LMFDB - Embedded newform 1764.3.z.g.901.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(325,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.325"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,12,0,0,0,0,0,18,0,0,0,0,0,24,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.0655186332$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.901
Dual form			1764.3.z.g.325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.00000 + 3.46410i) q^{5} +(9.00000 + 15.5885i) q^{11} +20.7846i q^{13} +(12.0000 - 6.92820i) q^{17} +(18.0000 + 10.3923i) q^{19} +(9.00000 - 15.5885i) q^{23} +(11.5000 + 19.9186i) q^{25} -18.0000 q^{29} +(-36.0000 + 20.7846i) q^{31} +(-5.00000 + 8.66025i) q^{37} -55.4256i q^{41} -38.0000 q^{43} +(24.0000 + 13.8564i) q^{47} +(9.00000 + 15.5885i) q^{53} +124.708i q^{55} +(6.00000 - 3.46410i) q^{59} +(-18.0000 - 10.3923i) q^{61} +(-72.0000 + 124.708i) q^{65} +(-13.0000 - 22.5167i) q^{67} -18.0000 q^{71} +(-36.0000 + 20.7846i) q^{73} +(-1.00000 + 1.73205i) q^{79} -34.6410i q^{83} +96.0000 q^{85} +(72.0000 + 124.708i) q^{95} -166.277i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} + 18 q^{11} + 24 q^{17} + 36 q^{19} + 18 q^{23} + 23 q^{25} - 36 q^{29} - 72 q^{31} - 10 q^{37} - 76 q^{43} + 48 q^{47} + 18 q^{53} + 12 q^{59} - 36 q^{61} - 144 q^{65} - 26 q^{67} - 36 q^{71}+ \cdots + 144 q^{95}+O(q^{100}) $ 2 * q + 12 * q^5 + 18 * q^11 + 24 * q^17 + 36 * q^19 + 18 * q^23 + 23 * q^25 - 36 * q^29 - 72 * q^31 - 10 * q^37 - 76 * q^43 + 48 * q^47 + 18 * q^53 + 12 * q^59 - 36 * q^61 - 144 * q^65 - 26 * q^67 - 36 * q^71 - 72 * q^73 - 2 * q^79 + 192 * q^85 + 144 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	6.00000	+	3.46410i	1.20000	+	0.692820i	0.960555	−	0.278090i	$-0.0897012\pi$
$5$	6.00000	+	3.46410i	1.20000	+	0.692820i	0.239445	+	0.970910i	$0.423035\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.00000	+	15.5885i	0.818182	+	1.41713i	0.907021	+	0.421086i	$0.138351\pi$
$11$	9.00000	+	15.5885i	0.818182	+	1.41713i	−0.0888387	+	0.996046i	$0.528316\pi$
$12$		0			0
$13$			20.7846i			1.59882i	0.600788	+	0.799408i	$0.294853\pi$
$13$			20.7846i			1.59882i	−0.600788	+	0.799408i	$0.705147\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.0000	−	6.92820i	0.705882	−	0.407541i	−0.103652	−	0.994614i	$-0.533053\pi$
$17$	12.0000	−	6.92820i	0.705882	−	0.407541i	0.809535	+	0.587072i	$0.199720\pi$
$18$		0			0
$19$	18.0000	+	10.3923i	0.947368	+	0.546963i	0.892262	−	0.451517i	$-0.149117\pi$
$19$	18.0000	+	10.3923i	0.947368	+	0.546963i	0.0551060	+	0.998481i	$0.482450\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.00000	−	15.5885i	0.391304	−	0.677759i	−0.601318	−	0.799010i	$-0.705357\pi$
$23$	9.00000	−	15.5885i	0.391304	−	0.677759i	0.992622	+	0.121251i	$0.0386906\pi$
$24$		0			0
$25$	11.5000	+	19.9186i	0.460000	+	0.796743i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.0000			−0.620690			−0.310345	−	0.950624i	$-0.600445\pi$
$29$	−18.0000			−0.620690			−0.310345	+	0.950624i	$0.600445\pi$
$30$		0			0
$31$	−36.0000	+	20.7846i	−1.16129	+	0.670471i	−0.951613	−	0.307299i	$-0.900575\pi$
$31$	−36.0000	+	20.7846i	−1.16129	+	0.670471i	−0.209677	+	0.977771i	$0.567241\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	8.66025i	−0.135135	+	0.234061i	−0.925649	−	0.378383i	$-0.876480\pi$
$37$	−5.00000	+	8.66025i	−0.135135	+	0.234061i	0.790514	+	0.612444i	$0.209814\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	55.4256i		−	1.35184i	−0.736973	−	0.675922i	$-0.763745\pi$
$41$		−	55.4256i		−	1.35184i	0.736973	−	0.675922i	$-0.236255\pi$
$42$		0			0
$43$	−38.0000			−0.883721			−0.441860	−	0.897084i	$-0.645681\pi$
$43$	−38.0000			−0.883721			−0.441860	+	0.897084i	$0.645681\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	24.0000	+	13.8564i	0.510638	+	0.294817i	0.733096	−	0.680125i	$-0.238075\pi$
$47$	24.0000	+	13.8564i	0.510638	+	0.294817i	−0.222458	+	0.974942i	$0.571408\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000	+	15.5885i	0.169811	+	0.294122i	0.938353	−	0.345677i	$-0.112351\pi$
$53$	9.00000	+	15.5885i	0.169811	+	0.294122i	−0.768542	+	0.639799i	$0.779018\pi$
$54$		0			0
$55$			124.708i			2.26741i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	−	3.46410i	0.101695	−	0.0587136i	−0.448290	−	0.893888i	$-0.647967\pi$
$59$	6.00000	−	3.46410i	0.101695	−	0.0587136i	0.549985	+	0.835175i	$0.314633\pi$
$60$		0			0
$61$	−18.0000	−	10.3923i	−0.295082	−	0.170366i	0.345149	−	0.938548i	$-0.387828\pi$
$61$	−18.0000	−	10.3923i	−0.295082	−	0.170366i	−0.640231	+	0.768182i	$0.721162\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−72.0000	+	124.708i	−1.10769	+	1.91858i
$66$		0			0
$67$	−13.0000	−	22.5167i	−0.194030	−	0.336070i	0.752552	−	0.658533i	$-0.228823\pi$
$67$	−13.0000	−	22.5167i	−0.194030	−	0.336070i	−0.946582	+	0.322463i	$0.895489\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−18.0000			−0.253521			−0.126761	−	0.991933i	$-0.540458\pi$
$71$	−18.0000			−0.253521			−0.126761	+	0.991933i	$0.540458\pi$
$72$		0			0
$73$	−36.0000	+	20.7846i	−0.493151	+	0.284721i	−0.725881	−	0.687821i	$-0.758568\pi$
$73$	−36.0000	+	20.7846i	−0.493151	+	0.284721i	0.232730	+	0.972541i	$0.425234\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.0126582	+	0.0219247i	−0.872285	−	0.488998i	$-0.837363\pi$
$79$	−1.00000	+	1.73205i	−0.0126582	+	0.0219247i	0.859627	+	0.510922i	$0.170696\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	34.6410i		−	0.417362i	−0.977984	−	0.208681i	$-0.933083\pi$
