LMFDB - Embedded newform 6561.2.a.c.1.42

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [72,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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			1.42
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)

$f(q)$	$=$	\(q+0.317274 q^{2} -1.89934 q^{4} +2.12131 q^{5} -2.54355 q^{7} -1.23716 q^{8} +0.673038 q^{10} -3.06004 q^{11} +5.51901 q^{13} -0.807002 q^{14} +3.40616 q^{16} -4.29040 q^{17} -0.0609980 q^{19} -4.02909 q^{20} -0.970873 q^{22} -4.28084 q^{23} -0.500032 q^{25} +1.75104 q^{26} +4.83105 q^{28} +10.6788 q^{29} +4.61351 q^{31} +3.55500 q^{32} -1.36123 q^{34} -5.39566 q^{35} +3.51955 q^{37} -0.0193531 q^{38} -2.62440 q^{40} +2.52003 q^{41} -0.775799 q^{43} +5.81205 q^{44} -1.35820 q^{46} -4.54654 q^{47} -0.530372 q^{49} -0.158647 q^{50} -10.4825 q^{52} -1.97270 q^{53} -6.49131 q^{55} +3.14677 q^{56} +3.38812 q^{58} +7.21340 q^{59} -11.6790 q^{61} +1.46375 q^{62} -5.68440 q^{64} +11.7075 q^{65} -1.21297 q^{67} +8.14892 q^{68} -1.71190 q^{70} -14.2681 q^{71} +9.71508 q^{73} +1.11666 q^{74} +0.115856 q^{76} +7.78336 q^{77} +1.61116 q^{79} +7.22552 q^{80} +0.799542 q^{82} +3.30224 q^{83} -9.10129 q^{85} -0.246141 q^{86} +3.78576 q^{88} -14.0469 q^{89} -14.0379 q^{91} +8.13076 q^{92} -1.44250 q^{94} -0.129396 q^{95} +7.02316 q^{97} -0.168273 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	0.317274	0.224347	0.112173	−	0.993689i	$-0.464219\pi$
$2$	0.317274	0.224347	0.112173	+	0.993689i	$0.464219\pi$
$3$	0	0
$3$	0	0
$4$	−1.89934	−0.949669
$5$	2.12131	0.948680	0.474340	−	0.880342i	$-0.342687\pi$
$5$	2.12131	0.948680	0.474340	+	0.880342i	$0.342687\pi$
$6$	0	0
$7$	−2.54355	−0.961370	−0.480685	−	0.876893i	$-0.659612\pi$
$7$	−2.54355	−0.961370	−0.480685	+	0.876893i	$0.659612\pi$
$8$	−1.23716	−0.437402
$9$	0	0
$10$	0.673038	0.212833
$11$	−3.06004	−0.922638	−0.461319	−	0.887234i	$-0.652624\pi$
$11$	−3.06004	−0.922638	−0.461319	+	0.887234i	$0.652624\pi$
$12$	0	0
$13$	5.51901	1.53070	0.765349	−	0.643616i	$-0.222567\pi$
$13$	5.51901	1.53070	0.765349	+	0.643616i	$0.222567\pi$
$14$	−0.807002	−0.215680
$15$	0	0
$16$	3.40616	0.851539
$17$	−4.29040	−1.04058	−0.520288	−	0.853991i	$-0.674175\pi$
$17$	−4.29040	−1.04058	−0.520288	+	0.853991i	$0.674175\pi$
$18$	0	0
$19$	−0.0609980	−0.0139939	−0.00699695	−	0.999976i	$-0.502227\pi$
$19$	−0.0609980	−0.0139939	−0.00699695	+	0.999976i	$0.502227\pi$
$20$	−4.02909	−0.900931
$21$	0	0
$22$	−0.970873	−0.206991
$23$	−4.28084	−0.892617	−0.446308	−	0.894879i	$-0.647262\pi$
$23$	−4.28084	−0.892617	−0.446308	+	0.894879i	$0.647262\pi$
$24$	0	0
$25$	−0.500032	−0.100006
$26$	1.75104	0.343407
$27$	0	0
$28$	4.83105	0.912983
$29$	10.6788	1.98301	0.991504	−	0.130077i	$-0.0415223\pi$
$29$	10.6788	1.98301	0.991504	+	0.130077i	$0.0415223\pi$
$30$	0	0
$31$	4.61351	0.828610	0.414305	−	0.910138i	$-0.364025\pi$
$31$	4.61351	0.828610	0.414305	+	0.910138i	$0.364025\pi$
$32$	3.55500	0.628442
$33$	0	0
$34$	−1.36123	−0.233450
$35$	−5.39566	−0.912033
$36$	0	0
$37$	3.51955	0.578611	0.289305	−	0.957237i	$-0.406576\pi$
$37$	3.51955	0.578611	0.289305	+	0.957237i	$0.406576\pi$
$38$	−0.0193531	−0.00313949
$39$	0	0
$40$	−2.62440	−0.414954
$41$	2.52003	0.393563	0.196782	−	0.980447i	$-0.436951\pi$
$41$	2.52003	0.393563	0.196782	+	0.980447i	$0.436951\pi$
$42$	0	0
$43$	−0.775799	−0.118308	−0.0591541	−	0.998249i	$-0.518840\pi$
$43$	−0.775799	−0.118308	−0.0591541	+	0.998249i	$0.518840\pi$
$44$	5.81205	0.876200
$45$	0	0
$46$	−1.35820	−0.200256
$47$	−4.54654	−0.663182	−0.331591	−	0.943423i	$-0.607585\pi$
$47$	−4.54654	−0.663182	−0.331591	+	0.943423i	$0.607585\pi$
$48$	0	0
$49$	−0.530372	−0.0757675
$50$	−0.158647	−0.0224361
$51$	0	0
$52$	−10.4825	−1.45366
$53$	−1.97270	−0.270971	−0.135485	−	0.990779i	$-0.543259\pi$
$53$	−1.97270	−0.270971	−0.135485	+	0.990779i	$0.543259\pi$
$54$	0	0
$55$	−6.49131	−0.875288
$56$	3.14677	0.420505
$57$	0	0
$58$	3.38812	0.444882
$59$	7.21340	0.939104	0.469552	−	0.882905i	$-0.344415\pi$
$59$	7.21340	0.939104	0.469552	+	0.882905i	$0.344415\pi$
$60$	0	0
$61$	−11.6790	−1.49534	−0.747671	−	0.664070i	$-0.768828\pi$
$61$	−11.6790	−1.49534	−0.747671	+	0.664070i	$0.768828\pi$
$62$	1.46375	0.185896
$63$	0	0
$64$	−5.68440	−0.710550
$65$	11.7075	1.45214
$66$	0	0
$67$	−1.21297	−0.148188	−0.0740940	−	0.997251i	$-0.523606\pi$
$67$	−1.21297	−0.148188	−0.0740940	+	0.997251i	$0.523606\pi$
$68$	8.14892	0.988202
$69$	0	0
$70$	−1.71190	−0.204612
$71$	−14.2681	−1.69331	−0.846657	−	0.532139i	$-0.821389\pi$
$71$	−14.2681	−1.69331	−0.846657	+	0.532139i	$0.821389\pi$
$72$	0	0
$73$	9.71508	1.13706	0.568532	−	0.822661i	$-0.307512\pi$
$73$	9.71508	1.13706	0.568532	+	0.822661i	$0.307512\pi$
$74$	1.11666	0.129809
$75$	0	0
$76$	0.115856	0.0132896
$77$	7.78336	0.886996
$78$	0	0
$79$	1.61116	0.181269	0.0906347	−	0.995884i	$-0.471110\pi$
$79$	1.61116	0.181269	0.0906347	+	0.995884i	$0.471110\pi$
$80$	7.22552	0.807838
$81$	0	0
