LMFDB - Embedded newform 8752.2.a.s.1.17

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			1.17
Root			$0.523506$ 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)

$f(q)$	$=$	$q+3.08736 q^{3} -1.35183 q^{5} -3.60927 q^{7} +6.53179 q^{9} +O(q^{10})$	$q+3.08736 q^{3} -1.35183 q^{5} -3.60927 q^{7} +6.53179 q^{9} +0.337880 q^{11} -0.968153 q^{13} -4.17358 q^{15} +1.24740 q^{17} +5.50049 q^{19} -11.1431 q^{21} -3.13569 q^{23} -3.17256 q^{25} +10.9039 q^{27} -7.18364 q^{29} -10.2089 q^{31} +1.04316 q^{33} +4.87912 q^{35} +5.45502 q^{37} -2.98904 q^{39} -7.24577 q^{41} +6.48922 q^{43} -8.82987 q^{45} +7.68104 q^{47} +6.02683 q^{49} +3.85117 q^{51} -4.50651 q^{53} -0.456756 q^{55} +16.9820 q^{57} -2.41312 q^{59} +6.37922 q^{61} -23.5750 q^{63} +1.30878 q^{65} -0.799497 q^{67} -9.68099 q^{69} -2.93852 q^{71} -15.3091 q^{73} -9.79483 q^{75} -1.21950 q^{77} -10.3913 q^{79} +14.0690 q^{81} +13.3213 q^{83} -1.68627 q^{85} -22.1785 q^{87} -4.42608 q^{89} +3.49433 q^{91} -31.5185 q^{93} -7.43573 q^{95} -6.98217 q^{97} +2.20696 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.08736	1.78249	0.891244	−	0.453524i	$-0.149833\pi$
$3$	3.08736	1.78249	0.891244	+	0.453524i	$0.149833\pi$
$4$	0	0
$5$	−1.35183	−0.604556	−0.302278	−	0.953220i	$-0.597747\pi$
$5$	−1.35183	−0.604556	−0.302278	+	0.953220i	$0.597747\pi$
$6$	0	0
$7$	−3.60927	−1.36418	−0.682088	−	0.731270i	$-0.738928\pi$
$7$	−3.60927	−1.36418	−0.682088	+	0.731270i	$0.738928\pi$
$8$	0	0
$9$	6.53179	2.17726
$10$	0	0
$11$	0.337880	0.101875	0.0509374	−	0.998702i	$-0.483779\pi$
$11$	0.337880	0.101875	0.0509374	+	0.998702i	$0.483779\pi$
$12$	0	0
$13$	−0.968153	−0.268517	−0.134259	−	0.990946i	$-0.542865\pi$
$13$	−0.968153	−0.268517	−0.134259	+	0.990946i	$0.542865\pi$
$14$	0	0
$15$	−4.17358	−1.07761
$16$	0	0
$17$	1.24740	0.302539	0.151270	−	0.988493i	$-0.451664\pi$
$17$	1.24740	0.302539	0.151270	+	0.988493i	$0.451664\pi$
$18$	0	0
$19$	5.50049	1.26190	0.630950	−	0.775824i	$-0.282665\pi$
$19$	5.50049	1.26190	0.630950	+	0.775824i	$0.282665\pi$
$20$	0	0
$21$	−11.1431	−2.43163
$22$	0	0
$23$	−3.13569	−0.653836	−0.326918	−	0.945053i	$-0.606010\pi$
$23$	−3.13569	−0.653836	−0.326918	+	0.945053i	$0.606010\pi$
$24$	0	0
$25$	−3.17256	−0.634511
$26$	0	0
$27$	10.9039	2.09846
$28$	0	0
$29$	−7.18364	−1.33397	−0.666984	−	0.745072i	$-0.732415\pi$
$29$	−7.18364	−1.33397	−0.666984	+	0.745072i	$0.732415\pi$
$30$	0	0
$31$	−10.2089	−1.83357	−0.916784	−	0.399384i	$-0.869224\pi$
$31$	−10.2089	−1.83357	−0.916784	+	0.399384i	$0.869224\pi$
$32$	0	0
$33$	1.04316	0.181591
$34$	0	0
$35$	4.87912	0.824721
$36$	0	0
$37$	5.45502	0.896799	0.448400	−	0.893833i	$-0.351994\pi$
$37$	5.45502	0.896799	0.448400	+	0.893833i	$0.351994\pi$
$38$	0	0
$39$	−2.98904	−0.478629
$40$	0	0
$41$	−7.24577	−1.13160	−0.565800	−	0.824543i	$-0.691432\pi$
$41$	−7.24577	−1.13160	−0.565800	+	0.824543i	$0.691432\pi$
$42$	0	0
$43$	6.48922	0.989597	0.494798	−	0.869008i	$-0.335242\pi$
$43$	6.48922	0.989597	0.494798	+	0.869008i	$0.335242\pi$
$44$	0	0
$45$	−8.82987	−1.31628
$46$	0	0
$47$	7.68104	1.12039	0.560197	−	0.828359i	$-0.310725\pi$
$47$	7.68104	1.12039	0.560197	+	0.828359i	$0.310725\pi$
$48$	0	0
$49$	6.02683	0.860976
$50$	0	0
$51$	3.85117	0.539272
$52$	0	0
$53$	−4.50651	−0.619017	−0.309509	−	0.950897i	$-0.600165\pi$
$53$	−4.50651	−0.619017	−0.309509	+	0.950897i	$0.600165\pi$
$54$	0	0
$55$	−0.456756	−0.0615890
$56$	0	0
$57$	16.9820	2.24932
$58$	0	0
$59$	−2.41312	−0.314161	−0.157081	−	0.987586i	$-0.550208\pi$
$59$	−2.41312	−0.314161	−0.157081	+	0.987586i	$0.550208\pi$
$60$	0	0
$61$	6.37922	0.816775	0.408387	−	0.912809i	$-0.366091\pi$
$61$	6.37922	0.816775	0.408387	+	0.912809i	$0.366091\pi$
$62$	0	0
$63$	−23.5750	−2.97017
$64$	0	0
$65$	1.30878	0.162334
$66$	0	0
$67$	−0.799497	−0.0976741	−0.0488370	−	0.998807i	$-0.515551\pi$
$67$	−0.799497	−0.0976741	−0.0488370	+	0.998807i	$0.515551\pi$
$68$	0	0
$69$	−9.68099	−1.16545
$70$	0	0
$71$	−2.93852	−0.348738	−0.174369	−	0.984680i	$-0.555789\pi$
$71$	−2.93852	−0.348738	−0.174369	+	0.984680i	$0.555789\pi$
$72$	0	0
$73$	−15.3091	−1.79180	−0.895899	−	0.444257i	$-0.853468\pi$
$73$	−15.3091	−1.79180	−0.895899	+	0.444257i	$0.853468\pi$
$74$	0	0
$75$	−9.79483	−1.13101
$76$	0	0
$77$	−1.21950	−0.138975
$78$	0	0
$79$	−10.3913	−1.16911	−0.584557	−	0.811353i	$-0.698732\pi$
$79$	−10.3913	−1.16911	−0.584557	+	0.811353i	$0.698732\pi$
$80$	0	0
$81$	14.0690	1.56322
$82$	0	0
