LMFDB - Embedded newform 891.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("891.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.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:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.519120$ of defining polynomial
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73051 q^{2} +5.45571 q^{4} -0.936586 q^{5} -0.519120 q^{7} -9.43585 q^{8} +O(q^{10})$	\(q-2.73051 q^{2} +5.45571 q^{4} -0.936586 q^{5} -0.519120 q^{7} -9.43585 q^{8} +2.55736 q^{10} +1.00000 q^{11} -4.70534 q^{13} +1.41747 q^{14} +14.8533 q^{16} -2.69227 q^{17} +3.41747 q^{19} -5.10974 q^{20} -2.73051 q^{22} +6.97483 q^{23} -4.12281 q^{25} +12.8480 q^{26} -2.83217 q^{28} +4.18622 q^{29} +5.18622 q^{31} -21.6855 q^{32} +7.35129 q^{34} +0.486201 q^{35} +2.06874 q^{37} -9.33144 q^{38} +8.83749 q^{40} +0.173153 q^{41} -2.26949 q^{43} +5.45571 q^{44} -19.0449 q^{46} -0.307727 q^{47} -6.73051 q^{49} +11.2574 q^{50} -25.6710 q^{52} -1.89835 q^{53} -0.936586 q^{55} +4.89835 q^{56} -11.4305 q^{58} +3.97483 q^{59} +4.51912 q^{61} -14.1610 q^{62} +29.5059 q^{64} +4.40696 q^{65} +3.37646 q^{67} -14.6883 q^{68} -1.32758 q^{70} +2.90367 q^{71} +9.52444 q^{73} -5.64871 q^{74} +18.6447 q^{76} -0.519120 q^{77} -2.04356 q^{79} -13.9114 q^{80} -0.472796 q^{82} +14.0594 q^{83} +2.52155 q^{85} +6.19686 q^{86} -9.43585 q^{88} +7.53751 q^{89} +2.44264 q^{91} +38.0526 q^{92} +0.840253 q^{94} -3.20075 q^{95} +16.3342 q^{97} +18.3778 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7}+O(q^{10}) $ 4 * q + q^2 + 11 * q^4 + 4 * q^5 + q^7	$ 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7} + q^{10} + 4 q^{11} + 7 q^{13} + q^{14} + 17 q^{16} - 5 q^{17} + 9 q^{19} - 10 q^{20} + q^{22} + 14 q^{23} + 14 q^{25} + 22 q^{26} - q^{28} - 6 q^{29} - 2 q^{31} - 34 q^{32} + 16 q^{34} - 8 q^{35} + 3 q^{37} + 3 q^{38} + 12 q^{40} - 2 q^{41} - 21 q^{43} + 11 q^{44} + 2 q^{46} - 7 q^{47} - 15 q^{49} + 23 q^{50} - 10 q^{52} - 6 q^{53} + 4 q^{55} + 18 q^{56} - 21 q^{58} + 2 q^{59} + 15 q^{61} - 20 q^{62} + 16 q^{64} + 19 q^{65} + 14 q^{67} - 7 q^{68} - 38 q^{70} - 3 q^{71} + 22 q^{73} - 36 q^{74} + 42 q^{76} + q^{77} + 11 q^{79} - 34 q^{80} - 17 q^{82} + 18 q^{83} + 13 q^{85} - 24 q^{86} - 6 q^{89} + 19 q^{91} + 67 q^{92} - 19 q^{94} - 30 q^{95} + 26 q^{97} + 15 q^{98}+O(q^{100}) $ 4 * q + q^2 + 11 * q^4 + 4 * q^5 + q^7 + q^10 + 4 * q^11 + 7 * q^13 + q^14 + 17 * q^16 - 5 * q^17 + 9 * q^19 - 10 * q^20 + q^22 + 14 * q^23 + 14 * q^25 + 22 * q^26 - q^28 - 6 * q^29 - 2 * q^31 - 34 * q^32 + 16 * q^34 - 8 * q^35 + 3 * q^37 + 3 * q^38 + 12 * q^40 - 2 * q^41 - 21 * q^43 + 11 * q^44 + 2 * q^46 - 7 * q^47 - 15 * q^49 + 23 * q^50 - 10 * q^52 - 6 * q^53 + 4 * q^55 + 18 * q^56 - 21 * q^58 + 2 * q^59 + 15 * q^61 - 20 * q^62 + 16 * q^64 + 19 * q^65 + 14 * q^67 - 7 * q^68 - 38 * q^70 - 3 * q^71 + 22 * q^73 - 36 * q^74 + 42 * q^76 + q^77 + 11 * q^79 - 34 * q^80 - 17 * q^82 + 18 * q^83 + 13 * q^85 - 24 * q^86 - 6 * q^89 + 19 * q^91 + 67 * q^92 - 19 * q^94 - 30 * q^95 + 26 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.73051	−1.93076	−0.965382	−	0.260838i	$-0.916001\pi$
$2$	−2.73051	−1.93076	−0.965382	+	0.260838i	$0.916001\pi$
$3$	0	0
$3$	0	0
$4$	5.45571	2.72785
$5$	−0.936586	−0.418854	−0.209427	−	0.977824i	$-0.567160\pi$
$5$	−0.936586	−0.418854	−0.209427	+	0.977824i	$0.567160\pi$
$6$	0	0
$7$	−0.519120	−0.196209	−0.0981045	−	0.995176i	$-0.531278\pi$
$7$	−0.519120	−0.196209	−0.0981045	+	0.995176i	$0.531278\pi$
$8$	−9.43585	−3.33608
$9$	0	0
$10$	2.55736	0.808709
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.70534	−1.30503	−0.652513	−	0.757777i	$-0.726285\pi$
$13$	−4.70534	−1.30503	−0.652513	+	0.757777i	$0.726285\pi$
$14$	1.41747	0.378834
$15$	0	0
$16$	14.8533	3.71333
$17$	−2.69227	−0.652972	−0.326486	−	0.945202i	$-0.605865\pi$
$17$	−2.69227	−0.652972	−0.326486	+	0.945202i	$0.605865\pi$
$18$	0	0
$19$	3.41747	0.784020	0.392010	−	0.919961i	$-0.371780\pi$
$19$	3.41747	0.784020	0.392010	+	0.919961i	$0.371780\pi$
$20$	−5.10974	−1.14257
$21$	0	0
$22$	−2.73051	−0.582148
$23$	6.97483	1.45435	0.727176	−	0.686451i	$-0.240832\pi$
$23$	6.97483	1.45435	0.727176	+	0.686451i	$0.240832\pi$
$24$	0	0
$25$	−4.12281	−0.824561
$26$	12.8480	2.51970
$27$	0	0
$28$	−2.83217	−0.535230
$29$	4.18622	0.777362	0.388681	−	0.921372i	$-0.372931\pi$
$29$	4.18622	0.777362	0.388681	+	0.921372i	$0.372931\pi$
$30$	0	0
$31$	5.18622	0.931473	0.465736	−	0.884924i	$-0.345790\pi$
$31$	5.18622	0.931473	0.465736	+	0.884924i	$0.345790\pi$
$32$	−21.6855	−3.83349
$33$	0	0
$34$	7.35129	1.26074
$35$	0.486201	0.0821830
$36$	0	0
$37$	2.06874	0.340098	0.170049	−	0.985436i	$-0.445607\pi$
$37$	2.06874	0.340098	0.170049	+	0.985436i	$0.445607\pi$
$38$	−9.33144	−1.51376
$39$	0	0
$40$	8.83749	1.39733
$41$	0.173153	0.0270419	0.0135209	−	0.999909i	$-0.495696\pi$
$41$	0.173153	0.0270419	0.0135209	+	0.999909i	$0.495696\pi$
$42$	0	0
$43$	−2.26949	−0.346093	−0.173047	−	0.984914i	$-0.555361\pi$
$43$	−2.26949	−0.346093	−0.173047	+	0.984914i	$0.555361\pi$
$44$	5.45571	0.822479
$45$	0	0
$46$	−19.0449	−2.80801
$47$	−0.307727	−0.0448866	−0.0224433	−	0.999748i	$-0.507145\pi$
$47$	−0.307727	−0.0448866	−0.0224433	+	0.999748i	$0.507145\pi$
$48$	0	0
$49$	−6.73051	−0.961502
$50$	11.2574	1.59203
