LMFDB - Embedded newform 648.4.i.r.217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.2
Root			$0.809017 + 1.40126i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.r.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.70820 + 13.3510i) q^{5} +(-11.9164 + 20.6398i) q^{7} +O(q^{10})$	$q+(7.70820 + 13.3510i) q^{5} +(-11.9164 + 20.6398i) q^{7} +(-7.12461 + 12.3402i) q^{11} +(6.91641 + 11.9796i) q^{13} -80.5836 q^{17} -144.331 q^{19} +(70.5410 + 122.181i) q^{23} +(-56.3328 + 97.5713i) q^{25} +(125.666 - 217.659i) q^{29} +(8.33437 + 14.4355i) q^{31} -367.416 q^{35} +305.164 q^{37} +(-214.663 - 371.806i) q^{41} +(90.8328 - 157.327i) q^{43} +(39.7082 - 68.7766i) q^{47} +(-112.502 - 194.858i) q^{49} -663.830 q^{53} -219.672 q^{55} +(-110.128 - 190.747i) q^{59} +(236.579 - 409.767i) q^{61} +(-106.626 + 184.682i) q^{65} +(323.831 + 560.892i) q^{67} +14.4922 q^{71} +776.003 q^{73} +(-169.800 - 294.101i) q^{77} +(128.913 - 223.284i) q^{79} +(-642.577 + 1112.98i) q^{83} +(-621.155 - 1075.87i) q^{85} -156.255 q^{89} -329.675 q^{91} +(-1112.53 - 1926.97i) q^{95} +(-580.822 + 1006.01i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 6 * q^7	$ 4 q + 4 q^{5} + 6 q^{7} + 52 q^{11} - 26 q^{13} - 376 q^{17} - 148 q^{19} + 148 q^{23} - 118 q^{25} + 288 q^{29} + 248 q^{31} - 1416 q^{35} + 684 q^{37} + 256 q^{43} + 132 q^{47} - 772 q^{49} - 1904 q^{53} - 1952 q^{55} - 1004 q^{59} + 34 q^{61} - 668 q^{65} + 866 q^{67} - 1552 q^{71} + 3748 q^{73} - 2316 q^{77} - 182 q^{79} - 1336 q^{83} - 16 q^{85} - 1752 q^{89} - 3036 q^{91} - 3028 q^{95} + 38 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 6 * q^7 + 52 * q^11 - 26 * q^13 - 376 * q^17 - 148 * q^19 + 148 * q^23 - 118 * q^25 + 288 * q^29 + 248 * q^31 - 1416 * q^35 + 684 * q^37 + 256 * q^43 + 132 * q^47 - 772 * q^49 - 1904 * q^53 - 1952 * q^55 - 1004 * q^59 + 34 * q^61 - 668 * q^65 + 866 * q^67 - 1552 * q^71 + 3748 * q^73 - 2316 * q^77 - 182 * q^79 - 1336 * q^83 - 16 * q^85 - 1752 * q^89 - 3036 * q^91 - 3028 * q^95 + 38 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.70820	+	13.3510i	0.689443	+	1.19415i	0.972018	+	0.234905i	$0.0754778\pi$
$5$	7.70820	+	13.3510i	0.689443	+	1.19415i	−0.282576	+	0.959245i	$0.591189\pi$
$6$		0			0
$7$	−11.9164	+	20.6398i	−0.643426	+	1.11445i	0.341237	+	0.939977i	$0.389154\pi$
$7$	−11.9164	+	20.6398i	−0.643426	+	1.11445i	−0.984663	+	0.174469i	$0.944179\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.12461	+	12.3402i	−0.195286	+	0.338246i	−0.946994	−	0.321250i	$-0.895897\pi$
$11$	−7.12461	+	12.3402i	−0.195286	+	0.338246i	0.751708	+	0.659496i	$0.229230\pi$
$12$		0			0
$13$	6.91641	+	11.9796i	0.147559	+	0.255580i	0.930325	−	0.366737i	$-0.119525\pi$
$13$	6.91641	+	11.9796i	0.147559	+	0.255580i	−0.782766	+	0.622316i	$0.786192\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−80.5836			−1.14967			−0.574835	−	0.818269i	$-0.694934\pi$
$17$	−80.5836			−1.14967			−0.574835	+	0.818269i	$0.694934\pi$
$18$		0			0
$19$	−144.331			−1.74273			−0.871365	−	0.490636i	$-0.836765\pi$
$19$	−144.331			−1.74273			−0.871365	+	0.490636i	$0.836765\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	70.5410	+	122.181i	0.639514	+	1.10767i	0.985540	+	0.169445i	$0.0541977\pi$
$23$	70.5410	+	122.181i	0.639514	+	1.10767i	−0.346026	+	0.938225i	$0.612469\pi$
$24$		0			0
$25$	−56.3328	+	97.5713i	−0.450663	+	0.780570i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	125.666	−	217.659i	0.804673	−	1.39373i	−0.111838	−	0.993726i	$-0.535674\pi$
$29$	125.666	−	217.659i	0.804673	−	1.39373i	0.916511	−	0.400008i	$-0.130993\pi$
$30$		0			0
$31$	8.33437	+	14.4355i	0.0482870	+	0.0836355i	0.889159	−	0.457599i	$-0.151290\pi$
$31$	8.33437	+	14.4355i	0.0482870	+	0.0836355i	−0.840872	+	0.541235i	$0.817957\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−367.416			−1.77442
$36$		0			0
$37$	305.164			1.35591			0.677955	−	0.735103i	$-0.262866\pi$
$37$	305.164			1.35591			0.677955	+	0.735103i	$0.262866\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−214.663	−	371.806i	−0.817674	−	1.41625i	−0.907391	−	0.420286i	$-0.861930\pi$
$41$	−214.663	−	371.806i	−0.817674	−	1.41625i	0.0897170	−	0.995967i	$-0.471404\pi$
$42$		0			0
$43$	90.8328	−	157.327i	0.322137	−	0.557957i	−0.658792	−	0.752325i	$-0.728932\pi$
$43$	90.8328	−	157.327i	0.322137	−	0.557957i	0.980929	+	0.194368i	$0.0622656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	39.7082	−	68.7766i	0.123235	−	0.213449i	−0.797807	−	0.602913i	$-0.794006\pi$
$47$	39.7082	−	68.7766i	0.123235	−	0.213449i	0.921042	+	0.389464i	$0.127340\pi$
$48$		0			0
$49$	−112.502	−	194.858i	−0.327993	−	0.568100i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−663.830			−1.72045			−0.860227	−	0.509912i	$-0.829678\pi$
$53$	−663.830			−1.72045			−0.860227	+	0.509912i	$0.829678\pi$
$54$		0			0
$55$	−219.672			−0.538555
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−110.128	−	190.747i	−0.243007	−	0.420900i	0.718562	−	0.695462i	$-0.244800\pi$
$59$	−110.128	−	190.747i	−0.243007	−	0.420900i	−0.961569	+	0.274562i	$0.911467\pi$
$60$		0			0
$61$	236.579	−	409.767i	0.496571	−	0.860086i	−0.503421	−	0.864041i	$-0.667926\pi$
$61$	236.579	−	409.767i	0.496571	−	0.860086i	0.999992	+	0.00395500i	$0.00125892\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−106.626	+	184.682i	−0.203467	+	0.352415i
$66$		0			0
$67$	323.831	+	560.892i	0.590482	+	1.02274i	0.994168	+	0.107847i	$0.0343956\pi$
$67$	323.831	+	560.892i	0.590482	+	1.02274i	−0.403686	+	0.914898i	$0.632271\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.4922			0.0242241			0.0121121	−	0.999927i	$-0.496145\pi$
$71$	14.4922			0.0242241			0.0121121	+	0.999927i	$0.496145\pi$
$72$		0			0
$73$	776.003			1.24417			0.622084	−	0.782950i	$-0.286286\pi$
$73$	776.003			1.24417			0.622084	+	0.782950i	$0.286286\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−169.800	−	294.101i	−0.251305	−	0.435272i
