LMFDB - Embedded newform 8752.2.a.s.1.14

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			1.14
Root			$-1.52216$ 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)

$f(q)$	$=$	$q+2.58636 q^{3} +1.24712 q^{5} +0.899316 q^{7} +3.68924 q^{9} +O(q^{10})$	$q+2.58636 q^{3} +1.24712 q^{5} +0.899316 q^{7} +3.68924 q^{9} -3.77827 q^{11} -4.29029 q^{13} +3.22549 q^{15} -7.86362 q^{17} +6.61960 q^{19} +2.32595 q^{21} -3.37854 q^{23} -3.44470 q^{25} +1.78261 q^{27} -2.81309 q^{29} +1.72157 q^{31} -9.77196 q^{33} +1.12155 q^{35} -4.86769 q^{37} -11.0962 q^{39} +5.86463 q^{41} -7.30517 q^{43} +4.60091 q^{45} -2.08319 q^{47} -6.19123 q^{49} -20.3381 q^{51} +3.26386 q^{53} -4.71195 q^{55} +17.1206 q^{57} +6.21926 q^{59} +12.5220 q^{61} +3.31779 q^{63} -5.35050 q^{65} +10.3052 q^{67} -8.73812 q^{69} -16.3692 q^{71} -15.5143 q^{73} -8.90922 q^{75} -3.39786 q^{77} -5.15466 q^{79} -6.45724 q^{81} -6.55843 q^{83} -9.80686 q^{85} -7.27565 q^{87} -4.60031 q^{89} -3.85833 q^{91} +4.45258 q^{93} +8.25541 q^{95} +9.16155 q^{97} -13.9389 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.58636	1.49323	0.746617	−	0.665254i	$-0.231677\pi$
$3$	2.58636	1.49323	0.746617	+	0.665254i	$0.231677\pi$
$4$	0	0
$5$	1.24712	0.557728	0.278864	−	0.960331i	$-0.410042\pi$
$5$	1.24712	0.557728	0.278864	+	0.960331i	$0.410042\pi$
$6$	0	0
$7$	0.899316	0.339909	0.169955	−	0.985452i	$-0.445638\pi$
$7$	0.899316	0.339909	0.169955	+	0.985452i	$0.445638\pi$
$8$	0	0
$9$	3.68924	1.22975
$10$	0	0
$11$	−3.77827	−1.13919	−0.569596	−	0.821925i	$-0.692900\pi$
$11$	−3.77827	−1.13919	−0.569596	+	0.821925i	$0.692900\pi$
$12$	0	0
$13$	−4.29029	−1.18991	−0.594957	−	0.803758i	$-0.702831\pi$
$13$	−4.29029	−1.18991	−0.594957	+	0.803758i	$0.702831\pi$
$14$	0	0
$15$	3.22549	0.832818
$16$	0	0
$17$	−7.86362	−1.90721	−0.953604	−	0.301063i	$-0.902658\pi$
$17$	−7.86362	−1.90721	−0.953604	+	0.301063i	$0.902658\pi$
$18$	0	0
$19$	6.61960	1.51864	0.759320	−	0.650718i	$-0.225532\pi$
$19$	6.61960	1.51864	0.759320	+	0.650718i	$0.225532\pi$
$20$	0	0
$21$	2.32595	0.507564
$22$	0	0
$23$	−3.37854	−0.704475	−0.352238	−	0.935911i	$-0.614579\pi$
$23$	−3.37854	−0.704475	−0.352238	+	0.935911i	$0.614579\pi$
$24$	0	0
$25$	−3.44470	−0.688940
$26$	0	0
$27$	1.78261	0.343064
$28$	0	0
$29$	−2.81309	−0.522377	−0.261189	−	0.965288i	$-0.584114\pi$
$29$	−2.81309	−0.522377	−0.261189	+	0.965288i	$0.584114\pi$
$30$	0	0
$31$	1.72157	0.309202	0.154601	−	0.987977i	$-0.450591\pi$
$31$	1.72157	0.309202	0.154601	+	0.987977i	$0.450591\pi$
$32$	0	0
$33$	−9.77196	−1.70108
$34$	0	0
$35$	1.12155	0.189577
$36$	0	0
$37$	−4.86769	−0.800243	−0.400122	−	0.916462i	$-0.631032\pi$
$37$	−4.86769	−0.800243	−0.400122	+	0.916462i	$0.631032\pi$
$38$	0	0
$39$	−11.0962	−1.77682
$40$	0	0
$41$	5.86463	0.915901	0.457951	−	0.888978i	$-0.348584\pi$
$41$	5.86463	0.915901	0.457951	+	0.888978i	$0.348584\pi$
$42$	0	0
$43$	−7.30517	−1.11403	−0.557014	−	0.830503i	$-0.688053\pi$
$43$	−7.30517	−1.11403	−0.557014	+	0.830503i	$0.688053\pi$
$44$	0	0
$45$	4.60091	0.685863
$46$	0	0
$47$	−2.08319	−0.303865	−0.151932	−	0.988391i	$-0.548550\pi$
$47$	−2.08319	−0.303865	−0.151932	+	0.988391i	$0.548550\pi$
$48$	0	0
$49$	−6.19123	−0.884462
$50$	0	0
$51$	−20.3381	−2.84791
$52$	0	0
$53$	3.26386	0.448325	0.224163	−	0.974552i	$-0.428035\pi$
$53$	3.26386	0.448325	0.224163	+	0.974552i	$0.428035\pi$
$54$	0	0
$55$	−4.71195	−0.635359
$56$	0	0
$57$	17.1206	2.26768
$58$	0	0
$59$	6.21926	0.809679	0.404840	−	0.914388i	$-0.367327\pi$
$59$	6.21926	0.809679	0.404840	+	0.914388i	$0.367327\pi$
$60$	0	0
$61$	12.5220	1.60327	0.801636	−	0.597812i	$-0.203963\pi$
$61$	12.5220	1.60327	0.801636	+	0.597812i	$0.203963\pi$
$62$	0	0
$63$	3.31779	0.418002
$64$	0	0
$65$	−5.35050	−0.663648
$66$	0	0
$67$	10.3052	1.25898	0.629491	−	0.777008i	$-0.283263\pi$
$67$	10.3052	1.25898	0.629491	+	0.777008i	$0.283263\pi$
$68$	0	0
$69$	−8.73812	−1.05195
$70$	0	0
$71$	−16.3692	−1.94267	−0.971334	−	0.237718i	$-0.923601\pi$
$71$	−16.3692	−1.94267	−0.971334	+	0.237718i	$0.923601\pi$
$72$	0	0
$73$	−15.5143	−1.81581	−0.907907	−	0.419172i	$-0.862320\pi$
$73$	−15.5143	−1.81581	−0.907907	+	0.419172i	$0.862320\pi$
$74$	0	0
$75$	−8.90922	−1.02875
$76$	0	0
$77$	−3.39786	−0.387222
$78$	0	0
$79$	−5.15466	−0.579945	−0.289973	−	0.957035i	$-0.593646\pi$
$79$	−5.15466	−0.579945	−0.289973	+	0.957035i	$0.593646\pi$
$80$	0	0
$81$	−6.45724	−0.717471
$82$	0	0
$83$	−6.55843	−0.719881	−0.359941	−	0.932975i	$-0.617203\pi$
