LMFDB - Embedded newform 6561.2.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.3898487662$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-1.73329 q^{2} +1.00429 q^{4} +1.91493 q^{5} +4.47719 q^{7} +1.72585 q^{8} +O(q^{10})$	\(q-1.73329 q^{2} +1.00429 q^{4} +1.91493 q^{5} +4.47719 q^{7} +1.72585 q^{8} -3.31913 q^{10} -5.07155 q^{11} -1.52947 q^{13} -7.76027 q^{14} -4.99998 q^{16} -4.05206 q^{17} -2.47829 q^{19} +1.92315 q^{20} +8.79047 q^{22} -1.69677 q^{23} -1.33304 q^{25} +2.65102 q^{26} +4.49642 q^{28} +2.91406 q^{29} +5.87465 q^{31} +5.21472 q^{32} +7.02340 q^{34} +8.57351 q^{35} +3.53677 q^{37} +4.29560 q^{38} +3.30488 q^{40} -4.48087 q^{41} +4.47215 q^{43} -5.09333 q^{44} +2.94099 q^{46} -3.08838 q^{47} +13.0452 q^{49} +2.31055 q^{50} -1.53604 q^{52} +2.99890 q^{53} -9.71168 q^{55} +7.72695 q^{56} -5.05092 q^{58} -2.07115 q^{59} -8.42990 q^{61} -10.1825 q^{62} +0.961336 q^{64} -2.92884 q^{65} +10.9228 q^{67} -4.06946 q^{68} -14.8604 q^{70} -0.415698 q^{71} -8.96715 q^{73} -6.13024 q^{74} -2.48894 q^{76} -22.7063 q^{77} -15.7220 q^{79} -9.57462 q^{80} +7.76664 q^{82} -7.80984 q^{83} -7.75942 q^{85} -7.75154 q^{86} -8.75273 q^{88} +3.76684 q^{89} -6.84775 q^{91} -1.70406 q^{92} +5.35306 q^{94} -4.74576 q^{95} +10.0287 q^{97} -22.6112 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.73329	−1.22562	−0.612811	−	0.790230i	$-0.709961\pi$
$2$	−1.73329	−1.22562	−0.612811	+	0.790230i	$0.709961\pi$
$3$	0	0
$3$	0	0
$4$	1.00429	0.502147
$5$	1.91493	0.856383	0.428192	−	0.903688i	$-0.359151\pi$
$5$	1.91493	0.856383	0.428192	+	0.903688i	$0.359151\pi$
$6$	0	0
$7$	4.47719	1.69222	0.846109	−	0.533009i	$-0.178939\pi$
$7$	4.47719	1.69222	0.846109	+	0.533009i	$0.178939\pi$
$8$	1.72585	0.610179
$9$	0	0
$10$	−3.31913	−1.04960
$11$	−5.07155	−1.52913	−0.764566	−	0.644546i	$-0.777046\pi$
$11$	−5.07155	−1.52913	−0.764566	+	0.644546i	$0.777046\pi$
$12$	0	0
$13$	−1.52947	−0.424200	−0.212100	−	0.977248i	$-0.568030\pi$
$13$	−1.52947	−0.424200	−0.212100	+	0.977248i	$0.568030\pi$
$14$	−7.76027	−2.07402
$15$	0	0
$16$	−4.99998	−1.25000
$17$	−4.05206	−0.982770	−0.491385	−	0.870943i	$-0.663509\pi$
$17$	−4.05206	−0.982770	−0.491385	+	0.870943i	$0.663509\pi$
$18$	0	0
$19$	−2.47829	−0.568560	−0.284280	−	0.958741i	$-0.591754\pi$
$19$	−2.47829	−0.568560	−0.284280	+	0.958741i	$0.591754\pi$
$20$	1.92315	0.430030
$21$	0	0
$22$	8.79047	1.87414
$23$	−1.69677	−0.353801	−0.176900	−	0.984229i	$-0.556607\pi$
$23$	−1.69677	−0.353801	−0.176900	+	0.984229i	$0.556607\pi$
$24$	0	0
$25$	−1.33304	−0.266608
$26$	2.65102	0.519908
$27$	0	0
$28$	4.49642	0.849743
$29$	2.91406	0.541128	0.270564	−	0.962702i	$-0.412790\pi$
$29$	2.91406	0.541128	0.270564	+	0.962702i	$0.412790\pi$
$30$	0	0
$31$	5.87465	1.05512	0.527559	−	0.849518i	$-0.323107\pi$
$31$	5.87465	1.05512	0.527559	+	0.849518i	$0.323107\pi$
$32$	5.21472	0.921842
$33$	0	0
$34$	7.02340	1.20450
$35$	8.57351	1.44919
$36$	0	0
$37$	3.53677	0.581441	0.290721	−	0.956808i	$-0.406105\pi$
$37$	3.53677	0.581441	0.290721	+	0.956808i	$0.406105\pi$
$38$	4.29560	0.696839
$39$	0	0
$40$	3.30488	0.522547
$41$	−4.48087	−0.699794	−0.349897	−	0.936788i	$-0.613783\pi$
$41$	−4.48087	−0.699794	−0.349897	+	0.936788i	$0.613783\pi$
$42$	0	0
$43$	4.47215	0.681997	0.340998	−	0.940064i	$-0.389235\pi$
$43$	4.47215	0.681997	0.340998	+	0.940064i	$0.389235\pi$
$44$	−5.09333	−0.767849
$45$	0	0
$46$	2.94099	0.433626
$47$	−3.08838	−0.450486	−0.225243	−	0.974303i	$-0.572318\pi$
$47$	−3.08838	−0.450486	−0.225243	+	0.974303i	$0.572318\pi$
$48$	0	0
$49$	13.0452	1.86360
$50$	2.31055	0.326760
$51$	0	0
$52$	−1.53604	−0.213011
$53$	2.99890	0.411931	0.205965	−	0.978559i	$-0.433967\pi$
$53$	2.99890	0.411931	0.205965	+	0.978559i	$0.433967\pi$
$54$	0	0
$55$	−9.71168	−1.30952
$56$	7.72695	1.03256
$57$	0	0
$58$	−5.05092	−0.663218
$59$	−2.07115	−0.269641	−0.134820	−	0.990870i	$-0.543046\pi$
$59$	−2.07115	−0.269641	−0.134820	+	0.990870i	$0.543046\pi$
$60$	0	0
$61$	−8.42990	−1.07934	−0.539669	−	0.841877i	$-0.681451\pi$
$61$	−8.42990	−1.07934	−0.539669	+	0.841877i	$0.681451\pi$
$62$	−10.1825	−1.29318
$63$	0	0
$64$	0.961336	0.120167
$65$	−2.92884	−0.363278
$66$	0	0
$67$	10.9228	1.33443	0.667213	−	0.744867i	$-0.267487\pi$
$67$	10.9228	1.33443	0.667213	+	0.744867i	$0.267487\pi$
$68$	−4.06946	−0.493495
$69$	0	0
$70$	−14.8604	−1.77615
$71$	−0.415698	−0.0493343	−0.0246671	−	0.999696i	$-0.507853\pi$
$71$	−0.415698	−0.0493343	−0.0246671	+	0.999696i	$0.507853\pi$
$72$	0	0
$73$	−8.96715	−1.04953	−0.524763	−	0.851249i	$-0.675846\pi$
$73$	−8.96715	−1.04953	−0.524763	+	0.851249i	$0.675846\pi$
$74$	−6.13024	−0.712626
$75$	0	0
$76$	−2.48894	−0.285501
$77$	−22.7063	−2.58762
$78$	0	0
$79$	−15.7220	−1.76887	−0.884434	−	0.466666i	$-0.845455\pi$
$79$	−15.7220	−1.76887	−0.884434	+	0.466666i	$0.845455\pi$
$80$	−9.57462	−1.07047
$81$	0	0
$82$	7.76664	0.857682
$83$	−7.80984	−0.857241	−0.428621	−	0.903485i	$-0.641000\pi$
