LMFDB - Embedded newform 1458.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(1,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, 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("1458.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.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:	$11.6421886147$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.96962$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.852666 q^{5} +1.37581 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.852666 q^{5} +1.37581 q^{7} -1.00000 q^{8} +0.852666 q^{10} -0.216780 q^{11} +1.95778 q^{13} -1.37581 q^{14} +1.00000 q^{16} -7.71783 q^{17} -6.12087 q^{19} -0.852666 q^{20} +0.216780 q^{22} +2.41293 q^{23} -4.27296 q^{25} -1.95778 q^{26} +1.37581 q^{28} -3.94668 q^{29} +8.84930 q^{31} -1.00000 q^{32} +7.71783 q^{34} -1.17310 q^{35} +6.63961 q^{37} +6.12087 q^{38} +0.852666 q^{40} +3.73396 q^{41} -10.6495 q^{43} -0.216780 q^{44} -2.41293 q^{46} +1.52861 q^{47} -5.10716 q^{49} +4.27296 q^{50} +1.95778 q^{52} -10.2304 q^{53} +0.184841 q^{55} -1.37581 q^{56} +3.94668 q^{58} -4.62382 q^{59} +5.24719 q^{61} -8.84930 q^{62} +1.00000 q^{64} -1.66933 q^{65} +4.45299 q^{67} -7.71783 q^{68} +1.17310 q^{70} -15.8545 q^{71} -5.39860 q^{73} -6.63961 q^{74} -6.12087 q^{76} -0.298248 q^{77} -8.02895 q^{79} -0.852666 q^{80} -3.73396 q^{82} +12.0654 q^{83} +6.58073 q^{85} +10.6495 q^{86} +0.216780 q^{88} -3.61359 q^{89} +2.69352 q^{91} +2.41293 q^{92} -1.52861 q^{94} +5.21905 q^{95} -10.6495 q^{97} +5.10716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{16} - 12 q^{17} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 12 q^{23} + 6 q^{25} - 6 q^{26} - 6 q^{29} - 6 q^{31} - 6 q^{32} + 12 q^{34} - 12 q^{35} - 6 q^{38} + 6 q^{40} - 24 q^{41} - 6 q^{43} - 6 q^{44} + 12 q^{46} - 18 q^{47} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 24 q^{53} - 18 q^{55} + 6 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 6 q^{64} - 12 q^{65} - 24 q^{67} - 12 q^{68} + 12 q^{70} + 6 q^{71} - 24 q^{73} + 6 q^{76} - 12 q^{77} - 12 q^{79} - 6 q^{80} + 24 q^{82} - 18 q^{83} + 6 q^{86} + 6 q^{88} - 12 q^{89} - 30 q^{91} - 12 q^{92} + 18 q^{94} + 6 q^{95} + 6 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^16 - 12 * q^17 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 12 * q^23 + 6 * q^25 - 6 * q^26 - 6 * q^29 - 6 * q^31 - 6 * q^32 + 12 * q^34 - 12 * q^35 - 6 * q^38 + 6 * q^40 - 24 * q^41 - 6 * q^43 - 6 * q^44 + 12 * q^46 - 18 * q^47 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 24 * q^53 - 18 * q^55 + 6 * q^58 - 12 * q^59 - 6 * q^61 + 6 * q^62 + 6 * q^64 - 12 * q^65 - 24 * q^67 - 12 * q^68 + 12 * q^70 + 6 * q^71 - 24 * q^73 + 6 * q^76 - 12 * q^77 - 12 * q^79 - 6 * q^80 + 24 * q^82 - 18 * q^83 + 6 * q^86 + 6 * q^88 - 12 * q^89 - 30 * q^91 - 12 * q^92 + 18 * q^94 + 6 * q^95 + 6 * q^97 - 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.852666	−0.381324	−0.190662	−	0.981656i	$-0.561063\pi$
$5$	−0.852666	−0.381324	−0.190662	+	0.981656i	$0.561063\pi$
$6$	0	0
$7$	1.37581	0.520006	0.260003	−	0.965608i	$-0.416277\pi$
$7$	1.37581	0.520006	0.260003	+	0.965608i	$0.416277\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.852666	0.269637
$11$	−0.216780	−0.0653618	−0.0326809	−	0.999466i	$-0.510405\pi$
$11$	−0.216780	−0.0653618	−0.0326809	+	0.999466i	$0.510405\pi$
$12$	0	0
$13$	1.95778	0.542991	0.271495	−	0.962440i	$-0.412482\pi$
$13$	1.95778	0.542991	0.271495	+	0.962440i	$0.412482\pi$
$14$	−1.37581	−0.367699
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.71783	−1.87185	−0.935925	−	0.352200i	$-0.885434\pi$
$17$	−7.71783	−1.87185	−0.935925	+	0.352200i	$0.885434\pi$
$18$	0	0
$19$	−6.12087	−1.40422	−0.702111	−	0.712067i	$-0.747759\pi$
$19$	−6.12087	−1.40422	−0.702111	+	0.712067i	$0.747759\pi$
$20$	−0.852666	−0.190662
$21$	0	0
$22$	0.216780	0.0462178
$23$	2.41293	0.503131	0.251566	−	0.967840i	$-0.419055\pi$
$23$	2.41293	0.503131	0.251566	+	0.967840i	$0.419055\pi$
$24$	0	0
$25$	−4.27296	−0.854592
$26$	−1.95778	−0.383952
$27$	0	0
$28$	1.37581	0.260003
$29$	−3.94668	−0.732879	−0.366440	−	0.930442i	$-0.619423\pi$
$29$	−3.94668	−0.732879	−0.366440	+	0.930442i	$0.619423\pi$
$30$	0	0
$31$	8.84930	1.58938	0.794691	−	0.607015i	$-0.207633\pi$
$31$	8.84930	1.58938	0.794691	+	0.607015i	$0.207633\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.71783	1.32360
$35$	−1.17310	−0.198290
$36$	0	0
$37$	6.63961	1.09155	0.545773	−	0.837933i	$-0.316236\pi$
$37$	6.63961	1.09155	0.545773	+	0.837933i	$0.316236\pi$
$38$	6.12087	0.992935
$39$	0	0
$40$	0.852666	0.134818
$41$	3.73396	0.583146	0.291573	−	0.956549i	$-0.405821\pi$
$41$	3.73396	0.583146	0.291573	+	0.956549i	$0.405821\pi$
$42$	0	0
$43$	−10.6495	−1.62404	−0.812018	−	0.583632i	$-0.801631\pi$
$43$	−10.6495	−1.62404	−0.812018	+	0.583632i	$0.801631\pi$
$44$	−0.216780	−0.0326809
$45$	0	0
$46$	−2.41293	−0.355768
$47$	1.52861	0.222971	0.111485	−	0.993766i	$-0.464439\pi$
$47$	1.52861	0.222971	0.111485	+	0.993766i	$0.464439\pi$
$48$	0	0
$49$	−5.10716	−0.729594
$50$	4.27296	0.604288
$51$	0	0
$52$	1.95778	0.271495
$53$	−10.2304	−1.40525	−0.702624	−	0.711561i	$-0.747988\pi$
$53$	−10.2304	−1.40525	−0.702624	+	0.711561i	$0.747988\pi$