$51$	0	0
$52$	−25.6710	−3.55992
$53$	−1.89835	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$53$	−1.89835	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$54$	0	0
$55$	−0.936586	−0.126289
$56$	4.89835	0.654569
$57$	0	0
$58$	−11.4305	−1.50090
$59$	3.97483	0.517478	0.258739	−	0.965947i	$-0.416693\pi$
$59$	3.97483	0.517478	0.258739	+	0.965947i	$0.416693\pi$
$60$	0	0
$61$	4.51912	0.578614	0.289307	−	0.957236i	$-0.406575\pi$
$61$	4.51912	0.578614	0.289307	+	0.957236i	$0.406575\pi$
$62$	−14.1610	−1.79845
$63$	0	0
$64$	29.5059	3.68824
$65$	4.40696	0.546616
$66$	0	0
$67$	3.37646	0.412501	0.206250	−	0.978499i	$-0.433874\pi$
$67$	3.37646	0.412501	0.206250	+	0.978499i	$0.433874\pi$
$68$	−14.6883	−1.78121
$69$	0	0
$70$	−1.32758	−0.158676
$71$	2.90367	0.344602	0.172301	−	0.985044i	$-0.444880\pi$
$71$	2.90367	0.344602	0.172301	+	0.985044i	$0.444880\pi$
$72$	0	0
$73$	9.52444	1.11475	0.557376	−	0.830260i	$-0.311808\pi$
$73$	9.52444	1.11475	0.557376	+	0.830260i	$0.311808\pi$
$74$	−5.64871	−0.656649
$75$	0	0
$76$	18.6447	2.13869
$77$	−0.519120	−0.0591593
$78$	0	0
$79$	−2.04356	−0.229919	−0.114959	−	0.993370i	$-0.536674\pi$
$79$	−2.04356	−0.229919	−0.114959	+	0.993370i	$0.536674\pi$
$80$	−13.9114	−1.55534
$81$	0	0
$82$	−0.472796	−0.0522115
$83$	14.0594	1.54322	0.771609	−	0.636097i	$-0.219452\pi$
$83$	14.0594	1.54322	0.771609	+	0.636097i	$0.219452\pi$
$84$	0	0
$85$	2.52155	0.273500
$86$	6.19686	0.668225
$87$	0	0
$88$	−9.43585	−1.00587
$89$	7.53751	0.798974	0.399487	−	0.916739i	$-0.369188\pi$
$89$	7.53751	0.798974	0.399487	+	0.916739i	$0.369188\pi$
$90$	0	0
$91$	2.44264	0.256058
$92$	38.0526	3.96726
$93$	0	0
$94$	0.840253	0.0866654
$95$	−3.20075	−0.328390
$96$	0	0
$97$	16.3342	1.65849	0.829243	−	0.558888i	$-0.188772\pi$
$97$	16.3342	1.65849	0.829243	+	0.558888i	$0.188772\pi$
$98$	18.3778	1.85643
$99$	0	0
$100$	−22.4928	−2.24928
$101$	−9.98257	−0.993303	−0.496652	−	0.867950i	$-0.665437\pi$
$101$	−9.98257	−0.993303	−0.496652	+	0.867950i	$0.665437\pi$
$102$	0	0
$103$	6.55494	0.645877	0.322939	−	0.946420i	$-0.395329\pi$
$103$	6.55494	0.645877	0.322939	+	0.946420i	$0.395329\pi$
$104$	44.3989	4.35367
$105$	0	0
$106$	5.18346	0.503462
$107$	14.5151	1.40323	0.701614	−	0.712557i	$-0.252463\pi$
$107$	14.5151	1.40323	0.701614	+	0.712557i	$0.252463\pi$
$108$	0	0
$109$	11.6802	1.11876	0.559379	−	0.828912i	$-0.311040\pi$
$109$	11.6802	1.11876	0.559379	+	0.828912i	$0.311040\pi$
$110$	2.55736	0.243835
$111$	0	0
$112$	−7.71066	−0.728589
$113$	−1.20607	−0.113458	−0.0567289	−	0.998390i	$-0.518067\pi$
$113$	−1.20607	−0.113458	−0.0567289	+	0.998390i	$0.518067\pi$
$114$	0	0
$115$	−6.53253	−0.609161
$116$	22.8388	2.12053
$117$	0	0
$118$	−10.8533	−0.999129
$119$	1.39761	0.128119
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.3395	−1.11717
$123$	0	0
$124$	28.2945	2.54092
$125$	8.54429	0.764225
$126$	0	0
$127$	−10.4533	−0.927579	−0.463789	−	0.885945i	$-0.653511\pi$
$127$	−10.4533	−0.927579	−0.463789	+	0.885945i	$0.653511\pi$
$128$	−37.1953	−3.28763
$129$	0	0
$130$	−12.0333	−1.05539
$131$	−13.4610	−1.17610	−0.588048	−	0.808826i	$-0.700103\pi$
$131$	−13.4610	−1.17610	−0.588048	+	0.808826i	$0.700103\pi$
$132$	0	0
$133$	−1.77408	−0.153832
$134$	−9.21948	−0.796442
$135$	0	0
$136$	25.4039	2.17837
$137$	20.8636	1.78250	0.891250	−	0.453512i	$-0.149829\pi$
$137$	20.8636	1.78250	0.891250	+	0.453512i	$0.149829\pi$
$138$	0	0
$139$	0.867851	0.0736101	0.0368051	−	0.999322i	$-0.488282\pi$
$139$	0.867851	0.0736101	0.0368051	+	0.999322i	$0.488282\pi$
$140$	2.65257	0.224183
$141$	0	0
$142$	−7.92850	−0.665345
$143$	−4.70534	−0.393480
$144$	0	0
$145$	−3.92076	−0.325601
$146$	−26.0066	−2.15232
$147$	0	0
$148$	11.2864	0.927737
$149$	6.56545	0.537862	0.268931	−	0.963159i	$-0.413330\pi$
$149$	6.56545	0.537862	0.268931	+	0.963159i	$0.413330\pi$
$150$	0	0
$151$	12.6419	1.02879	0.514393	−	0.857555i	$-0.328017\pi$
$151$	12.6419	1.02879	0.514393	+	0.857555i	$0.328017\pi$
$152$	−32.2467	−2.61555
$153$	0	0
$154$	1.41747	0.114223
$155$	−4.85734	−0.390151
$156$	0	0
$157$	14.0184	1.11879	0.559395	−	0.828901i	$-0.311034\pi$
$157$	14.0184	1.11879	0.559395	+	0.828901i	$0.311034\pi$
$158$	5.57998	0.443919
$159$	0	0
$160$	20.3103	1.60567
$161$	−3.62078	−0.285357
$162$	0	0
$163$	−21.6245	−1.69376	−0.846881	−	0.531783i	$-0.821522\pi$
$163$	−21.6245	−1.69376	−0.846881	+	0.531783i	$0.821522\pi$
$164$	0.944670	0.0737663
$165$	0	0
$166$	−38.3894	−2.97959
$167$	5.02115	0.388548	0.194274	−	0.980947i	$-0.437765\pi$
$167$	5.02115	0.388548	0.194274	+	0.980947i	$0.437765\pi$
$168$	0	0
$169$	9.14023	0.703095
$170$	−6.88512	−0.528064
$171$	0	0
$172$	−12.3816	−0.944092
$173$	0.458132	0.0348311	0.0174155	−	0.999848i	$-0.494456\pi$
$173$	0.458132	0.0348311	0.0174155	+	0.999848i	$0.494456\pi$
$174$	0	0
$175$	2.14023	0.161786
$176$	14.8533	1.11961
$177$	0	0
$178$	−20.5813	−1.54263
$179$	9.19929	0.687587	0.343794	−	0.939045i	$-0.388288\pi$
$179$	9.19929	0.687587	0.343794	+	0.939045i	$0.388288\pi$
$180$	0	0
$181$	15.1557	1.12652	0.563258	−	0.826281i	$-0.309548\pi$
$181$	15.1557	1.12652	0.563258	+	0.826281i	$0.309548\pi$
$182$	−6.66966	−0.494388
$183$	0	0
$184$	−65.8135	−4.85183
$185$	−1.93755	−0.142451
$186$	0	0
$187$	−2.69227	−0.196879
$188$	−1.67887	−0.122444
$189$	0	0
$190$	8.73969	0.634044
