LMFDB - Embedded newform 1764.3.z.e.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)

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);

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")

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 = [2,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,-30,0,-18] 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:	$\Z[a_1, \ldots, a_{5}]$
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.e.325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 0.866025i) q^{5} +(-1.50000 - 2.59808i) q^{11} +6.92820i q^{13} +(-15.0000 + 8.66025i) q^{17} +(-9.00000 - 5.19615i) q^{19} +(18.0000 - 31.1769i) q^{23} +(-11.0000 - 19.0526i) q^{25} +51.0000 q^{29} +(10.5000 - 6.06218i) q^{31} +(-11.0000 + 19.0526i) q^{37} -24.2487i q^{41} +10.0000 q^{43} +(78.0000 + 45.0333i) q^{47} +(25.5000 + 44.1673i) q^{53} -5.19615i q^{55} +(64.5000 - 37.2391i) q^{59} +(-60.0000 - 34.6410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(-34.0000 - 58.8897i) q^{67} +(18.0000 - 10.3923i) q^{73} +(-62.5000 + 108.253i) q^{79} -154.153i q^{83} -30.0000 q^{85} +(63.0000 + 36.3731i) q^{89} +(-9.00000 - 15.5885i) q^{95} -147.224i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 3 q^{11} - 30 q^{17} - 18 q^{19} + 36 q^{23} - 22 q^{25} + 102 q^{29} + 21 q^{31} - 22 q^{37} + 20 q^{43} + 156 q^{47} + 51 q^{53} + 129 q^{59} - 120 q^{61} - 12 q^{65} - 68 q^{67} + 36 q^{73}+ \cdots - 18 q^{95}+O(q^{100}) $ 2 * q + 3 * q^5 - 3 * q^11 - 30 * q^17 - 18 * q^19 + 36 * q^23 - 22 * q^25 + 102 * q^29 + 21 * q^31 - 22 * q^37 + 20 * q^43 + 156 * q^47 + 51 * q^53 + 129 * q^59 - 120 * q^61 - 12 * q^65 - 68 * q^67 + 36 * q^73 - 125 * q^79 - 60 * q^85 + 126 * q^89 - 18 * 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
$5$	1.50000	+	0.866025i	0.300000	+	0.173205i	0.642443	−	0.766334i	$-0.277921\pi$
$5$	1.50000	+	0.866025i	0.300000	+	0.173205i	−0.342443	+	0.939539i	$0.611254\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.136364	−	0.236189i	0.789754	−	0.613424i	$-0.210208\pi$
$11$	−1.50000	−	2.59808i	−0.136364	−	0.236189i	−0.926118	+	0.377235i	$0.876875\pi$
$12$		0			0
$13$			6.92820i			0.532939i	0.963843	+	0.266469i	$0.0858571\pi$
$13$			6.92820i			0.532939i	−0.963843	+	0.266469i	$0.914143\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.0000	+	8.66025i	−0.882353	+	0.509427i	−0.871433	−	0.490514i	$-0.836809\pi$
$17$	−15.0000	+	8.66025i	−0.882353	+	0.509427i	−0.0109194	+	0.999940i	$0.503476\pi$
$18$		0			0
$19$	−9.00000	−	5.19615i	−0.473684	−	0.273482i	0.244096	−	0.969751i	$-0.421509\pi$
$19$	−9.00000	−	5.19615i	−0.473684	−	0.273482i	−0.717781	+	0.696269i	$0.754842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	18.0000	−	31.1769i	0.782609	−	1.35552i	−0.147809	−	0.989016i	$-0.547222\pi$
$23$	18.0000	−	31.1769i	0.782609	−	1.35552i	0.930417	−	0.366502i	$-0.119445\pi$
$24$		0			0
$25$	−11.0000	−	19.0526i	−0.440000	−	0.762102i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	51.0000			1.75862			0.879310	−	0.476249i	$-0.158004\pi$
$29$	51.0000			1.75862			0.879310	+	0.476249i	$0.158004\pi$
$30$		0			0
$31$	10.5000	−	6.06218i	0.338710	−	0.195554i	−0.320992	−	0.947082i	$-0.604016\pi$
$31$	10.5000	−	6.06218i	0.338710	−	0.195554i	0.659701	+	0.751528i	$0.270683\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.0000	+	19.0526i	−0.297297	+	0.514934i	−0.975517	−	0.219925i	$-0.929419\pi$
$37$	−11.0000	+	19.0526i	−0.297297	+	0.514934i	0.678219	+	0.734859i	$0.262752\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	24.2487i		−	0.591432i	−0.955276	−	0.295716i	$-0.904442\pi$
$41$		−	24.2487i		−	0.591432i	0.955276	−	0.295716i	$-0.0955582\pi$
$42$		0			0
$43$	10.0000			0.232558			0.116279	−	0.993217i	$-0.462903\pi$
$43$	10.0000			0.232558			0.116279	+	0.993217i	$0.462903\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	78.0000	+	45.0333i	1.65957	+	0.958156i	0.972911	+	0.231180i	$0.0742588\pi$
$47$	78.0000	+	45.0333i	1.65957	+	0.958156i	0.686664	+	0.726975i	$0.259075\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	25.5000	+	44.1673i	0.481132	+	0.833345i	0.999766	−	0.0216515i	$-0.00689241\pi$
$53$	25.5000	+	44.1673i	0.481132	+	0.833345i	−0.518634	+	0.854997i	$0.673559\pi$
$54$		0			0
$55$		−	5.19615i		−	0.0944755i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	64.5000	−	37.2391i	1.09322	−	0.631171i	0.158788	−	0.987313i	$-0.449241\pi$
$59$	64.5000	−	37.2391i	1.09322	−	0.631171i	0.934432	+	0.356142i	$0.115908\pi$
$60$		0			0
$61$	−60.0000	−	34.6410i	−0.983607	−	0.567886i	−0.0802495	−	0.996775i	$-0.525572\pi$
$61$	−60.0000	−	34.6410i	−0.983607	−	0.567886i	−0.903357	+	0.428889i	$0.858905\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	+	10.3923i	−0.0923077	+	0.159882i
$66$		0			0
$67$	−34.0000	−	58.8897i	−0.507463	−	0.878951i	−0.999963	−	0.00863871i	$-0.997250\pi$
$67$	−34.0000	−	58.8897i	−0.507463	−	0.878951i	0.492500	−	0.870312i	$-0.336083\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	18.0000	−	10.3923i	0.246575	−	0.142360i	−0.371620	−	0.928385i	$-0.621197\pi$
$73$	18.0000	−	10.3923i	0.246575	−	0.142360i	0.618195	+	0.786025i	$0.287864\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−62.5000	+	108.253i	−0.791139	+	1.37029i	0.134123	+	0.990965i	$0.457178\pi$
$79$	−62.5000	+	108.253i	−0.791139	+	1.37029i	−0.925262	+	0.379329i	$0.876155\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	154.153i		−	1.85726i	−0.371008	−	0.928630i	$-0.620988\pi$
$83$		−	154.153i		−	1.85726i	0.371008	−	0.928630i	$-0.379012\pi$
