LMFDB - Embedded newform 3330.2.d.i.1999.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3330.1999");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.d (of order $2$, degree $1$, 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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3330.1999
Dual form			3330.2.d.i.1999.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -4.46410i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -4.46410i q^{7} +1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +1.73205 q^{11} -4.00000i q^{13} -4.46410 q^{14} +1.00000 q^{16} +6.46410i q^{17} +6.92820 q^{19} +(1.00000 - 2.00000i) q^{20} -1.73205i q^{22} -1.46410i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000 q^{26} +4.46410i q^{28} +3.00000 q^{29} -2.26795 q^{31} -1.00000i q^{32} +6.46410 q^{34} +(8.92820 + 4.46410i) q^{35} +1.00000i q^{37} -6.92820i q^{38} +(-2.00000 - 1.00000i) q^{40} -1.19615 q^{41} +7.19615i q^{43} -1.73205 q^{44} -1.46410 q^{46} -4.53590i q^{47} -12.9282 q^{49} +(-4.00000 + 3.00000i) q^{50} +4.00000i q^{52} -3.73205i q^{53} +(-1.73205 + 3.46410i) q^{55} +4.46410 q^{56} -3.00000i q^{58} +2.53590 q^{59} +5.19615 q^{61} +2.26795i q^{62} -1.00000 q^{64} +(8.00000 + 4.00000i) q^{65} -5.46410i q^{67} -6.46410i q^{68} +(4.46410 - 8.92820i) q^{70} +0.928203 q^{71} -14.3923i q^{73} +1.00000 q^{74} -6.92820 q^{76} -7.73205i q^{77} +11.4641 q^{79} +(-1.00000 + 2.00000i) q^{80} +1.19615i q^{82} -9.46410i q^{83} +(-12.9282 - 6.46410i) q^{85} +7.19615 q^{86} +1.73205i q^{88} +2.53590 q^{89} -17.8564 q^{91} +1.46410i q^{92} -4.53590 q^{94} +(-6.92820 + 13.8564i) q^{95} +13.5885i q^{97} +12.9282i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^5	$ 4 q - 4 q^{4} - 4 q^{5} + 8 q^{10} - 4 q^{14} + 4 q^{16} + 4 q^{20} - 12 q^{25} - 16 q^{26} + 12 q^{29} - 16 q^{31} + 12 q^{34} + 8 q^{35} - 8 q^{40} + 16 q^{41} + 8 q^{46} - 24 q^{49} - 16 q^{50} + 4 q^{56} + 24 q^{59} - 4 q^{64} + 32 q^{65} + 4 q^{70} - 24 q^{71} + 4 q^{74} + 32 q^{79} - 4 q^{80} - 24 q^{85} + 8 q^{86} + 24 q^{89} - 16 q^{91} - 32 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^10 - 4 * q^14 + 4 * q^16 + 4 * q^20 - 12 * q^25 - 16 * q^26 + 12 * q^29 - 16 * q^31 + 12 * q^34 + 8 * q^35 - 8 * q^40 + 16 * q^41 + 8 * q^46 - 24 * q^49 - 16 * q^50 + 4 * q^56 + 24 * q^59 - 4 * q^64 + 32 * q^65 + 4 * q^70 - 24 * q^71 + 4 * q^74 + 32 * q^79 - 4 * q^80 - 24 * q^85 + 8 * q^86 + 24 * q^89 - 16 * q^91 - 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$		−	4.46410i		−	1.68727i	−0.536916	−	0.843636i	$-0.680411\pi$
$7$		−	4.46410i		−	1.68727i	0.536916	−	0.843636i	$-0.319589\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	1.73205			0.522233			0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205			0.522233			0.261116	+	0.965307i	$0.415909\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−4.46410			−1.19308
$15$		0			0
$16$	1.00000			0.250000
$17$			6.46410i			1.56777i	0.620903	+	0.783887i	$0.286766\pi$
$17$			6.46410i			1.56777i	−0.620903	+	0.783887i	$0.713234\pi$
$18$		0			0
$19$	6.92820			1.58944			0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820			1.58944			0.794719	+	0.606977i	$0.207618\pi$
$20$	1.00000	−	2.00000i	0.223607	−	0.447214i
$21$		0			0
$22$		−	1.73205i		−	0.369274i
$23$		−	1.46410i		−	0.305286i	−0.988281	−	0.152643i	$-0.951221\pi$
$23$		−	1.46410i		−	0.305286i	0.988281	−	0.152643i	$-0.0487785\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$			4.46410i			0.843636i
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$	−2.26795			−0.407336			−0.203668	−	0.979040i	$-0.565286\pi$
$31$	−2.26795			−0.407336			−0.203668	+	0.979040i	$0.565286\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	6.46410			1.10858
$35$	8.92820	+	4.46410i	1.50914	+	0.754571i
$36$		0			0
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		−	6.92820i		−	1.12390i
$39$		0			0
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	−1.19615			−0.186808			−0.0934038	−	0.995628i	$-0.529775\pi$
$41$	−1.19615			−0.186808			−0.0934038	+	0.995628i	$0.529775\pi$
$42$		0			0
$43$			7.19615i			1.09740i	0.836018	+	0.548701i	$0.184878\pi$
$43$			7.19615i			1.09740i	−0.836018	+	0.548701i	$0.815122\pi$
$44$	−1.73205			−0.261116
$45$		0			0
$46$	−1.46410			−0.215870
$47$		−	4.53590i		−	0.661629i	−0.943696	−	0.330814i	$-0.892677\pi$
$47$		−	4.53590i		−	0.661629i	0.943696	−	0.330814i	$-0.107323\pi$
$48$		0			0
$49$	−12.9282			−1.84689
$50$	−4.00000	+	3.00000i	−0.565685	+	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$		−	3.73205i		−	0.512637i	−0.966592	−	0.256318i	$-0.917490\pi$
$53$		−	3.73205i		−	0.512637i	0.966592	−	0.256318i	$-0.0825095\pi$
$54$		0			0
$55$	−1.73205	+	3.46410i	−0.233550	+	0.467099i
$56$	4.46410			0.596541
$57$		0			0
$58$		−	3.00000i		−	0.393919i
$59$	2.53590			0.330146			0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590			0.330146			0.165073	+	0.986281i	$0.447214\pi$
$60$		0			0
$61$	5.19615			0.665299			0.332650	−	0.943051i	$-0.392057\pi$
$61$	5.19615			0.665299			0.332650	+	0.943051i	$0.392057\pi$
$62$			2.26795i			0.288030i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	8.00000	+	4.00000i	0.992278	+	0.496139i
$66$		0			0
$67$		−	5.46410i		−	0.667546i	−0.942653	−	0.333773i	$-0.891678\pi$
$67$		−	5.46410i		−	0.667546i	0.942653	−	0.333773i	$-0.108322\pi$
$68$		−	6.46410i		−	0.783887i
$69$		0			0
$70$	4.46410	−	8.92820i	0.533562	−	1.06712i
$71$	0.928203			0.110157			0.0550787	−	0.998482i	$-0.482459\pi$
$71$	0.928203			0.110157			0.0550787	+	0.998482i	$0.482459\pi$
$72$		0			0
$73$		−	14.3923i		−	1.68449i	−0.539093	−	0.842246i	$-0.681233\pi$
$73$		−	14.3923i		−	1.68449i	0.539093	−	0.842246i	$-0.318767\pi$
$74$	1.00000			0.116248
$75$		0			0
$76$	−6.92820			−0.794719
$77$		−	7.73205i		−	0.881149i
$78$		0			0
$79$	11.4641			1.28981			0.644906	−	0.764262i	$-0.276896\pi$
