LMFDB - Embedded newform 4046.2.a.p.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	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	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	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	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	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−4.00000	−0.730297
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	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	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	3.00000	0.486664
$39$	−2.00000	−0.320256
$40$	−4.00000	−0.632456
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	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	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	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	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−2.00000	−0.235702
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	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	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−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	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	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	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−5.00000	−0.481125
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	1.00000	0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	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	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	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	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−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	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	−3.00000	−0.232845
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	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	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−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	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	18.0000	1.29232
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	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	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	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	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	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	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	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	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	−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	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	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	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−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	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	−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	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	−2.00000	−0.113228
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−14.0000	−0.790066
$315$	8.00000	0.450749
$316$	−6.00000	−0.337526
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	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	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	−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	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−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	$-0.830345\pi$
$367$	−33.0000	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$368$	−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	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−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	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−16.0000	−0.802008
$399$	3.00000	0.150188
$400$	11.0000	0.550000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	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	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	−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	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−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	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	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	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	−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	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	−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	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	−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	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	−24.0000	−1.07331
$501$	17.0000	0.759504
$502$	0	0
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	−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	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	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	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	−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	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	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	$-0.158330\pi$
$571$	42.0000	1.75765	0.878823	+	0.477149i	$0.158330\pi$
$572$	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	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	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	$-0.283413\pi$
$607$	31.0000	1.25825	0.629126	+	0.777304i	$0.283413\pi$
$608$	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	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	7.00000	0.281581
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−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	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	−8.00000	−0.315735
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	−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	$-0.351413\pi$
$653$	23.0000	0.900060	0.450030	+	0.893014i	$0.351413\pi$
$654$	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	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	5.00000	0.192593
$675$	−55.0000	−2.11695
$676$	−9.00000	−0.346154
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−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	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−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	$-0.748503\pi$
$691$	−37.0000	−1.40755	−0.703773	+	0.710425i	$0.748503\pi$
$692$	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	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	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	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	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	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	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	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	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	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−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	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	−4.00000	−0.140546
$811$	43.0000	1.50993	0.754967	−	0.655763i	$-0.227653\pi$
$811$	43.0000	1.50993	0.754967	+	0.655763i	$0.227653\pi$
$812$	−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	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	17.0000	0.592943
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	5.00000	0.173972
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	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	$-0.157302\pi$
$839$	51.0000	1.76072	0.880358	+	0.474310i	$0.157302\pi$
$840$	−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	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−8.00000	−0.273754
$855$	24.0000	0.820783
$856$	−8.00000	−0.273434
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	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	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−25.0000	−0.843709
$879$	12.0000	0.404750
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−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	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−11.0000	−0.365048
$909$	12.0000	0.398015
$910$	8.00000	0.265197
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	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	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−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	$-0.416016\pi$
$941$	16.0000	0.521585	0.260793	+	0.965395i	$0.416016\pi$
$942$	−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	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−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	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	−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	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	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	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	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.p.1.1	yes	1
17.16	even	2		4046.2.a.l.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4046.2.a.l.1.1	✓	1	17.16	even	2
4046.2.a.p.1.1	yes	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 $\F_{q}$
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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