LMFDB - Embedded newform 4046.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4046,2,Mod(1,4046)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4046.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4046, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4046 = 2 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4046.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,4,-1,-1,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.3074726578$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4046.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} +3.00000 q^{19} +4.00000 q^{20} +1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +1.00000 q^{29} -4.00000 q^{30} -9.00000 q^{31} +1.00000 q^{32} -4.00000 q^{35} -2.00000 q^{36} -10.0000 q^{37} +3.00000 q^{38} +2.00000 q^{39} +4.00000 q^{40} +8.00000 q^{41} +1.00000 q^{42} +10.0000 q^{43} -8.00000 q^{45} +4.00000 q^{46} +13.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} -2.00000 q^{52} +9.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} -3.00000 q^{57} +1.00000 q^{58} +5.00000 q^{59} -4.00000 q^{60} +8.00000 q^{61} -9.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} +8.00000 q^{67} -4.00000 q^{69} -4.00000 q^{70} +2.00000 q^{71} -2.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -11.0000 q^{75} +3.00000 q^{76} +2.00000 q^{78} +6.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} -3.00000 q^{83} +1.00000 q^{84} +10.0000 q^{86} -1.00000 q^{87} -8.00000 q^{89} -8.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +9.00000 q^{93} +13.0000 q^{94} +12.0000 q^{95} -1.00000 q^{96} -18.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	4.00000	1.26491
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−2.00000	−0.471405
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	4.00000	0.894427
$21$	1.00000	0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−2.00000	−0.392232
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	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
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	−2.00000	−0.333333
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	3.00000	0.486664
$39$	2.00000	0.320256
$40$	4.00000	0.632456
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	1.00000	0.154303
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−8.00000	−1.19257
$46$	4.00000	0.589768
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	0	0
$52$	−2.00000	−0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−3.00000	−0.397360
$58$	1.00000	0.131306
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	−4.00000	−0.516398
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−9.00000	−1.14300
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	−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	$0.462134\pi$
$72$	−2.00000	−0.235702
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	−11.0000	−1.27017
$76$	3.00000	0.344124
$77$	0	0
$78$	2.00000	0.226455
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	10.0000	1.07833
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−8.00000	−0.843274
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	9.00000	0.933257
$94$	13.0000	1.34085
$95$	12.0000	1.23117
$96$	−1.00000	−0.102062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	9.00000	0.874157
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	5.00000	0.481125
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−3.00000	−0.280976
$115$	16.0000	1.49201
$116$	1.00000	0.0928477
$117$	4.00000	0.369800
$118$	5.00000	0.460287
$119$	0	0
$120$	−4.00000	−0.365148
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	−8.00000	−0.721336
$124$	−9.00000	−0.808224
$125$	24.0000	2.14663
$126$	2.00000	0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	−8.00000	−0.701646
