LMFDB - Embedded newform 1620.4.i.p.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 11x^{2} + 10x + 100 $ x^4 - x^3 + 11x^2 + 10x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$-1.35078 + 2.33962i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.p.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +(27.7617 - 48.0847i) q^{11} +(-21.7094 - 37.6017i) q^{13} +25.5234 q^{17} -103.361 q^{19} +(2.23828 + 3.87682i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(2.23828 - 3.87682i) q^{29} +(22.5523 + 39.0618i) q^{31} -80.5234 q^{35} +69.5703 q^{37} +(241.570 + 418.412i) q^{41} +(-75.8664 + 131.404i) q^{43} +(-33.8086 + 58.5582i) q^{47} +(41.8195 + 72.4336i) q^{49} -278.953 q^{53} +277.617 q^{55} +(128.953 + 223.353i) q^{59} +(244.670 - 423.780i) q^{61} +(108.547 - 188.009i) q^{65} +(386.298 + 669.087i) q^{67} -536.859 q^{71} -65.5703 q^{73} +(447.094 + 774.389i) q^{77} +(-374.641 + 648.897i) q^{79} +(630.234 - 1091.60i) q^{83} +(63.8086 + 110.520i) q^{85} -1326.28 q^{89} +699.245 q^{91} +(-258.402 - 447.566i) q^{95} +(16.4820 - 28.5477i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{5} - 13 q^{7}+O(q^{10}) $ 4 * q + 10 * q^5 - 13 * q^7	$ 4 q + 10 q^{5} - 13 q^{7} + 15 q^{11} - 10 q^{13} - 90 q^{17} + 86 q^{19} + 105 q^{23} - 50 q^{25} + 105 q^{29} + 71 q^{31} - 130 q^{35} - 298 q^{37} + 390 q^{41} - 169 q^{43} + 345 q^{47} + 417 q^{49} - 1500 q^{53} + 150 q^{55} + 900 q^{59} - q^{61} + 50 q^{65} + 335 q^{67} - 3300 q^{71} + 314 q^{73} + 1020 q^{77} - 346 q^{79} + 600 q^{83} - 225 q^{85} - 3000 q^{89} + 1606 q^{91} + 215 q^{95} + 623 q^{97}+O(q^{100}) $ 4 * q + 10 * q^5 - 13 * q^7 + 15 * q^11 - 10 * q^13 - 90 * q^17 + 86 * q^19 + 105 * q^23 - 50 * q^25 + 105 * q^29 + 71 * q^31 - 130 * q^35 - 298 * q^37 + 390 * q^41 - 169 * q^43 + 345 * q^47 + 417 * q^49 - 1500 * q^53 + 150 * q^55 + 900 * q^59 - q^61 + 50 * q^65 + 335 * q^67 - 3300 * q^71 + 314 * q^73 + 1020 * q^77 - 346 * q^79 + 600 * q^83 - 225 * q^85 - 3000 * q^89 + 1606 * q^91 + 215 * q^95 + 623 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−8.05234	+	13.9471i	−0.434786	+	0.753071i	−0.997278	−	0.0737316i	$-0.976509\pi$
$7$	−8.05234	+	13.9471i	−0.434786	+	0.753071i	0.562492	+	0.826802i	$0.309843\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	27.7617	−	48.0847i	0.760952	−	1.31801i	−0.181408	−	0.983408i	$-0.558066\pi$
$11$	27.7617	−	48.0847i	0.760952	−	1.31801i	0.942360	−	0.334600i	$-0.108601\pi$
$12$		0			0
$13$	−21.7094	−	37.6017i	−0.463161	−	0.802219i	0.535955	−	0.844246i	$-0.319952\pi$
$13$	−21.7094	−	37.6017i	−0.463161	−	0.802219i	−0.999116	+	0.0420276i	$0.986618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	25.5234			0.364138			0.182069	−	0.983286i	$-0.441721\pi$
$17$	25.5234			0.364138			0.182069	+	0.983286i	$0.441721\pi$
$18$		0			0
$19$	−103.361			−1.24803			−0.624016	−	0.781411i	$-0.714500\pi$
$19$	−103.361			−1.24803			−0.624016	+	0.781411i	$0.714500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.23828	+	3.87682i	0.0202919	+	0.0351467i	0.875993	−	0.482324i	$-0.160207\pi$
$23$	2.23828	+	3.87682i	0.0202919	+	0.0351467i	−0.855701	+	0.517470i	$0.826874\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.23828	−	3.87682i	0.0143324	−	0.0248244i	−0.858770	−	0.512361i	$-0.828771\pi$
$29$	2.23828	−	3.87682i	0.0143324	−	0.0248244i	0.873103	+	0.487536i	$0.162104\pi$
$30$		0			0
$31$	22.5523	+	39.0618i	0.130662	+	0.226313i	0.923932	−	0.382557i	$-0.124956\pi$
$31$	22.5523	+	39.0618i	0.130662	+	0.226313i	−0.793270	+	0.608870i	$0.791623\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−80.5234			−0.388884
$36$		0			0
$37$	69.5703			0.309116			0.154558	−	0.987984i	$-0.450605\pi$
$37$	69.5703			0.309116			0.154558	+	0.987984i	$0.450605\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	241.570	+	418.412i	0.920169	+	1.59378i	0.799152	+	0.601130i	$0.205282\pi$
$41$	241.570	+	418.412i	0.920169	+	1.59378i	0.121018	+	0.992650i	$0.461384\pi$
$42$		0			0
$43$	−75.8664	+	131.404i	−0.269059	+	0.466023i	−0.968619	−	0.248550i	$-0.920046\pi$
$43$	−75.8664	+	131.404i	−0.269059	+	0.466023i	0.699560	+	0.714573i	$0.253379\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.8086	+	58.5582i	−0.104925	+	0.181736i	−0.913708	−	0.406372i	$-0.866794\pi$
$47$	−33.8086	+	58.5582i	−0.104925	+	0.181736i	0.808782	+	0.588108i	$0.200127\pi$
$48$		0			0
$49$	41.8195	+	72.4336i	0.121923	+	0.211177i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−278.953			−0.722965			−0.361483	−	0.932379i	$-0.617729\pi$
$53$	−278.953			−0.722965			−0.361483	+	0.932379i	$0.617729\pi$
$54$		0			0
$55$	277.617			0.680616
$56$		0			0
$57$		0			0
$58$		0			0
$59$	128.953	+	223.353i	0.284547	+	0.492850i	0.972499	−	0.232906i	$-0.0748235\pi$
$59$	128.953	+	223.353i	0.284547	+	0.492850i	−0.687952	+	0.725756i	$0.741490\pi$
$60$		0			0
$61$	244.670	−	423.780i	0.513553	−	0.889500i	−0.486324	−	0.873779i	$-0.661662\pi$
$61$	244.670	−	423.780i	0.513553	−	0.889500i	0.999876	−	0.0157208i	$-0.00500428\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	108.547	−	188.009i	0.207132	−	0.358763i
$66$		0			0
$67$	386.298	+	669.087i	0.704385	+	1.22003i	0.966913	+	0.255106i	$0.0821103\pi$
$67$	386.298	+	669.087i	0.704385	+	1.22003i	−0.262529	+	0.964924i	$0.584556\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−536.859			−0.897373			−0.448687	−	0.893689i	$-0.648108\pi$
$71$	−536.859			−0.897373			−0.448687	+	0.893689i	$0.648108\pi$
$72$		0			0
$73$	−65.5703			−0.105129			−0.0525645	−	0.998618i	$-0.516740\pi$
$73$	−65.5703			−0.105129			−0.0525645	+	0.998618i	$0.516740\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	447.094	+	774.389i	0.661702	+	1.14610i
$78$		0			0
$79$	−374.641	+	648.897i	−0.533549	+	0.924134i	0.465683	+	0.884951i	$0.345809\pi$
