LMFDB - Embedded newform 6561.2.a.d.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:	$0$
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)

$f(q)$	$=$	\(q-1.38039 q^{2} -0.0945210 q^{4} -4.03747 q^{5} -3.55546 q^{7} +2.89126 q^{8} +5.57329 q^{10} -0.162486 q^{11} +0.349140 q^{13} +4.90792 q^{14} -3.80202 q^{16} +2.23489 q^{17} -1.24091 q^{19} +0.381626 q^{20} +0.224294 q^{22} -3.02974 q^{23} +11.3012 q^{25} -0.481950 q^{26} +0.336065 q^{28} +7.10850 q^{29} -0.865285 q^{31} -0.534236 q^{32} -3.08503 q^{34} +14.3551 q^{35} -4.50435 q^{37} +1.71294 q^{38} -11.6734 q^{40} -7.06898 q^{41} -9.43643 q^{43} +0.0153583 q^{44} +4.18223 q^{46} -4.37311 q^{47} +5.64127 q^{49} -15.6001 q^{50} -0.0330011 q^{52} +4.65293 q^{53} +0.656033 q^{55} -10.2797 q^{56} -9.81251 q^{58} -5.06541 q^{59} -9.22834 q^{61} +1.19443 q^{62} +8.34150 q^{64} -1.40965 q^{65} -12.9970 q^{67} -0.211244 q^{68} -19.8156 q^{70} +1.61606 q^{71} -11.6784 q^{73} +6.21777 q^{74} +0.117292 q^{76} +0.577712 q^{77} -7.21301 q^{79} +15.3506 q^{80} +9.75795 q^{82} +2.29803 q^{83} -9.02333 q^{85} +13.0260 q^{86} -0.469789 q^{88} -11.6550 q^{89} -1.24135 q^{91} +0.286374 q^{92} +6.03660 q^{94} +5.01013 q^{95} -4.20173 q^{97} -7.78716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.38039	−0.976084	−0.488042	−	0.872820i	$-0.662289\pi$
$2$	−1.38039	−0.976084	−0.488042	+	0.872820i	$0.662289\pi$
$3$	0	0
$3$	0	0
$4$	−0.0945210	−0.0472605
$5$	−4.03747	−1.80561	−0.902806	−	0.430047i	$-0.858497\pi$
$5$	−4.03747	−1.80561	−0.902806	+	0.430047i	$0.858497\pi$
$6$	0	0
$7$	−3.55546	−1.34384	−0.671918	−	0.740625i	$-0.734529\pi$
$7$	−3.55546	−1.34384	−0.671918	+	0.740625i	$0.734529\pi$
$8$	2.89126	1.02221
$9$	0	0
$10$	5.57329	1.76243
$11$	−0.162486	−0.0489914	−0.0244957	−	0.999700i	$-0.507798\pi$
$11$	−0.162486	−0.0489914	−0.0244957	+	0.999700i	$0.507798\pi$
$12$	0	0
$13$	0.349140	0.0968341	0.0484171	−	0.998827i	$-0.484582\pi$
$13$	0.349140	0.0968341	0.0484171	+	0.998827i	$0.484582\pi$
$14$	4.90792	1.31170
$15$	0	0
$16$	−3.80202	−0.950506
$17$	2.23489	0.542042	0.271021	−	0.962573i	$-0.412639\pi$
$17$	2.23489	0.542042	0.271021	+	0.962573i	$0.412639\pi$
$18$	0	0
$19$	−1.24091	−0.284683	−0.142342	−	0.989818i	$-0.545463\pi$
$19$	−1.24091	−0.284683	−0.142342	+	0.989818i	$0.545463\pi$
$20$	0.381626	0.0853341
$21$	0	0
$22$	0.224294	0.0478197
$23$	−3.02974	−0.631745	−0.315872	−	0.948802i	$-0.602297\pi$
$23$	−3.02974	−0.631745	−0.315872	+	0.948802i	$0.602297\pi$
$24$	0	0
$25$	11.3012	2.26024
$26$	−0.481950	−0.0945182
$27$	0	0
$28$	0.336065	0.0635103
$29$	7.10850	1.32002	0.660008	−	0.751259i	$-0.270553\pi$
$29$	7.10850	1.32002	0.660008	+	0.751259i	$0.270553\pi$
$30$	0	0
$31$	−0.865285	−0.155410	−0.0777049	−	0.996976i	$-0.524759\pi$
$31$	−0.865285	−0.155410	−0.0777049	+	0.996976i	$0.524759\pi$
$32$	−0.534236	−0.0944405
$33$	0	0
$34$	−3.08503	−0.529078
$35$	14.3551	2.42645
$36$	0	0
$37$	−4.50435	−0.740511	−0.370256	−	0.928930i	$-0.620730\pi$
$37$	−4.50435	−0.740511	−0.370256	+	0.928930i	$0.620730\pi$
$38$	1.71294	0.277875
$39$	0	0
$40$	−11.6734	−1.84572
$41$	−7.06898	−1.10399	−0.551994	−	0.833848i	$-0.686133\pi$
$41$	−7.06898	−1.10399	−0.551994	+	0.833848i	$0.686133\pi$
$42$	0	0
$43$	−9.43643	−1.43904	−0.719521	−	0.694471i	$-0.755639\pi$
$43$	−9.43643	−1.43904	−0.719521	+	0.694471i	$0.755639\pi$
$44$	0.0153583	0.00231536
$45$	0	0
$46$	4.18223	0.616636
$47$	−4.37311	−0.637884	−0.318942	−	0.947774i	$-0.603328\pi$
$47$	−4.37311	−0.637884	−0.318942	+	0.947774i	$0.603328\pi$
$48$	0	0
$49$	5.64127	0.805896
$50$	−15.6001	−2.20618
$51$	0	0
$52$	−0.0330011	−0.00457643
$53$	4.65293	0.639129	0.319564	−	0.947565i	$-0.396463\pi$
$53$	4.65293	0.639129	0.319564	+	0.947565i	$0.396463\pi$
$54$	0	0
$55$	0.656033	0.0884594
$56$	−10.2797	−1.37369
$57$	0	0
$58$	−9.81251	−1.28845
$59$	−5.06541	−0.659460	−0.329730	−	0.944075i	$-0.606958\pi$
$59$	−5.06541	−0.659460	−0.329730	+	0.944075i	$0.606958\pi$
$60$	0	0
$61$	−9.22834	−1.18157	−0.590784	−	0.806830i	$-0.701181\pi$
$61$	−9.22834	−1.18157	−0.590784	+	0.806830i	$0.701181\pi$
$62$	1.19443	0.151693
$63$	0	0
$64$	8.34150	1.04269
$65$	−1.40965	−0.174845
$66$	0	0
$67$	−12.9970	−1.58783	−0.793916	−	0.608027i	$-0.791961\pi$
$67$	−12.9970	−1.58783	−0.793916	+	0.608027i	$0.791961\pi$
$68$	−0.211244	−0.0256171
$69$	0	0
$70$	−19.8156	−2.36842
$71$	1.61606	0.191791	0.0958956	−	0.995391i	$-0.469429\pi$
$71$	1.61606	0.191791	0.0958956	+	0.995391i	$0.469429\pi$
$72$	0	0
$73$	−11.6784	−1.36686	−0.683429	−	0.730017i	$-0.739512\pi$
$73$	−11.6784	−1.36686	−0.683429	+	0.730017i	$0.739512\pi$
$74$	6.21777	0.722801
$75$	0	0
$76$	0.117292	0.0134543
$77$	0.577712	0.0658364
$78$	0	0
$79$	−7.21301	−0.811527	−0.405763	−	0.913978i	$-0.632994\pi$
$79$	−7.21301	−0.811527	−0.405763	+	0.913978i	$0.632994\pi$
$80$	15.3506	1.71625
$81$	0	0
$82$	9.75795	1.07759
$83$	2.29803	0.252242	0.126121	−	0.992015i	$-0.459747\pi$
