LMFDB - Embedded newform 6561.2.a.c.1.10

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			1.10
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)

$f(q)$	$=$	$q-2.36209 q^{2} +3.57945 q^{4} -2.65977 q^{5} +4.59390 q^{7} -3.73079 q^{8} +O(q^{10})$	\(q-2.36209 q^{2} +3.57945 q^{4} -2.65977 q^{5} +4.59390 q^{7} -3.73079 q^{8} +6.28260 q^{10} +0.484075 q^{11} -4.35502 q^{13} -10.8512 q^{14} +1.65355 q^{16} -0.996857 q^{17} -0.870812 q^{19} -9.52050 q^{20} -1.14343 q^{22} -2.05269 q^{23} +2.07436 q^{25} +10.2869 q^{26} +16.4436 q^{28} +0.830241 q^{29} +4.00712 q^{31} +3.55575 q^{32} +2.35466 q^{34} -12.2187 q^{35} +1.09671 q^{37} +2.05693 q^{38} +9.92303 q^{40} +0.292601 q^{41} +11.2232 q^{43} +1.73272 q^{44} +4.84862 q^{46} -4.03819 q^{47} +14.1039 q^{49} -4.89981 q^{50} -15.5886 q^{52} -1.61353 q^{53} -1.28753 q^{55} -17.1389 q^{56} -1.96110 q^{58} +5.27001 q^{59} +4.03415 q^{61} -9.46516 q^{62} -11.7061 q^{64} +11.5833 q^{65} -14.4228 q^{67} -3.56820 q^{68} +28.8616 q^{70} -12.6775 q^{71} +5.80341 q^{73} -2.59053 q^{74} -3.11703 q^{76} +2.22379 q^{77} -6.47412 q^{79} -4.39806 q^{80} -0.691147 q^{82} +12.7005 q^{83} +2.65141 q^{85} -26.5101 q^{86} -1.80598 q^{88} -15.8161 q^{89} -20.0065 q^{91} -7.34749 q^{92} +9.53855 q^{94} +2.31616 q^{95} +2.02665 q^{97} -33.3147 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.36209	−1.67025	−0.835123	−	0.550063i	$-0.814604\pi$
$2$	−2.36209	−1.67025	−0.835123	+	0.550063i	$0.814604\pi$
$3$	0	0
$3$	0	0
$4$	3.57945	1.78972
$5$	−2.65977	−1.18948	−0.594742	−	0.803917i	$-0.702746\pi$
$5$	−2.65977	−1.18948	−0.594742	+	0.803917i	$0.702746\pi$
$6$	0	0
$7$	4.59390	1.73633	0.868166	−	0.496274i	$-0.165299\pi$
$7$	4.59390	1.73633	0.868166	+	0.496274i	$0.165299\pi$
$8$	−3.73079	−1.31903
$9$	0	0
$10$	6.28260	1.98673
$11$	0.484075	0.145954	0.0729770	−	0.997334i	$-0.476750\pi$
$11$	0.484075	0.145954	0.0729770	+	0.997334i	$0.476750\pi$
$12$	0	0
$13$	−4.35502	−1.20787	−0.603933	−	0.797035i	$-0.706400\pi$
$13$	−4.35502	−1.20787	−0.603933	+	0.797035i	$0.706400\pi$
$14$	−10.8512	−2.90010
$15$	0	0
$16$	1.65355	0.413388
$17$	−0.996857	−0.241773	−0.120887	−	0.992666i	$-0.538574\pi$
$17$	−0.996857	−0.241773	−0.120887	+	0.992666i	$0.538574\pi$
$18$	0	0
$19$	−0.870812	−0.199778	−0.0998890	−	0.994999i	$-0.531849\pi$
$19$	−0.870812	−0.199778	−0.0998890	+	0.994999i	$0.531849\pi$
$20$	−9.52050	−2.12885
$21$	0	0
$22$	−1.14343	−0.243779
$23$	−2.05269	−0.428015	−0.214007	−	0.976832i	$-0.568652\pi$
$23$	−2.05269	−0.428015	−0.214007	+	0.976832i	$0.568652\pi$
$24$	0	0
$25$	2.07436	0.414872
$26$	10.2869	2.01743
$27$	0	0
$28$	16.4436	3.10755
$29$	0.830241	0.154172	0.0770859	−	0.997024i	$-0.475438\pi$
$29$	0.830241	0.154172	0.0770859	+	0.997024i	$0.475438\pi$
$30$	0	0
$31$	4.00712	0.719700	0.359850	−	0.933010i	$-0.382828\pi$
$31$	4.00712	0.719700	0.359850	+	0.933010i	$0.382828\pi$
$32$	3.55575	0.628574
$33$	0	0
$34$	2.35466	0.403821
$35$	−12.2187	−2.06534
$36$	0	0
$37$	1.09671	0.180298	0.0901492	−	0.995928i	$-0.471266\pi$
$37$	1.09671	0.180298	0.0901492	+	0.995928i	$0.471266\pi$
$38$	2.05693	0.333679
$39$	0	0
$40$	9.92303	1.56897
$41$	0.292601	0.0456965	0.0228483	−	0.999739i	$-0.492727\pi$
$41$	0.292601	0.0456965	0.0228483	+	0.999739i	$0.492727\pi$
$42$	0	0
$43$	11.2232	1.71152	0.855760	−	0.517373i	$-0.173090\pi$
$43$	11.2232	1.71152	0.855760	+	0.517373i	$0.173090\pi$
$44$	1.73272	0.261217
$45$	0	0
$46$	4.84862	0.714891
$47$	−4.03819	−0.589031	−0.294515	−	0.955647i	$-0.595158\pi$
$47$	−4.03819	−0.589031	−0.294515	+	0.955647i	$0.595158\pi$
$48$	0	0
$49$	14.1039	2.01485
$50$	−4.89981	−0.692938
$51$	0	0
$52$	−15.5886	−2.16175
$53$	−1.61353	−0.221635	−0.110818	−	0.993841i	$-0.535347\pi$
$53$	−1.61353	−0.221635	−0.110818	+	0.993841i	$0.535347\pi$
$54$	0	0
$55$	−1.28753	−0.173610
$56$	−17.1389	−2.29028
$57$	0	0
$58$	−1.96110	−0.257505
$59$	5.27001	0.686097	0.343048	−	0.939318i	$-0.388541\pi$
$59$	5.27001	0.686097	0.343048	+	0.939318i	$0.388541\pi$
$60$	0	0
$61$	4.03415	0.516520	0.258260	−	0.966075i	$-0.416851\pi$
$61$	4.03415	0.516520	0.258260	+	0.966075i	$0.416851\pi$
$62$	−9.46516	−1.20208
$63$	0	0
$64$	−11.7061	−1.46326
$65$	11.5833	1.43674
$66$	0	0
$67$	−14.4228	−1.76203	−0.881015	−	0.473088i	$-0.843139\pi$
$67$	−14.4228	−1.76203	−0.881015	+	0.473088i	$0.843139\pi$
$68$	−3.56820	−0.432708
$69$	0	0
$70$	28.8616	3.44962
$71$	−12.6775	−1.50454	−0.752272	−	0.658853i	$-0.771042\pi$
$71$	−12.6775	−1.50454	−0.752272	+	0.658853i	$0.771042\pi$
$72$	0	0
$73$	5.80341	0.679237	0.339619	−	0.940563i	$-0.389702\pi$
$73$	5.80341	0.679237	0.339619	+	0.940563i	$0.389702\pi$
$74$	−2.59053	−0.301143
$75$	0	0
$76$	−3.11703	−0.357547
$77$	2.22379	0.253425
$78$	0	0
$79$	−6.47412	−0.728396	−0.364198	−	0.931322i	$-0.618657\pi$
$79$	−6.47412	−0.728396	−0.364198	+	0.931322i	$0.618657\pi$
$80$	−4.39806	−0.491718
$81$	0	0
$82$	−0.691147	−0.0763245