$84$		0			0
$85$	−30.0000			−0.352941
$86$		0			0
$87$		0			0
$88$		0			0
$89$	63.0000	+	36.3731i	0.707865	+	0.408686i	0.810270	−	0.586057i	$-0.199320\pi$
$89$	63.0000	+	36.3731i	0.707865	+	0.408686i	−0.102405	+	0.994743i	$0.532654\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.00000	−	15.5885i	−0.0947368	−	0.164089i
$96$		0			0
$97$		−	147.224i		−	1.51778i	−0.651221	−	0.758888i	$-0.725743\pi$
$97$		−	147.224i		−	1.51778i	0.651221	−	0.758888i	$-0.274257\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000	−	10.3923i	0.178218	−	0.102894i	−0.408237	−	0.912876i	$-0.633856\pi$
$101$	18.0000	−	10.3923i	0.178218	−	0.102894i	0.586455	+	0.809982i	$0.300523\pi$
$102$		0			0
$103$	132.000	+	76.2102i	1.28155	+	0.739905i	0.977132	−	0.212632i	$-0.0682037\pi$
$103$	132.000	+	76.2102i	1.28155	+	0.739905i	0.304421	+	0.952538i	$0.401537\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.5000	+	28.5788i	−0.154206	+	0.267092i	−0.932769	−	0.360473i	$-0.882615\pi$
$107$	−16.5000	+	28.5788i	−0.154206	+	0.267092i	0.778564	+	0.627565i	$0.215948\pi$
$108$		0			0
$109$	16.0000	+	27.7128i	0.146789	+	0.254246i	0.930039	−	0.367461i	$-0.119773\pi$
$109$	16.0000	+	27.7128i	0.146789	+	0.254246i	−0.783250	+	0.621707i	$0.786440\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	186.000			1.64602			0.823009	−	0.568029i	$-0.192294\pi$
$113$	186.000			1.64602			0.823009	+	0.568029i	$0.192294\pi$
$114$		0			0
$115$	54.0000	−	31.1769i	0.469565	−	0.271104i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	56.0000	−	96.9948i	0.462810	−	0.801610i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	81.4064i		−	0.651251i
$126$		0			0
$127$	131.000			1.03150			0.515748	−	0.856740i	$-0.327514\pi$
$127$	131.000			1.03150			0.515748	+	0.856740i	$0.327514\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	85.5000	+	49.3634i	0.652672	+	0.376820i	0.789479	−	0.613777i	$-0.210351\pi$
$131$	85.5000	+	49.3634i	0.652672	+	0.376820i	−0.136807	+	0.990598i	$0.543684\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−72.0000	−	124.708i	−0.525547	−	0.910275i	−0.999557	−	0.0297553i	$-0.990527\pi$
$137$	−72.0000	−	124.708i	−0.525547	−	0.910275i	0.474010	−	0.880520i	$-0.342806\pi$
$138$		0			0
$139$			148.956i			1.07163i	0.844336	+	0.535814i	$0.179995\pi$
$139$			148.956i			1.07163i	−0.844336	+	0.535814i	$0.820005\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.0000	−	10.3923i	0.125874	−	0.0726735i
$144$		0			0
$145$	76.5000	+	44.1673i	0.527586	+	0.304602i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−75.0000	+	129.904i	−0.503356	+	0.871838i	0.496637	+	0.867958i	$0.334568\pi$
$149$	−75.0000	+	129.904i	−0.503356	+	0.871838i	−0.999992	+	0.00387918i	$0.998765\pi$
$150$		0			0
$151$	116.500	+	201.784i	0.771523	+	1.33632i	0.936728	+	0.350058i	$0.113838\pi$
$151$	116.500	+	201.784i	0.771523	+	1.33632i	−0.165205	+	0.986259i	$0.552829\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	21.0000			0.135484
$156$		0			0
$157$	−126.000	+	72.7461i	−0.802548	+	0.463351i	−0.844361	−	0.535774i	$-0.820020\pi$
$157$	−126.000	+	72.7461i	−0.802548	+	0.463351i	0.0418135	+	0.999125i	$0.486686\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	74.0000	−	128.172i	0.453988	−	0.786330i	−0.544642	−	0.838669i	$-0.683334\pi$
$163$	74.0000	−	128.172i	0.453988	−	0.786330i	0.998629	+	0.0523391i	$0.0166677\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	27.7128i		−	0.165945i	−0.996552	−	0.0829725i	$-0.973559\pi$
$167$		−	27.7128i		−	0.165945i	0.996552	−	0.0829725i	$-0.0264414\pi$
$168$		0			0
$169$	121.000			0.715976
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−162.000	−	93.5307i	−0.936416	−	0.540640i	−0.0475811	−	0.998867i	$-0.515151\pi$
$173$	−162.000	−	93.5307i	−0.936416	−	0.540640i	−0.888835	+	0.458227i	$0.848485\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−81.0000	−	140.296i	−0.452514	−	0.783777i	0.546028	−	0.837767i	$-0.316139\pi$
$179$	−81.0000	−	140.296i	−0.452514	−	0.783777i	−0.998541	+	0.0539901i	$0.982806\pi$
$180$		0			0
$181$		−	259.808i		−	1.43540i	−0.696352	−	0.717701i	$-0.745195\pi$
$181$		−	259.808i		−	1.43540i	0.696352	−	0.717701i	$-0.254805\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−33.0000	+	19.0526i	−0.178378	+	0.102987i
$186$		0			0
$187$	45.0000	+	25.9808i	0.240642	+	0.138935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	48.0000	−	83.1384i	0.251309	−	0.435280i	−0.712578	−	0.701593i	$-0.752472\pi$
$191$	48.0000	−	83.1384i	0.251309	−	0.435280i	0.963886	+	0.266314i	$0.0858057\pi$
$192$		0			0
$193$	−147.500	−	255.477i	−0.764249	−	1.32372i	−0.940643	−	0.339398i	$-0.889777\pi$
$193$	−147.500	−	255.477i	−0.764249	−	1.32372i	0.176394	−	0.984320i	$-0.443557\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	330.000			1.67513			0.837563	−	0.546340i	$-0.183979\pi$
$197$	330.000			1.67513			0.837563	+	0.546340i	$0.183979\pi$
$198$		0			0
$199$	102.000	−	58.8897i	0.512563	−	0.295928i	−0.221324	−	0.975200i	$-0.571038\pi$
$199$	102.000	−	58.8897i	0.512563	−	0.295928i	0.733886	+	0.679272i	$0.237704\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	21.0000	−	36.3731i	0.102439	−	0.177430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			31.1769i			0.149172i
$210$		0			0
$211$	220.000			1.04265			0.521327	−	0.853357i	$-0.325437\pi$
$211$	220.000			1.04265			0.521327	+	0.853357i	$0.325437\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.0000	+	8.66025i	0.0697674	+	0.0402803i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−60.0000	−	103.923i	−0.271493	−	0.470240i