$83$	2.29803	0.252242	0.126121	+	0.992015i	$0.459747\pi$
$84$	0	0
$85$	−9.02333	−0.978717
$86$	13.0260	1.40463
$87$	0	0
$88$	−0.469789	−0.0500797
$89$	−11.6550	−1.23543	−0.617715	−	0.786402i	$-0.711942\pi$
$89$	−11.6550	−1.23543	−0.617715	+	0.786402i	$0.711942\pi$
$90$	0	0
$91$	−1.24135	−0.130129
$92$	0.286374	0.0298566
$93$	0	0
$94$	6.03660	0.622628
$95$	5.01013	0.514028
$96$	0	0
$97$	−4.20173	−0.426621	−0.213310	−	0.976984i	$-0.568425\pi$
$97$	−4.20173	−0.426621	−0.213310	+	0.976984i	$0.568425\pi$
$98$	−7.78716	−0.786622
$99$	0	0
$100$	−1.06820	−0.106820
$101$	1.67527	0.166695	0.0833476	−	0.996521i	$-0.473439\pi$
$101$	1.67527	0.166695	0.0833476	+	0.996521i	$0.473439\pi$
$102$	0	0
$103$	3.35829	0.330902	0.165451	−	0.986218i	$-0.447092\pi$
$103$	3.35829	0.330902	0.165451	+	0.986218i	$0.447092\pi$
$104$	1.00946	0.0989852
$105$	0	0
$106$	−6.42286	−0.623843
$107$	11.8074	1.14146	0.570730	−	0.821137i	$-0.306660\pi$
$107$	11.8074	1.14146	0.570730	+	0.821137i	$0.306660\pi$
$108$	0	0
$109$	−10.0631	−0.963873	−0.481937	−	0.876206i	$-0.660066\pi$
$109$	−10.0631	−0.963873	−0.481937	+	0.876206i	$0.660066\pi$
$110$	−0.905582	−0.0863438
$111$	0	0
$112$	13.5179	1.27732
$113$	−3.90230	−0.367098	−0.183549	−	0.983011i	$-0.558759\pi$
$113$	−3.90230	−0.367098	−0.183549	+	0.983011i	$0.558759\pi$
$114$	0	0
$115$	12.2325	1.14069
$116$	−0.671902	−0.0623846
$117$	0	0
$118$	6.99224	0.643688
$119$	−7.94607	−0.728415
$120$	0	0
$121$	−10.9736	−0.997600
$122$	12.7387	1.15331
$123$	0	0
$124$	0.0817876	0.00734474
$125$	−25.4409	−2.27550
$126$	0	0
$127$	8.53278	0.757162	0.378581	−	0.925568i	$-0.376412\pi$
$127$	8.53278	0.757162	0.378581	+	0.925568i	$0.376412\pi$
$128$	−10.4461	−0.923310
$129$	0	0
$130$	1.94586	0.170663
$131$	16.1742	1.41314	0.706572	−	0.707641i	$-0.250241\pi$
$131$	16.1742	1.41314	0.706572	+	0.707641i	$0.250241\pi$
$132$	0	0
$133$	4.41199	0.382568
$134$	17.9409	1.54986
$135$	0	0
$136$	6.46166	0.554082
$137$	3.41862	0.292073	0.146036	−	0.989279i	$-0.453348\pi$
$137$	3.41862	0.292073	0.146036	+	0.989279i	$0.453348\pi$
$138$	0	0
$139$	−11.2768	−0.956485	−0.478243	−	0.878228i	$-0.658726\pi$
$139$	−11.2768	−0.956485	−0.478243	+	0.878228i	$0.658726\pi$
$140$	−1.35685	−0.114675
$141$	0	0
$142$	−2.23080	−0.187204
$143$	−0.0567304	−0.00474404
$144$	0	0
$145$	−28.7004	−2.38344
$146$	16.1208	1.33417
$147$	0	0
$148$	0.425756	0.0349969
$149$	0.597717	0.0489669	0.0244834	−	0.999700i	$-0.492206\pi$
$149$	0.597717	0.0489669	0.0244834	+	0.999700i	$0.492206\pi$
$150$	0	0
$151$	1.67721	0.136489	0.0682447	−	0.997669i	$-0.478260\pi$
$151$	1.67721	0.136489	0.0682447	+	0.997669i	$0.478260\pi$
$152$	−3.58778	−0.291007
$153$	0	0
$154$	−0.797468	−0.0642618
$155$	3.49356	0.280610
$156$	0	0
$157$	9.12223	0.728033	0.364017	−	0.931392i	$-0.381405\pi$
$157$	9.12223	0.728033	0.364017	+	0.931392i	$0.381405\pi$
$158$	9.95677	0.792118
$159$	0	0
$160$	2.15696	0.170523
$161$	10.7721	0.848962
$162$	0	0
$163$	−21.8751	−1.71339	−0.856697	−	0.515821i	$-0.827487\pi$
$163$	−21.8751	−1.71339	−0.856697	+	0.515821i	$0.827487\pi$
$164$	0.668167	0.0521750
$165$	0	0
$166$	−3.17218	−0.246209
$167$	5.20882	0.403070	0.201535	−	0.979481i	$-0.435407\pi$
$167$	5.20882	0.403070	0.201535	+	0.979481i	$0.435407\pi$
$168$	0	0
$169$	−12.8781	−0.990623
$170$	12.4557	0.955310
$171$	0	0
$172$	0.891940	0.0680098
$173$	−20.8604	−1.58599	−0.792994	−	0.609229i	$-0.791479\pi$
$173$	−20.8604	−1.58599	−0.792994	+	0.609229i	$0.791479\pi$
$174$	0	0
$175$	−40.1809	−3.03739
$176$	0.617776	0.0465666
$177$	0	0
$178$	16.0885	1.20588
$179$	4.08307	0.305183	0.152592	−	0.988289i	$-0.451238\pi$
$179$	4.08307	0.305183	0.152592	+	0.988289i	$0.451238\pi$
$180$	0	0
$181$	16.4423	1.22214	0.611072	−	0.791575i	$-0.290738\pi$
$181$	16.4423	1.22214	0.611072	+	0.791575i	$0.290738\pi$
$182$	1.71355	0.127017
$183$	0	0
$184$	−8.75977	−0.645778
$185$	18.1862	1.33708
$186$	0	0
$187$	−0.363139	−0.0265554
$188$	0.413351	0.0301467
$189$	0	0
$190$	−6.91593	−0.501735
$191$	0.00416915	0.000301669 0	0.000150835	−	1.00000i	$-0.499952\pi$
$191$	0.00416915	0.000301669 0	0.000150835		1.00000i	$0.499952\pi$
$192$	0	0
$193$	−20.9875	−1.51071	−0.755356	−	0.655314i	$-0.772536\pi$
$193$	−20.9875	−1.51071	−0.755356	+	0.655314i	$0.772536\pi$
$194$	5.80003	0.416418
$195$	0	0
$196$	−0.533218	−0.0380870
$197$	−9.73198	−0.693375	−0.346687	−	0.937981i	$-0.612694\pi$
$197$	−9.73198	−0.693375	−0.346687	+	0.937981i	$0.612694\pi$
$198$	0	0
$199$	−24.1221	−1.70997	−0.854987	−	0.518650i	$-0.826435\pi$
$199$	−24.1221	−1.70997	−0.854987	+	0.518650i	$0.826435\pi$
$200$	32.6746	2.31045
$201$	0	0
$202$	−2.31252	−0.162709
$203$	−25.2740	−1.77388
$204$	0	0
$205$	28.5408	1.99338
$206$	−4.63575	−0.322988
$207$	0	0
$208$	−1.32744	−0.0920414
$209$	0.201630	0.0139470
$210$	0	0
$211$	15.0475	1.03592	0.517958	−	0.855406i	$-0.326692\pi$
$211$	15.0475	1.03592	0.517958	+	0.855406i	$0.326692\pi$
$212$	−0.439799	−0.0302055
$213$	0	0
$214$	−16.2988	−1.11416