$78$		0			0
$79$	128.913	−	223.284i	0.183593	−	0.317993i	−0.759508	−	0.650498i	$-0.774560\pi$
$79$	128.913	−	223.284i	0.183593	−	0.317993i	0.943102	+	0.332505i	$0.107894\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−642.577	+	1112.98i	−0.849784	+	1.47187i	0.0316178	+	0.999500i	$0.489934\pi$
$83$	−642.577	+	1112.98i	−0.849784	+	1.47187i	−0.881401	+	0.472368i	$0.843399\pi$
$84$		0			0
$85$	−621.155	−	1075.87i	−0.792632	−	1.37288i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−156.255			−0.186102			−0.0930508	−	0.995661i	$-0.529662\pi$
$89$	−156.255			−0.186102			−0.0930508	+	0.995661i	$0.529662\pi$
$90$		0			0
$91$	−329.675			−0.379773
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1112.53	−	1926.97i	−1.20151	−	2.08108i
$96$		0			0
$97$	−580.822	+	1006.01i	−0.607975	+	1.05304i	0.383599	+	0.923500i	$0.374684\pi$
$97$	−580.822	+	1006.01i	−0.607975	+	1.05304i	−0.991574	+	0.129543i	$0.958649\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−842.991	+	1460.10i	−0.830502	+	1.43847i	0.0671386	+	0.997744i	$0.478613\pi$
$101$	−842.991	+	1460.10i	−0.830502	+	1.43847i	−0.897641	+	0.440728i	$0.854720\pi$
$102$		0			0
$103$	382.910	+	663.220i	0.366304	+	0.634456i	0.988984	−	0.148020i	$-0.0472899\pi$
$103$	382.910	+	663.220i	0.366304	+	0.634456i	−0.622681	+	0.782476i	$0.713957\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1747.89			−1.57920			−0.789602	−	0.613619i	$-0.789713\pi$
$107$	−1747.89			−1.57920			−0.789602	+	0.613619i	$0.789713\pi$
$108$		0			0
$109$	−1211.64			−1.06472			−0.532358	−	0.846519i	$-0.678694\pi$
$109$	−1211.64			−1.06472			−0.532358	+	0.846519i	$0.678694\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−482.213	−	835.217i	−0.401440	−	0.695315i	0.592460	−	0.805600i	$-0.298157\pi$
$113$	−482.213	−	835.217i	−0.401440	−	0.695315i	−0.993900	+	0.110285i	$0.964824\pi$
$114$		0			0
$115$	−1087.49	+	1883.59i	−0.881816	+	1.52735i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	960.267	−	1663.23i	0.739727	−	1.28124i
$120$		0			0
$121$	563.980	+	976.842i	0.423726	+	0.733916i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	190.152			0.136061
$126$		0			0
$127$	−2238.31			−1.56392			−0.781960	−	0.623329i	$-0.785780\pi$
$127$	−2238.31			−1.56392			−0.781960	+	0.623329i	$0.785780\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−837.155	−	1449.99i	−0.558340	−	0.967073i	−0.997635	−	0.0687306i	$-0.978105\pi$
$131$	−837.155	−	1449.99i	−0.558340	−	0.967073i	0.439295	−	0.898343i	$-0.355228\pi$
$132$		0			0
$133$	1719.91	−	2978.97i	1.12132	−	1.94218i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	47.5379	−	82.3381i	0.0296455	−	0.0513476i	−0.850822	−	0.525454i	$-0.823895\pi$
$137$	47.5379	−	82.3381i	0.0296455	−	0.0513476i	0.880468	+	0.474106i	$0.157229\pi$
$138$		0			0
$139$	−585.320	−	1013.80i	−0.357167	−	0.618632i	0.630319	−	0.776336i	$-0.282924\pi$
$139$	−585.320	−	1013.80i	−0.357167	−	0.618632i	−0.987486	+	0.157704i	$0.949591\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−197.107			−0.115265
$144$		0			0
$145$	3874.63			2.21910
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−429.568	−	744.034i	−0.236185	−	0.409085i	0.723431	−	0.690396i	$-0.242564\pi$
$149$	−429.568	−	744.034i	−0.236185	−	0.409085i	−0.959616	+	0.281312i	$0.909231\pi$
$150$		0			0
$151$	−1449.41	+	2510.46i	−0.781137	+	1.35297i	0.150143	+	0.988664i	$0.452027\pi$
$151$	−1449.41	+	2510.46i	−0.781137	+	1.35297i	−0.931280	+	0.364305i	$0.881307\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−128.486	+	222.544i	−0.0665822	+	0.115324i
$156$		0			0
$157$	654.508	+	1133.64i	0.332710	+	0.576270i	0.983042	−	0.183380i	$-0.0587038\pi$
$157$	654.508	+	1133.64i	0.332710	+	0.576270i	−0.650333	+	0.759650i	$0.725370\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3362.38			−1.64592
$162$		0			0
$163$	−190.988			−0.0917749			−0.0458874	−	0.998947i	$-0.514612\pi$
$163$	−190.988			−0.0917749			−0.0458874	+	0.998947i	$0.514612\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	720.365	+	1247.71i	0.333793	+	0.578147i	0.983252	−	0.182250i	$-0.0583379\pi$
$167$	720.365	+	1247.71i	0.333793	+	0.578147i	−0.649459	+	0.760397i	$0.725005\pi$
$168$		0			0
$169$	1002.83	−	1736.95i	0.456453	−	0.790599i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	790.663	−	1369.47i	0.347474	−	0.601842i	−0.638326	−	0.769766i	$-0.720373\pi$
$173$	790.663	−	1369.47i	0.347474	−	0.601842i	0.985800	+	0.167924i	$0.0537062\pi$
$174$		0			0
$175$	−1342.57	−	2325.40i	−0.579936	−	1.00448i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−741.836			−0.309762			−0.154881	−	0.987933i	$-0.549499\pi$
$179$	−741.836			−0.309762			−0.154881	+	0.987933i	$0.549499\pi$
$180$		0			0
$181$	626.786			0.257396			0.128698	−	0.991684i	$-0.458920\pi$
$181$	626.786			0.257396			0.128698	+	0.991684i	$0.458920\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2352.27	+	4074.25i	0.934822	+	1.61916i
$186$		0			0
$187$	574.127	−	994.417i	0.224515	−	0.388871i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	91.3073	−	158.149i	0.0345904	−	0.0599123i	−0.848212	−	0.529657i	$-0.822321\pi$
$191$	91.3073	−	158.149i	0.0345904	−	0.0599123i	0.882802	+	0.469745i	$0.155654\pi$
$192$		0			0
$193$	−859.330	−	1488.40i	−0.320497	−	0.555117i	0.660094	−	0.751183i	$-0.270517\pi$
$193$	−859.330	−	1488.40i	−0.320497	−	0.555117i	−0.980591	+	0.196066i	$0.937183\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	119.587			0.0432497			0.0216249	−	0.999766i	$-0.493116\pi$
$197$	119.587			0.0432497			0.0216249	+	0.999766i	$0.493116\pi$
$198$		0			0
$199$	707.467			0.252015			0.126008	−	0.992029i	$-0.459784\pi$
$199$	707.467			0.252015			0.126008	+	0.992029i	$0.459784\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2994.97	+	5187.43i	1.03549	+	1.79353i
$204$		0			0
