LMFDB - Embedded newform 50.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,4,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	50.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:	$2.95009550029$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -16.0000 q^{6} +4.00000 q^{7} -8.00000 q^{8} +37.0000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -16.0000 q^{6} +4.00000 q^{7} -8.00000 q^{8} +37.0000 q^{9} +12.0000 q^{11} +32.0000 q^{12} +58.0000 q^{13} -8.00000 q^{14} +16.0000 q^{16} -66.0000 q^{17} -74.0000 q^{18} -100.000 q^{19} +32.0000 q^{21} -24.0000 q^{22} -132.000 q^{23} -64.0000 q^{24} -116.000 q^{26} +80.0000 q^{27} +16.0000 q^{28} -90.0000 q^{29} +152.000 q^{31} -32.0000 q^{32} +96.0000 q^{33} +132.000 q^{34} +148.000 q^{36} +34.0000 q^{37} +200.000 q^{38} +464.000 q^{39} -438.000 q^{41} -64.0000 q^{42} -32.0000 q^{43} +48.0000 q^{44} +264.000 q^{46} +204.000 q^{47} +128.000 q^{48} -327.000 q^{49} -528.000 q^{51} +232.000 q^{52} -222.000 q^{53} -160.000 q^{54} -32.0000 q^{56} -800.000 q^{57} +180.000 q^{58} +420.000 q^{59} +902.000 q^{61} -304.000 q^{62} +148.000 q^{63} +64.0000 q^{64} -192.000 q^{66} +1024.00 q^{67} -264.000 q^{68} -1056.00 q^{69} +432.000 q^{71} -296.000 q^{72} -362.000 q^{73} -68.0000 q^{74} -400.000 q^{76} +48.0000 q^{77} -928.000 q^{78} -160.000 q^{79} -359.000 q^{81} +876.000 q^{82} -72.0000 q^{83} +128.000 q^{84} +64.0000 q^{86} -720.000 q^{87} -96.0000 q^{88} +810.000 q^{89} +232.000 q^{91} -528.000 q^{92} +1216.00 q^{93} -408.000 q^{94} -256.000 q^{96} -1106.00 q^{97} +654.000 q^{98} +444.000 q^{99} +O(q^{100})\)

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−16.0000	−1.08866
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	−8.00000	−0.353553
$9$	37.0000	1.37037
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	32.0000	0.769800
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	−8.00000	−0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	−74.0000	−0.968998
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	32.0000	0.332522
$22$	−24.0000	−0.232583
$23$	−132.000	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$23$	−132.000	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$24$	−64.0000	−0.544331
$25$	0	0
$26$	−116.000	−0.874980
$27$	80.0000	0.570222
$28$	16.0000	0.107990
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	−32.0000	−0.176777
$33$	96.0000	0.506408
$34$	132.000	0.665818
$35$	0	0
$36$	148.000	0.685185
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	200.000	0.853797
$39$	464.000	1.90511
$40$	0	0
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−438.000	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	−64.0000	−0.235129
$43$	−32.0000	−0.113487	−0.0567437	−	0.998389i	$-0.518072\pi$
$43$	−32.0000	−0.113487	−0.0567437	+	0.998389i	$0.518072\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	264.000	0.846189
$47$	204.000	0.633116	0.316558	−	0.948573i	$-0.397473\pi$
$47$	204.000	0.633116	0.316558	+	0.948573i	$0.397473\pi$
$48$	128.000	0.384900
$49$	−327.000	−0.953353
$50$	0	0
$51$	−528.000	−1.44970
$52$	232.000	0.618704
$53$	−222.000	−0.575359	−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000	−0.575359	−0.287680	+	0.957727i	$0.592884\pi$
$54$	−160.000	−0.403208
$55$	0	0
$56$	−32.0000	−0.0763604
$57$	−800.000	−1.85899
$58$	180.000	0.407503
$59$	420.000	0.926769	0.463384	−	0.886157i	$-0.346635\pi$
$59$	420.000	0.926769	0.463384	+	0.886157i	$0.346635\pi$
$60$	0	0
$61$	902.000	1.89327	0.946633	−	0.322312i	$-0.104460\pi$
$61$	902.000	1.89327	0.946633	+	0.322312i	$0.104460\pi$
$62$	−304.000	−0.622710
$63$	148.000	0.295972
$64$	64.0000	0.125000
$65$	0	0
$66$	−192.000	−0.358084
$67$	1024.00	1.86719	0.933593	−	0.358334i	$-0.116655\pi$
$67$	1024.00	1.86719	0.933593	+	0.358334i	$0.116655\pi$
$68$	−264.000	−0.470804
$69$	−1056.00	−1.84243
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	−296.000	−0.484499
$73$	−362.000	−0.580396	−0.290198	−	0.956967i	$-0.593721\pi$
$73$	−362.000	−0.580396	−0.290198	+	0.956967i	$0.593721\pi$
$74$	−68.0000	−0.106822
$75$	0	0
$76$	−400.000	−0.603726
$77$	48.0000	0.0710404
$78$	−928.000	−1.34712
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	876.000	1.17973
$83$	−72.0000	−0.0952172	−0.0476086	−	0.998866i	$-0.515160\pi$
$83$	−72.0000	−0.0952172	−0.0476086	+	0.998866i	$0.515160\pi$
$84$	128.000	0.166261
$85$	0	0
$86$	64.0000	0.0802476
$87$	−720.000	−0.887266
$88$	−96.0000	−0.116291
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	232.000	0.267255
$92$	−528.000	−0.598346
$93$	1216.00	1.35584
$94$	−408.000	−0.447681
$95$	0	0
$96$	−256.000	−0.272166
$97$	−1106.00	−1.15770	−0.578852	−	0.815433i	$-0.696499\pi$
$97$	−1106.00	−1.15770	−0.578852	+	0.815433i	$0.696499\pi$
$98$	654.000	0.674122
$99$	444.000	0.450744
$100$	0	0
$101$	−258.000	−0.254178	−0.127089	−	0.991891i	$-0.540563\pi$
$101$	−258.000	−0.254178	−0.127089	+	0.991891i	$0.540563\pi$
$102$	1056.00	1.02509
