LMFDB - Embedded newform 1600.2.o.e.607.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(543,1600)]

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, 2, 3]))

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

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

chi := DirichletCharacter("1600.543");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.o (of order $4$, degree $2$, 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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			607.2
Root			$0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	1600.607
Dual form			1600.2.o.e.543.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.456850 + 0.456850i) q^{3} +(-2.79129 + 2.79129i) q^{7} +2.58258i q^{9} +O(q^{10})$	$q+(-0.456850 + 0.456850i) q^{3} +(-2.79129 + 2.79129i) q^{7} +2.58258i q^{9} -4.37780 q^{11} +(1.73205 + 1.73205i) q^{13} +(-3.00000 - 3.00000i) q^{17} -3.46410i q^{19} -2.55040i q^{21} +(-0.791288 - 0.791288i) q^{23} +(-2.55040 - 2.55040i) q^{27} +5.29150 q^{29} -1.58258i q^{31} +(2.00000 - 2.00000i) q^{33} +(5.19615 - 5.19615i) q^{37} -1.58258 q^{39} -7.58258 q^{41} +(8.29875 - 8.29875i) q^{43} +(0.791288 - 0.791288i) q^{47} -8.58258i q^{49} +2.74110 q^{51} +(2.64575 + 2.64575i) q^{53} +(1.58258 + 1.58258i) q^{57} +5.29150i q^{59} -9.66930i q^{61} +(-7.20871 - 7.20871i) q^{63} +(-8.29875 - 8.29875i) q^{67} +0.723000 q^{69} +13.5826i q^{71} +(-0.582576 + 0.582576i) q^{73} +(12.2197 - 12.2197i) q^{77} -12.0000 q^{79} -5.41742 q^{81} +(-4.83465 + 4.83465i) q^{83} +(-2.41742 + 2.41742i) q^{87} -3.16515i q^{89} -9.66930 q^{91} +(0.723000 + 0.723000i) q^{93} +(-0.582576 - 0.582576i) q^{97} -11.3060i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} - 24 q^{17} + 12 q^{23} + 16 q^{33} + 24 q^{39} - 24 q^{41} - 12 q^{47} - 24 q^{57} - 76 q^{63} + 32 q^{73} - 96 q^{79} - 80 q^{81} - 56 q^{87} + 32 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 24 * q^17 + 12 * q^23 + 16 * q^33 + 24 * q^39 - 24 * q^41 - 12 * q^47 - 24 * q^57 - 76 * q^63 + 32 * q^73 - 96 * q^79 - 80 * q^81 - 56 * q^87 + 32 * q^97

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
$3$	−0.456850	+	0.456850i	−0.263763	+	0.263763i	−0.826581	−	0.562818i	$-0.809717\pi$
$3$	−0.456850	+	0.456850i	−0.263763	+	0.263763i	0.562818	+	0.826581i	$0.309717\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.79129	+	2.79129i	−1.05501	+	1.05501i	−0.0566113	+	0.998396i	$0.518030\pi$
$7$	−2.79129	+	2.79129i	−1.05501	+	1.05501i	−0.998396	+	0.0566113i	$0.981970\pi$
$8$		0			0
$9$			2.58258i			0.860859i
$10$		0			0
$11$	−4.37780			−1.31996			−0.659979	−	0.751284i	$-0.729435\pi$
$11$	−4.37780			−1.31996			−0.659979	+	0.751284i	$0.729435\pi$
$12$		0			0
$13$	1.73205	+	1.73205i	0.480384	+	0.480384i	0.905254	−	0.424870i	$-0.139680\pi$
$13$	1.73205	+	1.73205i	0.480384	+	0.480384i	−0.424870	+	0.905254i	$0.639680\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	−	3.00000i	−0.727607	−	0.727607i	0.242536	−	0.970143i	$-0.422021\pi$
$17$	−3.00000	−	3.00000i	−0.727607	−	0.727607i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		−	2.55040i		−	0.556543i
$22$		0			0
$23$	−0.791288	−	0.791288i	−0.164995	−	0.164995i	0.619780	−	0.784775i	$-0.287222\pi$
$23$	−0.791288	−	0.791288i	−0.164995	−	0.164995i	−0.784775	+	0.619780i	$0.787222\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.55040	−	2.55040i	−0.490825	−	0.490825i
$28$		0			0
$29$	5.29150			0.982607			0.491304	−	0.870988i	$-0.336521\pi$
$29$	5.29150			0.982607			0.491304	+	0.870988i	$0.336521\pi$
$30$		0			0
$31$		−	1.58258i		−	0.284239i	−0.989850	−	0.142119i	$-0.954608\pi$
$31$		−	1.58258i		−	0.284239i	0.989850	−	0.142119i	$-0.0453917\pi$
$32$		0			0
$33$	2.00000	−	2.00000i	0.348155	−	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.19615	−	5.19615i	0.854242	−	0.854242i	−0.136410	−	0.990652i	$-0.543557\pi$
$37$	5.19615	−	5.19615i	0.854242	−	0.854242i	0.990652	+	0.136410i	$0.0435565\pi$
$38$		0			0
$39$	−1.58258			−0.253415
$40$		0			0
$41$	−7.58258			−1.18420			−0.592100	−	0.805865i	$-0.701701\pi$
$41$	−7.58258			−1.18420			−0.592100	+	0.805865i	$0.701701\pi$
$42$		0			0
$43$	8.29875	−	8.29875i	1.26555	−	1.26555i	0.317184	−	0.948364i	$-0.397263\pi$
$43$	8.29875	−	8.29875i	1.26555	−	1.26555i	0.948364	−	0.317184i	$-0.102737\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.791288	−	0.791288i	0.115421	−	0.115421i	−0.647037	−	0.762458i	$-0.723992\pi$
$47$	0.791288	−	0.791288i	0.115421	−	0.115421i	0.762458	+	0.647037i	$0.223992\pi$
$48$		0			0
$49$		−	8.58258i		−	1.22608i
$50$		0			0
$51$	2.74110			0.383831
$52$		0			0
$53$	2.64575	+	2.64575i	0.363422	+	0.363422i	0.865071	−	0.501649i	$-0.167273\pi$
$53$	2.64575	+	2.64575i	0.363422	+	0.363422i	−0.501649	+	0.865071i	$0.667273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.58258	+	1.58258i	0.209617	+	0.209617i
$58$		0			0
$59$			5.29150i			0.688895i	0.938806	+	0.344447i	$0.111934\pi$
$59$			5.29150i			0.688895i	−0.938806	+	0.344447i	$0.888066\pi$
$60$		0			0
$61$		−	9.66930i		−	1.23803i	−0.785380	−	0.619014i	$-0.787532\pi$
$61$		−	9.66930i		−	1.23803i	0.785380	−	0.619014i	$-0.212468\pi$
$62$		0			0
$63$	−7.20871	−	7.20871i	−0.908212	−	0.908212i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.29875	−	8.29875i	−1.01385	−	1.01385i	−0.999903	−	0.0139515i	$-0.995559\pi$
$67$	−8.29875	−	8.29875i	−1.01385	−	1.01385i	−0.0139515	−	0.999903i	$-0.504441\pi$
$68$		0			0
$69$	0.723000			0.0870390
$70$		0			0
$71$			13.5826i			1.61196i	0.591946	+	0.805978i	$0.298360\pi$
$71$			13.5826i			1.61196i	−0.591946	+	0.805978i	$0.701640\pi$
$72$		0			0
$73$	−0.582576	+	0.582576i	−0.0681853	+	0.0681853i	−0.740377	−	0.672192i	$-0.765353\pi$
$73$	−0.582576	+	0.582576i	−0.0681853	+	0.0681853i	0.672192	+	0.740377i	$0.265353\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.2197	−	12.2197i	1.39256	−	1.39256i