$83$		−	34.6410i		−	0.417362i	0.977984	−	0.208681i	$-0.0669170\pi$
$84$		0			0
$85$	96.0000			1.12941
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	72.0000	+	124.708i	0.757895	+	1.31271i
$96$		0			0
$97$		−	166.277i		−	1.71419i	−0.515155	−	0.857097i	$-0.672265\pi$
$97$		−	166.277i		−	1.71419i	0.515155	−	0.857097i	$-0.327735\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	162.000	−	93.5307i	1.60396	−	0.926047i	0.613276	−	0.789869i	$-0.289851\pi$
$101$	162.000	−	93.5307i	1.60396	−	0.926047i	0.990684	−	0.136178i	$-0.0434820\pi$
$102$		0			0
$103$	−72.0000	−	41.5692i	−0.699029	−	0.403585i	0.107957	−	0.994156i	$-0.465569\pi$
$103$	−72.0000	−	41.5692i	−0.699029	−	0.403585i	−0.806986	+	0.590571i	$0.798903\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−63.0000	+	109.119i	−0.588785	+	1.01981i	0.405607	+	0.914048i	$0.367060\pi$
$107$	−63.0000	+	109.119i	−0.588785	+	1.01981i	−0.994392	+	0.105758i	$0.966273\pi$
$108$		0			0
$109$	67.0000	+	116.047i	0.614679	+	1.06466i	0.990441	+	0.137939i	$0.0440477\pi$
$109$	67.0000	+	116.047i	0.614679	+	1.06466i	−0.375762	+	0.926716i	$0.622619\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	126.000			1.11504			0.557522	−	0.830162i	$-0.311752\pi$
$113$	126.000			1.11504			0.557522	+	0.830162i	$0.311752\pi$
$114$		0			0
$115$	108.000	−	62.3538i	0.939130	−	0.542207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−101.500	+	175.803i	−0.838843	+	1.45292i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	13.8564i		−	0.110851i
$126$		0			0
$127$	−46.0000			−0.362205			−0.181102	−	0.983464i	$-0.557967\pi$
$127$	−46.0000			−0.362205			−0.181102	+	0.983464i	$0.557967\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−162.000	−	93.5307i	−1.23664	−	0.713975i	−0.268235	−	0.963353i	$-0.586440\pi$
$131$	−162.000	−	93.5307i	−1.23664	−	0.713975i	−0.968406	+	0.249378i	$0.919774\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	81.0000	+	140.296i	0.591241	+	1.02406i	0.994066	+	0.108782i	$0.0346951\pi$
$137$	81.0000	+	140.296i	0.591241	+	1.02406i	−0.402825	+	0.915277i	$0.631972\pi$
$138$		0			0
$139$		−	62.3538i		−	0.448589i	−0.974521	−	0.224294i	$-0.927992\pi$
$139$		−	62.3538i		−	0.448589i	0.974521	−	0.224294i	$-0.0720077\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−324.000	+	187.061i	−2.26573	+	1.30812i
$144$		0			0
$145$	−108.000	−	62.3538i	−0.744828	−	0.430026i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	81.0000	−	140.296i	0.543624	−	0.941585i	−0.455068	−	0.890457i	$-0.650385\pi$
$149$	81.0000	−	140.296i	0.543624	−	0.941585i	0.998692	−	0.0511280i	$-0.0162816\pi$
$150$		0			0
$151$	127.000	+	219.970i	0.841060	+	1.45676i	0.889000	+	0.457908i	$0.151401\pi$
$151$	127.000	+	219.970i	0.841060	+	1.45676i	−0.0479401	+	0.998850i	$0.515266\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−288.000			−1.85806
$156$		0			0
$157$	18.0000	−	10.3923i	0.114650	−	0.0661930i	−0.441579	−	0.897223i	$-0.645581\pi$
$157$	18.0000	−	10.3923i	0.114650	−	0.0661930i	0.556228	+	0.831030i	$0.312248\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−13.0000	+	22.5167i	−0.0797546	+	0.138139i	−0.903144	−	0.429338i	$-0.858747\pi$
$163$	−13.0000	+	22.5167i	−0.0797546	+	0.138139i	0.823389	+	0.567477i	$0.192080\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			180.133i			1.07864i	0.842100	+	0.539321i	$0.181319\pi$
$167$			180.133i			1.07864i	−0.842100	+	0.539321i	$0.818681\pi$
$168$		0			0
$169$	−263.000			−1.55621
$170$		0			0
$171$		0			0
$172$		0			0
$173$	162.000	+	93.5307i	0.936416	+	0.540640i	0.888835	−	0.458227i	$-0.151515\pi$
$173$	162.000	+	93.5307i	0.936416	+	0.540640i	0.0475811	+	0.998867i	$0.484849\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	153.000	+	265.004i	0.854749	+	1.48047i	0.876878	+	0.480713i	$0.159622\pi$
$179$	153.000	+	265.004i	0.854749	+	1.48047i	−0.0221294	+	0.999755i	$0.507045\pi$
$180$		0			0
$181$		−	187.061i		−	1.03349i	−0.856140	−	0.516744i	$-0.827144\pi$
$181$		−	187.061i		−	1.03349i	0.856140	−	0.516744i	$-0.172856\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−60.0000	+	34.6410i	−0.324324	+	0.187249i
$186$		0			0
$187$	216.000	+	124.708i	1.15508	+	0.666886i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−135.000	+	233.827i	−0.706806	+	1.22422i	0.259229	+	0.965816i	$0.416531\pi$
$191$	−135.000	+	233.827i	−0.706806	+	1.22422i	−0.966036	+	0.258409i	$0.916802\pi$
$192$		0			0
$193$	−137.000	−	237.291i	−0.709845	−	1.22949i	−0.964915	−	0.262564i	$-0.915432\pi$
$193$	−137.000	−	237.291i	−0.709845	−	1.22949i	0.255070	−	0.966923i	$-0.417901\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−162.000			−0.822335			−0.411168	−	0.911560i	$-0.634879\pi$
$197$	−162.000			−0.822335			−0.411168	+	0.911560i	$0.634879\pi$
$198$		0			0
$199$	72.0000	−	41.5692i	0.361809	−	0.208891i	−0.308065	−	0.951365i	$-0.599681\pi$
$199$	72.0000	−	41.5692i	0.361809	−	0.208891i	0.669874	+	0.742475i	$0.266348\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	192.000	−	332.554i	0.936585	−	1.62221i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			374.123i			1.79006i
$210$		0			0
$211$	10.0000			0.0473934			0.0236967	−	0.999719i	$-0.492456\pi$
$211$	10.0000			0.0473934			0.0236967	+	0.999719i	$0.492456\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−228.000	−	131.636i	−1.06047	−	0.612260i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	144.000	+	249.415i	0.651584	+	1.12858i
$222$		0			0
