LMFDB - Embedded newform 6561.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55207 q^{2} +0.408920 q^{4} +3.09765 q^{5} +1.06604 q^{7} +2.46947 q^{8} +O(q^{10})$	\(q-1.55207 q^{2} +0.408920 q^{4} +3.09765 q^{5} +1.06604 q^{7} +2.46947 q^{8} -4.80776 q^{10} +1.38445 q^{11} -6.28168 q^{13} -1.65457 q^{14} -4.65062 q^{16} -3.23982 q^{17} -3.54256 q^{19} +1.26669 q^{20} -2.14876 q^{22} +7.51514 q^{23} +4.59541 q^{25} +9.74960 q^{26} +0.435926 q^{28} -2.55724 q^{29} +3.29445 q^{31} +2.27916 q^{32} +5.02842 q^{34} +3.30222 q^{35} -1.98133 q^{37} +5.49831 q^{38} +7.64953 q^{40} -11.1828 q^{41} +3.75184 q^{43} +0.566129 q^{44} -11.6640 q^{46} +1.83828 q^{47} -5.86355 q^{49} -7.13240 q^{50} -2.56870 q^{52} -4.00461 q^{53} +4.28853 q^{55} +2.63256 q^{56} +3.96902 q^{58} +4.86024 q^{59} +9.87736 q^{61} -5.11321 q^{62} +5.76384 q^{64} -19.4584 q^{65} +2.18336 q^{67} -1.32483 q^{68} -5.12528 q^{70} -4.22175 q^{71} -11.1668 q^{73} +3.07516 q^{74} -1.44863 q^{76} +1.47588 q^{77} +15.2790 q^{79} -14.4060 q^{80} +17.3566 q^{82} +2.84828 q^{83} -10.0358 q^{85} -5.82311 q^{86} +3.41885 q^{88} -5.24231 q^{89} -6.69653 q^{91} +3.07309 q^{92} -2.85314 q^{94} -10.9736 q^{95} -12.0440 q^{97} +9.10064 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

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 $

$2$	−1.55207	−1.09748	−0.548739	−	0.835993i	$-0.684892\pi$
$2$	−1.55207	−1.09748	−0.548739	+	0.835993i	$0.684892\pi$
$3$	0	0
$3$	0	0
$4$	0.408920	0.204460
$5$	3.09765	1.38531	0.692655	−	0.721269i	$-0.256441\pi$
$5$	3.09765	1.38531	0.692655	+	0.721269i	$0.256441\pi$
$6$	0	0
$7$	1.06604	0.402926	0.201463	−	0.979496i	$-0.435430\pi$
$7$	1.06604	0.402926	0.201463	+	0.979496i	$0.435430\pi$
$8$	2.46947	0.873088
$9$	0	0
$10$	−4.80776	−1.52035
$11$	1.38445	0.417427	0.208713	−	0.977977i	$-0.433072\pi$
$11$	1.38445	0.417427	0.208713	+	0.977977i	$0.433072\pi$
$12$	0	0
$13$	−6.28168	−1.74222	−0.871112	−	0.491084i	$-0.836601\pi$
$13$	−6.28168	−1.74222	−0.871112	+	0.491084i	$0.836601\pi$
$14$	−1.65457	−0.442203
$15$	0	0
$16$	−4.65062	−1.16266
$17$	−3.23982	−0.785771	−0.392885	−	0.919587i	$-0.628523\pi$
$17$	−3.23982	−0.785771	−0.392885	+	0.919587i	$0.628523\pi$
$18$	0	0
$19$	−3.54256	−0.812720	−0.406360	−	0.913713i	$-0.633202\pi$
$19$	−3.54256	−0.812720	−0.406360	+	0.913713i	$0.633202\pi$
$20$	1.26669	0.283240
$21$	0	0
$22$	−2.14876	−0.458117
$23$	7.51514	1.56702	0.783508	−	0.621382i	$-0.213428\pi$
$23$	7.51514	1.56702	0.783508	+	0.621382i	$0.213428\pi$
$24$	0	0
$25$	4.59541	0.919082
$26$	9.74960	1.91205
$27$	0	0
$28$	0.435926	0.0823823
$29$	−2.55724	−0.474868	−0.237434	−	0.971404i	$-0.576306\pi$
$29$	−2.55724	−0.474868	−0.237434	+	0.971404i	$0.576306\pi$
$30$	0	0
$31$	3.29445	0.591700	0.295850	−	0.955234i	$-0.404397\pi$
$31$	3.29445	0.591700	0.295850	+	0.955234i	$0.404397\pi$
$32$	2.27916	0.402902
$33$	0	0
$34$	5.02842	0.862367
$35$	3.30222	0.558177
$36$	0	0
$37$	−1.98133	−0.325728	−0.162864	−	0.986649i	$-0.552073\pi$
$37$	−1.98133	−0.325728	−0.162864	+	0.986649i	$0.552073\pi$
$38$	5.49831	0.891943
$39$	0	0
$40$	7.64953	1.20950
$41$	−11.1828	−1.74647	−0.873233	−	0.487302i	$-0.837981\pi$
$41$	−11.1828	−1.74647	−0.873233	+	0.487302i	$0.837981\pi$
$42$	0	0
$43$	3.75184	0.572150	0.286075	−	0.958207i	$-0.407649\pi$
$43$	3.75184	0.572150	0.286075	+	0.958207i	$0.407649\pi$
$44$	0.566129	0.0853471
$45$	0	0
$46$	−11.6640	−1.71977
$47$	1.83828	0.268141	0.134071	−	0.990972i	$-0.457195\pi$
$47$	1.83828	0.268141	0.134071	+	0.990972i	$0.457195\pi$
$48$	0	0
$49$	−5.86355	−0.837651
$50$	−7.13240	−1.00867
$51$	0	0
$52$	−2.56870	−0.356215
$53$	−4.00461	−0.550076	−0.275038	−	0.961433i	$-0.588690\pi$
$53$	−4.00461	−0.550076	−0.275038	+	0.961433i	$0.588690\pi$
$54$	0	0
$55$	4.28853	0.578265
$56$	2.63256	0.351790
$57$	0	0
$58$	3.96902	0.521157
$59$	4.86024	0.632749	0.316375	−	0.948634i	$-0.397534\pi$
$59$	4.86024	0.632749	0.316375	+	0.948634i	$0.397534\pi$
$60$	0	0
$61$	9.87736	1.26467	0.632333	−	0.774697i	$-0.282097\pi$
$61$	9.87736	1.26467	0.632333	+	0.774697i	$0.282097\pi$
$62$	−5.11321	−0.649378
$63$	0	0
$64$	5.76384	0.720479
$65$	−19.4584	−2.41352
$66$	0	0
$67$	2.18336	0.266740	0.133370	−	0.991066i	$-0.457420\pi$
$67$	2.18336	0.266740	0.133370	+	0.991066i	$0.457420\pi$
$68$	−1.32483	−0.160659
$69$	0	0
$70$	−5.12528	−0.612588
$71$	−4.22175	−0.501030	−0.250515	−	0.968113i	$-0.580600\pi$
$71$	−4.22175	−0.501030	−0.250515	+	0.968113i	$0.580600\pi$
$72$	0	0
$73$	−11.1668	−1.30698	−0.653488	−	0.756937i	$-0.726695\pi$
$73$	−11.1668	−1.30698	−0.653488	+	0.756937i	$0.726695\pi$
$74$	3.07516	0.357480
$75$	0	0
$76$	−1.44863	−0.166169
$77$	1.47588	0.168192
$78$	0	0
$79$	15.2790	1.71902	0.859511	−	0.511116i	$-0.170768\pi$
$79$	15.2790	1.71902	0.859511	+	0.511116i	$0.170768\pi$
$80$	−14.4060	−1.61064
$81$	0	0
$82$	17.3566	1.91671
$83$	2.84828	0.312639	0.156319	−	0.987707i	$-0.450037\pi$
$83$	2.84828	0.312639	0.156319	+	0.987707i	$0.450037\pi$