$222$		0			0
$223$		−	150.688i		−	0.675733i	−0.941194	−	0.337866i	$-0.890295\pi$
$223$		−	150.688i		−	0.675733i	0.941194	−	0.337866i	$-0.109705\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	94.5000	−	54.5596i	0.416300	−	0.240351i	−0.277193	−	0.960814i	$-0.589404\pi$
$227$	94.5000	−	54.5596i	0.416300	−	0.240351i	0.693493	+	0.720464i	$0.256071\pi$
$228$		0			0
$229$	327.000	+	188.794i	1.42795	+	0.824426i	0.996959	−	0.0779313i	$-0.0248315\pi$
$229$	327.000	+	188.794i	1.42795	+	0.824426i	0.430989	+	0.902357i	$0.358165\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−105.000	+	181.865i	−0.450644	+	0.780538i	−0.998426	−	0.0560830i	$-0.982139\pi$
$233$	−105.000	+	181.865i	−0.450644	+	0.780538i	0.547782	+	0.836621i	$0.315472\pi$
$234$		0			0
$235$	78.0000	+	135.100i	0.331915	+	0.574893i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−120.000			−0.502092			−0.251046	−	0.967975i	$-0.580775\pi$
$239$	−120.000			−0.502092			−0.251046	+	0.967975i	$0.580775\pi$
$240$		0			0
$241$	10.5000	−	6.06218i	0.0435685	−	0.0251543i	−0.478058	−	0.878329i	$-0.658659\pi$
$241$	10.5000	−	6.06218i	0.0435685	−	0.0251543i	0.521626	+	0.853174i	$0.325326\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	36.0000	−	62.3538i	0.145749	−	0.252445i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	136.832i		−	0.545147i	−0.962135	−	0.272574i	$-0.912125\pi$
$251$		−	136.832i		−	0.545147i	0.962135	−	0.272574i	$-0.0878749\pi$
$252$		0			0
$253$	−108.000			−0.426877
$254$		0			0
$255$		0			0
$256$		0			0
$257$	117.000	+	67.5500i	0.455253	+	0.262840i	0.710046	−	0.704155i	$-0.248674\pi$
$257$	117.000	+	67.5500i	0.455253	+	0.262840i	−0.254793	+	0.966996i	$0.582007\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	81.0000	+	140.296i	0.307985	+	0.533445i	0.977921	−	0.208973i	$-0.0670121\pi$
$263$	81.0000	+	140.296i	0.307985	+	0.533445i	−0.669937	+	0.742418i	$0.733679\pi$
$264$		0			0
$265$			88.3346i			0.333338i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	334.500	−	193.124i	1.24349	−	0.717932i	0.273691	−	0.961818i	$-0.411756\pi$
$269$	334.500	−	193.124i	1.24349	−	0.717932i	0.969804	+	0.243886i	$0.0784223\pi$
$270$		0			0
$271$	−187.500	−	108.253i	−0.691882	−	0.399458i	0.112435	−	0.993659i	$-0.464135\pi$
$271$	−187.500	−	108.253i	−0.691882	−	0.399458i	−0.804317	+	0.594201i	$0.797468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−33.0000	+	57.1577i	−0.120000	+	0.207846i
$276$		0			0
$277$	50.0000	+	86.6025i	0.180505	+	0.312645i	0.942053	−	0.335465i	$-0.108893\pi$
$277$	50.0000	+	86.6025i	0.180505	+	0.312645i	−0.761547	+	0.648109i	$0.775560\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−228.000			−0.811388			−0.405694	−	0.914009i	$-0.632970\pi$
$281$	−228.000			−0.811388			−0.405694	+	0.914009i	$0.632970\pi$
$282$		0			0
$283$	−15.0000	+	8.66025i	−0.0530035	+	0.0306016i	−0.526268	−	0.850319i	$-0.676409\pi$
$283$	−15.0000	+	8.66025i	−0.0530035	+	0.0306016i	0.473264	+	0.880921i	$0.343076\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.50000	−	9.52628i	0.0190311	−	0.0329629i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	278.860i		−	0.951741i	−0.879515	−	0.475871i	$-0.842133\pi$
$293$		−	278.860i		−	0.951741i	0.879515	−	0.475871i	$-0.157867\pi$
$294$		0			0
$295$	129.000			0.437288
$296$		0			0
$297$		0			0
$298$		0			0
$299$	216.000	+	124.708i	0.722408	+	0.417082i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−60.0000	−	103.923i	−0.196721	−	0.340731i
$306$		0			0
$307$			27.7128i			0.0902697i	0.998981	+	0.0451349i	$0.0143718\pi$
$307$			27.7128i			0.0902697i	−0.998981	+	0.0451349i	$0.985628\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	57.0000	−	32.9090i	0.183280	−	0.105817i	−0.405553	−	0.914072i	$-0.632921\pi$
$311$	57.0000	−	32.9090i	0.183280	−	0.105817i	0.588833	+	0.808255i	$0.299588\pi$
$312$		0			0
$313$	−103.500	−	59.7558i	−0.330671	−	0.190913i	0.325468	−	0.945553i	$-0.394478\pi$
$313$	−103.500	−	59.7558i	−0.330671	−	0.190913i	−0.656139	+	0.754640i	$0.727811\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−100.500	+	174.071i	−0.317035	+	0.549120i	−0.979868	−	0.199647i	$-0.936021\pi$
$317$	−100.500	+	174.071i	−0.317035	+	0.549120i	0.662833	+	0.748767i	$0.269354\pi$
$318$		0			0
$319$	−76.5000	−	132.502i	−0.239812	−	0.415366i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	180.000			0.557276
$324$		0			0
$325$	132.000	−	76.2102i	0.406154	−	0.234493i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−224.000	+	387.979i	−0.676737	+	1.17214i	0.299221	+	0.954184i	$0.403273\pi$
$331$	−224.000	+	387.979i	−0.676737	+	1.17214i	−0.975958	+	0.217959i	$0.930060\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	117.779i		−	0.351580i
$336$		0			0
$337$	−293.000			−0.869436			−0.434718	−	0.900567i	$-0.643152\pi$
$337$	−293.000			−0.869436			−0.434718	+	0.900567i	$0.643152\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−31.5000	−	18.1865i	−0.0923754	−	0.0533329i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	195.000	+	337.750i	0.561960	+	0.973343i	0.997325	+	0.0730890i	$0.0232857\pi$
$347$	195.000	+	337.750i	0.561960	+	0.973343i	−0.435366	+	0.900254i	$0.643381\pi$
$348$		0			0
$349$		−	138.564i		−	0.397032i	−0.980098	−	0.198516i	$-0.936388\pi$
$349$		−	138.564i		−	0.397032i	0.980098	−	0.198516i	$-0.0636121\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−438.000	+	252.879i	−1.24079	+	0.716372i	−0.969256	−	0.246056i	$-0.920865\pi$
