LMFDB - Embedded newform 8752.2.a.s.1.4

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			1.4
Root			$0.763493$ 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)

$f(q)$	$=$	$q-1.83524 q^{3} -1.51218 q^{5} -1.20167 q^{7} +0.368111 q^{9} +O(q^{10})$	$q-1.83524 q^{3} -1.51218 q^{5} -1.20167 q^{7} +0.368111 q^{9} +5.83960 q^{11} -5.40780 q^{13} +2.77522 q^{15} +1.92787 q^{17} -0.965947 q^{19} +2.20536 q^{21} +0.470051 q^{23} -2.71331 q^{25} +4.83015 q^{27} -3.67331 q^{29} -0.675583 q^{31} -10.7171 q^{33} +1.81715 q^{35} +5.66915 q^{37} +9.92463 q^{39} -5.00667 q^{41} -3.18500 q^{43} -0.556651 q^{45} +10.1509 q^{47} -5.55598 q^{49} -3.53810 q^{51} +8.25219 q^{53} -8.83053 q^{55} +1.77275 q^{57} +5.23947 q^{59} -3.53188 q^{61} -0.442349 q^{63} +8.17758 q^{65} -8.70291 q^{67} -0.862657 q^{69} +9.50861 q^{71} +9.68659 q^{73} +4.97957 q^{75} -7.01728 q^{77} +5.23633 q^{79} -9.96883 q^{81} -14.9853 q^{83} -2.91529 q^{85} +6.74142 q^{87} +10.2143 q^{89} +6.49841 q^{91} +1.23986 q^{93} +1.46069 q^{95} -6.07529 q^{97} +2.14962 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.83524	−1.05958	−0.529789	−	0.848130i	$-0.677729\pi$
$3$	−1.83524	−1.05958	−0.529789	+	0.848130i	$0.677729\pi$
$4$	0	0
$5$	−1.51218	−0.676268	−0.338134	−	0.941098i	$-0.609796\pi$
$5$	−1.51218	−0.676268	−0.338134	+	0.941098i	$0.609796\pi$
$6$	0	0
$7$	−1.20167	−0.454189	−0.227095	−	0.973873i	$-0.572923\pi$
$7$	−1.20167	−0.454189	−0.227095	+	0.973873i	$0.572923\pi$
$8$	0	0
$9$	0.368111	0.122704
$10$	0	0
$11$	5.83960	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$11$	5.83960	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$12$	0	0
$13$	−5.40780	−1.49986	−0.749928	−	0.661520i	$-0.769912\pi$
$13$	−5.40780	−1.49986	−0.749928	+	0.661520i	$0.769912\pi$
$14$	0	0
$15$	2.77522	0.716558
$16$	0	0
$17$	1.92787	0.467577	0.233788	−	0.972288i	$-0.424888\pi$
$17$	1.92787	0.467577	0.233788	+	0.972288i	$0.424888\pi$
$18$	0	0
$19$	−0.965947	−0.221603	−0.110802	−	0.993843i	$-0.535342\pi$
$19$	−0.965947	−0.221603	−0.110802	+	0.993843i	$0.535342\pi$
$20$	0	0
$21$	2.20536	0.481249
$22$	0	0
$23$	0.470051	0.0980124	0.0490062	−	0.998798i	$-0.484395\pi$
$23$	0.470051	0.0980124	0.0490062	+	0.998798i	$0.484395\pi$
$24$	0	0
$25$	−2.71331	−0.542661
$26$	0	0
$27$	4.83015	0.929563
$28$	0	0
$29$	−3.67331	−0.682117	−0.341059	−	0.940042i	$-0.610785\pi$
$29$	−3.67331	−0.682117	−0.341059	+	0.940042i	$0.610785\pi$
$30$	0	0
$31$	−0.675583	−0.121338	−0.0606691	−	0.998158i	$-0.519323\pi$
$31$	−0.675583	−0.121338	−0.0606691	+	0.998158i	$0.519323\pi$
$32$	0	0
$33$	−10.7171	−1.86560
$34$	0	0
$35$	1.81715	0.307154
$36$	0	0
$37$	5.66915	0.932003	0.466002	−	0.884784i	$-0.345694\pi$
$37$	5.66915	0.932003	0.466002	+	0.884784i	$0.345694\pi$
$38$	0	0
$39$	9.92463	1.58921
$40$	0	0
$41$	−5.00667	−0.781911	−0.390955	−	0.920410i	$-0.627855\pi$
$41$	−5.00667	−0.781911	−0.390955	+	0.920410i	$0.627855\pi$
$42$	0	0
$43$	−3.18500	−0.485707	−0.242854	−	0.970063i	$-0.578083\pi$
$43$	−3.18500	−0.485707	−0.242854	+	0.970063i	$0.578083\pi$
$44$	0	0
$45$	−0.556651	−0.0829806
$46$	0	0
$47$	10.1509	1.48066	0.740331	−	0.672243i	$-0.234669\pi$
$47$	10.1509	1.48066	0.740331	+	0.672243i	$0.234669\pi$
$48$	0	0
$49$	−5.55598	−0.793712
$50$	0	0
$51$	−3.53810	−0.495433
$52$	0	0
$53$	8.25219	1.13353	0.566763	−	0.823881i	$-0.308196\pi$
$53$	8.25219	1.13353	0.566763	+	0.823881i	$0.308196\pi$
$54$	0	0
$55$	−8.83053	−1.19071
$56$	0	0
$57$	1.77275	0.234806
$58$	0	0
$59$	5.23947	0.682121	0.341060	−	0.940041i	$-0.389214\pi$
$59$	5.23947	0.682121	0.341060	+	0.940041i	$0.389214\pi$
$60$	0	0
$61$	−3.53188	−0.452211	−0.226106	−	0.974103i	$-0.572599\pi$
$61$	−3.53188	−0.452211	−0.226106	+	0.974103i	$0.572599\pi$
$62$	0	0
$63$	−0.442349	−0.0557307
$64$	0	0
$65$	8.17758	1.01430
$66$	0	0
$67$	−8.70291	−1.06323	−0.531615	−	0.846986i	$-0.678415\pi$
$67$	−8.70291	−1.06323	−0.531615	+	0.846986i	$0.678415\pi$
$68$	0	0
$69$	−0.862657	−0.103852
$70$	0	0
$71$	9.50861	1.12846	0.564232	−	0.825616i	$-0.309172\pi$
$71$	9.50861	1.12846	0.564232	+	0.825616i	$0.309172\pi$
$72$	0	0
$73$	9.68659	1.13373	0.566865	−	0.823811i	$-0.308156\pi$
$73$	9.68659	1.13373	0.566865	+	0.823811i	$0.308156\pi$
$74$	0	0
$75$	4.97957	0.574991
$76$	0	0
$77$	−7.01728	−0.799693
$78$	0	0
$79$	5.23633	0.589133	0.294567	−	0.955631i	$-0.404825\pi$
$79$	5.23633	0.589133	0.294567	+	0.955631i	$0.404825\pi$
$80$	0	0
$81$	−9.96883	−1.10765
$82$	0	0
