LMFDB - Embedded newform 4851.2.a.bn.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4851,2,Mod(1,4851)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4851.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4851, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,4,-2,0,0,9,0,9,3,0,11,0,0,2,-3,0,11,14,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.7354300205$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49086 q^{2} +4.20440 q^{4} +2.20440 q^{5} +5.49086 q^{8} +5.49086 q^{10} +1.00000 q^{11} +3.28646 q^{13} +5.26819 q^{16} +1.49086 q^{17} +6.91794 q^{19} +9.26819 q^{20} +2.49086 q^{22} -6.49086 q^{23} -0.140614 q^{25} +8.18613 q^{26} +1.64975 q^{29} +2.35025 q^{31} +2.14061 q^{32} +3.71354 q^{34} -5.55465 q^{37} +17.2316 q^{38} +12.1041 q^{40} -11.2499 q^{41} +5.26819 q^{43} +4.20440 q^{44} -16.1679 q^{46} +1.49086 q^{47} -0.350250 q^{50} +13.8176 q^{52} -0.304735 q^{53} +2.20440 q^{55} +4.10930 q^{58} -12.6587 q^{59} +12.9817 q^{61} +5.85415 q^{62} -5.20440 q^{64} +7.24468 q^{65} -4.57292 q^{67} +6.26819 q^{68} -11.3267 q^{71} +8.56769 q^{73} -13.8359 q^{74} +29.0858 q^{76} +4.63148 q^{79} +11.6132 q^{80} -28.0220 q^{82} -1.93621 q^{83} +3.28646 q^{85} +13.1223 q^{86} +5.49086 q^{88} +3.20440 q^{89} -27.2902 q^{92} +3.71354 q^{94} +15.2499 q^{95} -1.85939 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8} + 9 q^{10} + 3 q^{11} + 11 q^{13} + 2 q^{16} - 3 q^{17} + 11 q^{19} + 14 q^{20} - 12 q^{23} + 3 q^{25} + q^{26} + 9 q^{29} + 3 q^{31} + 3 q^{32} + 10 q^{34} - 4 q^{37}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8 + 9 * q^10 + 3 * q^11 + 11 * q^13 + 2 * q^16 - 3 * q^17 + 11 * q^19 + 14 * q^20 - 12 * q^23 + 3 * q^25 + q^26 + 9 * q^29 + 3 * q^31 + 3 * q^32 + 10 * q^34 - 4 * q^37 + 8 * q^38 + 3 * q^40 - 5 * q^41 + 2 * q^43 + 4 * q^44 - 10 * q^46 - 3 * q^47 + 3 * q^50 + 7 * q^52 - 17 * q^53 - 2 * q^55 - 13 * q^58 + 8 * q^59 + 24 * q^61 + 13 * q^62 - 7 * q^64 - 15 * q^65 - 16 * q^67 + 5 * q^68 - 7 * q^71 + 20 * q^73 - 22 * q^74 + 39 * q^76 + 3 * q^79 + 9 * q^80 - 41 * q^82 - 11 * q^83 + 11 * q^85 + 21 * q^86 + 9 * q^88 + q^89 - 25 * q^92 + 10 * q^94 + 17 * q^95 - 9 * q^97

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 $

$2$	2.49086	1.76131	0.880653	−	0.473761i	$-0.157104\pi$
$2$	2.49086	1.76131	0.880653	+	0.473761i	$0.157104\pi$
$3$	0	0
$3$	0	0
$4$	4.20440	2.10220
$5$	2.20440	0.985838	0.492919	−	0.870075i	$-0.335930\pi$
$5$	2.20440	0.985838	0.492919	+	0.870075i	$0.335930\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	5.49086	1.94131
$9$	0	0
$10$	5.49086	1.73636
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.28646	0.911501	0.455750	−	0.890108i	$-0.349371\pi$
$13$	3.28646	0.911501	0.455750	+	0.890108i	$0.349371\pi$
$14$	0	0
$15$	0	0
$16$	5.26819	1.31705
$17$	1.49086	0.361588	0.180794	−	0.983521i	$-0.442133\pi$
$17$	1.49086	0.361588	0.180794	+	0.983521i	$0.442133\pi$
$18$	0	0
$19$	6.91794	1.58708	0.793542	−	0.608515i	$-0.208235\pi$
$19$	6.91794	1.58708	0.793542	+	0.608515i	$0.208235\pi$
$20$	9.26819	2.07243
$21$	0	0
$22$	2.49086	0.531054
$23$	−6.49086	−1.35344	−0.676719	−	0.736241i	$-0.736599\pi$
$23$	−6.49086	−1.35344	−0.676719	+	0.736241i	$0.736599\pi$
$24$	0	0
$25$	−0.140614	−0.0281228
$26$	8.18613	1.60543
$27$	0	0
$28$	0	0
$29$	1.64975	0.306351	0.153175	−	0.988199i	$-0.451050\pi$
$29$	1.64975	0.306351	0.153175	+	0.988199i	$0.451050\pi$
$30$	0	0
$31$	2.35025	0.422117	0.211059	−	0.977473i	$-0.432309\pi$
$31$	2.35025	0.422117	0.211059	+	0.977473i	$0.432309\pi$
$32$	2.14061	0.378411
$33$	0	0
$34$	3.71354	0.636867
$35$	0	0
$36$	0	0
$37$	−5.55465	−0.913179	−0.456590	−	0.889677i	$-0.650929\pi$
$37$	−5.55465	−0.913179	−0.456590	+	0.889677i	$0.650929\pi$
$38$	17.2316	2.79534
$39$	0	0
$40$	12.1041	1.91382
$41$	−11.2499	−1.75694	−0.878471	−	0.477796i	$-0.841436\pi$
$41$	−11.2499	−1.75694	−0.878471	+	0.477796i	$0.841436\pi$
$42$	0	0
$43$	5.26819	0.803391	0.401696	−	0.915773i	$-0.368421\pi$
$43$	5.26819	0.803391	0.401696	+	0.915773i	$0.368421\pi$
$44$	4.20440	0.633837
$45$	0	0
$46$	−16.1679	−2.38382
$47$	1.49086	0.217465	0.108732	−	0.994071i	$-0.465321\pi$
$47$	1.49086	0.217465	0.108732	+	0.994071i	$0.465321\pi$
$48$	0	0
$49$	0	0
$50$	−0.350250	−0.0495328
$51$	0	0
$52$	13.8176	1.91616
$53$	−0.304735	−0.0418585	−0.0209293	−	0.999781i	$-0.506662\pi$
$53$	−0.304735	−0.0418585	−0.0209293	+	0.999781i	$0.506662\pi$
$54$	0	0
$55$	2.20440	0.297241
$56$	0	0
$57$	0	0
$58$	4.10930	0.539578
$59$	−12.6587	−1.64802	−0.824012	−	0.566572i	$-0.808269\pi$
$59$	−12.6587	−1.64802	−0.824012	+	0.566572i	$0.808269\pi$
$60$	0	0
$61$	12.9817	1.66214	0.831070	−	0.556168i	$-0.187729\pi$
$61$	12.9817	1.66214	0.831070	+	0.556168i	$0.187729\pi$
$62$	5.85415	0.743478
$63$	0	0
$64$	−5.20440	−0.650550
$65$	7.24468	0.898592
$66$	0	0
$67$	−4.57292	−0.558672	−0.279336	−	0.960193i	$-0.590114\pi$
$67$	−4.57292	−0.558672	−0.279336	+	0.960193i	$0.590114\pi$
$68$	6.26819	0.760130
$69$	0	0
$70$	0	0
$71$	−11.3267	−1.34424	−0.672119	−	0.740444i	$-0.734615\pi$