$83$	12.7005	1.39407	0.697033	−	0.717039i	$-0.254503\pi$
$83$	12.7005	1.39407	0.697033	+	0.717039i	$0.254503\pi$
$84$	0	0
$85$	2.65141	0.287586
$86$	−26.5101	−2.85866
$87$	0	0
$88$	−1.80598	−0.192518
$89$	−15.8161	−1.67650	−0.838250	−	0.545287i	$-0.816421\pi$
$89$	−15.8161	−1.67650	−0.838250	+	0.545287i	$0.816421\pi$
$90$	0	0
$91$	−20.0065	−2.09725
$92$	−7.34749	−0.766029
$93$	0	0
$94$	9.53855	0.983826
$95$	2.31616	0.237633
$96$	0	0
$97$	2.02665	0.205776	0.102888	−	0.994693i	$-0.467192\pi$
$97$	2.02665	0.205776	0.102888	+	0.994693i	$0.467192\pi$
$98$	−33.3147	−3.36529
$99$	0	0
$100$	7.42506	0.742506
$101$	−7.21959	−0.718376	−0.359188	−	0.933265i	$-0.616946\pi$
$101$	−7.21959	−0.718376	−0.359188	+	0.933265i	$0.616946\pi$
$102$	0	0
$103$	4.38665	0.432229	0.216115	−	0.976368i	$-0.430662\pi$
$103$	4.38665	0.432229	0.216115	+	0.976368i	$0.430662\pi$
$104$	16.2477	1.59322
$105$	0	0
$106$	3.81129	0.370185
$107$	−6.35670	−0.614525	−0.307263	−	0.951625i	$-0.599413\pi$
$107$	−6.35670	−0.614525	−0.307263	+	0.951625i	$0.599413\pi$
$108$	0	0
$109$	−8.80430	−0.843299	−0.421649	−	0.906759i	$-0.638549\pi$
$109$	−8.80430	−0.843299	−0.421649	+	0.906759i	$0.638549\pi$
$110$	3.04125	0.289971
$111$	0	0
$112$	7.59626	0.717779
$113$	−17.7156	−1.66654	−0.833272	−	0.552863i	$-0.813535\pi$
$113$	−17.7156	−1.66654	−0.833272	+	0.552863i	$0.813535\pi$
$114$	0	0
$115$	5.45967	0.509117
$116$	2.97180	0.275925
$117$	0	0
$118$	−12.4482	−1.14595
$119$	−4.57946	−0.419799
$120$	0	0
$121$	−10.7657	−0.978697
$122$	−9.52901	−0.862716
$123$	0	0
$124$	14.3433	1.28806
$125$	7.78152	0.696001
$126$	0	0
$127$	5.55125	0.492594	0.246297	−	0.969194i	$-0.420786\pi$
$127$	5.55125	0.492594	0.246297	+	0.969194i	$0.420786\pi$
$128$	20.5393	1.81543
$129$	0	0
$130$	−27.3608	−2.39970
$131$	18.3995	1.60757	0.803786	−	0.594919i	$-0.202816\pi$
$131$	18.3995	1.60757	0.803786	+	0.594919i	$0.202816\pi$
$132$	0	0
$133$	−4.00042	−0.346881
$134$	34.0680	2.94302
$135$	0	0
$136$	3.71907	0.318907
$137$	5.71285	0.488082	0.244041	−	0.969765i	$-0.421527\pi$
$137$	5.71285	0.488082	0.244041	+	0.969765i	$0.421527\pi$
$138$	0	0
$139$	−9.59352	−0.813712	−0.406856	−	0.913492i	$-0.633375\pi$
$139$	−9.59352	−0.813712	−0.406856	+	0.913492i	$0.633375\pi$
$140$	−43.7362	−3.69639
$141$	0	0
$142$	29.9454	2.51296
$143$	−2.10815	−0.176293
$144$	0	0
$145$	−2.20825	−0.183385
$146$	−13.7081	−1.13449
$147$	0	0
$148$	3.92563	0.322684
$149$	16.4730	1.34952	0.674760	−	0.738037i	$-0.264247\pi$
$149$	16.4730	1.34952	0.674760	+	0.738037i	$0.264247\pi$
$150$	0	0
$151$	11.7681	0.957676	0.478838	−	0.877903i	$-0.341058\pi$
$151$	11.7681	0.957676	0.478838	+	0.877903i	$0.341058\pi$
$152$	3.24882	0.263514
$153$	0	0
$154$	−5.25279	−0.423282
$155$	−10.6580	−0.856072
$156$	0	0
$157$	−13.5436	−1.08089	−0.540447	−	0.841378i	$-0.681745\pi$
$157$	−13.5436	−1.08089	−0.540447	+	0.841378i	$0.681745\pi$
$158$	15.2924	1.21660
$159$	0	0
$160$	−9.45747	−0.747678
$161$	−9.42984	−0.743176
$162$	0	0
$163$	−13.5555	−1.06175	−0.530874	−	0.847451i	$-0.678136\pi$
$163$	−13.5555	−1.06175	−0.530874	+	0.847451i	$0.678136\pi$
$164$	1.04735	0.0817842
$165$	0	0
$166$	−29.9998	−2.32843
$167$	11.0731	0.856862	0.428431	−	0.903574i	$-0.359067\pi$
$167$	11.0731	0.856862	0.428431	+	0.903574i	$0.359067\pi$
$168$	0	0
$169$	5.96619	0.458938
$170$	−6.26285	−0.480339
$171$	0	0
$172$	40.1728	3.06315
$173$	12.9653	0.985737	0.492868	−	0.870104i	$-0.335948\pi$
$173$	12.9653	0.985737	0.492868	+	0.870104i	$0.335948\pi$
$174$	0	0
$175$	9.52940	0.720355
$176$	0.800443	0.0603357
$177$	0	0
$178$	37.3589	2.80017
$179$	25.2956	1.89069	0.945343	−	0.326079i	$-0.105727\pi$
$179$	25.2956	1.89069	0.945343	+	0.326079i	$0.105727\pi$
$180$	0	0
$181$	−10.8829	−0.808919	−0.404460	−	0.914556i	$-0.632540\pi$
$181$	−10.8829	−0.808919	−0.404460	+	0.914556i	$0.632540\pi$
$182$	47.2571	3.50293
$183$	0	0
$184$	7.65815	0.564566
$185$	−2.91700	−0.214462
$186$	0	0
$187$	−0.482553	−0.0352878
$188$	−14.4545	−1.05420
$189$	0	0
$190$	−5.47096	−0.396905
$191$	11.8157	0.854955	0.427477	−	0.904026i	$-0.359402\pi$
$191$	11.8157	0.854955	0.427477	+	0.904026i	$0.359402\pi$
$192$	0	0
$193$	−8.61147	−0.619867	−0.309934	−	0.950758i	$-0.600307\pi$
$193$	−8.61147	−0.619867	−0.309934	+	0.950758i	$0.600307\pi$
$194$	−4.78713	−0.343696
$195$	0	0
$196$	50.4843	3.60602
$197$	14.6952	1.04699	0.523495	−	0.852029i	$-0.324628\pi$
$197$	14.6952	1.04699	0.523495	+	0.852029i	$0.324628\pi$
$198$	0	0
$199$	−14.2838	−1.01255	−0.506274	−	0.862373i	$-0.668978\pi$
$199$	−14.2838	−1.01255	−0.506274	+	0.862373i	$0.668978\pi$
$200$	−7.73900	−0.547230
$201$	0	0
$202$	17.0533	1.19987
$203$	3.81404	0.267693
$204$	0	0
$205$	−0.778249	−0.0543553
$206$	−10.3616	−0.721929
$207$	0	0
$208$	−7.20125	−0.499317
$209$	−0.421538	−0.0291584
$210$	0	0
$211$	2.78048	0.191416	0.0957081	−	0.995409i	$-0.469488\pi$
$211$	2.78048	0.191416	0.0957081	+	0.995409i	$0.469488\pi$
$212$	−5.77554	−0.396666
$213$	0	0
