LMFDB - Embedded newform 1638.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +7.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} -1.00000 q^{35} +7.00000 q^{37} +7.00000 q^{38} +1.00000 q^{40} -8.00000 q^{41} +7.00000 q^{43} +1.00000 q^{44} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +3.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} +1.00000 q^{68} -1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{73} +7.00000 q^{74} +7.00000 q^{76} -1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{80} -8.00000 q^{82} +6.00000 q^{83} +1.00000 q^{85} +7.00000 q^{86} +1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} +7.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	7.00000	1.13555
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	7.00000	0.802955
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.00000	−0.883452
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	7.00000	0.754829
$87$	0	0
$88$	1.00000	0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−8.00000	−0.825137
$95$	7.00000	0.718185
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	10.0000	0.971286
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	3.00000	0.278543
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.0000	−0.909091
$122$	7.00000	0.633750
$123$	0	0
$124$	8.00000	0.718421
$125$	−9.00000	−0.804984
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	2.00000	0.172774
$135$	0	0
$136$	1.00000	0.0857493
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−4.00000	−0.335673
$143$	1.00000	0.0836242
$144$	0	0
$145$	3.00000	0.249136
$146$	−1.00000	−0.0827606
$147$	0	0
$148$	7.00000	0.575396
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	8.00000	0.642575
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	1.00000	0.0790569
$161$	3.00000	0.236433
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	6.00000	0.465690
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.00000	0.0766965
$171$	0	0
$172$	7.00000	0.533745
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	1.00000	0.0753778
$177$	0	0
$178$	−14.0000	−1.04934
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−3.00000	−0.221163
$185$	7.00000	0.514650
$186$	0	0
$187$	1.00000	0.0731272
$188$	−8.00000	−0.583460
$189$	0	0
$190$	7.00000	0.507833
$191$	−19.0000	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$191$	−19.0000	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−4.00000	−0.281439
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−5.00000	−0.348367
$207$	0	0
$208$	1.00000	0.0693375
$209$	7.00000	0.484200
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	18.0000	1.23045
$215$	7.00000	0.477396
$216$	0	0
$217$	−8.00000	−0.543075
$218$	−11.0000	−0.745014
$219$	0	0
$220$	1.00000	0.0674200
$221$	1.00000	0.0672673
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	7.00000	0.448129
$245$	1.00000	0.0638877
$246$	0	0
$247$	7.00000	0.445399
$248$	8.00000	0.508001
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	18.0000	1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	1.00000	0.0620174
$261$	0	0
$262$	5.00000	0.308901
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	−7.00000	−0.429198
$267$	0	0
$268$	2.00000	0.122169
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	19.0000	1.14783
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	1.00000	0.0591312
$287$	8.00000	0.472225
$288$	0	0
$289$	−16.0000	−0.941176
$290$	3.00000	0.176166
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	7.00000	0.406867
$297$	0	0
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−7.00000	−0.403473
$302$	−9.00000	−0.517892
$303$	0	0
$304$	7.00000	0.401478
$305$	7.00000	0.400819
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	8.00000	0.454369
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	2.00000	0.112509
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	1.00000	0.0559017
$321$	0	0
$322$	3.00000	0.167183
$323$	7.00000	0.389490
$324$	0	0
$325$	−4.00000	−0.221880
$326$	2.00000	0.110770
$327$	0	0
$328$	−8.00000	−0.441726
$329$	8.00000	0.441054
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−15.0000	−0.820763
$335$	2.00000	0.109272
$336$	0	0
$337$	−21.0000	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	1.00000	0.0542326
$341$	8.00000	0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.00000	0.377415
$345$	0	0
$346$	−14.0000	−0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	1.00000	0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−22.0000	−1.15629
