LMFDB - Embedded newform 5445.2.a.b.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:	yes
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)

$f(q)$

$=$

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

$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{10} +2.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} -6.00000 q^{17} +5.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +6.00000 q^{28} -6.00000 q^{29} -5.00000 q^{31} +8.00000 q^{32} +12.0000 q^{34} +3.00000 q^{35} -11.0000 q^{37} -10.0000 q^{38} -4.00000 q^{41} +4.00000 q^{43} -8.00000 q^{46} -6.00000 q^{47} +2.00000 q^{49} -2.00000 q^{50} +4.00000 q^{52} -8.00000 q^{53} +12.0000 q^{58} -2.00000 q^{59} -1.00000 q^{61} +10.0000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -7.00000 q^{67} -12.0000 q^{68} -6.00000 q^{70} -10.0000 q^{71} +11.0000 q^{73} +22.0000 q^{74} +10.0000 q^{76} -1.00000 q^{79} -4.00000 q^{80} +8.00000 q^{82} -10.0000 q^{83} -6.00000 q^{85} -8.00000 q^{86} -14.0000 q^{89} +6.00000 q^{91} +8.00000 q^{92} +12.0000 q^{94} +5.00000 q^{95} -17.0000 q^{97} -4.00000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	0	0
$10$	−2.00000	−0.632456
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−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
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	6.00000	1.13389
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	12.0000	2.05798
$35$	3.00000	0.507093
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	−10.0000	−1.62221
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−2.00000	−0.282843
$51$	0	0
$52$	4.00000	0.554700
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	12.0000	1.57568
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	10.0000	1.27000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	−6.00000	−0.717137
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	22.0000	2.55745
$75$	0	0
$76$	10.0000	1.14708
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	8.00000	0.883452
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	8.00000	0.834058
$93$	0	0
$94$	12.0000	1.23771
$95$	5.00000	0.512989
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−4.00000	−0.404061
$99$	0	0
$100$	2.00000	0.200000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	0	0
$106$	16.0000	1.55406
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	0	0
$112$	−12.0000	−1.13389
$113$	−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
$116$	−12.0000	−1.11417
$117$	0	0
$118$	4.00000	0.368230
$119$	−18.0000	−1.65006
$120$	0	0
$121$	0	0
$122$	2.00000	0.181071
$123$	0	0
$124$	−10.0000	−0.898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	0	0
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	20.0000	1.67836
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−22.0000	−1.82073
$147$	0	0
$148$	−22.0000	−1.80839
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	8.00000	0.632456
$161$	12.0000	0.945732
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	20.0000	1.55230
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	0	0
$172$	8.00000	0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	0	0
$178$	28.0000	2.09869
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−12.0000	−0.889499
$183$	0	0
$184$	0	0
$185$	−11.0000	−0.808736
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−10.0000	−0.725476
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	34.0000	2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	27.0000	1.91398	0.956990	−	0.290122i	$-0.0936959\pi$
$199$	27.0000	1.91398	0.956990	+	0.290122i	$0.0936959\pi$
$200$	0	0
$201$	0	0
$202$	20.0000	1.40720
$203$	−18.0000	−1.26335
$204$	0	0
$205$	−4.00000	−0.279372
$206$	26.0000	1.81151
$207$	0	0
$208$	−8.00000	−0.554700
$209$	0	0
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−16.0000	−1.09888
$213$	0	0
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	0	0
$217$	−15.0000	−1.01827
$218$	30.0000	2.03186
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	12.0000	0.798228
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	0	0
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−4.00000	−0.260378
$237$	0	0
$238$	36.0000	2.33353
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	0	0
$244$	−2.00000	−0.128037
$245$	2.00000	0.127775
$246$	0	0
$247$	10.0000	0.636285
$248$	0	0
$249$	0	0
$250$	−2.00000	−0.126491
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	−26.0000	−1.63139
$255$	0	0
$256$	16.0000	1.00000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−33.0000	−2.05052
$260$	4.00000	0.248069
$261$	0	0
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−30.0000	−1.83942
$267$	0	0
$268$	−14.0000	−0.855186
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	−24.0000	−1.44989
$275$	0	0
