LMFDB - Embedded newform 50.4.a.e.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:	yes
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)

$f(q)$

$=$

\(q+2.00000 q^{2} +7.00000 q^{3} +4.00000 q^{4} +14.0000 q^{6} -34.0000 q^{7} +8.00000 q^{8} +22.0000 q^{9} +27.0000 q^{11} +28.0000 q^{12} -28.0000 q^{13} -68.0000 q^{14} +16.0000 q^{16} +21.0000 q^{17} +44.0000 q^{18} +35.0000 q^{19} -238.000 q^{21} +54.0000 q^{22} -78.0000 q^{23} +56.0000 q^{24} -56.0000 q^{26} -35.0000 q^{27} -136.000 q^{28} -120.000 q^{29} +182.000 q^{31} +32.0000 q^{32} +189.000 q^{33} +42.0000 q^{34} +88.0000 q^{36} +146.000 q^{37} +70.0000 q^{38} -196.000 q^{39} +357.000 q^{41} -476.000 q^{42} -148.000 q^{43} +108.000 q^{44} -156.000 q^{46} -84.0000 q^{47} +112.000 q^{48} +813.000 q^{49} +147.000 q^{51} -112.000 q^{52} +702.000 q^{53} -70.0000 q^{54} -272.000 q^{56} +245.000 q^{57} -240.000 q^{58} -840.000 q^{59} -238.000 q^{61} +364.000 q^{62} -748.000 q^{63} +64.0000 q^{64} +378.000 q^{66} +461.000 q^{67} +84.0000 q^{68} -546.000 q^{69} -708.000 q^{71} +176.000 q^{72} -133.000 q^{73} +292.000 q^{74} +140.000 q^{76} -918.000 q^{77} -392.000 q^{78} +650.000 q^{79} -839.000 q^{81} +714.000 q^{82} -903.000 q^{83} -952.000 q^{84} -296.000 q^{86} -840.000 q^{87} +216.000 q^{88} +735.000 q^{89} +952.000 q^{91} -312.000 q^{92} +1274.00 q^{93} -168.000 q^{94} +224.000 q^{96} +1106.00 q^{97} +1626.00 q^{98} +594.000 q^{99} +O(q^{100})\)

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
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	14.0000	0.952579
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	8.00000	0.353553
$9$	22.0000	0.814815
$10$	0	0
$11$	27.0000	0.740073	0.370037	−	0.929017i	$-0.379345\pi$
$11$	27.0000	0.740073	0.370037	+	0.929017i	$0.379345\pi$
$12$	28.0000	0.673575
$13$	−28.0000	−0.597369	−0.298685	−	0.954352i	$-0.596548\pi$
$13$	−28.0000	−0.597369	−0.298685	+	0.954352i	$0.596548\pi$
$14$	−68.0000	−1.29813
$15$	0	0
$16$	16.0000	0.250000
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	44.0000	0.576161
$19$	35.0000	0.422608	0.211304	−	0.977420i	$-0.432229\pi$
$19$	35.0000	0.422608	0.211304	+	0.977420i	$0.432229\pi$
$20$	0	0
$21$	−238.000	−2.47314
$22$	54.0000	0.523311
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	56.0000	0.476290
$25$	0	0
$26$	−56.0000	−0.422404
$27$	−35.0000	−0.249472
$28$	−136.000	−0.917914
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	32.0000	0.176777
$33$	189.000	0.996990
$34$	42.0000	0.211851
$35$	0	0
$36$	88.0000	0.407407
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	70.0000	0.298829
$39$	−196.000	−0.804747
$40$	0	0
$41$	357.000	1.35985	0.679927	−	0.733280i	$-0.262011\pi$
$41$	357.000	1.35985	0.679927	+	0.733280i	$0.262011\pi$
$42$	−476.000	−1.74877
$43$	−148.000	−0.524879	−0.262439	−	0.964948i	$-0.584527\pi$
$43$	−148.000	−0.524879	−0.262439	+	0.964948i	$0.584527\pi$
$44$	108.000	0.370037
$45$	0	0
$46$	−156.000	−0.500021
$47$	−84.0000	−0.260695	−0.130347	−	0.991468i	$-0.541609\pi$
$47$	−84.0000	−0.260695	−0.130347	+	0.991468i	$0.541609\pi$
$48$	112.000	0.336788
$49$	813.000	2.37026
$50$	0	0
$51$	147.000	0.403610
$52$	−112.000	−0.298685
$53$	702.000	1.81938	0.909690	−	0.415288i	$-0.136319\pi$
$53$	702.000	1.81938	0.909690	+	0.415288i	$0.136319\pi$
$54$	−70.0000	−0.176404
$55$	0	0
$56$	−272.000	−0.649063
$57$	245.000	0.569317
$58$	−240.000	−0.543337
$59$	−840.000	−1.85354	−0.926769	−	0.375633i	$-0.877425\pi$
$59$	−840.000	−1.85354	−0.926769	+	0.375633i	$0.877425\pi$
$60$	0	0
$61$	−238.000	−0.499554	−0.249777	−	0.968303i	$-0.580357\pi$
$61$	−238.000	−0.499554	−0.249777	+	0.968303i	$0.580357\pi$
$62$	364.000	0.745614
$63$	−748.000	−1.49586
$64$	64.0000	0.125000
$65$	0	0
$66$	378.000	0.704979
$67$	461.000	0.840599	0.420299	−	0.907386i	$-0.361925\pi$
$67$	461.000	0.840599	0.420299	+	0.907386i	$0.361925\pi$
$68$	84.0000	0.149801
$69$	−546.000	−0.952618
$70$	0	0
$71$	−708.000	−1.18344	−0.591719	−	0.806144i	$-0.701551\pi$
$71$	−708.000	−1.18344	−0.591719	+	0.806144i	$0.701551\pi$
$72$	176.000	0.288081
$73$	−133.000	−0.213239	−0.106620	−	0.994300i	$-0.534003\pi$
$73$	−133.000	−0.213239	−0.106620	+	0.994300i	$0.534003\pi$
$74$	292.000	0.458707
$75$	0	0
$76$	140.000	0.211304
$77$	−918.000	−1.35865
$78$	−392.000	−0.569042
$79$	650.000	0.925705	0.462853	−	0.886435i	$-0.346826\pi$
$79$	650.000	0.925705	0.462853	+	0.886435i	$0.346826\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	714.000	0.961562
$83$	−903.000	−1.19418	−0.597091	−	0.802173i	$-0.703677\pi$
$83$	−903.000	−1.19418	−0.597091	+	0.802173i	$0.703677\pi$
$84$	−952.000	−1.23657
$85$	0	0
$86$	−296.000	−0.371145
$87$	−840.000	−1.03514
$88$	216.000	0.261655
$89$	735.000	0.875392	0.437696	−	0.899123i	$-0.355795\pi$
$89$	735.000	0.875392	0.437696	+	0.899123i	$0.355795\pi$
$90$	0	0
$91$	952.000	1.09667
$92$	−312.000	−0.353568
$93$	1274.00	1.42051
$94$	−168.000	−0.184339
$95$	0	0
