LMFDB - Embedded newform 2166.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +19.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +19.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +32.0000 q^{11} +12.0000 q^{12} +81.0000 q^{13} -38.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} -124.000 q^{17} -18.0000 q^{18} -24.0000 q^{20} +57.0000 q^{21} -64.0000 q^{22} +98.0000 q^{23} -24.0000 q^{24} -89.0000 q^{25} -162.000 q^{26} +27.0000 q^{27} +76.0000 q^{28} -300.000 q^{29} +36.0000 q^{30} -225.000 q^{31} -32.0000 q^{32} +96.0000 q^{33} +248.000 q^{34} -114.000 q^{35} +36.0000 q^{36} -293.000 q^{37} +243.000 q^{39} +48.0000 q^{40} +176.000 q^{41} -114.000 q^{42} -111.000 q^{43} +128.000 q^{44} -54.0000 q^{45} -196.000 q^{46} -550.000 q^{47} +48.0000 q^{48} +18.0000 q^{49} +178.000 q^{50} -372.000 q^{51} +324.000 q^{52} -482.000 q^{53} -54.0000 q^{54} -192.000 q^{55} -152.000 q^{56} +600.000 q^{58} -496.000 q^{59} -72.0000 q^{60} +155.000 q^{61} +450.000 q^{62} +171.000 q^{63} +64.0000 q^{64} -486.000 q^{65} -192.000 q^{66} +465.000 q^{67} -496.000 q^{68} +294.000 q^{69} +228.000 q^{70} -110.000 q^{71} -72.0000 q^{72} +817.000 q^{73} +586.000 q^{74} -267.000 q^{75} +608.000 q^{77} -486.000 q^{78} +259.000 q^{79} -96.0000 q^{80} +81.0000 q^{81} -352.000 q^{82} -56.0000 q^{83} +228.000 q^{84} +744.000 q^{85} +222.000 q^{86} -900.000 q^{87} -256.000 q^{88} +308.000 q^{89} +108.000 q^{90} +1539.00 q^{91} +392.000 q^{92} -675.000 q^{93} +1100.00 q^{94} -96.0000 q^{96} -1150.00 q^{97} -36.0000 q^{98} +288.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	−6.00000	−0.408248
$7$	19.0000	1.02590	0.512952	−	0.858417i	$-0.328552\pi$
$7$	19.0000	1.02590	0.512952	+	0.858417i	$0.328552\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	12.0000	0.379473
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	12.0000	0.288675
$13$	81.0000	1.72810	0.864052	−	0.503402i	$-0.167919\pi$
$13$	81.0000	1.72810	0.864052	+	0.503402i	$0.167919\pi$
$14$	−38.0000	−0.725423
$15$	−18.0000	−0.309839
$16$	16.0000	0.250000
$17$	−124.000	−1.76908	−0.884542	−	0.466461i	$-0.845529\pi$
$17$	−124.000	−1.76908	−0.884542	+	0.466461i	$0.845529\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−24.0000	−0.268328
$21$	57.0000	0.592306
$22$	−64.0000	−0.620220
$23$	98.0000	0.888453	0.444226	−	0.895915i	$-0.353479\pi$
$23$	98.0000	0.888453	0.444226	+	0.895915i	$0.353479\pi$
$24$	−24.0000	−0.204124
$25$	−89.0000	−0.712000
$26$	−162.000	−1.22195
$27$	27.0000	0.192450
$28$	76.0000	0.512952
$29$	−300.000	−1.92099	−0.960493	−	0.278304i	$-0.910228\pi$
$29$	−300.000	−1.92099	−0.960493	+	0.278304i	$0.910228\pi$
$30$	36.0000	0.219089
$31$	−225.000	−1.30359	−0.651793	−	0.758397i	$-0.725983\pi$
$31$	−225.000	−1.30359	−0.651793	+	0.758397i	$0.725983\pi$
$32$	−32.0000	−0.176777
$33$	96.0000	0.506408
$34$	248.000	1.25093
$35$	−114.000	−0.550558
$36$	36.0000	0.166667
$37$	−293.000	−1.30186	−0.650931	−	0.759137i	$-0.725621\pi$
$37$	−293.000	−1.30186	−0.650931	+	0.759137i	$0.725621\pi$
$38$	0	0
$39$	243.000	0.997722
$40$	48.0000	0.189737
$41$	176.000	0.670404	0.335202	−	0.942146i	$-0.391195\pi$
$41$	176.000	0.670404	0.335202	+	0.942146i	$0.391195\pi$
$42$	−114.000	−0.418823
$43$	−111.000	−0.393659	−0.196830	−	0.980438i	$-0.563065\pi$
$43$	−111.000	−0.393659	−0.196830	+	0.980438i	$0.563065\pi$
$44$	128.000	0.438562
$45$	−54.0000	−0.178885
$46$	−196.000	−0.628231
$47$	−550.000	−1.70693	−0.853465	−	0.521150i	$-0.825503\pi$
$47$	−550.000	−1.70693	−0.853465	+	0.521150i	$0.825503\pi$
$48$	48.0000	0.144338
$49$	18.0000	0.0524781
$50$	178.000	0.503460
$51$	−372.000	−1.02138
$52$	324.000	0.864052
$53$	−482.000	−1.24920	−0.624602	−	0.780943i	$-0.714739\pi$
$53$	−482.000	−1.24920	−0.624602	+	0.780943i	$0.714739\pi$
$54$	−54.0000	−0.136083
$55$	−192.000	−0.470714
$56$	−152.000	−0.362712
$57$	0	0
$58$	600.000	1.35834
$59$	−496.000	−1.09447	−0.547235	−	0.836979i	$-0.684320\pi$
$59$	−496.000	−1.09447	−0.547235	+	0.836979i	$0.684320\pi$
$60$	−72.0000	−0.154919
$61$	155.000	0.325340	0.162670	−	0.986681i	$-0.447989\pi$
$61$	155.000	0.325340	0.162670	+	0.986681i	$0.447989\pi$
$62$	450.000	0.921775
$63$	171.000	0.341968
$64$	64.0000	0.125000
$65$	−486.000	−0.927398
$66$	−192.000	−0.358084
$67$	465.000	0.847892	0.423946	−	0.905687i	$-0.360645\pi$
$67$	465.000	0.847892	0.423946	+	0.905687i	$0.360645\pi$
$68$	−496.000	−0.884542
$69$	294.000	0.512948
$70$	228.000	0.389303
$71$	−110.000	−0.183868	−0.0919338	−	0.995765i	$-0.529305\pi$
$71$	−110.000	−0.183868	−0.0919338	+	0.995765i	$0.529305\pi$
$72$	−72.0000	−0.117851
$73$	817.000	1.30990	0.654949	−	0.755673i	$-0.272690\pi$
$73$	817.000	1.30990	0.654949	+	0.755673i	$0.272690\pi$
$74$	586.000	0.920556
$75$	−267.000	−0.411073
$76$	0	0
$77$	608.000	0.899845
$78$	−486.000	−0.705496
$79$	259.000	0.368858	0.184429	−	0.982846i	$-0.440956\pi$
$79$	259.000	0.368858	0.184429	+	0.982846i	$0.440956\pi$
$80$	−96.0000	−0.134164
$81$	81.0000	0.111111
$82$	−352.000	−0.474048
$83$	−56.0000	−0.0740578	−0.0370289	−	0.999314i	$-0.511789\pi$
$83$	−56.0000	−0.0740578	−0.0370289	+	0.999314i	$0.511789\pi$
$84$	228.000	0.296153
$85$	744.000	0.949390
