LMFDB - Embedded newform 1872.2.by.f.1297.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(433,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1872.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.by (of order $6$, 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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1297.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1872.1297
Dual form			1872.2.by.f.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{5} +(1.50000 + 0.866025i) q^{7} +(-3.00000 + 1.73205i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-3.00000 - 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} -7.00000 q^{25} +(3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(-3.00000 + 5.19615i) q^{35} +(6.00000 - 3.46410i) q^{41} +(-0.500000 + 0.866025i) q^{43} -3.46410i q^{47} +(-2.00000 - 3.46410i) q^{49} -12.0000 q^{53} +(-6.00000 - 10.3923i) q^{55} +(-3.00000 - 1.73205i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-3.00000 + 12.1244i) q^{65} +(-7.50000 + 4.33013i) q^{67} +(-9.00000 - 5.19615i) q^{71} -1.73205i q^{73} -6.00000 q^{77} +11.0000 q^{79} +13.8564i q^{83} +(-6.00000 + 3.46410i) q^{89} +(4.50000 + 4.33013i) q^{91} +(6.00000 - 10.3923i) q^{95} +(-4.50000 - 2.59808i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{7} - 6 q^{11} + 7 q^{13} - 6 q^{19} + 6 q^{23} - 14 q^{25} + 6 q^{29} - 6 q^{35} + 12 q^{41} - q^{43} - 4 q^{49} - 24 q^{53} - 12 q^{55} - 6 q^{59} - q^{61} - 6 q^{65} - 15 q^{67} - 18 q^{71}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q + 3 * q^7 - 6 * q^11 + 7 * q^13 - 6 * q^19 + 6 * q^23 - 14 * q^25 + 6 * q^29 - 6 * q^35 + 12 * q^41 - q^43 - 4 * q^49 - 24 * q^53 - 12 * q^55 - 6 * q^59 - q^61 - 6 * q^65 - 15 * q^67 - 18 * q^71 - 12 * q^77 + 22 * q^79 - 12 * q^89 + 9 * q^91 + 12 * q^95 - 9 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$	1.50000	+	0.866025i	0.566947	+	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	1.50000	+	0.866025i	0.566947	+	0.327327i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.0258656	+	0.999665i	$0.508234\pi$
$12$		0			0
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−3.00000	−	1.73205i	−0.688247	−	0.397360i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	−3.00000	−	1.73205i	−0.688247	−	0.397360i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	$0.0214140\pi$
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	−0.440652	+	0.897678i	$0.645253\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	+	5.19615i	−0.507093	+	0.878310i
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.0480106	−	0.998847i	$-0.484712\pi$
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.889032	+	0.457845i	$0.151379\pi$
$42$		0			0
$43$	−0.500000	+	0.866025i	−0.0762493	+	0.132068i	−0.901629	−	0.432511i	$-0.857628\pi$
$43$	−0.500000	+	0.866025i	−0.0762493	+	0.132068i	0.825380	+	0.564578i	$0.190961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	−2.00000	−	3.46410i	−0.285714	−	0.494872i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$	−6.00000	−	10.3923i	−0.809040	−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000	−	1.73205i	−0.390567	−	0.225494i	0.291839	−	0.956467i	$-0.405733\pi$
$59$	−3.00000	−	1.73205i	−0.390567	−	0.225494i	−0.682406	+	0.730974i	$0.739066\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	12.1244i	−0.372104	+	1.50384i
$66$		0			0
$67$	−7.50000	+	4.33013i	−0.916271	+	0.529009i	−0.882443	−	0.470418i	$-0.844103\pi$
$67$	−7.50000	+	4.33013i	−0.916271	+	0.529009i	−0.0338274	+	0.999428i	$0.510770\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.00000	−	5.19615i	−1.06810	−	0.616670i	−0.140441	−	0.990089i	$-0.544852\pi$
$71$	−9.00000	−	5.19615i	−1.06810	−	0.616670i	−0.927663	+	0.373419i	$0.878185\pi$
$72$		0			0
$73$		−	1.73205i		−	0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$		−	1.73205i		−	0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	11.0000			1.23760			0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000			1.23760			0.618798	+	0.785550i	$0.287620\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			13.8564i			1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$			13.8564i			1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	+	3.46410i	−0.635999	+	0.367194i	−0.783072	−	0.621932i	$-0.786348\pi$
$89$	−6.00000	+	3.46410i	−0.635999	+	0.367194i	0.147073	+	0.989126i	$0.453015\pi$
$90$		0			0
$91$	4.50000	+	4.33013i	0.471728	+	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	−	10.3923i	0.615587	−	1.06623i
$96$		0			0
