LMFDB - Embedded newform 4761.2.a.bn.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4761,2,Mod(1,4761)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [5,-2,0,4,-7,0,8,9,0,5,-13,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
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 23)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.30972$ 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)

$f(q)$	$=$	$q-2.20362 q^{2} +2.85592 q^{4} -1.23648 q^{5} +2.37279 q^{7} -1.88612 q^{8} +2.72472 q^{10} -1.91184 q^{11} +0.196474 q^{13} -5.22871 q^{14} -1.55555 q^{16} -1.55991 q^{17} +7.98446 q^{19} -3.53129 q^{20} +4.21297 q^{22} -3.47112 q^{25} -0.432953 q^{26} +6.77649 q^{28} -4.97732 q^{29} +1.78268 q^{31} +7.20009 q^{32} +3.43743 q^{34} -2.93390 q^{35} -3.88323 q^{37} -17.5947 q^{38} +2.33215 q^{40} -0.426496 q^{41} -4.45317 q^{43} -5.46008 q^{44} +2.58842 q^{47} -1.36989 q^{49} +7.64901 q^{50} +0.561114 q^{52} +9.81939 q^{53} +2.36395 q^{55} -4.47536 q^{56} +10.9681 q^{58} +7.21890 q^{59} -7.42966 q^{61} -3.92834 q^{62} -12.7551 q^{64} -0.242936 q^{65} -7.26650 q^{67} -4.45497 q^{68} +6.46519 q^{70} +0.730284 q^{71} -6.44434 q^{73} +8.55714 q^{74} +22.8030 q^{76} -4.53640 q^{77} +5.67808 q^{79} +1.92341 q^{80} +0.939833 q^{82} -12.8897 q^{83} +1.92879 q^{85} +9.81308 q^{86} +3.60597 q^{88} -13.9002 q^{89} +0.466190 q^{91} -5.70388 q^{94} -9.87261 q^{95} -4.32377 q^{97} +3.01871 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 2 q^{2} + 4 q^{4} - 7 q^{5} + 8 q^{7} + 9 q^{8} + 5 q^{10} - 13 q^{11} + 4 q^{13} - 12 q^{14} + 6 q^{16} - 16 q^{17} + 10 q^{19} - 10 q^{20} + 3 q^{22} - 2 q^{25} - 6 q^{26} - 9 q^{28} - 7 q^{29}+ \cdots - 3 q^{98}+O(q^{100})$ 5 * q - 2 * q^2 + 4 * q^4 - 7 * q^5 + 8 * q^7 + 9 * q^8 + 5 * q^10 - 13 * q^11 + 4 * q^13 - 12 * q^14 + 6 * q^16 - 16 * q^17 + 10 * q^19 - 10 * q^20 + 3 * q^22 - 2 * q^25 - 6 * q^26 - 9 * q^28 - 7 * q^29 + 5 * q^31 + 6 * q^32 - 9 * q^34 - 9 * q^35 - 7 * q^37 - 4 * q^38 - 17 * q^40 - 9 * q^41 - 17 * q^44 + 9 * q^47 - 9 * q^49 - 8 * q^50 + q^52 - 2 * q^53 + 16 * q^55 - q^56 - 6 * q^58 + 16 * q^59 + 4 * q^61 - 2 * q^62 - 21 * q^64 + q^65 - 17 * q^67 - 15 * q^68 + 19 * q^70 + 7 * q^71 - 18 * q^73 + 5 * q^74 + 30 * q^76 - 12 * q^77 + 13 * q^79 + 7 * q^80 - 3 * q^82 - 13 * q^83 + 7 * q^85 + 11 * q^86 - 30 * q^88 - 15 * q^89 + 2 * q^91 - 19 * q^94 + 8 * q^95 - 16 * q^97 - 3 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.20362	−1.55819	−0.779096	−	0.626905i	$-0.784321\pi$
$2$	−2.20362	−1.55819	−0.779096	+	0.626905i	$0.784321\pi$
$3$	0	0
$3$	0	0
$4$	2.85592	1.42796
$5$	−1.23648	−0.552970	−0.276485	−	0.961018i	$-0.589170\pi$
$5$	−1.23648	−0.552970	−0.276485	+	0.961018i	$0.589170\pi$
$6$	0	0
$7$	2.37279	0.896829	0.448414	−	0.893826i	$-0.351989\pi$
$7$	2.37279	0.896829	0.448414	+	0.893826i	$0.351989\pi$
$8$	−1.88612	−0.666845
$9$	0	0
$10$	2.72472	0.861633
$11$	−1.91184	−0.576443	−0.288221	−	0.957564i	$-0.593064\pi$
$11$	−1.91184	−0.576443	−0.288221	+	0.957564i	$0.593064\pi$
$12$	0	0
$13$	0.196474	0.0544920	0.0272460	−	0.999629i	$-0.491326\pi$
$13$	0.196474	0.0544920	0.0272460	+	0.999629i	$0.491326\pi$
$14$	−5.22871	−1.39743
$15$	0	0
$16$	−1.55555	−0.388889
$17$	−1.55991	−0.378333	−0.189166	−	0.981945i	$-0.560579\pi$
$17$	−1.55991	−0.378333	−0.189166	+	0.981945i	$0.560579\pi$
$18$	0	0
$19$	7.98446	1.83176	0.915880	−	0.401452i	$-0.131494\pi$
$19$	7.98446	1.83176	0.915880	+	0.401452i	$0.131494\pi$
$20$	−3.53129	−0.789620
$21$	0	0
$22$	4.21297	0.898208
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.47112	−0.694224
$26$	−0.432953	−0.0849090
$27$	0	0
$28$	6.77649	1.28064
$29$	−4.97732	−0.924264	−0.462132	−	0.886811i	$-0.652915\pi$
$29$	−4.97732	−0.924264	−0.462132	+	0.886811i	$0.652915\pi$
$30$	0	0
$31$	1.78268	0.320179	0.160089	−	0.987103i	$-0.448822\pi$
$31$	1.78268	0.320179	0.160089	+	0.987103i	$0.448822\pi$
$32$	7.20009	1.27281
$33$	0	0
$34$	3.43743	0.589515
$35$	−2.93390	−0.495919
$36$	0	0
$37$	−3.88323	−0.638398	−0.319199	−	0.947688i	$-0.603414\pi$
$37$	−3.88323	−0.638398	−0.319199	+	0.947688i	$0.603414\pi$
$38$	−17.5947	−2.85423
$39$	0	0
$40$	2.33215	0.368745
$41$	−0.426496	−0.0666075	−0.0333037	−	0.999445i	$-0.510603\pi$
$41$	−0.426496	−0.0666075	−0.0333037	+	0.999445i	$0.510603\pi$
$42$	0	0
$43$	−4.45317	−0.679102	−0.339551	−	0.940588i	$-0.610275\pi$
$43$	−4.45317	−0.679102	−0.339551	+	0.940588i	$0.610275\pi$
$44$	−5.46008	−0.823138
$45$	0	0
$46$	0	0
$47$	2.58842	0.377559	0.188780	−	0.982019i	$-0.439547\pi$
$47$	2.58842	0.377559	0.188780	+	0.982019i	$0.439547\pi$
$48$	0	0
$49$	−1.36989	−0.195698
$50$	7.64901	1.08173
$51$	0	0
$52$	0.561114	0.0778125
$53$	9.81939	1.34880	0.674399	−	0.738367i	$-0.264403\pi$
$53$	9.81939	1.34880	0.674399	+	0.738367i	$0.264403\pi$
$54$	0	0
$55$	2.36395	0.318756
$56$	−4.47536	−0.598046
$57$	0	0
$58$	10.9681	1.44018
$59$	7.21890	0.939821	0.469911	−	0.882714i	$-0.344286\pi$
$59$	7.21890	0.939821	0.469911	+	0.882714i	$0.344286\pi$
$60$	0	0
$61$	−7.42966	−0.951271	−0.475635	−	0.879643i	$-0.657782\pi$
