LMFDB - Embedded newform 2112.2.k.j.287.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,2,Mod(287,2112)] 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("2112.287"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.3
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2112.287
Dual form			2112.2.k.j.287.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.41421 + 1.00000i) q^{3} +2.00000 q^{5} -4.44949i q^{7} +(1.00000 - 2.82843i) q^{9} -1.00000i q^{11} +3.46410i q^{13} +(-2.82843 + 2.00000i) q^{15} +4.87832i q^{17} -0.778539 q^{19} +(4.44949 + 6.29253i) q^{21} +6.29253 q^{23} -1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +5.34847 q^{29} -6.89898i q^{31} +(1.00000 + 1.41421i) q^{33} -8.89898i q^{35} +2.19275i q^{37} +(-3.46410 - 4.89898i) q^{39} -7.70674i q^{41} +2.04989 q^{43} +(2.00000 - 5.65685i) q^{45} -2.82843 q^{47} -12.7980 q^{49} +(-4.87832 - 6.89898i) q^{51} +2.89898 q^{53} -2.00000i q^{55} +(1.10102 - 0.778539i) q^{57} -9.79796i q^{59} -9.12096i q^{61} +(-12.5851 - 4.44949i) q^{63} +6.92820i q^{65} +9.75663 q^{67} +(-8.89898 + 6.29253i) q^{69} -8.48528 q^{71} +6.89898 q^{73} +(1.41421 - 1.00000i) q^{75} -4.44949 q^{77} +9.34847i q^{79} +(-7.00000 - 5.65685i) q^{81} -9.79796i q^{83} +9.75663i q^{85} +(-7.56388 + 5.34847i) q^{87} -13.8564i q^{89} +15.4135 q^{91} +(6.89898 + 9.75663i) q^{93} -1.55708 q^{95} +12.0000 q^{97} +(-2.82843 - 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} + 8 q^{9} + 16 q^{21} - 8 q^{25} - 16 q^{29} + 8 q^{33} + 16 q^{45} - 24 q^{49} - 16 q^{53} + 48 q^{57} - 32 q^{69} + 16 q^{73} - 16 q^{77} - 56 q^{81} + 16 q^{93} + 96 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 + 8 * q^9 + 16 * q^21 - 8 * q^25 - 16 * q^29 + 8 * q^33 + 16 * q^45 - 24 * q^49 - 16 * q^53 + 48 * q^57 - 32 * q^69 + 16 * q^73 - 16 * q^77 - 56 * q^81 + 16 * q^93 + 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$	2.00000			0.894427			0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000			0.894427			0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		−	4.44949i		−	1.68175i	−0.541230	−	0.840875i	$-0.682041\pi$
$7$		−	4.44949i		−	1.68175i	0.541230	−	0.840875i	$-0.317959\pi$
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$	−2.82843	+	2.00000i	−0.730297	+	0.516398i
$16$		0			0
$17$			4.87832i			1.18317i	0.806244	+	0.591583i	$0.201497\pi$
$17$			4.87832i			1.18317i	−0.806244	+	0.591583i	$0.798503\pi$
$18$		0			0
$19$	−0.778539			−0.178609			−0.0893046	−	0.996004i	$-0.528464\pi$
$19$	−0.778539			−0.178609			−0.0893046	+	0.996004i	$0.528464\pi$
$20$		0			0
$21$	4.44949	+	6.29253i	0.970958	+	1.37314i
$22$		0			0
$23$	6.29253			1.31208			0.656041	−	0.754725i	$-0.272230\pi$
$23$	6.29253			1.31208			0.656041	+	0.754725i	$0.272230\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$	5.34847			0.993186			0.496593	−	0.867984i	$-0.334584\pi$
$29$	5.34847			0.993186			0.496593	+	0.867984i	$0.334584\pi$
$30$		0			0
$31$		−	6.89898i		−	1.23909i	−0.784960	−	0.619547i	$-0.787316\pi$
$31$		−	6.89898i		−	1.23909i	0.784960	−	0.619547i	$-0.212684\pi$
$32$		0			0
$33$	1.00000	+	1.41421i	0.174078	+	0.246183i
$34$		0			0
$35$		−	8.89898i		−	1.50420i
$36$		0			0
$37$			2.19275i			0.360486i	0.983622	+	0.180243i	$0.0576885\pi$
$37$			2.19275i			0.360486i	−0.983622	+	0.180243i	$0.942312\pi$
$38$		0			0
$39$	−3.46410	−	4.89898i	−0.554700	−	0.784465i
$40$		0			0
$41$		−	7.70674i		−	1.20359i	−0.798650	−	0.601795i	$-0.794452\pi$
$41$		−	7.70674i		−	1.20359i	0.798650	−	0.601795i	$-0.205548\pi$
$42$		0			0
$43$	2.04989			0.312605			0.156302	−	0.987709i	$-0.450043\pi$
$43$	2.04989			0.312605			0.156302	+	0.987709i	$0.450043\pi$
$44$		0			0
$45$	2.00000	−	5.65685i	0.298142	−	0.843274i
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−12.7980			−1.82828
$50$		0			0
$51$	−4.87832	−	6.89898i	−0.683101	−	0.966050i
$52$		0			0
$53$	2.89898			0.398205			0.199103	−	0.979979i	$-0.436197\pi$
$53$	2.89898			0.398205			0.199103	+	0.979979i	$0.436197\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$	1.10102	−	0.778539i	0.145834	−	0.103120i
$58$		0			0
$59$		−	9.79796i		−	1.27559i	−0.770208	−	0.637793i	$-0.779848\pi$
$59$		−	9.79796i		−	1.27559i	0.770208	−	0.637793i	$-0.220152\pi$
$60$		0			0
$61$		−	9.12096i		−	1.16782i	−0.811819	−	0.583909i	$-0.801522\pi$
$61$		−	9.12096i		−	1.16782i	0.811819	−	0.583909i	$-0.198478\pi$
$62$		0			0
$63$	−12.5851	−	4.44949i	−1.58557	−	0.560583i
$64$		0			0
$65$			6.92820i			0.859338i
$66$		0			0
$67$	9.75663			1.19196			0.595981	−	0.802998i	$-0.296763\pi$
$67$	9.75663			1.19196			0.595981	+	0.802998i	$0.296763\pi$
$68$		0			0
$69$	−8.89898	+	6.29253i	−1.07131	+	0.757531i
$70$		0			0
$71$	−8.48528			−1.00702			−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528			−1.00702			−0.503509	+	0.863990i	$0.667958\pi$
$72$		0			0
$73$	6.89898			0.807464			0.403732	−	0.914877i	$-0.367713\pi$
$73$	6.89898			0.807464			0.403732	+	0.914877i	$0.367713\pi$
$74$		0			0
$75$	1.41421	−	1.00000i	0.163299	−	0.115470i
$76$		0			0
$77$	−4.44949			−0.507066
$78$		0			0
$79$			9.34847i			1.05178i	0.850551	+	0.525892i	$0.176269\pi$
$79$			9.34847i			1.05178i	−0.850551	+	0.525892i	$0.823731\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	9.79796i		−	1.07547i	−0.843115	−	0.537733i	$-0.819281\pi$
$83$		−	9.79796i		−	1.07547i	0.843115	−	0.537733i	$-0.180719\pi$
$84$		0			0
$85$			9.75663i			1.05826i
$86$		0			0
$87$	−7.56388	+	5.34847i	−0.810933	+	0.573416i
$88$		0			0
