LMFDB - Embedded newform 6046.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
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	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} +4.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} +2.00000 q^{28} +1.00000 q^{29} +4.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -4.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -6.00000 q^{38} +4.00000 q^{39} +2.00000 q^{40} -10.0000 q^{41} -4.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -2.00000 q^{45} +6.00000 q^{46} -8.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +6.00000 q^{55} -2.00000 q^{56} +12.0000 q^{57} -1.00000 q^{58} +4.00000 q^{59} -4.00000 q^{60} +1.00000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +6.00000 q^{66} -2.00000 q^{67} -12.0000 q^{69} +4.00000 q^{70} +2.00000 q^{71} -1.00000 q^{72} +16.0000 q^{73} -1.00000 q^{74} -2.00000 q^{75} +6.00000 q^{76} -6.00000 q^{77} -4.00000 q^{78} -1.00000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +10.0000 q^{82} -14.0000 q^{83} +4.00000 q^{84} +1.00000 q^{86} +2.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +4.00000 q^{91} -6.00000 q^{92} +6.00000 q^{93} +8.00000 q^{94} -12.0000 q^{95} -2.00000 q^{96} +3.00000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−2.00000	−0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	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
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−2.00000	−0.447214
$21$	4.00000	0.872872
$22$	3.00000	0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−2.00000	−0.408248
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−4.00000	−0.769800
$28$	2.00000	0.377964
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	4.00000	0.730297
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−6.00000	−0.973329
$39$	4.00000	0.640513
$40$	2.00000	0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−4.00000	−0.617213
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	−2.00000	−0.298142
$46$	6.00000	0.884652
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	4.00000	0.544331
$55$	6.00000	0.809040
$56$	−2.00000	−0.267261
$57$	12.0000	1.58944
$58$	−1.00000	−0.131306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−4.00000	−0.516398
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−3.00000	−0.381000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	6.00000	0.738549
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	4.00000	0.478091
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−1.00000	−0.117851
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	−1.00000	−0.116248
$75$	−2.00000	−0.230940
$76$	6.00000	0.688247
$77$	−6.00000	−0.683763
$78$	−4.00000	−0.452911
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	10.0000	1.10432
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	1.00000	0.107833
$87$	2.00000	0.214423
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	6.00000	0.622171
$94$	8.00000	0.825137
$95$	−12.0000	−1.23117
$96$	−2.00000	−0.204124
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	3.00000	0.303046
$99$	−3.00000	−0.301511
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	−2.00000	−0.196116
$105$	−8.00000	−0.780720
$106$	6.00000	0.582772
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−4.00000	−0.384900
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	−6.00000	−0.572078
$111$	2.00000	0.189832
$112$	2.00000	0.188982
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−12.0000	−1.12390
$115$	12.0000	1.11901
$116$	1.00000	0.0928477
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	4.00000	0.365148
$121$	−2.00000	−0.181818
$122$	−1.00000	−0.0905357
$123$	−20.0000	−1.80334
$124$	3.00000	0.269408
$125$	12.0000	1.07331
$126$	−2.00000	−0.178174
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	−2.00000	−0.176090
$130$	4.00000	0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−6.00000	−0.522233
$133$	12.0000	1.04053
$134$	2.00000	0.172774