$97$	−4.50000	−	2.59808i	−0.456906	−	0.263795i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	−4.50000	−	2.59808i	−0.456906	−	0.263795i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	1.00000			0.0985329			0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000			0.0985329			0.0492665	+	0.998786i	$0.484312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	−0.683793	+	0.729676i	$0.739671\pi$
$108$		0			0
$109$			15.5885i			1.49310i	0.665327	+	0.746552i	$0.268292\pi$
$109$			15.5885i			1.49310i	−0.665327	+	0.746552i	$0.731708\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	−18.0000	+	10.3923i	−1.67851	+	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$	6.50000	+	11.2583i	0.576782	+	0.999015i	0.995846	+	0.0910585i	$0.0290250\pi$
$127$	6.50000	+	11.2583i	0.576782	+	0.999015i	−0.419064	+	0.907957i	$0.637642\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$	−3.00000	−	5.19615i	−0.260133	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	−0.740308	−	0.672268i	$-0.765320\pi$
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	0.952355	+	0.304991i	$0.0986536\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000	+	3.46410i	−1.00349	+	0.289683i
$144$		0			0
$145$	−18.0000	+	10.3923i	−1.49482	+	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	+	3.46410i	0.491539	+	0.283790i	0.725213	−	0.688525i	$-0.241741\pi$
$149$	6.00000	+	3.46410i	0.491539	+	0.283790i	−0.233674	+	0.972315i	$0.575075\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	11.0000			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			10.3923i			0.819028i
$162$		0			0
$163$	16.5000	+	9.52628i	1.29238	+	0.746156i	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	16.5000	+	9.52628i	1.29238	+	0.746156i	0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	3.46410i	−0.464294	+	0.268060i	−0.713848	−	0.700301i	$-0.753049\pi$
$167$	−6.00000	+	3.46410i	−0.464294	+	0.268060i	0.249554	+	0.968361i	$0.419716\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−10.5000	−	6.06218i	−0.793725	−	0.458258i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	0.549825	−	0.835280i	$-0.314694\pi$
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	−0.998286	+	0.0585225i	$0.981361\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		0			0
$193$	−13.5000	+	7.79423i	−0.971751	+	0.561041i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−13.5000	+	7.79423i	−0.971751	+	0.561041i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000	+	6.92820i	−0.854965	+	0.493614i	−0.862323	−	0.506359i	$-0.830991\pi$
$197$	−12.0000	+	6.92820i	−0.854965	+	0.493614i	0.00735824	+	0.999973i	$0.497658\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			10.3923i			0.729397i
$204$		0			0
$205$	12.0000	+	20.7846i	0.838116	+	1.45166i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	0.998221	−	0.0596196i	$-0.0189888\pi$
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.00000	−	1.73205i	−0.204598	−	0.118125i
$216$		0			0
$217$	−1.50000	+	2.59808i	−0.101827	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−15.0000	+	8.66025i	−1.00447	+	0.579934i	−0.909569	−	0.415553i	$-0.863588\pi$
$223$	−15.0000	+	8.66025i	−1.00447	+	0.579934i	−0.0949052	+	0.995486i	$0.530255\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.0000	+	10.3923i	1.19470	+	0.689761i	0.959369	−	0.282153i	$-0.0910487\pi$
$227$	18.0000	+	10.3923i	1.19470	+	0.689761i	0.235333	+	0.971915i	$0.424382\pi$
$228$		0			0
$229$		−	27.7128i		−	1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$		−	27.7128i		−	1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	18.0000	+	10.3923i	1.15948	+	0.669427i	0.951180	−	0.308637i	$-0.0998729\pi$
$241$	18.0000	+	10.3923i	1.15948	+	0.669427i	0.208302	+	0.978065i	$0.433206\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.0000	−	6.92820i	0.766652	−	0.442627i
$246$		0			0
$247$	−9.00000	−	8.66025i	−0.572656	−	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000	+	10.3923i	−0.378717	+	0.655956i	−0.990876	−	0.134778i	$-0.956968\pi$
$251$	−6.00000	+	10.3923i	−0.378717	+	0.655956i	0.612159	+	0.790735i	$0.290301\pi$
$252$		0			0
$253$	−18.0000	−	10.3923i	−1.13165	−	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$		−	41.5692i		−	2.55358i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	4.50000	−	2.59808i	0.273356	−	0.157822i	−0.357056	−	0.934083i	$-0.616219\pi$
$271$	4.50000	−	2.59808i	0.273356	−	0.157822i	0.630412	+	0.776261i	$0.282886\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.0000	−	12.1244i	1.26635	−	0.731126i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	24.2487i		−	1.44656i	−0.690557	−	0.723278i	$-0.742634\pi$
