LMFDB - Embedded newform 234.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [234,4,Mod(1,234)] 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("234.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	234.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,20,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.8064469413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	234.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +20.0000 q^{5} -32.0000 q^{7} -8.00000 q^{8} -40.0000 q^{10} -50.0000 q^{11} -13.0000 q^{13} +64.0000 q^{14} +16.0000 q^{16} +30.0000 q^{17} -120.000 q^{19} +80.0000 q^{20} +100.000 q^{22} +20.0000 q^{23} +275.000 q^{25} +26.0000 q^{26} -128.000 q^{28} -82.0000 q^{29} -44.0000 q^{31} -32.0000 q^{32} -60.0000 q^{34} -640.000 q^{35} -306.000 q^{37} +240.000 q^{38} -160.000 q^{40} -108.000 q^{41} -356.000 q^{43} -200.000 q^{44} -40.0000 q^{46} +178.000 q^{47} +681.000 q^{49} -550.000 q^{50} -52.0000 q^{52} -198.000 q^{53} -1000.00 q^{55} +256.000 q^{56} +164.000 q^{58} -94.0000 q^{59} -62.0000 q^{61} +88.0000 q^{62} +64.0000 q^{64} -260.000 q^{65} -140.000 q^{67} +120.000 q^{68} +1280.00 q^{70} +778.000 q^{71} +62.0000 q^{73} +612.000 q^{74} -480.000 q^{76} +1600.00 q^{77} -1096.00 q^{79} +320.000 q^{80} +216.000 q^{82} +462.000 q^{83} +600.000 q^{85} +712.000 q^{86} +400.000 q^{88} -1224.00 q^{89} +416.000 q^{91} +80.0000 q^{92} -356.000 q^{94} -2400.00 q^{95} +614.000 q^{97} -1362.00 q^{98} +O(q^{100})\)

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
$5$	20.0000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	20.0000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−40.0000	−1.26491
$11$	−50.0000	−1.37051	−0.685253	−	0.728305i	$-0.740308\pi$
$11$	−50.0000	−1.37051	−0.685253	+	0.728305i	$0.740308\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	64.0000	1.22177
$15$	0	0
$16$	16.0000	0.250000
$17$	30.0000	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$17$	30.0000	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$18$	0	0
$19$	−120.000	−1.44894	−0.724471	−	0.689306i	$-0.757916\pi$
$19$	−120.000	−1.44894	−0.724471	+	0.689306i	$0.757916\pi$
$20$	80.0000	0.894427
$21$	0	0
$22$	100.000	0.969094
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	0	0
$25$	275.000	2.20000
$26$	26.0000	0.196116
$27$	0	0
$28$	−128.000	−0.863919
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	0	0
$31$	−44.0000	−0.254924	−0.127462	−	0.991843i	$-0.540683\pi$
$31$	−44.0000	−0.254924	−0.127462	+	0.991843i	$0.540683\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−60.0000	−0.302645
$35$	−640.000	−3.09085
$36$	0	0
$37$	−306.000	−1.35962	−0.679812	−	0.733386i	$-0.737939\pi$
$37$	−306.000	−1.35962	−0.679812	+	0.733386i	$0.737939\pi$
$38$	240.000	1.02456
$39$	0	0
$40$	−160.000	−0.632456
$41$	−108.000	−0.411385	−0.205692	−	0.978617i	$-0.565945\pi$
$41$	−108.000	−0.411385	−0.205692	+	0.978617i	$0.565945\pi$
$42$	0	0
$43$	−356.000	−1.26255	−0.631273	−	0.775561i	$-0.717467\pi$
$43$	−356.000	−1.26255	−0.631273	+	0.775561i	$0.717467\pi$
$44$	−200.000	−0.685253
$45$	0	0
$46$	−40.0000	−0.128210
$47$	178.000	0.552425	0.276212	−	0.961097i	$-0.410921\pi$
$47$	178.000	0.552425	0.276212	+	0.961097i	$0.410921\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	−550.000	−1.55563
$51$	0	0
$52$	−52.0000	−0.138675
$53$	−198.000	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$53$	−198.000	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$54$	0	0
$55$	−1000.00	−2.45164
$56$	256.000	0.610883
$57$	0	0
$58$	164.000	0.371280
$59$	−94.0000	−0.207420	−0.103710	−	0.994608i	$-0.533071\pi$
$59$	−94.0000	−0.207420	−0.103710	+	0.994608i	$0.533071\pi$
$60$	0	0
$61$	−62.0000	−0.130136	−0.0650679	−	0.997881i	$-0.520726\pi$
$61$	−62.0000	−0.130136	−0.0650679	+	0.997881i	$0.520726\pi$
$62$	88.0000	0.180258
$63$	0	0
$64$	64.0000	0.125000
$65$	−260.000	−0.496139
$66$	0	0
$67$	−140.000	−0.255279	−0.127640	−	0.991821i	$-0.540740\pi$
$67$	−140.000	−0.255279	−0.127640	+	0.991821i	$0.540740\pi$
$68$	120.000	0.214002
$69$	0	0
$70$	1280.00	2.18556
$71$	778.000	1.30045	0.650223	−	0.759744i	$-0.274676\pi$
$71$	778.000	1.30045	0.650223	+	0.759744i	$0.274676\pi$
$72$	0	0
$73$	62.0000	0.0994048	0.0497024	−	0.998764i	$-0.484173\pi$
$73$	62.0000	0.0994048	0.0497024	+	0.998764i	$0.484173\pi$
$74$	612.000	0.961399
$75$	0	0
$76$	−480.000	−0.724471
$77$	1600.00	2.36801
$78$	0	0
$79$	−1096.00	−1.56088	−0.780441	−	0.625230i	$-0.785005\pi$
$79$	−1096.00	−1.56088	−0.780441	+	0.625230i	$0.785005\pi$
$80$	320.000	0.447214
$81$	0	0
$82$	216.000	0.290893
$83$	462.000	0.610977	0.305488	−	0.952196i	$-0.401180\pi$
$83$	462.000	0.610977	0.305488	+	0.952196i	$0.401180\pi$