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	−1.00000	−0.0523424
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	7.00000	0.363913
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−8.00000	−0.412568
$377$	3.00000	0.154508
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	7.00000	0.359092
$381$	0	0
$382$	−19.0000	−0.972125
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	16.0000	0.814379
$387$	0	0
$388$	−14.0000	−0.710742
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	1.00000	0.0505076
$393$	0	0
$394$	12.0000	0.604551
$395$	2.00000	0.100631
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	1.00000	0.0501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	−4.00000	−0.199007
$405$	0	0
$406$	−3.00000	−0.148888
$407$	7.00000	0.346977
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	−5.00000	−0.246332
$413$	4.00000	0.196827
$414$	0	0
$415$	6.00000	0.294528
$416$	1.00000	0.0490290
$417$	0	0
$418$	7.00000	0.342381
$419$	−1.00000	−0.0488532	−0.0244266	−	0.999702i	$-0.507776\pi$
$419$	−1.00000	−0.0488532	−0.0244266	+	0.999702i	$0.507776\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−3.00000	−0.146038
$423$	0	0
$424$	10.0000	0.485643
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−7.00000	−0.338754
$428$	18.0000	0.870063
$429$	0	0
$430$	7.00000	0.337570
$431$	38.0000	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$431$	38.0000	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$432$	0	0
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−11.0000	−0.526804
$437$	−21.0000	−1.00457
$438$	0	0
$439$	−17.0000	−0.811366	−0.405683	−	0.914014i	$-0.632966\pi$
$439$	−17.0000	−0.811366	−0.405683	+	0.914014i	$0.632966\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	1.00000	0.0475651
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	14.0000	0.662919
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	−6.00000	−0.282216
$453$	0	0
$454$	22.0000	1.03251
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	−3.00000	−0.139876
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−17.0000	−0.790057	−0.395029	−	0.918669i	$-0.629265\pi$
$463$	−17.0000	−0.790057	−0.395029	+	0.918669i	$0.629265\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	25.0000	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−4.00000	−0.184115
$473$	7.00000	0.321860
$474$	0	0
$475$	−28.0000	−1.28473
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−5.00000	−0.228456	−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000	−0.228456	−0.114228	+	0.993455i	$0.536439\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	10.0000	0.455488
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	7.00000	0.316875
$489$	0	0
$490$	1.00000	0.0451754
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	7.00000	0.314945
$495$	0	0
$496$	8.00000	0.359211
$497$	4.00000	0.179425
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	−7.00000	−0.312425
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	−3.00000	−0.133366
$507$	0	0
$508$	18.0000	0.798621
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−5.00000	−0.220326
$516$	0	0
$517$	−8.00000	−0.351840
$518$	−7.00000	−0.307562
$519$	0	0
$520$	1.00000	0.0438529
$521$	35.0000	1.53338	0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000	1.53338	0.766689	+	0.642019i	$0.221903\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	5.00000	0.218426
$525$	0	0
$526$	4.00000	0.174408
$527$	8.00000	0.348485
$528$	0	0
$529$	−14.0000	−0.608696
$530$	10.0000	0.434372
$531$	0	0
$532$	−7.00000	−0.303488
$533$	−8.00000	−0.346518
$534$	0	0
$535$	18.0000	0.778208
$536$	2.00000	0.0863868
$537$	0	0
$538$	2.00000	0.0862261
$539$	1.00000	0.0430730
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	1.00000	0.0428746
$545$	−11.0000	−0.471188
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	19.0000	0.811640
$549$	0	0
$550$	−4.00000	−0.170561
$551$	21.0000	0.894630
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	−14.0000	−0.594803
$555$	0	0
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	7.00000	0.296068
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	32.0000	1.34506
$567$	0	0
$568$	−4.00000	−0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	8.00000	0.333914
$575$	12.0000	0.500435
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	3.00000	0.124568
$581$	−6.00000	−0.248922
$582$	0	0
$583$	10.0000	0.414158
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−14.0000	−0.578335
$587$	38.0000	1.56843	0.784214	−	0.620491i	$-0.213066\pi$