$83$	−7.80984	−0.857241	−0.428621	+	0.903485i	$0.641000\pi$
$84$	0	0
$85$	−7.75942	−0.841627
$86$	−7.75154	−0.835870
$87$	0	0
$88$	−8.75273	−0.933044
$89$	3.76684	0.399284	0.199642	−	0.979869i	$-0.436022\pi$
$89$	3.76684	0.399284	0.199642	+	0.979869i	$0.436022\pi$
$90$	0	0
$91$	−6.84775	−0.717839
$92$	−1.70406	−0.177660
$93$	0	0
$94$	5.35306	0.552126
$95$	−4.74576	−0.486905
$96$	0	0
$97$	10.0287	1.01826	0.509132	−	0.860689i	$-0.329967\pi$
$97$	10.0287	1.01826	0.509132	+	0.860689i	$0.329967\pi$
$98$	−22.6112	−2.28407
$99$	0	0
$100$	−1.33876	−0.133876
$101$	−1.97498	−0.196518	−0.0982588	−	0.995161i	$-0.531327\pi$
$101$	−1.97498	−0.196518	−0.0982588	+	0.995161i	$0.531327\pi$
$102$	0	0
$103$	14.8362	1.46185	0.730925	−	0.682458i	$-0.239089\pi$
$103$	14.8362	1.46185	0.730925	+	0.682458i	$0.239089\pi$
$104$	−2.63964	−0.258838
$105$	0	0
$106$	−5.19796	−0.504871
$107$	−15.9894	−1.54576	−0.772878	−	0.634555i	$-0.781183\pi$
$107$	−15.9894	−1.54576	−0.772878	+	0.634555i	$0.781183\pi$
$108$	0	0
$109$	−5.33372	−0.510877	−0.255439	−	0.966825i	$-0.582220\pi$
$109$	−5.33372	−0.510877	−0.255439	+	0.966825i	$0.582220\pi$
$110$	16.8332	1.60498
$111$	0	0
$112$	−22.3859	−2.11527
$113$	2.93275	0.275890	0.137945	−	0.990440i	$-0.455950\pi$
$113$	2.93275	0.275890	0.137945	+	0.990440i	$0.455950\pi$
$114$	0	0
$115$	−3.24920	−0.302989
$116$	2.92658	0.271726
$117$	0	0
$118$	3.58991	0.330478
$119$	−18.1419	−1.66306
$120$	0	0
$121$	14.7207	1.33824
$122$	14.6115	1.32286
$123$	0	0
$124$	5.89988	0.529825
$125$	−12.1273	−1.08470
$126$	0	0
$127$	−10.6332	−0.943540	−0.471770	−	0.881722i	$-0.656385\pi$
$127$	−10.6332	−0.943540	−0.471770	+	0.881722i	$0.656385\pi$
$128$	−12.0957	−1.06912
$129$	0	0
$130$	5.07653	0.445241
$131$	11.3274	0.989683	0.494841	−	0.868983i	$-0.335226\pi$
$131$	11.3274	0.989683	0.494841	+	0.868983i	$0.335226\pi$
$132$	0	0
$133$	−11.0958	−0.962127
$134$	−18.9323	−1.63550
$135$	0	0
$136$	−6.99324	−0.599666
$137$	−17.4013	−1.48669	−0.743345	−	0.668908i	$-0.766762\pi$
$137$	−17.4013	−1.48669	−0.743345	+	0.668908i	$0.766762\pi$
$138$	0	0
$139$	−13.5069	−1.14564	−0.572822	−	0.819680i	$-0.694151\pi$
$139$	−13.5069	−1.14564	−0.572822	+	0.819680i	$0.694151\pi$
$140$	8.61032	0.727705
$141$	0	0
$142$	0.720525	0.0604651
$143$	7.75682	0.648657
$144$	0	0
$145$	5.58023	0.463413
$146$	15.5427	1.28632
$147$	0	0
$148$	3.55196	0.291969
$149$	5.33782	0.437291	0.218646	−	0.975804i	$-0.429836\pi$
$149$	5.33782	0.437291	0.218646	+	0.975804i	$0.429836\pi$
$150$	0	0
$151$	6.88562	0.560344	0.280172	−	0.959950i	$-0.409609\pi$
$151$	6.88562	0.560344	0.280172	+	0.959950i	$0.409609\pi$
$152$	−4.27716	−0.346923
$153$	0	0
$154$	39.3566	3.17145
$155$	11.2496	0.903586
$156$	0	0
$157$	14.2270	1.13544	0.567719	−	0.823223i	$-0.307826\pi$
$157$	14.2270	1.13544	0.567719	+	0.823223i	$0.307826\pi$
$158$	27.2508	2.16796
$159$	0	0
$160$	9.98583	0.789450
$161$	−7.59676	−0.598709
$162$	0	0
$163$	−4.92481	−0.385741	−0.192870	−	0.981224i	$-0.561780\pi$
$163$	−4.92481	−0.385741	−0.192870	+	0.981224i	$0.561780\pi$
$164$	−4.50011	−0.351400
$165$	0	0
$166$	13.5367	1.05065
$167$	−10.6201	−0.821806	−0.410903	−	0.911679i	$-0.634787\pi$
$167$	−10.6201	−0.821806	−0.410903	+	0.911679i	$0.634787\pi$
$168$	0	0
$169$	−10.6607	−0.820054
$170$	13.4493	1.03152
$171$	0	0
$172$	4.49136	0.342463
$173$	−8.86804	−0.674224	−0.337112	−	0.941464i	$-0.609450\pi$
$173$	−8.86804	−0.674224	−0.337112	+	0.941464i	$0.609450\pi$
$174$	0	0
$175$	−5.96827	−0.451159
$176$	25.3577	1.91141
$177$	0	0
$178$	−6.52902	−0.489371
$179$	7.47114	0.558419	0.279210	−	0.960230i	$-0.409928\pi$
$179$	7.47114	0.558419	0.279210	+	0.960230i	$0.409928\pi$
$180$	0	0
$181$	2.41862	0.179775	0.0898874	−	0.995952i	$-0.471349\pi$
$181$	2.41862	0.179775	0.0898874	+	0.995952i	$0.471349\pi$
$182$	11.8691	0.879799
$183$	0	0
$184$	−2.92836	−0.215882
$185$	6.77267	0.497936
$186$	0	0
$187$	20.5503	1.50278
$188$	−3.10164	−0.226210
$189$	0	0
$190$	8.22578	0.596761
$191$	−25.8023	−1.86699	−0.933494	−	0.358593i	$-0.883257\pi$
$191$	−25.8023	−1.86699	−0.933494	+	0.358593i	$0.883257\pi$
$192$	0	0
$193$	−14.4219	−1.03811	−0.519055	−	0.854741i	$-0.673716\pi$
$193$	−14.4219	−1.03811	−0.519055	+	0.854741i	$0.673716\pi$
$194$	−17.3827	−1.24801
$195$	0	0
$196$	13.1012	0.935804
$197$	−25.9638	−1.84984	−0.924922	−	0.380157i	$-0.875870\pi$
$197$	−25.9638	−1.84984	−0.924922	+	0.380157i	$0.875870\pi$
$198$	0	0
$199$	−4.78401	−0.339130	−0.169565	−	0.985519i	$-0.554236\pi$
$199$	−4.78401	−0.339130	−0.169565	+	0.985519i	$0.554236\pi$
$200$	−2.30062	−0.162679
$201$	0	0
$202$	3.42321	0.240856
$203$	13.0468	0.915707
$204$	0	0
$205$	−8.58055	−0.599292
$206$	−25.7154	−1.79167
$207$	0	0
$208$	7.64735	0.530248
$209$	12.5688	0.869402
$210$	0	0
$211$	16.8987	1.16335	0.581676	−	0.813421i	$-0.302397\pi$
$211$	16.8987	1.16335	0.581676	+	0.813421i	$0.302397\pi$
$212$	3.01178	0.206850
$213$	0	0
