LMFDB - Embedded newform 1638.4.a.h.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,-2,0,4,14,0,-7,-8,0,-28,-8] 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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} -8.00000 q^{8} -28.0000 q^{10} -8.00000 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +98.0000 q^{17} -28.0000 q^{19} +56.0000 q^{20} +16.0000 q^{22} +52.0000 q^{23} +71.0000 q^{25} -26.0000 q^{26} -28.0000 q^{28} +2.00000 q^{29} -168.000 q^{31} -32.0000 q^{32} -196.000 q^{34} -98.0000 q^{35} -146.000 q^{37} +56.0000 q^{38} -112.000 q^{40} +514.000 q^{41} -236.000 q^{43} -32.0000 q^{44} -104.000 q^{46} +216.000 q^{47} +49.0000 q^{49} -142.000 q^{50} +52.0000 q^{52} +66.0000 q^{53} -112.000 q^{55} +56.0000 q^{56} -4.00000 q^{58} +84.0000 q^{59} +446.000 q^{61} +336.000 q^{62} +64.0000 q^{64} +182.000 q^{65} +292.000 q^{67} +392.000 q^{68} +196.000 q^{70} -100.000 q^{71} +450.000 q^{73} +292.000 q^{74} -112.000 q^{76} +56.0000 q^{77} +392.000 q^{79} +224.000 q^{80} -1028.00 q^{82} +292.000 q^{83} +1372.00 q^{85} +472.000 q^{86} +64.0000 q^{88} +402.000 q^{89} -91.0000 q^{91} +208.000 q^{92} -432.000 q^{94} -392.000 q^{95} +314.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−28.0000	−0.885438
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	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
$17$	98.0000	1.39815	0.699073	−	0.715050i	$-0.253596\pi$
$17$	98.0000	1.39815	0.699073	+	0.715050i	$0.253596\pi$
$18$	0	0
$19$	−28.0000	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$20$	56.0000	0.626099
$21$	0	0
$22$	16.0000	0.155055
$23$	52.0000	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$23$	52.0000	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	−26.0000	−0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	2.00000	0.0128066	0.00640329	−	0.999979i	$-0.497962\pi$
$29$	2.00000	0.0128066	0.00640329	+	0.999979i	$0.497962\pi$
$30$	0	0
$31$	−168.000	−0.973345	−0.486672	−	0.873585i	$-0.661789\pi$
$31$	−168.000	−0.973345	−0.486672	+	0.873585i	$0.661789\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−196.000	−0.988639
$35$	−98.0000	−0.473286
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	56.0000	0.239063
$39$	0	0
$40$	−112.000	−0.442719
$41$	514.000	1.95789	0.978943	−	0.204135i	$-0.0654380\pi$
$41$	514.000	1.95789	0.978943	+	0.204135i	$0.0654380\pi$
$42$	0	0
$43$	−236.000	−0.836969	−0.418484	−	0.908224i	$-0.637439\pi$
$43$	−236.000	−0.836969	−0.418484	+	0.908224i	$0.637439\pi$
$44$	−32.0000	−0.109640
$45$	0	0
$46$	−104.000	−0.333347
$47$	216.000	0.670358	0.335179	−	0.942154i	$-0.391203\pi$
$47$	216.000	0.670358	0.335179	+	0.942154i	$0.391203\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−142.000	−0.401637
$51$	0	0
$52$	52.0000	0.138675
$53$	66.0000	0.171053	0.0855264	−	0.996336i	$-0.472743\pi$
$53$	66.0000	0.171053	0.0855264	+	0.996336i	$0.472743\pi$
$54$	0	0
$55$	−112.000	−0.274583
$56$	56.0000	0.133631
$57$	0	0
$58$	−4.00000	−0.00905562
$59$	84.0000	0.185354	0.0926769	−	0.995696i	$-0.470458\pi$
$59$	84.0000	0.185354	0.0926769	+	0.995696i	$0.470458\pi$
$60$	0	0
$61$	446.000	0.936138	0.468069	−	0.883692i	$-0.344950\pi$
$61$	446.000	0.936138	0.468069	+	0.883692i	$0.344950\pi$
$62$	336.000	0.688259
$63$	0	0
$64$	64.0000	0.125000
$65$	182.000	0.347297
$66$	0	0
$67$	292.000	0.532440	0.266220	−	0.963912i	$-0.414225\pi$
$67$	292.000	0.532440	0.266220	+	0.963912i	$0.414225\pi$
$68$	392.000	0.699073
$69$	0	0
$70$	196.000	0.334664
$71$	−100.000	−0.167152	−0.0835762	−	0.996501i	$-0.526634\pi$
$71$	−100.000	−0.167152	−0.0835762	+	0.996501i	$0.526634\pi$
$72$	0	0
$73$	450.000	0.721487	0.360743	−	0.932665i	$-0.382523\pi$
$73$	450.000	0.721487	0.360743	+	0.932665i	$0.382523\pi$
$74$	292.000	0.458707
$75$	0	0
$76$	−112.000	−0.169043
$77$	56.0000	0.0828804
$78$	0	0
$79$	392.000	0.558271	0.279136	−	0.960252i	$-0.409952\pi$
$79$	392.000	0.558271	0.279136	+	0.960252i	$0.409952\pi$
$80$	224.000	0.313050
$81$	0	0
$82$	−1028.00	−1.38443
$83$	292.000	0.386159	0.193079	−	0.981183i	$-0.438153\pi$
$83$	292.000	0.386159	0.193079	+	0.981183i	$0.438153\pi$
$84$	0	0
$85$	1372.00	1.75076
$86$	472.000	0.591826
$87$	0	0
$88$	64.0000	0.0775275
$89$	402.000	0.478786	0.239393	−	0.970923i	$-0.423052\pi$
$89$	402.000	0.478786	0.239393	+	0.970923i	$0.423052\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	208.000	0.235712
$93$	0	0
$94$	−432.000	−0.474015
$95$	−392.000	−0.423351
$96$	0	0
$97$	314.000	0.328679	0.164340	−	0.986404i	$-0.447451\pi$
