LMFDB - Embedded newform 4013.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.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:	$32.0439663311$
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	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} -4.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} -4.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +8.00000 q^{10} +3.00000 q^{11} -4.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -8.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} +8.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} +11.0000 q^{25} +4.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +8.00000 q^{29} -16.0000 q^{30} -5.00000 q^{31} -8.00000 q^{32} -6.00000 q^{33} -4.00000 q^{34} +4.00000 q^{35} +2.00000 q^{36} -4.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +5.00000 q^{41} -4.00000 q^{42} +5.00000 q^{43} +6.00000 q^{44} +4.00000 q^{45} +12.0000 q^{47} +8.00000 q^{48} -6.00000 q^{49} +22.0000 q^{50} +4.00000 q^{51} +4.00000 q^{52} +1.00000 q^{53} +8.00000 q^{54} +12.0000 q^{55} -2.00000 q^{57} +16.0000 q^{58} +7.00000 q^{59} -16.0000 q^{60} -7.00000 q^{61} -10.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} +8.00000 q^{65} -12.0000 q^{66} -7.00000 q^{67} -4.00000 q^{68} +8.00000 q^{70} +4.00000 q^{71} -13.0000 q^{73} -8.00000 q^{74} -22.0000 q^{75} +2.00000 q^{76} +3.00000 q^{77} -8.00000 q^{78} +4.00000 q^{79} -16.0000 q^{80} -11.0000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -8.00000 q^{85} +10.0000 q^{86} -16.0000 q^{87} -10.0000 q^{89} +8.00000 q^{90} +2.00000 q^{91} +10.0000 q^{93} +24.0000 q^{94} +4.00000 q^{95} +16.0000 q^{96} -17.0000 q^{97} -12.0000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	2.00000	1.00000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−4.00000	−1.63299
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	8.00000	2.52982
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−4.00000	−1.15470
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	2.00000	0.534522
$15$	−8.00000	−2.06559
$16$	−4.00000	−1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	2.00000	0.471405
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	8.00000	1.78885
$21$	−2.00000	−0.436436
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	2.00000	0.377964
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	−16.0000	−2.92119
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−8.00000	−1.41421
$33$	−6.00000	−1.04447
$34$	−4.00000	−0.685994
$35$	4.00000	0.676123
$36$	2.00000	0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	−4.00000	−0.617213
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	6.00000	0.904534
$45$	4.00000	0.596285
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	8.00000	1.15470
$49$	−6.00000	−0.857143
$50$	22.0000	3.11127
$51$	4.00000	0.560112
$52$	4.00000	0.554700
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	8.00000	1.08866
$55$	12.0000	1.61808
$56$	0	0
$57$	−2.00000	−0.264906
$58$	16.0000	2.10090
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	−16.0000	−2.06559
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−10.0000	−1.27000
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	8.00000	0.992278
$66$	−12.0000	−1.47710
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	8.00000	0.956183
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	−8.00000	−0.929981
$75$	−22.0000	−2.54034
$76$	2.00000	0.229416
$77$	3.00000	0.341882
$78$	−8.00000	−0.905822
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−16.0000	−1.78885
$81$	−11.0000	−1.22222
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000	−0.436436
$85$	−8.00000	−0.867722
$86$	10.0000	1.07833
$87$	−16.0000	−1.71538
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	8.00000	0.843274
$91$	2.00000	0.209657
$92$	0	0
$93$	10.0000	1.03695
$94$	24.0000	2.47541
$95$	4.00000	0.410391
$96$	16.0000	1.63299
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−12.0000	−1.21218
