LMFDB - Embedded newform 6042.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
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	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -5.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +7.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -5.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{42} -11.0000 q^{43} -5.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -5.00000 q^{51} +2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} -2.00000 q^{56} -1.00000 q^{57} -4.00000 q^{58} +3.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} +7.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -5.00000 q^{66} -8.00000 q^{67} -5.00000 q^{68} -1.00000 q^{69} -2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} +10.0000 q^{77} +2.00000 q^{78} -3.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{84} -5.00000 q^{85} -11.0000 q^{86} -4.00000 q^{87} -5.00000 q^{88} +7.00000 q^{89} +1.00000 q^{90} -4.00000 q^{91} -1.00000 q^{92} +7.00000 q^{93} +6.00000 q^{94} -1.00000 q^{95} +1.00000 q^{96} -7.00000 q^{97} -3.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−5.00000	−1.06600
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−5.00000	−0.857493
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−0.308607
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−5.00000	−0.753778
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	−5.00000	−0.700140
$52$	2.00000	0.277350
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	−5.00000	−0.674200
$56$	−2.00000	−0.267261
$57$	−1.00000	−0.132453
$58$	−4.00000	−0.525226
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	7.00000	0.889001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−5.00000	−0.615457
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−5.00000	−0.606339
$69$	−1.00000	−0.120386
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	10.0000	1.13961
$78$	2.00000	0.226455
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−2.00000	−0.218218
$85$	−5.00000	−0.542326
$86$	−11.0000	−1.18616
$87$	−4.00000	−0.428845
$88$	−5.00000	−0.533002
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	1.00000	0.105409
$91$	−4.00000	−0.419314
$92$	−1.00000	−0.104257
$93$	7.00000	0.725866
$94$	6.00000	0.618853
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−3.00000	−0.303046
$99$	−5.00000	−0.502519
$100$	−4.00000	−0.400000
$101$	13.0000	1.29355	0.646774	−	0.762682i	$-0.276118\pi$
$101$	13.0000	1.29355	0.646774	+	0.762682i	$0.276118\pi$
$102$	−5.00000	−0.495074
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	−2.00000	−0.195180
$106$	−1.00000	−0.0971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	−5.00000	−0.476731
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	−1.00000	−0.0936586
$115$	−1.00000	−0.0932505
$116$	−4.00000	−0.371391
$117$	2.00000	0.184900
$118$	3.00000	0.276172
$119$	10.0000	0.916698
$120$	1.00000	0.0912871
$121$	14.0000	1.27273
$122$	−6.00000	−0.543214
$123$	0	0
$124$	7.00000	0.628619
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	1.00000	0.0883883
$129$	−11.0000	−0.968496
$130$	2.00000	0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	−5.00000	−0.435194
$133$	2.00000	0.173422
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−1.00000	−0.0851257
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	−2.00000	−0.169031
$141$	6.00000	0.505291
$142$	8.00000	0.671345
$143$	−10.0000	−0.836242
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−8.00000	−0.662085
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−4.00000	−0.326599
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−1.00000	−0.0811107
$153$	−5.00000	−0.404226
$154$	10.0000	0.805823
$155$	7.00000	0.562254
$156$	2.00000	0.160128
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−3.00000	−0.238667
$159$	−1.00000	−0.0793052
$160$	1.00000	0.0790569
$161$	2.00000	0.157622