$83$	−6.55843	−0.719881	−0.359941	+	0.932975i	$0.617203\pi$
$84$	0	0
$85$	−9.80686	−1.06370
$86$	0	0
$87$	−7.27565	−0.780031
$88$	0	0
$89$	−4.60031	−0.487632	−0.243816	−	0.969822i	$-0.578399\pi$
$89$	−4.60031	−0.487632	−0.243816	+	0.969822i	$0.578399\pi$
$90$	0	0
$91$	−3.85833	−0.404463
$92$	0	0
$93$	4.45258	0.461711
$94$	0	0
$95$	8.25541	0.846987
$96$	0	0
$97$	9.16155	0.930215	0.465107	−	0.885254i	$-0.346016\pi$
$97$	9.16155	0.930215	0.465107	+	0.885254i	$0.346016\pi$
$98$	0	0
$99$	−13.9389	−1.40092
$100$	0	0
$101$	−11.0854	−1.10304	−0.551521	−	0.834161i	$-0.685952\pi$
$101$	−11.0854	−1.10304	−0.551521	+	0.834161i	$0.685952\pi$
$102$	0	0
$103$	4.38943	0.432503	0.216252	−	0.976338i	$-0.430617\pi$
$103$	4.38943	0.432503	0.216252	+	0.976338i	$0.430617\pi$
$104$	0	0
$105$	2.90073	0.283082
$106$	0	0
$107$	19.0618	1.84277	0.921385	−	0.388650i	$-0.127059\pi$
$107$	19.0618	1.84277	0.921385	+	0.388650i	$0.127059\pi$
$108$	0	0
$109$	4.54306	0.435147	0.217573	−	0.976044i	$-0.430186\pi$
$109$	4.54306	0.435147	0.217573	+	0.976044i	$0.430186\pi$
$110$	0	0
$111$	−12.5896	−1.19495
$112$	0	0
$113$	11.6865	1.09937	0.549685	−	0.835372i	$-0.314748\pi$
$113$	11.6865	1.09937	0.549685	+	0.835372i	$0.314748\pi$
$114$	0	0
$115$	−4.21344	−0.392905
$116$	0	0
$117$	−15.8279	−1.46329
$118$	0	0
$119$	−7.07188	−0.648278
$120$	0	0
$121$	3.27535	0.297759
$122$	0	0
$123$	15.1680	1.36765
$124$	0	0
$125$	−10.5315	−0.941968
$126$	0	0
$127$	13.7079	1.21638	0.608189	−	0.793792i	$-0.291896\pi$
$127$	13.7079	1.21638	0.608189	+	0.793792i	$0.291896\pi$
$128$	0	0
$129$	−18.8938	−1.66350
$130$	0	0
$131$	−15.4415	−1.34913	−0.674563	−	0.738217i	$-0.735668\pi$
$131$	−15.4415	−1.34913	−0.674563	+	0.738217i	$0.735668\pi$
$132$	0	0
$133$	5.95311	0.516200
$134$	0	0
$135$	2.22313	0.191336
$136$	0	0
$137$	0.634397	0.0542002	0.0271001	−	0.999633i	$-0.491373\pi$
$137$	0.634397	0.0542002	0.0271001	+	0.999633i	$0.491373\pi$
$138$	0	0
$139$	−8.90062	−0.754941	−0.377470	−	0.926022i	$-0.623206\pi$
$139$	−8.90062	−0.754941	−0.377470	+	0.926022i	$0.623206\pi$
$140$	0	0
$141$	−5.38787	−0.453741
$142$	0	0
$143$	16.2099	1.35554
$144$	0	0
$145$	−3.50825	−0.291344
$146$	0	0
$147$	−16.0127	−1.32071
$148$	0	0
$149$	0.864000	0.0707816	0.0353908	−	0.999374i	$-0.488732\pi$
$149$	0.864000	0.0707816	0.0353908	+	0.999374i	$0.488732\pi$
$150$	0	0
$151$	−9.82722	−0.799728	−0.399864	−	0.916575i	$-0.630943\pi$
$151$	−9.82722	−0.799728	−0.399864	+	0.916575i	$0.630943\pi$
$152$	0	0
$153$	−29.0108	−2.34538
$154$	0	0
$155$	2.14699	0.172451
$156$	0	0
$157$	−11.4355	−0.912656	−0.456328	−	0.889812i	$-0.650836\pi$
$157$	−11.4355	−0.912656	−0.456328	+	0.889812i	$0.650836\pi$
$158$	0	0
$159$	8.44150	0.669454
$160$	0	0
$161$	−3.03838	−0.239458
$162$	0	0
$163$	19.1215	1.49771	0.748854	−	0.662735i	$-0.230604\pi$
$163$	19.1215	1.49771	0.748854	+	0.662735i	$0.230604\pi$
$164$	0	0
$165$	−12.1868	−0.948739
$166$	0	0
$167$	−13.2573	−1.02588	−0.512942	−	0.858423i	$-0.671444\pi$
$167$	−13.2573	−1.02588	−0.512942	+	0.858423i	$0.671444\pi$
$168$	0	0
$169$	5.40662	0.415893
$170$	0	0
$171$	24.4213	1.86754
$172$	0	0
$173$	−1.24666	−0.0947815	−0.0473907	−	0.998876i	$-0.515091\pi$
$173$	−1.24666	−0.0947815	−0.0473907	+	0.998876i	$0.515091\pi$
$174$	0	0
$175$	−3.09787	−0.234177
$176$	0	0
$177$	16.0852	1.20904
$178$	0	0
$179$	−0.595954	−0.0445437	−0.0222718	−	0.999752i	$-0.507090\pi$
$179$	−0.595954	−0.0445437	−0.0222718	+	0.999752i	$0.507090\pi$
$180$	0	0
$181$	−9.48635	−0.705115	−0.352557	−	0.935790i	$-0.614688\pi$
$181$	−9.48635	−0.705115	−0.352557	+	0.935790i	$0.614688\pi$
$182$	0	0
$183$	32.3862	2.39406
$184$	0	0
$185$	−6.07058	−0.446318
$186$	0	0
$187$	29.7109	2.17268
$188$	0	0
$189$	1.60313	0.116611
$190$	0	0
$191$	−7.23592	−0.523573	−0.261786	−	0.965126i	$-0.584312\pi$
$191$	−7.23592	−0.523573	−0.261786	+	0.965126i	$0.584312\pi$
$192$	0	0
$193$	4.36945	0.314520	0.157260	−	0.987557i	$-0.449734\pi$
$193$	4.36945	0.314520	0.157260	+	0.987557i	$0.449734\pi$
$194$	0	0
$195$	−13.8383	−0.990981
$196$	0	0
$197$	−19.5262	−1.39118	−0.695592	−	0.718437i	$-0.744858\pi$
$197$	−19.5262	−1.39118	−0.695592	+	0.718437i	$0.744858\pi$
$198$	0	0
$199$	−0.0647319	−0.00458872	−0.00229436	−	0.999997i	$-0.500730\pi$
$199$	−0.0647319	−0.00458872	−0.00229436	+	0.999997i	$0.500730\pi$
$200$	0	0
$201$	26.6529	1.87995
$202$	0	0
$203$	−2.52985	−0.177561
$204$	0	0
$205$	7.31388	0.510824
$206$	0	0
