LMFDB - Embedded newform 78.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [78,4,Mod(1,78)] 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("78.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	78.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,-3,4,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.60214898045$
Analytic rank:	$0$
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)

$f(q)$

$=$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +32.0000 q^{10} +34.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -56.0000 q^{14} +48.0000 q^{15} +16.0000 q^{16} +138.000 q^{17} -18.0000 q^{18} +108.000 q^{19} -64.0000 q^{20} -84.0000 q^{21} -68.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} +131.000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +112.000 q^{28} -190.000 q^{29} -96.0000 q^{30} -176.000 q^{31} -32.0000 q^{32} -102.000 q^{33} -276.000 q^{34} -448.000 q^{35} +36.0000 q^{36} +342.000 q^{37} -216.000 q^{38} +39.0000 q^{39} +128.000 q^{40} +240.000 q^{41} +168.000 q^{42} -140.000 q^{43} +136.000 q^{44} -144.000 q^{45} +104.000 q^{46} +454.000 q^{47} -48.0000 q^{48} +441.000 q^{49} -262.000 q^{50} -414.000 q^{51} -52.0000 q^{52} +198.000 q^{53} +54.0000 q^{54} -544.000 q^{55} -224.000 q^{56} -324.000 q^{57} +380.000 q^{58} -154.000 q^{59} +192.000 q^{60} +34.0000 q^{61} +352.000 q^{62} +252.000 q^{63} +64.0000 q^{64} +208.000 q^{65} +204.000 q^{66} -656.000 q^{67} +552.000 q^{68} +156.000 q^{69} +896.000 q^{70} +550.000 q^{71} -72.0000 q^{72} +614.000 q^{73} -684.000 q^{74} -393.000 q^{75} +432.000 q^{76} +952.000 q^{77} -78.0000 q^{78} +8.00000 q^{79} -256.000 q^{80} +81.0000 q^{81} -480.000 q^{82} +762.000 q^{83} -336.000 q^{84} -2208.00 q^{85} +280.000 q^{86} +570.000 q^{87} -272.000 q^{88} -444.000 q^{89} +288.000 q^{90} -364.000 q^{91} -208.000 q^{92} +528.000 q^{93} -908.000 q^{94} -1728.00 q^{95} +96.0000 q^{96} +1022.00 q^{97} -882.000 q^{98} +306.000 q^{99} +O(q^{100})\)

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−16.0000	−1.43108	−0.715542	−	0.698570i	$-0.753820\pi$
$5$	−16.0000	−1.43108	−0.715542	+	0.698570i	$0.753820\pi$
$6$	6.00000	0.408248
$7$	28.0000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	28.0000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	32.0000	1.01193
$11$	34.0000	0.931944	0.465972	−	0.884799i	$-0.345705\pi$
$11$	34.0000	0.931944	0.465972	+	0.884799i	$0.345705\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−56.0000	−1.06904
$15$	48.0000	0.826236
$16$	16.0000	0.250000
$17$	138.000	1.96882	0.984409	−	0.175893i	$-0.0562813\pi$
$17$	138.000	1.96882	0.984409	+	0.175893i	$0.0562813\pi$
$18$	−18.0000	−0.235702
$19$	108.000	1.30405	0.652024	−	0.758199i	$-0.273920\pi$
$19$	108.000	1.30405	0.652024	+	0.758199i	$0.273920\pi$
$20$	−64.0000	−0.715542
$21$	−84.0000	−0.872872
$22$	−68.0000	−0.658984
$23$	−52.0000	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$23$	−52.0000	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$24$	24.0000	0.204124
$25$	131.000	1.04800
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	112.000	0.755929
$29$	−190.000	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$29$	−190.000	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$30$	−96.0000	−0.584237
$31$	−176.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$31$	−176.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$32$	−32.0000	−0.176777
$33$	−102.000	−0.538058
$34$	−276.000	−1.39216
$35$	−448.000	−2.16359
$36$	36.0000	0.166667
$37$	342.000	1.51958	0.759790	−	0.650169i	$-0.225302\pi$
$37$	342.000	1.51958	0.759790	+	0.650169i	$0.225302\pi$
$38$	−216.000	−0.922101
$39$	39.0000	0.160128
$40$	128.000	0.505964
$41$	240.000	0.914188	0.457094	−	0.889418i	$-0.348890\pi$
$41$	240.000	0.914188	0.457094	+	0.889418i	$0.348890\pi$
$42$	168.000	0.617213
$43$	−140.000	−0.496507	−0.248253	−	0.968695i	$-0.579857\pi$
$43$	−140.000	−0.496507	−0.248253	+	0.968695i	$0.579857\pi$
$44$	136.000	0.465972
$45$	−144.000	−0.477028
$46$	104.000	0.333347
$47$	454.000	1.40899	0.704497	−	0.709707i	$-0.251173\pi$
$47$	454.000	1.40899	0.704497	+	0.709707i	$0.251173\pi$
$48$	−48.0000	−0.144338
$49$	441.000	1.28571
$50$	−262.000	−0.741048
$51$	−414.000	−1.13670
$52$	−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
$55$	−544.000	−1.33369
$56$	−224.000	−0.534522
$57$	−324.000	−0.752892
$58$	380.000	0.860284
$59$	−154.000	−0.339815	−0.169908	−	0.985460i	$-0.554347\pi$
$59$	−154.000	−0.339815	−0.169908	+	0.985460i	$0.554347\pi$
$60$	192.000	0.413118
$61$	34.0000	0.0713648	0.0356824	−	0.999363i	$-0.488640\pi$
$61$	34.0000	0.0713648	0.0356824	+	0.999363i	$0.488640\pi$
$62$	352.000	0.721033
$63$	252.000	0.503953
$64$	64.0000	0.125000
$65$	208.000	0.396911
$66$	204.000	0.380465
$67$	−656.000	−1.19617	−0.598083	−	0.801434i	$-0.704071\pi$
$67$	−656.000	−1.19617	−0.598083	+	0.801434i	$0.704071\pi$
$68$	552.000	0.984409
$69$	156.000	0.272177
$70$	896.000	1.52989
$71$	550.000	0.919338	0.459669	−	0.888090i	$-0.347968\pi$
$71$	550.000	0.919338	0.459669	+	0.888090i	$0.347968\pi$
$72$	−72.0000	−0.117851
$73$	614.000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$73$	614.000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$74$	−684.000	−1.07451
$75$	−393.000	−0.605063
$76$	432.000	0.652024
$77$	952.000	1.40897
$78$	−78.0000	−0.113228