$97$	314.000	0.328679	0.164340	+	0.986404i	$0.447451\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	284.000	0.284000
$101$	−858.000	−0.845289	−0.422645	−	0.906296i	$-0.638898\pi$
$101$	−858.000	−0.845289	−0.422645	+	0.906296i	$0.638898\pi$
$102$	0	0
$103$	−944.000	−0.903059	−0.451530	−	0.892256i	$-0.649121\pi$
$103$	−944.000	−0.903059	−0.451530	+	0.892256i	$0.649121\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	−132.000	−0.120953
$107$	−840.000	−0.758933	−0.379467	−	0.925205i	$-0.623893\pi$
$107$	−840.000	−0.758933	−0.379467	+	0.925205i	$0.623893\pi$
$108$	0	0
$109$	1334.00	1.17224	0.586119	−	0.810225i	$-0.300655\pi$
$109$	1334.00	1.17224	0.586119	+	0.810225i	$0.300655\pi$
$110$	224.000	0.194160
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−946.000	−0.787542	−0.393771	−	0.919209i	$-0.628830\pi$
$113$	−946.000	−0.787542	−0.393771	+	0.919209i	$0.628830\pi$
$114$	0	0
$115$	728.000	0.590316
$116$	8.00000	0.00640329
$117$	0	0
$118$	−168.000	−0.131065
$119$	−686.000	−0.528450
$120$	0	0
$121$	−1267.00	−0.951916
$122$	−892.000	−0.661950
$123$	0	0
$124$	−672.000	−0.486672
$125$	−756.000	−0.540950
$126$	0	0
$127$	904.000	0.631630	0.315815	−	0.948821i	$-0.397722\pi$
$127$	904.000	0.631630	0.315815	+	0.948821i	$0.397722\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−364.000	−0.245576
$131$	1476.00	0.984418	0.492209	−	0.870477i	$-0.336190\pi$
$131$	1476.00	0.984418	0.492209	+	0.870477i	$0.336190\pi$
$132$	0	0
$133$	196.000	0.127785
$134$	−584.000	−0.376492
$135$	0	0
$136$	−784.000	−0.494319
$137$	62.0000	0.0386644	0.0193322	−	0.999813i	$-0.493846\pi$
$137$	62.0000	0.0386644	0.0193322	+	0.999813i	$0.493846\pi$
$138$	0	0
$139$	−908.000	−0.554069	−0.277034	−	0.960860i	$-0.589352\pi$
$139$	−908.000	−0.554069	−0.277034	+	0.960860i	$0.589352\pi$
$140$	−392.000	−0.236643
$141$	0	0
$142$	200.000	0.118195
$143$	−104.000	−0.0608176
$144$	0	0
$145$	28.0000	0.0160364
$146$	−900.000	−0.510168
$147$	0	0
$148$	−584.000	−0.324355
$149$	2514.00	1.38225	0.691124	−	0.722736i	$-0.257116\pi$
$149$	2514.00	1.38225	0.691124	+	0.722736i	$0.257116\pi$
$150$	0	0
$151$	−1048.00	−0.564802	−0.282401	−	0.959297i	$-0.591131\pi$
$151$	−1048.00	−0.564802	−0.282401	+	0.959297i	$0.591131\pi$
$152$	224.000	0.119532
$153$	0	0
$154$	−112.000	−0.0586053
$155$	−2352.00	−1.21882
$156$	0	0
$157$	3166.00	1.60939	0.804695	−	0.593688i	$-0.202329\pi$
$157$	3166.00	1.60939	0.804695	+	0.593688i	$0.202329\pi$
$158$	−784.000	−0.394758
$159$	0	0
$160$	−448.000	−0.221359
$161$	−364.000	−0.178181
$162$	0	0
$163$	−492.000	−0.236420	−0.118210	−	0.992989i	$-0.537716\pi$
$163$	−492.000	−0.236420	−0.118210	+	0.992989i	$0.537716\pi$
$164$	2056.00	0.978943
$165$	0	0
$166$	−584.000	−0.273055
$167$	1104.00	0.511557	0.255779	−	0.966735i	$-0.417668\pi$
$167$	1104.00	0.511557	0.255779	+	0.966735i	$0.417668\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−2744.00	−1.23797
$171$	0	0
$172$	−944.000	−0.418484
$173$	2686.00	1.18042	0.590210	−	0.807249i	$-0.299045\pi$
$173$	2686.00	1.18042	0.590210	+	0.807249i	$0.299045\pi$
$174$	0	0
$175$	−497.000	−0.214684
$176$	−128.000	−0.0548202
$177$	0	0
$178$	−804.000	−0.338553
$179$	−2088.00	−0.871868	−0.435934	−	0.899979i	$-0.643582\pi$
$179$	−2088.00	−0.871868	−0.435934	+	0.899979i	$0.643582\pi$
$180$	0	0
$181$	−242.000	−0.0993797	−0.0496898	−	0.998765i	$-0.515823\pi$
$181$	−242.000	−0.0993797	−0.0496898	+	0.998765i	$0.515823\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	−416.000	−0.166674
$185$	−2044.00	−0.812313
$186$	0	0
$187$	−784.000	−0.306587
$188$	864.000	0.335179
$189$	0	0
$190$	784.000	0.299354
$191$	1284.00	0.486424	0.243212	−	0.969973i	$-0.421799\pi$
$191$	1284.00	0.486424	0.243212	+	0.969973i	$0.421799\pi$
$192$	0	0
$193$	−2734.00	−1.01968	−0.509838	−	0.860270i	$-0.670295\pi$
$193$	−2734.00	−1.01968	−0.509838	+	0.860270i	$0.670295\pi$
$194$	−628.000	−0.232411
$195$	0	0
$196$	196.000	0.0714286
$197$	−5118.00	−1.85098	−0.925488	−	0.378776i	$-0.876345\pi$
$197$	−5118.00	−1.85098	−0.925488	+	0.378776i	$0.876345\pi$
$198$	0	0
$199$	1528.00	0.544307	0.272153	−	0.962254i	$-0.412264\pi$
$199$	1528.00	0.544307	0.272153	+	0.962254i	$0.412264\pi$
$200$	−568.000	−0.200818
$201$	0	0
$202$	1716.00	0.597710
$203$	−14.0000	−0.00484043
$204$	0	0
$205$	7196.00	2.45166
$206$	1888.00	0.638559
$207$	0	0
$208$	208.000	0.0693375
$209$	224.000	0.0741359
$210$	0	0
$211$	268.000	0.0874402	0.0437201	−	0.999044i	$-0.486079\pi$
$211$	268.000	0.0874402	0.0437201	+	0.999044i	$0.486079\pi$
$212$	264.000	0.0855264
$213$	0	0
$214$	1680.00	0.536647
$215$	−3304.00	−1.04805
$216$	0	0
