LMFDB - Embedded newform 4998.2.a.cp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4998,2,Mod(1,4998)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4998, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4998.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4998.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,4,4,2,4,0,4,4,2,10,4,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.9092309302$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.87996$ of defining polynomial
Character	$\chi$	$=$	4998.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.46575 q^{10} +2.22129 q^{11} +1.00000 q^{12} -2.29417 q^{13} +1.46575 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.07288 q^{19} +1.46575 q^{20} +2.22129 q^{22} +2.65867 q^{23} +1.00000 q^{24} -2.85158 q^{25} -2.29417 q^{26} +1.00000 q^{27} +8.90131 q^{29} +1.46575 q^{30} +2.07288 q^{31} +1.00000 q^{32} +2.22129 q^{33} +1.00000 q^{34} +1.00000 q^{36} +0.364492 q^{37} -4.07288 q^{38} -2.29417 q^{39} +1.46575 q^{40} +7.83280 q^{41} -4.91271 q^{43} +2.22129 q^{44} +1.46575 q^{45} +2.65867 q^{46} +10.2186 q^{47} +1.00000 q^{48} -2.85158 q^{50} +1.00000 q^{51} -2.29417 q^{52} +4.09166 q^{53} +1.00000 q^{54} +3.25586 q^{55} -4.07288 q^{57} +8.90131 q^{58} +6.16976 q^{59} +1.46575 q^{60} +2.38584 q^{61} +2.07288 q^{62} +1.00000 q^{64} -3.36268 q^{65} +2.22129 q^{66} +1.28980 q^{67} +1.00000 q^{68} +2.65867 q^{69} +9.17414 q^{71} +1.00000 q^{72} +0.191107 q^{73} +0.364492 q^{74} -2.85158 q^{75} -4.07288 q^{76} -2.29417 q^{78} +8.99297 q^{79} +1.46575 q^{80} +1.00000 q^{81} +7.83280 q^{82} -16.2967 q^{83} +1.46575 q^{85} -4.91271 q^{86} +8.90131 q^{87} +2.22129 q^{88} -14.7831 q^{89} +1.46575 q^{90} +2.65867 q^{92} +2.07288 q^{93} +10.2186 q^{94} -5.96981 q^{95} +1.00000 q^{96} -12.5430 q^{97} +2.22129 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 10 q^{11} + 4 q^{12} + 6 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 2 q^{20} + 10 q^{22} + 4 q^{24} + 6 q^{25}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 4 * q^8 + 4 * q^9 + 2 * q^10 + 10 * q^11 + 4 * q^12 + 6 * q^13 + 2 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 + 2 * q^20 + 10 * q^22 + 4 * q^24 + 6 * q^25 + 6 * q^26 + 4 * q^27 + 8 * q^29 + 2 * q^30 - 8 * q^31 + 4 * q^32 + 10 * q^33 + 4 * q^34 + 4 * q^36 + 6 * q^37 + 6 * q^39 + 2 * q^40 - 4 * q^41 + 6 * q^43 + 10 * q^44 + 2 * q^45 - 8 * q^47 + 4 * q^48 + 6 * q^50 + 4 * q^51 + 6 * q^52 + 18 * q^53 + 4 * q^54 + 10 * q^55 + 8 * q^58 + 24 * q^59 + 2 * q^60 - 4 * q^61 - 8 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^66 + 14 * q^67 + 4 * q^68 + 12 * q^71 + 4 * q^72 - 18 * q^73 + 6 * q^74 + 6 * q^75 + 6 * q^78 + 10 * q^79 + 2 * q^80 + 4 * q^81 - 4 * q^82 - 14 * q^83 + 2 * q^85 + 6 * q^86 + 8 * q^87 + 10 * q^88 - 34 * q^89 + 2 * q^90 - 8 * q^93 - 8 * q^94 - 4 * q^95 + 4 * q^96 - 6 * q^97 + 10 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.46575	0.655502	0.327751	−	0.944764i	$-0.393709\pi$
$5$	1.46575	0.655502	0.327751	+	0.944764i	$0.393709\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.46575	0.463510
$11$	2.22129	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$11$	2.22129	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$12$	1.00000	0.288675
$13$	−2.29417	−0.636289	−0.318145	−	0.948042i	$-0.603060\pi$
$13$	−2.29417	−0.636289	−0.318145	+	0.948042i	$0.603060\pi$
$14$	0	0
$15$	1.46575	0.378454
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	−4.07288	−0.934383	−0.467191	−	0.884156i	$-0.654734\pi$
$19$	−4.07288	−0.934383	−0.467191	+	0.884156i	$0.654734\pi$
$20$	1.46575	0.327751
$21$	0	0
$22$	2.22129	0.473582
$23$	2.65867	0.554370	0.277185	−	0.960817i	$-0.410598\pi$
$23$	2.65867	0.554370	0.277185	+	0.960817i	$0.410598\pi$
$24$	1.00000	0.204124
$25$	−2.85158	−0.570317
$26$	−2.29417	−0.449925
$27$	1.00000	0.192450
$28$	0	0
$29$	8.90131	1.65293	0.826466	−	0.562987i	$-0.190348\pi$
$29$	8.90131	1.65293	0.826466	+	0.562987i	$0.190348\pi$
$30$	1.46575	0.267608
$31$	2.07288	0.372300	0.186150	−	0.982521i	$-0.440399\pi$
$31$	2.07288	0.372300	0.186150	+	0.982521i	$0.440399\pi$
$32$	1.00000	0.176777
$33$	2.22129	0.386678
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	0.364492	0.0599221	0.0299610	−	0.999551i	$-0.490462\pi$
$37$	0.364492	0.0599221	0.0299610	+	0.999551i	$0.490462\pi$
$38$	−4.07288	−0.660708
$39$	−2.29417	−0.367362
$40$	1.46575	0.231755
$41$	7.83280	1.22328	0.611639	−	0.791137i	$-0.290511\pi$
$41$	7.83280	1.22328	0.611639	+	0.791137i	$0.290511\pi$
$42$	0	0
$43$	−4.91271	−0.749181	−0.374591	−	0.927190i	$-0.622217\pi$
$43$	−4.91271	−0.749181	−0.374591	+	0.927190i	$0.622217\pi$
$44$	2.22129	0.334873
$45$	1.46575	0.218501
$46$	2.65867	0.391999
$47$	10.2186	1.49054	0.745271	−	0.666762i	$-0.232320\pi$
$47$	10.2186	1.49054	0.745271	+	0.666762i	$0.232320\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−2.85158	−0.403275
$51$	1.00000	0.140028
$52$	−2.29417	−0.318145
$53$	4.09166	0.562033	0.281017	−	0.959703i	$-0.409328\pi$
$53$	4.09166	0.562033	0.281017	+	0.959703i	$0.409328\pi$
$54$	1.00000	0.136083
$55$	3.25586	0.439020
$56$	0	0
$57$	−4.07288	−0.539466
$58$	8.90131	1.16880
$59$	6.16976	0.803234	0.401617	−	0.915808i	$-0.368448\pi$
$59$	6.16976	0.803234	0.401617	+	0.915808i	$0.368448\pi$
$60$	1.46575	0.189227
$61$	2.38584	0.305475	0.152738	−	0.988267i	$-0.451191\pi$
$61$	2.38584	0.305475	0.152738	+	0.988267i	$0.451191\pi$
$62$	2.07288	0.263256
