LMFDB - Embedded newform 1600.2.n.p.1407.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1407.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1600.1407
Dual form			1600.2.n.p.1343.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.224745 + 0.224745i) q^{3} +(-2.00000 + 2.00000i) q^{7} -2.89898i q^{9} +O(q^{10})$	$q+(0.224745 + 0.224745i) q^{3} +(-2.00000 + 2.00000i) q^{7} -2.89898i q^{9} -1.44949i q^{11} +(-0.449490 + 0.449490i) q^{13} +(3.22474 + 3.22474i) q^{17} -7.44949 q^{19} -0.898979 q^{21} +(4.44949 + 4.44949i) q^{23} +(1.32577 - 1.32577i) q^{27} +6.89898i q^{29} +6.89898i q^{31} +(0.325765 - 0.325765i) q^{33} +(-6.00000 - 6.00000i) q^{37} -0.202041 q^{39} +5.00000 q^{41} +(6.89898 + 6.89898i) q^{43} +(-7.34847 + 7.34847i) q^{47} -1.00000i q^{49} +1.44949i q^{51} +(-1.55051 + 1.55051i) q^{53} +(-1.67423 - 1.67423i) q^{57} -11.7980 q^{59} -8.89898 q^{61} +(5.79796 + 5.79796i) q^{63} +(-1.77526 + 1.77526i) q^{67} +2.00000i q^{69} -2.00000i q^{71} +(1.67423 - 1.67423i) q^{73} +(2.89898 + 2.89898i) q^{77} -8.89898 q^{79} -8.10102 q^{81} +(2.67423 + 2.67423i) q^{83} +(-1.55051 + 1.55051i) q^{87} -2.10102i q^{89} -1.79796i q^{91} +(-1.55051 + 1.55051i) q^{93} +(4.89898 + 4.89898i) q^{97} -4.20204 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7	$ 4 q - 4 q^{3} - 8 q^{7} + 8 q^{13} + 8 q^{17} - 20 q^{19} + 16 q^{21} + 8 q^{23} + 20 q^{27} + 16 q^{33} - 24 q^{37} - 40 q^{39} + 20 q^{41} + 8 q^{43} - 16 q^{53} + 8 q^{57} - 8 q^{59} - 16 q^{61} - 16 q^{63} - 12 q^{67} - 8 q^{73} - 8 q^{77} - 16 q^{79} - 52 q^{81} - 4 q^{83} - 16 q^{87} - 16 q^{93} - 56 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 8 * q^13 + 8 * q^17 - 20 * q^19 + 16 * q^21 + 8 * q^23 + 20 * q^27 + 16 * q^33 - 24 * q^37 - 40 * q^39 + 20 * q^41 + 8 * q^43 - 16 * q^53 + 8 * q^57 - 8 * q^59 - 16 * q^61 - 16 * q^63 - 12 * q^67 - 8 * q^73 - 8 * q^77 - 16 * q^79 - 52 * q^81 - 4 * q^83 - 16 * q^87 - 16 * q^93 - 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.224745	+	0.224745i	0.129757	+	0.129757i	0.769002	−	0.639246i	$-0.220753\pi$
$3$	0.224745	+	0.224745i	0.129757	+	0.129757i	−0.639246	+	0.769002i	$0.720753\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	0.219650	+	0.975579i	$0.429509\pi$
$8$		0			0
$9$		−	2.89898i		−	0.966326i
$10$		0			0
$11$		−	1.44949i		−	0.437038i	−0.975833	−	0.218519i	$-0.929878\pi$
$11$		−	1.44949i		−	0.437038i	0.975833	−	0.218519i	$-0.0701225\pi$
$12$		0			0
$13$	−0.449490	+	0.449490i	−0.124666	+	0.124666i	−0.766687	−	0.642021i	$-0.778096\pi$
$13$	−0.449490	+	0.449490i	−0.124666	+	0.124666i	0.642021	+	0.766687i	$0.278096\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.22474	+	3.22474i	0.782116	+	0.782116i	0.980187	−	0.198072i	$-0.0634680\pi$
$17$	3.22474	+	3.22474i	0.782116	+	0.782116i	−0.198072	+	0.980187i	$0.563468\pi$
$18$		0			0
$19$	−7.44949			−1.70903			−0.854515	−	0.519427i	$-0.826146\pi$
$19$	−7.44949			−1.70903			−0.854515	+	0.519427i	$0.826146\pi$
$20$		0			0
$21$	−0.898979			−0.196173
$22$		0			0
$23$	4.44949	+	4.44949i	0.927783	+	0.927783i	0.997562	−	0.0697797i	$-0.0222296\pi$
$23$	4.44949	+	4.44949i	0.927783	+	0.927783i	−0.0697797	+	0.997562i	$0.522230\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.32577	−	1.32577i	0.255144	−	0.255144i
$28$		0			0
$29$			6.89898i			1.28111i	0.767913	+	0.640554i	$0.221295\pi$
$29$			6.89898i			1.28111i	−0.767913	+	0.640554i	$0.778705\pi$
$30$		0			0
$31$			6.89898i			1.23909i	0.784960	+	0.619547i	$0.212684\pi$
$31$			6.89898i			1.23909i	−0.784960	+	0.619547i	$0.787316\pi$
$32$		0			0
$33$	0.325765	−	0.325765i	0.0567085	−	0.0567085i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.00000	−	6.00000i	−0.986394	−	0.986394i	0.0135147	−	0.999909i	$-0.495698\pi$
$37$	−6.00000	−	6.00000i	−0.986394	−	0.986394i	−0.999909	+	0.0135147i	$0.995698\pi$
$38$		0			0
$39$	−0.202041			−0.0323525
$40$		0			0
$41$	5.00000			0.780869			0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000			0.780869			0.390434	+	0.920631i	$0.372325\pi$
$42$		0			0
$43$	6.89898	+	6.89898i	1.05208	+	1.05208i	0.998567	+	0.0535176i	$0.0170433\pi$
$43$	6.89898	+	6.89898i	1.05208	+	1.05208i	0.0535176	+	0.998567i	$0.482957\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.0746766	+	0.997208i	$0.523792\pi$
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.997208	+	0.0746766i	$0.976208\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$			1.44949i			0.202969i
$52$		0			0
$53$	−1.55051	+	1.55051i	−0.212979	+	0.212979i	−0.805532	−	0.592553i	$-0.798120\pi$
$53$	−1.55051	+	1.55051i	−0.212979	+	0.212979i	0.592553	+	0.805532i	$0.298120\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.67423	−	1.67423i	−0.221758	−	0.221758i
$58$		0			0
$59$	−11.7980			−1.53596			−0.767982	−	0.640472i	$-0.778739\pi$
$59$	−11.7980			−1.53596			−0.767982	+	0.640472i	$0.778739\pi$
$60$		0			0
$61$	−8.89898			−1.13940			−0.569699	−	0.821854i	$-0.692940\pi$
$61$	−8.89898			−1.13940			−0.569699	+	0.821854i	$0.692940\pi$
$62$		0			0
$63$	5.79796	+	5.79796i	0.730474	+	0.730474i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.77526	+	1.77526i	−0.216882	+	0.216882i	−0.807183	−	0.590301i	$-0.799009\pi$
$67$	−1.77526	+	1.77526i	−0.216882	+	0.216882i	0.590301	+	0.807183i	$0.299009\pi$
$68$		0			0
$69$			2.00000i			0.240772i
$70$		0			0
$71$		−	2.00000i		−	0.237356i	−0.992933	−	0.118678i	$-0.962134\pi$
$71$		−	2.00000i		−	0.237356i	0.992933	−	0.118678i	$-0.0378657\pi$
$72$		0			0
$73$	1.67423	−	1.67423i	0.195954	−	0.195954i	−0.602309	−	0.798263i	$-0.705752\pi$
$73$	1.67423	−	1.67423i	0.195954	−	0.195954i	0.798263	+	0.602309i	$0.205752\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.89898	+	2.89898i	0.330369	+	0.330369i
$78$		0			0
$79$	−8.89898			−1.00121			−0.500607	−	0.865675i	$-0.666890\pi$
