LMFDB - Embedded newform 5202.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, 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("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.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:	$41.5381791315$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5202.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -0.828427 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -0.828427 q^{7} -1.00000 q^{8} +3.41421 q^{10} -4.00000 q^{11} -2.82843 q^{13} +0.828427 q^{14} +1.00000 q^{16} -1.17157 q^{19} -3.41421 q^{20} +4.00000 q^{22} -8.82843 q^{23} +6.65685 q^{25} +2.82843 q^{26} -0.828427 q^{28} +3.41421 q^{29} +4.82843 q^{31} -1.00000 q^{32} +2.82843 q^{35} -6.24264 q^{37} +1.17157 q^{38} +3.41421 q^{40} -11.0711 q^{41} -9.65685 q^{43} -4.00000 q^{44} +8.82843 q^{46} -4.48528 q^{47} -6.31371 q^{49} -6.65685 q^{50} -2.82843 q^{52} +2.82843 q^{53} +13.6569 q^{55} +0.828427 q^{56} -3.41421 q^{58} -4.48528 q^{59} -13.0711 q^{61} -4.82843 q^{62} +1.00000 q^{64} +9.65685 q^{65} +6.82843 q^{67} -2.82843 q^{70} -10.4853 q^{71} +6.58579 q^{73} +6.24264 q^{74} -1.17157 q^{76} +3.31371 q^{77} -2.48528 q^{79} -3.41421 q^{80} +11.0711 q^{82} +12.4853 q^{83} +9.65685 q^{86} +4.00000 q^{88} +2.34315 q^{91} -8.82843 q^{92} +4.48528 q^{94} +4.00000 q^{95} -10.5858 q^{97} +6.31371 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 4 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8} + 4 q^{10} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{20} + 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{37} + 8 q^{38} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 8 q^{44} + 12 q^{46} + 8 q^{47} + 10 q^{49} - 2 q^{50} + 16 q^{55} - 4 q^{56} - 4 q^{58} + 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} + 8 q^{65} + 8 q^{67} - 4 q^{71} + 16 q^{73} + 4 q^{74} - 8 q^{76} - 16 q^{77} + 12 q^{79} - 4 q^{80} + 8 q^{82} + 8 q^{83} + 8 q^{86} + 8 q^{88} + 16 q^{91} - 12 q^{92} - 8 q^{94} + 8 q^{95} - 24 q^{97} - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 4 * q^7 - 2 * q^8 + 4 * q^10 - 8 * q^11 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^20 + 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^28 + 4 * q^29 + 4 * q^31 - 2 * q^32 - 4 * q^37 + 8 * q^38 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 8 * q^44 + 12 * q^46 + 8 * q^47 + 10 * q^49 - 2 * q^50 + 16 * q^55 - 4 * q^56 - 4 * q^58 + 8 * q^59 - 12 * q^61 - 4 * q^62 + 2 * q^64 + 8 * q^65 + 8 * q^67 - 4 * q^71 + 16 * q^73 + 4 * q^74 - 8 * q^76 - 16 * q^77 + 12 * q^79 - 4 * q^80 + 8 * q^82 + 8 * q^83 + 8 * q^86 + 8 * q^88 + 16 * q^91 - 12 * q^92 - 8 * q^94 + 8 * q^95 - 24 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.41421	1.07967
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0.828427	0.221406
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.17157	−0.268777	−0.134389	−	0.990929i	$-0.542907\pi$
$19$	−1.17157	−0.268777	−0.134389	+	0.990929i	$0.542907\pi$
$20$	−3.41421	−0.763441
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	2.82843	0.554700
$27$	0	0
$28$	−0.828427	−0.156558
$29$	3.41421	0.634004	0.317002	−	0.948425i	$-0.397324\pi$
$29$	3.41421	0.634004	0.317002	+	0.948425i	$0.397324\pi$
$30$	0	0
$31$	4.82843	0.867211	0.433606	−	0.901103i	$-0.357241\pi$
$31$	4.82843	0.867211	0.433606	+	0.901103i	$0.357241\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−6.24264	−1.02628	−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264	−1.02628	−0.513142	+	0.858304i	$0.671519\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	3.41421	0.539835
$41$	−11.0711	−1.72901	−0.864505	−	0.502624i	$-0.832368\pi$
$41$	−11.0711	−1.72901	−0.864505	+	0.502624i	$0.832368\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	8.82843	1.30168
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	−6.65685	−0.941421
$51$	0	0
$52$	−2.82843	−0.392232
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	13.6569	1.84149
$56$	0.828427	0.110703
$57$	0	0
$58$	−3.41421	−0.448308
$59$	−4.48528	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$59$	−4.48528	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$60$	0	0
$61$	−13.0711	−1.67358	−0.836789	−	0.547525i	$-0.815570\pi$
$61$	−13.0711	−1.67358	−0.836789	+	0.547525i	$0.815570\pi$
$62$	−4.82843	−0.613211
$63$	0	0
$64$	1.00000	0.125000
$65$	9.65685	1.19779
$66$	0	0
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	0	0
$69$	0	0
$70$	−2.82843	−0.338062
$71$	−10.4853	−1.24437	−0.622187	−	0.782869i	$-0.713756\pi$
$71$	−10.4853	−1.24437	−0.622187	+	0.782869i	$0.713756\pi$
$72$	0	0
$73$	6.58579	0.770808	0.385404	−	0.922748i	$-0.374062\pi$
$73$	6.58579	0.770808	0.385404	+	0.922748i	$0.374062\pi$
$74$	6.24264	0.725692
$75$	0	0
$76$	−1.17157	−0.134389
$77$	3.31371	0.377632
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	−3.41421	−0.381721
