LMFDB - Embedded newform 1638.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,5,0,-7,-8,0,-10,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
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)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +36.0000 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -26.0000 q^{17} -47.0000 q^{19} +20.0000 q^{20} -72.0000 q^{22} +99.0000 q^{23} -100.000 q^{25} +26.0000 q^{26} -28.0000 q^{28} +61.0000 q^{29} -23.0000 q^{31} -32.0000 q^{32} +52.0000 q^{34} -35.0000 q^{35} -50.0000 q^{37} +94.0000 q^{38} -40.0000 q^{40} -70.0000 q^{41} -19.0000 q^{43} +144.000 q^{44} -198.000 q^{46} -191.000 q^{47} +49.0000 q^{49} +200.000 q^{50} -52.0000 q^{52} -195.000 q^{53} +180.000 q^{55} +56.0000 q^{56} -122.000 q^{58} -264.000 q^{59} +310.000 q^{61} +46.0000 q^{62} +64.0000 q^{64} -65.0000 q^{65} -190.000 q^{67} -104.000 q^{68} +70.0000 q^{70} +166.000 q^{71} +873.000 q^{73} +100.000 q^{74} -188.000 q^{76} -252.000 q^{77} -1191.00 q^{79} +80.0000 q^{80} +140.000 q^{82} -259.000 q^{83} -130.000 q^{85} +38.0000 q^{86} -288.000 q^{88} +635.000 q^{89} +91.0000 q^{91} +396.000 q^{92} +382.000 q^{94} -235.000 q^{95} +133.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	5.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−26.0000	−0.370937	−0.185468	−	0.982650i	$-0.559380\pi$
$17$	−26.0000	−0.370937	−0.185468	+	0.982650i	$0.559380\pi$
$18$	0	0
$19$	−47.0000	−0.567502	−0.283751	−	0.958898i	$-0.591579\pi$
$19$	−47.0000	−0.567502	−0.283751	+	0.958898i	$0.591579\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−72.0000	−0.697748
$23$	99.0000	0.897519	0.448759	−	0.893653i	$-0.351866\pi$
$23$	99.0000	0.897519	0.448759	+	0.893653i	$0.351866\pi$
$24$	0	0
$25$	−100.000	−0.800000
$26$	26.0000	0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	61.0000	0.390601	0.195300	−	0.980743i	$-0.437432\pi$
$29$	61.0000	0.390601	0.195300	+	0.980743i	$0.437432\pi$
$30$	0	0
$31$	−23.0000	−0.133256	−0.0666278	−	0.997778i	$-0.521224\pi$
$31$	−23.0000	−0.133256	−0.0666278	+	0.997778i	$0.521224\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	52.0000	0.262292
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−50.0000	−0.222161	−0.111080	−	0.993811i	$-0.535431\pi$
$37$	−50.0000	−0.222161	−0.111080	+	0.993811i	$0.535431\pi$
$38$	94.0000	0.401285
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−70.0000	−0.266638	−0.133319	−	0.991073i	$-0.542564\pi$
$41$	−70.0000	−0.266638	−0.133319	+	0.991073i	$0.542564\pi$
$42$	0	0
$43$	−19.0000	−0.0673831	−0.0336915	−	0.999432i	$-0.510726\pi$
$43$	−19.0000	−0.0673831	−0.0336915	+	0.999432i	$0.510726\pi$
$44$	144.000	0.493382
$45$	0	0
$46$	−198.000	−0.634641
$47$	−191.000	−0.592770	−0.296385	−	0.955068i	$-0.595781\pi$
$47$	−191.000	−0.592770	−0.296385	+	0.955068i	$0.595781\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	200.000	0.565685
$51$	0	0
$52$	−52.0000	−0.138675
$53$	−195.000	−0.505383	−0.252692	−	0.967547i	$-0.581316\pi$
$53$	−195.000	−0.505383	−0.252692	+	0.967547i	$0.581316\pi$
$54$	0	0
$55$	180.000	0.441294
$56$	56.0000	0.133631
$57$	0	0
$58$	−122.000	−0.276196
$59$	−264.000	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−264.000	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	0	0
$61$	310.000	0.650679	0.325340	−	0.945597i	$-0.394521\pi$
$61$	310.000	0.650679	0.325340	+	0.945597i	$0.394521\pi$
$62$	46.0000	0.0942259
$63$	0	0
$64$	64.0000	0.125000
$65$	−65.0000	−0.124035
$66$	0	0
$67$	−190.000	−0.346451	−0.173225	−	0.984882i	$-0.555419\pi$
$67$	−190.000	−0.346451	−0.173225	+	0.984882i	$0.555419\pi$
$68$	−104.000	−0.185468
$69$	0	0
$70$	70.0000	0.119523
$71$	166.000	0.277473	0.138736	−	0.990329i	$-0.455696\pi$
$71$	166.000	0.277473	0.138736	+	0.990329i	$0.455696\pi$
$72$	0	0
$73$	873.000	1.39968	0.699842	−	0.714298i	$-0.253254\pi$
$73$	873.000	1.39968	0.699842	+	0.714298i	$0.253254\pi$
$74$	100.000	0.157091
$75$	0	0
$76$	−188.000	−0.283751
$77$	−252.000	−0.372962
$78$	0	0
$79$	−1191.00	−1.69618	−0.848088	−	0.529855i	$-0.822246\pi$
$79$	−1191.00	−1.69618	−0.848088	+	0.529855i	$0.822246\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	140.000	0.188542
$83$	−259.000	−0.342517	−0.171259	−	0.985226i	$-0.554783\pi$
$83$	−259.000	−0.342517	−0.171259	+	0.985226i	$0.554783\pi$
$84$	0	0
$85$	−130.000	−0.165888
$86$	38.0000	0.0476470
$87$	0	0
$88$	−288.000	−0.348874
$89$	635.000	0.756291	0.378145	−	0.925746i	$-0.376562\pi$
$89$	635.000	0.756291	0.378145	+	0.925746i	$0.376562\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	396.000	0.448759
$93$	0	0
$94$	382.000	0.419152
$95$	−235.000	−0.253795
