LMFDB - Embedded newform 2550.2.d.n.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.n.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +8.00000 q^{41} +2.00000i q^{42} -6.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} +4.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{57} +2.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} -4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -2.00000i q^{67} -1.00000i q^{68} +8.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -8.00000i q^{77} -4.00000i q^{78} +1.00000 q^{81} -8.00000i q^{82} +16.0000i q^{83} +2.00000 q^{84} -6.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +2.00000 q^{89} +8.00000 q^{91} +8.00000i q^{92} +4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 4 q^{21} - 2 q^{24} - 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 16 q^{41} + 8 q^{44} - 16 q^{46} + 6 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} + 4 q^{71} + 12 q^{74} - 8 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 - 4 * q^21 - 2 * q^24 - 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 16 * q^41 + 8 * q^44 - 16 * q^46 + 6 * q^49 - 2 * q^51 - 2 * q^54 - 4 * q^56 + 12 * q^59 + 28 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 4 * q^71 + 12 * q^74 - 8 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 + 4 * q^89 + 16 * q^91 + 16 * q^94 + 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.00000i	0.852803i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	− 1.00000i	− 0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	4.00000	0.640513
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	2.00000i	0.308607i
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	4.00000i	0.554700i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	4.00000i	0.529813i
$58$	2.00000i	0.262613i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	− 4.00000i	− 0.508001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 1.00000i	− 0.121268i
$69$	8.00000	0.963087
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 8.00000i	− 0.911685i
$78$	− 4.00000i	− 0.452911i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 8.00000i	− 0.883452i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−6.00000	−0.646997
$87$	− 2.00000i	− 0.214423i
$88$	− 4.00000i	− 0.426401i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	8.00000i	0.834058i
$93$	4.00000i	0.414781i
$94$	8.00000	0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 3.00000i	− 0.303046i
$99$	4.00000	0.402015
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	1.00000i	0.0990148i
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	2.00000	0.194257
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	2.00000i	0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	4.00000i	0.369800i
$118$	− 6.00000i	− 0.552345i
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	− 14.0000i	− 1.26750i
$123$	8.00000i	0.721336i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	6.00000	0.528271
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	4.00000i	0.348155i
$133$	8.00000i	0.693688i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 8.00000i	− 0.681005i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	− 2.00000i	− 0.167836i
$143$	16.0000i	1.33799i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	3.00000i	0.247436i
$148$	− 6.00000i	− 0.493197i
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000i	0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	16.0000	1.26098
$162$	− 1.00000i	− 0.0785674i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	6.00000i	0.457496i
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	6.00000i	0.450988i
$178$	− 2.00000i	− 0.149906i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	− 8.00000i	− 0.592999i
$183$	14.0000i	1.03491i
$184$	8.00000	0.589768
$185$	0	0
$186$	4.00000	0.293294
$187$	− 4.00000i	− 0.292509i
$188$	− 8.00000i	− 0.583460i
$189$	2.00000	0.145479
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	0	0
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	− 4.00000i	− 0.284268i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	− 8.00000i	− 0.562878i
$203$	− 4.00000i	− 0.280745i
$204$	1.00000	0.0700140
$205$	0	0
$206$	12.0000	0.836080
$207$	8.00000i	0.556038i
$208$	− 4.00000i	− 0.277350i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 2.00000i	− 0.137361i
$213$	2.00000i	0.137038i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000i	0.543075i
$218$	− 14.0000i	− 0.948200i
$219$	4.00000	0.270295
$220$	0	0
$221$	4.00000	0.269069
$222$	6.00000i	0.402694i
