LMFDB - Embedded newform 1344.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -4.00000 q^{11} +46.0000 q^{13} +18.0000 q^{15} -82.0000 q^{17} +84.0000 q^{19} +21.0000 q^{21} -44.0000 q^{23} -89.0000 q^{25} -27.0000 q^{27} -70.0000 q^{29} +152.000 q^{31} +12.0000 q^{33} +42.0000 q^{35} +146.000 q^{37} -138.000 q^{39} +94.0000 q^{41} +488.000 q^{43} -54.0000 q^{45} +32.0000 q^{47} +49.0000 q^{49} +246.000 q^{51} +562.000 q^{53} +24.0000 q^{55} -252.000 q^{57} -476.000 q^{59} -34.0000 q^{61} -63.0000 q^{63} -276.000 q^{65} -520.000 q^{67} +132.000 q^{69} +36.0000 q^{71} -654.000 q^{73} +267.000 q^{75} +28.0000 q^{77} +608.000 q^{79} +81.0000 q^{81} +284.000 q^{83} +492.000 q^{85} +210.000 q^{87} -954.000 q^{89} -322.000 q^{91} -456.000 q^{93} -504.000 q^{95} -1694.00 q^{97} -36.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−4.00000	−0.109640	−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000	−0.109640	−0.0548202	+	0.998496i	$0.517459\pi$
$12$	0	0
$13$	46.0000	0.981393	0.490696	−	0.871331i	$-0.336742\pi$
$13$	46.0000	0.981393	0.490696	+	0.871331i	$0.336742\pi$
$14$	0	0
$15$	18.0000	0.309839
$16$	0	0
$17$	−82.0000	−1.16988	−0.584939	−	0.811077i	$-0.698882\pi$
$17$	−82.0000	−1.16988	−0.584939	+	0.811077i	$0.698882\pi$
$18$	0	0
$19$	84.0000	1.01426	0.507130	−	0.861870i	$-0.330707\pi$
$19$	84.0000	1.01426	0.507130	+	0.861870i	$0.330707\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−44.0000	−0.398897	−0.199449	−	0.979908i	$-0.563915\pi$
$23$	−44.0000	−0.398897	−0.199449	+	0.979908i	$0.563915\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−70.0000	−0.448230	−0.224115	−	0.974563i	$-0.571949\pi$
$29$	−70.0000	−0.448230	−0.224115	+	0.974563i	$0.571949\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	0	0
$33$	12.0000	0.0633010
$34$	0	0
$35$	42.0000	0.202837
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	0	0
$39$	−138.000	−0.566607
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	0	0
$43$	488.000	1.73068	0.865341	−	0.501184i	$-0.167102\pi$
$43$	488.000	1.73068	0.865341	+	0.501184i	$0.167102\pi$
$44$	0	0
$45$	−54.0000	−0.178885
$46$	0	0
$47$	32.0000	0.0993123	0.0496562	−	0.998766i	$-0.484187\pi$
$47$	32.0000	0.0993123	0.0496562	+	0.998766i	$0.484187\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	246.000	0.675429
$52$	0	0
$53$	562.000	1.45654	0.728270	−	0.685290i	$-0.240325\pi$
$53$	562.000	1.45654	0.728270	+	0.685290i	$0.240325\pi$
$54$	0	0
$55$	24.0000	0.0588393
$56$	0	0
$57$	−252.000	−0.585583
$58$	0	0
$59$	−476.000	−1.05034	−0.525169	−	0.850998i	$-0.675998\pi$
$59$	−476.000	−1.05034	−0.525169	+	0.850998i	$0.675998\pi$
$60$	0	0
$61$	−34.0000	−0.0713648	−0.0356824	−	0.999363i	$-0.511360\pi$
$61$	−34.0000	−0.0713648	−0.0356824	+	0.999363i	$0.511360\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−276.000	−0.526671
$66$	0	0
$67$	−520.000	−0.948181	−0.474090	−	0.880476i	$-0.657223\pi$
$67$	−520.000	−0.948181	−0.474090	+	0.880476i	$0.657223\pi$
$68$	0	0
$69$	132.000	0.230303
$70$	0	0
$71$	36.0000	0.0601748	0.0300874	−	0.999547i	$-0.490421\pi$
$71$	36.0000	0.0601748	0.0300874	+	0.999547i	$0.490421\pi$
$72$	0	0
$73$	−654.000	−1.04856	−0.524280	−	0.851546i	$-0.675666\pi$
$73$	−654.000	−1.04856	−0.524280	+	0.851546i	$0.675666\pi$
$74$	0	0
$75$	267.000	0.411073
$76$	0	0
$77$	28.0000	0.0414402
$78$	0	0
$79$	608.000	0.865890	0.432945	−	0.901420i	$-0.357474\pi$
$79$	608.000	0.865890	0.432945	+	0.901420i	$0.357474\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	284.000	0.375579	0.187789	−	0.982209i	$-0.439868\pi$
$83$	284.000	0.375579	0.187789	+	0.982209i	$0.439868\pi$
$84$	0	0
$85$	492.000	0.627822
$86$	0	0
$87$	210.000	0.258786
$88$	0	0
$89$	−954.000	−1.13622	−0.568111	−	0.822952i	$-0.692326\pi$
$89$	−954.000	−1.13622	−0.568111	+	0.822952i	$0.692326\pi$
$90$	0	0
$91$	−322.000	−0.370932
$92$	0	0
$93$	−456.000	−0.508441
$94$	0	0
$95$	−504.000	−0.544309
$96$	0	0
$97$	−1694.00	−1.77319	−0.886596	−	0.462544i	$-0.846937\pi$
$97$	−1694.00	−1.77319	−0.886596	+	0.462544i	$0.846937\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	−318.000	−0.313289	−0.156644	−	0.987655i	$-0.550068\pi$
$101$	−318.000	−0.313289	−0.156644	+	0.987655i	$0.550068\pi$
$102$	0	0
$103$	872.000	0.834182	0.417091	−	0.908865i	$-0.363050\pi$
