LMFDB - Embedded newform 1638.4.a.b.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,-9,0,-7,-8,0,18,-62] 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 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)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} -9.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} +18.0000 q^{10} -62.0000 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +16.0000 q^{17} +79.0000 q^{19} -36.0000 q^{20} +124.000 q^{22} +155.000 q^{23} -44.0000 q^{25} +26.0000 q^{26} -28.0000 q^{28} -51.0000 q^{29} +243.000 q^{31} -32.0000 q^{32} -32.0000 q^{34} +63.0000 q^{35} +412.000 q^{37} -158.000 q^{38} +72.0000 q^{40} +406.000 q^{41} -103.000 q^{43} -248.000 q^{44} -310.000 q^{46} -429.000 q^{47} +49.0000 q^{49} +88.0000 q^{50} -52.0000 q^{52} +169.000 q^{53} +558.000 q^{55} +56.0000 q^{56} +102.000 q^{58} -320.000 q^{59} -614.000 q^{61} -486.000 q^{62} +64.0000 q^{64} +117.000 q^{65} +258.000 q^{67} +64.0000 q^{68} -126.000 q^{70} +264.000 q^{71} -121.000 q^{73} -824.000 q^{74} +316.000 q^{76} +434.000 q^{77} -967.000 q^{79} -144.000 q^{80} -812.000 q^{82} +679.000 q^{83} -144.000 q^{85} +206.000 q^{86} +496.000 q^{88} -1059.00 q^{89} +91.0000 q^{91} +620.000 q^{92} +858.000 q^{94} -711.000 q^{95} -21.0000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	18.0000	0.569210
$11$	−62.0000	−1.69943	−0.849714	−	0.527244i	$-0.823225\pi$
$11$	−62.0000	−1.69943	−0.849714	+	0.527244i	$0.823225\pi$
$12$	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
$17$	16.0000	0.228269	0.114134	−	0.993465i	$-0.463591\pi$
$17$	16.0000	0.228269	0.114134	+	0.993465i	$0.463591\pi$
$18$	0	0
$19$	79.0000	0.953886	0.476943	−	0.878934i	$-0.341745\pi$
$19$	79.0000	0.953886	0.476943	+	0.878934i	$0.341745\pi$
$20$	−36.0000	−0.402492
$21$	0	0
$22$	124.000	1.20168
$23$	155.000	1.40521	0.702603	−	0.711582i	$-0.252021\pi$
$23$	155.000	1.40521	0.702603	+	0.711582i	$0.252021\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	26.0000	0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−51.0000	−0.326568	−0.163284	−	0.986579i	$-0.552209\pi$
$29$	−51.0000	−0.326568	−0.163284	+	0.986579i	$0.552209\pi$
$30$	0	0
$31$	243.000	1.40787	0.703937	−	0.710263i	$-0.251424\pi$
$31$	243.000	1.40787	0.703937	+	0.710263i	$0.251424\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−32.0000	−0.161410
$35$	63.0000	0.304256
$36$	0	0
$37$	412.000	1.83060	0.915302	−	0.402767i	$-0.131952\pi$
$37$	412.000	1.83060	0.915302	+	0.402767i	$0.131952\pi$
$38$	−158.000	−0.674500
$39$	0	0
$40$	72.0000	0.284605
$41$	406.000	1.54650	0.773251	−	0.634101i	$-0.218629\pi$
$41$	406.000	1.54650	0.773251	+	0.634101i	$0.218629\pi$
$42$	0	0
$43$	−103.000	−0.365287	−0.182644	−	0.983179i	$-0.558465\pi$
$43$	−103.000	−0.365287	−0.182644	+	0.983179i	$0.558465\pi$
$44$	−248.000	−0.849714
$45$	0	0
$46$	−310.000	−0.993631
$47$	−429.000	−1.33141	−0.665703	−	0.746217i	$-0.731868\pi$
$47$	−429.000	−1.33141	−0.665703	+	0.746217i	$0.731868\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	88.0000	0.248902
$51$	0	0
$52$	−52.0000	−0.138675
$53$	169.000	0.437999	0.218999	−	0.975725i	$-0.429721\pi$
$53$	169.000	0.437999	0.218999	+	0.975725i	$0.429721\pi$
$54$	0	0
$55$	558.000	1.36801
$56$	56.0000	0.133631
$57$	0	0
$58$	102.000	0.230918
$59$	−320.000	−0.706109	−0.353055	−	0.935603i	$-0.614857\pi$
$59$	−320.000	−0.706109	−0.353055	+	0.935603i	$0.614857\pi$
$60$	0	0
$61$	−614.000	−1.28876	−0.644382	−	0.764703i	$-0.722885\pi$
$61$	−614.000	−1.28876	−0.644382	+	0.764703i	$0.722885\pi$
$62$	−486.000	−0.995517
$63$	0	0
$64$	64.0000	0.125000
$65$	117.000	0.223263
$66$	0	0
$67$	258.000	0.470444	0.235222	−	0.971942i	$-0.424418\pi$
$67$	258.000	0.470444	0.235222	+	0.971942i	$0.424418\pi$
$68$	64.0000	0.114134
$69$	0	0
$70$	−126.000	−0.215141
$71$	264.000	0.441282	0.220641	−	0.975355i	$-0.429185\pi$
$71$	264.000	0.441282	0.220641	+	0.975355i	$0.429185\pi$
$72$	0	0
$73$	−121.000	−0.194000	−0.0969999	−	0.995284i	$-0.530925\pi$
$73$	−121.000	−0.194000	−0.0969999	+	0.995284i	$0.530925\pi$
$74$	−824.000	−1.29443
$75$	0	0
$76$	316.000	0.476943
$77$	434.000	0.642323
$78$	0	0
$79$	−967.000	−1.37716	−0.688582	−	0.725158i	$-0.741767\pi$
$79$	−967.000	−1.37716	−0.688582	+	0.725158i	$0.741767\pi$
$80$	−144.000	−0.201246
$81$	0	0
$82$	−812.000	−1.09354
$83$	679.000	0.897951	0.448975	−	0.893544i	$-0.351789\pi$
$83$	679.000	0.897951	0.448975	+	0.893544i	$0.351789\pi$
$84$	0	0
$85$	−144.000	−0.183753
$86$	206.000	0.258297
$87$	0	0
$88$	496.000	0.600838
$89$	−1059.00	−1.26128	−0.630639	−	0.776076i	$-0.717207\pi$