$162$	1.00000	0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	−12.0000	−0.931381
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	−5.00000	−0.383482
$171$	−1.00000	−0.0764719
$172$	−11.0000	−0.838742
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	−4.00000	−0.303239
$175$	8.00000	0.604743
$176$	−5.00000	−0.376889
$177$	3.00000	0.225494
$178$	7.00000	0.524672
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.00000	0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−4.00000	−0.296500
$183$	−6.00000	−0.443533
$184$	−1.00000	−0.0737210
$185$	−2.00000	−0.147043
$186$	7.00000	0.513265
$187$	25.0000	1.82818
$188$	6.00000	0.437595
$189$	−2.00000	−0.145479
$190$	−1.00000	−0.0725476
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−7.00000	−0.502571
$195$	2.00000	0.143223
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−5.00000	−0.355335
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	−4.00000	−0.282843
$201$	−8.00000	−0.564276
$202$	13.0000	0.914677
$203$	8.00000	0.561490
$204$	−5.00000	−0.350070
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	2.00000	0.138675
$209$	5.00000	0.345857
$210$	−2.00000	−0.138013
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−1.00000	−0.0686803
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	−11.0000	−0.750194
$216$	1.00000	0.0680414
$217$	−14.0000	−0.950382
$218$	1.00000	0.0677285
$219$	−8.00000	−0.540590
$220$	−5.00000	−0.337100
$221$	−10.0000	−0.672673
$222$	−2.00000	−0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−2.00000	−0.133631
$225$	−4.00000	−0.266667
$226$	−15.0000	−0.997785
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−1.00000	−0.0662266
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	−1.00000	−0.0659380
$231$	10.0000	0.657952
$232$	−4.00000	−0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	6.00000	0.391397
$236$	3.00000	0.195283
$237$	−3.00000	−0.194871
$238$	10.0000	0.648204
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	1.00000	0.0645497
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−2.00000	−0.127257
$248$	7.00000	0.444500
$249$	−12.0000	−0.760469
$250$	−9.00000	−0.569210
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	−2.00000	−0.125988
$253$	5.00000	0.314347
$254$	3.00000	0.188237
$255$	−5.00000	−0.313112
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−11.0000	−0.684830
$259$	4.00000	0.248548
$260$	2.00000	0.124035
$261$	−4.00000	−0.247594
$262$	20.0000	1.23560
$263$	1.00000	0.0616626	0.0308313	−	0.999525i	$-0.490185\pi$
$263$	1.00000	0.0616626	0.0308313	+	0.999525i	$0.490185\pi$
$264$	−5.00000	−0.307729
$265$	−1.00000	−0.0614295
$266$	2.00000	0.122628
$267$	7.00000	0.428393
$268$	−8.00000	−0.488678
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	1.00000	0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−5.00000	−0.303170
$273$	−4.00000	−0.242091
$274$	−12.0000	−0.724947
$275$	20.0000	1.20605
$276$	−1.00000	−0.0601929
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	2.00000	0.119952
$279$	7.00000	0.419079
$280$	−2.00000	−0.119523
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	6.00000	0.357295
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	8.00000	0.474713
$285$	−1.00000	−0.0592349
$286$	−10.0000	−0.591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	−4.00000	−0.234888
$291$	−7.00000	−0.410347
$292$	−8.00000	−0.468165
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−3.00000	−0.174964
$295$	3.00000	0.174667
$296$	−2.00000	−0.116248
$297$	−5.00000	−0.290129
$298$	6.00000	0.347571
$299$	−2.00000	−0.115663
$300$	−4.00000	−0.230940
$301$	22.0000	1.26806
$302$	−16.0000	−0.920697
$303$	13.0000	0.746830
$304$	−1.00000	−0.0573539
$305$	−6.00000	−0.343559
$306$	−5.00000	−0.285831
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	10.0000	0.569803
$309$	0	0
$310$	7.00000	0.397573
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	2.00000	0.113228
