LMFDB - Embedded newform 6534.2.a.ck.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, 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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +3.60555 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +3.60555 q^{7} +1.00000 q^{8} -1.30278 q^{10} -3.30278 q^{13} +3.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} -3.69722 q^{19} -1.30278 q^{20} -6.90833 q^{23} -3.30278 q^{25} -3.30278 q^{26} +3.60555 q^{28} -0.394449 q^{29} -7.51388 q^{31} +1.00000 q^{32} +2.60555 q^{34} -4.69722 q^{35} -3.60555 q^{37} -3.69722 q^{38} -1.30278 q^{40} +0.394449 q^{41} -10.2111 q^{43} -6.90833 q^{46} -4.69722 q^{47} +6.00000 q^{49} -3.30278 q^{50} -3.30278 q^{52} +8.21110 q^{53} +3.60555 q^{56} -0.394449 q^{58} -12.9083 q^{59} -2.39445 q^{61} -7.51388 q^{62} +1.00000 q^{64} +4.30278 q^{65} -4.39445 q^{67} +2.60555 q^{68} -4.69722 q^{70} +16.4222 q^{71} +2.30278 q^{73} -3.60555 q^{74} -3.69722 q^{76} -9.30278 q^{79} -1.30278 q^{80} +0.394449 q^{82} +9.51388 q^{83} -3.39445 q^{85} -10.2111 q^{86} -1.69722 q^{89} -11.9083 q^{91} -6.90833 q^{92} -4.69722 q^{94} +4.81665 q^{95} +13.2111 q^{97} +6.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8} + q^{10} - 3 q^{13} + 2 q^{16} - 2 q^{17} - 11 q^{19} + q^{20} - 3 q^{23} - 3 q^{25} - 3 q^{26} - 8 q^{29} + 3 q^{31} + 2 q^{32} - 2 q^{34} - 13 q^{35} - 11 q^{38} + q^{40} + 8 q^{41} - 6 q^{43} - 3 q^{46} - 13 q^{47} + 12 q^{49} - 3 q^{50} - 3 q^{52} + 2 q^{53} - 8 q^{58} - 15 q^{59} - 12 q^{61} + 3 q^{62} + 2 q^{64} + 5 q^{65} - 16 q^{67} - 2 q^{68} - 13 q^{70} + 4 q^{71} + q^{73} - 11 q^{76} - 15 q^{79} + q^{80} + 8 q^{82} + q^{83} - 14 q^{85} - 6 q^{86} - 7 q^{89} - 13 q^{91} - 3 q^{92} - 13 q^{94} - 12 q^{95} + 12 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8 + q^10 - 3 * q^13 + 2 * q^16 - 2 * q^17 - 11 * q^19 + q^20 - 3 * q^23 - 3 * q^25 - 3 * q^26 - 8 * q^29 + 3 * q^31 + 2 * q^32 - 2 * q^34 - 13 * q^35 - 11 * q^38 + q^40 + 8 * q^41 - 6 * q^43 - 3 * q^46 - 13 * q^47 + 12 * q^49 - 3 * q^50 - 3 * q^52 + 2 * q^53 - 8 * q^58 - 15 * q^59 - 12 * q^61 + 3 * q^62 + 2 * q^64 + 5 * q^65 - 16 * q^67 - 2 * q^68 - 13 * q^70 + 4 * q^71 + q^73 - 11 * q^76 - 15 * q^79 + q^80 + 8 * q^82 + q^83 - 14 * q^85 - 6 * q^86 - 7 * q^89 - 13 * q^91 - 3 * q^92 - 13 * q^94 - 12 * q^95 + 12 * q^97 + 12 * 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
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	0	0
$7$	3.60555	1.36277	0.681385	−	0.731925i	$-0.261378\pi$
$7$	3.60555	1.36277	0.681385	+	0.731925i	$0.261378\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.30278	−0.411974
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.30278	−0.916025	−0.458013	−	0.888946i	$-0.651439\pi$
$13$	−3.30278	−0.916025	−0.458013	+	0.888946i	$0.651439\pi$
$14$	3.60555	0.963624
$15$	0	0
$16$	1.00000	0.250000
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	−3.69722	−0.848201	−0.424101	−	0.905615i	$-0.639410\pi$
$19$	−3.69722	−0.848201	−0.424101	+	0.905615i	$0.639410\pi$
$20$	−1.30278	−0.291309
$21$	0	0
$22$	0	0
$23$	−6.90833	−1.44049	−0.720243	−	0.693722i	$-0.755970\pi$
$23$	−6.90833	−1.44049	−0.720243	+	0.693722i	$0.755970\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	−3.30278	−0.647728
$27$	0	0
$28$	3.60555	0.681385
$29$	−0.394449	−0.0732473	−0.0366236	−	0.999329i	$-0.511660\pi$
$29$	−0.394449	−0.0732473	−0.0366236	+	0.999329i	$0.511660\pi$
$30$	0	0
$31$	−7.51388	−1.34953	−0.674766	−	0.738032i	$-0.735756\pi$
$31$	−7.51388	−1.34953	−0.674766	+	0.738032i	$0.735756\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.60555	0.446848
$35$	−4.69722	−0.793976
$36$	0	0
$37$	−3.60555	−0.592749	−0.296374	−	0.955072i	$-0.595778\pi$
$37$	−3.60555	−0.592749	−0.296374	+	0.955072i	$0.595778\pi$
$38$	−3.69722	−0.599769
$39$	0	0
$40$	−1.30278	−0.205987
$41$	0.394449	0.0616025	0.0308013	−	0.999526i	$-0.490194\pi$
$41$	0.394449	0.0616025	0.0308013	+	0.999526i	$0.490194\pi$
$42$	0	0
$43$	−10.2111	−1.55718	−0.778589	−	0.627534i	$-0.784064\pi$
$43$	−10.2111	−1.55718	−0.778589	+	0.627534i	$0.784064\pi$
$44$	0	0
$45$	0	0
$46$	−6.90833	−1.01858
$47$	−4.69722	−0.685161	−0.342580	−	0.939489i	$-0.611301\pi$
$47$	−4.69722	−0.685161	−0.342580	+	0.939489i	$0.611301\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	−3.30278	−0.467083
$51$	0	0
$52$	−3.30278	−0.458013
$53$	8.21110	1.12788	0.563941	−	0.825815i	$-0.309285\pi$
$53$	8.21110	1.12788	0.563941	+	0.825815i	$0.309285\pi$
$54$	0	0
$55$	0	0
$56$	3.60555	0.481812
$57$	0	0
$58$	−0.394449	−0.0517937
$59$	−12.9083	−1.68052	−0.840261	−	0.542183i	$-0.817598\pi$
$59$	−12.9083	−1.68052	−0.840261	+	0.542183i	$0.817598\pi$
$60$	0	0
$61$	−2.39445	−0.306578	−0.153289	−	0.988181i	$-0.548986\pi$
$61$	−2.39445	−0.306578	−0.153289	+	0.988181i	$0.548986\pi$
$62$	−7.51388	−0.954263
$63$	0	0
$64$	1.00000	0.125000