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	−24.0000	−1.43942
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	7.00000	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$283$	7.00000	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$284$	−20.0000	−1.18678
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	0	0
$292$	22.0000	1.28745
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	0	0
$297$	0	0
$298$	−28.0000	−1.62200
$299$	8.00000	0.462652
$300$	0	0
$301$	12.0000	0.691669
$302$	−40.0000	−2.30174
$303$	0	0
$304$	−20.0000	−1.14708
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	0	0
$309$	0	0
$310$	10.0000	0.567962
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−26.0000	−1.46726
$315$	0	0
$316$	−2.00000	−0.112509
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	−8.00000	−0.447214
$321$	0	0
$322$	−24.0000	−1.33747
$323$	−30.0000	−1.66924
$324$	0	0
$325$	2.00000	0.110940
$326$	−14.0000	−0.775388
$327$	0	0
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	−20.0000	−1.09764
$333$	0	0
$334$	16.0000	0.875481
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	18.0000	0.979071
$339$	0	0
$340$	−12.0000	−0.650791
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	−6.00000	−0.320713
$351$	0	0
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	−28.0000	−1.48400
$357$	0	0
$358$	4.00000	0.211407
$359$	34.0000	1.79445	0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000	1.79445	0.897226	+	0.441572i	$0.145579\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−10.0000	−0.525588
$363$	0	0
$364$	12.0000	0.628971
$365$	11.0000	0.575766
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−16.0000	−0.834058
$369$	0	0
$370$	22.0000	1.14373
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−15.0000	−0.776671	−0.388335	−	0.921518i	$-0.626950\pi$
$373$	−15.0000	−0.776671	−0.388335	+	0.921518i	$0.626950\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	10.0000	0.512989
$381$	0	0
$382$	−40.0000	−2.04658
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	46.0000	2.34134
$387$	0	0
$388$	−34.0000	−1.72609
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	32.0000	1.61214
$395$	−1.00000	−0.0503155
$396$	0	0
$397$	−27.0000	−1.35509	−0.677546	−	0.735481i	$-0.736956\pi$
$397$	−27.0000	−1.35509	−0.677546	+	0.735481i	$0.736956\pi$
$398$	−54.0000	−2.70678
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	−20.0000	−0.995037
$405$	0	0
$406$	36.0000	1.78665
$407$	0	0
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	8.00000	0.395092
$411$	0	0
$412$	−26.0000	−1.28093
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−10.0000	−0.490881
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	−34.0000	−1.66101	−0.830504	−	0.557012i	$-0.811948\pi$
$419$	−34.0000	−1.66101	−0.830504	+	0.557012i	$0.811948\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−3.00000	−0.145180
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	30.0000	1.44005
$435$	0	0
$436$	−30.0000	−1.43674
$437$	20.0000	0.956730
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	−22.0000	−1.04525	−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000	−1.04525	−0.522626	+	0.852562i	$0.675047\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	−10.0000	−0.473514
$447$	0	0
$448$	−24.0000	−1.13389
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	0	0
$452$	−12.0000	−0.564433
$453$	0	0
$454$	−8.00000	−0.375459
$455$	6.00000	0.281284
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	8.00000	0.373002
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	60.0000	2.77945
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	12.0000	0.553519
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.00000	0.229416
$476$	−36.0000	−1.65006
$477$	0	0
$478$	−28.0000	−1.28069
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−22.0000	−1.00311
$482$	−20.0000	−0.910975
$483$	0	0
$484$	0	0
$485$	−17.0000	−0.771930
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	−4.00000	−0.180702
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−20.0000	−0.899843
$495$	0	0
$496$	20.0000	0.898027
$497$	−30.0000	−1.34568
$498$	0	0
$499$	39.0000	1.74588	0.872940	−	0.487828i	$-0.162211\pi$
$499$	39.0000	1.74588	0.872940	+	0.487828i	$0.162211\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	−48.0000	−2.14234
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	0	0
$508$	26.0000	1.15356
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	33.0000	1.45983
$512$	−32.0000	−1.41421
$513$	0	0
$514$	24.0000	1.05859
$515$	−13.0000	−0.572848
$516$	0	0
$517$	0	0
$518$	66.0000	2.89987
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	0	0
$525$	0	0
$526$	−48.0000	−2.09290
$527$	30.0000	1.30682
$528$	0	0
$529$	−7.00000	−0.304348
$530$	16.0000	0.694996
$531$	0	0
$532$	30.0000	1.30066