$135$	8.00000	0.688530
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	12.0000	1.02151
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	−4.00000	−0.338062
$141$	−16.0000	−1.34744
$142$	−2.00000	−0.167836
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−16.0000	−1.32417
$147$	−6.00000	−0.494872
$148$	1.00000	0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	2.00000	0.163299
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	6.00000	0.483494
$155$	−6.00000	−0.481932
$156$	4.00000	0.320256
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	1.00000	0.0795557
$159$	−12.0000	−0.951662
$160$	2.00000	0.158114
$161$	−12.0000	−0.945732
$162$	11.0000	0.864242
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−10.0000	−0.780869
$165$	12.0000	0.934199
$166$	14.0000	1.08661
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	−1.00000	−0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−2.00000	−0.151620
$175$	−2.00000	−0.151186
$176$	−3.00000	−0.226134
$177$	8.00000	0.601317
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−2.00000	−0.149071
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−4.00000	−0.296500
$183$	2.00000	0.147844
$184$	6.00000	0.442326
$185$	−2.00000	−0.147043
$186$	−6.00000	−0.439941
$187$	0	0
$188$	−8.00000	−0.583460
$189$	−8.00000	−0.581914
$190$	12.0000	0.870572
$191$	27.0000	1.95365	0.976826	−	0.214036i	$-0.0686611\pi$
$191$	27.0000	1.95365	0.976826	+	0.214036i	$0.0686611\pi$
$192$	2.00000	0.144338
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	−3.00000	−0.215387
$195$	−8.00000	−0.572892
$196$	−3.00000	−0.214286
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	3.00000	0.213201
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	2.00000	0.140372
$204$	0	0
$205$	20.0000	1.39686
$206$	−1.00000	−0.0696733
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	−18.0000	−1.24509
$210$	8.00000	0.552052
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−6.00000	−0.412082
$213$	4.00000	0.274075
$214$	−8.00000	−0.546869
$215$	2.00000	0.136399
$216$	4.00000	0.272166
$217$	6.00000	0.407307
$218$	19.0000	1.28684
$219$	32.0000	2.16236
$220$	6.00000	0.404520
$221$	0	0
$222$	−2.00000	−0.134231
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	−2.00000	−0.133631
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	12.0000	0.794719
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−12.0000	−0.791257
$231$	−12.0000	−0.789542
$232$	−1.00000	−0.0656532
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	16.0000	1.04372
$236$	4.00000	0.260378
$237$	−2.00000	−0.129914
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	−4.00000	−0.258199
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	2.00000	0.128565
$243$	−10.0000	−0.641500
$244$	1.00000	0.0640184
$245$	6.00000	0.383326
$246$	20.0000	1.27515
$247$	12.0000	0.763542
$248$	−3.00000	−0.190500
$249$	−28.0000	−1.77443
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000	0.125988
$253$	18.0000	1.13165
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	2.00000	0.124515
$259$	2.00000	0.124274
$260$	−4.00000	−0.248069
$261$	1.00000	0.0618984
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	6.00000	0.369274
$265$	12.0000	0.737154
$266$	−12.0000	−0.735767
$267$	−12.0000	−0.734388
$268$	−2.00000	−0.122169
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	−8.00000	−0.486864
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	3.00000	0.181237
$275$	3.00000	0.180907
$276$	−12.0000	−0.722315
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	6.00000	0.359856
$279$	3.00000	0.179605
$280$	4.00000	0.239046
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	16.0000	0.952786
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	2.00000	0.118678
$285$	−24.0000	−1.42164
$286$	6.00000	0.354787
$287$	−20.0000	−1.18056
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	2.00000	0.117444
$291$	6.00000	0.351726
$292$	16.0000	0.936329
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	6.00000	0.349927