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	8.00000	0.691095
$135$	20.0000	1.72133
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	−4.00000	−0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−4.00000	−0.338062
$141$	−13.0000	−1.09480
$142$	2.00000	0.167836
$143$	0	0
$144$	−2.00000	−0.166667
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	−1.00000	−0.0824786
$148$	−10.0000	−0.821995
$149$	13.0000	1.06500	0.532501	−	0.846430i	$-0.321252\pi$
$149$	13.0000	1.06500	0.532501	+	0.846430i	$0.321252\pi$
$150$	−11.0000	−0.898146
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	0	0
$155$	−36.0000	−2.89159
$156$	2.00000	0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	6.00000	0.477334
$159$	−9.00000	−0.713746
$160$	4.00000	0.316228
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−3.00000	−0.232845
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	10.0000	0.762493
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	−1.00000	−0.0758098
$175$	−11.0000	−0.831522
$176$	0	0
$177$	−5.00000	−0.375823
$178$	−8.00000	−0.599625
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−8.00000	−0.596285
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	2.00000	0.148250
$183$	−8.00000	−0.591377
$184$	4.00000	0.294884
$185$	−40.0000	−2.94086
$186$	9.00000	0.659912
$187$	0	0
$188$	13.0000	0.948122
$189$	−5.00000	−0.363696
$190$	12.0000	0.870572
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−1.00000	−0.0721688
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	−18.0000	−1.29232
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	11.0000	0.777817
$201$	−8.00000	−0.564276
$202$	−6.00000	−0.422159
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	32.0000	2.23498
$206$	7.00000	0.487713
$207$	−8.00000	−0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	4.00000	0.276026
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	9.00000	0.618123
$213$	−2.00000	−0.137038
$214$	8.00000	0.546869
$215$	40.0000	2.72798
$216$	5.00000	0.340207
$217$	9.00000	0.610960
$218$	−19.0000	−1.28684
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	10.0000	0.671156
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−1.00000	−0.0668153
$225$	−22.0000	−1.46667
$226$	2.00000	0.133038
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	−3.00000	−0.198680
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	1.00000	0.0656532
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	4.00000	0.261488
$235$	52.0000	3.39211
$236$	5.00000	0.325472
$237$	−6.00000	−0.389742
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	−4.00000	−0.258199
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	8.00000	0.512148
$245$	4.00000	0.255551
$246$	−8.00000	−0.510061
$247$	−6.00000	−0.381771
$248$	−9.00000	−0.571501
$249$	3.00000	0.190117
$250$	24.0000	1.51789
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−10.0000	−0.622573
$259$	10.0000	0.621370
$260$	−8.00000	−0.496139
$261$	−2.00000	−0.123797
$262$	7.00000	0.432461
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	−3.00000	−0.183942
$267$	8.00000	0.489592
$268$	8.00000	0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	20.0000	1.21716
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	17.0000	1.02701
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	4.00000	0.239904
$279$	18.0000	1.07763
$280$	−4.00000	−0.239046
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−13.0000	−0.774139
$283$	9.00000	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$283$	9.00000	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$284$	2.00000	0.118678
$285$	−12.0000	−0.710819
$286$	0	0
$287$	−8.00000	−0.472225
$288$	−2.00000	−0.117851
$289$	0	0
$290$	4.00000	0.234888
$291$	18.0000	1.05518
$292$	2.00000	0.117041
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	−1.00000	−0.0583212
