LMFDB - Embedded newform 1638.2.a.m.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{19} -3.00000 q^{20} -1.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -7.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} -3.00000 q^{35} +1.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} -3.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} -2.00000 q^{53} +3.00000 q^{55} +1.00000 q^{56} -7.00000 q^{58} +6.00000 q^{59} -5.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +8.00000 q^{67} -5.00000 q^{68} -3.00000 q^{70} +6.00000 q^{71} -11.0000 q^{73} +1.00000 q^{74} -1.00000 q^{76} -1.00000 q^{77} -14.0000 q^{79} -3.00000 q^{80} -6.00000 q^{82} +2.00000 q^{83} +15.0000 q^{85} -3.00000 q^{86} -1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{91} +1.00000 q^{92} -8.00000 q^{94} +3.00000 q^{95} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	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
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	−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
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	1.00000	0.133631
$57$	0	0
$58$	−7.00000	−0.919145
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	−3.00000	−0.358569
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−6.00000	−0.662589
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	15.0000	1.62698
$86$	−3.00000	−0.323498
$87$	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
$92$	1.00000	0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	3.00000	0.307794
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	1.00000	0.0944911
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−7.00000	−0.649934
$117$	0	0
$118$	6.00000	0.552345
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−5.00000	−0.452679
$123$	0	0
$124$	−2.00000	−0.179605
$125$	3.00000	0.268328
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	8.00000	0.691095
$135$	0	0
$136$	−5.00000	−0.428746
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	6.00000	0.503509
$143$	1.00000	0.0836242
$144$	0	0
$145$	21.0000	1.74396
$146$	−11.0000	−0.910366
$147$	0	0
$148$	1.00000	0.0821995
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	6.00000	0.481932
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−14.0000	−1.11378
$159$	0	0
$160$	−3.00000	−0.237171
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	2.00000	0.155230
$167$	−5.00000	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$167$	−5.00000	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	15.0000	1.15045
$171$	0	0
$172$	−3.00000	−0.228748
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−14.0000	−1.04934
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	1.00000	0.0737210
$185$	−3.00000	−0.220564
$186$	0	0
$187$	5.00000	0.365636
$188$	−8.00000	−0.583460
$189$	0	0
$190$	3.00000	0.217643
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	6.00000	0.422159
$203$	−7.00000	−0.491304
$204$	0	0
$205$	18.0000	1.25717
$206$	9.00000	0.627060
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	1.00000	0.0691714
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−10.0000	−0.683586
$215$	9.00000	0.613795
$216$	0	0
$217$	−2.00000	−0.135769
$218$	15.0000	1.01593
$219$	0	0
$220$	3.00000	0.202260
$221$	5.00000	0.336336
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−4.00000	−0.266076
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	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
$232$	−7.00000	−0.459573
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	6.00000	0.390567
$237$	0	0
$238$	−5.00000	−0.324102
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	−5.00000	−0.320092
$245$	−3.00000	−0.191663
$246$	0	0
$247$	1.00000	0.0636285
$248$	−2.00000	−0.127000
$249$	0	0
$250$	3.00000	0.189737
$251$	25.0000	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$251$	25.0000	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	3.00000	0.186052
$261$	0	0
$262$	1.00000	0.0617802
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	8.00000	0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	3.00000	0.181237
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	10.0000	0.599760
$279$	0	0
$280$	−3.00000	−0.179284
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	1.00000	0.0591312
$287$	−6.00000	−0.354169
$288$	0	0
$289$	8.00000	0.470588
$290$	21.0000	1.23316
$291$	0	0
$292$	−11.0000	−0.643726
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	1.00000	0.0581238
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−3.00000	−0.172917
$302$	3.00000	0.172631
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	15.0000	0.858898
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	6.00000	0.340777