$587$	38.0000	1.56843	0.784214	+	0.620491i	$0.213066\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	−4.00000	−0.164677
$591$	0	0
$592$	7.00000	0.287698
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	0	0
$597$	0	0
$598$	−3.00000	−0.122679
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−7.00000	−0.285299
$603$	0	0
$604$	−9.00000	−0.366205
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	7.00000	0.283887
$609$	0	0
$610$	7.00000	0.283422
$611$	−8.00000	−0.323645
$612$	0	0
$613$	37.0000	1.49442	0.747208	−	0.664590i	$-0.231394\pi$
$613$	37.0000	1.49442	0.747208	+	0.664590i	$0.231394\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	0	0
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−30.0000	−1.20289
$623$	14.0000	0.560898
$624$	0	0
$625$	11.0000	0.440000
$626$	−24.0000	−0.959233
$627$	0	0
$628$	−13.0000	−0.518756
$629$	7.00000	0.279108
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	−12.0000	−0.476581
$635$	18.0000	0.714308
$636$	0	0
$637$	1.00000	0.0396214
$638$	3.00000	0.118771
$639$	0	0
$640$	1.00000	0.0395285
$641$	−50.0000	−1.97488	−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000	−1.97488	−0.987441	+	0.157991i	$0.949498\pi$
$642$	0	0
$643$	17.0000	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$643$	17.0000	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	7.00000	0.275411
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	−4.00000	−0.156893
$651$	0	0
$652$	2.00000	0.0783260
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	0	0
$655$	5.00000	0.195366
$656$	−8.00000	−0.312348
$657$	0	0
$658$	8.00000	0.311872
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	6.00000	0.232845
$665$	−7.00000	−0.271448
$666$	0	0
$667$	−9.00000	−0.348481
$668$	−15.0000	−0.580367
$669$	0	0
$670$	2.00000	0.0772667
$671$	7.00000	0.270232
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−21.0000	−0.808890
$675$	0	0
$676$	1.00000	0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	1.00000	0.0383482
$681$	0	0
$682$	8.00000	0.306336
$683$	19.0000	0.727015	0.363507	−	0.931591i	$-0.381579\pi$
$683$	19.0000	0.727015	0.363507	+	0.931591i	$0.381579\pi$
$684$	0	0
$685$	19.0000	0.725953
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	7.00000	0.266872
$689$	10.0000	0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	−18.0000	−0.681310
$699$	0	0
$700$	4.00000	0.151186
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	49.0000	1.84807
$704$	1.00000	0.0376889
$705$	0	0
$706$	−14.0000	−0.526897
$707$	4.00000	0.150435
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	−14.0000	−0.524672
$713$	−24.0000	−0.898807
$714$	0	0
$715$	1.00000	0.0373979
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	30.0000	1.11648
$723$	0	0
$724$	−22.0000	−0.817624
$725$	−12.0000	−0.445669
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−1.00000	−0.0370117
$731$	7.00000	0.258904
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	2.00000	0.0736709
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	7.00000	0.257325
$741$	0	0
$742$	−10.0000	−0.367112
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	0	0
$746$	−12.0000	−0.439351
$747$	0	0
$748$	1.00000	0.0365636
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	3.00000	0.109254
$755$	−9.00000	−0.327544
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	7.00000	0.253917
$761$	−16.0000	−0.580000	−0.290000	−	0.957027i	$-0.593655\pi$
$761$	−16.0000	−0.580000	−0.290000	+	0.957027i	$0.593655\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	−19.0000	−0.687396
$765$	0	0
$766$	9.00000	0.325183
$767$	−4.00000	−0.144432
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	16.0000	0.575853
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	−32.0000	−1.14947
$776$	−14.0000	−0.502571
$777$	0	0
$778$	2.00000	0.0717035
$779$	−56.0000	−2.00641
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−3.00000	−0.107280
$783$	0	0
$784$	1.00000	0.0357143
$785$	−13.0000	−0.463990
$786$	0	0
$787$	55.0000	1.96054	0.980269	−	0.197667i	$-0.0633366\pi$
$787$	55.0000	1.96054	0.980269	+	0.197667i	$0.0633366\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	2.00000	0.0711568
$791$	6.00000	0.213335
$792$	0	0
$793$	7.00000	0.248577
$794$	18.0000	0.638796
$795$	0	0
$796$	1.00000	0.0354441
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−18.0000	−0.635602
$803$	−1.00000	−0.0352892
$804$	0	0
$805$	3.00000	0.105736
$806$	8.00000	0.281788
$807$	0	0
$808$	−4.00000	−0.140720