$71$	−11.3267	−1.34424	−0.672119	+	0.740444i	$0.734615\pi$
$72$	0	0
$73$	8.56769	1.00277	0.501386	−	0.865224i	$-0.332824\pi$
$73$	8.56769	1.00277	0.501386	+	0.865224i	$0.332824\pi$
$74$	−13.8359	−1.60839
$75$	0	0
$76$	29.0858	3.33637
$77$	0	0
$78$	0	0
$79$	4.63148	0.521082	0.260541	−	0.965463i	$-0.416099\pi$
$79$	4.63148	0.521082	0.260541	+	0.965463i	$0.416099\pi$
$80$	11.6132	1.29840
$81$	0	0
$82$	−28.0220	−3.09451
$83$	−1.93621	−0.212527	−0.106263	−	0.994338i	$-0.533889\pi$
$83$	−1.93621	−0.212527	−0.106263	+	0.994338i	$0.533889\pi$
$84$	0	0
$85$	3.28646	0.356467
$86$	13.1223	1.41502
$87$	0	0
$88$	5.49086	0.585328
$89$	3.20440	0.339666	0.169833	−	0.985473i	$-0.445677\pi$
$89$	3.20440	0.339666	0.169833	+	0.985473i	$0.445677\pi$
$90$	0	0
$91$	0	0
$92$	−27.2902	−2.84520
$93$	0	0
$94$	3.71354	0.383022
$95$	15.2499	1.56461
$96$	0	0
$97$	−1.85939	−0.188792	−0.0943960	−	0.995535i	$-0.530092\pi$
$97$	−1.85939	−0.188792	−0.0943960	+	0.995535i	$0.530092\pi$
$98$	0	0
$99$	0	0
$100$	−0.591197	−0.0591197
$101$	6.04551	0.601551	0.300776	−	0.953695i	$-0.402754\pi$
$101$	6.04551	0.601551	0.300776	+	0.953695i	$0.402754\pi$
$102$	0	0
$103$	1.06379	0.104818	0.0524091	−	0.998626i	$-0.483310\pi$
$103$	1.06379	0.104818	0.0524091	+	0.998626i	$0.483310\pi$
$104$	18.0455	1.76951
$105$	0	0
$106$	−0.759053	−0.0737257
$107$	6.33198	0.612135	0.306068	−	0.952010i	$-0.400987\pi$
$107$	6.33198	0.612135	0.306068	+	0.952010i	$0.400987\pi$
$108$	0	0
$109$	−2.81387	−0.269520	−0.134760	−	0.990878i	$-0.543026\pi$
$109$	−2.81387	−0.269520	−0.134760	+	0.990878i	$0.543026\pi$
$110$	5.49086	0.523533
$111$	0	0
$112$	0	0
$113$	12.7538	1.19978	0.599889	−	0.800083i	$-0.295211\pi$
$113$	12.7538	1.19978	0.599889	+	0.800083i	$0.295211\pi$
$114$	0	0
$115$	−14.3085	−1.33427
$116$	6.93621	0.644011
$117$	0	0
$118$	−31.5311	−2.90268
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	32.3357	2.92754
$123$	0	0
$124$	9.88139	0.887375
$125$	−11.3320	−1.01356
$126$	0	0
$127$	−12.3775	−1.09832	−0.549162	−	0.835716i	$-0.685053\pi$
$127$	−12.3775	−1.09832	−0.549162	+	0.835716i	$0.685053\pi$
$128$	−17.2447	−1.52423
$129$	0	0
$130$	18.0455	1.58270
$131$	0.759053	0.0663188	0.0331594	−	0.999450i	$-0.489443\pi$
$131$	0.759053	0.0663188	0.0331594	+	0.999450i	$0.489443\pi$
$132$	0	0
$133$	0	0
$134$	−11.3905	−0.983992
$135$	0	0
$136$	8.18613	0.701955
$137$	5.84111	0.499040	0.249520	−	0.968370i	$-0.419727\pi$
$137$	5.84111	0.499040	0.249520	+	0.968370i	$0.419727\pi$
$138$	0	0
$139$	5.57292	0.472689	0.236345	−	0.971669i	$-0.424051\pi$
$139$	5.57292	0.472689	0.236345	+	0.971669i	$0.424051\pi$
$140$	0	0
$141$	0	0
$142$	−28.2134	−2.36761
$143$	3.28646	0.274828
$144$	0	0
$145$	3.63671	0.302012
$146$	21.3409	1.76619
$147$	0	0
$148$	−23.3540	−1.91969
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	14.8411	1.20775	0.603876	−	0.797078i	$-0.293622\pi$
$151$	14.8411	1.20775	0.603876	+	0.797078i	$0.293622\pi$
$152$	37.9855	3.08103
$153$	0	0
$154$	0	0
$155$	5.18089	0.416139
$156$	0	0
$157$	−6.39053	−0.510020	−0.255010	−	0.966938i	$-0.582079\pi$
$157$	−6.39053	−0.510020	−0.255010	+	0.966938i	$0.582079\pi$
$158$	11.5364	0.917785
$159$	0	0
$160$	4.71877	0.373052
$161$	0	0
$162$	0	0
$163$	9.94518	0.778967	0.389483	−	0.921033i	$-0.372654\pi$
$163$	9.94518	0.778967	0.389483	+	0.921033i	$0.372654\pi$
$164$	−47.2992	−3.69344
$165$	0	0
$166$	−4.82284	−0.374325
$167$	−1.94145	−0.150234	−0.0751168	−	0.997175i	$-0.523933\pi$
$167$	−1.94145	−0.150234	−0.0751168	+	0.997175i	$0.523933\pi$
$168$	0	0
$169$	−2.19917	−0.169167
$170$	8.18613	0.627847
$171$	0	0
$172$	22.1496	1.68889
$173$	6.43231	0.489039	0.244520	−	0.969644i	$-0.421370\pi$
$173$	6.43231	0.489039	0.244520	+	0.969644i	$0.421370\pi$
$174$	0	0
$175$	0	0
$176$	5.26819	0.397105
$177$	0	0
$178$	7.98173	0.598256
$179$	3.58596	0.268027	0.134014	−	0.990979i	$-0.457213\pi$
$179$	3.58596	0.268027	0.134014	+	0.990979i	$0.457213\pi$
$180$	0	0
$181$	−12.2134	−0.907813	−0.453906	−	0.891049i	$-0.649970\pi$
$181$	−12.2134	−0.907813	−0.453906	+	0.891049i	$0.649970\pi$
$182$	0	0
$183$	0	0
$184$	−35.6404	−2.62745
$185$	−12.2447	−0.900247
$186$	0	0
$187$	1.49086	0.109023
$188$	6.26819	0.457155
$189$	0	0
$190$	37.9855	2.75576
$191$	−12.1093	−0.876198	−0.438099	−	0.898927i	$-0.644348\pi$
$191$	−12.1093	−0.876198	−0.438099	+	0.898927i	$0.644348\pi$
$192$	0	0
$193$	−11.9399	−0.859456	−0.429728	−	0.902958i	$-0.641391\pi$
$193$	−11.9399	−0.859456	−0.429728	+	0.902958i	$0.641391\pi$
$194$	−4.63148	−0.332521
$195$	0	0
$196$	0	0
$197$	−12.1626	−0.866551	−0.433275	−	0.901262i	$-0.642642\pi$
$197$	−12.1626	−0.866551	−0.433275	+	0.901262i	$0.642642\pi$
$198$	0	0
$199$	−1.90490	−0.135035	−0.0675174	−	0.997718i	$-0.521508\pi$
$199$	−1.90490	−0.135035	−0.0675174	+	0.997718i	$0.521508\pi$
$200$	−0.772091	−0.0545951
$201$	0	0
$202$	15.0586	1.05952
$203$	0	0
$204$	0	0
$205$	−24.7993	−1.73206
$206$	2.64975	0.184617
$207$	0	0
$208$	17.3137	1.20049
$209$	6.91794	0.478524
$210$	0	0
$211$	−16.2447	−1.11833	−0.559165	−	0.829056i	$-0.688878\pi$
$211$	−16.2447	−1.11833	−0.559165	+	0.829056i	$0.688878\pi$
