LMFDB - Embedded newform 5202.2.a.o.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.585786 q^{5} +4.82843 q^{7} -1.00000 q^{8} +0.585786 q^{10} -4.00000 q^{11} +2.82843 q^{13} -4.82843 q^{14} +1.00000 q^{16} -6.82843 q^{19} -0.585786 q^{20} +4.00000 q^{22} -3.17157 q^{23} -4.65685 q^{25} -2.82843 q^{26} +4.82843 q^{28} +0.585786 q^{29} -0.828427 q^{31} -1.00000 q^{32} -2.82843 q^{35} +2.24264 q^{37} +6.82843 q^{38} +0.585786 q^{40} +3.07107 q^{41} +1.65685 q^{43} -4.00000 q^{44} +3.17157 q^{46} +12.4853 q^{47} +16.3137 q^{49} +4.65685 q^{50} +2.82843 q^{52} -2.82843 q^{53} +2.34315 q^{55} -4.82843 q^{56} -0.585786 q^{58} +12.4853 q^{59} +1.07107 q^{61} +0.828427 q^{62} +1.00000 q^{64} -1.65685 q^{65} +1.17157 q^{67} +2.82843 q^{70} +6.48528 q^{71} +9.41421 q^{73} -2.24264 q^{74} -6.82843 q^{76} -19.3137 q^{77} +14.4853 q^{79} -0.585786 q^{80} -3.07107 q^{82} -4.48528 q^{83} -1.65685 q^{86} +4.00000 q^{88} +13.6569 q^{91} -3.17157 q^{92} -12.4853 q^{94} +4.00000 q^{95} -13.4142 q^{97} -16.3137 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8} + 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}+ \cdots - 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

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
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.585786	0.185242
$11$	−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	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	−4.82843	−1.29045
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	−0.585786	−0.130986
$21$	0	0
$22$	4.00000	0.852803
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	−2.82843	−0.554700
$27$	0	0
$28$	4.82843	0.912487
$29$	0.585786	0.108778	0.0543889	−	0.998520i	$-0.482679\pi$
$29$	0.585786	0.108778	0.0543889	+	0.998520i	$0.482679\pi$
$30$	0	0
$31$	−0.828427	−0.148790	−0.0743950	−	0.997229i	$-0.523703\pi$
$31$	−0.828427	−0.148790	−0.0743950	+	0.997229i	$0.523703\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	2.24264	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$37$	2.24264	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$38$	6.82843	1.10772
$39$	0	0
$40$	0.585786	0.0926210
$41$	3.07107	0.479620	0.239810	−	0.970820i	$-0.422915\pi$
$41$	3.07107	0.479620	0.239810	+	0.970820i	$0.422915\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	3.17157	0.467623
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	4.65685	0.658579
$51$	0	0
$52$	2.82843	0.392232
$53$	−2.82843	−0.388514	−0.194257	−	0.980951i	$-0.562230\pi$
$53$	−2.82843	−0.388514	−0.194257	+	0.980951i	$0.562230\pi$
$54$	0	0
$55$	2.34315	0.315950
$56$	−4.82843	−0.645226
$57$	0	0
$58$	−0.585786	−0.0769175
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	0	0
$61$	1.07107	0.137136	0.0685681	−	0.997646i	$-0.478157\pi$
$61$	1.07107	0.137136	0.0685681	+	0.997646i	$0.478157\pi$
$62$	0.828427	0.105210
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.65685	−0.205507
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	0	0
$69$	0	0
$70$	2.82843	0.338062
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	−2.24264	−0.260702
$75$	0	0
$76$	−6.82843	−0.783274
$77$	−19.3137	−2.20100
$78$	0	0
$79$	14.4853	1.62972	0.814861	−	0.579657i	$-0.196813\pi$
$79$	14.4853	1.62972	0.814861	+	0.579657i	$0.196813\pi$
$80$	−0.585786	−0.0654929
$81$	0	0
$82$	−3.07107	−0.339143
$83$	−4.48528	−0.492324	−0.246162	−	0.969229i	$-0.579169\pi$
$83$	−4.48528	−0.492324	−0.246162	+	0.969229i	$0.579169\pi$
$84$	0	0
$85$	0	0
$86$	−1.65685	−0.178663
$87$	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
$91$	13.6569	1.43163
$92$	−3.17157	−0.330659
$93$	0	0
$94$	−12.4853	−1.28776
$95$	4.00000	0.410391
$96$	0	0
$97$	−13.4142	−1.36201	−0.681004	−	0.732280i	$-0.738456\pi$
$97$	−13.4142	−1.36201	−0.681004	+	0.732280i	$0.738456\pi$
$98$	−16.3137	−1.64793
$99$	0	0
$100$	−4.65685	−0.465685
$101$	−5.17157	−0.514591	−0.257295	−	0.966333i	$-0.582831\pi$
$101$	−5.17157	−0.514591	−0.257295	+	0.966333i	$0.582831\pi$
$102$	0	0
$103$	14.8284	1.46109	0.730544	−	0.682865i	$-0.239266\pi$
$103$	14.8284	1.46109	0.730544	+	0.682865i	$0.239266\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	2.82843	0.274721
$107$	−13.6569	−1.32026	−0.660129	−	0.751152i	$-0.729498\pi$
$107$	−13.6569	−1.32026	−0.660129	+	0.751152i	$0.729498\pi$
$108$	0	0
$109$	−3.89949	−0.373504	−0.186752	−	0.982407i	$-0.559796\pi$
$109$	−3.89949	−0.373504	−0.186752	+	0.982407i	$0.559796\pi$
$110$	−2.34315	−0.223410
$111$	0	0
$112$	4.82843	0.456243
