LMFDB - Embedded newform 78.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,4,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("78.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	78.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:	$4.60214898045$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	78.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -20.0000 q^{5} -6.00000 q^{6} -32.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -20.0000 q^{5} -6.00000 q^{6} -32.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -40.0000 q^{10} +50.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -64.0000 q^{14} +60.0000 q^{15} +16.0000 q^{16} -30.0000 q^{17} +18.0000 q^{18} -120.000 q^{19} -80.0000 q^{20} +96.0000 q^{21} +100.000 q^{22} -20.0000 q^{23} -24.0000 q^{24} +275.000 q^{25} -26.0000 q^{26} -27.0000 q^{27} -128.000 q^{28} +82.0000 q^{29} +120.000 q^{30} -44.0000 q^{31} +32.0000 q^{32} -150.000 q^{33} -60.0000 q^{34} +640.000 q^{35} +36.0000 q^{36} -306.000 q^{37} -240.000 q^{38} +39.0000 q^{39} -160.000 q^{40} +108.000 q^{41} +192.000 q^{42} -356.000 q^{43} +200.000 q^{44} -180.000 q^{45} -40.0000 q^{46} -178.000 q^{47} -48.0000 q^{48} +681.000 q^{49} +550.000 q^{50} +90.0000 q^{51} -52.0000 q^{52} +198.000 q^{53} -54.0000 q^{54} -1000.00 q^{55} -256.000 q^{56} +360.000 q^{57} +164.000 q^{58} +94.0000 q^{59} +240.000 q^{60} -62.0000 q^{61} -88.0000 q^{62} -288.000 q^{63} +64.0000 q^{64} +260.000 q^{65} -300.000 q^{66} -140.000 q^{67} -120.000 q^{68} +60.0000 q^{69} +1280.00 q^{70} -778.000 q^{71} +72.0000 q^{72} +62.0000 q^{73} -612.000 q^{74} -825.000 q^{75} -480.000 q^{76} -1600.00 q^{77} +78.0000 q^{78} -1096.00 q^{79} -320.000 q^{80} +81.0000 q^{81} +216.000 q^{82} -462.000 q^{83} +384.000 q^{84} +600.000 q^{85} -712.000 q^{86} -246.000 q^{87} +400.000 q^{88} +1224.00 q^{89} -360.000 q^{90} +416.000 q^{91} -80.0000 q^{92} +132.000 q^{93} -356.000 q^{94} +2400.00 q^{95} -96.0000 q^{96} +614.000 q^{97} +1362.00 q^{98} +450.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−20.0000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−20.0000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−6.00000	−0.408248
$7$	−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
$9$	9.00000	0.333333
$10$	−40.0000	−1.26491
$11$	50.0000	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−64.0000	−1.22177
$15$	60.0000	1.03280
$16$	16.0000	0.250000
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	18.0000	0.235702
$19$	−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
$21$	96.0000	0.997567
$22$	100.000	0.969094
$23$	−20.0000	−0.181317	−0.0906584	−	0.995882i	$-0.528897\pi$
$23$	−20.0000	−0.181317	−0.0906584	+	0.995882i	$0.528897\pi$
$24$	−24.0000	−0.204124
$25$	275.000	2.20000
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	−128.000	−0.863919
$29$	82.0000	0.525070	0.262535	−	0.964923i	$-0.415442\pi$
$29$	82.0000	0.525070	0.262535	+	0.964923i	$0.415442\pi$
$30$	120.000	0.730297
$31$	−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
$33$	−150.000	−0.791262
$34$	−60.0000	−0.302645
$35$	640.000	3.09085
$36$	36.0000	0.166667
$37$	−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
$39$	39.0000	0.160128
$40$	−160.000	−0.632456
$41$	108.000	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$41$	108.000	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$42$	192.000	0.705387
$43$	−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
$45$	−180.000	−0.596285
$46$	−40.0000	−0.128210
$47$	−178.000	−0.552425	−0.276212	−	0.961097i	$-0.589079\pi$
$47$	−178.000	−0.552425	−0.276212	+	0.961097i	$0.589079\pi$
$48$	−48.0000	−0.144338
$49$	681.000	1.98542
$50$	550.000	1.55563
$51$	90.0000	0.247108
$52$	−52.0000	−0.138675
$53$	198.000	0.513158	0.256579	−	0.966523i	$-0.417405\pi$
$53$	198.000	0.513158	0.256579	+	0.966523i	$0.417405\pi$
$54$	−54.0000	−0.136083
$55$	−1000.00	−2.45164
$56$	−256.000	−0.610883
$57$	360.000	0.836547
$58$	164.000	0.371280
$59$	94.0000	0.207420	0.103710	−	0.994608i	$-0.466929\pi$
$59$	94.0000	0.207420	0.103710	+	0.994608i	$0.466929\pi$
$60$	240.000	0.516398
$61$	−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
$63$	−288.000	−0.575946
$64$	64.0000	0.125000
$65$	260.000	0.496139
$66$	−300.000	−0.559507
$67$	−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
$69$	60.0000	0.104683
$70$	1280.00	2.18556
$71$	−778.000	−1.30045	−0.650223	−	0.759744i	$-0.725324\pi$
$71$	−778.000	−1.30045	−0.650223	+	0.759744i	$0.725324\pi$