$205$	3309.33	−	5731.92i	1.12748	−	1.95285i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1028.30	−	1781.08i	0.340332	−	0.589472i
$210$		0			0
$211$	1194.31	+	2068.60i	0.389666	+	0.674922i	0.992405	−	0.123017i	$-0.0392571\pi$
$211$	1194.31	+	2068.60i	0.389666	+	0.674922i	−0.602738	+	0.797939i	$0.705924\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2800.63			0.888379
$216$		0			0
$217$	−397.263			−0.124276
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−557.349	−	965.357i	−0.169644	−	0.293832i
$222$		0			0
$223$	−2811.30	+	4869.31i	−0.844209	+	1.46221i	0.0420975	+	0.999114i	$0.486596\pi$
$223$	−2811.30	+	4869.31i	−0.844209	+	1.46221i	−0.886306	+	0.463099i	$0.846737\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1732.23	+	3000.31i	−0.506485	+	0.877258i	0.493486	+	0.869753i	$0.335722\pi$
$227$	−1732.23	+	3000.31i	−0.506485	+	0.877258i	−0.999972	+	0.00750494i	$0.997611\pi$
$228$		0			0
$229$	3041.31	+	5267.70i	0.877622	+	1.52009i	0.853943	+	0.520366i	$0.174204\pi$
$229$	3041.31	+	5267.70i	0.877622	+	1.52009i	0.0236784	+	0.999720i	$0.492462\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5024.29			1.41267			0.706335	−	0.707877i	$-0.250347\pi$
$233$	5024.29			1.41267			0.706335	+	0.707877i	$0.250347\pi$
$234$		0			0
$235$	1224.32			0.339853
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1335.10	+	2312.46i	0.361341	+	0.625861i	0.988182	−	0.153286i	$-0.0489857\pi$
$239$	1335.10	+	2312.46i	0.361341	+	0.625861i	−0.626841	+	0.779147i	$0.715652\pi$
$240$		0			0
$241$	−377.327	+	653.549i	−0.100854	+	0.174684i	−0.912037	−	0.410109i	$-0.865491\pi$
$241$	−377.327	+	653.549i	−0.100854	+	0.174684i	0.811183	+	0.584793i	$0.198824\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1734.37	−	3004.02i	0.452265	−	0.783345i
$246$		0			0
$247$	−998.254	−	1729.03i	−0.257155	−	0.445406i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1588.91			−0.399566			−0.199783	−	0.979840i	$-0.564024\pi$
$251$	−1588.91			−0.399566			−0.199783	+	0.979840i	$0.564024\pi$
$252$		0			0
$253$	−2010.31			−0.499554
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2455.88	−	4253.70i	−0.596083	−	1.03245i	−0.993393	−	0.114762i	$-0.963389\pi$
$257$	−2455.88	−	4253.70i	−0.596083	−	1.03245i	0.397310	−	0.917685i	$-0.369944\pi$
$258$		0			0
$259$	−3636.46	+	6298.53i	−0.872427	+	1.51109i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2700.21	−	4676.89i	0.633087	−	1.09654i	−0.353830	−	0.935310i	$-0.615121\pi$
$263$	2700.21	−	4676.89i	0.633087	−	1.09654i	0.986917	−	0.161229i	$-0.0515457\pi$
$264$		0			0
$265$	−5116.93	−	8862.79i	−1.18615	−	2.05448i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3234.09			0.733034			0.366517	−	0.930411i	$-0.380550\pi$
$269$	3234.09			0.733034			0.366517	+	0.930411i	$0.380550\pi$
$270$		0			0
$271$	−6205.83			−1.39106			−0.695530	−	0.718497i	$-0.744830\pi$
$271$	−6205.83			−1.39106			−0.695530	+	0.718497i	$0.744830\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−802.699	−	1390.32i	−0.176017	−	0.304870i
$276$		0			0
$277$	−515.455	+	892.794i	−0.111808	+	0.193656i	−0.916499	−	0.400037i	$-0.868997\pi$
$277$	−515.455	+	892.794i	−0.111808	+	0.193656i	0.804692	+	0.593693i	$0.202331\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1910.66	+	3309.36i	−0.405625	+	0.702563i	−0.994394	−	0.105738i	$-0.966279\pi$
$281$	−1910.66	+	3309.36i	−0.405625	+	0.702563i	0.588769	+	0.808301i	$0.299613\pi$
$282$		0			0
$283$	2201.17	+	3812.53i	0.462353	+	0.800818i	0.999078	−	0.0429389i	$-0.0136721\pi$
$283$	2201.17	+	3812.53i	0.462353	+	0.800818i	−0.536725	+	0.843757i	$0.680339\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10232.0			2.10445
$288$		0			0
$289$	1580.72			0.321741
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2986.77	+	5173.24i	0.595526	+	1.03148i	0.993472	+	0.114072i	$0.0363896\pi$
$293$	2986.77	+	5173.24i	0.595526	+	1.03148i	−0.397947	+	0.917409i	$0.630277\pi$
$294$		0			0
$295$	1697.77	−	2940.63i	0.335079	−	0.580373i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−975.781	+	1690.10i	−0.188732	+	0.326893i
$300$		0			0
$301$	2164.80	+	3749.55i	0.414542	+	0.718008i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7294.39			1.36943
$306$		0			0
$307$	−219.622			−0.0408290			−0.0204145	−	0.999792i	$-0.506499\pi$
$307$	−219.622			−0.0408290			−0.0204145	+	0.999792i	$0.506499\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	478.614	+	828.983i	0.0872659	+	0.151149i	0.906355	−	0.422518i	$-0.138854\pi$
$311$	478.614	+	828.983i	0.0872659	+	0.151149i	−0.819089	+	0.573667i	$0.805520\pi$
$312$		0			0
$313$	−937.026	+	1622.98i	−0.169214	+	0.293087i	−0.938144	−	0.346246i	$-0.887456\pi$
$313$	−937.026	+	1622.98i	−0.169214	+	0.293087i	0.768930	+	0.639333i	$0.220789\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1459.89	+	2528.60i	−0.258661	+	0.448014i	−0.965884	−	0.258977i	$-0.916615\pi$
$317$	−1459.89	+	2528.60i	−0.258661	+	0.448014i	0.707222	+	0.706991i	$0.249948\pi$
$318$		0			0
$319$	1790.64	+	3101.48i	0.314284	+	0.544355i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11630.7			2.00356
$324$		0			0
$325$	−1558.48			−0.265997
$326$		0			0
$327$		0			0
$328$		0			0
$329$	946.358	+	1639.14i	0.158585	+	0.274677i
$330$		0			0
$331$	3324.33	−	5757.91i	0.552030	−	0.956144i	−0.446098	−	0.894984i	$-0.647187\pi$
$331$	3324.33	−	5757.91i	0.552030	−	0.956144i	0.998128	−	0.0611596i	$-0.0194799\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4992.31	+	8646.94i	−0.814207	+	1.41025i
$336$		0			0
$337$	−1943.17	−	3365.67i	−0.314098	−	0.544034i	0.665147	−	0.746712i	$-0.268369\pi$
$337$	−1943.17	−	3365.67i	−0.314098	−	0.544034i	−0.979245	+	0.202678i	$0.935036\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−237.517			−0.0377192
$342$		0			0
$343$	−2812.20			−0.442695
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2103.88	−	3644.02i	−0.325481	−	0.563750i	0.656128	−	0.754649i	$-0.272193\pi$