$83$	−14.9853	−1.64485	−0.822427	−	0.568871i	$-0.807380\pi$
$83$	−14.9853	−1.64485	−0.822427	+	0.568871i	$0.807380\pi$
$84$	0	0
$85$	−2.91529	−0.316207
$86$	0	0
$87$	6.74142	0.722756
$88$	0	0
$89$	10.2143	1.08271	0.541357	−	0.840793i	$-0.317911\pi$
$89$	10.2143	1.08271	0.541357	+	0.840793i	$0.317911\pi$
$90$	0	0
$91$	6.49841	0.681218
$92$	0	0
$93$	1.23986	0.128567
$94$	0	0
$95$	1.46069	0.149863
$96$	0	0
$97$	−6.07529	−0.616853	−0.308426	−	0.951248i	$-0.599802\pi$
$97$	−6.07529	−0.616853	−0.308426	+	0.951248i	$0.599802\pi$
$98$	0	0
$99$	2.14962	0.216045
$100$	0	0
$101$	−2.36723	−0.235549	−0.117774	−	0.993040i	$-0.537576\pi$
$101$	−2.36723	−0.235549	−0.117774	+	0.993040i	$0.537576\pi$
$102$	0	0
$103$	−12.4607	−1.22779	−0.613895	−	0.789388i	$-0.710398\pi$
$103$	−12.4607	−1.22779	−0.613895	+	0.789388i	$0.710398\pi$
$104$	0	0
$105$	−3.33490	−0.325453
$106$	0	0
$107$	17.8981	1.73027	0.865137	−	0.501536i	$-0.167232\pi$
$107$	17.8981	1.73027	0.865137	+	0.501536i	$0.167232\pi$
$108$	0	0
$109$	−2.89181	−0.276985	−0.138492	−	0.990363i	$-0.544226\pi$
$109$	−2.89181	−0.276985	−0.138492	+	0.990363i	$0.544226\pi$
$110$	0	0
$111$	−10.4043	−0.987529
$112$	0	0
$113$	5.34990	0.503276	0.251638	−	0.967821i	$-0.419031\pi$
$113$	5.34990	0.503276	0.251638	+	0.967821i	$0.419031\pi$
$114$	0	0
$115$	−0.710803	−0.0662827
$116$	0	0
$117$	−1.99067	−0.184038
$118$	0	0
$119$	−2.31666	−0.212368
$120$	0	0
$121$	23.1009	2.10008
$122$	0	0
$123$	9.18845	0.828495
$124$	0	0
$125$	11.6639	1.04325
$126$	0	0
$127$	−14.6006	−1.29559	−0.647795	−	0.761815i	$-0.724309\pi$
$127$	−14.6006	−1.29559	−0.647795	+	0.761815i	$0.724309\pi$
$128$	0	0
$129$	5.84524	0.514644
$130$	0	0
$131$	19.8795	1.73688	0.868441	−	0.495792i	$-0.165122\pi$
$131$	19.8795	1.73688	0.868441	+	0.495792i	$0.165122\pi$
$132$	0	0
$133$	1.16075	0.100650
$134$	0	0
$135$	−7.30407	−0.628634
$136$	0	0
$137$	19.2841	1.64755	0.823777	−	0.566914i	$-0.191863\pi$
$137$	19.2841	1.64755	0.823777	+	0.566914i	$0.191863\pi$
$138$	0	0
$139$	−11.2982	−0.958297	−0.479149	−	0.877734i	$-0.659054\pi$
$139$	−11.2982	−0.958297	−0.479149	+	0.877734i	$0.659054\pi$
$140$	0	0
$141$	−18.6294	−1.56887
$142$	0	0
$143$	−31.5794	−2.64080
$144$	0	0
$145$	5.55472	0.461294
$146$	0	0
$147$	10.1966	0.840999
$148$	0	0
$149$	0.559854	0.0458650	0.0229325	−	0.999737i	$-0.492700\pi$
$149$	0.559854	0.0458650	0.0229325	+	0.999737i	$0.492700\pi$
$150$	0	0
$151$	7.77394	0.632634	0.316317	−	0.948653i	$-0.397554\pi$
$151$	7.77394	0.632634	0.316317	+	0.948653i	$0.397554\pi$
$152$	0	0
$153$	0.709670	0.0573734
$154$	0	0
$155$	1.02160	0.0820572
$156$	0	0
$157$	−0.453089	−0.0361604	−0.0180802	−	0.999837i	$-0.505755\pi$
$157$	−0.453089	−0.0361604	−0.0180802	+	0.999837i	$0.505755\pi$
$158$	0	0
$159$	−15.1448	−1.20106
$160$	0	0
$161$	−0.564847	−0.0445162
$162$	0	0
$163$	7.12748	0.558267	0.279134	−	0.960252i	$-0.409953\pi$
$163$	7.12748	0.558267	0.279134	+	0.960252i	$0.409953\pi$
$164$	0	0
$165$	16.2062	1.26165
$166$	0	0
$167$	10.5555	0.816808	0.408404	−	0.912801i	$-0.366086\pi$
$167$	10.5555	0.816808	0.408404	+	0.912801i	$0.366086\pi$
$168$	0	0
$169$	16.2443	1.24957
$170$	0	0
$171$	−0.355576	−0.0271916
$172$	0	0
$173$	−16.7583	−1.27411	−0.637055	−	0.770818i	$-0.719848\pi$
$173$	−16.7583	−1.27411	−0.637055	+	0.770818i	$0.719848\pi$
$174$	0	0
$175$	3.26050	0.246471
$176$	0	0
$177$	−9.61569	−0.722760
$178$	0	0
$179$	−4.75191	−0.355174	−0.177587	−	0.984105i	$-0.556829\pi$
$179$	−4.75191	−0.355174	−0.177587	+	0.984105i	$0.556829\pi$
$180$	0	0
$181$	−9.02008	−0.670457	−0.335229	−	0.942137i	$-0.608814\pi$
$181$	−9.02008	−0.670457	−0.335229	+	0.942137i	$0.608814\pi$
$182$	0	0
$183$	6.48186	0.479153
$184$	0	0
$185$	−8.57279	−0.630284
$186$	0	0
$187$	11.2580	0.823264
$188$	0	0
$189$	−5.80426	−0.422198
$190$	0	0
$191$	24.6001	1.78000	0.890000	−	0.455960i	$-0.150704\pi$
$191$	24.6001	1.78000	0.890000	+	0.455960i	$0.150704\pi$
$192$	0	0
$193$	15.5317	1.11800	0.558998	−	0.829169i	$-0.311186\pi$
$193$	15.5317	1.11800	0.558998	+	0.829169i	$0.311186\pi$
$194$	0	0
$195$	−15.0078	−1.07473
$196$	0	0
$197$	−15.1172	−1.07705	−0.538527	−	0.842608i	$-0.681019\pi$
$197$	−15.1172	−1.07705	−0.538527	+	0.842608i	$0.681019\pi$
$198$	0	0
$199$	−13.6303	−0.966227	−0.483114	−	0.875558i	$-0.660494\pi$
$199$	−13.6303	−0.966227	−0.483114	+	0.875558i	$0.660494\pi$
$200$	0	0
$201$	15.9719	1.12657
$202$	0	0
$203$	4.41412	0.309810
$204$	0	0
$205$	7.57100	0.528781
$206$	0	0
$207$	0.173031	0.0120265
$208$	0	0