$99$	3.00000	0.301511
$100$	22.0000	2.20000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	8.00000	0.792118
$103$	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
$105$	−8.00000	−0.780720
$106$	2.00000	0.194257
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	8.00000	0.769800
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	24.0000	2.28831
$111$	8.00000	0.759326
$112$	−4.00000	−0.377964
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	16.0000	1.48556
$117$	2.00000	0.184900
$118$	14.0000	1.28880
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−14.0000	−1.26750
$123$	−10.0000	−0.901670
$124$	−10.0000	−0.898027
$125$	24.0000	2.14663
$126$	2.00000	0.178174
$127$	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
$129$	−10.0000	−0.880451
$130$	16.0000	1.40329
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−12.0000	−1.04447
$133$	1.00000	0.0867110
$134$	−14.0000	−1.20942
$135$	16.0000	1.37706
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	8.00000	0.676123
$141$	−24.0000	−2.02116
$142$	8.00000	0.671345
$143$	6.00000	0.501745
$144$	−4.00000	−0.333333
$145$	32.0000	2.65746
$146$	−26.0000	−2.15178
$147$	12.0000	0.989743
$148$	−8.00000	−0.657596
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−44.0000	−3.59258
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	6.00000	0.483494
$155$	−20.0000	−1.60644
$156$	−8.00000	−0.640513
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	8.00000	0.636446
$159$	−2.00000	−0.158610
$160$	−32.0000	−2.52982
$161$	0	0
$162$	−22.0000	−1.72848
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	10.0000	0.780869
$165$	−24.0000	−1.86840
$166$	24.0000	1.86276
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−16.0000	−1.22714
$171$	1.00000	0.0764719
$172$	10.0000	0.762493
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−32.0000	−2.42591
$175$	11.0000	0.831522
$176$	−12.0000	−0.904534
$177$	−14.0000	−1.05230
$178$	−20.0000	−1.49906
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	8.00000	0.596285
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	4.00000	0.296500
$183$	14.0000	1.03491
$184$	0	0
$185$	−16.0000	−1.17634
$186$	20.0000	1.46647
$187$	−6.00000	−0.438763
$188$	24.0000	1.75038
$189$	4.00000	0.290957
$190$	8.00000	0.580381
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	16.0000	1.15470
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−34.0000	−2.44106
$195$	−16.0000	−1.14578
$196$	−12.0000	−0.857143
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	6.00000	0.426401
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	24.0000	1.68863
$203$	8.00000	0.561490
$204$	8.00000	0.560112
$205$	20.0000	1.39686
$206$	28.0000	1.95085
$207$	0	0
$208$	−8.00000	−0.554700
$209$	3.00000	0.207514
$210$	−16.0000	−1.10410
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	2.00000	0.137361
$213$	−8.00000	−0.548151
$214$	−16.0000	−1.09374
$215$	20.0000	1.36399
$216$	0	0
$217$	−5.00000	−0.339422
$218$	22.0000	1.49003
$219$	26.0000	1.75692
$220$	24.0000	1.61808
$221$	−4.00000	−0.269069
$222$	16.0000	1.07385
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−8.00000	−0.534522
$225$	11.0000	0.733333
$226$	8.00000	0.532152
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	4.00000	0.261488
$235$	48.0000	3.13117
$236$	14.0000	0.911322
$237$	−8.00000	−0.519656
$238$	−4.00000	−0.259281
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	32.0000	2.06559
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	−4.00000	−0.257130
$243$	10.0000	0.641500
$244$	−14.0000	−0.896258
$245$	−24.0000	−1.53330
$246$	−20.0000	−1.27515
$247$	2.00000	0.127257
$248$	0	0
$249$	−24.0000	−1.52094
$250$	48.0000	3.03579
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	14.0000	0.878438
$255$	16.0000	1.00196
$256$	16.0000	1.00000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	−20.0000	−1.24515
$259$	−4.00000	−0.248548
$260$	16.0000	0.992278
