LMFDB - Embedded newform 6561.2.a.d.1.37

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			1.37
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)

$f(q)$	$=$	\(q+0.202096 q^{2} -1.95916 q^{4} -0.911319 q^{5} +0.164043 q^{7} -0.800128 q^{8} -0.184174 q^{10} +0.750003 q^{11} -3.53376 q^{13} +0.0331524 q^{14} +3.75661 q^{16} +4.03805 q^{17} +2.13035 q^{19} +1.78542 q^{20} +0.151572 q^{22} -8.20471 q^{23} -4.16950 q^{25} -0.714158 q^{26} -0.321387 q^{28} +6.58114 q^{29} +5.59557 q^{31} +2.35945 q^{32} +0.816071 q^{34} -0.149496 q^{35} +4.51438 q^{37} +0.430534 q^{38} +0.729172 q^{40} -10.9436 q^{41} -3.46024 q^{43} -1.46937 q^{44} -1.65813 q^{46} -8.93063 q^{47} -6.97309 q^{49} -0.842637 q^{50} +6.92320 q^{52} +6.13971 q^{53} -0.683492 q^{55} -0.131256 q^{56} +1.33002 q^{58} +2.91600 q^{59} -7.06159 q^{61} +1.13084 q^{62} -7.03639 q^{64} +3.22039 q^{65} +8.88178 q^{67} -7.91117 q^{68} -0.0302124 q^{70} -9.52060 q^{71} -8.81387 q^{73} +0.912337 q^{74} -4.17369 q^{76} +0.123033 q^{77} -3.22456 q^{79} -3.42347 q^{80} -2.21165 q^{82} +16.8284 q^{83} -3.67995 q^{85} -0.699298 q^{86} -0.600098 q^{88} +11.4302 q^{89} -0.579690 q^{91} +16.0743 q^{92} -1.80484 q^{94} -1.94143 q^{95} -4.53042 q^{97} -1.40923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.202096	0.142903	0.0714516	−	0.997444i	$-0.477237\pi$
$2$	0.202096	0.142903	0.0714516	+	0.997444i	$0.477237\pi$
$3$	0	0
$3$	0	0
$4$	−1.95916	−0.979579
$5$	−0.911319	−0.407554	−0.203777	−	0.979017i	$-0.565322\pi$
$5$	−0.911319	−0.407554	−0.203777	+	0.979017i	$0.565322\pi$
$6$	0	0
$7$	0.164043	0.0620025	0.0310013	−	0.999519i	$-0.490130\pi$
$7$	0.164043	0.0620025	0.0310013	+	0.999519i	$0.490130\pi$
$8$	−0.800128	−0.282888
$9$	0	0
$10$	−0.184174	−0.0582408
$11$	0.750003	0.226134	0.113067	−	0.993587i	$-0.463932\pi$
$11$	0.750003	0.226134	0.113067	+	0.993587i	$0.463932\pi$
$12$	0	0
$13$	−3.53376	−0.980090	−0.490045	−	0.871697i	$-0.663020\pi$
$13$	−3.53376	−0.980090	−0.490045	+	0.871697i	$0.663020\pi$
$14$	0.0331524	0.00886035
$15$	0	0
$16$	3.75661	0.939153
$17$	4.03805	0.979370	0.489685	−	0.871899i	$-0.337112\pi$
$17$	4.03805	0.979370	0.489685	+	0.871899i	$0.337112\pi$
$18$	0	0
$19$	2.13035	0.488736	0.244368	−	0.969683i	$-0.421419\pi$
$19$	2.13035	0.488736	0.244368	+	0.969683i	$0.421419\pi$
$20$	1.78542	0.399232
$21$	0	0
$22$	0.151572	0.0323153
$23$	−8.20471	−1.71080	−0.855400	−	0.517968i	$-0.826689\pi$
$23$	−8.20471	−1.71080	−0.855400	+	0.517968i	$0.826689\pi$
$24$	0	0
$25$	−4.16950	−0.833899
$26$	−0.714158	−0.140058
$27$	0	0
$28$	−0.321387	−0.0607364
$29$	6.58114	1.22209	0.611044	−	0.791597i	$-0.290750\pi$
$29$	6.58114	1.22209	0.611044	+	0.791597i	$0.290750\pi$
$30$	0	0
$31$	5.59557	1.00499	0.502497	−	0.864579i	$-0.332415\pi$
$31$	5.59557	1.00499	0.502497	+	0.864579i	$0.332415\pi$
$32$	2.35945	0.417096
$33$	0	0
$34$	0.816071	0.139955
$35$	−0.149496	−0.0252694
$36$	0	0
$37$	4.51438	0.742160	0.371080	−	0.928601i	$-0.378988\pi$
$37$	4.51438	0.742160	0.371080	+	0.928601i	$0.378988\pi$
$38$	0.430534	0.0698419
$39$	0	0
$40$	0.729172	0.115292
$41$	−10.9436	−1.70910	−0.854550	−	0.519369i	$-0.826167\pi$
$41$	−10.9436	−1.70910	−0.854550	+	0.519369i	$0.826167\pi$
$42$	0	0
$43$	−3.46024	−0.527681	−0.263841	−	0.964566i	$-0.584989\pi$
$43$	−3.46024	−0.527681	−0.263841	+	0.964566i	$0.584989\pi$
$44$	−1.46937	−0.221516
$45$	0	0
$46$	−1.65813	−0.244479
$47$	−8.93063	−1.30267	−0.651333	−	0.758792i	$-0.725790\pi$
$47$	−8.93063	−1.30267	−0.651333	+	0.758792i	$0.725790\pi$
$48$	0	0
$49$	−6.97309	−0.996156
$50$	−0.842637	−0.119167
$51$	0	0
$52$	6.92320	0.960075
$53$	6.13971	0.843355	0.421677	−	0.906746i	$-0.361442\pi$
$53$	6.13971	0.843355	0.421677	+	0.906746i	$0.361442\pi$
$54$	0	0
$55$	−0.683492	−0.0921621
$56$	−0.131256	−0.0175398
$57$	0	0
$58$	1.33002	0.174640
$59$	2.91600	0.379631	0.189816	−	0.981820i	$-0.439211\pi$
$59$	2.91600	0.379631	0.189816	+	0.981820i	$0.439211\pi$
$60$	0	0
$61$	−7.06159	−0.904143	−0.452072	−	0.891982i	$-0.649315\pi$
$61$	−7.06159	−0.904143	−0.452072	+	0.891982i	$0.649315\pi$
$62$	1.13084	0.143617
$63$	0	0
$64$	−7.03639	−0.879549
$65$	3.22039	0.399440
$66$	0	0
$67$	8.88178	1.08508	0.542541	−	0.840029i	$-0.317462\pi$
$67$	8.88178	1.08508	0.542541	+	0.840029i	$0.317462\pi$
$68$	−7.91117	−0.959370
$69$	0	0
$70$	−0.0302124	−0.00361108
$71$	−9.52060	−1.12989	−0.564944	−	0.825129i	$-0.691102\pi$
$71$	−9.52060	−1.12989	−0.564944	+	0.825129i	$0.691102\pi$
$72$	0	0
$73$	−8.81387	−1.03159	−0.515793	−	0.856714i	$-0.672502\pi$
$73$	−8.81387	−1.03159	−0.515793	+	0.856714i	$0.672502\pi$
$74$	0.912337	0.106057
$75$	0	0
$76$	−4.17369	−0.478755
$77$	0.123033	0.0140209
$78$	0	0
$79$	−3.22456	−0.362791	−0.181396	−	0.983410i	$-0.558061\pi$
$79$	−3.22456	−0.362791	−0.181396	+	0.983410i	$0.558061\pi$
$80$	−3.42347	−0.382756
$81$	0	0
$82$	−2.21165	−0.244236
$83$	16.8284	1.84716	0.923581	−	0.383404i	$-0.125248\pi$
$83$	16.8284	1.84716	0.923581	+	0.383404i	$0.125248\pi$
