LMFDB - Embedded newform 8001.2.a.t.1.7

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 = [16,2,0,12,9] 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} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.191617$ 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)

$f(q)$	$=$	\(q-0.191617 q^{2} -1.96328 q^{4} -3.92161 q^{5} -1.00000 q^{7} +0.759431 q^{8} +0.751445 q^{10} +5.26691 q^{11} +6.07655 q^{13} +0.191617 q^{14} +3.78105 q^{16} +4.12322 q^{17} +0.0374083 q^{19} +7.69922 q^{20} -1.00923 q^{22} +3.57588 q^{23} +10.3790 q^{25} -1.16437 q^{26} +1.96328 q^{28} +2.19905 q^{29} +4.14885 q^{31} -2.24337 q^{32} -0.790078 q^{34} +3.92161 q^{35} -9.31219 q^{37} -0.00716804 q^{38} -2.97819 q^{40} +9.48327 q^{41} -11.4342 q^{43} -10.3404 q^{44} -0.685199 q^{46} +3.87164 q^{47} +1.00000 q^{49} -1.98879 q^{50} -11.9300 q^{52} +9.97552 q^{53} -20.6547 q^{55} -0.759431 q^{56} -0.421374 q^{58} +9.91129 q^{59} -0.382013 q^{61} -0.794988 q^{62} -7.13223 q^{64} -23.8298 q^{65} +4.40596 q^{67} -8.09505 q^{68} -0.751445 q^{70} +10.3123 q^{71} -3.37816 q^{73} +1.78437 q^{74} -0.0734430 q^{76} -5.26691 q^{77} +9.73339 q^{79} -14.8278 q^{80} -1.81715 q^{82} -16.5257 q^{83} -16.1697 q^{85} +2.19098 q^{86} +3.99985 q^{88} +2.53028 q^{89} -6.07655 q^{91} -7.02047 q^{92} -0.741870 q^{94} -0.146700 q^{95} -5.85331 q^{97} -0.191617 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * 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 $

$2$	−0.191617	−0.135493	−0.0677467	−	0.997703i	$-0.521581\pi$
$2$	−0.191617	−0.135493	−0.0677467	+	0.997703i	$0.521581\pi$
$3$	0	0
$3$	0	0
$4$	−1.96328	−0.981642
$5$	−3.92161	−1.75380	−0.876898	−	0.480677i	$-0.840391\pi$
$5$	−3.92161	−1.75380	−0.876898	+	0.480677i	$0.840391\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.759431	0.268499
$9$	0	0
$10$	0.751445	0.237628
$11$	5.26691	1.58803	0.794016	−	0.607897i	$-0.207986\pi$
$11$	5.26691	1.58803	0.794016	+	0.607897i	$0.207986\pi$
$12$	0	0
$13$	6.07655	1.68533	0.842665	−	0.538438i	$-0.180985\pi$
$13$	6.07655	1.68533	0.842665	+	0.538438i	$0.180985\pi$
$14$	0.191617	0.0512117
$15$	0	0
$16$	3.78105	0.945262
$17$	4.12322	1.00003	0.500014	−	0.866017i	$-0.333328\pi$
$17$	4.12322	1.00003	0.500014	+	0.866017i	$0.333328\pi$
$18$	0	0
$19$	0.0374083	0.00858204	0.00429102	−	0.999991i	$-0.498634\pi$
$19$	0.0374083	0.00858204	0.00429102	+	0.999991i	$0.498634\pi$
$20$	7.69922	1.72160
$21$	0	0
$22$	−1.00923	−0.215168
$23$	3.57588	0.745623	0.372812	−	0.927907i	$-0.378394\pi$
$23$	3.57588	0.745623	0.372812	+	0.927907i	$0.378394\pi$
$24$	0	0
$25$	10.3790	2.07580
$26$	−1.16437	−0.228351
$27$	0	0
$28$	1.96328	0.371026
$29$	2.19905	0.408353	0.204177	−	0.978934i	$-0.434548\pi$
$29$	2.19905	0.408353	0.204177	+	0.978934i	$0.434548\pi$
$30$	0	0
$31$	4.14885	0.745155	0.372577	−	0.928001i	$-0.378474\pi$
$31$	4.14885	0.745155	0.372577	+	0.928001i	$0.378474\pi$
$32$	−2.24337	−0.396576
$33$	0	0
$34$	−0.790078	−0.135497
$35$	3.92161	0.662872
$36$	0	0
$37$	−9.31219	−1.53091	−0.765457	−	0.643487i	$-0.777487\pi$
$37$	−9.31219	−1.53091	−0.765457	+	0.643487i	$0.777487\pi$
$38$	−0.00716804	−0.00116281
$39$	0	0
$40$	−2.97819	−0.470893
$41$	9.48327	1.48104	0.740519	−	0.672035i	$-0.234580\pi$
$41$	9.48327	1.48104	0.740519	+	0.672035i	$0.234580\pi$
$42$	0	0
$43$	−11.4342	−1.74370	−0.871850	−	0.489774i	$-0.837079\pi$
$43$	−11.4342	−1.74370	−0.871850	+	0.489774i	$0.837079\pi$
$44$	−10.3404	−1.55888
$45$	0	0
$46$	−0.685199	−0.101027
$47$	3.87164	0.564737	0.282368	−	0.959306i	$-0.408880\pi$
$47$	3.87164	0.564737	0.282368	+	0.959306i	$0.408880\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.98879	−0.281257
$51$	0	0
$52$	−11.9300	−1.65439
$53$	9.97552	1.37024	0.685122	−	0.728429i	$-0.259749\pi$
$53$	9.97552	1.37024	0.685122	+	0.728429i	$0.259749\pi$
$54$	0	0
$55$	−20.6547	−2.78508
$56$	−0.759431	−0.101483
$57$	0	0
$58$	−0.421374	−0.0553292
$59$	9.91129	1.29034	0.645170	−	0.764039i	$-0.276786\pi$
$59$	9.91129	1.29034	0.645170	+	0.764039i	$0.276786\pi$
$60$	0	0
$61$	−0.382013	−0.0489118	−0.0244559	−	0.999701i	$-0.507785\pi$
$61$	−0.382013	−0.0489118	−0.0244559	+	0.999701i	$0.507785\pi$
$62$	−0.794988	−0.100964
$63$	0	0
$64$	−7.13223	−0.891528
$65$	−23.8298	−2.95573
$66$	0	0
$67$	4.40596	0.538273	0.269137	−	0.963102i	$-0.413262\pi$
$67$	4.40596	0.538273	0.269137	+	0.963102i	$0.413262\pi$
$68$	−8.09505	−0.981669
$69$	0	0
$70$	−0.751445	−0.0898148
$71$	10.3123	1.22385	0.611925	−	0.790916i	$-0.290396\pi$
$71$	10.3123	1.22385	0.611925	+	0.790916i	$0.290396\pi$
$72$	0	0
$73$	−3.37816	−0.395383	−0.197692	−	0.980264i	$-0.563344\pi$
$73$	−3.37816	−0.395383	−0.197692	+	0.980264i	$0.563344\pi$
$74$	1.78437	0.207429
$75$	0	0
$76$	−0.0734430	−0.00842449
$77$	−5.26691	−0.600220
$78$	0	0
$79$	9.73339	1.09509	0.547546	−	0.836776i	$-0.315562\pi$