$82$	0.799542	0.0882947
$83$	3.30224	0.362468	0.181234	−	0.983440i	$-0.441991\pi$
$83$	3.30224	0.362468	0.181234	+	0.983440i	$0.441991\pi$
$84$	0	0
$85$	−9.10129	−0.987173
$86$	−0.246141	−0.0265421
$87$	0	0
$88$	3.78576	0.403563
$89$	−14.0469	−1.48897	−0.744486	−	0.667639i	$-0.767305\pi$
$89$	−14.0469	−1.48897	−0.744486	+	0.667639i	$0.767305\pi$
$90$	0	0
$91$	−14.0379	−1.47157
$92$	8.13076	0.847690
$93$	0	0
$94$	−1.44250	−0.148783
$95$	−0.129396	−0.0132757
$96$	0	0
$97$	7.02316	0.713094	0.356547	−	0.934277i	$-0.383954\pi$
$97$	7.02316	0.713094	0.356547	+	0.934277i	$0.383954\pi$
$98$	−0.168273	−0.0169982
$99$	0	0
$100$	0.949730	0.0949730
$101$	10.8538	1.07999	0.539996	−	0.841668i	$-0.318426\pi$
$101$	10.8538	1.07999	0.539996	+	0.841668i	$0.318426\pi$
$102$	0	0
$103$	−12.4368	−1.22543	−0.612715	−	0.790304i	$-0.709923\pi$
$103$	−12.4368	−1.22543	−0.612715	+	0.790304i	$0.709923\pi$
$104$	−6.82789	−0.669530
$105$	0	0
$106$	−0.625887	−0.0607915
$107$	4.97831	0.481271	0.240636	−	0.970616i	$-0.422644\pi$
$107$	4.97831	0.481271	0.240636	+	0.970616i	$0.422644\pi$
$108$	0	0
$109$	0.361252	0.0346016	0.0173008	−	0.999850i	$-0.494493\pi$
$109$	0.361252	0.0346016	0.0173008	+	0.999850i	$0.494493\pi$
$110$	−2.05953	−0.196368
$111$	0	0
$112$	−8.66371	−0.818644
$113$	−19.7419	−1.85716	−0.928582	−	0.371126i	$-0.878972\pi$
$113$	−19.7419	−1.85716	−0.928582	+	0.371126i	$0.878972\pi$
$114$	0	0
$115$	−9.08100	−0.846807
$116$	−20.2827	−1.88320
$117$	0	0
$118$	2.28863	0.210685
$119$	10.9128	1.00038
$120$	0	0
$121$	−1.63614	−0.148740
$122$	−3.70544	−0.335475
$123$	0	0
$124$	−8.76260	−0.786905
$125$	−11.6673	−1.04355
$126$	0	0
$127$	−14.4255	−1.28006	−0.640029	−	0.768351i	$-0.721078\pi$
$127$	−14.4255	−1.28006	−0.640029	+	0.768351i	$0.721078\pi$
$128$	−8.91352	−0.787852
$129$	0	0
$130$	3.71450	0.325783
$131$	3.05479	0.266899	0.133449	−	0.991056i	$-0.457395\pi$
$131$	3.05479	0.266899	0.133449	+	0.991056i	$0.457395\pi$
$132$	0	0
$133$	0.155151	0.0134533
$134$	−0.384845	−0.0332455
$135$	0	0
$136$	5.30791	0.455150
$137$	2.60498	0.222559	0.111279	−	0.993789i	$-0.464505\pi$
$137$	2.60498	0.222559	0.111279	+	0.993789i	$0.464505\pi$
$138$	0	0
$139$	−18.9841	−1.61021	−0.805104	−	0.593134i	$-0.797890\pi$
$139$	−18.9841	−1.61021	−0.805104	+	0.593134i	$0.797890\pi$
$140$	10.2482	0.866129
$141$	0	0
$142$	−4.52691	−0.379890
$143$	−16.8884	−1.41228
$144$	0	0
$145$	22.6531	1.88124
$146$	3.08235	0.255097
$147$	0	0
$148$	−6.68481	−0.549488
$149$	4.09763	0.335691	0.167845	−	0.985813i	$-0.446319\pi$
$149$	4.09763	0.335691	0.167845	+	0.985813i	$0.446319\pi$
$150$	0	0
$151$	9.94125	0.809007	0.404504	−	0.914536i	$-0.367444\pi$
$151$	9.94125	0.809007	0.404504	+	0.914536i	$0.367444\pi$
$152$	0.0754643	0.00612096
$153$	0	0
$154$	2.46946	0.198995
$155$	9.78669	0.786086
$156$	0	0
$157$	11.5496	0.921755	0.460878	−	0.887464i	$-0.347535\pi$
$157$	11.5496	0.921755	0.460878	+	0.887464i	$0.347535\pi$
$158$	0.511179	0.0406672
$159$	0	0
$160$	7.54128	0.596190
$161$	10.8885	0.858135
$162$	0	0
$163$	−14.1333	−1.10700	−0.553502	−	0.832848i	$-0.686709\pi$
$163$	−14.1333	−1.10700	−0.553502	+	0.832848i	$0.686709\pi$
$164$	−4.78640	−0.373755
$165$	0	0
$166$	1.04772	0.0813186
$167$	−6.70352	−0.518734	−0.259367	−	0.965779i	$-0.583514\pi$
$167$	−6.70352	−0.518734	−0.259367	+	0.965779i	$0.583514\pi$
$168$	0	0
$169$	17.4594	1.34303
$170$	−2.88760	−0.221469
$171$	0	0
$172$	1.47350	0.112354
$173$	−14.7888	−1.12437	−0.562185	−	0.827011i	$-0.690039\pi$
$173$	−14.7888	−1.12437	−0.562185	+	0.827011i	$0.690039\pi$
$174$	0	0
$175$	1.27186	0.0961432
$176$	−10.4230	−0.785662
$177$	0	0
$178$	−4.45673	−0.334046
$179$	−0.633753	−0.0473689	−0.0236844	−	0.999719i	$-0.507540\pi$
$179$	−0.633753	−0.0473689	−0.0236844	+	0.999719i	$0.507540\pi$
$180$	0	0
$181$	−13.9541	−1.03720	−0.518601	−	0.855016i	$-0.673547\pi$
$181$	−13.9541	−1.03720	−0.518601	+	0.855016i	$0.673547\pi$
$182$	−4.45385	−0.330141
$183$	0	0
$184$	5.29608	0.390432
$185$	7.46607	0.548916
$186$	0	0
$187$	13.1288	0.960074
$188$	8.63542	0.629803
$189$	0	0
$190$	−0.0410540	−0.00297837
$191$	8.49326	0.614551	0.307275	−	0.951621i	$-0.400583\pi$
$191$	8.49326	0.614551	0.307275	+	0.951621i	$0.400583\pi$
$192$	0	0
$193$	13.2510	0.953831	0.476915	−	0.878949i	$-0.341755\pi$
$193$	13.2510	0.953831	0.476915	+	0.878949i	$0.341755\pi$
$194$	2.22827	0.159980
$195$	0	0
$196$	1.00736	0.0719540
$197$	−7.54829	−0.537793	−0.268897	−	0.963169i	$-0.586659\pi$
$197$	−7.54829	−0.537793	−0.268897	+	0.963169i	$0.586659\pi$
$198$	0	0
$199$	−2.11341	−0.149815	−0.0749077	−	0.997190i	$-0.523866\pi$
$199$	−2.11341	−0.149815	−0.0749077	+	0.997190i	$0.523866\pi$
$200$	0.618620	0.0437430
$201$	0	0
$202$	3.44362	0.242293
$203$	−27.1621	−1.90640
$204$	0	0
$205$	5.34578	0.373366
$206$	−3.94586	−0.274921
$207$	0	0
$208$	18.7986	1.30345
$209$	0.186657	0.0129113
$210$	0	0
$211$	0.656477	0.0451937	0.0225969	−	0.999745i	$-0.492807\pi$