$65$	4.30278	0.533694
$66$	0	0
$67$	−4.39445	−0.536867	−0.268434	−	0.963298i	$-0.586506\pi$
$67$	−4.39445	−0.536867	−0.268434	+	0.963298i	$0.586506\pi$
$68$	2.60555	0.315970
$69$	0	0
$70$	−4.69722	−0.561426
$71$	16.4222	1.94896	0.974479	−	0.224480i	$-0.0720685\pi$
$71$	16.4222	1.94896	0.974479	+	0.224480i	$0.0720685\pi$
$72$	0	0
$73$	2.30278	0.269520	0.134760	−	0.990878i	$-0.456974\pi$
$73$	2.30278	0.269520	0.134760	+	0.990878i	$0.456974\pi$
$74$	−3.60555	−0.419137
$75$	0	0
$76$	−3.69722	−0.424101
$77$	0	0
$78$	0	0
$79$	−9.30278	−1.04664	−0.523322	−	0.852135i	$-0.675308\pi$
$79$	−9.30278	−1.04664	−0.523322	+	0.852135i	$0.675308\pi$
$80$	−1.30278	−0.145655
$81$	0	0
$82$	0.394449	0.0435596
$83$	9.51388	1.04428	0.522142	−	0.852859i	$-0.325133\pi$
$83$	9.51388	1.04428	0.522142	+	0.852859i	$0.325133\pi$
$84$	0	0
$85$	−3.39445	−0.368180
$86$	−10.2111	−1.10109
$87$	0	0
$88$	0	0
$89$	−1.69722	−0.179905	−0.0899527	−	0.995946i	$-0.528672\pi$
$89$	−1.69722	−0.179905	−0.0899527	+	0.995946i	$0.528672\pi$
$90$	0	0
$91$	−11.9083	−1.24833
$92$	−6.90833	−0.720243
$93$	0	0
$94$	−4.69722	−0.484482
$95$	4.81665	0.494178
$96$	0	0
$97$	13.2111	1.34138	0.670692	−	0.741736i	$-0.265997\pi$
$97$	13.2111	1.34138	0.670692	+	0.741736i	$0.265997\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	−3.30278	−0.330278
$101$	−2.60555	−0.259262	−0.129631	−	0.991562i	$-0.541379\pi$
$101$	−2.60555	−0.259262	−0.129631	+	0.991562i	$0.541379\pi$
$102$	0	0
$103$	−15.6056	−1.53766	−0.768830	−	0.639453i	$-0.779161\pi$
$103$	−15.6056	−1.53766	−0.768830	+	0.639453i	$0.779161\pi$
$104$	−3.30278	−0.323864
$105$	0	0
$106$	8.21110	0.797533
$107$	13.8167	1.33571	0.667853	−	0.744293i	$-0.267213\pi$
$107$	13.8167	1.33571	0.667853	+	0.744293i	$0.267213\pi$
$108$	0	0
$109$	4.90833	0.470132	0.235066	−	0.971979i	$-0.424469\pi$
$109$	4.90833	0.470132	0.235066	+	0.971979i	$0.424469\pi$
$110$	0	0
$111$	0	0
$112$	3.60555	0.340693
$113$	−3.90833	−0.367664	−0.183832	−	0.982958i	$-0.558850\pi$
$113$	−3.90833	−0.367664	−0.183832	+	0.982958i	$0.558850\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	−0.394449	−0.0366236
$117$	0	0
$118$	−12.9083	−1.18831
$119$	9.39445	0.861188
$120$	0	0
$121$	0	0
$122$	−2.39445	−0.216783
$123$	0	0
$124$	−7.51388	−0.674766
$125$	10.8167	0.967471
$126$	0	0
$127$	−2.90833	−0.258072	−0.129036	−	0.991640i	$-0.541188\pi$
$127$	−2.90833	−0.258072	−0.129036	+	0.991640i	$0.541188\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.30278	0.377378
$131$	16.8167	1.46928	0.734639	−	0.678458i	$-0.237352\pi$
$131$	16.8167	1.46928	0.734639	+	0.678458i	$0.237352\pi$
$132$	0	0
$133$	−13.3305	−1.15590
$134$	−4.39445	−0.379623
$135$	0	0
$136$	2.60555	0.223424
$137$	−21.5139	−1.83805	−0.919027	−	0.394194i	$-0.871024\pi$
$137$	−21.5139	−1.83805	−0.919027	+	0.394194i	$0.871024\pi$
$138$	0	0
$139$	−9.42221	−0.799181	−0.399591	−	0.916694i	$-0.630848\pi$
$139$	−9.42221	−0.799181	−0.399591	+	0.916694i	$0.630848\pi$
$140$	−4.69722	−0.396988
$141$	0	0
$142$	16.4222	1.37812
$143$	0	0
$144$	0	0
$145$	0.513878	0.0426753
$146$	2.30278	0.190579
$147$	0	0
$148$	−3.60555	−0.296374
$149$	2.48612	0.203671	0.101836	−	0.994801i	$-0.467528\pi$
$149$	2.48612	0.203671	0.101836	+	0.994801i	$0.467528\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	−3.69722	−0.299884
$153$	0	0
$154$	0	0
$155$	9.78890	0.786263
$156$	0	0
$157$	−10.3944	−0.829567	−0.414784	−	0.909920i	$-0.636143\pi$
$157$	−10.3944	−0.829567	−0.414784	+	0.909920i	$0.636143\pi$
$158$	−9.30278	−0.740089
$159$	0	0
$160$	−1.30278	−0.102993
$161$	−24.9083	−1.96305
$162$	0	0
$163$	21.3028	1.66856	0.834281	−	0.551339i	$-0.185883\pi$
$163$	21.3028	1.66856	0.834281	+	0.551339i	$0.185883\pi$
$164$	0.394449	0.0308013
$165$	0	0
$166$	9.51388	0.738420
$167$	−2.09167	−0.161859	−0.0809293	−	0.996720i	$-0.525789\pi$
$167$	−2.09167	−0.161859	−0.0809293	+	0.996720i	$0.525789\pi$
$168$	0	0
$169$	−2.09167	−0.160898
$170$	−3.39445	−0.260342
$171$	0	0
$172$	−10.2111	−0.778589
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	−11.9083	−0.900185
$176$	0	0
$177$	0	0
$178$	−1.69722	−0.127212
$179$	−5.09167	−0.380570	−0.190285	−	0.981729i	$-0.560941\pi$
$179$	−5.09167	−0.380570	−0.190285	+	0.981729i	$0.560941\pi$
$180$	0	0
$181$	8.90833	0.662151	0.331075	−	0.943604i	$-0.392589\pi$
$181$	8.90833	0.662151	0.331075	+	0.943604i	$0.392589\pi$
$182$	−11.9083	−0.882704
$183$	0	0
$184$	−6.90833	−0.509289
$185$	4.69722	0.345347
$186$	0	0
$187$	0	0
$188$	−4.69722	−0.342580
$189$	0	0
$190$	4.81665	0.349437
$191$	−10.6972	−0.774024	−0.387012	−	0.922075i	$-0.626493\pi$
$191$	−10.6972	−0.774024	−0.387012	+	0.922075i	$0.626493\pi$
$192$	0	0
$193$	−25.7250	−1.85172	−0.925862	−	0.377861i	$-0.876660\pi$
$193$	−25.7250	−1.85172	−0.925862	+	0.377861i	$0.876660\pi$
$194$	13.2111	0.948502
$195$	0	0
$196$	6.00000	0.428571
$197$	−1.81665	−0.129431	−0.0647156	−	0.997904i	$-0.520614\pi$