$212$	−1.28123	−0.0879951
$213$	0	0
$214$	15.7721	1.07816
$215$	11.6132	0.792014
$216$	0	0
$217$	0	0
$218$	−7.00897	−0.474707
$219$	0	0
$220$	9.26819	0.624861
$221$	4.89967	0.329587
$222$	0	0
$223$	−3.03655	−0.203342	−0.101671	−	0.994818i	$-0.532419\pi$
$223$	−3.03655	−0.203342	−0.101671	+	0.994818i	$0.532419\pi$
$224$	0	0
$225$	0	0
$226$	31.7680	2.11318
$227$	18.4491	1.22451	0.612254	−	0.790661i	$-0.290263\pi$
$227$	18.4491	1.22451	0.612254	+	0.790661i	$0.290263\pi$
$228$	0	0
$229$	25.4998	1.68508	0.842538	−	0.538637i	$-0.181060\pi$
$229$	25.4998	1.68508	0.842538	+	0.538637i	$0.181060\pi$
$230$	−35.6404	−2.35006
$231$	0	0
$232$	9.05855	0.594723
$233$	−3.81761	−0.250100	−0.125050	−	0.992150i	$-0.539909\pi$
$233$	−3.81761	−0.250100	−0.125050	+	0.992150i	$0.539909\pi$
$234$	0	0
$235$	3.28646	0.214385
$236$	−53.2223	−3.46448
$237$	0	0
$238$	0	0
$239$	−13.0037	−0.841142	−0.420571	−	0.907260i	$-0.638170\pi$
$239$	−13.0037	−0.841142	−0.420571	+	0.907260i	$0.638170\pi$
$240$	0	0
$241$	0.450583	0.0290246	0.0145123	−	0.999895i	$-0.495380\pi$
$241$	0.450583	0.0290246	0.0145123	+	0.999895i	$0.495380\pi$
$242$	2.49086	0.160119
$243$	0	0
$244$	54.5804	3.49415
$245$	0	0
$246$	0	0
$247$	22.7355	1.44663
$248$	12.9049	0.819462
$249$	0	0
$250$	−28.2264	−1.78519
$251$	1.11861	0.0706058	0.0353029	−	0.999377i	$-0.488760\pi$
$251$	1.11861	0.0706058	0.0353029	+	0.999377i	$0.488760\pi$
$252$	0	0
$253$	−6.49086	−0.408077
$254$	−30.8306	−1.93449
$255$	0	0
$256$	−32.5453	−2.03408
$257$	−22.8396	−1.42470	−0.712348	−	0.701826i	$-0.752368\pi$
$257$	−22.8396	−1.42470	−0.712348	+	0.701826i	$0.752368\pi$
$258$	0	0
$259$	0	0
$260$	30.4596	1.88902
$261$	0	0
$262$	1.89070	0.116808
$263$	9.19136	0.566764	0.283382	−	0.959007i	$-0.408544\pi$
$263$	9.19136	0.566764	0.283382	+	0.959007i	$0.408544\pi$
$264$	0	0
$265$	−0.671758	−0.0412658
$266$	0	0
$267$	0	0
$268$	−19.2264	−1.17444
$269$	−15.6132	−0.951954	−0.475977	−	0.879458i	$-0.657905\pi$
$269$	−15.6132	−0.951954	−0.475977	+	0.879458i	$0.657905\pi$
$270$	0	0
$271$	−27.7445	−1.68536	−0.842680	−	0.538415i	$-0.819023\pi$
$271$	−27.7445	−1.68536	−0.842680	+	0.538415i	$0.819023\pi$
$272$	7.85415	0.476228
$273$	0	0
$274$	14.5494	0.878962
$275$	−0.140614	−0.00847933
$276$	0	0
$277$	14.8124	0.889989	0.444995	−	0.895533i	$-0.353206\pi$
$277$	14.8124	0.889989	0.444995	+	0.895533i	$0.353206\pi$
$278$	13.8814	0.832551
$279$	0	0
$280$	0	0
$281$	−15.2227	−0.908109	−0.454054	−	0.890974i	$-0.650023\pi$
$281$	−15.2227	−0.908109	−0.454054	+	0.890974i	$0.650023\pi$
$282$	0	0
$283$	−21.7173	−1.29096	−0.645479	−	0.763778i	$-0.723342\pi$
$283$	−21.7173	−1.29096	−0.645479	+	0.763778i	$0.723342\pi$
$284$	−47.6222	−2.82586
$285$	0	0
$286$	8.18613	0.484056
$287$	0	0
$288$	0	0
$289$	−14.7773	−0.869254
$290$	9.05855	0.531937
$291$	0	0
$292$	36.0220	2.10803
$293$	11.1276	0.650080	0.325040	−	0.945700i	$-0.394622\pi$
$293$	11.1276	0.650080	0.325040	+	0.945700i	$0.394622\pi$
$294$	0	0
$295$	−27.9049	−1.62469
$296$	−30.4998	−1.77277
$297$	0	0
$298$	2.49086	0.144292
$299$	−21.3320	−1.23366
$300$	0	0
$301$	0	0
$302$	36.9672	2.12722
$303$	0	0
$304$	36.4450	2.09026
$305$	28.6169	1.63860
$306$	0	0
$307$	−24.9855	−1.42600	−0.712998	−	0.701166i	$-0.752663\pi$
$307$	−24.9855	−1.42600	−0.712998	+	0.701166i	$0.752663\pi$
$308$	0	0
$309$	0	0
$310$	12.9049	0.732949
$311$	34.7408	1.96997	0.984984	−	0.172643i	$-0.0552307\pi$
$311$	34.7408	1.96997	0.984984	+	0.172643i	$0.0552307\pi$
$312$	0	0
$313$	−23.7095	−1.34014	−0.670069	−	0.742299i	$-0.733736\pi$
$313$	−23.7095	−1.34014	−0.670069	+	0.742299i	$0.733736\pi$
$314$	−15.9179	−0.898301
$315$	0	0
$316$	19.4726	1.09542
$317$	−19.6770	−1.10517	−0.552585	−	0.833457i	$-0.686359\pi$
$317$	−19.6770	−1.10517	−0.552585	+	0.833457i	$0.686359\pi$
$318$	0	0
$319$	1.64975	0.0923683
$320$	−11.4726	−0.641337
$321$	0	0
$322$	0	0
$323$	10.3137	0.573870
$324$	0	0
$325$	−0.462122	−0.0256339
$326$	24.7721	1.37200
$327$	0	0
$328$	−61.7718	−3.41077
$329$	0	0
$330$	0	0
$331$	−28.1899	−1.54946	−0.774728	−	0.632295i	$-0.782113\pi$
$331$	−28.1899	−1.54946	−0.774728	+	0.632295i	$0.782113\pi$
$332$	−8.14061	−0.446774
$333$	0	0
$334$	−4.83588	−0.264608
$335$	−10.0806	−0.550760
$336$	0	0
$337$	21.7460	1.18458	0.592290	−	0.805725i	$-0.298224\pi$
$337$	21.7460	1.18458	0.592290	+	0.805725i	$0.298224\pi$
$338$	−5.47783	−0.297954
$339$	0	0
$340$	13.8176	0.749365
$341$	2.35025	0.127273
$342$	0	0
$343$	0	0
$344$	28.9269	1.55963
$345$	0	0
$346$	16.0220	0.861348
$347$	3.94145	0.211588	0.105794	−	0.994388i	$-0.466262\pi$
$347$	3.94145	0.211588	0.105794	+	0.994388i	$0.466262\pi$
$348$	0	0
$349$	14.1093	0.755254	0.377627	−	0.925958i	$-0.376740\pi$
$349$	14.1093	0.755254	0.377627	+	0.925958i	$0.376740\pi$
$350$	0	0
$351$	0	0
$352$	2.14061	0.114095
$353$	−4.96869	−0.264457	−0.132228	−	0.991219i	$-0.542213\pi$
$353$	−4.96869	−0.264457	−0.132228	+	0.991219i	$0.542213\pi$
$354$	0	0
$355$	−24.9687	−1.32520
$356$	13.4726	0.714046
$357$	0	0
$358$	8.93214	0.472078
$359$	4.07159	0.214890	0.107445	−	0.994211i	$-0.465733\pi$