$79$	11.4641			1.28981			0.644906	+	0.764262i	$0.276896\pi$
$80$	−1.00000	+	2.00000i	−0.111803	+	0.223607i
$81$		0			0
$82$			1.19615i			0.132093i
$83$		−	9.46410i		−	1.03882i	−0.854525	−	0.519410i	$-0.826152\pi$
$83$		−	9.46410i		−	1.03882i	0.854525	−	0.519410i	$-0.173848\pi$
$84$		0			0
$85$	−12.9282	−	6.46410i	−1.40226	−	0.701130i
$86$	7.19615			0.775981
$87$		0			0
$88$			1.73205i			0.184637i
$89$	2.53590			0.268805			0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590			0.268805			0.134402	+	0.990927i	$0.457089\pi$
$90$		0			0
$91$	−17.8564			−1.87186
$92$			1.46410i			0.152643i
$93$		0			0
$94$	−4.53590			−0.467842
$95$	−6.92820	+	13.8564i	−0.710819	+	1.42164i
$96$		0			0
$97$			13.5885i			1.37970i	0.723953	+	0.689849i	$0.242323\pi$
$97$			13.5885i			1.37970i	−0.723953	+	0.689849i	$0.757677\pi$
$98$			12.9282i			1.30595i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	−6.39230			−0.636058			−0.318029	−	0.948081i	$-0.603021\pi$
$101$	−6.39230			−0.636058			−0.318029	+	0.948081i	$0.603021\pi$
$102$		0			0
$103$		−	2.53590i		−	0.249869i	−0.992165	−	0.124935i	$-0.960128\pi$
$103$		−	2.53590i		−	0.249869i	0.992165	−	0.124935i	$-0.0398722\pi$
$104$	4.00000			0.392232
$105$		0			0
$106$	−3.73205			−0.362489
$107$		−	0.535898i		−	0.0518073i	−0.999664	−	0.0259036i	$-0.991754\pi$
$107$		−	0.535898i		−	0.0518073i	0.999664	−	0.0259036i	$-0.00824630\pi$
$108$		0			0
$109$	−12.6603			−1.21263			−0.606316	−	0.795224i	$-0.707353\pi$
$109$	−12.6603			−1.21263			−0.606316	+	0.795224i	$0.707353\pi$
$110$	3.46410	+	1.73205i	0.330289	+	0.165145i
$111$		0			0
$112$		−	4.46410i		−	0.421818i
$113$		−	7.53590i		−	0.708918i	−0.935072	−	0.354459i	$-0.884665\pi$
$113$		−	7.53590i		−	0.708918i	0.935072	−	0.354459i	$-0.115335\pi$
$114$		0			0
$115$	2.92820	+	1.46410i	0.273056	+	0.136528i
$116$	−3.00000			−0.278543
$117$		0			0
$118$		−	2.53590i		−	0.233448i
$119$	28.8564			2.64526
$120$		0			0
$121$	−8.00000			−0.727273
$122$		−	5.19615i		−	0.470438i
$123$		0			0
$124$	2.26795			0.203668
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	4.00000	−	8.00000i	0.350823	−	0.701646i
$131$	−4.53590			−0.396303			−0.198152	−	0.980171i	$-0.563494\pi$
$131$	−4.53590			−0.396303			−0.198152	+	0.980171i	$0.563494\pi$
$132$		0			0
$133$		−	30.9282i		−	2.68182i
$134$	−5.46410			−0.472026
$135$		0			0
$136$	−6.46410			−0.554292
$137$		−	19.8564i		−	1.69645i	−0.529638	−	0.848224i	$-0.677672\pi$
$137$		−	19.8564i		−	1.69645i	0.529638	−	0.848224i	$-0.322328\pi$
$138$		0			0
$139$	5.92820			0.502824			0.251412	−	0.967880i	$-0.419105\pi$
$139$	5.92820			0.502824			0.251412	+	0.967880i	$0.419105\pi$
$140$	−8.92820	−	4.46410i	−0.754571	−	0.377285i
$141$		0			0
$142$		−	0.928203i		−	0.0778931i
$143$		−	6.92820i		−	0.579365i
$144$		0			0
$145$	−3.00000	+	6.00000i	−0.249136	+	0.498273i
$146$	−14.3923			−1.19112
$147$		0			0
$148$		−	1.00000i		−	0.0821995i
$149$	8.00000			0.655386			0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000			0.655386			0.327693	+	0.944784i	$0.393729\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$			6.92820i			0.561951i
$153$		0			0
$154$	−7.73205			−0.623066
$155$	2.26795	−	4.53590i	0.182166	−	0.364332i
$156$		0			0
$157$		−	9.53590i		−	0.761048i	−0.924771	−	0.380524i	$-0.875744\pi$
$157$		−	9.53590i		−	0.761048i	0.924771	−	0.380524i	$-0.124256\pi$
$158$		−	11.4641i		−	0.912035i
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$	−6.53590			−0.515101
$162$		0			0
$163$		−	4.26795i		−	0.334292i	−0.985932	−	0.167146i	$-0.946545\pi$
$163$		−	4.26795i		−	0.334292i	0.985932	−	0.167146i	$-0.0534551\pi$
$164$	1.19615			0.0934038
$165$		0			0
$166$	−9.46410			−0.734557
$167$			14.3923i			1.11371i	0.830610	+	0.556855i	$0.187992\pi$
$167$			14.3923i			1.11371i	−0.830610	+	0.556855i	$0.812008\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	−6.46410	+	12.9282i	−0.495774	+	0.991548i
$171$		0			0
$172$		−	7.19615i		−	0.548701i
$173$		−	21.0526i		−	1.60060i	−0.599602	−	0.800298i	$-0.704675\pi$
$173$		−	21.0526i		−	1.60060i	0.599602	−	0.800298i	$-0.295325\pi$
$174$		0			0
$175$	−17.8564	+	13.3923i	−1.34982	+	1.01236i
$176$	1.73205			0.130558
$177$		0			0
$178$		−	2.53590i		−	0.190074i
$179$	26.3923			1.97265			0.986327	−	0.164802i	$-0.0526984\pi$
$179$	26.3923			1.97265			0.986327	+	0.164802i	$0.0526984\pi$
$180$		0			0
$181$	12.3923			0.921113			0.460556	−	0.887630i	$-0.347650\pi$
$181$	12.3923			0.921113			0.460556	+	0.887630i	$0.347650\pi$
$182$			17.8564i			1.32360i
$183$		0			0
$184$	1.46410			0.107935
$185$	−2.00000	−	1.00000i	−0.147043	−	0.0735215i
$186$		0			0
$187$			11.1962i			0.818744i
$188$			4.53590i			0.330814i
$189$		0			0
$190$	13.8564	+	6.92820i	1.00525	+	0.502625i
$191$	−21.7846			−1.57628			−0.788140	−	0.615497i	$-0.788955\pi$
$191$	−21.7846			−1.57628			−0.788140	+	0.615497i	$0.788955\pi$
$192$		0			0
$193$		−	20.0000i		−	1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$		−	20.0000i		−	1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$	13.5885			0.975594
$195$		0			0
$196$	12.9282			0.923443
$197$		−	24.7846i		−	1.76583i	−0.469533	−	0.882915i	$-0.655578\pi$
$197$		−	24.7846i		−	1.76583i	0.469533	−	0.882915i	$-0.344422\pi$
$198$		0			0
$199$	−9.32051			−0.660713			−0.330357	−	0.943856i	$-0.607169\pi$
$199$	−9.32051			−0.660713			−0.330357	+	0.943856i	$0.607169\pi$
$200$	4.00000	−	3.00000i	0.282843	−	0.212132i
$201$		0			0
$202$			6.39230i			0.449761i
$203$		−	13.3923i		−	0.939956i
$204$		0			0
$205$	1.19615	−	2.39230i	0.0835429	−	0.167086i
$206$	−2.53590			−0.176684
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$	12.0000			0.830057
$210$		0			0
$211$	13.7846			0.948972			0.474486	−	0.880263i	$-0.342634\pi$
$211$	13.7846			0.948972			0.474486	+	0.880263i	$0.342634\pi$
$212$			3.73205i			0.256318i
$213$		0			0