$84$	0	0
$85$	600.000	0.765637
$86$	712.000	0.892755
$87$	0	0
$88$	400.000	0.484547
$89$	−1224.00	−1.45779	−0.728897	−	0.684623i	$-0.759967\pi$
$89$	−1224.00	−1.45779	−0.728897	+	0.684623i	$0.759967\pi$
$90$	0	0
$91$	416.000	0.479216
$92$	80.0000	0.0906584
$93$	0	0
$94$	−356.000	−0.390623
$95$	−2400.00	−2.59195
$96$	0	0
$97$	614.000	0.642704	0.321352	−	0.946960i	$-0.395863\pi$
$97$	614.000	0.642704	0.321352	+	0.946960i	$0.395863\pi$
$98$	−1362.00	−1.40391
$99$	0	0
$100$	1100.00	1.10000
$101$	−1058.00	−1.04233	−0.521163	−	0.853457i	$-0.674502\pi$
$101$	−1058.00	−1.04233	−0.521163	+	0.853457i	$0.674502\pi$
$102$	0	0
$103$	1768.00	1.69132	0.845661	−	0.533720i	$-0.179206\pi$
$103$	1768.00	1.69132	0.845661	+	0.533720i	$0.179206\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	396.000	0.362858
$107$	1808.00	1.63351	0.816757	−	0.576982i	$-0.195770\pi$
$107$	1808.00	1.63351	0.816757	+	0.576982i	$0.195770\pi$
$108$	0	0
$109$	−1886.00	−1.65730	−0.828652	−	0.559765i	$-0.810891\pi$
$109$	−1886.00	−1.65730	−0.828652	+	0.559765i	$0.810891\pi$
$110$	2000.00	1.73357
$111$	0	0
$112$	−512.000	−0.431959
$113$	−1246.00	−1.03729	−0.518645	−	0.854990i	$-0.673563\pi$
$113$	−1246.00	−1.03729	−0.518645	+	0.854990i	$0.673563\pi$
$114$	0	0
$115$	400.000	0.324349
$116$	−328.000	−0.262535
$117$	0	0
$118$	188.000	0.146668
$119$	−960.000	−0.739521
$120$	0	0
$121$	1169.00	0.878287
$122$	124.000	0.0920199
$123$	0	0
$124$	−176.000	−0.127462
$125$	3000.00	2.14663
$126$	0	0
$127$	1624.00	1.13470	0.567349	−	0.823477i	$-0.307969\pi$
$127$	1624.00	1.13470	0.567349	+	0.823477i	$0.307969\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	520.000	0.350823
$131$	2072.00	1.38192	0.690960	−	0.722893i	$-0.257188\pi$
$131$	2072.00	1.38192	0.690960	+	0.722893i	$0.257188\pi$
$132$	0	0
$133$	3840.00	2.50354
$134$	280.000	0.180510
$135$	0	0
$136$	−240.000	−0.151322
$137$	756.000	0.471456	0.235728	−	0.971819i	$-0.424253\pi$
$137$	756.000	0.471456	0.235728	+	0.971819i	$0.424253\pi$
$138$	0	0
$139$	172.000	0.104956	0.0524779	−	0.998622i	$-0.483288\pi$
$139$	172.000	0.104956	0.0524779	+	0.998622i	$0.483288\pi$
$140$	−2560.00	−1.54542
$141$	0	0
$142$	−1556.00	−0.919554
$143$	650.000	0.380110
$144$	0	0
$145$	−1640.00	−0.939273
$146$	−124.000	−0.0702898
$147$	0	0
$148$	−1224.00	−0.679812
$149$	−1272.00	−0.699371	−0.349686	−	0.936867i	$-0.613712\pi$
$149$	−1272.00	−0.699371	−0.349686	+	0.936867i	$0.613712\pi$
$150$	0	0
$151$	1404.00	0.756662	0.378331	−	0.925670i	$-0.376498\pi$
$151$	1404.00	0.756662	0.378331	+	0.925670i	$0.376498\pi$
$152$	960.000	0.512278
$153$	0	0
$154$	−3200.00	−1.67444
$155$	−880.000	−0.456021
$156$	0	0
$157$	−2170.00	−1.10309	−0.551544	−	0.834146i	$-0.685961\pi$
$157$	−2170.00	−1.10309	−0.551544	+	0.834146i	$0.685961\pi$
$158$	2192.00	1.10371
$159$	0	0
$160$	−640.000	−0.316228
$161$	−640.000	−0.313286
$162$	0	0
$163$	248.000	0.119171	0.0595855	−	0.998223i	$-0.481022\pi$
$163$	248.000	0.119171	0.0595855	+	0.998223i	$0.481022\pi$
$164$	−432.000	−0.205692
$165$	0	0
$166$	−924.000	−0.432026
$167$	−102.000	−0.0472635	−0.0236317	−	0.999721i	$-0.507523\pi$
$167$	−102.000	−0.0472635	−0.0236317	+	0.999721i	$0.507523\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−1200.00	−0.541387
$171$	0	0
$172$	−1424.00	−0.631273
$173$	−682.000	−0.299720	−0.149860	−	0.988707i	$-0.547882\pi$
$173$	−682.000	−0.299720	−0.149860	+	0.988707i	$0.547882\pi$
$174$	0	0
$175$	−8800.00	−3.80124
$176$	−800.000	−0.342627
$177$	0	0
$178$	2448.00	1.03082
$179$	612.000	0.255548	0.127774	−	0.991803i	$-0.459217\pi$
$179$	612.000	0.255548	0.127774	+	0.991803i	$0.459217\pi$
$180$	0	0
$181$	−66.0000	−0.0271035	−0.0135518	−	0.999908i	$-0.504314\pi$
$181$	−66.0000	−0.0271035	−0.0135518	+	0.999908i	$0.504314\pi$
$182$	−832.000	−0.338857
$183$	0	0
$184$	−160.000	−0.0641052
$185$	−6120.00	−2.43217
$186$	0	0
$187$	−1500.00	−0.586582
$188$	712.000	0.276212
$189$	0	0
$190$	4800.00	1.83278
$191$	−608.000	−0.230332	−0.115166	−	0.993346i	$-0.536740\pi$
$191$	−608.000	−0.230332	−0.115166	+	0.993346i	$0.536740\pi$
$192$	0	0
$193$	1370.00	0.510957	0.255479	−	0.966815i	$-0.417767\pi$
$193$	1370.00	0.510957	0.255479	+	0.966815i	$0.417767\pi$
$194$	−1228.00	−0.454460
$195$	0	0
$196$	2724.00	0.992711
$197$	4908.00	1.77503	0.887514	−	0.460781i	$-0.152431\pi$
$197$	4908.00	1.77503	0.887514	+	0.460781i	$0.152431\pi$
$198$	0	0
$199$	−328.000	−0.116841	−0.0584204	−	0.998292i	$-0.518606\pi$
$199$	−328.000	−0.116841	−0.0584204	+	0.998292i	$0.518606\pi$
$200$	−2200.00	−0.777817
$201$	0	0
$202$	2116.00	0.737036
$203$	2624.00	0.907235
$204$	0	0
$205$	−2160.00	−0.735907
$206$	−3536.00	−1.19595
$207$	0	0
$208$	−208.000	−0.0693375