$281$		−	24.2487i		−	1.44656i	0.690557	−	0.723278i	$-0.257366\pi$
$282$		0			0
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	0.654962	−	0.755662i	$-0.272685\pi$
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	−0.981903	+	0.189383i	$0.939351\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.0000	+	8.66025i	0.876309	+	0.505937i	0.869440	−	0.494039i	$-0.164480\pi$
$293$	15.0000	+	8.66025i	0.876309	+	0.505937i	0.00686959	+	0.999976i	$0.497813\pi$
$294$		0			0
$295$	6.00000	−	10.3923i	0.349334	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	20.7846i	0.346989	+	1.20201i
$300$		0			0
$301$	−1.50000	+	0.866025i	−0.0864586	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.00000	−	1.73205i	−0.171780	−	0.0991769i
$306$		0			0
$307$		−	1.73205i		−	0.0988534i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$		−	1.73205i		−	0.0988534i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.92820i			0.389127i	0.980890	+	0.194563i	$0.0623290\pi$
$317$			6.92820i			0.389127i	−0.980890	+	0.194563i	$0.937671\pi$
$318$		0			0
$319$	−18.0000	−	10.3923i	−1.00781	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.5000	−	6.06218i	−1.35902	−	0.336269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000	−	5.19615i	0.165395	−	0.286473i
$330$		0			0
$331$	−4.50000	−	2.59808i	−0.247342	−	0.142803i	0.371204	−	0.928551i	$-0.378945\pi$
$331$	−4.50000	−	2.59808i	−0.247342	−	0.142803i	−0.618547	+	0.785748i	$0.712278\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.0000	−	25.9808i	−0.819538	−	1.41948i
$336$		0			0
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.00000	−	5.19615i	−0.162459	−	0.281387i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	0.984487	−	0.175457i	$-0.0561403\pi$
$348$		0			0
$349$	−16.5000	+	9.52628i	−0.883225	+	0.509930i	−0.871720	−	0.490004i	$-0.836995\pi$
$349$	−16.5000	+	9.52628i	−0.883225	+	0.509930i	−0.0115044	+	0.999934i	$0.503662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	+	5.19615i	−0.479022	+	0.276563i	−0.720009	−	0.693965i	$-0.755862\pi$
$353$	−9.00000	+	5.19615i	−0.479022	+	0.276563i	0.240987	+	0.970528i	$0.422529\pi$
$354$		0			0
$355$	18.0000	−	31.1769i	0.955341	−	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.92820i		−	0.365657i	−0.983145	−	0.182828i	$-0.941475\pi$
$359$		−	6.92820i		−	0.365657i	0.983145	−	0.182828i	$-0.0585252\pi$
$360$		0			0
$361$	−3.50000	−	6.06218i	−0.184211	−	0.319062i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000			0.314054
$366$		0			0
$367$	8.50000	+	14.7224i	0.443696	+	0.768505i	0.997960	−	0.0638362i	$-0.0203335\pi$
$367$	8.50000	+	14.7224i	0.443696	+	0.768505i	−0.554264	+	0.832341i	$0.687000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000	−	10.3923i	−0.934513	−	0.539542i
$372$		0			0
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	−0.687776	−	0.725923i	$-0.741413\pi$
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	0.972556	+	0.232671i	$0.0747464\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000	+	20.7846i	0.309016	+	1.07046i
$378$		0			0
$379$	19.5000	−	11.2583i	1.00165	−	0.578302i	0.0929123	−	0.995674i	$-0.470382\pi$
$379$	19.5000	−	11.2583i	1.00165	−	0.578302i	0.908735	+	0.417373i	$0.137049\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.0000	+	13.8564i	1.22634	+	0.708029i	0.966263	−	0.257558i	$-0.0829178\pi$
$383$	24.0000	+	13.8564i	1.22634	+	0.708029i	0.260080	+	0.965587i	$0.416251\pi$
$384$		0			0
$385$		−	20.7846i		−	1.05928i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000			−0.608424			−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000			−0.608424			−0.304212	+	0.952604i	$0.598393\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			38.1051i			1.91728i
$396$		0			0
$397$	13.5000	+	7.79423i	0.677546	+	0.391181i	0.798930	−	0.601424i	$-0.205400\pi$
$397$	13.5000	+	7.79423i	0.677546	+	0.391181i	−0.121384	+	0.992606i	$0.538733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−1.50000	+	6.06218i	−0.0747203	+	0.301979i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	0.673830	−	0.738886i	$-0.264648\pi$
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	−0.302979	+	0.952997i	$0.597981\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.00000	−	5.19615i	−0.147620	−	0.255686i
$414$		0			0
$415$	−48.0000			−2.35623
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$			12.1244i			0.590905i	0.955357	+	0.295452i	$0.0954704\pi$
$421$			12.1244i			0.590905i	−0.955357	+	0.295452i	$0.904530\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.50000	+	0.866025i	−0.0725901	+	0.0419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.0000	−	10.3923i	0.867029	−	0.500580i	0.000669521	−	1.00000i	$-0.499787\pi$
