LMFDB - Embedded newform 4046.2.a.b.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,-2,1,-4,2,-1,-1,1] 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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 238)
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)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +4.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +8.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} -4.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} +2.00000 q^{24} +11.0000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -8.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} +8.00000 q^{39} +4.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{44} -4.00000 q^{45} +4.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} -4.00000 q^{52} +14.0000 q^{53} -4.00000 q^{54} -16.0000 q^{55} +1.00000 q^{56} +12.0000 q^{57} +6.00000 q^{58} -6.00000 q^{59} +8.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +16.0000 q^{65} +8.00000 q^{66} +4.00000 q^{67} -4.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -10.0000 q^{74} -22.0000 q^{75} -6.00000 q^{76} -4.00000 q^{77} -8.00000 q^{78} -4.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +10.0000 q^{83} +2.00000 q^{84} +12.0000 q^{87} -4.00000 q^{88} +10.0000 q^{89} +4.00000 q^{90} +4.00000 q^{91} +8.00000 q^{93} -4.00000 q^{94} +24.0000 q^{95} +2.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	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	$0.852416\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	8.00000	2.06559
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−4.00000	−0.894427
$21$	2.00000	0.436436
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.00000	0.408248
$25$	11.0000	2.20000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−8.00000	−1.46059
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−8.00000	−1.39262
$34$	0	0
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	6.00000	0.973329
$39$	8.00000	1.28103
$40$	4.00000	0.632456
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−2.00000	−0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	−4.00000	−0.596285
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−4.00000	−0.554700
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−4.00000	−0.544331
$55$	−16.0000	−2.15744
$56$	1.00000	0.133631
$57$	12.0000	1.58944
$58$	6.00000	0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	8.00000	1.03280
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	16.0000	1.98456
$66$	8.00000	0.984732
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	−4.00000	−0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−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
$75$	−22.0000	−2.54034
$76$	−6.00000	−0.688247
$77$	−4.00000	−0.455842
$78$	−8.00000	−0.905822
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	0	0
$87$	12.0000	1.28654
$88$	−4.00000	−0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	4.00000	0.421637
$91$	4.00000	0.419314
$92$	0	0
$93$	8.00000	0.829561
$94$	−4.00000	−0.412568
$95$	24.0000	2.46235
$96$	2.00000	0.204124
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−1.00000	−0.101015
$99$	4.00000	0.402015
$100$	11.0000	1.10000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	4.00000	0.392232
$105$	−8.00000	−0.780720
$106$	−14.0000	−1.35980
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	4.00000	0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	16.0000	1.52554
$111$	−20.0000	−1.89832
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	0	0
$120$	−8.00000	−0.730297
$121$	5.00000	0.454545
$122$	−12.0000	−1.08643
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	−24.0000	−2.14663
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−16.0000	−1.40329
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−8.00000	−0.696311
$133$	6.00000	0.520266
$134$	−4.00000	−0.345547
$135$	−16.0000	−1.37706
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	4.00000	0.338062
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	24.0000	1.99309
$146$	2.00000	0.165521
$147$	−2.00000	−0.164957
$148$	10.0000	0.821995