$83$	13.3213	1.46220	0.731102	−	0.682269i	$-0.239007\pi$
$83$	13.3213	1.46220	0.731102	+	0.682269i	$0.239007\pi$
$84$	0	0
$85$	−1.68627	−0.182902
$86$	0	0
$87$	−22.1785	−2.37778
$88$	0	0
$89$	−4.42608	−0.469163	−0.234582	−	0.972096i	$-0.575372\pi$
$89$	−4.42608	−0.469163	−0.234582	+	0.972096i	$0.575372\pi$
$90$	0	0
$91$	3.49433	0.366305
$92$	0	0
$93$	−31.5185	−3.26831
$94$	0	0
$95$	−7.43573	−0.762890
$96$	0	0
$97$	−6.98217	−0.708932	−0.354466	−	0.935069i	$-0.615337\pi$
$97$	−6.98217	−0.708932	−0.354466	+	0.935069i	$0.615337\pi$
$98$	0	0
$99$	2.20696	0.221808
$100$	0	0
$101$	−3.92586	−0.390638	−0.195319	−	0.980740i	$-0.562574\pi$
$101$	−3.92586	−0.390638	−0.195319	+	0.980740i	$0.562574\pi$
$102$	0	0
$103$	14.0937	1.38870	0.694348	−	0.719639i	$-0.255693\pi$
$103$	14.0937	1.38870	0.694348	+	0.719639i	$0.255693\pi$
$104$	0	0
$105$	15.0636	1.47006
$106$	0	0
$107$	−7.65150	−0.739699	−0.369849	−	0.929092i	$-0.620591\pi$
$107$	−7.65150	−0.739699	−0.369849	+	0.929092i	$0.620591\pi$
$108$	0	0
$109$	−4.57688	−0.438386	−0.219193	−	0.975682i	$-0.570342\pi$
$109$	−4.57688	−0.438386	−0.219193	+	0.975682i	$0.570342\pi$
$110$	0	0
$111$	16.8416	1.59853
$112$	0	0
$113$	−15.9981	−1.50497	−0.752487	−	0.658607i	$-0.771146\pi$
$113$	−15.9981	−1.50497	−0.752487	+	0.658607i	$0.771146\pi$
$114$	0	0
$115$	4.23891	0.395281
$116$	0	0
$117$	−6.32378	−0.584633
$118$	0	0
$119$	−4.50220	−0.412716
$120$	0	0
$121$	−10.8858	−0.989622
$122$	0	0
$123$	−22.3703	−2.01706
$124$	0	0
$125$	11.0479	0.988154
$126$	0	0
$127$	−7.02400	−0.623279	−0.311640	−	0.950200i	$-0.600878\pi$
$127$	−7.02400	−0.623279	−0.311640	+	0.950200i	$0.600878\pi$
$128$	0	0
$129$	20.0346	1.76394
$130$	0	0
$131$	3.36814	0.294276	0.147138	−	0.989116i	$-0.452994\pi$
$131$	3.36814	0.294276	0.147138	+	0.989116i	$0.452994\pi$
$132$	0	0
$133$	−19.8528	−1.72145
$134$	0	0
$135$	−14.7402	−1.26864
$136$	0	0
$137$	−21.8903	−1.87022	−0.935108	−	0.354362i	$-0.884698\pi$
$137$	−21.8903	−1.87022	−0.935108	+	0.354362i	$0.884698\pi$
$138$	0	0
$139$	7.81159	0.662571	0.331285	−	0.943531i	$-0.392518\pi$
$139$	7.81159	0.662571	0.331285	+	0.943531i	$0.392518\pi$
$140$	0	0
$141$	23.7141	1.99709
$142$	0	0
$143$	−0.327120	−0.0273551
$144$	0	0
$145$	9.71105	0.806459
$146$	0	0
$147$	18.6070	1.53468
$148$	0	0
$149$	−1.73671	−0.142277	−0.0711385	−	0.997466i	$-0.522663\pi$
$149$	−1.73671	−0.142277	−0.0711385	+	0.997466i	$0.522663\pi$
$150$	0	0
$151$	−9.10500	−0.740955	−0.370477	−	0.928841i	$-0.620806\pi$
$151$	−9.10500	−0.740955	−0.370477	+	0.928841i	$0.620806\pi$
$152$	0	0
$153$	8.14776	0.658708
$154$	0	0
$155$	13.8007	1.10850
$156$	0	0
$157$	10.4340	0.832727	0.416364	−	0.909198i	$-0.363304\pi$
$157$	10.4340	0.832727	0.416364	+	0.909198i	$0.363304\pi$
$158$	0	0
$159$	−13.9132	−1.10339
$160$	0	0
$161$	11.3175	0.891947
$162$	0	0
$163$	−23.6947	−1.85591	−0.927954	−	0.372694i	$-0.878434\pi$
$163$	−23.6947	−1.85591	−0.927954	+	0.372694i	$0.878434\pi$
$164$	0	0
$165$	−1.41017	−0.109782
$166$	0	0
$167$	14.5777	1.12806	0.564029	−	0.825755i	$-0.309251\pi$
$167$	14.5777	1.12806	0.564029	+	0.825755i	$0.309251\pi$
$168$	0	0
$169$	−12.0627	−0.927898
$170$	0	0
$171$	35.9281	2.74749
$172$	0	0
$173$	−16.5647	−1.25939	−0.629695	−	0.776843i	$-0.716820\pi$
$173$	−16.5647	−1.25939	−0.629695	+	0.776843i	$0.716820\pi$
$174$	0	0
$175$	11.4506	0.865585
$176$	0	0
$177$	−7.45017	−0.559989
$178$	0	0
$179$	−14.0810	−1.05246	−0.526230	−	0.850342i	$-0.676395\pi$
$179$	−14.0810	−1.05246	−0.526230	+	0.850342i	$0.676395\pi$
$180$	0	0
$181$	−5.40061	−0.401424	−0.200712	−	0.979650i	$-0.564326\pi$
$181$	−5.40061	−0.401424	−0.200712	+	0.979650i	$0.564326\pi$
$182$	0	0
$183$	19.6949	1.45589
$184$	0	0
$185$	−7.37425	−0.542166
$186$	0	0
$187$	0.421472	0.0308211
$188$	0	0
$189$	−39.3552	−2.86267
$190$	0	0
$191$	−7.86102	−0.568803	−0.284402	−	0.958705i	$-0.591795\pi$
$191$	−7.86102	−0.568803	−0.284402	+	0.958705i	$0.591795\pi$
$192$	0	0
$193$	23.5348	1.69407	0.847036	−	0.531536i	$-0.178385\pi$
$193$	23.5348	1.69407	0.847036	+	0.531536i	$0.178385\pi$
$194$	0	0
$195$	4.04067	0.289358
$196$	0	0
$197$	−1.91517	−0.136450	−0.0682251	−	0.997670i	$-0.521734\pi$
$197$	−1.91517	−0.136450	−0.0682251	+	0.997670i	$0.521734\pi$
$198$	0	0
$199$	−2.24700	−0.159286	−0.0796430	−	0.996823i	$-0.525378\pi$
$199$	−2.24700	−0.159286	−0.0796430	+	0.996823i	$0.525378\pi$
$200$	0	0
$201$	−2.46833	−0.174103
$202$	0	0
$203$	25.9277	1.81977
$204$	0	0
$205$	9.79505	0.684116
$206$	0	0