$103$	872.000	0.834182	0.417091	+	0.908865i	$0.363050\pi$
$104$	0	0
$105$	−126.000	−0.117108
$106$	0	0
$107$	−364.000	−0.328871	−0.164436	−	0.986388i	$-0.552580\pi$
$107$	−364.000	−0.328871	−0.164436	+	0.986388i	$0.552580\pi$
$108$	0	0
$109$	−838.000	−0.736384	−0.368192	−	0.929750i	$-0.620023\pi$
$109$	−838.000	−0.736384	−0.368192	+	0.929750i	$0.620023\pi$
$110$	0	0
$111$	−438.000	−0.374533
$112$	0	0
$113$	−366.000	−0.304694	−0.152347	−	0.988327i	$-0.548683\pi$
$113$	−366.000	−0.304694	−0.152347	+	0.988327i	$0.548683\pi$
$114$	0	0
$115$	264.000	0.214071
$116$	0	0
$117$	414.000	0.327131
$118$	0	0
$119$	574.000	0.442172
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	−282.000	−0.206724
$124$	0	0
$125$	1284.00	0.918756
$126$	0	0
$127$	536.000	0.374506	0.187253	−	0.982312i	$-0.440042\pi$
$127$	536.000	0.374506	0.187253	+	0.982312i	$0.440042\pi$
$128$	0	0
$129$	−1464.00	−0.999209
$130$	0	0
$131$	1980.00	1.32056	0.660280	−	0.751019i	$-0.270438\pi$
$131$	1980.00	1.32056	0.660280	+	0.751019i	$0.270438\pi$
$132$	0	0
$133$	−588.000	−0.383354
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	−2878.00	−1.79477	−0.897387	−	0.441244i	$-0.854537\pi$
$137$	−2878.00	−1.79477	−0.897387	+	0.441244i	$0.854537\pi$
$138$	0	0
$139$	−1436.00	−0.876258	−0.438129	−	0.898912i	$-0.644359\pi$
$139$	−1436.00	−0.876258	−0.438129	+	0.898912i	$0.644359\pi$
$140$	0	0
$141$	−96.0000	−0.0573380
$142$	0	0
$143$	−184.000	−0.107600
$144$	0	0
$145$	420.000	0.240546
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1374.00	−0.755453	−0.377726	−	0.925917i	$-0.623294\pi$
$149$	−1374.00	−0.755453	−0.377726	+	0.925917i	$0.623294\pi$
$150$	0	0
$151$	−688.000	−0.370786	−0.185393	−	0.982664i	$-0.559356\pi$
$151$	−688.000	−0.370786	−0.185393	+	0.982664i	$0.559356\pi$
$152$	0	0
$153$	−738.000	−0.389959
$154$	0	0
$155$	−912.000	−0.472604
$156$	0	0
$157$	6.00000	0.00305001	0.00152501	−	0.999999i	$-0.499515\pi$
$157$	6.00000	0.00305001	0.00152501	+	0.999999i	$0.499515\pi$
$158$	0	0
$159$	−1686.00	−0.840934
$160$	0	0
$161$	308.000	0.150769
$162$	0	0
$163$	−16.0000	−0.00768845	−0.00384422	−	0.999993i	$-0.501224\pi$
$163$	−16.0000	−0.00768845	−0.00384422	+	0.999993i	$0.501224\pi$
$164$	0	0
$165$	−72.0000	−0.0339709
$166$	0	0
$167$	−1952.00	−0.904493	−0.452246	−	0.891893i	$-0.649377\pi$
$167$	−1952.00	−0.904493	−0.452246	+	0.891893i	$0.649377\pi$
$168$	0	0
$169$	−81.0000	−0.0368685
$170$	0	0
$171$	756.000	0.338086
$172$	0	0
$173$	642.000	0.282141	0.141070	−	0.990000i	$-0.454946\pi$
$173$	642.000	0.282141	0.141070	+	0.990000i	$0.454946\pi$
$174$	0	0
$175$	623.000	0.269111
$176$	0	0
$177$	1428.00	0.606413
$178$	0	0
$179$	−468.000	−0.195419	−0.0977094	−	0.995215i	$-0.531152\pi$
$179$	−468.000	−0.195419	−0.0977094	+	0.995215i	$0.531152\pi$
$180$	0	0
$181$	−138.000	−0.0566710	−0.0283355	−	0.999598i	$-0.509021\pi$
$181$	−138.000	−0.0566710	−0.0283355	+	0.999598i	$0.509021\pi$
$182$	0	0
$183$	102.000	0.0412025
$184$	0	0
$185$	−876.000	−0.348134
$186$	0	0
$187$	328.000	0.128266
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−1980.00	−0.750093	−0.375047	−	0.927006i	$-0.622373\pi$
$191$	−1980.00	−0.750093	−0.375047	+	0.927006i	$0.622373\pi$
$192$	0	0
$193$	−1294.00	−0.482612	−0.241306	−	0.970449i	$-0.577576\pi$
$193$	−1294.00	−0.482612	−0.241306	+	0.970449i	$0.577576\pi$
$194$	0	0
$195$	828.000	0.304073
$196$	0	0
$197$	−4542.00	−1.64266	−0.821330	−	0.570453i	$-0.806768\pi$
$197$	−4542.00	−1.64266	−0.821330	+	0.570453i	$0.806768\pi$
$198$	0	0
$199$	2992.00	1.06582	0.532908	−	0.846173i	$-0.321099\pi$
$199$	2992.00	1.06582	0.532908	+	0.846173i	$0.321099\pi$
$200$	0	0
$201$	1560.00	0.547432
$202$	0	0
$203$	490.000	0.169415
$204$	0	0
$205$	−564.000	−0.192154
$206$	0	0
$207$	−396.000	−0.132966
$208$	0	0
$209$	−336.000	−0.111204
$210$	0	0
$211$	−3344.00	−1.09104	−0.545522	−	0.838096i	$-0.683669\pi$
$211$	−3344.00	−1.09104	−0.545522	+	0.838096i	$0.683669\pi$
$212$	0	0
$213$	−108.000	−0.0347420
$214$	0	0
$215$	−2928.00	−0.928781
$216$	0	0
$217$	−1064.00	−0.332853
$218$	0	0
$219$	1962.00	0.605387
$220$	0	0
$221$	−3772.00	−1.14811
$222$	0	0
$223$	5640.00	1.69364	0.846821	−	0.531877i	$-0.178513\pi$
