LMFDB - Embedded newform 1458.2.a.e.1.6

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			1.6
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.61144 q^{5} -5.13458 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.61144 q^{5} -5.13458 q^{7} -1.00000 q^{8} -2.61144 q^{10} -2.47781 q^{11} +3.80099 q^{13} +5.13458 q^{14} +1.00000 q^{16} -2.41052 q^{17} +5.05669 q^{19} +2.61144 q^{20} +2.47781 q^{22} -5.71834 q^{23} +1.81960 q^{25} -3.80099 q^{26} -5.13458 q^{28} +3.62167 q^{29} -4.02635 q^{31} -1.00000 q^{32} +2.41052 q^{34} -13.4086 q^{35} -1.49166 q^{37} -5.05669 q^{38} -2.61144 q^{40} -10.3448 q^{41} -0.257211 q^{43} -2.47781 q^{44} +5.71834 q^{46} -7.24280 q^{47} +19.3639 q^{49} -1.81960 q^{50} +3.80099 q^{52} -14.4797 q^{53} -6.47065 q^{55} +5.13458 q^{56} -3.62167 q^{58} -3.42075 q^{59} -9.61678 q^{61} +4.02635 q^{62} +1.00000 q^{64} +9.92604 q^{65} -3.26046 q^{67} -2.41052 q^{68} +13.4086 q^{70} +7.55858 q^{71} +3.93574 q^{73} +1.49166 q^{74} +5.05669 q^{76} +12.7225 q^{77} +0.964777 q^{79} +2.61144 q^{80} +10.3448 q^{82} -4.41948 q^{83} -6.29492 q^{85} +0.257211 q^{86} +2.47781 q^{88} -6.51476 q^{89} -19.5165 q^{91} -5.71834 q^{92} +7.24280 q^{94} +13.2052 q^{95} +9.29949 q^{97} -19.3639 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.61144	1.16787	0.583935	−	0.811801i	$-0.301512\pi$
$5$	2.61144	1.16787	0.583935	+	0.811801i	$0.301512\pi$
$6$	0	0
$7$	−5.13458	−1.94069	−0.970344	−	0.241730i	$-0.922285\pi$
$7$	−5.13458	−1.94069	−0.970344	+	0.241730i	$0.922285\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.61144	−0.825809
$11$	−2.47781	−0.747088	−0.373544	−	0.927612i	$-0.621858\pi$
$11$	−2.47781	−0.747088	−0.373544	+	0.927612i	$0.621858\pi$
$12$	0	0
$13$	3.80099	1.05421	0.527103	−	0.849802i	$-0.323278\pi$
$13$	3.80099	1.05421	0.527103	+	0.849802i	$0.323278\pi$
$14$	5.13458	1.37227
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.41052	−0.584637	−0.292319	−	0.956321i	$-0.594427\pi$
$17$	−2.41052	−0.584637	−0.292319	+	0.956321i	$0.594427\pi$
$18$	0	0
$19$	5.05669	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$19$	5.05669	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$20$	2.61144	0.583935
$21$	0	0
$22$	2.47781	0.528271
$23$	−5.71834	−1.19236	−0.596178	−	0.802852i	$-0.703315\pi$
$23$	−5.71834	−1.19236	−0.596178	+	0.802852i	$0.703315\pi$
$24$	0	0
$25$	1.81960	0.363920
$26$	−3.80099	−0.745436
$27$	0	0
$28$	−5.13458	−0.970344
$29$	3.62167	0.672527	0.336263	−	0.941768i	$-0.390837\pi$
$29$	3.62167	0.672527	0.336263	+	0.941768i	$0.390837\pi$
$30$	0	0
$31$	−4.02635	−0.723154	−0.361577	−	0.932342i	$-0.617762\pi$
$31$	−4.02635	−0.723154	−0.361577	+	0.932342i	$0.617762\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.41052	0.413401
$35$	−13.4086	−2.26647
$36$	0	0
$37$	−1.49166	−0.245227	−0.122614	−	0.992454i	$-0.539128\pi$
$37$	−1.49166	−0.245227	−0.122614	+	0.992454i	$0.539128\pi$
$38$	−5.05669	−0.820303
$39$	0	0
$40$	−2.61144	−0.412904
$41$	−10.3448	−1.61558	−0.807791	−	0.589469i	$-0.799337\pi$
$41$	−10.3448	−1.61558	−0.807791	+	0.589469i	$0.799337\pi$
$42$	0	0
$43$	−0.257211	−0.0392243	−0.0196121	−	0.999808i	$-0.506243\pi$
$43$	−0.257211	−0.0392243	−0.0196121	+	0.999808i	$0.506243\pi$
$44$	−2.47781	−0.373544
$45$	0	0
$46$	5.71834	0.843123
$47$	−7.24280	−1.05647	−0.528236	−	0.849098i	$-0.677146\pi$
$47$	−7.24280	−1.05647	−0.528236	+	0.849098i	$0.677146\pi$
$48$	0	0
$49$	19.3639	2.76627
$50$	−1.81960	−0.257330
$51$	0	0
$52$	3.80099	0.527103
$53$	−14.4797	−1.98894	−0.994471	−	0.105014i	$-0.966511\pi$
$53$	−14.4797	−1.98894	−0.994471	+	0.105014i	$0.966511\pi$
$54$	0	0
$55$	−6.47065	−0.872502
$56$	5.13458	0.686137
$57$	0	0
$58$	−3.62167	−0.475548
$59$	−3.42075	−0.445344	−0.222672	−	0.974893i	$-0.571478\pi$
$59$	−3.42075	−0.445344	−0.222672	+	0.974893i	$0.571478\pi$
$60$	0	0
$61$	−9.61678	−1.23130	−0.615651	−	0.788019i	$-0.711107\pi$
$61$	−9.61678	−1.23130	−0.615651	+	0.788019i	$0.711107\pi$
$62$	4.02635	0.511347
$63$	0	0
$64$	1.00000	0.125000
$65$	9.92604	1.23117
$66$	0	0
$67$	−3.26046	−0.398328	−0.199164	−	0.979966i	$-0.563823\pi$
$67$	−3.26046	−0.398328	−0.199164	+	0.979966i	$0.563823\pi$
$68$	−2.41052	−0.292319
$69$	0	0
$70$	13.4086	1.60264
$71$	7.55858	0.897038	0.448519	−	0.893773i	$-0.351952\pi$
$71$	7.55858	0.897038	0.448519	+	0.893773i	$0.351952\pi$
$72$	0	0
$73$	3.93574	0.460644	0.230322	−	0.973114i	$-0.426022\pi$
$73$	3.93574	0.460644	0.230322	+	0.973114i	$0.426022\pi$
$74$	1.49166	0.173402
$75$	0	0
$76$	5.05669	0.580042
$77$	12.7225	1.44987
$78$	0	0
$79$	0.964777	0.108546	0.0542729	−	0.998526i	$-0.482716\pi$
$79$	0.964777	0.108546	0.0542729	+	0.998526i	$0.482716\pi$
$80$	2.61144	0.291967
$81$	0	0
$82$	10.3448	1.14239
$83$	−4.41948	−0.485101	−0.242551	−	0.970139i	$-0.577984\pi$
$83$	−4.41948	−0.485101	−0.242551	+	0.970139i	$0.577984\pi$