$207$	−20.4817	−1.42357
$208$	0	0
$209$	1.85851	0.128556
$210$	0	0
$211$	−14.5078	−0.998758	−0.499379	−	0.866384i	$-0.666439\pi$
$211$	−14.5078	−0.998758	−0.499379	+	0.866384i	$0.666439\pi$
$212$	0	0
$213$	−9.07226	−0.621621
$214$	0	0
$215$	−8.77232	−0.598267
$216$	0	0
$217$	36.8466	2.50131
$218$	0	0
$219$	−47.2648	−3.19386
$220$	0	0
$221$	−1.20767	−0.0812370
$222$	0	0
$223$	25.0935	1.68038	0.840191	−	0.542290i	$-0.182443\pi$
$223$	25.0935	1.68038	0.840191	+	0.542290i	$0.182443\pi$
$224$	0	0
$225$	−20.7225	−1.38150
$226$	0	0
$227$	20.1566	1.33784	0.668922	−	0.743333i	$-0.266756\pi$
$227$	20.1566	1.33784	0.668922	+	0.743333i	$0.266756\pi$
$228$	0	0
$229$	24.6507	1.62897	0.814483	−	0.580188i	$-0.197021\pi$
$229$	24.6507	1.62897	0.814483	+	0.580188i	$0.197021\pi$
$230$	0	0
$231$	−3.76504	−0.247721
$232$	0	0
$233$	−22.2305	−1.45637	−0.728185	−	0.685380i	$-0.759636\pi$
$233$	−22.2305	−1.45637	−0.728185	+	0.685380i	$0.759636\pi$
$234$	0	0
$235$	−10.3834	−0.677342
$236$	0	0
$237$	−32.0817	−2.08393
$238$	0	0
$239$	5.88759	0.380837	0.190418	−	0.981703i	$-0.439016\pi$
$239$	5.88759	0.380837	0.190418	+	0.981703i	$0.439016\pi$
$240$	0	0
$241$	7.97674	0.513827	0.256913	−	0.966434i	$-0.417294\pi$
$241$	7.97674	0.513827	0.256913	+	0.966434i	$0.417294\pi$
$242$	0	0
$243$	10.7242	0.687955
$244$	0	0
$245$	−8.14725	−0.520508
$246$	0	0
$247$	−5.32532	−0.338842
$248$	0	0
$249$	41.1277	2.60636
$250$	0	0
$251$	20.2547	1.27847	0.639233	−	0.769013i	$-0.279252\pi$
$251$	20.2547	1.27847	0.639233	+	0.769013i	$0.279252\pi$
$252$	0	0
$253$	−1.05949	−0.0666093
$254$	0	0
$255$	−5.20613	−0.326021
$256$	0	0
$257$	−15.9624	−0.995705	−0.497853	−	0.867262i	$-0.665878\pi$
$257$	−15.9624	−0.995705	−0.497853	+	0.867262i	$0.665878\pi$
$258$	0	0
$259$	−19.6886	−1.22339
$260$	0	0
$261$	−46.9220	−2.90440
$262$	0	0
$263$	−26.2647	−1.61955	−0.809774	−	0.586741i	$-0.800411\pi$
$263$	−26.2647	−1.61955	−0.809774	+	0.586741i	$0.800411\pi$
$264$	0	0
$265$	6.09204	0.374231
$266$	0	0
$267$	−13.6649	−0.836278
$268$	0	0
$269$	7.56302	0.461125	0.230563	−	0.973057i	$-0.425943\pi$
$269$	7.56302	0.461125	0.230563	+	0.973057i	$0.425943\pi$
$270$	0	0
$271$	−0.855122	−0.0519450	−0.0259725	−	0.999663i	$-0.508268\pi$
$271$	−0.855122	−0.0519450	−0.0259725	+	0.999663i	$0.508268\pi$
$272$	0	0
$273$	10.7882	0.652934
$274$	0	0
$275$	−1.07194	−0.0646407
$276$	0	0
$277$	−7.71774	−0.463714	−0.231857	−	0.972750i	$-0.574480\pi$
$277$	−7.71774	−0.463714	−0.231857	+	0.972750i	$0.574480\pi$
$278$	0	0
$279$	−66.6823	−3.99216
$280$	0	0
$281$	5.04416	0.300910	0.150455	−	0.988617i	$-0.451926\pi$
$281$	5.04416	0.300910	0.150455	+	0.988617i	$0.451926\pi$
$282$	0	0
$283$	−1.44488	−0.0858894	−0.0429447	−	0.999077i	$-0.513674\pi$
$283$	−1.44488	−0.0858894	−0.0429447	+	0.999077i	$0.513674\pi$
$284$	0	0
$285$	−22.9568	−1.35984
$286$	0	0
$287$	26.1519	1.54370
$288$	0	0
$289$	−15.4440	−0.908470
$290$	0	0
$291$	−21.5565	−1.26366
$292$	0	0
$293$	11.9975	0.700902	0.350451	−	0.936581i	$-0.386028\pi$
$293$	11.9975	0.700902	0.350451	+	0.936581i	$0.386028\pi$
$294$	0	0
$295$	3.26213	0.189928
$296$	0	0
$297$	3.68422	0.213780
$298$	0	0
$299$	3.03582	0.175566
$300$	0	0
$301$	−23.4213	−1.34998
$302$	0	0
$303$	−12.1205	−0.696307
$304$	0	0
$305$	−8.62361	−0.493787
$306$	0	0
$307$	−14.7380	−0.841140	−0.420570	−	0.907260i	$-0.638170\pi$
$307$	−14.7380	−0.841140	−0.420570	+	0.907260i	$0.638170\pi$
$308$	0	0
$309$	43.5124	2.47534
$310$	0	0
$311$	25.6282	1.45324	0.726621	−	0.687038i	$-0.241090\pi$
$311$	25.6282	1.45324	0.726621	+	0.687038i	$0.241090\pi$
$312$	0	0
$313$	−3.29616	−0.186310	−0.0931551	−	0.995652i	$-0.529695\pi$
$313$	−3.29616	−0.186310	−0.0931551	+	0.995652i	$0.529695\pi$
$314$	0	0
$315$	31.8694	1.79564
$316$	0	0
$317$	21.5851	1.21234	0.606170	−	0.795335i	$-0.292705\pi$
$317$	21.5851	1.21234	0.606170	+	0.795335i	$0.292705\pi$
$318$	0	0
$319$	−2.42721	−0.135898
$320$	0	0
$321$	−23.6230	−1.31850
$322$	0	0
$323$	6.86132	0.381774
$324$	0	0
$325$	3.07152	0.170377
$326$	0	0
$327$	−14.1305	−0.781417
$328$	0	0
$329$	−27.7229	−1.52841
$330$	0	0
$331$	19.2871	1.06011	0.530057	−	0.847962i	$-0.322171\pi$
$331$	19.2871	1.06011	0.530057	+	0.847962i	$0.322171\pi$
$332$	0	0
$333$	35.6310	1.95257
$334$	0	0
$335$	1.08078	0.0590495
$336$	0	0
$337$	0.183297	0.00998481	0.00499241	−	0.999988i	$-0.498411\pi$
$337$	0.183297	0.00998481	0.00499241	+	0.999988i	$0.498411\pi$
$338$	0	0
$339$	−49.3919	−2.68260
$340$	0	0