$261$	8.00000	0.495188
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	2.00000	0.122628
$267$	20.0000	1.22398
$268$	−14.0000	−0.855186
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	32.0000	1.94746
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	8.00000	0.485071
$273$	−4.00000	−0.242091
$274$	36.0000	2.17484
$275$	33.0000	1.98997
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	−8.00000	−0.479808
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	−48.0000	−2.85836
$283$	−3.00000	−0.178331	−0.0891657	−	0.996017i	$-0.528420\pi$
$283$	−3.00000	−0.178331	−0.0891657	+	0.996017i	$0.528420\pi$
$284$	8.00000	0.474713
$285$	−8.00000	−0.473879
$286$	12.0000	0.709575
$287$	5.00000	0.295141
$288$	−8.00000	−0.471405
$289$	−13.0000	−0.764706
$290$	64.0000	3.75821
$291$	34.0000	1.99312
$292$	−26.0000	−1.52153
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	24.0000	1.39971
$295$	28.0000	1.63022
$296$	0	0
$297$	12.0000	0.696311
$298$	−20.0000	−1.15857
$299$	0	0
$300$	−44.0000	−2.54034
$301$	5.00000	0.288195
$302$	12.0000	0.690522
$303$	−24.0000	−1.37876
$304$	−4.00000	−0.229416
$305$	−28.0000	−1.60328
$306$	−4.00000	−0.228665
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	6.00000	0.341882
$309$	−28.0000	−1.59286
$310$	−40.0000	−2.27185
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	16.0000	0.902932
$315$	4.00000	0.225374
$316$	8.00000	0.450035
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−4.00000	−0.224309
$319$	24.0000	1.34374
$320$	−32.0000	−1.78885
$321$	16.0000	0.893033
$322$	0	0
$323$	−2.00000	−0.111283
$324$	−22.0000	−1.22222
$325$	22.0000	1.22034
$326$	−32.0000	−1.77232
$327$	−22.0000	−1.21660
$328$	0	0
$329$	12.0000	0.661581
$330$	−48.0000	−2.64231
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	24.0000	1.31717
$333$	−4.00000	−0.219199
$334$	28.0000	1.53209
$335$	−28.0000	−1.52980
$336$	8.00000	0.436436
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−18.0000	−0.979071
$339$	−8.00000	−0.434500
$340$	−16.0000	−0.867722
$341$	−15.0000	−0.812296
$342$	2.00000	0.108148
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	−28.0000	−1.50529
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	−32.0000	−1.71538
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	22.0000	1.17595
$351$	8.00000	0.427008
$352$	−24.0000	−1.27920
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	−28.0000	−1.48818
$355$	16.0000	0.849192
$356$	−20.0000	−1.06000
$357$	4.00000	0.211702
$358$	−8.00000	−0.422813
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−40.0000	−2.10235
$363$	4.00000	0.209946
$364$	4.00000	0.209657
$365$	−52.0000	−2.72180
$366$	28.0000	1.46358
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	0	0
$369$	5.00000	0.260290
$370$	−32.0000	−1.66360
$371$	1.00000	0.0519174
$372$	20.0000	1.03695
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	−12.0000	−0.620505
$375$	−48.0000	−2.47871
$376$	0	0
$377$	16.0000	0.824042
$378$	8.00000	0.411476
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	8.00000	0.410391
$381$	−14.0000	−0.717242
$382$	42.0000	2.14891
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	44.0000	2.23954
$387$	5.00000	0.254164
$388$	−34.0000	−1.72609
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−32.0000	−1.62038
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−20.0000	−1.00759
$395$	16.0000	0.805047
$396$	6.00000	0.301511
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	8.00000	0.401004
$399$	−2.00000	−0.100125
$400$	−44.0000	−2.20000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	28.0000	1.39651
$403$	−10.0000	−0.498135
$404$	24.0000	1.19404
$405$	−44.0000	−2.18638
$406$	16.0000	0.794067
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−29.0000	−1.43396	−0.716979	−	0.697095i	$-0.754476\pi$
$409$	−29.0000	−1.43396	−0.716979	+	0.697095i	$0.754476\pi$
$410$	40.0000	1.97546
$411$	−36.0000	−1.77575
$412$	28.0000	1.37946
$413$	7.00000	0.344447