$84$	0	0
$85$	−6.29492	−0.682780
$86$	0.257211	0.0277358
$87$	0	0
$88$	2.47781	0.264136
$89$	−6.51476	−0.690564	−0.345282	−	0.938499i	$-0.612217\pi$
$89$	−6.51476	−0.690564	−0.345282	+	0.938499i	$0.612217\pi$
$90$	0	0
$91$	−19.5165	−2.04588
$92$	−5.71834	−0.596178
$93$	0	0
$94$	7.24280	0.747038
$95$	13.2052	1.35483
$96$	0	0
$97$	9.29949	0.944220	0.472110	−	0.881540i	$-0.343492\pi$
$97$	9.29949	0.944220	0.472110	+	0.881540i	$0.343492\pi$
$98$	−19.3639	−1.95605
$99$	0	0
$100$	1.81960	0.181960
$101$	8.26590	0.822488	0.411244	−	0.911525i	$-0.365094\pi$
$101$	8.26590	0.822488	0.411244	+	0.911525i	$0.365094\pi$
$102$	0	0
$103$	−13.2940	−1.30989	−0.654947	−	0.755675i	$-0.727309\pi$
$103$	−13.2940	−1.30989	−0.654947	+	0.755675i	$0.727309\pi$
$104$	−3.80099	−0.372718
$105$	0	0
$106$	14.4797	1.40639
$107$	−8.55207	−0.826760	−0.413380	−	0.910559i	$-0.635652\pi$
$107$	−8.55207	−0.826760	−0.413380	+	0.910559i	$0.635652\pi$
$108$	0	0
$109$	6.66939	0.638812	0.319406	−	0.947618i	$-0.396517\pi$
$109$	6.66939	0.638812	0.319406	+	0.947618i	$0.396517\pi$
$110$	6.47065	0.616952
$111$	0	0
$112$	−5.13458	−0.485172
$113$	14.0036	1.31734	0.658672	−	0.752430i	$-0.271118\pi$
$113$	14.0036	1.31734	0.658672	+	0.752430i	$0.271118\pi$
$114$	0	0
$115$	−14.9331	−1.39252
$116$	3.62167	0.336263
$117$	0	0
$118$	3.42075	0.314906
$119$	12.3770	1.13460
$120$	0	0
$121$	−4.86045	−0.441859
$122$	9.61678	0.870662
$123$	0	0
$124$	−4.02635	−0.361577
$125$	−8.30542	−0.742859
$126$	0	0
$127$	3.58566	0.318176	0.159088	−	0.987264i	$-0.449145\pi$
$127$	3.58566	0.318176	0.159088	+	0.987264i	$0.449145\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.92604	−0.870572
$131$	5.85528	0.511578	0.255789	−	0.966733i	$-0.417665\pi$
$131$	5.85528	0.511578	0.255789	+	0.966733i	$0.417665\pi$
$132$	0	0
$133$	−25.9639	−2.25136
$134$	3.26046	0.281661
$135$	0	0
$136$	2.41052	0.206701
$137$	−12.7820	−1.09204	−0.546020	−	0.837772i	$-0.683858\pi$
$137$	−12.7820	−1.09204	−0.546020	+	0.837772i	$0.683858\pi$
$138$	0	0
$139$	−6.92890	−0.587702	−0.293851	−	0.955851i	$-0.594937\pi$
$139$	−6.92890	−0.587702	−0.293851	+	0.955851i	$0.594937\pi$
$140$	−13.4086	−1.13323
$141$	0	0
$142$	−7.55858	−0.634302
$143$	−9.41814	−0.787584
$144$	0	0
$145$	9.45775	0.785424
$146$	−3.93574	−0.325724
$147$	0	0
$148$	−1.49166	−0.122614
$149$	−14.4291	−1.18208	−0.591040	−	0.806642i	$-0.701283\pi$
$149$	−14.4291	−1.18208	−0.591040	+	0.806642i	$0.701283\pi$
$150$	0	0
$151$	−17.1645	−1.39683	−0.698415	−	0.715693i	$-0.746111\pi$
$151$	−17.1645	−1.39683	−0.698415	+	0.715693i	$0.746111\pi$
$152$	−5.05669	−0.410152
$153$	0	0
$154$	−12.7225	−1.02521
$155$	−10.5146	−0.844550
$156$	0	0
$157$	−2.52314	−0.201368	−0.100684	−	0.994918i	$-0.532103\pi$
$157$	−2.52314	−0.201368	−0.100684	+	0.994918i	$0.532103\pi$
$158$	−0.964777	−0.0767535
$159$	0	0
$160$	−2.61144	−0.206452
$161$	29.3613	2.31399
$162$	0	0
$163$	−22.6015	−1.77029	−0.885144	−	0.465317i	$-0.845940\pi$
$163$	−22.6015	−1.77029	−0.885144	+	0.465317i	$0.845940\pi$
$164$	−10.3448	−0.807791
$165$	0	0
$166$	4.41948	0.343018
$167$	−0.833449	−0.0644942	−0.0322471	−	0.999480i	$-0.510266\pi$
$167$	−0.833449	−0.0644942	−0.0322471	+	0.999480i	$0.510266\pi$
$168$	0	0
$169$	1.44753	0.111348
$170$	6.29492	0.482799
$171$	0	0
$172$	−0.257211	−0.0196121
$173$	−1.89424	−0.144016	−0.0720081	−	0.997404i	$-0.522941\pi$
$173$	−1.89424	−0.144016	−0.0720081	+	0.997404i	$0.522941\pi$
$174$	0	0
$175$	−9.34287	−0.706254
$176$	−2.47781	−0.186772
$177$	0	0
$178$	6.51476	0.488302
$179$	−0.437066	−0.0326678	−0.0163339	−	0.999867i	$-0.505199\pi$
$179$	−0.437066	−0.0326678	−0.0163339	+	0.999867i	$0.505199\pi$
$180$	0	0
$181$	10.6133	0.788880	0.394440	−	0.918922i	$-0.370939\pi$
$181$	10.6133	0.788880	0.394440	+	0.918922i	$0.370939\pi$
$182$	19.5165	1.44666
$183$	0	0
$184$	5.71834	0.421562
$185$	−3.89537	−0.286393
$186$	0	0
$187$	5.97282	0.436776
$188$	−7.24280	−0.528236
$189$	0	0
$190$	−13.2052	−0.958007
$191$	−16.8395	−1.21847	−0.609233	−	0.792992i	$-0.708522\pi$
$191$	−16.8395	−1.21847	−0.609233	+	0.792992i	$0.708522\pi$
$192$	0	0
$193$	6.99809	0.503734	0.251867	−	0.967762i	$-0.418955\pi$
$193$	6.99809	0.503734	0.251867	+	0.967762i	$0.418955\pi$
$194$	−9.29949	−0.667665
$195$	0	0
$196$	19.3639	1.38313
$197$	20.1550	1.43598	0.717992	−	0.696051i	$-0.245061\pi$
$197$	20.1550	1.43598	0.717992	+	0.696051i	$0.245061\pi$
$198$	0	0
$199$	14.6510	1.03859	0.519293	−	0.854596i	$-0.326195\pi$
$199$	14.6510	1.03859	0.519293	+	0.854596i	$0.326195\pi$
$200$	−1.81960	−0.128665
$201$	0	0
$202$	−8.26590	−0.581587
$203$	−18.5957	−1.30516
$204$	0	0
$205$	−27.0147	−1.88679
$206$	13.2940	0.926235
$207$	0	0
$208$	3.80099	0.263551
$209$	−12.5295	−0.866685
$210$	0	0
$211$	7.44295	0.512394	0.256197	−	0.966625i	$-0.417530\pi$
$211$	7.44295	0.512394	0.256197	+	0.966625i	$0.417530\pi$