$207$	−12.4643	−0.866325
$208$	0	0
$209$	−25.0106	−1.73002
$210$	0	0
$211$	−9.65449	−0.664642	−0.332321	−	0.943166i	$-0.607832\pi$
$211$	−9.65449	−0.664642	−0.332321	+	0.943166i	$0.607832\pi$
$212$	0	0
$213$	−42.3366	−2.90086
$214$	0	0
$215$	−9.11040	−0.621324
$216$	0	0
$217$	1.54823	0.105101
$218$	0	0
$219$	−40.1256	−2.71143
$220$	0	0
$221$	33.7372	2.26941
$222$	0	0
$223$	−12.7892	−0.856429	−0.428215	−	0.903677i	$-0.640857\pi$
$223$	−12.7892	−0.856429	−0.428215	+	0.903677i	$0.640857\pi$
$224$	0	0
$225$	−12.7083	−0.847221
$226$	0	0
$227$	−4.94823	−0.328426	−0.164213	−	0.986425i	$-0.552508\pi$
$227$	−4.94823	−0.328426	−0.164213	+	0.986425i	$0.552508\pi$
$228$	0	0
$229$	−26.7697	−1.76899	−0.884496	−	0.466548i	$-0.845497\pi$
$229$	−26.7697	−1.76899	−0.884496	+	0.466548i	$0.845497\pi$
$230$	0	0
$231$	−8.78808	−0.578213
$232$	0	0
$233$	−22.2664	−1.45872	−0.729361	−	0.684129i	$-0.760182\pi$
$233$	−22.2664	−1.45872	−0.729361	+	0.684129i	$0.760182\pi$
$234$	0	0
$235$	−2.59798	−0.169474
$236$	0	0
$237$	−13.3318	−0.865993
$238$	0	0
$239$	18.3982	1.19008	0.595041	−	0.803695i	$-0.297136\pi$
$239$	18.3982	1.19008	0.595041	+	0.803695i	$0.297136\pi$
$240$	0	0
$241$	21.7977	1.40411	0.702056	−	0.712122i	$-0.252266\pi$
$241$	21.7977	1.40411	0.702056	+	0.712122i	$0.252266\pi$
$242$	0	0
$243$	−22.0486	−1.41442
$244$	0	0
$245$	−7.72119	−0.493289
$246$	0	0
$247$	−28.4000	−1.80705
$248$	0	0
$249$	−16.9624	−1.07495
$250$	0	0
$251$	22.0114	1.38935	0.694674	−	0.719325i	$-0.255549\pi$
$251$	22.0114	1.38935	0.694674	+	0.719325i	$0.255549\pi$
$252$	0	0
$253$	12.7651	0.802533
$254$	0	0
$255$	−25.3640	−1.58836
$256$	0	0
$257$	9.18964	0.573234	0.286617	−	0.958045i	$-0.407469\pi$
$257$	9.18964	0.573234	0.286617	+	0.958045i	$0.407469\pi$
$258$	0	0
$259$	−4.37759	−0.272010
$260$	0	0
$261$	−10.3781	−0.642391
$262$	0	0
$263$	15.6117	0.962656	0.481328	−	0.876540i	$-0.340155\pi$
$263$	15.6117	0.962656	0.481328	+	0.876540i	$0.340155\pi$
$264$	0	0
$265$	4.07041	0.250043
$266$	0	0
$267$	−11.8980	−0.728148
$268$	0	0
$269$	−17.8631	−1.08913	−0.544566	−	0.838718i	$-0.683306\pi$
$269$	−17.8631	−1.08913	−0.544566	+	0.838718i	$0.683306\pi$
$270$	0	0
$271$	16.5337	1.00435	0.502174	−	0.864767i	$-0.332534\pi$
$271$	16.5337	1.00435	0.502174	+	0.864767i	$0.332534\pi$
$272$	0	0
$273$	−9.97901	−0.603957
$274$	0	0
$275$	13.0150	0.784835
$276$	0	0
$277$	28.3672	1.70442	0.852210	−	0.523199i	$-0.175262\pi$
$277$	28.3672	1.70442	0.852210	+	0.523199i	$0.175262\pi$
$278$	0	0
$279$	6.35126	0.380240
$280$	0	0
$281$	1.04933	0.0625977	0.0312988	−	0.999510i	$-0.490036\pi$
$281$	1.04933	0.0625977	0.0312988	+	0.999510i	$0.490036\pi$
$282$	0	0
$283$	13.8367	0.822505	0.411253	−	0.911521i	$-0.365091\pi$
$283$	13.8367	0.822505	0.411253	+	0.911521i	$0.365091\pi$
$284$	0	0
$285$	21.3514	1.26475
$286$	0	0
$287$	5.27415	0.311323
$288$	0	0
$289$	44.8365	2.63744
$290$	0	0
$291$	23.6950	1.38903
$292$	0	0
$293$	8.25156	0.482061	0.241031	−	0.970517i	$-0.422515\pi$
$293$	8.25156	0.482061	0.241031	+	0.970517i	$0.422515\pi$
$294$	0	0
$295$	7.75615	0.451581
$296$	0	0
$297$	−6.73520	−0.390816
$298$	0	0
$299$	14.4949	0.838264
$300$	0	0
$301$	−6.56965	−0.378669
$302$	0	0
$303$	−28.6709	−1.64710
$304$	0	0
$305$	15.6163	0.894189
$306$	0	0
$307$	5.46378	0.311835	0.155917	−	0.987770i	$-0.450167\pi$
$307$	5.46378	0.311835	0.155917	+	0.987770i	$0.450167\pi$
$308$	0	0
$309$	11.3526	0.645829
$310$	0	0
$311$	18.4424	1.04577	0.522887	−	0.852402i	$-0.324855\pi$
$311$	18.4424	1.04577	0.522887	+	0.852402i	$0.324855\pi$
$312$	0	0
$313$	−21.9260	−1.23933	−0.619664	−	0.784867i	$-0.712731\pi$
$313$	−21.9260	−1.23933	−0.619664	+	0.784867i	$0.712731\pi$
$314$	0	0
$315$	4.13767	0.233131
$316$	0	0
$317$	−7.73837	−0.434630	−0.217315	−	0.976102i	$-0.569730\pi$
$317$	−7.73837	−0.434630	−0.217315	+	0.976102i	$0.569730\pi$
$318$	0	0
$319$	10.6286	0.595088
$320$	0	0
$321$	49.3005	2.75169
$322$	0	0
$323$	−52.0540	−2.89636
$324$	0	0
$325$	14.7788	0.819779
$326$	0	0
$327$	11.7500	0.649775
$328$	0	0
$329$	−1.87345	−0.103286
$330$	0	0
$331$	21.2914	1.17028	0.585140	−	0.810932i	$-0.301040\pi$
$331$	21.2914	1.17028	0.585140	+	0.810932i	$0.301040\pi$
$332$	0	0
$333$	−17.9581	−0.984096
$334$	0	0
$335$	12.8518	0.702169
$336$	0	0
$337$	21.3815	1.16472	0.582362	−	0.812930i	$-0.302129\pi$
$337$	21.3815	1.16472	0.582362	+	0.812930i	$0.302129\pi$
$338$	0	0
$339$	30.2254	1.64162
$340$	0	0
$341$	−6.50455	−0.352241
$342$	0	0