$96$	0	0
$97$	133.000	0.139218	0.0696088	−	0.997574i	$-0.477825\pi$
$97$	133.000	0.139218	0.0696088	+	0.997574i	$0.477825\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−400.000	−0.400000
$101$	−152.000	−0.149748	−0.0748741	−	0.997193i	$-0.523855\pi$
$101$	−152.000	−0.149748	−0.0748741	+	0.997193i	$0.523855\pi$
$102$	0	0
$103$	−1664.00	−1.59183	−0.795916	−	0.605406i	$-0.793011\pi$
$103$	−1664.00	−1.59183	−0.795916	+	0.605406i	$0.793011\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	390.000	0.357360
$107$	36.0000	0.0325257	0.0162629	−	0.999868i	$-0.494823\pi$
$107$	36.0000	0.0325257	0.0162629	+	0.999868i	$0.494823\pi$
$108$	0	0
$109$	−232.000	−0.203868	−0.101934	−	0.994791i	$-0.532503\pi$
$109$	−232.000	−0.203868	−0.101934	+	0.994791i	$0.532503\pi$
$110$	−360.000	−0.312042
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−1353.00	−1.12637	−0.563184	−	0.826332i	$-0.690424\pi$
$113$	−1353.00	−1.12637	−0.563184	+	0.826332i	$0.690424\pi$
$114$	0	0
$115$	495.000	0.401383
$116$	244.000	0.195300
$117$	0	0
$118$	528.000	0.411918
$119$	182.000	0.140201
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	−620.000	−0.460100
$123$	0	0
$124$	−92.0000	−0.0666278
$125$	−1125.00	−0.804984
$126$	0	0
$127$	576.000	0.402455	0.201227	−	0.979545i	$-0.435507\pi$
$127$	576.000	0.402455	0.201227	+	0.979545i	$0.435507\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	130.000	0.0877058
$131$	2056.00	1.37125	0.685624	−	0.727956i	$-0.259529\pi$
$131$	2056.00	1.37125	0.685624	+	0.727956i	$0.259529\pi$
$132$	0	0
$133$	329.000	0.214496
$134$	380.000	0.244978
$135$	0	0
$136$	208.000	0.131146
$137$	1842.00	1.14871	0.574353	−	0.818608i	$-0.305254\pi$
$137$	1842.00	1.14871	0.574353	+	0.818608i	$0.305254\pi$
$138$	0	0
$139$	−1288.00	−0.785948	−0.392974	−	0.919550i	$-0.628554\pi$
$139$	−1288.00	−0.785948	−0.392974	+	0.919550i	$0.628554\pi$
$140$	−140.000	−0.0845154
$141$	0	0
$142$	−332.000	−0.196203
$143$	−468.000	−0.273679
$144$	0	0
$145$	305.000	0.174682
$146$	−1746.00	−0.989726
$147$	0	0
$148$	−200.000	−0.111080
$149$	−1196.00	−0.657585	−0.328792	−	0.944402i	$-0.606642\pi$
$149$	−1196.00	−0.657585	−0.328792	+	0.944402i	$0.606642\pi$
$150$	0	0
$151$	−890.000	−0.479650	−0.239825	−	0.970816i	$-0.577090\pi$
$151$	−890.000	−0.479650	−0.239825	+	0.970816i	$0.577090\pi$
$152$	376.000	0.200642
$153$	0	0
$154$	504.000	0.263724
$155$	−115.000	−0.0595937
$156$	0	0
$157$	138.000	0.0701503	0.0350752	−	0.999385i	$-0.488833\pi$
$157$	138.000	0.0701503	0.0350752	+	0.999385i	$0.488833\pi$
$158$	2382.00	1.19938
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−693.000	−0.339230
$162$	0	0
$163$	−3956.00	−1.90097	−0.950484	−	0.310773i	$-0.899412\pi$
$163$	−3956.00	−1.90097	−0.950484	+	0.310773i	$0.899412\pi$
$164$	−280.000	−0.133319
$165$	0	0
$166$	518.000	0.242196
$167$	3675.00	1.70287	0.851437	−	0.524457i	$-0.175731\pi$
$167$	3675.00	1.70287	0.851437	+	0.524457i	$0.175731\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	260.000	0.117301
$171$	0	0
$172$	−76.0000	−0.0336915
$173$	2826.00	1.24195	0.620973	−	0.783832i	$-0.286737\pi$
$173$	2826.00	1.24195	0.620973	+	0.783832i	$0.286737\pi$
$174$	0	0
$175$	700.000	0.302372
$176$	576.000	0.246691
$177$	0	0
$178$	−1270.00	−0.534778
$179$	3235.00	1.35081	0.675406	−	0.737446i	$-0.263969\pi$
$179$	3235.00	1.35081	0.675406	+	0.737446i	$0.263969\pi$
$180$	0	0
$181$	1260.00	0.517431	0.258716	−	0.965954i	$-0.416701\pi$
$181$	1260.00	0.517431	0.258716	+	0.965954i	$0.416701\pi$
$182$	−182.000	−0.0741249
$183$	0	0
$184$	−792.000	−0.317321
$185$	−250.000	−0.0993533
$186$	0	0
$187$	−936.000	−0.366027
$188$	−764.000	−0.296385
$189$	0	0
$190$	470.000	0.179460
$191$	232.000	0.0878897	0.0439448	−	0.999034i	$-0.486007\pi$
$191$	232.000	0.0878897	0.0439448	+	0.999034i	$0.486007\pi$
$192$	0	0
$193$	−5342.00	−1.99236	−0.996180	−	0.0873208i	$-0.972169\pi$
$193$	−5342.00	−1.99236	−0.996180	+	0.0873208i	$0.972169\pi$
$194$	−266.000	−0.0984417
$195$	0	0
$196$	196.000	0.0714286
$197$	−1542.00	−0.557680	−0.278840	−	0.960338i	$-0.589950\pi$
$197$	−1542.00	−0.557680	−0.278840	+	0.960338i	$0.589950\pi$
$198$	0	0
$199$	2182.00	0.777276	0.388638	−	0.921391i	$-0.372946\pi$
$199$	2182.00	0.777276	0.388638	+	0.921391i	$0.372946\pi$
$200$	800.000	0.282843
$201$	0	0
$202$	304.000	0.105888
$203$	−427.000	−0.147633
$204$	0	0
$205$	−350.000	−0.119244
$206$	3328.00	1.12560
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−1692.00	−0.559991
$210$	0	0
$211$	−523.000	−0.170639	−0.0853194	−	0.996354i	$-0.527191\pi$
$211$	−523.000	−0.170639	−0.0853194	+	0.996354i	$0.527191\pi$
$212$	−780.000	−0.252692