$347$	−2103.88	−	3644.02i	−0.325481	−	0.563750i	−0.981610	+	0.190899i	$0.938860\pi$
$348$		0			0
$349$	−1505.71	+	2607.97i	−0.230942	+	0.400004i	−0.958086	−	0.286482i	$-0.907514\pi$
$349$	−1505.71	+	2607.97i	−0.230942	+	0.400004i	0.727143	+	0.686486i	$0.240848\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1249.90	+	2164.88i	−0.188457	+	0.326417i	−0.944736	−	0.327832i	$-0.893682\pi$
$353$	−1249.90	+	2164.88i	−0.188457	+	0.326417i	0.756279	+	0.654249i	$0.227015\pi$
$354$		0			0
$355$	111.709	+	193.486i	0.0167011	+	0.0289272i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8319.17			−1.22303			−0.611517	−	0.791231i	$-0.709440\pi$
$359$	−8319.17			−1.22303			−0.611517	+	0.791231i	$0.709440\pi$
$360$		0			0
$361$	13972.5			2.03711
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5981.59	+	10360.4i	0.857783	+	1.48572i
$366$		0			0
$367$	2229.93	−	3862.34i	0.317169	−	0.549354i	−0.662727	−	0.748861i	$-0.730601\pi$
$367$	2229.93	−	3862.34i	0.317169	−	0.549354i	0.979896	+	0.199508i	$0.0639343\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7910.47	−	13701.3i	1.10698	−	1.91735i
$372$		0			0
$373$	589.703	+	1021.39i	0.0818596	+	0.141785i	0.904049	−	0.427429i	$-0.140581\pi$
$373$	589.703	+	1021.39i	0.0818596	+	0.141785i	−0.822189	+	0.569214i	$0.807247\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3476.62			0.474947
$378$		0			0
$379$	8413.88			1.14035			0.570174	−	0.821524i	$-0.306876\pi$
$379$	8413.88			1.14035			0.570174	+	0.821524i	$0.306876\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	800.972	+	1387.32i	0.106861	+	0.185089i	0.914497	−	0.404593i	$-0.132587\pi$
$383$	800.972	+	1387.32i	0.106861	+	0.185089i	−0.807636	+	0.589681i	$0.799253\pi$
$384$		0			0
$385$	2617.70	−	4533.99i	0.346520	−	0.600191i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2601.21	+	4505.44i	−0.339041	+	0.587236i	−0.984253	−	0.176768i	$-0.943436\pi$
$389$	−2601.21	+	4505.44i	−0.339041	+	0.587236i	0.645212	+	0.764004i	$0.276769\pi$
$390$		0			0
$391$	−5684.45	−	9845.75i	−0.735230	−	1.27346i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3974.76			0.506309
$396$		0			0
$397$	4310.87			0.544979			0.272489	−	0.962159i	$-0.412153\pi$
$397$	4310.87			0.544979			0.272489	+	0.962159i	$0.412153\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3120.29	+	5404.51i	0.388579	+	0.673038i	0.992259	−	0.124189i	$-0.0396329\pi$
$401$	3120.29	+	5404.51i	0.388579	+	0.673038i	−0.603680	+	0.797227i	$0.706300\pi$
$402$		0			0
$403$	−115.288	+	199.684i	−0.0142504	+	0.0246823i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2174.18	+	3765.78i	−0.264791	+	0.458631i
$408$		0			0
$409$	−1023.83	−	1773.33i	−0.123778	−	0.214390i	0.797477	−	0.603350i	$-0.206168\pi$
$409$	−1023.83	−	1773.33i	−0.123778	−	0.214390i	−0.921255	+	0.388960i	$0.872834\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5249.31			0.625427
$414$		0			0
$415$	−19812.5			−2.34351
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2980.06	+	5161.62i	0.347459	+	0.601817i	0.985797	−	0.167939i	$-0.0537111\pi$
$419$	2980.06	+	5161.62i	0.347459	+	0.601817i	−0.638338	+	0.769756i	$0.720378\pi$
$420$		0			0
$421$	−155.783	+	269.825i	−0.0180342	+	0.0312362i	−0.874902	−	0.484301i	$-0.839074\pi$
$421$	−155.783	+	269.825i	−0.0180342	+	0.0312362i	0.856867	+	0.515537i	$0.172407\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4539.50	−	7862.65i	0.518113	−	0.897398i
$426$		0			0
$427$	5638.34	+	9765.90i	0.639013	+	1.10680i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17105.0			−1.91164			−0.955821	−	0.293950i	$-0.905030\pi$
$431$	−17105.0			−1.91164			−0.955821	+	0.293950i	$0.905030\pi$
$432$		0			0
$433$	2582.96			0.286672			0.143336	−	0.989674i	$-0.454217\pi$
$433$	2582.96			0.286672			0.143336	+	0.989674i	$0.454217\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10181.3	−	17634.5i	−1.11450	−	1.93037i
$438$		0			0
$439$	3398.93	−	5887.13i	0.369527	−	0.640039i	−0.619965	−	0.784630i	$-0.712853\pi$
$439$	3398.93	−	5887.13i	0.369527	−	0.640039i	0.989492	+	0.144590i	$0.0461864\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5227.04	−	9053.50i	0.560596	−	0.970981i	−0.436848	−	0.899535i	$-0.643905\pi$
$443$	5227.04	−	9053.50i	0.560596	−	0.970981i	0.997444	−	0.0714458i	$-0.0227613\pi$
$444$		0			0
$445$	−1204.45	−	2086.17i	−0.128306	−	0.222233i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17208.6			−1.80873			−0.904367	−	0.426755i	$-0.859657\pi$
$449$	−17208.6			−1.80873			−0.904367	+	0.426755i	$0.859657\pi$
$450$		0			0
$451$	6117.55			0.638723
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2541.20	−	4401.49i	−0.261832	−	0.453506i
$456$		0			0
$457$	−7894.04	+	13672.9i	−0.808025	+	1.39954i	0.106205	+	0.994344i	$0.466130\pi$
$457$	−7894.04	+	13672.9i	−0.808025	+	1.39954i	−0.914230	+	0.405196i	$0.867203\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2591.59	−	4488.76i	0.261827	−	0.453497i	−0.704901	−	0.709306i	$-0.749008\pi$
$461$	2591.59	−	4488.76i	0.261827	−	0.453497i	0.966727	+	0.255809i	$0.0823417\pi$
$462$		0			0
$463$	2916.69	+	5051.86i	0.292765	+	0.507083i	0.974462	−	0.224551i	$-0.0720914\pi$
$463$	2916.69	+	5051.86i	0.292765	+	0.507083i	−0.681698	+	0.731634i	$0.738758\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6315.33			−0.625779			−0.312889	−	0.949790i	$-0.601297\pi$
$467$	−6315.33			−0.625779			−0.312889	+	0.949790i	$0.601297\pi$
$468$		0			0
$469$	−15435.6			−1.51972
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1294.30	+	2241.79i	0.125818	+	0.217923i
$474$		0			0
$475$	8130.59	−	14082.6i	0.785383	−	1.36032i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4867.94	−	8431.52i	0.464346	−	0.804271i	−0.534826	−	0.844963i	$-0.679623\pi$
$479$	4867.94	−	8431.52i	0.464346	−	0.804271i	0.999172	+	0.0406913i	$0.0129560\pi$