$212$	−14.4797	−0.994471
$213$	0	0
$214$	8.55207	0.584607
$215$	−0.671689	−0.0458088
$216$	0	0
$217$	20.6736	1.40342
$218$	−6.66939	−0.451708
$219$	0	0
$220$	−6.47065	−0.436251
$221$	−9.16237	−0.616328
$222$	0	0
$223$	25.0760	1.67921	0.839606	−	0.543196i	$-0.182786\pi$
$223$	25.0760	1.67921	0.839606	+	0.543196i	$0.182786\pi$
$224$	5.13458	0.343068
$225$	0	0
$226$	−14.0036	−0.931503
$227$	3.91361	0.259755	0.129878	−	0.991530i	$-0.458542\pi$
$227$	3.91361	0.259755	0.129878	+	0.991530i	$0.458542\pi$
$228$	0	0
$229$	−2.51025	−0.165882	−0.0829409	−	0.996554i	$-0.526431\pi$
$229$	−2.51025	−0.165882	−0.0829409	+	0.996554i	$0.526431\pi$
$230$	14.9331	0.984658
$231$	0	0
$232$	−3.62167	−0.237774
$233$	−4.23677	−0.277560	−0.138780	−	0.990323i	$-0.544318\pi$
$233$	−4.23677	−0.277560	−0.138780	+	0.990323i	$0.544318\pi$
$234$	0	0
$235$	−18.9141	−1.23382
$236$	−3.42075	−0.222672
$237$	0	0
$238$	−12.3770	−0.802282
$239$	13.1844	0.852828	0.426414	−	0.904528i	$-0.359777\pi$
$239$	13.1844	0.852828	0.426414	+	0.904528i	$0.359777\pi$
$240$	0	0
$241$	9.20561	0.592985	0.296493	−	0.955035i	$-0.404183\pi$
$241$	9.20561	0.592985	0.296493	+	0.955035i	$0.404183\pi$
$242$	4.86045	0.312441
$243$	0	0
$244$	−9.61678	−0.615651
$245$	50.5675	3.23064
$246$	0	0
$247$	19.2204	1.22297
$248$	4.02635	0.255674
$249$	0	0
$250$	8.30542	0.525281
$251$	16.4497	1.03830	0.519149	−	0.854683i	$-0.326249\pi$
$251$	16.4497	1.03830	0.519149	+	0.854683i	$0.326249\pi$
$252$	0	0
$253$	14.1690	0.890796
$254$	−3.58566	−0.224985
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.89481	−0.430086	−0.215043	−	0.976605i	$-0.568989\pi$
$257$	−6.89481	−0.430086	−0.215043	+	0.976605i	$0.568989\pi$
$258$	0	0
$259$	7.65903	0.475909
$260$	9.92604	0.615587
$261$	0	0
$262$	−5.85528	−0.361740
$263$	4.99673	0.308112	0.154056	−	0.988062i	$-0.450766\pi$
$263$	4.99673	0.308112	0.154056	+	0.988062i	$0.450766\pi$
$264$	0	0
$265$	−37.8128	−2.32282
$266$	25.9639	1.59195
$267$	0	0
$268$	−3.26046	−0.199164
$269$	9.95633	0.607048	0.303524	−	0.952824i	$-0.401837\pi$
$269$	9.95633	0.607048	0.303524	+	0.952824i	$0.401837\pi$
$270$	0	0
$271$	21.1021	1.28186	0.640932	−	0.767598i	$-0.278548\pi$
$271$	21.1021	1.28186	0.640932	+	0.767598i	$0.278548\pi$
$272$	−2.41052	−0.146159
$273$	0	0
$274$	12.7820	0.772190
$275$	−4.50862	−0.271880
$276$	0	0
$277$	0.134669	0.00809148	0.00404574	−	0.999992i	$-0.498712\pi$
$277$	0.134669	0.00809148	0.00404574	+	0.999992i	$0.498712\pi$
$278$	6.92890	0.415568
$279$	0	0
$280$	13.4086	0.801318
$281$	6.66923	0.397853	0.198927	−	0.980014i	$-0.436254\pi$
$281$	6.66923	0.397853	0.198927	+	0.980014i	$0.436254\pi$
$282$	0	0
$283$	27.6301	1.64244	0.821220	−	0.570611i	$-0.193294\pi$
$283$	27.6301	1.64244	0.821220	+	0.570611i	$0.193294\pi$
$284$	7.55858	0.448519
$285$	0	0
$286$	9.41814	0.556906
$287$	53.1160	3.13534
$288$	0	0
$289$	−11.1894	−0.658199
$290$	−9.45775	−0.555378
$291$	0	0
$292$	3.93574	0.230322
$293$	−16.3976	−0.957959	−0.478980	−	0.877826i	$-0.658993\pi$
$293$	−16.3976	−0.957959	−0.478980	+	0.877826i	$0.658993\pi$
$294$	0	0
$295$	−8.93308	−0.520104
$296$	1.49166	0.0867009
$297$	0	0
$298$	14.4291	0.835857
$299$	−21.7354	−1.25699
$300$	0	0
$301$	1.32067	0.0761221
$302$	17.1645	0.987709
$303$	0	0
$304$	5.05669	0.290021
$305$	−25.1136	−1.43800
$306$	0	0
$307$	−0.312008	−0.0178073	−0.00890363	−	0.999960i	$-0.502834\pi$
$307$	−0.312008	−0.0178073	−0.00890363	+	0.999960i	$0.502834\pi$
$308$	12.7225	0.724933
$309$	0	0
$310$	10.5146	0.597187
$311$	13.6782	0.775620	0.387810	−	0.921739i	$-0.373232\pi$
$311$	13.6782	0.775620	0.387810	+	0.921739i	$0.373232\pi$
$312$	0	0
$313$	23.6002	1.33396	0.666981	−	0.745075i	$-0.267586\pi$
$313$	23.6002	1.33396	0.666981	+	0.745075i	$0.267586\pi$
$314$	2.52314	0.142389
$315$	0	0
$316$	0.964777	0.0542729
$317$	−32.1852	−1.80770	−0.903851	−	0.427847i	$-0.859272\pi$
$317$	−32.1852	−1.80770	−0.903851	+	0.427847i	$0.859272\pi$
$318$	0	0
$319$	−8.97381	−0.502437
$320$	2.61144	0.145984
$321$	0	0
$322$	−29.3613	−1.63624
$323$	−12.1893	−0.678228
$324$	0	0
$325$	6.91628	0.383646
$326$	22.6015	1.25178
$327$	0	0
$328$	10.3448	0.571194
$329$	37.1887	2.05028
$330$	0	0
$331$	−2.30602	−0.126750	−0.0633750	−	0.997990i	$-0.520186\pi$
$331$	−2.30602	−0.126750	−0.0633750	+	0.997990i	$0.520186\pi$
$332$	−4.41948	−0.242551
$333$	0	0
$334$	0.833449	0.0456043
$335$	−8.51448	−0.465196
$336$	0	0
$337$	3.72848	0.203103	0.101552	−	0.994830i	$-0.467619\pi$
$337$	3.72848	0.203103	0.101552	+	0.994830i	$0.467619\pi$
$338$	−1.44753	−0.0787351
$339$	0	0
$340$	−6.29492	−0.341390
$341$	9.97655	0.540260
$342$	0	0
$343$	−63.4832	−3.42777
$344$	0.257211	0.0138679
$345$	0	0
$346$	1.89424	0.101835
$347$	10.2623	0.550907	0.275454	−	0.961314i	$-0.411172\pi$
$347$	10.2623	0.550907	0.275454	+	0.961314i	$0.411172\pi$
$348$	0	0