$341$	−3.44938	−0.186794
$342$	0	0
$343$	3.51243	0.189654
$344$	0	0
$345$	13.0871	0.704583
$346$	0	0
$347$	24.5220	1.31641	0.658206	−	0.752838i	$-0.271316\pi$
$347$	24.5220	1.31641	0.658206	+	0.752838i	$0.271316\pi$
$348$	0	0
$349$	−8.19072	−0.438439	−0.219220	−	0.975676i	$-0.570351\pi$
$349$	−8.19072	−0.438439	−0.219220	+	0.975676i	$0.570351\pi$
$350$	0	0
$351$	−10.5567	−0.563473
$352$	0	0
$353$	17.5573	0.934483	0.467242	−	0.884130i	$-0.345248\pi$
$353$	17.5573	0.934483	0.467242	+	0.884130i	$0.345248\pi$
$354$	0	0
$355$	3.97237	0.210832
$356$	0	0
$357$	−13.8999	−0.735662
$358$	0	0
$359$	−20.7724	−1.09632	−0.548162	−	0.836372i	$-0.684672\pi$
$359$	−20.7724	−1.09632	−0.548162	+	0.836372i	$0.684672\pi$
$360$	0	0
$361$	11.2554	0.592391
$362$	0	0
$363$	−33.6085	−1.76399
$364$	0	0
$365$	20.6953	1.08324
$366$	0	0
$367$	34.8797	1.82070	0.910352	−	0.413835i	$-0.135811\pi$
$367$	34.8797	1.82070	0.910352	+	0.413835i	$0.135811\pi$
$368$	0	0
$369$	−47.3279	−2.46379
$370$	0	0
$371$	16.2652	0.844448
$372$	0	0
$373$	−9.73592	−0.504107	−0.252054	−	0.967713i	$-0.581106\pi$
$373$	−9.73592	−0.504107	−0.252054	+	0.967713i	$0.581106\pi$
$374$	0	0
$375$	34.1089	1.76137
$376$	0	0
$377$	6.95486	0.358193
$378$	0	0
$379$	−17.7671	−0.912634	−0.456317	−	0.889817i	$-0.650832\pi$
$379$	−17.7671	−0.912634	−0.456317	+	0.889817i	$0.650832\pi$
$380$	0	0
$381$	−21.6856	−1.11099
$382$	0	0
$383$	−12.7845	−0.653257	−0.326628	−	0.945153i	$-0.605913\pi$
$383$	−12.7845	−0.653257	−0.326628	+	0.945153i	$0.605913\pi$
$384$	0	0
$385$	1.64856	0.0840183
$386$	0	0
$387$	42.3862	2.15461
$388$	0	0
$389$	−10.8081	−0.547992	−0.273996	−	0.961731i	$-0.588345\pi$
$389$	−10.8081	−0.547992	−0.273996	+	0.961731i	$0.588345\pi$
$390$	0	0
$391$	−3.91146	−0.197811
$392$	0	0
$393$	10.3987	0.524543
$394$	0	0
$395$	14.0473	0.706796
$396$	0	0
$397$	−2.02404	−0.101584	−0.0507919	−	0.998709i	$-0.516175\pi$
$397$	−2.02404	−0.101584	−0.0507919	+	0.998709i	$0.516175\pi$
$398$	0	0
$399$	−61.2926	−3.06847
$400$	0	0
$401$	−7.98951	−0.398977	−0.199488	−	0.979900i	$-0.563928\pi$
$401$	−7.98951	−0.398977	−0.199488	+	0.979900i	$0.563928\pi$
$402$	0	0
$403$	9.88375	0.492345
$404$	0	0
$405$	−19.0188	−0.945053
$406$	0	0
$407$	1.84314	0.0913612
$408$	0	0
$409$	−17.7234	−0.876366	−0.438183	−	0.898886i	$-0.644378\pi$
$409$	−17.7234	−0.876366	−0.438183	+	0.898886i	$0.644378\pi$
$410$	0	0
$411$	−67.5833	−3.33364
$412$	0	0
$413$	8.70960	0.428571
$414$	0	0
$415$	−18.0081	−0.883984
$416$	0	0
$417$	24.1172	1.18102
$418$	0	0
$419$	−23.2376	−1.13523	−0.567615	−	0.823294i	$-0.692134\pi$
$419$	−23.2376	−1.13523	−0.567615	+	0.823294i	$0.692134\pi$
$420$	0	0
$421$	16.0445	0.781961	0.390980	−	0.920399i	$-0.372136\pi$
$421$	16.0445	0.781961	0.390980	+	0.920399i	$0.372136\pi$
$422$	0	0
$423$	50.1709	2.43939
$424$	0	0
$425$	−3.95745	−0.191964
$426$	0	0
$427$	−23.0243	−1.11422
$428$	0	0
$429$	−1.00994	−0.0487602
$430$	0	0
$431$	−16.8476	−0.811519	−0.405759	−	0.913980i	$-0.632993\pi$
$431$	−16.8476	−0.811519	−0.405759	+	0.913980i	$0.632993\pi$
$432$	0	0
$433$	13.9110	0.668518	0.334259	−	0.942481i	$-0.391514\pi$
$433$	13.9110	0.668518	0.334259	+	0.942481i	$0.391514\pi$
$434$	0	0
$435$	29.9815	1.43750
$436$	0	0
$437$	−17.2478	−0.825075
$438$	0	0
$439$	−36.1726	−1.72643	−0.863213	−	0.504840i	$-0.831552\pi$
$439$	−36.1726	−1.72643	−0.863213	+	0.504840i	$0.831552\pi$
$440$	0	0
$441$	39.3660	1.87457
$442$	0	0
$443$	33.8300	1.60731	0.803655	−	0.595096i	$-0.202886\pi$
$443$	33.8300	1.60731	0.803655	+	0.595096i	$0.202886\pi$
$444$	0	0
$445$	5.98330	0.283636
$446$	0	0
$447$	−5.36186	−0.253607
$448$	0	0
$449$	−35.4973	−1.67522	−0.837611	−	0.546267i	$-0.816049\pi$
$449$	−35.4973	−1.67522	−0.837611	+	0.546267i	$0.816049\pi$
$450$	0	0
$451$	−2.44820	−0.115281
$452$	0	0
$453$	−28.1104	−1.32074
$454$	0	0
$455$	−4.72373	−0.221452
$456$	0	0
$457$	−27.6267	−1.29232	−0.646162	−	0.763201i	$-0.723627\pi$
$457$	−27.6267	−1.29232	−0.646162	+	0.763201i	$0.723627\pi$
$458$	0	0
$459$	13.6016	0.634866
$460$	0	0
$461$	−1.39808	−0.0651150	−0.0325575	−	0.999470i	$-0.510365\pi$
$461$	−1.39808	−0.0651150	−0.0325575	+	0.999470i	$0.510365\pi$
$462$	0	0
$463$	16.4657	0.765225	0.382613	−	0.923909i	$-0.375024\pi$
$463$	16.4657	0.765225	0.382613	+	0.923909i	$0.375024\pi$
$464$	0	0
$465$	42.6076	1.97588
$466$	0	0
$467$	−19.2949	−0.892863	−0.446432	−	0.894818i	$-0.647305\pi$
$467$	−19.2949	−0.892863	−0.446432	+	0.894818i	$0.647305\pi$
$468$	0	0