$86$	222.000	0.278359
$87$	−900.000	−1.10908
$88$	−256.000	−0.310110
$89$	308.000	0.366831	0.183415	−	0.983036i	$-0.441285\pi$
$89$	308.000	0.366831	0.183415	+	0.983036i	$0.441285\pi$
$90$	108.000	0.126491
$91$	1539.00	1.77287
$92$	392.000	0.444226
$93$	−675.000	−0.752626
$94$	1100.00	1.20698
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1150.00	−1.20376	−0.601880	−	0.798586i	$-0.705582\pi$
$97$	−1150.00	−1.20376	−0.601880	+	0.798586i	$0.705582\pi$
$98$	−36.0000	−0.0371076
$99$	288.000	0.292375
$100$	−356.000	−0.356000
$101$	−578.000	−0.569437	−0.284719	−	0.958611i	$-0.591900\pi$
$101$	−578.000	−0.569437	−0.284719	+	0.958611i	$0.591900\pi$
$102$	744.000	0.722225
$103$	819.000	0.783480	0.391740	−	0.920076i	$-0.371873\pi$
$103$	819.000	0.783480	0.391740	+	0.920076i	$0.371873\pi$
$104$	−648.000	−0.610977
$105$	−342.000	−0.317865
$106$	964.000	0.883320
$107$	262.000	0.236715	0.118357	−	0.992971i	$-0.462237\pi$
$107$	262.000	0.236715	0.118357	+	0.992971i	$0.462237\pi$
$108$	108.000	0.0962250
$109$	1034.00	0.908617	0.454308	−	0.890844i	$-0.349886\pi$
$109$	1034.00	0.908617	0.454308	+	0.890844i	$0.349886\pi$
$110$	384.000	0.332845
$111$	−879.000	−0.751631
$112$	304.000	0.256476
$113$	−1324.00	−1.10223	−0.551113	−	0.834431i	$-0.685797\pi$
$113$	−1324.00	−1.10223	−0.551113	+	0.834431i	$0.685797\pi$
$114$	0	0
$115$	−588.000	−0.476794
$116$	−1200.00	−0.960493
$117$	729.000	0.576035
$118$	992.000	0.773907
$119$	−2356.00	−1.81491
$120$	144.000	0.109545
$121$	−307.000	−0.230654
$122$	−310.000	−0.230050
$123$	528.000	0.387058
$124$	−900.000	−0.651793
$125$	1284.00	0.918756
$126$	−342.000	−0.241808
$127$	−1436.00	−1.00334	−0.501671	−	0.865059i	$-0.667281\pi$
$127$	−1436.00	−1.00334	−0.501671	+	0.865059i	$0.667281\pi$
$128$	−128.000	−0.0883883
$129$	−333.000	−0.227279
$130$	972.000	0.655770
$131$	−378.000	−0.252107	−0.126053	−	0.992023i	$-0.540231\pi$
$131$	−378.000	−0.252107	−0.126053	+	0.992023i	$0.540231\pi$
$132$	384.000	0.253204
$133$	0	0
$134$	−930.000	−0.599550
$135$	−162.000	−0.103280
$136$	992.000	0.625465
$137$	1062.00	0.662283	0.331142	−	0.943581i	$-0.392566\pi$
$137$	1062.00	0.662283	0.331142	+	0.943581i	$0.392566\pi$
$138$	−588.000	−0.362709
$139$	−1873.00	−1.14292	−0.571460	−	0.820630i	$-0.693623\pi$
$139$	−1873.00	−1.14292	−0.571460	+	0.820630i	$0.693623\pi$
$140$	−456.000	−0.275279
$141$	−1650.00	−0.985497
$142$	220.000	0.130014
$143$	2592.00	1.51576
$144$	144.000	0.0833333
$145$	1800.00	1.03091
$146$	−1634.00	−0.926238
$147$	54.0000	0.0302983
$148$	−1172.00	−0.650931
$149$	1372.00	0.754353	0.377177	−	0.926141i	$-0.376895\pi$
$149$	1372.00	0.754353	0.377177	+	0.926141i	$0.376895\pi$
$150$	534.000	0.290673
$151$	−3296.00	−1.77632	−0.888161	−	0.459532i	$-0.848017\pi$
$151$	−3296.00	−1.77632	−0.888161	+	0.459532i	$0.848017\pi$
$152$	0	0
$153$	−1116.00	−0.589694
$154$	−1216.00	−0.636286
$155$	1350.00	0.699578
$156$	972.000	0.498861
$157$	1767.00	0.898229	0.449114	−	0.893474i	$-0.351740\pi$
$157$	1767.00	0.898229	0.449114	+	0.893474i	$0.351740\pi$
$158$	−518.000	−0.260822
$159$	−1446.00	−0.721228
$160$	192.000	0.0948683
$161$	1862.00	0.911467
$162$	−162.000	−0.0785674
$163$	1709.00	0.821222	0.410611	−	0.911811i	$-0.365315\pi$
$163$	1709.00	0.821222	0.410611	+	0.911811i	$0.365315\pi$
$164$	704.000	0.335202
$165$	−576.000	−0.271767
$166$	112.000	0.0523668
$167$	−2760.00	−1.27889	−0.639447	−	0.768835i	$-0.720837\pi$
$167$	−2760.00	−1.27889	−0.639447	+	0.768835i	$0.720837\pi$
$168$	−456.000	−0.209412
$169$	4364.00	1.98635
$170$	−1488.00	−0.671320
$171$	0	0
$172$	−444.000	−0.196830
$173$	−1476.00	−0.648660	−0.324330	−	0.945944i	$-0.605139\pi$
$173$	−1476.00	−0.648660	−0.324330	+	0.945944i	$0.605139\pi$
$174$	1800.00	0.784239
$175$	−1691.00	−0.730443
$176$	512.000	0.219281
$177$	−1488.00	−0.631892
$178$	−616.000	−0.259388
$179$	26.0000	0.0108566	0.00542830	−	0.999985i	$-0.498272\pi$
$179$	26.0000	0.0108566	0.00542830	+	0.999985i	$0.498272\pi$
$180$	−216.000	−0.0894427
$181$	−3594.00	−1.47591	−0.737956	−	0.674849i	$-0.764209\pi$
$181$	−3594.00	−1.47591	−0.737956	+	0.674849i	$0.764209\pi$
$182$	−3078.00	−1.25361
$183$	465.000	0.187835
$184$	−784.000	−0.314115
$185$	1758.00	0.698653
$186$	1350.00	0.532187
$187$	−3968.00	−1.55171
$188$	−2200.00	−0.853465
$189$	513.000	0.197435
$190$	0	0
$191$	314.000	0.118954	0.0594771	−	0.998230i	$-0.481057\pi$
$191$	314.000	0.118954	0.0594771	+	0.998230i	$0.481057\pi$
$192$	192.000	0.0721688
$193$	−925.000	−0.344989	−0.172495	−	0.985010i	$-0.555183\pi$
$193$	−925.000	−0.344989	−0.172495	+	0.985010i	$0.555183\pi$
$194$	2300.00	0.851188
$195$	−1458.00	−0.535434
$196$	72.0000	0.0262391
$197$	1244.00	0.449905	0.224953	−	0.974370i	$-0.427777\pi$
$197$	1244.00	0.449905	0.224953	+	0.974370i	$0.427777\pi$
$198$	−576.000	−0.206740
$199$	3491.00	1.24357	0.621785	−	0.783188i	$-0.286408\pi$
$199$	3491.00	1.24357	0.621785	+	0.783188i	$0.286408\pi$
$200$	712.000	0.251730
$201$	1395.00	0.489531
$202$	1156.00	0.402653
$203$	−5700.00	−1.97075
$204$	−1488.00	−0.510690
$205$	−1056.00	−0.359777
$206$	−1638.00	−0.554004
$207$	882.000	0.296151
$208$	1296.00	0.432026
$209$	0	0
$210$	684.000	0.224764