$103$	988.000	0.945151	0.472575	−	0.881290i	$-0.343324\pi$
$103$	988.000	0.945151	0.472575	+	0.881290i	$0.343324\pi$
$104$	−464.000	−0.437490
$105$	0	0
$106$	444.000	0.406840
$107$	24.0000	0.0216838	0.0108419	−	0.999941i	$-0.496549\pi$
$107$	24.0000	0.0216838	0.0108419	+	0.999941i	$0.496549\pi$
$108$	320.000	0.285111
$109$	950.000	0.834803	0.417401	−	0.908722i	$-0.362941\pi$
$109$	950.000	0.834803	0.417401	+	0.908722i	$0.362941\pi$
$110$	0	0
$111$	272.000	0.232586
$112$	64.0000	0.0539949
$113$	1038.00	0.864131	0.432066	−	0.901842i	$-0.357785\pi$
$113$	1038.00	0.864131	0.432066	+	0.901842i	$0.357785\pi$
$114$	1600.00	1.31451
$115$	0	0
$116$	−360.000	−0.288148
$117$	2146.00	1.69571
$118$	−840.000	−0.655324
$119$	−264.000	−0.203368
$120$	0	0
$121$	−1187.00	−0.891811
$122$	−1804.00	−1.33874
$123$	−3504.00	−2.56866
$124$	608.000	0.440323
$125$	0	0
$126$	−296.000	−0.209284
$127$	124.000	0.0866395	0.0433198	−	0.999061i	$-0.486207\pi$
$127$	124.000	0.0866395	0.0433198	+	0.999061i	$0.486207\pi$
$128$	−128.000	−0.0883883
$129$	−256.000	−0.174725
$130$	0	0
$131$	132.000	0.0880374	0.0440187	−	0.999031i	$-0.485984\pi$
$131$	132.000	0.0880374	0.0440187	+	0.999031i	$0.485984\pi$
$132$	384.000	0.253204
$133$	−400.000	−0.260785
$134$	−2048.00	−1.32030
$135$	0	0
$136$	528.000	0.332909
$137$	1254.00	0.782018	0.391009	−	0.920387i	$-0.372126\pi$
$137$	1254.00	0.782018	0.391009	+	0.920387i	$0.372126\pi$
$138$	2112.00	1.30279
$139$	−2860.00	−1.74519	−0.872597	−	0.488440i	$-0.837566\pi$
$139$	−2860.00	−1.74519	−0.872597	+	0.488440i	$0.837566\pi$
$140$	0	0
$141$	1632.00	0.974746
$142$	−864.000	−0.510600
$143$	696.000	0.407010
$144$	592.000	0.342593
$145$	0	0
$146$	724.000	0.410402
$147$	−2616.00	−1.46778
$148$	136.000	0.0755347
$149$	750.000	0.412365	0.206183	−	0.978514i	$-0.433896\pi$
$149$	750.000	0.412365	0.206183	+	0.978514i	$0.433896\pi$
$150$	0	0
$151$	−448.000	−0.241442	−0.120721	−	0.992686i	$-0.538521\pi$
$151$	−448.000	−0.241442	−0.120721	+	0.992686i	$0.538521\pi$
$152$	800.000	0.426898
$153$	−2442.00	−1.29035
$154$	−96.0000	−0.0502331
$155$	0	0
$156$	1856.00	0.952557
$157$	−2246.00	−1.14172	−0.570861	−	0.821047i	$-0.693390\pi$
$157$	−2246.00	−1.14172	−0.570861	+	0.821047i	$0.693390\pi$
$158$	320.000	0.161126
$159$	−1776.00	−0.885824
$160$	0	0
$161$	−528.000	−0.258461
$162$	718.000	0.348219
$163$	568.000	0.272940	0.136470	−	0.990644i	$-0.456424\pi$
$163$	568.000	0.272940	0.136470	+	0.990644i	$0.456424\pi$
$164$	−1752.00	−0.834196
$165$	0	0
$166$	144.000	0.0673287
$167$	1524.00	0.706172	0.353086	−	0.935591i	$-0.385132\pi$
$167$	1524.00	0.706172	0.353086	+	0.935591i	$0.385132\pi$
$168$	−256.000	−0.117564
$169$	1167.00	0.531179
$170$	0	0
$171$	−3700.00	−1.65466
$172$	−128.000	−0.0567437
$173$	−3702.00	−1.62692	−0.813462	−	0.581618i	$-0.802420\pi$
$173$	−3702.00	−1.62692	−0.813462	+	0.581618i	$0.802420\pi$
$174$	1440.00	0.627391
$175$	0	0
$176$	192.000	0.0822304
$177$	3360.00	1.42685
$178$	−1620.00	−0.682158
$179$	3180.00	1.32785	0.663923	−	0.747801i	$-0.268890\pi$
$179$	3180.00	1.32785	0.663923	+	0.747801i	$0.268890\pi$
$180$	0	0
$181$	−2098.00	−0.861564	−0.430782	−	0.902456i	$-0.641762\pi$
$181$	−2098.00	−0.861564	−0.430782	+	0.902456i	$0.641762\pi$
$182$	−464.000	−0.188978
$183$	7216.00	2.91487
$184$	1056.00	0.423094
$185$	0	0
$186$	−2432.00	−0.958725
$187$	−792.000	−0.309715
$188$	816.000	0.316558
$189$	320.000	0.123156
$190$	0	0
$191$	4392.00	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$191$	4392.00	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$192$	512.000	0.192450
$193$	2158.00	0.804851	0.402425	−	0.915453i	$-0.368167\pi$
$193$	2158.00	0.804851	0.402425	+	0.915453i	$0.368167\pi$
$194$	2212.00	0.818620
$195$	0	0
$196$	−1308.00	−0.476676
$197$	1074.00	0.388423	0.194212	−	0.980960i	$-0.437785\pi$
$197$	1074.00	0.388423	0.194212	+	0.980960i	$0.437785\pi$
$198$	−888.000	−0.318724
$199$	2840.00	1.01167	0.505835	−	0.862630i	$-0.331185\pi$
$199$	2840.00	1.01167	0.505835	+	0.862630i	$0.331185\pi$
$200$	0	0
$201$	8192.00	2.87472
$202$	516.000	0.179731
$203$	−360.000	−0.124468
$204$	−2112.00	−0.724851
$205$	0	0
$206$	−1976.00	−0.668323
$207$	−4884.00	−1.63991
$208$	928.000	0.309352
$209$	−1200.00	−0.397157
$210$	0	0
$211$	−2668.00	−0.870487	−0.435243	−	0.900313i	$-0.643338\pi$
$211$	−2668.00	−0.870487	−0.435243	+	0.900313i	$0.643338\pi$
$212$	−888.000	−0.287680
$213$	3456.00	1.11174
$214$	−48.0000	−0.0153328
$215$	0	0
$216$	−640.000	−0.201604
$217$	608.000	0.190202
$218$	−1900.00	−0.590295
$219$	−2896.00	−0.893578
$220$	0	0
$221$	−3828.00	−1.16515
$222$	−544.000	−0.164463
$223$	−1772.00	−0.532116	−0.266058	−	0.963957i	$-0.585721\pi$
$223$	−1772.00	−0.532116	−0.266058	+	0.963957i	$0.585721\pi$
$224$	−128.000	−0.0381802
$225$	0	0
$226$	−2076.00	−0.611033
$227$	2784.00	0.814011	0.407006	−	0.913426i	$-0.366573\pi$
$227$	2784.00	0.814011	0.407006	+	0.913426i	$0.366573\pi$
$228$	−3200.00	−0.929496
$229$	350.000	0.100998	0.0504992	−	0.998724i	$-0.483919\pi$