$81$	0	0
$82$	11.0711	1.22259
$83$	12.4853	1.37044	0.685219	−	0.728337i	$-0.259707\pi$
$83$	12.4853	1.37044	0.685219	+	0.728337i	$0.259707\pi$
$84$	0	0
$85$	0	0
$86$	9.65685	1.04133
$87$	0	0
$88$	4.00000	0.426401
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	−8.82843	−0.920427
$93$	0	0
$94$	4.48528	0.462621
$95$	4.00000	0.410391
$96$	0	0
$97$	−10.5858	−1.07482	−0.537412	−	0.843320i	$-0.680598\pi$
$97$	−10.5858	−1.07482	−0.537412	+	0.843320i	$0.680598\pi$
$98$	6.31371	0.637781
$99$	0	0
$100$	6.65685	0.665685
$101$	−10.8284	−1.07747	−0.538734	−	0.842476i	$-0.681097\pi$
$101$	−10.8284	−1.07747	−0.538734	+	0.842476i	$0.681097\pi$
$102$	0	0
$103$	9.17157	0.903702	0.451851	−	0.892093i	$-0.350764\pi$
$103$	9.17157	0.903702	0.451851	+	0.892093i	$0.350764\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−2.82843	−0.274721
$107$	−2.34315	−0.226520	−0.113260	−	0.993565i	$-0.536129\pi$
$107$	−2.34315	−0.226520	−0.113260	+	0.993565i	$0.536129\pi$
$108$	0	0
$109$	15.8995	1.52290	0.761448	−	0.648226i	$-0.224489\pi$
$109$	15.8995	1.52290	0.761448	+	0.648226i	$0.224489\pi$
$110$	−13.6569	−1.30213
$111$	0	0
$112$	−0.828427	−0.0782790
$113$	−12.7279	−1.19734	−0.598671	−	0.800995i	$-0.704304\pi$
$113$	−12.7279	−1.19734	−0.598671	+	0.800995i	$0.704304\pi$
$114$	0	0
$115$	30.1421	2.81077
$116$	3.41421	0.317002
$117$	0	0
$118$	4.48528	0.412904
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	13.0711	1.18340
$123$	0	0
$124$	4.82843	0.433606
$125$	−5.65685	−0.505964
$126$	0	0
$127$	2.34315	0.207921	0.103960	−	0.994581i	$-0.466849\pi$
$127$	2.34315	0.207921	0.103960	+	0.994581i	$0.466849\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.65685	−0.846962
$131$	−3.31371	−0.289520	−0.144760	−	0.989467i	$-0.546241\pi$
$131$	−3.31371	−0.289520	−0.144760	+	0.989467i	$0.546241\pi$
$132$	0	0
$133$	0.970563	0.0841585
$134$	−6.82843	−0.589886
$135$	0	0
$136$	0	0
$137$	6.34315	0.541932	0.270966	−	0.962589i	$-0.412657\pi$
$137$	6.34315	0.541932	0.270966	+	0.962589i	$0.412657\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	2.82843	0.239046
$141$	0	0
$142$	10.4853	0.879905
$143$	11.3137	0.946100
$144$	0	0
$145$	−11.6569	−0.968049
$146$	−6.58579	−0.545044
$147$	0	0
$148$	−6.24264	−0.513142
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	2.34315	0.190682	0.0953412	−	0.995445i	$-0.469606\pi$
$151$	2.34315	0.190682	0.0953412	+	0.995445i	$0.469606\pi$
$152$	1.17157	0.0950271
$153$	0	0
$154$	−3.31371	−0.267026
$155$	−16.4853	−1.32413
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	2.48528	0.197718
$159$	0	0
$160$	3.41421	0.269917
$161$	7.31371	0.576401
$162$	0	0
$163$	13.6569	1.06969	0.534844	−	0.844951i	$-0.320370\pi$
$163$	13.6569	1.06969	0.534844	+	0.844951i	$0.320370\pi$
$164$	−11.0711	−0.864505
$165$	0	0
$166$	−12.4853	−0.969046
$167$	17.7990	1.37733	0.688664	−	0.725081i	$-0.258198\pi$
$167$	17.7990	1.37733	0.688664	+	0.725081i	$0.258198\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	−9.65685	−0.736328
$173$	23.8995	1.81704	0.908522	−	0.417836i	$-0.137211\pi$
$173$	23.8995	1.81704	0.908522	+	0.417836i	$0.137211\pi$
$174$	0	0
$175$	−5.51472	−0.416874
$176$	−4.00000	−0.301511
$177$	0	0
$178$	0	0
$179$	−20.9706	−1.56741	−0.783707	−	0.621131i	$-0.786673\pi$
$179$	−20.9706	−1.56741	−0.783707	+	0.621131i	$0.786673\pi$
$180$	0	0
$181$	−11.8995	−0.884482	−0.442241	−	0.896896i	$-0.645816\pi$
$181$	−11.8995	−0.884482	−0.442241	+	0.896896i	$0.645816\pi$
$182$	−2.34315	−0.173686
$183$	0	0
$184$	8.82843	0.650840
$185$	21.3137	1.56702
$186$	0	0
$187$	0	0
$188$	−4.48528	−0.327123
$189$	0	0
$190$	−4.00000	−0.290191
$191$	6.82843	0.494088	0.247044	−	0.969004i	$-0.420541\pi$
$191$	6.82843	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$192$	0	0
$193$	−13.8995	−1.00051	−0.500254	−	0.865879i	$-0.666760\pi$
$193$	−13.8995	−1.00051	−0.500254	+	0.865879i	$0.666760\pi$
$194$	10.5858	0.760015
$195$	0	0
$196$	−6.31371	−0.450979
$197$	1.75736	0.125207	0.0626033	−	0.998038i	$-0.480060\pi$
$197$	1.75736	0.125207	0.0626033	+	0.998038i	$0.480060\pi$
$198$	0	0
$199$	4.82843	0.342278	0.171139	−	0.985247i	$-0.445255\pi$
$199$	4.82843	0.342278	0.171139	+	0.985247i	$0.445255\pi$
$200$	−6.65685	−0.470711
$201$	0	0
$202$	10.8284	0.761885
$203$	−2.82843	−0.198517
$204$	0	0
$205$	37.7990	2.64000
$206$	−9.17157	−0.639014
$207$	0	0
$208$	−2.82843	−0.196116
$209$	4.68629	0.324158
$210$	0	0
$211$	4.97056	0.342188	0.171094	−	0.985255i	$-0.445270\pi$
$211$	4.97056	0.342188	0.171094	+	0.985255i	$0.445270\pi$
$212$	2.82843	0.194257
$213$	0	0
$214$	2.34315	0.160174
$215$	32.9706	2.24857
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−15.8995	−1.07685
$219$	0	0
$220$	13.6569	0.920745
$221$	0	0
$222$	0	0
$223$	1.17157	0.0784543	0.0392272	−	0.999230i	$-0.487510\pi$