$295$	20.0000	1.16445
$296$	−10.0000	−0.581238
$297$	0	0
$298$	13.0000	0.753070
$299$	−8.00000	−0.462652
$300$	−11.0000	−0.635085
$301$	−10.0000	−0.576390
$302$	−2.00000	−0.115087
$303$	6.00000	0.344691
$304$	3.00000	0.172062
$305$	32.0000	1.83231
$306$	0	0
$307$	−35.0000	−1.99756	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$	−35.0000	−1.99756	−0.998778	+	0.0494267i	$0.984261\pi$
$308$	0	0
$309$	−7.00000	−0.398216
$310$	−36.0000	−2.04466
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	2.00000	0.113228
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−14.0000	−0.790066
$315$	8.00000	0.450749
$316$	6.00000	0.337526
$317$	11.0000	0.617822	0.308911	−	0.951091i	$-0.400036\pi$
$317$	11.0000	0.617822	0.308911	+	0.951091i	$0.400036\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	4.00000	0.223607
$321$	−8.00000	−0.446516
$322$	−4.00000	−0.222911
$323$	0	0
$324$	1.00000	0.0555556
$325$	−22.0000	−1.22034
$326$	2.00000	0.110770
$327$	19.0000	1.05070
$328$	8.00000	0.441726
$329$	−13.0000	−0.716713
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−3.00000	−0.164646
$333$	20.0000	1.09599
$334$	−17.0000	−0.930199
$335$	32.0000	1.74835
$336$	1.00000	0.0545545
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	−9.00000	−0.489535
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	−6.00000	−0.324443
$343$	−1.00000	−0.0539949
$344$	10.0000	0.539164
$345$	−16.0000	−0.861411
$346$	−4.00000	−0.215041
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	−1.00000	−0.0536056
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	−11.0000	−0.587975
$351$	−10.0000	−0.533761
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−5.00000	−0.265747
$355$	8.00000	0.424596
$356$	−8.00000	−0.423999
$357$	0	0
$358$	−18.0000	−0.951330
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	−8.00000	−0.421637
$361$	−10.0000	−0.526316
$362$	0	0
$363$	11.0000	0.577350
$364$	2.00000	0.104828
$365$	8.00000	0.418739
$366$	−8.00000	−0.418167
$367$	33.0000	1.72259	0.861293	−	0.508109i	$-0.169655\pi$
$367$	33.0000	1.72259	0.861293	+	0.508109i	$0.169655\pi$
$368$	4.00000	0.208514
$369$	−16.0000	−0.832927
$370$	−40.0000	−2.07950
$371$	−9.00000	−0.467257
$372$	9.00000	0.466628
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	13.0000	0.670424
$377$	−2.00000	−0.103005
$378$	−5.00000	−0.257172
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	12.0000	0.615587
$381$	4.00000	0.204926
$382$	6.00000	0.306987
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−23.0000	−1.17067
$387$	−20.0000	−1.01666
$388$	−18.0000	−0.913812
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	8.00000	0.405096
$391$	0	0
$392$	1.00000	0.0505076
$393$	−7.00000	−0.353103
$394$	3.00000	0.151138
$395$	24.0000	1.20757
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	16.0000	0.802008
$399$	3.00000	0.150188
$400$	11.0000	0.550000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	−8.00000	−0.399004
$403$	18.0000	0.896644
$404$	−6.00000	−0.298511
$405$	4.00000	0.198762
$406$	−1.00000	−0.0496292
$407$	0	0
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	32.0000	1.58037
$411$	−17.0000	−0.838548
$412$	7.00000	0.344865
$413$	−5.00000	−0.246034
$414$	−8.00000	−0.393179
$415$	−12.0000	−0.589057
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	4.00000	0.195180
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	−26.0000	−1.26566
$423$	−26.0000	−1.26416
$424$	9.00000	0.437079
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	−8.00000	−0.387147
$428$	8.00000	0.386695
$429$	0	0
$430$	40.0000	1.92897
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	5.00000	0.240563
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	9.00000	0.432014
$435$	−4.00000	−0.191785
$436$	−19.0000	−0.909935
$437$	12.0000	0.574038