$84$	0	0
$85$	−10.0358	−1.08854
$86$	−5.82311	−0.627922
$87$	0	0
$88$	3.41885	0.364450
$89$	−5.24231	−0.555684	−0.277842	−	0.960627i	$-0.589619\pi$
$89$	−5.24231	−0.555684	−0.277842	+	0.960627i	$0.589619\pi$
$90$	0	0
$91$	−6.69653	−0.701987
$92$	3.07309	0.320392
$93$	0	0
$94$	−2.85314	−0.294279
$95$	−10.9736	−1.12587
$96$	0	0
$97$	−12.0440	−1.22288	−0.611441	−	0.791290i	$-0.709410\pi$
$97$	−12.0440	−1.22288	−0.611441	+	0.791290i	$0.709410\pi$
$98$	9.10064	0.919304
$99$	0	0
$100$	1.87916	0.187916
$101$	−8.48194	−0.843985	−0.421992	−	0.906599i	$-0.638669\pi$
$101$	−8.48194	−0.843985	−0.421992	+	0.906599i	$0.638669\pi$
$102$	0	0
$103$	−9.53349	−0.939363	−0.469681	−	0.882836i	$-0.655631\pi$
$103$	−9.53349	−0.939363	−0.469681	+	0.882836i	$0.655631\pi$
$104$	−15.5124	−1.52112
$105$	0	0
$106$	6.21544	0.603697
$107$	−10.0918	−0.975615	−0.487807	−	0.872951i	$-0.662203\pi$
$107$	−10.0918	−0.975615	−0.487807	+	0.872951i	$0.662203\pi$
$108$	0	0
$109$	−18.1075	−1.73438	−0.867190	−	0.497978i	$-0.834076\pi$
$109$	−18.1075	−1.73438	−0.867190	+	0.497978i	$0.834076\pi$
$110$	−6.65610	−0.634634
$111$	0	0
$112$	−4.95776	−0.468464
$113$	0.638269	0.0600433	0.0300217	−	0.999549i	$-0.490442\pi$
$113$	0.638269	0.0600433	0.0300217	+	0.999549i	$0.490442\pi$
$114$	0	0
$115$	23.2793	2.17080
$116$	−1.04571	−0.0970914
$117$	0	0
$118$	−7.54343	−0.694429
$119$	−3.45378	−0.316607
$120$	0	0
$121$	−9.08330	−0.825755
$122$	−15.3303	−1.38794
$123$	0	0
$124$	1.34717	0.120979
$125$	−1.25327	−0.112096
$126$	0	0
$127$	1.33334	0.118315	0.0591576	−	0.998249i	$-0.481159\pi$
$127$	1.33334	0.118315	0.0591576	+	0.998249i	$0.481159\pi$
$128$	−13.5042	−1.19361
$129$	0	0
$130$	30.2008	2.64879
$131$	−18.3925	−1.60696	−0.803481	−	0.595330i	$-0.797021\pi$
$131$	−18.3925	−1.60696	−0.803481	+	0.595330i	$0.797021\pi$
$132$	0	0
$133$	−3.77652	−0.327466
$134$	−3.38873	−0.292742
$135$	0	0
$136$	−8.00062	−0.686047
$137$	4.66159	0.398266	0.199133	−	0.979972i	$-0.436187\pi$
$137$	4.66159	0.398266	0.199133	+	0.979972i	$0.436187\pi$
$138$	0	0
$139$	1.01858	0.0863944	0.0431972	−	0.999067i	$-0.486246\pi$
$139$	1.01858	0.0863944	0.0431972	+	0.999067i	$0.486246\pi$
$140$	1.35034	0.114125
$141$	0	0
$142$	6.55245	0.549870
$143$	−8.69666	−0.727251
$144$	0	0
$145$	−7.92143	−0.657839
$146$	17.3317	1.43438
$147$	0	0
$148$	−0.810204	−0.0665984
$149$	−1.16008	−0.0950372	−0.0475186	−	0.998870i	$-0.515131\pi$
$149$	−1.16008	−0.0950372	−0.0475186	+	0.998870i	$0.515131\pi$
$150$	0	0
$151$	12.9477	1.05367	0.526836	−	0.849967i	$-0.323378\pi$
$151$	12.9477	1.05367	0.526836	+	0.849967i	$0.323378\pi$
$152$	−8.74824	−0.709576
$153$	0	0
$154$	−2.29067	−0.184587
$155$	10.2050	0.819688
$156$	0	0
$157$	−2.17658	−0.173710	−0.0868550	−	0.996221i	$-0.527682\pi$
$157$	−2.17658	−0.173710	−0.0868550	+	0.996221i	$0.527682\pi$
$158$	−23.7141	−1.88659
$159$	0	0
$160$	7.06003	0.558144
$161$	8.01146	0.631392
$162$	0	0
$163$	−15.2864	−1.19732	−0.598660	−	0.801003i	$-0.704300\pi$
$163$	−15.2864	−1.19732	−0.598660	+	0.801003i	$0.704300\pi$
$164$	−4.57289	−0.357083
$165$	0	0
$166$	−4.42072	−0.343115
$167$	−7.85948	−0.608185	−0.304092	−	0.952643i	$-0.598353\pi$
$167$	−7.85948	−0.608185	−0.304092	+	0.952643i	$0.598353\pi$
$168$	0	0
$169$	26.4595	2.03535
$170$	15.5763	1.19464
$171$	0	0
$172$	1.53420	0.116982
$173$	−4.52881	−0.344319	−0.172160	−	0.985069i	$-0.555075\pi$
$173$	−4.52881	−0.344319	−0.172160	+	0.985069i	$0.555075\pi$
$174$	0	0
$175$	4.89890	0.370322
$176$	−6.43855	−0.485324
$177$	0	0
$178$	8.13643	0.609851
$179$	4.60986	0.344557	0.172279	−	0.985048i	$-0.444887\pi$
$179$	4.60986	0.344557	0.172279	+	0.985048i	$0.444887\pi$
$180$	0	0
$181$	8.16985	0.607260	0.303630	−	0.952790i	$-0.401801\pi$
$181$	8.16985	0.607260	0.303630	+	0.952790i	$0.401801\pi$
$182$	10.3935	0.770416
$183$	0	0
$184$	18.5584	1.36814
$185$	−6.13745	−0.451234
$186$	0	0
$187$	−4.48536	−0.328002
$188$	0.751711	0.0548241
$189$	0	0
$190$	17.0318	1.23562
$191$	3.94379	0.285363	0.142681	−	0.989769i	$-0.454428\pi$
$191$	3.94379	0.285363	0.142681	+	0.989769i	$0.454428\pi$
$192$	0	0
$193$	7.20121	0.518354	0.259177	−	0.965830i	$-0.416549\pi$
$193$	7.20121	0.518354	0.259177	+	0.965830i	$0.416549\pi$
$194$	18.6931	1.34209
$195$	0	0
$196$	−2.39772	−0.171266
$197$	10.2533	0.730521	0.365260	−	0.930905i	$-0.380980\pi$
$197$	10.2533	0.730521	0.365260	+	0.930905i	$0.380980\pi$
$198$	0	0
$199$	26.5122	1.87940	0.939702	−	0.341995i	$-0.111103\pi$
$199$	26.5122	1.87940	0.939702	+	0.341995i	$0.111103\pi$
$200$	11.3482	0.802440
$201$	0	0
$202$	13.1646	0.926256
$203$	−2.72613	−0.191337
$204$	0	0
$205$	−34.6405	−2.41940
$206$	14.7966	1.03093
$207$	0	0
$208$	29.2137	2.02561
$209$	−4.90450	−0.339251
$210$	0	0
$211$	−9.19575	−0.633062	−0.316531	−	0.948582i	$-0.602518\pi$
$211$	−9.19575	−0.633062	−0.316531	+	0.948582i	$0.602518\pi$
$212$	−1.63757	−0.112469
$213$	0	0
$214$	15.6632	1.07072