$353$	−438.000	+	252.879i	−1.24079	+	0.716372i	−0.271537	+	0.962428i	$0.587532\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	150.000	−	259.808i	0.417827	−	0.723698i	−0.577893	−	0.816112i	$-0.696125\pi$
$359$	150.000	−	259.808i	0.417827	−	0.723698i	0.995721	+	0.0924142i	$0.0294584\pi$
$360$		0			0
$361$	−126.500	−	219.104i	−0.350416	−	0.606937i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	36.0000			0.0986301
$366$		0			0
$367$	175.500	−	101.325i	0.478202	−	0.276090i	−0.241465	−	0.970410i	$-0.577628\pi$
$367$	175.500	−	101.325i	0.478202	−	0.276090i	0.719667	+	0.694320i	$0.244295\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−205.000	+	355.070i	−0.549598	+	0.951931i	0.448704	+	0.893680i	$0.351886\pi$
$373$	−205.000	+	355.070i	−0.549598	+	0.951931i	−0.998302	+	0.0582510i	$0.981448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			353.338i			0.937237i
$378$		0			0
$379$	−374.000			−0.986807			−0.493404	−	0.869800i	$-0.664247\pi$
$379$	−374.000			−0.986807			−0.493404	+	0.869800i	$0.664247\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−249.000	−	143.760i	−0.650131	−	0.375353i	0.138376	−	0.990380i	$-0.455812\pi$
$383$	−249.000	−	143.760i	−0.650131	−	0.375353i	−0.788506	+	0.615027i	$0.789145\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−69.0000	−	119.512i	−0.177378	−	0.307228i	0.763604	−	0.645685i	$-0.223428\pi$
$389$	−69.0000	−	119.512i	−0.177378	−	0.307228i	−0.940982	+	0.338458i	$0.890095\pi$
$390$		0			0
$391$			623.538i			1.59473i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−187.500	+	108.253i	−0.474684	+	0.274059i
$396$		0			0
$397$	−534.000	−	308.305i	−1.34509	−	0.776587i	−0.357539	−	0.933898i	$-0.616384\pi$
$397$	−534.000	−	308.305i	−1.34509	−	0.776587i	−0.987549	+	0.157311i	$0.949717\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	258.000	−	446.869i	0.643392	−	1.11439i	−0.341279	−	0.939962i	$-0.610860\pi$
$401$	258.000	−	446.869i	0.643392	−	1.11439i	0.984671	−	0.174425i	$-0.0558066\pi$
$402$		0			0
$403$	42.0000	+	72.7461i	0.104218	+	0.180511i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	66.0000			0.162162
$408$		0			0
$409$	−226.500	+	130.770i	−0.553790	+	0.319731i	−0.750649	−	0.660701i	$-0.770259\pi$
$409$	−226.500	+	130.770i	−0.553790	+	0.319731i	0.196859	+	0.980432i	$0.436926\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	133.500	−	231.229i	0.321687	−	0.557178i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			533.472i			1.27320i	0.771193	+	0.636601i	$0.219660\pi$
$419$			533.472i			1.27320i	−0.771193	+	0.636601i	$0.780340\pi$
$420$		0			0
$421$	−76.0000			−0.180523			−0.0902613	−	0.995918i	$-0.528770\pi$
$421$	−76.0000			−0.180523			−0.0902613	+	0.995918i	$0.528770\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	330.000	+	190.526i	0.776471	+	0.448296i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	303.000	+	524.811i	0.703016	+	1.21766i	0.967403	+	0.253243i	$0.0814972\pi$
$431$	303.000	+	524.811i	0.703016	+	1.21766i	−0.264386	+	0.964417i	$0.585169\pi$
$432$		0			0
$433$			34.6410i			0.0800023i	0.999200	+	0.0400012i	$0.0127362\pi$
$433$			34.6410i			0.0800023i	−0.999200	+	0.0400012i	$0.987264\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−324.000	+	187.061i	−0.741419	+	0.428058i
$438$		0			0
$439$	−85.5000	−	49.3634i	−0.194761	−	0.112445i	0.399449	−	0.916756i	$-0.369202\pi$
$439$	−85.5000	−	49.3634i	−0.194761	−	0.112445i	−0.594209	+	0.804310i	$0.702535\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	310.500	−	537.802i	0.700903	−	1.21400i	−0.267247	−	0.963628i	$-0.586114\pi$
$443$	310.500	−	537.802i	0.700903	−	1.21400i	0.968150	−	0.250371i	$-0.0805527\pi$
$444$		0			0
$445$	63.0000	+	109.119i	0.141573	+	0.245212i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−504.000			−1.12249			−0.561247	−	0.827648i	$-0.689678\pi$
$449$	−504.000			−1.12249			−0.561247	+	0.827648i	$0.689678\pi$
$450$		0			0
$451$	−63.0000	+	36.3731i	−0.139690	+	0.0806498i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	221.500	−	383.649i	0.484683	−	0.839495i	−0.515162	−	0.857093i	$-0.672268\pi$
$457$	221.500	−	383.649i	0.484683	−	0.839495i	0.999845	+	0.0175975i	$0.00560174\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	180.133i		−	0.390745i	−0.980729	−	0.195372i	$-0.937408\pi$
$461$		−	180.133i		−	0.390745i	0.980729	−	0.195372i	$-0.0625915\pi$
$462$		0			0
$463$	158.000			0.341253			0.170626	−	0.985336i	$-0.445421\pi$
$463$	158.000			0.341253			0.170626	+	0.985336i	$0.445421\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.0000	+	10.3923i	0.0385439	+	0.0222533i	0.519148	−	0.854684i	$-0.326249\pi$
$467$	18.0000	+	10.3923i	0.0385439	+	0.0222533i	−0.480604	+	0.876938i	$0.659583\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.0000	−	25.9808i	−0.0317125	−	0.0549276i
$474$		0			0
$475$			228.631i			0.481328i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	357.000	−	206.114i	0.745303	−	0.430301i	−0.0786914	−	0.996899i	$-0.525074\pi$
$479$	357.000	−	206.114i	0.745303	−	0.430301i	0.823994	+	0.566598i	$0.191741\pi$
$480$		0			0
$481$	−132.000	−	76.2102i	−0.274428	−	0.158441i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	127.500	−	220.836i	0.262887	−	0.455333i
$486$		0			0
$487$	−57.5000	−	99.5929i	−0.118070	−	0.204503i	0.800933	−	0.598754i	$-0.204337\pi$
$487$	−57.5000	−	99.5929i	−0.118070	−	0.204503i	−0.919003	+	0.394251i	$0.871004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−273.000			−0.556008			−0.278004	−	0.960580i	$-0.589673\pi$