$359$	4.07159	0.214890	0.107445	+	0.994211i	$0.465733\pi$
$360$	0	0
$361$	28.8579	1.51884
$362$	−30.4218	−1.59894
$363$	0	0
$364$	0	0
$365$	18.8866	0.988571
$366$	0	0
$367$	18.0220	0.940741	0.470371	−	0.882469i	$-0.344120\pi$
$367$	18.0220	0.940741	0.470371	+	0.882469i	$0.344120\pi$
$368$	−34.1951	−1.78254
$369$	0	0
$370$	−30.4998	−1.58561
$371$	0	0
$372$	0	0
$373$	28.9164	1.49724	0.748618	−	0.663001i	$-0.230718\pi$
$373$	28.9164	1.49724	0.748618	+	0.663001i	$0.230718\pi$
$374$	3.71354	0.192022
$375$	0	0
$376$	8.18613	0.422167
$377$	5.42184	0.279239
$378$	0	0
$379$	4.30847	0.221311	0.110656	−	0.993859i	$-0.464705\pi$
$379$	4.30847	0.221311	0.110656	+	0.993859i	$0.464705\pi$
$380$	64.1168	3.28912
$381$	0	0
$382$	−30.1626	−1.54325
$383$	−12.3488	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$383$	−12.3488	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$384$	0	0
$385$	0	0
$386$	−29.7408	−1.51377
$387$	0	0
$388$	−7.81761	−0.396879
$389$	−29.4230	−1.49181	−0.745903	−	0.666055i	$-0.767982\pi$
$389$	−29.4230	−1.49181	−0.745903	+	0.666055i	$0.767982\pi$
$390$	0	0
$391$	−9.67699	−0.489387
$392$	0	0
$393$	0	0
$394$	−30.2954	−1.52626
$395$	10.2096	0.513703
$396$	0	0
$397$	17.2995	0.868237	0.434119	−	0.900856i	$-0.357060\pi$
$397$	17.2995	0.868237	0.434119	+	0.900856i	$0.357060\pi$
$398$	−4.74485	−0.237838
$399$	0	0
$400$	−0.740780	−0.0370390
$401$	24.9504	1.24596	0.622982	−	0.782236i	$-0.285921\pi$
$401$	24.9504	1.24596	0.622982	+	0.782236i	$0.285921\pi$
$402$	0	0
$403$	7.72401	0.384760
$404$	25.4178	1.26458
$405$	0	0
$406$	0	0
$407$	−5.55465	−0.275334
$408$	0	0
$409$	38.5584	1.90659	0.953295	−	0.302042i	$-0.0976682\pi$
$409$	38.5584	1.90659	0.953295	+	0.302042i	$0.0976682\pi$
$410$	−61.7718	−3.05069
$411$	0	0
$412$	4.47259	0.220349
$413$	0	0
$414$	0	0
$415$	−4.26819	−0.209517
$416$	7.03505	0.344922
$417$	0	0
$418$	17.2316	0.842827
$419$	0.908970	0.0444061	0.0222030	−	0.999753i	$-0.492932\pi$
$419$	0.908970	0.0444061	0.0222030	+	0.999753i	$0.492932\pi$
$420$	0	0
$421$	15.5532	0.758014	0.379007	−	0.925394i	$-0.376266\pi$
$421$	15.5532	0.758014	0.379007	+	0.925394i	$0.376266\pi$
$422$	−40.4633	−1.96972
$423$	0	0
$424$	−1.67326	−0.0812606
$425$	−0.209636	−0.0101688
$426$	0	0
$427$	0	0
$428$	26.6222	1.28683
$429$	0	0
$430$	28.9269	1.39498
$431$	−3.53638	−0.170341	−0.0851707	−	0.996366i	$-0.527144\pi$
$431$	−3.53638	−0.170341	−0.0851707	+	0.996366i	$0.527144\pi$
$432$	0	0
$433$	−17.6457	−0.847997	−0.423999	−	0.905663i	$-0.639374\pi$
$433$	−17.6457	−0.847997	−0.423999	+	0.905663i	$0.639374\pi$
$434$	0	0
$435$	0	0
$436$	−11.8306	−0.566585
$437$	−44.9034	−2.14802
$438$	0	0
$439$	−15.0272	−0.717211	−0.358606	−	0.933489i	$-0.616748\pi$
$439$	−15.0272	−0.717211	−0.358606	+	0.933489i	$0.616748\pi$
$440$	12.1041	0.577039
$441$	0	0
$442$	12.2044	0.580504
$443$	−13.2227	−0.628228	−0.314114	−	0.949385i	$-0.601707\pi$
$443$	−13.2227	−0.628228	−0.314114	+	0.949385i	$0.601707\pi$
$444$	0	0
$445$	7.06379	0.334856
$446$	−7.56362	−0.358148
$447$	0	0
$448$	0	0
$449$	9.90864	0.467617	0.233809	−	0.972283i	$-0.424881\pi$
$449$	9.90864	0.467617	0.233809	+	0.972283i	$0.424881\pi$
$450$	0	0
$451$	−11.2499	−0.529738
$452$	53.6222	2.52217
$453$	0	0
$454$	45.9542	2.15674
$455$	0	0
$456$	0	0
$457$	10.3450	0.483919	0.241960	−	0.970286i	$-0.422210\pi$
$457$	10.3450	0.483919	0.241960	+	0.970286i	$0.422210\pi$
$458$	63.5166	2.96794
$459$	0	0
$460$	−60.1586	−2.80491
$461$	−15.3372	−0.714325	−0.357163	−	0.934042i	$-0.616256\pi$
$461$	−15.3372	−0.714325	−0.357163	+	0.934042i	$0.616256\pi$
$462$	0	0
$463$	−25.1313	−1.16795	−0.583976	−	0.811771i	$-0.698504\pi$
$463$	−25.1313	−1.16795	−0.583976	+	0.811771i	$0.698504\pi$
$464$	8.69120	0.403479
$465$	0	0
$466$	−9.50914	−0.440502
$467$	−0.746015	−0.0345214	−0.0172607	−	0.999851i	$-0.505495\pi$
$467$	−0.746015	−0.0345214	−0.0172607	+	0.999851i	$0.505495\pi$
$468$	0	0
$469$	0	0
$470$	8.18613	0.377598
$471$	0	0
$472$	−69.5073	−3.19933
$473$	5.26819	0.242232
$474$	0	0
$475$	−0.972758	−0.0446332
$476$	0	0
$477$	0	0
$478$	−32.3905	−1.48151
$479$	−20.1130	−0.918988	−0.459494	−	0.888181i	$-0.651969\pi$
$479$	−20.1130	−0.918988	−0.459494	+	0.888181i	$0.651969\pi$
$480$	0	0
$481$	−18.2552	−0.832363
$482$	1.12234	0.0511212
$483$	0	0
$484$	4.20440	0.191109
$485$	−4.09883	−0.186118
$486$	0	0
$487$	−4.24992	−0.192582	−0.0962911	−	0.995353i	$-0.530698\pi$
$487$	−4.24992	−0.192582	−0.0962911	+	0.995353i	$0.530698\pi$
$488$	71.2809	3.22673
$489$	0	0
$490$	0	0
$491$	−30.3279	−1.36868	−0.684340	−	0.729163i	$-0.739909\pi$
$491$	−30.3279	−1.36868	−0.684340	+	0.729163i	$0.739909\pi$
$492$	0	0
$493$	2.45955	0.110773
$494$	56.6311	2.54796
$495$	0	0
$496$	12.3816	0.555948
$497$	0	0
$498$	0	0
$499$	14.4413	0.646480	0.323240	−	0.946317i	$-0.395228\pi$
$499$	14.4413	0.646480	0.323240	+	0.946317i	$0.395228\pi$
$500$	−47.6442	−2.13071
$501$	0	0
$502$	2.78630	0.124358
$503$	−2.95822	−0.131901	−0.0659503	−	0.997823i	$-0.521008\pi$
$503$	−2.95822	−0.131901	−0.0659503	+	0.997823i	$0.521008\pi$