$223$	1.17157	0.0784543	0.0392272	+	0.999230i	$0.487510\pi$
$224$	0.828427	0.0553516
$225$	0	0
$226$	12.7279	0.846649
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−30.1421	−1.98751
$231$	0	0
$232$	−3.41421	−0.224154
$233$	12.7279	0.833834	0.416917	−	0.908945i	$-0.363111\pi$
$233$	12.7279	0.833834	0.416917	+	0.908945i	$0.363111\pi$
$234$	0	0
$235$	15.3137	0.998956
$236$	−4.48528	−0.291967
$237$	0	0
$238$	0	0
$239$	4.48528	0.290129	0.145064	−	0.989422i	$-0.453661\pi$
$239$	4.48528	0.290129	0.145064	+	0.989422i	$0.453661\pi$
$240$	0	0
$241$	−0.242641	−0.0156299	−0.00781493	−	0.999969i	$-0.502488\pi$
$241$	−0.242641	−0.0156299	−0.00781493	+	0.999969i	$0.502488\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−13.0711	−0.836789
$245$	21.5563	1.37718
$246$	0	0
$247$	3.31371	0.210846
$248$	−4.82843	−0.306605
$249$	0	0
$250$	5.65685	0.357771
$251$	6.34315	0.400376	0.200188	−	0.979758i	$-0.435845\pi$
$251$	6.34315	0.400376	0.200188	+	0.979758i	$0.435845\pi$
$252$	0	0
$253$	35.3137	2.22015
$254$	−2.34315	−0.147022
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.68629	0.417079	0.208540	−	0.978014i	$-0.433129\pi$
$257$	6.68629	0.417079	0.208540	+	0.978014i	$0.433129\pi$
$258$	0	0
$259$	5.17157	0.321346
$260$	9.65685	0.598893
$261$	0	0
$262$	3.31371	0.204722
$263$	18.1421	1.11869	0.559346	−	0.828934i	$-0.311052\pi$
$263$	18.1421	1.11869	0.559346	+	0.828934i	$0.311052\pi$
$264$	0	0
$265$	−9.65685	−0.593216
$266$	−0.970563	−0.0595090
$267$	0	0
$268$	6.82843	0.417113
$269$	9.07107	0.553073	0.276536	−	0.961003i	$-0.410813\pi$
$269$	9.07107	0.553073	0.276536	+	0.961003i	$0.410813\pi$
$270$	0	0
$271$	10.3431	0.628301	0.314151	−	0.949373i	$-0.398280\pi$
$271$	10.3431	0.628301	0.314151	+	0.949373i	$0.398280\pi$
$272$	0	0
$273$	0	0
$274$	−6.34315	−0.383203
$275$	−26.6274	−1.60569
$276$	0	0
$277$	−18.2426	−1.09609	−0.548047	−	0.836448i	$-0.684629\pi$
$277$	−18.2426	−1.09609	−0.548047	+	0.836448i	$0.684629\pi$
$278$	17.6569	1.05899
$279$	0	0
$280$	−2.82843	−0.169031
$281$	−3.31371	−0.197679	−0.0988396	−	0.995103i	$-0.531513\pi$
$281$	−3.31371	−0.197679	−0.0988396	+	0.995103i	$0.531513\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−10.4853	−0.622187
$285$	0	0
$286$	−11.3137	−0.668994
$287$	9.17157	0.541381
$288$	0	0
$289$	0	0
$290$	11.6569	0.684514
$291$	0	0
$292$	6.58579	0.385404
$293$	−13.3137	−0.777795	−0.388898	−	0.921281i	$-0.627144\pi$
$293$	−13.3137	−0.777795	−0.388898	+	0.921281i	$0.627144\pi$
$294$	0	0
$295$	15.3137	0.891599
$296$	6.24264	0.362846
$297$	0	0
$298$	−10.0000	−0.579284
$299$	24.9706	1.44408
$300$	0	0
$301$	8.00000	0.461112
$302$	−2.34315	−0.134833
$303$	0	0
$304$	−1.17157	−0.0671943
$305$	44.6274	2.55536
$306$	0	0
$307$	−17.6569	−1.00773	−0.503865	−	0.863782i	$-0.668089\pi$
$307$	−17.6569	−1.00773	−0.503865	+	0.863782i	$0.668089\pi$
$308$	3.31371	0.188816
$309$	0	0
$310$	16.4853	0.936301
$311$	4.82843	0.273795	0.136897	−	0.990585i	$-0.456287\pi$
$311$	4.82843	0.273795	0.136897	+	0.990585i	$0.456287\pi$
$312$	0	0
$313$	29.8995	1.69002	0.845010	−	0.534751i	$-0.179595\pi$
$313$	29.8995	1.69002	0.845010	+	0.534751i	$0.179595\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−2.48528	−0.139808
$317$	9.55635	0.536738	0.268369	−	0.963316i	$-0.413515\pi$
$317$	9.55635	0.536738	0.268369	+	0.963316i	$0.413515\pi$
$318$	0	0
$319$	−13.6569	−0.764637
$320$	−3.41421	−0.190860
$321$	0	0
$322$	−7.31371	−0.407577
$323$	0	0
$324$	0	0
$325$	−18.8284	−1.04441
$326$	−13.6569	−0.756383
$327$	0	0
$328$	11.0711	0.611297
$329$	3.71573	0.204855
$330$	0	0
$331$	−26.6274	−1.46358	−0.731788	−	0.681533i	$-0.761314\pi$
$331$	−26.6274	−1.46358	−0.731788	+	0.681533i	$0.761314\pi$
$332$	12.4853	0.685219
$333$	0	0
$334$	−17.7990	−0.973917
$335$	−23.3137	−1.27376
$336$	0	0
$337$	12.7279	0.693334	0.346667	−	0.937988i	$-0.387313\pi$
$337$	12.7279	0.693334	0.346667	+	0.937988i	$0.387313\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	0	0
$341$	−19.3137	−1.04590
$342$	0	0
$343$	11.0294	0.595534
$344$	9.65685	0.520663
$345$	0	0
$346$	−23.8995	−1.28484
$347$	−11.3137	−0.607352	−0.303676	−	0.952775i	$-0.598214\pi$
$347$	−11.3137	−0.607352	−0.303676	+	0.952775i	$0.598214\pi$
$348$	0	0
$349$	−5.17157	−0.276828	−0.138414	−	0.990374i	$-0.544200\pi$
$349$	−5.17157	−0.276828	−0.138414	+	0.990374i	$0.544200\pi$
$350$	5.51472	0.294774
$351$	0	0
$352$	4.00000	0.213201
$353$	32.6274	1.73658	0.868291	−	0.496055i	$-0.165219\pi$
$353$	32.6274	1.73658	0.868291	+	0.496055i	$0.165219\pi$
$354$	0	0
$355$	35.7990	1.90001
$356$	0	0
$357$	0	0
$358$	20.9706	1.10833
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	11.8995	0.625424
$363$	0	0
$364$	2.34315	0.122814
$365$	−22.4853	−1.17693
$366$	0	0
$367$	−34.4853	−1.80012	−0.900059	−	0.435769i	$-0.856476\pi$
$367$	−34.4853	−1.80012	−0.900059	+	0.435769i	$0.856476\pi$
$368$	−8.82843	−0.460214