$295$	−8.00000	−0.465778
$296$	−1.00000	−0.0581238
$297$	12.0000	0.696311
$298$	18.0000	1.04271
$299$	−12.0000	−0.693978
$300$	−2.00000	−0.115470
$301$	−2.00000	−0.115278
$302$	16.0000	0.920697
$303$	20.0000	1.14897
$304$	6.00000	0.344124
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	−6.00000	−0.341882
$309$	2.00000	0.113776
$310$	6.00000	0.340777
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−4.00000	−0.226455
$313$	−15.0000	−0.847850	−0.423925	−	0.905697i	$-0.639348\pi$
$313$	−15.0000	−0.847850	−0.423925	+	0.905697i	$0.639348\pi$
$314$	11.0000	0.620766
$315$	−4.00000	−0.225374
$316$	−1.00000	−0.0562544
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	12.0000	0.672927
$319$	−3.00000	−0.167968
$320$	−2.00000	−0.111803
$321$	16.0000	0.893033
$322$	12.0000	0.668734
$323$	0	0
$324$	−11.0000	−0.611111
$325$	−2.00000	−0.110940
$326$	6.00000	0.332309
$327$	−38.0000	−2.10140
$328$	10.0000	0.552158
$329$	−16.0000	−0.882109
$330$	−12.0000	−0.660578
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	−14.0000	−0.768350
$333$	1.00000	0.0547997
$334$	8.00000	0.437741
$335$	4.00000	0.218543
$336$	4.00000	0.218218
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	9.00000	0.489535
$339$	28.0000	1.52075
$340$	0	0
$341$	−9.00000	−0.487377
$342$	−6.00000	−0.324443
$343$	−20.0000	−1.07990
$344$	1.00000	0.0539164
$345$	24.0000	1.29212
$346$	−6.00000	−0.322562
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	2.00000	0.107211
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	2.00000	0.106904
$351$	−8.00000	−0.427008
$352$	3.00000	0.159901
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	−8.00000	−0.425195
$355$	−4.00000	−0.212298
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−13.0000	−0.686114	−0.343057	−	0.939315i	$-0.611462\pi$
$359$	−13.0000	−0.686114	−0.343057	+	0.939315i	$0.611462\pi$
$360$	2.00000	0.105409
$361$	17.0000	0.894737
$362$	12.0000	0.630706
$363$	−4.00000	−0.209946
$364$	4.00000	0.209657
$365$	−32.0000	−1.67496
$366$	−2.00000	−0.104542
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−6.00000	−0.312772
$369$	−10.0000	−0.520579
$370$	2.00000	0.103975
$371$	−12.0000	−0.623009
$372$	6.00000	0.311086
$373$	33.0000	1.70868	0.854338	−	0.519718i	$-0.173963\pi$
$373$	33.0000	1.70868	0.854338	+	0.519718i	$0.173963\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	8.00000	0.412568
$377$	2.00000	0.103005
$378$	8.00000	0.411476
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−12.0000	−0.615587
$381$	−4.00000	−0.204926
$382$	−27.0000	−1.38144
$383$	33.0000	1.68622	0.843111	−	0.537740i	$-0.180722\pi$
$383$	33.0000	1.68622	0.843111	+	0.537740i	$0.180722\pi$
$384$	−2.00000	−0.102062
$385$	12.0000	0.611577
$386$	−9.00000	−0.458088
$387$	−1.00000	−0.0508329
$388$	3.00000	0.152302
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	8.00000	0.405096
$391$	0	0
$392$	3.00000	0.151523
$393$	0	0
$394$	14.0000	0.705310
$395$	2.00000	0.100631
$396$	−3.00000	−0.150756
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−4.00000	−0.200502
$399$	24.0000	1.20150
$400$	−1.00000	−0.0500000
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	4.00000	0.199502
$403$	6.00000	0.298881
$404$	10.0000	0.497519
$405$	22.0000	1.09319
$406$	−2.00000	−0.0992583
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	−20.0000	−0.987730
$411$	−6.00000	−0.295958
$412$	1.00000	0.0492665
$413$	8.00000	0.393654
$414$	6.00000	0.294884
$415$	28.0000	1.37447
$416$	−2.00000	−0.0980581
$417$	−12.0000	−0.587643
$418$	18.0000	0.880409
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−8.00000	−0.390360
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	12.0000	0.584151
$423$	−8.00000	−0.388973
$424$	6.00000	0.291386
$425$	0	0
$426$	−4.00000	−0.193801
$427$	2.00000	0.0967868
$428$	8.00000	0.386695
$429$	−12.0000	−0.579365
$430$	−2.00000	−0.0964486
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	−4.00000	−0.192450
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	−6.00000	−0.288009
