LMFDB - Embedded newform 8001.2.a.s.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8001, 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("8001.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,-4,0,20,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.284398$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.284398 q^{2} -1.91912 q^{4} +3.66581 q^{5} -1.00000 q^{7} +1.11459 q^{8} -1.04255 q^{10} +2.89343 q^{11} +1.88240 q^{13} +0.284398 q^{14} +3.52125 q^{16} +5.41792 q^{17} -4.15704 q^{19} -7.03512 q^{20} -0.822888 q^{22} +4.94630 q^{23} +8.43817 q^{25} -0.535352 q^{26} +1.91912 q^{28} +6.42248 q^{29} +8.67066 q^{31} -3.23062 q^{32} -1.54085 q^{34} -3.66581 q^{35} -7.13997 q^{37} +1.18226 q^{38} +4.08588 q^{40} -3.80151 q^{41} +6.40053 q^{43} -5.55284 q^{44} -1.40672 q^{46} -6.07023 q^{47} +1.00000 q^{49} -2.39980 q^{50} -3.61255 q^{52} -3.63552 q^{53} +10.6068 q^{55} -1.11459 q^{56} -1.82654 q^{58} +7.78156 q^{59} +6.41002 q^{61} -2.46592 q^{62} -6.12371 q^{64} +6.90052 q^{65} +13.3619 q^{67} -10.3976 q^{68} +1.04255 q^{70} -8.96501 q^{71} -5.77303 q^{73} +2.03060 q^{74} +7.97785 q^{76} -2.89343 q^{77} -1.95064 q^{79} +12.9082 q^{80} +1.08114 q^{82} +8.20767 q^{83} +19.8611 q^{85} -1.82030 q^{86} +3.22499 q^{88} +0.196140 q^{89} -1.88240 q^{91} -9.49253 q^{92} +1.72637 q^{94} -15.2389 q^{95} -4.42233 q^{97} -0.284398 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

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 $

$2$	−0.284398	−0.201100	−0.100550	−	0.994932i	$-0.532060\pi$
$2$	−0.284398	−0.201100	−0.100550	+	0.994932i	$0.532060\pi$
$3$	0	0
$3$	0	0
$4$	−1.91912	−0.959559
$5$	3.66581	1.63940	0.819700	−	0.572793i	$-0.194140\pi$
$5$	3.66581	1.63940	0.819700	+	0.572793i	$0.194140\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.11459	0.394067
$9$	0	0
$10$	−1.04255	−0.329684
$11$	2.89343	0.872403	0.436201	−	0.899849i	$-0.356324\pi$
$11$	2.89343	0.872403	0.436201	+	0.899849i	$0.356324\pi$
$12$	0	0
$13$	1.88240	0.522084	0.261042	−	0.965327i	$-0.415934\pi$
$13$	1.88240	0.522084	0.261042	+	0.965327i	$0.415934\pi$
$14$	0.284398	0.0760087
$15$	0	0
$16$	3.52125	0.880312
$17$	5.41792	1.31404	0.657019	−	0.753874i	$-0.271817\pi$
$17$	5.41792	1.31404	0.657019	+	0.753874i	$0.271817\pi$
$18$	0	0
$19$	−4.15704	−0.953691	−0.476845	−	0.878987i	$-0.658220\pi$
$19$	−4.15704	−0.953691	−0.476845	+	0.878987i	$0.658220\pi$
$20$	−7.03512	−1.57310
$21$	0	0
$22$	−0.822888	−0.175440
$23$	4.94630	1.03137	0.515687	−	0.856777i	$-0.327537\pi$
$23$	4.94630	1.03137	0.515687	+	0.856777i	$0.327537\pi$
$24$	0	0
$25$	8.43817	1.68763
$26$	−0.535352	−0.104991
$27$	0	0
$28$	1.91912	0.362679
$29$	6.42248	1.19262	0.596312	−	0.802753i	$-0.296632\pi$
$29$	6.42248	1.19262	0.596312	+	0.802753i	$0.296632\pi$
$30$	0	0
$31$	8.67066	1.55730	0.778648	−	0.627461i	$-0.215906\pi$
$31$	8.67066	1.55730	0.778648	+	0.627461i	$0.215906\pi$
$32$	−3.23062	−0.571098
$33$	0	0
$34$	−1.54085	−0.264253
$35$	−3.66581	−0.619635
$36$	0	0
$37$	−7.13997	−1.17380	−0.586902	−	0.809658i	$-0.699653\pi$
$37$	−7.13997	−1.17380	−0.586902	+	0.809658i	$0.699653\pi$
$38$	1.18226	0.191787
$39$	0	0
$40$	4.08588	0.646034
$41$	−3.80151	−0.593697	−0.296848	−	0.954925i	$-0.595936\pi$
$41$	−3.80151	−0.593697	−0.296848	+	0.954925i	$0.595936\pi$
$42$	0	0
$43$	6.40053	0.976072	0.488036	−	0.872823i	$-0.337713\pi$
$43$	6.40053	0.976072	0.488036	+	0.872823i	$0.337713\pi$
$44$	−5.55284	−0.837122
$45$	0	0
$46$	−1.40672	−0.207410
$47$	−6.07023	−0.885435	−0.442717	−	0.896661i	$-0.645986\pi$
$47$	−6.07023	−0.885435	−0.442717	+	0.896661i	$0.645986\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.39980	−0.339383
$51$	0	0
$52$	−3.61255	−0.500970
$53$	−3.63552	−0.499377	−0.249689	−	0.968326i	$-0.580328\pi$
$53$	−3.63552	−0.499377	−0.249689	+	0.968326i	$0.580328\pi$
$54$	0	0
$55$	10.6068	1.43022
$56$	−1.11459	−0.148943
$57$	0	0
$58$	−1.82654	−0.239837
$59$	7.78156	1.01307	0.506536	−	0.862219i	$-0.330926\pi$
$59$	7.78156	1.01307	0.506536	+	0.862219i	$0.330926\pi$
$60$	0	0
$61$	6.41002	0.820719	0.410360	−	0.911924i	$-0.365403\pi$
$61$	6.41002	0.820719	0.410360	+	0.911924i	$0.365403\pi$
$62$	−2.46592	−0.313172
$63$	0	0
$64$	−6.12371	−0.765464
$65$	6.90052	0.855904
$66$	0	0
$67$	13.3619	1.63242	0.816208	−	0.577758i	$-0.196072\pi$
$67$	13.3619	1.63242	0.816208	+	0.577758i	$0.196072\pi$
$68$	−10.3976	−1.26090
$69$	0	0
$70$	1.04255	0.124609
$71$	−8.96501	−1.06395	−0.531975	−	0.846760i	$-0.678550\pi$
$71$	−8.96501	−1.06395	−0.531975	+	0.846760i	$0.678550\pi$
$72$	0	0
$73$	−5.77303	−0.675682	−0.337841	−	0.941203i	$-0.609697\pi$
$73$	−5.77303	−0.675682	−0.337841	+	0.941203i	$0.609697\pi$
$74$	2.03060	0.236052
$75$	0	0
$76$	7.97785	0.915123
$77$	−2.89343	−0.329737
$78$	0	0
$79$	−1.95064	−0.219464	−0.109732	−	0.993961i	$-0.534999\pi$