$349$	1.46351	0.0783400	0.0391700	−	0.999233i	$-0.487529\pi$
$349$	1.46351	0.0783400	0.0391700	+	0.999233i	$0.487529\pi$
$350$	9.34287	0.499397
$351$	0	0
$352$	2.47781	0.132068
$353$	−1.97617	−0.105181	−0.0525905	−	0.998616i	$-0.516748\pi$
$353$	−1.97617	−0.105181	−0.0525905	+	0.998616i	$0.516748\pi$
$354$	0	0
$355$	19.7387	1.04762
$356$	−6.51476	−0.345282
$357$	0	0
$358$	0.437066	0.0230996
$359$	33.5870	1.77266	0.886328	−	0.463058i	$-0.153248\pi$
$359$	33.5870	1.77266	0.886328	+	0.463058i	$0.153248\pi$
$360$	0	0
$361$	6.57009	0.345794
$362$	−10.6133	−0.557822
$363$	0	0
$364$	−19.5165	−1.02294
$365$	10.2779	0.537972
$366$	0	0
$367$	−6.82332	−0.356174	−0.178087	−	0.984015i	$-0.556991\pi$
$367$	−6.82332	−0.356174	−0.178087	+	0.984015i	$0.556991\pi$
$368$	−5.71834	−0.298089
$369$	0	0
$370$	3.89537	0.202511
$371$	74.3472	3.85991
$372$	0	0
$373$	−2.83888	−0.146992	−0.0734958	−	0.997296i	$-0.523416\pi$
$373$	−2.83888	−0.146992	−0.0734958	+	0.997296i	$0.523416\pi$
$374$	−5.97282	−0.308847
$375$	0	0
$376$	7.24280	0.373519
$377$	13.7659	0.708981
$378$	0	0
$379$	−10.7922	−0.554356	−0.277178	−	0.960819i	$-0.589399\pi$
$379$	−10.7922	−0.554356	−0.277178	+	0.960819i	$0.589399\pi$
$380$	13.2052	0.677413
$381$	0	0
$382$	16.8395	0.861585
$383$	−27.0660	−1.38301	−0.691505	−	0.722372i	$-0.743052\pi$
$383$	−27.0660	−1.38301	−0.691505	+	0.722372i	$0.743052\pi$
$384$	0	0
$385$	33.2240	1.69325
$386$	−6.99809	−0.356194
$387$	0	0
$388$	9.29949	0.472110
$389$	−32.3882	−1.64215	−0.821074	−	0.570821i	$-0.806625\pi$
$389$	−32.3882	−1.64215	−0.821074	+	0.570821i	$0.806625\pi$
$390$	0	0
$391$	13.7842	0.697096
$392$	−19.3639	−0.978023
$393$	0	0
$394$	−20.1550	−1.01539
$395$	2.51945	0.126767
$396$	0	0
$397$	7.18123	0.360415	0.180208	−	0.983629i	$-0.442323\pi$
$397$	7.18123	0.360415	0.180208	+	0.983629i	$0.442323\pi$
$398$	−14.6510	−0.734391
$399$	0	0
$400$	1.81960	0.0909799
$401$	−29.3177	−1.46405	−0.732027	−	0.681275i	$-0.761426\pi$
$401$	−29.3177	−1.46405	−0.732027	+	0.681275i	$0.761426\pi$
$402$	0	0
$403$	−15.3041	−0.762353
$404$	8.26590	0.411244
$405$	0	0
$406$	18.5957	0.922891
$407$	3.69605	0.183206
$408$	0	0
$409$	1.22575	0.0606094	0.0303047	−	0.999541i	$-0.490352\pi$
$409$	1.22575	0.0606094	0.0303047	+	0.999541i	$0.490352\pi$
$410$	27.0147	1.33416
$411$	0	0
$412$	−13.2940	−0.654947
$413$	17.5641	0.864274
$414$	0	0
$415$	−11.5412	−0.566535
$416$	−3.80099	−0.186359
$417$	0	0
$418$	12.5295	0.612839
$419$	22.2987	1.08936	0.544681	−	0.838643i	$-0.316651\pi$
$419$	22.2987	1.08936	0.544681	+	0.838643i	$0.316651\pi$
$420$	0	0
$421$	−39.8226	−1.94083	−0.970417	−	0.241434i	$-0.922382\pi$
$421$	−39.8226	−1.94083	−0.970417	+	0.241434i	$0.922382\pi$
$422$	−7.44295	−0.362317
$423$	0	0
$424$	14.4797	0.703197
$425$	−4.38618	−0.212761
$426$	0	0
$427$	49.3781	2.38957
$428$	−8.55207	−0.413380
$429$	0	0
$430$	0.671689	0.0323917
$431$	14.5119	0.699015	0.349507	−	0.936934i	$-0.386349\pi$
$431$	14.5119	0.699015	0.349507	+	0.936934i	$0.386349\pi$
$432$	0	0
$433$	−13.8876	−0.667393	−0.333697	−	0.942680i	$-0.608296\pi$
$433$	−13.8876	−0.667393	−0.333697	+	0.942680i	$0.608296\pi$
$434$	−20.6736	−0.992365
$435$	0	0
$436$	6.66939	0.319406
$437$	−28.9159	−1.38323
$438$	0	0
$439$	10.7609	0.513590	0.256795	−	0.966466i	$-0.417333\pi$
$439$	10.7609	0.513590	0.256795	+	0.966466i	$0.417333\pi$
$440$	6.47065	0.308476
$441$	0	0
$442$	9.16237	0.435810
$443$	19.4970	0.926332	0.463166	−	0.886272i	$-0.346713\pi$
$443$	19.4970	0.926332	0.463166	+	0.886272i	$0.346713\pi$
$444$	0	0
$445$	−17.0129	−0.806488
$446$	−25.0760	−1.18738
$447$	0	0
$448$	−5.13458	−0.242586
$449$	7.65616	0.361317	0.180658	−	0.983546i	$-0.442177\pi$
$449$	7.65616	0.361317	0.180658	+	0.983546i	$0.442177\pi$
$450$	0	0
$451$	25.6324	1.20698
$452$	14.0036	0.658672
$453$	0	0
$454$	−3.91361	−0.183675
$455$	−50.9660	−2.38932
$456$	0	0
$457$	15.6787	0.733420	0.366710	−	0.930335i	$-0.380484\pi$
$457$	15.6787	0.733420	0.366710	+	0.930335i	$0.380484\pi$
$458$	2.51025	0.117296
$459$	0	0
$460$	−14.9331	−0.696258
$461$	14.6885	0.684114	0.342057	−	0.939679i	$-0.388876\pi$
$461$	14.6885	0.684114	0.342057	+	0.939679i	$0.388876\pi$
$462$	0	0
$463$	0.614733	0.0285691	0.0142845	−	0.999898i	$-0.495453\pi$
$463$	0.614733	0.0285691	0.0142845	+	0.999898i	$0.495453\pi$
$464$	3.62167	0.168132
$465$	0	0
$466$	4.23677	0.196265
$467$	−18.1301	−0.838959	−0.419479	−	0.907765i	$-0.637787\pi$
$467$	−18.1301	−0.838959	−0.419479	+	0.907765i	$0.637787\pi$
$468$	0	0
$469$	16.7411	0.773031
$470$	18.9141	0.872443
$471$	0	0
$472$	3.42075	0.157453
$473$	0.637320	0.0293040
$474$	0	0
$475$	9.20114	0.422177
$476$	12.3770	0.567299
$477$	0	0
$478$	−13.1844	−0.603040
$479$	8.98748	0.410648	0.205324	−	0.978694i	$-0.434175\pi$
$479$	8.98748	0.410648	0.205324	+	0.978694i	$0.434175\pi$
$480$	0	0
$481$	−5.66978	−0.258520
$482$	−9.20561	−0.419304
$483$	0	0