$209$	6000.00	1.98578
$210$	0	0
$211$	1316.00	0.429371	0.214685	−	0.976683i	$-0.431127\pi$
$211$	1316.00	0.429371	0.214685	+	0.976683i	$0.431127\pi$
$212$	−792.000	−0.256579
$213$	0	0
$214$	−3616.00	−1.15507
$215$	−7120.00	−2.25851
$216$	0	0
$217$	1408.00	0.440467
$218$	3772.00	1.17189
$219$	0	0
$220$	−4000.00	−1.22582
$221$	−390.000	−0.118707
$222$	0	0
$223$	−1932.00	−0.580163	−0.290081	−	0.957002i	$-0.593682\pi$
$223$	−1932.00	−0.580163	−0.290081	+	0.957002i	$0.593682\pi$
$224$	1024.00	0.305441
$225$	0	0
$226$	2492.00	0.733475
$227$	−4998.00	−1.46136	−0.730680	−	0.682720i	$-0.760797\pi$
$227$	−4998.00	−1.46136	−0.730680	+	0.682720i	$0.760797\pi$
$228$	0	0
$229$	−78.0000	−0.0225082	−0.0112541	−	0.999937i	$-0.503582\pi$
$229$	−78.0000	−0.0225082	−0.0112541	+	0.999937i	$0.503582\pi$
$230$	−800.000	−0.229350
$231$	0	0
$232$	656.000	0.185640
$233$	1282.00	0.360458	0.180229	−	0.983625i	$-0.442316\pi$
$233$	1282.00	0.360458	0.180229	+	0.983625i	$0.442316\pi$
$234$	0	0
$235$	3560.00	0.988208
$236$	−376.000	−0.103710
$237$	0	0
$238$	1920.00	0.522921
$239$	−294.000	−0.0795702	−0.0397851	−	0.999208i	$-0.512667\pi$
$239$	−294.000	−0.0795702	−0.0397851	+	0.999208i	$0.512667\pi$
$240$	0	0
$241$	−4962.00	−1.32627	−0.663134	−	0.748501i	$-0.730774\pi$
$241$	−4962.00	−1.32627	−0.663134	+	0.748501i	$0.730774\pi$
$242$	−2338.00	−0.621043
$243$	0	0
$244$	−248.000	−0.0650679
$245$	13620.0	3.55163
$246$	0	0
$247$	1560.00	0.401864
$248$	352.000	0.0901291
$249$	0	0
$250$	−6000.00	−1.51789
$251$	−744.000	−0.187095	−0.0935475	−	0.995615i	$-0.529821\pi$
$251$	−744.000	−0.187095	−0.0935475	+	0.995615i	$0.529821\pi$
$252$	0	0
$253$	−1000.00	−0.248496
$254$	−3248.00	−0.802353
$255$	0	0
$256$	256.000	0.0625000
$257$	1026.00	0.249028	0.124514	−	0.992218i	$-0.460263\pi$
$257$	1026.00	0.249028	0.124514	+	0.992218i	$0.460263\pi$
$258$	0	0
$259$	9792.00	2.34921
$260$	−1040.00	−0.248069
$261$	0	0
$262$	−4144.00	−0.977165
$263$	5532.00	1.29703	0.648513	−	0.761204i	$-0.275391\pi$
$263$	5532.00	1.29703	0.648513	+	0.761204i	$0.275391\pi$
$264$	0	0
$265$	−3960.00	−0.917966
$266$	−7680.00	−1.77027
$267$	0	0
$268$	−560.000	−0.127640
$269$	3534.00	0.801010	0.400505	−	0.916294i	$-0.368835\pi$
$269$	3534.00	0.801010	0.400505	+	0.916294i	$0.368835\pi$
$270$	0	0
$271$	2392.00	0.536176	0.268088	−	0.963394i	$-0.413608\pi$
$271$	2392.00	0.536176	0.268088	+	0.963394i	$0.413608\pi$
$272$	480.000	0.107001
$273$	0	0
$274$	−1512.00	−0.333370
$275$	−13750.0	−3.01511
$276$	0	0
$277$	6102.00	1.32359	0.661794	−	0.749686i	$-0.269796\pi$
$277$	6102.00	1.32359	0.661794	+	0.749686i	$0.269796\pi$
$278$	−344.000	−0.0742149
$279$	0	0
$280$	5120.00	1.09278
$281$	7540.00	1.60071	0.800354	−	0.599528i	$-0.204645\pi$
$281$	7540.00	1.60071	0.800354	+	0.599528i	$0.204645\pi$
$282$	0	0
$283$	−2756.00	−0.578895	−0.289447	−	0.957194i	$-0.593472\pi$
$283$	−2756.00	−0.578895	−0.289447	+	0.957194i	$0.593472\pi$
$284$	3112.00	0.650223
$285$	0	0
$286$	−1300.00	−0.268778
$287$	3456.00	0.710806
$288$	0	0
$289$	−4013.00	−0.816813
$290$	3280.00	0.664166
$291$	0	0
$292$	248.000	0.0497024
$293$	−968.000	−0.193007	−0.0965037	−	0.995333i	$-0.530766\pi$
$293$	−968.000	−0.193007	−0.0965037	+	0.995333i	$0.530766\pi$
$294$	0	0
$295$	−1880.00	−0.371043
$296$	2448.00	0.480700
$297$	0	0
$298$	2544.00	0.494530
$299$	−260.000	−0.0502883
$300$	0	0
$301$	11392.0	2.18147
$302$	−2808.00	−0.535041
$303$	0	0
$304$	−1920.00	−0.362235
$305$	−1240.00	−0.232794
$306$	0	0
$307$	−6436.00	−1.19649	−0.598244	−	0.801314i	$-0.704135\pi$
$307$	−6436.00	−1.19649	−0.598244	+	0.801314i	$0.704135\pi$
$308$	6400.00	1.18401
$309$	0	0
$310$	1760.00	0.322456
$311$	−7932.00	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$311$	−7932.00	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$312$	0	0
$313$	10358.0	1.87051	0.935254	−	0.353978i	$-0.115171\pi$
$313$	10358.0	1.87051	0.935254	+	0.353978i	$0.115171\pi$
$314$	4340.00	0.780001
$315$	0	0
$316$	−4384.00	−0.780441
$317$	2820.00	0.499643	0.249822	−	0.968292i	$-0.419628\pi$
$317$	2820.00	0.499643	0.249822	+	0.968292i	$0.419628\pi$
$318$	0	0
$319$	4100.00	0.719611
$320$	1280.00	0.223607
$321$	0	0
$322$	1280.00	0.221527
$323$	−3600.00	−0.620153
$324$	0	0
$325$	−3575.00	−0.610170
$326$	−496.000	−0.0842666
$327$	0	0
$328$	864.000	0.145446
$329$	−5696.00	−0.954500
$330$	0	0
$331$	−4180.00	−0.694120	−0.347060	−	0.937843i	$-0.612820\pi$
$331$	−4180.00	−0.694120	−0.347060	+	0.937843i	$0.612820\pi$
$332$	1848.00	0.305488
$333$	0	0
$334$	204.000	0.0334203
$335$	−2800.00	−0.456658
$336$	0	0
$337$	−5026.00	−0.812414	−0.406207	−	0.913781i	$-0.633149\pi$
$337$	−5026.00	−0.812414	−0.406207	+	0.913781i	$0.633149\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	2400.00	0.382818