$223$	5640.00	1.69364	0.846821	+	0.531877i	$0.178513\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	2660.00	0.777755	0.388878	−	0.921289i	$-0.372863\pi$
$227$	2660.00	0.777755	0.388878	+	0.921289i	$0.372863\pi$
$228$	0	0
$229$	−5626.00	−1.62348	−0.811739	−	0.584020i	$-0.801479\pi$
$229$	−5626.00	−1.62348	−0.811739	+	0.584020i	$0.801479\pi$
$230$	0	0
$231$	−84.0000	−0.0239255
$232$	0	0
$233$	−606.000	−0.170388	−0.0851939	−	0.996364i	$-0.527151\pi$
$233$	−606.000	−0.170388	−0.0851939	+	0.996364i	$0.527151\pi$
$234$	0	0
$235$	−192.000	−0.0532966
$236$	0	0
$237$	−1824.00	−0.499922
$238$	0	0
$239$	−6300.00	−1.70508	−0.852538	−	0.522665i	$-0.824938\pi$
$239$	−6300.00	−1.70508	−0.852538	+	0.522665i	$0.824938\pi$
$240$	0	0
$241$	2210.00	0.590700	0.295350	−	0.955389i	$-0.404564\pi$
$241$	2210.00	0.590700	0.295350	+	0.955389i	$0.404564\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−294.000	−0.0766652
$246$	0	0
$247$	3864.00	0.995386
$248$	0	0
$249$	−852.000	−0.216841
$250$	0	0
$251$	−5044.00	−1.26842	−0.634212	−	0.773159i	$-0.718675\pi$
$251$	−5044.00	−1.26842	−0.634212	+	0.773159i	$0.718675\pi$
$252$	0	0
$253$	176.000	0.0437353
$254$	0	0
$255$	−1476.00	−0.362473
$256$	0	0
$257$	−98.0000	−0.0237863	−0.0118931	−	0.999929i	$-0.503786\pi$
$257$	−98.0000	−0.0237863	−0.0118931	+	0.999929i	$0.503786\pi$
$258$	0	0
$259$	−1022.00	−0.245189
$260$	0	0
$261$	−630.000	−0.149410
$262$	0	0
$263$	−5732.00	−1.34392	−0.671959	−	0.740589i	$-0.734547\pi$
$263$	−5732.00	−1.34392	−0.671959	+	0.740589i	$0.734547\pi$
$264$	0	0
$265$	−3372.00	−0.781662
$266$	0	0
$267$	2862.00	0.655998
$268$	0	0
$269$	−630.000	−0.142795	−0.0713974	−	0.997448i	$-0.522746\pi$
$269$	−630.000	−0.142795	−0.0713974	+	0.997448i	$0.522746\pi$
$270$	0	0
$271$	−2560.00	−0.573834	−0.286917	−	0.957955i	$-0.592630\pi$
$271$	−2560.00	−0.573834	−0.286917	+	0.957955i	$0.592630\pi$
$272$	0	0
$273$	966.000	0.214157
$274$	0	0
$275$	356.000	0.0780640
$276$	0	0
$277$	2266.00	0.491519	0.245759	−	0.969331i	$-0.420963\pi$
$277$	2266.00	0.491519	0.245759	+	0.969331i	$0.420963\pi$
$278$	0	0
$279$	1368.00	0.293548
$280$	0	0
$281$	4338.00	0.920937	0.460469	−	0.887676i	$-0.347681\pi$
$281$	4338.00	0.920937	0.460469	+	0.887676i	$0.347681\pi$
$282$	0	0
$283$	−6364.00	−1.33675	−0.668376	−	0.743824i	$-0.733010\pi$
$283$	−6364.00	−1.33675	−0.668376	+	0.743824i	$0.733010\pi$
$284$	0	0
$285$	1512.00	0.314257
$286$	0	0
$287$	−658.000	−0.135333
$288$	0	0
$289$	1811.00	0.368614
$290$	0	0
$291$	5082.00	1.02375
$292$	0	0
$293$	1778.00	0.354511	0.177256	−	0.984165i	$-0.443278\pi$
$293$	1778.00	0.354511	0.177256	+	0.984165i	$0.443278\pi$
$294$	0	0
$295$	2856.00	0.563670
$296$	0	0
$297$	108.000	0.0211003
$298$	0	0
$299$	−2024.00	−0.391475
$300$	0	0
$301$	−3416.00	−0.654136
$302$	0	0
$303$	954.000	0.180877
$304$	0	0
$305$	204.000	0.0382984
$306$	0	0
$307$	1364.00	0.253575	0.126788	−	0.991930i	$-0.459533\pi$
$307$	1364.00	0.253575	0.126788	+	0.991930i	$0.459533\pi$
$308$	0	0
$309$	−2616.00	−0.481615
$310$	0	0
$311$	7512.00	1.36967	0.684834	−	0.728700i	$-0.259875\pi$
$311$	7512.00	1.36967	0.684834	+	0.728700i	$0.259875\pi$
$312$	0	0
$313$	2914.00	0.526227	0.263113	−	0.964765i	$-0.415251\pi$
$313$	2914.00	0.526227	0.263113	+	0.964765i	$0.415251\pi$
$314$	0	0
$315$	378.000	0.0676123
$316$	0	0
$317$	−7846.00	−1.39014	−0.695071	−	0.718941i	$-0.744627\pi$
$317$	−7846.00	−1.39014	−0.695071	+	0.718941i	$0.744627\pi$
$318$	0	0
$319$	280.000	0.0491442
$320$	0	0
$321$	1092.00	0.189874
$322$	0	0
$323$	−6888.00	−1.18656
$324$	0	0
$325$	−4094.00	−0.698752
$326$	0	0
$327$	2514.00	0.425151
$328$	0	0
$329$	−224.000	−0.0375365
$330$	0	0
$331$	24.0000	0.00398538	0.00199269	−	0.999998i	$-0.499366\pi$
$331$	24.0000	0.00398538	0.00199269	+	0.999998i	$0.499366\pi$
$332$	0	0
$333$	1314.00	0.216237
$334$	0	0
$335$	3120.00	0.508847
$336$	0	0
$337$	5970.00	0.965005	0.482502	−	0.875895i	$-0.339728\pi$
$337$	5970.00	0.965005	0.482502	+	0.875895i	$0.339728\pi$
$338$	0	0
$339$	1098.00	0.175915
$340$	0	0
$341$	−608.000	−0.0965544
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−792.000	−0.123594
$346$	0	0
$347$	10212.0	1.57985	0.789926	−	0.613202i	$-0.210119\pi$
$347$	10212.0	1.57985	0.789926	+	0.613202i	$0.210119\pi$
$348$	0	0
$349$	−3914.00	−0.600320	−0.300160	−	0.953889i	$-0.597040\pi$