$223$			41.5692i			0.186409i	0.995647	+	0.0932045i	$0.0297110\pi$
$223$			41.5692i			0.186409i	−0.995647	+	0.0932045i	$0.970289\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	162.000	−	93.5307i	0.713656	−	0.412030i	−0.0987570	−	0.995112i	$-0.531487\pi$
$227$	162.000	−	93.5307i	0.713656	−	0.412030i	0.812413	+	0.583082i	$0.198153\pi$
$228$		0			0
$229$	234.000	+	135.100i	1.02183	+	0.589956i	0.914635	−	0.404281i	$-0.132478\pi$
$229$	234.000	+	135.100i	1.02183	+	0.589956i	0.107199	+	0.994238i	$0.465812\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−63.0000	+	109.119i	−0.270386	+	0.468323i	−0.968961	−	0.247215i	$-0.920485\pi$
$233$	−63.0000	+	109.119i	−0.270386	+	0.468323i	0.698574	+	0.715537i	$0.253818\pi$
$234$		0			0
$235$	96.0000	+	166.277i	0.408511	+	0.707561i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.0000			−0.0753138			−0.0376569	−	0.999291i	$-0.511989\pi$
$239$	−18.0000			−0.0753138			−0.0376569	+	0.999291i	$0.511989\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−216.000	+	374.123i	−0.874494	+	1.51467i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	48.4974i		−	0.193217i	−0.995322	−	0.0966084i	$-0.969201\pi$
$251$		−	48.4974i		−	0.193217i	0.995322	−	0.0966084i	$-0.0307994\pi$
$252$		0			0
$253$	324.000			1.28063
$254$		0			0
$255$		0			0
$256$		0			0
$257$	324.000	+	187.061i	1.26070	+	0.727866i	0.973210	−	0.229918i	$-0.0738457\pi$
$257$	324.000	+	187.061i	1.26070	+	0.727866i	0.287490	+	0.957784i	$0.407179\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−135.000	−	233.827i	−0.513308	−	0.889076i	−0.999881	−	0.0154355i	$-0.995087\pi$
$263$	−135.000	−	233.827i	−0.513308	−	0.889076i	0.486573	−	0.873640i	$-0.338247\pi$
$264$		0			0
$265$			124.708i			0.470595i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−66.0000	+	38.1051i	−0.245353	+	0.141655i	−0.617635	−	0.786465i	$-0.711909\pi$
$269$	−66.0000	+	38.1051i	−0.245353	+	0.141655i	0.372281	+	0.928120i	$0.378576\pi$
$270$		0			0
$271$	180.000	+	103.923i	0.664207	+	0.383480i	0.793878	−	0.608077i	$-0.208059\pi$
$271$	180.000	+	103.923i	0.664207	+	0.383480i	−0.129671	+	0.991557i	$0.541392\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−207.000	+	358.535i	−0.752727	+	1.30376i
$276$		0			0
$277$	179.000	+	310.037i	0.646209	+	1.11927i	0.984021	+	0.178054i	$0.0569801\pi$
$277$	179.000	+	310.037i	0.646209	+	1.11927i	−0.337811	+	0.941214i	$0.609687\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	414.000			1.47331			0.736655	−	0.676269i	$-0.236404\pi$
$281$	414.000			1.47331			0.736655	+	0.676269i	$0.236404\pi$
$282$		0			0
$283$	234.000	−	135.100i	0.826855	−	0.477385i	−0.0259195	−	0.999664i	$-0.508251\pi$
$283$	234.000	−	135.100i	0.826855	−	0.477385i	0.852775	+	0.522279i	$0.174918\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−48.5000	+	84.0045i	−0.167820	+	0.290673i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	117.779i		−	0.401978i	−0.979594	−	0.200989i	$-0.935585\pi$
$293$		−	117.779i		−	0.401978i	0.979594	−	0.200989i	$-0.0644155\pi$
$294$		0			0
$295$	48.0000			0.162712
$296$		0			0
$297$		0			0
$298$		0			0
$299$	324.000	+	187.061i	1.08361	+	0.625624i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−72.0000	−	124.708i	−0.236066	−	0.408878i
$306$		0			0
$307$			62.3538i			0.203107i	0.994830	+	0.101553i	$0.0323813\pi$
$307$			62.3538i			0.203107i	−0.994830	+	0.101553i	$0.967619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−276.000	+	159.349i	−0.887460	+	0.512375i	−0.873111	−	0.487522i	$-0.837901\pi$
$311$	−276.000	+	159.349i	−0.887460	+	0.512375i	−0.0143490	+	0.999897i	$0.504568\pi$
$312$		0			0
$313$	−324.000	−	187.061i	−1.03514	−	0.597641i	−0.116690	−	0.993168i	$-0.537228\pi$
$313$	−324.000	−	187.061i	−1.03514	−	0.597641i	−0.918454	+	0.395528i	$0.870562\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	225.000	−	389.711i	0.709779	−	1.22937i	−0.255160	−	0.966899i	$-0.582128\pi$
$317$	225.000	−	389.711i	0.709779	−	1.22937i	0.964939	−	0.262475i	$-0.0845386\pi$
$318$		0			0
$319$	−162.000	−	280.592i	−0.507837	−	0.879599i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	288.000			0.891641
$324$		0			0
$325$	−414.000	+	239.023i	−1.27385	+	0.735455i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−125.000	+	216.506i	−0.377644	+	0.654098i	−0.990719	−	0.135927i	$-0.956599\pi$
$331$	−125.000	+	216.506i	−0.377644	+	0.654098i	0.613075	+	0.790024i	$0.289932\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	180.133i		−	0.537711i
$336$		0			0
$337$	−14.0000			−0.0415430			−0.0207715	−	0.999784i	$-0.506612\pi$
$337$	−14.0000			−0.0415430			−0.0207715	+	0.999784i	$0.506612\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−648.000	−	374.123i	−1.90029	−	1.09713i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−207.000	−	358.535i	−0.596542	−	1.03324i	−0.993327	−	0.115329i	$-0.963208\pi$
$347$	−207.000	−	358.535i	−0.596542	−	1.03324i	0.396786	−	0.917911i	$-0.370126\pi$
$348$		0			0
$349$			436.477i			1.25065i	0.780364	+	0.625325i	$0.215034\pi$
$349$			436.477i			1.25065i	−0.780364	+	0.625325i	$0.784966\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	516.000	−	297.913i	1.46176	−	0.843945i	0.462664	−	0.886534i	$-0.346894\pi$
$353$	516.000	−	297.913i	1.46176	−	0.843945i	0.999093	+	0.0425884i	$0.0135604\pi$
$354$		0			0
$355$	−108.000	−	62.3538i	−0.304225	−	0.175645i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.00000	−	15.5885i	0.0250696	−	0.0434219i	−0.853218	−	0.521554i	$-0.825353\pi$