$79$	−8.89898			−1.00121			−0.500607	+	0.865675i	$0.666890\pi$
$80$		0			0
$81$	−8.10102			−0.900113
$82$		0			0
$83$	2.67423	+	2.67423i	0.293535	+	0.293535i	0.838475	−	0.544940i	$-0.183447\pi$
$83$	2.67423	+	2.67423i	0.293535	+	0.293535i	−0.544940	+	0.838475i	$0.683447\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.55051	+	1.55051i	−0.166232	+	0.166232i
$88$		0			0
$89$		−	2.10102i		−	0.222708i	−0.993781	−	0.111354i	$-0.964481\pi$
$89$		−	2.10102i		−	0.222708i	0.993781	−	0.111354i	$-0.0355187\pi$
$90$		0			0
$91$		−	1.79796i		−	0.188477i
$92$		0			0
$93$	−1.55051	+	1.55051i	−0.160780	+	0.160780i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.89898	+	4.89898i	0.497416	+	0.497416i	0.910633	−	0.413217i	$-0.135595\pi$
$97$	4.89898	+	4.89898i	0.497416	+	0.497416i	−0.413217	+	0.910633i	$0.635595\pi$
$98$		0			0
$99$	−4.20204			−0.422321
$100$		0			0
$101$	14.6969			1.46240			0.731200	−	0.682163i	$-0.238961\pi$
$101$	14.6969			1.46240			0.731200	+	0.682163i	$0.238961\pi$
$102$		0			0
$103$	2.44949	+	2.44949i	0.241355	+	0.241355i	0.817411	−	0.576055i	$-0.195409\pi$
$103$	2.44949	+	2.44949i	0.241355	+	0.241355i	−0.576055	+	0.817411i	$0.695409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.5732	+	11.5732i	−1.11882	+	1.11882i	−0.126911	+	0.991914i	$0.540506\pi$
$107$	−11.5732	+	11.5732i	−1.11882	+	1.11882i	−0.991914	+	0.126911i	$0.959494\pi$
$108$		0			0
$109$		−	12.8990i		−	1.23550i	−0.786375	−	0.617749i	$-0.788045\pi$
$109$		−	12.8990i		−	1.23550i	0.786375	−	0.617749i	$-0.211955\pi$
$110$		0			0
$111$		−	2.69694i		−	0.255982i
$112$		0			0
$113$	−4.32577	+	4.32577i	−0.406934	+	0.406934i	−0.880668	−	0.473734i	$-0.842906\pi$
$113$	−4.32577	+	4.32577i	−0.406934	+	0.406934i	0.473734	+	0.880668i	$0.342906\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.30306	+	1.30306i	0.120468	+	0.120468i
$118$		0			0
$119$	−12.8990			−1.18245
$120$		0			0
$121$	8.89898			0.808998
$122$		0			0
$123$	1.12372	+	1.12372i	0.101323	+	0.101323i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.55051	−	3.55051i	0.315057	−	0.315057i	−0.531808	−	0.846865i	$-0.678487\pi$
$127$	3.55051	−	3.55051i	0.315057	−	0.315057i	0.846865	+	0.531808i	$0.178487\pi$
$128$		0			0
$129$			3.10102i			0.273030i
$130$		0			0
$131$			3.79796i			0.331829i	0.986140	+	0.165915i	$0.0530576\pi$
$131$			3.79796i			0.331829i	−0.986140	+	0.165915i	$0.946942\pi$
$132$		0			0
$133$	14.8990	−	14.8990i	1.29191	−	1.29191i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.1237	−	10.1237i	−0.864928	−	0.864928i	0.126977	−	0.991906i	$-0.459472\pi$
$137$	−10.1237	−	10.1237i	−0.864928	−	0.864928i	−0.991906	+	0.126977i	$0.959472\pi$
$138$		0			0
$139$	15.2474			1.29327			0.646636	−	0.762799i	$-0.276175\pi$
$139$	15.2474			1.29327			0.646636	+	0.762799i	$0.276175\pi$
$140$		0			0
$141$	−3.30306			−0.278168
$142$		0			0
$143$	0.651531	+	0.651531i	0.0544837	+	0.0544837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.224745	−	0.224745i	0.0185366	−	0.0185366i
$148$		0			0
$149$		−	9.10102i		−	0.745585i	−0.927915	−	0.372792i	$-0.878400\pi$
$149$		−	9.10102i		−	0.745585i	0.927915	−	0.372792i	$-0.121600\pi$
$150$		0			0
$151$			6.69694i			0.544989i	0.962157	+	0.272495i	$0.0878487\pi$
$151$			6.69694i			0.544989i	−0.962157	+	0.272495i	$0.912151\pi$
$152$		0			0
$153$	9.34847	−	9.34847i	0.755779	−	0.755779i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.2474	+	16.2474i	1.29669	+	1.29669i	0.930568	+	0.366120i	$0.119314\pi$
$157$	16.2474	+	16.2474i	1.29669	+	1.29669i	0.366120	+	0.930568i	$0.380686\pi$
$158$		0			0
$159$	−0.696938			−0.0552708
$160$		0			0
$161$	−17.7980			−1.40268
$162$		0			0
$163$	10.2247	+	10.2247i	0.800864	+	0.800864i	0.983231	−	0.182367i	$-0.0583758\pi$
$163$	10.2247	+	10.2247i	0.800864	+	0.800864i	−0.182367	+	0.983231i	$0.558376\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.55051	−	5.55051i	0.429511	−	0.429511i	−0.458950	−	0.888462i	$-0.651774\pi$
$167$	5.55051	−	5.55051i	0.429511	−	0.429511i	0.888462	+	0.458950i	$0.151774\pi$
$168$		0			0
$169$			12.5959i			0.968917i
$170$		0			0
$171$			21.5959i			1.65148i
$172$		0			0
$173$	−13.7980	+	13.7980i	−1.04904	+	1.04904i	−0.0503055	+	0.998734i	$0.516020\pi$
$173$	−13.7980	+	13.7980i	−1.04904	+	1.04904i	−0.998734	+	0.0503055i	$0.983980\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.65153	−	2.65153i	−0.199301	−	0.199301i
$178$		0			0
$179$	9.24745			0.691187			0.345593	−	0.938384i	$-0.387678\pi$
$179$	9.24745			0.691187			0.345593	+	0.938384i	$0.387678\pi$
$180$		0			0
$181$	4.20204			0.312335			0.156168	−	0.987731i	$-0.450086\pi$
$181$	4.20204			0.312335			0.156168	+	0.987731i	$0.450086\pi$
$182$		0			0
$183$	−2.00000	−	2.00000i	−0.147844	−	0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.67423	−	4.67423i	0.341814	−	0.341814i
$188$		0			0
$189$			5.30306i			0.385741i
$190$		0			0
$191$		−	22.8990i		−	1.65691i	−0.560054	−	0.828456i	$-0.689220\pi$
$191$		−	22.8990i		−	1.65691i	0.560054	−	0.828456i	$-0.310780\pi$
$192$		0			0
$193$	−16.1237	+	16.1237i	−1.16061	+	1.16061i	−0.176269	+	0.984342i	$0.556403\pi$
$193$	−16.1237	+	16.1237i	−1.16061	+	1.16061i	−0.984342	+	0.176269i	$0.943597\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.44949	−	6.44949i	−0.459507	−	0.459507i	0.438987	−	0.898494i	$-0.355338\pi$
$197$	−6.44949	−	6.44949i	−0.459507	−	0.459507i	−0.898494	+	0.438987i	$0.855338\pi$
$198$		0			0
$199$	−9.79796			−0.694559			−0.347279	−	0.937762i	$-0.612894\pi$
$199$	−9.79796			−0.694559			−0.347279	+	0.937762i	$0.612894\pi$
$200$		0			0
$201$	−0.797959			−0.0562837
$202$		0			0
$203$	−13.7980	−	13.7980i	−0.968427	−	0.968427i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	12.8990	−	12.8990i	0.896541	−	0.896541i
$208$		0			0
$209$			10.7980i			0.746910i
$210$		0			0
$211$		−	11.2474i		−	0.774306i	−0.922015	−	0.387153i	$-0.873458\pi$