$54$	0	0
$55$	0.184841	0.0249240
$56$	−1.37581	−0.183850
$57$	0	0
$58$	3.94668	0.518224
$59$	−4.62382	−0.601971	−0.300985	−	0.953629i	$-0.597316\pi$
$59$	−4.62382	−0.601971	−0.300985	+	0.953629i	$0.597316\pi$
$60$	0	0
$61$	5.24719	0.671834	0.335917	−	0.941892i	$-0.390954\pi$
$61$	5.24719	0.671834	0.335917	+	0.941892i	$0.390954\pi$
$62$	−8.84930	−1.12386
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.66933	−0.207055
$66$	0	0
$67$	4.45299	0.544020	0.272010	−	0.962294i	$-0.412312\pi$
$67$	4.45299	0.544020	0.272010	+	0.962294i	$0.412312\pi$
$68$	−7.71783	−0.935925
$69$	0	0
$70$	1.17310	0.140213
$71$	−15.8545	−1.88158	−0.940791	−	0.338987i	$-0.889916\pi$
$71$	−15.8545	−1.88158	−0.940791	+	0.338987i	$0.889916\pi$
$72$	0	0
$73$	−5.39860	−0.631858	−0.315929	−	0.948783i	$-0.602316\pi$
$73$	−5.39860	−0.631858	−0.315929	+	0.948783i	$0.602316\pi$
$74$	−6.63961	−0.771840
$75$	0	0
$76$	−6.12087	−0.702111
$77$	−0.298248	−0.0339885
$78$	0	0
$79$	−8.02895	−0.903328	−0.451664	−	0.892188i	$-0.649169\pi$
$79$	−8.02895	−0.903328	−0.451664	+	0.892188i	$0.649169\pi$
$80$	−0.852666	−0.0953309
$81$	0	0
$82$	−3.73396	−0.412346
$83$	12.0654	1.32435	0.662174	−	0.749350i	$-0.269634\pi$
$83$	12.0654	1.32435	0.662174	+	0.749350i	$0.269634\pi$
$84$	0	0
$85$	6.58073	0.713780
$86$	10.6495	1.14837
$87$	0	0
$88$	0.216780	0.0231089
$89$	−3.61359	−0.383040	−0.191520	−	0.981489i	$-0.561342\pi$
$89$	−3.61359	−0.383040	−0.191520	+	0.981489i	$0.561342\pi$
$90$	0	0
$91$	2.69352	0.282358
$92$	2.41293	0.251566
$93$	0	0
$94$	−1.52861	−0.157664
$95$	5.21905	0.535463
$96$	0	0
$97$	−10.6495	−1.08129	−0.540645	−	0.841251i	$-0.681820\pi$
$97$	−10.6495	−1.08129	−0.540645	+	0.841251i	$0.681820\pi$
$98$	5.10716	0.515901
$99$	0	0
$100$	−4.27296	−0.427296
$101$	13.3509	1.32846	0.664232	−	0.747527i	$-0.268759\pi$
$101$	13.3509	1.32846	0.664232	+	0.747527i	$0.268759\pi$
$102$	0	0
$103$	−14.3519	−1.41414	−0.707069	−	0.707145i	$-0.749983\pi$
$103$	−14.3519	−1.41414	−0.707069	+	0.707145i	$0.749983\pi$
$104$	−1.95778	−0.191976
$105$	0	0
$106$	10.2304	0.993661
$107$	−3.04925	−0.294782	−0.147391	−	0.989078i	$-0.547088\pi$
$107$	−3.04925	−0.294782	−0.147391	+	0.989078i	$0.547088\pi$
$108$	0	0
$109$	9.71567	0.930593	0.465296	−	0.885155i	$-0.345948\pi$
$109$	9.71567	0.930593	0.465296	+	0.885155i	$0.345948\pi$
$110$	−0.184841	−0.0176239
$111$	0	0
$112$	1.37581	0.130001
$113$	−9.26439	−0.871521	−0.435760	−	0.900063i	$-0.643521\pi$
$113$	−9.26439	−0.871521	−0.435760	+	0.900063i	$0.643521\pi$
$114$	0	0
$115$	−2.05742	−0.191856
$116$	−3.94668	−0.366440
$117$	0	0
$118$	4.62382	0.425657
$119$	−10.6182	−0.973372
$120$	0	0
$121$	−10.9530	−0.995728
$122$	−5.24719	−0.475059
$123$	0	0
$124$	8.84930	0.794691
$125$	7.90673	0.707200
$126$	0	0
$127$	−0.0235456	−0.00208933	−0.00104467	−	0.999999i	$-0.500333\pi$
$127$	−0.0235456	−0.00208933	−0.00104467	+	0.999999i	$0.500333\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.66933	0.146410
$131$	−18.5654	−1.62206	−0.811031	−	0.585003i	$-0.801093\pi$
$131$	−18.5654	−1.62206	−0.811031	+	0.585003i	$0.801093\pi$
$132$	0	0
$133$	−8.42112	−0.730204
$134$	−4.45299	−0.384680
$135$	0	0
$136$	7.71783	0.661799
$137$	−7.47470	−0.638607	−0.319303	−	0.947653i	$-0.603449\pi$
$137$	−7.47470	−0.638607	−0.319303	+	0.947653i	$0.603449\pi$
$138$	0	0
$139$	−7.56904	−0.641998	−0.320999	−	0.947080i	$-0.604019\pi$
$139$	−7.56904	−0.641998	−0.320999	+	0.947080i	$0.604019\pi$
$140$	−1.17310	−0.0991452
$141$	0	0
$142$	15.8545	1.33048
$143$	−0.424409	−0.0354908
$144$	0	0
$145$	3.36519	0.279464
$146$	5.39860	0.446791
$147$	0	0
$148$	6.63961	0.545773
$149$	12.6704	1.03800	0.518998	−	0.854775i	$-0.326305\pi$
$149$	12.6704	1.03800	0.518998	+	0.854775i	$0.326305\pi$
$150$	0	0
$151$	3.98749	0.324498	0.162249	−	0.986750i	$-0.448125\pi$
$151$	3.98749	0.324498	0.162249	+	0.986750i	$0.448125\pi$
$152$	6.12087	0.496468
$153$	0	0
$154$	0.298248	0.0240335
$155$	−7.54549	−0.606069
$156$	0	0
$157$	0.523140	0.0417511	0.0208756	−	0.999782i	$-0.493355\pi$
$157$	0.523140	0.0417511	0.0208756	+	0.999782i	$0.493355\pi$
$158$	8.02895	0.638749
$159$	0	0
$160$	0.852666	0.0674091
$161$	3.31973	0.261631
$162$	0	0
$163$	−18.0795	−1.41609	−0.708046	−	0.706166i	$-0.750423\pi$
$163$	−18.0795	−1.41609	−0.708046	+	0.706166i	$0.750423\pi$
$164$	3.73396	0.291573
$165$	0	0
$166$	−12.0654	−0.936455
$167$	19.7557	1.52874	0.764369	−	0.644779i	$-0.223050\pi$
$167$	19.7557	1.52874	0.764369	+	0.644779i	$0.223050\pi$
$168$	0	0
$169$	−9.16710	−0.705161
$170$	−6.58073	−0.504719
$171$	0	0
$172$	−10.6495	−0.812018
$173$	−18.5746	−1.41220	−0.706101	−	0.708112i	$-0.749547\pi$
$173$	−18.5746	−1.41220	−0.706101	+	0.708112i	$0.749547\pi$
$174$	0	0
$175$	−5.87876	−0.444393
$176$	−0.216780	−0.0163404
$177$	0	0
$178$	3.61359	0.270850
$179$	−10.1892	−0.761579	−0.380789	−	0.924662i	$-0.624348\pi$
$179$	−10.1892	−0.761579	−0.380789	+	0.924662i	$0.624348\pi$
$180$	0	0
$181$	3.19006	0.237115	0.118558	−	0.992947i	$-0.462173\pi$
$181$	3.19006	0.237115	0.118558	+	0.992947i	$0.462173\pi$
$182$	−2.69352	−0.199657
$183$	0	0
$184$	−2.41293	−0.177884
$185$	−5.66137	−0.416232
$186$	0	0
$187$	1.67308	0.122347
$188$	1.52861	0.111485
$189$	0	0
$190$	−5.21905	−0.378630