$369$	0	0
$370$	−21.3137	−1.10805
$371$	−2.34315	−0.121650
$372$	0	0
$373$	−19.7990	−1.02515	−0.512576	−	0.858642i	$-0.671309\pi$
$373$	−19.7990	−1.02515	−0.512576	+	0.858642i	$0.671309\pi$
$374$	0	0
$375$	0	0
$376$	4.48528	0.231311
$377$	−9.65685	−0.497353
$378$	0	0
$379$	−8.97056	−0.460787	−0.230393	−	0.973098i	$-0.574001\pi$
$379$	−8.97056	−0.460787	−0.230393	+	0.973098i	$0.574001\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−6.82843	−0.349373
$383$	15.7990	0.807291	0.403645	−	0.914916i	$-0.367743\pi$
$383$	15.7990	0.807291	0.403645	+	0.914916i	$0.367743\pi$
$384$	0	0
$385$	−11.3137	−0.576600
$386$	13.8995	0.707466
$387$	0	0
$388$	−10.5858	−0.537412
$389$	−8.62742	−0.437427	−0.218714	−	0.975789i	$-0.570186\pi$
$389$	−8.62742	−0.437427	−0.218714	+	0.975789i	$0.570186\pi$
$390$	0	0
$391$	0	0
$392$	6.31371	0.318890
$393$	0	0
$394$	−1.75736	−0.0885345
$395$	8.48528	0.426941
$396$	0	0
$397$	−18.9289	−0.950016	−0.475008	−	0.879982i	$-0.657555\pi$
$397$	−18.9289	−0.950016	−0.475008	+	0.879982i	$0.657555\pi$
$398$	−4.82843	−0.242027
$399$	0	0
$400$	6.65685	0.332843
$401$	−16.7279	−0.835353	−0.417676	−	0.908596i	$-0.637155\pi$
$401$	−16.7279	−0.835353	−0.417676	+	0.908596i	$0.637155\pi$
$402$	0	0
$403$	−13.6569	−0.680296
$404$	−10.8284	−0.538734
$405$	0	0
$406$	2.82843	0.140372
$407$	24.9706	1.23774
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−37.7990	−1.86676
$411$	0	0
$412$	9.17157	0.451851
$413$	3.71573	0.182839
$414$	0	0
$415$	−42.6274	−2.09250
$416$	2.82843	0.138675
$417$	0	0
$418$	−4.68629	−0.229214
$419$	−9.65685	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$419$	−9.65685	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$420$	0	0
$421$	32.4853	1.58324	0.791618	−	0.611016i	$-0.209239\pi$
$421$	32.4853	1.58324	0.791618	+	0.611016i	$0.209239\pi$
$422$	−4.97056	−0.241963
$423$	0	0
$424$	−2.82843	−0.137361
$425$	0	0
$426$	0	0
$427$	10.8284	0.524024
$428$	−2.34315	−0.113260
$429$	0	0
$430$	−32.9706	−1.58998
$431$	8.14214	0.392193	0.196096	−	0.980585i	$-0.437173\pi$
$431$	8.14214	0.392193	0.196096	+	0.980585i	$0.437173\pi$
$432$	0	0
$433$	28.6274	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$433$	28.6274	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	15.8995	0.761448
$437$	10.3431	0.494780
$438$	0	0
$439$	30.4853	1.45498	0.727492	−	0.686117i	$-0.240686\pi$
$439$	30.4853	1.45498	0.727492	+	0.686117i	$0.240686\pi$
$440$	−13.6569	−0.651065
$441$	0	0
$442$	0	0
$443$	−4.97056	−0.236159	−0.118079	−	0.993004i	$-0.537674\pi$
$443$	−4.97056	−0.236159	−0.118079	+	0.993004i	$0.537674\pi$
$444$	0	0
$445$	0	0
$446$	−1.17157	−0.0554756
$447$	0	0
$448$	−0.828427	−0.0391395
$449$	36.0416	1.70091	0.850455	−	0.526048i	$-0.176327\pi$
$449$	36.0416	1.70091	0.850455	+	0.526048i	$0.176327\pi$
$450$	0	0
$451$	44.2843	2.08526
$452$	−12.7279	−0.598671
$453$	0	0
$454$	6.34315	0.297699
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−25.3137	−1.18413	−0.592063	−	0.805892i	$-0.701686\pi$
$457$	−25.3137	−1.18413	−0.592063	+	0.805892i	$0.701686\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	30.1421	1.40538
$461$	25.4558	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$462$	0	0
$463$	−1.17157	−0.0544476	−0.0272238	−	0.999629i	$-0.508667\pi$
$463$	−1.17157	−0.0544476	−0.0272238	+	0.999629i	$0.508667\pi$
$464$	3.41421	0.158501
$465$	0	0
$466$	−12.7279	−0.589610
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	−5.65685	−0.261209
$470$	−15.3137	−0.706369
$471$	0	0
$472$	4.48528	0.206452
$473$	38.6274	1.77609
$474$	0	0
$475$	−7.79899	−0.357842
$476$	0	0
$477$	0	0
$478$	−4.48528	−0.205152
$479$	−12.8284	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$479$	−12.8284	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$480$	0	0
$481$	17.6569	0.805083
$482$	0.242641	0.0110520
$483$	0	0
$484$	5.00000	0.227273
$485$	36.1421	1.64113
$486$	0	0
$487$	21.1127	0.956708	0.478354	−	0.878167i	$-0.341234\pi$
$487$	21.1127	0.956708	0.478354	+	0.878167i	$0.341234\pi$
$488$	13.0711	0.591699
$489$	0	0
$490$	−21.5563	−0.973817
$491$	−40.2843	−1.81800	−0.909002	−	0.416792i	$-0.863154\pi$
$491$	−40.2843	−1.81800	−0.909002	+	0.416792i	$0.863154\pi$
$492$	0	0
$493$	0	0
$494$	−3.31371	−0.149091
$495$	0	0
$496$	4.82843	0.216803
$497$	8.68629	0.389633
$498$	0	0
$499$	−8.97056	−0.401578	−0.200789	−	0.979635i	$-0.564350\pi$
$499$	−8.97056	−0.401578	−0.200789	+	0.979635i	$0.564350\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	−6.34315	−0.283108
$503$	−26.4853	−1.18092	−0.590460	−	0.807067i	$-0.701054\pi$
$503$	−26.4853	−1.18092	−0.590460	+	0.807067i	$0.701054\pi$
$504$	0	0
$505$	36.9706	1.64517
$506$	−35.3137	−1.56989
$507$	0	0
$508$	2.34315	0.103960
$509$	−25.3137	−1.12201	−0.561005	−	0.827812i	$-0.689585\pi$
$509$	−25.3137	−1.12201	−0.561005	+	0.827812i	$0.689585\pi$