$113$	12.7279	1.19734	0.598671	−	0.800995i	$-0.295696\pi$
$113$	12.7279	1.19734	0.598671	+	0.800995i	$0.295696\pi$
$114$	0	0
$115$	1.85786	0.173247
$116$	0.585786	0.0543889
$117$	0	0
$118$	−12.4853	−1.14936
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−1.07107	−0.0969699
$123$	0	0
$124$	−0.828427	−0.0743950
$125$	5.65685	0.505964
$126$	0	0
$127$	13.6569	1.21185	0.605925	−	0.795522i	$-0.292803\pi$
$127$	13.6569	1.21185	0.605925	+	0.795522i	$0.292803\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.65685	0.145316
$131$	19.3137	1.68745	0.843723	−	0.536778i	$-0.180359\pi$
$131$	19.3137	1.68745	0.843723	+	0.536778i	$0.180359\pi$
$132$	0	0
$133$	−32.9706	−2.85891
$134$	−1.17157	−0.101208
$135$	0	0
$136$	0	0
$137$	17.6569	1.50853	0.754263	−	0.656572i	$-0.227994\pi$
$137$	17.6569	1.50853	0.754263	+	0.656572i	$0.227994\pi$
$138$	0	0
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	−2.82843	−0.239046
$141$	0	0
$142$	−6.48528	−0.544233
$143$	−11.3137	−0.946100
$144$	0	0
$145$	−0.343146	−0.0284967
$146$	−9.41421	−0.779126
$147$	0	0
$148$	2.24264	0.184344
$149$	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
$151$	13.6569	1.11138	0.555690	−	0.831390i	$-0.312454\pi$
$151$	13.6569	1.11138	0.555690	+	0.831390i	$0.312454\pi$
$152$	6.82843	0.553859
$153$	0	0
$154$	19.3137	1.55634
$155$	0.485281	0.0389787
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−14.4853	−1.15239
$159$	0	0
$160$	0.585786	0.0463105
$161$	−15.3137	−1.20689
$162$	0	0
$163$	2.34315	0.183529	0.0917647	−	0.995781i	$-0.470749\pi$
$163$	2.34315	0.183529	0.0917647	+	0.995781i	$0.470749\pi$
$164$	3.07107	0.239810
$165$	0	0
$166$	4.48528	0.348125
$167$	−21.7990	−1.68686	−0.843428	−	0.537242i	$-0.819466\pi$
$167$	−21.7990	−1.68686	−0.843428	+	0.537242i	$0.819466\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	1.65685	0.126334
$173$	4.10051	0.311756	0.155878	−	0.987776i	$-0.450179\pi$
$173$	4.10051	0.311756	0.155878	+	0.987776i	$0.450179\pi$
$174$	0	0
$175$	−22.4853	−1.69973
$176$	−4.00000	−0.301511
$177$	0	0
$178$	0	0
$179$	12.9706	0.969465	0.484733	−	0.874662i	$-0.338917\pi$
$179$	12.9706	0.969465	0.484733	+	0.874662i	$0.338917\pi$
$180$	0	0
$181$	7.89949	0.587165	0.293582	−	0.955934i	$-0.405152\pi$
$181$	7.89949	0.587165	0.293582	+	0.955934i	$0.405152\pi$
$182$	−13.6569	−1.01231
$183$	0	0
$184$	3.17157	0.233811
$185$	−1.31371	−0.0965858
$186$	0	0
$187$	0	0
$188$	12.4853	0.910583
$189$	0	0
$190$	−4.00000	−0.290191
$191$	1.17157	0.0847720	0.0423860	−	0.999101i	$-0.486504\pi$
$191$	1.17157	0.0847720	0.0423860	+	0.999101i	$0.486504\pi$
$192$	0	0
$193$	5.89949	0.424655	0.212327	−	0.977199i	$-0.431896\pi$
$193$	5.89949	0.424655	0.212327	+	0.977199i	$0.431896\pi$
$194$	13.4142	0.963084
$195$	0	0
$196$	16.3137	1.16526
$197$	10.2426	0.729758	0.364879	−	0.931055i	$-0.381110\pi$
$197$	10.2426	0.729758	0.364879	+	0.931055i	$0.381110\pi$
$198$	0	0
$199$	−0.828427	−0.0587256	−0.0293628	−	0.999569i	$-0.509348\pi$
$199$	−0.828427	−0.0587256	−0.0293628	+	0.999569i	$0.509348\pi$
$200$	4.65685	0.329289
$201$	0	0
$202$	5.17157	0.363871
$203$	2.82843	0.198517
$204$	0	0
$205$	−1.79899	−0.125647
$206$	−14.8284	−1.03315
$207$	0	0
$208$	2.82843	0.196116
$209$	27.3137	1.88933
$210$	0	0
$211$	−28.9706	−1.99442	−0.997208	−	0.0746754i	$-0.976208\pi$
$211$	−28.9706	−1.99442	−0.997208	+	0.0746754i	$0.976208\pi$
$212$	−2.82843	−0.194257
$213$	0	0
$214$	13.6569	0.933563
$215$	−0.970563	−0.0661918
$216$	0	0
$217$	−4.00000	−0.271538
$218$	3.89949	0.264107
$219$	0	0
$220$	2.34315	0.157975
$221$	0	0
$222$	0	0
$223$	6.82843	0.457265	0.228633	−	0.973513i	$-0.426575\pi$
$223$	6.82843	0.457265	0.228633	+	0.973513i	$0.426575\pi$
$224$	−4.82843	−0.322613
$225$	0	0
$226$	−12.7279	−0.846649
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.85786	−0.122504
$231$	0	0
$232$	−0.585786	−0.0384588
$233$	−12.7279	−0.833834	−0.416917	−	0.908945i	$-0.636889\pi$
$233$	−12.7279	−0.833834	−0.416917	+	0.908945i	$0.636889\pi$
$234$	0	0
$235$	−7.31371	−0.477094
$236$	12.4853	0.812723
$237$	0	0
$238$	0	0
$239$	−12.4853	−0.807606	−0.403803	−	0.914846i	$-0.632312\pi$
$239$	−12.4853	−0.807606	−0.403803	+	0.914846i	$0.632312\pi$
$240$	0	0
$241$	8.24264	0.530955	0.265478	−	0.964117i	$-0.414470\pi$
$241$	8.24264	0.530955	0.265478	+	0.964117i	$0.414470\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	1.07107	0.0685681
$245$	−9.55635	−0.610533
$246$	0	0
$247$	−19.3137	−1.22890
$248$	0.828427	0.0526052
$249$	0	0
$250$	−5.65685	−0.357771