$214$	−0.535898			−0.0366333
$215$	−14.3923	−	7.19615i	−0.981547	−	0.490774i
$216$		0			0
$217$			10.1244i			0.687286i
$218$			12.6603i			0.857461i
$219$		0			0
$220$	1.73205	−	3.46410i	0.116775	−	0.233550i
$221$	25.8564			1.73929
$222$		0			0
$223$		−	24.3205i		−	1.62862i	−0.580429	−	0.814311i	$-0.697115\pi$
$223$		−	24.3205i		−	1.62862i	0.580429	−	0.814311i	$-0.302885\pi$
$224$	−4.46410			−0.298270
$225$		0			0
$226$	−7.53590			−0.501281
$227$			14.4641i			0.960016i	0.877264	+	0.480008i	$0.159366\pi$
$227$			14.4641i			0.960016i	−0.877264	+	0.480008i	$0.840634\pi$
$228$		0			0
$229$	−3.60770			−0.238403			−0.119202	−	0.992870i	$-0.538033\pi$
$229$	−3.60770			−0.238403			−0.119202	+	0.992870i	$0.538033\pi$
$230$	1.46410	−	2.92820i	0.0965400	−	0.193080i
$231$		0			0
$232$			3.00000i			0.196960i
$233$			17.3205i			1.13470i	0.823475	+	0.567352i	$0.192032\pi$
$233$			17.3205i			1.13470i	−0.823475	+	0.567352i	$0.807968\pi$
$234$		0			0
$235$	9.07180	+	4.53590i	0.591779	+	0.295889i
$236$	−2.53590			−0.165073
$237$		0			0
$238$		−	28.8564i		−	1.87048i
$239$	−5.00000			−0.323423			−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000			−0.323423			−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$	−7.07180			−0.455534			−0.227767	−	0.973716i	$-0.573143\pi$
$241$	−7.07180			−0.455534			−0.227767	+	0.973716i	$0.573143\pi$
$242$			8.00000i			0.514259i
$243$		0			0
$244$	−5.19615			−0.332650
$245$	12.9282	−	25.8564i	0.825953	−	1.65191i
$246$		0			0
$247$		−	27.7128i		−	1.76332i
$248$		−	2.26795i		−	0.144015i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		−	2.53590i		−	0.159431i
$254$	−16.0000			−1.00393
$255$		0			0
$256$	1.00000			0.0625000
$257$			19.8564i			1.23861i	0.785151	+	0.619304i	$0.212585\pi$
$257$			19.8564i			1.23861i	−0.785151	+	0.619304i	$0.787415\pi$
$258$		0			0
$259$	4.46410			0.277386
$260$	−8.00000	−	4.00000i	−0.496139	−	0.248069i
$261$		0			0
$262$			4.53590i			0.280229i
$263$			14.1244i			0.870945i	0.900202	+	0.435473i	$0.143419\pi$
$263$			14.1244i			0.870945i	−0.900202	+	0.435473i	$0.856581\pi$
$264$		0			0
$265$	7.46410	+	3.73205i	0.458516	+	0.229258i
$266$	−30.9282			−1.89633
$267$		0			0
$268$			5.46410i			0.333773i
$269$	−16.0000			−0.975537			−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000			−0.975537			−0.487769	+	0.872973i	$0.662189\pi$
$270$		0			0
$271$	23.8564			1.44917			0.724587	−	0.689184i	$-0.242031\pi$
$271$	23.8564			1.44917			0.724587	+	0.689184i	$0.242031\pi$
$272$			6.46410i			0.391944i
$273$		0			0
$274$	−19.8564			−1.19957
$275$	−5.19615	−	6.92820i	−0.313340	−	0.417786i
$276$		0			0
$277$		−	27.7128i		−	1.66510i	−0.553949	−	0.832551i	$-0.686880\pi$
$277$		−	27.7128i		−	1.66510i	0.553949	−	0.832551i	$-0.313120\pi$
$278$		−	5.92820i		−	0.355550i
$279$		0			0
$280$	−4.46410	+	8.92820i	−0.266781	+	0.533562i
$281$	−12.9282			−0.771232			−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282			−0.771232			−0.385616	+	0.922659i	$0.626011\pi$
$282$		0			0
$283$		−	11.4641i		−	0.681470i	−0.940159	−	0.340735i	$-0.889324\pi$
$283$		−	11.4641i		−	0.681470i	0.940159	−	0.340735i	$-0.110676\pi$
$284$	−0.928203			−0.0550787
$285$		0			0
$286$	−6.92820			−0.409673
$287$			5.33975i			0.315195i
$288$		0			0
$289$	−24.7846			−1.45792
$290$	6.00000	+	3.00000i	0.352332	+	0.176166i
$291$		0			0
$292$			14.3923i			0.842246i
$293$		−	30.1244i		−	1.75988i	−0.475081	−	0.879942i	$-0.657581\pi$
$293$		−	30.1244i		−	1.75988i	0.475081	−	0.879942i	$-0.342419\pi$
$294$		0			0
$295$	−2.53590	+	5.07180i	−0.147646	+	0.295291i
$296$	−1.00000			−0.0581238
$297$		0			0
$298$		−	8.00000i		−	0.463428i
$299$	−5.85641			−0.338685
$300$		0			0
$301$	32.1244			1.85162
$302$			2.00000i			0.115087i
$303$		0			0
$304$	6.92820			0.397360
$305$	−5.19615	+	10.3923i	−0.297531	+	0.595062i
$306$		0			0
$307$		−	13.4641i		−	0.768437i	−0.923242	−	0.384218i	$-0.874471\pi$
$307$		−	13.4641i		−	0.768437i	0.923242	−	0.384218i	$-0.125529\pi$
$308$			7.73205i			0.440574i
$309$		0			0
$310$	−4.53590	−	2.26795i	−0.257622	−	0.128811i
$311$	17.7846			1.00847			0.504236	−	0.863566i	$-0.331774\pi$
$311$	17.7846			1.00847			0.504236	+	0.863566i	$0.331774\pi$
$312$		0			0
$313$			17.8564i			1.00930i	0.863323	+	0.504652i	$0.168379\pi$
$313$			17.8564i			1.00930i	−0.863323	+	0.504652i	$0.831621\pi$
$314$	−9.53590			−0.538142
$315$		0			0
$316$	−11.4641			−0.644906
$317$			0.803848i			0.0451486i	0.999745	+	0.0225743i	$0.00718623\pi$
$317$			0.803848i			0.0451486i	−0.999745	+	0.0225743i	$0.992814\pi$
$318$		0			0
$319$	5.19615			0.290929
$320$	1.00000	−	2.00000i	0.0559017	−	0.111803i
$321$		0			0
$322$			6.53590i			0.364231i
$323$			44.7846i			2.49188i
$324$		0			0
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$	−4.26795			−0.236380
$327$		0			0
$328$		−	1.19615i		−	0.0660465i
$329$	−20.2487			−1.11635
$330$		0			0
$331$	7.07180			0.388701			0.194351	−	0.980932i	$-0.437740\pi$
$331$	7.07180			0.388701			0.194351	+	0.980932i	$0.437740\pi$
$332$			9.46410i			0.519410i
$333$		0			0
$334$	14.3923			0.787512
$335$	10.9282	+	5.46410i	0.597072	+	0.298536i
$336$		0			0
$337$			3.32051i			0.180880i	0.995902	+	0.0904398i	$0.0288273\pi$
$337$			3.32051i			0.180880i	−0.995902	+	0.0904398i	$0.971173\pi$
$338$			3.00000i			0.163178i
$339$		0			0
$340$	12.9282	+	6.46410i	0.701130	+	0.350565i
$341$	−3.92820			−0.212724
$342$		0			0
$343$			26.4641i			1.42893i
$344$	−7.19615			−0.387991
$345$		0			0
$346$	−21.0526			−1.13179
$347$			30.9282i			1.66031i	0.557530	+	0.830156i	$0.311749\pi$
$347$			30.9282i			1.66031i	−0.557530	+	0.830156i	$0.688251\pi$
$348$		0			0
$349$	15.4641			0.827774			0.413887	−	0.910328i	$-0.364171\pi$
$349$	15.4641			0.827774			0.413887	+	0.910328i	$0.364171\pi$
$350$	13.3923	+	17.8564i	0.715849	+	0.954465i
$351$		0			0
$352$		−	1.73205i		−	0.0923186i