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−7.00000	−0.395033
$315$	−2.00000	−0.112687
$316$	−3.00000	−0.168763
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−1.00000	−0.0560772
$319$	20.0000	1.11979
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	2.00000	0.111456
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	−8.00000	−0.443760
$326$	−1.00000	−0.0553849
$327$	1.00000	0.0553001
$328$	0	0
$329$	−12.0000	−0.661581
$330$	−5.00000	−0.275241
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−12.0000	−0.658586
$333$	−2.00000	−0.109599
$334$	3.00000	0.164153
$335$	−8.00000	−0.437087
$336$	−2.00000	−0.109109
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−9.00000	−0.489535
$339$	−15.0000	−0.814688
$340$	−5.00000	−0.271163
$341$	−35.0000	−1.89536
$342$	−1.00000	−0.0540738
$343$	20.0000	1.07990
$344$	−11.0000	−0.593080
$345$	−1.00000	−0.0538382
$346$	7.00000	0.376322
$347$	−11.0000	−0.590511	−0.295255	−	0.955418i	$-0.595405\pi$
$347$	−11.0000	−0.590511	−0.295255	+	0.955418i	$0.595405\pi$
$348$	−4.00000	−0.214423
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	8.00000	0.427618
$351$	2.00000	0.106752
$352$	−5.00000	−0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	3.00000	0.159448
$355$	8.00000	0.424596
$356$	7.00000	0.370999
$357$	10.0000	0.529256
$358$	0	0
$359$	13.0000	0.686114	0.343057	−	0.939315i	$-0.388538\pi$
$359$	13.0000	0.686114	0.343057	+	0.939315i	$0.388538\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	14.0000	0.735824
$363$	14.0000	0.734809
$364$	−4.00000	−0.209657
$365$	−8.00000	−0.418739
$366$	−6.00000	−0.313625
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−2.00000	−0.103975
$371$	2.00000	0.103835
$372$	7.00000	0.362933
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	25.0000	1.29272
$375$	−9.00000	−0.464758
$376$	6.00000	0.309426
$377$	−8.00000	−0.412021
$378$	−2.00000	−0.102869
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−1.00000	−0.0512989
$381$	3.00000	0.153695
$382$	−3.00000	−0.153493
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	1.00000	0.0510310
$385$	10.0000	0.509647
$386$	−4.00000	−0.203595
$387$	−11.0000	−0.559161
$388$	−7.00000	−0.355371
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	2.00000	0.101274
$391$	5.00000	0.252861
$392$	−3.00000	−0.151523
$393$	20.0000	1.00887
$394$	18.0000	0.906827
$395$	−3.00000	−0.150946
$396$	−5.00000	−0.251259
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	26.0000	1.30326
$399$	2.00000	0.100125
$400$	−4.00000	−0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−8.00000	−0.399004
$403$	14.0000	0.697390
$404$	13.0000	0.646774
$405$	1.00000	0.0496904
$406$	8.00000	0.397033
$407$	10.0000	0.495682
$408$	−5.00000	−0.247537
$409$	−9.00000	−0.445021	−0.222511	−	0.974930i	$-0.571425\pi$
$409$	−9.00000	−0.445021	−0.222511	+	0.974930i	$0.571425\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−6.00000	−0.295241
$414$	−1.00000	−0.0491473
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	2.00000	0.0979404
$418$	5.00000	0.244558
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	−2.00000	−0.0975900
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	−5.00000	−0.243396
$423$	6.00000	0.291730
$424$	−1.00000	−0.0485643
$425$	20.0000	0.970143
$426$	8.00000	0.387601
$427$	12.0000	0.580721
$428$	−12.0000	−0.580042
$429$	−10.0000	−0.482805
$430$	−11.0000	−0.530467
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	1.00000	0.0481125
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−14.0000	−0.672022
$435$	−4.00000	−0.191785
$436$	1.00000	0.0478913
$437$	1.00000	0.0478365
$438$	−8.00000	−0.382255
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	−5.00000	−0.238366
$441$	−3.00000	−0.142857
$442$	−10.0000	−0.475651
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−2.00000	−0.0949158
$445$	7.00000	0.331832
$446$	4.00000	0.189405
$447$	6.00000	0.283790
$448$	−2.00000	−0.0944911