$229$	350.000	0.100998	0.0504992	+	0.998724i	$0.483919\pi$
$230$	0	0
$231$	384.000	0.109374
$232$	720.000	0.203751
$233$	−1962.00	−0.551652	−0.275826	−	0.961208i	$-0.588951\pi$
$233$	−1962.00	−0.551652	−0.275826	+	0.961208i	$0.588951\pi$
$234$	−4292.00	−1.19905
$235$	0	0
$236$	1680.00	0.463384
$237$	−1280.00	−0.350823
$238$	528.000	0.143803
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	2374.00	0.630605
$243$	−5032.00	−1.32841
$244$	3608.00	0.946633
$245$	0	0
$246$	7008.00	1.81632
$247$	−5800.00	−1.49411
$248$	−1216.00	−0.311355
$249$	−576.000	−0.146596
$250$	0	0
$251$	2652.00	0.666903	0.333452	−	0.942767i	$-0.391787\pi$
$251$	2652.00	0.666903	0.333452	+	0.942767i	$0.391787\pi$
$252$	592.000	0.147986
$253$	−1584.00	−0.393617
$254$	−248.000	−0.0612634
$255$	0	0
$256$	256.000	0.0625000
$257$	2334.00	0.566502	0.283251	−	0.959046i	$-0.408587\pi$
$257$	2334.00	0.566502	0.283251	+	0.959046i	$0.408587\pi$
$258$	512.000	0.123549
$259$	136.000	0.0326279
$260$	0	0
$261$	−3330.00	−0.789739
$262$	−264.000	−0.0622518
$263$	3948.00	0.925643	0.462822	−	0.886451i	$-0.346837\pi$
$263$	3948.00	0.925643	0.462822	+	0.886451i	$0.346837\pi$
$264$	−768.000	−0.179042
$265$	0	0
$266$	800.000	0.184403
$267$	6480.00	1.48528
$268$	4096.00	0.933593
$269$	1590.00	0.360387	0.180193	−	0.983631i	$-0.442328\pi$
$269$	1590.00	0.360387	0.180193	+	0.983631i	$0.442328\pi$
$270$	0	0
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	−1056.00	−0.235402
$273$	1856.00	0.411466
$274$	−2508.00	−0.552970
$275$	0	0
$276$	−4224.00	−0.921213
$277$	−1646.00	−0.357034	−0.178517	−	0.983937i	$-0.557130\pi$
$277$	−1646.00	−0.357034	−0.178517	+	0.983937i	$0.557130\pi$
$278$	5720.00	1.23404
$279$	5624.00	1.20681
$280$	0	0
$281$	−1158.00	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$281$	−1158.00	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$282$	−3264.00	−0.689250
$283$	−6992.00	−1.46866	−0.734331	−	0.678792i	$-0.762504\pi$
$283$	−6992.00	−1.46866	−0.734331	+	0.678792i	$0.762504\pi$
$284$	1728.00	0.361049
$285$	0	0
$286$	−1392.00	−0.287800
$287$	−1752.00	−0.360339
$288$	−1184.00	−0.242250
$289$	−557.000	−0.113373
$290$	0	0
$291$	−8848.00	−1.78240
$292$	−1448.00	−0.290198
$293$	258.000	0.0514421	0.0257210	−	0.999669i	$-0.491812\pi$
$293$	258.000	0.0514421	0.0257210	+	0.999669i	$0.491812\pi$
$294$	5232.00	1.03788
$295$	0	0
$296$	−272.000	−0.0534111
$297$	960.000	0.187558
$298$	−1500.00	−0.291586
$299$	−7656.00	−1.48080
$300$	0	0
$301$	−128.000	−0.0245110
$302$	896.000	0.170725
$303$	−2064.00	−0.391332
$304$	−1600.00	−0.301863
$305$	0	0
$306$	4884.00	0.912417
$307$	8944.00	1.66274	0.831370	−	0.555720i	$-0.187557\pi$
$307$	8944.00	1.66274	0.831370	+	0.555720i	$0.187557\pi$
$308$	192.000	0.0355202
$309$	7904.00	1.45515
$310$	0	0
$311$	1392.00	0.253804	0.126902	−	0.991915i	$-0.459497\pi$
$311$	1392.00	0.253804	0.126902	+	0.991915i	$0.459497\pi$
$312$	−3712.00	−0.673560
$313$	5878.00	1.06148	0.530742	−	0.847534i	$-0.321913\pi$
$313$	5878.00	1.06148	0.530742	+	0.847534i	$0.321913\pi$
$314$	4492.00	0.807319
$315$	0	0
$316$	−640.000	−0.113933
$317$	−10326.0	−1.82955	−0.914773	−	0.403969i	$-0.867630\pi$
$317$	−10326.0	−1.82955	−0.914773	+	0.403969i	$0.867630\pi$
$318$	3552.00	0.626372
$319$	−1080.00	−0.189556
$320$	0	0
$321$	192.000	0.0333844
$322$	1056.00	0.182760
$323$	6600.00	1.13695
$324$	−1436.00	−0.246228
$325$	0	0
$326$	−1136.00	−0.192998
$327$	7600.00	1.28526
$328$	3504.00	0.589866
$329$	816.000	0.136740
$330$	0	0
$331$	−4228.00	−0.702090	−0.351045	−	0.936359i	$-0.614174\pi$
$331$	−4228.00	−0.702090	−0.351045	+	0.936359i	$0.614174\pi$
$332$	−288.000	−0.0476086
$333$	1258.00	0.207021
$334$	−3048.00	−0.499339
$335$	0	0
$336$	512.000	0.0831306
$337$	−1106.00	−0.178776	−0.0893882	−	0.995997i	$-0.528491\pi$
$337$	−1106.00	−0.178776	−0.0893882	+	0.995997i	$0.528491\pi$
$338$	−2334.00	−0.375600
$339$	8304.00	1.33042
$340$	0	0
$341$	1824.00	0.289663
$342$	7400.00	1.17002
$343$	−2680.00	−0.421885
$344$	256.000	0.0401238
$345$	0	0
$346$	7404.00	1.15041
$347$	−9336.00	−1.44433	−0.722165	−	0.691720i	$-0.756853\pi$
$347$	−9336.00	−1.44433	−0.722165	+	0.691720i	$0.756853\pi$
$348$	−2880.00	−0.443633
$349$	−11770.0	−1.80525	−0.902627	−	0.430424i	$-0.858364\pi$
$349$	−11770.0	−1.80525	−0.902627	+	0.430424i	$0.858364\pi$
$350$	0	0
$351$	4640.00	0.705598
$352$	−384.000	−0.0581456
$353$	−8322.00	−1.25477	−0.627387	−	0.778707i	$-0.715876\pi$
$353$	−8322.00	−1.25477	−0.627387	+	0.778707i	$0.715876\pi$
$354$	−6720.00	−1.00894
$355$	0	0
$356$	3240.00	0.482359
$357$	−2112.00	−0.313106
$358$	−6360.00	−0.938929
$359$	10680.0	1.57011	0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0	1.57011	0.785054	+	0.619427i	$0.212635\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	4196.00	0.609218
$363$	−9496.00	−1.37303
$364$	928.000	0.133628
$365$	0	0
$366$	−14432.0	−2.06113
$367$	5884.00	0.836900	0.418450	−	0.908240i	$-0.362574\pi$
$367$	5884.00	0.836900	0.418450	+	0.908240i	$0.362574\pi$