$79$	−1.95064	−0.219464	−0.109732	+	0.993961i	$0.534999\pi$
$80$	12.9082	1.44318
$81$	0	0
$82$	1.08114	0.119392
$83$	8.20767	0.900909	0.450454	−	0.892799i	$-0.351262\pi$
$83$	8.20767	0.900909	0.450454	+	0.892799i	$0.351262\pi$
$84$	0	0
$85$	19.8611	2.15424
$86$	−1.82030	−0.196288
$87$	0	0
$88$	3.22499	0.343785
$89$	0.196140	0.0207908	0.0103954	−	0.999946i	$-0.496691\pi$
$89$	0.196140	0.0207908	0.0103954	+	0.999946i	$0.496691\pi$
$90$	0	0
$91$	−1.88240	−0.197329
$92$	−9.49253	−0.989665
$93$	0	0
$94$	1.72637	0.178061
$95$	−15.2389	−1.56348
$96$	0	0
$97$	−4.42233	−0.449020	−0.224510	−	0.974472i	$-0.572078\pi$
$97$	−4.42233	−0.449020	−0.224510	+	0.974472i	$0.572078\pi$
$98$	−0.284398	−0.0287286
$99$	0	0
$100$	−16.1938	−1.61938
$101$	−15.2503	−1.51746	−0.758732	−	0.651403i	$-0.774181\pi$
$101$	−15.2503	−1.51746	−0.758732	+	0.651403i	$0.774181\pi$
$102$	0	0
$103$	−10.2948	−1.01437	−0.507187	−	0.861836i	$-0.669315\pi$
$103$	−10.2948	−1.01437	−0.507187	+	0.861836i	$0.669315\pi$
$104$	2.09811	0.205736
$105$	0	0
$106$	1.03394	0.100425
$107$	−5.29148	−0.511546	−0.255773	−	0.966737i	$-0.582330\pi$
$107$	−5.29148	−0.511546	−0.255773	+	0.966737i	$0.582330\pi$
$108$	0	0
$109$	1.14135	0.109322	0.0546608	−	0.998505i	$-0.482592\pi$
$109$	1.14135	0.109322	0.0546608	+	0.998505i	$0.482592\pi$
$110$	−3.01655	−0.287617
$111$	0	0
$112$	−3.52125	−0.332727
$113$	−5.87786	−0.552942	−0.276471	−	0.961022i	$-0.589165\pi$
$113$	−5.87786	−0.552942	−0.276471	+	0.961022i	$0.589165\pi$
$114$	0	0
$115$	18.1322	1.69084
$116$	−12.3255	−1.14439
$117$	0	0
$118$	−2.21306	−0.203729
$119$	−5.41792	−0.496660
$120$	0	0
$121$	−2.62805	−0.238913
$122$	−1.82300	−0.165047
$123$	0	0
$124$	−16.6400	−1.49432
$125$	12.6037	1.12731
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.20281	0.725033
$129$	0	0
$130$	−1.96250	−0.172122
$131$	8.12871	0.710209	0.355104	−	0.934827i	$-0.384445\pi$
$131$	8.12871	0.710209	0.355104	+	0.934827i	$0.384445\pi$
$132$	0	0
$133$	4.15704	0.360461
$134$	−3.80011	−0.328279
$135$	0	0
$136$	6.03876	0.517820
$137$	−19.3581	−1.65387	−0.826935	−	0.562297i	$-0.809918\pi$
$137$	−19.3581	−1.65387	−0.826935	+	0.562297i	$0.809918\pi$
$138$	0	0
$139$	9.82585	0.833417	0.416709	−	0.909040i	$-0.363184\pi$
$139$	9.82585	0.833417	0.416709	+	0.909040i	$0.363184\pi$
$140$	7.03512	0.594576
$141$	0	0
$142$	2.54963	0.213961
$143$	5.44660	0.455467
$144$	0	0
$145$	23.5436	1.95519
$146$	1.64184	0.135880
$147$	0	0
$148$	13.7024	1.12633
$149$	8.10651	0.664111	0.332056	−	0.943260i	$-0.392258\pi$
$149$	8.10651	0.664111	0.332056	+	0.943260i	$0.392258\pi$
$150$	0	0
$151$	−1.16810	−0.0950590	−0.0475295	−	0.998870i	$-0.515135\pi$
$151$	−1.16810	−0.0950590	−0.0475295	+	0.998870i	$0.515135\pi$
$152$	−4.63340	−0.375819
$153$	0	0
$154$	0.822888	0.0663102
$155$	31.7850	2.55303
$156$	0	0
$157$	18.3308	1.46296	0.731479	−	0.681864i	$-0.238830\pi$
$157$	18.3308	1.46296	0.731479	+	0.681864i	$0.238830\pi$
$158$	0.554758	0.0441342
$159$	0	0
$160$	−11.8428	−0.936259
$161$	−4.94630	−0.389823
$162$	0	0
$163$	24.4471	1.91485	0.957423	−	0.288689i	$-0.0932193\pi$
$163$	24.4471	1.91485	0.957423	+	0.288689i	$0.0932193\pi$
$164$	7.29555	0.569687
$165$	0	0
$166$	−2.33425	−0.181173
$167$	−4.54592	−0.351774	−0.175887	−	0.984410i	$-0.556279\pi$
$167$	−4.54592	−0.351774	−0.175887	+	0.984410i	$0.556279\pi$
$168$	0	0
$169$	−9.45657	−0.727429
$170$	−5.64846	−0.433217
$171$	0	0
$172$	−12.2834	−0.936599
$173$	6.42599	0.488559	0.244279	−	0.969705i	$-0.421449\pi$
$173$	6.42599	0.488559	0.244279	+	0.969705i	$0.421449\pi$
$174$	0	0
$175$	−8.43817	−0.637866
$176$	10.1885	0.767986
$177$	0	0
$178$	−0.0557818	−0.00418102
$179$	−16.7986	−1.25559	−0.627793	−	0.778381i	$-0.716041\pi$
$179$	−16.7986	−1.25559	−0.627793	+	0.778381i	$0.716041\pi$
$180$	0	0
$181$	15.9915	1.18864	0.594318	−	0.804230i	$-0.297422\pi$
$181$	15.9915	1.18864	0.594318	+	0.804230i	$0.297422\pi$
$182$	0.535352	0.0396829
$183$	0	0
$184$	5.51310	0.406431
$185$	−26.1738	−1.92433
$186$	0	0
$187$	15.6764	1.14637
$188$	11.6495	0.849627
$189$	0	0
$190$	4.33393	0.314416
$191$	21.5802	1.56149	0.780746	−	0.624849i	$-0.214839\pi$
$191$	21.5802	1.56149	0.780746	+	0.624849i	$0.214839\pi$
$192$	0	0
$193$	−13.5153	−0.972851	−0.486425	−	0.873722i	$-0.661699\pi$
$193$	−13.5153	−0.972851	−0.486425	+	0.873722i	$0.661699\pi$
$194$	1.25770	0.0902979
$195$	0	0
$196$	−1.91912	−0.137080
$197$	−13.4193	−0.956084	−0.478042	−	0.878337i	$-0.658653\pi$
$197$	−13.4193	−0.956084	−0.478042	+	0.878337i	$0.658653\pi$
$198$	0	0
$199$	−9.63107	−0.682728	−0.341364	−	0.939931i	$-0.610889\pi$
$199$	−9.63107	−0.682728	−0.341364	+	0.939931i	$0.610889\pi$
$200$	9.40510	0.665041
$201$	0	0
$202$	4.33717	0.305162
$203$	−6.42248	−0.450770
$204$	0	0
$205$	−13.9356	−0.973307
$206$	2.92782	0.203991