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	−15.0000	−0.705541
$453$	−16.0000	−0.751746
$454$	−20.0000	−0.938647
$455$	−4.00000	−0.187523
$456$	−1.00000	−0.0468293
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	16.0000	0.747631
$459$	−5.00000	−0.233380
$460$	−1.00000	−0.0466252
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	10.0000	0.465242
$463$	33.0000	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$463$	33.0000	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$464$	−4.00000	−0.185695
$465$	7.00000	0.324617
$466$	−6.00000	−0.277945
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	2.00000	0.0924500
$469$	16.0000	0.738811
$470$	6.00000	0.276759
$471$	−7.00000	−0.322543
$472$	3.00000	0.138086
$473$	55.0000	2.52890
$474$	−3.00000	−0.137795
$475$	4.00000	0.183533
$476$	10.0000	0.458349
$477$	−1.00000	−0.0457869
$478$	−9.00000	−0.411650
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	1.00000	0.0456435
$481$	−4.00000	−0.182384
$482$	−23.0000	−1.04762
$483$	2.00000	0.0910032
$484$	14.0000	0.636364
$485$	−7.00000	−0.317854
$486$	1.00000	0.0453609
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	−6.00000	−0.271607
$489$	−1.00000	−0.0452216
$490$	−3.00000	−0.135526
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	−2.00000	−0.0899843
$495$	−5.00000	−0.224733
$496$	7.00000	0.314309
$497$	−16.0000	−0.717698
$498$	−12.0000	−0.537733
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−9.00000	−0.402492
$501$	3.00000	0.134030
$502$	30.0000	1.33897
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000	−0.0890871
$505$	13.0000	0.578492
$506$	5.00000	0.222277
$507$	−9.00000	−0.399704
$508$	3.00000	0.133103
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−5.00000	−0.221404
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	6.00000	0.264649
$515$	0	0
$516$	−11.0000	−0.484248
$517$	−30.0000	−1.31940
$518$	4.00000	0.175750
$519$	7.00000	0.307266
$520$	2.00000	0.0877058
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	−4.00000	−0.175075
$523$	−3.00000	−0.131181	−0.0655904	−	0.997847i	$-0.520893\pi$
$523$	−3.00000	−0.131181	−0.0655904	+	0.997847i	$0.520893\pi$
$524$	20.0000	0.873704
$525$	8.00000	0.349149
$526$	1.00000	0.0436021
$527$	−35.0000	−1.52462
$528$	−5.00000	−0.217597
$529$	−22.0000	−0.956522
$530$	−1.00000	−0.0434372
$531$	3.00000	0.130189
$532$	2.00000	0.0867110
$533$	0	0
$534$	7.00000	0.302920
$535$	−12.0000	−0.518805
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−12.0000	−0.517357
$539$	15.0000	0.646096
$540$	1.00000	0.0430331
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−20.0000	−0.859074
$543$	14.0000	0.600798
$544$	−5.00000	−0.214373
$545$	1.00000	0.0428353
$546$	−4.00000	−0.171184
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	−12.0000	−0.512615
$549$	−6.00000	−0.256074
$550$	20.0000	0.852803
$551$	4.00000	0.170406
$552$	−1.00000	−0.0425628
$553$	6.00000	0.255146
$554$	−9.00000	−0.382373
$555$	−2.00000	−0.0848953
$556$	2.00000	0.0848189
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	7.00000	0.296334
$559$	−22.0000	−0.930501
$560$	−2.00000	−0.0845154
$561$	25.0000	1.05550
$562$	−9.00000	−0.379642
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	6.00000	0.252646
$565$	−15.0000	−0.631055
$566$	20.0000	0.840663
$567$	−2.00000	−0.0839921
$568$	8.00000	0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−1.00000	−0.0418854
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	−10.0000	−0.418121
$573$	−3.00000	−0.125327
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	45.0000	1.87337	0.936687	−	0.350167i	$-0.113875\pi$
$577$	45.0000	1.87337	0.936687	+	0.350167i	$0.113875\pi$
$578$	8.00000	0.332756
$579$	−4.00000	−0.166234
$580$	−4.00000	−0.166091
$581$	24.0000	0.995688
$582$	−7.00000	−0.290159
$583$	5.00000	0.207079
$584$	−8.00000	−0.331042
$585$	2.00000	0.0826898
$586$	0	0
$587$	−37.0000	−1.52715	−0.763577	−	0.645717i	$-0.776559\pi$