$84$	0	0
$85$	−3.67995	−0.399147
$86$	−0.699298	−0.0754073
$87$	0	0
$88$	−0.600098	−0.0639707
$89$	11.4302	1.21160	0.605798	−	0.795618i	$-0.292854\pi$
$89$	11.4302	1.21160	0.605798	+	0.795618i	$0.292854\pi$
$90$	0	0
$91$	−0.579690	−0.0607680
$92$	16.0743	1.67586
$93$	0	0
$94$	−1.80484	−0.186155
$95$	−1.94143	−0.199187
$96$	0	0
$97$	−4.53042	−0.459994	−0.229997	−	0.973191i	$-0.573872\pi$
$97$	−4.53042	−0.459994	−0.229997	+	0.973191i	$0.573872\pi$
$98$	−1.40923	−0.142354
$99$	0	0
$100$	8.16870	0.816870
$101$	−2.30166	−0.229024	−0.114512	−	0.993422i	$-0.536530\pi$
$101$	−2.30166	−0.229024	−0.114512	+	0.993422i	$0.536530\pi$
$102$	0	0
$103$	2.60246	0.256428	0.128214	−	0.991746i	$-0.459076\pi$
$103$	2.60246	0.256428	0.128214	+	0.991746i	$0.459076\pi$
$104$	2.82746	0.277256
$105$	0	0
$106$	1.24081	0.120518
$107$	−11.9544	−1.15567	−0.577837	−	0.816152i	$-0.696103\pi$
$107$	−11.9544	−1.15567	−0.577837	+	0.816152i	$0.696103\pi$
$108$	0	0
$109$	18.1311	1.73664	0.868320	−	0.496005i	$-0.165200\pi$
$109$	18.1311	1.73664	0.868320	+	0.496005i	$0.165200\pi$
$110$	−0.138131	−0.0131702
$111$	0	0
$112$	0.616247	0.0582299
$113$	11.9001	1.11947	0.559733	−	0.828673i	$-0.310904\pi$
$113$	11.9001	1.11947	0.559733	+	0.828673i	$0.310904\pi$
$114$	0	0
$115$	7.47711	0.697244
$116$	−12.8935	−1.19713
$117$	0	0
$118$	0.589311	0.0542505
$119$	0.662414	0.0607234
$120$	0	0
$121$	−10.4375	−0.948863
$122$	−1.42711	−0.129205
$123$	0	0
$124$	−10.9626	−0.984471
$125$	8.35634	0.747414
$126$	0	0
$127$	14.5915	1.29478	0.647392	−	0.762158i	$-0.275860\pi$
$127$	14.5915	1.29478	0.647392	+	0.762158i	$0.275860\pi$
$128$	−6.14092	−0.542786
$129$	0	0
$130$	0.650826	0.0570812
$131$	6.66713	0.582510	0.291255	−	0.956645i	$-0.405927\pi$
$131$	6.66713	0.582510	0.291255	+	0.956645i	$0.405927\pi$
$132$	0	0
$133$	0.349470	0.0303029
$134$	1.79497	0.155062
$135$	0	0
$136$	−3.23095	−0.277052
$137$	14.6275	1.24971	0.624854	−	0.780741i	$-0.285158\pi$
$137$	14.6275	1.24971	0.624854	+	0.780741i	$0.285158\pi$
$138$	0	0
$139$	−3.31539	−0.281208	−0.140604	−	0.990066i	$-0.544904\pi$
$139$	−3.31539	−0.281208	−0.140604	+	0.990066i	$0.544904\pi$
$140$	0.292886	0.0247534
$141$	0	0
$142$	−1.92407	−0.161464
$143$	−2.65033	−0.221632
$144$	0	0
$145$	−5.99752	−0.498067
$146$	−1.78124	−0.147417
$147$	0	0
$148$	−8.84439	−0.727004
$149$	−4.90208	−0.401594	−0.200797	−	0.979633i	$-0.564353\pi$
$149$	−4.90208	−0.401594	−0.200797	+	0.979633i	$0.564353\pi$
$150$	0	0
$151$	16.1570	1.31484	0.657419	−	0.753526i	$-0.271648\pi$
$151$	16.1570	1.31484	0.657419	+	0.753526i	$0.271648\pi$
$152$	−1.70455	−0.138258
$153$	0	0
$154$	0.0248644	0.00200363
$155$	−5.09935	−0.409590
$156$	0	0
$157$	12.3635	0.986718	0.493359	−	0.869826i	$-0.335769\pi$
$157$	12.3635	0.986718	0.493359	+	0.869826i	$0.335769\pi$
$158$	−0.651669	−0.0518440
$159$	0	0
$160$	−2.15021	−0.169989
$161$	−1.34593	−0.106074
$162$	0	0
$163$	−12.1435	−0.951151	−0.475576	−	0.879675i	$-0.657760\pi$
$163$	−12.1435	−0.951151	−0.475576	+	0.879675i	$0.657760\pi$
$164$	21.4402	1.67420
$165$	0	0
$166$	3.40095	0.263965
$167$	−2.54113	−0.196639	−0.0983194	−	0.995155i	$-0.531347\pi$
$167$	−2.54113	−0.196639	−0.0983194	+	0.995155i	$0.531347\pi$
$168$	0	0
$169$	−0.512518	−0.0394245
$170$	−0.743701	−0.0570393
$171$	0	0
$172$	6.77915	0.516905
$173$	9.73670	0.740268	0.370134	−	0.928978i	$-0.379312\pi$
$173$	9.73670	0.740268	0.370134	+	0.928978i	$0.379312\pi$
$174$	0	0
$175$	−0.683978	−0.0517039
$176$	2.81747	0.212375
$177$	0	0
$178$	2.30999	0.173141
$179$	3.72297	0.278268	0.139134	−	0.990274i	$-0.455568\pi$
$179$	3.72297	0.278268	0.139134	+	0.990274i	$0.455568\pi$
$180$	0	0
$181$	−7.30595	−0.543047	−0.271524	−	0.962432i	$-0.587527\pi$
$181$	−7.30595	−0.543047	−0.271524	+	0.962432i	$0.587527\pi$
$182$	−0.117153	−0.00868394
$183$	0	0
$184$	6.56482	0.483965
$185$	−4.11405	−0.302471
$186$	0	0
$187$	3.02855	0.221469
$188$	17.4965	1.27606
$189$	0	0
$190$	−0.392354	−0.0284644
$191$	20.7764	1.50333	0.751663	−	0.659548i	$-0.229252\pi$
$191$	20.7764	1.50333	0.751663	+	0.659548i	$0.229252\pi$
$192$	0	0
$193$	−3.49292	−0.251426	−0.125713	−	0.992067i	$-0.540122\pi$
$193$	−3.49292	−0.251426	−0.125713	+	0.992067i	$0.540122\pi$
$194$	−0.915577	−0.0657346
$195$	0	0
$196$	13.6614	0.975813
$197$	1.18478	0.0844123	0.0422062	−	0.999109i	$-0.486561\pi$
$197$	1.18478	0.0844123	0.0422062	+	0.999109i	$0.486561\pi$
$198$	0	0
$199$	−1.10157	−0.0780881	−0.0390441	−	0.999237i	$-0.512431\pi$
$199$	−1.10157	−0.0780881	−0.0390441	+	0.999237i	$0.512431\pi$
$200$	3.33613	0.235900
$201$	0	0
$202$	−0.465155	−0.0327282
$203$	1.07959	0.0757725
$204$	0	0
$205$	9.97310	0.696551
$206$	0.525946	0.0366444
$207$	0	0
$208$	−13.2750	−0.920454
$209$	1.59777	0.110520
$210$	0	0
$211$	−14.9836	−1.03151	−0.515757	−	0.856735i	$-0.672489\pi$
$211$	−14.9836	−1.03151	−0.515757	+	0.856735i	$0.672489\pi$
$212$	−12.0287	−0.826132
$213$	0	0
$214$	−2.41593	−0.165150
$215$	3.15338	0.215059