$343$	−11.8631	−0.640546
$344$	0	0
$345$	−10.8975	−0.586699
$346$	0	0
$347$	7.25286	0.389354	0.194677	−	0.980867i	$-0.437634\pi$
$347$	7.25286	0.389354	0.194677	+	0.980867i	$0.437634\pi$
$348$	0	0
$349$	24.7806	1.32647	0.663236	−	0.748410i	$-0.269182\pi$
$349$	24.7806	1.32647	0.663236	+	0.748410i	$0.269182\pi$
$350$	0	0
$351$	−7.64793	−0.408217
$352$	0	0
$353$	−24.3581	−1.29645	−0.648227	−	0.761447i	$-0.724489\pi$
$353$	−24.3581	−1.29645	−0.648227	+	0.761447i	$0.724489\pi$
$354$	0	0
$355$	−20.4143	−1.08348
$356$	0	0
$357$	−18.2904	−0.968030
$358$	0	0
$359$	−24.9267	−1.31558	−0.657790	−	0.753201i	$-0.728509\pi$
$359$	−24.9267	−1.31558	−0.657790	+	0.753201i	$0.728509\pi$
$360$	0	0
$361$	24.8191	1.30627
$362$	0	0
$363$	8.47122	0.444624
$364$	0	0
$365$	−19.3482	−1.01273
$366$	0	0
$367$	2.52376	0.131739	0.0658695	−	0.997828i	$-0.479018\pi$
$367$	2.52376	0.131739	0.0658695	+	0.997828i	$0.479018\pi$
$368$	0	0
$369$	21.6360	1.12633
$370$	0	0
$371$	2.93524	0.152390
$372$	0	0
$373$	27.2959	1.41333	0.706664	−	0.707549i	$-0.250199\pi$
$373$	27.2959	1.41333	0.706664	+	0.707549i	$0.250199\pi$
$374$	0	0
$375$	−27.2383	−1.40658
$376$	0	0
$377$	12.0690	0.621584
$378$	0	0
$379$	8.84527	0.454351	0.227175	−	0.973854i	$-0.427051\pi$
$379$	8.84527	0.454351	0.227175	+	0.973854i	$0.427051\pi$
$380$	0	0
$381$	35.4535	1.81634
$382$	0	0
$383$	9.56433	0.488715	0.244357	−	0.969685i	$-0.421423\pi$
$383$	9.56433	0.488715	0.244357	+	0.969685i	$0.421423\pi$
$384$	0	0
$385$	−4.23753	−0.215964
$386$	0	0
$387$	−26.9505	−1.36997
$388$	0	0
$389$	−0.0942924	−0.00478081	−0.00239041	−	0.999997i	$-0.500761\pi$
$389$	−0.0942924	−0.00478081	−0.00239041	+	0.999997i	$0.500761\pi$
$390$	0	0
$391$	26.5676	1.34358
$392$	0	0
$393$	−39.9371	−2.01456
$394$	0	0
$395$	−6.42847	−0.323451
$396$	0	0
$397$	−29.4428	−1.47769	−0.738847	−	0.673873i	$-0.764629\pi$
$397$	−29.4428	−1.47769	−0.738847	+	0.673873i	$0.764629\pi$
$398$	0	0
$399$	15.3969	0.770807
$400$	0	0
$401$	−28.1453	−1.40551	−0.702756	−	0.711431i	$-0.748047\pi$
$401$	−28.1453	−1.40551	−0.702756	+	0.711431i	$0.748047\pi$
$402$	0	0
$403$	−7.38602	−0.367924
$404$	0	0
$405$	−8.05293	−0.400153
$406$	0	0
$407$	18.3915	0.911631
$408$	0	0
$409$	10.4165	0.515062	0.257531	−	0.966270i	$-0.417091\pi$
$409$	10.4165	0.515062	0.257531	+	0.966270i	$0.417091\pi$
$410$	0	0
$411$	1.64078	0.0809336
$412$	0	0
$413$	5.59308	0.275218
$414$	0	0
$415$	−8.17913	−0.401498
$416$	0	0
$417$	−23.0202	−1.12730
$418$	0	0
$419$	10.1329	0.495026	0.247513	−	0.968885i	$-0.420387\pi$
$419$	10.1329	0.495026	0.247513	+	0.968885i	$0.420387\pi$
$420$	0	0
$421$	−34.9879	−1.70521	−0.852603	−	0.522559i	$-0.824977\pi$
$421$	−34.9879	−1.70521	−0.852603	+	0.522559i	$0.824977\pi$
$422$	0	0
$423$	−7.68539	−0.373676
$424$	0	0
$425$	27.0878	1.31395
$426$	0	0
$427$	11.2612	0.544967
$428$	0	0
$429$	41.9246	2.02414
$430$	0	0
$431$	−5.20902	−0.250910	−0.125455	−	0.992099i	$-0.540039\pi$
$431$	−5.20902	−0.250910	−0.125455	+	0.992099i	$0.540039\pi$
$432$	0	0
$433$	15.2360	0.732196	0.366098	−	0.930576i	$-0.380693\pi$
$433$	15.2360	0.732196	0.366098	+	0.930576i	$0.380693\pi$
$434$	0	0
$435$	−9.07358	−0.435045
$436$	0	0
$437$	−22.3646	−1.06984
$438$	0	0
$439$	−1.92960	−0.0920947	−0.0460473	−	0.998939i	$-0.514663\pi$
$439$	−1.92960	−0.0920947	−0.0460473	+	0.998939i	$0.514663\pi$
$440$	0	0
$441$	−22.8409	−1.08766
$442$	0	0
$443$	−12.2961	−0.584204	−0.292102	−	0.956387i	$-0.594355\pi$
$443$	−12.2961	−0.584204	−0.292102	+	0.956387i	$0.594355\pi$
$444$	0	0
$445$	−5.73712	−0.271966
$446$	0	0
$447$	2.23461	0.105693
$448$	0	0
$449$	26.8719	1.26816	0.634082	−	0.773266i	$-0.281378\pi$
$449$	26.8719	1.26816	0.634082	+	0.773266i	$0.281378\pi$
$450$	0	0
$451$	−22.1582	−1.04339
$452$	0	0
$453$	−25.4167	−1.19418
$454$	0	0
$455$	−4.81179	−0.225580
$456$	0	0
$457$	−0.906919	−0.0424239	−0.0212119	−	0.999775i	$-0.506752\pi$
$457$	−0.906919	−0.0424239	−0.0212119	+	0.999775i	$0.506752\pi$
$458$	0	0
$459$	−14.0178	−0.654295
$460$	0	0
$461$	4.46280	0.207853	0.103927	−	0.994585i	$-0.466859\pi$
$461$	4.46280	0.207853	0.103927	+	0.994585i	$0.466859\pi$
$462$	0	0
$463$	−23.4974	−1.09202	−0.546009	−	0.837779i	$-0.683854\pi$
$463$	−23.4974	−1.09202	−0.546009	+	0.837779i	$0.683854\pi$
$464$	0	0
$465$	5.55289	0.257509
$466$	0	0
$467$	−8.31369	−0.384712	−0.192356	−	0.981325i	$-0.561613\pi$
$467$	−8.31369	−0.384712	−0.192356	+	0.981325i	$0.561613\pi$
$468$	0	0
$469$	9.26764	0.427940
$470$	0	0