$61$	−7.42966	−0.951271	−0.475635	+	0.879643i	$0.657782\pi$
$62$	−3.92834	−0.498900
$63$	0	0
$64$	−12.7551	−1.59439
$65$	−0.242936	−0.0301325
$66$	0	0
$67$	−7.26650	−0.887744	−0.443872	−	0.896090i	$-0.646396\pi$
$67$	−7.26650	−0.887744	−0.443872	+	0.896090i	$0.646396\pi$
$68$	−4.45497	−0.540244
$69$	0	0
$70$	6.46519	0.772738
$71$	0.730284	0.0866687	0.0433344	−	0.999061i	$-0.486202\pi$
$71$	0.730284	0.0866687	0.0433344	+	0.999061i	$0.486202\pi$
$72$	0	0
$73$	−6.44434	−0.754253	−0.377126	−	0.926162i	$-0.623088\pi$
$73$	−6.44434	−0.754253	−0.377126	+	0.926162i	$0.623088\pi$
$74$	8.55714	0.994747
$75$	0	0
$76$	22.8030	2.61568
$77$	−4.53640	−0.516970
$78$	0	0
$79$	5.67808	0.638834	0.319417	−	0.947614i	$-0.396513\pi$
$79$	5.67808	0.638834	0.319417	+	0.947614i	$0.396513\pi$
$80$	1.92341	0.215044
$81$	0	0
$82$	0.939833	0.103787
$83$	−12.8897	−1.41483	−0.707416	−	0.706797i	$-0.750139\pi$
$83$	−12.8897	−1.41483	−0.707416	+	0.706797i	$0.750139\pi$
$84$	0	0
$85$	1.92879	0.209207
$86$	9.81308	1.05817
$87$	0	0
$88$	3.60597	0.384398
$89$	−13.9002	−1.47342	−0.736711	−	0.676208i	$-0.763622\pi$
$89$	−13.9002	−1.47342	−0.736711	+	0.676208i	$0.763622\pi$
$90$	0	0
$91$	0.466190	0.0488700
$92$	0	0
$93$	0	0
$94$	−5.70388	−0.588310
$95$	−9.87261	−1.01291
$96$	0	0
$97$	−4.32377	−0.439012	−0.219506	−	0.975611i	$-0.570445\pi$
$97$	−4.32377	−0.439012	−0.219506	+	0.975611i	$0.570445\pi$
$98$	3.01871	0.304936
$99$	0	0
$100$	−9.91325	−0.991325
$101$	1.82028	0.181124	0.0905622	−	0.995891i	$-0.471134\pi$
$101$	1.82028	0.181124	0.0905622	+	0.995891i	$0.471134\pi$
$102$	0	0
$103$	15.3132	1.50886	0.754429	−	0.656382i	$-0.227914\pi$
$103$	15.3132	1.50886	0.754429	+	0.656382i	$0.227914\pi$
$104$	−0.370574	−0.0363377
$105$	0	0
$106$	−21.6382	−2.10168
$107$	1.98833	0.192219	0.0961094	−	0.995371i	$-0.469360\pi$
$107$	1.98833	0.192219	0.0961094	+	0.995371i	$0.469360\pi$
$108$	0	0
$109$	−15.5948	−1.49371	−0.746855	−	0.664987i	$-0.768437\pi$
$109$	−15.5948	−1.49371	−0.746855	+	0.664987i	$0.768437\pi$
$110$	−5.20925	−0.496682
$111$	0	0
$112$	−3.69100	−0.348766
$113$	−1.48825	−0.140002	−0.0700012	−	0.997547i	$-0.522300\pi$
$113$	−1.48825	−0.140002	−0.0700012	+	0.997547i	$0.522300\pi$
$114$	0	0
$115$	0	0
$116$	−14.2148	−1.31981
$117$	0	0
$118$	−15.9077	−1.46442
$119$	−3.70132	−0.339300
$120$	0	0
$121$	−7.34485	−0.667714
$122$	16.3721	1.48226
$123$	0	0
$124$	5.09119	0.457203
$125$	10.4744	0.936855
$126$	0	0
$127$	5.99437	0.531914	0.265957	−	0.963985i	$-0.414312\pi$
$127$	5.99437	0.531914	0.265957	+	0.963985i	$0.414312\pi$
$128$	13.7072	1.21156
$129$	0	0
$130$	0.535337	0.0469521
$131$	12.7688	1.11562	0.557809	−	0.829970i	$-0.311642\pi$
$131$	12.7688	1.11562	0.557809	+	0.829970i	$0.311642\pi$
$132$	0	0
$133$	18.9454	1.64277
$134$	16.0126	1.38328
$135$	0	0
$136$	2.94217	0.252289
$137$	0.697560	0.0595966	0.0297983	−	0.999556i	$-0.490514\pi$
$137$	0.697560	0.0595966	0.0297983	+	0.999556i	$0.490514\pi$
$138$	0	0
$139$	13.9559	1.18372	0.591862	−	0.806039i	$-0.298393\pi$
$139$	13.9559	1.18372	0.591862	+	0.806039i	$0.298393\pi$
$140$	−8.37899	−0.708154
$141$	0	0
$142$	−1.60926	−0.135046
$143$	−0.375627	−0.0314115
$144$	0	0
$145$	6.15435	0.511091
$146$	14.2008	1.17527
$147$	0	0
$148$	−11.0902	−0.911608
$149$	−16.6390	−1.36312	−0.681561	−	0.731761i	$-0.738699\pi$
$149$	−16.6390	−1.36312	−0.681561	+	0.731761i	$0.738699\pi$
$150$	0	0
$151$	13.1900	1.07338	0.536692	−	0.843778i	$-0.319674\pi$
$151$	13.1900	1.07338	0.536692	+	0.843778i	$0.319674\pi$
$152$	−15.0597	−1.22150
$153$	0	0
$154$	9.99647	0.805539
$155$	−2.20425	−0.177049
$156$	0	0
$157$	−4.12867	−0.329504	−0.164752	−	0.986335i	$-0.552682\pi$
$157$	−4.12867	−0.329504	−0.164752	+	0.986335i	$0.552682\pi$
$158$	−12.5123	−0.995426
$159$	0	0
$160$	−8.90276	−0.703825
$161$	0	0
$162$	0	0
$163$	−14.7893	−1.15839	−0.579194	−	0.815190i	$-0.696633\pi$
$163$	−14.7893	−1.15839	−0.579194	+	0.815190i	$0.696633\pi$
$164$	−1.21804	−0.0951129
$165$	0	0
$166$	28.4040	2.20458
$167$	−1.32974	−0.102899	−0.0514493	−	0.998676i	$-0.516384\pi$
$167$	−1.32974	−0.102899	−0.0514493	+	0.998676i	$0.516384\pi$
$168$	0	0
$169$	−12.9614	−0.997031
$170$	−4.25031	−0.325984
$171$	0	0
$172$	−12.7179	−0.969731
$173$	16.9361	1.28762	0.643812	−	0.765184i	$-0.277352\pi$
$173$	16.9361	1.28762	0.643812	+	0.765184i	$0.277352\pi$
$174$	0	0
$175$	−8.23622	−0.622600
$176$	2.97398	0.224172
$177$	0	0
$178$	30.6308	2.29587
$179$	1.43135	0.106984	0.0534920	−	0.998568i	$-0.482965\pi$
$179$	1.43135	0.106984	0.0534920	+	0.998568i	$0.482965\pi$
$180$	0	0
$181$	16.1314	1.19903	0.599517	−	0.800362i	$-0.295359\pi$
$181$	16.1314	1.19903	0.599517	+	0.800362i	$0.295359\pi$
$182$	−1.02730	−0.0761488
$183$	0	0
$184$	0	0
$185$	4.80153	0.353015
$186$	0	0
$187$	2.98230	0.218087
$188$	7.39232	0.539140
$189$	0	0
$190$	21.7554	1.57831
$191$	−20.7445	−1.50102	−0.750509	−	0.660860i	$-0.770192\pi$
$191$	−20.7445	−1.50102	−0.750509	+	0.660860i	$0.770192\pi$
$192$	0	0
$193$	14.6523	1.05469	0.527346	−	0.849651i	$-0.323187\pi$
$193$	14.6523	1.05469	0.527346	+	0.849651i	$0.323187\pi$