$72$	72.0000	0.117851
$73$	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
$75$	−825.000	−1.27017
$76$	−480.000	−0.724471
$77$	−1600.00	−2.36801
$78$	78.0000	0.113228
$79$	−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
$81$	81.0000	0.111111
$82$	216.000	0.290893
$83$	−462.000	−0.610977	−0.305488	−	0.952196i	$-0.598820\pi$
$83$	−462.000	−0.610977	−0.305488	+	0.952196i	$0.598820\pi$
$84$	384.000	0.498784
$85$	600.000	0.765637
$86$	−712.000	−0.892755
$87$	−246.000	−0.303149
$88$	400.000	0.484547
$89$	1224.00	1.45779	0.728897	−	0.684623i	$-0.240033\pi$
$89$	1224.00	1.45779	0.728897	+	0.684623i	$0.240033\pi$
$90$	−360.000	−0.421637
$91$	416.000	0.479216
$92$	−80.0000	−0.0906584
$93$	132.000	0.147180
$94$	−356.000	−0.390623
$95$	2400.00	2.59195
$96$	−96.0000	−0.102062
$97$	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
$99$	450.000	0.456835
$100$	1100.00	1.10000
$101$	1058.00	1.04233	0.521163	−	0.853457i	$-0.325498\pi$
$101$	1058.00	1.04233	0.521163	+	0.853457i	$0.325498\pi$
$102$	180.000	0.174732
$103$	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
$105$	−1920.00	−1.78450
$106$	396.000	0.362858
$107$	−1808.00	−1.63351	−0.816757	−	0.576982i	$-0.804230\pi$
$107$	−1808.00	−1.63351	−0.816757	+	0.576982i	$0.804230\pi$
$108$	−108.000	−0.0962250
$109$	−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
$111$	918.000	0.784979
$112$	−512.000	−0.431959
$113$	1246.00	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$113$	1246.00	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$114$	720.000	0.591528
$115$	400.000	0.324349
$116$	328.000	0.262535
$117$	−117.000	−0.0924500
$118$	188.000	0.146668
$119$	960.000	0.739521
$120$	480.000	0.365148
$121$	1169.00	0.878287
$122$	−124.000	−0.0920199
$123$	−324.000	−0.237513
$124$	−176.000	−0.127462
$125$	−3000.00	−2.14663
$126$	−576.000	−0.407255
$127$	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
$129$	1068.00	0.728931
$130$	520.000	0.350823
$131$	−2072.00	−1.38192	−0.690960	−	0.722893i	$-0.742812\pi$
$131$	−2072.00	−1.38192	−0.690960	+	0.722893i	$0.742812\pi$
$132$	−600.000	−0.395631
$133$	3840.00	2.50354
$134$	−280.000	−0.180510
$135$	540.000	0.344265
$136$	−240.000	−0.151322
$137$	−756.000	−0.471456	−0.235728	−	0.971819i	$-0.575747\pi$
$137$	−756.000	−0.471456	−0.235728	+	0.971819i	$0.575747\pi$
$138$	120.000	0.0740223
$139$	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
$141$	534.000	0.318943
$142$	−1556.00	−0.919554
$143$	−650.000	−0.380110
$144$	144.000	0.0833333
$145$	−1640.00	−0.939273
$146$	124.000	0.0702898
$147$	−2043.00	−1.14628
$148$	−1224.00	−0.679812
$149$	1272.00	0.699371	0.349686	−	0.936867i	$-0.386288\pi$
$149$	1272.00	0.699371	0.349686	+	0.936867i	$0.386288\pi$
$150$	−1650.00	−0.898146
$151$	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
$153$	−270.000	−0.142668
$154$	−3200.00	−1.67444
$155$	880.000	0.456021
$156$	156.000	0.0800641
$157$	−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
$159$	−594.000	−0.296272
$160$	−640.000	−0.316228
$161$	640.000	0.313286
$162$	162.000	0.0785674
$163$	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
$165$	3000.00	1.41545
$166$	−924.000	−0.432026
$167$	102.000	0.0472635	0.0236317	−	0.999721i	$-0.492477\pi$
$167$	102.000	0.0472635	0.0236317	+	0.999721i	$0.492477\pi$
$168$	768.000	0.352693
$169$	169.000	0.0769231
$170$	1200.00	0.541387
$171$	−1080.00	−0.482980
$172$	−1424.00	−0.631273
$173$	682.000	0.299720	0.149860	−	0.988707i	$-0.452118\pi$
$173$	682.000	0.299720	0.149860	+	0.988707i	$0.452118\pi$
$174$	−492.000	−0.214359
$175$	−8800.00	−3.80124
$176$	800.000	0.342627
$177$	−282.000	−0.119754
$178$	2448.00	1.03082
$179$	−612.000	−0.255548	−0.127774	−	0.991803i	$-0.540783\pi$
$179$	−612.000	−0.255548	−0.127774	+	0.991803i	$0.540783\pi$
$180$	−720.000	−0.298142
$181$	−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
$183$	186.000	0.0751340
$184$	−160.000	−0.0641052
$185$	6120.00	2.43217
$186$	264.000	0.104072
$187$	−1500.00	−0.586582
$188$	−712.000	−0.276212
$189$	864.000	0.332522
$190$	4800.00	1.83278
$191$	608.000	0.230332	0.115166	−	0.993346i	$-0.463260\pi$
$191$	608.000	0.230332	0.115166	+	0.993346i	$0.463260\pi$
$192$	−192.000	−0.0721688
$193$	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
$195$	−780.000	−0.286446
$196$	2724.00	0.992711
$197$	−4908.00	−1.77503	−0.887514	−	0.460781i	$-0.847569\pi$
$197$	−4908.00	−1.77503	−0.887514	+	0.460781i	$0.847569\pi$
$198$	900.000	0.323031
$199$	−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