$89$	−1059.00	−1.26128	−0.630639	+	0.776076i	$0.717207\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	620.000	0.702603
$93$	0	0
$94$	858.000	0.941446
$95$	−711.000	−0.767864
$96$	0	0
$97$	−21.0000	−0.0219817	−0.0109909	−	0.999940i	$-0.503499\pi$
$97$	−21.0000	−0.0219817	−0.0109909	+	0.999940i	$0.503499\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−176.000	−0.176000
$101$	−894.000	−0.880756	−0.440378	−	0.897813i	$-0.645156\pi$
$101$	−894.000	−0.880756	−0.440378	+	0.897813i	$0.645156\pi$
$102$	0	0
$103$	128.000	0.122449	0.0612243	−	0.998124i	$-0.480499\pi$
$103$	128.000	0.122449	0.0612243	+	0.998124i	$0.480499\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−338.000	−0.309712
$107$	708.000	0.639672	0.319836	−	0.947473i	$-0.396372\pi$
$107$	708.000	0.639672	0.319836	+	0.947473i	$0.396372\pi$
$108$	0	0
$109$	1238.00	1.08788	0.543940	−	0.839124i	$-0.316932\pi$
$109$	1238.00	1.08788	0.543940	+	0.839124i	$0.316932\pi$
$110$	−1116.00	−0.967331
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−1073.00	−0.893269	−0.446634	−	0.894717i	$-0.647377\pi$
$113$	−1073.00	−0.893269	−0.446634	+	0.894717i	$0.647377\pi$
$114$	0	0
$115$	−1395.00	−1.13117
$116$	−204.000	−0.163284
$117$	0	0
$118$	640.000	0.499295
$119$	−112.000	−0.0862775
$120$	0	0
$121$	2513.00	1.88805
$122$	1228.00	0.911294
$123$	0	0
$124$	972.000	0.703937
$125$	1521.00	1.08834
$126$	0	0
$127$	−1748.00	−1.22134	−0.610669	−	0.791886i	$-0.709099\pi$
$127$	−1748.00	−1.22134	−0.610669	+	0.791886i	$0.709099\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−234.000	−0.157870
$131$	936.000	0.624265	0.312132	−	0.950039i	$-0.398957\pi$
$131$	936.000	0.624265	0.312132	+	0.950039i	$0.398957\pi$
$132$	0	0
$133$	−553.000	−0.360535
$134$	−516.000	−0.332654
$135$	0	0
$136$	−128.000	−0.0807052
$137$	1128.00	0.703442	0.351721	−	0.936105i	$-0.385597\pi$
$137$	1128.00	0.703442	0.351721	+	0.936105i	$0.385597\pi$
$138$	0	0
$139$	−854.000	−0.521118	−0.260559	−	0.965458i	$-0.583907\pi$
$139$	−854.000	−0.521118	−0.260559	+	0.965458i	$0.583907\pi$
$140$	252.000	0.152128
$141$	0	0
$142$	−528.000	−0.312034
$143$	806.000	0.471336
$144$	0	0
$145$	459.000	0.262882
$146$	242.000	0.137179
$147$	0	0
$148$	1648.00	0.915302
$149$	442.000	0.243020	0.121510	−	0.992590i	$-0.461226\pi$
$149$	442.000	0.243020	0.121510	+	0.992590i	$0.461226\pi$
$150$	0	0
$151$	−1016.00	−0.547556	−0.273778	−	0.961793i	$-0.588273\pi$
$151$	−1016.00	−0.547556	−0.273778	+	0.961793i	$0.588273\pi$
$152$	−632.000	−0.337250
$153$	0	0
$154$	−868.000	−0.454191
$155$	−2187.00	−1.13332
$156$	0	0
$157$	−3880.00	−1.97234	−0.986171	−	0.165731i	$-0.947002\pi$
$157$	−3880.00	−1.97234	−0.986171	+	0.165731i	$0.947002\pi$
$158$	1934.00	0.973802
$159$	0	0
$160$	288.000	0.142302
$161$	−1085.00	−0.531118
$162$	0	0
$163$	−1604.00	−0.770767	−0.385383	−	0.922757i	$-0.625931\pi$
$163$	−1604.00	−0.770767	−0.385383	+	0.922757i	$0.625931\pi$
$164$	1624.00	0.773251
$165$	0	0
$166$	−1358.00	−0.634947
$167$	−959.000	−0.444369	−0.222185	−	0.975005i	$-0.571319\pi$
$167$	−959.000	−0.444369	−0.222185	+	0.975005i	$0.571319\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	288.000	0.129933
$171$	0	0
$172$	−412.000	−0.182644
$173$	488.000	0.214462	0.107231	−	0.994234i	$-0.465802\pi$
$173$	488.000	0.214462	0.107231	+	0.994234i	$0.465802\pi$
$174$	0	0
$175$	308.000	0.133043
$176$	−992.000	−0.424857
$177$	0	0
$178$	2118.00	0.891858
$179$	−2533.00	−1.05768	−0.528842	−	0.848721i	$-0.677373\pi$
$179$	−2533.00	−1.05768	−0.528842	+	0.848721i	$0.677373\pi$
$180$	0	0
$181$	−3626.00	−1.48905	−0.744526	−	0.667593i	$-0.767325\pi$
$181$	−3626.00	−1.48905	−0.744526	+	0.667593i	$0.767325\pi$
$182$	−182.000	−0.0741249
$183$	0	0
$184$	−1240.00	−0.496815
$185$	−3708.00	−1.47361
$186$	0	0
$187$	−992.000	−0.387926
$188$	−1716.00	−0.665703
$189$	0	0
$190$	1422.00	0.542962
$191$	4992.00	1.89114	0.945572	−	0.325413i	$-0.105503\pi$
$191$	4992.00	1.89114	0.945572	+	0.325413i	$0.105503\pi$
$192$	0	0
$193$	−470.000	−0.175292	−0.0876460	−	0.996152i	$-0.527934\pi$
$193$	−470.000	−0.175292	−0.0876460	+	0.996152i	$0.527934\pi$
$194$	42.0000	0.0155434
$195$	0	0
$196$	196.000	0.0714286
$197$	3190.00	1.15370	0.576848	−	0.816852i	$-0.304283\pi$
$197$	3190.00	1.15370	0.576848	+	0.816852i	$0.304283\pi$
$198$	0	0
$199$	1944.00	0.692495	0.346248	−	0.938143i	$-0.387456\pi$
$199$	1944.00	0.692495	0.346248	+	0.938143i	$0.387456\pi$
$200$	352.000	0.124451
$201$	0	0
$202$	1788.00	0.622788
$203$	357.000	0.123431
$204$	0	0
$205$	−3654.00	−1.24491
$206$	−256.000	−0.0865843
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−4898.00	−1.62106
$210$	0	0