$211$	0.656477	0.0451937	0.0225969	+	0.999745i	$0.492807\pi$
$212$	3.74682	0.257333
$213$	0	0
$214$	1.57949	0.107972
$215$	−1.64571	−0.112237
$216$	0	0
$217$	−11.7347	−0.796601
$218$	0.114616	0.00776277
$219$	0	0
$220$	12.3292	0.831233
$221$	−23.6788	−1.59281
$222$	0	0
$223$	22.5110	1.50745	0.753724	−	0.657191i	$-0.228255\pi$
$223$	22.5110	1.50745	0.753724	+	0.657191i	$0.228255\pi$
$224$	−9.04232	−0.604165
$225$	0	0
$226$	−6.26361	−0.416649
$227$	−13.4086	−0.889959	−0.444980	−	0.895541i	$-0.646789\pi$
$227$	−13.4086	−0.889959	−0.444980	+	0.895541i	$0.646789\pi$
$228$	0	0
$229$	6.22339	0.411253	0.205627	−	0.978631i	$-0.434077\pi$
$229$	6.22339	0.411253	0.205627	+	0.978631i	$0.434077\pi$
$230$	−2.88117	−0.189979
$231$	0	0
$232$	−13.2114	−0.867372
$233$	−12.3897	−0.811677	−0.405839	−	0.913945i	$-0.633021\pi$
$233$	−12.3897	−0.811677	−0.405839	+	0.913945i	$0.633021\pi$
$234$	0	0
$235$	−9.64464	−0.629147
$236$	−13.7007	−0.891838
$237$	0	0
$238$	3.46236	0.224432
$239$	−9.83205	−0.635983	−0.317991	−	0.948094i	$-0.603008\pi$
$239$	−9.83205	−0.635983	−0.317991	+	0.948094i	$0.603008\pi$
$240$	0	0
$241$	−17.3372	−1.11678	−0.558392	−	0.829577i	$-0.688582\pi$
$241$	−17.3372	−1.11678	−0.558392	+	0.829577i	$0.688582\pi$
$242$	−0.519105	−0.0333693
$243$	0	0
$244$	22.1823	1.42008
$245$	−1.12509	−0.0718791
$246$	0	0
$247$	−0.336649	−0.0214204
$248$	−5.70764	−0.362436
$249$	0	0
$250$	−3.70173	−0.234118
$251$	−25.0330	−1.58007	−0.790034	−	0.613063i	$-0.789937\pi$
$251$	−25.0330	−1.58007	−0.790034	+	0.613063i	$0.789937\pi$
$252$	0	0
$253$	13.0996	0.823562
$254$	−4.57685	−0.287177
$255$	0	0
$256$	8.54076	0.533798
$257$	2.41046	0.150360	0.0751801	−	0.997170i	$-0.476047\pi$
$257$	2.41046	0.150360	0.0751801	+	0.997170i	$0.476047\pi$
$258$	0	0
$259$	−8.95214	−0.556259
$260$	−22.2366	−1.37905
$261$	0	0
$262$	0.969208	0.0598778
$263$	−3.92278	−0.241889	−0.120944	−	0.992659i	$-0.538592\pi$
$263$	−3.92278	−0.241889	−0.120944	+	0.992659i	$0.538592\pi$
$264$	0	0
$265$	−4.18471	−0.257065
$266$	0.0492255	0.00301821
$267$	0	0
$268$	2.30384	0.140730
$269$	11.0903	0.676185	0.338093	−	0.941113i	$-0.390218\pi$
$269$	11.0903	0.676185	0.338093	+	0.941113i	$0.390218\pi$
$270$	0	0
$271$	0.794617	0.0482695	0.0241348	−	0.999709i	$-0.492317\pi$
$271$	0.794617	0.0482695	0.0241348	+	0.999709i	$0.492317\pi$
$272$	−14.6138	−0.886091
$273$	0	0
$274$	0.826494	0.0499303
$275$	1.53012	0.0922697
$276$	0	0
$277$	29.1878	1.75373	0.876863	−	0.480741i	$-0.159632\pi$
$277$	29.1878	1.75373	0.876863	+	0.480741i	$0.159632\pi$
$278$	−6.02316	−0.361245
$279$	0	0
$280$	6.67529	0.398925
$281$	−27.5889	−1.64582	−0.822909	−	0.568173i	$-0.807650\pi$
$281$	−27.5889	−1.64582	−0.822909	+	0.568173i	$0.807650\pi$
$282$	0	0
$283$	−5.10861	−0.303676	−0.151838	−	0.988405i	$-0.548519\pi$
$283$	−5.10861	−0.303676	−0.151838	+	0.988405i	$0.548519\pi$
$284$	27.1000	1.60809
$285$	0	0
$286$	−5.35826	−0.316840
$287$	−6.40983	−0.378360
$288$	0	0
$289$	1.40756	0.0827979
$290$	7.18726	0.422050
$291$	0	0
$292$	−18.4522	−1.07983
$293$	−2.34345	−0.136906	−0.0684529	−	0.997654i	$-0.521806\pi$
$293$	−2.34345	−0.136906	−0.0684529	+	0.997654i	$0.521806\pi$
$294$	0	0
$295$	15.3019	0.890909
$296$	−4.35425	−0.253085
$297$	0	0
$298$	1.30007	0.0753111
$299$	−23.6260	−1.36633
$300$	0	0
$301$	1.97328	0.113738
$302$	3.15410	0.181498
$303$	0	0
$304$	−0.207769	−0.0119164
$305$	−24.7748	−1.41860
$306$	0	0
$307$	−16.1673	−0.922718	−0.461359	−	0.887213i	$-0.652638\pi$
$307$	−16.1673	−0.922718	−0.461359	+	0.887213i	$0.652638\pi$
$308$	−14.7832	−0.842352
$309$	0	0
$310$	3.10507	0.176356
$311$	−21.1989	−1.20208	−0.601041	−	0.799218i	$-0.705247\pi$
$311$	−21.1989	−1.20208	−0.601041	+	0.799218i	$0.705247\pi$
$312$	0	0
$313$	−5.41141	−0.305871	−0.152936	−	0.988236i	$-0.548873\pi$
$313$	−5.41141	−0.305871	−0.152936	+	0.988236i	$0.548873\pi$
$314$	3.66438	0.206793
$315$	0	0
$316$	−3.06013	−0.172146
$317$	8.72733	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$317$	8.72733	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$318$	0	0
$319$	−32.6777	−1.82960
$320$	−12.0584	−0.674084
$321$	0	0
$322$	3.45465	0.192520
$323$	0.261706	0.0145617
$324$	0	0
$325$	−2.75968	−0.153080
$326$	−4.48412	−0.248353
$327$	0	0
$328$	−3.11769	−0.172145
$329$	11.5643	0.637563
$330$	0	0
$331$	−19.6506	−1.08009	−0.540047	−	0.841635i	$-0.681593\pi$
$331$	−19.6506	−1.08009	−0.540047	+	0.841635i	$0.681593\pi$
$332$	−6.27207	−0.344225
$333$	0	0
$334$	−2.12686	−0.116376
$335$	−2.57309	−0.140583
$336$	0	0
$337$	−7.34551	−0.400136	−0.200068	−	0.979782i	$-0.564116\pi$
$337$	−7.34551	−0.400136	−0.200068	+	0.979782i	$0.564116\pi$
$338$	5.53943	0.301305
$339$	0	0
$340$	17.2864	0.937487
$341$	−14.1175	−0.764507
$342$	0	0
$343$	19.1539	1.03421
$344$	0.959788	0.0517483
$345$	0	0
$346$	−4.69210	−0.252249
$347$	18.2185	0.978020	0.489010	−	0.872278i	$-0.337358\pi$
$347$	18.2185	0.978020	0.489010	+	0.872278i	$0.337358\pi$
$348$	0	0