$349$	−3914.00	−0.600320	−0.300160	+	0.953889i	$0.597040\pi$
$350$	0	0
$351$	−1242.00	−0.188869
$352$	0	0
$353$	3222.00	0.485807	0.242903	−	0.970050i	$-0.421900\pi$
$353$	3222.00	0.485807	0.242903	+	0.970050i	$0.421900\pi$
$354$	0	0
$355$	−216.000	−0.0322932
$356$	0	0
$357$	−1722.00	−0.255288
$358$	0	0
$359$	−3140.00	−0.461624	−0.230812	−	0.972998i	$-0.574138\pi$
$359$	−3140.00	−0.461624	−0.230812	+	0.972998i	$0.574138\pi$
$360$	0	0
$361$	197.000	0.0287214
$362$	0	0
$363$	3945.00	0.570410
$364$	0	0
$365$	3924.00	0.562717
$366$	0	0
$367$	6680.00	0.950118	0.475059	−	0.879954i	$-0.342427\pi$
$367$	6680.00	0.950118	0.475059	+	0.879954i	$0.342427\pi$
$368$	0	0
$369$	846.000	0.119352
$370$	0	0
$371$	−3934.00	−0.550520
$372$	0	0
$373$	−7830.00	−1.08692	−0.543461	−	0.839434i	$-0.682886\pi$
$373$	−7830.00	−1.08692	−0.543461	+	0.839434i	$0.682886\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	−3220.00	−0.439890
$378$	0	0
$379$	−11368.0	−1.54073	−0.770363	−	0.637606i	$-0.779925\pi$
$379$	−11368.0	−1.54073	−0.770363	+	0.637606i	$0.779925\pi$
$380$	0	0
$381$	−1608.00	−0.216221
$382$	0	0
$383$	136.000	0.0181443	0.00907216	−	0.999959i	$-0.497112\pi$
$383$	136.000	0.0181443	0.00907216	+	0.999959i	$0.497112\pi$
$384$	0	0
$385$	−168.000	−0.0222392
$386$	0	0
$387$	4392.00	0.576894
$388$	0	0
$389$	−13374.0	−1.74316	−0.871579	−	0.490255i	$-0.836904\pi$
$389$	−13374.0	−1.74316	−0.871579	+	0.490255i	$0.836904\pi$
$390$	0	0
$391$	3608.00	0.466661
$392$	0	0
$393$	−5940.00	−0.762426
$394$	0	0
$395$	−3648.00	−0.464686
$396$	0	0
$397$	6406.00	0.809844	0.404922	−	0.914351i	$-0.367299\pi$
$397$	6406.00	0.809844	0.404922	+	0.914351i	$0.367299\pi$
$398$	0	0
$399$	1764.00	0.221329
$400$	0	0
$401$	−3958.00	−0.492900	−0.246450	−	0.969155i	$-0.579264\pi$
$401$	−3958.00	−0.492900	−0.246450	+	0.969155i	$0.579264\pi$
$402$	0	0
$403$	6992.00	0.864259
$404$	0	0
$405$	−486.000	−0.0596285
$406$	0	0
$407$	−584.000	−0.0711248
$408$	0	0
$409$	−12614.0	−1.52499	−0.762497	−	0.646992i	$-0.776027\pi$
$409$	−12614.0	−1.52499	−0.762497	+	0.646992i	$0.776027\pi$
$410$	0	0
$411$	8634.00	1.03621
$412$	0	0
$413$	3332.00	0.396990
$414$	0	0
$415$	−1704.00	−0.201557
$416$	0	0
$417$	4308.00	0.505908
$418$	0	0
$419$	1404.00	0.163699	0.0818495	−	0.996645i	$-0.473917\pi$
$419$	1404.00	0.163699	0.0818495	+	0.996645i	$0.473917\pi$
$420$	0	0
$421$	1482.00	0.171564	0.0857818	−	0.996314i	$-0.472661\pi$
$421$	1482.00	0.171564	0.0857818	+	0.996314i	$0.472661\pi$
$422$	0	0
$423$	288.000	0.0331041
$424$	0	0
$425$	7298.00	0.832953
$426$	0	0
$427$	238.000	0.0269734
$428$	0	0
$429$	552.000	0.0621231
$430$	0	0
$431$	−14228.0	−1.59011	−0.795056	−	0.606535i	$-0.792559\pi$
$431$	−14228.0	−1.59011	−0.795056	+	0.606535i	$0.792559\pi$
$432$	0	0
$433$	11674.0	1.29565	0.647825	−	0.761789i	$-0.275679\pi$
$433$	11674.0	1.29565	0.647825	+	0.761789i	$0.275679\pi$
$434$	0	0
$435$	−1260.00	−0.138879
$436$	0	0
$437$	−3696.00	−0.404585
$438$	0	0
$439$	14712.0	1.59947	0.799733	−	0.600356i	$-0.204974\pi$
$439$	14712.0	1.59947	0.799733	+	0.600356i	$0.204974\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	2940.00	0.315313	0.157656	−	0.987494i	$-0.449606\pi$
$443$	2940.00	0.315313	0.157656	+	0.987494i	$0.449606\pi$
$444$	0	0
$445$	5724.00	0.609761
$446$	0	0
$447$	4122.00	0.436161
$448$	0	0
$449$	4818.00	0.506404	0.253202	−	0.967413i	$-0.418516\pi$
$449$	4818.00	0.506404	0.253202	+	0.967413i	$0.418516\pi$
$450$	0	0
$451$	−376.000	−0.0392575
$452$	0	0
$453$	2064.00	0.214073
$454$	0	0
$455$	1932.00	0.199063
$456$	0	0
$457$	2122.00	0.217206	0.108603	−	0.994085i	$-0.465362\pi$
$457$	2122.00	0.217206	0.108603	+	0.994085i	$0.465362\pi$
$458$	0	0
$459$	2214.00	0.225143
$460$	0	0
$461$	15010.0	1.51645	0.758227	−	0.651991i	$-0.226066\pi$
$461$	15010.0	1.51645	0.758227	+	0.651991i	$0.226066\pi$
$462$	0	0
$463$	5712.00	0.573346	0.286673	−	0.958029i	$-0.407451\pi$
$463$	5712.00	0.573346	0.286673	+	0.958029i	$0.407451\pi$
$464$	0	0
$465$	2736.00	0.272858
$466$	0	0
$467$	10356.0	1.02616	0.513082	−	0.858340i	$-0.328504\pi$
$467$	10356.0	1.02616	0.513082	+	0.858340i	$0.328504\pi$
$468$	0	0
$469$	3640.00	0.358379
$470$	0	0
$471$	−18.0000	−0.00176093
$472$	0	0
$473$	−1952.00	−0.189753
$474$	0	0
$475$	−7476.00	−0.722152
$476$	0	0