$368$	−2112.00	−0.299173
$369$	−16206.0	−2.28632
$370$	0	0
$371$	−888.000	−0.124266
$372$	4864.00	0.677921
$373$	2098.00	0.291234	0.145617	−	0.989341i	$-0.453483\pi$
$373$	2098.00	0.291234	0.145617	+	0.989341i	$0.453483\pi$
$374$	1584.00	0.219002
$375$	0	0
$376$	−1632.00	−0.223840
$377$	−5220.00	−0.713113
$378$	−640.000	−0.0870848
$379$	3860.00	0.523153	0.261576	−	0.965183i	$-0.415758\pi$
$379$	3860.00	0.523153	0.261576	+	0.965183i	$0.415758\pi$
$380$	0	0
$381$	992.000	0.133390
$382$	−8784.00	−1.17651
$383$	9588.00	1.27917	0.639587	−	0.768718i	$-0.279105\pi$
$383$	9588.00	1.27917	0.639587	+	0.768718i	$0.279105\pi$
$384$	−1024.00	−0.136083
$385$	0	0
$386$	−4316.00	−0.569116
$387$	−1184.00	−0.155520
$388$	−4424.00	−0.578852
$389$	−13410.0	−1.74785	−0.873925	−	0.486060i	$-0.838434\pi$
$389$	−13410.0	−1.74785	−0.873925	+	0.486060i	$0.838434\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	2616.00	0.337061
$393$	1056.00	0.135542
$394$	−2148.00	−0.274657
$395$	0	0
$396$	1776.00	0.225372
$397$	13114.0	1.65787	0.828933	−	0.559348i	$-0.188948\pi$
$397$	13114.0	1.65787	0.828933	+	0.559348i	$0.188948\pi$
$398$	−5680.00	−0.715358
$399$	−3200.00	−0.401505
$400$	0	0
$401$	−5838.00	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$401$	−5838.00	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$402$	−16384.0	−2.03274
$403$	8816.00	1.08972
$404$	−1032.00	−0.127089
$405$	0	0
$406$	720.000	0.0880123
$407$	408.000	0.0496899
$408$	4224.00	0.512547
$409$	9530.00	1.15215	0.576074	−	0.817398i	$-0.304584\pi$
$409$	9530.00	1.15215	0.576074	+	0.817398i	$0.304584\pi$
$410$	0	0
$411$	10032.0	1.20400
$412$	3952.00	0.472575
$413$	1680.00	0.200163
$414$	9768.00	1.15959
$415$	0	0
$416$	−1856.00	−0.218745
$417$	−22880.0	−2.68690
$418$	2400.00	0.280832
$419$	7260.00	0.846478	0.423239	−	0.906018i	$-0.360893\pi$
$419$	7260.00	0.846478	0.423239	+	0.906018i	$0.360893\pi$
$420$	0	0
$421$	12062.0	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$421$	12062.0	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$422$	5336.00	0.615527
$423$	7548.00	0.867604
$424$	1776.00	0.203420
$425$	0	0
$426$	−6912.00	−0.786121
$427$	3608.00	0.408907
$428$	96.0000	0.0108419
$429$	5568.00	0.626633
$430$	0	0
$431$	−13608.0	−1.52082	−0.760411	−	0.649442i	$-0.775002\pi$
$431$	−13608.0	−1.52082	−0.760411	+	0.649442i	$0.775002\pi$
$432$	1280.00	0.142556
$433$	3838.00	0.425964	0.212982	−	0.977056i	$-0.431682\pi$
$433$	3838.00	0.425964	0.212982	+	0.977056i	$0.431682\pi$
$434$	−1216.00	−0.134493
$435$	0	0
$436$	3800.00	0.417401
$437$	13200.0	1.44495
$438$	5792.00	0.631855
$439$	7400.00	0.804516	0.402258	−	0.915526i	$-0.368225\pi$
$439$	7400.00	0.804516	0.402258	+	0.915526i	$0.368225\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	7656.00	0.823889
$443$	−8352.00	−0.895746	−0.447873	−	0.894097i	$-0.647818\pi$
$443$	−8352.00	−0.895746	−0.447873	+	0.894097i	$0.647818\pi$
$444$	1088.00	0.116293
$445$	0	0
$446$	3544.00	0.376263
$447$	6000.00	0.634878
$448$	256.000	0.0269975
$449$	10770.0	1.13200	0.566000	−	0.824405i	$-0.308490\pi$
$449$	10770.0	1.13200	0.566000	+	0.824405i	$0.308490\pi$
$450$	0	0
$451$	−5256.00	−0.548770
$452$	4152.00	0.432066
$453$	−3584.00	−0.371724
$454$	−5568.00	−0.575593
$455$	0	0
$456$	6400.00	0.657253
$457$	6694.00	0.685191	0.342595	−	0.939483i	$-0.388694\pi$
$457$	6694.00	0.685191	0.342595	+	0.939483i	$0.388694\pi$
$458$	−700.000	−0.0714167
$459$	−5280.00	−0.536927
$460$	0	0
$461$	−3018.00	−0.304907	−0.152454	−	0.988311i	$-0.548717\pi$
$461$	−3018.00	−0.304907	−0.152454	+	0.988311i	$0.548717\pi$
$462$	−768.000	−0.0773389
$463$	−14492.0	−1.45464	−0.727322	−	0.686296i	$-0.759235\pi$
$463$	−14492.0	−1.45464	−0.727322	+	0.686296i	$0.759235\pi$
$464$	−1440.00	−0.144074
$465$	0	0
$466$	3924.00	0.390077
$467$	−7776.00	−0.770515	−0.385257	−	0.922809i	$-0.625887\pi$
$467$	−7776.00	−0.770515	−0.385257	+	0.922809i	$0.625887\pi$
$468$	8584.00	0.847854
$469$	4096.00	0.403274
$470$	0	0
$471$	−17968.0	−1.75780
$472$	−3360.00	−0.327662
$473$	−384.000	−0.0373284
$474$	2560.00	0.248069
$475$	0	0
$476$	−1056.00	−0.101684
$477$	−8214.00	−0.788455
$478$	8640.00	0.826746
$479$	−13680.0	−1.30492	−0.652458	−	0.757825i	$-0.726262\pi$
$479$	−13680.0	−1.30492	−0.652458	+	0.757825i	$0.726262\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	956.000	0.0903415
$483$	−4224.00	−0.397927
$484$	−4748.00	−0.445905
$485$	0	0
$486$	10064.0	0.939326
$487$	−7916.00	−0.736567	−0.368284	−	0.929714i	$-0.620054\pi$
$487$	−7916.00	−0.736567	−0.368284	+	0.929714i	$0.620054\pi$
$488$	−7216.00	−0.669371
$489$	4544.00	0.420218
$490$	0	0
$491$	13932.0	1.28053	0.640267	−	0.768152i	$-0.278824\pi$
$491$	13932.0	1.28053	0.640267	+	0.768152i	$0.278824\pi$
$492$	−14016.0	−1.28433
$493$	5940.00	0.542645
$494$	11600.0	1.05650
$495$	0	0
$496$	2432.00	0.220161
$497$	1728.00	0.155959
$498$	1152.00	0.103659
$499$	−8260.00	−0.741019	−0.370509	−	0.928829i	$-0.620817\pi$
$499$	−8260.00	−0.741019	−0.370509	+	0.928829i	$0.620817\pi$
