LMFDB - Embedded newform 5202.2.a.bu.1.4

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:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*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			1.4
Root			$-0.765367$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.26197 q^{5} +2.94495 q^{7} +1.00000 q^{8} +1.26197 q^{10} -3.29769 q^{11} -3.36370 q^{13} +2.94495 q^{14} +1.00000 q^{16} +7.44155 q^{19} +1.26197 q^{20} -3.29769 q^{22} +1.41421 q^{23} -3.40743 q^{25} -3.36370 q^{26} +2.94495 q^{28} -0.522726 q^{29} +7.10973 q^{31} +1.00000 q^{32} +3.71644 q^{35} -0.792706 q^{37} +7.44155 q^{38} +1.26197 q^{40} -4.06306 q^{41} +0.867091 q^{43} -3.29769 q^{44} +1.41421 q^{46} +10.3086 q^{47} +1.67271 q^{49} -3.40743 q^{50} -3.36370 q^{52} +9.98868 q^{53} -4.16160 q^{55} +2.94495 q^{56} -0.522726 q^{58} +7.31543 q^{59} +10.7035 q^{61} +7.10973 q^{62} +1.00000 q^{64} -4.24489 q^{65} +13.5349 q^{67} +3.71644 q^{70} -7.38144 q^{71} -5.23880 q^{73} -0.792706 q^{74} +7.44155 q^{76} -9.71153 q^{77} +10.9449 q^{79} +1.26197 q^{80} -4.06306 q^{82} -12.3633 q^{83} +0.867091 q^{86} -3.29769 q^{88} +17.3975 q^{89} -9.90591 q^{91} +1.41421 q^{92} +10.3086 q^{94} +9.39104 q^{95} -6.22466 q^{97} +1.67271 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 8 q^{5} + 4 q^{8} - 8 q^{10} - 8 q^{11} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{22} + 12 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{32} + 8 q^{35} + 16 q^{37} + 8 q^{38} - 8 q^{40} - 8 q^{41}+ \cdots + 12 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 8 * q^5 + 4 * q^8 - 8 * q^10 - 8 * q^11 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 8 * q^22 + 12 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^32 + 8 * q^35 + 16 * q^37 + 8 * q^38 - 8 * q^40 - 8 * q^41 - 8 * q^44 + 16 * q^47 + 12 * q^49 + 12 * q^50 + 16 * q^53 - 16 * q^58 + 24 * q^59 + 8 * q^61 + 8 * q^62 + 4 * q^64 - 16 * q^65 + 8 * q^67 + 8 * q^70 - 16 * q^71 + 16 * q^73 + 16 * q^74 + 8 * q^76 + 16 * q^77 + 32 * q^79 - 8 * q^80 - 8 * q^82 + 8 * q^83 - 8 * q^88 + 16 * q^89 + 24 * q^91 + 16 * q^94 + 8 * q^95 + 12 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.26197	0.564371	0.282186	−	0.959360i	$-0.408941\pi$
$5$	1.26197	0.564371	0.282186	+	0.959360i	$0.408941\pi$
$6$	0	0
$7$	2.94495	1.11309	0.556543	−	0.830819i	$-0.312128\pi$
$7$	2.94495	1.11309	0.556543	+	0.830819i	$0.312128\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.26197	0.399071
$11$	−3.29769	−0.994292	−0.497146	−	0.867667i	$-0.665619\pi$
$11$	−3.29769	−0.994292	−0.497146	+	0.867667i	$0.665619\pi$
$12$	0	0
$13$	−3.36370	−0.932922	−0.466461	−	0.884542i	$-0.654471\pi$
$13$	−3.36370	−0.932922	−0.466461	+	0.884542i	$0.654471\pi$
$14$	2.94495	0.787070
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	7.44155	1.70721	0.853605	−	0.520921i	$-0.174412\pi$
$19$	7.44155	1.70721	0.853605	+	0.520921i	$0.174412\pi$
$20$	1.26197	0.282186
$21$	0	0
$22$	−3.29769	−0.703071
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	−3.40743	−0.681485
$26$	−3.36370	−0.659675
$27$	0	0
$28$	2.94495	0.556543
$29$	−0.522726	−0.0970678	−0.0485339	−	0.998822i	$-0.515455\pi$
$29$	−0.522726	−0.0970678	−0.0485339	+	0.998822i	$0.515455\pi$
$30$	0	0
$31$	7.10973	1.27695	0.638473	−	0.769644i	$-0.279566\pi$
$31$	7.10973	1.27695	0.638473	+	0.769644i	$0.279566\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	3.71644	0.628194
$36$	0	0
$37$	−0.792706	−0.130320	−0.0651601	−	0.997875i	$-0.520756\pi$
$37$	−0.792706	−0.130320	−0.0651601	+	0.997875i	$0.520756\pi$
$38$	7.44155	1.20718
$39$	0	0
$40$	1.26197	0.199535
$41$	−4.06306	−0.634543	−0.317272	−	0.948335i	$-0.602767\pi$
$41$	−4.06306	−0.634543	−0.317272	+	0.948335i	$0.602767\pi$
$42$	0	0
$43$	0.867091	0.132230	0.0661151	−	0.997812i	$-0.478940\pi$
$43$	0.867091	0.132230	0.0661151	+	0.997812i	$0.478940\pi$
$44$	−3.29769	−0.497146
$45$	0	0
$46$	1.41421	0.208514
$47$	10.3086	1.50367	0.751835	−	0.659351i	$-0.229169\pi$
$47$	10.3086	1.50367	0.751835	+	0.659351i	$0.229169\pi$
$48$	0	0
$49$	1.67271	0.238959
$50$	−3.40743	−0.481883
$51$	0	0
$52$	−3.36370	−0.466461
$53$	9.98868	1.37205	0.686025	−	0.727578i	$-0.259354\pi$
$53$	9.98868	1.37205	0.686025	+	0.727578i	$0.259354\pi$
$54$	0	0
$55$	−4.16160	−0.561150
$56$	2.94495	0.393535
$57$	0	0
$58$	−0.522726	−0.0686373
$59$	7.31543	0.952388	0.476194	−	0.879340i	$-0.342016\pi$
$59$	7.31543	0.952388	0.476194	+	0.879340i	$0.342016\pi$
$60$	0	0
$61$	10.7035	1.37045	0.685223	−	0.728333i	$-0.259705\pi$
$61$	10.7035	1.37045	0.685223	+	0.728333i	$0.259705\pi$
$62$	7.10973	0.902937
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.24489	−0.526514
$66$	0	0
$67$	13.5349	1.65355	0.826775	−	0.562532i	$-0.190173\pi$
$67$	13.5349	1.65355	0.826775	+	0.562532i	$0.190173\pi$
$68$	0	0
$69$	0	0
$70$	3.71644	0.444200
$71$	−7.38144	−0.876015	−0.438008	−	0.898971i	$-0.644316\pi$
$71$	−7.38144	−0.876015	−0.438008	+	0.898971i	$0.644316\pi$
$72$	0	0
$73$	−5.23880	−0.613155	−0.306577	−	0.951846i	$-0.599184\pi$
