LMFDB - Embedded newform 2541.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.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:	$20.2899871536$
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	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{12} +4.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{19} +6.00000 q^{20} -1.00000 q^{21} +4.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} +3.00000 q^{35} -2.00000 q^{36} -10.0000 q^{37} +4.00000 q^{39} -6.00000 q^{41} -11.0000 q^{43} -3.00000 q^{45} -3.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -8.00000 q^{52} +12.0000 q^{53} -2.00000 q^{57} -3.00000 q^{59} +6.00000 q^{60} -8.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -12.0000 q^{65} +5.00000 q^{67} -6.00000 q^{68} -12.0000 q^{71} -8.00000 q^{73} +4.00000 q^{75} +4.00000 q^{76} -8.00000 q^{79} -12.0000 q^{80} +1.00000 q^{81} -15.0000 q^{83} +2.00000 q^{84} -9.00000 q^{85} +6.00000 q^{87} +15.0000 q^{89} -4.00000 q^{91} +2.00000 q^{93} +6.00000 q^{95} +14.0000 q^{97} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	4.00000	1.00000
$17$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	6.00000	1.34164
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	−2.00000	−0.333333
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	−8.00000	−1.10940
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	6.00000	0.774597
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	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	$0.648589\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	2.00000	0.218218
$85$	−9.00000	−0.976187
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−2.00000	−0.192450
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	−4.00000	−0.377964
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	4.00000	0.369800
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−6.00000	−0.541002
$124$	−4.00000	−0.359211
$125$	3.00000	0.268328
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	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
$133$	2.00000	0.173422
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−6.00000	−0.507093
$141$	−3.00000	−0.252646
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	−18.0000	−1.49482
$146$	0	0
$147$	1.00000	0.0824786
$148$	20.0000	1.64399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	−6.00000	−0.481932
$156$	−8.00000	−0.640513
$157$	−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
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−2.00000	−0.152944
$172$	22.0000	1.67748
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	6.00000	0.447214
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	30.0000	2.20564
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−8.00000	−0.577350
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	−2.00000	−0.142857
$197$	−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
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	−6.00000	−0.421117
$204$	−6.00000	−0.420084
$205$	18.0000	1.25717
$206$	0	0
$207$	0	0
$208$	16.0000	1.10940
$209$	0	0
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	−24.0000	−1.64833
$213$	−12.0000	−0.822226
$214$	0	0
$215$	33.0000	2.25058
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	−8.00000	−0.540590
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−9.00000	−0.597351	−0.298675	−	0.954355i	$-0.596545\pi$
$227$	−9.00000	−0.597351	−0.298675	+	0.954355i	$0.596545\pi$
$228$	4.00000	0.264906
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	6.00000	0.390567
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−12.0000	−0.774597
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	16.0000	1.02430
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	0	0
$255$	−9.00000	−0.563602
$256$	16.0000	1.00000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	24.0000	1.48842
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	0	0
$267$	15.0000	0.917985
$268$	−10.0000	−0.610847
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	12.0000	0.727607
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	24.0000	1.42414
$285$	6.00000	0.355409
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	16.0000	0.936329
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−8.00000	−0.461880
$301$	11.0000	0.634029
$302$	0	0
$303$	−3.00000	−0.172345
$304$	−8.00000	−0.458831
$305$	24.0000	1.37424
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	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
$315$	3.00000	0.169031
$316$	16.0000	0.900070
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	0	0
$320$	24.0000	1.34164
$321$	12.0000	0.669775
$322$	0	0
$323$	−6.00000	−0.333849
$324$	−2.00000	−0.111111
$325$	16.0000	0.887520
$326$	0	0
$327$	1.00000	0.0553001
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	30.0000	1.64646
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−15.0000	−0.819538
$336$	−4.00000	−0.218218
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	18.0000	0.976187
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	−12.0000	−0.643268
