−	0.744238i	$-0.732813\pi$
$607$	−32.9113	−1.33583	−0.667914	+	0.744238i	$0.732813\pi$
$608$	−25.0371	−1.01539
$609$	0	0
$610$	97.1407	3.93311
$611$	25.5518	1.03371
$612$	0	0
$613$	40.4100	1.63214	0.816072	−	0.577951i	$-0.196147\pi$
$613$	40.4100	1.63214	0.816072	+	0.577951i	$0.196147\pi$
$614$	27.6767	1.11694
$615$	0	0
$616$	−2.23384	−0.0900040
$617$	30.0009	1.20779	0.603896	−	0.797063i	$-0.293614\pi$
$617$	30.0009	1.20779	0.603896	+	0.797063i	$0.293614\pi$
$618$	0	0
$619$	−26.4907	−1.06475	−0.532375	−	0.846509i	$-0.678700\pi$
$619$	−26.4907	−1.06475	−0.532375	+	0.846509i	$0.678700\pi$
$620$	84.4517	3.39166
$621$	0	0
$622$	−35.0876	−1.40688
$623$	30.5089	1.22231
$624$	0	0
$625$	−30.3570	−1.21428
$626$	36.5989	1.46278
$627$	0	0
$628$	−38.3780	−1.53145
$629$	−17.7492	−0.707709
$630$	0	0
$631$	−16.0502	−0.638947	−0.319474	−	0.947595i	$-0.603506\pi$
$631$	−16.0502	−0.638947	−0.319474	+	0.947595i	$0.603506\pi$
$632$	60.4459	2.40441
$633$	0	0
$634$	38.1453	1.51495
$635$	5.83699	0.231634
$636$	0	0
$637$	7.45158	0.295242
$638$	−0.451050	−0.0178572
$639$	0	0
$640$	−34.4604	−1.36217
$641$	−46.8565	−1.85072	−0.925360	−	0.379090i	$-0.876237\pi$
$641$	−46.8565	−1.85072	−0.925360	+	0.379090i	$0.876237\pi$
$642$	0	0
$643$	27.2328	1.07396	0.536979	−	0.843596i	$-0.319565\pi$
$643$	27.2328	1.07396	0.536979	+	0.843596i	$0.319565\pi$
$644$	0	0
$645$	0	0
$646$	−11.8271	−0.465333
$647$	−24.8045	−0.975165	−0.487583	−	0.873077i	$-0.662121\pi$
$647$	−24.8045	−0.975165	−0.487583	+	0.873077i	$0.662121\pi$
$648$	0	0
$649$	0.322835	0.0126724
$650$	−39.5558	−1.55151
$651$	0	0
$652$	8.46998	0.331710
$653$	29.4481	1.15239	0.576197	−	0.817311i	$-0.304536\pi$
$653$	29.4481	1.15239	0.576197	+	0.817311i	$0.304536\pi$
$654$	0	0
$655$	−14.2743	−0.557744
$656$	75.5790	2.95086
$657$	0	0
$658$	36.1524	1.40937
$659$	13.9794	0.544559	0.272279	−	0.962218i	$-0.412223\pi$
$659$	13.9794	0.544559	0.272279	+	0.962218i	$0.412223\pi$
$660$	0	0
$661$	24.8201	0.965390	0.482695	−	0.875789i	$-0.339658\pi$
$661$	24.8201	0.965390	0.482695	+	0.875789i	$0.339658\pi$
$662$	−31.4764	−1.22336
$663$	0	0
$664$	22.7337	0.882237
$665$	−11.5263	−0.446972
$666$	0	0
$667$	0	0
$668$	−108.758	−4.20799
$669$	0	0
$670$	−0.650912	−0.0251469
$671$	−1.49517	−0.0577205
$672$	0	0
$673$	8.90233	0.343160	0.171580	−	0.985170i	$-0.445113\pi$
$673$	8.90233	0.343160	0.171580	+	0.985170i	$0.445113\pi$
$674$	−71.1827	−2.74185
$675$	0	0
$676$	28.6027	1.10011
$677$	2.82095	0.108418	0.0542089	−	0.998530i	$-0.482736\pi$
