LMFDB - Embedded newform 57.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("57.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	57.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:	$0.455147291521$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	57.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} -6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} +6.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} +12.0000 q^{26} +1.00000 q^{27} +6.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} +2.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} -6.00000 q^{39} -8.00000 q^{41} -6.00000 q^{42} -1.00000 q^{43} -6.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +3.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +8.00000 q^{50} +3.00000 q^{51} -12.0000 q^{52} -6.00000 q^{53} -2.00000 q^{54} -3.00000 q^{55} -1.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} +7.00000 q^{61} -4.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +6.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -6.00000 q^{70} +12.0000 q^{71} -11.0000 q^{73} -16.0000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -9.00000 q^{77} +12.0000 q^{78} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} +4.00000 q^{83} +6.00000 q^{84} +3.00000 q^{85} +2.00000 q^{86} -10.0000 q^{87} +10.0000 q^{89} -2.00000 q^{90} -18.0000 q^{91} +8.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} -1.00000 q^{95} +8.00000 q^{96} -2.00000 q^{97} -4.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−2.00000	−0.816497
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−6.00000	−1.60357
$15$	1.00000	0.258199
$16$	−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	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.00000	0.447214
$21$	3.00000	0.654654
$22$	6.00000	1.27920
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	12.0000	2.35339
$27$	1.00000	0.192450
$28$	6.00000	1.13389
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	−2.00000	−0.365148
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	8.00000	1.41421
$33$	−3.00000	−0.522233
$34$	−6.00000	−1.02899
$35$	3.00000	0.507093
$36$	2.00000	0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	2.00000	0.324443
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−6.00000	−0.925820
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−6.00000	−0.904534
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	8.00000	1.13137
$51$	3.00000	0.420084
$52$	−12.0000	−1.66410
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−2.00000	−0.272166
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−1.00000	−0.132453
$58$	20.0000	2.62613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−4.00000	−0.508001
$63$	3.00000	0.377964
$64$	−8.00000	−1.00000
$65$	−6.00000	−0.744208
$66$	6.00000	0.738549
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000	0.727607
$69$	4.00000	0.481543
$70$	−6.00000	−0.717137
$71$	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
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−16.0000	−1.85996
$75$	−4.00000	−0.461880
$76$	−2.00000	−0.229416
$77$	−9.00000	−1.02565
$78$	12.0000	1.35873
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	16.0000	1.76690
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	6.00000	0.654654
$85$	3.00000	0.325396
$86$	2.00000	0.215666
$87$	−10.0000	−1.07211
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−2.00000	−0.210819
$91$	−18.0000	−1.88691
$92$	8.00000	0.834058
$93$	2.00000	0.207390
$94$	−6.00000	−0.618853
$95$	−1.00000	−0.102598
$96$	8.00000	0.816497
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−4.00000	−0.404061
$99$	−3.00000	−0.301511
$100$	−8.00000	−0.800000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−6.00000	−0.594089
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	12.0000	1.16554
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	2.00000	0.192450
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	6.00000	0.572078