$215$	38.0993	2.59835
$216$	0	0
$217$	3.07648	0.208845
$218$	13.8911	0.940821
$219$	0	0
$220$	−0.0620088	−0.00418064
$221$	0.780292	0.0524881
$222$	0	0
$223$	9.52530	0.637861	0.318931	−	0.947778i	$-0.396676\pi$
$223$	9.52530	0.637861	0.318931	+	0.947778i	$0.396676\pi$
$224$	1.89945	0.126913
$225$	0	0
$226$	5.38670	0.358318
$227$	−12.2498	−0.813050	−0.406525	−	0.913640i	$-0.633260\pi$
$227$	−12.2498	−0.813050	−0.406525	+	0.913640i	$0.633260\pi$
$228$	0	0
$229$	3.73359	0.246723	0.123361	−	0.992362i	$-0.460633\pi$
$229$	3.73359	0.246723	0.123361	+	0.992362i	$0.460633\pi$
$230$	−16.8856	−1.11341
$231$	0	0
$232$	20.5525	1.34934
$233$	−20.5033	−1.34322	−0.671609	−	0.740906i	$-0.734397\pi$
$233$	−20.5033	−1.34322	−0.671609	+	0.740906i	$0.734397\pi$
$234$	0	0
$235$	17.6563	1.15177
$236$	0.478787	0.0311664
$237$	0	0
$238$	10.9687	0.710994
$239$	12.9521	0.837800	0.418900	−	0.908032i	$-0.362416\pi$
$239$	12.9521	0.837800	0.418900	+	0.908032i	$0.362416\pi$
$240$	0	0
$241$	12.1261	0.781114	0.390557	−	0.920579i	$-0.372282\pi$
$241$	12.1261	0.781114	0.390557	+	0.920579i	$0.372282\pi$
$242$	15.1479	0.973741
$243$	0	0
$244$	0.872272	0.0558415
$245$	−22.7765	−1.45514
$246$	0	0
$247$	−0.433251	−0.0275671
$248$	−2.50176	−0.158862
$249$	0	0
$250$	35.1184	2.22108
$251$	28.3870	1.79177	0.895885	−	0.444287i	$-0.146543\pi$
$251$	28.3870	1.79177	0.895885	+	0.444287i	$0.146543\pi$
$252$	0	0
$253$	0.492291	0.0309501
$254$	−11.7786	−0.739053
$255$	0	0
$256$	−2.26336	−0.141460
$257$	−27.4880	−1.71465	−0.857327	−	0.514772i	$-0.827876\pi$
$257$	−27.4880	−1.71465	−0.857327	+	0.514772i	$0.827876\pi$
$258$	0	0
$259$	16.0150	0.995126
$260$	0.133241	0.00826326
$261$	0	0
$262$	−22.3267	−1.37935
$263$	−4.64377	−0.286347	−0.143174	−	0.989698i	$-0.545731\pi$
$263$	−4.64377	−0.286347	−0.143174	+	0.989698i	$0.545731\pi$
$264$	0	0
$265$	−18.7861	−1.15402
$266$	−6.09027	−0.373418
$267$	0	0
$268$	1.22849	0.0750417
$269$	5.01537	0.305792	0.152896	−	0.988242i	$-0.451140\pi$
$269$	5.01537	0.305792	0.152896	+	0.988242i	$0.451140\pi$
$270$	0	0
$271$	16.0871	0.977223	0.488612	−	0.872501i	$-0.337504\pi$
$271$	16.0871	0.977223	0.488612	+	0.872501i	$0.337504\pi$
$272$	−8.49712	−0.515214
$273$	0	0
$274$	−4.71903	−0.285087
$275$	−1.83628	−0.110732
$276$	0	0
$277$	9.84120	0.591300	0.295650	−	0.955296i	$-0.404464\pi$
$277$	9.84120	0.591300	0.295650	+	0.955296i	$0.404464\pi$
$278$	15.5664	0.933610
$279$	0	0
$280$	41.5042	2.48035
$281$	−21.7423	−1.29704	−0.648519	−	0.761199i	$-0.724611\pi$
$281$	−21.7423	−1.29704	−0.648519	+	0.761199i	$0.724611\pi$
$282$	0	0
$283$	−4.80315	−0.285518	−0.142759	−	0.989758i	$-0.545597\pi$
$283$	−4.80315	−0.285518	−0.142759	+	0.989758i	$0.545597\pi$
$284$	−0.152752	−0.00906414
$285$	0	0
$286$	0.0783102	0.00463058
$287$	25.1334	1.48358
$288$	0	0
$289$	−12.0052	−0.706191
$290$	39.6177	2.32643
$291$	0	0
$292$	1.10386	0.0645984
$293$	5.65004	0.330079	0.165040	−	0.986287i	$-0.447225\pi$
$293$	5.65004	0.330079	0.165040	+	0.986287i	$0.447225\pi$
$294$	0	0
$295$	20.4514	1.19073
$296$	−13.0232	−0.756961
$297$	0	0
$298$	−0.825083	−0.0477958
$299$	−1.05781	−0.0611745
$300$	0	0
$301$	33.5508	1.93384
$302$	−2.31521	−0.133225
$303$	0	0
$304$	4.71796	0.270593
$305$	37.2592	2.13345
$306$	0	0
$307$	7.98273	0.455599	0.227799	−	0.973708i	$-0.426847\pi$
$307$	7.98273	0.455599	0.227799	+	0.973708i	$0.426847\pi$
$308$	−0.0546059	−0.00311146
$309$	0	0
$310$	−4.82249	−0.273899
$311$	20.6448	1.17066	0.585329	−	0.810796i	$-0.300965\pi$
$311$	20.6448	1.17066	0.585329	+	0.810796i	$0.300965\pi$
$312$	0	0
$313$	19.4329	1.09841	0.549207	−	0.835686i	$-0.314930\pi$
$313$	19.4329	1.09841	0.549207	+	0.835686i	$0.314930\pi$
$314$	−12.5922	−0.710621
$315$	0	0
$316$	0.681780	0.0383531
$317$	−29.0866	−1.63367	−0.816834	−	0.576873i	$-0.804273\pi$
$317$	−29.0866	−1.63367	−0.816834	+	0.576873i	$0.804273\pi$
$318$	0	0
$319$	−1.15503	−0.0646694
$320$	−33.6786	−1.88269
$321$	0	0
$322$	−14.8697	−0.828658
$323$	−2.77330	−0.154310
$324$	0	0
$325$	3.94570	0.218868
$326$	30.1962	1.67242
$327$	0	0
$328$	−20.4382	−1.12851
$329$	15.5484	0.857212
$330$	0	0
$331$	3.07120	0.168808	0.0844042	−	0.996432i	$-0.473101\pi$
$331$	3.07120	0.168808	0.0844042	+	0.996432i	$0.473101\pi$
$332$	−0.217212	−0.0119211
$333$	0	0
$334$	−7.19020	−0.393430
$335$	52.4749	2.86701
$336$	0	0
$337$	27.8022	1.51448	0.757240	−	0.653137i	$-0.226547\pi$
$337$	27.8022	1.51448	0.757240	+	0.653137i	$0.226547\pi$
$338$	17.7768	0.966931
$339$	0	0
$340$	0.852893	0.0462546
$341$	0.140597	0.00761374
$342$	0	0
$343$	4.83091	0.260844
$344$	−27.2832	−1.47101
$345$	0	0
$346$	28.7955	1.54806
$347$	3.50450	0.188131	0.0940657	−	0.995566i	$-0.470014\pi$
$347$	3.50450	0.188131	0.0940657	+	0.995566i	$0.470014\pi$
$348$	0	0
$349$	−11.4402	−0.612381	−0.306190	−	0.951970i	$-0.599054\pi$
$349$	−11.4402	−0.612381	−0.306190	+	0.951970i	$0.599054\pi$
$350$	55.4653	2.96475
$351$	0	0
$352$	0.0868059	0.00462677