$201$	420.000	0.147386
$202$	2116.00	0.737036
$203$	−2624.00	−0.907235
$204$	360.000	0.123554
$205$	−2160.00	−0.735907
$206$	3536.00	1.19595
$207$	−180.000	−0.0604390
$208$	−208.000	−0.0693375
$209$	−6000.00	−1.98578
$210$	−3840.00	−1.26183
$211$	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
$213$	2334.00	0.750812
$214$	−3616.00	−1.15507
$215$	7120.00	2.25851
$216$	−216.000	−0.0680414
$217$	1408.00	0.440467
$218$	−3772.00	−1.17189
$219$	−186.000	−0.0573914
$220$	−4000.00	−1.22582
$221$	390.000	0.118707
$222$	1836.00	0.555064
$223$	−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
$225$	2475.00	0.733333
$226$	2492.00	0.733475
$227$	4998.00	1.46136	0.730680	−	0.682720i	$-0.239203\pi$
$227$	4998.00	1.46136	0.730680	+	0.682720i	$0.239203\pi$
$228$	1440.00	0.418273
$229$	−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
$231$	4800.00	1.36717
$232$	656.000	0.185640
$233$	−1282.00	−0.360458	−0.180229	−	0.983625i	$-0.557684\pi$
$233$	−1282.00	−0.360458	−0.180229	+	0.983625i	$0.557684\pi$
$234$	−234.000	−0.0653720
$235$	3560.00	0.988208
$236$	376.000	0.103710
$237$	3288.00	0.901175
$238$	1920.00	0.522921
$239$	294.000	0.0795702	0.0397851	−	0.999208i	$-0.487333\pi$
$239$	294.000	0.0795702	0.0397851	+	0.999208i	$0.487333\pi$
$240$	960.000	0.258199
$241$	−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
$243$	−243.000	−0.0641500
$244$	−248.000	−0.0650679
$245$	−13620.0	−3.55163
$246$	−648.000	−0.167947
$247$	1560.00	0.401864
$248$	−352.000	−0.0901291
$249$	1386.00	0.352748
$250$	−6000.00	−1.51789
$251$	744.000	0.187095	0.0935475	−	0.995615i	$-0.470179\pi$
$251$	744.000	0.187095	0.0935475	+	0.995615i	$0.470179\pi$
$252$	−1152.00	−0.287973
$253$	−1000.00	−0.248496
$254$	3248.00	0.802353
$255$	−1800.00	−0.442041
$256$	256.000	0.0625000
$257$	−1026.00	−0.249028	−0.124514	−	0.992218i	$-0.539737\pi$
$257$	−1026.00	−0.249028	−0.124514	+	0.992218i	$0.539737\pi$
$258$	2136.00	0.515432
$259$	9792.00	2.34921
$260$	1040.00	0.248069
$261$	738.000	0.175023
$262$	−4144.00	−0.977165
$263$	−5532.00	−1.29703	−0.648513	−	0.761204i	$-0.724609\pi$
$263$	−5532.00	−1.29703	−0.648513	+	0.761204i	$0.724609\pi$
$264$	−1200.00	−0.279753
$265$	−3960.00	−0.917966
$266$	7680.00	1.77027
$267$	−3672.00	−0.841658
$268$	−560.000	−0.127640
$269$	−3534.00	−0.801010	−0.400505	−	0.916294i	$-0.631165\pi$
$269$	−3534.00	−0.801010	−0.400505	+	0.916294i	$0.631165\pi$
$270$	1080.00	0.243432
$271$	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
$273$	−1248.00	−0.276675
$274$	−1512.00	−0.333370
$275$	13750.0	3.01511
$276$	240.000	0.0523417
$277$	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
$279$	−396.000	−0.0849746
$280$	5120.00	1.09278
$281$	−7540.00	−1.60071	−0.800354	−	0.599528i	$-0.795355\pi$
$281$	−7540.00	−1.60071	−0.800354	+	0.599528i	$0.795355\pi$
$282$	1068.00	0.225527
$283$	−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
$285$	−7200.00	−1.49646
$286$	−1300.00	−0.268778
$287$	−3456.00	−0.710806
$288$	288.000	0.0589256
$289$	−4013.00	−0.816813
$290$	−3280.00	−0.664166
$291$	−1842.00	−0.371065
$292$	248.000	0.0497024
$293$	968.000	0.193007	0.0965037	−	0.995333i	$-0.469234\pi$
$293$	968.000	0.193007	0.0965037	+	0.995333i	$0.469234\pi$
$294$	−4086.00	−0.810545
$295$	−1880.00	−0.371043
$296$	−2448.00	−0.480700
$297$	−1350.00	−0.263754
$298$	2544.00	0.494530
$299$	260.000	0.0502883
$300$	−3300.00	−0.635085
$301$	11392.0	2.18147
$302$	2808.00	0.535041
$303$	−3174.00	−0.601787
$304$	−1920.00	−0.362235
$305$	1240.00	0.232794
$306$	−540.000	−0.100882
$307$	−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
$309$	−5304.00	−0.976485
$310$	1760.00	0.322456
$311$	7932.00	1.44625	0.723123	−	0.690719i	$-0.242706\pi$
$311$	7932.00	1.44625	0.723123	+	0.690719i	$0.242706\pi$
$312$	312.000	0.0566139
$313$	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
$315$	5760.00	1.03028
$316$	−4384.00	−0.780441
$317$	−2820.00	−0.499643	−0.249822	−	0.968292i	$-0.580372\pi$
$317$	−2820.00	−0.499643	−0.249822	+	0.968292i	$0.580372\pi$
$318$	−1188.00	−0.209496
$319$	4100.00	0.719611
$320$	−1280.00	−0.223607
$321$	5424.00	0.943110
$322$	1280.00	0.221527
$323$	3600.00	0.620153
$324$	324.000	0.0555556
$325$	−3575.00	−0.610170
$326$	496.000	0.0842666
$327$	5658.00	0.956844
$328$	864.000	0.145446
$329$	5696.00	0.954500
$330$	6000.00	1.00088
$331$	−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
$333$	−2754.00	−0.453208
$334$	204.000	0.0334203
$335$	2800.00	0.456658
$336$	1536.00	0.249392