$96$	224.000	0.238145
$97$	1106.00	1.15770	0.578852	−	0.815433i	$-0.303501\pi$
$97$	1106.00	1.15770	0.578852	+	0.815433i	$0.303501\pi$
$98$	1626.00	1.67603
$99$	594.000	0.603023
$100$	0	0
$101$	462.000	0.455156	0.227578	−	0.973760i	$-0.426919\pi$
$101$	462.000	0.455156	0.227578	+	0.973760i	$0.426919\pi$
$102$	294.000	0.285395
$103$	812.000	0.776784	0.388392	−	0.921494i	$-0.373031\pi$
$103$	812.000	0.776784	0.388392	+	0.921494i	$0.373031\pi$
$104$	−224.000	−0.211202
$105$	0	0
$106$	1404.00	1.28650
$107$	−789.000	−0.712855	−0.356428	−	0.934323i	$-0.616005\pi$
$107$	−789.000	−0.712855	−0.356428	+	0.934323i	$0.616005\pi$
$108$	−140.000	−0.124736
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	0	0
$111$	1022.00	0.873909
$112$	−544.000	−0.458957
$113$	−2073.00	−1.72576	−0.862882	−	0.505405i	$-0.831343\pi$
$113$	−2073.00	−1.72576	−0.862882	+	0.505405i	$0.831343\pi$
$114$	490.000	0.402568
$115$	0	0
$116$	−480.000	−0.384197
$117$	−616.000	−0.486745
$118$	−1680.00	−1.31065
$119$	−714.000	−0.550019
$120$	0	0
$121$	−602.000	−0.452292
$122$	−476.000	−0.353238
$123$	2499.00	1.83193
$124$	728.000	0.527228
$125$	0	0
$126$	−1496.00	−1.05773
$127$	−1114.00	−0.778358	−0.389179	−	0.921162i	$-0.627241\pi$
$127$	−1114.00	−0.778358	−0.389179	+	0.921162i	$0.627241\pi$
$128$	128.000	0.0883883
$129$	−1036.00	−0.707091
$130$	0	0
$131$	252.000	0.168071	0.0840357	−	0.996463i	$-0.473219\pi$
$131$	252.000	0.168071	0.0840357	+	0.996463i	$0.473219\pi$
$132$	756.000	0.498495
$133$	−1190.00	−0.775835
$134$	922.000	0.594393
$135$	0	0
$136$	168.000	0.105926
$137$	1941.00	1.21044	0.605222	−	0.796057i	$-0.293084\pi$
$137$	1941.00	1.21044	0.605222	+	0.796057i	$0.293084\pi$
$138$	−1092.00	−0.673603
$139$	−1645.00	−1.00379	−0.501896	−	0.864928i	$-0.667364\pi$
$139$	−1645.00	−1.00379	−0.501896	+	0.864928i	$0.667364\pi$
$140$	0	0
$141$	−588.000	−0.351195
$142$	−1416.00	−0.836817
$143$	−756.000	−0.442097
$144$	352.000	0.203704
$145$	0	0
$146$	−266.000	−0.150783
$147$	5691.00	3.19310
$148$	584.000	0.324355
$149$	−1800.00	−0.989676	−0.494838	−	0.868985i	$-0.664773\pi$
$149$	−1800.00	−0.989676	−0.494838	+	0.868985i	$0.664773\pi$
$150$	0	0
$151$	−3298.00	−1.77740	−0.888700	−	0.458489i	$-0.848391\pi$
$151$	−3298.00	−1.77740	−0.888700	+	0.458489i	$0.848391\pi$
$152$	280.000	0.149414
$153$	462.000	0.244121
$154$	−1836.00	−0.960708
$155$	0	0
$156$	−784.000	−0.402373
$157$	266.000	0.135217	0.0676086	−	0.997712i	$-0.478463\pi$
$157$	266.000	0.135217	0.0676086	+	0.997712i	$0.478463\pi$
$158$	1300.00	0.654572
$159$	4914.00	2.45098
$160$	0	0
$161$	2652.00	1.29818
$162$	−1678.00	−0.813803
$163$	1157.00	0.555971	0.277985	−	0.960585i	$-0.410333\pi$
$163$	1157.00	0.555971	0.277985	+	0.960585i	$0.410333\pi$
$164$	1428.00	0.679927
$165$	0	0
$166$	−1806.00	−0.844414
$167$	−1764.00	−0.817380	−0.408690	−	0.912673i	$-0.634014\pi$
$167$	−1764.00	−0.817380	−0.408690	+	0.912673i	$0.634014\pi$
$168$	−1904.00	−0.874386
$169$	−1413.00	−0.643150
$170$	0	0
$171$	770.000	0.344347
$172$	−592.000	−0.262439
$173$	−1848.00	−0.812144	−0.406072	−	0.913841i	$-0.633102\pi$
$173$	−1848.00	−0.812144	−0.406072	+	0.913841i	$0.633102\pi$
$174$	−1680.00	−0.731957
$175$	0	0
$176$	432.000	0.185018
$177$	−5880.00	−2.49699
$178$	1470.00	0.618995
$179$	135.000	0.0563708	0.0281854	−	0.999603i	$-0.491027\pi$
$179$	135.000	0.0563708	0.0281854	+	0.999603i	$0.491027\pi$
$180$	0	0
$181$	2282.00	0.937126	0.468563	−	0.883430i	$-0.344772\pi$
$181$	2282.00	0.937126	0.468563	+	0.883430i	$0.344772\pi$
$182$	1904.00	0.775461
$183$	−1666.00	−0.672974
$184$	−624.000	−0.250010
$185$	0	0
$186$	2548.00	1.00445
$187$	567.000	0.221728
$188$	−336.000	−0.130347
$189$	1190.00	0.457988
$190$	0	0
$191$	−1398.00	−0.529611	−0.264806	−	0.964302i	$-0.585308\pi$
$191$	−1398.00	−0.529611	−0.264806	+	0.964302i	$0.585308\pi$
$192$	448.000	0.168394
$193$	3317.00	1.23711	0.618557	−	0.785740i	$-0.287718\pi$
$193$	3317.00	1.23711	0.618557	+	0.785740i	$0.287718\pi$
$194$	2212.00	0.818620
$195$	0	0
$196$	3252.00	1.18513
$197$	1686.00	0.609759	0.304880	−	0.952391i	$-0.401384\pi$
$197$	1686.00	0.609759	0.304880	+	0.952391i	$0.401384\pi$
$198$	1188.00	0.426401
$199$	−1540.00	−0.548581	−0.274291	−	0.961647i	$-0.588443\pi$
$199$	−1540.00	−0.548581	−0.274291	+	0.961647i	$0.588443\pi$
$200$	0	0
$201$	3227.00	1.13241
$202$	924.000	0.321844
$203$	4080.00	1.41064
$204$	588.000	0.201805
$205$	0	0
$206$	1624.00	0.549269
$207$	−1716.00	−0.576185
$208$	−448.000	−0.149342
$209$	945.000	0.312761
$210$	0	0
$211$	−3043.00	−0.992838	−0.496419	−	0.868083i	$-0.665352\pi$
$211$	−3043.00	−0.992838	−0.496419	+	0.868083i	$0.665352\pi$
$212$	2808.00	0.909690
$213$	−4956.00	−1.59427
$214$	−1578.00	−0.504065
$215$	0	0
$216$	−280.000	−0.0882018
$217$	−6188.00	−1.93580
$218$	460.000	0.142913
$219$	−931.000	−0.287266
$220$	0	0
$221$	−588.000	−0.178974
$222$	2044.00	0.617947
$223$	3332.00	1.00057	0.500285	−	0.865861i	$-0.333228\pi$