$214$	15.0151	1.02641
$215$	−29.8511	−2.03583
$216$	0	0
$217$	18.4083	1.24964
$218$	20.7965	1.40852
$219$	0	0
$220$	−4.60863	−0.310714
$221$	4.34133	0.292030
$222$	0	0
$223$	−23.8397	−1.59642	−0.798212	−	0.602377i	$-0.794220\pi$
$223$	−23.8397	−1.59642	−0.798212	+	0.602377i	$0.794220\pi$
$224$	16.3348	1.09141
$225$	0	0
$226$	41.8458	2.78354
$227$	3.36595	0.223406	0.111703	−	0.993742i	$-0.464370\pi$
$227$	3.36595	0.223406	0.111703	+	0.993742i	$0.464370\pi$
$228$	0	0
$229$	9.22555	0.609641	0.304821	−	0.952410i	$-0.401403\pi$
$229$	9.22555	0.609641	0.304821	+	0.952410i	$0.401403\pi$
$230$	−12.8962	−0.850351
$231$	0	0
$232$	−3.09745	−0.203358
$233$	−4.22267	−0.276636	−0.138318	−	0.990388i	$-0.544170\pi$
$233$	−4.22267	−0.276636	−0.138318	+	0.990388i	$0.544170\pi$
$234$	0	0
$235$	10.7406	0.700642
$236$	18.8637	1.22792
$237$	0	0
$238$	10.8171	0.701167
$239$	−4.51239	−0.291883	−0.145941	−	0.989293i	$-0.546621\pi$
$239$	−4.51239	−0.291883	−0.145941	+	0.989293i	$0.546621\pi$
$240$	0	0
$241$	8.42204	0.542511	0.271256	−	0.962507i	$-0.412561\pi$
$241$	8.42204	0.542511	0.271256	+	0.962507i	$0.412561\pi$
$242$	25.4294	1.63467
$243$	0	0
$244$	14.4400	0.924428
$245$	−37.5132	−2.39663
$246$	0	0
$247$	3.79240	0.241305
$248$	−14.9497	−0.949309
$249$	0	0
$250$	−18.3806	−1.16249
$251$	9.18621	0.579829	0.289914	−	0.957053i	$-0.406373\pi$
$251$	9.18621	0.579829	0.289914	+	0.957053i	$0.406373\pi$
$252$	0	0
$253$	−0.993654	−0.0624705
$254$	−13.1125	−0.822753
$255$	0	0
$256$	−25.1034	−1.56896
$257$	−8.23248	−0.513528	−0.256764	−	0.966474i	$-0.582656\pi$
$257$	−8.23248	−0.513528	−0.256764	+	0.966474i	$0.582656\pi$
$258$	0	0
$259$	5.03819	0.313058
$260$	41.4619	2.57136
$261$	0	0
$262$	−43.4612	−2.68504
$263$	0.431080	0.0265815	0.0132908	−	0.999912i	$-0.495769\pi$
$263$	0.431080	0.0265815	0.0132908	+	0.999912i	$0.495769\pi$
$264$	0	0
$265$	4.29161	0.263631
$266$	9.44935	0.579377
$267$	0	0
$268$	−51.6258	−3.15355
$269$	10.6777	0.651030	0.325515	−	0.945537i	$-0.394462\pi$
$269$	10.6777	0.651030	0.325515	+	0.945537i	$0.394462\pi$
$270$	0	0
$271$	−15.3058	−0.929759	−0.464880	−	0.885374i	$-0.653902\pi$
$271$	−15.3058	−0.929759	−0.464880	+	0.885374i	$0.653902\pi$
$272$	−1.64836	−0.0999462
$273$	0	0
$274$	−13.4942	−0.815217
$275$	1.00414	0.0605522
$276$	0	0
$277$	−12.4206	−0.746283	−0.373142	−	0.927774i	$-0.621719\pi$
$277$	−12.4206	−0.746283	−0.373142	+	0.927774i	$0.621719\pi$
$278$	22.6607	1.35910
$279$	0	0
$280$	45.5854	2.72425
$281$	−31.4134	−1.87397	−0.936985	−	0.349371i	$-0.886395\pi$
$281$	−31.4134	−1.87397	−0.936985	+	0.349371i	$0.886395\pi$
$282$	0	0
$283$	−0.586993	−0.0348931	−0.0174466	−	0.999848i	$-0.505554\pi$
$283$	−0.586993	−0.0348931	−0.0174466	+	0.999848i	$0.505554\pi$
$284$	−45.3785	−2.69272
$285$	0	0
$286$	4.97964	0.294452
$287$	1.34418	0.0793443
$288$	0	0
$289$	−16.0063	−0.941546
$290$	5.21607	0.306298
$291$	0	0
$292$	20.7730	1.21565
$293$	−29.9545	−1.74996	−0.874979	−	0.484161i	$-0.839125\pi$
$293$	−29.9545	−1.74996	−0.874979	+	0.484161i	$0.839125\pi$
$294$	0	0
$295$	−14.0170	−0.816101
$296$	−4.09161	−0.237820
$297$	0	0
$298$	−38.9106	−2.25403
$299$	8.93949	0.516984
$300$	0	0
$301$	51.5582	2.97177
$302$	−27.7973	−1.59956
$303$	0	0
$304$	−1.43993	−0.0825858
$305$	−10.7299	−0.614392
$306$	0	0
$307$	−2.38538	−0.136141	−0.0680704	−	0.997681i	$-0.521684\pi$
$307$	−2.38538	−0.136141	−0.0680704	+	0.997681i	$0.521684\pi$
$308$	7.95995	0.453560
$309$	0	0
$310$	25.1751	1.42985
$311$	−26.7176	−1.51502	−0.757508	−	0.652826i	$-0.773583\pi$
$311$	−26.7176	−1.51502	−0.757508	+	0.652826i	$0.773583\pi$
$312$	0	0
$313$	7.00877	0.396159	0.198080	−	0.980186i	$-0.436530\pi$
$313$	7.00877	0.396159	0.198080	+	0.980186i	$0.436530\pi$
$314$	31.9910	1.80536
$315$	0	0
$316$	−23.1738	−1.30363
$317$	−14.0833	−0.790999	−0.395500	−	0.918466i	$-0.629429\pi$
$317$	−14.0833	−0.790999	−0.395500	+	0.918466i	$0.629429\pi$
$318$	0	0
$319$	0.401898	0.0225020
$320$	31.1355	1.74053
$321$	0	0
$322$	22.2741	1.24129
$323$	0.868075	0.0483010
$324$	0	0
$325$	−9.03387	−0.501109
$326$	32.0192	1.77338
$327$	0	0
$328$	−1.09163	−0.0602753
$329$	−18.5510	−1.02275
$330$	0	0
$331$	29.1813	1.60395	0.801975	−	0.597358i	$-0.203783\pi$
$331$	29.1813	1.60395	0.801975	+	0.597358i	$0.203783\pi$
$332$	45.4609	2.49499
$333$	0	0
$334$	−26.1556	−1.43117
$335$	38.3614	2.09591
$336$	0	0
$337$	0.712784	0.0388278	0.0194139	−	0.999812i	$-0.493820\pi$
$337$	0.712784	0.0388278	0.0194139	+	0.999812i	$0.493820\pi$
$338$	−14.0927	−0.766540
$339$	0	0
$340$	9.49058	0.514699
$341$	1.93975	0.105043
$342$	0	0
$343$	32.6348	1.76211
$344$	−41.8714	−2.25755
$345$	0	0
$346$	−30.6253	−1.64642
$347$	13.8038	0.741026	0.370513	−	0.928827i	$-0.379182\pi$
$347$	13.8038	0.741026	0.370513	+	0.928827i	$0.379182\pi$
$348$	0	0
$349$	−17.3587	−0.929190	−0.464595	−	0.885523i	$-0.653800\pi$
$349$	−17.3587	−0.929190	−0.464595	+	0.885523i	$0.653800\pi$