$337$	−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
$339$	−3738.00	−0.598880
$340$	2400.00	0.382818
$341$	−2200.00	−0.349374
$342$	−2160.00	−0.341519
$343$	−10816.0	−1.70265
$344$	−2848.00	−0.446378
$345$	−1200.00	−0.187263
$346$	1364.00	0.211934
$347$	−7332.00	−1.13430	−0.567150	−	0.823614i	$-0.691954\pi$
$347$	−7332.00	−1.13430	−0.567150	+	0.823614i	$0.691954\pi$
$348$	−984.000	−0.151575
$349$	−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
$351$	351.000	0.0533761
$352$	1600.00	0.242274
$353$	1244.00	0.187568	0.0937839	−	0.995593i	$-0.470104\pi$
$353$	1244.00	0.187568	0.0937839	+	0.995593i	$0.470104\pi$
$354$	−564.000	−0.0846787
$355$	15560.0	2.32631
$356$	4896.00	0.728897
$357$	−2880.00	−0.426963
$358$	−1224.00	−0.180699
$359$	9558.00	1.40516	0.702579	−	0.711605i	$-0.252032\pi$
$359$	9558.00	1.40516	0.702579	+	0.711605i	$0.252032\pi$
$360$	−1440.00	−0.210819
$361$	7541.00	1.09943
$362$	−132.000	−0.0191651
$363$	−3507.00	−0.507079
$364$	1664.00	0.239608
$365$	−1240.00	−0.177821
$366$	372.000	0.0531277
$367$	−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
$369$	972.000	0.137128
$370$	12240.0	1.71980
$371$	−6336.00	−0.886654
$372$	528.000	0.0735901
$373$	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
$375$	9000.00	1.23935
$376$	−1424.00	−0.195312
$377$	−1066.00	−0.145628
$378$	1728.00	0.235129
$379$	−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
$381$	−4872.00	−0.655118
$382$	1216.00	0.162869
$383$	−1830.00	−0.244148	−0.122074	−	0.992521i	$-0.538955\pi$
$383$	−1830.00	−0.244148	−0.122074	+	0.992521i	$0.538955\pi$
$384$	−384.000	−0.0510310
$385$	32000.0	4.23603
$386$	2740.00	0.361301
$387$	−3204.00	−0.420849
$388$	2456.00	0.321352
$389$	10158.0	1.32399	0.661994	−	0.749509i	$-0.269711\pi$
$389$	10158.0	1.32399	0.661994	+	0.749509i	$0.269711\pi$
$390$	−1560.00	−0.202548
$391$	600.000	0.0776044
$392$	5448.00	0.701953
$393$	6216.00	0.797852
$394$	−9816.00	−1.25513
$395$	21920.0	2.79219
$396$	1800.00	0.228418
$397$	−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
$399$	−11520.0	−1.44542
$400$	4400.00	0.550000
$401$	15720.0	1.95765	0.978827	−	0.204689i	$-0.0656182\pi$
$401$	15720.0	1.95765	0.978827	+	0.204689i	$0.0656182\pi$
$402$	840.000	0.104217
$403$	572.000	0.0707031
$404$	4232.00	0.521163
$405$	−1620.00	−0.198762
$406$	−5248.00	−0.641512
$407$	−15300.0	−1.86337
$408$	720.000	0.0873660
$409$	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
$411$	2268.00	0.272195
$412$	7072.00	0.845661
$413$	−3008.00	−0.358387
$414$	−360.000	−0.0427368
$415$	9240.00	1.09295
$416$	−416.000	−0.0490290
$417$	−516.000	−0.0605962
$418$	−12000.0	−1.40416
$419$	−1848.00	−0.215467	−0.107734	−	0.994180i	$-0.534359\pi$
$419$	−1848.00	−0.215467	−0.107734	+	0.994180i	$0.534359\pi$
$420$	−7680.00	−0.892251
$421$	−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
$423$	−1602.00	−0.184142
$424$	1584.00	0.181429
$425$	−8250.00	−0.941609
$426$	4668.00	0.530905
$427$	1984.00	0.224854
$428$	−7232.00	−0.816757
$429$	1950.00	0.219457
$430$	14240.0	1.59701
$431$	−5238.00	−0.585396	−0.292698	−	0.956205i	$-0.594553\pi$
$431$	−5238.00	−0.585396	−0.292698	+	0.956205i	$0.594553\pi$
$432$	−432.000	−0.0481125
$433$	−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
$435$	4920.00	0.542290
$436$	−7544.00	−0.828652
$437$	2400.00	0.262718
$438$	−372.000	−0.0405818
$439$	−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
$441$	6129.00	0.661808
$442$	780.000	0.0839385
$443$	12744.0	1.36678	0.683392	−	0.730051i	$-0.260504\pi$
$443$	12744.0	1.36678	0.683392	+	0.730051i	$0.260504\pi$
$444$	3672.00	0.392490
$445$	−24480.0	−2.60778
$446$	−3864.00	−0.410237
$447$	−3816.00	−0.403782
$448$	−2048.00	−0.215980
$449$	−11776.0	−1.23774	−0.618868	−	0.785495i	$-0.712409\pi$
$449$	−11776.0	−1.23774	−0.618868	+	0.785495i	$0.712409\pi$
$450$	4950.00	0.518545
$451$	5400.00	0.563805
$452$	4984.00	0.518645
$453$	−4212.00	−0.436859
$454$	9996.00	1.03334
$455$	−8320.00	−0.857248
$456$	2880.00	0.295764
$457$	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
$459$	810.000	0.0823694
$460$	1600.00	0.162175
$461$	2724.00	0.275205	0.137602	−	0.990488i	$-0.456060\pi$
$461$	2724.00	0.275205	0.137602	+	0.990488i	$0.456060\pi$
$462$	9600.00	0.966737
$463$	−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
$465$	−2640.00	−0.263284
$466$	−2564.00	−0.254882
$467$	−18224.0	−1.80579	−0.902897	−	0.429856i	$-0.858564\pi$
$467$	−18224.0	−1.80579	−0.902897	+	0.429856i	$0.858564\pi$
$468$	−468.000	−0.0462250
$469$	4480.00	0.441081