$341$	2200.00	0.349374
$342$	0	0
$343$	−10816.0	−1.70265
$344$	2848.00	0.446378
$345$	0	0
$346$	1364.00	0.211934
$347$	7332.00	1.13430	0.567150	−	0.823614i	$-0.308046\pi$
$347$	7332.00	1.13430	0.567150	+	0.823614i	$0.308046\pi$
$348$	0	0
$349$	−8162.00	−1.25187	−0.625934	−	0.779876i	$-0.715282\pi$
$349$	−8162.00	−1.25187	−0.625934	+	0.779876i	$0.715282\pi$
$350$	17600.0	2.68788
$351$	0	0
$352$	1600.00	0.242274
$353$	−1244.00	−0.187568	−0.0937839	−	0.995593i	$-0.529896\pi$
$353$	−1244.00	−0.187568	−0.0937839	+	0.995593i	$0.529896\pi$
$354$	0	0
$355$	15560.0	2.32631
$356$	−4896.00	−0.728897
$357$	0	0
$358$	−1224.00	−0.180699
$359$	−9558.00	−1.40516	−0.702579	−	0.711605i	$-0.747968\pi$
$359$	−9558.00	−1.40516	−0.702579	+	0.711605i	$0.747968\pi$
$360$	0	0
$361$	7541.00	1.09943
$362$	132.000	0.0191651
$363$	0	0
$364$	1664.00	0.239608
$365$	1240.00	0.177821
$366$	0	0
$367$	−11032.0	−1.56912	−0.784558	−	0.620055i	$-0.787110\pi$
$367$	−11032.0	−1.56912	−0.784558	+	0.620055i	$0.787110\pi$
$368$	320.000	0.0453292
$369$	0	0
$370$	12240.0	1.71980
$371$	6336.00	0.886654
$372$	0	0
$373$	5474.00	0.759874	0.379937	−	0.925012i	$-0.375946\pi$
$373$	5474.00	0.759874	0.379937	+	0.925012i	$0.375946\pi$
$374$	3000.00	0.414776
$375$	0	0
$376$	−1424.00	−0.195312
$377$	1066.00	0.145628
$378$	0	0
$379$	−7040.00	−0.954144	−0.477072	−	0.878864i	$-0.658302\pi$
$379$	−7040.00	−0.954144	−0.477072	+	0.878864i	$0.658302\pi$
$380$	−9600.00	−1.29597
$381$	0	0
$382$	1216.00	0.162869
$383$	1830.00	0.244148	0.122074	−	0.992521i	$-0.461045\pi$
$383$	1830.00	0.244148	0.122074	+	0.992521i	$0.461045\pi$
$384$	0	0
$385$	32000.0	4.23603
$386$	−2740.00	−0.361301
$387$	0	0
$388$	2456.00	0.321352
$389$	−10158.0	−1.32399	−0.661994	−	0.749509i	$-0.730289\pi$
$389$	−10158.0	−1.32399	−0.661994	+	0.749509i	$0.730289\pi$
$390$	0	0
$391$	600.000	0.0776044
$392$	−5448.00	−0.701953
$393$	0	0
$394$	−9816.00	−1.25513
$395$	−21920.0	−2.79219
$396$	0	0
$397$	−12658.0	−1.60022	−0.800109	−	0.599854i	$-0.795225\pi$
$397$	−12658.0	−1.60022	−0.800109	+	0.599854i	$0.795225\pi$
$398$	656.000	0.0826189
$399$	0	0
$400$	4400.00	0.550000
$401$	−15720.0	−1.95765	−0.978827	−	0.204689i	$-0.934382\pi$
$401$	−15720.0	−1.95765	−0.978827	+	0.204689i	$0.934382\pi$
$402$	0	0
$403$	572.000	0.0707031
$404$	−4232.00	−0.521163
$405$	0	0
$406$	−5248.00	−0.641512
$407$	15300.0	1.86337
$408$	0	0
$409$	7654.00	0.925345	0.462672	−	0.886529i	$-0.346891\pi$
$409$	7654.00	0.925345	0.462672	+	0.886529i	$0.346891\pi$
$410$	4320.00	0.520365
$411$	0	0
$412$	7072.00	0.845661
$413$	3008.00	0.358387
$414$	0	0
$415$	9240.00	1.09295
$416$	416.000	0.0490290
$417$	0	0
$418$	−12000.0	−1.40416
$419$	1848.00	0.215467	0.107734	−	0.994180i	$-0.465641\pi$
$419$	1848.00	0.215467	0.107734	+	0.994180i	$0.465641\pi$
$420$	0	0
$421$	−12542.0	−1.45192	−0.725962	−	0.687735i	$-0.758605\pi$
$421$	−12542.0	−1.45192	−0.725962	+	0.687735i	$0.758605\pi$
$422$	−2632.00	−0.303611
$423$	0	0
$424$	1584.00	0.181429
$425$	8250.00	0.941609
$426$	0	0
$427$	1984.00	0.224854
$428$	7232.00	0.816757
$429$	0	0
$430$	14240.0	1.59701
$431$	5238.00	0.585396	0.292698	−	0.956205i	$-0.405447\pi$
$431$	5238.00	0.585396	0.292698	+	0.956205i	$0.405447\pi$
$432$	0	0
$433$	−8258.00	−0.916522	−0.458261	−	0.888818i	$-0.651528\pi$
$433$	−8258.00	−0.916522	−0.458261	+	0.888818i	$0.651528\pi$
$434$	−2816.00	−0.311457
$435$	0	0
$436$	−7544.00	−0.828652
$437$	−2400.00	−0.262718
$438$	0	0
$439$	−6304.00	−0.685361	−0.342681	−	0.939452i	$-0.611335\pi$
$439$	−6304.00	−0.685361	−0.342681	+	0.939452i	$0.611335\pi$
$440$	8000.00	0.866784
$441$	0	0
$442$	780.000	0.0839385
$443$	−12744.0	−1.36678	−0.683392	−	0.730051i	$-0.739496\pi$
$443$	−12744.0	−1.36678	−0.683392	+	0.730051i	$0.739496\pi$
$444$	0	0
$445$	−24480.0	−2.60778
$446$	3864.00	0.410237
$447$	0	0
$448$	−2048.00	−0.215980
$449$	11776.0	1.23774	0.618868	−	0.785495i	$-0.287591\pi$
$449$	11776.0	1.23774	0.618868	+	0.785495i	$0.287591\pi$
$450$	0	0
$451$	5400.00	0.563805
$452$	−4984.00	−0.518645
$453$	0	0
$454$	9996.00	1.03334
$455$	8320.00	0.857248
$456$	0	0
$457$	2134.00	0.218434	0.109217	−	0.994018i	$-0.465166\pi$
$457$	2134.00	0.218434	0.109217	+	0.994018i	$0.465166\pi$
$458$	156.000	0.0159157
$459$	0	0
$460$	1600.00	0.162175
$461$	−2724.00	−0.275205	−0.137602	−	0.990488i	$-0.543940\pi$
$461$	−2724.00	−0.275205	−0.137602	+	0.990488i	$0.543940\pi$
$462$	0	0
$463$	−5648.00	−0.566922	−0.283461	−	0.958984i	$-0.591483\pi$
$463$	−5648.00	−0.566922	−0.283461	+	0.958984i	$0.591483\pi$
$464$	−1312.00	−0.131267
$465$	0	0
$466$	−2564.00	−0.254882
$467$	18224.0	1.80579	0.902897	−	0.429856i	$-0.141436\pi$