$350$	−22.5093	−1.20317
$351$	0	0
$352$	1.72125	0.0917429
$353$	−10.5057	−0.559164	−0.279582	−	0.960122i	$-0.590196\pi$
$353$	−10.5057	−0.559164	−0.279582	+	0.960122i	$0.590196\pi$
$354$	0	0
$355$	33.7192	1.78963
$356$	−56.6128	−3.00047
$357$	0	0
$358$	−59.7505	−3.15791
$359$	−2.58228	−0.136288	−0.0681438	−	0.997676i	$-0.521708\pi$
$359$	−2.58228	−0.136288	−0.0681438	+	0.997676i	$0.521708\pi$
$360$	0	0
$361$	−18.2417	−0.960089
$362$	25.7063	1.35109
$363$	0	0
$364$	−71.6123	−3.75351
$365$	−15.4357	−0.807942
$366$	0	0
$367$	22.1049	1.15386	0.576932	−	0.816792i	$-0.304250\pi$
$367$	22.1049	1.15386	0.576932	+	0.816792i	$0.304250\pi$
$368$	−3.39423	−0.176936
$369$	0	0
$370$	6.89020	0.358205
$371$	−7.41239	−0.384832
$372$	0	0
$373$	−10.3711	−0.536994	−0.268497	−	0.963281i	$-0.586527\pi$
$373$	−10.3711	−0.536994	−0.268497	+	0.963281i	$0.586527\pi$
$374$	1.13983	0.0589393
$375$	0	0
$376$	15.0656	0.776951
$377$	−3.61571	−0.186219
$378$	0	0
$379$	−7.92961	−0.407317	−0.203658	−	0.979042i	$-0.565283\pi$
$379$	−7.92961	−0.407317	−0.203658	+	0.979042i	$0.565283\pi$
$380$	8.29056	0.425297
$381$	0	0
$382$	−27.9097	−1.42799
$383$	4.38771	0.224202	0.112101	−	0.993697i	$-0.464242\pi$
$383$	4.38771	0.224202	0.112101	+	0.993697i	$0.464242\pi$
$384$	0	0
$385$	−5.91477	−0.301444
$386$	20.3410	1.03533
$387$	0	0
$388$	7.25430	0.368282
$389$	−3.10587	−0.157474	−0.0787370	−	0.996895i	$-0.525089\pi$
$389$	−3.10587	−0.157474	−0.0787370	+	0.996895i	$0.525089\pi$
$390$	0	0
$391$	2.04624	0.103483
$392$	−52.6188	−2.65765
$393$	0	0
$394$	−34.7113	−1.74873
$395$	17.2197	0.866415
$396$	0	0
$397$	24.1194	1.21052	0.605260	−	0.796028i	$-0.293069\pi$
$397$	24.1194	1.21052	0.605260	+	0.796028i	$0.293069\pi$
$398$	33.7394	1.69121
$399$	0	0
$400$	3.43006	0.171503
$401$	−30.1922	−1.50773	−0.753863	−	0.657031i	$-0.771812\pi$
$401$	−30.1922	−1.50773	−0.753863	+	0.657031i	$0.771812\pi$
$402$	0	0
$403$	−17.4511	−0.869301
$404$	−25.8422	−1.28570
$405$	0	0
$406$	−9.00910	−0.447114
$407$	0.530891	0.0263153
$408$	0	0
$409$	4.95249	0.244885	0.122443	−	0.992476i	$-0.460927\pi$
$409$	4.95249	0.244885	0.122443	+	0.992476i	$0.460927\pi$
$410$	1.83829	0.0907867
$411$	0	0
$412$	15.7018	0.773571
$413$	24.2099	1.19129
$414$	0	0
$415$	−33.7805	−1.65822
$416$	−15.4854	−0.759232
$417$	0	0
$418$	0.995709	0.0487017
$419$	−25.0498	−1.22376	−0.611882	−	0.790949i	$-0.709587\pi$
$419$	−25.0498	−1.22376	−0.611882	+	0.790949i	$0.709587\pi$
$420$	0	0
$421$	−31.2604	−1.52354	−0.761769	−	0.647848i	$-0.775669\pi$
$421$	−31.2604	−1.52354	−0.761769	+	0.647848i	$0.775669\pi$
$422$	−6.56773	−0.319712
$423$	0	0
$424$	6.01973	0.292344
$425$	−2.06784	−0.100305
$426$	0	0
$427$	18.5325	0.896850
$428$	−22.7535	−1.09983
$429$	0	0
$430$	70.5108	3.40033
$431$	1.25510	0.0604562	0.0302281	−	0.999543i	$-0.490377\pi$
$431$	1.25510	0.0604562	0.0302281	+	0.999543i	$0.490377\pi$
$432$	0	0
$433$	−32.8817	−1.58019	−0.790097	−	0.612982i	$-0.789970\pi$
$433$	−32.8817	−1.58019	−0.790097	+	0.612982i	$0.789970\pi$
$434$	−43.4820	−2.08720
$435$	0	0
$436$	−31.5145	−1.50927
$437$	1.78750	0.0855080
$438$	0	0
$439$	25.6579	1.22458	0.612292	−	0.790632i	$-0.290248\pi$
$439$	25.6579	1.22458	0.612292	+	0.790632i	$0.290248\pi$
$440$	4.80349	0.228997
$441$	0	0
$442$	−10.2546	−0.487761
$443$	19.4636	0.924742	0.462371	−	0.886686i	$-0.346999\pi$
$443$	19.4636	0.924742	0.462371	+	0.886686i	$0.346999\pi$
$444$	0	0
$445$	42.0670	1.99417
$446$	56.3114	2.66642
$447$	0	0
$448$	−53.7766	−2.54071
$449$	−1.33875	−0.0631796	−0.0315898	−	0.999501i	$-0.510057\pi$
$449$	−1.33875	−0.0631796	−0.0315898	+	0.999501i	$0.510057\pi$
$450$	0	0
$451$	0.141641	0.00666959
$452$	−63.4121	−2.98265
$453$	0	0
$454$	−7.95066	−0.373143
$455$	53.2127	2.49465
$456$	0	0
$457$	19.8287	0.927548	0.463774	−	0.885953i	$-0.346495\pi$
$457$	19.8287	0.927548	0.463774	+	0.885953i	$0.346495\pi$
$458$	−21.7915	−1.01825
$459$	0	0
$460$	19.5426	0.911179
$461$	−31.0694	−1.44704	−0.723522	−	0.690301i	$-0.757478\pi$
$461$	−31.0694	−1.44704	−0.723522	+	0.690301i	$0.757478\pi$
$462$	0	0
$463$	4.21971	0.196107	0.0980533	−	0.995181i	$-0.468738\pi$
$463$	4.21971	0.196107	0.0980533	+	0.995181i	$0.468738\pi$
$464$	1.37285	0.0637328
$465$	0	0
$466$	9.97430	0.462050
$467$	29.2942	1.35557	0.677787	−	0.735259i	$-0.262939\pi$
$467$	29.2942	1.35557	0.677787	+	0.735259i	$0.262939\pi$
$468$	0	0
$469$	−66.2571	−3.05947
$470$	−25.3703	−1.17025
$471$	0	0
$472$	−19.6613	−0.904985
$473$	5.43286	0.249803
$474$	0	0
$475$	−1.80638	−0.0828823
$476$	−16.3920	−0.751324
$477$	0	0
$478$	10.6587	0.487516
$479$	−25.5592	−1.16783	−0.583914	−	0.811815i	$-0.698480\pi$
$479$	−25.5592	−1.16783	−0.583914	+	0.811815i	$0.698480\pi$
$480$	0	0
$481$	−4.77620	−0.217776
$482$	−19.8936	−0.906128
$483$	0	0
$484$	−38.5352	−1.75160
$485$	−5.39043	−0.244767
$486$	0	0
$487$	−3.18751	−0.144440	−0.0722199	−	0.997389i	$-0.523008\pi$
$487$	−3.18751	−0.144440	−0.0722199	+	0.997389i	$0.523008\pi$