$213$	0	0
$214$	−72.0000	−0.0229992
$215$	−95.0000	−0.0301346
$216$	0	0
$217$	161.000	0.0503659
$218$	464.000	0.144156
$219$	0	0
$220$	720.000	0.220647
$221$	338.000	0.102879
$222$	0	0
$223$	−1981.00	−0.594877	−0.297439	−	0.954741i	$-0.596132\pi$
$223$	−1981.00	−0.594877	−0.297439	+	0.954741i	$0.596132\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	2706.00	0.796462
$227$	−4352.00	−1.27248	−0.636239	−	0.771492i	$-0.719511\pi$
$227$	−4352.00	−1.27248	−0.636239	+	0.771492i	$0.719511\pi$
$228$	0	0
$229$	2130.00	0.614648	0.307324	−	0.951605i	$-0.400566\pi$
$229$	2130.00	0.614648	0.307324	+	0.951605i	$0.400566\pi$
$230$	−990.000	−0.283820
$231$	0	0
$232$	−488.000	−0.138098
$233$	−2687.00	−0.755499	−0.377749	−	0.925908i	$-0.623302\pi$
$233$	−2687.00	−0.755499	−0.377749	+	0.925908i	$0.623302\pi$
$234$	0	0
$235$	−955.000	−0.265095
$236$	−1056.00	−0.291270
$237$	0	0
$238$	−364.000	−0.0991370
$239$	−3852.00	−1.04253	−0.521266	−	0.853394i	$-0.674540\pi$
$239$	−3852.00	−1.04253	−0.521266	+	0.853394i	$0.674540\pi$
$240$	0	0
$241$	1069.00	0.285728	0.142864	−	0.989742i	$-0.454369\pi$
$241$	1069.00	0.285728	0.142864	+	0.989742i	$0.454369\pi$
$242$	70.0000	0.0185941
$243$	0	0
$244$	1240.00	0.325340
$245$	245.000	0.0638877
$246$	0	0
$247$	611.000	0.157397
$248$	184.000	0.0471130
$249$	0	0
$250$	2250.00	0.569210
$251$	−5460.00	−1.37304	−0.686518	−	0.727113i	$-0.740862\pi$
$251$	−5460.00	−1.37304	−0.686518	+	0.727113i	$0.740862\pi$
$252$	0	0
$253$	3564.00	0.885639
$254$	−1152.00	−0.284578
$255$	0	0
$256$	256.000	0.0625000
$257$	−2172.00	−0.527181	−0.263591	−	0.964635i	$-0.584907\pi$
$257$	−2172.00	−0.527181	−0.263591	+	0.964635i	$0.584907\pi$
$258$	0	0
$259$	350.000	0.0839689
$260$	−260.000	−0.0620174
$261$	0	0
$262$	−4112.00	−0.969619
$263$	3417.00	0.801145	0.400573	−	0.916265i	$-0.368811\pi$
$263$	3417.00	0.801145	0.400573	+	0.916265i	$0.368811\pi$
$264$	0	0
$265$	−975.000	−0.226014
$266$	−658.000	−0.151671
$267$	0	0
$268$	−760.000	−0.173225
$269$	−3792.00	−0.859488	−0.429744	−	0.902951i	$-0.641396\pi$
$269$	−3792.00	−0.859488	−0.429744	+	0.902951i	$0.641396\pi$
$270$	0	0
$271$	−4408.00	−0.988070	−0.494035	−	0.869442i	$-0.664479\pi$
$271$	−4408.00	−0.988070	−0.494035	+	0.869442i	$0.664479\pi$
$272$	−416.000	−0.0927342
$273$	0	0
$274$	−3684.00	−0.812258
$275$	−3600.00	−0.789412
$276$	0	0
$277$	−1023.00	−0.221899	−0.110950	−	0.993826i	$-0.535389\pi$
$277$	−1023.00	−0.221899	−0.110950	+	0.993826i	$0.535389\pi$
$278$	2576.00	0.555749
$279$	0	0
$280$	280.000	0.0597614
$281$	−7912.00	−1.67968	−0.839840	−	0.542833i	$-0.817352\pi$
$281$	−7912.00	−1.67968	−0.839840	+	0.542833i	$0.817352\pi$
$282$	0	0
$283$	−5336.00	−1.12082	−0.560410	−	0.828215i	$-0.689357\pi$
$283$	−5336.00	−1.12082	−0.560410	+	0.828215i	$0.689357\pi$
$284$	664.000	0.138736
$285$	0	0
$286$	936.000	0.193520
$287$	490.000	0.100780
$288$	0	0
$289$	−4237.00	−0.862406
$290$	−610.000	−0.123519
$291$	0	0
$292$	3492.00	0.699842
$293$	−6615.00	−1.31895	−0.659475	−	0.751726i	$-0.729221\pi$
$293$	−6615.00	−1.31895	−0.659475	+	0.751726i	$0.729221\pi$
$294$	0	0
$295$	−1320.00	−0.260520
$296$	400.000	0.0785457
$297$	0	0
$298$	2392.00	0.464983
$299$	−1287.00	−0.248927
$300$	0	0
$301$	133.000	0.0254684
$302$	1780.00	0.339164
$303$	0	0
$304$	−752.000	−0.141876
$305$	1550.00	0.290993
$306$	0	0
$307$	3843.00	0.714435	0.357218	−	0.934021i	$-0.383725\pi$
$307$	3843.00	0.714435	0.357218	+	0.934021i	$0.383725\pi$
$308$	−1008.00	−0.186481
$309$	0	0
$310$	230.000	0.0421391
$311$	−1748.00	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$311$	−1748.00	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$312$	0	0
$313$	−9392.00	−1.69606	−0.848031	−	0.529947i	$-0.822212\pi$
$313$	−9392.00	−1.69606	−0.848031	+	0.529947i	$0.822212\pi$
$314$	−276.000	−0.0496038
$315$	0	0
$316$	−4764.00	−0.848088
$317$	−10380.0	−1.83911	−0.919557	−	0.392958i	$-0.871452\pi$
$317$	−10380.0	−1.83911	−0.919557	+	0.392958i	$0.871452\pi$
$318$	0	0
$319$	2196.00	0.385431
$320$	320.000	0.0559017
$321$	0	0
$322$	1386.00	0.239872
$323$	1222.00	0.210507
$324$	0	0
$325$	1300.00	0.221880
$326$	7912.00	1.34419
$327$	0	0
$328$	560.000	0.0942708
$329$	1337.00	0.224046
$330$	0	0
$331$	6250.00	1.03786	0.518929	−	0.854817i	$-0.326331\pi$
$331$	6250.00	1.03786	0.518929	+	0.854817i	$0.326331\pi$
$332$	−1036.00	−0.171259
$333$	0	0
$334$	−7350.00	−1.20411
$335$	−950.000	−0.154937
$336$	0	0
$337$	−2315.00	−0.374202	−0.187101	−	0.982341i	$-0.559909\pi$
$337$	−2315.00	−0.374202	−0.187101	+	0.982341i	$0.559909\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−520.000	−0.0829440
$341$	−828.000	−0.131492