$467$	18224.0	1.80579	0.902897	+	0.429856i	$0.141436\pi$
$468$	0	0
$469$	4480.00	0.441081
$470$	−7120.00	−0.698768
$471$	0	0
$472$	752.000	0.0733339
$473$	17800.0	1.73033
$474$	0	0
$475$	−33000.0	−3.18767
$476$	−3840.00	−0.369761
$477$	0	0
$478$	588.000	0.0562646
$479$	−9066.00	−0.864794	−0.432397	−	0.901683i	$-0.642332\pi$
$479$	−9066.00	−0.864794	−0.432397	+	0.901683i	$0.642332\pi$
$480$	0	0
$481$	3978.00	0.377092
$482$	9924.00	0.937813
$483$	0	0
$484$	4676.00	0.439144
$485$	12280.0	1.14970
$486$	0	0
$487$	8948.00	0.832593	0.416296	−	0.909229i	$-0.363328\pi$
$487$	8948.00	0.832593	0.416296	+	0.909229i	$0.363328\pi$
$488$	496.000	0.0460100
$489$	0	0
$490$	−27240.0	−2.51138
$491$	−8720.00	−0.801483	−0.400741	−	0.916191i	$-0.631247\pi$
$491$	−8720.00	−0.801483	−0.400741	+	0.916191i	$0.631247\pi$
$492$	0	0
$493$	−2460.00	−0.224732
$494$	−3120.00	−0.284161
$495$	0	0
$496$	−704.000	−0.0637309
$497$	−24896.0	−2.24696
$498$	0	0
$499$	6604.00	0.592456	0.296228	−	0.955117i	$-0.404271\pi$
$499$	6604.00	0.592456	0.296228	+	0.955117i	$0.404271\pi$
$500$	12000.0	1.07331
$501$	0	0
$502$	1488.00	0.132296
$503$	−3404.00	−0.301743	−0.150872	−	0.988553i	$-0.548208\pi$
$503$	−3404.00	−0.301743	−0.150872	+	0.988553i	$0.548208\pi$
$504$	0	0
$505$	−21160.0	−1.86457
$506$	2000.00	0.175713
$507$	0	0
$508$	6496.00	0.567349
$509$	76.0000	0.00661815	0.00330908	−	0.999995i	$-0.498947\pi$
$509$	76.0000	0.00661815	0.00330908	+	0.999995i	$0.498947\pi$
$510$	0	0
$511$	−1984.00	−0.171755
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−2052.00	−0.176089
$515$	35360.0	3.02553
$516$	0	0
$517$	−8900.00	−0.757102
$518$	−19584.0	−1.66114
$519$	0	0
$520$	2080.00	0.175412
$521$	−12054.0	−1.01362	−0.506809	−	0.862058i	$-0.669175\pi$
$521$	−12054.0	−1.01362	−0.506809	+	0.862058i	$0.669175\pi$
$522$	0	0
$523$	276.000	0.0230758	0.0115379	−	0.999933i	$-0.496327\pi$
$523$	276.000	0.0230758	0.0115379	+	0.999933i	$0.496327\pi$
$524$	8288.00	0.690960
$525$	0	0
$526$	−11064.0	−0.917136
$527$	−1320.00	−0.109108
$528$	0	0
$529$	−11767.0	−0.967124
$530$	7920.00	0.649100
$531$	0	0
$532$	15360.0	1.25177
$533$	1404.00	0.114098
$534$	0	0
$535$	36160.0	2.92212
$536$	1120.00	0.0902549
$537$	0	0
$538$	−7068.00	−0.566400
$539$	−34050.0	−2.72103
$540$	0	0
$541$	13778.0	1.09494	0.547470	−	0.836825i	$-0.315591\pi$
$541$	13778.0	1.09494	0.547470	+	0.836825i	$0.315591\pi$
$542$	−4784.00	−0.379134
$543$	0	0
$544$	−960.000	−0.0756611
$545$	−37720.0	−2.96467
$546$	0	0
$547$	−10844.0	−0.847634	−0.423817	−	0.905748i	$-0.639310\pi$
$547$	−10844.0	−0.847634	−0.423817	+	0.905748i	$0.639310\pi$
$548$	3024.00	0.235728
$549$	0	0
$550$	27500.0	2.13201
$551$	9840.00	0.760795
$552$	0	0
$553$	35072.0	2.69695
$554$	−12204.0	−0.935917
$555$	0	0
$556$	688.000	0.0524779
$557$	−20544.0	−1.56280	−0.781398	−	0.624033i	$-0.785493\pi$
$557$	−20544.0	−1.56280	−0.781398	+	0.624033i	$0.785493\pi$
$558$	0	0
$559$	4628.00	0.350167
$560$	−10240.0	−0.772712
$561$	0	0
$562$	−15080.0	−1.13187
$563$	−6988.00	−0.523107	−0.261553	−	0.965189i	$-0.584235\pi$
$563$	−6988.00	−0.523107	−0.261553	+	0.965189i	$0.584235\pi$
$564$	0	0
$565$	−24920.0	−1.85556
$566$	5512.00	0.409340
$567$	0	0
$568$	−6224.00	−0.459777
$569$	706.000	0.0520159	0.0260080	−	0.999662i	$-0.491720\pi$
$569$	706.000	0.0520159	0.0260080	+	0.999662i	$0.491720\pi$
$570$	0	0
$571$	−17532.0	−1.28492	−0.642462	−	0.766318i	$-0.722087\pi$
$571$	−17532.0	−1.28492	−0.642462	+	0.766318i	$0.722087\pi$
$572$	2600.00	0.190055
$573$	0	0
$574$	−6912.00	−0.502616
$575$	5500.00	0.398897
$576$	0	0
$577$	−14814.0	−1.06883	−0.534415	−	0.845222i	$-0.679468\pi$
$577$	−14814.0	−1.06883	−0.534415	+	0.845222i	$0.679468\pi$
$578$	8026.00	0.577574
$579$	0	0
$580$	−6560.00	−0.469637
$581$	−14784.0	−1.05567
$582$	0	0
$583$	9900.00	0.703287
$584$	−496.000	−0.0351449
$585$	0	0
$586$	1936.00	0.136477
$587$	−14170.0	−0.996352	−0.498176	−	0.867076i	$-0.665997\pi$
$587$	−14170.0	−0.996352	−0.498176	+	0.867076i	$0.665997\pi$
$588$	0	0
$589$	5280.00	0.369369
$590$	3760.00	0.262367
$591$	0	0
$592$	−4896.00	−0.339906
$593$	11744.0	0.813269	0.406634	−	0.913591i	$-0.366702\pi$
$593$	11744.0	0.813269	0.406634	+	0.913591i	$0.366702\pi$
$594$	0	0
$595$	−19200.0	−1.32290
$596$	−5088.00	−0.349686
$597$	0	0
$598$	520.000	0.0355592
$599$	15076.0	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$599$	15076.0	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$600$	0	0
$601$	20230.0	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$601$	20230.0	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$602$	−22784.0	−1.54254
$603$	0	0
$604$	5616.00	0.378331
$605$	23380.0	1.57113
$606$	0	0
