LMFDB - Embedded newform 8752.2.a.s.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$69.8850718490$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$1.04467$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71791 q^{3} +0.714085 q^{5} +2.03236 q^{7} +4.38705 q^{9} +O(q^{10})$	$q+2.71791 q^{3} +0.714085 q^{5} +2.03236 q^{7} +4.38705 q^{9} -5.43392 q^{11} -3.96770 q^{13} +1.94082 q^{15} -1.30793 q^{17} -8.24470 q^{19} +5.52379 q^{21} +2.44824 q^{23} -4.49008 q^{25} +3.76989 q^{27} +1.49405 q^{29} +6.02301 q^{31} -14.7689 q^{33} +1.45128 q^{35} -6.36158 q^{37} -10.7839 q^{39} -2.87189 q^{41} +5.76713 q^{43} +3.13273 q^{45} -6.42848 q^{47} -2.86949 q^{49} -3.55483 q^{51} -0.932545 q^{53} -3.88028 q^{55} -22.4084 q^{57} -1.57369 q^{59} -7.93215 q^{61} +8.91609 q^{63} -2.83328 q^{65} -10.5294 q^{67} +6.65410 q^{69} +12.8929 q^{71} +9.81833 q^{73} -12.2037 q^{75} -11.0437 q^{77} +2.80129 q^{79} -2.91492 q^{81} +15.3935 q^{83} -0.933971 q^{85} +4.06069 q^{87} -1.89811 q^{89} -8.06382 q^{91} +16.3700 q^{93} -5.88742 q^{95} +0.276333 q^{97} -23.8389 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

Display 100 coefficients 10 coefficients

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	0
$2$	0	0
$3$	2.71791	1.56919	0.784594	−	0.620010i	$-0.212871\pi$
$3$	2.71791	1.56919	0.784594	+	0.620010i	$0.212871\pi$
$4$	0	0
$5$	0.714085	0.319349	0.159674	−	0.987170i	$-0.448956\pi$
$5$	0.714085	0.319349	0.159674	+	0.987170i	$0.448956\pi$
$6$	0	0
$7$	2.03236	0.768162	0.384081	−	0.923299i	$-0.374518\pi$
$7$	2.03236	0.768162	0.384081	+	0.923299i	$0.374518\pi$
$8$	0	0
$9$	4.38705	1.46235
$10$	0	0
$11$	−5.43392	−1.63839	−0.819194	−	0.573516i	$-0.805579\pi$
$11$	−5.43392	−1.63839	−0.819194	+	0.573516i	$0.805579\pi$
$12$	0	0
$13$	−3.96770	−1.10044	−0.550221	−	0.835019i	$-0.685457\pi$
$13$	−3.96770	−1.10044	−0.550221	+	0.835019i	$0.685457\pi$
$14$	0	0
$15$	1.94082	0.501118
$16$	0	0
$17$	−1.30793	−0.317219	−0.158609	−	0.987341i	$-0.550701\pi$
$17$	−1.30793	−0.317219	−0.158609	+	0.987341i	$0.550701\pi$
$18$	0	0
$19$	−8.24470	−1.89146	−0.945732	−	0.324947i	$-0.894654\pi$
$19$	−8.24470	−1.89146	−0.945732	+	0.324947i	$0.894654\pi$
$20$	0	0
$21$	5.52379	1.20539
$22$	0	0
$23$	2.44824	0.510493	0.255246	−	0.966876i	$-0.417843\pi$
$23$	2.44824	0.510493	0.255246	+	0.966876i	$0.417843\pi$
$24$	0	0
$25$	−4.49008	−0.898016
$26$	0	0
$27$	3.76989	0.725517
$28$	0	0
$29$	1.49405	0.277438	0.138719	−	0.990332i	$-0.455702\pi$
$29$	1.49405	0.277438	0.138719	+	0.990332i	$0.455702\pi$
$30$	0	0
$31$	6.02301	1.08176	0.540882	−	0.841098i	$-0.318091\pi$
$31$	6.02301	1.08176	0.540882	+	0.841098i	$0.318091\pi$
$32$	0	0
$33$	−14.7689	−2.57094
$34$	0	0
$35$	1.45128	0.245311
$36$	0	0
$37$	−6.36158	−1.04584	−0.522919	−	0.852383i	$-0.675157\pi$
$37$	−6.36158	−1.04584	−0.522919	+	0.852383i	$0.675157\pi$
$38$	0	0
$39$	−10.7839	−1.72680
$40$	0	0
$41$	−2.87189	−0.448514	−0.224257	−	0.974530i	$-0.571995\pi$
$41$	−2.87189	−0.448514	−0.224257	+	0.974530i	$0.571995\pi$
$42$	0	0
$43$	5.76713	0.879478	0.439739	−	0.898125i	$-0.355071\pi$
$43$	5.76713	0.879478	0.439739	+	0.898125i	$0.355071\pi$
$44$	0	0
$45$	3.13273	0.467000
$46$	0	0
$47$	−6.42848	−0.937689	−0.468845	−	0.883281i	$-0.655330\pi$
$47$	−6.42848	−0.937689	−0.468845	+	0.883281i	$0.655330\pi$
$48$	0	0
$49$	−2.86949	−0.409928
$50$	0	0
$51$	−3.55483	−0.497776
$52$	0	0
$53$	−0.932545	−0.128095	−0.0640474	−	0.997947i	$-0.520401\pi$
$53$	−0.932545	−0.128095	−0.0640474	+	0.997947i	$0.520401\pi$
$54$	0	0
$55$	−3.88028	−0.523217
$56$	0	0
$57$	−22.4084	−2.96806
$58$	0	0
$59$	−1.57369	−0.204877	−0.102438	−	0.994739i	$-0.532664\pi$
$59$	−1.57369	−0.204877	−0.102438	+	0.994739i	$0.532664\pi$
$60$	0	0
$61$	−7.93215	−1.01561	−0.507804	−	0.861473i	$-0.669542\pi$
$61$	−7.93215	−1.01561	−0.507804	+	0.861473i	$0.669542\pi$
$62$	0	0
$63$	8.91609	1.12332
$64$	0	0
$65$	−2.83328	−0.351425
$66$	0	0
$67$	−10.5294	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$67$	−10.5294	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$68$	0	0
$69$	6.65410	0.801059
$70$	0	0
$71$	12.8929	1.53011	0.765054	−	0.643966i	$-0.222712\pi$
$71$	12.8929	1.53011	0.765054	+	0.643966i	$0.222712\pi$
$72$	0	0
$73$	9.81833	1.14915	0.574574	−	0.818452i	$-0.305168\pi$
$73$	9.81833	1.14915	0.574574	+	0.818452i	$0.305168\pi$
$74$	0	0
$75$	−12.2037	−1.40916
$76$	0	0
$77$	−11.0437	−1.25855
$78$	0	0
$79$	2.80129	0.315170	0.157585	−	0.987505i	$-0.449629\pi$