$342$	0	0
$343$	−343.000	−0.0539949
$344$	152.000	0.0238235
$345$	0	0
$346$	−5652.00	−0.878189
$347$	−3296.00	−0.509909	−0.254955	−	0.966953i	$-0.582061\pi$
$347$	−3296.00	−0.509909	−0.254955	+	0.966953i	$0.582061\pi$
$348$	0	0
$349$	−5607.00	−0.859988	−0.429994	−	0.902832i	$-0.641484\pi$
$349$	−5607.00	−0.859988	−0.429994	+	0.902832i	$0.641484\pi$
$350$	−1400.00	−0.213809
$351$	0	0
$352$	−1152.00	−0.174437
$353$	−3722.00	−0.561196	−0.280598	−	0.959825i	$-0.590533\pi$
$353$	−3722.00	−0.561196	−0.280598	+	0.959825i	$0.590533\pi$
$354$	0	0
$355$	830.000	0.124090
$356$	2540.00	0.378145
$357$	0	0
$358$	−6470.00	−0.955168
$359$	2430.00	0.357244	0.178622	−	0.983918i	$-0.442836\pi$
$359$	2430.00	0.357244	0.178622	+	0.983918i	$0.442836\pi$
$360$	0	0
$361$	−4650.00	−0.677941
$362$	−2520.00	−0.365879
$363$	0	0
$364$	364.000	0.0524142
$365$	4365.00	0.625958
$366$	0	0
$367$	−8584.00	−1.22093	−0.610465	−	0.792043i	$-0.709017\pi$
$367$	−8584.00	−1.22093	−0.610465	+	0.792043i	$0.709017\pi$
$368$	1584.00	0.224380
$369$	0	0
$370$	500.000	0.0702534
$371$	1365.00	0.191017
$372$	0	0
$373$	−7274.00	−1.00974	−0.504871	−	0.863195i	$-0.668460\pi$
$373$	−7274.00	−1.00974	−0.504871	+	0.863195i	$0.668460\pi$
$374$	1872.00	0.258820
$375$	0	0
$376$	1528.00	0.209576
$377$	−793.000	−0.108333
$378$	0	0
$379$	−1160.00	−0.157217	−0.0786084	−	0.996906i	$-0.525048\pi$
$379$	−1160.00	−0.157217	−0.0786084	+	0.996906i	$0.525048\pi$
$380$	−940.000	−0.126897
$381$	0	0
$382$	−464.000	−0.0621474
$383$	5528.00	0.737513	0.368757	−	0.929526i	$-0.379784\pi$
$383$	5528.00	0.737513	0.368757	+	0.929526i	$0.379784\pi$
$384$	0	0
$385$	−1260.00	−0.166794
$386$	10684.0	1.40881
$387$	0	0
$388$	532.000	0.0696088
$389$	−9666.00	−1.25986	−0.629930	−	0.776652i	$-0.716916\pi$
$389$	−9666.00	−1.25986	−0.629930	+	0.776652i	$0.716916\pi$
$390$	0	0
$391$	−2574.00	−0.332923
$392$	−392.000	−0.0505076
$393$	0	0
$394$	3084.00	0.394339
$395$	−5955.00	−0.758553
$396$	0	0
$397$	4125.00	0.521481	0.260740	−	0.965409i	$-0.416033\pi$
$397$	4125.00	0.521481	0.260740	+	0.965409i	$0.416033\pi$
$398$	−4364.00	−0.549617
$399$	0	0
$400$	−1600.00	−0.200000
$401$	11796.0	1.46899	0.734494	−	0.678615i	$-0.237420\pi$
$401$	11796.0	1.46899	0.734494	+	0.678615i	$0.237420\pi$
$402$	0	0
$403$	299.000	0.0369584
$404$	−608.000	−0.0748741
$405$	0	0
$406$	854.000	0.104392
$407$	−1800.00	−0.219220
$408$	0	0
$409$	8181.00	0.989057	0.494529	−	0.869161i	$-0.335341\pi$
$409$	8181.00	0.989057	0.494529	+	0.869161i	$0.335341\pi$
$410$	700.000	0.0843184
$411$	0	0
$412$	−6656.00	−0.795916
$413$	1848.00	0.220180
$414$	0	0
$415$	−1295.00	−0.153178
$416$	416.000	0.0490290
$417$	0	0
$418$	3384.00	0.395973
$419$	9296.00	1.08386	0.541932	−	0.840422i	$-0.317693\pi$
$419$	9296.00	1.08386	0.541932	+	0.840422i	$0.317693\pi$
$420$	0	0
$421$	9074.00	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$421$	9074.00	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$422$	1046.00	0.120660
$423$	0	0
$424$	1560.00	0.178680
$425$	2600.00	0.296749
$426$	0	0
$427$	−2170.00	−0.245934
$428$	144.000	0.0162629
$429$	0	0
$430$	190.000	0.0213084
$431$	−9358.00	−1.04584	−0.522922	−	0.852380i	$-0.675158\pi$
$431$	−9358.00	−1.04584	−0.522922	+	0.852380i	$0.675158\pi$
$432$	0	0
$433$	−14392.0	−1.59731	−0.798655	−	0.601789i	$-0.794455\pi$
$433$	−14392.0	−1.59731	−0.798655	+	0.601789i	$0.794455\pi$
$434$	−322.000	−0.0356140
$435$	0	0
$436$	−928.000	−0.101934
$437$	−4653.00	−0.509344
$438$	0	0
$439$	−6074.00	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$439$	−6074.00	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$440$	−1440.00	−0.156021
$441$	0	0
$442$	−676.000	−0.0727467
$443$	6483.00	0.695297	0.347649	−	0.937625i	$-0.386980\pi$
$443$	6483.00	0.695297	0.347649	+	0.937625i	$0.386980\pi$
$444$	0	0
$445$	3175.00	0.338223
$446$	3962.00	0.420642
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−15388.0	−1.61738	−0.808691	−	0.588234i	$-0.799824\pi$
$449$	−15388.0	−1.61738	−0.808691	+	0.588234i	$0.799824\pi$
$450$	0	0
$451$	−2520.00	−0.263109
$452$	−5412.00	−0.563184
$453$	0	0
$454$	8704.00	0.899777
$455$	455.000	0.0468807
$456$	0	0
$457$	−4642.00	−0.475150	−0.237575	−	0.971369i	$-0.576353\pi$
$457$	−4642.00	−0.475150	−0.237575	+	0.971369i	$0.576353\pi$
$458$	−4260.00	−0.434622
$459$	0	0
$460$	1980.00	0.200691
$461$	14266.0	1.44129	0.720644	−	0.693305i	$-0.243846\pi$
$461$	14266.0	1.44129	0.720644	+	0.693305i	$0.243846\pi$
$462$	0	0
$463$	1472.00	0.147753	0.0738765	−	0.997267i	$-0.476463\pi$
$463$	1472.00	0.147753	0.0738765	+	0.997267i	$0.476463\pi$
$464$	976.000	0.0976501
$465$	0	0
$466$	5374.00	0.534218