$431$	18.0000	−	10.3923i	0.867029	−	0.500580i	0.866360	+	0.499420i	$0.166454\pi$
$432$		0			0
$433$	11.5000	−	19.9186i	0.552655	−	0.957226i	−0.445427	−	0.895318i	$-0.646948\pi$
$433$	11.5000	−	19.9186i	0.552655	−	0.957226i	0.998082	−	0.0619079i	$-0.0197185\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	20.7846i		−	0.994263i
$438$		0			0
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	0.893843	+	0.448379i	$0.147999\pi$
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	−0.0586141	+	0.998281i	$0.518668\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$	−12.0000	−	20.7846i	−0.568855	−	0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0000	−	19.0526i	−1.55737	−	0.899146i	−0.997508	−	0.0705577i	$-0.977522\pi$
$449$	−33.0000	−	19.0526i	−1.55737	−	0.899146i	−0.559859	−	0.828588i	$-0.689145\pi$
$450$		0			0
$451$	−12.0000	+	20.7846i	−0.565058	+	0.978709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−15.0000	+	15.5885i	−0.703211	+	0.730798i
$456$		0			0
$457$	−31.5000	+	18.1865i	−1.47351	+	0.850730i	−0.999555	−	0.0298202i	$-0.990507\pi$
$457$	−31.5000	+	18.1865i	−1.47351	+	0.850730i	−0.473953	+	0.880550i	$0.657173\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.0000	+	20.7846i	1.67669	+	0.968036i	0.963750	+	0.266808i	$0.0859690\pi$
$461$	36.0000	+	20.7846i	1.67669	+	0.968036i	0.712938	+	0.701228i	$0.247364\pi$
$462$		0			0
$463$		−	36.3731i		−	1.69040i	−0.534450	−	0.845200i	$-0.679481\pi$
$463$		−	36.3731i		−	1.69040i	0.534450	−	0.845200i	$-0.320519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$	−15.0000			−0.692636
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	3.46410i		−	0.159280i
$474$		0			0
$475$	21.0000	+	12.1244i	0.963546	+	0.556304i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.0000	−	17.3205i	1.37073	−	0.791394i	0.379714	−	0.925104i	$-0.376022\pi$
$479$	30.0000	−	17.3205i	1.37073	−	0.791394i	0.991021	+	0.133710i	$0.0426889\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.00000	−	15.5885i	0.408669	−	0.707835i
$486$		0			0
$487$	21.0000	+	12.1244i	0.951601	+	0.549407i	0.893578	−	0.448908i	$-0.148187\pi$
$487$	21.0000	+	12.1244i	0.951601	+	0.549407i	0.0580230	+	0.998315i	$0.481520\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	0.790358	−	0.612646i	$-0.209895\pi$
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	−0.925746	+	0.378147i	$0.876561\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	−	15.5885i	−0.403705	−	0.699238i
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.0000	−	25.9808i	0.668817	−	1.15842i	−0.309418	−	0.950926i	$-0.600134\pi$
$503$	15.0000	−	25.9808i	0.668817	−	1.15842i	0.978235	−	0.207499i	$-0.0665323\pi$
$504$		0			0
$505$	54.0000	−	31.1769i	2.40297	−	1.38735i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.0000	−	8.66025i	0.664863	−	0.383859i	−0.129264	−	0.991610i	$-0.541262\pi$
$509$	15.0000	−	8.66025i	0.664863	−	0.383859i	0.794128	+	0.607751i	$0.207928\pi$
$510$		0			0
$511$	1.50000	−	2.59808i	0.0663561	−	0.114932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			3.46410i			0.152647i
$516$		0			0
$517$	6.00000	+	10.3923i	0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	−0.990873	−	0.134801i	$-0.956961\pi$
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	0.378695	−	0.925521i	$-0.376373\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000	−	6.92820i	1.03956	−	0.300094i
$534$		0			0
$535$	−18.0000	+	10.3923i	−0.778208	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000	+	6.92820i	0.516877	+	0.298419i
$540$		0			0
$541$			29.4449i			1.26593i	0.774179	+	0.632967i	$0.218163\pi$
$541$			29.4449i			1.26593i	−0.774179	+	0.632967i	$0.781837\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−54.0000			−2.31311
$546$		0			0
$547$	19.0000			0.812381			0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000			0.812381			0.406191	+	0.913788i	$0.366857\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.7846i		−	0.885454i
$552$		0			0
$553$	16.5000	+	9.52628i	0.701651	+	0.405099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.0000	−	13.8564i	1.01691	−	0.587115i	0.103704	−	0.994608i	$-0.466930\pi$
$557$	24.0000	−	13.8564i	1.01691	−	0.587115i	0.913208	+	0.407493i	$0.133597\pi$
$558$		0			0
$559$	−2.50000	+	2.59808i	−0.105739	+	0.109887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.0000	−	36.3731i	0.885044	−	1.53294i	0.0393818	−	0.999224i	$-0.487461\pi$