$251$	17.6569	1.11449	0.557245	−	0.830348i	$-0.311858\pi$
$251$	17.6569	1.11449	0.557245	+	0.830348i	$0.311858\pi$
$252$	0	0
$253$	12.6863	0.797580
$254$	−13.6569	−0.856907
$255$	0	0
$256$	1.00000	0.0625000
$257$	29.3137	1.82854	0.914269	−	0.405107i	$-0.132766\pi$
$257$	29.3137	1.82854	0.914269	+	0.405107i	$0.132766\pi$
$258$	0	0
$259$	10.8284	0.672846
$260$	−1.65685	−0.102754
$261$	0	0
$262$	−19.3137	−1.19320
$263$	−10.1421	−0.625391	−0.312695	−	0.949853i	$-0.601232\pi$
$263$	−10.1421	−0.625391	−0.312695	+	0.949853i	$0.601232\pi$
$264$	0	0
$265$	1.65685	0.101780
$266$	32.9706	2.02155
$267$	0	0
$268$	1.17157	0.0715652
$269$	−5.07107	−0.309188	−0.154594	−	0.987978i	$-0.549407\pi$
$269$	−5.07107	−0.309188	−0.154594	+	0.987978i	$0.549407\pi$
$270$	0	0
$271$	21.6569	1.31556	0.657780	−	0.753210i	$-0.271496\pi$
$271$	21.6569	1.31556	0.657780	+	0.753210i	$0.271496\pi$
$272$	0	0
$273$	0	0
$274$	−17.6569	−1.06669
$275$	18.6274	1.12328
$276$	0	0
$277$	−9.75736	−0.586263	−0.293131	−	0.956072i	$-0.594697\pi$
$277$	−9.75736	−0.586263	−0.293131	+	0.956072i	$0.594697\pi$
$278$	6.34315	0.380437
$279$	0	0
$280$	2.82843	0.169031
$281$	19.3137	1.15216	0.576080	−	0.817394i	$-0.304582\pi$
$281$	19.3137	1.15216	0.576080	+	0.817394i	$0.304582\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	6.48528	0.384831
$285$	0	0
$286$	11.3137	0.668994
$287$	14.8284	0.875294
$288$	0	0
$289$	0	0
$290$	0.343146	0.0201502
$291$	0	0
$292$	9.41421	0.550925
$293$	9.31371	0.544113	0.272056	−	0.962281i	$-0.412296\pi$
$293$	9.31371	0.544113	0.272056	+	0.962281i	$0.412296\pi$
$294$	0	0
$295$	−7.31371	−0.425821
$296$	−2.24264	−0.130351
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−8.97056	−0.518781
$300$	0	0
$301$	8.00000	0.461112
$302$	−13.6569	−0.785864
$303$	0	0
$304$	−6.82843	−0.391637
$305$	−0.627417	−0.0359258
$306$	0	0
$307$	−6.34315	−0.362022	−0.181011	−	0.983481i	$-0.557937\pi$
$307$	−6.34315	−0.362022	−0.181011	+	0.983481i	$0.557937\pi$
$308$	−19.3137	−1.10050
$309$	0	0
$310$	−0.485281	−0.0275621
$311$	−0.828427	−0.0469758	−0.0234879	−	0.999724i	$-0.507477\pi$
$311$	−0.828427	−0.0469758	−0.0234879	+	0.999724i	$0.507477\pi$
$312$	0	0
$313$	10.1005	0.570914	0.285457	−	0.958391i	$-0.407855\pi$
$313$	10.1005	0.570914	0.285457	+	0.958391i	$0.407855\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	14.4853	0.814861
$317$	−21.5563	−1.21073	−0.605363	−	0.795950i	$-0.706972\pi$
$317$	−21.5563	−1.21073	−0.605363	+	0.795950i	$0.706972\pi$
$318$	0	0
$319$	−2.34315	−0.131191
$320$	−0.585786	−0.0327465
$321$	0	0
$322$	15.3137	0.853400
$323$	0	0
$324$	0	0
$325$	−13.1716	−0.730627
$326$	−2.34315	−0.129775
$327$	0	0
$328$	−3.07107	−0.169571
$329$	60.2843	3.32358
$330$	0	0
$331$	18.6274	1.02386	0.511928	−	0.859029i	$-0.328932\pi$
$331$	18.6274	1.02386	0.511928	+	0.859029i	$0.328932\pi$
$332$	−4.48528	−0.246162
$333$	0	0
$334$	21.7990	1.19279
$335$	−0.686292	−0.0374961
$336$	0	0
$337$	−12.7279	−0.693334	−0.346667	−	0.937988i	$-0.612687\pi$
$337$	−12.7279	−0.693334	−0.346667	+	0.937988i	$0.612687\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	0	0
$341$	3.31371	0.179447
$342$	0	0
$343$	44.9706	2.42818
$344$	−1.65685	−0.0893316
$345$	0	0
$346$	−4.10051	−0.220445
$347$	11.3137	0.607352	0.303676	−	0.952775i	$-0.401786\pi$
$347$	11.3137	0.607352	0.303676	+	0.952775i	$0.401786\pi$
$348$	0	0
$349$	−10.8284	−0.579632	−0.289816	−	0.957082i	$-0.593594\pi$
$349$	−10.8284	−0.579632	−0.289816	+	0.957082i	$0.593594\pi$
$350$	22.4853	1.20189
$351$	0	0
$352$	4.00000	0.213201
$353$	−12.6274	−0.672090	−0.336045	−	0.941846i	$-0.609089\pi$
$353$	−12.6274	−0.672090	−0.336045	+	0.941846i	$0.609089\pi$
$354$	0	0
$355$	−3.79899	−0.201629
$356$	0	0
$357$	0	0
$358$	−12.9706	−0.685516
$359$	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
$361$	27.6274	1.45407
$362$	−7.89949	−0.415188
$363$	0	0
$364$	13.6569	0.715814
$365$	−5.51472	−0.288654
$366$	0	0
$367$	−17.5147	−0.914261	−0.457130	−	0.889400i	$-0.651123\pi$
$367$	−17.5147	−0.914261	−0.457130	+	0.889400i	$0.651123\pi$
$368$	−3.17157	−0.165330
$369$	0	0
$370$	1.31371	0.0682965
$371$	−13.6569	−0.709029
$372$	0	0
$373$	19.7990	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$373$	19.7990	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$374$	0	0
$375$	0	0
$376$	−12.4853	−0.643879
$377$	1.65685	0.0853323
$378$	0	0
$379$	24.9706	1.28265	0.641326	−	0.767269i	$-0.278385\pi$
$379$	24.9706	1.28265	0.641326	+	0.767269i	$0.278385\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−1.17157	−0.0599429