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	−30.0000	−1.59000
$357$	−3.00000	−0.158777
$358$	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
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	8.00000	0.419314
$365$	24.0000	1.25622
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	−4.00000	−0.207390
$373$	−35.0000	−1.81223	−0.906116	−	0.423030i	$-0.860966\pi$
$373$	−35.0000	−1.81223	−0.906116	+	0.423030i	$0.860966\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	−12.0000	−0.615587
$381$	1.00000	0.0512316
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.0000	−0.559161
$388$	−28.0000	−1.42148
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	24.0000	1.20757
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	16.0000	0.800000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	3.00000	0.147620
$414$	0	0
$415$	45.0000	2.20896
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	−6.00000	−0.292770
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	8.00000	0.387147
$428$	−24.0000	−1.16008
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	4.00000	0.192450
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	20.0000	0.949158
$445$	−45.0000	−2.13320
$446$	0	0
$447$	−6.00000	−0.283790
$448$	8.00000	0.377964
$449$	−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
$452$	36.0000	1.69330
$453$	−17.0000	−0.798730
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	24.0000	1.11417
$465$	−6.00000	−0.278243
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−8.00000	−0.369800
$469$	−5.00000	−0.230879
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	6.00000	0.275010
$477$	12.0000	0.549442
$478$	0	0
$479$	9.00000	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$479$	9.00000	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−42.0000	−1.90712
$486$	0	0
$487$	−7.00000	−0.317200	−0.158600	−	0.987343i	$-0.550698\pi$
$487$	−7.00000	−0.317200	−0.158600	+	0.987343i	$0.550698\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	12.0000	0.541002
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	12.0000	0.538274
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	−6.00000	−0.268328
$501$	21.0000	0.938211
$502$	0	0
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	0	0
$507$	3.00000	0.133235
$508$	−2.00000	−0.0887357
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	12.0000	0.528783
$516$	22.0000	0.968496
$517$	0	0
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	30.0000	1.31056
$525$	−4.00000	−0.174574
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−3.00000	−0.130189
$532$	−4.00000	−0.173422
$533$	−24.0000	−1.03956
$534$	0	0
$535$	−36.0000	−1.55642
$536$	0	0
$537$	24.0000	1.03568
$538$	0	0
$539$	0	0
$540$	6.00000	0.258199
$541$	−5.00000	−0.214967	−0.107483	−	0.994207i	$-0.534279\pi$
$541$	−5.00000	−0.214967	−0.107483	+	0.994207i	$0.534279\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−12.0000	−0.512615
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	30.0000	1.27343
$556$	40.0000	1.69638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−44.0000	−1.86100
$560$	12.0000	0.507093
$561$	0	0
$562$	0	0
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	6.00000	0.252646
$565$	54.0000	2.27180
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	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
$579$	−23.0000	−0.955847
$580$	36.0000	1.49482
$581$	15.0000	0.622305
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	−2.00000	−0.0824786
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−40.0000	−1.64399
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	12.0000	0.491539
$597$	8.00000	0.327418
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	34.0000	1.38344
$605$	0	0
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−12.0000	−0.485468
$612$	−6.00000	−0.242536
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	0	0
$623$	−15.0000	−0.600962
$624$	16.0000	0.640513
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	8.00000	0.319235
$629$	−30.0000	−1.19618
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	7.00000	0.278225
$634$	0	0
$635$	−3.00000	−0.119051
$636$	−24.0000	−0.951662
$637$	4.00000	0.158486
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	33.0000	1.29937
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	32.0000	1.25322
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	−24.0000	−0.937043
$657$	−8.00000	−0.312110
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	−42.0000	−1.62503
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	−6.00000	−0.230769
$677$	45.0000	1.72949	0.864745	−	0.502211i	$-0.167480\pi$
$677$	45.0000	1.72949	0.864745	+	0.502211i	$0.167480\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−9.00000	−0.344881
$682$	0	0
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	4.00000	0.152944
$685$	−18.0000	−0.687745
$686$	0	0
$687$	8.00000	0.305219
$688$	−44.0000	−1.67748
$689$	48.0000	1.82865
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	0	0
$695$	60.0000	2.27593
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	18.0000	0.680823
$700$	8.00000	0.302372