$73$	−5.23880	−0.613155	−0.306577	+	0.951846i	$0.599184\pi$
$74$	−0.792706	−0.0921502
$75$	0	0
$76$	7.44155	0.853605
$77$	−9.71153	−1.10673
$78$	0	0
$79$	10.9449	1.23140	0.615701	−	0.787980i	$-0.288873\pi$
$79$	10.9449	1.23140	0.615701	+	0.787980i	$0.288873\pi$
$80$	1.26197	0.141093
$81$	0	0
$82$	−4.06306	−0.448690
$83$	−12.3633	−1.35705	−0.678526	−	0.734577i	$-0.737381\pi$
$83$	−12.3633	−1.35705	−0.678526	+	0.734577i	$0.737381\pi$
$84$	0	0
$85$	0	0
$86$	0.867091	0.0935008
$87$	0	0
$88$	−3.29769	−0.351535
$89$	17.3975	1.84413	0.922063	−	0.387040i	$-0.126502\pi$
$89$	17.3975	1.84413	0.922063	+	0.387040i	$0.126502\pi$
$90$	0	0
$91$	−9.90591	−1.03842
$92$	1.41421	0.147442
$93$	0	0
$94$	10.3086	1.06326
$95$	9.39104	0.963500
$96$	0	0
$97$	−6.22466	−0.632018	−0.316009	−	0.948756i	$-0.602343\pi$
$97$	−6.22466	−0.632018	−0.316009	+	0.948756i	$0.602343\pi$
$98$	1.67271	0.168970
$99$	0	0
$100$	−3.40743	−0.340743
$101$	−9.88764	−0.983857	−0.491929	−	0.870636i	$-0.663708\pi$
$101$	−9.88764	−0.983857	−0.491929	+	0.870636i	$0.663708\pi$
$102$	0	0
$103$	6.54168	0.644571	0.322286	−	0.946642i	$-0.395549\pi$
$103$	6.54168	0.644571	0.322286	+	0.946642i	$0.395549\pi$
$104$	−3.36370	−0.329838
$105$	0	0
$106$	9.98868	0.970186
$107$	3.49207	0.337591	0.168796	−	0.985651i	$-0.446012\pi$
$107$	3.49207	0.337591	0.168796	+	0.985651i	$0.446012\pi$
$108$	0	0
$109$	15.8751	1.52056	0.760279	−	0.649596i	$-0.225062\pi$
$109$	15.8751	1.52056	0.760279	+	0.649596i	$0.225062\pi$
$110$	−4.16160	−0.396793
$111$	0	0
$112$	2.94495	0.278271
$113$	−19.3827	−1.82337	−0.911683	−	0.410893i	$-0.865217\pi$
$113$	−19.3827	−1.82337	−0.911683	+	0.410893i	$0.865217\pi$
$114$	0	0
$115$	1.78470	0.166424
$116$	−0.522726	−0.0485339
$117$	0	0
$118$	7.31543	0.673440
$119$	0	0
$120$	0	0
$121$	−0.125218	−0.0113835
$122$	10.7035	0.969052
$123$	0	0
$124$	7.10973	0.638473
$125$	−10.6099	−0.948982
$126$	0	0
$127$	17.7830	1.57798	0.788992	−	0.614404i	$-0.210603\pi$
$127$	17.7830	1.57798	0.788992	+	0.614404i	$0.210603\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.24489	−0.372302
$131$	−15.2127	−1.32914	−0.664569	−	0.747227i	$-0.731385\pi$
$131$	−15.2127	−1.32914	−0.664569	+	0.747227i	$0.731385\pi$
$132$	0	0
$133$	21.9150	1.90027
$134$	13.5349	1.16924
$135$	0	0
$136$	0	0
$137$	−3.61766	−0.309078	−0.154539	−	0.987987i	$-0.549389\pi$
$137$	−3.61766	−0.309078	−0.154539	+	0.987987i	$0.549389\pi$
$138$	0	0
$139$	−20.2446	−1.71712	−0.858560	−	0.512713i	$-0.828641\pi$
$139$	−20.2446	−1.71712	−0.858560	+	0.512713i	$0.828641\pi$
$140$	3.71644	0.314097
$141$	0	0
$142$	−7.38144	−0.619436
$143$	11.0924	0.927596
$144$	0	0
$145$	−0.659666	−0.0547823
$146$	−5.23880	−0.433566
$147$	0	0
$148$	−0.792706	−0.0651601
$149$	−7.73418	−0.633609	−0.316804	−	0.948491i	$-0.602610\pi$
$149$	−7.73418	−0.633609	−0.316804	+	0.948491i	$0.602610\pi$
$150$	0	0
$151$	1.02280	0.0832346	0.0416173	−	0.999134i	$-0.486749\pi$
$151$	1.02280	0.0832346	0.0416173	+	0.999134i	$0.486749\pi$
$152$	7.44155	0.603590
$153$	0	0
$154$	−9.71153	−0.782578
$155$	8.97229	0.720671
$156$	0	0
$157$	23.7502	1.89547	0.947736	−	0.319055i	$-0.103366\pi$
$157$	23.7502	1.89547	0.947736	+	0.319055i	$0.103366\pi$
$158$	10.9449	0.870733
$159$	0	0
$160$	1.26197	0.0997677
$161$	4.16478	0.328231
$162$	0	0
$163$	−3.22037	−0.252239	−0.126119	−	0.992015i	$-0.540252\pi$
$163$	−3.22037	−0.252239	−0.126119	+	0.992015i	$0.540252\pi$
$164$	−4.06306	−0.317272
$165$	0	0
$166$	−12.3633	−0.959580
$167$	−4.68231	−0.362328	−0.181164	−	0.983453i	$-0.557987\pi$
$167$	−4.68231	−0.362328	−0.181164	+	0.983453i	$0.557987\pi$
$168$	0	0
$169$	−1.68554	−0.129657
$170$	0	0
$171$	0	0
$172$	0.867091	0.0661151
$173$	−17.4428	−1.32615	−0.663075	−	0.748553i	$-0.730749\pi$
$173$	−17.4428	−1.32615	−0.663075	+	0.748553i	$0.730749\pi$
$174$	0	0
$175$	−10.0347	−0.758551
$176$	−3.29769	−0.248573
$177$	0	0
$178$	17.3975	1.30399
$179$	8.76606	0.655206	0.327603	−	0.944815i	$-0.393759\pi$
$179$	8.76606	0.655206	0.327603	+	0.944815i	$0.393759\pi$
$180$	0	0
$181$	−3.78273	−0.281168	−0.140584	−	0.990069i	$-0.544898\pi$
$181$	−3.78273	−0.281168	−0.140584	+	0.990069i	$0.544898\pi$
$182$	−9.90591	−0.734275
$183$	0	0
$184$	1.41421	0.104257
$185$	−1.00037	−0.0735489
$186$	0	0
$187$	0	0
$188$	10.3086	0.751835
$189$	0	0
$190$	9.39104	0.681297
$191$	0.371014	0.0268456	0.0134228	−	0.999910i	$-0.495727\pi$
$191$	0.371014	0.0268456	0.0134228	+	0.999910i	$0.495727\pi$
$192$	0	0
$193$	1.04026	0.0748793	0.0374397	−	0.999299i	$-0.488080\pi$
$193$	1.04026	0.0748793	0.0374397	+	0.999299i	$0.488080\pi$
$194$	−6.22466	−0.446904
$195$	0	0
$196$	1.67271	0.119480
$197$	5.29573	0.377305	0.188652	−	0.982044i	$-0.439588\pi$
$197$	5.29573	0.377305	0.188652	+	0.982044i	$0.439588\pi$
$198$	0	0
$199$	−21.2738	−1.50806	−0.754029	−	0.656841i	$-0.771892\pi$
$199$	−21.2738	−1.50806	−0.754029	+	0.656841i	$0.771892\pi$
$200$	−3.40743	−0.240941
$201$	0	0
$202$	−9.88764	−0.695692
$203$	−1.53940	−0.108045
$204$	0	0
$205$	−5.12747	−0.358118
$206$	6.54168	0.455781