$214$	27.7143	1.89451
$215$	8.56386	0.584050
$216$	0	0
$217$	26.3019	1.78549
$218$	9.24488	0.626142
$219$	0	0
$220$	−9.75338	−0.657573
$221$	6.19753	0.416891
$222$	0	0
$223$	16.9907	1.13778	0.568892	−	0.822413i	$-0.307372\pi$
$223$	16.9907	1.13778	0.568892	+	0.822413i	$0.307372\pi$
$224$	23.3473	1.55996
$225$	0	0
$226$	−5.08331	−0.338137
$227$	7.96918	0.528933	0.264466	−	0.964395i	$-0.414804\pi$
$227$	7.96918	0.528933	0.264466	+	0.964395i	$0.414804\pi$
$228$	0	0
$229$	18.9622	1.25306	0.626529	−	0.779398i	$-0.284475\pi$
$229$	18.9622	1.25306	0.626529	+	0.779398i	$0.284475\pi$
$230$	5.63180	0.371350
$231$	0	0
$232$	5.02923	0.330185
$233$	−22.8864	−1.49934	−0.749668	−	0.661815i	$-0.769787\pi$
$233$	−22.8864	−1.49934	−0.749668	+	0.661815i	$0.769787\pi$
$234$	0	0
$235$	−5.91403	−0.385789
$236$	−2.08004	−0.135399
$237$	0	0
$238$	31.4451	2.03828
$239$	3.23018	0.208943	0.104472	−	0.994528i	$-0.466685\pi$
$239$	3.23018	0.208943	0.104472	+	0.994528i	$0.466685\pi$
$240$	0	0
$241$	2.81775	0.181507	0.0907536	−	0.995873i	$-0.471072\pi$
$241$	2.81775	0.181507	0.0907536	+	0.995873i	$0.471072\pi$
$242$	−25.5152	−1.64018
$243$	0	0
$244$	−8.46610	−0.541986
$245$	24.9807	1.59596
$246$	0	0
$247$	3.79049	0.241183
$248$	10.1388	0.643811
$249$	0	0
$250$	21.0202	1.32943
$251$	7.26581	0.458614	0.229307	−	0.973354i	$-0.426354\pi$
$251$	7.26581	0.458614	0.229307	+	0.973354i	$0.426354\pi$
$252$	0	0
$253$	8.60526	0.541008
$254$	18.4303	1.15642
$255$	0	0
$256$	19.0427	1.19017
$257$	−8.56813	−0.534465	−0.267233	−	0.963632i	$-0.586109\pi$
$257$	−8.56813	−0.534465	−0.267233	+	0.963632i	$0.586109\pi$
$258$	0	0
$259$	15.8348	0.983926
$260$	−2.94141	−0.182419
$261$	0	0
$262$	−19.6337	−1.21298
$263$	−17.6054	−1.08560	−0.542798	−	0.839863i	$-0.682635\pi$
$263$	−17.6054	−1.08560	−0.542798	+	0.839863i	$0.682635\pi$
$264$	0	0
$265$	5.74269	0.352770
$266$	19.2322	1.17920
$267$	0	0
$268$	10.9697	0.670078
$269$	−8.48826	−0.517538	−0.258769	−	0.965939i	$-0.583317\pi$
$269$	−8.48826	−0.517538	−0.258769	+	0.965939i	$0.583317\pi$
$270$	0	0
$271$	−0.607024	−0.0368741	−0.0184370	−	0.999830i	$-0.505869\pi$
$271$	−0.607024	−0.0368741	−0.0184370	+	0.999830i	$0.505869\pi$
$272$	20.2602	1.22846
$273$	0	0
$274$	30.1614	1.82212
$275$	6.76059	0.407679
$276$	0	0
$277$	−7.81273	−0.469421	−0.234711	−	0.972065i	$-0.575414\pi$
$277$	−7.81273	−0.469421	−0.234711	+	0.972065i	$0.575414\pi$
$278$	23.4114	1.40412
$279$	0	0
$280$	14.7966	0.884264
$281$	19.5514	1.16634	0.583170	−	0.812350i	$-0.301812\pi$
$281$	19.5514	1.16634	0.583170	+	0.812350i	$0.301812\pi$
$282$	0	0
$283$	−26.9996	−1.60496	−0.802479	−	0.596681i	$-0.796486\pi$
$283$	−26.9996	−1.60496	−0.802479	+	0.596681i	$0.796486\pi$
$284$	−0.417483	−0.0247731
$285$	0	0
$286$	−13.4448	−0.795008
$287$	−20.0617	−1.18420
$288$	0	0
$289$	−0.580778	−0.0341634
$290$	−9.67215	−0.567968
$291$	0	0
$292$	−9.00565	−0.527016
$293$	−7.48248	−0.437131	−0.218566	−	0.975822i	$-0.570138\pi$
$293$	−7.48248	−0.437131	−0.218566	+	0.975822i	$0.570138\pi$
$294$	0	0
$295$	−3.96611	−0.230916
$296$	6.10392	0.354783
$297$	0	0
$298$	−9.25199	−0.535953
$299$	2.59517	0.150082
$300$	0	0
$301$	20.0227	1.15409
$302$	−11.9348	−0.686769
$303$	0	0
$304$	12.3914	0.710697
$305$	−16.1427	−0.924327
$306$	0	0
$307$	−32.5362	−1.85694	−0.928469	−	0.371409i	$-0.878875\pi$
$307$	−32.5362	−1.85694	−0.928469	+	0.371409i	$0.878875\pi$
$308$	−22.8038	−1.29937
$309$	0	0
$310$	−19.4987	−1.10745
$311$	−18.0824	−1.02536	−0.512681	−	0.858579i	$-0.671347\pi$
$311$	−18.0824	−1.02536	−0.512681	+	0.858579i	$0.671347\pi$
$312$	0	0
$313$	−5.86518	−0.331520	−0.165760	−	0.986166i	$-0.553008\pi$
$313$	−5.86518	−0.331520	−0.165760	+	0.986166i	$0.553008\pi$
$314$	−24.6595	−1.39162
$315$	0	0
$316$	−15.7895	−0.888231
$317$	11.2867	0.633923	0.316961	−	0.948438i	$-0.397337\pi$
$317$	11.2867	0.633923	0.316961	+	0.948438i	$0.397337\pi$
$318$	0	0
$319$	−14.7788	−0.827456
$320$	1.84089	0.102909
$321$	0	0
$322$	13.1674	0.733790
$323$	10.0422	0.558763
$324$	0	0
$325$	2.03885	0.113095
$326$	8.53612	0.472772
$327$	0	0
$328$	−7.73329	−0.427000
$329$	−13.8273	−0.762322
$330$	0	0
$331$	−20.0067	−1.09967	−0.549835	−	0.835273i	$-0.685309\pi$
$331$	−20.0067	−1.09967	−0.549835	+	0.835273i	$0.685309\pi$
$332$	−7.84338	−0.430461
$333$	0	0
$334$	18.4077	1.00722
$335$	20.9163	1.14278
$336$	0	0
$337$	10.5321	0.573718	0.286859	−	0.957973i	$-0.407389\pi$
$337$	10.5321	0.573718	0.286859	+	0.957973i	$0.407389\pi$
$338$	18.4781	1.00508
$339$	0	0
$340$	−7.79274	−0.422621
$341$	−29.7936	−1.61341
$342$	0	0
$343$	27.0657	1.46141
$344$	7.71825	0.416140
$345$	0	0
$346$	15.3709	0.826344
$347$	−26.1523	−1.40393	−0.701965	−	0.712211i	$-0.747694\pi$
$347$	−26.1523	−1.40393	−0.701965	+	0.712211i	$0.747694\pi$
$348$	0	0
$349$	−0.767616	−0.0410895	−0.0205448	−	0.999789i	$-0.506540\pi$
$349$	−0.767616	−0.0410895	−0.0205448	+	0.999789i	$0.506540\pi$