$477$	5058.00	0.485513
$478$	0	0
$479$	−16992.0	−1.62084	−0.810422	−	0.585847i	$-0.800762\pi$
$479$	−16992.0	−1.62084	−0.810422	+	0.585847i	$0.800762\pi$
$480$	0	0
$481$	6716.00	0.636639
$482$	0	0
$483$	−924.000	−0.0870465
$484$	0	0
$485$	10164.0	0.951595
$486$	0	0
$487$	−12000.0	−1.11657	−0.558287	−	0.829648i	$-0.688541\pi$
$487$	−12000.0	−1.11657	−0.558287	+	0.829648i	$0.688541\pi$
$488$	0	0
$489$	48.0000	0.00443893
$490$	0	0
$491$	−4804.00	−0.441551	−0.220775	−	0.975325i	$-0.570859\pi$
$491$	−4804.00	−0.441551	−0.220775	+	0.975325i	$0.570859\pi$
$492$	0	0
$493$	5740.00	0.524374
$494$	0	0
$495$	216.000	0.0196131
$496$	0	0
$497$	−252.000	−0.0227440
$498$	0	0
$499$	11696.0	1.04927	0.524634	−	0.851328i	$-0.324202\pi$
$499$	11696.0	1.04927	0.524634	+	0.851328i	$0.324202\pi$
$500$	0	0
$501$	5856.00	0.522209
$502$	0	0
$503$	3328.00	0.295006	0.147503	−	0.989062i	$-0.452876\pi$
$503$	3328.00	0.295006	0.147503	+	0.989062i	$0.452876\pi$
$504$	0	0
$505$	1908.00	0.168128
$506$	0	0
$507$	243.000	0.0212860
$508$	0	0
$509$	15290.0	1.33147	0.665734	−	0.746189i	$-0.268119\pi$
$509$	15290.0	1.33147	0.665734	+	0.746189i	$0.268119\pi$
$510$	0	0
$511$	4578.00	0.396319
$512$	0	0
$513$	−2268.00	−0.195194
$514$	0	0
$515$	−5232.00	−0.447669
$516$	0	0
$517$	−128.000	−0.0108887
$518$	0	0
$519$	−1926.00	−0.162894
$520$	0	0
$521$	−1946.00	−0.163639	−0.0818194	−	0.996647i	$-0.526073\pi$
$521$	−1946.00	−0.163639	−0.0818194	+	0.996647i	$0.526073\pi$
$522$	0	0
$523$	11468.0	0.958816	0.479408	−	0.877592i	$-0.340851\pi$
$523$	11468.0	0.958816	0.479408	+	0.877592i	$0.340851\pi$
$524$	0	0
$525$	−1869.00	−0.155371
$526$	0	0
$527$	−12464.0	−1.03025
$528$	0	0
$529$	−10231.0	−0.840881
$530$	0	0
$531$	−4284.00	−0.350113
$532$	0	0
$533$	4324.00	0.351394
$534$	0	0
$535$	2184.00	0.176491
$536$	0	0
$537$	1404.00	0.112825
$538$	0	0
$539$	−196.000	−0.0156629
$540$	0	0
$541$	20178.0	1.60355	0.801774	−	0.597627i	$-0.203890\pi$
$541$	20178.0	1.60355	0.801774	+	0.597627i	$0.203890\pi$
$542$	0	0
$543$	414.000	0.0327190
$544$	0	0
$545$	5028.00	0.395185
$546$	0	0
$547$	−4864.00	−0.380200	−0.190100	−	0.981765i	$-0.560881\pi$
$547$	−4864.00	−0.380200	−0.190100	+	0.981765i	$0.560881\pi$
$548$	0	0
$549$	−306.000	−0.0237883
$550$	0	0
$551$	−5880.00	−0.454621
$552$	0	0
$553$	−4256.00	−0.327276
$554$	0	0
$555$	2628.00	0.200995
$556$	0	0
$557$	13722.0	1.04384	0.521921	−	0.852994i	$-0.325216\pi$
$557$	13722.0	1.04384	0.521921	+	0.852994i	$0.325216\pi$
$558$	0	0
$559$	22448.0	1.69848
$560$	0	0
$561$	−984.000	−0.0740544
$562$	0	0
$563$	−26468.0	−1.98134	−0.990669	−	0.136290i	$-0.956482\pi$
$563$	−26468.0	−1.98134	−0.990669	+	0.136290i	$0.956482\pi$
$564$	0	0
$565$	2196.00	0.163516
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−7518.00	−0.553903	−0.276952	−	0.960884i	$-0.589324\pi$
$569$	−7518.00	−0.553903	−0.276952	+	0.960884i	$0.589324\pi$
$570$	0	0
$571$	16432.0	1.20430	0.602152	−	0.798381i	$-0.294310\pi$
$571$	16432.0	1.20430	0.602152	+	0.798381i	$0.294310\pi$
$572$	0	0
$573$	5940.00	0.433066
$574$	0	0
$575$	3916.00	0.284015
$576$	0	0
$577$	−5822.00	−0.420057	−0.210029	−	0.977695i	$-0.567356\pi$
$577$	−5822.00	−0.420057	−0.210029	+	0.977695i	$0.567356\pi$
$578$	0	0
$579$	3882.00	0.278636
$580$	0	0
$581$	−1988.00	−0.141955
$582$	0	0
$583$	−2248.00	−0.159696
$584$	0	0
$585$	−2484.00	−0.175557
$586$	0	0
$587$	−21540.0	−1.51457	−0.757284	−	0.653086i	$-0.773474\pi$
$587$	−21540.0	−1.51457	−0.757284	+	0.653086i	$0.773474\pi$
$588$	0	0
$589$	12768.0	0.893203
$590$	0	0
$591$	13626.0	0.948390
$592$	0	0
$593$	−3066.00	−0.212320	−0.106160	−	0.994349i	$-0.533856\pi$
$593$	−3066.00	−0.212320	−0.106160	+	0.994349i	$0.533856\pi$
$594$	0	0
$595$	−3444.00	−0.237295
$596$	0	0
$597$	−8976.00	−0.615349
$598$	0	0
$599$	−20652.0	−1.40871	−0.704355	−	0.709847i	$-0.748764\pi$
$599$	−20652.0	−1.40871	−0.704355	+	0.709847i	$0.748764\pi$
$600$	0	0
$601$	6074.00	0.412252	0.206126	−	0.978525i	$-0.433914\pi$
$601$	6074.00	0.412252	0.206126	+	0.978525i	$0.433914\pi$
$602$	0	0
$603$	−4680.00	−0.316060
$604$	0	0
$605$	7890.00	0.530205
$606$	0	0
$607$	−8136.00	−0.544036	−0.272018	−	0.962292i	$-0.587691\pi$
$607$	−8136.00	−0.544036	−0.272018	+	0.962292i	$0.587691\pi$
$608$	0	0
$609$	−1470.00	−0.0978118
$610$	0	0