$211$	−1139.00	−0.371621	−0.185810	−	0.982586i	$-0.559491\pi$
$211$	−1139.00	−0.371621	−0.185810	+	0.982586i	$0.559491\pi$
$212$	676.000	0.218999
$213$	0	0
$214$	−1416.00	−0.452317
$215$	927.000	0.294051
$216$	0	0
$217$	−1701.00	−0.532126
$218$	−2476.00	−0.769247
$219$	0	0
$220$	2232.00	0.684006
$221$	−208.000	−0.0633104
$222$	0	0
$223$	833.000	0.250143	0.125071	−	0.992148i	$-0.460084\pi$
$223$	833.000	0.250143	0.125071	+	0.992148i	$0.460084\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	2146.00	0.631636
$227$	3516.00	1.02804	0.514020	−	0.857778i	$-0.328156\pi$
$227$	3516.00	1.02804	0.514020	+	0.857778i	$0.328156\pi$
$228$	0	0
$229$	3222.00	0.929763	0.464882	−	0.885373i	$-0.346097\pi$
$229$	3222.00	0.929763	0.464882	+	0.885373i	$0.346097\pi$
$230$	2790.00	0.799857
$231$	0	0
$232$	408.000	0.115459
$233$	477.000	0.134117	0.0670586	−	0.997749i	$-0.478639\pi$
$233$	477.000	0.134117	0.0670586	+	0.997749i	$0.478639\pi$
$234$	0	0
$235$	3861.00	1.07176
$236$	−1280.00	−0.353055
$237$	0	0
$238$	224.000	0.0610074
$239$	6368.00	1.72348	0.861740	−	0.507350i	$-0.169375\pi$
$239$	6368.00	1.72348	0.861740	+	0.507350i	$0.169375\pi$
$240$	0	0
$241$	747.000	0.199662	0.0998309	−	0.995004i	$-0.468170\pi$
$241$	747.000	0.199662	0.0998309	+	0.995004i	$0.468170\pi$
$242$	−5026.00	−1.33506
$243$	0	0
$244$	−2456.00	−0.644382
$245$	−441.000	−0.114998
$246$	0	0
$247$	−1027.00	−0.264561
$248$	−1944.00	−0.497759
$249$	0	0
$250$	−3042.00	−0.769572
$251$	−7518.00	−1.89057	−0.945283	−	0.326252i	$-0.894214\pi$
$251$	−7518.00	−1.89057	−0.945283	+	0.326252i	$0.894214\pi$
$252$	0	0
$253$	−9610.00	−2.38805
$254$	3496.00	0.863616
$255$	0	0
$256$	256.000	0.0625000
$257$	−1094.00	−0.265532	−0.132766	−	0.991147i	$-0.542386\pi$
$257$	−1094.00	−0.265532	−0.132766	+	0.991147i	$0.542386\pi$
$258$	0	0
$259$	−2884.00	−0.691904
$260$	468.000	0.111631
$261$	0	0
$262$	−1872.00	−0.441422
$263$	1289.00	0.302217	0.151109	−	0.988517i	$-0.451716\pi$
$263$	1289.00	0.302217	0.151109	+	0.988517i	$0.451716\pi$
$264$	0	0
$265$	−1521.00	−0.352582
$266$	1106.00	0.254937
$267$	0	0
$268$	1032.00	0.235222
$269$	2060.00	0.466916	0.233458	−	0.972367i	$-0.424996\pi$
$269$	2060.00	0.466916	0.233458	+	0.972367i	$0.424996\pi$
$270$	0	0
$271$	576.000	0.129113	0.0645563	−	0.997914i	$-0.479437\pi$
$271$	576.000	0.129113	0.0645563	+	0.997914i	$0.479437\pi$
$272$	256.000	0.0570672
$273$	0	0
$274$	−2256.00	−0.497409
$275$	2728.00	0.598199
$276$	0	0
$277$	−5699.00	−1.23617	−0.618086	−	0.786110i	$-0.712092\pi$
$277$	−5699.00	−1.23617	−0.618086	+	0.786110i	$0.712092\pi$
$278$	1708.00	0.368486
$279$	0	0
$280$	−504.000	−0.107571
$281$	−4062.00	−0.862344	−0.431172	−	0.902270i	$-0.641900\pi$
$281$	−4062.00	−0.862344	−0.431172	+	0.902270i	$0.641900\pi$
$282$	0	0
$283$	−772.000	−0.162158	−0.0810789	−	0.996708i	$-0.525837\pi$
$283$	−772.000	−0.162158	−0.0810789	+	0.996708i	$0.525837\pi$
$284$	1056.00	0.220641
$285$	0	0
$286$	−1612.00	−0.333285
$287$	−2842.00	−0.584522
$288$	0	0
$289$	−4657.00	−0.947893
$290$	−918.000	−0.185886
$291$	0	0
$292$	−484.000	−0.0969999
$293$	−7833.00	−1.56180	−0.780902	−	0.624653i	$-0.785240\pi$
$293$	−7833.00	−1.56180	−0.780902	+	0.624653i	$0.785240\pi$
$294$	0	0
$295$	2880.00	0.568407
$296$	−3296.00	−0.647217
$297$	0	0
$298$	−884.000	−0.171841
$299$	−2015.00	−0.389734
$300$	0	0
$301$	721.000	0.138066
$302$	2032.00	0.387180
$303$	0	0
$304$	1264.00	0.238472
$305$	5526.00	1.03744
$306$	0	0
$307$	−2891.00	−0.537453	−0.268727	−	0.963217i	$-0.586603\pi$
$307$	−2891.00	−0.537453	−0.268727	+	0.963217i	$0.586603\pi$
$308$	1736.00	0.321162
$309$	0	0
$310$	4374.00	0.801376
$311$	3530.00	0.643627	0.321813	−	0.946803i	$-0.395708\pi$
$311$	3530.00	0.643627	0.321813	+	0.946803i	$0.395708\pi$
$312$	0	0
$313$	5490.00	0.991416	0.495708	−	0.868489i	$-0.334909\pi$
$313$	5490.00	0.991416	0.495708	+	0.868489i	$0.334909\pi$
$314$	7760.00	1.39466
$315$	0	0
$316$	−3868.00	−0.688582
$317$	5636.00	0.998578	0.499289	−	0.866435i	$-0.333595\pi$
$317$	5636.00	0.998578	0.499289	+	0.866435i	$0.333595\pi$
$318$	0	0
$319$	3162.00	0.554978
$320$	−576.000	−0.100623
$321$	0	0
$322$	2170.00	0.375557
$323$	1264.00	0.217743
$324$	0	0
$325$	572.000	0.0976272
$326$	3208.00	0.545014
$327$	0	0
$328$	−3248.00	−0.546771
$329$	3003.00	0.503224
$330$	0	0
$331$	−9962.00	−1.65426	−0.827131	−	0.562008i	$-0.810029\pi$
$331$	−9962.00	−1.65426	−0.827131	+	0.562008i	$0.810029\pi$
$332$	2716.00	0.448975
$333$	0	0
$334$	1918.00	0.314216
$335$	−2322.00	−0.378700
$336$	0	0
$337$	−11415.0	−1.84515	−0.922574	−	0.385821i	$-0.873918\pi$