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$		0			0
$81$	−5.41742			−0.601936
$82$		0			0
$83$	−4.83465	+	4.83465i	−0.530672	+	0.530672i	−0.920772	−	0.390100i	$-0.872440\pi$
$83$	−4.83465	+	4.83465i	−0.530672	+	0.530672i	0.390100	+	0.920772i	$0.372440\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.41742	+	2.41742i	−0.259175	+	0.259175i
$88$		0			0
$89$		−	3.16515i		−	0.335505i	−0.985829	−	0.167753i	$-0.946349\pi$
$89$		−	3.16515i		−	0.335505i	0.985829	−	0.167753i	$-0.0536510\pi$
$90$		0			0
$91$	−9.66930			−1.01362
$92$		0			0
$93$	0.723000	+	0.723000i	0.0749716	+	0.0749716i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.582576	−	0.582576i	−0.0591516	−	0.0591516i	0.676912	−	0.736064i	$-0.263318\pi$
$97$	−0.582576	−	0.582576i	−0.0591516	−	0.0591516i	−0.736064	+	0.676912i	$0.763318\pi$
$98$		0			0
$99$		−	11.3060i		−	1.13630i
$100$		0			0
$101$			3.65480i			0.363666i	0.983329	+	0.181833i	$0.0582031\pi$
$101$			3.65480i			0.363666i	−0.983329	+	0.181833i	$0.941797\pi$
$102$		0			0
$103$	10.3739	+	10.3739i	1.02217	+	1.02217i	0.999749	+	0.0224185i	$0.00713662\pi$
$103$	10.3739	+	10.3739i	1.02217	+	1.02217i	0.0224185	+	0.999749i	$0.492863\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.29875	−	8.29875i	−0.802271	−	0.802271i	0.181179	−	0.983450i	$-0.442009\pi$
$107$	−8.29875	−	8.29875i	−0.802271	−	0.802271i	−0.983450	+	0.181179i	$0.942009\pi$
$108$		0			0
$109$	−6.20520			−0.594351			−0.297175	−	0.954823i	$-0.596045\pi$
$109$	−6.20520			−0.594351			−0.297175	+	0.954823i	$0.596045\pi$
$110$		0			0
$111$			4.74773i			0.450634i
$112$		0			0
$113$	4.58258	−	4.58258i	0.431092	−	0.431092i	−0.457907	−	0.889000i	$-0.651401\pi$
$113$	4.58258	−	4.58258i	0.431092	−	0.431092i	0.889000	+	0.457907i	$0.151401\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.47315	+	4.47315i	−0.413543	+	0.413543i
$118$		0			0
$119$	16.7477			1.53526
$120$		0			0
$121$	8.16515			0.742286
$122$		0			0
$123$	3.46410	−	3.46410i	0.312348	−	0.312348i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.20871	−	1.20871i	0.107256	−	0.107256i	−0.651442	−	0.758698i	$-0.725836\pi$
$127$	1.20871	−	1.20871i	0.107256	−	0.107256i	0.758698	+	0.651442i	$0.225836\pi$
$128$		0			0
$129$			7.58258i			0.667609i
$130$		0			0
$131$	−11.3060			−0.987810			−0.493905	−	0.869516i	$-0.664431\pi$
$131$	−11.3060			−0.987810			−0.493905	+	0.869516i	$0.664431\pi$
$132$		0			0
$133$	9.66930	+	9.66930i	0.838435	+	0.838435i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.1652	−	12.1652i	−1.03934	−	1.03934i	−0.999194	−	0.0401452i	$-0.987218\pi$
$137$	−12.1652	−	12.1652i	−1.03934	−	1.03934i	−0.0401452	−	0.999194i	$-0.512782\pi$
$138$		0			0
$139$		−	8.94630i		−	0.758816i	−0.925230	−	0.379408i	$-0.876128\pi$
$139$		−	8.94630i		−	0.758816i	0.925230	−	0.379408i	$-0.123872\pi$
$140$		0			0
$141$			0.723000i			0.0608876i
$142$		0			0
$143$	−7.58258	−	7.58258i	−0.634087	−	0.634087i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.92095	+	3.92095i	0.323395	+	0.323395i
$148$		0			0
$149$	4.37780			0.358644			0.179322	−	0.983790i	$-0.442610\pi$
$149$	4.37780			0.358644			0.179322	+	0.983790i	$0.442610\pi$
$150$		0			0
$151$		−	12.7477i		−	1.03740i	−0.854958	−	0.518698i	$-0.826417\pi$
$151$		−	12.7477i		−	1.03740i	0.854958	−	0.518698i	$-0.173583\pi$
$152$		0			0
$153$	7.74773	−	7.74773i	0.626367	−	0.626367i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.4014	+	11.4014i	−0.909927	+	0.909927i	−0.996266	−	0.0863386i	$-0.972483\pi$
$157$	−11.4014	+	11.4014i	−0.909927	+	0.909927i	0.0863386	+	0.996266i	$0.472483\pi$
$158$		0			0
$159$	−2.41742			−0.191714
$160$		0			0
$161$	4.41742			0.348142
$162$		0			0
$163$	8.29875	−	8.29875i	0.650009	−	0.650009i	−0.302986	−	0.952995i	$-0.597984\pi$
$163$	8.29875	−	8.29875i	0.650009	−	0.650009i	0.952995	+	0.302986i	$0.0979837\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.2087	+	11.2087i	−0.867356	+	0.867356i	−0.992179	−	0.124823i	$-0.960164\pi$
$167$	−11.2087	+	11.2087i	−0.867356	+	0.867356i	0.124823	+	0.992179i	$0.460164\pi$
$168$		0			0
$169$		−	7.00000i		−	0.538462i
$170$		0			0
$171$	8.94630			0.684141
$172$		0			0
$173$	−8.66025	−	8.66025i	−0.658427	−	0.658427i	0.296581	−	0.955008i	$-0.404154\pi$
$173$	−8.66025	−	8.66025i	−0.658427	−	0.658427i	−0.955008	+	0.296581i	$0.904154\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.41742	−	2.41742i	−0.181705	−	0.181705i
$178$		0			0
$179$			8.56490i			0.640171i	0.947389	+	0.320085i	$0.103712\pi$
$179$			8.56490i			0.640171i	−0.947389	+	0.320085i	$0.896288\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	4.41742	+	4.41742i	0.326545	+	0.326545i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	13.1334	+	13.1334i	0.960410	+	0.960410i
$188$		0			0
$189$	14.2378			1.03565
$190$		0			0
$191$		−	13.5826i		−	0.982801i	−0.870934	−	0.491400i	$-0.836485\pi$
$191$		−	13.5826i		−	0.982801i	0.870934	−	0.491400i	$-0.163515\pi$
$192$		0			0
$193$	−14.1652	+	14.1652i	−1.01963	+	1.01963i	−0.0198265	+	0.999803i	$0.506311\pi$
$193$	−14.1652	+	14.1652i	−1.01963	+	1.01963i	−0.999803	+	0.0198265i	$0.993689\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.64575	−	2.64575i	0.188502	−	0.188502i	−0.606546	−	0.795048i	$-0.707446\pi$
$197$	2.64575	−	2.64575i	0.188502	−	0.188502i	0.795048	+	0.606546i	$0.207446\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$	7.58258			0.534834
$202$		0			0
$203$	−14.7701	+	14.7701i	−1.03666	+	1.03666i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.04356	−	2.04356i	0.142037	−	0.142037i
$208$		0			0
$209$			15.1652i			1.04900i
$210$		0			0
$211$	7.65120			0.526731			0.263365	−	0.964696i	$-0.415168\pi$
$211$	7.65120			0.526731			0.263365	+	0.964696i	$0.415168\pi$
$212$		0			0
$213$	−6.20520	−	6.20520i	−0.425174	−	0.425174i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.41742	+	4.41742i	0.299874	+	0.299874i