$211$		−	11.2474i		−	0.774306i	0.922015	−	0.387153i	$-0.126542\pi$
$212$		0			0
$213$	0.449490	−	0.449490i	0.0307985	−	0.0307985i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.7980	−	13.7980i	−0.936666	−	0.936666i
$218$		0			0
$219$	0.752551			0.0508527
$220$		0			0
$221$	−2.89898			−0.195006
$222$		0			0
$223$	−7.10102	−	7.10102i	−0.475520	−	0.475520i	0.428176	−	0.903695i	$-0.359156\pi$
$223$	−7.10102	−	7.10102i	−0.475520	−	0.475520i	−0.903695	+	0.428176i	$0.859156\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0000	+	10.0000i	−0.663723	+	0.663723i	−0.956256	−	0.292532i	$-0.905502\pi$
$227$	−10.0000	+	10.0000i	−0.663723	+	0.663723i	0.292532	+	0.956256i	$0.405502\pi$
$228$		0			0
$229$		−	5.79796i		−	0.383140i	−0.981479	−	0.191570i	$-0.938642\pi$
$229$		−	5.79796i		−	0.383140i	0.981479	−	0.191570i	$-0.0613579\pi$
$230$		0			0
$231$			1.30306i			0.0857352i
$232$		0			0
$233$	7.10102	−	7.10102i	0.465203	−	0.465203i	−0.435153	−	0.900356i	$-0.643306\pi$
$233$	7.10102	−	7.10102i	0.465203	−	0.465203i	0.900356	+	0.435153i	$0.143306\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.00000	−	2.00000i	−0.129914	−	0.129914i
$238$		0			0
$239$	13.5959			0.879447			0.439723	−	0.898133i	$-0.355076\pi$
$239$	13.5959			0.879447			0.439723	+	0.898133i	$0.355076\pi$
$240$		0			0
$241$	−3.89898			−0.251155			−0.125578	−	0.992084i	$-0.540078\pi$
$241$	−3.89898			−0.251155			−0.125578	+	0.992084i	$0.540078\pi$
$242$		0			0
$243$	−5.79796	−	5.79796i	−0.371939	−	0.371939i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.34847	−	3.34847i	0.213058	−	0.213058i
$248$		0			0
$249$			1.20204i			0.0761763i
$250$		0			0
$251$		−	12.5505i		−	0.792181i	−0.918211	−	0.396091i	$-0.870367\pi$
$251$		−	12.5505i		−	0.792181i	0.918211	−	0.396091i	$-0.129633\pi$
$252$		0			0
$253$	6.44949	−	6.44949i	0.405476	−	0.405476i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.6969	+	10.6969i	0.667257	+	0.667257i	0.957080	−	0.289823i	$-0.0935966\pi$
$257$	10.6969	+	10.6969i	0.667257	+	0.667257i	−0.289823	+	0.957080i	$0.593597\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$		0			0
$261$	20.0000			1.23797
$262$		0			0
$263$	−18.2474	−	18.2474i	−1.12519	−	1.12519i	−0.990949	−	0.134237i	$-0.957142\pi$
$263$	−18.2474	−	18.2474i	−1.12519	−	1.12519i	−0.134237	−	0.990949i	$-0.542858\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.472194	−	0.472194i	0.0288978	−	0.0288978i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			17.1010i			1.03881i	0.854527	+	0.519407i	$0.173847\pi$
$271$			17.1010i			1.03881i	−0.854527	+	0.519407i	$0.826153\pi$
$272$		0			0
$273$	0.404082	−	0.404082i	0.0244562	−	0.0244562i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−12.4495	−	12.4495i	−0.748017	−	0.748017i	0.226089	−	0.974107i	$-0.427406\pi$
$277$	−12.4495	−	12.4495i	−0.748017	−	0.748017i	−0.974107	+	0.226089i	$0.927406\pi$
$278$		0			0
$279$	20.0000			1.19737
$280$		0			0
$281$	12.2020			0.727913			0.363956	−	0.931416i	$-0.381426\pi$
$281$	12.2020			0.727913			0.363956	+	0.931416i	$0.381426\pi$
$282$		0			0
$283$	−1.57321	−	1.57321i	−0.0935179	−	0.0935179i	0.658800	−	0.752318i	$-0.271064\pi$
$283$	−1.57321	−	1.57321i	−0.0935179	−	0.0935179i	−0.752318	+	0.658800i	$0.771064\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0000	+	10.0000i	−0.590281	+	0.590281i
$288$		0			0
$289$			3.79796i			0.223409i
$290$		0			0
$291$			2.20204i			0.129086i
$292$		0			0
$293$	2.89898	−	2.89898i	0.169360	−	0.169360i	−0.617338	−	0.786698i	$-0.711789\pi$
$293$	2.89898	−	2.89898i	0.169360	−	0.169360i	0.786698	+	0.617338i	$0.211789\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.92168	−	1.92168i	−0.111507	−	0.111507i
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$	−27.5959			−1.59060
$302$		0			0
$303$	3.30306	+	3.30306i	0.189756	+	0.189756i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	17.5732	−	17.5732i	1.00296	−	1.00296i	0.00296069	−	0.999996i	$-0.499058\pi$
$307$	17.5732	−	17.5732i	1.00296	−	1.00296i	0.999996	−	0.00296069i	$-0.000942417\pi$
$308$		0			0
$309$			1.10102i			0.0626349i
$310$		0			0
$311$		−	24.8990i		−	1.41189i	−0.708266	−	0.705946i	$-0.750522\pi$
$311$		−	24.8990i		−	1.41189i	0.708266	−	0.705946i	$-0.249478\pi$
$312$		0			0
$313$	−7.10102	+	7.10102i	−0.401373	+	0.401373i	−0.878717	−	0.477343i	$-0.841600\pi$
$313$	−7.10102	+	7.10102i	−0.401373	+	0.401373i	0.477343	+	0.878717i	$0.341600\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.6969	+	22.6969i	1.27479	+	1.27479i	0.943546	+	0.331241i	$0.107467\pi$
$317$	22.6969	+	22.6969i	1.27479	+	1.27479i	0.331241	+	0.943546i	$0.392533\pi$
$318$		0			0
$319$	10.0000			0.559893
$320$		0			0
$321$	−5.20204			−0.290350
$322$		0			0
$323$	−24.0227	−	24.0227i	−1.33666	−	1.33666i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.89898	−	2.89898i	0.160314	−	0.160314i
$328$		0			0
$329$		−	29.3939i		−	1.62054i
$330$		0			0
$331$		−	4.34847i		−	0.239013i	−0.992833	−	0.119507i	$-0.961869\pi$
$331$		−	4.34847i		−	0.239013i	0.992833	−	0.119507i	$-0.0381313\pi$
$332$		0			0
$333$	−17.3939	+	17.3939i	−0.953179	+	0.953179i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.5732	+	24.5732i	1.33859	+	1.33859i	0.897429	+	0.441160i	$0.145433\pi$
$337$	24.5732	+	24.5732i	1.33859	+	1.33859i	0.441160	+	0.897429i	$0.354567\pi$
$338$		0			0
$339$	−1.94439			−0.105605
$340$		0			0
$341$	10.0000			0.541530
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.3258	−	13.3258i	0.715365	−	0.715365i	−0.252288	−	0.967652i	$-0.581183\pi$
$347$	13.3258	−	13.3258i	0.715365	−	0.715365i	0.967652	+	0.252288i	$0.0811829\pi$
$348$		0			0
$349$			18.6969i			1.00082i	0.865787	+	0.500412i	$0.166818\pi$
$349$			18.6969i			1.00082i	−0.865787	+	0.500412i	$0.833182\pi$
$350$		0			0
$351$			1.19184i			0.0636155i
$352$		0			0