$111$	8.00000	0.759326
$112$	−12.0000	−1.13389
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	2.00000	0.187317
$115$	4.00000	0.373002
$116$	−20.0000	−1.85695
$117$	−6.00000	−0.554700
$118$	0	0
$119$	9.00000	0.825029
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−14.0000	−1.26750
$123$	−8.00000	−0.721336
$124$	4.00000	0.359211
$125$	−9.00000	−0.804984
$126$	−6.00000	−0.534522
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	12.0000	1.05247
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	−6.00000	−0.522233
$133$	−3.00000	−0.260133
$134$	−16.0000	−1.38219
$135$	1.00000	0.0860663
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	−8.00000	−0.681005
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	6.00000	0.507093
$141$	3.00000	0.252646
$142$	−24.0000	−2.01404
$143$	18.0000	1.50524
$144$	−4.00000	−0.333333
$145$	−10.0000	−0.830455
$146$	22.0000	1.82073
$147$	2.00000	0.164957
$148$	16.0000	1.31519
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	8.00000	0.653197
$151$	−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
$153$	3.00000	0.242536
$154$	18.0000	1.45048
$155$	2.00000	0.160644
$156$	−12.0000	−0.960769
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	8.00000	0.632456
$161$	12.0000	0.945732
$162$	−2.00000	−0.157135
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−16.0000	−1.24939
$165$	−3.00000	−0.233550
$166$	−8.00000	−0.620920
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−6.00000	−0.460179
$171$	−1.00000	−0.0764719
$172$	−2.00000	−0.152499
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	20.0000	1.51620
$175$	−12.0000	−0.907115
$176$	12.0000	0.904534
$177$	0	0
$178$	−20.0000	−1.49906
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	2.00000	0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	36.0000	2.66850
$183$	7.00000	0.517455
$184$	0	0
$185$	8.00000	0.588172
$186$	−4.00000	−0.293294
$187$	−9.00000	−0.658145
$188$	6.00000	0.437595
$189$	3.00000	0.218218
$190$	2.00000	0.145095
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−8.00000	−0.577350
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	4.00000	0.287183
$195$	−6.00000	−0.429669
$196$	4.00000	0.285714
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	6.00000	0.426401
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−4.00000	−0.281439
$203$	−30.0000	−2.10559
$204$	6.00000	0.420084
$205$	−8.00000	−0.558744
$206$	−28.0000	−1.95085
$207$	4.00000	0.278019
$208$	24.0000	1.66410
$209$	3.00000	0.207514
$210$	−6.00000	−0.414039
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	−12.0000	−0.824163
$213$	12.0000	0.822226
$214$	4.00000	0.273434
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	6.00000	0.407307
$218$	−40.0000	−2.70914
$219$	−11.0000	−0.743311
$220$	−6.00000	−0.404520
$221$	−18.0000	−1.21081
$222$	−16.0000	−1.07385
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	24.0000	1.60357
$225$	−4.00000	−0.266667
$226$	12.0000	0.798228
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−2.00000	−0.132453
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	−8.00000	−0.527504
$231$	−9.00000	−0.592157
$232$	0	0
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	12.0000	0.784465
$235$	3.00000	0.195698
$236$	0	0
$237$	0	0
$238$	−18.0000	−1.16677
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−4.00000	−0.258199
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	4.00000	0.257130
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	2.00000	0.127775
$246$	16.0000	1.02012
$247$	6.00000	0.381771
$248$	0	0
$249$	4.00000	0.253490
$250$	18.0000	1.13842
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	6.00000	0.377964
$253$	−12.0000	−0.754434
$254$	4.00000	0.250982
$255$	3.00000	0.187867
$256$	16.0000	1.00000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	2.00000	0.124515
$259$	24.0000	1.49129
$260$	−12.0000	−0.744208
$261$	−10.0000	−0.618984
$262$	26.0000	1.60629
$263$	−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
$265$	−6.00000	−0.368577