$488$	−15.0506	−0.681307
$489$	0	0
$490$	88.6093	4.00296
$491$	16.7126	0.754228	0.377114	−	0.926167i	$-0.376916\pi$
$491$	16.7126	0.754228	0.377114	+	0.926167i	$0.376916\pi$
$492$	0	0
$493$	−0.827631	−0.0372746
$494$	−8.95798	−0.403039
$495$	0	0
$496$	6.62598	0.297515
$497$	−58.2392	−2.61239
$498$	0	0
$499$	18.3483	0.821381	0.410690	−	0.911775i	$-0.365288\pi$
$499$	18.3483	0.821381	0.410690	+	0.911775i	$0.365288\pi$
$500$	27.8536	1.24565
$501$	0	0
$502$	−21.6986	−0.968457
$503$	−22.7070	−1.01245	−0.506227	−	0.862400i	$-0.668960\pi$
$503$	−22.7070	−1.01245	−0.506227	+	0.862400i	$0.668960\pi$
$504$	0	0
$505$	19.2024	0.854497
$506$	2.34710	0.104341
$507$	0	0
$508$	19.8704	0.881606
$509$	−26.4710	−1.17331	−0.586653	−	0.809838i	$-0.699555\pi$
$509$	−26.4710	−1.17331	−0.586653	+	0.809838i	$0.699555\pi$
$510$	0	0
$511$	26.6603	1.17938
$512$	18.2177	0.805118
$513$	0	0
$514$	19.4458	0.857719
$515$	−11.6675	−0.514130
$516$	0	0
$517$	−1.95479	−0.0859714
$518$	−11.9006	−0.522884
$519$	0	0
$520$	−43.2150	−1.89510
$521$	34.2038	1.49850	0.749249	−	0.662289i	$-0.230415\pi$
$521$	34.2038	1.49850	0.749249	+	0.662289i	$0.230415\pi$
$522$	0	0
$523$	23.0166	1.00645	0.503223	−	0.864157i	$-0.332147\pi$
$523$	23.0166	1.00645	0.503223	+	0.864157i	$0.332147\pi$
$524$	65.8600	2.87711
$525$	0	0
$526$	−1.01825	−0.0443977
$527$	−3.99453	−0.174004
$528$	0	0
$529$	−18.7865	−0.816803
$530$	−10.1371	−0.440329
$531$	0	0
$532$	−14.3193	−0.620821
$533$	−1.27428	−0.0551952
$534$	0	0
$535$	16.9073	0.730968
$536$	53.8086	2.32418
$537$	0	0
$538$	−25.2216	−1.08738
$539$	6.82736	0.294075
$540$	0	0
$541$	−28.2528	−1.21468	−0.607340	−	0.794442i	$-0.707764\pi$
$541$	−28.2528	−1.21468	−0.607340	+	0.794442i	$0.707764\pi$
$542$	36.1535	1.55293
$543$	0	0
$544$	−3.54458	−0.151972
$545$	23.4174	1.00309
$546$	0	0
$547$	18.1666	0.776749	0.388375	−	0.921502i	$-0.373037\pi$
$547$	18.1666	0.776749	0.388375	+	0.921502i	$0.373037\pi$
$548$	20.4488	0.873531
$549$	0	0
$550$	−2.37188	−0.101137
$551$	−0.722983	−0.0308001
$552$	0	0
$553$	−29.7415	−1.26474
$554$	29.3386	1.24648
$555$	0	0
$556$	−34.3395	−1.45632
$557$	5.90453	0.250183	0.125091	−	0.992145i	$-0.460078\pi$
$557$	5.90453	0.250183	0.125091	+	0.992145i	$0.460078\pi$
$558$	0	0
$559$	−48.8772	−2.06729
$560$	−20.2043	−0.853786
$561$	0	0
$562$	74.2013	3.12999
$563$	23.0921	0.973218	0.486609	−	0.873620i	$-0.338234\pi$
$563$	23.0921	0.973218	0.486609	+	0.873620i	$0.338234\pi$
$564$	0	0
$565$	47.1194	1.98233
$566$	1.38653	0.0582802
$567$	0	0
$568$	47.2971	1.98454
$569$	−9.15316	−0.383721	−0.191860	−	0.981422i	$-0.561452\pi$
$569$	−9.15316	−0.383721	−0.191860	+	0.981422i	$0.561452\pi$
$570$	0	0
$571$	11.3769	0.476109	0.238054	−	0.971252i	$-0.423490\pi$
$571$	11.3769	0.476109	0.238054	+	0.971252i	$0.423490\pi$
$572$	−7.54603	−0.315515
$573$	0	0
$574$	−3.17506	−0.132525
$575$	−4.25801	−0.177571
$576$	0	0
$577$	−22.5526	−0.938877	−0.469438	−	0.882965i	$-0.655544\pi$
$577$	−22.5526	−0.938877	−0.469438	+	0.882965i	$0.655544\pi$
$578$	37.8082	1.57261
$579$	0	0
$580$	−7.90430	−0.328208
$581$	58.3450	2.42056
$582$	0	0
$583$	−0.781068	−0.0323485
$584$	−21.6513	−0.895937
$585$	0	0
$586$	70.7550	2.92286
$587$	−25.2597	−1.04258	−0.521290	−	0.853380i	$-0.674549\pi$
$587$	−25.2597	−1.04258	−0.521290	+	0.853380i	$0.674549\pi$
$588$	0	0
$589$	−3.48945	−0.143780
$590$	33.1093	1.36309
$591$	0	0
$592$	1.81347	0.0745332
$593$	26.1733	1.07481	0.537404	−	0.843325i	$-0.319405\pi$
$593$	26.1733	1.07481	0.537404	+	0.843325i	$0.319405\pi$
$594$	0	0
$595$	12.1803	0.499344
$596$	58.9642	2.41527
$597$	0	0
$598$	−21.1158	−0.863491
$599$	−14.0171	−0.572723	−0.286362	−	0.958122i	$-0.592446\pi$
$599$	−14.0171	−0.572723	−0.286362	+	0.958122i	$0.592446\pi$
$600$	0	0
$601$	−13.6342	−0.556151	−0.278075	−	0.960559i	$-0.589697\pi$
$601$	−13.6342	−0.556151	−0.278075	+	0.960559i	$0.589697\pi$
$602$	−121.785	−4.96358
$603$	0	0
$604$	42.1234	1.71398
$605$	28.6342	1.16414
$606$	0	0
$607$	21.3111	0.864992	0.432496	−	0.901636i	$-0.357633\pi$
$607$	21.3111	0.864992	0.432496	+	0.901636i	$0.357633\pi$
$608$	−3.09639	−0.125575
$609$	0	0
$610$	25.3449	1.02619
$611$	17.5864	0.711469
$612$	0	0
$613$	25.6266	1.03505	0.517525	−	0.855668i	$-0.326854\pi$
$613$	25.6266	1.03505	0.517525	+	0.855668i	$0.326854\pi$
$614$	5.63447	0.227389
$615$	0	0
$616$	−8.29650	−0.334276
$617$	−11.8179	−0.475771	−0.237886	−	0.971293i	$-0.576454\pi$
$617$	−11.8179	−0.475771	−0.237886	+	0.971293i	$0.576454\pi$
$618$	0	0
$619$	30.2419	1.21552	0.607762	−	0.794119i	$-0.292067\pi$
$619$	30.2419	1.21552	0.607762	+	0.794119i	$0.292067\pi$
$620$	−38.1498	−1.53213
$621$	0	0
$622$	63.1092	2.53045
$623$	−72.6574	−2.91096
$624$	0	0
$625$	−31.0688	−1.24275
$626$	−16.5553	−0.661684
$627$	0	0
$628$	−48.4785	−1.93450
$629$	−1.09327	−0.0435914
$630$	0	0
$631$	0.896907	0.0357053	0.0178526	−	0.999841i	$-0.494317\pi$