$79$	2.80129	0.315170	0.157585	+	0.987505i	$0.449629\pi$
$80$	0	0
$81$	−2.91492	−0.323880
$82$	0	0
$83$	15.3935	1.68966	0.844829	−	0.535037i	$-0.179702\pi$
$83$	15.3935	1.68966	0.844829	+	0.535037i	$0.179702\pi$
$84$	0	0
$85$	−0.933971	−0.101303
$86$	0	0
$87$	4.06069	0.435352
$88$	0	0
$89$	−1.89811	−0.201199	−0.100600	−	0.994927i	$-0.532076\pi$
$89$	−1.89811	−0.201199	−0.100600	+	0.994927i	$0.532076\pi$
$90$	0	0
$91$	−8.06382	−0.845318
$92$	0	0
$93$	16.3700	1.69749
$94$	0	0
$95$	−5.88742	−0.604037
$96$	0	0
$97$	0.276333	0.0280574	0.0140287	−	0.999902i	$-0.495534\pi$
$97$	0.276333	0.0280574	0.0140287	+	0.999902i	$0.495534\pi$
$98$	0	0
$99$	−23.8389	−2.39590
$100$	0	0
$101$	−17.0451	−1.69605	−0.848024	−	0.529958i	$-0.822208\pi$
$101$	−17.0451	−1.69605	−0.848024	+	0.529958i	$0.822208\pi$
$102$	0	0
$103$	−6.40415	−0.631020	−0.315510	−	0.948922i	$-0.602176\pi$
$103$	−6.40415	−0.631020	−0.315510	+	0.948922i	$0.602176\pi$
$104$	0	0
$105$	3.94446	0.384940
$106$	0	0
$107$	−6.34857	−0.613739	−0.306870	−	0.951752i	$-0.599282\pi$
$107$	−6.34857	−0.613739	−0.306870	+	0.951752i	$0.599282\pi$
$108$	0	0
$109$	9.33636	0.894261	0.447130	−	0.894469i	$-0.352446\pi$
$109$	9.33636	0.894261	0.447130	+	0.894469i	$0.352446\pi$
$110$	0	0
$111$	−17.2902	−1.64112
$112$	0	0
$113$	−1.89369	−0.178144	−0.0890718	−	0.996025i	$-0.528390\pi$
$113$	−1.89369	−0.178144	−0.0890718	+	0.996025i	$0.528390\pi$
$114$	0	0
$115$	1.74825	0.163025
$116$	0	0
$117$	−17.4065	−1.60923
$118$	0	0
$119$	−2.65818	−0.243675
$120$	0	0
$121$	18.5275	1.68432
$122$	0	0
$123$	−7.80555	−0.703802
$124$	0	0
$125$	−6.77673	−0.606129
$126$	0	0
$127$	−10.2086	−0.905867	−0.452933	−	0.891544i	$-0.649622\pi$
$127$	−10.2086	−0.905867	−0.452933	+	0.891544i	$0.649622\pi$
$128$	0	0
$129$	15.6746	1.38007
$130$	0	0
$131$	13.9590	1.21960	0.609801	−	0.792554i	$-0.291249\pi$
$131$	13.9590	1.21960	0.609801	+	0.792554i	$0.291249\pi$
$132$	0	0
$133$	−16.7562	−1.45295
$134$	0	0
$135$	2.69203	0.231693
$136$	0	0
$137$	−2.19310	−0.187369	−0.0936845	−	0.995602i	$-0.529864\pi$
$137$	−2.19310	−0.187369	−0.0936845	+	0.995602i	$0.529864\pi$
$138$	0	0
$139$	19.6789	1.66914	0.834572	−	0.550899i	$-0.185715\pi$
$139$	19.6789	1.66914	0.834572	+	0.550899i	$0.185715\pi$
$140$	0	0
$141$	−17.4720	−1.47141
$142$	0	0
$143$	21.5602	1.80295
$144$	0	0
$145$	1.06688	0.0885994
$146$	0	0
$147$	−7.79904	−0.643254
$148$	0	0
$149$	−11.0882	−0.908379	−0.454189	−	0.890905i	$-0.650071\pi$
$149$	−11.0882	−0.908379	−0.454189	+	0.890905i	$0.650071\pi$
$150$	0	0
$151$	−7.03425	−0.572439	−0.286219	−	0.958164i	$-0.592399\pi$
$151$	−7.03425	−0.572439	−0.286219	+	0.958164i	$0.592399\pi$
$152$	0	0
$153$	−5.73795	−0.463885
$154$	0	0
$155$	4.30094	0.345460
$156$	0	0
$157$	−24.2151	−1.93257	−0.966287	−	0.257468i	$-0.917112\pi$
$157$	−24.2151	−1.93257	−0.966287	+	0.257468i	$0.917112\pi$
$158$	0	0
$159$	−2.53458	−0.201005
$160$	0	0
$161$	4.97571	0.392141
$162$	0	0
$163$	−0.672203	−0.0526510	−0.0263255	−	0.999653i	$-0.508381\pi$
$163$	−0.672203	−0.0526510	−0.0263255	+	0.999653i	$0.508381\pi$
$164$	0	0
$165$	−10.5463	−0.821026
$166$	0	0
$167$	13.2184	1.02287	0.511436	−	0.859321i	$-0.329114\pi$
$167$	13.2184	1.02287	0.511436	+	0.859321i	$0.329114\pi$
$168$	0	0
$169$	2.74265	0.210973
$170$	0	0
$171$	−36.1700	−2.76599
$172$	0	0
$173$	−18.2674	−1.38884	−0.694421	−	0.719569i	$-0.744339\pi$
$173$	−18.2674	−1.38884	−0.694421	+	0.719569i	$0.744339\pi$
$174$	0	0
$175$	−9.12548	−0.689822
$176$	0	0
$177$	−4.27715	−0.321490
$178$	0	0
$179$	−21.6343	−1.61702	−0.808511	−	0.588481i	$-0.799726\pi$
$179$	−21.6343	−1.61702	−0.808511	+	0.588481i	$0.799726\pi$
$180$	0	0
$181$	0.765562	0.0569038	0.0284519	−	0.999595i	$-0.490942\pi$
$181$	0.765562	0.0569038	0.0284519	+	0.999595i	$0.490942\pi$
$182$	0	0
$183$	−21.5589	−1.59368
$184$	0	0
$185$	−4.54271	−0.333987
$186$	0	0
$187$	7.10717	0.519728
$188$	0	0
$189$	7.66180	0.557314
$190$	0	0
$191$	3.52515	0.255071	0.127535	−	0.991834i	$-0.459293\pi$
$191$	3.52515	0.255071	0.127535	+	0.991834i	$0.459293\pi$
$192$	0	0
$193$	−8.94657	−0.643988	−0.321994	−	0.946742i	$-0.604353\pi$
$193$	−8.94657	−0.643988	−0.321994	+	0.946742i	$0.604353\pi$
$194$	0	0
$195$	−7.70060	−0.551452
$196$	0	0
$197$	−20.1670	−1.43684	−0.718418	−	0.695612i	$-0.755133\pi$
$197$	−20.1670	−1.43684	−0.718418	+	0.695612i	$0.755133\pi$
$198$	0	0
$199$	−3.72704	−0.264203	−0.132101	−	0.991236i	$-0.542172\pi$
$199$	−3.72704	−0.264203	−0.132101	+	0.991236i	$0.542172\pi$
$200$	0	0