$191$	−20.9432	−1.51540	−0.757699	−	0.652605i	$-0.773676\pi$
$191$	−20.9432	−1.51540	−0.757699	+	0.652605i	$0.773676\pi$
$192$	0	0
$193$	−10.4489	−0.752130	−0.376065	−	0.926593i	$-0.622723\pi$
$193$	−10.4489	−0.752130	−0.376065	+	0.926593i	$0.622723\pi$
$194$	−44.6008	−3.20215
$195$	0	0
$196$	−36.7197	−2.62284
$197$	−20.9855	−1.49515	−0.747576	−	0.664176i	$-0.768783\pi$
$197$	−20.9855	−1.49515	−0.747576	+	0.664176i	$0.768783\pi$
$198$	0	0
$199$	−19.0502	−1.35043	−0.675216	−	0.737620i	$-0.735950\pi$
$199$	−19.0502	−1.35043	−0.675216	+	0.737620i	$0.735950\pi$
$200$	38.9022	2.75080
$201$	0	0
$202$	27.2576	1.91783
$203$	−2.17315	−0.152525
$204$	0	0
$205$	−0.162172	−0.0113266
$206$	−17.8983	−1.24704
$207$	0	0
$208$	−69.8899	−4.84600
$209$	3.41747	0.236391
$210$	0	0
$211$	16.2892	1.12139	0.560697	−	0.828021i	$-0.310533\pi$
$211$	16.2892	1.12139	0.560697	+	0.828021i	$0.310533\pi$
$212$	−10.3568	−0.711309
$213$	0	0
$214$	−39.6337	−2.70930
$215$	2.12557	0.144963
$216$	0	0
$217$	−2.69227	−0.182763
$218$	−31.8929	−2.16006
$219$	0	0
$220$	−5.10974	−0.344499
$221$	12.6681	0.852146
$222$	0	0
$223$	−9.04502	−0.605700	−0.302850	−	0.953038i	$-0.597938\pi$
$223$	−9.04502	−0.605700	−0.302850	+	0.953038i	$0.597938\pi$
$224$	11.2574	0.752165
$225$	0	0
$226$	3.29320	0.219060
$227$	−5.17169	−0.343257	−0.171629	−	0.985162i	$-0.554903\pi$
$227$	−5.17169	−0.343257	−0.171629	+	0.985162i	$0.554903\pi$
$228$	0	0
$229$	16.0237	1.05888	0.529438	−	0.848348i	$-0.322403\pi$
$229$	16.0237	1.05888	0.529438	+	0.848348i	$0.322403\pi$
$230$	17.8372	1.17615
$231$	0	0
$232$	−39.5006	−2.59334
$233$	−16.9177	−1.10832	−0.554158	−	0.832412i	$-0.686960\pi$
$233$	−16.9177	−1.10832	−0.554158	+	0.832412i	$0.686960\pi$
$234$	0	0
$235$	0.288213	0.0188009
$236$	21.6855	1.41161
$237$	0	0
$238$	−3.81620	−0.247368
$239$	0.722430	0.0467301	0.0233651	−	0.999727i	$-0.492562\pi$
$239$	0.722430	0.0467301	0.0233651	+	0.999727i	$0.492562\pi$
$240$	0	0
$241$	13.6469	0.879075	0.439537	−	0.898224i	$-0.355142\pi$
$241$	13.6469	0.879075	0.439537	+	0.898224i	$0.355142\pi$
$242$	−2.73051	−0.175524
$243$	0	0
$244$	24.6550	1.57837
$245$	6.30371	0.402729
$246$	0	0
$247$	−16.0803	−1.02317
$248$	−48.9364	−3.10747
$249$	0	0
$250$	−23.3303	−1.47554
$251$	20.5733	1.29858	0.649288	−	0.760542i	$-0.275067\pi$
$251$	20.5733	1.29858	0.649288	+	0.760542i	$0.275067\pi$
$252$	0	0
$253$	6.97483	0.438504
$254$	28.5428	1.79094
$255$	0	0
$256$	42.5504	2.65940
$257$	5.26416	0.328370	0.164185	−	0.986430i	$-0.447501\pi$
$257$	5.26416	0.328370	0.164185	+	0.986430i	$0.447501\pi$
$258$	0	0
$259$	−1.07392	−0.0667303
$260$	24.0431	1.49109
$261$	0	0
$262$	36.7555	2.27076
$263$	2.59836	0.160222	0.0801110	−	0.996786i	$-0.474473\pi$
$263$	2.59836	0.160222	0.0801110	+	0.996786i	$0.474473\pi$
$264$	0	0
$265$	1.77796	0.109219
$266$	4.84414	0.297013
$267$	0	0
$268$	18.4210	1.12524
$269$	−10.0952	−0.615516	−0.307758	−	0.951465i	$-0.599579\pi$
$269$	−10.0952	−0.615516	−0.307758	+	0.951465i	$0.599579\pi$
$270$	0	0
$271$	32.3022	1.96222	0.981111	−	0.193447i	$-0.0619667\pi$
$271$	32.3022	1.96222	0.981111	+	0.193447i	$0.0619667\pi$
$272$	−39.9892	−2.42470
$273$	0	0
$274$	−56.9684	−3.44159
$275$	−4.12281	−0.248615
$276$	0	0
$277$	32.3379	1.94300	0.971499	−	0.237044i	$-0.0761787\pi$
$277$	32.3379	1.94300	0.971499	+	0.237044i	$0.0761787\pi$
$278$	−2.36968	−0.142124
$279$	0	0
$280$	−4.58772	−0.274169
$281$	−9.54270	−0.569270	−0.284635	−	0.958636i	$-0.591872\pi$
$281$	−9.54270	−0.569270	−0.284635	+	0.958636i	$0.591872\pi$
$282$	0	0
$283$	−5.92076	−0.351952	−0.175976	−	0.984394i	$-0.556308\pi$
$283$	−5.92076	−0.351952	−0.175976	+	0.984394i	$0.556308\pi$
$284$	15.8416	0.940023
$285$	0	0
$286$	12.8480	0.759718
$287$	−0.0898871	−0.00530587
$288$	0	0
$289$	−9.75167	−0.573627
$290$	10.7057	0.628659
$291$	0	0
$292$	51.9626	3.04088
$293$	14.4291	0.842955	0.421478	−	0.906839i	$-0.361512\pi$
$293$	14.4291	0.842955	0.421478	+	0.906839i	$0.361512\pi$
$294$	0	0
$295$	−3.72277	−0.216748
$296$	−19.5203	−1.13459
$297$	0	0
$298$	−17.9270	−1.03849
$299$	−32.8189	−1.89797
$300$	0	0
$301$	1.17814	0.0679067
$302$	−34.5190	−1.98634
$303$	0	0
$304$	50.7607	2.91133
$305$	−4.23255	−0.242355
$306$	0	0
$307$	9.60611	0.548250	0.274125	−	0.961694i	$-0.411612\pi$
$307$	9.60611	0.548250	0.274125	+	0.961694i	$0.411612\pi$
$308$	−2.83217	−0.161378
$309$	0	0
$310$	13.2630	0.753290
$311$	0.343409	0.0194729	0.00973646	−	0.999953i	$-0.496901\pi$
$311$	0.343409	0.0194729	0.00973646	+	0.999953i	$0.496901\pi$
$312$	0	0
$313$	−8.35904	−0.472481	−0.236240	−	0.971695i	$-0.575915\pi$
$313$	−8.35904	−0.472481	−0.236240	+	0.971695i	$0.575915\pi$
$314$	−38.2774	−2.16012
$315$	0	0
$316$	−11.1491	−0.627185
$317$	−13.4083	−0.753083	−0.376541	−	0.926400i	$-0.622887\pi$
$317$	−13.4083	−0.753083	−0.376541	+	0.926400i	$0.622887\pi$
$318$	0	0
$319$	4.18622	0.234383
$320$	−27.6348	−1.54483
$321$	0	0
$322$	9.88658	0.550957
$323$	−9.20075	−0.511943
$324$	0	0
$325$	19.3992	1.07607
$326$	59.0460	3.27026
$327$	0	0
$328$	−1.63384	−0.0902139
$329$	0.159747	0.00880716
$330$	0	0
$331$	−29.8494	−1.64067	−0.820337	−	0.571881i	$-0.806214\pi$
$331$	−29.8494	−1.64067	−0.820337	+	0.571881i	$0.806214\pi$
$332$	76.7039	4.20967
$333$	0	0
$334$	−13.7103	−0.750196
$335$	−3.16235	−0.172778