$484$	−4.86045	−0.220929
$485$	24.2850	1.10273
$486$	0	0
$487$	−35.3720	−1.60286	−0.801430	−	0.598089i	$-0.795927\pi$
$487$	−35.3720	−1.60286	−0.801430	+	0.598089i	$0.795927\pi$
$488$	9.61678	0.435331
$489$	0	0
$490$	−50.5675	−2.28441
$491$	5.50209	0.248306	0.124153	−	0.992263i	$-0.460379\pi$
$491$	5.50209	0.248306	0.124153	+	0.992263i	$0.460379\pi$
$492$	0	0
$493$	−8.73011	−0.393184
$494$	−19.2204	−0.864768
$495$	0	0
$496$	−4.02635	−0.180789
$497$	−38.8101	−1.74087
$498$	0	0
$499$	−6.86687	−0.307403	−0.153702	−	0.988117i	$-0.549119\pi$
$499$	−6.86687	−0.307403	−0.153702	+	0.988117i	$0.549119\pi$
$500$	−8.30542	−0.371429
$501$	0	0
$502$	−16.4497	−0.734188
$503$	−25.1598	−1.12182	−0.560911	−	0.827876i	$-0.689549\pi$
$503$	−25.1598	−1.12182	−0.560911	+	0.827876i	$0.689549\pi$
$504$	0	0
$505$	21.5859	0.960559
$506$	−14.1690	−0.629888
$507$	0	0
$508$	3.58566	0.159088
$509$	35.0446	1.55333	0.776663	−	0.629917i	$-0.216911\pi$
$509$	35.0446	1.55333	0.776663	+	0.629917i	$0.216911\pi$
$510$	0	0
$511$	−20.2084	−0.893966
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.89481	0.304117
$515$	−34.7163	−1.52978
$516$	0	0
$517$	17.9463	0.789278
$518$	−7.65903	−0.336519
$519$	0	0
$520$	−9.92604	−0.435286
$521$	9.36975	0.410496	0.205248	−	0.978710i	$-0.434200\pi$
$521$	9.36975	0.410496	0.205248	+	0.978710i	$0.434200\pi$
$522$	0	0
$523$	−31.4938	−1.37713	−0.688565	−	0.725175i	$-0.741759\pi$
$523$	−31.4938	−1.37713	−0.688565	+	0.725175i	$0.741759\pi$
$524$	5.85528	0.255789
$525$	0	0
$526$	−4.99673	−0.217868
$527$	9.70561	0.422783
$528$	0	0
$529$	9.69941	0.421714
$530$	37.8128	1.64248
$531$	0	0
$532$	−25.9639	−1.12568
$533$	−39.3204	−1.70315
$534$	0	0
$535$	−22.3332	−0.965547
$536$	3.26046	0.140830
$537$	0	0
$538$	−9.95633	−0.429248
$539$	−47.9800	−2.06665
$540$	0	0
$541$	18.3497	0.788917	0.394459	−	0.918914i	$-0.370932\pi$
$541$	18.3497	0.788917	0.394459	+	0.918914i	$0.370932\pi$
$542$	−21.1021	−0.906414
$543$	0	0
$544$	2.41052	0.103350
$545$	17.4167	0.746049
$546$	0	0
$547$	3.95373	0.169049	0.0845247	−	0.996421i	$-0.473063\pi$
$547$	3.95373	0.169049	0.0845247	+	0.996421i	$0.473063\pi$
$548$	−12.7820	−0.546020
$549$	0	0
$550$	4.50862	0.192248
$551$	18.3136	0.780187
$552$	0	0
$553$	−4.95372	−0.210654
$554$	−0.134669	−0.00572154
$555$	0	0
$556$	−6.92890	−0.293851
$557$	−40.3173	−1.70830	−0.854149	−	0.520029i	$-0.825921\pi$
$557$	−40.3173	−1.70830	−0.854149	+	0.520029i	$0.825921\pi$
$558$	0	0
$559$	−0.977656	−0.0413504
$560$	−13.4086	−0.566617
$561$	0	0
$562$	−6.66923	−0.281325
$563$	23.1559	0.975906	0.487953	−	0.872870i	$-0.337744\pi$
$563$	23.1559	0.975906	0.487953	+	0.872870i	$0.337744\pi$
$564$	0	0
$565$	36.5694	1.53849
$566$	−27.6301	−1.16138
$567$	0	0
$568$	−7.55858	−0.317151
$569$	19.2788	0.808209	0.404105	−	0.914713i	$-0.367583\pi$
$569$	19.2788	0.808209	0.404105	+	0.914713i	$0.367583\pi$
$570$	0	0
$571$	−1.86807	−0.0781761	−0.0390881	−	0.999236i	$-0.512445\pi$
$571$	−1.86807	−0.0781761	−0.0390881	+	0.999236i	$0.512445\pi$
$572$	−9.41814	−0.393792
$573$	0	0
$574$	−53.1160	−2.21702
$575$	−10.4051	−0.433922
$576$	0	0
$577$	46.2523	1.92551	0.962754	−	0.270381i	$-0.0871495\pi$
$577$	46.2523	1.92551	0.962754	+	0.270381i	$0.0871495\pi$
$578$	11.1894	0.465417
$579$	0	0
$580$	9.45775	0.392712
$581$	22.6922	0.941430
$582$	0	0
$583$	35.8780	1.48592
$584$	−3.93574	−0.162862
$585$	0	0
$586$	16.3976	0.677380
$587$	1.16246	0.0479800	0.0239900	−	0.999712i	$-0.492363\pi$
$587$	1.16246	0.0479800	0.0239900	+	0.999712i	$0.492363\pi$
$588$	0	0
$589$	−20.3600	−0.838920
$590$	8.93308	0.367769
$591$	0	0
$592$	−1.49166	−0.0613068
$593$	−17.5010	−0.718678	−0.359339	−	0.933207i	$-0.616998\pi$
$593$	−17.5010	−0.718678	−0.359339	+	0.933207i	$0.616998\pi$
$594$	0	0
$595$	32.3218	1.32506
$596$	−14.4291	−0.591040
$597$	0	0
$598$	21.7354	0.888825
$599$	−1.32656	−0.0542017	−0.0271008	−	0.999633i	$-0.508628\pi$
$599$	−1.32656	−0.0542017	−0.0271008	+	0.999633i	$0.508628\pi$
$600$	0	0
$601$	6.01059	0.245177	0.122589	−	0.992458i	$-0.460880\pi$
$601$	6.01059	0.245177	0.122589	+	0.992458i	$0.460880\pi$
$602$	−1.32067	−0.0538264
$603$	0	0
$604$	−17.1645	−0.698415
$605$	−12.6927	−0.516033
$606$	0	0
$607$	−27.7947	−1.12815	−0.564075	−	0.825723i	$-0.690767\pi$
$607$	−27.7947	−1.12815	−0.564075	+	0.825723i	$0.690767\pi$
$608$	−5.05669	−0.205076
$609$	0	0
$610$	25.1136	1.01682
$611$	−27.5298	−1.11374
$612$	0	0
$613$	46.4294	1.87527	0.937633	−	0.347626i	$-0.113012\pi$
$613$	46.4294	1.87527	0.937633	+	0.347626i	$0.113012\pi$
$614$	0.312008	0.0125916
$615$	0	0
$616$	−12.7225	−0.512605
$617$	30.8174	1.24066	0.620331	−	0.784340i	$-0.286998\pi$
$617$	30.8174	1.24066	0.620331	+	0.784340i	$0.286998\pi$
$618$	0	0
$619$	36.2723	1.45791	0.728953	−	0.684563i	$-0.240007\pi$
$619$	36.2723	1.45791	0.728953	+	0.684563i	$0.240007\pi$
$620$	−10.5146	−0.422275