$500$	0	0
$501$	12192.0	1.08722
$502$	−5304.00	−0.471572
$503$	11148.0	0.988200	0.494100	−	0.869405i	$-0.335498\pi$
$503$	11148.0	0.988200	0.494100	+	0.869405i	$0.335498\pi$
$504$	−1184.00	−0.104642
$505$	0	0
$506$	3168.00	0.278330
$507$	9336.00	0.817803
$508$	496.000	0.0433198
$509$	−9690.00	−0.843815	−0.421907	−	0.906639i	$-0.638639\pi$
$509$	−9690.00	−0.843815	−0.421907	+	0.906639i	$0.638639\pi$
$510$	0	0
$511$	−1448.00	−0.125354
$512$	−512.000	−0.0441942
$513$	−8000.00	−0.688516
$514$	−4668.00	−0.400577
$515$	0	0
$516$	−1024.00	−0.0873626
$517$	2448.00	0.208245
$518$	−272.000	−0.0230714
$519$	−29616.0	−2.50481
$520$	0	0
$521$	−16038.0	−1.34863	−0.674316	−	0.738443i	$-0.735562\pi$
$521$	−16038.0	−1.34863	−0.674316	+	0.738443i	$0.735562\pi$
$522$	6660.00	0.558430
$523$	−992.000	−0.0829391	−0.0414695	−	0.999140i	$-0.513204\pi$
$523$	−992.000	−0.0829391	−0.0414695	+	0.999140i	$0.513204\pi$
$524$	528.000	0.0440187
$525$	0	0
$526$	−7896.00	−0.654528
$527$	−10032.0	−0.829223
$528$	1536.00	0.126602
$529$	5257.00	0.432070
$530$	0	0
$531$	15540.0	1.27002
$532$	−1600.00	−0.130392
$533$	−25404.0	−2.06448
$534$	−12960.0	−1.05025
$535$	0	0
$536$	−8192.00	−0.660150
$537$	25440.0	2.04435
$538$	−3180.00	−0.254832
$539$	−3924.00	−0.313578
$540$	0	0
$541$	7142.00	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$541$	7142.00	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$542$	−9904.00	−0.784895
$543$	−16784.0	−1.32646
$544$	2112.00	0.166455
$545$	0	0
$546$	−3712.00	−0.290950
$547$	−7616.00	−0.595314	−0.297657	−	0.954673i	$-0.596205\pi$
$547$	−7616.00	−0.595314	−0.297657	+	0.954673i	$0.596205\pi$
$548$	5016.00	0.391009
$549$	33374.0	2.59448
$550$	0	0
$551$	9000.00	0.695849
$552$	8448.00	0.651396
$553$	−640.000	−0.0492144
$554$	3292.00	0.252462
$555$	0	0
$556$	−11440.0	−0.872597
$557$	10314.0	0.784593	0.392296	−	0.919839i	$-0.371681\pi$
$557$	10314.0	0.784593	0.392296	+	0.919839i	$0.371681\pi$
$558$	−11248.0	−0.853344
$559$	−1856.00	−0.140430
$560$	0	0
$561$	−6336.00	−0.476838
$562$	2316.00	0.173834
$563$	7128.00	0.533587	0.266793	−	0.963754i	$-0.414036\pi$
$563$	7128.00	0.533587	0.266793	+	0.963754i	$0.414036\pi$
$564$	6528.00	0.487373
$565$	0	0
$566$	13984.0	1.03850
$567$	−1436.00	−0.106360
$568$	−3456.00	−0.255300
$569$	2010.00	0.148091	0.0740453	−	0.997255i	$-0.476409\pi$
$569$	2010.00	0.148091	0.0740453	+	0.997255i	$0.476409\pi$
$570$	0	0
$571$	−23188.0	−1.69945	−0.849726	−	0.527224i	$-0.823233\pi$
$571$	−23188.0	−1.69945	−0.849726	+	0.527224i	$0.823233\pi$
$572$	2784.00	0.203505
$573$	35136.0	2.56165
$574$	3504.00	0.254798
$575$	0	0
$576$	2368.00	0.171296
$577$	−22466.0	−1.62092	−0.810461	−	0.585793i	$-0.800783\pi$
$577$	−22466.0	−1.62092	−0.810461	+	0.585793i	$0.800783\pi$
$578$	1114.00	0.0801666
$579$	17264.0	1.23915
$580$	0	0
$581$	−288.000	−0.0205650
$582$	17696.0	1.26035
$583$	−2664.00	−0.189248
$584$	2896.00	0.205201
$585$	0	0
$586$	−516.000	−0.0363750
$587$	−22776.0	−1.60148	−0.800738	−	0.599015i	$-0.795559\pi$
$587$	−22776.0	−1.60148	−0.800738	+	0.599015i	$0.795559\pi$
$588$	−10464.0	−0.733891
$589$	−15200.0	−1.06334
$590$	0	0
$591$	8592.00	0.598016
$592$	544.000	0.0377673
$593$	21198.0	1.46796	0.733978	−	0.679174i	$-0.237662\pi$
$593$	21198.0	1.46796	0.733978	+	0.679174i	$0.237662\pi$
$594$	−1920.00	−0.132624
$595$	0	0
$596$	3000.00	0.206183
$597$	22720.0	1.55757
$598$	15312.0	1.04708
$599$	15960.0	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$599$	15960.0	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$600$	0	0
$601$	5882.00	0.399221	0.199610	−	0.979875i	$-0.436032\pi$
$601$	5882.00	0.399221	0.199610	+	0.979875i	$0.436032\pi$
$602$	256.000	0.0173319
$603$	37888.0	2.55874
$604$	−1792.00	−0.120721
$605$	0	0
$606$	4128.00	0.276714
$607$	−8516.00	−0.569446	−0.284723	−	0.958610i	$-0.591902\pi$
$607$	−8516.00	−0.569446	−0.284723	+	0.958610i	$0.591902\pi$
$608$	3200.00	0.213449
$609$	−2880.00	−0.191631
$610$	0	0
$611$	11832.0	0.783423
$612$	−9768.00	−0.645176
$613$	−8462.00	−0.557548	−0.278774	−	0.960357i	$-0.589928\pi$
$613$	−8462.00	−0.557548	−0.278774	+	0.960357i	$0.589928\pi$
$614$	−17888.0	−1.17573
$615$	0	0
$616$	−384.000	−0.0251166
$617$	11094.0	0.723870	0.361935	−	0.932203i	$-0.382116\pi$
$617$	11094.0	0.723870	0.361935	+	0.932203i	$0.382116\pi$
$618$	−15808.0	−1.02895
$619$	2180.00	0.141553	0.0707767	−	0.997492i	$-0.477452\pi$
$619$	2180.00	0.141553	0.0707767	+	0.997492i	$0.477452\pi$
$620$	0	0
$621$	−10560.0	−0.682380
$622$	−2784.00	−0.179467
$623$	3240.00	0.208359
$624$	7424.00	0.476279
$625$	0	0
$626$	−11756.0	−0.750582
$627$	−9600.00	−0.611463
$628$	−8984.00	−0.570861
$629$	−2244.00	−0.142248
$630$	0	0
$631$	−26848.0	−1.69382	−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0	−1.69382	−0.846911	+	0.531734i	$0.821541\pi$
$632$	1280.00	0.0805628
$633$	−21344.0	−1.34020
$634$	20652.0	1.29368
$635$	0	0
$636$	−7104.00	−0.442912
$637$	−18966.0	−1.17969
$638$	2160.00	0.134036
$639$	15984.0	0.989542
