LMFDB - Embedded newform 4840.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} -4.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} -3.00000 q^{19} +3.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -1.00000 q^{29} -5.00000 q^{31} -3.00000 q^{35} -5.00000 q^{37} -4.00000 q^{39} -2.00000 q^{41} -8.00000 q^{43} +2.00000 q^{45} -10.0000 q^{47} +2.00000 q^{49} +3.00000 q^{51} +11.0000 q^{53} -3.00000 q^{57} +1.00000 q^{61} -6.00000 q^{63} +4.00000 q^{65} +4.00000 q^{67} +4.00000 q^{69} +5.00000 q^{71} +10.0000 q^{73} +1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -1.00000 q^{87} -15.0000 q^{89} -12.0000 q^{91} -5.00000 q^{93} +3.00000 q^{95} -6.00000 q^{97} +O(q^{100})$

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	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$63$	−6.00000	−0.755929
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	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	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	8.00000	0.739600
$118$	0	0
$119$	9.00000	0.825029
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−9.00000	−0.780399
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−19.0000	−1.55654	−0.778270	−	0.627929i	$-0.783903\pi$
$149$	−19.0000	−1.55654	−0.778270	+	0.627929i	$0.783903\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823	−0.0386912	−	0.999251i	$-0.512319\pi$
$167$	−1.00000	−0.0773823	−0.0386912	+	0.999251i	$0.512319\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	5.00000	0.367607
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−15.0000	−1.09109
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	0	0
$213$	5.00000	0.342594
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−15.0000	−1.01827
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	10.0000	0.652328
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	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
$259$	−15.0000	−0.932055
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	0	0
$267$	−15.0000	−0.917985
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	−18.0000	−1.03407
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	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
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	−5.00000	−0.265372
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	33.0000	1.71327
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	0	0
$399$	−9.00000	−0.450564
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	20.0000	0.972433
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	3.00000	0.145180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	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
$435$	1.00000	0.0479463
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−38.0000	−1.81364	−0.906821	−	0.421517i	$-0.861498\pi$
$439$	−38.0000	−1.81364	−0.906821	+	0.421517i	$0.861498\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	0	0
$447$	−19.0000	−0.898669
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	29.0000	1.35066	0.675332	−	0.737514i	$-0.264000\pi$
$461$	29.0000	1.35066	0.675332	+	0.737514i	$0.264000\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	11.0000	0.506853
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−3.00000	−0.137649
$476$	0	0
$477$	−22.0000	−1.00731
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	5.00000	0.226108
$490$	0	0
$491$	−11.0000	−0.496423	−0.248212	−	0.968706i	$-0.579843\pi$
$491$	−11.0000	−0.496423	−0.248212	+	0.968706i	$0.579843\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	0	0
$513$	15.0000	0.662266
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−18.0000	−0.787085	−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000	−0.787085	−0.393543	+	0.919306i	$0.628751\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	−15.0000	−0.653410
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−2.00000	−0.0863064
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	5.00000	0.212238
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	24.0000	0.999133	0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000	0.999133	0.499567	+	0.866276i	$0.333493\pi$
$578$	0	0
$579$	−5.00000	−0.207793
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	0	0
$591$	−26.0000	−1.06950
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	17.0000	0.695764
$598$	0	0
$599$	19.0000	0.776319	0.388159	−	0.921592i	$-0.373111\pi$
$599$	19.0000	0.776319	0.388159	+	0.921592i	$0.373111\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	0	0
$609$	−3.00000	−0.121566
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	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
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	−20.0000	−0.802572
$622$	0	0
$623$	−45.0000	−1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.0000	−0.598089
$630$	0	0
$631$	49.0000	1.95066	0.975330	−	0.220754i	$-0.0708517\pi$
$631$	49.0000	1.95066	0.975330	+	0.220754i	$0.0708517\pi$
$632$	0	0
$633$	−23.0000	−0.914168
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−8.00000	−0.316972
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	19.0000	0.750455	0.375227	−	0.926933i	$-0.377565\pi$
$641$	19.0000	0.750455	0.375227	+	0.926933i	$0.377565\pi$
$642$	0	0
$643$	33.0000	1.30139	0.650696	−	0.759338i	$-0.274477\pi$
$643$	33.0000	1.30139	0.650696	+	0.759338i	$0.274477\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−15.0000	−0.587896
$652$	0	0