$491$	−273.000			−0.556008			−0.278004	+	0.960580i	$0.589673\pi$
$492$		0			0
$493$	−765.000	+	441.673i	−1.55172	+	0.895888i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−233.000	+	403.568i	−0.466934	+	0.808753i	−0.999286	−	0.0377695i	$-0.987975\pi$
$499$	−233.000	+	403.568i	−0.466934	+	0.808753i	0.532353	+	0.846523i	$0.321308\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			723.997i			1.43936i	0.694307	+	0.719679i	$0.255711\pi$
$503$			723.997i			1.43936i	−0.694307	+	0.719679i	$0.744289\pi$
$504$		0			0
$505$	36.0000			0.0712871
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−307.500	−	177.535i	−0.604126	−	0.348792i	0.166537	−	0.986035i	$-0.446741\pi$
$509$	−307.500	−	177.535i	−0.604126	−	0.348792i	−0.770663	+	0.637243i	$0.780075\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	132.000	+	228.631i	0.256311	+	0.443943i
$516$		0			0
$517$		−	270.200i		−	0.522630i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−795.000	+	458.993i	−1.52591	+	0.880986i	−0.526384	+	0.850247i	$0.676453\pi$
$521$	−795.000	+	458.993i	−1.52591	+	0.880986i	−0.999527	+	0.0307387i	$0.990214\pi$
$522$		0			0
$523$	−420.000	−	242.487i	−0.803059	−	0.463646i	0.0414805	−	0.999139i	$-0.486793\pi$
$523$	−420.000	−	242.487i	−0.803059	−	0.463646i	−0.844540	+	0.535493i	$0.820126\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−105.000	+	181.865i	−0.199241	+	0.345096i
$528$		0			0
$529$	−383.500	−	664.241i	−0.724953	−	1.25565i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	168.000			0.315197
$534$		0			0
$535$	−49.5000	+	28.5788i	−0.0925234	+	0.0534184i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−31.0000	+	53.6936i	−0.0573013	+	0.0992488i	−0.893253	−	0.449554i	$-0.851583\pi$
$541$	−31.0000	+	53.6936i	−0.0573013	+	0.0992488i	0.835952	+	0.548803i	$0.184916\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			55.4256i			0.101698i
$546$		0			0
$547$	−802.000			−1.46618			−0.733090	−	0.680132i	$-0.761922\pi$
$547$	−802.000			−1.46618			−0.733090	+	0.680132i	$0.761922\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−459.000	−	265.004i	−0.833031	−	0.480951i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−55.5000	−	96.1288i	−0.0996409	−	0.172583i	0.811895	−	0.583803i	$-0.198436\pi$
$557$	−55.5000	−	96.1288i	−0.0996409	−	0.172583i	−0.911536	+	0.411220i	$0.865103\pi$
$558$		0			0
$559$			69.2820i			0.123939i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	166.500	−	96.1288i	0.295737	−	0.170744i	−0.344789	−	0.938680i	$-0.612050\pi$
$563$	166.500	−	96.1288i	0.295737	−	0.170744i	0.640526	+	0.767936i	$0.278716\pi$
$564$		0			0
$565$	279.000	+	161.081i	0.493805	+	0.285099i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−213.000	+	368.927i	−0.374341	+	0.648378i	−0.990228	−	0.139457i	$-0.955464\pi$
$569$	−213.000	+	368.927i	−0.374341	+	0.648378i	0.615887	+	0.787834i	$0.288798\pi$
$570$		0			0
$571$	290.000	+	502.295i	0.507881	+	0.879676i	0.999958	+	0.00912412i	$0.00290434\pi$
$571$	290.000	+	502.295i	0.507881	+	0.879676i	−0.492077	+	0.870551i	$0.663762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−792.000			−1.37739
$576$		0			0
$577$	835.500	−	482.376i	1.44801	−	0.836007i	0.449644	−	0.893208i	$-0.351551\pi$
$577$	835.500	−	482.376i	1.44801	−	0.836007i	0.998363	+	0.0572006i	$0.0182174\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	76.5000	−	132.502i	0.131218	−	0.227276i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1002.86i			1.70845i	0.519907	+	0.854223i	$0.325966\pi$
$587$			1002.86i			1.70845i	−0.519907	+	0.854223i	$0.674034\pi$
$588$		0			0
$589$	−126.000			−0.213922
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−762.000	−	439.941i	−1.28499	−	0.741890i	−0.307235	−	0.951634i	$-0.599404\pi$
$593$	−762.000	−	439.941i	−1.28499	−	0.741890i	−0.977756	+	0.209743i	$0.932737\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−78.0000	−	135.100i	−0.130217	−	0.225543i	0.793543	−	0.608514i	$-0.208234\pi$
$599$	−78.0000	−	135.100i	−0.130217	−	0.225543i	−0.923760	+	0.382972i	$0.874901\pi$
$600$		0			0
$601$			1124.10i			1.87038i	0.354141	+	0.935192i	$0.384773\pi$
$601$			1124.10i			1.87038i	−0.354141	+	0.935192i	$0.615227\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	168.000	−	96.9948i	0.277686	−	0.160322i
$606$		0			0
$607$	346.500	+	200.052i	0.570840	+	0.329575i	0.757485	−	0.652853i	$-0.226428\pi$
$607$	346.500	+	200.052i	0.570840	+	0.329575i	−0.186645	+	0.982427i	$0.559761\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−312.000	+	540.400i	−0.510638	+	0.884451i
$612$		0			0
$613$	428.000	+	741.318i	0.698206	+	1.20933i	0.969088	+	0.246715i	$0.0793512\pi$
$613$	428.000	+	741.318i	0.698206	+	1.20933i	−0.270883	+	0.962612i	$0.587315\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−444.000			−0.719611			−0.359806	−	0.933027i	$-0.617157\pi$
$617$	−444.000			−0.719611			−0.359806	+	0.933027i	$0.617157\pi$
$618$		0			0
$619$	222.000	−	128.172i	0.358643	−	0.207063i	−0.309842	−	0.950788i	$-0.600276\pi$
$619$	222.000	−	128.172i	0.358643	−	0.207063i	0.668485	+	0.743725i	$0.266943\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−204.500	+	354.204i	−0.327200	+	0.566727i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	381.051i		−	0.605805i
$630$		0			0
$631$	−85.0000			−0.134707			−0.0673534	−	0.997729i	$-0.521455\pi$
$631$	−85.0000			−0.134707			−0.0673534	+	0.997729i	$0.521455\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	196.500	+	113.449i	0.309449	+	0.178660i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−66.0000	−	114.315i	−0.102964	−	0.178339i	0.809940	−	0.586512i	$-0.199499\pi$