$223$	3332.00	1.00057	0.500285	+	0.865861i	$0.333228\pi$
$224$	−1088.00	−0.324532
$225$	0	0
$226$	−4146.00	−1.22030
$227$	1596.00	0.466653	0.233327	−	0.972398i	$-0.425039\pi$
$227$	1596.00	0.466653	0.233327	+	0.972398i	$0.425039\pi$
$228$	980.000	0.284658
$229$	4340.00	1.25238	0.626191	−	0.779670i	$-0.284613\pi$
$229$	4340.00	1.25238	0.626191	+	0.779670i	$0.284613\pi$
$230$	0	0
$231$	−6426.00	−1.83030
$232$	−960.000	−0.271668
$233$	−3018.00	−0.848565	−0.424283	−	0.905530i	$-0.639474\pi$
$233$	−3018.00	−0.848565	−0.424283	+	0.905530i	$0.639474\pi$
$234$	−1232.00	−0.344181
$235$	0	0
$236$	−3360.00	−0.926769
$237$	4550.00	1.24706
$238$	−1428.00	−0.388922
$239$	4440.00	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$239$	4440.00	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$240$	0	0
$241$	−3703.00	−0.989756	−0.494878	−	0.868962i	$-0.664787\pi$
$241$	−3703.00	−0.989756	−0.494878	+	0.868962i	$0.664787\pi$
$242$	−1204.00	−0.319818
$243$	−4928.00	−1.30095
$244$	−952.000	−0.249777
$245$	0	0
$246$	4998.00	1.29537
$247$	−980.000	−0.252453
$248$	1456.00	0.372807
$249$	−6321.00	−1.60874
$250$	0	0
$251$	7077.00	1.77967	0.889833	−	0.456286i	$-0.150821\pi$
$251$	7077.00	1.77967	0.889833	+	0.456286i	$0.150821\pi$
$252$	−2992.00	−0.747930
$253$	−2106.00	−0.523332
$254$	−2228.00	−0.550382
$255$	0	0
$256$	256.000	0.0625000
$257$	6846.00	1.66164	0.830821	−	0.556540i	$-0.187872\pi$
$257$	6846.00	1.66164	0.830821	+	0.556540i	$0.187872\pi$
$258$	−2072.00	−0.499989
$259$	−4964.00	−1.19092
$260$	0	0
$261$	−2640.00	−0.626099
$262$	504.000	0.118844
$263$	−3438.00	−0.806069	−0.403035	−	0.915185i	$-0.632045\pi$
$263$	−3438.00	−0.806069	−0.403035	+	0.915185i	$0.632045\pi$
$264$	1512.00	0.352489
$265$	0	0
$266$	−2380.00	−0.548598
$267$	5145.00	1.17928
$268$	1844.00	0.420299
$269$	−1680.00	−0.380786	−0.190393	−	0.981708i	$-0.560976\pi$
$269$	−1680.00	−0.380786	−0.190393	+	0.981708i	$0.560976\pi$
$270$	0	0
$271$	5222.00	1.17053	0.585266	−	0.810842i	$-0.300990\pi$
$271$	5222.00	1.17053	0.585266	+	0.810842i	$0.300990\pi$
$272$	336.000	0.0749007
$273$	6664.00	1.47738
$274$	3882.00	0.855913
$275$	0	0
$276$	−2184.00	−0.476309
$277$	−1384.00	−0.300204	−0.150102	−	0.988671i	$-0.547960\pi$
$277$	−1384.00	−0.300204	−0.150102	+	0.988671i	$0.547960\pi$
$278$	−3290.00	−0.709788
$279$	4004.00	0.859187
$280$	0	0
$281$	−3858.00	−0.819036	−0.409518	−	0.912302i	$-0.634303\pi$
$281$	−3858.00	−0.819036	−0.409518	+	0.912302i	$0.634303\pi$
$282$	−1176.00	−0.248333
$283$	4277.00	0.898379	0.449190	−	0.893437i	$-0.351713\pi$
$283$	4277.00	0.898379	0.449190	+	0.893437i	$0.351713\pi$
$284$	−2832.00	−0.591719
$285$	0	0
$286$	−1512.00	−0.312610
$287$	−12138.0	−2.49646
$288$	704.000	0.144040
$289$	−4472.00	−0.910238
$290$	0	0
$291$	7742.00	1.55960
$292$	−532.000	−0.106620
$293$	6342.00	1.26452	0.632259	−	0.774757i	$-0.282128\pi$
$293$	6342.00	1.26452	0.632259	+	0.774757i	$0.282128\pi$
$294$	11382.0	2.25786
$295$	0	0
$296$	1168.00	0.229353
$297$	−945.000	−0.184628
$298$	−3600.00	−0.699807
$299$	2184.00	0.422421
$300$	0	0
$301$	5032.00	0.963587
$302$	−6596.00	−1.25681
$303$	3234.00	0.613163
$304$	560.000	0.105652
$305$	0	0
$306$	924.000	0.172619
$307$	5831.00	1.08402	0.542008	−	0.840373i	$-0.317664\pi$
$307$	5831.00	1.08402	0.542008	+	0.840373i	$0.317664\pi$
$308$	−3672.00	−0.679323
$309$	5684.00	1.04644
$310$	0	0
$311$	−2478.00	−0.451815	−0.225908	−	0.974149i	$-0.572535\pi$
$311$	−2478.00	−0.451815	−0.225908	+	0.974149i	$0.572535\pi$
$312$	−1568.00	−0.284521
$313$	−2758.00	−0.498056	−0.249028	−	0.968496i	$-0.580111\pi$
$313$	−2758.00	−0.498056	−0.249028	+	0.968496i	$0.580111\pi$
$314$	532.000	0.0956130
$315$	0	0
$316$	2600.00	0.462853
$317$	636.000	0.112686	0.0563428	−	0.998411i	$-0.482056\pi$
$317$	636.000	0.112686	0.0563428	+	0.998411i	$0.482056\pi$
$318$	9828.00	1.73310
$319$	−3240.00	−0.568668
$320$	0	0
$321$	−5523.00	−0.960323
$322$	5304.00	0.917951
$323$	735.000	0.126615
$324$	−3356.00	−0.575446
$325$	0	0
$326$	2314.00	0.393131
$327$	1610.00	0.272273
$328$	2856.00	0.480781
$329$	2856.00	0.478591
$330$	0	0
$331$	6887.00	1.14364	0.571818	−	0.820380i	$-0.306238\pi$
$331$	6887.00	1.14364	0.571818	+	0.820380i	$0.306238\pi$
$332$	−3612.00	−0.597091
$333$	3212.00	0.528578
$334$	−3528.00	−0.577975
$335$	0	0
$336$	−3808.00	−0.618284
$337$	7331.00	1.18500	0.592500	−	0.805570i	$-0.298141\pi$
$337$	7331.00	1.18500	0.592500	+	0.805570i	$0.298141\pi$
$338$	−2826.00	−0.454776
$339$	−14511.0	−2.32487
$340$	0	0
$341$	4914.00	0.780375
$342$	1540.00	0.243490
$343$	−15980.0	−2.51557
$344$	−1184.00	−0.185573
$345$	0	0
$346$	−3696.00	−0.574272
$347$	−5349.00	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$347$	−5349.00	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$348$	−3360.00	−0.517572
$349$	11270.0	1.72857	0.864283	−	0.503007i	$-0.167773\pi$
$349$	11270.0	1.72857	0.864283	+	0.503007i	$0.167773\pi$
$350$	0	0
$351$	980.000	0.149027
$352$	864.000	0.130828
