LMFDB - Embedded newform 8048.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -4.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -4.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} -3.00000 q^{13} +4.00000 q^{15} -4.00000 q^{17} +1.00000 q^{21} +3.00000 q^{23} +11.0000 q^{25} +5.00000 q^{27} -10.0000 q^{29} -10.0000 q^{31} -3.00000 q^{33} +4.00000 q^{35} +2.00000 q^{37} +3.00000 q^{39} +2.00000 q^{41} -5.00000 q^{43} +8.00000 q^{45} -11.0000 q^{47} -6.00000 q^{49} +4.00000 q^{51} -14.0000 q^{53} -12.0000 q^{55} +4.00000 q^{59} -7.00000 q^{61} +2.00000 q^{63} +12.0000 q^{65} +3.00000 q^{67} -3.00000 q^{69} -6.00000 q^{71} -10.0000 q^{73} -11.0000 q^{75} -3.00000 q^{77} +1.00000 q^{81} -9.00000 q^{83} +16.0000 q^{85} +10.0000 q^{87} +6.00000 q^{89} +3.00000 q^{91} +10.0000 q^{93} -14.0000 q^{97} -6.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	8.00000	1.19257
$46$	0	0
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	12.0000	1.48842
$66$	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
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	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
$91$	3.00000	0.314485
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−20.0000	−1.99007	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−20.0000	−1.99007	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	21.0000	1.97551	0.987757	−	0.156001i	$-0.0498603\pi$
$113$	21.0000	1.97551	0.987757	+	0.156001i	$0.0498603\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	5.00000	0.440225
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−20.0000	−1.72133
$136$	0	0
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	0	0
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	11.0000	0.926367
$142$	0	0
$143$	−9.00000	−0.752618
$144$	0	0
$145$	40.0000	3.32182
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	40.0000	3.21288
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	7.00000	0.517455
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	0	0
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	10.0000	0.701862
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	20.0000	1.36399
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	0	0
$225$	−22.0000	−1.46667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−13.0000	−0.851658	−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000	−0.851658	−0.425829	+	0.904804i	$0.640018\pi$
$234$	0	0
$235$	44.0000	2.87024
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	24.0000	1.53330
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	−16.0000	−1.00196
$256$	0	0
$257$	−5.00000	−0.311891	−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000	−0.311891	−0.155946	+	0.987766i	$0.549842\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	20.0000	1.23797
$262$	0	0
$263$	7.00000	0.431638	0.215819	−	0.976433i	$-0.430758\pi$
$263$	7.00000	0.431638	0.215819	+	0.976433i	$0.430758\pi$
$264$	0	0
$265$	56.0000	3.44005
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	33.0000	1.98997
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	20.0000	1.19737
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	5.00000	0.288195
$302$	0	0
$303$	20.0000	1.14897
$304$	0	0
$305$	28.0000	1.60328
$306$	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
$309$	12.0000	0.682656
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−33.0000	−1.83051
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	11.0000	0.606450
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−21.0000	−1.14056
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−15.0000	−0.800641
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	40.0000	2.09370
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	14.0000	0.726844
$372$	0	0
$373$	−3.00000	−0.155334	−0.0776671	−	0.996979i	$-0.524747\pi$
$373$	−3.00000	−0.155334	−0.0776671	+	0.996979i	$0.524747\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	9.00000	0.453990
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	0	0
$403$	30.0000	1.49441
$404$	0	0
$405$	−4.00000	−0.198762
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	22.0000	1.07734
$418$	0	0
$419$	22.0000	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	22.0000	1.06968
$424$	0	0
$425$	−44.0000	−2.13431
$426$	0	0
$427$	7.00000	0.338754
$428$	0	0
$429$	9.00000	0.434524
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	−40.0000	−1.91785
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	0	0
$459$	−20.0000	−0.933520
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	−15.0000	−0.697109	−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000	−0.697109	−0.348555	+	0.937288i	$0.613327\pi$
$464$	0	0
$465$	−40.0000	−1.85496
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−3.00000	−0.138527
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−15.0000	−0.689701
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.0000	1.28203
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	56.0000	2.54283
$486$	0	0
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	40.0000	1.80151
$494$	0	0
$495$	24.0000	1.07872
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	30.0000	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	80.0000	3.55995
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.0000	2.11513
$516$	0	0
$517$	−33.0000	−1.45134
$518$	0	0
$519$	13.0000	0.570637
$520$	0	0
$521$	−5.00000	−0.219054	−0.109527	−	0.993984i	$-0.534934\pi$
$521$	−5.00000	−0.219054	−0.109527	+	0.993984i	$0.534934\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	11.0000	0.480079
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	32.0000	1.38348
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	15.0000	0.634432
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	0	0
$565$	−84.0000	−3.53391
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	−2.00000	−0.0835512
$574$	0	0