$211$	−4577.00	−1.49333	−0.746667	−	0.665197i	$-0.768347\pi$
$211$	−4577.00	−1.49333	−0.746667	+	0.665197i	$0.768347\pi$
$212$	−1928.00	−0.624602
$213$	−330.000	−0.106156
$214$	−524.000	−0.167383
$215$	666.000	0.211260
$216$	−216.000	−0.0680414
$217$	−4275.00	−1.33735
$218$	−2068.00	−0.642489
$219$	2451.00	0.756270
$220$	−768.000	−0.235357
$221$	−10044.0	−3.05716
$222$	1758.00	0.531483
$223$	2295.00	0.689168	0.344584	−	0.938755i	$-0.388020\pi$
$223$	2295.00	0.689168	0.344584	+	0.938755i	$0.388020\pi$
$224$	−608.000	−0.181356
$225$	−801.000	−0.237333
$226$	2648.00	0.779391
$227$	2518.00	0.736236	0.368118	−	0.929779i	$-0.380002\pi$
$227$	2518.00	0.736236	0.368118	+	0.929779i	$0.380002\pi$
$228$	0	0
$229$	5401.00	1.55855	0.779275	−	0.626682i	$-0.215587\pi$
$229$	5401.00	1.55855	0.779275	+	0.626682i	$0.215587\pi$
$230$	1176.00	0.337144
$231$	1824.00	0.519525
$232$	2400.00	0.679171
$233$	−6078.00	−1.70894	−0.854470	−	0.519501i	$-0.826118\pi$
$233$	−6078.00	−1.70894	−0.854470	+	0.519501i	$0.826118\pi$
$234$	−1458.00	−0.407318
$235$	3300.00	0.916035
$236$	−1984.00	−0.547235
$237$	777.000	0.212960
$238$	4712.00	1.28333
$239$	1062.00	0.287427	0.143714	−	0.989619i	$-0.454096\pi$
$239$	1062.00	0.287427	0.143714	+	0.989619i	$0.454096\pi$
$240$	−288.000	−0.0774597
$241$	−4619.00	−1.23459	−0.617294	−	0.786732i	$-0.711771\pi$
$241$	−4619.00	−1.23459	−0.617294	+	0.786732i	$0.711771\pi$
$242$	614.000	0.163097
$243$	243.000	0.0641500
$244$	620.000	0.162670
$245$	−108.000	−0.0281627
$246$	−1056.00	−0.273691
$247$	0	0
$248$	1800.00	0.460888
$249$	−168.000	−0.0427573
$250$	−2568.00	−0.649658
$251$	2178.00	0.547706	0.273853	−	0.961772i	$-0.411702\pi$
$251$	2178.00	0.547706	0.273853	+	0.961772i	$0.411702\pi$
$252$	684.000	0.170984
$253$	3136.00	0.779283
$254$	2872.00	0.709470
$255$	2232.00	0.548130
$256$	256.000	0.0625000
$257$	−630.000	−0.152912	−0.0764559	−	0.997073i	$-0.524360\pi$
$257$	−630.000	−0.152912	−0.0764559	+	0.997073i	$0.524360\pi$
$258$	666.000	0.160711
$259$	−5567.00	−1.33559
$260$	−1944.00	−0.463699
$261$	−2700.00	−0.640329
$262$	756.000	0.178267
$263$	2742.00	0.642886	0.321443	−	0.946929i	$-0.395832\pi$
$263$	2742.00	0.642886	0.321443	+	0.946929i	$0.395832\pi$
$264$	−768.000	−0.179042
$265$	2892.00	0.670393
$266$	0	0
$267$	924.000	0.211790
$268$	1860.00	0.423946
$269$	2092.00	0.474169	0.237085	−	0.971489i	$-0.423808\pi$
$269$	2092.00	0.474169	0.237085	+	0.971489i	$0.423808\pi$
$270$	324.000	0.0730297
$271$	5432.00	1.21760	0.608802	−	0.793322i	$-0.291651\pi$
$271$	5432.00	1.21760	0.608802	+	0.793322i	$0.291651\pi$
$272$	−1984.00	−0.442271
$273$	4617.00	1.02357
$274$	−2124.00	−0.468305
$275$	−2848.00	−0.624512
$276$	1176.00	0.256474
$277$	−2834.00	−0.614724	−0.307362	−	0.951593i	$-0.599446\pi$
$277$	−2834.00	−0.614724	−0.307362	+	0.951593i	$0.599446\pi$
$278$	3746.00	0.808166
$279$	−2025.00	−0.434529
$280$	912.000	0.194652
$281$	5690.00	1.20796	0.603980	−	0.796999i	$-0.293581\pi$
$281$	5690.00	1.20796	0.603980	+	0.796999i	$0.293581\pi$
$282$	3300.00	0.696852
$283$	−4388.00	−0.921694	−0.460847	−	0.887479i	$-0.652454\pi$
$283$	−4388.00	−0.921694	−0.460847	+	0.887479i	$0.652454\pi$
$284$	−440.000	−0.0919338
$285$	0	0
$286$	−5184.00	−1.07181
$287$	3344.00	0.687770
$288$	−288.000	−0.0589256
$289$	10463.0	2.12966
$290$	−3600.00	−0.728963
$291$	−3450.00	−0.694992
$292$	3268.00	0.654949
$293$	−1658.00	−0.330585	−0.165292	−	0.986245i	$-0.552857\pi$
$293$	−1658.00	−0.330585	−0.165292	+	0.986245i	$0.552857\pi$
$294$	−108.000	−0.0214241
$295$	2976.00	0.587354
$296$	2344.00	0.460278
$297$	864.000	0.168803
$298$	−2744.00	−0.533408
$299$	7938.00	1.53534
$300$	−1068.00	−0.205537
$301$	−2109.00	−0.403856
$302$	6592.00	1.25605
$303$	−1734.00	−0.328765
$304$	0	0
$305$	−930.000	−0.174596
$306$	2232.00	0.416977
$307$	804.000	0.149468	0.0747340	−	0.997204i	$-0.476189\pi$
$307$	804.000	0.149468	0.0747340	+	0.997204i	$0.476189\pi$
$308$	2432.00	0.449922
$309$	2457.00	0.452343
$310$	−2700.00	−0.494676
$311$	−3172.00	−0.578352	−0.289176	−	0.957276i	$-0.593381\pi$
$311$	−3172.00	−0.578352	−0.289176	+	0.957276i	$0.593381\pi$
$312$	−1944.00	−0.352748
$313$	3834.00	0.692366	0.346183	−	0.938167i	$-0.387478\pi$
$313$	3834.00	0.692366	0.346183	+	0.938167i	$0.387478\pi$
$314$	−3534.00	−0.635144
$315$	−1026.00	−0.183519
$316$	1036.00	0.184429
$317$	2922.00	0.517716	0.258858	−	0.965915i	$-0.416654\pi$
$317$	2922.00	0.517716	0.258858	+	0.965915i	$0.416654\pi$
$318$	2892.00	0.509985
$319$	−9600.00	−1.68494
$320$	−384.000	−0.0670820
$321$	786.000	0.136667
$322$	−3724.00	−0.644504
$323$	0	0
$324$	324.000	0.0555556
$325$	−7209.00	−1.23041
$326$	−3418.00	−0.580692
$327$	3102.00	0.524590
$328$	−1408.00	−0.237024
$329$	−10450.0	−1.75115
$330$	1152.00	0.192168
$331$	−4359.00	−0.723844	−0.361922	−	0.932208i	$-0.617879\pi$
$331$	−4359.00	−0.723844	−0.361922	+	0.932208i	$0.617879\pi$
$332$	−224.000	−0.0370289
$333$	−2637.00	−0.433954
$334$	5520.00	0.904314
$335$	−2790.00	−0.455027
$336$	912.000	0.148076
$337$	8729.00	1.41098	0.705488	−	0.708722i	$-0.250728\pi$
$337$	8729.00	1.41098	0.705488	+	0.708722i	$0.250728\pi$
$338$	−8728.00	−1.40456