$353$	1302.00	0.196313	0.0981565	−	0.995171i	$-0.468705\pi$
$353$	1302.00	0.196313	0.0981565	+	0.995171i	$0.468705\pi$
$354$	−11760.0	−1.76564
$355$	0	0
$356$	2940.00	0.437696
$357$	−4998.00	−0.740959
$358$	270.000	0.0398602
$359$	−750.000	−0.110260	−0.0551302	−	0.998479i	$-0.517557\pi$
$359$	−750.000	−0.110260	−0.0551302	+	0.998479i	$0.517557\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	4564.00	0.662648
$363$	−4214.00	−0.609305
$364$	3808.00	0.548334
$365$	0	0
$366$	−3332.00	−0.475865
$367$	−1204.00	−0.171249	−0.0856244	−	0.996327i	$-0.527288\pi$
$367$	−1204.00	−0.171249	−0.0856244	+	0.996327i	$0.527288\pi$
$368$	−1248.00	−0.176784
$369$	7854.00	1.10803
$370$	0	0
$371$	−23868.0	−3.34007
$372$	5096.00	0.710256
$373$	−1198.00	−0.166301	−0.0831503	−	0.996537i	$-0.526498\pi$
$373$	−1198.00	−0.166301	−0.0831503	+	0.996537i	$0.526498\pi$
$374$	1134.00	0.156785
$375$	0	0
$376$	−672.000	−0.0921696
$377$	3360.00	0.459015
$378$	2380.00	0.323847
$379$	8105.00	1.09849	0.549243	−	0.835663i	$-0.314916\pi$
$379$	8105.00	1.09849	0.549243	+	0.835663i	$0.314916\pi$
$380$	0	0
$381$	−7798.00	−1.04857
$382$	−2796.00	−0.374492
$383$	−3318.00	−0.442668	−0.221334	−	0.975198i	$-0.571041\pi$
$383$	−3318.00	−0.442668	−0.221334	+	0.975198i	$0.571041\pi$
$384$	896.000	0.119072
$385$	0	0
$386$	6634.00	0.874771
$387$	−3256.00	−0.427679
$388$	4424.00	0.578852
$389$	−13770.0	−1.79477	−0.897387	−	0.441245i	$-0.854537\pi$
$389$	−13770.0	−1.79477	−0.897387	+	0.441245i	$0.854537\pi$
$390$	0	0
$391$	−1638.00	−0.211860
$392$	6504.00	0.838014
$393$	1764.00	0.226417
$394$	3372.00	0.431165
$395$	0	0
$396$	2376.00	0.301511
$397$	−3724.00	−0.470786	−0.235393	−	0.971900i	$-0.575638\pi$
$397$	−3724.00	−0.470786	−0.235393	+	0.971900i	$0.575638\pi$
$398$	−3080.00	−0.387906
$399$	−8330.00	−1.04517
$400$	0	0
$401$	6117.00	0.761767	0.380883	−	0.924623i	$-0.375620\pi$
$401$	6117.00	0.761767	0.380883	+	0.924623i	$0.375620\pi$
$402$	6454.00	0.800737
$403$	−5096.00	−0.629900
$404$	1848.00	0.227578
$405$	0	0
$406$	8160.00	0.997473
$407$	3942.00	0.480093
$408$	1176.00	0.142698
$409$	6125.00	0.740493	0.370247	−	0.928933i	$-0.379273\pi$
$409$	6125.00	0.740493	0.370247	+	0.928933i	$0.379273\pi$
$410$	0	0
$411$	13587.0	1.63065
$412$	3248.00	0.388392
$413$	28560.0	3.40277
$414$	−3432.00	−0.407424
$415$	0	0
$416$	−896.000	−0.105601
$417$	−11515.0	−1.35226
$418$	1890.00	0.221155
$419$	1575.00	0.183637	0.0918184	−	0.995776i	$-0.470732\pi$
$419$	1575.00	0.183637	0.0918184	+	0.995776i	$0.470732\pi$
$420$	0	0
$421$	−988.000	−0.114376	−0.0571879	−	0.998363i	$-0.518213\pi$
$421$	−988.000	−0.114376	−0.0571879	+	0.998363i	$0.518213\pi$
$422$	−6086.00	−0.702042
$423$	−1848.00	−0.212418
$424$	5616.00	0.643248
$425$	0	0
$426$	−9912.00	−1.12732
$427$	8092.00	0.917094
$428$	−3156.00	−0.356428
$429$	−5292.00	−0.595571
$430$	0	0
$431$	−558.000	−0.0623617	−0.0311809	−	0.999514i	$-0.509927\pi$
$431$	−558.000	−0.0623617	−0.0311809	+	0.999514i	$0.509927\pi$
$432$	−560.000	−0.0623681
$433$	−2443.00	−0.271139	−0.135569	−	0.990768i	$-0.543286\pi$
$433$	−2443.00	−0.271139	−0.135569	+	0.990768i	$0.543286\pi$
$434$	−12376.0	−1.36882
$435$	0	0
$436$	920.000	0.101055
$437$	−2730.00	−0.298841
$438$	−1862.00	−0.203127
$439$	12320.0	1.33941	0.669706	−	0.742627i	$-0.266420\pi$
$439$	12320.0	1.33941	0.669706	+	0.742627i	$0.266420\pi$
$440$	0	0
$441$	17886.0	1.93132
$442$	−1176.00	−0.126553
$443$	−2343.00	−0.251285	−0.125643	−	0.992076i	$-0.540099\pi$
$443$	−2343.00	−0.251285	−0.125643	+	0.992076i	$0.540099\pi$
$444$	4088.00	0.436955
$445$	0	0
$446$	6664.00	0.707510
$447$	−12600.0	−1.33324
$448$	−2176.00	−0.229478
$449$	10905.0	1.14619	0.573094	−	0.819489i	$-0.305743\pi$
$449$	10905.0	1.14619	0.573094	+	0.819489i	$0.305743\pi$
$450$	0	0
$451$	9639.00	1.00639
$452$	−8292.00	−0.862882
$453$	−23086.0	−2.39443
$454$	3192.00	0.329974
$455$	0	0
$456$	1960.00	0.201284
$457$	−3319.00	−0.339729	−0.169865	−	0.985467i	$-0.554333\pi$
$457$	−3319.00	−0.339729	−0.169865	+	0.985467i	$0.554333\pi$
$458$	8680.00	0.885567
$459$	−735.000	−0.0747426
$460$	0	0
$461$	−6468.00	−0.653459	−0.326730	−	0.945118i	$-0.605947\pi$
$461$	−6468.00	−0.653459	−0.326730	+	0.945118i	$0.605947\pi$
$462$	−12852.0	−1.29422
$463$	11972.0	1.20170	0.600849	−	0.799363i	$-0.294829\pi$
$463$	11972.0	1.20170	0.600849	+	0.799363i	$0.294829\pi$
$464$	−1920.00	−0.192099
$465$	0	0
$466$	−6036.00	−0.600026
$467$	6636.00	0.657553	0.328777	−	0.944408i	$-0.393364\pi$
$467$	6636.00	0.657553	0.328777	+	0.944408i	$0.393364\pi$
$468$	−2464.00	−0.243373
$469$	−15674.0	−1.54319
$470$	0	0
$471$	1862.00	0.182158
$472$	−6720.00	−0.655324
$473$	−3996.00	−0.388449
$474$	9100.00	0.881808
$475$	0	0
$476$	−2856.00	−0.275010
$477$	15444.0	1.48246
$478$	8880.00	0.849711
$479$	−630.000	−0.0600949	−0.0300474	−	0.999548i	$-0.509566\pi$
$479$	−630.000	−0.0600949	−0.0300474	+	0.999548i	$0.509566\pi$
$480$	0	0
$481$	−4088.00	−0.387519
$482$	−7406.00	−0.699863