$191$	10.6274	0.768971	0.384485	−	0.923131i	$-0.374379\pi$
$191$	10.6274	0.768971	0.384485	+	0.923131i	$0.374379\pi$
$192$	0	0
$193$	−10.5447	−0.759026	−0.379513	−	0.925186i	$-0.623909\pi$
$193$	−10.5447	−0.759026	−0.379513	+	0.925186i	$0.623909\pi$
$194$	10.6495	0.764588
$195$	0	0
$196$	−5.10716	−0.364797
$197$	14.2848	1.01775	0.508873	−	0.860841i	$-0.330062\pi$
$197$	14.2848	1.01775	0.508873	+	0.860841i	$0.330062\pi$
$198$	0	0
$199$	−14.0693	−0.997348	−0.498674	−	0.866790i	$-0.666180\pi$
$199$	−14.0693	−0.997348	−0.498674	+	0.866790i	$0.666180\pi$
$200$	4.27296	0.302144
$201$	0	0
$202$	−13.3509	−0.939366
$203$	−5.42986	−0.381101
$204$	0	0
$205$	−3.18381	−0.222367
$206$	14.3519	0.999946
$207$	0	0
$208$	1.95778	0.135748
$209$	1.32688	0.0917825
$210$	0	0
$211$	21.1542	1.45632	0.728159	−	0.685408i	$-0.240376\pi$
$211$	21.1542	1.45632	0.728159	+	0.685408i	$0.240376\pi$
$212$	−10.2304	−0.702624
$213$	0	0
$214$	3.04925	0.208443
$215$	9.08048	0.619283
$216$	0	0
$217$	12.1749	0.826487
$218$	−9.71567	−0.658028
$219$	0	0
$220$	0.184841	0.0124620
$221$	−15.1098	−1.01640
$222$	0	0
$223$	−7.33082	−0.490908	−0.245454	−	0.969408i	$-0.578937\pi$
$223$	−7.33082	−0.490908	−0.245454	+	0.969408i	$0.578937\pi$
$224$	−1.37581	−0.0919249
$225$	0	0
$226$	9.26439	0.616258
$227$	−6.28319	−0.417030	−0.208515	−	0.978019i	$-0.566863\pi$
$227$	−6.28319	−0.417030	−0.208515	+	0.978019i	$0.566863\pi$
$228$	0	0
$229$	−0.0268956	−0.00177731	−0.000888656	−	1.00000i	$-0.500283\pi$
$229$	−0.0268956	−0.00177731	−0.000888656		1.00000i	$0.500283\pi$
$230$	2.05742	0.135663
$231$	0	0
$232$	3.94668	0.259112
$233$	23.8435	1.56204	0.781019	−	0.624508i	$-0.214700\pi$
$233$	23.8435	1.56204	0.781019	+	0.624508i	$0.214700\pi$
$234$	0	0
$235$	−1.30339	−0.0850240
$236$	−4.62382	−0.300985
$237$	0	0
$238$	10.6182	0.688278
$239$	15.3682	0.994088	0.497044	−	0.867725i	$-0.334419\pi$
$239$	15.3682	0.994088	0.497044	+	0.867725i	$0.334419\pi$
$240$	0	0
$241$	−0.964379	−0.0621211	−0.0310605	−	0.999518i	$-0.509888\pi$
$241$	−0.964379	−0.0621211	−0.0310605	+	0.999518i	$0.509888\pi$
$242$	10.9530	0.704086
$243$	0	0
$244$	5.24719	0.335917
$245$	4.35470	0.278212
$246$	0	0
$247$	−11.9833	−0.762480
$248$	−8.84930	−0.561931
$249$	0	0
$250$	−7.90673	−0.500066
$251$	18.2157	1.14977	0.574884	−	0.818235i	$-0.305047\pi$
$251$	18.2157	1.14977	0.574884	+	0.818235i	$0.305047\pi$
$252$	0	0
$253$	−0.523077	−0.0328856
$254$	0.0235456	0.00147738
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.3732	1.02133	0.510665	−	0.859780i	$-0.329399\pi$
$257$	16.3732	1.02133	0.510665	+	0.859780i	$0.329399\pi$
$258$	0	0
$259$	9.13482	0.567610
$260$	−1.66933	−0.103528
$261$	0	0
$262$	18.5654	1.14697
$263$	13.4275	0.827977	0.413989	−	0.910282i	$-0.364135\pi$
$263$	13.4275	0.827977	0.413989	+	0.910282i	$0.364135\pi$
$264$	0	0
$265$	8.72308	0.535854
$266$	8.42112	0.516332
$267$	0	0
$268$	4.45299	0.272010
$269$	−1.91176	−0.116562	−0.0582809	−	0.998300i	$-0.518562\pi$
$269$	−1.91176	−0.116562	−0.0582809	+	0.998300i	$0.518562\pi$
$270$	0	0
$271$	−4.71707	−0.286542	−0.143271	−	0.989684i	$-0.545762\pi$
$271$	−4.71707	−0.286542	−0.143271	+	0.989684i	$0.545762\pi$
$272$	−7.71783	−0.467962
$273$	0	0
$274$	7.47470	0.451563
$275$	0.926295	0.0558577
$276$	0	0
$277$	−21.8798	−1.31463	−0.657316	−	0.753615i	$-0.728308\pi$
$277$	−21.8798	−1.31463	−0.657316	+	0.753615i	$0.728308\pi$
$278$	7.56904	0.453961
$279$	0	0
$280$	1.17310	0.0701063
$281$	23.7942	1.41945	0.709723	−	0.704481i	$-0.248820\pi$
$281$	23.7942	1.41945	0.709723	+	0.704481i	$0.248820\pi$
$282$	0	0
$283$	6.28259	0.373461	0.186731	−	0.982411i	$-0.440211\pi$
$283$	6.28259	0.373461	0.186731	+	0.982411i	$0.440211\pi$
$284$	−15.8545	−0.940791
$285$	0	0
$286$	0.424409	0.0250958
$287$	5.13720	0.303239
$288$	0	0
$289$	42.5650	2.50382
$290$	−3.36519	−0.197611
$291$	0	0
$292$	−5.39860	−0.315929
$293$	−2.73672	−0.159881	−0.0799404	−	0.996800i	$-0.525473\pi$
$293$	−2.73672	−0.159881	−0.0799404	+	0.996800i	$0.525473\pi$
$294$	0	0
$295$	3.94258	0.229546
$296$	−6.63961	−0.385920
$297$	0	0
$298$	−12.6704	−0.733974
$299$	4.72399	0.273196
$300$	0	0
$301$	−14.6517	−0.844508
$302$	−3.98749	−0.229454
$303$	0	0
$304$	−6.12087	−0.351056
$305$	−4.47410	−0.256186
$306$	0	0
$307$	−28.4426	−1.62331	−0.811654	−	0.584139i	$-0.801432\pi$
$307$	−28.4426	−1.62331	−0.811654	+	0.584139i	$0.801432\pi$
$308$	−0.298248	−0.0169942
$309$	0	0
$310$	7.54549	0.428555
$311$	−29.0437	−1.64692	−0.823458	−	0.567377i	$-0.807958\pi$
$311$	−29.0437	−1.64692	−0.823458	+	0.567377i	$0.807958\pi$
$312$	0	0
$313$	16.6720	0.942357	0.471178	−	0.882038i	$-0.343829\pi$
$313$	16.6720	0.942357	0.471178	+	0.882038i	$0.343829\pi$
$314$	−0.523140	−0.0295225
$315$	0	0
$316$	−8.02895	−0.451664
$317$	−13.1666	−0.739511	−0.369756	−	0.929129i	$-0.620559\pi$
$317$	−13.1666	−0.739511	−0.369756	+	0.929129i	$0.620559\pi$
$318$	0	0
$319$	0.855562	0.0479023
$320$	−0.852666	−0.0476655
$321$	0	0
$322$	−3.31973	−0.185001
$323$	47.2398	2.62849
$324$	0	0
$325$	−8.36552	−0.464036
$326$	18.0795	1.00133
$327$	0	0
$328$	−3.73396	−0.206173
$329$	2.10307	0.115946
$330$	0	0
$331$	−23.6804	−1.30159	−0.650795	−	0.759253i	$-0.725564\pi$
$331$	−23.6804	−1.30159	−0.650795	+	0.759253i	$0.725564\pi$