$216$	0	0
$217$	0.917916	0.0623122
$218$	3.66420	0.248171
$219$	0	0
$220$	1.33907	0.0902800
$221$	−14.2695	−0.959870
$222$	0	0
$223$	5.10101	0.341589	0.170795	−	0.985307i	$-0.445367\pi$
$223$	5.10101	0.341589	0.170795	+	0.985307i	$0.445367\pi$
$224$	0.387052	0.0258610
$225$	0	0
$226$	2.40495	0.159975
$227$	16.7632	1.11261	0.556305	−	0.830978i	$-0.312218\pi$
$227$	16.7632	1.11261	0.556305	+	0.830978i	$0.312218\pi$
$228$	0	0
$229$	16.1230	1.06544	0.532720	−	0.846291i	$-0.321170\pi$
$229$	16.1230	1.06544	0.532720	+	0.846291i	$0.321170\pi$
$230$	1.51109	0.0996384
$231$	0	0
$232$	−5.26576	−0.345714
$233$	10.4877	0.687075	0.343537	−	0.939139i	$-0.388375\pi$
$233$	10.4877	0.687075	0.343537	+	0.939139i	$0.388375\pi$
$234$	0	0
$235$	8.13866	0.530907
$236$	−5.71291	−0.371879
$237$	0	0
$238$	0.133871	0.00867757
$239$	−0.0543615	−0.00351635	−0.00175818	−	0.999998i	$-0.500560\pi$
$239$	−0.0543615	−0.00351635	−0.00175818	+	0.999998i	$0.500560\pi$
$240$	0	0
$241$	11.8962	0.766303	0.383151	−	0.923686i	$-0.374839\pi$
$241$	11.8962	0.766303	0.383151	+	0.923686i	$0.374839\pi$
$242$	−2.10937	−0.135596
$243$	0	0
$244$	13.8348	0.885680
$245$	6.35471	0.405988
$246$	0	0
$247$	−7.52816	−0.479005
$248$	−4.47717	−0.284301
$249$	0	0
$250$	1.68878	0.106808
$251$	4.45032	0.280902	0.140451	−	0.990088i	$-0.455145\pi$
$251$	4.45032	0.280902	0.140451	+	0.990088i	$0.455145\pi$
$252$	0	0
$253$	−6.15356	−0.386871
$254$	2.94887	0.185029
$255$	0	0
$256$	12.8317	0.801983
$257$	15.4743	0.965262	0.482631	−	0.875824i	$-0.339681\pi$
$257$	15.4743	0.965262	0.482631	+	0.875824i	$0.339681\pi$
$258$	0	0
$259$	0.740554	0.0460158
$260$	−6.30925	−0.391283
$261$	0	0
$262$	1.34740	0.0832425
$263$	−14.3353	−0.883952	−0.441976	−	0.897027i	$-0.645722\pi$
$263$	−14.3353	−0.883952	−0.441976	+	0.897027i	$0.645722\pi$
$264$	0	0
$265$	−5.59524	−0.343713
$266$	0.0706263	0.00433037
$267$	0	0
$268$	−17.4008	−1.06292
$269$	−1.82045	−0.110995	−0.0554974	−	0.998459i	$-0.517674\pi$
$269$	−1.82045	−0.110995	−0.0554974	+	0.998459i	$0.517674\pi$
$270$	0	0
$271$	−13.5864	−0.825313	−0.412656	−	0.910887i	$-0.635399\pi$
$271$	−13.5864	−0.825313	−0.412656	+	0.910887i	$0.635399\pi$
$272$	15.1694	0.919779
$273$	0	0
$274$	2.95615	0.178587
$275$	−3.12713	−0.188573
$276$	0	0
$277$	−21.8862	−1.31501	−0.657507	−	0.753448i	$-0.728389\pi$
$277$	−21.8862	−1.31501	−0.657507	+	0.753448i	$0.728389\pi$
$278$	−0.670025	−0.0401854
$279$	0	0
$280$	0.119616	0.00714841
$281$	25.2626	1.50704	0.753522	−	0.657423i	$-0.228354\pi$
$281$	25.2626	1.50704	0.753522	+	0.657423i	$0.228354\pi$
$282$	0	0
$283$	22.3824	1.33049	0.665247	−	0.746623i	$-0.268326\pi$
$283$	22.3824	1.33049	0.665247	+	0.746623i	$0.268326\pi$
$284$	18.6524	1.10681
$285$	0	0
$286$	−0.535620	−0.0316719
$287$	−1.79522	−0.105969
$288$	0	0
$289$	−0.694179	−0.0408341
$290$	−1.21207	−0.0711753
$291$	0	0
$292$	17.2678	1.01052
$293$	13.0937	0.764945	0.382472	−	0.923967i	$-0.375073\pi$
$293$	13.0937	0.764945	0.382472	+	0.923967i	$0.375073\pi$
$294$	0	0
$295$	−2.65741	−0.154720
$296$	−3.61208	−0.209948
$297$	0	0
$298$	−0.990689	−0.0573891
$299$	28.9935	1.67674
$300$	0	0
$301$	−0.567628	−0.0327176
$302$	3.26525	0.187894
$303$	0	0
$304$	8.00290	0.458998
$305$	6.43536	0.368488
$306$	0	0
$307$	14.6940	0.838632	0.419316	−	0.907840i	$-0.362270\pi$
$307$	14.6940	0.838632	0.419316	+	0.907840i	$0.362270\pi$
$308$	−0.241041	−0.0137346
$309$	0	0
$310$	−1.03056	−0.0585317
$311$	33.3819	1.89291	0.946457	−	0.322831i	$-0.104635\pi$
$311$	33.3819	1.89291	0.946457	+	0.322831i	$0.104635\pi$
$312$	0	0
$313$	−22.8598	−1.29211	−0.646056	−	0.763290i	$-0.723583\pi$
$313$	−22.8598	−1.29211	−0.646056	+	0.763290i	$0.723583\pi$
$314$	2.49862	0.141005
$315$	0	0
$316$	6.31742	0.355383
$317$	−18.7851	−1.05508	−0.527539	−	0.849531i	$-0.676885\pi$
$317$	−18.7851	−1.05508	−0.527539	+	0.849531i	$0.676885\pi$
$318$	0	0
$319$	4.93587	0.276356
$320$	6.41240	0.358464
$321$	0	0
$322$	−0.272006	−0.0151583
$323$	8.60246	0.478654
$324$	0	0
$325$	14.7340	0.817296
$326$	−2.45414	−0.135922
$327$	0	0
$328$	8.75626	0.483484
$329$	−1.46501	−0.0807686
$330$	0	0
$331$	−5.91925	−0.325351	−0.162676	−	0.986680i	$-0.552012\pi$
$331$	−5.91925	−0.325351	−0.162676	+	0.986680i	$0.552012\pi$
$332$	−32.9696	−1.80944
$333$	0	0
$334$	−0.513552	−0.0281003
$335$	−8.09414	−0.442230
$336$	0	0
$337$	−12.1352	−0.661049	−0.330525	−	0.943797i	$-0.607226\pi$
$337$	−12.1352	−0.661049	−0.330525	+	0.943797i	$0.607226\pi$
$338$	−0.103578	−0.00563388
$339$	0	0
$340$	7.20960	0.390996
$341$	4.19670	0.227264
$342$	0	0
$343$	−2.29219	−0.123767
$344$	2.76863	0.149275
$345$	0	0
$346$	1.96774	0.105787
$347$	2.43917	0.130942	0.0654709	−	0.997854i	$-0.479145\pi$
$347$	2.43917	0.130942	0.0654709	+	0.997854i	$0.479145\pi$
$348$	0	0
$349$	22.9813	1.23016	0.615080	−	0.788464i	$-0.289124\pi$
$349$	22.9813	1.23016	0.615080	+	0.788464i	$0.289124\pi$
$350$	−0.138229	−0.00738864
$351$	0	0
$352$	1.76959	0.0943197