$207$	0	0
$208$	−3.36370	−0.233230
$209$	−24.5400	−1.69746
$210$	0	0
$211$	12.9900	0.894270	0.447135	−	0.894466i	$-0.352444\pi$
$211$	12.9900	0.894270	0.447135	+	0.894466i	$0.352444\pi$
$212$	9.98868	0.686025
$213$	0	0
$214$	3.49207	0.238713
$215$	1.09425	0.0746269
$216$	0	0
$217$	20.9378	1.42135
$218$	15.8751	1.07520
$219$	0	0
$220$	−4.16160	−0.280575
$221$	0	0
$222$	0	0
$223$	−3.88573	−0.260208	−0.130104	−	0.991500i	$-0.541531\pi$
$223$	−3.88573	−0.260208	−0.130104	+	0.991500i	$0.541531\pi$
$224$	2.94495	0.196768
$225$	0	0
$226$	−19.3827	−1.28932
$227$	−4.38875	−0.291292	−0.145646	−	0.989337i	$-0.546526\pi$
$227$	−4.38875	−0.291292	−0.145646	+	0.989337i	$0.546526\pi$
$228$	0	0
$229$	−3.42063	−0.226041	−0.113021	−	0.993593i	$-0.536053\pi$
$229$	−3.42063	−0.226041	−0.113021	+	0.993593i	$0.536053\pi$
$230$	1.78470	0.117680
$231$	0	0
$232$	−0.522726	−0.0343187
$233$	0.893211	0.0585162	0.0292581	−	0.999572i	$-0.490686\pi$
$233$	0.893211	0.0585162	0.0292581	+	0.999572i	$0.490686\pi$
$234$	0	0
$235$	13.0092	0.848628
$236$	7.31543	0.476194
$237$	0	0
$238$	0	0
$239$	−0.389157	−0.0251725	−0.0125863	−	0.999921i	$-0.504006\pi$
$239$	−0.389157	−0.0251725	−0.0125863	+	0.999921i	$0.504006\pi$
$240$	0	0
$241$	15.7554	1.01489	0.507447	−	0.861683i	$-0.330589\pi$
$241$	15.7554	1.01489	0.507447	+	0.861683i	$0.330589\pi$
$242$	−0.125218	−0.00804934
$243$	0	0
$244$	10.7035	0.685223
$245$	2.11092	0.134862
$246$	0	0
$247$	−25.0311	−1.59269
$248$	7.10973	0.451468
$249$	0	0
$250$	−10.6099	−0.671032
$251$	−15.2423	−0.962083	−0.481042	−	0.876698i	$-0.659741\pi$
$251$	−15.2423	−0.962083	−0.481042	+	0.876698i	$0.659741\pi$
$252$	0	0
$253$	−4.66364	−0.293201
$254$	17.7830	1.11580
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.05696	0.440201	0.220101	−	0.975477i	$-0.429361\pi$
$257$	7.05696	0.440201	0.220101	+	0.975477i	$0.429361\pi$
$258$	0	0
$259$	−2.33448	−0.145057
$260$	−4.24489	−0.263257
$261$	0	0
$262$	−15.2127	−0.939842
$263$	−5.53808	−0.341493	−0.170746	−	0.985315i	$-0.554618\pi$
$263$	−5.53808	−0.341493	−0.170746	+	0.985315i	$0.554618\pi$
$264$	0	0
$265$	12.6054	0.774346
$266$	21.9150	1.34369
$267$	0	0
$268$	13.5349	0.826775
$269$	4.31477	0.263076	0.131538	−	0.991311i	$-0.458008\pi$
$269$	4.31477	0.263076	0.131538	+	0.991311i	$0.458008\pi$
$270$	0	0
$271$	−4.45569	−0.270664	−0.135332	−	0.990800i	$-0.543210\pi$
$271$	−4.45569	−0.270664	−0.135332	+	0.990800i	$0.543210\pi$
$272$	0	0
$273$	0	0
$274$	−3.61766	−0.218551
$275$	11.2366	0.677595
$276$	0	0
$277$	14.0382	0.843471	0.421736	−	0.906719i	$-0.361421\pi$
$277$	14.0382	0.843471	0.421736	+	0.906719i	$0.361421\pi$
$278$	−20.2446	−1.21419
$279$	0	0
$280$	3.71644	0.222100
$281$	−8.67913	−0.517753	−0.258877	−	0.965910i	$-0.583352\pi$
$281$	−8.67913	−0.517753	−0.258877	+	0.965910i	$0.583352\pi$
$282$	0	0
$283$	−14.3623	−0.853753	−0.426876	−	0.904310i	$-0.640386\pi$
$283$	−14.3623	−0.853753	−0.426876	+	0.904310i	$0.640386\pi$
$284$	−7.38144	−0.438008
$285$	0	0
$286$	11.0924	0.655910
$287$	−11.9655	−0.706301
$288$	0	0
$289$	0	0
$290$	−0.659666	−0.0387369
$291$	0	0
$292$	−5.23880	−0.306577
$293$	−22.0243	−1.28668	−0.643338	−	0.765582i	$-0.722451\pi$
$293$	−22.0243	−1.28668	−0.643338	+	0.765582i	$0.722451\pi$
$294$	0	0
$295$	9.23188	0.537501
$296$	−0.792706	−0.0460751
$297$	0	0
$298$	−7.73418	−0.448029
$299$	−4.75699	−0.275104
$300$	0	0
$301$	2.55354	0.147183
$302$	1.02280	0.0588557
$303$	0	0
$304$	7.44155	0.426802
$305$	13.5076	0.773440
$306$	0	0
$307$	−7.13291	−0.407097	−0.203548	−	0.979065i	$-0.565247\pi$
$307$	−7.13291	−0.407097	−0.203548	+	0.979065i	$0.565247\pi$
$308$	−9.71153	−0.553366
$309$	0	0
$310$	8.97229	0.509592
$311$	−19.1212	−1.08427	−0.542133	−	0.840293i	$-0.682383\pi$
$311$	−19.1212	−1.08427	−0.542133	+	0.840293i	$0.682383\pi$
$312$	0	0
$313$	−13.0225	−0.736076	−0.368038	−	0.929811i	$-0.619970\pi$
$313$	−13.0225	−0.736076	−0.368038	+	0.929811i	$0.619970\pi$
$314$	23.7502	1.34030
$315$	0	0
$316$	10.9449	0.615701
$317$	−4.55648	−0.255917	−0.127959	−	0.991779i	$-0.540843\pi$
$317$	−4.55648	−0.255917	−0.127959	+	0.991779i	$0.540843\pi$
$318$	0	0
$319$	1.72379	0.0965137
$320$	1.26197	0.0705464
$321$	0	0
$322$	4.16478	0.232094
$323$	0	0
$324$	0	0
$325$	11.4615	0.635772
$326$	−3.22037	−0.178360
$327$	0	0
$328$	−4.06306	−0.224345
$329$	30.3584	1.67371
$330$	0	0
$331$	−18.7047	−1.02811	−0.514053	−	0.857758i	$-0.671856\pi$
$331$	−18.7047	−1.02811	−0.514053	+	0.857758i	$0.671856\pi$
$332$	−12.3633	−0.678526
$333$	0	0
$334$	−4.68231	−0.256205
$335$	17.0807	0.933217
$336$	0	0
$337$	−23.8756	−1.30059	−0.650294	−	0.759683i	$-0.725354\pi$
$337$	−23.8756	−1.30059	−0.650294	+	0.759683i	$0.725354\pi$
$338$	−1.68554	−0.0916815
$339$	0	0
$340$	0	0
$341$	−23.4457	−1.26966
$342$	0	0
$343$	−15.6886	−0.847103
$344$	0.867091	0.0467504
$345$	0	0
$346$	−17.4428	−0.937729
$347$	24.9127	1.33738	0.668692	−	0.743540i	$-0.266855\pi$
$347$	24.9127	1.33738	0.668692	+	0.743540i	$0.266855\pi$
$348$	0	0
$349$	−19.5194	−1.04485	−0.522425	−	0.852685i	$-0.674973\pi$