$339$	−3972.00	−0.636370
$340$	2976.00	0.474695
$341$	−7200.00	−1.14341
$342$	0	0
$343$	−6175.00	−0.972066
$344$	888.000	0.139180
$345$	−1764.00	−0.275277
$346$	2952.00	0.458672
$347$	38.0000	0.00587881	0.00293940	−	0.999996i	$-0.499064\pi$
$347$	38.0000	0.00587881	0.00293940	+	0.999996i	$0.499064\pi$
$348$	−3600.00	−0.554541
$349$	6103.00	0.936063	0.468032	−	0.883712i	$-0.344963\pi$
$349$	6103.00	0.936063	0.468032	+	0.883712i	$0.344963\pi$
$350$	3382.00	0.516501
$351$	2187.00	0.332574
$352$	−1024.00	−0.155055
$353$	−12198.0	−1.83919	−0.919595	−	0.392868i	$-0.871483\pi$
$353$	−12198.0	−1.83919	−0.919595	+	0.392868i	$0.871483\pi$
$354$	2976.00	0.446815
$355$	660.000	0.0986737
$356$	1232.00	0.183415
$357$	−7068.00	−1.04784
$358$	−52.0000	−0.00767677
$359$	2988.00	0.439277	0.219639	−	0.975581i	$-0.429512\pi$
$359$	2988.00	0.439277	0.219639	+	0.975581i	$0.429512\pi$
$360$	432.000	0.0632456
$361$	0	0
$362$	7188.00	1.04363
$363$	−921.000	−0.133168
$364$	6156.00	0.886434
$365$	−4902.00	−0.702965
$366$	−930.000	−0.132819
$367$	2105.00	0.299401	0.149700	−	0.988731i	$-0.452169\pi$
$367$	2105.00	0.299401	0.149700	+	0.988731i	$0.452169\pi$
$368$	1568.00	0.222113
$369$	1584.00	0.223468
$370$	−3516.00	−0.494022
$371$	−9158.00	−1.28156
$372$	−2700.00	−0.376313
$373$	−2494.00	−0.346205	−0.173102	−	0.984904i	$-0.555379\pi$
$373$	−2494.00	−0.346205	−0.173102	+	0.984904i	$0.555379\pi$
$374$	7936.00	1.09722
$375$	3852.00	0.530444
$376$	4400.00	0.603491
$377$	−24300.0	−3.31966
$378$	−1026.00	−0.139608
$379$	5729.00	0.776462	0.388231	−	0.921562i	$-0.373086\pi$
$379$	5729.00	0.776462	0.388231	+	0.921562i	$0.373086\pi$
$380$	0	0
$381$	−4308.00	−0.579280
$382$	−628.000	−0.0841133
$383$	−10016.0	−1.33628	−0.668138	−	0.744037i	$-0.732908\pi$
$383$	−10016.0	−1.33628	−0.668138	+	0.744037i	$0.732908\pi$
$384$	−384.000	−0.0510310
$385$	−3648.00	−0.482907
$386$	1850.00	0.243944
$387$	−999.000	−0.131220
$388$	−4600.00	−0.601880
$389$	5818.00	0.758314	0.379157	−	0.925332i	$-0.376214\pi$
$389$	5818.00	0.758314	0.379157	+	0.925332i	$0.376214\pi$
$390$	2916.00	0.378609
$391$	−12152.0	−1.57175
$392$	−144.000	−0.0185538
$393$	−1134.00	−0.145554
$394$	−2488.00	−0.318131
$395$	−1554.00	−0.197950
$396$	1152.00	0.146187
$397$	−4149.00	−0.524515	−0.262257	−	0.964998i	$-0.584467\pi$
$397$	−4149.00	−0.524515	−0.262257	+	0.964998i	$0.584467\pi$
$398$	−6982.00	−0.879337
$399$	0	0
$400$	−1424.00	−0.178000
$401$	9072.00	1.12976	0.564880	−	0.825173i	$-0.308922\pi$
$401$	9072.00	1.12976	0.564880	+	0.825173i	$0.308922\pi$
$402$	−2790.00	−0.346151
$403$	−18225.0	−2.25273
$404$	−2312.00	−0.284719
$405$	−486.000	−0.0596285
$406$	11400.0	1.39353
$407$	−9376.00	−1.14189
$408$	2976.00	0.361113
$409$	8694.00	1.05108	0.525539	−	0.850770i	$-0.323864\pi$
$409$	8694.00	1.05108	0.525539	+	0.850770i	$0.323864\pi$
$410$	2112.00	0.254401
$411$	3186.00	0.382369
$412$	3276.00	0.391740
$413$	−9424.00	−1.12282
$414$	−1764.00	−0.209410
$415$	336.000	0.0397436
$416$	−2592.00	−0.305489
$417$	−5619.00	−0.659865
$418$	0	0
$419$	−4392.00	−0.512084	−0.256042	−	0.966666i	$-0.582419\pi$
$419$	−4392.00	−0.512084	−0.256042	+	0.966666i	$0.582419\pi$
$420$	−1368.00	−0.158932
$421$	6910.00	0.799935	0.399968	−	0.916529i	$-0.369021\pi$
$421$	6910.00	0.799935	0.399968	+	0.916529i	$0.369021\pi$
$422$	9154.00	1.05595
$423$	−4950.00	−0.568977
$424$	3856.00	0.441660
$425$	11036.0	1.25959
$426$	660.000	0.0750636
$427$	2945.00	0.333767
$428$	1048.00	0.118357
$429$	7776.00	0.875125
$430$	−1332.00	−0.149383
$431$	−8882.00	−0.992647	−0.496324	−	0.868138i	$-0.665317\pi$
$431$	−8882.00	−0.992647	−0.496324	+	0.868138i	$0.665317\pi$
$432$	432.000	0.0481125
$433$	3699.00	0.410537	0.205269	−	0.978706i	$-0.434193\pi$
$433$	3699.00	0.410537	0.205269	+	0.978706i	$0.434193\pi$
$434$	8550.00	0.945652
$435$	5400.00	0.595196
$436$	4136.00	0.454308
$437$	0	0
$438$	−4902.00	−0.534764
$439$	−16907.0	−1.83810	−0.919051	−	0.394138i	$-0.871043\pi$
$439$	−16907.0	−1.83810	−0.919051	+	0.394138i	$0.871043\pi$
$440$	1536.00	0.166423
$441$	162.000	0.0174927
$442$	20088.0	2.16174
$443$	−9972.00	−1.06949	−0.534745	−	0.845014i	$-0.679592\pi$
$443$	−9972.00	−1.06949	−0.534745	+	0.845014i	$0.679592\pi$
$444$	−3516.00	−0.375815
$445$	−1848.00	−0.196862
$446$	−4590.00	−0.487316
$447$	4116.00	0.435526
$448$	1216.00	0.128238
$449$	−10368.0	−1.08975	−0.544873	−	0.838518i	$-0.683422\pi$
$449$	−10368.0	−1.08975	−0.544873	+	0.838518i	$0.683422\pi$
$450$	1602.00	0.167820
$451$	5632.00	0.588028
$452$	−5296.00	−0.551113
$453$	−9888.00	−1.02556
$454$	−5036.00	−0.520597
$455$	−9234.00	−0.951421
$456$	0	0
$457$	4481.00	0.458670	0.229335	−	0.973348i	$-0.426345\pi$
$457$	4481.00	0.458670	0.229335	+	0.973348i	$0.426345\pi$
$458$	−10802.0	−1.10206
$459$	−3348.00	−0.340460
$460$	−2352.00	−0.238397
$461$	−2112.00	−0.213375	−0.106687	−	0.994293i	$-0.534024\pi$
$461$	−2112.00	−0.213375	−0.106687	+	0.994293i	$0.534024\pi$
$462$	−3648.00	−0.367360
$463$	5817.00	0.583885	0.291943	−	0.956436i	$-0.405698\pi$
$463$	5817.00	0.583885	0.291943	+	0.956436i	$0.405698\pi$
$464$	−4800.00	−0.480247
$465$	4050.00	0.403902
$466$	12156.0	1.20840
$467$	9100.00	0.901708	0.450854	−	0.892598i	$-0.351119\pi$