$353$	16.8990	−	16.8990i	0.899442	−	0.899442i	−0.0959447	−	0.995387i	$-0.530587\pi$
$353$	16.8990	−	16.8990i	0.899442	−	0.899442i	0.995387	+	0.0959447i	$0.0305872\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.89898	−	2.89898i	−0.153430	−	0.153430i
$358$		0			0
$359$	28.8990			1.52523			0.762615	−	0.646853i	$-0.223915\pi$
$359$	28.8990			1.52523			0.762615	+	0.646853i	$0.223915\pi$
$360$		0			0
$361$	36.4949			1.92078
$362$		0			0
$363$	2.00000	+	2.00000i	0.104973	+	0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	10.6969	−	10.6969i	0.558376	−	0.558376i	−0.370469	−	0.928845i	$-0.620803\pi$
$367$	10.6969	−	10.6969i	0.558376	−	0.558376i	0.928845	+	0.370469i	$0.120803\pi$
$368$		0			0
$369$		−	14.4949i		−	0.754574i
$370$		0			0
$371$		−	6.20204i		−	0.321994i
$372$		0			0
$373$	−2.44949	+	2.44949i	−0.126830	+	0.126830i	−0.767672	−	0.640843i	$-0.778585\pi$
$373$	−2.44949	+	2.44949i	−0.126830	+	0.126830i	0.640843	+	0.767672i	$0.278585\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.10102	−	3.10102i	−0.159711	−	0.159711i
$378$		0			0
$379$	11.0454			0.567364			0.283682	−	0.958918i	$-0.408444\pi$
$379$	11.0454			0.567364			0.283682	+	0.958918i	$0.408444\pi$
$380$		0			0
$381$	1.59592			0.0817614
$382$		0			0
$383$	−7.79796	−	7.79796i	−0.398457	−	0.398457i	0.479231	−	0.877689i	$-0.340915\pi$
$383$	−7.79796	−	7.79796i	−0.398457	−	0.398457i	−0.877689	+	0.479231i	$0.840915\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.0000	−	20.0000i	1.01666	−	1.01666i
$388$		0			0
$389$			7.79796i			0.395372i	0.980265	+	0.197686i	$0.0633427\pi$
$389$			7.79796i			0.395372i	−0.980265	+	0.197686i	$0.936657\pi$
$390$		0			0
$391$			28.6969i			1.45127i
$392$		0			0
$393$	−0.853572	+	0.853572i	−0.0430570	+	0.0430570i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13.7980	−	13.7980i	−0.692500	−	0.692500i	0.270282	−	0.962781i	$-0.412883\pi$
$397$	−13.7980	−	13.7980i	−0.692500	−	0.692500i	−0.962781	+	0.270282i	$0.912883\pi$
$398$		0			0
$399$	6.69694			0.335266
$400$		0			0
$401$	−13.8990			−0.694082			−0.347041	−	0.937850i	$-0.612814\pi$
$401$	−13.8990			−0.694082			−0.347041	+	0.937850i	$0.612814\pi$
$402$		0			0
$403$	−3.10102	−	3.10102i	−0.154473	−	0.154473i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.69694	+	8.69694i	−0.431091	+	0.431091i
$408$		0			0
$409$		−	8.79796i		−	0.435031i	−0.976057	−	0.217516i	$-0.930205\pi$
$409$		−	8.79796i		−	0.435031i	0.976057	−	0.217516i	$-0.0697953\pi$
$410$		0			0
$411$		−	4.55051i		−	0.224460i
$412$		0			0
$413$	23.5959	−	23.5959i	1.16108	−	1.16108i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.42679	+	3.42679i	0.167810	+	0.167810i
$418$		0			0
$419$	−12.1464			−0.593392			−0.296696	−	0.954972i	$-0.595885\pi$
$419$	−12.1464			−0.593392			−0.296696	+	0.954972i	$0.595885\pi$
$420$		0			0
$421$	−11.5959			−0.565150			−0.282575	−	0.959245i	$-0.591189\pi$
$421$	−11.5959			−0.565150			−0.282575	+	0.959245i	$0.591189\pi$
$422$		0			0
$423$	21.3031	+	21.3031i	1.03579	+	1.03579i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.7980	−	17.7980i	0.861304	−	0.861304i
$428$		0			0
$429$			0.292856i			0.0141392i
$430$		0			0
$431$			3.10102i			0.149371i	0.997207	+	0.0746855i	$0.0237953\pi$
$431$			3.10102i			0.149371i	−0.997207	+	0.0746855i	$0.976205\pi$
$432$		0			0
$433$	−13.6742	+	13.6742i	−0.657142	+	0.657142i	−0.954703	−	0.297561i	$-0.903827\pi$
$433$	−13.6742	+	13.6742i	−0.657142	+	0.657142i	0.297561	+	0.954703i	$0.403827\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−33.1464	−	33.1464i	−1.58561	−	1.58561i
$438$		0			0
$439$	26.0000			1.24091			0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000			1.24091			0.620456	+	0.784241i	$0.286947\pi$
$440$		0			0
$441$	−2.89898			−0.138047
$442$		0			0
$443$	−14.9217	−	14.9217i	−0.708951	−	0.708951i	0.257364	−	0.966315i	$-0.417146\pi$
$443$	−14.9217	−	14.9217i	−0.708951	−	0.708951i	−0.966315	+	0.257364i	$0.917146\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	2.04541	−	2.04541i	0.0967445	−	0.0967445i
$448$		0			0
$449$		−	3.89898i		−	0.184004i	−0.995759	−	0.0920021i	$-0.970673\pi$
$449$		−	3.89898i		−	0.184004i	0.995759	−	0.0920021i	$-0.0293267\pi$
$450$		0			0
$451$		−	7.24745i		−	0.341269i
$452$		0			0
$453$	−1.50510	+	1.50510i	−0.0707159	+	0.0707159i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.977296	−	0.977296i	−0.0457160	−	0.0457160i	0.683879	−	0.729595i	$-0.260292\pi$
$457$	−0.977296	−	0.977296i	−0.0457160	−	0.0457160i	−0.729595	+	0.683879i	$0.760292\pi$
$458$		0			0
$459$	8.55051			0.399104
$460$		0			0
$461$	−21.7980			−1.01523			−0.507616	−	0.861583i	$-0.669473\pi$
$461$	−21.7980			−1.01523			−0.507616	+	0.861583i	$0.669473\pi$
$462$		0			0
$463$	−26.6969	−	26.6969i	−1.24071	−	1.24071i	−0.959706	−	0.281006i	$-0.909332\pi$
$463$	−26.6969	−	26.6969i	−1.24071	−	1.24071i	−0.281006	−	0.959706i	$-0.590668\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.4949	+	22.4949i	−1.04094	+	1.04094i	−0.0418143	+	0.999125i	$0.513314\pi$
$467$	−22.4949	+	22.4949i	−1.04094	+	1.04094i	−0.999125	+	0.0418143i	$0.986686\pi$
$468$		0			0
$469$		−	7.10102i		−	0.327895i
$470$		0			0
$471$			7.30306i			0.336507i
$472$		0			0
$473$	10.0000	−	10.0000i	0.459800	−	0.459800i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.49490	+	4.49490i	0.205807	+	0.205807i
$478$		0			0
$479$	6.89898			0.315222			0.157611	−	0.987501i	$-0.449621\pi$
$479$	6.89898			0.315222			0.157611	+	0.987501i	$0.449621\pi$
$480$		0			0
$481$	5.39388			0.245940
$482$		0			0
$483$	−4.00000	−	4.00000i	−0.182006	−	0.182006i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.55051	+	7.55051i	−0.342146	+	0.342146i	−0.857174	−	0.515027i	$-0.827782\pi$
$487$	−7.55051	+	7.55051i	−0.342146	+	0.342146i	0.515027	+	0.857174i	$0.327782\pi$
$488$		0			0
$489$			4.59592i			0.207835i
$490$		0			0