$471$	−29.5764	−1.36281
$472$	0	0
$473$	27.6009	1.26909
$474$	0	0
$475$	−22.8025	−1.04625
$476$	0	0
$477$	12.0411	0.551326
$478$	0	0
$479$	−21.3313	−0.974651	−0.487325	−	0.873220i	$-0.662027\pi$
$479$	−21.3313	−0.974651	−0.487325	+	0.873220i	$0.662027\pi$
$480$	0	0
$481$	20.8838	0.952220
$482$	0	0
$483$	−7.85832	−0.357566
$484$	0	0
$485$	11.4255	0.518806
$486$	0	0
$487$	−15.0513	−0.682038	−0.341019	−	0.940056i	$-0.610772\pi$
$487$	−15.0513	−0.682038	−0.341019	+	0.940056i	$0.610772\pi$
$488$	0	0
$489$	49.4549	2.23643
$490$	0	0
$491$	43.1675	1.94812	0.974060	−	0.226289i	$-0.0726595\pi$
$491$	43.1675	1.94812	0.974060	+	0.226289i	$0.0726595\pi$
$492$	0	0
$493$	22.1211	0.996282
$494$	0	0
$495$	−17.3835	−0.781330
$496$	0	0
$497$	−14.7211	−0.660331
$498$	0	0
$499$	−16.8824	−0.755759	−0.377880	−	0.925855i	$-0.623347\pi$
$499$	−16.8824	−0.755759	−0.377880	+	0.925855i	$0.623347\pi$
$500$	0	0
$501$	−34.2882	−1.53188
$502$	0	0
$503$	13.4026	0.597592	0.298796	−	0.954317i	$-0.403415\pi$
$503$	13.4026	0.597592	0.298796	+	0.954317i	$0.403415\pi$
$504$	0	0
$505$	−13.8248	−0.615197
$506$	0	0
$507$	13.9834	0.621026
$508$	0	0
$509$	−17.5986	−0.780045	−0.390022	−	0.920805i	$-0.627533\pi$
$509$	−17.5986	−0.780045	−0.390022	+	0.920805i	$0.627533\pi$
$510$	0	0
$511$	−13.9523	−0.617212
$512$	0	0
$513$	11.8002	0.520991
$514$	0	0
$515$	5.47413	0.241219
$516$	0	0
$517$	7.87086	0.346160
$518$	0	0
$519$	−3.22430	−0.141531
$520$	0	0
$521$	−9.08215	−0.397896	−0.198948	−	0.980010i	$-0.563753\pi$
$521$	−9.08215	−0.397896	−0.198948	+	0.980010i	$0.563753\pi$
$522$	0	0
$523$	−12.7052	−0.555561	−0.277780	−	0.960645i	$-0.589599\pi$
$523$	−12.7052	−0.555561	−0.277780	+	0.960645i	$0.589599\pi$
$524$	0	0
$525$	−8.01220	−0.349681
$526$	0	0
$527$	−13.5377	−0.589713
$528$	0	0
$529$	−11.5854	−0.503715
$530$	0	0
$531$	22.9443	0.995700
$532$	0	0
$533$	−25.1610	−1.08984
$534$	0	0
$535$	23.7723	1.02776
$536$	0	0
$537$	−1.54135	−0.0665141
$538$	0	0
$539$	23.3922	1.00757
$540$	0	0
$541$	37.0432	1.59261	0.796306	−	0.604894i	$-0.206785\pi$
$541$	37.0432	1.59261	0.796306	+	0.604894i	$0.206785\pi$
$542$	0	0
$543$	−24.5351	−1.05290
$544$	0	0
$545$	5.66573	0.242693
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	46.1965	1.97162
$550$	0	0
$551$	−18.6215	−0.793303
$552$	0	0
$553$	−4.63567	−0.197129
$554$	0	0
$555$	−15.7007	−0.666456
$556$	0	0
$557$	−46.3948	−1.96581	−0.982905	−	0.184111i	$-0.941059\pi$
$557$	−46.3948	−1.96581	−0.982905	+	0.184111i	$0.941059\pi$
$558$	0	0
$559$	31.3413	1.32560
$560$	0	0
$561$	76.8430	3.24431
$562$	0	0
$563$	25.5383	1.07631	0.538155	−	0.842846i	$-0.319122\pi$
$563$	25.5383	1.07631	0.538155	+	0.842846i	$0.319122\pi$
$564$	0	0
$565$	14.5744	0.613149
$566$	0	0
$567$	−5.80710	−0.243875
$568$	0	0
$569$	34.7766	1.45791	0.728956	−	0.684560i	$-0.240006\pi$
$569$	34.7766	1.45791	0.728956	+	0.684560i	$0.240006\pi$
$570$	0	0
$571$	18.0804	0.756643	0.378322	−	0.925674i	$-0.376501\pi$
$571$	18.0804	0.756643	0.378322	+	0.925674i	$0.376501\pi$
$572$	0	0
$573$	−18.7147	−0.781816
$574$	0	0
$575$	11.6381	0.485341
$576$	0	0
$577$	17.7760	0.740025	0.370012	−	0.929027i	$-0.379353\pi$
$577$	17.7760	0.740025	0.370012	+	0.929027i	$0.379353\pi$
$578$	0	0
$579$	11.3010	0.469652
$580$	0	0
$581$	−5.89810	−0.244694
$582$	0	0
$583$	−12.3317	−0.510729
$584$	0	0
$585$	−19.7393	−0.816118
$586$	0	0
$587$	12.4600	0.514279	0.257139	−	0.966374i	$-0.417220\pi$
$587$	12.4600	0.514279	0.257139	+	0.966374i	$0.417220\pi$
$588$	0	0
$589$	11.3961	0.469567
$590$	0	0
$591$	−50.5017	−2.07736
$592$	0	0
$593$	−5.62287	−0.230904	−0.115452	−	0.993313i	$-0.536832\pi$
$593$	−5.62287	−0.230904	−0.115452	+	0.993313i	$0.536832\pi$
$594$	0	0
$595$	−8.81946	−0.361563
$596$	0	0
$597$	−0.167420	−0.00685203
$598$	0	0
$599$	−19.3513	−0.790673	−0.395337	−	0.918536i	$-0.629372\pi$
$599$	−19.3513	−0.790673	−0.395337	+	0.918536i	$0.629372\pi$
$600$	0	0
$601$	−25.3645	−1.03464	−0.517321	−	0.855792i	$-0.673071\pi$
$601$	−25.3645	−1.03464	−0.517321	+	0.855792i	$0.673071\pi$
$602$	0	0
$603$	38.0184	1.54823
$604$	0	0
$605$	4.08474	0.166068
$606$	0	0
$607$	−13.9505	−0.566235	−0.283117	−	0.959085i	$-0.591369\pi$
$607$	−13.9505	−0.566235	−0.283117	+	0.959085i	$0.591369\pi$
$608$	0	0
$609$	−6.54310	−0.265140
$610$	0	0
$611$	8.93750	0.361573
$612$	0	0
$613$	25.5058	1.03017	0.515084	−	0.857139i	$-0.327761\pi$
$613$	25.5058	1.03017	0.515084	+	0.857139i	$0.327761\pi$
$614$	0	0
$615$	18.9163	0.762779