$438$	−2.00000	−0.0955637
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	10.0000	0.474579
$445$	−32.0000	−1.51695
$446$	−9.00000	−0.426162
$447$	−13.0000	−0.614879
$448$	−1.00000	−0.0472456
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	−22.0000	−1.03709
$451$	0	0
$452$	2.00000	0.0940721
$453$	2.00000	0.0939682
$454$	11.0000	0.516256
$455$	8.00000	0.375046
$456$	−3.00000	−0.140488
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	16.0000	0.746004
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	1.00000	0.0464238
$465$	36.0000	1.66946
$466$	−25.0000	−1.15810
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	4.00000	0.184900
$469$	−8.00000	−0.369406
$470$	52.0000	2.39858
$471$	14.0000	0.645086
$472$	5.00000	0.230144
$473$	0	0
$474$	−6.00000	−0.275589
$475$	33.0000	1.51414
$476$	0	0
$477$	−18.0000	−0.824163
$478$	14.0000	0.640345
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	−4.00000	−0.182574
$481$	20.0000	0.911922
$482$	20.0000	0.910975
$483$	4.00000	0.182006
$484$	−11.0000	−0.500000
$485$	−72.0000	−3.26935
$486$	−16.0000	−0.725775
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	8.00000	0.362143
$489$	−2.00000	−0.0904431
$490$	4.00000	0.180702
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	−8.00000	−0.360668
$493$	0	0
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−9.00000	−0.404112
$497$	−2.00000	−0.0897123
$498$	3.00000	0.134433
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	24.0000	1.07331
$501$	17.0000	0.759504
$502$	0	0
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	2.00000	0.0890871
$505$	−24.0000	−1.06799
$506$	0	0
$507$	9.00000	0.399704
$508$	−4.00000	−0.177471
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	15.0000	0.662266
$514$	30.0000	1.32324
$515$	28.0000	1.23383
$516$	−10.0000	−0.440225
$517$	0	0
$518$	10.0000	0.439375
$519$	4.00000	0.175581
$520$	−8.00000	−0.350823
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−2.00000	−0.0875376
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	7.00000	0.305796
$525$	11.0000	0.480079
$526$	16.0000	0.697633
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	36.0000	1.56374
$531$	−10.0000	−0.433963
$532$	−3.00000	−0.130066
$533$	−16.0000	−0.693037
$534$	8.00000	0.346194
$535$	32.0000	1.38348
$536$	8.00000	0.345547
$537$	18.0000	0.776757
$538$	−18.0000	−0.776035
$539$	0	0
$540$	20.0000	0.860663
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	−9.00000	−0.386583
$543$	0	0
$544$	0	0
$545$	−76.0000	−3.25548
$546$	−2.00000	−0.0855921
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	17.0000	0.726204
$549$	−16.0000	−0.682863
$550$	0	0
$551$	3.00000	0.127804
$552$	−4.00000	−0.170251
$553$	−6.00000	−0.255146
$554$	−17.0000	−0.722261
$555$	40.0000	1.69791
$556$	4.00000	0.169638
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	18.0000	0.762001
$559$	−20.0000	−0.845910
$560$	−4.00000	−0.169031
$561$	0	0
$562$	2.00000	0.0843649
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−13.0000	−0.547399
$565$	8.00000	0.336563
$566$	9.00000	0.378298
$567$	−1.00000	−0.0419961
$568$	2.00000	0.0839181
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	−12.0000	−0.502625
$571$	−42.0000	−1.75765	−0.878823	−	0.477149i	$-0.841670\pi$
$571$	−42.0000	−1.75765	−0.878823	+	0.477149i	$0.841670\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	−8.00000	−0.333914
$575$	44.0000	1.83493
$576$	−2.00000	−0.0833333
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	0	0
$579$	23.0000	0.955847
$580$	4.00000	0.166091
$581$	3.00000	0.124461
$582$	18.0000	0.746124
$583$	0	0
$584$	2.00000	0.0827606
$585$	16.0000	0.661519
$586$	12.0000	0.495715
$587$	−11.0000	−0.454019	−0.227009	−	0.973893i	$-0.572895\pi$
$587$	−11.0000	−0.454019	−0.227009	+	0.973893i	$0.572895\pi$
$588$	−1.00000	−0.0412393