$207$	0	0
$208$	6.62840	0.459597
$209$	−12.0281	−0.832003
$210$	0	0
$211$	5.69553	0.392096	0.196048	−	0.980594i	$-0.437189\pi$
$211$	5.69553	0.392096	0.196048	+	0.980594i	$0.437189\pi$
$212$	6.97699	0.479182
$213$	0	0
$214$	1.50489	0.102872
$215$	23.4631	1.60017
$216$	0	0
$217$	−8.67066	−0.588602
$218$	−0.324598	−0.0219846
$219$	0	0
$220$	−20.3557	−1.37238
$221$	10.1987	0.686038
$222$	0	0
$223$	−21.8851	−1.46554	−0.732768	−	0.680478i	$-0.761772\pi$
$223$	−21.8851	−1.46554	−0.732768	+	0.680478i	$0.761772\pi$
$224$	3.23062	0.215855
$225$	0	0
$226$	1.67165	0.111197
$227$	8.01519	0.531987	0.265993	−	0.963975i	$-0.414300\pi$
$227$	8.01519	0.531987	0.265993	+	0.963975i	$0.414300\pi$
$228$	0	0
$229$	−12.6832	−0.838126	−0.419063	−	0.907957i	$-0.637641\pi$
$229$	−12.6832	−0.838126	−0.419063	+	0.907957i	$0.637641\pi$
$230$	−5.15677	−0.340027
$231$	0	0
$232$	7.15844	0.469974
$233$	5.87223	0.384702	0.192351	−	0.981326i	$-0.438389\pi$
$233$	5.87223	0.384702	0.192351	+	0.981326i	$0.438389\pi$
$234$	0	0
$235$	−22.2523	−1.45158
$236$	−14.9337	−0.972103
$237$	0	0
$238$	1.54085	0.0998783
$239$	−23.2627	−1.50474	−0.752369	−	0.658742i	$-0.771089\pi$
$239$	−23.2627	−1.50474	−0.752369	+	0.658742i	$0.771089\pi$
$240$	0	0
$241$	−9.27487	−0.597447	−0.298723	−	0.954340i	$-0.596561\pi$
$241$	−9.27487	−0.597447	−0.298723	+	0.954340i	$0.596561\pi$
$242$	0.747413	0.0480455
$243$	0	0
$244$	−12.3016	−0.787528
$245$	3.66581	0.234200
$246$	0	0
$247$	−7.82522	−0.497907
$248$	9.66423	0.613679
$249$	0	0
$250$	−3.58446	−0.226701
$251$	−4.99183	−0.315082	−0.157541	−	0.987512i	$-0.550357\pi$
$251$	−4.99183	−0.315082	−0.157541	+	0.987512i	$0.550357\pi$
$252$	0	0
$253$	14.3118	0.899774
$254$	0.284398	0.0178447
$255$	0	0
$256$	9.91456	0.619660
$257$	−2.58595	−0.161307	−0.0806535	−	0.996742i	$-0.525701\pi$
$257$	−2.58595	−0.161307	−0.0806535	+	0.996742i	$0.525701\pi$
$258$	0	0
$259$	7.13997	0.443656
$260$	−13.2429	−0.821291
$261$	0	0
$262$	−2.31179	−0.142823
$263$	10.0488	0.619637	0.309819	−	0.950796i	$-0.399732\pi$
$263$	10.0488	0.619637	0.309819	+	0.950796i	$0.399732\pi$
$264$	0	0
$265$	−13.3271	−0.818679
$266$	−1.18226	−0.0724888
$267$	0	0
$268$	−25.6431	−1.56640
$269$	25.2677	1.54060	0.770298	−	0.637684i	$-0.220107\pi$
$269$	25.2677	1.54060	0.770298	+	0.637684i	$0.220107\pi$
$270$	0	0
$271$	5.90645	0.358791	0.179396	−	0.983777i	$-0.442586\pi$
$271$	5.90645	0.358791	0.179396	+	0.983777i	$0.442586\pi$
$272$	19.0778	1.15676
$273$	0	0
$274$	5.50540	0.332593
$275$	24.4153	1.47230
$276$	0	0
$277$	11.9163	0.715980	0.357990	−	0.933725i	$-0.383462\pi$
$277$	11.9163	0.715980	0.357990	+	0.933725i	$0.383462\pi$
$278$	−2.79446	−0.167600
$279$	0	0
$280$	−4.08588	−0.244178
$281$	2.78115	0.165910	0.0829548	−	0.996553i	$-0.473564\pi$
$281$	2.78115	0.165910	0.0829548	+	0.996553i	$0.473564\pi$
$282$	0	0
$283$	−3.24384	−0.192826	−0.0964131	−	0.995341i	$-0.530737\pi$
$283$	−3.24384	−0.192826	−0.0964131	+	0.995341i	$0.530737\pi$
$284$	17.2049	1.02092
$285$	0	0
$286$	−1.54900	−0.0915945
$287$	3.80151	0.224396
$288$	0	0
$289$	12.3539	0.726697
$290$	−6.69576	−0.393189
$291$	0	0
$292$	11.0791	0.648357
$293$	−13.7305	−0.802145	−0.401073	−	0.916046i	$-0.631362\pi$
$293$	−13.7305	−0.802145	−0.401073	+	0.916046i	$0.631362\pi$
$294$	0	0
$295$	28.5257	1.66083
$296$	−7.95814	−0.462558
$297$	0	0
$298$	−2.30548	−0.133553
$299$	9.31092	0.538464
$300$	0	0
$301$	−6.40053	−0.368921
$302$	0.332207	0.0191164
$303$	0	0
$304$	−14.6380	−0.839545
$305$	23.4979	1.34549
$306$	0	0
$307$	1.50941	0.0861464	0.0430732	−	0.999072i	$-0.486285\pi$
$307$	1.50941	0.0861464	0.0430732	+	0.999072i	$0.486285\pi$
$308$	5.55284	0.316402
$309$	0	0
$310$	−9.03960	−0.513415
$311$	25.9571	1.47189	0.735947	−	0.677040i	$-0.236737\pi$
$311$	25.9571	1.47189	0.735947	+	0.677040i	$0.236737\pi$
$312$	0	0
$313$	−19.6113	−1.10850	−0.554249	−	0.832351i	$-0.686994\pi$
$313$	−19.6113	−1.10850	−0.554249	+	0.832351i	$0.686994\pi$
$314$	−5.21325	−0.294201
$315$	0	0
$316$	3.74350	0.210588
$317$	−6.56414	−0.368679	−0.184339	−	0.982863i	$-0.559015\pi$
$317$	−6.56414	−0.368679	−0.184339	+	0.982863i	$0.559015\pi$
$318$	0	0
$319$	18.5830	1.04045
$320$	−22.4484	−1.25490
$321$	0	0
$322$	1.40672	0.0783934
$323$	−22.5225	−1.25319
$324$	0	0
$325$	15.8840	0.881086
$326$	−6.95272	−0.385076
$327$	0	0
$328$	−4.23713	−0.233957
$329$	6.07023	0.334663
$330$	0	0
$331$	6.45257	0.354666	0.177333	−	0.984151i	$-0.443253\pi$
$331$	6.45257	0.354666	0.177333	+	0.984151i	$0.443253\pi$
$332$	−15.7515	−0.864475
$333$	0	0
$334$	1.29285	0.0707418
$335$	48.9822	2.67618
$336$	0	0
$337$	−35.0026	−1.90672	−0.953358	−	0.301843i	$-0.902398\pi$
$337$	−35.0026	−1.90672	−0.953358	+	0.301843i	$0.902398\pi$
$338$	2.68943	0.146286
$339$	0	0
$340$	−38.1157	−2.06712
$341$	25.0880	1.35859