$89$		−	13.8564i		−	1.46878i	−0.678730	−	0.734388i	$-0.737469\pi$
$89$		−	13.8564i		−	1.46878i	0.678730	−	0.734388i	$-0.262531\pi$
$90$		0			0
$91$	15.4135			1.61577
$92$		0			0
$93$	6.89898	+	9.75663i	0.715391	+	1.01172i
$94$		0			0
$95$	−1.55708			−0.159753
$96$		0			0
$97$	12.0000			1.21842			0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000			1.21842			0.609208	+	0.793011i	$0.291488\pi$
$98$		0			0
$99$	−2.82843	−	1.00000i	−0.284268	−	0.100504i
$100$		0			0
$101$	10.2474			1.01966			0.509830	−	0.860275i	$-0.329708\pi$
$101$	10.2474			1.01966			0.509830	+	0.860275i	$0.329708\pi$
$102$		0			0
$103$			7.79796i			0.768356i	0.923259	+	0.384178i	$0.125515\pi$
$103$			7.79796i			0.768356i	−0.923259	+	0.384178i	$0.874485\pi$
$104$		0			0
$105$	8.89898	+	12.5851i	0.868451	+	1.22818i
$106$		0			0
$107$		−	4.89898i		−	0.473602i	−0.971558	−	0.236801i	$-0.923901\pi$
$107$		−	4.89898i		−	0.473602i	0.971558	−	0.236801i	$-0.0760990\pi$
$108$		0			0
$109$			0.635674i			0.0608866i	0.999536	+	0.0304433i	$0.00969190\pi$
$109$			0.635674i			0.0608866i	−0.999536	+	0.0304433i	$0.990308\pi$
$110$		0			0
$111$	−2.19275	−	3.10102i	−0.208127	−	0.294336i
$112$		0			0
$113$		−	16.6848i		−	1.56958i	−0.619764	−	0.784789i	$-0.712772\pi$
$113$		−	16.6848i		−	1.56958i	0.619764	−	0.784789i	$-0.287228\pi$
$114$		0			0
$115$	12.5851			1.17356
$116$		0			0
$117$	9.79796	+	3.46410i	0.905822	+	0.320256i
$118$		0			0
$119$	21.7060			1.98979
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	7.70674	+	10.8990i	0.694894	+	0.982728i
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$		−	20.4495i		−	1.81460i	−0.420485	−	0.907299i	$-0.638140\pi$
$127$		−	20.4495i		−	1.81460i	0.420485	−	0.907299i	$-0.361860\pi$
$128$		0			0
$129$	−2.89898	+	2.04989i	−0.255241	+	0.180483i
$130$		0			0
$131$			18.6969i			1.63356i	0.576950	+	0.816780i	$0.304243\pi$
$131$			18.6969i			1.63356i	−0.576950	+	0.816780i	$0.695757\pi$
$132$		0			0
$133$			3.46410i			0.300376i
$134$		0			0
$135$	2.82843	+	10.0000i	0.243432	+	0.860663i
$136$		0			0
$137$		−	8.48528i		−	0.724947i	−0.931994	−	0.362473i	$-0.881932\pi$
$137$		−	8.48528i		−	0.724947i	0.931994	−	0.362473i	$-0.118068\pi$
$138$		0			0
$139$	−2.04989			−0.173869			−0.0869346	−	0.996214i	$-0.527707\pi$
$139$	−2.04989			−0.173869			−0.0869346	+	0.996214i	$0.527707\pi$
$140$		0			0
$141$	4.00000	−	2.82843i	0.336861	−	0.238197i
$142$		0			0
$143$	3.46410			0.289683
$144$		0			0
$145$	10.6969			0.888332
$146$		0			0
$147$	18.0990	−	12.7980i	1.49278	−	1.05556i
$148$		0			0
$149$	20.4495			1.67529			0.837644	−	0.546217i	$-0.183933\pi$
$149$	20.4495			1.67529			0.837644	+	0.546217i	$0.183933\pi$
$150$		0			0
$151$			7.55051i			0.614452i	0.951637	+	0.307226i	$0.0994007\pi$
$151$			7.55051i			0.614452i	−0.951637	+	0.307226i	$0.900599\pi$
$152$		0			0
$153$	13.7980	+	4.87832i	1.11550	+	0.394388i
$154$		0			0
$155$		−	13.7980i		−	1.10828i
$156$		0			0
$157$			8.19955i			0.654396i	0.944956	+	0.327198i	$0.106104\pi$
$157$			8.19955i			0.654396i	−0.944956	+	0.327198i	$0.893896\pi$
$158$		0			0
$159$	−4.09978	+	2.89898i	−0.325133	+	0.229904i
$160$		0			0
$161$		−	27.9985i		−	2.20659i
$162$		0			0
$163$	−23.8988			−1.87190			−0.935948	−	0.352138i	$-0.885455\pi$
$163$	−23.8988			−1.87190			−0.935948	+	0.352138i	$0.885455\pi$
$164$		0			0
$165$	2.00000	+	2.82843i	0.155700	+	0.220193i
$166$		0			0
$167$	2.82843			0.218870			0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843			0.218870			0.109435	+	0.993994i	$0.465096\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−0.778539	+	2.20204i	−0.0595364	+	0.168394i
$172$		0			0
$173$	6.24745			0.474985			0.237492	−	0.971389i	$-0.423675\pi$
$173$	6.24745			0.474985			0.237492	+	0.971389i	$0.423675\pi$
$174$		0			0
$175$			4.44949i			0.336350i
$176$		0			0
$177$	9.79796	+	13.8564i	0.736460	+	1.04151i
$178$		0			0
$179$			17.5959i			1.31518i	0.753376	+	0.657590i	$0.228424\pi$
$179$			17.5959i			1.31518i	−0.753376	+	0.657590i	$0.771576\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	9.12096	+	12.8990i	0.674240	+	0.953520i
$184$		0			0
$185$			4.38551i			0.322429i
$186$		0			0
$187$	4.87832			0.356738
$188$		0			0
$189$	22.2474	−	6.29253i	1.61826	−	0.457714i
$190$		0			0
$191$	−19.7990			−1.43260			−0.716302	−	0.697790i	$-0.754167\pi$
$191$	−19.7990			−1.43260			−0.716302	+	0.697790i	$0.754167\pi$
$192$		0			0
$193$	24.6969			1.77772			0.888862	−	0.458175i	$-0.151497\pi$
$193$	24.6969			1.77772			0.888862	+	0.458175i	$0.151497\pi$
$194$		0			0
$195$	−6.92820	−	9.79796i	−0.496139	−	0.701646i
$196$		0			0
$197$	−18.2474			−1.30008			−0.650038	−	0.759901i	$-0.725247\pi$
$197$	−18.2474			−1.30008			−0.650038	+	0.759901i	$0.725247\pi$
$198$		0			0
$199$			14.8990i			1.05616i	0.849194	+	0.528080i	$0.177088\pi$
$199$			14.8990i			1.05616i	−0.849194	+	0.528080i	$0.822912\pi$
$200$		0			0
$201$	−13.7980	+	9.75663i	−0.973233	+	0.688180i
$202$		0			0
$203$		−	23.7980i		−	1.67029i
$204$		0			0
$205$		−	15.4135i		−	1.07652i
$206$		0			0
$207$	6.29253	−	17.7980i	0.437361	−	1.23704i
$208$		0			0
$209$			0.778539i			0.0538527i
$210$		0			0
$211$	−22.8345			−1.57199			−0.785996	−	0.618232i	$-0.787849\pi$
$211$	−22.8345			−1.57199			−0.785996	+	0.618232i	$0.787849\pi$
$212$		0			0
$213$	12.0000	−	8.48528i	0.822226	−	0.581402i
$214$		0			0
$215$	4.09978			0.279602
$216$		0			0
$217$	−30.6969			−2.08384
$218$		0			0
$219$	−9.75663	+	6.89898i	−0.659292	+	0.466190i
$220$		0			0
$221$	−16.8990			−1.13675
$222$		0			0