$491$		−	37.5959i		−	1.69668i	−0.529452	−	0.848340i	$-0.677602\pi$
$491$		−	37.5959i		−	1.69668i	0.529452	−	0.848340i	$-0.322398\pi$
$492$		0			0
$493$	−22.2474	+	22.2474i	−1.00197	+	1.00197i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.00000	+	4.00000i	0.179425	+	0.179425i
$498$		0			0
$499$	−11.7980			−0.528149			−0.264075	−	0.964502i	$-0.585067\pi$
$499$	−11.7980			−0.528149			−0.264075	+	0.964502i	$0.585067\pi$
$500$		0			0
$501$	2.49490			0.111464
$502$		0			0
$503$	5.79796	+	5.79796i	0.258518	+	0.258518i	0.824451	−	0.565933i	$-0.191484\pi$
$503$	5.79796	+	5.79796i	0.258518	+	0.258518i	−0.565933	+	0.824451i	$0.691484\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.83087	+	2.83087i	−0.125723	+	0.125723i
$508$		0			0
$509$			39.7980i			1.76401i	0.471237	+	0.882007i	$0.343808\pi$
$509$			39.7980i			1.76401i	−0.471237	+	0.882007i	$0.656192\pi$
$510$		0			0
$511$			6.69694i			0.296255i
$512$		0			0
$513$	−9.87628	+	9.87628i	−0.436048	+	0.436048i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	10.6515	+	10.6515i	0.468454	+	0.468454i
$518$		0			0
$519$	−6.20204			−0.272239
$520$		0			0
$521$	−19.0000			−0.832405			−0.416203	−	0.909272i	$-0.636639\pi$
$521$	−19.0000			−0.832405			−0.416203	+	0.909272i	$0.636639\pi$
$522$		0			0
$523$	5.77526	+	5.77526i	0.252534	+	0.252534i	0.822009	−	0.569475i	$-0.192853\pi$
$523$	5.77526	+	5.77526i	0.252534	+	0.252534i	−0.569475	+	0.822009i	$0.692853\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.2474	+	22.2474i	−0.969114	+	0.969114i
$528$		0			0
$529$			16.5959i			0.721562i
$530$		0			0
$531$			34.2020i			1.48424i
$532$		0			0
$533$	−2.24745	+	2.24745i	−0.0973478	+	0.0973478i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.07832	+	2.07832i	0.0896860	+	0.0896860i
$538$		0			0
$539$	−1.44949			−0.0624339
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	0.944387	+	0.944387i	0.0405275	+	0.0405275i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.67423	−	8.67423i	0.370883	−	0.370883i	−0.496915	−	0.867799i	$-0.665534\pi$
$547$	8.67423	−	8.67423i	0.370883	−	0.370883i	0.867799	+	0.496915i	$0.165534\pi$
$548$		0			0
$549$			25.7980i			1.10103i
$550$		0			0
$551$		−	51.3939i		−	2.18945i
$552$		0			0
$553$	17.7980	−	17.7980i	0.756846	−	0.756846i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.55051	+	3.55051i	0.150440	+	0.150440i	0.778315	−	0.627875i	$-0.216075\pi$
$557$	3.55051	+	3.55051i	0.150440	+	0.150440i	−0.627875	+	0.778315i	$0.716075\pi$
$558$		0			0
$559$	−6.20204			−0.262318
$560$		0			0
$561$	2.10102			0.0887052
$562$		0			0
$563$	1.10102	+	1.10102i	0.0464025	+	0.0464025i	0.729927	−	0.683525i	$-0.239554\pi$
$563$	1.10102	+	1.10102i	0.0464025	+	0.0464025i	−0.683525	+	0.729927i	$0.739554\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	16.2020	−	16.2020i	0.680422	−	0.680422i
$568$		0			0
$569$		−	5.40408i		−	0.226551i	−0.993564	−	0.113275i	$-0.963866\pi$
$569$		−	5.40408i		−	0.226551i	0.993564	−	0.113275i	$-0.0361343\pi$
$570$		0			0
$571$			14.0000i			0.585882i	0.956131	+	0.292941i	$0.0946339\pi$
$571$			14.0000i			0.585882i	−0.956131	+	0.292941i	$0.905366\pi$
$572$		0			0
$573$	5.14643	−	5.14643i	0.214995	−	0.214995i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−6.77526	−	6.77526i	−0.282058	−	0.282058i	0.551872	−	0.833929i	$-0.313914\pi$
$577$	−6.77526	−	6.77526i	−0.282058	−	0.282058i	−0.833929	+	0.551872i	$0.813914\pi$
$578$		0			0
$579$	−7.24745			−0.301194
$580$		0			0
$581$	−10.6969			−0.443784
$582$		0			0
$583$	2.24745	+	2.24745i	0.0930798	+	0.0930798i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.57321	+	1.57321i	−0.0649335	+	0.0649335i	−0.738828	−	0.673894i	$-0.764620\pi$
$587$	−1.57321	+	1.57321i	−0.0649335	+	0.0649335i	0.673894	+	0.738828i	$0.264620\pi$
$588$		0			0
$589$		−	51.3939i		−	2.11765i
$590$		0			0
$591$		−	2.89898i		−	0.119248i
$592$		0			0
$593$	2.77526	−	2.77526i	0.113966	−	0.113966i	−0.647824	−	0.761790i	$-0.724321\pi$
$593$	2.77526	−	2.77526i	0.113966	−	0.113966i	0.761790	+	0.647824i	$0.224321\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.20204	−	2.20204i	−0.0901235	−	0.0901235i
$598$		0			0
$599$	−30.6969			−1.25424			−0.627121	−	0.778921i	$-0.715767\pi$
$599$	−30.6969			−1.25424			−0.627121	+	0.778921i	$0.715767\pi$
$600$		0			0
$601$	15.6969			0.640291			0.320146	−	0.947368i	$-0.396268\pi$
$601$	15.6969			0.640291			0.320146	+	0.947368i	$0.396268\pi$
$602$		0			0
$603$	5.14643	+	5.14643i	0.209579	+	0.209579i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	13.7980	−	13.7980i	0.560042	−	0.560042i	−0.369277	−	0.929319i	$-0.620395\pi$
$607$	13.7980	−	13.7980i	0.560042	−	0.560042i	0.929319	+	0.369277i	$0.120395\pi$
$608$		0			0
$609$		−	6.20204i		−	0.251319i
$610$		0			0
$611$		−	6.60612i		−	0.267255i
$612$		0			0
$613$	29.1464	−	29.1464i	1.17721	−	1.17721i	0.196762	−	0.980451i	$-0.436957\pi$
$613$	29.1464	−	29.1464i	1.17721	−	1.17721i	0.980451	−	0.196762i	$-0.0630426\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.5959	−	15.5959i	−0.627868	−	0.627868i	0.319663	−	0.947531i	$-0.396430\pi$
$617$	−15.5959	−	15.5959i	−0.627868	−	0.627868i	−0.947531	+	0.319663i	$0.896430\pi$
$618$		0			0
$619$	−18.0000			−0.723481			−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000			−0.723481			−0.361741	+	0.932279i	$0.617817\pi$
$620$		0			0
$621$	11.7980			0.473436
$622$		0			0
$623$	4.20204	+	4.20204i	0.168351	+	0.168351i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−2.42679	+	2.42679i	−0.0969165	+	0.0969165i
$628$		0			0
$629$		−	38.6969i		−	1.54295i
$630$		0			0
$631$			24.6969i			0.983170i	0.870830	+	0.491585i	$0.163582\pi$
$631$			24.6969i			0.983170i	−0.870830	+	0.491585i	$0.836418\pi$
$632$		0			0
$633$	2.52781	−	2.52781i	0.100471	−	0.100471i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.449490	+	0.449490i	0.0178094	+	0.0178094i