$337$	−11415.0	−1.84515	−0.922574	+	0.385821i	$0.873918\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−576.000	−0.0918764
$341$	−15066.0	−2.39258
$342$	0	0
$343$	−343.000	−0.0539949
$344$	824.000	0.129149
$345$	0	0
$346$	−976.000	−0.151648
$347$	−4864.00	−0.752488	−0.376244	−	0.926521i	$-0.622784\pi$
$347$	−4864.00	−0.752488	−0.376244	+	0.926521i	$0.622784\pi$
$348$	0	0
$349$	−469.000	−0.0719341	−0.0359670	−	0.999353i	$-0.511451\pi$
$349$	−469.000	−0.0719341	−0.0359670	+	0.999353i	$0.511451\pi$
$350$	−616.000	−0.0940760
$351$	0	0
$352$	1984.00	0.300419
$353$	−4198.00	−0.632966	−0.316483	−	0.948598i	$-0.602502\pi$
$353$	−4198.00	−0.632966	−0.316483	+	0.948598i	$0.602502\pi$
$354$	0	0
$355$	−2376.00	−0.355225
$356$	−4236.00	−0.630639
$357$	0	0
$358$	5066.00	0.747895
$359$	−3408.00	−0.501023	−0.250512	−	0.968114i	$-0.580599\pi$
$359$	−3408.00	−0.501023	−0.250512	+	0.968114i	$0.580599\pi$
$360$	0	0
$361$	−618.000	−0.0901006
$362$	7252.00	1.05292
$363$	0	0
$364$	364.000	0.0524142
$365$	1089.00	0.156167
$366$	0	0
$367$	−8290.00	−1.17911	−0.589557	−	0.807727i	$-0.700697\pi$
$367$	−8290.00	−1.17911	−0.589557	+	0.807727i	$0.700697\pi$
$368$	2480.00	0.351301
$369$	0	0
$370$	7416.00	1.04200
$371$	−1183.00	−0.165548
$372$	0	0
$373$	−9906.00	−1.37510	−0.687551	−	0.726136i	$-0.741314\pi$
$373$	−9906.00	−1.37510	−0.687551	+	0.726136i	$0.741314\pi$
$374$	1984.00	0.274305
$375$	0	0
$376$	3432.00	0.470723
$377$	663.000	0.0905736
$378$	0	0
$379$	−9238.00	−1.25204	−0.626021	−	0.779806i	$-0.715318\pi$
$379$	−9238.00	−1.25204	−0.626021	+	0.779806i	$0.715318\pi$
$380$	−2844.00	−0.383932
$381$	0	0
$382$	−9984.00	−1.33724
$383$	−4972.00	−0.663335	−0.331668	−	0.943396i	$-0.607611\pi$
$383$	−4972.00	−0.663335	−0.331668	+	0.943396i	$0.607611\pi$
$384$	0	0
$385$	−3906.00	−0.517060
$386$	940.000	0.123950
$387$	0	0
$388$	−84.0000	−0.0109909
$389$	−7454.00	−0.971550	−0.485775	−	0.874084i	$-0.661462\pi$
$389$	−7454.00	−0.971550	−0.485775	+	0.874084i	$0.661462\pi$
$390$	0	0
$391$	2480.00	0.320765
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−6380.00	−0.815786
$395$	8703.00	1.10860
$396$	0	0
$397$	7219.00	0.912623	0.456311	−	0.889820i	$-0.349170\pi$
$397$	7219.00	0.912623	0.456311	+	0.889820i	$0.349170\pi$
$398$	−3888.00	−0.489668
$399$	0	0
$400$	−704.000	−0.0880000
$401$	−10772.0	−1.34147	−0.670733	−	0.741699i	$-0.734020\pi$
$401$	−10772.0	−1.34147	−0.670733	+	0.741699i	$0.734020\pi$
$402$	0	0
$403$	−3159.00	−0.390474
$404$	−3576.00	−0.440378
$405$	0	0
$406$	−714.000	−0.0872789
$407$	−25544.0	−3.11098
$408$	0	0
$409$	13151.0	1.58991	0.794957	−	0.606665i	$-0.207493\pi$
$409$	13151.0	1.58991	0.794957	+	0.606665i	$0.207493\pi$
$410$	7308.00	0.880284
$411$	0	0
$412$	512.000	0.0612243
$413$	2240.00	0.266884
$414$	0	0
$415$	−6111.00	−0.722837
$416$	416.000	0.0490290
$417$	0	0
$418$	9796.00	1.14626
$419$	−6034.00	−0.703533	−0.351766	−	0.936088i	$-0.614419\pi$
$419$	−6034.00	−0.703533	−0.351766	+	0.936088i	$0.614419\pi$
$420$	0	0
$421$	8864.00	1.02614	0.513070	−	0.858347i	$-0.328508\pi$
$421$	8864.00	1.02614	0.513070	+	0.858347i	$0.328508\pi$
$422$	2278.00	0.262776
$423$	0	0
$424$	−1352.00	−0.154856
$425$	−704.000	−0.0803506
$426$	0	0
$427$	4298.00	0.487107
$428$	2832.00	0.319836
$429$	0	0
$430$	−1854.00	−0.207925
$431$	134.000	0.0149758	0.00748788	−	0.999972i	$-0.497617\pi$
$431$	134.000	0.0149758	0.00748788	+	0.999972i	$0.497617\pi$
$432$	0	0
$433$	12376.0	1.37356	0.686781	−	0.726864i	$-0.259023\pi$
$433$	12376.0	1.37356	0.686781	+	0.726864i	$0.259023\pi$
$434$	3402.00	0.376270
$435$	0	0
$436$	4952.00	0.543940
$437$	12245.0	1.34041
$438$	0	0
$439$	14198.0	1.54358	0.771792	−	0.635875i	$-0.219361\pi$
$439$	14198.0	1.54358	0.771792	+	0.635875i	$0.219361\pi$
$440$	−4464.00	−0.483666
$441$	0	0
$442$	416.000	0.0447672
$443$	−7657.00	−0.821208	−0.410604	−	0.911814i	$-0.634682\pi$
$443$	−7657.00	−0.821208	−0.410604	+	0.911814i	$0.634682\pi$
$444$	0	0
$445$	9531.00	1.01531
$446$	−1666.00	−0.176878
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−15500.0	−1.62915	−0.814577	−	0.580055i	$-0.803031\pi$
$449$	−15500.0	−1.62915	−0.814577	+	0.580055i	$0.803031\pi$
$450$	0	0
$451$	−25172.0	−2.62817
$452$	−4292.00	−0.446634
$453$	0	0
$454$	−7032.00	−0.726934
$455$	−819.000	−0.0843853
$456$	0	0
$457$	10184.0	1.04242	0.521212	−	0.853427i	$-0.325480\pi$
$457$	10184.0	1.04242	0.521212	+	0.853427i	$0.325480\pi$
$458$	−6444.00	−0.657442
$459$	0	0
$460$	−5580.00	−0.565584
$461$	−19250.0	−1.94482	−0.972410	−	0.233279i	$-0.925054\pi$
$461$	−19250.0	−1.94482	−0.972410	+	0.233279i	$0.925054\pi$
$462$	0	0
$463$	−15622.0	−1.56807	−0.784034	−	0.620717i	$-0.786841\pi$