$359$	9.00000	−	15.5885i	0.0250696	−	0.0434219i	0.878288	+	0.478132i	$0.158686\pi$
$360$		0			0
$361$	35.5000	+	61.4878i	0.0983380	+	0.170326i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−288.000			−0.789041
$366$		0			0
$367$	−396.000	+	228.631i	−1.07902	+	0.622972i	−0.930632	−	0.365957i	$-0.880742\pi$
$367$	−396.000	+	228.631i	−1.07902	+	0.622972i	−0.148387	+	0.988929i	$0.547408\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	227.000	−	393.176i	0.608579	−	1.05409i	−0.382896	−	0.923792i	$-0.625073\pi$
$373$	227.000	−	393.176i	0.608579	−	1.05409i	0.991475	−	0.130298i	$-0.0415935\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	374.123i		−	0.992369i
$378$		0			0
$379$	−134.000			−0.353562			−0.176781	−	0.984250i	$-0.556568\pi$
$379$	−134.000			−0.353562			−0.176781	+	0.984250i	$0.556568\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−168.000	−	96.9948i	−0.438642	−	0.253250i	0.264379	−	0.964419i	$-0.414833\pi$
$383$	−168.000	−	96.9948i	−0.438642	−	0.253250i	−0.703022	+	0.711169i	$0.748166\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	297.000	+	514.419i	0.763496	+	1.32241i	0.941038	+	0.338301i	$0.109852\pi$
$389$	297.000	+	514.419i	0.763496	+	1.32241i	−0.177542	+	0.984113i	$0.556815\pi$
$390$		0			0
$391$		−	249.415i		−	0.637891i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000	+	6.92820i	−0.0303797	+	0.0175398i
$396$		0			0
$397$	−306.000	−	176.669i	−0.770781	−	0.445011i	0.0623722	−	0.998053i	$-0.480133\pi$
$397$	−306.000	−	176.669i	−0.770781	−	0.445011i	−0.833153	+	0.553042i	$0.813467\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	81.0000	−	140.296i	0.201995	−	0.349866i	−0.747176	−	0.664626i	$-0.768591\pi$
$401$	81.0000	−	140.296i	0.201995	−	0.349866i	0.949171	+	0.314760i	$0.101924\pi$
$402$		0			0
$403$	−432.000	−	748.246i	−1.07196	−	1.85669i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−180.000			−0.442260
$408$		0			0
$409$	−108.000	+	62.3538i	−0.264059	+	0.152454i	−0.626185	−	0.779675i	$-0.715384\pi$
$409$	−108.000	+	62.3538i	−0.264059	+	0.152454i	0.362126	+	0.932129i	$0.382051\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	120.000	−	207.846i	0.289157	−	0.500834i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	505.759i		−	1.20706i	−0.797340	−	0.603531i	$-0.793760\pi$
$419$		−	505.759i		−	1.20706i	0.797340	−	0.603531i	$-0.206240\pi$
$420$		0			0
$421$	−454.000			−1.07838			−0.539192	−	0.842183i	$-0.681270\pi$
$421$	−454.000			−1.07838			−0.539192	+	0.842183i	$0.681270\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	276.000	+	159.349i	0.649412	+	0.374938i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−135.000	−	233.827i	−0.313225	−	0.542522i	0.665834	−	0.746100i	$-0.268076\pi$
$431$	−135.000	−	233.827i	−0.313225	−	0.542522i	−0.979059	+	0.203579i	$0.934743\pi$
$432$		0			0
$433$		−	166.277i		−	0.384011i	−0.981394	−	0.192006i	$-0.938501\pi$
$433$		−	166.277i		−	0.384011i	0.981394	−	0.192006i	$-0.0614992\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	324.000	−	187.061i	0.741419	−	0.428058i
$438$		0			0
$439$	−432.000	−	249.415i	−0.984055	−	0.568144i	−0.0805628	−	0.996750i	$-0.525672\pi$
$439$	−432.000	−	249.415i	−0.984055	−	0.568144i	−0.903492	+	0.428605i	$0.859005\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	153.000	−	265.004i	0.345372	−	0.598203i	−0.640049	−	0.768334i	$-0.721086\pi$
$443$	153.000	−	265.004i	0.345372	−	0.598203i	0.985421	+	0.170132i	$0.0544193\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−162.000			−0.360802			−0.180401	−	0.983593i	$-0.557740\pi$
$449$	−162.000			−0.360802			−0.180401	+	0.983593i	$0.557740\pi$
$450$		0			0
$451$	864.000	−	498.831i	1.91574	−	1.10605i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	79.0000	−	136.832i	0.172867	−	0.299414i	−0.766554	−	0.642179i	$-0.778030\pi$
$457$	79.0000	−	136.832i	0.172867	−	0.299414i	0.939421	+	0.342766i	$0.111364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	408.764i		−	0.886690i	−0.896351	−	0.443345i	$-0.853792\pi$
$461$		−	408.764i		−	0.886690i	0.896351	−	0.443345i	$-0.146208\pi$
$462$		0			0
$463$	−382.000			−0.825054			−0.412527	−	0.910945i	$-0.635354\pi$
$463$	−382.000			−0.825054			−0.412527	+	0.910945i	$0.635354\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	486.000	+	280.592i	1.04069	+	0.600840i	0.920027	−	0.391854i	$-0.128166\pi$
$467$	486.000	+	280.592i	1.04069	+	0.600840i	0.120658	+	0.992694i	$0.461500\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−342.000	−	592.361i	−0.723044	−	1.25235i
$474$		0			0
$475$			478.046i			1.00641i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	672.000	−	387.979i	1.40292	−	0.809978i	0.408231	−	0.912879i	$-0.366146\pi$
$479$	672.000	−	387.979i	1.40292	−	0.809978i	0.994692	+	0.102901i	$0.0328124\pi$
$480$		0			0
$481$	−180.000	−	103.923i	−0.374220	−	0.216056i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	576.000	−	997.661i	1.18763	−	2.05703i
$486$		0			0
$487$	271.000	+	469.386i	0.556468	+	0.963831i	0.997788	+	0.0664811i	$0.0211772\pi$
$487$	271.000	+	469.386i	0.556468	+	0.963831i	−0.441320	+	0.897350i	$0.645489\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−594.000			−1.20978			−0.604888	−	0.796311i	$-0.706782\pi$
$491$	−594.000			−1.20978			−0.604888	+	0.796311i	$0.706782\pi$
$492$		0			0
$493$	−216.000	+	124.708i	−0.438134	+	0.252957i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	331.000	−	573.309i	0.663327	−	1.14892i	−0.316409	−	0.948623i	$-0.602477\pi$