$336$	0	0
$337$	−21.7237	−1.18337	−0.591683	−	0.806170i	$-0.701536\pi$
$337$	−21.7237	−1.18337	−0.591683	+	0.806170i	$0.701536\pi$
$338$	−24.9575	−1.35751
$339$	0	0
$340$	13.7568	0.746068
$341$	5.18622	0.280850
$342$	0	0
$343$	7.12779	0.384865
$344$	21.4145	1.15459
$345$	0	0
$346$	−1.25093	−0.0672507
$347$	−28.2298	−1.51545	−0.757727	−	0.652572i	$-0.773690\pi$
$347$	−28.2298	−1.51545	−0.757727	+	0.652572i	$0.773690\pi$
$348$	0	0
$349$	−7.15703	−0.383107	−0.191553	−	0.981482i	$-0.561353\pi$
$349$	−7.15703	−0.383107	−0.191553	+	0.981482i	$0.561353\pi$
$350$	−5.84394	−0.312372
$351$	0	0
$352$	−21.6855	−1.15584
$353$	−29.6298	−1.57704	−0.788518	−	0.615011i	$-0.789151\pi$
$353$	−29.6298	−1.57704	−0.788518	+	0.615011i	$0.789151\pi$
$354$	0	0
$355$	−2.71953	−0.144338
$356$	41.1224	2.17949
$357$	0	0
$358$	−25.1188	−1.32757
$359$	−13.0116	−0.686726	−0.343363	−	0.939203i	$-0.611566\pi$
$359$	−13.0116	−0.686726	−0.343363	+	0.939203i	$0.611566\pi$
$360$	0	0
$361$	−7.32093	−0.385312
$362$	−41.3829	−2.17504
$363$	0	0
$364$	13.3263	0.698489
$365$	−8.92046	−0.466918
$366$	0	0
$367$	32.1162	1.67645	0.838225	−	0.545325i	$-0.183594\pi$
$367$	32.1162	1.67645	0.838225	+	0.545325i	$0.183594\pi$
$368$	103.599	5.40049
$369$	0	0
$370$	5.29050	0.275040
$371$	0.985470	0.0511630
$372$	0	0
$373$	34.9156	1.80786	0.903931	−	0.427679i	$-0.140668\pi$
$373$	34.9156	1.80786	0.903931	+	0.427679i	$0.140668\pi$
$374$	7.35129	0.380126
$375$	0	0
$376$	2.90367	0.149745
$377$	−19.6976	−1.01448
$378$	0	0
$379$	−11.4728	−0.589320	−0.294660	−	0.955602i	$-0.595206\pi$
$379$	−11.4728	−0.589320	−0.294660	+	0.955602i	$0.595206\pi$
$380$	−17.4624	−0.895800
$381$	0	0
$382$	57.1857	2.92588
$383$	30.2359	1.54498	0.772492	−	0.635024i	$-0.219010\pi$
$383$	30.2359	1.54498	0.772492	+	0.635024i	$0.219010\pi$
$384$	0	0
$385$	0.486201	0.0247791
$386$	28.5309	1.45219
$387$	0	0
$388$	89.1146	4.52411
$389$	−26.1079	−1.32373	−0.661863	−	0.749625i	$-0.730234\pi$
$389$	−26.1079	−1.32373	−0.661863	+	0.749625i	$0.730234\pi$
$390$	0	0
$391$	−18.7781	−0.949651
$392$	63.5082	3.20765
$393$	0	0
$394$	57.3011	2.88679
$395$	1.91397	0.0963024
$396$	0	0
$397$	−9.94467	−0.499109	−0.249554	−	0.968361i	$-0.580284\pi$
$397$	−9.94467	−0.499109	−0.249554	+	0.968361i	$0.580284\pi$
$398$	52.0168	2.60737
$399$	0	0
$400$	−61.2374	−3.06187
$401$	−23.5968	−1.17837	−0.589183	−	0.807999i	$-0.700550\pi$
$401$	−23.5968	−1.17837	−0.589183	+	0.807999i	$0.700550\pi$
$402$	0	0
$403$	−24.4029	−1.21560
$404$	−54.4620	−2.70959
$405$	0	0
$406$	5.93382	0.294491
$407$	2.06874	0.102543
$408$	0	0
$409$	−2.60658	−0.128887	−0.0644436	−	0.997921i	$-0.520527\pi$
$409$	−2.60658	−0.128887	−0.0644436	+	0.997921i	$0.520527\pi$
$410$	0.442814	0.0218690
$411$	0	0
$412$	35.7618	1.76186
$413$	−2.06341	−0.101534
$414$	0	0
$415$	−13.1678	−0.646383
$416$	102.038	5.00281
$417$	0	0
$418$	−9.33144	−0.456416
$419$	−17.6046	−0.860043	−0.430022	−	0.902819i	$-0.641494\pi$
$419$	−17.6046	−0.860043	−0.430022	+	0.902819i	$0.641494\pi$
$420$	0	0
$421$	6.35648	0.309796	0.154898	−	0.987930i	$-0.450495\pi$
$421$	6.35648	0.309796	0.154898	+	0.987930i	$0.450495\pi$
$422$	−44.4778	−2.16515
$423$	0	0
$424$	17.9125	0.869908
$425$	11.0997	0.538416
$426$	0	0
$427$	−2.34597	−0.113529
$428$	79.1901	3.82780
$429$	0	0
$430$	−5.80390	−0.279889
$431$	29.1820	1.40565	0.702824	−	0.711364i	$-0.251922\pi$
$431$	29.1820	1.40565	0.702824	+	0.711364i	$0.251922\pi$
$432$	0	0
$433$	13.7210	0.659388	0.329694	−	0.944088i	$-0.393054\pi$
$433$	13.7210	0.659388	0.329694	+	0.944088i	$0.393054\pi$
$434$	7.35129	0.352873
$435$	0	0
$436$	63.7236	3.05181
$437$	23.8362	1.14024
$438$	0	0
$439$	−2.00485	−0.0956863	−0.0478431	−	0.998855i	$-0.515235\pi$
$439$	−2.00485	−0.0956863	−0.0478431	+	0.998855i	$0.515235\pi$
$440$	8.83749	0.421311
$441$	0	0
$442$	−34.5903	−1.64529
$443$	34.5945	1.64363	0.821817	−	0.569752i	$-0.192961\pi$
$443$	34.5945	1.64363	0.821817	+	0.569752i	$0.192961\pi$
$444$	0	0
$445$	−7.05953	−0.334654
$446$	24.6976	1.16946
$447$	0	0
$448$	−15.3171	−0.723665
$449$	−2.43875	−0.115092	−0.0575459	−	0.998343i	$-0.518328\pi$
$449$	−2.43875	−0.115092	−0.0575459	+	0.998343i	$0.518328\pi$
$450$	0	0
$451$	0.173153	0.00815344
$452$	−6.57998	−0.309496
$453$	0	0
$454$	14.1214	0.662749
$455$	−2.28774	−0.107251
$456$	0	0
$457$	−41.8995	−1.95998	−0.979988	−	0.199059i	$-0.936212\pi$
$457$	−41.8995	−1.95998	−0.979988	+	0.199059i	$0.936212\pi$
$458$	−43.7530	−2.04444
$459$	0	0
$460$	−35.6395	−1.66170
$461$	38.3881	1.78791	0.893956	−	0.448154i	$-0.147918\pi$
$461$	38.3881	1.78791	0.893956	+	0.448154i	$0.147918\pi$
$462$	0	0
$463$	−0.243350	−0.0113094	−0.00565472	−	0.999984i	$-0.501800\pi$
$463$	−0.243350	−0.0113094	−0.00565472	+	0.999984i	$0.501800\pi$
$464$	62.1793	2.88660
$465$	0	0
$466$	46.1940	2.13990
$467$	−15.6918	−0.726129	−0.363064	−	0.931764i	$-0.618269\pi$
$467$	−15.6918	−0.726129	−0.363064	+	0.931764i	$0.618269\pi$
$468$	0	0
$469$	−1.75279	−0.0809363
$470$	−0.786969	−0.0363002
$471$	0	0
$472$	−37.5059	−1.72635
$473$	−2.26949	−0.104351
$474$	0	0
$475$	−14.0895	−0.646473
$476$	7.62497	0.349490
$477$	0	0
$478$	−1.97261	−0.0902249
$479$	−16.2805	−0.743873	−0.371937	−	0.928258i	$-0.621306\pi$
$479$	−16.2805	−0.743873	−0.371937	+	0.928258i	$0.621306\pi$