$223$		−	0.202041i		−	0.0135297i	−0.999977	−	0.00676483i	$-0.997847\pi$
$223$		−	0.202041i		−	0.0135297i	0.999977	−	0.00676483i	$-0.00215333\pi$
$224$		0			0
$225$	−1.00000	+	2.82843i	−0.0666667	+	0.188562i
$226$		0			0
$227$		−	13.7980i		−	0.915803i	−0.889003	−	0.457901i	$-0.848601\pi$
$227$		−	13.7980i		−	0.915803i	0.889003	−	0.457901i	$-0.151399\pi$
$228$		0			0
$229$			27.3629i			1.80819i	0.427332	+	0.904095i	$0.359453\pi$
$229$			27.3629i			1.80819i	−0.427332	+	0.904095i	$0.640547\pi$
$230$		0			0
$231$	6.29253	−	4.44949i	0.414018	−	0.292755i
$232$		0			0
$233$		−	19.0205i		−	1.24607i	−0.782193	−	0.623036i	$-0.785899\pi$
$233$		−	19.0205i		−	1.24607i	0.782193	−	0.623036i	$-0.214101\pi$
$234$		0			0
$235$	−5.65685			−0.369012
$236$		0			0
$237$	−9.34847	−	13.2207i	−0.607248	−	0.858779i
$238$		0			0
$239$	12.2993			0.795577			0.397789	−	0.917477i	$-0.369778\pi$
$239$	12.2993			0.795577			0.397789	+	0.917477i	$0.369778\pi$
$240$		0			0
$241$	10.8990			0.702065			0.351032	−	0.936363i	$-0.385831\pi$
$241$	10.8990			0.702065			0.351032	+	0.936363i	$0.385831\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$	−25.5959			−1.63526
$246$		0			0
$247$		−	2.69694i		−	0.171602i
$248$		0			0
$249$	9.79796	+	13.8564i	0.620920	+	0.878114i
$250$		0			0
$251$		−	8.20204i		−	0.517708i	−0.965916	−	0.258854i	$-0.916655\pi$
$251$		−	8.20204i		−	0.517708i	0.965916	−	0.258854i	$-0.0833449\pi$
$252$		0			0
$253$		−	6.29253i		−	0.395608i
$254$		0			0
$255$	−9.75663	−	13.7980i	−0.610984	−	0.864062i
$256$		0			0
$257$		−	1.27135i		−	0.0793046i	−0.999214	−	0.0396523i	$-0.987375\pi$
$257$		−	1.27135i		−	0.0793046i	0.999214	−	0.0396523i	$-0.0126250\pi$
$258$		0			0
$259$	9.75663			0.606248
$260$		0			0
$261$	5.34847	−	15.1278i	0.331062	−	0.936385i
$262$		0			0
$263$	14.1421			0.872041			0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421			0.872041			0.436021	+	0.899937i	$0.356387\pi$
$264$		0			0
$265$	5.79796			0.356166
$266$		0			0
$267$	13.8564	+	19.5959i	0.847998	+	1.19925i
$268$		0			0
$269$	−14.8990			−0.908407			−0.454203	−	0.890898i	$-0.650076\pi$
$269$	−14.8990			−0.908407			−0.454203	+	0.890898i	$0.650076\pi$
$270$		0			0
$271$		−	8.44949i		−	0.513270i	−0.966508	−	0.256635i	$-0.917386\pi$
$271$		−	8.44949i		−	0.513270i	0.966508	−	0.256635i	$-0.0826138\pi$
$272$		0			0
$273$	−21.7980	+	15.4135i	−1.31927	+	0.932867i
$274$		0			0
$275$			1.00000i			0.0603023i
$276$		0			0
$277$		−	5.02118i		−	0.301693i	−0.988557	−	0.150847i	$-0.951800\pi$
$277$		−	5.02118i		−	0.301693i	0.988557	−	0.150847i	$-0.0482000\pi$
$278$		0			0
$279$	−19.5133	−	6.89898i	−1.16823	−	0.413031i
$280$		0			0
$281$			20.2918i			1.21051i	0.796033	+	0.605254i	$0.206928\pi$
$281$			20.2918i			1.21051i	−0.796033	+	0.605254i	$0.793072\pi$
$282$		0			0
$283$	10.5352			0.626251			0.313125	−	0.949712i	$-0.398624\pi$
$283$	10.5352			0.626251			0.313125	+	0.949712i	$0.398624\pi$
$284$		0			0
$285$	2.20204	−	1.55708i	0.130438	−	0.0922333i
$286$		0			0
$287$	−34.2911			−2.02414
$288$		0			0
$289$	−6.79796			−0.399880
$290$		0			0
$291$	−16.9706	+	12.0000i	−0.994832	+	0.703452i
$292$		0			0
$293$	11.1464			0.651181			0.325591	−	0.945511i	$-0.394437\pi$
$293$	11.1464			0.651181			0.325591	+	0.945511i	$0.394437\pi$
$294$		0			0
$295$		−	19.5959i		−	1.14092i
$296$		0			0
$297$	5.00000	−	1.41421i	0.290129	−	0.0820610i
$298$		0			0
$299$			21.7980i			1.26061i
$300$		0			0
$301$		−	9.12096i		−	0.525723i
$302$		0			0
$303$	−14.4921	+	10.2474i	−0.832548	+	0.588701i
$304$		0			0
$305$		−	18.2419i		−	1.04453i
$306$		0			0
$307$	30.3342			1.73126			0.865631	−	0.500683i	$-0.166918\pi$
$307$	30.3342			1.73126			0.865631	+	0.500683i	$0.166918\pi$
$308$		0			0
$309$	−7.79796	−	11.0280i	−0.443610	−	0.627360i
$310$		0			0
$311$	28.9199			1.63990			0.819950	−	0.572435i	$-0.194001\pi$
$311$	28.9199			1.63990			0.819950	+	0.572435i	$0.194001\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$	−25.1701	−	8.89898i	−1.41818	−	0.501401i
$316$		0			0
$317$	−15.7980			−0.887302			−0.443651	−	0.896200i	$-0.646317\pi$
$317$	−15.7980			−0.887302			−0.443651	+	0.896200i	$0.646317\pi$
$318$		0			0
$319$		−	5.34847i		−	0.299457i
$320$		0			0
$321$	4.89898	+	6.92820i	0.273434	+	0.386695i
$322$		0			0
$323$		−	3.79796i		−	0.211324i
$324$		0			0
$325$		−	3.46410i		−	0.192154i
$326$		0			0
$327$	−0.635674	−	0.898979i	−0.0351529	−	0.0497137i
$328$		0			0
$329$			12.5851i			0.693837i
$330$		0			0
$331$	−30.8270			−1.69440			−0.847202	−	0.531271i	$-0.821714\pi$
$331$	−30.8270			−1.69440			−0.847202	+	0.531271i	$0.821714\pi$
$332$		0			0
$333$	6.20204	+	2.19275i	0.339870	+	0.120162i
$334$		0			0
$335$	19.5133			1.06612
$336$		0			0
$337$	−14.4949			−0.789587			−0.394794	−	0.918770i	$-0.629184\pi$
$337$	−14.4949			−0.789587			−0.394794	+	0.918770i	$0.629184\pi$
$338$		0			0
$339$	16.6848	+	23.5959i	0.906196	+	1.28155i
$340$		0			0
$341$	−6.89898			−0.373601
$342$		0			0
$343$			25.7980i			1.39296i
$344$		0			0
$345$	−17.7980	+	12.5851i	−0.958210	+	0.677557i
$346$		0			0
$347$		−	6.20204i		−	0.332943i	−0.986046	−	0.166472i	$-0.946763\pi$
$347$		−	6.20204i		−	0.332943i	0.986046	−	0.166472i	$-0.0532374\pi$
$348$		0			0
$349$			28.9199i			1.54805i	0.633156	+	0.774025i	$0.281759\pi$
$349$			28.9199i			1.54805i	−0.633156	+	0.774025i	$0.718241\pi$
$350$		0			0
$351$	−17.3205	+	4.89898i	−0.924500	+	0.261488i
$352$		0			0
$353$			18.2419i			0.970919i	0.874259	+	0.485459i	$0.161348\pi$
$353$			18.2419i			0.970919i	−0.874259	+	0.485459i	$0.838652\pi$
