LMFDB - Embedded newform 1488.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1488,2,Mod(1,1488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1488, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1488 = 2^{4} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1488.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:	$11.8817398208$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 186)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +3.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -7.00000 q^{13} -3.00000 q^{15} -1.00000 q^{17} -7.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{31} +5.00000 q^{33} +6.00000 q^{35} -6.00000 q^{37} +7.00000 q^{39} -2.00000 q^{41} +10.0000 q^{43} +3.00000 q^{45} +1.00000 q^{47} -3.00000 q^{49} +1.00000 q^{51} +6.00000 q^{53} -15.0000 q^{55} +7.00000 q^{57} +10.0000 q^{59} +1.00000 q^{61} +2.00000 q^{63} -21.0000 q^{65} +3.00000 q^{67} +4.00000 q^{69} -3.00000 q^{71} +14.0000 q^{73} -4.00000 q^{75} -10.0000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -7.00000 q^{83} -3.00000 q^{85} +8.00000 q^{87} -6.00000 q^{89} -14.0000 q^{91} -1.00000 q^{93} -21.0000 q^{95} -3.00000 q^{97} -5.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	7.00000	1.12090
$40$	0	0
$41$	−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
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−15.0000	−2.02260
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−21.0000	−2.60473
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−14.0000	−1.46760
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−21.0000	−2.15455
$96$	0	0
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−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
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	−7.00000	−0.647150
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	−14.0000	−1.21395
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	0	0
$143$	35.0000	2.92685
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	15.0000	1.16775
$166$	0	0
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	−7.00000	−0.535303
$172$	0	0
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	−19.0000	−1.42013	−0.710063	−	0.704138i	$-0.751334\pi$
$179$	−19.0000	−1.42013	−0.710063	+	0.704138i	$0.751334\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	−18.0000	−1.32339
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	0	0
$195$	21.0000	1.50384
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−16.0000	−1.12298
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	35.0000	2.42100
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	3.00000	0.205557
$214$	0	0
$215$	30.0000	2.04598
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	7.00000	0.470871
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	10.0000	0.657952
$232$	0	0
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.00000	−0.574989
$246$	0	0
$247$	49.0000	3.11780
$248$	0	0
$249$	7.00000	0.443607
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	20.0000	1.25739
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−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
$259$	−12.0000	−0.745644
$260$	0	0
$261$	−8.00000	−0.495188
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	0	0
$273$	14.0000	0.847319
$274$	0	0
$275$	−20.0000	−1.20605
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	0	0
$285$	21.0000	1.24393
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	3.00000	0.175863
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	30.0000	1.74667
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	28.0000	1.61928
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	3.00000	0.171780
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	0	0
$319$	40.0000	2.23957
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	−28.0000	−1.55316
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	9.00000	0.491723
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−5.00000	−0.270765
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−17.0000	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$347$	−17.0000	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$348$	0	0
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	7.00000	0.373632
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	42.0000	2.19838
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	56.0000	2.88415
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	10.0000	0.510976	0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000	0.510976	0.255488	+	0.966812i	$0.417764\pi$
$384$	0	0
$385$	−30.0000	−1.52894
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	33.0000	1.66041