$217$	1176.00	0.367890
$218$	−2668.00	−0.828898
$219$	0	0
$220$	−448.000	−0.137292
$221$	1274.00	0.387776
$222$	0	0
$223$	1392.00	0.418005	0.209003	−	0.977915i	$-0.432978\pi$
$223$	1392.00	0.418005	0.209003	+	0.977915i	$0.432978\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	1892.00	0.556876
$227$	6140.00	1.79527	0.897635	−	0.440740i	$-0.145284\pi$
$227$	6140.00	1.79527	0.897635	+	0.440740i	$0.145284\pi$
$228$	0	0
$229$	1870.00	0.539620	0.269810	−	0.962914i	$-0.413039\pi$
$229$	1870.00	0.539620	0.269810	+	0.962914i	$0.413039\pi$
$230$	−1456.00	−0.417417
$231$	0	0
$232$	−16.0000	−0.00452781
$233$	−50.0000	−0.0140584	−0.00702920	−	0.999975i	$-0.502237\pi$
$233$	−50.0000	−0.0140584	−0.00702920	+	0.999975i	$0.502237\pi$
$234$	0	0
$235$	3024.00	0.839421
$236$	336.000	0.0926769
$237$	0	0
$238$	1372.00	0.373670
$239$	3228.00	0.873648	0.436824	−	0.899547i	$-0.356103\pi$
$239$	3228.00	0.873648	0.436824	+	0.899547i	$0.356103\pi$
$240$	0	0
$241$	5978.00	1.59783	0.798915	−	0.601444i	$-0.205408\pi$
$241$	5978.00	1.59783	0.798915	+	0.601444i	$0.205408\pi$
$242$	2534.00	0.673106
$243$	0	0
$244$	1784.00	0.468069
$245$	686.000	0.178885
$246$	0	0
$247$	−364.000	−0.0937683
$248$	1344.00	0.344129
$249$	0	0
$250$	1512.00	0.382509
$251$	4084.00	1.02701	0.513506	−	0.858086i	$-0.328347\pi$
$251$	4084.00	1.02701	0.513506	+	0.858086i	$0.328347\pi$
$252$	0	0
$253$	−416.000	−0.103374
$254$	−1808.00	−0.446630
$255$	0	0
$256$	256.000	0.0625000
$257$	−2646.00	−0.642229	−0.321115	−	0.947040i	$-0.604057\pi$
$257$	−2646.00	−0.642229	−0.321115	+	0.947040i	$0.604057\pi$
$258$	0	0
$259$	1022.00	0.245189
$260$	728.000	0.173649
$261$	0	0
$262$	−2952.00	−0.696088
$263$	−5164.00	−1.21074	−0.605372	−	0.795942i	$-0.706976\pi$
$263$	−5164.00	−1.21074	−0.605372	+	0.795942i	$0.706976\pi$
$264$	0	0
$265$	924.000	0.214192
$266$	−392.000	−0.0903574
$267$	0	0
$268$	1168.00	0.266220
$269$	4710.00	1.06756	0.533780	−	0.845623i	$-0.320771\pi$
$269$	4710.00	1.06756	0.533780	+	0.845623i	$0.320771\pi$
$270$	0	0
$271$	−3144.00	−0.704739	−0.352370	−	0.935861i	$-0.614624\pi$
$271$	−3144.00	−0.704739	−0.352370	+	0.935861i	$0.614624\pi$
$272$	1568.00	0.349537
$273$	0	0
$274$	−124.000	−0.0273398
$275$	−568.000	−0.124552
$276$	0	0
$277$	3310.00	0.717973	0.358987	−	0.933343i	$-0.383122\pi$
$277$	3310.00	0.717973	0.358987	+	0.933343i	$0.383122\pi$
$278$	1816.00	0.391786
$279$	0	0
$280$	784.000	0.167332
$281$	4910.00	1.04237	0.521185	−	0.853444i	$-0.325490\pi$
$281$	4910.00	1.04237	0.521185	+	0.853444i	$0.325490\pi$
$282$	0	0
$283$	7436.00	1.56192	0.780962	−	0.624579i	$-0.214729\pi$
$283$	7436.00	1.56192	0.780962	+	0.624579i	$0.214729\pi$
$284$	−400.000	−0.0835762
$285$	0	0
$286$	208.000	0.0430045
$287$	−3598.00	−0.740011
$288$	0	0
$289$	4691.00	0.954814
$290$	−56.0000	−0.0113394
$291$	0	0
$292$	1800.00	0.360743
$293$	310.000	0.0618102	0.0309051	−	0.999522i	$-0.490161\pi$
$293$	310.000	0.0618102	0.0309051	+	0.999522i	$0.490161\pi$
$294$	0	0
$295$	1176.00	0.232100
$296$	1168.00	0.229353
$297$	0	0
$298$	−5028.00	−0.977397
$299$	676.000	0.130749
$300$	0	0
$301$	1652.00	0.316345
$302$	2096.00	0.399375
$303$	0	0
$304$	−448.000	−0.0845216
$305$	6244.00	1.17223
$306$	0	0
$307$	−3516.00	−0.653644	−0.326822	−	0.945086i	$-0.605978\pi$
$307$	−3516.00	−0.653644	−0.326822	+	0.945086i	$0.605978\pi$
$308$	224.000	0.0414402
$309$	0	0
$310$	4704.00	0.861836
$311$	−3216.00	−0.586375	−0.293188	−	0.956055i	$-0.594716\pi$
$311$	−3216.00	−0.586375	−0.293188	+	0.956055i	$0.594716\pi$
$312$	0	0
$313$	−1398.00	−0.252459	−0.126229	−	0.992001i	$-0.540288\pi$
$313$	−1398.00	−0.252459	−0.126229	+	0.992001i	$0.540288\pi$
$314$	−6332.00	−1.13801
$315$	0	0
$316$	1568.00	0.279136
$317$	9034.00	1.60063	0.800315	−	0.599579i	$-0.204665\pi$
$317$	9034.00	1.60063	0.800315	+	0.599579i	$0.204665\pi$
$318$	0	0
$319$	−16.0000	−0.00280824
$320$	896.000	0.156525
$321$	0	0
$322$	728.000	0.125993
$323$	−2744.00	−0.472694
$324$	0	0
$325$	923.000	0.157535
$326$	984.000	0.167174
$327$	0	0
$328$	−4112.00	−0.692217
$329$	−1512.00	−0.253372
$330$	0	0
$331$	9108.00	1.51245	0.756225	−	0.654312i	$-0.227042\pi$
$331$	9108.00	1.51245	0.756225	+	0.654312i	$0.227042\pi$
$332$	1168.00	0.193079
$333$	0	0
$334$	−2208.00	−0.361726
$335$	4088.00	0.666720
$336$	0	0
$337$	−1566.00	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$337$	−1566.00	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	5488.00	0.875378
$341$	1344.00	0.213436
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1888.00	0.295913
$345$	0	0
$346$	−5372.00	−0.834684