$353$	23.1555	1.23245	0.616223	−	0.787572i	$-0.288662\pi$
$353$	23.1555	1.23245	0.616223	+	0.787572i	$0.288662\pi$
$354$	0	0
$355$	8.67631	0.460491
$356$	−22.3935	−1.18685
$357$	0	0
$358$	0.752395	0.0397653
$359$	31.3108	1.65252	0.826259	−	0.563290i	$-0.190465\pi$
$359$	31.3108	1.65252	0.826259	+	0.563290i	$0.190465\pi$
$360$	0	0
$361$	−14.4616	−0.761137
$362$	−1.47650	−0.0776031
$363$	0	0
$364$	1.13570	0.0595271
$365$	8.03225	0.420427
$366$	0	0
$367$	7.67467	0.400614	0.200307	−	0.979733i	$-0.435806\pi$
$367$	7.67467	0.400614	0.200307	+	0.979733i	$0.435806\pi$
$368$	−30.8219	−1.60670
$369$	0	0
$370$	−0.831430	−0.0432240
$371$	1.00718	0.0522901
$372$	0	0
$373$	25.9539	1.34384	0.671922	−	0.740622i	$-0.265469\pi$
$373$	25.9539	1.34384	0.671922	+	0.740622i	$0.265469\pi$
$374$	0.612056	0.0316486
$375$	0	0
$376$	7.14565	0.368509
$377$	−23.2562	−1.19776
$378$	0	0
$379$	26.5701	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$379$	26.5701	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$380$	3.80357	0.195119
$381$	0	0
$382$	4.19881	0.214830
$383$	2.69562	0.137740	0.0688699	−	0.997626i	$-0.478061\pi$
$383$	2.69562	0.137740	0.0688699	+	0.997626i	$0.478061\pi$
$384$	0	0
$385$	−0.112122	−0.00571428
$386$	−0.705903	−0.0359295
$387$	0	0
$388$	8.87580	0.450600
$389$	23.6943	1.20135	0.600674	−	0.799494i	$-0.294899\pi$
$389$	23.6943	1.20135	0.600674	+	0.799494i	$0.294899\pi$
$390$	0	0
$391$	−33.1310	−1.67551
$392$	5.57936	0.281800
$393$	0	0
$394$	0.239439	0.0120628
$395$	2.93860	0.147857
$396$	0	0
$397$	0.109546	0.00549795	0.00274897	−	0.999996i	$-0.499125\pi$
$397$	0.109546	0.00549795	0.00274897	+	0.999996i	$0.499125\pi$
$398$	−0.222622	−0.0111590
$399$	0	0
$400$	−15.6632	−0.783159
$401$	8.44683	0.421814	0.210907	−	0.977506i	$-0.432358\pi$
$401$	8.44683	0.421814	0.210907	+	0.977506i	$0.432358\pi$
$402$	0	0
$403$	−19.7734	−0.984985
$404$	4.50932	0.224347
$405$	0	0
$406$	0.218181	0.0108281
$407$	3.38580	0.167828
$408$	0	0
$409$	−5.05552	−0.249979	−0.124990	−	0.992158i	$-0.539890\pi$
$409$	−5.05552	−0.249979	−0.124990	+	0.992158i	$0.539890\pi$
$410$	2.01552	0.0995393
$411$	0	0
$412$	−5.09864	−0.251192
$413$	0.478351	0.0235381
$414$	0	0
$415$	−15.3361	−0.752819
$416$	−8.33774	−0.408791
$417$	0	0
$418$	0.322902	0.0157937
$419$	−12.7750	−0.624099	−0.312049	−	0.950066i	$-0.601015\pi$
$419$	−12.7750	−0.624099	−0.312049	+	0.950066i	$0.601015\pi$
$420$	0	0
$421$	−18.2073	−0.887371	−0.443686	−	0.896183i	$-0.646329\pi$
$421$	−18.2073	−0.887371	−0.443686	+	0.896183i	$0.646329\pi$
$422$	−3.02812	−0.147407
$423$	0	0
$424$	−4.91256	−0.238575
$425$	−16.8366	−0.816696
$426$	0	0
$427$	−1.15841	−0.0560592
$428$	23.4205	1.13207
$429$	0	0
$430$	0.637284	0.0307326
$431$	−23.1350	−1.11438	−0.557188	−	0.830387i	$-0.688120\pi$
$431$	−23.1350	−1.11438	−0.557188	+	0.830387i	$0.688120\pi$
$432$	0	0
$433$	−33.5063	−1.61021	−0.805104	−	0.593133i	$-0.797891\pi$
$433$	−33.5063	−1.61021	−0.805104	+	0.593133i	$0.797891\pi$
$434$	0.185507	0.00890461
$435$	0	0
$436$	−35.5216	−1.70118
$437$	−17.4789	−0.836130
$438$	0	0
$439$	−31.6536	−1.51074	−0.755371	−	0.655297i	$-0.772543\pi$
$439$	−31.6536	−1.51074	−0.755371	+	0.655297i	$0.772543\pi$
$440$	0.546881	0.0260715
$441$	0	0
$442$	−2.88380	−0.137168
$443$	22.6104	1.07425	0.537127	−	0.843502i	$-0.319510\pi$
$443$	22.6104	1.07425	0.537127	+	0.843502i	$0.319510\pi$
$444$	0	0
$445$	−10.4165	−0.493791
$446$	1.03089	0.0488142
$447$	0	0
$448$	−1.15427	−0.0545343
$449$	18.0204	0.850435	0.425218	−	0.905091i	$-0.360198\pi$
$449$	18.0204	0.850435	0.425218	+	0.905091i	$0.360198\pi$
$450$	0	0
$451$	−8.20771	−0.386486
$452$	−23.3141	−1.09661
$453$	0	0
$454$	3.38776	0.158996
$455$	0.528283	0.0247663
$456$	0	0
$457$	−23.5593	−1.10206	−0.551029	−	0.834486i	$-0.685765\pi$
$457$	−23.5593	−1.10206	−0.551029	+	0.834486i	$0.685765\pi$
$458$	3.25839	0.152255
$459$	0	0
$460$	−14.6488	−0.683006
$461$	40.1716	1.87098	0.935489	−	0.353356i	$-0.114960\pi$
$461$	40.1716	1.87098	0.935489	+	0.353356i	$0.114960\pi$
$462$	0	0
$463$	−10.9824	−0.510394	−0.255197	−	0.966889i	$-0.582140\pi$
$463$	−10.9824	−0.510394	−0.255197	+	0.966889i	$0.582140\pi$
$464$	24.7228	1.14773
$465$	0	0
$466$	2.11953	0.0981851
$467$	1.68549	0.0779953	0.0389976	−	0.999239i	$-0.487584\pi$
$467$	1.68549	0.0779953	0.0389976	+	0.999239i	$0.487584\pi$
$468$	0	0
$469$	1.45700	0.0672779
$470$	1.64479	0.0758683
$471$	0	0
$472$	−2.33318	−0.107393
$473$	−2.59519	−0.119327
$474$	0	0
$475$	−8.88249	−0.407557
$476$	−1.29777	−0.0594834
$477$	0	0
$478$	−0.0109862	−0.000502498 0
$479$	19.8895	0.908776	0.454388	−	0.890804i	$-0.349858\pi$
$479$	19.8895	0.908776	0.454388	+	0.890804i	$0.349858\pi$
$480$	0	0
$481$	−15.9528	−0.727383
$482$	2.40417	0.109507
$483$	0	0
$484$	20.4487	0.929486
$485$	4.12866	0.187473
$486$	0	0
$487$	−16.9386	−0.767563	−0.383782	−	0.923424i	$-0.625378\pi$
$487$	−16.9386	−0.767563	−0.383782	+	0.923424i	$0.625378\pi$
$488$	5.65017	0.255771