$79$	9.73339	1.09509	0.547546	+	0.836776i	$0.315562\pi$
$80$	−14.8278	−1.65780
$81$	0	0
$82$	−1.81715	−0.200671
$83$	−16.5257	−1.81393	−0.906964	−	0.421209i	$-0.861606\pi$
$83$	−16.5257	−1.81393	−0.906964	+	0.421209i	$0.861606\pi$
$84$	0	0
$85$	−16.1697	−1.75385
$86$	2.19098	0.236260
$87$	0	0
$88$	3.99985	0.426386
$89$	2.53028	0.268209	0.134105	−	0.990967i	$-0.457184\pi$
$89$	2.53028	0.268209	0.134105	+	0.990967i	$0.457184\pi$
$90$	0	0
$91$	−6.07655	−0.636995
$92$	−7.02047	−0.731935
$93$	0	0
$94$	−0.741870	−0.0765181
$95$	−0.146700	−0.0150512
$96$	0	0
$97$	−5.85331	−0.594314	−0.297157	−	0.954829i	$-0.596038\pi$
$97$	−5.85331	−0.594314	−0.297157	+	0.954829i	$0.596038\pi$
$98$	−0.191617	−0.0193562
$99$	0	0
$100$	−20.3769	−2.03769
$101$	1.78412	0.177527	0.0887633	−	0.996053i	$-0.471709\pi$
$101$	1.78412	0.177527	0.0887633	+	0.996053i	$0.471709\pi$
$102$	0	0
$103$	−8.17058	−0.805071	−0.402536	−	0.915404i	$-0.631871\pi$
$103$	−8.17058	−0.805071	−0.402536	+	0.915404i	$0.631871\pi$
$104$	4.61471	0.452510
$105$	0	0
$106$	−1.91148	−0.185659
$107$	−3.22757	−0.312021	−0.156010	−	0.987755i	$-0.549863\pi$
$107$	−3.22757	−0.312021	−0.156010	+	0.987755i	$0.549863\pi$
$108$	0	0
$109$	16.9092	1.61961	0.809805	−	0.586700i	$-0.199573\pi$
$109$	16.9092	1.61961	0.809805	+	0.586700i	$0.199573\pi$
$110$	3.95779	0.377360
$111$	0	0
$112$	−3.78105	−0.357275
$113$	−5.58588	−0.525476	−0.262738	−	0.964867i	$-0.584625\pi$
$113$	−5.58588	−0.525476	−0.262738	+	0.964867i	$0.584625\pi$
$114$	0	0
$115$	−14.0232	−1.30767
$116$	−4.31736	−0.400857
$117$	0	0
$118$	−1.89917	−0.174833
$119$	−4.12322	−0.377975
$120$	0	0
$121$	16.7403	1.52185
$122$	0.0732001	0.00662722
$123$	0	0
$124$	−8.14536	−0.731475
$125$	−21.0943	−1.88673
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	5.85340	0.517372
$129$	0	0
$130$	4.56619	0.400481
$131$	15.7005	1.37176	0.685879	−	0.727716i	$-0.259418\pi$
$131$	15.7005	1.37176	0.685879	+	0.727716i	$0.259418\pi$
$132$	0	0
$133$	−0.0374083	−0.00324371
$134$	−0.844254	−0.0729325
$135$	0	0
$136$	3.13130	0.268507
$137$	−7.97414	−0.681277	−0.340639	−	0.940194i	$-0.610643\pi$
$137$	−7.97414	−0.681277	−0.340639	+	0.940194i	$0.610643\pi$
$138$	0	0
$139$	−3.40873	−0.289125	−0.144562	−	0.989496i	$-0.546177\pi$
$139$	−3.40873	−0.289125	−0.144562	+	0.989496i	$0.546177\pi$
$140$	−7.69922	−0.650703
$141$	0	0
$142$	−1.97601	−0.165823
$143$	32.0046	2.67636
$144$	0	0
$145$	−8.62381	−0.716169
$146$	0.647310	0.0535718
$147$	0	0
$148$	18.2825	1.50281
$149$	2.34920	0.192454	0.0962271	−	0.995359i	$-0.469322\pi$
$149$	2.34920	0.192454	0.0962271	+	0.995359i	$0.469322\pi$
$150$	0	0
$151$	−3.42902	−0.279050	−0.139525	−	0.990219i	$-0.544558\pi$
$151$	−3.42902	−0.279050	−0.139525	+	0.990219i	$0.544558\pi$
$152$	0.0284090	0.00230427
$153$	0	0
$154$	1.00923	0.0813258
$155$	−16.2701	−1.30685
$156$	0	0
$157$	−19.8947	−1.58777	−0.793886	−	0.608067i	$-0.791945\pi$
$157$	−19.8947	−1.58777	−0.793886	+	0.608067i	$0.791945\pi$
$158$	−1.86508	−0.148378
$159$	0	0
$160$	8.79762	0.695513
$161$	−3.57588	−0.281819
$162$	0	0
$163$	6.68481	0.523595	0.261797	−	0.965123i	$-0.415685\pi$
$163$	6.68481	0.523595	0.261797	+	0.965123i	$0.415685\pi$
$164$	−18.6183	−1.45385
$165$	0	0
$166$	3.16659	0.245775
$167$	−16.0276	−1.24025	−0.620126	−	0.784502i	$-0.712918\pi$
$167$	−16.0276	−1.24025	−0.620126	+	0.784502i	$0.712918\pi$
$168$	0	0
$169$	23.9244	1.84034
$170$	3.09837	0.237634
$171$	0	0
$172$	22.4486	1.71169
$173$	14.8583	1.12965	0.564827	−	0.825209i	$-0.308943\pi$
$173$	14.8583	1.12965	0.564827	+	0.825209i	$0.308943\pi$
$174$	0	0
$175$	−10.3790	−0.784578
$176$	19.9144	1.50111
$177$	0	0
$178$	−0.484844	−0.0363406
$179$	10.4016	0.777456	0.388728	−	0.921353i	$-0.372915\pi$
$179$	10.4016	0.777456	0.388728	+	0.921353i	$0.372915\pi$
$180$	0	0
$181$	−6.10810	−0.454011	−0.227006	−	0.973893i	$-0.572894\pi$
$181$	−6.10810	−0.454011	−0.227006	+	0.973893i	$0.572894\pi$
$182$	1.16437	0.0863086
$183$	0	0
$184$	2.71564	0.200199
$185$	36.5187	2.68491
$186$	0	0
$187$	21.7166	1.58808
$188$	−7.60113	−0.554369
$189$	0	0
$190$	0.0281102	0.00203933
$191$	−8.67944	−0.628022	−0.314011	−	0.949419i	$-0.601673\pi$
$191$	−8.67944	−0.628022	−0.314011	+	0.949419i	$0.601673\pi$
$192$	0	0
$193$	−4.28438	−0.308396	−0.154198	−	0.988040i	$-0.549279\pi$
$193$	−4.28438	−0.308396	−0.154198	+	0.988040i	$0.549279\pi$
$194$	1.12159	0.0805255
$195$	0	0
$196$	−1.96328	−0.140235
$197$	−20.8740	−1.48721	−0.743606	−	0.668618i	$-0.766886\pi$
$197$	−20.8740	−1.48721	−0.743606	+	0.668618i	$0.766886\pi$
$198$	0	0
$199$	23.3191	1.65305	0.826525	−	0.562900i	$-0.190314\pi$
$199$	23.3191	1.65305	0.826525	+	0.562900i	$0.190314\pi$
$200$	7.88213	0.557350
$201$	0	0
$202$	−0.341867	−0.0240537
$203$	−2.19905	−0.154343
$204$	0	0
$205$	−37.1897	−2.59744
$206$	1.56562	0.109082
$207$	0	0
$208$	22.9757	1.59308