$587$	−37.0000	−1.52715	−0.763577	+	0.645717i	$0.776559\pi$
$588$	−3.00000	−0.123718
$589$	−7.00000	−0.288430
$590$	3.00000	0.123508
$591$	18.0000	0.740421
$592$	−2.00000	−0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−5.00000	−0.205152
$595$	10.0000	0.409960
$596$	6.00000	0.245770
$597$	26.0000	1.06411
$598$	−2.00000	−0.0817861
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−4.00000	−0.163299
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	22.0000	0.896653
$603$	−8.00000	−0.325785
$604$	−16.0000	−0.651031
$605$	14.0000	0.569181
$606$	13.0000	0.528089
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−1.00000	−0.0405554
$609$	8.00000	0.324176
$610$	−6.00000	−0.242933
$611$	12.0000	0.485468
$612$	−5.00000	−0.202113
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	25.0000	1.00892
$615$	0	0
$616$	10.0000	0.402911
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	7.00000	0.281127
$621$	−1.00000	−0.0401286
$622$	20.0000	0.801927
$623$	−14.0000	−0.560898
$624$	2.00000	0.0800641
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	5.00000	0.199681
$628$	−7.00000	−0.279330
$629$	10.0000	0.398726
$630$	−2.00000	−0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−3.00000	−0.119334
$633$	−5.00000	−0.198732
$634$	−10.0000	−0.397151
$635$	3.00000	0.119051
$636$	−1.00000	−0.0396526
$637$	−6.00000	−0.237729
$638$	20.0000	0.791808
$639$	8.00000	0.316475
$640$	1.00000	0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−12.0000	−0.473602
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	2.00000	0.0788110
$645$	−11.0000	−0.433125
$646$	5.00000	0.196722
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	1.00000	0.0392837
$649$	−15.0000	−0.588802
$650$	−8.00000	−0.313786
$651$	−14.0000	−0.548703
$652$	−1.00000	−0.0391630
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	1.00000	0.0391031
$655$	20.0000	0.781465
$656$	0	0
$657$	−8.00000	−0.312110
$658$	−12.0000	−0.467809
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	−5.00000	−0.194625
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	−8.00000	−0.310929
$663$	−10.0000	−0.388368
$664$	−12.0000	−0.465690
$665$	2.00000	0.0775567
$666$	−2.00000	−0.0774984
$667$	4.00000	0.154881
$668$	3.00000	0.116073
$669$	4.00000	0.154649
$670$	−8.00000	−0.309067
$671$	30.0000	1.15814
$672$	−2.00000	−0.0771517
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	26.0000	1.00148
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−15.0000	−0.576072
$679$	14.0000	0.537271
$680$	−5.00000	−0.191741
$681$	−20.0000	−0.766402
$682$	−35.0000	−1.34022
$683$	−5.00000	−0.191320	−0.0956598	−	0.995414i	$-0.530496\pi$
$683$	−5.00000	−0.191320	−0.0956598	+	0.995414i	$0.530496\pi$
$684$	−1.00000	−0.0382360
$685$	−12.0000	−0.458496
$686$	20.0000	0.763604
$687$	16.0000	0.610438
$688$	−11.0000	−0.419371
$689$	−2.00000	−0.0761939
$690$	−1.00000	−0.0380693
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	7.00000	0.266100
$693$	10.0000	0.379869
$694$	−11.0000	−0.417554
$695$	2.00000	0.0758643
$696$	−4.00000	−0.151620
$697$	0	0
$698$	1.00000	0.0378506
$699$	−6.00000	−0.226941
$700$	8.00000	0.302372
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	2.00000	0.0754851
$703$	2.00000	0.0754314
$704$	−5.00000	−0.188445
$705$	6.00000	0.225973
$706$	−18.0000	−0.677439
$707$	−26.0000	−0.977831
$708$	3.00000	0.112747
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	8.00000	0.300235
$711$	−3.00000	−0.112509
$712$	7.00000	0.262336
$713$	−7.00000	−0.262152
$714$	10.0000	0.374241
$715$	−10.0000	−0.373979
$716$	0	0
$717$	−9.00000	−0.336111
$718$	13.0000	0.485156
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	1.00000	0.0372161
$723$	−23.0000	−0.855379
$724$	14.0000	0.520306
$725$	16.0000	0.594225
$726$	14.0000	0.519589
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	55.0000	2.03425
$732$	−6.00000	−0.221766