$463$	−15622.0	−1.56807	−0.784034	+	0.620717i	$0.786841\pi$
$464$	−816.000	−0.0816419
$465$	0	0
$466$	−954.000	−0.0948352
$467$	15594.0	1.54519	0.772596	−	0.634898i	$-0.218958\pi$
$467$	15594.0	1.54519	0.772596	+	0.634898i	$0.218958\pi$
$468$	0	0
$469$	−1806.00	−0.177811
$470$	−7722.00	−0.757850
$471$	0	0
$472$	2560.00	0.249647
$473$	6386.00	0.620779
$474$	0	0
$475$	−3476.00	−0.335768
$476$	−448.000	−0.0431388
$477$	0	0
$478$	−12736.0	−1.21868
$479$	−3239.00	−0.308964	−0.154482	−	0.987996i	$-0.549371\pi$
$479$	−3239.00	−0.308964	−0.154482	+	0.987996i	$0.549371\pi$
$480$	0	0
$481$	−5356.00	−0.507718
$482$	−1494.00	−0.141182
$483$	0	0
$484$	10052.0	0.944027
$485$	189.000	0.0176949
$486$	0	0
$487$	1798.00	0.167300	0.0836501	−	0.996495i	$-0.473342\pi$
$487$	1798.00	0.167300	0.0836501	+	0.996495i	$0.473342\pi$
$488$	4912.00	0.455647
$489$	0	0
$490$	882.000	0.0813157
$491$	−13708.0	−1.25995	−0.629973	−	0.776617i	$-0.716934\pi$
$491$	−13708.0	−1.25995	−0.629973	+	0.776617i	$0.716934\pi$
$492$	0	0
$493$	−816.000	−0.0745452
$494$	2054.00	0.187073
$495$	0	0
$496$	3888.00	0.351968
$497$	−1848.00	−0.166789
$498$	0	0
$499$	−11936.0	−1.07080	−0.535400	−	0.844599i	$-0.679839\pi$
$499$	−11936.0	−1.07080	−0.535400	+	0.844599i	$0.679839\pi$
$500$	6084.00	0.544170
$501$	0	0
$502$	15036.0	1.33683
$503$	−2562.00	−0.227105	−0.113553	−	0.993532i	$-0.536223\pi$
$503$	−2562.00	−0.227105	−0.113553	+	0.993532i	$0.536223\pi$
$504$	0	0
$505$	8046.00	0.708995
$506$	19220.0	1.68860
$507$	0	0
$508$	−6992.00	−0.610669
$509$	8055.00	0.701437	0.350719	−	0.936481i	$-0.385937\pi$
$509$	8055.00	0.701437	0.350719	+	0.936481i	$0.385937\pi$
$510$	0	0
$511$	847.000	0.0733250
$512$	−512.000	−0.0441942
$513$	0	0
$514$	2188.00	0.187760
$515$	−1152.00	−0.0985693
$516$	0	0
$517$	26598.0	2.26263
$518$	5768.00	0.489250
$519$	0	0
$520$	−936.000	−0.0789352
$521$	5980.00	0.502857	0.251429	−	0.967876i	$-0.419100\pi$
$521$	5980.00	0.502857	0.251429	+	0.967876i	$0.419100\pi$
$522$	0	0
$523$	19238.0	1.60845	0.804225	−	0.594325i	$-0.202581\pi$
$523$	19238.0	1.60845	0.804225	+	0.594325i	$0.202581\pi$
$524$	3744.00	0.312132
$525$	0	0
$526$	−2578.00	−0.213700
$527$	3888.00	0.321374
$528$	0	0
$529$	11858.0	0.974603
$530$	3042.00	0.249313
$531$	0	0
$532$	−2212.00	−0.180268
$533$	−5278.00	−0.428922
$534$	0	0
$535$	−6372.00	−0.514926
$536$	−2064.00	−0.166327
$537$	0	0
$538$	−4120.00	−0.330160
$539$	−3038.00	−0.242775
$540$	0	0
$541$	13012.0	1.03407	0.517033	−	0.855966i	$-0.327036\pi$
$541$	13012.0	1.03407	0.517033	+	0.855966i	$0.327036\pi$
$542$	−1152.00	−0.0912964
$543$	0	0
$544$	−512.000	−0.0403526
$545$	−11142.0	−0.875726
$546$	0	0
$547$	18055.0	1.41129	0.705645	−	0.708565i	$-0.250657\pi$
$547$	18055.0	1.41129	0.705645	+	0.708565i	$0.250657\pi$
$548$	4512.00	0.351721
$549$	0	0
$550$	−5456.00	−0.422990
$551$	−4029.00	−0.311508
$552$	0	0
$553$	6769.00	0.520519
$554$	11398.0	0.874106
$555$	0	0
$556$	−3416.00	−0.260559
$557$	19608.0	1.49159	0.745797	−	0.666174i	$-0.232069\pi$
$557$	19608.0	1.49159	0.745797	+	0.666174i	$0.232069\pi$
$558$	0	0
$559$	1339.00	0.101312
$560$	1008.00	0.0760639
$561$	0	0
$562$	8124.00	0.609769
$563$	13512.0	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	13512.0	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	9657.00	0.719067
$566$	1544.00	0.114663
$567$	0	0
$568$	−2112.00	−0.156017
$569$	4061.00	0.299202	0.149601	−	0.988746i	$-0.452201\pi$
$569$	4061.00	0.299202	0.149601	+	0.988746i	$0.452201\pi$
$570$	0	0
$571$	−2255.00	−0.165269	−0.0826347	−	0.996580i	$-0.526333\pi$
$571$	−2255.00	−0.165269	−0.0826347	+	0.996580i	$0.526333\pi$
$572$	3224.00	0.235668
$573$	0	0
$574$	5684.00	0.413320
$575$	−6820.00	−0.494632
$576$	0	0
$577$	−254.000	−0.0183261	−0.00916305	−	0.999958i	$-0.502917\pi$
$577$	−254.000	−0.0183261	−0.00916305	+	0.999958i	$0.502917\pi$
$578$	9314.00	0.670262
$579$	0	0
$580$	1836.00	0.131441
$581$	−4753.00	−0.339394
$582$	0	0
$583$	−10478.0	−0.744347
$584$	968.000	0.0685893
$585$	0	0
$586$	15666.0	1.10436
$587$	−15057.0	−1.05872	−0.529360	−	0.848397i	$-0.677568\pi$
$587$	−15057.0	−1.05872	−0.529360	+	0.848397i	$0.677568\pi$
$588$	0	0
$589$	19197.0	1.34295
$590$	−5760.00	−0.401924
$591$	0	0
$592$	6592.00	0.457651
$593$	−19917.0	−1.37925	−0.689623	−	0.724168i	$-0.742224\pi$
$593$	−19917.0	−1.37925	−0.689623	+	0.724168i	$0.742224\pi$
$594$	0	0
$595$	1008.00	0.0694521
$596$	1768.00	0.121510
$597$	0	0
$598$	4030.00	0.275584
$599$	15107.0	1.03048	0.515238	−	0.857047i	$-0.327703\pi$
$599$	15107.0	1.03048	0.515238	+	0.857047i	$0.327703\pi$
$600$	0	0