$677$	2.82095	0.108418	0.0542089	+	0.998530i	$0.482736\pi$
$678$	0	0
$679$	−19.5703	−0.751038
$680$	−61.1754	−2.34597
$681$	0	0
$682$	−1.81221	−0.0693933
$683$	−20.7357	−0.793431	−0.396716	−	0.917942i	$-0.629850\pi$
$683$	−20.7357	−0.793431	−0.396716	+	0.917942i	$0.629850\pi$
$684$	0	0
$685$	−23.0492	−0.880664
$686$	53.2994	2.03498
$687$	0	0
$688$	−100.137	−3.81769
$689$	−15.6191	−0.595040
$690$	0	0
$691$	1.36601	0.0519655	0.0259827	−	0.999662i	$-0.491729\pi$
$691$	1.36601	0.0519655	0.0259827	+	0.999662i	$0.491729\pi$
$692$	32.1215	1.22107
$693$	0	0
$694$	23.5271	0.893075
$695$	−31.3813	−1.19036
$696$	0	0
$697$	16.7770	0.635473
$698$	−79.2038	−2.99791
$699$	0	0
$700$	−40.1434	−1.51728
$701$	30.1012	1.13691	0.568454	−	0.822715i	$-0.307542\pi$
$701$	30.1012	1.13691	0.568454	+	0.822715i	$0.307542\pi$
$702$	0	0
$703$	11.9066	0.449066
$704$	1.82722	0.0688659
$705$	0	0
$706$	45.7784	1.72289
$707$	16.0583	0.603935
$708$	0	0
$709$	39.7000	1.49097	0.745483	−	0.666524i	$-0.232219\pi$
$709$	39.7000	1.49097	0.745483	+	0.666524i	$0.232219\pi$
$710$	119.741	4.49382
$711$	0	0
$712$	−108.623	−4.07080
$713$	0	0
$714$	0	0
$715$	1.49249	0.0558161
$716$	−41.5702	−1.55355
$717$	0	0
$718$	−18.9566	−0.707455
$719$	9.17245	0.342075	0.171037	−	0.985265i	$-0.445288\pi$
$719$	9.17245	0.342075	0.171037	+	0.985265i	$0.445288\pi$
$720$	0	0
$721$	31.7288	1.18164
$722$	−42.6008	−1.58544
$723$	0	0
$724$	−32.0298	−1.19038
$725$	−4.91075	−0.182381
$726$	0	0
$727$	3.80369	0.141071	0.0705355	−	0.997509i	$-0.477529\pi$
$727$	3.80369	0.141071	0.0705355	+	0.997509i	$0.477529\pi$
$728$	81.0614	3.00434
$729$	0	0
$730$	5.37538	0.198952
$731$	−22.2284	−0.822147
$732$	0	0
$733$	−40.8693	−1.50954	−0.754771	−	0.655989i	$-0.772252\pi$
$733$	−40.8693	−1.50954	−0.754771	+	0.655989i	$0.772252\pi$
$734$	−36.1287	−1.33354
$735$	0	0
$736$	0	0
$737$	0.0100187	0.000369044 0
$738$	0	0
$739$	−29.2620	−1.07642	−0.538211	−	0.842810i	$-0.680900\pi$
$739$	−29.2620	−1.07642	−0.538211	+	0.842810i	$0.680900\pi$
$740$	101.654	3.73687
$741$	0	0
$742$	−22.0990	−0.811279
$743$	−8.66121	−0.317749	−0.158875	−	0.987299i	$-0.550787\pi$
$743$	−8.66121	−0.317749	−0.158875	+	0.987299i	$0.550787\pi$
$744$	0	0
$745$	−48.1018	−1.76231
$746$	−49.8095	−1.82366
$747$	0	0
$748$	1.55419	0.0568270
$749$	−18.8837	−0.689994
$750$	0	0
$751$	−6.87470	−0.250861	−0.125431	−	0.992102i	$-0.540031\pi$
$751$	−6.87470	−0.250861	−0.125431	+	0.992102i	$0.540031\pi$
$752$	−68.6497	−2.50340
