LMFDB - Embedded newform 5712.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1428)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{17} -5.00000 q^{19} +1.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +10.0000 q^{31} -3.00000 q^{33} +3.00000 q^{35} -4.00000 q^{37} +1.00000 q^{39} -9.00000 q^{41} +1.00000 q^{43} -3.00000 q^{45} +6.00000 q^{47} +1.00000 q^{49} -1.00000 q^{51} +6.00000 q^{53} -9.00000 q^{55} +5.00000 q^{57} +6.00000 q^{59} +8.00000 q^{61} -1.00000 q^{63} +3.00000 q^{65} +4.00000 q^{67} -3.00000 q^{69} +12.0000 q^{71} -10.0000 q^{73} -4.00000 q^{75} -3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -3.00000 q^{85} +6.00000 q^{87} +12.0000 q^{89} +1.00000 q^{91} -10.0000 q^{93} +15.0000 q^{95} +8.00000 q^{97} +3.00000 q^{99} +O(q^{100})$

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
$5$	−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
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	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
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	0	0
$57$	5.00000	0.662266
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	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
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	15.0000	1.53897
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	9.00000	0.811503
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	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
$133$	5.00000	0.433555
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−30.0000	−2.40966
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	27.0000	1.88576
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	−15.0000	−1.03757
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	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
$231$	3.00000	0.197386
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−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
$265$	−18.0000	−1.10573
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	−15.0000	−0.888523
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	0	0
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	−5.00000	−0.278207
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	0	0
$341$	30.0000	1.62459
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.00000	0.484544
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	1.00000	0.0529256
$358$	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
$363$	2.00000	0.104973
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	15.0000	0.756650
$394$	0	0
$395$	−30.0000	−1.50946
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	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
$423$	6.00000	0.291730
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	3.00000	0.144841
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	35.0000	1.68199	0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000	1.68199	0.840996	+	0.541041i	$0.181970\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	0	0
$437$	−15.0000	−0.717547
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	30.0000	1.39122
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	−20.0000	−0.917663
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	10.0000	0.453143	0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000	0.453143	0.226572	+	0.973995i	$0.427248\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−9.00000	−0.404520
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	33.0000	1.45415
$516$	0	0
$517$	18.0000	0.791639
$518$	0	0
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	9.00000	0.389833
$534$	0	0
$535$	27.0000	1.16731
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	45.0000	1.89316
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	22.0000	0.914289
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	0	0
$589$	−50.0000	−2.06021
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	6.00000	0.243935
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	0	0
$615$	−27.0000	−1.08875
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	15.0000	0.599042
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	14.0000	0.556450
$634$	0	0
$635$	−21.0000	−0.833360
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	3.00000	0.118125
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	10.0000	0.391931
$652$	0	0
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	0	0
$663$	1.00000	0.0388368
$664$	0	0
$665$	−15.0000	−0.581675
$666$	0	0
$667$	−18.0000	−0.696963
$668$	0	0
$669$	17.0000	0.657258
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	15.0000	0.574801
$682$	0	0
$683$	−51.0000	−1.95146	−0.975730	−	0.218975i	$-0.929729\pi$
$683$	−51.0000	−1.95146	−0.975730	+	0.218975i	$0.929729\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	42.0000	1.59315
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	−3.00000	−0.113470
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	0	0
$721$	11.0000	0.409661
$722$	0	0
$723$	28.0000	1.04133
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.00000	0.0369863
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	31.0000	1.14035	0.570177	−	0.821522i	$-0.306875\pi$
$739$	31.0000	1.14035	0.570177	+	0.821522i	$0.306875\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	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
$745$	−54.0000	−1.97841
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	0	0
$759$	−9.00000	−0.326679
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	45.0000	1.61229
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	39.0000	1.39197
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.0000	0.533339
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−30.0000	−1.05868
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	29.0000	1.01707
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−5.00000	−0.174928
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−9.00000	−0.311458
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−39.0000	−1.34643	−0.673215	−	0.739447i	$-0.735087\pi$
$839$	−39.0000	−1.34643	−0.673215	+	0.739447i	$0.735087\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	15.0000	0.512989
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	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
$861$	−9.00000	−0.306719
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	−27.0000	−0.918028
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	30.0000	1.01768
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	43.0000	1.44707	0.723533	−	0.690290i	$-0.242517\pi$
$883$	43.0000	1.44707	0.723533	+	0.690290i	$0.242517\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	−30.0000	−1.00391
$894$	0	0
$895$	−54.0000	−1.80502
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	−60.0000	−2.00111
$900$	0	0
$901$	6.00000	0.199889
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	30.0000	0.997234
$906$	0	0
$907$	34.0000	1.12895	0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000	1.12895	0.564476	+	0.825450i	$0.309078\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	15.0000	0.495344
$918$	0	0
$919$	7.00000	0.230909	0.115454	−	0.993313i	$-0.463168\pi$
$919$	7.00000	0.230909	0.115454	+	0.993313i	$0.463168\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−16.0000	−0.526077
$926$	0	0
$927$	−11.0000	−0.361287
$928$	0	0
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−9.00000	−0.294331
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	4.00000	0.130535
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	−9.00000	−0.290021
$964$	0	0
$965$	66.0000	2.12462
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	8.00000	0.255420
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	3.00000	0.0953945
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	29.0000	0.920287
$994$	0	0
$995$	60.0000	1.90213
$996$	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
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.b.1.1		1
4.3	odd	2		1428.2.a.d.1.1	✓	1
12.11	even	2		4284.2.a.i.1.1		1
28.27	even	2		9996.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.a.d.1.1	✓	1	4.3	odd	2
4284.2.a.i.1.1		1	12.11	even	2
5712.2.a.b.1.1		1	1.1	even	1	trivial
9996.2.a.g.1.1		1	28.27	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:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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