$79$	8.00000	0.0113933	0.00569665	−	0.999984i	$-0.498187\pi$
$79$	8.00000	0.0113933	0.00569665	+	0.999984i	$0.498187\pi$
$80$	−256.000	−0.357771
$81$	81.0000	0.111111
$82$	−480.000	−0.646428
$83$	762.000	1.00772	0.503858	−	0.863787i	$-0.331914\pi$
$83$	762.000	1.00772	0.503858	+	0.863787i	$0.331914\pi$
$84$	−336.000	−0.436436
$85$	−2208.00	−2.81754
$86$	280.000	0.351083
$87$	570.000	0.702419
$88$	−272.000	−0.329492
$89$	−444.000	−0.528808	−0.264404	−	0.964412i	$-0.585175\pi$
$89$	−444.000	−0.528808	−0.264404	+	0.964412i	$0.585175\pi$
$90$	288.000	0.337310
$91$	−364.000	−0.419314
$92$	−208.000	−0.235712
$93$	528.000	0.588721
$94$	−908.000	−0.996309
$95$	−1728.00	−1.86620
$96$	96.0000	0.102062
$97$	1022.00	1.06978	0.534889	−	0.844923i	$-0.320354\pi$
$97$	1022.00	1.06978	0.534889	+	0.844923i	$0.320354\pi$
$98$	−882.000	−0.909137
$99$	306.000	0.310648
$100$	524.000	0.524000
$101$	−1190.00	−1.17237	−0.586185	−	0.810177i	$-0.699371\pi$
$101$	−1190.00	−1.17237	−0.586185	+	0.810177i	$0.699371\pi$
$102$	828.000	0.803767
$103$	−224.000	−0.214285	−0.107143	−	0.994244i	$-0.534170\pi$
$103$	−224.000	−0.214285	−0.107143	+	0.994244i	$0.534170\pi$
$104$	104.000	0.0980581
$105$	1344.00	1.24915
$106$	−396.000	−0.362858
$107$	−640.000	−0.578235	−0.289117	−	0.957294i	$-0.593362\pi$
$107$	−640.000	−0.578235	−0.289117	+	0.957294i	$0.593362\pi$
$108$	−108.000	−0.0962250
$109$	−1934.00	−1.69948	−0.849741	−	0.527200i	$-0.823242\pi$
$109$	−1934.00	−1.69948	−0.849741	+	0.527200i	$0.823242\pi$
$110$	1088.00	0.943061
$111$	−1026.00	−0.877330
$112$	448.000	0.377964
$113$	−418.000	−0.347983	−0.173992	−	0.984747i	$-0.555667\pi$
$113$	−418.000	−0.347983	−0.173992	+	0.984747i	$0.555667\pi$
$114$	648.000	0.532375
$115$	832.000	0.674647
$116$	−760.000	−0.608312
$117$	−117.000	−0.0924500
$118$	308.000	0.240286
$119$	3864.00	2.97657
$120$	−384.000	−0.292119
$121$	−175.000	−0.131480
$122$	−68.0000	−0.0504625
$123$	−720.000	−0.527807
$124$	−704.000	−0.509847
$125$	−96.0000	−0.0686920
$126$	−504.000	−0.356348
$127$	−1040.00	−0.726654	−0.363327	−	0.931662i	$-0.618359\pi$
$127$	−1040.00	−0.726654	−0.363327	+	0.931662i	$0.618359\pi$
$128$	−128.000	−0.0883883
$129$	420.000	0.286658
$130$	−416.000	−0.280659
$131$	−568.000	−0.378827	−0.189414	−	0.981897i	$-0.560659\pi$
$131$	−568.000	−0.378827	−0.189414	+	0.981897i	$0.560659\pi$
$132$	−408.000	−0.269029
$133$	3024.00	1.97153
$134$	1312.00	0.845817
$135$	432.000	0.275412
$136$	−1104.00	−0.696082
$137$	528.000	0.329271	0.164635	−	0.986355i	$-0.447355\pi$
$137$	528.000	0.329271	0.164635	+	0.986355i	$0.447355\pi$
$138$	−312.000	−0.192458
$139$	−1556.00	−0.949483	−0.474742	−	0.880125i	$-0.657459\pi$
$139$	−1556.00	−0.949483	−0.474742	+	0.880125i	$0.657459\pi$
$140$	−1792.00	−1.08180
$141$	−1362.00	−0.813483
$142$	−1100.00	−0.650070
$143$	−442.000	−0.258475
$144$	144.000	0.0833333
$145$	3040.00	1.74109
$146$	−1228.00	−0.696096
$147$	−1323.00	−0.742307
$148$	1368.00	0.759790
$149$	1524.00	0.837926	0.418963	−	0.908003i	$-0.362394\pi$
$149$	1524.00	0.837926	0.418963	+	0.908003i	$0.362394\pi$
$150$	786.000	0.427844
$151$	−3024.00	−1.62973	−0.814866	−	0.579649i	$-0.803190\pi$
$151$	−3024.00	−1.62973	−0.814866	+	0.579649i	$0.803190\pi$
$152$	−864.000	−0.461050
$153$	1242.00	0.656273
$154$	−1904.00	−0.996290
$155$	2816.00	1.45927
$156$	156.000	0.0800641
$157$	2198.00	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	2198.00	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−16.0000	−0.00805628
$159$	−594.000	−0.296272
$160$	512.000	0.252982
$161$	−1456.00	−0.712726
$162$	−162.000	−0.0785674
$163$	−268.000	−0.128781	−0.0643907	−	0.997925i	$-0.520510\pi$
$163$	−268.000	−0.128781	−0.0643907	+	0.997925i	$0.520510\pi$
$164$	960.000	0.457094
$165$	1632.00	0.770006
$166$	−1524.00	−0.712562
$167$	702.000	0.325284	0.162642	−	0.986685i	$-0.447998\pi$
$167$	702.000	0.325284	0.162642	+	0.986685i	$0.447998\pi$
$168$	672.000	0.308607
$169$	169.000	0.0769231
$170$	4416.00	1.99230
$171$	972.000	0.434682
$172$	−560.000	−0.248253
$173$	2066.00	0.907948	0.453974	−	0.891015i	$-0.350006\pi$
$173$	2066.00	0.907948	0.453974	+	0.891015i	$0.350006\pi$
$174$	−1140.00	−0.496685
$175$	3668.00	1.58443
$176$	544.000	0.232986
$177$	462.000	0.196192
$178$	888.000	0.373924
$179$	−276.000	−0.115247	−0.0576235	−	0.998338i	$-0.518352\pi$
$179$	−276.000	−0.115247	−0.0576235	+	0.998338i	$0.518352\pi$
$180$	−576.000	−0.238514
$181$	−3474.00	−1.42663	−0.713316	−	0.700843i	$-0.752808\pi$
$181$	−3474.00	−1.42663	−0.713316	+	0.700843i	$0.752808\pi$
$182$	728.000	0.296500
$183$	−102.000	−0.0412025
$184$	416.000	0.166674
$185$	−5472.00	−2.17465
$186$	−1056.00	−0.416289
$187$	4692.00	1.83483
$188$	1816.00	0.704497
$189$	−756.000	−0.290957
$190$	3456.00	1.31960
$191$	−3920.00	−1.48503	−0.742516	−	0.669828i	$-0.766368\pi$
$191$	−3920.00	−1.48503	−0.742516	+	0.669828i	$0.766368\pi$
$192$	−192.000	−0.0721688
$193$	2186.00	0.815294	0.407647	−	0.913140i	$-0.366349\pi$
$193$	2186.00	0.815294	0.407647	+	0.913140i	$0.366349\pi$
$194$	−2044.00	−0.756447
$195$	−624.000	−0.229157
$196$	1764.00	0.642857
$197$	−1368.00	−0.494751	−0.247376	−	0.968920i	$-0.579568\pi$
$197$	−1368.00	−0.494751	−0.247376	+	0.968920i	$0.579568\pi$
$198$	−612.000	−0.219661
$199$	−1072.00	−0.381870	−0.190935	−	0.981603i	$-0.561152\pi$
$199$	−1072.00	−0.381870	−0.190935	+	0.981603i	$0.561152\pi$