$266$	6.00000	0.367884
$267$	10.0000	0.611990
$268$	16.0000	0.977356
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	−2.00000	−0.121716
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−12.0000	−0.727607
$273$	−18.0000	−1.08941
$274$	−6.00000	−0.362473
$275$	12.0000	0.723627
$276$	8.00000	0.481543
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	10.0000	0.599760
$279$	2.00000	0.119737
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−6.00000	−0.357295
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	24.0000	1.42414
$285$	−1.00000	−0.0592349
$286$	−36.0000	−2.12872
$287$	−24.0000	−1.41668
$288$	8.00000	0.471405
$289$	−8.00000	−0.470588
$290$	20.0000	1.17444
$291$	−2.00000	−0.117242
$292$	−22.0000	−1.28745
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	−4.00000	−0.233285
$295$	0	0
$296$	0	0
$297$	−3.00000	−0.174078
$298$	−30.0000	−1.73785
$299$	−24.0000	−1.38796
$300$	−8.00000	−0.461880
$301$	−3.00000	−0.172917
$302$	16.0000	0.920697
$303$	2.00000	0.114897
$304$	4.00000	0.229416
$305$	7.00000	0.400819
$306$	−6.00000	−0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−18.0000	−1.02565
$309$	14.0000	0.796432
$310$	−4.00000	−0.227185
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	4.00000	0.225733
$315$	3.00000	0.169031
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	12.0000	0.672927
$319$	30.0000	1.67968
$320$	−8.00000	−0.447214
$321$	−2.00000	−0.111629
$322$	−24.0000	−1.33747
$323$	−3.00000	−0.166924
$324$	2.00000	0.111111
$325$	24.0000	1.33128
$326$	32.0000	1.77232
$327$	20.0000	1.10600
$328$	0	0
$329$	9.00000	0.496186
$330$	6.00000	0.330289
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	8.00000	0.439057
$333$	8.00000	0.438397
$334$	−36.0000	−1.96983
$335$	8.00000	0.437087
$336$	−12.0000	−0.654654
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−46.0000	−2.50207
$339$	−6.00000	−0.325875
$340$	6.00000	0.325396
$341$	−6.00000	−0.324918
$342$	2.00000	0.108148
$343$	−15.0000	−0.809924
$344$	0	0
$345$	4.00000	0.215353
$346$	−28.0000	−1.50529
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	−20.0000	−1.07211
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	24.0000	1.28285
$351$	−6.00000	−0.320256
$352$	−24.0000	−1.27920
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	20.0000	1.06000
$357$	9.00000	0.476331
$358$	20.0000	1.05703
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.00000	−0.210235
$363$	−2.00000	−0.104973
$364$	−36.0000	−1.88691
$365$	−11.0000	−0.575766
$366$	−14.0000	−0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−16.0000	−0.834058
$369$	−8.00000	−0.416463
$370$	−16.0000	−0.831800
$371$	−18.0000	−0.934513
$372$	4.00000	0.207390
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	18.0000	0.930758
$375$	−9.00000	−0.464758
$376$	0	0
$377$	60.0000	3.09016
$378$	−6.00000	−0.308607
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−2.00000	−0.102598
$381$	−2.00000	−0.102463
$382$	6.00000	0.306987
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	−8.00000	−0.407189
$387$	−1.00000	−0.0508329
$388$	−4.00000	−0.203069
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	12.0000	0.607644
$391$	12.0000	0.606866
$392$	0	0
$393$	−13.0000	−0.655763
$394$	4.00000	0.201517
$395$	0	0
$396$	−6.00000	−0.301511
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	10.0000	0.501255
$399$	−3.00000	−0.150188
$400$	16.0000	0.800000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−16.0000	−0.798007
$403$	−12.0000	−0.597763
$404$	4.00000	0.199007
$405$	1.00000	0.0496904
$406$	60.0000	2.97775
$407$	−24.0000	−1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	16.0000	0.790184
$411$	3.00000	0.147979
$412$	28.0000	1.37946
$413$	0	0
$414$	−8.00000	−0.393179
$415$	4.00000	0.196352
$416$	−48.0000	−2.35339
$417$	−5.00000	−0.244851
$418$	−6.00000	−0.293470
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	6.00000	0.292770
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	56.0000	2.72604
$423$	3.00000	0.145865
$424$	0	0