$510$	0	0
$511$	−5.45584	−0.241352
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−6.68629	−0.294920
$515$	−31.3137	−1.37985
$516$	0	0
$517$	17.9411	0.789050
$518$	−5.17157	−0.227226
$519$	0	0
$520$	−9.65685	−0.423481
$521$	−16.2426	−0.711603	−0.355802	−	0.934562i	$-0.615792\pi$
$521$	−16.2426	−0.711603	−0.355802	+	0.934562i	$0.615792\pi$
$522$	0	0
$523$	−10.1421	−0.443485	−0.221742	−	0.975105i	$-0.571174\pi$
$523$	−10.1421	−0.443485	−0.221742	+	0.975105i	$0.571174\pi$
$524$	−3.31371	−0.144760
$525$	0	0
$526$	−18.1421	−0.791035
$527$	0	0
$528$	0	0
$529$	54.9411	2.38874
$530$	9.65685	0.419467
$531$	0	0
$532$	0.970563	0.0420792
$533$	31.3137	1.35635
$534$	0	0
$535$	8.00000	0.345870
$536$	−6.82843	−0.294943
$537$	0	0
$538$	−9.07107	−0.391082
$539$	25.2548	1.08780
$540$	0	0
$541$	9.75736	0.419502	0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736	0.419502	0.209751	+	0.977755i	$0.432735\pi$
$542$	−10.3431	−0.444276
$543$	0	0
$544$	0	0
$545$	−54.2843	−2.32528
$546$	0	0
$547$	−26.6274	−1.13851	−0.569253	−	0.822162i	$-0.692768\pi$
$547$	−26.6274	−1.13851	−0.569253	+	0.822162i	$0.692768\pi$
$548$	6.34315	0.270966
$549$	0	0
$550$	26.6274	1.13540
$551$	−4.00000	−0.170406
$552$	0	0
$553$	2.05887	0.0875522
$554$	18.2426	0.775056
$555$	0	0
$556$	−17.6569	−0.748817
$557$	−8.48528	−0.359533	−0.179766	−	0.983709i	$-0.557534\pi$
$557$	−8.48528	−0.359533	−0.179766	+	0.983709i	$0.557534\pi$
$558$	0	0
$559$	27.3137	1.15525
$560$	2.82843	0.119523
$561$	0	0
$562$	3.31371	0.139780
$563$	20.4853	0.863352	0.431676	−	0.902029i	$-0.357922\pi$
$563$	20.4853	0.863352	0.431676	+	0.902029i	$0.357922\pi$
$564$	0	0
$565$	43.4558	1.82820
$566$	8.00000	0.336265
$567$	0	0
$568$	10.4853	0.439953
$569$	−20.9706	−0.879132	−0.439566	−	0.898210i	$-0.644868\pi$
$569$	−20.9706	−0.879132	−0.439566	+	0.898210i	$0.644868\pi$
$570$	0	0
$571$	31.3137	1.31044	0.655219	−	0.755439i	$-0.272576\pi$
$571$	31.3137	1.31044	0.655219	+	0.755439i	$0.272576\pi$
$572$	11.3137	0.473050
$573$	0	0
$574$	−9.17157	−0.382814
$575$	−58.7696	−2.45086
$576$	0	0
$577$	19.3137	0.804040	0.402020	−	0.915631i	$-0.368308\pi$
$577$	19.3137	0.804040	0.402020	+	0.915631i	$0.368308\pi$
$578$	0	0
$579$	0	0
$580$	−11.6569	−0.484025
$581$	−10.3431	−0.429106
$582$	0	0
$583$	−11.3137	−0.468566
$584$	−6.58579	−0.272522
$585$	0	0
$586$	13.3137	0.549984
$587$	3.02944	0.125038	0.0625191	−	0.998044i	$-0.480087\pi$
$587$	3.02944	0.125038	0.0625191	+	0.998044i	$0.480087\pi$
$588$	0	0
$589$	−5.65685	−0.233087
$590$	−15.3137	−0.630455
$591$	0	0
$592$	−6.24264	−0.256571
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	−24.9706	−1.02112
$599$	14.6274	0.597660	0.298830	−	0.954306i	$-0.403404\pi$
$599$	14.6274	0.597660	0.298830	+	0.954306i	$0.403404\pi$
$600$	0	0
$601$	9.41421	0.384014	0.192007	−	0.981394i	$-0.438500\pi$
$601$	9.41421	0.384014	0.192007	+	0.981394i	$0.438500\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	2.34315	0.0953412
$605$	−17.0711	−0.694038
$606$	0	0
$607$	−6.48528	−0.263229	−0.131615	−	0.991301i	$-0.542016\pi$
$607$	−6.48528	−0.263229	−0.131615	+	0.991301i	$0.542016\pi$
$608$	1.17157	0.0475136
$609$	0	0
$610$	−44.6274	−1.80691
$611$	12.6863	0.513232
$612$	0	0
$613$	1.31371	0.0530602	0.0265301	−	0.999648i	$-0.491554\pi$
$613$	1.31371	0.0530602	0.0265301	+	0.999648i	$0.491554\pi$
$614$	17.6569	0.712573
$615$	0	0
$616$	−3.31371	−0.133513
$617$	−0.928932	−0.0373974	−0.0186987	−	0.999825i	$-0.505952\pi$
$617$	−0.928932	−0.0373974	−0.0186987	+	0.999825i	$0.505952\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	−16.4853	−0.662065
$621$	0	0
$622$	−4.82843	−0.193602
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	−29.8995	−1.19502
$627$	0	0
$628$	−6.00000	−0.239426
$629$	0	0
$630$	0	0
$631$	−2.14214	−0.0852771	−0.0426385	−	0.999091i	$-0.513576\pi$
$631$	−2.14214	−0.0852771	−0.0426385	+	0.999091i	$0.513576\pi$
$632$	2.48528	0.0988592
$633$	0	0
$634$	−9.55635	−0.379531
$635$	−8.00000	−0.317470
$636$	0	0
$637$	17.8579	0.707554
$638$	13.6569	0.540680
$639$	0	0
$640$	3.41421	0.134959
$641$	−4.92893	−0.194681	−0.0973406	−	0.995251i	$-0.531034\pi$
$641$	−4.92893	−0.194681	−0.0973406	+	0.995251i	$0.531034\pi$
$642$	0	0
$643$	−23.3137	−0.919403	−0.459701	−	0.888074i	$-0.652044\pi$
$643$	−23.3137	−0.919403	−0.459701	+	0.888074i	$0.652044\pi$
$644$	7.31371	0.288200
$645$	0	0
$646$	0	0
$647$	46.4264	1.82521	0.912605	−	0.408842i	$-0.134067\pi$
$647$	46.4264	1.82521	0.912605	+	0.408842i	$0.134067\pi$
$648$	0	0
$649$	17.9411	0.704251
$650$	18.8284	0.738512
$651$	0	0
$652$	13.6569	0.534844
$653$	−28.5858	−1.11865	−0.559324	−	0.828949i	$-0.688939\pi$
$653$	−28.5858	−1.11865	−0.559324	+	0.828949i	$0.688939\pi$
$654$	0	0
$655$	11.3137	0.442063
$656$	−11.0711	−0.432253
$657$	0	0
$658$	−3.71573	−0.144854
$659$	20.4853	0.797993	0.398997	−	0.916952i	$-0.369359\pi$