$200$	−1048.00	−0.370524
$201$	1968.00	0.690607
$202$	2380.00	0.828991
$203$	−5320.00	−1.83936
$204$	−1656.00	−0.568349
$205$	−3840.00	−1.30828
$206$	448.000	0.151523
$207$	−468.000	−0.157141
$208$	−208.000	−0.0693375
$209$	3672.00	1.21530
$210$	−2688.00	−0.883284
$211$	5444.00	1.77621	0.888105	−	0.459640i	$-0.152022\pi$
$211$	5444.00	1.77621	0.888105	+	0.459640i	$0.152022\pi$
$212$	792.000	0.256579
$213$	−1650.00	−0.530780
$214$	1280.00	0.408874
$215$	2240.00	0.710543
$216$	216.000	0.0680414
$217$	−4928.00	−1.54163
$218$	3868.00	1.20172
$219$	−1842.00	−0.568360
$220$	−2176.00	−0.666845
$221$	−1794.00	−0.546052
$222$	2052.00	0.620366
$223$	96.0000	0.0288280	0.0144140	−	0.999896i	$-0.495412\pi$
$223$	96.0000	0.0288280	0.0144140	+	0.999896i	$0.495412\pi$
$224$	−896.000	−0.267261
$225$	1179.00	0.349333
$226$	836.000	0.246061
$227$	198.000	0.0578930	0.0289465	−	0.999581i	$-0.490785\pi$
$227$	198.000	0.0578930	0.0289465	+	0.999581i	$0.490785\pi$
$228$	−1296.00	−0.376446
$229$	5922.00	1.70889	0.854447	−	0.519538i	$-0.173896\pi$
$229$	5922.00	1.70889	0.854447	+	0.519538i	$0.173896\pi$
$230$	−1664.00	−0.477047
$231$	−2856.00	−0.813468
$232$	1520.00	0.430142
$233$	−5114.00	−1.43789	−0.718947	−	0.695065i	$-0.755376\pi$
$233$	−5114.00	−1.43789	−0.718947	+	0.695065i	$0.755376\pi$
$234$	234.000	0.0653720
$235$	−7264.00	−2.01639
$236$	−616.000	−0.169908
$237$	−24.0000	−0.00657792
$238$	−7728.00	−2.10476
$239$	−5226.00	−1.41440	−0.707200	−	0.707013i	$-0.750042\pi$
$239$	−5226.00	−1.41440	−0.707200	+	0.707013i	$0.750042\pi$
$240$	768.000	0.206559
$241$	−762.000	−0.203671	−0.101836	−	0.994801i	$-0.532472\pi$
$241$	−762.000	−0.203671	−0.101836	+	0.994801i	$0.532472\pi$
$242$	350.000	0.0929705
$243$	−243.000	−0.0641500
$244$	136.000	0.0356824
$245$	−7056.00	−1.83996
$246$	1440.00	0.373216
$247$	−1404.00	−0.361678
$248$	1408.00	0.360516
$249$	−2286.00	−0.581805
$250$	192.000	0.0485726
$251$	3240.00	0.814769	0.407384	−	0.913257i	$-0.366441\pi$
$251$	3240.00	0.814769	0.407384	+	0.913257i	$0.366441\pi$
$252$	1008.00	0.251976
$253$	−1768.00	−0.439341
$254$	2080.00	0.513822
$255$	6624.00	1.62671
$256$	256.000	0.0625000
$257$	−1386.00	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$257$	−1386.00	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$258$	−840.000	−0.202698
$259$	9576.00	2.29739
$260$	832.000	0.198456
$261$	−1710.00	−0.405542
$262$	1136.00	0.267871
$263$	3300.00	0.773714	0.386857	−	0.922140i	$-0.373561\pi$
$263$	3300.00	0.773714	0.386857	+	0.922140i	$0.373561\pi$
$264$	816.000	0.190232
$265$	−3168.00	−0.734372
$266$	−6048.00	−1.39409
$267$	1332.00	0.305307
$268$	−2624.00	−0.598083
$269$	4290.00	0.972364	0.486182	−	0.873858i	$-0.338389\pi$
$269$	4290.00	0.972364	0.486182	+	0.873858i	$0.338389\pi$
$270$	−864.000	−0.194746
$271$	2452.00	0.549625	0.274813	−	0.961498i	$-0.411384\pi$
$271$	2452.00	0.549625	0.274813	+	0.961498i	$0.411384\pi$
$272$	2208.00	0.492205
$273$	1092.00	0.242091
$274$	−1056.00	−0.232830
$275$	4454.00	0.976677
$276$	624.000	0.136088
$277$	−42.0000	−0.00911024	−0.00455512	−	0.999990i	$-0.501450\pi$
$277$	−42.0000	−0.00911024	−0.00455512	+	0.999990i	$0.501450\pi$
$278$	3112.00	0.671386
$279$	−1584.00	−0.339898
$280$	3584.00	0.764946
$281$	−2288.00	−0.485732	−0.242866	−	0.970060i	$-0.578088\pi$
$281$	−2288.00	−0.485732	−0.242866	+	0.970060i	$0.578088\pi$
$282$	2724.00	0.575219
$283$	1156.00	0.242816	0.121408	−	0.992603i	$-0.461259\pi$
$283$	1156.00	0.242816	0.121408	+	0.992603i	$0.461259\pi$
$284$	2200.00	0.459669
$285$	5184.00	1.07745
$286$	884.000	0.182769
$287$	6720.00	1.38212
$288$	−288.000	−0.0589256
$289$	14131.0	2.87625
$290$	−6080.00	−1.23114
$291$	−3066.00	−0.617636
$292$	2456.00	0.492214
$293$	−8684.00	−1.73148	−0.865742	−	0.500491i	$-0.833153\pi$
$293$	−8684.00	−1.73148	−0.865742	+	0.500491i	$0.833153\pi$
$294$	2646.00	0.524891
$295$	2464.00	0.486304
$296$	−2736.00	−0.537253
$297$	−918.000	−0.179353
$298$	−3048.00	−0.592503
$299$	676.000	0.130749
$300$	−1572.00	−0.302532
$301$	−3920.00	−0.750648
$302$	6048.00	1.15240
$303$	3570.00	0.676868
$304$	1728.00	0.326012
$305$	−544.000	−0.102129
$306$	−2484.00	−0.464055
$307$	−7552.00	−1.40396	−0.701979	−	0.712197i	$-0.747700\pi$
$307$	−7552.00	−1.40396	−0.701979	+	0.712197i	$0.747700\pi$
$308$	3808.00	0.704484
$309$	672.000	0.123718
$310$	−5632.00	−1.03186
$311$	2652.00	0.483541	0.241770	−	0.970334i	$-0.422272\pi$
$311$	2652.00	0.483541	0.241770	+	0.970334i	$0.422272\pi$
$312$	−312.000	−0.0566139
$313$	−4426.00	−0.799273	−0.399636	−	0.916674i	$-0.630864\pi$
$313$	−4426.00	−0.799273	−0.399636	+	0.916674i	$0.630864\pi$
$314$	−4396.00	−0.790066
$315$	−4032.00	−0.721198
$316$	32.0000	0.00569665
$317$	−4944.00	−0.875971	−0.437985	−	0.898982i	$-0.644308\pi$
$317$	−4944.00	−0.875971	−0.437985	+	0.898982i	$0.644308\pi$
$318$	1188.00	0.209496
$319$	−6460.00	−1.13383
$320$	−1024.00	−0.178885
$321$	1920.00	0.333844
$322$	2912.00	0.503973
$323$	14904.0	2.56743
$324$	324.000	0.0555556
$325$	−1703.00	−0.290663
$326$	536.000	0.0910623
$327$	5802.00	0.981197
$328$	−1920.00	−0.323214
$329$	12712.0	2.13020
$330$	−3264.00	−0.544477
$331$	−6088.00	−1.01096	−0.505478	−	0.862839i	$-0.668684\pi$
$331$	−6088.00	−1.01096	−0.505478	+	0.862839i	$0.668684\pi$
$332$	3048.00	0.503858
$333$	3078.00	0.506527
$334$	−1404.00	−0.230010
