LMFDB - Embedded newform 1848.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1848,2,Mod(1,1848)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1848, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1848.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.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:	$14.7563542935$
Analytic rank:	$0$
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	$\chi$	$=$	1848.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -3.00000 q^{13} +1.00000 q^{15} +4.00000 q^{17} +1.00000 q^{19} -1.00000 q^{21} +2.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{33} -1.00000 q^{35} -3.00000 q^{37} +3.00000 q^{39} +4.00000 q^{41} +2.00000 q^{43} -1.00000 q^{45} +5.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +2.00000 q^{53} +1.00000 q^{55} -1.00000 q^{57} +9.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} +3.00000 q^{65} -3.00000 q^{67} -2.00000 q^{69} +12.0000 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -4.00000 q^{85} +1.00000 q^{87} -2.00000 q^{89} -3.00000 q^{91} -4.00000 q^{93} -1.00000 q^{95} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	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
$15$	1.00000	0.258199
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	5.00000	0.729325	0.364662	−	0.931140i	$-0.381184\pi$
$47$	5.00000	0.729325	0.364662	+	0.931140i	$0.381184\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−1.00000	−0.113961
$78$	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
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	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
$159$	−2.00000	−0.158610
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	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
$171$	1.00000	0.0764719
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	0	0
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	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
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−5.00000	−0.326164
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	17.0000	1.09964	0.549819	−	0.835284i	$-0.314697\pi$
$239$	17.0000	1.09964	0.549819	+	0.835284i	$0.314697\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.00000	−0.190885
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	2.00000	0.122398
$268$	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
$271$	−31.0000	−1.88312	−0.941558	−	0.336851i	$-0.890638\pi$
$271$	−31.0000	−1.88312	−0.941558	+	0.336851i	$0.890638\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	5.00000	0.275659
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−3.00000	−0.164399
$334$	0	0
$335$	3.00000	0.163908
$336$	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
$339$	6.00000	0.325875
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	0	0
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−7.00000	−0.366397
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	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
$385$	1.00000	0.0509647
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.00000	0.148704
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−13.0000	−0.635092	−0.317546	−	0.948243i	$-0.602859\pi$
$419$	−13.0000	−0.635092	−0.317546	+	0.948243i	$0.602859\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	0	0
$423$	5.00000	0.243108
$424$	0	0
$425$	−16.0000	−0.776114
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	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
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	5.00000	0.236492
$448$	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
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	−7.00000	−0.325318	−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000	−0.325318	−0.162659	+	0.986682i	$0.552007\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	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
$469$	−3.00000	−0.138527
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−6.00000	−0.264392
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	9.00000	0.390567
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−15.0000	−0.648507
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	8.00000	0.342682
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−1.00000	−0.0426014
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	36.0000	1.49870	0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000	1.49870	0.749350	+	0.662174i	$0.230366\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−5.00000	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$587$	−5.00000	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	10.0000	0.409273
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	−15.0000	−0.606835
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	1.00000	0.0399362
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−2.00000	−0.0794929
$634$	0	0
$635$	−20.0000	−0.793676
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−32.0000	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$641$	−32.0000	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	−40.0000	−1.56532	−0.782660	−	0.622449i	$-0.786138\pi$
$653$	−40.0000	−1.56532	−0.782660	+	0.622449i	$0.786138\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	7.00000	0.273096
$658$	0	0
$659$	11.0000	0.428499	0.214250	−	0.976779i	$-0.431269\pi$
$659$	11.0000	0.428499	0.214250	+	0.976779i	$0.431269\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	18.0000	0.695920
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	0	0
$705$	5.00000	0.188311
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−3.00000	−0.112194
$716$	0	0
$717$	−17.0000	−0.634877
$718$	0	0
$719$	−45.0000	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	−1.00000	−0.0371904
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	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
$735$	1.00000	0.0368856
$736$	0	0
$737$	3.00000	0.110506
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	0	0
$743$	47.0000	1.72426	0.862131	−	0.506685i	$-0.169129\pi$
$743$	47.0000	1.72426	0.862131	+	0.506685i	$0.169129\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	0	0
$753$	−7.00000	−0.255094
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−27.0000	−0.974913
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	0	0
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−7.00000	−0.247025
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	31.0000	1.08722
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	27.0000	0.938882	0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000	0.938882	0.469441	+	0.882964i	$0.344455\pi$
$828$	0	0
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−3.00000	−0.103572	−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000	−0.103572	−0.0517858	+	0.998658i	$0.516491\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−3.00000	−0.103325
$844$	0	0
$845$	4.00000	0.137604
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	9.00000	0.304953
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.0000	−0.977035	−0.488517	−	0.872554i	$-0.662462\pi$
$881$	−29.0000	−0.977035	−0.488517	+	0.872554i	$0.662462\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	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
$889$	20.0000	0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	5.00000	0.167319
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	−36.0000	−1.18495
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	6.00000	0.197066
$928$	0	0
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	4.00000	0.130814
$936$	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
$939$	−8.00000	−0.261070
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	0	0
$949$	−21.0000	−0.681689
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−29.0000	−0.939402	−0.469701	−	0.882826i	$-0.655638\pi$
$953$	−29.0000	−0.939402	−0.469701	+	0.882826i	$0.655638\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	−1.00000	−0.0323254
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	15.0000	0.483368
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	19.0000	0.609739	0.304870	−	0.952394i	$-0.401387\pi$
$971$	19.0000	0.609739	0.304870	+	0.952394i	$0.401387\pi$
$972$	0	0
$973$	12.0000	0.384702
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	−8.00000	−0.255420
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−5.00000	−0.159152
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−51.0000	−1.62007	−0.810034	−	0.586383i	$-0.800552\pi$
$991$	−51.0000	−1.62007	−0.810034	+	0.586383i	$0.800552\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.a.b.1.1	✓	1
3.2	odd	2		5544.2.a.r.1.1		1
4.3	odd	2		3696.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.a.b.1.1	✓	1	1.1	even	1	trivial
3696.2.a.v.1.1		1	4.3	odd	2
5544.2.a.r.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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