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.36268	−0.417089
$66$	2.22129	0.273422
$67$	1.28980	0.157574	0.0787871	−	0.996891i	$-0.474895\pi$
$67$	1.28980	0.157574	0.0787871	+	0.996891i	$0.474895\pi$
$68$	1.00000	0.121268
$69$	2.65867	0.320066
$70$	0	0
$71$	9.17414	1.08877	0.544385	−	0.838836i	$-0.316763\pi$
$71$	9.17414	1.08877	0.544385	+	0.838836i	$0.316763\pi$
$72$	1.00000	0.117851
$73$	0.191107	0.0223674	0.0111837	−	0.999937i	$-0.496440\pi$
$73$	0.191107	0.0223674	0.0111837	+	0.999937i	$0.496440\pi$
$74$	0.364492	0.0423713
$75$	−2.85158	−0.329273
$76$	−4.07288	−0.467191
$77$	0	0
$78$	−2.29417	−0.259764
$79$	8.99297	1.01179	0.505894	−	0.862596i	$-0.331163\pi$
$79$	8.99297	1.01179	0.505894	+	0.862596i	$0.331163\pi$
$80$	1.46575	0.163876
$81$	1.00000	0.111111
$82$	7.83280	0.864988
$83$	−16.2967	−1.78880	−0.894400	−	0.447269i	$-0.852397\pi$
$83$	−16.2967	−1.78880	−0.894400	+	0.447269i	$0.852397\pi$
$84$	0	0
$85$	1.46575	0.158983
$86$	−4.91271	−0.529751
$87$	8.90131	0.954320
$88$	2.22129	0.236791
$89$	−14.7831	−1.56700	−0.783502	−	0.621390i	$-0.786568\pi$
$89$	−14.7831	−1.56700	−0.783502	+	0.621390i	$0.786568\pi$
$90$	1.46575	0.154503
$91$	0	0
$92$	2.65867	0.277185
$93$	2.07288	0.214948
$94$	10.2186	1.05397
$95$	−5.96981	−0.612490
$96$	1.00000	0.102062
$97$	−12.5430	−1.27355	−0.636774	−	0.771050i	$-0.719732\pi$
$97$	−12.5430	−1.27355	−0.636774	+	0.771050i	$0.719732\pi$
$98$	0	0
$99$	2.22129	0.223249
$100$	−2.85158	−0.285158
$101$	0.925308	0.0920716	0.0460358	−	0.998940i	$-0.485341\pi$
$101$	0.925308	0.0920716	0.0460358	+	0.998940i	$0.485341\pi$
$102$	1.00000	0.0990148
$103$	−2.87996	−0.283771	−0.141885	−	0.989883i	$-0.545316\pi$
$103$	−2.87996	−0.283771	−0.141885	+	0.989883i	$0.545316\pi$
$104$	−2.29417	−0.224962
$105$	0	0
$106$	4.09166	0.397417
$107$	17.5198	1.69371	0.846854	−	0.531826i	$-0.178494\pi$
$107$	17.5198	1.69371	0.846854	+	0.531826i	$0.178494\pi$
$108$	1.00000	0.0962250
$109$	5.04450	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$109$	5.04450	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$110$	3.25586	0.310434
$111$	0.364492	0.0345960
$112$	0	0
$113$	7.29236	0.686008	0.343004	−	0.939334i	$-0.388556\pi$
$113$	7.29236	0.686008	0.343004	+	0.939334i	$0.388556\pi$
$114$	−4.07288	−0.381460
$115$	3.89693	0.363391
$116$	8.90131	0.826466
$117$	−2.29417	−0.212096
$118$	6.16976	0.567973
$119$	0	0
$120$	1.46575	0.133804
$121$	−6.06585	−0.551441
$122$	2.38584	0.216004
$123$	7.83280	0.706260
$124$	2.07288	0.186150
$125$	−11.5084	−1.02935
$126$	0	0
$127$	0.0567518	0.00503591	0.00251795	−	0.999997i	$-0.499199\pi$
$127$	0.0567518	0.00503591	0.00251795	+	0.999997i	$0.499199\pi$
$128$	1.00000	0.0883883
$129$	−4.91271	−0.432540
$130$	−3.36268	−0.294927
$131$	−7.07725	−0.618343	−0.309171	−	0.951006i	$-0.600052\pi$
$131$	−7.07725	−0.618343	−0.309171	+	0.951006i	$0.600052\pi$
$132$	2.22129	0.193339
$133$	0	0
$134$	1.28980	0.111422
$135$	1.46575	0.126151
$136$	1.00000	0.0857493
$137$	1.79011	0.152939	0.0764697	−	0.997072i	$-0.475635\pi$
$137$	1.79011	0.152939	0.0764697	+	0.997072i	$0.475635\pi$
$138$	2.65867	0.226321
$139$	−7.77508	−0.659474	−0.329737	−	0.944073i	$-0.606960\pi$
$139$	−7.77508	−0.659474	−0.329737	+	0.944073i	$0.606960\pi$
$140$	0	0
$141$	10.2186	0.860565
$142$	9.17414	0.769876
$143$	−5.09604	−0.426152
$144$	1.00000	0.0833333
$145$	13.0471	1.08350
$146$	0.191107	0.0158161
$147$	0	0
$148$	0.364492	0.0299610
$149$	−13.0283	−1.06732	−0.533659	−	0.845700i	$-0.679184\pi$
$149$	−13.0283	−1.06732	−0.533659	+	0.845700i	$0.679184\pi$
$150$	−2.85158	−0.232831
$151$	−14.1384	−1.15057	−0.575283	−	0.817955i	$-0.695108\pi$
$151$	−14.1384	−1.15057	−0.575283	+	0.817955i	$0.695108\pi$
$152$	−4.07288	−0.330354
$153$	1.00000	0.0808452
$154$	0	0
$155$	3.03832	0.244044
$156$	−2.29417	−0.183681
$157$	5.46309	0.436002	0.218001	−	0.975949i	$-0.430046\pi$
$157$	5.46309	0.436002	0.218001	+	0.975949i	$0.430046\pi$
$158$	8.99297	0.715442
$159$	4.09166	0.324490
$160$	1.46575	0.115877
$161$	0	0
$162$	1.00000	0.0785674
$163$	8.82586	0.691295	0.345648	−	0.938364i	$-0.387659\pi$
$163$	8.82586	0.691295	0.345648	+	0.938364i	$0.387659\pi$
$164$	7.83280	0.611639
$165$	3.25586	0.253468
$166$	−16.2967	−1.26487
$167$	−4.35831	−0.337256	−0.168628	−	0.985680i	$-0.553934\pi$
$167$	−4.35831	−0.337256	−0.168628	+	0.985680i	$0.553934\pi$
$168$	0	0
$169$	−7.73676	−0.595136
$170$	1.46575	0.112418
$171$	−4.07288	−0.311461
$172$	−4.91271	−0.374591
$173$	−4.04632	−0.307636	−0.153818	−	0.988099i	$-0.549157\pi$
$173$	−4.04632	−0.307636	−0.153818	+	0.988099i	$0.549157\pi$
$174$	8.90131	0.674806
$175$	0	0
$176$	2.22129	0.167436
$177$	6.16976	0.463748
$178$	−14.7831	−1.10804
$179$	−19.0471	−1.42364	−0.711822	−	0.702360i	$-0.752130\pi$
$179$	−19.0471	−1.42364	−0.711822	+	0.702360i	$0.752130\pi$
$180$	1.46575	0.109250
$181$	−17.7952	−1.32271	−0.661355	−	0.750073i	$-0.730018\pi$
$181$	−17.7952	−1.32271	−0.661355	+	0.750073i	$0.730018\pi$
$182$	0	0
$183$	2.38584	0.176366
$184$	2.65867	0.195999
$185$	0.534253	0.0392790
$186$	2.07288	0.151991
$187$	2.22129	0.162437
$188$	10.2186	0.745271
$189$	0	0
$190$	−5.96981	−0.433096
$191$	3.43918	0.248851	0.124425	−	0.992229i	$-0.460291\pi$
$191$	3.43918	0.248851	0.124425	+	0.992229i	$0.460291\pi$
$192$	1.00000	0.0721688
$193$	16.4779	1.18611	0.593053	−	0.805164i	$-0.297923\pi$
$193$	16.4779	1.18611	0.593053	+	0.805164i	$0.297923\pi$
$194$	−12.5430	−0.900535
$195$	−3.36268	−0.240807
$196$	0	0