$499$	331.000	−	573.309i	0.663327	−	1.14892i	0.979736	−	0.200293i	$-0.0641893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.8564i		−	0.0275475i	−0.999905	−	0.0137738i	$-0.995616\pi$
$503$		−	13.8564i		−	0.0275475i	0.999905	−	0.0137738i	$-0.00438446\pi$
$504$		0			0
$505$	1296.00			2.56634
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−222.000	−	128.172i	−0.436149	−	0.251811i	0.265814	−	0.964024i	$-0.414359\pi$
$509$	−222.000	−	128.172i	−0.436149	−	0.251811i	−0.701963	+	0.712214i	$0.747693\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−288.000	−	498.831i	−0.559223	−	0.968603i
$516$		0			0
$517$			498.831i			0.964856i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−768.000	+	443.405i	−1.47409	+	0.851065i	−0.999574	−	0.0291826i	$-0.990710\pi$
$521$	−768.000	+	443.405i	−1.47409	+	0.851065i	−0.474514	+	0.880248i	$0.657376\pi$
$522$		0			0
$523$	558.000	+	322.161i	1.06692	+	0.615987i	0.927339	−	0.374223i	$-0.122091\pi$
$523$	558.000	+	322.161i	1.06692	+	0.615987i	0.139583	+	0.990210i	$0.455424\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−288.000	+	498.831i	−0.546490	+	0.946548i
$528$		0			0
$529$	102.500	+	177.535i	0.193762	+	0.335605i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1152.00			2.16135
$534$		0			0
$535$	−756.000	+	436.477i	−1.41308	+	0.815844i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−229.000	+	396.640i	−0.423290	+	0.733160i	−0.996259	−	0.0864170i	$-0.972458\pi$
$541$	−229.000	+	396.640i	−0.423290	+	0.733160i	0.572969	+	0.819577i	$0.305792\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			928.379i			1.70345i
$546$		0			0
$547$	506.000			0.925046			0.462523	−	0.886607i	$-0.346944\pi$
$547$	506.000			0.925046			0.462523	+	0.886607i	$0.346944\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−324.000	−	187.061i	−0.588022	−	0.339495i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−279.000	−	483.242i	−0.500898	−	0.867580i	−0.999999	−	0.00103685i	$-0.999670\pi$
$557$	−279.000	−	483.242i	−0.500898	−	0.867580i	0.499102	−	0.866543i	$-0.333663\pi$
$558$		0			0
$559$		−	789.815i		−	1.41291i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	162.000	−	93.5307i	0.287744	−	0.166129i	−0.349180	−	0.937056i	$-0.613540\pi$
$563$	162.000	−	93.5307i	0.287744	−	0.166129i	0.636924	+	0.770927i	$0.280206\pi$
$564$		0			0
$565$	756.000	+	436.477i	1.33805	+	0.772525i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	369.000	−	639.127i	0.648506	−	1.12325i	−0.334974	−	0.942228i	$-0.608727\pi$
$569$	369.000	−	639.127i	0.648506	−	1.12325i	0.983480	−	0.181018i	$-0.0579393\pi$
$570$		0			0
$571$	−133.000	−	230.363i	−0.232925	−	0.403437i	0.725743	−	0.687966i	$-0.241496\pi$
$571$	−133.000	−	230.363i	−0.232925	−	0.403437i	−0.958668	+	0.284529i	$0.908163\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	414.000			0.720000
$576$		0			0
$577$	72.0000	−	41.5692i	0.124783	−	0.0720437i	−0.436309	−	0.899797i	$-0.643715\pi$
$577$	72.0000	−	41.5692i	0.124783	−	0.0720437i	0.561092	+	0.827753i	$0.310381\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−162.000	+	280.592i	−0.277873	+	0.481290i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			561.184i			0.956021i	0.878354	+	0.478011i	$0.158642\pi$
$587$			561.184i			0.956021i	−0.878354	+	0.478011i	$0.841358\pi$
$588$		0			0
$589$	−864.000			−1.46689
$590$		0			0
$591$		0			0
$592$		0			0
$593$	228.000	+	131.636i	0.384486	+	0.221983i	0.679768	−	0.733427i	$-0.262080\pi$
$593$	228.000	+	131.636i	0.384486	+	0.221983i	−0.295282	+	0.955410i	$0.595414\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.00000	+	15.5885i	0.0150250	+	0.0260241i	0.873440	−	0.486931i	$-0.161884\pi$
$599$	9.00000	+	15.5885i	0.0150250	+	0.0260241i	−0.858415	+	0.512956i	$0.828551\pi$
$600$		0			0
$601$		−	41.5692i		−	0.0691668i	−0.999402	−	0.0345834i	$-0.988990\pi$
$601$		−	41.5692i		−	0.0691668i	0.999402	−	0.0345834i	$-0.0110104\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1218.00	+	703.213i	−2.01322	+	1.16233i
$606$		0			0
$607$	−324.000	−	187.061i	−0.533773	−	0.308174i	0.208779	−	0.977963i	$-0.433051\pi$
$607$	−324.000	−	187.061i	−0.533773	−	0.308174i	−0.742551	+	0.669789i	$0.766384\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−288.000	+	498.831i	−0.471358	+	0.816417i
$612$		0			0
$613$	−397.000	−	687.624i	−0.647635	−	1.12174i	−0.983686	−	0.179892i	$-0.942425\pi$
$613$	−397.000	−	687.624i	−0.647635	−	1.12174i	0.336052	−	0.941844i	$-0.390908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	702.000			1.13776			0.568882	−	0.822419i	$-0.307376\pi$
$617$	702.000			1.13776			0.568882	+	0.822419i	$0.307376\pi$
$618$		0			0
$619$	90.0000	−	51.9615i	0.145396	−	0.0839443i	−0.425537	−	0.904941i	$-0.639915\pi$
$619$	90.0000	−	51.9615i	0.145396	−	0.0839443i	0.570933	+	0.820997i	$0.306582\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	335.500	−	581.103i	0.536800	−	0.929765i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			138.564i			0.220293i
$630$		0			0
$631$	530.000			0.839937			0.419968	−	0.907539i	$-0.362041\pi$
$631$	530.000			0.839937			0.419968	+	0.907539i	$0.362041\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−276.000	−	159.349i	−0.434646	−	0.250943i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−495.000	−	857.365i	−0.772231	−	1.33754i	−0.936338	−	0.351101i	$-0.885808\pi$
$641$	−495.000	−	857.365i	−0.772231	−	1.33754i	0.164107	−	0.986443i	$-0.447526\pi$
$642$		0			0