$335$	10496.0	1.71181
$336$	−1344.00	−0.218218
$337$	6638.00	1.07298	0.536491	−	0.843906i	$-0.319750\pi$
$337$	6638.00	1.07298	0.536491	+	0.843906i	$0.319750\pi$
$338$	−338.000	−0.0543928
$339$	1254.00	0.200908
$340$	−8832.00	−1.40877
$341$	−5984.00	−0.950298
$342$	−1944.00	−0.307367
$343$	2744.00	0.431959
$344$	1120.00	0.175542
$345$	−2496.00	−0.389508
$346$	−4132.00	−0.642016
$347$	−2292.00	−0.354585	−0.177293	−	0.984158i	$-0.556734\pi$
$347$	−2292.00	−0.354585	−0.177293	+	0.984158i	$0.556734\pi$
$348$	2280.00	0.351209
$349$	−9866.00	−1.51322	−0.756612	−	0.653865i	$-0.773147\pi$
$349$	−9866.00	−1.51322	−0.756612	+	0.653865i	$0.773147\pi$
$350$	−7336.00	−1.12036
$351$	351.000	0.0533761
$352$	−1088.00	−0.164746
$353$	2368.00	0.357042	0.178521	−	0.983936i	$-0.442869\pi$
$353$	2368.00	0.357042	0.178521	+	0.983936i	$0.442869\pi$
$354$	−924.000	−0.138729
$355$	−8800.00	−1.31565
$356$	−1776.00	−0.264404
$357$	−11592.0	−1.71853
$358$	552.000	0.0814919
$359$	5070.00	0.745360	0.372680	−	0.927960i	$-0.378439\pi$
$359$	5070.00	0.745360	0.372680	+	0.927960i	$0.378439\pi$
$360$	1152.00	0.168655
$361$	4805.00	0.700539
$362$	6948.00	1.00878
$363$	525.000	0.0759101
$364$	−1456.00	−0.209657
$365$	−9824.00	−1.40880
$366$	204.000	0.0291346
$367$	−8584.00	−1.22093	−0.610465	−	0.792043i	$-0.709017\pi$
$367$	−8584.00	−1.22093	−0.610465	+	0.792043i	$0.709017\pi$
$368$	−832.000	−0.117856
$369$	2160.00	0.304729
$370$	10944.0	1.53771
$371$	5544.00	0.775822
$372$	2112.00	0.294360
$373$	4994.00	0.693243	0.346621	−	0.938005i	$-0.387329\pi$
$373$	4994.00	0.693243	0.346621	+	0.938005i	$0.387329\pi$
$374$	−9384.00	−1.29742
$375$	288.000	0.0396593
$376$	−3632.00	−0.498155
$377$	2470.00	0.337431
$378$	1512.00	0.205738
$379$	1300.00	0.176191	0.0880957	−	0.996112i	$-0.471922\pi$
$379$	1300.00	0.176191	0.0880957	+	0.996112i	$0.471922\pi$
$380$	−6912.00	−0.933100
$381$	3120.00	0.419534
$382$	7840.00	1.05008
$383$	−4590.00	−0.612371	−0.306185	−	0.951972i	$-0.599053\pi$
$383$	−4590.00	−0.612371	−0.306185	+	0.951972i	$0.599053\pi$
$384$	384.000	0.0510310
$385$	−15232.0	−2.01635
$386$	−4372.00	−0.576500
$387$	−1260.00	−0.165502
$388$	4088.00	0.534889
$389$	3510.00	0.457491	0.228746	−	0.973486i	$-0.426538\pi$
$389$	3510.00	0.457491	0.228746	+	0.973486i	$0.426538\pi$
$390$	1248.00	0.162038
$391$	−7176.00	−0.928148
$392$	−3528.00	−0.454569
$393$	1704.00	0.218716
$394$	2736.00	0.349842
$395$	−128.000	−0.0163048
$396$	1224.00	0.155324
$397$	6230.00	0.787594	0.393797	−	0.919197i	$-0.371161\pi$
$397$	6230.00	0.787594	0.393797	+	0.919197i	$0.371161\pi$
$398$	2144.00	0.270023
$399$	−9072.00	−1.13827
$400$	2096.00	0.262000
$401$	−7500.00	−0.933995	−0.466998	−	0.884259i	$-0.654664\pi$
$401$	−7500.00	−0.933995	−0.466998	+	0.884259i	$0.654664\pi$
$402$	−3936.00	−0.488333
$403$	2288.00	0.282812
$404$	−4760.00	−0.586185
$405$	−1296.00	−0.159009
$406$	10640.0	1.30063
$407$	11628.0	1.41616
$408$	3312.00	0.401883
$409$	8254.00	0.997883	0.498941	−	0.866636i	$-0.333722\pi$
$409$	8254.00	0.997883	0.498941	+	0.866636i	$0.333722\pi$
$410$	7680.00	0.925093
$411$	−1584.00	−0.190105
$412$	−896.000	−0.107143
$413$	−4312.00	−0.513752
$414$	936.000	0.111116
$415$	−12192.0	−1.44212
$416$	416.000	0.0490290
$417$	4668.00	0.548185
$418$	−7344.00	−0.859346
$419$	−14808.0	−1.72653	−0.863267	−	0.504747i	$-0.831586\pi$
$419$	−14808.0	−1.72653	−0.863267	+	0.504747i	$0.831586\pi$
$420$	5376.00	0.624576
$421$	10354.0	1.19863	0.599315	−	0.800513i	$-0.295440\pi$
$421$	10354.0	1.19863	0.599315	+	0.800513i	$0.295440\pi$
$422$	−10888.0	−1.25597
$423$	4086.00	0.469665
$424$	−1584.00	−0.181429
$425$	18078.0	2.06332
$426$	3300.00	0.375318
$427$	952.000	0.107893
$428$	−2560.00	−0.289117
$429$	1326.00	0.149230
$430$	−4480.00	−0.502430
$431$	−15486.0	−1.73071	−0.865353	−	0.501163i	$-0.832906\pi$
$431$	−15486.0	−1.73071	−0.865353	+	0.501163i	$0.832906\pi$
$432$	−432.000	−0.0481125
$433$	−2018.00	−0.223970	−0.111985	−	0.993710i	$-0.535721\pi$
$433$	−2018.00	−0.223970	−0.111985	+	0.993710i	$0.535721\pi$
$434$	9856.00	1.09010
$435$	−9120.00	−1.00522
$436$	−7736.00	−0.849741
$437$	−5616.00	−0.614759
$438$	3684.00	0.401891
$439$	8792.00	0.955853	0.477926	−	0.878400i	$-0.341389\pi$
$439$	8792.00	0.955853	0.477926	+	0.878400i	$0.341389\pi$
$440$	4352.00	0.471531
$441$	3969.00	0.428571
$442$	3588.00	0.386117
$443$	−2760.00	−0.296008	−0.148004	−	0.988987i	$-0.547285\pi$
$443$	−2760.00	−0.296008	−0.148004	+	0.988987i	$0.547285\pi$
$444$	−4104.00	−0.438665
$445$	7104.00	0.756768
$446$	−192.000	−0.0203844
$447$	−4572.00	−0.483777
$448$	1792.00	0.188982
$449$	9532.00	1.00188	0.500939	−	0.865483i	$-0.332988\pi$
$449$	9532.00	1.00188	0.500939	+	0.865483i	$0.332988\pi$
$450$	−2358.00	−0.247016
$451$	8160.00	0.851972
$452$	−1672.00	−0.173992
$453$	9072.00	0.940927
$454$	−396.000	−0.0409366
$455$	5824.00	0.600073
$456$	2592.00	0.266188
$457$	12862.0	1.31654	0.658270	−	0.752782i	$-0.271288\pi$
$457$	12862.0	1.31654	0.658270	+	0.752782i	$0.271288\pi$
$458$	−11844.0	−1.20837
$459$	−3726.00	−0.378899
$460$	3328.00	0.337323
$461$	−6744.00	−0.681344	−0.340672	−	0.940182i	$-0.610654\pi$
$461$	−6744.00	−0.681344	−0.340672	+	0.940182i	$0.610654\pi$
$462$	5712.00	0.575208
$463$	−9572.00	−0.960796	−0.480398	−	0.877051i	$-0.659508\pi$
$463$	−9572.00	−0.960796	−0.480398	+	0.877051i	$0.659508\pi$
$464$	−3040.00	−0.304156