$197$	14.0353	0.999975	0.499987	−	0.866033i	$-0.333338\pi$
$197$	14.0353	0.999975	0.499987	+	0.866033i	$0.333338\pi$
$198$	2.22129	0.157861
$199$	9.38659	0.665398	0.332699	−	0.943033i	$-0.392041\pi$
$199$	9.38659	0.665398	0.332699	+	0.943033i	$0.392041\pi$
$200$	−2.85158	−0.201638
$201$	1.28980	0.0909755
$202$	0.925308	0.0651045
$203$	0	0
$204$	1.00000	0.0700140
$205$	11.4809	0.801861
$206$	−2.87996	−0.200656
$207$	2.65867	0.184790
$208$	−2.29417	−0.159072
$209$	−9.04707	−0.625799
$210$	0	0
$211$	−21.3163	−1.46747	−0.733736	−	0.679434i	$-0.762225\pi$
$211$	−21.3163	−1.46747	−0.733736	+	0.679434i	$0.762225\pi$
$212$	4.09166	0.281017
$213$	9.17414	0.628601
$214$	17.5198	1.19763
$215$	−7.20079	−0.491090
$216$	1.00000	0.0680414
$217$	0	0
$218$	5.04450	0.341657
$219$	0.191107	0.0129138
$220$	3.25586	0.219510
$221$	−2.29417	−0.154323
$222$	0.364492	0.0244631
$223$	−18.1181	−1.21328	−0.606640	−	0.794977i	$-0.707483\pi$
$223$	−18.1181	−1.21328	−0.606640	+	0.794977i	$0.707483\pi$
$224$	0	0
$225$	−2.85158	−0.190106
$226$	7.29236	0.485081
$227$	12.9706	0.860886	0.430443	−	0.902618i	$-0.358357\pi$
$227$	12.9706	0.860886	0.430443	+	0.902618i	$0.358357\pi$
$228$	−4.07288	−0.269733
$229$	−5.81402	−0.384201	−0.192101	−	0.981375i	$-0.561530\pi$
$229$	−5.81402	−0.384201	−0.192101	+	0.981375i	$0.561530\pi$
$230$	3.89693	0.256956
$231$	0	0
$232$	8.90131	0.584399
$233$	1.25329	0.0821060	0.0410530	−	0.999157i	$-0.486929\pi$
$233$	1.25329	0.0821060	0.0410530	+	0.999157i	$0.486929\pi$
$234$	−2.29417	−0.149975
$235$	14.9779	0.977053
$236$	6.16976	0.401617
$237$	8.99297	0.584156
$238$	0	0
$239$	13.0960	0.847112	0.423556	−	0.905870i	$-0.360782\pi$
$239$	13.0960	0.847112	0.423556	+	0.905870i	$0.360782\pi$
$240$	1.46575	0.0946136
$241$	15.6193	1.00613	0.503063	−	0.864250i	$-0.332206\pi$
$241$	15.6193	1.00613	0.503063	+	0.864250i	$0.332206\pi$
$242$	−6.06585	−0.389928
$243$	1.00000	0.0641500
$244$	2.38584	0.152738
$245$	0	0
$246$	7.83280	0.499401
$247$	9.34390	0.594538
$248$	2.07288	0.131628
$249$	−16.2967	−1.03276
$250$	−11.5084	−0.727858
$251$	−20.2967	−1.28112	−0.640559	−	0.767909i	$-0.721297\pi$
$251$	−20.2967	−1.28112	−0.640559	+	0.767909i	$0.721297\pi$
$252$	0	0
$253$	5.90568	0.371287
$254$	0.0567518	0.00356092
$255$	1.46575	0.0917887
$256$	1.00000	0.0625000
$257$	−11.7058	−0.730189	−0.365095	−	0.930970i	$-0.618963\pi$
$257$	−11.7058	−0.730189	−0.365095	+	0.930970i	$0.618963\pi$
$258$	−4.91271	−0.305852
$259$	0	0
$260$	−3.36268	−0.208545
$261$	8.90131	0.550977
$262$	−7.07725	−0.437234
$263$	5.01578	0.309286	0.154643	−	0.987970i	$-0.450577\pi$
$263$	5.01578	0.309286	0.154643	+	0.987970i	$0.450577\pi$
$264$	2.22129	0.136711
$265$	5.99734	0.368414
$266$	0	0
$267$	−14.7831	−0.904710
$268$	1.28980	0.0787871
$269$	11.9057	0.725902	0.362951	−	0.931808i	$-0.381769\pi$
$269$	11.9057	0.725902	0.362951	+	0.931808i	$0.381769\pi$
$270$	1.46575	0.0892025
$271$	12.4941	0.758964	0.379482	−	0.925199i	$-0.376102\pi$
$271$	12.4941	0.758964	0.379482	+	0.925199i	$0.376102\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	1.79011	0.108144
$275$	−6.33421	−0.381967
$276$	2.65867	0.160033
$277$	7.58654	0.455831	0.227915	−	0.973681i	$-0.426809\pi$
$277$	7.58654	0.455831	0.227915	+	0.973681i	$0.426809\pi$
$278$	−7.77508	−0.466318
$279$	2.07288	0.124100
$280$	0	0
$281$	3.99638	0.238404	0.119202	−	0.992870i	$-0.461966\pi$
$281$	3.99638	0.238404	0.119202	+	0.992870i	$0.461966\pi$
$282$	10.2186	0.608511
$283$	−5.74476	−0.341491	−0.170745	−	0.985315i	$-0.554618\pi$
$283$	−5.74476	−0.341491	−0.170745	+	0.985315i	$0.554618\pi$
$284$	9.17414	0.544385
$285$	−5.96981	−0.353621
$286$	−5.09604	−0.301335
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	13.0471	0.766150
$291$	−12.5430	−0.735284
$292$	0.191107	0.0111837
$293$	−30.4805	−1.78069	−0.890344	−	0.455289i	$-0.849536\pi$
$293$	−30.4805	−1.78069	−0.890344	+	0.455289i	$0.849536\pi$
$294$	0	0
$295$	9.04331	0.526522
$296$	0.364492	0.0211857
$297$	2.22129	0.128893
$298$	−13.0283	−0.754708
$299$	−6.09944	−0.352740
$300$	−2.85158	−0.164636
$301$	0	0
$302$	−14.1384	−0.813572
$303$	0.925308	0.0531576
$304$	−4.07288	−0.233596
$305$	3.49704	0.200240
$306$	1.00000	0.0571662
$307$	1.13763	0.0649279	0.0324640	−	0.999473i	$-0.489665\pi$
$307$	1.13763	0.0649279	0.0324640	+	0.999473i	$0.489665\pi$
$308$	0	0
$309$	−2.87996	−0.163835
$310$	3.03832	0.172565
$311$	−9.39287	−0.532621	−0.266310	−	0.963887i	$-0.585805\pi$
$311$	−9.39287	−0.532621	−0.266310	+	0.963887i	$0.585805\pi$
$312$	−2.29417	−0.129882
$313$	2.98825	0.168906	0.0844528	−	0.996427i	$-0.473086\pi$
$313$	2.98825	0.168906	0.0844528	+	0.996427i	$0.473086\pi$
$314$	5.46309	0.308300
$315$	0	0
$316$	8.99297	0.505894
$317$	−10.6774	−0.599700	−0.299850	−	0.953986i	$-0.596937\pi$
$317$	−10.6774	−0.599700	−0.299850	+	0.953986i	$0.596937\pi$
$318$	4.09166	0.229449
$319$	19.7724	1.10704
$320$	1.46575	0.0819378
$321$	17.5198	0.977862
$322$	0	0
$323$	−4.07288	−0.226621
$324$	1.00000	0.0555556
$325$	6.54203	0.362887
$326$	8.82586	0.488819
$327$	5.04450	0.278962
$328$	7.83280	0.432494
$329$	0	0
$330$	3.25586	0.179229
$331$	6.12698	0.336769	0.168385	−	0.985721i	$-0.446145\pi$
$331$	6.12698	0.336769	0.168385	+	0.985721i	$0.446145\pi$
$332$	−16.2967	−0.894400
$333$	0.364492	0.0199740
$334$	−4.35831	−0.238476
$335$	1.89052	0.103290
$336$	0	0
$337$	−25.8114	−1.40603	−0.703017	−	0.711173i	$-0.748164\pi$
$337$	−25.8114	−1.40603	−0.703017	+	0.711173i	$0.748164\pi$
