LMFDB - Embedded newform 990.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [990,2,Mod(1,990)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(990, 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("990.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	990.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:	$7.90518980011$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	990.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -6.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +5.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} +3.00000 q^{28} -5.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} -3.00000 q^{35} +3.00000 q^{37} -5.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +2.00000 q^{47} +2.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +1.00000 q^{53} +1.00000 q^{55} -3.00000 q^{56} +5.00000 q^{58} +10.0000 q^{59} +7.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +7.00000 q^{68} +3.00000 q^{70} -7.00000 q^{71} +14.0000 q^{73} -3.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} +10.0000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +6.00000 q^{83} -7.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} +15.0000 q^{89} -18.0000 q^{91} +6.00000 q^{92} -2.00000 q^{94} -5.00000 q^{95} -12.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	3.00000	0.566947
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.00000	−1.20049
$35$	−3.00000	−0.507093
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−3.00000	−0.400892
$57$	0	0
$58$	5.00000	0.656532
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	7.00000	0.848875
$69$	0	0
$70$	3.00000	0.358569
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	5.00000	0.573539
$77$	−3.00000	−0.341882
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−7.00000	−0.759257
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	6.00000	0.625543
$93$	0	0
$94$	−2.00000	−0.206284
$95$	−5.00000	−0.512989
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	3.00000	0.283473
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−5.00000	−0.464238
$117$	0	0
$118$	−10.0000	−0.920575
$119$	21.0000	1.92507
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.00000	−0.633750
$123$	0	0
$124$	−3.00000	−0.269408
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.00000	−0.526235
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−7.00000	−0.600245
$137$	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
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	7.00000	0.587427
$143$	6.00000	0.501745
$144$	0	0
$145$	5.00000	0.415227
$146$	−14.0000	−1.15865
$147$	0	0
$148$	3.00000	0.246598
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	3.00000	0.241747
$155$	3.00000	0.240966
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	1.00000	0.0790569
$161$	18.0000	1.41860
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	7.00000	0.536875
$171$	0	0
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−15.0000	−1.12430
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	18.0000	1.33425
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−7.00000	−0.511891
$188$	2.00000	0.145865
$189$	0	0
$190$	5.00000	0.362738
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	2.00000	0.140720
$203$	−15.0000	−1.05279
$204$	0	0
$205$	2.00000	0.139686
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	8.00000	0.546869
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−9.00000	−0.610960
$218$	10.0000	0.677285
$219$	0	0
$220$	1.00000	0.0674200
$221$	−42.0000	−2.82523
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−16.0000	−1.06430
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	5.00000	0.328266
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	10.0000	0.650945
$237$	0	0
$238$	−21.0000	−1.36123
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	7.00000	0.448129
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−30.0000	−1.90885
$248$	3.00000	0.190500
$249$	0	0
$250$	1.00000	0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	6.00000	0.372104
$261$	0	0
$262$	17.0000	1.05026
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	−15.0000	−0.919709
$267$	0	0
$268$	8.00000	0.488678
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	3.00000	0.179284
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−6.00000	−0.354169
$288$	0	0
$289$	32.0000	1.88235
$290$	−5.00000	−0.293610
$291$	0	0
$292$	14.0000	0.819288
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	−3.00000	−0.174371
$297$	0	0
$298$	15.0000	0.868927
$299$	−36.0000	−2.08193
$300$	0	0
$301$	12.0000	0.691669
$302$	−2.00000	−0.115087
$303$	0	0
$304$	5.00000	0.286770
$305$	−7.00000	−0.400819
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	−3.00000	−0.170389