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−45.0000	−1.58016	−0.790082	−	0.613001i	$-0.789962\pi$
$811$	−45.0000	−1.58016	−0.790082	+	0.613001i	$0.789962\pi$
$812$	−3.00000	−0.105279
$813$	0	0
$814$	7.00000	0.245350
$815$	2.00000	0.0700569
$816$	0	0
$817$	49.0000	1.71429
$818$	5.00000	0.174821
$819$	0	0
$820$	−8.00000	−0.279372
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	−5.00000	−0.174183
$825$	0	0
$826$	4.00000	0.139178
$827$	41.0000	1.42571	0.712855	−	0.701312i	$-0.247402\pi$
$827$	41.0000	1.42571	0.712855	+	0.701312i	$0.247402\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	1.00000	0.0346688
$833$	1.00000	0.0346479
$834$	0	0
$835$	−15.0000	−0.519096
$836$	7.00000	0.242100
$837$	0	0
$838$	−1.00000	−0.0345444
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	34.0000	1.17172
$843$	0	0
$844$	−3.00000	−0.103264
$845$	1.00000	0.0344010
$846$	0	0
$847$	10.0000	0.343604
$848$	10.0000	0.343401
$849$	0	0
$850$	−4.00000	−0.137199
$851$	−21.0000	−0.719871
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−7.00000	−0.239535
$855$	0	0
$856$	18.0000	0.615227
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	7.00000	0.238698
$861$	0	0
$862$	38.0000	1.29429
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	12.0000	0.407777
$867$	0	0
$868$	−8.00000	−0.271538
$869$	2.00000	0.0678454
$870$	0	0
$871$	2.00000	0.0677674
$872$	−11.0000	−0.372507
$873$	0	0
$874$	−21.0000	−0.710336
$875$	9.00000	0.304256
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−17.0000	−0.573722
$879$	0	0
$880$	1.00000	0.0337100
$881$	49.0000	1.65085	0.825426	−	0.564510i	$-0.190935\pi$
$881$	49.0000	1.65085	0.825426	+	0.564510i	$0.190935\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	10.0000	0.335957
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	−14.0000	−0.469281
$891$	0	0
$892$	14.0000	0.468755
$893$	−56.0000	−1.87397
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	39.0000	1.30145
$899$	24.0000	0.800445
$900$	0	0
$901$	10.0000	0.333148
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−22.0000	−0.731305
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	22.0000	0.730096
$909$	0	0
$910$	−1.00000	−0.0331497
$911$	−13.0000	−0.430709	−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000	−0.430709	−0.215355	+	0.976536i	$0.569091\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−12.0000	−0.396925
$915$	0	0
$916$	10.0000	0.330409
$917$	−5.00000	−0.165115
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	−21.0000	−0.691598
$923$	−4.00000	−0.131662
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−17.0000	−0.558655
$927$	0	0
$928$	3.00000	0.0984798
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	−12.0000	−0.393073
$933$	0	0
$934$	25.0000	0.818025
$935$	1.00000	0.0327035
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	−8.00000	−0.260931
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	−4.00000	−0.130189
$945$	0	0
$946$	7.00000	0.227590
$947$	13.0000	0.422443	0.211222	−	0.977438i	$-0.432256\pi$
$947$	13.0000	0.422443	0.211222	+	0.977438i	$0.432256\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	−28.0000	−0.908440
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	−19.0000	−0.614826
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−5.00000	−0.161543
$959$	−19.0000	−0.613542
$960$	0	0
$961$	33.0000	1.06452
$962$	7.00000	0.225689
$963$	0	0
$964$	10.0000	0.322078
$965$	16.0000	0.515058
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	−14.0000	−0.449513
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	7.00000	0.224065
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	0	0
$979$	−14.0000	−0.447442
$980$	1.00000	0.0319438
$981$	0	0
$982$	30.0000	0.957338
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	3.00000	0.0955395
$987$	0	0
$988$	7.00000	0.222700
$989$	−21.0000	−0.667761
$990$	0	0
$991$	42.0000	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$991$	42.0000	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	4.00000	0.126872
$995$	1.00000	0.0317021
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	12.0000	0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.r.1.1		1
3.2	odd	2		546.2.a.a.1.1	✓	1
12.11	even	2		4368.2.a.s.1.1		1
21.20	even	2		3822.2.a.n.1.1		1
39.38	odd	2		7098.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.a.1.1	✓	1	3.2	odd	2
1638.2.a.r.1.1		1	1.1	even	1	trivial
3822.2.a.n.1.1		1	21.20	even	2
4368.2.a.s.1.1		1	12.11	even	2
7098.2.a.t.1.1		1	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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