$469$	2.88560	0.133245
$470$	0	0
$471$	32.2137	1.48433
$472$	0	0
$473$	2.19258	0.100815
$474$	0	0
$475$	−17.4506	−0.800690
$476$	0	0
$477$	−29.4356	−1.34776
$478$	0	0
$479$	26.2032	1.19725	0.598626	−	0.801028i	$-0.295713\pi$
$479$	26.2032	1.19725	0.598626	+	0.801028i	$0.295713\pi$
$480$	0	0
$481$	−5.28129	−0.240806
$482$	0	0
$483$	34.9413	1.58989
$484$	0	0
$485$	9.43870	0.428589
$486$	0	0
$487$	−30.4664	−1.38057	−0.690283	−	0.723539i	$-0.742514\pi$
$487$	−30.4664	−1.38057	−0.690283	+	0.723539i	$0.742514\pi$
$488$	0	0
$489$	−73.1539	−3.30814
$490$	0	0
$491$	−36.3956	−1.64251	−0.821255	−	0.570562i	$-0.806726\pi$
$491$	−36.3956	−1.64251	−0.821255	+	0.570562i	$0.806726\pi$
$492$	0	0
$493$	−8.96087	−0.403577
$494$	0	0
$495$	−2.98344	−0.134096
$496$	0	0
$497$	10.6059	0.475740
$498$	0	0
$499$	2.03599	0.0911435	0.0455717	−	0.998961i	$-0.485489\pi$
$499$	2.03599	0.0911435	0.0455717	+	0.998961i	$0.485489\pi$
$500$	0	0
$501$	45.0067	2.01075
$502$	0	0
$503$	−23.3684	−1.04195	−0.520973	−	0.853573i	$-0.674431\pi$
$503$	−23.3684	−1.04195	−0.520973	+	0.853573i	$0.674431\pi$
$504$	0	0
$505$	5.30709	0.236163
$506$	0	0
$507$	−37.2418	−1.65397
$508$	0	0
$509$	32.6744	1.44827	0.724133	−	0.689661i	$-0.242240\pi$
$509$	32.6744	1.44827	0.724133	+	0.689661i	$0.242240\pi$
$510$	0	0
$511$	55.2548	2.44433
$512$	0	0
$513$	59.9769	2.64805
$514$	0	0
$515$	−19.0523	−0.839546
$516$	0	0
$517$	2.59527	0.114140
$518$	0	0
$519$	−51.1412	−2.24485
$520$	0	0
$521$	8.73632	0.382745	0.191373	−	0.981517i	$-0.438706\pi$
$521$	8.73632	0.382745	0.191373	+	0.981517i	$0.438706\pi$
$522$	0	0
$523$	−15.6850	−0.685855	−0.342928	−	0.939362i	$-0.611419\pi$
$523$	−15.6850	−0.685855	−0.342928	+	0.939362i	$0.611419\pi$
$524$	0	0
$525$	35.3522	1.54290
$526$	0	0
$527$	−12.7346	−0.554726
$528$	0	0
$529$	−13.1675	−0.572499
$530$	0	0
$531$	−15.7620	−0.684013
$532$	0	0
$533$	7.01502	0.303854
$534$	0	0
$535$	10.3435	0.447190
$536$	0	0
$537$	−43.4730	−1.87600
$538$	0	0
$539$	2.03635	0.0877117
$540$	0	0
$541$	28.4519	1.22324	0.611621	−	0.791151i	$-0.290518\pi$
$541$	28.4519	1.22324	0.611621	+	0.791151i	$0.290518\pi$
$542$	0	0
$543$	−16.6736	−0.715534
$544$	0	0
$545$	6.18716	0.265029
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	41.6677	1.77834
$550$	0	0
$551$	−39.5135	−1.68333
$552$	0	0
$553$	37.5051	1.59488
$554$	0	0
$555$	−22.7670	−0.966404
$556$	0	0
$557$	32.4488	1.37490	0.687450	−	0.726231i	$-0.258730\pi$
$557$	32.4488	1.37490	0.687450	+	0.726231i	$0.258730\pi$
$558$	0	0
$559$	−6.28256	−0.265724
$560$	0	0
$561$	1.30124	0.0549382
$562$	0	0
$563$	−23.7803	−1.00222	−0.501111	−	0.865383i	$-0.667075\pi$
$563$	−23.7803	−1.00222	−0.501111	+	0.865383i	$0.667075\pi$
$564$	0	0
$565$	21.6267	0.909842
$566$	0	0
$567$	−50.7786	−2.13250
$568$	0	0
$569$	42.0011	1.76078	0.880389	−	0.474252i	$-0.157281\pi$
$569$	42.0011	1.76078	0.880389	+	0.474252i	$0.157281\pi$
$570$	0	0
$571$	39.1170	1.63699	0.818497	−	0.574510i	$-0.194807\pi$
$571$	39.1170	1.63699	0.818497	+	0.574510i	$0.194807\pi$
$572$	0	0
$573$	−24.2698	−1.01389
$574$	0	0
$575$	9.94814	0.414866
$576$	0	0
$577$	−20.9523	−0.872256	−0.436128	−	0.899885i	$-0.643651\pi$
$577$	−20.9523	−0.872256	−0.436128	+	0.899885i	$0.643651\pi$
$578$	0	0
$579$	72.6604	3.01966
$580$	0	0
$581$	−48.0802	−1.99470
$582$	0	0
$583$	−1.52266	−0.0630622
$584$	0	0
$585$	8.54867	0.353444
$586$	0	0
$587$	37.1311	1.53257	0.766283	−	0.642503i	$-0.222104\pi$
$587$	37.1311	1.53257	0.766283	+	0.642503i	$0.222104\pi$
$588$	0	0
$589$	−56.1538	−2.31378
$590$	0	0
$591$	−5.91282	−0.243221
$592$	0	0
$593$	19.1878	0.787949	0.393974	−	0.919121i	$-0.371100\pi$
$593$	19.1878	0.787949	0.393974	+	0.919121i	$0.371100\pi$
$594$	0	0
$595$	6.08621	0.249510
$596$	0	0
$597$	−6.93731	−0.283925
$598$	0	0
$599$	−31.5960	−1.29098	−0.645488	−	0.763770i	$-0.723346\pi$
$599$	−31.5960	−1.29098	−0.645488	+	0.763770i	$0.723346\pi$
$600$	0	0
$601$	16.9578	0.691725	0.345862	−	0.938285i	$-0.387586\pi$
$601$	16.9578	0.691725	0.345862	+	0.938285i	$0.387586\pi$
$602$	0	0
$603$	−5.22215	−0.212662
$604$	0	0
$605$	14.7158	0.598282
$606$	0	0
$607$	10.5096	0.426572	0.213286	−	0.976990i	$-0.431583\pi$
$607$	10.5096	0.426572	0.213286	+	0.976990i	$0.431583\pi$
$608$	0	0
$609$	80.0481	3.24371
$610$	0	0
$611$	−7.43642	−0.300845
$612$	0	0
$613$	−14.8071	−0.598055	−0.299027	−	0.954245i	$-0.596662\pi$
$613$	−14.8071	−0.598055	−0.299027	+	0.954245i	$0.596662\pi$
$614$	0	0
$615$	30.2408	1.21943