$215$	11.6219	0.792604
$216$	0	0
$217$	3.51202	0.238411
$218$	28.1040	1.90345
$219$	0	0
$220$	1.75367	0.118232
$221$	20.3515	1.36899
$222$	0	0
$223$	−13.1134	−0.878138	−0.439069	−	0.898453i	$-0.644692\pi$
$223$	−13.1134	−0.878138	−0.439069	+	0.898453i	$0.644692\pi$
$224$	2.42968	0.162340
$225$	0	0
$226$	−0.990638	−0.0658963
$227$	14.1466	0.938941	0.469471	−	0.882948i	$-0.344445\pi$
$227$	14.1466	0.938941	0.469471	+	0.882948i	$0.344445\pi$
$228$	0	0
$229$	12.7862	0.844933	0.422466	−	0.906379i	$-0.361164\pi$
$229$	12.7862	0.844933	0.422466	+	0.906379i	$0.361164\pi$
$230$	−36.1310	−2.38241
$231$	0	0
$232$	−6.31502	−0.414601
$233$	−6.87197	−0.450198	−0.225099	−	0.974336i	$-0.572271\pi$
$233$	−6.87197	−0.450198	−0.225099	+	0.974336i	$0.572271\pi$
$234$	0	0
$235$	5.69435	0.371458
$236$	1.98745	0.129372
$237$	0	0
$238$	5.36051	0.347470
$239$	−13.4794	−0.871909	−0.435955	−	0.899969i	$-0.643589\pi$
$239$	−13.4794	−0.871909	−0.435955	+	0.899969i	$0.643589\pi$
$240$	0	0
$241$	−0.352508	−0.0227071	−0.0113535	−	0.999936i	$-0.503614\pi$
$241$	−0.352508	−0.0227071	−0.0113535	+	0.999936i	$0.503614\pi$
$242$	14.0979	0.906249
$243$	0	0
$244$	4.03905	0.258574
$245$	−18.1632	−1.16041
$246$	0	0
$247$	22.2532	1.41594
$248$	8.13553	0.516606
$249$	0	0
$250$	1.94517	0.123023
$251$	−22.3482	−1.41060	−0.705302	−	0.708907i	$-0.749189\pi$
$251$	−22.3482	−1.41060	−0.705302	+	0.708907i	$0.749189\pi$
$252$	0	0
$253$	10.4043	0.654114
$254$	−2.06944	−0.129848
$255$	0	0
$256$	9.43178	0.589486
$257$	−18.4036	−1.14798	−0.573992	−	0.818861i	$-0.694606\pi$
$257$	−18.4036	−1.14798	−0.573992	+	0.818861i	$0.694606\pi$
$258$	0	0
$259$	−2.11218	−0.131244
$260$	−7.95694	−0.493468
$261$	0	0
$262$	28.5465	1.76361
$263$	2.33887	0.144221	0.0721105	−	0.997397i	$-0.477027\pi$
$263$	2.33887	0.144221	0.0721105	+	0.997397i	$0.477027\pi$
$264$	0	0
$265$	−12.4049	−0.762026
$266$	5.86143	0.359387
$267$	0	0
$268$	0.892821	0.0545377
$269$	15.2298	0.928575	0.464287	−	0.885685i	$-0.346310\pi$
$269$	15.2298	0.928575	0.464287	+	0.885685i	$0.346310\pi$
$270$	0	0
$271$	5.50367	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$271$	5.50367	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$272$	15.0672	0.913581
$273$	0	0
$274$	−7.23511	−0.437089
$275$	6.36211	0.383650
$276$	0	0
$277$	18.7195	1.12474	0.562371	−	0.826885i	$-0.309889\pi$
$277$	18.7195	1.12474	0.562371	+	0.826885i	$0.309889\pi$
$278$	−1.58090	−0.0948160
$279$	0	0
$280$	8.15473	0.487338
$281$	−18.4498	−1.10062	−0.550311	−	0.834960i	$-0.685491\pi$
$281$	−18.4498	−1.10062	−0.550311	+	0.834960i	$0.685491\pi$
$282$	0	0
$283$	−12.1191	−0.720407	−0.360203	−	0.932874i	$-0.617293\pi$
$283$	−12.1191	−0.720407	−0.360203	+	0.932874i	$0.617293\pi$
$284$	−1.72636	−0.102441
$285$	0	0
$286$	13.4978	0.798143
$287$	−11.9214	−0.703697
$288$	0	0
$289$	−6.50360	−0.382565
$290$	12.2946	0.721964
$291$	0	0
$292$	−4.56633	−0.267224
$293$	19.2223	1.12298	0.561488	−	0.827485i	$-0.310229\pi$
$293$	19.2223	1.12298	0.561488	+	0.827485i	$0.310229\pi$
$294$	0	0
$295$	15.0553	0.876553
$296$	−4.89282	−0.284389
$297$	0	0
$298$	1.80052	0.104301
$299$	−47.2077	−2.73009
$300$	0	0
$301$	3.99961	0.230534
$302$	−20.0958	−1.15638
$303$	0	0
$304$	16.4751	0.944914
$305$	30.5966	1.75195
$306$	0	0
$307$	11.0534	0.630851	0.315425	−	0.948950i	$-0.397853\pi$
$307$	11.0534	0.630851	0.315425	+	0.948950i	$0.397853\pi$
$308$	0.603517	0.0343886
$309$	0	0
$310$	−15.8389	−0.899590
$311$	−14.1072	−0.799947	−0.399974	−	0.916527i	$-0.630981\pi$
$311$	−14.1072	−0.799947	−0.399974	+	0.916527i	$0.630981\pi$
$312$	0	0
$313$	−1.60117	−0.0905034	−0.0452517	−	0.998976i	$-0.514409\pi$
$313$	−1.60117	−0.0905034	−0.0452517	+	0.998976i	$0.514409\pi$
$314$	3.37820	0.190643
$315$	0	0
$316$	6.24789	0.351471
$317$	−10.7421	−0.603334	−0.301667	−	0.953413i	$-0.597543\pi$
$317$	−10.7421	−0.603334	−0.301667	+	0.953413i	$0.597543\pi$
$318$	0	0
$319$	−3.54037	−0.198222
$320$	17.8543	0.998087
$321$	0	0
$322$	−12.4343	−0.692939
$323$	11.4773	0.638611
$324$	0	0
$325$	−28.8669	−1.60125
$326$	23.7255	1.31403
$327$	0	0
$328$	−27.6157	−1.52482
$329$	1.95969	0.108041
$330$	0	0
$331$	−28.4978	−1.56638	−0.783191	−	0.621782i	$-0.786409\pi$
$331$	−28.4978	−1.56638	−0.783191	+	0.621782i	$0.786409\pi$
$332$	1.16472	0.0639222
$333$	0	0
$334$	12.1985	0.667470
$335$	6.76328	0.369518
$336$	0	0
$337$	−1.26712	−0.0690242	−0.0345121	−	0.999404i	$-0.510988\pi$
$337$	−1.26712	−0.0690242	−0.0345121	+	0.999404i	$0.510988\pi$
$338$	−41.0670	−2.23375
$339$	0	0
$340$	−4.10384	−0.222562
$341$	4.56099	0.246991
$342$	0	0
$343$	−13.7131	−0.740437
$344$	9.26503	0.499537
$345$	0	0
$346$	7.02904	0.377883
$347$	−15.3859	−0.825960	−0.412980	−	0.910740i	$-0.635512\pi$
$347$	−15.3859	−0.825960	−0.412980	+	0.910740i	$0.635512\pi$
$348$	0	0
$349$	11.8590	0.634798	0.317399	−	0.948292i	$-0.397191\pi$
$349$	11.8590	0.634798	0.317399	+	0.948292i	$0.397191\pi$
$350$	−7.60344	−0.406421