$414$	0	0
$415$	48.0000	2.35623
$416$	−16.0000	−0.784465
$417$	8.00000	0.391762
$418$	6.00000	0.293470
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−16.0000	−0.780720
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	−2.00000	−0.0973585
$423$	12.0000	0.583460
$424$	0	0
$425$	−22.0000	−1.06716
$426$	−16.0000	−0.775203
$427$	−7.00000	−0.338754
$428$	−16.0000	−0.773389
$429$	−12.0000	−0.579365
$430$	40.0000	1.92897
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	−16.0000	−0.769800
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	−10.0000	−0.480015
$435$	−64.0000	−3.06857
$436$	22.0000	1.05361
$437$	0	0
$438$	52.0000	2.48466
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−8.00000	−0.380521
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	16.0000	0.759326
$445$	−40.0000	−1.89618
$446$	12.0000	0.568216
$447$	20.0000	0.945968
$448$	−8.00000	−0.377964
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	22.0000	1.03709
$451$	15.0000	0.706322
$452$	8.00000	0.376288
$453$	−12.0000	−0.563809
$454$	16.0000	0.750917
$455$	8.00000	0.375046
$456$	0	0
$457$	−39.0000	−1.82434	−0.912172	−	0.409809i	$-0.865595\pi$
$457$	−39.0000	−1.82434	−0.912172	+	0.409809i	$0.865595\pi$
$458$	28.0000	1.30835
$459$	−8.00000	−0.373408
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−12.0000	−0.558291
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	−32.0000	−1.48556
$465$	40.0000	1.85496
$466$	2.00000	0.0926482
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	4.00000	0.184900
$469$	−7.00000	−0.323230
$470$	96.0000	4.42815
$471$	−16.0000	−0.737241
$472$	0	0
$473$	15.0000	0.689701
$474$	−16.0000	−0.734904
$475$	11.0000	0.504715
$476$	−4.00000	−0.183340
$477$	1.00000	0.0457869
$478$	4.00000	0.182956
$479$	−37.0000	−1.69057	−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000	−1.69057	−0.845287	+	0.534313i	$0.820570\pi$
$480$	64.0000	2.92119
$481$	−8.00000	−0.364769
$482$	32.0000	1.45756
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−68.0000	−3.08772
$486$	20.0000	0.907218
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	32.0000	1.44709
$490$	−48.0000	−2.16842
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−20.0000	−0.901670
$493$	−16.0000	−0.720604
$494$	4.00000	0.179969
$495$	12.0000	0.539360
$496$	20.0000	0.898027
$497$	4.00000	0.179425
$498$	−48.0000	−2.15093
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	48.0000	2.14663
$501$	−28.0000	−1.25095
$502$	−4.00000	−0.178529
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	48.0000	2.13597
$506$	0	0
$507$	18.0000	0.799408
$508$	14.0000	0.621150
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	32.0000	1.41698
$511$	−13.0000	−0.575086
$512$	32.0000	1.41421
$513$	4.00000	0.176604
$514$	−30.0000	−1.32324
$515$	56.0000	2.46765
$516$	−20.0000	−0.880451
$517$	36.0000	1.58328
$518$	−8.00000	−0.351500
$519$	28.0000	1.22906
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	16.0000	0.700301
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−22.0000	−0.960159
$526$	−18.0000	−0.784837
$527$	10.0000	0.435607
$528$	24.0000	1.04447
$529$	−23.0000	−1.00000
$530$	8.00000	0.347498
$531$	7.00000	0.303774
$532$	2.00000	0.0867110
$533$	10.0000	0.433148
$534$	40.0000	1.73097
$535$	−32.0000	−1.38348
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	−18.0000	−0.775315
$540$	32.0000	1.37706
$541$	19.0000	0.816874	0.408437	−	0.912787i	$-0.366074\pi$
$541$	19.0000	0.816874	0.408437	+	0.912787i	$0.366074\pi$
$542$	−38.0000	−1.63224
$543$	40.0000	1.71656
$544$	16.0000	0.685994
$545$	44.0000	1.88475
$546$	−8.00000	−0.342368
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	36.0000	1.53784
$549$	−7.00000	−0.298753
$550$	66.0000	2.81425
$551$	8.00000	0.340811
$552$	0	0
$553$	4.00000	0.170097
$554$	−40.0000	−1.69944
$555$	32.0000	1.35832
$556$	−8.00000	−0.339276
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−10.0000	−0.423334
$559$	10.0000	0.422955