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	3.00000	0.112827
$708$	6.00000	0.225494
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−48.0000	−1.79384
$717$	−6.00000	−0.224074
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−12.0000	−0.447214
$721$	4.00000	0.148968
$722$	0	0
$723$	10.0000	0.371904
$724$	−40.0000	−1.48659
$725$	24.0000	0.891338
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.0000	−1.22055
$732$	16.0000	0.591377
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−60.0000	−2.20564
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	−15.0000	−0.548821
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−12.0000	−0.437595
$753$	−12.0000	−0.437304
$754$	0	0
$755$	51.0000	1.85608
$756$	2.00000	0.0727393
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	36.0000	1.30243
$765$	−9.00000	−0.325396
$766$	0	0
$767$	−12.0000	−0.433295
$768$	16.0000	0.577350
$769$	−44.0000	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$769$	−44.0000	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$770$	0	0
$771$	−21.0000	−0.756297
$772$	46.0000	1.65558
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	12.0000	0.429945
$780$	24.0000	0.859338
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	4.00000	0.142857
$785$	12.0000	0.428298
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	−36.0000	−1.27679
$796$	−16.0000	−0.567105
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	15.0000	0.529999
$802$	0	0
$803$	0	0
$804$	−10.0000	−0.352673
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	12.0000	0.421117
$813$	16.0000	0.561144
$814$	0	0
$815$	48.0000	1.68137
$816$	12.0000	0.420084
$817$	22.0000	0.769683
$818$	0	0
$819$	−4.00000	−0.139771
$820$	−36.0000	−1.25717
$821$	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
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	−17.0000	−0.589723
$832$	−32.0000	−1.10940
$833$	3.00000	0.103944
$834$	0	0
$835$	−63.0000	−2.18020
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	27.0000	0.932144	0.466072	−	0.884747i	$-0.345669\pi$
$839$	27.0000	0.932144	0.466072	+	0.884747i	$0.345669\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	24.0000	0.826604
$844$	−14.0000	−0.481900
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	48.0000	1.64833
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	24.0000	0.822226
$853$	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
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−66.0000	−2.25058
$861$	6.00000	0.204479
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	9.00000	0.306009
$866$	0	0
$867$	−8.00000	−0.271694
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	−3.00000	−0.101419
$876$	16.0000	0.540590
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	−24.0000	−0.807207
$885$	9.00000	0.302532
$886$	0	0
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	0	0
$892$	32.0000	1.07144
$893$	6.00000	0.200782
$894$	0	0
$895$	−72.0000	−2.40669
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000	0.400222
$900$	−8.00000	−0.266667
$901$	36.0000	1.19933
$902$	0	0
$903$	11.0000	0.366057
$904$	0	0
$905$	−60.0000	−1.99447
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	18.0000	0.597351
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	0	0
$915$	24.0000	0.793416
$916$	−16.0000	−0.528655
$917$	15.0000	0.495344
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−14.0000	−0.461316
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	−4.00000	−0.131377
$928$	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
$931$	−2.00000	−0.0655474
$932$	−36.0000	−1.17922
$933$	27.0000	0.883940
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	−18.0000	−0.587095
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	0	0
$944$	−12.0000	−0.390567
$945$	3.00000	0.0975900
$946$	0	0
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	16.0000	0.519656
$949$	−32.0000	−1.03876
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	54.0000	1.74740
$956$	12.0000	0.388108
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	24.0000	0.774597
$961$	−27.0000	−0.870968
$962$	0	0
$963$	12.0000	0.386695
$964$	−20.0000	−0.644157
$965$	69.0000	2.22119
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	−2.00000	−0.0641500
$973$	20.0000	0.641171
$974$	0	0
$975$	16.0000	0.512410
$976$	−32.0000	−1.02430
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	0	0
$980$	6.00000	0.191663
$981$	1.00000	0.0319275
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	3.00000	0.0954911
$988$	16.0000	0.509028
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	−24.0000	−0.760851
$996$	30.0000	0.950586
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.e.1.1	✓	1
3.2	odd	2		7623.2.a.k.1.1		1
11.10	odd	2		2541.2.a.f.1.1	yes	1
33.32	even	2		7623.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.e.1.1	✓	1	1.1	even	1	trivial
2541.2.a.f.1.1	yes	1	11.10	odd	2
7623.2.a.k.1.1		1	3.2	odd	2
7623.2.a.l.1.1		1	33.32	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:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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