LMFDB - Embedded newform 169.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, 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("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -17.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -23.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -17.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -23.0000 q^{9} +68.0000 q^{10} +32.0000 q^{11} +16.0000 q^{12} +80.0000 q^{14} -34.0000 q^{15} -64.0000 q^{16} -13.0000 q^{17} +92.0000 q^{18} -30.0000 q^{19} -136.000 q^{20} -40.0000 q^{21} -128.000 q^{22} +78.0000 q^{23} +164.000 q^{25} -100.000 q^{27} -160.000 q^{28} +197.000 q^{29} +136.000 q^{30} +74.0000 q^{31} +256.000 q^{32} +64.0000 q^{33} +52.0000 q^{34} +340.000 q^{35} -184.000 q^{36} +227.000 q^{37} +120.000 q^{38} +165.000 q^{41} +160.000 q^{42} -156.000 q^{43} +256.000 q^{44} +391.000 q^{45} -312.000 q^{46} +162.000 q^{47} -128.000 q^{48} +57.0000 q^{49} -656.000 q^{50} -26.0000 q^{51} +93.0000 q^{53} +400.000 q^{54} -544.000 q^{55} -60.0000 q^{57} -788.000 q^{58} +864.000 q^{59} -272.000 q^{60} +145.000 q^{61} -296.000 q^{62} +460.000 q^{63} -512.000 q^{64} -256.000 q^{66} -862.000 q^{67} -104.000 q^{68} +156.000 q^{69} -1360.00 q^{70} -654.000 q^{71} -215.000 q^{73} -908.000 q^{74} +328.000 q^{75} -240.000 q^{76} -640.000 q^{77} -76.0000 q^{79} +1088.00 q^{80} +421.000 q^{81} -660.000 q^{82} -628.000 q^{83} -320.000 q^{84} +221.000 q^{85} +624.000 q^{86} +394.000 q^{87} +266.000 q^{89} -1564.00 q^{90} +624.000 q^{92} +148.000 q^{93} -648.000 q^{94} +510.000 q^{95} +512.000 q^{96} -238.000 q^{97} -228.000 q^{98} -736.000 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	8.00000	1.00000
$5$	−17.0000	−1.52053	−0.760263	−	0.649615i	$-0.774930\pi$
$5$	−17.0000	−1.52053	−0.760263	+	0.649615i	$0.774930\pi$
$6$	−8.00000	−0.544331
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	68.0000	2.15035
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	16.0000	0.384900
$13$	0	0
$13$	0	0
$14$	80.0000	1.52721
$15$	−34.0000	−0.585251
$16$	−64.0000	−1.00000
$17$	−13.0000	−0.185468	−0.0927342	−	0.995691i	$-0.529561\pi$
$17$	−13.0000	−0.185468	−0.0927342	+	0.995691i	$0.529561\pi$
$18$	92.0000	1.20470
$19$	−30.0000	−0.362235	−0.181118	−	0.983461i	$-0.557971\pi$
$19$	−30.0000	−0.362235	−0.181118	+	0.983461i	$0.557971\pi$
$20$	−136.000	−1.52053
$21$	−40.0000	−0.415653
$22$	−128.000	−1.24044
$23$	78.0000	0.707136	0.353568	−	0.935409i	$-0.384968\pi$
$23$	78.0000	0.707136	0.353568	+	0.935409i	$0.384968\pi$
$24$	0	0
$25$	164.000	1.31200
$26$	0	0
$27$	−100.000	−0.712778
$28$	−160.000	−1.07990
$29$	197.000	1.26145	0.630724	−	0.776007i	$-0.282758\pi$
$29$	197.000	1.26145	0.630724	+	0.776007i	$0.282758\pi$
$30$	136.000	0.827670
$31$	74.0000	0.428735	0.214368	−	0.976753i	$-0.431231\pi$
$31$	74.0000	0.428735	0.214368	+	0.976753i	$0.431231\pi$
$32$	256.000	1.41421
$33$	64.0000	0.337605
$34$	52.0000	0.262292
$35$	340.000	1.64201
$36$	−184.000	−0.851852
$37$	227.000	1.00861	0.504305	−	0.863526i	$-0.331749\pi$
$37$	227.000	1.00861	0.504305	+	0.863526i	$0.331749\pi$
$38$	120.000	0.512278
$39$	0	0
$40$	0	0
$41$	165.000	0.628504	0.314252	−	0.949340i	$-0.398246\pi$
$41$	165.000	0.628504	0.314252	+	0.949340i	$0.398246\pi$
$42$	160.000	0.587822
$43$	−156.000	−0.553251	−0.276625	−	0.960978i	$-0.589216\pi$
$43$	−156.000	−0.553251	−0.276625	+	0.960978i	$0.589216\pi$
$44$	256.000	0.877124
$45$	391.000	1.29526
$46$	−312.000	−1.00004
$47$	162.000	0.502769	0.251384	−	0.967887i	$-0.419114\pi$
$47$	162.000	0.502769	0.251384	+	0.967887i	$0.419114\pi$
$48$	−128.000	−0.384900
$49$	57.0000	0.166181
$50$	−656.000	−1.85545
$51$	−26.0000	−0.0713868
$52$	0	0
$53$	93.0000	0.241029	0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000	0.241029	0.120514	+	0.992712i	$0.461546\pi$
$54$	400.000	1.00802
$55$	−544.000	−1.33369
$56$	0	0
$57$	−60.0000	−0.139424
$58$	−788.000	−1.78396
$59$	864.000	1.90650	0.953248	−	0.302190i	$-0.0977178\pi$
$59$	864.000	1.90650	0.953248	+	0.302190i	$0.0977178\pi$
$60$	−272.000	−0.585251
$61$	145.000	0.304350	0.152175	−	0.988354i	$-0.451372\pi$
$61$	145.000	0.304350	0.152175	+	0.988354i	$0.451372\pi$
$62$	−296.000	−0.606323
$63$	460.000	0.919914
$64$	−512.000	−1.00000
$65$	0	0
$66$	−256.000	−0.477446
$67$	−862.000	−1.57179	−0.785896	−	0.618359i	$-0.787798\pi$
$67$	−862.000	−1.57179	−0.785896	+	0.618359i	$0.787798\pi$
$68$	−104.000	−0.185468
$69$	156.000	0.272177
$70$	−1360.00	−2.32216
$71$	−654.000	−1.09318	−0.546588	−	0.837402i	$-0.684074\pi$
$71$	−654.000	−1.09318	−0.546588	+	0.837402i	$0.684074\pi$
$72$	0	0
$73$	−215.000	−0.344710	−0.172355	−	0.985035i	$-0.555138\pi$
$73$	−215.000	−0.344710	−0.172355	+	0.985035i	$0.555138\pi$
$74$	−908.000	−1.42639
$75$	328.000	0.504989
$76$	−240.000	−0.362235
$77$	−640.000	−0.947205
$78$	0	0
$79$	−76.0000	−0.108236	−0.0541182	−	0.998535i	$-0.517235\pi$
$79$	−76.0000	−0.108236	−0.0541182	+	0.998535i	$0.517235\pi$
$80$	1088.00	1.52053
$81$	421.000	0.577503
$82$	−660.000	−0.888839
$83$	−628.000	−0.830505	−0.415253	−	0.909706i	$-0.636307\pi$