$753$	0	0
$754$	16.3677	0.596075
$755$	−54.9254	−1.99894
$756$	0	0
$757$	3.15721	0.114751	0.0573754	−	0.998353i	$-0.481727\pi$
$757$	3.15721	0.114751	0.0573754	+	0.998353i	$0.481727\pi$
$758$	26.6430	0.967717
$759$	0	0
$760$	41.0378	1.48860
$761$	−17.1282	−0.620897	−0.310449	−	0.950590i	$-0.600479\pi$
$761$	−17.1282	−0.620897	−0.310449	+	0.950590i	$0.600479\pi$
$762$	0	0
$763$	−39.6640	−1.43593
$764$	65.6909	2.37661
$765$	0	0
$766$	−77.4590	−2.79871
$767$	−11.7150	−0.423005
$768$	0	0
$769$	21.1864	0.764001	0.382000	−	0.924162i	$-0.375235\pi$
$769$	21.1864	0.764001	0.382000	+	0.924162i	$0.375235\pi$
$770$	2.11168	0.0760996
$771$	0	0
$772$	63.9915	2.30310
$773$	5.34046	0.192083	0.0960415	−	0.995377i	$-0.469382\pi$
$773$	5.34046	0.192083	0.0960415	+	0.995377i	$0.469382\pi$
$774$	0	0
$775$	−19.7303	−0.708732
$776$	69.6770	2.50126
$777$	0	0
$778$	11.6582	0.417966
$779$	−11.2544	−0.403230
$780$	0	0
$781$	−1.84304	−0.0659491
$782$	0	0
$783$	0	0
$784$	−20.0201	−0.715003
$785$	21.9796	0.784485
$786$	0	0
$787$	−43.6090	−1.55449	−0.777247	−	0.629196i	$-0.783384\pi$
$787$	−43.6090	−1.55449	−0.777247	+	0.629196i	$0.783384\pi$
$788$	73.7403	2.62689
$789$	0	0
$790$	−57.1403	−2.03296
$791$	−17.2186	−0.612222
$792$	0	0
$793$	54.2567	1.92671
$794$	0.387583	0.0137548
$795$	0	0
$796$	−102.362	−3.62812
$797$	−11.5987	−0.410848	−0.205424	−	0.978673i	$-0.565857\pi$
$797$	−11.5987	−0.410848	−0.205424	+	0.978673i	$0.565857\pi$
$798$	0	0
$799$	−15.2388	−0.539110
$800$	49.9395	1.76563
$801$	0	0
$802$	−89.5316	−3.16147
$803$	−0.0827368	−0.00291972
$804$	0	0
$805$	0	0
$806$	65.7615	2.31635
$807$	0	0
$808$	−57.1733	−2.01135
$809$	7.81731	0.274842	0.137421	−	0.990513i	$-0.456119\pi$
$809$	7.81731	0.274842	0.137421	+	0.990513i	$0.456119\pi$
$810$	0	0
$811$	11.5174	0.404432	0.202216	−	0.979341i	$-0.435186\pi$
$811$	11.5174	0.404432	0.202216	+	0.979341i	$0.435186\pi$
$812$	16.6108	0.582926
$813$	0	0
$814$	−2.18134	−0.0764561
$815$	−4.85087	−0.169919
$816$	0	0
$817$	14.9113	0.521681
$818$	−80.5052	−2.81480
$819$	0	0
$820$	−96.0854	−3.35545
$821$	0.522107	0.0182217	0.00911084	−	0.999958i	$-0.497100\pi$
$821$	0.522107	0.0182217	0.00911084	+	0.999958i	$0.497100\pi$
$822$	0	0
$823$	−33.2942	−1.16056	−0.580281	−	0.814416i	$-0.697057\pi$
$823$	−33.2942	−1.16056	−0.580281	+	0.814416i	$0.697057\pi$
$824$	−112.966	−3.93535
$825$	0	0
$826$	−16.5752	−0.576725
$827$	24.2079	0.841791	0.420895	−	0.907109i	$-0.361716\pi$
$827$	24.2079	0.841791	0.420895	+	0.907109i	$0.361716\pi$