$79$	−374.641	+	648.897i	−0.533549	+	0.924134i	−0.999232	+	0.0391822i	$0.987525\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	630.234	−	1091.60i	0.833460	−	1.44360i	−0.0618176	−	0.998087i	$-0.519690\pi$
$83$	630.234	−	1091.60i	0.833460	−	1.44360i	0.895278	−	0.445508i	$-0.146977\pi$
$84$		0			0
$85$	63.8086	+	110.520i	0.0814237	+	0.141030i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1326.28			−1.57961			−0.789806	−	0.613356i	$-0.789819\pi$
$89$	−1326.28			−1.57961			−0.789806	+	0.613356i	$0.789819\pi$
$90$		0			0
$91$	699.245			0.805504
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−258.402	−	447.566i	−0.279069	−	0.483361i
$96$		0			0
$97$	16.4820	−	28.5477i	0.0172526	−	0.0298823i	−0.857270	−	0.514867i	$-0.827841\pi$
$97$	16.4820	−	28.5477i	0.0172526	−	0.0298823i	0.874523	+	0.484984i	$0.161175\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−910.090	+	1576.32i	−0.896607	+	1.55297i	−0.0648041	+	0.997898i	$0.520642\pi$
$101$	−910.090	+	1576.32i	−0.896607	+	1.55297i	−0.831803	+	0.555071i	$0.812691\pi$
$102$		0			0
$103$	−687.496	−	1190.78i	−0.657680	−	1.13913i	−0.981215	−	0.192918i	$-0.938205\pi$
$103$	−687.496	−	1190.78i	−0.657680	−	1.13913i	0.323535	−	0.946216i	$-0.395129\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1914.19			−1.72945			−0.864727	−	0.502243i	$-0.832508\pi$
$107$	−1914.19			−1.72945			−0.864727	+	0.502243i	$0.832508\pi$
$108$		0			0
$109$	−563.978			−0.495590			−0.247795	−	0.968812i	$-0.579706\pi$
$109$	−563.978			−0.495590			−0.247795	+	0.968812i	$0.579706\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−620.613	−	1074.93i	−0.516658	−	0.894878i	−0.999813	−	0.0193433i	$-0.993842\pi$
$113$	−620.613	−	1074.93i	−0.516658	−	0.894878i	0.483155	−	0.875535i	$-0.339491\pi$
$114$		0			0
$115$	−11.1914	+	19.3841i	−0.00907483	+	0.0157181i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−205.523	+	355.977i	−0.158322	+	0.274222i
$120$		0			0
$121$	−875.926	−	1517.15i	−0.658096	−	1.13986i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−1699.42			−1.18740			−0.593698	−	0.804688i	$-0.702333\pi$
$127$	−1699.42			−1.18740			−0.593698	+	0.804688i	$0.702333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	786.949	+	1363.04i	0.524855	+	0.909076i	0.999581	+	0.0289425i	$0.00921397\pi$
$131$	786.949	+	1363.04i	0.524855	+	0.909076i	−0.474726	+	0.880134i	$0.657453\pi$
$132$		0			0
$133$	832.298	−	1441.58i	0.542627	−	0.939857i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−611.859	+	1059.77i	−0.381567	+	0.660893i	−0.991286	−	0.131724i	$-0.957949\pi$
$137$	−611.859	+	1059.77i	−0.381567	+	0.660893i	0.609719	+	0.792617i	$0.291282\pi$
$138$		0			0
$139$	1072.64	+	1857.86i	0.654531	+	1.13368i	0.982011	+	0.188823i	$0.0604673\pi$
$139$	1072.64	+	1857.86i	0.654531	+	1.13368i	−0.327480	+	0.944858i	$0.606199\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2410.76			−1.40977
$144$		0			0
$145$	22.3828			0.0128193
$146$		0			0
$147$		0			0
$148$		0			0
$149$	417.996	+	723.990i	0.229823	+	0.398064i	0.957755	−	0.287584i	$-0.0928521\pi$
$149$	417.996	+	723.990i	0.229823	+	0.398064i	−0.727933	+	0.685648i	$0.759519\pi$
$150$		0			0
$151$	−418.244	+	724.419i	−0.225405	+	0.390413i	−0.956441	−	0.291926i	$-0.905704\pi$
$151$	−418.244	+	724.419i	−0.225405	+	0.390413i	0.731036	+	0.682339i	$0.239037\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−112.762	+	195.309i	−0.0584338	+	0.101210i
$156$		0			0
$157$	−705.487	−	1221.94i	−0.358624	−	0.621155i	0.629107	−	0.777319i	$-0.283421\pi$
$157$	−705.487	−	1221.94i	−0.358624	−	0.621155i	−0.987731	+	0.156164i	$0.950087\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−72.0937			−0.0352906
$162$		0			0
$163$	−1926.56			−0.925765			−0.462883	−	0.886420i	$-0.653185\pi$
$163$	−1926.56			−0.925765			−0.462883	+	0.886420i	$0.653185\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	312.094	+	540.562i	0.144614	+	0.250479i	0.929229	−	0.369505i	$-0.120473\pi$
$167$	312.094	+	540.562i	0.144614	+	0.250479i	−0.784615	+	0.619983i	$0.787139\pi$
$168$		0			0
$169$	155.906	−	270.038i	0.0709633	−	0.122912i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1795.52	+	3109.94i	−0.789082	+	1.36673i	0.137449	+	0.990509i	$0.456110\pi$
$173$	−1795.52	+	3109.94i	−0.789082	+	1.36673i	−0.926530	+	0.376221i	$0.877224\pi$
$174$		0			0
$175$	−201.309	−	348.677i	−0.0869571	−	0.150614i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	424.078			0.177079			0.0885393	−	0.996073i	$-0.471780\pi$
$179$	424.078			0.177079			0.0885393	+	0.996073i	$0.471780\pi$
$180$		0			0
$181$	3492.80			1.43435			0.717177	−	0.696891i	$-0.245434\pi$
$181$	3492.80			1.43435			0.717177	+	0.696891i	$0.245434\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	173.926	+	301.248i	0.0691204	+	0.119720i
$186$		0			0
$187$	708.574	−	1227.29i	0.277091	−	0.479936i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−598.195	+	1036.10i	−0.226617	+	0.392513i	−0.956803	−	0.290736i	$-0.906100\pi$
$191$	−598.195	+	1036.10i	−0.226617	+	0.392513i	0.730186	+	0.683248i	$0.239433\pi$
$192$		0			0
$193$	791.546	+	1371.00i	0.295216	+	0.511329i	0.975035	−	0.222051i	$-0.0712750\pi$
$193$	791.546	+	1371.00i	0.295216	+	0.511329i	−0.679819	+	0.733380i	$0.737942\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2803.03			1.01374			0.506872	−	0.862021i	$-0.330802\pi$
$197$	2803.03			1.01374			0.506872	+	0.862021i	$0.330802\pi$
$198$		0			0
$199$	2568.59			0.914989			0.457494	−	0.889213i	$-0.348747\pi$
$199$	2568.59			0.914989			0.457494	+	0.889213i	$0.348747\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	36.0469	+	62.4350i	0.0124630	+	0.0215866i
$204$		0			0
$205$	−1207.85	+	2092.06i	−0.411512	+	0.712760i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2869.48	+	4970.08i	−0.949693	+	1.64492i