$425$	−12.0000	−0.582086
$426$	−24.0000	−1.16280
$427$	21.0000	1.01626
$428$	−4.00000	−0.193347
$429$	18.0000	0.869048
$430$	2.00000	0.0964486
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−4.00000	−0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−12.0000	−0.576018
$435$	−10.0000	−0.479463
$436$	40.0000	1.91565
$437$	−4.00000	−0.191346
$438$	22.0000	1.05120
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	36.0000	1.71235
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	16.0000	0.759326
$445$	10.0000	0.474045
$446$	−8.00000	−0.378811
$447$	15.0000	0.709476
$448$	−24.0000	−1.13389
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	8.00000	0.377124
$451$	24.0000	1.13012
$452$	−12.0000	−0.564433
$453$	−8.00000	−0.375873
$454$	−36.0000	−1.68956
$455$	−18.0000	−0.843853
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	30.0000	1.40181
$459$	3.00000	0.140028
$460$	8.00000	0.373002
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	18.0000	0.837436
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	40.0000	1.85695
$465$	2.00000	0.0927478
$466$	22.0000	1.01913
$467$	−17.0000	−0.786666	−0.393333	−	0.919396i	$-0.628678\pi$
$467$	−17.0000	−0.786666	−0.393333	+	0.919396i	$0.628678\pi$
$468$	−12.0000	−0.554700
$469$	24.0000	1.10822
$470$	−6.00000	−0.276759
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	4.00000	0.183533
$476$	18.0000	0.825029
$477$	−6.00000	−0.274721
$478$	30.0000	1.37217
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	8.00000	0.365148
$481$	−48.0000	−2.18861
$482$	−24.0000	−1.09317
$483$	12.0000	0.546019
$484$	−4.00000	−0.181818
$485$	−2.00000	−0.0908153
$486$	−2.00000	−0.0907218
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	−4.00000	−0.180702
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	−16.0000	−0.721336
$493$	−30.0000	−1.35113
$494$	−12.0000	−0.539906
$495$	−3.00000	−0.134840
$496$	−8.00000	−0.359211
$497$	36.0000	1.61482
$498$	−8.00000	−0.358489
$499$	−35.0000	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$500$	−18.0000	−0.804984
$501$	18.0000	0.804181
$502$	−54.0000	−2.41014
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	24.0000	1.06693
$507$	23.0000	1.02147
$508$	−4.00000	−0.177471
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−6.00000	−0.265684
$511$	−33.0000	−1.45983
$512$	−32.0000	−1.41421
$513$	−1.00000	−0.0441511
$514$	−16.0000	−0.705730
$515$	14.0000	0.616914
$516$	−2.00000	−0.0880451
$517$	−9.00000	−0.395820
$518$	−48.0000	−2.10900
$519$	14.0000	0.614532
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	20.0000	0.875376
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−26.0000	−1.13582
$525$	−12.0000	−0.523723
$526$	42.0000	1.83129
$527$	6.00000	0.261364
$528$	12.0000	0.522233
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	0	0
$532$	−6.00000	−0.260133
$533$	48.0000	2.07911
$534$	−20.0000	−0.865485
$535$	−2.00000	−0.0864675
$536$	0	0
$537$	−10.0000	−0.431532
$538$	60.0000	2.58678
$539$	−6.00000	−0.258438
$540$	2.00000	0.0860663
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	−24.0000	−1.03089
$543$	2.00000	0.0858282
$544$	24.0000	1.02899
$545$	20.0000	0.856706
$546$	36.0000	1.54066
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	6.00000	0.256307
$549$	7.00000	0.298753
$550$	−24.0000	−1.02336
$551$	10.0000	0.426014
$552$	0	0
$553$	0	0
$554$	−26.0000	−1.10463
$555$	8.00000	0.339581
$556$	−10.0000	−0.424094
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−4.00000	−0.169334
$559$	6.00000	0.253773
$560$	−12.0000	−0.507093
$561$	−9.00000	−0.379980
$562$	−4.00000	−0.168730
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	6.00000	0.252646
$565$	−6.00000	−0.252422
$566$	−38.0000	−1.59726
$567$	3.00000	0.125988
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	2.00000	0.0837708
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	36.0000	1.50524
$573$	−3.00000	−0.125327
$574$	48.0000	2.00348
$575$	−16.0000	−0.667246
$576$	−8.00000	−0.333333
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	16.0000	0.665512