$560$	−16.0000	−0.676123
$561$	12.0000	0.506640
$562$	−34.0000	−1.43420
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−48.0000	−2.02116
$565$	16.0000	0.673125
$566$	−6.00000	−0.252199
$567$	−11.0000	−0.461957
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	−16.0000	−0.670166
$571$	−29.0000	−1.21361	−0.606806	−	0.794850i	$-0.707550\pi$
$571$	−29.0000	−1.21361	−0.606806	+	0.794850i	$0.707550\pi$
$572$	12.0000	0.501745
$573$	−42.0000	−1.75458
$574$	10.0000	0.417392
$575$	0	0
$576$	−8.00000	−0.333333
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−26.0000	−1.08146
$579$	−44.0000	−1.82858
$580$	64.0000	2.65746
$581$	12.0000	0.497844
$582$	68.0000	2.81869
$583$	3.00000	0.124247
$584$	0	0
$585$	8.00000	0.330759
$586$	4.00000	0.165238
$587$	23.0000	0.949312	0.474656	−	0.880172i	$-0.342573\pi$
$587$	23.0000	0.949312	0.474656	+	0.880172i	$0.342573\pi$
$588$	24.0000	0.989743
$589$	−5.00000	−0.206021
$590$	56.0000	2.30548
$591$	20.0000	0.822690
$592$	16.0000	0.657596
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	24.0000	0.984732
$595$	−8.00000	−0.327968
$596$	−20.0000	−0.819232
$597$	−8.00000	−0.327418
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	10.0000	0.407570
$603$	−7.00000	−0.285062
$604$	12.0000	0.488273
$605$	−8.00000	−0.325246
$606$	−48.0000	−1.94987
$607$	−31.0000	−1.25825	−0.629126	−	0.777304i	$-0.716587\pi$
$607$	−31.0000	−1.25825	−0.629126	+	0.777304i	$0.716587\pi$
$608$	−8.00000	−0.324443
$609$	−16.0000	−0.648353
$610$	−56.0000	−2.26737
$611$	24.0000	0.970936
$612$	−4.00000	−0.161690
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	−52.0000	−2.09855
$615$	−40.0000	−1.61296
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−56.0000	−2.25265
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−40.0000	−1.60644
$621$	0	0
$622$	32.0000	1.28308
$623$	−10.0000	−0.400642
$624$	16.0000	0.640513
$625$	41.0000	1.64000
$626$	−20.0000	−0.799361
$627$	−6.00000	−0.239617
$628$	16.0000	0.638470
$629$	8.00000	0.318981
$630$	8.00000	0.318728
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	24.0000	0.953162
$635$	28.0000	1.11115
$636$	−4.00000	−0.158610
$637$	−12.0000	−0.475457
$638$	48.0000	1.90034
$639$	4.00000	0.158238
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	32.0000	1.26294
$643$	38.0000	1.49857	0.749287	−	0.662246i	$-0.230396\pi$
$643$	38.0000	1.49857	0.749287	+	0.662246i	$0.230396\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	−4.00000	−0.157378
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	0	0
$649$	21.0000	0.824322
$650$	44.0000	1.72582
$651$	10.0000	0.391931
$652$	−32.0000	−1.25322
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−44.0000	−1.72054
$655$	0	0
$656$	−20.0000	−0.780869
$657$	−13.0000	−0.507178
$658$	24.0000	0.935617
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	−48.0000	−1.86840
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	−16.0000	−0.621858
$663$	8.00000	0.310694
$664$	0	0
$665$	4.00000	0.155113
$666$	−8.00000	−0.309994
$667$	0	0
$668$	28.0000	1.08335
$669$	−12.0000	−0.463947
$670$	−56.0000	−2.16347
$671$	−21.0000	−0.810696
$672$	16.0000	0.617213
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	4.00000	0.154074
$675$	44.0000	1.69356
$676$	−18.0000	−0.692308
$677$	−29.0000	−1.11456	−0.557280	−	0.830324i	$-0.688155\pi$
$677$	−29.0000	−1.11456	−0.557280	+	0.830324i	$0.688155\pi$
$678$	−16.0000	−0.614476
$679$	−17.0000	−0.652400
$680$	0	0
$681$	−16.0000	−0.613121
$682$	−30.0000	−1.14876
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	2.00000	0.0764719
$685$	72.0000	2.75098
$686$	−26.0000	−0.992685
$687$	−28.0000	−1.06827
$688$	−20.0000	−0.762493
$689$	2.00000	0.0761939
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−28.0000	−1.06440
$693$	3.00000	0.113961
$694$	18.0000	0.683271
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−10.0000	−0.378777
$698$	−44.0000	−1.66542
$699$	−2.00000	−0.0756469
$700$	22.0000	0.831522