$83$	−628.000	−0.830505	−0.415253	+	0.909706i	$0.636307\pi$
$84$	−320.000	−0.415653
$85$	221.000	0.282010
$86$	624.000	0.782415
$87$	394.000	0.485531
$88$	0	0
$89$	266.000	0.316808	0.158404	−	0.987374i	$-0.449365\pi$
$89$	266.000	0.316808	0.158404	+	0.987374i	$0.449365\pi$
$90$	−1564.00	−1.83178
$91$	0	0
$92$	624.000	0.707136
$93$	148.000	0.165020
$94$	−648.000	−0.711022
$95$	510.000	0.550788
$96$	512.000	0.544331
$97$	−238.000	−0.249126	−0.124563	−	0.992212i	$-0.539753\pi$
$97$	−238.000	−0.249126	−0.124563	+	0.992212i	$0.539753\pi$
$98$	−228.000	−0.235015
$99$	−736.000	−0.747180
$100$	1312.00	1.31200
$101$	−819.000	−0.806867	−0.403433	−	0.915009i	$-0.632183\pi$
$101$	−819.000	−0.806867	−0.403433	+	0.915009i	$0.632183\pi$
$102$	104.000	0.100956
$103$	1638.00	1.56696	0.783480	−	0.621417i	$-0.213443\pi$
$103$	1638.00	1.56696	0.783480	+	0.621417i	$0.213443\pi$
$104$	0	0
$105$	680.000	0.632011
$106$	−372.000	−0.340866
$107$	522.000	0.471623	0.235811	−	0.971799i	$-0.424225\pi$
$107$	522.000	0.471623	0.235811	+	0.971799i	$0.424225\pi$
$108$	−800.000	−0.712778
$109$	1634.00	1.43586	0.717930	−	0.696115i	$-0.245090\pi$
$109$	1634.00	1.43586	0.717930	+	0.696115i	$0.245090\pi$
$110$	2176.00	1.88612
$111$	454.000	0.388214
$112$	1280.00	1.07990
$113$	327.000	0.272226	0.136113	−	0.990693i	$-0.456539\pi$
$113$	327.000	0.272226	0.136113	+	0.990693i	$0.456539\pi$
$114$	240.000	0.197176
$115$	−1326.00	−1.07522
$116$	1576.00	1.26145
$117$	0	0
$118$	−3456.00	−2.69619
$119$	260.000	0.200287
$120$	0	0
$121$	−307.000	−0.230654
$122$	−580.000	−0.430416
$123$	330.000	0.241911
$124$	592.000	0.428735
$125$	−663.000	−0.474404
$126$	−1840.00	−1.30095
$127$	−2158.00	−1.50781	−0.753904	−	0.656985i	$-0.771831\pi$
$127$	−2158.00	−1.50781	−0.753904	+	0.656985i	$0.771831\pi$
$128$	0	0
$129$	−312.000	−0.212946
$130$	0	0
$131$	730.000	0.486873	0.243437	−	0.969917i	$-0.421725\pi$
$131$	730.000	0.486873	0.243437	+	0.969917i	$0.421725\pi$
$132$	512.000	0.337605
$133$	600.000	0.391177
$134$	3448.00	2.22285
$135$	1700.00	1.08380
$136$	0	0
$137$	−1671.00	−1.04207	−0.521033	−	0.853536i	$-0.674453\pi$
$137$	−1671.00	−1.04207	−0.521033	+	0.853536i	$0.674453\pi$
$138$	−624.000	−0.384916
$139$	912.000	0.556510	0.278255	−	0.960507i	$-0.410244\pi$
$139$	912.000	0.556510	0.278255	+	0.960507i	$0.410244\pi$
$140$	2720.00	1.64201
$141$	324.000	0.193516
$142$	2616.00	1.54598
$143$	0	0
$144$	1472.00	0.851852
$145$	−3349.00	−1.91806
$146$	860.000	0.487494
$147$	114.000	0.0639630
$148$	1816.00	1.00861
$149$	2115.00	1.16287	0.581435	−	0.813593i	$-0.302492\pi$
$149$	2115.00	1.16287	0.581435	+	0.813593i	$0.302492\pi$
$150$	−1312.00	−0.714162
$151$	−514.000	−0.277011	−0.138506	−	0.990362i	$-0.544230\pi$
$151$	−514.000	−0.277011	−0.138506	+	0.990362i	$0.544230\pi$
$152$	0	0
$153$	299.000	0.157992
$154$	2560.00	1.33955
$155$	−1258.00	−0.651903
$156$	0	0
$157$	2901.00	1.47468	0.737341	−	0.675521i	$-0.236081\pi$
$157$	2901.00	1.47468	0.737341	+	0.675521i	$0.236081\pi$
$158$	304.000	0.153069
$159$	186.000	0.0927721
$160$	−4352.00	−2.15035
$161$	−1560.00	−0.763635
$162$	−1684.00	−0.816713
$163$	−2360.00	−1.13405	−0.567023	−	0.823702i	$-0.691905\pi$
$163$	−2360.00	−1.13405	−0.567023	+	0.823702i	$0.691905\pi$
$164$	1320.00	0.628504
$165$	−1088.00	−0.513337
$166$	2512.00	1.17451
$167$	−280.000	−0.129743	−0.0648714	−	0.997894i	$-0.520664\pi$
$167$	−280.000	−0.129743	−0.0648714	+	0.997894i	$0.520664\pi$
$168$	0	0
$169$	0	0
$170$	−884.000	−0.398822
$171$	690.000	0.308571
$172$	−1248.00	−0.553251
$173$	1326.00	0.582739	0.291370	−	0.956611i	$-0.405889\pi$
$173$	1326.00	0.582739	0.291370	+	0.956611i	$0.405889\pi$
$174$	−1576.00	−0.686645
$175$	−3280.00	−1.41683
$176$	−2048.00	−0.877124
$177$	1728.00	0.733810
$178$	−1064.00	−0.448035
$179$	4264.00	1.78048	0.890241	−	0.455490i	$-0.150536\pi$
$179$	4264.00	1.78048	0.890241	+	0.455490i	$0.150536\pi$
$180$	3128.00	1.29526
$181$	−403.000	−0.165496	−0.0827479	−	0.996571i	$-0.526370\pi$
$181$	−403.000	−0.165496	−0.0827479	+	0.996571i	$0.526370\pi$
$182$	0	0
$183$	290.000	0.117144
$184$	0	0
$185$	−3859.00	−1.53362
$186$	−592.000	−0.233374
$187$	−416.000	−0.162679
$188$	1296.00	0.502769
$189$	2000.00	0.769728
$190$	−2040.00	−0.778932
$191$	−1246.00	−0.472028	−0.236014	−	0.971750i	$-0.575841\pi$
$191$	−1246.00	−0.472028	−0.236014	+	0.971750i	$0.575841\pi$
$192$	−1024.00	−0.384900
$193$	−267.000	−0.0995807	−0.0497904	−	0.998760i	$-0.515855\pi$
$193$	−267.000	−0.0995807	−0.0497904	+	0.998760i	$0.515855\pi$
$194$	952.000	0.352318
$195$	0	0
$196$	456.000	0.166181
$197$	−1278.00	−0.462202	−0.231101	−	0.972930i	$-0.574233\pi$
$197$	−1278.00	−0.462202	−0.231101	+	0.972930i	$0.574233\pi$
$198$	2944.00	1.05667
$199$	4238.00	1.50967	0.754834	−	0.655916i	$-0.227717\pi$
$199$	4238.00	1.50967	0.754834	+	0.655916i	$0.227717\pi$
$200$	0	0
$201$	−1724.00	−0.604983
$202$	3276.00	1.14108
$203$	−3940.00	−1.36224
$204$	−208.000	−0.0713868
$205$	−2805.00	−0.955657
$206$	−6552.00	−2.21602
$207$	−1794.00	−0.602375
$208$	0	0
$209$	−960.000	−0.317725
$210$	−2720.00	−0.893799
$211$	3070.00	1.00165	0.500823	−	0.865549i	$-0.333031\pi$