$353$			25.3923i			1.35150i	0.737133	+	0.675748i	$0.236179\pi$
$353$			25.3923i			1.35150i	−0.737133	+	0.675748i	$0.763821\pi$
$354$		0			0
$355$	−0.928203	+	1.85641i	−0.0492639	+	0.0985278i
$356$	−2.53590			−0.134402
$357$		0			0
$358$		−	26.3923i		−	1.39488i
$359$	−27.8564			−1.47020			−0.735102	−	0.677956i	$-0.762866\pi$
$359$	−27.8564			−1.47020			−0.735102	+	0.677956i	$0.762866\pi$
$360$		0			0
$361$	29.0000			1.52632
$362$		−	12.3923i		−	0.651325i
$363$		0			0
$364$	17.8564			0.935930
$365$	28.7846	+	14.3923i	1.50666	+	0.753328i
$366$		0			0
$367$			27.3923i			1.42987i	0.699193	+	0.714933i	$0.253543\pi$
$367$			27.3923i			1.42987i	−0.699193	+	0.714933i	$0.746457\pi$
$368$		−	1.46410i		−	0.0763216i
$369$		0			0
$370$	−1.00000	+	2.00000i	−0.0519875	+	0.103975i
$371$	−16.6603			−0.864957
$372$		0			0
$373$			29.7128i			1.53847i	0.638965	+	0.769236i	$0.279363\pi$
$373$			29.7128i			1.53847i	−0.638965	+	0.769236i	$0.720637\pi$
$374$	11.1962			0.578939
$375$		0			0
$376$	4.53590			0.233921
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	−6.92820			−0.355878			−0.177939	−	0.984042i	$-0.556943\pi$
$379$	−6.92820			−0.355878			−0.177939	+	0.984042i	$0.556943\pi$
$380$	6.92820	−	13.8564i	0.355409	−	0.710819i
$381$		0			0
$382$			21.7846i			1.11460i
$383$		−	3.60770i		−	0.184345i	−0.995743	−	0.0921723i	$-0.970619\pi$
$383$		−	3.60770i		−	0.184345i	0.995743	−	0.0921723i	$-0.0293811\pi$
$384$		0			0
$385$	15.4641	+	7.73205i	0.788124	+	0.394062i
$386$	−20.0000			−1.01797
$387$		0			0
$388$		−	13.5885i		−	0.689849i
$389$	12.8564			0.651846			0.325923	−	0.945396i	$-0.394325\pi$
$389$	12.8564			0.651846			0.325923	+	0.945396i	$0.394325\pi$
$390$		0			0
$391$	9.46410			0.478620
$392$		−	12.9282i		−	0.652973i
$393$		0			0
$394$	−24.7846			−1.24863
$395$	−11.4641	+	22.9282i	−0.576822	+	1.15364i
$396$		0			0
$397$			28.9282i			1.45186i	0.687766	+	0.725932i	$0.258592\pi$
$397$			28.9282i			1.45186i	−0.687766	+	0.725932i	$0.741408\pi$
$398$			9.32051i			0.467195i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	−3.07180			−0.153398			−0.0766991	−	0.997054i	$-0.524438\pi$
$401$	−3.07180			−0.153398			−0.0766991	+	0.997054i	$0.524438\pi$
$402$		0			0
$403$			9.07180i			0.451898i
$404$	6.39230			0.318029
$405$		0			0
$406$	−13.3923			−0.664649
$407$			1.73205i			0.0858546i
$408$		0			0
$409$	−24.3923			−1.20612			−0.603061	−	0.797695i	$-0.706052\pi$
$409$	−24.3923			−1.20612			−0.603061	+	0.797695i	$0.706052\pi$
$410$	−2.39230	−	1.19615i	−0.118148	−	0.0590738i
$411$		0			0
$412$			2.53590i			0.124935i
$413$		−	11.3205i		−	0.557046i
$414$		0			0
$415$	18.9282	+	9.46410i	0.929149	+	0.464574i
$416$	−4.00000			−0.196116
$417$		0			0
$418$		−	12.0000i		−	0.586939i
$419$	25.3205			1.23699			0.618494	−	0.785790i	$-0.287743\pi$
$419$	25.3205			1.23699			0.618494	+	0.785790i	$0.287743\pi$
$420$		0			0
$421$	−13.8564			−0.675320			−0.337660	−	0.941268i	$-0.609635\pi$
$421$	−13.8564			−0.675320			−0.337660	+	0.941268i	$0.609635\pi$
$422$		−	13.7846i		−	0.671024i
$423$		0			0
$424$	3.73205			0.181244
$425$	25.8564	−	19.3923i	1.25422	−	0.940665i
$426$		0			0
$427$		−	23.1962i		−	1.12254i
$428$			0.535898i			0.0259036i
$429$		0			0
$430$	−7.19615	+	14.3923i	−0.347029	+	0.694059i
$431$	−16.0718			−0.774151			−0.387076	−	0.922048i	$-0.626515\pi$
$431$	−16.0718			−0.774151			−0.387076	+	0.922048i	$0.626515\pi$
$432$		0			0
$433$			24.3923i			1.17222i	0.810232	+	0.586110i	$0.199341\pi$
$433$			24.3923i			1.17222i	−0.810232	+	0.586110i	$0.800659\pi$
$434$	10.1244			0.485985
$435$		0			0
$436$	12.6603			0.606316
$437$		−	10.1436i		−	0.485234i
$438$		0			0
$439$	2.80385			0.133820			0.0669102	−	0.997759i	$-0.478686\pi$
$439$	2.80385			0.133820			0.0669102	+	0.997759i	$0.478686\pi$
$440$	−3.46410	−	1.73205i	−0.165145	−	0.0825723i
$441$		0			0
$442$		−	25.8564i		−	1.22986i
$443$			26.0000i			1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$			26.0000i			1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$		0			0
$445$	−2.53590	+	5.07180i	−0.120213	+	0.240426i
$446$	−24.3205			−1.15161
$447$		0			0
$448$			4.46410i			0.210909i
$449$	33.4641			1.57927			0.789634	−	0.613578i	$-0.210270\pi$
$449$	33.4641			1.57927			0.789634	+	0.613578i	$0.210270\pi$
$450$		0			0
$451$	−2.07180			−0.0975571
$452$			7.53590i			0.354459i
$453$		0			0
$454$	14.4641			0.678834
$455$	17.8564	−	35.7128i	0.837121	−	1.67424i
$456$		0			0
$457$		−	8.80385i		−	0.411827i	−0.978570	−	0.205913i	$-0.933984\pi$
$457$		−	8.80385i		−	0.411827i	0.978570	−	0.205913i	$-0.0660165\pi$
$458$			3.60770i			0.168577i
$459$		0			0
$460$	−2.92820	−	1.46410i	−0.136528	−	0.0682641i
$461$	−2.07180			−0.0964932			−0.0482466	−	0.998835i	$-0.515363\pi$
$461$	−2.07180			−0.0964932			−0.0482466	+	0.998835i	$0.515363\pi$
$462$		0			0
$463$			7.32051i			0.340213i	0.985426	+	0.170106i	$0.0544112\pi$
$463$			7.32051i			0.340213i	−0.985426	+	0.170106i	$0.945589\pi$
$464$	3.00000			0.139272
$465$		0			0
$466$	17.3205			0.802357
$467$		−	19.3923i		−	0.897369i	−0.893690	−	0.448684i	$-0.851893\pi$
$467$		−	19.3923i		−	0.897369i	0.893690	−	0.448684i	$-0.148107\pi$
$468$		0			0
$469$	−24.3923			−1.12633
$470$	4.53590	−	9.07180i	0.209225	−	0.418451i
$471$		0			0
$472$			2.53590i			0.116724i
$473$			12.4641i			0.573100i
$474$		0			0
$475$	−20.7846	−	27.7128i	−0.953663	−	1.27155i
$476$	−28.8564			−1.32263
$477$		0			0
$478$			5.00000i			0.228695i
$479$	−1.85641			−0.0848214			−0.0424107	−	0.999100i	$-0.513504\pi$
$479$	−1.85641			−0.0848214			−0.0424107	+	0.999100i	$0.513504\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$			7.07180i			0.322112i
$483$		0			0
$484$	8.00000			0.363636
$485$	−27.1769	−	13.5885i	−1.23404	−	0.617020i
$486$		0			0
$487$			19.6077i			0.888509i	0.895901	+	0.444255i	$0.146531\pi$
$487$			19.6077i			0.888509i	−0.895901	+	0.444255i	$0.853469\pi$