$383$	−23.7990	−1.21607	−0.608036	−	0.793910i	$-0.708042\pi$
$383$	−23.7990	−1.21607	−0.608036	+	0.793910i	$0.708042\pi$
$384$	0	0
$385$	11.3137	0.576600
$386$	−5.89949	−0.300276
$387$	0	0
$388$	−13.4142	−0.681004
$389$	36.6274	1.85708	0.928542	−	0.371228i	$-0.121063\pi$
$389$	36.6274	1.85708	0.928542	+	0.371228i	$0.121063\pi$
$390$	0	0
$391$	0	0
$392$	−16.3137	−0.823967
$393$	0	0
$394$	−10.2426	−0.516017
$395$	−8.48528	−0.426941
$396$	0	0
$397$	−33.0711	−1.65979	−0.829895	−	0.557920i	$-0.811600\pi$
$397$	−33.0711	−1.65979	−0.829895	+	0.557920i	$0.811600\pi$
$398$	0.828427	0.0415253
$399$	0	0
$400$	−4.65685	−0.232843
$401$	8.72792	0.435852	0.217926	−	0.975965i	$-0.430071\pi$
$401$	8.72792	0.435852	0.217926	+	0.975965i	$0.430071\pi$
$402$	0	0
$403$	−2.34315	−0.116720
$404$	−5.17157	−0.257295
$405$	0	0
$406$	−2.82843	−0.140372
$407$	−8.97056	−0.444654
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	1.79899	0.0888458
$411$	0	0
$412$	14.8284	0.730544
$413$	60.2843	2.96640
$414$	0	0
$415$	2.62742	0.128975
$416$	−2.82843	−0.138675
$417$	0	0
$418$	−27.3137	−1.33596
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	15.5147	0.756141	0.378071	−	0.925777i	$-0.376588\pi$
$421$	15.5147	0.756141	0.378071	+	0.925777i	$0.376588\pi$
$422$	28.9706	1.41026
$423$	0	0
$424$	2.82843	0.137361
$425$	0	0
$426$	0	0
$427$	5.17157	0.250270
$428$	−13.6569	−0.660129
$429$	0	0
$430$	0.970563	0.0468047
$431$	−20.1421	−0.970213	−0.485106	−	0.874455i	$-0.661219\pi$
$431$	−20.1421	−0.970213	−0.485106	+	0.874455i	$0.661219\pi$
$432$	0	0
$433$	−16.6274	−0.799063	−0.399531	−	0.916720i	$-0.630827\pi$
$433$	−16.6274	−0.799063	−0.399531	+	0.916720i	$0.630827\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−3.89949	−0.186752
$437$	21.6569	1.03599
$438$	0	0
$439$	13.5147	0.645022	0.322511	−	0.946566i	$-0.395473\pi$
$439$	13.5147	0.645022	0.322511	+	0.946566i	$0.395473\pi$
$440$	−2.34315	−0.111705
$441$	0	0
$442$	0	0
$443$	28.9706	1.37643	0.688216	−	0.725505i	$-0.258394\pi$
$443$	28.9706	1.37643	0.688216	+	0.725505i	$0.258394\pi$
$444$	0	0
$445$	0	0
$446$	−6.82843	−0.323335
$447$	0	0
$448$	4.82843	0.228122
$449$	−12.0416	−0.568280	−0.284140	−	0.958783i	$-0.591708\pi$
$449$	−12.0416	−0.568280	−0.284140	+	0.958783i	$0.591708\pi$
$450$	0	0
$451$	−12.2843	−0.578444
$452$	12.7279	0.598671
$453$	0	0
$454$	17.6569	0.828677
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−2.68629	−0.125659	−0.0628297	−	0.998024i	$-0.520012\pi$
$457$	−2.68629	−0.125659	−0.0628297	+	0.998024i	$0.520012\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	1.85786	0.0866234
$461$	−25.4558	−1.18560	−0.592798	−	0.805351i	$-0.701977\pi$
$461$	−25.4558	−1.18560	−0.592798	+	0.805351i	$0.701977\pi$
$462$	0	0
$463$	−6.82843	−0.317344	−0.158672	−	0.987331i	$-0.550721\pi$
$463$	−6.82843	−0.317344	−0.158672	+	0.987331i	$0.550721\pi$
$464$	0.585786	0.0271945
$465$	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
$470$	7.31371	0.337356
$471$	0	0
$472$	−12.4853	−0.574682
$473$	−6.62742	−0.304729
$474$	0	0
$475$	31.7990	1.45904
$476$	0	0
$477$	0	0
$478$	12.4853	0.571063
$479$	−7.17157	−0.327678	−0.163839	−	0.986487i	$-0.552388\pi$
$479$	−7.17157	−0.327678	−0.163839	+	0.986487i	$0.552388\pi$
$480$	0	0
$481$	6.34315	0.289223
$482$	−8.24264	−0.375442
$483$	0	0
$484$	5.00000	0.227273
$485$	7.85786	0.356807
$486$	0	0
$487$	−41.1127	−1.86299	−0.931497	−	0.363749i	$-0.881497\pi$
$487$	−41.1127	−1.86299	−0.931497	+	0.363749i	$0.881497\pi$
$488$	−1.07107	−0.0484850
$489$	0	0
$490$	9.55635	0.431712
$491$	16.2843	0.734899	0.367449	−	0.930043i	$-0.380231\pi$
$491$	16.2843	0.734899	0.367449	+	0.930043i	$0.380231\pi$
$492$	0	0
$493$	0	0
$494$	19.3137	0.868965
$495$	0	0
$496$	−0.828427	−0.0371975
$497$	31.3137	1.40461
$498$	0	0
$499$	24.9706	1.11784	0.558918	−	0.829223i	$-0.311217\pi$
$499$	24.9706	1.11784	0.558918	+	0.829223i	$0.311217\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	−17.6569	−0.788064
$503$	−9.51472	−0.424240	−0.212120	−	0.977244i	$-0.568037\pi$
$503$	−9.51472	−0.424240	−0.212120	+	0.977244i	$0.568037\pi$
$504$	0	0
$505$	3.02944	0.134808
$506$	−12.6863	−0.563974
$507$	0	0
$508$	13.6569	0.605925
$509$	−2.68629	−0.119068	−0.0595339	−	0.998226i	$-0.518961\pi$
$509$	−2.68629	−0.119068	−0.0595339	+	0.998226i	$0.518961\pi$
$510$	0	0
$511$	45.4558	2.01085
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−29.3137	−1.29297
$515$	−8.68629	−0.382764
$516$	0	0
$517$	−49.9411	−2.19641
$518$	−10.8284	−0.475774