$201$	−28.6180	−2.01856
$202$	0	0
$203$	3.03645	0.213117
$204$	0	0
$205$	−2.05077	−0.143232
$206$	0	0
$207$	10.7406	0.746520
$208$	0	0
$209$	44.8011	3.09895
$210$	0	0
$211$	−17.9812	−1.23788	−0.618940	−	0.785438i	$-0.712438\pi$
$211$	−17.9812	−1.23788	−0.618940	+	0.785438i	$0.712438\pi$
$212$	0	0
$213$	35.0419	2.40103
$214$	0	0
$215$	4.11822	0.280860
$216$	0	0
$217$	12.2409	0.830970
$218$	0	0
$219$	26.6854	1.80323
$220$	0	0
$221$	5.18946	0.349081
$222$	0	0
$223$	19.2192	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$223$	19.2192	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$224$	0	0
$225$	−19.6982	−1.31322
$226$	0	0
$227$	−23.2733	−1.54470	−0.772350	−	0.635197i	$-0.780919\pi$
$227$	−23.2733	−1.54470	−0.772350	+	0.635197i	$0.780919\pi$
$228$	0	0
$229$	1.04868	0.0692987	0.0346494	−	0.999400i	$-0.488969\pi$
$229$	1.04868	0.0692987	0.0346494	+	0.999400i	$0.488969\pi$
$230$	0	0
$231$	−30.0158	−1.97490
$232$	0	0
$233$	10.4018	0.681441	0.340721	−	0.940165i	$-0.389329\pi$
$233$	10.4018	0.681441	0.340721	+	0.940165i	$0.389329\pi$
$234$	0	0
$235$	−4.59048	−0.299450
$236$	0	0
$237$	7.61367	0.494561
$238$	0	0
$239$	−5.21082	−0.337060	−0.168530	−	0.985697i	$-0.553902\pi$
$239$	−5.21082	−0.337060	−0.168530	+	0.985697i	$0.553902\pi$
$240$	0	0
$241$	−23.1270	−1.48974	−0.744871	−	0.667209i	$-0.767489\pi$
$241$	−23.1270	−1.48974	−0.744871	+	0.667209i	$0.767489\pi$
$242$	0	0
$243$	−19.2322	−1.23374
$244$	0	0
$245$	−2.04906	−0.130910
$246$	0	0
$247$	32.7125	2.08145
$248$	0	0
$249$	41.8382	2.65139
$250$	0	0
$251$	18.2828	1.15400	0.577001	−	0.816744i	$-0.304223\pi$
$251$	18.2828	1.15400	0.577001	+	0.816744i	$0.304223\pi$
$252$	0	0
$253$	−13.3035	−0.836386
$254$	0	0
$255$	−2.53845	−0.158964
$256$	0	0
$257$	30.4924	1.90207	0.951033	−	0.309091i	$-0.100025\pi$
$257$	30.4924	1.90207	0.951033	+	0.309091i	$0.100025\pi$
$258$	0	0
$259$	−12.9291	−0.803372
$260$	0	0
$261$	6.55447	0.405712
$262$	0	0
$263$	11.2828	0.695730	0.347865	−	0.937545i	$-0.386907\pi$
$263$	11.2828	0.695730	0.347865	+	0.937545i	$0.386907\pi$
$264$	0	0
$265$	−0.665916	−0.0409069
$266$	0	0
$267$	−5.15890	−0.315720
$268$	0	0
$269$	−1.54283	−0.0940679	−0.0470339	−	0.998893i	$-0.514977\pi$
$269$	−1.54283	−0.0940679	−0.0470339	+	0.998893i	$0.514977\pi$
$270$	0	0
$271$	23.6023	1.43374	0.716870	−	0.697207i	$-0.245574\pi$
$271$	23.6023	1.43374	0.716870	+	0.697207i	$0.245574\pi$
$272$	0	0
$273$	−21.9168	−1.32646
$274$	0	0
$275$	24.3987	1.47130
$276$	0	0
$277$	14.2694	0.857367	0.428684	−	0.903455i	$-0.358978\pi$
$277$	14.2694	0.857367	0.428684	+	0.903455i	$0.358978\pi$
$278$	0	0
$279$	26.4233	1.58192
$280$	0	0
$281$	22.6952	1.35388	0.676942	−	0.736037i	$-0.263305\pi$
$281$	22.6952	1.35388	0.676942	+	0.736037i	$0.263305\pi$
$282$	0	0
$283$	22.9783	1.36592	0.682958	−	0.730458i	$-0.260693\pi$
$283$	22.9783	1.36592	0.682958	+	0.730458i	$0.260693\pi$
$284$	0	0
$285$	−16.0015	−0.947847
$286$	0	0
$287$	−5.83672	−0.344531
$288$	0	0
$289$	−15.2893	−0.899372
$290$	0	0
$291$	0.751051	0.0440274
$292$	0	0
$293$	−25.1176	−1.46739	−0.733694	−	0.679480i	$-0.762205\pi$
$293$	−25.1176	−1.46739	−0.733694	+	0.679480i	$0.762205\pi$
$294$	0	0
$295$	−1.12375	−0.0654271
$296$	0	0
$297$	−20.4853	−1.18868
$298$	0	0
$299$	−9.71388	−0.561768
$300$	0	0
$301$	11.7209	0.675582
$302$	0	0
$303$	−46.3270	−2.66142
$304$	0	0
$305$	−5.66423	−0.324333
$306$	0	0
$307$	14.5944	0.832947	0.416473	−	0.909148i	$-0.363266\pi$
$307$	14.5944	0.832947	0.416473	+	0.909148i	$0.363266\pi$
$308$	0	0
$309$	−17.4059	−0.990188
$310$	0	0
$311$	−8.99147	−0.509860	−0.254930	−	0.966960i	$-0.582052\pi$
$311$	−8.99147	−0.509860	−0.254930	+	0.966960i	$0.582052\pi$
$312$	0	0
$313$	−4.56361	−0.257951	−0.128975	−	0.991648i	$-0.541169\pi$
$313$	−4.56361	−0.257951	−0.128975	+	0.991648i	$0.541169\pi$
$314$	0	0
$315$	6.36685	0.358731
$316$	0	0
$317$	−7.39682	−0.415447	−0.207723	−	0.978188i	$-0.566605\pi$
$317$	−7.39682	−0.415447	−0.207723	+	0.978188i	$0.566605\pi$
$318$	0	0
$319$	−8.11854	−0.454551
$320$	0	0
$321$	−17.2549	−0.963072
$322$	0	0
$323$	10.7835	0.600008
$324$	0	0
$325$	17.8153	0.988215
$326$	0	0
$327$	25.3754	1.40326
$328$	0	0
$329$	−13.0650	−0.720297
$330$	0	0
$331$	23.5847	1.29633	0.648166	−	0.761499i	$-0.275536\pi$
$331$	23.5847	1.29633	0.648166	+	0.761499i	$0.275536\pi$
$332$	0	0
$333$	−27.9086	−1.52938
$334$	0	0
$335$	−7.51890	−0.410801
$336$	0	0
$337$	25.0637	1.36531	0.682653	−	0.730743i	$-0.260826\pi$