$210$		0			0
$211$	−1168.60	−	2024.07i	−0.381278	−	0.660392i	0.609968	−	0.792426i	$-0.291182\pi$
$211$	−1168.60	−	2024.07i	−0.381278	−	0.660392i	−0.991245	+	0.132034i	$0.957849\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−758.664			−0.240653
$216$		0			0
$217$	−726.397			−0.227240
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−554.098	−	959.725i	−0.168654	−	0.292118i
$222$		0			0
$223$	1413.28	−	2447.88i	0.424397	−	0.735077i	−0.571967	−	0.820277i	$-0.693819\pi$
$223$	1413.28	−	2447.88i	0.424397	−	0.735077i	0.996364	+	0.0851996i	$0.0271528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3194.13	+	5532.40i	−0.933930	+	1.61761i	−0.157398	+	0.987535i	$0.550311\pi$
$227$	−3194.13	+	5532.40i	−0.933930	+	1.61761i	−0.776532	+	0.630078i	$0.783023\pi$
$228$		0			0
$229$	2380.57	+	4123.27i	0.686955	+	1.18984i	0.972818	+	0.231571i	$0.0743864\pi$
$229$	2380.57	+	4123.27i	0.686955	+	1.18984i	−0.285863	+	0.958270i	$0.592280\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1580.58			0.444408			0.222204	−	0.975000i	$-0.428675\pi$
$233$	1580.58			0.444408			0.222204	+	0.975000i	$0.428675\pi$
$234$		0			0
$235$	−338.086			−0.0938480
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−900.234	−	1559.25i	−0.243646	−	0.422007i	0.718104	−	0.695936i	$-0.245010\pi$
$239$	−900.234	−	1559.25i	−0.243646	−	0.422007i	−0.961750	+	0.273929i	$0.911677\pi$
$240$		0			0
$241$	−2621.83	+	4541.14i	−0.700775	+	1.21378i	0.267420	+	0.963580i	$0.413829\pi$
$241$	−2621.83	+	4541.14i	−0.700775	+	1.21378i	−0.968195	+	0.250198i	$0.919504\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−209.098	+	362.168i	−0.0545256	+	0.0944410i
$246$		0			0
$247$	2243.90	+	3886.55i	0.578040	+	1.00120i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4934.37			−1.24085			−0.620427	−	0.784264i	$-0.713041\pi$
$251$	−4934.37			−1.24085			−0.620427	+	0.784264i	$0.713041\pi$
$252$		0			0
$253$	248.554			0.0617648
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2758.46	−	4777.80i	−0.669526	−	1.15965i	−0.978037	−	0.208433i	$-0.933164\pi$
$257$	−2758.46	−	4777.80i	−0.669526	−	1.15965i	0.308510	−	0.951221i	$-0.400170\pi$
$258$		0			0
$259$	−560.204	+	970.302i	−0.134399	+	0.232786i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2598.37	+	4500.52i	−0.609212	+	1.05519i	0.382159	+	0.924097i	$0.375181\pi$
$263$	−2598.37	+	4500.52i	−0.609212	+	1.05519i	−0.991371	+	0.131089i	$0.958153\pi$
$264$		0			0
$265$	−697.383	−	1207.90i	−0.161660	−	0.280003i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	115.055			0.0260781			0.0130391	−	0.999915i	$-0.495849\pi$
$269$	115.055			0.0260781			0.0130391	+	0.999915i	$0.495849\pi$
$270$		0			0
$271$	2804.16			0.628564			0.314282	−	0.949330i	$-0.398236\pi$
$271$	2804.16			0.628564			0.314282	+	0.949330i	$0.398236\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	694.043	+	1202.12i	0.152190	+	0.263602i
$276$		0			0
$277$	3973.19	−	6881.77i	0.861827	−	1.49273i	−0.00833675	−	0.999965i	$-0.502654\pi$
$277$	3973.19	−	6881.77i	0.861827	−	1.49273i	0.870164	−	0.492763i	$-0.164013\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2806.10	+	4860.31i	−0.595722	+	1.03182i	0.397722	+	0.917506i	$0.369801\pi$
$281$	−2806.10	+	4860.31i	−0.595722	+	1.03182i	−0.993444	+	0.114316i	$0.963533\pi$
$282$		0			0
$283$	−3773.25	−	6535.45i	−0.792566	−	1.37276i	−0.924373	−	0.381489i	$-0.875411\pi$
$283$	−3773.25	−	6535.45i	−0.792566	−	1.37276i	0.131807	−	0.991275i	$-0.457922\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7780.83			−1.60031
$288$		0			0
$289$	−4261.55			−0.867404
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1506.98	+	2610.17i	0.300474	+	0.520437i	0.976243	−	0.216677i	$-0.0695217\pi$
$293$	1506.98	+	2610.17i	0.300474	+	0.520437i	−0.675769	+	0.737113i	$0.736188\pi$
$294$		0			0
$295$	−644.766	+	1116.77i	−0.127253	+	0.220409i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	97.1835	−	168.327i	0.0187969	−	0.0325571i
$300$		0			0
$301$	−1221.80	−	2116.23i	−0.233966	−	0.405240i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2446.70			0.459336
$306$		0			0
$307$	−4316.35			−0.802434			−0.401217	−	0.915983i	$-0.631413\pi$
$307$	−4316.35			−0.802434			−0.401217	+	0.915983i	$0.631413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3815.02	−	6607.80i	−0.695594	−	1.20480i	−0.969980	−	0.243184i	$-0.921808\pi$
$311$	−3815.02	−	6607.80i	−0.695594	−	1.20480i	0.274386	−	0.961620i	$-0.411525\pi$
$312$		0			0
$313$	−3351.41	+	5804.81i	−0.605217	+	1.04827i	0.386800	+	0.922164i	$0.373580\pi$
$313$	−3351.41	+	5804.81i	−0.605217	+	1.04827i	−0.992017	+	0.126103i	$0.959753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	892.453	−	1545.77i	0.158123	−	0.273878i	−0.776069	−	0.630649i	$-0.782789\pi$
$317$	892.453	−	1545.77i	0.158123	−	0.273878i	0.934192	+	0.356771i	$0.116122\pi$
$318$		0			0
$319$	−124.277	−	215.254i	−0.0218125	−	0.0377804i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2638.13			−0.454456
$324$		0			0
$325$	1085.47			0.185265
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−544.477	−	943.061i	−0.0912400	−	0.158032i
$330$		0			0
$331$	−3788.49	+	6561.85i	−0.629106	+	1.08964i	0.358625	+	0.933482i	$0.383246\pi$
$331$	−3788.49	+	6561.85i	−0.629106	+	1.08964i	−0.987731	+	0.156162i	$0.950088\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1931.49	+	3345.44i	−0.315010	+	0.545614i
$336$		0			0
$337$	5390.92	+	9337.35i	0.871401	+	1.50931i	0.860548	+	0.509370i	$0.170121\pi$
$337$	5390.92	+	9337.35i	0.871401	+	1.50931i	0.0108534	+	0.999941i	$0.496545\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2504.37			0.397710
$342$		0			0
$343$	−6870.89			−1.08161
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2420.41	+	4192.28i	0.374451	+	0.648569i	0.990245	−	0.139339i	$-0.0444977\pi$