$607$	−28056.0	−1.87604	−0.938021	−	0.346577i	$-0.887344\pi$
$607$	−28056.0	−1.87604	−0.938021	+	0.346577i	$0.887344\pi$
$608$	3840.00	0.256139
$609$	0	0
$610$	2480.00	0.164610
$611$	−2314.00	−0.153215
$612$	0	0
$613$	27446.0	1.80837	0.904187	−	0.427136i	$-0.140478\pi$
$613$	27446.0	1.80837	0.904187	+	0.427136i	$0.140478\pi$
$614$	12872.0	0.846045
$615$	0	0
$616$	−12800.0	−0.837219
$617$	−8804.00	−0.574450	−0.287225	−	0.957863i	$-0.592733\pi$
$617$	−8804.00	−0.574450	−0.287225	+	0.957863i	$0.592733\pi$
$618$	0	0
$619$	3508.00	0.227784	0.113892	−	0.993493i	$-0.463668\pi$
$619$	3508.00	0.227784	0.113892	+	0.993493i	$0.463668\pi$
$620$	−3520.00	−0.228011
$621$	0	0
$622$	15864.0	1.02265
$623$	39168.0	2.51883
$624$	0	0
$625$	25625.0	1.64000
$626$	−20716.0	−1.32265
$627$	0	0
$628$	−8680.00	−0.551544
$629$	−9180.00	−0.581925
$630$	0	0
$631$	22084.0	1.39326	0.696632	−	0.717428i	$-0.254681\pi$
$631$	22084.0	1.39326	0.696632	+	0.717428i	$0.254681\pi$
$632$	8768.00	0.551855
$633$	0	0
$634$	−5640.00	−0.353301
$635$	32480.0	2.02981
$636$	0	0
$637$	−8853.00	−0.550657
$638$	−8200.00	−0.508842
$639$	0	0
$640$	−2560.00	−0.158114
$641$	7342.00	0.452405	0.226202	−	0.974080i	$-0.427369\pi$
$641$	7342.00	0.452405	0.226202	+	0.974080i	$0.427369\pi$
$642$	0	0
$643$	2996.00	0.183749	0.0918746	−	0.995771i	$-0.470714\pi$
$643$	2996.00	0.183749	0.0918746	+	0.995771i	$0.470714\pi$
$644$	−2560.00	−0.156643
$645$	0	0
$646$	7200.00	0.438514
$647$	−9344.00	−0.567775	−0.283888	−	0.958858i	$-0.591624\pi$
$647$	−9344.00	−0.567775	−0.283888	+	0.958858i	$0.591624\pi$
$648$	0	0
$649$	4700.00	0.284270
$650$	7150.00	0.431455
$651$	0	0
$652$	992.000	0.0595855
$653$	16686.0	0.999960	0.499980	−	0.866037i	$-0.333341\pi$
$653$	16686.0	0.999960	0.499980	+	0.866037i	$0.333341\pi$
$654$	0	0
$655$	41440.0	2.47205
$656$	−1728.00	−0.102846
$657$	0	0
$658$	11392.0	0.674934
$659$	−31356.0	−1.85350	−0.926750	−	0.375679i	$-0.877410\pi$
$659$	−31356.0	−1.85350	−0.926750	+	0.375679i	$0.877410\pi$
$660$	0	0
$661$	590.000	0.0347176	0.0173588	−	0.999849i	$-0.494474\pi$
$661$	590.000	0.0347176	0.0173588	+	0.999849i	$0.494474\pi$
$662$	8360.00	0.490817
$663$	0	0
$664$	−3696.00	−0.216013
$665$	76800.0	4.47846
$666$	0	0
$667$	−1640.00	−0.0952040
$668$	−408.000	−0.0236317
$669$	0	0
$670$	5600.00	0.322906
$671$	3100.00	0.178352
$672$	0	0
$673$	5938.00	0.340109	0.170054	−	0.985435i	$-0.445606\pi$
$673$	5938.00	0.340109	0.170054	+	0.985435i	$0.445606\pi$
$674$	10052.0	0.574464
$675$	0	0
$676$	676.000	0.0384615
$677$	−9486.00	−0.538518	−0.269259	−	0.963068i	$-0.586779\pi$
$677$	−9486.00	−0.538518	−0.269259	+	0.963068i	$0.586779\pi$
$678$	0	0
$679$	−19648.0	−1.11049
$680$	−4800.00	−0.270694
$681$	0	0
$682$	−4400.00	−0.247045
$683$	−26162.0	−1.46568	−0.732841	−	0.680400i	$-0.761806\pi$
$683$	−26162.0	−1.46568	−0.732841	+	0.680400i	$0.761806\pi$
$684$	0	0
$685$	15120.0	0.843366
$686$	21632.0	1.20396
$687$	0	0
$688$	−5696.00	−0.315637
$689$	2574.00	0.142325
$690$	0	0
$691$	−17348.0	−0.955064	−0.477532	−	0.878614i	$-0.658468\pi$
$691$	−17348.0	−0.955064	−0.477532	+	0.878614i	$0.658468\pi$
$692$	−2728.00	−0.149860
$693$	0	0
$694$	−14664.0	−0.802072
$695$	3440.00	0.187751
$696$	0	0
$697$	−3240.00	−0.176074
$698$	16324.0	0.885204
$699$	0	0
$700$	−35200.0	−1.90062
$701$	−30.0000	−0.00161638	−0.000808191	−	1.00000i	$-0.500257\pi$
$701$	−30.0000	−0.00161638	−0.000808191		1.00000i	$0.500257\pi$
$702$	0	0
$703$	36720.0	1.97002
$704$	−3200.00	−0.171313
$705$	0	0
$706$	2488.00	0.132630
$707$	33856.0	1.80097
$708$	0	0
$709$	31466.0	1.66676	0.833378	−	0.552703i	$-0.186404\pi$
$709$	31466.0	1.66676	0.833378	+	0.552703i	$0.186404\pi$
$710$	−31120.0	−1.64495
$711$	0	0
$712$	9792.00	0.515408
$713$	−880.000	−0.0462220
$714$	0	0
$715$	13000.0	0.679961
$716$	2448.00	0.127774
$717$	0	0
$718$	19116.0	0.993597
$719$	28892.0	1.49859	0.749297	−	0.662234i	$-0.230391\pi$
$719$	28892.0	1.49859	0.749297	+	0.662234i	$0.230391\pi$
$720$	0	0
$721$	−56576.0	−2.92233
$722$	−15082.0	−0.777415
$723$	0	0
$724$	−264.000	−0.0135518
$725$	−22550.0	−1.15515
$726$	0	0
$727$	13384.0	0.682786	0.341393	−	0.939921i	$-0.389101\pi$
$727$	13384.0	0.682786	0.341393	+	0.939921i	$0.389101\pi$
$728$	−3328.00	−0.169428
$729$	0	0
$730$	−2480.00	−0.125738
$731$	−10680.0	−0.540375
$732$	0	0
$733$	7130.00	0.359280	0.179640	−	0.983732i	$-0.442507\pi$
$733$	7130.00	0.359280	0.179640	+	0.983732i	$0.442507\pi$
$734$	22064.0	1.10953
$735$	0	0
$736$	−640.000	−0.0320526
$737$	7000.00	0.349862
$738$	0	0
$739$	−29268.0	−1.45689	−0.728444	−	0.685105i	$-0.759756\pi$
$739$	−29268.0	−1.45689	−0.728444	+	0.685105i	$0.759756\pi$
$740$	−24480.0	−1.21608
$741$	0	0
$742$	−12672.0	−0.626959
$743$	9898.00	0.488725	0.244362	−	0.969684i	$-0.421421\pi$