$504$	0	0
$505$	13.3267	0.593032
$506$	−16.1679	−0.718749
$507$	0	0
$508$	−52.0399	−2.30890
$509$	−24.9907	−1.10769	−0.553847	−	0.832619i	$-0.686841\pi$
$509$	−24.9907	−1.10769	−0.553847	+	0.832619i	$0.686841\pi$
$510$	0	0
$511$	0	0
$512$	−46.5767	−2.05842
$513$	0	0
$514$	−56.8904	−2.50933
$515$	2.34502	0.103334
$516$	0	0
$517$	1.49086	0.0655681
$518$	0	0
$519$	0	0
$520$	39.7796	1.74445
$521$	28.8631	1.26452	0.632258	−	0.774758i	$-0.282128\pi$
$521$	28.8631	1.26452	0.632258	+	0.774758i	$0.282128\pi$
$522$	0	0
$523$	−0.472591	−0.0206650	−0.0103325	−	0.999947i	$-0.503289\pi$
$523$	−0.472591	−0.0206650	−0.0103325	+	0.999947i	$0.503289\pi$
$524$	3.19136	0.139415
$525$	0	0
$526$	22.8944	0.998245
$527$	3.50390	0.152632
$528$	0	0
$529$	19.1313	0.831796
$530$	−1.67326	−0.0726817
$531$	0	0
$532$	0	0
$533$	−36.9724	−1.60145
$534$	0	0
$535$	13.9582	0.603466
$536$	−25.1093	−1.08456
$537$	0	0
$538$	−38.8904	−1.67668
$539$	0	0
$540$	0	0
$541$	15.4596	0.664658	0.332329	−	0.943164i	$-0.392166\pi$
$541$	15.4596	0.664658	0.332329	+	0.943164i	$0.392166\pi$
$542$	−69.1078	−2.96843
$543$	0	0
$544$	3.19136	0.136829
$545$	−6.20290	−0.265703
$546$	0	0
$547$	−35.0440	−1.49837	−0.749187	−	0.662359i	$-0.769556\pi$
$547$	−35.0440	−1.49837	−0.749187	+	0.662359i	$0.769556\pi$
$548$	24.5584	1.04908
$549$	0	0
$550$	−0.350250	−0.0149347
$551$	11.4129	0.486205
$552$	0	0
$553$	0	0
$554$	36.8956	1.56754
$555$	0	0
$556$	23.4308	0.993688
$557$	−10.5326	−0.446282	−0.223141	−	0.974786i	$-0.571631\pi$
$557$	−10.5326	−0.446282	−0.223141	+	0.974786i	$0.571631\pi$
$558$	0	0
$559$	17.3137	0.732292
$560$	0	0
$561$	0	0
$562$	−37.9176	−1.59946
$563$	−30.7147	−1.29447	−0.647235	−	0.762290i	$-0.724075\pi$
$563$	−30.7147	−1.29447	−0.647235	+	0.762290i	$0.724075\pi$
$564$	0	0
$565$	28.1145	1.18279
$566$	−54.0948	−2.27377
$567$	0	0
$568$	−62.1936	−2.60959
$569$	30.7721	1.29003	0.645017	−	0.764169i	$-0.276850\pi$
$569$	30.7721	1.29003	0.645017	+	0.764169i	$0.276850\pi$
$570$	0	0
$571$	−7.51694	−0.314574	−0.157287	−	0.987553i	$-0.550275\pi$
$571$	−7.51694	−0.314574	−0.157287	+	0.987553i	$0.550275\pi$
$572$	13.8176	0.577743
$573$	0	0
$574$	0	0
$575$	0.912705	0.0380624
$576$	0	0
$577$	−27.6184	−1.14977	−0.574885	−	0.818234i	$-0.694953\pi$
$577$	−27.6184	−1.14977	−0.574885	+	0.818234i	$0.694953\pi$
$578$	−36.8083	−1.53102
$579$	0	0
$580$	15.2902	0.634891
$581$	0	0
$582$	0	0
$583$	−0.304735	−0.0126208
$584$	47.0440	1.94670
$585$	0	0
$586$	27.7173	1.14499
$587$	−29.9582	−1.23651	−0.618254	−	0.785978i	$-0.712160\pi$
$587$	−29.9582	−1.23651	−0.618254	+	0.785978i	$0.712160\pi$
$588$	0	0
$589$	16.2589	0.669936
$590$	−69.5073	−2.86157
$591$	0	0
$592$	−29.2630	−1.20270
$593$	12.6680	0.520213	0.260107	−	0.965580i	$-0.416242\pi$
$593$	12.6680	0.520213	0.260107	+	0.965580i	$0.416242\pi$
$594$	0	0
$595$	0	0
$596$	4.20440	0.172219
$597$	0	0
$598$	−53.1350	−2.17285
$599$	−1.29427	−0.0528823	−0.0264411	−	0.999650i	$-0.508417\pi$
$599$	−1.29427	−0.0528823	−0.0264411	+	0.999650i	$0.508417\pi$
$600$	0	0
$601$	41.0220	1.67332	0.836661	−	0.547721i	$-0.184504\pi$
$601$	41.0220	1.67332	0.836661	+	0.547721i	$0.184504\pi$
$602$	0	0
$603$	0	0
$604$	62.3980	2.53894
$605$	2.20440	0.0896217
$606$	0	0
$607$	21.1768	0.859541	0.429770	−	0.902938i	$-0.358594\pi$
$607$	21.1768	0.859541	0.429770	+	0.902938i	$0.358594\pi$
$608$	14.8086	0.600570
$609$	0	0
$610$	71.2809	2.88608
$611$	4.89967	0.198219
$612$	0	0
$613$	6.76312	0.273160	0.136580	−	0.990629i	$-0.456389\pi$
$613$	6.76312	0.273160	0.136580	+	0.990629i	$0.456389\pi$
$614$	−62.2354	−2.51162
$615$	0	0
$616$	0	0
$617$	41.0728	1.65353	0.826763	−	0.562550i	$-0.190180\pi$
$617$	41.0728	1.65353	0.826763	+	0.562550i	$0.190180\pi$
$618$	0	0
$619$	30.4267	1.22295	0.611477	−	0.791262i	$-0.290576\pi$
$619$	30.4267	1.22295	0.611477	+	0.791262i	$0.290576\pi$
$620$	21.7826	0.874809
$621$	0	0
$622$	86.5345	3.46972
$623$	0	0
$624$	0	0
$625$	−24.2772	−0.971086
$626$	−59.0571	−2.36039
$627$	0	0
$628$	−26.8684	−1.07216
$629$	−8.28123	−0.330194
$630$	0	0
$631$	12.3670	0.492323	0.246162	−	0.969229i	$-0.420831\pi$
$631$	12.3670	0.492323	0.246162	+	0.969229i	$0.420831\pi$
$632$	25.4308	1.01158
$633$	0	0
$634$	−49.0127	−1.94654
$635$	−27.2850	−1.08277
$636$	0	0
$637$	0	0
$638$	4.10930	0.162689
$639$	0	0
$640$	−38.0142	−1.50264
$641$	47.1313	1.86157	0.930787	−	0.365561i	$-0.119123\pi$
$641$	47.1313	1.86157	0.930787	+	0.365561i	$0.119123\pi$
$642$	0	0
$643$	−1.87242	−0.0738412	−0.0369206	−	0.999318i	$-0.511755\pi$
$643$	−1.87242	−0.0738412	−0.0369206	+	0.999318i	$0.511755\pi$
$644$	0	0
$645$	0	0
$646$	25.6900	1.01076
$647$	−1.83588	−0.0721758	−0.0360879	−	0.999349i	$-0.511490\pi$
$647$	−1.83588	−0.0721758	−0.0360879	+	0.999349i	$0.511490\pi$
$648$	0	0
$649$	−12.6587	−0.496898
$650$	−1.15108	−0.0451492
$651$	0	0
$652$	41.8135	1.63754
$653$	18.1772	0.711327	0.355664	−	0.934614i	$-0.384255\pi$
$653$	18.1772	0.711327	0.355664	+	0.934614i	$0.384255\pi$
$654$	0	0
$655$	1.67326	0.0653796
$656$	−59.2667	−2.31398
$657$	0	0
$658$	0	0