$349$	−16.1102	−0.862361	−0.431180	−	0.902266i	$-0.641903\pi$
$349$	−16.1102	−0.862361	−0.431180	+	0.902266i	$0.641903\pi$
$350$	0.403527	0.0215694
$351$	0	0
$352$	−10.8785	−0.579824
$353$	−17.1839	−0.914606	−0.457303	−	0.889311i	$-0.651185\pi$
$353$	−17.1839	−0.914606	−0.457303	+	0.889311i	$0.651185\pi$
$354$	0	0
$355$	−30.2671	−1.60641
$356$	26.6798	1.41403
$357$	0	0
$358$	−0.201073	−0.0106271
$359$	−28.4341	−1.50070	−0.750348	−	0.661043i	$-0.770114\pi$
$359$	−28.4341	−1.50070	−0.750348	+	0.661043i	$0.770114\pi$
$360$	0	0
$361$	−18.9963	−0.999804
$362$	−4.42729	−0.232693
$363$	0	0
$364$	26.6626	1.39750
$365$	20.6087	1.07871
$366$	0	0
$367$	10.3771	0.541679	0.270840	−	0.962624i	$-0.412699\pi$
$367$	10.3771	0.541679	0.270840	+	0.962624i	$0.412699\pi$
$368$	−14.5812	−0.760098
$369$	0	0
$370$	2.36879	0.123148
$371$	5.01765	0.260503
$372$	0	0
$373$	14.8925	0.771104	0.385552	−	0.922686i	$-0.374011\pi$
$373$	14.8925	0.771104	0.385552	+	0.922686i	$0.374011\pi$
$374$	4.16544	0.215390
$375$	0	0
$376$	5.62480	0.290077
$377$	58.9365	3.03538
$378$	0	0
$379$	11.6243	0.597103	0.298551	−	0.954394i	$-0.403497\pi$
$379$	11.6243	0.597103	0.298551	+	0.954394i	$0.403497\pi$
$380$	0.245766	0.0126075
$381$	0	0
$382$	2.69469	0.137872
$383$	−28.5160	−1.45710	−0.728550	−	0.684993i	$-0.759805\pi$
$383$	−28.5160	−1.45710	−0.728550	+	0.684993i	$0.759805\pi$
$384$	0	0
$385$	16.5109	0.841475
$386$	4.20421	0.213989
$387$	0	0
$388$	−13.3393	−0.677203
$389$	3.07037	0.155674	0.0778369	−	0.996966i	$-0.475199\pi$
$389$	3.07037	0.155674	0.0778369	+	0.996966i	$0.475199\pi$
$390$	0	0
$391$	18.3665	0.928835
$392$	0.656155	0.0331408
$393$	0	0
$394$	−2.39488	−0.120652
$395$	3.41777	0.171967
$396$	0	0
$397$	−10.5098	−0.527470	−0.263735	−	0.964595i	$-0.584954\pi$
$397$	−10.5098	−0.527470	−0.263735	+	0.964595i	$0.584954\pi$
$398$	−0.670529	−0.0336106
$399$	0	0
$400$	−1.70319	−0.0851594
$401$	26.4982	1.32326	0.661630	−	0.749831i	$-0.269865\pi$
$401$	26.4982	1.32326	0.661630	+	0.749831i	$0.269865\pi$
$402$	0	0
$403$	25.4620	1.26835
$404$	−20.6150	−1.02563
$405$	0	0
$406$	−8.61783	−0.427696
$407$	−10.7700	−0.533848
$408$	0	0
$409$	26.3525	1.30305	0.651523	−	0.758629i	$-0.274130\pi$
$409$	26.3525	1.30305	0.651523	+	0.758629i	$0.274130\pi$
$410$	1.69608	0.0837634
$411$	0	0
$412$	23.6216	1.16375
$413$	−18.3476	−0.902827
$414$	0	0
$415$	7.00509	0.343866
$416$	19.6201	0.961954
$417$	0	0
$418$	0.0592213	0.00289661
$419$	7.83156	0.382597	0.191298	−	0.981532i	$-0.438730\pi$
$419$	7.83156	0.382597	0.191298	+	0.981532i	$0.438730\pi$
$420$	0	0
$421$	−31.8518	−1.55236	−0.776182	−	0.630509i	$-0.782846\pi$
$421$	−31.8518	−1.55236	−0.776182	+	0.630509i	$0.782846\pi$
$422$	0.208283	0.0101391
$423$	0	0
$424$	2.44054	0.118523
$425$	2.14534	0.104064
$426$	0	0
$427$	29.7061	1.43758
$428$	−9.45548	−0.457048
$429$	0	0
$430$	−0.522143	−0.0251799
$431$	−33.3134	−1.60465	−0.802324	−	0.596889i	$-0.796403\pi$
$431$	−33.3134	−1.60465	−0.802324	+	0.596889i	$0.796403\pi$
$432$	0	0
$433$	−5.51586	−0.265075	−0.132538	−	0.991178i	$-0.542313\pi$
$433$	−5.51586	−0.265075	−0.132538	+	0.991178i	$0.542313\pi$
$434$	−3.72311	−0.178715
$435$	0	0
$436$	−0.686139	−0.0328601
$437$	0.261123	0.0124912
$438$	0	0
$439$	−3.31005	−0.157980	−0.0789900	−	0.996875i	$-0.525170\pi$
$439$	−3.31005	−0.157980	−0.0789900	+	0.996875i	$0.525170\pi$
$440$	8.03078	0.382853
$441$	0	0
$442$	−7.51267	−0.357341
$443$	17.1859	0.816525	0.408263	−	0.912865i	$-0.366135\pi$
$443$	17.1859	0.816525	0.408263	+	0.912865i	$0.366135\pi$
$444$	0	0
$445$	−29.7979	−1.41256
$446$	7.14216	0.338191
$447$	0	0
$448$	14.4585	0.683101
$449$	−17.6006	−0.830625	−0.415312	−	0.909679i	$-0.636328\pi$
$449$	−17.6006	−0.830625	−0.415312	+	0.909679i	$0.636328\pi$
$450$	0	0
$451$	−7.71141	−0.363116
$452$	37.4966	1.76369
$453$	0	0
$454$	−4.25420	−0.199660
$455$	−29.7787	−1.39605
$456$	0	0
$457$	17.8745	0.836135	0.418068	−	0.908416i	$-0.362707\pi$
$457$	17.8745	0.836135	0.418068	+	0.908416i	$0.362707\pi$
$458$	1.97452	0.0922634
$459$	0	0
$460$	17.2479	0.804186
$461$	−5.02719	−0.234140	−0.117070	−	0.993124i	$-0.537350\pi$
$461$	−5.02719	−0.234140	−0.117070	+	0.993124i	$0.537350\pi$
$462$	0	0
$463$	17.2052	0.799593	0.399796	−	0.916604i	$-0.369081\pi$
$463$	17.2052	0.799593	0.399796	+	0.916604i	$0.369081\pi$
$464$	36.3737	1.68861
$465$	0	0
$466$	−3.93094	−0.182097
$467$	−37.4574	−1.73332	−0.866661	−	0.498897i	$-0.833738\pi$
$467$	−37.4574	−1.73332	−0.866661	+	0.498897i	$0.833738\pi$
$468$	0	0
$469$	3.08525	0.142464
$470$	−3.06000	−0.141147
$471$	0	0
$472$	−8.92412	−0.410766
$473$	2.37398	0.109156
$474$	0	0
$475$	0.0305010	0.00139948
$476$	−20.7272	−0.950028
$477$	0	0
$478$	−3.11946	−0.142681
$479$	−3.42274	−0.156389	−0.0781945	−	0.996938i	$-0.524916\pi$
$479$	−3.42274	−0.156389	−0.0781945	+	0.996938i	$0.524916\pi$
$480$	0	0
$481$	19.4244	0.885678
$482$	−5.50064	−0.250547
$483$	0	0
$484$	3.10758	0.141254
$485$	14.8983	0.676498
$486$	0	0