$638$		0			0
$639$	−5.79796			−0.229364
$640$		0			0
$641$	−3.79796			−0.150010			−0.0750052	−	0.997183i	$-0.523897\pi$
$641$	−3.79796			−0.150010			−0.0750052	+	0.997183i	$0.523897\pi$
$642$		0			0
$643$	28.6969	+	28.6969i	1.13170	+	1.13170i	0.989895	+	0.141802i	$0.0452895\pi$
$643$	28.6969	+	28.6969i	1.13170	+	1.13170i	0.141802	+	0.989895i	$0.454710\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.5959	+	11.5959i	−0.455883	+	0.455883i	−0.897301	−	0.441419i	$-0.854475\pi$
$647$	−11.5959	+	11.5959i	−0.455883	+	0.455883i	0.441419	+	0.897301i	$0.354475\pi$
$648$		0			0
$649$			17.1010i			0.671274i
$650$		0			0
$651$		−	6.20204i		−	0.243077i
$652$		0			0
$653$	−26.0000	+	26.0000i	−1.01746	+	1.01746i	−0.0176138	+	0.999845i	$0.505607\pi$
$653$	−26.0000	+	26.0000i	−1.01746	+	1.01746i	−0.999845	+	0.0176138i	$0.994393\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−4.85357	−	4.85357i	−0.189356	−	0.189356i
$658$		0			0
$659$	30.1464			1.17434			0.587169	−	0.809464i	$-0.300242\pi$
$659$	30.1464			1.17434			0.587169	+	0.809464i	$0.300242\pi$
$660$		0			0
$661$	28.2929			1.10046			0.550232	−	0.835012i	$-0.314539\pi$
$661$	28.2929			1.10046			0.550232	+	0.835012i	$0.314539\pi$
$662$		0			0
$663$	−0.651531	−	0.651531i	−0.0253034	−	0.0253034i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−30.6969	+	30.6969i	−1.18859	+	1.18859i
$668$		0			0
$669$		−	3.19184i		−	0.123404i
$670$		0			0
$671$			12.8990i			0.497960i
$672$		0			0
$673$	−5.79796	+	5.79796i	−0.223495	+	0.223495i	−0.809968	−	0.586474i	$-0.800516\pi$
$673$	−5.79796	+	5.79796i	−0.223495	+	0.223495i	0.586474	+	0.809968i	$0.300516\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.0000	+	28.0000i	1.07613	+	1.07613i	0.996853	+	0.0792746i	$0.0252604\pi$
$677$	28.0000	+	28.0000i	1.07613	+	1.07613i	0.0792746	+	0.996853i	$0.474740\pi$
$678$		0			0
$679$	−19.5959			−0.752022
$680$		0			0
$681$	−4.49490			−0.172245
$682$		0			0
$683$	9.77526	+	9.77526i	0.374040	+	0.374040i	0.868946	−	0.494906i	$-0.164798\pi$
$683$	9.77526	+	9.77526i	0.374040	+	0.374040i	−0.494906	+	0.868946i	$0.664798\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.30306	−	1.30306i	0.0497149	−	0.0497149i
$688$		0			0
$689$		−	1.39388i		−	0.0531025i
$690$		0			0
$691$			13.9444i			0.530469i	0.964184	+	0.265235i	$0.0854495\pi$
$691$			13.9444i			0.530469i	−0.964184	+	0.265235i	$0.914551\pi$
$692$		0			0
$693$	8.40408	−	8.40408i	0.319245	−	0.319245i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	16.1237	+	16.1237i	0.610730	+	0.610730i
$698$		0			0
$699$	3.19184			0.120726
$700$		0			0
$701$	22.8990			0.864883			0.432441	−	0.901662i	$-0.357652\pi$
$701$	22.8990			0.864883			0.432441	+	0.901662i	$0.357652\pi$
$702$		0			0
$703$	44.6969	+	44.6969i	1.68578	+	1.68578i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.3939	+	29.3939i	−1.10547	+	1.10547i
$708$		0			0
$709$			12.0000i			0.450669i	0.974281	+	0.225335i	$0.0723476\pi$
$709$			12.0000i			0.450669i	−0.974281	+	0.225335i	$0.927652\pi$
$710$		0			0
$711$			25.7980i			0.967499i
$712$		0			0
$713$	−30.6969	+	30.6969i	−1.14961	+	1.14961i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	3.05561	+	3.05561i	0.114114	+	0.114114i
$718$		0			0
$719$	−40.8990			−1.52527			−0.762637	−	0.646826i	$-0.776096\pi$
$719$	−40.8990			−1.52527			−0.762637	+	0.646826i	$0.776096\pi$
$720$		0			0
$721$	−9.79796			−0.364895
$722$		0			0
$723$	−0.876276	−	0.876276i	−0.0325890	−	0.0325890i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	17.1464	−	17.1464i	0.635926	−	0.635926i	−0.313622	−	0.949548i	$-0.601542\pi$
$727$	17.1464	−	17.1464i	0.635926	−	0.635926i	0.949548	+	0.313622i	$0.101542\pi$
$728$		0			0
$729$			21.6969i			0.803590i
$730$		0			0
$731$			44.4949i			1.64570i
$732$		0			0
$733$	20.4495	−	20.4495i	0.755319	−	0.755319i	−0.220147	−	0.975467i	$-0.570654\pi$
$733$	20.4495	−	20.4495i	0.755319	−	0.755319i	0.975467	+	0.220147i	$0.0706539\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.57321	+	2.57321i	0.0947856	+	0.0947856i
$738$		0			0
$739$	5.59592			0.205849			0.102925	−	0.994689i	$-0.467180\pi$
$739$	5.59592			0.205849			0.102925	+	0.994689i	$0.467180\pi$
$740$		0			0
$741$	1.50510			0.0552913
$742$		0			0
$743$	−9.34847	−	9.34847i	−0.342962	−	0.342962i	0.514518	−	0.857480i	$-0.327971\pi$
$743$	−9.34847	−	9.34847i	−0.342962	−	0.342962i	−0.857480	+	0.514518i	$0.827971\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	7.75255	−	7.75255i	0.283651	−	0.283651i
$748$		0			0
$749$		−	46.2929i		−	1.69150i
$750$		0			0
$751$		−	37.1918i		−	1.35715i	−0.734531	−	0.678575i	$-0.762598\pi$
$751$		−	37.1918i		−	1.35715i	0.734531	−	0.678575i	$-0.237402\pi$
$752$		0			0
$753$	2.82066	−	2.82066i	0.102791	−	0.102791i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	14.6969	+	14.6969i	0.534169	+	0.534169i	0.921810	−	0.387641i	$-0.126710\pi$
$757$	14.6969	+	14.6969i	0.534169	+	0.534169i	−0.387641	+	0.921810i	$0.626710\pi$
$758$		0			0
$759$	2.89898			0.105226
$760$		0			0
$761$	−17.0000			−0.616250			−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000			−0.616250			−0.308125	+	0.951346i	$0.599701\pi$
$762$		0			0
$763$	25.7980	+	25.7980i	0.933949	+	0.933949i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.30306	−	5.30306i	0.191482	−	0.191482i
$768$		0			0
$769$		−	3.69694i		−	0.133315i	−0.997776	−	0.0666575i	$-0.978767\pi$
$769$		−	3.69694i		−	0.133315i	0.997776	−	0.0666575i	$-0.0212335\pi$
$770$		0			0
$771$			4.80816i			0.173162i
$772$		0			0
$773$	−24.9444	+	24.9444i	−0.897187	+	0.897187i	−0.995186	−	0.0979992i	$-0.968756\pi$
$773$	−24.9444	+	24.9444i	−0.897187	+	0.897187i	0.0979992	+	0.995186i	$0.468756\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.39388	+	5.39388i	0.193504	+	0.193504i
$778$		0			0
$779$	−37.2474			−1.33453