$611$	1472.00	0.0974644
$612$	0	0
$613$	−5102.00	−0.336163	−0.168081	−	0.985773i	$-0.553757\pi$
$613$	−5102.00	−0.336163	−0.168081	+	0.985773i	$0.553757\pi$
$614$	0	0
$615$	1692.00	0.110940
$616$	0	0
$617$	3074.00	0.200575	0.100287	−	0.994959i	$-0.468024\pi$
$617$	3074.00	0.200575	0.100287	+	0.994959i	$0.468024\pi$
$618$	0	0
$619$	−5156.00	−0.334793	−0.167397	−	0.985890i	$-0.553536\pi$
$619$	−5156.00	−0.334793	−0.167397	+	0.985890i	$0.553536\pi$
$620$	0	0
$621$	1188.00	0.0767678
$622$	0	0
$623$	6678.00	0.429452
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	1008.00	0.0642036
$628$	0	0
$629$	−11972.0	−0.758911
$630$	0	0
$631$	−20320.0	−1.28198	−0.640988	−	0.767551i	$-0.721475\pi$
$631$	−20320.0	−1.28198	−0.640988	+	0.767551i	$0.721475\pi$
$632$	0	0
$633$	10032.0	0.629915
$634$	0	0
$635$	−3216.00	−0.200981
$636$	0	0
$637$	2254.00	0.140199
$638$	0	0
$639$	324.000	0.0200583
$640$	0	0
$641$	14842.0	0.914546	0.457273	−	0.889326i	$-0.348826\pi$
$641$	14842.0	0.914546	0.457273	+	0.889326i	$0.348826\pi$
$642$	0	0
$643$	−14516.0	−0.890288	−0.445144	−	0.895459i	$-0.646848\pi$
$643$	−14516.0	−0.890288	−0.445144	+	0.895459i	$0.646848\pi$
$644$	0	0
$645$	8784.00	0.536232
$646$	0	0
$647$	−6096.00	−0.370415	−0.185207	−	0.982699i	$-0.559296\pi$
$647$	−6096.00	−0.370415	−0.185207	+	0.982699i	$0.559296\pi$
$648$	0	0
$649$	1904.00	0.115160
$650$	0	0
$651$	3192.00	0.192173
$652$	0	0
$653$	8650.00	0.518378	0.259189	−	0.965827i	$-0.416545\pi$
$653$	8650.00	0.518378	0.259189	+	0.965827i	$0.416545\pi$
$654$	0	0
$655$	−11880.0	−0.708687
$656$	0	0
$657$	−5886.00	−0.349520
$658$	0	0
$659$	−22260.0	−1.31582	−0.657911	−	0.753096i	$-0.728560\pi$
$659$	−22260.0	−1.31582	−0.657911	+	0.753096i	$0.728560\pi$
$660$	0	0
$661$	14406.0	0.847698	0.423849	−	0.905733i	$-0.360679\pi$
$661$	14406.0	0.847698	0.423849	+	0.905733i	$0.360679\pi$
$662$	0	0
$663$	11316.0	0.662861
$664$	0	0
$665$	3528.00	0.205729
$666$	0	0
$667$	3080.00	0.178798
$668$	0	0
$669$	−16920.0	−0.977825
$670$	0	0
$671$	136.000	0.00782447
$672$	0	0
$673$	−8334.00	−0.477343	−0.238672	−	0.971100i	$-0.576712\pi$
$673$	−8334.00	−0.477343	−0.238672	+	0.971100i	$0.576712\pi$
$674$	0	0
$675$	2403.00	0.137024
$676$	0	0
$677$	22314.0	1.26676	0.633380	−	0.773841i	$-0.281667\pi$
$677$	22314.0	1.26676	0.633380	+	0.773841i	$0.281667\pi$
$678$	0	0
$679$	11858.0	0.670204
$680$	0	0
$681$	−7980.00	−0.449037
$682$	0	0
$683$	13692.0	0.767071	0.383536	−	0.923526i	$-0.374706\pi$
$683$	13692.0	0.767071	0.383536	+	0.923526i	$0.374706\pi$
$684$	0	0
$685$	17268.0	0.963177
$686$	0	0
$687$	16878.0	0.937316
$688$	0	0
$689$	25852.0	1.42944
$690$	0	0
$691$	29020.0	1.59765	0.798823	−	0.601567i	$-0.205457\pi$
$691$	29020.0	1.59765	0.798823	+	0.601567i	$0.205457\pi$
$692$	0	0
$693$	252.000	0.0138134
$694$	0	0
$695$	8616.00	0.470250
$696$	0	0
$697$	−7708.00	−0.418883
$698$	0	0
$699$	1818.00	0.0983735
$700$	0	0
$701$	16410.0	0.884161	0.442081	−	0.896975i	$-0.354240\pi$
$701$	16410.0	0.884161	0.442081	+	0.896975i	$0.354240\pi$
$702$	0	0
$703$	12264.0	0.657959
$704$	0	0
$705$	576.000	0.0307708
$706$	0	0
$707$	2226.00	0.118412
$708$	0	0
$709$	11170.0	0.591676	0.295838	−	0.955238i	$-0.404401\pi$
$709$	11170.0	0.591676	0.295838	+	0.955238i	$0.404401\pi$
$710$	0	0
$711$	5472.00	0.288630
$712$	0	0
$713$	−6688.00	−0.351287
$714$	0	0
$715$	1104.00	0.0577444
$716$	0	0
$717$	18900.0	0.984426
$718$	0	0
$719$	−22728.0	−1.17888	−0.589438	−	0.807814i	$-0.700651\pi$
$719$	−22728.0	−1.17888	−0.589438	+	0.807814i	$0.700651\pi$
$720$	0	0
$721$	−6104.00	−0.315291
$722$	0	0
$723$	−6630.00	−0.341041
$724$	0	0
$725$	6230.00	0.319140
$726$	0	0
$727$	1288.00	0.0657074	0.0328537	−	0.999460i	$-0.489540\pi$
$727$	1288.00	0.0657074	0.0328537	+	0.999460i	$0.489540\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−40016.0	−2.02469
$732$	0	0
$733$	−266.000	−0.0134037	−0.00670187	−	0.999978i	$-0.502133\pi$
$733$	−266.000	−0.0134037	−0.00670187	+	0.999978i	$0.502133\pi$
$734$	0	0
$735$	882.000	0.0442627
$736$	0	0
$737$	2080.00	0.103959
$738$	0	0
$739$	25280.0	1.25838	0.629188	−	0.777253i	$-0.283388\pi$
$739$	25280.0	1.25838	0.629188	+	0.777253i	$0.283388\pi$
$740$	0	0
$741$	−11592.0	−0.574687
$742$	0	0
$743$	−6964.00	−0.343855	−0.171928	−	0.985110i	$-0.555000\pi$