$659$	16.8997	0.658318	0.329159	−	0.944275i	$-0.393235\pi$
$659$	16.8997	0.658318	0.329159	+	0.944275i	$0.393235\pi$
$660$	0	0
$661$	45.3032	1.76209	0.881046	−	0.473031i	$-0.156840\pi$
$661$	45.3032	1.76209	0.881046	+	0.473031i	$0.156840\pi$
$662$	−70.2171	−2.72907
$663$	0	0
$664$	−10.6315	−0.412581
$665$	0	0
$666$	0	0
$667$	−10.7083	−0.414627
$668$	−8.16262	−0.315821
$669$	0	0
$670$	−25.1093	−0.970057
$671$	12.9817	0.501154
$672$	0	0
$673$	39.5076	1.52291	0.761454	−	0.648219i	$-0.224486\pi$
$673$	39.5076	1.52291	0.761454	+	0.648219i	$0.224486\pi$
$674$	54.1664	2.08641
$675$	0	0
$676$	−9.24618	−0.355622
$677$	−34.1339	−1.31187	−0.655936	−	0.754817i	$-0.727726\pi$
$677$	−34.1339	−1.31187	−0.655936	+	0.754817i	$0.727726\pi$
$678$	0	0
$679$	0	0
$680$	18.0455	0.692014
$681$	0	0
$682$	5.85415	0.224167
$683$	−23.7863	−0.910157	−0.455079	−	0.890451i	$-0.650389\pi$
$683$	−23.7863	−0.910157	−0.455079	+	0.890451i	$0.650389\pi$
$684$	0	0
$685$	12.8762	0.491973
$686$	0	0
$687$	0	0
$688$	27.7538	1.05810
$689$	−1.00150	−0.0381541
$690$	0	0
$691$	20.5625	0.782233	0.391116	−	0.920341i	$-0.372089\pi$
$691$	20.5625	0.782233	0.391116	+	0.920341i	$0.372089\pi$
$692$	27.0440	1.02806
$693$	0	0
$694$	9.81761	0.372671
$695$	12.2850	0.465995
$696$	0	0
$697$	−16.7721	−0.635288
$698$	35.1443	1.33023
$699$	0	0
$700$	0	0
$701$	23.9907	0.906116	0.453058	−	0.891481i	$-0.350333\pi$
$701$	23.9907	0.906116	0.453058	+	0.891481i	$0.350333\pi$
$702$	0	0
$703$	−38.4267	−1.44929
$704$	−5.20440	−0.196148
$705$	0	0
$706$	−12.3763	−0.465789
$707$	0	0
$708$	0	0
$709$	−51.7117	−1.94207	−0.971037	−	0.238930i	$-0.923204\pi$
$709$	−51.7117	−1.94207	−0.971037	+	0.238930i	$0.923204\pi$
$710$	−62.1936	−2.33408
$711$	0	0
$712$	17.5949	0.659398
$713$	−15.2552	−0.571310
$714$	0	0
$715$	7.24468	0.270936
$716$	15.0768	0.563447
$717$	0	0
$718$	10.1418	0.378488
$719$	25.7851	0.961623	0.480812	−	0.876824i	$-0.340342\pi$
$719$	25.7851	0.961623	0.480812	+	0.876824i	$0.340342\pi$
$720$	0	0
$721$	0	0
$722$	71.8811	2.67514
$723$	0	0
$724$	−51.3499	−1.90840
$725$	−0.231978	−0.00861543
$726$	0	0
$727$	−32.7330	−1.21400	−0.606999	−	0.794702i	$-0.707627\pi$
$727$	−32.7330	−1.21400	−0.606999	+	0.794702i	$0.707627\pi$
$728$	0	0
$729$	0	0
$730$	47.0440	1.74118
$731$	7.85415	0.290496
$732$	0	0
$733$	9.28273	0.342865	0.171433	−	0.985196i	$-0.445160\pi$
$733$	9.28273	0.342865	0.171433	+	0.985196i	$0.445160\pi$
$734$	44.8904	1.65693
$735$	0	0
$736$	−13.8944	−0.512156
$737$	−4.57292	−0.168446
$738$	0	0
$739$	−50.0933	−1.84271	−0.921355	−	0.388722i	$-0.872917\pi$
$739$	−50.0933	−1.84271	−0.921355	+	0.388722i	$0.872917\pi$
$740$	−51.4816	−1.89250
$741$	0	0
$742$	0	0
$743$	−13.1679	−0.483082	−0.241541	−	0.970391i	$-0.577653\pi$
$743$	−13.1679	−0.483082	−0.241541	+	0.970391i	$0.577653\pi$
$744$	0	0
$745$	2.20440	0.0807630
$746$	72.0269	2.63709
$747$	0	0
$748$	6.26819	0.229188
$749$	0	0
$750$	0	0
$751$	41.3137	1.50756	0.753779	−	0.657128i	$-0.228229\pi$
$751$	41.3137	1.50756	0.753779	+	0.657128i	$0.228229\pi$
$752$	7.85415	0.286411
$753$	0	0
$754$	13.5051	0.491826
$755$	32.7158	1.19065
$756$	0	0
$757$	−45.5114	−1.65414	−0.827069	−	0.562100i	$-0.809994\pi$
$757$	−45.5114	−1.65414	−0.827069	+	0.562100i	$0.809994\pi$
$758$	10.7318	0.389797
$759$	0	0
$760$	83.7352	3.03740
$761$	31.2719	1.13361	0.566803	−	0.823853i	$-0.308180\pi$
$761$	31.2719	1.13361	0.566803	+	0.823853i	$0.308180\pi$
$762$	0	0
$763$	0	0
$764$	−50.9124	−1.84194
$765$	0	0
$766$	−30.7591	−1.11137
$767$	−41.6024	−1.50218
$768$	0	0
$769$	−42.0467	−1.51624	−0.758121	−	0.652114i	$-0.773882\pi$
$769$	−42.0467	−1.51624	−0.758121	+	0.652114i	$0.773882\pi$
$770$	0	0
$771$	0	0
$772$	−50.2003	−1.80675
$773$	7.64452	0.274954	0.137477	−	0.990505i	$-0.456101\pi$
$773$	7.64452	0.274954	0.137477	+	0.990505i	$0.456101\pi$
$774$	0	0
$775$	−0.330478	−0.0118711
$776$	−10.2096	−0.366505
$777$	0	0
$778$	−73.2887	−2.62753
$779$	−77.8262	−2.78841
$780$	0	0
$781$	−11.3267	−0.405303
$782$	−24.1041	−0.861960
$783$	0	0
$784$	0	0
$785$	−14.0873	−0.502797
$786$	0	0
$787$	27.9034	0.994649	0.497324	−	0.867565i	$-0.334316\pi$
$787$	27.9034	0.994649	0.497324	+	0.867565i	$0.334316\pi$
$788$	−51.1365	−1.82166
$789$	0	0
$790$	25.4308	0.904788
$791$	0	0
$792$	0	0
$793$	42.6640	1.51504
$794$	43.0907	1.52923
$795$	0	0
$796$	−8.00897	−0.283870
$797$	10.0560	0.356201	0.178101	−	0.984012i	$-0.443005\pi$
$797$	10.0560	0.356201	0.178101	+	0.984012i	$0.443005\pi$
$798$	0	0
$799$	2.22267	0.0786326
$800$	−0.301000	−0.0106420
$801$	0	0
$802$	62.1481	2.19453
$803$	8.56769	0.302347
$804$	0	0
$805$	0	0
$806$	19.2394	0.677681
$807$	0	0
$808$	33.1951	1.16780
$809$	−38.5726	−1.35614	−0.678070	−	0.734997i	$-0.737183\pi$
$809$	−38.5726	−1.35614	−0.678070	+	0.734997i	$0.737183\pi$
$810$	0	0
$811$	−12.3760	−0.434580	−0.217290	−	0.976107i	$-0.569722\pi$
$811$	−12.3760	−0.434580	−0.217290	+	0.976107i	$0.569722\pi$
$812$	0	0
$813$	0	0
$814$	−13.8359	−0.484947
$815$	21.9232	0.767935
$816$	0	0
$817$	36.4450	1.27505
$818$	96.0437	3.35809
$819$	0	0
$820$	−104.266	−3.64114