$435$	−4.00000	−0.191785
$436$	−19.0000	−0.909935
$437$	−36.0000	−1.72211
$438$	−32.0000	−1.52902
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	−6.00000	−0.286039
$441$	−3.00000	−0.142857
$442$	0	0
$443$	7.00000	0.332580	0.166290	−	0.986077i	$-0.446821\pi$
$443$	7.00000	0.332580	0.166290	+	0.986077i	$0.446821\pi$
$444$	2.00000	0.0949158
$445$	12.0000	0.568855
$446$	−11.0000	−0.520865
$447$	−36.0000	−1.70274
$448$	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	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	30.0000	1.41264
$452$	14.0000	0.658505
$453$	−32.0000	−1.50349
$454$	2.00000	0.0938647
$455$	−8.00000	−0.375046
$456$	−12.0000	−0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	12.0000	0.559503
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	12.0000	0.558291
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	1.00000	0.0464238
$465$	−12.0000	−0.556487
$466$	18.0000	0.833834
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	2.00000	0.0924500
$469$	−4.00000	−0.184703
$470$	−16.0000	−0.738025
$471$	−22.0000	−1.01371
$472$	−4.00000	−0.184115
$473$	3.00000	0.137940
$474$	2.00000	0.0918630
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−5.00000	−0.228695
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	4.00000	0.182574
$481$	2.00000	0.0911922
$482$	4.00000	0.182195
$483$	−24.0000	−1.09204
$484$	−2.00000	−0.0909091
$485$	−6.00000	−0.272446
$486$	10.0000	0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−1.00000	−0.0452679
$489$	−12.0000	−0.542659
$490$	−6.00000	−0.271052
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−20.0000	−0.901670
$493$	0	0
$494$	−12.0000	−0.539906
$495$	6.00000	0.269680
$496$	3.00000	0.134704
$497$	4.00000	0.179425
$498$	28.0000	1.25471
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	12.0000	0.536656
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	−2.00000	−0.0890871
$505$	−20.0000	−0.889988
$506$	−18.0000	−0.800198
$507$	−18.0000	−0.799408
$508$	−2.00000	−0.0887357
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	32.0000	1.41560
$512$	−1.00000	−0.0441942
$513$	−24.0000	−1.05963
$514$	3.00000	0.132324
$515$	−2.00000	−0.0881305
$516$	−2.00000	−0.0880451
$517$	24.0000	1.05552
$518$	−2.00000	−0.0878750
$519$	12.0000	0.526742
$520$	4.00000	0.175412
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	−1.00000	−0.0437688
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	16.0000	0.697633
$527$	0	0
$528$	−6.00000	−0.261116
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	4.00000	0.173585
$532$	12.0000	0.520266
$533$	−20.0000	−0.866296
$534$	12.0000	0.519291
$535$	−16.0000	−0.691740
$536$	2.00000	0.0863868
$537$	40.0000	1.72613
$538$	−3.00000	−0.129339
$539$	9.00000	0.387657
$540$	8.00000	0.344265
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	14.0000	0.601351
$543$	−24.0000	−1.02994
$544$	0	0
$545$	38.0000	1.62774
$546$	−8.00000	−0.342368
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−3.00000	−0.128154
$549$	1.00000	0.0426790
$550$	−3.00000	−0.127920
$551$	6.00000	0.255609
$552$	12.0000	0.510754
$553$	−2.00000	−0.0850487
$554$	2.00000	0.0849719
$555$	−4.00000	−0.169791
$556$	−6.00000	−0.254457
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−3.00000	−0.127000
$559$	−2.00000	−0.0845910
$560$	−4.00000	−0.169031
$561$	0	0
$562$	−17.0000	−0.717102
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	−16.0000	−0.673722
$565$	−28.0000	−1.17797
$566$	22.0000	0.924729
$567$	−22.0000	−0.923913
$568$	−2.00000	−0.0839181
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	24.0000	1.00525
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	−6.00000	−0.250873
$573$	54.0000	2.25588
$574$	20.0000	0.834784
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	17.0000	0.707107
$579$	18.0000	0.748054
$580$	−2.00000	−0.0830455
$581$	−28.0000	−1.16164
$582$	−6.00000	−0.248708
$583$	18.0000	0.745484
$584$	−16.0000	−0.662085
$585$	−4.00000	−0.165380
$586$	16.0000	0.660954