$197$	−1.81665	−0.129431	−0.0647156	+	0.997904i	$0.520614\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−3.30278	−0.233542
$201$	0	0
$202$	−2.60555	−0.183326
$203$	−1.42221	−0.0998192
$204$	0	0
$205$	−0.513878	−0.0358908
$206$	−15.6056	−1.08729
$207$	0	0
$208$	−3.30278	−0.229006
$209$	0	0
$210$	0	0
$211$	14.0278	0.965711	0.482855	−	0.875700i	$-0.339600\pi$
$211$	14.0278	0.965711	0.482855	+	0.875700i	$0.339600\pi$
$212$	8.21110	0.563941
$213$	0	0
$214$	13.8167	0.944487
$215$	13.3028	0.907242
$216$	0	0
$217$	−27.0917	−1.83910
$218$	4.90833	0.332434
$219$	0	0
$220$	0	0
$221$	−8.60555	−0.578872
$222$	0	0
$223$	7.72498	0.517303	0.258651	−	0.965971i	$-0.416722\pi$
$223$	7.72498	0.517303	0.258651	+	0.965971i	$0.416722\pi$
$224$	3.60555	0.240906
$225$	0	0
$226$	−3.90833	−0.259978
$227$	−3.90833	−0.259405	−0.129702	−	0.991553i	$-0.541402\pi$
$227$	−3.90833	−0.259405	−0.129702	+	0.991553i	$0.541402\pi$
$228$	0	0
$229$	29.1194	1.92427	0.962133	−	0.272580i	$-0.0878770\pi$
$229$	29.1194	1.92427	0.962133	+	0.272580i	$0.0878770\pi$
$230$	9.00000	0.593442
$231$	0	0
$232$	−0.394449	−0.0258968
$233$	−26.7250	−1.75081	−0.875406	−	0.483389i	$-0.839406\pi$
$233$	−26.7250	−1.75081	−0.875406	+	0.483389i	$0.839406\pi$
$234$	0	0
$235$	6.11943	0.399188
$236$	−12.9083	−0.840261
$237$	0	0
$238$	9.39445	0.608952
$239$	−16.0278	−1.03675	−0.518375	−	0.855154i	$-0.673463\pi$
$239$	−16.0278	−1.03675	−0.518375	+	0.855154i	$0.673463\pi$
$240$	0	0
$241$	−14.7889	−0.952637	−0.476318	−	0.879273i	$-0.658029\pi$
$241$	−14.7889	−0.952637	−0.476318	+	0.879273i	$0.658029\pi$
$242$	0	0
$243$	0	0
$244$	−2.39445	−0.153289
$245$	−7.81665	−0.499388
$246$	0	0
$247$	12.2111	0.776974
$248$	−7.51388	−0.477132
$249$	0	0
$250$	10.8167	0.684105
$251$	−10.6972	−0.675203	−0.337601	−	0.941289i	$-0.609616\pi$
$251$	−10.6972	−0.675203	−0.337601	+	0.941289i	$0.609616\pi$
$252$	0	0
$253$	0	0
$254$	−2.90833	−0.182485
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.8167	0.674724	0.337362	−	0.941375i	$-0.390465\pi$
$257$	10.8167	0.674724	0.337362	+	0.941375i	$0.390465\pi$
$258$	0	0
$259$	−13.0000	−0.807781
$260$	4.30278	0.266847
$261$	0	0
$262$	16.8167	1.03894
$263$	−20.6056	−1.27059	−0.635296	−	0.772268i	$-0.719122\pi$
$263$	−20.6056	−1.27059	−0.635296	+	0.772268i	$0.719122\pi$
$264$	0	0
$265$	−10.6972	−0.657125
$266$	−13.3305	−0.817347
$267$	0	0
$268$	−4.39445	−0.268434
$269$	14.4861	0.883234	0.441617	−	0.897204i	$-0.354405\pi$
$269$	14.4861	0.883234	0.441617	+	0.897204i	$0.354405\pi$
$270$	0	0
$271$	24.3305	1.47797	0.738987	−	0.673719i	$-0.235304\pi$
$271$	24.3305	1.47797	0.738987	+	0.673719i	$0.235304\pi$
$272$	2.60555	0.157985
$273$	0	0
$274$	−21.5139	−1.29970
$275$	0	0
$276$	0	0
$277$	5.18335	0.311437	0.155719	−	0.987801i	$-0.450231\pi$
$277$	5.18335	0.311437	0.155719	+	0.987801i	$0.450231\pi$
$278$	−9.42221	−0.565106
$279$	0	0
$280$	−4.69722	−0.280713
$281$	−19.9361	−1.18929	−0.594644	−	0.803989i	$-0.702707\pi$
$281$	−19.9361	−1.18929	−0.594644	+	0.803989i	$0.702707\pi$
$282$	0	0
$283$	−12.0278	−0.714976	−0.357488	−	0.933918i	$-0.616367\pi$
$283$	−12.0278	−0.714976	−0.357488	+	0.933918i	$0.616367\pi$
$284$	16.4222	0.974479
$285$	0	0
$286$	0	0
$287$	1.42221	0.0839501
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0.513878	0.0301760
$291$	0	0
$292$	2.30278	0.134760
$293$	−29.3305	−1.71351	−0.856754	−	0.515725i	$-0.827523\pi$
$293$	−29.3305	−1.71351	−0.856754	+	0.515725i	$0.827523\pi$
$294$	0	0
$295$	16.8167	0.979103
$296$	−3.60555	−0.209568
$297$	0	0
$298$	2.48612	0.144017
$299$	22.8167	1.31952
$300$	0	0
$301$	−36.8167	−2.12208
$302$	−17.0000	−0.978240
$303$	0	0
$304$	−3.69722	−0.212050
$305$	3.11943	0.178618
$306$	0	0
$307$	4.90833	0.280133	0.140067	−	0.990142i	$-0.455268\pi$
$307$	4.90833	0.280133	0.140067	+	0.990142i	$0.455268\pi$
$308$	0	0
$309$	0	0
$310$	9.78890	0.555972
$311$	5.60555	0.317862	0.158931	−	0.987290i	$-0.449195\pi$
$311$	5.60555	0.317862	0.158931	+	0.987290i	$0.449195\pi$
$312$	0	0
$313$	−14.9361	−0.844237	−0.422119	−	0.906541i	$-0.638713\pi$
$313$	−14.9361	−0.844237	−0.422119	+	0.906541i	$0.638713\pi$
$314$	−10.3944	−0.586593
$315$	0	0
$316$	−9.30278	−0.523322
$317$	21.3944	1.20163	0.600816	−	0.799387i	$-0.294842\pi$
$317$	21.3944	1.20163	0.600816	+	0.799387i	$0.294842\pi$
$318$	0	0
$319$	0	0
$320$	−1.30278	−0.0728274
$321$	0	0
$322$	−24.9083	−1.38809
$323$	−9.63331	−0.536012
$324$	0	0
$325$	10.9083	0.605085
$326$	21.3028	1.17985
$327$	0	0
$328$	0.394449	0.0217798
$329$	−16.9361	−0.933716
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	9.51388	0.522142
$333$	0	0
$334$	−2.09167	−0.114451
$335$	5.72498	0.312789
$336$	0	0
$337$	11.8167	0.643694	0.321847	−	0.946792i	$-0.395696\pi$
$337$	11.8167	0.643694	0.321847	+	0.946792i	$0.395696\pi$
$338$	−2.09167	−0.113772
$339$	0	0
$340$	−3.39445	−0.184090
$341$	0	0
$342$	0	0