$616$	0	0
$617$	7.43879	0.299474	0.149737	−	0.988726i	$-0.452157\pi$
$617$	7.43879	0.299474	0.149737	+	0.988726i	$0.452157\pi$
$618$	0	0
$619$	−10.5397	−0.423626	−0.211813	−	0.977310i	$-0.567937\pi$
$619$	−10.5397	−0.423626	−0.211813	+	0.977310i	$0.567937\pi$
$620$	0	0
$621$	−6.02264	−0.241680
$622$	0	0
$623$	−4.13713	−0.165751
$624$	0	0
$625$	4.08945	0.163578
$626$	0	0
$627$	−64.6864	−2.58333
$628$	0	0
$629$	38.2777	1.52623
$630$	0	0
$631$	−23.4904	−0.935139	−0.467570	−	0.883956i	$-0.654870\pi$
$631$	−23.4904	−0.935139	−0.467570	+	0.883956i	$0.654870\pi$
$632$	0	0
$633$	−24.9699	−0.992466
$634$	0	0
$635$	17.0953	0.678408
$636$	0	0
$637$	26.5622	1.05243
$638$	0	0
$639$	−60.3899	−2.38899
$640$	0	0
$641$	−12.1894	−0.481454	−0.240727	−	0.970593i	$-0.577386\pi$
$641$	−12.1894	−0.481454	−0.240727	+	0.970593i	$0.577386\pi$
$642$	0	0
$643$	45.5594	1.79669	0.898343	−	0.439295i	$-0.144772\pi$
$643$	45.5594	1.79669	0.898343	+	0.439295i	$0.144772\pi$
$644$	0	0
$645$	−23.5627	−0.927782
$646$	0	0
$647$	13.8400	0.544107	0.272054	−	0.962282i	$-0.412297\pi$
$647$	13.8400	0.544107	0.272054	+	0.962282i	$0.412297\pi$
$648$	0	0
$649$	−23.4981	−0.922380
$650$	0	0
$651$	4.00428	0.156940
$652$	0	0
$653$	−37.0134	−1.44845	−0.724224	−	0.689565i	$-0.757802\pi$
$653$	−37.0134	−1.44845	−0.724224	+	0.689565i	$0.757802\pi$
$654$	0	0
$655$	−19.2573	−0.752445
$656$	0	0
$657$	−57.2360	−2.23299
$658$	0	0
$659$	−4.58672	−0.178673	−0.0893366	−	0.996001i	$-0.528475\pi$
$659$	−4.58672	−0.178673	−0.0893366	+	0.996001i	$0.528475\pi$
$660$	0	0
$661$	33.9504	1.32052	0.660259	−	0.751038i	$-0.270446\pi$
$661$	33.9504	1.32052	0.660259	+	0.751038i	$0.270446\pi$
$662$	0	0
$663$	87.2565	3.38876
$664$	0	0
$665$	7.42422	0.287899
$666$	0	0
$667$	9.50414	0.368002
$668$	0	0
$669$	−33.0775	−1.27885
$670$	0	0
$671$	−47.3114	−1.82644
$672$	0	0
$673$	−19.4772	−0.750792	−0.375396	−	0.926865i	$-0.622493\pi$
$673$	−19.4772	−0.750792	−0.375396	+	0.926865i	$0.622493\pi$
$674$	0	0
$675$	−6.14057	−0.236351
$676$	0	0
$677$	4.07259	0.156523	0.0782613	−	0.996933i	$-0.475063\pi$
$677$	4.07259	0.156523	0.0782613	+	0.996933i	$0.475063\pi$
$678$	0	0
$679$	8.23913	0.316189
$680$	0	0
$681$	−12.7979	−0.490416
$682$	0	0
$683$	0.965668	0.0369503	0.0184751	−	0.999829i	$-0.494119\pi$
$683$	0.965668	0.0369503	0.0184751	+	0.999829i	$0.494119\pi$
$684$	0	0
$685$	0.791168	0.0302290
$686$	0	0
$687$	−69.2360	−2.64152
$688$	0	0
$689$	−14.0029	−0.533468
$690$	0	0
$691$	9.03055	0.343538	0.171769	−	0.985137i	$-0.445052\pi$
$691$	9.03055	0.343538	0.171769	+	0.985137i	$0.445052\pi$
$692$	0	0
$693$	−12.5355	−0.476185
$694$	0	0
$695$	−11.1001	−0.421051
$696$	0	0
$697$	−46.1172	−1.74681
$698$	0	0
$699$	−57.5889	−2.17821
$700$	0	0
$701$	9.28360	0.350637	0.175318	−	0.984512i	$-0.443905\pi$
$701$	9.28360	0.350637	0.175318	+	0.984512i	$0.443905\pi$
$702$	0	0
$703$	−32.2221	−1.21528
$704$	0	0
$705$	−6.71931	−0.253064
$706$	0	0
$707$	−9.96930	−0.374934
$708$	0	0
$709$	−42.7085	−1.60395	−0.801976	−	0.597357i	$-0.796218\pi$
$709$	−42.7085	−1.60395	−0.801976	+	0.597357i	$0.796218\pi$
$710$	0	0
$711$	−19.0168	−0.713185
$712$	0	0
$713$	−5.81639	−0.217825
$714$	0	0
$715$	20.2156	0.756022
$716$	0	0
$717$	47.5844	1.77707
$718$	0	0
$719$	−3.48301	−0.129895	−0.0649473	−	0.997889i	$-0.520688\pi$
$719$	−3.48301	−0.129895	−0.0649473	+	0.997889i	$0.520688\pi$
$720$	0	0
$721$	3.94748	0.147012
$722$	0	0
$723$	56.3766	2.09667
$724$	0	0
$725$	9.69024	0.359887
$726$	0	0
$727$	43.5536	1.61531	0.807657	−	0.589652i	$-0.200735\pi$
$727$	43.5536	1.61531	0.807657	+	0.589652i	$0.200735\pi$
$728$	0	0
$729$	−37.6537	−1.39458
$730$	0	0
$731$	57.4451	2.12468
$732$	0	0
$733$	35.9769	1.32884	0.664418	−	0.747361i	$-0.268679\pi$
$733$	35.9769	1.32884	0.664418	+	0.747361i	$0.268679\pi$
$734$	0	0
$735$	−19.9697	−0.736595
$736$	0	0
$737$	−38.9359	−1.43422
$738$	0	0
$739$	9.01253	0.331531	0.165766	−	0.986165i	$-0.446990\pi$
$739$	9.01253	0.331531	0.165766	+	0.986165i	$0.446990\pi$
$740$	0	0
$741$	−73.4525	−2.69835
$742$	0	0
$743$	27.7846	1.01932	0.509658	−	0.860377i	$-0.329772\pi$
$743$	27.7846	1.01932	0.509658	+	0.860377i	$0.329772\pi$
$744$	0	0
$745$	1.07751	0.0394769
$746$	0	0
$747$	−24.1956	−0.885271
$748$	0	0
$749$	17.1425	0.626375
$750$	0	0
$751$	35.3238	1.28898	0.644492	−	0.764611i	$-0.277069\pi$
$751$	35.3238	1.28898	0.644492	+	0.764611i	$0.277069\pi$
$752$	0	0
$753$	56.9293	2.07462
$754$	0	0
$755$	−12.2557	−0.446030