$659$	20.4853	0.797993	0.398997	+	0.916952i	$0.369359\pi$
$660$	0	0
$661$	−1.85786	−0.0722625	−0.0361313	−	0.999347i	$-0.511503\pi$
$661$	−1.85786	−0.0722625	−0.0361313	+	0.999347i	$0.511503\pi$
$662$	26.6274	1.03490
$663$	0	0
$664$	−12.4853	−0.484523
$665$	−3.31371	−0.128500
$666$	0	0
$667$	−30.1421	−1.16711
$668$	17.7990	0.688664
$669$	0	0
$670$	23.3137	0.900687
$671$	52.2843	2.01841
$672$	0	0
$673$	4.44365	0.171290	0.0856451	−	0.996326i	$-0.472705\pi$
$673$	4.44365	0.171290	0.0856451	+	0.996326i	$0.472705\pi$
$674$	−12.7279	−0.490261
$675$	0	0
$676$	−5.00000	−0.192308
$677$	−8.38478	−0.322253	−0.161127	−	0.986934i	$-0.551513\pi$
$677$	−8.38478	−0.322253	−0.161127	+	0.986934i	$0.551513\pi$
$678$	0	0
$679$	8.76955	0.336545
$680$	0	0
$681$	0	0
$682$	19.3137	0.739560
$683$	−36.2843	−1.38838	−0.694190	−	0.719792i	$-0.744237\pi$
$683$	−36.2843	−1.38838	−0.694190	+	0.719792i	$0.744237\pi$
$684$	0	0
$685$	−21.6569	−0.827466
$686$	−11.0294	−0.421106
$687$	0	0
$688$	−9.65685	−0.368164
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−20.9706	−0.797758	−0.398879	−	0.917004i	$-0.630601\pi$
$691$	−20.9706	−0.797758	−0.398879	+	0.917004i	$0.630601\pi$
$692$	23.8995	0.908522
$693$	0	0
$694$	11.3137	0.429463
$695$	60.2843	2.28671
$696$	0	0
$697$	0	0
$698$	5.17157	0.195747
$699$	0	0
$700$	−5.51472	−0.208437
$701$	−40.4853	−1.52911	−0.764554	−	0.644560i	$-0.777041\pi$
$701$	−40.4853	−1.52911	−0.764554	+	0.644560i	$0.777041\pi$
$702$	0	0
$703$	7.31371	0.275842
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−32.6274	−1.22795
$707$	8.97056	0.337373
$708$	0	0
$709$	37.3553	1.40291	0.701455	−	0.712714i	$-0.252534\pi$
$709$	37.3553	1.40291	0.701455	+	0.712714i	$0.252534\pi$
$710$	−35.7990	−1.34351
$711$	0	0
$712$	0	0
$713$	−42.6274	−1.59641
$714$	0	0
$715$	−38.6274	−1.44458
$716$	−20.9706	−0.783707
$717$	0	0
$718$	−16.0000	−0.597115
$719$	−49.7990	−1.85719	−0.928594	−	0.371096i	$-0.878982\pi$
$719$	−49.7990	−1.85719	−0.928594	+	0.371096i	$0.878982\pi$
$720$	0	0
$721$	−7.59798	−0.282964
$722$	17.6274	0.656025
$723$	0	0
$724$	−11.8995	−0.442241
$725$	22.7279	0.844094
$726$	0	0
$727$	21.6569	0.803208	0.401604	−	0.915813i	$-0.368453\pi$
$727$	21.6569	0.803208	0.401604	+	0.915813i	$0.368453\pi$
$728$	−2.34315	−0.0868428
$729$	0	0
$730$	22.4853	0.832218
$731$	0	0
$732$	0	0
$733$	−9.45584	−0.349260	−0.174630	−	0.984634i	$-0.555873\pi$
$733$	−9.45584	−0.349260	−0.174630	+	0.984634i	$0.555873\pi$
$734$	34.4853	1.27287
$735$	0	0
$736$	8.82843	0.325420
$737$	−27.3137	−1.00611
$738$	0	0
$739$	−14.8284	−0.545473	−0.272736	−	0.962089i	$-0.587929\pi$
$739$	−14.8284	−0.545473	−0.272736	+	0.962089i	$0.587929\pi$
$740$	21.3137	0.783508
$741$	0	0
$742$	2.34315	0.0860196
$743$	−19.4558	−0.713766	−0.356883	−	0.934149i	$-0.616160\pi$
$743$	−19.4558	−0.713766	−0.356883	+	0.934149i	$0.616160\pi$
$744$	0	0
$745$	−34.1421	−1.25087
$746$	19.7990	0.724893
$747$	0	0
$748$	0	0
$749$	1.94113	0.0709272
$750$	0	0
$751$	39.4558	1.43976	0.719882	−	0.694096i	$-0.244196\pi$
$751$	39.4558	1.43976	0.719882	+	0.694096i	$0.244196\pi$
$752$	−4.48528	−0.163561
$753$	0	0
$754$	9.65685	0.351682
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	8.97056	0.325826
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−9.65685	−0.350061	−0.175030	−	0.984563i	$-0.556002\pi$
$761$	−9.65685	−0.350061	−0.175030	+	0.984563i	$0.556002\pi$
$762$	0	0
$763$	−13.1716	−0.476843
$764$	6.82843	0.247044
$765$	0	0
$766$	−15.7990	−0.570841
$767$	12.6863	0.458075
$768$	0	0
$769$	−7.02944	−0.253488	−0.126744	−	0.991935i	$-0.540453\pi$
$769$	−7.02944	−0.253488	−0.126744	+	0.991935i	$0.540453\pi$
$770$	11.3137	0.407718
$771$	0	0
$772$	−13.8995	−0.500254
$773$	−0.627417	−0.0225666	−0.0112833	−	0.999936i	$-0.503592\pi$
$773$	−0.627417	−0.0225666	−0.0112833	+	0.999936i	$0.503592\pi$
$774$	0	0
$775$	32.1421	1.15458
$776$	10.5858	0.380008
$777$	0	0
$778$	8.62742	0.309308
$779$	12.9706	0.464719
$780$	0	0
$781$	41.9411	1.50077
$782$	0	0
$783$	0	0
$784$	−6.31371	−0.225490
$785$	20.4853	0.731151
$786$	0	0
$787$	45.9411	1.63762	0.818812	−	0.574062i	$-0.194633\pi$
$787$	45.9411	1.63762	0.818812	+	0.574062i	$0.194633\pi$
$788$	1.75736	0.0626033
$789$	0	0
$790$	−8.48528	−0.301893
$791$	10.5442	0.374907
$792$	0	0
$793$	36.9706	1.31286
$794$	18.9289	0.671763
$795$	0	0
$796$	4.82843	0.171139
$797$	−26.8284	−0.950312	−0.475156	−	0.879902i	$-0.657608\pi$
$797$	−26.8284	−0.950312	−0.475156	+	0.879902i	$0.657608\pi$
$798$	0	0
$799$	0	0
$800$	−6.65685	−0.235355
$801$	0	0
$802$	16.7279	0.590683
$803$	−26.3431	−0.929629
$804$	0	0
$805$	−24.9706	−0.880097
$806$	13.6569	0.481042
$807$	0	0
$808$	10.8284	0.380943
$809$	−20.5269	−0.721688	−0.360844	−	0.932626i	$-0.617511\pi$
$809$	−20.5269	−0.721688	−0.360844	+	0.932626i	$0.617511\pi$
$810$	0	0
$811$	−4.97056	−0.174540	−0.0872700	−	0.996185i	$-0.527814\pi$