$332$	12.0654	0.662174
$333$	0	0
$334$	−19.7557	−1.08098
$335$	−3.79691	−0.207448
$336$	0	0
$337$	26.5282	1.44508	0.722542	−	0.691327i	$-0.242973\pi$
$337$	26.5282	1.44508	0.722542	+	0.691327i	$0.242973\pi$
$338$	9.16710	0.498624
$339$	0	0
$340$	6.58073	0.356890
$341$	−1.91836	−0.103885
$342$	0	0
$343$	−16.6571	−0.899399
$344$	10.6495	0.574183
$345$	0	0
$346$	18.5746	0.998577
$347$	13.8715	0.744660	0.372330	−	0.928100i	$-0.378559\pi$
$347$	13.8715	0.744660	0.372330	+	0.928100i	$0.378559\pi$
$348$	0	0
$349$	23.7682	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$349$	23.7682	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$350$	5.87876	0.314233
$351$	0	0
$352$	0.216780	0.0115544
$353$	−31.9264	−1.69927	−0.849636	−	0.527369i	$-0.823179\pi$
$353$	−31.9264	−1.69927	−0.849636	+	0.527369i	$0.823179\pi$
$354$	0	0
$355$	13.5186	0.717492
$356$	−3.61359	−0.191520
$357$	0	0
$358$	10.1892	0.538518
$359$	35.0803	1.85147	0.925734	−	0.378174i	$-0.123448\pi$
$359$	35.0803	1.85147	0.925734	+	0.378174i	$0.123448\pi$
$360$	0	0
$361$	18.4650	0.971842
$362$	−3.19006	−0.167666
$363$	0	0
$364$	2.69352	0.141179
$365$	4.60320	0.240943
$366$	0	0
$367$	−14.5368	−0.758813	−0.379407	−	0.925230i	$-0.623872\pi$
$367$	−14.5368	−0.758813	−0.379407	+	0.925230i	$0.623872\pi$
$368$	2.41293	0.125783
$369$	0	0
$370$	5.66137	0.294321
$371$	−14.0750	−0.730737
$372$	0	0
$373$	23.2530	1.20400	0.601998	−	0.798497i	$-0.294371\pi$
$373$	23.2530	1.20400	0.601998	+	0.798497i	$0.294371\pi$
$374$	−1.67308	−0.0865127
$375$	0	0
$376$	−1.52861	−0.0788320
$377$	−7.72672	−0.397946
$378$	0	0
$379$	−8.25841	−0.424206	−0.212103	−	0.977247i	$-0.568031\pi$
$379$	−8.25841	−0.424206	−0.212103	+	0.977247i	$0.568031\pi$
$380$	5.21905	0.267732
$381$	0	0
$382$	−10.6274	−0.543745
$383$	−9.20617	−0.470413	−0.235207	−	0.971945i	$-0.575577\pi$
$383$	−9.20617	−0.470413	−0.235207	+	0.971945i	$0.575577\pi$
$384$	0	0
$385$	0.254306	0.0129606
$386$	10.5447	0.536713
$387$	0	0
$388$	−10.6495	−0.540645
$389$	34.8315	1.76603	0.883014	−	0.469348i	$-0.155511\pi$
$389$	34.8315	1.76603	0.883014	+	0.469348i	$0.155511\pi$
$390$	0	0
$391$	−18.6226	−0.941786
$392$	5.10716	0.257950
$393$	0	0
$394$	−14.2848	−0.719655
$395$	6.84601	0.344460
$396$	0	0
$397$	30.4492	1.52820	0.764100	−	0.645097i	$-0.223183\pi$
$397$	30.4492	1.52820	0.764100	+	0.645097i	$0.223183\pi$
$398$	14.0693	0.705232
$399$	0	0
$400$	−4.27296	−0.213648
$401$	6.84345	0.341745	0.170873	−	0.985293i	$-0.445341\pi$
$401$	6.84345	0.341745	0.170873	+	0.985293i	$0.445341\pi$
$402$	0	0
$403$	17.3250	0.863019
$404$	13.3509	0.664232
$405$	0	0
$406$	5.42986	0.269479
$407$	−1.43934	−0.0713454
$408$	0	0
$409$	16.2080	0.801435	0.400717	−	0.916202i	$-0.368761\pi$
$409$	16.2080	0.801435	0.400717	+	0.916202i	$0.368761\pi$
$410$	3.18381	0.157237
$411$	0	0
$412$	−14.3519	−0.707069
$413$	−6.36148	−0.313028
$414$	0	0
$415$	−10.2877	−0.505005
$416$	−1.95778	−0.0959881
$417$	0	0
$418$	−1.32688	−0.0649000
$419$	12.1019	0.591215	0.295608	−	0.955309i	$-0.404478\pi$
$419$	12.1019	0.591215	0.295608	+	0.955309i	$0.404478\pi$
$420$	0	0
$421$	25.4767	1.24166	0.620830	−	0.783945i	$-0.286796\pi$
$421$	25.4767	1.24166	0.620830	+	0.783945i	$0.286796\pi$
$422$	−21.1542	−1.02977
$423$	0	0
$424$	10.2304	0.496830
$425$	32.9780	1.59967
$426$	0	0
$427$	7.21912	0.349358
$428$	−3.04925	−0.147391
$429$	0	0
$430$	−9.08048	−0.437899
$431$	−10.5220	−0.506828	−0.253414	−	0.967358i	$-0.581554\pi$
$431$	−10.5220	−0.506828	−0.253414	+	0.967358i	$0.581554\pi$
$432$	0	0
$433$	6.89705	0.331451	0.165726	−	0.986172i	$-0.447003\pi$
$433$	6.89705	0.331451	0.165726	+	0.986172i	$0.447003\pi$
$434$	−12.1749	−0.584415
$435$	0	0
$436$	9.71567	0.465296
$437$	−14.7692	−0.706508
$438$	0	0
$439$	13.8072	0.658981	0.329491	−	0.944159i	$-0.393123\pi$
$439$	13.8072	0.658981	0.329491	+	0.944159i	$0.393123\pi$
$440$	−0.184841	−0.00881196
$441$	0	0
$442$	15.1098	0.718701
$443$	3.20767	0.152401	0.0762005	−	0.997093i	$-0.475721\pi$
$443$	3.20767	0.152401	0.0762005	+	0.997093i	$0.475721\pi$
$444$	0	0
$445$	3.08119	0.146062
$446$	7.33082	0.347124
$447$	0	0
$448$	1.37581	0.0650007
$449$	26.3248	1.24235	0.621173	−	0.783673i	$-0.286656\pi$
$449$	26.3248	1.24235	0.621173	+	0.783673i	$0.286656\pi$
$450$	0	0
$451$	−0.809449	−0.0381154
$452$	−9.26439	−0.435760
$453$	0	0
$454$	6.28319	0.294885
$455$	−2.29668	−0.107670
$456$	0	0
$457$	−3.98002	−0.186177	−0.0930887	−	0.995658i	$-0.529674\pi$
$457$	−3.98002	−0.186177	−0.0930887	+	0.995658i	$0.529674\pi$
$458$	0.0268956	0.00125675
$459$	0	0
$460$	−2.05742	−0.0959279
$461$	−7.93927	−0.369769	−0.184884	−	0.982760i	$-0.559191\pi$
$461$	−7.93927	−0.369769	−0.184884	+	0.982760i	$0.559191\pi$
$462$	0	0
$463$	−27.6106	−1.28317	−0.641587	−	0.767050i	$-0.721724\pi$
$463$	−27.6106	−1.28317	−0.641587	+	0.767050i	$0.721724\pi$
$464$	−3.94668	−0.183220
$465$	0	0
$466$	−23.8435	−1.10453
$467$	−20.6638	−0.956207	−0.478103	−	0.878304i	$-0.658676\pi$
$467$	−20.6638	−0.956207	−0.478103	+	0.878304i	$0.658676\pi$
$468$	0	0
$469$	6.12645	0.282893
$470$	1.30339	0.0601210
$471$	0	0
$472$	4.62382	0.212829
$473$	2.30861	0.106150
$474$	0	0
$475$	26.1542	1.20004
$476$	−10.6182	−0.486686
$477$	0	0