$643$		−	436.477i		−	0.678813i	−0.940640	−	0.339407i	$-0.889774\pi$
$643$		−	436.477i		−	0.678813i	0.940640	−	0.339407i	$-0.110226\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	324.000	−	187.061i	0.500773	−	0.289121i	−0.228260	−	0.973600i	$-0.573304\pi$
$647$	324.000	−	187.061i	0.500773	−	0.289121i	0.729033	+	0.684479i	$0.239970\pi$
$648$		0			0
$649$	108.000	+	62.3538i	0.166410	+	0.0960768i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.0137825	−	0.0238721i	−0.859052	−	0.511889i	$-0.828946\pi$
$653$	9.00000	−	15.5885i	0.0137825	−	0.0238721i	0.872834	+	0.488016i	$0.162279\pi$
$654$		0			0
$655$	−648.000	−	1122.37i	−0.989313	−	1.71354i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−882.000			−1.33839			−0.669196	−	0.743086i	$-0.733361\pi$
$659$	−882.000			−1.33839			−0.669196	+	0.743086i	$0.733361\pi$
$660$		0			0
$661$	1062.00	−	613.146i	1.60666	−	0.927604i	0.616545	−	0.787319i	$-0.288532\pi$
$661$	1062.00	−	613.146i	1.60666	−	0.927604i	0.990111	−	0.140284i	$-0.0448016\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−162.000	+	280.592i	−0.242879	+	0.420678i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	374.123i		−	0.557560i
$672$		0			0
$673$	706.000			1.04903			0.524517	−	0.851400i	$-0.324246\pi$
$673$	706.000			1.04903			0.524517	+	0.851400i	$0.324246\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−162.000	−	93.5307i	−0.239291	−	0.138155i	0.375560	−	0.926798i	$-0.377450\pi$
$677$	−162.000	−	93.5307i	−0.239291	−	0.138155i	−0.614851	+	0.788643i	$0.710784\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−279.000	−	483.242i	−0.408492	−	0.707529i	0.586229	−	0.810145i	$-0.300612\pi$
$683$	−279.000	−	483.242i	−0.408492	−	0.707529i	−0.994721	+	0.102617i	$0.967279\pi$
$684$		0			0
$685$			1122.37i			1.63849i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−324.000	+	187.061i	−0.470247	+	0.271497i
$690$		0			0
$691$	522.000	+	301.377i	0.755427	+	0.436146i	0.827651	−	0.561242i	$-0.189676\pi$
$691$	522.000	+	301.377i	0.755427	+	0.436146i	−0.0722245	+	0.997388i	$0.523010\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	216.000	−	374.123i	0.310791	−	0.538306i
$696$		0			0
$697$	−384.000	−	665.108i	−0.550933	−	0.954243i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	414.000			0.590585			0.295292	−	0.955407i	$-0.404583\pi$
$701$	414.000			0.590585			0.295292	+	0.955407i	$0.404583\pi$
$702$		0			0
$703$	−180.000	+	103.923i	−0.256046	+	0.147828i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	91.0000	−	157.617i	0.128350	−	0.222308i	−0.794688	−	0.607019i	$-0.792365\pi$
$709$	91.0000	−	157.617i	0.128350	−	0.222308i	0.923037	+	0.384710i	$0.125699\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			748.246i			1.04943i
$714$		0			0
$715$	−2592.00			−3.62517
$716$		0			0
$717$		0			0
$718$		0			0
$719$	984.000	+	568.113i	1.36857	+	0.790143i	0.990745	−	0.135735i	$-0.0433397\pi$
$719$	984.000	+	568.113i	1.36857	+	0.790143i	0.377822	+	0.925878i	$0.376673\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−207.000	−	358.535i	−0.285517	−	0.494530i
$726$		0			0
$727$		−	1247.08i		−	1.71537i	−0.514173	−	0.857687i	$-0.671901\pi$
$727$		−	1247.08i		−	1.71537i	0.514173	−	0.857687i	$-0.328099\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−456.000	+	263.272i	−0.623803	+	0.360153i
$732$		0			0
$733$	1206.00	+	696.284i	1.64529	+	0.949911i	0.978908	+	0.204303i	$0.0654929\pi$
$733$	1206.00	+	696.284i	1.64529	+	0.949911i	0.666386	+	0.745607i	$0.267840\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	234.000	−	405.300i	0.317503	−	0.549932i
$738$		0			0
$739$	611.000	+	1058.28i	0.826793	+	1.43205i	0.900541	+	0.434771i	$0.143171\pi$
$739$	611.000	+	1058.28i	0.826793	+	1.43205i	−0.0737483	+	0.997277i	$0.523496\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	270.000			0.363392			0.181696	−	0.983355i	$-0.441841\pi$
$743$	270.000			0.363392			0.181696	+	0.983355i	$0.441841\pi$
$744$		0			0
$745$	972.000	−	561.184i	1.30470	−	0.753268i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	103.000	−	178.401i	0.137150	−	0.237552i	−0.789266	−	0.614051i	$-0.789539\pi$
$751$	103.000	−	178.401i	0.137150	−	0.237552i	0.926417	+	0.376499i	$0.122872\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1759.76i			2.33081i
$756$		0			0
$757$	1162.00			1.53501			0.767503	−	0.641045i	$-0.221499\pi$
$757$	1162.00			1.53501			0.767503	+	0.641045i	$0.221499\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	480.000	+	277.128i	0.630749	+	0.364163i	0.781042	−	0.624478i	$-0.214688\pi$
$761$	480.000	+	277.128i	0.630749	+	0.364163i	−0.150293	+	0.988641i	$0.548022\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	72.0000	+	124.708i	0.0938722	+	0.162591i
$768$		0			0
$769$		−	1330.22i		−	1.72980i	−0.501946	−	0.864899i	$-0.667382\pi$
$769$		−	1330.22i		−	1.72980i	0.501946	−	0.864899i	$-0.332618\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1146.00	−	661.643i	1.48254	−	0.855942i	0.482732	−	0.875768i	$-0.339644\pi$
$773$	1146.00	−	661.643i	1.48254	−	0.855942i	0.999803	+	0.0198257i	$0.00631114\pi$
$774$		0			0
$775$	−828.000	−	478.046i	−1.06839	−	0.616834i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	576.000	−	997.661i	0.739409	−	1.28069i
$780$		0			0
$781$	−162.000	−	280.592i	−0.207426	−	0.359273i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	144.000			0.183439
$786$		0			0
$787$	−666.000	+	384.515i	−0.846252	+	0.488584i	−0.859384	−	0.511330i	$-0.829153\pi$
