LMFDB - Embedded newform 5445.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} -4.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} -6.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} +6.00000 q^{43} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -6.00000 q^{53} +6.00000 q^{56} -6.00000 q^{58} -4.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +12.0000 q^{67} +6.00000 q^{68} -2.00000 q^{70} -8.00000 q^{71} -16.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} -2.00000 q^{79} -1.00000 q^{80} +6.00000 q^{82} -6.00000 q^{85} -6.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} +6.00000 q^{95} -6.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	0	0
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.00000	0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−6.00000	−0.646997
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	4.00000	0.417029
$93$	0	0
$94$	8.00000	0.825137
$95$	6.00000	0.615587
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	6.00000	0.582772
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−6.00000	−0.557086
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	0	0
$122$	4.00000	0.362143
$123$	0	0
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−18.0000	−1.54349
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	16.0000	1.32417
$147$	0	0
$148$	6.00000	0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	18.0000	1.45999
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	6.00000	0.460179
$171$	0	0
$172$	−6.00000	−0.457496
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	10.0000	0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	8.00000	0.592999
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	−6.00000	−0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	3.00000	0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−14.0000	−0.985037
$203$	12.0000	0.842235
$204$	0	0
$205$	−6.00000	−0.419058
$206$	8.00000	0.557386
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−8.00000	−0.546869
$215$	6.00000	0.409197
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	6.00000	0.399114
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	18.0000	1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	0	0
$238$	12.0000	0.777844
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	0	0
$244$	4.00000	0.256074
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−24.0000	−1.52708
$248$	24.0000	1.52400
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	4.00000	0.248069
$261$	0	0
$262$	−16.0000	−0.988483
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−12.0000	−0.735767
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	14.0000	0.845771
$275$	0	0
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−22.0000	−1.31947
$279$	0	0
$280$	6.00000	0.358569
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	0	0
$292$	16.0000	0.936329
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	−18.0000	−1.04623
$297$	0	0
$298$	6.00000	0.347571
$299$	16.0000	0.925304
$300$	0	0
$301$	12.0000	0.691669
$302$	2.00000	0.115087
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	2.00000	0.112509
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	0	0
$320$	7.00000	0.391312
$321$	0	0
$322$	8.00000	0.445823
$323$	−36.0000	−2.00309
$324$	0	0
$325$	−4.00000	−0.221880
$326$	4.00000	0.221540
$327$	0	0
$328$	−18.0000	−0.993884
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	18.0000	0.970495
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−36.0000	−1.92704	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000	−1.92704	−0.963518	+	0.267644i	$0.913755\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	10.0000	0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	26.0000	1.36653
$363$	0	0
$364$	8.00000	0.419314
$365$	−16.0000	−0.837478
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	6.00000	0.311925
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	0	0
$375$	0	0
$376$	−24.0000	−1.23771
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	6.00000	0.304604
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−2.00000	−0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	0	0
$416$	20.0000	0.980581
$417$	0	0
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	−18.0000	−0.874157
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	16.0000	0.757622
$447$	0	0
$448$	14.0000	0.661438
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−16.0000	−0.750917
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	4.00000	0.186501
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	8.00000	0.369012
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.00000	0.275299
$476$	12.0000	0.550019
$477$	0	0
$478$	12.0000	0.548867
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	20.0000	0.910975
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−12.0000	−0.543214
$489$	0	0
$490$	3.00000	0.135526
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	24.0000	1.07981
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	20.0000	0.892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	11.0000	0.486136
$513$	0	0
$514$	18.0000	0.793946
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	12.0000	0.527250
$519$	0	0
$520$	−12.0000	−0.526235
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	24.0000	1.04645
$527$	−48.0000	−2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	6.00000	0.260623
$531$	0	0
$532$	−12.0000	−0.520266
$533$	24.0000	1.03956
$534$	0	0
$535$	8.00000	0.345870
$536$	36.0000	1.55496
$537$	0	0
$538$	6.00000	0.258678
$539$	0	0
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	30.0000	1.28624
$545$	0	0
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	−4.00000	−0.170097
$554$	4.00000	0.169944
$555$	0	0
$556$	−22.0000	−0.933008
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	22.0000	0.928014
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−14.0000	−0.588464
$567$	0	0
$568$	−24.0000	−1.00702
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	−4.00000	−0.166812
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−48.0000	−1.98625
$585$	0	0
$586$	18.0000	0.743573
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	0	0
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	6.00000	0.245770
$597$	0	0