$487$	−5.43342	−0.246212	−0.123106	−	0.992394i	$-0.539285\pi$
$487$	−5.43342	−0.246212	−0.123106	+	0.992394i	$0.539285\pi$
$488$	14.4488	0.654065
$489$	0	0
$490$	−0.356961	−0.0161258
$491$	3.97386	0.179338	0.0896690	−	0.995972i	$-0.471419\pi$
$491$	3.97386	0.179338	0.0896690	+	0.995972i	$0.471419\pi$
$492$	0	0
$493$	−45.8165	−2.06347
$494$	−0.106810	−0.00480561
$495$	0	0
$496$	15.7143	0.705594
$497$	36.2916	1.62790
$498$	0	0
$499$	−21.4564	−0.960523	−0.480261	−	0.877125i	$-0.659458\pi$
$499$	−21.4564	−0.960523	−0.480261	+	0.877125i	$0.659458\pi$
$500$	22.1601	0.991030
$501$	0	0
$502$	−7.94233	−0.354483
$503$	24.6671	1.09985	0.549926	−	0.835213i	$-0.314656\pi$
$503$	24.6671	1.09985	0.549926	+	0.835213i	$0.314656\pi$
$504$	0	0
$505$	23.0243	1.02457
$506$	4.15615	0.184763
$507$	0	0
$508$	27.3989	1.21563
$509$	−7.34691	−0.325646	−0.162823	−	0.986655i	$-0.552060\pi$
$509$	−7.34691	−0.325646	−0.162823	+	0.986655i	$0.552060\pi$
$510$	0	0
$511$	−24.7108	−1.09314
$512$	20.5368	0.907607
$513$	0	0
$514$	0.764776	0.0337328
$515$	−26.3823	−1.16254
$516$	0	0
$517$	13.9126	0.611876
$518$	−2.84028	−0.124795
$519$	0	0
$520$	−14.4841	−0.635170
$521$	27.2332	1.19311	0.596555	−	0.802572i	$-0.296536\pi$
$521$	27.2332	1.19311	0.596555	+	0.802572i	$0.296536\pi$
$522$	0	0
$523$	15.2184	0.665452	0.332726	−	0.943023i	$-0.392032\pi$
$523$	15.2184	0.665452	0.332726	+	0.943023i	$0.392032\pi$
$524$	−5.80208	−0.253465
$525$	0	0
$526$	−1.24460	−0.0542670
$527$	−19.7938	−0.862232
$528$	0	0
$529$	−4.67442	−0.203235
$530$	−1.32770	−0.0576716
$531$	0	0
$532$	−0.294685	−0.0127762
$533$	13.9081	0.602426
$534$	0	0
$535$	10.5605	0.456572
$536$	1.50064	0.0648177
$537$	0	0
$538$	3.51866	0.151700
$539$	1.62296	0.0699059
$540$	0	0
$541$	16.6232	0.714686	0.357343	−	0.933973i	$-0.383683\pi$
$541$	16.6232	0.714686	0.357343	+	0.933973i	$0.383683\pi$
$542$	0.252112	0.0108291
$543$	0	0
$544$	−15.2524	−0.653941
$545$	0.766328	0.0328259
$546$	0	0
$547$	11.2829	0.482422	0.241211	−	0.970473i	$-0.422455\pi$
$547$	11.2829	0.482422	0.241211	+	0.970473i	$0.422455\pi$
$548$	−4.94774	−0.211357
$549$	0	0
$550$	0.485468	0.0207004
$551$	−0.651387	−0.0277500
$552$	0	0
$553$	−4.09805	−0.174267
$554$	9.26054	0.393443
$555$	0	0
$556$	36.0571	1.52916
$557$	29.1548	1.23533	0.617665	−	0.786441i	$-0.288079\pi$
$557$	29.1548	1.23533	0.617665	+	0.786441i	$0.288079\pi$
$558$	0	0
$559$	−4.28164	−0.181094
$560$	−18.3784	−0.776631
$561$	0	0
$562$	−8.75326	−0.369234
$563$	−21.4660	−0.904685	−0.452342	−	0.891844i	$-0.649412\pi$
$563$	−21.4660	−0.904685	−0.452342	+	0.891844i	$0.649412\pi$
$564$	0	0
$565$	−41.8788	−1.76185
$566$	−1.62083	−0.0681286
$567$	0	0
$568$	17.6519	0.740659
$569$	−2.68116	−0.112400	−0.0562001	−	0.998420i	$-0.517898\pi$
$569$	−2.68116	−0.112400	−0.0562001	+	0.998420i	$0.517898\pi$
$570$	0	0
$571$	−2.84080	−0.118884	−0.0594420	−	0.998232i	$-0.518932\pi$
$571$	−2.84080	−0.118884	−0.0594420	+	0.998232i	$0.518932\pi$
$572$	32.0768	1.34120
$573$	0	0
$574$	−2.03367	−0.0848839
$575$	2.14056	0.0892674
$576$	0	0
$577$	5.92252	0.246558	0.123279	−	0.992372i	$-0.460659\pi$
$577$	5.92252	0.246558	0.123279	+	0.992372i	$0.460659\pi$
$578$	0.446584	0.0185754
$579$	0	0
$580$	−43.0259	−1.78655
$581$	−8.39940	−0.348466
$582$	0	0
$583$	6.03654	0.250008
$584$	−12.0191	−0.497354
$585$	0	0
$586$	−0.743517	−0.0307144
$587$	−11.2678	−0.465072	−0.232536	−	0.972588i	$-0.574702\pi$
$587$	−11.2678	−0.465072	−0.232536	+	0.972588i	$0.574702\pi$
$588$	0	0
$589$	−0.281415	−0.0115955
$590$	4.85489	0.199873
$591$	0	0
$592$	11.9881	0.492709
$593$	−9.49254	−0.389812	−0.194906	−	0.980822i	$-0.562440\pi$
$593$	−9.49254	−0.389812	−0.194906	+	0.980822i	$0.562440\pi$
$594$	0	0
$595$	23.1495	0.949039
$596$	−7.78277	−0.318795
$597$	0	0
$598$	−7.49592	−0.306531
$599$	24.9508	1.01946	0.509731	−	0.860334i	$-0.329745\pi$
$599$	24.9508	1.01946	0.509731	+	0.860334i	$0.329745\pi$
$600$	0	0
$601$	−11.4871	−0.468568	−0.234284	−	0.972168i	$-0.575275\pi$
$601$	−11.4871	−0.468568	−0.234284	+	0.972168i	$0.575275\pi$
$602$	0.626072	0.0255168
$603$	0	0
$604$	−18.8818	−0.768289
$605$	−3.47076	−0.141107
$606$	0	0
$607$	39.8542	1.61763	0.808816	−	0.588061i	$-0.200109\pi$
$607$	39.8542	1.61763	0.808816	+	0.588061i	$0.200109\pi$
$608$	−0.216848	−0.00879436
$609$	0	0
$610$	−7.86041	−0.318259
$611$	−25.0924	−1.01513
$612$	0	0
$613$	17.9916	0.726672	0.363336	−	0.931658i	$-0.381638\pi$
$613$	17.9916	0.726672	0.363336	+	0.931658i	$0.381638\pi$
$614$	−5.12948	−0.207009
$615$	0	0
$616$	−9.62926	−0.387974
$617$	18.3474	0.738640	0.369320	−	0.929302i	$-0.379591\pi$
$617$	18.3474	0.738640	0.369320	+	0.929302i	$0.379591\pi$
$618$	0	0
$619$	−44.9371	−1.80617	−0.903086	−	0.429459i	$-0.858704\pi$
$619$	−44.9371	−1.80617	−0.903086	+	0.429459i	$0.858704\pi$
$620$	−18.5882	−0.746521
$621$	0	0
$622$	−6.72588	−0.269683
$623$	35.7290	1.43145
$624$	0	0
$625$	−22.2498	−0.889992
$626$	−1.71690	−0.0686212
$627$	0	0
$628$	−21.9365	−0.875362