$347$	−4528.00	−0.700507	−0.350253	−	0.936655i	$-0.613904\pi$
$347$	−4528.00	−0.700507	−0.350253	+	0.936655i	$0.613904\pi$
$348$	0	0
$349$	−1090.00	−0.167182	−0.0835908	−	0.996500i	$-0.526639\pi$
$349$	−1090.00	−0.167182	−0.0835908	+	0.996500i	$0.526639\pi$
$350$	994.000	0.151804
$351$	0	0
$352$	256.000	0.0387638
$353$	11234.0	1.69384	0.846920	−	0.531720i	$-0.178454\pi$
$353$	11234.0	1.69384	0.846920	+	0.531720i	$0.178454\pi$
$354$	0	0
$355$	−1400.00	−0.209308
$356$	1608.00	0.239393
$357$	0	0
$358$	4176.00	0.616504
$359$	−3444.00	−0.506316	−0.253158	−	0.967425i	$-0.581469\pi$
$359$	−3444.00	−0.506316	−0.253158	+	0.967425i	$0.581469\pi$
$360$	0	0
$361$	−6075.00	−0.885698
$362$	484.000	0.0702720
$363$	0	0
$364$	−364.000	−0.0524142
$365$	6300.00	0.903444
$366$	0	0
$367$	11712.0	1.66583	0.832917	−	0.553397i	$-0.186669\pi$
$367$	11712.0	1.66583	0.832917	+	0.553397i	$0.186669\pi$
$368$	832.000	0.117856
$369$	0	0
$370$	4088.00	0.574392
$371$	−462.000	−0.0646519
$372$	0	0
$373$	10990.0	1.52558	0.762789	−	0.646647i	$-0.223829\pi$
$373$	10990.0	1.52558	0.762789	+	0.646647i	$0.223829\pi$
$374$	1568.00	0.216790
$375$	0	0
$376$	−1728.00	−0.237007
$377$	26.0000	0.00355190
$378$	0	0
$379$	−7004.00	−0.949265	−0.474632	−	0.880184i	$-0.657419\pi$
$379$	−7004.00	−0.949265	−0.474632	+	0.880184i	$0.657419\pi$
$380$	−1568.00	−0.211676
$381$	0	0
$382$	−2568.00	−0.343954
$383$	5472.00	0.730042	0.365021	−	0.930999i	$-0.381062\pi$
$383$	5472.00	0.730042	0.365021	+	0.930999i	$0.381062\pi$
$384$	0	0
$385$	784.000	0.103783
$386$	5468.00	0.721020
$387$	0	0
$388$	1256.00	0.164340
$389$	−1910.00	−0.248948	−0.124474	−	0.992223i	$-0.539724\pi$
$389$	−1910.00	−0.248948	−0.124474	+	0.992223i	$0.539724\pi$
$390$	0	0
$391$	5096.00	0.659120
$392$	−392.000	−0.0505076
$393$	0	0
$394$	10236.0	1.30884
$395$	5488.00	0.699066
$396$	0	0
$397$	−13234.0	−1.67304	−0.836518	−	0.547939i	$-0.815413\pi$
$397$	−13234.0	−1.67304	−0.836518	+	0.547939i	$0.815413\pi$
$398$	−3056.00	−0.384883
$399$	0	0
$400$	1136.00	0.142000
$401$	−4586.00	−0.571107	−0.285554	−	0.958363i	$-0.592177\pi$
$401$	−4586.00	−0.571107	−0.285554	+	0.958363i	$0.592177\pi$
$402$	0	0
$403$	−2184.00	−0.269957
$404$	−3432.00	−0.422645
$405$	0	0
$406$	28.0000	0.00342270
$407$	1168.00	0.142250
$408$	0	0
$409$	−4174.00	−0.504624	−0.252312	−	0.967646i	$-0.581191\pi$
$409$	−4174.00	−0.504624	−0.252312	+	0.967646i	$0.581191\pi$
$410$	−14392.0	−1.73359
$411$	0	0
$412$	−3776.00	−0.451530
$413$	−588.000	−0.0700571
$414$	0	0
$415$	4088.00	0.483547
$416$	−416.000	−0.0490290
$417$	0	0
$418$	−448.000	−0.0524220
$419$	10012.0	1.16735	0.583673	−	0.811989i	$-0.301615\pi$
$419$	10012.0	1.16735	0.583673	+	0.811989i	$0.301615\pi$
$420$	0	0
$421$	−1250.00	−0.144706	−0.0723531	−	0.997379i	$-0.523051\pi$
$421$	−1250.00	−0.144706	−0.0723531	+	0.997379i	$0.523051\pi$
$422$	−536.000	−0.0618296
$423$	0	0
$424$	−528.000	−0.0604763
$425$	6958.00	0.794147
$426$	0	0
$427$	−3122.00	−0.353827
$428$	−3360.00	−0.379467
$429$	0	0
$430$	6608.00	0.741084
$431$	−6236.00	−0.696932	−0.348466	−	0.937321i	$-0.613297\pi$
$431$	−6236.00	−0.696932	−0.348466	+	0.937321i	$0.613297\pi$
$432$	0	0
$433$	−3966.00	−0.440170	−0.220085	−	0.975481i	$-0.570634\pi$
$433$	−3966.00	−0.440170	−0.220085	+	0.975481i	$0.570634\pi$
$434$	−2352.00	−0.260137
$435$	0	0
$436$	5336.00	0.586119
$437$	−1456.00	−0.159382
$438$	0	0
$439$	8776.00	0.954113	0.477057	−	0.878873i	$-0.341704\pi$
$439$	8776.00	0.954113	0.477057	+	0.878873i	$0.341704\pi$
$440$	896.000	0.0970798
$441$	0	0
$442$	−2548.00	−0.274199
$443$	−6096.00	−0.653792	−0.326896	−	0.945060i	$-0.606003\pi$
$443$	−6096.00	−0.653792	−0.326896	+	0.945060i	$0.606003\pi$
$444$	0	0
$445$	5628.00	0.599534
$446$	−2784.00	−0.295574
$447$	0	0
$448$	−448.000	−0.0472456
$449$	9390.00	0.986952	0.493476	−	0.869759i	$-0.335726\pi$
$449$	9390.00	0.986952	0.493476	+	0.869759i	$0.335726\pi$
$450$	0	0
$451$	−4112.00	−0.429327
$452$	−3784.00	−0.393771
$453$	0	0
$454$	−12280.0	−1.26945
$455$	−1274.00	−0.131266
$456$	0	0
$457$	−6.00000	−0.000614154 0	−0.000307077	−	1.00000i	$-0.500098\pi$
$457$	−6.00000	−0.000614154 0	−0.000307077		1.00000i	$0.500098\pi$
$458$	−3740.00	−0.381569
$459$	0	0
$460$	2912.00	0.295158
$461$	9374.00	0.947051	0.473526	−	0.880780i	$-0.342981\pi$
$461$	9374.00	0.947051	0.473526	+	0.880780i	$0.342981\pi$
$462$	0	0
$463$	4008.00	0.402306	0.201153	−	0.979560i	$-0.435531\pi$
$463$	4008.00	0.402306	0.201153	+	0.979560i	$0.435531\pi$
$464$	32.0000	0.00320164
$465$	0	0
$466$	100.000	0.00994080
$467$	3260.00	0.323030	0.161515	−	0.986870i	$-0.448362\pi$