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−14.0000	−0.787562
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	7.00000	0.391925
$320$	−3.00000	−0.167705
$321$	0	0
$322$	1.00000	0.0557278
$323$	5.00000	0.278207
$324$	0	0
$325$	−4.00000	−0.221880
$326$	10.0000	0.553849
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	−5.00000	−0.273588
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	15.0000	0.813489
$341$	2.00000	0.108306
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.00000	−0.161749
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	−14.0000	−0.741999
$357$	0	0
$358$	8.00000	0.422813
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	22.0000	1.15629
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	33.0000	1.72730
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−3.00000	−0.155963
$371$	−2.00000	−0.103835
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	−8.00000	−0.412568
$377$	7.00000	0.360518
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	3.00000	0.153897
$381$	0	0
$382$	−11.0000	−0.562809
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	22.0000	1.11977
$387$	0	0
$388$	−10.0000	−0.507673
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	1.00000	0.0505076
$393$	0	0
$394$	18.0000	0.906827
$395$	42.0000	2.11325
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	4.00000	0.200000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	6.00000	0.298511
$405$	0	0
$406$	−7.00000	−0.347404
$407$	−1.00000	−0.0495682
$408$	0	0
$409$	39.0000	1.92843	0.964213	−	0.265129i	$-0.0854146\pi$
$409$	39.0000	1.92843	0.964213	+	0.265129i	$0.0854146\pi$
$410$	18.0000	0.888957
$411$	0	0
$412$	9.00000	0.443398
$413$	6.00000	0.295241
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	1.00000	0.0489116
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−17.0000	−0.827547
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−20.0000	−0.970143
$426$	0	0
$427$	−5.00000	−0.241967
$428$	−10.0000	−0.483368
$429$	0	0
$430$	9.00000	0.434019
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	15.0000	0.718370
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	5.00000	0.237826
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	42.0000	1.99099
$446$	−24.0000	−1.13643
$447$	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
$451$	6.00000	0.282529
$452$	−4.00000	−0.188144
$453$	0	0
$454$	18.0000	0.844782
$455$	3.00000	0.140642
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	−3.00000	−0.139876
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	−33.0000	−1.53364	−0.766820	−	0.641862i	$-0.778162\pi$
$463$	−33.0000	−1.53364	−0.766820	+	0.641862i	$0.778162\pi$
$464$	−7.00000	−0.324967
$465$	0	0
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	24.0000	1.10704
$471$	0	0
$472$	6.00000	0.276172
$473$	3.00000	0.137940
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−5.00000	−0.229175
$477$	0	0
$478$	24.0000	1.09773
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	6.00000	0.273293
$483$	0	0
$484$	−10.0000	−0.454545
$485$	30.0000	1.36223
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	35.0000	1.57632
$494$	1.00000	0.0449921
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	6.00000	0.269137
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	25.0000	1.11580
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	−1.00000	−0.0444554
$507$	0	0
$508$	12.0000	0.532414
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.0000	−0.970378
$515$	−27.0000	−1.18976
$516$	0	0
$517$	8.00000	0.351840
$518$	1.00000	0.0439375
$519$	0	0
$520$	3.00000	0.131559
$521$	−19.0000	−0.832405	−0.416203	−	0.909272i	$-0.636639\pi$
$521$	−19.0000	−0.832405	−0.416203	+	0.909272i	$0.636639\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	−22.0000	−0.956522
$530$	6.00000	0.260623
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	6.00000	0.259889
$534$	0	0
$535$	30.0000	1.29701
$536$	8.00000	0.345547
$537$	0	0
$538$	−10.0000	−0.431131
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−33.0000	−1.41878	−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000	−1.41878	−0.709390	+	0.704816i	$0.751030\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−5.00000	−0.214373
$545$	−45.0000	−1.92759
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	−4.00000	−0.170561
$551$	7.00000	0.298210
$552$	0	0
$553$	−14.0000	−0.595341
$554$	−16.0000	−0.679775
$555$	0	0
$556$	10.0000	0.424094
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	−3.00000	−0.126773
$561$	0	0
$562$	10.0000	0.421825
$563$	25.0000	1.05362	0.526812	−	0.849982i	$-0.323387\pi$
$563$	25.0000	1.05362	0.526812	+	0.849982i	$0.323387\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	−6.00000	−0.250435