$218$		0			0
$219$		−	0.532300i		−	0.0359695i
$220$		0			0
$221$		−	10.3923i		−	0.699062i
$222$		0			0
$223$	−12.3739	−	12.3739i	−0.828615	−	0.828615i	0.158710	−	0.987325i	$-0.449266\pi$
$223$	−12.3739	−	12.3739i	−0.828615	−	0.828615i	−0.987325	+	0.158710i	$0.949266\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.66205	+	6.66205i	0.442176	+	0.442176i	0.892743	−	0.450567i	$-0.148778\pi$
$227$	6.66205	+	6.66205i	0.442176	+	0.442176i	−0.450567	+	0.892743i	$0.648778\pi$
$228$		0			0
$229$	−3.46410			−0.228914			−0.114457	−	0.993428i	$-0.536513\pi$
$229$	−3.46410			−0.228914			−0.114457	+	0.993428i	$0.536513\pi$
$230$		0			0
$231$			11.1652i			0.734613i
$232$		0			0
$233$	−6.16515	+	6.16515i	−0.403892	+	0.403892i	−0.879602	−	0.475710i	$-0.842191\pi$
$233$	−6.16515	+	6.16515i	−0.403892	+	0.403892i	0.475710	+	0.879602i	$0.342191\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.48220	−	5.48220i	0.356107	−	0.356107i
$238$		0			0
$239$	−3.16515			−0.204737			−0.102368	−	0.994747i	$-0.532642\pi$
$239$	−3.16515			−0.204737			−0.102368	+	0.994747i	$0.532642\pi$
$240$		0			0
$241$	−10.7477			−0.692322			−0.346161	−	0.938175i	$-0.612515\pi$
$241$	−10.7477			−0.692322			−0.346161	+	0.938175i	$0.612515\pi$
$242$		0			0
$243$	10.1262	−	10.1262i	0.649593	−	0.649593i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.00000	−	6.00000i	0.381771	−	0.381771i
$248$		0			0
$249$		−	4.41742i		−	0.279943i
$250$		0			0
$251$	18.2342			1.15093			0.575467	−	0.817825i	$-0.304821\pi$
$251$	18.2342			1.15093			0.575467	+	0.817825i	$0.304821\pi$
$252$		0			0
$253$	3.46410	+	3.46410i	0.217786	+	0.217786i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.7477	+	19.7477i	1.23183	+	1.23183i	0.963260	+	0.268569i	$0.0865507\pi$
$257$	19.7477	+	19.7477i	1.23183	+	1.23183i	0.268569	+	0.963260i	$0.413449\pi$
$258$		0			0
$259$			29.0079i			1.80246i
$260$		0			0
$261$			13.6657i			0.845886i
$262$		0			0
$263$	18.7913	+	18.7913i	1.15872	+	1.15872i	0.984751	+	0.173969i	$0.0556594\pi$
$263$	18.7913	+	18.7913i	1.15872	+	1.15872i	0.173969	+	0.984751i	$0.444341\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.44600	+	1.44600i	0.0884938	+	0.0884938i
$268$		0			0
$269$	−23.7164			−1.44602			−0.723008	−	0.690840i	$-0.757241\pi$
$269$	−23.7164			−1.44602			−0.723008	+	0.690840i	$0.757241\pi$
$270$		0			0
$271$		−	8.74773i		−	0.531387i	−0.964058	−	0.265693i	$-0.914399\pi$
$271$		−	8.74773i		−	0.531387i	0.964058	−	0.265693i	$-0.0856008\pi$
$272$		0			0
$273$	4.41742	−	4.41742i	0.267355	−	0.267355i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.91915	+	5.91915i	−0.355647	+	0.355647i	−0.862206	−	0.506558i	$-0.830917\pi$
$277$	−5.91915	+	5.91915i	−0.355647	+	0.355647i	0.506558	+	0.862206i	$0.330917\pi$
$278$		0			0
$279$	4.08712			0.244690
$280$		0			0
$281$	−10.7477			−0.641156			−0.320578	−	0.947222i	$-0.603877\pi$
$281$	−10.7477			−0.641156			−0.320578	+	0.947222i	$0.603877\pi$
$282$		0			0
$283$	−17.2451	+	17.2451i	−1.02511	+	1.02511i	−0.0254359	+	0.999676i	$0.508097\pi$
$283$	−17.2451	+	17.2451i	−1.02511	+	1.02511i	−0.999676	+	0.0254359i	$0.991903\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.1652	−	21.1652i	1.24934	−	1.24934i
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$	0.532300			0.0312040
$292$		0			0
$293$	−23.4304	−	23.4304i	−1.36882	−	1.36882i	−0.862131	−	0.506685i	$-0.830871\pi$
$293$	−23.4304	−	23.4304i	−1.36882	−	1.36882i	−0.506685	−	0.862131i	$-0.669129\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	11.1652	+	11.1652i	0.647868	+	0.647868i
$298$		0			0
$299$		−	2.74110i		−	0.158522i
$300$		0			0
$301$			46.3284i			2.67033i
$302$		0			0
$303$	−1.66970	−	1.66970i	−0.0959216	−	0.0959216i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.37055	−	1.37055i	−0.0782215	−	0.0782215i	0.666914	−	0.745135i	$-0.267615\pi$
$307$	−1.37055	−	1.37055i	−0.0782215	−	0.0782215i	−0.745135	+	0.666914i	$0.767615\pi$
$308$		0			0
$309$	−9.47860			−0.539219
$310$		0			0
$311$			10.4174i			0.590718i	0.955386	+	0.295359i	$0.0954392\pi$
$311$			10.4174i			0.590718i	−0.955386	+	0.295359i	$0.904561\pi$
$312$		0			0
$313$	−2.58258	+	2.58258i	−0.145976	+	0.145976i	−0.776318	−	0.630342i	$-0.782915\pi$
$313$	−2.58258	+	2.58258i	−0.145976	+	0.145976i	0.630342	+	0.776318i	$0.282915\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.6066	+	17.6066i	−0.988883	+	0.988883i	−0.999939	−	0.0110560i	$-0.996481\pi$
$317$	−17.6066	+	17.6066i	−0.988883	+	0.988883i	0.0110560	+	0.999939i	$0.496481\pi$
$318$		0			0
$319$	−23.1652			−1.29700
$320$		0			0
$321$	7.58258			0.423218
$322$		0			0
$323$	−10.3923	+	10.3923i	−0.578243	+	0.578243i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.83485	−	2.83485i	0.156767	−	0.156767i
$328$		0			0
$329$			4.41742i			0.243540i
$330$		0			0
$331$	6.20520			0.341069			0.170534	−	0.985352i	$-0.445451\pi$
$331$	6.20520			0.341069			0.170534	+	0.985352i	$0.445451\pi$
$332$		0			0
$333$	13.4195	+	13.4195i	0.735382	+	0.735382i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.00000	−	1.00000i	−0.0544735	−	0.0544735i	0.679345	−	0.733819i	$-0.262264\pi$
$337$	−1.00000	−	1.00000i	−0.0544735	−	0.0544735i	−0.733819	+	0.679345i	$0.762264\pi$
$338$		0			0
$339$			4.18710i			0.227412i
$340$		0			0
$341$			6.92820i			0.375183i
$342$		0			0
$343$	4.41742	+	4.41742i	0.238518	+	0.238518i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.92095	−	3.92095i	−0.210488	−	0.210488i	0.593987	−	0.804475i	$-0.297553\pi$
$347$	−3.92095	−	3.92095i	−0.210488	−	0.210488i	−0.804475	+	0.593987i	$0.797553\pi$
$348$		0			0
$349$	10.3923			0.556287			0.278144	−	0.960539i	$-0.410281\pi$
$349$	10.3923			0.556287			0.278144	+	0.960539i	$0.410281\pi$
$350$		0			0
$351$		−	8.83485i		−	0.471569i
$352$		0			0
$353$	−5.83485	+	5.83485i	−0.310558	+	0.310558i	−0.845126	−	0.534568i	$-0.820474\pi$