$641$	−66.0000	−	114.315i	−0.102964	−	0.178339i	−0.912905	+	0.408173i	$0.866166\pi$
$642$		0			0
$643$		−	713.605i		−	1.10981i	−0.831915	−	0.554903i	$-0.812755\pi$
$643$		−	713.605i		−	1.10981i	0.831915	−	0.554903i	$-0.187245\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	864.000	−	498.831i	1.33539	−	0.770990i	0.349274	−	0.937021i	$-0.386428\pi$
$647$	864.000	−	498.831i	1.33539	−	0.770990i	0.986121	+	0.166031i	$0.0530951\pi$
$648$		0			0
$649$	−193.500	−	111.717i	−0.298151	−	0.172138i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−187.500	+	324.760i	−0.287136	+	0.497335i	−0.973125	−	0.230278i	$-0.926037\pi$
$653$	−187.500	+	324.760i	−0.287136	+	0.497335i	0.685989	+	0.727612i	$0.259370\pi$
$654$		0			0
$655$	85.5000	+	148.090i	0.130534	+	0.226092i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1038.00			−1.57511			−0.787557	−	0.616242i	$-0.788654\pi$
$659$	−1038.00			−1.57511			−0.787557	+	0.616242i	$0.788654\pi$
$660$		0			0
$661$	−387.000	+	223.435i	−0.585477	+	0.338025i	−0.763307	−	0.646036i	$-0.776426\pi$
$661$	−387.000	+	223.435i	−0.585477	+	0.338025i	0.177830	+	0.984061i	$0.443092\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	918.000	−	1590.02i	1.37631	−	2.38384i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			207.846i			0.309756i
$672$		0			0
$673$	1093.00			1.62407			0.812036	−	0.583608i	$-0.198359\pi$
$673$	1093.00			1.62407			0.812036	+	0.583608i	$0.198359\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1048.50	−	605.352i	−1.54874	−	0.894168i	−0.998238	−	0.0593371i	$-0.981101\pi$
$677$	−1048.50	−	605.352i	−1.54874	−	0.894168i	−0.550506	−	0.834831i	$-0.685565\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.50000	−	2.59808i	−0.00219619	−	0.00380392i	0.864925	−	0.501901i	$-0.167366\pi$
$683$	−1.50000	−	2.59808i	−0.00219619	−	0.00380392i	−0.867121	+	0.498097i	$0.834032\pi$
$684$		0			0
$685$		−	249.415i		−	0.364110i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−306.000	+	176.669i	−0.444122	+	0.256414i
$690$		0			0
$691$	987.000	+	569.845i	1.42836	+	0.824667i	0.996992	−	0.0775070i	$-0.0246960\pi$
$691$	987.000	+	569.845i	1.42836	+	0.824667i	0.431373	+	0.902174i	$0.358029\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−129.000	+	223.435i	−0.185612	+	0.321489i
$696$		0			0
$697$	210.000	+	363.731i	0.301291	+	0.521852i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−447.000			−0.637660			−0.318830	−	0.947812i	$-0.603290\pi$
$701$	−447.000			−0.637660			−0.318830	+	0.947812i	$0.603290\pi$
$702$		0			0
$703$	198.000	−	114.315i	0.281650	−	0.162611i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	211.000	−	365.463i	0.297602	−	0.515462i	−0.677985	−	0.735076i	$-0.737146\pi$
$709$	211.000	−	365.463i	0.297602	−	0.515462i	0.975587	+	0.219614i	$0.0704797\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	436.477i		−	0.612169i
$714$		0			0
$715$	36.0000			0.0503497
$716$		0			0
$717$		0			0
$718$		0			0
$719$	147.000	+	84.8705i	0.204451	+	0.118040i	0.598730	−	0.800951i	$-0.295672\pi$
$719$	147.000	+	84.8705i	0.204451	+	0.118040i	−0.394279	+	0.918991i	$0.629006\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−561.000	−	971.681i	−0.773793	−	1.34025i
$726$		0			0
$727$			50.2295i			0.0690914i	0.999403	+	0.0345457i	$0.0109984\pi$
$727$			50.2295i			0.0690914i	−0.999403	+	0.0345457i	$0.989002\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−150.000	+	86.6025i	−0.205198	+	0.118471i
$732$		0			0
$733$	−255.000	−	147.224i	−0.347885	−	0.200852i	0.315868	−	0.948803i	$-0.397704\pi$
$733$	−255.000	−	147.224i	−0.347885	−	0.200852i	−0.663754	+	0.747951i	$0.731038\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−102.000	+	176.669i	−0.138399	+	0.239714i
$738$		0			0
$739$	−49.0000	−	84.8705i	−0.0663058	−	0.114845i	0.830967	−	0.556322i	$-0.187788\pi$
$739$	−49.0000	−	84.8705i	−0.0663058	−	0.114845i	−0.897272	+	0.441477i	$0.854455\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	180.000			0.242261			0.121131	−	0.992637i	$-0.461348\pi$
$743$	180.000			0.242261			0.121131	+	0.992637i	$0.461348\pi$
$744$		0			0
$745$	−225.000	+	129.904i	−0.302013	+	0.174368i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−360.500	+	624.404i	−0.480027	+	0.831431i	−0.999737	−	0.0229117i	$-0.992706\pi$
$751$	−360.500	+	624.404i	−0.480027	+	0.831431i	0.519711	+	0.854342i	$0.326040\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			403.568i			0.534527i
$756$		0			0
$757$	−644.000			−0.850727			−0.425363	−	0.905023i	$-0.639854\pi$
$757$	−644.000			−0.850727			−0.425363	+	0.905023i	$0.639854\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−492.000	−	284.056i	−0.646518	−	0.373267i	0.140603	−	0.990066i	$-0.455096\pi$
$761$	−492.000	−	284.056i	−0.646518	−	0.373267i	−0.787121	+	0.616799i	$0.788429\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	258.000	+	446.869i	0.336375	+	0.582619i
$768$		0			0
$769$		−	275.396i		−	0.358122i	−0.983838	−	0.179061i	$-0.942694\pi$
$769$		−	275.396i		−	0.358122i	0.983838	−	0.179061i	$-0.0573060\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−870.000	+	502.295i	−1.12549	+	0.649799i	−0.942796	−	0.333372i	$-0.891814\pi$
$773$	−870.000	+	502.295i	−1.12549	+	0.649799i	−0.182690	+	0.983171i	$0.558480\pi$
$774$		0			0
$775$	−231.000	−	133.368i	−0.298065	−	0.172088i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−126.000	+	218.238i	−0.161746	+	0.280152i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−252.000			−0.321019