$489$	0	0
$490$	1.28426	0.0580169
$491$	34.3958	1.55226	0.776130	−	0.630574i	$-0.217180\pi$
$491$	34.3958	1.55226	0.776130	+	0.630574i	$0.217180\pi$
$492$	0	0
$493$	26.5750	1.19688
$494$	−1.52141	−0.0684513
$495$	0	0
$496$	21.0204	0.943844
$497$	−1.56179	−0.0700559
$498$	0	0
$499$	17.5931	0.787576	0.393788	−	0.919201i	$-0.371164\pi$
$499$	17.5931	0.787576	0.393788	+	0.919201i	$0.371164\pi$
$500$	−16.3714	−0.732151
$501$	0	0
$502$	0.899390	0.0401418
$503$	−23.0323	−1.02696	−0.513480	−	0.858102i	$-0.671644\pi$
$503$	−23.0323	−1.02696	−0.513480	+	0.858102i	$0.671644\pi$
$504$	0	0
$505$	2.09755	0.0933397
$506$	−1.24361	−0.0552850
$507$	0	0
$508$	−28.5870	−1.26834
$509$	32.0259	1.41952	0.709760	−	0.704443i	$-0.248803\pi$
$509$	32.0259	1.41952	0.709760	+	0.704443i	$0.248803\pi$
$510$	0	0
$511$	−1.44586	−0.0639609
$512$	14.8751	0.657392
$513$	0	0
$514$	3.12729	0.137939
$515$	−2.37168	−0.104509
$516$	0	0
$517$	−6.69800	−0.294578
$518$	0.149663	0.00657580
$519$	0	0
$520$	−2.57672	−0.112997
$521$	−2.51563	−0.110212	−0.0551060	−	0.998481i	$-0.517550\pi$
$521$	−2.51563	−0.110212	−0.0551060	+	0.998481i	$0.517550\pi$
$522$	0	0
$523$	27.8338	1.21709	0.608544	−	0.793520i	$-0.291754\pi$
$523$	27.8338	1.21709	0.608544	+	0.793520i	$0.291754\pi$
$524$	−13.0620	−0.570614
$525$	0	0
$526$	−2.89710	−0.126320
$527$	22.5952	0.984262
$528$	0	0
$529$	44.3173	1.92684
$530$	−1.13077	−0.0491176
$531$	0	0
$532$	−0.684666	−0.0296841
$533$	38.6720	1.67507
$534$	0	0
$535$	10.8943	0.471000
$536$	−7.10656	−0.306957
$537$	0	0
$538$	−0.367905	−0.0158615
$539$	−5.22984	−0.225265
$540$	0	0
$541$	−6.47702	−0.278469	−0.139234	−	0.990259i	$-0.544464\pi$
$541$	−6.47702	−0.278469	−0.139234	+	0.990259i	$0.544464\pi$
$542$	−2.74574	−0.117940
$543$	0	0
$544$	9.52757	0.408491
$545$	−16.5232	−0.707775
$546$	0	0
$547$	15.3996	0.658438	0.329219	−	0.944254i	$-0.393215\pi$
$547$	15.3996	0.658438	0.329219	+	0.944254i	$0.393215\pi$
$548$	−28.6575	−1.22419
$549$	0	0
$550$	−0.631980	−0.0269477
$551$	14.0201	0.597278
$552$	0	0
$553$	−0.528967	−0.0224940
$554$	−4.42310	−0.187920
$555$	0	0
$556$	6.49537	0.275465
$557$	22.6130	0.958142	0.479071	−	0.877776i	$-0.340974\pi$
$557$	22.6130	0.958142	0.479071	+	0.877776i	$0.340974\pi$
$558$	0	0
$559$	12.2277	0.517175
$560$	−0.561598	−0.0237318
$561$	0	0
$562$	5.10547	0.215361
$563$	−23.7258	−0.999923	−0.499961	−	0.866048i	$-0.666652\pi$
$563$	−23.7258	−0.999923	−0.499961	+	0.866048i	$0.666652\pi$
$564$	0	0
$565$	−10.8448	−0.456243
$566$	4.52338	0.190132
$567$	0	0
$568$	7.61770	0.319632
$569$	0.632660	0.0265225	0.0132612	−	0.999912i	$-0.495779\pi$
$569$	0.632660	0.0265225	0.0132612	+	0.999912i	$0.495779\pi$
$570$	0	0
$571$	40.0927	1.67783	0.838914	−	0.544263i	$-0.183191\pi$
$571$	40.0927	1.67783	0.838914	+	0.544263i	$0.183191\pi$
$572$	5.19242	0.217106
$573$	0	0
$574$	−0.362806	−0.0151432
$575$	34.2095	1.42664
$576$	0	0
$577$	−16.7429	−0.697015	−0.348508	−	0.937306i	$-0.613311\pi$
$577$	−16.7429	−0.697015	−0.348508	+	0.937306i	$0.613311\pi$
$578$	−0.140290	−0.00583531
$579$	0	0
$580$	11.7501	0.487896
$581$	2.76059	0.114529
$582$	0	0
$583$	4.60480	0.190711
$584$	7.05222	0.291823
$585$	0	0
$586$	2.64619	0.109313
$587$	−16.4711	−0.679834	−0.339917	−	0.940455i	$-0.610399\pi$
$587$	−16.4711	−0.679834	−0.339917	+	0.940455i	$0.610399\pi$
$588$	0	0
$589$	11.9205	0.491177
$590$	−0.537051	−0.0221100
$591$	0	0
$592$	16.9588	0.697002
$593$	40.3500	1.65697	0.828487	−	0.560008i	$-0.189202\pi$
$593$	40.3500	1.65697	0.828487	+	0.560008i	$0.189202\pi$
$594$	0	0
$595$	−0.603671	−0.0247481
$596$	9.60395	0.393393
$597$	0	0
$598$	5.85946	0.239611
$599$	−5.63253	−0.230139	−0.115069	−	0.993357i	$-0.536709\pi$
$599$	−5.63253	−0.230139	−0.115069	+	0.993357i	$0.536709\pi$
$600$	0	0
$601$	17.0922	0.697204	0.348602	−	0.937271i	$-0.386657\pi$
$601$	17.0922	0.697204	0.348602	+	0.937271i	$0.386657\pi$
$602$	−0.114715	−0.00467544
$603$	0	0
$604$	−31.6541	−1.28799
$605$	9.51189	0.386713
$606$	0	0
$607$	18.1244	0.735646	0.367823	−	0.929896i	$-0.380103\pi$
$607$	18.1244	0.735646	0.367823	+	0.929896i	$0.380103\pi$
$608$	5.02646	0.203850
$609$	0	0
$610$	1.30056	0.0526580
$611$	31.5587	1.27673
$612$	0	0
$613$	22.1812	0.895890	0.447945	−	0.894061i	$-0.352156\pi$
$613$	22.1812	0.895890	0.447945	+	0.894061i	$0.352156\pi$
$614$	2.96960	0.119843
$615$	0	0
$616$	−0.0984421	−0.00396634
$617$	−10.0390	−0.404155	−0.202077	−	0.979370i	$-0.564769\pi$
$617$	−10.0390	−0.404155	−0.202077	+	0.979370i	$0.564769\pi$
$618$	0	0
$619$	−19.9368	−0.801328	−0.400664	−	0.916225i	$-0.631221\pi$
$619$	−19.9368	−0.801328	−0.400664	+	0.916225i	$0.631221\pi$
$620$	9.99044	0.401226
$621$	0	0
$622$	6.74633	0.270503
$623$	1.87504	0.0751220
$624$	0	0
$625$	13.2322	0.529288
$626$	−4.61986	−0.184647
$627$	0	0
$628$	−24.2221	−0.966568
$629$	18.2293	0.726849
$630$	0	0
$631$	2.03105	0.0808549	0.0404275	−	0.999182i	$-0.487128\pi$
$631$	2.03105	0.0808549	0.0404275	+	0.999182i	$0.487128\pi$