$209$	0.197026	0.0136286
$210$	0	0
$211$	−17.2768	−1.18938	−0.594691	−	0.803954i	$-0.702726\pi$
$211$	−17.2768	−1.18938	−0.594691	+	0.803954i	$0.702726\pi$
$212$	−19.5848	−1.34509
$213$	0	0
$214$	0.618456	0.0422768
$215$	44.8404	3.05809
$216$	0	0
$217$	−4.14885	−0.281642
$218$	−3.24009	−0.219446
$219$	0	0
$220$	40.5511	2.73395
$221$	25.0550	1.68538
$222$	0	0
$223$	26.8248	1.79632	0.898162	−	0.439665i	$-0.144903\pi$
$223$	26.8248	1.79632	0.898162	+	0.439665i	$0.144903\pi$
$224$	2.24337	0.149892
$225$	0	0
$226$	1.07035	0.0711985
$227$	−18.2819	−1.21341	−0.606707	−	0.794925i	$-0.707510\pi$
$227$	−18.2819	−1.21341	−0.606707	+	0.794925i	$0.707510\pi$
$228$	0	0
$229$	4.27326	0.282385	0.141192	−	0.989982i	$-0.454906\pi$
$229$	4.27326	0.282385	0.141192	+	0.989982i	$0.454906\pi$
$230$	2.68708	0.177181
$231$	0	0
$232$	1.67003	0.109643
$233$	0.982041	0.0643357	0.0321678	−	0.999482i	$-0.489759\pi$
$233$	0.982041	0.0643357	0.0321678	+	0.999482i	$0.489759\pi$
$234$	0	0
$235$	−15.1830	−0.990433
$236$	−19.4587	−1.26665
$237$	0	0
$238$	0.790078	0.0512131
$239$	8.88724	0.574868	0.287434	−	0.957800i	$-0.407198\pi$
$239$	8.88724	0.574868	0.287434	+	0.957800i	$0.407198\pi$
$240$	0	0
$241$	−27.4668	−1.76929	−0.884646	−	0.466264i	$-0.845600\pi$
$241$	−27.4668	−1.76929	−0.884646	+	0.466264i	$0.845600\pi$
$242$	−3.20772	−0.206200
$243$	0	0
$244$	0.750000	0.0480138
$245$	−3.92161	−0.250542
$246$	0	0
$247$	0.227313	0.0144636
$248$	3.15076	0.200074
$249$	0	0
$250$	4.04202	0.255640
$251$	7.65870	0.483413	0.241706	−	0.970349i	$-0.422293\pi$
$251$	7.65870	0.483413	0.241706	+	0.970349i	$0.422293\pi$
$252$	0	0
$253$	18.8339	1.18407
$254$	0.191617	0.0120231
$255$	0	0
$256$	13.1428	0.821428
$257$	19.3455	1.20674	0.603369	−	0.797462i	$-0.293824\pi$
$257$	19.3455	1.20674	0.603369	+	0.797462i	$0.293824\pi$
$258$	0	0
$259$	9.31219	0.578631
$260$	46.7847	2.90146
$261$	0	0
$262$	−3.00847	−0.185864
$263$	−12.7413	−0.785660	−0.392830	−	0.919611i	$-0.628504\pi$
$263$	−12.7413	−0.785660	−0.392830	+	0.919611i	$0.628504\pi$
$264$	0	0
$265$	−39.1201	−2.40313
$266$	0.00716804	0.000439501 0
$267$	0	0
$268$	−8.65014	−0.528392
$269$	−9.02960	−0.550545	−0.275272	−	0.961366i	$-0.588768\pi$
$269$	−9.02960	−0.550545	−0.275272	+	0.961366i	$0.588768\pi$
$270$	0	0
$271$	16.6046	1.00866	0.504328	−	0.863512i	$-0.331740\pi$
$271$	16.6046	1.00866	0.504328	+	0.863512i	$0.331740\pi$
$272$	15.5901	0.945289
$273$	0	0
$274$	1.52798	0.0923085
$275$	54.6652	3.29644
$276$	0	0
$277$	−7.79112	−0.468123	−0.234061	−	0.972222i	$-0.575202\pi$
$277$	−7.79112	−0.468123	−0.234061	+	0.972222i	$0.575202\pi$
$278$	0.653169	0.0391745
$279$	0	0
$280$	2.97819	0.177981
$281$	8.94643	0.533700	0.266850	−	0.963738i	$-0.414017\pi$
$281$	8.94643	0.533700	0.266850	+	0.963738i	$0.414017\pi$
$282$	0	0
$283$	−11.5107	−0.684243	−0.342121	−	0.939656i	$-0.611145\pi$
$283$	−11.5107	−0.684243	−0.342121	+	0.939656i	$0.611145\pi$
$284$	−20.2460	−1.20138
$285$	0	0
$286$	−6.13261	−0.362629
$287$	−9.48327	−0.559780
$288$	0	0
$289$	0.000967701 0	5.69236e−5 0
$290$	1.65246	0.0970361
$291$	0	0
$292$	6.63228	0.388125
$293$	4.41505	0.257930	0.128965	−	0.991649i	$-0.458834\pi$
$293$	4.41505	0.257930	0.128965	+	0.991649i	$0.458834\pi$
$294$	0	0
$295$	−38.8682	−2.26299
$296$	−7.07196	−0.411049
$297$	0	0
$298$	−0.450146	−0.0260763
$299$	21.7290	1.25662
$300$	0	0
$301$	11.4342	0.659056
$302$	0.657057	0.0378094
$303$	0	0
$304$	0.141442	0.00811228
$305$	1.49811	0.0857813
$306$	0	0
$307$	4.82910	0.275611	0.137806	−	0.990459i	$-0.455995\pi$
$307$	4.82910	0.275611	0.137806	+	0.990459i	$0.455995\pi$
$308$	10.3404	0.589201
$309$	0	0
$310$	3.11763	0.177069
$311$	15.7618	0.893772	0.446886	−	0.894591i	$-0.352533\pi$
$311$	15.7618	0.893772	0.446886	+	0.894591i	$0.352533\pi$
$312$	0	0
$313$	13.2573	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$313$	13.2573	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$314$	3.81216	0.215132
$315$	0	0
$316$	−19.1094	−1.07499
$317$	−5.19102	−0.291557	−0.145778	−	0.989317i	$-0.546569\pi$
$317$	−5.19102	−0.291557	−0.145778	+	0.989317i	$0.546569\pi$
$318$	0	0
$319$	11.5822	0.648479
$320$	27.9698	1.56356
$321$	0	0
$322$	0.685199	0.0381846
$323$	0.154243	0.00858229
$324$	0	0
$325$	63.0684	3.49841
$326$	−1.28092	−0.0709436
$327$	0	0
$328$	7.20189	0.397658
$329$	−3.87164	−0.213450
$330$	0	0
$331$	−11.1884	−0.614969	−0.307485	−	0.951553i	$-0.599487\pi$
$331$	−11.1884	−0.614969	−0.307485	+	0.951553i	$0.599487\pi$
$332$	32.4446	1.78063
$333$	0	0
$334$	3.07115	0.168046
$335$	−17.2784	−0.944022
$336$	0	0
$337$	17.1501	0.934223	0.467111	−	0.884198i	$-0.345295\pi$
$337$	17.1501	0.934223	0.467111	+	0.884198i	$0.345295\pi$
$338$	−4.58431	−0.249354
$339$	0	0
$340$	31.7456	1.72165
$341$	21.8516	1.18333
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.68348	−0.468182