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	22.0000	1.79629
$151$	−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
$154$	4.00000	0.322329
$155$	16.0000	1.28515
$156$	8.00000	0.640513
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	−28.0000	−2.22054
$160$	4.00000	0.316228
$161$	0	0
$162$	11.0000	0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.00000	−0.468521
$165$	32.0000	2.49120
$166$	−10.0000	−0.776151
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	−12.0000	−0.909718
$175$	−11.0000	−0.831522
$176$	4.00000	0.301511
$177$	12.0000	0.901975
$178$	−10.0000	−0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−4.00000	−0.298142
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−4.00000	−0.296500
$183$	−24.0000	−1.77413
$184$	0	0
$185$	−40.0000	−2.94086
$186$	−8.00000	−0.586588
$187$	0	0
$188$	4.00000	0.291730
$189$	−4.00000	−0.290957
$190$	−24.0000	−1.74114
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−2.00000	−0.144338
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	6.00000	0.430775
$195$	−32.0000	−2.29157
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−4.00000	−0.284268
$199$	−28.0000	−1.98487	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−28.0000	−1.98487	−0.992434	+	0.122782i	$0.960818\pi$
$200$	−11.0000	−0.777817
$201$	−8.00000	−0.564276
$202$	−16.0000	−1.12576
$203$	6.00000	0.421117
$204$	0	0
$205$	24.0000	1.67623
$206$	4.00000	0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−24.0000	−1.66011
$210$	8.00000	0.552052
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	14.0000	0.961524
$213$	−16.0000	−1.09630
$214$	0	0
$215$	0	0
$216$	−4.00000	−0.272166
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	4.00000	0.270295
$220$	−16.0000	−1.07872
$221$	0	0
$222$	20.0000	1.34231
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	1.00000	0.0668153
$225$	11.0000	0.733333
$226$	14.0000	0.931266
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	12.0000	0.794719
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	6.00000	0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	4.00000	0.261488
$235$	−16.0000	−1.04372
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	8.00000	0.516398
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	10.0000	0.641500
$244$	12.0000	0.768221
$245$	−4.00000	−0.255551
$246$	−12.0000	−0.765092
$247$	24.0000	1.52708
$248$	4.00000	0.254000
$249$	−20.0000	−1.26745
$250$	24.0000	1.51789
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	16.0000	0.992278
$261$	−6.00000	−0.371391
$262$	−18.0000	−1.11204
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	8.00000	0.492366
$265$	−56.0000	−3.44005
$266$	−6.00000	−0.367884
$267$	−20.0000	−1.22398
$268$	4.00000	0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	16.0000	0.973729
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	−2.00000	−0.120824
$275$	44.0000	2.65330
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	10.0000	0.599760
$279$	−4.00000	−0.239474
$280$	−4.00000	−0.239046
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	8.00000	0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	8.00000	0.474713
$285$	−48.0000	−2.84327
$286$	16.0000	0.946100
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−24.0000	−1.40933
$291$	12.0000	0.703452
$292$	−2.00000	−0.117041
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	2.00000	0.116642
$295$	24.0000	1.39733
$296$	−10.0000	−0.581238
$297$	16.0000	0.928414
$298$	2.00000	0.115857
$299$	0	0
$300$	−22.0000	−1.27017
$301$	0	0
$302$	16.0000	0.920697
$303$	−32.0000	−1.83835
$304$	−6.00000	−0.344124
$305$	−48.0000	−2.74847
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	−4.00000	−0.227921
$309$	8.00000	0.455104
$310$	−16.0000	−0.908739
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−8.00000	−0.452911
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−4.00000	−0.225733
$315$	4.00000	0.225374
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	28.0000	1.57016
$319$	−24.0000	−1.34374
$320$	−4.00000	−0.223607
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	−44.0000	−2.44068
$326$	4.00000	0.221540
$327$	−4.00000	−0.221201
$328$	6.00000	0.331295