$828$	0	0
$829$	26.4851	0.919866	0.459933	−	0.887954i	$-0.347873\pi$
$829$	26.4851	0.919866	0.459933	+	0.887954i	$0.347873\pi$
$830$	−21.4904	−0.745944
$831$	0	0
$832$	−66.3060	−2.29875
$833$	−4.44405	−0.153977
$834$	0	0
$835$	62.2874	2.15554
$836$	−1.04259	−0.0360587
$837$	0	0
$838$	49.8348	1.72151
$839$	−38.7473	−1.33771	−0.668853	−	0.743394i	$-0.733215\pi$
$839$	−38.7473	−1.33771	−0.668853	+	0.743394i	$0.733215\pi$
$840$	0	0
$841$	−26.9680	−0.929931
$842$	−104.251	−3.59272
$843$	0	0
$844$	37.8216	1.30187
$845$	−16.3812	−0.563529
$846$	0	0
$847$	25.2290	0.866879
$848$	41.9637	1.44104
$849$	0	0
$850$	23.5907	0.809153
$851$	0	0
$852$	0	0
$853$	8.82914	0.302304	0.151152	−	0.988511i	$-0.451702\pi$
$853$	8.82914	0.302304	0.151152	+	0.988511i	$0.451702\pi$
$854$	76.7660	2.62688
$855$	0	0
$856$	67.2325	2.29796
$857$	0.684430	0.0233797	0.0116898	−	0.999932i	$-0.496279\pi$
$857$	0.684430	0.0233797	0.0116898	+	0.999932i	$0.496279\pi$
$858$	0	0
$859$	12.6327	0.431023	0.215511	−	0.976501i	$-0.430858\pi$
$859$	12.6327	0.431023	0.215511	+	0.976501i	$0.430858\pi$
$860$	127.307	4.34112
$861$	0	0
$862$	57.1023	1.94491
$863$	5.68262	0.193439	0.0967193	−	0.995312i	$-0.469165\pi$
$863$	5.68262	0.193439	0.0967193	+	0.995312i	$0.469165\pi$
$864$	0	0
$865$	−18.3964	−0.625496
$866$	2.01700	0.0685405
$867$	0	0
$868$	66.7385	2.26525
$869$	0.879493	0.0298347
$870$	0	0
$871$	−0.363558	−0.0123187
$872$	141.218	4.78224
$873$	0	0
$874$	0	0
$875$	−10.3778	−0.350833
$876$	0	0
$877$	−22.6537	−0.764961	−0.382480	−	0.923964i	$-0.624930\pi$
$877$	−22.6537	−0.764961	−0.382480	+	0.923964i	$0.624930\pi$
$878$	34.8674	1.17672
$879$	0	0
$880$	−4.00986	−0.135172
$881$	−22.5951	−0.761250	−0.380625	−	0.924730i	$-0.624291\pi$
$881$	−22.5951	−0.761250	−0.380625	+	0.924730i	$0.624291\pi$
$882$	0	0
$883$	41.1095	1.38344	0.691722	−	0.722164i	$-0.256852\pi$
$883$	41.1095	1.38344	0.691722	+	0.722164i	$0.256852\pi$
$884$	−56.3985	−1.89689
$885$	0	0
$886$	8.13796	0.273400
$887$	5.45590	0.183191	0.0915955	−	0.995796i	$-0.470803\pi$
$887$	5.45590	0.183191	0.0915955	+	0.995796i	$0.470803\pi$
$888$	0	0
$889$	4.61271	0.154705
$890$	102.682	3.44192
$891$	0	0
$892$	8.53452	0.285757
$893$	10.2225	0.342085
$894$	0	0
$895$	23.8078	0.795807
$896$	−27.2326	−0.909776
$897$	0	0
$898$	20.3984	0.680703
$899$	8.16413	0.272289
$900$	0	0
$901$	9.31507	0.310330
$902$	2.06186	0.0686523
$903$	0	0
$904$	61.3042	2.03895
$905$	18.3439	0.609772
$906$	0	0