$354$		0			0
$355$	−16.9706			−0.900704
$356$		0			0
$357$	−30.6969	+	21.7060i	−1.62465	+	1.14880i
$358$		0			0
$359$	12.8708			0.679294			0.339647	−	0.940553i	$-0.389692\pi$
$359$	12.8708			0.679294			0.339647	+	0.940553i	$0.389692\pi$
$360$		0			0
$361$	−18.3939			−0.968099
$362$		0			0
$363$	1.41421	−	1.00000i	0.0742270	−	0.0524864i
$364$		0			0
$365$	13.7980			0.722218
$366$		0			0
$367$		−	11.7980i		−	0.615848i	−0.951411	−	0.307924i	$-0.900366\pi$
$367$		−	11.7980i		−	0.615848i	0.951411	−	0.307924i	$-0.0996343\pi$
$368$		0			0
$369$	−21.7980	−	7.70674i	−1.13476	−	0.401197i
$370$		0			0
$371$		−	12.8990i		−	0.669682i
$372$		0			0
$373$		−	10.6780i		−	0.552888i	−0.961030	−	0.276444i	$-0.910844\pi$
$373$		−	10.6780i		−	0.552888i	0.961030	−	0.276444i	$-0.0891560\pi$
$374$		0			0
$375$	16.9706	−	12.0000i	0.876356	−	0.619677i
$376$		0			0
$377$			18.5276i			0.954222i
$378$		0			0
$379$	−25.1701			−1.29290			−0.646451	−	0.762956i	$-0.723748\pi$
$379$	−25.1701			−1.29290			−0.646451	+	0.762956i	$0.723748\pi$
$380$		0			0
$381$	20.4495	+	28.9199i	1.04766	+	1.48161i
$382$		0			0
$383$	14.4921			0.740511			0.370255	−	0.928930i	$-0.379270\pi$
$383$	14.4921			0.740511			0.370255	+	0.928930i	$0.379270\pi$
$384$		0			0
$385$	−8.89898			−0.453534
$386$		0			0
$387$	2.04989	−	5.79796i	0.104202	−	0.294727i
$388$		0			0
$389$	8.20204			0.415860			0.207930	−	0.978144i	$-0.433327\pi$
$389$	8.20204			0.415860			0.207930	+	0.978144i	$0.433327\pi$
$390$		0			0
$391$			30.6969i			1.55241i
$392$		0			0
$393$	−18.6969	−	26.4415i	−0.943136	−	1.33380i
$394$		0			0
$395$			18.6969i			0.940745i
$396$		0			0
$397$			2.19275i			0.110051i	0.998485	+	0.0550256i	$0.0175240\pi$
$397$			2.19275i			0.110051i	−0.998485	+	0.0550256i	$0.982476\pi$
$398$		0			0
$399$	−3.46410	−	4.89898i	−0.173422	−	0.245256i
$400$		0			0
$401$		−	6.64247i		−	0.331709i	−0.986150	−	0.165855i	$-0.946962\pi$
$401$		−	6.64247i		−	0.331709i	0.986150	−	0.165855i	$-0.0530383\pi$
$402$		0			0
$403$	23.8988			1.19048
$404$		0			0
$405$	−14.0000	−	11.3137i	−0.695666	−	0.562183i
$406$		0			0
$407$	2.19275			0.108691
$408$		0			0
$409$	7.79796			0.385584			0.192792	−	0.981240i	$-0.438246\pi$
$409$	7.79796			0.385584			0.192792	+	0.981240i	$0.438246\pi$
$410$		0			0
$411$	8.48528	+	12.0000i	0.418548	+	0.591916i
$412$		0			0
$413$	−43.5959			−2.14521
$414$		0			0
$415$		−	19.5959i		−	0.961926i
$416$		0			0
$417$	2.89898	−	2.04989i	0.141964	−	0.100383i
$418$		0			0
$419$			37.5959i			1.83668i	0.395792	+	0.918340i	$0.370470\pi$
$419$			37.5959i			1.83668i	−0.395792	+	0.918340i	$0.629530\pi$
$420$		0			0
$421$			22.6274i			1.10279i	0.834243	+	0.551396i	$0.185905\pi$
$421$			22.6274i			1.10279i	−0.834243	+	0.551396i	$0.814095\pi$
$422$		0			0
$423$	−2.82843	+	8.00000i	−0.137523	+	0.388973i
$424$		0			0
$425$		−	4.87832i		−	0.236633i
$426$		0			0
$427$	−40.5836			−1.96398
$428$		0			0
$429$	−4.89898	+	3.46410i	−0.236525	+	0.167248i
$430$		0			0
$431$	−18.2419			−0.878682			−0.439341	−	0.898320i	$-0.644788\pi$
$431$	−18.2419			−0.878682			−0.439341	+	0.898320i	$0.644788\pi$
$432$		0			0
$433$	−2.20204			−0.105823			−0.0529117	−	0.998599i	$-0.516850\pi$
$433$	−2.20204			−0.105823			−0.0529117	+	0.998599i	$0.516850\pi$
$434$		0			0
$435$	−15.1278	+	10.6969i	−0.725320	+	0.512879i
$436$		0			0
$437$	−4.89898			−0.234350
$438$		0			0
$439$			24.0454i			1.14762i	0.818987	+	0.573812i	$0.194536\pi$
$439$			24.0454i			1.14762i	−0.818987	+	0.573812i	$0.805464\pi$
$440$		0			0
$441$	−12.7980	+	36.1981i	−0.609427	+	1.72372i
$442$		0			0
$443$			23.5959i			1.12108i	0.828129	+	0.560538i	$0.189406\pi$
$443$			23.5959i			1.12108i	−0.828129	+	0.560538i	$0.810594\pi$
$444$		0			0
$445$		−	27.7128i		−	1.31371i
$446$		0			0
$447$	−28.9199	+	20.4495i	−1.36787	+	0.967228i
$448$		0			0
$449$		−	39.0265i		−	1.84178i	−0.389828	−	0.920888i	$-0.627466\pi$
$449$		−	39.0265i		−	1.84178i	0.389828	−	0.920888i	$-0.372534\pi$
$450$		0			0
$451$	−7.70674			−0.362896
$452$		0			0
$453$	−7.55051	−	10.6780i	−0.354754	−	0.501698i
$454$		0			0
$455$	30.8270			1.44519
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	−24.3916	+	6.89898i	−1.13850	+	0.322017i
$460$		0			0
$461$	22.6515			1.05499			0.527493	−	0.849559i	$-0.323132\pi$
$461$	22.6515			1.05499			0.527493	+	0.849559i	$0.323132\pi$
$462$		0			0
$463$			21.5959i			1.00365i	0.864970	+	0.501824i	$0.167337\pi$
$463$			21.5959i			1.00365i	−0.864970	+	0.501824i	$0.832663\pi$
$464$		0			0
$465$	13.7980	+	19.5133i	0.639865	+	0.904906i
$466$		0			0
$467$		−	2.20204i		−	0.101898i	−0.998701	−	0.0509492i	$-0.983775\pi$
$467$		−	2.20204i		−	0.101898i	0.998701	−	0.0509492i	$-0.0162246\pi$
$468$		0			0
$469$		−	43.4120i		−	2.00458i
$470$		0			0
$471$	−8.19955	−	11.5959i	−0.377815	−	0.534312i
$472$		0			0
$473$		−	2.04989i		−	0.0942540i
$474$		0			0
$475$	0.778539			0.0357218
$476$		0			0
$477$	2.89898	−	8.19955i	0.132735	−	0.375432i
$478$		0			0
$479$	25.1701			1.15005			0.575026	−	0.818135i	$-0.304992\pi$
$479$	25.1701			1.15005			0.575026	+	0.818135i	$0.304992\pi$
$480$		0			0
$481$	−7.59592			−0.346344
$482$		0			0
$483$	27.9985	+	39.5959i	1.27398	+	1.80168i
$484$		0			0
$485$	24.0000			1.08978
$486$		0			0
$487$		−	10.8990i		−	0.493880i	−0.969031	−	0.246940i	$-0.920575\pi$
$487$		−	10.8990i		−	0.493880i	0.969031	−	0.246940i	$-0.0794250\pi$
$488$		0			0
$489$	33.7980	−	23.8988i	1.52840	−	1.08074i
$490$		0			0
$491$		−	2.69694i		−	0.121711i	−0.998147	−	0.0608556i	$-0.980617\pi$