$467$	9100.00	0.901708	0.450854	+	0.892598i	$0.351119\pi$
$468$	2916.00	0.288017
$469$	8835.00	0.869856
$470$	−6600.00	−0.647735
$471$	5301.00	0.518593
$472$	3968.00	0.386953
$473$	−3552.00	−0.345288
$474$	−1554.00	−0.150586
$475$	0	0
$476$	−9424.00	−0.907454
$477$	−4338.00	−0.416401
$478$	−2124.00	−0.203242
$479$	−30.0000	−0.00286166	−0.00143083	−	0.999999i	$-0.500455\pi$
$479$	−30.0000	−0.00286166	−0.00143083	+	0.999999i	$0.500455\pi$
$480$	576.000	0.0547723
$481$	−23733.0	−2.24975
$482$	9238.00	0.872986
$483$	5586.00	0.526236
$484$	−1228.00	−0.115327
$485$	6900.00	0.646006
$486$	−486.000	−0.0453609
$487$	−10924.0	−1.01646	−0.508228	−	0.861223i	$-0.669699\pi$
$487$	−10924.0	−1.01646	−0.508228	+	0.861223i	$0.669699\pi$
$488$	−1240.00	−0.115025
$489$	5127.00	0.474133
$490$	216.000	0.0199141
$491$	7338.00	0.674459	0.337229	−	0.941422i	$-0.390510\pi$
$491$	7338.00	0.674459	0.337229	+	0.941422i	$0.390510\pi$
$492$	2112.00	0.193529
$493$	37200.0	3.39838
$494$	0	0
$495$	−1728.00	−0.156905
$496$	−3600.00	−0.325897
$497$	−2090.00	−0.188630
$498$	336.000	0.0302340
$499$	−8521.00	−0.764434	−0.382217	−	0.924073i	$-0.624839\pi$
$499$	−8521.00	−0.764434	−0.382217	+	0.924073i	$0.624839\pi$
$500$	5136.00	0.459378
$501$	−8280.00	−0.738369
$502$	−4356.00	−0.387286
$503$	−5500.00	−0.487541	−0.243770	−	0.969833i	$-0.578384\pi$
$503$	−5500.00	−0.487541	−0.243770	+	0.969833i	$0.578384\pi$
$504$	−1368.00	−0.120904
$505$	3468.00	0.305592
$506$	−6272.00	−0.551036
$507$	13092.0	1.14682
$508$	−5744.00	−0.501671
$509$	5938.00	0.517087	0.258543	−	0.966000i	$-0.416757\pi$
$509$	5938.00	0.517087	0.258543	+	0.966000i	$0.416757\pi$
$510$	−4464.00	−0.387587
$511$	15523.0	1.34383
$512$	−512.000	−0.0441942
$513$	0	0
$514$	1260.00	0.108125
$515$	−4914.00	−0.420460
$516$	−1332.00	−0.113640
$517$	−17600.0	−1.49719
$518$	11134.0	0.944401
$519$	−4428.00	−0.374504
$520$	3888.00	0.327885
$521$	−2884.00	−0.242515	−0.121258	−	0.992621i	$-0.538693\pi$
$521$	−2884.00	−0.242515	−0.121258	+	0.992621i	$0.538693\pi$
$522$	5400.00	0.452781
$523$	−10661.0	−0.891344	−0.445672	−	0.895196i	$-0.647035\pi$
$523$	−10661.0	−0.891344	−0.445672	+	0.895196i	$0.647035\pi$
$524$	−1512.00	−0.126053
$525$	−5073.00	−0.421722
$526$	−5484.00	−0.454589
$527$	27900.0	2.30615
$528$	1536.00	0.126602
$529$	−2563.00	−0.210652
$530$	−5784.00	−0.474039
$531$	−4464.00	−0.364823
$532$	0	0
$533$	14256.0	1.15853
$534$	−1848.00	−0.149758
$535$	−1572.00	−0.127035
$536$	−3720.00	−0.299775
$537$	78.0000	0.00626806
$538$	−4184.00	−0.335288
$539$	576.000	0.0460298
$540$	−648.000	−0.0516398
$541$	5885.00	0.467682	0.233841	−	0.972275i	$-0.424870\pi$
$541$	5885.00	0.467682	0.233841	+	0.972275i	$0.424870\pi$
$542$	−10864.0	−0.860976
$543$	−10782.0	−0.852118
$544$	3968.00	0.312733
$545$	−6204.00	−0.487615
$546$	−9234.00	−0.723771
$547$	1637.00	0.127958	0.0639790	−	0.997951i	$-0.479621\pi$
$547$	1637.00	0.127958	0.0639790	+	0.997951i	$0.479621\pi$
$548$	4248.00	0.331142
$549$	1395.00	0.108447
$550$	5696.00	0.441597
$551$	0	0
$552$	−2352.00	−0.181355
$553$	4921.00	0.378413
$554$	5668.00	0.434676
$555$	5274.00	0.403367
$556$	−7492.00	−0.571460
$557$	6470.00	0.492177	0.246089	−	0.969247i	$-0.420855\pi$
$557$	6470.00	0.492177	0.246089	+	0.969247i	$0.420855\pi$
$558$	4050.00	0.307258
$559$	−8991.00	−0.680284
$560$	−1824.00	−0.137639
$561$	−11904.0	−0.895877
$562$	−11380.0	−0.854157
$563$	14898.0	1.11523	0.557616	−	0.830099i	$-0.311716\pi$
$563$	14898.0	1.11523	0.557616	+	0.830099i	$0.311716\pi$
$564$	−6600.00	−0.492748
$565$	7944.00	0.591516
$566$	8776.00	0.651736
$567$	1539.00	0.113989
$568$	880.000	0.0650070
$569$	−13260.0	−0.976956	−0.488478	−	0.872576i	$-0.662448\pi$
$569$	−13260.0	−0.976956	−0.488478	+	0.872576i	$0.662448\pi$
$570$	0	0
$571$	−6097.00	−0.446850	−0.223425	−	0.974721i	$-0.571724\pi$
$571$	−6097.00	−0.446850	−0.223425	+	0.974721i	$0.571724\pi$
$572$	10368.0	0.757881
$573$	942.000	0.0686782
$574$	−6688.00	−0.486327
$575$	−8722.00	−0.632578
$576$	576.000	0.0416667
$577$	−5594.00	−0.403607	−0.201804	−	0.979426i	$-0.564680\pi$
$577$	−5594.00	−0.403607	−0.201804	+	0.979426i	$0.564680\pi$
$578$	−20926.0	−1.50589
$579$	−2775.00	−0.199180
$580$	7200.00	0.515455
$581$	−1064.00	−0.0759762
$582$	6900.00	0.491433
$583$	−15424.0	−1.09571
$584$	−6536.00	−0.463119
$585$	−4374.00	−0.309133
$586$	3316.00	0.233759
$587$	−18076.0	−1.27100	−0.635499	−	0.772101i	$-0.719206\pi$
$587$	−18076.0	−1.27100	−0.635499	+	0.772101i	$0.719206\pi$
$588$	216.000	0.0151491
$589$	0	0
$590$	−5952.00	−0.415322
$591$	3732.00	0.259753
$592$	−4688.00	−0.325466
$593$	22928.0	1.58776	0.793879	−	0.608076i	$-0.208058\pi$
$593$	22928.0	1.58776	0.793879	+	0.608076i	$0.208058\pi$
$594$	−1728.00	−0.119361
$595$	14136.0	0.973982
$596$	5488.00	0.377177
$597$	10473.0	0.717975
$598$	−15876.0	−1.08565
$599$	−11792.0	−0.804354	−0.402177	−	0.915562i	$-0.631746\pi$
$599$	−11792.0	−0.804354	−0.402177	+	0.915562i	$0.631746\pi$
$600$	2136.00	0.145336
$601$	−15585.0	−1.05778	−0.528890	−	0.848691i	$-0.677391\pi$
$601$	−15585.0	−1.05778	−0.528890	+	0.848691i	$0.677391\pi$
$602$	4218.00	0.285570
$603$	4185.00	0.282631