$338$	−7.73676	−0.420824
$339$	7.29236	0.396067
$340$	1.46575	0.0794913
$341$	4.60448	0.249346
$342$	−4.07288	−0.220236
$343$	0	0
$344$	−4.91271	−0.264876
$345$	3.89693	0.209804
$346$	−4.04632	−0.217531
$347$	20.6726	1.10977	0.554883	−	0.831929i	$-0.312763\pi$
$347$	20.6726	1.10977	0.554883	+	0.831929i	$0.312763\pi$
$348$	8.90131	0.477160
$349$	−16.9715	−0.908465	−0.454233	−	0.890883i	$-0.650086\pi$
$349$	−16.9715	−0.908465	−0.454233	+	0.890883i	$0.650086\pi$
$350$	0	0
$351$	−2.29417	−0.122454
$352$	2.22129	0.118395
$353$	33.5136	1.78375	0.891874	−	0.452285i	$-0.149391\pi$
$353$	33.5136	1.78375	0.891874	+	0.452285i	$0.149391\pi$
$354$	6.16976	0.327919
$355$	13.4470	0.713691
$356$	−14.7831	−0.783502
$357$	0	0
$358$	−19.0471	−1.00667
$359$	1.60373	0.0846414	0.0423207	−	0.999104i	$-0.486525\pi$
$359$	1.60373	0.0846414	0.0423207	+	0.999104i	$0.486525\pi$
$360$	1.46575	0.0772517
$361$	−2.41165	−0.126929
$362$	−17.7952	−0.935297
$363$	−6.06585	−0.318375
$364$	0	0
$365$	0.280115	0.0146619
$366$	2.38584	0.124710
$367$	4.97056	0.259461	0.129731	−	0.991549i	$-0.458589\pi$
$367$	4.97056	0.259461	0.129731	+	0.991549i	$0.458589\pi$
$368$	2.65867	0.138593
$369$	7.83280	0.407759
$370$	0.534253	0.0277745
$371$	0	0
$372$	2.07288	0.107474
$373$	−6.07650	−0.314629	−0.157315	−	0.987549i	$-0.550284\pi$
$373$	−6.07650	−0.314629	−0.157315	+	0.987549i	$0.550284\pi$
$374$	2.22129	0.114860
$375$	−11.5084	−0.594293
$376$	10.2186	0.526986
$377$	−20.4212	−1.05174
$378$	0	0
$379$	−29.8016	−1.53080	−0.765401	−	0.643553i	$-0.777460\pi$
$379$	−29.8016	−1.53080	−0.765401	+	0.643553i	$0.777460\pi$
$380$	−5.96981	−0.306245
$381$	0.0567518	0.00290748
$382$	3.43918	0.175964
$383$	−23.1841	−1.18465	−0.592325	−	0.805699i	$-0.701790\pi$
$383$	−23.1841	−1.18465	−0.592325	+	0.805699i	$0.701790\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	16.4779	0.838703
$387$	−4.91271	−0.249727
$388$	−12.5430	−0.636774
$389$	0.837176	0.0424465	0.0212232	−	0.999775i	$-0.493244\pi$
$389$	0.837176	0.0424465	0.0212232	+	0.999775i	$0.493244\pi$
$390$	−3.36268	−0.170276
$391$	2.65867	0.134455
$392$	0	0
$393$	−7.07725	−0.357000
$394$	14.0353	0.707089
$395$	13.1814	0.663229
$396$	2.22129	0.111624
$397$	2.88880	0.144985	0.0724924	−	0.997369i	$-0.476905\pi$
$397$	2.88880	0.144985	0.0724924	+	0.997369i	$0.476905\pi$
$398$	9.38659	0.470507
$399$	0	0
$400$	−2.85158	−0.142579
$401$	−20.3119	−1.01433	−0.507164	−	0.861850i	$-0.669306\pi$
$401$	−20.3119	−1.01433	−0.507164	+	0.861850i	$0.669306\pi$
$402$	1.28980	0.0643294
$403$	−4.75555	−0.236891
$404$	0.925308	0.0460358
$405$	1.46575	0.0728336
$406$	0	0
$407$	0.809644	0.0401325
$408$	1.00000	0.0495074
$409$	−10.7367	−0.530894	−0.265447	−	0.964125i	$-0.585520\pi$
$409$	−10.7367	−0.530894	−0.265447	+	0.964125i	$0.585520\pi$
$410$	11.4809	0.567002
$411$	1.79011	0.0882996
$412$	−2.87996	−0.141885
$413$	0	0
$414$	2.65867	0.130666
$415$	−23.8869	−1.17256
$416$	−2.29417	−0.112481
$417$	−7.77508	−0.380747
$418$	−9.04707	−0.442506
$419$	−5.90431	−0.288445	−0.144222	−	0.989545i	$-0.546068\pi$
$419$	−5.90431	−0.288445	−0.144222	+	0.989545i	$0.546068\pi$
$420$	0	0
$421$	−30.6700	−1.49476	−0.747382	−	0.664395i	$-0.768689\pi$
$421$	−30.6700	−1.49476	−0.747382	+	0.664395i	$0.768689\pi$
$422$	−21.3163	−1.03766
$423$	10.2186	0.496847
$424$	4.09166	0.198709
$425$	−2.85158	−0.138322
$426$	9.17414	0.444488
$427$	0	0
$428$	17.5198	0.846854
$429$	−5.09604	−0.246039
$430$	−7.20079	−0.347253
$431$	24.7262	1.19102	0.595510	−	0.803348i	$-0.296950\pi$
$431$	24.7262	1.19102	0.595510	+	0.803348i	$0.296950\pi$
$432$	1.00000	0.0481125
$433$	18.8397	0.905380	0.452690	−	0.891668i	$-0.350464\pi$
$433$	18.8397	0.905380	0.452690	+	0.891668i	$0.350464\pi$
$434$	0	0
$435$	13.0471	0.625559
$436$	5.04450	0.241588
$437$	−10.8284	−0.517994
$438$	0.191107	0.00913144
$439$	4.75555	0.226970	0.113485	−	0.993540i	$-0.463799\pi$
$439$	4.75555	0.226970	0.113485	+	0.993540i	$0.463799\pi$
$440$	3.25586	0.155217
$441$	0	0
$442$	−2.29417	−0.109123
$443$	21.3386	1.01383	0.506913	−	0.861997i	$-0.330786\pi$
$443$	21.3386	1.01383	0.506913	+	0.861997i	$0.330786\pi$
$444$	0.364492	0.0172980
$445$	−21.6683	−1.02717
$446$	−18.1181	−0.857919
$447$	−13.0283	−0.616217
$448$	0	0
$449$	1.11266	0.0525097	0.0262548	−	0.999655i	$-0.491642\pi$
$449$	1.11266	0.0525097	0.0262548	+	0.999655i	$0.491642\pi$
$450$	−2.85158	−0.134425
$451$	17.3990	0.819285
$452$	7.29236	0.343004
$453$	−14.1384	−0.664279
$454$	12.9706	0.608739
$455$	0	0
$456$	−4.07288	−0.190730
$457$	31.4568	1.47149	0.735744	−	0.677260i	$-0.236833\pi$
$457$	31.4568	1.47149	0.735744	+	0.677260i	$0.236833\pi$
$458$	−5.81402	−0.271671
$459$	1.00000	0.0466760
$460$	3.89693	0.181695
$461$	1.60060	0.0745472	0.0372736	−	0.999305i	$-0.488133\pi$
$461$	1.60060	0.0745472	0.0372736	+	0.999305i	$0.488133\pi$
$462$	0	0
$463$	−29.5553	−1.37355	−0.686775	−	0.726870i	$-0.740974\pi$
$463$	−29.5553	−1.37355	−0.686775	+	0.726870i	$0.740974\pi$
$464$	8.90131	0.413233
$465$	3.03832	0.140899
$466$	1.25329	0.0580577
$467$	−10.1470	−0.469545	−0.234772	−	0.972050i	$-0.575435\pi$
$467$	−10.1470	−0.469545	−0.234772	+	0.972050i	$0.575435\pi$
$468$	−2.29417	−0.106048
$469$	0	0
$470$	14.9779	0.690881
$471$	5.46309	0.251726
$472$	6.16976	0.283986
$473$	−10.9126	−0.501761
$474$	8.99297	0.413061
$475$	11.6142	0.532894
$476$	0	0
$477$	4.09166	0.187344
$478$	13.0960	0.598999
$479$	0.800895	0.0365938	0.0182969	−	0.999833i	$-0.494176\pi$