$601$	23674.0	1.60679	0.803397	−	0.595444i	$-0.203024\pi$
$601$	23674.0	1.60679	0.803397	+	0.595444i	$0.203024\pi$
$602$	−1442.00	−0.0976271
$603$	0	0
$604$	−4064.00	−0.273778
$605$	−22617.0	−1.51985
$606$	0	0
$607$	23746.0	1.58784	0.793921	−	0.608021i	$-0.208036\pi$
$607$	23746.0	1.58784	0.793921	+	0.608021i	$0.208036\pi$
$608$	−2528.00	−0.168625
$609$	0	0
$610$	−11052.0	−0.733578
$611$	5577.00	0.369266
$612$	0	0
$613$	−19216.0	−1.26611	−0.633056	−	0.774106i	$-0.718200\pi$
$613$	−19216.0	−1.26611	−0.633056	+	0.774106i	$0.718200\pi$
$614$	5782.00	0.380037
$615$	0	0
$616$	−3472.00	−0.227096
$617$	−20862.0	−1.36122	−0.680610	−	0.732646i	$-0.738285\pi$
$617$	−20862.0	−1.36122	−0.680610	+	0.732646i	$0.738285\pi$
$618$	0	0
$619$	124.000	0.00805167	0.00402583	−	0.999992i	$-0.498719\pi$
$619$	124.000	0.00805167	0.00402583	+	0.999992i	$0.498719\pi$
$620$	−8748.00	−0.566658
$621$	0	0
$622$	−7060.00	−0.455113
$623$	7413.00	0.476718
$624$	0	0
$625$	−8189.00	−0.524096
$626$	−10980.0	−0.701037
$627$	0	0
$628$	−15520.0	−0.986171
$629$	6592.00	0.417870
$630$	0	0
$631$	23942.0	1.51048	0.755242	−	0.655446i	$-0.227519\pi$
$631$	23942.0	1.51048	0.755242	+	0.655446i	$0.227519\pi$
$632$	7736.00	0.486901
$633$	0	0
$634$	−11272.0	−0.706101
$635$	15732.0	0.983158
$636$	0	0
$637$	−637.000	−0.0396214
$638$	−6324.00	−0.392429
$639$	0	0
$640$	1152.00	0.0711512
$641$	15723.0	0.968832	0.484416	−	0.874838i	$-0.339032\pi$
$641$	15723.0	0.968832	0.484416	+	0.874838i	$0.339032\pi$
$642$	0	0
$643$	−5544.00	−0.340022	−0.170011	−	0.985442i	$-0.554380\pi$
$643$	−5544.00	−0.340022	−0.170011	+	0.985442i	$0.554380\pi$
$644$	−4340.00	−0.265559
$645$	0	0
$646$	−2528.00	−0.153967
$647$	6078.00	0.369321	0.184661	−	0.982802i	$-0.440881\pi$
$647$	6078.00	0.369321	0.184661	+	0.982802i	$0.440881\pi$
$648$	0	0
$649$	19840.0	1.19998
$650$	−1144.00	−0.0690329
$651$	0	0
$652$	−6416.00	−0.385383
$653$	−30778.0	−1.84447	−0.922233	−	0.386635i	$-0.873637\pi$
$653$	−30778.0	−1.84447	−0.922233	+	0.386635i	$0.873637\pi$
$654$	0	0
$655$	−8424.00	−0.502524
$656$	6496.00	0.386625
$657$	0	0
$658$	−6006.00	−0.355833
$659$	11205.0	0.662344	0.331172	−	0.943570i	$-0.392556\pi$
$659$	11205.0	0.662344	0.331172	+	0.943570i	$0.392556\pi$
$660$	0	0
$661$	−20785.0	−1.22306	−0.611530	−	0.791221i	$-0.709446\pi$
$661$	−20785.0	−1.22306	−0.611530	+	0.791221i	$0.709446\pi$
$662$	19924.0	1.16974
$663$	0	0
$664$	−5432.00	−0.317474
$665$	4977.00	0.290225
$666$	0	0
$667$	−7905.00	−0.458895
$668$	−3836.00	−0.222185
$669$	0	0
$670$	4644.00	0.267781
$671$	38068.0	2.19016
$672$	0	0
$673$	1297.00	0.0742878	0.0371439	−	0.999310i	$-0.488174\pi$
$673$	1297.00	0.0742878	0.0371439	+	0.999310i	$0.488174\pi$
$674$	22830.0	1.30472
$675$	0	0
$676$	676.000	0.0384615
$677$	−7310.00	−0.414987	−0.207493	−	0.978236i	$-0.566531\pi$
$677$	−7310.00	−0.414987	−0.207493	+	0.978236i	$0.566531\pi$
$678$	0	0
$679$	147.000	0.00830831
$680$	1152.00	0.0649664
$681$	0	0
$682$	30132.0	1.69181
$683$	26384.0	1.47812	0.739060	−	0.673640i	$-0.235270\pi$
$683$	26384.0	1.47812	0.739060	+	0.673640i	$0.235270\pi$
$684$	0	0
$685$	−10152.0	−0.566260
$686$	686.000	0.0381802
$687$	0	0
$688$	−1648.00	−0.0913218
$689$	−2197.00	−0.121479
$690$	0	0
$691$	−27823.0	−1.53175	−0.765873	−	0.642992i	$-0.777693\pi$
$691$	−27823.0	−1.53175	−0.765873	+	0.642992i	$0.777693\pi$
$692$	1952.00	0.107231
$693$	0	0
$694$	9728.00	0.532089
$695$	7686.00	0.419492
$696$	0	0
$697$	6496.00	0.353018
$698$	938.000	0.0508651
$699$	0	0
$700$	1232.00	0.0665217
$701$	−24117.0	−1.29941	−0.649705	−	0.760186i	$-0.725108\pi$
$701$	−24117.0	−1.29941	−0.649705	+	0.760186i	$0.725108\pi$
$702$	0	0
$703$	32548.0	1.74619
$704$	−3968.00	−0.212428
$705$	0	0
$706$	8396.00	0.447575
$707$	6258.00	0.332894
$708$	0	0
$709$	28286.0	1.49831	0.749156	−	0.662394i	$-0.230459\pi$
$709$	28286.0	1.49831	0.749156	+	0.662394i	$0.230459\pi$
$710$	4752.00	0.251182
$711$	0	0
$712$	8472.00	0.445929
$713$	37665.0	1.97835
$714$	0	0
$715$	−7254.00	−0.379418
$716$	−10132.0	−0.528842
$717$	0	0
$718$	6816.00	0.354277
$719$	26346.0	1.36654	0.683268	−	0.730167i	$-0.260558\pi$
$719$	26346.0	1.36654	0.683268	+	0.730167i	$0.260558\pi$
$720$	0	0
$721$	−896.000	−0.0462813
$722$	1236.00	0.0637107
$723$	0	0
$724$	−14504.0	−0.744526
$725$	2244.00	0.114952
$726$	0	0
$727$	−22862.0	−1.16631	−0.583153	−	0.812362i	$-0.698181\pi$
$727$	−22862.0	−1.16631	−0.583153	+	0.812362i	$0.698181\pi$
$728$	−728.000	−0.0370625
$729$	0	0
$730$	−2178.00	−0.110427
$731$	−1648.00	−0.0833837
$732$	0	0
$733$	18587.0	0.936598	0.468299	−	0.883570i	$-0.344867\pi$
$733$	18587.0	0.936598	0.468299	+	0.883570i	$0.344867\pi$