$478$	−15.3682	−0.702926
$479$	−30.4658	−1.39202	−0.696009	−	0.718033i	$-0.745043\pi$
$479$	−30.4658	−1.39202	−0.696009	+	0.718033i	$0.745043\pi$
$480$	0	0
$481$	12.9989	0.592699
$482$	0.964379	0.0439262
$483$	0	0
$484$	−10.9530	−0.497864
$485$	9.08044	0.412322
$486$	0	0
$487$	−11.9590	−0.541912	−0.270956	−	0.962592i	$-0.587340\pi$
$487$	−11.9590	−0.541912	−0.270956	+	0.962592i	$0.587340\pi$
$488$	−5.24719	−0.237529
$489$	0	0
$490$	−4.35470	−0.196725
$491$	40.9377	1.84749	0.923745	−	0.383007i	$-0.125111\pi$
$491$	40.9377	1.84749	0.923745	+	0.383007i	$0.125111\pi$
$492$	0	0
$493$	30.4598	1.37184
$494$	11.9833	0.539155
$495$	0	0
$496$	8.84930	0.397345
$497$	−21.8127	−0.978433
$498$	0	0
$499$	6.69932	0.299903	0.149951	−	0.988693i	$-0.452088\pi$
$499$	6.69932	0.299903	0.149951	+	0.988693i	$0.452088\pi$
$500$	7.90673	0.353600
$501$	0	0
$502$	−18.2157	−0.813009
$503$	−33.2407	−1.48213	−0.741065	−	0.671433i	$-0.765679\pi$
$503$	−33.2407	−1.48213	−0.741065	+	0.671433i	$0.765679\pi$
$504$	0	0
$505$	−11.3839	−0.506575
$506$	0.523077	0.0232536
$507$	0	0
$508$	−0.0235456	−0.00104467
$509$	11.8539	0.525414	0.262707	−	0.964876i	$-0.415385\pi$
$509$	11.8539	0.525414	0.262707	+	0.964876i	$0.415385\pi$
$510$	0	0
$511$	−7.42742	−0.328570
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.3732	−0.722189
$515$	12.2374	0.539244
$516$	0	0
$517$	−0.331373	−0.0145738
$518$	−9.13482	−0.401361
$519$	0	0
$520$	1.66933	0.0732050
$521$	−4.70898	−0.206304	−0.103152	−	0.994666i	$-0.532893\pi$
$521$	−4.70898	−0.206304	−0.103152	+	0.994666i	$0.532893\pi$
$522$	0	0
$523$	−16.4344	−0.718624	−0.359312	−	0.933217i	$-0.616989\pi$
$523$	−16.4344	−0.718624	−0.359312	+	0.933217i	$0.616989\pi$
$524$	−18.5654	−0.811031
$525$	0	0
$526$	−13.4275	−0.585468
$527$	−68.2974	−2.97508
$528$	0	0
$529$	−17.1778	−0.746859
$530$	−8.72308	−0.378906
$531$	0	0
$532$	−8.42112	−0.365102
$533$	7.31026	0.316643
$534$	0	0
$535$	2.59999	0.112407
$536$	−4.45299	−0.192340
$537$	0	0
$538$	1.91176	0.0824216
$539$	1.10713	0.0476876
$540$	0	0
$541$	26.0900	1.12170	0.560849	−	0.827918i	$-0.310475\pi$
$541$	26.0900	1.12170	0.560849	+	0.827918i	$0.310475\pi$
$542$	4.71707	0.202616
$543$	0	0
$544$	7.71783	0.330899
$545$	−8.28422	−0.354857
$546$	0	0
$547$	28.3476	1.21205	0.606027	−	0.795444i	$-0.292762\pi$
$547$	28.3476	1.21205	0.606027	+	0.795444i	$0.292762\pi$
$548$	−7.47470	−0.319303
$549$	0	0
$550$	−0.926295	−0.0394973
$551$	24.1571	1.02913
$552$	0	0
$553$	−11.0463	−0.469736
$554$	21.8798	0.929585
$555$	0	0
$556$	−7.56904	−0.320999
$557$	29.2139	1.23783	0.618916	−	0.785457i	$-0.287572\pi$
$557$	29.2139	1.23783	0.618916	+	0.785457i	$0.287572\pi$
$558$	0	0
$559$	−20.8494	−0.881836
$560$	−1.17310	−0.0495726
$561$	0	0
$562$	−23.7942	−1.00370
$563$	2.30342	0.0970776	0.0485388	−	0.998821i	$-0.484544\pi$
$563$	2.30342	0.0970776	0.0485388	+	0.998821i	$0.484544\pi$
$564$	0	0
$565$	7.89943	0.332332
$566$	−6.28259	−0.264077
$567$	0	0
$568$	15.8545	0.665240
$569$	4.26971	0.178996	0.0894978	−	0.995987i	$-0.471474\pi$
$569$	4.26971	0.178996	0.0894978	+	0.995987i	$0.471474\pi$
$570$	0	0
$571$	19.1153	0.799950	0.399975	−	0.916526i	$-0.369019\pi$
$571$	19.1153	0.799950	0.399975	+	0.916526i	$0.369019\pi$
$572$	−0.424409	−0.0177454
$573$	0	0
$574$	−5.13720	−0.214422
$575$	−10.3104	−0.429972
$576$	0	0
$577$	8.31584	0.346193	0.173096	−	0.984905i	$-0.444623\pi$
$577$	8.31584	0.346193	0.173096	+	0.984905i	$0.444623\pi$
$578$	−42.5650	−1.77047
$579$	0	0
$580$	3.36519	0.139732
$581$	16.5996	0.688668
$582$	0	0
$583$	2.21774	0.0918495
$584$	5.39860	0.223396
$585$	0	0
$586$	2.73672	0.113053
$587$	−36.5516	−1.50865	−0.754324	−	0.656502i	$-0.772035\pi$
$587$	−36.5516	−1.50865	−0.754324	+	0.656502i	$0.772035\pi$
$588$	0	0
$589$	−54.1654	−2.23185
$590$	−3.94258	−0.162313
$591$	0	0
$592$	6.63961	0.272886
$593$	−22.4408	−0.921535	−0.460768	−	0.887521i	$-0.652426\pi$
$593$	−22.4408	−0.921535	−0.460768	+	0.887521i	$0.652426\pi$
$594$	0	0
$595$	9.05381	0.371170
$596$	12.6704	0.518998
$597$	0	0
$598$	−4.72399	−0.193178
$599$	19.3129	0.789105	0.394553	−	0.918873i	$-0.370900\pi$
$599$	19.3129	0.789105	0.394553	+	0.918873i	$0.370900\pi$
$600$	0	0
$601$	−8.68146	−0.354124	−0.177062	−	0.984200i	$-0.556659\pi$
$601$	−8.68146	−0.354124	−0.177062	+	0.984200i	$0.556659\pi$
$602$	14.6517	0.597157
$603$	0	0
$604$	3.98749	0.162249
$605$	9.33925	0.379695
$606$	0	0
$607$	8.27174	0.335740	0.167870	−	0.985809i	$-0.446311\pi$
$607$	8.27174	0.335740	0.167870	+	0.985809i	$0.446311\pi$
$608$	6.12087	0.248234
$609$	0	0
$610$	4.47410	0.181151
$611$	2.99268	0.121071
$612$	0	0
$613$	−15.0384	−0.607394	−0.303697	−	0.952769i	$-0.598221\pi$
$613$	−15.0384	−0.607394	−0.303697	+	0.952769i	$0.598221\pi$
$614$	28.4426	1.14785
$615$	0	0
$616$	0.298248	0.0120167
$617$	−32.6891	−1.31601	−0.658006	−	0.753012i	$-0.728600\pi$
$617$	−32.6891	−1.31601	−0.658006	+	0.753012i	$0.728600\pi$
$618$	0	0
$619$	−29.2318	−1.17493	−0.587463	−	0.809251i	$-0.699873\pi$
$619$	−29.2318	−1.17493	−0.587463	+	0.809251i	$0.699873\pi$
$620$	−7.54549	−0.303034
$621$	0	0
$622$	29.0437	1.16455
$623$	−4.97160	−0.199183
$624$	0	0
$625$	14.6230	0.584920
$626$	−16.6720	−0.666347
$627$	0	0
$628$	0.523140	0.0208756