$563$	21.0000	−	36.3731i	0.885044	−	1.53294i	0.845663	−	0.533718i	$-0.179206\pi$
$564$		0			0
$565$	18.0000	+	10.3923i	0.757266	+	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.0000	−	36.3731i	−0.875761	−	1.51686i
$576$		0			0
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000	+	20.7846i	−0.497844	+	0.862291i
$582$		0			0
$583$	36.0000	−	20.7846i	1.49097	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−27.0000	+	15.5885i	−1.11441	+	0.643404i	−0.939968	−	0.341263i	$-0.889145\pi$
$587$	−27.0000	+	15.5885i	−1.11441	+	0.643404i	−0.174441	+	0.984668i	$0.555812\pi$
$588$		0			0
$589$	3.00000	−	5.19615i	0.123613	−	0.214104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3.46410i			0.142254i	0.997467	+	0.0711268i	$0.0226595\pi$
$593$			3.46410i			0.142254i	−0.997467	+	0.0711268i	$0.977341\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.00000	+	1.73205i	0.121967	+	0.0704179i
$606$		0			0
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	−0.773358	−	0.633970i	$-0.781424\pi$
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	0.935713	+	0.352763i	$0.114758\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.00000	−	12.1244i	0.121367	−	0.490499i
$612$		0			0
$613$	7.50000	−	4.33013i	0.302922	−	0.174892i	−0.340833	−	0.940124i	$-0.610709\pi$
$613$	7.50000	−	4.33013i	0.302922	−	0.174892i	0.643755	+	0.765232i	$0.277376\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	+	5.19615i	0.362326	+	0.209189i	0.670101	−	0.742270i	$-0.266251\pi$
$617$	9.00000	+	5.19615i	0.362326	+	0.209189i	−0.307774	+	0.951459i	$0.599584\pi$
$618$		0			0
$619$			25.9808i			1.04425i	0.852867	+	0.522127i	$0.174861\pi$
$619$			25.9808i			1.04425i	−0.852867	+	0.522127i	$0.825139\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.50000	−	0.866025i	−0.0597141	−	0.0344759i	0.469846	−	0.882749i	$-0.344310\pi$
$631$	−1.50000	−	0.866025i	−0.0597141	−	0.0344759i	−0.529560	+	0.848273i	$0.677643\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−39.0000	+	22.5167i	−1.54767	+	0.893546i
$636$		0			0
$637$	−4.00000	−	13.8564i	−0.158486	−	0.549011i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	−0.631721	−	0.775196i	$-0.717651\pi$
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	0.987200	+	0.159489i	$0.0509845\pi$
$642$		0			0
$643$	−16.5000	−	9.52628i	−0.650696	−	0.375680i	0.138027	−	0.990429i	$-0.455924\pi$
$643$	−16.5000	−	9.52628i	−0.650696	−	0.375680i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	0.986916	−	0.161233i	$-0.0515470\pi$
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	−0.633090	+	0.774078i	$0.718214\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000	+	31.1769i	0.704394	+	1.22005i	0.966910	+	0.255119i	$0.0821147\pi$
$653$	18.0000	+	31.1769i	0.704394	+	1.22005i	−0.262515	+	0.964928i	$0.584552\pi$
$654$		0			0
$655$			20.7846i			0.812122i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.0000	+	41.5692i	−0.934907	+	1.61931i	−0.160108	+	0.987099i	$0.551184\pi$
$659$	−24.0000	+	41.5692i	−0.934907	+	1.61931i	−0.774799	+	0.632207i	$0.782149\pi$
$660$		0			0
$661$	−22.5000	+	12.9904i	−0.875149	+	0.505267i	−0.869056	−	0.494714i	$-0.835273\pi$
$661$	−22.5000	+	12.9904i	−0.875149	+	0.505267i	−0.00609283	+	0.999981i	$0.501939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.0000	−	10.3923i	0.698010	−	0.402996i
$666$		0			0
$667$	−18.0000	+	31.1769i	−0.696963	+	1.20717i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.46410i		−	0.133730i
$672$		0			0
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	0.875501	−	0.483216i	$-0.160531\pi$
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	−0.856228	+	0.516599i	$0.827198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−48.0000			−1.84479			−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000			−1.84479			−0.922395	+	0.386248i	$0.873771\pi$
$678$		0			0
$679$	−4.50000	−	7.79423i	−0.172694	−	0.299115i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.0000	+	12.1244i	0.803543	+	0.463926i	0.844708	−	0.535227i	$-0.179774\pi$
$683$	21.0000	+	12.1244i	0.803543	+	0.463926i	−0.0411658	+	0.999152i	$0.513107\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−42.0000	−	10.3923i	−1.60007	−	0.395915i
$690$		0			0
$691$	−37.5000	+	21.6506i	−1.42657	+	0.823629i	−0.996848	−	0.0793336i	$-0.974721\pi$
$691$	−37.5000	+	21.6506i	−1.42657	+	0.823629i	−0.429719	+	0.902963i	$0.641387\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.0000	+	8.66025i	0.568982	+	0.328502i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	31.1769i		−	1.17253i
$708$		0			0
$709$	−16.5000	−	9.52628i	−0.619671	−	0.357767i	0.157070	−	0.987587i	$-0.449795\pi$