$343$	−3.60555	−0.194681
$344$	−10.2111	−0.550546
$345$	0	0
$346$	21.0000	1.12897
$347$	29.7250	1.59572	0.797860	−	0.602842i	$-0.205965\pi$
$347$	29.7250	1.59572	0.797860	+	0.602842i	$0.205965\pi$
$348$	0	0
$349$	1.51388	0.0810360	0.0405180	−	0.999179i	$-0.487099\pi$
$349$	1.51388	0.0810360	0.0405180	+	0.999179i	$0.487099\pi$
$350$	−11.9083	−0.636527
$351$	0	0
$352$	0	0
$353$	33.6333	1.79012	0.895060	−	0.445945i	$-0.147132\pi$
$353$	33.6333	1.79012	0.895060	+	0.445945i	$0.147132\pi$
$354$	0	0
$355$	−21.3944	−1.13550
$356$	−1.69722	−0.0899527
$357$	0	0
$358$	−5.09167	−0.269103
$359$	−12.3944	−0.654154	−0.327077	−	0.944998i	$-0.606064\pi$
$359$	−12.3944	−0.654154	−0.327077	+	0.944998i	$0.606064\pi$
$360$	0	0
$361$	−5.33053	−0.280554
$362$	8.90833	0.468211
$363$	0	0
$364$	−11.9083	−0.624166
$365$	−3.00000	−0.157027
$366$	0	0
$367$	−6.09167	−0.317983	−0.158991	−	0.987280i	$-0.550824\pi$
$367$	−6.09167	−0.317983	−0.158991	+	0.987280i	$0.550824\pi$
$368$	−6.90833	−0.360121
$369$	0	0
$370$	4.69722	0.244197
$371$	29.6056	1.53704
$372$	0	0
$373$	−7.48612	−0.387617	−0.193808	−	0.981039i	$-0.562084\pi$
$373$	−7.48612	−0.387617	−0.193808	+	0.981039i	$0.562084\pi$
$374$	0	0
$375$	0	0
$376$	−4.69722	−0.242241
$377$	1.30278	0.0670964
$378$	0	0
$379$	8.11943	0.417067	0.208534	−	0.978015i	$-0.433131\pi$
$379$	8.11943	0.417067	0.208534	+	0.978015i	$0.433131\pi$
$380$	4.81665	0.247089
$381$	0	0
$382$	−10.6972	−0.547318
$383$	−22.8167	−1.16588	−0.582938	−	0.812516i	$-0.698097\pi$
$383$	−22.8167	−1.16588	−0.582938	+	0.812516i	$0.698097\pi$
$384$	0	0
$385$	0	0
$386$	−25.7250	−1.30937
$387$	0	0
$388$	13.2111	0.670692
$389$	8.21110	0.416319	0.208160	−	0.978095i	$-0.433253\pi$
$389$	8.21110	0.416319	0.208160	+	0.978095i	$0.433253\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	6.00000	0.303046
$393$	0	0
$394$	−1.81665	−0.0915217
$395$	12.1194	0.609795
$396$	0	0
$397$	38.2389	1.91915	0.959577	−	0.281447i	$-0.0908145\pi$
$397$	38.2389	1.91915	0.959577	+	0.281447i	$0.0908145\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−3.30278	−0.165139
$401$	33.9083	1.69330	0.846651	−	0.532149i	$-0.178616\pi$
$401$	33.9083	1.69330	0.846651	+	0.532149i	$0.178616\pi$
$402$	0	0
$403$	24.8167	1.23621
$404$	−2.60555	−0.129631
$405$	0	0
$406$	−1.42221	−0.0705829
$407$	0	0
$408$	0	0
$409$	−33.5416	−1.65853	−0.829264	−	0.558858i	$-0.811240\pi$
$409$	−33.5416	−1.65853	−0.829264	+	0.558858i	$0.811240\pi$
$410$	−0.513878	−0.0253786
$411$	0	0
$412$	−15.6056	−0.768830
$413$	−46.5416	−2.29016
$414$	0	0
$415$	−12.3944	−0.608420
$416$	−3.30278	−0.161932
$417$	0	0
$418$	0	0
$419$	24.6333	1.20342	0.601708	−	0.798716i	$-0.294487\pi$
$419$	24.6333	1.20342	0.601708	+	0.798716i	$0.294487\pi$
$420$	0	0
$421$	−12.2111	−0.595133	−0.297566	−	0.954701i	$-0.596175\pi$
$421$	−12.2111	−0.595133	−0.297566	+	0.954701i	$0.596175\pi$
$422$	14.0278	0.682860
$423$	0	0
$424$	8.21110	0.398766
$425$	−8.60555	−0.417431
$426$	0	0
$427$	−8.63331	−0.417795
$428$	13.8167	0.667853
$429$	0	0
$430$	13.3028	0.641517
$431$	−32.6056	−1.57055	−0.785277	−	0.619145i	$-0.787480\pi$
$431$	−32.6056	−1.57055	−0.785277	+	0.619145i	$0.787480\pi$
$432$	0	0
$433$	−23.9361	−1.15029	−0.575147	−	0.818050i	$-0.695055\pi$
$433$	−23.9361	−1.15029	−0.575147	+	0.818050i	$0.695055\pi$
$434$	−27.0917	−1.30044
$435$	0	0
$436$	4.90833	0.235066
$437$	25.5416	1.22182
$438$	0	0
$439$	−24.0278	−1.14678	−0.573391	−	0.819282i	$-0.694372\pi$
$439$	−24.0278	−1.14678	−0.573391	+	0.819282i	$0.694372\pi$
$440$	0	0
$441$	0	0
$442$	−8.60555	−0.409324
$443$	−10.1833	−0.483825	−0.241913	−	0.970298i	$-0.577775\pi$
$443$	−10.1833	−0.483825	−0.241913	+	0.970298i	$0.577775\pi$
$444$	0	0
$445$	2.21110	0.104816
$446$	7.72498	0.365788
$447$	0	0
$448$	3.60555	0.170346
$449$	−10.8167	−0.510469	−0.255235	−	0.966879i	$-0.582153\pi$
$449$	−10.8167	−0.510469	−0.255235	+	0.966879i	$0.582153\pi$
$450$	0	0
$451$	0	0
$452$	−3.90833	−0.183832
$453$	0	0
$454$	−3.90833	−0.183427
$455$	15.5139	0.727302
$456$	0	0
$457$	32.4222	1.51665	0.758323	−	0.651879i	$-0.226019\pi$
$457$	32.4222	1.51665	0.758323	+	0.651879i	$0.226019\pi$
$458$	29.1194	1.36066
$459$	0	0
$460$	9.00000	0.419627
$461$	−13.1833	−0.614010	−0.307005	−	0.951708i	$-0.599327\pi$
$461$	−13.1833	−0.614010	−0.307005	+	0.951708i	$0.599327\pi$
$462$	0	0
$463$	11.1194	0.516764	0.258382	−	0.966043i	$-0.416811\pi$
$463$	11.1194	0.516764	0.258382	+	0.966043i	$0.416811\pi$
$464$	−0.394449	−0.0183118
$465$	0	0
$466$	−26.7250	−1.23801
$467$	23.4861	1.08681	0.543404	−	0.839471i	$-0.317135\pi$
$467$	23.4861	1.08681	0.543404	+	0.839471i	$0.317135\pi$
$468$	0	0
$469$	−15.8444	−0.731627
$470$	6.11943	0.282268
$471$	0	0
$472$	−12.9083	−0.594154
$473$	0	0
$474$	0	0
$475$	12.2111	0.560284
$476$	9.39445	0.430594
$477$	0	0
$478$	−16.0278	−0.733093
$479$	28.6972	1.31121	0.655605	−	0.755104i	$-0.272414\pi$