$465$	−8448.00	−0.842509
$466$	10228.0	1.01674
$467$	9104.00	0.902105	0.451052	−	0.892498i	$-0.351049\pi$
$467$	9104.00	0.902105	0.451052	+	0.892498i	$0.351049\pi$
$468$	−468.000	−0.0462250
$469$	−18368.0	−1.80843
$470$	14528.0	1.42580
$471$	−6594.00	−0.645086
$472$	1232.00	0.120143
$473$	−4760.00	−0.462717
$474$	48.0000	0.00465129
$475$	14148.0	1.36664
$476$	15456.0	1.48829
$477$	1782.00	0.171053
$478$	10452.0	1.00013
$479$	−18870.0	−1.79998	−0.899992	−	0.435906i	$-0.856428\pi$
$479$	−18870.0	−1.79998	−0.899992	+	0.435906i	$0.856428\pi$
$480$	−1536.00	−0.146059
$481$	−4446.00	−0.421456
$482$	1524.00	0.144017
$483$	4368.00	0.411493
$484$	−700.000	−0.0657400
$485$	−16352.0	−1.53094
$486$	486.000	0.0453609
$487$	−1744.00	−0.162276	−0.0811378	−	0.996703i	$-0.525855\pi$
$487$	−1744.00	−0.162276	−0.0811378	+	0.996703i	$0.525855\pi$
$488$	−272.000	−0.0252313
$489$	804.000	0.0743520
$490$	14112.0	1.30105
$491$	13360.0	1.22796	0.613980	−	0.789322i	$-0.289568\pi$
$491$	13360.0	1.22796	0.613980	+	0.789322i	$0.289568\pi$
$492$	−2880.00	−0.263903
$493$	−26220.0	−2.39531
$494$	2808.00	0.255745
$495$	−4896.00	−0.444563
$496$	−2816.00	−0.254924
$497$	15400.0	1.38991
$498$	4572.00	0.411398
$499$	17368.0	1.55811	0.779057	−	0.626954i	$-0.215698\pi$
$499$	17368.0	1.55811	0.779057	+	0.626954i	$0.215698\pi$
$500$	−384.000	−0.0343460
$501$	−2106.00	−0.187803
$502$	−6480.00	−0.576129
$503$	−5828.00	−0.516616	−0.258308	−	0.966063i	$-0.583165\pi$
$503$	−5828.00	−0.516616	−0.258308	+	0.966063i	$0.583165\pi$
$504$	−2016.00	−0.178174
$505$	19040.0	1.67776
$506$	3536.00	0.310661
$507$	−507.000	−0.0444116
$508$	−4160.00	−0.363327
$509$	10744.0	0.935598	0.467799	−	0.883835i	$-0.345047\pi$
$509$	10744.0	0.935598	0.467799	+	0.883835i	$0.345047\pi$
$510$	−13248.0	−1.15026
$511$	17192.0	1.48832
$512$	−512.000	−0.0441942
$513$	−2916.00	−0.250964
$514$	2772.00	0.237875
$515$	3584.00	0.306660
$516$	1680.00	0.143329
$517$	15436.0	1.31310
$518$	−19152.0	−1.62450
$519$	−6198.00	−0.524204
$520$	−1664.00	−0.140329
$521$	−12234.0	−1.02875	−0.514377	−	0.857564i	$-0.671977\pi$
$521$	−12234.0	−1.02875	−0.514377	+	0.857564i	$0.671977\pi$
$522$	3420.00	0.286761
$523$	1812.00	0.151498	0.0757488	−	0.997127i	$-0.475865\pi$
$523$	1812.00	0.151498	0.0757488	+	0.997127i	$0.475865\pi$
$524$	−2272.00	−0.189414
$525$	−11004.0	−0.914769
$526$	−6600.00	−0.547098
$527$	−24288.0	−2.00759
$528$	−1632.00	−0.134515
$529$	−9463.00	−0.777760
$530$	6336.00	0.519280
$531$	−1386.00	−0.113272
$532$	12096.0	0.985767
$533$	−3120.00	−0.253550
$534$	−2664.00	−0.215885
$535$	10240.0	0.827502
$536$	5248.00	0.422909
$537$	828.000	0.0665379
$538$	−8580.00	−0.687565
$539$	14994.0	1.19821
$540$	1728.00	0.137706
$541$	6098.00	0.484609	0.242305	−	0.970200i	$-0.422097\pi$
$541$	6098.00	0.484609	0.242305	+	0.970200i	$0.422097\pi$
$542$	−4904.00	−0.388644
$543$	10422.0	0.823666
$544$	−4416.00	−0.348041
$545$	30944.0	2.43210
$546$	−2184.00	−0.171184
$547$	−18332.0	−1.43294	−0.716471	−	0.697616i	$-0.754244\pi$
$547$	−18332.0	−1.43294	−0.716471	+	0.697616i	$0.754244\pi$
$548$	2112.00	0.164635
$549$	306.000	0.0237883
$550$	−8908.00	−0.690615
$551$	−20520.0	−1.58654
$552$	−1248.00	−0.0962290
$553$	224.000	0.0172250
$554$	84.0000	0.00644191
$555$	16416.0	1.25553
$556$	−6224.00	−0.474742
$557$	−20004.0	−1.52172	−0.760859	−	0.648917i	$-0.775222\pi$
$557$	−20004.0	−1.52172	−0.760859	+	0.648917i	$0.775222\pi$
$558$	3168.00	0.240344
$559$	1820.00	0.137706
$560$	−7168.00	−0.540899
$561$	−14076.0	−1.05934
$562$	4576.00	0.343464
$563$	10988.0	0.822538	0.411269	−	0.911514i	$-0.365086\pi$
$563$	10988.0	0.822538	0.411269	+	0.911514i	$0.365086\pi$
$564$	−5448.00	−0.406741
$565$	6688.00	0.497993
$566$	−2312.00	−0.171697
$567$	2268.00	0.167984
$568$	−4400.00	−0.325035
$569$	11062.0	0.815014	0.407507	−	0.913202i	$-0.366398\pi$
$569$	11062.0	0.815014	0.407507	+	0.913202i	$0.366398\pi$
$570$	−10368.0	−0.761873
$571$	−708.000	−0.0518895	−0.0259447	−	0.999663i	$-0.508259\pi$
$571$	−708.000	−0.0518895	−0.0259447	+	0.999663i	$0.508259\pi$
$572$	−1768.00	−0.129237
$573$	11760.0	0.857384
$574$	−13440.0	−0.977308
$575$	−6812.00	−0.494052
$576$	576.000	0.0416667
$577$	−2094.00	−0.151082	−0.0755410	−	0.997143i	$-0.524068\pi$
$577$	−2094.00	−0.151082	−0.0755410	+	0.997143i	$0.524068\pi$
$578$	−28262.0	−2.03381
$579$	−6558.00	−0.470710
$580$	12160.0	0.870546
$581$	21336.0	1.52352
$582$	6132.00	0.436735
$583$	6732.00	0.478235
$584$	−4912.00	−0.348048
$585$	1872.00	0.132304
$586$	17368.0	1.22434
$587$	−17854.0	−1.25539	−0.627695	−	0.778460i	$-0.716001\pi$
$587$	−17854.0	−1.25539	−0.627695	+	0.778460i	$0.716001\pi$
$588$	−5292.00	−0.371154
$589$	−19008.0	−1.32973
$590$	−4928.00	−0.343869
$591$	4104.00	0.285645
$592$	5472.00	0.379895
$593$	23948.0	1.65839	0.829196	−	0.558958i	$-0.188799\pi$
$593$	23948.0	1.65839	0.829196	+	0.558958i	$0.188799\pi$
$594$	1836.00	0.126822
$595$	−61824.0	−4.25973
$596$	6096.00	0.418963
$597$	3216.00	0.220473
$598$	−1352.00	−0.0924538
$599$	−18068.0	−1.23245	−0.616226	−	0.787570i	$-0.711339\pi$
$599$	−18068.0	−1.23245	−0.616226	+	0.787570i	$0.711339\pi$
$600$	3144.00	0.213922
$601$	19942.0	1.35350	0.676748	−	0.736215i	$-0.263389\pi$
$601$	19942.0	1.35350	0.676748	+	0.736215i	$0.263389\pi$
$602$	7840.00	0.530788
$603$	−5904.00	−0.398722
$604$	−12096.0	−0.814866