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	16.0000	0.603881
$703$	−4.00000	−0.150863
$704$	−24.0000	−0.904534
$705$	−96.0000	−3.61557
$706$	42.0000	1.58069
$707$	12.0000	0.451306
$708$	−28.0000	−1.05230
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	32.0000	1.20094
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	8.00000	0.299392
$715$	24.0000	0.897549
$716$	−8.00000	−0.298974
$717$	−4.00000	−0.149383
$718$	30.0000	1.11959
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	−16.0000	−0.596285
$721$	14.0000	0.521387
$722$	−36.0000	−1.33978
$723$	−32.0000	−1.19009
$724$	−40.0000	−1.48659
$725$	88.0000	3.26824
$726$	8.00000	0.296908
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−104.000	−3.84921
$731$	−10.0000	−0.369863
$732$	28.0000	1.03491
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−42.0000	−1.55025
$735$	48.0000	1.77051
$736$	0	0
$737$	−21.0000	−0.773545
$738$	10.0000	0.368105
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−32.0000	−1.17634
$741$	−4.00000	−0.146944
$742$	2.00000	0.0734223
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−40.0000	−1.46549
$746$	38.0000	1.39128
$747$	12.0000	0.439057
$748$	−12.0000	−0.438763
$749$	−8.00000	−0.292314
$750$	−96.0000	−3.50542
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	−48.0000	−1.75038
$753$	4.00000	0.145768
$754$	32.0000	1.16537
$755$	24.0000	0.873449
$756$	8.00000	0.290957
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	−40.0000	−1.45287
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−28.0000	−1.01433
$763$	11.0000	0.398227
$764$	42.0000	1.51951
$765$	−8.00000	−0.289241
$766$	16.0000	0.578103
$767$	14.0000	0.505511
$768$	−32.0000	−1.15470
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	24.0000	0.864900
$771$	30.0000	1.08042
$772$	44.0000	1.58359
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	10.0000	0.359443
$775$	−55.0000	−1.97566
$776$	0	0
$777$	8.00000	0.286998
$778$	−12.0000	−0.430221
$779$	5.00000	0.179144
$780$	−32.0000	−1.14578
$781$	12.0000	0.429394
$782$	0	0
$783$	32.0000	1.14359
$784$	24.0000	0.857143
$785$	32.0000	1.14213
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−20.0000	−0.712470
$789$	18.0000	0.640817
$790$	32.0000	1.13851
$791$	4.00000	0.142224
$792$	0	0
$793$	−14.0000	−0.497155
$794$	36.0000	1.27759
$795$	−8.00000	−0.283731
$796$	8.00000	0.283552
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	−4.00000	−0.141598
$799$	−24.0000	−0.849059
$800$	−88.0000	−3.11127
$801$	−10.0000	−0.353333
$802$	64.0000	2.25992
$803$	−39.0000	−1.37628
$804$	28.0000	0.987484
$805$	0	0
$806$	−20.0000	−0.704470
$807$	0	0
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	−88.0000	−3.09200
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	16.0000	0.561490
$813$	38.0000	1.33272
$814$	−24.0000	−0.841200
$815$	−64.0000	−2.24182
$816$	−16.0000	−0.560112
$817$	5.00000	0.174928
$818$	−58.0000	−2.02792
$819$	2.00000	0.0698857
$820$	40.0000	1.39686
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	−72.0000	−2.51129
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	0	0
$825$	−66.0000	−2.29783
$826$	14.0000	0.487122
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	96.0000	3.33221
$831$	40.0000	1.38758
$832$	−16.0000	−0.554700
$833$	12.0000	0.415775
$834$	16.0000	0.554035
$835$	56.0000	1.93796
$836$	6.00000	0.207514
$837$	−20.0000	−0.691301
$838$	−60.0000	−2.07267
$839$	51.0000	1.76072	0.880358	−	0.474310i	$-0.157302\pi$
$839$	51.0000	1.76072	0.880358	+	0.474310i	$0.157302\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−32.0000	−1.10279
$843$	34.0000	1.17102
$844$	−2.00000	−0.0688428
$845$	−36.0000	−1.23844
$846$	24.0000	0.825137
$847$	−2.00000	−0.0687208
$848$	−4.00000	−0.137361
$849$	6.00000	0.205919
$850$	−44.0000	−1.50919
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	−14.0000	−0.479070
$855$	4.00000	0.136797
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	−24.0000	−0.819346