$786$		0			0
$787$	−690.000	+	398.372i	−0.876747	+	0.506190i	−0.869585	−	0.493784i	$-0.835613\pi$
$787$	−690.000	+	398.372i	−0.876747	+	0.506190i	−0.00716264	+	0.999974i	$0.502280\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	240.000	−	415.692i	0.302648	−	0.524202i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			545.596i			0.684562i	0.939598	+	0.342281i	$0.111199\pi$
$797$			545.596i			0.684562i	−0.939598	+	0.342281i	$0.888801\pi$
$798$		0			0
$799$	−1560.00			−1.95244
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−54.0000	−	31.1769i	−0.0672478	−	0.0388255i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.0000	+	41.5692i	0.0296663	+	0.0513835i	0.880477	−	0.474088i	$-0.157222\pi$
$809$	24.0000	+	41.5692i	0.0296663	+	0.0513835i	−0.850811	+	0.525472i	$0.823889\pi$
$810$		0			0
$811$			436.477i			0.538196i	0.963113	+	0.269098i	$0.0867255\pi$
$811$			436.477i			0.538196i	−0.963113	+	0.269098i	$0.913274\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	222.000	−	128.172i	0.272393	−	0.157266i
$816$		0			0
$817$	−90.0000	−	51.9615i	−0.110159	−	0.0636004i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	595.500	−	1031.44i	0.725335	−	1.25632i	−0.233501	−	0.972357i	$-0.575018\pi$
$821$	595.500	−	1031.44i	0.725335	−	1.25632i	0.958836	−	0.283960i	$-0.0916484\pi$
$822$		0			0
$823$	125.000	+	216.506i	0.151883	+	0.263070i	0.931920	−	0.362664i	$-0.118133\pi$
$823$	125.000	+	216.506i	0.151883	+	0.263070i	−0.780036	+	0.625734i	$0.784800\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1137.00			1.37485			0.687424	−	0.726256i	$-0.258741\pi$
$827$	1137.00			1.37485			0.687424	+	0.726256i	$0.258741\pi$
$828$		0			0
$829$	−63.0000	+	36.3731i	−0.0759952	+	0.0438758i	−0.537516	−	0.843253i	$-0.680637\pi$
$829$	−63.0000	+	36.3731i	−0.0759952	+	0.0438758i	0.461521	+	0.887129i	$0.347304\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	24.0000	−	41.5692i	0.0287425	−	0.0497835i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1416.82i			1.68870i	0.535794	+	0.844349i	$0.320012\pi$
$839$			1416.82i			1.68870i	−0.535794	+	0.844349i	$0.679988\pi$
$840$		0			0
$841$	1760.00			2.09275
$842$		0			0
$843$		0			0
$844$		0			0
$845$	181.500	+	104.789i	0.214793	+	0.124011i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	396.000	+	685.892i	0.465335	+	0.805984i
$852$		0			0
$853$			872.954i			1.02339i	0.859166	+	0.511696i	$0.170983\pi$
$853$			872.954i			1.02339i	−0.859166	+	0.511696i	$0.829017\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−600.000	+	346.410i	−0.700117	+	0.404213i	−0.807391	−	0.590017i	$-0.799121\pi$
$857$	−600.000	+	346.410i	−0.700117	+	0.404213i	0.107274	+	0.994229i	$0.465788\pi$
$858$		0			0
$859$	771.000	+	445.137i	0.897555	+	0.518204i	0.876406	−	0.481573i	$-0.159934\pi$
$859$	771.000	+	445.137i	0.897555	+	0.518204i	0.0211491	+	0.999776i	$0.493268\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	315.000	−	545.596i	0.365006	−	0.632209i	−0.623771	−	0.781607i	$-0.714400\pi$
$863$	315.000	−	545.596i	0.365006	−	0.632209i	0.988777	+	0.149398i	$0.0477337\pi$
$864$		0			0
$865$	−162.000	−	280.592i	−0.187283	−	0.324384i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	375.000			0.431530
$870$		0			0
$871$	408.000	−	235.559i	0.468427	−	0.270447i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−820.000	+	1420.28i	−0.935006	+	1.61948i	−0.160382	+	0.987055i	$0.551273\pi$
$877$	−820.000	+	1420.28i	−0.935006	+	1.61948i	−0.774624	+	0.632422i	$0.782061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1267.86i		−	1.43912i	−0.694432	−	0.719558i	$-0.744344\pi$
$881$		−	1267.86i		−	1.43912i	0.694432	−	0.719558i	$-0.255656\pi$
$882$		0			0
$883$	226.000			0.255946			0.127973	−	0.991778i	$-0.459153\pi$
$883$	226.000			0.255946			0.127973	+	0.991778i	$0.459153\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000	+	3.46410i	0.00676437	+	0.00390541i	0.503378	−	0.864066i	$-0.332090\pi$
$887$	6.00000	+	3.46410i	0.00676437	+	0.00390541i	−0.496614	+	0.867972i	$0.665424\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−468.000	−	810.600i	−0.524076	−	0.907727i
$894$		0			0
$895$		−	280.592i		−	0.313511i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	535.500	−	309.171i	0.595662	−	0.343906i
$900$		0			0
$901$	−765.000	−	441.673i	−0.849057	−	0.490203i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	225.000	−	389.711i	0.248619	−	0.430620i
$906$		0			0
$907$	−572.000	−	990.733i	−0.630650	−	1.09232i	−0.987419	−	0.158126i	$-0.949455\pi$
$907$	−572.000	−	990.733i	−0.630650	−	1.09232i	0.356768	−	0.934193i	$-0.383879\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	480.000			0.526894			0.263447	−	0.964674i	$-0.415141\pi$
$911$	480.000			0.526894			0.263447	+	0.964674i	$0.415141\pi$
$912$		0			0
$913$	−400.500	+	231.229i	−0.438664	+	0.253263i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	871.000	−	1508.62i	0.947769	−	1.64158i	0.197660	−	0.980271i	$-0.436666\pi$
$919$	871.000	−	1508.62i	0.947769	−	1.64158i	0.750109	−	0.661314i	$-0.230001\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	484.000			0.523243
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1005.00	+	580.237i	1.08181	+	0.624582i	0.931384	−	0.364039i	$-0.118603\pi$
$929$	1005.00	+	580.237i	1.08181	+	0.624582i	0.150425	+	0.988621i	$0.451936\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	45.0000	+	77.9423i	0.0481283	+	0.0833607i
$936$		0			0