$756$	0	0
$757$	−7.21058	−0.262073	−0.131036	−	0.991378i	$-0.541830\pi$
$757$	−7.21058	−0.262073	−0.131036	+	0.991378i	$0.541830\pi$
$758$	0	0
$759$	33.0150	1.19837
$760$	0	0
$761$	15.5783	0.564712	0.282356	−	0.959310i	$-0.408884\pi$
$761$	15.5783	0.564712	0.282356	+	0.959310i	$0.408884\pi$
$762$	0	0
$763$	4.08565	0.147910
$764$	0	0
$765$	−36.1798	−1.30808
$766$	0	0
$767$	−26.6825	−0.963448
$768$	0	0
$769$	24.1433	0.870629	0.435314	−	0.900279i	$-0.356637\pi$
$769$	24.1433	0.870629	0.435314	+	0.900279i	$0.356637\pi$
$770$	0	0
$771$	23.7677	0.855972
$772$	0	0
$773$	−23.9888	−0.862818	−0.431409	−	0.902156i	$-0.641983\pi$
$773$	−23.9888	−0.862818	−0.431409	+	0.902156i	$0.641983\pi$
$774$	0	0
$775$	−5.93028	−0.213022
$776$	0	0
$777$	−11.3220	−0.406175
$778$	0	0
$779$	38.8215	1.39092
$780$	0	0
$781$	61.8474	2.21307
$782$	0	0
$783$	−5.01465	−0.179209
$784$	0	0
$785$	−14.2615	−0.509013
$786$	0	0
$787$	5.35961	0.191050	0.0955248	−	0.995427i	$-0.469547\pi$
$787$	5.35961	0.191050	0.0955248	+	0.995427i	$0.469547\pi$
$788$	0	0
$789$	40.3773	1.43747
$790$	0	0
$791$	10.5098	0.373686
$792$	0	0
$793$	−53.7229	−1.90776
$794$	0	0
$795$	10.5275	0.373373
$796$	0	0
$797$	−19.3882	−0.686766	−0.343383	−	0.939195i	$-0.611573\pi$
$797$	−19.3882	−0.686766	−0.343383	+	0.939195i	$0.611573\pi$
$798$	0	0
$799$	16.3814	0.579533
$800$	0	0
$801$	−16.9716	−0.599663
$802$	0	0
$803$	58.6173	2.06856
$804$	0	0
$805$	−3.78921	−0.133552
$806$	0	0
$807$	−46.2003	−1.62633
$808$	0	0
$809$	−35.6076	−1.25190	−0.625948	−	0.779865i	$-0.715288\pi$
$809$	−35.6076	−1.25190	−0.625948	+	0.779865i	$0.715288\pi$
$810$	0	0
$811$	−7.02922	−0.246829	−0.123415	−	0.992355i	$-0.539385\pi$
$811$	−7.02922	−0.246829	−0.123415	+	0.992355i	$0.539385\pi$
$812$	0	0
$813$	42.7619	1.49973
$814$	0	0
$815$	23.8467	0.835314
$816$	0	0
$817$	−48.3573	−1.69181
$818$	0	0
$819$	−14.2343	−0.497386
$820$	0	0
$821$	−11.9108	−0.415691	−0.207845	−	0.978162i	$-0.566645\pi$
$821$	−11.9108	−0.415691	−0.207845	+	0.978162i	$0.566645\pi$
$822$	0	0
$823$	−38.1506	−1.32985	−0.664924	−	0.746911i	$-0.731536\pi$
$823$	−38.1506	−1.32985	−0.664924	+	0.746911i	$0.731536\pi$
$824$	0	0
$825$	33.6615	1.17194
$826$	0	0
$827$	−22.1837	−0.771404	−0.385702	−	0.922623i	$-0.626041\pi$
$827$	−22.1837	−0.771404	−0.385702	+	0.922623i	$0.626041\pi$
$828$	0	0
$829$	−26.7091	−0.927646	−0.463823	−	0.885928i	$-0.653523\pi$
$829$	−26.7091	−0.927646	−0.463823	+	0.885928i	$0.653523\pi$
$830$	0	0
$831$	73.3677	2.54510
$832$	0	0
$833$	48.6855	1.68685
$834$	0	0
$835$	−16.5335	−0.572164
$836$	0	0
$837$	3.06889	0.106076
$838$	0	0
$839$	24.3713	0.841392	0.420696	−	0.907202i	$-0.361786\pi$
$839$	24.3713	0.841392	0.420696	+	0.907202i	$0.361786\pi$
$840$	0	0
$841$	−21.0865	−0.727122
$842$	0	0
$843$	2.71394	0.0934730
$844$	0	0
$845$	6.74268	0.231955
$846$	0	0
$847$	2.94557	0.101211
$848$	0	0
$849$	35.7866	1.22819
$850$	0	0
$851$	16.4457	0.563751
$852$	0	0
$853$	−21.5357	−0.737369	−0.368685	−	0.929555i	$-0.620192\pi$
$853$	−21.5357	−0.737369	−0.368685	+	0.929555i	$0.620192\pi$
$854$	0	0
$855$	30.4562	1.04158
$856$	0	0
$857$	10.1092	0.345325	0.172663	−	0.984981i	$-0.444763\pi$
$857$	10.1092	0.345325	0.172663	+	0.984981i	$0.444763\pi$
$858$	0	0
$859$	−29.3539	−1.00154	−0.500771	−	0.865580i	$-0.666950\pi$
$859$	−29.3539	−1.00154	−0.500771	+	0.865580i	$0.666950\pi$
$860$	0	0
$861$	13.6408	0.464878
$862$	0	0
$863$	−9.84744	−0.335211	−0.167605	−	0.985854i	$-0.553603\pi$
$863$	−9.84744	−0.335211	−0.167605	+	0.985854i	$0.553603\pi$
$864$	0	0
$865$	−1.55473	−0.0528622
$866$	0	0
$867$	115.963	3.93832
$868$	0	0
$869$	19.4757	0.660669
$870$	0	0
$871$	−44.2124	−1.49808
$872$	0	0
$873$	33.7991	1.14393
$874$	0	0
$875$	−9.47117	−0.320184
$876$	0	0
$877$	−32.9822	−1.11373	−0.556864	−	0.830603i	$-0.687996\pi$
$877$	−32.9822	−1.11373	−0.556864	+	0.830603i	$0.687996\pi$
$878$	0	0
$879$	21.3415	0.719830
$880$	0	0
$881$	34.3375	1.15686	0.578429	−	0.815733i	$-0.303666\pi$
$881$	34.3375	1.15686	0.578429	+	0.815733i	$0.303666\pi$
$882$	0	0
$883$	−12.5688	−0.422974	−0.211487	−	0.977381i	$-0.567831\pi$
$883$	−12.5688	−0.422974	−0.211487	+	0.977381i	$0.567831\pi$
$884$	0	0
$885$	20.0602	0.674315
$886$	0	0
$887$	−34.7329	−1.16622	−0.583109	−	0.812394i	$-0.698164\pi$
$887$	−34.7329	−1.16622	−0.583109	+	0.812394i	$0.698164\pi$
$888$	0	0
$889$	12.3277	0.413458
$890$	0	0
$891$	24.3972	0.817337
$892$	0	0
$893$	−13.7899	−0.461461
$894$	0	0