$709$	−16.5000	−	9.52628i	−0.619671	−	0.357767i	−0.776741	+	0.629821i	$0.783128\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.00000	+	5.19615i	−0.337053	+	0.194597i
$714$		0			0
$715$	−12.0000	−	41.5692i	−0.448775	−	1.55460i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.00000	−	5.19615i	0.111881	−	0.193784i	−0.804648	−	0.593753i	$-0.797646\pi$
$719$	3.00000	−	5.19615i	0.111881	−	0.193784i	0.916529	+	0.399969i	$0.130979\pi$
$720$		0			0
$721$	1.50000	+	0.866025i	0.0558629	+	0.0322525i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.0000	−	36.3731i	−0.779920	−	1.35086i
$726$		0			0
$727$	35.0000			1.29808			0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000			1.29808			0.649039	+	0.760755i	$0.275171\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	39.8372i		−	1.47142i	−0.677297	−	0.735710i	$-0.736849\pi$
$733$		−	39.8372i		−	1.47142i	0.677297	−	0.735710i	$-0.263151\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000	−	25.9808i	0.552532	−	0.957014i
$738$		0			0
$739$	−39.0000	+	22.5167i	−1.43464	+	0.828289i	−0.997470	−	0.0710909i	$-0.977352\pi$
$739$	−39.0000	+	22.5167i	−1.43464	+	0.828289i	−0.437168	+	0.899380i	$0.644019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.00000	+	5.19615i	−0.330178	+	0.190628i	−0.655920	−	0.754830i	$-0.727719\pi$
$743$	−9.00000	+	5.19615i	−0.330178	+	0.190628i	0.325742	+	0.945459i	$0.394386\pi$
$744$		0			0
$745$	−12.0000	+	20.7846i	−0.439646	+	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			10.3923i			0.379727i
$750$		0			0
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	0.929731	−	0.368238i	$-0.120039\pi$
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	−0.783769	+	0.621052i	$0.786706\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−17.0000	−	29.4449i	−0.617876	−	1.07019i	−0.989873	−	0.141958i	$-0.954660\pi$
$757$	−17.0000	−	29.4449i	−0.617876	−	1.07019i	0.371997	−	0.928234i	$-0.378673\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000	+	10.3923i	0.652499	+	0.376721i	0.789413	−	0.613862i	$-0.210385\pi$
$761$	18.0000	+	10.3923i	0.652499	+	0.376721i	−0.136914	+	0.990583i	$0.543718\pi$
$762$		0			0
$763$	−13.5000	+	23.3827i	−0.488733	+	0.846510i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.00000	−	8.66025i	−0.324971	−	0.312704i
$768$		0			0
$769$	−6.00000	+	3.46410i	−0.216366	+	0.124919i	−0.604266	−	0.796782i	$-0.706534\pi$
$769$	−6.00000	+	3.46410i	−0.216366	+	0.124919i	0.387901	+	0.921701i	$0.373200\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	45.0000	+	25.9808i	1.61854	+	0.934463i	0.987299	+	0.158874i	$0.0507865\pi$
$773$	45.0000	+	25.9808i	1.61854	+	0.934463i	0.631239	+	0.775589i	$0.282547\pi$
$774$		0			0
$775$		−	12.1244i		−	0.435520i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$			38.1051i			1.36003i
$786$		0			0
$787$	−28.5000	−	16.4545i	−1.01592	−	0.586539i	−0.102997	−	0.994682i	$-0.532843\pi$
$787$	−28.5000	−	16.4545i	−1.01592	−	0.586539i	−0.912918	+	0.408143i	$0.866177\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.00000	−	5.19615i	0.320003	−	0.184754i
$792$		0			0
$793$	−2.50000	+	2.59808i	−0.0887776	+	0.0922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.00000	−	10.3923i	0.212531	−	0.368114i	−0.739975	−	0.672634i	$-0.765163\pi$
$797$	6.00000	−	10.3923i	0.212531	−	0.368114i	0.952506	+	0.304520i	$0.0984960\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$	−36.0000			−1.26883
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000	+	10.3923i	0.210949	+	0.365374i	0.952012	−	0.306062i	$-0.0990113\pi$
$809$	6.00000	+	10.3923i	0.210949	+	0.365374i	−0.741063	+	0.671436i	$0.765678\pi$
$810$		0			0
$811$		−	25.9808i		−	0.912308i	−0.889901	−	0.456154i	$-0.849227\pi$
$811$		−	25.9808i		−	0.912308i	0.889901	−	0.456154i	$-0.150773\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.0000	+	57.1577i	−1.15594	+	2.00215i
$816$		0			0
$817$	3.00000	−	1.73205i	0.104957	−	0.0605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.0000	+	12.1244i	−0.732905	+	0.423143i	−0.819484	−	0.573102i	$-0.805740\pi$
$821$	−21.0000	+	12.1244i	−0.732905	+	0.423143i	0.0865789	+	0.996245i	$0.472407\pi$
$822$		0			0
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	0.997686	−	0.0679910i	$-0.0216589\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.4974i		−	1.68642i	−0.537584	−	0.843210i	$-0.680663\pi$
$827$		−	48.4974i		−	1.68642i	0.537584	−	0.843210i	$-0.319337\pi$
$828$		0			0
$829$	−15.5000	−	26.8468i	−0.538337	−	0.932427i	−0.998994	−	0.0448490i	$-0.985719\pi$