$579$	4.00000	0.166234
$580$	−20.0000	−0.830455
$581$	12.0000	0.497844
$582$	4.00000	0.165805
$583$	18.0000	0.745484
$584$	0	0
$585$	−6.00000	−0.248069
$586$	−8.00000	−0.330477
$587$	−37.0000	−1.52715	−0.763577	−	0.645717i	$-0.776559\pi$
$587$	−37.0000	−1.52715	−0.763577	+	0.645717i	$0.776559\pi$
$588$	4.00000	0.164957
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	−32.0000	−1.31519
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	6.00000	0.246183
$595$	9.00000	0.368964
$596$	30.0000	1.22885
$597$	−5.00000	−0.204636
$598$	48.0000	1.96287
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	6.00000	0.244542
$603$	8.00000	0.325785
$604$	−16.0000	−0.651031
$605$	−2.00000	−0.0813116
$606$	−4.00000	−0.162489
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	−8.00000	−0.324443
$609$	−30.0000	−1.21566
$610$	−14.0000	−0.566843
$611$	−18.0000	−0.728202
$612$	6.00000	0.242536
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	24.0000	0.968561
$615$	−8.00000	−0.322591
$616$	0	0
$617$	23.0000	0.925945	0.462973	−	0.886373i	$-0.346783\pi$
$617$	23.0000	0.925945	0.462973	+	0.886373i	$0.346783\pi$
$618$	−28.0000	−1.12633
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	4.00000	0.160644
$621$	4.00000	0.160514
$622$	−14.0000	−0.561349
$623$	30.0000	1.20192
$624$	24.0000	0.960769
$625$	11.0000	0.440000
$626$	−28.0000	−1.11911
$627$	3.00000	0.119808
$628$	−4.00000	−0.159617
$629$	24.0000	0.956943
$630$	−6.00000	−0.239046
$631$	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
$633$	−28.0000	−1.11290
$634$	24.0000	0.953162
$635$	−2.00000	−0.0793676
$636$	−12.0000	−0.475831
$637$	−12.0000	−0.475457
$638$	−60.0000	−2.37542
$639$	12.0000	0.474713
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000	0.157867
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	24.0000	0.945732
$645$	−1.00000	−0.0393750
$646$	6.00000	0.236067
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	0	0
$650$	−48.0000	−1.88271
$651$	6.00000	0.235159
$652$	−32.0000	−1.25322
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	−40.0000	−1.56412
$655$	−13.0000	−0.507952
$656$	32.0000	1.24939
$657$	−11.0000	−0.429151
$658$	−18.0000	−0.701713
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	−6.00000	−0.233550
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−24.0000	−0.932786
$663$	−18.0000	−0.699062
$664$	0	0
$665$	−3.00000	−0.116335
$666$	−16.0000	−0.619987
$667$	−40.0000	−1.54881
$668$	36.0000	1.39288
$669$	4.00000	0.154649
$670$	−16.0000	−0.618134
$671$	−21.0000	−0.810696
$672$	24.0000	0.925820
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	44.0000	1.69482
$675$	−4.00000	−0.153960
$676$	46.0000	1.76923
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	12.0000	0.460857
$679$	−6.00000	−0.230259
$680$	0	0
$681$	18.0000	0.689761
$682$	12.0000	0.459504
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	−2.00000	−0.0764719
$685$	3.00000	0.114624
$686$	30.0000	1.14541
$687$	−15.0000	−0.572286
$688$	4.00000	0.152499
$689$	36.0000	1.37149
$690$	−8.00000	−0.304555
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	28.0000	1.06440
$693$	−9.00000	−0.341882
$694$	−6.00000	−0.227757
$695$	−5.00000	−0.189661
$696$	0	0
$697$	−24.0000	−0.909065
$698$	−50.0000	−1.89253
$699$	−11.0000	−0.416058
$700$	−24.0000	−0.907115
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	12.0000	0.452911
$703$	−8.00000	−0.301726
$704$	24.0000	0.904534
$705$	3.00000	0.112987
$706$	−28.0000	−1.05379
$707$	6.00000	0.225653
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	−18.0000	−0.673633
$715$	18.0000	0.673162
$716$	−20.0000	−0.747435
$717$	−15.0000	−0.560185
$718$	−50.0000	−1.86598
$719$	−35.0000	−1.30528	−0.652640	−	0.757668i	$-0.726339\pi$
$719$	−35.0000	−1.30528	−0.652640	+	0.757668i	$0.726339\pi$
$720$	−4.00000	−0.149071
$721$	42.0000	1.56416
$722$	−2.00000	−0.0744323
$723$	12.0000	0.446285
$724$	4.00000	0.148659
$725$	40.0000	1.48556
$726$	4.00000	0.148454