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	10.0000	0.562544
$317$	7.00000	0.393159	0.196580	−	0.980488i	$-0.437017\pi$
$317$	7.00000	0.393159	0.196580	+	0.980488i	$0.437017\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−18.0000	−1.00310
$323$	35.0000	1.94745
$324$	0	0
$325$	−6.00000	−0.332820
$326$	−19.0000	−1.05231
$327$	0	0
$328$	2.00000	0.110432
$329$	6.00000	0.330791
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	3.00000	0.164153
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	−7.00000	−0.379628
$341$	3.00000	0.162459
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	1.00000	0.0533002
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	7.00000	0.371521
$356$	15.0000	0.794998
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−2.00000	−0.105118
$363$	0	0
$364$	−18.0000	−0.943456
$365$	−14.0000	−0.732793
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	3.00000	0.155963
$371$	3.00000	0.155752
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	7.00000	0.361961
$375$	0	0
$376$	−2.00000	−0.103142
$377$	30.0000	1.54508
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−5.00000	−0.256495
$381$	0	0
$382$	12.0000	0.613973
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	11.0000	0.559885
$387$	0	0
$388$	−12.0000	−0.609208
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	42.0000	2.12403
$392$	−2.00000	−0.101015
$393$	0	0
$394$	−12.0000	−0.604551
$395$	−10.0000	−0.503155
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	25.0000	1.25314
$399$	0	0
$400$	1.00000	0.0500000
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	15.0000	0.744438
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	4.00000	0.197066
$413$	30.0000	1.47620
$414$	0	0
$415$	−6.00000	−0.294528
$416$	6.00000	0.294174
$417$	0	0
$418$	5.00000	0.244558
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	23.0000	1.11962
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	7.00000	0.339550
$426$	0	0
$427$	21.0000	1.01626
$428$	−8.00000	−0.386695
$429$	0	0
$430$	4.00000	0.192897
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	−10.0000	−0.478913
$437$	30.0000	1.43509
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	42.0000	1.99774
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−15.0000	−0.711068
$446$	6.00000	0.284108
$447$	0	0
$448$	3.00000	0.141737
$449$	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
$451$	2.00000	0.0941763
$452$	16.0000	0.752577
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	18.0000	0.843853
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	−6.00000	−0.279751
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	9.00000	0.416917
$467$	−23.0000	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	2.00000	0.0922531
$471$	0	0
$472$	−10.0000	−0.460287
$473$	−4.00000	−0.183920
$474$	0	0
$475$	5.00000	0.229416
$476$	21.0000	0.962533
$477$	0	0
$478$	10.0000	0.457389
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	18.0000	0.819878
$483$	0	0
$484$	1.00000	0.0454545
$485$	12.0000	0.544892
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−7.00000	−0.316875
$489$	0	0
$490$	2.00000	0.0903508
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	−35.0000	−1.57632
$494$	30.0000	1.34976
$495$	0	0
$496$	−3.00000	−0.134704
$497$	−21.0000	−0.941979
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	2.00000	0.0892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	6.00000	0.266733
$507$	0	0
$508$	8.00000	0.354943
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	42.0000	1.85797
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−4.00000	−0.176261
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	−9.00000	−0.395437
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−17.0000	−0.742648
$525$	0	0
$526$	9.00000	0.392419
$527$	−21.0000	−0.914774
$528$	0	0
$529$	13.0000	0.565217
$530$	1.00000	0.0434372
$531$	0	0
$532$	15.0000	0.650332
$533$	12.0000	0.519778
$534$	0	0
$535$	8.00000	0.345870
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−20.0000	−0.862261
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−7.00000	−0.300123
$545$	10.0000	0.428353
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	1.00000	0.0426401
$551$	−25.0000	−1.06504
$552$	0	0
$553$	30.0000	1.27573
$554$	12.0000	0.509831
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−18.0000	−0.759284
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	6.00000	0.252199
$567$	0	0
$568$	7.00000	0.293713
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	27.0000	1.12991	0.564957	−	0.825120i	$-0.308893\pi$
$571$	27.0000	1.12991	0.564957	+	0.825120i	$0.308893\pi$
$572$	6.00000	0.250873
$573$	0	0
$574$	6.00000	0.250435
$575$	6.00000	0.250217
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	5.00000	0.207614