$907$	−28.1199	−0.933706	−0.466853	−	0.884335i	$-0.654612\pi$
$907$	−28.1199	−0.933706	−0.466853	+	0.884335i	$0.654612\pi$
$908$	54.7063	1.81549
$909$	0	0
$910$	−76.6285	−2.54021
$911$	−57.1742	−1.89427	−0.947133	−	0.320840i	$-0.896035\pi$
$911$	−57.1742	−1.89427	−0.947133	+	0.320840i	$0.896035\pi$
$912$	0	0
$913$	0.330777	0.0109471
$914$	−23.8412	−0.788596
$915$	0	0
$916$	−111.163	−3.67292
$917$	−11.2804	−0.372510
$918$	0	0
$919$	−16.3674	−0.539909	−0.269955	−	0.962873i	$-0.587009\pi$
$919$	−16.3674	−0.539909	−0.269955	+	0.962873i	$0.587009\pi$
$920$	0	0
$921$	0	0
$922$	20.0464	0.660195
$923$	66.8800	2.20138
$924$	0	0
$925$	−23.7491	−0.780867
$926$	−87.6763	−2.88122
$927$	0	0
$928$	−20.6643	−0.678339
$929$	−38.2037	−1.25342	−0.626711	−	0.779252i	$-0.715599\pi$
$929$	−38.2037	−1.25342	−0.626711	+	0.779252i	$0.715599\pi$
$930$	0	0
$931$	2.98117	0.0977039
$932$	−126.200	−4.13381
$933$	0	0
$934$	−35.8643	−1.17352
$935$	−0.890107	−0.0291096
$936$	0	0
$937$	−14.0763	−0.459852	−0.229926	−	0.973208i	$-0.573848\pi$
$937$	−14.0763	−0.459852	−0.229926	+	0.973208i	$0.573848\pi$
$938$	−0.514387	−0.0167953
$939$	0	0
$940$	87.2760	2.84663
$941$	−43.5448	−1.41952	−0.709759	−	0.704444i	$-0.751196\pi$
$941$	−43.5448	−1.41952	−0.709759	+	0.704444i	$0.751196\pi$
$942$	0	0
$943$	0	0
$944$	31.4746	1.02441
$945$	0	0
$946$	−2.73182	−0.0888192
$947$	−30.7988	−1.00083	−0.500414	−	0.865786i	$-0.666819\pi$
$947$	−30.7988	−1.00083	−0.500414	+	0.865786i	$0.666819\pi$
$948$	0	0
$949$	3.00235	0.0974603
$950$	−15.8252	−0.513436
$951$	0	0
$952$	−48.3442	−1.56684
$953$	53.6218	1.73698	0.868490	−	0.495708i	$-0.165091\pi$
$953$	53.6218	1.73698	0.868490	+	0.495708i	$0.165091\pi$
$954$	0	0
$955$	−37.6220	−1.21742
$956$	−21.2953	−0.688739
$957$	0	0
$958$	−55.5143	−1.79359
$959$	−18.2148	−0.588185
$960$	0	0
$961$	1.80157	0.0581151
$962$	79.1565	2.55211
$963$	0	0
$964$	139.181	4.48271
$965$	−36.6487	−1.17976
$966$	0	0
$967$	−35.8006	−1.15127	−0.575635	−	0.817707i	$-0.695245\pi$
$967$	−35.8006	−1.15127	−0.575635	+	0.817707i	$0.695245\pi$
$968$	−89.8242	−2.88706
$969$	0	0
$970$	−65.8666	−2.11485
$971$	−21.0599	−0.675846	−0.337923	−	0.941174i	$-0.609724\pi$
$971$	−21.0599	−0.675846	−0.337923	+	0.941174i	$0.609724\pi$
$972$	0	0
$973$	−24.7993	−0.795029
$974$	−21.9840	−0.704411
$975$	0	0
$976$	−145.771	−4.66601
$977$	−42.3706	−1.35556	−0.677778	−	0.735266i	$-0.737057\pi$
$977$	−42.3706	−1.35556	−0.677778	+	0.735266i	$0.737057\pi$
$978$	0	0
$979$	−1.58047	−0.0505120