$194$	9.52793	0.684065
$195$	0	0
$196$	−3.91229	−0.279450
$197$	−11.6779	−0.832016	−0.416008	−	0.909361i	$-0.636571\pi$
$197$	−11.6779	−0.832016	−0.416008	+	0.909361i	$0.636571\pi$
$198$	0	0
$199$	5.72341	0.405722	0.202861	−	0.979208i	$-0.434976\pi$
$199$	5.72341	0.405722	0.202861	+	0.979208i	$0.434976\pi$
$200$	6.54696	0.462940
$201$	0	0
$202$	−4.01119	−0.282226
$203$	−11.8101	−0.828907
$204$	0	0
$205$	0.527353	0.0368319
$206$	−33.7445	−2.35109
$207$	0	0
$208$	−0.305626	−0.0211913
$209$	−15.2650	−1.05590
$210$	0	0
$211$	−14.0677	−0.968463	−0.484232	−	0.874940i	$-0.660901\pi$
$211$	−14.0677	−0.968463	−0.484232	+	0.874940i	$0.660901\pi$
$212$	28.0434	1.92603
$213$	0	0
$214$	−4.38151	−0.299514
$215$	5.50625	0.375523
$216$	0	0
$217$	4.22992	0.287145
$218$	34.3649	2.32748
$219$	0	0
$220$	6.75127	0.455171
$221$	−0.306480	−0.0206161
$222$	0	0
$223$	−4.20359	−0.281493	−0.140746	−	0.990046i	$-0.544950\pi$
$223$	−4.20359	−0.281493	−0.140746	+	0.990046i	$0.544950\pi$
$224$	17.0843	1.14149
$225$	0	0
$226$	3.27952	0.218150
$227$	18.4308	1.22329	0.611647	−	0.791131i	$-0.290507\pi$
$227$	18.4308	1.22329	0.611647	+	0.791131i	$0.290507\pi$
$228$	0	0
$229$	−10.9247	−0.721923	−0.360961	−	0.932581i	$-0.617551\pi$
$229$	−10.9247	−0.721923	−0.360961	+	0.932581i	$0.617551\pi$
$230$	0	0
$231$	0	0
$232$	9.38783	0.616341
$233$	−4.16483	−0.272847	−0.136423	−	0.990651i	$-0.543561\pi$
$233$	−4.16483	−0.272847	−0.136423	+	0.990651i	$0.543561\pi$
$234$	0	0
$235$	−3.20052	−0.208779
$236$	20.6166	1.34203
$237$	0	0
$238$	8.15629	0.528694
$239$	22.5171	1.45651	0.728257	−	0.685305i	$-0.240331\pi$
$239$	22.5171	1.45651	0.728257	+	0.685305i	$0.240331\pi$
$240$	0	0
$241$	−5.92913	−0.381929	−0.190965	−	0.981597i	$-0.561162\pi$
$241$	−5.92913	−0.381929	−0.190965	+	0.981597i	$0.561162\pi$
$242$	16.1852	1.04043
$243$	0	0
$244$	−21.2185	−1.35838
$245$	1.69384	0.108215
$246$	0	0
$247$	1.56874	0.0998163
$248$	−3.36235	−0.213510
$249$	0	0
$250$	−23.0815	−1.45980
$251$	−1.55061	−0.0978738	−0.0489369	−	0.998802i	$-0.515583\pi$
$251$	−1.55061	−0.0978738	−0.0489369	+	0.998802i	$0.515583\pi$
$252$	0	0
$253$	0	0
$254$	−13.2093	−0.828824
$255$	0	0
$256$	−4.69517	−0.293448
$257$	−27.4579	−1.71278	−0.856388	−	0.516333i	$-0.827297\pi$
$257$	−27.4579	−1.71278	−0.856388	+	0.516333i	$0.827297\pi$
$258$	0	0
$259$	−9.21406	−0.572534
$260$	−0.693805	−0.0430280
$261$	0	0
$262$	−28.1376	−1.73835
$263$	−12.9407	−0.797958	−0.398979	−	0.916960i	$-0.630635\pi$
$263$	−12.9407	−0.797958	−0.398979	+	0.916960i	$0.630635\pi$
$264$	0	0
$265$	−12.1415	−0.745845
$266$	−41.7484	−2.55976
$267$	0	0
$268$	−20.7526	−1.26766
$269$	9.99766	0.609568	0.304784	−	0.952421i	$-0.401416\pi$
$269$	9.99766	0.609568	0.304784	+	0.952421i	$0.401416\pi$
$270$	0	0
$271$	4.41139	0.267973	0.133986	−	0.990983i	$-0.457222\pi$
$271$	4.41139	0.267973	0.133986	+	0.990983i	$0.457222\pi$
$272$	2.42652	0.147129
$273$	0	0
$274$	−1.53715	−0.0928629
$275$	6.63624	0.400180
$276$	0	0
$277$	−30.8042	−1.85085	−0.925423	−	0.378935i	$-0.876290\pi$
$277$	−30.8042	−1.85085	−0.925423	+	0.378935i	$0.876290\pi$
$278$	−30.7535	−1.84447
$279$	0	0
$280$	5.53369	0.330701
$281$	−3.57931	−0.213524	−0.106762	−	0.994285i	$-0.534048\pi$
$281$	−3.57931	−0.213524	−0.106762	+	0.994285i	$0.534048\pi$
$282$	0	0
$283$	−14.7804	−0.878604	−0.439302	−	0.898339i	$-0.644774\pi$
$283$	−14.7804	−0.878604	−0.439302	+	0.898339i	$0.644774\pi$
$284$	2.08563	0.123760
$285$	0	0
$286$	0.827738	0.0489452
$287$	−1.01198	−0.0597355
$288$	0	0
$289$	−14.5667	−0.856864
$290$	−13.5618	−0.796377
$291$	0	0
$292$	−18.4045	−1.07704
$293$	21.5135	1.25683	0.628415	−	0.777878i	$-0.283704\pi$
$293$	21.5135	1.25683	0.628415	+	0.777878i	$0.283704\pi$
$294$	0	0
$295$	−8.92602	−0.519693
$296$	7.32424	0.425713
$297$	0	0
$298$	36.6660	2.12401
$299$	0	0
$300$	0	0
$301$	−10.5664	−0.609038
$302$	−29.0656	−1.67254
$303$	0	0
$304$	−12.4203	−0.712351
$305$	9.18662	0.526024
$306$	0	0
$307$	14.1449	0.807294	0.403647	−	0.914915i	$-0.367742\pi$
$307$	14.1449	0.807294	0.403647	+	0.914915i	$0.367742\pi$
$308$	−12.9556	−0.738213
$309$	0	0
$310$	4.85731	0.275877
$311$	−0.570139	−0.0323296	−0.0161648	−	0.999869i	$-0.505146\pi$
$311$	−0.570139	−0.0323296	−0.0161648	+	0.999869i	$0.505146\pi$
$312$	0	0
$313$	21.1327	1.19449	0.597247	−	0.802058i	$-0.296261\pi$
$313$	21.1327	1.19449	0.597247	+	0.802058i	$0.296261\pi$
$314$	9.09801	0.513431
$315$	0	0
$316$	16.2162	0.912230
$317$	8.81213	0.494939	0.247469	−	0.968896i	$-0.420401\pi$
$317$	8.81213	0.494939	0.247469	+	0.968896i	$0.420401\pi$
$318$	0	0
$319$	9.51585	0.532785
$320$	15.7714	0.881650
$321$	0	0
$322$	0	0
$323$	−12.4550	−0.693015
$324$	0	0
$325$	−0.681984	−0.0378297
$326$	32.5900	1.80499
$327$	0	0
$328$	0.804424	0.0444169
$329$	6.14176	0.338606
$330$	0	0
$331$	30.1875	1.65925	0.829627	−	0.558317i	$-0.188553\pi$
$331$	30.1875	1.65925	0.829627	+	0.558317i	$0.188553\pi$
$332$	−36.8121	−2.02033
$333$	0	0
$334$	2.93024	0.160336
$335$	8.98487	0.490896
$336$	0	0
$337$	4.95577	0.269958	0.134979	−	0.990848i	$-0.456903\pi$