$351$	0	0
$352$	3.15538	0.168182
$353$	12.3740	0.658603	0.329302	−	0.944225i	$-0.393187\pi$
$353$	12.3740	0.658603	0.329302	+	0.944225i	$0.393187\pi$
$354$	0	0
$355$	−13.0775	−0.694081
$356$	−2.14369	−0.113615
$357$	0	0
$358$	−7.15483	−0.378145
$359$	−13.0244	−0.687403	−0.343702	−	0.939079i	$-0.611681\pi$
$359$	−13.0244	−0.687403	−0.343702	+	0.939079i	$0.611681\pi$
$360$	0	0
$361$	−6.45024	−0.339486
$362$	−12.6802	−0.666455
$363$	0	0
$364$	−2.73835	−0.143528
$365$	−34.5908	−1.81057
$366$	0	0
$367$	7.13638	0.372516	0.186258	−	0.982501i	$-0.440364\pi$
$367$	7.13638	0.372516	0.186258	+	0.982501i	$0.440364\pi$
$368$	−34.9501	−1.82190
$369$	0	0
$370$	9.52575	0.495220
$371$	−4.26909	−0.221640
$372$	0	0
$373$	−12.6365	−0.654294	−0.327147	−	0.944973i	$-0.606087\pi$
$373$	−12.6365	−0.654294	−0.327147	+	0.944973i	$0.606087\pi$
$374$	6.96158	0.359975
$375$	0	0
$376$	4.53958	0.234111
$377$	16.0638	0.827326
$378$	0	0
$379$	−24.1944	−1.24278	−0.621391	−	0.783501i	$-0.713432\pi$
$379$	−24.1944	−1.24278	−0.621391	+	0.783501i	$0.713432\pi$
$380$	−4.48733	−0.230195
$381$	0	0
$382$	−6.12104	−0.313180
$383$	25.8249	1.31959	0.659795	−	0.751446i	$-0.270643\pi$
$383$	25.8249	1.31959	0.659795	+	0.751446i	$0.270643\pi$
$384$	0	0
$385$	4.57175	0.232998
$386$	−11.1768	−0.568883
$387$	0	0
$388$	−4.92503	−0.250031
$389$	−16.4106	−0.832051	−0.416026	−	0.909353i	$-0.636577\pi$
$389$	−16.4106	−0.832051	−0.416026	+	0.909353i	$0.636577\pi$
$390$	0	0
$391$	−24.3477	−1.23132
$392$	−14.4799	−0.731343
$393$	0	0
$394$	−15.9139	−0.801731
$395$	47.3290	2.38138
$396$	0	0
$397$	35.8123	1.79737	0.898685	−	0.438595i	$-0.144524\pi$
$397$	35.8123	1.79737	0.898685	+	0.438595i	$0.144524\pi$
$398$	−41.1489	−2.06261
$399$	0	0
$400$	−21.3715	−1.06858
$401$	−5.99513	−0.299382	−0.149691	−	0.988733i	$-0.547828\pi$
$401$	−5.99513	−0.299382	−0.149691	+	0.988733i	$0.547828\pi$
$402$	0	0
$403$	−20.6947	−1.03087
$404$	−3.46844	−0.172561
$405$	0	0
$406$	4.23114	0.209988
$407$	−2.74304	−0.135968
$408$	0	0
$409$	24.9644	1.23441	0.617204	−	0.786803i	$-0.288265\pi$
$409$	24.9644	1.23441	0.617204	+	0.786803i	$0.288265\pi$
$410$	53.7645	2.65524
$411$	0	0
$412$	−3.89844	−0.192062
$413$	5.18122	0.254951
$414$	0	0
$415$	8.82295	0.433102
$416$	−14.3169	−0.701946
$417$	0	0
$418$	7.61212	0.372321
$419$	9.65291	0.471575	0.235788	−	0.971805i	$-0.424233\pi$
$419$	9.65291	0.471575	0.235788	+	0.971805i	$0.424233\pi$
$420$	0	0
$421$	12.0046	0.585070	0.292535	−	0.956255i	$-0.405501\pi$
$421$	12.0046	0.585070	0.292535	+	0.956255i	$0.405501\pi$
$422$	14.2724	0.694772
$423$	0	0
$424$	−9.88926	−0.480265
$425$	−14.8883	−0.722188
$426$	0	0
$427$	10.5297	0.509567
$428$	−4.12675	−0.199474
$429$	0	0
$430$	−18.0379	−0.869867
$431$	−39.4201	−1.89880	−0.949399	−	0.314073i	$-0.898306\pi$
$431$	−39.4201	−1.89880	−0.949399	+	0.314073i	$0.898306\pi$
$432$	0	0
$433$	−16.1605	−0.776623	−0.388312	−	0.921528i	$-0.626942\pi$
$433$	−16.1605	−0.776623	−0.388312	+	0.921528i	$0.626942\pi$
$434$	−5.45090	−0.261651
$435$	0	0
$436$	−7.40450	−0.354611
$437$	−26.6229	−1.27355
$438$	0	0
$439$	−31.7176	−1.51380	−0.756899	−	0.653532i	$-0.773286\pi$
$439$	−31.7176	−1.51380	−0.756899	+	0.653532i	$0.773286\pi$
$440$	10.5904	0.504877
$441$	0	0
$442$	−31.5869	−1.50244
$443$	26.3536	1.25210	0.626050	−	0.779783i	$-0.284671\pi$
$443$	26.3536	1.25210	0.626050	+	0.779783i	$0.284671\pi$
$444$	0	0
$445$	−16.2388	−0.769794
$446$	20.3529	0.963738
$447$	0	0
$448$	6.14449	0.290300
$449$	31.3644	1.48018	0.740088	−	0.672510i	$-0.234784\pi$
$449$	31.3644	1.48018	0.740088	+	0.672510i	$0.234784\pi$
$450$	0	0
$451$	−15.4821	−0.729022
$452$	0.261001	0.0122765
$453$	0	0
$454$	−21.9565	−1.03047
$455$	−20.7435	−0.972470
$456$	0	0
$457$	0.181065	0.00846986	0.00423493	−	0.999991i	$-0.498652\pi$
$457$	0.181065	0.00846986	0.00423493	+	0.999991i	$0.498652\pi$
$458$	−19.8450	−0.927296
$459$	0	0
$460$	9.51936	0.443842
$461$	0.650426	0.0302934	0.0151467	−	0.999885i	$-0.495178\pi$
$461$	0.650426	0.0302934	0.0151467	+	0.999885i	$0.495178\pi$
$462$	0	0
$463$	−7.97915	−0.370822	−0.185411	−	0.982661i	$-0.559362\pi$
$463$	−7.97915	−0.370822	−0.185411	+	0.982661i	$0.559362\pi$
$464$	11.8928	0.552108
$465$	0	0
$466$	10.6658	0.494083
$467$	−4.82104	−0.223091	−0.111546	−	0.993759i	$-0.535580\pi$
$467$	−4.82104	−0.223091	−0.111546	+	0.993759i	$0.535580\pi$
$468$	0	0
$469$	2.32756	0.107477
$470$	−8.83803	−0.407668
$471$	0	0
$472$	12.0022	0.552446
$473$	5.19422	0.238831
$474$	0	0
$475$	−16.2795	−0.746957
$476$	−1.41232	−0.0647336
$477$	0	0
$478$	20.9210	0.956902
$479$	31.4409	1.43657	0.718285	−	0.695749i	$-0.244928\pi$
$479$	31.4409	1.43657	0.718285	+	0.695749i	$0.244928\pi$
$480$	0	0
$481$	12.4461	0.567491
$482$	0.547117	0.0249205
$483$	0	0
$484$	−3.71434	−0.168834
$485$	−37.3080	−1.69407
$486$	0	0
$487$	4.28326	0.194093	0.0970465	−	0.995280i	$-0.469060\pi$
$487$	4.28326	0.194093	0.0970465	+	0.995280i	$0.469060\pi$