$734$	16580.0	0.833759
$735$	0	0
$736$	−4960.00	−0.248408
$737$	−15996.0	−0.799485
$738$	0	0
$739$	5102.00	0.253965	0.126982	−	0.991905i	$-0.459471\pi$
$739$	5102.00	0.253965	0.126982	+	0.991905i	$0.459471\pi$
$740$	−14832.0	−0.736804
$741$	0	0
$742$	2366.00	0.117060
$743$	−27428.0	−1.35429	−0.677144	−	0.735851i	$-0.736783\pi$
$743$	−27428.0	−1.35429	−0.677144	+	0.735851i	$0.736783\pi$
$744$	0	0
$745$	−3978.00	−0.195628
$746$	19812.0	0.972344
$747$	0	0
$748$	−3968.00	−0.193963
$749$	−4956.00	−0.241773
$750$	0	0
$751$	125.000	0.00607365	0.00303683	−	0.999995i	$-0.499033\pi$
$751$	125.000	0.00607365	0.00303683	+	0.999995i	$0.499033\pi$
$752$	−6864.00	−0.332851
$753$	0	0
$754$	−1326.00	−0.0640452
$755$	9144.00	0.440774
$756$	0	0
$757$	−27907.0	−1.33989	−0.669945	−	0.742410i	$-0.733682\pi$
$757$	−27907.0	−1.33989	−0.669945	+	0.742410i	$0.733682\pi$
$758$	18476.0	0.885328
$759$	0	0
$760$	5688.00	0.271481
$761$	−7793.00	−0.371217	−0.185608	−	0.982624i	$-0.559426\pi$
$761$	−7793.00	−0.371217	−0.185608	+	0.982624i	$0.559426\pi$
$762$	0	0
$763$	−8666.00	−0.411180
$764$	19968.0	0.945572
$765$	0	0
$766$	9944.00	0.469049
$767$	4160.00	0.195839
$768$	0	0
$769$	29393.0	1.37833	0.689167	−	0.724603i	$-0.257977\pi$
$769$	29393.0	1.37833	0.689167	+	0.724603i	$0.257977\pi$
$770$	7812.00	0.365617
$771$	0	0
$772$	−1880.00	−0.0876460
$773$	−20326.0	−0.945764	−0.472882	−	0.881126i	$-0.656786\pi$
$773$	−20326.0	−0.945764	−0.472882	+	0.881126i	$0.656786\pi$
$774$	0	0
$775$	−10692.0	−0.495572
$776$	168.000	0.00777171
$777$	0	0
$778$	14908.0	0.686989
$779$	32074.0	1.47519
$780$	0	0
$781$	−16368.0	−0.749927
$782$	−4960.00	−0.226815
$783$	0	0
$784$	784.000	0.0357143
$785$	34920.0	1.58770
$786$	0	0
$787$	16679.0	0.755454	0.377727	−	0.925917i	$-0.376706\pi$
$787$	16679.0	0.755454	0.377727	+	0.925917i	$0.376706\pi$
$788$	12760.0	0.576848
$789$	0	0
$790$	−17406.0	−0.783896
$791$	7511.00	0.337624
$792$	0	0
$793$	7982.00	0.357439
$794$	−14438.0	−0.645322
$795$	0	0
$796$	7776.00	0.346248
$797$	9282.00	0.412529	0.206264	−	0.978496i	$-0.433869\pi$
$797$	9282.00	0.412529	0.206264	+	0.978496i	$0.433869\pi$
$798$	0	0
$799$	−6864.00	−0.303918
$800$	1408.00	0.0622254
$801$	0	0
$802$	21544.0	0.948560
$803$	7502.00	0.329688
$804$	0	0
$805$	9765.00	0.427542
$806$	6318.00	0.276107
$807$	0	0
$808$	7152.00	0.311394
$809$	883.000	0.0383741	0.0191870	−	0.999816i	$-0.493892\pi$
$809$	883.000	0.0383741	0.0191870	+	0.999816i	$0.493892\pi$
$810$	0	0
$811$	−12684.0	−0.549193	−0.274596	−	0.961560i	$-0.588544\pi$
$811$	−12684.0	−0.549193	−0.274596	+	0.961560i	$0.588544\pi$
$812$	1428.00	0.0617155
$813$	0	0
$814$	51088.0	2.19980
$815$	14436.0	0.620455
$816$	0	0
$817$	−8137.00	−0.348443
$818$	−26302.0	−1.12424
$819$	0	0
$820$	−14616.0	−0.622455
$821$	34126.0	1.45068	0.725338	−	0.688393i	$-0.241683\pi$
$821$	34126.0	1.45068	0.725338	+	0.688393i	$0.241683\pi$
$822$	0	0
$823$	−7568.00	−0.320539	−0.160270	−	0.987073i	$-0.551236\pi$
$823$	−7568.00	−0.320539	−0.160270	+	0.987073i	$0.551236\pi$
$824$	−1024.00	−0.0432921
$825$	0	0
$826$	−4480.00	−0.188716
$827$	−32076.0	−1.34872	−0.674360	−	0.738403i	$-0.735580\pi$
$827$	−32076.0	−1.34872	−0.674360	+	0.738403i	$0.735580\pi$
$828$	0	0
$829$	−22598.0	−0.946756	−0.473378	−	0.880859i	$-0.656966\pi$
$829$	−22598.0	−0.946756	−0.473378	+	0.880859i	$0.656966\pi$
$830$	12222.0	0.511123
$831$	0	0
$832$	−832.000	−0.0346688
$833$	784.000	0.0326098
$834$	0	0
$835$	8631.00	0.357710
$836$	−19592.0	−0.810530
$837$	0	0
$838$	12068.0	0.497473
$839$	−25480.0	−1.04847	−0.524236	−	0.851573i	$-0.675649\pi$
$839$	−25480.0	−1.04847	−0.524236	+	0.851573i	$0.675649\pi$
$840$	0	0
$841$	−21788.0	−0.893354
$842$	−17728.0	−0.725591
$843$	0	0
$844$	−4556.00	−0.185810
$845$	−1521.00	−0.0619219
$846$	0	0
$847$	−17591.0	−0.713617
$848$	2704.00	0.109500
$849$	0	0
$850$	1408.00	0.0568165
$851$	63860.0	2.57238
$852$	0	0
$853$	−22939.0	−0.920770	−0.460385	−	0.887719i	$-0.652289\pi$
$853$	−22939.0	−0.920770	−0.460385	+	0.887719i	$0.652289\pi$
$854$	−8596.00	−0.344437
$855$	0	0
$856$	−5664.00	−0.226158
$857$	38782.0	1.54582	0.772910	−	0.634516i	$-0.218800\pi$
$857$	38782.0	1.54582	0.772910	+	0.634516i	$0.218800\pi$
$858$	0	0
$859$	−13970.0	−0.554890	−0.277445	−	0.960742i	$-0.589488\pi$
$859$	−13970.0	−0.554890	−0.277445	+	0.960742i	$0.589488\pi$
$860$	3708.00	0.147025
$861$	0	0
$862$	−268.000	−0.0105895
$863$	22212.0	0.876136	0.438068	−	0.898942i	$-0.355663\pi$
$863$	22212.0	0.876136	0.438068	+	0.898942i	$0.355663\pi$
$864$	0	0
$865$	−4392.00	−0.172639
$866$	−24752.0	−0.971255
$867$	0	0
$868$	−6804.00	−0.266063