$629$	−15.1003	−0.602088
$630$	0	0
$631$	34.0139	1.35407	0.677036	−	0.735950i	$-0.263264\pi$
$631$	34.0139	1.35407	0.677036	+	0.735950i	$0.263264\pi$
$632$	−1.99326	−0.0792876
$633$	0	0
$634$	2.76896	0.109969
$635$	−30.6010	−1.21437
$636$	0	0
$637$	−2.92713	−0.115977
$638$	−10.3678	−0.410464
$639$	0	0
$640$	−18.9084	−0.747419
$641$	−0.314618	−0.0124267	−0.00621333	−	0.999981i	$-0.501978\pi$
$641$	−0.314618	−0.0124267	−0.00621333	+	0.999981i	$0.501978\pi$
$642$	0	0
$643$	−33.1338	−1.30667	−0.653335	−	0.757069i	$-0.726631\pi$
$643$	−33.1338	−1.30667	−0.653335	+	0.757069i	$0.726631\pi$
$644$	−20.6810	−0.814944
$645$	0	0
$646$	0.0830326	0.00326688
$647$	14.0510	0.552403	0.276201	−	0.961100i	$-0.410924\pi$
$647$	14.0510	0.552403	0.276201	+	0.961100i	$0.410924\pi$
$648$	0	0
$649$	−22.0733	−0.866453
$650$	−0.875576	−0.0343429
$651$	0	0
$652$	26.8438	1.05129
$653$	−44.9131	−1.75759	−0.878793	−	0.477204i	$-0.841650\pi$
$653$	−44.9131	−1.75759	−0.878793	+	0.477204i	$0.841650\pi$
$654$	0	0
$655$	6.48017	0.253201
$656$	8.58363	0.335134
$657$	0	0
$658$	3.66907	0.143035
$659$	−33.5200	−1.30575	−0.652876	−	0.757465i	$-0.726438\pi$
$659$	−33.5200	−1.30575	−0.652876	+	0.757465i	$0.726438\pi$
$660$	0	0
$661$	16.2273	0.631169	0.315584	−	0.948897i	$-0.397799\pi$
$661$	16.2273	0.631169	0.315584	+	0.948897i	$0.397799\pi$
$662$	−6.23463	−0.242316
$663$	0	0
$664$	−4.08540	−0.158544
$665$	0.329124	0.0127629
$666$	0	0
$667$	−45.7143	−1.77007
$668$	12.7323	0.492626
$669$	0	0
$670$	−0.816376	−0.0315394
$671$	35.7382	1.37966
$672$	0	0
$673$	1.96712	0.0758269	0.0379134	−	0.999281i	$-0.487929\pi$
$673$	1.96712	0.0758269	0.0379134	+	0.999281i	$0.487929\pi$
$674$	−2.33054	−0.0897691
$675$	0	0
$676$	−33.1614	−1.27544
$677$	20.5194	0.788623	0.394311	−	0.918977i	$-0.370983\pi$
$677$	20.5194	0.788623	0.394311	+	0.918977i	$0.370983\pi$
$678$	0	0
$679$	−17.8637	−0.685547
$680$	11.2597	0.431792
$681$	0	0
$682$	−4.47913	−0.171515
$683$	−21.4331	−0.820115	−0.410057	−	0.912060i	$-0.634491\pi$
$683$	−21.4331	−0.820115	−0.410057	+	0.912060i	$0.634491\pi$
$684$	0	0
$685$	5.52598	0.211137
$686$	6.07702	0.232022
$687$	0	0
$688$	−2.64249	−0.100744
$689$	−10.8873	−0.414775
$690$	0	0
$691$	27.7607	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$691$	27.7607	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$692$	28.0889	1.06778
$693$	0	0
$694$	5.78026	0.219416
$695$	−40.2711	−1.52757
$696$	0	0
$697$	−10.8120	−0.409532
$698$	−5.11136	−0.193468
$699$	0	0
$700$	−2.41568	−0.0913042
$701$	0.709747	0.0268068	0.0134034	−	0.999910i	$-0.495733\pi$
$701$	0.709747	0.0268068	0.0134034	+	0.999910i	$0.495733\pi$
$702$	0	0
$703$	−0.214686	−0.00809702
$704$	17.3945	0.655580
$705$	0	0
$706$	−5.45201	−0.205189
$707$	−27.6071	−1.03827
$708$	0	0
$709$	−20.1907	−0.758279	−0.379139	−	0.925340i	$-0.623780\pi$
$709$	−20.1907	−0.758279	−0.379139	+	0.925340i	$0.623780\pi$
$710$	−9.60299	−0.360394
$711$	0	0
$712$	17.3783	0.651279
$713$	−19.7497	−0.739631
$714$	0	0
$715$	−35.8256	−1.33980
$716$	1.20371	0.0449847
$717$	0	0
$718$	−9.02142	−0.336676
$719$	16.7747	0.625592	0.312796	−	0.949820i	$-0.398734\pi$
$719$	16.7747	0.625592	0.312796	+	0.949820i	$0.398734\pi$
$720$	0	0
$721$	31.6335	1.17809
$722$	−6.02703	−0.224303
$723$	0	0
$724$	26.5036	0.984999
$725$	−5.33976	−0.198314
$726$	0	0
$727$	−22.2022	−0.823435	−0.411717	−	0.911312i	$-0.635071\pi$
$727$	−22.2022	−0.823435	−0.411717	+	0.911312i	$0.635071\pi$
$728$	17.3671	0.643666
$729$	0	0
$730$	6.53862	0.242005
$731$	3.32849	0.123109
$732$	0	0
$733$	3.89468	0.143853	0.0719267	−	0.997410i	$-0.477085\pi$
$733$	3.89468	0.143853	0.0719267	+	0.997410i	$0.477085\pi$
$734$	3.29238	0.121524
$735$	0	0
$736$	−15.2184	−0.560958
$737$	3.71174	0.136724
$738$	0	0
$739$	−19.8726	−0.731025	−0.365512	−	0.930806i	$-0.619106\pi$
$739$	−19.8726	−0.731025	−0.365512	+	0.930806i	$0.619106\pi$
$740$	−14.1806	−0.521288
$741$	0	0
$742$	1.59197	0.0584431
$743$	6.21847	0.228133	0.114067	−	0.993473i	$-0.463612\pi$
$743$	6.21847	0.228133	0.114067	+	0.993473i	$0.463612\pi$
$744$	0	0
$745$	8.69235	0.318463
$746$	4.72501	0.172995
$747$	0	0
$748$	−24.9361	−0.911752
$749$	−12.6626	−0.462680
$750$	0	0
$751$	−16.1144	−0.588023	−0.294012	−	0.955802i	$-0.594990\pi$
$751$	−16.1144	−0.588023	−0.294012	+	0.955802i	$0.594990\pi$
$752$	−15.4862	−0.564725
$753$	0	0
$754$	18.6990	0.680979
$755$	21.0885	0.767489
$756$	0	0
$757$	−35.5236	−1.29113	−0.645563	−	0.763707i	$-0.723377\pi$
$757$	−35.5236	−1.29113	−0.645563	+	0.763707i	$0.723377\pi$
$758$	3.68811	0.133958
$759$	0	0
$760$	0.160083	0.00580683
$761$	−20.3537	−0.737820	−0.368910	−	0.929465i	$-0.620269\pi$
$761$	−20.3537	−0.737820	−0.368910	+	0.929465i	$0.620269\pi$
$762$	0	0
$763$	−0.918861	−0.0332650
$764$	−16.1316	−0.583619
$765$	0	0
$766$	−9.04739	−0.326896
$767$	39.8108	1.43748
$768$	0	0
$769$	−18.6207	−0.671479	−0.335739	−	0.941955i	$-0.608986\pi$
$769$	−18.6207	−0.671479	−0.335739	+	0.941955i	$0.608986\pi$