$640$	0	0
$641$	26322.0	1.62193	0.810965	−	0.585095i	$-0.198943\pi$
$641$	26322.0	1.62193	0.810965	+	0.585095i	$0.198943\pi$
$642$	−384.000	−0.0236063
$643$	10168.0	0.623619	0.311809	−	0.950145i	$-0.399065\pi$
$643$	10168.0	0.623619	0.311809	+	0.950145i	$0.399065\pi$
$644$	−2112.00	−0.129231
$645$	0	0
$646$	−13200.0	−0.803943
$647$	23604.0	1.43426	0.717132	−	0.696937i	$-0.245454\pi$
$647$	23604.0	1.43426	0.717132	+	0.696937i	$0.245454\pi$
$648$	2872.00	0.174109
$649$	5040.00	0.304834
$650$	0	0
$651$	4864.00	0.292834
$652$	2272.00	0.136470
$653$	−16422.0	−0.984139	−0.492069	−	0.870556i	$-0.663759\pi$
$653$	−16422.0	−0.984139	−0.492069	+	0.870556i	$0.663759\pi$
$654$	−15200.0	−0.908818
$655$	0	0
$656$	−7008.00	−0.417098
$657$	−13394.0	−0.795357
$658$	−1632.00	−0.0966899
$659$	−26100.0	−1.54281	−0.771405	−	0.636345i	$-0.780446\pi$
$659$	−26100.0	−1.54281	−0.771405	+	0.636345i	$0.780446\pi$
$660$	0	0
$661$	−3058.00	−0.179943	−0.0899716	−	0.995944i	$-0.528678\pi$
$661$	−3058.00	−0.179943	−0.0899716	+	0.995944i	$0.528678\pi$
$662$	8456.00	0.496453
$663$	−30624.0	−1.79387
$664$	576.000	0.0336644
$665$	0	0
$666$	−2516.00	−0.146386
$667$	11880.0	0.689648
$668$	6096.00	0.353086
$669$	−14176.0	−0.819246
$670$	0	0
$671$	10824.0	0.622736
$672$	−1024.00	−0.0587822
$673$	−10802.0	−0.618702	−0.309351	−	0.950948i	$-0.600112\pi$
$673$	−10802.0	−0.618702	−0.309351	+	0.950948i	$0.600112\pi$
$674$	2212.00	0.126414
$675$	0	0
$676$	4668.00	0.265589
$677$	10674.0	0.605960	0.302980	−	0.952997i	$-0.402018\pi$
$677$	10674.0	0.605960	0.302980	+	0.952997i	$0.402018\pi$
$678$	−16608.0	−0.940747
$679$	−4424.00	−0.250041
$680$	0	0
$681$	22272.0	1.25325
$682$	−3648.00	−0.204823
$683$	28608.0	1.60272	0.801358	−	0.598185i	$-0.204111\pi$
$683$	28608.0	1.60272	0.801358	+	0.598185i	$0.204111\pi$
$684$	−14800.0	−0.827328
$685$	0	0
$686$	5360.00	0.298317
$687$	2800.00	0.155497
$688$	−512.000	−0.0283718
$689$	−12876.0	−0.711954
$690$	0	0
$691$	−2428.00	−0.133669	−0.0668346	−	0.997764i	$-0.521290\pi$
$691$	−2428.00	−0.133669	−0.0668346	+	0.997764i	$0.521290\pi$
$692$	−14808.0	−0.813462
$693$	1776.00	0.0973516
$694$	18672.0	1.02130
$695$	0	0
$696$	5760.00	0.313696
$697$	28908.0	1.57097
$698$	23540.0	1.27651
$699$	−15696.0	−0.849324
$700$	0	0
$701$	−6618.00	−0.356574	−0.178287	−	0.983979i	$-0.557056\pi$
$701$	−6618.00	−0.356574	−0.178287	+	0.983979i	$0.557056\pi$
$702$	−9280.00	−0.498933
$703$	−3400.00	−0.182409
$704$	768.000	0.0411152
$705$	0	0
$706$	16644.0	0.887259
$707$	−1032.00	−0.0548972
$708$	13440.0	0.713427
$709$	20510.0	1.08642	0.543208	−	0.839598i	$-0.317209\pi$
$709$	20510.0	1.08642	0.543208	+	0.839598i	$0.317209\pi$
$710$	0	0
$711$	−5920.00	−0.312261
$712$	−6480.00	−0.341079
$713$	−20064.0	−1.05386
$714$	4224.00	0.221399
$715$	0	0
$716$	12720.0	0.663923
$717$	−34560.0	−1.80009
$718$	−21360.0	−1.11023
$719$	31680.0	1.64321	0.821603	−	0.570061i	$-0.193080\pi$
$719$	31680.0	1.64321	0.821603	+	0.570061i	$0.193080\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	−6282.00	−0.323811
$723$	−3824.00	−0.196703
$724$	−8392.00	−0.430782
$725$	0	0
$726$	18992.0	0.970880
$727$	−13196.0	−0.673195	−0.336597	−	0.941649i	$-0.609276\pi$
$727$	−13196.0	−0.673195	−0.336597	+	0.941649i	$0.609276\pi$
$728$	−1856.00	−0.0944889
$729$	−30563.0	−1.55276
$730$	0	0
$731$	2112.00	0.106861
$732$	28864.0	1.45744
$733$	−8102.00	−0.408259	−0.204130	−	0.978944i	$-0.565436\pi$
$733$	−8102.00	−0.408259	−0.204130	+	0.978944i	$0.565436\pi$
$734$	−11768.0	−0.591778
$735$	0	0
$736$	4224.00	0.211547
$737$	12288.0	0.614158
$738$	32412.0	1.61667
$739$	−12580.0	−0.626201	−0.313101	−	0.949720i	$-0.601368\pi$
$739$	−12580.0	−0.626201	−0.313101	+	0.949720i	$0.601368\pi$
$740$	0	0
$741$	−46400.0	−2.30033
$742$	1776.00	0.0878693
$743$	−29892.0	−1.47595	−0.737975	−	0.674828i	$-0.764218\pi$
$743$	−29892.0	−1.47595	−0.737975	+	0.674828i	$0.764218\pi$
$744$	−9728.00	−0.479363
$745$	0	0
$746$	−4196.00	−0.205934
$747$	−2664.00	−0.130483
$748$	−3168.00	−0.154858
$749$	96.0000	0.00468326
$750$	0	0
$751$	−40408.0	−1.96339	−0.981697	−	0.190450i	$-0.939005\pi$
$751$	−40408.0	−1.96339	−0.981697	+	0.190450i	$0.939005\pi$
$752$	3264.00	0.158279
$753$	21216.0	1.02676
$754$	10440.0	0.504247
$755$	0	0
$756$	1280.00	0.0615782
$757$	−32366.0	−1.55398	−0.776990	−	0.629513i	$-0.783254\pi$
$757$	−32366.0	−1.55398	−0.776990	+	0.629513i	$0.783254\pi$
$758$	−7720.00	−0.369925
$759$	−12672.0	−0.606014
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	−1984.00	−0.0943212
$763$	3800.00	0.180300
$764$	17568.0	0.831921
$765$	0	0
$766$	−19176.0	−0.904513
$767$	24360.0	1.14679
$768$	2048.00	0.0962250
$769$	10850.0	0.508792	0.254396	−	0.967100i	$-0.418123\pi$
$769$	10850.0	0.508792	0.254396	+	0.967100i	$0.418123\pi$
$770$	0	0
$771$	18672.0	0.872186
$772$	8632.00	0.402425
$773$	−9102.00	−0.423514	−0.211757	−	0.977322i	$-0.567919\pi$
$773$	−9102.00	−0.423514	−0.211757	+	0.977322i	$0.567919\pi$
$774$	2368.00	0.109969
$775$	0	0
$776$	8848.00	0.409310