$211$	3070.00	1.00165	0.500823	+	0.865549i	$0.333031\pi$
$212$	744.000	0.241029
$213$	−1308.00	−0.420764
$214$	−2088.00	−0.666975
$215$	2652.00	0.841232
$216$	0	0
$217$	−1480.00	−0.462991
$218$	−6536.00	−2.03061
$219$	−430.000	−0.132679
$220$	−4352.00	−1.33369
$221$	0	0
$222$	−1816.00	−0.549018
$223$	5378.00	1.61497	0.807483	−	0.589891i	$-0.200829\pi$
$223$	5378.00	1.61497	0.807483	+	0.589891i	$0.200829\pi$
$224$	−5120.00	−1.52721
$225$	−3772.00	−1.11763
$226$	−1308.00	−0.384986
$227$	3974.00	1.16195	0.580977	−	0.813920i	$-0.302671\pi$
$227$	3974.00	1.16195	0.580977	+	0.813920i	$0.302671\pi$
$228$	−480.000	−0.139424
$229$	6298.00	1.81740	0.908698	−	0.417455i	$-0.137078\pi$
$229$	6298.00	1.81740	0.908698	+	0.417455i	$0.137078\pi$
$230$	5304.00	1.52059
$231$	−1280.00	−0.364579
$232$	0	0
$233$	4030.00	1.13311	0.566554	−	0.824025i	$-0.308276\pi$
$233$	4030.00	1.13311	0.566554	+	0.824025i	$0.308276\pi$
$234$	0	0
$235$	−2754.00	−0.764473
$236$	6912.00	1.90650
$237$	−152.000	−0.0416602
$238$	−1040.00	−0.283249
$239$	984.000	0.266317	0.133158	−	0.991095i	$-0.457488\pi$
$239$	984.000	0.266317	0.133158	+	0.991095i	$0.457488\pi$
$240$	2176.00	0.585251
$241$	−943.000	−0.252050	−0.126025	−	0.992027i	$-0.540222\pi$
$241$	−943.000	−0.252050	−0.126025	+	0.992027i	$0.540222\pi$
$242$	1228.00	0.326194
$243$	3542.00	0.935059
$244$	1160.00	0.304350
$245$	−969.000	−0.252682
$246$	−1320.00	−0.342114
$247$	0	0
$248$	0	0
$249$	−1256.00	−0.319662
$250$	2652.00	0.670909
$251$	−2730.00	−0.686518	−0.343259	−	0.939241i	$-0.611531\pi$
$251$	−2730.00	−0.686518	−0.343259	+	0.939241i	$0.611531\pi$
$252$	3680.00	0.919914
$253$	2496.00	0.620246
$254$	8632.00	2.13236
$255$	442.000	0.108546
$256$	4096.00	1.00000
$257$	−1885.00	−0.457522	−0.228761	−	0.973483i	$-0.573467\pi$
$257$	−1885.00	−0.457522	−0.228761	+	0.973483i	$0.573467\pi$
$258$	1248.00	0.301151
$259$	−4540.00	−1.08920
$260$	0	0
$261$	−4531.00	−1.07457
$262$	−2920.00	−0.688543
$263$	4032.00	0.945338	0.472669	−	0.881240i	$-0.343291\pi$
$263$	4032.00	0.945338	0.472669	+	0.881240i	$0.343291\pi$
$264$	0	0
$265$	−1581.00	−0.366491
$266$	−2400.00	−0.553208
$267$	532.000	0.121940
$268$	−6896.00	−1.57179
$269$	4006.00	0.907993	0.453997	−	0.891003i	$-0.349998\pi$
$269$	4006.00	0.907993	0.453997	+	0.891003i	$0.349998\pi$
$270$	−6800.00	−1.53272
$271$	4296.00	0.962965	0.481482	−	0.876456i	$-0.340099\pi$
$271$	4296.00	0.962965	0.481482	+	0.876456i	$0.340099\pi$
$272$	832.000	0.185468
$273$	0	0
$274$	6684.00	1.47371
$275$	5248.00	1.15079
$276$	1248.00	0.272177
$277$	−5551.00	−1.20407	−0.602035	−	0.798470i	$-0.705643\pi$
$277$	−5551.00	−1.20407	−0.602035	+	0.798470i	$0.705643\pi$
$278$	−3648.00	−0.787023
$279$	−1702.00	−0.365219
$280$	0	0
$281$	5557.00	1.17973	0.589863	−	0.807504i	$-0.299182\pi$
$281$	5557.00	1.17973	0.589863	+	0.807504i	$0.299182\pi$
$282$	−1296.00	−0.273673
$283$	3120.00	0.655352	0.327676	−	0.944790i	$-0.393734\pi$
$283$	3120.00	0.655352	0.327676	+	0.944790i	$0.393734\pi$
$284$	−5232.00	−1.09318
$285$	1020.00	0.211999
$286$	0	0
$287$	−3300.00	−0.678721
$288$	−5888.00	−1.20470
$289$	−4744.00	−0.965601
$290$	13396.0	2.71255
$291$	−476.000	−0.0958887
$292$	−1720.00	−0.344710
$293$	−8301.00	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$293$	−8301.00	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$294$	−456.000	−0.0904573
$295$	−14688.0	−2.89888
$296$	0	0
$297$	−3200.00	−0.625195
$298$	−8460.00	−1.64455
$299$	0	0
$300$	2624.00	0.504989
$301$	3120.00	0.597455
$302$	2056.00	0.391753
$303$	−1638.00	−0.310563
$304$	1920.00	0.362235
$305$	−2465.00	−0.462772
$306$	−1196.00	−0.223434
$307$	−8678.00	−1.61329	−0.806644	−	0.591037i	$-0.798719\pi$
$307$	−8678.00	−1.61329	−0.806644	+	0.591037i	$0.798719\pi$
$308$	−5120.00	−0.947205
$309$	3276.00	0.603123
$310$	5032.00	0.921930
$311$	8658.00	1.57862	0.789309	−	0.613996i	$-0.210439\pi$
$311$	8658.00	1.57862	0.789309	+	0.613996i	$0.210439\pi$
$312$	0	0
$313$	−5250.00	−0.948075	−0.474038	−	0.880505i	$-0.657204\pi$
$313$	−5250.00	−0.948075	−0.474038	+	0.880505i	$0.657204\pi$
$314$	−11604.0	−2.08551
$315$	−7820.00	−1.39875
$316$	−608.000	−0.108236
$317$	−6413.00	−1.13625	−0.568123	−	0.822944i	$-0.692330\pi$
$317$	−6413.00	−1.13625	−0.568123	+	0.822944i	$0.692330\pi$
$318$	−744.000	−0.131200
$319$	6304.00	1.10645
$320$	8704.00	1.52053
$321$	1044.00	0.181528
$322$	6240.00	1.07994
$323$	390.000	0.0671832
$324$	3368.00	0.577503
$325$	0	0
$326$	9440.00	1.60378
$327$	3268.00	0.552663
$328$	0	0
$329$	−3240.00	−0.542939
$330$	4352.00	0.725969
$331$	−3488.00	−0.579208	−0.289604	−	0.957147i	$-0.593524\pi$
$331$	−3488.00	−0.579208	−0.289604	+	0.957147i	$0.593524\pi$
$332$	−5024.00	−0.830505
$333$	−5221.00	−0.859186
$334$	1120.00	0.183484
$335$	14654.0	2.38995
$336$	2560.00	0.415653
$337$	−1833.00	−0.296290	−0.148145	−	0.988966i	$-0.547330\pi$
$337$	−1833.00	−0.296290	−0.148145	+	0.988966i	$0.547330\pi$
$338$	0	0
$339$	654.000	0.104780
$340$	1768.00	0.282010
$341$	2368.00	0.376054
$342$	−2760.00	−0.436385
$343$	5720.00	0.900440
$344$	0	0
$345$	−2652.00	−0.413852
$346$	−5304.00	−0.824118