$353$	−5.83485	+	5.83485i	−0.310558	+	0.310558i	0.534568	+	0.845126i	$0.320474\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−7.65120	+	7.65120i	−0.404945	+	0.404945i
$358$		0			0
$359$	8.83485			0.466285			0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485			0.466285			0.233143	+	0.972443i	$0.425099\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−3.73025	+	3.73025i	−0.195787	+	0.195787i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.5390	−	15.5390i	0.811130	−	0.811130i	−0.173673	−	0.984803i	$-0.555564\pi$
$367$	15.5390	−	15.5390i	0.811130	−	0.811130i	0.984803	+	0.173673i	$0.0555637\pi$
$368$		0			0
$369$		−	19.5826i		−	1.01943i
$370$		0			0
$371$	−14.7701			−0.766826
$372$		0			0
$373$	5.19615	+	5.19615i	0.269047	+	0.269047i	0.828716	−	0.559669i	$-0.189072\pi$
$373$	5.19615	+	5.19615i	0.269047	+	0.269047i	−0.559669	+	0.828716i	$0.689072\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.16515	+	9.16515i	0.472029	+	0.472029i
$378$		0			0
$379$		−	10.3923i		−	0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$		−	10.3923i		−	0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$		0			0
$381$			1.10440i			0.0565802i
$382$		0			0
$383$	3.62614	+	3.62614i	0.185287	+	0.185287i	0.793655	−	0.608368i	$-0.208176\pi$
$383$	3.62614	+	3.62614i	0.185287	+	0.185287i	−0.608368	+	0.793655i	$0.708176\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	21.4322	+	21.4322i	1.08946	+	1.08946i
$388$		0			0
$389$	−30.2632			−1.53441			−0.767203	−	0.641404i	$-0.778352\pi$
$389$	−30.2632			−1.53441			−0.767203	+	0.641404i	$0.778352\pi$
$390$		0			0
$391$			4.74773i			0.240103i
$392$		0			0
$393$	5.16515	−	5.16515i	0.260547	−	0.260547i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.93725	+	7.93725i	−0.398359	+	0.398359i	−0.877654	−	0.479295i	$-0.840893\pi$
$397$	−7.93725	+	7.93725i	−0.398359	+	0.398359i	0.479295	+	0.877654i	$0.340893\pi$
$398$		0			0
$399$	−8.83485			−0.442296
$400$		0			0
$401$	12.3303			0.615746			0.307873	−	0.951427i	$-0.400383\pi$
$401$	12.3303			0.615746			0.307873	+	0.951427i	$0.400383\pi$
$402$		0			0
$403$	2.74110	−	2.74110i	0.136544	−	0.136544i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−22.7477	+	22.7477i	−1.12756	+	1.12756i
$408$		0			0
$409$			25.5826i			1.26498i	0.774570	+	0.632488i	$0.217966\pi$
$409$			25.5826i			1.26498i	−0.774570	+	0.632488i	$0.782034\pi$
$410$		0			0
$411$	11.1153			0.548278
$412$		0			0
$413$	−14.7701	−	14.7701i	−0.726789	−	0.726789i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.08712	+	4.08712i	0.200147	+	0.200147i
$418$		0			0
$419$		−	0.190700i		−	0.00931632i	−0.999989	−	0.00465816i	$-0.998517\pi$
$419$		−	0.190700i		−	0.00931632i	0.999989	−	0.00465816i	$-0.00148274\pi$
$420$		0			0
$421$		−	2.74110i		−	0.133593i	−0.997767	−	0.0667966i	$-0.978722\pi$
$421$		−	2.74110i		−	0.133593i	0.997767	−	0.0667966i	$-0.0212778\pi$
$422$		0			0
$423$	2.04356	+	2.04356i	0.0993613	+	0.0993613i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	26.9898	+	26.9898i	1.30613	+	1.30613i
$428$		0			0
$429$	6.92820			0.334497
$430$		0			0
$431$		−	16.7477i		−	0.806710i	−0.915044	−	0.403355i	$-0.867844\pi$
$431$		−	16.7477i		−	0.806710i	0.915044	−	0.403355i	$-0.132156\pi$
$432$		0			0
$433$	14.5826	−	14.5826i	0.700794	−	0.700794i	−0.263787	−	0.964581i	$-0.584972\pi$
$433$	14.5826	−	14.5826i	0.700794	−	0.700794i	0.964581	+	0.263787i	$0.0849716\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.74110	+	2.74110i	−0.131125	+	0.131125i
$438$		0			0
$439$	29.4955			1.40774			0.703871	−	0.710328i	$-0.251453\pi$
$439$	29.4955			1.40774			0.703871	+	0.710328i	$0.251453\pi$
$440$		0			0
$441$	22.1652			1.05548
$442$		0			0
$443$	−1.17985	+	1.17985i	−0.0560564	+	0.0560564i	−0.734579	−	0.678523i	$-0.762620\pi$
$443$	−1.17985	+	1.17985i	−0.0560564	+	0.0560564i	0.678523	+	0.734579i	$0.262620\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.00000	+	2.00000i	−0.0945968	+	0.0945968i
$448$		0			0
$449$			10.4174i			0.491629i	0.969317	+	0.245814i	$0.0790553\pi$
$449$			10.4174i			0.491629i	−0.969317	+	0.245814i	$0.920945\pi$
$450$		0			0
$451$	33.1950			1.56309
$452$		0			0
$453$	5.82380	+	5.82380i	0.273626	+	0.273626i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.41742	−	3.41742i	−0.159860	−	0.159860i	0.622644	−	0.782505i	$-0.286058\pi$
$457$	−3.41742	−	3.41742i	−0.159860	−	0.159860i	−0.782505	+	0.622644i	$0.786058\pi$
$458$		0			0
$459$			15.3024i			0.714255i
$460$		0			0
$461$		−	17.5112i		−	0.815578i	−0.913076	−	0.407789i	$-0.866300\pi$
$461$		−	17.5112i		−	0.815578i	0.913076	−	0.407789i	$-0.133700\pi$
$462$		0			0
$463$	−16.7913	−	16.7913i	−0.780357	−	0.780357i	0.199534	−	0.979891i	$-0.436057\pi$
$463$	−16.7913	−	16.7913i	−0.780357	−	0.780357i	−0.979891	+	0.199534i	$0.936057\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.57575	−	7.57575i	−0.350564	−	0.350564i	0.509755	−	0.860319i	$-0.329736\pi$
$467$	−7.57575	−	7.57575i	−0.350564	−	0.350564i	−0.860319	+	0.509755i	$0.829736\pi$
$468$		0			0
$469$	46.3284			2.13925
$470$		0			0
$471$		−	10.4174i		−	0.480010i
$472$		0			0
$473$	−36.3303	+	36.3303i	−1.67047	+	1.67047i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.83285	+	6.83285i	−0.312855	+	0.312855i
$478$		0			0
$479$	21.4955			0.982152			0.491076	−	0.871117i	$-0.336604\pi$
$479$	21.4955			0.982152			0.491076	+	0.871117i	$0.336604\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$	−2.01810	+	2.01810i	−0.0918268	+	0.0918268i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.373864	−	0.373864i	0.0169414	−	0.0169414i	−0.698585	−	0.715527i	$-0.746187\pi$
$487$	0.373864	−	0.373864i	0.0169414	−	0.0169414i	0.715527	+	0.698585i	$0.246187\pi$
$488$		0			0
$489$			7.58258i			0.342896i
$490$		0			0
$491$	9.86001			0.444976			0.222488	−	0.974935i	$-0.428582\pi$