$519$	0	0
$520$	1.65685	0.0726579
$521$	−7.75736	−0.339856	−0.169928	−	0.985456i	$-0.554354\pi$
$521$	−7.75736	−0.339856	−0.169928	+	0.985456i	$0.554354\pi$
$522$	0	0
$523$	18.1421	0.793300	0.396650	−	0.917970i	$-0.370173\pi$
$523$	18.1421	0.793300	0.396650	+	0.917970i	$0.370173\pi$
$524$	19.3137	0.843723
$525$	0	0
$526$	10.1421	0.442218
$527$	0	0
$528$	0	0
$529$	−12.9411	−0.562658
$530$	−1.65685	−0.0719691
$531$	0	0
$532$	−32.9706	−1.42946
$533$	8.68629	0.376245
$534$	0	0
$535$	8.00000	0.345870
$536$	−1.17157	−0.0506042
$537$	0	0
$538$	5.07107	0.218629
$539$	−65.2548	−2.81072
$540$	0	0
$541$	18.2426	0.784312	0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426	0.784312	0.392156	+	0.919899i	$0.371729\pi$
$542$	−21.6569	−0.930242
$543$	0	0
$544$	0	0
$545$	2.28427	0.0978474
$546$	0	0
$547$	18.6274	0.796451	0.398225	−	0.917288i	$-0.369626\pi$
$547$	18.6274	0.796451	0.398225	+	0.917288i	$0.369626\pi$
$548$	17.6569	0.754263
$549$	0	0
$550$	−18.6274	−0.794276
$551$	−4.00000	−0.170406
$552$	0	0
$553$	69.9411	2.97420
$554$	9.75736	0.414550
$555$	0	0
$556$	−6.34315	−0.269009
$557$	8.48528	0.359533	0.179766	−	0.983709i	$-0.442466\pi$
$557$	8.48528	0.359533	0.179766	+	0.983709i	$0.442466\pi$
$558$	0	0
$559$	4.68629	0.198209
$560$	−2.82843	−0.119523
$561$	0	0
$562$	−19.3137	−0.814700
$563$	3.51472	0.148128	0.0740639	−	0.997254i	$-0.476403\pi$
$563$	3.51472	0.148128	0.0740639	+	0.997254i	$0.476403\pi$
$564$	0	0
$565$	−7.45584	−0.313670
$566$	8.00000	0.336265
$567$	0	0
$568$	−6.48528	−0.272116
$569$	12.9706	0.543754	0.271877	−	0.962332i	$-0.412356\pi$
$569$	12.9706	0.543754	0.271877	+	0.962332i	$0.412356\pi$
$570$	0	0
$571$	8.68629	0.363510	0.181755	−	0.983344i	$-0.441822\pi$
$571$	8.68629	0.363510	0.181755	+	0.983344i	$0.441822\pi$
$572$	−11.3137	−0.473050
$573$	0	0
$574$	−14.8284	−0.618927
$575$	14.7696	0.615933
$576$	0	0
$577$	−3.31371	−0.137951	−0.0689757	−	0.997618i	$-0.521973\pi$
$577$	−3.31371	−0.137951	−0.0689757	+	0.997618i	$0.521973\pi$
$578$	0	0
$579$	0	0
$580$	−0.343146	−0.0142484
$581$	−21.6569	−0.898478
$582$	0	0
$583$	11.3137	0.468566
$584$	−9.41421	−0.389563
$585$	0	0
$586$	−9.31371	−0.384746
$587$	36.9706	1.52594	0.762969	−	0.646435i	$-0.223741\pi$
$587$	36.9706	1.52594	0.762969	+	0.646435i	$0.223741\pi$
$588$	0	0
$589$	5.65685	0.233087
$590$	7.31371	0.301101
$591$	0	0
$592$	2.24264	0.0921720
$593$	−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
$598$	8.97056	0.366834
$599$	−30.6274	−1.25140	−0.625701	−	0.780063i	$-0.715187\pi$
$599$	−30.6274	−1.25140	−0.625701	+	0.780063i	$0.715187\pi$
$600$	0	0
$601$	6.58579	0.268640	0.134320	−	0.990938i	$-0.457115\pi$
$601$	6.58579	0.268640	0.134320	+	0.990938i	$0.457115\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	13.6569	0.555690
$605$	−2.92893	−0.119078
$606$	0	0
$607$	10.4853	0.425584	0.212792	−	0.977097i	$-0.431744\pi$
$607$	10.4853	0.425584	0.212792	+	0.977097i	$0.431744\pi$
$608$	6.82843	0.276929
$609$	0	0
$610$	0.627417	0.0254034
$611$	35.3137	1.42864
$612$	0	0
$613$	−21.3137	−0.860853	−0.430426	−	0.902626i	$-0.641637\pi$
$613$	−21.3137	−0.860853	−0.430426	+	0.902626i	$0.641637\pi$
$614$	6.34315	0.255989
$615$	0	0
$616$	19.3137	0.778171
$617$	−15.0711	−0.606738	−0.303369	−	0.952873i	$-0.598112\pi$
$617$	−15.0711	−0.606738	−0.303369	+	0.952873i	$0.598112\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0.485281	0.0194894
$621$	0	0
$622$	0.828427	0.0332169
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	−10.1005	−0.403697
$627$	0	0
$628$	−6.00000	−0.239426
$629$	0	0
$630$	0	0
$631$	26.1421	1.04070	0.520351	−	0.853952i	$-0.325801\pi$
$631$	26.1421	1.04070	0.520351	+	0.853952i	$0.325801\pi$
$632$	−14.4853	−0.576194
$633$	0	0
$634$	21.5563	0.856112
$635$	−8.00000	−0.317470
$636$	0	0
$637$	46.1421	1.82822
$638$	2.34315	0.0927660
$639$	0	0
$640$	0.585786	0.0231552
$641$	−19.0711	−0.753262	−0.376631	−	0.926363i	$-0.622918\pi$
$641$	−19.0711	−0.753262	−0.376631	+	0.926363i	$0.622918\pi$
$642$	0	0
$643$	−0.686292	−0.0270647	−0.0135323	−	0.999908i	$-0.504308\pi$
$643$	−0.686292	−0.0270647	−0.0135323	+	0.999908i	$0.504308\pi$
$644$	−15.3137	−0.603445
$645$	0	0
$646$	0	0
$647$	−38.4264	−1.51070	−0.755349	−	0.655323i	$-0.772533\pi$
$647$	−38.4264	−1.51070	−0.755349	+	0.655323i	$0.772533\pi$
$648$	0	0
$649$	−49.9411	−1.96036
$650$	13.1716	0.516632
$651$	0	0
$652$	2.34315	0.0917647
$653$	−31.4142	−1.22933	−0.614667	−	0.788787i	$-0.710709\pi$
$653$	−31.4142	−1.22933	−0.614667	+	0.788787i	$0.710709\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	3.07107	0.119905