$479$	28.6972	1.31121	0.655605	+	0.755104i	$0.272414\pi$
$480$	0	0
$481$	11.9083	0.542973
$482$	−14.7889	−0.673616
$483$	0	0
$484$	0	0
$485$	−17.2111	−0.781516
$486$	0	0
$487$	−3.60555	−0.163383	−0.0816916	−	0.996658i	$-0.526032\pi$
$487$	−3.60555	−0.163383	−0.0816916	+	0.996658i	$0.526032\pi$
$488$	−2.39445	−0.108392
$489$	0	0
$490$	−7.81665	−0.353120
$491$	−4.81665	−0.217373	−0.108686	−	0.994076i	$-0.534664\pi$
$491$	−4.81665	−0.217373	−0.108686	+	0.994076i	$0.534664\pi$
$492$	0	0
$493$	−1.02776	−0.0462878
$494$	12.2111	0.549403
$495$	0	0
$496$	−7.51388	−0.337383
$497$	59.2111	2.65598
$498$	0	0
$499$	−13.7889	−0.617276	−0.308638	−	0.951180i	$-0.599873\pi$
$499$	−13.7889	−0.617276	−0.308638	+	0.951180i	$0.599873\pi$
$500$	10.8167	0.483735
$501$	0	0
$502$	−10.6972	−0.477440
$503$	−33.7527	−1.50496	−0.752480	−	0.658615i	$-0.771143\pi$
$503$	−33.7527	−1.50496	−0.752480	+	0.658615i	$0.771143\pi$
$504$	0	0
$505$	3.39445	0.151051
$506$	0	0
$507$	0	0
$508$	−2.90833	−0.129036
$509$	−7.42221	−0.328983	−0.164492	−	0.986378i	$-0.552598\pi$
$509$	−7.42221	−0.328983	−0.164492	+	0.986378i	$0.552598\pi$
$510$	0	0
$511$	8.30278	0.367293
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.8167	0.477102
$515$	20.3305	0.895870
$516$	0	0
$517$	0	0
$518$	−13.0000	−0.571187
$519$	0	0
$520$	4.30278	0.188689
$521$	6.27502	0.274914	0.137457	−	0.990508i	$-0.456107\pi$
$521$	6.27502	0.274914	0.137457	+	0.990508i	$0.456107\pi$
$522$	0	0
$523$	27.7250	1.21233	0.606164	−	0.795339i	$-0.292707\pi$
$523$	27.7250	1.21233	0.606164	+	0.795339i	$0.292707\pi$
$524$	16.8167	0.734639
$525$	0	0
$526$	−20.6056	−0.898445
$527$	−19.5778	−0.852822
$528$	0	0
$529$	24.7250	1.07500
$530$	−10.6972	−0.464658
$531$	0	0
$532$	−13.3305	−0.577952
$533$	−1.30278	−0.0564295
$534$	0	0
$535$	−18.0000	−0.778208
$536$	−4.39445	−0.189811
$537$	0	0
$538$	14.4861	0.624541
$539$	0	0
$540$	0	0
$541$	−33.4222	−1.43693	−0.718466	−	0.695562i	$-0.755156\pi$
$541$	−33.4222	−1.43693	−0.718466	+	0.695562i	$0.755156\pi$
$542$	24.3305	1.04509
$543$	0	0
$544$	2.60555	0.111712
$545$	−6.39445	−0.273908
$546$	0	0
$547$	−41.1194	−1.75814	−0.879070	−	0.476693i	$-0.841835\pi$
$547$	−41.1194	−1.75814	−0.879070	+	0.476693i	$0.841835\pi$
$548$	−21.5139	−0.919027
$549$	0	0
$550$	0	0
$551$	1.45837	0.0621285
$552$	0	0
$553$	−33.5416	−1.42634
$554$	5.18335	0.220219
$555$	0	0
$556$	−9.42221	−0.399591
$557$	22.9361	0.971833	0.485917	−	0.874005i	$-0.338486\pi$
$557$	22.9361	0.971833	0.485917	+	0.874005i	$0.338486\pi$
$558$	0	0
$559$	33.7250	1.42641
$560$	−4.69722	−0.198494
$561$	0	0
$562$	−19.9361	−0.840953
$563$	13.8167	0.582303	0.291151	−	0.956677i	$-0.405962\pi$
$563$	13.8167	0.582303	0.291151	+	0.956677i	$0.405962\pi$
$564$	0	0
$565$	5.09167	0.214208
$566$	−12.0278	−0.505564
$567$	0	0
$568$	16.4222	0.689060
$569$	−27.2389	−1.14191	−0.570956	−	0.820981i	$-0.693427\pi$
$569$	−27.2389	−1.14191	−0.570956	+	0.820981i	$0.693427\pi$
$570$	0	0
$571$	1.00000	0.0418487	0.0209243	−	0.999781i	$-0.493339\pi$
$571$	1.00000	0.0418487	0.0209243	+	0.999781i	$0.493339\pi$
$572$	0	0
$573$	0	0
$574$	1.42221	0.0593617
$575$	22.8167	0.951520
$576$	0	0
$577$	−10.2389	−0.426249	−0.213125	−	0.977025i	$-0.568364\pi$
$577$	−10.2389	−0.426249	−0.213125	+	0.977025i	$0.568364\pi$
$578$	−10.2111	−0.424726
$579$	0	0
$580$	0.513878	0.0213376
$581$	34.3028	1.42312
$582$	0	0
$583$	0	0
$584$	2.30278	0.0952895
$585$	0	0
$586$	−29.3305	−1.21163
$587$	−30.7527	−1.26930	−0.634651	−	0.772799i	$-0.718856\pi$
$587$	−30.7527	−1.26930	−0.634651	+	0.772799i	$0.718856\pi$
$588$	0	0
$589$	27.7805	1.14468
$590$	16.8167	0.692331
$591$	0	0
$592$	−3.60555	−0.148187
$593$	−10.6972	−0.439282	−0.219641	−	0.975581i	$-0.570489\pi$
$593$	−10.6972	−0.439282	−0.219641	+	0.975581i	$0.570489\pi$
$594$	0	0
$595$	−12.2389	−0.501744
$596$	2.48612	0.101836
$597$	0	0
$598$	22.8167	0.933042
$599$	39.3583	1.60814	0.804068	−	0.594537i	$-0.202665\pi$
$599$	39.3583	1.60814	0.804068	+	0.594537i	$0.202665\pi$
$600$	0	0
$601$	−1.60555	−0.0654918	−0.0327459	−	0.999464i	$-0.510425\pi$
$601$	−1.60555	−0.0654918	−0.0327459	+	0.999464i	$0.510425\pi$
$602$	−36.8167	−1.50053
$603$	0	0
$604$	−17.0000	−0.691720
$605$	0	0
$606$	0	0
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	−3.69722	−0.149942
$609$	0	0
$610$	3.11943	0.126302
$611$	15.5139	0.627624
$612$	0	0
$613$	−29.6333	−1.19688	−0.598439	−	0.801168i	$-0.704212\pi$
$613$	−29.6333	−1.19688	−0.598439	+	0.801168i	$0.704212\pi$
$614$	4.90833	0.198084
$615$	0	0
$616$	0	0
$617$	13.6972	0.551429	0.275715	−	0.961240i	$-0.411085\pi$
$617$	13.6972	0.551429	0.275715	+	0.961240i	$0.411085\pi$
$618$	0	0
$619$	−10.5139	−0.422588	−0.211294	−	0.977423i	$-0.567768\pi$
$619$	−10.5139	−0.422588	−0.211294	+	0.977423i	$0.567768\pi$
$620$	9.78890	0.393132
$621$	0	0
$622$	5.60555	0.224762
$623$	−6.11943	−0.245170
$624$	0	0
$625$	2.42221	0.0968882