$598$	−16.0000	−0.654289
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	4.00000	0.161955
$611$	32.0000	1.29458
$612$	0	0
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−28.0000	−1.12270
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−18.0000	−0.718278
$629$	36.0000	1.43541
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	−30.0000	−1.19145
$635$	2.00000	0.0793676
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	3.00000	0.118585
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	36.0000	1.41640
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	0	0
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	4.00000	0.156652
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	6.00000	0.234261
$657$	0	0
$658$	16.0000	0.623745
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	−12.0000	−0.463600
$671$	0	0
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	−3.00000	−0.115385
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	−18.0000	−0.690268
$681$	0	0
$682$	0	0
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	20.0000	0.763604
$687$	0	0
$688$	−6.00000	−0.228748
$689$	24.0000	0.914327
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	8.00000	0.303676
$695$	22.0000	0.834508
$696$	0	0
$697$	36.0000	1.36360
$698$	36.0000	1.36262
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	28.0000	1.05305
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	−30.0000	−1.12430
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−8.00000	−0.298557
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−17.0000	−0.632674
$723$	0	0
$724$	26.0000	0.966282
$725$	6.00000	0.222834
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−24.0000	−0.889499
$729$	0	0
$730$	16.0000	0.592187
$731$	−36.0000	−1.33151
$732$	0	0
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	12.0000	0.440534
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	24.0000	0.878702
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	24.0000	0.874028
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	18.0000	0.652929
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	36.0000	1.30073
$767$	0	0
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	20.0000	0.719816
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−18.0000	−0.646162
$777$	0	0
$778$	−6.00000	−0.215110
$779$	−36.0000	−1.28983
$780$	0	0
$781$	0	0
$782$	−24.0000	−0.858238
$783$	0	0
$784$	3.00000	0.107143
$785$	18.0000	0.642448
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	2.00000	0.0711568
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000	0.568177
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−10.0000	−0.353112
$803$	0	0
$804$	0	0
$805$	−8.00000	−0.281963
$806$	32.0000	1.12715
$807$	0	0
$808$	42.0000	1.47755
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	−50.0000	−1.75574	−0.877869	−	0.478901i	$-0.841035\pi$
$811$	−50.0000	−1.75574	−0.877869	+	0.478901i	$0.841035\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	36.0000	1.25948
$818$	−4.00000	−0.139857
$819$	0	0
$820$	6.00000	0.209529
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	−28.0000	−0.970725
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−28.0000	−0.967244
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	6.00000	0.206774
$843$	0	0
$844$	−22.0000	−0.757271
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	6.00000	0.205798
$851$	24.0000	0.822709
$852$	0	0
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	24.0000	0.820303
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	12.0000	0.408722
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	−18.0000	−0.611665
$867$	0	0
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	0	0
$874$	24.0000	0.811812
$875$	2.00000	0.0676123
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−26.0000	−0.877457
$879$	0	0
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	16.0000	0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	10.0000	0.335201
$891$	0	0
$892$	16.0000	0.535720
$893$	−48.0000	−1.60626
$894$	0	0
$895$	−12.0000	−0.401116
$896$	6.00000	0.200446
$897$	0	0
$898$	−30.0000	−1.00111
$899$	48.0000	1.60089
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−26.0000	−0.864269
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−16.0000	−0.530979
$909$	0	0
$910$	8.00000	0.265197
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	0	0
$914$	8.00000	0.264616
$915$	0	0
$916$	22.0000	0.726900
$917$	32.0000	1.05673
$918$	0	0
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	22.0000	0.724531
$923$	32.0000	1.05329
$924$	0	0
$925$	−6.00000	−0.197279
$926$	8.00000	0.262896
$927$	0	0
$928$	−30.0000	−0.984798
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−6.00000	−0.196537
$933$	0	0
$934$	16.0000	0.523536
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	−24.0000	−0.783628
$939$	0	0
$940$	8.00000	0.260931
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	−6.00000	−0.194666
$951$	0	0
$952$	−36.0000	−1.16677
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	−28.0000	−0.904639
$959$	−28.0000	−0.904167
$960$	0	0
$961$	33.0000	1.06452
$962$	−24.0000	−0.773791
$963$	0	0
$964$	20.0000	0.644157
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	0	0
$969$	0	0
$970$	6.00000	0.192648
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	44.0000	1.41058
$974$	16.0000	0.512673
$975$	0	0
$976$	4.00000	0.128037
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	0	0
$980$	3.00000	0.0958315
$981$	0	0
$982$	36.0000	1.14881
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	36.0000	1.14647
$987$	0	0
$988$	24.0000	0.763542
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	16.0000	0.507489
$995$	16.0000	0.507234
$996$	0	0
$997$	−24.0000	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.e.1.1		1
3.2	odd	2		1815.2.a.e.1.1	yes	1
11.10	odd	2		5445.2.a.j.1.1		1
15.14	odd	2		9075.2.a.d.1.1		1
33.32	even	2		1815.2.a.a.1.1	✓	1
165.164	even	2		9075.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.a.1.1	✓	1	33.32	even	2
1815.2.a.e.1.1	yes	1	3.2	odd	2
5445.2.a.e.1.1		1	1.1	even	1	trivial
5445.2.a.j.1.1		1	11.10	odd	2
9075.2.a.d.1.1		1	15.14	odd	2
9075.2.a.n.1.1		1	165.164	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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