$479$	0.800895	0.0365938	0.0182969	+	0.999833i	$0.494176\pi$
$480$	1.46575	0.0669019
$481$	−0.836208	−0.0381278
$482$	15.6193	0.711439
$483$	0	0
$484$	−6.06585	−0.275720
$485$	−18.3849	−0.834814
$486$	1.00000	0.0453609
$487$	1.35018	0.0611823	0.0305912	−	0.999532i	$-0.490261\pi$
$487$	1.35018	0.0611823	0.0305912	+	0.999532i	$0.490261\pi$
$488$	2.38584	0.108002
$489$	8.82586	0.399119
$490$	0	0
$491$	−17.5360	−0.791387	−0.395694	−	0.918383i	$-0.629496\pi$
$491$	−17.5360	−0.791387	−0.395694	+	0.918383i	$0.629496\pi$
$492$	7.83280	0.353130
$493$	8.90131	0.400895
$494$	9.34390	0.420402
$495$	3.25586	0.146340
$496$	2.07288	0.0930750
$497$	0	0
$498$	−16.2967	−0.730274
$499$	44.2529	1.98103	0.990516	−	0.137399i	$-0.0438744\pi$
$499$	44.2529	1.98103	0.990516	+	0.137399i	$0.0438744\pi$
$500$	−11.5084	−0.514673
$501$	−4.35831	−0.194715
$502$	−20.2967	−0.905888
$503$	−36.3255	−1.61967	−0.809836	−	0.586656i	$-0.800444\pi$
$503$	−36.3255	−1.61967	−0.809836	+	0.586656i	$0.800444\pi$
$504$	0	0
$505$	1.35627	0.0603532
$506$	5.90568	0.262540
$507$	−7.73676	−0.343602
$508$	0.0567518	0.00251795
$509$	20.3531	0.902135	0.451067	−	0.892490i	$-0.351043\pi$
$509$	20.3531	0.902135	0.451067	+	0.892490i	$0.351043\pi$
$510$	1.46575	0.0649044
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.07288	−0.179822
$514$	−11.7058	−0.516322
$515$	−4.22129	−0.186012
$516$	−4.91271	−0.216270
$517$	22.6986	0.998284
$518$	0	0
$519$	−4.04632	−0.177614
$520$	−3.36268	−0.147463
$521$	−19.8328	−0.868891	−0.434445	−	0.900698i	$-0.643056\pi$
$521$	−19.8328	−0.868891	−0.434445	+	0.900698i	$0.643056\pi$
$522$	8.90131	0.389600
$523$	−5.90206	−0.258079	−0.129039	−	0.991639i	$-0.541189\pi$
$523$	−5.90206	−0.258079	−0.129039	+	0.991639i	$0.541189\pi$
$524$	−7.07725	−0.309171
$525$	0	0
$526$	5.01578	0.218698
$527$	2.07288	0.0902961
$528$	2.22129	0.0966694
$529$	−15.9315	−0.692674
$530$	5.99734	0.260508
$531$	6.16976	0.267745
$532$	0	0
$533$	−17.9698	−0.778359
$534$	−14.7831	−0.639726
$535$	25.6797	1.11023
$536$	1.28980	0.0557109
$537$	−19.0471	−0.821942
$538$	11.9057	0.513290
$539$	0	0
$540$	1.46575	0.0630757
$541$	−18.4639	−0.793827	−0.396913	−	0.917856i	$-0.629919\pi$
$541$	−18.4639	−0.793827	−0.396913	+	0.917856i	$0.629919\pi$
$542$	12.4941	0.536669
$543$	−17.7952	−0.763667
$544$	1.00000	0.0428746
$545$	7.39397	0.316723
$546$	0	0
$547$	−1.24701	−0.0533185	−0.0266593	−	0.999645i	$-0.508487\pi$
$547$	−1.24701	−0.0533185	−0.0266593	+	0.999645i	$0.508487\pi$
$548$	1.79011	0.0764697
$549$	2.38584	0.101825
$550$	−6.33421	−0.270092
$551$	−36.2540	−1.54447
$552$	2.65867	0.113160
$553$	0	0
$554$	7.58654	0.322321
$555$	0.534253	0.0226778
$556$	−7.77508	−0.329737
$557$	−35.3775	−1.49899	−0.749497	−	0.662008i	$-0.769705\pi$
$557$	−35.3775	−1.49899	−0.749497	+	0.662008i	$0.769705\pi$
$558$	2.07288	0.0877520
$559$	11.2706	0.476696
$560$	0	0
$561$	2.22129	0.0937831
$562$	3.99638	0.168577
$563$	30.7304	1.29513	0.647566	−	0.762010i	$-0.275787\pi$
$563$	30.7304	1.29513	0.647566	+	0.762010i	$0.275787\pi$
$564$	10.2186	0.430282
$565$	10.6888	0.449680
$566$	−5.74476	−0.241470
$567$	0	0
$568$	9.17414	0.384938
$569$	−34.4249	−1.44317	−0.721584	−	0.692327i	$-0.756585\pi$
$569$	−34.4249	−1.44317	−0.721584	+	0.692327i	$0.756585\pi$
$570$	−5.96981	−0.250048
$571$	35.6873	1.49347	0.746735	−	0.665122i	$-0.231621\pi$
$571$	35.6873	1.49347	0.746735	+	0.665122i	$0.231621\pi$
$572$	−5.09604	−0.213076
$573$	3.43918	0.143674
$574$	0	0
$575$	−7.58141	−0.316167
$576$	1.00000	0.0416667
$577$	−1.16676	−0.0485727	−0.0242863	−	0.999705i	$-0.507731\pi$
$577$	−1.16676	−0.0485727	−0.0242863	+	0.999705i	$0.507731\pi$
$578$	1.00000	0.0415945
$579$	16.4779	0.684798
$580$	13.0471	0.541750
$581$	0	0
$582$	−12.5430	−0.519924
$583$	9.08879	0.376419
$584$	0.191107	0.00790806
$585$	−3.36268	−0.139030
$586$	−30.4805	−1.25914
$587$	22.3911	0.924178	0.462089	−	0.886834i	$-0.347100\pi$
$587$	22.3911	0.924178	0.462089	+	0.886834i	$0.347100\pi$
$588$	0	0
$589$	−8.44259	−0.347871
$590$	9.04331	0.372307
$591$	14.0353	0.577336
$592$	0.364492	0.0149805
$593$	37.9467	1.55828	0.779142	−	0.626848i	$-0.215655\pi$
$593$	37.9467	1.55828	0.779142	+	0.626848i	$0.215655\pi$
$594$	2.22129	0.0911408
$595$	0	0
$596$	−13.0283	−0.533659
$597$	9.38659	0.384167
$598$	−6.09944	−0.249425
$599$	−1.70679	−0.0697377	−0.0348688	−	0.999392i	$-0.511101\pi$
$599$	−1.70679	−0.0697377	−0.0348688	+	0.999392i	$0.511101\pi$
$600$	−2.85158	−0.116415
$601$	30.5830	1.24751	0.623754	−	0.781621i	$-0.285607\pi$
$601$	30.5830	1.24751	0.623754	+	0.781621i	$0.285607\pi$
$602$	0	0
$603$	1.28980	0.0525247
$604$	−14.1384	−0.575283
$605$	−8.89100	−0.361471
$606$	0.925308	0.0375881
$607$	−9.33346	−0.378833	−0.189417	−	0.981897i	$-0.560660\pi$
$607$	−9.33346	−0.378833	−0.189417	+	0.981897i	$0.560660\pi$
$608$	−4.07288	−0.165177
$609$	0	0
$610$	3.49704	0.141591
$611$	−23.4433	−0.948416
$612$	1.00000	0.0404226
$613$	−19.0347	−0.768804	−0.384402	−	0.923166i	$-0.625592\pi$
$613$	−19.0347	−0.768804	−0.384402	+	0.923166i	$0.625592\pi$
$614$	1.13763	0.0459110
$615$	11.4809	0.462955
$616$	0	0
$617$	−5.50725	−0.221713	−0.110857	−	0.993836i	$-0.535359\pi$
$617$	−5.50725	−0.221713	−0.110857	+	0.993836i	$0.535359\pi$
$618$	−2.87996	−0.115849
$619$	−19.4407	−0.781387	−0.390694	−	0.920521i	$-0.627765\pi$
$619$	−19.4407	−0.781387	−0.390694	+	0.920521i	$0.627765\pi$
$620$	3.03832	0.122022
$621$	2.65867	0.106689
$622$	−9.39287	−0.376620
$623$	0	0
$624$	−2.29417	−0.0918405