$347$	2420.41	+	4192.28i	0.374451	+	0.648569i	−0.615793	+	0.787908i	$0.711164\pi$
$348$		0			0
$349$	4254.21	−	7368.50i	0.652500	−	1.13016i	−0.330014	−	0.943976i	$-0.607054\pi$
$349$	4254.21	−	7368.50i	0.652500	−	1.13016i	0.982514	−	0.186187i	$-0.0596131\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4719.69	+	8174.74i	−0.711626	+	1.23257i	0.252621	+	0.967565i	$0.418707\pi$
$353$	−4719.69	+	8174.74i	−0.711626	+	1.23257i	−0.964247	+	0.265006i	$0.914626\pi$
$354$		0			0
$355$	−1342.15	−	2324.67i	−0.200659	−	0.347551i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10696.6			−1.57255			−0.786277	−	0.617874i	$-0.787994\pi$
$359$	−10696.6			−1.57255			−0.786277	+	0.617874i	$0.787994\pi$
$360$		0			0
$361$	3824.48			0.557586
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−163.926	−	283.928i	−0.0235076	−	0.0407163i
$366$		0			0
$367$	3129.69	−	5420.79i	0.445146	−	0.771016i	−0.552916	−	0.833237i	$-0.686485\pi$
$367$	3129.69	−	5420.79i	0.445146	−	0.771016i	0.998062	+	0.0622208i	$0.0198183\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2246.23	−	3890.58i	0.314335	−	0.544444i
$372$		0			0
$373$	−305.050	−	528.363i	−0.0423456	−	0.0733447i	0.844076	−	0.536224i	$-0.180150\pi$
$373$	−305.050	−	528.363i	−0.0423456	−	0.0733447i	−0.886421	+	0.462879i	$0.846816\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−194.367			−0.0265528
$378$		0			0
$379$	−2025.74			−0.274552			−0.137276	−	0.990533i	$-0.543835\pi$
$379$	−2025.74			−0.274552			−0.137276	+	0.990533i	$0.543835\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4562.07	+	7901.74i	0.608645	+	1.05420i	0.991464	+	0.130381i	$0.0416200\pi$
$383$	4562.07	+	7901.74i	0.608645	+	1.05420i	−0.382819	+	0.923823i	$0.625047\pi$
$384$		0			0
$385$	−2235.47	+	3871.95i	−0.295922	+	0.512552i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2547.60	+	4412.57i	−0.332052	+	0.575131i	−0.982914	−	0.184065i	$-0.941074\pi$
$389$	−2547.60	+	4412.57i	−0.332052	+	0.575131i	0.650862	+	0.759196i	$0.274408\pi$
$390$		0			0
$391$	57.1287	+	98.9498i	0.00738906	+	0.0127982i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3746.41			−0.477221
$396$		0			0
$397$	6140.93			0.776334			0.388167	−	0.921589i	$-0.373108\pi$
$397$	6140.93			0.776334			0.388167	+	0.921589i	$0.373108\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1090.99	−	1889.65i	−0.135864	−	0.235324i	0.790063	−	0.613026i	$-0.210048\pi$
$401$	−1090.99	−	1889.65i	−0.135864	−	0.235324i	−0.925927	+	0.377702i	$0.876714\pi$
$402$		0			0
$403$	979.194	−	1696.01i	0.121035	−	0.209639i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1931.39	−	3345.27i	0.235222	−	0.407417i
$408$		0			0
$409$	−3546.37	−	6142.49i	−0.428745	−	0.742607i	0.568017	−	0.823017i	$-0.307711\pi$
$409$	−3546.37	−	6142.49i	−0.428745	−	0.742607i	−0.996762	+	0.0804092i	$0.974377\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4153.50			−0.494868
$414$		0			0
$415$	6302.34			0.745470
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2883.81	+	4994.90i	0.336237	+	0.582379i	0.983722	−	0.179699i	$-0.0575124\pi$
$419$	2883.81	+	4994.90i	0.336237	+	0.582379i	−0.647485	+	0.762078i	$0.724179\pi$
$420$		0			0
$421$	−2233.95	+	3869.31i	−0.258613	+	0.447931i	−0.965871	−	0.259025i	$-0.916599\pi$
$421$	−2233.95	+	3869.31i	−0.258613	+	0.447931i	0.707258	+	0.706956i	$0.249932\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−319.043	+	552.599i	−0.0364138	+	0.0630705i
$426$		0			0
$427$	3940.33	+	6824.84i	0.446571	+	0.773483i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4560.33			−0.509660			−0.254830	−	0.966986i	$-0.582019\pi$
$431$	−4560.33			−0.509660			−0.254830	+	0.966986i	$0.582019\pi$
$432$		0			0
$433$	−1412.82			−0.156803			−0.0784016	−	0.996922i	$-0.524982\pi$
$433$	−1412.82			−0.156803			−0.0784016	+	0.996922i	$0.524982\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−231.351	−	400.712i	−0.0253250	−	0.0438642i
$438$		0			0
$439$	976.231	−	1690.88i	0.106134	−	0.183830i	−0.808067	−	0.589091i	$-0.799486\pi$
$439$	976.231	−	1690.88i	0.106134	−	0.183830i	0.914201	+	0.405261i	$0.132819\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2078.95	−	3600.85i	0.222966	−	0.386189i	−0.732741	−	0.680508i	$-0.761759\pi$
$443$	2078.95	−	3600.85i	0.222966	−	0.386189i	0.955707	+	0.294319i	$0.0950927\pi$
$444$		0			0
$445$	−3315.70	−	5742.97i	−0.353212	−	0.611781i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4352.20			0.457446			0.228723	−	0.973492i	$-0.426545\pi$
$449$	4352.20			0.457446			0.228723	+	0.973492i	$0.426545\pi$
$450$		0			0
$451$	26825.6			2.80082
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1748.11	+	3027.82i	0.180116	+	0.311970i
$456$		0			0
$457$	1910.16	−	3308.50i	0.195522	−	0.338655i	−0.751549	−	0.659677i	$-0.770693\pi$
$457$	1910.16	−	3308.50i	0.195522	−	0.338655i	0.947072	+	0.321022i	$0.104026\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5932.85	−	10276.0i	0.599394	−	1.03818i	−0.393517	−	0.919317i	$-0.628742\pi$
$461$	5932.85	−	10276.0i	0.599394	−	1.03818i	0.992911	−	0.118863i	$-0.0379249\pi$
$462$		0			0
$463$	2653.61	+	4596.19i	0.266358	+	0.461346i	0.967919	−	0.251264i	$-0.0808463\pi$
$463$	2653.61	+	4596.19i	0.266358	+	0.461346i	−0.701560	+	0.712610i	$0.747513\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2810.91			−0.278529			−0.139265	−	0.990255i	$-0.544474\pi$
$467$	−2810.91			−0.278529			−0.139265	+	0.990255i	$0.544474\pi$
$468$		0			0
$469$	−12442.4			−1.22503
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4212.36	+	7296.03i	0.409481	+	0.709242i
$474$		0			0
$475$	1292.01	−	2237.83i	0.124803	−	0.216166i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9370.19	−	16229.7i	0.893810	−	1.54812i	0.0585401	−	0.998285i	$-0.481355\pi$