$353$	1.83458	0.0976447	0.0488223	−	0.998807i	$-0.484453\pi$
$353$	1.83458	0.0976447	0.0488223	+	0.998807i	$0.484453\pi$
$354$	0	0
$355$	−6.52480	−0.346301
$356$	1.10164	0.0583870
$357$	0	0
$358$	−5.63624	−0.297884
$359$	20.1156	1.06166	0.530831	−	0.847478i	$-0.321880\pi$
$359$	20.1156	1.06166	0.530831	+	0.847478i	$0.321880\pi$
$360$	0	0
$361$	−17.4602	−0.918955
$362$	−22.6968	−1.19292
$363$	0	0
$364$	0.117334	0.00614997
$365$	47.1514	2.46802
$366$	0	0
$367$	26.4105	1.37862	0.689308	−	0.724468i	$-0.257915\pi$
$367$	26.4105	1.37862	0.689308	+	0.724468i	$0.257915\pi$
$368$	11.5192	0.600477
$369$	0	0
$370$	−25.1041	−1.30510
$371$	−16.5433	−0.858884
$372$	0	0
$373$	−29.6387	−1.53463	−0.767317	−	0.641268i	$-0.778409\pi$
$373$	−29.6387	−1.53463	−0.767317	+	0.641268i	$0.778409\pi$
$374$	0.501274	0.0259203
$375$	0	0
$376$	−12.6438	−0.652054
$377$	2.48187	0.127823
$378$	0	0
$379$	−11.5058	−0.591016	−0.295508	−	0.955340i	$-0.595489\pi$
$379$	−11.5058	−0.591016	−0.295508	+	0.955340i	$0.595489\pi$
$380$	−0.473562	−0.0242932
$381$	0	0
$382$	−0.00575506	−0.000294455 0
$383$	28.7641	1.46978	0.734888	−	0.678188i	$-0.237235\pi$
$383$	28.7641	1.46978	0.734888	+	0.678188i	$0.237235\pi$
$384$	0	0
$385$	−2.33250	−0.118875
$386$	28.9709	1.47458
$387$	0	0
$388$	0.397151	0.0201623
$389$	11.4553	0.580808	0.290404	−	0.956904i	$-0.406210\pi$
$389$	11.4553	0.580808	0.290404	+	0.956904i	$0.406210\pi$
$390$	0	0
$391$	−6.77115	−0.342432
$392$	16.3104	0.823798
$393$	0	0
$394$	13.4339	0.676792
$395$	29.1223	1.46530
$396$	0	0
$397$	10.9419	0.549158	0.274579	−	0.961565i	$-0.411462\pi$
$397$	10.9419	0.549158	0.274579	+	0.961565i	$0.411462\pi$
$398$	33.2980	1.66908
$399$	0	0
$400$	−42.9674	−2.14837
$401$	−0.347567	−0.0173567	−0.00867834	−	0.999962i	$-0.502762\pi$
$401$	−0.347567	−0.0173567	−0.00867834	+	0.999962i	$0.502762\pi$
$402$	0	0
$403$	−0.302106	−0.0150490
$404$	−0.158348	−0.00787810
$405$	0	0
$406$	34.8880	1.73146
$407$	0.731894	0.0362787
$408$	0	0
$409$	26.6015	1.31536	0.657679	−	0.753298i	$-0.271538\pi$
$409$	26.6015	1.31536	0.657679	+	0.753298i	$0.271538\pi$
$410$	−39.3975	−1.94570
$411$	0	0
$412$	−0.317428	−0.0156386
$413$	18.0098	0.886206
$414$	0	0
$415$	−9.27823	−0.455450
$416$	−0.186523	−0.00914506
$417$	0	0
$418$	−0.278328	−0.0136135
$419$	−12.7912	−0.624893	−0.312447	−	0.949935i	$-0.601149\pi$
$419$	−12.7912	−0.624893	−0.312447	+	0.949935i	$0.601149\pi$
$420$	0	0
$421$	−10.8749	−0.530008	−0.265004	−	0.964247i	$-0.585373\pi$
$421$	−10.8749	−0.530008	−0.265004	+	0.964247i	$0.585373\pi$
$422$	−20.7715	−1.01114
$423$	0	0
$424$	13.4528	0.653326
$425$	25.2570	1.22514
$426$	0	0
$427$	32.8110	1.58783
$428$	−1.11604	−0.0539460
$429$	0	0
$430$	−52.5920	−2.53621
$431$	2.45939	0.118465	0.0592323	−	0.998244i	$-0.481135\pi$
$431$	2.45939	0.118465	0.0592323	+	0.998244i	$0.481135\pi$
$432$	0	0
$433$	−12.8867	−0.619296	−0.309648	−	0.950851i	$-0.600211\pi$
$433$	−12.8867	−0.619296	−0.309648	+	0.950851i	$0.600211\pi$
$434$	−4.24675	−0.203850
$435$	0	0
$436$	0.951177	0.0455531
$437$	3.75963	0.179847
$438$	0	0
$439$	−27.5664	−1.31567	−0.657837	−	0.753160i	$-0.728528\pi$
$439$	−27.5664	−1.31567	−0.657837	+	0.753160i	$0.728528\pi$
$440$	1.89676	0.0904245
$441$	0	0
$442$	−1.07711	−0.0512328
$443$	−5.73362	−0.272413	−0.136206	−	0.990681i	$-0.543491\pi$
$443$	−5.73362	−0.272413	−0.136206	+	0.990681i	$0.543491\pi$
$444$	0	0
$445$	47.0569	2.23071
$446$	−13.1486	−0.622606
$447$	0	0
$448$	−29.6578	−1.40120
$449$	−34.4301	−1.62486	−0.812429	−	0.583060i	$-0.801855\pi$
$449$	−34.4301	−1.62486	−0.812429	+	0.583060i	$0.801855\pi$
$450$	0	0
$451$	1.14861	0.0540859
$452$	0.368849	0.0173492
$453$	0	0
$454$	16.9096	0.793605
$455$	5.01193	0.234963
$456$	0	0
$457$	−7.58715	−0.354912	−0.177456	−	0.984129i	$-0.556787\pi$
$457$	−7.58715	−0.354912	−0.177456	+	0.984129i	$0.556787\pi$
$458$	−5.15381	−0.240822
$459$	0	0
$460$	−1.15623	−0.0539094
$461$	1.84118	0.0857521	0.0428761	−	0.999080i	$-0.486348\pi$
$461$	1.84118	0.0857521	0.0428761	+	0.999080i	$0.486348\pi$
$462$	0	0
$463$	−22.5036	−1.04583	−0.522915	−	0.852385i	$-0.675156\pi$
$463$	−22.5036	−1.04583	−0.522915	+	0.852385i	$0.675156\pi$
$464$	−27.0267	−1.25468
$465$	0	0
$466$	28.3026	1.31109
$467$	−39.4224	−1.82425	−0.912126	−	0.409909i	$-0.865560\pi$
$467$	−39.4224	−1.82425	−0.912126	+	0.409909i	$0.865560\pi$
$468$	0	0
$469$	46.2101	2.13379
$470$	−24.3726	−1.12423
$471$	0	0
$472$	−14.6454	−0.674109
$473$	1.53329	0.0705006
$474$	0	0
$475$	−14.0237	−0.643452
$476$	0.751070	0.0344252
$477$	0	0
$478$	−17.8789	−0.817763
$479$	2.63929	0.120592	0.0602961	−	0.998181i	$-0.480795\pi$
$479$	2.63929	0.120592	0.0602961	+	0.998181i	$0.480795\pi$
$480$	0	0
$481$	−1.57265	−0.0717068
$482$	−16.7388	−0.762432
$483$	0	0
$484$	1.03723	0.0471470
$485$	16.9644	0.770312
$486$	0	0
$487$	−25.0040	−1.13304	−0.566519	−	0.824048i	$-0.691710\pi$
$487$	−25.0040	−1.13304	−0.566519	+	0.824048i	$0.691710\pi$