$626$	−14.9361	−0.596966
$627$	0	0
$628$	−10.3944	−0.414784
$629$	−9.39445	−0.374581
$630$	0	0
$631$	32.2389	1.28341	0.641704	−	0.766952i	$-0.278228\pi$
$631$	32.2389	1.28341	0.641704	+	0.766952i	$0.278228\pi$
$632$	−9.30278	−0.370045
$633$	0	0
$634$	21.3944	0.849682
$635$	3.78890	0.150358
$636$	0	0
$637$	−19.8167	−0.785164
$638$	0	0
$639$	0	0
$640$	−1.30278	−0.0514967
$641$	−2.09167	−0.0826161	−0.0413081	−	0.999146i	$-0.513153\pi$
$641$	−2.09167	−0.0826161	−0.0413081	+	0.999146i	$0.513153\pi$
$642$	0	0
$643$	−31.6333	−1.24750	−0.623748	−	0.781626i	$-0.714391\pi$
$643$	−31.6333	−1.24750	−0.623748	+	0.781626i	$0.714391\pi$
$644$	−24.9083	−0.981526
$645$	0	0
$646$	−9.63331	−0.379017
$647$	−47.8444	−1.88096	−0.940479	−	0.339852i	$-0.889623\pi$
$647$	−47.8444	−1.88096	−0.940479	+	0.339852i	$0.889623\pi$
$648$	0	0
$649$	0	0
$650$	10.9083	0.427860
$651$	0	0
$652$	21.3028	0.834281
$653$	10.1833	0.398505	0.199253	−	0.979948i	$-0.436149\pi$
$653$	10.1833	0.398505	0.199253	+	0.979948i	$0.436149\pi$
$654$	0	0
$655$	−21.9083	−0.856029
$656$	0.394449	0.0154006
$657$	0	0
$658$	−16.9361	−0.660237
$659$	−42.7527	−1.66541	−0.832705	−	0.553717i	$-0.813209\pi$
$659$	−42.7527	−1.66541	−0.832705	+	0.553717i	$0.813209\pi$
$660$	0	0
$661$	−38.0278	−1.47911	−0.739554	−	0.673097i	$-0.764964\pi$
$661$	−38.0278	−1.47911	−0.739554	+	0.673097i	$0.764964\pi$
$662$	−7.00000	−0.272063
$663$	0	0
$664$	9.51388	0.369210
$665$	17.3667	0.673451
$666$	0	0
$667$	2.72498	0.105512
$668$	−2.09167	−0.0809293
$669$	0	0
$670$	5.72498	0.221175
$671$	0	0
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	11.8167	0.455160
$675$	0	0
$676$	−2.09167	−0.0804490
$677$	36.7527	1.41252	0.706261	−	0.707951i	$-0.250380\pi$
$677$	36.7527	1.41252	0.706261	+	0.707951i	$0.250380\pi$
$678$	0	0
$679$	47.6333	1.82800
$680$	−3.39445	−0.130171
$681$	0	0
$682$	0	0
$683$	−25.9361	−0.992417	−0.496208	−	0.868203i	$-0.665275\pi$
$683$	−25.9361	−0.992417	−0.496208	+	0.868203i	$0.665275\pi$
$684$	0	0
$685$	28.0278	1.07089
$686$	−3.60555	−0.137661
$687$	0	0
$688$	−10.2111	−0.389295
$689$	−27.1194	−1.03317
$690$	0	0
$691$	−23.6972	−0.901485	−0.450742	−	0.892654i	$-0.648841\pi$
$691$	−23.6972	−0.901485	−0.450742	+	0.892654i	$0.648841\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	29.7250	1.12834
$695$	12.2750	0.465618
$696$	0	0
$697$	1.02776	0.0389290
$698$	1.51388	0.0573011
$699$	0	0
$700$	−11.9083	−0.450092
$701$	−20.7250	−0.782772	−0.391386	−	0.920227i	$-0.628004\pi$
$701$	−20.7250	−0.782772	−0.391386	+	0.920227i	$0.628004\pi$
$702$	0	0
$703$	13.3305	0.502771
$704$	0	0
$705$	0	0
$706$	33.6333	1.26581
$707$	−9.39445	−0.353315
$708$	0	0
$709$	43.8444	1.64661	0.823306	−	0.567598i	$-0.192127\pi$
$709$	43.8444	1.64661	0.823306	+	0.567598i	$0.192127\pi$
$710$	−21.3944	−0.802919
$711$	0	0
$712$	−1.69722	−0.0636062
$713$	51.9083	1.94398
$714$	0	0
$715$	0	0
$716$	−5.09167	−0.190285
$717$	0	0
$718$	−12.3944	−0.462557
$719$	38.2111	1.42503	0.712517	−	0.701655i	$-0.247555\pi$
$719$	38.2111	1.42503	0.712517	+	0.701655i	$0.247555\pi$
$720$	0	0
$721$	−56.2666	−2.09548
$722$	−5.33053	−0.198382
$723$	0	0
$724$	8.90833	0.331075
$725$	1.30278	0.0483839
$726$	0	0
$727$	−8.97224	−0.332762	−0.166381	−	0.986062i	$-0.553208\pi$
$727$	−8.97224	−0.332762	−0.166381	+	0.986062i	$0.553208\pi$
$728$	−11.9083	−0.441352
$729$	0	0
$730$	−3.00000	−0.111035
$731$	−26.6056	−0.984042
$732$	0	0
$733$	21.3305	0.787861	0.393931	−	0.919140i	$-0.371115\pi$
$733$	21.3305	0.787861	0.393931	+	0.919140i	$0.371115\pi$
$734$	−6.09167	−0.224848
$735$	0	0
$736$	−6.90833	−0.254644
$737$	0	0
$738$	0	0
$739$	−4.09167	−0.150515	−0.0752573	−	0.997164i	$-0.523978\pi$
$739$	−4.09167	−0.150515	−0.0752573	+	0.997164i	$0.523978\pi$
$740$	4.69722	0.172673
$741$	0	0
$742$	29.6056	1.08685
$743$	31.9361	1.17162	0.585811	−	0.810448i	$-0.300776\pi$
$743$	31.9361	1.17162	0.585811	+	0.810448i	$0.300776\pi$
$744$	0	0
$745$	−3.23886	−0.118663
$746$	−7.48612	−0.274086
$747$	0	0
$748$	0	0
$749$	49.8167	1.82026
$750$	0	0
$751$	34.8444	1.27149	0.635745	−	0.771899i	$-0.280693\pi$
$751$	34.8444	1.27149	0.635745	+	0.771899i	$0.280693\pi$
$752$	−4.69722	−0.171290
$753$	0	0
$754$	1.30278	0.0474443
$755$	22.1472	0.806019
$756$	0	0
$757$	−34.3944	−1.25009	−0.625044	−	0.780590i	$-0.714919\pi$
$757$	−34.3944	−1.25009	−0.625044	+	0.780590i	$0.714919\pi$
$758$	8.11943	0.294911
$759$	0	0
$760$	4.81665	0.174718
$761$	19.4222	0.704054	0.352027	−	0.935990i	$-0.385492\pi$
$761$	19.4222	0.704054	0.352027	+	0.935990i	$0.385492\pi$
$762$	0	0
$763$	17.6972	0.640683
$764$	−10.6972	−0.387012
$765$	0	0
$766$	−22.8167	−0.824399
$767$	42.6333	1.53940
$768$	0	0
$769$	28.5139	1.02824	0.514118	−	0.857719i	$-0.328119\pi$
$769$	28.5139	1.02824	0.514118	+	0.857719i	$0.328119\pi$
$770$	0	0
$771$	0	0
$772$	−25.7250	−0.925862
$773$	29.8444	1.07343	0.536714	−	0.843764i	$-0.319665\pi$
