LMFDB - Embedded newform 1848.2.a.j.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:	$1$
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	$\chi$	$=$	1848.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} -5.00000 q^{19} -1.00000 q^{21} -6.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{33} -1.00000 q^{35} +5.00000 q^{37} -5.00000 q^{39} +6.00000 q^{43} +1.00000 q^{45} +7.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +2.00000 q^{53} -1.00000 q^{55} -5.00000 q^{57} +3.00000 q^{59} -14.0000 q^{61} -1.00000 q^{63} -5.00000 q^{65} -7.00000 q^{67} -6.00000 q^{69} +9.00000 q^{73} -4.00000 q^{75} +1.00000 q^{77} +1.00000 q^{81} +12.0000 q^{83} -4.00000 q^{85} -9.00000 q^{87} +10.0000 q^{89} +5.00000 q^{91} +4.00000 q^{93} -5.00000 q^{95} -14.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	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
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	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
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	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
$57$	−5.00000	−0.662266
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	1.00000	0.113961
$78$	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
$83$	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
$87$	−9.00000	−0.964901
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−5.00000	−0.512989
$96$	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
$99$	−1.00000	−0.100504
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	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
$141$	7.00000	0.589506
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	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
$153$	−4.00000	−0.323381
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−15.0000	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	3.00000	0.225494
$178$	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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	5.00000	0.367607
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−5.00000	−0.358057
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.00000	−0.194054	−0.0970269	−	0.995282i	$-0.530933\pi$
$239$	−3.00000	−0.194054	−0.0970269	+	0.995282i	$0.530933\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	25.0000	1.59071
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	27.0000	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$272$	0	0
$273$	5.00000	0.302614
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−32.0000	−1.92269	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−32.0000	−1.92269	−0.961347	+	0.275340i	$0.911209\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	0	0
$285$	−5.00000	−0.296174
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	30.0000	1.73494
$300$	0	0
$301$	−6.00000	−0.345834
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	−16.0000	−0.884802
$328$	0	0
$329$	−7.00000	−0.385922
$330$	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
$333$	5.00000	0.273998
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	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
$345$	−6.00000	−0.323029
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	45.0000	2.31762
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	6.00000	0.304997
$388$	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
$391$	24.0000	1.21373
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	5.00000	0.250313
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	25.0000	1.22133	0.610665	−	0.791889i	$-0.290902\pi$
$419$	25.0000	1.22133	0.610665	+	0.791889i	$0.290902\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	0	0
$423$	7.00000	0.340352
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	14.0000	0.677507
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	−21.0000	−1.01153	−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000	−1.01153	−0.505767	+	0.862670i	$0.668791\pi$
$432$	0	0
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	0	0
$437$	30.0000	1.43509
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	3.00000	0.141895
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	5.00000	0.234404
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	21.0000	0.975953	0.487976	−	0.872857i	$-0.337735\pi$
$463$	21.0000	0.975953	0.487976	+	0.872857i	$0.337735\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−15.0000	−0.678323
$490$	0	0
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	−7.00000	−0.307860
$518$	0	0
$519$	−8.00000	−0.351161
$520$	0	0
$521$	25.0000	1.09527	0.547635	−	0.836717i	$-0.315528\pi$
$521$	25.0000	1.09527	0.547635	+	0.836717i	$0.315528\pi$
$522$	0	0
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3.00000	0.129701
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	45.0000	1.91706
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.00000	0.212238
$556$	0	0
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	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
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−5.00000	−0.206725
$586$	0	0
$587$	−7.00000	−0.288921	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000	−0.288921	−0.144460	+	0.989511i	$0.546145\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	9.00000	0.367118	0.183559	−	0.983009i	$-0.441238\pi$
$601$	9.00000	0.367118	0.183559	+	0.983009i	$0.441238\pi$
$602$	0	0