$223$	− 28.0000i	− 1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	− 28.0000i	− 1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	− 2.00000i	− 0.131306i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	2.00000i	0.129641i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 5.00000i	− 0.321412i
$243$	1.00000i	0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	8.00000	0.510061
$247$	− 16.0000i	− 1.01806i
$248$	4.00000i	0.254000i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	2.00000i	0.125988i
$253$	32.0000i	2.01182i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	− 6.00000i	− 0.373544i
$259$	−12.0000	−0.745644
$260$	0	0
$261$	2.00000	0.123797
$262$	16.0000i	0.988483i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	8.00000	0.490511
$267$	2.00000i	0.122398i
$268$	2.00000i	0.122169i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	1.00000i	0.0606339i
$273$	8.00000i	0.484182i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 20.0000i	− 1.19952i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	8.00000i	0.476393i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	16.0000	0.946100
$287$	16.0000i	0.944450i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000i	0.232104i
$298$	16.0000i	0.926855i
$299$	−32.0000	−1.85061
$300$	0	0
$301$	12.0000	0.691669
$302$	8.00000i	0.460348i
$303$	8.00000i	0.459588i
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	8.00000i	0.455842i
$309$	−12.0000	−0.682656
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	4.00000i	0.226455i
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	2.00000i	0.112154i
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 16.0000i	− 0.891645i
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	14.0000i	0.774202i
$328$	8.00000i	0.441726i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	− 16.0000i	− 0.878114i
$333$	− 6.00000i	− 0.328798i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−2.00000	−0.109109
$337$	20.0000i	1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$	20.0000i	1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	3.00000i	0.163178i
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	4.00000i	0.216295i
$343$	20.0000i	1.07990i
$344$	6.00000	0.323498
$345$	0	0
$346$	−22.0000	−1.18273
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	2.00000i	0.107211i
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	4.00000i	0.213201i
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−2.00000	−0.106000
$357$	− 2.00000i	− 0.105851i
$358$	6.00000i	0.317110i
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 26.0000i	− 1.36653i
$363$	5.00000i	0.262432i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	14.0000	0.731792
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	− 8.00000i	− 0.417029i
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−4.00000	−0.207670
$372$	− 4.00000i	− 0.207390i
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−8.00000	−0.412568
$377$	8.00000i	0.412021i
$378$	− 2.00000i	− 0.102869i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	− 4.00000i	− 0.204658i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−24.0000	−1.22157
$387$	6.00000i	0.304997i
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	3.00000i	0.151523i
$393$	− 16.0000i	− 0.807093i
$394$	−26.0000	−1.30986
$395$	0	0
$396$	−4.00000	−0.201008
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	− 20.0000i	− 1.00251i
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	− 16.0000i	− 0.797017i
$404$	−8.00000	−0.398015
$405$	0	0
$406$	−4.00000	−0.198517
$407$	− 24.0000i	− 1.18964i
$408$	− 1.00000i	− 0.0495074i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 12.0000i	− 0.591198i
$413$	12.0000i	0.590481i
$414$	8.00000	0.393179
$415$	0	0
$416$	−4.00000	−0.196116
$417$	20.0000i	0.979404i
$418$	16.0000i	0.782586i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 8.00000i	− 0.388973i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	2.00000	0.0969003
$427$	28.0000i	1.35501i
$428$	− 12.0000i	− 0.580042i
$429$	−16.0000	−0.772487
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−14.0000	−0.670478
$437$	− 32.0000i	− 1.53077i
$438$	− 4.00000i	− 0.191127i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 4.00000i	− 0.190261i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−28.0000	−1.32584
$447$	− 16.0000i	− 0.756774i
$448$	− 2.00000i	− 0.0944911i
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	−32.0000	−1.50682
$452$	6.00000i	0.282216i
$453$	− 8.00000i	− 0.375873i
$454$	28.0000	1.31411
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	6.00000i	0.280362i
$459$	1.00000	0.0466760
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	− 8.00000i	− 0.372194i
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	− 4.00000i	− 0.184900i
$469$	4.00000	0.184703