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	14.0000	0.700877
$400$	0	0
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	0	0
$403$	−7.00000	−0.348695
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	20.0000	0.984136
$414$	0	0
$415$	−21.0000	−1.03085
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	1.00000	0.0486217
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−35.0000	−1.68982
$430$	0	0
$431$	−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
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	24.0000	1.15071
$436$	0	0
$437$	28.0000	1.33942
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	−42.0000	−1.96899
$456$	0	0
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−33.0000	−1.53364	−0.766820	−	0.641862i	$-0.778162\pi$
$463$	−33.0000	−1.53364	−0.766820	+	0.641862i	$0.778162\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	−50.0000	−2.29900
$474$	0	0
$475$	−28.0000	−1.28473
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−29.0000	−1.32504	−0.662522	−	0.749043i	$-0.730514\pi$
$479$	−29.0000	−1.32504	−0.662522	+	0.749043i	$0.730514\pi$
$480$	0	0
$481$	42.0000	1.91504
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−9.00000	−0.408669
$486$	0	0
$487$	27.0000	1.22349	0.611743	−	0.791056i	$-0.290469\pi$
$487$	27.0000	1.22349	0.611743	+	0.791056i	$0.290469\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	−15.0000	−0.674200
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	10.0000	0.446767
$502$	0	0
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	−36.0000	−1.59882
$508$	0	0
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	7.00000	0.309058
$514$	0	0
$515$	−30.0000	−1.32196
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	−1.00000	−0.0435607
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	14.0000	0.606407
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	19.0000	0.819911
$538$	0	0
$539$	15.0000	0.646096
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	56.0000	2.38568
$552$	0	0
$553$	22.0000	0.935535
$554$	0	0
$555$	18.0000	0.764057
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−70.0000	−2.96068
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	0	0
$579$	−19.0000	−0.789613
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	−30.0000	−1.24247
$584$	0	0
$585$	−21.0000	−0.868243
$586$	0	0
$587$	17.0000	0.701665	0.350833	−	0.936438i	$-0.385899\pi$
$587$	17.0000	0.701665	0.350833	+	0.936438i	$0.385899\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	3.00000	0.122169
$604$	0	0
$605$	42.0000	1.70754
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	0	0
$611$	−7.00000	−0.283190
$612$	0	0
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−35.0000	−1.39777
$628$	0	0
$629$	6.00000	0.239236
$630$	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
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	21.0000	0.832050
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	23.0000	0.908445	0.454223	−	0.890888i	$-0.349917\pi$
$641$	23.0000	0.908445	0.454223	+	0.890888i	$0.349917\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	−30.0000	−1.18125
$646$	0	0
$647$	−14.0000	−0.550397	−0.275198	−	0.961387i	$-0.588744\pi$
$647$	−14.0000	−0.550397	−0.275198	+	0.961387i	$0.588744\pi$
$648$	0	0
$649$	−50.0000	−1.96267
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−30.0000	−1.17220
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	−7.00000	−0.271857
$664$	0	0
$665$	−42.0000	−1.62869
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	−5.00000	−0.193311
$670$	0	0
$671$	−5.00000	−0.193023
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	−54.0000	−2.06323
$686$	0	0
$687$	−13.0000	−0.495981
$688$	0	0
$689$	−42.0000	−1.60007
$690$	0	0
$691$	−13.0000	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$692$	0	0
$693$	−10.0000	−0.379869
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	42.0000	1.58406
$704$	0	0
$705$	−3.00000	−0.112987
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−21.0000	−0.788672	−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000	−0.788672	−0.394336	+	0.918966i	$0.629025\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	105.000	3.92678
$716$	0	0
$717$	−4.00000	−0.149383
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	−32.0000	−1.18845
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	−15.0000	−0.552532
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−49.0000	−1.80006
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	45.0000	1.64867
$746$	0	0
$747$	−7.00000	−0.256117
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	3.00000	0.109181