$629$	−51.2434	−2.04321
$630$	0	0
$631$	37.6857	1.50024	0.750121	−	0.661300i	$-0.229995\pi$
$631$	37.6857	1.50024	0.750121	+	0.661300i	$0.229995\pi$
$632$	8.02895	0.319375
$633$	0	0
$634$	13.1666	0.522913
$635$	0.0200765	0.000796713 0
$636$	0	0
$637$	−9.99870	−0.396163
$638$	−0.855562	−0.0338720
$639$	0	0
$640$	0.852666	0.0337046
$641$	−8.66494	−0.342244	−0.171122	−	0.985250i	$-0.554739\pi$
$641$	−8.66494	−0.342244	−0.171122	+	0.985250i	$0.554739\pi$
$642$	0	0
$643$	−10.0722	−0.397210	−0.198605	−	0.980080i	$-0.563641\pi$
$643$	−10.0722	−0.397210	−0.198605	+	0.980080i	$0.563641\pi$
$644$	3.31973	0.130816
$645$	0	0
$646$	−47.2398	−1.85863
$647$	−25.0736	−0.985745	−0.492873	−	0.870101i	$-0.664053\pi$
$647$	−25.0736	−0.985745	−0.492873	+	0.870101i	$0.664053\pi$
$648$	0	0
$649$	1.00235	0.0393459
$650$	8.36552	0.328123
$651$	0	0
$652$	−18.0795	−0.708046
$653$	−12.3050	−0.481530	−0.240765	−	0.970583i	$-0.577398\pi$
$653$	−12.3050	−0.481530	−0.240765	+	0.970583i	$0.577398\pi$
$654$	0	0
$655$	15.8300	0.618531
$656$	3.73396	0.145786
$657$	0	0
$658$	−2.10307	−0.0819862
$659$	−29.0125	−1.13017	−0.565084	−	0.825033i	$-0.691156\pi$
$659$	−29.0125	−1.13017	−0.565084	+	0.825033i	$0.691156\pi$
$660$	0	0
$661$	−40.9322	−1.59208	−0.796039	−	0.605245i	$-0.793075\pi$
$661$	−40.9322	−1.59208	−0.796039	+	0.605245i	$0.793075\pi$
$662$	23.6804	0.920364
$663$	0	0
$664$	−12.0654	−0.468228
$665$	7.18040	0.278444
$666$	0	0
$667$	−9.52306	−0.368734
$668$	19.7557	0.764369
$669$	0	0
$670$	3.79691	0.146688
$671$	−1.13749	−0.0439123
$672$	0	0
$673$	10.9601	0.422482	0.211241	−	0.977434i	$-0.432249\pi$
$673$	10.9601	0.422482	0.211241	+	0.977434i	$0.432249\pi$
$674$	−26.5282	−1.02183
$675$	0	0
$676$	−9.16710	−0.352581
$677$	−29.9509	−1.15111	−0.575554	−	0.817764i	$-0.695214\pi$
$677$	−29.9509	−1.15111	−0.575554	+	0.817764i	$0.695214\pi$
$678$	0	0
$679$	−14.6516	−0.562277
$680$	−6.58073	−0.252360
$681$	0	0
$682$	1.91836	0.0734576
$683$	−41.8638	−1.60187	−0.800937	−	0.598749i	$-0.795665\pi$
$683$	−41.8638	−1.60187	−0.800937	+	0.598749i	$0.795665\pi$
$684$	0	0
$685$	6.37342	0.243516
$686$	16.6571	0.635971
$687$	0	0
$688$	−10.6495	−0.406009
$689$	−20.0288	−0.763037
$690$	0	0
$691$	16.0023	0.608758	0.304379	−	0.952551i	$-0.401551\pi$
$691$	16.0023	0.608758	0.304379	+	0.952551i	$0.401551\pi$
$692$	−18.5746	−0.706101
$693$	0	0
$694$	−13.8715	−0.526554
$695$	6.45386	0.244809
$696$	0	0
$697$	−28.8180	−1.09156
$698$	−23.7682	−0.899640
$699$	0	0
$700$	−5.87876	−0.222196
$701$	−31.8940	−1.20462	−0.602309	−	0.798263i	$-0.705753\pi$
$701$	−31.8940	−1.20462	−0.602309	+	0.798263i	$0.705753\pi$
$702$	0	0
$703$	−40.6402	−1.53277
$704$	−0.216780	−0.00817022
$705$	0	0
$706$	31.9264	1.20157
$707$	18.3682	0.690809
$708$	0	0
$709$	28.5427	1.07194	0.535972	−	0.844236i	$-0.319945\pi$
$709$	28.5427	1.07194	0.535972	+	0.844236i	$0.319945\pi$
$710$	−13.5186	−0.507343
$711$	0	0
$712$	3.61359	0.135425
$713$	21.3528	0.799668
$714$	0	0
$715$	0.361879	0.0135335
$716$	−10.1892	−0.380789
$717$	0	0
$718$	−35.0803	−1.30919
$719$	−26.5082	−0.988590	−0.494295	−	0.869294i	$-0.664574\pi$
$719$	−26.5082	−0.988590	−0.494295	+	0.869294i	$0.664574\pi$
$720$	0	0
$721$	−19.7455	−0.735359
$722$	−18.4650	−0.687196
$723$	0	0
$724$	3.19006	0.118558
$725$	16.8640	0.626313
$726$	0	0
$727$	37.9038	1.40577	0.702887	−	0.711302i	$-0.251894\pi$
$727$	37.9038	1.40577	0.702887	+	0.711302i	$0.251894\pi$
$728$	−2.69352	−0.0998287
$729$	0	0
$730$	−4.60320	−0.170372
$731$	82.1912	3.03995
$732$	0	0
$733$	18.3195	0.676647	0.338324	−	0.941030i	$-0.390140\pi$
$733$	18.3195	0.676647	0.338324	+	0.941030i	$0.390140\pi$
$734$	14.5368	0.536562
$735$	0	0
$736$	−2.41293	−0.0889419
$737$	−0.965322	−0.0355581
$738$	0	0
$739$	−6.16227	−0.226683	−0.113341	−	0.993556i	$-0.536155\pi$
$739$	−6.16227	−0.226683	−0.113341	+	0.993556i	$0.536155\pi$
$740$	−5.66137	−0.208116
$741$	0	0
$742$	14.0750	0.516709
$743$	−0.542730	−0.0199108	−0.00995542	−	0.999950i	$-0.503169\pi$
$743$	−0.542730	−0.0199108	−0.00995542	+	0.999950i	$0.503169\pi$
$744$	0	0
$745$	−10.8036	−0.395812
$746$	−23.2530	−0.851354
$747$	0	0
$748$	1.67308	0.0611737
$749$	−4.19518	−0.153288
$750$	0	0
$751$	−39.9221	−1.45678	−0.728389	−	0.685164i	$-0.759730\pi$
$751$	−39.9221	−1.45678	−0.728389	+	0.685164i	$0.759730\pi$
$752$	1.52861	0.0557427
$753$	0	0
$754$	7.72672	0.281391
$755$	−3.40000	−0.123739
$756$	0	0
$757$	24.6220	0.894903	0.447452	−	0.894308i	$-0.352332\pi$
$757$	24.6220	0.894903	0.447452	+	0.894308i	$0.352332\pi$
$758$	8.25841	0.299959
$759$	0	0
$760$	−5.21905	−0.189315
$761$	18.8199	0.682221	0.341111	−	0.940023i	$-0.389197\pi$
$761$	18.8199	0.682221	0.341111	+	0.940023i	$0.389197\pi$
$762$	0	0
$763$	13.3669	0.483913
$764$	10.6274	0.384485
$765$	0	0
$766$	9.20617	0.332632
$767$	−9.05243	−0.326864
$768$	0	0
$769$	10.2179	0.368468	0.184234	−	0.982882i	$-0.441020\pi$
$769$	10.2179	0.368468	0.184234	+	0.982882i	$0.441020\pi$
$770$	−0.254306	−0.00916454
$771$	0	0
$772$	−10.5447	−0.379513
$773$	19.8681	0.714605	0.357302	−	0.933989i	$-0.383697\pi$
$773$	19.8681	0.714605	0.357302	+	0.933989i	$0.383697\pi$
$774$	0	0
$775$	−37.8127	−1.35827
$776$	10.6495	0.382294
$777$	0	0
$778$	−34.8315	−1.24877