$329$	−4.00000	−0.220527
$330$	−32.0000	−1.76154
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	10.0000	0.548821
$333$	10.0000	0.547997
$334$	−20.0000	−1.09435
$335$	−16.0000	−0.874173
$336$	2.00000	0.109109
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−3.00000	−0.163178
$339$	28.0000	1.52075
$340$	0	0
$341$	−16.0000	−0.866449
$342$	6.00000	0.324443
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−8.00000	−0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	12.0000	0.643268
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	11.0000	0.587975
$351$	−16.0000	−0.854017
$352$	−4.00000	−0.213201
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	−12.0000	−0.637793
$355$	−32.0000	−1.69838
$356$	10.0000	0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	4.00000	0.210819
$361$	17.0000	0.894737
$362$	0	0
$363$	−10.0000	−0.524864
$364$	4.00000	0.209657
$365$	8.00000	0.418739
$366$	24.0000	1.25450
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	40.0000	2.07950
$371$	−14.0000	−0.726844
$372$	8.00000	0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	48.0000	2.47871
$376$	−4.00000	−0.206284
$377$	24.0000	1.23606
$378$	4.00000	0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	24.0000	1.23117
$381$	16.0000	0.819705
$382$	24.0000	1.22795
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	2.00000	0.102062
$385$	16.0000	0.815436
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−6.00000	−0.304604
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	32.0000	1.62038
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−36.0000	−1.81596
$394$	2.00000	0.100759
$395$	0	0
$396$	4.00000	0.201008
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	28.0000	1.40351
$399$	−12.0000	−0.600751
$400$	11.0000	0.550000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	8.00000	0.399004
$403$	16.0000	0.797017
$404$	16.0000	0.796030
$405$	44.0000	2.18638
$406$	−6.00000	−0.297775
$407$	40.0000	1.98273
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−24.0000	−1.18528
$411$	−4.00000	−0.197305
$412$	−4.00000	−0.197066
$413$	6.00000	0.295241
$414$	0	0
$415$	−40.0000	−1.96352
$416$	4.00000	0.196116
$417$	20.0000	0.979404
$418$	24.0000	1.17388
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−8.00000	−0.390360
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−8.00000	−0.389434
$423$	4.00000	0.194487
$424$	−14.0000	−0.679900
$425$	0	0
$426$	16.0000	0.775203
$427$	−12.0000	−0.580721
$428$	0	0
$429$	32.0000	1.54497
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	4.00000	0.192450
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−4.00000	−0.192006
$435$	−48.0000	−2.30142
$436$	2.00000	0.0957826
$437$	0	0
$438$	−4.00000	−0.191127
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	16.0000	0.762770
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−20.0000	−0.949158
$445$	−40.0000	−1.89618
$446$	−24.0000	−1.13643
$447$	4.00000	0.189194
$448$	−1.00000	−0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−11.0000	−0.518545
$451$	−24.0000	−1.13012
$452$	−14.0000	−0.658505
$453$	32.0000	1.50349
$454$	−2.00000	−0.0938647
$455$	−16.0000	−0.750092
$456$	−12.0000	−0.561951
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	28.0000	1.30835
$459$	0	0
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	−8.00000	−0.372194
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−6.00000	−0.278543
$465$	−32.0000	−1.48396
$466$	26.0000	1.20443
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	−4.00000	−0.184900
$469$	−4.00000	−0.184703
$470$	16.0000	0.738025
$471$	−8.00000	−0.368621
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	−66.0000	−3.02829
$476$	0	0
$477$	14.0000	0.641016
$478$	8.00000	0.365911
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	−8.00000	−0.365148
$481$	−40.0000	−1.82384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	24.0000	1.08978
$486$	−10.0000	−0.453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−12.0000	−0.543214
$489$	8.00000	0.361773
$490$	4.00000	0.180702
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	−24.0000	−1.07981
$495$	−16.0000	−0.719147
$496$	−4.00000	−0.179605