$657$	0	0
$658$	−60.2843	−2.35013
$659$	3.51472	0.136914	0.0684570	−	0.997654i	$-0.478192\pi$
$659$	3.51472	0.136914	0.0684570	+	0.997654i	$0.478192\pi$
$660$	0	0
$661$	−30.1421	−1.17239	−0.586197	−	0.810169i	$-0.699375\pi$
$661$	−30.1421	−1.17239	−0.586197	+	0.810169i	$0.699375\pi$
$662$	−18.6274	−0.723975
$663$	0	0
$664$	4.48528	0.174063
$665$	19.3137	0.748953
$666$	0	0
$667$	−1.85786	−0.0719368
$668$	−21.7990	−0.843428
$669$	0	0
$670$	0.686292	0.0265138
$671$	−4.28427	−0.165392
$672$	0	0
$673$	35.5563	1.37060	0.685298	−	0.728263i	$-0.259672\pi$
$673$	35.5563	1.37060	0.685298	+	0.728263i	$0.259672\pi$
$674$	12.7279	0.490261
$675$	0	0
$676$	−5.00000	−0.192308
$677$	28.3848	1.09092	0.545458	−	0.838138i	$-0.316356\pi$
$677$	28.3848	1.09092	0.545458	+	0.838138i	$0.316356\pi$
$678$	0	0
$679$	−64.7696	−2.48563
$680$	0	0
$681$	0	0
$682$	−3.31371	−0.126888
$683$	20.2843	0.776156	0.388078	−	0.921627i	$-0.373139\pi$
$683$	20.2843	0.776156	0.388078	+	0.921627i	$0.373139\pi$
$684$	0	0
$685$	−10.3431	−0.395191
$686$	−44.9706	−1.71698
$687$	0	0
$688$	1.65685	0.0631670
$689$	−8.00000	−0.304776
$690$	0	0
$691$	12.9706	0.493423	0.246712	−	0.969089i	$-0.420650\pi$
$691$	12.9706	0.493423	0.246712	+	0.969089i	$0.420650\pi$
$692$	4.10051	0.155878
$693$	0	0
$694$	−11.3137	−0.429463
$695$	3.71573	0.140946
$696$	0	0
$697$	0	0
$698$	10.8284	0.409862
$699$	0	0
$700$	−22.4853	−0.849864
$701$	−23.5147	−0.888139	−0.444069	−	0.895992i	$-0.646466\pi$
$701$	−23.5147	−0.888139	−0.444069	+	0.895992i	$0.646466\pi$
$702$	0	0
$703$	−15.3137	−0.577567
$704$	−4.00000	−0.150756
$705$	0	0
$706$	12.6274	0.475239
$707$	−24.9706	−0.939115
$708$	0	0
$709$	−33.3553	−1.25269	−0.626343	−	0.779548i	$-0.715449\pi$
$709$	−33.3553	−1.25269	−0.626343	+	0.779548i	$0.715449\pi$
$710$	3.79899	0.142574
$711$	0	0
$712$	0	0
$713$	2.62742	0.0983975
$714$	0	0
$715$	6.62742	0.247851
$716$	12.9706	0.484733
$717$	0	0
$718$	−16.0000	−0.597115
$719$	−10.2010	−0.380433	−0.190217	−	0.981742i	$-0.560919\pi$
$719$	−10.2010	−0.380433	−0.190217	+	0.981742i	$0.560919\pi$
$720$	0	0
$721$	71.5980	2.66645
$722$	−27.6274	−1.02819
$723$	0	0
$724$	7.89949	0.293582
$725$	−2.72792	−0.101312
$726$	0	0
$727$	10.3431	0.383606	0.191803	−	0.981433i	$-0.438567\pi$
$727$	10.3431	0.383606	0.191803	+	0.981433i	$0.438567\pi$
$728$	−13.6569	−0.506157
$729$	0	0
$730$	5.51472	0.204109
$731$	0	0
$732$	0	0
$733$	41.4558	1.53121	0.765603	−	0.643313i	$-0.222441\pi$
$733$	41.4558	1.53121	0.765603	+	0.643313i	$0.222441\pi$
$734$	17.5147	0.646480
$735$	0	0
$736$	3.17157	0.116906
$737$	−4.68629	−0.172622
$738$	0	0
$739$	−9.17157	−0.337382	−0.168691	−	0.985669i	$-0.553954\pi$
$739$	−9.17157	−0.337382	−0.168691	+	0.985669i	$0.553954\pi$
$740$	−1.31371	−0.0482929
$741$	0	0
$742$	13.6569	0.501359
$743$	31.4558	1.15400	0.577001	−	0.816743i	$-0.304223\pi$
$743$	31.4558	1.15400	0.577001	+	0.816743i	$0.304223\pi$
$744$	0	0
$745$	−5.85786	−0.214616
$746$	−19.7990	−0.724893
$747$	0	0
$748$	0	0
$749$	−65.9411	−2.40944
$750$	0	0
$751$	−11.4558	−0.418030	−0.209015	−	0.977912i	$-0.567026\pi$
$751$	−11.4558	−0.418030	−0.209015	+	0.977912i	$0.567026\pi$
$752$	12.4853	0.455291
$753$	0	0
$754$	−1.65685	−0.0603391
$755$	−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	$0.534777\pi$
$758$	−24.9706	−0.906972
$759$	0	0
$760$	−4.00000	−0.145095
$761$	1.65685	0.0600609	0.0300305	−	0.999549i	$-0.490440\pi$
$761$	1.65685	0.0600609	0.0300305	+	0.999549i	$0.490440\pi$
$762$	0	0
$763$	−18.8284	−0.681635
$764$	1.17157	0.0423860
$765$	0	0
$766$	23.7990	0.859892
$767$	35.3137	1.27510
$768$	0	0
$769$	−40.9706	−1.47744	−0.738718	−	0.674014i	$-0.764569\pi$
$769$	−40.9706	−1.47744	−0.738718	+	0.674014i	$0.764569\pi$
$770$	−11.3137	−0.407718
$771$	0	0
$772$	5.89949	0.212327
$773$	44.6274	1.60514	0.802568	−	0.596560i	$-0.203466\pi$
$773$	44.6274	1.60514	0.802568	+	0.596560i	$0.203466\pi$
$774$	0	0
$775$	3.85786	0.138579
$776$	13.4142	0.481542
$777$	0	0
$778$	−36.6274	−1.31316
$779$	−20.9706	−0.751348
$780$	0	0
$781$	−25.9411	−0.928246
$782$	0	0
$783$	0	0
$784$	16.3137	0.582632
$785$	3.51472	0.125446
$786$	0	0
$787$	−21.9411	−0.782117	−0.391058	−	0.920366i	$-0.627891\pi$
$787$	−21.9411	−0.782117	−0.391058	+	0.920366i	$0.627891\pi$
$788$	10.2426	0.364879
$789$	0	0
$790$	8.48528	0.301893
$791$	61.4558	2.18512
$792$	0	0
$793$	3.02944	0.107578
$794$	33.0711	1.17365
$795$	0	0
$796$	−0.828427	−0.0293628
$797$	−21.1716	−0.749936	−0.374968	−	0.927038i	$-0.622346\pi$