$491$	9.86001			0.444976			0.222488	+	0.974935i	$0.428582\pi$
$492$		0			0
$493$	−15.8745	−	15.8745i	−0.714952	−	0.714952i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−37.9129	−	37.9129i	−1.70063	−	1.70063i
$498$		0			0
$499$		−	24.2487i		−	1.08552i	−0.839887	−	0.542761i	$-0.817379\pi$
$499$		−	24.2487i		−	1.08552i	0.839887	−	0.542761i	$-0.182621\pi$
$500$		0			0
$501$		−	10.2414i		−	0.457552i
$502$		0			0
$503$	17.5390	+	17.5390i	0.782026	+	0.782026i	0.980172	−	0.198146i	$-0.0634922\pi$
$503$	17.5390	+	17.5390i	0.782026	+	0.782026i	−0.198146	+	0.980172i	$0.563492\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.19795	+	3.19795i	0.142026	+	0.142026i
$508$		0			0
$509$	−38.4865			−1.70588			−0.852942	−	0.522005i	$-0.825184\pi$
$509$	−38.4865			−1.70588			−0.852942	+	0.522005i	$0.825184\pi$
$510$		0			0
$511$		−	3.25227i		−	0.143872i
$512$		0			0
$513$	−8.83485	+	8.83485i	−0.390068	+	0.390068i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3.46410	+	3.46410i	−0.152351	+	0.152351i
$518$		0			0
$519$	7.91288			0.347337
$520$		0			0
$521$	−33.1652			−1.45299			−0.726496	−	0.687171i	$-0.758852\pi$
$521$	−33.1652			−1.45299			−0.726496	+	0.687171i	$0.758852\pi$
$522$		0			0
$523$	15.9500	−	15.9500i	0.697443	−	0.697443i	−0.266415	−	0.963858i	$-0.585839\pi$
$523$	15.9500	−	15.9500i	0.697443	−	0.697443i	0.963858	+	0.266415i	$0.0858393\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.74773	+	4.74773i	−0.206814	+	0.206814i
$528$		0			0
$529$		−	21.7477i		−	0.945553i
$530$		0			0
$531$	−13.6657			−0.593041
$532$		0			0
$533$	−13.1334	−	13.1334i	−0.568871	−	0.568871i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−3.91288	−	3.91288i	−0.168853	−	0.168853i
$538$		0			0
$539$			37.5728i			1.61838i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	3.16515	+	3.16515i	0.135830	+	0.135830i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.647551	−	0.647551i	−0.0276873	−	0.0276873i	0.693128	−	0.720815i	$-0.256232\pi$
$547$	−0.647551	−	0.647551i	−0.0276873	−	0.0276873i	−0.720815	+	0.693128i	$0.756232\pi$
$548$		0			0
$549$	24.9717			1.06577
$550$		0			0
$551$		−	18.3303i		−	0.780897i
$552$		0			0
$553$	33.4955	−	33.4955i	1.42437	−	1.42437i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.2970	−	10.2970i	0.436296	−	0.436296i	−0.454467	−	0.890763i	$-0.650170\pi$
$557$	10.2970	−	10.2970i	0.436296	−	0.436296i	0.890763	+	0.454467i	$0.150170\pi$
$558$		0			0
$559$	28.7477			1.21590
$560$		0			0
$561$	−12.0000			−0.506640
$562$		0			0
$563$	−14.3133	+	14.3133i	−0.603232	+	0.603232i	−0.941169	−	0.337937i	$-0.890271\pi$
$563$	−14.3133	+	14.3133i	−0.603232	+	0.603232i	0.337937	+	0.941169i	$0.390271\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.1216	−	15.1216i	0.635047	−	0.635047i
$568$		0			0
$569$			4.74773i			0.199035i	0.995036	+	0.0995175i	$0.0317299\pi$
$569$			4.74773i			0.199035i	−0.995036	+	0.0995175i	$0.968270\pi$
$570$		0			0
$571$	−39.4002			−1.64885			−0.824424	−	0.565973i	$-0.808501\pi$
$571$	−39.4002			−1.64885			−0.824424	+	0.565973i	$0.808501\pi$
$572$		0			0
$573$	6.20520	+	6.20520i	0.259226	+	0.259226i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.7477	+	11.7477i	0.489064	+	0.489064i	0.908011	−	0.418947i	$-0.137601\pi$
$577$	11.7477	+	11.7477i	0.489064	+	0.489064i	−0.418947	+	0.908011i	$0.637601\pi$
$578$		0			0
$579$		−	12.9427i		−	0.537881i
$580$		0			0
$581$		−	26.9898i		−	1.11973i
$582$		0			0
$583$	−11.5826	−	11.5826i	−0.479701	−	0.479701i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−28.3604	−	28.3604i	−1.17056	−	1.17056i	−0.982077	−	0.188481i	$-0.939644\pi$
$587$	−28.3604	−	28.3604i	−1.17056	−	1.17056i	−0.188481	−	0.982077i	$-0.560356\pi$
$588$		0			0
$589$	−5.48220			−0.225890
$590$		0			0
$591$			2.41742i			0.0994395i
$592$		0			0
$593$	−0.165151	+	0.165151i	−0.00678195	+	0.00678195i	−0.710490	−	0.703708i	$-0.751526\pi$
$593$	−0.165151	+	0.165151i	−0.00678195	+	0.00678195i	0.703708	+	0.710490i	$0.251526\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.82740	+	1.82740i	−0.0747905	+	0.0747905i
$598$		0			0
$599$	−42.3303			−1.72957			−0.864785	−	0.502143i	$-0.832545\pi$
$599$	−42.3303			−1.72957			−0.864785	+	0.502143i	$0.832545\pi$
$600$		0			0
$601$	−2.74773			−0.112082			−0.0560411	−	0.998428i	$-0.517848\pi$
$601$	−2.74773			−0.112082			−0.0560411	+	0.998428i	$0.517848\pi$
$602$		0			0
$603$	21.4322	−	21.4322i	0.872785	−	0.872785i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−6.37386	+	6.37386i	−0.258707	+	0.258707i	−0.824528	−	0.565821i	$-0.808559\pi$
$607$	−6.37386	+	6.37386i	−0.258707	+	0.258707i	0.565821	+	0.824528i	$0.308559\pi$
$608$		0			0
$609$		−	13.4955i		−	0.546863i
$610$		0			0
$611$	2.74110			0.110893
$612$		0			0
$613$	8.66025	+	8.66025i	0.349784	+	0.349784i	0.860029	−	0.510245i	$-0.170445\pi$
$613$	8.66025	+	8.66025i	0.349784	+	0.349784i	−0.510245	+	0.860029i	$0.670445\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	0.644136	−	0.764911i	$-0.277217\pi$
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		−	29.7309i		−	1.19499i	−0.801874	−	0.597493i	$-0.796164\pi$
$619$		−	29.7309i		−	1.19499i	0.801874	−	0.597493i	$-0.203836\pi$
$620$		0			0
$621$			4.03620i			0.161967i
$622$		0			0
$623$	8.83485	+	8.83485i	0.353961	+	0.353961i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−6.92820	−	6.92820i	−0.276686	−	0.276686i
$628$		0			0
$629$	−31.1769			−1.24310
$630$		0			0
$631$			47.0780i			1.87415i	0.349132	+	0.937073i	$0.386476\pi$
$631$			47.0780i			1.87415i	−0.349132	+	0.937073i	$0.613524\pi$
$632$		0			0
$633$	−3.49545	+	3.49545i	−0.138932	+	0.138932i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.8655	−	14.8655i	0.588991	−	0.588991i
$638$		0			0
$639$	−35.0780			−1.38767
$640$		0			0
$641$	25.9129			1.02350			0.511749	−	0.859135i	$-0.328998\pi$
$641$	25.9129			1.02350			0.511749	+	0.859135i	$0.328998\pi$