$773$	29.8444	1.07343	0.536714	+	0.843764i	$0.319665\pi$
$774$	0	0
$775$	24.8167	0.891441
$776$	13.2111	0.474251
$777$	0	0
$778$	8.21110	0.294382
$779$	−1.45837	−0.0522514
$780$	0	0
$781$	0	0
$782$	−18.0000	−0.643679
$783$	0	0
$784$	6.00000	0.214286
$785$	13.5416	0.483322
$786$	0	0
$787$	5.18335	0.184766	0.0923832	−	0.995724i	$-0.470552\pi$
$787$	5.18335	0.184766	0.0923832	+	0.995724i	$0.470552\pi$
$788$	−1.81665	−0.0647156
$789$	0	0
$790$	12.1194	0.431190
$791$	−14.0917	−0.501042
$792$	0	0
$793$	7.90833	0.280833
$794$	38.2389	1.35705
$795$	0	0
$796$	20.0000	0.708881
$797$	23.2111	0.822179	0.411090	−	0.911595i	$-0.365148\pi$
$797$	23.2111	0.822179	0.411090	+	0.911595i	$0.365148\pi$
$798$	0	0
$799$	−12.2389	−0.432980
$800$	−3.30278	−0.116771
$801$	0	0
$802$	33.9083	1.19734
$803$	0	0
$804$	0	0
$805$	32.4500	1.14371
$806$	24.8167	0.874129
$807$	0	0
$808$	−2.60555	−0.0916630
$809$	17.6056	0.618978	0.309489	−	0.950903i	$-0.399842\pi$
$809$	17.6056	0.618978	0.309489	+	0.950903i	$0.399842\pi$
$810$	0	0
$811$	−8.51388	−0.298963	−0.149481	−	0.988765i	$-0.547760\pi$
$811$	−8.51388	−0.298963	−0.149481	+	0.988765i	$0.547760\pi$
$812$	−1.42221	−0.0499096
$813$	0	0
$814$	0	0
$815$	−27.7527	−0.972136
$816$	0	0
$817$	37.7527	1.32080
$818$	−33.5416	−1.17276
$819$	0	0
$820$	−0.513878	−0.0179454
$821$	42.5139	1.48374	0.741872	−	0.670541i	$-0.233938\pi$
$821$	42.5139	1.48374	0.741872	+	0.670541i	$0.233938\pi$
$822$	0	0
$823$	13.3305	0.464673	0.232337	−	0.972635i	$-0.425363\pi$
$823$	13.3305	0.464673	0.232337	+	0.972635i	$0.425363\pi$
$824$	−15.6056	−0.543645
$825$	0	0
$826$	−46.5416	−1.61939
$827$	−2.36669	−0.0822980	−0.0411490	−	0.999153i	$-0.513102\pi$
$827$	−2.36669	−0.0822980	−0.0411490	+	0.999153i	$0.513102\pi$
$828$	0	0
$829$	22.8806	0.794675	0.397338	−	0.917673i	$-0.369934\pi$
$829$	22.8806	0.794675	0.397338	+	0.917673i	$0.369934\pi$
$830$	−12.3944	−0.430218
$831$	0	0
$832$	−3.30278	−0.114503
$833$	15.6333	0.541662
$834$	0	0
$835$	2.72498	0.0943018
$836$	0	0
$837$	0	0
$838$	24.6333	0.850943
$839$	23.0917	0.797213	0.398607	−	0.917122i	$-0.369494\pi$
$839$	23.0917	0.797213	0.398607	+	0.917122i	$0.369494\pi$
$840$	0	0
$841$	−28.8444	−0.994635
$842$	−12.2111	−0.420822
$843$	0	0
$844$	14.0278	0.482855
$845$	2.72498	0.0937422
$846$	0	0
$847$	0	0
$848$	8.21110	0.281970
$849$	0	0
$850$	−8.60555	−0.295168
$851$	24.9083	0.853846
$852$	0	0
$853$	−31.0555	−1.06332	−0.531660	−	0.846958i	$-0.678432\pi$
$853$	−31.0555	−1.06332	−0.531660	+	0.846958i	$0.678432\pi$
$854$	−8.63331	−0.295426
$855$	0	0
$856$	13.8167	0.472244
$857$	30.7527	1.05049	0.525247	−	0.850950i	$-0.323973\pi$
$857$	30.7527	1.05049	0.525247	+	0.850950i	$0.323973\pi$
$858$	0	0
$859$	−1.78890	−0.0610364	−0.0305182	−	0.999534i	$-0.509716\pi$
$859$	−1.78890	−0.0610364	−0.0305182	+	0.999534i	$0.509716\pi$
$860$	13.3028	0.453621
$861$	0	0
$862$	−32.6056	−1.11055
$863$	31.8167	1.08305	0.541526	−	0.840684i	$-0.317847\pi$
$863$	31.8167	1.08305	0.541526	+	0.840684i	$0.317847\pi$
$864$	0	0
$865$	−27.3583	−0.930210
$866$	−23.9361	−0.813381
$867$	0	0
$868$	−27.0917	−0.919551
$869$	0	0
$870$	0	0
$871$	14.5139	0.491784
$872$	4.90833	0.166217
$873$	0	0
$874$	25.5416	0.863959
$875$	39.0000	1.31844
$876$	0	0
$877$	9.60555	0.324356	0.162178	−	0.986761i	$-0.448148\pi$
$877$	9.60555	0.324356	0.162178	+	0.986761i	$0.448148\pi$
$878$	−24.0278	−0.810898
$879$	0	0
$880$	0	0
$881$	−4.06392	−0.136917	−0.0684584	−	0.997654i	$-0.521808\pi$
$881$	−4.06392	−0.136917	−0.0684584	+	0.997654i	$0.521808\pi$
$882$	0	0
$883$	23.0000	0.774012	0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000	0.774012	0.387006	+	0.922077i	$0.373509\pi$
$884$	−8.60555	−0.289436
$885$	0	0
$886$	−10.1833	−0.342116
$887$	−16.8167	−0.564648	−0.282324	−	0.959319i	$-0.591105\pi$
$887$	−16.8167	−0.564648	−0.282324	+	0.959319i	$0.591105\pi$
$888$	0	0
$889$	−10.4861	−0.351693
$890$	2.21110	0.0741163
$891$	0	0
$892$	7.72498	0.258651
$893$	17.3667	0.581154
$894$	0	0
$895$	6.63331	0.221727
$896$	3.60555	0.120453
$897$	0	0
$898$	−10.8167	−0.360956
$899$	2.96384	0.0988496
$900$	0	0
$901$	21.3944	0.712752
$902$	0	0
$903$	0	0
$904$	−3.90833	−0.129989
$905$	−11.6056	−0.385782
$906$	0	0
$907$	8.51388	0.282699	0.141349	−	0.989960i	$-0.454856\pi$
$907$	8.51388	0.282699	0.141349	+	0.989960i	$0.454856\pi$
$908$	−3.90833	−0.129702
$909$	0	0
$910$	15.5139	0.514280
$911$	−16.0278	−0.531023	−0.265512	−	0.964108i	$-0.585541\pi$
$911$	−16.0278	−0.531023	−0.265512	+	0.964108i	$0.585541\pi$
$912$	0	0
$913$	0	0
$914$	32.4222	1.07243
$915$	0	0
$916$	29.1194	0.962133
$917$	60.6333	2.00229
$918$	0	0
$919$	39.7250	1.31041	0.655203	−	0.755453i	$-0.272583\pi$
$919$	39.7250	1.31041	0.655203	+	0.755453i	$0.272583\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	−13.1833	−0.434170
$923$	−54.2389	−1.78529
$924$	0	0
$925$	11.9083	0.391543
$926$	11.1194	0.365407