$616$	0	0
$617$	−27.3637	−1.10162	−0.550811	−	0.834630i	$-0.685681\pi$
$617$	−27.3637	−1.10162	−0.550811	+	0.834630i	$0.685681\pi$
$618$	0	0
$619$	−4.81738	−0.193627	−0.0968134	−	0.995303i	$-0.530865\pi$
$619$	−4.81738	−0.193627	−0.0968134	+	0.995303i	$0.530865\pi$
$620$	0	0
$621$	−34.1913	−1.37205
$622$	0	0
$623$	15.9749	0.640021
$624$	0	0
$625$	0.927908	0.0371163
$626$	0	0
$627$	5.73788	0.229149
$628$	0	0
$629$	6.80459	0.271317
$630$	0	0
$631$	−29.9291	−1.19146	−0.595730	−	0.803185i	$-0.703137\pi$
$631$	−29.9291	−1.19146	−0.595730	+	0.803185i	$0.703137\pi$
$632$	0	0
$633$	−44.7908	−1.78028
$634$	0	0
$635$	9.49525	0.376808
$636$	0	0
$637$	−5.83489	−0.231187
$638$	0	0
$639$	−19.1938	−0.759294
$640$	0	0
$641$	2.88534	0.113964	0.0569821	−	0.998375i	$-0.481852\pi$
$641$	2.88534	0.113964	0.0569821	+	0.998375i	$0.481852\pi$
$642$	0	0
$643$	32.7470	1.29142	0.645708	−	0.763584i	$-0.276562\pi$
$643$	32.7470	1.29142	0.645708	+	0.763584i	$0.276562\pi$
$644$	0	0
$645$	−27.0833	−1.06640
$646$	0	0
$647$	−20.0383	−0.787787	−0.393894	−	0.919156i	$-0.628872\pi$
$647$	−20.0383	−0.787787	−0.393894	+	0.919156i	$0.628872\pi$
$648$	0	0
$649$	−0.815346	−0.0320051
$650$	0	0
$651$	113.759	4.45855
$652$	0	0
$653$	−8.58315	−0.335885	−0.167942	−	0.985797i	$-0.553712\pi$
$653$	−8.58315	−0.335885	−0.167942	+	0.985797i	$0.553712\pi$
$654$	0	0
$655$	−4.55315	−0.177906
$656$	0	0
$657$	−99.9961	−3.90122
$658$	0	0
$659$	34.5534	1.34601	0.673004	−	0.739638i	$-0.265003\pi$
$659$	34.5534	1.34601	0.673004	+	0.739638i	$0.265003\pi$
$660$	0	0
$661$	−18.6496	−0.725386	−0.362693	−	0.931909i	$-0.618143\pi$
$661$	−18.6496	−0.725386	−0.362693	+	0.931909i	$0.618143\pi$
$662$	0	0
$663$	−3.72853	−0.144804
$664$	0	0
$665$	26.8375	1.04072
$666$	0	0
$667$	22.5256	0.872196
$668$	0	0
$669$	77.4726	2.99526
$670$	0	0
$671$	2.15541	0.0832087
$672$	0	0
$673$	23.0907	0.890083	0.445041	−	0.895510i	$-0.353189\pi$
$673$	23.0907	0.890083	0.445041	+	0.895510i	$0.353189\pi$
$674$	0	0
$675$	−34.5933	−1.33150
$676$	0	0
$677$	−41.7475	−1.60449	−0.802243	−	0.596997i	$-0.796360\pi$
$677$	−41.7475	−1.60449	−0.802243	+	0.596997i	$0.796360\pi$
$678$	0	0
$679$	25.2005	0.967108
$680$	0	0
$681$	62.2308	2.38469
$682$	0	0
$683$	46.2392	1.76930	0.884648	−	0.466260i	$-0.154399\pi$
$683$	46.2392	1.76930	0.884648	+	0.466260i	$0.154399\pi$
$684$	0	0
$685$	29.5920	1.13065
$686$	0	0
$687$	76.1056	2.90361
$688$	0	0
$689$	4.36299	0.166217
$690$	0	0
$691$	51.8556	1.97268	0.986340	−	0.164724i	$-0.0526732\pi$
$691$	51.8556	1.97268	0.986340	+	0.164724i	$0.0526732\pi$
$692$	0	0
$693$	−7.96553	−0.302585
$694$	0	0
$695$	−10.5599	−0.400561
$696$	0	0
$697$	−9.03838	−0.342353
$698$	0	0
$699$	−68.6337	−2.59596
$700$	0	0
$701$	46.8098	1.76798	0.883992	−	0.467503i	$-0.154846\pi$
$701$	46.8098	1.76798	0.883992	+	0.467503i	$0.154846\pi$
$702$	0	0
$703$	30.0053	1.13167
$704$	0	0
$705$	−32.0574	−1.20735
$706$	0	0
$707$	14.1695	0.532899
$708$	0	0
$709$	2.30760	0.0866637	0.0433318	−	0.999061i	$-0.486203\pi$
$709$	2.30760	0.0866637	0.0433318	+	0.999061i	$0.486203\pi$
$710$	0	0
$711$	−67.8739	−2.54547
$712$	0	0
$713$	32.0118	1.19885
$714$	0	0
$715$	0.442210	0.0165377
$716$	0	0
$717$	18.1771	0.678837
$718$	0	0
$719$	−1.47404	−0.0549723	−0.0274862	−	0.999622i	$-0.508750\pi$
$719$	−1.47404	−0.0549723	−0.0274862	+	0.999622i	$0.508750\pi$
$720$	0	0
$721$	−50.8681	−1.89443
$722$	0	0
$723$	24.6271	0.915890
$724$	0	0
$725$	22.7905	0.846418
$726$	0	0
$727$	11.7670	0.436412	0.218206	−	0.975903i	$-0.429979\pi$
$727$	11.7670	0.436412	0.218206	+	0.975903i	$0.429979\pi$
$728$	0	0
$729$	−9.09751	−0.336945
$730$	0	0
$731$	8.09466	0.299392
$732$	0	0
$733$	20.5086	0.757502	0.378751	−	0.925499i	$-0.376354\pi$
$733$	20.5086	0.757502	0.378751	+	0.925499i	$0.376354\pi$
$734$	0	0
$735$	−25.1535	−0.927800
$736$	0	0
$737$	−0.270134	−0.00995052
$738$	0	0
$739$	31.6389	1.16385	0.581927	−	0.813241i	$-0.302299\pi$
$739$	31.6389	1.16385	0.581927	+	0.813241i	$0.302299\pi$
$740$	0	0
$741$	−16.4412	−0.603982
$742$	0	0
$743$	40.5825	1.48883	0.744414	−	0.667718i	$-0.232729\pi$
$743$	40.5825	1.48883	0.744414	+	0.667718i	$0.232729\pi$
$744$	0	0
$745$	2.34774	0.0860145
$746$	0	0
$747$	87.0120	3.18360
$748$	0	0
$749$	27.6163	1.00908
$750$	0	0
$751$	−27.4293	−1.00091	−0.500454	−	0.865763i	$-0.666834\pi$
$751$	−27.4293	−1.00091	−0.500454	+	0.865763i	$0.666834\pi$
$752$	0	0
$753$	62.5336	2.27885
$754$	0	0
$755$	12.3084	0.447949
$756$	0	0