$347$	7230.00	1.11852	0.559260	−	0.828992i	$-0.311085\pi$
$347$	7230.00	1.11852	0.559260	+	0.828992i	$0.311085\pi$
$348$	3152.00	0.485531
$349$	5258.00	0.806459	0.403230	−	0.915099i	$-0.367888\pi$
$349$	5258.00	0.806459	0.403230	+	0.915099i	$0.367888\pi$
$350$	13120.0	2.00370
$351$	0	0
$352$	8192.00	1.24044
$353$	−3163.00	−0.476911	−0.238455	−	0.971153i	$-0.576641\pi$
$353$	−3163.00	−0.476911	−0.238455	+	0.971153i	$0.576641\pi$
$354$	−6912.00	−1.03776
$355$	11118.0	1.66220
$356$	2128.00	0.316808
$357$	520.000	0.0770905
$358$	−17056.0	−2.51798
$359$	10068.0	1.48014	0.740068	−	0.672532i	$-0.234793\pi$
$359$	10068.0	1.48014	0.740068	+	0.672532i	$0.234793\pi$
$360$	0	0
$361$	−5959.00	−0.868786
$362$	1612.00	0.234047
$363$	−614.000	−0.0887786
$364$	0	0
$365$	3655.00	0.524141
$366$	−1160.00	−0.165667
$367$	7438.00	1.05793	0.528965	−	0.848644i	$-0.322580\pi$
$367$	7438.00	1.05793	0.528965	+	0.848644i	$0.322580\pi$
$368$	−4992.00	−0.707136
$369$	−3795.00	−0.535392
$370$	15436.0	2.16886
$371$	−1860.00	−0.260287
$372$	1184.00	0.165020
$373$	−9683.00	−1.34415	−0.672073	−	0.740485i	$-0.734596\pi$
$373$	−9683.00	−1.34415	−0.672073	+	0.740485i	$0.734596\pi$
$374$	1664.00	0.230063
$375$	−1326.00	−0.182598
$376$	0	0
$377$	0	0
$378$	−8000.00	−1.08856
$379$	1062.00	0.143935	0.0719674	−	0.997407i	$-0.477072\pi$
$379$	1062.00	0.143935	0.0719674	+	0.997407i	$0.477072\pi$
$380$	4080.00	0.550788
$381$	−4316.00	−0.580355
$382$	4984.00	0.667549
$383$	3532.00	0.471219	0.235609	−	0.971848i	$-0.424291\pi$
$383$	3532.00	0.471219	0.235609	+	0.971848i	$0.424291\pi$
$384$	0	0
$385$	10880.0	1.44025
$386$	1068.00	0.140828
$387$	3588.00	0.471288
$388$	−1904.00	−0.249126
$389$	−11063.0	−1.44194	−0.720972	−	0.692964i	$-0.756304\pi$
$389$	−11063.0	−1.44194	−0.720972	+	0.692964i	$0.756304\pi$
$390$	0	0
$391$	−1014.00	−0.131151
$392$	0	0
$393$	1460.00	0.187398
$394$	5112.00	0.653652
$395$	1292.00	0.164576
$396$	−5888.00	−0.747180
$397$	5986.00	0.756747	0.378374	−	0.925653i	$-0.376483\pi$
$397$	5986.00	0.756747	0.378374	+	0.925653i	$0.376483\pi$
$398$	−16952.0	−2.13499
$399$	1200.00	0.150564
$400$	−10496.0	−1.31200
$401$	−5935.00	−0.739102	−0.369551	−	0.929211i	$-0.620488\pi$
$401$	−5935.00	−0.739102	−0.369551	+	0.929211i	$0.620488\pi$
$402$	6896.00	0.855575
$403$	0	0
$404$	−6552.00	−0.806867
$405$	−7157.00	−0.878109
$406$	15760.0	1.92649
$407$	7264.00	0.884676
$408$	0	0
$409$	15089.0	1.82421	0.912106	−	0.409954i	$-0.134455\pi$
$409$	15089.0	1.82421	0.912106	+	0.409954i	$0.134455\pi$
$410$	11220.0	1.35150
$411$	−3342.00	−0.401092
$412$	13104.0	1.56696
$413$	−17280.0	−2.05882
$414$	7176.00	0.851887
$415$	10676.0	1.26281
$416$	0	0
$417$	1824.00	0.214201
$418$	3840.00	0.449331
$419$	−10814.0	−1.26086	−0.630428	−	0.776248i	$-0.717120\pi$
$419$	−10814.0	−1.26086	−0.630428	+	0.776248i	$0.717120\pi$
$420$	5440.00	0.632011
$421$	6535.00	0.756524	0.378262	−	0.925699i	$-0.376522\pi$
$421$	6535.00	0.756524	0.378262	+	0.925699i	$0.376522\pi$
$422$	−12280.0	−1.41654
$423$	−3726.00	−0.428284
$424$	0	0
$425$	−2132.00	−0.243335
$426$	5232.00	0.595050
$427$	−2900.00	−0.328667
$428$	4176.00	0.471623
$429$	0	0
$430$	−10608.0	−1.18968
$431$	−1980.00	−0.221284	−0.110642	−	0.993860i	$-0.535291\pi$
$431$	−1980.00	−0.221284	−0.110642	+	0.993860i	$0.535291\pi$
$432$	6400.00	0.712778
$433$	−6929.00	−0.769022	−0.384511	−	0.923120i	$-0.625630\pi$
$433$	−6929.00	−0.769022	−0.384511	+	0.923120i	$0.625630\pi$
$434$	5920.00	0.654767
$435$	−6698.00	−0.738263
$436$	13072.0	1.43586
$437$	−2340.00	−0.256150
$438$	1720.00	0.187636
$439$	−4576.00	−0.497496	−0.248748	−	0.968568i	$-0.580019\pi$
$439$	−4576.00	−0.497496	−0.248748	+	0.968568i	$0.580019\pi$
$440$	0	0
$441$	−1311.00	−0.141561
$442$	0	0
$443$	−8812.00	−0.945081	−0.472540	−	0.881309i	$-0.656663\pi$
$443$	−8812.00	−0.945081	−0.472540	+	0.881309i	$0.656663\pi$
$444$	3632.00	0.388214
$445$	−4522.00	−0.481715
$446$	−21512.0	−2.28391
$447$	4230.00	0.447589
$448$	10240.0	1.07990
$449$	−1918.00	−0.201595	−0.100797	−	0.994907i	$-0.532139\pi$
$449$	−1918.00	−0.201595	−0.100797	+	0.994907i	$0.532139\pi$
$450$	15088.0	1.58057
$451$	5280.00	0.551276
$452$	2616.00	0.272226
$453$	−1028.00	−0.106622
$454$	−15896.0	−1.64325
$455$	0	0
$456$	0	0
$457$	11761.0	1.20384	0.601922	−	0.798555i	$-0.294402\pi$
$457$	11761.0	1.20384	0.601922	+	0.798555i	$0.294402\pi$
$458$	−25192.0	−2.57019
$459$	1300.00	0.132198
$460$	−10608.0	−1.07522
$461$	−901.000	−0.0910277	−0.0455138	−	0.998964i	$-0.514493\pi$
$461$	−901.000	−0.0910277	−0.0455138	+	0.998964i	$0.514493\pi$
$462$	5120.00	0.515593
$463$	−1372.00	−0.137715	−0.0688577	−	0.997626i	$-0.521935\pi$
$463$	−1372.00	−0.137715	−0.0688577	+	0.997626i	$0.521935\pi$
$464$	−12608.0	−1.26145
$465$	−2516.00	−0.250918
$466$	−16120.0	−1.60246
$467$	−6396.00	−0.633772	−0.316886	−	0.948464i	$-0.602637\pi$
$467$	−6396.00	−0.633772	−0.316886	+	0.948464i	$0.602637\pi$
$468$	0	0
$469$	17240.0	1.69738
$470$	11016.0	1.08113
$471$	5802.00	0.567605
$472$	0	0
$473$	−4992.00	−0.485269
$474$	608.000	0.0589164
$475$	−4920.00	−0.475253
$476$	2080.00	0.200287
$477$	−2139.00	−0.205321