$603$	−7.00000	−0.285062
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−35.0000	−1.41595
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	−10.0000	−0.400642
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	5.00000	0.199681
$628$	0	0
$629$	−20.0000	−0.797452
$630$	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
$633$	18.0000	0.715436
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−5.00000	−0.198107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.0000	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$642$	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
$645$	6.00000	0.236250
$646$	0	0
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	−49.0000	−1.90877	−0.954384	−	0.298580i	$-0.903487\pi$
$659$	−49.0000	−1.90877	−0.954384	+	0.298580i	$0.903487\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	20.0000	0.776736
$664$	0	0
$665$	5.00000	0.193892
$666$	0	0
$667$	54.0000	2.09089
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	0	0
$687$	−30.0000	−1.14457
$688$	0	0
$689$	−10.0000	−0.380970
$690$	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
$693$	1.00000	0.0379869
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−25.0000	−0.942893
$704$	0	0
$705$	7.00000	0.263635
$706$	0	0
$707$	−8.00000	−0.300871
$708$	0	0
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	5.00000	0.186989
$716$	0	0
$717$	−3.00000	−0.112037
$718$	0	0
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	0	0
$721$	2.00000	0.0744839
$722$	0	0
$723$	−25.0000	−0.929760
$724$	0	0
$725$	36.0000	1.33701
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	7.00000	0.257848
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	25.0000	0.918398
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	0	0
$753$	21.0000	0.765283
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	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
$763$	16.0000	0.579239
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−15.0000	−0.541619
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	0	0
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−5.00000	−0.179374
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−27.0000	−0.962446	−0.481223	−	0.876598i	$-0.659807\pi$
$787$	−27.0000	−0.962446	−0.481223	+	0.876598i	$0.659807\pi$
$788$	0	0
$789$	−21.0000	−0.747620
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	70.0000	2.48577
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	47.0000	1.66483	0.832413	−	0.554156i	$-0.186959\pi$
$797$	47.0000	1.66483	0.832413	+	0.554156i	$0.186959\pi$
$798$	0	0
$799$	−28.0000	−0.990569
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	−9.00000	−0.317603
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	17.0000	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$811$	17.0000	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$812$	0	0
$813$	27.0000	0.946931
$814$	0	0
$815$	−15.0000	−0.525427
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	5.00000	0.174714
$820$	0	0
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	0	0
$829$	8.00000	0.277851	0.138926	−	0.990303i	$-0.455635\pi$
$829$	8.00000	0.277851	0.138926	+	0.990303i	$0.455635\pi$
$830$	0	0
$831$	−32.0000	−1.11007
$832$	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
$839$	39.0000	1.34643	0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000	1.34643	0.673215	+	0.739447i	$0.264913\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	19.0000	0.654395
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−13.0000	−0.446159
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	−5.00000	−0.170996
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	35.0000	1.18593
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.0000	0.977035	0.488517	−	0.872554i	$-0.337538\pi$
$881$	29.0000	0.977035	0.488517	+	0.872554i	$0.337538\pi$
$882$	0	0
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	0	0
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−35.0000	−1.17123
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	30.0000	1.00167
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	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
$909$	8.00000	0.265343
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	−49.0000	−1.60764	−0.803819	−	0.594874i	$-0.797202\pi$
$929$	−49.0000	−1.60764	−0.803819	+	0.594874i	$0.797202\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	0	0
$949$	−45.0000	−1.46076
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	51.0000	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	9.00000	0.290929
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	3.00000	0.0966736
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	6.00000	0.192947	0.0964735	−	0.995336i	$-0.469244\pi$
$967$	6.00000	0.192947	0.0964735	+	0.995336i	$0.469244\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	49.0000	1.57248	0.786242	−	0.617918i	$-0.212024\pi$
$971$	49.0000	1.57248	0.786242	+	0.617918i	$0.212024\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−7.00000	−0.222812
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−2.00000	−0.0634043
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	5.00000	0.158193

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.a.j.1.1	✓	1	1.1	even	1	trivial
3696.2.a.k.1.1		1	4.3	odd	2
5544.2.a.g.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

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