$467$	−1332.00	−0.131986	−0.0659932	−	0.997820i	$-0.521022\pi$
$467$	−1332.00	−0.131986	−0.0659932	+	0.997820i	$0.521022\pi$
$468$	0	0
$469$	1330.00	0.130946
$470$	1910.00	0.187450
$471$	0	0
$472$	2112.00	0.205959
$473$	−684.000	−0.0664912
$474$	0	0
$475$	4700.00	0.454002
$476$	728.000	0.0701005
$477$	0	0
$478$	7704.00	0.737182
$479$	−1629.00	−0.155388	−0.0776941	−	0.996977i	$-0.524756\pi$
$479$	−1629.00	−0.155388	−0.0776941	+	0.996977i	$0.524756\pi$
$480$	0	0
$481$	650.000	0.0616163
$482$	−2138.00	−0.202040
$483$	0	0
$484$	−140.000	−0.0131480
$485$	665.000	0.0622600
$486$	0	0
$487$	13754.0	1.27978	0.639890	−	0.768466i	$-0.278980\pi$
$487$	13754.0	1.27978	0.639890	+	0.768466i	$0.278980\pi$
$488$	−2480.00	−0.230050
$489$	0	0
$490$	−490.000	−0.0451754
$491$	10904.0	1.00222	0.501111	−	0.865383i	$-0.332925\pi$
$491$	10904.0	1.00222	0.501111	+	0.865383i	$0.332925\pi$
$492$	0	0
$493$	−1586.00	−0.144888
$494$	−1222.00	−0.111296
$495$	0	0
$496$	−368.000	−0.0333139
$497$	−1162.00	−0.104875
$498$	0	0
$499$	10394.0	0.932464	0.466232	−	0.884663i	$-0.345611\pi$
$499$	10394.0	0.932464	0.466232	+	0.884663i	$0.345611\pi$
$500$	−4500.00	−0.402492
$501$	0	0
$502$	10920.0	0.970883
$503$	5754.00	0.510056	0.255028	−	0.966934i	$-0.417915\pi$
$503$	5754.00	0.510056	0.255028	+	0.966934i	$0.417915\pi$
$504$	0	0
$505$	−760.000	−0.0669694
$506$	−7128.00	−0.626242
$507$	0	0
$508$	2304.00	0.201227
$509$	−13547.0	−1.17969	−0.589843	−	0.807518i	$-0.700810\pi$
$509$	−13547.0	−1.17969	−0.589843	+	0.807518i	$0.700810\pi$
$510$	0	0
$511$	−6111.00	−0.529031
$512$	−512.000	−0.0441942
$513$	0	0
$514$	4344.00	0.372774
$515$	−8320.00	−0.711889
$516$	0	0
$517$	−6876.00	−0.584925
$518$	−700.000	−0.0593750
$519$	0	0
$520$	520.000	0.0438529
$521$	7044.00	0.592329	0.296164	−	0.955137i	$-0.404292\pi$
$521$	7044.00	0.592329	0.296164	+	0.955137i	$0.404292\pi$
$522$	0	0
$523$	9858.00	0.824207	0.412103	−	0.911137i	$-0.364794\pi$
$523$	9858.00	0.824207	0.412103	+	0.911137i	$0.364794\pi$
$524$	8224.00	0.685624
$525$	0	0
$526$	−6834.00	−0.566495
$527$	598.000	0.0494294
$528$	0	0
$529$	−2366.00	−0.194460
$530$	1950.00	0.159816
$531$	0	0
$532$	1316.00	0.107248
$533$	910.000	0.0739521
$534$	0	0
$535$	180.000	0.0145459
$536$	1520.00	0.122489
$537$	0	0
$538$	7584.00	0.607750
$539$	1764.00	0.140966
$540$	0	0
$541$	−5650.00	−0.449006	−0.224503	−	0.974473i	$-0.572076\pi$
$541$	−5650.00	−0.449006	−0.224503	+	0.974473i	$0.572076\pi$
$542$	8816.00	0.698671
$543$	0	0
$544$	832.000	0.0655730
$545$	−1160.00	−0.0911724
$546$	0	0
$547$	−14285.0	−1.11660	−0.558302	−	0.829638i	$-0.688547\pi$
$547$	−14285.0	−1.11660	−0.558302	+	0.829638i	$0.688547\pi$
$548$	7368.00	0.574353
$549$	0	0
$550$	7200.00	0.558198
$551$	−2867.00	−0.221667
$552$	0	0
$553$	8337.00	0.641095
$554$	2046.00	0.156907
$555$	0	0
$556$	−5152.00	−0.392974
$557$	2472.00	0.188047	0.0940233	−	0.995570i	$-0.470027\pi$
$557$	2472.00	0.188047	0.0940233	+	0.995570i	$0.470027\pi$
$558$	0	0
$559$	247.000	0.0186887
$560$	−560.000	−0.0422577
$561$	0	0
$562$	15824.0	1.18771
$563$	3936.00	0.294641	0.147320	−	0.989089i	$-0.452935\pi$
$563$	3936.00	0.294641	0.147320	+	0.989089i	$0.452935\pi$
$564$	0	0
$565$	−6765.00	−0.503727
$566$	10672.0	0.792540
$567$	0	0
$568$	−1328.00	−0.0981015
$569$	−2995.00	−0.220662	−0.110331	−	0.993895i	$-0.535191\pi$
$569$	−2995.00	−0.220662	−0.110331	+	0.993895i	$0.535191\pi$
$570$	0	0
$571$	19221.0	1.40871	0.704355	−	0.709848i	$-0.251236\pi$
$571$	19221.0	1.40871	0.704355	+	0.709848i	$0.251236\pi$
$572$	−1872.00	−0.136840
$573$	0	0
$574$	−980.000	−0.0712620
$575$	−9900.00	−0.718015
$576$	0	0
$577$	−3166.00	−0.228427	−0.114213	−	0.993456i	$-0.536435\pi$
$577$	−3166.00	−0.228427	−0.114213	+	0.993456i	$0.536435\pi$
$578$	8474.00	0.609813
$579$	0	0
$580$	1220.00	0.0873409
$581$	1813.00	0.129459
$582$	0	0
$583$	−7020.00	−0.498694
$584$	−6984.00	−0.494863
$585$	0	0
$586$	13230.0	0.932639
$587$	−13951.0	−0.980953	−0.490476	−	0.871454i	$-0.663177\pi$
$587$	−13951.0	−0.980953	−0.490476	+	0.871454i	$0.663177\pi$
$588$	0	0
$589$	1081.00	0.0756228
$590$	2640.00	0.184215
$591$	0	0
$592$	−800.000	−0.0555402
$593$	10645.0	0.737163	0.368582	−	0.929595i	$-0.379844\pi$
$593$	10645.0	0.737163	0.368582	+	0.929595i	$0.379844\pi$
$594$	0	0
$595$	910.000	0.0626998
$596$	−4784.00	−0.328792
$597$	0	0
$598$	2574.00	0.176018
$599$	−6313.00	−0.430621	−0.215311	−	0.976546i	$-0.569076\pi$
$599$	−6313.00	−0.430621	−0.215311	+	0.976546i	$0.569076\pi$
$600$	0	0
$601$	−22162.0	−1.50417	−0.752086	−	0.659065i	$-0.770952\pi$
$601$	−22162.0	−1.50417	−0.752086	+	0.659065i	$0.770952\pi$
$602$	−266.000	−0.0180089