$605$	2800.00	0.188159
$606$	−7140.00	−0.478618
$607$	26376.0	1.76370	0.881852	−	0.471526i	$-0.156296\pi$
$607$	26376.0	1.76370	0.881852	+	0.471526i	$0.156296\pi$
$608$	−3456.00	−0.230525
$609$	15960.0	1.06196
$610$	1088.00	0.0722161
$611$	−5902.00	−0.390785
$612$	4968.00	0.328136
$613$	−19426.0	−1.27995	−0.639975	−	0.768396i	$-0.721055\pi$
$613$	−19426.0	−1.27995	−0.639975	+	0.768396i	$0.721055\pi$
$614$	15104.0	0.992749
$615$	11520.0	0.755335
$616$	−7616.00	−0.498145
$617$	−8024.00	−0.523556	−0.261778	−	0.965128i	$-0.584309\pi$
$617$	−8024.00	−0.523556	−0.261778	+	0.965128i	$0.584309\pi$
$618$	−1344.00	−0.0874816
$619$	−20648.0	−1.34073	−0.670366	−	0.742031i	$-0.733863\pi$
$619$	−20648.0	−1.34073	−0.670366	+	0.742031i	$0.733863\pi$
$620$	11264.0	0.729634
$621$	1404.00	0.0907256
$622$	−5304.00	−0.341915
$623$	−12432.0	−0.799482
$624$	624.000	0.0400320
$625$	−14839.0	−0.949696
$626$	8852.00	0.565171
$627$	−11016.0	−0.701653
$628$	8792.00	0.558661
$629$	47196.0	2.99178
$630$	8064.00	0.509964
$631$	12280.0	0.774737	0.387369	−	0.921925i	$-0.373384\pi$
$631$	12280.0	0.774737	0.387369	+	0.921925i	$0.373384\pi$
$632$	−64.0000	−0.00402814
$633$	−16332.0	−1.02550
$634$	9888.00	0.619405
$635$	16640.0	1.03990
$636$	−2376.00	−0.148136
$637$	−5733.00	−0.356593
$638$	12920.0	0.801736
$639$	4950.00	0.306446
$640$	2048.00	0.126491
$641$	−15878.0	−0.978383	−0.489191	−	0.872176i	$-0.662708\pi$
$641$	−15878.0	−0.978383	−0.489191	+	0.872176i	$0.662708\pi$
$642$	−3840.00	−0.236063
$643$	−21520.0	−1.31985	−0.659927	−	0.751330i	$-0.729413\pi$
$643$	−21520.0	−1.31985	−0.659927	+	0.751330i	$0.729413\pi$
$644$	−5824.00	−0.356363
$645$	−6720.00	−0.410232
$646$	−29808.0	−1.81545
$647$	7312.00	0.444304	0.222152	−	0.975012i	$-0.428692\pi$
$647$	7312.00	0.444304	0.222152	+	0.975012i	$0.428692\pi$
$648$	−648.000	−0.0392837
$649$	−5236.00	−0.316689
$650$	3406.00	0.205530
$651$	14784.0	0.890062
$652$	−1072.00	−0.0643907
$653$	3090.00	0.185178	0.0925889	−	0.995704i	$-0.470486\pi$
$653$	3090.00	0.185178	0.0925889	+	0.995704i	$0.470486\pi$
$654$	−11604.0	−0.693811
$655$	9088.00	0.542134
$656$	3840.00	0.228547
$657$	5526.00	0.328143
$658$	−25424.0	−1.50628
$659$	−13428.0	−0.793749	−0.396875	−	0.917873i	$-0.629905\pi$
$659$	−13428.0	−0.793749	−0.396875	+	0.917873i	$0.629905\pi$
$660$	6528.00	0.385003
$661$	22598.0	1.32974	0.664872	−	0.746958i	$-0.268486\pi$
$661$	22598.0	1.32974	0.664872	+	0.746958i	$0.268486\pi$
$662$	12176.0	0.714854
$663$	5382.00	0.315263
$664$	−6096.00	−0.356281
$665$	−48384.0	−2.82143
$666$	−6156.00	−0.358168
$667$	9880.00	0.573546
$668$	2808.00	0.162642
$669$	−288.000	−0.0166438
$670$	−20992.0	−1.21044
$671$	1156.00	0.0665080
$672$	2688.00	0.154303
$673$	6178.00	0.353855	0.176927	−	0.984224i	$-0.443384\pi$
$673$	6178.00	0.353855	0.176927	+	0.984224i	$0.443384\pi$
$674$	−13276.0	−0.758713
$675$	−3537.00	−0.201688
$676$	676.000	0.0384615
$677$	22398.0	1.27153	0.635764	−	0.771883i	$-0.280685\pi$
$677$	22398.0	1.27153	0.635764	+	0.771883i	$0.280685\pi$
$678$	−2508.00	−0.142064
$679$	28616.0	1.61735
$680$	17664.0	0.996152
$681$	−594.000	−0.0334246
$682$	11968.0	0.671962
$683$	11410.0	0.639226	0.319613	−	0.947548i	$-0.396447\pi$
$683$	11410.0	0.639226	0.319613	+	0.947548i	$0.396447\pi$
$684$	3888.00	0.217341
$685$	−8448.00	−0.471214
$686$	−5488.00	−0.305441
$687$	−17766.0	−0.986631
$688$	−2240.00	−0.124127
$689$	−2574.00	−0.142325
$690$	4992.00	0.275423
$691$	32488.0	1.78857	0.894285	−	0.447498i	$-0.147685\pi$
$691$	32488.0	1.78857	0.894285	+	0.447498i	$0.147685\pi$
$692$	8264.00	0.453974
$693$	8568.00	0.469656
$694$	4584.00	0.250729
$695$	24896.0	1.35879
$696$	−4560.00	−0.248342
$697$	33120.0	1.79987
$698$	19732.0	1.07001
$699$	15342.0	0.830168
$700$	14672.0	0.792214
$701$	5094.00	0.274462	0.137231	−	0.990539i	$-0.456180\pi$
$701$	5094.00	0.274462	0.137231	+	0.990539i	$0.456180\pi$
$702$	−702.000	−0.0377426
$703$	36936.0	1.98160
$704$	2176.00	0.116493
$705$	21792.0	1.16416
$706$	−4736.00	−0.252467
$707$	−33320.0	−1.77246
$708$	1848.00	0.0980962
$709$	25418.0	1.34639	0.673197	−	0.739463i	$-0.264921\pi$
$709$	25418.0	1.34639	0.673197	+	0.739463i	$0.264921\pi$
$710$	17600.0	0.930305
$711$	72.0000	0.00379777
$712$	3552.00	0.186962
$713$	9152.00	0.480708
$714$	23184.0	1.21518
$715$	7072.00	0.369899
$716$	−1104.00	−0.0576235
$717$	15678.0	0.816605
$718$	−10140.0	−0.527049
$719$	−20428.0	−1.05958	−0.529788	−	0.848130i	$-0.677729\pi$
$719$	−20428.0	−1.05958	−0.529788	+	0.848130i	$0.677729\pi$
$720$	−2304.00	−0.119257
$721$	−6272.00	−0.323969
$722$	−9610.00	−0.495356
$723$	2286.00	0.117590
$724$	−13896.0	−0.713316
$725$	−24890.0	−1.27502
$726$	−1050.00	−0.0536765
$727$	−38336.0	−1.95571	−0.977857	−	0.209276i	$-0.932889\pi$
$727$	−38336.0	−1.95571	−0.977857	+	0.209276i	$0.932889\pi$
$728$	2912.00	0.148250
$729$	729.000	0.0370370
$730$	19648.0	0.996171
$731$	−19320.0	−0.977532
$732$	−408.000	−0.0206012
$733$	−166.000	−0.00836473	−0.00418237	−	0.999991i	$-0.501331\pi$
$733$	−166.000	−0.00836473	−0.00418237	+	0.999991i	$0.501331\pi$
$734$	17168.0	0.863328
$735$	21168.0	1.06230
$736$	1664.00	0.0833368
$737$	−22304.0	−1.11476
$738$	−4320.00	−0.215476
$739$	−25248.0	−1.25678	−0.628392	−	0.777897i	$-0.716286\pi$
$739$	−25248.0	−1.25678	−0.628392	+	0.777897i	$0.716286\pi$
$740$	−21888.0	−1.08732
$741$	4212.00	0.208815
$742$	−11088.0	−0.548589