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.13398	0.384638
$345$	0	0
$346$	−1.82754	−0.0982492
$347$	−25.6545	−1.37721	−0.688603	−	0.725139i	$-0.741776\pi$
$347$	−25.6545	−1.37721	−0.688603	+	0.725139i	$0.741776\pi$
$348$	0	0
$349$	19.5960	1.04895	0.524474	−	0.851426i	$-0.324262\pi$
$349$	19.5960	1.04895	0.524474	+	0.851426i	$0.324262\pi$
$350$	2.39980	0.128275
$351$	0	0
$352$	−9.34758	−0.498228
$353$	0.516604	0.0274961	0.0137480	−	0.999905i	$-0.495624\pi$
$353$	0.516604	0.0274961	0.0137480	+	0.999905i	$0.495624\pi$
$354$	0	0
$355$	−32.8640	−1.74424
$356$	−0.376415	−0.0199499
$357$	0	0
$358$	4.77749	0.252498
$359$	29.2134	1.54182	0.770912	−	0.636942i	$-0.219801\pi$
$359$	29.2134	1.54182	0.770912	+	0.636942i	$0.219801\pi$
$360$	0	0
$361$	−1.71900	−0.0904736
$362$	−4.54795	−0.239035
$363$	0	0
$364$	3.61255	0.189349
$365$	−21.1628	−1.10771
$366$	0	0
$367$	−28.1968	−1.47186	−0.735931	−	0.677057i	$-0.763255\pi$
$367$	−28.1968	−1.47186	−0.735931	+	0.677057i	$0.763255\pi$
$368$	17.4171	0.907932
$369$	0	0
$370$	7.44378	0.386984
$371$	3.63552	0.188747
$372$	0	0
$373$	33.4097	1.72989	0.864945	−	0.501866i	$-0.167353\pi$
$373$	33.4097	1.72989	0.864945	+	0.501866i	$0.167353\pi$
$374$	−4.45834	−0.230535
$375$	0	0
$376$	−6.76583	−0.348921
$377$	12.0897	0.622650
$378$	0	0
$379$	−7.56506	−0.388591	−0.194296	−	0.980943i	$-0.562242\pi$
$379$	−7.56506	−0.388591	−0.194296	+	0.980943i	$0.562242\pi$
$380$	29.2453	1.50025
$381$	0	0
$382$	−6.13739	−0.314016
$383$	13.9909	0.714900	0.357450	−	0.933932i	$-0.383646\pi$
$383$	13.9909	0.714900	0.357450	+	0.933932i	$0.383646\pi$
$384$	0	0
$385$	−10.6068	−0.540571
$386$	3.84372	0.195640
$387$	0	0
$388$	8.48697	0.430861
$389$	20.1522	1.02176	0.510879	−	0.859653i	$-0.329320\pi$
$389$	20.1522	1.02176	0.510879	+	0.859653i	$0.329320\pi$
$390$	0	0
$391$	26.7987	1.35527
$392$	1.11459	0.0562953
$393$	0	0
$394$	3.81642	0.192269
$395$	−7.15066	−0.359789
$396$	0	0
$397$	4.91282	0.246568	0.123284	−	0.992371i	$-0.460657\pi$
$397$	4.91282	0.246568	0.123284	+	0.992371i	$0.460657\pi$
$398$	2.73906	0.137297
$399$	0	0
$400$	29.7129	1.48564
$401$	−10.2127	−0.509998	−0.254999	−	0.966941i	$-0.582075\pi$
$401$	−10.2127	−0.509998	−0.254999	+	0.966941i	$0.582075\pi$
$402$	0	0
$403$	16.3216	0.813039
$404$	29.2672	1.45610
$405$	0	0
$406$	1.82654	0.0906498
$407$	−20.6590	−1.02403
$408$	0	0
$409$	−23.1233	−1.14337	−0.571687	−	0.820472i	$-0.693711\pi$
$409$	−23.1233	−1.14337	−0.571687	+	0.820472i	$0.693711\pi$
$410$	3.96327	0.195732
$411$	0	0
$412$	19.7569	0.973352
$413$	−7.78156	−0.382906
$414$	0	0
$415$	30.0878	1.47695
$416$	−6.08132	−0.298161
$417$	0	0
$418$	3.42078	0.167316
$419$	−25.5851	−1.24991	−0.624957	−	0.780659i	$-0.714883\pi$
$419$	−25.5851	−1.24991	−0.624957	+	0.780659i	$0.714883\pi$
$420$	0	0
$421$	−13.6451	−0.665023	−0.332512	−	0.943099i	$-0.607896\pi$
$421$	−13.6451	−0.665023	−0.332512	+	0.943099i	$0.607896\pi$
$422$	−1.61980	−0.0788506
$423$	0	0
$424$	−4.05212	−0.196788
$425$	45.7173	2.21762
$426$	0	0
$427$	−6.41002	−0.310203
$428$	10.1550	0.490859
$429$	0	0
$430$	−6.67288	−0.321795
$431$	−22.7266	−1.09470	−0.547352	−	0.836903i	$-0.684364\pi$
$431$	−22.7266	−1.09470	−0.547352	+	0.836903i	$0.684364\pi$
$432$	0	0
$433$	−2.49007	−0.119665	−0.0598325	−	0.998208i	$-0.519057\pi$
$433$	−2.49007	−0.119665	−0.0598325	+	0.998208i	$0.519057\pi$
$434$	2.46592	0.118368
$435$	0	0
$436$	−2.19039	−0.104900
$437$	−20.5620	−0.983613
$438$	0	0
$439$	36.1038	1.72314	0.861570	−	0.507639i	$-0.169482\pi$
$439$	36.1038	1.72314	0.861570	+	0.507639i	$0.169482\pi$
$440$	11.8222	0.563602
$441$	0	0
$442$	−2.90049	−0.137962
$443$	41.2105	1.95797	0.978984	−	0.203935i	$-0.0653731\pi$
$443$	41.2105	1.95797	0.978984	+	0.203935i	$0.0653731\pi$
$444$	0	0
$445$	0.719011	0.0340844
$446$	6.22410	0.294720
$447$	0	0
$448$	6.12371	0.289318
$449$	12.4015	0.585262	0.292631	−	0.956225i	$-0.405469\pi$
$449$	12.4015	0.585262	0.292631	+	0.956225i	$0.405469\pi$
$450$	0	0
$451$	−10.9994	−0.517943
$452$	11.2803	0.530581
$453$	0	0
$454$	−2.27951	−0.106983
$455$	−6.90052	−0.323501
$456$	0	0
$457$	0.710700	0.0332451	0.0166226	−	0.999862i	$-0.494709\pi$
$457$	0.710700	0.0332451	0.0166226	+	0.999862i	$0.494709\pi$
$458$	3.60707	0.168547
$459$	0	0
$460$	−34.7978	−1.62246
$461$	−35.3687	−1.64728	−0.823642	−	0.567110i	$-0.808061\pi$
$461$	−35.3687	−1.64728	−0.823642	+	0.567110i	$0.808061\pi$
$462$	0	0
$463$	34.5748	1.60683	0.803413	−	0.595423i	$-0.203015\pi$
$463$	34.5748	1.60683	0.803413	+	0.595423i	$0.203015\pi$
$464$	22.6151	1.04988
$465$	0	0
$466$	−1.67005	−0.0773637
$467$	3.12852	0.144771	0.0723853	−	0.997377i	$-0.476939\pi$
$467$	3.12852	0.144771	0.0723853	+	0.997377i	$0.476939\pi$
$468$	0	0
$469$	−13.3619	−0.616996
$470$	6.32853	0.291913
$471$	0	0
$472$	8.67326	0.399219
$473$	18.5195	0.851528