$470$	0	0
$471$	4.00000	0.184310
$472$	6.00000i	0.276172i
$473$	24.0000i	1.10352i
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	14.0000i	0.637683i
$483$	16.0000i	0.728025i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	14.0000i	0.633750i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	− 8.00000i	− 0.360668i
$493$	− 2.00000i	− 0.0900755i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	4.00000	0.179605
$497$	4.00000i	0.179425i
$498$	16.0000i	0.716977i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	− 26.0000i	− 1.16044i
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	32.0000	1.42257
$507$	− 3.00000i	− 0.133235i
$508$	4.00000i	0.177471i
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−6.00000	−0.264135
$517$	− 32.0000i	− 1.40736i
$518$	12.0000i	0.527250i
$519$	22.0000	0.965693
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000i	0.174243i
$528$	− 4.00000i	− 0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−6.00000	−0.260378
$532$	− 8.00000i	− 0.346844i
$533$	− 32.0000i	− 1.38607i
$534$	2.00000	0.0865485
$535$	0	0
$536$	2.00000	0.0863868
$537$	− 6.00000i	− 0.258919i
$538$	− 30.0000i	− 1.29339i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	24.0000i	1.03089i
$543$	26.0000i	1.11577i
$544$	1.00000	0.0428746
$545$	0	0
$546$	8.00000	0.342368
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	6.00000i	0.256307i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−8.00000	−0.340811
$552$	8.00000i	0.340503i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	4.00000i	0.169334i
$559$	−24.0000	−1.01509
$560$	0	0
$561$	4.00000	0.168880
$562$	2.00000i	0.0843649i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	20.0000	0.840663
$567$	2.00000i	0.0839921i
$568$	2.00000i	0.0839181i
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 16.0000i	− 0.668994i
$573$	4.00000i	0.167102i
$574$	16.0000	0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	1.00000i	0.0415945i
$579$	24.0000	0.997406
$580$	0	0
$581$	−32.0000	−1.32758
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	4.00000	0.165521
$585$	0	0
$586$	14.0000	0.578335
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	− 3.00000i	− 0.123718i
$589$	16.0000	0.659269
$590$	0	0
$591$	26.0000	1.06950
$592$	6.00000i	0.246598i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	16.0000	0.655386
$597$	20.0000i	0.818546i
$598$	32.0000i	1.30858i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	− 12.0000i	− 0.489083i
$603$	2.00000i	0.0814463i
$604$	8.00000	0.325515
$605$	0	0
$606$	8.00000	0.324978
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	− 4.00000i	− 0.162221i
$609$	4.00000	0.162088
$610$	0	0
$611$	32.0000	1.29458
$612$	1.00000i	0.0404226i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	8.00000	0.322329
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	12.0000i	0.482711i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	− 14.0000i	− 0.561349i
$623$	4.00000i	0.160257i
$624$	4.00000	0.160128
$625$	0	0
$626$	−16.0000	−0.639489
$627$	− 16.0000i	− 0.638978i
$628$	4.00000i	0.159617i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	2.00000	0.0793052
$637$	− 12.0000i	− 0.475457i
$638$	− 8.00000i	− 0.316723i
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	12.0000i	0.473602i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	4.00000	0.157378
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	1.00000i	0.0392837i
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 8.00000i	− 0.313304i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	8.00000	0.312348
$657$	4.00000i	0.156055i
$658$	16.0000i	0.623745i
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	8.00000i	0.310929i
$663$	4.00000i	0.155347i
$664$	−16.0000	−0.620920
$665$	0	0
$666$	−6.00000	−0.232495
$667$	16.0000i	0.619522i
$668$	12.0000i	0.464294i
$669$	28.0000	1.08254
$670$	0	0
$671$	−56.0000	−2.16186
$672$	2.00000i	0.0771517i
$673$	8.00000i	0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$	8.00000i	0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	3.00000	0.115385
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	− 6.00000i	− 0.230429i
$679$	0	0
$680$	0	0
$681$	−28.0000	−1.07296
$682$	16.0000i	0.612672i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	20.0000	0.763604
$687$	− 6.00000i	− 0.228914i
$688$	− 6.00000i	− 0.228748i
$689$	8.00000	0.304776
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	22.0000i	0.836315i
$693$	8.00000i	0.303895i
$694$	−28.0000	−1.06287
$695$	0	0
$696$	2.00000	0.0758098
$697$	8.00000i	0.303022i
$698$	− 14.0000i	− 0.529908i
$699$	18.0000	0.680823
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	4.00000i	0.150970i
$703$	24.0000i	0.905177i