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−6.00000	−0.247436
$589$	18.0000	0.741677
$590$	8.00000	0.329355
$591$	−28.0000	−1.15177
$592$	1.00000	0.0410997
$593$	43.0000	1.76580	0.882899	−	0.469563i	$-0.155588\pi$
$593$	43.0000	1.76580	0.882899	+	0.469563i	$0.155588\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	−18.0000	−0.737309
$597$	8.00000	0.327418
$598$	12.0000	0.490716
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	2.00000	0.0816497
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	2.00000	0.0815139
$603$	−2.00000	−0.0814463
$604$	−16.0000	−0.651031
$605$	4.00000	0.162623
$606$	−20.0000	−0.812444
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	−6.00000	−0.243332
$609$	4.00000	0.162088
$610$	2.00000	0.0809776
$611$	−16.0000	−0.647291
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	11.0000	0.443924
$615$	40.0000	1.61296
$616$	6.00000	0.241747
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	−2.00000	−0.0804518
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−6.00000	−0.240966
$621$	24.0000	0.963087
$622$	4.00000	0.160385
$623$	−12.0000	−0.480770
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	15.0000	0.599521
$627$	−36.0000	−1.43770
$628$	−11.0000	−0.438948
$629$	0	0
$630$	4.00000	0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	1.00000	0.0397779
$633$	−24.0000	−0.953914
$634$	2.00000	0.0794301
$635$	4.00000	0.158735
$636$	−12.0000	−0.475831
$637$	−6.00000	−0.237729
$638$	3.00000	0.118771
$639$	2.00000	0.0791188
$640$	2.00000	0.0790569
$641$	19.0000	0.750455	0.375227	−	0.926933i	$-0.377565\pi$
$641$	19.0000	0.750455	0.375227	+	0.926933i	$0.377565\pi$
$642$	−16.0000	−0.631470
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−12.0000	−0.472866
$645$	4.00000	0.157500
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	11.0000	0.432121
$649$	−12.0000	−0.471041
$650$	2.00000	0.0784465
$651$	12.0000	0.470317
$652$	−6.00000	−0.234978
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	38.0000	1.48592
$655$	0	0
$656$	−10.0000	−0.390434
$657$	16.0000	0.624219
$658$	16.0000	0.623745
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	12.0000	0.467099
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	36.0000	1.39918
$663$	0	0
$664$	14.0000	0.543305
$665$	−24.0000	−0.930680
$666$	−1.00000	−0.0387492
$667$	−6.00000	−0.232321
$668$	−8.00000	−0.309529
$669$	22.0000	0.850569
$670$	−4.00000	−0.154533
$671$	−3.00000	−0.115814
$672$	−4.00000	−0.154303
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−18.0000	−0.693334
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	−28.0000	−1.07533
$679$	6.00000	0.230259
$680$	0	0
$681$	−4.00000	−0.153280
$682$	9.00000	0.344628
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	6.00000	0.229416
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	−52.0000	−1.98392
$688$	−1.00000	−0.0381246
$689$	−12.0000	−0.457164
$690$	−24.0000	−0.913664
$691$	45.0000	1.71188	0.855940	−	0.517075i	$-0.172979\pi$
$691$	45.0000	1.71188	0.855940	+	0.517075i	$0.172979\pi$
$692$	6.00000	0.228086
$693$	−6.00000	−0.227921
$694$	30.0000	1.13878
$695$	12.0000	0.455186
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	−14.0000	−0.529908
$699$	−36.0000	−1.36165
$700$	−2.00000	−0.0755929
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	8.00000	0.301941
$703$	6.00000	0.226294
$704$	−3.00000	−0.113067
$705$	32.0000	1.20519
$706$	0	0
$707$	20.0000	0.752177
$708$	8.00000	0.300658
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	4.00000	0.150117
$711$	−1.00000	−0.0375029
$712$	6.00000	0.224860
$713$	−18.0000	−0.674105
$714$	0	0
$715$	12.0000	0.448775
$716$	20.0000	0.747435
$717$	10.0000	0.373457
$718$	13.0000	0.485156
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−2.00000	−0.0745356
$721$	2.00000	0.0744839
$722$	−17.0000	−0.632674
$723$	−8.00000	−0.297523
$724$	−12.0000	−0.445976
$725$	−1.00000	−0.0371391
$726$	4.00000	0.148454
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	32.0000	1.18437
$731$	0	0
$732$	2.00000	0.0739221