$621$	0	0
$622$	−13.6782	−0.548446
$623$	33.4505	1.34017
$624$	0	0
$625$	−30.7871	−1.23148
$626$	−23.6002	−0.943254
$627$	0	0
$628$	−2.52314	−0.100684
$629$	3.59568	0.143369
$630$	0	0
$631$	−44.7890	−1.78302	−0.891512	−	0.452997i	$-0.850355\pi$
$631$	−44.7890	−1.78302	−0.891512	+	0.452997i	$0.850355\pi$
$632$	−0.964777	−0.0383768
$633$	0	0
$634$	32.1852	1.27824
$635$	9.36373	0.371588
$636$	0	0
$637$	73.6019	2.91621
$638$	8.97381	0.355277
$639$	0	0
$640$	−2.61144	−0.103226
$641$	19.3649	0.764868	0.382434	−	0.923983i	$-0.375086\pi$
$641$	19.3649	0.764868	0.382434	+	0.923983i	$0.375086\pi$
$642$	0	0
$643$	19.8512	0.782855	0.391428	−	0.920209i	$-0.371981\pi$
$643$	19.8512	0.782855	0.391428	+	0.920209i	$0.371981\pi$
$644$	29.3613	1.15700
$645$	0	0
$646$	12.1893	0.479580
$647$	17.7255	0.696861	0.348431	−	0.937335i	$-0.386715\pi$
$647$	17.7255	0.696861	0.348431	+	0.937335i	$0.386715\pi$
$648$	0	0
$649$	8.47599	0.332712
$650$	−6.91628	−0.271279
$651$	0	0
$652$	−22.6015	−0.885144
$653$	38.0893	1.49055	0.745275	−	0.666757i	$-0.232318\pi$
$653$	38.0893	1.49055	0.745275	+	0.666757i	$0.232318\pi$
$654$	0	0
$655$	15.2907	0.597457
$656$	−10.3448	−0.403895
$657$	0	0
$658$	−37.1887	−1.44977
$659$	−20.6589	−0.804758	−0.402379	−	0.915473i	$-0.631817\pi$
$659$	−20.6589	−0.804758	−0.402379	+	0.915473i	$0.631817\pi$
$660$	0	0
$661$	39.5038	1.53652	0.768260	−	0.640138i	$-0.221123\pi$
$661$	39.5038	1.53652	0.768260	+	0.640138i	$0.221123\pi$
$662$	2.30602	0.0896258
$663$	0	0
$664$	4.41948	0.171509
$665$	−67.8032	−2.62929
$666$	0	0
$667$	−20.7099	−0.801892
$668$	−0.833449	−0.0322471
$669$	0	0
$670$	8.51448	0.328943
$671$	23.8286	0.919892
$672$	0	0
$673$	24.4255	0.941535	0.470768	−	0.882257i	$-0.343977\pi$
$673$	24.4255	0.941535	0.470768	+	0.882257i	$0.343977\pi$
$674$	−3.72848	−0.143616
$675$	0	0
$676$	1.44753	0.0556741
$677$	−15.8043	−0.607409	−0.303705	−	0.952766i	$-0.598224\pi$
$677$	−15.8043	−0.607409	−0.303705	+	0.952766i	$0.598224\pi$
$678$	0	0
$679$	−47.7489	−1.83244
$680$	6.29492	0.241399
$681$	0	0
$682$	−9.97655	−0.382022
$683$	33.3421	1.27580	0.637900	−	0.770119i	$-0.279803\pi$
$683$	33.3421	1.27580	0.637900	+	0.770119i	$0.279803\pi$
$684$	0	0
$685$	−33.3794	−1.27536
$686$	63.4832	2.42380
$687$	0	0
$688$	−0.257211	−0.00980607
$689$	−55.0372	−2.09675
$690$	0	0
$691$	5.17471	0.196855	0.0984277	−	0.995144i	$-0.468619\pi$
$691$	5.17471	0.196855	0.0984277	+	0.995144i	$0.468619\pi$
$692$	−1.89424	−0.0720081
$693$	0	0
$694$	−10.2623	−0.389550
$695$	−18.0944	−0.686359
$696$	0	0
$697$	24.9363	0.944530
$698$	−1.46351	−0.0553947
$699$	0	0
$700$	−9.34287	−0.353127
$701$	−11.0146	−0.416017	−0.208009	−	0.978127i	$-0.566698\pi$
$701$	−11.0146	−0.416017	−0.208009	+	0.978127i	$0.566698\pi$
$702$	0	0
$703$	−7.54285	−0.284484
$704$	−2.47781	−0.0933861
$705$	0	0
$706$	1.97617	0.0743742
$707$	−42.4419	−1.59619
$708$	0	0
$709$	−34.0603	−1.27916	−0.639580	−	0.768725i	$-0.720892\pi$
$709$	−34.0603	−1.27916	−0.639580	+	0.768725i	$0.720892\pi$
$710$	−19.7387	−0.740782
$711$	0	0
$712$	6.51476	0.244151
$713$	23.0241	0.862258
$714$	0	0
$715$	−24.5949	−0.919796
$716$	−0.437066	−0.0163339
$717$	0	0
$718$	−33.5870	−1.25346
$719$	37.2474	1.38909	0.694547	−	0.719447i	$-0.255605\pi$
$719$	37.2474	1.38909	0.694547	+	0.719447i	$0.255605\pi$
$720$	0	0
$721$	68.2589	2.54209
$722$	−6.57009	−0.244513
$723$	0	0
$724$	10.6133	0.394440
$725$	6.58998	0.244746
$726$	0	0
$727$	3.11767	0.115628	0.0578140	−	0.998327i	$-0.481587\pi$
$727$	3.11767	0.115628	0.0578140	+	0.998327i	$0.481587\pi$
$728$	19.5165	0.723329
$729$	0	0
$730$	−10.2779	−0.380404
$731$	0.620012	0.0229320
$732$	0	0
$733$	−7.43179	−0.274499	−0.137250	−	0.990536i	$-0.543826\pi$
$733$	−7.43179	−0.274499	−0.137250	+	0.990536i	$0.543826\pi$
$734$	6.82332	0.251853
$735$	0	0
$736$	5.71834	0.210781
$737$	8.07881	0.297587
$738$	0	0
$739$	20.7149	0.762010	0.381005	−	0.924573i	$-0.375578\pi$
$739$	20.7149	0.762010	0.381005	+	0.924573i	$0.375578\pi$
$740$	−3.89537	−0.143197
$741$	0	0
$742$	−74.3472	−2.72937
$743$	39.2101	1.43848	0.719239	−	0.694762i	$-0.244490\pi$
$743$	39.2101	1.43848	0.719239	+	0.694762i	$0.244490\pi$
$744$	0	0
$745$	−37.6807	−1.38052
$746$	2.83888	0.103939
$747$	0	0
$748$	5.97282	0.218388
$749$	43.9112	1.60448
$750$	0	0
$751$	−22.3020	−0.813813	−0.406906	−	0.913470i	$-0.633392\pi$
$751$	−22.3020	−0.813813	−0.406906	+	0.913470i	$0.633392\pi$
$752$	−7.24280	−0.264118
$753$	0	0
$754$	−13.7659	−0.501325
$755$	−44.8241	−1.63132
$756$	0	0
$757$	−14.3362	−0.521059	−0.260530	−	0.965466i	$-0.583897\pi$
$757$	−14.3362	−0.521059	−0.260530	+	0.965466i	$0.583897\pi$
$758$	10.7922	0.391989
$759$	0	0
$760$	−13.2052	−0.479004
$761$	−29.9441	−1.08547	−0.542737	−	0.839903i	$-0.682612\pi$
$761$	−29.9441	−1.08547	−0.542737	+	0.839903i	$0.682612\pi$
$762$	0	0
$763$	−34.2445	−1.23973