$488$			5.19615i			0.235219i
$489$		0			0
$490$	−25.8564	−	12.9282i	−1.16807	−	0.584037i
$491$	−34.3923			−1.55210			−0.776051	−	0.630670i	$-0.782780\pi$
$491$	−34.3923			−1.55210			−0.776051	+	0.630670i	$0.782780\pi$
$492$		0			0
$493$			19.3923i			0.873385i
$494$	−27.7128			−1.24686
$495$		0			0
$496$	−2.26795			−0.101834
$497$		−	4.14359i		−	0.185866i
$498$		0			0
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$	−11.0000	+	2.00000i	−0.491935	+	0.0894427i
$501$		0			0
$502$			20.0000i			0.892644i
$503$		−	40.9282i		−	1.82490i	−0.409190	−	0.912449i	$-0.634189\pi$
$503$		−	40.9282i		−	1.82490i	0.409190	−	0.912449i	$-0.365811\pi$
$504$		0			0
$505$	6.39230	−	12.7846i	0.284454	−	0.568908i
$506$	−2.53590			−0.112734
$507$		0			0
$508$			16.0000i			0.709885i
$509$	13.6077			0.603150			0.301575	−	0.953442i	$-0.402488\pi$
$509$	13.6077			0.603150			0.301575	+	0.953442i	$0.402488\pi$
$510$		0			0
$511$	−64.2487			−2.84220
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	19.8564			0.875829
$515$	5.07180	+	2.53590i	0.223490	+	0.111745i
$516$		0			0
$517$		−	7.85641i		−	0.345524i
$518$		−	4.46410i		−	0.196141i
$519$		0			0
$520$	−4.00000	+	8.00000i	−0.175412	+	0.350823i
$521$	33.4449			1.46525			0.732623	−	0.680635i	$-0.238296\pi$
$521$	33.4449			1.46525			0.732623	+	0.680635i	$0.238296\pi$
$522$		0			0
$523$			28.2487i			1.23523i	0.786481	+	0.617615i	$0.211901\pi$
$523$			28.2487i			1.23523i	−0.786481	+	0.617615i	$0.788099\pi$
$524$	4.53590			0.198152
$525$		0			0
$526$	14.1244			0.615851
$527$		−	14.6603i		−	0.638611i
$528$		0			0
$529$	20.8564			0.906800
$530$	3.73205	−	7.46410i	0.162110	−	0.324220i
$531$		0			0
$532$			30.9282i			1.34091i
$533$			4.78461i			0.207244i
$534$		0			0
$535$	1.07180	+	0.535898i	0.0463378	+	0.0231689i
$536$	5.46410			0.236013
$537$		0			0
$538$			16.0000i			0.689809i
$539$	−22.3923			−0.964505
$540$		0			0
$541$	−29.8564			−1.28363			−0.641814	−	0.766861i	$-0.721818\pi$
$541$	−29.8564			−1.28363			−0.641814	+	0.766861i	$0.721818\pi$
$542$		−	23.8564i		−	1.02472i
$543$		0			0
$544$	6.46410			0.277146
$545$	12.6603	−	25.3205i	0.542306	−	1.08461i
$546$		0			0
$547$			1.58846i			0.0679175i	0.999423	+	0.0339588i	$0.0108115\pi$
$547$			1.58846i			0.0679175i	−0.999423	+	0.0339588i	$0.989189\pi$
$548$			19.8564i			0.848224i
$549$		0			0
$550$	−6.92820	+	5.19615i	−0.295420	+	0.221565i
$551$	20.7846			0.885454
$552$		0			0
$553$		−	51.1769i		−	2.17626i
$554$	−27.7128			−1.17740
$555$		0			0
$556$	−5.92820			−0.251412
$557$			1.60770i			0.0681202i	0.999420	+	0.0340601i	$0.0108438\pi$
$557$			1.60770i			0.0681202i	−0.999420	+	0.0340601i	$0.989156\pi$
$558$		0			0
$559$	28.7846			1.21746
$560$	8.92820	+	4.46410i	0.377285	+	0.188643i
$561$		0			0
$562$			12.9282i			0.545343i
$563$			44.1769i			1.86183i	0.365230	+	0.930917i	$0.380990\pi$
$563$			44.1769i			1.86183i	−0.365230	+	0.930917i	$0.619010\pi$
$564$		0			0
$565$	15.0718	+	7.53590i	0.634075	+	0.317038i
$566$	−11.4641			−0.481872
$567$		0			0
$568$			0.928203i			0.0389465i
$569$	11.4641			0.480600			0.240300	−	0.970699i	$-0.422754\pi$
$569$	11.4641			0.480600			0.240300	+	0.970699i	$0.422754\pi$
$570$		0			0
$571$	−37.7846			−1.58124			−0.790618	−	0.612309i	$-0.790241\pi$
$571$	−37.7846			−1.58124			−0.790618	+	0.612309i	$0.790241\pi$
$572$			6.92820i			0.289683i
$573$		0			0
$574$	5.33975			0.222877
$575$	−5.85641	+	4.39230i	−0.244229	+	0.183172i
$576$		0			0
$577$		−	36.7846i		−	1.53136i	−0.643220	−	0.765682i	$-0.722402\pi$
$577$		−	36.7846i		−	1.53136i	0.643220	−	0.765682i	$-0.277598\pi$
$578$			24.7846i			1.03090i
$579$		0			0
$580$	3.00000	−	6.00000i	0.124568	−	0.249136i
$581$	−42.2487			−1.75277
$582$		0			0
$583$		−	6.46410i		−	0.267716i
$584$	14.3923			0.595558
$585$		0			0
$586$	−30.1244			−1.24443
$587$		−	22.6077i		−	0.933119i	−0.884490	−	0.466560i	$-0.845493\pi$
$587$		−	22.6077i		−	0.933119i	0.884490	−	0.466560i	$-0.154507\pi$
$588$		0			0
$589$	−15.7128			−0.647435
$590$	5.07180	+	2.53590i	0.208803	+	0.104401i
$591$		0			0
$592$			1.00000i			0.0410997i
$593$			25.1769i			1.03389i	0.856018	+	0.516946i	$0.172931\pi$
$593$			25.1769i			1.03389i	−0.856018	+	0.516946i	$0.827069\pi$
$594$		0			0
$595$	−28.8564	+	57.7128i	−1.18300	+	2.36599i
$596$	−8.00000			−0.327693
$597$		0			0
$598$			5.85641i			0.239486i
$599$	22.2487			0.909058			0.454529	−	0.890732i	$-0.349808\pi$
$599$	22.2487			0.909058			0.454529	+	0.890732i	$0.349808\pi$
$600$		0			0
$601$	25.0000			1.01977			0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000			1.01977			0.509886	+	0.860242i	$0.329688\pi$
$602$		−	32.1244i		−	1.30929i
$603$		0			0
$604$	2.00000			0.0813788
$605$	8.00000	−	16.0000i	0.325246	−	0.650493i
$606$		0			0
$607$			6.00000i			0.243532i	0.992559	+	0.121766i	$0.0388558\pi$
$607$			6.00000i			0.243532i	−0.992559	+	0.121766i	$0.961144\pi$
$608$		−	6.92820i		−	0.280976i
$609$		0			0
$610$	10.3923	+	5.19615i	0.420772	+	0.210386i
$611$	−18.1436			−0.734011
$612$		0			0
$613$		−	8.32051i		−	0.336062i	−0.985782	−	0.168031i	$-0.946259\pi$
$613$		−	8.32051i		−	0.336062i	0.985782	−	0.168031i	$-0.0537409\pi$
$614$	−13.4641			−0.543367
$615$		0			0
$616$	7.73205			0.311533
$617$			21.7128i			0.874125i	0.899431	+	0.437062i	$0.143981\pi$
$617$			21.7128i			0.874125i	−0.899431	+	0.437062i	$0.856019\pi$
$618$		0			0
$619$	21.6410			0.869826			0.434913	−	0.900473i	$-0.356779\pi$
$619$	21.6410			0.869826			0.434913	+	0.900473i	$0.356779\pi$
$620$	−2.26795	+	4.53590i	−0.0910830	+	0.182166i
$621$		0			0
$622$		−	17.7846i		−	0.713098i
$623$		−	11.3205i		−	0.453547i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	17.8564			0.713686
$627$		0			0
$628$			9.53590i			0.380524i
$629$	−6.46410			−0.257741
$630$		0			0
$631$	26.5167			1.05561			0.527806	−	0.849365i	$-0.323015\pi$