$209$	−5.64074	−0.390178
$210$	0	0
$211$	−11.3226	−0.779478	−0.389739	−	0.920925i	$-0.627435\pi$
$211$	−11.3226	−0.779478	−0.389739	+	0.920925i	$0.627435\pi$
$212$	0	0
$213$	−17.4506	−1.19570
$214$	0	0
$215$	4.81629	0.328468
$216$	0	0
$217$	0.811829	0.0551106
$218$	0	0
$219$	−17.7772	−1.20127
$220$	0	0
$221$	−10.4255	−0.701297
$222$	0	0
$223$	−5.49894	−0.368236	−0.184118	−	0.982904i	$-0.558943\pi$
$223$	−5.49894	−0.368236	−0.184118	+	0.982904i	$0.558943\pi$
$224$	0	0
$225$	−0.998798	−0.0665866
$226$	0	0
$227$	2.09244	0.138880	0.0694402	−	0.997586i	$-0.477879\pi$
$227$	2.09244	0.138880	0.0694402	+	0.997586i	$0.477879\pi$
$228$	0	0
$229$	−16.4218	−1.08518	−0.542592	−	0.839996i	$-0.682557\pi$
$229$	−16.4218	−1.08518	−0.542592	+	0.839996i	$0.682557\pi$
$230$	0	0
$231$	12.8784	0.847337
$232$	0	0
$233$	−10.0033	−0.655338	−0.327669	−	0.944793i	$-0.606263\pi$
$233$	−10.0033	−0.655338	−0.327669	+	0.944793i	$0.606263\pi$
$234$	0	0
$235$	−15.3500	−1.00132
$236$	0	0
$237$	−9.60993	−0.624232
$238$	0	0
$239$	−12.8814	−0.833227	−0.416614	−	0.909084i	$-0.636783\pi$
$239$	−12.8814	−0.833227	−0.416614	+	0.909084i	$0.636783\pi$
$240$	0	0
$241$	9.58394	0.617356	0.308678	−	0.951167i	$-0.400114\pi$
$241$	9.58394	0.617356	0.308678	+	0.951167i	$0.400114\pi$
$242$	0	0
$243$	3.80475	0.244075
$244$	0	0
$245$	8.40166	0.536762
$246$	0	0
$247$	5.22365	0.332373
$248$	0	0
$249$	27.5017	1.74285
$250$	0	0
$251$	7.09126	0.447596	0.223798	−	0.974636i	$-0.428154\pi$
$251$	7.09126	0.447596	0.223798	+	0.974636i	$0.428154\pi$
$252$	0	0
$253$	2.74491	0.172571
$254$	0	0
$255$	5.35025	0.335046
$256$	0	0
$257$	−7.64221	−0.476708	−0.238354	−	0.971178i	$-0.576608\pi$
$257$	−7.64221	−0.476708	−0.238354	+	0.971178i	$0.576608\pi$
$258$	0	0
$259$	−6.81246	−0.423306
$260$	0	0
$261$	−1.35219	−0.0836983
$262$	0	0
$263$	18.7853	1.15835	0.579176	−	0.815202i	$-0.303374\pi$
$263$	18.7853	1.15835	0.579176	+	0.815202i	$0.303374\pi$
$264$	0	0
$265$	−12.4788	−0.766567
$266$	0	0
$267$	−18.7457	−1.14722
$268$	0	0
$269$	−17.1967	−1.04850	−0.524251	−	0.851564i	$-0.675654\pi$
$269$	−17.1967	−1.04850	−0.524251	+	0.851564i	$0.675654\pi$
$270$	0	0
$271$	13.7691	0.836416	0.418208	−	0.908351i	$-0.362658\pi$
$271$	13.7691	0.836416	0.418208	+	0.908351i	$0.362658\pi$
$272$	0	0
$273$	−11.9261	−0.721803
$274$	0	0
$275$	−15.8446	−0.955466
$276$	0	0
$277$	−18.7759	−1.12813	−0.564066	−	0.825730i	$-0.690764\pi$
$277$	−18.7759	−1.12813	−0.564066	+	0.825730i	$0.690764\pi$
$278$	0	0
$279$	−0.248690	−0.0148887
$280$	0	0
$281$	9.00392	0.537129	0.268564	−	0.963262i	$-0.413451\pi$
$281$	9.00392	0.537129	0.268564	+	0.963262i	$0.413451\pi$
$282$	0	0
$283$	−14.0938	−0.837789	−0.418894	−	0.908035i	$-0.637582\pi$
$283$	−14.0938	−0.837789	−0.418894	+	0.908035i	$0.637582\pi$
$284$	0	0
$285$	−2.68071	−0.158792
$286$	0	0
$287$	6.01638	0.355135
$288$	0	0
$289$	−13.2833	−0.781372
$290$	0	0
$291$	11.1496	0.653603
$292$	0	0
$293$	−25.2452	−1.47484	−0.737421	−	0.675434i	$-0.763956\pi$
$293$	−25.2452	−1.47484	−0.737421	+	0.675434i	$0.763956\pi$
$294$	0	0
$295$	−7.92303	−0.461297
$296$	0	0
$297$	28.2061	1.63669
$298$	0	0
$299$	−2.54194	−0.147004
$300$	0	0
$301$	3.82732	0.220603
$302$	0	0
$303$	4.34445	0.249582
$304$	0	0
$305$	5.34085	0.305816
$306$	0	0
$307$	−16.6513	−0.950342	−0.475171	−	0.879893i	$-0.657614\pi$
$307$	−16.6513	−0.950342	−0.475171	+	0.879893i	$0.657614\pi$
$308$	0	0
$309$	22.8684	1.30094
$310$	0	0
$311$	3.26046	0.184884	0.0924418	−	0.995718i	$-0.470533\pi$
$311$	3.26046	0.184884	0.0924418	+	0.995718i	$0.470533\pi$
$312$	0	0
$313$	−5.48703	−0.310145	−0.155073	−	0.987903i	$-0.549561\pi$
$313$	−5.48703	−0.310145	−0.155073	+	0.987903i	$0.549561\pi$
$314$	0	0
$315$	0.668912	0.0376889
$316$	0	0
$317$	−22.3722	−1.25655	−0.628275	−	0.777992i	$-0.716239\pi$
$317$	−22.3722	−1.25655	−0.628275	+	0.777992i	$0.716239\pi$
$318$	0	0
$319$	−21.4507	−1.20101
$320$	0	0
$321$	−32.8473	−1.83336
$322$	0	0
$323$	−1.86222	−0.103617
$324$	0	0
$325$	14.6730	0.813913
$326$	0	0
$327$	5.30716	0.293487
$328$	0	0
$329$	−12.1981	−0.672501
$330$	0	0
$331$	33.2750	1.82896	0.914479	−	0.404634i	$-0.132601\pi$
$331$	33.2750	1.82896	0.914479	+	0.404634i	$0.132601\pi$
$332$	0	0
$333$	2.08688	0.114360
$334$	0	0
$335$	13.1604	0.719028
$336$	0	0
$337$	4.21753	0.229743	0.114872	−	0.993380i	$-0.463354\pi$
$337$	4.21753	0.229743	0.114872	+	0.993380i	$0.463354\pi$
$338$	0	0
$339$	−9.81835	−0.533260
$340$	0	0
$341$	−3.94513	−0.213641
$342$	0	0