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	22.0000	0.814257
$731$	−3.00000	−0.110959
$732$	14.0000	0.517455
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−16.0000	−0.590571
$735$	2.00000	0.0737711
$736$	32.0000	1.17954
$737$	−24.0000	−0.884051
$738$	16.0000	0.588968
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	16.0000	0.588172
$741$	6.00000	0.220416
$742$	36.0000	1.32160
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	32.0000	1.17160
$747$	4.00000	0.146352
$748$	−18.0000	−0.658145
$749$	−6.00000	−0.219235
$750$	18.0000	0.657267
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−12.0000	−0.437595
$753$	27.0000	0.983935
$754$	−120.000	−4.37014
$755$	−8.00000	−0.291150
$756$	6.00000	0.218218
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	60.0000	2.17930
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	4.00000	0.144905
$763$	60.0000	2.17215
$764$	−6.00000	−0.217072
$765$	3.00000	0.108465
$766$	−28.0000	−1.01168
$767$	0	0
$768$	16.0000	0.577350
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	18.0000	0.648675
$771$	8.00000	0.288113
$772$	8.00000	0.287926
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	2.00000	0.0718885
$775$	−8.00000	−0.287368
$776$	0	0
$777$	24.0000	0.860995
$778$	30.0000	1.07555
$779$	8.00000	0.286630
$780$	−12.0000	−0.429669
$781$	−36.0000	−1.28818
$782$	−24.0000	−0.858238
$783$	−10.0000	−0.357371
$784$	−8.00000	−0.285714
$785$	−2.00000	−0.0713831
$786$	26.0000	0.927389
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−4.00000	−0.142494
$789$	−21.0000	−0.747620
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−42.0000	−1.49146
$794$	14.0000	0.496841
$795$	−6.00000	−0.212798
$796$	−10.0000	−0.354441
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	6.00000	0.212398
$799$	9.00000	0.318397
$800$	−32.0000	−1.13137
$801$	10.0000	0.353333
$802$	56.0000	1.97743
$803$	33.0000	1.16454
$804$	16.0000	0.564276
$805$	12.0000	0.422944
$806$	24.0000	0.845364
$807$	−30.0000	−1.05605
$808$	0	0
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	−2.00000	−0.0702728
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−60.0000	−2.10559
$813$	12.0000	0.420858
$814$	48.0000	1.68240
$815$	−16.0000	−0.560456
$816$	−12.0000	−0.420084
$817$	1.00000	0.0349856
$818$	−20.0000	−0.699284
$819$	−18.0000	−0.628971
$820$	−16.0000	−0.558744
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	−6.00000	−0.209274
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	8.00000	0.278019
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−8.00000	−0.277684
$831$	13.0000	0.450965
$832$	48.0000	1.66410
$833$	6.00000	0.207888
$834$	10.0000	0.346272
$835$	18.0000	0.622916
$836$	6.00000	0.207514
$837$	2.00000	0.0691301
$838$	−40.0000	−1.38178
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−4.00000	−0.137849
$843$	2.00000	0.0688837
$844$	−56.0000	−1.92760
$845$	23.0000	0.791224
$846$	−6.00000	−0.206284
$847$	−6.00000	−0.206162
$848$	24.0000	0.824163
$849$	19.0000	0.652078
$850$	24.0000	0.823193
$851$	32.0000	1.09695
$852$	24.0000	0.822226
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	−42.0000	−1.43721
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	−36.0000	−1.22902
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	−2.00000	−0.0681994
$861$	−24.0000	−0.817918
$862$	36.0000	1.22616
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	8.00000	0.272166
$865$	14.0000	0.476014
$866$	52.0000	1.76703
$867$	−8.00000	−0.271694
$868$	12.0000	0.407307
$869$	0	0
$870$	20.0000	0.678064
$871$	−48.0000	−1.62642
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	8.00000	0.270604
$875$	−27.0000	−0.912767
$876$	−22.0000	−0.743311
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−20.0000	−0.674967
$879$	4.00000	0.134917
$880$	12.0000	0.404520
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	−4.00000	−0.134687
$883$	−21.0000	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$883$	−21.0000	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	−78.0000	−2.62046