$780$		0			0
$781$	−2.89898			−0.103734
$782$		0			0
$783$	9.14643	+	9.14643i	0.326867	+	0.326867i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	2.89898	−	2.89898i	0.103337	−	0.103337i	−0.653548	−	0.756885i	$-0.726720\pi$
$787$	2.89898	−	2.89898i	0.103337	−	0.103337i	0.756885	+	0.653548i	$0.226720\pi$
$788$		0			0
$789$		−	8.20204i		−	0.292000i
$790$		0			0
$791$		−	17.3031i		−	0.615226i
$792$		0			0
$793$	4.00000	−	4.00000i	0.142044	−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.34847	+	3.34847i	0.118609	+	0.118609i	0.763920	−	0.645311i	$-0.223272\pi$
$797$	3.34847	+	3.34847i	0.118609	+	0.118609i	−0.645311	+	0.763920i	$0.723272\pi$
$798$		0			0
$799$	−47.3939			−1.67667
$800$		0			0
$801$	−6.09082			−0.215208
$802$		0			0
$803$	−2.42679	−	2.42679i	−0.0856394	−	0.0856394i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			19.7980i			0.696059i	0.937484	+	0.348030i	$0.113149\pi$
$809$			19.7980i			0.696059i	−0.937484	+	0.348030i	$0.886851\pi$
$810$		0			0
$811$			26.0000i			0.912983i	0.889728	+	0.456492i	$0.150894\pi$
$811$			26.0000i			0.912983i	−0.889728	+	0.456492i	$0.849106\pi$
$812$		0			0
$813$	−3.84337	+	3.84337i	−0.134793	+	0.134793i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−51.3939	−	51.3939i	−1.79804	−	1.79804i
$818$		0			0
$819$	−5.21225			−0.182131
$820$		0			0
$821$	3.79796			0.132550			0.0662748	−	0.997801i	$-0.478889\pi$
$821$	3.79796			0.132550			0.0662748	+	0.997801i	$0.478889\pi$
$822$		0			0
$823$	18.6969	+	18.6969i	0.651734	+	0.651734i	0.953410	−	0.301676i	$-0.0975461\pi$
$823$	18.6969	+	18.6969i	0.651734	+	0.651734i	−0.301676	+	0.953410i	$0.597546\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−34.9217	+	34.9217i	−1.21435	+	1.21435i	−0.244763	+	0.969583i	$0.578710\pi$
$827$	−34.9217	+	34.9217i	−1.21435	+	1.21435i	−0.969583	+	0.244763i	$0.921290\pi$
$828$		0			0
$829$			46.8990i			1.62887i	0.580256	+	0.814434i	$0.302953\pi$
$829$			46.8990i			1.62887i	−0.580256	+	0.814434i	$0.697047\pi$
$830$		0			0
$831$		−	5.59592i		−	0.194120i
$832$		0			0
$833$	3.22474	−	3.22474i	0.111731	−	0.111731i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	9.14643	+	9.14643i	0.316147	+	0.316147i
$838$		0			0
$839$	−39.1918			−1.35305			−0.676526	−	0.736419i	$-0.736515\pi$
$839$	−39.1918			−1.35305			−0.676526	+	0.736419i	$0.736515\pi$
$840$		0			0
$841$	−18.5959			−0.641239
$842$		0			0
$843$	2.74235	+	2.74235i	0.0944514	+	0.0944514i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−17.7980	+	17.7980i	−0.611545	+	0.611545i
$848$		0			0
$849$		−	0.707144i		−	0.0242691i
$850$		0			0
$851$		−	53.3939i		−	1.83032i
$852$		0			0
$853$	−19.3485	+	19.3485i	−0.662479	+	0.662479i	−0.955964	−	0.293485i	$-0.905185\pi$
$853$	−19.3485	+	19.3485i	−0.662479	+	0.662479i	0.293485	+	0.955964i	$0.405185\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.5732	−	24.5732i	−0.839405	−	0.839405i	0.149376	−	0.988781i	$-0.452274\pi$
$857$	−24.5732	−	24.5732i	−0.839405	−	0.839405i	−0.988781	+	0.149376i	$0.952274\pi$
$858$		0			0
$859$	27.6515			0.943458			0.471729	−	0.881744i	$-0.343630\pi$
$859$	27.6515			0.943458			0.471729	+	0.881744i	$0.343630\pi$
$860$		0			0
$861$	−4.49490			−0.153186
$862$		0			0
$863$	−10.4495	−	10.4495i	−0.355705	−	0.355705i	0.506522	−	0.862227i	$-0.330931\pi$
$863$	−10.4495	−	10.4495i	−0.355705	−	0.355705i	−0.862227	+	0.506522i	$0.830931\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.853572	+	0.853572i	−0.0289888	+	0.0289888i
$868$		0			0
$869$			12.8990i			0.437568i
$870$		0			0
$871$		−	1.59592i		−	0.0540756i
$872$		0			0
$873$	14.2020	−	14.2020i	0.480666	−	0.480666i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−18.6969	−	18.6969i	−0.631351	−	0.631351i	0.317056	−	0.948407i	$-0.397306\pi$
$877$	−18.6969	−	18.6969i	−0.631351	−	0.631351i	−0.948407	+	0.317056i	$0.897306\pi$
$878$		0			0
$879$	1.30306			0.0439512
$880$		0			0
$881$	35.7980			1.20606			0.603032	−	0.797717i	$-0.293959\pi$
$881$	35.7980			1.20606			0.603032	+	0.797717i	$0.293959\pi$
$882$		0			0
$883$	19.8207	+	19.8207i	0.667018	+	0.667018i	0.957025	−	0.290006i	$-0.0936574\pi$
$883$	19.8207	+	19.8207i	0.667018	+	0.667018i	−0.290006	+	0.957025i	$0.593657\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−40.0000	+	40.0000i	−1.34307	+	1.34307i	−0.450081	+	0.892988i	$0.648605\pi$
$887$	−40.0000	+	40.0000i	−1.34307	+	1.34307i	−0.892988	+	0.450081i	$0.851395\pi$
$888$		0			0
$889$			14.2020i			0.476321i
$890$		0			0
$891$			11.7423i			0.393383i
$892$		0			0
$893$	54.7423	−	54.7423i	1.83188	−	1.83188i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.898979	−	0.898979i	−0.0300161	−	0.0300161i
$898$		0			0
$899$	−47.5959			−1.58741
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$	−6.20204	−	6.20204i	−0.206391	−	0.206391i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.10102	+	1.10102i	−0.0365588	+	0.0365588i	−0.725150	−	0.688591i	$-0.758230\pi$
$907$	−1.10102	+	1.10102i	−0.0365588	+	0.0365588i	0.688591	+	0.725150i	$0.258230\pi$
$908$		0			0
$909$		−	42.6061i		−	1.41316i
$910$		0			0
$911$		−	10.2020i		−	0.338009i	−0.985615	−	0.169004i	$-0.945945\pi$
$911$		−	10.2020i		−	0.338009i	0.985615	−	0.169004i	$-0.0540552\pi$
$912$		0			0
$913$	3.87628	−	3.87628i	0.128286	−	0.128286i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.59592	−	7.59592i	−0.250839	−	0.250839i
$918$		0			0
$919$	−27.3031			−0.900645			−0.450322	−	0.892866i	$-0.648691\pi$
$919$	−27.3031			−0.900645			−0.450322	+	0.892866i	$0.648691\pi$
$920$		0			0
$921$	7.89898			0.260280
$922$		0			0
$923$	0.898979	+	0.898979i	0.0295903	+	0.0295903i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.10102	−	7.10102i	0.233228	−	0.233228i
$928$		0			0
$929$			45.5959i			1.49595i	0.663725	+	0.747977i	$0.268975\pi$
$929$			45.5959i			1.49595i	−0.663725	+	0.747977i	$0.731025\pi$