$337$	4.95577	0.269958	0.134979	+	0.990848i	$0.456903\pi$
$338$	28.5619	1.55356
$339$	0	0
$340$	5.50847	0.298739
$341$	−3.40820	−0.184565
$342$	0	0
$343$	−19.8600	−1.07234
$344$	8.39923	0.452856
$345$	0	0
$346$	−37.3205	−2.00636
$347$	6.76696	0.363269	0.181635	−	0.983366i	$-0.441861\pi$
$347$	6.76696	0.363269	0.181635	+	0.983366i	$0.441861\pi$
$348$	0	0
$349$	10.0918	0.540204	0.270102	−	0.962832i	$-0.412943\pi$
$349$	10.0918	0.540204	0.270102	+	0.962832i	$0.412943\pi$
$350$	18.1495	0.970130
$351$	0	0
$352$	−13.7654	−0.733701
$353$	−24.4693	−1.30237	−0.651184	−	0.758920i	$-0.725727\pi$
$353$	−24.4693	−1.30237	−0.651184	+	0.758920i	$0.725727\pi$
$354$	0	0
$355$	−0.902980	−0.0479252
$356$	−39.6980	−2.10399
$357$	0	0
$358$	−3.15414	−0.166702
$359$	0.418813	0.0221041	0.0110520	−	0.999939i	$-0.496482\pi$
$359$	0.418813	0.0221041	0.0110520	+	0.999939i	$0.496482\pi$
$360$	0	0
$361$	44.7516	2.35534
$362$	−35.5473	−1.86833
$363$	0	0
$364$	1.33140	0.0697845
$365$	7.96829	0.417079
$366$	0	0
$367$	−28.5682	−1.49125	−0.745624	−	0.666367i	$-0.767849\pi$
$367$	−28.5682	−1.49125	−0.745624	+	0.666367i	$0.767849\pi$
$368$	0	0
$369$	0	0
$370$	−10.5807	−0.550065
$371$	23.2993	1.20964
$372$	0	0
$373$	−14.4078	−0.746006	−0.373003	−	0.927830i	$-0.621672\pi$
$373$	−14.4078	−0.746006	−0.373003	+	0.927830i	$0.621672\pi$
$374$	−6.57183	−0.339821
$375$	0	0
$376$	−4.88207	−0.251774
$377$	−0.977912	−0.0503650
$378$	0	0
$379$	5.62712	0.289046	0.144523	−	0.989501i	$-0.453835\pi$
$379$	5.62712	0.289046	0.144523	+	0.989501i	$0.453835\pi$
$380$	−28.1954	−1.44639
$381$	0	0
$382$	45.7129	2.33887
$383$	0.231660	0.0118373	0.00591864	−	0.999982i	$-0.498116\pi$
$383$	0.231660	0.0118373	0.00591864	+	0.999982i	$0.498116\pi$
$384$	0	0
$385$	5.60916	0.285869
$386$	−32.2879	−1.64341
$387$	0	0
$388$	−12.3484	−0.626893
$389$	−30.1698	−1.52967	−0.764835	−	0.644226i	$-0.777180\pi$
$389$	−30.1698	−1.52967	−0.764835	+	0.644226i	$0.777180\pi$
$390$	0	0
$391$	0	0
$392$	2.58378	0.130500
$393$	0	0
$394$	25.7336	1.29644
$395$	−7.02083	−0.353256
$396$	0	0
$397$	−13.0453	−0.654723	−0.327362	−	0.944899i	$-0.606160\pi$
$397$	−13.0453	−0.654723	−0.327362	+	0.944899i	$0.606160\pi$
$398$	−12.6122	−0.632192
$399$	0	0
$400$	5.39952	0.269976
$401$	−21.3257	−1.06496	−0.532479	−	0.846444i	$-0.678739\pi$
$401$	−21.3257	−1.06496	−0.532479	+	0.846444i	$0.678739\pi$
$402$	0	0
$403$	0.350250	0.0174472
$404$	5.19857	0.258639
$405$	0	0
$406$	26.0249	1.29160
$407$	7.42412	0.368000
$408$	0	0
$409$	−9.14902	−0.452390	−0.226195	−	0.974082i	$-0.572629\pi$
$409$	−9.14902	−0.452390	−0.226195	+	0.974082i	$0.572629\pi$
$410$	−1.16208	−0.0573912
$411$	0	0
$412$	43.7334	2.15459
$413$	17.1289	0.842859
$414$	0	0
$415$	15.9379	0.782360
$416$	1.41463	0.0693579
$417$	0	0
$418$	33.6383	1.64530
$419$	−8.97420	−0.438418	−0.219209	−	0.975678i	$-0.570348\pi$
$419$	−8.97420	−0.438418	−0.219209	+	0.975678i	$0.570348\pi$
$420$	0	0
$421$	−31.4723	−1.53387	−0.766933	−	0.641728i	$-0.778218\pi$
$421$	−31.4723	−1.53387	−0.766933	+	0.641728i	$0.778218\pi$
$422$	30.9999	1.50905
$423$	0	0
$424$	−18.5206	−0.899439
$425$	5.41462	0.262648
$426$	0	0
$427$	−17.6290	−0.853127
$428$	5.67851	0.274481
$429$	0	0
$430$	−12.1337	−0.585137
$431$	−22.1024	−1.06464	−0.532318	−	0.846545i	$-0.678679\pi$
$431$	−22.1024	−1.06464	−0.532318	+	0.846545i	$0.678679\pi$
$432$	0	0
$433$	−12.6952	−0.610095	−0.305047	−	0.952337i	$-0.598672\pi$
$433$	−12.6952	−0.610095	−0.305047	+	0.952337i	$0.598672\pi$
$434$	−9.32111	−0.447428
$435$	0	0
$436$	−44.5375	−2.13296
$437$	0	0
$438$	0	0
$439$	18.4210	0.879187	0.439593	−	0.898197i	$-0.355123\pi$
$439$	18.4210	0.879187	0.439593	+	0.898197i	$0.355123\pi$
$440$	−4.45871	−0.212561
$441$	0	0
$442$	0.675365	0.0321238
$443$	13.9128	0.661016	0.330508	−	0.943803i	$-0.392780\pi$
$443$	13.9128	0.661016	0.330508	+	0.943803i	$0.392780\pi$
$444$	0	0
$445$	17.1873	0.814758
$446$	9.26309	0.438620
$447$	0	0
$448$	−30.2652	−1.42989
$449$	−30.0925	−1.42015	−0.710076	−	0.704125i	$-0.751339\pi$
$449$	−30.0925	−1.42015	−0.710076	+	0.704125i	$0.751339\pi$
$450$	0	0
$451$	0.815394	0.0383954
$452$	−4.25031	−0.199918
$453$	0	0
$454$	−40.6143	−1.90612
$455$	−0.576434	−0.0270237
$456$	0	0
$457$	−38.0475	−1.77979	−0.889894	−	0.456166i	$-0.849222\pi$
$457$	−38.0475	−1.77979	−0.889894	+	0.456166i	$0.849222\pi$
$458$	24.0738	1.12489
$459$	0	0
$460$	0	0
$461$	−8.76016	−0.408001	−0.204001	−	0.978971i	$-0.565394\pi$
$461$	−8.76016	−0.408001	−0.204001	+	0.978971i	$0.565394\pi$
$462$	0	0
$463$	−27.8367	−1.29368	−0.646841	−	0.762625i	$-0.723910\pi$
$463$	−27.8367	−1.29368	−0.646841	+	0.762625i	$0.723910\pi$
$464$	7.74248	0.359436
$465$	0	0
$466$	9.17768	0.425148
$467$	−17.3663	−0.803619	−0.401809	−	0.915723i	$-0.631619\pi$
$467$	−17.3663	−0.803619	−0.401809	+	0.915723i	$0.631619\pi$
$468$	0	0
$469$	−17.2418	−0.796155
$470$	7.05272	0.325318
$471$	0	0
$472$	−13.6157	−0.626715
$473$	8.51377	0.391463
$474$	0	0
$475$	−27.7150	−1.27165
$476$	−10.5707	−0.484507
$477$	0	0
$478$	−49.6191	−2.26953