$467$	3260.00	0.323030	0.161515	+	0.986870i	$0.448362\pi$
$468$	0	0
$469$	−2044.00	−0.201243
$470$	−6048.00	−0.593561
$471$	0	0
$472$	−672.000	−0.0655324
$473$	1888.00	0.183531
$474$	0	0
$475$	−1988.00	−0.192033
$476$	−2744.00	−0.264225
$477$	0	0
$478$	−6456.00	−0.617763
$479$	−12696.0	−1.21105	−0.605527	−	0.795825i	$-0.707038\pi$
$479$	−12696.0	−1.21105	−0.605527	+	0.795825i	$0.707038\pi$
$480$	0	0
$481$	−1898.00	−0.179920
$482$	−11956.0	−1.12984
$483$	0	0
$484$	−5068.00	−0.475958
$485$	4396.00	0.411571
$486$	0	0
$487$	10760.0	1.00120	0.500598	−	0.865680i	$-0.333114\pi$
$487$	10760.0	1.00120	0.500598	+	0.865680i	$0.333114\pi$
$488$	−3568.00	−0.330975
$489$	0	0
$490$	−1372.00	−0.126491
$491$	11600.0	1.06619	0.533096	−	0.846055i	$-0.321028\pi$
$491$	11600.0	1.06619	0.533096	+	0.846055i	$0.321028\pi$
$492$	0	0
$493$	196.000	0.0179055
$494$	728.000	0.0663042
$495$	0	0
$496$	−2688.00	−0.243336
$497$	700.000	0.0631776
$498$	0	0
$499$	−15908.0	−1.42713	−0.713567	−	0.700587i	$-0.752922\pi$
$499$	−15908.0	−1.42713	−0.713567	+	0.700587i	$0.752922\pi$
$500$	−3024.00	−0.270475
$501$	0	0
$502$	−8168.00	−0.726207
$503$	12232.0	1.08429	0.542145	−	0.840285i	$-0.317612\pi$
$503$	12232.0	1.08429	0.542145	+	0.840285i	$0.317612\pi$
$504$	0	0
$505$	−12012.0	−1.05847
$506$	832.000	0.0730967
$507$	0	0
$508$	3616.00	0.315815
$509$	13350.0	1.16253	0.581266	−	0.813714i	$-0.302558\pi$
$509$	13350.0	1.16253	0.581266	+	0.813714i	$0.302558\pi$
$510$	0	0
$511$	−3150.00	−0.272696
$512$	−512.000	−0.0441942
$513$	0	0
$514$	5292.00	0.454125
$515$	−13216.0	−1.13081
$516$	0	0
$517$	−1728.00	−0.146997
$518$	−2044.00	−0.173375
$519$	0	0
$520$	−1456.00	−0.122788
$521$	18282.0	1.53733	0.768665	−	0.639652i	$-0.220921\pi$
$521$	18282.0	1.53733	0.768665	+	0.639652i	$0.220921\pi$
$522$	0	0
$523$	−6108.00	−0.510677	−0.255339	−	0.966852i	$-0.582187\pi$
$523$	−6108.00	−0.510677	−0.255339	+	0.966852i	$0.582187\pi$
$524$	5904.00	0.492209
$525$	0	0
$526$	10328.0	0.856126
$527$	−16464.0	−1.36088
$528$	0	0
$529$	−9463.00	−0.777760
$530$	−1848.00	−0.151457
$531$	0	0
$532$	784.000	0.0638923
$533$	6682.00	0.543020
$534$	0	0
$535$	−11760.0	−0.950335
$536$	−2336.00	−0.188246
$537$	0	0
$538$	−9420.00	−0.754879
$539$	−392.000	−0.0313259
$540$	0	0
$541$	−14906.0	−1.18458	−0.592291	−	0.805724i	$-0.701776\pi$
$541$	−14906.0	−1.18458	−0.592291	+	0.805724i	$0.701776\pi$
$542$	6288.00	0.498326
$543$	0	0
$544$	−3136.00	−0.247160
$545$	18676.0	1.46788
$546$	0	0
$547$	5492.00	0.429289	0.214644	−	0.976692i	$-0.431141\pi$
$547$	5492.00	0.429289	0.214644	+	0.976692i	$0.431141\pi$
$548$	248.000	0.0193322
$549$	0	0
$550$	1136.00	0.0880713
$551$	−56.0000	−0.00432973
$552$	0	0
$553$	−2744.00	−0.211007
$554$	−6620.00	−0.507684
$555$	0	0
$556$	−3632.00	−0.277034
$557$	6746.00	0.513173	0.256586	−	0.966521i	$-0.417402\pi$
$557$	6746.00	0.513173	0.256586	+	0.966521i	$0.417402\pi$
$558$	0	0
$559$	−3068.00	−0.232133
$560$	−1568.00	−0.118322
$561$	0	0
$562$	−9820.00	−0.737067
$563$	−596.000	−0.0446153	−0.0223076	−	0.999751i	$-0.507101\pi$
$563$	−596.000	−0.0446153	−0.0223076	+	0.999751i	$0.507101\pi$
$564$	0	0
$565$	−13244.0	−0.986158
$566$	−14872.0	−1.10445
$567$	0	0
$568$	800.000	0.0590973
$569$	2638.00	0.194360	0.0971799	−	0.995267i	$-0.469018\pi$
$569$	2638.00	0.194360	0.0971799	+	0.995267i	$0.469018\pi$
$570$	0	0
$571$	−22612.0	−1.65724	−0.828619	−	0.559813i	$-0.810873\pi$
$571$	−22612.0	−1.65724	−0.828619	+	0.559813i	$0.810873\pi$
$572$	−416.000	−0.0304088
$573$	0	0
$574$	7196.00	0.523267
$575$	3692.00	0.267769
$576$	0	0
$577$	−1806.00	−0.130303	−0.0651514	−	0.997875i	$-0.520753\pi$
$577$	−1806.00	−0.130303	−0.0651514	+	0.997875i	$0.520753\pi$
$578$	−9382.00	−0.675155
$579$	0	0
$580$	112.000	0.00801818
$581$	−2044.00	−0.145954
$582$	0	0
$583$	−528.000	−0.0375086
$584$	−3600.00	−0.255084
$585$	0	0
$586$	−620.000	−0.0437064
$587$	27156.0	1.90945	0.954726	−	0.297487i	$-0.0961486\pi$
$587$	27156.0	1.90945	0.954726	+	0.297487i	$0.0961486\pi$
$588$	0	0
$589$	4704.00	0.329075
$590$	−2352.00	−0.164119
$591$	0	0
$592$	−2336.00	−0.162177
$593$	6034.00	0.417853	0.208926	−	0.977931i	$-0.433003\pi$
$593$	6034.00	0.417853	0.208926	+	0.977931i	$0.433003\pi$
$594$	0	0
$595$	−9604.00	−0.661724
$596$	10056.0	0.691124
$597$	0	0
$598$	−1352.00	−0.0924538
$599$	−19284.0	−1.31540	−0.657699	−	0.753281i	$-0.728470\pi$
$599$	−19284.0	−1.31540	−0.657699	+	0.753281i	$0.728470\pi$
$600$	0	0
$601$	10490.0	0.711973	0.355987	−	0.934491i	$-0.384145\pi$
$601$	10490.0	0.711973	0.355987	+	0.934491i	$0.384145\pi$