$811$	−4.97056	−0.174540	−0.0872700	+	0.996185i	$0.527814\pi$
$812$	−2.82843	−0.0992583
$813$	0	0
$814$	−24.9706	−0.875218
$815$	−46.6274	−1.63329
$816$	0	0
$817$	11.3137	0.395817
$818$	14.0000	0.489499
$819$	0	0
$820$	37.7990	1.32000
$821$	−22.2426	−0.776274	−0.388137	−	0.921602i	$-0.626881\pi$
$821$	−22.2426	−0.776274	−0.388137	+	0.921602i	$0.626881\pi$
$822$	0	0
$823$	36.8284	1.28376	0.641879	−	0.766806i	$-0.278155\pi$
$823$	36.8284	1.28376	0.641879	+	0.766806i	$0.278155\pi$
$824$	−9.17157	−0.319507
$825$	0	0
$826$	−3.71573	−0.129287
$827$	29.9411	1.04115	0.520577	−	0.853814i	$-0.325717\pi$
$827$	29.9411	1.04115	0.520577	+	0.853814i	$0.325717\pi$
$828$	0	0
$829$	−39.9411	−1.38721	−0.693606	−	0.720354i	$-0.743979\pi$
$829$	−39.9411	−1.38721	−0.693606	+	0.720354i	$0.743979\pi$
$830$	42.6274	1.47962
$831$	0	0
$832$	−2.82843	−0.0980581
$833$	0	0
$834$	0	0
$835$	−60.7696	−2.10302
$836$	4.68629	0.162079
$837$	0	0
$838$	9.65685	0.333590
$839$	−14.4853	−0.500087	−0.250044	−	0.968235i	$-0.580445\pi$
$839$	−14.4853	−0.500087	−0.250044	+	0.968235i	$0.580445\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	−32.4853	−1.11952
$843$	0	0
$844$	4.97056	0.171094
$845$	17.0711	0.587263
$846$	0	0
$847$	−4.14214	−0.142325
$848$	2.82843	0.0971286
$849$	0	0
$850$	0	0
$851$	55.1127	1.88924
$852$	0	0
$853$	−11.6152	−0.397698	−0.198849	−	0.980030i	$-0.563720\pi$
$853$	−11.6152	−0.397698	−0.198849	+	0.980030i	$0.563720\pi$
$854$	−10.8284	−0.370541
$855$	0	0
$856$	2.34315	0.0800871
$857$	24.2426	0.828113	0.414056	−	0.910251i	$-0.364112\pi$
$857$	24.2426	0.828113	0.414056	+	0.910251i	$0.364112\pi$
$858$	0	0
$859$	9.65685	0.329488	0.164744	−	0.986336i	$-0.447320\pi$
$859$	9.65685	0.329488	0.164744	+	0.986336i	$0.447320\pi$
$860$	32.9706	1.12429
$861$	0	0
$862$	−8.14214	−0.277322
$863$	−5.65685	−0.192562	−0.0962808	−	0.995354i	$-0.530695\pi$
$863$	−5.65685	−0.192562	−0.0962808	+	0.995354i	$0.530695\pi$
$864$	0	0
$865$	−81.5980	−2.77441
$866$	−28.6274	−0.972799
$867$	0	0
$868$	−4.00000	−0.135769
$869$	9.94113	0.337230
$870$	0	0
$871$	−19.3137	−0.654420
$872$	−15.8995	−0.538425
$873$	0	0
$874$	−10.3431	−0.349862
$875$	4.68629	0.158426
$876$	0	0
$877$	31.8995	1.07717	0.538585	−	0.842571i	$-0.318959\pi$
$877$	31.8995	1.07717	0.538585	+	0.842571i	$0.318959\pi$
$878$	−30.4853	−1.02883
$879$	0	0
$880$	13.6569	0.460372
$881$	−13.2132	−0.445164	−0.222582	−	0.974914i	$-0.571449\pi$
$881$	−13.2132	−0.445164	−0.222582	+	0.974914i	$0.571449\pi$
$882$	0	0
$883$	24.7696	0.833562	0.416781	−	0.909007i	$-0.363158\pi$
$883$	24.7696	0.833562	0.416781	+	0.909007i	$0.363158\pi$
$884$	0	0
$885$	0	0
$886$	4.97056	0.166989
$887$	37.7990	1.26917	0.634583	−	0.772855i	$-0.281172\pi$
$887$	37.7990	1.26917	0.634583	+	0.772855i	$0.281172\pi$
$888$	0	0
$889$	−1.94113	−0.0651033
$890$	0	0
$891$	0	0
$892$	1.17157	0.0392272
$893$	5.25483	0.175846
$894$	0	0
$895$	71.5980	2.39326
$896$	0.828427	0.0276758
$897$	0	0
$898$	−36.0416	−1.20272
$899$	16.4853	0.549815
$900$	0	0
$901$	0	0
$902$	−44.2843	−1.47450
$903$	0	0
$904$	12.7279	0.423324
$905$	40.6274	1.35050
$906$	0	0
$907$	35.3137	1.17257	0.586286	−	0.810104i	$-0.300589\pi$
$907$	35.3137	1.17257	0.586286	+	0.810104i	$0.300589\pi$
$908$	−6.34315	−0.210505
$909$	0	0
$910$	8.00000	0.265197
$911$	33.1127	1.09707	0.548536	−	0.836127i	$-0.315185\pi$
$911$	33.1127	1.09707	0.548536	+	0.836127i	$0.315185\pi$
$912$	0	0
$913$	−49.9411	−1.65281
$914$	25.3137	0.837303
$915$	0	0
$916$	−14.0000	−0.462573
$917$	2.74517	0.0906534
$918$	0	0
$919$	−29.6569	−0.978289	−0.489145	−	0.872203i	$-0.662691\pi$
$919$	−29.6569	−0.978289	−0.489145	+	0.872203i	$0.662691\pi$
$920$	−30.1421	−0.993757
$921$	0	0
$922$	−25.4558	−0.838344
$923$	29.6569	0.976167
$924$	0	0
$925$	−41.5563	−1.36636
$926$	1.17157	0.0385003
$927$	0	0
$928$	−3.41421	−0.112077
$929$	45.4142	1.48999	0.744996	−	0.667069i	$-0.232451\pi$
$929$	45.4142	1.48999	0.744996	+	0.667069i	$0.232451\pi$
$930$	0	0
$931$	7.39697	0.242426
$932$	12.7279	0.416917
$933$	0	0
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	5.65685	0.184703
$939$	0	0
$940$	15.3137	0.499478
$941$	−25.7574	−0.839666	−0.419833	−	0.907601i	$-0.637911\pi$
$941$	−25.7574	−0.839666	−0.419833	+	0.907601i	$0.637911\pi$
$942$	0	0
$943$	97.7401	3.18286
$944$	−4.48528	−0.145983
$945$	0	0
$946$	−38.6274	−1.25589
$947$	−45.9411	−1.49289	−0.746443	−	0.665449i	$-0.768240\pi$
$947$	−45.9411	−1.49289	−0.746443	+	0.665449i	$0.768240\pi$
$948$	0	0
$949$	−18.6274	−0.604672
$950$	7.79899	0.253033
$951$	0	0
$952$	0	0
$953$	−40.9706	−1.32717	−0.663583	−	0.748102i	$-0.730965\pi$
$953$	−40.9706	−1.32717	−0.663583	+	0.748102i	$0.730965\pi$
$954$	0	0
$955$	−23.3137	−0.754414
$956$	4.48528	0.145064
$957$	0	0
$958$	12.8284	0.414468
$959$	−5.25483	−0.169687
$960$	0	0
$961$	−7.68629	−0.247945