$777$	1088.00	0.0502340
$778$	26820.0	1.23592
$779$	43800.0	2.01450
$780$	0	0
$781$	5184.00	0.237514
$782$	−17424.0	−0.796779
$783$	−7200.00	−0.328617
$784$	−5232.00	−0.238338
$785$	0	0
$786$	−2112.00	−0.0958429
$787$	25504.0	1.15517	0.577585	−	0.816330i	$-0.303995\pi$
$787$	25504.0	1.15517	0.577585	+	0.816330i	$0.303995\pi$
$788$	4296.00	0.194212
$789$	31584.0	1.42512
$790$	0	0
$791$	4152.00	0.186635
$792$	−3552.00	−0.159362
$793$	52316.0	2.34274
$794$	−26228.0	−1.17229
$795$	0	0
$796$	11360.0	0.505835
$797$	−14166.0	−0.629593	−0.314796	−	0.949159i	$-0.601936\pi$
$797$	−14166.0	−0.629593	−0.314796	+	0.949159i	$0.601936\pi$
$798$	6400.00	0.283907
$799$	−13464.0	−0.596148
$800$	0	0
$801$	29970.0	1.32202
$802$	11676.0	0.514082
$803$	−4344.00	−0.190905
$804$	32768.0	1.43736
$805$	0	0
$806$	−17632.0	−0.770547
$807$	12720.0	0.554852
$808$	2064.00	0.0898654
$809$	33210.0	1.44327	0.721633	−	0.692276i	$-0.243392\pi$
$809$	33210.0	1.44327	0.721633	+	0.692276i	$0.243392\pi$
$810$	0	0
$811$	39212.0	1.69780	0.848902	−	0.528550i	$-0.177264\pi$
$811$	39212.0	1.69780	0.848902	+	0.528550i	$0.177264\pi$
$812$	−1440.00	−0.0622341
$813$	39616.0	1.70897
$814$	−816.000	−0.0351361
$815$	0	0
$816$	−8448.00	−0.362425
$817$	3200.00	0.137030
$818$	−19060.0	−0.814691
$819$	8584.00	0.366238
$820$	0	0
$821$	6222.00	0.264494	0.132247	−	0.991217i	$-0.457781\pi$
$821$	6222.00	0.264494	0.132247	+	0.991217i	$0.457781\pi$
$822$	−20064.0	−0.851353
$823$	−31172.0	−1.32028	−0.660138	−	0.751144i	$-0.729502\pi$
$823$	−31172.0	−1.32028	−0.660138	+	0.751144i	$0.729502\pi$
$824$	−7904.00	−0.334161
$825$	0	0
$826$	−3360.00	−0.141537
$827$	264.000	0.0111006	0.00555029	−	0.999985i	$-0.498233\pi$
$827$	264.000	0.0111006	0.00555029	+	0.999985i	$0.498233\pi$
$828$	−19536.0	−0.819955
$829$	−29050.0	−1.21707	−0.608533	−	0.793528i	$-0.708242\pi$
$829$	−29050.0	−1.21707	−0.608533	+	0.793528i	$0.708242\pi$
$830$	0	0
$831$	−13168.0	−0.549691
$832$	3712.00	0.154676
$833$	21582.0	0.897685
$834$	45760.0	1.89993
$835$	0	0
$836$	−4800.00	−0.198578
$837$	12160.0	0.502164
$838$	−14520.0	−0.598550
$839$	−21720.0	−0.893752	−0.446876	−	0.894596i	$-0.647463\pi$
$839$	−21720.0	−0.893752	−0.446876	+	0.894596i	$0.647463\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	−24124.0	−0.987373
$843$	−9264.00	−0.378492
$844$	−10672.0	−0.435243
$845$	0	0
$846$	−15096.0	−0.613488
$847$	−4748.00	−0.192613
$848$	−3552.00	−0.143840
$849$	−55936.0	−2.26115
$850$	0	0
$851$	−4488.00	−0.180783
$852$	13824.0	0.555871
$853$	6658.00	0.267252	0.133626	−	0.991032i	$-0.457338\pi$
$853$	6658.00	0.267252	0.133626	+	0.991032i	$0.457338\pi$
$854$	−7216.00	−0.289141
$855$	0	0
$856$	−192.000	−0.00766638
$857$	13974.0	0.556993	0.278496	−	0.960437i	$-0.410164\pi$
$857$	13974.0	0.556993	0.278496	+	0.960437i	$0.410164\pi$
$858$	−11136.0	−0.443096
$859$	23780.0	0.944544	0.472272	−	0.881453i	$-0.343434\pi$
$859$	23780.0	0.944544	0.472272	+	0.881453i	$0.343434\pi$
$860$	0	0
$861$	−14016.0	−0.554778
$862$	27216.0	1.07538
$863$	12228.0	0.482324	0.241162	−	0.970485i	$-0.422471\pi$
$863$	12228.0	0.482324	0.241162	+	0.970485i	$0.422471\pi$
$864$	−2560.00	−0.100802
$865$	0	0
$866$	−7676.00	−0.301202
$867$	−4456.00	−0.174549
$868$	2432.00	0.0951008
$869$	−1920.00	−0.0749500
$870$	0	0
$871$	59392.0	2.31047
$872$	−7600.00	−0.295147
$873$	−40922.0	−1.58648
$874$	−26400.0	−1.02173
$875$	0	0
$876$	−11584.0	−0.446789
$877$	−11606.0	−0.446872	−0.223436	−	0.974719i	$-0.571727\pi$
$877$	−11606.0	−0.446872	−0.223436	+	0.974719i	$0.571727\pi$
$878$	−14800.0	−0.568879
$879$	2064.00	0.0792002
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	24198.0	0.923797
$883$	−8072.00	−0.307638	−0.153819	−	0.988099i	$-0.549157\pi$
$883$	−8072.00	−0.307638	−0.153819	+	0.988099i	$0.549157\pi$
$884$	−15312.0	−0.582577
$885$	0	0
$886$	16704.0	0.633388
$887$	−15756.0	−0.596431	−0.298216	−	0.954498i	$-0.596391\pi$
$887$	−15756.0	−0.596431	−0.298216	+	0.954498i	$0.596391\pi$
$888$	−2176.00	−0.0822317
$889$	496.000	0.0187124
$890$	0	0
$891$	−4308.00	−0.161979
$892$	−7088.00	−0.266058
$893$	−20400.0	−0.764457
$894$	−12000.0	−0.448926
$895$	0	0
$896$	−512.000	−0.0190901
$897$	−61248.0	−2.27983
$898$	−21540.0	−0.800444
$899$	−13680.0	−0.507512
$900$	0	0
$901$	14652.0	0.541763
$902$	10512.0	0.388039
$903$	−1024.00	−0.0377371
$904$	−8304.00	−0.305517
$905$	0	0
$906$	7168.00	0.262849
$907$	−18776.0	−0.687372	−0.343686	−	0.939085i	$-0.611676\pi$
$907$	−18776.0	−0.687372	−0.343686	+	0.939085i	$0.611676\pi$
$908$	11136.0	0.407006
$909$	−9546.00	−0.348318
$910$	0	0
$911$	−20568.0	−0.748022	−0.374011	−	0.927424i	$-0.622018\pi$
$911$	−20568.0	−0.748022	−0.374011	+	0.927424i	$0.622018\pi$
$912$	−12800.0	−0.464748
$913$	−864.000	−0.0313190
$914$	−13388.0	−0.484503
$915$	0	0
$916$	1400.00	0.0504992
$917$	528.000	0.0190143
$918$	10560.0	0.379664
$919$	−6280.00	−0.225417	−0.112708	−	0.993628i	$-0.535953\pi$
$919$	−6280.00	−0.225417	−0.112708	+	0.993628i	$0.535953\pi$
$920$	0	0