$337$	25.0637	1.36531	0.682653	+	0.730743i	$0.260826\pi$
$338$	0	0
$339$	−5.14689	−0.279541
$340$	0	0
$341$	−32.7285	−1.77235
$342$	0	0
$343$	−20.0584	−1.08305
$344$	0	0
$345$	4.75159	0.255817
$346$	0	0
$347$	−11.1374	−0.597888	−0.298944	−	0.954271i	$-0.596634\pi$
$347$	−11.1374	−0.597888	−0.298944	+	0.954271i	$0.596634\pi$
$348$	0	0
$349$	34.3068	1.83640	0.918202	−	0.396114i	$-0.129641\pi$
$349$	34.3068	1.83640	0.918202	+	0.396114i	$0.129641\pi$
$350$	0	0
$351$	−14.9578	−0.798389
$352$	0	0
$353$	18.3887	0.978731	0.489366	−	0.872079i	$-0.337228\pi$
$353$	18.3887	0.978731	0.489366	+	0.872079i	$0.337228\pi$
$354$	0	0
$355$	9.20665	0.488638
$356$	0	0
$357$	−7.22471	−0.382372
$358$	0	0
$359$	−16.8572	−0.889688	−0.444844	−	0.895608i	$-0.646741\pi$
$359$	−16.8572	−0.889688	−0.444844	+	0.895608i	$0.646741\pi$
$360$	0	0
$361$	48.9751	2.57764
$362$	0	0
$363$	50.3561	2.64301
$364$	0	0
$365$	7.01112	0.366979
$366$	0	0
$367$	−20.9807	−1.09518	−0.547592	−	0.836745i	$-0.684456\pi$
$367$	−20.9807	−1.09518	−0.547592	+	0.836745i	$0.684456\pi$
$368$	0	0
$369$	−12.5991	−0.655885
$370$	0	0
$371$	−1.89527	−0.0983975
$372$	0	0
$373$	8.57191	0.443837	0.221918	−	0.975065i	$-0.428768\pi$
$373$	8.57191	0.443837	0.221918	+	0.975065i	$0.428768\pi$
$374$	0	0
$375$	−18.4186	−0.951130
$376$	0	0
$377$	−5.92794	−0.305304
$378$	0	0
$379$	24.0731	1.23655	0.618275	−	0.785962i	$-0.287832\pi$
$379$	24.0731	1.23655	0.618275	+	0.785962i	$0.287832\pi$
$380$	0	0
$381$	−27.7461	−1.42148
$382$	0	0
$383$	28.2472	1.44336	0.721682	−	0.692225i	$-0.243369\pi$
$383$	28.2472	1.44336	0.721682	+	0.692225i	$0.243369\pi$
$384$	0	0
$385$	−7.88615	−0.401915
$386$	0	0
$387$	25.3007	1.28611
$388$	0	0
$389$	20.2718	1.02782	0.513911	−	0.857843i	$-0.328196\pi$
$389$	20.2718	1.02782	0.513911	+	0.857843i	$0.328196\pi$
$390$	0	0
$391$	−3.20211	−0.161938
$392$	0	0
$393$	37.9393	1.91379
$394$	0	0
$395$	2.00036	0.100649
$396$	0	0
$397$	9.39156	0.471349	0.235674	−	0.971832i	$-0.424270\pi$
$397$	9.39156	0.471349	0.235674	+	0.971832i	$0.424270\pi$
$398$	0	0
$399$	−45.5420	−2.27995
$400$	0	0
$401$	8.35327	0.417143	0.208571	−	0.978007i	$-0.433119\pi$
$401$	8.35327	0.417143	0.208571	+	0.978007i	$0.433119\pi$
$402$	0	0
$403$	−23.8975	−1.19042
$404$	0	0
$405$	−2.08150	−0.103431
$406$	0	0
$407$	34.5683	1.71349
$408$	0	0
$409$	−9.04325	−0.447160	−0.223580	−	0.974686i	$-0.571774\pi$
$409$	−9.04325	−0.447160	−0.223580	+	0.974686i	$0.571774\pi$
$410$	0	0
$411$	−5.96065	−0.294017
$412$	0	0
$413$	−3.19831	−0.157378
$414$	0	0
$415$	10.9923	0.539590
$416$	0	0
$417$	53.4856	2.61920
$418$	0	0
$419$	21.2265	1.03698	0.518491	−	0.855083i	$-0.326494\pi$
$419$	21.2265	1.03698	0.518491	+	0.855083i	$0.326494\pi$
$420$	0	0
$421$	16.3175	0.795269	0.397634	−	0.917544i	$-0.369831\pi$
$421$	16.3175	0.795269	0.397634	+	0.917544i	$0.369831\pi$
$422$	0	0
$423$	−28.2021	−1.37123
$424$	0	0
$425$	5.87270	0.284868
$426$	0	0
$427$	−16.1210	−0.780151
$428$	0	0
$429$	58.5987	2.82917
$430$	0	0
$431$	−17.2192	−0.829420	−0.414710	−	0.909954i	$-0.636117\pi$
$431$	−17.2192	−0.829420	−0.414710	+	0.909954i	$0.636117\pi$
$432$	0	0
$433$	0.155714	0.00748312	0.00374156	−	0.999993i	$-0.498809\pi$
$433$	0.155714	0.00748312	0.00374156	+	0.999993i	$0.498809\pi$
$434$	0	0
$435$	2.89968	0.139029
$436$	0	0
$437$	−20.1850	−0.965579
$438$	0	0
$439$	−23.1300	−1.10394	−0.551968	−	0.833865i	$-0.686123\pi$
$439$	−23.1300	−1.10394	−0.551968	+	0.833865i	$0.686123\pi$
$440$	0	0
$441$	−12.5886	−0.599459
$442$	0	0
$443$	−12.5268	−0.595168	−0.297584	−	0.954696i	$-0.596181\pi$
$443$	−12.5268	−0.595168	−0.297584	+	0.954696i	$0.596181\pi$
$444$	0	0
$445$	−1.35541	−0.0642527
$446$	0	0
$447$	−30.1367	−1.42542
$448$	0	0
$449$	−38.0487	−1.79563	−0.897814	−	0.440375i	$-0.854846\pi$
$449$	−38.0487	−1.79563	−0.897814	+	0.440375i	$0.854846\pi$
$450$	0	0
$451$	15.6056	0.734840
$452$	0	0
$453$	−19.1185	−0.898264
$454$	0	0
$455$	−5.75825	−0.269951
$456$	0	0
$457$	25.4956	1.19263	0.596317	−	0.802749i	$-0.296630\pi$
$457$	25.4956	1.19263	0.596317	+	0.802749i	$0.296630\pi$
$458$	0	0
$459$	−4.93075	−0.230147
$460$	0	0
$461$	22.5869	1.05198	0.525988	−	0.850492i	$-0.323696\pi$
$461$	22.5869	1.05198	0.525988	+	0.850492i	$0.323696\pi$
$462$	0	0
$463$	−42.2153	−1.96191	−0.980956	−	0.194232i	$-0.937779\pi$
$463$	−42.2153	−1.96191	−0.980956	+	0.194232i	$0.937779\pi$
$464$	0	0
$465$	11.6896	0.542092
$466$	0	0
$467$	−4.29006	−0.198520	−0.0992601	−	0.995062i	$-0.531648\pi$
$467$	−4.29006	−0.198520	−0.0992601	+	0.995062i	$0.531648\pi$