$349$	−19.5194	−1.04485	−0.522425	+	0.852685i	$0.674973\pi$
$350$	−10.0347	−0.536377
$351$	0	0
$352$	−3.29769	−0.175768
$353$	25.7115	1.36849	0.684243	−	0.729254i	$-0.260133\pi$
$353$	25.7115	1.36849	0.684243	+	0.729254i	$0.260133\pi$
$354$	0	0
$355$	−9.31517	−0.494398
$356$	17.3975	0.922063
$357$	0	0
$358$	8.76606	0.463301
$359$	−19.2482	−1.01588	−0.507939	−	0.861393i	$-0.669593\pi$
$359$	−19.2482	−1.01588	−0.507939	+	0.861393i	$0.669593\pi$
$360$	0	0
$361$	36.3767	1.91456
$362$	−3.78273	−0.198816
$363$	0	0
$364$	−9.90591	−0.519211
$365$	−6.61122	−0.346047
$366$	0	0
$367$	5.08708	0.265544	0.132772	−	0.991147i	$-0.457612\pi$
$367$	5.08708	0.265544	0.132772	+	0.991147i	$0.457612\pi$
$368$	1.41421	0.0737210
$369$	0	0
$370$	−1.00037	−0.0520070
$371$	29.4161	1.52721
$372$	0	0
$373$	21.9719	1.13766	0.568831	−	0.822454i	$-0.307396\pi$
$373$	21.9719	1.13766	0.568831	+	0.822454i	$0.307396\pi$
$374$	0	0
$375$	0	0
$376$	10.3086	0.531628
$377$	1.75829	0.0905567
$378$	0	0
$379$	−10.0798	−0.517763	−0.258881	−	0.965909i	$-0.583354\pi$
$379$	−10.0798	−0.517763	−0.258881	+	0.965909i	$0.583354\pi$
$380$	9.39104	0.481750
$381$	0	0
$382$	0.371014	0.0189827
$383$	27.9738	1.42940	0.714698	−	0.699433i	$-0.246564\pi$
$383$	27.9738	1.42940	0.714698	+	0.699433i	$0.246564\pi$
$384$	0	0
$385$	−12.2557	−0.624608
$386$	1.04026	0.0529477
$387$	0	0
$388$	−6.22466	−0.316009
$389$	12.3355	0.625432	0.312716	−	0.949847i	$-0.398761\pi$
$389$	12.3355	0.625432	0.312716	+	0.949847i	$0.398761\pi$
$390$	0	0
$391$	0	0
$392$	1.67271	0.0844848
$393$	0	0
$394$	5.29573	0.266795
$395$	13.8122	0.694968
$396$	0	0
$397$	7.10626	0.356653	0.178326	−	0.983971i	$-0.442932\pi$
$397$	7.10626	0.356653	0.178326	+	0.983971i	$0.442932\pi$
$398$	−21.2738	−1.06636
$399$	0	0
$400$	−3.40743	−0.170371
$401$	29.6691	1.48160	0.740801	−	0.671725i	$-0.234446\pi$
$401$	29.6691	1.48160	0.740801	+	0.671725i	$0.234446\pi$
$402$	0	0
$403$	−23.9150	−1.19129
$404$	−9.88764	−0.491929
$405$	0	0
$406$	−1.53940	−0.0763992
$407$	2.61410	0.129576
$408$	0	0
$409$	14.8922	0.736371	0.368185	−	0.929752i	$-0.379979\pi$
$409$	14.8922	0.736371	0.368185	+	0.929752i	$0.379979\pi$
$410$	−5.12747	−0.253228
$411$	0	0
$412$	6.54168	0.322286
$413$	21.5436	1.06009
$414$	0	0
$415$	−15.6022	−0.765881
$416$	−3.36370	−0.164919
$417$	0	0
$418$	−24.5400	−1.20029
$419$	26.2549	1.28264	0.641319	−	0.767274i	$-0.278388\pi$
$419$	26.2549	1.28264	0.641319	+	0.767274i	$0.278388\pi$
$420$	0	0
$421$	16.6683	0.812365	0.406182	−	0.913792i	$-0.366860\pi$
$421$	16.6683	0.812365	0.406182	+	0.913792i	$0.366860\pi$
$422$	12.9900	0.632345
$423$	0	0
$424$	9.98868	0.485093
$425$	0	0
$426$	0	0
$427$	31.5213	1.52542
$428$	3.49207	0.168796
$429$	0	0
$430$	1.09425	0.0527692
$431$	−4.86443	−0.234312	−0.117156	−	0.993114i	$-0.537378\pi$
$431$	−4.86443	−0.234312	−0.117156	+	0.993114i	$0.537378\pi$
$432$	0	0
$433$	2.20663	0.106044	0.0530220	−	0.998593i	$-0.483115\pi$
$433$	2.20663	0.106044	0.0530220	+	0.998593i	$0.483115\pi$
$434$	20.9378	1.00505
$435$	0	0
$436$	15.8751	0.760279
$437$	10.5239	0.503429
$438$	0	0
$439$	9.89180	0.472110	0.236055	−	0.971740i	$-0.424145\pi$
$439$	9.89180	0.472110	0.236055	+	0.971740i	$0.424145\pi$
$440$	−4.16160	−0.198396
$441$	0	0
$442$	0	0
$443$	−5.50565	−0.261581	−0.130791	−	0.991410i	$-0.541752\pi$
$443$	−5.50565	−0.261581	−0.130791	+	0.991410i	$0.541752\pi$
$444$	0	0
$445$	21.9551	1.04077
$446$	−3.88573	−0.183995
$447$	0	0
$448$	2.94495	0.139136
$449$	33.2572	1.56951	0.784753	−	0.619808i	$-0.212790\pi$
$449$	33.2572	1.56951	0.784753	+	0.619808i	$0.212790\pi$
$450$	0	0
$451$	13.3987	0.630921
$452$	−19.3827	−0.911683
$453$	0	0
$454$	−4.38875	−0.205974
$455$	−12.5010	−0.586055
$456$	0	0
$457$	−17.4812	−0.817736	−0.408868	−	0.912594i	$-0.634076\pi$
$457$	−17.4812	−0.817736	−0.408868	+	0.912594i	$0.634076\pi$
$458$	−3.42063	−0.159835
$459$	0	0
$460$	1.78470	0.0832120
$461$	14.5385	0.677123	0.338562	−	0.940944i	$-0.390060\pi$
$461$	14.5385	0.677123	0.338562	+	0.940944i	$0.390060\pi$
$462$	0	0
$463$	−1.47703	−0.0686435	−0.0343217	−	0.999411i	$-0.510927\pi$
$463$	−1.47703	−0.0686435	−0.0343217	+	0.999411i	$0.510927\pi$
$464$	−0.522726	−0.0242670
$465$	0	0
$466$	0.893211	0.0413772
$467$	−8.68873	−0.402066	−0.201033	−	0.979584i	$-0.564430\pi$
$467$	−8.68873	−0.402066	−0.201033	+	0.979584i	$0.564430\pi$
$468$	0	0
$469$	39.8596	1.84054
$470$	13.0092	0.600071
$471$	0	0
$472$	7.31543	0.336720
$473$	−2.85940	−0.131475
$474$	0	0
$475$	−25.3565	−1.16344
$476$	0	0
$477$	0	0
$478$	−0.389157	−0.0177996
$479$	−10.9463	−0.500151	−0.250075	−	0.968226i	$-0.580455\pi$
$479$	−10.9463	−0.500151	−0.250075	+	0.968226i	$0.580455\pi$
$480$	0	0
$481$	2.66642	0.121578
$482$	15.7554	0.717638
$483$	0	0
$484$	−0.125218	−0.00569174
$485$	−7.85535	−0.356693
$486$	0	0
$487$	−8.75751	−0.396841	−0.198420	−	0.980117i	$-0.563581\pi$
$487$	−8.75751	−0.396841	−0.198420	+	0.980117i	$0.563581\pi$
$488$	10.7035	0.484526
$489$	0	0
$490$	2.11092	0.0953616