$632$	2.58006	0.102629
$633$	0	0
$634$	−3.79639	−0.150774
$635$	−13.2975	−0.527695
$636$	0	0
$637$	24.6412	0.976322
$638$	0.997518	0.0394921
$639$	0	0
$640$	5.59634	0.221215
$641$	−26.7525	−1.05666	−0.528329	−	0.849040i	$-0.677181\pi$
$641$	−26.7525	−1.05666	−0.528329	+	0.849040i	$0.677181\pi$
$642$	0	0
$643$	−8.75409	−0.345228	−0.172614	−	0.984990i	$-0.555221\pi$
$643$	−8.75409	−0.345228	−0.172614	+	0.984990i	$0.555221\pi$
$644$	2.63688	0.103908
$645$	0	0
$646$	1.73852	0.0684011
$647$	−1.73387	−0.0681653	−0.0340827	−	0.999419i	$-0.510851\pi$
$647$	−1.73387	−0.0681653	−0.0340827	+	0.999419i	$0.510851\pi$
$648$	0	0
$649$	2.18701	0.0858477
$650$	2.97768	0.116794
$651$	0	0
$652$	23.7910	0.931727
$653$	−25.8655	−1.01220	−0.506098	−	0.862476i	$-0.668913\pi$
$653$	−25.8655	−1.01220	−0.506098	+	0.862476i	$0.668913\pi$
$654$	0	0
$655$	−6.07589	−0.237405
$656$	−41.1108	−1.60511
$657$	0	0
$658$	−0.296072	−0.0115421
$659$	7.78344	0.303200	0.151600	−	0.988442i	$-0.451557\pi$
$659$	7.78344	0.303200	0.151600	+	0.988442i	$0.451557\pi$
$660$	0	0
$661$	7.82579	0.304388	0.152194	−	0.988351i	$-0.451366\pi$
$661$	7.82579	0.304388	0.152194	+	0.988351i	$0.451366\pi$
$662$	−1.19625	−0.0464937
$663$	0	0
$664$	−13.4649	−0.522540
$665$	−0.318479	−0.0123501
$666$	0	0
$667$	−53.9964	−2.09075
$668$	4.97848	0.192623
$669$	0	0
$670$	−1.63579	−0.0631961
$671$	−5.29621	−0.204458
$672$	0	0
$673$	6.32541	0.243827	0.121913	−	0.992541i	$-0.461097\pi$
$673$	6.32541	0.243827	0.121913	+	0.992541i	$0.461097\pi$
$674$	−2.45248	−0.0944660
$675$	0	0
$676$	1.00410	0.0386194
$677$	−24.6661	−0.947993	−0.473997	−	0.880527i	$-0.657189\pi$
$677$	−24.6661	−0.947993	−0.473997	+	0.880527i	$0.657189\pi$
$678$	0	0
$679$	−0.743184	−0.0285208
$680$	2.94443	0.112914
$681$	0	0
$682$	0.848133	0.0324767
$683$	21.7390	0.831820	0.415910	−	0.909406i	$-0.363463\pi$
$683$	21.7390	0.831820	0.415910	+	0.909406i	$0.363463\pi$
$684$	0	0
$685$	−13.3303	−0.509324
$686$	−0.463242	−0.0176866
$687$	0	0
$688$	−12.9988	−0.495573
$689$	−21.6963	−0.826563
$690$	0	0
$691$	−48.5869	−1.84833	−0.924166	−	0.381991i	$-0.875238\pi$
$691$	−48.5869	−1.84833	−0.924166	+	0.381991i	$0.875238\pi$
$692$	−19.0757	−0.725151
$693$	0	0
$694$	0.492946	0.0187120
$695$	3.02138	0.114607
$696$	0	0
$697$	−44.1907	−1.67384
$698$	4.64442	0.175794
$699$	0	0
$700$	1.34002	0.0506480
$701$	−25.9876	−0.981540	−0.490770	−	0.871289i	$-0.663284\pi$
$701$	−25.9876	−0.981540	−0.490770	+	0.871289i	$0.663284\pi$
$702$	0	0
$703$	9.61722	0.362720
$704$	−5.27731	−0.198896
$705$	0	0
$706$	4.67963	0.176120
$707$	−0.377572	−0.0142001
$708$	0	0
$709$	36.2646	1.36195	0.680973	−	0.732309i	$-0.261557\pi$
$709$	36.2646	1.36195	0.680973	+	0.732309i	$0.261557\pi$
$710$	1.75344	0.0658056
$711$	0	0
$712$	−9.14560	−0.342746
$713$	−45.9100	−1.71934
$714$	0	0
$715$	2.41530	0.0903271
$716$	−7.29388	−0.272585
$717$	0	0
$718$	6.32776	0.236150
$719$	0.582371	0.0217188	0.0108594	−	0.999941i	$-0.496543\pi$
$719$	0.582371	0.0217188	0.0108594	+	0.999941i	$0.496543\pi$
$720$	0	0
$721$	0.426917	0.0158992
$722$	−2.92263	−0.108769
$723$	0	0
$724$	14.3135	0.531957
$725$	−27.4401	−1.01910
$726$	0	0
$727$	20.0001	0.741764	0.370882	−	0.928680i	$-0.379055\pi$
$727$	20.0001	0.741764	0.370882	+	0.928680i	$0.379055\pi$
$728$	0.463826	0.0171905
$729$	0	0
$730$	1.62328	0.0600803
$731$	−13.9726	−0.516795
$732$	0	0
$733$	−42.8774	−1.58371	−0.791857	−	0.610707i	$-0.790885\pi$
$733$	−42.8774	−1.58371	−0.791857	+	0.610707i	$0.790885\pi$
$734$	1.55102	0.0572490
$735$	0	0
$736$	−19.3586	−0.713568
$737$	6.66136	0.245374
$738$	0	0
$739$	32.3852	1.19131	0.595654	−	0.803241i	$-0.296893\pi$
$739$	32.3852	1.19131	0.595654	+	0.803241i	$0.296893\pi$
$740$	8.06006	0.296294
$741$	0	0
$742$	0.203546	0.00747242
$743$	−12.5358	−0.459895	−0.229948	−	0.973203i	$-0.573855\pi$
$743$	−12.5358	−0.459895	−0.229948	+	0.973203i	$0.573855\pi$
$744$	0	0
$745$	4.46736	0.163672
$746$	5.24517	0.192039
$747$	0	0
$748$	−5.93340	−0.216947
$749$	−1.96104	−0.0716548
$750$	0	0
$751$	−24.5016	−0.894077	−0.447039	−	0.894515i	$-0.647521\pi$
$751$	−24.5016	−0.894077	−0.447039	+	0.894515i	$0.647521\pi$
$752$	−33.5489	−1.22340
$753$	0	0
$754$	−4.69997	−0.171163
$755$	−14.7242	−0.535868
$756$	0	0
$757$	21.4627	0.780074	0.390037	−	0.920799i	$-0.372462\pi$
$757$	21.4627	0.780074	0.390037	+	0.920799i	$0.372462\pi$
$758$	5.36971	0.195037
$759$	0	0
$760$	1.55339	0.0563475
$761$	35.6077	1.29078	0.645390	−	0.763853i	$-0.276695\pi$
$761$	35.6077	1.29078	0.645390	+	0.763853i	$0.276695\pi$
$762$	0	0
$763$	2.97428	0.107676
$764$	−40.7042	−1.47263
$765$	0	0
$766$	0.544773	0.0196834
$767$	−10.3045	−0.372073
$768$	0	0
$769$	−16.5914	−0.598299	−0.299150	−	0.954206i	$-0.596703\pi$
$769$	−16.5914	−0.598299	−0.299150	+	0.954206i	$0.596703\pi$
$770$	−0.0226594	−0.000816588 0
$771$	0	0
$772$	6.84318	0.246291
$773$	−11.3858	−0.409519	−0.204759	−	0.978812i	$-0.565641\pi$
$773$	−11.3858	−0.409519	−0.204759	+	0.978812i	$0.565641\pi$