$350$	10.3447	0.552950
$351$	0	0
$352$	−26.4468	−1.40962
$353$	−18.3224	−0.975202	−0.487601	−	0.873067i	$-0.662128\pi$
$353$	−18.3224	−0.975202	−0.487601	+	0.873067i	$0.662128\pi$
$354$	0	0
$355$	−0.796033	−0.0422490
$356$	3.78301	0.200499
$357$	0	0
$358$	−12.9497	−0.684411
$359$	28.1010	1.48311	0.741556	−	0.670891i	$-0.234088\pi$
$359$	28.1010	1.48311	0.741556	+	0.670891i	$0.234088\pi$
$360$	0	0
$361$	−12.8581	−0.676740
$362$	−4.19218	−0.220336
$363$	0	0
$364$	−6.87715	−0.360461
$365$	−17.1715	−0.898795
$366$	0	0
$367$	10.0057	0.522291	0.261145	−	0.965299i	$-0.415900\pi$
$367$	10.0057	0.522291	0.261145	+	0.965299i	$0.415900\pi$
$368$	8.48382	0.442250
$369$	0	0
$370$	−11.7390	−0.610281
$371$	13.4266	0.697077
$372$	0	0
$373$	−1.93039	−0.0999516	−0.0499758	−	0.998750i	$-0.515914\pi$
$373$	−1.93039	−0.0999516	−0.0499758	+	0.998750i	$0.515914\pi$
$374$	−35.6196	−1.84184
$375$	0	0
$376$	−5.33007	−0.274877
$377$	−4.45699	−0.229546
$378$	0	0
$379$	7.87391	0.404456	0.202228	−	0.979339i	$-0.435182\pi$
$379$	7.87391	0.404456	0.202228	+	0.979339i	$0.435182\pi$
$380$	−4.76614	−0.244498
$381$	0	0
$382$	44.7228	2.28822
$383$	3.27765	0.167480	0.0837401	−	0.996488i	$-0.473313\pi$
$383$	3.27765	0.167480	0.0837401	+	0.996488i	$0.473313\pi$
$384$	0	0
$385$	−43.4810	−2.21600
$386$	24.9973	1.27233
$387$	0	0
$388$	10.0718	0.511318
$389$	−7.18434	−0.364260	−0.182130	−	0.983274i	$-0.558299\pi$
$389$	−7.18434	−0.364260	−0.182130	+	0.983274i	$0.558299\pi$
$390$	0	0
$391$	6.87542	0.347705
$392$	22.5141	1.13713
$393$	0	0
$394$	45.0028	2.26721
$395$	−30.1066	−1.51483
$396$	0	0
$397$	−33.6636	−1.68953	−0.844764	−	0.535138i	$-0.820259\pi$
$397$	−33.6636	−1.68953	−0.844764	+	0.535138i	$0.820259\pi$
$398$	8.29208	0.415645
$399$	0	0
$400$	6.66518	0.333259
$401$	7.83739	0.391380	0.195690	−	0.980666i	$-0.437305\pi$
$401$	7.83739	0.391380	0.195690	+	0.980666i	$0.437305\pi$
$402$	0	0
$403$	−8.98513	−0.447581
$404$	−1.98346	−0.0986808
$405$	0	0
$406$	−22.6139	−1.12231
$407$	−17.9369	−0.889100
$408$	0	0
$409$	−9.25057	−0.457411	−0.228706	−	0.973496i	$-0.573449\pi$
$409$	−9.25057	−0.457411	−0.228706	+	0.973496i	$0.573449\pi$
$410$	14.8726	0.734505
$411$	0	0
$412$	14.8999	0.734063
$413$	−9.27294	−0.456291
$414$	0	0
$415$	−14.9553	−0.734127
$416$	−7.97579	−0.391045
$417$	0	0
$418$	−21.7854	−1.06556
$419$	−20.7972	−1.01601	−0.508005	−	0.861354i	$-0.669617\pi$
$419$	−20.7972	−1.01601	−0.508005	+	0.861354i	$0.669617\pi$
$420$	0	0
$421$	1.80019	0.0877359	0.0438679	−	0.999037i	$-0.486032\pi$
$421$	1.80019	0.0877359	0.0438679	+	0.999037i	$0.486032\pi$
$422$	−29.2903	−1.42583
$423$	0	0
$424$	5.17564	0.251351
$425$	5.40156	0.262014
$426$	0	0
$427$	−37.7423	−1.82648
$428$	−16.0581	−0.776196
$429$	0	0
$430$	−14.8437	−0.715825
$431$	−4.24473	−0.204462	−0.102231	−	0.994761i	$-0.532598\pi$
$431$	−4.24473	−0.204462	−0.102231	+	0.994761i	$0.532598\pi$
$432$	0	0
$433$	24.2142	1.16366	0.581830	−	0.813311i	$-0.302337\pi$
$433$	24.2142	1.16366	0.581830	+	0.813311i	$0.302337\pi$
$434$	−45.5889	−2.18834
$435$	0	0
$436$	−5.35662	−0.256536
$437$	4.20509	0.201157
$438$	0	0
$439$	−39.9413	−1.90630	−0.953148	−	0.302504i	$-0.902178\pi$
$439$	−39.9413	−1.90630	−0.953148	+	0.302504i	$0.902178\pi$
$440$	−16.7609	−0.799043
$441$	0	0
$442$	−10.7421	−0.510950
$443$	15.0166	0.713462	0.356731	−	0.934207i	$-0.383891\pi$
$443$	15.0166	0.713462	0.356731	+	0.934207i	$0.383891\pi$
$444$	0	0
$445$	7.21324	0.341940
$446$	−29.4499	−1.39449
$447$	0	0
$448$	4.30408	0.203349
$449$	18.1292	0.855568	0.427784	−	0.903881i	$-0.359294\pi$
$449$	18.1292	0.855568	0.427784	+	0.903881i	$0.359294\pi$
$450$	0	0
$451$	22.7250	1.07008
$452$	2.94535	0.138538
$453$	0	0
$454$	−13.8129	−0.648271
$455$	−13.1130	−0.614745
$456$	0	0
$457$	−4.26540	−0.199527	−0.0997635	−	0.995011i	$-0.531809\pi$
$457$	−4.26540	−0.199527	−0.0997635	+	0.995011i	$0.531809\pi$
$458$	−32.8670	−1.53577
$459$	0	0
$460$	−3.26315	−0.152145
$461$	−4.28290	−0.199474	−0.0997372	−	0.995014i	$-0.531800\pi$
$461$	−4.28290	−0.199474	−0.0997372	+	0.995014i	$0.531800\pi$
$462$	0	0
$463$	−18.0384	−0.838314	−0.419157	−	0.907914i	$-0.637674\pi$
$463$	−18.0384	−0.838314	−0.419157	+	0.907914i	$0.637674\pi$
$464$	−14.5703	−0.676407
$465$	0	0
$466$	39.6687	1.83762
$467$	25.4015	1.17544	0.587720	−	0.809064i	$-0.300026\pi$
$467$	25.4015	1.17544	0.587720	+	0.809064i	$0.300026\pi$
$468$	0	0
$469$	48.9032	2.25814
$470$	10.2507	0.472831
$471$	0	0
$472$	−3.57449	−0.164529
$473$	−22.6808	−1.04286
$474$	0	0
$475$	3.30367	0.151583
$476$	−18.2198	−0.835101
$477$	0	0
$478$	−5.59884	−0.256085
$479$	−12.7273	−0.581523	−0.290762	−	0.956796i	$-0.593909\pi$
$479$	−12.7273	−0.581523	−0.290762	+	0.956796i	$0.593909\pi$
$480$	0	0
$481$	−5.40940	−0.246647
$482$	−4.88398	−0.222459
$483$	0	0
$484$	14.7839	0.671994
$485$	19.2043	0.872024
$486$	0	0
$487$	22.0924	1.00110	0.500551	−	0.865707i	$-0.333131\pi$