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	40.0000	1.36399
$861$	−10.0000	−0.340799
$862$	−4.00000	−0.136241
$863$	2.00000	0.0680808	0.0340404	−	0.999420i	$-0.489163\pi$
$863$	2.00000	0.0680808	0.0340404	+	0.999420i	$0.489163\pi$
$864$	−32.0000	−1.08866
$865$	−56.0000	−1.90406
$866$	20.0000	0.679628
$867$	26.0000	0.883006
$868$	−10.0000	−0.339422
$869$	12.0000	0.407072
$870$	−128.000	−4.33961
$871$	−14.0000	−0.474372
$872$	0	0
$873$	−17.0000	−0.575363
$874$	0	0
$875$	24.0000	0.811348
$876$	52.0000	1.75692
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−4.00000	−0.134993
$879$	−4.00000	−0.134917
$880$	−48.0000	−1.61808
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	−12.0000	−0.404061
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−8.00000	−0.269069
$885$	−56.0000	−1.88242
$886$	−48.0000	−1.61259
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	−80.0000	−2.68161
$891$	−33.0000	−1.10554
$892$	12.0000	0.401790
$893$	12.0000	0.401565
$894$	40.0000	1.33780
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	−40.0000	−1.33407
$900$	22.0000	0.733333
$901$	−2.00000	−0.0666297
$902$	30.0000	0.998891
$903$	−10.0000	−0.332779
$904$	0	0
$905$	−80.0000	−2.65929
$906$	−24.0000	−0.797347
$907$	−50.0000	−1.66022	−0.830111	−	0.557598i	$-0.811723\pi$
$907$	−50.0000	−1.66022	−0.830111	+	0.557598i	$0.811723\pi$
$908$	16.0000	0.530979
$909$	12.0000	0.398015
$910$	16.0000	0.530395
$911$	49.0000	1.62344	0.811721	−	0.584045i	$-0.198531\pi$
$911$	49.0000	1.62344	0.811721	+	0.584045i	$0.198531\pi$
$912$	8.00000	0.264906
$913$	36.0000	1.19143
$914$	−78.0000	−2.58001
$915$	56.0000	1.85130
$916$	28.0000	0.925146
$917$	0	0
$918$	−16.0000	−0.528079
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	52.0000	1.71346
$922$	12.0000	0.395199
$923$	8.00000	0.263323
$924$	−12.0000	−0.394771
$925$	−44.0000	−1.44671
$926$	52.0000	1.70883
$927$	14.0000	0.459820
$928$	−64.0000	−2.10090
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	80.0000	2.62330
$931$	−6.00000	−0.196642
$932$	2.00000	0.0655122
$933$	−32.0000	−1.04763
$934$	−66.0000	−2.15959
$935$	−24.0000	−0.784884
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	−14.0000	−0.457116
$939$	20.0000	0.652675
$940$	96.0000	3.13117
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	−32.0000	−1.04262
$943$	0	0
$944$	−28.0000	−0.911322
$945$	16.0000	0.520480
$946$	30.0000	0.975384
$947$	39.0000	1.26733	0.633665	−	0.773608i	$-0.281550\pi$
$947$	39.0000	1.26733	0.633665	+	0.773608i	$0.281550\pi$
$948$	−16.0000	−0.519656
$949$	−26.0000	−0.843996
$950$	22.0000	0.713774
$951$	−24.0000	−0.778253
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	2.00000	0.0647524
$955$	84.0000	2.71818
$956$	4.00000	0.129369
$957$	−48.0000	−1.55162
$958$	−74.0000	−2.39083
$959$	18.0000	0.581250
$960$	64.0000	2.06559
$961$	−6.00000	−0.193548
$962$	−16.0000	−0.515861
$963$	−8.00000	−0.257796
$964$	32.0000	1.03065
$965$	88.0000	2.83282
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	−136.000	−4.36670
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	20.0000	0.641500
$973$	−4.00000	−0.128234
$974$	−56.0000	−1.79436
$975$	−44.0000	−1.40913
$976$	28.0000	0.896258
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	64.0000	2.04649
$979$	−30.0000	−0.958804
$980$	−48.0000	−1.53330
$981$	11.0000	0.351203
$982$	30.0000	0.957338
$983$	19.0000	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$983$	19.0000	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$984$	0	0
$985$	−40.0000	−1.27451
$986$	−32.0000	−1.01909
$987$	−24.0000	−0.763928
$988$	4.00000	0.127257
$989$	0	0
$990$	24.0000	0.762770
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	40.0000	1.27000
$993$	16.0000	0.507745
$994$	8.00000	0.253745
$995$	16.0000	0.507234
$996$	−48.0000	−1.52094
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	40.0000	1.26618
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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