$937$		−	1331.95i		−	1.42150i	−0.703444	−	0.710751i	$-0.748355\pi$
$937$		−	1331.95i		−	1.42150i	0.703444	−	0.710751i	$-0.251645\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	703.500	−	406.166i	0.747609	−	0.431632i	−0.0772204	−	0.997014i	$-0.524605\pi$
$941$	703.500	−	406.166i	0.747609	−	0.431632i	0.824829	+	0.565382i	$0.191271\pi$
$942$		0			0
$943$	−756.000	−	436.477i	−0.801697	−	0.462860i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−645.000	+	1117.17i	−0.681098	+	1.17970i	0.293548	+	0.955944i	$0.405164\pi$
$947$	−645.000	+	1117.17i	−0.681098	+	1.17970i	−0.974646	+	0.223752i	$0.928169\pi$
$948$		0			0
$949$	72.0000	+	124.708i	0.0758693	+	0.131410i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−174.000			−0.182581			−0.0912907	−	0.995824i	$-0.529099\pi$
$953$	−174.000			−0.182581			−0.0912907	+	0.995824i	$0.529099\pi$
$954$		0			0
$955$	144.000	−	83.1384i	0.150785	−	0.0870560i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−407.000	+	704.945i	−0.423517	+	0.733553i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	510.955i		−	0.529487i
$966$		0			0
$967$	41.0000			0.0423992			0.0211996	−	0.999775i	$-0.493251\pi$
$967$	41.0000			0.0423992			0.0211996	+	0.999775i	$0.493251\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	853.500	+	492.768i	0.878991	+	0.507486i	0.870326	−	0.492477i	$-0.163908\pi$
$971$	853.500	+	492.768i	0.878991	+	0.507486i	0.00866523	+	0.999962i	$0.497242\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	639.000	+	1106.78i	0.654043	+	1.13284i	0.982133	+	0.188189i	$0.0602618\pi$
$977$	639.000	+	1106.78i	0.654043	+	1.13284i	−0.328090	+	0.944647i	$0.606405\pi$
$978$		0			0
$979$		−	218.238i		−	0.222920i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−201.000	+	116.047i	−0.204476	+	0.118054i	−0.598742	−	0.800942i	$-0.704332\pi$
$983$	−201.000	+	116.047i	−0.204476	+	0.118054i	0.394266	+	0.918997i	$0.370999\pi$
$984$		0			0
$985$	495.000	+	285.788i	0.502538	+	0.290140i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	180.000	−	311.769i	0.182002	−	0.315237i
$990$		0			0
$991$	416.500	+	721.399i	0.420283	+	0.727951i	0.995967	−	0.0897213i	$-0.0285976\pi$
$991$	416.500	+	721.399i	0.420283	+	0.727951i	−0.575684	+	0.817672i	$0.695264\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	204.000			0.205025
$996$		0			0
$997$	−1014.00	+	585.433i	−1.01705	+	0.587195i	−0.913248	−	0.407403i	$-0.866434\pi$
$997$	−1014.00	+	585.433i	−1.01705	+	0.587195i	−0.103803	+	0.994598i	$0.533101\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.e.901.1		2
3.2	odd	2		588.3.m.a.313.1		2
7.2	even	3		1764.3.d.c.685.1		2
7.3	odd	6	inner	1764.3.z.e.325.1		2
7.4	even	3		252.3.z.b.73.1		2
7.5	odd	6		1764.3.d.c.685.2		2
7.6	odd	2		252.3.z.b.145.1		2
21.2	odd	6		588.3.d.a.97.1		2
21.5	even	6		588.3.d.a.97.2		2
21.11	odd	6		84.3.m.a.73.1	yes	2
21.17	even	6		588.3.m.a.325.1		2
21.20	even	2		84.3.m.a.61.1	✓	2
28.11	odd	6		1008.3.cg.b.577.1		2
28.27	even	2		1008.3.cg.b.145.1		2
84.11	even	6		336.3.bh.b.241.1		2
84.23	even	6		2352.3.f.c.97.2		2
84.47	odd	6		2352.3.f.c.97.1		2
84.83	odd	2		336.3.bh.b.145.1		2
105.32	even	12		2100.3.be.c.1249.2		4
105.53	even	12		2100.3.be.c.1249.1		4
105.62	odd	4		2100.3.be.c.649.1		4
105.74	odd	6		2100.3.bd.b.1501.1		2
105.83	odd	4		2100.3.be.c.649.2		4
105.104	even	2		2100.3.bd.b.901.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.a.61.1	✓	2	21.20	even	2
84.3.m.a.73.1	yes	2	21.11	odd	6
252.3.z.b.73.1		2	7.4	even	3
252.3.z.b.145.1		2	7.6	odd	2
336.3.bh.b.145.1		2	84.83	odd	2
336.3.bh.b.241.1		2	84.11	even	6
588.3.d.a.97.1		2	21.2	odd	6
588.3.d.a.97.2		2	21.5	even	6
588.3.m.a.313.1		2	3.2	odd	2
588.3.m.a.325.1		2	21.17	even	6
1008.3.cg.b.145.1		2	28.27	even	2
1008.3.cg.b.577.1		2	28.11	odd	6
1764.3.d.c.685.1		2	7.2	even	3
1764.3.d.c.685.2		2	7.5	odd	6
1764.3.z.e.325.1		2	7.3	odd	6	inner
1764.3.z.e.901.1		2	1.1	even	1	trivial
2100.3.bd.b.901.1		2	105.104	even	2
2100.3.bd.b.1501.1		2	105.74	odd	6
2100.3.be.c.649.1		4	105.62	odd	4
2100.3.be.c.649.2		4	105.83	odd	4
2100.3.be.c.1249.1		4	105.53	even	12
2100.3.be.c.1249.2		4	105.32	even	12
2352.3.f.c.97.1		2	84.47	odd	6
2352.3.f.c.97.2		2	84.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{5} +(-1.50000 - 2.59808i) q^{11} +6.92820i q^{13} +(-15.0000 + 8.66025i) q^{17} +(-9.00000 - 5.19615i) q^{19} +(18.0000 - 31.1769i) q^{23} +(-11.0000 - 19.0526i) q^{25} +51.0000 q^{29} +(10.5000 - 6.06218i) q^{31} +(-11.0000 + 19.0526i) q^{37} -24.2487i q^{41} +10.0000 q^{43} +(78.0000 + 45.0333i) q^{47} +(25.5000 + 44.1673i) q^{53} -5.19615i q^{55} +(64.5000 - 37.2391i) q^{59} +(-60.0000 - 34.6410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(-34.0000 - 58.8897i) q^{67} +(18.0000 - 10.3923i) q^{73} +(-62.5000 + 108.253i) q^{79} -154.153i q^{83} -30.0000 q^{85} +(63.0000 + 36.3731i) q^{89} +(-9.00000 - 15.5885i) q^{95} -147.224i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 3 q^{11} - 30 q^{17} - 18 q^{19} + 36 q^{23} - 22 q^{25} + 102 q^{29} + 21 q^{31} - 22 q^{37} + 20 q^{43} + 156 q^{47} + 51 q^{53} + 129 q^{59} - 120 q^{61} - 12 q^{65} - 68 q^{67} + 36 q^{73}+ \cdots - 18 q^{95}+O(q^{100}) \) 2 * q + 3 * q^5 - 3 * q^11 - 30 * q^17 - 18 * q^19 + 36 * q^23 - 22 * q^25 + 102 * q^29 + 21 * q^31 - 22 * q^37 + 20 * q^43 + 156 * q^47 + 51 * q^53 + 129 * q^59 - 120 * q^61 - 12 * q^65 - 68 * q^67 + 36 * q^73 - 125 * q^79 - 60 * q^85 + 126 * q^89 - 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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