$625$	−2.61054	−0.104422
$626$	2.98825	0.119434
$627$	−9.04707	−0.361305
$628$	5.46309	0.218001
$629$	0.364492	0.0145332
$630$	0	0
$631$	−17.0735	−0.679685	−0.339843	−	0.940482i	$-0.610374\pi$
$631$	−17.0735	−0.679685	−0.339843	+	0.940482i	$0.610374\pi$
$632$	8.99297	0.357721
$633$	−21.3163	−0.847246
$634$	−10.6774	−0.424052
$635$	0.0831838	0.00330105
$636$	4.09166	0.162245
$637$	0	0
$638$	19.7724	0.782798
$639$	9.17414	0.362923
$640$	1.46575	0.0579387
$641$	−15.7350	−0.621493	−0.310747	−	0.950493i	$-0.600579\pi$
$641$	−15.7350	−0.621493	−0.310747	+	0.950493i	$0.600579\pi$
$642$	17.5198	0.691453
$643$	−21.3412	−0.841616	−0.420808	−	0.907150i	$-0.638253\pi$
$643$	−21.3412	−0.841616	−0.420808	+	0.907150i	$0.638253\pi$
$644$	0	0
$645$	−7.20079	−0.283531
$646$	−4.07288	−0.160245
$647$	−47.9732	−1.88602	−0.943012	−	0.332760i	$-0.892020\pi$
$647$	−47.9732	−1.88602	−0.943012	+	0.332760i	$0.892020\pi$
$648$	1.00000	0.0392837
$649$	13.7049	0.537963
$650$	6.54203	0.256600
$651$	0	0
$652$	8.82586	0.345648
$653$	−12.7290	−0.498124	−0.249062	−	0.968488i	$-0.580122\pi$
$653$	−12.7290	−0.498124	−0.249062	+	0.968488i	$0.580122\pi$
$654$	5.04450	0.197256
$655$	−10.3735	−0.405325
$656$	7.83280	0.305820
$657$	0.191107	0.00745579
$658$	0	0
$659$	34.5132	1.34444	0.672222	−	0.740350i	$-0.265340\pi$
$659$	34.5132	1.34444	0.672222	+	0.740350i	$0.265340\pi$
$660$	3.25586	0.126734
$661$	−32.0102	−1.24505	−0.622527	−	0.782598i	$-0.713894\pi$
$661$	−32.0102	−1.24505	−0.622527	+	0.782598i	$0.713894\pi$
$662$	6.12698	0.238132
$663$	−2.29417	−0.0890984
$664$	−16.2967	−0.632436
$665$	0	0
$666$	0.364492	0.0141238
$667$	23.6656	0.916336
$668$	−4.35831	−0.168628
$669$	−18.1181	−0.700488
$670$	1.89052	0.0730372
$671$	5.29965	0.204591
$672$	0	0
$673$	10.3358	0.398415	0.199207	−	0.979957i	$-0.436163\pi$
$673$	10.3358	0.398415	0.199207	+	0.979957i	$0.436163\pi$
$674$	−25.8114	−0.994217
$675$	−2.85158	−0.109758
$676$	−7.73676	−0.297568
$677$	−4.21824	−0.162120	−0.0810600	−	0.996709i	$-0.525831\pi$
$677$	−4.21824	−0.162120	−0.0810600	+	0.996709i	$0.525831\pi$
$678$	7.29236	0.280061
$679$	0	0
$680$	1.46575	0.0562088
$681$	12.9706	0.497033
$682$	4.60448	0.176315
$683$	35.6193	1.36293	0.681467	−	0.731849i	$-0.261342\pi$
$683$	35.6193	1.36293	0.681467	+	0.731849i	$0.261342\pi$
$684$	−4.07288	−0.155730
$685$	2.62385	0.100252
$686$	0	0
$687$	−5.81402	−0.221819
$688$	−4.91271	−0.187295
$689$	−9.38699	−0.357616
$690$	3.89693	0.148354
$691$	−13.1264	−0.499350	−0.249675	−	0.968330i	$-0.580324\pi$
$691$	−13.1264	−0.499350	−0.249675	+	0.968330i	$0.580324\pi$
$692$	−4.04632	−0.153818
$693$	0	0
$694$	20.6726	0.784722
$695$	−11.3963	−0.432286
$696$	8.90131	0.337403
$697$	7.83280	0.296689
$698$	−16.9715	−0.642382
$699$	1.25329	0.0474039
$700$	0	0
$701$	−19.1139	−0.721920	−0.360960	−	0.932581i	$-0.617551\pi$
$701$	−19.1139	−0.721920	−0.360960	+	0.932581i	$0.617551\pi$
$702$	−2.29417	−0.0865880
$703$	−1.48453	−0.0559901
$704$	2.22129	0.0837182
$705$	14.9779	0.564102
$706$	33.5136	1.26130
$707$	0	0
$708$	6.16976	0.231874
$709$	−43.0418	−1.61647	−0.808235	−	0.588860i	$-0.799577\pi$
$709$	−43.0418	−1.61647	−0.808235	+	0.588860i	$0.799577\pi$
$710$	13.4470	0.504656
$711$	8.99297	0.337263
$712$	−14.7831	−0.554019
$713$	5.51109	0.206392
$714$	0	0
$715$	−7.46950	−0.279344
$716$	−19.0471	−0.711822
$717$	13.0960	0.489080
$718$	1.60373	0.0598505
$719$	30.4992	1.13743	0.568713	−	0.822536i	$-0.307441\pi$
$719$	30.4992	1.13743	0.568713	+	0.822536i	$0.307441\pi$
$720$	1.46575	0.0546252
$721$	0	0
$722$	−2.41165	−0.0897524
$723$	15.6193	0.580888
$724$	−17.7952	−0.661355
$725$	−25.3828	−0.942695
$726$	−6.06585	−0.225125
$727$	47.1215	1.74764	0.873821	−	0.486248i	$-0.161635\pi$
$727$	47.1215	1.74764	0.873821	+	0.486248i	$0.161635\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.280115	0.0103675
$731$	−4.91271	−0.181703
$732$	2.38584	0.0881831
$733$	−29.6317	−1.09447	−0.547235	−	0.836979i	$-0.684320\pi$
$733$	−29.6317	−1.09447	−0.547235	+	0.836979i	$0.684320\pi$
$734$	4.97056	0.183467
$735$	0	0
$736$	2.65867	0.0979997
$737$	2.86503	0.105535
$738$	7.83280	0.288329
$739$	8.36233	0.307613	0.153807	−	0.988101i	$-0.450847\pi$
$739$	8.36233	0.307613	0.153807	+	0.988101i	$0.450847\pi$
$740$	0.534253	0.0196395
$741$	9.34390	0.343257
$742$	0	0
$743$	−19.2970	−0.707937	−0.353968	−	0.935257i	$-0.615168\pi$
$743$	−19.2970	−0.707937	−0.353968	+	0.935257i	$0.615168\pi$
$744$	2.07288	0.0759955
$745$	−19.0962	−0.699630
$746$	−6.07650	−0.222477
$747$	−16.2967	−0.596266
$748$	2.22129	0.0812186
$749$	0	0
$750$	−11.5084	−0.420229
$751$	0.0123721	0.000451465 0	0.000225733	−	1.00000i	$-0.499928\pi$
$751$	0.0123721	0.000451465 0	0.000225733		1.00000i	$0.499928\pi$
$752$	10.2186	0.372635
$753$	−20.2967	−0.739654
$754$	−20.4212	−0.743694
$755$	−20.7233	−0.754198
$756$	0	0
$757$	49.0006	1.78096	0.890478	−	0.455027i	$-0.150370\pi$
$757$	49.0006	1.78096	0.890478	+	0.455027i	$0.150370\pi$
$758$	−29.8016	−1.08244
$759$	5.90568	0.214363
$760$	−5.96981	−0.216548
$761$	8.96278	0.324901	0.162450	−	0.986717i	$-0.448060\pi$
$761$	8.96278	0.324901	0.162450	+	0.986717i	$0.448060\pi$
$762$	0.0567518	0.00205590
$763$	0	0
$764$	3.43918	0.124425
$765$	1.46575	0.0529942
$766$	−23.1841	−0.837675
$767$	−14.1545	−0.511090
$768$	1.00000	0.0360844
$769$	−15.9834	−0.576375	−0.288188	−	0.957574i	$-0.593053\pi$
$769$	−15.9834	−0.576375	−0.288188	+	0.957574i	$0.593053\pi$
$770$	0	0
$771$	−11.7058	−0.421575