$797$	−21.1716	−0.749936	−0.374968	+	0.927038i	$0.622346\pi$
$798$	0	0
$799$	0	0
$800$	4.65685	0.164645
$801$	0	0
$802$	−8.72792	−0.308194
$803$	−37.6569	−1.32888
$804$	0	0
$805$	8.97056	0.316171
$806$	2.34315	0.0825338
$807$	0	0
$808$	5.17157	0.181935
$809$	44.5269	1.56548	0.782741	−	0.622347i	$-0.213821\pi$
$809$	44.5269	1.56548	0.782741	+	0.622347i	$0.213821\pi$
$810$	0	0
$811$	28.9706	1.01729	0.508647	−	0.860975i	$-0.330146\pi$
$811$	28.9706	1.01729	0.508647	+	0.860975i	$0.330146\pi$
$812$	2.82843	0.0992583
$813$	0	0
$814$	8.97056	0.314418
$815$	−1.37258	−0.0480795
$816$	0	0
$817$	−11.3137	−0.395817
$818$	14.0000	0.489499
$819$	0	0
$820$	−1.79899	−0.0628235
$821$	−13.7574	−0.480135	−0.240068	−	0.970756i	$-0.577170\pi$
$821$	−13.7574	−0.480135	−0.240068	+	0.970756i	$0.577170\pi$
$822$	0	0
$823$	31.1716	1.08657	0.543286	−	0.839547i	$-0.317180\pi$
$823$	31.1716	1.08657	0.543286	+	0.839547i	$0.317180\pi$
$824$	−14.8284	−0.516573
$825$	0	0
$826$	−60.2843	−2.09756
$827$	−37.9411	−1.31934	−0.659671	−	0.751554i	$-0.729304\pi$
$827$	−37.9411	−1.31934	−0.659671	+	0.751554i	$0.729304\pi$
$828$	0	0
$829$	27.9411	0.970435	0.485218	−	0.874393i	$-0.338740\pi$
$829$	27.9411	0.970435	0.485218	+	0.874393i	$0.338740\pi$
$830$	−2.62742	−0.0911990
$831$	0	0
$832$	2.82843	0.0980581
$833$	0	0
$834$	0	0
$835$	12.7696	0.441909
$836$	27.3137	0.944664
$837$	0	0
$838$	−1.65685	−0.0572351
$839$	2.48528	0.0858014	0.0429007	−	0.999079i	$-0.486340\pi$
$839$	2.48528	0.0858014	0.0429007	+	0.999079i	$0.486340\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	−15.5147	−0.534673
$843$	0	0
$844$	−28.9706	−0.997208
$845$	2.92893	0.100758
$846$	0	0
$847$	24.1421	0.829534
$848$	−2.82843	−0.0971286
$849$	0	0
$850$	0	0
$851$	−7.11270	−0.243820
$852$	0	0
$853$	−48.3848	−1.65666	−0.828332	−	0.560238i	$-0.810710\pi$
$853$	−48.3848	−1.65666	−0.828332	+	0.560238i	$0.810710\pi$
$854$	−5.17157	−0.176968
$855$	0	0
$856$	13.6569	0.466782
$857$	15.7574	0.538261	0.269131	−	0.963104i	$-0.413264\pi$
$857$	15.7574	0.538261	0.269131	+	0.963104i	$0.413264\pi$
$858$	0	0
$859$	−1.65685	−0.0565311	−0.0282656	−	0.999600i	$-0.508998\pi$
$859$	−1.65685	−0.0565311	−0.0282656	+	0.999600i	$0.508998\pi$
$860$	−0.970563	−0.0330959
$861$	0	0
$862$	20.1421	0.686044
$863$	5.65685	0.192562	0.0962808	−	0.995354i	$-0.469305\pi$
$863$	5.65685	0.192562	0.0962808	+	0.995354i	$0.469305\pi$
$864$	0	0
$865$	−2.40202	−0.0816711
$866$	16.6274	0.565023
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−57.9411	−1.96552
$870$	0	0
$871$	3.31371	0.112281
$872$	3.89949	0.132054
$873$	0	0
$874$	−21.6569	−0.732554
$875$	27.3137	0.923372
$876$	0	0
$877$	12.1005	0.408605	0.204303	−	0.978908i	$-0.434507\pi$
$877$	12.1005	0.408605	0.204303	+	0.978908i	$0.434507\pi$
$878$	−13.5147	−0.456100
$879$	0	0
$880$	2.34315	0.0789874
$881$	29.2132	0.984218	0.492109	−	0.870534i	$-0.336226\pi$
$881$	29.2132	0.984218	0.492109	+	0.870534i	$0.336226\pi$
$882$	0	0
$883$	−48.7696	−1.64123	−0.820613	−	0.571484i	$-0.806368\pi$
$883$	−48.7696	−1.64123	−0.820613	+	0.571484i	$0.806368\pi$
$884$	0	0
$885$	0	0
$886$	−28.9706	−0.973285
$887$	−1.79899	−0.0604042	−0.0302021	−	0.999544i	$-0.509615\pi$
$887$	−1.79899	−0.0604042	−0.0302021	+	0.999544i	$0.509615\pi$
$888$	0	0
$889$	65.9411	2.21159
$890$	0	0
$891$	0	0
$892$	6.82843	0.228633
$893$	−85.2548	−2.85294
$894$	0	0
$895$	−7.59798	−0.253972
$896$	−4.82843	−0.161306
$897$	0	0
$898$	12.0416	0.401834
$899$	−0.485281	−0.0161850
$900$	0	0
$901$	0	0
$902$	12.2843	0.409021
$903$	0	0
$904$	−12.7279	−0.423324
$905$	−4.62742	−0.153821
$906$	0	0
$907$	12.6863	0.421241	0.210621	−	0.977568i	$-0.432452\pi$
$907$	12.6863	0.421241	0.210621	+	0.977568i	$0.432452\pi$
$908$	−17.6569	−0.585963
$909$	0	0
$910$	8.00000	0.265197
$911$	−29.1127	−0.964547	−0.482273	−	0.876021i	$-0.660189\pi$
$911$	−29.1127	−0.964547	−0.482273	+	0.876021i	$0.660189\pi$
$912$	0	0
$913$	17.9411	0.593765
$914$	2.68629	0.0888546
$915$	0	0
$916$	−14.0000	−0.462573
$917$	93.2548	3.07955
$918$	0	0
$919$	−18.3431	−0.605085	−0.302542	−	0.953136i	$-0.597835\pi$
$919$	−18.3431	−0.605085	−0.302542	+	0.953136i	$0.597835\pi$
$920$	−1.85786	−0.0612520
$921$	0	0
$922$	25.4558	0.838344
$923$	18.3431	0.603772
$924$	0	0
$925$	−10.4437	−0.343385
$926$	6.82843	0.224396
$927$	0	0
$928$	−0.585786	−0.0192294
$929$	42.5858	1.39719	0.698597	−	0.715515i	$-0.253808\pi$
$929$	42.5858	1.39719	0.698597	+	0.715515i	$0.253808\pi$
$930$	0	0
$931$	−111.397	−3.65089