$704$	4.00000	0.150756
$705$	0	0
$706$	10.0000	0.376355
$707$	16.0000i	0.601742i
$708$	− 6.00000i	− 0.225494i
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	0	0
$712$	2.00000i	0.0749532i
$713$	− 32.0000i	− 1.19841i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	36.0000i	1.34351i
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	3.00000i	0.111648i
$723$	− 14.0000i	− 0.520666i
$724$	−26.0000	−0.966282
$725$	0	0
$726$	5.00000	0.185567
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	− 14.0000i	− 0.517455i
$733$	− 48.0000i	− 1.77292i	−0.462805	−	0.886460i	$-0.653157\pi$
$733$	− 48.0000i	− 1.77292i	0.462805	−	0.886460i	$-0.346843\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−8.00000	−0.294884
$737$	8.00000i	0.294684i
$738$	8.00000i	0.294484i
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	4.00000i	0.146845i
$743$	20.0000i	0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$	20.0000i	0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	16.0000	0.585802
$747$	− 16.0000i	− 0.585409i
$748$	4.00000i	0.146254i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	8.00000i	0.291730i
$753$	26.0000i	0.947493i
$754$	8.00000	0.291343
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	− 28.0000i	− 1.01768i	−0.860862	−	0.508839i	$-0.830075\pi$
$757$	− 28.0000i	− 1.01768i	0.860862	−	0.508839i	$-0.169925\pi$
$758$	− 4.00000i	− 0.145287i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	− 4.00000i	− 0.144905i
$763$	28.0000i	1.01367i
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−8.00000	−0.289052
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	24.0000i	0.863779i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	6.00000	0.215666
$775$	0	0
$776$	0	0
$777$	− 12.0000i	− 0.430498i
$778$	24.0000i	0.860442i
$779$	32.0000	1.14652
$780$	0	0
$781$	−8.00000	−0.286263
$782$	− 8.00000i	− 0.286079i
$783$	2.00000i	0.0714742i
$784$	3.00000	0.107143
$785$	0	0
$786$	−16.0000	−0.570701
$787$	24.0000i	0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$	24.0000i	0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	26.0000i	0.926212i
$789$	8.00000	0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	4.00000i	0.142134i
$793$	− 56.0000i	− 1.98862i
$794$	6.00000	0.212932
$795$	0	0
$796$	−20.0000	−0.708881
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	8.00000i	0.283197i
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	12.0000i	0.423735i
$803$	16.0000i	0.564628i
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−16.0000	−0.563576
$807$	30.0000i	1.05605i
$808$	8.00000i	0.281439i
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	4.00000i	0.140372i
$813$	− 24.0000i	− 0.841717i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	− 24.0000i	− 0.839654i
$818$	− 26.0000i	− 0.909069i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	12.0000	0.417533
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	− 8.00000i	− 0.278019i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	4.00000i	0.138675i
$833$	3.00000i	0.103944i
$834$	20.0000	0.692543
$835$	0	0
$836$	16.0000	0.553372
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	18.0000i	0.620321i
$843$	− 2.00000i	− 0.0688837i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−8.00000	−0.275046
$847$	10.0000i	0.343604i
$848$	2.00000i	0.0686803i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	48.0000	1.64542
$852$	− 2.00000i	− 0.0685189i
$853$	− 2.00000i	− 0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 22.0000i	− 0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$	− 22.0000i	− 0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$	16.0000i	0.546231i
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	− 14.0000i	− 0.476842i
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	− 1.00000i	− 0.0339618i
$868$	− 8.00000i	− 0.271538i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	14.0000i	0.474100i
$873$	0	0
$874$	−32.0000	−1.08242
$875$	0	0
$876$	−4.00000	−0.135147
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	− 24.0000i	− 0.809961i
$879$	−14.0000	−0.472208
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	3.00000i	0.101015i
$883$	46.0000i	1.54802i	0.633171	+	0.774012i	$0.281753\pi$
$883$	46.0000i	1.54802i	−0.633171	+	0.774012i	$0.718247\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−24.0000	−0.806296
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 6.00000i	− 0.201347i
$889$	8.00000	0.268311
$890$	0	0
$891$	−4.00000	−0.134005
$892$	28.0000i	0.937509i
$893$	32.0000i	1.07084i
$894$	−16.0000	−0.535120
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 32.0000i	− 1.06845i
$898$	− 12.0000i	− 0.400445i
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	32.0000i	1.06548i