$491$	−30.4170	−1.37270	−0.686351	−	0.727271i	$-0.740788\pi$
$491$	−30.4170	−1.37270	−0.686351	+	0.727271i	$0.740788\pi$
$492$	0	0
$493$	0	0
$494$	−25.0311	−1.12620
$495$	0	0
$496$	7.10973	0.319236
$497$	−21.7379	−0.975080
$498$	0	0
$499$	22.3333	0.999777	0.499889	−	0.866090i	$-0.333374\pi$
$499$	22.3333	0.999777	0.499889	+	0.866090i	$0.333374\pi$
$500$	−10.6099	−0.474491
$501$	0	0
$502$	−15.2423	−0.680296
$503$	42.3520	1.88838	0.944191	−	0.329398i	$-0.106846\pi$
$503$	42.3520	1.88838	0.944191	+	0.329398i	$0.106846\pi$
$504$	0	0
$505$	−12.4779	−0.555261
$506$	−4.66364	−0.207324
$507$	0	0
$508$	17.7830	0.788992
$509$	9.03882	0.400639	0.200319	−	0.979731i	$-0.435802\pi$
$509$	9.03882	0.400639	0.200319	+	0.979731i	$0.435802\pi$
$510$	0	0
$511$	−15.4280	−0.682493
$512$	1.00000	0.0441942
$513$	0	0
$514$	7.05696	0.311269
$515$	8.25543	0.363778
$516$	0	0
$517$	−33.9947	−1.49509
$518$	−2.33448	−0.102571
$519$	0	0
$520$	−4.24489	−0.186151
$521$	−33.4356	−1.46484	−0.732420	−	0.680853i	$-0.761609\pi$
$521$	−33.4356	−1.46484	−0.732420	+	0.680853i	$0.761609\pi$
$522$	0	0
$523$	−23.9751	−1.04836	−0.524180	−	0.851608i	$-0.675628\pi$
$523$	−23.9751	−1.04836	−0.524180	+	0.851608i	$0.675628\pi$
$524$	−15.2127	−0.664569
$525$	0	0
$526$	−5.53808	−0.241472
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	12.6054	0.547545
$531$	0	0
$532$	21.9150	0.950135
$533$	13.6669	0.591979
$534$	0	0
$535$	4.40690	0.190527
$536$	13.5349	0.584618
$537$	0	0
$538$	4.31477	0.186023
$539$	−5.51610	−0.237595
$540$	0	0
$541$	−17.5884	−0.756182	−0.378091	−	0.925768i	$-0.623419\pi$
$541$	−17.5884	−0.756182	−0.378091	+	0.925768i	$0.623419\pi$
$542$	−4.45569	−0.191388
$543$	0	0
$544$	0	0
$545$	20.0339	0.858160
$546$	0	0
$547$	4.79884	0.205183	0.102592	−	0.994724i	$-0.467286\pi$
$547$	4.79884	0.205183	0.102592	+	0.994724i	$0.467286\pi$
$548$	−3.61766	−0.154539
$549$	0	0
$550$	11.2366	0.479132
$551$	−3.88989	−0.165715
$552$	0	0
$553$	32.2323	1.37066
$554$	14.0382	0.596424
$555$	0	0
$556$	−20.2446	−0.858560
$557$	27.0366	1.14558	0.572788	−	0.819704i	$-0.305862\pi$
$557$	27.0366	1.14558	0.572788	+	0.819704i	$0.305862\pi$
$558$	0	0
$559$	−2.91663	−0.123360
$560$	3.71644	0.157048
$561$	0	0
$562$	−8.67913	−0.366107
$563$	−38.3143	−1.61475	−0.807377	−	0.590036i	$-0.799114\pi$
$563$	−38.3143	−1.61475	−0.807377	+	0.590036i	$0.799114\pi$
$564$	0	0
$565$	−24.4604	−1.02906
$566$	−14.3623	−0.603694
$567$	0	0
$568$	−7.38144	−0.309718
$569$	−18.6210	−0.780633	−0.390317	−	0.920681i	$-0.627634\pi$
$569$	−18.6210	−0.780633	−0.390317	+	0.920681i	$0.627634\pi$
$570$	0	0
$571$	3.86349	0.161682	0.0808410	−	0.996727i	$-0.474239\pi$
$571$	3.86349	0.161682	0.0808410	+	0.996727i	$0.474239\pi$
$572$	11.0924	0.463798
$573$	0	0
$574$	−11.9655	−0.499430
$575$	−4.81883	−0.200959
$576$	0	0
$577$	−25.4772	−1.06063	−0.530315	−	0.847801i	$-0.677926\pi$
$577$	−25.4772	−1.06063	−0.530315	+	0.847801i	$0.677926\pi$
$578$	0	0
$579$	0	0
$580$	−0.659666	−0.0273911
$581$	−36.4093	−1.51051
$582$	0	0
$583$	−32.9396	−1.36422
$584$	−5.23880	−0.216783
$585$	0	0
$586$	−22.0243	−0.909817
$587$	0.531636	0.0219430	0.0109715	−	0.999940i	$-0.496508\pi$
$587$	0.531636	0.0219430	0.0109715	+	0.999940i	$0.496508\pi$
$588$	0	0
$589$	52.9074	2.18001
$590$	9.23188	0.380070
$591$	0	0
$592$	−0.792706	−0.0325800
$593$	12.1266	0.497982	0.248991	−	0.968506i	$-0.419901\pi$
$593$	12.1266	0.497982	0.248991	+	0.968506i	$0.419901\pi$
$594$	0	0
$595$	0	0
$596$	−7.73418	−0.316804
$597$	0	0
$598$	−4.75699	−0.194528
$599$	−14.6476	−0.598486	−0.299243	−	0.954177i	$-0.596734\pi$
$599$	−14.6476	−0.598486	−0.299243	+	0.954177i	$0.596734\pi$
$600$	0	0
$601$	−26.2961	−1.07264	−0.536320	−	0.844015i	$-0.680186\pi$
$601$	−26.2961	−1.07264	−0.536320	+	0.844015i	$0.680186\pi$
$602$	2.55354	0.104074
$603$	0	0
$604$	1.02280	0.0416173
$605$	−0.158022	−0.00642451
$606$	0	0
$607$	24.3315	0.987584	0.493792	−	0.869580i	$-0.335610\pi$
$607$	24.3315	0.987584	0.493792	+	0.869580i	$0.335610\pi$
$608$	7.44155	0.301795
$609$	0	0
$610$	13.5076	0.546905
$611$	−34.6752	−1.40281
$612$	0	0
$613$	−8.21114	−0.331645	−0.165822	−	0.986156i	$-0.553028\pi$
$613$	−8.21114	−0.331645	−0.165822	+	0.986156i	$0.553028\pi$
$614$	−7.13291	−0.287861
$615$	0	0
$616$	−9.71153	−0.391289
$617$	−44.5518	−1.79359	−0.896793	−	0.442450i	$-0.854109\pi$
$617$	−44.5518	−1.79359	−0.896793	+	0.442450i	$0.854109\pi$
$618$	0	0
$619$	21.8112	0.876668	0.438334	−	0.898812i	$-0.355569\pi$
$619$	21.8112	0.876668	0.438334	+	0.898812i	$0.355569\pi$
$620$	8.97229	0.360336
$621$	0	0
$622$	−19.1212	−0.766692
$623$	51.2346	2.05267
$624$	0	0
$625$	3.64767	0.145907
$626$	−13.0225	−0.520484
$627$	0	0
$628$	23.7502	0.947736
$629$	0	0
$630$	0	0
$631$	−41.7021	−1.66014	−0.830068	−	0.557663i	$-0.811698\pi$
$631$	−41.7021	−1.66014	−0.830068	+	0.557663i	$0.811698\pi$
$632$	10.9449	0.435367
$633$	0	0
$634$	−4.55648	−0.180961
$635$	22.4416	0.890569
$636$	0	0
$637$	−5.62650	−0.222930
$638$	1.72379	0.0682455
$639$	0	0
$640$	1.26197	0.0498838
$641$	−13.7067	−0.541381	−0.270691	−	0.962666i	$-0.587252\pi$