$488$	−26.6815	−1.20782
$489$	0	0
$490$	31.4404	1.42033
$491$	−30.1066	−1.35869	−0.679347	−	0.733817i	$-0.737737\pi$
$491$	−30.1066	−1.35869	−0.679347	+	0.733817i	$0.737737\pi$
$492$	0	0
$493$	15.8868	0.715503
$494$	0.598055	0.0269078
$495$	0	0
$496$	3.28983	0.147718
$497$	−5.74584	−0.257736
$498$	0	0
$499$	−5.63783	−0.252384	−0.126192	−	0.992006i	$-0.540276\pi$
$499$	−5.63783	−0.252384	−0.126192	+	0.992006i	$0.540276\pi$
$500$	2.40470	0.107541
$501$	0	0
$502$	−39.1851	−1.74892
$503$	−11.5799	−0.516324	−0.258162	−	0.966102i	$-0.583117\pi$
$503$	−11.5799	−0.516324	−0.258162	+	0.966102i	$0.583117\pi$
$504$	0	0
$505$	−6.76384	−0.300987
$506$	−0.679554	−0.0302098
$507$	0	0
$508$	−0.806526	−0.0357838
$509$	−35.8631	−1.58960	−0.794802	−	0.606869i	$-0.792425\pi$
$509$	−35.8631	−1.58960	−0.794802	+	0.606869i	$0.792425\pi$
$510$	0	0
$511$	41.5222	1.83683
$512$	24.0164	1.06139
$513$	0	0
$514$	37.9442	1.67365
$515$	−13.5590	−0.597481
$516$	0	0
$517$	0.710570	0.0312508
$518$	−22.1070	−0.971326
$519$	0	0
$520$	−4.07565	−0.178729
$521$	8.90540	0.390153	0.195076	−	0.980788i	$-0.437505\pi$
$521$	8.90540	0.390153	0.195076	+	0.980788i	$0.437505\pi$
$522$	0	0
$523$	16.1552	0.706417	0.353209	−	0.935545i	$-0.385091\pi$
$523$	16.1552	0.706417	0.353209	+	0.935545i	$0.385091\pi$
$524$	−1.52880	−0.0667859
$525$	0	0
$526$	6.41022	0.279499
$527$	−1.93382	−0.0842386
$528$	0	0
$529$	−13.8207	−0.600898
$530$	25.9321	1.12642
$531$	0	0
$532$	−0.417025	−0.0180803
$533$	−2.46807	−0.106904
$534$	0	0
$535$	−47.6719	−2.06104
$536$	−37.5776	−1.62310
$537$	0	0
$538$	−6.92316	−0.298479
$539$	−0.916627	−0.0394819
$540$	0	0
$541$	−9.50515	−0.408658	−0.204329	−	0.978902i	$-0.565501\pi$
$541$	−9.50515	−0.408658	−0.204329	+	0.978902i	$0.565501\pi$
$542$	−22.2065	−0.953852
$543$	0	0
$544$	−1.19396	−0.0511907
$545$	40.6296	1.74038
$546$	0	0
$547$	5.30600	0.226868	0.113434	−	0.993546i	$-0.463815\pi$
$547$	5.30600	0.226868	0.113434	+	0.993546i	$0.463815\pi$
$548$	−0.323131	−0.0138035
$549$	0	0
$550$	2.53479	0.108084
$551$	−8.82099	−0.375787
$552$	0	0
$553$	25.6455	1.09056
$554$	−13.5847	−0.577159
$555$	0	0
$556$	1.06589	0.0452040
$557$	−32.8789	−1.39312	−0.696561	−	0.717498i	$-0.745287\pi$
$557$	−32.8789	−1.39312	−0.696561	+	0.717498i	$0.745287\pi$
$558$	0	0
$559$	−3.29464	−0.139348
$560$	−54.5783	−2.30635
$561$	0	0
$562$	30.0129	1.26602
$563$	−12.7672	−0.538074	−0.269037	−	0.963130i	$-0.586705\pi$
$563$	−12.7672	−0.538074	−0.269037	+	0.963130i	$0.586705\pi$
$564$	0	0
$565$	15.7554	0.662836
$566$	6.63022	0.278689
$567$	0	0
$568$	4.67245	0.196052
$569$	42.9771	1.80169	0.900847	−	0.434137i	$-0.142947\pi$
$569$	42.9771	1.80169	0.900847	+	0.434137i	$0.142947\pi$
$570$	0	0
$571$	−3.80525	−0.159245	−0.0796223	−	0.996825i	$-0.525371\pi$
$571$	−3.80525	−0.159245	−0.0796223	+	0.996825i	$0.525371\pi$
$572$	0.00536221	0.000224205 0
$573$	0	0
$574$	−34.6940	−1.44810
$575$	−34.2397	−1.42789
$576$	0	0
$577$	15.0528	0.626655	0.313327	−	0.949645i	$-0.398556\pi$
$577$	15.0528	0.626655	0.313327	+	0.949645i	$0.398556\pi$
$578$	16.5719	0.689302
$579$	0	0
$580$	2.71279	0.112642
$581$	−8.17054	−0.338971
$582$	0	0
$583$	−0.756036	−0.0313118
$584$	−33.7654	−1.39722
$585$	0	0
$586$	−7.79927	−0.322185
$587$	38.1289	1.57375	0.786874	−	0.617113i	$-0.211698\pi$
$587$	38.1289	1.57375	0.786874	+	0.617113i	$0.211698\pi$
$588$	0	0
$589$	1.07374	0.0442426
$590$	−28.2310	−1.16225
$591$	0	0
$592$	17.1257	0.703860
$593$	11.6275	0.477484	0.238742	−	0.971083i	$-0.423265\pi$
$593$	11.6275	0.477484	0.238742	+	0.971083i	$0.423265\pi$
$594$	0	0
$595$	32.0820	1.31524
$596$	−0.0564968	−0.00231420
$597$	0	0
$598$	1.46019	0.0597114
$599$	40.3757	1.64971	0.824853	−	0.565347i	$-0.191258\pi$
$599$	40.3757	1.64971	0.824853	+	0.565347i	$0.191258\pi$
$600$	0	0
$601$	−19.7159	−0.804230	−0.402115	−	0.915589i	$-0.631725\pi$
$601$	−19.7159	−0.804230	−0.402115	+	0.915589i	$0.631725\pi$
$602$	−46.3132	−1.88759
$603$	0	0
$604$	−0.158531	−0.00645055
$605$	44.3056	1.80128
$606$	0	0
$607$	1.83891	0.0746392	0.0373196	−	0.999303i	$-0.488118\pi$
$607$	1.83891	0.0746392	0.0373196	+	0.999303i	$0.488118\pi$
$608$	0.662937	0.0268856
$609$	0	0
$610$	−51.4322	−2.08243
$611$	−1.52683	−0.0617690
$612$	0	0
$613$	35.4287	1.43095	0.715476	−	0.698638i	$-0.246210\pi$
$613$	35.4287	1.43095	0.715476	+	0.698638i	$0.246210\pi$
$614$	−11.0193	−0.444703
$615$	0	0
$616$	1.67031	0.0672989
$617$	37.1192	1.49436	0.747181	−	0.664621i	$-0.231407\pi$
$617$	37.1192	1.49436	0.747181	+	0.664621i	$0.231407\pi$
$618$	0	0
$619$	33.9746	1.36555	0.682777	−	0.730627i	$-0.260772\pi$
$619$	33.9746	1.36555	0.682777	+	0.730627i	$0.260772\pi$
$620$	−0.330215	−0.0132618
$621$	0	0
$622$	−28.4979	−1.14266
$623$	41.4389	1.66022
$624$	0	0
$625$	46.2109	1.84844
$626$	−26.8251	−1.07214
$627$	0	0
$628$	−0.862242	−0.0344072
$629$	−10.0668	−0.401388
$630$	0	0
$631$	−11.0506	−0.439919	−0.219960	−	0.975509i	$-0.570593\pi$