$491$		−	2.69694i		−	0.121711i	0.998147	−	0.0608556i	$-0.0193829\pi$
$492$		0			0
$493$			26.0915i			1.17510i
$494$		0			0
$495$	−5.65685	−	2.00000i	−0.254257	−	0.0898933i
$496$		0			0
$497$			37.7552i			1.69355i
$498$		0			0
$499$	−31.8126			−1.42413			−0.712064	−	0.702115i	$-0.752239\pi$
$499$	−31.8126			−1.42413			−0.712064	+	0.702115i	$0.752239\pi$
$500$		0			0
$501$	−4.00000	+	2.82843i	−0.178707	+	0.126365i
$502$		0			0
$503$	16.3991			0.731200			0.365600	−	0.930772i	$-0.380864\pi$
$503$	16.3991			0.731200			0.365600	+	0.930772i	$0.380864\pi$
$504$		0			0
$505$	20.4949			0.912011
$506$		0			0
$507$	−1.41421	+	1.00000i	−0.0628074	+	0.0444116i
$508$		0			0
$509$	−30.4949			−1.35166			−0.675831	−	0.737056i	$-0.736215\pi$
$509$	−30.4949			−1.35166			−0.675831	+	0.737056i	$0.736215\pi$
$510$		0			0
$511$		−	30.6969i		−	1.35795i
$512$		0			0
$513$	−1.10102	−	3.89270i	−0.0486112	−	0.171867i
$514$		0			0
$515$			15.5959i			0.687238i
$516$		0			0
$517$			2.82843i			0.124394i
$518$		0			0
$519$	−8.83523	+	6.24745i	−0.387823	+	0.274233i
$520$		0			0
$521$		−	26.7272i		−	1.17094i	−0.810694	−	0.585470i	$-0.800910\pi$
$521$		−	26.7272i		−	1.17094i	0.810694	−	0.585470i	$-0.199090\pi$
$522$		0			0
$523$	28.7771			1.25833			0.629167	−	0.777270i	$-0.283396\pi$
$523$	28.7771			1.25833			0.629167	+	0.777270i	$0.283396\pi$
$524$		0			0
$525$	−4.44949	−	6.29253i	−0.194192	−	0.274628i
$526$		0			0
$527$	33.6554			1.46605
$528$		0			0
$529$	16.5959			0.721562
$530$		0			0
$531$	−27.7128	−	9.79796i	−1.20263	−	0.425195i
$532$		0			0
$533$	26.6969			1.15637
$534$		0			0
$535$		−	9.79796i		−	0.423603i
$536$		0			0
$537$	−17.5959	−	24.8844i	−0.759320	−	1.07384i
$538$		0			0
$539$			12.7980i			0.551247i
$540$		0			0
$541$			2.47848i			0.106558i	0.998580	+	0.0532791i	$0.0169673\pi$
$541$			2.47848i			0.106558i	−0.998580	+	0.0532791i	$0.983033\pi$
$542$		0			0
$543$	6.92820	+	9.79796i	0.297318	+	0.420471i
$544$		0			0
$545$			1.27135i			0.0544586i
$546$		0			0
$547$	−33.1626			−1.41793			−0.708965	−	0.705244i	$-0.750838\pi$
$547$	−33.1626			−1.41793			−0.708965	+	0.705244i	$0.750838\pi$
$548$		0			0
$549$	−25.7980	−	9.12096i	−1.10103	−	0.389273i
$550$		0			0
$551$	−4.16399			−0.177392
$552$		0			0
$553$	41.5959			1.76884
$554$		0			0
$555$	−4.38551	−	6.20204i	−0.186154	−	0.263262i
$556$		0			0
$557$	−6.65153			−0.281834			−0.140917	−	0.990021i	$-0.545005\pi$
$557$	−6.65153			−0.281834			−0.140917	+	0.990021i	$0.545005\pi$
$558$		0			0
$559$			7.10102i			0.300341i
$560$		0			0
$561$	−6.89898	+	4.87832i	−0.291275	+	0.205963i
$562$		0			0
$563$		−	25.7980i		−	1.08725i	−0.839327	−	0.543627i	$-0.817051\pi$
$563$		−	25.7980i		−	1.08725i	0.839327	−	0.543627i	$-0.182949\pi$
$564$		0			0
$565$		−	33.3697i		−	1.40387i
$566$		0			0
$567$	−25.1701	+	31.1464i	−1.05705	+	1.30803i
$568$		0			0
$569$			24.9630i			1.04650i	0.852178	+	0.523252i	$0.175282\pi$
$569$			24.9630i			1.04650i	−0.852178	+	0.523252i	$0.824718\pi$
$570$		0			0
$571$	7.70674			0.322517			0.161259	−	0.986912i	$-0.448445\pi$
$571$	7.70674			0.322517			0.161259	+	0.986912i	$0.448445\pi$
$572$		0			0
$573$	28.0000	−	19.7990i	1.16972	−	0.827115i
$574$		0			0
$575$	−6.29253			−0.262417
$576$		0			0
$577$	−43.7980			−1.82333			−0.911666	−	0.410931i	$-0.865204\pi$
$577$	−43.7980			−1.82333			−0.911666	+	0.410931i	$0.865204\pi$
$578$		0			0
$579$	−34.9267	+	24.6969i	−1.45151	+	1.02637i
$580$		0			0
$581$	−43.5959			−1.80866
$582$		0			0
$583$		−	2.89898i		−	0.120063i
$584$		0			0
$585$	19.5959	+	6.92820i	0.810191	+	0.286446i
$586$		0			0
$587$		−	8.00000i		−	0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$		−	8.00000i		−	0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$		0			0
$589$			5.37113i			0.221313i
$590$		0			0
$591$	25.8058	−	18.2474i	1.06151	−	0.750600i
$592$		0			0
$593$			37.2624i			1.53018i	0.643922	+	0.765091i	$0.277306\pi$
$593$			37.2624i			1.53018i	−0.643922	+	0.765091i	$0.722694\pi$
$594$		0			0
$595$	43.4120			1.77972
$596$		0			0
$597$	−14.8990	−	21.0703i	−0.609775	−	0.862352i
$598$		0			0
$599$	45.3191			1.85169			0.925843	−	0.377908i	$-0.123356\pi$
$599$	45.3191			1.85169			0.925843	+	0.377908i	$0.123356\pi$
$600$		0			0
$601$	−38.8990			−1.58672			−0.793361	−	0.608751i	$-0.791671\pi$
$601$	−38.8990			−1.58672			−0.793361	+	0.608751i	$0.791671\pi$
$602$		0			0
$603$	9.75663	−	27.5959i	0.397321	−	1.12379i
$604$		0			0
$605$	−2.00000			−0.0813116
$606$		0			0
$607$			24.0454i			0.975973i	0.872851	+	0.487986i	$0.162268\pi$
$607$			24.0454i			0.975973i	−0.872851	+	0.487986i	$0.837732\pi$
$608$		0			0
$609$	23.7980	+	33.6554i	0.964342	+	1.36379i
$610$		0			0
$611$		−	9.79796i		−	0.396383i
$612$		0			0
$613$		−	30.4770i		−	1.23096i	−0.788154	−	0.615478i	$-0.788963\pi$
$613$		−	30.4770i		−	1.23096i	0.788154	−	0.615478i	$-0.211037\pi$
$614$		0			0
$615$	15.4135	+	21.7980i	0.621532	+	0.878979i
$616$		0			0
$617$			5.37113i			0.216233i	0.994138	+	0.108117i	$0.0344820\pi$
$617$			5.37113i			0.216233i	−0.994138	+	0.108117i	$0.965518\pi$
$618$		0			0
$619$	6.92820			0.278468			0.139234	−	0.990260i	$-0.455536\pi$
$619$	6.92820			0.278468			0.139234	+	0.990260i	$0.455536\pi$
$620$		0			0
$621$	8.89898	+	31.4626i	0.357104	+	1.26255i
$622$		0			0
$623$	−61.6539			−2.47011
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−0.778539	−	1.10102i	−0.0310919	−	0.0439705i
$628$		0			0
$629$	−10.6969			−0.426515
$630$		0			0
$631$			17.1010i			0.680781i	0.940284	+	0.340390i	$0.110559\pi$