$602$	−3304.00	−0.223689
$603$	0	0
$604$	−4192.00	−0.282401
$605$	−17738.0	−1.19199
$606$	0	0
$607$	−13216.0	−0.883725	−0.441862	−	0.897083i	$-0.645682\pi$
$607$	−13216.0	−0.883725	−0.441862	+	0.897083i	$0.645682\pi$
$608$	896.000	0.0597658
$609$	0	0
$610$	−12488.0	−0.828892
$611$	2808.00	0.185924
$612$	0	0
$613$	−8242.00	−0.543053	−0.271526	−	0.962431i	$-0.587528\pi$
$613$	−8242.00	−0.543053	−0.271526	+	0.962431i	$0.587528\pi$
$614$	7032.00	0.462196
$615$	0	0
$616$	−448.000	−0.0293027
$617$	18206.0	1.18792	0.593959	−	0.804495i	$-0.297564\pi$
$617$	18206.0	1.18792	0.593959	+	0.804495i	$0.297564\pi$
$618$	0	0
$619$	16580.0	1.07659	0.538293	−	0.842758i	$-0.319069\pi$
$619$	16580.0	1.07659	0.538293	+	0.842758i	$0.319069\pi$
$620$	−9408.00	−0.609410
$621$	0	0
$622$	6432.00	0.414630
$623$	−2814.00	−0.180964
$624$	0	0
$625$	−19459.0	−1.24538
$626$	2796.00	0.178515
$627$	0	0
$628$	12664.0	0.804695
$629$	−14308.0	−0.906991
$630$	0	0
$631$	21040.0	1.32740	0.663700	−	0.747999i	$-0.268985\pi$
$631$	21040.0	1.32740	0.663700	+	0.747999i	$0.268985\pi$
$632$	−3136.00	−0.197379
$633$	0	0
$634$	−18068.0	−1.13182
$635$	12656.0	0.790926
$636$	0	0
$637$	637.000	0.0396214
$638$	32.0000	0.00198572
$639$	0	0
$640$	−1792.00	−0.110680
$641$	−170.000	−0.0104752	−0.00523759	−	0.999986i	$-0.501667\pi$
$641$	−170.000	−0.0104752	−0.00523759	+	0.999986i	$0.501667\pi$
$642$	0	0
$643$	23924.0	1.46729	0.733647	−	0.679530i	$-0.237816\pi$
$643$	23924.0	1.46729	0.733647	+	0.679530i	$0.237816\pi$
$644$	−1456.00	−0.0890907
$645$	0	0
$646$	5488.00	0.334245
$647$	−22992.0	−1.39708	−0.698538	−	0.715572i	$-0.746166\pi$
$647$	−22992.0	−1.39708	−0.698538	+	0.715572i	$0.746166\pi$
$648$	0	0
$649$	−672.000	−0.0406445
$650$	−1846.00	−0.111394
$651$	0	0
$652$	−1968.00	−0.118210
$653$	−12990.0	−0.778466	−0.389233	−	0.921139i	$-0.627260\pi$
$653$	−12990.0	−0.778466	−0.389233	+	0.921139i	$0.627260\pi$
$654$	0	0
$655$	20664.0	1.23269
$656$	8224.00	0.489471
$657$	0	0
$658$	3024.00	0.179161
$659$	−11088.0	−0.655428	−0.327714	−	0.944777i	$-0.606278\pi$
$659$	−11088.0	−0.655428	−0.327714	+	0.944777i	$0.606278\pi$
$660$	0	0
$661$	6886.00	0.405196	0.202598	−	0.979262i	$-0.435062\pi$
$661$	6886.00	0.405196	0.202598	+	0.979262i	$0.435062\pi$
$662$	−18216.0	−1.06946
$663$	0	0
$664$	−2336.00	−0.136528
$665$	2744.00	0.160012
$666$	0	0
$667$	104.000	0.00603733
$668$	4416.00	0.255779
$669$	0	0
$670$	−8176.00	−0.471442
$671$	−3568.00	−0.205277
$672$	0	0
$673$	−8238.00	−0.471845	−0.235922	−	0.971772i	$-0.575811\pi$
$673$	−8238.00	−0.471845	−0.235922	+	0.971772i	$0.575811\pi$
$674$	3132.00	0.178991
$675$	0	0
$676$	676.000	0.0384615
$677$	−778.000	−0.0441669	−0.0220834	−	0.999756i	$-0.507030\pi$
$677$	−778.000	−0.0441669	−0.0220834	+	0.999756i	$0.507030\pi$
$678$	0	0
$679$	−2198.00	−0.124229
$680$	−10976.0	−0.618986
$681$	0	0
$682$	−2688.00	−0.150922
$683$	−19552.0	−1.09537	−0.547684	−	0.836685i	$-0.684490\pi$
$683$	−19552.0	−1.09537	−0.547684	+	0.836685i	$0.684490\pi$
$684$	0	0
$685$	868.000	0.0484154
$686$	686.000	0.0381802
$687$	0	0
$688$	−3776.00	−0.209242
$689$	858.000	0.0474415
$690$	0	0
$691$	−31700.0	−1.74519	−0.872594	−	0.488446i	$-0.837564\pi$
$691$	−31700.0	−1.74519	−0.872594	+	0.488446i	$0.837564\pi$
$692$	10744.0	0.590210
$693$	0	0
$694$	9056.00	0.495333
$695$	−12712.0	−0.693804
$696$	0	0
$697$	50372.0	2.73741
$698$	2180.00	0.118215
$699$	0	0
$700$	−1988.00	−0.107342
$701$	−1470.00	−0.0792028	−0.0396014	−	0.999216i	$-0.512609\pi$
$701$	−1470.00	−0.0792028	−0.0396014	+	0.999216i	$0.512609\pi$
$702$	0	0
$703$	4088.00	0.219320
$704$	−512.000	−0.0274101
$705$	0	0
$706$	−22468.0	−1.19773
$707$	6006.00	0.319489
$708$	0	0
$709$	26718.0	1.41525	0.707627	−	0.706586i	$-0.249766\pi$
$709$	26718.0	1.41525	0.707627	+	0.706586i	$0.249766\pi$
$710$	2800.00	0.148003
$711$	0	0
$712$	−3216.00	−0.169276
$713$	−8736.00	−0.458858
$714$	0	0
$715$	−1456.00	−0.0761557
$716$	−8352.00	−0.435934
$717$	0	0
$718$	6888.00	0.358019
$719$	7408.00	0.384244	0.192122	−	0.981371i	$-0.438463\pi$
$719$	7408.00	0.384244	0.192122	+	0.981371i	$0.438463\pi$
$720$	0	0
$721$	6608.00	0.341324
$722$	12150.0	0.626283
$723$	0	0
$724$	−968.000	−0.0496898
$725$	142.000	0.00727413
$726$	0	0
$727$	−21120.0	−1.07744	−0.538719	−	0.842486i	$-0.681092\pi$
$727$	−21120.0	−1.07744	−0.538719	+	0.842486i	$0.681092\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	−12600.0	−0.638831
$731$	−23128.0	−1.17021
$732$	0	0
$733$	−39354.0	−1.98305	−0.991523	−	0.129929i	$-0.958525\pi$
$733$	−39354.0	−1.98305	−0.991523	+	0.129929i	$0.958525\pi$