$474$	0	0
$475$	−35.0778	−1.60948
$476$	10.3976	0.476574
$477$	0	0
$478$	6.61587	0.302603
$479$	−19.1146	−0.873368	−0.436684	−	0.899615i	$-0.643847\pi$
$479$	−19.1146	−0.873368	−0.436684	+	0.899615i	$0.643847\pi$
$480$	0	0
$481$	−13.4403	−0.612824
$482$	2.63776	0.120147
$483$	0	0
$484$	5.04353	0.229251
$485$	−16.2114	−0.736123
$486$	0	0
$487$	37.2169	1.68646	0.843230	−	0.537552i	$-0.180651\pi$
$487$	37.2169	1.68646	0.843230	+	0.537552i	$0.180651\pi$
$488$	7.14455	0.323419
$489$	0	0
$490$	−1.04255	−0.0470976
$491$	14.3956	0.649664	0.324832	−	0.945772i	$-0.394692\pi$
$491$	14.3956	0.649664	0.324832	+	0.945772i	$0.394692\pi$
$492$	0	0
$493$	34.7965	1.56715
$494$	2.22548	0.100129
$495$	0	0
$496$	30.5315	1.37091
$497$	8.96501	0.402136
$498$	0	0
$499$	30.4124	1.36145	0.680723	−	0.732541i	$-0.261666\pi$
$499$	30.4124	1.36145	0.680723	+	0.732541i	$0.261666\pi$
$500$	−24.1879	−1.08172
$501$	0	0
$502$	1.41967	0.0633629
$503$	−17.2114	−0.767419	−0.383710	−	0.923454i	$-0.625354\pi$
$503$	−17.2114	−0.767419	−0.383710	+	0.923454i	$0.625354\pi$
$504$	0	0
$505$	−55.9048	−2.48773
$506$	−4.07025	−0.180945
$507$	0	0
$508$	1.91912	0.0851471
$509$	−0.686999	−0.0304507	−0.0152253	−	0.999884i	$-0.504847\pi$
$509$	−0.686999	−0.0304507	−0.0152253	+	0.999884i	$0.504847\pi$
$510$	0	0
$511$	5.77303	0.255384
$512$	−19.2253	−0.849647
$513$	0	0
$514$	0.735440	0.0324389
$515$	−37.7387	−1.66297
$516$	0	0
$517$	−17.5638	−0.772456
$518$	−2.03060	−0.0892193
$519$	0	0
$520$	7.69126	0.337284
$521$	−8.77448	−0.384417	−0.192209	−	0.981354i	$-0.561565\pi$
$521$	−8.77448	−0.384417	−0.192209	+	0.981354i	$0.561565\pi$
$522$	0	0
$523$	32.2063	1.40828	0.704142	−	0.710059i	$-0.251332\pi$
$523$	32.2063	1.40828	0.704142	+	0.710059i	$0.251332\pi$
$524$	−15.5999	−0.681487
$525$	0	0
$526$	−2.85787	−0.124609
$527$	46.9769	2.04635
$528$	0	0
$529$	1.46589	0.0637343
$530$	3.79021	0.164636
$531$	0	0
$532$	−7.97785	−0.345884
$533$	−7.15597	−0.309959
$534$	0	0
$535$	−19.3976	−0.838629
$536$	14.8931	0.643282
$537$	0	0
$538$	−7.18608	−0.309814
$539$	2.89343	0.124629
$540$	0	0
$541$	17.3738	0.746957	0.373478	−	0.927639i	$-0.378165\pi$
$541$	17.3738	0.746957	0.373478	+	0.927639i	$0.378165\pi$
$542$	−1.67978	−0.0721529
$543$	0	0
$544$	−17.5032	−0.750445
$545$	4.18398	0.179222
$546$	0	0
$547$	35.7583	1.52891	0.764457	−	0.644674i	$-0.223007\pi$
$547$	35.7583	1.52891	0.764457	+	0.644674i	$0.223007\pi$
$548$	37.1504	1.58699
$549$	0	0
$550$	−6.94366	−0.296079
$551$	−26.6985	−1.13740
$552$	0	0
$553$	1.95064	0.0829495
$554$	−3.38897	−0.143984
$555$	0	0
$556$	−18.8570	−0.799713
$557$	−27.9404	−1.18387	−0.591936	−	0.805985i	$-0.701636\pi$
$557$	−27.9404	−1.18387	−0.591936	+	0.805985i	$0.701636\pi$
$558$	0	0
$559$	12.0484	0.509592
$560$	−12.9082	−0.545472
$561$	0	0
$562$	−0.790955	−0.0333644
$563$	27.3336	1.15197	0.575987	−	0.817459i	$-0.304618\pi$
$563$	27.3336	1.15197	0.575987	+	0.817459i	$0.304618\pi$
$564$	0	0
$565$	−21.5471	−0.906494
$566$	0.922542	0.0387774
$567$	0	0
$568$	−9.99232	−0.419268
$569$	−16.5911	−0.695536	−0.347768	−	0.937581i	$-0.613060\pi$
$569$	−16.5911	−0.695536	−0.347768	+	0.937581i	$0.613060\pi$
$570$	0	0
$571$	−2.71786	−0.113739	−0.0568694	−	0.998382i	$-0.518112\pi$
$571$	−2.71786	−0.113739	−0.0568694	+	0.998382i	$0.518112\pi$
$572$	−10.4527	−0.437048
$573$	0	0
$574$	−1.08114	−0.0451261
$575$	41.7377	1.74058
$576$	0	0
$577$	27.9955	1.16547	0.582733	−	0.812663i	$-0.301983\pi$
$577$	27.9955	1.16547	0.582733	+	0.812663i	$0.301983\pi$
$578$	−3.51342	−0.146139
$579$	0	0
$580$	−45.1829	−1.87612
$581$	−8.20767	−0.340512
$582$	0	0
$583$	−10.5191	−0.435658
$584$	−6.43457	−0.266264
$585$	0	0
$586$	3.90494	0.161311
$587$	36.2581	1.49653	0.748266	−	0.663399i	$-0.230887\pi$
$587$	36.2581	1.49653	0.748266	+	0.663399i	$0.230887\pi$
$588$	0	0
$589$	−36.0443	−1.48518
$590$	−8.11267	−0.333993
$591$	0	0
$592$	−25.1416	−1.03331
$593$	20.6633	0.848538	0.424269	−	0.905536i	$-0.360531\pi$
$593$	20.6633	0.848538	0.424269	+	0.905536i	$0.360531\pi$
$594$	0	0
$595$	−19.8611	−0.814224
$596$	−15.5573	−0.637254
$597$	0	0
$598$	−2.64801	−0.108285
$599$	−42.3558	−1.73061	−0.865305	−	0.501246i	$-0.832875\pi$
$599$	−42.3558	−1.73061	−0.865305	+	0.501246i	$0.832875\pi$
$600$	0	0
$601$	−12.4973	−0.509776	−0.254888	−	0.966971i	$-0.582039\pi$
$601$	−12.4973	−0.509776	−0.254888	+	0.966971i	$0.582039\pi$
$602$	1.82030	0.0741900
$603$	0	0
$604$	2.24173	0.0912147
$605$	−9.63393	−0.391675
$606$	0	0
$607$	−33.9932	−1.37974	−0.689871	−	0.723932i	$-0.742333\pi$
$607$	−33.9932	−1.37974	−0.689871	+	0.723932i	$0.742333\pi$
$608$	13.4298	0.544651
$609$	0	0
$610$	−6.68277	−0.270578
$611$	−11.4266	−0.462271
$612$	0	0
$613$	41.5833	1.67953	0.839766	−	0.542948i	$-0.182692\pi$
$613$	41.5833	1.67953	0.839766	+	0.542948i	$0.182692\pi$