$604$	−13184.0	−0.888161
$605$	1842.00	0.123782
$606$	3468.00	0.232472
$607$	−27181.0	−1.81753	−0.908767	−	0.417305i	$-0.862975\pi$
$607$	−27181.0	−1.81753	−0.908767	+	0.417305i	$0.862975\pi$
$608$	0	0
$609$	−17100.0	−1.13781
$610$	1860.00	0.123458
$611$	−44550.0	−2.94975
$612$	−4464.00	−0.294847
$613$	13826.0	0.910974	0.455487	−	0.890243i	$-0.349465\pi$
$613$	13826.0	0.910974	0.455487	+	0.890243i	$0.349465\pi$
$614$	−1608.00	−0.105690
$615$	−3168.00	−0.207717
$616$	−4864.00	−0.318143
$617$	−12444.0	−0.811956	−0.405978	−	0.913883i	$-0.633069\pi$
$617$	−12444.0	−0.811956	−0.405978	+	0.913883i	$0.633069\pi$
$618$	−4914.00	−0.319854
$619$	−137.000	−0.00889579	−0.00444790	−	0.999990i	$-0.501416\pi$
$619$	−137.000	−0.00889579	−0.00444790	+	0.999990i	$0.501416\pi$
$620$	5400.00	0.349789
$621$	2646.00	0.170983
$622$	6344.00	0.408957
$623$	5852.00	0.376333
$624$	3888.00	0.249430
$625$	3421.00	0.218944
$626$	−7668.00	−0.489577
$627$	0	0
$628$	7068.00	0.449114
$629$	36332.0	2.30310
$630$	2052.00	0.129768
$631$	−10261.0	−0.647360	−0.323680	−	0.946167i	$-0.604920\pi$
$631$	−10261.0	−0.647360	−0.323680	+	0.946167i	$0.604920\pi$
$632$	−2072.00	−0.130411
$633$	−13731.0	−0.862177
$634$	−5844.00	−0.366080
$635$	8616.00	0.538450
$636$	−5784.00	−0.360614
$637$	1458.00	0.0906877
$638$	19200.0	1.19143
$639$	−990.000	−0.0612892
$640$	768.000	0.0474342
$641$	−9250.00	−0.569974	−0.284987	−	0.958531i	$-0.591989\pi$
$641$	−9250.00	−0.569974	−0.284987	+	0.958531i	$0.591989\pi$
$642$	−1572.00	−0.0966385
$643$	−16529.0	−1.01375	−0.506874	−	0.862020i	$-0.669199\pi$
$643$	−16529.0	−1.01375	−0.506874	+	0.862020i	$0.669199\pi$
$644$	7448.00	0.455733
$645$	1998.00	0.121971
$646$	0	0
$647$	13884.0	0.843642	0.421821	−	0.906679i	$-0.361391\pi$
$647$	13884.0	0.843642	0.421821	+	0.906679i	$0.361391\pi$
$648$	−648.000	−0.0392837
$649$	−15872.0	−0.959985
$650$	14418.0	0.870032
$651$	−12825.0	−0.772122
$652$	6836.00	0.410611
$653$	−20538.0	−1.23080	−0.615401	−	0.788214i	$-0.711006\pi$
$653$	−20538.0	−1.23080	−0.615401	+	0.788214i	$0.711006\pi$
$654$	−6204.00	−0.370941
$655$	2268.00	0.135295
$656$	2816.00	0.167601
$657$	7353.00	0.436633
$658$	20900.0	1.23825
$659$	15248.0	0.901332	0.450666	−	0.892693i	$-0.351187\pi$
$659$	15248.0	0.901332	0.450666	+	0.892693i	$0.351187\pi$
$660$	−2304.00	−0.135883
$661$	24826.0	1.46085	0.730423	−	0.682995i	$-0.239323\pi$
$661$	24826.0	1.46085	0.730423	+	0.682995i	$0.239323\pi$
$662$	8718.00	0.511835
$663$	−30132.0	−1.76505
$664$	448.000	0.0261834
$665$	0	0
$666$	5274.00	0.306852
$667$	−29400.0	−1.70671
$668$	−11040.0	−0.639447
$669$	6885.00	0.397892
$670$	5580.00	0.321753
$671$	4960.00	0.285363
$672$	−1824.00	−0.104706
$673$	10907.0	0.624716	0.312358	−	0.949964i	$-0.398881\pi$
$673$	10907.0	0.624716	0.312358	+	0.949964i	$0.398881\pi$
$674$	−17458.0	−0.997711
$675$	−2403.00	−0.137024
$676$	17456.0	0.993173
$677$	7386.00	0.419301	0.209651	−	0.977776i	$-0.432767\pi$
$677$	7386.00	0.419301	0.209651	+	0.977776i	$0.432767\pi$
$678$	7944.00	0.449982
$679$	−21850.0	−1.23494
$680$	−5952.00	−0.335660
$681$	7554.00	0.425066
$682$	14400.0	0.808511
$683$	16542.0	0.926738	0.463369	−	0.886165i	$-0.346640\pi$
$683$	16542.0	0.926738	0.463369	+	0.886165i	$0.346640\pi$
$684$	0	0
$685$	−6372.00	−0.355418
$686$	12350.0	0.687355
$687$	16203.0	0.899830
$688$	−1776.00	−0.0984148
$689$	−39042.0	−2.15875
$690$	3528.00	0.194650
$691$	1972.00	0.108565	0.0542825	−	0.998526i	$-0.482713\pi$
$691$	1972.00	0.108565	0.0542825	+	0.998526i	$0.482713\pi$
$692$	−5904.00	−0.324330
$693$	5472.00	0.299948
$694$	−76.0000	−0.00415695
$695$	11238.0	0.613355
$696$	7200.00	0.392120
$697$	−21824.0	−1.18600
$698$	−12206.0	−0.661897
$699$	−18234.0	−0.986657
$700$	−6764.00	−0.365222
$701$	−4414.00	−0.237824	−0.118912	−	0.992905i	$-0.537941\pi$
$701$	−4414.00	−0.237824	−0.118912	+	0.992905i	$0.537941\pi$
$702$	−4374.00	−0.235165
$703$	0	0
$704$	2048.00	0.109640
$705$	9900.00	0.528873
$706$	24396.0	1.30050
$707$	−10982.0	−0.584188
$708$	−5952.00	−0.315946
$709$	30689.0	1.62560	0.812799	−	0.582544i	$-0.197942\pi$
$709$	30689.0	1.62560	0.812799	+	0.582544i	$0.197942\pi$
$710$	−1320.00	−0.0697728
$711$	2331.00	0.122953
$712$	−2464.00	−0.129694
$713$	−22050.0	−1.15818
$714$	14136.0	0.740933
$715$	−15552.0	−0.813443
$716$	104.000	0.00542830
$717$	3186.00	0.165946
$718$	−5976.00	−0.310616
$719$	−21730.0	−1.12711	−0.563555	−	0.826078i	$-0.690567\pi$
$719$	−21730.0	−1.12711	−0.563555	+	0.826078i	$0.690567\pi$
$720$	−864.000	−0.0447214
$721$	15561.0	0.803775
$722$	0	0
$723$	−13857.0	−0.712790
$724$	−14376.0	−0.737956
$725$	26700.0	1.36774
$726$	1842.00	0.0941640
$727$	2723.00	0.138914	0.0694570	−	0.997585i	$-0.477873\pi$
$727$	2723.00	0.138914	0.0694570	+	0.997585i	$0.477873\pi$
$728$	−12312.0	−0.626804
$729$	729.000	0.0370370
$730$	9804.00	0.497072
$731$	13764.0	0.696416
$732$	1860.00	0.0939175
$733$	−26186.0	−1.31951	−0.659756	−	0.751480i	$-0.729340\pi$
$733$	−26186.0	−1.31951	−0.659756	+	0.751480i	$0.729340\pi$
$734$	−4210.00	−0.211708
$735$	−324.000	−0.0162598
$736$	−3136.00	−0.157058
$737$	14880.0	0.743707
$738$	−3168.00	−0.158016
$739$	−9791.00	−0.487372	−0.243686	−	0.969854i	$-0.578357\pi$