$734$	−23424.0	−1.17792
$735$	0	0
$736$	−1664.00	−0.0833368
$737$	−2336.00	−0.116754
$738$	0	0
$739$	−15596.0	−0.776330	−0.388165	−	0.921590i	$-0.626891\pi$
$739$	−15596.0	−0.776330	−0.388165	+	0.921590i	$0.626891\pi$
$740$	−8176.00	−0.406156
$741$	0	0
$742$	924.000	0.0457158
$743$	−4468.00	−0.220612	−0.110306	−	0.993898i	$-0.535183\pi$
$743$	−4468.00	−0.220612	−0.110306	+	0.993898i	$0.535183\pi$
$744$	0	0
$745$	35196.0	1.73085
$746$	−21980.0	−1.07875
$747$	0	0
$748$	−3136.00	−0.153293
$749$	5880.00	0.286850
$750$	0	0
$751$	5440.00	0.264325	0.132163	−	0.991228i	$-0.457808\pi$
$751$	5440.00	0.264325	0.132163	+	0.991228i	$0.457808\pi$
$752$	3456.00	0.167590
$753$	0	0
$754$	−52.0000	−0.00251158
$755$	−14672.0	−0.707243
$756$	0	0
$757$	14638.0	0.702810	0.351405	−	0.936224i	$-0.385704\pi$
$757$	14638.0	0.702810	0.351405	+	0.936224i	$0.385704\pi$
$758$	14008.0	0.671231
$759$	0	0
$760$	3136.00	0.149677
$761$	−35950.0	−1.71247	−0.856233	−	0.516590i	$-0.827201\pi$
$761$	−35950.0	−1.71247	−0.856233	+	0.516590i	$0.827201\pi$
$762$	0	0
$763$	−9338.00	−0.443065
$764$	5136.00	0.243212
$765$	0	0
$766$	−10944.0	−0.516218
$767$	1092.00	0.0514079
$768$	0	0
$769$	−24974.0	−1.17111	−0.585556	−	0.810632i	$-0.699124\pi$
$769$	−24974.0	−1.17111	−0.585556	+	0.810632i	$0.699124\pi$
$770$	−1568.00	−0.0733855
$771$	0	0
$772$	−10936.0	−0.509838
$773$	41254.0	1.91954	0.959769	−	0.280790i	$-0.0905964\pi$
$773$	41254.0	1.91954	0.959769	+	0.280790i	$0.0905964\pi$
$774$	0	0
$775$	−11928.0	−0.552860
$776$	−2512.00	−0.116206
$777$	0	0
$778$	3820.00	0.176033
$779$	−14392.0	−0.661934
$780$	0	0
$781$	800.000	0.0366533
$782$	−10192.0	−0.466068
$783$	0	0
$784$	784.000	0.0357143
$785$	44324.0	2.01528
$786$	0	0
$787$	26780.0	1.21297	0.606483	−	0.795097i	$-0.292580\pi$
$787$	26780.0	1.21297	0.606483	+	0.795097i	$0.292580\pi$
$788$	−20472.0	−0.925488
$789$	0	0
$790$	−10976.0	−0.494315
$791$	6622.00	0.297663
$792$	0	0
$793$	5798.00	0.259638
$794$	26468.0	1.18302
$795$	0	0
$796$	6112.00	0.272153
$797$	18222.0	0.809857	0.404929	−	0.914348i	$-0.367296\pi$
$797$	18222.0	0.809857	0.404929	+	0.914348i	$0.367296\pi$
$798$	0	0
$799$	21168.0	0.937259
$800$	−2272.00	−0.100409
$801$	0	0
$802$	9172.00	0.403834
$803$	−3600.00	−0.158208
$804$	0	0
$805$	−5096.00	−0.223119
$806$	4368.00	0.190889
$807$	0	0
$808$	6864.00	0.298855
$809$	−7338.00	−0.318900	−0.159450	−	0.987206i	$-0.550972\pi$
$809$	−7338.00	−0.318900	−0.159450	+	0.987206i	$0.550972\pi$
$810$	0	0
$811$	12500.0	0.541226	0.270613	−	0.962688i	$-0.412774\pi$
$811$	12500.0	0.541226	0.270613	+	0.962688i	$0.412774\pi$
$812$	−56.0000	−0.00242022
$813$	0	0
$814$	−2336.00	−0.100586
$815$	−6888.00	−0.296044
$816$	0	0
$817$	6608.00	0.282968
$818$	8348.00	0.356823
$819$	0	0
$820$	28784.0	1.22583
$821$	−44830.0	−1.90570	−0.952849	−	0.303445i	$-0.901863\pi$
$821$	−44830.0	−1.90570	−0.952849	+	0.303445i	$0.901863\pi$
$822$	0	0
$823$	11152.0	0.472338	0.236169	−	0.971712i	$-0.424108\pi$
$823$	11152.0	0.472338	0.236169	+	0.971712i	$0.424108\pi$
$824$	7552.00	0.319280
$825$	0	0
$826$	1176.00	0.0495379
$827$	2408.00	0.101251	0.0506254	−	0.998718i	$-0.483879\pi$
$827$	2408.00	0.101251	0.0506254	+	0.998718i	$0.483879\pi$
$828$	0	0
$829$	−44922.0	−1.88203	−0.941017	−	0.338360i	$-0.890128\pi$
$829$	−44922.0	−1.88203	−0.941017	+	0.338360i	$0.890128\pi$
$830$	−8176.00	−0.341919
$831$	0	0
$832$	832.000	0.0346688
$833$	4802.00	0.199735
$834$	0	0
$835$	15456.0	0.640571
$836$	896.000	0.0370680
$837$	0	0
$838$	−20024.0	−0.825439
$839$	−36720.0	−1.51098	−0.755492	−	0.655158i	$-0.772602\pi$
$839$	−36720.0	−1.51098	−0.755492	+	0.655158i	$0.772602\pi$
$840$	0	0
$841$	−24385.0	−0.999836
$842$	2500.00	0.102323
$843$	0	0
$844$	1072.00	0.0437201
$845$	2366.00	0.0963229
$846$	0	0
$847$	8869.00	0.359790
$848$	1056.00	0.0427632
$849$	0	0
$850$	−13916.0	−0.561547
$851$	−7592.00	−0.305817
$852$	0	0
$853$	16006.0	0.642479	0.321240	−	0.946998i	$-0.395900\pi$
$853$	16006.0	0.642479	0.321240	+	0.946998i	$0.395900\pi$
$854$	6244.00	0.250194
$855$	0	0
$856$	6720.00	0.268323
$857$	−37278.0	−1.48587	−0.742936	−	0.669363i	$-0.766567\pi$
$857$	−37278.0	−1.48587	−0.742936	+	0.669363i	$0.766567\pi$
$858$	0	0
$859$	−45236.0	−1.79678	−0.898389	−	0.439201i	$-0.855262\pi$
$859$	−45236.0	−1.79678	−0.898389	+	0.439201i	$0.855262\pi$
$860$	−13216.0	−0.524025
$861$	0	0
$862$	12472.0	0.492805
$863$	23916.0	0.943349	0.471674	−	0.881773i	$-0.343650\pi$
$863$	23916.0	0.943349	0.471674	+	0.881773i	$0.343650\pi$
$864$	0	0
$865$	37604.0	1.47812
$866$	7932.00	0.311247
$867$	0	0