$581$	18.0000	0.746766
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	10.0000	0.411693
$591$	0	0
$592$	3.00000	0.123299
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−21.0000	−0.860916
$596$	−15.0000	−0.614424
$597$	0	0
$598$	36.0000	1.47215
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	2.00000	0.0813788
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−47.0000	−1.90767	−0.953836	−	0.300329i	$-0.902903\pi$
$607$	−47.0000	−1.90767	−0.953836	+	0.300329i	$0.902903\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	7.00000	0.283422
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	3.00000	0.120873
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	3.00000	0.120483
$621$	0	0
$622$	−3.00000	−0.120289
$623$	45.0000	1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	3.00000	0.119713
$629$	21.0000	0.837325
$630$	0	0
$631$	−33.0000	−1.31371	−0.656855	−	0.754017i	$-0.728113\pi$
$631$	−33.0000	−1.31371	−0.656855	+	0.754017i	$0.728113\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	−7.00000	−0.278006
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−12.0000	−0.475457
$638$	−5.00000	−0.197952
$639$	0	0
$640$	1.00000	0.0395285
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	−35.0000	−1.37706
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	6.00000	0.235339
$651$	0	0
$652$	19.0000	0.744097
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	0	0
$655$	17.0000	0.664245
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−15.0000	−0.581675
$666$	0	0
$667$	−30.0000	−1.16160
$668$	−3.00000	−0.116073
$669$	0	0
$670$	8.00000	0.309067
$671$	−7.00000	−0.270232
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	17.0000	0.654816
$675$	0	0
$676$	23.0000	0.884615
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	7.00000	0.268438
$681$	0	0
$682$	−3.00000	−0.114876
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000	0.152499
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−38.0000	−1.44559	−0.722794	−	0.691063i	$-0.757142\pi$
$691$	−38.0000	−1.44559	−0.722794	+	0.691063i	$0.757142\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	18.0000	0.683271
$695$	20.0000	0.758643
$696$	0	0
$697$	−14.0000	−0.530288
$698$	−30.0000	−1.13552
$699$	0	0
$700$	3.00000	0.113389
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	15.0000	0.565736
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	34.0000	1.27961
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−7.00000	−0.262705
$711$	0	0
$712$	−15.0000	−0.562149
$713$	−18.0000	−0.674105
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	0	0
$718$	−20.0000	−0.746393
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−6.00000	−0.223297
$723$	0	0
$724$	2.00000	0.0743294
$725$	−5.00000	−0.185695
$726$	0	0
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	18.0000	0.667124
$729$	0	0
$730$	14.0000	0.518163
$731$	28.0000	1.03562
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−8.00000	−0.294684
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−3.00000	−0.110133
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	6.00000	0.219676
$747$	0	0
$748$	−7.00000	−0.255945
$749$	−24.0000	−0.876941
$750$	0	0
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	−30.0000	−1.09254
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	30.0000	1.08965
$759$	0	0
$760$	5.00000	0.181369
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−30.0000	−1.08607
$764$	−12.0000	−0.434145
$765$	0	0
$766$	34.0000	1.22847
$767$	−60.0000	−2.16647
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	−3.00000	−0.108112
$771$	0	0
$772$	−11.0000	−0.395899
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	12.0000	0.430775
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−10.0000	−0.358287
$780$	0	0
$781$	7.00000	0.250480
$782$	−42.0000	−1.50192
$783$	0	0
$784$	2.00000	0.0714286
$785$	−3.00000	−0.107075
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	10.0000	0.355784
$791$	48.0000	1.70668
$792$	0	0
$793$	−42.0000	−1.49146
$794$	2.00000	0.0709773
$795$	0	0
$796$	−25.0000	−0.886102
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−13.0000	−0.459046
$803$	−14.0000	−0.494049
$804$	0	0
$805$	−18.0000	−0.634417
$806$	−18.0000	−0.634023
$807$	0	0
$808$	2.00000	0.0703598
$809$	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
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	−15.0000	−0.526397
$813$	0	0
$814$	3.00000	0.105150
$815$	−19.0000	−0.665541
$816$	0	0
$817$	20.0000	0.699711
$818$	20.0000	0.699284
$819$	0	0
$820$	2.00000	0.0698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−30.0000	−1.04383
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	−6.00000	−0.208013
$833$	14.0000	0.485071
$834$	0	0
$835$	3.00000	0.103819
$836$	−5.00000	−0.172929