$343$	15.0882	0.814685
$344$	0	0
$345$	1.30449	0.0702316
$346$	0	0
$347$	17.8458	0.958013	0.479007	−	0.877811i	$-0.340997\pi$
$347$	17.8458	0.958013	0.479007	+	0.877811i	$0.340997\pi$
$348$	0	0
$349$	−4.70118	−0.251648	−0.125824	−	0.992053i	$-0.540157\pi$
$349$	−4.70118	−0.251648	−0.125824	+	0.992053i	$0.540157\pi$
$350$	0	0
$351$	−26.1205	−1.39421
$352$	0	0
$353$	−25.5243	−1.35852	−0.679261	−	0.733896i	$-0.737700\pi$
$353$	−25.5243	−1.35852	−0.679261	+	0.733896i	$0.737700\pi$
$354$	0	0
$355$	−14.3787	−0.763145
$356$	0	0
$357$	4.25164	0.225021
$358$	0	0
$359$	−32.0017	−1.68898	−0.844492	−	0.535568i	$-0.820097\pi$
$359$	−32.0017	−1.68898	−0.844492	+	0.535568i	$0.820097\pi$
$360$	0	0
$361$	−18.0669	−0.950892
$362$	0	0
$363$	−42.3957	−2.22520
$364$	0	0
$365$	−14.6479	−0.766705
$366$	0	0
$367$	−14.7807	−0.771546	−0.385773	−	0.922594i	$-0.626065\pi$
$367$	−14.7807	−0.771546	−0.385773	+	0.922594i	$0.626065\pi$
$368$	0	0
$369$	−1.84301	−0.0959433
$370$	0	0
$371$	−9.91643	−0.514835
$372$	0	0
$373$	−30.5239	−1.58047	−0.790235	−	0.612804i	$-0.790042\pi$
$373$	−30.5239	−1.58047	−0.790235	+	0.612804i	$0.790042\pi$
$374$	0	0
$375$	−21.4061	−1.10541
$376$	0	0
$377$	19.8646	1.02308
$378$	0	0
$379$	13.3030	0.683329	0.341665	−	0.939822i	$-0.389009\pi$
$379$	13.3030	0.683329	0.341665	+	0.939822i	$0.389009\pi$
$380$	0	0
$381$	26.7956	1.37278
$382$	0	0
$383$	−20.6416	−1.05474	−0.527369	−	0.849636i	$-0.676821\pi$
$383$	−20.6416	−1.05474	−0.527369	+	0.849636i	$0.676821\pi$
$384$	0	0
$385$	10.6114	0.540807
$386$	0	0
$387$	−1.17243	−0.0595981
$388$	0	0
$389$	18.0385	0.914590	0.457295	−	0.889315i	$-0.348818\pi$
$389$	18.0385	0.914590	0.457295	+	0.889315i	$0.348818\pi$
$390$	0	0
$391$	0.906196	0.0458283
$392$	0	0
$393$	−36.4837	−1.84036
$394$	0	0
$395$	−7.91828	−0.398412
$396$	0	0
$397$	−24.2893	−1.21904	−0.609521	−	0.792770i	$-0.708638\pi$
$397$	−24.2893	−1.21904	−0.609521	+	0.792770i	$0.708638\pi$
$398$	0	0
$399$	−2.13026	−0.106646
$400$	0	0
$401$	11.4436	0.571464	0.285732	−	0.958310i	$-0.407763\pi$
$401$	11.4436	0.571464	0.285732	+	0.958310i	$0.407763\pi$
$402$	0	0
$403$	3.65342	0.181990
$404$	0	0
$405$	15.0747	0.749067
$406$	0	0
$407$	33.1056	1.64098
$408$	0	0
$409$	4.34200	0.214698	0.107349	−	0.994221i	$-0.465764\pi$
$409$	4.34200	0.214698	0.107349	+	0.994221i	$0.465764\pi$
$410$	0	0
$411$	−35.3910	−1.74571
$412$	0	0
$413$	−6.29613	−0.309812
$414$	0	0
$415$	22.6605	1.11236
$416$	0	0
$417$	20.7348	1.01539
$418$	0	0
$419$	35.0918	1.71434	0.857172	−	0.515030i	$-0.172219\pi$
$419$	35.0918	1.71434	0.857172	+	0.515030i	$0.172219\pi$
$420$	0	0
$421$	−36.2929	−1.76881	−0.884405	−	0.466720i	$-0.845436\pi$
$421$	−36.2929	−1.76881	−0.884405	+	0.466720i	$0.845436\pi$
$422$	0	0
$423$	3.73666	0.181683
$424$	0	0
$425$	−5.23089	−0.253736
$426$	0	0
$427$	4.24417	0.205390
$428$	0	0
$429$	57.9558	2.79813
$430$	0	0
$431$	−8.92616	−0.429958	−0.214979	−	0.976619i	$-0.568968\pi$
$431$	−8.92616	−0.429958	−0.214979	+	0.976619i	$0.568968\pi$
$432$	0	0
$433$	40.1476	1.92937	0.964684	−	0.263408i	$-0.0848466\pi$
$433$	40.1476	1.92937	0.964684	+	0.263408i	$0.0848466\pi$
$434$	0	0
$435$	−10.1942	−0.488777
$436$	0	0
$437$	−0.454045	−0.0217199
$438$	0	0
$439$	16.4516	0.785194	0.392597	−	0.919711i	$-0.371577\pi$
$439$	16.4516	0.785194	0.392597	+	0.919711i	$0.371577\pi$
$440$	0	0
$441$	−2.04522	−0.0973914
$442$	0	0
$443$	−14.1442	−0.672012	−0.336006	−	0.941860i	$-0.609076\pi$
$443$	−14.1442	−0.672012	−0.336006	+	0.941860i	$0.609076\pi$
$444$	0	0
$445$	−15.4459	−0.732205
$446$	0	0
$447$	−1.02747	−0.0485975
$448$	0	0
$449$	6.96741	0.328813	0.164406	−	0.986393i	$-0.447429\pi$
$449$	6.96741	0.328813	0.164406	+	0.986393i	$0.447429\pi$
$450$	0	0
$451$	−29.2369	−1.37671
$452$	0	0
$453$	−14.2671	−0.670325
$454$	0	0
$455$	−9.82677	−0.460686
$456$	0	0
$457$	−16.6490	−0.778809	−0.389404	−	0.921067i	$-0.627319\pi$
$457$	−16.6490	−0.778809	−0.389404	+	0.921067i	$0.627319\pi$
$458$	0	0
$459$	9.31189	0.434642
$460$	0	0
$461$	18.1015	0.843072	0.421536	−	0.906812i	$-0.361491\pi$
$461$	18.1015	0.843072	0.421536	+	0.906812i	$0.361491\pi$
$462$	0	0
$463$	−33.1519	−1.54070	−0.770350	−	0.637621i	$-0.779919\pi$
$463$	−33.1519	−1.54070	−0.770350	+	0.637621i	$0.779919\pi$
$464$	0	0
$465$	−1.87489	−0.0869460
$466$	0	0
$467$	−3.70462	−0.171430	−0.0857148	−	0.996320i	$-0.527317\pi$
$467$	−3.70462	−0.171430	−0.0857148	+	0.996320i	$0.527317\pi$
$468$	0	0
$469$	10.4580	0.482907
$470$	0	0