$483$	18564.0	1.74884
$484$	−2408.00	−0.226146
$485$	0	0
$486$	−9856.00	−0.919912
$487$	10646.0	0.990588	0.495294	−	0.868725i	$-0.335060\pi$
$487$	10646.0	0.990588	0.495294	+	0.868725i	$0.335060\pi$
$488$	−1904.00	−0.176619
$489$	8099.00	0.748976
$490$	0	0
$491$	−2388.00	−0.219489	−0.109744	−	0.993960i	$-0.535003\pi$
$491$	−2388.00	−0.219489	−0.109744	+	0.993960i	$0.535003\pi$
$492$	9996.00	0.915964
$493$	−2520.00	−0.230213
$494$	−1960.00	−0.178511
$495$	0	0
$496$	2912.00	0.263614
$497$	24072.0	2.17259
$498$	−12642.0	−1.13755
$499$	−10540.0	−0.945562	−0.472781	−	0.881180i	$-0.656750\pi$
$499$	−10540.0	−0.945562	−0.472781	+	0.881180i	$0.656750\pi$
$500$	0	0
$501$	−12348.0	−1.10113
$502$	14154.0	1.25841
$503$	6972.00	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$503$	6972.00	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$504$	−5984.00	−0.528866
$505$	0	0
$506$	−4212.00	−0.370052
$507$	−9891.00	−0.866420
$508$	−4456.00	−0.389179
$509$	−9030.00	−0.786341	−0.393171	−	0.919466i	$-0.628622\pi$
$509$	−9030.00	−0.786341	−0.393171	+	0.919466i	$0.628622\pi$
$510$	0	0
$511$	4522.00	0.391471
$512$	512.000	0.0441942
$513$	−1225.00	−0.105429
$514$	13692.0	1.17496
$515$	0	0
$516$	−4144.00	−0.353545
$517$	−2268.00	−0.192933
$518$	−9928.00	−0.842107
$519$	−12936.0	−1.09408
$520$	0	0
$521$	10437.0	0.877645	0.438823	−	0.898574i	$-0.355396\pi$
$521$	10437.0	0.877645	0.438823	+	0.898574i	$0.355396\pi$
$522$	−5280.00	−0.442719
$523$	−8113.00	−0.678311	−0.339156	−	0.940730i	$-0.610141\pi$
$523$	−8113.00	−0.678311	−0.339156	+	0.940730i	$0.610141\pi$
$524$	1008.00	0.0840357
$525$	0	0
$526$	−6876.00	−0.569977
$527$	3822.00	0.315918
$528$	3024.00	0.249248
$529$	−6083.00	−0.499959
$530$	0	0
$531$	−18480.0	−1.51029
$532$	−4760.00	−0.387918
$533$	−9996.00	−0.812336
$534$	10290.0	0.833880
$535$	0	0
$536$	3688.00	0.297197
$537$	945.000	0.0759400
$538$	−3360.00	−0.269256
$539$	21951.0	1.75417
$540$	0	0
$541$	−14848.0	−1.17997	−0.589986	−	0.807413i	$-0.700867\pi$
$541$	−14848.0	−1.17997	−0.589986	+	0.807413i	$0.700867\pi$
$542$	10444.0	0.827690
$543$	15974.0	1.26245
$544$	672.000	0.0529628
$545$	0	0
$546$	13328.0	1.04466
$547$	−20329.0	−1.58904	−0.794520	−	0.607238i	$-0.792278\pi$
$547$	−20329.0	−1.58904	−0.794520	+	0.607238i	$0.792278\pi$
$548$	7764.00	0.605222
$549$	−5236.00	−0.407044
$550$	0	0
$551$	−4200.00	−0.324730
$552$	−4368.00	−0.336801
$553$	−22100.0	−1.69944
$554$	−2768.00	−0.212276
$555$	0	0
$556$	−6580.00	−0.501896
$557$	−15324.0	−1.16571	−0.582853	−	0.812577i	$-0.698064\pi$
$557$	−15324.0	−1.16571	−0.582853	+	0.812577i	$0.698064\pi$
$558$	8008.00	0.607537
$559$	4144.00	0.313547
$560$	0	0
$561$	3969.00	0.298701
$562$	−7716.00	−0.579146
$563$	−9408.00	−0.704263	−0.352131	−	0.935951i	$-0.614543\pi$
$563$	−9408.00	−0.704263	−0.352131	+	0.935951i	$0.614543\pi$
$564$	−2352.00	−0.175598
$565$	0	0
$566$	8554.00	0.635250
$567$	28526.0	2.11284
$568$	−5664.00	−0.418409
$569$	−24375.0	−1.79588	−0.897938	−	0.440122i	$-0.854935\pi$
$569$	−24375.0	−1.79588	−0.897938	+	0.440122i	$0.854935\pi$
$570$	0	0
$571$	−988.000	−0.0724107	−0.0362054	−	0.999344i	$-0.511527\pi$
$571$	−988.000	−0.0724107	−0.0362054	+	0.999344i	$0.511527\pi$
$572$	−3024.00	−0.221049
$573$	−9786.00	−0.713466
$574$	−24276.0	−1.76526
$575$	0	0
$576$	1408.00	0.101852
$577$	−16429.0	−1.18535	−0.592676	−	0.805441i	$-0.701929\pi$
$577$	−16429.0	−1.18535	−0.592676	+	0.805441i	$0.701929\pi$
$578$	−8944.00	−0.643636
$579$	23219.0	1.66658
$580$	0	0
$581$	30702.0	2.19231
$582$	15484.0	1.10280
$583$	18954.0	1.34647
$584$	−1064.00	−0.0753915
$585$	0	0
$586$	12684.0	0.894149
$587$	−13839.0	−0.973078	−0.486539	−	0.873659i	$-0.661741\pi$
$587$	−13839.0	−0.973078	−0.486539	+	0.873659i	$0.661741\pi$
$588$	22764.0	1.59655
$589$	6370.00	0.445622
$590$	0	0
$591$	11802.0	0.821437
$592$	2336.00	0.162177
$593$	14007.0	0.969981	0.484990	−	0.874520i	$-0.338823\pi$
$593$	14007.0	0.969981	0.484990	+	0.874520i	$0.338823\pi$
$594$	−1890.00	−0.130552
$595$	0	0
$596$	−7200.00	−0.494838
$597$	−10780.0	−0.739022
$598$	4368.00	0.298697
$599$	21090.0	1.43859	0.719294	−	0.694706i	$-0.244465\pi$
$599$	21090.0	1.43859	0.719294	+	0.694706i	$0.244465\pi$
$600$	0	0
$601$	5747.00	0.390058	0.195029	−	0.980797i	$-0.437520\pi$
$601$	5747.00	0.390058	0.195029	+	0.980797i	$0.437520\pi$
$602$	10064.0	0.681359
$603$	10142.0	0.684932
$604$	−13192.0	−0.888700
$605$	0	0
$606$	6468.00	0.433572
$607$	17696.0	1.18329	0.591646	−	0.806198i	$-0.298478\pi$
$607$	17696.0	1.18329	0.591646	+	0.806198i	$0.298478\pi$
$608$	1120.00	0.0747072
$609$	28560.0	1.90034
$610$	0	0
$611$	2352.00	0.155731
$612$	1848.00	0.122060
$613$	26102.0	1.71982	0.859910	−	0.510445i	$-0.170519\pi$
$613$	26102.0	1.71982	0.859910	+	0.510445i	$0.170519\pi$
$614$	11662.0	0.766515
$615$	0	0
$616$	−7344.00	−0.480354
$617$	−4194.00	−0.273653	−0.136827	−	0.990595i	$-0.543690\pi$
$617$	−4194.00	−0.273653	−0.136827	+	0.990595i	$0.543690\pi$