$829$	−15.5000	−	26.8468i	−0.538337	−	0.932427i	0.460657	−	0.887578i	$-0.347614\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−12.0000	−	20.7846i	−0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27.0000	+	15.5885i	0.932144	+	0.538173i	0.887489	−	0.460829i	$-0.152448\pi$
$839$	27.0000	+	15.5885i	0.932144	+	0.538173i	0.0446547	+	0.999002i	$0.485781\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.0000	+	39.8372i	−0.722422	+	1.37044i
$846$		0			0
$847$	1.50000	−	0.866025i	0.0515406	−	0.0297570i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			25.9808i			0.889564i	0.895639	+	0.444782i	$0.146719\pi$
$853$			25.9808i			0.889564i	−0.895639	+	0.444782i	$0.853281\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18.0000			−0.614868			−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000			−0.614868			−0.307434	+	0.951569i	$0.599470\pi$
$858$		0			0
$859$	−13.0000			−0.443554			−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000			−0.443554			−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17.3205i			0.589597i	0.955559	+	0.294798i	$0.0952525\pi$
$863$			17.3205i			0.589597i	−0.955559	+	0.294798i	$0.904747\pi$
$864$		0			0
$865$	−18.0000	−	10.3923i	−0.612018	−	0.353349i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−33.0000	+	19.0526i	−1.11945	+	0.646314i
$870$		0			0
$871$	−30.0000	+	8.66025i	−1.01651	+	0.293442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.00000	−	10.3923i	0.202837	−	0.351324i
$876$		0			0
$877$	36.0000	+	20.7846i	1.21563	+	0.701846i	0.963981	−	0.265971i	$-0.0856926\pi$
$877$	36.0000	+	20.7846i	1.21563	+	0.701846i	0.251653	+	0.967818i	$0.419026\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.00000	−	10.3923i	−0.202145	−	0.350126i	0.747074	−	0.664741i	$-0.231458\pi$
$881$	−6.00000	−	10.3923i	−0.202145	−	0.350126i	−0.949219	+	0.314615i	$0.898125\pi$
$882$		0			0
$883$	5.00000			0.168263			0.0841317	−	0.996455i	$-0.473188\pi$
$883$	5.00000			0.168263			0.0841317	+	0.996455i	$0.473188\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	0.966651	+	0.256096i	$0.0824362\pi$
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	−0.261540	+	0.965193i	$0.584230\pi$
$888$		0			0
$889$			22.5167i			0.755185i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.00000	+	10.3923i	−0.200782	+	0.347765i
$894$		0			0
$895$	36.0000	−	20.7846i	1.20335	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.00000	+	5.19615i	−0.300167	+	0.173301i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			48.4974i			1.61211i
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	−24.0000	−	41.5692i	−0.794284	−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.00000	+	5.19615i	0.297206	+	0.171592i
$918$		0			0
$919$	−16.0000	+	27.7128i	−0.527791	+	0.914161i	0.471684	+	0.881768i	$0.343646\pi$
$919$	−16.0000	+	27.7128i	−0.527791	+	0.914161i	−0.999475	+	0.0323936i	$0.989687\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−27.0000	−	25.9808i	−0.888716	−	0.855167i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.0000	+	10.3923i	0.590561	+	0.340960i	0.765319	−	0.643651i	$-0.222581\pi$
$929$	18.0000	+	10.3923i	0.590561	+	0.340960i	−0.174758	+	0.984611i	$0.555914\pi$
$930$		0			0
$931$			13.8564i			0.454125i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−14.0000			−0.457360			−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000			−0.457360			−0.228680	+	0.973502i	$0.573441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	10.3923i		−	0.338779i	−0.985549	−	0.169390i	$-0.945820\pi$
$941$		−	10.3923i		−	0.338779i	0.985549	−	0.169390i	$-0.0541797\pi$
$942$		0			0
$943$	36.0000	+	20.7846i	1.17232	+	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−42.0000	+	24.2487i	−1.36482	+	0.787977i	−0.990260	−	0.139227i	$-0.955538\pi$
$947$	−42.0000	+	24.2487i	−1.36482	+	0.787977i	−0.374556	+	0.927204i	$0.622205\pi$
$948$		0			0
$949$	1.50000	−	6.06218i	0.0486921	−	0.196787i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.00000	−	5.19615i	0.0971795	−	0.168320i	−0.813337	−	0.581793i	$-0.802351\pi$
$953$	3.00000	−	5.19615i	0.0971795	−	0.168320i	0.910516	+	0.413473i	$0.135685\pi$
$954$		0			0
$955$	54.0000	+	31.1769i	1.74740	+	1.00886i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−27.0000	−	46.7654i	−0.869161	−	1.50543i
$966$		0			0
$967$		−	24.2487i		−	0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$		−	24.2487i		−	0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	0.418101	+	0.908401i	$0.362696\pi$
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	−0.995748	+	0.0921142i	$0.970638\pi$
$972$		0			0