$869$	59954.0	2.34039
$870$	0	0
$871$	−3354.00	−0.130478
$872$	−9904.00	−0.384624
$873$	0	0
$874$	−24490.0	−0.947811
$875$	−10647.0	−0.411353
$876$	0	0
$877$	−24816.0	−0.955504	−0.477752	−	0.878495i	$-0.658548\pi$
$877$	−24816.0	−0.955504	−0.477752	+	0.878495i	$0.658548\pi$
$878$	−28396.0	−1.09148
$879$	0	0
$880$	8928.00	0.342003
$881$	−7490.00	−0.286430	−0.143215	−	0.989692i	$-0.545744\pi$
$881$	−7490.00	−0.286430	−0.143215	+	0.989692i	$0.545744\pi$
$882$	0	0
$883$	6092.00	0.232177	0.116088	−	0.993239i	$-0.462964\pi$
$883$	6092.00	0.232177	0.116088	+	0.993239i	$0.462964\pi$
$884$	−832.000	−0.0316552
$885$	0	0
$886$	15314.0	0.580682
$887$	6480.00	0.245295	0.122648	−	0.992450i	$-0.460861\pi$
$887$	6480.00	0.245295	0.122648	+	0.992450i	$0.460861\pi$
$888$	0	0
$889$	12236.0	0.461622
$890$	−19062.0	−0.717932
$891$	0	0
$892$	3332.00	0.125071
$893$	−33891.0	−1.27001
$894$	0	0
$895$	22797.0	0.851419
$896$	896.000	0.0334077
$897$	0	0
$898$	31000.0	1.15199
$899$	−12393.0	−0.459766
$900$	0	0
$901$	2704.00	0.0999815
$902$	50344.0	1.85839
$903$	0	0
$904$	8584.00	0.315818
$905$	32634.0	1.19866
$906$	0	0
$907$	−12111.0	−0.443373	−0.221686	−	0.975118i	$-0.571156\pi$
$907$	−12111.0	−0.443373	−0.221686	+	0.975118i	$0.571156\pi$
$908$	14064.0	0.514020
$909$	0	0
$910$	1638.00	0.0596694
$911$	−7191.00	−0.261524	−0.130762	−	0.991414i	$-0.541742\pi$
$911$	−7191.00	−0.261524	−0.130762	+	0.991414i	$0.541742\pi$
$912$	0	0
$913$	−42098.0	−1.52600
$914$	−20368.0	−0.737105
$915$	0	0
$916$	12888.0	0.464882
$917$	−6552.00	−0.235950
$918$	0	0
$919$	−12272.0	−0.440496	−0.220248	−	0.975444i	$-0.570687\pi$
$919$	−12272.0	−0.440496	−0.220248	+	0.975444i	$0.570687\pi$
$920$	11160.0	0.399929
$921$	0	0
$922$	38500.0	1.37520
$923$	−3432.00	−0.122390
$924$	0	0
$925$	−18128.0	−0.644373
$926$	31244.0	1.10879
$927$	0	0
$928$	1632.00	0.0577296
$929$	−3285.00	−0.116014	−0.0580072	−	0.998316i	$-0.518475\pi$
$929$	−3285.00	−0.116014	−0.0580072	+	0.998316i	$0.518475\pi$
$930$	0	0
$931$	3871.00	0.136269
$932$	1908.00	0.0670586
$933$	0	0
$934$	−31188.0	−1.09262
$935$	8928.00	0.312275
$936$	0	0
$937$	−5068.00	−0.176696	−0.0883481	−	0.996090i	$-0.528159\pi$
$937$	−5068.00	−0.176696	−0.0883481	+	0.996090i	$0.528159\pi$
$938$	3612.00	0.125731
$939$	0	0
$940$	15444.0	0.535881
$941$	−13571.0	−0.470140	−0.235070	−	0.971978i	$-0.575532\pi$
$941$	−13571.0	−0.470140	−0.235070	+	0.971978i	$0.575532\pi$
$942$	0	0
$943$	62930.0	2.17315
$944$	−5120.00	−0.176527
$945$	0	0
$946$	−12772.0	−0.438957
$947$	17662.0	0.606059	0.303030	−	0.952981i	$-0.402002\pi$
$947$	17662.0	0.606059	0.303030	+	0.952981i	$0.402002\pi$
$948$	0	0
$949$	1573.00	0.0538058
$950$	6952.00	0.237424
$951$	0	0
$952$	896.000	0.0305037
$953$	−37893.0	−1.28801	−0.644006	−	0.765021i	$-0.722729\pi$
$953$	−37893.0	−1.28801	−0.644006	+	0.765021i	$0.722729\pi$
$954$	0	0
$955$	−44928.0	−1.52234
$956$	25472.0	0.861740
$957$	0	0
$958$	6478.00	0.218470
$959$	−7896.00	−0.265876
$960$	0	0
$961$	29258.0	0.982109
$962$	10712.0	0.359011
$963$	0	0
$964$	2988.00	0.0998309
$965$	4230.00	0.141107
$966$	0	0
$967$	−48914.0	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$967$	−48914.0	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$968$	−20104.0	−0.667528
$969$	0	0
$970$	−378.000	−0.0125122
$971$	−31110.0	−1.02818	−0.514092	−	0.857735i	$-0.671871\pi$
$971$	−31110.0	−1.02818	−0.514092	+	0.857735i	$0.671871\pi$
$972$	0	0
$973$	5978.00	0.196964
$974$	−3596.00	−0.118299
$975$	0	0
$976$	−9824.00	−0.322191
$977$	−22294.0	−0.730039	−0.365020	−	0.931000i	$-0.618938\pi$
$977$	−22294.0	−0.730039	−0.365020	+	0.931000i	$0.618938\pi$
$978$	0	0
$979$	65658.0	2.14345
$980$	−1764.00	−0.0574989
$981$	0	0
$982$	27416.0	0.890916
$983$	317.000	0.0102856	0.00514279	−	0.999987i	$-0.498363\pi$
$983$	317.000	0.0102856	0.00514279	+	0.999987i	$0.498363\pi$
$984$	0	0
$985$	−28710.0	−0.928707
$986$	1632.00	0.0527114
$987$	0	0
$988$	−4108.00	−0.132280
$989$	−15965.0	−0.513304
$990$	0	0
$991$	40564.0	1.30026	0.650130	−	0.759823i	$-0.274714\pi$
$991$	40564.0	1.30026	0.650130	+	0.759823i	$0.274714\pi$
$992$	−7776.00	−0.248879
$993$	0	0
$994$	3696.00	0.117938
$995$	−17496.0	−0.557448
$996$	0	0
$997$	−57528.0	−1.82741	−0.913706	−	0.406376i	$-0.866792\pi$
$997$	−57528.0	−1.82741	−0.913706	+	0.406376i	$0.866792\pi$
$998$	23872.0	0.757169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.b.1.1		1
3.2	odd	2		546.4.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.e.1.1	✓	1	3.2	odd	2
1638.4.a.b.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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