$468$	0	0
$469$	−21.3996	−0.988141
$470$	0	0
$471$	−65.8145	−3.03257
$472$	0	0
$473$	−31.3381	−1.44093
$474$	0	0
$475$	37.0194	1.69857
$476$	0	0
$477$	−4.09112	−0.187320
$478$	0	0
$479$	7.74802	0.354016	0.177008	−	0.984209i	$-0.443358\pi$
$479$	7.74802	0.354016	0.177008	+	0.984209i	$0.443358\pi$
$480$	0	0
$481$	25.2409	1.15088
$482$	0	0
$483$	13.5236	0.615343
$484$	0	0
$485$	0.197326	0.00896010
$486$	0	0
$487$	−3.12687	−0.141692	−0.0708459	−	0.997487i	$-0.522570\pi$
$487$	−3.12687	−0.141692	−0.0708459	+	0.997487i	$0.522570\pi$
$488$	0	0
$489$	−1.82699	−0.0826193
$490$	0	0
$491$	17.8547	0.805771	0.402886	−	0.915250i	$-0.368007\pi$
$491$	17.8547	0.805771	0.402886	+	0.915250i	$0.368007\pi$
$492$	0	0
$493$	−1.95411	−0.0880085
$494$	0	0
$495$	−17.0230	−0.765127
$496$	0	0
$497$	26.2031	1.17537
$498$	0	0
$499$	−2.04004	−0.0913246	−0.0456623	−	0.998957i	$-0.514540\pi$
$499$	−2.04004	−0.0913246	−0.0456623	+	0.998957i	$0.514540\pi$
$500$	0	0
$501$	35.9265	1.60508
$502$	0	0
$503$	−7.07972	−0.315669	−0.157835	−	0.987466i	$-0.550451\pi$
$503$	−7.07972	−0.315669	−0.157835	+	0.987466i	$0.550451\pi$
$504$	0	0
$505$	−12.1716	−0.541631
$506$	0	0
$507$	7.45430	0.331057
$508$	0	0
$509$	17.9368	0.795037	0.397518	−	0.917594i	$-0.369871\pi$
$509$	17.9368	0.795037	0.397518	+	0.917594i	$0.369871\pi$
$510$	0	0
$511$	19.9544	0.882732
$512$	0	0
$513$	−31.0817	−1.37229
$514$	0	0
$515$	−4.57311	−0.201515
$516$	0	0
$517$	34.9318	1.53630
$518$	0	0
$519$	−49.6491	−2.17935
$520$	0	0
$521$	−3.19375	−0.139921	−0.0699603	−	0.997550i	$-0.522287\pi$
$521$	−3.19375	−0.139921	−0.0699603	+	0.997550i	$0.522287\pi$
$522$	0	0
$523$	15.7713	0.689630	0.344815	−	0.938671i	$-0.387942\pi$
$523$	15.7713	0.689630	0.344815	+	0.938671i	$0.387942\pi$
$524$	0	0
$525$	−24.8023	−1.08246
$526$	0	0
$527$	−7.87765	−0.343156
$528$	0	0
$529$	−17.0061	−0.739397
$530$	0	0
$531$	−6.90386	−0.299602
$532$	0	0
$533$	11.3948	0.493563
$534$	0	0
$535$	−4.53342	−0.195997
$536$	0	0
$537$	−58.8001	−2.53741
$538$	0	0
$539$	15.5926	0.671621
$540$	0	0
$541$	28.1775	1.21144	0.605722	−	0.795676i	$-0.292884\pi$
$541$	28.1775	1.21144	0.605722	+	0.795676i	$0.292884\pi$
$542$	0	0
$543$	2.08073	0.0892927
$544$	0	0
$545$	6.66696	0.285581
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	−34.7988	−1.48517
$550$	0	0
$551$	−12.3180	−0.524764
$552$	0	0
$553$	5.69324	0.242101
$554$	0	0
$555$	−12.3467	−0.524088
$556$	0	0
$557$	−11.6421	−0.493290	−0.246645	−	0.969106i	$-0.579328\pi$
$557$	−11.6421	−0.493290	−0.246645	+	0.969106i	$0.579328\pi$
$558$	0	0
$559$	−22.8822	−0.967815
$560$	0	0
$561$	19.3167	0.815550
$562$	0	0
$563$	−35.1631	−1.48195	−0.740974	−	0.671534i	$-0.765636\pi$
$563$	−35.1631	−1.48195	−0.740974	+	0.671534i	$0.765636\pi$
$564$	0	0
$565$	−1.35226	−0.0568899
$566$	0	0
$567$	−5.92417	−0.248792
$568$	0	0
$569$	−5.54994	−0.232665	−0.116333	−	0.993210i	$-0.537114\pi$
$569$	−5.54994	−0.232665	−0.116333	+	0.993210i	$0.537114\pi$
$570$	0	0
$571$	12.2975	0.514633	0.257317	−	0.966327i	$-0.417162\pi$
$571$	12.2975	0.514633	0.257317	+	0.966327i	$0.417162\pi$
$572$	0	0
$573$	9.58105	0.400254
$574$	0	0
$575$	−10.9928	−0.458431
$576$	0	0
$577$	−21.7654	−0.906105	−0.453052	−	0.891484i	$-0.649665\pi$
$577$	−21.7654	−0.906105	−0.453052	+	0.891484i	$0.649665\pi$
$578$	0	0
$579$	−24.3160	−1.01054
$580$	0	0
$581$	31.2852	1.29793
$582$	0	0
$583$	5.06737	0.209869
$584$	0	0
$585$	−12.4297	−0.513907
$586$	0	0
$587$	1.32577	0.0547206	0.0273603	−	0.999626i	$-0.491290\pi$
$587$	1.32577	0.0547206	0.0273603	+	0.999626i	$0.491290\pi$
$588$	0	0
$589$	−49.6579	−2.04612
$590$	0	0
$591$	−54.8120	−2.25467
$592$	0	0
$593$	−0.00527691	−0.000216697 0	−0.000108348	−	1.00000i	$-0.500034\pi$
$593$	−0.00527691	−0.000216697 0	−0.000108348		1.00000i	$0.500034\pi$
$594$	0	0
$595$	−1.89817	−0.0778174
$596$	0	0
$597$	−10.1298	−0.414584
$598$	0	0
$599$	26.4628	1.08124	0.540620	−	0.841267i	$-0.318190\pi$
$599$	26.4628	1.08124	0.540620	+	0.841267i	$0.318190\pi$
$600$	0	0
$601$	−17.3745	−0.708719	−0.354360	−	0.935109i	$-0.615301\pi$
$601$	−17.3745	−0.708719	−0.354360	+	0.935109i	$0.615301\pi$
$602$	0	0
$603$	−46.1931	−1.88113
$604$	0	0
$605$	13.2302	0.537884
$606$	0	0
$607$	28.3811	1.15195	0.575977	−	0.817466i	$-0.304622\pi$
$607$	28.3811	1.15195	0.575977	+	0.817466i	$0.304622\pi$
$608$	0	0
$609$	8.25281	0.334421
$610$	0	0
$611$	25.5063	1.03187
$612$	0	0
$613$	−31.8417	−1.28607	−0.643036	−	0.765836i	$-0.722326\pi$