$631$			17.1010i			0.680781i	−0.940284	+	0.340390i	$0.889441\pi$
$632$		0			0
$633$	32.2929	−	22.8345i	1.28353	−	0.907590i
$634$		0			0
$635$		−	40.8990i		−	1.62303i
$636$		0			0
$637$		−	44.3334i		−	1.75655i
$638$		0			0
$639$	−8.48528	+	24.0000i	−0.335673	+	0.949425i
$640$		0			0
$641$			20.7846i			0.820943i	0.911873	+	0.410471i	$0.134636\pi$
$641$			20.7846i			0.820943i	−0.911873	+	0.410471i	$0.865364\pi$
$642$		0			0
$643$	4.38551			0.172947			0.0864737	−	0.996254i	$-0.472440\pi$
$643$	4.38551			0.172947			0.0864737	+	0.996254i	$0.472440\pi$
$644$		0			0
$645$	−5.79796	+	4.09978i	−0.228294	+	0.161429i
$646$		0			0
$647$	29.6198			1.16448			0.582238	−	0.813018i	$-0.302177\pi$
$647$	29.6198			1.16448			0.582238	+	0.813018i	$0.302177\pi$
$648$		0			0
$649$	−9.79796			−0.384604
$650$		0			0
$651$	43.4120	−	30.6969i	1.70145	−	1.20311i
$652$		0			0
$653$	2.49490			0.0976329			0.0488164	−	0.998808i	$-0.484455\pi$
$653$	2.49490			0.0976329			0.0488164	+	0.998808i	$0.484455\pi$
$654$		0			0
$655$			37.3939i			1.46110i
$656$		0			0
$657$	6.89898	−	19.5133i	0.269155	−	0.761285i
$658$		0			0
$659$		−	3.10102i		−	0.120799i	−0.998174	−	0.0603993i	$-0.980763\pi$
$659$		−	3.10102i		−	0.120799i	0.998174	−	0.0603993i	$-0.0192374\pi$
$660$		0			0
$661$			10.7423i			0.417825i	0.977934	+	0.208913i	$0.0669924\pi$
$661$			10.7423i			0.417825i	−0.977934	+	0.208913i	$0.933008\pi$
$662$		0			0
$663$	23.8988	−	16.8990i	0.928151	−	0.656302i
$664$		0			0
$665$			6.92820i			0.268664i
$666$		0			0
$667$	33.6554			1.30314
$668$		0			0
$669$	0.202041	+	0.285729i	0.00781136	+	0.0110469i
$670$		0			0
$671$	−9.12096			−0.352111
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$	−1.41421	−	5.00000i	−0.0544331	−	0.192450i
$676$		0			0
$677$	15.5505			0.597655			0.298827	−	0.954307i	$-0.403405\pi$
$677$	15.5505			0.597655			0.298827	+	0.954307i	$0.403405\pi$
$678$		0			0
$679$		−	53.3939i		−	2.04907i
$680$		0			0
$681$	13.7980	+	19.5133i	0.528739	+	0.747750i
$682$		0			0
$683$			27.7980i			1.06366i	0.846851	+	0.531830i	$0.178495\pi$
$683$			27.7980i			1.06366i	−0.846851	+	0.531830i	$0.821505\pi$
$684$		0			0
$685$		−	16.9706i		−	0.648412i
$686$		0			0
$687$	−27.3629	−	38.6969i	−1.04396	−	1.47638i
$688$		0			0
$689$			10.0424i			0.382583i
$690$		0			0
$691$	20.7846			0.790684			0.395342	−	0.918534i	$-0.370626\pi$
$691$	20.7846			0.790684			0.395342	+	0.918534i	$0.370626\pi$
$692$		0			0
$693$	−4.44949	+	12.5851i	−0.169022	+	0.478067i
$694$		0			0
$695$	−4.09978			−0.155513
$696$		0			0
$697$	37.5959			1.42405
$698$		0			0
$699$	19.0205	+	26.8990i	0.719420	+	1.01741i
$700$		0			0
$701$	−24.4495			−0.923444			−0.461722	−	0.887025i	$-0.652768\pi$
$701$	−24.4495			−0.923444			−0.461722	+	0.887025i	$0.652768\pi$
$702$		0			0
$703$		−	1.70714i		−	0.0643861i
$704$		0			0
$705$	8.00000	−	5.65685i	0.301297	−	0.213049i
$706$		0			0
$707$		−	45.5959i		−	1.71481i
$708$		0			0
$709$		−	41.2193i		−	1.54802i	−0.633172	−	0.774011i	$-0.718247\pi$
$709$		−	41.2193i		−	1.54802i	0.633172	−	0.774011i	$-0.281753\pi$
$710$		0			0
$711$	26.4415	+	9.34847i	0.991632	+	0.350595i
$712$		0			0
$713$		−	43.4120i		−	1.62579i
$714$		0			0
$715$	6.92820			0.259100
$716$		0			0
$717$	−17.3939	+	12.2993i	−0.649586	+	0.459327i
$718$		0			0
$719$	−11.0280			−0.411274			−0.205637	−	0.978628i	$-0.565927\pi$
$719$	−11.0280			−0.411274			−0.205637	+	0.978628i	$0.565927\pi$
$720$		0			0
$721$	34.6969			1.29218
$722$		0			0
$723$	−15.4135	+	10.8990i	−0.573234	+	0.405337i
$724$		0			0
$725$	−5.34847			−0.198637
$726$		0			0
$727$			10.4949i			0.389234i	0.980879	+	0.194617i	$0.0623464\pi$
$727$			10.4949i			0.389234i	−0.980879	+	0.194617i	$0.937654\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$			10.0000i			0.369863i
$732$		0			0
$733$			41.5050i			1.53302i	0.642231	+	0.766511i	$0.278009\pi$
$733$			41.5050i			1.53302i	−0.642231	+	0.766511i	$0.721991\pi$
$734$		0			0
$735$	36.1981	−	25.5959i	1.33519	−	0.944120i
$736$		0			0
$737$		−	9.75663i		−	0.359390i
$738$		0			0
$739$	−20.2918			−0.746446			−0.373223	−	0.927742i	$-0.621747\pi$
$739$	−20.2918			−0.746446			−0.373223	+	0.927742i	$0.621747\pi$
$740$		0			0
$741$	2.69694	+	3.81405i	0.0990745	+	0.140113i
$742$		0			0
$743$	−35.2125			−1.29182			−0.645910	−	0.763413i	$-0.723522\pi$
$743$	−35.2125			−1.29182			−0.645910	+	0.763413i	$0.723522\pi$
$744$		0			0
$745$	40.8990			1.49842
$746$		0			0
$747$	−27.7128	−	9.79796i	−1.01396	−	0.358489i
$748$		0			0
$749$	−21.7980			−0.796480
$750$		0			0
$751$		−	42.0908i		−	1.53592i	−0.640500	−	0.767958i	$-0.721273\pi$
$751$		−	42.0908i		−	1.53592i	0.640500	−	0.767958i	$-0.278727\pi$
$752$		0			0
$753$	8.20204	+	11.5994i	0.298899	+	0.422707i
$754$		0			0
$755$			15.1010i			0.549582i
$756$		0			0
$757$			20.0847i			0.729992i	0.931009	+	0.364996i	$0.118930\pi$
$757$			20.0847i			0.729992i	−0.931009	+	0.364996i	$0.881070\pi$
$758$		0			0
$759$	6.29253	+	8.89898i	0.228404	+	0.323012i
$760$		0			0
$761$		−	21.8489i		−	0.792021i	−0.918246	−	0.396011i	$-0.870394\pi$
$761$		−	21.8489i		−	0.792021i	0.918246	−	0.396011i	$-0.129606\pi$
$762$		0			0
$763$	2.82843			0.102396
$764$		0			0
$765$	27.5959	+	9.75663i	0.997733	+	0.352752i
$766$		0			0
$767$	33.9411			1.22554
$768$		0			0
$769$	20.2929			0.731779			0.365890	−	0.930658i	$-0.380765\pi$
$769$	20.2929			0.731779			0.365890	+	0.930658i	$0.380765\pi$
$770$		0			0
$771$	1.27135	+	1.79796i	0.0457865	+	0.0647519i
$772$		0			0
$773$	4.69694			0.168937			0.0844686	−	0.996426i	$-0.473081\pi$