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	0	0
$538$	−60.0000	−2.58678
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	48.0000	2.06178
$543$	0	0
$544$	−48.0000	−2.05798
$545$	−15.0000	−0.642529
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	24.0000	1.02523
$549$	0	0
$550$	0	0
$551$	−30.0000	−1.27804
$552$	0	0
$553$	−3.00000	−0.127573
$554$	34.0000	1.44452
$555$	0	0
$556$	24.0000	1.01783
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−12.0000	−0.507093
$561$	0	0
$562$	−12.0000	−0.506189
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−14.0000	−0.588464
$567$	0	0
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−45.0000	−1.88319	−0.941596	−	0.336746i	$-0.890674\pi$
$571$	−45.0000	−1.88319	−0.941596	+	0.336746i	$0.890674\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	4.00000	0.166812
$576$	0	0
$577$	29.0000	1.20729	0.603643	−	0.797255i	$-0.293715\pi$
$577$	29.0000	1.20729	0.603643	+	0.797255i	$0.293715\pi$
$578$	−38.0000	−1.58059
$579$	0	0
$580$	−12.0000	−0.498273
$581$	−30.0000	−1.24461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	4.00000	0.165238
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−25.0000	−1.03011
$590$	4.00000	0.164677
$591$	0	0
$592$	44.0000	1.80839
$593$	40.0000	1.64260	0.821302	−	0.570494i	$-0.193248\pi$
$593$	40.0000	1.64260	0.821302	+	0.570494i	$0.193248\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	28.0000	1.14692
$597$	0	0
$598$	−16.0000	−0.654289
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	−24.0000	−0.978167
$603$	0	0
$604$	40.0000	1.62758
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	2.00000	0.0809776
$611$	−12.0000	−0.485468
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	−46.0000	−1.85641
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−10.0000	−0.401610
$621$	0	0
$622$	−36.0000	−1.44347
$623$	−42.0000	−1.68269
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.00000	−0.159872
$627$	0	0
$628$	26.0000	1.03751
$629$	66.0000	2.63159
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	−12.0000	−0.476581
$635$	13.0000	0.515889
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	60.0000	2.36067
$647$	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
$652$	14.0000	0.548282
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	0	0
$655$	0	0
$656$	16.0000	0.624695
$657$	0	0
$658$	36.0000	1.40343
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	46.0000	1.78784
$663$	0	0
$664$	0	0
$665$	15.0000	0.581675
$666$	0	0
$667$	−24.0000	−0.929284
$668$	−16.0000	−0.619059
$669$	0	0
$670$	14.0000	0.540867
$671$	0	0
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−18.0000	−0.692308
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	0	0
$679$	−51.0000	−1.95720
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	30.0000	1.14541
$687$	0	0
$688$	−16.0000	−0.609994
$689$	−16.0000	−0.609551
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−64.0000	−2.42941
$695$	12.0000	0.455186
$696$	0	0
$697$	24.0000	0.909065
$698$	−46.0000	−1.74113
$699$	0	0
$700$	6.00000	0.226779
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	−55.0000	−2.07436
$704$	0	0
$705$	0	0
$706$	−60.0000	−2.25813
$707$	−30.0000	−1.12827
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	20.0000	0.750587
$711$	0	0
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−68.0000	−2.53774
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	−39.0000	−1.45244
$722$	−12.0000	−0.446594
$723$	0	0
$724$	10.0000	0.371647
$725$	−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	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	−22.0000	−0.814257
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	64.0000	2.36228
$735$	0	0
$736$	32.0000	1.17954
$737$	0	0
$738$	0	0
$739$	13.0000	0.478213	0.239106	−	0.970993i	$-0.423146\pi$
$739$	13.0000	0.478213	0.239106	+	0.970993i	$0.423146\pi$
$740$	−22.0000	−0.808736
$741$	0	0
$742$	48.0000	1.76214
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	30.0000	1.09838
$747$	0	0
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	24.0000	0.875190
$753$	0	0
$754$	24.0000	0.874028
$755$	20.0000	0.727875
$756$	0	0
$757$	−33.0000	−1.19941	−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000	−1.19941	−0.599703	+	0.800223i	$0.704714\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	−45.0000	−1.62911
$764$	40.0000	1.44715
$765$	0	0
$766$	−48.0000	−1.73431
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	0	0
$772$	−46.0000	−1.65558
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	0	0
$778$	−72.0000	−2.58133
$779$	−20.0000	−0.716574
$780$	0	0
$781$	0	0
$782$	48.0000	1.71648
$783$	0	0
$784$	−8.00000	−0.285714
$785$	13.0000	0.463990
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−32.0000	−1.13995
$789$	0	0
$790$	2.00000	0.0711568