$631$	26.5167			1.05561			0.527806	+	0.849365i	$0.323015\pi$
$632$			11.4641i			0.456017i
$633$		0			0
$634$	0.803848			0.0319249
$635$	32.0000	+	16.0000i	1.26988	+	0.634941i
$636$		0			0
$637$			51.7128i			2.04894i
$638$		−	5.19615i		−	0.205718i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	2.26795			0.0895786			0.0447893	−	0.998996i	$-0.485738\pi$
$641$	2.26795			0.0895786			0.0447893	+	0.998996i	$0.485738\pi$
$642$		0			0
$643$		−	0.267949i		−	0.0105669i	−0.999986	−	0.00528344i	$-0.998318\pi$
$643$		−	0.267949i		−	0.0105669i	0.999986	−	0.00528344i	$-0.00168178\pi$
$644$	6.53590			0.257550
$645$		0			0
$646$	44.7846			1.76203
$647$		−	23.1769i		−	0.911179i	−0.890190	−	0.455589i	$-0.849429\pi$
$647$		−	23.1769i		−	0.911179i	0.890190	−	0.455589i	$-0.150571\pi$
$648$		0			0
$649$	4.39230			0.172413
$650$	12.0000	+	16.0000i	0.470679	+	0.627572i
$651$		0			0
$652$			4.26795i			0.167146i
$653$			47.1769i			1.84618i	0.384590	+	0.923088i	$0.374343\pi$
$653$			47.1769i			1.84618i	−0.384590	+	0.923088i	$0.625657\pi$
$654$		0			0
$655$	4.53590	−	9.07180i	0.177232	−	0.354464i
$656$	−1.19615			−0.0467019
$657$		0			0
$658$			20.2487i			0.789377i
$659$	−20.5359			−0.799965			−0.399983	−	0.916523i	$-0.630984\pi$
$659$	−20.5359			−0.799965			−0.399983	+	0.916523i	$0.630984\pi$
$660$		0			0
$661$	35.8372			1.39390			0.696952	−	0.717118i	$-0.254539\pi$
$661$	35.8372			1.39390			0.696952	+	0.717118i	$0.254539\pi$
$662$		−	7.07180i		−	0.274853i
$663$		0			0
$664$	9.46410			0.367278
$665$	61.8564	+	30.9282i	2.39869	+	1.19934i
$666$		0			0
$667$		−	4.39230i		−	0.170071i
$668$		−	14.3923i		−	0.556855i
$669$		0			0
$670$	5.46410	−	10.9282i	0.211097	−	0.422193i
$671$	9.00000			0.347441
$672$		0			0
$673$		−	13.1769i		−	0.507933i	−0.967213	−	0.253966i	$-0.918265\pi$
$673$		−	13.1769i		−	0.507933i	0.967213	−	0.253966i	$-0.0817353\pi$
$674$	3.32051			0.127901
$675$		0			0
$676$	3.00000			0.115385
$677$		−	32.7846i		−	1.26001i	−0.776589	−	0.630007i	$-0.783052\pi$
$677$		−	32.7846i		−	1.26001i	0.776589	−	0.630007i	$-0.216948\pi$
$678$		0			0
$679$	60.6603			2.32793
$680$	6.46410	−	12.9282i	0.247887	−	0.495774i
$681$		0			0
$682$			3.92820i			0.150419i
$683$		−	33.1051i		−	1.26673i	−0.773852	−	0.633366i	$-0.781673\pi$
$683$		−	33.1051i		−	1.26673i	0.773852	−	0.633366i	$-0.218327\pi$
$684$		0			0
$685$	39.7128	+	19.8564i	1.51735	+	0.758674i
$686$	26.4641			1.01040
$687$		0			0
$688$			7.19615i			0.274351i
$689$	−14.9282			−0.568719
$690$		0			0
$691$	−0.856406			−0.0325792			−0.0162896	−	0.999867i	$-0.505185\pi$
$691$	−0.856406			−0.0325792			−0.0162896	+	0.999867i	$0.505185\pi$
$692$			21.0526i			0.800298i
$693$		0			0
$694$	30.9282			1.17402
$695$	−5.92820	+	11.8564i	−0.224870	+	0.449739i
$696$		0			0
$697$		−	7.73205i		−	0.292872i
$698$		−	15.4641i		−	0.585325i
$699$		0			0
$700$	17.8564	−	13.3923i	0.674909	−	0.506182i
$701$	28.9282			1.09260			0.546302	−	0.837589i	$-0.316035\pi$
$701$	28.9282			1.09260			0.546302	+	0.837589i	$0.316035\pi$
$702$		0			0
$703$			6.92820i			0.261302i
$704$	−1.73205			−0.0652791
$705$		0			0
$706$	25.3923			0.955652
$707$			28.5359i			1.07320i
$708$		0			0
$709$	21.4449			0.805379			0.402689	−	0.915337i	$-0.368075\pi$
$709$	21.4449			0.805379			0.402689	+	0.915337i	$0.368075\pi$
$710$	1.85641	+	0.928203i	0.0696697	+	0.0348348i
$711$		0			0
$712$			2.53590i			0.0950368i
$713$			3.32051i			0.124354i
$714$		0			0
$715$	13.8564	+	6.92820i	0.518200	+	0.259100i
$716$	−26.3923			−0.986327
$717$		0			0
$718$			27.8564i			1.03959i
$719$	29.4641			1.09883			0.549413	−	0.835551i	$-0.314851\pi$
$719$	29.4641			1.09883			0.549413	+	0.835551i	$0.314851\pi$
$720$		0			0
$721$	−11.3205			−0.421598
$722$		−	29.0000i		−	1.07927i
$723$		0			0
$724$	−12.3923			−0.460556
$725$	−9.00000	−	12.0000i	−0.334252	−	0.445669i
$726$		0			0
$727$		−	12.5359i		−	0.464931i	−0.972605	−	0.232465i	$-0.925321\pi$
$727$		−	12.5359i		−	0.464931i	0.972605	−	0.232465i	$-0.0746792\pi$
$728$		−	17.8564i		−	0.661802i
$729$		0			0
$730$	14.3923	−	28.7846i	0.532683	−	1.06537i
$731$	−46.5167			−1.72048
$732$		0			0
$733$		−	22.3205i		−	0.824427i	−0.911087	−	0.412213i	$-0.864756\pi$
$733$		−	22.3205i		−	0.824427i	0.911087	−	0.412213i	$-0.135244\pi$
$734$	27.3923			1.01107
$735$		0			0
$736$	−1.46410			−0.0539675
$737$		−	9.46410i		−	0.348615i
$738$		0			0
$739$	−2.07180			−0.0762123			−0.0381061	−	0.999274i	$-0.512132\pi$
$739$	−2.07180			−0.0762123			−0.0381061	+	0.999274i	$0.512132\pi$
$740$	2.00000	+	1.00000i	0.0735215	+	0.0367607i
$741$		0			0
$742$			16.6603i			0.611617i
$743$		−	3.73205i		−	0.136916i	−0.997654	−	0.0684578i	$-0.978192\pi$
$743$		−	3.73205i		−	0.136916i	0.997654	−	0.0684578i	$-0.0218079\pi$
$744$		0			0
$745$	−8.00000	+	16.0000i	−0.293097	+	0.586195i
$746$	29.7128			1.08786
$747$		0			0
$748$		−	11.1962i		−	0.409372i
$749$	−2.39230			−0.0874129
$750$		0			0
$751$	−9.17691			−0.334870			−0.167435	−	0.985883i	$-0.553548\pi$
$751$	−9.17691			−0.334870			−0.167435	+	0.985883i	$0.553548\pi$
$752$		−	4.53590i		−	0.165407i
$753$		0			0
$754$	−12.0000			−0.437014
$755$	2.00000	−	4.00000i	0.0727875	−	0.145575i
$756$		0			0
$757$		−	41.4641i		−	1.50704i	−0.657426	−	0.753519i	$-0.728355\pi$
$757$		−	41.4641i		−	1.50704i	0.657426	−	0.753519i	$-0.271645\pi$
$758$			6.92820i			0.251644i
$759$		0			0
$760$	−13.8564	−	6.92820i	−0.502625	−	0.251312i
$761$	18.2679			0.662213			0.331106	−	0.943593i	$-0.392578\pi$
$761$	18.2679			0.662213			0.331106	+	0.943593i	$0.392578\pi$
$762$		0			0
$763$			56.5167i			2.04604i
$764$	21.7846			0.788140
$765$		0			0
$766$	−3.60770			−0.130351
$767$		−	10.1436i		−	0.366264i
$768$		0			0
$769$	−13.4641			−0.485528			−0.242764	−	0.970085i	$-0.578054\pi$
$769$	−13.4641			−0.485528			−0.242764	+	0.970085i	$0.578054\pi$
$770$	7.73205	−	15.4641i	0.278644	−	0.557288i
$771$		0			0