$631$	−11.0506	−0.439919	−0.219960	+	0.975509i	$0.570593\pi$
$632$	−20.8547	−0.829554
$633$	0	0
$634$	40.1509	1.59460
$635$	−34.4509	−1.36714
$636$	0	0
$637$	1.96960	0.0780382
$638$	1.59440	0.0631227
$639$	0	0
$640$	42.1757	1.66714
$641$	−16.2618	−0.642304	−0.321152	−	0.947028i	$-0.604070\pi$
$641$	−16.2618	−0.642304	−0.321152	+	0.947028i	$0.604070\pi$
$642$	0	0
$643$	14.2733	0.562884	0.281442	−	0.959578i	$-0.409187\pi$
$643$	14.2733	0.562884	0.281442	+	0.959578i	$0.409187\pi$
$644$	−1.01819	−0.0401223
$645$	0	0
$646$	3.82823	0.150620
$647$	40.7899	1.60362	0.801808	−	0.597582i	$-0.203872\pi$
$647$	40.7899	1.60362	0.801808	+	0.597582i	$0.203872\pi$
$648$	0	0
$649$	0.823057	0.0323078
$650$	−5.44661	−0.213634
$651$	0	0
$652$	2.06766	0.0809758
$653$	−2.57356	−0.100711	−0.0503555	−	0.998731i	$-0.516035\pi$
$653$	−2.57356	−0.100711	−0.0503555	+	0.998731i	$0.516035\pi$
$654$	0	0
$655$	−65.3028	−2.55159
$656$	26.8764	1.04935
$657$	0	0
$658$	−21.4629	−0.836710
$659$	8.58546	0.334442	0.167221	−	0.985919i	$-0.446521\pi$
$659$	8.58546	0.334442	0.167221	+	0.985919i	$0.446521\pi$
$660$	0	0
$661$	37.0919	1.44271	0.721353	−	0.692567i	$-0.243520\pi$
$661$	37.0919	1.44271	0.721353	+	0.692567i	$0.243520\pi$
$662$	−4.23946	−0.164771
$663$	0	0
$664$	6.64420	0.257845
$665$	−17.8133	−0.690770
$666$	0	0
$667$	−21.5369	−0.833913
$668$	−0.492342	−0.0190493
$669$	0	0
$670$	−72.4359	−2.79844
$671$	1.49948	0.0578866
$672$	0	0
$673$	22.1804	0.854992	0.427496	−	0.904017i	$-0.359396\pi$
$673$	22.1804	0.854992	0.427496	+	0.904017i	$0.359396\pi$
$674$	−38.3778	−1.47826
$675$	0	0
$676$	1.21725	0.0468173
$677$	−4.33174	−0.166482	−0.0832411	−	0.996529i	$-0.526527\pi$
$677$	−4.33174	−0.166482	−0.0832411	+	0.996529i	$0.526527\pi$
$678$	0	0
$679$	14.9391	0.573309
$680$	−26.0888	−1.00046
$681$	0	0
$682$	−0.194078	−0.00743165
$683$	−40.4681	−1.54847	−0.774234	−	0.632900i	$-0.781864\pi$
$683$	−40.4681	−1.54847	−0.774234	+	0.632900i	$0.781864\pi$
$684$	0	0
$685$	−13.8026	−0.527370
$686$	−6.66854	−0.254606
$687$	0	0
$688$	35.8775	1.36782
$689$	1.62453	0.0618895
$690$	0	0
$691$	−23.3106	−0.886778	−0.443389	−	0.896329i	$-0.646224\pi$
$691$	−23.3106	−0.886778	−0.443389	+	0.896329i	$0.646224\pi$
$692$	1.97175	0.0749546
$693$	0	0
$694$	−4.83758	−0.183632
$695$	45.5298	1.72704
$696$	0	0
$697$	−15.7984	−0.598408
$698$	15.7920	0.597735
$699$	0	0
$700$	3.79793	0.143548
$701$	−11.9520	−0.451420	−0.225710	−	0.974194i	$-0.572470\pi$
$701$	−11.9520	−0.451420	−0.225710	+	0.974194i	$0.572470\pi$
$702$	0	0
$703$	5.58948	0.210811
$704$	−1.35538	−0.0510827
$705$	0	0
$706$	−2.53243	−0.0953094
$707$	−5.95634	−0.224011
$708$	0	0
$709$	−14.2925	−0.536766	−0.268383	−	0.963312i	$-0.586489\pi$
$709$	−14.2925	−0.536766	−0.268383	+	0.963312i	$0.586489\pi$
$710$	9.00678	0.338018
$711$	0	0
$712$	−33.6977	−1.26287
$713$	2.62159	0.0981793
$714$	0	0
$715$	0.229048	0.00856589
$716$	−0.385936	−0.0144231
$717$	0	0
$718$	−27.7674	−1.03627
$719$	13.3415	0.497556	0.248778	−	0.968561i	$-0.419971\pi$
$719$	13.3415	0.497556	0.248778	+	0.968561i	$0.419971\pi$
$720$	0	0
$721$	−11.9402	−0.444678
$722$	24.1018	0.896977
$723$	0	0
$724$	−1.55414	−0.0577591
$725$	80.3345	2.98355
$726$	0	0
$727$	41.2259	1.52899	0.764493	−	0.644632i	$-0.222990\pi$
$727$	41.2259	1.52899	0.764493	+	0.644632i	$0.222990\pi$
$728$	−3.58907	−0.133020
$729$	0	0
$730$	−65.0874	−2.40899
$731$	−21.0894	−0.780021
$732$	0	0
$733$	11.6114	0.428876	0.214438	−	0.976738i	$-0.431208\pi$
$733$	11.6114	0.428876	0.214438	+	0.976738i	$0.431208\pi$
$734$	−36.4568	−1.34564
$735$	0	0
$736$	1.61860	0.0596623
$737$	2.11183	0.0777901
$738$	0	0
$739$	−36.1839	−1.33105	−0.665524	−	0.746377i	$-0.731792\pi$
$739$	−36.1839	−1.33105	−0.665524	+	0.746377i	$0.731792\pi$
$740$	−1.71898	−0.0631909
$741$	0	0
$742$	22.8362	0.838343
$743$	50.5839	1.85574	0.927871	−	0.372902i	$-0.121637\pi$
$743$	50.5839	1.85574	0.927871	+	0.372902i	$0.121637\pi$
$744$	0	0
$745$	−2.41327	−0.0884152
$746$	40.9130	1.49793
$747$	0	0
$748$	0.0343243	0.00125502
$749$	−41.9806	−1.53394
$750$	0	0
$751$	40.3590	1.47272	0.736360	−	0.676590i	$-0.236543\pi$
$751$	40.3590	1.47272	0.736360	+	0.676590i	$0.236543\pi$
$752$	16.6267	0.606313
$753$	0	0
$754$	−3.42594	−0.124766
$755$	−6.77169	−0.246447
$756$	0	0
$757$	−46.4998	−1.69006	−0.845032	−	0.534715i	$-0.820419\pi$
$757$	−46.4998	−1.69006	−0.845032	+	0.534715i	$0.820419\pi$
$758$	15.8826	0.576881
$759$	0	0
$760$	14.4856	0.525447
$761$	6.72968	0.243951	0.121975	−	0.992533i	$-0.461077\pi$
$761$	6.72968	0.243951	0.121975	+	0.992533i	$0.461077\pi$
$762$	0	0
$763$	35.7790	1.29529
$764$	−0.000394072 0	−1.42570e−5 0
$765$	0	0
$766$	−39.7057	−1.43462
$767$	−1.76854	−0.0638582
$768$	0	0
$769$	26.1965	0.944671	0.472336	−	0.881419i	$-0.343411\pi$
$769$	26.1965	0.944671	0.472336	+	0.881419i	$0.343411\pi$
$770$	3.21976	0.116032
$771$	0	0
$772$	1.98376	0.0713970
$773$	16.5410	0.594938	0.297469	−	0.954731i	$-0.403857\pi$