$980$	25.4520	0.813035
$981$	0	0
$982$	29.5418	0.942716
$983$	27.6419	0.881639	0.440820	−	0.897596i	$-0.354688\pi$
$983$	27.6419	0.881639	0.440820	+	0.897596i	$0.354688\pi$
$984$	0	0
$985$	−42.2320	−1.34562
$986$	−9.76151	−0.310870
$987$	0	0
$988$	37.8334	1.20364
$989$	0	0
$990$	0	0
$991$	−45.4334	−1.44324	−0.721619	−	0.692290i	$-0.756602\pi$
$991$	−45.4334	−1.44324	−0.721619	+	0.692290i	$0.756602\pi$
$992$	−83.0244	−2.63603
$993$	0	0
$994$	94.6264	3.00137
$995$	58.6240	1.85851
$996$	0	0
$997$	−13.0846	−0.414393	−0.207196	−	0.978299i	$-0.566434\pi$
$997$	−13.0846	−0.414393	−0.207196	+	0.978299i	$0.566434\pi$
$998$	−16.1802	−0.512176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bu.1.10		10
3.2	odd	2		1587.2.a.t.1.1		10
23.11	odd	22		207.2.i.d.190.2		20
23.21	odd	22		207.2.i.d.73.2		20
23.22	odd	2		4761.2.a.bt.1.10		10
69.11	even	22		69.2.e.c.52.1	yes	20
69.44	even	22		69.2.e.c.4.1	✓	20
69.68	even	2		1587.2.a.u.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.4.1	✓	20	69.44	even	22
69.2.e.c.52.1	yes	20	69.11	even	22
207.2.i.d.73.2		20	23.21	odd	22
207.2.i.d.190.2		20	23.11	odd	22
1587.2.a.t.1.1		10	3.2	odd	2
1587.2.a.u.1.1		10	69.68	even	2
4761.2.a.bt.1.10		10	23.22	odd	2
4761.2.a.bu.1.10		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)

\(f(q)\)	\(=\)	\(q+2.65972 q^{2} +5.07414 q^{4} -2.90602 q^{5} -2.29650 q^{7} +8.17636 q^{8} -7.72922 q^{10} +0.118967 q^{11} -4.31706 q^{13} -6.10806 q^{14} +11.5986 q^{16} +2.57465 q^{17} -1.72713 q^{19} -14.7456 q^{20} +0.316419 q^{22} +3.44497 q^{25} -11.4822 q^{26} -11.6528 q^{28} -1.42548 q^{29} -5.72727 q^{31} +14.4963 q^{32} +6.84785 q^{34} +6.67369 q^{35} -6.89385 q^{37} -4.59370 q^{38} -23.7607 q^{40} +6.51623 q^{41} -8.63357 q^{43} +0.603653 q^{44} -5.91880 q^{47} -1.72608 q^{49} +9.16268 q^{50} -21.9053 q^{52} +3.61800 q^{53} -0.345720 q^{55} -18.7770 q^{56} -3.79140 q^{58} +2.71366 q^{59} -12.5680 q^{61} -15.2329 q^{62} +15.3591 q^{64} +12.5455 q^{65} +0.0842144 q^{67} +13.0641 q^{68} +17.7502 q^{70} -15.4920 q^{71} -0.695461 q^{73} -18.3358 q^{74} -8.76371 q^{76} -0.273207 q^{77} +7.39276 q^{79} -33.7058 q^{80} +17.3314 q^{82} +2.78042 q^{83} -7.48198 q^{85} -22.9629 q^{86} +0.972714 q^{88} -13.2850 q^{89} +9.91413 q^{91} -15.7424 q^{94} +5.01909 q^{95} +8.52177 q^{97} -4.59090 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

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Groups

Database

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