$575$	4.00000	0.166812
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	21.0000	0.871978
$581$	2.00000	0.0829740
$582$	0	0
$583$	2.00000	0.0828315
$584$	−11.0000	−0.455183
$585$	0	0
$586$	18.0000	0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	−18.0000	−0.741048
$591$	0	0
$592$	1.00000	0.0410997
$593$	−32.0000	−1.31408	−0.657041	−	0.753855i	$-0.728192\pi$
$593$	−32.0000	−1.31408	−0.657041	+	0.753855i	$0.728192\pi$
$594$	0	0
$595$	15.0000	0.614940
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−1.00000	−0.0408930
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−3.00000	−0.122271
$603$	0	0
$604$	3.00000	0.122068
$605$	30.0000	1.21967
$606$	0	0
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	15.0000	0.607332
$611$	8.00000	0.323645
$612$	0	0
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	0	0
$619$	−21.0000	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$619$	−21.0000	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−14.0000	−0.560898
$624$	0	0
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	0	0
$628$	7.00000	0.279330
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	−14.0000	−0.556890
$633$	0	0
$634$	−26.0000	−1.03259
$635$	−36.0000	−1.42862
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	7.00000	0.277133
$639$	0	0
$640$	−3.00000	−0.118585
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	45.0000	1.77463	0.887313	−	0.461167i	$-0.152569\pi$
$643$	45.0000	1.77463	0.887313	+	0.461167i	$0.152569\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	5.00000	0.196722
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	−4.00000	−0.156893
$651$	0	0
$652$	10.0000	0.391630
$653$	−45.0000	−1.76099	−0.880493	−	0.474059i	$-0.842788\pi$
$653$	−45.0000	−1.76099	−0.880493	+	0.474059i	$0.842788\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−8.00000	−0.311872
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	2.00000	0.0776151
$665$	3.00000	0.116335
$666$	0	0
$667$	−7.00000	−0.271041
$668$	−5.00000	−0.193456
$669$	0	0
$670$	−24.0000	−0.927201
$671$	5.00000	0.193023
$672$	0	0
$673$	27.0000	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$673$	27.0000	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$674$	−9.00000	−0.346667
$675$	0	0
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	15.0000	0.575224
$681$	0	0
$682$	2.00000	0.0765840
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	1.00000	0.0381802
$687$	0	0
$688$	−3.00000	−0.114374
$689$	2.00000	0.0761939
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−16.0000	−0.607352
$695$	−30.0000	−1.13796
$696$	0	0
$697$	30.0000	1.13633
$698$	−28.0000	−1.05982
$699$	0	0
$700$	4.00000	0.151186
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	6.00000	0.225653
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−18.0000	−0.675528
$711$	0	0
$712$	−14.0000	−0.524672
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−3.00000	−0.112194
$716$	8.00000	0.298974
$717$	0	0
$718$	10.0000	0.373197
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	−18.0000	−0.669891
$723$	0	0
$724$	22.0000	0.817624
$725$	−28.0000	−1.03989
$726$	0	0
$727$	−43.0000	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$727$	−43.0000	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	33.0000	1.22138
$731$	15.0000	0.554795
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	1.00000	0.0368605
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	10.0000	0.366126
$747$	0	0
$748$	5.00000	0.182818
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	7.00000	0.254925
$755$	−9.00000	−0.327544
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	3.00000	0.108821
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	15.0000	0.543036
$764$	−11.0000	−0.397966
$765$	0	0
$766$	15.0000	0.541972
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	3.00000	0.108112
$771$	0	0
$772$	22.0000	0.791797
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−10.0000	−0.358979
$777$	0	0
$778$	22.0000	0.788738
$779$	6.00000	0.214972
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−5.00000	−0.178800
$783$	0	0
$784$	1.00000	0.0357143
$785$	−21.0000	−0.749522
$786$	0	0
$787$	−37.0000	−1.31891	−0.659454	−	0.751745i	$-0.729212\pi$
$787$	−37.0000	−1.31891	−0.659454	+	0.751745i	$0.729212\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	42.0000	1.49429
$791$	−4.00000	−0.142224
$792$	0	0
$793$	5.00000	0.177555
$794$	0	0
$795$	0	0
$796$	−17.0000	−0.602549
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	4.00000	0.141421
$801$	0	0
$802$	−14.0000	−0.494357
$803$	11.0000	0.388182
$804$	0	0
$805$	−3.00000	−0.105736
$806$	2.00000	0.0704470
$807$	0	0