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	54.0000	1.91639
$795$	0	0
$796$	54.0000	1.91398
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	8.00000	0.282843
$801$	0	0
$802$	20.0000	0.706225
$803$	0	0
$804$	0	0
$805$	12.0000	0.422944
$806$	20.0000	0.704470
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−9.00000	−0.316033	−0.158016	−	0.987436i	$-0.550510\pi$
$811$	−9.00000	−0.316033	−0.158016	+	0.987436i	$0.550510\pi$
$812$	−36.0000	−1.26335
$813$	0	0
$814$	0	0
$815$	7.00000	0.245199
$816$	0	0
$817$	20.0000	0.699711
$818$	6.00000	0.209785
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	0	0
$823$	−57.0000	−1.98690	−0.993448	−	0.114289i	$-0.963541\pi$
$823$	−57.0000	−1.98690	−0.993448	+	0.114289i	$0.963541\pi$
$824$	0	0
$825$	0	0
$826$	12.0000	0.417533
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	0	0
$829$	−35.0000	−1.21560	−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000	−1.21560	−0.607800	+	0.794090i	$0.707948\pi$
$830$	20.0000	0.694210
$831$	0	0
$832$	−16.0000	−0.554700
$833$	−12.0000	−0.415775
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	68.0000	2.34902
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−12.0000	−0.413547
$843$	0	0
$844$	−10.0000	−0.344214
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	32.0000	1.09888
$849$	0	0
$850$	12.0000	0.411597
$851$	−44.0000	−1.50830
$852$	0	0
$853$	29.0000	0.992941	0.496471	−	0.868054i	$-0.334629\pi$
$853$	29.0000	0.992941	0.496471	+	0.868054i	$0.334629\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	48.0000	1.63489
$863$	−14.0000	−0.476566	−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000	−0.476566	−0.238283	+	0.971196i	$0.576585\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	14.0000	0.475739
$867$	0	0
$868$	−30.0000	−1.01827
$869$	0	0
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	0	0
$874$	−40.0000	−1.35302
$875$	3.00000	0.101419
$876$	0	0
$877$	5.00000	0.168838	0.0844190	−	0.996430i	$-0.473097\pi$
$877$	5.00000	0.168838	0.0844190	+	0.996430i	$0.473097\pi$
$878$	−2.00000	−0.0674967
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	44.0000	1.47821
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	0	0
$889$	39.0000	1.30802
$890$	28.0000	0.938562
$891$	0	0
$892$	10.0000	0.334825
$893$	−30.0000	−1.00391
$894$	0	0
$895$	−2.00000	−0.0668526
$896$	0	0
$897$	0	0
$898$	20.0000	0.667409
$899$	30.0000	1.00056
$900$	0	0
$901$	48.0000	1.59911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.00000	0.166206
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	−12.0000	−0.397796
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	−20.0000	−0.661541
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	0	0
$919$	27.0000	0.890648	0.445324	−	0.895370i	$-0.353089\pi$
$919$	27.0000	0.890648	0.445324	+	0.895370i	$0.353089\pi$
$920$	0	0
$921$	0	0
$922$	−40.0000	−1.31733
$923$	−20.0000	−0.658308
$924$	0	0
$925$	−11.0000	−0.361678
$926$	48.0000	1.57738
$927$	0	0
$928$	−48.0000	−1.57568
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	0	0
$931$	10.0000	0.327737
$932$	−60.0000	−1.96537
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−3.00000	−0.0980057	−0.0490029	−	0.998799i	$-0.515604\pi$
$937$	−3.00000	−0.0980057	−0.0490029	+	0.998799i	$0.515604\pi$
$938$	42.0000	1.37135
$939$	0	0
$940$	−12.0000	−0.391397
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	8.00000	0.260378
$945$	0	0
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	22.0000	0.714150
$950$	−10.0000	−0.324443
$951$	0	0
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	28.0000	0.905585
$957$	0	0
$958$	−12.0000	−0.387702
$959$	36.0000	1.16250
$960$	0	0
$961$	−6.00000	−0.193548
$962$	44.0000	1.41862
$963$	0	0
$964$	20.0000	0.644157
$965$	−23.0000	−0.740396
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	0	0
$969$	0	0
$970$	34.0000	1.09167
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	36.0000	1.15411
$974$	16.0000	0.512673
$975$	0	0
$976$	4.00000	0.128037
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	0	0
$980$	4.00000	0.127775
$981$	0	0
$982$	−16.0000	−0.510581
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	−72.0000	−2.29295
$987$	0	0
$988$	20.0000	0.636285
$989$	16.0000	0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	60.0000	1.90308
$995$	27.0000	0.855958
$996$	0	0
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	−78.0000	−2.46905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.b.1.1	yes	1
3.2	odd	2		5445.2.a.k.1.1	yes	1
11.10	odd	2		5445.2.a.l.1.1	yes	1
33.32	even	2		5445.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.a.1.1	✓	1	33.32	even	2
5445.2.a.b.1.1	yes	1	1.1	even	1	trivial
5445.2.a.k.1.1	yes	1	3.2	odd	2
5445.2.a.l.1.1	yes	1	11.10	odd	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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