$480$	0	0
$481$	−9.73411	−0.443837
$482$	−37.2631	−1.69729
$483$	0	0
$484$	5.45571	0.247987
$485$	−15.2984	−0.694664
$486$	0	0
$487$	−32.0421	−1.45197	−0.725983	−	0.687713i	$-0.758615\pi$
$487$	−32.0421	−1.45197	−0.725983	+	0.687713i	$0.758615\pi$
$488$	−42.6418	−1.93030
$489$	0	0
$490$	−17.2124	−0.777575
$491$	−20.9458	−0.945269	−0.472635	−	0.881258i	$-0.656697\pi$
$491$	−20.9458	−0.945269	−0.472635	+	0.881258i	$0.656697\pi$
$492$	0	0
$493$	−11.2704	−0.507595
$494$	43.9076	1.97550
$495$	0	0
$496$	77.0326	3.45887
$497$	−1.50735	−0.0676140
$498$	0	0
$499$	7.23104	0.323706	0.161853	−	0.986815i	$-0.448253\pi$
$499$	7.23104	0.323706	0.161853	+	0.986815i	$0.448253\pi$
$500$	46.6152	2.08469
$501$	0	0
$502$	−56.1758	−2.50725
$503$	−5.90333	−0.263216	−0.131608	−	0.991302i	$-0.542014\pi$
$503$	−5.90333	−0.263216	−0.131608	+	0.991302i	$0.542014\pi$
$504$	0	0
$505$	9.34954	0.416049
$506$	−19.0449	−0.846647
$507$	0	0
$508$	−57.0300	−2.53030
$509$	−10.8400	−0.480477	−0.240238	−	0.970714i	$-0.577226\pi$
$509$	−10.8400	−0.480477	−0.240238	+	0.970714i	$0.577226\pi$
$510$	0	0
$511$	−4.94433	−0.218724
$512$	−41.7940	−1.84705
$513$	0	0
$514$	−14.3739	−0.634004
$515$	−6.13926	−0.270528
$516$	0	0
$517$	−0.307727	−0.0135338
$518$	2.93236	0.128841
$519$	0	0
$520$	−41.5834	−1.82355
$521$	14.7412	0.645822	0.322911	−	0.946429i	$-0.395339\pi$
$521$	14.7412	0.645822	0.322911	+	0.946429i	$0.395339\pi$
$522$	0	0
$523$	−16.8230	−0.735617	−0.367808	−	0.929902i	$-0.619892\pi$
$523$	−16.8230	−0.735617	−0.367808	+	0.929902i	$0.619892\pi$
$524$	−73.4394	−3.20822
$525$	0	0
$526$	−7.09487	−0.309351
$527$	−13.9627	−0.608226
$528$	0	0
$529$	25.6482	1.11514
$530$	−4.85475	−0.210877
$531$	0	0
$532$	−9.67884	−0.419631
$533$	−0.814742	−0.0352904
$534$	0	0
$535$	−13.5946	−0.587748
$536$	−31.8598	−1.37613
$537$	0	0
$538$	27.5651	1.18842
$539$	−6.73051	−0.289904
$540$	0	0
$541$	−22.5357	−0.968886	−0.484443	−	0.874823i	$-0.660978\pi$
$541$	−22.5357	−0.968886	−0.484443	+	0.874823i	$0.660978\pi$
$542$	−88.2017	−3.78859
$543$	0	0
$544$	58.3833	2.50316
$545$	−10.9395	−0.468596
$546$	0	0
$547$	28.4278	1.21549	0.607743	−	0.794134i	$-0.292075\pi$
$547$	28.4278	1.21549	0.607743	+	0.794134i	$0.292075\pi$
$548$	113.826	4.86240
$549$	0	0
$550$	11.2574	0.480016
$551$	14.3063	0.609467
$552$	0	0
$553$	1.06085	0.0451121
$554$	−88.2992	−3.75147
$555$	0	0
$556$	4.73474	0.200798
$557$	43.1863	1.82986	0.914930	−	0.403612i	$-0.132245\pi$
$557$	43.1863	1.82986	0.914930	+	0.403612i	$0.132245\pi$
$558$	0	0
$559$	10.6787	0.451661
$560$	7.22170	0.305172
$561$	0	0
$562$	26.0565	1.09913
$563$	−0.964522	−0.0406497	−0.0203249	−	0.999793i	$-0.506470\pi$
$563$	−0.964522	−0.0406497	−0.0203249	+	0.999793i	$0.506470\pi$
$564$	0	0
$565$	1.12959	0.0475222
$566$	16.1667	0.679537
$567$	0	0
$568$	−27.3986	−1.14962
$569$	8.52833	0.357526	0.178763	−	0.983892i	$-0.442790\pi$
$569$	8.52833	0.357526	0.178763	+	0.983892i	$0.442790\pi$
$570$	0	0
$571$	17.6035	0.736685	0.368342	−	0.929690i	$-0.379925\pi$
$571$	17.6035	0.736685	0.368342	+	0.929690i	$0.379925\pi$
$572$	−25.6710	−1.07336
$573$	0	0
$574$	0.245438	0.0102444
$575$	−28.7559	−1.19920
$576$	0	0
$577$	−11.1810	−0.465472	−0.232736	−	0.972540i	$-0.574768\pi$
$577$	−11.1810	−0.465472	−0.232736	+	0.972540i	$0.574768\pi$
$578$	26.6271	1.10754
$579$	0	0
$580$	−21.3905	−0.888192
$581$	−7.29852	−0.302794
$582$	0	0
$583$	−1.89835	−0.0786214
$584$	−89.8713	−3.71890
$585$	0	0
$586$	−39.3988	−1.62755
$587$	21.6747	0.894610	0.447305	−	0.894381i	$-0.352384\pi$
$587$	21.6747	0.894610	0.447305	+	0.894381i	$0.352384\pi$
$588$	0	0
$589$	17.7237	0.730294
$590$	10.1651	0.418489
$591$	0	0
$592$	30.7276	1.26290
$593$	−7.82200	−0.321211	−0.160605	−	0.987019i	$-0.551345\pi$
$593$	−7.82200	−0.321211	−0.160605	+	0.987019i	$0.551345\pi$
$594$	0	0
$595$	−1.30899	−0.0536632
$596$	35.8191	1.46721
$597$	0	0
$598$	89.6126	3.66453
$599$	−10.5080	−0.429344	−0.214672	−	0.976686i	$-0.568868\pi$
$599$	−10.5080	−0.429344	−0.214672	+	0.976686i	$0.568868\pi$
$600$	0	0
$601$	37.0976	1.51324	0.756622	−	0.653853i	$-0.226848\pi$
$601$	37.0976	1.51324	0.756622	+	0.653853i	$0.226848\pi$
$602$	−3.21692	−0.131112
$603$	0	0
$604$	68.9706	2.80638
$605$	−0.936586	−0.0380776
$606$	0	0
$607$	−41.5115	−1.68490	−0.842451	−	0.538773i	$-0.818888\pi$
$607$	−41.5115	−1.68490	−0.842451	+	0.538773i	$0.818888\pi$
$608$	−74.1094	−3.00553
$609$	0	0
$610$	11.5570	0.467930
$611$	1.44796	0.0585782
$612$	0	0
$613$	33.7344	1.36252	0.681259	−	0.732042i	$-0.261433\pi$
$613$	33.7344	1.36252	0.681259	+	0.732042i	$0.261433\pi$
$614$	−26.2296	−1.05854
$615$	0	0
$616$	4.89835	0.197360
$617$	−28.0595	−1.12963	−0.564817	−	0.825216i	$-0.691053\pi$
$617$	−28.0595	−1.12963	−0.564817	+	0.825216i	$0.691053\pi$
$618$	0	0
$619$	−31.8097	−1.27854	−0.639270	−	0.768982i	$-0.720764\pi$
$619$	−31.8097	−1.27854	−0.639270	+	0.768982i	$0.720764\pi$
$620$	−26.5002	−1.06427
$621$	0	0
$622$	−0.937683	−0.0375976
$623$	−3.91288	−0.156766
$624$	0	0
$625$	12.6116	0.504463
$626$	22.8245	0.912249
$627$	0	0
$628$	76.4802	3.05189
$629$	−5.56960	−0.222075
$630$	0	0
$631$	−23.4280	−0.932653	−0.466326	−	0.884613i	$-0.654423\pi$
$631$	−23.4280	−0.932653	−0.466326	+	0.884613i	$0.654423\pi$
$632$	19.2828	0.767027
$633$	0	0
$634$	36.6114	1.45403