$487$	22.0924	1.00110	0.500551	+	0.865707i	$0.333131\pi$
$488$	−14.5487	−0.658590
$489$	0	0
$490$	−43.2988	−1.95604
$491$	40.6096	1.83269	0.916343	−	0.400393i	$-0.131127\pi$
$491$	40.6096	1.83269	0.916343	+	0.400393i	$0.131127\pi$
$492$	0	0
$493$	−11.8080	−0.531804
$494$	−6.57002	−0.295599
$495$	0	0
$496$	−29.3731	−1.31889
$497$	−1.86116	−0.0834844
$498$	0	0
$499$	−5.80018	−0.259652	−0.129826	−	0.991537i	$-0.541442\pi$
$499$	−5.80018	−0.259652	−0.129826	+	0.991537i	$0.541442\pi$
$500$	−12.1794	−0.544680
$501$	0	0
$502$	−12.5938	−0.562087
$503$	39.1256	1.74452	0.872261	−	0.489040i	$-0.162653\pi$
$503$	39.1256	1.74452	0.872261	+	0.489040i	$0.162653\pi$
$504$	0	0
$505$	−3.78195	−0.168294
$506$	−14.9154	−0.663071
$507$	0	0
$508$	−10.6788	−0.473796
$509$	1.18079	0.0523378	0.0261689	−	0.999658i	$-0.491669\pi$
$509$	1.18079	0.0523378	0.0261689	+	0.999658i	$0.491669\pi$
$510$	0	0
$511$	−40.1476	−1.77603
$512$	−8.81511	−0.389577
$513$	0	0
$514$	14.8511	0.655052
$515$	28.4102	1.25190
$516$	0	0
$517$	15.6629	0.688853
$518$	−27.4463	−1.20592
$519$	0	0
$520$	−5.05473	−0.221664
$521$	18.0666	0.791512	0.395756	−	0.918356i	$-0.370483\pi$
$521$	18.0666	0.791512	0.395756	+	0.918356i	$0.370483\pi$
$522$	0	0
$523$	−21.7525	−0.951171	−0.475585	−	0.879670i	$-0.657764\pi$
$523$	−21.7525	−0.951171	−0.475585	+	0.879670i	$0.657764\pi$
$524$	11.3761	0.496966
$525$	0	0
$526$	30.5153	1.33053
$527$	−23.8045	−1.03694
$528$	0	0
$529$	−20.1210	−0.874825
$530$	−9.95374	−0.432363
$531$	0	0
$532$	−11.1434	−0.483129
$533$	6.85338	0.296853
$534$	0	0
$535$	−30.6186	−1.32376
$536$	18.8510	0.814239
$537$	0	0
$538$	14.7126	0.634306
$539$	−66.1596	−2.84970
$540$	0	0
$541$	9.57243	0.411551	0.205775	−	0.978599i	$-0.434028\pi$
$541$	9.57243	0.411551	0.205775	+	0.978599i	$0.434028\pi$
$542$	1.05215	0.0451936
$543$	0	0
$544$	−21.1304	−0.905958
$545$	−10.2137	−0.437507
$546$	0	0
$547$	−42.1656	−1.80287	−0.901436	−	0.432912i	$-0.857486\pi$
$547$	−42.1656	−1.80287	−0.901436	+	0.432912i	$0.857486\pi$
$548$	−17.4760	−0.746537
$549$	0	0
$550$	−11.7181	−0.499660
$551$	−7.22191	−0.307664
$552$	0	0
$553$	−70.3905	−2.99331
$554$	13.5417	0.575332
$555$	0	0
$556$	−13.5649	−0.575281
$557$	12.9322	0.547955	0.273978	−	0.961736i	$-0.411661\pi$
$557$	12.9322	0.547955	0.273978	+	0.961736i	$0.411661\pi$
$558$	0	0
$559$	−6.84004	−0.289303
$560$	−42.8674	−1.81148
$561$	0	0
$562$	−33.8883	−1.42949
$563$	−18.0093	−0.759001	−0.379501	−	0.925192i	$-0.623904\pi$
$563$	−18.0093	−0.759001	−0.379501	+	0.925192i	$0.623904\pi$
$564$	0	0
$565$	5.61602	0.236268
$566$	46.7981	1.96707
$567$	0	0
$568$	−0.717431	−0.0301027
$569$	31.1659	1.30654	0.653272	−	0.757124i	$-0.273396\pi$
$569$	31.1659	1.30654	0.653272	+	0.757124i	$0.273396\pi$
$570$	0	0
$571$	16.1290	0.674976	0.337488	−	0.941330i	$-0.390423\pi$
$571$	16.1290	0.674976	0.337488	+	0.941330i	$0.390423\pi$
$572$	7.79012	0.325721
$573$	0	0
$574$	34.7727	1.45139
$575$	2.26186	0.0943262
$576$	0	0
$577$	23.0274	0.958644	0.479322	−	0.877639i	$-0.340883\pi$
$577$	23.0274	0.958644	0.479322	+	0.877639i	$0.340883\pi$
$578$	1.00666	0.0418714
$579$	0	0
$580$	5.60419	0.232701
$581$	−34.9661	−1.45064
$582$	0	0
$583$	−15.2091	−0.629896
$584$	−15.4759	−0.640398
$585$	0	0
$586$	12.9693	0.535757
$587$	16.1468	0.666452	0.333226	−	0.942847i	$-0.391863\pi$
$587$	16.1468	0.666452	0.333226	+	0.942847i	$0.391863\pi$
$588$	0	0
$589$	−14.5591	−0.599898
$590$	6.87442	0.283015
$591$	0	0
$592$	−17.6838	−0.726799
$593$	−24.5852	−1.00959	−0.504796	−	0.863239i	$-0.668432\pi$
$593$	−24.5852	−1.00959	−0.504796	+	0.863239i	$0.668432\pi$
$594$	0	0
$595$	−34.7404	−1.42422
$596$	5.36074	0.219584
$597$	0	0
$598$	−4.49818	−0.183944
$599$	46.7153	1.90874	0.954368	−	0.298634i	$-0.0965309\pi$
$599$	46.7153	1.90874	0.954368	+	0.298634i	$0.0965309\pi$
$600$	0	0
$601$	28.4459	1.16033	0.580165	−	0.814499i	$-0.302988\pi$
$601$	28.4459	1.16033	0.580165	+	0.814499i	$0.302988\pi$
$602$	−34.7051	−1.41447
$603$	0	0
$604$	6.91518	0.281375
$605$	28.1891	1.14605
$606$	0	0
$607$	0.916476	0.0371986	0.0185993	−	0.999827i	$-0.494079\pi$
$607$	0.916476	0.0371986	0.0185993	+	0.999827i	$0.494079\pi$
$608$	−12.9236	−0.524122
$609$	0	0
$610$	27.9799	1.13287
$611$	4.72360	0.191096
$612$	0	0
$613$	−3.56201	−0.143868	−0.0719341	−	0.997409i	$-0.522917\pi$
$613$	−3.56201	−0.143868	−0.0719341	+	0.997409i	$0.522917\pi$
$614$	56.3947	2.27590
$615$	0	0
$616$	−39.1876	−1.57891
$617$	12.6206	0.508087	0.254044	−	0.967193i	$-0.418239\pi$
$617$	12.6206	0.508087	0.254044	+	0.967193i	$0.418239\pi$
$618$	0	0
$619$	−34.0824	−1.36989	−0.684943	−	0.728597i	$-0.740173\pi$
$619$	−34.0824	−1.36989	−0.684943	+	0.728597i	$0.740173\pi$
$620$	11.2979	0.453733
$621$	0	0
$622$	31.3421	1.25670
$623$	16.8649	0.675676
$624$	0	0
$625$	−16.5578	−0.662312
$626$	10.1661	0.406318
$627$	0	0
$628$	14.2881	0.570156
$629$	−14.3312	−0.571423
$630$	0	0
$631$	−7.02385	−0.279615	−0.139808	−	0.990179i	$-0.544648\pi$