$779$	−22.8550	−0.818867
$780$	0	0
$781$	3.43694	0.122984
$782$	18.6226	0.665943
$783$	0	0
$784$	−5.10716	−0.182399
$785$	−0.446063	−0.0159207
$786$	0	0
$787$	31.9487	1.13885	0.569424	−	0.822044i	$-0.307166\pi$
$787$	31.9487	1.13885	0.569424	+	0.822044i	$0.307166\pi$
$788$	14.2848	0.508873
$789$	0	0
$790$	−6.84601	−0.243570
$791$	−12.7460	−0.453196
$792$	0	0
$793$	10.2729	0.364800
$794$	−30.4492	−1.08060
$795$	0	0
$796$	−14.0693	−0.498674
$797$	9.66226	0.342255	0.171127	−	0.985249i	$-0.445259\pi$
$797$	9.66226	0.342255	0.171127	+	0.985249i	$0.445259\pi$
$798$	0	0
$799$	−11.7976	−0.417368
$800$	4.27296	0.151072
$801$	0	0
$802$	−6.84345	−0.241651
$803$	1.17031	0.0412994
$804$	0	0
$805$	−2.83062	−0.0997661
$806$	−17.3250	−0.610247
$807$	0	0
$808$	−13.3509	−0.469683
$809$	−10.0580	−0.353621	−0.176811	−	0.984245i	$-0.556578\pi$
$809$	−10.0580	−0.353621	−0.176811	+	0.984245i	$0.556578\pi$
$810$	0	0
$811$	5.87482	0.206293	0.103146	−	0.994666i	$-0.467109\pi$
$811$	5.87482	0.206293	0.103146	+	0.994666i	$0.467109\pi$
$812$	−5.42986	−0.190551
$813$	0	0
$814$	1.43934	0.0504488
$815$	15.4157	0.539989
$816$	0	0
$817$	65.1843	2.28051
$818$	−16.2080	−0.566700
$819$	0	0
$820$	−3.18381	−0.111184
$821$	24.8647	0.867785	0.433893	−	0.900965i	$-0.357140\pi$
$821$	24.8647	0.867785	0.433893	+	0.900965i	$0.357140\pi$
$822$	0	0
$823$	−7.16718	−0.249832	−0.124916	−	0.992167i	$-0.539866\pi$
$823$	−7.16718	−0.249832	−0.124916	+	0.992167i	$0.539866\pi$
$824$	14.3519	0.499973
$825$	0	0
$826$	6.36148	0.221344
$827$	−10.5317	−0.366223	−0.183112	−	0.983092i	$-0.558617\pi$
$827$	−10.5317	−0.366223	−0.183112	+	0.983092i	$0.558617\pi$
$828$	0	0
$829$	20.5499	0.713728	0.356864	−	0.934156i	$-0.383846\pi$
$829$	20.5499	0.713728	0.356864	+	0.934156i	$0.383846\pi$
$830$	10.2877	0.357092
$831$	0	0
$832$	1.95778	0.0678738
$833$	39.4162	1.36569
$834$	0	0
$835$	−16.8450	−0.582944
$836$	1.32688	0.0458912
$837$	0	0
$838$	−12.1019	−0.418052
$839$	−42.6483	−1.47238	−0.736192	−	0.676773i	$-0.763378\pi$
$839$	−42.6483	−1.47238	−0.736192	+	0.676773i	$0.763378\pi$
$840$	0	0
$841$	−13.4238	−0.462888
$842$	−25.4767	−0.877986
$843$	0	0
$844$	21.1542	0.728159
$845$	7.81647	0.268895
$846$	0	0
$847$	−15.0692	−0.517784
$848$	−10.2304	−0.351312
$849$	0	0
$850$	−32.9780	−1.13114
$851$	16.0209	0.549191
$852$	0	0
$853$	−21.3810	−0.732071	−0.366036	−	0.930601i	$-0.619285\pi$
$853$	−21.3810	−0.732071	−0.366036	+	0.930601i	$0.619285\pi$
$854$	−7.21912	−0.247033
$855$	0	0
$856$	3.04925	0.104221
$857$	2.73552	0.0934434	0.0467217	−	0.998908i	$-0.485123\pi$
$857$	2.73552	0.0934434	0.0467217	+	0.998908i	$0.485123\pi$
$858$	0	0
$859$	11.9332	0.407156	0.203578	−	0.979059i	$-0.434743\pi$
$859$	11.9332	0.407156	0.203578	+	0.979059i	$0.434743\pi$
$860$	9.08048	0.309642
$861$	0	0
$862$	10.5220	0.358382
$863$	−9.23050	−0.314210	−0.157105	−	0.987582i	$-0.550216\pi$
$863$	−9.23050	−0.314210	−0.157105	+	0.987582i	$0.550216\pi$
$864$	0	0
$865$	15.8379	0.538506
$866$	−6.89705	−0.234371
$867$	0	0
$868$	12.1749	0.413244
$869$	1.74052	0.0590431
$870$	0	0
$871$	8.71798	0.295397
$872$	−9.71567	−0.329014
$873$	0	0
$874$	14.7692	0.499577
$875$	10.8781	0.367748
$876$	0	0
$877$	32.0380	1.08185	0.540924	−	0.841072i	$-0.318075\pi$
$877$	32.0380	1.08185	0.540924	+	0.841072i	$0.318075\pi$
$878$	−13.8072	−0.465970
$879$	0	0
$880$	0.184841	0.00623100
$881$	−20.5135	−0.691116	−0.345558	−	0.938397i	$-0.612310\pi$
$881$	−20.5135	−0.691116	−0.345558	+	0.938397i	$0.612310\pi$
$882$	0	0
$883$	2.02711	0.0682176	0.0341088	−	0.999418i	$-0.489141\pi$
$883$	2.02711	0.0682176	0.0341088	+	0.999418i	$0.489141\pi$
$884$	−15.1098	−0.508198
$885$	0	0
$886$	−3.20767	−0.107764
$887$	−0.862252	−0.0289516	−0.0144758	−	0.999895i	$-0.504608\pi$
$887$	−0.862252	−0.0289516	−0.0144758	+	0.999895i	$0.504608\pi$
$888$	0	0
$889$	−0.0323942	−0.00108647
$890$	−3.08119	−0.103282
$891$	0	0
$892$	−7.33082	−0.245454
$893$	−9.35642	−0.313101
$894$	0	0
$895$	8.68801	0.290408
$896$	−1.37581	−0.0459624
$897$	0	0
$898$	−26.3248	−0.878472
$899$	−34.9253	−1.16482
$900$	0	0
$901$	78.9562	2.63041
$902$	0.809449	0.0269517
$903$	0	0
$904$	9.26439	0.308129
$905$	−2.72005	−0.0904176
$906$	0	0
$907$	−21.7579	−0.722458	−0.361229	−	0.932477i	$-0.617643\pi$
$907$	−21.7579	−0.722458	−0.361229	+	0.932477i	$0.617643\pi$
$908$	−6.28319	−0.208515
$909$	0	0
$910$	2.29668	0.0761341
$911$	49.0649	1.62559	0.812797	−	0.582547i	$-0.197944\pi$
$911$	49.0649	1.62559	0.812797	+	0.582547i	$0.197944\pi$
$912$	0	0
$913$	−2.61554	−0.0865617
$914$	3.98002	0.131647
$915$	0	0
$916$	−0.0268956	−0.000888656 0
$917$	−25.5423	−0.843482
$918$	0	0
$919$	20.1308	0.664053	0.332026	−	0.943270i	$-0.392268\pi$
$919$	20.1308	0.664053	0.332026	+	0.943270i	$0.392268\pi$
$920$	2.05742	0.0678313
$921$	0	0
$922$	7.93927	0.261466
$923$	−31.0396	−1.02168
$924$	0	0
$925$	−28.3708	−0.932827
$926$	27.6106	0.907341
$927$	0	0
$928$	3.94668	0.129556
$929$	7.60365	0.249468	0.124734	−	0.992190i	$-0.460192\pi$
$929$	7.60365	0.249468	0.124734	+	0.992190i	$0.460192\pi$
$930$	0	0
$931$	31.2602	1.02451
$932$	23.8435	0.781019
$933$	0	0
$934$	20.6638	0.676140
$935$	−1.42657	−0.0466540
$936$	0	0
$937$	22.0037	0.718830	0.359415	−	0.933178i	$-0.382976\pi$