$635$	9.79040	0.388520
$636$	0	0
$637$	31.6694	1.25479
$638$	−11.4305	−0.452539
$639$	0	0
$640$	34.8366	1.37704
$641$	15.5156	0.612828	0.306414	−	0.951898i	$-0.400871\pi$
$641$	15.5156	0.612828	0.306414	+	0.951898i	$0.400871\pi$
$642$	0	0
$643$	−10.9474	−0.431725	−0.215862	−	0.976424i	$-0.569256\pi$
$643$	−10.9474	−0.431725	−0.215862	+	0.976424i	$0.569256\pi$
$644$	−19.7539	−0.778412
$645$	0	0
$646$	25.1228	0.988442
$647$	16.9732	0.667287	0.333643	−	0.942699i	$-0.391722\pi$
$647$	16.9732	0.667287	0.333643	+	0.942699i	$0.391722\pi$
$648$	0	0
$649$	3.97483	0.156026
$650$	−52.9698	−2.07765
$651$	0	0
$652$	−117.977	−4.62033
$653$	0.454582	0.0177892	0.00889458	−	0.999960i	$-0.497169\pi$
$653$	0.454582	0.0177892	0.00889458	+	0.999960i	$0.497169\pi$
$654$	0	0
$655$	12.6074	0.492612
$656$	2.57189	0.100415
$657$	0	0
$658$	−0.436192	−0.0170045
$659$	−3.15703	−0.122980	−0.0614901	−	0.998108i	$-0.519585\pi$
$659$	−3.15703	−0.122980	−0.0614901	+	0.998108i	$0.519585\pi$
$660$	0	0
$661$	−26.5572	−1.03296	−0.516478	−	0.856301i	$-0.672757\pi$
$661$	−26.5572	−1.03296	−0.516478	+	0.856301i	$0.672757\pi$
$662$	81.5043	3.16775
$663$	0	0
$664$	−132.662	−5.14830
$665$	1.66158	0.0644331
$666$	0	0
$667$	29.1982	1.13056
$668$	27.3939	1.05990
$669$	0	0
$670$	8.63483	0.333593
$671$	4.51912	0.174459
$672$	0	0
$673$	−27.2969	−1.05222	−0.526110	−	0.850417i	$-0.676350\pi$
$673$	−27.2969	−1.05222	−0.526110	+	0.850417i	$0.676350\pi$
$674$	59.3169	2.28480
$675$	0	0
$676$	49.8664	1.91794
$677$	−0.400335	−0.0153861	−0.00769307	−	0.999970i	$-0.502449\pi$
$677$	−0.400335	−0.0153861	−0.00769307	+	0.999970i	$0.502449\pi$
$678$	0	0
$679$	−8.47942	−0.325410
$680$	−23.7929	−0.912417
$681$	0	0
$682$	−14.1610	−0.542255
$683$	−26.2154	−1.00310	−0.501552	−	0.865127i	$-0.667238\pi$
$683$	−26.2154	−1.00310	−0.501552	+	0.865127i	$0.667238\pi$
$684$	0	0
$685$	−19.5406	−0.746607
$686$	−19.4625	−0.743083
$687$	0	0
$688$	−33.7094	−1.28516
$689$	8.93236	0.340296
$690$	0	0
$691$	−15.5963	−0.593310	−0.296655	−	0.954985i	$-0.595871\pi$
$691$	−15.5963	−0.593310	−0.296655	+	0.954985i	$0.595871\pi$
$692$	2.49943	0.0950141
$693$	0	0
$694$	77.0818	2.92599
$695$	−0.812817	−0.0308319
$696$	0	0
$697$	−0.466174	−0.0176576
$698$	19.5424	0.739689
$699$	0	0
$700$	11.6765	0.441330
$701$	27.4965	1.03853	0.519265	−	0.854613i	$-0.326206\pi$
$701$	27.4965	1.03853	0.519265	+	0.854613i	$0.326206\pi$
$702$	0	0
$703$	7.06983	0.266644
$704$	29.5059	1.11205
$705$	0	0
$706$	80.9046	3.04489
$707$	5.18216	0.194895
$708$	0	0
$709$	−30.6514	−1.15114	−0.575570	−	0.817753i	$-0.695220\pi$
$709$	−30.6514	−1.15114	−0.575570	+	0.817753i	$0.695220\pi$
$710$	7.42572	0.278682
$711$	0	0
$712$	−71.1228	−2.66544
$713$	36.1730	1.35469
$714$	0	0
$715$	4.40696	0.164811
$716$	50.1886	1.87564
$717$	0	0
$718$	35.5284	1.32591
$719$	−19.7250	−0.735620	−0.367810	−	0.929901i	$-0.619892\pi$
$719$	−19.7250	−0.735620	−0.367810	+	0.929901i	$0.619892\pi$
$720$	0	0
$721$	−3.40280	−0.126727
$722$	19.9899	0.743947
$723$	0	0
$724$	82.6852	3.07297
$725$	−17.2590	−0.640982
$726$	0	0
$727$	−5.41214	−0.200725	−0.100363	−	0.994951i	$-0.532000\pi$
$727$	−5.41214	−0.200725	−0.100363	+	0.994951i	$0.532000\pi$
$728$	−23.0484	−0.854230
$729$	0	0
$730$	24.3574	0.901509
$731$	6.11008	0.225989
$732$	0	0
$733$	−1.63804	−0.0605024	−0.0302512	−	0.999542i	$-0.509631\pi$
$733$	−1.63804	−0.0605024	−0.0302512	+	0.999542i	$0.509631\pi$
$734$	−87.6936	−3.23683
$735$	0	0
$736$	−151.253	−5.57524
$737$	3.37646	0.124374
$738$	0	0
$739$	−12.8306	−0.471980	−0.235990	−	0.971755i	$-0.575833\pi$
$739$	−12.8306	−0.471980	−0.235990	+	0.971755i	$0.575833\pi$
$740$	−10.5707	−0.388587
$741$	0	0
$742$	−2.69084	−0.0987838
$743$	36.3117	1.33215	0.666074	−	0.745886i	$-0.267974\pi$
$743$	36.3117	1.33215	0.666074	+	0.745886i	$0.267974\pi$
$744$	0	0
$745$	−6.14910	−0.225286
$746$	−95.3376	−3.49056
$747$	0	0
$748$	−14.6883	−0.537056
$749$	−7.53508	−0.275326
$750$	0	0
$751$	20.6326	0.752894	0.376447	−	0.926438i	$-0.377146\pi$
$751$	20.6326	0.752894	0.376447	+	0.926438i	$0.377146\pi$
$752$	−4.57077	−0.166679
$753$	0	0
$754$	53.7846	1.95872
$755$	−11.8403	−0.430911
$756$	0	0
$757$	13.5638	0.492986	0.246493	−	0.969145i	$-0.420722\pi$
$757$	13.5638	0.492986	0.246493	+	0.969145i	$0.420722\pi$
$758$	31.3267	1.13784
$759$	0	0
$760$	30.2018	1.09554
$761$	15.9960	0.579854	0.289927	−	0.957049i	$-0.406369\pi$
$761$	15.9960	0.579854	0.289927	+	0.957049i	$0.406369\pi$
$762$	0	0
$763$	−6.06341	−0.219510
$764$	−114.260	−4.13378
$765$	0	0
$766$	−82.5596	−2.98300
$767$	−18.7029	−0.675323
$768$	0	0
$769$	26.5677	0.958056	0.479028	−	0.877800i	$-0.340989\pi$
$769$	26.5677	0.958056	0.479028	+	0.877800i	$0.340989\pi$
$770$	−1.32758	−0.0478426
$771$	0	0
$772$	−57.0063	−2.05170
$773$	24.2035	0.870539	0.435269	−	0.900300i	$-0.356653\pi$
$773$	24.2035	0.870539	0.435269	+	0.900300i	$0.356653\pi$
$774$	0	0
$775$	−21.3818	−0.768056
$776$	−154.127	−5.53284
$777$	0	0
$778$	71.2881	2.55580
$779$	0.591743	0.0212014
$780$	0	0
$781$	2.90367	0.103901
$782$	51.2740	1.83355
$783$	0	0
$784$	−99.9705	−3.57037
$785$	−13.1294	−0.468609
$786$	0	0
$787$	−27.2323	−0.970728	−0.485364	−	0.874312i	$-0.661313\pi$
$787$	−27.2323	−0.970728	−0.485364	+	0.874312i	$0.661313\pi$
$788$	−114.491	−4.07856
$789$	0	0