$739$	−9791.00	−0.487372	−0.243686	+	0.969854i	$0.578357\pi$
$740$	7032.00	0.349326
$741$	0	0
$742$	18316.0	0.906202
$743$	5532.00	0.273149	0.136574	−	0.990630i	$-0.456391\pi$
$743$	5532.00	0.273149	0.136574	+	0.990630i	$0.456391\pi$
$744$	5400.00	0.266094
$745$	−8232.00	−0.404828
$746$	4988.00	0.244804
$747$	−504.000	−0.0246859
$748$	−15872.0	−0.775853
$749$	4978.00	0.242847
$750$	−7704.00	−0.375080
$751$	27931.0	1.35715	0.678573	−	0.734533i	$-0.262599\pi$
$751$	27931.0	1.35715	0.678573	+	0.734533i	$0.262599\pi$
$752$	−8800.00	−0.426733
$753$	6534.00	0.316218
$754$	48600.0	2.34736
$755$	19776.0	0.953275
$756$	2052.00	0.0987176
$757$	25661.0	1.23205	0.616027	−	0.787725i	$-0.288741\pi$
$757$	25661.0	1.23205	0.616027	+	0.787725i	$0.288741\pi$
$758$	−11458.0	−0.549041
$759$	9408.00	0.449919
$760$	0	0
$761$	−16868.0	−0.803501	−0.401751	−	0.915749i	$-0.631598\pi$
$761$	−16868.0	−0.803501	−0.401751	+	0.915749i	$0.631598\pi$
$762$	8616.00	0.409613
$763$	19646.0	0.932153
$764$	1256.00	0.0594771
$765$	6696.00	0.316463
$766$	20032.0	0.944890
$767$	−40176.0	−1.89136
$768$	768.000	0.0360844
$769$	5691.00	0.266870	0.133435	−	0.991058i	$-0.457399\pi$
$769$	5691.00	0.266870	0.133435	+	0.991058i	$0.457399\pi$
$770$	7296.00	0.341467
$771$	−1890.00	−0.0882836
$772$	−3700.00	−0.172495
$773$	−18918.0	−0.880250	−0.440125	−	0.897937i	$-0.645066\pi$
$773$	−18918.0	−0.880250	−0.440125	+	0.897937i	$0.645066\pi$
$774$	1998.00	0.0927863
$775$	20025.0	0.928154
$776$	9200.00	0.425594
$777$	−16701.0	−0.771100
$778$	−11636.0	−0.536209
$779$	0	0
$780$	−5832.00	−0.267717
$781$	−3520.00	−0.161275
$782$	24304.0	1.11139
$783$	−8100.00	−0.369694
$784$	288.000	0.0131195
$785$	−10602.0	−0.482040
$786$	2268.00	0.102922
$787$	29191.0	1.32217	0.661084	−	0.750312i	$-0.270097\pi$
$787$	29191.0	1.32217	0.661084	+	0.750312i	$0.270097\pi$
$788$	4976.00	0.224953
$789$	8226.00	0.371170
$790$	3108.00	0.139972
$791$	−25156.0	−1.13078
$792$	−2304.00	−0.103370
$793$	12555.0	0.562221
$794$	8298.00	0.370888
$795$	8676.00	0.387052
$796$	13964.0	0.621785
$797$	22594.0	1.00417	0.502083	−	0.864819i	$-0.332567\pi$
$797$	22594.0	1.00417	0.502083	+	0.864819i	$0.332567\pi$
$798$	0	0
$799$	68200.0	3.01970
$800$	2848.00	0.125865
$801$	2772.00	0.122277
$802$	−18144.0	−0.798861
$803$	26144.0	1.14894
$804$	5580.00	0.244765
$805$	−11172.0	−0.489144
$806$	36450.0	1.59292
$807$	6276.00	0.273762
$808$	4624.00	0.201326
$809$	522.000	0.0226855	0.0113427	−	0.999936i	$-0.496389\pi$
$809$	522.000	0.0226855	0.0113427	+	0.999936i	$0.496389\pi$
$810$	972.000	0.0421637
$811$	−12080.0	−0.523041	−0.261520	−	0.965198i	$-0.584224\pi$
$811$	−12080.0	−0.523041	−0.261520	+	0.965198i	$0.584224\pi$
$812$	−22800.0	−0.985373
$813$	16296.0	0.702984
$814$	18752.0	0.807441
$815$	−10254.0	−0.440714
$816$	−5952.00	−0.255345
$817$	0	0
$818$	−17388.0	−0.743224
$819$	13851.0	0.590956
$820$	−4224.00	−0.179888
$821$	−36468.0	−1.55023	−0.775117	−	0.631818i	$-0.782309\pi$
$821$	−36468.0	−1.55023	−0.775117	+	0.631818i	$0.782309\pi$
$822$	−6372.00	−0.270376
$823$	18688.0	0.791522	0.395761	−	0.918354i	$-0.370481\pi$
$823$	18688.0	0.791522	0.395761	+	0.918354i	$0.370481\pi$
$824$	−6552.00	−0.277002
$825$	−8544.00	−0.360562
$826$	18848.0	0.793954
$827$	5976.00	0.251277	0.125638	−	0.992076i	$-0.459902\pi$
$827$	5976.00	0.251277	0.125638	+	0.992076i	$0.459902\pi$
$828$	3528.00	0.148075
$829$	−11167.0	−0.467848	−0.233924	−	0.972255i	$-0.575157\pi$
$829$	−11167.0	−0.467848	−0.233924	+	0.972255i	$0.575157\pi$
$830$	−672.000	−0.0281030
$831$	−8502.00	−0.354911
$832$	5184.00	0.216013
$833$	−2232.00	−0.0928382
$834$	11238.0	0.466595
$835$	16560.0	0.686326
$836$	0	0
$837$	−6075.00	−0.250875
$838$	8784.00	0.362098
$839$	20070.0	0.825856	0.412928	−	0.910764i	$-0.364506\pi$
$839$	20070.0	0.825856	0.412928	+	0.910764i	$0.364506\pi$
$840$	2736.00	0.112382
$841$	65611.0	2.69019
$842$	−13820.0	−0.565640
$843$	17070.0	0.697416
$844$	−18308.0	−0.746667
$845$	−26184.0	−1.06598
$846$	9900.00	0.402327
$847$	−5833.00	−0.236628
$848$	−7712.00	−0.312301
$849$	−13164.0	−0.532141
$850$	−22072.0	−0.890663
$851$	−28714.0	−1.15664
$852$	−1320.00	−0.0530780
$853$	10827.0	0.434595	0.217297	−	0.976105i	$-0.430276\pi$
$853$	10827.0	0.434595	0.217297	+	0.976105i	$0.430276\pi$
$854$	−5890.00	−0.236009
$855$	0	0
$856$	−2096.00	−0.0836914
$857$	−22614.0	−0.901376	−0.450688	−	0.892681i	$-0.648821\pi$
$857$	−22614.0	−0.901376	−0.450688	+	0.892681i	$0.648821\pi$
$858$	−15552.0	−0.618807
$859$	−26339.0	−1.04619	−0.523094	−	0.852275i	$-0.675222\pi$
$859$	−26339.0	−1.04619	−0.523094	+	0.852275i	$0.675222\pi$
$860$	2664.00	0.105630
$861$	10032.0	0.397084
$862$	17764.0	0.701907
$863$	20458.0	0.806951	0.403475	−	0.914991i	$-0.367802\pi$
$863$	20458.0	0.806951	0.403475	+	0.914991i	$0.367802\pi$
$864$	−864.000	−0.0340207
$865$	8856.00	0.348108
$866$	−7398.00	−0.290294
$867$	31389.0	1.22956
$868$	−17100.0	−0.668677
$869$	8288.00	0.323534
$870$	−10800.0	−0.420867
$871$	37665.0	1.46525
$872$	−8272.00	−0.321245
$873$	−10350.0	−0.401254
$874$	0	0
$875$	24396.0	0.942555
$876$	9804.00	0.378135
$877$	−16321.0	−0.628416	−0.314208	−	0.949354i	$-0.601739\pi$