$743$	−4442.00	−0.219329	−0.109664	−	0.993969i	$-0.534978\pi$
$743$	−4442.00	−0.219329	−0.109664	+	0.993969i	$0.534978\pi$
$744$	−4224.00	−0.208144
$745$	−24384.0	−1.19914
$746$	−9988.00	−0.490197
$747$	6858.00	0.335905
$748$	18768.0	0.917414
$749$	−17920.0	−0.874209
$750$	−576.000	−0.0280434
$751$	−19848.0	−0.964399	−0.482200	−	0.876061i	$-0.660162\pi$
$751$	−19848.0	−0.964399	−0.482200	+	0.876061i	$0.660162\pi$
$752$	7264.00	0.352248
$753$	−9720.00	−0.470407
$754$	−4940.00	−0.238600
$755$	48384.0	2.33228
$756$	−3024.00	−0.145479
$757$	−29166.0	−1.40034	−0.700169	−	0.713977i	$-0.746892\pi$
$757$	−29166.0	−1.40034	−0.700169	+	0.713977i	$0.746892\pi$
$758$	−2600.00	−0.124586
$759$	5304.00	0.253653
$760$	13824.0	0.659802
$761$	−6240.00	−0.297240	−0.148620	−	0.988894i	$-0.547483\pi$
$761$	−6240.00	−0.297240	−0.148620	+	0.988894i	$0.547483\pi$
$762$	−6240.00	−0.296655
$763$	−54152.0	−2.56938
$764$	−15680.0	−0.742516
$765$	−19872.0	−0.939181
$766$	9180.00	0.433012
$767$	2002.00	0.0942478
$768$	−768.000	−0.0360844
$769$	−39750.0	−1.86401	−0.932004	−	0.362449i	$-0.881941\pi$
$769$	−39750.0	−1.86401	−0.932004	+	0.362449i	$0.881941\pi$
$770$	30464.0	1.42577
$771$	4158.00	0.194224
$772$	8744.00	0.407647
$773$	9764.00	0.454317	0.227158	−	0.973858i	$-0.427057\pi$
$773$	9764.00	0.454317	0.227158	+	0.973858i	$0.427057\pi$
$774$	2520.00	0.117028
$775$	−23056.0	−1.06864
$776$	−8176.00	−0.378223
$777$	−28728.0	−1.32640
$778$	−7020.00	−0.323495
$779$	25920.0	1.19214
$780$	−2496.00	−0.114578
$781$	18700.0	0.856772
$782$	14352.0	0.656300
$783$	5130.00	0.234140
$784$	7056.00	0.321429
$785$	−35168.0	−1.59898
$786$	−3408.00	−0.154656
$787$	−36016.0	−1.63130	−0.815649	−	0.578547i	$-0.803620\pi$
$787$	−36016.0	−1.63130	−0.815649	+	0.578547i	$0.803620\pi$
$788$	−5472.00	−0.247376
$789$	−9900.00	−0.446704
$790$	256.000	0.0115292
$791$	−11704.0	−0.526102
$792$	−2448.00	−0.109831
$793$	−442.000	−0.0197930
$794$	−12460.0	−0.556913
$795$	9504.00	0.423990
$796$	−4288.00	−0.190935
$797$	−22290.0	−0.990655	−0.495328	−	0.868706i	$-0.664952\pi$
$797$	−22290.0	−0.990655	−0.495328	+	0.868706i	$0.664952\pi$
$798$	18144.0	0.804875
$799$	62652.0	2.77405
$800$	−4192.00	−0.185262
$801$	−3996.00	−0.176269
$802$	15000.0	0.660434
$803$	20876.0	0.917432
$804$	7872.00	0.345304
$805$	23296.0	1.01997
$806$	−4576.00	−0.199979
$807$	−12870.0	−0.561395
$808$	9520.00	0.414496
$809$	−25578.0	−1.11159	−0.555794	−	0.831320i	$-0.687586\pi$
$809$	−25578.0	−1.11159	−0.555794	+	0.831320i	$0.687586\pi$
$810$	2592.00	0.112437
$811$	29900.0	1.29461	0.647306	−	0.762230i	$-0.275895\pi$
$811$	29900.0	1.29461	0.647306	+	0.762230i	$0.275895\pi$
$812$	−21280.0	−0.919682
$813$	−7356.00	−0.317326
$814$	−23256.0	−1.00138
$815$	4288.00	0.184297
$816$	−6624.00	−0.284174
$817$	−15120.0	−0.647469
$818$	−16508.0	−0.705610
$819$	−3276.00	−0.139771
$820$	−15360.0	−0.654140
$821$	16412.0	0.697665	0.348832	−	0.937185i	$-0.386578\pi$
$821$	16412.0	0.697665	0.348832	+	0.937185i	$0.386578\pi$
$822$	3168.00	0.134424
$823$	18552.0	0.785762	0.392881	−	0.919589i	$-0.371478\pi$
$823$	18552.0	0.785762	0.392881	+	0.919589i	$0.371478\pi$
$824$	1792.00	0.0757613
$825$	−13362.0	−0.563885
$826$	8624.00	0.363278
$827$	−28662.0	−1.20517	−0.602585	−	0.798055i	$-0.705863\pi$
$827$	−28662.0	−1.20517	−0.602585	+	0.798055i	$0.705863\pi$
$828$	−1872.00	−0.0785706
$829$	−3686.00	−0.154427	−0.0772136	−	0.997015i	$-0.524602\pi$
$829$	−3686.00	−0.154427	−0.0772136	+	0.997015i	$0.524602\pi$
$830$	24384.0	1.01974
$831$	126.000	0.00525980
$832$	−832.000	−0.0346688
$833$	60858.0	2.53134
$834$	−9336.00	−0.387625
$835$	−11232.0	−0.465508
$836$	14688.0	0.607650
$837$	4752.00	0.196240
$838$	29616.0	1.22084
$839$	13370.0	0.550159	0.275080	−	0.961421i	$-0.411296\pi$
$839$	13370.0	0.550159	0.275080	+	0.961421i	$0.411296\pi$
$840$	−10752.0	−0.441642
$841$	11711.0	0.480175
$842$	−20708.0	−0.847559
$843$	6864.00	0.280437
$844$	21776.0	0.888105
$845$	−2704.00	−0.110083
$846$	−8172.00	−0.332103
$847$	−4900.00	−0.198779
$848$	3168.00	0.128290
$849$	−3468.00	−0.140190
$850$	−36156.0	−1.45899
$851$	−17784.0	−0.716366
$852$	−6600.00	−0.265390
$853$	11398.0	0.457515	0.228757	−	0.973483i	$-0.426534\pi$
$853$	11398.0	0.457515	0.228757	+	0.973483i	$0.426534\pi$
$854$	−1904.00	−0.0762922
$855$	−15552.0	−0.622067
$856$	5120.00	0.204437
$857$	7990.00	0.318475	0.159238	−	0.987240i	$-0.449096\pi$
$857$	7990.00	0.318475	0.159238	+	0.987240i	$0.449096\pi$
$858$	−2652.00	−0.105522
$859$	7652.00	0.303938	0.151969	−	0.988385i	$-0.451439\pi$
$859$	7652.00	0.303938	0.151969	+	0.988385i	$0.451439\pi$
$860$	8960.00	0.355271
$861$	−20160.0	−0.797969
$862$	30972.0	1.22379
$863$	−1022.00	−0.0403120	−0.0201560	−	0.999797i	$-0.506416\pi$
$863$	−1022.00	−0.0403120	−0.0201560	+	0.999797i	$0.506416\pi$
$864$	864.000	0.0340207
$865$	−33056.0	−1.29935
$866$	4036.00	0.158371
$867$	−42393.0	−1.66060
$868$	−19712.0	−0.770817
$869$	272.000	0.0106179
$870$	18240.0	0.710798
$871$	8528.00	0.331757
$872$	15472.0	0.600858
$873$	9198.00	0.356592
$874$	11232.0	0.434700
$875$	−2688.00	−0.103853
$876$	−7368.00	−0.284180
$877$	−15546.0	−0.598576	−0.299288	−	0.954163i	$-0.596749\pi$
$877$	−15546.0	−0.598576	−0.299288	+	0.954163i	$0.596749\pi$
$878$	−17584.0	−0.675890
$879$	26052.0	0.999673
$880$	−8704.00	−0.333422
$881$	11310.0	0.432513	0.216256	−	0.976337i	$-0.430615\pi$