$479$	9370.19	−	16229.7i	0.893810	−	1.54812i	0.835270	−	0.549840i	$-0.185311\pi$
$480$		0			0
$481$	−1510.33	−	2615.96i	−0.143170	−	0.247979i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	164.820			0.0154312
$486$		0			0
$487$	−8260.64			−0.768635			−0.384318	−	0.923201i	$-0.625563\pi$
$487$	−8260.64			−0.768635			−0.384318	+	0.923201i	$0.625563\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7462.29	+	12925.1i	0.685883	+	1.18798i	0.973159	+	0.230136i	$0.0739170\pi$
$491$	7462.29	+	12925.1i	0.685883	+	1.18798i	−0.287276	+	0.957848i	$0.592750\pi$
$492$		0			0
$493$	57.1287	−	98.9498i	0.00521896	−	0.00903950i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4322.98	−	7487.61i	0.390165	−	0.675786i
$498$		0			0
$499$	2967.06	+	5139.10i	0.266180	+	0.461037i	0.967872	−	0.251443i	$-0.0809051\pi$
$499$	2967.06	+	5139.10i	0.266180	+	0.461037i	−0.701692	+	0.712480i	$0.747572\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2704.62			0.239747			0.119874	−	0.992789i	$-0.461751\pi$
$503$	2704.62			0.239747			0.119874	+	0.992789i	$0.461751\pi$
$504$		0			0
$505$	−9100.90			−0.801950
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1383.25	−	2395.85i	−0.120454	−	0.208633i	0.799493	−	0.600676i	$-0.205102\pi$
$509$	−1383.25	−	2395.85i	−0.120454	−	0.208633i	−0.919947	+	0.392043i	$0.871768\pi$
$510$		0			0
$511$	527.995	−	914.513i	0.0457086	−	0.0791696i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3437.48	−	5953.89i	0.294123	−	0.509436i
$516$		0			0
$517$	1877.17	+	3251.35i	0.159686	+	0.276585i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1502.48			−0.126344			−0.0631718	−	0.998003i	$-0.520122\pi$
$521$	−1502.48			−0.126344			−0.0631718	+	0.998003i	$0.520122\pi$
$522$		0			0
$523$	−23380.0			−1.95475			−0.977377	−	0.211504i	$-0.932164\pi$
$523$	−23380.0			−1.95475			−0.977377	+	0.211504i	$0.932164\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	575.613	+	996.991i	0.0475789	+	0.0824092i
$528$		0			0
$529$	6073.48	−	10519.6i	0.499176	−	0.864599i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10488.7	−	18166.9i	0.852374	−	1.47635i
$534$		0			0
$535$	−4785.47	−	8288.67i	−0.386717	−	0.669814i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4643.93			0.371110
$540$		0			0
$541$	18343.1			1.45773			0.728865	−	0.684657i	$-0.240048\pi$
$541$	18343.1			1.45773			0.728865	+	0.684657i	$0.240048\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1409.95	−	2442.10i	−0.110817	−	0.191941i
$546$		0			0
$547$	−2856.60	+	4947.78i	−0.223290	+	0.386749i	−0.955805	−	0.294002i	$-0.905013\pi$
$547$	−2856.60	+	4947.78i	−0.223290	+	0.386749i	0.732515	+	0.680751i	$0.238346\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−231.351	+	400.712i	−0.0178873	+	0.0309817i
$552$		0			0
$553$	−6033.47	−	10450.3i	−0.463959	−	0.803600i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20519.3			1.56092			0.780460	−	0.625206i	$-0.214985\pi$
$557$	20519.3			1.56092			0.780460	+	0.625206i	$0.214985\pi$
$558$		0			0
$559$	6588.05			0.498470
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12722.9	−	22036.7i	−0.952408	−	1.64962i	−0.740192	−	0.672396i	$-0.765265\pi$
$563$	−12722.9	−	22036.7i	−0.952408	−	1.64962i	−0.212216	−	0.977223i	$-0.568068\pi$
$564$		0			0
$565$	3103.07	−	5374.67i	0.231057	−	0.400202i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3657.47	+	6334.92i	−0.269471	+	0.466738i	−0.968725	−	0.248135i	$-0.920182\pi$
$569$	−3657.47	+	6334.92i	−0.269471	+	0.466738i	0.699254	+	0.714873i	$0.253516\pi$
$570$		0			0
$571$	−3796.52	−	6575.76i	−0.278247	−	0.481939i	0.692702	−	0.721224i	$-0.256420\pi$
$571$	−3796.52	−	6575.76i	−0.278247	−	0.481939i	−0.970949	+	0.239285i	$0.923087\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−111.914			−0.00811677
$576$		0			0
$577$	−19291.2			−1.39186			−0.695929	−	0.718110i	$-0.745007\pi$
$577$	−19291.2			−1.39186			−0.695929	+	0.718110i	$0.745007\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10149.7	+	17579.8i	0.724753	+	1.25531i
$582$		0			0
$583$	−7744.22	+	13413.4i	−0.550142	+	0.952874i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10151.2	+	17582.4i	−0.713773	+	1.23629i	0.249657	+	0.968334i	$0.419682\pi$
$587$	−10151.2	+	17582.4i	−0.713773	+	1.23629i	−0.963431	+	0.267957i	$0.913651\pi$
$588$		0			0
$589$	−2331.03	−	4037.46i	−0.163070	−	0.282446i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11878.2			0.822562			0.411281	−	0.911509i	$-0.365082\pi$
$593$	11878.2			0.822562			0.411281	+	0.911509i	$0.365082\pi$
$594$		0			0
$595$	−2055.23			−0.141607
$596$		0			0
$597$		0			0
$598$		0			0
$599$	804.093	+	1392.73i	0.0548487	+	0.0950007i	0.892146	−	0.451747i	$-0.149199\pi$
$599$	804.093	+	1392.73i	0.0548487	+	0.0950007i	−0.837297	+	0.546748i	$0.815866\pi$
$600$		0			0
$601$	−9517.75	+	16485.2i	−0.645985	+	1.11888i	0.338088	+	0.941115i	$0.390220\pi$
$601$	−9517.75	+	16485.2i	−0.645985	+	1.11888i	−0.984073	+	0.177765i	$0.943113\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4379.63	−	7585.74i	0.294309	−	0.509759i
$606$		0			0
$607$	3082.24	+	5338.59i	0.206102	+	0.356980i	0.950483	−	0.310775i	$-0.100589\pi$
$607$	3082.24	+	5338.59i	0.206102	+	0.356980i	−0.744381	+	0.667755i	$0.767255\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2935.85			0.194389
$612$		0			0
$613$	−10601.8			−0.698538			−0.349269	−	0.937023i	$-0.613570\pi$
$613$	−10601.8			−0.698538			−0.349269	+	0.937023i	$0.613570\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7086.92	+	12274.9i	0.462413	+	0.800923i	0.999081	−	0.0428709i	$-0.0136504\pi$
$617$	7086.92	+	12274.9i	0.462413	+	0.800923i	−0.536668	+	0.843794i	$0.680317\pi$
$618$		0			0
$619$	1111.12	−	1924.51i	0.0721479	−	0.124964i	−0.827695	−	0.561179i	$-0.810348\pi$