$470$	7120.00	0.698768
$471$	6510.00	0.636868
$472$	752.000	0.0733339
$473$	−17800.0	−1.73033
$474$	6576.00	0.637227
$475$	−33000.0	−3.18767
$476$	3840.00	0.369761
$477$	1782.00	0.171053
$478$	588.000	0.0562646
$479$	9066.00	0.864794	0.432397	−	0.901683i	$-0.357668\pi$
$479$	9066.00	0.864794	0.432397	+	0.901683i	$0.357668\pi$
$480$	1920.00	0.182574
$481$	3978.00	0.377092
$482$	−9924.00	−0.937813
$483$	−1920.00	−0.180876
$484$	4676.00	0.439144
$485$	−12280.0	−1.14970
$486$	−486.000	−0.0453609
$487$	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
$489$	−744.000	−0.0688034
$490$	−27240.0	−2.51138
$491$	8720.00	0.801483	0.400741	−	0.916191i	$-0.368753\pi$
$491$	8720.00	0.801483	0.400741	+	0.916191i	$0.368753\pi$
$492$	−1296.00	−0.118756
$493$	−2460.00	−0.224732
$494$	3120.00	0.284161
$495$	−9000.00	−0.817212
$496$	−704.000	−0.0637309
$497$	24896.0	2.24696
$498$	2772.00	0.249430
$499$	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
$501$	−306.000	−0.0272876
$502$	1488.00	0.132296
$503$	3404.00	0.301743	0.150872	−	0.988553i	$-0.451792\pi$
$503$	3404.00	0.301743	0.150872	+	0.988553i	$0.451792\pi$
$504$	−2304.00	−0.203628
$505$	−21160.0	−1.86457
$506$	−2000.00	−0.175713
$507$	−507.000	−0.0444116
$508$	6496.00	0.567349
$509$	−76.0000	−0.00661815	−0.00330908	−	0.999995i	$-0.501053\pi$
$509$	−76.0000	−0.00661815	−0.00330908	+	0.999995i	$0.501053\pi$
$510$	−3600.00	−0.312570
$511$	−1984.00	−0.171755
$512$	512.000	0.0441942
$513$	3240.00	0.278849
$514$	−2052.00	−0.176089
$515$	−35360.0	−3.02553
$516$	4272.00	0.364466
$517$	−8900.00	−0.757102
$518$	19584.0	1.66114
$519$	−2046.00	−0.173043
$520$	2080.00	0.175412
$521$	12054.0	1.01362	0.506809	−	0.862058i	$-0.330825\pi$
$521$	12054.0	1.01362	0.506809	+	0.862058i	$0.330825\pi$
$522$	1476.00	0.123760
$523$	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
$525$	26400.0	2.19465
$526$	−11064.0	−0.917136
$527$	1320.00	0.109108
$528$	−2400.00	−0.197816
$529$	−11767.0	−0.967124
$530$	−7920.00	−0.649100
$531$	846.000	0.0691399
$532$	15360.0	1.25177
$533$	−1404.00	−0.114098
$534$	−7344.00	−0.595142
$535$	36160.0	2.92212
$536$	−1120.00	−0.0902549
$537$	1836.00	0.147540
$538$	−7068.00	−0.566400
$539$	34050.0	2.72103
$540$	2160.00	0.172133
$541$	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
$543$	198.000	0.0156482
$544$	−960.000	−0.0756611
$545$	37720.0	2.96467
$546$	−2496.00	−0.195639
$547$	−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
$549$	−558.000	−0.0433786
$550$	27500.0	2.13201
$551$	−9840.00	−0.760795
$552$	480.000	0.0370112
$553$	35072.0	2.69695
$554$	12204.0	0.935917
$555$	−18360.0	−1.40421
$556$	688.000	0.0524779
$557$	20544.0	1.56280	0.781398	−	0.624033i	$-0.214507\pi$
$557$	20544.0	1.56280	0.781398	+	0.624033i	$0.214507\pi$
$558$	−792.000	−0.0600861
$559$	4628.00	0.350167
$560$	10240.0	0.772712
$561$	4500.00	0.338663
$562$	−15080.0	−1.13187
$563$	6988.00	0.523107	0.261553	−	0.965189i	$-0.415765\pi$
$563$	6988.00	0.523107	0.261553	+	0.965189i	$0.415765\pi$
$564$	2136.00	0.159471
$565$	−24920.0	−1.85556
$566$	−5512.00	−0.409340
$567$	−2592.00	−0.191982
$568$	−6224.00	−0.459777
$569$	−706.000	−0.0520159	−0.0260080	−	0.999662i	$-0.508280\pi$
$569$	−706.000	−0.0520159	−0.0260080	+	0.999662i	$0.508280\pi$
$570$	−14400.0	−1.05816
$571$	−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
$573$	−1824.00	−0.132982
$574$	−6912.00	−0.502616
$575$	−5500.00	−0.398897
$576$	576.000	0.0416667
$577$	−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
$579$	−4110.00	−0.295001
$580$	−6560.00	−0.469637
$581$	14784.0	1.05567
$582$	−3684.00	−0.262383
$583$	9900.00	0.703287
$584$	496.000	0.0351449
$585$	2340.00	0.165380
$586$	1936.00	0.136477
$587$	14170.0	0.996352	0.498176	−	0.867076i	$-0.334003\pi$
$587$	14170.0	0.996352	0.498176	+	0.867076i	$0.334003\pi$
$588$	−8172.00	−0.573142
$589$	5280.00	0.369369
$590$	−3760.00	−0.262367
$591$	14724.0	1.02481
$592$	−4896.00	−0.339906
$593$	−11744.0	−0.813269	−0.406634	−	0.913591i	$-0.633298\pi$
$593$	−11744.0	−0.813269	−0.406634	+	0.913591i	$0.633298\pi$
$594$	−2700.00	−0.186502
$595$	−19200.0	−1.32290
$596$	5088.00	0.349686
$597$	984.000	0.0674580
$598$	520.000	0.0355592
$599$	−15076.0	−1.02836	−0.514181	−	0.857682i	$-0.671904\pi$
$599$	−15076.0	−1.02836	−0.514181	+	0.857682i	$0.671904\pi$
$600$	−6600.00	−0.449073
$601$	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
$603$	−1260.00	−0.0850931
$604$	5616.00	0.378331
$605$	−23380.0	−1.57113
$606$	−6348.00	−0.425528