$773$	16.5410	0.594938	0.297469	+	0.954731i	$0.403857\pi$
$774$	0	0
$775$	−9.77875	−0.351263
$776$	−12.1483	−0.436098
$777$	0	0
$778$	−15.8128	−0.566917
$779$	8.77194	0.314287
$780$	0	0
$781$	−0.262587	−0.00939612
$782$	9.34684	0.334242
$783$	0	0
$784$	−21.4482	−0.766009
$785$	−36.8308	−1.31455
$786$	0	0
$787$	−13.4634	−0.479918	−0.239959	−	0.970783i	$-0.577134\pi$
$787$	−13.4634	−0.479918	−0.239959	+	0.970783i	$0.577134\pi$
$788$	0.919876	0.0327692
$789$	0	0
$790$	−40.2002	−1.43026
$791$	13.8745	0.493319
$792$	0	0
$793$	−3.22199	−0.114416
$794$	−15.1041	−0.536024
$795$	0	0
$796$	2.28005	0.0808142
$797$	0.496045	0.0175708	0.00878540	−	0.999961i	$-0.497203\pi$
$797$	0.496045	0.0175708	0.00878540	+	0.999961i	$0.497203\pi$
$798$	0	0
$799$	−9.77345	−0.345760
$800$	−6.03750	−0.213458
$801$	0	0
$802$	0.479778	0.0169416
$803$	1.89758	0.0669643
$804$	0	0
$805$	−43.4921	−1.53290
$806$	0.417024	0.0146891
$807$	0	0
$808$	4.84363	0.170398
$809$	−20.5900	−0.723906	−0.361953	−	0.932196i	$-0.617890\pi$
$809$	−20.5900	−0.723906	−0.361953	+	0.932196i	$0.617890\pi$
$810$	0	0
$811$	27.9946	0.983022	0.491511	−	0.870871i	$-0.336445\pi$
$811$	27.9946	0.983022	0.491511	+	0.870871i	$0.336445\pi$
$812$	2.38892	0.0838346
$813$	0	0
$814$	−1.01030	−0.0354110
$815$	88.3203	3.09372
$816$	0	0
$817$	11.7097	0.409671
$818$	−36.7205	−1.28390
$819$	0	0
$820$	−2.69770	−0.0942079
$821$	−44.3263	−1.54700	−0.773499	−	0.633797i	$-0.781495\pi$
$821$	−44.3263	−1.54700	−0.773499	+	0.633797i	$0.781495\pi$
$822$	0	0
$823$	−13.9321	−0.485642	−0.242821	−	0.970071i	$-0.578073\pi$
$823$	−13.9321	−0.485642	−0.242821	+	0.970071i	$0.578073\pi$
$824$	9.70967	0.338253
$825$	0	0
$826$	−24.8606	−0.865011
$827$	11.3432	0.394443	0.197222	−	0.980359i	$-0.436808\pi$
$827$	11.3432	0.394443	0.197222	+	0.980359i	$0.436808\pi$
$828$	0	0
$829$	37.7431	1.31087	0.655436	−	0.755251i	$-0.272485\pi$
$829$	37.7431	1.31087	0.655436	+	0.755251i	$0.272485\pi$
$830$	12.8076	0.444558
$831$	0	0
$832$	2.91236	0.100968
$833$	12.6076	0.436829
$834$	0	0
$835$	−21.0305	−0.727789
$836$	−0.0190583	−0.000659143 0
$837$	0	0
$838$	17.6569	0.609948
$839$	−28.4991	−0.983899	−0.491949	−	0.870624i	$-0.663716\pi$
$839$	−28.4991	−0.983899	−0.491949	+	0.870624i	$0.663716\pi$
$840$	0	0
$841$	21.5308	0.742441
$842$	15.0115	0.517332
$843$	0	0
$844$	−1.42231	−0.0489579
$845$	51.9950	1.78868
$846$	0	0
$847$	39.0162	1.34061
$848$	−17.6905	−0.607496
$849$	0	0
$850$	−34.8645	−1.19584
$851$	13.6470	0.467814
$852$	0	0
$853$	−28.4465	−0.973991	−0.486995	−	0.873405i	$-0.661907\pi$
$853$	−28.4465	−0.973991	−0.486995	+	0.873405i	$0.661907\pi$
$854$	−45.2920	−1.54986
$855$	0	0
$856$	34.1381	1.16682
$857$	−16.2767	−0.556003	−0.278002	−	0.960581i	$-0.589672\pi$
$857$	−16.2767	−0.556003	−0.278002	+	0.960581i	$0.589672\pi$
$858$	0	0
$859$	56.7736	1.93709	0.968545	−	0.248840i	$-0.0800493\pi$
$859$	56.7736	1.93709	0.968545	+	0.248840i	$0.0800493\pi$
$860$	−3.60118	−0.122799
$861$	0	0
$862$	−3.39492	−0.115631
$863$	7.15351	0.243508	0.121754	−	0.992560i	$-0.461148\pi$
$863$	7.15351	0.243508	0.121754	+	0.992560i	$0.461148\pi$
$864$	0	0
$865$	84.2234	2.86368
$866$	17.7887	0.604485
$867$	0	0
$868$	−0.290792	−0.00987013
$869$	1.17201	0.0397578
$870$	0	0
$871$	−4.53777	−0.153756
$872$	−29.0951	−0.985285
$873$	0	0
$874$	−5.18975	−0.175546
$875$	90.4539	3.05790
$876$	0	0
$877$	18.5366	0.625935	0.312968	−	0.949764i	$-0.398677\pi$
$877$	18.5366	0.625935	0.312968	+	0.949764i	$0.398677\pi$
$878$	38.0524	1.28421
$879$	0	0
$880$	−2.49425	−0.0840812
$881$	46.8249	1.57757	0.788785	−	0.614669i	$-0.210711\pi$
$881$	46.8249	1.57757	0.788785	+	0.614669i	$0.210711\pi$
$882$	0	0
$883$	−46.0472	−1.54961	−0.774806	−	0.632199i	$-0.782152\pi$
$883$	−46.0472	−1.54961	−0.774806	+	0.632199i	$0.782152\pi$
$884$	−0.0737540	−0.00248061
$885$	0	0
$886$	7.91464	0.265897
$887$	−3.67441	−0.123375	−0.0616873	−	0.998096i	$-0.519648\pi$
$887$	−3.67441	−0.123375	−0.0616873	+	0.998096i	$0.519648\pi$
$888$	0	0
$889$	−30.3379	−1.01750
$890$	−64.9569	−2.17736
$891$	0	0
$892$	−0.900341	−0.0301456
$893$	5.42662	0.181595
$894$	0	0
$895$	−16.4853	−0.551043
$896$	37.1405	1.24078
$897$	0	0
$898$	47.5270	1.58600
$899$	−6.15088	−0.205143
$900$	0	0
$901$	10.3988	0.346434
$902$	−1.58553	−0.0527924
$903$	0	0
$904$	−11.2826	−0.375252
$905$	−66.3852	−2.20672
$906$	0	0
$907$	13.6026	0.451668	0.225834	−	0.974166i	$-0.427489\pi$
$907$	13.6026	0.451668	0.225834	+	0.974166i	$0.427489\pi$
$908$	1.15787	0.0384251
$909$	0	0
$910$	−6.91843	−0.229344
$911$	4.53153	0.150136	0.0750682	−	0.997178i	$-0.476083\pi$
$911$	4.53153	0.150136	0.0750682	+	0.997178i	$0.476083\pi$
$912$	0	0
$913$	−0.373398	−0.0123577
$914$	10.4732	0.346424
$915$	0	0
$916$	−0.352902	−0.0116602
$917$	−57.5066	−1.89903
$918$	0	0
$919$	−24.8263	−0.818944	−0.409472	−	0.912323i	$-0.634287\pi$
$919$	−24.8263	−0.818944	−0.409472	+	0.912323i	$0.634287\pi$
$920$	35.3673	1.16603