$614$	−0.429273	−0.0173240
$615$	0	0
$616$	−3.22499	−0.129939
$617$	−20.4430	−0.823005	−0.411503	−	0.911409i	$-0.634996\pi$
$617$	−20.4430	−0.823005	−0.411503	+	0.911409i	$0.634996\pi$
$618$	0	0
$619$	34.4729	1.38558	0.692791	−	0.721139i	$-0.256381\pi$
$619$	34.4729	1.38558	0.692791	+	0.721139i	$0.256381\pi$
$620$	−60.9991	−2.44978
$621$	0	0
$622$	−7.38216	−0.295998
$623$	−0.196140	−0.00785817
$624$	0	0
$625$	4.01184	0.160473
$626$	5.57743	0.222919
$627$	0	0
$628$	−35.1790	−1.40379
$629$	−38.6838	−1.54242
$630$	0	0
$631$	42.8278	1.70495	0.852473	−	0.522771i	$-0.175102\pi$
$631$	42.8278	1.70495	0.852473	+	0.522771i	$0.175102\pi$
$632$	−2.17416	−0.0864835
$633$	0	0
$634$	1.86683	0.0741414
$635$	−3.66581	−0.145473
$636$	0	0
$637$	1.88240	0.0745834
$638$	−5.28498	−0.209234
$639$	0	0
$640$	30.0700	1.18862
$641$	2.54643	0.100578	0.0502889	−	0.998735i	$-0.483986\pi$
$641$	2.54643	0.100578	0.0502889	+	0.998735i	$0.483986\pi$
$642$	0	0
$643$	−7.78508	−0.307014	−0.153507	−	0.988148i	$-0.549057\pi$
$643$	−7.78508	−0.307014	−0.153507	+	0.988148i	$0.549057\pi$
$644$	9.49253	0.374058
$645$	0	0
$646$	6.40537	0.252016
$647$	−27.6341	−1.08641	−0.543205	−	0.839600i	$-0.682789\pi$
$647$	−27.6341	−1.08641	−0.543205	+	0.839600i	$0.682789\pi$
$648$	0	0
$649$	22.5154	0.883807
$650$	−4.51739	−0.177186
$651$	0	0
$652$	−46.9169	−1.83741
$653$	35.9520	1.40691	0.703454	−	0.710741i	$-0.251640\pi$
$653$	35.9520	1.40691	0.703454	+	0.710741i	$0.251640\pi$
$654$	0	0
$655$	29.7983	1.16432
$656$	−13.3861	−0.522638
$657$	0	0
$658$	−1.72637	−0.0673007
$659$	15.3056	0.596222	0.298111	−	0.954531i	$-0.403644\pi$
$659$	15.3056	0.596222	0.298111	+	0.954531i	$0.403644\pi$
$660$	0	0
$661$	11.6672	0.453801	0.226900	−	0.973918i	$-0.427141\pi$
$661$	11.6672	0.453801	0.226900	+	0.973918i	$0.427141\pi$
$662$	−1.83510	−0.0713233
$663$	0	0
$664$	9.14819	0.355019
$665$	15.2389	0.590940
$666$	0	0
$667$	31.7675	1.23004
$668$	8.72416	0.337548
$669$	0	0
$670$	−13.9305	−0.538181
$671$	18.5470	0.715998
$672$	0	0
$673$	39.0480	1.50519	0.752594	−	0.658484i	$-0.228802\pi$
$673$	39.0480	1.50519	0.752594	+	0.658484i	$0.228802\pi$
$674$	9.95470	0.383441
$675$	0	0
$676$	18.1483	0.698010
$677$	−6.62387	−0.254576	−0.127288	−	0.991866i	$-0.540627\pi$
$677$	−6.62387	−0.254576	−0.127288	+	0.991866i	$0.540627\pi$
$678$	0	0
$679$	4.42233	0.169714
$680$	22.1370	0.848914
$681$	0	0
$682$	−7.13498	−0.273212
$683$	13.6588	0.522639	0.261320	−	0.965252i	$-0.415842\pi$
$683$	13.6588	0.522639	0.261320	+	0.965252i	$0.415842\pi$
$684$	0	0
$685$	−70.9630	−2.71136
$686$	0.284398	0.0108584
$687$	0	0
$688$	22.5379	0.859248
$689$	−6.84350	−0.260717
$690$	0	0
$691$	−42.2624	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$691$	−42.2624	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$692$	−12.3322	−0.468801
$693$	0	0
$694$	7.29610	0.276956
$695$	36.0197	1.36630
$696$	0	0
$697$	−20.5963	−0.780140
$698$	−5.57307	−0.210944
$699$	0	0
$700$	16.1938	0.612069
$701$	−38.9352	−1.47056	−0.735281	−	0.677763i	$-0.762950\pi$
$701$	−38.9352	−1.47056	−0.735281	+	0.677763i	$0.762950\pi$
$702$	0	0
$703$	29.6812	1.11945
$704$	−17.7185	−0.667793
$705$	0	0
$706$	−0.146921	−0.00552946
$707$	15.2503	0.573548
$708$	0	0
$709$	0.580801	0.0218125	0.0109062	−	0.999941i	$-0.496528\pi$
$709$	0.580801	0.0218125	0.0109062	+	0.999941i	$0.496528\pi$
$710$	9.34648	0.350767
$711$	0	0
$712$	0.218615	0.00819296
$713$	42.8877	1.60616
$714$	0	0
$715$	19.9662	0.746693
$716$	32.2385	1.20481
$717$	0	0
$718$	−8.30824	−0.310061
$719$	−4.96075	−0.185005	−0.0925025	−	0.995712i	$-0.529487\pi$
$719$	−4.96075	−0.185005	−0.0925025	+	0.995712i	$0.529487\pi$
$720$	0	0
$721$	10.2948	0.383397
$722$	0.488880	0.0181942
$723$	0	0
$724$	−30.6895	−1.14057
$725$	54.1940	2.01271
$726$	0	0
$727$	8.47781	0.314425	0.157212	−	0.987565i	$-0.449749\pi$
$727$	8.47781	0.314425	0.157212	+	0.987565i	$0.449749\pi$
$728$	−2.09811	−0.0777610
$729$	0	0
$730$	6.01868	0.222761
$731$	34.6776	1.28260
$732$	0	0
$733$	−1.31188	−0.0484553	−0.0242276	−	0.999706i	$-0.507713\pi$
$733$	−1.31188	−0.0484553	−0.0242276	+	0.999706i	$0.507713\pi$
$734$	8.01913	0.295991
$735$	0	0
$736$	−15.9796	−0.589016
$737$	38.6618	1.42413
$738$	0	0
$739$	0.732368	0.0269406	0.0134703	−	0.999909i	$-0.495712\pi$
$739$	0.732368	0.0269406	0.0134703	+	0.999909i	$0.495712\pi$
$740$	50.2305	1.84651
$741$	0	0
$742$	−1.03394	−0.0379570
$743$	−4.48518	−0.164545	−0.0822726	−	0.996610i	$-0.526218\pi$
$743$	−4.48518	−0.164545	−0.0822726	+	0.996610i	$0.526218\pi$
$744$	0	0
$745$	29.7169	1.08874
$746$	−9.50168	−0.347881
$747$	0	0
$748$	−30.0848	−1.10001
$749$	5.29148	0.193346
$750$	0	0
$751$	42.4145	1.54773	0.773864	−	0.633352i	$-0.218321\pi$
$751$	42.4145	1.54773	0.773864	+	0.633352i	$0.218321\pi$
$752$	−21.3748	−0.779459
$753$	0	0