$743$	−6964.00	−0.343855	−0.171928	+	0.985110i	$0.555000\pi$
$744$	0	0
$745$	8244.00	0.405419
$746$	0	0
$747$	2556.00	0.125193
$748$	0	0
$749$	2548.00	0.124302
$750$	0	0
$751$	20208.0	0.981891	0.490946	−	0.871190i	$-0.336651\pi$
$751$	20208.0	0.981891	0.490946	+	0.871190i	$0.336651\pi$
$752$	0	0
$753$	15132.0	0.732325
$754$	0	0
$755$	4128.00	0.198985
$756$	0	0
$757$	−24766.0	−1.18908	−0.594541	−	0.804065i	$-0.702666\pi$
$757$	−24766.0	−1.18908	−0.594541	+	0.804065i	$0.702666\pi$
$758$	0	0
$759$	−528.000	−0.0252506
$760$	0	0
$761$	−34338.0	−1.63568	−0.817839	−	0.575447i	$-0.804828\pi$
$761$	−34338.0	−1.63568	−0.817839	+	0.575447i	$0.804828\pi$
$762$	0	0
$763$	5866.00	0.278327
$764$	0	0
$765$	4428.00	0.209274
$766$	0	0
$767$	−21896.0	−1.03079
$768$	0	0
$769$	41178.0	1.93097	0.965485	−	0.260457i	$-0.0838732\pi$
$769$	41178.0	1.93097	0.965485	+	0.260457i	$0.0838732\pi$
$770$	0	0
$771$	294.000	0.0137330
$772$	0	0
$773$	−21918.0	−1.01984	−0.509920	−	0.860222i	$-0.670325\pi$
$773$	−21918.0	−1.01984	−0.509920	+	0.860222i	$0.670325\pi$
$774$	0	0
$775$	−13528.0	−0.627019
$776$	0	0
$777$	3066.00	0.141560
$778$	0	0
$779$	7896.00	0.363162
$780$	0	0
$781$	−144.000	−0.00659760
$782$	0	0
$783$	1890.00	0.0862619
$784$	0	0
$785$	−36.0000	−0.00163681
$786$	0	0
$787$	−17972.0	−0.814019	−0.407009	−	0.913424i	$-0.633428\pi$
$787$	−17972.0	−0.814019	−0.407009	+	0.913424i	$0.633428\pi$
$788$	0	0
$789$	17196.0	0.775911
$790$	0	0
$791$	2562.00	0.115163
$792$	0	0
$793$	−1564.00	−0.0700369
$794$	0	0
$795$	10116.0	0.451293
$796$	0	0
$797$	−27310.0	−1.21376	−0.606882	−	0.794792i	$-0.707580\pi$
$797$	−27310.0	−1.21376	−0.606882	+	0.794792i	$0.707580\pi$
$798$	0	0
$799$	−2624.00	−0.116183
$800$	0	0
$801$	−8586.00	−0.378741
$802$	0	0
$803$	2616.00	0.114965
$804$	0	0
$805$	−1848.00	−0.0809111
$806$	0	0
$807$	1890.00	0.0824426
$808$	0	0
$809$	−5734.00	−0.249192	−0.124596	−	0.992208i	$-0.539764\pi$
$809$	−5734.00	−0.249192	−0.124596	+	0.992208i	$0.539764\pi$
$810$	0	0
$811$	40556.0	1.75600	0.877999	−	0.478663i	$-0.158878\pi$
$811$	40556.0	1.75600	0.877999	+	0.478663i	$0.158878\pi$
$812$	0	0
$813$	7680.00	0.331303
$814$	0	0
$815$	96.0000	0.00412605
$816$	0	0
$817$	40992.0	1.75536
$818$	0	0
$819$	−2898.00	−0.123644
$820$	0	0
$821$	−28046.0	−1.19222	−0.596110	−	0.802903i	$-0.703288\pi$
$821$	−28046.0	−1.19222	−0.596110	+	0.802903i	$0.703288\pi$
$822$	0	0
$823$	−31744.0	−1.34450	−0.672252	−	0.740323i	$-0.734673\pi$
$823$	−31744.0	−1.34450	−0.672252	+	0.740323i	$0.734673\pi$
$824$	0	0
$825$	−1068.00	−0.0450703
$826$	0	0
$827$	−41244.0	−1.73421	−0.867107	−	0.498123i	$-0.834023\pi$
$827$	−41244.0	−1.73421	−0.867107	+	0.498123i	$0.834023\pi$
$828$	0	0
$829$	−20298.0	−0.850396	−0.425198	−	0.905100i	$-0.639796\pi$
$829$	−20298.0	−0.850396	−0.425198	+	0.905100i	$0.639796\pi$
$830$	0	0
$831$	−6798.00	−0.283779
$832$	0	0
$833$	−4018.00	−0.167125
$834$	0	0
$835$	11712.0	0.485402
$836$	0	0
$837$	−4104.00	−0.169480
$838$	0	0
$839$	40048.0	1.64793	0.823963	−	0.566643i	$-0.191758\pi$
$839$	40048.0	1.64793	0.823963	+	0.566643i	$0.191758\pi$
$840$	0	0
$841$	−19489.0	−0.799090
$842$	0	0
$843$	−13014.0	−0.531703
$844$	0	0
$845$	486.000	0.0197857
$846$	0	0
$847$	9205.00	0.373421
$848$	0	0
$849$	19092.0	0.771774
$850$	0	0
$851$	−6424.00	−0.258768
$852$	0	0
$853$	7758.00	0.311405	0.155703	−	0.987804i	$-0.450236\pi$
$853$	7758.00	0.311405	0.155703	+	0.987804i	$0.450236\pi$
$854$	0	0
$855$	−4536.00	−0.181436
$856$	0	0
$857$	−28434.0	−1.13336	−0.566678	−	0.823939i	$-0.691772\pi$
$857$	−28434.0	−1.13336	−0.566678	+	0.823939i	$0.691772\pi$
$858$	0	0
$859$	16028.0	0.636634	0.318317	−	0.947984i	$-0.396882\pi$
$859$	16028.0	0.636634	0.318317	+	0.947984i	$0.396882\pi$
$860$	0	0
$861$	1974.00	0.0781344
$862$	0	0
$863$	40188.0	1.58519	0.792593	−	0.609751i	$-0.208731\pi$
$863$	40188.0	1.58519	0.792593	+	0.609751i	$0.208731\pi$
$864$	0	0
$865$	−3852.00	−0.151413
$866$	0	0
$867$	−5433.00	−0.212819
$868$	0	0
$869$	−2432.00	−0.0949367
$870$	0	0
$871$	−23920.0	−0.930538
$872$	0	0
$873$	−15246.0	−0.591064
$874$	0	0
$875$	−8988.00	−0.347257
$876$	0	0
$877$	32810.0	1.26330	0.631651	−	0.775253i	$-0.282378\pi$
$877$	32810.0	1.26330	0.631651	+	0.775253i	$0.282378\pi$
$878$	0	0
$879$	−5334.00	−0.204677