$480$		0			0
$481$	2110.64	+	3655.73i	0.200077	+	0.346543i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17908.4			−1.67665
$486$		0			0
$487$	13447.9			1.25130			0.625649	−	0.780104i	$-0.284834\pi$
$487$	13447.9			1.25130			0.625649	+	0.780104i	$0.284834\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3514.04	−	6086.50i	−0.322987	−	0.559430i	0.658116	−	0.752917i	$-0.271354\pi$
$491$	−3514.04	−	6086.50i	−0.322987	−	0.559430i	−0.981103	+	0.193487i	$0.938020\pi$
$492$		0			0
$493$	−10126.6	+	17539.8i	−0.925109	+	1.60234i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−172.695	+	299.117i	−0.0155864	+	0.0269965i
$498$		0			0
$499$	−827.722	−	1433.66i	−0.0742563	−	0.128616i	0.826506	−	0.562927i	$-0.190325\pi$
$499$	−827.722	−	1433.66i	−0.0742563	−	0.128616i	−0.900763	+	0.434312i	$0.856992\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1909.16			−0.169235			−0.0846174	−	0.996414i	$-0.526967\pi$
$503$	−1909.16			−0.169235			−0.0846174	+	0.996414i	$0.526967\pi$
$504$		0			0
$505$	−25991.8			−2.29033
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4432.85	+	7677.92i	0.386017	+	0.668601i	0.991910	−	0.126945i	$-0.0405173\pi$
$509$	4432.85	+	7677.92i	0.386017	+	0.668601i	−0.605893	+	0.795546i	$0.707184\pi$
$510$		0			0
$511$	−9247.17	+	16016.6i	−0.800530	+	1.38656i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5903.10	+	10224.5i	−0.505091	+	0.874843i
$516$		0			0
$517$	565.811	+	980.014i	0.0481322	+	0.0833674i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9256.80			−0.778403			−0.389201	−	0.921153i	$-0.627249\pi$
$521$	−9256.80			−0.778403			−0.389201	+	0.921153i	$0.627249\pi$
$522$		0			0
$523$	13607.5			1.13770			0.568849	−	0.822442i	$-0.307389\pi$
$523$	13607.5			1.13770			0.568849	+	0.822442i	$0.307389\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−671.613	−	1163.27i	−0.0555141	−	0.0961533i
$528$		0			0
$529$	−3868.57	+	6700.56i	−0.317956	+	0.550716i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2969.39	−	5143.13i	0.241310	−	0.417962i
$534$		0			0
$535$	−13473.1	−	23336.1i	−1.08877	−	1.88581i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3206.12			0.256210
$540$		0			0
$541$	−483.548			−0.0384276			−0.0192138	−	0.999815i	$-0.506116\pi$
$541$	−483.548			−0.0384276			−0.0192138	+	0.999815i	$0.506116\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9339.57	−	16176.6i	−0.734061	−	1.27143i
$546$		0			0
$547$	2355.64	−	4080.09i	0.184131	−	0.318925i	−0.759152	−	0.650913i	$-0.774386\pi$
$547$	2355.64	−	4080.09i	0.184131	−	0.318925i	0.943283	+	0.331988i	$0.107719\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18137.5	+	31415.0i	−1.40233	+	2.42890i
$552$		0			0
$553$	3072.37	+	5321.50i	0.236257	+	0.409210i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21650.7			1.64698			0.823492	−	0.567327i	$-0.192023\pi$
$557$	21650.7			1.64698			0.823492	+	0.567327i	$0.192023\pi$
$558$		0			0
$559$	2512.95			0.190137
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7011.50	+	12144.3i	0.524866	+	0.909094i	0.999581	+	0.0289545i	$0.00921780\pi$
$563$	7011.50	+	12144.3i	0.524866	+	0.909094i	−0.474715	+	0.880140i	$0.657449\pi$
$564$		0			0
$565$	7433.99	−	12876.0i	0.553540	−	0.958760i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2437.67	−	4222.16i	0.179600	−	0.311076i	−0.762144	−	0.647408i	$-0.775853\pi$
$569$	2437.67	−	4222.16i	0.179600	−	0.311076i	0.941744	+	0.336332i	$0.109186\pi$
$570$		0			0
$571$	12984.6	+	22489.9i	0.951640	+	1.64829i	0.741876	+	0.670537i	$0.233936\pi$
$571$	12984.6	+	22489.9i	0.951640	+	1.64829i	0.209765	+	0.977752i	$0.432730\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15895.1			−1.15282
$576$		0			0
$577$	−23113.1			−1.66761			−0.833806	−	0.552058i	$-0.813843\pi$
$577$	−23113.1			−1.66761			−0.833806	+	0.552058i	$0.813843\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15314.4	−	26525.4i	−1.09354	−	1.89408i
$582$		0			0
$583$	4729.53	−	8191.78i	0.335981	−	0.581937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2735.83	+	4738.59i	−0.192368	+	0.333190i	−0.946034	−	0.324066i	$-0.894950\pi$
$587$	−2735.83	+	4738.59i	−0.192368	+	0.333190i	0.753667	+	0.657257i	$0.228283\pi$
$588$		0			0
$589$	−1202.91	−	2083.50i	−0.0841512	−	0.145754i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22609.2			−1.56568			−0.782841	−	0.622222i	$-0.786230\pi$
$593$	−22609.2			−1.56568			−0.782841	+	0.622222i	$0.786230\pi$
$594$		0			0
$595$	29607.7			2.04000
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1854.45	−	3211.99i	−0.126495	−	0.219096i	0.795821	−	0.605532i	$-0.207039\pi$
$599$	−1854.45	−	3211.99i	−0.126495	−	0.219096i	−0.922316	+	0.386436i	$0.873706\pi$
$600$		0			0
$601$	5239.08	−	9074.36i	0.355585	−	0.615892i	−0.631633	−	0.775268i	$-0.717615\pi$
$601$	5239.08	−	9074.36i	0.355585	−	0.615892i	0.987218	+	0.159376i	$0.0509483\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8694.54	+	15059.4i	−0.584270	+	1.01199i
$606$		0			0
$607$	−5356.78	−	9278.22i	−0.358196	−	0.620414i	0.629463	−	0.777030i	$-0.283275\pi$
$607$	−5356.78	−	9278.22i	−0.358196	−	0.620414i	−0.987660	+	0.156616i	$0.949941\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1098.55			0.0727376
$612$		0			0
$613$	7860.62			0.517924			0.258962	−	0.965887i	$-0.416619\pi$
$613$	7860.62			0.517924			0.258962	+	0.965887i	$0.416619\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4264.00	−	7385.47i	−0.278221	−	0.481893i	0.692722	−	0.721205i	$-0.256411\pi$
$617$	−4264.00	−	7385.47i	−0.278221	−	0.481893i	−0.970943	+	0.239312i	$0.923078\pi$
$618$		0			0
$619$	4011.45	−	6948.03i	0.260474	−	0.451155i	−0.705894	−	0.708318i	$-0.749454\pi$
$619$	4011.45	−	6948.03i	0.260474	−	0.451155i	0.966368	+	0.257163i	$0.0827877\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1862.00	−	3225.08i	0.119743	−	0.207400i
$624$		0			0