$754$	−3.43828	−0.125215
$755$	−4.28205	−0.155840
$756$	0	0
$757$	−19.0296	−0.691642	−0.345821	−	0.938301i	$-0.612400\pi$
$757$	−19.0296	−0.691642	−0.345821	+	0.938301i	$0.612400\pi$
$758$	2.15149	0.0781457
$759$	0	0
$760$	−16.9852	−0.616117
$761$	4.59091	0.166420	0.0832102	−	0.996532i	$-0.473483\pi$
$761$	4.59091	0.166420	0.0832102	+	0.996532i	$0.473483\pi$
$762$	0	0
$763$	−1.14135	−0.0413197
$764$	−41.4150	−1.49834
$765$	0	0
$766$	−3.97898	−0.143766
$767$	14.6480	0.528909
$768$	0	0
$769$	−12.1471	−0.438037	−0.219019	−	0.975721i	$-0.570286\pi$
$769$	−12.1471	−0.438037	−0.219019	+	0.975721i	$0.570286\pi$
$770$	3.01655	0.108709
$771$	0	0
$772$	25.9374	0.933507
$773$	−30.0148	−1.07956	−0.539778	−	0.841807i	$-0.681492\pi$
$773$	−30.0148	−1.07956	−0.539778	+	0.841807i	$0.681492\pi$
$774$	0	0
$775$	73.1644	2.62814
$776$	−4.92909	−0.176944
$777$	0	0
$778$	−5.73126	−0.205476
$779$	15.8031	0.566203
$780$	0	0
$781$	−25.9396	−0.928194
$782$	−7.62150	−0.272544
$783$	0	0
$784$	3.52125	0.125759
$785$	67.1973	2.39837
$786$	0	0
$787$	−38.4452	−1.37042	−0.685211	−	0.728344i	$-0.740290\pi$
$787$	−38.4452	−1.37042	−0.685211	+	0.728344i	$0.740290\pi$
$788$	25.7532	0.917419
$789$	0	0
$790$	2.03364	0.0723536
$791$	5.87786	0.208993
$792$	0	0
$793$	12.0662	0.428484
$794$	−1.39720	−0.0495847
$795$	0	0
$796$	18.4832	0.655118
$797$	−36.2045	−1.28243	−0.641214	−	0.767362i	$-0.721569\pi$
$797$	−36.2045	−1.28243	−0.641214	+	0.767362i	$0.721569\pi$
$798$	0	0
$799$	−32.8880	−1.16350
$800$	−27.2605	−0.963804
$801$	0	0
$802$	2.90448	0.102561
$803$	−16.7039	−0.589467
$804$	0	0
$805$	−18.1322	−0.639076
$806$	−4.64185	−0.163502
$807$	0	0
$808$	−16.9979	−0.597983
$809$	4.38270	0.154088	0.0770438	−	0.997028i	$-0.475452\pi$
$809$	4.38270	0.154088	0.0770438	+	0.997028i	$0.475452\pi$
$810$	0	0
$811$	10.5122	0.369133	0.184566	−	0.982820i	$-0.440912\pi$
$811$	10.5122	0.369133	0.184566	+	0.982820i	$0.440912\pi$
$812$	12.3255	0.432540
$813$	0	0
$814$	5.87539	0.205932
$815$	89.6185	3.13920
$816$	0	0
$817$	−26.6073	−0.930871
$818$	6.57623	0.229932
$819$	0	0
$820$	26.7441	0.933945
$821$	−2.32555	−0.0811623	−0.0405812	−	0.999176i	$-0.512921\pi$
$821$	−2.32555	−0.0811623	−0.0405812	+	0.999176i	$0.512921\pi$
$822$	0	0
$823$	−12.2091	−0.425581	−0.212790	−	0.977098i	$-0.568255\pi$
$823$	−12.2091	−0.425581	−0.212790	+	0.977098i	$0.568255\pi$
$824$	−11.4745	−0.399732
$825$	0	0
$826$	2.21306	0.0770023
$827$	35.3401	1.22890	0.614448	−	0.788958i	$-0.289379\pi$
$827$	35.3401	1.22890	0.614448	+	0.788958i	$0.289379\pi$
$828$	0	0
$829$	−38.3442	−1.33175	−0.665875	−	0.746063i	$-0.731942\pi$
$829$	−38.3442	−1.33175	−0.665875	+	0.746063i	$0.731942\pi$
$830$	−8.55691	−0.297015
$831$	0	0
$832$	−11.5273	−0.399636
$833$	5.41792	0.187720
$834$	0	0
$835$	−16.6645	−0.576699
$836$	23.0834	0.798355
$837$	0	0
$838$	7.27636	0.251358
$839$	24.1433	0.833521	0.416760	−	0.909016i	$-0.363165\pi$
$839$	24.1433	0.833521	0.416760	+	0.909016i	$0.363165\pi$
$840$	0	0
$841$	12.2482	0.422353
$842$	3.88065	0.133736
$843$	0	0
$844$	−10.9304	−0.376240
$845$	−34.6660	−1.19255
$846$	0	0
$847$	2.62805	0.0903008
$848$	−12.8016	−0.439607
$849$	0	0
$850$	−13.0019	−0.445963
$851$	−35.3164	−1.21063
$852$	0	0
$853$	17.0066	0.582294	0.291147	−	0.956678i	$-0.405963\pi$
$853$	17.0066	0.582294	0.291147	+	0.956678i	$0.405963\pi$
$854$	1.82300	0.0623818
$855$	0	0
$856$	−5.89783	−0.201584
$857$	−20.6638	−0.705862	−0.352931	−	0.935649i	$-0.614815\pi$
$857$	−20.6638	−0.705862	−0.352931	+	0.935649i	$0.614815\pi$
$858$	0	0
$859$	−5.64045	−0.192450	−0.0962248	−	0.995360i	$-0.530677\pi$
$859$	−5.64045	−0.192450	−0.0962248	+	0.995360i	$0.530677\pi$
$860$	−45.0285	−1.53546
$861$	0	0
$862$	6.46342	0.220145
$863$	−40.7214	−1.38617	−0.693086	−	0.720855i	$-0.743749\pi$
$863$	−40.7214	−1.38617	−0.693086	+	0.720855i	$0.743749\pi$
$864$	0	0
$865$	23.5565	0.800944
$866$	0.708171	0.0240646
$867$	0	0
$868$	16.6400	0.564799
$869$	−5.64404	−0.191461
$870$	0	0
$871$	25.1525	0.852258
$872$	1.27214	0.0430801
$873$	0	0
$874$	5.84780	0.197805
$875$	−12.6037	−0.426082
$876$	0	0
$877$	36.1198	1.21968	0.609840	−	0.792525i	$-0.291234\pi$
$877$	36.1198	1.21968	0.609840	+	0.792525i	$0.291234\pi$
$878$	−10.2679	−0.346523
$879$	0	0
$880$	37.3491	1.25904
$881$	47.4231	1.59773	0.798863	−	0.601513i	$-0.205435\pi$
$881$	47.4231	1.59773	0.798863	+	0.601513i	$0.205435\pi$
$882$	0	0
$883$	−27.5922	−0.928551	−0.464276	−	0.885691i	$-0.653685\pi$
$883$	−27.5922	−0.928551	−0.464276	+	0.885691i	$0.653685\pi$
$884$	−19.5725	−0.658294
$885$	0	0
$886$	−11.7202	−0.393748
$887$	−34.4617	−1.15711	−0.578556	−	0.815643i	$-0.696384\pi$
$887$	−34.4617	−1.15711	−0.578556	+	0.815643i	$0.696384\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.204485	−0.00685437
$891$	0	0
$892$	42.0001	1.40627