$787$	−666.000	+	384.515i	−0.846252	+	0.488584i	0.0131328	+	0.999914i	$0.495820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	216.000	−	374.123i	0.272383	−	0.471782i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	187.061i		−	0.234707i	−0.993090	−	0.117354i	$-0.962559\pi$
$797$		−	187.061i		−	0.234707i	0.993090	−	0.117354i	$-0.0374411\pi$
$798$		0			0
$799$	384.000			0.480601
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−648.000	−	374.123i	−0.806974	−	0.465907i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	225.000	+	389.711i	0.278121	+	0.481720i	0.970918	−	0.239413i	$-0.0769550\pi$
$809$	225.000	+	389.711i	0.278121	+	0.481720i	−0.692797	+	0.721133i	$0.743622\pi$
$810$		0			0
$811$			852.169i			1.05076i	0.850867	+	0.525382i	$0.176077\pi$
$811$			852.169i			1.05076i	−0.850867	+	0.525382i	$0.823923\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−156.000	+	90.0666i	−0.191411	+	0.110511i
$816$		0			0
$817$	−684.000	−	394.908i	−0.837209	−	0.483363i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−423.000	+	732.657i	−0.515225	+	0.892396i	0.484619	+	0.874726i	$0.338958\pi$
$821$	−423.000	+	732.657i	−0.515225	+	0.892396i	−0.999844	+	0.0176709i	$0.994375\pi$
$822$		0			0
$823$	695.000	+	1203.78i	0.844471	+	1.46267i	0.886079	+	0.463533i	$0.153419\pi$
$823$	695.000	+	1203.78i	0.844471	+	1.46267i	−0.0416080	+	0.999134i	$0.513248\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1314.00			−1.58888			−0.794438	−	0.607346i	$-0.792234\pi$
$827$	−1314.00			−1.58888			−0.794438	+	0.607346i	$0.792234\pi$
$828$		0			0
$829$	−414.000	+	239.023i	−0.499397	+	0.288327i	−0.728464	−	0.685084i	$-0.759766\pi$
$829$	−414.000	+	239.023i	−0.499397	+	0.288327i	0.229068	+	0.973411i	$0.426432\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−624.000	+	1080.80i	−0.747305	+	1.29437i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	900.666i		−	1.07350i	−0.843741	−	0.536750i	$-0.819652\pi$
$839$		−	900.666i		−	1.07350i	0.843741	−	0.536750i	$-0.180348\pi$
$840$		0			0
$841$	−517.000			−0.614744
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1578.00	−	911.059i	−1.86746	−	1.07818i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	90.0000	+	155.885i	0.105758	+	0.183178i
$852$		0			0
$853$		−	1517.28i		−	1.77875i	−0.457175	−	0.889377i	$-0.651139\pi$
$853$		−	1517.28i		−	1.77875i	0.457175	−	0.889377i	$-0.348861\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−672.000	+	387.979i	−0.784131	+	0.452718i	−0.837892	−	0.545836i	$-0.816212\pi$
$857$	−672.000	+	387.979i	−0.784131	+	0.452718i	0.0537615	+	0.998554i	$0.482879\pi$
$858$		0			0
$859$	−1026.00	−	592.361i	−1.19441	−	0.689594i	−0.235108	−	0.971969i	$-0.575544\pi$
$859$	−1026.00	−	592.361i	−1.19441	−	0.689594i	−0.959304	+	0.282375i	$0.908878\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−279.000	+	483.242i	−0.323291	+	0.559956i	−0.981165	−	0.193172i	$-0.938123\pi$
$863$	−279.000	+	483.242i	−0.323291	+	0.559956i	0.657874	+	0.753128i	$0.271456\pi$
$864$		0			0
$865$	648.000	+	1122.37i	0.749133	+	1.29754i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−36.0000			−0.0414269
$870$		0			0
$871$	468.000	−	270.200i	0.537313	−	0.310218i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	587.000	−	1016.71i	0.669327	−	1.15931i	−0.308765	−	0.951138i	$-0.599916\pi$
$877$	587.000	−	1016.71i	0.669327	−	1.15931i	0.978093	−	0.208171i	$-0.0667509\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1122.37i		−	1.27397i	−0.770876	−	0.636986i	$-0.780181\pi$
$881$		−	1122.37i		−	1.27397i	0.770876	−	0.636986i	$-0.219819\pi$
$882$		0			0
$883$	1450.00			1.64213			0.821065	−	0.570835i	$-0.193381\pi$
$883$	1450.00			1.64213			0.821065	+	0.570835i	$0.193381\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	60.0000	+	34.6410i	0.0676437	+	0.0390541i	0.533440	−	0.845838i	$-0.320899\pi$
$887$	60.0000	+	34.6410i	0.0676437	+	0.0390541i	−0.465797	+	0.884892i	$0.654232\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	288.000	+	498.831i	0.322508	+	0.558601i
$894$		0			0
$895$			2120.03i			2.36875i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	648.000	−	374.123i	0.720801	−	0.416155i
$900$		0			0
$901$	216.000	+	124.708i	0.239734	+	0.138410i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	648.000	−	1122.37i	0.716022	−	1.24019i
$906$		0			0
$907$	−413.000	−	715.337i	−0.455347	−	0.788685i	0.543361	−	0.839499i	$-0.317152\pi$
$907$	−413.000	−	715.337i	−0.455347	−	0.788685i	−0.998708	+	0.0508146i	$0.983818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−594.000			−0.652031			−0.326015	−	0.945364i	$-0.605706\pi$
$911$	−594.000			−0.652031			−0.326015	+	0.945364i	$0.605706\pi$
$912$		0			0
$913$	540.000	−	311.769i	0.591457	−	0.341478i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−41.0000	+	71.0141i	−0.0446137	+	0.0772732i	−0.887470	−	0.460866i	$-0.847539\pi$
$919$	−41.0000	+	71.0141i	−0.0446137	+	0.0772732i	0.842856	+	0.538139i	$0.180872\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	374.123i		−	0.405334i
$924$		0			0
$925$	−230.000			−0.248649
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−732.000	−	422.620i	−0.787944	−	0.454920i	0.0512942	−	0.998684i	$-0.483665\pi$
$929$	−732.000	−	422.620i	−0.787944	−	0.454920i	−0.839238	+	0.543764i	$0.816999\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	864.000	+	1496.49i	0.924064	+	1.60053i
$936$		0			0
$937$		−	872.954i		−	0.931647i	−0.884878	−	0.465824i	$-0.845758\pi$