$345$	0	0
$346$	−2.84709	−0.153061
$347$	−0.0256912	−0.00137917	−0.000689587	−	1.00000i	$-0.500220\pi$
$347$	−0.0256912	−0.00137917	−0.000689587		1.00000i	$0.500220\pi$
$348$	0	0
$349$	23.9479	1.28190	0.640950	−	0.767583i	$-0.278541\pi$
$349$	23.9479	1.28190	0.640950	+	0.767583i	$0.278541\pi$
$350$	1.98879	0.106305
$351$	0	0
$352$	−11.8156	−0.629775
$353$	6.33830	0.337354	0.168677	−	0.985671i	$-0.446051\pi$
$353$	6.33830	0.337354	0.168677	+	0.985671i	$0.446051\pi$
$354$	0	0
$355$	−40.4409	−2.14638
$356$	−4.96766	−0.263285
$357$	0	0
$358$	−1.99313	−0.105340
$359$	10.9811	0.579563	0.289781	−	0.957093i	$-0.406417\pi$
$359$	10.9811	0.579563	0.289781	+	0.957093i	$0.406417\pi$
$360$	0	0
$361$	−18.9986	−0.999926
$362$	1.17041	0.0615155
$363$	0	0
$364$	11.9300	0.625301
$365$	13.2478	0.693421
$366$	0	0
$367$	21.0669	1.09968	0.549840	−	0.835270i	$-0.314689\pi$
$367$	21.0669	1.09968	0.549840	+	0.835270i	$0.314689\pi$
$368$	13.5206	0.704809
$369$	0	0
$370$	−6.99759	−0.363787
$371$	−9.97552	−0.517903
$372$	0	0
$373$	10.6031	0.549009	0.274505	−	0.961586i	$-0.411486\pi$
$373$	10.6031	0.549009	0.274505	+	0.961586i	$0.411486\pi$
$374$	−4.16127	−0.215174
$375$	0	0
$376$	2.94024	0.151631
$377$	13.3626	0.688211
$378$	0	0
$379$	−5.57679	−0.286460	−0.143230	−	0.989689i	$-0.545749\pi$
$379$	−5.57679	−0.286460	−0.143230	+	0.989689i	$0.545749\pi$
$380$	0.288015	0.0147748
$381$	0	0
$382$	1.66312	0.0850928
$383$	−19.7345	−1.00838	−0.504192	−	0.863592i	$-0.668209\pi$
$383$	−19.7345	−1.00838	−0.504192	+	0.863592i	$0.668209\pi$
$384$	0	0
$385$	20.6547	1.05266
$386$	0.820958	0.0417857
$387$	0	0
$388$	11.4917	0.583403
$389$	−16.1098	−0.816799	−0.408399	−	0.912803i	$-0.633913\pi$
$389$	−16.1098	−0.816799	−0.408399	+	0.912803i	$0.633913\pi$
$390$	0	0
$391$	14.7442	0.745645
$392$	0.759431	0.0383570
$393$	0	0
$394$	3.99981	0.201507
$395$	−38.1705	−1.92057
$396$	0	0
$397$	−6.80662	−0.341614	−0.170807	−	0.985304i	$-0.554638\pi$
$397$	−6.80662	−0.341614	−0.170807	+	0.985304i	$0.554638\pi$
$398$	−4.46833	−0.223977
$399$	0	0
$400$	39.2435	1.96217
$401$	17.1989	0.858873	0.429437	−	0.903097i	$-0.358712\pi$
$401$	17.1989	0.858873	0.429437	+	0.903097i	$0.358712\pi$
$402$	0	0
$403$	25.2107	1.25583
$404$	−3.50273	−0.174267
$405$	0	0
$406$	0.421374	0.0209125
$407$	−49.0464	−2.43114
$408$	0	0
$409$	3.13066	0.154801	0.0774005	−	0.997000i	$-0.475338\pi$
$409$	3.13066	0.154801	0.0774005	+	0.997000i	$0.475338\pi$
$410$	7.12615	0.351936
$411$	0	0
$412$	16.0412	0.790291
$413$	−9.91129	−0.487703
$414$	0	0
$415$	64.8071	3.18126
$416$	−13.6320	−0.668362
$417$	0	0
$418$	−0.0377534	−0.00184658
$419$	−18.8437	−0.920576	−0.460288	−	0.887770i	$-0.652254\pi$
$419$	−18.8437	−0.920576	−0.460288	+	0.887770i	$0.652254\pi$
$420$	0	0
$421$	−13.1531	−0.641045	−0.320523	−	0.947241i	$-0.603859\pi$
$421$	−13.1531	−0.641045	−0.320523	+	0.947241i	$0.603859\pi$
$422$	3.31052	0.161153
$423$	0	0
$424$	7.57572	0.367909
$425$	42.7949	2.07586
$426$	0	0
$427$	0.382013	0.0184869
$428$	6.33663	0.306293
$429$	0	0
$430$	−8.59217	−0.414351
$431$	38.5459	1.85669	0.928346	−	0.371716i	$-0.121231\pi$
$431$	38.5459	1.85669	0.928346	+	0.371716i	$0.121231\pi$
$432$	0	0
$433$	27.5890	1.32584	0.662922	−	0.748689i	$-0.269316\pi$
$433$	27.5890	1.32584	0.662922	+	0.748689i	$0.269316\pi$
$434$	0.794988	0.0381606
$435$	0	0
$436$	−33.1976	−1.58988
$437$	0.133768	0.00639897
$438$	0	0
$439$	−18.5614	−0.885887	−0.442943	−	0.896550i	$-0.646066\pi$
$439$	−18.5614	−0.885887	−0.442943	+	0.896550i	$0.646066\pi$
$440$	−15.6858	−0.747793
$441$	0	0
$442$	−4.80094	−0.228358
$443$	−4.02148	−0.191066	−0.0955332	−	0.995426i	$-0.530456\pi$
$443$	−4.02148	−0.191066	−0.0955332	+	0.995426i	$0.530456\pi$
$444$	0	0
$445$	−9.92277	−0.470384
$446$	−5.14008	−0.243390
$447$	0	0
$448$	7.13223	0.336966
$449$	25.0013	1.17989	0.589943	−	0.807445i	$-0.299150\pi$
$449$	25.0013	1.17989	0.589943	+	0.807445i	$0.299150\pi$
$450$	0	0
$451$	49.9475	2.35194
$452$	10.9667	0.515829
$453$	0	0
$454$	3.50312	0.164410
$455$	23.8298	1.11716
$456$	0	0
$457$	−10.2474	−0.479354	−0.239677	−	0.970853i	$-0.577042\pi$
$457$	−10.2474	−0.479354	−0.239677	+	0.970853i	$0.577042\pi$
$458$	−0.818826	−0.0382612
$459$	0	0
$460$	27.5315	1.28366
$461$	−10.5197	−0.489951	−0.244975	−	0.969529i	$-0.578780\pi$
$461$	−10.5197	−0.489951	−0.244975	+	0.969529i	$0.578780\pi$
$462$	0	0
$463$	25.2692	1.17436	0.587179	−	0.809457i	$-0.300238\pi$
$463$	25.2692	1.17436	0.587179	+	0.809457i	$0.300238\pi$
$464$	8.31471	0.386001
$465$	0	0
$466$	−0.188175	−0.00871705
$467$	−4.08966	−0.189247	−0.0946234	−	0.995513i	$-0.530165\pi$
$467$	−4.08966	−0.189247	−0.0946234	+	0.995513i	$0.530165\pi$
$468$	0	0
$469$	−4.40596	−0.203448
$470$	2.90932	0.134197
$471$	0	0
$472$	7.52694	0.346455
$473$	−60.2229	−2.76905
$474$	0	0
$475$	0.388260	0.0178146
$476$	8.09505	0.371036