$589$	−27.0000	−1.11252
$590$	20.0000	0.823387
$591$	−3.00000	−0.123404
$592$	−10.0000	−0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	13.0000	0.532501
$597$	−16.0000	−0.654836
$598$	−8.00000	−0.327144
$599$	38.0000	1.55264	0.776319	−	0.630340i	$-0.217085\pi$
$599$	38.0000	1.55264	0.776319	+	0.630340i	$0.217085\pi$
$600$	−11.0000	−0.449073
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	−10.0000	−0.407570
$603$	−16.0000	−0.651570
$604$	−2.00000	−0.0813788
$605$	−44.0000	−1.78885
$606$	6.00000	0.243733
$607$	−31.0000	−1.25825	−0.629126	−	0.777304i	$-0.716587\pi$
$607$	−31.0000	−1.25825	−0.629126	+	0.777304i	$0.716587\pi$
$608$	3.00000	0.121666
$609$	1.00000	0.0405220
$610$	32.0000	1.29564
$611$	−26.0000	−1.05185
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−35.0000	−1.41249
$615$	−32.0000	−1.29036
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	−7.00000	−0.281581
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	−36.0000	−1.44579
$621$	20.0000	0.802572
$622$	7.00000	0.280674
$623$	8.00000	0.320513
$624$	2.00000	0.0800641
$625$	41.0000	1.64000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	−14.0000	−0.558661
$629$	0	0
$630$	8.00000	0.318728
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	6.00000	0.238667
$633$	26.0000	1.03341
$634$	11.0000	0.436866
$635$	−16.0000	−0.634941
$636$	−9.00000	−0.356873
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−4.00000	−0.158238
$640$	4.00000	0.158114
$641$	−13.0000	−0.513469	−0.256735	−	0.966482i	$-0.582647\pi$
$641$	−13.0000	−0.513469	−0.256735	+	0.966482i	$0.582647\pi$
$642$	−8.00000	−0.315735
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	−4.00000	−0.157622
$645$	−40.0000	−1.57500
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−22.0000	−0.862911
$651$	−9.00000	−0.352738
$652$	2.00000	0.0783260
$653$	−23.0000	−0.900060	−0.450030	−	0.893014i	$-0.648587\pi$
$653$	−23.0000	−0.900060	−0.450030	+	0.893014i	$0.648587\pi$
$654$	19.0000	0.742959
$655$	28.0000	1.09405
$656$	8.00000	0.312348
$657$	−4.00000	−0.156055
$658$	−13.0000	−0.506793
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−3.00000	−0.116423
$665$	−12.0000	−0.465340
$666$	20.0000	0.774984
$667$	4.00000	0.154881
$668$	−17.0000	−0.657750
$669$	9.00000	0.347960
$670$	32.0000	1.23627
$671$	0	0
$672$	1.00000	0.0385758
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	−5.00000	−0.192593
$675$	55.0000	2.11695
$676$	−9.00000	−0.346154
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−2.00000	−0.0768095
$679$	18.0000	0.690777
$680$	0	0
$681$	−11.0000	−0.421521
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−6.00000	−0.229416
$685$	68.0000	2.59815
$686$	−1.00000	−0.0381802
$687$	−26.0000	−0.991962
$688$	10.0000	0.381246
$689$	−18.0000	−0.685745
$690$	−16.0000	−0.609110
$691$	37.0000	1.40755	0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000	1.40755	0.703773	+	0.710425i	$0.251497\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	24.0000	0.911028
$695$	16.0000	0.606915
$696$	−1.00000	−0.0379049
$697$	0	0
$698$	4.00000	0.151402
$699$	25.0000	0.945587
$700$	−11.0000	−0.415761
$701$	−29.0000	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$702$	−10.0000	−0.377426
$703$	−30.0000	−1.13147
$704$	0	0
$705$	−52.0000	−1.95843
$706$	−24.0000	−0.903252
$707$	6.00000	0.225653
$708$	−5.00000	−0.187912
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	8.00000	0.300235
$711$	−12.0000	−0.450035
$712$	−8.00000	−0.299813
$713$	−36.0000	−1.34821
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−14.0000	−0.522840
$718$	4.00000	0.149279
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−8.00000	−0.298142
$721$	−7.00000	−0.260694
$722$	−10.0000	−0.372161
$723$	−20.0000	−0.743808
$724$	0	0
$725$	11.0000	0.408530
$726$	11.0000	0.408248