$757$	−19.8595	−0.721805	−0.360903	−	0.932603i	$-0.617531\pi$
$757$	−19.8595	−0.721805	−0.360903	+	0.932603i	$0.617531\pi$
$758$	0	0
$759$	−3.27102	−0.118730
$760$	0	0
$761$	−30.0646	−1.08984	−0.544920	−	0.838488i	$-0.683440\pi$
$761$	−30.0646	−1.08984	−0.544920	+	0.838488i	$0.683440\pi$
$762$	0	0
$763$	16.5192	0.598035
$764$	0	0
$765$	−11.0144	−0.398226
$766$	0	0
$767$	2.33627	0.0843578
$768$	0	0
$769$	−27.2458	−0.982510	−0.491255	−	0.871016i	$-0.663462\pi$
$769$	−27.2458	−0.982510	−0.491255	+	0.871016i	$0.663462\pi$
$770$	0	0
$771$	−49.2816	−1.77483
$772$	0	0
$773$	−42.4735	−1.52767	−0.763833	−	0.645414i	$-0.776685\pi$
$773$	−42.4735	−1.52767	−0.763833	+	0.645414i	$0.776685\pi$
$774$	0	0
$775$	32.3882	1.16342
$776$	0	0
$777$	−60.7859	−2.18068
$778$	0	0
$779$	−39.8553	−1.42796
$780$	0	0
$781$	−0.992867	−0.0355276
$782$	0	0
$783$	−78.3298	−2.79928
$784$	0	0
$785$	−14.1050	−0.503431
$786$	0	0
$787$	−45.4020	−1.61841	−0.809203	−	0.587529i	$-0.800101\pi$
$787$	−45.4020	−1.61841	−0.809203	+	0.587529i	$0.800101\pi$
$788$	0	0
$789$	−81.0885	−2.88683
$790$	0	0
$791$	57.7414	2.05305
$792$	0	0
$793$	−6.17606	−0.219318
$794$	0	0
$795$	18.8083	0.667062
$796$	0	0
$797$	27.6932	0.980944	0.490472	−	0.871457i	$-0.336824\pi$
$797$	27.6932	0.980944	0.490472	+	0.871457i	$0.336824\pi$
$798$	0	0
$799$	9.58133	0.338963
$800$	0	0
$801$	−28.9102	−1.02149
$802$	0	0
$803$	−5.17265	−0.182539
$804$	0	0
$805$	−15.2994	−0.539232
$806$	0	0
$807$	23.3498	0.821950
$808$	0	0
$809$	12.6422	0.444475	0.222238	−	0.974993i	$-0.428664\pi$
$809$	12.6422	0.444475	0.222238	+	0.974993i	$0.428664\pi$
$810$	0	0
$811$	7.30807	0.256621	0.128310	−	0.991734i	$-0.459045\pi$
$811$	7.30807	0.256621	0.128310	+	0.991734i	$0.459045\pi$
$812$	0	0
$813$	−2.64007	−0.0925913
$814$	0	0
$815$	32.0311	1.12200
$816$	0	0
$817$	35.6939	1.24877
$818$	0	0
$819$	22.8242	0.797543
$820$	0	0
$821$	36.6884	1.28043	0.640217	−	0.768194i	$-0.278844\pi$
$821$	36.6884	1.28043	0.640217	+	0.768194i	$0.278844\pi$
$822$	0	0
$823$	4.83046	0.168379	0.0841897	−	0.996450i	$-0.473170\pi$
$823$	4.83046	0.168379	0.0841897	+	0.996450i	$0.473170\pi$
$824$	0	0
$825$	−3.30948	−0.115221
$826$	0	0
$827$	23.1753	0.805883	0.402942	−	0.915226i	$-0.367988\pi$
$827$	23.1753	0.805883	0.402942	+	0.915226i	$0.367988\pi$
$828$	0	0
$829$	27.0086	0.938046	0.469023	−	0.883186i	$-0.344606\pi$
$829$	27.0086	0.938046	0.469023	+	0.883186i	$0.344606\pi$
$830$	0	0
$831$	−23.8274	−0.826565
$832$	0	0
$833$	7.51787	0.260479
$834$	0	0
$835$	−19.7066	−0.681974
$836$	0	0
$837$	−111.317	−3.84767
$838$	0	0
$839$	19.5336	0.674373	0.337187	−	0.941438i	$-0.390525\pi$
$839$	19.5336	0.674373	0.337187	+	0.941438i	$0.390525\pi$
$840$	0	0
$841$	22.6046	0.779470
$842$	0	0
$843$	15.5732	0.536368
$844$	0	0
$845$	16.3067	0.560967
$846$	0	0
$847$	39.2899	1.35002
$848$	0	0
$849$	−4.46088	−0.153097
$850$	0	0
$851$	−17.1052	−0.586359
$852$	0	0
$853$	38.7260	1.32595	0.662977	−	0.748640i	$-0.269293\pi$
$853$	38.7260	1.32595	0.662977	+	0.748640i	$0.269293\pi$
$854$	0	0
$855$	−48.5686	−1.66101
$856$	0	0
$857$	35.3753	1.20840	0.604199	−	0.796834i	$-0.293493\pi$
$857$	35.3753	1.20840	0.604199	+	0.796834i	$0.293493\pi$
$858$	0	0
$859$	−7.81176	−0.266534	−0.133267	−	0.991080i	$-0.542547\pi$
$859$	−7.81176	−0.266534	−0.133267	+	0.991080i	$0.542547\pi$
$860$	0	0
$861$	80.7405	2.75163
$862$	0	0
$863$	−7.24097	−0.246486	−0.123243	−	0.992377i	$-0.539329\pi$
$863$	−7.24097	−0.246486	−0.123243	+	0.992377i	$0.539329\pi$
$864$	0	0
$865$	22.3926	0.761372
$866$	0	0
$867$	−47.6812	−1.61934
$868$	0	0
$869$	−3.51102	−0.119103
$870$	0	0
$871$	0.774035	0.0262272
$872$	0	0
$873$	−45.6061	−1.54353
$874$	0	0
$875$	−39.8749	−1.34802
$876$	0	0
$877$	−27.0100	−0.912062	−0.456031	−	0.889964i	$-0.650729\pi$
$877$	−27.0100	−0.912062	−0.456031	+	0.889964i	$0.650729\pi$
$878$	0	0
$879$	37.0406	1.24935
$880$	0	0
$881$	20.0865	0.676731	0.338366	−	0.941015i	$-0.390126\pi$
$881$	20.0865	0.676731	0.338366	+	0.941015i	$0.390126\pi$
$882$	0	0
$883$	−11.1980	−0.376842	−0.188421	−	0.982088i	$-0.560337\pi$
$883$	−11.1980	−0.376842	−0.188421	+	0.982088i	$0.560337\pi$
$884$	0	0
$885$	10.0714	0.338545
$886$	0	0
$887$	−56.2189	−1.88765	−0.943824	−	0.330450i	$-0.892800\pi$
$887$	−56.2189	−1.88765	−0.943824	+	0.330450i	$0.892800\pi$
$888$	0	0
$889$	25.3515	0.850263
$890$	0	0
$891$	4.75362	0.159252
$892$	0	0
$893$	42.2495	1.41382
$894$	0	0
$895$	19.0351	0.636272
$896$	0	0
$897$	9.37268	0.312945