$631$	0.896907	0.0357053	0.0178526	+	0.999841i	$0.494317\pi$
$632$	24.1536	0.960779
$633$	0	0
$634$	33.2661	1.32116
$635$	−14.7650	−0.585932
$636$	0	0
$637$	−61.4229	−2.43366
$638$	−0.949319	−0.0375839
$639$	0	0
$640$	−54.6297	−2.15943
$641$	6.39989	0.252780	0.126390	−	0.991981i	$-0.459661\pi$
$641$	6.39989	0.252780	0.126390	+	0.991981i	$0.459661\pi$
$642$	0	0
$643$	−14.9728	−0.590468	−0.295234	−	0.955425i	$-0.595398\pi$
$643$	−14.9728	−0.590468	−0.295234	+	0.955425i	$0.595398\pi$
$644$	−33.7536	−1.33008
$645$	0	0
$646$	−2.05047	−0.0806746
$647$	6.78181	0.266620	0.133310	−	0.991074i	$-0.457439\pi$
$647$	6.78181	0.266620	0.133310	+	0.991074i	$0.457439\pi$
$648$	0	0
$649$	2.55108	0.100139
$650$	21.3388	0.836976
$651$	0	0
$652$	−48.5212	−1.90024
$653$	−35.7396	−1.39860	−0.699299	−	0.714830i	$-0.746504\pi$
$653$	−35.7396	−1.39860	−0.699299	+	0.714830i	$0.746504\pi$
$654$	0	0
$655$	−48.9384	−1.91218
$656$	0.483830	0.0188904
$657$	0	0
$658$	43.8192	1.70825
$659$	−13.4086	−0.522326	−0.261163	−	0.965295i	$-0.584106\pi$
$659$	−13.4086	−0.522326	−0.261163	+	0.965295i	$0.584106\pi$
$660$	0	0
$661$	−7.95097	−0.309257	−0.154628	−	0.987973i	$-0.549418\pi$
$661$	−7.95097	−0.309257	−0.154628	+	0.987973i	$0.549418\pi$
$662$	−68.9287	−2.67899
$663$	0	0
$664$	−47.3831	−1.83882
$665$	10.6402	0.412609
$666$	0	0
$667$	−1.70422	−0.0659878
$668$	39.6356	1.53355
$669$	0	0
$670$	−90.6129	−3.50068
$671$	1.95283	0.0753882
$672$	0	0
$673$	−29.5690	−1.13980	−0.569900	−	0.821714i	$-0.693018\pi$
$673$	−29.5690	−1.13980	−0.569900	+	0.821714i	$0.693018\pi$
$674$	−1.68366	−0.0648520
$675$	0	0
$676$	21.3557	0.821372
$677$	−25.9484	−0.997279	−0.498639	−	0.866810i	$-0.666167\pi$
$677$	−25.9484	−0.997279	−0.498639	+	0.866810i	$0.666167\pi$
$678$	0	0
$679$	9.31025	0.357295
$680$	−9.89185	−0.379335
$681$	0	0
$682$	−4.58185	−0.175448
$683$	−14.8760	−0.569215	−0.284608	−	0.958644i	$-0.591863\pi$
$683$	−14.8760	−0.569215	−0.284608	+	0.958644i	$0.591863\pi$
$684$	0	0
$685$	−15.1948	−0.580565
$686$	−77.0861	−2.94316
$687$	0	0
$688$	18.5581	0.707522
$689$	7.02694	0.267705
$690$	0	0
$691$	−21.3073	−0.810568	−0.405284	−	0.914191i	$-0.632827\pi$
$691$	−21.3073	−0.810568	−0.405284	+	0.914191i	$0.632827\pi$
$692$	46.4088	1.76420
$693$	0	0
$694$	−32.6057	−1.23770
$695$	25.5165	0.967897
$696$	0	0
$697$	−0.291681	−0.0110482
$698$	41.0027	1.55198
$699$	0	0
$700$	34.1100	1.28924
$701$	23.8864	0.902176	0.451088	−	0.892479i	$-0.351036\pi$
$701$	23.8864	0.902176	0.451088	+	0.892479i	$0.351036\pi$
$702$	0	0
$703$	−0.955031	−0.0360197
$704$	−5.66662	−0.213569
$705$	0	0
$706$	24.8155	0.933943
$707$	−33.1661	−1.24734
$708$	0	0
$709$	−30.4671	−1.14422	−0.572108	−	0.820178i	$-0.693874\pi$
$709$	−30.4671	−1.14422	−0.572108	+	0.820178i	$0.693874\pi$
$710$	−79.6477	−2.98912
$711$	0	0
$712$	59.0064	2.21136
$713$	−8.22537	−0.308042
$714$	0	0
$715$	5.60720	0.209697
$716$	90.5444	3.38380
$717$	0	0
$718$	6.09957	0.227634
$719$	−41.8968	−1.56249	−0.781244	−	0.624226i	$-0.785414\pi$
$719$	−41.8968	−1.56249	−0.781244	+	0.624226i	$0.785414\pi$
$720$	0	0
$721$	20.1518	0.750493
$722$	43.0884	1.60359
$723$	0	0
$724$	−38.9548	−1.44774
$725$	1.72222	0.0639615
$726$	0	0
$727$	−20.7062	−0.767952	−0.383976	−	0.923343i	$-0.625445\pi$
$727$	−20.7062	−0.767952	−0.383976	+	0.923343i	$0.625445\pi$
$728$	74.6402	2.76635
$729$	0	0
$730$	36.4605	1.34946
$731$	−11.1879	−0.413800
$732$	0	0
$733$	−40.1316	−1.48230	−0.741148	−	0.671342i	$-0.765718\pi$
$733$	−40.1316	−1.48230	−0.741148	+	0.671342i	$0.765718\pi$
$734$	−52.2136	−1.92724
$735$	0	0
$736$	−7.29885	−0.269039
$737$	−6.98173	−0.257175
$738$	0	0
$739$	34.4385	1.26684	0.633421	−	0.773808i	$-0.281650\pi$
$739$	34.4385	1.26684	0.633421	+	0.773808i	$0.281650\pi$
$740$	−10.4412	−0.383828
$741$	0	0
$742$	17.5087	0.642764
$743$	12.2060	0.447795	0.223898	−	0.974613i	$-0.428122\pi$
$743$	12.2060	0.447795	0.223898	+	0.974613i	$0.428122\pi$
$744$	0	0
$745$	−43.8143	−1.60523
$746$	24.4974	0.896912
$747$	0	0
$748$	−1.72727	−0.0631554
$749$	−29.2020	−1.06702
$750$	0	0
$751$	−39.6527	−1.44695	−0.723474	−	0.690352i	$-0.757456\pi$
$751$	−39.6527	−1.44695	−0.723474	+	0.690352i	$0.757456\pi$
$752$	−6.67736	−0.243498
$753$	0	0
$754$	8.54062	0.311031
$755$	−31.3005	−1.13914
$756$	0	0
$757$	19.4719	0.707719	0.353859	−	0.935299i	$-0.384869\pi$
$757$	19.4719	0.707719	0.353859	+	0.935299i	$0.384869\pi$
$758$	18.7304	0.680320
$759$	0	0
$760$	−8.64110	−0.313446
$761$	5.70156	0.206682	0.103341	−	0.994646i	$-0.467047\pi$
$761$	5.70156	0.206682	0.103341	+	0.994646i	$0.467047\pi$
$762$	0	0
$763$	−40.4461	−1.46425
$764$	42.2937	1.53013
$765$	0	0
$766$	−10.3642	−0.374472
$767$	−22.9510	−0.828712
$768$	0	0
$769$	−25.5498	−0.921348	−0.460674	−	0.887569i	$-0.652392\pi$
$769$	−25.5498	−0.921348	−0.460674	+	0.887569i	$0.652392\pi$
$770$	13.9712	0.503487
$771$	0	0
$772$	−30.8243	−1.10939
$773$	−42.7583	−1.53791	−0.768954	−	0.639304i	$-0.779223\pi$