$937$	22.0037	0.718830	0.359415	+	0.933178i	$0.382976\pi$
$938$	−6.12645	−0.200036
$939$	0	0
$940$	−1.30339	−0.0425120
$941$	−32.1377	−1.04766	−0.523830	−	0.851823i	$-0.675497\pi$
$941$	−32.1377	−1.04766	−0.523830	+	0.851823i	$0.675497\pi$
$942$	0	0
$943$	9.00978	0.293399
$944$	−4.62382	−0.150493
$945$	0	0
$946$	−2.30861	−0.0750593
$947$	−2.73709	−0.0889435	−0.0444717	−	0.999011i	$-0.514160\pi$
$947$	−2.73709	−0.0889435	−0.0444717	+	0.999011i	$0.514160\pi$
$948$	0	0
$949$	−10.5693	−0.343093
$950$	−26.1542	−0.848555
$951$	0	0
$952$	10.6182	0.344139
$953$	−32.8770	−1.06499	−0.532495	−	0.846433i	$-0.678746\pi$
$953$	−32.8770	−1.06499	−0.532495	+	0.846433i	$0.678746\pi$
$954$	0	0
$955$	−9.06161	−0.293227
$956$	15.3682	0.497044
$957$	0	0
$958$	30.4658	0.984306
$959$	−10.2837	−0.332079
$960$	0	0
$961$	47.3101	1.52613
$962$	−12.9989	−0.419102
$963$	0	0
$964$	−0.964379	−0.0310605
$965$	8.99113	0.289435
$966$	0	0
$967$	−18.4425	−0.593071	−0.296535	−	0.955022i	$-0.595831\pi$
$967$	−18.4425	−0.593071	−0.296535	+	0.955022i	$0.595831\pi$
$968$	10.9530	0.352043
$969$	0	0
$970$	−9.08044	−0.291555
$971$	57.1595	1.83434	0.917168	−	0.398500i	$-0.130469\pi$
$971$	57.1595	1.83434	0.917168	+	0.398500i	$0.130469\pi$
$972$	0	0
$973$	−10.4135	−0.333842
$974$	11.9590	0.383190
$975$	0	0
$976$	5.24719	0.167959
$977$	3.67102	0.117446	0.0587232	−	0.998274i	$-0.481297\pi$
$977$	3.67102	0.117446	0.0587232	+	0.998274i	$0.481297\pi$
$978$	0	0
$979$	0.783356	0.0250362
$980$	4.35470	0.139106
$981$	0	0
$982$	−40.9377	−1.30637
$983$	−6.68787	−0.213310	−0.106655	−	0.994296i	$-0.534014\pi$
$983$	−6.68787	−0.213310	−0.106655	+	0.994296i	$0.534014\pi$
$984$	0	0
$985$	−12.1801	−0.388091
$986$	−30.4598	−0.970037
$987$	0	0
$988$	−11.9833	−0.381240
$989$	−25.6966	−0.817103
$990$	0	0
$991$	−51.4790	−1.63528	−0.817642	−	0.575728i	$-0.804719\pi$
$991$	−51.4790	−1.63528	−0.817642	+	0.575728i	$0.804719\pi$
$992$	−8.84930	−0.280966
$993$	0	0
$994$	21.8127	0.691857
$995$	11.9964	0.380313
$996$	0	0
$997$	−44.2781	−1.40230	−0.701150	−	0.713014i	$-0.747330\pi$
$997$	−44.2781	−1.40230	−0.701150	+	0.713014i	$0.747330\pi$
$998$	−6.69932	−0.212063
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.e.1.3	✓	6
3.2	odd	2		1458.2.a.h.1.4	yes	6
9.2	odd	6		1458.2.c.e.973.3		12
9.4	even	3		1458.2.c.h.487.4		12
9.5	odd	6		1458.2.c.e.487.3		12
9.7	even	3		1458.2.c.h.973.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1458.2.a.e.1.3	✓	6	1.1	even	1	trivial
1458.2.a.h.1.4	yes	6	3.2	odd	2
1458.2.c.e.487.3		12	9.5	odd	6
1458.2.c.e.973.3		12	9.2	odd	6
1458.2.c.h.487.4		12	9.4	even	3
1458.2.c.h.973.4		12	9.7	even	3

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.852666 q^{5} +1.37581 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.852666 q^{5} +1.37581 q^{7} -1.00000 q^{8} +0.852666 q^{10} -0.216780 q^{11} +1.95778 q^{13} -1.37581 q^{14} +1.00000 q^{16} -7.71783 q^{17} -6.12087 q^{19} -0.852666 q^{20} +0.216780 q^{22} +2.41293 q^{23} -4.27296 q^{25} -1.95778 q^{26} +1.37581 q^{28} -3.94668 q^{29} +8.84930 q^{31} -1.00000 q^{32} +7.71783 q^{34} -1.17310 q^{35} +6.63961 q^{37} +6.12087 q^{38} +0.852666 q^{40} +3.73396 q^{41} -10.6495 q^{43} -0.216780 q^{44} -2.41293 q^{46} +1.52861 q^{47} -5.10716 q^{49} +4.27296 q^{50} +1.95778 q^{52} -10.2304 q^{53} +0.184841 q^{55} -1.37581 q^{56} +3.94668 q^{58} -4.62382 q^{59} +5.24719 q^{61} -8.84930 q^{62} +1.00000 q^{64} -1.66933 q^{65} +4.45299 q^{67} -7.71783 q^{68} +1.17310 q^{70} -15.8545 q^{71} -5.39860 q^{73} -6.63961 q^{74} -6.12087 q^{76} -0.298248 q^{77} -8.02895 q^{79} -0.852666 q^{80} -3.73396 q^{82} +12.0654 q^{83} +6.58073 q^{85} +10.6495 q^{86} +0.216780 q^{88} -3.61359 q^{89} +2.69352 q^{91} +2.41293 q^{92} -1.52861 q^{94} +5.21905 q^{95} -10.6495 q^{97} +5.10716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{16} - 12 q^{17} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 12 q^{23} + 6 q^{25} - 6 q^{26} - 6 q^{29} - 6 q^{31} - 6 q^{32} + 12 q^{34} - 12 q^{35} - 6 q^{38} + 6 q^{40} - 24 q^{41} - 6 q^{43} - 6 q^{44} + 12 q^{46} - 18 q^{47} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 24 q^{53} - 18 q^{55} + 6 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 6 q^{64} - 12 q^{65} - 24 q^{67} - 12 q^{68} + 12 q^{70} + 6 q^{71} - 24 q^{73} + 6 q^{76} - 12 q^{77} - 12 q^{79} - 6 q^{80} + 24 q^{82} - 18 q^{83} + 6 q^{86} + 6 q^{88} - 12 q^{89} - 30 q^{91} - 12 q^{92} + 18 q^{94} + 6 q^{95} + 6 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^16 - 12 * q^17 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 12 * q^23 + 6 * q^25 - 6 * q^26 - 6 * q^29 - 6 * q^31 - 6 * q^32 + 12 * q^34 - 12 * q^35 - 6 * q^38 + 6 * q^40 - 24 * q^41 - 6 * q^43 - 6 * q^44 + 12 * q^46 - 18 * q^47 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 24 * q^53 - 18 * q^55 + 6 * q^58 - 12 * q^59 - 6 * q^61 + 6 * q^62 + 6 * q^64 - 12 * q^65 - 24 * q^67 - 12 * q^68 + 12 * q^70 + 6 * q^71 - 24 * q^73 + 6 * q^76 - 12 * q^77 - 12 * q^79 - 6 * q^80 + 24 * q^82 - 18 * q^83 + 6 * q^86 + 6 * q^88 - 12 * q^89 - 30 * q^91 - 12 * q^92 + 18 * q^94 + 6 * q^95 + 6 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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