$790$	−5.22613	−0.185937
$791$	0.626097	0.0222614
$792$	0	0
$793$	−21.2640	−0.755107
$794$	27.1541	0.963662
$795$	0	0
$796$	−103.932	−3.68378
$797$	38.5692	1.36619	0.683095	−	0.730329i	$-0.260633\pi$
$797$	38.5692	1.36619	0.683095	+	0.730329i	$0.260633\pi$
$798$	0	0
$799$	0.828485	0.0293097
$800$	89.4051	3.16095
$801$	0	0
$802$	64.4313	2.27515
$803$	9.52444	0.336110
$804$	0	0
$805$	3.39117	0.119523
$806$	66.6326	2.34703
$807$	0	0
$808$	94.1941	3.31374
$809$	−12.0951	−0.425240	−0.212620	−	0.977135i	$-0.568200\pi$
$809$	−12.0951	−0.425240	−0.212620	+	0.977135i	$0.568200\pi$
$810$	0	0
$811$	35.0487	1.23073	0.615364	−	0.788243i	$-0.289009\pi$
$811$	35.0487	1.23073	0.615364	+	0.788243i	$0.289009\pi$
$812$	−11.8561	−0.416067
$813$	0	0
$814$	−5.64871	−0.197987
$815$	20.2532	0.709439
$816$	0	0
$817$	−7.75589	−0.271344
$818$	7.11731	0.248851
$819$	0	0
$820$	−0.884765	−0.0308973
$821$	20.0796	0.700782	0.350391	−	0.936603i	$-0.386049\pi$
$821$	20.0796	0.700782	0.350391	+	0.936603i	$0.386049\pi$
$822$	0	0
$823$	−25.6116	−0.892763	−0.446382	−	0.894843i	$-0.647288\pi$
$823$	−25.6116	−0.892763	−0.446382	+	0.894843i	$0.647288\pi$
$824$	−61.8514	−2.15470
$825$	0	0
$826$	5.63418	0.196038
$827$	−0.864167	−0.0300500	−0.0150250	−	0.999887i	$-0.504783\pi$
$827$	−0.864167	−0.0300500	−0.0150250	+	0.999887i	$0.504783\pi$
$828$	0	0
$829$	38.1823	1.32613	0.663064	−	0.748563i	$-0.269256\pi$
$829$	38.1823	1.32613	0.663064	+	0.748563i	$0.269256\pi$
$830$	35.9549	1.24801
$831$	0	0
$832$	−138.835	−4.81325
$833$	18.1204	0.627834
$834$	0	0
$835$	−4.70274	−0.162745
$836$	18.6447	0.644840
$837$	0	0
$838$	48.0697	1.66054
$839$	11.7024	0.404013	0.202007	−	0.979384i	$-0.435254\pi$
$839$	11.7024	0.404013	0.202007	+	0.979384i	$0.435254\pi$
$840$	0	0
$841$	−11.4756	−0.395709
$842$	−17.3564	−0.598143
$843$	0	0
$844$	88.8690	3.05900
$845$	−8.56062	−0.294494
$846$	0	0
$847$	−0.519120	−0.0178372
$848$	−28.1967	−0.968280
$849$	0	0
$850$	−30.3079	−1.03955
$851$	14.4291	0.494622
$852$	0	0
$853$	−17.4755	−0.598351	−0.299175	−	0.954198i	$-0.596712\pi$
$853$	−17.4755	−0.598351	−0.299175	+	0.954198i	$0.596712\pi$
$854$	6.40570	0.219198
$855$	0	0
$856$	−136.962	−4.68128
$857$	−35.6260	−1.21696	−0.608480	−	0.793569i	$-0.708220\pi$
$857$	−35.6260	−1.21696	−0.608480	+	0.793569i	$0.708220\pi$
$858$	0	0
$859$	−11.9179	−0.406632	−0.203316	−	0.979113i	$-0.565172\pi$
$859$	−11.9179	−0.406632	−0.203316	+	0.979113i	$0.565172\pi$
$860$	11.5965	0.395437
$861$	0	0
$862$	−79.6818	−2.71397
$863$	−1.22689	−0.0417637	−0.0208818	−	0.999782i	$-0.506647\pi$
$863$	−1.22689	−0.0417637	−0.0208818	+	0.999782i	$0.506647\pi$
$864$	0	0
$865$	−0.429080	−0.0145891
$866$	−37.4653	−1.27312
$867$	0	0
$868$	−14.6883	−0.498552
$869$	−2.04356	−0.0693231
$870$	0	0
$871$	−15.8874	−0.538324
$872$	−110.212	−3.73226
$873$	0	0
$874$	−65.0852	−2.20154
$875$	−4.43552	−0.149948
$876$	0	0
$877$	25.9734	0.877058	0.438529	−	0.898717i	$-0.355500\pi$
$877$	25.9734	0.877058	0.438529	+	0.898717i	$0.355500\pi$
$878$	5.47427	0.184748
$879$	0	0
$880$	−13.9114	−0.468954
$881$	53.5351	1.80364	0.901822	−	0.432107i	$-0.142230\pi$
$881$	53.5351	1.80364	0.901822	+	0.432107i	$0.142230\pi$
$882$	0	0
$883$	17.3818	0.584945	0.292472	−	0.956274i	$-0.405522\pi$
$883$	17.3818	0.584945	0.292472	+	0.956274i	$0.405522\pi$
$884$	69.1132	2.32453
$885$	0	0
$886$	−94.4607	−3.17347
$887$	40.0160	1.34361	0.671803	−	0.740730i	$-0.265520\pi$
$887$	40.0160	1.34361	0.671803	+	0.740730i	$0.265520\pi$
$888$	0	0
$889$	5.42651	0.181999
$890$	19.2761	0.646138
$891$	0	0
$892$	−49.3470	−1.65226
$893$	−1.05165	−0.0351920
$894$	0	0
$895$	−8.61593	−0.287999
$896$	19.3088	0.645063
$897$	0	0
$898$	6.65904	0.222215
$899$	21.7107	0.724091
$900$	0	0
$901$	5.11086	0.170268
$902$	−0.472796	−0.0157424
$903$	0	0
$904$	11.3803	0.378504
$905$	−14.1946	−0.471846
$906$	0	0
$907$	−29.8093	−0.989800	−0.494900	−	0.868950i	$-0.664795\pi$
$907$	−29.8093	−0.989800	−0.494900	+	0.868950i	$0.664795\pi$
$908$	−28.2152	−0.936355
$909$	0	0
$910$	6.24671	0.207076
$911$	7.51912	0.249120	0.124560	−	0.992212i	$-0.460248\pi$
$911$	7.51912	0.249120	0.124560	+	0.992212i	$0.460248\pi$
$912$	0	0
$913$	14.0594	0.465298
$914$	114.407	3.78425
$915$	0	0
$916$	87.4207	2.88846
$917$	6.98789	0.230761
$918$	0	0
$919$	−48.1441	−1.58813	−0.794064	−	0.607835i	$-0.792038\pi$
$919$	−48.1441	−1.58813	−0.794064	+	0.607835i	$0.792038\pi$
$920$	61.6400	2.03221
$921$	0	0
$922$	−104.819	−3.45204
$923$	−13.6627	−0.449715
$924$	0	0
$925$	−8.52900	−0.280432
$926$	0.664472	0.0218359
$927$	0	0
$928$	−90.7802	−2.98001
$929$	−39.1770	−1.28536	−0.642678	−	0.766136i	$-0.722177\pi$
$929$	−39.1770	−1.28536	−0.642678	+	0.766136i	$0.722177\pi$
$930$	0	0
$931$	−23.0013	−0.753837
$932$	−92.2980	−3.02332
$933$	0	0
$934$	42.8466	1.40198
$935$	2.52155	0.0824634
$936$	0	0
$937$	36.6480	1.19724	0.598620	−	0.801033i	$-0.295716\pi$
$937$	36.6480	1.19724	0.598620	+	0.801033i	$0.295716\pi$
$938$	4.78602	0.156269
$939$	0	0
$940$	1.57240	0.0512862
$941$	3.89722	0.127046	0.0635229	−	0.997980i	$-0.479766\pi$
$941$	3.89722	0.127046	0.0635229	+	0.997980i	$0.479766\pi$
$942$	0	0
$943$	1.20771	0.0393284
$944$	59.0394	1.92157
$945$	0	0
$946$	6.19686	0.201477
$947$	44.4041	1.44294	0.721469	−	0.692446i	$-0.243467\pi$