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	−20.0000	−0.670402
$891$	−3.00000	−0.100504
$892$	8.00000	0.267860
$893$	−3.00000	−0.100391
$894$	−30.0000	−1.00335
$895$	−10.0000	−0.334263
$896$	0	0
$897$	−24.0000	−0.801337
$898$	40.0000	1.33482
$899$	−20.0000	−0.667037
$900$	−8.00000	−0.266667
$901$	−18.0000	−0.599667
$902$	−48.0000	−1.59823
$903$	−3.00000	−0.0998337
$904$	0	0
$905$	2.00000	0.0664822
$906$	16.0000	0.531564
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	36.0000	1.19470
$909$	2.00000	0.0663358
$910$	36.0000	1.19339
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	4.00000	0.132453
$913$	−12.0000	−0.397142
$914$	−6.00000	−0.198462
$915$	7.00000	0.231413
$916$	−30.0000	−0.991228
$917$	−39.0000	−1.28789
$918$	−6.00000	−0.198030
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	66.0000	2.17359
$923$	−72.0000	−2.36991
$924$	−18.0000	−0.592157
$925$	−32.0000	−1.05215
$926$	62.0000	2.03745
$927$	14.0000	0.459820
$928$	−80.0000	−2.62613
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	−4.00000	−0.131165
$931$	−2.00000	−0.0655474
$932$	−22.0000	−0.720634
$933$	7.00000	0.229170
$934$	34.0000	1.11251
$935$	−9.00000	−0.294331
$936$	0	0
$937$	53.0000	1.73143	0.865717	−	0.500533i	$-0.166863\pi$
$937$	53.0000	1.73143	0.865717	+	0.500533i	$0.166863\pi$
$938$	−48.0000	−1.56726
$939$	14.0000	0.456873
$940$	6.00000	0.195698
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	4.00000	0.130327
$943$	−32.0000	−1.04206
$944$	0	0
$945$	3.00000	0.0975900
$946$	−6.00000	−0.195077
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	66.0000	2.14245
$950$	−8.00000	−0.259554
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	12.0000	0.388514
$955$	−3.00000	−0.0970777
$956$	−30.0000	−0.970269
$957$	30.0000	0.969762
$958$	80.0000	2.58468
$959$	9.00000	0.290625
$960$	−8.00000	−0.258199
$961$	−27.0000	−0.870968
$962$	96.0000	3.09516
$963$	−2.00000	−0.0644491
$964$	24.0000	0.772988
$965$	4.00000	0.128765
$966$	−24.0000	−0.772187
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	−3.00000	−0.0963739
$970$	4.00000	0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	2.00000	0.0641500
$973$	−15.0000	−0.480878
$974$	−16.0000	−0.512673
$975$	24.0000	0.768615
$976$	−28.0000	−0.896258
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	32.0000	1.02325
$979$	−30.0000	−0.958804
$980$	4.00000	0.127775
$981$	20.0000	0.638551
$982$	16.0000	0.510581
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	60.0000	1.91079
$987$	9.00000	0.286473
$988$	12.0000	0.381771
$989$	−4.00000	−0.127193
$990$	6.00000	0.190693
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	16.0000	0.508001
$993$	12.0000	0.380808
$994$	−72.0000	−2.28370
$995$	−5.00000	−0.158511
$996$	8.00000	0.253490
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	70.0000	2.21581
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.2.a.b.1.1	✓	1
3.2	odd	2		171.2.a.c.1.1		1
4.3	odd	2		912.2.a.d.1.1		1
5.2	odd	4		1425.2.c.a.799.1		2
5.3	odd	4		1425.2.c.a.799.2		2
5.4	even	2		1425.2.a.i.1.1		1
7.6	odd	2		2793.2.a.a.1.1		1
8.3	odd	2		3648.2.a.y.1.1		1
8.5	even	2		3648.2.a.h.1.1		1
11.10	odd	2		6897.2.a.g.1.1		1
12.11	even	2		2736.2.a.h.1.1		1
13.12	even	2		9633.2.a.p.1.1		1
15.14	odd	2		4275.2.a.a.1.1		1
19.18	odd	2		1083.2.a.d.1.1		1
21.20	even	2		8379.2.a.q.1.1		1
57.56	even	2		3249.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.b.1.1	✓	1	1.1	even	1	trivial
171.2.a.c.1.1		1	3.2	odd	2
912.2.a.d.1.1		1	4.3	odd	2
1083.2.a.d.1.1		1	19.18	odd	2
1425.2.a.i.1.1		1	5.4	even	2
1425.2.c.a.799.1		2	5.2	odd	4
1425.2.c.a.799.2		2	5.3	odd	4
2736.2.a.h.1.1		1	12.11	even	2
2793.2.a.a.1.1		1	7.6	odd	2
3249.2.a.a.1.1		1	57.56	even	2
3648.2.a.h.1.1		1	8.5	even	2
3648.2.a.y.1.1		1	8.3	odd	2
4275.2.a.a.1.1		1	15.14	odd	2
6897.2.a.g.1.1		1	11.10	odd	2
8379.2.a.q.1.1		1	21.20	even	2
9633.2.a.p.1.1		1	13.12	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:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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