$488$	24.3918	1.10417
$489$	0	0
$490$	28.1906	1.27352
$491$	6.25146	0.282125	0.141062	−	0.990001i	$-0.454948\pi$
$491$	6.25146	0.282125	0.141062	+	0.990001i	$0.454948\pi$
$492$	0	0
$493$	8.28499	0.373137
$494$	−34.5386	−1.55396
$495$	0	0
$496$	−15.3212	−0.687944
$497$	−4.50056	−0.201878
$498$	0	0
$499$	3.31703	0.148491	0.0742453	−	0.997240i	$-0.476345\pi$
$499$	3.31703	0.148491	0.0742453	+	0.997240i	$0.476345\pi$
$500$	−0.512488	−0.0229192
$501$	0	0
$502$	34.6859	1.54811
$503$	−33.0307	−1.47277	−0.736384	−	0.676564i	$-0.763468\pi$
$503$	−33.0307	−1.47277	−0.736384	+	0.676564i	$0.763468\pi$
$504$	0	0
$505$	−26.2741	−1.16918
$506$	−16.1482	−0.717877
$507$	0	0
$508$	0.545231	0.0241907
$509$	−24.3641	−1.07992	−0.539961	−	0.841690i	$-0.681561\pi$
$509$	−24.3641	−1.07992	−0.539961	+	0.841690i	$0.681561\pi$
$510$	0	0
$511$	−11.9043	−0.526614
$512$	12.3696	0.546665
$513$	0	0
$514$	28.5637	1.25989
$515$	−29.5314	−1.30131
$516$	0	0
$517$	2.54501	0.111929
$518$	3.27825	0.144038
$519$	0	0
$520$	−48.0519	−2.10722
$521$	−33.9428	−1.48706	−0.743531	−	0.668701i	$-0.766851\pi$
$521$	−33.9428	−1.48706	−0.743531	+	0.668701i	$0.766851\pi$
$522$	0	0
$523$	−11.6261	−0.508372	−0.254186	−	0.967155i	$-0.581808\pi$
$523$	−11.6261	−0.508372	−0.254186	+	0.967155i	$0.581808\pi$
$524$	−7.52107	−0.328560
$525$	0	0
$526$	−3.63009	−0.158279
$527$	−10.6734	−0.464941
$528$	0	0
$529$	33.4774	1.45554
$530$	19.2532	0.836307
$531$	0	0
$532$	−1.54430	−0.0669537
$533$	70.2470	3.04274
$534$	0	0
$535$	−31.2609	−1.35153
$536$	5.39174	0.232888
$537$	0	0
$538$	−23.6376	−1.01909
$539$	−8.11779	−0.349658
$540$	0	0
$541$	9.89502	0.425420	0.212710	−	0.977115i	$-0.431771\pi$
$541$	9.89502	0.425420	0.212710	+	0.977115i	$0.431771\pi$
$542$	−8.54207	−0.366913
$543$	0	0
$544$	−7.38406	−0.316589
$545$	−56.0905	−2.40265
$546$	0	0
$547$	−38.7955	−1.65878	−0.829388	−	0.558672i	$-0.811311\pi$
$547$	−38.7955	−1.65878	−0.829388	+	0.558672i	$0.811311\pi$
$548$	1.90622	0.0814295
$549$	0	0
$550$	−9.87444	−0.421047
$551$	9.05919	0.385934
$552$	0	0
$553$	16.2881	0.692639
$554$	−29.0539	−1.23438
$555$	0	0
$556$	0.416516	0.0176642
$557$	11.7470	0.497735	0.248867	−	0.968538i	$-0.419942\pi$
$557$	11.7470	0.497735	0.248867	+	0.968538i	$0.419942\pi$
$558$	0	0
$559$	−23.5678	−0.996813
$560$	−15.3574	−0.648968
$561$	0	0
$562$	28.6353	1.20791
$563$	−14.6494	−0.617397	−0.308699	−	0.951160i	$-0.599893\pi$
$563$	−14.6494	−0.617397	−0.308699	+	0.951160i	$0.599893\pi$
$564$	0	0
$565$	1.97713	0.0831786
$566$	18.8097	0.790631
$567$	0	0
$568$	−10.4255	−0.437443
$569$	−32.3915	−1.35792	−0.678962	−	0.734174i	$-0.737570\pi$
$569$	−32.3915	−1.35792	−0.678962	+	0.734174i	$0.737570\pi$
$570$	0	0
$571$	7.93896	0.332235	0.166118	−	0.986106i	$-0.446877\pi$
$571$	7.93896	0.332235	0.166118	+	0.986106i	$0.446877\pi$
$572$	−3.55624	−0.148694
$573$	0	0
$574$	18.5028	0.772293
$575$	34.5352	1.44022
$576$	0	0
$577$	20.4467	0.851209	0.425605	−	0.904909i	$-0.360061\pi$
$577$	20.4467	0.851209	0.425605	+	0.904909i	$0.360061\pi$
$578$	10.0940	0.419856
$579$	0	0
$580$	−3.23923	−0.134502
$581$	3.03638	0.125970
$582$	0	0
$583$	−5.54418	−0.229616
$584$	−27.5760	−1.14111
$585$	0	0
$586$	−29.8343	−1.23244
$587$	17.0854	0.705189	0.352595	−	0.935776i	$-0.385299\pi$
$587$	17.0854	0.705189	0.352595	+	0.935776i	$0.385299\pi$
$588$	0	0
$589$	−11.6708	−0.480886
$590$	−23.3669	−0.961999
$591$	0	0
$592$	9.21440	0.378710
$593$	−5.11946	−0.210231	−0.105115	−	0.994460i	$-0.533521\pi$
$593$	−5.11946	−0.210231	−0.105115	+	0.994460i	$0.533521\pi$
$594$	0	0
$595$	−10.6986	−0.438599
$596$	−0.474379	−0.0194313
$597$	0	0
$598$	73.2697	2.99622
$599$	17.4709	0.713842	0.356921	−	0.934135i	$-0.383827\pi$
$599$	17.4709	0.713842	0.356921	+	0.934135i	$0.383827\pi$
$600$	0	0
$601$	−16.6665	−0.679840	−0.339920	−	0.940454i	$-0.610400\pi$
$601$	−16.6665	−0.679840	−0.339920	+	0.940454i	$0.610400\pi$
$602$	−6.20768	−0.253006
$603$	0	0
$604$	5.29459	0.215434
$605$	−28.1369	−1.14393
$606$	0	0
$607$	38.1348	1.54784	0.773922	−	0.633281i	$-0.218292\pi$
$607$	38.1348	1.54784	0.773922	+	0.633281i	$0.218292\pi$
$608$	−8.07407	−0.327447
$609$	0	0
$610$	−47.4880	−1.92273
$611$	−11.5475	−0.467162
$612$	0	0
$613$	−36.2223	−1.46301	−0.731503	−	0.681838i	$-0.761181\pi$
$613$	−36.2223	−1.46301	−0.731503	+	0.681838i	$0.761181\pi$
$614$	−17.1556	−0.692346
$615$	0	0
$616$	3.64464	0.146847
$617$	−37.4001	−1.50567	−0.752835	−	0.658209i	$-0.771314\pi$
$617$	−37.4001	−1.50567	−0.752835	+	0.658209i	$0.771314\pi$
$618$	0	0
$619$	−29.9624	−1.20429	−0.602146	−	0.798386i	$-0.705687\pi$
$619$	−29.9624	−1.20429	−0.602146	+	0.798386i	$0.705687\pi$
$620$	4.17304	0.167593
$621$	0	0
$622$	21.8954	0.877925
$623$	−5.58852	−0.223899
$624$	0	0
$625$	−26.8593	−1.07437
$626$	2.48513	0.0993256
$627$	0	0
$628$	−0.890047	−0.0355168
$629$	6.41913	0.255948
$630$	0	0
$631$	12.2317	0.486938	0.243469	−	0.969909i	$-0.421715\pi$