$743$	9898.00	0.488725	0.244362	+	0.969684i	$0.421421\pi$
$744$	0	0
$745$	−25440.0	−1.25107
$746$	−10948.0	−0.537312
$747$	0	0
$748$	−6000.00	−0.293291
$749$	−57856.0	−2.82245
$750$	0	0
$751$	−15120.0	−0.734669	−0.367335	−	0.930089i	$-0.619730\pi$
$751$	−15120.0	−0.734669	−0.367335	+	0.930089i	$0.619730\pi$
$752$	2848.00	0.138106
$753$	0	0
$754$	−2132.00	−0.102975
$755$	28080.0	1.35356
$756$	0	0
$757$	−5454.00	−0.261861	−0.130931	−	0.991392i	$-0.541797\pi$
$757$	−5454.00	−0.261861	−0.130931	+	0.991392i	$0.541797\pi$
$758$	14080.0	0.674682
$759$	0	0
$760$	19200.0	0.916391
$761$	11988.0	0.571044	0.285522	−	0.958372i	$-0.407833\pi$
$761$	11988.0	0.571044	0.285522	+	0.958372i	$0.407833\pi$
$762$	0	0
$763$	60352.0	2.86355
$764$	−2432.00	−0.115166
$765$	0	0
$766$	−3660.00	−0.172639
$767$	1222.00	0.0575279
$768$	0	0
$769$	1338.00	0.0627432	0.0313716	−	0.999508i	$-0.490012\pi$
$769$	1338.00	0.0627432	0.0313716	+	0.999508i	$0.490012\pi$
$770$	−64000.0	−2.99532
$771$	0	0
$772$	5480.00	0.255479
$773$	14408.0	0.670401	0.335200	−	0.942147i	$-0.391196\pi$
$773$	14408.0	0.670401	0.335200	+	0.942147i	$0.391196\pi$
$774$	0	0
$775$	−12100.0	−0.560832
$776$	−4912.00	−0.227230
$777$	0	0
$778$	20316.0	0.936200
$779$	12960.0	0.596072
$780$	0	0
$781$	−38900.0	−1.78227
$782$	−1200.00	−0.0548746
$783$	0	0
$784$	10896.0	0.496356
$785$	−43400.0	−1.97326
$786$	0	0
$787$	−10660.0	−0.482831	−0.241415	−	0.970422i	$-0.577612\pi$
$787$	−10660.0	−0.482831	−0.241415	+	0.970422i	$0.577612\pi$
$788$	19632.0	0.887514
$789$	0	0
$790$	43840.0	1.97438
$791$	39872.0	1.79227
$792$	0	0
$793$	806.000	0.0360932
$794$	25316.0	1.13153
$795$	0	0
$796$	−1312.00	−0.0584204
$797$	−1974.00	−0.0877323	−0.0438662	−	0.999037i	$-0.513968\pi$
$797$	−1974.00	−0.0877323	−0.0438662	+	0.999037i	$0.513968\pi$
$798$	0	0
$799$	5340.00	0.236440
$800$	−8800.00	−0.388909
$801$	0	0
$802$	31440.0	1.38427
$803$	−3100.00	−0.136235
$804$	0	0
$805$	−12800.0	−0.560423
$806$	−1144.00	−0.0499946
$807$	0	0
$808$	8464.00	0.368518
$809$	−31734.0	−1.37912	−0.689560	−	0.724229i	$-0.742196\pi$
$809$	−31734.0	−1.37912	−0.689560	+	0.724229i	$0.742196\pi$
$810$	0	0
$811$	−38824.0	−1.68100	−0.840502	−	0.541808i	$-0.817740\pi$
$811$	−38824.0	−1.68100	−0.840502	+	0.541808i	$0.817740\pi$
$812$	10496.0	0.453617
$813$	0	0
$814$	−30600.0	−1.31760
$815$	4960.00	0.213179
$816$	0	0
$817$	42720.0	1.82936
$818$	−15308.0	−0.654317
$819$	0	0
$820$	−8640.00	−0.367954
$821$	16736.0	0.711438	0.355719	−	0.934593i	$-0.384236\pi$
$821$	16736.0	0.711438	0.355719	+	0.934593i	$0.384236\pi$
$822$	0	0
$823$	42096.0	1.78296	0.891479	−	0.453062i	$-0.149668\pi$
$823$	42096.0	1.78296	0.891479	+	0.453062i	$0.149668\pi$
$824$	−14144.0	−0.597973
$825$	0	0
$826$	−6016.00	−0.253418
$827$	−24858.0	−1.04522	−0.522610	−	0.852572i	$-0.675042\pi$
$827$	−24858.0	−1.04522	−0.522610	+	0.852572i	$0.675042\pi$
$828$	0	0
$829$	922.000	0.0386277	0.0193139	−	0.999813i	$-0.493852\pi$
$829$	922.000	0.0386277	0.0193139	+	0.999813i	$0.493852\pi$
$830$	−18480.0	−0.772832
$831$	0	0
$832$	−832.000	−0.0346688
$833$	20430.0	0.849769
$834$	0	0
$835$	−2040.00	−0.0845474
$836$	24000.0	0.992892
$837$	0	0
$838$	−3696.00	−0.152358
$839$	14294.0	0.588181	0.294090	−	0.955778i	$-0.404983\pi$
$839$	14294.0	0.588181	0.294090	+	0.955778i	$0.404983\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	25084.0	1.02666
$843$	0	0
$844$	5264.00	0.214685
$845$	3380.00	0.137604
$846$	0	0
$847$	−37408.0	−1.51754
$848$	−3168.00	−0.128290
$849$	0	0
$850$	−16500.0	−0.665818
$851$	−6120.00	−0.246523
$852$	0	0
$853$	37966.0	1.52395	0.761976	−	0.647605i	$-0.224229\pi$
$853$	37966.0	1.52395	0.761976	+	0.647605i	$0.224229\pi$
$854$	−3968.00	−0.158996
$855$	0	0
$856$	−14464.0	−0.577534
$857$	−39038.0	−1.55602	−0.778012	−	0.628249i	$-0.783772\pi$
$857$	−39038.0	−1.55602	−0.778012	+	0.628249i	$0.783772\pi$
$858$	0	0
$859$	20564.0	0.816804	0.408402	−	0.912802i	$-0.366086\pi$
$859$	20564.0	0.816804	0.408402	+	0.912802i	$0.366086\pi$
$860$	−28480.0	−1.12926
$861$	0	0
$862$	−10476.0	−0.413937
$863$	−39866.0	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$863$	−39866.0	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$864$	0	0
$865$	−13640.0	−0.536155
$866$	16516.0	0.648079
$867$	0	0
$868$	5632.00	0.220233
$869$	54800.0	2.13920
$870$	0	0
$871$	1820.00	0.0708018
$872$	15088.0	0.585945
$873$	0	0
$874$	4800.00	0.185769
$875$	−96000.0	−3.70902
$876$	0	0
$877$	30990.0	1.19322	0.596612	−	0.802530i	$-0.296513\pi$
$877$	30990.0	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$878$	12608.0	0.484623
$879$	0	0
$880$	−16000.0	−0.612909
$881$	4458.00	0.170481	0.0852405	−	0.996360i	$-0.472834\pi$