$478$	−3936.00	−0.376629
$479$	−3270.00	−0.311921	−0.155960	−	0.987763i	$-0.549847\pi$
$479$	−3270.00	−0.311921	−0.155960	+	0.987763i	$0.549847\pi$
$480$	−8704.00	−0.827670
$481$	0	0
$482$	3772.00	0.356452
$483$	−3120.00	−0.293923
$484$	−2456.00	−0.230654
$485$	4046.00	0.378803
$486$	−14168.0	−1.32237
$487$	−19920.0	−1.85351	−0.926757	−	0.375661i	$-0.877416\pi$
$487$	−19920.0	−1.85351	−0.926757	+	0.375661i	$0.877416\pi$
$488$	0	0
$489$	−4720.00	−0.436494
$490$	3876.00	0.357347
$491$	6552.00	0.602215	0.301108	−	0.953590i	$-0.402644\pi$
$491$	6552.00	0.602215	0.301108	+	0.953590i	$0.402644\pi$
$492$	2640.00	0.241911
$493$	−2561.00	−0.233959
$494$	0	0
$495$	12512.0	1.13611
$496$	−4736.00	−0.428735
$497$	13080.0	1.18052
$498$	5024.00	0.452070
$499$	−1746.00	−0.156637	−0.0783183	−	0.996928i	$-0.524955\pi$
$499$	−1746.00	−0.156637	−0.0783183	+	0.996928i	$0.524955\pi$
$500$	−5304.00	−0.474404
$501$	−560.000	−0.0499380
$502$	10920.0	0.970883
$503$	14692.0	1.30235	0.651177	−	0.758926i	$-0.274276\pi$
$503$	14692.0	1.30235	0.651177	+	0.758926i	$0.274276\pi$
$504$	0	0
$505$	13923.0	1.22686
$506$	−9984.00	−0.877160
$507$	0	0
$508$	−17264.0	−1.50781
$509$	−8077.00	−0.703353	−0.351677	−	0.936122i	$-0.614388\pi$
$509$	−8077.00	−0.703353	−0.351677	+	0.936122i	$0.614388\pi$
$510$	−1768.00	−0.153507
$511$	4300.00	0.372252
$512$	−16384.0	−1.41421
$513$	3000.00	0.258193
$514$	7540.00	0.647033
$515$	−27846.0	−2.38260
$516$	−2496.00	−0.212946
$517$	5184.00	0.440990
$518$	18160.0	1.54036
$519$	2652.00	0.224296
$520$	0	0
$521$	11247.0	0.945758	0.472879	−	0.881127i	$-0.343215\pi$
$521$	11247.0	0.945758	0.472879	+	0.881127i	$0.343215\pi$
$522$	18124.0	1.51967
$523$	2732.00	0.228417	0.114208	−	0.993457i	$-0.463567\pi$
$523$	2732.00	0.228417	0.114208	+	0.993457i	$0.463567\pi$
$524$	5840.00	0.486873
$525$	−6560.00	−0.545337
$526$	−16128.0	−1.33691
$527$	−962.000	−0.0795168
$528$	−4096.00	−0.337605
$529$	−6083.00	−0.499959
$530$	6324.00	0.518296
$531$	−19872.0	−1.62405
$532$	4800.00	0.391177
$533$	0	0
$534$	−2128.00	−0.172449
$535$	−8874.00	−0.717115
$536$	0	0
$537$	8528.00	0.685308
$538$	−16024.0	−1.28410
$539$	1824.00	0.145761
$540$	13600.0	1.08380
$541$	18375.0	1.46026	0.730132	−	0.683306i	$-0.239458\pi$
$541$	18375.0	1.46026	0.730132	+	0.683306i	$0.239458\pi$
$542$	−17184.0	−1.36184
$543$	−806.000	−0.0636994
$544$	−3328.00	−0.262292
$545$	−27778.0	−2.18326
$546$	0	0
$547$	−10346.0	−0.808708	−0.404354	−	0.914603i	$-0.632504\pi$
$547$	−10346.0	−0.808708	−0.404354	+	0.914603i	$0.632504\pi$
$548$	−13368.0	−1.04207
$549$	−3335.00	−0.259261
$550$	−20992.0	−1.62746
$551$	−5910.00	−0.456941
$552$	0	0
$553$	1520.00	0.116884
$554$	22204.0	1.70281
$555$	−7718.00	−0.590290
$556$	7296.00	0.556510
$557$	−345.000	−0.0262444	−0.0131222	−	0.999914i	$-0.504177\pi$
$557$	−345.000	−0.0262444	−0.0131222	+	0.999914i	$0.504177\pi$
$558$	6808.00	0.516498
$559$	0	0
$560$	−21760.0	−1.64201
$561$	−832.000	−0.0626151
$562$	−22228.0	−1.66838
$563$	−8580.00	−0.642280	−0.321140	−	0.947032i	$-0.604066\pi$
$563$	−8580.00	−0.642280	−0.321140	+	0.947032i	$0.604066\pi$
$564$	2592.00	0.193516
$565$	−5559.00	−0.413927
$566$	−12480.0	−0.926808
$567$	−8420.00	−0.623645
$568$	0	0
$569$	−19682.0	−1.45011	−0.725055	−	0.688691i	$-0.758186\pi$
$569$	−19682.0	−1.45011	−0.725055	+	0.688691i	$0.758186\pi$
$570$	−4080.00	−0.299811
$571$	26624.0	1.95128	0.975639	−	0.219382i	$-0.0704042\pi$
$571$	26624.0	1.95128	0.975639	+	0.219382i	$0.0704042\pi$
$572$	0	0
$573$	−2492.00	−0.181684
$574$	13200.0	0.959856
$575$	12792.0	0.927762
$576$	11776.0	0.851852
$577$	14101.0	1.01739	0.508694	−	0.860948i	$-0.330129\pi$
$577$	14101.0	1.01739	0.508694	+	0.860948i	$0.330129\pi$
$578$	18976.0	1.36557
$579$	−534.000	−0.0383286
$580$	−26792.0	−1.91806
$581$	12560.0	0.896862
$582$	1904.00	0.135607
$583$	2976.00	0.211412
$584$	0	0
$585$	0	0
$586$	33204.0	2.34069
$587$	−1408.00	−0.0990023	−0.0495012	−	0.998774i	$-0.515763\pi$
$587$	−1408.00	−0.0990023	−0.0495012	+	0.998774i	$0.515763\pi$
$588$	912.000	0.0639630
$589$	−2220.00	−0.155303
$590$	58752.0	4.09963
$591$	−2556.00	−0.177902
$592$	−14528.0	−1.00861
$593$	1241.00	0.0859389	0.0429694	−	0.999076i	$-0.486318\pi$
$593$	1241.00	0.0859389	0.0429694	+	0.999076i	$0.486318\pi$
$594$	12800.0	0.884159
$595$	−4420.00	−0.304542
$596$	16920.0	1.16287
$597$	8476.00	0.581071
$598$	0	0
$599$	11078.0	0.755651	0.377825	−	0.925877i	$-0.376672\pi$
$599$	11078.0	0.755651	0.377825	+	0.925877i	$0.376672\pi$
$600$	0	0
$601$	−13817.0	−0.937782	−0.468891	−	0.883256i	$-0.655346\pi$
$601$	−13817.0	−0.937782	−0.468891	+	0.883256i	$0.655346\pi$
$602$	−12480.0	−0.844928
$603$	19826.0	1.33893
$604$	−4112.00	−0.277011
$605$	5219.00	0.350715
$606$	6552.00	0.439203
$607$	8270.00	0.552997	0.276498	−	0.961014i	$-0.410826\pi$
$607$	8270.00	0.552997	0.276498	+	0.961014i	$0.410826\pi$
$608$	−7680.00	−0.512278
$609$	−7880.00	−0.524325
$610$	9860.00	0.654459
$611$	0	0
$612$	2392.00	0.157992
$613$	−22273.0	−1.46753	−0.733767	−	0.679402i	$-0.762239\pi$
$613$	−22273.0	−1.46753	−0.733767	+	0.679402i	$0.762239\pi$
$614$	34712.0	2.28153