$772$			20.0000i			0.719816i
$773$			31.9808i			1.15027i	0.818059	+	0.575134i	$0.195050\pi$
$773$			31.9808i			1.15027i	−0.818059	+	0.575134i	$0.804950\pi$
$774$		0			0
$775$	6.80385	+	9.07180i	0.244401	+	0.325869i
$776$	−13.5885			−0.487797
$777$		0			0
$778$		−	12.8564i		−	0.460925i
$779$	−8.28719			−0.296919
$780$		0			0
$781$	1.60770			0.0575279
$782$		−	9.46410i		−	0.338436i
$783$		0			0
$784$	−12.9282			−0.461722
$785$	19.0718	+	9.53590i	0.680702	+	0.340351i
$786$		0			0
$787$		−	26.3923i		−	0.940784i	−0.882458	−	0.470392i	$-0.844113\pi$
$787$		−	26.3923i		−	0.940784i	0.882458	−	0.470392i	$-0.155887\pi$
$788$			24.7846i			0.882915i
$789$		0			0
$790$	22.9282	+	11.4641i	0.815749	+	0.407874i
$791$	−33.6410			−1.19614
$792$		0			0
$793$		−	20.7846i		−	0.738083i
$794$	28.9282			1.02662
$795$		0			0
$796$	9.32051			0.330357
$797$			5.07180i			0.179652i	0.995957	+	0.0898261i	$0.0286311\pi$
$797$			5.07180i			0.179652i	−0.995957	+	0.0898261i	$0.971369\pi$
$798$		0			0
$799$	29.3205			1.03729
$800$	−4.00000	+	3.00000i	−0.141421	+	0.106066i
$801$		0			0
$802$			3.07180i			0.108469i
$803$		−	24.9282i		−	0.879697i
$804$		0			0
$805$	6.53590	−	13.0718i	0.230360	−	0.460720i
$806$	9.07180			0.319540
$807$		0			0
$808$		−	6.39230i		−	0.224880i
$809$	33.4641			1.17654			0.588268	−	0.808666i	$-0.299810\pi$
$809$	33.4641			1.17654			0.588268	+	0.808666i	$0.299810\pi$
$810$		0			0
$811$	37.8564			1.32932			0.664659	−	0.747147i	$-0.268577\pi$
$811$	37.8564			1.32932			0.664659	+	0.747147i	$0.268577\pi$
$812$			13.3923i			0.469978i
$813$		0			0
$814$	1.73205			0.0607083
$815$	8.53590	+	4.26795i	0.298999	+	0.149500i
$816$		0			0
$817$			49.8564i			1.74425i
$818$			24.3923i			0.852857i
$819$		0			0
$820$	−1.19615	+	2.39230i	−0.0417715	+	0.0835429i
$821$	−23.8564			−0.832594			−0.416297	−	0.909229i	$-0.636672\pi$
$821$	−23.8564			−0.832594			−0.416297	+	0.909229i	$0.636672\pi$
$822$		0			0
$823$		−	49.5692i		−	1.72787i	−0.503600	−	0.863937i	$-0.667991\pi$
$823$		−	49.5692i		−	1.72787i	0.503600	−	0.863937i	$-0.332009\pi$
$824$	2.53590			0.0883422
$825$		0			0
$826$	−11.3205			−0.393891
$827$		−	51.2487i		−	1.78209i	−0.453913	−	0.891046i	$-0.649972\pi$
$827$		−	51.2487i		−	1.78209i	0.453913	−	0.891046i	$-0.350028\pi$
$828$		0			0
$829$	40.6603			1.41219			0.706094	−	0.708118i	$-0.250455\pi$
$829$	40.6603			1.41219			0.706094	+	0.708118i	$0.250455\pi$
$830$	9.46410	−	18.9282i	0.328504	−	0.657008i
$831$		0			0
$832$			4.00000i			0.138675i
$833$		−	83.5692i		−	2.89550i
$834$		0			0
$835$	−28.7846	−	14.3923i	−0.996132	−	0.498066i
$836$	−12.0000			−0.415029
$837$		0			0
$838$		−	25.3205i		−	0.874682i
$839$	37.8564			1.30695			0.653474	−	0.756949i	$-0.273311\pi$
$839$	37.8564			1.30695			0.653474	+	0.756949i	$0.273311\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$			13.8564i			0.477523i
$843$		0			0
$844$	−13.7846			−0.474486
$845$	3.00000	−	6.00000i	0.103203	−	0.206406i
$846$		0			0
$847$			35.7128i			1.22711i
$848$		−	3.73205i		−	0.128159i
$849$		0			0
$850$	−19.3923	−	25.8564i	−0.665151	−	0.886867i
$851$	1.46410			0.0501888
$852$		0			0
$853$			57.0333i			1.95278i	0.216007	+	0.976392i	$0.430697\pi$
$853$			57.0333i			1.95278i	−0.216007	+	0.976392i	$0.569303\pi$
$854$	−23.1962			−0.793756
$855$		0			0
$856$	0.535898			0.0183166
$857$			24.3205i			0.830773i	0.909645	+	0.415386i	$0.136354\pi$
$857$			24.3205i			0.830773i	−0.909645	+	0.415386i	$0.863646\pi$
$858$		0			0
$859$	−0.287187			−0.00979870			−0.00489935	−	0.999988i	$-0.501560\pi$
$859$	−0.287187			−0.00979870			−0.00489935	+	0.999988i	$0.501560\pi$
$860$	14.3923	+	7.19615i	0.490774	+	0.245387i
$861$		0			0
$862$			16.0718i			0.547408i
$863$			43.9808i			1.49712i	0.663066	+	0.748561i	$0.269255\pi$
$863$			43.9808i			1.49712i	−0.663066	+	0.748561i	$0.730745\pi$
$864$		0			0
$865$	42.1051	+	21.0526i	1.43162	+	0.715808i
$866$	24.3923			0.828884
$867$		0			0
$868$		−	10.1244i		−	0.343643i
$869$	19.8564			0.673582
$870$		0			0
$871$	−21.8564			−0.740576
$872$		−	12.6603i		−	0.428730i
$873$		0			0
$874$	−10.1436			−0.343112
$875$	−8.92820	−	49.1051i	−0.301828	−	1.66006i
$876$		0			0
$877$		−	30.1769i		−	1.01900i	−0.860470	−	0.509501i	$-0.829830\pi$
$877$		−	30.1769i		−	1.01900i	0.860470	−	0.509501i	$-0.170170\pi$
$878$		−	2.80385i		−	0.0946253i
$879$		0			0
$880$	−1.73205	+	3.46410i	−0.0583874	+	0.116775i
$881$	−53.1962			−1.79222			−0.896112	−	0.443828i	$-0.853620\pi$
$881$	−53.1962			−1.79222			−0.896112	+	0.443828i	$0.853620\pi$
$882$		0			0
$883$			8.80385i			0.296273i	0.988967	+	0.148137i	$0.0473275\pi$
$883$			8.80385i			0.296273i	−0.988967	+	0.148137i	$0.952672\pi$
$884$	−25.8564			−0.869645
$885$		0			0
$886$	26.0000			0.873487
$887$			4.80385i			0.161297i	0.996743	+	0.0806487i	$0.0256992\pi$
$887$			4.80385i			0.161297i	−0.996743	+	0.0806487i	$0.974301\pi$
$888$		0			0
$889$	−71.4256			−2.39554
$890$	5.07180	+	2.53590i	0.170007	+	0.0850035i
$891$		0			0
$892$			24.3205i			0.814311i
$893$		−	31.4256i		−	1.05162i
$894$		0			0
$895$	−26.3923	+	52.7846i	−0.882197	+	1.76439i
$896$	4.46410			0.149135
$897$		0			0
$898$		−	33.4641i		−	1.11671i
$899$	−6.80385			−0.226921
$900$		0			0
$901$	24.1244			0.803699
$902$			2.07180i			0.0689833i
$903$		0			0
$904$	7.53590			0.250640
$905$	−12.3923	+	24.7846i	−0.411934	+	0.823868i
$906$		0			0
$907$			32.2487i			1.07080i	0.844598	+	0.535400i	$0.179839\pi$
$907$			32.2487i			1.07080i	−0.844598	+	0.535400i	$0.820161\pi$
$908$		−	14.4641i		−	0.480008i
$909$		0			0
$910$	−35.7128	−	17.8564i	−1.18387	−	0.591934i
$911$	39.7128			1.31574			0.657872	−	0.753130i	$-0.271457\pi$
$911$	39.7128			1.31574			0.657872	+	0.753130i	$0.271457\pi$
$912$		0			0
$913$		−	16.3923i		−	0.542506i
$914$	−8.80385			−0.291205
$915$		0			0
$916$	3.60770			0.119202
$917$			20.2487i			0.668671i
$918$		0			0