$497$	−8.00000	−0.358849
$498$	20.0000	0.896221
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−24.0000	−1.07331
$501$	−40.0000	−1.78707
$502$	−14.0000	−0.624851
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	1.00000	0.0445435
$505$	−64.0000	−2.84796
$506$	0	0
$507$	−6.00000	−0.266469
$508$	−8.00000	−0.354943
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−1.00000	−0.0441942
$513$	−24.0000	−1.05963
$514$	6.00000	0.264649
$515$	16.0000	0.705044
$516$	0	0
$517$	16.0000	0.703679
$518$	10.0000	0.439375
$519$	−16.0000	−0.702322
$520$	−16.0000	−0.701646
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	6.00000	0.262613
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	18.0000	0.786334
$525$	22.0000	0.960159
$526$	−24.0000	−1.04645
$527$	0	0
$528$	−8.00000	−0.348155
$529$	−23.0000	−1.00000
$530$	56.0000	2.43248
$531$	−6.00000	−0.260378
$532$	6.00000	0.260133
$533$	24.0000	1.03956
$534$	20.0000	0.865485
$535$	0	0
$536$	−4.00000	−0.172774
$537$	24.0000	1.03568
$538$	−24.0000	−1.03471
$539$	4.00000	0.172292
$540$	−16.0000	−0.688530
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.00000	−0.342682
$546$	8.00000	0.342368
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	2.00000	0.0854358
$549$	12.0000	0.512148
$550$	−44.0000	−1.87617
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	80.0000	3.39581
$556$	−10.0000	−0.424094
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	6.00000	0.253095
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−8.00000	−0.336861
$565$	56.0000	2.35594
$566$	14.0000	0.588464
$567$	11.0000	0.461957
$568$	−8.00000	−0.335673
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	48.0000	2.01050
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	−16.0000	−0.668994
$573$	48.0000	2.00523
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	24.0000	0.996546
$581$	−10.0000	−0.414870
$582$	−12.0000	−0.497416
$583$	56.0000	2.31928
$584$	2.00000	0.0827606
$585$	16.0000	0.661519
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	−2.00000	−0.0824786
$589$	24.0000	0.988903
$590$	−24.0000	−0.988064
$591$	4.00000	0.164538
$592$	10.0000	0.410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	56.0000	2.29193
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	22.0000	0.898146
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−16.0000	−0.651031
$605$	−20.0000	−0.813116
$606$	32.0000	1.29991
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	6.00000	0.243332
$609$	−12.0000	−0.486265
$610$	48.0000	1.94346
$611$	−16.0000	−0.647291
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−10.0000	−0.403567
$615$	−48.0000	−1.93555
$616$	4.00000	0.161165
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	−8.00000	−0.321807
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	8.00000	0.320771
$623$	−10.0000	−0.400642
$624$	8.00000	0.320256
$625$	41.0000	1.64000
$626$	6.00000	0.239808
$627$	48.0000	1.91694
$628$	4.00000	0.159617
$629$	0	0
$630$	−4.00000	−0.159364
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	−22.0000	−0.873732
$635$	32.0000	1.26988
$636$	−28.0000	−1.11027
$637$	−4.00000	−0.158486
$638$	24.0000	0.950169
$639$	8.00000	0.316475
$640$	4.00000	0.158114
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	11.0000	0.432121
$649$	−24.0000	−0.942082
$650$	44.0000	1.72582
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	4.00000	0.156412
$655$	−72.0000	−2.81327
$656$	−6.00000	−0.234261
$657$	−2.00000	−0.0780274
$658$	4.00000	0.155936
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	32.0000	1.24560
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	0	0
$664$	−10.0000	−0.388075
$665$	−24.0000	−0.930680
$666$	−10.0000	−0.387492
$667$	0	0
$668$	20.0000	0.773823
$669$	−48.0000	−1.85579
$670$	16.0000	0.618134
$671$	48.0000	1.85302
$672$	−2.00000	−0.0771517
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	22.0000	0.847408
$675$	44.0000	1.69356
$676$	3.00000	0.115385
$677$	−16.0000	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$678$	−28.0000	−1.07533
$679$	6.00000	0.230259
$680$	0	0
$681$	−4.00000	−0.153280
$682$	16.0000	0.612672
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	−6.00000	−0.229416