$631$	12.2317	0.486938	0.243469	+	0.969909i	$0.421715\pi$
$632$	37.7310	1.50086
$633$	0	0
$634$	16.6724	0.662146
$635$	4.13023	0.163903
$636$	0	0
$637$	36.8330	1.45938
$638$	5.49490	0.217545
$639$	0	0
$640$	−41.8312	−1.65352
$641$	4.37305	0.172725	0.0863626	−	0.996264i	$-0.472476\pi$
$641$	4.37305	0.172725	0.0863626	+	0.996264i	$0.472476\pi$
$642$	0	0
$643$	13.9521	0.550218	0.275109	−	0.961413i	$-0.411286\pi$
$643$	13.9521	0.550218	0.275109	+	0.961413i	$0.411286\pi$
$644$	3.27605	0.129094
$645$	0	0
$646$	−17.8135	−0.700863
$647$	32.6853	1.28499	0.642496	−	0.766289i	$-0.277899\pi$
$647$	32.6853	1.28499	0.642496	+	0.766289i	$0.277899\pi$
$648$	0	0
$649$	6.72875	0.264126
$650$	44.8034	1.75734
$651$	0	0
$652$	−6.25090	−0.244804
$653$	3.18833	0.124769	0.0623845	−	0.998052i	$-0.480129\pi$
$653$	3.18833	0.124769	0.0623845	+	0.998052i	$0.480129\pi$
$654$	0	0
$655$	−56.9735	−2.22614
$656$	52.0072	2.03054
$657$	0	0
$658$	−3.04157	−0.118573
$659$	13.8463	0.539374	0.269687	−	0.962948i	$-0.413080\pi$
$659$	13.8463	0.539374	0.269687	+	0.962948i	$0.413080\pi$
$660$	0	0
$661$	−26.1181	−1.01588	−0.507938	−	0.861394i	$-0.669592\pi$
$661$	−26.1181	−1.01588	−0.507938	+	0.861394i	$0.669592\pi$
$662$	44.2306	1.71907
$663$	0	0
$664$	7.03372	0.272961
$665$	−11.6983	−0.453642
$666$	0	0
$667$	−19.2180	−0.744125
$668$	−3.21390	−0.124349
$669$	0	0
$670$	−10.4971	−0.405538
$671$	13.6747	0.527906
$672$	0	0
$673$	13.0801	0.504201	0.252101	−	0.967701i	$-0.418879\pi$
$673$	13.0801	0.504201	0.252101	+	0.967701i	$0.418879\pi$
$674$	1.96665	0.0757526
$675$	0	0
$676$	10.8198	0.416147
$677$	−9.24387	−0.355271	−0.177635	−	0.984096i	$-0.556845\pi$
$677$	−9.24387	−0.355271	−0.177635	+	0.984096i	$0.556845\pi$
$678$	0	0
$679$	−12.8394	−0.492731
$680$	−24.7831	−0.950388
$681$	0	0
$682$	−7.07897	−0.271068
$683$	−32.8657	−1.25757	−0.628786	−	0.777579i	$-0.716448\pi$
$683$	−32.8657	−1.25757	−0.628786	+	0.777579i	$0.716448\pi$
$684$	0	0
$685$	14.4399	0.551722
$686$	21.2837	0.812614
$687$	0	0
$688$	−17.4484	−0.665213
$689$	25.1557	0.958356
$690$	0	0
$691$	39.1055	1.48764	0.743822	−	0.668377i	$-0.233011\pi$
$691$	39.1055	1.48764	0.743822	+	0.668377i	$0.233011\pi$
$692$	−1.85192	−0.0703995
$693$	0	0
$694$	23.8800	0.906474
$695$	3.15519	0.119683
$696$	0	0
$697$	36.2304	1.37232
$698$	−18.4060	−0.696677
$699$	0	0
$700$	2.00326	0.0757161
$701$	26.3237	0.994233	0.497117	−	0.867684i	$-0.334392\pi$
$701$	26.3237	0.994233	0.497117	+	0.867684i	$0.334392\pi$
$702$	0	0
$703$	7.01898	0.264726
$704$	7.97973	0.300747
$705$	0	0
$706$	−19.2054	−0.722803
$707$	−9.04211	−0.340063
$708$	0	0
$709$	21.9606	0.824747	0.412374	−	0.911015i	$-0.364700\pi$
$709$	21.9606	0.824747	0.412374	+	0.911015i	$0.364700\pi$
$710$	20.2972	0.761740
$711$	0	0
$712$	−12.9457	−0.485161
$713$	24.7582	0.927203
$714$	0	0
$715$	−26.9392	−1.00747
$716$	1.88507	0.0704482
$717$	0	0
$718$	20.2148	0.754410
$719$	−50.5293	−1.88442	−0.942212	−	0.335016i	$-0.891258\pi$
$719$	−50.5293	−1.88442	−0.942212	+	0.335016i	$0.891258\pi$
$720$	0	0
$721$	−10.1631	−0.378494
$722$	10.0112	0.372579
$723$	0	0
$724$	3.34081	0.124160
$725$	−11.7516	−0.436442
$726$	0	0
$727$	−3.79927	−0.140907	−0.0704536	−	0.997515i	$-0.522445\pi$
$727$	−3.79927	−0.140907	−0.0704536	+	0.997515i	$0.522445\pi$
$728$	−16.5369	−0.612897
$729$	0	0
$730$	53.6873	1.98706
$731$	−12.1553	−0.449578
$732$	0	0
$733$	−45.9886	−1.69863	−0.849315	−	0.527887i	$-0.822984\pi$
$733$	−45.9886	−1.69863	−0.849315	+	0.527887i	$0.822984\pi$
$734$	−11.0762	−0.408828
$735$	0	0
$736$	17.1282	0.631354
$737$	3.02275	0.111344
$738$	0	0
$739$	−18.1148	−0.666364	−0.333182	−	0.942863i	$-0.608122\pi$
$739$	−18.1148	−0.666364	−0.333182	+	0.942863i	$0.608122\pi$
$740$	−2.50973	−0.0922593
$741$	0	0
$742$	6.62592	0.243245
$743$	8.15490	0.299174	0.149587	−	0.988749i	$-0.452206\pi$
$743$	8.15490	0.299174	0.149587	+	0.988749i	$0.452206\pi$
$744$	0	0
$745$	−3.59351	−0.131656
$746$	19.6128	0.718074
$747$	0	0
$748$	−1.83415	−0.0670632
$749$	−10.7583	−0.393100
$750$	0	0
$751$	41.1249	1.50067	0.750334	−	0.661059i	$-0.229893\pi$
$751$	41.1249	1.50067	0.750334	+	0.661059i	$0.229893\pi$
$752$	−8.54916	−0.311756
$753$	0	0
$754$	−24.9321	−0.907973
$755$	40.1075	1.45966
$756$	0	0
$757$	−15.1095	−0.549166	−0.274583	−	0.961563i	$-0.588540\pi$
$757$	−15.1095	−0.549166	−0.274583	+	0.961563i	$0.588540\pi$
$758$	37.5513	1.36393
$759$	0	0
$760$	−27.0990	−0.982983
$761$	17.1651	0.622235	0.311117	−	0.950371i	$-0.399297\pi$
$761$	17.1651	0.622235	0.311117	+	0.950371i	$0.399297\pi$
$762$	0	0
$763$	−19.3033	−0.698827
$764$	1.61270	0.0583453
$765$	0	0
$766$	−40.0820	−1.44822
$767$	−30.5305	−1.10239
$768$	0	0
$769$	−36.6544	−1.32179	−0.660896	−	0.750478i	$-0.729824\pi$
$769$	−36.6544	−1.32179	−0.660896	+	0.750478i	$0.729824\pi$
$770$	−7.09568	−0.255711
$771$	0	0
$772$	2.94472	0.105983
$773$	3.45042	0.124103	0.0620515	−	0.998073i	$-0.480236\pi$
$773$	3.45042	0.124103	0.0620515	+	0.998073i	$0.480236\pi$