$773$	4.69694			0.168937			0.0844686	+	0.996426i	$0.473081\pi$
$774$		0			0
$775$			6.89898i			0.247819i
$776$		0			0
$777$	−13.7980	+	9.75663i	−0.494999	+	0.350017i
$778$		0			0
$779$			6.00000i			0.214972i
$780$		0			0
$781$			8.48528i			0.303627i
$782$		0			0
$783$	7.56388	+	26.7423i	0.270311	+	0.955693i
$784$		0			0
$785$			16.3991i			0.585309i
$786$		0			0
$787$	6.14966			0.219212			0.109606	−	0.993975i	$-0.465041\pi$
$787$	6.14966			0.219212			0.109606	+	0.993975i	$0.465041\pi$
$788$		0			0
$789$	−20.0000	+	14.1421i	−0.712019	+	0.503473i
$790$		0			0
$791$	−74.2390			−2.63963
$792$		0			0
$793$	31.5959			1.12200
$794$		0			0
$795$	−8.19955	+	5.79796i	−0.290808	+	0.205632i
$796$		0			0
$797$	−8.20204			−0.290531			−0.145266	−	0.989393i	$-0.546404\pi$
$797$	−8.20204			−0.290531			−0.145266	+	0.989393i	$0.546404\pi$
$798$		0			0
$799$		−	13.7980i		−	0.488137i
$800$		0			0
$801$	−39.1918	−	13.8564i	−1.38478	−	0.489592i
$802$		0			0
$803$		−	6.89898i		−	0.243460i
$804$		0			0
$805$		−	55.9971i		−	1.97364i
$806$		0			0
$807$	21.0703	−	14.8990i	0.741711	−	0.524469i
$808$		0			0
$809$		−	19.0205i		−	0.668723i	−0.942445	−	0.334362i	$-0.891479\pi$
$809$		−	19.0205i		−	0.668723i	0.942445	−	0.334362i	$-0.108521\pi$
$810$		0			0
$811$	7.13528			0.250554			0.125277	−	0.992122i	$-0.460018\pi$
$811$	7.13528			0.250554			0.125277	+	0.992122i	$0.460018\pi$
$812$		0			0
$813$	8.44949	+	11.9494i	0.296337	+	0.419083i
$814$		0			0
$815$	−47.7975			−1.67427
$816$		0			0
$817$	−1.59592			−0.0558341
$818$		0			0
$819$	15.4135	−	43.5959i	0.538591	−	1.52336i
$820$		0			0
$821$	5.75255			0.200765			0.100383	−	0.994949i	$-0.467993\pi$
$821$	5.75255			0.200765			0.100383	+	0.994949i	$0.467993\pi$
$822$		0			0
$823$			14.0000i			0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$			14.0000i			0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$		0			0
$825$	−1.00000	−	1.41421i	−0.0348155	−	0.0492366i
$826$		0			0
$827$			35.1010i			1.22058i	0.792177	+	0.610291i	$0.208947\pi$
$827$			35.1010i			1.22058i	−0.792177	+	0.610291i	$0.791053\pi$
$828$		0			0
$829$			46.8761i			1.62808i	0.580812	+	0.814038i	$0.302735\pi$
$829$			46.8761i			1.62808i	−0.580812	+	0.814038i	$0.697265\pi$
$830$		0			0
$831$	5.02118	+	7.10102i	0.174183	+	0.246332i
$832$		0			0
$833$		−	62.4325i		−	2.16316i
$834$		0			0
$835$	5.65685			0.195764
$836$		0			0
$837$	34.4949	−	9.75663i	1.19232	−	0.337238i
$838$		0			0
$839$	−7.56388			−0.261134			−0.130567	−	0.991439i	$-0.541680\pi$
$839$	−7.56388			−0.261134			−0.130567	+	0.991439i	$0.541680\pi$
$840$		0			0
$841$	−0.393877			−0.0135820
$842$		0			0
$843$	−20.2918	−	28.6969i	−0.698887	−	0.988375i
$844$		0			0
$845$	2.00000			0.0688021
$846$		0			0
$847$			4.44949i			0.152886i
$848$		0			0
$849$	−14.8990	+	10.5352i	−0.511332	+	0.361566i
$850$		0			0
$851$			13.7980i			0.472988i
$852$		0			0
$853$			22.6916i			0.776947i	0.921460	+	0.388473i	$0.126997\pi$
$853$			22.6916i			0.776947i	−0.921460	+	0.388473i	$0.873003\pi$
$854$		0			0
$855$	−1.55708	+	4.40408i	−0.0532509	+	0.150616i
$856$		0			0
$857$			18.4490i			0.630206i	0.949058	+	0.315103i	$0.102039\pi$
$857$			18.4490i			0.630206i	−0.949058	+	0.315103i	$0.897961\pi$
$858$		0			0
$859$	1.84281			0.0628758			0.0314379	−	0.999506i	$-0.489991\pi$
$859$	1.84281			0.0628758			0.0314379	+	0.999506i	$0.489991\pi$
$860$		0			0
$861$	48.4949	−	34.2911i	1.65270	−	1.16864i
$862$		0			0
$863$	52.2473			1.77852			0.889259	−	0.457405i	$-0.151221\pi$
$863$	52.2473			1.77852			0.889259	+	0.457405i	$0.151221\pi$
$864$		0			0
$865$	12.4949			0.424839
$866$		0			0
$867$	9.61377	−	6.79796i	0.326501	−	0.230871i
$868$		0			0
$869$	9.34847			0.317125
$870$		0			0
$871$			33.7980i			1.14520i
$872$		0			0
$873$	12.0000	−	33.9411i	0.406138	−	1.14873i
$874$		0			0
$875$			53.3939i			1.80504i
$876$		0			0
$877$			30.4770i			1.02914i	0.857450	+	0.514568i	$0.172048\pi$
$877$			30.4770i			1.02914i	−0.857450	+	0.514568i	$0.827952\pi$
$878$		0			0
$879$	−15.7634	+	11.1464i	−0.531687	+	0.375960i
$880$		0			0
$881$		−	39.3123i		−	1.32446i	−0.749299	−	0.662232i	$-0.769609\pi$
$881$		−	39.3123i		−	1.32446i	0.749299	−	0.662232i	$-0.230391\pi$
$882$		0			0
$883$	12.2993			0.413905			0.206953	−	0.978351i	$-0.433645\pi$
$883$	12.2993			0.413905			0.206953	+	0.978351i	$0.433645\pi$
$884$		0			0
$885$	19.5959	+	27.7128i	0.658710	+	0.931556i
$886$		0			0
$887$	−34.6410			−1.16313			−0.581566	−	0.813499i	$-0.697560\pi$
$887$	−34.6410			−1.16313			−0.581566	+	0.813499i	$0.697560\pi$
$888$		0			0
$889$	−90.9898			−3.05170
$890$		0			0
$891$	−5.65685	+	7.00000i	−0.189512	+	0.234509i
$892$		0			0
$893$	2.20204			0.0736885
$894$		0			0
$895$			35.1918i			1.17633i
$896$		0			0
$897$	−21.7980	−	30.8270i	−0.727813	−	1.02928i
$898$		0			0
$899$		−	36.8990i		−	1.23065i
$900$		0			0
$901$			14.1421i			0.471143i
$902$		0			0
$903$	9.12096	+	12.8990i	0.303526	+	0.429251i
$904$		0			0
$905$		−	13.8564i		−	0.460603i
$906$		0			0
$907$	−30.2555			−1.00462			−0.502309	−	0.864688i	$-0.667516\pi$
$907$	−30.2555			−1.00462			−0.502309	+	0.864688i	$0.667516\pi$
$908$		0			0
$909$	10.2474	−	28.9842i	0.339886	−	0.961344i
$910$		0			0
$911$	−27.9985			−0.927633			−0.463817	−	0.885931i	$-0.653520\pi$
$911$	−27.9985			−0.927633			−0.463817	+	0.885931i	$0.653520\pi$
$912$		0			0
$913$	−9.79796			−0.324265
$914$		0			0
$915$	18.2419	+	25.7980i	0.603059	+	0.852854i
$916$		0			0
$917$	83.1918			2.74724
$918$		0			0
$919$		−	2.65153i		−	0.0874659i	−0.999043	−	0.0437330i	$-0.986075\pi$