$575$	33.0000	1.37620
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	26.0000	1.08052
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	−42.0000	−1.73946
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	17.0000	0.699287
$592$	0	0
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	0	0
$611$	33.0000	1.33504
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	0	0
$619$	42.0000	1.68812	0.844061	−	0.536247i	$-0.180158\pi$
$619$	42.0000	1.68812	0.844061	+	0.536247i	$0.180158\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−43.0000	−1.71180	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000	−1.71180	−0.855901	+	0.517139i	$0.826997\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	0	0
$635$	−72.0000	−2.85723
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	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
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−20.0000	−0.787499
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	−15.0000	−0.586995	−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000	−0.586995	−0.293498	+	0.955960i	$0.594819\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	0	0
$657$	20.0000	0.780274
$658$	0	0
$659$	−17.0000	−0.662226	−0.331113	−	0.943591i	$-0.607424\pi$
$659$	−17.0000	−0.662226	−0.331113	+	0.943591i	$0.607424\pi$
$660$	0	0
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−30.0000	−1.16160
$668$	0	0
$669$	−7.00000	−0.270636
$670$	0	0
$671$	−21.0000	−0.810696
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	55.0000	2.11695
$676$	0	0
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−80.0000	−3.05664
$686$	0	0
$687$	−7.00000	−0.267067
$688$	0	0
$689$	42.0000	1.60007
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	88.0000	3.33803
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	13.0000	0.491705
$700$	0	0
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−44.0000	−1.65714
$706$	0	0
$707$	20.0000	0.752177
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.0000	−1.12351
$714$	0	0
$715$	36.0000	1.34632
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	−110.000	−4.08530
$726$	0	0
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	−24.0000	−0.885253
$736$	0	0
$737$	9.00000	0.331519
$738$	0	0
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	0	0
$755$	−80.0000	−2.91150
$756$	0	0
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	0	0
$759$	−9.00000	−0.326679
$760$	0	0
$761$	−17.0000	−0.616250	−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000	−0.616250	−0.308125	+	0.951346i	$0.599701\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−32.0000	−1.15696
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	5.00000	0.180071
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	−110.000	−3.95132
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	−50.0000	−1.78685
$784$	0	0
$785$	−48.0000	−1.71319
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	0	0
$789$	−7.00000	−0.249207
$790$	0	0
$791$	−21.0000	−0.746674
$792$	0	0
$793$	21.0000	0.745732
$794$	0	0
$795$	−56.0000	−1.98612
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	44.0000	1.55661
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	−30.0000	−1.05868
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	7.00000	0.245501
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	−33.0000	−1.14891
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	−72.0000	−2.49166
$836$	0	0
$837$	−50.0000	−1.72825
$838$	0	0
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	−15.0000	−0.516627
$844$	0	0
$845$	16.0000	0.550417
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	52.0000	1.76805
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−9.00000	−0.304953
$872$	0	0
$873$	28.0000	0.947656
$874$	0	0
$875$	24.0000	0.811348
$876$	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
$879$	11.0000	0.371021
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	100.000	3.33519
$900$	0	0
$901$	56.0000	1.86563
$902$	0	0
$903$	−5.00000	−0.166390
$904$	0	0
$905$	−80.0000	−2.65929
$906$	0	0
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	0	0
$909$	40.0000	1.32672
$910$	0	0
$911$	10.0000	0.331315	0.165657	−	0.986183i	$-0.447025\pi$
$911$	10.0000	0.331315	0.165657	+	0.986183i	$0.447025\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	0	0
$915$	−28.0000	−0.925651
$916$	0	0
$917$	9.00000	0.297206
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	18.0000	0.592477
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	24.0000	0.788263
$928$	0	0
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.0000	0.589294
$934$	0	0
$935$	48.0000	1.56977
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	20.0000	0.650600
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	17.0000	0.551263
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	30.0000	0.969762
$958$	0	0
$959$	−20.0000	−0.645834
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	104.000	3.34788
$966$	0	0
$967$	−26.0000	−0.836104	−0.418052	−	0.908423i	$-0.637287\pi$
$967$	−26.0000	−0.836104	−0.418052	+	0.908423i	$0.637287\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	0	0
$973$	22.0000	0.705288
$974$	0	0
$975$	33.0000	1.05685
$976$	0	0
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−34.0000	−1.08443	−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000	−1.08443	−0.542216	+	0.840239i	$0.682414\pi$
$984$	0	0
$985$	68.0000	2.16666
$986$	0	0
$987$	−11.0000	−0.350134
$988$	0	0
$989$	−15.0000	−0.476972
$990$	0	0
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	−64.0000	−2.02894
$996$	0	0
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.b.1.1		1
4.3	odd	2		1006.2.a.e.1.1	✓	1
12.11	even	2		9054.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.e.1.1	✓	1	4.3	odd	2
8048.2.a.b.1.1		1	1.1	even	1	trivial
9054.2.a.k.1.1		1	12.11	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more