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	8.00000	0.296093
$731$	0	0
$732$	−8.00000	−0.295689
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	33.0000	1.21805
$735$	−4.00000	−0.147542
$736$	4.00000	0.147442
$737$	0	0
$738$	−16.0000	−0.588968
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	−40.0000	−1.47043
$741$	6.00000	0.220416
$742$	−9.00000	−0.330400
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	9.00000	0.329956
$745$	52.0000	1.90513
$746$	−11.0000	−0.402739
$747$	6.00000	0.219529
$748$	0	0
$749$	−8.00000	−0.292314
$750$	−24.0000	−0.876356
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	13.0000	0.474061
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	−8.00000	−0.291150
$756$	−5.00000	−0.181848
$757$	−33.0000	−1.19941	−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000	−1.19941	−0.599703	+	0.800223i	$0.704714\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	12.0000	0.435286
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	4.00000	0.144905
$763$	19.0000	0.687846
$764$	6.00000	0.217072
$765$	0	0
$766$	11.0000	0.397446
$767$	−10.0000	−0.361079
$768$	−1.00000	−0.0360844
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	−23.0000	−0.827788
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−20.0000	−0.718885
$775$	−99.0000	−3.55618
$776$	−18.0000	−0.646162
$777$	−10.0000	−0.358748
$778$	15.0000	0.537776
$779$	24.0000	0.859889
$780$	8.00000	0.286446
$781$	0	0
$782$	0	0
$783$	5.00000	0.178685
$784$	1.00000	0.0357143
$785$	−56.0000	−1.99873
$786$	−7.00000	−0.249682
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	3.00000	0.106871
$789$	−16.0000	−0.569615
$790$	24.0000	0.853882
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−16.0000	−0.568177
$794$	−8.00000	−0.283909
$795$	−36.0000	−1.27679
$796$	16.0000	0.567105
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	3.00000	0.106199
$799$	0	0
$800$	11.0000	0.388909
$801$	16.0000	0.565332
$802$	15.0000	0.529668
$803$	0	0
$804$	−8.00000	−0.282138
$805$	−16.0000	−0.563926
$806$	18.0000	0.634023
$807$	18.0000	0.633630
$808$	−6.00000	−0.211079
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	4.00000	0.140546
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	−1.00000	−0.0350931
$813$	9.00000	0.315644
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	30.0000	1.04957
$818$	8.00000	0.279713
$819$	−4.00000	−0.139771
$820$	32.0000	1.11749
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−17.0000	−0.592943
$823$	54.0000	1.88232	0.941161	−	0.337959i	$-0.109737\pi$
$823$	54.0000	1.88232	0.941161	+	0.337959i	$0.109737\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−5.00000	−0.173972
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−12.0000	−0.416526
$831$	17.0000	0.589723
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−68.0000	−2.35324
$836$	0	0
$837$	−45.0000	−1.55543
$838$	−4.00000	−0.138178
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	4.00000	0.138013
$841$	−28.0000	−0.965517
$842$	−15.0000	−0.516934
$843$	−2.00000	−0.0688837
$844$	−26.0000	−0.894957
$845$	−36.0000	−1.23844
$846$	−26.0000	−0.893898
$847$	11.0000	0.377964
$848$	9.00000	0.309061
$849$	−9.00000	−0.308879
$850$	0	0
$851$	−40.0000	−1.37118
$852$	−2.00000	−0.0685189
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	−8.00000	−0.273754
$855$	−24.0000	−0.820783
$856$	8.00000	0.273434
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	9.00000	0.307076	0.153538	−	0.988143i	$-0.450933\pi$
$859$	9.00000	0.307076	0.153538	+	0.988143i	$0.450933\pi$
$860$	40.0000	1.36399
$861$	8.00000	0.272639
$862$	−12.0000	−0.408722
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	5.00000	0.170103
$865$	−16.0000	−0.544016
$866$	−16.0000	−0.543702
$867$	0	0
$868$	9.00000	0.305480
$869$	0	0
$870$	−4.00000	−0.135613
$871$	−16.0000	−0.542139