$770$	5.23850	0.188782
$771$	0	0
$772$	−25.1682	−0.905823
$773$	−3.93462	−0.141518	−0.0707592	−	0.997493i	$-0.522542\pi$
$773$	−3.93462	−0.141518	−0.0707592	+	0.997493i	$0.522542\pi$
$774$	0	0
$775$	−2.30690	−0.0828664
$776$	−8.68877	−0.311909
$777$	0	0
$778$	0.974148	0.0349249
$779$	−0.153717	−0.00550749
$780$	0	0
$781$	43.6611	1.56232
$782$	5.82723	0.208381
$783$	0	0
$784$	−1.80653	−0.0645189
$785$	24.5002	0.874451
$786$	0	0
$787$	7.37679	0.262954	0.131477	−	0.991319i	$-0.458028\pi$
$787$	7.37679	0.262954	0.131477	+	0.991319i	$0.458028\pi$
$788$	14.3367	0.510725
$789$	0	0
$790$	1.08437	0.0385802
$791$	50.2145	1.78542
$792$	0	0
$793$	−64.4564	−2.28892
$794$	−3.33448	−0.118336
$795$	0	0
$796$	4.01407	0.142275
$797$	14.9474	0.529464	0.264732	−	0.964322i	$-0.414717\pi$
$797$	14.9474	0.529464	0.264732	+	0.964322i	$0.414717\pi$
$798$	0	0
$799$	19.5065	0.690091
$800$	−1.77762	−0.0628483
$801$	0	0
$802$	8.40721	0.296869
$803$	−29.7286	−1.04910
$804$	0	0
$805$	23.0979	0.814095
$806$	8.07843	0.284551
$807$	0	0
$808$	−13.4279	−0.472390
$809$	−28.6228	−1.00633	−0.503163	−	0.864192i	$-0.667830\pi$
$809$	−28.6228	−1.00633	−0.503163	+	0.864192i	$0.667830\pi$
$810$	0	0
$811$	15.2720	0.536273	0.268136	−	0.963381i	$-0.413592\pi$
$811$	15.2720	0.536273	0.268136	+	0.963381i	$0.413592\pi$
$812$	51.5900	1.81045
$813$	0	0
$814$	−3.41704	−0.119767
$815$	−29.9811	−1.05019
$816$	0	0
$817$	0.0473222	0.00165559
$818$	8.36097	0.292334
$819$	0	0
$820$	−10.1534	−0.354574
$821$	−3.11253	−0.108628	−0.0543141	−	0.998524i	$-0.517297\pi$
$821$	−3.11253	−0.108628	−0.0543141	+	0.998524i	$0.517297\pi$
$822$	0	0
$823$	38.5247	1.34289	0.671444	−	0.741055i	$-0.265674\pi$
$823$	38.5247	1.34289	0.671444	+	0.741055i	$0.265674\pi$
$824$	15.3863	0.536006
$825$	0	0
$826$	−5.82122	−0.202546
$827$	−7.92730	−0.275659	−0.137830	−	0.990456i	$-0.544013\pi$
$827$	−7.92730	−0.275659	−0.137830	+	0.990456i	$0.544013\pi$
$828$	0	0
$829$	31.7471	1.10262	0.551312	−	0.834299i	$-0.314128\pi$
$829$	31.7471	1.10262	0.551312	+	0.834299i	$0.314128\pi$
$830$	2.22253	0.0771453
$831$	0	0
$832$	−31.3722	−1.08764
$833$	2.27551	0.0788418
$834$	0	0
$835$	−14.2203	−0.492113
$836$	−0.354524	−0.0122615
$837$	0	0
$838$	2.48475	0.0858343
$839$	45.2670	1.56279	0.781396	−	0.624036i	$-0.214508\pi$
$839$	45.2670	1.56279	0.781396	+	0.624036i	$0.214508\pi$
$840$	0	0
$841$	85.0373	2.93232
$842$	−10.1058	−0.348268
$843$	0	0
$844$	−1.24687	−0.0429191
$845$	37.0369	1.27411
$846$	0	0
$847$	4.16160	0.142994
$848$	−6.71932	−0.230742
$849$	0	0
$850$	0.680662	0.0233465
$851$	−15.0666	−0.516477
$852$	0	0
$853$	46.6312	1.59662	0.798312	−	0.602245i	$-0.205727\pi$
$853$	46.6312	1.59662	0.798312	+	0.602245i	$0.205727\pi$
$854$	9.42497	0.322516
$855$	0	0
$856$	−6.15896	−0.210509
$857$	−6.82673	−0.233197	−0.116598	−	0.993179i	$-0.537199\pi$
$857$	−6.82673	−0.233197	−0.116598	+	0.993179i	$0.537199\pi$
$858$	0	0
$859$	10.9872	0.374878	0.187439	−	0.982276i	$-0.439981\pi$
$859$	10.9872	0.374878	0.187439	+	0.982276i	$0.439981\pi$
$860$	3.12576	0.106588
$861$	0	0
$862$	−10.5695	−0.359998
$863$	−55.6284	−1.89361	−0.946807	−	0.321802i	$-0.895711\pi$
$863$	−55.6284	−1.89361	−0.946807	+	0.321802i	$0.895711\pi$
$864$	0	0
$865$	−31.3716	−1.06667
$866$	−1.75004	−0.0594688
$867$	0	0
$868$	22.2881	0.756507
$869$	−4.93021	−0.167246
$870$	0	0
$871$	−6.69440	−0.226831
$872$	−0.446926	−0.0151348
$873$	0	0
$874$	0.0828475	0.00280236
$875$	29.6763	1.00324
$876$	0	0
$877$	24.6736	0.833168	0.416584	−	0.909097i	$-0.363227\pi$
$877$	24.6736	0.833168	0.416584	+	0.909097i	$0.363227\pi$
$878$	−1.05019	−0.0354423
$879$	0	0
$880$	−22.1104	−0.745341
$881$	−17.2629	−0.581601	−0.290800	−	0.956784i	$-0.593922\pi$
$881$	−17.2629	−0.581601	−0.290800	+	0.956784i	$0.593922\pi$
$882$	0	0
$883$	−31.4366	−1.05793	−0.528963	−	0.848645i	$-0.677419\pi$
$883$	−31.4366	−1.05793	−0.528963	+	0.848645i	$0.677419\pi$
$884$	44.9740	1.51264
$885$	0	0
$886$	5.45263	0.183185
$887$	44.2247	1.48492	0.742460	−	0.669890i	$-0.233659\pi$
$887$	44.2247	1.48492	0.742460	+	0.669890i	$0.233659\pi$
$888$	0	0
$889$	36.6920	1.23061
$890$	−9.45412	−0.316903
$891$	0	0
$892$	−42.7560	−1.43158
$893$	0.277330	0.00928050
$894$	0	0
$895$	−1.34439	−0.0449379
$896$	22.6720	0.757417
$897$	0	0
$898$	−5.58423	−0.186348
$899$	49.2668	1.64314
$900$	0	0
$901$	8.46367	0.281966
$902$	−2.44663	−0.0814640
$903$	0	0
$904$	24.4239	0.812328
$905$	−29.6011	−0.983974
$906$	0	0
$907$	14.7813	0.490805	0.245402	−	0.969421i	$-0.421080\pi$
$907$	14.7813	0.490805	0.245402	+	0.969421i	$0.421080\pi$
$908$	25.4674	0.845166
$909$	0	0
$910$	−9.44801	−0.313198
$911$	−38.7546	−1.28400	−0.641998	−	0.766707i	$-0.721894\pi$
$911$	−38.7546	−1.28400	−0.641998	+	0.766707i	$0.721894\pi$
$912$	0	0
$913$	−10.1050	−0.334427
$914$	5.67113	0.187584
$915$	0	0
$916$	−11.8203	−0.390554
$917$	−7.77001	−0.256588
$918$	0	0
$919$	41.7879	1.37846	0.689228	−	0.724545i	$-0.257950\pi$