$919$	−4.53590			−0.149625			−0.0748127	−	0.997198i	$-0.523836\pi$
$919$	−4.53590			−0.149625			−0.0748127	+	0.997198i	$0.523836\pi$
$920$	−1.46410	+	2.92820i	−0.0482700	+	0.0965400i
$921$		0			0
$922$			2.07180i			0.0682310i
$923$		−	3.71281i		−	0.122209i
$924$		0			0
$925$	4.00000	−	3.00000i	0.131519	−	0.0986394i
$926$	7.32051			0.240567
$927$		0			0
$928$		−	3.00000i		−	0.0984798i
$929$	48.1244			1.57891			0.789454	−	0.613809i	$-0.210364\pi$
$929$	48.1244			1.57891			0.789454	+	0.613809i	$0.210364\pi$
$930$		0			0
$931$	−89.5692			−2.93551
$932$		−	17.3205i		−	0.567352i
$933$		0			0
$934$	−19.3923			−0.634536
$935$	−22.3923	−	11.1962i	−0.732307	−	0.366153i
$936$		0			0
$937$		−	12.7846i		−	0.417655i	−0.977952	−	0.208827i	$-0.933035\pi$
$937$		−	12.7846i		−	0.417655i	0.977952	−	0.208827i	$-0.0669647\pi$
$938$			24.3923i			0.796437i
$939$		0			0
$940$	−9.07180	−	4.53590i	−0.295889	−	0.147945i
$941$	−38.4974			−1.25498			−0.627490	−	0.778625i	$-0.715918\pi$
$941$	−38.4974			−1.25498			−0.627490	+	0.778625i	$0.715918\pi$
$942$		0			0
$943$			1.75129i			0.0570298i
$944$	2.53590			0.0825365
$945$		0			0
$946$	12.4641			0.405243
$947$			29.3923i			0.955122i	0.878599	+	0.477561i	$0.158479\pi$
$947$			29.3923i			0.955122i	−0.878599	+	0.477561i	$0.841521\pi$
$948$		0			0
$949$	−57.5692			−1.86878
$950$	−27.7128	+	20.7846i	−0.899122	+	0.674342i
$951$		0			0
$952$			28.8564i			0.935242i
$953$			31.8564i			1.03193i	0.856610	+	0.515965i	$0.172567\pi$
$953$			31.8564i			1.03193i	−0.856610	+	0.515965i	$0.827433\pi$
$954$		0			0
$955$	21.7846	−	43.5692i	0.704933	−	1.40987i
$956$	5.00000			0.161712
$957$		0			0
$958$			1.85641i			0.0599778i
$959$	−88.6410			−2.86237
$960$		0			0
$961$	−25.8564			−0.834078
$962$		−	4.00000i		−	0.128965i
$963$		0			0
$964$	7.07180			0.227767
$965$	40.0000	+	20.0000i	1.28765	+	0.643823i
$966$		0			0
$967$			8.64102i			0.277876i	0.990301	+	0.138938i	$0.0443689\pi$
$967$			8.64102i			0.277876i	−0.990301	+	0.138938i	$0.955631\pi$
$968$		−	8.00000i		−	0.257130i
$969$		0			0
$970$	−13.5885	+	27.1769i	−0.436299	+	0.872598i
$971$	−59.8372			−1.92027			−0.960133	−	0.279543i	$-0.909817\pi$
$971$	−59.8372			−1.92027			−0.960133	+	0.279543i	$0.909817\pi$
$972$		0			0
$973$		−	26.4641i		−	0.848400i
$974$	19.6077			0.628271
$975$		0			0
$976$	5.19615			0.166325
$977$			5.53590i			0.177109i	0.996071	+	0.0885545i	$0.0282248\pi$
$977$			5.53590i			0.177109i	−0.996071	+	0.0885545i	$0.971775\pi$
$978$		0			0
$979$	4.39230			0.140379
$980$	−12.9282	+	25.8564i	−0.412976	+	0.825953i
$981$		0			0
$982$			34.3923i			1.09750i
$983$		−	53.3013i		−	1.70005i	−0.526745	−	0.850023i	$-0.676588\pi$
$983$		−	53.3013i		−	1.70005i	0.526745	−	0.850023i	$-0.323412\pi$
$984$		0			0
$985$	49.5692	+	24.7846i	1.57941	+	0.789703i
$986$	19.3923			0.617577
$987$		0			0
$988$			27.7128i			0.881662i
$989$	10.5359			0.335022
$990$		0			0
$991$	44.6603			1.41868			0.709340	−	0.704867i	$-0.248993\pi$
$991$	44.6603			1.41868			0.709340	+	0.704867i	$0.248993\pi$
$992$			2.26795i			0.0720075i
$993$		0			0
$994$	−4.14359			−0.131427
$995$	9.32051	−	18.6410i	0.295480	−	0.590960i
$996$		0			0
$997$			12.3923i			0.392468i	0.980557	+	0.196234i	$0.0628713\pi$
$997$			12.3923i			0.392468i	−0.980557	+	0.196234i	$0.937129\pi$
$998$			6.00000i			0.189927i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.d.i.1999.1	✓	4
3.2	odd	2		3330.2.d.l.1999.3	yes	4
5.4	even	2	inner	3330.2.d.i.1999.4	yes	4
15.14	odd	2		3330.2.d.l.1999.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.d.i.1999.1	✓	4	1.1	even	1	trivial
3330.2.d.i.1999.4	yes	4	5.4	even	2	inner
3330.2.d.l.1999.2	yes	4	15.14	odd	2
3330.2.d.l.1999.3	yes	4	3.2	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -4.46410i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -4.46410i q^{7} +1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +1.73205 q^{11} -4.00000i q^{13} -4.46410 q^{14} +1.00000 q^{16} +6.46410i q^{17} +6.92820 q^{19} +(1.00000 - 2.00000i) q^{20} -1.73205i q^{22} -1.46410i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000 q^{26} +4.46410i q^{28} +3.00000 q^{29} -2.26795 q^{31} -1.00000i q^{32} +6.46410 q^{34} +(8.92820 + 4.46410i) q^{35} +1.00000i q^{37} -6.92820i q^{38} +(-2.00000 - 1.00000i) q^{40} -1.19615 q^{41} +7.19615i q^{43} -1.73205 q^{44} -1.46410 q^{46} -4.53590i q^{47} -12.9282 q^{49} +(-4.00000 + 3.00000i) q^{50} +4.00000i q^{52} -3.73205i q^{53} +(-1.73205 + 3.46410i) q^{55} +4.46410 q^{56} -3.00000i q^{58} +2.53590 q^{59} +5.19615 q^{61} +2.26795i q^{62} -1.00000 q^{64} +(8.00000 + 4.00000i) q^{65} -5.46410i q^{67} -6.46410i q^{68} +(4.46410 - 8.92820i) q^{70} +0.928203 q^{71} -14.3923i q^{73} +1.00000 q^{74} -6.92820 q^{76} -7.73205i q^{77} +11.4641 q^{79} +(-1.00000 + 2.00000i) q^{80} +1.19615i q^{82} -9.46410i q^{83} +(-12.9282 - 6.46410i) q^{85} +7.19615 q^{86} +1.73205i q^{88} +2.53590 q^{89} -17.8564 q^{91} +1.46410i q^{92} -4.53590 q^{94} +(-6.92820 + 13.8564i) q^{95} +13.5885i q^{97} +12.9282i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^5	\( 4 q - 4 q^{4} - 4 q^{5} + 8 q^{10} - 4 q^{14} + 4 q^{16} + 4 q^{20} - 12 q^{25} - 16 q^{26} + 12 q^{29} - 16 q^{31} + 12 q^{34} + 8 q^{35} - 8 q^{40} + 16 q^{41} + 8 q^{46} - 24 q^{49} - 16 q^{50} + 4 q^{56} + 24 q^{59} - 4 q^{64} + 32 q^{65} + 4 q^{70} - 24 q^{71} + 4 q^{74} + 32 q^{79} - 4 q^{80} - 24 q^{85} + 8 q^{86} + 24 q^{89} - 16 q^{91} - 32 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^10 - 4 * q^14 + 4 * q^16 + 4 * q^20 - 12 * q^25 - 16 * q^26 + 12 * q^29 - 16 * q^31 + 12 * q^34 + 8 * q^35 - 8 * q^40 + 16 * q^41 + 8 * q^46 - 24 * q^49 - 16 * q^50 + 4 * q^56 + 24 * q^59 - 4 * q^64 + 32 * q^65 + 4 * q^70 - 24 * q^71 + 4 * q^74 + 32 * q^79 - 4 * q^80 - 24 * q^85 + 8 * q^86 + 24 * q^89 - 16 * q^91 - 32 * q^94

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