$774$	0	0
$775$	15.1393	0.543821
$776$	−29.7423	−1.06768
$777$	0	0
$778$	25.4704	0.913159
$779$	39.6159	1.41939
$780$	0	0
$781$	−5.84480	−0.209143
$782$	37.7893	1.35134
$783$	0	0
$784$	27.2692	0.973900
$785$	−6.74228	−0.240642
$786$	0	0
$787$	8.55556	0.304973	0.152486	−	0.988306i	$-0.451272\pi$
$787$	8.55556	0.304973	0.152486	+	0.988306i	$0.451272\pi$
$788$	4.19280	0.149362
$789$	0	0
$790$	−73.4579	−2.61351
$791$	0.680422	0.0241930
$792$	0	0
$793$	−62.0464	−2.20333
$794$	−55.5832	−1.97258
$795$	0	0
$796$	10.8414	0.384263
$797$	−9.21308	−0.326344	−0.163172	−	0.986598i	$-0.552173\pi$
$797$	−9.21308	−0.326344	−0.163172	+	0.986598i	$0.552173\pi$
$798$	0	0
$799$	−5.95570	−0.210697
$800$	10.4737	0.370300
$801$	0	0
$802$	9.30485	0.328566
$803$	−15.4599	−0.545567
$804$	0	0
$805$	24.8167	0.874673
$806$	32.1195	1.13136
$807$	0	0
$808$	−20.9459	−0.736873
$809$	−37.5769	−1.32113	−0.660567	−	0.750767i	$-0.729684\pi$
$809$	−37.5769	−1.32113	−0.660567	+	0.750767i	$0.729684\pi$
$810$	0	0
$811$	−42.7728	−1.50196	−0.750979	−	0.660327i	$-0.770418\pi$
$811$	−42.7728	−1.50196	−0.750979	+	0.660327i	$0.770418\pi$
$812$	−1.11477	−0.0391207
$813$	0	0
$814$	4.25739	0.149222
$815$	−47.3517	−1.65866
$816$	0	0
$817$	−13.2911	−0.464997
$818$	−38.7464	−1.35474
$819$	0	0
$820$	−14.1652	−0.494670
$821$	−6.38778	−0.222935	−0.111468	−	0.993768i	$-0.535555\pi$
$821$	−6.38778	−0.222935	−0.111468	+	0.993768i	$0.535555\pi$
$822$	0	0
$823$	45.7263	1.59392	0.796960	−	0.604033i	$-0.206440\pi$
$823$	45.7263	1.59392	0.796960	+	0.604033i	$0.206440\pi$
$824$	−23.5426	−0.820147
$825$	0	0
$826$	−8.04161	−0.279803
$827$	15.5401	0.540382	0.270191	−	0.962807i	$-0.412913\pi$
$827$	15.5401	0.540382	0.270191	+	0.962807i	$0.412913\pi$
$828$	0	0
$829$	11.0103	0.382403	0.191201	−	0.981551i	$-0.438762\pi$
$829$	11.0103	0.382403	0.191201	+	0.981551i	$0.438762\pi$
$830$	−13.6938	−0.475320
$831$	0	0
$832$	−36.2066	−1.25524
$833$	18.9968	0.658201
$834$	0	0
$835$	−24.3459	−0.842524
$836$	−2.00555	−0.0693633
$837$	0	0
$838$	−14.9820	−0.517544
$839$	−44.4487	−1.53454	−0.767270	−	0.641325i	$-0.778385\pi$
$839$	−44.4487	−1.53454	−0.767270	+	0.641325i	$0.778385\pi$
$840$	0	0
$841$	−22.4605	−0.774501
$842$	−18.6320	−0.642102
$843$	0	0
$844$	−3.76033	−0.129436
$845$	81.9621	2.81958
$846$	0	0
$847$	−9.68318	−0.332718
$848$	18.6240	0.639549
$849$	0	0
$850$	23.1077	0.792586
$851$	−14.8900	−0.510421
$852$	0	0
$853$	−18.0717	−0.618764	−0.309382	−	0.950938i	$-0.600122\pi$
$853$	−18.0717	−0.618764	−0.309382	+	0.950938i	$0.600122\pi$
$854$	−16.3428	−0.559239
$855$	0	0
$856$	−24.9215	−0.851798
$857$	−42.1769	−1.44073	−0.720367	−	0.693593i	$-0.756027\pi$
$857$	−42.1769	−1.44073	−0.720367	+	0.693593i	$0.756027\pi$
$858$	0	0
$859$	−11.8565	−0.404540	−0.202270	−	0.979330i	$-0.564832\pi$
$859$	−11.8565	−0.404540	−0.202270	+	0.979330i	$0.564832\pi$
$860$	4.75241	0.162056
$861$	0	0
$862$	61.1827	2.08389
$863$	−18.8577	−0.641924	−0.320962	−	0.947092i	$-0.604006\pi$
$863$	−18.8577	−0.641924	−0.320962	+	0.947092i	$0.604006\pi$
$864$	0	0
$865$	−14.0287	−0.476989
$866$	25.0822	0.852328
$867$	0	0
$868$	1.43613	0.0487456
$869$	21.1530	0.717566
$870$	0	0
$871$	−13.7152	−0.464721
$872$	−44.7158	−1.51427
$873$	0	0
$874$	41.3206	1.39769
$875$	−1.33604	−0.0451664
$876$	0	0
$877$	10.9959	0.371307	0.185653	−	0.982615i	$-0.440560\pi$
$877$	10.9959	0.371307	0.185653	+	0.982615i	$0.440560\pi$
$878$	49.2279	1.66136
$879$	0	0
$880$	−19.9443	−0.672324
$881$	−8.18806	−0.275863	−0.137931	−	0.990442i	$-0.544045\pi$
$881$	−8.18806	−0.275863	−0.137931	+	0.990442i	$0.544045\pi$
$882$	0	0
$883$	−14.5462	−0.489518	−0.244759	−	0.969584i	$-0.578709\pi$
$883$	−14.5462	−0.489518	−0.244759	+	0.969584i	$0.578709\pi$
$884$	8.32213	0.279903
$885$	0	0
$886$	−40.9027	−1.37415
$887$	39.2569	1.31812	0.659058	−	0.752092i	$-0.270955\pi$
$887$	39.2569	1.31812	0.659058	+	0.752092i	$0.270955\pi$
$888$	0	0
$889$	1.42140	0.0476722
$890$	25.2038	0.844833
$891$	0	0
$892$	−5.36233	−0.179544
$893$	−6.51223	−0.217924
$894$	0	0
$895$	14.2797	0.477319
$896$	−14.3960	−0.480938
$897$	0	0
$898$	−48.6797	−1.62446
$899$	−8.42469	−0.280979
$900$	0	0
$901$	12.9742	0.432234
$902$	24.0292	0.800086
$903$	0	0
$904$	1.57618	0.0524231
$905$	25.3073	0.841243
$906$	0	0
$907$	−0.280162	−0.00930263	−0.00465132	−	0.999989i	$-0.501481\pi$
$907$	−0.280162	−0.00930263	−0.00465132	+	0.999989i	$0.501481\pi$
$908$	5.78482	0.191976
$909$	0	0
$910$	32.1953	1.06727
$911$	24.1986	0.801735	0.400868	−	0.916136i	$-0.368709\pi$
$911$	24.1986	0.801735	0.400868	+	0.916136i	$0.368709\pi$
$912$	0	0
$913$	3.94329	0.130504
$914$	−0.281025	−0.00929550
$915$	0	0
$916$	5.22851	0.172755
$917$	−19.6072	−0.647487
$918$	0	0
$919$	16.7088	0.551174	0.275587	−	0.961276i	$-0.411128\pi$
$919$	16.7088	0.551174	0.275587	+	0.961276i	$0.411128\pi$
$920$	57.4874	1.89530
$921$	0	0
$922$	−1.00951	−0.0332463
$923$	26.5197	0.872906
$924$	0	0