$937$		−	872.954i		−	0.931647i	0.884878	−	0.465824i	$-0.154242\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−426.000	+	245.951i	−0.452710	+	0.261372i	−0.708974	−	0.705235i	$-0.750842\pi$
$941$	−426.000	+	245.951i	−0.452710	+	0.261372i	0.256264	+	0.966607i	$0.417508\pi$
$942$		0			0
$943$	−864.000	−	498.831i	−0.916225	−	0.528983i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−855.000	+	1480.90i	−0.902851	+	1.56378i	−0.0790750	+	0.996869i	$0.525197\pi$
$947$	−855.000	+	1480.90i	−0.902851	+	1.56378i	−0.823776	+	0.566915i	$0.808137\pi$
$948$		0			0
$949$	−432.000	−	748.246i	−0.455216	−	0.788457i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	126.000			0.132214			0.0661070	−	0.997813i	$-0.478942\pi$
$953$	126.000			0.132214			0.0661070	+	0.997813i	$0.478942\pi$
$954$		0			0
$955$	−1620.00	+	935.307i	−1.69634	+	0.979380i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	383.500	−	664.241i	0.399063	−	0.691198i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1898.33i		−	1.96718i
$966$		0			0
$967$	−766.000			−0.792141			−0.396070	−	0.918220i	$-0.629626\pi$
$967$	−766.000			−0.792141			−0.396070	+	0.918220i	$0.629626\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1362.00	+	786.351i	1.40268	+	0.809836i	0.994667	−	0.103141i	$-0.0328892\pi$
$971$	1362.00	+	786.351i	1.40268	+	0.809836i	0.408011	+	0.912977i	$0.366223\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	801.000	+	1387.37i	0.819857	+	1.42003i	0.905787	+	0.423732i	$0.139280\pi$
$977$	801.000	+	1387.37i	0.819857	+	1.42003i	−0.0859307	+	0.996301i	$0.527386\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−732.000	+	422.620i	−0.744659	+	0.429929i	−0.823761	−	0.566937i	$-0.808128\pi$
$983$	−732.000	+	422.620i	−0.744659	+	0.429929i	0.0791017	+	0.996867i	$0.474795\pi$
$984$		0			0
$985$	−972.000	−	561.184i	−0.986802	−	0.569730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−342.000	+	592.361i	−0.345804	+	0.598950i
$990$		0			0
$991$	31.0000	+	53.6936i	0.0312815	+	0.0541812i	0.881242	−	0.472665i	$-0.156708\pi$
$991$	31.0000	+	53.6936i	0.0312815	+	0.0541812i	−0.849961	+	0.526846i	$0.823374\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	576.000			0.578894
$996$		0			0
$997$	54.0000	−	31.1769i	0.0541625	−	0.0312707i	−0.472674	−	0.881237i	$-0.656711\pi$
$997$	54.0000	−	31.1769i	0.0541625	−	0.0312707i	0.526837	+	0.849967i	$0.323378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.g.901.1		2
3.2	odd	2		588.3.m.c.313.1		2
7.2	even	3		252.3.d.a.181.1		2
7.3	odd	6	inner	1764.3.z.g.325.1		2
7.4	even	3		1764.3.z.a.325.1		2
7.5	odd	6		252.3.d.a.181.2		2
7.6	odd	2		1764.3.z.a.901.1		2
21.2	odd	6		84.3.d.a.13.2	yes	2
21.5	even	6		84.3.d.a.13.1	✓	2
21.11	odd	6		588.3.m.b.325.1		2
21.17	even	6		588.3.m.c.325.1		2
21.20	even	2		588.3.m.b.313.1		2
28.19	even	6		1008.3.f.f.433.2		2
28.23	odd	6		1008.3.f.f.433.1		2
84.23	even	6		336.3.f.b.97.1		2
84.47	odd	6		336.3.f.b.97.2		2
105.2	even	12		2100.3.p.b.349.3		4
105.23	even	12		2100.3.p.b.349.2		4
105.44	odd	6		2100.3.j.c.601.1		2
105.47	odd	12		2100.3.p.b.349.1		4
105.68	odd	12		2100.3.p.b.349.4		4
105.89	even	6		2100.3.j.c.601.2		2
168.5	even	6		1344.3.f.a.769.2		2
168.107	even	6		1344.3.f.d.769.2		2
168.131	odd	6		1344.3.f.d.769.1		2
168.149	odd	6		1344.3.f.a.769.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.d.a.13.1	✓	2	21.5	even	6
84.3.d.a.13.2	yes	2	21.2	odd	6
252.3.d.a.181.1		2	7.2	even	3
252.3.d.a.181.2		2	7.5	odd	6
336.3.f.b.97.1		2	84.23	even	6
336.3.f.b.97.2		2	84.47	odd	6
588.3.m.b.313.1		2	21.20	even	2
588.3.m.b.325.1		2	21.11	odd	6
588.3.m.c.313.1		2	3.2	odd	2
588.3.m.c.325.1		2	21.17	even	6
1008.3.f.f.433.1		2	28.23	odd	6
1008.3.f.f.433.2		2	28.19	even	6
1344.3.f.a.769.1		2	168.149	odd	6
1344.3.f.a.769.2		2	168.5	even	6
1344.3.f.d.769.1		2	168.131	odd	6
1344.3.f.d.769.2		2	168.107	even	6
1764.3.z.a.325.1		2	7.4	even	3
1764.3.z.a.901.1		2	7.6	odd	2
1764.3.z.g.325.1		2	7.3	odd	6	inner
1764.3.z.g.901.1		2	1.1	even	1	trivial
2100.3.j.c.601.1		2	105.44	odd	6
2100.3.j.c.601.2		2	105.89	even	6
2100.3.p.b.349.1		4	105.47	odd	12
2100.3.p.b.349.2		4	105.23	even	12
2100.3.p.b.349.3		4	105.2	even	12
2100.3.p.b.349.4		4	105.68	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.00000 + 3.46410i) q^{5} +(9.00000 + 15.5885i) q^{11} +20.7846i q^{13} +(12.0000 - 6.92820i) q^{17} +(18.0000 + 10.3923i) q^{19} +(9.00000 - 15.5885i) q^{23} +(11.5000 + 19.9186i) q^{25} -18.0000 q^{29} +(-36.0000 + 20.7846i) q^{31} +(-5.00000 + 8.66025i) q^{37} -55.4256i q^{41} -38.0000 q^{43} +(24.0000 + 13.8564i) q^{47} +(9.00000 + 15.5885i) q^{53} +124.708i q^{55} +(6.00000 - 3.46410i) q^{59} +(-18.0000 - 10.3923i) q^{61} +(-72.0000 + 124.708i) q^{65} +(-13.0000 - 22.5167i) q^{67} -18.0000 q^{71} +(-36.0000 + 20.7846i) q^{73} +(-1.00000 + 1.73205i) q^{79} -34.6410i q^{83} +96.0000 q^{85} +(72.0000 + 124.708i) q^{95} -166.277i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} + 18 q^{11} + 24 q^{17} + 36 q^{19} + 18 q^{23} + 23 q^{25} - 36 q^{29} - 72 q^{31} - 10 q^{37} - 76 q^{43} + 48 q^{47} + 18 q^{53} + 12 q^{59} - 36 q^{61} - 144 q^{65} - 26 q^{67} - 36 q^{71}+ \cdots + 144 q^{95}+O(q^{100}) \) 2 * q + 12 * q^5 + 18 * q^11 + 24 * q^17 + 36 * q^19 + 18 * q^23 + 23 * q^25 - 36 * q^29 - 72 * q^31 - 10 * q^37 - 76 * q^43 + 48 * q^47 + 18 * q^53 + 12 * q^59 - 36 * q^61 - 144 * q^65 - 26 * q^67 - 36 * q^71 - 72 * q^73 - 2 * q^79 + 192 * q^85 + 144 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more