$642$		0			0
$643$	−18.6911	+	18.6911i	−0.737103	+	0.737103i	−0.972016	−	0.234913i	$-0.924519\pi$
$643$	−18.6911	+	18.6911i	−0.737103	+	0.737103i	0.234913	+	0.972016i	$0.424519\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.460985	+	0.460985i	−0.0181232	+	0.0181232i	−0.716110	−	0.697987i	$-0.754079\pi$
$647$	−0.460985	+	0.460985i	−0.0181232	+	0.0181232i	0.697987	+	0.716110i	$0.254079\pi$
$648$		0			0
$649$		−	23.1652i		−	0.909312i
$650$		0			0
$651$	−4.03620			−0.158191
$652$		0			0
$653$	11.4014	+	11.4014i	0.446170	+	0.446170i	0.894079	−	0.447909i	$-0.147831\pi$
$653$	11.4014	+	11.4014i	0.446170	+	0.446170i	−0.447909	+	0.894079i	$0.647831\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.50455	−	1.50455i	−0.0586979	−	0.0586979i
$658$		0			0
$659$		−	3.84550i		−	0.149800i	−0.997191	−	0.0748998i	$-0.976136\pi$
$659$		−	3.84550i		−	0.149800i	0.997191	−	0.0748998i	$-0.0238637\pi$
$660$		0			0
$661$			41.4183i			1.61099i	0.592605	+	0.805493i	$0.298099\pi$
$661$			41.4183i			1.61099i	−0.592605	+	0.805493i	$0.701901\pi$
$662$		0			0
$663$	4.74773	+	4.74773i	0.184386	+	0.184386i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.18710	−	4.18710i	−0.162125	−	0.162125i
$668$		0			0
$669$	11.3060			0.437115
$670$		0			0
$671$			42.3303i			1.63414i
$672$		0			0
$673$	18.5826	−	18.5826i	0.716306	−	0.716306i	−0.251541	−	0.967847i	$-0.580937\pi$
$673$	18.5826	−	18.5826i	0.716306	−	0.716306i	0.967847	+	0.251541i	$0.0809373\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.3296	−	18.3296i	0.704462	−	0.704462i	−0.260903	−	0.965365i	$-0.584020\pi$
$677$	18.3296	−	18.3296i	0.704462	−	0.704462i	0.965365	+	0.260903i	$0.0840202\pi$
$678$		0			0
$679$	3.25227			0.124811
$680$		0			0
$681$	−6.08712			−0.233259
$682$		0			0
$683$	6.85275	−	6.85275i	0.262213	−	0.262213i	−0.563739	−	0.825953i	$-0.690638\pi$
$683$	6.85275	−	6.85275i	0.262213	−	0.262213i	0.825953	+	0.563739i	$0.190638\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.58258	−	1.58258i	0.0603790	−	0.0603790i
$688$		0			0
$689$			9.16515i			0.349164i
$690$		0			0
$691$	32.4720			1.23529			0.617647	−	0.786456i	$-0.288086\pi$
$691$	32.4720			1.23529			0.617647	+	0.786456i	$0.288086\pi$
$692$		0			0
$693$	31.5583	+	31.5583i	1.19880	+	1.19880i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	22.7477	+	22.7477i	0.861632	+	0.861632i
$698$		0			0
$699$		−	5.63310i		−	0.213063i
$700$		0			0
$701$			19.8709i			0.750514i	0.926921	+	0.375257i	$0.122446\pi$
$701$			19.8709i			0.750514i	−0.926921	+	0.375257i	$0.877554\pi$
$702$		0			0
$703$	−18.0000	−	18.0000i	−0.678883	−	0.678883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.2016	−	10.2016i	−0.383671	−	0.383671i
$708$		0			0
$709$	−4.91010			−0.184403			−0.0922014	−	0.995740i	$-0.529390\pi$
$709$	−4.91010			−0.184403			−0.0922014	+	0.995740i	$0.529390\pi$
$710$		0			0
$711$		−	30.9909i		−	1.16225i
$712$		0			0
$713$	−1.25227	+	1.25227i	−0.0468980	+	0.0468980i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.44600	−	1.44600i	0.0540019	−	0.0540019i
$718$		0			0
$719$	−39.8258			−1.48525			−0.742625	−	0.669707i	$-0.766420\pi$
$719$	−39.8258			−1.48525			−0.742625	+	0.669707i	$0.766420\pi$
$720$		0			0
$721$	−57.9129			−2.15679
$722$		0			0
$723$	4.91010	−	4.91010i	0.182609	−	0.182609i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.95644	+	5.95644i	−0.220912	+	0.220912i	−0.808882	−	0.587970i	$-0.799927\pi$
$727$	−5.95644	+	5.95644i	−0.220912	+	0.220912i	0.587970	+	0.808882i	$0.299927\pi$
$728$		0			0
$729$		−	7.00000i		−	0.259259i
$730$		0			0
$731$	−49.7925			−1.84164
$732$		0			0
$733$	17.6066	+	17.6066i	0.650313	+	0.650313i	0.953068	−	0.302755i	$-0.0979065\pi$
$733$	17.6066	+	17.6066i	0.650313	+	0.650313i	−0.302755	+	0.953068i	$0.597906\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.3303	+	36.3303i	1.33824	+	1.33824i
$738$		0			0
$739$			2.01810i			0.0742371i	0.999311	+	0.0371185i	$0.0118179\pi$
$739$			2.01810i			0.0742371i	−0.999311	+	0.0371185i	$0.988182\pi$
$740$		0			0
$741$			5.48220i			0.201394i
$742$		0			0
$743$	−17.2087	−	17.2087i	−0.631326	−	0.631326i	0.317074	−	0.948401i	$-0.397300\pi$
$743$	−17.2087	−	17.2087i	−0.631326	−	0.631326i	−0.948401	+	0.317074i	$0.897300\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−12.4859	−	12.4859i	−0.456834	−	0.456834i
$748$		0			0
$749$	46.3284			1.69280
$750$		0			0
$751$			23.0780i			0.842129i	0.907031	+	0.421065i	$0.138343\pi$
$751$			23.0780i			0.842129i	−0.907031	+	0.421065i	$0.861657\pi$
$752$		0			0
$753$	−8.33030	+	8.33030i	−0.303573	+	0.303573i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.3477	−	20.3477i	0.739548	−	0.739548i	−0.232942	−	0.972491i	$-0.574835\pi$
$757$	20.3477	−	20.3477i	0.739548	−	0.739548i	0.972491	+	0.232942i	$0.0748353\pi$
$758$		0			0
$759$	−3.16515			−0.114888
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	17.3205	−	17.3205i	0.627044	−	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.16515	+	9.16515i	−0.330934	+	0.330934i
$768$		0			0
$769$			37.4955i			1.35212i	0.736846	+	0.676060i	$0.236314\pi$
$769$			37.4955i			1.35212i	−0.736846	+	0.676060i	$0.763686\pi$
$770$		0			0
$771$	−18.0435			−0.649821
$772$		0			0
$773$	0.286051	+	0.286051i	0.0102885	+	0.0102885i	0.712232	−	0.701944i	$-0.247684\pi$
$773$	0.286051	+	0.286051i	0.0102885	+	0.0102885i	−0.701944	+	0.712232i	$0.747684\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−13.2523	−	13.2523i	−0.475423	−	0.475423i
$778$		0			0
$779$			26.2668i			0.941106i
$780$		0			0
$781$		−	59.4618i		−	2.12771i
$782$		0			0
$783$	−13.4955	−	13.4955i	−0.482288	−	0.482288i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−20.7092	−	20.7092i	−0.738202	−	0.738202i	0.234028	−	0.972230i	$-0.424809\pi$
$787$	−20.7092	−	20.7092i	−0.738202	−	0.738202i	−0.972230	+	0.234028i	$0.924809\pi$
$788$		0			0