$613$	−31.8417	−1.28607	−0.643036	+	0.765836i	$0.722326\pi$
$614$	0	0
$615$	−5.57383	−0.224758
$616$	0	0
$617$	−41.2687	−1.66141	−0.830707	−	0.556710i	$-0.812064\pi$
$617$	−41.2687	−1.66141	−0.830707	+	0.556710i	$0.812064\pi$
$618$	0	0
$619$	−36.6519	−1.47316	−0.736582	−	0.676348i	$-0.763561\pi$
$619$	−36.6519	−1.47316	−0.736582	+	0.676348i	$0.763561\pi$
$620$	0	0
$621$	9.22960	0.370371
$622$	0	0
$623$	−3.85765	−0.154554
$624$	0	0
$625$	17.6112	0.704450
$626$	0	0
$627$	121.765	4.86284
$628$	0	0
$629$	8.32048	0.331759
$630$	0	0
$631$	33.4006	1.32966	0.664829	−	0.746996i	$-0.268505\pi$
$631$	33.4006	1.32966	0.664829	+	0.746996i	$0.268505\pi$
$632$	0	0
$633$	−48.8715	−1.94247
$634$	0	0
$635$	−7.28981	−0.289287
$636$	0	0
$637$	11.3853	0.451102
$638$	0	0
$639$	56.5620	2.23756
$640$	0	0
$641$	6.94918	0.274476	0.137238	−	0.990538i	$-0.456177\pi$
$641$	6.94918	0.274476	0.137238	+	0.990538i	$0.456177\pi$
$642$	0	0
$643$	−4.52617	−0.178495	−0.0892474	−	0.996009i	$-0.528446\pi$
$643$	−4.52617	−0.178495	−0.0892474	+	0.996009i	$0.528446\pi$
$644$	0	0
$645$	11.1930	0.440723
$646$	0	0
$647$	−25.3439	−0.996371	−0.498185	−	0.867071i	$-0.666000\pi$
$647$	−25.3439	−0.996371	−0.498185	+	0.867071i	$0.666000\pi$
$648$	0	0
$649$	8.55130	0.335668
$650$	0	0
$651$	33.2698	1.30395
$652$	0	0
$653$	−8.49304	−0.332358	−0.166179	−	0.986096i	$-0.553143\pi$
$653$	−8.49304	−0.332358	−0.166179	+	0.986096i	$0.553143\pi$
$654$	0	0
$655$	9.96791	0.389478
$656$	0	0
$657$	43.0735	1.68046
$658$	0	0
$659$	−18.2273	−0.710036	−0.355018	−	0.934859i	$-0.615525\pi$
$659$	−18.2273	−0.710036	−0.355018	+	0.934859i	$0.615525\pi$
$660$	0	0
$661$	−16.2494	−0.632027	−0.316013	−	0.948755i	$-0.602344\pi$
$661$	−16.2494	−0.632027	−0.316013	+	0.948755i	$0.602344\pi$
$662$	0	0
$663$	14.1045	0.547774
$664$	0	0
$665$	−11.9654	−0.463998
$666$	0	0
$667$	3.65779	0.141630
$668$	0	0
$669$	52.2362	2.01957
$670$	0	0
$671$	43.1027	1.66396
$672$	0	0
$673$	39.3776	1.51789	0.758947	−	0.651153i	$-0.225714\pi$
$673$	39.3776	1.51789	0.758947	+	0.651153i	$0.225714\pi$
$674$	0	0
$675$	−16.9271	−0.651526
$676$	0	0
$677$	−39.4458	−1.51602	−0.758012	−	0.652241i	$-0.773829\pi$
$677$	−39.4458	−1.51602	−0.758012	+	0.652241i	$0.773829\pi$
$678$	0	0
$679$	0.561610	0.0215526
$680$	0	0
$681$	−63.2547	−2.42393
$682$	0	0
$683$	−25.8506	−0.989144	−0.494572	−	0.869137i	$-0.664675\pi$
$683$	−25.8506	−0.989144	−0.494572	+	0.869137i	$0.664675\pi$
$684$	0	0
$685$	−1.56606	−0.0598360
$686$	0	0
$687$	2.85022	0.108743
$688$	0	0
$689$	3.70006	0.140961
$690$	0	0
$691$	34.6845	1.31946	0.659729	−	0.751503i	$-0.270671\pi$
$691$	34.6845	1.31946	0.659729	+	0.751503i	$0.270671\pi$
$692$	0	0
$693$	−48.4493	−1.84044
$694$	0	0
$695$	14.0524	0.533039
$696$	0	0
$697$	3.75622	0.142277
$698$	0	0
$699$	28.2711	1.06931
$700$	0	0
$701$	−38.7984	−1.46540	−0.732698	−	0.680553i	$-0.761739\pi$
$701$	−38.7984	−1.46540	−0.732698	+	0.680553i	$0.761739\pi$
$702$	0	0
$703$	52.4493	1.97816
$704$	0	0
$705$	−12.4765	−0.469893
$706$	0	0
$707$	−34.6418	−1.30284
$708$	0	0
$709$	−40.9056	−1.53624	−0.768121	−	0.640305i	$-0.778808\pi$
$709$	−40.9056	−1.53624	−0.768121	+	0.640305i	$0.778808\pi$
$710$	0	0
$711$	12.2894	0.460889
$712$	0	0
$713$	14.7458	0.552233
$714$	0	0
$715$	15.3958	0.575770
$716$	0	0
$717$	−14.1626	−0.528910
$718$	0	0
$719$	−4.65393	−0.173562	−0.0867812	−	0.996227i	$-0.527658\pi$
$719$	−4.65393	−0.173562	−0.0867812	+	0.996227i	$0.527658\pi$
$720$	0	0
$721$	−13.0156	−0.484725
$722$	0	0
$723$	−62.8572	−2.33768
$724$	0	0
$725$	−6.70840	−0.249144
$726$	0	0
$727$	−8.00848	−0.297018	−0.148509	−	0.988911i	$-0.547447\pi$
$727$	−8.00848	−0.297018	−0.148509	+	0.988911i	$0.547447\pi$
$728$	0	0
$729$	−43.5266	−1.61210
$730$	0	0
$731$	−7.54298	−0.278987
$732$	0	0
$733$	20.7022	0.764652	0.382326	−	0.924028i	$-0.375123\pi$
$733$	20.7022	0.764652	0.382326	+	0.924028i	$0.375123\pi$
$734$	0	0
$735$	−5.56918	−0.205422
$736$	0	0
$737$	57.2160	2.10758
$738$	0	0
$739$	−10.3160	−0.379479	−0.189740	−	0.981834i	$-0.560764\pi$
$739$	−10.3160	−0.379479	−0.189740	+	0.981834i	$0.560764\pi$
$740$	0	0
$741$	88.9098	3.26618
$742$	0	0
$743$	27.5334	1.01010	0.505051	−	0.863090i	$-0.331474\pi$
$743$	27.5334	1.01010	0.505051	+	0.863090i	$0.331474\pi$
$744$	0	0
$745$	−7.91790	−0.290090
$746$	0	0
$747$	67.5322	2.47087
$748$	0	0
$749$	−12.9026	−0.471451
$750$	0	0
$751$	−30.6714	−1.11922	−0.559608	−	0.828758i	$-0.689048\pi$
$751$	−30.6714	−1.11922	−0.559608	+	0.828758i	$0.689048\pi$