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−32.0000	−1.10279
$843$	0	0
$844$	−23.0000	−0.791693
$845$	−23.0000	−0.791224
$846$	0	0
$847$	3.00000	0.103081
$848$	1.00000	0.0343401
$849$	0	0
$850$	−7.00000	−0.240098
$851$	18.0000	0.617032
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−21.0000	−0.718605
$855$	0	0
$856$	8.00000	0.273434
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−8.00000	−0.272481
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	16.0000	0.543702
$867$	0	0
$868$	−9.00000	−0.305480
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−48.0000	−1.62642
$872$	10.0000	0.338643
$873$	0	0
$874$	−30.0000	−1.01477
$875$	−3.00000	−0.101419
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−20.0000	−0.674967
$879$	0	0
$880$	1.00000	0.0337100
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	−42.0000	−1.41261
$885$	0	0
$886$	4.00000	0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	15.0000	0.502801
$891$	0	0
$892$	−6.00000	−0.200895
$893$	10.0000	0.334637
$894$	0	0
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−30.0000	−1.00111
$899$	15.0000	0.500278
$900$	0	0
$901$	7.00000	0.233204
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	−16.0000	−0.532152
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−57.0000	−1.89265	−0.946327	−	0.323211i	$-0.895238\pi$
$907$	−57.0000	−1.89265	−0.946327	+	0.323211i	$0.895238\pi$
$908$	2.00000	0.0663723
$909$	0	0
$910$	−18.0000	−0.596694
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	10.0000	0.330409
$917$	−51.0000	−1.68417
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	27.0000	0.889198
$923$	42.0000	1.38245
$924$	0	0
$925$	3.00000	0.0986394
$926$	−34.0000	−1.11731
$927$	0	0
$928$	5.00000	0.164133
$929$	−35.0000	−1.14831	−0.574156	−	0.818746i	$-0.694670\pi$
$929$	−35.0000	−1.14831	−0.574156	+	0.818746i	$0.694670\pi$
$930$	0	0
$931$	10.0000	0.327737
$932$	−9.00000	−0.294805
$933$	0	0
$934$	23.0000	0.752583
$935$	7.00000	0.228924
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−24.0000	−0.783628
$939$	0	0
$940$	−2.00000	−0.0652328
$941$	−17.0000	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	10.0000	0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	−84.0000	−2.72676
$950$	−5.00000	−0.162221
$951$	0	0
$952$	−21.0000	−0.680614
$953$	−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
$955$	12.0000	0.388311
$956$	−10.0000	−0.323423
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	18.0000	0.580343
$963$	0	0
$964$	−18.0000	−0.579741
$965$	11.0000	0.354103
$966$	0	0
$967$	−27.0000	−0.868261	−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000	−0.868261	−0.434131	+	0.900850i	$0.642944\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−12.0000	−0.385297
$971$	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
$973$	−60.0000	−1.92351
$974$	12.0000	0.384505
$975$	0	0
$976$	7.00000	0.224065
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−15.0000	−0.479402
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	−3.00000	−0.0957338
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	35.0000	1.11463
$987$	0	0
$988$	−30.0000	−0.954427
$989$	24.0000	0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	3.00000	0.0952501
$993$	0	0
$994$	21.0000	0.666080
$995$	25.0000	0.792553
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	990.2.a.d.1.1		1
3.2	odd	2		110.2.a.b.1.1	✓	1
4.3	odd	2		7920.2.a.d.1.1		1
5.2	odd	4		4950.2.c.m.199.1		2
5.3	odd	4		4950.2.c.m.199.2		2
5.4	even	2		4950.2.a.bc.1.1		1
12.11	even	2		880.2.a.i.1.1		1
15.2	even	4		550.2.b.a.199.2		2
15.8	even	4		550.2.b.a.199.1		2
15.14	odd	2		550.2.a.f.1.1		1
21.20	even	2		5390.2.a.bf.1.1		1
24.5	odd	2		3520.2.a.y.1.1		1
24.11	even	2		3520.2.a.h.1.1		1
33.32	even	2		1210.2.a.b.1.1		1
60.23	odd	4		4400.2.b.i.4049.2		2
60.47	odd	4		4400.2.b.i.4049.1		2
60.59	even	2		4400.2.a.l.1.1		1
132.131	odd	2		9680.2.a.x.1.1		1
165.164	even	2		6050.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.b.1.1	✓	1	3.2	odd	2
550.2.a.f.1.1		1	15.14	odd	2
550.2.b.a.199.1		2	15.8	even	4
550.2.b.a.199.2		2	15.2	even	4
880.2.a.i.1.1		1	12.11	even	2
990.2.a.d.1.1		1	1.1	even	1	trivial
1210.2.a.b.1.1		1	33.32	even	2
3520.2.a.h.1.1		1	24.11	even	2
3520.2.a.y.1.1		1	24.5	odd	2
4400.2.a.l.1.1		1	60.59	even	2
4400.2.b.i.4049.1		2	60.47	odd	4
4400.2.b.i.4049.2		2	60.23	odd	4
4950.2.a.bc.1.1		1	5.4	even	2
4950.2.c.m.199.1		2	5.2	odd	4
4950.2.c.m.199.2		2	5.3	odd	4
5390.2.a.bf.1.1		1	21.20	even	2
6050.2.a.bj.1.1		1	165.164	even	2
7920.2.a.d.1.1		1	4.3	odd	2
9680.2.a.x.1.1		1	132.131	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:	\( N \)	\(=\)	\( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	990.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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