$615$	−5610.00	−0.367833
$616$	0	0
$617$	18989.0	1.23901	0.619504	−	0.784993i	$-0.287334\pi$
$617$	18989.0	1.23901	0.619504	+	0.784993i	$0.287334\pi$
$618$	−13104.0	−0.852945
$619$	−72.0000	−0.00467516	−0.00233758	−	0.999997i	$-0.500744\pi$
$619$	−72.0000	−0.00467516	−0.00233758	+	0.999997i	$0.500744\pi$
$620$	−10064.0	−0.651903
$621$	−7800.00	−0.504031
$622$	−34632.0	−2.23250
$623$	−5320.00	−0.342121
$624$	0	0
$625$	−9229.00	−0.590656
$626$	21000.0	1.34078
$627$	−1920.00	−0.122293
$628$	23208.0	1.47468
$629$	−2951.00	−0.187065
$630$	31280.0	1.97813
$631$	23380.0	1.47503	0.737514	−	0.675331i	$-0.235999\pi$
$631$	23380.0	1.47503	0.737514	+	0.675331i	$0.235999\pi$
$632$	0	0
$633$	6140.00	0.385534
$634$	25652.0	1.60689
$635$	36686.0	2.29266
$636$	1488.00	0.0927721
$637$	0	0
$638$	−25216.0	−1.56475
$639$	15042.0	0.931224
$640$	0	0
$641$	6383.00	0.393313	0.196656	−	0.980472i	$-0.436992\pi$
$641$	6383.00	0.393313	0.196656	+	0.980472i	$0.436992\pi$
$642$	−4176.00	−0.256719
$643$	17104.0	1.04901	0.524507	−	0.851406i	$-0.324250\pi$
$643$	17104.0	1.04901	0.524507	+	0.851406i	$0.324250\pi$
$644$	−12480.0	−0.763635
$645$	5304.00	0.323790
$646$	−1560.00	−0.0950114
$647$	6994.00	0.424981	0.212490	−	0.977163i	$-0.431843\pi$
$647$	6994.00	0.424981	0.212490	+	0.977163i	$0.431843\pi$
$648$	0	0
$649$	27648.0	1.67223
$650$	0	0
$651$	−2960.00	−0.178205
$652$	−18880.0	−1.13405
$653$	−5250.00	−0.314622	−0.157311	−	0.987549i	$-0.550283\pi$
$653$	−5250.00	−0.314622	−0.157311	+	0.987549i	$0.550283\pi$
$654$	−13072.0	−0.781584
$655$	−12410.0	−0.740304
$656$	−10560.0	−0.628504
$657$	4945.00	0.293642
$658$	12960.0	0.767832
$659$	−4340.00	−0.256544	−0.128272	−	0.991739i	$-0.540943\pi$
$659$	−4340.00	−0.256544	−0.128272	+	0.991739i	$0.540943\pi$
$660$	−8704.00	−0.513337
$661$	4179.00	0.245907	0.122953	−	0.992412i	$-0.460763\pi$
$661$	4179.00	0.245907	0.122953	+	0.992412i	$0.460763\pi$
$662$	13952.0	0.819124
$663$	0	0
$664$	0	0
$665$	−10200.0	−0.594796
$666$	20884.0	1.21507
$667$	15366.0	0.892015
$668$	−2240.00	−0.129743
$669$	10756.0	0.621601
$670$	−58616.0	−3.37990
$671$	4640.00	0.266953
$672$	−10240.0	−0.587822
$673$	22867.0	1.30974	0.654872	−	0.755740i	$-0.272722\pi$
$673$	22867.0	1.30974	0.654872	+	0.755740i	$0.272722\pi$
$674$	7332.00	0.419018
$675$	−16400.0	−0.935165
$676$	0	0
$677$	5410.00	0.307124	0.153562	−	0.988139i	$-0.450925\pi$
$677$	5410.00	0.307124	0.153562	+	0.988139i	$0.450925\pi$
$678$	−2616.00	−0.148181
$679$	4760.00	0.269031
$680$	0	0
$681$	7948.00	0.447236
$682$	−9472.00	−0.531821
$683$	13578.0	0.760685	0.380342	−	0.924846i	$-0.375806\pi$
$683$	13578.0	0.760685	0.380342	+	0.924846i	$0.375806\pi$
$684$	5520.00	0.308571
$685$	28407.0	1.58449
$686$	−22880.0	−1.27341
$687$	12596.0	0.699516
$688$	9984.00	0.553251
$689$	0	0
$690$	10608.0	0.585275
$691$	−12744.0	−0.701599	−0.350799	−	0.936451i	$-0.614090\pi$
$691$	−12744.0	−0.701599	−0.350799	+	0.936451i	$0.614090\pi$
$692$	10608.0	0.582739
$693$	14720.0	0.806878
$694$	−28920.0	−1.58183
$695$	−15504.0	−0.846187
$696$	0	0
$697$	−2145.00	−0.116568
$698$	−21032.0	−1.14051
$699$	8060.00	0.436133
$700$	−26240.0	−1.41683
$701$	16406.0	0.883946	0.441973	−	0.897028i	$-0.354279\pi$
$701$	16406.0	0.883946	0.441973	+	0.897028i	$0.354279\pi$
$702$	0	0
$703$	−6810.00	−0.365354
$704$	−16384.0	−0.877124
$705$	−5508.00	−0.294246
$706$	12652.0	0.674454
$707$	16380.0	0.871334
$708$	13824.0	0.733810
$709$	−709.000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−709.000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	−44472.0	−2.35071
$711$	1748.00	0.0922013
$712$	0	0
$713$	5772.00	0.303174
$714$	−2080.00	−0.109022
$715$	0	0
$716$	34112.0	1.78048
$717$	1968.00	0.102505
$718$	−40272.0	−2.09323
$719$	−7644.00	−0.396486	−0.198243	−	0.980153i	$-0.563523\pi$
$719$	−7644.00	−0.396486	−0.198243	+	0.980153i	$0.563523\pi$
$720$	−25024.0	−1.29526
$721$	−32760.0	−1.69216
$722$	23836.0	1.22865
$723$	−1886.00	−0.0970140
$724$	−3224.00	−0.165496
$725$	32308.0	1.65502
$726$	2456.00	0.125552
$727$	−15808.0	−0.806446	−0.403223	−	0.915102i	$-0.632110\pi$
$727$	−15808.0	−0.806446	−0.403223	+	0.915102i	$0.632110\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	−14620.0	−0.741247
$731$	2028.00	0.102611
$732$	2320.00	0.117144
$733$	2583.00	0.130157	0.0650786	−	0.997880i	$-0.479270\pi$
$733$	2583.00	0.130157	0.0650786	+	0.997880i	$0.479270\pi$
$734$	−29752.0	−1.49614
$735$	−1938.00	−0.0972574
$736$	19968.0	1.00004
$737$	−27584.0	−1.37866
$738$	15180.0	0.757159
$739$	−4076.00	−0.202893	−0.101447	−	0.994841i	$-0.532347\pi$
$739$	−4076.00	−0.202893	−0.101447	+	0.994841i	$0.532347\pi$
$740$	−30872.0	−1.53362
$741$	0	0
$742$	7440.00	0.368101
$743$	34056.0	1.68155	0.840776	−	0.541383i	$-0.182099\pi$
$743$	34056.0	1.68155	0.840776	+	0.541383i	$0.182099\pi$
$744$	0	0
$745$	−35955.0	−1.76817
$746$	38732.0	1.90091
$747$	14444.0	0.707468
$748$	−3328.00	−0.162679
$749$	−10440.0	−0.509305
$750$	5304.00	0.258233
$751$	−364.000	−0.0176865	−0.00884324	−	0.999961i	$-0.502815\pi$
$751$	−364.000	−0.0176865	−0.00884324	+	0.999961i	$0.502815\pi$