$631$	−7.02385	−0.279615	−0.139808	+	0.990179i	$0.544648\pi$
$632$	−27.1338	−1.07933
$633$	0	0
$634$	−19.5631	−0.776949
$635$	−20.3617	−0.808031
$636$	0	0
$637$	−19.9524	−0.790541
$638$	25.6160	1.01415
$639$	0	0
$640$	−23.1625	−0.915577
$641$	−23.4866	−0.927666	−0.463833	−	0.885923i	$-0.653526\pi$
$641$	−23.4866	−0.927666	−0.463833	+	0.885923i	$0.653526\pi$
$642$	0	0
$643$	−36.3516	−1.43357	−0.716784	−	0.697295i	$-0.754387\pi$
$643$	−36.3516	−1.43357	−0.716784	+	0.697295i	$0.754387\pi$
$644$	−7.62938	−0.300640
$645$	0	0
$646$	−17.4061	−0.684832
$647$	−46.1669	−1.81501	−0.907505	−	0.420042i	$-0.862015\pi$
$647$	−46.1669	−1.81501	−0.907505	+	0.420042i	$0.862015\pi$
$648$	0	0
$649$	10.5040	0.412316
$650$	−3.53392	−0.138612
$651$	0	0
$652$	−4.94595	−0.193698
$653$	18.7022	0.731872	0.365936	−	0.930640i	$-0.380749\pi$
$653$	18.7022	0.731872	0.365936	+	0.930640i	$0.380749\pi$
$654$	0	0
$655$	21.6913	0.847548
$656$	22.4043	0.874739
$657$	0	0
$658$	23.9667	0.934317
$659$	−16.7988	−0.654389	−0.327194	−	0.944957i	$-0.606103\pi$
$659$	−16.7988	−0.654389	−0.327194	+	0.944957i	$0.606103\pi$
$660$	0	0
$661$	29.8636	1.16156	0.580779	−	0.814061i	$-0.302748\pi$
$661$	29.8636	1.16156	0.580779	+	0.814061i	$0.302748\pi$
$662$	34.6775	1.34778
$663$	0	0
$664$	−13.4786	−0.523071
$665$	−21.2477	−0.823950
$666$	0	0
$667$	−4.94449	−0.191452
$668$	−10.6657	−0.412668
$669$	0	0
$670$	−36.2540	−1.40062
$671$	42.7527	1.65045
$672$	0	0
$673$	7.82858	0.301770	0.150885	−	0.988551i	$-0.451788\pi$
$673$	7.82858	0.301770	0.150885	+	0.988551i	$0.451788\pi$
$674$	−18.2551	−0.703161
$675$	0	0
$676$	−10.7065	−0.411788
$677$	36.4551	1.40108	0.700542	−	0.713611i	$-0.252942\pi$
$677$	36.4551	1.40108	0.700542	+	0.713611i	$0.252942\pi$
$678$	0	0
$679$	44.9005	1.72312
$680$	−13.3916	−0.513544
$681$	0	0
$682$	51.6410	1.97744
$683$	33.2955	1.27402	0.637008	−	0.770857i	$-0.280172\pi$
$683$	33.2955	1.27402	0.637008	+	0.770857i	$0.280172\pi$
$684$	0	0
$685$	−33.3222	−1.27318
$686$	−46.9126	−1.79113
$687$	0	0
$688$	−22.3607	−0.852493
$689$	−4.58674	−0.174741
$690$	0	0
$691$	−18.0893	−0.688151	−0.344075	−	0.938942i	$-0.611808\pi$
$691$	−18.0893	−0.688151	−0.344075	+	0.938942i	$0.611808\pi$
$692$	−8.90612	−0.338560
$693$	0	0
$694$	45.3296	1.72069
$695$	−25.8648	−0.981109
$696$	0	0
$697$	18.1568	0.687737
$698$	1.33050	0.0503602
$699$	0	0
$700$	−5.99390	−0.226548
$701$	49.1435	1.85612	0.928062	−	0.372427i	$-0.121474\pi$
$701$	49.1435	1.85612	0.928062	+	0.372427i	$0.121474\pi$
$702$	0	0
$703$	−8.76515	−0.330584
$704$	−4.87547	−0.183751
$705$	0	0
$706$	31.7580	1.19523
$707$	−8.84235	−0.332551
$708$	0	0
$709$	15.2421	0.572430	0.286215	−	0.958165i	$-0.407603\pi$
$709$	15.2421	0.572430	0.286215	+	0.958165i	$0.407603\pi$
$710$	1.37976	0.0517813
$711$	0	0
$712$	6.50099	0.243635
$713$	−9.96793	−0.373302
$714$	0	0
$715$	14.8538	0.555499
$716$	7.50322	0.280409
$717$	0	0
$718$	−48.7071	−1.81773
$719$	4.44369	0.165722	0.0828609	−	0.996561i	$-0.473594\pi$
$719$	4.44369	0.165722	0.0828609	+	0.996561i	$0.473594\pi$
$720$	0	0
$721$	66.4243	2.47377
$722$	22.2867	0.829427
$723$	0	0
$724$	2.42901	0.0902734
$725$	−3.88456	−0.144269
$726$	0	0
$727$	52.3928	1.94314	0.971571	−	0.236748i	$-0.0760816\pi$
$727$	52.3928	1.94314	0.971571	+	0.236748i	$0.0760816\pi$
$728$	−11.8182	−0.438011
$729$	0	0
$730$	29.7631	1.10158
$731$	−18.1214	−0.670246
$732$	0	0
$733$	21.4454	0.792105	0.396053	−	0.918228i	$-0.370380\pi$
$733$	21.4454	0.792105	0.396053	+	0.918228i	$0.370380\pi$
$734$	−17.3427	−0.640131
$735$	0	0
$736$	−8.84818	−0.326148
$737$	−55.3953	−2.04051
$738$	0	0
$739$	15.2159	0.559725	0.279863	−	0.960040i	$-0.409711\pi$
$739$	15.2159	0.559725	0.279863	+	0.960040i	$0.409711\pi$
$740$	6.80175	0.250037
$741$	0	0
$742$	−23.2723	−0.854352
$743$	27.7392	1.01765	0.508827	−	0.860869i	$-0.330079\pi$
$743$	27.7392	1.01765	0.508827	+	0.860869i	$0.330079\pi$
$744$	0	0
$745$	10.2216	0.374489
$746$	3.34592	0.122503
$747$	0	0
$748$	20.6385	0.754619
$749$	−71.5877	−2.61576
$750$	0	0
$751$	−30.6207	−1.11737	−0.558683	−	0.829381i	$-0.688693\pi$
$751$	−30.6207	−1.11737	−0.558683	+	0.829381i	$0.688693\pi$
$752$	15.4418	0.563106
$753$	0	0
$754$	7.72525	0.281337
$755$	13.1855	0.479869
$756$	0	0
$757$	−20.7098	−0.752711	−0.376355	−	0.926475i	$-0.622823\pi$
$757$	−20.7098	−0.752711	−0.376355	+	0.926475i	$0.622823\pi$
$758$	−13.6478	−0.495709
$759$	0	0
$760$	−8.19046	−0.297099
$761$	−47.7400	−1.73057	−0.865286	−	0.501278i	$-0.832863\pi$
$761$	−47.7400	−1.73057	−0.865286	+	0.501278i	$0.832863\pi$
$762$	0	0
$763$	−23.8801	−0.864516
$764$	−25.9131	−0.937502
$765$	0	0
$766$	−5.68112	−0.205267
$767$	3.16777	0.114382
$768$	0	0
$769$	−28.2142	−1.01743	−0.508715	−	0.860935i	$-0.669879\pi$
$769$	−28.2142	−1.01743	−0.508715	+	0.860935i	$0.669879\pi$
$770$	75.3652	2.71597
$771$	0	0
$772$	−14.4838	−0.521284
$773$	−41.6866	−1.49936	−0.749681	−	0.661800i	$-0.769793\pi$