$603$	0	0
$604$	−3560.00	−0.239825
$605$	−175.000	−0.0117599
$606$	0	0
$607$	28716.0	1.92018	0.960088	−	0.279699i	$-0.0902347\pi$
$607$	28716.0	1.92018	0.960088	+	0.279699i	$0.0902347\pi$
$608$	1504.00	0.100321
$609$	0	0
$610$	−3100.00	−0.205763
$611$	2483.00	0.164405
$612$	0	0
$613$	−5860.00	−0.386106	−0.193053	−	0.981188i	$-0.561839\pi$
$613$	−5860.00	−0.386106	−0.193053	+	0.981188i	$0.561839\pi$
$614$	−7686.00	−0.505182
$615$	0	0
$616$	2016.00	0.131862
$617$	16154.0	1.05403	0.527014	−	0.849856i	$-0.323311\pi$
$617$	16154.0	1.05403	0.527014	+	0.849856i	$0.323311\pi$
$618$	0	0
$619$	9644.00	0.626212	0.313106	−	0.949718i	$-0.398631\pi$
$619$	9644.00	0.626212	0.313106	+	0.949718i	$0.398631\pi$
$620$	−460.000	−0.0297968
$621$	0	0
$622$	3496.00	0.225365
$623$	−4445.00	−0.285851
$624$	0	0
$625$	6875.00	0.440000
$626$	18784.0	1.19930
$627$	0	0
$628$	552.000	0.0350752
$629$	1300.00	0.0824076
$630$	0	0
$631$	8682.00	0.547742	0.273871	−	0.961766i	$-0.411696\pi$
$631$	8682.00	0.547742	0.273871	+	0.961766i	$0.411696\pi$
$632$	9528.00	0.599689
$633$	0	0
$634$	20760.0	1.30045
$635$	2880.00	0.179983
$636$	0	0
$637$	−637.000	−0.0396214
$638$	−4392.00	−0.272541
$639$	0	0
$640$	−640.000	−0.0395285
$641$	13791.0	0.849784	0.424892	−	0.905244i	$-0.360312\pi$
$641$	13791.0	0.849784	0.424892	+	0.905244i	$0.360312\pi$
$642$	0	0
$643$	−15316.0	−0.939353	−0.469677	−	0.882839i	$-0.655629\pi$
$643$	−15316.0	−0.939353	−0.469677	+	0.882839i	$0.655629\pi$
$644$	−2772.00	−0.169615
$645$	0	0
$646$	−2444.00	−0.148851
$647$	−15244.0	−0.926280	−0.463140	−	0.886285i	$-0.653277\pi$
$647$	−15244.0	−0.926280	−0.463140	+	0.886285i	$0.653277\pi$
$648$	0	0
$649$	−9504.00	−0.574830
$650$	−2600.00	−0.156893
$651$	0	0
$652$	−15824.0	−0.950484
$653$	−90.0000	−0.00539353	−0.00269676	−	0.999996i	$-0.500858\pi$
$653$	−90.0000	−0.00539353	−0.00269676	+	0.999996i	$0.500858\pi$
$654$	0	0
$655$	10280.0	0.613241
$656$	−1120.00	−0.0666595
$657$	0	0
$658$	−2674.00	−0.158425
$659$	−10887.0	−0.643547	−0.321773	−	0.946817i	$-0.604279\pi$
$659$	−10887.0	−0.643547	−0.321773	+	0.946817i	$0.604279\pi$
$660$	0	0
$661$	−4475.00	−0.263324	−0.131662	−	0.991295i	$-0.542031\pi$
$661$	−4475.00	−0.263324	−0.131662	+	0.991295i	$0.542031\pi$
$662$	−12500.0	−0.733877
$663$	0	0
$664$	2072.00	0.121098
$665$	1645.00	0.0959254
$666$	0	0
$667$	6039.00	0.350571
$668$	14700.0	0.851437
$669$	0	0
$670$	1900.00	0.109557
$671$	11160.0	0.642067
$672$	0	0
$673$	−33451.0	−1.91596	−0.957980	−	0.286834i	$-0.907397\pi$
$673$	−33451.0	−1.91596	−0.957980	+	0.286834i	$0.907397\pi$
$674$	4630.00	0.264601
$675$	0	0
$676$	676.000	0.0384615
$677$	5556.00	0.315413	0.157706	−	0.987486i	$-0.449590\pi$
$677$	5556.00	0.315413	0.157706	+	0.987486i	$0.449590\pi$
$678$	0	0
$679$	−931.000	−0.0526193
$680$	1040.00	0.0586503
$681$	0	0
$682$	1656.00	0.0929788
$683$	6504.00	0.364376	0.182188	−	0.983264i	$-0.441682\pi$
$683$	6504.00	0.364376	0.182188	+	0.983264i	$0.441682\pi$
$684$	0	0
$685$	9210.00	0.513717
$686$	686.000	0.0381802
$687$	0	0
$688$	−304.000	−0.0168458
$689$	2535.00	0.140168
$690$	0	0
$691$	16963.0	0.933868	0.466934	−	0.884292i	$-0.345359\pi$
$691$	16963.0	0.933868	0.466934	+	0.884292i	$0.345359\pi$
$692$	11304.0	0.620973
$693$	0	0
$694$	6592.00	0.360560
$695$	−6440.00	−0.351487
$696$	0	0
$697$	1820.00	0.0989059
$698$	11214.0	0.608103
$699$	0	0
$700$	2800.00	0.151186
$701$	−12805.0	−0.689926	−0.344963	−	0.938616i	$-0.612109\pi$
$701$	−12805.0	−0.689926	−0.344963	+	0.938616i	$0.612109\pi$
$702$	0	0
$703$	2350.00	0.126077
$704$	2304.00	0.123346
$705$	0	0
$706$	7444.00	0.396825
$707$	1064.00	0.0565995
$708$	0	0
$709$	10772.0	0.570594	0.285297	−	0.958439i	$-0.407908\pi$
$709$	10772.0	0.570594	0.285297	+	0.958439i	$0.407908\pi$
$710$	−1660.00	−0.0877446
$711$	0	0
$712$	−5080.00	−0.267389
$713$	−2277.00	−0.119599
$714$	0	0
$715$	−2340.00	−0.122393
$716$	12940.0	0.675406
$717$	0	0
$718$	−4860.00	−0.252609
$719$	22524.0	1.16829	0.584147	−	0.811648i	$-0.301429\pi$
$719$	22524.0	1.16829	0.584147	+	0.811648i	$0.301429\pi$
$720$	0	0
$721$	11648.0	0.601656
$722$	9300.00	0.479377
$723$	0	0
$724$	5040.00	0.258716
$725$	−6100.00	−0.312480
$726$	0	0
$727$	−5754.00	−0.293541	−0.146770	−	0.989171i	$-0.546888\pi$
$727$	−5754.00	−0.293541	−0.146770	+	0.989171i	$0.546888\pi$
$728$	−728.000	−0.0370625
$729$	0	0
$730$	−8730.00	−0.442619
$731$	494.000	0.0249949
$732$	0	0
$733$	31817.0	1.60326	0.801629	−	0.597822i	$-0.203967\pi$
$733$	31817.0	1.60326	0.801629	+	0.597822i	$0.203967\pi$
$734$	17168.0	0.863328
$735$	0	0
$736$	−3168.00	−0.158660
$737$	−6840.00	−0.341865
$738$	0	0