$752$	0	0
$753$	49.6911	1.81084
$754$	0	0
$755$	−5.02305	−0.182808
$756$	0	0
$757$	29.0720	1.05664	0.528320	−	0.849046i	$-0.322822\pi$
$757$	29.0720	1.05664	0.528320	+	0.849046i	$0.322822\pi$
$758$	0	0
$759$	−36.1578	−1.31245
$760$	0	0
$761$	−46.1332	−1.67233	−0.836163	−	0.548481i	$-0.815206\pi$
$761$	−46.1332	−1.67233	−0.836163	+	0.548481i	$0.815206\pi$
$762$	0	0
$763$	18.9749	0.686937
$764$	0	0
$765$	−4.09738	−0.148141
$766$	0	0
$767$	6.24393	0.225455
$768$	0	0
$769$	39.6177	1.42865	0.714326	−	0.699813i	$-0.246733\pi$
$769$	39.6177	1.42865	0.714326	+	0.699813i	$0.246733\pi$
$770$	0	0
$771$	82.8758	2.98470
$772$	0	0
$773$	−2.07610	−0.0746722	−0.0373361	−	0.999303i	$-0.511887\pi$
$773$	−2.07610	−0.0746722	−0.0373361	+	0.999303i	$0.511887\pi$
$774$	0	0
$775$	−27.0438	−0.971442
$776$	0	0
$777$	−35.1400	−1.26064
$778$	0	0
$779$	23.6779	0.848348
$780$	0	0
$781$	−70.0591	−2.50691
$782$	0	0
$783$	5.63241	0.201286
$784$	0	0
$785$	−17.2916	−0.617165
$786$	0	0
$787$	7.33613	0.261505	0.130752	−	0.991415i	$-0.458261\pi$
$787$	7.33613	0.261505	0.130752	+	0.991415i	$0.458261\pi$
$788$	0	0
$789$	30.6658	1.09173
$790$	0	0
$791$	−3.84867	−0.136843
$792$	0	0
$793$	31.4724	1.11762
$794$	0	0
$795$	−1.80990	−0.0641907
$796$	0	0
$797$	−19.2052	−0.680283	−0.340141	−	0.940374i	$-0.610475\pi$
$797$	−19.2052	−0.680283	−0.340141	+	0.940374i	$0.610475\pi$
$798$	0	0
$799$	8.40797	0.297453
$800$	0	0
$801$	−8.32712	−0.294224
$802$	0	0
$803$	−53.3520	−1.88275
$804$	0	0
$805$	3.55308	0.125230
$806$	0	0
$807$	−4.19327	−0.147610
$808$	0	0
$809$	15.0621	0.529554	0.264777	−	0.964310i	$-0.414702\pi$
$809$	15.0621	0.529554	0.264777	+	0.964310i	$0.414702\pi$
$810$	0	0
$811$	3.10727	0.109111	0.0545554	−	0.998511i	$-0.482626\pi$
$811$	3.10727	0.109111	0.0545554	+	0.998511i	$0.482626\pi$
$812$	0	0
$813$	64.1491	2.24981
$814$	0	0
$815$	−0.480010	−0.0168140
$816$	0	0
$817$	−47.5482	−1.66350
$818$	0	0
$819$	−35.3764	−1.23615
$820$	0	0
$821$	−39.3045	−1.37174	−0.685869	−	0.727725i	$-0.740577\pi$
$821$	−39.3045	−1.37174	−0.685869	+	0.727725i	$0.740577\pi$
$822$	0	0
$823$	24.7300	0.862033	0.431017	−	0.902344i	$-0.358155\pi$
$823$	24.7300	0.862033	0.431017	+	0.902344i	$0.358155\pi$
$824$	0	0
$825$	66.3137	2.30875
$826$	0	0
$827$	37.2644	1.29581	0.647905	−	0.761721i	$-0.275645\pi$
$827$	37.2644	1.29581	0.647905	+	0.761721i	$0.275645\pi$
$828$	0	0
$829$	52.3014	1.81650	0.908251	−	0.418425i	$-0.137418\pi$
$829$	52.3014	1.81650	0.908251	+	0.418425i	$0.137418\pi$
$830$	0	0
$831$	38.7831	1.34537
$832$	0	0
$833$	3.75309	0.130037
$834$	0	0
$835$	9.43908	0.326653
$836$	0	0
$837$	22.7061	0.784838
$838$	0	0
$839$	−18.0759	−0.624050	−0.312025	−	0.950074i	$-0.601007\pi$
$839$	−18.0759	−0.624050	−0.312025	+	0.950074i	$0.601007\pi$
$840$	0	0
$841$	−26.7678	−0.923028
$842$	0	0
$843$	61.6836	2.12450
$844$	0	0
$845$	1.95849	0.0673741
$846$	0	0
$847$	37.6546	1.29383
$848$	0	0
$849$	62.4529	2.14338
$850$	0	0
$851$	−15.5747	−0.533892
$852$	0	0
$853$	2.26471	0.0775422	0.0387711	−	0.999248i	$-0.487656\pi$
$853$	2.26471	0.0775422	0.0387711	+	0.999248i	$0.487656\pi$
$854$	0	0
$855$	−25.8284	−0.883314
$856$	0	0
$857$	−34.0377	−1.16271	−0.581353	−	0.813652i	$-0.697476\pi$
$857$	−34.0377	−1.16271	−0.581353	+	0.813652i	$0.697476\pi$
$858$	0	0
$859$	−41.4218	−1.41329	−0.706646	−	0.707567i	$-0.749793\pi$
$859$	−41.4218	−1.41329	−0.706646	+	0.707567i	$0.749793\pi$
$860$	0	0
$861$	−15.8637	−0.540634
$862$	0	0
$863$	−27.5129	−0.936551	−0.468275	−	0.883583i	$-0.655124\pi$
$863$	−27.5129	−0.936551	−0.468275	+	0.883583i	$0.655124\pi$
$864$	0	0
$865$	−13.0445	−0.443525
$866$	0	0
$867$	−41.5551	−1.41128
$868$	0	0
$869$	−15.2220	−0.516371
$870$	0	0
$871$	41.7775	1.41558
$872$	0	0
$873$	1.21229	0.0410298
$874$	0	0
$875$	−13.7728	−0.465605
$876$	0	0
$877$	24.8045	0.837589	0.418794	−	0.908081i	$-0.362453\pi$
$877$	24.8045	0.837589	0.418794	+	0.908081i	$0.362453\pi$
$878$	0	0
$879$	−68.2675	−2.30261
$880$	0	0
$881$	25.2629	0.851128	0.425564	−	0.904928i	$-0.360076\pi$
$881$	25.2629	0.851128	0.425564	+	0.904928i	$0.360076\pi$
$882$	0	0
$883$	19.5417	0.657631	0.328815	−	0.944394i	$-0.393351\pi$
$883$	19.5417	0.657631	0.328815	+	0.944394i	$0.393351\pi$
$884$	0	0
$885$	−3.05425	−0.102667
$886$	0	0
$887$	34.5222	1.15914	0.579571	−	0.814922i	$-0.303220\pi$
$887$	34.5222	1.15914	0.579571	+	0.814922i	$0.303220\pi$
$888$	0	0
$889$	−20.7476	−0.695852
$890$	0	0
$891$	15.8394	0.530641
$892$	0	0
$893$	53.0009	1.77361
$894$	0	0