$607$	−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
$609$	7872.00	0.523792
$610$	2480.00	0.164610
$611$	2314.00	0.153215
$612$	−1080.00	−0.0713340
$613$	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
$615$	6480.00	0.424876
$616$	−12800.0	−0.837219
$617$	8804.00	0.574450	0.287225	−	0.957863i	$-0.407267\pi$
$617$	8804.00	0.574450	0.287225	+	0.957863i	$0.407267\pi$
$618$	−10608.0	−0.690480
$619$	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
$621$	540.000	0.0348945
$622$	15864.0	1.02265
$623$	−39168.0	−2.51883
$624$	624.000	0.0400320
$625$	25625.0	1.64000
$626$	20716.0	1.32265
$627$	18000.0	1.14649
$628$	−8680.00	−0.551544
$629$	9180.00	0.581925
$630$	11520.0	0.728520
$631$	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
$633$	−3948.00	−0.247897
$634$	−5640.00	−0.353301
$635$	−32480.0	−2.02981
$636$	−2376.00	−0.148136
$637$	−8853.00	−0.550657
$638$	8200.00	0.508842
$639$	−7002.00	−0.433482
$640$	−2560.00	−0.158114
$641$	−7342.00	−0.452405	−0.226202	−	0.974080i	$-0.572631\pi$
$641$	−7342.00	−0.452405	−0.226202	+	0.974080i	$0.572631\pi$
$642$	10848.0	0.666879
$643$	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
$645$	−21360.0	−1.30395
$646$	7200.00	0.438514
$647$	9344.00	0.567775	0.283888	−	0.958858i	$-0.408376\pi$
$647$	9344.00	0.567775	0.283888	+	0.958858i	$0.408376\pi$
$648$	648.000	0.0392837
$649$	4700.00	0.284270
$650$	−7150.00	−0.431455
$651$	−4224.00	−0.254304
$652$	992.000	0.0595855
$653$	−16686.0	−0.999960	−0.499980	−	0.866037i	$-0.666659\pi$
$653$	−16686.0	−0.999960	−0.499980	+	0.866037i	$0.666659\pi$
$654$	11316.0	0.676591
$655$	41440.0	2.47205
$656$	1728.00	0.102846
$657$	558.000	0.0331349
$658$	11392.0	0.674934
$659$	31356.0	1.85350	0.926750	−	0.375679i	$-0.122590\pi$
$659$	31356.0	1.85350	0.926750	+	0.375679i	$0.122590\pi$
$660$	12000.0	0.707726
$661$	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
$663$	−1170.00	−0.0685355
$664$	−3696.00	−0.216013
$665$	−76800.0	−4.47846
$666$	−5508.00	−0.320466
$667$	−1640.00	−0.0952040
$668$	408.000	0.0236317
$669$	5796.00	0.334957
$670$	5600.00	0.322906
$671$	−3100.00	−0.178352
$672$	3072.00	0.176347
$673$	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
$675$	−7425.00	−0.423390
$676$	676.000	0.0384615
$677$	9486.00	0.538518	0.269259	−	0.963068i	$-0.413221\pi$
$677$	9486.00	0.538518	0.269259	+	0.963068i	$0.413221\pi$
$678$	−7476.00	−0.423472
$679$	−19648.0	−1.11049
$680$	4800.00	0.270694
$681$	−14994.0	−0.843717
$682$	−4400.00	−0.247045
$683$	26162.0	1.46568	0.732841	−	0.680400i	$-0.238194\pi$
$683$	26162.0	1.46568	0.732841	+	0.680400i	$0.238194\pi$
$684$	−4320.00	−0.241490
$685$	15120.0	0.843366
$686$	−21632.0	−1.20396
$687$	234.000	0.0129951
$688$	−5696.00	−0.315637
$689$	−2574.00	−0.142325
$690$	−2400.00	−0.132415
$691$	−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
$693$	−14400.0	−0.789337
$694$	−14664.0	−0.802072
$695$	−3440.00	−0.187751
$696$	−1968.00	−0.107179
$697$	−3240.00	−0.176074
$698$	−16324.0	−0.885204
$699$	3846.00	0.208110
$700$	−35200.0	−1.90062
$701$	30.0000	0.00161638	0.000808191	−	1.00000i	$-0.499743\pi$
$701$	30.0000	0.00161638	0.000808191		1.00000i	$0.499743\pi$
$702$	702.000	0.0377426
$703$	36720.0	1.97002
$704$	3200.00	0.171313
$705$	−10680.0	−0.570542
$706$	2488.00	0.132630
$707$	−33856.0	−1.80097
$708$	−1128.00	−0.0598769
$709$	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
$711$	−9864.00	−0.520294
$712$	9792.00	0.515408
$713$	880.000	0.0462220
$714$	−5760.00	−0.301908
$715$	13000.0	0.679961
$716$	−2448.00	−0.127774
$717$	−882.000	−0.0459399
$718$	19116.0	0.993597
$719$	−28892.0	−1.49859	−0.749297	−	0.662234i	$-0.769609\pi$
$719$	−28892.0	−1.49859	−0.749297	+	0.662234i	$0.769609\pi$
$720$	−2880.00	−0.149071
$721$	−56576.0	−2.92233
$722$	15082.0	0.777415
$723$	14886.0	0.765721
$724$	−264.000	−0.0135518
$725$	22550.0	1.15515
$726$	−7014.00	−0.358559
$727$	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
$729$	729.000	0.0370370
$730$	−2480.00	−0.125738
$731$	10680.0	0.540375
$732$	744.000	0.0375670
$733$	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
$735$	40860.0	2.05054
$736$	−640.000	−0.0320526
$737$	−7000.00	−0.349862
$738$	1944.00	0.0969643
$739$	−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
$741$	−4680.00	−0.232016
$742$	−12672.0	−0.626959
$743$	−9898.00	−0.488725	−0.244362	−	0.969684i	$-0.578579\pi$
$743$	−9898.00	−0.488725	−0.244362	+	0.969684i	$0.578579\pi$
$744$	1056.00	0.0520361
$745$	−25440.0	−1.25107
$746$	10948.0	0.537312
$747$	−4158.00	−0.203659