$685$	−8.00000	−0.305664
$686$	1.00000	0.0381802
$687$	56.0000	2.13653
$688$	0	0
$689$	−56.0000	−2.13343
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	8.00000	0.304114
$693$	−4.00000	−0.151947
$694$	12.0000	0.455514
$695$	40.0000	1.51729
$696$	−12.0000	−0.454859
$697$	0	0
$698$	−20.0000	−0.757011
$699$	52.0000	1.96682
$700$	−11.0000	−0.415761
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	16.0000	0.603881
$703$	−60.0000	−2.26294
$704$	4.00000	0.150756
$705$	32.0000	1.20519
$706$	−26.0000	−0.978523
$707$	−16.0000	−0.601742
$708$	12.0000	0.450988
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	32.0000	1.20094
$711$	0	0
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	64.0000	2.39346
$716$	−12.0000	−0.448461
$717$	16.0000	0.597531
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−4.00000	−0.149071
$721$	4.00000	0.148968
$722$	−17.0000	−0.632674
$723$	−4.00000	−0.148762
$724$	0	0
$725$	−66.0000	−2.45118
$726$	10.0000	0.371135
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	−8.00000	−0.296093
$731$	0	0
$732$	−24.0000	−0.887066
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−24.0000	−0.885856
$735$	8.00000	0.295084
$736$	0	0
$737$	16.0000	0.589368
$738$	6.00000	0.220863
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−40.0000	−1.47043
$741$	−48.0000	−1.76332
$742$	14.0000	0.513956
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	−8.00000	−0.293294
$745$	8.00000	0.293097
$746$	10.0000	0.366126
$747$	10.0000	0.365881
$748$	0	0
$749$	0	0
$750$	−48.0000	−1.75271
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	4.00000	0.145865
$753$	−28.0000	−1.02038
$754$	−24.0000	−0.874028
$755$	64.0000	2.32920
$756$	−4.00000	−0.145479
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−24.0000	−0.870572
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−16.0000	−0.579619
$763$	−2.00000	−0.0724049
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−4.00000	−0.144526
$767$	24.0000	0.866590
$768$	−2.00000	−0.0721688
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−16.0000	−0.576600
$771$	12.0000	0.432169
$772$	2.00000	0.0719816
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	0	0
$775$	−44.0000	−1.58053
$776$	6.00000	0.215387
$777$	20.0000	0.717496
$778$	30.0000	1.07555
$779$	36.0000	1.28983
$780$	−32.0000	−1.14578
$781$	32.0000	1.14505
$782$	0	0
$783$	−24.0000	−0.857690
$784$	1.00000	0.0357143
$785$	−16.0000	−0.571064
$786$	36.0000	1.28408
$787$	10.0000	0.356462	0.178231	−	0.983989i	$-0.442963\pi$
$787$	10.0000	0.356462	0.178231	+	0.983989i	$0.442963\pi$
$788$	−2.00000	−0.0712470
$789$	−48.0000	−1.70885
$790$	0	0
$791$	14.0000	0.497783
$792$	−4.00000	−0.142134
$793$	−48.0000	−1.70453
$794$	−16.0000	−0.567819
$795$	112.000	3.97223
$796$	−28.0000	−0.992434
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	12.0000	0.424795
$799$	0	0
$800$	−11.0000	−0.388909
$801$	10.0000	0.353333
$802$	−18.0000	−0.635602
$803$	−8.00000	−0.282314
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−16.0000	−0.563576
$807$	−48.0000	−1.68968
$808$	−16.0000	−0.562878
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−44.0000	−1.54600
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−40.0000	−1.40200
$815$	16.0000	0.560456
$816$	0	0
$817$	0	0
$818$	26.0000	0.909069
$819$	4.00000	0.139771
$820$	24.0000	0.838116
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	4.00000	0.139516
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	4.00000	0.139347
$825$	−88.0000	−3.06377
$826$	−6.00000	−0.208767
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	40.0000	1.38842
$831$	36.0000	1.24883
$832$	−4.00000	−0.138675
$833$	0	0
$834$	−20.0000	−0.692543
$835$	−80.0000	−2.76851
$836$	−24.0000	−0.830057
$837$	−16.0000	−0.553041
$838$	18.0000	0.621800
$839$	−52.0000	−1.79524	−0.897620	−	0.440771i	$-0.854705\pi$
$839$	−52.0000	−1.79524	−0.897620	+	0.440771i	$0.854705\pi$
$840$	8.00000	0.276026
$841$	7.00000	0.241379
$842$	18.0000	0.620321
$843$	12.0000	0.413302
$844$	8.00000	0.275371
$845$	−12.0000	−0.412813
$846$	−4.00000	−0.137523
$847$	−5.00000	−0.171802
$848$	14.0000	0.480762
$849$	28.0000	0.960958