$808$	6.00000	0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	−7.00000	−0.245652
$813$	0	0
$814$	−1.00000	−0.0350500
$815$	−30.0000	−1.05085
$816$	0	0
$817$	3.00000	0.104957
$818$	39.0000	1.36360
$819$	0	0
$820$	18.0000	0.628587
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	9.00000	0.313530
$825$	0	0
$826$	6.00000	0.208767
$827$	−33.0000	−1.14752	−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000	−1.14752	−0.573761	+	0.819023i	$0.694516\pi$
$828$	0	0
$829$	51.0000	1.77130	0.885652	−	0.464350i	$-0.153712\pi$
$829$	51.0000	1.77130	0.885652	+	0.464350i	$0.153712\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−5.00000	−0.173240
$834$	0	0
$835$	15.0000	0.519096
$836$	1.00000	0.0345857
$837$	0	0
$838$	−21.0000	−0.725433
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	10.0000	0.344623
$843$	0	0
$844$	−17.0000	−0.585164
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−10.0000	−0.343604
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−20.0000	−0.685994
$851$	1.00000	0.0342796
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−10.0000	−0.341793
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	9.00000	0.306897
$861$	0	0
$862$	30.0000	1.02180
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	−22.0000	−0.747590
$867$	0	0
$868$	−2.00000	−0.0678844
$869$	14.0000	0.474917
$870$	0	0
$871$	−8.00000	−0.271070
$872$	15.0000	0.507964
$873$	0	0
$874$	−1.00000	−0.0338255
$875$	3.00000	0.101419
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	1.00000	0.0337484
$879$	0	0
$880$	3.00000	0.101130
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	−15.0000	−0.504790	−0.252395	−	0.967624i	$-0.581218\pi$
$883$	−15.0000	−0.504790	−0.252395	+	0.967624i	$0.581218\pi$
$884$	5.00000	0.168168
$885$	0	0
$886$	−6.00000	−0.201574
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	42.0000	1.40784
$891$	0	0
$892$	−24.0000	−0.803579
$893$	8.00000	0.267710
$894$	0	0
$895$	−24.0000	−0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	39.0000	1.30145
$899$	14.0000	0.466926
$900$	0	0
$901$	10.0000	0.333148
$902$	6.00000	0.199778
$903$	0	0
$904$	−4.00000	−0.133038
$905$	−66.0000	−2.19391
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	3.00000	0.0994490
$911$	23.0000	0.762024	0.381012	−	0.924570i	$-0.375576\pi$
$911$	23.0000	0.762024	0.381012	+	0.924570i	$0.375576\pi$
$912$	0	0
$913$	−2.00000	−0.0661903
$914$	8.00000	0.264616
$915$	0	0
$916$	10.0000	0.330409
$917$	1.00000	0.0330229
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	15.0000	0.493999
$923$	−6.00000	−0.197492
$924$	0	0
$925$	4.00000	0.131519
$926$	−33.0000	−1.08445
$927$	0	0
$928$	−7.00000	−0.229786
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	−27.0000	−0.883467
$935$	−15.0000	−0.490552
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	24.0000	0.782794
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	6.00000	0.195283
$945$	0	0
$946$	3.00000	0.0975384
$947$	−41.0000	−1.33232	−0.666160	−	0.745808i	$-0.732063\pi$
$947$	−41.0000	−1.33232	−0.666160	+	0.745808i	$0.732063\pi$
$948$	0	0
$949$	11.0000	0.357075
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−5.00000	−0.162051
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	0	0
$955$	33.0000	1.06785
$956$	24.0000	0.776215
$957$	0	0
$958$	25.0000	0.807713
$959$	3.00000	0.0968751
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−1.00000	−0.0322413
$963$	0	0
$964$	6.00000	0.193247
$965$	−66.0000	−2.12462
$966$	0	0
$967$	17.0000	0.546683	0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000	0.546683	0.273342	+	0.961917i	$0.411871\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	30.0000	0.963242
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−5.00000	−0.160046
$977$	−43.0000	−1.37569	−0.687846	−	0.725857i	$-0.741444\pi$
$977$	−43.0000	−1.37569	−0.687846	+	0.725857i	$0.741444\pi$
$978$	0	0
$979$	14.0000	0.447442
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−4.00000	−0.127645
$983$	−43.0000	−1.37149	−0.685744	−	0.727843i	$-0.740523\pi$
$983$	−43.0000	−1.37149	−0.685744	+	0.727843i	$0.740523\pi$
$984$	0	0
$985$	−54.0000	−1.72058
$986$	35.0000	1.11463
$987$	0	0
$988$	1.00000	0.0318142
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	62.0000	1.96949	0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000	1.96949	0.984747	+	0.173990i	$0.0556660\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	6.00000	0.190308
$995$	51.0000	1.61681
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.m.1.1	yes	1
3.2	odd	2		1638.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.a.i.1.1	✓	1	3.2	odd	2
1638.2.a.m.1.1	yes	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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