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	14.0000	0.516749
$735$	−3.00000	−0.110657
$736$	−1.00000	−0.0368605
$737$	40.0000	1.47342
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	−2.00000	−0.0735215
$741$	−2.00000	−0.0734718
$742$	2.00000	0.0734223
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	7.00000	0.256632
$745$	6.00000	0.219823
$746$	1.00000	0.0366126
$747$	−12.0000	−0.439057
$748$	25.0000	0.914091
$749$	24.0000	0.876941
$750$	−9.00000	−0.328634
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	6.00000	0.218797
$753$	30.0000	1.09326
$754$	−8.00000	−0.291343
$755$	−16.0000	−0.582300
$756$	−2.00000	−0.0727393
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−4.00000	−0.145287
$759$	5.00000	0.181489
$760$	−1.00000	−0.0362738
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	3.00000	0.108679
$763$	−2.00000	−0.0724049
$764$	−3.00000	−0.108536
$765$	−5.00000	−0.180775
$766$	15.0000	0.541972
$767$	6.00000	0.216647
$768$	1.00000	0.0360844
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	10.0000	0.360375
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	−11.0000	−0.395387
$775$	−28.0000	−1.00579
$776$	−7.00000	−0.251285
$777$	4.00000	0.143499
$778$	15.0000	0.537776
$779$	0	0
$780$	2.00000	0.0716115
$781$	−40.0000	−1.43131
$782$	5.00000	0.178800
$783$	−4.00000	−0.142948
$784$	−3.00000	−0.107143
$785$	−7.00000	−0.249841
$786$	20.0000	0.713376
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	18.0000	0.641223
$789$	1.00000	0.0356009
$790$	−3.00000	−0.106735
$791$	30.0000	1.06668
$792$	−5.00000	−0.177667
$793$	−12.0000	−0.426132
$794$	−15.0000	−0.532330
$795$	−1.00000	−0.0354663
$796$	26.0000	0.921546
$797$	−7.00000	−0.247953	−0.123976	−	0.992285i	$-0.539565\pi$
$797$	−7.00000	−0.247953	−0.123976	+	0.992285i	$0.539565\pi$
$798$	2.00000	0.0707992
$799$	−30.0000	−1.06132
$800$	−4.00000	−0.141421
$801$	7.00000	0.247333
$802$	−2.00000	−0.0706225
$803$	40.0000	1.41157
$804$	−8.00000	−0.282138
$805$	2.00000	0.0704907
$806$	14.0000	0.493129
$807$	−12.0000	−0.422420
$808$	13.0000	0.457338
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	1.00000	0.0351364
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	8.00000	0.280745
$813$	−20.0000	−0.701431
$814$	10.0000	0.350500
$815$	−1.00000	−0.0350285
$816$	−5.00000	−0.175035
$817$	11.0000	0.384841
$818$	−9.00000	−0.314678
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	−12.0000	−0.418548
$823$	50.0000	1.74289	0.871445	−	0.490493i	$-0.163183\pi$
$823$	50.0000	1.74289	0.871445	+	0.490493i	$0.163183\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	−6.00000	−0.208767
$827$	−38.0000	−1.32139	−0.660695	−	0.750655i	$-0.729738\pi$
$827$	−38.0000	−1.32139	−0.660695	+	0.750655i	$0.729738\pi$
$828$	−1.00000	−0.0347524
$829$	23.0000	0.798823	0.399412	−	0.916772i	$-0.369214\pi$
$829$	23.0000	0.798823	0.399412	+	0.916772i	$0.369214\pi$
$830$	−12.0000	−0.416526
$831$	−9.00000	−0.312207
$832$	2.00000	0.0693375
$833$	15.0000	0.519719
$834$	2.00000	0.0692543
$835$	3.00000	0.103819
$836$	5.00000	0.172929
$837$	7.00000	0.241955
$838$	18.0000	0.621800
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	−2.00000	−0.0690066
$841$	−13.0000	−0.448276
$842$	1.00000	0.0344623
$843$	−9.00000	−0.309976
$844$	−5.00000	−0.172107
$845$	−9.00000	−0.309609
$846$	6.00000	0.206284
$847$	−28.0000	−0.962091
$848$	−1.00000	−0.0343401
$849$	20.0000	0.686398
$850$	20.0000	0.685994
$851$	2.00000	0.0685591
$852$	8.00000	0.274075
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	12.0000	0.410632
$855$	−1.00000	−0.0341993
$856$	−12.0000	−0.410152
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	−10.0000	−0.341394
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	−11.0000	−0.375097
$861$	0	0
$862$	−4.00000	−0.136241
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	1.00000	0.0340207
$865$	7.00000	0.238007
$866$	−22.0000	−0.747590
$867$	8.00000	0.271694
$868$	−14.0000	−0.475191