$477$	0	0
$478$	−1.70294	−0.0778908
$479$	36.7606	1.67963	0.839817	−	0.542869i	$-0.182662\pi$
$479$	36.7606	1.67963	0.839817	+	0.542869i	$0.182662\pi$
$480$	0	0
$481$	−56.5859	−2.58010
$482$	5.26309	0.239727
$483$	0	0
$484$	−32.8660	−1.49391
$485$	22.9544	1.04230
$486$	0	0
$487$	−33.7554	−1.52960	−0.764802	−	0.644265i	$-0.777163\pi$
$487$	−33.7554	−1.52960	−0.764802	+	0.644265i	$0.777163\pi$
$488$	−0.290113	−0.0131328
$489$	0	0
$490$	0.751445	0.0339468
$491$	−2.85086	−0.128657	−0.0643286	−	0.997929i	$-0.520491\pi$
$491$	−2.85086	−0.128657	−0.0643286	+	0.997929i	$0.520491\pi$
$492$	0	0
$493$	9.06718	0.408365
$494$	−0.0435569	−0.00195972
$495$	0	0
$496$	15.6870	0.704366
$497$	−10.3123	−0.462572
$498$	0	0
$499$	−14.9560	−0.669524	−0.334762	−	0.942303i	$-0.608656\pi$
$499$	−14.9560	−0.669524	−0.334762	+	0.942303i	$0.608656\pi$
$500$	41.4141	1.85209
$501$	0	0
$502$	−1.46753	−0.0654992
$503$	8.56487	0.381889	0.190944	−	0.981601i	$-0.438845\pi$
$503$	8.56487	0.381889	0.190944	+	0.981601i	$0.438845\pi$
$504$	0	0
$505$	−6.99661	−0.311345
$506$	−3.60888	−0.160434
$507$	0	0
$508$	1.96328	0.0871066
$509$	−24.3428	−1.07898	−0.539489	−	0.841993i	$-0.681382\pi$
$509$	−24.3428	−1.07898	−0.539489	+	0.841993i	$0.681382\pi$
$510$	0	0
$511$	3.37816	0.149441
$512$	−14.2252	−0.628670
$513$	0	0
$514$	−3.70692	−0.163505
$515$	32.0418	1.41193
$516$	0	0
$517$	20.3916	0.896820
$518$	−1.78437	−0.0784007
$519$	0	0
$520$	−18.0971	−0.793610
$521$	33.9526	1.48749	0.743745	−	0.668464i	$-0.233048\pi$
$521$	33.9526	1.48749	0.743745	+	0.668464i	$0.233048\pi$
$522$	0	0
$523$	33.7243	1.47466	0.737331	−	0.675532i	$-0.236086\pi$
$523$	33.7243	1.47466	0.737331	+	0.675532i	$0.236086\pi$
$524$	−30.8245	−1.34657
$525$	0	0
$526$	2.44144	0.106452
$527$	17.1066	0.745176
$528$	0	0
$529$	−10.2131	−0.444046
$530$	7.49605	0.325608
$531$	0	0
$532$	0.0734430	0.00318416
$533$	57.6255	2.49604
$534$	0	0
$535$	12.6573	0.547221
$536$	3.34602	0.144526
$537$	0	0
$538$	1.73022	0.0745951
$539$	5.26691	0.226862
$540$	0	0
$541$	−22.8612	−0.982879	−0.491439	−	0.870912i	$-0.663529\pi$
$541$	−22.8612	−0.982879	−0.491439	+	0.870912i	$0.663529\pi$
$542$	−3.18171	−0.136666
$543$	0	0
$544$	−9.24992	−0.396587
$545$	−66.3113	−2.84046
$546$	0	0
$547$	−30.4179	−1.30058	−0.650288	−	0.759688i	$-0.725352\pi$
$547$	−30.4179	−1.30058	−0.650288	+	0.759688i	$0.725352\pi$
$548$	15.6555	0.668770
$549$	0	0
$550$	−10.4748	−0.446645
$551$	0.0822627	0.00350451
$552$	0	0
$553$	−9.73339	−0.413906
$554$	1.49291	0.0634275
$555$	0	0
$556$	6.69230	0.283817
$557$	−24.3671	−1.03247	−0.516233	−	0.856448i	$-0.672666\pi$
$557$	−24.3671	−1.03247	−0.516233	+	0.856448i	$0.672666\pi$
$558$	0	0
$559$	−69.4804	−2.93871
$560$	14.8278	0.626588
$561$	0	0
$562$	−1.71428	−0.0723127
$563$	39.1345	1.64932	0.824661	−	0.565627i	$-0.191366\pi$
$563$	39.1345	1.64932	0.824661	+	0.565627i	$0.191366\pi$
$564$	0	0
$565$	21.9056	0.921577
$566$	2.20565	0.0927103
$567$	0	0
$568$	7.83151	0.328603
$569$	−41.9315	−1.75786	−0.878930	−	0.476951i	$-0.841742\pi$
$569$	−41.9315	−1.75786	−0.878930	+	0.476951i	$0.841742\pi$
$570$	0	0
$571$	1.63409	0.0683845	0.0341923	−	0.999415i	$-0.489114\pi$
$571$	1.63409	0.0683845	0.0341923	+	0.999415i	$0.489114\pi$
$572$	−62.8341	−2.62723
$573$	0	0
$574$	1.81715	0.0758464
$575$	37.1141	1.54776
$576$	0	0
$577$	−3.28569	−0.136785	−0.0683925	−	0.997658i	$-0.521787\pi$
$577$	−3.28569	−0.136785	−0.0683925	+	0.997658i	$0.521787\pi$
$578$	−0.000185428 0	−7.71277e−6 0
$579$	0	0
$580$	16.9310	0.703021
$581$	16.5257	0.685600
$582$	0	0
$583$	52.5402	2.17599
$584$	−2.56547	−0.106160
$585$	0	0
$586$	−0.845997	−0.0349478
$587$	37.3079	1.53986	0.769931	−	0.638127i	$-0.220291\pi$
$587$	37.3079	1.53986	0.769931	+	0.638127i	$0.220291\pi$
$588$	0	0
$589$	0.155201	0.00639495
$590$	7.44779	0.306621
$591$	0	0
$592$	−35.2098	−1.44711
$593$	10.6325	0.436626	0.218313	−	0.975879i	$-0.429945\pi$
$593$	10.6325	0.436626	0.218313	+	0.975879i	$0.429945\pi$
$594$	0	0
$595$	16.1697	0.662891
$596$	−4.61215	−0.188921
$597$	0	0
$598$	−4.16364	−0.170264
$599$	−42.0174	−1.71678	−0.858391	−	0.512995i	$-0.828536\pi$
$599$	−42.0174	−1.71678	−0.858391	+	0.512995i	$0.828536\pi$
$600$	0	0
$601$	−16.1101	−0.657144	−0.328572	−	0.944479i	$-0.606567\pi$
$601$	−16.1101	−0.657144	−0.328572	+	0.944479i	$0.606567\pi$
$602$	−2.19098	−0.0892977
$603$	0	0
$604$	6.73214	0.273927
$605$	−65.6489	−2.66901
$606$	0	0
$607$	−14.1660	−0.574979	−0.287489	−	0.957784i	$-0.592821\pi$
$607$	−14.1660	−0.574979	−0.287489	+	0.957784i	$0.592821\pi$
$608$	−0.0839207	−0.00340343
$609$	0	0
$610$	−0.287062	−0.0116228
$611$	23.5262	0.951768
$612$	0	0
$613$	17.1420	0.692361	0.346180	−	0.938168i	$-0.387479\pi$
$613$	17.1420	0.692361	0.346180	+	0.938168i	$0.387479\pi$
$614$	−0.925335	−0.0373435
$615$	0	0
$616$	−3.99985	−0.161159