$927$	0	0
$928$	−0.394449	−0.0129484
$929$	17.7250	0.581538	0.290769	−	0.956793i	$-0.406089\pi$
$929$	17.7250	0.581538	0.290769	+	0.956793i	$0.406089\pi$
$930$	0	0
$931$	−22.1833	−0.727030
$932$	−26.7250	−0.875406
$933$	0	0
$934$	23.4861	0.768489
$935$	0	0
$936$	0	0
$937$	9.60555	0.313800	0.156900	−	0.987615i	$-0.449850\pi$
$937$	9.60555	0.313800	0.156900	+	0.987615i	$0.449850\pi$
$938$	−15.8444	−0.517338
$939$	0	0
$940$	6.11943	0.199594
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−2.72498	−0.0887376
$944$	−12.9083	−0.420130
$945$	0	0
$946$	0	0
$947$	52.0278	1.69067	0.845337	−	0.534233i	$-0.179399\pi$
$947$	52.0278	1.69067	0.845337	+	0.534233i	$0.179399\pi$
$948$	0	0
$949$	−7.60555	−0.246887
$950$	12.2111	0.396180
$951$	0	0
$952$	9.39445	0.304476
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	0	0
$955$	13.9361	0.450961
$956$	−16.0278	−0.518375
$957$	0	0
$958$	28.6972	0.927165
$959$	−77.5694	−2.50485
$960$	0	0
$961$	25.4584	0.821238
$962$	11.9083	0.383940
$963$	0	0
$964$	−14.7889	−0.476318
$965$	33.5139	1.07885
$966$	0	0
$967$	36.0917	1.16063	0.580315	−	0.814392i	$-0.302929\pi$
$967$	36.0917	1.16063	0.580315	+	0.814392i	$0.302929\pi$
$968$	0	0
$969$	0	0
$970$	−17.2111	−0.552615
$971$	−38.8444	−1.24658	−0.623288	−	0.781992i	$-0.714204\pi$
$971$	−38.8444	−1.24658	−0.623288	+	0.781992i	$0.714204\pi$
$972$	0	0
$973$	−33.9722	−1.08910
$974$	−3.60555	−0.115529
$975$	0	0
$976$	−2.39445	−0.0766444
$977$	−17.2111	−0.550632	−0.275316	−	0.961354i	$-0.588782\pi$
$977$	−17.2111	−0.550632	−0.275316	+	0.961354i	$0.588782\pi$
$978$	0	0
$979$	0	0
$980$	−7.81665	−0.249694
$981$	0	0
$982$	−4.81665	−0.153706
$983$	1.42221	0.0453613	0.0226806	−	0.999743i	$-0.492780\pi$
$983$	1.42221	0.0453613	0.0226806	+	0.999743i	$0.492780\pi$
$984$	0	0
$985$	2.36669	0.0754091
$986$	−1.02776	−0.0327304
$987$	0	0
$988$	12.2111	0.388487
$989$	70.5416	2.24309
$990$	0	0
$991$	−37.2389	−1.18293	−0.591466	−	0.806330i	$-0.701450\pi$
$991$	−37.2389	−1.18293	−0.591466	+	0.806330i	$0.701450\pi$
$992$	−7.51388	−0.238566
$993$	0	0
$994$	59.2111	1.87806
$995$	−26.0555	−0.826015
$996$	0	0
$997$	−17.9083	−0.567162	−0.283581	−	0.958948i	$-0.591523\pi$
$997$	−17.9083	−0.567162	−0.283581	+	0.958948i	$0.591523\pi$
$998$	−13.7889	−0.436480
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.ck.1.1	yes	2
3.2	odd	2		6534.2.a.bj.1.2	✓	2
11.10	odd	2		6534.2.a.bq.1.1	yes	2
33.32	even	2		6534.2.a.cd.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6534.2.a.bj.1.2	✓	2	3.2	odd	2
6534.2.a.bq.1.1	yes	2	11.10	odd	2
6534.2.a.cd.1.2	yes	2	33.32	even	2
6534.2.a.ck.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +3.60555 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +3.60555 q^{7} +1.00000 q^{8} -1.30278 q^{10} -3.30278 q^{13} +3.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} -3.69722 q^{19} -1.30278 q^{20} -6.90833 q^{23} -3.30278 q^{25} -3.30278 q^{26} +3.60555 q^{28} -0.394449 q^{29} -7.51388 q^{31} +1.00000 q^{32} +2.60555 q^{34} -4.69722 q^{35} -3.60555 q^{37} -3.69722 q^{38} -1.30278 q^{40} +0.394449 q^{41} -10.2111 q^{43} -6.90833 q^{46} -4.69722 q^{47} +6.00000 q^{49} -3.30278 q^{50} -3.30278 q^{52} +8.21110 q^{53} +3.60555 q^{56} -0.394449 q^{58} -12.9083 q^{59} -2.39445 q^{61} -7.51388 q^{62} +1.00000 q^{64} +4.30278 q^{65} -4.39445 q^{67} +2.60555 q^{68} -4.69722 q^{70} +16.4222 q^{71} +2.30278 q^{73} -3.60555 q^{74} -3.69722 q^{76} -9.30278 q^{79} -1.30278 q^{80} +0.394449 q^{82} +9.51388 q^{83} -3.39445 q^{85} -10.2111 q^{86} -1.69722 q^{89} -11.9083 q^{91} -6.90833 q^{92} -4.69722 q^{94} +4.81665 q^{95} +13.2111 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8} + q^{10} - 3 q^{13} + 2 q^{16} - 2 q^{17} - 11 q^{19} + q^{20} - 3 q^{23} - 3 q^{25} - 3 q^{26} - 8 q^{29} + 3 q^{31} + 2 q^{32} - 2 q^{34} - 13 q^{35} - 11 q^{38} + q^{40} + 8 q^{41} - 6 q^{43} - 3 q^{46} - 13 q^{47} + 12 q^{49} - 3 q^{50} - 3 q^{52} + 2 q^{53} - 8 q^{58} - 15 q^{59} - 12 q^{61} + 3 q^{62} + 2 q^{64} + 5 q^{65} - 16 q^{67} - 2 q^{68} - 13 q^{70} + 4 q^{71} + q^{73} - 11 q^{76} - 15 q^{79} + q^{80} + 8 q^{82} + q^{83} - 14 q^{85} - 6 q^{86} - 7 q^{89} - 13 q^{91} - 3 q^{92} - 13 q^{94} - 12 q^{95} + 12 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8 + q^10 - 3 * q^13 + 2 * q^16 - 2 * q^17 - 11 * q^19 + q^20 - 3 * q^23 - 3 * q^25 - 3 * q^26 - 8 * q^29 + 3 * q^31 + 2 * q^32 - 2 * q^34 - 13 * q^35 - 11 * q^38 + q^40 + 8 * q^41 - 6 * q^43 - 3 * q^46 - 13 * q^47 + 12 * q^49 - 3 * q^50 - 3 * q^52 + 2 * q^53 - 8 * q^58 - 15 * q^59 - 12 * q^61 + 3 * q^62 + 2 * q^64 + 5 * q^65 - 16 * q^67 - 2 * q^68 - 13 * q^70 + 4 * q^71 + q^73 - 11 * q^76 - 15 * q^79 + q^80 + 8 * q^82 + q^83 - 14 * q^85 - 6 * q^86 - 7 * q^89 - 13 * q^91 - 3 * q^92 - 13 * q^94 - 12 * q^95 + 12 * q^97 + 12 * q^98

Introduction

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