$869$	15.0000	0.508840
$870$	−4.00000	−0.135613
$871$	−16.0000	−0.542139
$872$	1.00000	0.0338643
$873$	−7.00000	−0.236914
$874$	1.00000	0.0338255
$875$	18.0000	0.608511
$876$	−8.00000	−0.270295
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	6.00000	0.202490
$879$	0	0
$880$	−5.00000	−0.168550
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	−3.00000	−0.101015
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	−10.0000	−0.336336
$885$	3.00000	0.100844
$886$	0	0
$887$	45.0000	1.51095	0.755476	−	0.655176i	$-0.227406\pi$
$887$	45.0000	1.51095	0.755476	+	0.655176i	$0.227406\pi$
$888$	−2.00000	−0.0671156
$889$	−6.00000	−0.201234
$890$	7.00000	0.234641
$891$	−5.00000	−0.167506
$892$	4.00000	0.133930
$893$	−6.00000	−0.200782
$894$	6.00000	0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	−2.00000	−0.0667781
$898$	30.0000	1.00111
$899$	−28.0000	−0.933852
$900$	−4.00000	−0.133333
$901$	5.00000	0.166574
$902$	0	0
$903$	22.0000	0.732114
$904$	−15.0000	−0.498893
$905$	14.0000	0.465376
$906$	−16.0000	−0.531564
$907$	−51.0000	−1.69343	−0.846714	−	0.532049i	$-0.821422\pi$
$907$	−51.0000	−1.69343	−0.846714	+	0.532049i	$0.821422\pi$
$908$	−20.0000	−0.663723
$909$	13.0000	0.431183
$910$	−4.00000	−0.132599
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	−1.00000	−0.0331133
$913$	60.0000	1.98571
$914$	−12.0000	−0.396925
$915$	−6.00000	−0.198354
$916$	16.0000	0.528655
$917$	−40.0000	−1.32092
$918$	−5.00000	−0.165025
$919$	37.0000	1.22052	0.610259	−	0.792202i	$-0.291065\pi$
$919$	37.0000	1.22052	0.610259	+	0.792202i	$0.291065\pi$
$920$	−1.00000	−0.0329690
$921$	25.0000	0.823778
$922$	−28.0000	−0.922131
$923$	16.0000	0.526646
$924$	10.0000	0.328976
$925$	8.00000	0.263038
$926$	33.0000	1.08445
$927$	0	0
$928$	−4.00000	−0.131306
$929$	37.0000	1.21393	0.606965	−	0.794728i	$-0.292387\pi$
$929$	37.0000	1.21393	0.606965	+	0.794728i	$0.292387\pi$
$930$	7.00000	0.229539
$931$	3.00000	0.0983210
$932$	−6.00000	−0.196537
$933$	20.0000	0.654771
$934$	16.0000	0.523536
$935$	25.0000	0.817587
$936$	2.00000	0.0653720
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	16.0000	0.522419
$939$	−22.0000	−0.717943
$940$	6.00000	0.195698
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	−7.00000	−0.228072
$943$	0	0
$944$	3.00000	0.0976417
$945$	−2.00000	−0.0650600
$946$	55.0000	1.78820
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−3.00000	−0.0974355
$949$	−16.0000	−0.519382
$950$	4.00000	0.129777
$951$	−10.0000	−0.324272
$952$	10.0000	0.324102
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−1.00000	−0.0323762
$955$	−3.00000	−0.0970777
$956$	−9.00000	−0.291081
$957$	20.0000	0.646508
$958$	16.0000	0.516937
$959$	24.0000	0.775000
$960$	1.00000	0.0322749
$961$	18.0000	0.580645
$962$	−4.00000	−0.128965
$963$	−12.0000	−0.386695
$964$	−23.0000	−0.740780
$965$	−4.00000	−0.128765
$966$	2.00000	0.0643489
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	14.0000	0.449977
$969$	5.00000	0.160623
$970$	−7.00000	−0.224756
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	−26.0000	−0.833094
$975$	−8.00000	−0.256205
$976$	−6.00000	−0.192055
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	−1.00000	−0.0319765
$979$	−35.0000	−1.11860
$980$	−3.00000	−0.0958315
$981$	1.00000	0.0319275
$982$	−16.0000	−0.510581
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	20.0000	0.636930
$987$	−12.0000	−0.381964
$988$	−2.00000	−0.0636285
$989$	11.0000	0.349780
$990$	−5.00000	−0.158910
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	7.00000	0.222250
$993$	−8.00000	−0.253872
$994$	−16.0000	−0.507489
$995$	26.0000	0.824255
$996$	−12.0000	−0.380235
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	−16.0000	−0.506471
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.k.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.k.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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