$895$	−15.4487	−0.516394
$896$	0	0
$897$	−26.4015	−0.881520
$898$	0	0
$899$	8.99867	0.300122
$900$	0	0
$901$	1.21970	0.0406341
$902$	0	0
$903$	31.8564	1.06011
$904$	0	0
$905$	0.546677	0.0181721
$906$	0	0
$907$	−48.4647	−1.60924	−0.804622	−	0.593787i	$-0.797632\pi$
$907$	−48.4647	−1.60924	−0.804622	+	0.593787i	$0.797632\pi$
$908$	0	0
$909$	−74.7777	−2.48022
$910$	0	0
$911$	−14.4271	−0.477990	−0.238995	−	0.971021i	$-0.576818\pi$
$911$	−14.4271	−0.477990	−0.238995	+	0.971021i	$0.576818\pi$
$912$	0	0
$913$	−83.6471	−2.76832
$914$	0	0
$915$	−15.3949	−0.508939
$916$	0	0
$917$	28.3698	0.936852
$918$	0	0
$919$	−14.9071	−0.491741	−0.245871	−	0.969303i	$-0.579074\pi$
$919$	−14.9071	−0.491741	−0.245871	+	0.969303i	$0.579074\pi$
$920$	0	0
$921$	39.6663	1.30705
$922$	0	0
$923$	−51.1553	−1.68380
$924$	0	0
$925$	28.5640	0.939179
$926$	0	0
$927$	−28.0954	−0.922772
$928$	0	0
$929$	−19.0507	−0.625033	−0.312517	−	0.949912i	$-0.601172\pi$
$929$	−19.0507	−0.625033	−0.312517	+	0.949912i	$0.601172\pi$
$930$	0	0
$931$	23.6581	0.775364
$932$	0	0
$933$	−24.4380	−0.800066
$934$	0	0
$935$	5.07512	0.165974
$936$	0	0
$937$	1.62214	0.0529931	0.0264965	−	0.999649i	$-0.491565\pi$
$937$	1.62214	0.0529931	0.0264965	+	0.999649i	$0.491565\pi$
$938$	0	0
$939$	−12.4035	−0.404773
$940$	0	0
$941$	−1.42371	−0.0464115	−0.0232058	−	0.999731i	$-0.507387\pi$
$941$	−1.42371	−0.0464115	−0.0232058	+	0.999731i	$0.507387\pi$
$942$	0	0
$943$	−7.03107	−0.228963
$944$	0	0
$945$	5.47118	0.177977
$946$	0	0
$947$	42.2934	1.37435	0.687175	−	0.726492i	$-0.258850\pi$
$947$	42.2934	1.37435	0.687175	+	0.726492i	$0.258850\pi$
$948$	0	0
$949$	−38.9562	−1.26457
$950$	0	0
$951$	−20.1039	−0.651914
$952$	0	0
$953$	−30.0275	−0.972686	−0.486343	−	0.873768i	$-0.661670\pi$
$953$	−30.0275	−0.972686	−0.486343	+	0.873768i	$0.661670\pi$
$954$	0	0
$955$	2.51726	0.0814565
$956$	0	0
$957$	−22.0655	−0.713276
$958$	0	0
$959$	−4.45717	−0.143930
$960$	0	0
$961$	5.27662	0.170213
$962$	0	0
$963$	−27.8515	−0.897502
$964$	0	0
$965$	−6.38862	−0.205657
$966$	0	0
$967$	26.6788	0.857933	0.428966	−	0.903320i	$-0.358878\pi$
$967$	26.6788	0.857933	0.428966	+	0.903320i	$0.358878\pi$
$968$	0	0
$969$	29.3085	0.941526
$970$	0	0
$971$	−11.1042	−0.356351	−0.178175	−	0.983999i	$-0.557019\pi$
$971$	−11.1042	−0.356351	−0.178175	+	0.983999i	$0.557019\pi$
$972$	0	0
$973$	39.9947	1.28217
$974$	0	0
$975$	48.4205	1.55070
$976$	0	0
$977$	16.4245	0.525467	0.262733	−	0.964868i	$-0.415376\pi$
$977$	16.4245	0.525467	0.262733	+	0.964868i	$0.415376\pi$
$978$	0	0
$979$	10.3142	0.329643
$980$	0	0
$981$	40.9591	1.30772
$982$	0	0
$983$	−8.61161	−0.274668	−0.137334	−	0.990525i	$-0.543853\pi$
$983$	−8.61161	−0.274668	−0.137334	+	0.990525i	$0.543853\pi$
$984$	0	0
$985$	−14.4009	−0.458852
$986$	0	0
$987$	−35.5096	−1.13028
$988$	0	0
$989$	14.1193	0.448967
$990$	0	0
$991$	−30.1651	−0.958227	−0.479114	−	0.877753i	$-0.659042\pi$
$991$	−30.1651	−0.958227	−0.479114	+	0.877753i	$0.659042\pi$
$992$	0	0
$993$	64.1012	2.03419
$994$	0	0
$995$	−2.66142	−0.0843728
$996$	0	0
$997$	6.84091	0.216654	0.108327	−	0.994115i	$-0.465451\pi$
$997$	6.84091	0.216654	0.108327	+	0.994115i	$0.465451\pi$
$998$	0	0
$999$	−23.9825	−0.758772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.15		18
4.3	odd	2		547.2.a.b.1.7	✓	18
12.11	even	2		4923.2.a.l.1.12		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.7	✓	18	4.3	odd	2
4923.2.a.l.1.12		18	12.11	even	2
8752.2.a.s.1.15		18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71791 q^{3} +0.714085 q^{5} +2.03236 q^{7} +4.38705 q^{9} +O(q^{10})\)	\(q+2.71791 q^{3} +0.714085 q^{5} +2.03236 q^{7} +4.38705 q^{9} -5.43392 q^{11} -3.96770 q^{13} +1.94082 q^{15} -1.30793 q^{17} -8.24470 q^{19} +5.52379 q^{21} +2.44824 q^{23} -4.49008 q^{25} +3.76989 q^{27} +1.49405 q^{29} +6.02301 q^{31} -14.7689 q^{33} +1.45128 q^{35} -6.36158 q^{37} -10.7839 q^{39} -2.87189 q^{41} +5.76713 q^{43} +3.13273 q^{45} -6.42848 q^{47} -2.86949 q^{49} -3.55483 q^{51} -0.932545 q^{53} -3.88028 q^{55} -22.4084 q^{57} -1.57369 q^{59} -7.93215 q^{61} +8.91609 q^{63} -2.83328 q^{65} -10.5294 q^{67} +6.65410 q^{69} +12.8929 q^{71} +9.81833 q^{73} -12.2037 q^{75} -11.0437 q^{77} +2.80129 q^{79} -2.91492 q^{81} +15.3935 q^{83} -0.933971 q^{85} +4.06069 q^{87} -1.89811 q^{89} -8.06382 q^{91} +16.3700 q^{93} -5.88742 q^{95} +0.276333 q^{97} -23.8389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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