$903$	12.0000i	0.399335i
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	− 28.0000i	− 0.929213i
$909$	−8.00000	−0.265343
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	4.00000i	0.132453i
$913$	− 64.0000i	− 2.11809i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	6.00000	0.198246
$917$	− 32.0000i	− 1.05673i
$918$	− 1.00000i	− 0.0330049i
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	28.0000i	0.922131i
$923$	− 8.00000i	− 0.263323i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−4.00000	−0.131448
$927$	− 12.0000i	− 0.394132i
$928$	2.00000i	0.0656532i
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	18.0000i	0.589610i
$933$	14.0000i	0.458339i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−4.00000	−0.130744
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	− 4.00000i	− 0.130605i
$939$	16.0000	0.522140
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	− 4.00000i	− 0.130327i
$943$	− 64.0000i	− 2.08413i
$944$	6.00000	0.195283
$945$	0	0
$946$	24.0000	0.780307
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	2.00000	0.0648544
$952$	− 2.00000i	− 0.0648204i
$953$	− 42.0000i	− 1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	− 42.0000i	− 1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	0	0
$957$	8.00000i	0.258603i
$958$	− 22.0000i	− 0.710788i
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 24.0000i	− 0.773791i
$963$	− 12.0000i	− 0.386695i
$964$	14.0000	0.450910
$965$	0	0
$966$	16.0000	0.514792
$967$	56.0000i	1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$	56.0000i	1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$	5.00000i	0.160706i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−50.0000	−1.60458	−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000	−1.60458	−0.802288	+	0.596937i	$0.796384\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	40.0000i	1.28234i
$974$	−22.0000	−0.704925
$975$	0	0
$976$	14.0000	0.448129
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	8.00000i	0.255812i
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−14.0000	−0.446986
$982$	18.0000i	0.574403i
$983$	52.0000i	1.65854i	0.558846	+	0.829271i	$0.311244\pi$
$983$	52.0000i	1.65854i	−0.558846	+	0.829271i	$0.688756\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	− 16.0000i	− 0.509286i
$988$	16.0000i	0.509028i
$989$	−48.0000	−1.52631
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	− 4.00000i	− 0.127000i
$993$	− 8.00000i	− 0.253872i
$994$	4.00000	0.126872
$995$	0	0
$996$	16.0000	0.506979
$997$	− 50.0000i	− 1.58352i	−0.610835	−	0.791758i	$-0.709166\pi$
$997$	− 50.0000i	− 1.58352i	0.610835	−	0.791758i	$-0.290834\pi$
$998$	4.00000i	0.126618i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.n.2449.1		2
5.2	odd	4		2550.2.a.bc.1.1		1
5.3	odd	4		510.2.a.a.1.1	✓	1
5.4	even	2	inner	2550.2.d.n.2449.2		2
15.2	even	4		7650.2.a.k.1.1		1
15.8	even	4		1530.2.a.p.1.1		1
20.3	even	4		4080.2.a.s.1.1		1
85.33	odd	4		8670.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.a.1.1	✓	1	5.3	odd	4
1530.2.a.p.1.1		1	15.8	even	4
2550.2.a.bc.1.1		1	5.2	odd	4
2550.2.d.n.2449.1		2	1.1	even	1	trivial
2550.2.d.n.2449.2		2	5.4	even	2	inner
4080.2.a.s.1.1		1	20.3	even	4
7650.2.a.k.1.1		1	15.2	even	4
8670.2.a.k.1.1		1	85.33	odd	4

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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +8.00000 q^{41} +2.00000i q^{42} -6.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} +4.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{57} +2.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} -4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -2.00000i q^{67} -1.00000i q^{68} +8.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -8.00000i q^{77} -4.00000i q^{78} +1.00000 q^{81} -8.00000i q^{82} +16.0000i q^{83} +2.00000 q^{84} -6.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +2.00000 q^{89} +8.00000 q^{91} +8.00000i q^{92} +4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 4 q^{21} - 2 q^{24} - 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 16 q^{41} + 8 q^{44} - 16 q^{46} + 6 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} + 4 q^{71} + 12 q^{74} - 8 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 - 4 * q^21 - 2 * q^24 - 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 16 * q^41 + 8 * q^44 - 16 * q^46 + 6 * q^49 - 2 * q^51 - 2 * q^54 - 4 * q^56 + 12 * q^59 + 28 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 4 * q^71 + 12 * q^74 - 8 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 + 4 * q^89 + 16 * q^91 + 16 * q^94 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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