$471$	0.831527	0.0383148
$472$	0	0
$473$	−18.5991	−0.855187
$474$	0	0
$475$	2.62091	0.120256
$476$	0	0
$477$	3.03772	0.139088
$478$	0	0
$479$	−13.6667	−0.624449	−0.312225	−	0.950008i	$-0.601074\pi$
$479$	−13.6667	−0.624449	−0.312225	+	0.950008i	$0.601074\pi$
$480$	0	0
$481$	−30.6577	−1.39787
$482$	0	0
$483$	1.03663	0.0471683
$484$	0	0
$485$	9.18695	0.417158
$486$	0	0
$487$	−26.6571	−1.20795	−0.603974	−	0.797004i	$-0.706417\pi$
$487$	−26.6571	−1.20795	−0.603974	+	0.797004i	$0.706417\pi$
$488$	0	0
$489$	−13.0806	−0.591527
$490$	0	0
$491$	−8.27214	−0.373317	−0.186658	−	0.982425i	$-0.559766\pi$
$491$	−8.27214	−0.373317	−0.186658	+	0.982425i	$0.559766\pi$
$492$	0	0
$493$	−7.08166	−0.318942
$494$	0	0
$495$	−3.25062	−0.146104
$496$	0	0
$497$	−11.4262	−0.512537
$498$	0	0
$499$	32.1496	1.43921	0.719607	−	0.694382i	$-0.244322\pi$
$499$	32.1496	1.43921	0.719607	+	0.694382i	$0.244322\pi$
$500$	0	0
$501$	−19.3719	−0.865471
$502$	0	0
$503$	4.95942	0.221130	0.110565	−	0.993869i	$-0.464734\pi$
$503$	4.95942	0.221130	0.110565	+	0.993869i	$0.464734\pi$
$504$	0	0
$505$	3.57969	0.159294
$506$	0	0
$507$	−29.8123	−1.32401
$508$	0	0
$509$	13.2112	0.585577	0.292788	−	0.956177i	$-0.405417\pi$
$509$	13.2112	0.585577	0.292788	+	0.956177i	$0.405417\pi$
$510$	0	0
$511$	−11.6401	−0.514928
$512$	0	0
$513$	−4.66567	−0.205994
$514$	0	0
$515$	18.8429	0.830315
$516$	0	0
$517$	59.2772	2.60701
$518$	0	0
$519$	30.7555	1.35002
$520$	0	0
$521$	−7.65030	−0.335166	−0.167583	−	0.985858i	$-0.553596\pi$
$521$	−7.65030	−0.335166	−0.167583	+	0.985858i	$0.553596\pi$
$522$	0	0
$523$	−38.3821	−1.67833	−0.839165	−	0.543876i	$-0.816956\pi$
$523$	−38.3821	−1.67833	−0.839165	+	0.543876i	$0.816956\pi$
$524$	0	0
$525$	−5.98381	−0.261155
$526$	0	0
$527$	−1.30243	−0.0567349
$528$	0	0
$529$	−22.7791	−0.990394
$530$	0	0
$531$	1.92871	0.0836988
$532$	0	0
$533$	27.0751	1.17275
$534$	0	0
$535$	−27.0652	−1.17013
$536$	0	0
$537$	8.72090	0.376335
$538$	0	0
$539$	−32.4447	−1.39749
$540$	0	0
$541$	−15.6994	−0.674969	−0.337484	−	0.941331i	$-0.609576\pi$
$541$	−15.6994	−0.674969	−0.337484	+	0.941331i	$0.609576\pi$
$542$	0	0
$543$	16.5540	0.710401
$544$	0	0
$545$	4.37294	0.187316
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	−1.30013	−0.0554880
$550$	0	0
$551$	3.54823	0.151160
$552$	0	0
$553$	−6.29235	−0.267578
$554$	0	0
$555$	15.7331	0.667835
$556$	0	0
$557$	−25.9854	−1.10104	−0.550518	−	0.834823i	$-0.685570\pi$
$557$	−25.9854	−1.10104	−0.550518	+	0.834823i	$0.685570\pi$
$558$	0	0
$559$	17.2238	0.728491
$560$	0	0
$561$	−20.6611	−0.872312
$562$	0	0
$563$	−11.0040	−0.463762	−0.231881	−	0.972744i	$-0.574488\pi$
$563$	−11.0040	−0.463762	−0.231881	+	0.972744i	$0.574488\pi$
$564$	0	0
$565$	−8.09002	−0.340350
$566$	0	0
$567$	11.9793	0.503082
$568$	0	0
$569$	−2.32372	−0.0974152	−0.0487076	−	0.998813i	$-0.515510\pi$
$569$	−2.32372	−0.0974152	−0.0487076	+	0.998813i	$0.515510\pi$
$570$	0	0
$571$	6.08147	0.254502	0.127251	−	0.991871i	$-0.459385\pi$
$571$	6.08147	0.254502	0.127251	+	0.991871i	$0.459385\pi$
$572$	0	0
$573$	−45.1471	−1.88605
$574$	0	0
$575$	−1.27539	−0.0531875
$576$	0	0
$577$	−43.2965	−1.80246	−0.901228	−	0.433346i	$-0.857333\pi$
$577$	−43.2965	−1.80246	−0.901228	+	0.433346i	$0.857333\pi$
$578$	0	0
$579$	−28.5044	−1.18460
$580$	0	0
$581$	18.0075	0.747075
$582$	0	0
$583$	48.1895	1.99580
$584$	0	0
$585$	3.01026	0.124459
$586$	0	0
$587$	−34.3433	−1.41750	−0.708749	−	0.705460i	$-0.750740\pi$
$587$	−34.3433	−1.41750	−0.708749	+	0.705460i	$0.750740\pi$
$588$	0	0
$589$	0.652578	0.0268890
$590$	0	0
$591$	27.7437	1.14122
$592$	0	0
$593$	12.4576	0.511572	0.255786	−	0.966733i	$-0.417666\pi$
$593$	12.4576	0.511572	0.255786	+	0.966733i	$0.417666\pi$
$594$	0	0
$595$	3.50322	0.143618
$596$	0	0
$597$	25.0149	1.02379
$598$	0	0
$599$	−41.2358	−1.68485	−0.842424	−	0.538815i	$-0.818872\pi$
$599$	−41.2358	−1.68485	−0.842424	+	0.538815i	$0.818872\pi$
$600$	0	0
$601$	22.9872	0.937668	0.468834	−	0.883286i	$-0.344674\pi$
$601$	22.9872	0.937668	0.468834	+	0.883286i	$0.344674\pi$
$602$	0	0
$603$	−3.20364	−0.130462
$604$	0	0
$605$	−34.9327	−1.42022
$606$	0	0
$607$	44.6075	1.81056	0.905282	−	0.424812i	$-0.139660\pi$
$607$	44.6075	1.81056	0.905282	+	0.424812i	$0.139660\pi$
$608$	0	0
$609$	−8.10097	−0.328268
$610$	0	0
$611$	−54.8941	−2.22078
$612$	0	0
$613$	−21.3775	−0.863428	−0.431714	−	0.902010i	$-0.642091\pi$
$613$	−21.3775	−0.863428	−0.431714	+	0.902010i	$0.642091\pi$
$614$	0	0
$615$	−13.8946	−0.560285