$619$	1111.12	−	1924.51i	0.0721479	−	0.124964i	0.899842	+	0.436215i	$0.143681\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10679.7	−	18497.7i	0.686793	−	1.18956i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1775.67			0.112561
$630$		0			0
$631$	20385.1			1.28608			0.643041	−	0.765832i	$-0.277672\pi$
$631$	20385.1			1.28608			0.643041	+	0.765832i	$0.277672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4248.55	−	7358.71i	−0.265510	−	0.459877i
$636$		0			0
$637$	1815.75	−	3144.97i	0.112940	−	0.195618i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6379.38	+	11049.4i	−0.393090	+	0.680851i	−0.992855	−	0.119324i	$-0.961927\pi$
$641$	−6379.38	+	11049.4i	−0.393090	+	0.680851i	0.599766	+	0.800176i	$0.295260\pi$
$642$		0			0
$643$	6652.21	+	11522.0i	0.407990	+	0.706660i	0.994664	−	0.103163i	$-0.0328964\pi$
$643$	6652.21	+	11522.0i	0.407990	+	0.706660i	−0.586674	+	0.809823i	$0.699563\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24988.8			−1.51841			−0.759206	−	0.650851i	$-0.774412\pi$
$647$	−24988.8			−1.51841			−0.759206	+	0.650851i	$0.774412\pi$
$648$		0			0
$649$	14319.8			0.866106
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7394.11	−	12807.0i	−0.443115	−	0.767497i	0.554804	−	0.831981i	$-0.312793\pi$
$653$	−7394.11	−	12807.0i	−0.443115	−	0.767497i	−0.997919	+	0.0644840i	$0.979460\pi$
$654$		0			0
$655$	−3934.75	+	6815.18i	−0.234723	+	0.406551i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1500.94	−	2599.70i	0.0887226	−	0.153672i	−0.818249	−	0.574864i	$-0.805055\pi$
$659$	1500.94	−	2599.70i	0.0887226	−	0.153672i	0.906971	+	0.421192i	$0.138388\pi$
$660$		0			0
$661$	9242.46	+	16008.4i	0.543858	+	0.941989i	0.998678	+	0.0514058i	$0.0163702\pi$
$661$	9242.46	+	16008.4i	0.543858	+	0.941989i	−0.454820	+	0.890583i	$0.650296\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8322.98			0.485340
$666$		0			0
$667$	20.0397			0.00116333
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13584.9	−	23529.7i	−0.781578	−	1.35373i
$672$		0			0
$673$	−463.411	+	802.651i	−0.0265426	+	0.0459732i	−0.878992	−	0.476837i	$-0.841783\pi$
$673$	−463.411	+	802.651i	−0.0265426	+	0.0459732i	0.852449	+	0.522810i	$0.175116\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16802.3	−	29102.5i	0.953865	−	1.65214i	0.216920	−	0.976189i	$-0.430399\pi$
$677$	16802.3	−	29102.5i	0.953865	−	1.65214i	0.736945	−	0.675953i	$-0.236268\pi$
$678$		0			0
$679$	265.438	+	459.752i	0.0150023	+	0.0259848i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6468.33			−0.362377			−0.181189	−	0.983448i	$-0.557994\pi$
$683$	−6468.33			−0.362377			−0.181189	+	0.983448i	$0.557994\pi$
$684$		0			0
$685$	−6118.59			−0.341284
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6055.90	+	10489.1i	0.334850	+	0.579976i
$690$		0			0
$691$	15760.2	−	27297.4i	0.867649	−	1.50281i	0.00325629	−	0.999995i	$-0.498963\pi$
$691$	15760.2	−	27297.4i	0.867649	−	1.50281i	0.864393	−	0.502817i	$-0.167703\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5363.18	+	9289.31i	−0.292715	+	0.506998i
$696$		0			0
$697$	6165.70	+	10679.3i	0.335068	+	0.580355i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	32088.9			1.72893			0.864465	−	0.502693i	$-0.167657\pi$
$701$	32088.9			1.72893			0.864465	+	0.502693i	$0.167657\pi$
$702$		0			0
$703$	−7190.85			−0.385787
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14656.7	−	25386.2i	−0.779664	−	1.35042i
$708$		0			0
$709$	−4338.98	+	7515.34i	−0.229836	+	0.398088i	−0.957759	−	0.287571i	$-0.907152\pi$
$709$	−4338.98	+	7515.34i	−0.229836	+	0.398088i	0.727923	+	0.685659i	$0.240486\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−100.957	+	174.863i	−0.00530277	+	0.00918466i
$714$		0			0
$715$	−6026.89	−	10438.9i	−0.315235	−	0.546003i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22564.2			1.17038			0.585190	−	0.810896i	$-0.301020\pi$
$719$	22564.2			1.17038			0.585190	+	0.810896i	$0.301020\pi$
$720$		0			0
$721$	22143.8			1.14380
$722$		0			0
$723$		0			0
$724$		0			0
$725$	55.9571	+	96.9205i	0.00286648	+	0.00496488i
$726$		0			0
$727$	423.322	−	733.215i	0.0215958	−	0.0374050i	−0.855026	−	0.518586i	$-0.826459\pi$
$727$	423.322	−	733.215i	0.0215958	−	0.0374050i	0.876621	+	0.481181i	$0.159792\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1936.37	+	3353.89i	−0.0979744	+	0.169697i
$732$		0			0
$733$	−14590.7	−	25271.8i	−0.735225	−	1.27345i	−0.954625	−	0.297812i	$-0.903743\pi$
$733$	−14590.7	−	25271.8i	−0.735225	−	1.27345i	0.219400	−	0.975635i	$-0.429590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	42897.1			2.14401
$738$		0			0
$739$	5544.59			0.275996			0.137998	−	0.990432i	$-0.455933\pi$
$739$	5544.59			0.275996			0.137998	+	0.990432i	$0.455933\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10048.0	−	17403.6i	−0.496131	−	0.859324i	0.503859	−	0.863786i	$-0.331913\pi$
$743$	−10048.0	−	17403.6i	−0.496131	−	0.859324i	−0.999990	+	0.00446196i	$0.998580\pi$
$744$		0			0
$745$	−2089.98	+	3619.95i	−0.102780	+	0.178020i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15413.7	−	26697.3i	0.751941	−	1.30240i
$750$		0			0
$751$	19589.0	+	33929.2i	0.951817	+	1.64860i	0.741490	+	0.670964i	$0.234120\pi$
$751$	19589.0	+	33929.2i	0.951817	+	1.64860i	0.210327	+	0.977631i	$0.432547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4182.44			−0.201609
$756$		0			0
$757$	37739.2			1.81196			0.905980	−	0.423322i	$-0.139136\pi$
$757$	37739.2			1.81196			0.905980	+	0.423322i	$0.139136\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4556.30	−	7891.74i	−0.217038	−	0.375920i	0.736863	−	0.676042i	$-0.236306\pi$
$761$	−4556.30	−	7891.74i	−0.217038	−	0.375920i	−0.953901	+	0.300122i	$0.902973\pi$
$762$		0			0
$763$	4541.35	−	7865.84i	0.215475	−	0.373214i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5598.98	−	9697.72i	0.263582	−	0.456538i