$739$	30820.0	1.53414	0.767072	−	0.641561i	$-0.221713\pi$
$739$	30820.0	1.53414	0.767072	+	0.641561i	$0.221713\pi$
$740$	−1000.00	−0.0496767
$741$	0	0
$742$	−2730.00	−0.135069
$743$	5724.00	0.282629	0.141314	−	0.989965i	$-0.454867\pi$
$743$	5724.00	0.282629	0.141314	+	0.989965i	$0.454867\pi$
$744$	0	0
$745$	−5980.00	−0.294081
$746$	14548.0	0.713995
$747$	0	0
$748$	−3744.00	−0.183014
$749$	−252.000	−0.0122936
$750$	0	0
$751$	20397.0	0.991075	0.495537	−	0.868587i	$-0.334971\pi$
$751$	20397.0	0.991075	0.495537	+	0.868587i	$0.334971\pi$
$752$	−3056.00	−0.148193
$753$	0	0
$754$	1586.00	0.0766031
$755$	−4450.00	−0.214506
$756$	0	0
$757$	−21103.0	−1.01321	−0.506606	−	0.862178i	$-0.669100\pi$
$757$	−21103.0	−1.01321	−0.506606	+	0.862178i	$0.669100\pi$
$758$	2320.00	0.111169
$759$	0	0
$760$	1880.00	0.0897300
$761$	22209.0	1.05792	0.528959	−	0.848647i	$-0.322583\pi$
$761$	22209.0	1.05792	0.528959	+	0.848647i	$0.322583\pi$
$762$	0	0
$763$	1624.00	0.0770547
$764$	928.000	0.0439448
$765$	0	0
$766$	−11056.0	−0.521501
$767$	3432.00	0.161568
$768$	0	0
$769$	27895.0	1.30809	0.654044	−	0.756457i	$-0.273071\pi$
$769$	27895.0	1.30809	0.654044	+	0.756457i	$0.273071\pi$
$770$	2520.00	0.117941
$771$	0	0
$772$	−21368.0	−0.996180
$773$	−24722.0	−1.15031	−0.575154	−	0.818045i	$-0.695058\pi$
$773$	−24722.0	−1.15031	−0.575154	+	0.818045i	$0.695058\pi$
$774$	0	0
$775$	2300.00	0.106604
$776$	−1064.00	−0.0492208
$777$	0	0
$778$	19332.0	0.890856
$779$	3290.00	0.151318
$780$	0	0
$781$	5976.00	0.273800
$782$	5148.00	0.235412
$783$	0	0
$784$	784.000	0.0357143
$785$	690.000	0.0313722
$786$	0	0
$787$	−40091.0	−1.81587	−0.907935	−	0.419111i	$-0.862342\pi$
$787$	−40091.0	−1.81587	−0.907935	+	0.419111i	$0.862342\pi$
$788$	−6168.00	−0.278840
$789$	0	0
$790$	11910.0	0.536378
$791$	9471.00	0.425727
$792$	0	0
$793$	−4030.00	−0.180466
$794$	−8250.00	−0.368742
$795$	0	0
$796$	8728.00	0.388638
$797$	−602.000	−0.0267552	−0.0133776	−	0.999911i	$-0.504258\pi$
$797$	−602.000	−0.0267552	−0.0133776	+	0.999911i	$0.504258\pi$
$798$	0	0
$799$	4966.00	0.219880
$800$	3200.00	0.141421
$801$	0	0
$802$	−23592.0	−1.03873
$803$	31428.0	1.38116
$804$	0	0
$805$	−3465.00	−0.151708
$806$	−598.000	−0.0261336
$807$	0	0
$808$	1216.00	0.0529440
$809$	38963.0	1.69328	0.846642	−	0.532163i	$-0.178621\pi$
$809$	38963.0	1.69328	0.846642	+	0.532163i	$0.178621\pi$
$810$	0	0
$811$	18116.0	0.784388	0.392194	−	0.919882i	$-0.371716\pi$
$811$	18116.0	0.784388	0.392194	+	0.919882i	$0.371716\pi$
$812$	−1708.00	−0.0738166
$813$	0	0
$814$	3600.00	0.155012
$815$	−19780.0	−0.850139
$816$	0	0
$817$	893.000	0.0382400
$818$	−16362.0	−0.699369
$819$	0	0
$820$	−1400.00	−0.0596221
$821$	−8070.00	−0.343051	−0.171526	−	0.985180i	$-0.554870\pi$
$821$	−8070.00	−0.343051	−0.171526	+	0.985180i	$0.554870\pi$
$822$	0	0
$823$	14664.0	0.621087	0.310544	−	0.950559i	$-0.399489\pi$
$823$	14664.0	0.621087	0.310544	+	0.950559i	$0.399489\pi$
$824$	13312.0	0.562798
$825$	0	0
$826$	−3696.00	−0.155690
$827$	12836.0	0.539724	0.269862	−	0.962899i	$-0.413022\pi$
$827$	12836.0	0.539724	0.269862	+	0.962899i	$0.413022\pi$
$828$	0	0
$829$	22664.0	0.949521	0.474761	−	0.880115i	$-0.342535\pi$
$829$	22664.0	0.949521	0.474761	+	0.880115i	$0.342535\pi$
$830$	2590.00	0.108314
$831$	0	0
$832$	−832.000	−0.0346688
$833$	−1274.00	−0.0529910
$834$	0	0
$835$	18375.0	0.761549
$836$	−6768.00	−0.279995
$837$	0	0
$838$	−18592.0	−0.766408
$839$	420.000	0.0172825	0.00864125	−	0.999963i	$-0.497249\pi$
$839$	420.000	0.0172825	0.00864125	+	0.999963i	$0.497249\pi$
$840$	0	0
$841$	−20668.0	−0.847431
$842$	−18148.0	−0.742781
$843$	0	0
$844$	−2092.00	−0.0853194
$845$	845.000	0.0344010
$846$	0	0
$847$	245.000	0.00993896
$848$	−3120.00	−0.126346
$849$	0	0
$850$	−5200.00	−0.209834
$851$	−4950.00	−0.199393
$852$	0	0
$853$	−5425.00	−0.217759	−0.108880	−	0.994055i	$-0.534726\pi$
$853$	−5425.00	−0.217759	−0.108880	+	0.994055i	$0.534726\pi$
$854$	4340.00	0.173901
$855$	0	0
$856$	−288.000	−0.0114996
$857$	33294.0	1.32707	0.663536	−	0.748144i	$-0.269055\pi$
$857$	33294.0	1.32707	0.663536	+	0.748144i	$0.269055\pi$
$858$	0	0
$859$	27386.0	1.08777	0.543887	−	0.839158i	$-0.316952\pi$
$859$	27386.0	1.08777	0.543887	+	0.839158i	$0.316952\pi$
$860$	−380.000	−0.0150673
$861$	0	0
$862$	18716.0	0.739524
$863$	−12004.0	−0.473489	−0.236744	−	0.971572i	$-0.576080\pi$
$863$	−12004.0	−0.473489	−0.236744	+	0.971572i	$0.576080\pi$
$864$	0	0
$865$	14130.0	0.555416
$866$	28784.0	1.12947
$867$	0	0
$868$	644.000	0.0251829
$869$	−42876.0	−1.67373
$870$	0	0
$871$	2470.00	0.0960881
$872$	1856.00	0.0720781
$873$	0	0
$874$	9306.00	0.360160
$875$	7875.00	0.304256
$876$	0	0