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	17.0000	0.627481
$735$	12.0000	0.442627
$736$	6.00000	0.221163
$737$	6.00000	0.221013
$738$	10.0000	0.368105
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	−2.00000	−0.0735215
$741$	24.0000	0.881662
$742$	12.0000	0.440534
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−6.00000	−0.219971
$745$	36.0000	1.31894
$746$	−33.0000	−1.20822
$747$	−14.0000	−0.512233
$748$	0	0
$749$	16.0000	0.584627
$750$	−24.0000	−0.876356
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	32.0000	1.16460
$756$	−8.00000	−0.290957
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−6.00000	−0.217930
$759$	36.0000	1.30672
$760$	12.0000	0.435286
$761$	53.0000	1.92125	0.960624	−	0.277851i	$-0.0896221\pi$
$761$	53.0000	1.92125	0.960624	+	0.277851i	$0.0896221\pi$
$762$	4.00000	0.144905
$763$	−38.0000	−1.37569
$764$	27.0000	0.976826
$765$	0	0
$766$	−33.0000	−1.19234
$767$	8.00000	0.288863
$768$	2.00000	0.0721688
$769$	−44.0000	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$769$	−44.0000	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$770$	−12.0000	−0.432450
$771$	−6.00000	−0.216085
$772$	9.00000	0.323917
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	1.00000	0.0359443
$775$	−3.00000	−0.107763
$776$	−3.00000	−0.107694
$777$	4.00000	0.143499
$778$	−4.00000	−0.143407
$779$	−60.0000	−2.14972
$780$	−8.00000	−0.286446
$781$	−6.00000	−0.214697
$782$	0	0
$783$	−4.00000	−0.142948
$784$	−3.00000	−0.107143
$785$	22.0000	0.785214
$786$	0	0
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	−14.0000	−0.498729
$789$	−32.0000	−1.13923
$790$	−2.00000	−0.0711568
$791$	28.0000	0.995565
$792$	3.00000	0.106600
$793$	2.00000	0.0710221
$794$	−14.0000	−0.496841
$795$	24.0000	0.851192
$796$	4.00000	0.141776
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	−24.0000	−0.849591
$799$	0	0
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	−8.00000	−0.282490
$803$	−48.0000	−1.69388
$804$	−4.00000	−0.141069
$805$	24.0000	0.845889
$806$	−6.00000	−0.211341
$807$	6.00000	0.211210
$808$	−10.0000	−0.351799
$809$	1.00000	0.0351581	0.0175791	−	0.999845i	$-0.494404\pi$
$809$	1.00000	0.0351581	0.0175791	+	0.999845i	$0.494404\pi$
$810$	−22.0000	−0.773001
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	2.00000	0.0701862
$813$	−28.0000	−0.982003
$814$	3.00000	0.105150
$815$	12.0000	0.420342
$816$	0	0
$817$	−6.00000	−0.209913
$818$	23.0000	0.804176
$819$	4.00000	0.139771
$820$	20.0000	0.698430
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	6.00000	0.209274
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−1.00000	−0.0348367
$825$	6.00000	0.208893
$826$	−8.00000	−0.278356
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	−6.00000	−0.208514
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	−28.0000	−0.971894
$831$	−4.00000	−0.138758
$832$	2.00000	0.0693375
$833$	0	0
$834$	12.0000	0.415526
$835$	16.0000	0.553703
$836$	−18.0000	−0.622543
$837$	−12.0000	−0.414781
$838$	30.0000	1.03633
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	8.00000	0.276026
$841$	−28.0000	−0.965517
$842$	−14.0000	−0.482472
$843$	34.0000	1.17102
$844$	−12.0000	−0.413057
$845$	18.0000	0.619219
$846$	8.00000	0.275046
$847$	−4.00000	−0.137442
$848$	−6.00000	−0.206041
$849$	−44.0000	−1.51008
$850$	0	0
$851$	−6.00000	−0.205677
$852$	4.00000	0.137038
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	−2.00000	−0.0684386
$855$	−12.0000	−0.410391
$856$	−8.00000	−0.273434
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	12.0000	0.409673
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	2.00000	0.0681994
$861$	−40.0000	−1.36320
$862$	−33.0000	−1.12398
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	4.00000	0.136083
$865$	−12.0000	−0.408012
$866$	−29.0000	−0.985460
$867$	−34.0000	−1.15470
$868$	6.00000	0.203653
$869$	3.00000	0.101768
$870$	4.00000	0.135613
$871$	−4.00000	−0.135535
$872$	19.0000	0.643421
$873$	3.00000	0.101535
$874$	36.0000	1.21772