$748$	−6000.00	−0.293291
$749$	57856.0	2.82245
$750$	18000.0	0.876356
$751$	−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
$753$	−2232.00	−0.108019
$754$	−2132.00	−0.102975
$755$	−28080.0	−1.35356
$756$	3456.00	0.166261
$757$	−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
$759$	3000.00	0.143469
$760$	19200.0	0.916391
$761$	−11988.0	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$761$	−11988.0	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$762$	−9744.00	−0.463239
$763$	60352.0	2.86355
$764$	2432.00	0.115166
$765$	5400.00	0.255212
$766$	−3660.00	−0.172639
$767$	−1222.00	−0.0575279
$768$	−768.000	−0.0360844
$769$	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
$771$	3078.00	0.143776
$772$	5480.00	0.255479
$773$	−14408.0	−0.670401	−0.335200	−	0.942147i	$-0.608804\pi$
$773$	−14408.0	−0.670401	−0.335200	+	0.942147i	$0.608804\pi$
$774$	−6408.00	−0.297585
$775$	−12100.0	−0.560832
$776$	4912.00	0.227230
$777$	−29376.0	−1.35632
$778$	20316.0	0.936200
$779$	−12960.0	−0.596072
$780$	−3120.00	−0.143223
$781$	−38900.0	−1.78227
$782$	1200.00	0.0548746
$783$	−2214.00	−0.101050
$784$	10896.0	0.496356
$785$	43400.0	1.97326
$786$	12432.0	0.564166
$787$	−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
$789$	16596.0	0.748838
$790$	43840.0	1.97438
$791$	−39872.0	−1.79227
$792$	3600.00	0.161516
$793$	806.000	0.0360932
$794$	−25316.0	−1.13153
$795$	11880.0	0.529988
$796$	−1312.00	−0.0584204
$797$	1974.00	0.0877323	0.0438662	−	0.999037i	$-0.486032\pi$
$797$	1974.00	0.0877323	0.0438662	+	0.999037i	$0.486032\pi$
$798$	−23040.0	−1.02206
$799$	5340.00	0.236440
$800$	8800.00	0.388909
$801$	11016.0	0.485932
$802$	31440.0	1.38427
$803$	3100.00	0.136235
$804$	1680.00	0.0736928
$805$	−12800.0	−0.560423
$806$	1144.00	0.0499946
$807$	10602.0	0.462464
$808$	8464.00	0.368518
$809$	31734.0	1.37912	0.689560	−	0.724229i	$-0.257804\pi$
$809$	31734.0	1.37912	0.689560	+	0.724229i	$0.257804\pi$
$810$	−3240.00	−0.140546
$811$	−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
$813$	−7176.00	−0.309561
$814$	−30600.0	−1.31760
$815$	−4960.00	−0.213179
$816$	1440.00	0.0617771
$817$	42720.0	1.82936
$818$	15308.0	0.654317
$819$	3744.00	0.159739
$820$	−8640.00	−0.367954
$821$	−16736.0	−0.711438	−0.355719	−	0.934593i	$-0.615764\pi$
$821$	−16736.0	−0.711438	−0.355719	+	0.934593i	$0.615764\pi$
$822$	4536.00	0.192471
$823$	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
$825$	−41250.0	−1.74078
$826$	−6016.00	−0.253418
$827$	24858.0	1.04522	0.522610	−	0.852572i	$-0.324958\pi$
$827$	24858.0	1.04522	0.522610	+	0.852572i	$0.324958\pi$
$828$	−720.000	−0.0302195
$829$	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
$831$	−18306.0	−0.764173
$832$	−832.000	−0.0346688
$833$	−20430.0	−0.849769
$834$	−1032.00	−0.0428480
$835$	−2040.00	−0.0845474
$836$	−24000.0	−0.992892
$837$	1188.00	0.0490601
$838$	−3696.00	−0.152358
$839$	−14294.0	−0.588181	−0.294090	−	0.955778i	$-0.595017\pi$
$839$	−14294.0	−0.588181	−0.294090	+	0.955778i	$0.595017\pi$
$840$	−15360.0	−0.630917
$841$	−17665.0	−0.724302
$842$	−25084.0	−1.02666
$843$	22620.0	0.924169
$844$	5264.00	0.214685
$845$	−3380.00	−0.137604
$846$	−3204.00	−0.130208
$847$	−37408.0	−1.51754
$848$	3168.00	0.128290
$849$	8268.00	0.334225
$850$	−16500.0	−0.665818
$851$	6120.00	0.246523
$852$	9336.00	0.375406
$853$	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
$855$	21600.0	0.863982
$856$	−14464.0	−0.577534
$857$	39038.0	1.55602	0.778012	−	0.628249i	$-0.216228\pi$
$857$	39038.0	1.55602	0.778012	+	0.628249i	$0.216228\pi$
$858$	3900.00	0.155179
$859$	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
$861$	10368.0	0.410384
$862$	−10476.0	−0.413937
$863$	39866.0	1.57248	0.786242	−	0.617918i	$-0.212024\pi$
$863$	39866.0	1.57248	0.786242	+	0.617918i	$0.212024\pi$
$864$	−864.000	−0.0340207
$865$	−13640.0	−0.536155
$866$	−16516.0	−0.648079
$867$	12039.0	0.471587
$868$	5632.00	0.220233
$869$	−54800.0	−2.13920
$870$	9840.00	0.383457
$871$	1820.00	0.0708018
$872$	−15088.0	−0.585945
$873$	5526.00	0.214235
$874$	4800.00	0.185769
$875$	96000.0	3.70902
$876$	−744.000	−0.0286957
$877$	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
$879$	−2904.00	−0.111433
$880$	−16000.0	−0.612909
$881$	−4458.00	−0.170481	−0.0852405	−	0.996360i	$-0.527166\pi$
$881$	−4458.00	−0.170481	−0.0852405	+	0.996360i	$0.527166\pi$
$882$	12258.0	0.467969
$883$	−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
$885$	5640.00	0.214222
$886$	25488.0	0.966463