$618$	11368.0	0.739948
$619$	−7420.00	−0.481801	−0.240901	−	0.970550i	$-0.577443\pi$
$619$	−7420.00	−0.481801	−0.240901	+	0.970550i	$0.577443\pi$
$620$	0	0
$621$	2730.00	0.176411
$622$	−4956.00	−0.319482
$623$	−24990.0	−1.60707
$624$	−3136.00	−0.201187
$625$	0	0
$626$	−5516.00	−0.352178
$627$	6615.00	0.421336
$628$	1064.00	0.0676086
$629$	3066.00	0.194355
$630$	0	0
$631$	−5818.00	−0.367054	−0.183527	−	0.983015i	$-0.558751\pi$
$631$	−5818.00	−0.367054	−0.183527	+	0.983015i	$0.558751\pi$
$632$	5200.00	0.327286
$633$	−21301.0	−1.33750
$634$	1272.00	0.0796807
$635$	0	0
$636$	19656.0	1.22549
$637$	−22764.0	−1.41592
$638$	−6480.00	−0.402109
$639$	−15576.0	−0.964283
$640$	0	0
$641$	−29478.0	−1.81640	−0.908199	−	0.418539i	$-0.862542\pi$
$641$	−29478.0	−1.81640	−0.908199	+	0.418539i	$0.862542\pi$
$642$	−11046.0	−0.679051
$643$	5852.00	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$643$	5852.00	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$644$	10608.0	0.649090
$645$	0	0
$646$	1470.00	0.0895300
$647$	−29484.0	−1.79155	−0.895777	−	0.444503i	$-0.853380\pi$
$647$	−29484.0	−1.79155	−0.895777	+	0.444503i	$0.853380\pi$
$648$	−6712.00	−0.406902
$649$	−22680.0	−1.37175
$650$	0	0
$651$	−43316.0	−2.60782
$652$	4628.00	0.277985
$653$	−3498.00	−0.209628	−0.104814	−	0.994492i	$-0.533425\pi$
$653$	−3498.00	−0.209628	−0.104814	+	0.994492i	$0.533425\pi$
$654$	3220.00	0.192526
$655$	0	0
$656$	5712.00	0.339964
$657$	−2926.00	−0.173751
$658$	5712.00	0.338415
$659$	7905.00	0.467276	0.233638	−	0.972324i	$-0.424937\pi$
$659$	7905.00	0.467276	0.233638	+	0.972324i	$0.424937\pi$
$660$	0	0
$661$	27272.0	1.60478	0.802389	−	0.596802i	$-0.203562\pi$
$661$	27272.0	1.60478	0.802389	+	0.596802i	$0.203562\pi$
$662$	13774.0	0.808673
$663$	−4116.00	−0.241104
$664$	−7224.00	−0.422207
$665$	0	0
$666$	6424.00	0.373761
$667$	9360.00	0.543359
$668$	−7056.00	−0.408690
$669$	23324.0	1.34792
$670$	0	0
$671$	−6426.00	−0.369706
$672$	−7616.00	−0.437193
$673$	12602.0	0.721800	0.360900	−	0.932605i	$-0.382470\pi$
$673$	12602.0	0.721800	0.360900	+	0.932605i	$0.382470\pi$
$674$	14662.0	0.837922
$675$	0	0
$676$	−5652.00	−0.321575
$677$	25536.0	1.44967	0.724836	−	0.688921i	$-0.241915\pi$
$677$	25536.0	1.44967	0.724836	+	0.688921i	$0.241915\pi$
$678$	−29022.0	−1.64393
$679$	−37604.0	−2.12534
$680$	0	0
$681$	11172.0	0.628652
$682$	9828.00	0.551809
$683$	2127.00	0.119162	0.0595808	−	0.998223i	$-0.481024\pi$
$683$	2127.00	0.119162	0.0595808	+	0.998223i	$0.481024\pi$
$684$	3080.00	0.172174
$685$	0	0
$686$	−31960.0	−1.77877
$687$	30380.0	1.68715
$688$	−2368.00	−0.131220
$689$	−19656.0	−1.08684
$690$	0	0
$691$	−8953.00	−0.492892	−0.246446	−	0.969157i	$-0.579263\pi$
$691$	−8953.00	−0.492892	−0.246446	+	0.969157i	$0.579263\pi$
$692$	−7392.00	−0.406072
$693$	−20196.0	−1.10705
$694$	−10698.0	−0.585145
$695$	0	0
$696$	−6720.00	−0.365978
$697$	7497.00	0.407416
$698$	22540.0	1.22228
$699$	−21126.0	−1.14315
$700$	0	0
$701$	23172.0	1.24849	0.624247	−	0.781227i	$-0.285406\pi$
$701$	23172.0	1.24849	0.624247	+	0.781227i	$0.285406\pi$
$702$	1960.00	0.105378
$703$	5110.00	0.274150
$704$	1728.00	0.0925092
$705$	0	0
$706$	2604.00	0.138814
$707$	−15708.0	−0.835587
$708$	−23520.0	−1.24850
$709$	9620.00	0.509572	0.254786	−	0.966997i	$-0.417995\pi$
$709$	9620.00	0.509572	0.254786	+	0.966997i	$0.417995\pi$
$710$	0	0
$711$	14300.0	0.754278
$712$	5880.00	0.309498
$713$	−14196.0	−0.745644
$714$	−9996.00	−0.523937
$715$	0	0
$716$	540.000	0.0281854
$717$	31080.0	1.61883
$718$	−1500.00	−0.0779659
$719$	4830.00	0.250527	0.125263	−	0.992124i	$-0.460022\pi$
$719$	4830.00	0.250527	0.125263	+	0.992124i	$0.460022\pi$
$720$	0	0
$721$	−27608.0	−1.42604
$722$	−11268.0	−0.580819
$723$	−25921.0	−1.33335
$724$	9128.00	0.468563
$725$	0	0
$726$	−8428.00	−0.430844
$727$	11816.0	0.602794	0.301397	−	0.953499i	$-0.402547\pi$
$727$	11816.0	0.602794	0.301397	+	0.953499i	$0.402547\pi$
$728$	7616.00	0.387730
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−3108.00	−0.157255
$732$	−6664.00	−0.336487
$733$	23492.0	1.18376	0.591881	−	0.806026i	$-0.298386\pi$
$733$	23492.0	1.18376	0.591881	+	0.806026i	$0.298386\pi$
$734$	−2408.00	−0.121091
$735$	0	0
$736$	−2496.00	−0.125005
$737$	12447.0	0.622105
$738$	15708.0	0.783495
$739$	35300.0	1.75715	0.878573	−	0.477607i	$-0.158496\pi$
$739$	35300.0	1.75715	0.878573	+	0.477607i	$0.158496\pi$
$740$	0	0
$741$	−6860.00	−0.340092
$742$	−47736.0	−2.36178
$743$	16242.0	0.801967	0.400983	−	0.916085i	$-0.368669\pi$
$743$	16242.0	0.801967	0.400983	+	0.916085i	$0.368669\pi$
$744$	10192.0	0.502227
$745$	0	0
$746$	−2396.00	−0.117592
$747$	−19866.0	−0.973037
$748$	2268.00	0.110864
$749$	26826.0	1.30868
$750$	0	0
$751$	10712.0	0.520488	0.260244	−	0.965543i	$-0.416197\pi$
$751$	10712.0	0.520488	0.260244	+	0.965543i	$0.416197\pi$
$752$	−1344.00	−0.0651737
$753$	49539.0	2.39748
$754$	6720.00	0.324573
$755$	0	0
$756$	4760.00	0.228994