$877$	−16321.0	−0.628416	−0.314208	+	0.949354i	$0.601739\pi$
$878$	33814.0	1.29973
$879$	−4974.00	−0.190863
$880$	−3072.00	−0.117679
$881$	−15990.0	−0.611483	−0.305742	−	0.952115i	$-0.598904\pi$
$881$	−15990.0	−0.611483	−0.305742	+	0.952115i	$0.598904\pi$
$882$	−324.000	−0.0123692
$883$	12825.0	0.488783	0.244392	−	0.969677i	$-0.421412\pi$
$883$	12825.0	0.488783	0.244392	+	0.969677i	$0.421412\pi$
$884$	−40176.0	−1.52858
$885$	8928.00	0.339109
$886$	19944.0	0.756244
$887$	22476.0	0.850812	0.425406	−	0.905003i	$-0.360131\pi$
$887$	22476.0	0.850812	0.425406	+	0.905003i	$0.360131\pi$
$888$	7032.00	0.265742
$889$	−27284.0	−1.02933
$890$	3696.00	0.139202
$891$	2592.00	0.0974582
$892$	9180.00	0.344584
$893$	0	0
$894$	−8232.00	−0.307963
$895$	−156.000	−0.00582626
$896$	−2432.00	−0.0906779
$897$	23814.0	0.886428
$898$	20736.0	0.770567
$899$	67500.0	2.50417
$900$	−3204.00	−0.118667
$901$	59768.0	2.20995
$902$	−11264.0	−0.415798
$903$	−6327.00	−0.233167
$904$	10592.0	0.389695
$905$	21564.0	0.792057
$906$	19776.0	0.725181
$907$	−2204.00	−0.0806865	−0.0403432	−	0.999186i	$-0.512845\pi$
$907$	−2204.00	−0.0806865	−0.0403432	+	0.999186i	$0.512845\pi$
$908$	10072.0	0.368118
$909$	−5202.00	−0.189812
$910$	18468.0	0.672756
$911$	−32472.0	−1.18095	−0.590475	−	0.807056i	$-0.701060\pi$
$911$	−32472.0	−1.18095	−0.590475	+	0.807056i	$0.701060\pi$
$912$	0	0
$913$	−1792.00	−0.0649579
$914$	−8962.00	−0.324329
$915$	−2790.00	−0.100803
$916$	21604.0	0.779275
$917$	−7182.00	−0.258637
$918$	6696.00	0.240742
$919$	−6095.00	−0.218776	−0.109388	−	0.993999i	$-0.534889\pi$
$919$	−6095.00	−0.218776	−0.109388	+	0.993999i	$0.534889\pi$
$920$	4704.00	0.168572
$921$	2412.00	0.0862954
$922$	4224.00	0.150879
$923$	−8910.00	−0.317742
$924$	7296.00	0.259763
$925$	26077.0	0.926926
$926$	−11634.0	−0.412869
$927$	7371.00	0.261160
$928$	9600.00	0.339586
$929$	43986.0	1.55343	0.776714	−	0.629854i	$-0.216885\pi$
$929$	43986.0	1.55343	0.776714	+	0.629854i	$0.216885\pi$
$930$	−8100.00	−0.285602
$931$	0	0
$932$	−24312.0	−0.854470
$933$	−9516.00	−0.333912
$934$	−18200.0	−0.637604
$935$	23808.0	0.832732
$936$	−5832.00	−0.203659
$937$	22151.0	0.772296	0.386148	−	0.922437i	$-0.373805\pi$
$937$	22151.0	0.772296	0.386148	+	0.922437i	$0.373805\pi$
$938$	−17670.0	−0.615081
$939$	11502.0	0.399738
$940$	13200.0	0.458018
$941$	13400.0	0.464216	0.232108	−	0.972690i	$-0.425438\pi$
$941$	13400.0	0.464216	0.232108	+	0.972690i	$0.425438\pi$
$942$	−10602.0	−0.366700
$943$	17248.0	0.595623
$944$	−7936.00	−0.273617
$945$	−3078.00	−0.105955
$946$	7104.00	0.244155
$947$	−54394.0	−1.86649	−0.933246	−	0.359239i	$-0.883036\pi$
$947$	−54394.0	−1.86649	−0.933246	+	0.359239i	$0.883036\pi$
$948$	3108.00	0.106480
$949$	66177.0	2.26364
$950$	0	0
$951$	8766.00	0.298903
$952$	18848.0	0.641667
$953$	13656.0	0.464178	0.232089	−	0.972695i	$-0.425444\pi$
$953$	13656.0	0.464178	0.232089	+	0.972695i	$0.425444\pi$
$954$	8676.00	0.294440
$955$	−1884.00	−0.0638375
$956$	4248.00	0.143714
$957$	−28800.0	−0.972802
$958$	60.0000	0.00202350
$959$	20178.0	0.679439
$960$	−1152.00	−0.0387298
$961$	20834.0	0.699339
$962$	47466.0	1.59082
$963$	2358.00	0.0789050
$964$	−18476.0	−0.617294
$965$	5550.00	0.185141
$966$	−11172.0	−0.372105
$967$	24681.0	0.820773	0.410386	−	0.911912i	$-0.365394\pi$
$967$	24681.0	0.820773	0.410386	+	0.911912i	$0.365394\pi$
$968$	2456.00	0.0815484
$969$	0	0
$970$	−13800.0	−0.456795
$971$	46422.0	1.53425	0.767123	−	0.641500i	$-0.221688\pi$
$971$	46422.0	1.53425	0.767123	+	0.641500i	$0.221688\pi$
$972$	972.000	0.0320750
$973$	−35587.0	−1.17253
$974$	21848.0	0.718742
$975$	−21627.0	−0.710378
$976$	2480.00	0.0813349
$977$	10266.0	0.336170	0.168085	−	0.985772i	$-0.446242\pi$
$977$	10266.0	0.336170	0.168085	+	0.985772i	$0.446242\pi$
$978$	−10254.0	−0.335263
$979$	9856.00	0.321756
$980$	−432.000	−0.0140814
$981$	9306.00	0.302872
$982$	−14676.0	−0.476914
$983$	−28212.0	−0.915385	−0.457692	−	0.889111i	$-0.651324\pi$
$983$	−28212.0	−0.915385	−0.457692	+	0.889111i	$0.651324\pi$
$984$	−4224.00	−0.136846
$985$	−7464.00	−0.241444
$986$	−74400.0	−2.40302
$987$	−31350.0	−1.01102
$988$	0	0
$989$	−10878.0	−0.349747
$990$	3456.00	0.110948
$991$	40897.0	1.31093	0.655467	−	0.755224i	$-0.272472\pi$
$991$	40897.0	1.31093	0.655467	+	0.755224i	$0.272472\pi$
$992$	7200.00	0.230444
$993$	−13077.0	−0.417911
$994$	4180.00	0.133382
$995$	−20946.0	−0.667370
$996$	−672.000	−0.0213786
$997$	−14821.0	−0.470798	−0.235399	−	0.971899i	$-0.575640\pi$
$997$	−14821.0	−0.470798	−0.235399	+	0.971899i	$0.575640\pi$
$998$	17042.0	0.540536
$999$	−7911.00	−0.250544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.b.1.1		1
19.7	even	3		114.4.e.b.49.1	yes	2
19.11	even	3		114.4.e.b.7.1	✓	2
19.18	odd	2		2166.4.a.e.1.1		1
57.11	odd	6		342.4.g.a.235.1		2
57.26	odd	6		342.4.g.a.163.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.b.7.1	✓	2	19.11	even	3
114.4.e.b.49.1	yes	2	19.7	even	3
342.4.g.a.163.1		2	57.26	odd	6
342.4.g.a.235.1		2	57.11	odd	6
2166.4.a.b.1.1		1	1.1	even	1	trivial
2166.4.a.e.1.1		1	19.18	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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