$479$	−25.4778	−1.16411	−0.582055	−	0.813150i	$-0.697751\pi$
$479$	−25.4778	−1.16411	−0.582055	+	0.813150i	$0.697751\pi$
$480$	0	0
$481$	−0.762952	−0.0347876
$482$	13.0655	0.595119
$483$	0	0
$484$	−20.9763	−0.953469
$485$	5.34625	0.242761
$486$	0	0
$487$	−6.16107	−0.279185	−0.139592	−	0.990209i	$-0.544579\pi$
$487$	−6.16107	−0.279185	−0.139592	+	0.990209i	$0.544579\pi$
$488$	14.0133	0.634350
$489$	0	0
$490$	−3.73257	−0.168620
$491$	32.6476	1.47337	0.736683	−	0.676238i	$-0.236391\pi$
$491$	32.6476	1.47337	0.736683	+	0.676238i	$0.236391\pi$
$492$	0	0
$493$	7.76414	0.349679
$494$	−3.45689	−0.155533
$495$	0	0
$496$	−2.77305	−0.124514
$497$	1.73281	0.0777270
$498$	0	0
$499$	−8.57369	−0.383811	−0.191906	−	0.981413i	$-0.561467\pi$
$499$	−8.57369	−0.383811	−0.191906	+	0.981413i	$0.561467\pi$
$500$	29.9140	1.33779
$501$	0	0
$502$	3.41695	0.152506
$503$	6.75554	0.301214	0.150607	−	0.988594i	$-0.451877\pi$
$503$	6.75554	0.301214	0.150607	+	0.988594i	$0.451877\pi$
$504$	0	0
$505$	−2.25073	−0.100156
$506$	0	0
$507$	0	0
$508$	17.1194	0.759553
$509$	−40.9726	−1.81608	−0.908038	−	0.418887i	$-0.862420\pi$
$509$	−40.9726	−1.81608	−0.908038	+	0.418887i	$0.862420\pi$
$510$	0	0
$511$	−15.2910	−0.676436
$512$	−17.0681	−0.754309
$513$	0	0
$514$	60.5066	2.66883
$515$	−18.9345	−0.834353
$516$	0	0
$517$	−4.94865	−0.217641
$518$	20.3043	0.892118
$519$	0	0
$520$	0.458206	0.0200937
$521$	−18.7268	−0.820438	−0.410219	−	0.911987i	$-0.634548\pi$
$521$	−18.7268	−0.820438	−0.410219	+	0.911987i	$0.634548\pi$
$522$	0	0
$523$	−23.5640	−1.03038	−0.515191	−	0.857075i	$-0.672279\pi$
$523$	−23.5640	−1.03038	−0.515191	+	0.857075i	$0.672279\pi$
$524$	36.4668	1.59306
$525$	0	0
$526$	28.5163	1.24337
$527$	−2.78081	−0.121134
$528$	0	0
$529$	0	0
$530$	26.7551	1.16217
$531$	0	0
$532$	54.1066	2.34582
$533$	−0.0837952	−0.00362957
$534$	0	0
$535$	−2.45853	−0.106291
$536$	13.7055	0.591988
$537$	0	0
$538$	−22.0310	−0.949824
$539$	2.61901	0.112809
$540$	0	0
$541$	0.953288	0.0409850	0.0204925	−	0.999790i	$-0.493477\pi$
$541$	0.953288	0.0409850	0.0204925	+	0.999790i	$0.493477\pi$
$542$	−9.72101	−0.417553
$543$	0	0
$544$	−11.2315	−0.481545
$545$	19.2826	0.825977
$546$	0	0
$547$	29.7860	1.27356	0.636779	−	0.771046i	$-0.280266\pi$
$547$	29.7860	1.27356	0.636779	+	0.771046i	$0.280266\pi$
$548$	1.99218	0.0851016
$549$	0	0
$550$	−14.6237	−0.623558
$551$	−39.7412	−1.69303
$552$	0	0
$553$	13.4729	0.572925
$554$	67.8807	2.88397
$555$	0	0
$556$	39.8570	1.69031
$557$	−6.00871	−0.254597	−0.127299	−	0.991864i	$-0.540631\pi$
$557$	−6.00871	−0.254597	−0.127299	+	0.991864i	$0.540631\pi$
$558$	0	0
$559$	−0.874931	−0.0370056
$560$	4.56384	0.192857
$561$	0	0
$562$	7.88742	0.332711
$563$	0.0891900	0.00375891	0.00187946	−	0.999998i	$-0.499402\pi$
$563$	0.0891900	0.00375891	0.00187946	+	0.999998i	$0.499402\pi$
$564$	0	0
$565$	1.84018	0.0774171
$566$	32.5703	1.36903
$567$	0	0
$568$	−1.37740	−0.0577946
$569$	−41.3596	−1.73388	−0.866942	−	0.498410i	$-0.833918\pi$
$569$	−41.3596	−1.73388	−0.866942	+	0.498410i	$0.833918\pi$
$570$	0	0
$571$	−25.7905	−1.07930	−0.539649	−	0.841890i	$-0.681443\pi$
$571$	−25.7905	−1.07930	−0.539649	+	0.841890i	$0.681443\pi$
$572$	−1.07276	−0.0448544
$573$	0	0
$574$	2.23002	0.0930793
$575$	0	0
$576$	0	0
$577$	27.1442	1.13003	0.565014	−	0.825081i	$-0.308871\pi$
$577$	27.1442	1.13003	0.565014	+	0.825081i	$0.308871\pi$
$578$	32.0994	1.33516
$579$	0	0
$580$	17.5763	0.729817
$581$	−30.5846	−1.26886
$582$	0	0
$583$	−18.7731	−0.777504
$584$	12.1548	0.502970
$585$	0	0
$586$	−47.4074	−1.95838
$587$	−31.6348	−1.30571	−0.652854	−	0.757484i	$-0.726428\pi$
$587$	−31.6348	−1.30571	−0.652854	+	0.757484i	$0.726428\pi$
$588$	0	0
$589$	14.2337	0.586490
$590$	19.6695	0.809781
$591$	0	0
$592$	6.04057	0.248266
$593$	25.7460	1.05726	0.528630	−	0.848852i	$-0.322706\pi$
$593$	25.7460	1.05726	0.528630	+	0.848852i	$0.322706\pi$
$594$	0	0
$595$	4.57661	0.187623
$596$	−47.5198	−1.94649
$597$	0	0
$598$	0	0
$599$	−16.9434	−0.692289	−0.346144	−	0.938181i	$-0.612509\pi$
$599$	−16.9434	−0.692289	−0.346144	+	0.938181i	$0.612509\pi$
$600$	0	0
$601$	−9.65496	−0.393834	−0.196917	−	0.980420i	$-0.563093\pi$
$601$	−9.65496	−0.393834	−0.196917	+	0.980420i	$0.563093\pi$
$602$	23.2843	0.948998
$603$	0	0
$604$	37.6695	1.53275
$605$	9.08175	0.369226
$606$	0	0
$607$	3.24527	0.131721	0.0658607	−	0.997829i	$-0.479021\pi$
$607$	3.24527	0.131721	0.0658607	+	0.997829i	$0.479021\pi$
$608$	57.4888	2.33148
$609$	0	0
$610$	−20.2438	−0.819647
$611$	0.508556	0.0205740
$612$	0	0
$613$	32.9143	1.32940	0.664698	−	0.747112i	$-0.268560\pi$
$613$	32.9143	1.32940	0.664698	+	0.747112i	$0.268560\pi$
$614$	−31.1700	−1.25792
$615$	0	0
$616$	8.55620	0.344739
$617$	3.02442	0.121759	0.0608794	−	0.998145i	$-0.480610\pi$
$617$	3.02442	0.121759	0.0608794	+	0.998145i	$0.480610\pi$
$618$	0	0
$619$	−10.4224	−0.418911	−0.209455	−	0.977818i	$-0.567169\pi$
$619$	−10.4224	−0.418911	−0.209455	+	0.977818i	$0.567169\pi$
$620$	−6.29515	−0.252819
$621$	0	0
$622$	1.25637	0.0503758
$623$	−32.9823	−1.32141
$624$	0	0
$625$	4.40427	0.176171