$789$	−17.1696			−0.611254
$790$		0			0
$791$			25.5826i			0.909612i
$792$		0			0
$793$	16.7477	−	16.7477i	0.594729	−	0.594729i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.9271	−	34.9271i	1.23718	−	1.23718i	0.276032	−	0.961149i	$-0.410981\pi$
$797$	34.9271	−	34.9271i	1.23718	−	1.23718i	0.961149	−	0.276032i	$-0.0890195\pi$
$798$		0			0
$799$	−4.74773			−0.167963
$800$		0			0
$801$	8.17424			0.288823
$802$		0			0
$803$	2.55040	−	2.55040i	0.0900017	−	0.0900017i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	10.8348	−	10.8348i	0.381405	−	0.381405i
$808$		0			0
$809$		−	30.3303i		−	1.06636i	−0.846003	−	0.533178i	$-0.820997\pi$
$809$		−	30.3303i		−	1.06636i	0.846003	−	0.533178i	$-0.179003\pi$
$810$		0			0
$811$	−2.16900			−0.0761639			−0.0380820	−	0.999275i	$-0.512125\pi$
$811$	−2.16900			−0.0761639			−0.0380820	+	0.999275i	$0.512125\pi$
$812$		0			0
$813$	3.99640	+	3.99640i	0.140160	+	0.140160i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−28.7477	−	28.7477i	−1.00576	−	1.00576i
$818$		0			0
$819$		−	24.9717i		−	0.872582i
$820$		0			0
$821$		−	54.8933i		−	1.91579i	−0.287118	−	0.957895i	$-0.592697\pi$
$821$		−	54.8933i		−	1.91579i	0.287118	−	0.957895i	$-0.407303\pi$
$822$		0			0
$823$	3.20871	+	3.20871i	0.111849	+	0.111849i	0.760816	−	0.648967i	$-0.224799\pi$
$823$	3.20871	+	3.20871i	0.111849	+	0.111849i	−0.648967	+	0.760816i	$0.724799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.1226	−	14.1226i	−0.491089	−	0.491089i	0.417560	−	0.908649i	$-0.362885\pi$
$827$	−14.1226	−	14.1226i	−0.491089	−	0.491089i	−0.908649	+	0.417560i	$0.862885\pi$
$828$		0			0
$829$	7.65120			0.265737			0.132869	−	0.991134i	$-0.457581\pi$
$829$	7.65120			0.265737			0.132869	+	0.991134i	$0.457581\pi$
$830$		0			0
$831$		−	5.40833i		−	0.187613i
$832$		0			0
$833$	−25.7477	+	25.7477i	−0.892106	+	0.892106i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.03620	+	4.03620i	−0.139512	+	0.139512i
$838$		0			0
$839$	30.3303			1.04712			0.523559	−	0.851989i	$-0.324604\pi$
$839$	30.3303			1.04712			0.523559	+	0.851989i	$0.324604\pi$
$840$		0			0
$841$	−1.00000			−0.0344828
$842$		0			0
$843$	4.91010	−	4.91010i	0.169113	−	0.169113i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−22.7913	+	22.7913i	−0.783118	+	0.783118i
$848$		0			0
$849$		−	15.7568i		−	0.540773i
$850$		0			0
$851$	−8.22330			−0.281891
$852$		0			0
$853$	−36.9452	−	36.9452i	−1.26498	−	1.26498i	−0.948650	−	0.316329i	$-0.897550\pi$
$853$	−36.9452	−	36.9452i	−1.26498	−	1.26498i	−0.316329	−	0.948650i	$-0.602450\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−39.3303	−	39.3303i	−1.34350	−	1.34350i	−0.892551	−	0.450947i	$-0.851086\pi$
$857$	−39.3303	−	39.3303i	−1.34350	−	1.34350i	−0.450947	−	0.892551i	$-0.648914\pi$
$858$		0			0
$859$		−	2.01810i		−	0.0688567i	−0.999407	−	0.0344284i	$-0.989039\pi$
$859$		−	2.01810i		−	0.0688567i	0.999407	−	0.0344284i	$-0.0109611\pi$
$860$		0			0
$861$			19.3386i			0.659058i
$862$		0			0
$863$	32.7042	+	32.7042i	1.11326	+	1.11326i	0.992706	+	0.120556i	$0.0384678\pi$
$863$	32.7042	+	32.7042i	1.11326	+	1.11326i	0.120556	+	0.992706i	$0.461532\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.456850	−	0.456850i	−0.0155154	−	0.0155154i
$868$		0			0
$869$	52.5336			1.78208
$870$		0			0
$871$		−	28.7477i		−	0.974080i
$872$		0			0
$873$	1.50455	−	1.50455i	0.0509212	−	0.0509212i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−19.7756	+	19.7756i	−0.667773	+	0.667773i	−0.957200	−	0.289427i	$-0.906535\pi$
$877$	−19.7756	+	19.7756i	−0.667773	+	0.667773i	0.289427	+	0.957200i	$0.406535\pi$
$878$		0			0
$879$	21.4083			0.722085
$880$		0			0
$881$	−55.5826			−1.87262			−0.936312	−	0.351169i	$-0.885784\pi$
$881$	−55.5826			−1.87262			−0.936312	+	0.351169i	$0.885784\pi$
$882$		0			0
$883$	22.1552	−	22.1552i	0.745581	−	0.745581i	−0.228065	−	0.973646i	$-0.573240\pi$
$883$	22.1552	−	22.1552i	0.745581	−	0.745581i	0.973646	+	0.228065i	$0.0732400\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.9564	+	21.9564i	−0.737225	+	0.737225i	−0.972040	−	0.234815i	$-0.924552\pi$
$887$	−21.9564	+	21.9564i	−0.737225	+	0.737225i	0.234815	+	0.972040i	$0.424552\pi$
$888$		0			0
$889$			6.74773i			0.226312i
$890$		0			0
$891$	23.7164			0.794530
$892$		0			0
$893$	−2.74110	−	2.74110i	−0.0917275	−	0.0917275i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.25227	+	1.25227i	0.0418122	+	0.0418122i
$898$		0			0
$899$		−	8.37420i		−	0.279295i
$900$		0			0
$901$		−	15.8745i		−	0.528857i
$902$		0			0
$903$	−21.1652	−	21.1652i	−0.704332	−	0.704332i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11.0399	+	11.0399i	0.366572	+	0.366572i	0.866226	−	0.499653i	$-0.166539\pi$
$907$	11.0399	+	11.0399i	0.366572	+	0.366572i	−0.499653	+	0.866226i	$0.666539\pi$
$908$		0			0
$909$	−9.43880			−0.313065
$910$		0			0
$911$		−	49.5826i		−	1.64274i	−0.570393	−	0.821372i	$-0.693209\pi$
$911$		−	49.5826i		−	1.64274i	0.570393	−	0.821372i	$-0.306791\pi$
$912$		0			0
$913$	21.1652	−	21.1652i	0.700464	−	0.700464i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	31.5583	−	31.5583i	1.04215	−	1.04215i
$918$		0			0
$919$	−42.3303			−1.39635			−0.698174	−	0.715928i	$-0.746004\pi$
$919$	−42.3303			−1.39635			−0.698174	+	0.715928i	$0.746004\pi$
$920$		0			0
$921$	1.25227			0.0412638
$922$		0			0
$923$	−23.5257	+	23.5257i	−0.774358	+	0.774358i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−26.7913	+	26.7913i	−0.879941	+	0.879941i
$928$		0			0
$929$			19.9129i			0.653320i	0.945142	+	0.326660i	$0.105923\pi$
$929$			19.9129i			0.653320i	−0.945142	+	0.326660i	$0.894077\pi$
$930$		0			0
$931$	−29.7309			−0.974391
$932$		0			0
$933$	−4.75920	−	4.75920i	−0.155809	−	0.155809i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.1652	−	34.1652i	−1.11613	−	1.11613i	−0.992304	−	0.123822i	$-0.960485\pi$