$773$	−42.7583	−1.53791	−0.768954	+	0.639304i	$0.779223\pi$
$774$	0	0
$775$	8.31221	0.298583
$776$	−7.56103	−0.271425
$777$	0	0
$778$	7.33634	0.263021
$779$	−0.254800	−0.00912916
$780$	0	0
$781$	−6.13686	−0.219594
$782$	−4.83338	−0.172841
$783$	0	0
$784$	23.3216	0.832914
$785$	36.0227	1.28571
$786$	0	0
$787$	−43.6243	−1.55504	−0.777520	−	0.628859i	$-0.783522\pi$
$787$	−43.6243	−1.55504	−0.777520	+	0.628859i	$0.783522\pi$
$788$	52.6007	1.87382
$789$	0	0
$790$	−40.6743	−1.44713
$791$	−81.3838	−2.89367
$792$	0	0
$793$	−17.5688	−0.623886
$794$	−56.9722	−2.02187
$795$	0	0
$796$	−51.1279	−1.81218
$797$	−32.1020	−1.13711	−0.568556	−	0.822645i	$-0.692498\pi$
$797$	−32.1020	−1.13711	−0.568556	+	0.822645i	$0.692498\pi$
$798$	0	0
$799$	4.02550	0.142412
$800$	7.37590	0.260778
$801$	0	0
$802$	71.3165	2.51827
$803$	2.80928	0.0991374
$804$	0	0
$805$	25.0812	0.883996
$806$	41.2210	1.45195
$807$	0	0
$808$	26.9348	0.947563
$809$	40.7617	1.43310	0.716552	−	0.697533i	$-0.245719\pi$
$809$	40.7617	1.43310	0.716552	+	0.697533i	$0.245719\pi$
$810$	0	0
$811$	22.4762	0.789245	0.394623	−	0.918843i	$-0.370875\pi$
$811$	22.4762	0.789245	0.394623	+	0.918843i	$0.370875\pi$
$812$	13.6522	0.479097
$813$	0	0
$814$	−1.25401	−0.0439530
$815$	36.0544	1.26293
$816$	0	0
$817$	−9.77329	−0.341924
$818$	−11.6982	−0.409018
$819$	0	0
$820$	−2.78570	−0.0972809
$821$	51.0196	1.78060	0.890299	−	0.455376i	$-0.150495\pi$
$821$	51.0196	1.78060	0.890299	+	0.455376i	$0.150495\pi$
$822$	0	0
$823$	−39.6403	−1.38177	−0.690886	−	0.722963i	$-0.742779\pi$
$823$	−39.6403	−1.38177	−0.690886	+	0.722963i	$0.742779\pi$
$824$	−16.3657	−0.570125
$825$	0	0
$826$	−57.1859	−1.98975
$827$	−25.5073	−0.886977	−0.443488	−	0.896280i	$-0.646259\pi$
$827$	−25.5073	−0.886977	−0.443488	+	0.896280i	$0.646259\pi$
$828$	0	0
$829$	−28.4148	−0.986888	−0.493444	−	0.869778i	$-0.664262\pi$
$829$	−28.4148	−0.986888	−0.493444	+	0.869778i	$0.664262\pi$
$830$	79.7924	2.76963
$831$	0	0
$832$	50.9803	1.76742
$833$	−14.0596	−0.487136
$834$	0	0
$835$	−29.4519	−1.01922
$836$	−1.50887	−0.0521855
$837$	0	0
$838$	59.1699	2.04399
$839$	28.6330	0.988519	0.494260	−	0.869314i	$-0.335439\pi$
$839$	28.6330	0.988519	0.494260	+	0.869314i	$0.335439\pi$
$840$	0	0
$841$	−28.3107	−0.976231
$842$	73.8398	2.54469
$843$	0	0
$844$	9.95258	0.342582
$845$	−15.8687	−0.545899
$846$	0	0
$847$	−49.4564	−1.69934
$848$	−2.66805	−0.0916213
$849$	0	0
$850$	4.88441	0.167534
$851$	−2.25121	−0.0771704
$852$	0	0
$853$	11.6656	0.399421	0.199711	−	0.979855i	$-0.436000\pi$
$853$	11.6656	0.399421	0.199711	+	0.979855i	$0.436000\pi$
$854$	−43.7753	−1.49796
$855$	0	0
$856$	23.7155	0.810580
$857$	−34.0291	−1.16241	−0.581206	−	0.813757i	$-0.697419\pi$
$857$	−34.0291	−1.16241	−0.581206	+	0.813757i	$0.697419\pi$
$858$	0	0
$859$	−33.5474	−1.14462	−0.572312	−	0.820036i	$-0.693953\pi$
$859$	−33.5474	−1.14462	−0.572312	+	0.820036i	$0.693953\pi$
$860$	−106.850	−3.64357
$861$	0	0
$862$	−2.96466	−0.100977
$863$	57.2486	1.94877	0.974383	−	0.224897i	$-0.0722045\pi$
$863$	57.2486	1.94877	0.974383	+	0.224897i	$0.0722045\pi$
$864$	0	0
$865$	−34.4848	−1.17252
$866$	77.6694	2.63931
$867$	0	0
$868$	65.8916	2.23651
$869$	−3.13396	−0.106312
$870$	0	0
$871$	62.8117	2.12829
$872$	32.8470	1.11234
$873$	0	0
$874$	−4.22224	−0.142819
$875$	35.7475	1.20849
$876$	0	0
$877$	38.9014	1.31361	0.656804	−	0.754062i	$-0.271908\pi$
$877$	38.9014	1.31361	0.656804	+	0.754062i	$0.271908\pi$
$878$	−60.6061	−2.04536
$879$	0	0
$880$	−2.12899	−0.0717683
$881$	−30.6355	−1.03214	−0.516069	−	0.856547i	$-0.672605\pi$
$881$	−30.6355	−1.03214	−0.516069	+	0.856547i	$0.672605\pi$
$882$	0	0
$883$	17.7425	0.597082	0.298541	−	0.954397i	$-0.403500\pi$
$883$	17.7425	0.597082	0.298541	+	0.954397i	$0.403500\pi$
$884$	15.5396	0.522652
$885$	0	0
$886$	−45.9746	−1.54455
$887$	−7.93629	−0.266474	−0.133237	−	0.991084i	$-0.542537\pi$
$887$	−7.93629	−0.266474	−0.133237	+	0.991084i	$0.542537\pi$
$888$	0	0
$889$	25.5019	0.855306
$890$	−99.3659	−3.33075
$891$	0	0
$892$	−85.3329	−2.85716
$893$	3.51650	0.117675
$894$	0	0
$895$	−67.2805	−2.24894
$896$	94.3555	3.15220
$897$	0	0
$898$	3.16224	0.105525
$899$	3.32687	0.110957
$900$	0	0
$901$	1.60846	0.0535855
$902$	−0.334567	−0.0111399
$903$	0	0
$904$	66.0932	2.19823
$905$	28.9460	0.962196
$906$	0	0
$907$	48.3605	1.60578	0.802892	−	0.596124i	$-0.203293\pi$
$907$	48.3605	1.60578	0.802892	+	0.596124i	$0.203293\pi$
$908$	12.0482	0.399835
$909$	0	0
$910$	−125.693	−4.16668
$911$	−52.8788	−1.75195	−0.875976	−	0.482355i	$-0.839782\pi$
$911$	−52.8788	−1.75195	−0.875976	+	0.482355i	$0.839782\pi$
$912$	0	0
$913$	6.14801	0.203470
$914$	−46.8371	−1.54923
$915$	0	0
$916$	33.0224	1.09109
$917$	84.5255	2.79128
$918$	0	0
$919$	−48.2466	−1.59151	−0.795754	−	0.605620i	$-0.792925\pi$
$919$	−48.2466	−1.59151	−0.795754	+	0.605620i	$0.792925\pi$
$920$	−20.3689	−0.671542
$921$	0	0
$922$	73.3885	2.41692
$923$	55.2108	1.81729
$924$	0	0