$626$	−46.5684	−1.86125
$627$	0	0
$628$	−11.7912	−0.470519
$629$	6.05746	0.241527
$630$	0	0
$631$	−34.1013	−1.35755	−0.678776	−	0.734346i	$-0.737489\pi$
$631$	−34.1013	−1.35755	−0.678776	+	0.734346i	$0.737489\pi$
$632$	−10.7096	−0.426003
$633$	0	0
$634$	−19.4186	−0.771209
$635$	−7.41191	−0.294133
$636$	0	0
$637$	−0.269147	−0.0106640
$638$	−20.9693	−0.830182
$639$	0	0
$640$	−16.9487	−0.669955
$641$	−9.00567	−0.355702	−0.177851	−	0.984057i	$-0.556915\pi$
$641$	−9.00567	−0.355702	−0.177851	+	0.984057i	$0.556915\pi$
$642$	0	0
$643$	22.9522	0.905146	0.452573	−	0.891727i	$-0.350506\pi$
$643$	22.9522	0.905146	0.452573	+	0.891727i	$0.350506\pi$
$644$	0	0
$645$	0	0
$646$	27.4460	1.07985
$647$	44.9543	1.76734	0.883669	−	0.468113i	$-0.155066\pi$
$647$	44.9543	1.76734	0.883669	+	0.468113i	$0.155066\pi$
$648$	0	0
$649$	−13.8014	−0.541753
$650$	1.50283	0.0589459
$651$	0	0
$652$	−42.2371	−1.65413
$653$	29.3634	1.14908	0.574539	−	0.818477i	$-0.305181\pi$
$653$	29.3634	1.14908	0.574539	+	0.818477i	$0.305181\pi$
$654$	0	0
$655$	−15.7884	−0.616903
$656$	0.663437	0.0259029
$657$	0	0
$658$	−13.5341	−0.527613
$659$	−40.1537	−1.56417	−0.782083	−	0.623174i	$-0.785843\pi$
$659$	−40.1537	−1.56417	−0.782083	+	0.623174i	$0.785843\pi$
$660$	0	0
$661$	−9.96660	−0.387656	−0.193828	−	0.981036i	$-0.562090\pi$
$661$	−9.96660	−0.387656	−0.193828	+	0.981036i	$0.562090\pi$
$662$	−66.5216	−2.58544
$663$	0	0
$664$	24.3116	0.943474
$665$	−23.4256	−0.908406
$666$	0	0
$667$	0	0
$668$	−3.79764	−0.146935
$669$	0	0
$670$	−19.7992	−0.764910
$671$	14.2044	0.548353
$672$	0	0
$673$	−34.0567	−1.31279	−0.656395	−	0.754417i	$-0.727920\pi$
$673$	−34.0567	−1.31279	−0.656395	+	0.754417i	$0.727920\pi$
$674$	−10.9206	−0.420646
$675$	0	0
$676$	−37.0167	−1.42372
$677$	−24.9911	−0.960486	−0.480243	−	0.877135i	$-0.659452\pi$
$677$	−24.9911	−0.960486	−0.480243	+	0.877135i	$0.659452\pi$
$678$	0	0
$679$	−10.2594	−0.393719
$680$	−3.63793	−0.139508
$681$	0	0
$682$	7.51037	0.287587
$683$	−34.4762	−1.31919	−0.659597	−	0.751619i	$-0.729273\pi$
$683$	−34.4762	−1.31919	−0.659597	+	0.751619i	$0.729273\pi$
$684$	0	0
$685$	−0.862518	−0.0329551
$686$	43.7637	1.67091
$687$	0	0
$688$	6.92715	0.264095
$689$	1.92925	0.0734987
$690$	0	0
$691$	−21.2259	−0.807470	−0.403735	−	0.914876i	$-0.632288\pi$
$691$	−21.2259	−0.807470	−0.403735	+	0.914876i	$0.632288\pi$
$692$	48.3680	1.83868
$693$	0	0
$694$	−14.9118	−0.566043
$695$	−17.2562	−0.654565
$696$	0	0
$697$	0.665293	0.0251998
$698$	−22.2386	−0.841742
$699$	0	0
$700$	−23.5220	−0.889048
$701$	−24.1261	−0.911232	−0.455616	−	0.890176i	$-0.650581\pi$
$701$	−24.1261	−0.911232	−0.455616	+	0.890176i	$0.650581\pi$
$702$	0	0
$703$	−31.0054	−1.16939
$704$	24.3858	0.919074
$705$	0	0
$706$	53.9208	2.02934
$707$	4.31913	0.162438
$708$	0	0
$709$	−7.03693	−0.264277	−0.132139	−	0.991231i	$-0.542184\pi$
$709$	−7.03693	−0.264277	−0.132139	+	0.991231i	$0.542184\pi$
$710$	1.98982	0.0746767
$711$	0	0
$712$	26.2175	0.982544
$713$	0	0
$714$	0	0
$715$	0.464455	0.0173696
$716$	4.08782	0.152769
$717$	0	0
$718$	−0.922903	−0.0344424
$719$	−26.5682	−0.990829	−0.495414	−	0.868657i	$-0.664984\pi$
$719$	−26.5682	−0.990829	−0.495414	+	0.868657i	$0.664984\pi$
$720$	0	0
$721$	36.3350	1.35319
$722$	−98.6152	−3.67008
$723$	0	0
$724$	46.0699	1.71217
$725$	17.2769	0.641646
$726$	0	0
$727$	5.22756	0.193879	0.0969397	−	0.995290i	$-0.469095\pi$
$727$	5.22756	0.193879	0.0969397	+	0.995290i	$0.469095\pi$
$728$	−0.879292	−0.0325887
$729$	0	0
$730$	−17.5590	−0.649889
$731$	6.94652	0.256926
$732$	0	0
$733$	26.6889	0.985776	0.492888	−	0.870093i	$-0.335941\pi$
$733$	26.6889	0.985776	0.492888	+	0.870093i	$0.335941\pi$
$734$	62.9533	2.32365
$735$	0	0
$736$	0	0
$737$	13.8924	0.511734
$738$	0	0
$739$	−2.23955	−0.0823831	−0.0411916	−	0.999151i	$-0.513115\pi$
$739$	−2.23955	−0.0823831	−0.0411916	+	0.999151i	$0.513115\pi$
$740$	13.7128	0.504092
$741$	0	0
$742$	−51.3427	−1.88485
$743$	18.1401	0.665497	0.332748	−	0.943016i	$-0.392024\pi$
$743$	18.1401	0.665497	0.332748	+	0.943016i	$0.392024\pi$
$744$	0	0
$745$	20.5738	0.753766
$746$	31.7492	1.16242
$747$	0	0
$748$	8.51720	0.311420
$749$	4.71788	0.172387
$750$	0	0
$751$	−28.2267	−1.03001	−0.515003	−	0.857188i	$-0.672209\pi$
$751$	−28.2267	−1.03001	−0.515003	+	0.857188i	$0.672209\pi$
$752$	−4.02642	−0.146829
$753$	0	0
$754$	2.15494	0.0784783
$755$	−16.3091	−0.593549
$756$	0	0
$757$	−0.122014	−0.00443466	−0.00221733	−	0.999998i	$-0.500706\pi$
$757$	−0.122014	−0.00443466	−0.00221733	+	0.999998i	$0.500706\pi$
$758$	−12.4000	−0.450389
$759$	0	0
$760$	18.6210	0.675453
$761$	26.7054	0.968070	0.484035	−	0.875049i	$-0.339171\pi$
$761$	26.7054	0.968070	0.484035	+	0.875049i	$0.339171\pi$
$762$	0	0
$763$	−37.0031	−1.33960
$764$	−59.2446	−2.14340
$765$	0	0
$766$	−0.510490	−0.0184447
$767$	1.41833	0.0512127
$768$	0	0
$769$	7.85104	0.283116	0.141558	−	0.989930i	$-0.454789\pi$
$769$	7.85104	0.283116	0.141558	+	0.989930i	$0.454789\pi$
$770$	−12.3604	−0.445439
$771$	0	0
$772$	41.8457	1.50606
$773$	−34.2817	−1.23303	−0.616513	−	0.787345i	$-0.711455\pi$