$625$	8507.33	+	14735.1i	0.544469	+	0.943048i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24591.2			−1.55885
$630$		0			0
$631$	−3984.40			−0.251373			−0.125687	−	0.992070i	$-0.540113\pi$
$631$	−3984.40			−0.251373			−0.125687	+	0.992070i	$0.540113\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17253.3	−	29883.7i	−1.07823	−	1.86755i
$636$		0			0
$637$	1556.21	−	2695.44i	0.0967966	−	0.167657i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7375.12	+	12774.1i	−0.454446	+	0.787123i	−0.998656	−	0.0518255i	$-0.983496\pi$
$641$	−7375.12	+	12774.1i	−0.454446	+	0.787123i	0.544210	+	0.838949i	$0.316829\pi$
$642$		0			0
$643$	−4905.37	−	8496.35i	−0.300854	−	0.521094i	0.675476	−	0.737382i	$-0.263938\pi$
$643$	−4905.37	−	8496.35i	−0.300854	−	0.521094i	−0.976330	+	0.216288i	$0.930605\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5683.55			0.345353			0.172677	−	0.984979i	$-0.444758\pi$
$647$	5683.55			0.345353			0.172677	+	0.984979i	$0.444758\pi$
$648$		0			0
$649$	3138.47			0.189824
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1702.76	−	2949.27i	−0.102043	−	0.176744i	0.810483	−	0.585762i	$-0.199205\pi$
$653$	−1702.76	−	2949.27i	−0.102043	−	0.176744i	−0.912526	+	0.409018i	$0.865871\pi$
$654$		0			0
$655$	12905.9	−	22353.7i	0.769887	−	1.33348i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9524.57	−	16497.0i	0.563011	−	0.975164i	−0.434220	−	0.900807i	$-0.642976\pi$
$659$	9524.57	−	16497.0i	0.563011	−	0.975164i	0.997232	−	0.0743575i	$-0.0236906\pi$
$660$		0			0
$661$	−10211.2	−	17686.4i	−0.600864	−	1.04073i	−0.992691	−	0.120688i	$-0.961490\pi$
$661$	−10211.2	−	17686.4i	−0.600864	−	1.04073i	0.391827	−	0.920039i	$-0.371843\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	53029.7			3.09233
$666$		0			0
$667$	35458.3			2.05840
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3371.07	+	5838.86i	0.193947	+	0.335926i
$672$		0			0
$673$	−4306.73	+	7459.47i	−0.246675	+	0.427253i	−0.962601	−	0.270923i	$-0.912671\pi$
$673$	−4306.73	+	7459.47i	−0.246675	+	0.427253i	0.715926	+	0.698176i	$0.246005\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5234.56	+	9066.52i	−0.297164	+	0.514704i	−0.975486	−	0.220061i	$-0.929374\pi$
$677$	−5234.56	+	9066.52i	−0.297164	+	0.514704i	0.678322	+	0.734765i	$0.262708\pi$
$678$		0			0
$679$	−13842.6	−	23976.1i	−0.782373	−	1.35511i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9490.28			−0.531677			−0.265839	−	0.964018i	$-0.585649\pi$
$683$	−9490.28			−0.531677			−0.265839	+	0.964018i	$0.585649\pi$
$684$		0			0
$685$	1465.73			0.0817556
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4591.32	−	7952.39i	−0.253868	−	0.439713i
$690$		0			0
$691$	−7907.36	+	13696.0i	−0.435326	+	0.754007i	−0.997322	−	0.0731330i	$-0.976700\pi$
$691$	−7907.36	+	13696.0i	−0.435326	+	0.754007i	0.561996	+	0.827140i	$0.310034\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9023.54	−	15629.2i	0.492493	−	0.853022i
$696$		0			0
$697$	17298.3	+	29961.5i	0.940056	+	1.62822i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15790.6			−0.850791			−0.425395	−	0.905008i	$-0.639865\pi$
$701$	−15790.6			−0.850791			−0.425395	+	0.905008i	$0.639865\pi$
$702$		0			0
$703$	−44044.7			−2.36298
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20090.8	−	34798.4i	−1.06873	−	1.85110i
$708$		0			0
$709$	−13771.4	+	23852.8i	−0.729474	+	1.26349i	0.227632	+	0.973747i	$0.426902\pi$
$709$	−13771.4	+	23852.8i	−0.729474	+	1.26349i	−0.957106	+	0.289738i	$0.906432\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1175.83	+	2036.60i	−0.0617604	+	0.106972i
$714$		0			0
$715$	−1519.34	−	2631.57i	−0.0794687	−	0.137644i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19578.6			1.01552			0.507761	−	0.861498i	$-0.330473\pi$
$719$	19578.6			1.01552			0.507761	+	0.861498i	$0.330473\pi$
$720$		0			0
$721$	−18251.7			−0.942756
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14158.2	+	24522.7i	0.725272	+	1.25621i
$726$		0			0
$727$	−10474.0	+	18141.5i	−0.534330	+	0.925487i	0.464865	+	0.885382i	$0.346103\pi$
$727$	−10474.0	+	18141.5i	−0.534330	+	0.925487i	−0.999195	+	0.0401059i	$0.987230\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7319.63	+	12678.0i	−0.370351	+	0.641466i
$732$		0			0
$733$	412.674	+	714.773i	0.0207946	+	0.0360174i	0.876235	−	0.481883i	$-0.160047\pi$
$733$	412.674	+	714.773i	0.0207946	+	0.0360174i	−0.855441	+	0.517901i	$0.826714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9228.69			−0.461253
$738$		0			0
$739$	22821.1			1.13598			0.567988	−	0.823037i	$-0.307722\pi$
$739$	22821.1			1.13598			0.567988	+	0.823037i	$0.307722\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6028.23	−	10441.2i	−0.297650	−	0.515545i	0.677948	−	0.735110i	$-0.262870\pi$
$743$	−6028.23	−	10441.2i	−0.297650	−	0.515545i	−0.975598	+	0.219565i	$0.929536\pi$
$744$		0			0
$745$	6622.40	−	11470.3i	0.325672	−	0.564081i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20828.6	−	36076.1i	1.01610	−	1.75994i
$750$		0			0
$751$	3861.16	+	6687.72i	0.187611	+	0.324951i	0.944453	−	0.328646i	$-0.106592\pi$
$751$	3861.16	+	6687.72i	0.187611	+	0.324951i	−0.756842	+	0.653597i	$0.773259\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−44689.5			−2.15420
$756$		0			0
$757$	−1440.60			−0.0691671			−0.0345835	−	0.999402i	$-0.511010\pi$
$757$	−1440.60			−0.0691671			−0.0345835	+	0.999402i	$0.511010\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15361.3	+	26606.5i	0.731729	+	1.26739i	0.956144	+	0.292899i	$0.0946198\pi$
$761$	15361.3	+	26606.5i	0.731729	+	1.26739i	−0.224414	+	0.974494i	$0.572047\pi$
$762$		0			0
$763$	14438.4	−	25008.1i	0.685066	−	1.18657i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1523.38	−	2638.57i	0.0717157	−	0.124215i
$768$		0			0
$769$	20140.4	+	34884.3i	0.944451	+	1.63584i	0.756847	+	0.653592i	$0.226739\pi$