$875$	24.0000	0.811348
$876$	32.0000	1.08118
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	−14.0000	−0.472477
$879$	−32.0000	−1.07933
$880$	6.00000	0.202260
$881$	−25.0000	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$882$	3.00000	0.101015
$883$	−47.0000	−1.58168	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000	−1.58168	−0.790838	+	0.612026i	$0.790355\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	−7.00000	−0.235170
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	−2.00000	−0.0671156
$889$	−4.00000	−0.134156
$890$	−12.0000	−0.402241
$891$	33.0000	1.10554
$892$	11.0000	0.368307
$893$	−48.0000	−1.60626
$894$	36.0000	1.20402
$895$	−40.0000	−1.33705
$896$	−2.00000	−0.0668153
$897$	−24.0000	−0.801337
$898$	−30.0000	−1.00111
$899$	3.00000	0.100056
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	−30.0000	−0.998891
$903$	−4.00000	−0.133112
$904$	−14.0000	−0.465633
$905$	24.0000	0.797787
$906$	32.0000	1.06313
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−2.00000	−0.0663723
$909$	10.0000	0.331679
$910$	8.00000	0.265197
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	12.0000	0.397360
$913$	42.0000	1.39000
$914$	−18.0000	−0.595387
$915$	−4.00000	−0.132236
$916$	−26.0000	−0.859064
$917$	0	0
$918$	0	0
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	−12.0000	−0.395628
$921$	−22.0000	−0.724925
$922$	−12.0000	−0.395199
$923$	4.00000	0.131662
$924$	−12.0000	−0.394771
$925$	−1.00000	−0.0328798
$926$	26.0000	0.854413
$927$	1.00000	0.0328443
$928$	−1.00000	−0.0328266
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	12.0000	0.393496
$931$	−18.0000	−0.589926
$932$	−18.0000	−0.589610
$933$	−8.00000	−0.261908
$934$	−13.0000	−0.425373
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	21.0000	0.686040	0.343020	−	0.939328i	$-0.388550\pi$
$937$	21.0000	0.686040	0.343020	+	0.939328i	$0.388550\pi$
$938$	4.00000	0.130605
$939$	−30.0000	−0.979013
$940$	16.0000	0.521862
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	22.0000	0.716799
$943$	60.0000	1.95387
$944$	4.00000	0.130189
$945$	16.0000	0.520480
$946$	−3.00000	−0.0975384
$947$	54.0000	1.75476	0.877382	−	0.479792i	$-0.159288\pi$
$947$	54.0000	1.75476	0.877382	+	0.479792i	$0.159288\pi$
$948$	−2.00000	−0.0649570
$949$	32.0000	1.03876
$950$	6.00000	0.194666
$951$	−4.00000	−0.129709
$952$	0	0
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	6.00000	0.194257
$955$	−54.0000	−1.74740
$956$	5.00000	0.161712
$957$	−6.00000	−0.193952
$958$	−30.0000	−0.969256
$959$	−6.00000	−0.193750
$960$	−4.00000	−0.129099
$961$	−22.0000	−0.709677
$962$	−2.00000	−0.0644826
$963$	8.00000	0.257796
$964$	−4.00000	−0.128831
$965$	−18.0000	−0.579441
$966$	24.0000	0.772187
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	6.00000	0.192648
$971$	−57.0000	−1.82922	−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000	−1.82922	−0.914609	+	0.404341i	$0.867501\pi$
$972$	−10.0000	−0.320750
$973$	−12.0000	−0.384702
$974$	−2.00000	−0.0640841
$975$	−4.00000	−0.128103
$976$	1.00000	0.0320092
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	12.0000	0.383718
$979$	18.0000	0.575282
$980$	6.00000	0.191663
$981$	−19.0000	−0.606623
$982$	12.0000	0.382935
$983$	−1.00000	−0.0318950	−0.0159475	−	0.999873i	$-0.505076\pi$
$983$	−1.00000	−0.0318950	−0.0159475	+	0.999873i	$0.505076\pi$
$984$	20.0000	0.637577
$985$	28.0000	0.892154
$986$	0	0
$987$	−32.0000	−1.01857
$988$	12.0000	0.381771
$989$	6.00000	0.190789
$990$	−6.00000	−0.190693
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	−3.00000	−0.0952501
$993$	−72.0000	−2.28485
$994$	−4.00000	−0.126872
$995$	−8.00000	−0.253617
$996$	−28.0000	−0.887214
$997$	60.0000	1.90022	0.950110	−	0.311916i	$-0.100971\pi$
$997$	60.0000	1.90022	0.950110	+	0.311916i	$0.100971\pi$
$998$	40.0000	1.26618
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.b.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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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