$881$	11310.0	0.432513	0.216256	+	0.976337i	$0.430615\pi$
$882$	−7938.00	−0.303046
$883$	17260.0	0.657809	0.328904	−	0.944363i	$-0.393321\pi$
$883$	17260.0	0.657809	0.328904	+	0.944363i	$0.393321\pi$
$884$	−7176.00	−0.273026
$885$	−7392.00	−0.280768
$886$	5520.00	0.209309
$887$	832.000	0.0314947	0.0157474	−	0.999876i	$-0.494987\pi$
$887$	832.000	0.0314947	0.0157474	+	0.999876i	$0.494987\pi$
$888$	8208.00	0.310183
$889$	−29120.0	−1.09860
$890$	−14208.0	−0.535116
$891$	2754.00	0.103549
$892$	384.000	0.0144140
$893$	49032.0	1.83739
$894$	9144.00	0.342082
$895$	4416.00	0.164928
$896$	−3584.00	−0.133631
$897$	−2028.00	−0.0754882
$898$	−19064.0	−0.708434
$899$	33440.0	1.24059
$900$	4716.00	0.174667
$901$	27324.0	1.01032
$902$	−16320.0	−0.602435
$903$	11760.0	0.433387
$904$	3344.00	0.123031
$905$	55584.0	2.04163
$906$	−18144.0	−0.665336
$907$	31740.0	1.16197	0.580986	−	0.813913i	$-0.302667\pi$
$907$	31740.0	1.16197	0.580986	+	0.813913i	$0.302667\pi$
$908$	792.000	0.0289465
$909$	−10710.0	−0.390790
$910$	−11648.0	−0.424316
$911$	−23568.0	−0.857127	−0.428563	−	0.903512i	$-0.640980\pi$
$911$	−23568.0	−0.857127	−0.428563	+	0.903512i	$0.640980\pi$
$912$	−5184.00	−0.188223
$913$	25908.0	0.939134
$914$	−25724.0	−0.930935
$915$	1632.00	0.0589642
$916$	23688.0	0.854447
$917$	−15904.0	−0.572733
$918$	7452.00	0.267922
$919$	−18864.0	−0.677112	−0.338556	−	0.940946i	$-0.609938\pi$
$919$	−18864.0	−0.677112	−0.338556	+	0.940946i	$0.609938\pi$
$920$	−6656.00	−0.238524
$921$	22656.0	0.810576
$922$	13488.0	0.481783
$923$	−7150.00	−0.254978
$924$	−11424.0	−0.406734
$925$	44802.0	1.59252
$926$	19144.0	0.679385
$927$	−2016.00	−0.0714284
$928$	6080.00	0.215071
$929$	−19536.0	−0.689941	−0.344971	−	0.938613i	$-0.612111\pi$
$929$	−19536.0	−0.689941	−0.344971	+	0.938613i	$0.612111\pi$
$930$	16896.0	0.595744
$931$	47628.0	1.67663
$932$	−20456.0	−0.718947
$933$	−7956.00	−0.279172
$934$	−18208.0	−0.637884
$935$	−75072.0	−2.62579
$936$	936.000	0.0326860
$937$	18174.0	0.633638	0.316819	−	0.948486i	$-0.397385\pi$
$937$	18174.0	0.633638	0.316819	+	0.948486i	$0.397385\pi$
$938$	36736.0	1.27876
$939$	13278.0	0.461460
$940$	−29056.0	−1.00819
$941$	51172.0	1.77275	0.886376	−	0.462966i	$-0.153215\pi$
$941$	51172.0	1.77275	0.886376	+	0.462966i	$0.153215\pi$
$942$	13188.0	0.456145
$943$	−12480.0	−0.430970
$944$	−2464.00	−0.0849538
$945$	12096.0	0.416384
$946$	9520.00	0.327190
$947$	3726.00	0.127855	0.0639275	−	0.997955i	$-0.479637\pi$
$947$	3726.00	0.127855	0.0639275	+	0.997955i	$0.479637\pi$
$948$	−96.0000	−0.00328896
$949$	−7982.00	−0.273031
$950$	−28296.0	−0.966362
$951$	14832.0	0.505742
$952$	−30912.0	−1.05238
$953$	−40498.0	−1.37656	−0.688279	−	0.725447i	$-0.741633\pi$
$953$	−40498.0	−1.37656	−0.688279	+	0.725447i	$0.741633\pi$
$954$	−3564.00	−0.120953
$955$	62720.0	2.12521
$956$	−20904.0	−0.707200
$957$	19380.0	0.654615
$958$	37740.0	1.27278
$959$	14784.0	0.497810
$960$	3072.00	0.103280
$961$	1185.00	0.0397771
$962$	8892.00	0.298014
$963$	−5760.00	−0.192745
$964$	−3048.00	−0.101836
$965$	−34976.0	−1.16675
$966$	−8736.00	−0.290969
$967$	28568.0	0.950036	0.475018	−	0.879976i	$-0.342442\pi$
$967$	28568.0	0.950036	0.475018	+	0.879976i	$0.342442\pi$
$968$	1400.00	0.0464852
$969$	−44712.0	−1.48231
$970$	32704.0	1.08254
$971$	−8676.00	−0.286742	−0.143371	−	0.989669i	$-0.545794\pi$
$971$	−8676.00	−0.286742	−0.143371	+	0.989669i	$0.545794\pi$
$972$	−972.000	−0.0320750
$973$	−43568.0	−1.43548
$974$	3488.00	0.114746
$975$	5109.00	0.167814
$976$	544.000	0.0178412
$977$	−2796.00	−0.0915578	−0.0457789	−	0.998952i	$-0.514577\pi$
$977$	−2796.00	−0.0915578	−0.0457789	+	0.998952i	$0.514577\pi$
$978$	−1608.00	−0.0525748
$979$	−15096.0	−0.492819
$980$	−28224.0	−0.919982
$981$	−17406.0	−0.566494
$982$	−26720.0	−0.868299
$983$	−406.000	−0.0131733	−0.00658667	−	0.999978i	$-0.502097\pi$
$983$	−406.000	−0.0131733	−0.00658667	+	0.999978i	$0.502097\pi$
$984$	5760.00	0.186608
$985$	21888.0	0.708030
$986$	52440.0	1.69374
$987$	−38136.0	−1.22987
$988$	−5616.00	−0.180839
$989$	7280.00	0.234065
$990$	9792.00	0.314354
$991$	−23232.0	−0.744691	−0.372346	−	0.928094i	$-0.621446\pi$
$991$	−23232.0	−0.744691	−0.372346	+	0.928094i	$0.621446\pi$
$992$	5632.00	0.180258
$993$	18264.0	0.583676
$994$	−30800.0	−0.982814
$995$	17152.0	0.546487
$996$	−9144.00	−0.290902
$997$	6110.00	0.194088	0.0970440	−	0.995280i	$-0.469061\pi$
$997$	6110.00	0.194088	0.0970440	+	0.995280i	$0.469061\pi$
$998$	−34736.0	−1.10175
$999$	−9234.00	−0.292443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.4.a.a.1.1	✓	1
3.2	odd	2		234.4.a.k.1.1		1
4.3	odd	2		624.4.a.f.1.1		1
5.4	even	2		1950.4.a.o.1.1		1
8.3	odd	2		2496.4.a.g.1.1		1
8.5	even	2		2496.4.a.q.1.1		1
12.11	even	2		1872.4.a.o.1.1		1
13.5	odd	4		1014.4.b.a.337.2		2
13.8	odd	4		1014.4.b.a.337.1		2
13.12	even	2		1014.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.a.1.1	✓	1	1.1	even	1	trivial
234.4.a.k.1.1		1	3.2	odd	2
624.4.a.f.1.1		1	4.3	odd	2
1014.4.a.i.1.1		1	13.12	even	2
1014.4.b.a.337.1		2	13.8	odd	4
1014.4.b.a.337.2		2	13.5	odd	4
1872.4.a.o.1.1		1	12.11	even	2
1950.4.a.o.1.1		1	5.4	even	2
2496.4.a.g.1.1		1	8.3	odd	2
2496.4.a.q.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 $\F_{q}$
Belyi maps

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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