$752$	−10368.0	−0.502769
$753$	−5460.00	−0.264241
$754$	0	0
$755$	8738.00	0.421203
$756$	16000.0	0.769728
$757$	−6914.00	−0.331960	−0.165980	−	0.986129i	$-0.553079\pi$
$757$	−6914.00	−0.331960	−0.165980	+	0.986129i	$0.553079\pi$
$758$	−4248.00	−0.203554
$759$	4992.00	0.238733
$760$	0	0
$761$	−13982.0	−0.666028	−0.333014	−	0.942922i	$-0.608066\pi$
$761$	−13982.0	−0.666028	−0.333014	+	0.942922i	$0.608066\pi$
$762$	17264.0	0.820746
$763$	−32680.0	−1.55058
$764$	−9968.00	−0.472028
$765$	−5083.00	−0.240230
$766$	−14128.0	−0.666404
$767$	0	0
$768$	8192.00	0.384900
$769$	18066.0	0.847174	0.423587	−	0.905855i	$-0.360771\pi$
$769$	18066.0	0.847174	0.423587	+	0.905855i	$0.360771\pi$
$770$	−43520.0	−2.03682
$771$	−3770.00	−0.176100
$772$	−2136.00	−0.0995807
$773$	−14434.0	−0.671610	−0.335805	−	0.941931i	$-0.609008\pi$
$773$	−14434.0	−0.671610	−0.335805	+	0.941931i	$0.609008\pi$
$774$	−14352.0	−0.666501
$775$	12136.0	0.562501
$776$	0	0
$777$	−9080.00	−0.419232
$778$	44252.0	2.03922
$779$	−4950.00	−0.227666
$780$	0	0
$781$	−20928.0	−0.958851
$782$	4056.00	0.185476
$783$	−19700.0	−0.899132
$784$	−3648.00	−0.166181
$785$	−49317.0	−2.24229
$786$	−5840.00	−0.265020
$787$	15398.0	0.697433	0.348716	−	0.937228i	$-0.386618\pi$
$787$	15398.0	0.697433	0.348716	+	0.937228i	$0.386618\pi$
$788$	−10224.0	−0.462202
$789$	8064.00	0.363861
$790$	−5168.00	−0.232746
$791$	−6540.00	−0.293977
$792$	0	0
$793$	0	0
$794$	−23944.0	−1.07020
$795$	−3162.00	−0.141062
$796$	33904.0	1.50967
$797$	−36842.0	−1.63740	−0.818702	−	0.574219i	$-0.805306\pi$
$797$	−36842.0	−1.63740	−0.818702	+	0.574219i	$0.805306\pi$
$798$	−4800.00	−0.212930
$799$	−2106.00	−0.0932477
$800$	41984.0	1.85545
$801$	−6118.00	−0.269874
$802$	23740.0	1.04525
$803$	−6880.00	−0.302354
$804$	−13792.0	−0.604983
$805$	26520.0	1.16113
$806$	0	0
$807$	8012.00	0.349487
$808$	0	0
$809$	41511.0	1.80402	0.902008	−	0.431719i	$-0.142093\pi$
$809$	41511.0	1.80402	0.902008	+	0.431719i	$0.142093\pi$
$810$	28628.0	1.24183
$811$	−23066.0	−0.998714	−0.499357	−	0.866396i	$-0.666430\pi$
$811$	−23066.0	−0.998714	−0.499357	+	0.866396i	$0.666430\pi$
$812$	−31520.0	−1.36224
$813$	8592.00	0.370645
$814$	−29056.0	−1.25112
$815$	40120.0	1.72435
$816$	1664.00	0.0713868
$817$	4680.00	0.200407
$818$	−60356.0	−2.57983
$819$	0	0
$820$	−22440.0	−0.955657
$821$	−28838.0	−1.22589	−0.612943	−	0.790127i	$-0.710015\pi$
$821$	−28838.0	−1.22589	−0.612943	+	0.790127i	$0.710015\pi$
$822$	13368.0	0.567229
$823$	27456.0	1.16289	0.581443	−	0.813587i	$-0.302488\pi$
$823$	27456.0	1.16289	0.581443	+	0.813587i	$0.302488\pi$
$824$	0	0
$825$	10496.0	0.442938
$826$	69120.0	2.91161
$827$	33572.0	1.41162	0.705812	−	0.708399i	$-0.250582\pi$
$827$	33572.0	1.41162	0.705812	+	0.708399i	$0.250582\pi$
$828$	−14352.0	−0.602375
$829$	−45799.0	−1.91878	−0.959388	−	0.282090i	$-0.908972\pi$
$829$	−45799.0	−1.91878	−0.959388	+	0.282090i	$0.908972\pi$
$830$	−42704.0	−1.78588
$831$	−11102.0	−0.463447
$832$	0	0
$833$	−741.000	−0.0308213
$834$	−7296.00	−0.302925
$835$	4760.00	0.197277
$836$	−7680.00	−0.317725
$837$	−7400.00	−0.305593
$838$	43256.0	1.78312
$839$	−32286.0	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$839$	−32286.0	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$840$	0	0
$841$	14420.0	0.591250
$842$	−26140.0	−1.06989
$843$	11114.0	0.454077
$844$	24560.0	1.00165
$845$	0	0
$846$	14904.0	0.605686
$847$	6140.00	0.249083
$848$	−5952.00	−0.241029
$849$	6240.00	0.252245
$850$	8528.00	0.344127
$851$	17706.0	0.713224
$852$	−10464.0	−0.420764
$853$	−20937.0	−0.840409	−0.420205	−	0.907429i	$-0.638042\pi$
$853$	−20937.0	−0.840409	−0.420205	+	0.907429i	$0.638042\pi$
$854$	11600.0	0.464805
$855$	−11730.0	−0.469190
$856$	0	0
$857$	−7189.00	−0.286548	−0.143274	−	0.989683i	$-0.545763\pi$
$857$	−7189.00	−0.286548	−0.143274	+	0.989683i	$0.545763\pi$
$858$	0	0
$859$	−32498.0	−1.29082	−0.645412	−	0.763835i	$-0.723314\pi$
$859$	−32498.0	−1.29082	−0.645412	+	0.763835i	$0.723314\pi$
$860$	21216.0	0.841232
$861$	−6600.00	−0.261240
$862$	7920.00	0.312942
$863$	−8428.00	−0.332436	−0.166218	−	0.986089i	$-0.553156\pi$
$863$	−8428.00	−0.332436	−0.166218	+	0.986089i	$0.553156\pi$
$864$	−25600.0	−1.00802
$865$	−22542.0	−0.886071
$866$	27716.0	1.08756
$867$	−9488.00	−0.371660
$868$	−11840.0	−0.462991
$869$	−2432.00	−0.0949367
$870$	26792.0	1.04406
$871$	0	0
$872$	0	0
$873$	5474.00	0.212219
$874$	9360.00	0.362250
$875$	13260.0	0.512308
$876$	−3440.00	−0.132679
$877$	6847.00	0.263634	0.131817	−	0.991274i	$-0.457919\pi$
$877$	6847.00	0.263634	0.131817	+	0.991274i	$0.457919\pi$
$878$	18304.0	0.703565
$879$	−16602.0	−0.637055
$880$	34816.0	1.33369
$881$	29731.0	1.13696	0.568481	−	0.822697i	$-0.307531\pi$
$881$	29731.0	1.13696	0.568481	+	0.822697i	$0.307531\pi$
$882$	5244.00	0.200198
$883$	−23738.0	−0.904697	−0.452348	−	0.891841i	$-0.649414\pi$
$883$	−23738.0	−0.904697	−0.452348	+	0.891841i	$0.649414\pi$
$884$	0	0
$885$	−29376.0	−1.11578
$886$	35248.0	1.33655
$887$	27588.0	1.04432	0.522161	−	0.852847i	$-0.325126\pi$
$887$	27588.0	1.04432	0.522161	+	0.852847i	$0.325126\pi$
$888$	0	0
$889$	43160.0	1.62828