$764$	−16.8395	−0.609233
$765$	0	0
$766$	27.0660	0.977935
$767$	−13.0023	−0.469484
$768$	0	0
$769$	−10.0213	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$769$	−10.0213	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$770$	−33.2240	−1.19731
$771$	0	0
$772$	6.99809	0.251867
$773$	−36.0701	−1.29735	−0.648676	−	0.761065i	$-0.724677\pi$
$773$	−36.0701	−1.29735	−0.648676	+	0.761065i	$0.724677\pi$
$774$	0	0
$775$	−7.32635	−0.263170
$776$	−9.29949	−0.333832
$777$	0	0
$778$	32.3882	1.16117
$779$	−52.3103	−1.87421
$780$	0	0
$781$	−18.7287	−0.670167
$782$	−13.7842	−0.492921
$783$	0	0
$784$	19.3639	0.691567
$785$	−6.58902	−0.235172
$786$	0	0
$787$	−40.2345	−1.43421	−0.717103	−	0.696968i	$-0.754532\pi$
$787$	−40.2345	−1.43421	−0.717103	+	0.696968i	$0.754532\pi$
$788$	20.1550	0.717992
$789$	0	0
$790$	−2.51945	−0.0896381
$791$	−71.9024	−2.55655
$792$	0	0
$793$	−36.5533	−1.29805
$794$	−7.18123	−0.254852
$795$	0	0
$796$	14.6510	0.519293
$797$	−25.2279	−0.893618	−0.446809	−	0.894629i	$-0.647440\pi$
$797$	−25.2279	−0.893618	−0.446809	+	0.894629i	$0.647440\pi$
$798$	0	0
$799$	17.4589	0.617653
$800$	−1.81960	−0.0643325
$801$	0	0
$802$	29.3177	1.03524
$803$	−9.75203	−0.344142
$804$	0	0
$805$	76.6750	2.70244
$806$	15.3041	0.539065
$807$	0	0
$808$	−8.26590	−0.290793
$809$	11.9297	0.419425	0.209712	−	0.977763i	$-0.432747\pi$
$809$	11.9297	0.419425	0.209712	+	0.977763i	$0.432747\pi$
$810$	0	0
$811$	−39.9706	−1.40356	−0.701778	−	0.712395i	$-0.747610\pi$
$811$	−39.9706	−1.40356	−0.701778	+	0.712395i	$0.747610\pi$
$812$	−18.5957	−0.652582
$813$	0	0
$814$	−3.69605	−0.129546
$815$	−59.0224	−2.06747
$816$	0	0
$817$	−1.30063	−0.0455034
$818$	−1.22575	−0.0428573
$819$	0	0
$820$	−27.0147	−0.943394
$821$	3.66230	0.127815	0.0639076	−	0.997956i	$-0.479644\pi$
$821$	3.66230	0.127815	0.0639076	+	0.997956i	$0.479644\pi$
$822$	0	0
$823$	−50.1793	−1.74914	−0.874570	−	0.484899i	$-0.838856\pi$
$823$	−50.1793	−1.74914	−0.874570	+	0.484899i	$0.838856\pi$
$824$	13.2940	0.463117
$825$	0	0
$826$	−17.5641	−0.611134
$827$	−30.6526	−1.06590	−0.532948	−	0.846148i	$-0.678916\pi$
$827$	−30.6526	−1.06590	−0.532948	+	0.846148i	$0.678916\pi$
$828$	0	0
$829$	−37.8809	−1.31566	−0.657829	−	0.753167i	$-0.728525\pi$
$829$	−37.8809	−1.31566	−0.657829	+	0.753167i	$0.728525\pi$
$830$	11.5412	0.400601
$831$	0	0
$832$	3.80099	0.131776
$833$	−46.6770	−1.61726
$834$	0	0
$835$	−2.17650	−0.0753209
$836$	−12.5295	−0.433343
$837$	0	0
$838$	−22.2987	−0.770295
$839$	7.69230	0.265568	0.132784	−	0.991145i	$-0.457608\pi$
$839$	7.69230	0.265568	0.132784	+	0.991145i	$0.457608\pi$
$840$	0	0
$841$	−15.8835	−0.547708
$842$	39.8226	1.37238
$843$	0	0
$844$	7.44295	0.256197
$845$	3.78012	0.130040
$846$	0	0
$847$	24.9563	0.857510
$848$	−14.4797	−0.497235
$849$	0	0
$850$	4.38618	0.150445
$851$	8.52981	0.292398
$852$	0	0
$853$	−4.33320	−0.148366	−0.0741830	−	0.997245i	$-0.523635\pi$
$853$	−4.33320	−0.148366	−0.0741830	+	0.997245i	$0.523635\pi$
$854$	−49.3781	−1.68968
$855$	0	0
$856$	8.55207	0.292304
$857$	4.72383	0.161363	0.0806815	−	0.996740i	$-0.474290\pi$
$857$	4.72383	0.161363	0.0806815	+	0.996740i	$0.474290\pi$
$858$	0	0
$859$	16.6776	0.569032	0.284516	−	0.958671i	$-0.408167\pi$
$859$	16.6776	0.569032	0.284516	+	0.958671i	$0.408167\pi$
$860$	−0.671689	−0.0229044
$861$	0	0
$862$	−14.5119	−0.494278
$863$	33.8413	1.15197	0.575986	−	0.817460i	$-0.304618\pi$
$863$	33.8413	1.15197	0.575986	+	0.817460i	$0.304618\pi$
$864$	0	0
$865$	−4.94668	−0.168192
$866$	13.8876	0.471918
$867$	0	0
$868$	20.6736	0.701708
$869$	−2.39054	−0.0810934
$870$	0	0
$871$	−12.3930	−0.419920
$872$	−6.66939	−0.225854
$873$	0	0
$874$	28.9159	0.978094
$875$	42.6448	1.44166
$876$	0	0
$877$	6.92687	0.233904	0.116952	−	0.993138i	$-0.462688\pi$
$877$	6.92687	0.233904	0.116952	+	0.993138i	$0.462688\pi$
$878$	−10.7609	−0.363163
$879$	0	0
$880$	−6.47065	−0.218126
$881$	−12.8175	−0.431833	−0.215917	−	0.976412i	$-0.569274\pi$
$881$	−12.8175	−0.431833	−0.215917	+	0.976412i	$0.569274\pi$
$882$	0	0
$883$	10.8932	0.366586	0.183293	−	0.983058i	$-0.441324\pi$
$883$	10.8932	0.366586	0.183293	+	0.983058i	$0.441324\pi$
$884$	−9.16237	−0.308164
$885$	0	0
$886$	−19.4970	−0.655016
$887$	−12.2857	−0.412514	−0.206257	−	0.978498i	$-0.566128\pi$
$887$	−12.2857	−0.412514	−0.206257	+	0.978498i	$0.566128\pi$
$888$	0	0
$889$	−18.4109	−0.617481
$890$	17.0129	0.570273
$891$	0	0
$892$	25.0760	0.839606
$893$	−36.6246	−1.22560
$894$	0	0
$895$	−1.14137	−0.0381518
$896$	5.13458	0.171534
$897$	0	0
$898$	−7.65616	−0.255489
$899$	−14.5821	−0.486341
$900$	0	0
$901$	34.9037	1.16281
$902$	−25.6324	−0.853465
$903$	0	0
$904$	−14.0036	−0.465752
$905$	27.7159	0.921309
$906$	0	0
$907$	39.2653	1.30378	0.651891	−	0.758312i	$-0.273976\pi$
$907$	39.2653	1.30378	0.651891	+	0.758312i	$0.273976\pi$
$908$	3.91361	0.129878
$909$	0	0