$895$	−0.743224	−0.0248432
$896$	0	0
$897$	37.4891	1.25172
$898$	0	0
$899$	−4.84292	−0.161520
$900$	0	0
$901$	−25.6657	−0.855050
$902$	0	0
$903$	−16.9915	−0.565441
$904$	0	0
$905$	−11.8306	−0.393262
$906$	0	0
$907$	−18.5744	−0.616753	−0.308376	−	0.951264i	$-0.599786\pi$
$907$	−18.5744	−0.616753	−0.308376	+	0.951264i	$0.599786\pi$
$908$	0	0
$909$	−40.8968	−1.35646
$910$	0	0
$911$	4.83712	0.160261	0.0801304	−	0.996784i	$-0.474466\pi$
$911$	4.83712	0.160261	0.0801304	+	0.996784i	$0.474466\pi$
$912$	0	0
$913$	24.7795	0.820083
$914$	0	0
$915$	40.3894	1.33523
$916$	0	0
$917$	−13.8867	−0.458580
$918$	0	0
$919$	−57.3488	−1.89176	−0.945881	−	0.324513i	$-0.894800\pi$
$919$	−57.3488	−1.89176	−0.945881	+	0.324513i	$0.894800\pi$
$920$	0	0
$921$	14.1313	0.465642
$922$	0	0
$923$	70.2287	2.31161
$924$	0	0
$925$	16.7677	0.551319
$926$	0	0
$927$	16.1937	0.531869
$928$	0	0
$929$	46.6546	1.53069	0.765344	−	0.643622i	$-0.222569\pi$
$929$	46.6546	1.53069	0.765344	+	0.643622i	$0.222569\pi$
$930$	0	0
$931$	−40.9835	−1.34318
$932$	0	0
$933$	47.6987	1.56159
$934$	0	0
$935$	37.0530	1.21176
$936$	0	0
$937$	20.5835	0.672433	0.336217	−	0.941785i	$-0.390853\pi$
$937$	20.5835	0.672433	0.336217	+	0.941785i	$0.390853\pi$
$938$	0	0
$939$	−56.7083	−1.85061
$940$	0	0
$941$	26.4195	0.861251	0.430625	−	0.902531i	$-0.358293\pi$
$941$	26.4195	0.861251	0.430625	+	0.902531i	$0.358293\pi$
$942$	0	0
$943$	−19.8139	−0.645230
$944$	0	0
$945$	1.99929	0.0650370
$946$	0	0
$947$	−30.9279	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$947$	−30.9279	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$948$	0	0
$949$	66.5610	2.16066
$950$	0	0
$951$	−20.0142	−0.649004
$952$	0	0
$953$	31.9144	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$953$	31.9144	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$954$	0	0
$955$	−9.02404	−0.292011
$956$	0	0
$957$	27.4894	0.888606
$958$	0	0
$959$	0.570523	0.0184232
$960$	0	0
$961$	−28.0362	−0.904394
$962$	0	0
$963$	70.3234	2.26614
$964$	0	0
$965$	5.44922	0.175416
$966$	0	0
$967$	−25.3550	−0.815363	−0.407682	−	0.913124i	$-0.633663\pi$
$967$	−25.3550	−0.815363	−0.407682	+	0.913124i	$0.633663\pi$
$968$	0	0
$969$	−134.630	−4.32495
$970$	0	0
$971$	−4.38779	−0.140811	−0.0704054	−	0.997518i	$-0.522429\pi$
$971$	−4.38779	−0.140811	−0.0704054	+	0.997518i	$0.522429\pi$
$972$	0	0
$973$	−8.00447	−0.256611
$974$	0	0
$975$	38.2232	1.22412
$976$	0	0
$977$	3.45588	0.110563	0.0552817	−	0.998471i	$-0.482394\pi$
$977$	3.45588	0.110563	0.0552817	+	0.998471i	$0.482394\pi$
$978$	0	0
$979$	17.3812	0.555506
$980$	0	0
$981$	16.7604	0.535120
$982$	0	0
$983$	−56.4424	−1.80023	−0.900117	−	0.435648i	$-0.856519\pi$
$983$	−56.4424	−1.80023	−0.900117	+	0.435648i	$0.856519\pi$
$984$	0	0
$985$	−24.3515	−0.775902
$986$	0	0
$987$	−4.84540	−0.154231
$988$	0	0
$989$	24.6808	0.784805
$990$	0	0
$991$	55.5706	1.76526	0.882629	−	0.470069i	$-0.155771\pi$
$991$	55.5706	1.76526	0.882629	+	0.470069i	$0.155771\pi$
$992$	0	0
$993$	55.0671	1.74750
$994$	0	0
$995$	−0.0807282	−0.00255926
$996$	0	0
$997$	−20.1408	−0.637866	−0.318933	−	0.947777i	$-0.603324\pi$
$997$	−20.1408	−0.637866	−0.318933	+	0.947777i	$0.603324\pi$
$998$	0	0
$999$	−8.67721	−0.274535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.14		18
4.3	odd	2		547.2.a.b.1.15	✓	18
12.11	even	2		4923.2.a.l.1.4		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.15	✓	18	4.3	odd	2
4923.2.a.l.1.4		18	12.11	even	2
8752.2.a.s.1.14		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)

\(f(q)\)	\(=\)	\(q+2.58636 q^{3} +1.24712 q^{5} +0.899316 q^{7} +3.68924 q^{9} +O(q^{10})\)	\(q+2.58636 q^{3} +1.24712 q^{5} +0.899316 q^{7} +3.68924 q^{9} -3.77827 q^{11} -4.29029 q^{13} +3.22549 q^{15} -7.86362 q^{17} +6.61960 q^{19} +2.32595 q^{21} -3.37854 q^{23} -3.44470 q^{25} +1.78261 q^{27} -2.81309 q^{29} +1.72157 q^{31} -9.77196 q^{33} +1.12155 q^{35} -4.86769 q^{37} -11.0962 q^{39} +5.86463 q^{41} -7.30517 q^{43} +4.60091 q^{45} -2.08319 q^{47} -6.19123 q^{49} -20.3381 q^{51} +3.26386 q^{53} -4.71195 q^{55} +17.1206 q^{57} +6.21926 q^{59} +12.5220 q^{61} +3.31779 q^{63} -5.35050 q^{65} +10.3052 q^{67} -8.73812 q^{69} -16.3692 q^{71} -15.5143 q^{73} -8.90922 q^{75} -3.39786 q^{77} -5.15466 q^{79} -6.45724 q^{81} -6.55843 q^{83} -9.80686 q^{85} -7.27565 q^{87} -4.60031 q^{89} -3.85833 q^{91} +4.45258 q^{93} +8.25541 q^{95} +9.16155 q^{97} -13.9389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more