$773$	−41.6866	−1.49936	−0.749681	+	0.661800i	$0.769793\pi$
$774$	0	0
$775$	−7.83115	−0.281303
$776$	17.3081	0.621323
$777$	0	0
$778$	12.4525	0.446445
$779$	11.1049	0.397875
$780$	0	0
$781$	2.10823	0.0754386
$782$	−11.9171	−0.426154
$783$	0	0
$784$	−65.2259	−2.32950
$785$	27.2437	0.972369
$786$	0	0
$787$	12.6241	0.449999	0.225000	−	0.974359i	$-0.427762\pi$
$787$	12.6241	0.449999	0.225000	+	0.974359i	$0.427762\pi$
$788$	−26.0753	−0.928894
$789$	0	0
$790$	52.1835	1.85661
$791$	13.1305	0.466867
$792$	0	0
$793$	12.8933	0.457855
$794$	58.3488	2.07072
$795$	0	0
$796$	−4.80456	−0.170293
$797$	26.0733	0.923564	0.461782	−	0.886993i	$-0.347210\pi$
$797$	26.0733	0.923564	0.461782	+	0.886993i	$0.347210\pi$
$798$	0	0
$799$	12.5143	0.442724
$800$	−6.95144	−0.245770
$801$	0	0
$802$	−13.5845	−0.479684
$803$	45.4774	1.60486
$804$	0	0
$805$	−14.5473	−0.512724
$806$	15.5738	0.548565
$807$	0	0
$808$	−3.40851	−0.119911
$809$	22.3622	0.786213	0.393107	−	0.919493i	$-0.371400\pi$
$809$	22.3622	0.786213	0.393107	+	0.919493i	$0.371400\pi$
$810$	0	0
$811$	−41.9608	−1.47344	−0.736722	−	0.676196i	$-0.763627\pi$
$811$	−41.9608	−1.47344	−0.736722	+	0.676196i	$0.763627\pi$
$812$	13.1028	0.459819
$813$	0	0
$814$	31.0899	1.08970
$815$	−9.43066	−0.330342
$816$	0	0
$817$	−11.0833	−0.387756
$818$	16.0339	0.560613
$819$	0	0
$820$	−8.61740	−0.300933
$821$	21.2478	0.741554	0.370777	−	0.928722i	$-0.379091\pi$
$821$	21.2478	0.741554	0.370777	+	0.928722i	$0.379091\pi$
$822$	0	0
$823$	−41.4444	−1.44466	−0.722331	−	0.691547i	$-0.756929\pi$
$823$	−41.4444	−1.44466	−0.722331	+	0.691547i	$0.756929\pi$
$824$	25.6049	0.891990
$825$	0	0
$826$	16.0727	0.559240
$827$	40.8206	1.41947	0.709736	−	0.704467i	$-0.248814\pi$
$827$	40.8206	1.41947	0.709736	+	0.704467i	$0.248814\pi$
$828$	0	0
$829$	−30.9187	−1.07385	−0.536925	−	0.843630i	$-0.680414\pi$
$829$	−30.9187	−1.07385	−0.536925	+	0.843630i	$0.680414\pi$
$830$	25.9219	0.899761
$831$	0	0
$832$	−1.47034	−0.0509748
$833$	−52.8601	−1.83149
$834$	0	0
$835$	−20.3367	−0.703781
$836$	12.6228	0.436568
$837$	0	0
$838$	36.0476	1.24524
$839$	−13.9778	−0.482567	−0.241284	−	0.970455i	$-0.577568\pi$
$839$	−13.9778	−0.482567	−0.241284	+	0.970455i	$0.577568\pi$
$840$	0	0
$841$	−20.5082	−0.707181
$842$	−3.12025	−0.107531
$843$	0	0
$844$	16.9712	0.584174
$845$	−20.4145	−0.702281
$846$	0	0
$847$	65.9072	2.26460
$848$	−14.9944	−0.514911
$849$	0	0
$850$	−9.36248	−0.321130
$851$	−6.00108	−0.205714
$852$	0	0
$853$	24.8130	0.849581	0.424791	−	0.905292i	$-0.360348\pi$
$853$	24.8130	0.849581	0.424791	+	0.905292i	$0.360348\pi$
$854$	65.4183	2.23857
$855$	0	0
$856$	−27.5953	−0.943187
$857$	0.608737	0.0207941	0.0103970	−	0.999946i	$-0.496690\pi$
$857$	0.608737	0.0207941	0.0103970	+	0.999946i	$0.496690\pi$
$858$	0	0
$859$	−22.5649	−0.769905	−0.384953	−	0.922936i	$-0.625782\pi$
$859$	−22.5649	−0.769905	−0.384953	+	0.922936i	$0.625782\pi$
$860$	8.60063	0.293279
$861$	0	0
$862$	7.35736	0.250593
$863$	27.0122	0.919506	0.459753	−	0.888047i	$-0.347938\pi$
$863$	27.0122	0.919506	0.459753	+	0.888047i	$0.347938\pi$
$864$	0	0
$865$	−16.9817	−0.577394
$866$	−41.9702	−1.42621
$867$	0	0
$868$	26.4149	0.896579
$869$	79.7352	2.70483
$870$	0	0
$871$	−16.7061	−0.566064
$872$	−9.20518	−0.311727
$873$	0	0
$874$	−7.28865	−0.246542
$875$	−54.2964	−1.83555
$876$	0	0
$877$	18.2291	0.615552	0.307776	−	0.951459i	$-0.400415\pi$
$877$	18.2291	0.615552	0.307776	+	0.951459i	$0.400415\pi$
$878$	69.2299	2.33640
$879$	0	0
$880$	48.5582	1.63690
$881$	29.0952	0.980243	0.490122	−	0.871654i	$-0.336952\pi$
$881$	29.0952	0.980243	0.490122	+	0.871654i	$0.336952\pi$
$882$	0	0
$883$	6.45423	0.217202	0.108601	−	0.994085i	$-0.465363\pi$
$883$	6.45423	0.217202	0.108601	+	0.994085i	$0.465363\pi$
$884$	6.22414	0.209341
$885$	0	0
$886$	−26.0282	−0.874434
$887$	−0.671837	−0.0225581	−0.0112790	−	0.999936i	$-0.503590\pi$
$887$	−0.671837	−0.0225581	−0.0112790	+	0.999936i	$0.503590\pi$
$888$	0	0
$889$	−47.6066	−1.59668
$890$	−12.5026	−0.419089
$891$	0	0
$892$	17.0637	0.571334
$893$	7.65391	0.256128
$894$	0	0
$895$	14.3067	0.478221
$896$	−54.1548	−1.80919
$897$	0	0
$898$	−31.4231	−1.04860
$899$	17.1191	0.570954
$900$	0	0
$901$	−12.1517	−0.404833
$902$	−39.3890	−1.31151
$903$	0	0
$904$	5.06149	0.168343
$905$	4.63150	0.153956
$906$	0	0
$907$	0.918046	0.0304832	0.0152416	−	0.999884i	$-0.495148\pi$
$907$	0.918046	0.0304832	0.0152416	+	0.999884i	$0.495148\pi$
$908$	8.00340	0.265602
$909$	0	0
$910$	22.7286	0.753445
$911$	49.1125	1.62717	0.813584	−	0.581447i	$-0.197513\pi$
$911$	49.1125	1.62717	0.813584	+	0.581447i	$0.197513\pi$
$912$	0	0
$913$	39.6080	1.31083
$914$	7.39318	0.244545
$915$	0	0
$916$	19.0436	0.629219
$917$	50.7151	1.67476
$918$	0	0
$919$	22.4857	0.741734	0.370867	−	0.928686i	$-0.379060\pi$
$919$	22.4857	0.741734	0.370867	+	0.928686i	$0.379060\pi$
$920$	−5.60762	−0.184878
$921$	0	0
$922$	7.42351	0.244480