$868$	4704.00	0.183945
$869$	−3136.00	−0.122418
$870$	0	0
$871$	3796.00	0.147672
$872$	−10672.0	−0.414449
$873$	0	0
$874$	2912.00	0.112700
$875$	5292.00	0.204460
$876$	0	0
$877$	−24538.0	−0.944800	−0.472400	−	0.881384i	$-0.656612\pi$
$877$	−24538.0	−0.944800	−0.472400	+	0.881384i	$0.656612\pi$
$878$	−17552.0	−0.674660
$879$	0	0
$880$	−1792.00	−0.0686458
$881$	−910.000	−0.0347999	−0.0173999	−	0.999849i	$-0.505539\pi$
$881$	−910.000	−0.0347999	−0.0173999	+	0.999849i	$0.505539\pi$
$882$	0	0
$883$	10460.0	0.398649	0.199324	−	0.979934i	$-0.436125\pi$
$883$	10460.0	0.398649	0.199324	+	0.979934i	$0.436125\pi$
$884$	5096.00	0.193888
$885$	0	0
$886$	12192.0	0.462301
$887$	−20048.0	−0.758902	−0.379451	−	0.925212i	$-0.623887\pi$
$887$	−20048.0	−0.758902	−0.379451	+	0.925212i	$0.623887\pi$
$888$	0	0
$889$	−6328.00	−0.238734
$890$	−11256.0	−0.423935
$891$	0	0
$892$	5568.00	0.209003
$893$	−6048.00	−0.226639
$894$	0	0
$895$	−29232.0	−1.09175
$896$	896.000	0.0334077
$897$	0	0
$898$	−18780.0	−0.697881
$899$	−336.000	−0.0124652
$900$	0	0
$901$	6468.00	0.239157
$902$	8224.00	0.303580
$903$	0	0
$904$	7568.00	0.278438
$905$	−3388.00	−0.124443
$906$	0	0
$907$	−876.000	−0.0320696	−0.0160348	−	0.999871i	$-0.505104\pi$
$907$	−876.000	−0.0320696	−0.0160348	+	0.999871i	$0.505104\pi$
$908$	24560.0	0.897635
$909$	0	0
$910$	2548.00	0.0928191
$911$	−14796.0	−0.538105	−0.269052	−	0.963126i	$-0.586710\pi$
$911$	−14796.0	−0.538105	−0.269052	+	0.963126i	$0.586710\pi$
$912$	0	0
$913$	−2336.00	−0.0846772
$914$	12.0000	0.000434272 0
$915$	0	0
$916$	7480.00	0.269810
$917$	−10332.0	−0.372075
$918$	0	0
$919$	728.000	0.0261311	0.0130656	−	0.999915i	$-0.495841\pi$
$919$	728.000	0.0261311	0.0130656	+	0.999915i	$0.495841\pi$
$920$	−5824.00	−0.208708
$921$	0	0
$922$	−18748.0	−0.669666
$923$	−1300.00	−0.0463597
$924$	0	0
$925$	−10366.0	−0.368467
$926$	−8016.00	−0.284473
$927$	0	0
$928$	−64.0000	−0.00226390
$929$	−39894.0	−1.40891	−0.704456	−	0.709747i	$-0.748809\pi$
$929$	−39894.0	−1.40891	−0.704456	+	0.709747i	$0.748809\pi$
$930$	0	0
$931$	−1372.00	−0.0482980
$932$	−200.000	−0.00702920
$933$	0	0
$934$	−6520.00	−0.228416
$935$	−10976.0	−0.383908
$936$	0	0
$937$	18906.0	0.659159	0.329580	−	0.944128i	$-0.393093\pi$
$937$	18906.0	0.659159	0.329580	+	0.944128i	$0.393093\pi$
$938$	4088.00	0.142301
$939$	0	0
$940$	12096.0	0.419711
$941$	−49514.0	−1.71531	−0.857657	−	0.514222i	$-0.828081\pi$
$941$	−49514.0	−1.71531	−0.857657	+	0.514222i	$0.828081\pi$
$942$	0	0
$943$	26728.0	0.922994
$944$	1344.00	0.0463384
$945$	0	0
$946$	−3776.00	−0.129776
$947$	−16384.0	−0.562205	−0.281103	−	0.959678i	$-0.590700\pi$
$947$	−16384.0	−0.562205	−0.281103	+	0.959678i	$0.590700\pi$
$948$	0	0
$949$	5850.00	0.200104
$950$	3976.00	0.135788
$951$	0	0
$952$	5488.00	0.186835
$953$	30742.0	1.04494	0.522472	−	0.852657i	$-0.325010\pi$
$953$	30742.0	1.04494	0.522472	+	0.852657i	$0.325010\pi$
$954$	0	0
$955$	17976.0	0.609099
$956$	12912.0	0.436824
$957$	0	0
$958$	25392.0	0.856345
$959$	−434.000	−0.0146138
$960$	0	0
$961$	−1567.00	−0.0525998
$962$	3796.00	0.127222
$963$	0	0
$964$	23912.0	0.798915
$965$	−38276.0	−1.27684
$966$	0	0
$967$	−41376.0	−1.37597	−0.687985	−	0.725725i	$-0.741504\pi$
$967$	−41376.0	−1.37597	−0.687985	+	0.725725i	$0.741504\pi$
$968$	10136.0	0.336553
$969$	0	0
$970$	−8792.00	−0.291025
$971$	−57300.0	−1.89376	−0.946882	−	0.321582i	$-0.895786\pi$
$971$	−57300.0	−1.89376	−0.946882	+	0.321582i	$0.895786\pi$
$972$	0	0
$973$	6356.00	0.209418
$974$	−21520.0	−0.707952
$975$	0	0
$976$	7136.00	0.234035
$977$	7734.00	0.253258	0.126629	−	0.991950i	$-0.459584\pi$
$977$	7734.00	0.253258	0.126629	+	0.991950i	$0.459584\pi$
$978$	0	0
$979$	−3216.00	−0.104989
$980$	2744.00	0.0894427
$981$	0	0
$982$	−23200.0	−0.753912
$983$	31432.0	1.01986	0.509931	−	0.860215i	$-0.329671\pi$
$983$	31432.0	1.01986	0.509931	+	0.860215i	$0.329671\pi$
$984$	0	0
$985$	−71652.0	−2.31779
$986$	−392.000	−0.0126611
$987$	0	0
$988$	−1456.00	−0.0468841
$989$	−12272.0	−0.394567
$990$	0	0
$991$	−24432.0	−0.783156	−0.391578	−	0.920145i	$-0.628071\pi$
$991$	−24432.0	−0.783156	−0.391578	+	0.920145i	$0.628071\pi$
$992$	5376.00	0.172065
$993$	0	0
$994$	−1400.00	−0.0446733
$995$	21392.0	0.681580
$996$	0	0
$997$	3094.00	0.0982828	0.0491414	−	0.998792i	$-0.484352\pi$
$997$	3094.00	0.0982828	0.0491414	+	0.998792i	$0.484352\pi$
$998$	31816.0	1.00914
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.c.1.1	✓	1	3.2	odd	2
1638.4.a.h.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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