$925$	2.27498	0.0748007
$926$	−9.96732	−0.327546
$927$	0	0
$928$	2.95213	0.0969084
$929$	35.3222	1.15888	0.579442	−	0.815014i	$-0.303271\pi$
$929$	35.3222	1.15888	0.579442	+	0.815014i	$0.303271\pi$
$930$	0	0
$931$	−12.2819	−0.402522
$932$	−15.1148	−0.495102
$933$	0	0
$934$	−69.1954	−2.26414
$935$	1.28348	0.0419743
$936$	0	0
$937$	26.0392	0.850665	0.425332	−	0.905037i	$-0.360157\pi$
$937$	26.0392	0.850665	0.425332	+	0.905037i	$0.360157\pi$
$938$	156.505	5.11007
$939$	0	0
$940$	38.4456	1.25396
$941$	−46.8637	−1.52771	−0.763856	−	0.645386i	$-0.776696\pi$
$941$	−46.8637	−1.52771	−0.763856	+	0.645386i	$0.776696\pi$
$942$	0	0
$943$	−0.600617	−0.0195588
$944$	8.71424	0.283624
$945$	0	0
$946$	−12.8329	−0.417233
$947$	−23.3176	−0.757720	−0.378860	−	0.925454i	$-0.623684\pi$
$947$	−23.3176	−0.757720	−0.378860	+	0.925454i	$0.623684\pi$
$948$	0	0
$949$	−25.2739	−0.820427
$950$	4.26682	0.138434
$951$	0	0
$952$	17.0850	0.553729
$953$	−25.9058	−0.839171	−0.419585	−	0.907716i	$-0.637825\pi$
$953$	−25.9058	−0.839171	−0.419585	+	0.907716i	$0.637825\pi$
$954$	0	0
$955$	−31.4270	−1.01695
$956$	−16.1519	−0.522389
$957$	0	0
$958$	60.3729	1.95056
$959$	26.2443	0.847471
$960$	0	0
$961$	−14.9430	−0.482032
$962$	11.2818	0.363740
$963$	0	0
$964$	30.1462	0.970945
$965$	22.9045	0.737322
$966$	0	0
$967$	15.0788	0.484901	0.242450	−	0.970164i	$-0.422049\pi$
$967$	15.0788	0.484901	0.242450	+	0.970164i	$0.422049\pi$
$968$	40.1645	1.29094
$969$	0	0
$970$	12.7327	0.408821
$971$	−8.71600	−0.279710	−0.139855	−	0.990172i	$-0.544664\pi$
$971$	−8.71600	−0.279710	−0.139855	+	0.990172i	$0.544664\pi$
$972$	0	0
$973$	−44.0717	−1.41287
$974$	7.52917	0.241250
$975$	0	0
$976$	6.67068	0.213523
$977$	13.1373	0.420301	0.210150	−	0.977669i	$-0.432605\pi$
$977$	13.1373	0.420301	0.210150	+	0.977669i	$0.432605\pi$
$978$	0	0
$979$	−7.65616	−0.244692
$980$	−134.276	−4.28930
$981$	0	0
$982$	−39.4766	−1.25975
$983$	−8.89260	−0.283630	−0.141815	−	0.989893i	$-0.545294\pi$
$983$	−8.89260	−0.283630	−0.141815	+	0.989893i	$0.545294\pi$
$984$	0	0
$985$	−39.0858	−1.24538
$986$	1.95494	0.0622578
$987$	0	0
$988$	13.5747	0.431869
$989$	−23.0377	−0.732556
$990$	0	0
$991$	55.3256	1.75747	0.878737	−	0.477306i	$-0.158387\pi$
$991$	55.3256	1.75747	0.878737	+	0.477306i	$0.158387\pi$
$992$	14.2483	0.452385
$993$	0	0
$994$	137.566	4.36333
$995$	37.9914	1.20441
$996$	0	0
$997$	26.4501	0.837683	0.418841	−	0.908059i	$-0.362436\pi$
$997$	26.4501	0.837683	0.418841	+	0.908059i	$0.362436\pi$
$998$	−43.3401	−1.37191
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.10		72
3.2	odd	2		6561.2.a.d.1.63		72
81.5	odd	54		729.2.g.b.460.7		144
81.11	odd	54		243.2.g.a.91.2		144
81.16	even	27		729.2.g.c.271.2		144
81.22	even	27		81.2.g.a.79.7	yes	144
81.32	odd	54		729.2.g.a.703.2		144
81.38	odd	54		729.2.g.a.28.2		144
81.43	even	27		729.2.g.d.28.7		144
81.49	even	27		729.2.g.d.703.7		144
81.59	odd	54		243.2.g.a.235.2		144
81.65	odd	54		729.2.g.b.271.7		144
81.70	even	27		81.2.g.a.40.7	✓	144
81.76	even	27		729.2.g.c.460.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.40.7	✓	144	81.70	even	27
81.2.g.a.79.7	yes	144	81.22	even	27
243.2.g.a.91.2		144	81.11	odd	54
243.2.g.a.235.2		144	81.59	odd	54
729.2.g.a.28.2		144	81.38	odd	54
729.2.g.a.703.2		144	81.32	odd	54
729.2.g.b.271.7		144	81.65	odd	54
729.2.g.b.460.7		144	81.5	odd	54
729.2.g.c.271.2		144	81.16	even	27
729.2.g.c.460.2		144	81.76	even	27
729.2.g.d.28.7		144	81.43	even	27
729.2.g.d.703.7		144	81.49	even	27
6561.2.a.c.1.10		72	1.1	even	1	trivial
6561.2.a.d.1.63		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)

\(f(q)\)	\(=\)	\(q-2.36209 q^{2} +3.57945 q^{4} -2.65977 q^{5} +4.59390 q^{7} -3.73079 q^{8} +O(q^{10})\)	\(q-2.36209 q^{2} +3.57945 q^{4} -2.65977 q^{5} +4.59390 q^{7} -3.73079 q^{8} +6.28260 q^{10} +0.484075 q^{11} -4.35502 q^{13} -10.8512 q^{14} +1.65355 q^{16} -0.996857 q^{17} -0.870812 q^{19} -9.52050 q^{20} -1.14343 q^{22} -2.05269 q^{23} +2.07436 q^{25} +10.2869 q^{26} +16.4436 q^{28} +0.830241 q^{29} +4.00712 q^{31} +3.55575 q^{32} +2.35466 q^{34} -12.2187 q^{35} +1.09671 q^{37} +2.05693 q^{38} +9.92303 q^{40} +0.292601 q^{41} +11.2232 q^{43} +1.73272 q^{44} +4.84862 q^{46} -4.03819 q^{47} +14.1039 q^{49} -4.89981 q^{50} -15.5886 q^{52} -1.61353 q^{53} -1.28753 q^{55} -17.1389 q^{56} -1.96110 q^{58} +5.27001 q^{59} +4.03415 q^{61} -9.46516 q^{62} -11.7061 q^{64} +11.5833 q^{65} -14.4228 q^{67} -3.56820 q^{68} +28.8616 q^{70} -12.6775 q^{71} +5.80341 q^{73} -2.59053 q^{74} -3.11703 q^{76} +2.22379 q^{77} -6.47412 q^{79} -4.39806 q^{80} -0.691147 q^{82} +12.7005 q^{83} +2.65141 q^{85} -26.5101 q^{86} -1.80598 q^{88} -15.8161 q^{89} -20.0065 q^{91} -7.34749 q^{92} +9.53855 q^{94} +2.31616 q^{95} +2.02665 q^{97} -33.3147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 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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