$923$	0.635800	0.0209276
$924$	0	0
$925$	−4.71465	−0.155017
$926$	31.2657	1.02746
$927$	0	0
$928$	15.1960	0.498834
$929$	−36.4334	−1.19534	−0.597670	−	0.801742i	$-0.703907\pi$
$929$	−36.4334	−1.19534	−0.597670	+	0.801742i	$0.703907\pi$
$930$	0	0
$931$	−32.3299	−1.05957
$932$	−22.9846	−0.752887
$933$	0	0
$934$	−44.0281	−1.44064
$935$	39.3523	1.28696
$936$	0	0
$937$	−41.7231	−1.36303	−0.681517	−	0.731803i	$-0.738679\pi$
$937$	−41.7231	−1.36303	−0.681517	+	0.731803i	$0.738679\pi$
$938$	−84.7635	−2.76763
$939$	0	0
$940$	−5.93943	−0.193723
$941$	−39.6227	−1.29166	−0.645832	−	0.763480i	$-0.723489\pi$
$941$	−39.6227	−1.29166	−0.645832	+	0.763480i	$0.723489\pi$
$942$	0	0
$943$	7.60300	0.247588
$944$	10.3557	0.337050
$945$	0	0
$946$	39.3123	1.27815
$947$	−26.8873	−0.873721	−0.436861	−	0.899529i	$-0.643910\pi$
$947$	−26.8873	−0.873721	−0.436861	+	0.899529i	$0.643910\pi$
$948$	0	0
$949$	13.7150	0.445209
$950$	−5.72621	−0.185783
$951$	0	0
$952$	−31.3101	−1.01477
$953$	−25.8927	−0.838746	−0.419373	−	0.907814i	$-0.637750\pi$
$953$	−25.8927	−0.838746	−0.419373	+	0.907814i	$0.637750\pi$
$954$	0	0
$955$	−49.4096	−1.59886
$956$	3.24405	0.104920
$957$	0	0
$958$	22.0600	0.712727
$959$	−77.9088	−2.51581
$960$	0	0
$961$	3.51153	0.113275
$962$	9.37606	0.302296
$963$	0	0
$964$	2.82985	0.0911433
$965$	−27.6169	−0.889020
$966$	0	0
$967$	15.6657	0.503775	0.251887	−	0.967757i	$-0.418949\pi$
$967$	15.6657	0.503775	0.251887	+	0.967757i	$0.418949\pi$
$968$	25.4056	0.816568
$969$	0	0
$970$	−33.2867	−1.06877
$971$	44.9509	1.44254	0.721272	−	0.692652i	$-0.243558\pi$
$971$	44.9509	1.44254	0.721272	+	0.692652i	$0.243558\pi$
$972$	0	0
$973$	−60.4731	−1.93868
$974$	−38.2925	−1.22697
$975$	0	0
$976$	42.1493	1.34917
$977$	−38.0230	−1.21646	−0.608232	−	0.793759i	$-0.708121\pi$
$977$	−38.0230	−1.21646	−0.608232	+	0.793759i	$0.708121\pi$
$978$	0	0
$979$	−19.1037	−0.610558
$980$	25.0880	0.801406
$981$	0	0
$982$	−70.3883	−2.24618
$983$	−17.6253	−0.562161	−0.281081	−	0.959684i	$-0.590693\pi$
$983$	−17.6253	−0.562161	−0.281081	+	0.959684i	$0.590693\pi$
$984$	0	0
$985$	−49.7189	−1.58418
$986$	20.4666	0.651790
$987$	0	0
$988$	3.80677	0.121109
$989$	−7.58821	−0.241291
$990$	0	0
$991$	−20.9899	−0.666766	−0.333383	−	0.942792i	$-0.608190\pi$
$991$	−20.9899	−0.666766	−0.333383	+	0.942792i	$0.608190\pi$
$992$	30.6347	0.972652
$993$	0	0
$994$	3.22593	0.102320
$995$	−9.16105	−0.290425
$996$	0	0
$997$	−19.8464	−0.628540	−0.314270	−	0.949334i	$-0.601760\pi$
$997$	−19.8464	−0.628540	−0.314270	+	0.949334i	$0.601760\pi$
$998$	10.0534	0.318234
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.16		72
3.2	odd	2		6561.2.a.d.1.57		72
81.4	even	27		729.2.g.c.136.2		144
81.7	even	27		729.2.g.d.352.7		144
81.20	odd	54		729.2.g.b.595.7		144
81.23	odd	54		729.2.g.a.379.2		144
81.31	even	27		81.2.g.a.70.7	yes	144
81.34	even	27		81.2.g.a.22.7	✓	144
81.47	odd	54		243.2.g.a.199.2		144
81.50	odd	54		243.2.g.a.127.2		144
81.58	even	27		729.2.g.d.379.7		144
81.61	even	27		729.2.g.c.595.2		144
81.74	odd	54		729.2.g.a.352.2		144
81.77	odd	54		729.2.g.b.136.7		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.22.7	✓	144	81.34	even	27
81.2.g.a.70.7	yes	144	81.31	even	27
243.2.g.a.127.2		144	81.50	odd	54
243.2.g.a.199.2		144	81.47	odd	54
729.2.g.a.352.2		144	81.74	odd	54
729.2.g.a.379.2		144	81.23	odd	54
729.2.g.b.136.7		144	81.77	odd	54
729.2.g.b.595.7		144	81.20	odd	54
729.2.g.c.136.2		144	81.4	even	27
729.2.g.c.595.2		144	81.61	even	27
729.2.g.d.352.7		144	81.7	even	27
729.2.g.d.379.7		144	81.58	even	27
6561.2.a.c.1.16		72	1.1	even	1	trivial
6561.2.a.d.1.57		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.73329 q^{2} +1.00429 q^{4} +1.91493 q^{5} +4.47719 q^{7} +1.72585 q^{8} +O(q^{10})\)	\(q-1.73329 q^{2} +1.00429 q^{4} +1.91493 q^{5} +4.47719 q^{7} +1.72585 q^{8} -3.31913 q^{10} -5.07155 q^{11} -1.52947 q^{13} -7.76027 q^{14} -4.99998 q^{16} -4.05206 q^{17} -2.47829 q^{19} +1.92315 q^{20} +8.79047 q^{22} -1.69677 q^{23} -1.33304 q^{25} +2.65102 q^{26} +4.49642 q^{28} +2.91406 q^{29} +5.87465 q^{31} +5.21472 q^{32} +7.02340 q^{34} +8.57351 q^{35} +3.53677 q^{37} +4.29560 q^{38} +3.30488 q^{40} -4.48087 q^{41} +4.47215 q^{43} -5.09333 q^{44} +2.94099 q^{46} -3.08838 q^{47} +13.0452 q^{49} +2.31055 q^{50} -1.53604 q^{52} +2.99890 q^{53} -9.71168 q^{55} +7.72695 q^{56} -5.05092 q^{58} -2.07115 q^{59} -8.42990 q^{61} -10.1825 q^{62} +0.961336 q^{64} -2.92884 q^{65} +10.9228 q^{67} -4.06946 q^{68} -14.8604 q^{70} -0.415698 q^{71} -8.96715 q^{73} -6.13024 q^{74} -2.48894 q^{76} -22.7063 q^{77} -15.7220 q^{79} -9.57462 q^{80} +7.76664 q^{82} -7.80984 q^{83} -7.75942 q^{85} -7.75154 q^{86} -8.75273 q^{88} +3.76684 q^{89} -6.84775 q^{91} -1.70406 q^{92} +5.35306 q^{94} -4.74576 q^{95} +10.0287 q^{97} -22.6112 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

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