$893$	25.2342	0.844431
$894$	0	0
$895$	−61.5804	−2.05841
$896$	−8.20281	−0.274037
$897$	0	0
$898$	−3.52696	−0.117696
$899$	55.6871	1.85727
$900$	0	0
$901$	−19.6970	−0.656201
$902$	3.12822	0.104158
$903$	0	0
$904$	−6.55141	−0.217897
$905$	58.6217	1.94865
$906$	0	0
$907$	−49.9156	−1.65742	−0.828711	−	0.559677i	$-0.810925\pi$
$907$	−49.9156	−1.65742	−0.828711	+	0.559677i	$0.810925\pi$
$908$	−15.3821	−0.510473
$909$	0	0
$910$	1.96250	0.0650562
$911$	14.2520	0.472191	0.236096	−	0.971730i	$-0.424132\pi$
$911$	14.2520	0.472191	0.236096	+	0.971730i	$0.424132\pi$
$912$	0	0
$913$	23.7483	0.785955
$914$	−0.202122	−0.00668560
$915$	0	0
$916$	24.3405	0.804231
$917$	−8.12871	−0.268434
$918$	0	0
$919$	50.6386	1.67041	0.835207	−	0.549936i	$-0.185348\pi$
$919$	50.6386	1.67041	0.835207	+	0.549936i	$0.185348\pi$
$920$	20.2100	0.666304
$921$	0	0
$922$	10.0588	0.331269
$923$	−16.8757	−0.555471
$924$	0	0
$925$	−60.2483	−1.98095
$926$	−9.83301	−0.323133
$927$	0	0
$928$	−20.7486	−0.681106
$929$	13.4169	0.440195	0.220097	−	0.975478i	$-0.429362\pi$
$929$	13.4169	0.440195	0.220097	+	0.975478i	$0.429362\pi$
$930$	0	0
$931$	−4.15704	−0.136242
$932$	−11.2695	−0.369145
$933$	0	0
$934$	−0.889746	−0.0291134
$935$	57.4667	1.87936
$936$	0	0
$937$	28.6008	0.934349	0.467174	−	0.884165i	$-0.345272\pi$
$937$	28.6008	0.934349	0.467174	+	0.884165i	$0.345272\pi$
$938$	3.80011	0.124078
$939$	0	0
$940$	42.7048	1.39288
$941$	17.9194	0.584154	0.292077	−	0.956395i	$-0.405654\pi$
$941$	17.9194	0.584154	0.292077	+	0.956395i	$0.405654\pi$
$942$	0	0
$943$	−18.8034	−0.612324
$944$	27.4008	0.891820
$945$	0	0
$946$	−5.26692	−0.171242
$947$	45.2544	1.47057	0.735285	−	0.677758i	$-0.237048\pi$
$947$	45.2544	1.47057	0.735285	+	0.677758i	$0.237048\pi$
$948$	0	0
$949$	−10.8672	−0.352763
$950$	9.97608	0.323667
$951$	0	0
$952$	−6.03876	−0.195717
$953$	−26.0590	−0.844135	−0.422067	−	0.906564i	$-0.638695\pi$
$953$	−26.0590	−0.844135	−0.422067	+	0.906564i	$0.638695\pi$
$954$	0	0
$955$	79.1091	2.55991
$956$	44.6438	1.44388
$957$	0	0
$958$	5.43616	0.175634
$959$	19.3581	0.625104
$960$	0	0
$961$	44.1803	1.42517
$962$	3.82239	0.123239
$963$	0	0
$964$	17.7996	0.573285
$965$	−49.5444	−1.59489
$966$	0	0
$967$	37.1768	1.19553	0.597763	−	0.801673i	$-0.296056\pi$
$967$	37.1768	1.19553	0.597763	+	0.801673i	$0.296056\pi$
$968$	−2.92920	−0.0941480
$969$	0	0
$970$	4.61051	0.148034
$971$	10.6319	0.341194	0.170597	−	0.985341i	$-0.445430\pi$
$971$	10.6319	0.341194	0.170597	+	0.985341i	$0.445430\pi$
$972$	0	0
$973$	−9.82585	−0.315002
$974$	−10.5844	−0.339147
$975$	0	0
$976$	22.5713	0.722489
$977$	−34.6940	−1.10996	−0.554979	−	0.831864i	$-0.687274\pi$
$977$	−34.6940	−1.10996	−0.554979	+	0.831864i	$0.687274\pi$
$978$	0	0
$979$	0.567517	0.0181379
$980$	−7.03512	−0.224729
$981$	0	0
$982$	−4.09409	−0.130648
$983$	29.1744	0.930518	0.465259	−	0.885175i	$-0.345961\pi$
$983$	29.1744	0.930518	0.465259	+	0.885175i	$0.345961\pi$
$984$	0	0
$985$	−49.1925	−1.56740
$986$	−9.89606	−0.315155
$987$	0	0
$988$	15.0175	0.477771
$989$	31.6590	1.00670
$990$	0	0
$991$	−37.4717	−1.19033	−0.595163	−	0.803605i	$-0.702913\pi$
$991$	−37.4717	−1.19033	−0.595163	+	0.803605i	$0.702913\pi$
$992$	−28.0116	−0.889369
$993$	0	0
$994$	−2.54963	−0.0808695
$995$	−35.3057	−1.11927
$996$	0	0
$997$	−20.6670	−0.654530	−0.327265	−	0.944933i	$-0.606127\pi$
$997$	−20.6670	−0.654530	−0.327265	+	0.944933i	$0.606127\pi$
$998$	−8.64924	−0.273787
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.9		16
3.2	odd	2		2667.2.a.n.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.8	✓	16	3.2	odd	2
8001.2.a.s.1.9		16	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.284398 q^{2} -1.91912 q^{4} +3.66581 q^{5} -1.00000 q^{7} +1.11459 q^{8} -1.04255 q^{10} +2.89343 q^{11} +1.88240 q^{13} +0.284398 q^{14} +3.52125 q^{16} +5.41792 q^{17} -4.15704 q^{19} -7.03512 q^{20} -0.822888 q^{22} +4.94630 q^{23} +8.43817 q^{25} -0.535352 q^{26} +1.91912 q^{28} +6.42248 q^{29} +8.67066 q^{31} -3.23062 q^{32} -1.54085 q^{34} -3.66581 q^{35} -7.13997 q^{37} +1.18226 q^{38} +4.08588 q^{40} -3.80151 q^{41} +6.40053 q^{43} -5.55284 q^{44} -1.40672 q^{46} -6.07023 q^{47} +1.00000 q^{49} -2.39980 q^{50} -3.61255 q^{52} -3.63552 q^{53} +10.6068 q^{55} -1.11459 q^{56} -1.82654 q^{58} +7.78156 q^{59} +6.41002 q^{61} -2.46592 q^{62} -6.12371 q^{64} +6.90052 q^{65} +13.3619 q^{67} -10.3976 q^{68} +1.04255 q^{70} -8.96501 q^{71} -5.77303 q^{73} +2.03060 q^{74} +7.97785 q^{76} -2.89343 q^{77} -1.95064 q^{79} +12.9082 q^{80} +1.08114 q^{82} +8.20767 q^{83} +19.8611 q^{85} -1.82030 q^{86} +3.22499 q^{88} +0.196140 q^{89} -1.88240 q^{91} -9.49253 q^{92} +1.72637 q^{94} -15.2389 q^{95} -4.42233 q^{97} -0.284398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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