$921$	0	0
$922$	−2.54154	−0.0837012
$923$	0.564232	0.0185719
$924$	0	0
$925$	−50.9045	−1.67373
$926$	31.0637	1.02082
$927$	0	0
$928$	−3.79762	−0.124663
$929$	−12.9296	−0.424206	−0.212103	−	0.977247i	$-0.568031\pi$
$929$	−12.9296	−0.424206	−0.212103	+	0.977247i	$0.568031\pi$
$930$	0	0
$931$	−7.00029	−0.229425
$932$	1.93799	0.0634811
$933$	0	0
$934$	54.4184	1.78062
$935$	1.46616	0.0479487
$936$	0	0
$937$	−57.5836	−1.88117	−0.940587	−	0.339552i	$-0.889724\pi$
$937$	−57.5836	−1.88117	−0.940587	+	0.339552i	$0.889724\pi$
$938$	−63.7881	−2.08275
$939$	0	0
$940$	−1.66889	−0.0544333
$941$	16.5032	0.537987	0.268994	−	0.963142i	$-0.413309\pi$
$941$	16.5032	0.537987	0.268994	+	0.963142i	$0.413309\pi$
$942$	0	0
$943$	21.4172	0.697439
$944$	19.2588	0.626820
$945$	0	0
$946$	−2.11654	−0.0688145
$947$	−18.1425	−0.589553	−0.294776	−	0.955566i	$-0.595245\pi$
$947$	−18.1425	−0.589553	−0.294776	+	0.955566i	$0.595245\pi$
$948$	0	0
$949$	−4.07742	−0.132359
$950$	19.3582	0.628063
$951$	0	0
$952$	−22.9741	−0.744596
$953$	25.1595	0.814998	0.407499	−	0.913206i	$-0.366401\pi$
$953$	25.1595	0.814998	0.407499	+	0.913206i	$0.366401\pi$
$954$	0	0
$955$	−0.0168328	−0.000544698 0
$956$	−1.22424	−0.0395948
$957$	0	0
$958$	−3.64325	−0.117708
$959$	−12.1548	−0.392498
$960$	0	0
$961$	−30.2513	−0.975848
$962$	2.17087	0.0699918
$963$	0	0
$964$	−1.14617	−0.0369158
$965$	84.7364	2.72776
$966$	0	0
$967$	−47.2478	−1.51939	−0.759693	−	0.650281i	$-0.774651\pi$
$967$	−47.2478	−1.51939	−0.759693	+	0.650281i	$0.774651\pi$
$968$	−31.7275	−1.01976
$969$	0	0
$970$	−23.4175	−0.751889
$971$	−40.3149	−1.29377	−0.646883	−	0.762589i	$-0.723928\pi$
$971$	−40.3149	−1.29377	−0.646883	+	0.762589i	$0.723928\pi$
$972$	0	0
$973$	40.0942	1.28536
$974$	34.5153	1.10594
$975$	0	0
$976$	35.0864	1.12309
$977$	2.76461	0.0884476	0.0442238	−	0.999022i	$-0.485919\pi$
$977$	2.76461	0.0884476	0.0442238	+	0.999022i	$0.485919\pi$
$978$	0	0
$979$	1.89378	0.0605254
$980$	2.15285	0.0687704
$981$	0	0
$982$	41.5589	1.32620
$983$	−1.68060	−0.0536029	−0.0268015	−	0.999641i	$-0.508532\pi$
$983$	−1.68060	−0.0536029	−0.0268015	+	0.999641i	$0.508532\pi$
$984$	0	0
$985$	39.2926	1.25197
$986$	−21.9299	−0.698391
$987$	0	0
$988$	0.0409513	0.00130283
$989$	28.5899	0.909108
$990$	0	0
$991$	−7.34449	−0.233305	−0.116653	−	0.993173i	$-0.537216\pi$
$991$	−7.34449	−0.233305	−0.116653	+	0.993173i	$0.537216\pi$
$992$	0.462267	0.0146770
$993$	0	0
$994$	7.93150	0.251572
$995$	97.3925	3.08755
$996$	0	0
$997$	−19.2605	−0.609987	−0.304993	−	0.952354i	$-0.598654\pi$
$997$	−19.2605	−0.609987	−0.304993	+	0.952354i	$0.598654\pi$
$998$	7.78241	0.246348
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.19		72
3.2	odd	2		6561.2.a.c.1.54		72
81.4	even	27		729.2.g.a.136.3		144
81.7	even	27		243.2.g.a.37.6		144
81.20	odd	54		729.2.g.d.595.6		144
81.23	odd	54		81.2.g.a.43.3	✓	144
81.31	even	27		729.2.g.b.622.6		144
81.34	even	27		729.2.g.b.109.6		144
81.47	odd	54		729.2.g.c.109.3		144
81.50	odd	54		729.2.g.c.622.3		144
81.58	even	27		243.2.g.a.46.6		144
81.61	even	27		729.2.g.a.595.3		144
81.74	odd	54		81.2.g.a.49.3	yes	144
81.77	odd	54		729.2.g.d.136.6		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.43.3	✓	144	81.23	odd	54
81.2.g.a.49.3	yes	144	81.74	odd	54
243.2.g.a.37.6		144	81.7	even	27
243.2.g.a.46.6		144	81.58	even	27
729.2.g.a.136.3		144	81.4	even	27
729.2.g.a.595.3		144	81.61	even	27
729.2.g.b.109.6		144	81.34	even	27
729.2.g.b.622.6		144	81.31	even	27
729.2.g.c.109.3		144	81.47	odd	54
729.2.g.c.622.3		144	81.50	odd	54
729.2.g.d.136.6		144	81.77	odd	54
729.2.g.d.595.6		144	81.20	odd	54
6561.2.a.c.1.54		72	3.2	odd	2
6561.2.a.d.1.19		72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.38039 q^{2} -0.0945210 q^{4} -4.03747 q^{5} -3.55546 q^{7} +2.89126 q^{8} +5.57329 q^{10} -0.162486 q^{11} +0.349140 q^{13} +4.90792 q^{14} -3.80202 q^{16} +2.23489 q^{17} -1.24091 q^{19} +0.381626 q^{20} +0.224294 q^{22} -3.02974 q^{23} +11.3012 q^{25} -0.481950 q^{26} +0.336065 q^{28} +7.10850 q^{29} -0.865285 q^{31} -0.534236 q^{32} -3.08503 q^{34} +14.3551 q^{35} -4.50435 q^{37} +1.71294 q^{38} -11.6734 q^{40} -7.06898 q^{41} -9.43643 q^{43} +0.0153583 q^{44} +4.18223 q^{46} -4.37311 q^{47} +5.64127 q^{49} -15.6001 q^{50} -0.0330011 q^{52} +4.65293 q^{53} +0.656033 q^{55} -10.2797 q^{56} -9.81251 q^{58} -5.06541 q^{59} -9.22834 q^{61} +1.19443 q^{62} +8.34150 q^{64} -1.40965 q^{65} -12.9970 q^{67} -0.211244 q^{68} -19.8156 q^{70} +1.61606 q^{71} -11.6784 q^{73} +6.21777 q^{74} +0.117292 q^{76} +0.577712 q^{77} -7.21301 q^{79} +15.3506 q^{80} +9.75795 q^{82} +2.29803 q^{83} -9.02333 q^{85} +13.0260 q^{86} -0.469789 q^{88} -11.6550 q^{89} -1.24135 q^{91} +0.286374 q^{92} +6.03660 q^{94} +5.01013 q^{95} -4.20173 q^{97} -7.78716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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

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