$872$	−19.0000	−0.643421
$873$	36.0000	1.21842
$874$	12.0000	0.405906
$875$	−24.0000	−0.811348
$876$	−2.00000	−0.0675737
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	25.0000	0.843709
$879$	−12.0000	−0.404750
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	−2.00000	−0.0673435
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−20.0000	−0.672293
$886$	2.00000	0.0671913
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	10.0000	0.335578
$889$	4.00000	0.134156
$890$	−32.0000	−1.07264
$891$	0	0
$892$	−9.00000	−0.301342
$893$	39.0000	1.30509
$894$	−13.0000	−0.434785
$895$	−72.0000	−2.40669
$896$	−1.00000	−0.0334077
$897$	8.00000	0.267112
$898$	33.0000	1.10122
$899$	−9.00000	−0.300167
$900$	−22.0000	−0.733333
$901$	0	0
$902$	0	0
$903$	10.0000	0.332779
$904$	2.00000	0.0665190
$905$	0	0
$906$	2.00000	0.0664455
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	11.0000	0.365048
$909$	12.0000	0.398015
$910$	8.00000	0.265197
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	−3.00000	−0.0993399
$913$	0	0
$914$	19.0000	0.628464
$915$	−32.0000	−1.05789
$916$	26.0000	0.859064
$917$	−7.00000	−0.231160
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	16.0000	0.527504
$921$	35.0000	1.15329
$922$	42.0000	1.38320
$923$	−4.00000	−0.131662
$924$	0	0
$925$	−110.000	−3.61678
$926$	−40.0000	−1.31448
$927$	−14.0000	−0.459820
$928$	1.00000	0.0328266
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	36.0000	1.18049
$931$	3.00000	0.0983210
$932$	−25.0000	−0.818902
$933$	−7.00000	−0.229170
$934$	−28.0000	−0.916188
$935$	0	0
$936$	4.00000	0.130744
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	−8.00000	−0.261209
$939$	6.00000	0.195803
$940$	52.0000	1.69605
$941$	−16.0000	−0.521585	−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000	−0.521585	−0.260793	+	0.965395i	$0.583984\pi$
$942$	14.0000	0.456145
$943$	32.0000	1.04206
$944$	5.00000	0.162736
$945$	−20.0000	−0.650600
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−6.00000	−0.194871
$949$	−4.00000	−0.129845
$950$	33.0000	1.07066
$951$	−11.0000	−0.356699
$952$	0	0
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	−18.0000	−0.582772
$955$	24.0000	0.776622
$956$	14.0000	0.452792
$957$	0	0
$958$	4.00000	0.129234
$959$	−17.0000	−0.548959
$960$	−4.00000	−0.129099
$961$	50.0000	1.61290
$962$	20.0000	0.644826
$963$	−16.0000	−0.515593
$964$	20.0000	0.644157
$965$	−92.0000	−2.96158
$966$	4.00000	0.128698
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−72.0000	−2.31178
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−16.0000	−0.513200
$973$	−4.00000	−0.128234
$974$	−34.0000	−1.08943
$975$	22.0000	0.704564
$976$	8.00000	0.256074
$977$	−35.0000	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$977$	−35.0000	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$978$	−2.00000	−0.0639529
$979$	0	0
$980$	4.00000	0.127775
$981$	38.0000	1.21325
$982$	−18.0000	−0.574403
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	−8.00000	−0.255031
$985$	12.0000	0.382352
$986$	0	0
$987$	13.0000	0.413795
$988$	−6.00000	−0.190885
$989$	40.0000	1.27193
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−9.00000	−0.285750
$993$	12.0000	0.380808
$994$	−2.00000	−0.0634361
$995$	64.0000	2.02894
$996$	3.00000	0.0950586
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−30.0000	−0.949633
$999$	−50.0000	−1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4046.2.a.l.1.1	✓	1
17.16	even	2		4046.2.a.p.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4046.2.a.l.1.1	✓	1	1.1	even	1	trivial
4046.2.a.p.1.1	yes	1	17.16	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4046 = 2 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4046.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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