$910$	50.9660	1.68951
$911$	41.4194	1.37229	0.686143	−	0.727467i	$-0.259303\pi$
$911$	41.4194	1.37229	0.686143	+	0.727467i	$0.259303\pi$
$912$	0	0
$913$	10.9507	0.362414
$914$	−15.6787	−0.518606
$915$	0	0
$916$	−2.51025	−0.0829409
$917$	−30.0644	−0.992813
$918$	0	0
$919$	−12.8390	−0.423518	−0.211759	−	0.977322i	$-0.567919\pi$
$919$	−12.8390	−0.423518	−0.211759	+	0.977322i	$0.567919\pi$
$920$	14.9331	0.492329
$921$	0	0
$922$	−14.6885	−0.483742
$923$	28.7301	0.945662
$924$	0	0
$925$	−2.71422	−0.0892430
$926$	−0.614733	−0.0202014
$927$	0	0
$928$	−3.62167	−0.118887
$929$	−25.9887	−0.852662	−0.426331	−	0.904567i	$-0.640194\pi$
$929$	−25.9887	−0.852662	−0.426331	+	0.904567i	$0.640194\pi$
$930$	0	0
$931$	97.9170	3.20910
$932$	−4.23677	−0.138780
$933$	0	0
$934$	18.1301	0.593233
$935$	15.5976	0.510097
$936$	0	0
$937$	−29.0543	−0.949162	−0.474581	−	0.880212i	$-0.657400\pi$
$937$	−29.0543	−0.949162	−0.474581	+	0.880212i	$0.657400\pi$
$938$	−16.7411	−0.546615
$939$	0	0
$940$	−18.9141	−0.616910
$941$	−29.3817	−0.957815	−0.478908	−	0.877865i	$-0.658967\pi$
$941$	−29.3817	−0.957815	−0.478908	+	0.877865i	$0.658967\pi$
$942$	0	0
$943$	59.1549	1.92635
$944$	−3.42075	−0.111336
$945$	0	0
$946$	−0.637320	−0.0207211
$947$	−42.4899	−1.38074	−0.690368	−	0.723458i	$-0.742552\pi$
$947$	−42.4899	−1.38074	−0.690368	+	0.723458i	$0.742552\pi$
$948$	0	0
$949$	14.9597	0.485613
$950$	−9.20114	−0.298524
$951$	0	0
$952$	−12.3770	−0.401141
$953$	57.9513	1.87723	0.938613	−	0.344971i	$-0.112111\pi$
$953$	57.9513	1.87723	0.938613	+	0.344971i	$0.112111\pi$
$954$	0	0
$955$	−43.9753	−1.42301
$956$	13.1844	0.426414
$957$	0	0
$958$	−8.98748	−0.290372
$959$	65.6302	2.11931
$960$	0	0
$961$	−14.7885	−0.477048
$962$	5.66978	0.182801
$963$	0	0
$964$	9.20561	0.296493
$965$	18.2751	0.588296
$966$	0	0
$967$	−31.3181	−1.00712	−0.503562	−	0.863959i	$-0.667977\pi$
$967$	−31.3181	−1.00712	−0.503562	+	0.863959i	$0.667977\pi$
$968$	4.86045	0.156221
$969$	0	0
$970$	−24.2850	−0.779745
$971$	−23.9670	−0.769138	−0.384569	−	0.923096i	$-0.625650\pi$
$971$	−23.9670	−0.769138	−0.384569	+	0.923096i	$0.625650\pi$
$972$	0	0
$973$	35.5770	1.14055
$974$	35.3720	1.13339
$975$	0	0
$976$	−9.61678	−0.307826
$977$	6.23158	0.199366	0.0996830	−	0.995019i	$-0.468217\pi$
$977$	6.23158	0.199366	0.0996830	+	0.995019i	$0.468217\pi$
$978$	0	0
$979$	16.1424	0.515912
$980$	50.5675	1.61532
$981$	0	0
$982$	−5.50209	−0.175579
$983$	−44.1797	−1.40911	−0.704556	−	0.709648i	$-0.748854\pi$
$983$	−44.1797	−1.40911	−0.704556	+	0.709648i	$0.748854\pi$
$984$	0	0
$985$	52.6335	1.67704
$986$	8.73011	0.278023
$987$	0	0
$988$	19.2204	0.611483
$989$	1.47082	0.0467693
$990$	0	0
$991$	26.7053	0.848322	0.424161	−	0.905587i	$-0.360569\pi$
$991$	26.7053	0.848322	0.424161	+	0.905587i	$0.360569\pi$
$992$	4.02635	0.127837
$993$	0	0
$994$	38.8101	1.23098
$995$	38.2603	1.21293
$996$	0	0
$997$	−9.36433	−0.296571	−0.148286	−	0.988945i	$-0.547375\pi$
$997$	−9.36433	−0.296571	−0.148286	+	0.988945i	$0.547375\pi$
$998$	6.86687	0.217367
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.e.1.6	✓	6
3.2	odd	2		1458.2.a.h.1.1	yes	6
9.2	odd	6		1458.2.c.e.973.6		12
9.4	even	3		1458.2.c.h.487.1		12
9.5	odd	6		1458.2.c.e.487.6		12
9.7	even	3		1458.2.c.h.973.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1458.2.a.e.1.6	✓	6	1.1	even	1	trivial
1458.2.a.h.1.1	yes	6	3.2	odd	2
1458.2.c.e.487.6		12	9.5	odd	6
1458.2.c.e.973.6		12	9.2	odd	6
1458.2.c.h.487.1		12	9.4	even	3
1458.2.c.h.973.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.61144 q^{5} -5.13458 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.61144 q^{5} -5.13458 q^{7} -1.00000 q^{8} -2.61144 q^{10} -2.47781 q^{11} +3.80099 q^{13} +5.13458 q^{14} +1.00000 q^{16} -2.41052 q^{17} +5.05669 q^{19} +2.61144 q^{20} +2.47781 q^{22} -5.71834 q^{23} +1.81960 q^{25} -3.80099 q^{26} -5.13458 q^{28} +3.62167 q^{29} -4.02635 q^{31} -1.00000 q^{32} +2.41052 q^{34} -13.4086 q^{35} -1.49166 q^{37} -5.05669 q^{38} -2.61144 q^{40} -10.3448 q^{41} -0.257211 q^{43} -2.47781 q^{44} +5.71834 q^{46} -7.24280 q^{47} +19.3639 q^{49} -1.81960 q^{50} +3.80099 q^{52} -14.4797 q^{53} -6.47065 q^{55} +5.13458 q^{56} -3.62167 q^{58} -3.42075 q^{59} -9.61678 q^{61} +4.02635 q^{62} +1.00000 q^{64} +9.92604 q^{65} -3.26046 q^{67} -2.41052 q^{68} +13.4086 q^{70} +7.55858 q^{71} +3.93574 q^{73} +1.49166 q^{74} +5.05669 q^{76} +12.7225 q^{77} +0.964777 q^{79} +2.61144 q^{80} +10.3448 q^{82} -4.41948 q^{83} -6.29492 q^{85} +0.257211 q^{86} +2.47781 q^{88} -6.51476 q^{89} -19.5165 q^{91} -5.71834 q^{92} +7.24280 q^{94} +13.2052 q^{95} +9.29949 q^{97} -19.3639 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 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

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Database

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