$887$	−32512.0	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$887$	−32512.0	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$888$	7344.00	0.277532
$889$	−51968.0	−1.96057
$890$	−48960.0	−1.84398
$891$	4050.00	0.152278
$892$	−7728.00	−0.290081
$893$	21360.0	0.800431
$894$	−7632.00	−0.285517
$895$	12240.0	0.457138
$896$	−4096.00	−0.152721
$897$	−780.000	−0.0290339
$898$	−23552.0	−0.875212
$899$	−3608.00	−0.133853
$900$	9900.00	0.366667
$901$	−5940.00	−0.219634
$902$	10800.0	0.398670
$903$	−34176.0	−1.25948
$904$	9968.00	0.366738
$905$	1320.00	0.0484843
$906$	−8424.00	−0.308906
$907$	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
$909$	9522.00	0.347442
$910$	−16640.0	−0.606166
$911$	9840.00	0.357864	0.178932	−	0.983861i	$-0.442736\pi$
$911$	9840.00	0.357864	0.178932	+	0.983861i	$0.442736\pi$
$912$	5760.00	0.209137
$913$	−23100.0	−0.837348
$914$	4268.00	0.154456
$915$	−3720.00	−0.134404
$916$	−312.000	−0.0112541
$917$	66304.0	2.38773
$918$	1620.00	0.0582440
$919$	−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
$921$	19308.0	0.690793
$922$	5448.00	0.194599
$923$	10114.0	0.360679
$924$	19200.0	0.683586
$925$	−84150.0	−2.99117
$926$	−11296.0	−0.400874
$927$	15912.0	0.563774
$928$	2624.00	0.0928201
$929$	44172.0	1.56000	0.779998	−	0.625782i	$-0.215220\pi$
$929$	44172.0	1.56000	0.779998	+	0.625782i	$0.215220\pi$
$930$	−5280.00	−0.186170
$931$	−81720.0	−2.87676
$932$	−5128.00	−0.180229
$933$	−23796.0	−0.834990
$934$	−36448.0	−1.27689
$935$	30000.0	1.04931
$936$	−936.000	−0.0326860
$937$	−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
$939$	−31074.0	−1.07994
$940$	14240.0	0.494104
$941$	−1672.00	−0.0579231	−0.0289616	−	0.999581i	$-0.509220\pi$
$941$	−1672.00	−0.0579231	−0.0289616	+	0.999581i	$0.509220\pi$
$942$	13020.0	0.450334
$943$	−2160.00	−0.0745910
$944$	1504.00	0.0518549
$945$	−17280.0	−0.594834
$946$	−35600.0	−1.22353
$947$	5238.00	0.179738	0.0898691	−	0.995954i	$-0.471355\pi$
$947$	5238.00	0.179738	0.0898691	+	0.995954i	$0.471355\pi$
$948$	13152.0	0.450588
$949$	−806.000	−0.0275699
$950$	−66000.0	−2.25402
$951$	8460.00	0.288469
$952$	7680.00	0.261460
$953$	−50042.0	−1.70096	−0.850482	−	0.526004i	$-0.823690\pi$
$953$	−50042.0	−1.70096	−0.850482	+	0.526004i	$0.823690\pi$
$954$	3564.00	0.120953
$955$	−12160.0	−0.412030
$956$	1176.00	0.0397851
$957$	−12300.0	−0.415468
$958$	18132.0	0.611501
$959$	24192.0	0.814599
$960$	3840.00	0.129099
$961$	−27855.0	−0.935014
$962$	7956.00	0.266644
$963$	−16272.0	−0.544505
$964$	−19848.0	−0.663134
$965$	−27400.0	−0.914028
$966$	−3840.00	−0.127899
$967$	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
$969$	−10800.0	−0.358045
$970$	−24560.0	−0.812963
$971$	−17364.0	−0.573880	−0.286940	−	0.957949i	$-0.592638\pi$
$971$	−17364.0	−0.573880	−0.286940	+	0.957949i	$0.592638\pi$
$972$	−972.000	−0.0320750
$973$	−5504.00	−0.181346
$974$	17896.0	0.588732
$975$	10725.0	0.352282
$976$	−992.000	−0.0325340
$977$	−14904.0	−0.488046	−0.244023	−	0.969769i	$-0.578467\pi$
$977$	−14904.0	−0.488046	−0.244023	+	0.969769i	$0.578467\pi$
$978$	−1488.00	−0.0486513
$979$	61200.0	1.99792
$980$	−54480.0	−1.77582
$981$	−16974.0	−0.552434
$982$	17440.0	0.566734
$983$	−18038.0	−0.585272	−0.292636	−	0.956224i	$-0.594533\pi$
$983$	−18038.0	−0.585272	−0.292636	+	0.956224i	$0.594533\pi$
$984$	−2592.00	−0.0839735
$985$	98160.0	3.17527
$986$	−4920.00	−0.158909
$987$	−17088.0	−0.551081
$988$	6240.00	0.200932
$989$	7120.00	0.228921
$990$	−18000.0	−0.577856
$991$	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
$993$	12540.0	0.400750
$994$	49792.0	1.58884
$995$	6560.00	0.209011
$996$	5544.00	0.176374
$997$	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
$999$	8262.00	0.261660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.4.a.d.1.1	✓	1
3.2	odd	2		234.4.a.f.1.1		1
4.3	odd	2		624.4.a.e.1.1		1
5.4	even	2		1950.4.a.h.1.1		1
8.3	odd	2		2496.4.a.i.1.1		1
8.5	even	2		2496.4.a.r.1.1		1
12.11	even	2		1872.4.a.r.1.1		1
13.5	odd	4		1014.4.b.e.337.1		2
13.8	odd	4		1014.4.b.e.337.2		2
13.12	even	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	1.1	even	1	trivial
234.4.a.f.1.1		1	3.2	odd	2
624.4.a.e.1.1		1	4.3	odd	2
1014.4.a.d.1.1		1	13.12	even	2
1014.4.b.e.337.1		2	13.5	odd	4
1014.4.b.e.337.2		2	13.8	odd	4
1872.4.a.r.1.1		1	12.11	even	2
1950.4.a.h.1.1		1	5.4	even	2
2496.4.a.i.1.1		1	8.3	odd	2
2496.4.a.r.1.1		1	8.5	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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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