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	15.0000	0.586098
$656$	0	0
$657$	−20.0000	−0.780274
$658$	0	0
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	9.00000	0.349005
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−18.0000	−0.690777
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	0	0
$687$	−16.0000	−0.610438
$688$	0	0
$689$	−44.0000	−1.67627
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	−13.0000	−0.491705
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	0	0
$703$	15.0000	0.565736
$704$	0	0
$705$	10.0000	0.376622
$706$	0	0
$707$	−54.0000	−2.03088
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.0000	0.522840
$718$	0	0
$719$	31.0000	1.15610	0.578052	−	0.816000i	$-0.303813\pi$
$719$	31.0000	1.15610	0.578052	+	0.816000i	$0.303813\pi$
$720$	0	0
$721$	30.0000	1.11726
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	0	0
$738$	0	0
$739$	48.0000	1.76571	0.882854	−	0.469647i	$-0.155619\pi$
$739$	48.0000	1.76571	0.882854	+	0.469647i	$0.155619\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	23.0000	0.843788	0.421894	−	0.906645i	$-0.361365\pi$
$743$	23.0000	0.843788	0.421894	+	0.906645i	$0.361365\pi$
$744$	0	0
$745$	19.0000	0.696106
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−54.0000	−1.97312
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	−42.0000	−1.52050
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	0	0
$768$	0	0
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	15.0000	0.539513	0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000	0.539513	0.269756	+	0.962929i	$0.413057\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	−15.0000	−0.538122
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	−11.0000	−0.392607
$786$	0	0
$787$	6.00000	0.213877	0.106938	−	0.994266i	$-0.465895\pi$
$787$	6.00000	0.213877	0.106938	+	0.994266i	$0.465895\pi$
$788$	0	0
$789$	−11.0000	−0.391610
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	−11.0000	−0.390130
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	−30.0000	−1.06132
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	0	0
$815$	−5.00000	−0.175142
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	24.0000	0.838628
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	1.00000	0.0346064
$836$	0	0
$837$	25.0000	0.864126
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	−20.0000	−0.685591
$852$	0	0
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−25.0000	−0.853984	−0.426992	−	0.904255i	$-0.640427\pi$
$857$	−25.0000	−0.853984	−0.426992	+	0.904255i	$0.640427\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	50.0000	1.70202	0.851010	−	0.525150i	$-0.175991\pi$
$863$	50.0000	1.70202	0.851010	+	0.525150i	$0.175991\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	0	0
$879$	−10.0000	−0.337292
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	−57.0000	−1.91820	−0.959101	−	0.283063i	$-0.908649\pi$
$883$	−57.0000	−1.91820	−0.959101	+	0.283063i	$0.908649\pi$
$884$	0	0
$885$	0	0
$886$	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
$889$	−48.0000	−1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.0000	1.00391
$894$	0	0
$895$	2.00000	0.0668526
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	5.00000	0.166759
$900$	0	0
$901$	33.0000	1.09939
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	−16.0000	−0.531858
$906$	0	0
$907$	−35.0000	−1.16216	−0.581078	−	0.813848i	$-0.697369\pi$
$907$	−35.0000	−1.16216	−0.581078	+	0.813848i	$0.697369\pi$
$908$	0	0
$909$	36.0000	1.19404
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.00000	−0.0330590
$916$	0	0
$917$	−45.0000	−1.48603
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	0	0
$923$	−20.0000	−0.658308
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	−20.0000	−0.656886
$928$	0	0
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−21.0000	−0.687509
$934$	0	0
$935$	0	0
$936$	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
$939$	−34.0000	−1.10955
$940$	0	0
$941$	39.0000	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$941$	39.0000	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	15.0000	0.487950
$946$	0	0
$947$	59.0000	1.91724	0.958621	−	0.284685i	$-0.0918889\pi$
$947$	59.0000	1.91724	0.958621	+	0.284685i	$0.0918889\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	17.0000	0.551263
$952$	0	0
$953$	−43.0000	−1.39291	−0.696453	−	0.717602i	$-0.745240\pi$
$953$	−43.0000	−1.39291	−0.696453	+	0.717602i	$0.745240\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	36.0000	1.16008
$964$	0	0
$965$	5.00000	0.160956
$966$	0	0
$967$	−55.0000	−1.76868	−0.884340	−	0.466843i	$-0.845391\pi$
$967$	−55.0000	−1.76868	−0.884340	+	0.466843i	$0.845391\pi$
$968$	0	0
$969$	−9.00000	−0.289122
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	28.0000	0.893971
$982$	0	0
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	0	0
$987$	−30.0000	−0.954911
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−17.0000	−0.538936
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	0	0
$999$	25.0000	0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.h.1.1	yes	1
4.3	odd	2		9680.2.a.e.1.1		1
11.10	odd	2		4840.2.a.g.1.1	✓	1
44.43	even	2		9680.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.g.1.1	✓	1	11.10	odd	2
4840.2.a.h.1.1	yes	1	1.1	even	1	trivial
9680.2.a.e.1.1		1	4.3	odd	2
9680.2.a.i.1.1		1	44.43	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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