$930$		0			0
$931$			7.44949i			0.244147i
$932$		0			0
$933$	5.59592	−	5.59592i	0.183202	−	0.183202i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	21.6742	+	21.6742i	0.708066	+	0.708066i	0.966128	−	0.258062i	$-0.0830839\pi$
$937$	21.6742	+	21.6742i	0.708066	+	0.708066i	−0.258062	+	0.966128i	$0.583084\pi$
$938$		0			0
$939$	−3.19184			−0.104162
$940$		0			0
$941$	18.4949			0.602916			0.301458	−	0.953479i	$-0.402527\pi$
$941$	18.4949			0.602916			0.301458	+	0.953479i	$0.402527\pi$
$942$		0			0
$943$	22.2474	+	22.2474i	0.724477	+	0.724477i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.1010	+	17.1010i	−0.555708	+	0.555708i	−0.928083	−	0.372374i	$-0.878544\pi$
$947$	−17.1010	+	17.1010i	−0.555708	+	0.555708i	0.372374	+	0.928083i	$0.378544\pi$
$948$		0			0
$949$			1.50510i			0.0488577i
$950$		0			0
$951$			10.2020i			0.330824i
$952$		0			0
$953$	−30.1237	+	30.1237i	−0.975803	+	0.975803i	−0.999714	−	0.0239109i	$-0.992388\pi$
$953$	−30.1237	+	30.1237i	−0.975803	+	0.975803i	0.0239109	+	0.999714i	$0.492388\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	2.24745	+	2.24745i	0.0726497	+	0.0726497i
$958$		0			0
$959$	40.4949			1.30765
$960$		0			0
$961$	−16.5959			−0.535352
$962$		0			0
$963$	33.5505	+	33.5505i	1.08115	+	1.08115i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	7.10102	−	7.10102i	0.228353	−	0.228353i	−0.583651	−	0.812005i	$-0.698376\pi$
$967$	7.10102	−	7.10102i	0.228353	−	0.228353i	0.812005	+	0.583651i	$0.198376\pi$
$968$		0			0
$969$		−	10.7980i		−	0.346880i
$970$		0			0
$971$		−	2.14643i		−	0.0688822i	−0.999407	−	0.0344411i	$-0.989035\pi$
$971$		−	2.14643i		−	0.0688822i	0.999407	−	0.0344411i	$-0.0109651\pi$
$972$		0			0
$973$	−30.4949	+	30.4949i	−0.977622	+	0.977622i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	26.3258	+	26.3258i	0.842236	+	0.842236i	0.989149	−	0.146913i	$-0.0469339\pi$
$977$	26.3258	+	26.3258i	0.842236	+	0.842236i	−0.146913	+	0.989149i	$0.546934\pi$
$978$		0			0
$979$	−3.04541			−0.0973317
$980$		0			0
$981$	−37.3939			−1.19389
$982$		0			0
$983$	6.89898	+	6.89898i	0.220043	+	0.220043i	0.808517	−	0.588473i	$-0.200271\pi$
$983$	6.89898	+	6.89898i	0.220043	+	0.220043i	−0.588473	+	0.808517i	$0.700271\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	6.60612	−	6.60612i	0.210275	−	0.210275i
$988$		0			0
$989$			61.3939i			1.95221i
$990$		0			0
$991$			23.1010i			0.733828i	0.930255	+	0.366914i	$0.119586\pi$
$991$			23.1010i			0.733828i	−0.930255	+	0.366914i	$0.880414\pi$
$992$		0			0
$993$	0.977296	−	0.977296i	0.0310136	−	0.0310136i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.79796	+	1.79796i	0.0569419	+	0.0569419i	0.735004	−	0.678062i	$-0.237180\pi$
$997$	1.79796	+	1.79796i	0.0569419	+	0.0569419i	−0.678062	+	0.735004i	$0.737180\pi$
$998$		0			0
$999$	−15.9092			−0.503344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.n.p.1407.2		4
4.3	odd	2		1600.2.n.t.1407.1		4
5.2	odd	4		1600.2.n.o.1343.2		4
5.3	odd	4		1600.2.n.t.1343.1		4
5.4	even	2		1600.2.n.s.1407.1		4
8.3	odd	2		800.2.n.k.607.2	yes	4
8.5	even	2		800.2.n.m.607.1	yes	4
20.3	even	4	inner	1600.2.n.p.1343.2		4
20.7	even	4		1600.2.n.s.1343.1		4
20.19	odd	2		1600.2.n.o.1407.2		4
40.3	even	4		800.2.n.m.543.1	yes	4
40.13	odd	4		800.2.n.k.543.2	✓	4
40.19	odd	2		800.2.n.n.607.1	yes	4
40.27	even	4		800.2.n.l.543.2	yes	4
40.29	even	2		800.2.n.l.607.2	yes	4
40.37	odd	4		800.2.n.n.543.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.n.k.543.2	✓	4	40.13	odd	4
800.2.n.k.607.2	yes	4	8.3	odd	2
800.2.n.l.543.2	yes	4	40.27	even	4
800.2.n.l.607.2	yes	4	40.29	even	2
800.2.n.m.543.1	yes	4	40.3	even	4
800.2.n.m.607.1	yes	4	8.5	even	2
800.2.n.n.543.1	yes	4	40.37	odd	4
800.2.n.n.607.1	yes	4	40.19	odd	2
1600.2.n.o.1343.2		4	5.2	odd	4
1600.2.n.o.1407.2		4	20.19	odd	2
1600.2.n.p.1343.2		4	20.3	even	4	inner
1600.2.n.p.1407.2		4	1.1	even	1	trivial
1600.2.n.s.1343.1		4	20.7	even	4
1600.2.n.s.1407.1		4	5.4	even	2
1600.2.n.t.1343.1		4	5.3	odd	4
1600.2.n.t.1407.1		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.224745 + 0.224745i) q^{3} +(-2.00000 + 2.00000i) q^{7} -2.89898i q^{9} +O(q^{10})\)	\(q+(0.224745 + 0.224745i) q^{3} +(-2.00000 + 2.00000i) q^{7} -2.89898i q^{9} -1.44949i q^{11} +(-0.449490 + 0.449490i) q^{13} +(3.22474 + 3.22474i) q^{17} -7.44949 q^{19} -0.898979 q^{21} +(4.44949 + 4.44949i) q^{23} +(1.32577 - 1.32577i) q^{27} +6.89898i q^{29} +6.89898i q^{31} +(0.325765 - 0.325765i) q^{33} +(-6.00000 - 6.00000i) q^{37} -0.202041 q^{39} +5.00000 q^{41} +(6.89898 + 6.89898i) q^{43} +(-7.34847 + 7.34847i) q^{47} -1.00000i q^{49} +1.44949i q^{51} +(-1.55051 + 1.55051i) q^{53} +(-1.67423 - 1.67423i) q^{57} -11.7980 q^{59} -8.89898 q^{61} +(5.79796 + 5.79796i) q^{63} +(-1.77526 + 1.77526i) q^{67} +2.00000i q^{69} -2.00000i q^{71} +(1.67423 - 1.67423i) q^{73} +(2.89898 + 2.89898i) q^{77} -8.89898 q^{79} -8.10102 q^{81} +(2.67423 + 2.67423i) q^{83} +(-1.55051 + 1.55051i) q^{87} -2.10102i q^{89} -1.79796i q^{91} +(-1.55051 + 1.55051i) q^{93} +(4.89898 + 4.89898i) q^{97} -4.20204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7	\( 4 q - 4 q^{3} - 8 q^{7} + 8 q^{13} + 8 q^{17} - 20 q^{19} + 16 q^{21} + 8 q^{23} + 20 q^{27} + 16 q^{33} - 24 q^{37} - 40 q^{39} + 20 q^{41} + 8 q^{43} - 16 q^{53} + 8 q^{57} - 8 q^{59} - 16 q^{61} - 16 q^{63} - 12 q^{67} - 8 q^{73} - 8 q^{77} - 16 q^{79} - 52 q^{81} - 4 q^{83} - 16 q^{87} - 16 q^{93} - 56 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 8 * q^13 + 8 * q^17 - 20 * q^19 + 16 * q^21 + 8 * q^23 + 20 * q^27 + 16 * q^33 - 24 * q^37 - 40 * q^39 + 20 * q^41 + 8 * q^43 - 16 * q^53 + 8 * q^57 - 8 * q^59 - 16 * q^61 - 16 * q^63 - 12 * q^67 - 8 * q^73 - 8 * q^77 - 16 * q^79 - 52 * q^81 - 4 * q^83 - 16 * q^87 - 16 * q^93 - 56 * q^99

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