$617$	−30.3241	−1.22080	−0.610401	−	0.792092i	$-0.708992\pi$
$617$	−30.3241	−1.22080	−0.610401	+	0.792092i	$0.708992\pi$
$618$	0	0
$619$	−9.91575	−0.398547	−0.199274	−	0.979944i	$-0.563858\pi$
$619$	−9.91575	−0.398547	−0.199274	+	0.979944i	$0.563858\pi$
$620$	31.9429	1.28286
$621$	0	0
$622$	−3.02023	−0.121100
$623$	−2.53028	−0.101374
$624$	0	0
$625$	30.8286	1.23314
$626$	−2.54033	−0.101532
$627$	0	0
$628$	39.0590	1.55862
$629$	−38.3962	−1.53096
$630$	0	0
$631$	−4.94297	−0.196777	−0.0983884	−	0.995148i	$-0.531369\pi$
$631$	−4.94297	−0.196777	−0.0983884	+	0.995148i	$0.531369\pi$
$632$	7.39183	0.294031
$633$	0	0
$634$	0.994685	0.0395040
$635$	3.92161	0.155624
$636$	0	0
$637$	6.07655	0.240762
$638$	−2.21934	−0.0878645
$639$	0	0
$640$	−22.9547	−0.907365
$641$	−8.92412	−0.352481	−0.176241	−	0.984347i	$-0.556394\pi$
$641$	−8.92412	−0.352481	−0.176241	+	0.984347i	$0.556394\pi$
$642$	0	0
$643$	8.44158	0.332903	0.166452	−	0.986050i	$-0.446769\pi$
$643$	8.44158	0.332903	0.166452	+	0.986050i	$0.446769\pi$
$644$	7.02047	0.276645
$645$	0	0
$646$	−0.0295554	−0.00116284
$647$	−50.2006	−1.97359	−0.986795	−	0.161976i	$-0.948213\pi$
$647$	−50.2006	−1.97359	−0.986795	+	0.161976i	$0.948213\pi$
$648$	0	0
$649$	52.2019	2.04910
$650$	−12.0850	−0.474011
$651$	0	0
$652$	−13.1242	−0.513982
$653$	−40.5316	−1.58612	−0.793062	−	0.609141i	$-0.791514\pi$
$653$	−40.5316	−1.58612	−0.793062	+	0.609141i	$0.791514\pi$
$654$	0	0
$655$	−61.5711	−2.40578
$656$	35.8567	1.39997
$657$	0	0
$658$	0.741870	0.0289211
$659$	−41.9115	−1.63264	−0.816320	−	0.577600i	$-0.803989\pi$
$659$	−41.9115	−1.63264	−0.816320	+	0.577600i	$0.803989\pi$
$660$	0	0
$661$	27.5407	1.07121	0.535605	−	0.844469i	$-0.320084\pi$
$661$	27.5407	1.07121	0.535605	+	0.844469i	$0.320084\pi$
$662$	2.14388	0.0833242
$663$	0	0
$664$	−12.5501	−0.487038
$665$	0.146700	0.00568880
$666$	0	0
$667$	7.86355	0.304478
$668$	31.4667	1.21748
$669$	0	0
$670$	3.31083	0.127909
$671$	−2.01203	−0.0776735
$672$	0	0
$673$	6.81901	0.262853	0.131427	−	0.991326i	$-0.458044\pi$
$673$	6.81901	0.262853	0.131427	+	0.991326i	$0.458044\pi$
$674$	−3.28623	−0.126581
$675$	0	0
$676$	−46.9704	−1.80655
$677$	6.66448	0.256137	0.128068	−	0.991765i	$-0.459122\pi$
$677$	6.66448	0.256137	0.128068	+	0.991765i	$0.459122\pi$
$678$	0	0
$679$	5.85331	0.224629
$680$	−12.2797	−0.470906
$681$	0	0
$682$	−4.18713	−0.160333
$683$	−23.2496	−0.889623	−0.444811	−	0.895624i	$-0.646729\pi$
$683$	−23.2496	−0.889623	−0.444811	+	0.895624i	$0.646729\pi$
$684$	0	0
$685$	31.2715	1.19482
$686$	0.191617	0.00731595
$687$	0	0
$688$	−43.2332	−1.64825
$689$	60.6167	2.30931
$690$	0	0
$691$	7.85702	0.298895	0.149447	−	0.988770i	$-0.452251\pi$
$691$	7.85702	0.298895	0.149447	+	0.988770i	$0.452251\pi$
$692$	−29.1710	−1.10892
$693$	0	0
$694$	0.00492285	0.000186869 0
$695$	13.3677	0.507066
$696$	0	0
$697$	39.1016	1.48108
$698$	−4.58881	−0.173689
$699$	0	0
$700$	20.3769	0.770175
$701$	0.243911	0.00921239	0.00460619	−	0.999989i	$-0.498534\pi$
$701$	0.243911	0.00921239	0.00460619	+	0.999989i	$0.498534\pi$
$702$	0	0
$703$	−0.348353	−0.0131384
$704$	−37.5648	−1.41578
$705$	0	0
$706$	−1.21452	−0.0457092
$707$	−1.78412	−0.0670987
$708$	0	0
$709$	27.6428	1.03815	0.519073	−	0.854730i	$-0.326277\pi$
$709$	27.6428	1.03815	0.519073	+	0.854730i	$0.326277\pi$
$710$	7.74915	0.290820
$711$	0	0
$712$	1.92157	0.0720140
$713$	14.8358	0.555605
$714$	0	0
$715$	−125.509	−4.69379
$716$	−20.4214	−0.763183
$717$	0	0
$718$	−2.10417	−0.0785269
$719$	−13.2320	−0.493470	−0.246735	−	0.969083i	$-0.579358\pi$
$719$	−13.2320	−0.493470	−0.246735	+	0.969083i	$0.579358\pi$
$720$	0	0
$721$	8.17058	0.304288
$722$	3.64045	0.135483
$723$	0	0
$724$	11.9919	0.445677
$725$	22.8239	0.847660
$726$	0	0
$727$	−19.0077	−0.704958	−0.352479	−	0.935820i	$-0.614661\pi$
$727$	−19.0077	−0.704958	−0.352479	+	0.935820i	$0.614661\pi$
$728$	−4.61471	−0.171033
$729$	0	0
$730$	−2.53850	−0.0939540
$731$	−47.1458	−1.74375
$732$	0	0
$733$	−49.8815	−1.84242	−0.921208	−	0.389071i	$-0.872796\pi$
$733$	−49.8815	−1.84242	−0.921208	+	0.389071i	$0.872796\pi$
$734$	−4.03676	−0.148999
$735$	0	0
$736$	−8.02204	−0.295696
$737$	23.2058	0.854796
$738$	0	0
$739$	−29.8966	−1.09976	−0.549882	−	0.835242i	$-0.685327\pi$
$739$	−29.8966	−1.09976	−0.549882	+	0.835242i	$0.685327\pi$
$740$	−71.6966	−2.63562
$741$	0	0
$742$	1.91148	0.0701725
$743$	−7.33058	−0.268933	−0.134467	−	0.990918i	$-0.542932\pi$
$743$	−7.33058	−0.268933	−0.134467	+	0.990918i	$0.542932\pi$
$744$	0	0
$745$	−9.21265	−0.337525
$746$	−2.03174	−0.0743871
$747$	0	0
$748$	−42.6359	−1.55892
$749$	3.22757	0.117933
$750$	0	0
$751$	−37.8737	−1.38203	−0.691015	−	0.722840i	$-0.742836\pi$
$751$	−37.8737	−1.38203	−0.691015	+	0.722840i	$0.742836\pi$
$752$	14.6389	0.533824
$753$	0	0
$754$	−2.56050	−0.0932480
$755$	13.4473	0.489396
$756$	0	0
$757$	22.7557	0.827070	0.413535	−	0.910488i	$-0.364294\pi$