$616$	0	0
$617$	12.0119	0.483583	0.241791	−	0.970328i	$-0.422265\pi$
$617$	12.0119	0.483583	0.241791	+	0.970328i	$0.422265\pi$
$618$	0	0
$619$	−19.5487	−0.785727	−0.392863	−	0.919597i	$-0.628516\pi$
$619$	−19.5487	−0.785727	−0.392863	+	0.919597i	$0.628516\pi$
$620$	0	0
$621$	2.27042	0.0911087
$622$	0	0
$623$	−12.2742	−0.491757
$624$	0	0
$625$	−4.07144	−0.162857
$626$	0	0
$627$	10.3521	0.413424
$628$	0	0
$629$	10.9294	0.435783
$630$	0	0
$631$	−40.4106	−1.60872	−0.804360	−	0.594142i	$-0.797492\pi$
$631$	−40.4106	−1.60872	−0.804360	+	0.594142i	$0.797492\pi$
$632$	0	0
$633$	20.7797	0.825917
$634$	0	0
$635$	22.0787	0.876166
$636$	0	0
$637$	30.0457	1.19045
$638$	0	0
$639$	3.50023	0.138467
$640$	0	0
$641$	22.6243	0.893607	0.446803	−	0.894632i	$-0.352562\pi$
$641$	22.6243	0.893607	0.446803	+	0.894632i	$0.352562\pi$
$642$	0	0
$643$	−40.2337	−1.58666	−0.793330	−	0.608792i	$-0.791655\pi$
$643$	−40.2337	−1.58666	−0.793330	+	0.608792i	$0.791655\pi$
$644$	0	0
$645$	−8.83906	−0.348038
$646$	0	0
$647$	26.0029	1.02228	0.511139	−	0.859498i	$-0.329224\pi$
$647$	26.0029	1.02228	0.511139	+	0.859498i	$0.329224\pi$
$648$	0	0
$649$	30.5964	1.20101
$650$	0	0
$651$	−1.48990	−0.0583939
$652$	0	0
$653$	−20.6624	−0.808584	−0.404292	−	0.914630i	$-0.632482\pi$
$653$	−20.6624	−0.808584	−0.404292	+	0.914630i	$0.632482\pi$
$654$	0	0
$655$	−30.0615	−1.17460
$656$	0	0
$657$	3.56574	0.139113
$658$	0	0
$659$	−5.80002	−0.225937	−0.112968	−	0.993599i	$-0.536036\pi$
$659$	−5.80002	−0.225937	−0.112968	+	0.993599i	$0.536036\pi$
$660$	0	0
$661$	2.81113	0.109340	0.0546702	−	0.998504i	$-0.482589\pi$
$661$	2.81113	0.109340	0.0546702	+	0.998504i	$0.482589\pi$
$662$	0	0
$663$	19.1334	0.743078
$664$	0	0
$665$	−1.75527	−0.0680664
$666$	0	0
$667$	−1.72665	−0.0668560
$668$	0	0
$669$	10.0919	0.390175
$670$	0	0
$671$	−20.6248	−0.796210
$672$	0	0
$673$	−4.40770	−0.169904	−0.0849521	−	0.996385i	$-0.527074\pi$
$673$	−4.40770	−0.169904	−0.0849521	+	0.996385i	$0.527074\pi$
$674$	0	0
$675$	−13.1057	−0.504438
$676$	0	0
$677$	−10.0686	−0.386969	−0.193484	−	0.981103i	$-0.561979\pi$
$677$	−10.0686	−0.386969	−0.193484	+	0.981103i	$0.561979\pi$
$678$	0	0
$679$	7.30051	0.280168
$680$	0	0
$681$	−3.84014	−0.147154
$682$	0	0
$683$	−43.3757	−1.65973	−0.829863	−	0.557968i	$-0.811581\pi$
$683$	−43.3757	−1.65973	−0.829863	+	0.557968i	$0.811581\pi$
$684$	0	0
$685$	−29.1611	−1.11419
$686$	0	0
$687$	30.1380	1.14984
$688$	0	0
$689$	−44.6262	−1.70012
$690$	0	0
$691$	−11.2784	−0.429052	−0.214526	−	0.976718i	$-0.568821\pi$
$691$	−11.2784	−0.429052	−0.214526	+	0.976718i	$0.568821\pi$
$692$	0	0
$693$	−2.58314	−0.0981253
$694$	0	0
$695$	17.0849	0.648066
$696$	0	0
$697$	−9.65220	−0.365603
$698$	0	0
$699$	18.3585	0.694382
$700$	0	0
$701$	24.4031	0.921691	0.460845	−	0.887480i	$-0.347546\pi$
$701$	24.4031	0.921691	0.460845	+	0.887480i	$0.347546\pi$
$702$	0	0
$703$	−5.47610	−0.206535
$704$	0	0
$705$	28.1710	1.06098
$706$	0	0
$707$	2.84464	0.106984
$708$	0	0
$709$	20.1044	0.755037	0.377518	−	0.926002i	$-0.376777\pi$
$709$	20.1044	0.755037	0.377518	+	0.926002i	$0.376777\pi$
$710$	0	0
$711$	1.92755	0.0722888
$712$	0	0
$713$	−0.317559	−0.0118927
$714$	0	0
$715$	47.7538	1.78589
$716$	0	0
$717$	23.6404	0.882869
$718$	0	0
$719$	26.8520	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$719$	26.8520	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$720$	0	0
$721$	14.9737	0.557649
$722$	0	0
$723$	−17.5888	−0.654136
$724$	0	0
$725$	9.96683	0.370159
$726$	0	0
$727$	−43.5422	−1.61489	−0.807446	−	0.589941i	$-0.799151\pi$
$727$	−43.5422	−1.61489	−0.807446	+	0.589941i	$0.799151\pi$
$728$	0	0
$729$	22.9238	0.849031
$730$	0	0
$731$	−6.14025	−0.227105
$732$	0	0
$733$	−29.4062	−1.08614	−0.543072	−	0.839686i	$-0.682739\pi$
$733$	−29.4062	−1.08614	−0.543072	+	0.839686i	$0.682739\pi$
$734$	0	0
$735$	−15.4191	−0.568741
$736$	0	0
$737$	−50.8215	−1.87203
$738$	0	0
$739$	10.0339	0.369103	0.184552	−	0.982823i	$-0.440917\pi$
$739$	10.0339	0.369103	0.184552	+	0.982823i	$0.440917\pi$
$740$	0	0
$741$	−9.58667	−0.352175
$742$	0	0
$743$	−32.7364	−1.20098	−0.600492	−	0.799631i	$-0.705028\pi$
$743$	−32.7364	−1.20098	−0.600492	+	0.799631i	$0.705028\pi$
$744$	0	0
$745$	−0.846601	−0.0310171
$746$	0	0
$747$	−5.51627	−0.201830
$748$	0	0
$749$	−21.5076	−0.785872
$750$	0	0
$751$	29.0656	1.06062	0.530309	−	0.847804i	$-0.322076\pi$
$751$	29.0656	1.06062	0.530309	+	0.847804i	$0.322076\pi$
$752$	0	0
$753$	−13.0142	−0.474263
$754$	0	0