$919$		−	2.65153i		−	0.0874659i	0.999043	−	0.0437330i	$-0.0139251\pi$
$920$		0			0
$921$	−42.8990	+	30.3342i	−1.41357	+	0.999545i
$922$		0			0
$923$		−	29.3939i		−	0.967511i
$924$		0			0
$925$		−	2.19275i		−	0.0720973i
$926$		0			0
$927$	22.0560	+	7.79796i	0.724413	+	0.256119i
$928$		0			0
$929$			34.9267i			1.14591i	0.819587	+	0.572955i	$0.194203\pi$
$929$			34.9267i			1.14591i	−0.819587	+	0.572955i	$0.805797\pi$
$930$		0			0
$931$	9.96371			0.326547
$932$		0			0
$933$	−40.8990	+	28.9199i	−1.33897	+	0.946797i
$934$		0			0
$935$	9.75663			0.319076
$936$		0			0
$937$	27.7980			0.908120			0.454060	−	0.890971i	$-0.349975\pi$
$937$	27.7980			0.908120			0.454060	+	0.890971i	$0.349975\pi$
$938$		0			0
$939$	5.65685	−	4.00000i	0.184604	−	0.130535i
$940$		0			0
$941$	46.7423			1.52376			0.761878	−	0.647720i	$-0.224277\pi$
$941$	46.7423			1.52376			0.761878	+	0.647720i	$0.224277\pi$
$942$		0			0
$943$		−	48.4949i		−	1.57921i
$944$		0			0
$945$	44.4949	−	12.5851i	1.44742	−	0.409392i
$946$		0			0
$947$			14.4041i			0.468070i	0.972228	+	0.234035i	$0.0751930\pi$
$947$			14.4041i			0.468070i	−0.972228	+	0.234035i	$0.924807\pi$
$948$		0			0
$949$			23.8988i			0.775787i
$950$		0			0
$951$	22.3417	−	15.7980i	0.724479	−	0.512284i
$952$		0			0
$953$		−	0.492810i		−	0.0159637i	−0.999968	−	0.00798184i	$-0.997459\pi$
$953$		−	0.492810i		−	0.0159637i	0.999968	−	0.00798184i	$-0.00254073\pi$
$954$		0			0
$955$	−39.5980			−1.28136
$956$		0			0
$957$	5.34847	+	7.56388i	0.172891	+	0.244505i
$958$		0			0
$959$	−37.7552			−1.21918
$960$		0			0
$961$	−16.5959			−0.535352
$962$		0			0
$963$	−13.8564	−	4.89898i	−0.446516	−	0.157867i
$964$		0			0
$965$	49.3939			1.59005
$966$		0			0
$967$			38.2474i			1.22996i	0.788545	+	0.614978i	$0.210835\pi$
$967$			38.2474i			1.22996i	−0.788545	+	0.614978i	$0.789165\pi$
$968$		0			0
$969$	3.79796	+	5.37113i	0.122008	+	0.172545i
$970$		0			0
$971$			34.2020i			1.09760i	0.835955	+	0.548798i	$0.184914\pi$
$971$			34.2020i			1.09760i	−0.835955	+	0.548798i	$0.815086\pi$
$972$		0			0
$973$			9.12096i			0.292404i
$974$		0			0
$975$	3.46410	+	4.89898i	0.110940	+	0.156893i
$976$		0			0
$977$			17.2563i			0.552078i	0.961146	+	0.276039i	$0.0890218\pi$
$977$			17.2563i			0.552078i	−0.961146	+	0.276039i	$0.910978\pi$
$978$		0			0
$979$	−13.8564			−0.442853
$980$		0			0
$981$	1.79796	+	0.635674i	0.0574044	+	0.0202955i
$982$		0			0
$983$	−32.7340			−1.04405			−0.522026	−	0.852930i	$-0.674824\pi$
$983$	−32.7340			−1.04405			−0.522026	+	0.852930i	$0.674824\pi$
$984$		0			0
$985$	−36.4949			−1.16282
$986$		0			0
$987$	−12.5851	−	17.7980i	−0.400587	−	0.566515i
$988$		0			0
$989$	12.8990			0.410164
$990$		0			0
$991$		−	5.10102i		−	0.162039i	−0.996713	−	0.0810196i	$-0.974182\pi$
$991$		−	5.10102i		−	0.162039i	0.996713	−	0.0810196i	$-0.0258176\pi$
$992$		0			0
$993$	43.5959	−	30.8270i	1.38347	−	0.978264i
$994$		0			0
$995$			29.7980i			0.944659i
$996$		0			0
$997$			11.3779i			0.360342i	0.983635	+	0.180171i	$0.0576652\pi$
$997$			11.3779i			0.360342i	−0.983635	+	0.180171i	$0.942335\pi$
$998$		0			0
$999$	−10.9638	+	3.10102i	−0.346878	+	0.0981119i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.k.j.287.3	yes	8
3.2	odd	2		2112.2.k.i.287.1	✓	8
4.3	odd	2	inner	2112.2.k.j.287.6	yes	8
8.3	odd	2		2112.2.k.i.287.4	yes	8
8.5	even	2		2112.2.k.i.287.5	yes	8
12.11	even	2		2112.2.k.i.287.8	yes	8
24.5	odd	2	inner	2112.2.k.j.287.7	yes	8
24.11	even	2	inner	2112.2.k.j.287.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.k.i.287.1	✓	8	3.2	odd	2
2112.2.k.i.287.4	yes	8	8.3	odd	2
2112.2.k.i.287.5	yes	8	8.5	even	2
2112.2.k.i.287.8	yes	8	12.11	even	2
2112.2.k.j.287.2	yes	8	24.11	even	2	inner
2112.2.k.j.287.3	yes	8	1.1	even	1	trivial
2112.2.k.j.287.6	yes	8	4.3	odd	2	inner
2112.2.k.j.287.7	yes	8	24.5	odd	2	inner

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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} +2.00000 q^{5} -4.44949i q^{7} +(1.00000 - 2.82843i) q^{9} -1.00000i q^{11} +3.46410i q^{13} +(-2.82843 + 2.00000i) q^{15} +4.87832i q^{17} -0.778539 q^{19} +(4.44949 + 6.29253i) q^{21} +6.29253 q^{23} -1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +5.34847 q^{29} -6.89898i q^{31} +(1.00000 + 1.41421i) q^{33} -8.89898i q^{35} +2.19275i q^{37} +(-3.46410 - 4.89898i) q^{39} -7.70674i q^{41} +2.04989 q^{43} +(2.00000 - 5.65685i) q^{45} -2.82843 q^{47} -12.7980 q^{49} +(-4.87832 - 6.89898i) q^{51} +2.89898 q^{53} -2.00000i q^{55} +(1.10102 - 0.778539i) q^{57} -9.79796i q^{59} -9.12096i q^{61} +(-12.5851 - 4.44949i) q^{63} +6.92820i q^{65} +9.75663 q^{67} +(-8.89898 + 6.29253i) q^{69} -8.48528 q^{71} +6.89898 q^{73} +(1.41421 - 1.00000i) q^{75} -4.44949 q^{77} +9.34847i q^{79} +(-7.00000 - 5.65685i) q^{81} -9.79796i q^{83} +9.75663i q^{85} +(-7.56388 + 5.34847i) q^{87} -13.8564i q^{89} +15.4135 q^{91} +(6.89898 + 9.75663i) q^{93} -1.55708 q^{95} +12.0000 q^{97} +(-2.82843 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} + 8 q^{9} + 16 q^{21} - 8 q^{25} - 16 q^{29} + 8 q^{33} + 16 q^{45} - 24 q^{49} - 16 q^{53} + 48 q^{57} - 32 q^{69} + 16 q^{73} - 16 q^{77} - 56 q^{81} + 16 q^{93} + 96 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 + 8 * q^9 + 16 * q^21 - 8 * q^25 - 16 * q^29 + 8 * q^33 + 16 * q^45 - 24 * q^49 - 16 * q^53 + 48 * q^57 - 32 * q^69 + 16 * q^73 - 16 * q^77 - 56 * q^81 + 16 * q^93 + 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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