$932$	−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
$940$	−7.31371	−0.238547
$941$	−34.2426	−1.11628	−0.558139	−	0.829747i	$-0.688484\pi$
$941$	−34.2426	−1.11628	−0.558139	+	0.829747i	$0.688484\pi$
$942$	0	0
$943$	−9.74012	−0.317182
$944$	12.4853	0.406361
$945$	0	0
$946$	6.62742	0.215476
$947$	21.9411	0.712991	0.356495	−	0.934297i	$-0.383972\pi$
$947$	21.9411	0.712991	0.356495	+	0.934297i	$0.383972\pi$
$948$	0	0
$949$	26.6274	0.864363
$950$	−31.7990	−1.03170
$951$	0	0
$952$	0	0
$953$	−7.02944	−0.227706	−0.113853	−	0.993498i	$-0.536319\pi$
$953$	−7.02944	−0.227706	−0.113853	+	0.993498i	$0.536319\pi$
$954$	0	0
$955$	−0.686292	−0.0222079
$956$	−12.4853	−0.403803
$957$	0	0
$958$	7.17157	0.231703
$959$	85.2548	2.75302
$960$	0	0
$961$	−30.3137	−0.977862
$962$	−6.34315	−0.204511
$963$	0	0
$964$	8.24264	0.265478
$965$	−3.45584	−0.111248
$966$	0	0
$967$	−18.1421	−0.583412	−0.291706	−	0.956508i	$-0.594223\pi$
$967$	−18.1421	−0.583412	−0.291706	+	0.956508i	$0.594223\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−7.85786	−0.252301
$971$	20.4853	0.657404	0.328702	−	0.944434i	$-0.393389\pi$
$971$	20.4853	0.657404	0.328702	+	0.944434i	$0.393389\pi$
$972$	0	0
$973$	−30.6274	−0.981870
$974$	41.1127	1.31734
$975$	0	0
$976$	1.07107	0.0342840
$977$	0.284271	0.00909464	0.00454732	−	0.999990i	$-0.498553\pi$
$977$	0.284271	0.00909464	0.00454732	+	0.999990i	$0.498553\pi$
$978$	0	0
$979$	0	0
$980$	−9.55635	−0.305266
$981$	0	0
$982$	−16.2843	−0.519652
$983$	4.82843	0.154003	0.0770015	−	0.997031i	$-0.475465\pi$
$983$	4.82843	0.154003	0.0770015	+	0.997031i	$0.475465\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	−19.3137	−0.614451
$989$	−5.25483	−0.167094
$990$	0	0
$991$	54.4853	1.73078	0.865391	−	0.501097i	$-0.167070\pi$
$991$	54.4853	1.73078	0.865391	+	0.501097i	$0.167070\pi$
$992$	0.828427	0.0263026
$993$	0	0
$994$	−31.3137	−0.993211
$995$	0.485281	0.0153845
$996$	0	0
$997$	−46.5269	−1.47352	−0.736761	−	0.676153i	$-0.763646\pi$
$997$	−46.5269	−1.47352	−0.736761	+	0.676153i	$0.763646\pi$
$998$	−24.9706	−0.790429
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.o.1.2		2
3.2	odd	2		1734.2.a.n.1.1		2
17.8	even	8		306.2.g.g.217.1		4
17.15	even	8		306.2.g.g.55.1		4
17.16	even	2		5202.2.a.x.1.1		2
51.2	odd	8		1734.2.f.i.1483.1		4
51.8	odd	8		102.2.f.a.13.2	✓	4
51.26	odd	8		1734.2.f.i.829.1		4
51.32	odd	8		102.2.f.a.55.2	yes	4
51.38	odd	4		1734.2.b.h.577.4		4
51.47	odd	4		1734.2.b.h.577.1		4
51.50	odd	2		1734.2.a.o.1.2		2
68.15	odd	8		2448.2.be.t.1585.1		4
68.59	odd	8		2448.2.be.t.1441.1		4
204.59	even	8		816.2.bd.a.625.1		4
204.83	even	8		816.2.bd.a.769.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.f.a.13.2	✓	4	51.8	odd	8
102.2.f.a.55.2	yes	4	51.32	odd	8
306.2.g.g.55.1		4	17.15	even	8
306.2.g.g.217.1		4	17.8	even	8
816.2.bd.a.625.1		4	204.59	even	8
816.2.bd.a.769.1		4	204.83	even	8
1734.2.a.n.1.1		2	3.2	odd	2
1734.2.a.o.1.2		2	51.50	odd	2
1734.2.b.h.577.1		4	51.47	odd	4
1734.2.b.h.577.4		4	51.38	odd	4
1734.2.f.i.829.1		4	51.26	odd	8
1734.2.f.i.1483.1		4	51.2	odd	8
2448.2.be.t.1441.1		4	68.59	odd	8
2448.2.be.t.1585.1		4	68.15	odd	8
5202.2.a.o.1.2		2	1.1	even	1	trivial
5202.2.a.x.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.585786 q^{5} +4.82843 q^{7} -1.00000 q^{8} +0.585786 q^{10} -4.00000 q^{11} +2.82843 q^{13} -4.82843 q^{14} +1.00000 q^{16} -6.82843 q^{19} -0.585786 q^{20} +4.00000 q^{22} -3.17157 q^{23} -4.65685 q^{25} -2.82843 q^{26} +4.82843 q^{28} +0.585786 q^{29} -0.828427 q^{31} -1.00000 q^{32} -2.82843 q^{35} +2.24264 q^{37} +6.82843 q^{38} +0.585786 q^{40} +3.07107 q^{41} +1.65685 q^{43} -4.00000 q^{44} +3.17157 q^{46} +12.4853 q^{47} +16.3137 q^{49} +4.65685 q^{50} +2.82843 q^{52} -2.82843 q^{53} +2.34315 q^{55} -4.82843 q^{56} -0.585786 q^{58} +12.4853 q^{59} +1.07107 q^{61} +0.828427 q^{62} +1.00000 q^{64} -1.65685 q^{65} +1.17157 q^{67} +2.82843 q^{70} +6.48528 q^{71} +9.41421 q^{73} -2.24264 q^{74} -6.82843 q^{76} -19.3137 q^{77} +14.4853 q^{79} -0.585786 q^{80} -3.07107 q^{82} -4.48528 q^{83} -1.65685 q^{86} +4.00000 q^{88} +13.6569 q^{91} -3.17157 q^{92} -12.4853 q^{94} +4.00000 q^{95} -13.4142 q^{97} -16.3137 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 4 q^{7} - 2 q^{8} + 4 q^{10} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{20} + 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{37}+ \cdots - 10 q^{98}+O(q^{100}) \) 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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