$641$	−13.7067	−0.541381	−0.270691	+	0.962666i	$0.587252\pi$
$642$	0	0
$643$	−21.8390	−0.861247	−0.430624	−	0.902532i	$-0.641706\pi$
$643$	−21.8390	−0.861247	−0.430624	+	0.902532i	$0.641706\pi$
$644$	4.16478	0.164115
$645$	0	0
$646$	0	0
$647$	21.2760	0.836447	0.418223	−	0.908344i	$-0.362653\pi$
$647$	21.2760	0.836447	0.418223	+	0.908344i	$0.362653\pi$
$648$	0	0
$649$	−24.1241	−0.946952
$650$	11.4615	0.449559
$651$	0	0
$652$	−3.22037	−0.126119
$653$	−33.1843	−1.29860	−0.649301	−	0.760531i	$-0.724938\pi$
$653$	−33.1843	−1.29860	−0.649301	+	0.760531i	$0.724938\pi$
$654$	0	0
$655$	−19.1980	−0.750127
$656$	−4.06306	−0.158636
$657$	0	0
$658$	30.3584	1.18349
$659$	0.649089	0.0252849	0.0126425	−	0.999920i	$-0.495976\pi$
$659$	0.649089	0.0252849	0.0126425	+	0.999920i	$0.495976\pi$
$660$	0	0
$661$	−30.8996	−1.20185	−0.600927	−	0.799304i	$-0.705202\pi$
$661$	−30.8996	−1.20185	−0.600927	+	0.799304i	$0.705202\pi$
$662$	−18.7047	−0.726981
$663$	0	0
$664$	−12.3633	−0.479790
$665$	27.6561	1.07246
$666$	0	0
$667$	−0.739246	−0.0286237
$668$	−4.68231	−0.181164
$669$	0	0
$670$	17.0807	0.659884
$671$	−35.2969	−1.36262
$672$	0	0
$673$	−30.3508	−1.16994	−0.584968	−	0.811056i	$-0.698893\pi$
$673$	−30.3508	−1.16994	−0.584968	+	0.811056i	$0.698893\pi$
$674$	−23.8756	−0.919655
$675$	0	0
$676$	−1.68554	−0.0648286
$677$	6.53702	0.251238	0.125619	−	0.992079i	$-0.459908\pi$
$677$	6.53702	0.251238	0.125619	+	0.992079i	$0.459908\pi$
$678$	0	0
$679$	−18.3313	−0.703490
$680$	0	0
$681$	0	0
$682$	−23.4457	−0.897783
$683$	−10.6469	−0.407392	−0.203696	−	0.979034i	$-0.565295\pi$
$683$	−10.6469	−0.407392	−0.203696	+	0.979034i	$0.565295\pi$
$684$	0	0
$685$	−4.56539	−0.174435
$686$	−15.6886	−0.598993
$687$	0	0
$688$	0.867091	0.0330575
$689$	−33.5989	−1.28002
$690$	0	0
$691$	−19.7603	−0.751718	−0.375859	−	0.926677i	$-0.622652\pi$
$691$	−19.7603	−0.751718	−0.375859	+	0.926677i	$0.622652\pi$
$692$	−17.4428	−0.663075
$693$	0	0
$694$	24.9127	0.945673
$695$	−25.5481	−0.969093
$696$	0	0
$697$	0	0
$698$	−19.5194	−0.738821
$699$	0	0
$700$	−10.0347	−0.379276
$701$	−39.7169	−1.50009	−0.750043	−	0.661390i	$-0.769967\pi$
$701$	−39.7169	−1.50009	−0.750043	+	0.661390i	$0.769967\pi$
$702$	0	0
$703$	−5.89897	−0.222484
$704$	−3.29769	−0.124286
$705$	0	0
$706$	25.7115	0.967666
$707$	−29.1186	−1.09512
$708$	0	0
$709$	44.9863	1.68950	0.844749	−	0.535163i	$-0.179750\pi$
$709$	44.9863	1.68950	0.844749	+	0.535163i	$0.179750\pi$
$710$	−9.31517	−0.349592
$711$	0	0
$712$	17.3975	0.651997
$713$	10.0547	0.376551
$714$	0	0
$715$	13.9984	0.523509
$716$	8.76606	0.327603
$717$	0	0
$718$	−19.2482	−0.718335
$719$	34.9648	1.30397	0.651983	−	0.758233i	$-0.273937\pi$
$719$	34.9648	1.30397	0.651983	+	0.758233i	$0.273937\pi$
$720$	0	0
$721$	19.2649	0.717463
$722$	36.3767	1.35380
$723$	0	0
$724$	−3.78273	−0.140584
$725$	1.78115	0.0661503
$726$	0	0
$727$	−9.34543	−0.346603	−0.173301	−	0.984869i	$-0.555443\pi$
$727$	−9.34543	−0.346603	−0.173301	+	0.984869i	$0.555443\pi$
$728$	−9.90591	−0.367137
$729$	0	0
$730$	−6.61122	−0.244692
$731$	0	0
$732$	0	0
$733$	35.0025	1.29285	0.646423	−	0.762979i	$-0.276264\pi$
$733$	35.0025	1.29285	0.646423	+	0.762979i	$0.276264\pi$
$734$	5.08708	0.187768
$735$	0	0
$736$	1.41421	0.0521286
$737$	−44.6339	−1.64411
$738$	0	0
$739$	20.3466	0.748460	0.374230	−	0.927336i	$-0.377907\pi$
$739$	20.3466	0.748460	0.374230	+	0.927336i	$0.377907\pi$
$740$	−1.00037	−0.0367745
$741$	0	0
$742$	29.4161	1.07990
$743$	−19.1359	−0.702027	−0.351014	−	0.936370i	$-0.614163\pi$
$743$	−19.1359	−0.702027	−0.351014	+	0.936370i	$0.614163\pi$
$744$	0	0
$745$	−9.76033	−0.357591
$746$	21.9719	0.804449
$747$	0	0
$748$	0	0
$749$	10.2840	0.375768
$750$	0	0
$751$	−42.2934	−1.54331	−0.771654	−	0.636043i	$-0.780570\pi$
$751$	−42.2934	−1.54331	−0.771654	+	0.636043i	$0.780570\pi$
$752$	10.3086	0.375918
$753$	0	0
$754$	1.75829	0.0640332
$755$	1.29075	0.0469752
$756$	0	0
$757$	46.3076	1.68308	0.841538	−	0.540197i	$-0.181650\pi$
$757$	46.3076	1.68308	0.841538	+	0.540197i	$0.181650\pi$
$758$	−10.0798	−0.366114
$759$	0	0
$760$	9.39104	0.340649
$761$	2.15386	0.0780775	0.0390388	−	0.999238i	$-0.487570\pi$
$761$	2.15386	0.0780775	0.0390388	+	0.999238i	$0.487570\pi$
$762$	0	0
$763$	46.7513	1.69251
$764$	0.371014	0.0134228
$765$	0	0
$766$	27.9738	1.01074
$767$	−24.6069	−0.888504
$768$	0	0
$769$	−22.8143	−0.822705	−0.411353	−	0.911476i	$-0.634944\pi$
$769$	−22.8143	−0.822705	−0.411353	+	0.911476i	$0.634944\pi$
$770$	−12.2557	−0.441664
$771$	0	0
$772$	1.04026	0.0374397
$773$	−14.4131	−0.518403	−0.259201	−	0.965823i	$-0.583459\pi$
$773$	−14.4131	−0.518403	−0.259201	+	0.965823i	$0.583459\pi$
$774$	0	0
$775$	−24.2259	−0.870219
$776$	−6.22466	−0.223452
$777$	0	0
$778$	12.3355	0.442247
$779$	−30.2355	−1.08330
$780$	0	0
$781$	24.3417	0.871015
$782$	0	0
$783$	0	0
$784$	1.67271	0.0597398
$785$	29.9721	1.06975
$786$	0	0
$787$	−1.65367	−0.0589469	−0.0294735	−	0.999566i	$-0.509383\pi$
$787$	−1.65367	−0.0589469	−0.0294735	+	0.999566i	$0.509383\pi$