$921$	71552.0	2.55996
$922$	6036.00	0.215602
$923$	25056.0	0.893530
$924$	1536.00	0.0546869
$925$	0	0
$926$	28984.0	1.02859
$927$	36556.0	1.29521
$928$	2880.00	0.101876
$929$	−20430.0	−0.721514	−0.360757	−	0.932660i	$-0.617482\pi$
$929$	−20430.0	−0.721514	−0.360757	+	0.932660i	$0.617482\pi$
$930$	0	0
$931$	32700.0	1.15113
$932$	−7848.00	−0.275826
$933$	11136.0	0.390757
$934$	15552.0	0.544836
$935$	0	0
$936$	−17168.0	−0.599523
$937$	−8906.00	−0.310508	−0.155254	−	0.987875i	$-0.549620\pi$
$937$	−8906.00	−0.310508	−0.155254	+	0.987875i	$0.549620\pi$
$938$	−8192.00	−0.285158
$939$	47024.0	1.63426
$940$	0	0
$941$	−17418.0	−0.603412	−0.301706	−	0.953401i	$-0.597556\pi$
$941$	−17418.0	−0.603412	−0.301706	+	0.953401i	$0.597556\pi$
$942$	35936.0	1.24295
$943$	57816.0	1.99655
$944$	6720.00	0.231692
$945$	0	0
$946$	768.000	0.0263952
$947$	2544.00	0.0872956	0.0436478	−	0.999047i	$-0.486102\pi$
$947$	2544.00	0.0872956	0.0436478	+	0.999047i	$0.486102\pi$
$948$	−5120.00	−0.175411
$949$	−20996.0	−0.718187
$950$	0	0
$951$	−82608.0	−2.81677
$952$	2112.00	0.0719016
$953$	−15402.0	−0.523525	−0.261763	−	0.965132i	$-0.584304\pi$
$953$	−15402.0	−0.523525	−0.261763	+	0.965132i	$0.584304\pi$
$954$	16428.0	0.557522
$955$	0	0
$956$	−17280.0	−0.584597
$957$	−8640.00	−0.291841
$958$	27360.0	0.922716
$959$	5016.00	0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	−3944.00	−0.132183
$963$	888.000	0.0297148
$964$	−1912.00	−0.0638811
$965$	0	0
$966$	8448.00	0.281377
$967$	49444.0	1.64427	0.822136	−	0.569291i	$-0.192782\pi$
$967$	49444.0	1.64427	0.822136	+	0.569291i	$0.192782\pi$
$968$	9496.00	0.315303
$969$	52800.0	1.75044
$970$	0	0
$971$	−25188.0	−0.832463	−0.416231	−	0.909259i	$-0.636649\pi$
$971$	−25188.0	−0.832463	−0.416231	+	0.909259i	$0.636649\pi$
$972$	−20128.0	−0.664204
$973$	−11440.0	−0.376927
$974$	15832.0	0.520832
$975$	0	0
$976$	14432.0	0.473317
$977$	−2946.00	−0.0964697	−0.0482348	−	0.998836i	$-0.515360\pi$
$977$	−2946.00	−0.0964697	−0.0482348	+	0.998836i	$0.515360\pi$
$978$	−9088.00	−0.297139
$979$	9720.00	0.317316
$980$	0	0
$981$	35150.0	1.14399
$982$	−27864.0	−0.905475
$983$	−15012.0	−0.487089	−0.243544	−	0.969890i	$-0.578310\pi$
$983$	−15012.0	−0.487089	−0.243544	+	0.969890i	$0.578310\pi$
$984$	28032.0	0.908158
$985$	0	0
$986$	−11880.0	−0.383708
$987$	6528.00	0.210525
$988$	−23200.0	−0.747055
$989$	4224.00	0.135809
$990$	0	0
$991$	−5128.00	−0.164376	−0.0821878	−	0.996617i	$-0.526191\pi$
$991$	−5128.00	−0.164376	−0.0821878	+	0.996617i	$0.526191\pi$
$992$	−4864.00	−0.155678
$993$	−33824.0	−1.08094
$994$	−3456.00	−0.110279
$995$	0	0
$996$	−2304.00	−0.0732982
$997$	49714.0	1.57920	0.789598	−	0.613625i	$-0.210289\pi$
$997$	49714.0	1.57920	0.789598	+	0.613625i	$0.210289\pi$
$998$	16520.0	0.523979
$999$	2720.00	0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.4.a.c.1.1		1
3.2	odd	2		450.4.a.q.1.1		1
4.3	odd	2		400.4.a.b.1.1		1
5.2	odd	4		50.4.b.a.49.1		2
5.3	odd	4		50.4.b.a.49.2		2
5.4	even	2		10.4.a.a.1.1	✓	1
7.6	odd	2		2450.4.a.b.1.1		1
8.3	odd	2		1600.4.a.bx.1.1		1
8.5	even	2		1600.4.a.d.1.1		1
15.2	even	4		450.4.c.d.199.2		2
15.8	even	4		450.4.c.d.199.1		2
15.14	odd	2		90.4.a.a.1.1		1
20.3	even	4		400.4.c.c.49.1		2
20.7	even	4		400.4.c.c.49.2		2
20.19	odd	2		80.4.a.f.1.1		1
35.4	even	6		490.4.e.i.471.1		2
35.9	even	6		490.4.e.i.361.1		2
35.19	odd	6		490.4.e.a.361.1		2
35.24	odd	6		490.4.e.a.471.1		2
35.34	odd	2		490.4.a.o.1.1		1
40.19	odd	2		320.4.a.b.1.1		1
40.29	even	2		320.4.a.m.1.1		1
45.4	even	6		810.4.e.c.541.1		2
45.14	odd	6		810.4.e.w.541.1		2
45.29	odd	6		810.4.e.w.271.1		2
45.34	even	6		810.4.e.c.271.1		2
55.54	odd	2		1210.4.a.b.1.1		1
60.59	even	2		720.4.a.j.1.1		1
65.64	even	2		1690.4.a.a.1.1		1
80.19	odd	4		1280.4.d.g.641.2		2
80.29	even	4		1280.4.d.j.641.1		2
80.59	odd	4		1280.4.d.g.641.1		2
80.69	even	4		1280.4.d.j.641.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	5.4	even	2
50.4.a.c.1.1		1	1.1	even	1	trivial
50.4.b.a.49.1		2	5.2	odd	4
50.4.b.a.49.2		2	5.3	odd	4
80.4.a.f.1.1		1	20.19	odd	2
90.4.a.a.1.1		1	15.14	odd	2
320.4.a.b.1.1		1	40.19	odd	2
320.4.a.m.1.1		1	40.29	even	2
400.4.a.b.1.1		1	4.3	odd	2
400.4.c.c.49.1		2	20.3	even	4
400.4.c.c.49.2		2	20.7	even	4
450.4.a.q.1.1		1	3.2	odd	2
450.4.c.d.199.1		2	15.8	even	4
450.4.c.d.199.2		2	15.2	even	4
490.4.a.o.1.1		1	35.34	odd	2
490.4.e.a.361.1		2	35.19	odd	6
490.4.e.a.471.1		2	35.24	odd	6
490.4.e.i.361.1		2	35.9	even	6
490.4.e.i.471.1		2	35.4	even	6
720.4.a.j.1.1		1	60.59	even	2
810.4.e.c.271.1		2	45.34	even	6
810.4.e.c.541.1		2	45.4	even	6
810.4.e.w.271.1		2	45.29	odd	6
810.4.e.w.541.1		2	45.14	odd	6
1210.4.a.b.1.1		1	55.54	odd	2
1280.4.d.g.641.1		2	80.59	odd	4
1280.4.d.g.641.2		2	80.19	odd	4
1280.4.d.j.641.1		2	80.29	even	4
1280.4.d.j.641.2		2	80.69	even	4
1600.4.a.d.1.1		1	8.5	even	2
1600.4.a.bx.1.1		1	8.3	odd	2
1690.4.a.a.1.1		1	65.64	even	2
2450.4.a.b.1.1		1	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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