$850$	0	0
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	12.0000	0.410632
$855$	24.0000	0.820783
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	−32.0000	−1.09246
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	−8.00000	−0.272481
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	−4.00000	−0.136083
$865$	−32.0000	−1.08803
$866$	34.0000	1.15537
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	48.0000	1.62735
$871$	−16.0000	−0.542139
$872$	−2.00000	−0.0677285
$873$	−6.00000	−0.203069
$874$	0	0
$875$	24.0000	0.811348
$876$	4.00000	0.135147
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	40.0000	1.34993
$879$	0	0
$880$	−16.0000	−0.539360
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−1.00000	−0.0336718
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	−48.0000	−1.61350
$886$	−12.0000	−0.403148
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	20.0000	0.671156
$889$	8.00000	0.268311
$890$	40.0000	1.34080
$891$	−44.0000	−1.47406
$892$	24.0000	0.803579
$893$	−24.0000	−0.803129
$894$	−4.00000	−0.133780
$895$	48.0000	1.60446
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	24.0000	0.800445
$900$	11.0000	0.366667
$901$	0	0
$902$	24.0000	0.799113
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	−32.0000	−1.06313
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	2.00000	0.0663723
$909$	16.0000	0.530687
$910$	16.0000	0.530395
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	12.0000	0.397360
$913$	40.0000	1.32381
$914$	−22.0000	−0.727695
$915$	96.0000	3.17366
$916$	−28.0000	−0.925146
$917$	−18.0000	−0.594412
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	20.0000	0.658665
$923$	−32.0000	−1.05329
$924$	8.00000	0.263181
$925$	110.000	3.61678
$926$	40.0000	1.31448
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	32.0000	1.04932
$931$	−6.00000	−0.196642
$932$	−26.0000	−0.851658
$933$	16.0000	0.523816
$934$	−18.0000	−0.588978
$935$	0	0
$936$	4.00000	0.130744
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	4.00000	0.130605
$939$	12.0000	0.391605
$940$	−16.0000	−0.521862
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	8.00000	0.260654
$943$	0	0
$944$	−6.00000	−0.195283
$945$	16.0000	0.520480
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	66.0000	2.14132
$951$	−44.0000	−1.42680
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−14.0000	−0.453267
$955$	96.0000	3.10649
$956$	−8.00000	−0.258738
$957$	48.0000	1.55162
$958$	4.00000	0.129234
$959$	−2.00000	−0.0645834
$960$	8.00000	0.258199
$961$	−15.0000	−0.483871
$962$	40.0000	1.28965
$963$	0	0
$964$	2.00000	0.0644157
$965$	−8.00000	−0.257529
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−24.0000	−0.770594
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	10.0000	0.320750
$973$	10.0000	0.320585
$974$	−32.0000	−1.02535
$975$	88.0000	2.81826
$976$	12.0000	0.384111
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	−8.00000	−0.255812
$979$	40.0000	1.27841
$980$	−4.00000	−0.127775
$981$	2.00000	0.0638551
$982$	28.0000	0.893516
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	−12.0000	−0.382546
$985$	8.00000	0.254901
$986$	0	0
$987$	8.00000	0.254643
$988$	24.0000	0.763542
$989$	0	0
$990$	16.0000	0.508513
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	8.00000	0.253745
$995$	112.000	3.55064
$996$	−20.0000	−0.633724
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−24.0000	−0.759707
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4046.2.a.b.1.1		1
17.16	even	2		238.2.a.b.1.1	✓	1
51.50	odd	2		2142.2.a.l.1.1		1
68.67	odd	2		1904.2.a.b.1.1		1
85.84	even	2		5950.2.a.k.1.1		1
119.118	odd	2		1666.2.a.b.1.1		1
136.67	odd	2		7616.2.a.i.1.1		1
136.101	even	2		7616.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.b.1.1	✓	1	17.16	even	2
1666.2.a.b.1.1		1	119.118	odd	2
1904.2.a.b.1.1		1	68.67	odd	2
2142.2.a.l.1.1		1	51.50	odd	2
4046.2.a.b.1.1		1	1.1	even	1	trivial
5950.2.a.k.1.1		1	85.84	even	2
7616.2.a.a.1.1		1	136.101	even	2
7616.2.a.i.1.1		1	136.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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

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