$768$		0			0
$769$	15504.9	+	26855.3i	0.727076	+	1.25933i	0.958114	+	0.286388i	$0.0924548\pi$
$769$	15504.9	+	26855.3i	0.727076	+	1.25933i	−0.231037	+	0.972945i	$0.574212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13117.9			0.610372			0.305186	−	0.952293i	$-0.401281\pi$
$773$	13117.9			0.610372			0.305186	+	0.952293i	$0.401281\pi$
$774$		0			0
$775$	−1127.62			−0.0522648
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24968.9	−	43247.5i	−1.14840	−	1.98909i
$780$		0			0
$781$	−14904.1	+	25814.7i	−0.682858	+	1.18274i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3527.43	−	6109.69i	0.160382	−	0.277789i
$786$		0			0
$787$	−4334.37	−	7507.35i	−0.196320	−	0.340036i	0.751013	−	0.660288i	$-0.229566\pi$
$787$	−4334.37	−	7507.35i	−0.196320	−	0.340036i	−0.947332	+	0.320252i	$0.896232\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	19989.6			0.898542
$792$		0			0
$793$	−21246.5			−0.951431
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12589.3	+	21805.3i	0.559519	+	0.969115i	0.997537	+	0.0701484i	$0.0223473\pi$
$797$	12589.3	+	21805.3i	0.559519	+	0.969115i	−0.438018	+	0.898966i	$0.644319\pi$
$798$		0			0
$799$	−862.911	+	1494.61i	−0.0382073	+	0.0661769i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1820.34	+	3152.93i	−0.0799982	+	0.138561i
$804$		0			0
$805$	−180.234	−	312.175i	−0.00789121	−	0.0136680i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21442.3			0.931856			0.465928	−	0.884823i	$-0.345721\pi$
$809$	21442.3			0.931856			0.465928	+	0.884823i	$0.345721\pi$
$810$		0			0
$811$	−23957.5			−1.03731			−0.518656	−	0.854983i	$-0.673568\pi$
$811$	−23957.5			−1.03731			−0.518656	+	0.854983i	$0.673568\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4816.40	−	8342.25i	−0.207007	−	0.358547i
$816$		0			0
$817$	7841.62	−	13582.1i	0.335794	−	0.581612i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10332.2	−	17895.8i	0.439214	−	0.760742i	−0.558415	−	0.829562i	$-0.688590\pi$
$821$	10332.2	−	17895.8i	0.439214	−	0.760742i	0.997629	+	0.0688203i	$0.0219235\pi$
$822$		0			0
$823$	−22979.2	−	39801.2i	−0.973276	−	1.68576i	−0.685512	−	0.728062i	$-0.740421\pi$
$823$	−22979.2	−	39801.2i	−0.973276	−	1.68576i	−0.287764	−	0.957701i	$-0.592912\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22841.3			0.960423			0.480211	−	0.877153i	$-0.340560\pi$
$827$	22841.3			0.960423			0.480211	+	0.877153i	$0.340560\pi$
$828$		0			0
$829$	−28412.5			−1.19036			−0.595179	−	0.803594i	$-0.702919\pi$
$829$	−28412.5			−1.19036			−0.595179	+	0.803594i	$0.702919\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1067.38	+	1848.75i	0.0443967	+	0.0768974i
$834$		0			0
$835$	−1560.47	+	2702.81i	−0.0646733	+	0.112018i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10331.0	+	17893.9i	−0.425110	+	0.736312i	−0.996431	−	0.0844151i	$-0.973098\pi$
$839$	−10331.0	+	17893.9i	−0.425110	+	0.736312i	0.571321	+	0.820727i	$0.306431\pi$
$840$		0			0
$841$	12184.5	+	21104.1i	0.499589	+	0.865314i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1559.06			0.0634715
$846$		0			0
$847$	28213.0			1.14452
$848$		0			0
$849$		0			0
$850$		0			0
$851$	155.718	+	269.712i	0.00627256	+	0.0108644i
$852$		0			0
$853$	−11777.1	+	20398.6i	−0.472733	+	0.818798i	−0.999513	−	0.0312037i	$-0.990066\pi$
$853$	−11777.1	+	20398.6i	−0.472733	+	0.818798i	0.526780	+	0.850002i	$0.323399\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19445.5	+	33680.6i	−0.775084	+	1.34248i	0.159664	+	0.987171i	$0.448959\pi$
$857$	−19445.5	+	33680.6i	−0.775084	+	1.34248i	−0.934747	+	0.355313i	$0.884374\pi$
$858$		0			0
$859$	−14971.0	−	25930.5i	−0.594649	−	1.02996i	−0.993596	−	0.112988i	$-0.963958\pi$
$859$	−14971.0	−	25930.5i	−0.594649	−	1.02996i	0.398947	−	0.916974i	$-0.369376\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11036.3			0.435318			0.217659	−	0.976025i	$-0.430158\pi$
$863$	11036.3			0.435318			0.217659	+	0.976025i	$0.430158\pi$
$864$		0			0
$865$	−17955.2			−0.705776
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20801.3	+	36029.0i	0.812010	+	1.40644i
$870$		0			0
$871$	16772.6	−	29050.9i	0.652487	−	1.13014i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1006.54	−	1743.38i	0.0388884	−	0.0673567i
$876$		0			0
$877$	12433.7	+	21535.8i	0.478742	+	0.829205i	0.999703	−	0.0243754i	$-0.00775970\pi$
$877$	12433.7	+	21535.8i	0.478742	+	0.829205i	−0.520961	+	0.853580i	$0.674426\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10250.2			0.391983			0.195991	−	0.980606i	$-0.437208\pi$
$881$	10250.2			0.391983			0.195991	+	0.980606i	$0.437208\pi$
$882$		0			0
$883$	−39991.9			−1.52416			−0.762081	−	0.647481i	$-0.775822\pi$
$883$	−39991.9			−1.52416			−0.762081	+	0.647481i	$0.775822\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12121.9	−	20995.8i	−0.458867	−	0.794780i	0.540035	−	0.841643i	$-0.318411\pi$
$887$	−12121.9	−	20995.8i	−0.458867	−	0.794780i	−0.998901	+	0.0468624i	$0.985078\pi$
$888$		0			0
$889$	13684.3	−	23702.0i	0.516263	−	0.894193i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3494.49	−	6052.63i	0.130950	−	0.226812i
$894$		0			0
$895$	1060.19	+	1836.31i	0.0395960	+	0.0685822i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	201.914			0.00749079
$900$		0			0
$901$	−7119.84			−0.263259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8732.01	+	15124.3i	0.320731	+	0.555523i
$906$		0			0
$907$	−5503.94	+	9533.10i	−0.201494	+	0.348998i	−0.949010	−	0.315246i	$-0.897913\pi$
$907$	−5503.94	+	9533.10i	−0.201494	+	0.348998i	0.747516	+	0.664244i	$0.231246\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6056.53	−	10490.2i	0.220265	−	0.381511i	−0.734623	−	0.678475i	$-0.762641\pi$
$911$	6056.53	−	10490.2i	0.220265	−	0.381511i	0.954889	+	0.296965i	$0.0959743\pi$