$757$	22.7557	0.827070	0.413535	+	0.910488i	$0.364294\pi$
$758$	1.06860	0.0388135
$759$	0	0
$760$	−0.111409	−0.00404122
$761$	−7.34081	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$761$	−7.34081	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$762$	0	0
$763$	−16.9092	−0.612155
$764$	17.0402	0.616493
$765$	0	0
$766$	3.78145	0.136629
$767$	60.2264	2.17465
$768$	0	0
$769$	−39.6967	−1.43150	−0.715749	−	0.698358i	$-0.753915\pi$
$769$	−39.6967	−1.43150	−0.715749	+	0.698358i	$0.753915\pi$
$770$	−3.95779	−0.142629
$771$	0	0
$772$	8.41145	0.302735
$773$	9.92476	0.356969	0.178484	−	0.983943i	$-0.442881\pi$
$773$	9.92476	0.356969	0.178484	+	0.983943i	$0.442881\pi$
$774$	0	0
$775$	43.0609	1.54679
$776$	−4.44518	−0.159573
$777$	0	0
$778$	3.08690	0.110671
$779$	0.354753	0.0127103
$780$	0	0
$781$	54.3141	1.94351
$782$	−2.82523	−0.101030
$783$	0	0
$784$	3.78105	0.135037
$785$	78.0193	2.78463
$786$	0	0
$787$	17.2831	0.616077	0.308038	−	0.951374i	$-0.400327\pi$
$787$	17.2831	0.616077	0.308038	+	0.951374i	$0.400327\pi$
$788$	40.9816	1.45991
$789$	0	0
$790$	7.31410	0.260224
$791$	5.58588	0.198611
$792$	0	0
$793$	−2.32132	−0.0824325
$794$	1.30426	0.0462865
$795$	0	0
$796$	−45.7821	−1.62270
$797$	−3.27024	−0.115838	−0.0579190	−	0.998321i	$-0.518447\pi$
$797$	−3.27024	−0.115838	−0.0579190	+	0.998321i	$0.518447\pi$
$798$	0	0
$799$	15.9636	0.564753
$800$	−23.2839	−0.823212
$801$	0	0
$802$	−3.29560	−0.116372
$803$	−17.7924	−0.627881
$804$	0	0
$805$	14.0232	0.494253
$806$	−4.83078	−0.170157
$807$	0	0
$808$	1.35491	0.0476657
$809$	45.7392	1.60810	0.804052	−	0.594558i	$-0.202673\pi$
$809$	45.7392	1.60810	0.804052	+	0.594558i	$0.202673\pi$
$810$	0	0
$811$	20.1033	0.705923	0.352962	−	0.935638i	$-0.385175\pi$
$811$	20.1033	0.705923	0.352962	+	0.935638i	$0.385175\pi$
$812$	4.31736	0.151510
$813$	0	0
$814$	9.39811	0.329403
$815$	−26.2152	−0.918278
$816$	0	0
$817$	−0.427734	−0.0149645
$818$	−0.599886	−0.0209745
$819$	0	0
$820$	73.0138	2.54975
$821$	19.1738	0.669170	0.334585	−	0.942366i	$-0.391404\pi$
$821$	19.1738	0.669170	0.334585	+	0.942366i	$0.391404\pi$
$822$	0	0
$823$	8.70002	0.303264	0.151632	−	0.988437i	$-0.451547\pi$
$823$	8.70002	0.303264	0.151632	+	0.988437i	$0.451547\pi$
$824$	−6.20499	−0.216161
$825$	0	0
$826$	1.89917	0.0660805
$827$	−11.0112	−0.382898	−0.191449	−	0.981503i	$-0.561319\pi$
$827$	−11.0112	−0.382898	−0.191449	+	0.981503i	$0.561319\pi$
$828$	0	0
$829$	40.0764	1.39191	0.695955	−	0.718086i	$-0.254981\pi$
$829$	40.0764	1.39191	0.695955	+	0.718086i	$0.254981\pi$
$830$	−12.4181	−0.431039
$831$	0	0
$832$	−43.3393	−1.50252
$833$	4.12322	0.142861
$834$	0	0
$835$	62.8539	2.17515
$836$	−0.386818	−0.0133784
$837$	0	0
$838$	3.61077	0.124732
$839$	5.37182	0.185456	0.0927279	−	0.995691i	$-0.470441\pi$
$839$	5.37182	0.185456	0.0927279	+	0.995691i	$0.470441\pi$
$840$	0	0
$841$	−24.1642	−0.833247
$842$	2.52036	0.0868574
$843$	0	0
$844$	33.9192	1.16755
$845$	−93.8221	−3.22758
$846$	0	0
$847$	−16.7403	−0.575204
$848$	37.7179	1.29524
$849$	0	0
$850$	−8.20021	−0.281265
$851$	−33.2993	−1.14149
$852$	0	0
$853$	37.5087	1.28427	0.642137	−	0.766590i	$-0.278048\pi$
$853$	37.5087	1.28427	0.642137	+	0.766590i	$0.278048\pi$
$854$	−0.0732001	−0.00250485
$855$	0	0
$856$	−2.45111	−0.0837774
$857$	−17.2234	−0.588339	−0.294169	−	0.955753i	$-0.595043\pi$
$857$	−17.2234	−0.588339	−0.294169	+	0.955753i	$0.595043\pi$
$858$	0	0
$859$	−26.0045	−0.887262	−0.443631	−	0.896210i	$-0.646310\pi$
$859$	−26.0045	−0.887262	−0.443631	+	0.896210i	$0.646310\pi$
$860$	−88.0345	−3.00195
$861$	0	0
$862$	−7.38604	−0.251569
$863$	−32.4210	−1.10362	−0.551812	−	0.833969i	$-0.686063\pi$
$863$	−32.4210	−1.10362	−0.551812	+	0.833969i	$0.686063\pi$
$864$	0	0
$865$	−58.2683	−1.98118
$866$	−5.28651	−0.179643
$867$	0	0
$868$	8.14536	0.276472
$869$	51.2649	1.73904
$870$	0	0
$871$	26.7730	0.907169
$872$	12.8414	0.434864
$873$	0	0
$874$	−0.0256321	−0.000867018 0
$875$	21.0943	0.713117
$876$	0	0
$877$	31.8805	1.07653	0.538264	−	0.842776i	$-0.319080\pi$
$877$	31.8805	1.07653	0.538264	+	0.842776i	$0.319080\pi$
$878$	3.55667	0.120032
$879$	0	0
$880$	−78.0965	−2.63263
$881$	10.9692	0.369563	0.184782	−	0.982780i	$-0.440842\pi$
$881$	10.9692	0.369563	0.184782	+	0.982780i	$0.440842\pi$
$882$	0	0
$883$	−43.5368	−1.46513	−0.732564	−	0.680698i	$-0.761677\pi$
$883$	−43.5368	−1.46513	−0.732564	+	0.680698i	$0.761677\pi$
$884$	−49.1900	−1.65444
$885$	0	0
$886$	0.770582	0.0258882
$887$	12.9767	0.435715	0.217857	−	0.975981i	$-0.430093\pi$
$887$	12.9767	0.435715	0.217857	+	0.975981i	$0.430093\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	1.90137	0.0637339
$891$	0	0
$892$	−52.6647	−1.76335
$893$	0.144831	0.00484660
$894$	0	0
$895$	−40.7912	−1.36350
$896$	−5.85340	−0.195548
$897$	0	0
$898$	−4.79067	−0.159867
$899$	9.12353	0.304287