$821$	−29.7381	−1.03787	−0.518934	−	0.854814i	$-0.673671\pi$
$821$	−29.7381	−1.03787	−0.518934	+	0.854814i	$0.673671\pi$
$822$	0	0
$823$	40.7770	1.42140	0.710698	−	0.703497i	$-0.248379\pi$
$823$	40.7770	1.42140	0.710698	+	0.703497i	$0.248379\pi$
$824$	5.84111	0.203485
$825$	0	0
$826$	0	0
$827$	18.8411	0.655170	0.327585	−	0.944822i	$-0.393765\pi$
$827$	18.8411	0.655170	0.327585	+	0.944822i	$0.393765\pi$
$828$	0	0
$829$	31.1406	1.08156	0.540779	−	0.841165i	$-0.318129\pi$
$829$	31.1406	1.08156	0.540779	+	0.841165i	$0.318129\pi$
$830$	−10.6315	−0.369024
$831$	0	0
$832$	−17.1041	−0.592977
$833$	0	0
$834$	0	0
$835$	−4.27973	−0.148106
$836$	29.0858	1.00595
$837$	0	0
$838$	2.26412	0.0782127
$839$	10.2589	0.354176	0.177088	−	0.984195i	$-0.443332\pi$
$839$	10.2589	0.354176	0.177088	+	0.984195i	$0.443332\pi$
$840$	0	0
$841$	−26.2783	−0.906149
$842$	38.7408	1.33510
$843$	0	0
$844$	−68.2992	−2.35095
$845$	−4.84785	−0.166771
$846$	0	0
$847$	0	0
$848$	−1.60540	−0.0551297
$849$	0	0
$850$	−0.522175	−0.0179104
$851$	36.0545	1.23593
$852$	0	0
$853$	−3.04435	−0.104237	−0.0521183	−	0.998641i	$-0.516597\pi$
$853$	−3.04435	−0.104237	−0.0521183	+	0.998641i	$0.516597\pi$
$854$	0	0
$855$	0	0
$856$	34.7680	1.18835
$857$	49.5532	1.69270	0.846352	−	0.532624i	$-0.178794\pi$
$857$	49.5532	1.69270	0.846352	+	0.532624i	$0.178794\pi$
$858$	0	0
$859$	−36.8747	−1.25815	−0.629074	−	0.777346i	$-0.716566\pi$
$859$	−36.8747	−1.25815	−0.629074	+	0.777346i	$0.716566\pi$
$860$	48.8266	1.66497
$861$	0	0
$862$	−8.80864	−0.300023
$863$	46.4230	1.58026	0.790129	−	0.612941i	$-0.210014\pi$
$863$	46.4230	1.58026	0.790129	+	0.612941i	$0.210014\pi$
$864$	0	0
$865$	14.1794	0.482114
$866$	−43.9530	−1.49358
$867$	0	0
$868$	0	0
$869$	4.63148	0.157112
$870$	0	0
$871$	−15.0287	−0.509229
$872$	−15.4506	−0.523223
$873$	0	0
$874$	−111.848	−3.78332
$875$	0	0
$876$	0	0
$877$	39.0310	1.31798	0.658991	−	0.752151i	$-0.270983\pi$
$877$	39.0310	1.31798	0.658991	+	0.752151i	$0.270983\pi$
$878$	−37.4308	−1.26323
$879$	0	0
$880$	11.6132	0.391481
$881$	−44.0049	−1.48256	−0.741281	−	0.671194i	$-0.765782\pi$
$881$	−44.0049	−1.48256	−0.741281	+	0.671194i	$0.765782\pi$
$882$	0	0
$883$	−23.3596	−0.786112	−0.393056	−	0.919515i	$-0.628582\pi$
$883$	−23.3596	−0.786112	−0.393056	+	0.919515i	$0.628582\pi$
$884$	20.6002	0.692859
$885$	0	0
$886$	−32.9359	−1.10650
$887$	41.9176	1.40746	0.703728	−	0.710470i	$-0.251517\pi$
$887$	41.9176	1.40746	0.703728	+	0.710470i	$0.251517\pi$
$888$	0	0
$889$	0	0
$890$	17.5949	0.589783
$891$	0	0
$892$	−12.7669	−0.427466
$893$	10.3137	0.345135
$894$	0	0
$895$	7.90490	0.264232
$896$	0	0
$897$	0	0
$898$	24.6811	0.823618
$899$	3.87733	0.129316
$900$	0	0
$901$	−0.454318	−0.0151355
$902$	−28.0220	−0.933031
$903$	0	0
$904$	70.0295	2.32915
$905$	−26.9232	−0.894957
$906$	0	0
$907$	−20.5181	−0.681293	−0.340646	−	0.940192i	$-0.610646\pi$
$907$	−20.5181	−0.681293	−0.340646	+	0.940192i	$0.610646\pi$
$908$	77.5674	2.57416
$909$	0	0
$910$	0	0
$911$	−27.6755	−0.916930	−0.458465	−	0.888712i	$-0.651601\pi$
$911$	−27.6755	−0.916930	−0.458465	+	0.888712i	$0.651601\pi$
$912$	0	0
$913$	−1.93621	−0.0640793
$914$	25.7680	0.852330
$915$	0	0
$916$	107.212	3.54237
$917$	0	0
$918$	0	0
$919$	0.963454	0.0317814	0.0158907	−	0.999874i	$-0.494942\pi$
$919$	0.963454	0.0317814	0.0158907	+	0.999874i	$0.494942\pi$
$920$	−78.5659	−2.59024
$921$	0	0
$922$	−38.2029	−1.25815
$923$	−37.2249	−1.22527
$924$	0	0
$925$	0.781061	0.0256811
$926$	−62.5987	−2.05712
$927$	0	0
$928$	3.53148	0.115926
$929$	26.3320	0.863924	0.431962	−	0.901892i	$-0.357821\pi$
$929$	26.3320	0.863924	0.431962	+	0.901892i	$0.357821\pi$
$930$	0	0
$931$	0	0
$932$	−16.0507	−0.525760
$933$	0	0
$934$	−1.85822	−0.0608028
$935$	3.28646	0.107479
$936$	0	0
$937$	21.3865	0.698665	0.349333	−	0.936999i	$-0.386408\pi$
$937$	21.3865	0.698665	0.349333	+	0.936999i	$0.386408\pi$
$938$	0	0
$939$	0	0
$940$	13.8176	0.450681
$941$	14.8228	0.483211	0.241605	−	0.970375i	$-0.422326\pi$
$941$	14.8228	0.483211	0.241605	+	0.970375i	$0.422326\pi$
$942$	0	0
$943$	73.0217	2.37791
$944$	−66.6885	−2.17053
$945$	0	0
$946$	13.1223	0.426644
$947$	−31.7120	−1.03050	−0.515251	−	0.857039i	$-0.672301\pi$
$947$	−31.7120	−1.03050	−0.515251	+	0.857039i	$0.672301\pi$
$948$	0	0
$949$	28.1574	0.914027
$950$	−2.42301	−0.0786127
$951$	0	0
$952$	0	0
$953$	−49.9620	−1.61843	−0.809213	−	0.587515i	$-0.800106\pi$
$953$	−49.9620	−1.61843	−0.809213	+	0.587515i	$0.800106\pi$
$954$	0	0
$955$	−26.6938	−0.863790
$956$	−54.6729	−1.76825
$957$	0	0
$958$	−50.0988	−1.61862
$959$	0	0
$960$	0	0
$961$	−25.4763	−0.821817
$962$	−45.4711	−1.46605
$963$	0	0
$964$	1.89443	0.0610156
$965$	−26.3204	−0.847285
$966$	0	0
$967$	−52.4581	−1.68694	−0.843469	−	0.537178i	$-0.819490\pi$
$967$	−52.4581	−1.68694	−0.843469	+	0.537178i	$0.819490\pi$
$968$	5.49086	0.176483
$969$	0	0
$970$	−10.2096	−0.327812
$971$	30.8266	0.989272	0.494636	−	0.869100i	$-0.335301\pi$
$971$	30.8266	0.989272	0.494636	+	0.869100i	$0.335301\pi$
$972$	0	0
$973$	0	0
$974$	−10.5860	−0.339196
$975$	0	0
$976$	68.3902	2.18912