$947$	44.4041	1.44294	0.721469	+	0.692446i	$0.243467\pi$
$948$	0	0
$949$	−44.8157	−1.45478
$950$	38.4717	1.24819
$951$	0	0
$952$	−13.1877	−0.427415
$953$	20.1218	0.651810	0.325905	−	0.945403i	$-0.394331\pi$
$953$	20.1218	0.651810	0.325905	+	0.945403i	$0.394331\pi$
$954$	0	0
$955$	19.6151	0.634730
$956$	3.94137	0.127473
$957$	0	0
$958$	44.4540	1.43624
$959$	−10.8307	−0.349743
$960$	0	0
$961$	−4.10312	−0.132359
$962$	26.5791	0.856945
$963$	0	0
$964$	74.4535	2.39799
$965$	9.78632	0.315033
$966$	0	0
$967$	23.1384	0.744082	0.372041	−	0.928216i	$-0.378658\pi$
$967$	23.1384	0.744082	0.372041	+	0.928216i	$0.378658\pi$
$968$	−9.43585	−0.303280
$969$	0	0
$970$	41.7725	1.34123
$971$	28.3629	0.910209	0.455104	−	0.890438i	$-0.349602\pi$
$971$	28.3629	0.910209	0.455104	+	0.890438i	$0.349602\pi$
$972$	0	0
$973$	−0.450519	−0.0144430
$974$	87.4914	2.80341
$975$	0	0
$976$	67.1239	2.14859
$977$	15.6896	0.501956	0.250978	−	0.967993i	$-0.419248\pi$
$977$	15.6896	0.501956	0.250978	+	0.967993i	$0.419248\pi$
$978$	0	0
$979$	7.53751	0.240900
$980$	34.3912	1.09859
$981$	0	0
$982$	57.1927	1.82509
$983$	36.1316	1.15242	0.576210	−	0.817301i	$-0.304531\pi$
$983$	36.1316	1.15242	0.576210	+	0.817301i	$0.304531\pi$
$984$	0	0
$985$	19.6547	0.626251
$986$	30.7741	0.980048
$987$	0	0
$988$	−87.7296	−2.79105
$989$	−15.8293	−0.503342
$990$	0	0
$991$	−8.69420	−0.276180	−0.138090	−	0.990420i	$-0.544096\pi$
$991$	−8.69420	−0.276180	−0.138090	+	0.990420i	$0.544096\pi$
$992$	−112.466	−3.57079
$993$	0	0
$994$	4.11585	0.130547
$995$	17.8421	0.565634
$996$	0	0
$997$	35.5296	1.12523	0.562616	−	0.826718i	$-0.309795\pi$
$997$	35.5296	1.12523	0.562616	+	0.826718i	$0.309795\pi$
$998$	−19.7445	−0.625000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.q.1.1		4
3.2	odd	2		891.2.a.p.1.4		4
9.2	odd	6		297.2.e.e.199.1		8
9.4	even	3		99.2.e.e.34.4	✓	8
9.5	odd	6		297.2.e.e.100.1		8
9.7	even	3		99.2.e.e.67.4	yes	8
11.10	odd	2		9801.2.a.bi.1.4		4
33.32	even	2		9801.2.a.bl.1.1		4
99.43	odd	6		1089.2.e.i.364.1		8
99.76	odd	6		1089.2.e.i.727.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.4	✓	8	9.4	even	3
99.2.e.e.67.4	yes	8	9.7	even	3
297.2.e.e.100.1		8	9.5	odd	6
297.2.e.e.199.1		8	9.2	odd	6
891.2.a.p.1.4		4	3.2	odd	2
891.2.a.q.1.1		4	1.1	even	1	trivial
1089.2.e.i.364.1		8	99.43	odd	6
1089.2.e.i.727.1		8	99.76	odd	6
9801.2.a.bi.1.4		4	11.10	odd	2
9801.2.a.bl.1.1		4	33.32	even	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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73051 q^{2} +5.45571 q^{4} -0.936586 q^{5} -0.519120 q^{7} -9.43585 q^{8} +O(q^{10})\)	\(q-2.73051 q^{2} +5.45571 q^{4} -0.936586 q^{5} -0.519120 q^{7} -9.43585 q^{8} +2.55736 q^{10} +1.00000 q^{11} -4.70534 q^{13} +1.41747 q^{14} +14.8533 q^{16} -2.69227 q^{17} +3.41747 q^{19} -5.10974 q^{20} -2.73051 q^{22} +6.97483 q^{23} -4.12281 q^{25} +12.8480 q^{26} -2.83217 q^{28} +4.18622 q^{29} +5.18622 q^{31} -21.6855 q^{32} +7.35129 q^{34} +0.486201 q^{35} +2.06874 q^{37} -9.33144 q^{38} +8.83749 q^{40} +0.173153 q^{41} -2.26949 q^{43} +5.45571 q^{44} -19.0449 q^{46} -0.307727 q^{47} -6.73051 q^{49} +11.2574 q^{50} -25.6710 q^{52} -1.89835 q^{53} -0.936586 q^{55} +4.89835 q^{56} -11.4305 q^{58} +3.97483 q^{59} +4.51912 q^{61} -14.1610 q^{62} +29.5059 q^{64} +4.40696 q^{65} +3.37646 q^{67} -14.6883 q^{68} -1.32758 q^{70} +2.90367 q^{71} +9.52444 q^{73} -5.64871 q^{74} +18.6447 q^{76} -0.519120 q^{77} -2.04356 q^{79} -13.9114 q^{80} -0.472796 q^{82} +14.0594 q^{83} +2.52155 q^{85} +6.19686 q^{86} -9.43585 q^{88} +7.53751 q^{89} +2.44264 q^{91} +38.0526 q^{92} +0.840253 q^{94} -3.20075 q^{95} +16.3342 q^{97} +18.3778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7}+O(q^{10}) \) 4 * q + q^2 + 11 * q^4 + 4 * q^5 + q^7	\( 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7} + q^{10} + 4 q^{11} + 7 q^{13} + q^{14} + 17 q^{16} - 5 q^{17} + 9 q^{19} - 10 q^{20} + q^{22} + 14 q^{23} + 14 q^{25} + 22 q^{26} - q^{28} - 6 q^{29} - 2 q^{31} - 34 q^{32} + 16 q^{34} - 8 q^{35} + 3 q^{37} + 3 q^{38} + 12 q^{40} - 2 q^{41} - 21 q^{43} + 11 q^{44} + 2 q^{46} - 7 q^{47} - 15 q^{49} + 23 q^{50} - 10 q^{52} - 6 q^{53} + 4 q^{55} + 18 q^{56} - 21 q^{58} + 2 q^{59} + 15 q^{61} - 20 q^{62} + 16 q^{64} + 19 q^{65} + 14 q^{67} - 7 q^{68} - 38 q^{70} - 3 q^{71} + 22 q^{73} - 36 q^{74} + 42 q^{76} + q^{77} + 11 q^{79} - 34 q^{80} - 17 q^{82} + 18 q^{83} + 13 q^{85} - 24 q^{86} - 6 q^{89} + 19 q^{91} + 67 q^{92} - 19 q^{94} - 30 q^{95} + 26 q^{97} + 15 q^{98}+O(q^{100}) \) 4 * q + q^2 + 11 * q^4 + 4 * q^5 + q^7 + q^10 + 4 * q^11 + 7 * q^13 + q^14 + 17 * q^16 - 5 * q^17 + 9 * q^19 - 10 * q^20 + q^22 + 14 * q^23 + 14 * q^25 + 22 * q^26 - q^28 - 6 * q^29 - 2 * q^31 - 34 * q^32 + 16 * q^34 - 8 * q^35 + 3 * q^37 + 3 * q^38 + 12 * q^40 - 2 * q^41 - 21 * q^43 + 11 * q^44 + 2 * q^46 - 7 * q^47 - 15 * q^49 + 23 * q^50 - 10 * q^52 - 6 * q^53 + 4 * q^55 + 18 * q^56 - 21 * q^58 + 2 * q^59 + 15 * q^61 - 20 * q^62 + 16 * q^64 + 19 * q^65 + 14 * q^67 - 7 * q^68 - 38 * q^70 - 3 * q^71 + 22 * q^73 - 36 * q^74 + 42 * q^76 + q^77 + 11 * q^79 - 34 * q^80 - 17 * q^82 + 18 * q^83 + 13 * q^85 - 24 * q^86 - 6 * q^89 + 19 * q^91 + 67 * q^92 - 19 * q^94 - 30 * q^95 + 26 * q^97 + 15 * q^98

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