$925$	−9.10501	−0.299371
$926$	12.3842	0.406970
$927$	0	0
$928$	−5.82836	−0.191325
$929$	6.12959	0.201105	0.100553	−	0.994932i	$-0.467939\pi$
$929$	6.12959	0.201105	0.100553	+	0.994932i	$0.467939\pi$
$930$	0	0
$931$	20.7720	0.680775
$932$	−2.81009	−0.0920474
$933$	0	0
$934$	7.48259	0.244838
$935$	−13.8940	−0.454384
$936$	0	0
$937$	−41.3386	−1.35047	−0.675236	−	0.737602i	$-0.735958\pi$
$937$	−41.3386	−1.35047	−0.675236	+	0.737602i	$0.735958\pi$
$938$	−3.61253	−0.117953
$939$	0	0
$940$	2.32853	0.0759484
$941$	55.2101	1.79980	0.899900	−	0.436097i	$-0.143640\pi$
$941$	55.2101	1.79980	0.899900	+	0.436097i	$0.143640\pi$
$942$	0	0
$943$	−84.0407	−2.73674
$944$	−22.6031	−0.735670
$945$	0	0
$946$	−8.06179	−0.262112
$947$	28.2391	0.917647	0.458823	−	0.888528i	$-0.348271\pi$
$947$	28.2391	0.917647	0.458823	+	0.888528i	$0.348271\pi$
$948$	0	0
$949$	70.1463	2.27704
$950$	25.2670	0.819769
$951$	0	0
$952$	−8.52899	−0.276426
$953$	26.5565	0.860249	0.430124	−	0.902770i	$-0.358470\pi$
$953$	26.5565	0.860249	0.430124	+	0.902770i	$0.358470\pi$
$954$	0	0
$955$	12.2165	0.395316
$956$	−5.51199	−0.178271
$957$	0	0
$958$	−48.7984	−1.57660
$959$	4.96945	0.160472
$960$	0	0
$961$	−20.1466	−0.649891
$962$	−19.3171	−0.622810
$963$	0	0
$964$	−0.144148	−0.00464269
$965$	22.3068	0.718081
$966$	0	0
$967$	−26.4092	−0.849262	−0.424631	−	0.905367i	$-0.639596\pi$
$967$	−26.4092	−0.849262	−0.424631	+	0.905367i	$0.639596\pi$
$968$	−22.4309	−0.720957
$969$	0	0
$970$	57.9047	1.85921
$971$	22.9842	0.737600	0.368800	−	0.929509i	$-0.379769\pi$
$971$	22.9842	0.737600	0.368800	+	0.929509i	$0.379769\pi$
$972$	0	0
$973$	1.08584	0.0348106
$974$	−6.64792	−0.213013
$975$	0	0
$976$	−45.9359	−1.47037
$977$	46.8752	1.49967	0.749835	−	0.661625i	$-0.230133\pi$
$977$	46.8752	1.49967	0.749835	+	0.661625i	$0.230133\pi$
$978$	0	0
$979$	−7.25771	−0.231957
$980$	−7.42730	−0.237257
$981$	0	0
$982$	−9.70270	−0.309626
$983$	−58.1599	−1.85501	−0.927506	−	0.373808i	$-0.878052\pi$
$983$	−58.1599	−1.85501	−0.927506	+	0.373808i	$0.878052\pi$
$984$	0	0
$985$	31.7612	1.01200
$986$	−12.8589	−0.409510
$987$	0	0
$988$	9.09980	0.289503
$989$	28.1956	0.896568
$990$	0	0
$991$	33.3871	1.06058	0.530288	−	0.847818i	$-0.322084\pi$
$991$	33.3871	1.06058	0.530288	+	0.847818i	$0.322084\pi$
$992$	7.50857	0.238397
$993$	0	0
$994$	6.98519	0.221557
$995$	82.1256	2.60356
$996$	0	0
$997$	−15.6656	−0.496134	−0.248067	−	0.968743i	$-0.579795\pi$
$997$	−15.6656	−0.496134	−0.248067	+	0.968743i	$0.579795\pi$
$998$	−5.14826	−0.162965
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.19		72
3.2	odd	2		6561.2.a.d.1.54		72
81.5	odd	54		729.2.g.a.460.7		144
81.11	odd	54		729.2.g.b.514.2		144
81.16	even	27		729.2.g.d.271.2		144
81.22	even	27		729.2.g.c.217.7		144
81.32	odd	54		243.2.g.a.154.2		144
81.38	odd	54		243.2.g.a.172.2		144
81.43	even	27		81.2.g.a.67.7	yes	144
81.49	even	27		81.2.g.a.52.7	✓	144
81.59	odd	54		729.2.g.b.217.2		144
81.65	odd	54		729.2.g.a.271.7		144
81.70	even	27		729.2.g.c.514.7		144
81.76	even	27		729.2.g.d.460.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.52.7	✓	144	81.49	even	27
81.2.g.a.67.7	yes	144	81.43	even	27
243.2.g.a.154.2		144	81.32	odd	54
243.2.g.a.172.2		144	81.38	odd	54
729.2.g.a.271.7		144	81.65	odd	54
729.2.g.a.460.7		144	81.5	odd	54
729.2.g.b.217.2		144	81.59	odd	54
729.2.g.b.514.2		144	81.11	odd	54
729.2.g.c.217.7		144	81.22	even	27
729.2.g.c.514.7		144	81.70	even	27
729.2.g.d.271.2		144	81.16	even	27
729.2.g.d.460.2		144	81.76	even	27
6561.2.a.c.1.19		72	1.1	even	1	trivial
6561.2.a.d.1.54		72	3.2	odd	2

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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-1.55207 q^{2} +0.408920 q^{4} +3.09765 q^{5} +1.06604 q^{7} +2.46947 q^{8} +O(q^{10})\)	\(q-1.55207 q^{2} +0.408920 q^{4} +3.09765 q^{5} +1.06604 q^{7} +2.46947 q^{8} -4.80776 q^{10} +1.38445 q^{11} -6.28168 q^{13} -1.65457 q^{14} -4.65062 q^{16} -3.23982 q^{17} -3.54256 q^{19} +1.26669 q^{20} -2.14876 q^{22} +7.51514 q^{23} +4.59541 q^{25} +9.74960 q^{26} +0.435926 q^{28} -2.55724 q^{29} +3.29445 q^{31} +2.27916 q^{32} +5.02842 q^{34} +3.30222 q^{35} -1.98133 q^{37} +5.49831 q^{38} +7.64953 q^{40} -11.1828 q^{41} +3.75184 q^{43} +0.566129 q^{44} -11.6640 q^{46} +1.83828 q^{47} -5.86355 q^{49} -7.13240 q^{50} -2.56870 q^{52} -4.00461 q^{53} +4.28853 q^{55} +2.63256 q^{56} +3.96902 q^{58} +4.86024 q^{59} +9.87736 q^{61} -5.11321 q^{62} +5.76384 q^{64} -19.4584 q^{65} +2.18336 q^{67} -1.32483 q^{68} -5.12528 q^{70} -4.22175 q^{71} -11.1668 q^{73} +3.07516 q^{74} -1.44863 q^{76} +1.47588 q^{77} +15.2790 q^{79} -14.4060 q^{80} +17.3566 q^{82} +2.84828 q^{83} -10.0358 q^{85} -5.82311 q^{86} +3.41885 q^{88} -5.24231 q^{89} -6.69653 q^{91} +3.07309 q^{92} -2.85314 q^{94} -10.9736 q^{95} -12.0440 q^{97} +9.10064 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

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