$900$	0	0
$901$	41.1313	1.37028
$902$	−9.57077	−0.318672
$903$	0	0
$904$	−4.24209	−0.141090
$905$	23.9536	0.796243
$906$	0	0
$907$	−2.23113	−0.0740833	−0.0370417	−	0.999314i	$-0.511793\pi$
$907$	−2.23113	−0.0740833	−0.0370417	+	0.999314i	$0.511793\pi$
$908$	35.8926	1.19114
$909$	0	0
$910$	−4.56619	−0.151368
$911$	−29.4118	−0.974457	−0.487228	−	0.873275i	$-0.661992\pi$
$911$	−29.4118	−0.974457	−0.487228	+	0.873275i	$0.661992\pi$
$912$	0	0
$913$	−87.0391	−2.88058
$914$	1.96358	0.0649493
$915$	0	0
$916$	−8.38961	−0.277201
$917$	−15.7005	−0.518476
$918$	0	0
$919$	3.92759	0.129559	0.0647796	−	0.997900i	$-0.479366\pi$
$919$	3.92759	0.129559	0.0647796	+	0.997900i	$0.479366\pi$
$920$	−10.6497	−0.351109
$921$	0	0
$922$	2.01575	0.0663851
$923$	62.6634	2.06259
$924$	0	0
$925$	−96.6511	−3.17787
$926$	−4.84199	−0.159118
$927$	0	0
$928$	−4.93329	−0.161943
$929$	−26.5871	−0.872293	−0.436147	−	0.899876i	$-0.643657\pi$
$929$	−26.5871	−0.872293	−0.436147	+	0.899876i	$0.643657\pi$
$930$	0	0
$931$	0.0374083	0.00122601
$932$	−1.92802	−0.0631546
$933$	0	0
$934$	0.783646	0.0256417
$935$	−85.1641	−2.78516
$936$	0	0
$937$	−46.5758	−1.52156	−0.760782	−	0.649007i	$-0.775185\pi$
$937$	−46.5758	−1.52156	−0.760782	+	0.649007i	$0.775185\pi$
$938$	0.844254	0.0275659
$939$	0	0
$940$	29.8086	0.972250
$941$	3.71923	0.121243	0.0606217	−	0.998161i	$-0.480692\pi$
$941$	3.71923	0.121243	0.0606217	+	0.998161i	$0.480692\pi$
$942$	0	0
$943$	33.9111	1.10430
$944$	37.4751	1.21971
$945$	0	0
$946$	11.5397	0.375188
$947$	1.23688	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$947$	1.23688	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$948$	0	0
$949$	−20.5275	−0.666351
$950$	−0.0743971	−0.00241376
$951$	0	0
$952$	−3.13130	−0.101486
$953$	52.2025	1.69101	0.845503	−	0.533971i	$-0.179301\pi$
$953$	52.2025	1.69101	0.845503	+	0.533971i	$0.179301\pi$
$954$	0	0
$955$	34.0373	1.10142
$956$	−17.4482	−0.564314
$957$	0	0
$958$	−7.04394	−0.227579
$959$	7.97414	0.257499
$960$	0	0
$961$	−13.7871	−0.444744
$962$	10.8428	0.349586
$963$	0	0
$964$	53.9251	1.73681
$965$	16.8017	0.540864
$966$	0	0
$967$	25.3043	0.813731	0.406866	−	0.913488i	$-0.366622\pi$
$967$	25.3043	0.813731	0.406866	+	0.913488i	$0.366622\pi$
$968$	12.7131	0.408615
$969$	0	0
$970$	−4.39844	−0.141225
$971$	−4.96055	−0.159192	−0.0795958	−	0.996827i	$-0.525363\pi$
$971$	−4.96055	−0.159192	−0.0795958	+	0.996827i	$0.525363\pi$
$972$	0	0
$973$	3.40873	0.109279
$974$	6.46810	0.207251
$975$	0	0
$976$	−1.44441	−0.0462344
$977$	3.61365	0.115611	0.0578055	−	0.998328i	$-0.481590\pi$
$977$	3.61365	0.115611	0.0578055	+	0.998328i	$0.481590\pi$
$978$	0	0
$979$	13.3268	0.425925
$980$	7.69922	0.245943
$981$	0	0
$982$	0.546271	0.0174322
$983$	51.8644	1.65422	0.827109	−	0.562042i	$-0.189984\pi$
$983$	51.8644	1.65422	0.827109	+	0.562042i	$0.189984\pi$
$984$	0	0
$985$	81.8597	2.60827
$986$	−1.73742	−0.0553308
$987$	0	0
$988$	−0.446280	−0.0141981
$989$	−40.8874	−1.30014
$990$	0	0
$991$	6.60298	0.209751	0.104875	−	0.994485i	$-0.466556\pi$
$991$	6.60298	0.209751	0.104875	+	0.994485i	$0.466556\pi$
$992$	−9.30741	−0.295511
$993$	0	0
$994$	1.97601	0.0626754
$995$	−91.4485	−2.89911
$996$	0	0
$997$	−0.888839	−0.0281498	−0.0140749	−	0.999901i	$-0.504480\pi$
$997$	−0.888839	−0.0281498	−0.0140749	+	0.999901i	$0.504480\pi$
$998$	2.86582	0.0907160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.7		16
3.2	odd	2		889.2.a.c.1.10	✓	16
21.20	even	2		6223.2.a.k.1.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.10	✓	16	3.2	odd	2
6223.2.a.k.1.10		16	21.20	even	2
8001.2.a.t.1.7		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)

\(f(q)\)	\(=\)	\(q-0.191617 q^{2} -1.96328 q^{4} -3.92161 q^{5} -1.00000 q^{7} +0.759431 q^{8} +0.751445 q^{10} +5.26691 q^{11} +6.07655 q^{13} +0.191617 q^{14} +3.78105 q^{16} +4.12322 q^{17} +0.0374083 q^{19} +7.69922 q^{20} -1.00923 q^{22} +3.57588 q^{23} +10.3790 q^{25} -1.16437 q^{26} +1.96328 q^{28} +2.19905 q^{29} +4.14885 q^{31} -2.24337 q^{32} -0.790078 q^{34} +3.92161 q^{35} -9.31219 q^{37} -0.00716804 q^{38} -2.97819 q^{40} +9.48327 q^{41} -11.4342 q^{43} -10.3404 q^{44} -0.685199 q^{46} +3.87164 q^{47} +1.00000 q^{49} -1.98879 q^{50} -11.9300 q^{52} +9.97552 q^{53} -20.6547 q^{55} -0.759431 q^{56} -0.421374 q^{58} +9.91129 q^{59} -0.382013 q^{61} -0.794988 q^{62} -7.13223 q^{64} -23.8298 q^{65} +4.40596 q^{67} -8.09505 q^{68} -0.751445 q^{70} +10.3123 q^{71} -3.37816 q^{73} +1.78437 q^{74} -0.0734430 q^{76} -5.26691 q^{77} +9.73339 q^{79} -14.8278 q^{80} -1.81715 q^{82} -16.5257 q^{83} -16.1697 q^{85} +2.19098 q^{86} +3.99985 q^{88} +2.53028 q^{89} -6.07655 q^{91} -7.02047 q^{92} -0.741870 q^{94} -0.146700 q^{95} -5.85331 q^{97} -0.191617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

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