$773$	−34.2817	−1.23303	−0.616513	+	0.787345i	$0.711455\pi$
$774$	0	0
$775$	−6.18789	−0.222276
$776$	8.15516	0.292753
$777$	0	0
$778$	66.4827	2.38352
$779$	−3.40534	−0.122009
$780$	0	0
$781$	−1.39619	−0.0499596
$782$	0	0
$783$	0	0
$784$	2.13094	0.0761049
$785$	5.10502	0.182206
$786$	0	0
$787$	−16.0125	−0.570785	−0.285392	−	0.958411i	$-0.592124\pi$
$787$	−16.0125	−0.570785	−0.285392	+	0.958411i	$0.592124\pi$
$788$	−33.3512	−1.18809
$789$	0	0
$790$	15.4712	0.550441
$791$	−3.53129	−0.125558
$792$	0	0
$793$	−1.45973	−0.0518367
$794$	28.7468	1.02018
$795$	0	0
$796$	16.3456	0.579355
$797$	21.5237	0.762409	0.381205	−	0.924491i	$-0.375509\pi$
$797$	21.5237	0.762409	0.381205	+	0.924491i	$0.375509\pi$
$798$	0	0
$799$	−4.03769	−0.142843
$800$	−24.9924	−0.883614
$801$	0	0
$802$	46.9938	1.65941
$803$	12.3206	0.434784
$804$	0	0
$805$	0	0
$806$	−0.771816	−0.0271860
$807$	0	0
$808$	−3.43327	−0.120782
$809$	−24.1251	−0.848194	−0.424097	−	0.905617i	$-0.639408\pi$
$809$	−24.1251	−0.848194	−0.424097	+	0.905617i	$0.639408\pi$
$810$	0	0
$811$	14.3383	0.503486	0.251743	−	0.967794i	$-0.418996\pi$
$811$	14.3383	0.503486	0.251743	+	0.967794i	$0.418996\pi$
$812$	−33.7287	−1.18365
$813$	0	0
$814$	−16.3599	−0.573415
$815$	18.2867	0.640554
$816$	0	0
$817$	−35.5561	−1.24395
$818$	20.1609	0.704910
$819$	0	0
$820$	1.50608	0.0525946
$821$	−14.2085	−0.495879	−0.247939	−	0.968776i	$-0.579753\pi$
$821$	−14.2085	−0.495879	−0.247939	+	0.968776i	$0.579753\pi$
$822$	0	0
$823$	16.0231	0.558530	0.279265	−	0.960214i	$-0.409909\pi$
$823$	16.0231	0.558530	0.279265	+	0.960214i	$0.409909\pi$
$824$	−28.8826	−1.00617
$825$	0	0
$826$	−37.7455	−1.31334
$827$	35.1240	1.22138	0.610690	−	0.791870i	$-0.290892\pi$
$827$	35.1240	1.22138	0.610690	+	0.791870i	$0.290892\pi$
$828$	0	0
$829$	−2.29604	−0.0797447	−0.0398723	−	0.999205i	$-0.512695\pi$
$829$	−2.29604	−0.0797447	−0.0398723	+	0.999205i	$0.512695\pi$
$830$	−35.1210	−1.21907
$831$	0	0
$832$	−2.50605	−0.0868815
$833$	2.13690	0.0740391
$834$	0	0
$835$	1.64420	0.0568999
$836$	−43.5958	−1.50779
$837$	0	0
$838$	19.7757	0.683140
$839$	38.6117	1.33302	0.666511	−	0.745495i	$-0.267787\pi$
$839$	38.6117	1.33302	0.666511	+	0.745495i	$0.267787\pi$
$840$	0	0
$841$	−4.22633	−0.145736
$842$	69.3528	2.39006
$843$	0	0
$844$	−40.1764	−1.38293
$845$	16.0265	0.551328
$846$	0	0
$847$	−17.4278	−0.598825
$848$	−15.2746	−0.524532
$849$	0	0
$850$	−11.9317	−0.409255
$851$	0	0
$852$	0	0
$853$	−38.4294	−1.31580	−0.657898	−	0.753107i	$-0.728554\pi$
$853$	−38.4294	−1.31580	−0.657898	+	0.753107i	$0.728554\pi$
$854$	38.8475	1.32933
$855$	0	0
$856$	−3.75023	−0.128180
$857$	50.8895	1.73835	0.869176	−	0.494504i	$-0.164650\pi$
$857$	50.8895	1.73835	0.869176	+	0.494504i	$0.164650\pi$
$858$	0	0
$859$	9.14711	0.312095	0.156048	−	0.987750i	$-0.450125\pi$
$859$	9.14711	0.312095	0.156048	+	0.987750i	$0.450125\pi$
$860$	15.7254	0.536232
$861$	0	0
$862$	48.7052	1.65891
$863$	−12.3389	−0.420019	−0.210010	−	0.977699i	$-0.567350\pi$
$863$	−12.3389	−0.420019	−0.210010	+	0.977699i	$0.567350\pi$
$864$	0	0
$865$	−20.9411	−0.712018
$866$	27.9754	0.950644
$867$	0	0
$868$	12.0803	0.410032
$869$	−10.8556	−0.368251
$870$	0	0
$871$	−1.42768	−0.0483750
$872$	29.4137	0.996073
$873$	0	0
$874$	0	0
$875$	24.8534	0.840199
$876$	0	0
$877$	2.30720	0.0779086	0.0389543	−	0.999241i	$-0.487597\pi$
$877$	2.30720	0.0779086	0.0389543	+	0.999241i	$0.487597\pi$
$878$	−40.5928	−1.36994
$879$	0	0
$880$	−3.67726	−0.123960
$881$	−35.6004	−1.19941	−0.599703	−	0.800222i	$-0.704715\pi$
$881$	−35.6004	−1.19941	−0.599703	+	0.800222i	$0.704715\pi$
$882$	0	0
$883$	30.4745	1.02555	0.512775	−	0.858523i	$-0.328618\pi$
$883$	30.4745	1.02555	0.512775	+	0.858523i	$0.328618\pi$
$884$	−0.875284	−0.0294390
$885$	0	0
$886$	−30.6584	−1.02999
$887$	−20.7912	−0.698099	−0.349049	−	0.937104i	$-0.613495\pi$
$887$	−20.7912	−0.698099	−0.349049	+	0.937104i	$0.613495\pi$
$888$	0	0
$889$	14.2233	0.477036
$890$	−37.8743	−1.26955
$891$	0	0
$892$	−12.0051	−0.401961
$893$	20.6671	0.691598
$894$	0	0
$895$	−1.76983	−0.0591590
$896$	32.5243	1.08656
$897$	0	0
$898$	66.3122	2.21287
$899$	−8.87296	−0.295930
$900$	0	0
$901$	−15.3173	−0.510294
$902$	−1.79681	−0.0598274
$903$	0	0
$904$	2.80701	0.0933599
$905$	−19.9461	−0.663030
$906$	0	0
$907$	22.7118	0.754133	0.377067	−	0.926186i	$-0.376933\pi$
$907$	22.7118	0.754133	0.377067	+	0.926186i	$0.376933\pi$
$908$	52.6368	1.74681
$909$	0	0
$910$	1.27024	0.0421080
$911$	0.834044	0.0276331	0.0138165	−	0.999905i	$-0.495602\pi$
$911$	0.834044	0.0276331	0.0138165	+	0.999905i	$0.495602\pi$
$912$	0	0
$913$	24.6432	0.815570
$914$	83.8422	2.77325
$915$	0	0
$916$	−31.2000	−1.03088
$917$	30.2977	1.00052
$918$	0	0
$919$	45.3320	1.49537	0.747683	−	0.664056i	$-0.231166\pi$
$919$	45.3320	1.49537	0.747683	+	0.664056i	$0.231166\pi$
$920$	0	0
$921$	0	0
$922$	19.3040	0.635744
$923$	0.143482	0.00472275
$924$	0	0
$925$	13.4791	0.443191
$926$	61.3414	2.01580
$927$	0	0
$928$	−35.8371	−1.17641
$929$	17.7072	0.580955	0.290477	−	0.956882i	$-0.406186\pi$
$929$	17.7072	0.580955	0.290477	+	0.956882i	$0.406186\pi$