$912$		0			0
$913$	−34992.8	−	60609.3i	−1.26845	−	2.19701i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25347.1			−0.912799
$918$		0			0
$919$	5343.49			0.191801			0.0959007	−	0.995391i	$-0.469427\pi$
$919$	5343.49			0.191801			0.0959007	+	0.995391i	$0.469427\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11654.9	+	20186.8i	0.415628	+	0.719890i
$924$		0			0
$925$	−869.629	+	1506.24i	−0.0309116	+	0.0535404i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3995.65	−	6920.67i	0.141112	−	0.244413i	−0.786804	−	0.617203i	$-0.788266\pi$
$929$	3995.65	−	6920.67i	0.141112	−	0.244413i	0.927916	+	0.372790i	$0.121599\pi$
$930$		0			0
$931$	−4322.51	−	7486.80i	−0.152164	−	0.263555i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7085.74			0.247838
$936$		0			0
$937$	33059.7			1.15263			0.576315	−	0.817227i	$-0.304490\pi$
$937$	33059.7			1.15263			0.576315	+	0.817227i	$0.304490\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11880.5	+	20577.6i	0.411576	+	0.712871i	0.995062	−	0.0992521i	$-0.0316450\pi$
$941$	11880.5	+	20577.6i	0.411576	+	0.712871i	−0.583486	+	0.812123i	$0.698312\pi$
$942$		0			0
$943$	−1081.41	+	1873.05i	−0.0373440	+	0.0646818i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17484.1	−	30283.3i	0.599953	−	1.03915i	−0.392874	−	0.919592i	$-0.628519\pi$
$947$	17484.1	−	30283.3i	0.599953	−	1.03915i	0.992827	−	0.119557i	$-0.0381476\pi$
$948$		0			0
$949$	1423.49	+	2465.56i	0.0486917	+	0.0843365i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	50724.6			1.72417			0.862084	−	0.506766i	$-0.169159\pi$
$953$	50724.6			1.72417			0.862084	+	0.506766i	$0.169159\pi$
$954$		0			0
$955$	−5981.95			−0.202693
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9853.80	−	17067.3i	−0.331800	−	0.574694i
$960$		0			0
$961$	13878.3	−	24037.9i	0.465855	−	0.806884i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3957.73	+	6854.99i	−0.132025	+	0.228673i
$966$		0			0
$967$	−7558.50	−	13091.7i	−0.251360	−	0.435368i	0.712541	−	0.701631i	$-0.247544\pi$
$967$	−7558.50	−	13091.7i	−0.251360	−	0.435368i	−0.963900	+	0.266263i	$0.914211\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48459.4			−1.60158			−0.800792	−	0.598943i	$-0.795588\pi$
$971$	−48459.4			−1.60158			−0.800792	+	0.598943i	$0.795588\pi$
$972$		0			0
$973$	−34549.0			−1.13832
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10306.0	−	17850.5i	−0.337481	−	0.584534i	0.646477	−	0.762933i	$-0.276241\pi$
$977$	−10306.0	−	17850.5i	−0.337481	−	0.584534i	−0.983958	+	0.178399i	$0.942908\pi$
$978$		0			0
$979$	−36819.8	+	63773.8i	−1.20201	+	2.08194i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4545.20	+	7872.52i	−0.147476	+	0.255437i	−0.930294	−	0.366814i	$-0.880448\pi$
$983$	−4545.20	+	7872.52i	−0.147476	+	0.255437i	0.782818	+	0.622251i	$0.213782\pi$
$984$		0			0
$985$	7007.58	+	12137.5i	0.226680	+	0.392622i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−679.242			−0.0218389
$990$		0			0
$991$	29334.8			0.940312			0.470156	−	0.882583i	$-0.344198\pi$
$991$	29334.8			0.940312			0.470156	+	0.882583i	$0.344198\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6421.48	+	11122.3i	0.204598	+	0.354374i
$996$		0			0
$997$	1141.47	−	1977.09i	0.0362595	−	0.0628034i	−0.847326	−	0.531073i	$-0.821789\pi$
$997$	1141.47	−	1977.09i	0.0362595	−	0.0628034i	0.883586	+	0.468270i	$0.155122\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.p.541.1		4
3.2	odd	2		1620.4.i.m.541.1		4
9.2	odd	6		540.4.a.j.1.2	yes	2
9.4	even	3	inner	1620.4.i.p.1081.1		4
9.5	odd	6		1620.4.i.m.1081.1		4
9.7	even	3		540.4.a.g.1.2	✓	2
36.7	odd	6		2160.4.a.u.1.1		2
36.11	even	6		2160.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.g.1.2	✓	2	9.7	even	3
540.4.a.j.1.2	yes	2	9.2	odd	6
1620.4.i.m.541.1		4	3.2	odd	2
1620.4.i.m.1081.1		4	9.5	odd	6
1620.4.i.p.541.1		4	1.1	even	1	trivial
1620.4.i.p.1081.1		4	9.4	even	3	inner
2160.4.a.u.1.1		2	36.7	odd	6
2160.4.a.z.1.1		2	36.11	even	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +(27.7617 - 48.0847i) q^{11} +(-21.7094 - 37.6017i) q^{13} +25.5234 q^{17} -103.361 q^{19} +(2.23828 + 3.87682i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(2.23828 - 3.87682i) q^{29} +(22.5523 + 39.0618i) q^{31} -80.5234 q^{35} +69.5703 q^{37} +(241.570 + 418.412i) q^{41} +(-75.8664 + 131.404i) q^{43} +(-33.8086 + 58.5582i) q^{47} +(41.8195 + 72.4336i) q^{49} -278.953 q^{53} +277.617 q^{55} +(128.953 + 223.353i) q^{59} +(244.670 - 423.780i) q^{61} +(108.547 - 188.009i) q^{65} +(386.298 + 669.087i) q^{67} -536.859 q^{71} -65.5703 q^{73} +(447.094 + 774.389i) q^{77} +(-374.641 + 648.897i) q^{79} +(630.234 - 1091.60i) q^{83} +(63.8086 + 110.520i) q^{85} -1326.28 q^{89} +699.245 q^{91} +(-258.402 - 447.566i) q^{95} +(16.4820 - 28.5477i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{5} - 13 q^{7}+O(q^{10}) \) 4 * q + 10 * q^5 - 13 * q^7	\( 4 q + 10 q^{5} - 13 q^{7} + 15 q^{11} - 10 q^{13} - 90 q^{17} + 86 q^{19} + 105 q^{23} - 50 q^{25} + 105 q^{29} + 71 q^{31} - 130 q^{35} - 298 q^{37} + 390 q^{41} - 169 q^{43} + 345 q^{47} + 417 q^{49} - 1500 q^{53} + 150 q^{55} + 900 q^{59} - q^{61} + 50 q^{65} + 335 q^{67} - 3300 q^{71} + 314 q^{73} + 1020 q^{77} - 346 q^{79} + 600 q^{83} - 225 q^{85} - 3000 q^{89} + 1606 q^{91} + 215 q^{95} + 623 q^{97}+O(q^{100}) \) 4 * q + 10 * q^5 - 13 * q^7 + 15 * q^11 - 10 * q^13 - 90 * q^17 + 86 * q^19 + 105 * q^23 - 50 * q^25 + 105 * q^29 + 71 * q^31 - 130 * q^35 - 298 * q^37 + 390 * q^41 - 169 * q^43 + 345 * q^47 + 417 * q^49 - 1500 * q^53 + 150 * q^55 + 900 * q^59 - q^61 + 50 * q^65 + 335 * q^67 - 3300 * q^71 + 314 * q^73 + 1020 * q^77 - 346 * q^79 + 600 * q^83 - 225 * q^85 - 3000 * q^89 + 1606 * q^91 + 215 * q^95 + 623 * q^97

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