$881$	4458.00	0.170481	0.0852405	+	0.996360i	$0.472834\pi$
$882$	0	0
$883$	−3164.00	−0.120586	−0.0602928	−	0.998181i	$-0.519203\pi$
$883$	−3164.00	−0.120586	−0.0602928	+	0.998181i	$0.519203\pi$
$884$	−1560.00	−0.0593535
$885$	0	0
$886$	25488.0	0.966463
$887$	32512.0	1.23072	0.615359	−	0.788247i	$-0.289011\pi$
$887$	32512.0	1.23072	0.615359	+	0.788247i	$0.289011\pi$
$888$	0	0
$889$	−51968.0	−1.96057
$890$	48960.0	1.84398
$891$	0	0
$892$	−7728.00	−0.290081
$893$	−21360.0	−0.800431
$894$	0	0
$895$	12240.0	0.457138
$896$	4096.00	0.152721
$897$	0	0
$898$	−23552.0	−0.875212
$899$	3608.00	0.133853
$900$	0	0
$901$	−5940.00	−0.219634
$902$	−10800.0	−0.398670
$903$	0	0
$904$	9968.00	0.366738
$905$	−1320.00	−0.0484843
$906$	0	0
$907$	10500.0	0.384396	0.192198	−	0.981356i	$-0.438438\pi$
$907$	10500.0	0.384396	0.192198	+	0.981356i	$0.438438\pi$
$908$	−19992.0	−0.730680
$909$	0	0
$910$	−16640.0	−0.606166
$911$	−9840.00	−0.357864	−0.178932	−	0.983861i	$-0.557264\pi$
$911$	−9840.00	−0.357864	−0.178932	+	0.983861i	$0.557264\pi$
$912$	0	0
$913$	−23100.0	−0.837348
$914$	−4268.00	−0.154456
$915$	0	0
$916$	−312.000	−0.0112541
$917$	−66304.0	−2.38773
$918$	0	0
$919$	−35040.0	−1.25774	−0.628870	−	0.777511i	$-0.716482\pi$
$919$	−35040.0	−1.25774	−0.628870	+	0.777511i	$0.716482\pi$
$920$	−3200.00	−0.114675
$921$	0	0
$922$	5448.00	0.194599
$923$	−10114.0	−0.360679
$924$	0	0
$925$	−84150.0	−2.99117
$926$	11296.0	0.400874
$927$	0	0
$928$	2624.00	0.0928201
$929$	−44172.0	−1.56000	−0.779998	−	0.625782i	$-0.784780\pi$
$929$	−44172.0	−1.56000	−0.779998	+	0.625782i	$0.784780\pi$
$930$	0	0
$931$	−81720.0	−2.87676
$932$	5128.00	0.180229
$933$	0	0
$934$	−36448.0	−1.27689
$935$	−30000.0	−1.04931
$936$	0	0
$937$	−54018.0	−1.88334	−0.941671	−	0.336535i	$-0.890745\pi$
$937$	−54018.0	−1.88334	−0.941671	+	0.336535i	$0.890745\pi$
$938$	−8960.00	−0.311892
$939$	0	0
$940$	14240.0	0.494104
$941$	1672.00	0.0579231	0.0289616	−	0.999581i	$-0.490780\pi$
$941$	1672.00	0.0579231	0.0289616	+	0.999581i	$0.490780\pi$
$942$	0	0
$943$	−2160.00	−0.0745910
$944$	−1504.00	−0.0518549
$945$	0	0
$946$	−35600.0	−1.22353
$947$	−5238.00	−0.179738	−0.0898691	−	0.995954i	$-0.528645\pi$
$947$	−5238.00	−0.179738	−0.0898691	+	0.995954i	$0.528645\pi$
$948$	0	0
$949$	−806.000	−0.0275699
$950$	66000.0	2.25402
$951$	0	0
$952$	7680.00	0.261460
$953$	50042.0	1.70096	0.850482	−	0.526004i	$-0.176310\pi$
$953$	50042.0	1.70096	0.850482	+	0.526004i	$0.176310\pi$
$954$	0	0
$955$	−12160.0	−0.412030
$956$	−1176.00	−0.0397851
$957$	0	0
$958$	18132.0	0.611501
$959$	−24192.0	−0.814599
$960$	0	0
$961$	−27855.0	−0.935014
$962$	−7956.00	−0.266644
$963$	0	0
$964$	−19848.0	−0.663134
$965$	27400.0	0.914028
$966$	0	0
$967$	37676.0	1.25293	0.626463	−	0.779452i	$-0.284502\pi$
$967$	37676.0	1.25293	0.626463	+	0.779452i	$0.284502\pi$
$968$	−9352.00	−0.310521
$969$	0	0
$970$	−24560.0	−0.812963
$971$	17364.0	0.573880	0.286940	−	0.957949i	$-0.407362\pi$
$971$	17364.0	0.573880	0.286940	+	0.957949i	$0.407362\pi$
$972$	0	0
$973$	−5504.00	−0.181346
$974$	−17896.0	−0.588732
$975$	0	0
$976$	−992.000	−0.0325340
$977$	14904.0	0.488046	0.244023	−	0.969769i	$-0.421533\pi$
$977$	14904.0	0.488046	0.244023	+	0.969769i	$0.421533\pi$
$978$	0	0
$979$	61200.0	1.99792
$980$	54480.0	1.77582
$981$	0	0
$982$	17440.0	0.566734
$983$	18038.0	0.585272	0.292636	−	0.956224i	$-0.405467\pi$
$983$	18038.0	0.585272	0.292636	+	0.956224i	$0.405467\pi$
$984$	0	0
$985$	98160.0	3.17527
$986$	4920.00	0.158909
$987$	0	0
$988$	6240.00	0.200932
$989$	−7120.00	−0.228921
$990$	0	0
$991$	46176.0	1.48015	0.740075	−	0.672524i	$-0.234790\pi$
$991$	46176.0	1.48015	0.740075	+	0.672524i	$0.234790\pi$
$992$	1408.00	0.0450646
$993$	0	0
$994$	49792.0	1.58884
$995$	−6560.00	−0.209011
$996$	0	0
$997$	55838.0	1.77373	0.886864	−	0.462030i	$-0.152879\pi$
$997$	55838.0	1.77373	0.886864	+	0.462030i	$0.152879\pi$
$998$	−13208.0	−0.418930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.4.a.f.1.1		1
3.2	odd	2		78.4.a.d.1.1	✓	1
4.3	odd	2		1872.4.a.r.1.1		1
12.11	even	2		624.4.a.e.1.1		1
15.14	odd	2		1950.4.a.h.1.1		1
24.5	odd	2		2496.4.a.r.1.1		1
24.11	even	2		2496.4.a.i.1.1		1
39.5	even	4		1014.4.b.e.337.1		2
39.8	even	4		1014.4.b.e.337.2		2
39.38	odd	2		1014.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.d.1.1	✓	1	3.2	odd	2
234.4.a.f.1.1		1	1.1	even	1	trivial
624.4.a.e.1.1		1	12.11	even	2
1014.4.a.d.1.1		1	39.38	odd	2
1014.4.b.e.337.1		2	39.5	even	4
1014.4.b.e.337.2		2	39.8	even	4
1872.4.a.r.1.1		1	4.3	odd	2
1950.4.a.h.1.1		1	15.14	odd	2
2496.4.a.i.1.1		1	24.11	even	2
2496.4.a.r.1.1		1	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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