$772$	16.4779	0.593053
$773$	4.49372	0.161628	0.0808140	−	0.996729i	$-0.474248\pi$
$773$	4.49372	0.161628	0.0808140	+	0.996729i	$0.474248\pi$
$774$	−4.91271	−0.176584
$775$	−5.91099	−0.212329
$776$	−12.5430	−0.450268
$777$	0	0
$778$	0.837176	0.0300142
$779$	−31.9021	−1.14301
$780$	−3.36268	−0.120403
$781$	20.3785	0.729199
$782$	2.65867	0.0950737
$783$	8.90131	0.318107
$784$	0	0
$785$	8.00751	0.285800
$786$	−7.07725	−0.252437
$787$	50.2970	1.79290	0.896448	−	0.443149i	$-0.146139\pi$
$787$	50.2970	1.79290	0.896448	+	0.443149i	$0.146139\pi$
$788$	14.0353	0.499987
$789$	5.01578	0.178566
$790$	13.1814	0.468974
$791$	0	0
$792$	2.22129	0.0789303
$793$	−5.47353	−0.194371
$794$	2.88880	0.102520
$795$	5.99734	0.212704
$796$	9.38659	0.332699
$797$	31.7551	1.12482	0.562412	−	0.826857i	$-0.309874\pi$
$797$	31.7551	1.12482	0.562412	+	0.826857i	$0.309874\pi$
$798$	0	0
$799$	10.2186	0.361509
$800$	−2.85158	−0.100819
$801$	−14.7831	−0.522334
$802$	−20.3119	−0.717238
$803$	0.424505	0.0149805
$804$	1.28980	0.0454877
$805$	0	0
$806$	−4.75555	−0.167507
$807$	11.9057	0.419100
$808$	0.925308	0.0325522
$809$	33.5812	1.18065	0.590327	−	0.807165i	$-0.298999\pi$
$809$	33.5812	1.18065	0.590327	+	0.807165i	$0.298999\pi$
$810$	1.46575	0.0515011
$811$	36.5039	1.28182	0.640912	−	0.767614i	$-0.278556\pi$
$811$	36.5039	1.28182	0.640912	+	0.767614i	$0.278556\pi$
$812$	0	0
$813$	12.4941	0.438188
$814$	0.809644	0.0283780
$815$	12.9365	0.453145
$816$	1.00000	0.0350070
$817$	20.0089	0.700022
$818$	−10.7367	−0.375399
$819$	0	0
$820$	11.4809	0.400931
$821$	−3.99624	−0.139470	−0.0697349	−	0.997566i	$-0.522215\pi$
$821$	−3.99624	−0.139470	−0.0697349	+	0.997566i	$0.522215\pi$
$822$	1.79011	0.0624372
$823$	27.2130	0.948585	0.474292	−	0.880367i	$-0.342704\pi$
$823$	27.2130	0.948585	0.474292	+	0.880367i	$0.342704\pi$
$824$	−2.87996	−0.100328
$825$	−6.33421	−0.220529
$826$	0	0
$827$	28.4289	0.988571	0.494285	−	0.869300i	$-0.335430\pi$
$827$	28.4289	0.988571	0.494285	+	0.869300i	$0.335430\pi$
$828$	2.65867	0.0923950
$829$	−11.9562	−0.415254	−0.207627	−	0.978208i	$-0.566574\pi$
$829$	−11.9562	−0.415254	−0.207627	+	0.978208i	$0.566574\pi$
$830$	−23.8869	−0.829126
$831$	7.58654	0.263174
$832$	−2.29417	−0.0795362
$833$	0	0
$834$	−7.77508	−0.269229
$835$	−6.38817	−0.221072
$836$	−9.04707	−0.312899
$837$	2.07288	0.0716492
$838$	−5.90431	−0.203961
$839$	−11.2714	−0.389133	−0.194566	−	0.980889i	$-0.562330\pi$
$839$	−11.2714	−0.389133	−0.194566	+	0.980889i	$0.562330\pi$
$840$	0	0
$841$	50.2333	1.73218
$842$	−30.6700	−1.05696
$843$	3.99638	0.137643
$844$	−21.3163	−0.733736
$845$	−11.3401	−0.390113
$846$	10.2186	0.351324
$847$	0	0
$848$	4.09166	0.140508
$849$	−5.74476	−0.197160
$850$	−2.85158	−0.0978086
$851$	0.969062	0.0332190
$852$	9.17414	0.314301
$853$	19.6618	0.673209	0.336604	−	0.941646i	$-0.390721\pi$
$853$	19.6618	0.673209	0.336604	+	0.941646i	$0.390721\pi$
$854$	0	0
$855$	−5.96981	−0.204163
$856$	17.5198	0.598816
$857$	−35.7650	−1.22171	−0.610855	−	0.791742i	$-0.709174\pi$
$857$	−35.7650	−1.22171	−0.610855	+	0.791742i	$0.709174\pi$
$858$	−5.09604	−0.173976
$859$	47.3423	1.61530	0.807650	−	0.589662i	$-0.200739\pi$
$859$	47.3423	1.61530	0.807650	+	0.589662i	$0.200739\pi$
$860$	−7.20079	−0.245545
$861$	0	0
$862$	24.7262	0.842179
$863$	−7.71833	−0.262735	−0.131368	−	0.991334i	$-0.541937\pi$
$863$	−7.71833	−0.262735	−0.131368	+	0.991334i	$0.541937\pi$
$864$	1.00000	0.0340207
$865$	−5.93088	−0.201656
$866$	18.8397	0.640200
$867$	1.00000	0.0339618
$868$	0	0
$869$	19.9760	0.677641
$870$	13.0471	0.442337
$871$	−2.95903	−0.100263
$872$	5.04450	0.170828
$873$	−12.5430	−0.424516
$874$	−10.8284	−0.366277
$875$	0	0
$876$	0.191107	0.00645691
$877$	−45.7600	−1.54520	−0.772602	−	0.634891i	$-0.781045\pi$
$877$	−45.7600	−1.54520	−0.772602	+	0.634891i	$0.781045\pi$
$878$	4.75555	0.160492
$879$	−30.4805	−1.02808
$880$	3.25586	0.109755
$881$	−46.2699	−1.55887	−0.779437	−	0.626481i	$-0.784495\pi$
$881$	−46.2699	−1.55887	−0.779437	+	0.626481i	$0.784495\pi$
$882$	0	0
$883$	−35.0771	−1.18044	−0.590219	−	0.807243i	$-0.700958\pi$
$883$	−35.0771	−1.18044	−0.590219	+	0.807243i	$0.700958\pi$
$884$	−2.29417	−0.0771614
$885$	9.04331	0.303988
$886$	21.3386	0.716884
$887$	−14.7381	−0.494858	−0.247429	−	0.968906i	$-0.579586\pi$
$887$	−14.7381	−0.494858	−0.247429	+	0.968906i	$0.579586\pi$
$888$	0.364492	0.0122315
$889$	0	0
$890$	−21.6683	−0.726322
$891$	2.22129	0.0744162
$892$	−18.1181	−0.606640
$893$	−41.6193	−1.39274
$894$	−13.0283	−0.435731
$895$	−27.9182	−0.933202
$896$	0	0
$897$	−6.09944	−0.203654
$898$	1.11266	0.0371299
$899$	18.4513	0.615387
$900$	−2.85158	−0.0950528
$901$	4.09166	0.136313
$902$	17.3990	0.579322
$903$	0	0
$904$	7.29236	0.242540
$905$	−26.0833	−0.867039
$906$	−14.1384	−0.469716
$907$	−52.4000	−1.73991	−0.869957	−	0.493128i	$-0.835854\pi$
$907$	−52.4000	−1.73991	−0.869957	+	0.493128i	$0.835854\pi$
$908$	12.9706	0.430443
$909$	0.925308	0.0306905
$910$	0	0
$911$	−0.484088	−0.0160385	−0.00801927	−	0.999968i	$-0.502553\pi$
$911$	−0.484088	−0.0160385	−0.00801927	+	0.999968i	$0.502553\pi$
$912$	−4.07288	−0.134867
$913$	−36.1999	−1.19804
$914$	31.4568	1.04050
$915$	3.49704	0.115608
$916$	−5.81402	−0.192101
$917$	0	0
$918$	1.00000	0.0330049
$919$	17.4699	0.576279	0.288139	−	0.957588i	$-0.406963\pi$
$919$	17.4699	0.576279	0.288139	+	0.957588i	$0.406963\pi$
$920$	3.89693	0.128478
$921$	1.13763	0.0374862
$922$	1.60060	0.0527128
$923$	−21.0471	−0.692773
$924$	0	0