$898$	0	0
$899$	73.3368	2.44592
$900$	0	0
$901$	−5.62143	−0.187277
$902$	0	0
$903$	−72.3101	−2.40633
$904$	0	0
$905$	7.30071	0.242684
$906$	0	0
$907$	41.1969	1.36792	0.683960	−	0.729520i	$-0.260256\pi$
$907$	41.1969	1.36792	0.683960	+	0.729520i	$0.260256\pi$
$908$	0	0
$909$	−25.6429	−0.850522
$910$	0	0
$911$	−26.4943	−0.877795	−0.438897	−	0.898537i	$-0.644631\pi$
$911$	−26.4943	−0.877795	−0.438897	+	0.898537i	$0.644631\pi$
$912$	0	0
$913$	4.50101	0.148962
$914$	0	0
$915$	−26.6242	−0.880169
$916$	0	0
$917$	−12.1565	−0.401444
$918$	0	0
$919$	29.7090	0.980008	0.490004	−	0.871720i	$-0.336995\pi$
$919$	29.7090	0.980008	0.490004	+	0.871720i	$0.336995\pi$
$920$	0	0
$921$	−45.5014	−1.49932
$922$	0	0
$923$	2.84493	0.0936421
$924$	0	0
$925$	−17.3064	−0.569029
$926$	0	0
$927$	92.0574	3.02356
$928$	0	0
$929$	31.6127	1.03718	0.518589	−	0.855024i	$-0.326457\pi$
$929$	31.6127	1.03718	0.518589	+	0.855024i	$0.326457\pi$
$930$	0	0
$931$	33.1505	1.08646
$932$	0	0
$933$	79.1235	2.59039
$934$	0	0
$935$	−0.569758	−0.0186331
$936$	0	0
$937$	25.5752	0.835506	0.417753	−	0.908561i	$-0.362818\pi$
$937$	25.5752	0.835506	0.417753	+	0.908561i	$0.362818\pi$
$938$	0	0
$939$	−10.1764	−0.332096
$940$	0	0
$941$	−38.4356	−1.25296	−0.626482	−	0.779436i	$-0.715506\pi$
$941$	−38.4356	−1.25296	−0.626482	+	0.779436i	$0.715506\pi$
$942$	0	0
$943$	22.7205	0.739880
$944$	0	0
$945$	53.2015	1.73065
$946$	0	0
$947$	−39.7014	−1.29012	−0.645060	−	0.764132i	$-0.723168\pi$
$947$	−39.7014	−1.29012	−0.645060	+	0.764132i	$0.723168\pi$
$948$	0	0
$949$	14.8216	0.481129
$950$	0	0
$951$	66.6409	2.16098
$952$	0	0
$953$	−22.9371	−0.743007	−0.371504	−	0.928432i	$-0.621158\pi$
$953$	−22.9371	−0.743007	−0.371504	+	0.928432i	$0.621158\pi$
$954$	0	0
$955$	10.6268	0.343874
$956$	0	0
$957$	−7.49367	−0.242236
$958$	0	0
$959$	79.0081	2.55130
$960$	0	0
$961$	73.2211	2.36197
$962$	0	0
$963$	−49.9780	−1.61052
$964$	0	0
$965$	−31.8150	−1.02416
$966$	0	0
$967$	26.6775	0.857891	0.428946	−	0.903330i	$-0.358885\pi$
$967$	26.6775	0.857891	0.428946	+	0.903330i	$0.358885\pi$
$968$	0	0
$969$	21.1834	0.680508
$970$	0	0
$971$	9.36106	0.300411	0.150205	−	0.988655i	$-0.452007\pi$
$971$	9.36106	0.300411	0.150205	+	0.988655i	$0.452007\pi$
$972$	0	0
$973$	−28.1942	−0.903863
$974$	0	0
$975$	9.48289	0.303696
$976$	0	0
$977$	17.6428	0.564445	0.282222	−	0.959349i	$-0.408928\pi$
$977$	17.6428	0.564445	0.282222	+	0.959349i	$0.408928\pi$
$978$	0	0
$979$	−1.49548	−0.0477959
$980$	0	0
$981$	−29.8952	−0.954481
$982$	0	0
$983$	1.76510	0.0562979	0.0281490	−	0.999604i	$-0.491039\pi$
$983$	1.76510	0.0562979	0.0281490	+	0.999604i	$0.491039\pi$
$984$	0	0
$985$	2.58898	0.0824919
$986$	0	0
$987$	−85.5907	−2.72438
$988$	0	0
$989$	−20.3482	−0.647034
$990$	0	0
$991$	33.2753	1.05702	0.528512	−	0.848926i	$-0.322750\pi$
$991$	33.2753	1.05702	0.528512	+	0.848926i	$0.322750\pi$
$992$	0	0
$993$	59.5462	1.88964
$994$	0	0
$995$	3.03757	0.0962973
$996$	0	0
$997$	10.1732	0.322187	0.161094	−	0.986939i	$-0.448498\pi$
$997$	10.1732	0.322187	0.161094	+	0.986939i	$0.448498\pi$
$998$	0	0
$999$	59.4811	1.88190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.17		18
4.3	odd	2		547.2.a.b.1.10	✓	18
12.11	even	2		4923.2.a.l.1.9		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.10	✓	18	4.3	odd	2
4923.2.a.l.1.9		18	12.11	even	2
8752.2.a.s.1.17		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)

\(f(q)\)	\(=\)	\(q+3.08736 q^{3} -1.35183 q^{5} -3.60927 q^{7} +6.53179 q^{9} +O(q^{10})\)	\(q+3.08736 q^{3} -1.35183 q^{5} -3.60927 q^{7} +6.53179 q^{9} +0.337880 q^{11} -0.968153 q^{13} -4.17358 q^{15} +1.24740 q^{17} +5.50049 q^{19} -11.1431 q^{21} -3.13569 q^{23} -3.17256 q^{25} +10.9039 q^{27} -7.18364 q^{29} -10.2089 q^{31} +1.04316 q^{33} +4.87912 q^{35} +5.45502 q^{37} -2.98904 q^{39} -7.24577 q^{41} +6.48922 q^{43} -8.82987 q^{45} +7.68104 q^{47} +6.02683 q^{49} +3.85117 q^{51} -4.50651 q^{53} -0.456756 q^{55} +16.9820 q^{57} -2.41312 q^{59} +6.37922 q^{61} -23.5750 q^{63} +1.30878 q^{65} -0.799497 q^{67} -9.68099 q^{69} -2.93852 q^{71} -15.3091 q^{73} -9.79483 q^{75} -1.21950 q^{77} -10.3913 q^{79} +14.0690 q^{81} +13.3213 q^{83} -1.68627 q^{85} -22.1785 q^{87} -4.42608 q^{89} +3.49433 q^{91} -31.5185 q^{93} -7.43573 q^{95} -6.98217 q^{97} +2.20696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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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