$788$	5.29573	0.188652
$789$	0	0
$790$	13.8122	0.491417
$791$	−57.0809	−2.02956
$792$	0	0
$793$	−36.0034	−1.27852
$794$	7.10626	0.252192
$795$	0	0
$796$	−21.2738	−0.754029
$797$	37.4699	1.32725	0.663625	−	0.748065i	$-0.269017\pi$
$797$	37.4699	1.32725	0.663625	+	0.748065i	$0.269017\pi$
$798$	0	0
$799$	0	0
$800$	−3.40743	−0.120471
$801$	0	0
$802$	29.6691	1.04765
$803$	17.2759	0.609655
$804$	0	0
$805$	5.25584	0.185244
$806$	−23.9150	−0.842369
$807$	0	0
$808$	−9.88764	−0.347846
$809$	−51.2959	−1.80347	−0.901734	−	0.432291i	$-0.857705\pi$
$809$	−51.2959	−1.80347	−0.901734	+	0.432291i	$0.857705\pi$
$810$	0	0
$811$	1.19528	0.0419719	0.0209860	−	0.999780i	$-0.493319\pi$
$811$	1.19528	0.0419719	0.0209860	+	0.999780i	$0.493319\pi$
$812$	−1.53940	−0.0540224
$813$	0	0
$814$	2.61410	0.0916242
$815$	−4.06401	−0.142356
$816$	0	0
$817$	6.45250	0.225745
$818$	14.8922	0.520693
$819$	0	0
$820$	−5.12747	−0.179059
$821$	14.3412	0.500511	0.250255	−	0.968180i	$-0.419485\pi$
$821$	14.3412	0.500511	0.250255	+	0.968180i	$0.419485\pi$
$822$	0	0
$823$	45.4114	1.58294	0.791471	−	0.611207i	$-0.209316\pi$
$823$	45.4114	1.58294	0.791471	+	0.611207i	$0.209316\pi$
$824$	6.54168	0.227890
$825$	0	0
$826$	21.5436	0.749596
$827$	−23.4799	−0.816475	−0.408238	−	0.912876i	$-0.633857\pi$
$827$	−23.4799	−0.816475	−0.408238	+	0.912876i	$0.633857\pi$
$828$	0	0
$829$	−4.28183	−0.148714	−0.0743571	−	0.997232i	$-0.523690\pi$
$829$	−4.28183	−0.148714	−0.0743571	+	0.997232i	$0.523690\pi$
$830$	−15.6022	−0.541559
$831$	0	0
$832$	−3.36370	−0.116615
$833$	0	0
$834$	0	0
$835$	−5.90895	−0.204488
$836$	−24.5400	−0.848732
$837$	0	0
$838$	26.2549	0.906962
$839$	29.6465	1.02351	0.511756	−	0.859131i	$-0.328995\pi$
$839$	29.6465	1.02351	0.511756	+	0.859131i	$0.328995\pi$
$840$	0	0
$841$	−28.7268	−0.990578
$842$	16.6683	0.574429
$843$	0	0
$844$	12.9900	0.447135
$845$	−2.12711	−0.0731748
$846$	0	0
$847$	−0.368761	−0.0126708
$848$	9.98868	0.343013
$849$	0	0
$850$	0	0
$851$	−1.12106	−0.0384293
$852$	0	0
$853$	−1.16274	−0.0398116	−0.0199058	−	0.999802i	$-0.506337\pi$
$853$	−1.16274	−0.0398116	−0.0199058	+	0.999802i	$0.506337\pi$
$854$	31.5213	1.07864
$855$	0	0
$856$	3.49207	0.119356
$857$	−38.8166	−1.32595	−0.662975	−	0.748642i	$-0.730706\pi$
$857$	−38.8166	−1.32595	−0.662975	+	0.748642i	$0.730706\pi$
$858$	0	0
$859$	11.0615	0.377412	0.188706	−	0.982034i	$-0.439571\pi$
$859$	11.0615	0.377412	0.188706	+	0.982034i	$0.439571\pi$
$860$	1.09425	0.0373134
$861$	0	0
$862$	−4.86443	−0.165683
$863$	−8.01331	−0.272776	−0.136388	−	0.990655i	$-0.543549\pi$
$863$	−8.01331	−0.272776	−0.136388	+	0.990655i	$0.543549\pi$
$864$	0	0
$865$	−22.0123	−0.748441
$866$	2.20663	0.0749845
$867$	0	0
$868$	20.9378	0.710675
$869$	−36.0931	−1.22437
$870$	0	0
$871$	−45.5273	−1.54263
$872$	15.8751	0.537599
$873$	0	0
$874$	10.5239	0.355978
$875$	−31.2457	−1.05630
$876$	0	0
$877$	14.8632	0.501894	0.250947	−	0.968001i	$-0.419258\pi$
$877$	14.8632	0.501894	0.250947	+	0.968001i	$0.419258\pi$
$878$	9.89180	0.333832
$879$	0	0
$880$	−4.16160	−0.140287
$881$	−9.72089	−0.327505	−0.163753	−	0.986501i	$-0.552360\pi$
$881$	−9.72089	−0.327505	−0.163753	+	0.986501i	$0.552360\pi$
$882$	0	0
$883$	21.2137	0.713896	0.356948	−	0.934124i	$-0.383817\pi$
$883$	21.2137	0.713896	0.356948	+	0.934124i	$0.383817\pi$
$884$	0	0
$885$	0	0
$886$	−5.50565	−0.184966
$887$	1.24492	0.0418005	0.0209002	−	0.999782i	$-0.493347\pi$
$887$	1.24492	0.0418005	0.0209002	+	0.999782i	$0.493347\pi$
$888$	0	0
$889$	52.3699	1.75643
$890$	21.9551	0.735937
$891$	0	0
$892$	−3.88573	−0.130104
$893$	76.7123	2.56708
$894$	0	0
$895$	11.0625	0.369779
$896$	2.94495	0.0983838
$897$	0	0
$898$	33.2572	1.10981
$899$	−3.71644	−0.123950
$900$	0	0
$901$	0	0
$902$	13.3987	0.446129
$903$	0	0
$904$	−19.3827	−0.644658
$905$	−4.77370	−0.158683
$906$	0	0
$907$	16.2330	0.539009	0.269505	−	0.962999i	$-0.413140\pi$
$907$	16.2330	0.539009	0.269505	+	0.962999i	$0.413140\pi$
$908$	−4.38875	−0.145646
$909$	0	0
$910$	−12.5010	−0.414404
$911$	−19.2058	−0.636316	−0.318158	−	0.948038i	$-0.603064\pi$
$911$	−19.2058	−0.636316	−0.318158	+	0.948038i	$0.603064\pi$
$912$	0	0
$913$	40.7704	1.34930
$914$	−17.4812	−0.578226
$915$	0	0
$916$	−3.42063	−0.113021
$917$	−44.8005	−1.47944
$918$	0	0
$919$	−35.7959	−1.18080	−0.590398	−	0.807112i	$-0.701029\pi$
$919$	−35.7959	−1.18080	−0.590398	+	0.807112i	$0.701029\pi$
$920$	1.78470	0.0588398
$921$	0	0
$922$	14.5385	0.478799
$923$	24.8289	0.817254
$924$	0	0
$925$	2.70109	0.0888112
$926$	−1.47703	−0.0485383
$927$	0	0
$928$	−0.522726	−0.0171593
$929$	−38.9490	−1.27788	−0.638938	−	0.769258i	$-0.720626\pi$
$929$	−38.9490	−1.27788	−0.638938	+	0.769258i	$0.720626\pi$
$930$	0	0
$931$	12.4476	0.407953
$932$	0.893211	0.0292581
$933$	0	0
$934$	−8.68873	−0.284304
$935$	0	0
$936$	0	0
$937$	−49.5538	−1.61885	−0.809426	−	0.587222i	$-0.800222\pi$
$937$	−49.5538	−1.61885	−0.809426	+	0.587222i	$0.800222\pi$
$938$	39.8596	1.30146
$939$	0	0
$940$	13.0092	0.424314
$941$	−3.44941	−0.112447	−0.0562237	−	0.998418i	$-0.517906\pi$