$769$	20140.4	+	34884.3i	0.944451	+	1.63584i	0.187604	+	0.982245i	$0.439928\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4282.45			−0.199261			−0.0996306	−	0.995024i	$-0.531766\pi$
$773$	−4282.45			−0.199261			−0.0996306	+	0.995024i	$0.531766\pi$
$774$		0			0
$775$	−1877.99			−0.0870446
$776$		0			0
$777$		0			0
$778$		0			0
$779$	30982.5	+	53663.3i	1.42499	+	2.46815i
$780$		0			0
$781$	−103.252	+	178.837i	−0.00473064	+	0.00819371i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10090.2	+	17476.7i	−0.458768	+	0.794610i
$786$		0			0
$787$	6285.82	+	10887.4i	0.284708	+	0.493129i	0.972538	−	0.232743i	$-0.0747700\pi$
$787$	6285.82	+	10887.4i	0.284708	+	0.493129i	−0.687830	+	0.725872i	$0.741437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22985.0			1.03319
$792$		0			0
$793$	6545.11			0.293094
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14037.0	+	24312.8i	0.623860	+	1.08056i	0.988760	+	0.149510i	$0.0477697\pi$
$797$	14037.0	+	24312.8i	0.623860	+	1.08056i	−0.364900	+	0.931047i	$0.618897\pi$
$798$		0			0
$799$	−3199.83	+	5542.27i	−0.141679	+	0.245396i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5528.72	+	9576.03i	−0.242969	+	0.420835i
$804$		0			0
$805$	−25917.9	−	44891.2i	−1.13477	−	1.96547i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−293.254			−0.0127445			−0.00637223	−	0.999980i	$-0.502028\pi$
$809$	−293.254			−0.0127445			−0.00637223	+	0.999980i	$0.502028\pi$
$810$		0			0
$811$	−45634.2			−1.97588			−0.987938	−	0.154851i	$-0.950510\pi$
$811$	−45634.2			−1.97588			−0.987938	+	0.154851i	$0.950510\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1472.17	−	2549.88i	−0.0632735	−	0.109593i
$816$		0			0
$817$	−13110.0	+	22707.2i	−0.561397	+	0.972368i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2703.34	+	4682.32i	−0.114917	+	0.199043i	−0.917747	−	0.397166i	$-0.869994\pi$
$821$	−2703.34	+	4682.32i	−0.114917	+	0.199043i	0.802829	+	0.596209i	$0.203327\pi$
$822$		0			0
$823$	−614.440	−	1064.24i	−0.0260243	−	0.0450755i	0.852720	−	0.522368i	$-0.174951\pi$
$823$	−614.440	−	1064.24i	−0.0260243	−	0.0450755i	−0.878744	+	0.477293i	$0.841618\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	45769.8			1.92451			0.962256	−	0.272147i	$-0.0877337\pi$
$827$	45769.8			1.92451			0.962256	+	0.272147i	$0.0877337\pi$
$828$		0			0
$829$	−13630.9			−0.571076			−0.285538	−	0.958367i	$-0.592172\pi$
$829$	−13630.9			−0.571076			−0.285538	+	0.958367i	$0.592172\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9065.78	+	15702.4i	0.377084	+	0.653128i
$834$		0			0
$835$	−11105.4	+	19235.2i	−0.460263	+	0.797198i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22493.8	−	38960.4i	0.925593	−	1.60317i	0.134988	−	0.990847i	$-0.456900\pi$
$839$	22493.8	−	38960.4i	0.925593	−	1.60317i	0.790605	−	0.612327i	$-0.209766\pi$
$840$		0			0
$841$	−19389.2	−	33583.1i	−0.794998	−	1.37698i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	30920.0			1.25879
$846$		0			0
$847$	−26882.5			−1.09055
$848$		0			0
$849$		0			0
$850$		0			0
$851$	21526.6	+	37285.1i	0.867123	+	1.50190i
$852$		0			0
$853$	−13021.6	+	22554.0i	−0.522685	+	0.905317i	0.476967	+	0.878921i	$0.341736\pi$
$853$	−13021.6	+	22554.0i	−0.522685	+	0.905317i	−0.999652	+	0.0263955i	$0.991597\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6853.20	−	11870.1i	0.273163	−	0.473133i	−0.696507	−	0.717550i	$-0.745263\pi$
$857$	6853.20	−	11870.1i	0.273163	−	0.473133i	0.969670	+	0.244417i	$0.0785967\pi$
$858$		0			0
$859$	22659.5	+	39247.5i	0.900039	+	1.55891i	0.827442	+	0.561552i	$0.189795\pi$
$859$	22659.5	+	39247.5i	0.900039	+	1.55891i	0.0725970	+	0.997361i	$0.476871\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28057.3			−1.10670			−0.553350	−	0.832949i	$-0.686651\pi$
$863$	−28057.3			−1.10670			−0.553350	+	0.832949i	$0.686651\pi$
$864$		0			0
$865$	24378.4			0.958253
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1836.91	+	3181.63i	0.0717066	+	0.124200i
$870$		0			0
$871$	−4479.50	+	7758.72i	−0.174262	+	0.301830i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2265.92	+	3924.70i	−0.0875454	+	0.151633i
$876$		0			0
$877$	−1961.59	−	3397.58i	−0.0755282	−	0.130819i	0.825788	−	0.563981i	$-0.190731\pi$
$877$	−1961.59	−	3397.58i	−0.0755282	−	0.130819i	−0.901316	+	0.433162i	$0.857398\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2441.97			0.0933849			0.0466924	−	0.998909i	$-0.485132\pi$
$881$	2441.97			0.0933849			0.0466924	+	0.998909i	$0.485132\pi$
$882$		0			0
$883$	44576.9			1.69890			0.849452	−	0.527667i	$-0.176933\pi$
$883$	44576.9			1.69890			0.849452	+	0.527667i	$0.176933\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8547.74	+	14805.1i	0.323568	+	0.560437i	0.981222	−	0.192884i	$-0.0617841\pi$
$887$	8547.74	+	14805.1i	0.323568	+	0.560437i	−0.657653	+	0.753321i	$0.728451\pi$
$888$		0			0
$889$	26672.6	−	46198.3i	1.00627	−	1.74290i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5731.14	+	9926.62i	−0.214765	+	0.371984i
$894$		0			0
$895$	−5718.22	−	9904.25i	−0.213563	−	0.369902i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4189.37			0.155421
$900$		0			0
$901$	53493.8			1.97795
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4831.40	+	8368.22i	0.177460	+	0.307369i
$906$		0			0
$907$	16571.3	−	28702.4i	0.606662	−	1.05077i	−0.385125	−	0.922865i	$-0.625842\pi$
$907$	16571.3	−	28702.4i	0.606662	−	1.05077i	0.991786	−	0.127905i	$-0.0408252\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16375.2	−	28362.7i	0.595539	−	1.03150i	−0.397932	−	0.917415i	$-0.630272\pi$
$911$	16375.2	−	28362.7i	0.595539	−	1.03150i	0.993471	−	0.114089i	$-0.0363948\pi$
$912$		0			0
$913$	−9156.23	−	15859.1i	−0.331902	−	0.574872i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	39903.5			1.43700
$918$		0			0
$919$	6095.76			0.218804			0.109402	−	0.993998i	$-0.465106\pi$