$774$	0	0
$775$	−23.3307	−0.838064
$776$	3.62491	0.130127
$777$	0	0
$778$	4.78851	0.171676
$779$	−23.3137	−0.835299
$780$	0	0
$781$	−7.14048	−0.255506
$782$	−6.69563	−0.239435
$783$	0	0
$784$	−26.1952	−0.935543
$785$	−11.2671	−0.402141
$786$	0	0
$787$	−43.0600	−1.53492	−0.767461	−	0.641095i	$-0.778480\pi$
$787$	−43.0600	−1.53492	−0.767461	+	0.641095i	$0.778480\pi$
$788$	−2.32118	−0.0826885
$789$	0	0
$790$	0.593879	0.0211293
$791$	1.95213	0.0694097
$792$	0	0
$793$	24.9540	0.886142
$794$	0.0221387	0.000785674 0
$795$	0	0
$796$	2.15815	0.0764935
$797$	−32.6980	−1.15822	−0.579111	−	0.815249i	$-0.696600\pi$
$797$	−32.6980	−1.15822	−0.579111	+	0.815249i	$0.696600\pi$
$798$	0	0
$799$	−36.0623	−1.27579
$800$	−9.83772	−0.347816
$801$	0	0
$802$	1.70707	0.0602786
$803$	−6.61042	−0.233277
$804$	0	0
$805$	1.22657	0.0432309
$806$	−3.99612	−0.140757
$807$	0	0
$808$	1.84162	0.0647881
$809$	−25.5159	−0.897092	−0.448546	−	0.893760i	$-0.648058\pi$
$809$	−25.5159	−0.897092	−0.448546	+	0.893760i	$0.648058\pi$
$810$	0	0
$811$	−8.80464	−0.309173	−0.154586	−	0.987979i	$-0.549404\pi$
$811$	−8.80464	−0.309173	−0.154586	+	0.987979i	$0.549404\pi$
$812$	−2.11509	−0.0742251
$813$	0	0
$814$	0.684255	0.0239831
$815$	11.0666	0.387646
$816$	0	0
$817$	−7.37152	−0.257897
$818$	−1.02170	−0.0357228
$819$	0	0
$820$	−19.5389	−0.682327
$821$	−5.89246	−0.205648	−0.102824	−	0.994700i	$-0.532788\pi$
$821$	−5.89246	−0.205648	−0.102824	+	0.994700i	$0.532788\pi$
$822$	0	0
$823$	−45.2067	−1.57581	−0.787904	−	0.615798i	$-0.788834\pi$
$823$	−45.2067	−1.57581	−0.787904	+	0.615798i	$0.788834\pi$
$824$	−2.08230	−0.0725405
$825$	0	0
$826$	0.0966725	0.00336367
$827$	4.30278	0.149622	0.0748112	−	0.997198i	$-0.476165\pi$
$827$	4.30278	0.149622	0.0748112	+	0.997198i	$0.476165\pi$
$828$	0	0
$829$	51.1600	1.77686	0.888430	−	0.459012i	$-0.151797\pi$
$829$	51.1600	1.77686	0.888430	+	0.459012i	$0.151797\pi$
$830$	−3.09935	−0.107580
$831$	0	0
$832$	24.8649	0.862037
$833$	−28.1577	−0.975605
$834$	0	0
$835$	2.31578	0.0801410
$836$	−3.13028	−0.108263
$837$	0	0
$838$	−2.58177	−0.0891857
$839$	−22.2666	−0.768730	−0.384365	−	0.923181i	$-0.625580\pi$
$839$	−22.2666	−0.768730	−0.384365	+	0.923181i	$0.625580\pi$
$840$	0	0
$841$	14.3114	0.493497
$842$	−3.67962	−0.126808
$843$	0	0
$844$	29.3553	1.01045
$845$	0.467068	0.0160676
$846$	0	0
$847$	−1.71220	−0.0588319
$848$	23.0645	0.792039
$849$	0	0
$850$	−3.40261	−0.116708
$851$	−37.0392	−1.26969
$852$	0	0
$853$	−6.67502	−0.228548	−0.114274	−	0.993449i	$-0.536454\pi$
$853$	−6.67502	−0.228548	−0.114274	+	0.993449i	$0.536454\pi$
$854$	−0.234109	−0.00801103
$855$	0	0
$856$	9.56504	0.326926
$857$	3.23621	0.110547	0.0552734	−	0.998471i	$-0.482397\pi$
$857$	3.23621	0.110547	0.0552734	+	0.998471i	$0.482397\pi$
$858$	0	0
$859$	−8.95134	−0.305416	−0.152708	−	0.988271i	$-0.548799\pi$
$859$	−8.95134	−0.305416	−0.152708	+	0.988271i	$0.548799\pi$
$860$	−6.17797	−0.210667
$861$	0	0
$862$	−4.67549	−0.159248
$863$	16.6061	0.565278	0.282639	−	0.959226i	$-0.408790\pi$
$863$	16.6061	0.565278	0.282639	+	0.959226i	$0.408790\pi$
$864$	0	0
$865$	−8.87325	−0.301699
$866$	−6.77147	−0.230104
$867$	0	0
$868$	−1.79834	−0.0610397
$869$	−2.41843	−0.0820396
$870$	0	0
$871$	−31.3861	−1.06348
$872$	−14.5072	−0.491274
$873$	0	0
$874$	−3.53241	−0.119486
$875$	1.37080	0.0463415
$876$	0	0
$877$	5.59472	0.188920	0.0944601	−	0.995529i	$-0.469888\pi$
$877$	5.59472	0.188920	0.0944601	+	0.995529i	$0.469888\pi$
$878$	−6.39704	−0.215890
$879$	0	0
$880$	−2.56762	−0.0865543
$881$	−8.94054	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$881$	−8.94054	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$882$	0	0
$883$	−14.7132	−0.495140	−0.247570	−	0.968870i	$-0.579632\pi$
$883$	−14.7132	−0.495140	−0.247570	+	0.968870i	$0.579632\pi$
$884$	27.9562	0.940269
$885$	0	0
$886$	4.56946	0.153514
$887$	−4.40891	−0.148037	−0.0740184	−	0.997257i	$-0.523582\pi$
$887$	−4.40891	−0.148037	−0.0740184	+	0.997257i	$0.523582\pi$
$888$	0	0
$889$	2.39363	0.0802798
$890$	−2.10514	−0.0705643
$891$	0	0
$892$	−9.99369	−0.334614
$893$	−19.0254	−0.636660
$894$	0	0
$895$	−3.39281	−0.113409
$896$	−1.00738	−0.0336541
$897$	0	0
$898$	3.64184	0.121530
$899$	36.8253	1.22819
$900$	0	0
$901$	24.7925	0.825956
$902$	−1.65874	−0.0552301
$903$	0	0
$904$	−9.52159	−0.316684
$905$	6.65806	0.221321
$906$	0	0
$907$	3.27708	0.108814	0.0544069	−	0.998519i	$-0.482673\pi$
$907$	3.27708	0.108814	0.0544069	+	0.998519i	$0.482673\pi$
$908$	−32.8417	−1.08989
$909$	0	0
$910$	0.106764	0.00353918
$911$	42.0943	1.39465	0.697323	−	0.716757i	$-0.254374\pi$
$911$	42.0943	1.39465	0.697323	+	0.716757i	$0.254374\pi$
$912$	0	0
$913$	12.6214	0.417707
$914$	−4.76123	−0.157488
$915$	0	0
$916$	−31.5876	−1.04368
$917$	1.09370	0.0361171
$918$	0	0
$919$	−30.1308	−0.993924	−0.496962	−	0.867772i	$-0.665551\pi$
$919$	−30.1308	−0.993924	−0.496962	+	0.867772i	$0.665551\pi$
$920$	−5.98265	−0.197242
$921$	0	0