$755$	−11.7556	−0.427831
$756$	0	0
$757$	−49.0178	−1.78158	−0.890791	−	0.454414i	$-0.849849\pi$
$757$	−49.0178	−1.78158	−0.890791	+	0.454414i	$0.849849\pi$
$758$	0	0
$759$	−5.03757	−0.182852
$760$	0	0
$761$	42.6524	1.54615	0.773073	−	0.634317i	$-0.218718\pi$
$761$	42.6524	1.54615	0.773073	+	0.634317i	$0.218718\pi$
$762$	0	0
$763$	3.47500	0.125804
$764$	0	0
$765$	−1.07315	−0.0387998
$766$	0	0
$767$	−28.3340	−1.02308
$768$	0	0
$769$	23.5755	0.850156	0.425078	−	0.905157i	$-0.360247\pi$
$769$	23.5755	0.850156	0.425078	+	0.905157i	$0.360247\pi$
$770$	0	0
$771$	14.0253	0.505109
$772$	0	0
$773$	−9.61811	−0.345939	−0.172970	−	0.984927i	$-0.555336\pi$
$773$	−9.61811	−0.345939	−0.172970	+	0.984927i	$0.555336\pi$
$774$	0	0
$775$	1.83306	0.0658456
$776$	0	0
$777$	12.5025	0.448525
$778$	0	0
$779$	4.83618	0.173274
$780$	0	0
$781$	55.5264	1.98689
$782$	0	0
$783$	−17.7427	−0.634071
$784$	0	0
$785$	0.685153	0.0244541
$786$	0	0
$787$	27.3656	0.975479	0.487740	−	0.872989i	$-0.337822\pi$
$787$	27.3656	0.975479	0.487740	+	0.872989i	$0.337822\pi$
$788$	0	0
$789$	−34.4756	−1.22736
$790$	0	0
$791$	−6.42882	−0.228583
$792$	0	0
$793$	19.0997	0.678251
$794$	0	0
$795$	22.9016	0.812237
$796$	0	0
$797$	−10.5153	−0.372471	−0.186236	−	0.982505i	$-0.559629\pi$
$797$	−10.5153	−0.372471	−0.186236	+	0.982505i	$0.559629\pi$
$798$	0	0
$799$	19.5696	0.692322
$800$	0	0
$801$	3.76000	0.132853
$802$	0	0
$803$	56.5657	1.99616
$804$	0	0
$805$	0.854152	0.0301049
$806$	0	0
$807$	31.5601	1.11097
$808$	0	0
$809$	9.73359	0.342215	0.171107	−	0.985252i	$-0.445266\pi$
$809$	9.73359	0.342215	0.171107	+	0.985252i	$0.445266\pi$
$810$	0	0
$811$	−11.8190	−0.415020	−0.207510	−	0.978233i	$-0.566536\pi$
$811$	−11.8190	−0.415020	−0.207510	+	0.978233i	$0.566536\pi$
$812$	0	0
$813$	−25.2697	−0.886247
$814$	0	0
$815$	−10.7780	−0.377538
$816$	0	0
$817$	3.07654	0.107634
$818$	0	0
$819$	2.39214	0.0835880
$820$	0	0
$821$	20.8740	0.728507	0.364253	−	0.931300i	$-0.381324\pi$
$821$	20.8740	0.728507	0.364253	+	0.931300i	$0.381324\pi$
$822$	0	0
$823$	−19.2428	−0.670762	−0.335381	−	0.942083i	$-0.608865\pi$
$823$	−19.2428	−0.670762	−0.335381	+	0.942083i	$0.608865\pi$
$824$	0	0
$825$	29.0787	1.01239
$826$	0	0
$827$	31.8820	1.10864	0.554322	−	0.832302i	$-0.312978\pi$
$827$	31.8820	1.10864	0.554322	+	0.832302i	$0.312978\pi$
$828$	0	0
$829$	−40.5150	−1.40714	−0.703572	−	0.710624i	$-0.748413\pi$
$829$	−40.5150	−1.40714	−0.703572	+	0.710624i	$0.748413\pi$
$830$	0	0
$831$	34.4582	1.19534
$832$	0	0
$833$	−10.7112	−0.371121
$834$	0	0
$835$	−15.9618	−0.552381
$836$	0	0
$837$	−3.26317	−0.112792
$838$	0	0
$839$	30.1260	1.04006	0.520032	−	0.854147i	$-0.325920\pi$
$839$	30.1260	1.04006	0.520032	+	0.854147i	$0.325920\pi$
$840$	0	0
$841$	−15.5068	−0.534716
$842$	0	0
$843$	−16.5244	−0.569129
$844$	0	0
$845$	−24.5644	−0.845041
$846$	0	0
$847$	−27.7597	−0.953834
$848$	0	0
$849$	25.8655	0.887702
$850$	0	0
$851$	2.66479	0.0913479
$852$	0	0
$853$	9.30329	0.318539	0.159269	−	0.987235i	$-0.449086\pi$
$853$	9.30329	0.318539	0.159269	+	0.987235i	$0.449086\pi$
$854$	0	0
$855$	0.537695	0.0183888
$856$	0	0
$857$	46.7828	1.59807	0.799035	−	0.601284i	$-0.205344\pi$
$857$	46.7828	1.59807	0.799035	+	0.601284i	$0.205344\pi$
$858$	0	0
$859$	11.8562	0.404528	0.202264	−	0.979331i	$-0.435170\pi$
$859$	11.8562	0.404528	0.202264	+	0.979331i	$0.435170\pi$
$860$	0	0
$861$	−11.0415	−0.376293
$862$	0	0
$863$	34.0665	1.15964	0.579819	−	0.814746i	$-0.303123\pi$
$863$	34.0665	1.15964	0.579819	+	0.814746i	$0.303123\pi$
$864$	0	0
$865$	25.3416	0.861640
$866$	0	0
$867$	24.3781	0.827924
$868$	0	0
$869$	30.5781	1.03729
$870$	0	0
$871$	47.0636	1.59469
$872$	0	0
$873$	−2.23638	−0.0756901
$874$	0	0
$875$	−14.0162	−0.473834
$876$	0	0
$877$	6.26125	0.211427	0.105714	−	0.994397i	$-0.466287\pi$
$877$	6.26125	0.211427	0.105714	+	0.994397i	$0.466287\pi$
$878$	0	0
$879$	46.3311	1.56271
$880$	0	0
$881$	3.16036	0.106475	0.0532377	−	0.998582i	$-0.483046\pi$
$881$	3.16036	0.106475	0.0532377	+	0.998582i	$0.483046\pi$
$882$	0	0
$883$	31.3346	1.05449	0.527247	−	0.849712i	$-0.323224\pi$
$883$	31.3346	1.05449	0.527247	+	0.849712i	$0.323224\pi$
$884$	0	0
$885$	14.5407	0.488779
$886$	0	0
$887$	12.6168	0.423632	0.211816	−	0.977310i	$-0.432062\pi$
$887$	12.6168	0.423632	0.211816	+	0.977310i	$0.432062\pi$
$888$	0	0
$889$	17.5451	0.588443
$890$	0	0
$891$	−58.2139	−1.95024
$892$	0	0
$893$	−9.80523	−0.328120
$894$	0	0
$895$	7.18575	0.240193
$896$	0	0
$897$	4.66508	0.155763