$930$	0	0
$931$	−10.9378	−0.358472
$932$	−11.8944	−0.389615
$933$	0	0
$934$	38.2687	1.25219
$935$	−3.68755	−0.120596
$936$	0	0
$937$	23.3260	0.762028	0.381014	−	0.924569i	$-0.375575\pi$
$937$	23.3260	0.762028	0.381014	+	0.924569i	$0.375575\pi$
$938$	37.9944	1.24056
$939$	0	0
$940$	−9.14044	−0.298128
$941$	11.6148	0.378630	0.189315	−	0.981916i	$-0.439373\pi$
$941$	11.6148	0.378630	0.189315	+	0.981916i	$0.439373\pi$
$942$	0	0
$943$	0	0
$944$	−11.2294	−0.365486
$945$	0	0
$946$	−18.7611	−0.609975
$947$	−5.99794	−0.194907	−0.0974534	−	0.995240i	$-0.531070\pi$
$947$	−5.99794	−0.194907	−0.0974534	+	0.995240i	$0.531070\pi$
$948$	0	0
$949$	−1.26614	−0.0411008
$950$	61.0732	1.98148
$951$	0	0
$952$	6.98115	0.226260
$953$	−17.9067	−0.580054	−0.290027	−	0.957018i	$-0.593664\pi$
$953$	−17.9067	−0.580054	−0.290027	+	0.957018i	$0.593664\pi$
$954$	0	0
$955$	25.6501	0.830019
$956$	64.3072	2.07984
$957$	0	0
$958$	56.1432	1.81391
$959$	1.65516	0.0534479
$960$	0	0
$961$	−27.8221	−0.897486
$962$	1.68125	0.0542058
$963$	0	0
$964$	−16.9331	−0.545380
$965$	−18.1172	−0.583213
$966$	0	0
$967$	33.5068	1.07751	0.538753	−	0.842464i	$-0.318896\pi$
$967$	33.5068	1.07751	0.538753	+	0.842464i	$0.318896\pi$
$968$	13.8533	0.445262
$969$	0	0
$970$	−11.7811	−0.378268
$971$	59.2860	1.90258	0.951289	−	0.308300i	$-0.0997602\pi$
$971$	59.2860	1.90258	0.951289	+	0.308300i	$0.0997602\pi$
$972$	0	0
$973$	33.1144	1.06160
$974$	13.5766	0.435024
$975$	0	0
$976$	11.5572	0.369938
$977$	−12.6420	−0.404452	−0.202226	−	0.979339i	$-0.564818\pi$
$977$	−12.6420	−0.404452	−0.202226	+	0.979339i	$0.564818\pi$
$978$	0	0
$979$	26.5751	0.849343
$980$	4.83747	0.154527
$981$	0	0
$982$	−71.9428	−2.29579
$983$	−58.3650	−1.86156	−0.930778	−	0.365586i	$-0.880869\pi$
$983$	−58.3650	−1.86156	−0.930778	+	0.365586i	$0.880869\pi$
$984$	0	0
$985$	14.4395	0.460080
$986$	−17.1092	−0.544867
$987$	0	0
$988$	4.48019	0.142534
$989$	0	0
$990$	0	0
$991$	−45.7911	−1.45460	−0.727301	−	0.686319i	$-0.759225\pi$
$991$	−45.7911	−1.45460	−0.727301	+	0.686319i	$0.759225\pi$
$992$	12.8354	0.407526
$993$	0	0
$994$	−3.81844	−0.121114
$995$	−7.07687	−0.224352
$996$	0	0
$997$	−39.3521	−1.24629	−0.623147	−	0.782105i	$-0.714146\pi$
$997$	−39.3521	−1.24629	−0.623147	+	0.782105i	$0.714146\pi$
$998$	18.8931	0.598051
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bn.1.1		5
3.2	odd	2		529.2.a.j.1.5		5
12.11	even	2		8464.2.a.bt.1.5		5
23.7	odd	22		207.2.i.c.118.1		10
23.10	odd	22		207.2.i.c.100.1		10
23.22	odd	2		4761.2.a.bo.1.1		5
69.2	odd	22		529.2.c.f.487.1		10
69.5	even	22		529.2.c.i.255.1		10
69.8	odd	22		529.2.c.c.501.1		10
69.11	even	22		529.2.c.g.466.1		10
69.14	even	22		529.2.c.i.334.1		10
69.17	even	22		529.2.c.d.266.1		10
69.20	even	22		529.2.c.b.170.1		10
69.26	odd	22		529.2.c.c.170.1		10
69.29	odd	22		529.2.c.e.266.1		10
69.32	odd	22		529.2.c.h.334.1		10
69.35	odd	22		529.2.c.f.466.1		10
69.38	even	22		529.2.c.b.501.1		10
69.41	odd	22		529.2.c.h.255.1		10
69.44	even	22		529.2.c.g.487.1		10
69.50	odd	22		529.2.c.e.177.1		10
69.53	even	22		23.2.c.a.3.1	✓	10
69.56	even	22		23.2.c.a.8.1	yes	10
69.59	odd	22		529.2.c.a.399.1		10
69.62	odd	22		529.2.c.a.118.1		10
69.65	even	22		529.2.c.d.177.1		10
69.68	even	2		529.2.a.i.1.5		5
276.191	odd	22		368.2.m.c.49.1		10
276.263	odd	22		368.2.m.c.353.1		10
276.275	odd	2		8464.2.a.bs.1.5		5
345.53	odd	44		575.2.p.b.49.1		20
345.122	odd	44		575.2.p.b.49.2		20
345.194	even	22		575.2.k.b.376.1		10
345.263	odd	44		575.2.p.b.399.2		20
345.329	even	22		575.2.k.b.26.1		10
345.332	odd	44		575.2.p.b.399.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.c.a.3.1	✓	10	69.53	even	22
23.2.c.a.8.1	yes	10	69.56	even	22
207.2.i.c.100.1		10	23.10	odd	22
207.2.i.c.118.1		10	23.7	odd	22
368.2.m.c.49.1		10	276.191	odd	22
368.2.m.c.353.1		10	276.263	odd	22
529.2.a.i.1.5		5	69.68	even	2
529.2.a.j.1.5		5	3.2	odd	2
529.2.c.a.118.1		10	69.62	odd	22
529.2.c.a.399.1		10	69.59	odd	22
529.2.c.b.170.1		10	69.20	even	22
529.2.c.b.501.1		10	69.38	even	22
529.2.c.c.170.1		10	69.26	odd	22
529.2.c.c.501.1		10	69.8	odd	22
529.2.c.d.177.1		10	69.65	even	22
529.2.c.d.266.1		10	69.17	even	22
529.2.c.e.177.1		10	69.50	odd	22
529.2.c.e.266.1		10	69.29	odd	22
529.2.c.f.466.1		10	69.35	odd	22
529.2.c.f.487.1		10	69.2	odd	22
529.2.c.g.466.1		10	69.11	even	22
529.2.c.g.487.1		10	69.44	even	22
529.2.c.h.255.1		10	69.41	odd	22
529.2.c.h.334.1		10	69.32	odd	22
529.2.c.i.255.1		10	69.5	even	22
529.2.c.i.334.1		10	69.14	even	22
575.2.k.b.26.1		10	345.329	even	22
575.2.k.b.376.1		10	345.194	even	22
575.2.p.b.49.1		20	345.53	odd	44
575.2.p.b.49.2		20	345.122	odd	44
575.2.p.b.399.1		20	345.332	odd	44
575.2.p.b.399.2		20	345.263	odd	44
4761.2.a.bn.1.1		5	1.1	even	1	trivial
4761.2.a.bo.1.1		5	23.22	odd	2
8464.2.a.bs.1.5		5	276.275	odd	2
8464.2.a.bt.1.5		5	12.11	even	2

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4761.2.a.bn.1.1

Level	$4761$
Weight	$2$
Character	4761.1
Self dual	yes
Analytic conductor	$38.017$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$