$919$	6095.76			0.218804			0.109402	+	0.993998i	$0.465106\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	100.234	+	173.611i	0.00357448	+	0.00619119i
$924$		0			0
$925$	−17190.8	+	29775.3i	−0.611058	+	1.05838i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12354.0	−	21397.7i	0.436298	−	0.755691i	−0.561102	−	0.827747i	$-0.689623\pi$
$929$	12354.0	−	21397.7i	0.436298	−	0.755691i	0.997401	+	0.0720555i	$0.0229559\pi$
$930$		0			0
$931$	16237.5	+	28124.2i	0.571603	+	0.990045i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	17701.9			0.619161
$936$		0			0
$937$	−44713.2			−1.55893			−0.779465	−	0.626446i	$-0.784509\pi$
$937$	−44713.2			−1.55893			−0.779465	+	0.626446i	$0.784509\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5086.37	−	8809.85i	−0.176207	−	0.305200i	0.764371	−	0.644776i	$-0.223049\pi$
$941$	−5086.37	−	8809.85i	−0.176207	−	0.305200i	−0.940578	+	0.339577i	$0.889716\pi$
$942$		0			0
$943$	30285.0	−	52455.2i	1.04583	−	1.81143i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6673.28	+	11558.5i	−0.228989	+	0.396621i	−0.957509	−	0.288404i	$-0.906875\pi$
$947$	−6673.28	+	11558.5i	−0.228989	+	0.396621i	0.728520	+	0.685025i	$0.240209\pi$
$948$		0			0
$949$	5367.15	+	9296.18i	0.183588	+	0.317984i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36232.4			−1.23157			−0.615783	−	0.787916i	$-0.711160\pi$
$953$	−36232.4			−1.23157			−0.615783	+	0.787916i	$0.711160\pi$
$954$		0			0
$955$	2815.26			0.0953924
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1132.96	+	1962.35i	0.0381494	+	0.0660767i
$960$		0			0
$961$	14756.6	−	25559.1i	0.495337	−	0.857948i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	13247.8	−	22945.8i	0.441929	−	0.765443i
$966$		0			0
$967$	−2714.51	−	4701.68i	−0.0902718	−	0.156355i	0.817354	−	0.576136i	$-0.195440\pi$
$967$	−2714.51	−	4701.68i	−0.0902718	−	0.156355i	−0.907625	+	0.419781i	$0.862107\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5645.37			0.186579			0.0932897	−	0.995639i	$-0.470262\pi$
$971$	5645.37			0.186579			0.0932897	+	0.995639i	$0.470262\pi$
$972$		0			0
$973$	27899.7			0.919242
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25551.8	−	44257.0i	−0.836719	−	1.44924i	−0.892623	−	0.450804i	$-0.851137\pi$
$977$	−25551.8	−	44257.0i	−0.836719	−	1.44924i	0.0559039	−	0.998436i	$-0.482196\pi$
$978$		0			0
$979$	1113.26	−	1928.22i	0.0363431	−	0.0629481i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16041.3	+	27784.3i	−0.520485	+	0.901507i	0.479231	+	0.877689i	$0.340916\pi$
$983$	−16041.3	+	27784.3i	−0.520485	+	0.901507i	−0.999716	+	0.0238179i	$0.992418\pi$
$984$		0			0
$985$	921.799	+	1596.60i	0.0298182	+	0.0516467i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25629.8			0.824043
$990$		0			0
$991$	−27696.1			−0.887785			−0.443893	−	0.896080i	$-0.646403\pi$
$991$	−27696.1			−0.887785			−0.443893	+	0.896080i	$0.646403\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5453.30	+	9445.40i	0.173750	+	0.300944i
$996$		0			0
$997$	20642.5	−	35753.8i	0.655720	−	1.13574i	−0.325992	−	0.945372i	$-0.605698\pi$
$997$	20642.5	−	35753.8i	0.655720	−	1.13574i	0.981713	−	0.190369i	$-0.0609683\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.r.217.2		4
3.2	odd	2		648.4.i.o.217.1		4
9.2	odd	6		216.4.a.g.1.2	yes	2
9.4	even	3	inner	648.4.i.r.433.2		4
9.5	odd	6		648.4.i.o.433.1		4
9.7	even	3		216.4.a.f.1.1	✓	2
36.7	odd	6		432.4.a.p.1.1		2
36.11	even	6		432.4.a.r.1.2		2
72.11	even	6		1728.4.a.bj.1.1		2
72.29	odd	6		1728.4.a.bi.1.1		2
72.43	odd	6		1728.4.a.br.1.2		2
72.61	even	6		1728.4.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.a.f.1.1	✓	2	9.7	even	3
216.4.a.g.1.2	yes	2	9.2	odd	6
432.4.a.p.1.1		2	36.7	odd	6
432.4.a.r.1.2		2	36.11	even	6
648.4.i.o.217.1		4	3.2	odd	2
648.4.i.o.433.1		4	9.5	odd	6
648.4.i.r.217.2		4	1.1	even	1	trivial
648.4.i.r.433.2		4	9.4	even	3	inner
1728.4.a.bi.1.1		2	72.29	odd	6
1728.4.a.bj.1.1		2	72.11	even	6
1728.4.a.bq.1.2		2	72.61	even	6
1728.4.a.br.1.2		2	72.43	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.70820 + 13.3510i) q^{5} +(-11.9164 + 20.6398i) q^{7} +O(q^{10})\)	\(q+(7.70820 + 13.3510i) q^{5} +(-11.9164 + 20.6398i) q^{7} +(-7.12461 + 12.3402i) q^{11} +(6.91641 + 11.9796i) q^{13} -80.5836 q^{17} -144.331 q^{19} +(70.5410 + 122.181i) q^{23} +(-56.3328 + 97.5713i) q^{25} +(125.666 - 217.659i) q^{29} +(8.33437 + 14.4355i) q^{31} -367.416 q^{35} +305.164 q^{37} +(-214.663 - 371.806i) q^{41} +(90.8328 - 157.327i) q^{43} +(39.7082 - 68.7766i) q^{47} +(-112.502 - 194.858i) q^{49} -663.830 q^{53} -219.672 q^{55} +(-110.128 - 190.747i) q^{59} +(236.579 - 409.767i) q^{61} +(-106.626 + 184.682i) q^{65} +(323.831 + 560.892i) q^{67} +14.4922 q^{71} +776.003 q^{73} +(-169.800 - 294.101i) q^{77} +(128.913 - 223.284i) q^{79} +(-642.577 + 1112.98i) q^{83} +(-621.155 - 1075.87i) q^{85} -156.255 q^{89} -329.675 q^{91} +(-1112.53 - 1926.97i) q^{95} +(-580.822 + 1006.01i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 6 * q^7	\( 4 q + 4 q^{5} + 6 q^{7} + 52 q^{11} - 26 q^{13} - 376 q^{17} - 148 q^{19} + 148 q^{23} - 118 q^{25} + 288 q^{29} + 248 q^{31} - 1416 q^{35} + 684 q^{37} + 256 q^{43} + 132 q^{47} - 772 q^{49} - 1904 q^{53} - 1952 q^{55} - 1004 q^{59} + 34 q^{61} - 668 q^{65} + 866 q^{67} - 1552 q^{71} + 3748 q^{73} - 2316 q^{77} - 182 q^{79} - 1336 q^{83} - 16 q^{85} - 1752 q^{89} - 3036 q^{91} - 3028 q^{95} + 38 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 6 * q^7 + 52 * q^11 - 26 * q^13 - 376 * q^17 - 148 * q^19 + 148 * q^23 - 118 * q^25 + 288 * q^29 + 248 * q^31 - 1416 * q^35 + 684 * q^37 + 256 * q^43 + 132 * q^47 - 772 * q^49 - 1904 * q^53 - 1952 * q^55 - 1004 * q^59 + 34 * q^61 - 668 * q^65 + 866 * q^67 - 1552 * q^71 + 3748 * q^73 - 2316 * q^77 - 182 * q^79 - 1336 * q^83 - 16 * q^85 - 1752 * q^89 - 3036 * q^91 - 3028 * q^95 + 38 * q^97

Introduction

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