$925$	−1.03938	−0.0341746
$926$	−29.5553	−0.971247
$927$	−2.87996	−0.0945903
$928$	8.90131	0.292200
$929$	8.71811	0.286032	0.143016	−	0.989720i	$-0.454320\pi$
$929$	8.71811	0.286032	0.143016	+	0.989720i	$0.454320\pi$
$930$	3.03832	0.0996304
$931$	0	0
$932$	1.25329	0.0410530
$933$	−9.39287	−0.307509
$934$	−10.1470	−0.332018
$935$	3.25586	0.106478
$936$	−2.29417	−0.0749874
$937$	−52.3418	−1.70993	−0.854965	−	0.518685i	$-0.826422\pi$
$937$	−52.3418	−1.70993	−0.854965	+	0.518685i	$0.826422\pi$
$938$	0	0
$939$	2.98825	0.0975177
$940$	14.9779	0.488527
$941$	50.0112	1.63032	0.815160	−	0.579236i	$-0.196649\pi$
$941$	50.0112	1.63032	0.815160	+	0.579236i	$0.196649\pi$
$942$	5.46309	0.177997
$943$	20.8248	0.678149
$944$	6.16976	0.200809
$945$	0	0
$946$	−10.9126	−0.354799
$947$	−6.87474	−0.223399	−0.111700	−	0.993742i	$-0.535629\pi$
$947$	−6.87474	−0.223399	−0.111700	+	0.993742i	$0.535629\pi$
$948$	8.99297	0.292078
$949$	−0.438433	−0.0142321
$950$	11.6142	0.376813
$951$	−10.6774	−0.346237
$952$	0	0
$953$	−0.103199	−0.00334296	−0.00167148	−	0.999999i	$-0.500532\pi$
$953$	−0.103199	−0.00334296	−0.00167148	+	0.999999i	$0.500532\pi$
$954$	4.09166	0.132472
$955$	5.04097	0.163122
$956$	13.0960	0.423556
$957$	19.7724	0.639152
$958$	0.800895	0.0258757
$959$	0	0
$960$	1.46575	0.0473068
$961$	−26.7032	−0.861393
$962$	−0.836208	−0.0269604
$963$	17.5198	0.564569
$964$	15.6193	0.503063
$965$	24.1524	0.777495
$966$	0	0
$967$	−16.3608	−0.526127	−0.263064	−	0.964779i	$-0.584733\pi$
$967$	−16.3608	−0.526127	−0.263064	+	0.964779i	$0.584733\pi$
$968$	−6.06585	−0.194964
$969$	−4.07288	−0.130840
$970$	−18.3849	−0.590303
$971$	−34.5368	−1.10834	−0.554170	−	0.832404i	$-0.686964\pi$
$971$	−34.5368	−1.10834	−0.554170	+	0.832404i	$0.686964\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	1.35018	0.0432624
$975$	6.54203	0.209513
$976$	2.38584	0.0763688
$977$	17.9732	0.575015	0.287508	−	0.957778i	$-0.407173\pi$
$977$	17.9732	0.575015	0.287508	+	0.957778i	$0.407173\pi$
$978$	8.82586	0.282220
$979$	−32.8376	−1.04949
$980$	0	0
$981$	5.04450	0.161059
$982$	−17.5360	−0.559595
$983$	−27.1304	−0.865325	−0.432662	−	0.901556i	$-0.642426\pi$
$983$	−27.1304	−0.865325	−0.432662	+	0.901556i	$0.642426\pi$
$984$	7.83280	0.249701
$985$	20.5722	0.655485
$986$	8.90131	0.283475
$987$	0	0
$988$	9.34390	0.297269
$989$	−13.0613	−0.415324
$990$	3.25586	0.103478
$991$	−41.3927	−1.31488	−0.657441	−	0.753506i	$-0.728361\pi$
$991$	−41.3927	−1.31488	−0.657441	+	0.753506i	$0.728361\pi$
$992$	2.07288	0.0658140
$993$	6.12698	0.194434
$994$	0	0
$995$	13.7584	0.436170
$996$	−16.2967	−0.516382
$997$	15.4741	0.490069	0.245035	−	0.969514i	$-0.421201\pi$
$997$	15.4741	0.490069	0.245035	+	0.969514i	$0.421201\pi$
$998$	44.2529	1.40080
$999$	0.364492	0.0115320

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.cp.1.3	yes	4
7.6	odd	2		4998.2.a.co.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4998.2.a.co.1.2	✓	4	7.6	odd	2
4998.2.a.cp.1.3	yes	4	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:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.46575 q^{10} +2.22129 q^{11} +1.00000 q^{12} -2.29417 q^{13} +1.46575 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.07288 q^{19} +1.46575 q^{20} +2.22129 q^{22} +2.65867 q^{23} +1.00000 q^{24} -2.85158 q^{25} -2.29417 q^{26} +1.00000 q^{27} +8.90131 q^{29} +1.46575 q^{30} +2.07288 q^{31} +1.00000 q^{32} +2.22129 q^{33} +1.00000 q^{34} +1.00000 q^{36} +0.364492 q^{37} -4.07288 q^{38} -2.29417 q^{39} +1.46575 q^{40} +7.83280 q^{41} -4.91271 q^{43} +2.22129 q^{44} +1.46575 q^{45} +2.65867 q^{46} +10.2186 q^{47} +1.00000 q^{48} -2.85158 q^{50} +1.00000 q^{51} -2.29417 q^{52} +4.09166 q^{53} +1.00000 q^{54} +3.25586 q^{55} -4.07288 q^{57} +8.90131 q^{58} +6.16976 q^{59} +1.46575 q^{60} +2.38584 q^{61} +2.07288 q^{62} +1.00000 q^{64} -3.36268 q^{65} +2.22129 q^{66} +1.28980 q^{67} +1.00000 q^{68} +2.65867 q^{69} +9.17414 q^{71} +1.00000 q^{72} +0.191107 q^{73} +0.364492 q^{74} -2.85158 q^{75} -4.07288 q^{76} -2.29417 q^{78} +8.99297 q^{79} +1.46575 q^{80} +1.00000 q^{81} +7.83280 q^{82} -16.2967 q^{83} +1.46575 q^{85} -4.91271 q^{86} +8.90131 q^{87} +2.22129 q^{88} -14.7831 q^{89} +1.46575 q^{90} +2.65867 q^{92} +2.07288 q^{93} +10.2186 q^{94} -5.96981 q^{95} +1.00000 q^{96} -12.5430 q^{97} +2.22129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 10 q^{11} + 4 q^{12} + 6 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 2 q^{20} + 10 q^{22} + 4 q^{24} + 6 q^{25}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 4 * q^8 + 4 * q^9 + 2 * q^10 + 10 * q^11 + 4 * q^12 + 6 * q^13 + 2 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 + 2 * q^20 + 10 * q^22 + 4 * q^24 + 6 * q^25 + 6 * q^26 + 4 * q^27 + 8 * q^29 + 2 * q^30 - 8 * q^31 + 4 * q^32 + 10 * q^33 + 4 * q^34 + 4 * q^36 + 6 * q^37 + 6 * q^39 + 2 * q^40 - 4 * q^41 + 6 * q^43 + 10 * q^44 + 2 * q^45 - 8 * q^47 + 4 * q^48 + 6 * q^50 + 4 * q^51 + 6 * q^52 + 18 * q^53 + 4 * q^54 + 10 * q^55 + 8 * q^58 + 24 * q^59 + 2 * q^60 - 4 * q^61 - 8 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^66 + 14 * q^67 + 4 * q^68 + 12 * q^71 + 4 * q^72 - 18 * q^73 + 6 * q^74 + 6 * q^75 + 6 * q^78 + 10 * q^79 + 2 * q^80 + 4 * q^81 - 4 * q^82 - 14 * q^83 + 2 * q^85 + 6 * q^86 + 8 * q^87 + 10 * q^88 - 34 * q^89 + 2 * q^90 - 8 * q^93 - 8 * q^94 - 4 * q^95 + 4 * q^96 - 6 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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