$919$	41.7879	1.37846	0.689228	+	0.724545i	$0.257950\pi$
$920$	11.2346	0.370395
$921$	0	0
$922$	−1.59500	−0.0525285
$923$	−78.7459	−2.59195
$924$	0	0
$925$	−1.75989	−0.0578648
$926$	5.45876	0.179386
$927$	0	0
$928$	37.9633	1.24621
$929$	−26.6246	−0.873525	−0.436763	−	0.899577i	$-0.643875\pi$
$929$	−26.6246	−0.873525	−0.436763	+	0.899577i	$0.643875\pi$
$930$	0	0
$931$	0.0323517	0.00106028
$932$	23.5322	0.770824
$933$	0	0
$934$	−11.8843	−0.388865
$935$	27.8503	0.910803
$936$	0	0
$937$	−11.2669	−0.368073	−0.184036	−	0.982919i	$-0.558916\pi$
$937$	−11.2669	−0.368073	−0.184036	+	0.982919i	$0.558916\pi$
$938$	0.978870	0.0319613
$939$	0	0
$940$	18.3184	0.597481
$941$	17.7098	0.577322	0.288661	−	0.957431i	$-0.406790\pi$
$941$	17.7098	0.577322	0.288661	+	0.957431i	$0.406790\pi$
$942$	0	0
$943$	−10.7879	−0.351301
$944$	24.5699	0.799684
$945$	0	0
$946$	0.753203	0.0244887
$947$	−59.4221	−1.93096	−0.965479	−	0.260479i	$-0.916119\pi$
$947$	−59.4221	−1.93096	−0.965479	+	0.260479i	$0.916119\pi$
$948$	0	0
$949$	53.6176	1.74050
$950$	0.00967718	0.000313969 0
$951$	0	0
$952$	−13.5009	−0.437567
$953$	54.5065	1.76564	0.882819	−	0.469712i	$-0.155642\pi$
$953$	54.5065	1.76564	0.882819	+	0.469712i	$0.155642\pi$
$954$	0	0
$955$	18.0169	0.583012
$956$	18.6744	0.603973
$957$	0	0
$958$	−1.08595	−0.0350854
$959$	−6.62589	−0.213961
$960$	0	0
$961$	−9.71556	−0.313405
$962$	6.16287	0.198699
$963$	0	0
$964$	32.9291	1.06058
$965$	28.1096	0.904880
$966$	0	0
$967$	28.3395	0.911337	0.455668	−	0.890150i	$-0.349400\pi$
$967$	28.3395	0.911337	0.455668	+	0.890150i	$0.349400\pi$
$968$	2.02416	0.0650591
$969$	0	0
$970$	4.72685	0.151770
$971$	−34.6038	−1.11049	−0.555244	−	0.831687i	$-0.687375\pi$
$971$	−34.6038	−1.11049	−0.555244	+	0.831687i	$0.687375\pi$
$972$	0	0
$973$	48.2868	1.54801
$974$	−1.72388	−0.0552368
$975$	0	0
$976$	−39.7805	−1.27334
$977$	−13.2704	−0.424557	−0.212278	−	0.977209i	$-0.568088\pi$
$977$	−13.2704	−0.424557	−0.212278	+	0.977209i	$0.568088\pi$
$978$	0	0
$979$	42.9842	1.37378
$980$	2.13692	0.0682613
$981$	0	0
$982$	1.26080	0.0402339
$983$	36.0019	1.14828	0.574141	−	0.818757i	$-0.305336\pi$
$983$	36.0019	1.14828	0.574141	+	0.818757i	$0.305336\pi$
$984$	0	0
$985$	−16.0123	−0.510193
$986$	−14.5364	−0.462933
$987$	0	0
$988$	0.639409	0.0203423
$989$	3.32107	0.105604
$990$	0	0
$991$	52.1849	1.65771	0.828854	−	0.559466i	$-0.188994\pi$
$991$	52.1849	1.65771	0.828854	+	0.559466i	$0.188994\pi$
$992$	16.4010	0.520733
$993$	0	0
$994$	11.5144	0.365215
$995$	−4.48319	−0.142127
$996$	0	0
$997$	18.0502	0.571654	0.285827	−	0.958281i	$-0.407732\pi$
$997$	18.0502	0.571654	0.285827	+	0.958281i	$0.407732\pi$
$998$	−6.80758	−0.215490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.42		72
3.2	odd	2		6561.2.a.d.1.31		72
81.4	even	27		81.2.g.a.16.5	✓	144
81.7	even	27		729.2.g.c.352.4		144
81.20	odd	54		243.2.g.a.118.4		144
81.23	odd	54		729.2.g.b.379.5		144
81.31	even	27		729.2.g.d.622.4		144
81.34	even	27		729.2.g.d.109.4		144
81.47	odd	54		729.2.g.a.109.5		144
81.50	odd	54		729.2.g.a.622.5		144
81.58	even	27		729.2.g.c.379.4		144
81.61	even	27		81.2.g.a.76.5	yes	144
81.74	odd	54		729.2.g.b.352.5		144
81.77	odd	54		243.2.g.a.208.4		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.5	✓	144	81.4	even	27
81.2.g.a.76.5	yes	144	81.61	even	27
243.2.g.a.118.4		144	81.20	odd	54
243.2.g.a.208.4		144	81.77	odd	54
729.2.g.a.109.5		144	81.47	odd	54
729.2.g.a.622.5		144	81.50	odd	54
729.2.g.b.352.5		144	81.74	odd	54
729.2.g.b.379.5		144	81.23	odd	54
729.2.g.c.352.4		144	81.7	even	27
729.2.g.c.379.4		144	81.58	even	27
729.2.g.d.109.4		144	81.34	even	27
729.2.g.d.622.4		144	81.31	even	27
6561.2.a.c.1.42		72	1.1	even	1	trivial
6561.2.a.d.1.31		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.317274 q^{2} -1.89934 q^{4} +2.12131 q^{5} -2.54355 q^{7} -1.23716 q^{8} +0.673038 q^{10} -3.06004 q^{11} +5.51901 q^{13} -0.807002 q^{14} +3.40616 q^{16} -4.29040 q^{17} -0.0609980 q^{19} -4.02909 q^{20} -0.970873 q^{22} -4.28084 q^{23} -0.500032 q^{25} +1.75104 q^{26} +4.83105 q^{28} +10.6788 q^{29} +4.61351 q^{31} +3.55500 q^{32} -1.36123 q^{34} -5.39566 q^{35} +3.51955 q^{37} -0.0193531 q^{38} -2.62440 q^{40} +2.52003 q^{41} -0.775799 q^{43} +5.81205 q^{44} -1.35820 q^{46} -4.54654 q^{47} -0.530372 q^{49} -0.158647 q^{50} -10.4825 q^{52} -1.97270 q^{53} -6.49131 q^{55} +3.14677 q^{56} +3.38812 q^{58} +7.21340 q^{59} -11.6790 q^{61} +1.46375 q^{62} -5.68440 q^{64} +11.7075 q^{65} -1.21297 q^{67} +8.14892 q^{68} -1.71190 q^{70} -14.2681 q^{71} +9.71508 q^{73} +1.11666 q^{74} +0.115856 q^{76} +7.78336 q^{77} +1.61116 q^{79} +7.22552 q^{80} +0.799542 q^{82} +3.30224 q^{83} -9.10129 q^{85} -0.246141 q^{86} +3.78576 q^{88} -14.0469 q^{89} -14.0379 q^{91} +8.13076 q^{92} -1.44250 q^{94} -0.129396 q^{95} +7.02316 q^{97} -0.168273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

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