LMFDB - Embedded newform 1989.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1989,2,Mod(1,1989)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1989, 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("1989.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1989 = 3^{2} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1989.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:	$15.8822449620$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 663)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1989.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−6.00000	−0.948683
$41$	−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	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	−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
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−4.00000	−0.431331
$87$	0	0
$88$	12.0000	1.27920
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	10.0000	0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	0	0
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	2.00000	0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	3.00000	0.257248
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	−14.0000	−1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	10.0000	0.790569
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.00000	−0.153393
$171$	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
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	4.00000	0.292509
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	7.00000	0.500000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	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
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	16.0000	1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.0000	−0.686803
$213$	0	0
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	−10.0000	−0.677285
$219$	0	0
$220$	8.00000	0.539360
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	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
$235$	−16.0000	−1.04372
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	−14.0000	−0.894427
$246$	0	0
$247$	−4.00000	−0.254514
$248$	24.0000	1.52400
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	0	0
$260$	−2.00000	−0.124035
$261$	0	0
$262$	4.00000	0.247121
$263$	−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
$265$	20.0000	1.22859
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	22.0000	1.32907
$275$	4.00000	0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	20.0000	1.19952
$279$	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
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	4.00000	0.234888
$291$	0	0
$292$	14.0000	0.819288
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	28.0000	1.60328
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	14.0000	0.782624
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−8.00000	−0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	2.00000	0.108465
$341$	32.0000	1.73290
$342$	0	0
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	−14.0000	−0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	−20.0000	−1.06600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	0	0
$364$	0	0
$365$	−28.0000	−1.46559
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	0	0
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	24.0000	1.23771
$377$	2.00000	0.103005
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−16.0000	−0.818631
$383$	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
$386$	−22.0000	−1.11977
$387$	0	0
$388$	6.00000	0.304604
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	0	0
$392$	21.0000	1.06066
$393$	0	0
$394$	2.00000	0.100759
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	5.00000	0.245145
$417$	0	0
$418$	16.0000	0.782586
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−30.0000	−1.45693
$425$	1.00000	0.0485071
$426$	0	0
$427$	0	0
$428$	−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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	24.0000	1.14416
$441$	0	0
$442$	−1.00000	−0.0475651
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	2.00000	0.0940721
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	0	0
$470$	−16.0000	−0.738025
$471$	0	0
$472$	12.0000	0.552345
$473$	16.0000	0.735681
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	10.0000	0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−12.0000	−0.544892
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−42.0000	−1.90125
$489$	0	0
$490$	−14.0000	−0.632456
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−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
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	30.0000	1.32324
$515$	−16.0000	−0.705044
$516$	0	0
$517$	32.0000	1.40736
$518$	0	0
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	8.00000	0.348485
$528$	0	0
$529$	−23.0000	−1.00000
$530$	20.0000	0.868744
$531$	0	0
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	24.0000	1.03761
$536$	12.0000	0.518321
$537$	0	0
$538$	18.0000	0.776035
$539$	28.0000	1.20605
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−5.00000	−0.214373
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−22.0000	−0.939793
$549$	0	0
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	20.0000	0.840663
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−4.00000	−0.166091
$581$	0	0
$582$	0	0
$583$	−40.0000	−1.65663
$584$	42.0000	1.73797
$585$	0	0
$586$	26.0000	1.07405
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	−8.00000	−0.329355
$591$	0	0
$592$	2.00000	0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	16.0000	0.651031
$605$	10.0000	0.406558
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	28.0000	1.13369
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	2.00000	0.0798087
$629$	2.00000	0.0797452
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	10.0000	0.397151
$635$	0	0
$636$	0	0
$637$	−7.00000	−0.277350
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−6.00000	−0.237171
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	16.0000	0.619059
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−56.0000	−2.16186
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	0	0
$680$	6.00000	0.230089
$681$	0	0
$682$	32.0000	1.22534
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	44.0000	1.68115
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	10.0000	0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	40.0000	1.51729
$696$	0	0
$697$	2.00000	0.0757554
$698$	30.0000	1.13552
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−28.0000	−1.05529
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	0	0
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	0	0
$730$	−28.0000	−1.03633
$731$	4.00000	0.147945
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	22.0000	0.805477
$747$	0	0
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	2.00000	0.0728357
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	24.0000	0.870572
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	0	0
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	22.0000	0.791797
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	18.0000	0.646162
$777$	0	0
$778$	−38.0000	−1.36237
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	−16.0000	−0.569254
$791$	0	0
$792$	0	0
$793$	14.0000	0.497155
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	−5.00000	−0.176777
$801$	0	0
$802$	6.00000	0.211867
$803$	56.0000	1.97620
$804$	0	0
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	18.0000	0.633238
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	0	0
$814$	8.00000	0.280400
$815$	−8.00000	−0.280228
$816$	0	0
$817$	16.0000	0.559769
$818$	26.0000	0.909069
$819$	0	0
$820$	4.00000	0.139686
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	7.00000	0.242681
$833$	7.00000	0.242536
$834$	0	0
$835$	−32.0000	−1.10741
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−28.0000	−0.967244
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	0	0
$844$	−12.0000	−0.413057
$845$	2.00000	0.0688021
$846$	0	0
$847$	0	0
$848$	−10.0000	−0.343401
$849$	0	0
$850$	1.00000	0.0342997
$851$	0	0
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	0	0
$855$	0	0
$856$	−36.0000	−1.23045
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	−14.0000	−0.475739
$867$	0	0
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−4.00000	−0.135535
$872$	30.0000	1.01593
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	16.0000	0.539974
$879$	0	0
$880$	8.00000	0.269680
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	20.0000	0.671913
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	0	0
$890$	12.0000	0.402241
$891$	0	0
$892$	−8.00000	−0.267860
$893$	32.0000	1.07084
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	22.0000	0.734150
$899$	−16.0000	−0.533630
$900$	0	0
$901$	−10.0000	−0.333148
$902$	8.00000	0.266371
$903$	0	0
$904$	6.00000	0.199557
$905$	12.0000	0.398893
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	24.0000	0.788689
$927$	0	0
$928$	10.0000	0.328266
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−4.00000	−0.130884
$935$	8.00000	0.261628
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	0	0
$940$	16.0000	0.521862
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	4.00000	0.129777
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−32.0000	−1.03550
$956$	−24.0000	−0.776215
$957$	0	0
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−44.0000	−1.41641
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−15.0000	−0.482118
$969$	0	0
$970$	−12.0000	−0.385297
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	32.0000	1.02535
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	14.0000	0.447214
$981$	0	0
$982$	−28.0000	−0.893516
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	0	0
$995$	32.0000	1.01447
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1989.2.a.e.1.1		1
3.2	odd	2		663.2.a.a.1.1	✓	1
39.38	odd	2		8619.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.a.a.1.1	✓	1	3.2	odd	2
1989.2.a.e.1.1		1	1.1	even	1	trivial
8619.2.a.i.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1989 = 3^{2} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1989.a (trivial)

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