$757$	−13504.0	−0.648364	−0.324182	−	0.945995i	$-0.605089\pi$
$757$	−13504.0	−0.648364	−0.324182	+	0.945995i	$0.605089\pi$
$758$	16210.0	0.776746
$759$	−14742.0	−0.705008
$760$	0	0
$761$	−18123.0	−0.863283	−0.431641	−	0.902045i	$-0.642065\pi$
$761$	−18123.0	−0.863283	−0.431641	+	0.902045i	$0.642065\pi$
$762$	−15596.0	−0.741448
$763$	−7820.00	−0.371039
$764$	−5592.00	−0.264806
$765$	0	0
$766$	−6636.00	−0.313014
$767$	23520.0	1.10725
$768$	1792.00	0.0841969
$769$	9485.00	0.444783	0.222391	−	0.974957i	$-0.428614\pi$
$769$	9485.00	0.444783	0.222391	+	0.974957i	$0.428614\pi$
$770$	0	0
$771$	47922.0	2.23848
$772$	13268.0	0.618557
$773$	−31248.0	−1.45396	−0.726981	−	0.686658i	$-0.759077\pi$
$773$	−31248.0	−1.45396	−0.726981	+	0.686658i	$0.759077\pi$
$774$	−6512.00	−0.302415
$775$	0	0
$776$	8848.00	0.409310
$777$	−34748.0	−1.60435
$778$	−27540.0	−1.26910
$779$	12495.0	0.574685
$780$	0	0
$781$	−19116.0	−0.875831
$782$	−3276.00	−0.149808
$783$	4200.00	0.191693
$784$	13008.0	0.592566
$785$	0	0
$786$	3528.00	0.160101
$787$	−15484.0	−0.701328	−0.350664	−	0.936501i	$-0.614044\pi$
$787$	−15484.0	−0.701328	−0.350664	+	0.936501i	$0.614044\pi$
$788$	6744.00	0.304880
$789$	−24066.0	−1.08590
$790$	0	0
$791$	70482.0	3.16821
$792$	4752.00	0.213201
$793$	6664.00	0.298418
$794$	−7448.00	−0.332896
$795$	0	0
$796$	−6160.00	−0.274291
$797$	34146.0	1.51758	0.758791	−	0.651334i	$-0.225790\pi$
$797$	34146.0	1.51758	0.758791	+	0.651334i	$0.225790\pi$
$798$	−16660.0	−0.739045
$799$	−1764.00	−0.0781049
$800$	0	0
$801$	16170.0	0.713282
$802$	12234.0	0.538650
$803$	−3591.00	−0.157813
$804$	12908.0	0.566207
$805$	0	0
$806$	−10192.0	−0.445407
$807$	−11760.0	−0.512976
$808$	3696.00	0.160922
$809$	−36030.0	−1.56582	−0.782909	−	0.622136i	$-0.786265\pi$
$809$	−36030.0	−1.56582	−0.782909	+	0.622136i	$0.786265\pi$
$810$	0	0
$811$	−6748.00	−0.292175	−0.146088	−	0.989272i	$-0.546668\pi$
$811$	−6748.00	−0.292175	−0.146088	+	0.989272i	$0.546668\pi$
$812$	16320.0	0.705320
$813$	36554.0	1.57688
$814$	7884.00	0.339477
$815$	0	0
$816$	2352.00	0.100903
$817$	−5180.00	−0.221818
$818$	12250.0	0.523608
$819$	20944.0	0.893581
$820$	0	0
$821$	−5598.00	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$821$	−5598.00	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$822$	27174.0	1.15304
$823$	5732.00	0.242776	0.121388	−	0.992605i	$-0.461265\pi$
$823$	5732.00	0.242776	0.121388	+	0.992605i	$0.461265\pi$
$824$	6496.00	0.274635
$825$	0	0
$826$	57120.0	2.40612
$827$	−39999.0	−1.68186	−0.840932	−	0.541141i	$-0.817993\pi$
$827$	−39999.0	−1.68186	−0.840932	+	0.541141i	$0.817993\pi$
$828$	−6864.00	−0.288092
$829$	16940.0	0.709711	0.354856	−	0.934921i	$-0.384530\pi$
$829$	16940.0	0.709711	0.354856	+	0.934921i	$0.384530\pi$
$830$	0	0
$831$	−9688.00	−0.404420
$832$	−1792.00	−0.0746712
$833$	17073.0	0.710137
$834$	−23030.0	−0.956191
$835$	0	0
$836$	3780.00	0.156380
$837$	−6370.00	−0.263058
$838$	3150.00	0.129851
$839$	−45360.0	−1.86651	−0.933255	−	0.359216i	$-0.883044\pi$
$839$	−45360.0	−1.86651	−0.933255	+	0.359216i	$0.883044\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	−1976.00	−0.0808758
$843$	−27006.0	−1.10336
$844$	−12172.0	−0.496419
$845$	0	0
$846$	−3696.00	−0.150202
$847$	20468.0	0.830329
$848$	11232.0	0.454845
$849$	29939.0	1.21025
$850$	0	0
$851$	−11388.0	−0.458726
$852$	−19824.0	−0.797135
$853$	−43918.0	−1.76286	−0.881432	−	0.472310i	$-0.843420\pi$
$853$	−43918.0	−1.76286	−0.881432	+	0.472310i	$0.843420\pi$
$854$	16184.0	0.648484
$855$	0	0
$856$	−6312.00	−0.252032
$857$	−3339.00	−0.133090	−0.0665450	−	0.997783i	$-0.521198\pi$
$857$	−3339.00	−0.133090	−0.0665450	+	0.997783i	$0.521198\pi$
$858$	−10584.0	−0.421133
$859$	−36925.0	−1.46666	−0.733332	−	0.679870i	$-0.762036\pi$
$859$	−36925.0	−1.46666	−0.733332	+	0.679870i	$0.762036\pi$
$860$	0	0
$861$	−84966.0	−3.36311
$862$	−1116.00	−0.0440964
$863$	−40608.0	−1.60175	−0.800876	−	0.598830i	$-0.795633\pi$
$863$	−40608.0	−1.60175	−0.800876	+	0.598830i	$0.795633\pi$
$864$	−1120.00	−0.0441009
$865$	0	0
$866$	−4886.00	−0.191724
$867$	−31304.0	−1.22623
$868$	−24752.0	−0.967900
$869$	17550.0	0.685090
$870$	0	0
$871$	−12908.0	−0.502148
$872$	1840.00	0.0714567
$873$	24332.0	0.943314
$874$	−5460.00	−0.211313
$875$	0	0
$876$	−3724.00	−0.143633
$877$	35156.0	1.35363	0.676815	−	0.736153i	$-0.263360\pi$
$877$	35156.0	1.35363	0.676815	+	0.736153i	$0.263360\pi$
$878$	24640.0	0.947107
$879$	44394.0	1.70350
$880$	0	0
$881$	2142.00	0.0819135	0.0409568	−	0.999161i	$-0.486959\pi$
$881$	2142.00	0.0819135	0.0409568	+	0.999161i	$0.486959\pi$
$882$	35772.0	1.36565
$883$	−19153.0	−0.729954	−0.364977	−	0.931016i	$-0.618923\pi$
$883$	−19153.0	−0.729954	−0.364977	+	0.931016i	$0.618923\pi$
$884$	−2352.00	−0.0894868
$885$	0	0
$886$	−4686.00	−0.177685
$887$	−24444.0	−0.925309	−0.462655	−	0.886539i	$-0.653103\pi$
$887$	−24444.0	−0.925309	−0.462655	+	0.886539i	$0.653103\pi$
$888$	8176.00	0.308974
$889$	37876.0	1.42893
$890$	0	0