$877$	39346.0	1.51496	0.757480	−	0.652858i	$-0.226430\pi$
$877$	39346.0	1.51496	0.757480	+	0.652858i	$0.226430\pi$
$878$	12148.0	0.466942
$879$	0	0
$880$	2880.00	0.110324
$881$	−50806.0	−1.94290	−0.971452	−	0.237238i	$-0.923758\pi$
$881$	−50806.0	−1.94290	−0.971452	+	0.237238i	$0.923758\pi$
$882$	0	0
$883$	−34592.0	−1.31836	−0.659181	−	0.751984i	$-0.729097\pi$
$883$	−34592.0	−1.31836	−0.659181	+	0.751984i	$0.729097\pi$
$884$	1352.00	0.0514397
$885$	0	0
$886$	−12966.0	−0.491649
$887$	−34624.0	−1.31067	−0.655333	−	0.755340i	$-0.727472\pi$
$887$	−34624.0	−1.31067	−0.655333	+	0.755340i	$0.727472\pi$
$888$	0	0
$889$	−4032.00	−0.152114
$890$	−6350.00	−0.239160
$891$	0	0
$892$	−7924.00	−0.297439
$893$	8977.00	0.336398
$894$	0	0
$895$	16175.0	0.604101
$896$	896.000	0.0334077
$897$	0	0
$898$	30776.0	1.14366
$899$	−1403.00	−0.0520497
$900$	0	0
$901$	5070.00	0.187465
$902$	5040.00	0.186046
$903$	0	0
$904$	10824.0	0.398231
$905$	6300.00	0.231402
$906$	0	0
$907$	10849.0	0.397172	0.198586	−	0.980083i	$-0.436365\pi$
$907$	10849.0	0.397172	0.198586	+	0.980083i	$0.436365\pi$
$908$	−17408.0	−0.636239
$909$	0	0
$910$	−910.000	−0.0331497
$911$	−39671.0	−1.44276	−0.721382	−	0.692537i	$-0.756493\pi$
$911$	−39671.0	−1.44276	−0.721382	+	0.692537i	$0.756493\pi$
$912$	0	0
$913$	−9324.00	−0.337984
$914$	9284.00	0.335982
$915$	0	0
$916$	8520.00	0.307324
$917$	−14392.0	−0.518283
$918$	0	0
$919$	27208.0	0.976615	0.488307	−	0.872672i	$-0.337614\pi$
$919$	27208.0	0.976615	0.488307	+	0.872672i	$0.337614\pi$
$920$	−3960.00	−0.141910
$921$	0	0
$922$	−28532.0	−1.01914
$923$	−2158.00	−0.0769571
$924$	0	0
$925$	5000.00	0.177729
$926$	−2944.00	−0.104477
$927$	0	0
$928$	−1952.00	−0.0690491
$929$	27193.0	0.960359	0.480179	−	0.877170i	$-0.340572\pi$
$929$	27193.0	0.960359	0.480179	+	0.877170i	$0.340572\pi$
$930$	0	0
$931$	−2303.00	−0.0810717
$932$	−10748.0	−0.377749
$933$	0	0
$934$	2664.00	0.0933284
$935$	−4680.00	−0.163692
$936$	0	0
$937$	24262.0	0.845896	0.422948	−	0.906154i	$-0.360995\pi$
$937$	24262.0	0.845896	0.422948	+	0.906154i	$0.360995\pi$
$938$	−2660.00	−0.0925928
$939$	0	0
$940$	−3820.00	−0.132548
$941$	303.000	0.0104968	0.00524842	−	0.999986i	$-0.498329\pi$
$941$	303.000	0.0104968	0.00524842	+	0.999986i	$0.498329\pi$
$942$	0	0
$943$	−6930.00	−0.239313
$944$	−4224.00	−0.145635
$945$	0	0
$946$	1368.00	0.0470164
$947$	20098.0	0.689649	0.344824	−	0.938667i	$-0.387938\pi$
$947$	20098.0	0.689649	0.344824	+	0.938667i	$0.387938\pi$
$948$	0	0
$949$	−11349.0	−0.388202
$950$	−9400.00	−0.321028
$951$	0	0
$952$	−1456.00	−0.0495685
$953$	−13477.0	−0.458093	−0.229047	−	0.973415i	$-0.573561\pi$
$953$	−13477.0	−0.458093	−0.229047	+	0.973415i	$0.573561\pi$
$954$	0	0
$955$	1160.00	0.0393055
$956$	−15408.0	−0.521266
$957$	0	0
$958$	3258.00	0.109876
$959$	−12894.0	−0.434170
$960$	0	0
$961$	−29262.0	−0.982243
$962$	−1300.00	−0.0435693
$963$	0	0
$964$	4276.00	0.142864
$965$	−26710.0	−0.891011
$966$	0	0
$967$	−14488.0	−0.481802	−0.240901	−	0.970550i	$-0.577443\pi$
$967$	−14488.0	−0.481802	−0.240901	+	0.970550i	$0.577443\pi$
$968$	280.000	0.00929705
$969$	0	0
$970$	−1330.00	−0.0440245
$971$	−2830.00	−0.0935314	−0.0467657	−	0.998906i	$-0.514891\pi$
$971$	−2830.00	−0.0935314	−0.0467657	+	0.998906i	$0.514891\pi$
$972$	0	0
$973$	9016.00	0.297060
$974$	−27508.0	−0.904942
$975$	0	0
$976$	4960.00	0.162670
$977$	36016.0	1.17938	0.589690	−	0.807630i	$-0.299250\pi$
$977$	36016.0	1.17938	0.589690	+	0.807630i	$0.299250\pi$
$978$	0	0
$979$	22860.0	0.746281
$980$	980.000	0.0319438
$981$	0	0
$982$	−21808.0	−0.708677
$983$	16907.0	0.548575	0.274288	−	0.961648i	$-0.411558\pi$
$983$	16907.0	0.548575	0.274288	+	0.961648i	$0.411558\pi$
$984$	0	0
$985$	−7710.00	−0.249402
$986$	3172.00	0.102451
$987$	0	0
$988$	2444.00	0.0786984
$989$	−1881.00	−0.0604776
$990$	0	0
$991$	−51528.0	−1.65171	−0.825853	−	0.563885i	$-0.809306\pi$
$991$	−51528.0	−1.65171	−0.825853	+	0.563885i	$0.809306\pi$
$992$	736.000	0.0235565
$993$	0	0
$994$	2324.00	0.0741578
$995$	10910.0	0.347608
$996$	0	0
$997$	46492.0	1.47685	0.738423	−	0.674337i	$-0.235571\pi$
$997$	46492.0	1.47685	0.738423	+	0.674337i	$0.235571\pi$
$998$	−20788.0	−0.659351
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.e.1.1		1
3.2	odd	2		182.4.a.c.1.1	✓	1
12.11	even	2		1456.4.a.f.1.1		1
21.20	even	2		1274.4.a.g.1.1		1
39.38	odd	2		2366.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.c.1.1	✓	1	3.2	odd	2
1274.4.a.g.1.1		1	21.20	even	2
1456.4.a.f.1.1		1	12.11	even	2
1638.4.a.e.1.1		1	1.1	even	1	trivial
2366.4.a.b.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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