$922$	8.11850	0.267369
$923$	33.6436	1.10739
$924$	0	0
$925$	−18.8227	−0.618887
$926$	−2.21949	−0.0729369
$927$	0	0
$928$	15.5279	0.509728
$929$	−35.6647	−1.17012	−0.585060	−	0.810990i	$-0.698929\pi$
$929$	−35.6647	−1.17012	−0.585060	+	0.810990i	$0.698929\pi$
$930$	0	0
$931$	−14.8551	−0.486857
$932$	−20.5471	−0.673044
$933$	0	0
$934$	0.340630	0.0111458
$935$	−2.75997	−0.0902608
$936$	0	0
$937$	−16.2042	−0.529368	−0.264684	−	0.964335i	$-0.585268\pi$
$937$	−16.2042	−0.529368	−0.264684	+	0.964335i	$0.585268\pi$
$938$	0.294453	0.00961422
$939$	0	0
$940$	−15.9449	−0.520066
$941$	15.6246	0.509348	0.254674	−	0.967027i	$-0.418032\pi$
$941$	15.6246	0.509348	0.254674	+	0.967027i	$0.418032\pi$
$942$	0	0
$943$	89.7889	2.92393
$944$	10.9543	0.356532
$945$	0	0
$946$	−0.524476	−0.0170522
$947$	−50.9846	−1.65678	−0.828389	−	0.560154i	$-0.810742\pi$
$947$	−50.9846	−1.65678	−0.828389	+	0.560154i	$0.810742\pi$
$948$	0	0
$949$	31.1461	1.01105
$950$	−1.79511	−0.0582411
$951$	0	0
$952$	−0.530016	−0.0171779
$953$	39.7922	1.28900	0.644498	−	0.764606i	$-0.277066\pi$
$953$	39.7922	1.28900	0.644498	+	0.764606i	$0.277066\pi$
$954$	0	0
$955$	−18.9339	−0.612687
$956$	0.106503	0.00344455
$957$	0	0
$958$	4.01959	0.129867
$959$	2.39954	0.0774851
$960$	0	0
$961$	0.310434	0.0100140
$962$	−3.22398	−0.103945
$963$	0	0
$964$	−23.3066	−0.750654
$965$	3.18316	0.102470
$966$	0	0
$967$	22.5777	0.726048	0.363024	−	0.931780i	$-0.381744\pi$
$967$	22.5777	0.726048	0.363024	+	0.931780i	$0.381744\pi$
$968$	8.35133	0.268422
$969$	0	0
$970$	0.834383	0.0267904
$971$	−7.94868	−0.255085	−0.127543	−	0.991833i	$-0.540709\pi$
$971$	−7.94868	−0.255085	−0.127543	+	0.991833i	$0.540709\pi$
$972$	0	0
$973$	−0.543867	−0.0174356
$974$	−3.42322	−0.109687
$975$	0	0
$976$	−26.5276	−0.849129
$977$	−38.7688	−1.24032	−0.620162	−	0.784474i	$-0.712933\pi$
$977$	−38.7688	−1.24032	−0.620162	+	0.784474i	$0.712933\pi$
$978$	0	0
$979$	8.57266	0.273984
$980$	−12.4499	−0.397697
$981$	0	0
$982$	6.95123	0.221823
$983$	−16.8463	−0.537314	−0.268657	−	0.963236i	$-0.586580\pi$
$983$	−16.8463	−0.537314	−0.268657	+	0.963236i	$0.586580\pi$
$984$	0	0
$985$	−1.07972	−0.0344026
$986$	5.37068	0.171037
$987$	0	0
$988$	14.7488	0.469223
$989$	28.3902	0.902757
$990$	0	0
$991$	−6.81915	−0.216618	−0.108309	−	0.994117i	$-0.534544\pi$
$991$	−6.81915	−0.216618	−0.108309	+	0.994117i	$0.534544\pi$
$992$	13.2025	0.419179
$993$	0	0
$994$	−0.315631	−0.0100112
$995$	1.00388	0.0318252
$996$	0	0
$997$	−11.8247	−0.374492	−0.187246	−	0.982313i	$-0.559956\pi$
$997$	−11.8247	−0.374492	−0.187246	+	0.982313i	$0.559956\pi$
$998$	3.55549	0.112547
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.37		72
3.2	odd	2		6561.2.a.c.1.36		72
81.2	odd	54		729.2.g.d.433.4		144
81.13	even	27		243.2.g.a.100.4		144
81.14	odd	54		729.2.g.c.55.5		144
81.25	even	27		243.2.g.a.226.4		144
81.29	odd	54		729.2.g.c.676.5		144
81.40	even	27		729.2.g.a.298.5		144
81.41	odd	54		729.2.g.d.298.4		144
81.52	even	27		729.2.g.b.676.4		144
81.56	odd	54		81.2.g.a.58.5	yes	144
81.67	even	27		729.2.g.b.55.4		144
81.68	odd	54		81.2.g.a.7.5	✓	144
81.79	even	27		729.2.g.a.433.5		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.7.5	✓	144	81.68	odd	54
81.2.g.a.58.5	yes	144	81.56	odd	54
243.2.g.a.100.4		144	81.13	even	27
243.2.g.a.226.4		144	81.25	even	27
729.2.g.a.298.5		144	81.40	even	27
729.2.g.a.433.5		144	81.79	even	27
729.2.g.b.55.4		144	81.67	even	27
729.2.g.b.676.4		144	81.52	even	27
729.2.g.c.55.5		144	81.14	odd	54
729.2.g.c.676.5		144	81.29	odd	54
729.2.g.d.298.4		144	81.41	odd	54
729.2.g.d.433.4		144	81.2	odd	54
6561.2.a.c.1.36		72	3.2	odd	2
6561.2.a.d.1.37		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)

\(f(q)\)	\(=\)	\(q+0.202096 q^{2} -1.95916 q^{4} -0.911319 q^{5} +0.164043 q^{7} -0.800128 q^{8} -0.184174 q^{10} +0.750003 q^{11} -3.53376 q^{13} +0.0331524 q^{14} +3.75661 q^{16} +4.03805 q^{17} +2.13035 q^{19} +1.78542 q^{20} +0.151572 q^{22} -8.20471 q^{23} -4.16950 q^{25} -0.714158 q^{26} -0.321387 q^{28} +6.58114 q^{29} +5.59557 q^{31} +2.35945 q^{32} +0.816071 q^{34} -0.149496 q^{35} +4.51438 q^{37} +0.430534 q^{38} +0.729172 q^{40} -10.9436 q^{41} -3.46024 q^{43} -1.46937 q^{44} -1.65813 q^{46} -8.93063 q^{47} -6.97309 q^{49} -0.842637 q^{50} +6.92320 q^{52} +6.13971 q^{53} -0.683492 q^{55} -0.131256 q^{56} +1.33002 q^{58} +2.91600 q^{59} -7.06159 q^{61} +1.13084 q^{62} -7.03639 q^{64} +3.22039 q^{65} +8.88178 q^{67} -7.91117 q^{68} -0.0302124 q^{70} -9.52060 q^{71} -8.81387 q^{73} +0.912337 q^{74} -4.17369 q^{76} +0.123033 q^{77} -3.22456 q^{79} -3.42347 q^{80} -2.21165 q^{82} +16.8284 q^{83} -3.67995 q^{85} -0.699298 q^{86} -0.600098 q^{88} +11.4302 q^{89} -0.579690 q^{91} +16.0743 q^{92} -1.80484 q^{94} -1.94143 q^{95} -4.53042 q^{97} -1.40923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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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