$891$	−22653.0	−0.851744
$892$	13328.0	0.500285
$893$	−2940.00	−0.110172
$894$	−25200.0	−0.942745
$895$	0	0
$896$	−4352.00	−0.162266
$897$	15288.0	0.569065
$898$	21810.0	0.810478
$899$	−21840.0	−0.810239
$900$	0	0
$901$	14742.0	0.545091
$902$	19278.0	0.711627
$903$	35224.0	1.29810
$904$	−16584.0	−0.610150
$905$	0	0
$906$	−46172.0	−1.69311
$907$	20036.0	0.733500	0.366750	−	0.930320i	$-0.380470\pi$
$907$	20036.0	0.733500	0.366750	+	0.930320i	$0.380470\pi$
$908$	6384.00	0.233327
$909$	10164.0	0.370868
$910$	0	0
$911$	−33468.0	−1.21717	−0.608586	−	0.793488i	$-0.708263\pi$
$911$	−33468.0	−1.21717	−0.608586	+	0.793488i	$0.708263\pi$
$912$	3920.00	0.142329
$913$	−24381.0	−0.883782
$914$	−6638.00	−0.240225
$915$	0	0
$916$	17360.0	0.626191
$917$	−8568.00	−0.308550
$918$	−1470.00	−0.0528510
$919$	35090.0	1.25953	0.629767	−	0.776784i	$-0.283150\pi$
$919$	35090.0	1.25953	0.629767	+	0.776784i	$0.283150\pi$
$920$	0	0
$921$	40817.0	1.46033
$922$	−12936.0	−0.462066
$923$	19824.0	0.706950
$924$	−25704.0	−0.915151
$925$	0	0
$926$	23944.0	0.849729
$927$	17864.0	0.632935
$928$	−3840.00	−0.135834
$929$	−32130.0	−1.13472	−0.567358	−	0.823471i	$-0.692034\pi$
$929$	−32130.0	−1.13472	−0.567358	+	0.823471i	$0.692034\pi$
$930$	0	0
$931$	28455.0	1.00169
$932$	−12072.0	−0.424283
$933$	−17346.0	−0.608663
$934$	13272.0	0.464960
$935$	0	0
$936$	−4928.00	−0.172091
$937$	20531.0	0.715815	0.357907	−	0.933757i	$-0.383490\pi$
$937$	20531.0	0.715815	0.357907	+	0.933757i	$0.383490\pi$
$938$	−31348.0	−1.09120
$939$	−19306.0	−0.670956
$940$	0	0
$941$	7812.00	0.270631	0.135316	−	0.990803i	$-0.456795\pi$
$941$	7812.00	0.270631	0.135316	+	0.990803i	$0.456795\pi$
$942$	3724.00	0.128805
$943$	−27846.0	−0.961602
$944$	−13440.0	−0.463384
$945$	0	0
$946$	−7992.00	−0.274675
$947$	23256.0	0.798013	0.399007	−	0.916948i	$-0.369355\pi$
$947$	23256.0	0.798013	0.399007	+	0.916948i	$0.369355\pi$
$948$	18200.0	0.623532
$949$	3724.00	0.127383
$950$	0	0
$951$	4452.00	0.151804
$952$	−5712.00	−0.194461
$953$	14097.0	0.479167	0.239584	−	0.970876i	$-0.422989\pi$
$953$	14097.0	0.479167	0.239584	+	0.970876i	$0.422989\pi$
$954$	30888.0	1.04826
$955$	0	0
$956$	17760.0	0.600836
$957$	−22680.0	−0.766082
$958$	−1260.00	−0.0424935
$959$	−65994.0	−2.22217
$960$	0	0
$961$	3333.00	0.111879
$962$	−8176.00	−0.274017
$963$	−17358.0	−0.580845
$964$	−14812.0	−0.494878
$965$	0	0
$966$	37128.0	1.23662
$967$	−34144.0	−1.13547	−0.567734	−	0.823212i	$-0.692180\pi$
$967$	−34144.0	−1.13547	−0.567734	+	0.823212i	$0.692180\pi$
$968$	−4816.00	−0.159909
$969$	5145.00	0.170569
$970$	0	0
$971$	−32613.0	−1.07786	−0.538929	−	0.842351i	$-0.681171\pi$
$971$	−32613.0	−1.07786	−0.538929	+	0.842351i	$0.681171\pi$
$972$	−19712.0	−0.650476
$973$	55930.0	1.84279
$974$	21292.0	0.700451
$975$	0	0
$976$	−3808.00	−0.124888
$977$	−43359.0	−1.41983	−0.709917	−	0.704286i	$-0.751267\pi$
$977$	−43359.0	−1.41983	−0.709917	+	0.704286i	$0.751267\pi$
$978$	16198.0	0.529606
$979$	19845.0	0.647854
$980$	0	0
$981$	5060.00	0.164682
$982$	−4776.00	−0.155202
$983$	−28518.0	−0.925313	−0.462657	−	0.886538i	$-0.653104\pi$
$983$	−28518.0	−0.925313	−0.462657	+	0.886538i	$0.653104\pi$
$984$	19992.0	0.647685
$985$	0	0
$986$	−5040.00	−0.162785
$987$	19992.0	0.644734
$988$	−3920.00	−0.126227
$989$	11544.0	0.371161
$990$	0	0
$991$	18122.0	0.580892	0.290446	−	0.956891i	$-0.406196\pi$
$991$	18122.0	0.580892	0.290446	+	0.956891i	$0.406196\pi$
$992$	5824.00	0.186403
$993$	48209.0	1.54065
$994$	48144.0	1.53625
$995$	0	0
$996$	−25284.0	−0.804372
$997$	−28924.0	−0.918789	−0.459394	−	0.888232i	$-0.651934\pi$
$997$	−28924.0	−0.918789	−0.459394	+	0.888232i	$0.651934\pi$
$998$	−21080.0	−0.668613
$999$	−5110.00	−0.161835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.4.a.e.1.1	yes	1
3.2	odd	2		450.4.a.a.1.1		1
4.3	odd	2		400.4.a.d.1.1		1
5.2	odd	4		50.4.b.b.49.2		2
5.3	odd	4		50.4.b.b.49.1		2
5.4	even	2		50.4.a.a.1.1	✓	1
7.6	odd	2		2450.4.a.y.1.1		1
8.3	odd	2		1600.4.a.bv.1.1		1
8.5	even	2		1600.4.a.f.1.1		1
15.2	even	4		450.4.c.c.199.1		2
15.8	even	4		450.4.c.c.199.2		2
15.14	odd	2		450.4.a.t.1.1		1
20.3	even	4		400.4.c.d.49.1		2
20.7	even	4		400.4.c.d.49.2		2
20.19	odd	2		400.4.a.r.1.1		1
35.34	odd	2		2450.4.a.t.1.1		1
40.19	odd	2		1600.4.a.g.1.1		1
40.29	even	2		1600.4.a.bu.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.a.a.1.1	✓	1	5.4	even	2
50.4.a.e.1.1	yes	1	1.1	even	1	trivial
50.4.b.b.49.1		2	5.3	odd	4
50.4.b.b.49.2		2	5.2	odd	4
400.4.a.d.1.1		1	4.3	odd	2
400.4.a.r.1.1		1	20.19	odd	2
400.4.c.d.49.1		2	20.3	even	4
400.4.c.d.49.2		2	20.7	even	4
450.4.a.a.1.1		1	3.2	odd	2
450.4.a.t.1.1		1	15.14	odd	2
450.4.c.c.199.1		2	15.2	even	4
450.4.c.c.199.2		2	15.8	even	4
1600.4.a.f.1.1		1	8.5	even	2
1600.4.a.g.1.1		1	40.19	odd	2
1600.4.a.bu.1.1		1	40.29	even	2
1600.4.a.bv.1.1		1	8.3	odd	2
2450.4.a.t.1.1		1	35.34	odd	2
2450.4.a.y.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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