$973$	7.50000	−	4.33013i	0.240439	−	0.138817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.0000	−	13.8564i	0.767828	−	0.443306i	−0.0642712	−	0.997932i	$-0.520472\pi$
$977$	24.0000	−	13.8564i	0.767828	−	0.443306i	0.832099	+	0.554627i	$0.187139\pi$
$978$		0			0
$979$	12.0000	−	20.7846i	0.383522	−	0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			10.3923i			0.331463i	0.986171	+	0.165732i	$0.0529985\pi$
$983$			10.3923i			0.331463i	−0.986171	+	0.165732i	$0.947001\pi$
$984$		0			0
$985$	−24.0000	−	41.5692i	−0.764704	−	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.00000			−0.190789
$990$		0			0
$991$	−8.00000	−	13.8564i	−0.254128	−	0.440163i	0.710530	−	0.703667i	$-0.248455\pi$
$991$	−8.00000	−	13.8564i	−0.254128	−	0.440163i	−0.964658	+	0.263504i	$0.915122\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	21.0000	+	12.1244i	0.665745	+	0.384368i
$996$		0			0
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	0.443732	+	0.896159i	$0.353654\pi$
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	−0.997963	+	0.0637961i	$0.979679\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.by.f.1297.1		2
3.2	odd	2		624.2.bv.b.49.1		2
4.3	odd	2		117.2.q.a.10.1		2
12.11	even	2		39.2.j.a.10.1	yes	2
13.4	even	6	inner	1872.2.by.f.433.1		2
39.2	even	12		8112.2.a.bu.1.1		2
39.11	even	12		8112.2.a.bu.1.2		2
39.17	odd	6		624.2.bv.b.433.1		2
52.3	odd	6		1521.2.b.f.1351.2		2
52.11	even	12		1521.2.a.h.1.1		2
52.15	even	12		1521.2.a.h.1.2		2
52.23	odd	6		1521.2.b.f.1351.1		2
52.43	odd	6		117.2.q.a.82.1		2
60.23	odd	4		975.2.w.d.49.1		4
60.47	odd	4		975.2.w.d.49.2		4
60.59	even	2		975.2.bc.c.751.1		2
156.11	odd	12		507.2.a.e.1.2		2
156.23	even	6		507.2.b.c.337.2		2
156.35	even	6		507.2.j.b.316.1		2
156.47	odd	4		507.2.e.f.484.2		4
156.59	odd	12		507.2.e.f.22.2		4
156.71	odd	12		507.2.e.f.22.1		4
156.83	odd	4		507.2.e.f.484.1		4
156.95	even	6		39.2.j.a.4.1	✓	2
156.107	even	6		507.2.b.c.337.1		2
156.119	odd	12		507.2.a.e.1.1		2
156.155	even	2		507.2.j.b.361.1		2
780.407	odd	12		975.2.w.d.199.1		4
780.563	odd	12		975.2.w.d.199.2		4
780.719	even	6		975.2.bc.c.901.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.j.a.4.1	✓	2	156.95	even	6
39.2.j.a.10.1	yes	2	12.11	even	2
117.2.q.a.10.1		2	4.3	odd	2
117.2.q.a.82.1		2	52.43	odd	6
507.2.a.e.1.1		2	156.119	odd	12
507.2.a.e.1.2		2	156.11	odd	12
507.2.b.c.337.1		2	156.107	even	6
507.2.b.c.337.2		2	156.23	even	6
507.2.e.f.22.1		4	156.71	odd	12
507.2.e.f.22.2		4	156.59	odd	12
507.2.e.f.484.1		4	156.83	odd	4
507.2.e.f.484.2		4	156.47	odd	4
507.2.j.b.316.1		2	156.35	even	6
507.2.j.b.361.1		2	156.155	even	2
624.2.bv.b.49.1		2	3.2	odd	2
624.2.bv.b.433.1		2	39.17	odd	6
975.2.w.d.49.1		4	60.23	odd	4
975.2.w.d.49.2		4	60.47	odd	4
975.2.w.d.199.1		4	780.407	odd	12
975.2.w.d.199.2		4	780.563	odd	12
975.2.bc.c.751.1		2	60.59	even	2
975.2.bc.c.901.1		2	780.719	even	6
1521.2.a.h.1.1		2	52.11	even	12
1521.2.a.h.1.2		2	52.15	even	12
1521.2.b.f.1351.1		2	52.23	odd	6
1521.2.b.f.1351.2		2	52.3	odd	6
1872.2.by.f.433.1		2	13.4	even	6	inner
1872.2.by.f.1297.1		2	1.1	even	1	trivial
8112.2.a.bu.1.1		2	39.2	even	12
8112.2.a.bu.1.2		2	39.11	even	12

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{5} +(1.50000 + 0.866025i) q^{7} +(-3.00000 + 1.73205i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-3.00000 - 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} -7.00000 q^{25} +(3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(-3.00000 + 5.19615i) q^{35} +(6.00000 - 3.46410i) q^{41} +(-0.500000 + 0.866025i) q^{43} -3.46410i q^{47} +(-2.00000 - 3.46410i) q^{49} -12.0000 q^{53} +(-6.00000 - 10.3923i) q^{55} +(-3.00000 - 1.73205i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-3.00000 + 12.1244i) q^{65} +(-7.50000 + 4.33013i) q^{67} +(-9.00000 - 5.19615i) q^{71} -1.73205i q^{73} -6.00000 q^{77} +11.0000 q^{79} +13.8564i q^{83} +(-6.00000 + 3.46410i) q^{89} +(4.50000 + 4.33013i) q^{91} +(6.00000 - 10.3923i) q^{95} +(-4.50000 - 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{7} - 6 q^{11} + 7 q^{13} - 6 q^{19} + 6 q^{23} - 14 q^{25} + 6 q^{29} - 6 q^{35} + 12 q^{41} - q^{43} - 4 q^{49} - 24 q^{53} - 12 q^{55} - 6 q^{59} - q^{61} - 6 q^{65} - 15 q^{67} - 18 q^{71}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q + 3 * q^7 - 6 * q^11 + 7 * q^13 - 6 * q^19 + 6 * q^23 - 14 * q^25 + 6 * q^29 - 6 * q^35 + 12 * q^41 - q^43 - 4 * q^49 - 24 * q^53 - 12 * q^55 - 6 * q^59 - q^61 - 6 * q^65 - 15 * q^67 - 18 * q^71 - 12 * q^77 + 22 * q^79 - 12 * q^89 + 9 * q^91 + 12 * q^95 - 9 * q^97

Introduction

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