$880$	0	0
$881$	17414.0	0.665939	0.332970	−	0.942938i	$-0.391949\pi$
$881$	17414.0	0.665939	0.332970	+	0.942938i	$0.391949\pi$
$882$	0	0
$883$	8656.00	0.329895	0.164948	−	0.986302i	$-0.447254\pi$
$883$	8656.00	0.329895	0.164948	+	0.986302i	$0.447254\pi$
$884$	0	0
$885$	−8568.00	−0.325435
$886$	0	0
$887$	3824.00	0.144755	0.0723773	−	0.997377i	$-0.476941\pi$
$887$	3824.00	0.144755	0.0723773	+	0.997377i	$0.476941\pi$
$888$	0	0
$889$	−3752.00	−0.141550
$890$	0	0
$891$	−324.000	−0.0121823
$892$	0	0
$893$	2688.00	0.100728
$894$	0	0
$895$	2808.00	0.104873
$896$	0	0
$897$	6072.00	0.226018
$898$	0	0
$899$	−10640.0	−0.394732
$900$	0	0
$901$	−46084.0	−1.70397
$902$	0	0
$903$	10248.0	0.377666
$904$	0	0
$905$	828.000	0.0304129
$906$	0	0
$907$	41008.0	1.50127	0.750633	−	0.660719i	$-0.229749\pi$
$907$	41008.0	1.50127	0.750633	+	0.660719i	$0.229749\pi$
$908$	0	0
$909$	−2862.00	−0.104430
$910$	0	0
$911$	−51172.0	−1.86104	−0.930518	−	0.366246i	$-0.880643\pi$
$911$	−51172.0	−1.86104	−0.930518	+	0.366246i	$0.880643\pi$
$912$	0	0
$913$	−1136.00	−0.0411787
$914$	0	0
$915$	−612.000	−0.0221116
$916$	0	0
$917$	−13860.0	−0.499125
$918$	0	0
$919$	−31432.0	−1.12823	−0.564116	−	0.825695i	$-0.690783\pi$
$919$	−31432.0	−1.12823	−0.564116	+	0.825695i	$0.690783\pi$
$920$	0	0
$921$	−4092.00	−0.146402
$922$	0	0
$923$	1656.00	0.0590552
$924$	0	0
$925$	−12994.0	−0.461881
$926$	0	0
$927$	7848.00	0.278061
$928$	0	0
$929$	38214.0	1.34958	0.674790	−	0.738009i	$-0.264234\pi$
$929$	38214.0	1.34958	0.674790	+	0.738009i	$0.264234\pi$
$930$	0	0
$931$	4116.00	0.144894
$932$	0	0
$933$	−22536.0	−0.790778
$934$	0	0
$935$	−1968.00	−0.0688347
$936$	0	0
$937$	−41446.0	−1.44502	−0.722509	−	0.691362i	$-0.757011\pi$
$937$	−41446.0	−1.44502	−0.722509	+	0.691362i	$0.757011\pi$
$938$	0	0
$939$	−8742.00	−0.303817
$940$	0	0
$941$	−2590.00	−0.0897254	−0.0448627	−	0.998993i	$-0.514285\pi$
$941$	−2590.00	−0.0897254	−0.0448627	+	0.998993i	$0.514285\pi$
$942$	0	0
$943$	−4136.00	−0.142828
$944$	0	0
$945$	−1134.00	−0.0390360
$946$	0	0
$947$	3980.00	0.136571	0.0682854	−	0.997666i	$-0.478247\pi$
$947$	3980.00	0.136571	0.0682854	+	0.997666i	$0.478247\pi$
$948$	0	0
$949$	−30084.0	−1.02905
$950$	0	0
$951$	23538.0	0.802599
$952$	0	0
$953$	6778.00	0.230389	0.115195	−	0.993343i	$-0.463251\pi$
$953$	6778.00	0.230389	0.115195	+	0.993343i	$0.463251\pi$
$954$	0	0
$955$	11880.0	0.402542
$956$	0	0
$957$	−840.000	−0.0283734
$958$	0	0
$959$	20146.0	0.678361
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	−3276.00	−0.109624
$964$	0	0
$965$	7764.00	0.258997
$966$	0	0
$967$	−296.000	−0.00984356	−0.00492178	−	0.999988i	$-0.501567\pi$
$967$	−296.000	−0.00984356	−0.00492178	+	0.999988i	$0.501567\pi$
$968$	0	0
$969$	20664.0	0.685060
$970$	0	0
$971$	25356.0	0.838015	0.419008	−	0.907983i	$-0.362378\pi$
$971$	25356.0	0.838015	0.419008	+	0.907983i	$0.362378\pi$
$972$	0	0
$973$	10052.0	0.331195
$974$	0	0
$975$	12282.0	0.403424
$976$	0	0
$977$	53290.0	1.74503	0.872517	−	0.488584i	$-0.162486\pi$
$977$	53290.0	1.74503	0.872517	+	0.488584i	$0.162486\pi$
$978$	0	0
$979$	3816.00	0.124576
$980$	0	0
$981$	−7542.00	−0.245461
$982$	0	0
$983$	28584.0	0.927455	0.463727	−	0.885978i	$-0.346512\pi$
$983$	28584.0	0.927455	0.463727	+	0.885978i	$0.346512\pi$
$984$	0	0
$985$	27252.0	0.881544
$986$	0	0
$987$	672.000	0.0216717
$988$	0	0
$989$	−21472.0	−0.690364
$990$	0	0
$991$	−3592.00	−0.115140	−0.0575699	−	0.998341i	$-0.518335\pi$
$991$	−3592.00	−0.115140	−0.0575699	+	0.998341i	$0.518335\pi$
$992$	0	0
$993$	−72.0000	−0.00230096
$994$	0	0
$995$	−17952.0	−0.571977
$996$	0	0
$997$	−9290.00	−0.295103	−0.147551	−	0.989054i	$-0.547139\pi$
$997$	−9290.00	−0.295103	−0.147551	+	0.989054i	$0.547139\pi$
$998$	0	0
$999$	−3942.00	−0.124844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.c.1.1		1
4.3	odd	2		1344.4.a.r.1.1		1
8.3	odd	2		672.4.a.b.1.1	✓	1
8.5	even	2		672.4.a.d.1.1	yes	1
24.5	odd	2		2016.4.a.a.1.1		1
24.11	even	2		2016.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.b.1.1	✓	1	8.3	odd	2
672.4.a.d.1.1	yes	1	8.5	even	2
1344.4.a.c.1.1		1	1.1	even	1	trivial
1344.4.a.r.1.1		1	4.3	odd	2
2016.4.a.a.1.1		1	24.5	odd	2
2016.4.a.b.1.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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