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	−20.0000	−0.725954
$760$	0	0
$761$	1.00000	0.0362500	0.0181250	−	0.999836i	$-0.494230\pi$
$761$	1.00000	0.0362500	0.0181250	+	0.999836i	$0.494230\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	−70.0000	−2.52755
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	−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
$775$	4.00000	0.143684
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	14.0000	0.501602
$780$	0	0
$781$	15.0000	0.536742
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−7.00000	−0.248577
$794$	0	0
$795$	−18.0000	−0.638394
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−1.00000	−0.0353775
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−70.0000	−2.47025
$804$	0	0
$805$	−24.0000	−0.845889
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	−29.0000	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$809$	−29.0000	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−19.0000	−0.666359
$814$	0	0
$815$	27.0000	0.945769
$816$	0	0
$817$	−70.0000	−2.44899
$818$	0	0
$819$	−14.0000	−0.489200
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−43.0000	−1.49889	−0.749443	−	0.662069i	$-0.769679\pi$
$823$	−43.0000	−1.49889	−0.749443	+	0.662069i	$0.769679\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	7.00000	0.242827
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	−30.0000	−1.03819
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	108.000	3.71531
$846$	0	0
$847$	28.0000	0.962091
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−21.0000	−0.718185
$856$	0	0
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−55.0000	−1.86575
$870$	0	0
$871$	−21.0000	−0.711558
$872$	0	0
$873$	−3.00000	−0.101535
$874$	0	0
$875$	−6.00000	−0.202837
$876$	0	0
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	51.0000	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$882$	0	0
$883$	46.0000	1.54802	0.774012	−	0.633171i	$-0.218247\pi$
$883$	46.0000	1.54802	0.774012	+	0.633171i	$0.218247\pi$
$884$	0	0
$885$	−30.0000	−1.00844
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−7.00000	−0.234246
$894$	0	0
$895$	−57.0000	−1.90530
$896$	0	0
$897$	−28.0000	−0.934893
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	54.0000	1.79502
$906$	0	0
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	35.0000	1.15833
$914$	0	0
$915$	−3.00000	−0.0991769
$916$	0	0
$917$	−20.0000	−0.660458
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	21.0000	0.691223
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	21.0000	0.688247
$932$	0	0
$933$	21.0000	0.687509
$934$	0	0
$935$	15.0000	0.490552
$936$	0	0
$937$	1.00000	0.0326686	0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000	0.0326686	0.0163343	+	0.999867i	$0.494800\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	−6.00000	−0.195180
$946$	0	0
$947$	5.00000	0.162478	0.0812391	−	0.996695i	$-0.474112\pi$
$947$	5.00000	0.162478	0.0812391	+	0.996695i	$0.474112\pi$
$948$	0	0
$949$	−98.0000	−3.18121
$950$	0	0
$951$	−5.00000	−0.162136
$952$	0	0
$953$	−5.00000	−0.161966	−0.0809829	−	0.996715i	$-0.525806\pi$
$953$	−5.00000	−0.161966	−0.0809829	+	0.996715i	$0.525806\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−40.0000	−1.29302
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	57.0000	1.83489
$966$	0	0
$967$	−33.0000	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$967$	−33.0000	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$968$	0	0
$969$	−7.00000	−0.224872
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	28.0000	0.896718
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	0	0
$989$	−40.0000	−1.27193
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.0000	1.04617
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1488.2.a.h.1.1		1
3.2	odd	2		4464.2.a.c.1.1		1
4.3	odd	2		186.2.a.b.1.1	✓	1
8.3	odd	2		5952.2.a.b.1.1		1
8.5	even	2		5952.2.a.s.1.1		1
12.11	even	2		558.2.a.f.1.1		1
20.3	even	4		4650.2.d.k.3349.2		2
20.7	even	4		4650.2.d.k.3349.1		2
20.19	odd	2		4650.2.a.bh.1.1		1
28.27	even	2		9114.2.a.b.1.1		1
124.123	even	2		5766.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.b.1.1	✓	1	4.3	odd	2
558.2.a.f.1.1		1	12.11	even	2
1488.2.a.h.1.1		1	1.1	even	1	trivial
4464.2.a.c.1.1		1	3.2	odd	2
4650.2.a.bh.1.1		1	20.19	odd	2
4650.2.d.k.3349.1		2	20.7	even	4
4650.2.d.k.3349.2		2	20.3	even	4
5766.2.a.c.1.1		1	124.123	even	2
5952.2.a.b.1.1		1	8.3	odd	2
5952.2.a.s.1.1		1	8.5	even	2
9114.2.a.b.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1488 = 2^{4} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1488.a (trivial)

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