$977$	−24.7355	−0.791360	−0.395680	−	0.918388i	$-0.629491\pi$
$977$	−24.7355	−0.791360	−0.395680	+	0.918388i	$0.629491\pi$
$978$	0	0
$979$	3.20440	0.102413
$980$	0	0
$981$	0	0
$982$	−75.5427	−2.41066
$983$	12.2238	0.389880	0.194940	−	0.980815i	$-0.437549\pi$
$983$	12.2238	0.389880	0.194940	+	0.980815i	$0.437549\pi$
$984$	0	0
$985$	−26.8113	−0.854279
$986$	6.12641	0.195105
$987$	0	0
$988$	95.5894	3.04110
$989$	−34.1951	−1.08734
$990$	0	0
$991$	4.60017	0.146129	0.0730645	−	0.997327i	$-0.476722\pi$
$991$	4.60017	0.146129	0.0730645	+	0.997327i	$0.476722\pi$
$992$	5.03098	0.159734
$993$	0	0
$994$	0	0
$995$	−4.19917	−0.133123
$996$	0	0
$997$	34.4763	1.09188	0.545938	−	0.837826i	$-0.316173\pi$
$997$	34.4763	1.09188	0.545938	+	0.837826i	$0.316173\pi$
$998$	35.9713	1.13865
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bn.1.3		3
3.2	odd	2		539.2.a.i.1.1		3
7.3	odd	6		693.2.i.g.100.1		6
7.5	odd	6		693.2.i.g.298.1		6
7.6	odd	2		4851.2.a.bo.1.3		3
12.11	even	2		8624.2.a.ck.1.2		3
21.2	odd	6		539.2.e.l.67.3		6
21.5	even	6		77.2.e.b.67.3	yes	6
21.11	odd	6		539.2.e.l.177.3		6
21.17	even	6		77.2.e.b.23.3	✓	6
21.20	even	2		539.2.a.h.1.1		3
33.32	even	2		5929.2.a.w.1.3		3
84.47	odd	6		1232.2.q.k.529.2		6
84.59	odd	6		1232.2.q.k.177.2		6
84.83	odd	2		8624.2.a.cl.1.2		3
231.5	even	30		847.2.n.e.487.1		24
231.17	odd	30		847.2.n.d.366.1		24
231.26	even	30		847.2.n.e.753.1		24
231.38	even	30		847.2.n.e.366.3		24
231.47	even	30		847.2.n.e.130.3		24
231.59	even	30		847.2.n.e.632.3		24
231.68	odd	30		847.2.n.d.81.1		24
231.80	even	30		847.2.n.e.9.1		24
231.101	odd	30		847.2.n.d.807.3		24
231.131	odd	6		847.2.e.d.606.1		6
231.152	even	30		847.2.n.e.81.3		24
231.164	odd	6		847.2.e.d.485.1		6
231.173	odd	30		847.2.n.d.130.1		24
231.185	even	30		847.2.n.e.807.1		24
231.194	odd	30		847.2.n.d.753.3		24
231.206	odd	30		847.2.n.d.9.3		24
231.215	odd	30		847.2.n.d.487.3		24
231.227	odd	30		847.2.n.d.632.1		24
231.230	odd	2		5929.2.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.3	✓	6	21.17	even	6
77.2.e.b.67.3	yes	6	21.5	even	6
539.2.a.h.1.1		3	21.20	even	2
539.2.a.i.1.1		3	3.2	odd	2
539.2.e.l.67.3		6	21.2	odd	6
539.2.e.l.177.3		6	21.11	odd	6
693.2.i.g.100.1		6	7.3	odd	6
693.2.i.g.298.1		6	7.5	odd	6
847.2.e.d.485.1		6	231.164	odd	6
847.2.e.d.606.1		6	231.131	odd	6
847.2.n.d.9.3		24	231.206	odd	30
847.2.n.d.81.1		24	231.68	odd	30
847.2.n.d.130.1		24	231.173	odd	30
847.2.n.d.366.1		24	231.17	odd	30
847.2.n.d.487.3		24	231.215	odd	30
847.2.n.d.632.1		24	231.227	odd	30
847.2.n.d.753.3		24	231.194	odd	30
847.2.n.d.807.3		24	231.101	odd	30
847.2.n.e.9.1		24	231.80	even	30
847.2.n.e.81.3		24	231.152	even	30
847.2.n.e.130.3		24	231.47	even	30
847.2.n.e.366.3		24	231.38	even	30
847.2.n.e.487.1		24	231.5	even	30
847.2.n.e.632.3		24	231.59	even	30
847.2.n.e.753.1		24	231.26	even	30
847.2.n.e.807.1		24	231.185	even	30
1232.2.q.k.177.2		6	84.59	odd	6
1232.2.q.k.529.2		6	84.47	odd	6
4851.2.a.bn.1.3		3	1.1	even	1	trivial
4851.2.a.bo.1.3		3	7.6	odd	2
5929.2.a.v.1.3		3	231.230	odd	2
5929.2.a.w.1.3		3	33.32	even	2
8624.2.a.ck.1.2		3	12.11	even	2
8624.2.a.cl.1.2		3	84.83	odd	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

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49086 q^{2} +4.20440 q^{4} +2.20440 q^{5} +5.49086 q^{8} +5.49086 q^{10} +1.00000 q^{11} +3.28646 q^{13} +5.26819 q^{16} +1.49086 q^{17} +6.91794 q^{19} +9.26819 q^{20} +2.49086 q^{22} -6.49086 q^{23} -0.140614 q^{25} +8.18613 q^{26} +1.64975 q^{29} +2.35025 q^{31} +2.14061 q^{32} +3.71354 q^{34} -5.55465 q^{37} +17.2316 q^{38} +12.1041 q^{40} -11.2499 q^{41} +5.26819 q^{43} +4.20440 q^{44} -16.1679 q^{46} +1.49086 q^{47} -0.350250 q^{50} +13.8176 q^{52} -0.304735 q^{53} +2.20440 q^{55} +4.10930 q^{58} -12.6587 q^{59} +12.9817 q^{61} +5.85415 q^{62} -5.20440 q^{64} +7.24468 q^{65} -4.57292 q^{67} +6.26819 q^{68} -11.3267 q^{71} +8.56769 q^{73} -13.8359 q^{74} +29.0858 q^{76} +4.63148 q^{79} +11.6132 q^{80} -28.0220 q^{82} -1.93621 q^{83} +3.28646 q^{85} +13.1223 q^{86} +5.49086 q^{88} +3.20440 q^{89} -27.2902 q^{92} +3.71354 q^{94} +15.2499 q^{95} -1.85939 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8} + 9 q^{10} + 3 q^{11} + 11 q^{13} + 2 q^{16} - 3 q^{17} + 11 q^{19} + 14 q^{20} - 12 q^{23} + 3 q^{25} + q^{26} + 9 q^{29} + 3 q^{31} + 3 q^{32} + 10 q^{34} - 4 q^{37}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8 + 9 * q^10 + 3 * q^11 + 11 * q^13 + 2 * q^16 - 3 * q^17 + 11 * q^19 + 14 * q^20 - 12 * q^23 + 3 * q^25 + q^26 + 9 * q^29 + 3 * q^31 + 3 * q^32 + 10 * q^34 - 4 * q^37 + 8 * q^38 + 3 * q^40 - 5 * q^41 + 2 * q^43 + 4 * q^44 - 10 * q^46 - 3 * q^47 + 3 * q^50 + 7 * q^52 - 17 * q^53 - 2 * q^55 - 13 * q^58 + 8 * q^59 + 24 * q^61 + 13 * q^62 - 7 * q^64 - 15 * q^65 - 16 * q^67 + 5 * q^68 - 7 * q^71 + 20 * q^73 - 22 * q^74 + 39 * q^76 + 3 * q^79 + 9 * q^80 - 41 * q^82 - 11 * q^83 + 11 * q^85 + 21 * q^86 + 9 * q^88 + q^89 - 25 * q^92 + 10 * q^94 + 17 * q^95 - 9 * q^97

Introduction

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