$937$	−34.1652	−	34.1652i	−1.11613	−	1.11613i	−0.123822	−	0.992304i	$-0.539515\pi$
$938$		0			0
$939$		−	2.35970i		−	0.0770059i
$940$		0			0
$941$		−	24.4394i		−	0.796702i	−0.917233	−	0.398351i	$-0.869583\pi$
$941$		−	24.4394i		−	0.796702i	0.917233	−	0.398351i	$-0.130417\pi$
$942$		0			0
$943$	6.00000	+	6.00000i	0.195387	+	0.195387i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.6947	+	14.6947i	0.477512	+	0.477512i	0.904335	−	0.426823i	$-0.140367\pi$
$947$	14.6947	+	14.6947i	0.477512	+	0.477512i	−0.426823	+	0.904335i	$0.640367\pi$
$948$		0			0
$949$	−2.01810			−0.0655103
$950$		0			0
$951$		−	16.0871i		−	0.521661i
$952$		0			0
$953$	10.9129	−	10.9129i	0.353503	−	0.353503i	−0.507908	−	0.861411i	$-0.669581\pi$
$953$	10.9129	−	10.9129i	0.353503	−	0.353503i	0.861411	+	0.507908i	$0.169581\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	10.5830	−	10.5830i	0.342100	−	0.342100i
$958$		0			0
$959$	67.9129			2.19302
$960$		0			0
$961$	28.4955			0.919208
$962$		0			0
$963$	21.4322	−	21.4322i	0.690642	−	0.690642i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	8.79129	−	8.79129i	0.282709	−	0.282709i	−0.551480	−	0.834188i	$-0.685937\pi$
$967$	8.79129	−	8.79129i	0.282709	−	0.282709i	0.834188	+	0.551480i	$0.185937\pi$
$968$		0			0
$969$		−	9.49545i		−	0.305038i
$970$		0			0
$971$	−37.5728			−1.20577			−0.602885	−	0.797828i	$-0.705982\pi$
$971$	−37.5728			−1.20577			−0.602885	+	0.797828i	$0.705982\pi$
$972$		0			0
$973$	24.9717	+	24.9717i	0.800556	+	0.800556i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.5826	−	28.5826i	−0.914438	−	0.914438i	0.0821799	−	0.996618i	$-0.473812\pi$
$977$	−28.5826	−	28.5826i	−0.914438	−	0.914438i	−0.996618	+	0.0821799i	$0.973812\pi$
$978$		0			0
$979$			13.8564i			0.442853i
$980$		0			0
$981$		−	16.0254i		−	0.511652i
$982$		0			0
$983$	−26.7042	−	26.7042i	−0.851731	−	0.851731i	0.138616	−	0.990346i	$-0.455735\pi$
$983$	−26.7042	−	26.7042i	−0.851731	−	0.851731i	−0.990346	+	0.138616i	$0.955735\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.01810	−	2.01810i	−0.0642369	−	0.0642369i
$988$		0			0
$989$	−13.1334			−0.417618
$990$		0			0
$991$			3.25227i			0.103312i	0.998665	+	0.0516559i	$0.0164499\pi$
$991$			3.25227i			0.103312i	−0.998665	+	0.0516559i	$0.983550\pi$
$992$		0			0
$993$	−2.83485	+	2.83485i	−0.0899612	+	0.0899612i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−17.6066	+	17.6066i	−0.557605	+	0.557605i	−0.928625	−	0.371020i	$-0.879008\pi$
$997$	−17.6066	+	17.6066i	−0.557605	+	0.557605i	0.371020	+	0.928625i	$0.379008\pi$
$998$		0			0
$999$	−26.5045			−0.838567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.o.e.607.2		8
4.3	odd	2		1600.2.o.l.607.3		8
5.2	odd	4		320.2.o.e.223.3	yes	8
5.3	odd	4		1600.2.o.l.543.2		8
5.4	even	2		320.2.o.f.287.3	yes	8
8.3	odd	2		1600.2.o.l.607.2		8
8.5	even	2	inner	1600.2.o.e.607.3		8
20.3	even	4	inner	1600.2.o.e.543.3		8
20.7	even	4		320.2.o.f.223.2	yes	8
20.19	odd	2		320.2.o.e.287.2	yes	8
40.3	even	4	inner	1600.2.o.e.543.2		8
40.13	odd	4		1600.2.o.l.543.3		8
40.19	odd	2		320.2.o.e.287.3	yes	8
40.27	even	4		320.2.o.f.223.3	yes	8
40.29	even	2		320.2.o.f.287.2	yes	8
40.37	odd	4		320.2.o.e.223.2	✓	8
80.19	odd	4		1280.2.n.p.767.2		8
80.27	even	4		1280.2.n.n.1023.2		8
80.29	even	4		1280.2.n.n.767.3		8
80.37	odd	4		1280.2.n.p.1023.3		8
80.59	odd	4		1280.2.n.p.767.3		8
80.67	even	4		1280.2.n.n.1023.3		8
80.69	even	4		1280.2.n.n.767.2		8
80.77	odd	4		1280.2.n.p.1023.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.o.e.223.2	✓	8	40.37	odd	4
320.2.o.e.223.3	yes	8	5.2	odd	4
320.2.o.e.287.2	yes	8	20.19	odd	2
320.2.o.e.287.3	yes	8	40.19	odd	2
320.2.o.f.223.2	yes	8	20.7	even	4
320.2.o.f.223.3	yes	8	40.27	even	4
320.2.o.f.287.2	yes	8	40.29	even	2
320.2.o.f.287.3	yes	8	5.4	even	2
1280.2.n.n.767.2		8	80.69	even	4
1280.2.n.n.767.3		8	80.29	even	4
1280.2.n.n.1023.2		8	80.27	even	4
1280.2.n.n.1023.3		8	80.67	even	4
1280.2.n.p.767.2		8	80.19	odd	4
1280.2.n.p.767.3		8	80.59	odd	4
1280.2.n.p.1023.2		8	80.77	odd	4
1280.2.n.p.1023.3		8	80.37	odd	4
1600.2.o.e.543.2		8	40.3	even	4	inner
1600.2.o.e.543.3		8	20.3	even	4	inner
1600.2.o.e.607.2		8	1.1	even	1	trivial
1600.2.o.e.607.3		8	8.5	even	2	inner
1600.2.o.l.543.2		8	5.3	odd	4
1600.2.o.l.543.3		8	40.13	odd	4
1600.2.o.l.607.2		8	8.3	odd	2
1600.2.o.l.607.3		8	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.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.456850 + 0.456850i) q^{3} +(-2.79129 + 2.79129i) q^{7} +2.58258i q^{9} +O(q^{10})\)	\(q+(-0.456850 + 0.456850i) q^{3} +(-2.79129 + 2.79129i) q^{7} +2.58258i q^{9} -4.37780 q^{11} +(1.73205 + 1.73205i) q^{13} +(-3.00000 - 3.00000i) q^{17} -3.46410i q^{19} -2.55040i q^{21} +(-0.791288 - 0.791288i) q^{23} +(-2.55040 - 2.55040i) q^{27} +5.29150 q^{29} -1.58258i q^{31} +(2.00000 - 2.00000i) q^{33} +(5.19615 - 5.19615i) q^{37} -1.58258 q^{39} -7.58258 q^{41} +(8.29875 - 8.29875i) q^{43} +(0.791288 - 0.791288i) q^{47} -8.58258i q^{49} +2.74110 q^{51} +(2.64575 + 2.64575i) q^{53} +(1.58258 + 1.58258i) q^{57} +5.29150i q^{59} -9.66930i q^{61} +(-7.20871 - 7.20871i) q^{63} +(-8.29875 - 8.29875i) q^{67} +0.723000 q^{69} +13.5826i q^{71} +(-0.582576 + 0.582576i) q^{73} +(12.2197 - 12.2197i) q^{77} -12.0000 q^{79} -5.41742 q^{81} +(-4.83465 + 4.83465i) q^{83} +(-2.41742 + 2.41742i) q^{87} -3.16515i q^{89} -9.66930 q^{91} +(0.723000 + 0.723000i) q^{93} +(-0.582576 - 0.582576i) q^{97} -11.3060i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} - 24 q^{17} + 12 q^{23} + 16 q^{33} + 24 q^{39} - 24 q^{41} - 12 q^{47} - 24 q^{57} - 76 q^{63} + 32 q^{73} - 96 q^{79} - 80 q^{81} - 56 q^{87} + 32 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 24 * q^17 + 12 * q^23 + 16 * q^33 + 24 * q^39 - 24 * q^41 - 12 * q^47 - 24 * q^57 - 76 * q^63 + 32 * q^73 - 96 * q^79 - 80 * q^81 - 56 * q^87 + 32 * q^97

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