$941$	−3.44941	−0.112447	−0.0562237	+	0.998418i	$0.517906\pi$
$942$	0	0
$943$	−5.74603	−0.187117
$944$	7.31543	0.238097
$945$	0	0
$946$	−2.85940	−0.0929671
$947$	−21.5300	−0.699631	−0.349815	−	0.936819i	$-0.613756\pi$
$947$	−21.5300	−0.699631	−0.349815	+	0.936819i	$0.613756\pi$
$948$	0	0
$949$	17.6217	0.572025
$950$	−25.3565	−0.822675
$951$	0	0
$952$	0	0
$953$	40.4171	1.30924	0.654619	−	0.755959i	$-0.272829\pi$
$953$	40.4171	1.30924	0.654619	+	0.755959i	$0.272829\pi$
$954$	0	0
$955$	0.468209	0.0151509
$956$	−0.389157	−0.0125863
$957$	0	0
$958$	−10.9463	−0.353660
$959$	−10.6538	−0.344030
$960$	0	0
$961$	19.5483	0.630590
$962$	2.66642	0.0859690
$963$	0	0
$964$	15.7554	0.507447
$965$	1.31278	0.0422597
$966$	0	0
$967$	−19.1408	−0.615528	−0.307764	−	0.951463i	$-0.599581\pi$
$967$	−19.1408	−0.615528	−0.307764	+	0.951463i	$0.599581\pi$
$968$	−0.125218	−0.00402467
$969$	0	0
$970$	−7.85535	−0.252220
$971$	52.0327	1.66981	0.834905	−	0.550394i	$-0.185522\pi$
$971$	52.0327	1.66981	0.834905	+	0.550394i	$0.185522\pi$
$972$	0	0
$973$	−59.6191	−1.91130
$974$	−8.75751	−0.280609
$975$	0	0
$976$	10.7035	0.342612
$977$	−50.2515	−1.60769	−0.803845	−	0.594839i	$-0.797216\pi$
$977$	−50.2515	−1.60769	−0.803845	+	0.594839i	$0.797216\pi$
$978$	0	0
$979$	−57.3715	−1.83360
$980$	2.11092	0.0674309
$981$	0	0
$982$	−30.4170	−0.970646
$983$	−36.8882	−1.17655	−0.588275	−	0.808661i	$-0.700193\pi$
$983$	−36.8882	−1.17655	−0.588275	+	0.808661i	$0.700193\pi$
$984$	0	0
$985$	6.68306	0.212940
$986$	0	0
$987$	0	0
$988$	−25.0311	−0.796346
$989$	1.22625	0.0389925
$990$	0	0
$991$	−0.650857	−0.0206751	−0.0103376	−	0.999947i	$-0.503291\pi$
$991$	−0.650857	−0.0206751	−0.0103376	+	0.999947i	$0.503291\pi$
$992$	7.10973	0.225734
$993$	0	0
$994$	−21.7379	−0.689486
$995$	−26.8469	−0.851104
$996$	0	0
$997$	−49.6033	−1.57095	−0.785476	−	0.618892i	$-0.787582\pi$
$997$	−49.6033	−1.57095	−0.785476	+	0.618892i	$0.787582\pi$
$998$	22.3333	0.706949
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.bu.1.4		4
3.2	odd	2		1734.2.a.t.1.1		4
17.5	odd	16		306.2.l.e.127.2		8
17.7	odd	16		306.2.l.e.253.2		8
17.16	even	2		5202.2.a.bx.1.1		4
51.2	odd	8		1734.2.f.l.1483.2		8
51.5	even	16		102.2.h.b.25.2	✓	8
51.8	odd	8		1734.2.f.k.829.3		8
51.26	odd	8		1734.2.f.l.829.2		8
51.32	odd	8		1734.2.f.k.1483.3		8
51.38	odd	4		1734.2.b.l.577.8		8
51.41	even	16		102.2.h.b.49.2	yes	8
51.47	odd	4		1734.2.b.l.577.1		8
51.50	odd	2		1734.2.a.u.1.4		4
204.107	odd	16		816.2.bq.c.433.1		8
204.143	odd	16		816.2.bq.c.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.h.b.25.2	✓	8	51.5	even	16
102.2.h.b.49.2	yes	8	51.41	even	16
306.2.l.e.127.2		8	17.5	odd	16
306.2.l.e.253.2		8	17.7	odd	16
816.2.bq.c.49.1		8	204.143	odd	16
816.2.bq.c.433.1		8	204.107	odd	16
1734.2.a.t.1.1		4	3.2	odd	2
1734.2.a.u.1.4		4	51.50	odd	2
1734.2.b.l.577.1		8	51.47	odd	4
1734.2.b.l.577.8		8	51.38	odd	4
1734.2.f.k.829.3		8	51.8	odd	8
1734.2.f.k.1483.3		8	51.32	odd	8
1734.2.f.l.829.2		8	51.26	odd	8
1734.2.f.l.1483.2		8	51.2	odd	8
5202.2.a.bu.1.4		4	1.1	even	1	trivial
5202.2.a.bx.1.1		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.26197 q^{5} +2.94495 q^{7} +1.00000 q^{8} +1.26197 q^{10} -3.29769 q^{11} -3.36370 q^{13} +2.94495 q^{14} +1.00000 q^{16} +7.44155 q^{19} +1.26197 q^{20} -3.29769 q^{22} +1.41421 q^{23} -3.40743 q^{25} -3.36370 q^{26} +2.94495 q^{28} -0.522726 q^{29} +7.10973 q^{31} +1.00000 q^{32} +3.71644 q^{35} -0.792706 q^{37} +7.44155 q^{38} +1.26197 q^{40} -4.06306 q^{41} +0.867091 q^{43} -3.29769 q^{44} +1.41421 q^{46} +10.3086 q^{47} +1.67271 q^{49} -3.40743 q^{50} -3.36370 q^{52} +9.98868 q^{53} -4.16160 q^{55} +2.94495 q^{56} -0.522726 q^{58} +7.31543 q^{59} +10.7035 q^{61} +7.10973 q^{62} +1.00000 q^{64} -4.24489 q^{65} +13.5349 q^{67} +3.71644 q^{70} -7.38144 q^{71} -5.23880 q^{73} -0.792706 q^{74} +7.44155 q^{76} -9.71153 q^{77} +10.9449 q^{79} +1.26197 q^{80} -4.06306 q^{82} -12.3633 q^{83} +0.867091 q^{86} -3.29769 q^{88} +17.3975 q^{89} -9.90591 q^{91} +1.41421 q^{92} +10.3086 q^{94} +9.39104 q^{95} -6.22466 q^{97} +1.67271 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 8 q^{5} + 4 q^{8} - 8 q^{10} - 8 q^{11} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{22} + 12 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{32} + 8 q^{35} + 16 q^{37} + 8 q^{38} - 8 q^{40} - 8 q^{41}+ \cdots + 12 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 8 * q^5 + 4 * q^8 - 8 * q^10 - 8 * q^11 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 8 * q^22 + 12 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^32 + 8 * q^35 + 16 * q^37 + 8 * q^38 - 8 * q^40 - 8 * q^41 - 8 * q^44 + 16 * q^47 + 12 * q^49 + 12 * q^50 + 16 * q^53 - 16 * q^58 + 24 * q^59 + 8 * q^61 + 8 * q^62 + 4 * q^64 - 16 * q^65 + 8 * q^67 + 8 * q^70 - 16 * q^71 + 16 * q^73 + 16 * q^74 + 8 * q^76 + 16 * q^77 + 32 * q^79 - 8 * q^80 - 8 * q^82 + 8 * q^83 - 8 * q^88 + 16 * q^89 + 24 * q^91 + 16 * q^94 + 8 * q^95 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more