$890$	18088.0	0.681248
$891$	13472.0	0.506542
$892$	43024.0	1.61497
$893$	−4860.00	−0.182121
$894$	−16920.0	−0.632986
$895$	−72488.0	−2.70727
$896$	0	0
$897$	0	0
$898$	7672.00	0.285098
$899$	14578.0	0.540827
$900$	−30176.0	−1.11763
$901$	−1209.00	−0.0447033
$902$	−21120.0	−0.779622
$903$	6240.00	0.229960
$904$	0	0
$905$	6851.00	0.251641
$906$	4112.00	0.150786
$907$	−37128.0	−1.35922	−0.679611	−	0.733572i	$-0.737852\pi$
$907$	−37128.0	−1.35922	−0.679611	+	0.733572i	$0.737852\pi$
$908$	31792.0	1.16195
$909$	18837.0	0.687331
$910$	0	0
$911$	20516.0	0.746131	0.373066	−	0.927805i	$-0.378307\pi$
$911$	20516.0	0.746131	0.373066	+	0.927805i	$0.378307\pi$
$912$	3840.00	0.139424
$913$	−20096.0	−0.728456
$914$	−47044.0	−1.70249
$915$	−4930.00	−0.178121
$916$	50384.0	1.81740
$917$	−14600.0	−0.525774
$918$	−5200.00	−0.186956
$919$	−21006.0	−0.753998	−0.376999	−	0.926214i	$-0.623044\pi$
$919$	−21006.0	−0.753998	−0.376999	+	0.926214i	$0.623044\pi$
$920$	0	0
$921$	−17356.0	−0.620955
$922$	3604.00	0.128733
$923$	0	0
$924$	−10240.0	−0.364579
$925$	37228.0	1.32330
$926$	5488.00	0.194759
$927$	−37674.0	−1.33482
$928$	50432.0	1.78396
$929$	−20427.0	−0.721408	−0.360704	−	0.932680i	$-0.617464\pi$
$929$	−20427.0	−0.721408	−0.360704	+	0.932680i	$0.617464\pi$
$930$	10064.0	0.354851
$931$	−1710.00	−0.0601965
$932$	32240.0	1.13311
$933$	17316.0	0.607610
$934$	25584.0	0.896289
$935$	7072.00	0.247357
$936$	0	0
$937$	33191.0	1.15721	0.578603	−	0.815609i	$-0.303598\pi$
$937$	33191.0	1.15721	0.578603	+	0.815609i	$0.303598\pi$
$938$	−68960.0	−2.40045
$939$	−10500.0	−0.364914
$940$	−22032.0	−0.764473
$941$	36422.0	1.26177	0.630884	−	0.775877i	$-0.282692\pi$
$941$	36422.0	1.26177	0.630884	+	0.775877i	$0.282692\pi$
$942$	−23208.0	−0.802715
$943$	12870.0	0.444438
$944$	−55296.0	−1.90650
$945$	−34000.0	−1.17039
$946$	19968.0	0.686275
$947$	39630.0	1.35988	0.679938	−	0.733270i	$-0.262007\pi$
$947$	39630.0	1.35988	0.679938	+	0.733270i	$0.262007\pi$
$948$	−1216.00	−0.0416602
$949$	0	0
$950$	19680.0	0.672109
$951$	−12826.0	−0.437341
$952$	0	0
$953$	57642.0	1.95929	0.979647	−	0.200727i	$-0.0643305\pi$
$953$	57642.0	1.95929	0.979647	+	0.200727i	$0.0643305\pi$
$954$	8556.00	0.290368
$955$	21182.0	0.717731
$956$	7872.00	0.266317
$957$	12608.0	0.425871
$958$	13080.0	0.441123
$959$	33420.0	1.12533
$960$	17408.0	0.585251
$961$	−24315.0	−0.816186
$962$	0	0
$963$	−12006.0	−0.401753
$964$	−7544.00	−0.252050
$965$	4539.00	0.151415
$966$	12480.0	0.415670
$967$	−2162.00	−0.0718979	−0.0359489	−	0.999354i	$-0.511445\pi$
$967$	−2162.00	−0.0718979	−0.0359489	+	0.999354i	$0.511445\pi$
$968$	0	0
$969$	780.000	0.0258588
$970$	−16184.0	−0.535708
$971$	−19758.0	−0.653001	−0.326501	−	0.945197i	$-0.605870\pi$
$971$	−19758.0	−0.653001	−0.326501	+	0.945197i	$0.605870\pi$
$972$	28336.0	0.935059
$973$	−18240.0	−0.600974
$974$	79680.0	2.62126
$975$	0	0
$976$	−9280.00	−0.304350
$977$	12489.0	0.408965	0.204482	−	0.978870i	$-0.434449\pi$
$977$	12489.0	0.408965	0.204482	+	0.978870i	$0.434449\pi$
$978$	18880.0	0.617296
$979$	8512.00	0.277880
$980$	−7752.00	−0.252682
$981$	−37582.0	−1.22314
$982$	−26208.0	−0.851661
$983$	28658.0	0.929856	0.464928	−	0.885349i	$-0.346080\pi$
$983$	28658.0	0.929856	0.464928	+	0.885349i	$0.346080\pi$
$984$	0	0
$985$	21726.0	0.702790
$986$	10244.0	0.330868
$987$	−6480.00	−0.208977
$988$	0	0
$989$	−12168.0	−0.391223
$990$	−50048.0	−1.60670
$991$	−42794.0	−1.37174	−0.685871	−	0.727723i	$-0.740579\pi$
$991$	−42794.0	−1.37174	−0.685871	+	0.727723i	$0.740579\pi$
$992$	18944.0	0.606323
$993$	−6976.00	−0.222937
$994$	−52320.0	−1.66951
$995$	−72046.0	−2.29549
$996$	−10048.0	−0.319662
$997$	−52583.0	−1.67033	−0.835166	−	0.549998i	$-0.814628\pi$
$997$	−52583.0	−1.67033	−0.835166	+	0.549998i	$0.814628\pi$
$998$	6984.00	0.221518
$999$	−22700.0	−0.718915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.a.1.1		1
3.2	odd	2		1521.4.a.k.1.1		1
13.2	odd	12		169.4.e.c.147.1		4
13.3	even	3		169.4.c.d.22.1		2
13.4	even	6		13.4.c.a.3.1	✓	2
13.5	odd	4		169.4.b.c.168.2		2
13.6	odd	12		169.4.e.c.23.2		4
13.7	odd	12		169.4.e.c.23.1		4
13.8	odd	4		169.4.b.c.168.1		2
13.9	even	3		169.4.c.d.146.1		2
13.10	even	6		13.4.c.a.9.1	yes	2
13.11	odd	12		169.4.e.c.147.2		4
13.12	even	2		169.4.a.d.1.1		1
39.17	odd	6		117.4.g.c.55.1		2
39.23	odd	6		117.4.g.c.100.1		2
39.38	odd	2		1521.4.a.b.1.1		1
52.23	odd	6		208.4.i.b.113.1		2
52.43	odd	6		208.4.i.b.81.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.a.3.1	✓	2	13.4	even	6
13.4.c.a.9.1	yes	2	13.10	even	6
117.4.g.c.55.1		2	39.17	odd	6
117.4.g.c.100.1		2	39.23	odd	6
169.4.a.a.1.1		1	1.1	even	1	trivial
169.4.a.d.1.1		1	13.12	even	2
169.4.b.c.168.1		2	13.8	odd	4
169.4.b.c.168.2		2	13.5	odd	4
169.4.c.d.22.1		2	13.3	even	3
169.4.c.d.146.1		2	13.9	even	3
169.4.e.c.23.1		4	13.7	odd	12
169.4.e.c.23.2		4	13.6	odd	12
169.4.e.c.147.1		4	13.2	odd	12
169.4.e.c.147.2		4	13.11	odd	12
208.4.i.b.81.1		2	52.43	odd	6
208.4.i.b.113.1		2	52.23	odd	6
1521.4.a.b.1.1		1	39.38	odd	2
1521.4.a.k.1.1		1	3.2	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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