$962$	−17.6569	−0.569280
$963$	0	0
$964$	−0.242641	−0.00781493
$965$	47.4558	1.52766
$966$	0	0
$967$	10.1421	0.326149	0.163075	−	0.986614i	$-0.447859\pi$
$967$	10.1421	0.326149	0.163075	+	0.986614i	$0.447859\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−36.1421	−1.16045
$971$	3.51472	0.112793	0.0563963	−	0.998408i	$-0.482039\pi$
$971$	3.51472	0.112793	0.0563963	+	0.998408i	$0.482039\pi$
$972$	0	0
$973$	14.6274	0.468933
$974$	−21.1127	−0.676494
$975$	0	0
$976$	−13.0711	−0.418395
$977$	−56.2843	−1.80069	−0.900347	−	0.435173i	$-0.856687\pi$
$977$	−56.2843	−1.80069	−0.900347	+	0.435173i	$0.856687\pi$
$978$	0	0
$979$	0	0
$980$	21.5563	0.688592
$981$	0	0
$982$	40.2843	1.28552
$983$	−0.828427	−0.0264227	−0.0132114	−	0.999913i	$-0.504205\pi$
$983$	−0.828427	−0.0264227	−0.0132114	+	0.999913i	$0.504205\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	3.31371	0.105423
$989$	85.2548	2.71095
$990$	0	0
$991$	37.5147	1.19169	0.595847	−	0.803098i	$-0.296816\pi$
$991$	37.5147	1.19169	0.595847	+	0.803098i	$0.296816\pi$
$992$	−4.82843	−0.153303
$993$	0	0
$994$	−8.68629	−0.275512
$995$	−16.4853	−0.522619
$996$	0	0
$997$	18.5269	0.586753	0.293377	−	0.955997i	$-0.405221\pi$
$997$	18.5269	0.586753	0.293377	+	0.955997i	$0.405221\pi$
$998$	8.97056	0.283958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.o.1.1		2
3.2	odd	2		1734.2.a.n.1.2		2
17.2	even	8		306.2.g.g.55.2		4
17.9	even	8		306.2.g.g.217.2		4
17.16	even	2		5202.2.a.x.1.2		2
51.2	odd	8		102.2.f.a.55.1	yes	4
51.8	odd	8		1734.2.f.i.829.2		4
51.26	odd	8		102.2.f.a.13.1	✓	4
51.32	odd	8		1734.2.f.i.1483.2		4
51.38	odd	4		1734.2.b.h.577.3		4
51.47	odd	4		1734.2.b.h.577.2		4
51.50	odd	2		1734.2.a.o.1.1		2
68.19	odd	8		2448.2.be.t.1585.2		4
68.43	odd	8		2448.2.be.t.1441.2		4
204.155	even	8		816.2.bd.a.769.2		4
204.179	even	8		816.2.bd.a.625.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.f.a.13.1	✓	4	51.26	odd	8
102.2.f.a.55.1	yes	4	51.2	odd	8
306.2.g.g.55.2		4	17.2	even	8
306.2.g.g.217.2		4	17.9	even	8
816.2.bd.a.625.2		4	204.179	even	8
816.2.bd.a.769.2		4	204.155	even	8
1734.2.a.n.1.2		2	3.2	odd	2
1734.2.a.o.1.1		2	51.50	odd	2
1734.2.b.h.577.2		4	51.47	odd	4
1734.2.b.h.577.3		4	51.38	odd	4
1734.2.f.i.829.2		4	51.8	odd	8
1734.2.f.i.1483.2		4	51.32	odd	8
2448.2.be.t.1441.2		4	68.43	odd	8
2448.2.be.t.1585.2		4	68.19	odd	8
5202.2.a.o.1.1		2	1.1	even	1	trivial
5202.2.a.x.1.2		2	17.16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -0.828427 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -0.828427 q^{7} -1.00000 q^{8} +3.41421 q^{10} -4.00000 q^{11} -2.82843 q^{13} +0.828427 q^{14} +1.00000 q^{16} -1.17157 q^{19} -3.41421 q^{20} +4.00000 q^{22} -8.82843 q^{23} +6.65685 q^{25} +2.82843 q^{26} -0.828427 q^{28} +3.41421 q^{29} +4.82843 q^{31} -1.00000 q^{32} +2.82843 q^{35} -6.24264 q^{37} +1.17157 q^{38} +3.41421 q^{40} -11.0711 q^{41} -9.65685 q^{43} -4.00000 q^{44} +8.82843 q^{46} -4.48528 q^{47} -6.31371 q^{49} -6.65685 q^{50} -2.82843 q^{52} +2.82843 q^{53} +13.6569 q^{55} +0.828427 q^{56} -3.41421 q^{58} -4.48528 q^{59} -13.0711 q^{61} -4.82843 q^{62} +1.00000 q^{64} +9.65685 q^{65} +6.82843 q^{67} -2.82843 q^{70} -10.4853 q^{71} +6.58579 q^{73} +6.24264 q^{74} -1.17157 q^{76} +3.31371 q^{77} -2.48528 q^{79} -3.41421 q^{80} +11.0711 q^{82} +12.4853 q^{83} +9.65685 q^{86} +4.00000 q^{88} +2.34315 q^{91} -8.82843 q^{92} +4.48528 q^{94} +4.00000 q^{95} -10.5858 q^{97} +6.31371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 4 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8} + 4 q^{10} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{20} + 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{37} + 8 q^{38} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 8 q^{44} + 12 q^{46} + 8 q^{47} + 10 q^{49} - 2 q^{50} + 16 q^{55} - 4 q^{56} - 4 q^{58} + 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} + 8 q^{65} + 8 q^{67} - 4 q^{71} + 16 q^{73} + 4 q^{74} - 8 q^{76} - 16 q^{77} + 12 q^{79} - 4 q^{80} + 8 q^{82} + 8 q^{83} + 8 q^{86} + 8 q^{88} + 16 q^{91} - 12 q^{92} - 8 q^{94} + 8 q^{95} - 24 q^{97} - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 4 * q^7 - 2 * q^8 + 4 * q^10 - 8 * q^11 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^20 + 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^28 + 4 * q^29 + 4 * q^31 - 2 * q^32 - 4 * q^37 + 8 * q^38 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 8 * q^44 + 12 * q^46 + 8 * q^47 + 10 * q^49 - 2 * q^50 + 16 * q^55 - 4 * q^56 - 4 * q^58 + 8 * q^59 - 12 * q^61 - 4 * q^62 + 2 * q^64 + 8 * q^65 + 8 * q^67 - 4 * q^71 + 16 * q^73 + 4 * q^74 - 8 * q^76 - 16 * q^77 + 12 * q^79 - 4 * q^80 + 8 * q^82 + 8 * q^83 + 8 * q^86 + 8 * q^88 + 16 * q^91 - 12 * q^92 - 8 * q^94 + 8 * q^95 - 24 * q^97 - 10 * q^98

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