$898$	0	0
$899$	2.48163	0.0827669
$900$	0	0
$901$	15.9091	0.530010
$902$	0	0
$903$	−7.02406	−0.233746
$904$	0	0
$905$	13.6400	0.453409
$906$	0	0
$907$	34.0345	1.13010	0.565049	−	0.825057i	$-0.308857\pi$
$907$	34.0345	1.13010	0.565049	+	0.825057i	$0.308857\pi$
$908$	0	0
$909$	−0.871405	−0.0289027
$910$	0	0
$911$	41.1212	1.36241	0.681203	−	0.732095i	$-0.261457\pi$
$911$	41.1212	1.36241	0.681203	+	0.732095i	$0.261457\pi$
$912$	0	0
$913$	−87.5083	−2.89610
$914$	0	0
$915$	−9.80175	−0.324036
$916$	0	0
$917$	−23.8887	−0.788874
$918$	0	0
$919$	0.391473	0.0129135	0.00645675	−	0.999979i	$-0.497945\pi$
$919$	0.391473	0.0129135	0.00645675	+	0.999979i	$0.497945\pi$
$920$	0	0
$921$	30.5592	1.00696
$922$	0	0
$923$	−51.4207	−1.69253
$924$	0	0
$925$	−15.3822	−0.505762
$926$	0	0
$927$	−4.58693	−0.150654
$928$	0	0
$929$	20.6457	0.677365	0.338682	−	0.940901i	$-0.390019\pi$
$929$	20.6457	0.677365	0.338682	+	0.940901i	$0.390019\pi$
$930$	0	0
$931$	5.36679	0.175889
$932$	0	0
$933$	−5.98373	−0.195898
$934$	0	0
$935$	−17.0241	−0.556747
$936$	0	0
$937$	−44.5315	−1.45478	−0.727390	−	0.686225i	$-0.759267\pi$
$937$	−44.5315	−1.45478	−0.727390	+	0.686225i	$0.759267\pi$
$938$	0	0
$939$	10.0700	0.328623
$940$	0	0
$941$	−40.1815	−1.30988	−0.654940	−	0.755681i	$-0.727306\pi$
$941$	−40.1815	−1.30988	−0.654940	+	0.755681i	$0.727306\pi$
$942$	0	0
$943$	−2.35339	−0.0766369
$944$	0	0
$945$	8.77709	0.285519
$946$	0	0
$947$	54.0978	1.75794	0.878972	−	0.476874i	$-0.158230\pi$
$947$	54.0978	1.75794	0.878972	+	0.476874i	$0.158230\pi$
$948$	0	0
$949$	−52.3832	−1.70043
$950$	0	0
$951$	41.0584	1.33141
$952$	0	0
$953$	−18.8291	−0.609933	−0.304967	−	0.952363i	$-0.598645\pi$
$953$	−18.8291	−0.609933	−0.304967	+	0.952363i	$0.598645\pi$
$954$	0	0
$955$	−37.1998	−1.20376
$956$	0	0
$957$	39.3672	1.27256
$958$	0	0
$959$	−23.1732	−0.748302
$960$	0	0
$961$	−30.5436	−0.985277
$962$	0	0
$963$	6.58849	0.212311
$964$	0	0
$965$	−23.4867	−0.756065
$966$	0	0
$967$	−53.0693	−1.70659	−0.853296	−	0.521426i	$-0.825400\pi$
$967$	−53.0693	−1.70659	−0.853296	+	0.521426i	$0.825400\pi$
$968$	0	0
$969$	3.41762	0.109790
$970$	0	0
$971$	−40.4523	−1.29818	−0.649088	−	0.760713i	$-0.724849\pi$
$971$	−40.4523	−1.29818	−0.649088	+	0.760713i	$0.724849\pi$
$972$	0	0
$973$	13.5767	0.435248
$974$	0	0
$975$	−26.9286	−0.862404
$976$	0	0
$977$	−10.6610	−0.341075	−0.170537	−	0.985351i	$-0.554550\pi$
$977$	−10.6610	−0.341075	−0.170537	+	0.985351i	$0.554550\pi$
$978$	0	0
$979$	59.6474	1.90634
$980$	0	0
$981$	−1.06451	−0.0339871
$982$	0	0
$983$	−50.7688	−1.61927	−0.809637	−	0.586931i	$-0.800336\pi$
$983$	−50.7688	−1.61927	−0.809637	+	0.586931i	$0.800336\pi$
$984$	0	0
$985$	22.8599	0.728377
$986$	0	0
$987$	22.3864	0.712566
$988$	0	0
$989$	−1.49711	−0.0476054
$990$	0	0
$991$	31.6640	1.00584	0.502919	−	0.864333i	$-0.332259\pi$
$991$	31.6640	1.00584	0.502919	+	0.864333i	$0.332259\pi$
$992$	0	0
$993$	−61.0676	−1.93792
$994$	0	0
$995$	20.6115	0.653429
$996$	0	0
$997$	46.8252	1.48297	0.741485	−	0.670970i	$-0.234122\pi$
$997$	46.8252	1.48297	0.741485	+	0.670970i	$0.234122\pi$
$998$	0	0
$999$	27.3829	0.866356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.4		18
4.3	odd	2		547.2.a.b.1.9	✓	18
12.11	even	2		4923.2.a.l.1.10		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.9	✓	18	4.3	odd	2
4923.2.a.l.1.10		18	12.11	even	2
8752.2.a.s.1.4		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)

\(f(q)\)	\(=\)	\(q-1.83524 q^{3} -1.51218 q^{5} -1.20167 q^{7} +0.368111 q^{9} +O(q^{10})\)	\(q-1.83524 q^{3} -1.51218 q^{5} -1.20167 q^{7} +0.368111 q^{9} +5.83960 q^{11} -5.40780 q^{13} +2.77522 q^{15} +1.92787 q^{17} -0.965947 q^{19} +2.20536 q^{21} +0.470051 q^{23} -2.71331 q^{25} +4.83015 q^{27} -3.67331 q^{29} -0.675583 q^{31} -10.7171 q^{33} +1.81715 q^{35} +5.66915 q^{37} +9.92463 q^{39} -5.00667 q^{41} -3.18500 q^{43} -0.556651 q^{45} +10.1509 q^{47} -5.55598 q^{49} -3.53810 q^{51} +8.25219 q^{53} -8.83053 q^{55} +1.77275 q^{57} +5.23947 q^{59} -3.53188 q^{61} -0.442349 q^{63} +8.17758 q^{65} -8.70291 q^{67} -0.862657 q^{69} +9.50861 q^{71} +9.68659 q^{73} +4.97957 q^{75} -7.01728 q^{77} +5.23633 q^{79} -9.96883 q^{81} -14.9853 q^{83} -2.91529 q^{85} +6.74142 q^{87} +10.2143 q^{89} +6.49841 q^{91} +1.23986 q^{93} +1.46069 q^{95} -6.07529 q^{97} +2.14962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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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