LMFDB - Embedded newform 3330.2.e.f.739.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3330,2,Mod(739,3330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3330.739"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64 $ x^16 - 2x^15 + 2x^14 + 8x^13 + 138x^12 - 220x^11 + 196x^10 + 744x^9 + 4241x^8 - 3018x^7 + 658x^6 - 1584x^5 + 16372x^4 - 18840x^3 + 10952x^2 - 1184*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			739.3
Root			$-1.32920 - 1.32920i$ of defining polynomial
Character	$\chi$	$=$	3330.739
Dual form			3330.2.e.f.739.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +(-1.32920 - 1.79812i) q^{5} +1.87266i q^{7} +1.00000 q^{8} +(-1.32920 - 1.79812i) q^{10} +1.51044 q^{11} -4.78011 q^{13} +1.87266i q^{14} +1.00000 q^{16} +3.99190 q^{17} -4.68621i q^{19} +(-1.32920 - 1.79812i) q^{20} +1.51044 q^{22} +6.51179 q^{23} +(-1.46648 + 4.78011i) q^{25} -4.78011 q^{26} +1.87266i q^{28} +0.200007i q^{29} -5.79625i q^{31} +1.00000 q^{32} +3.99190 q^{34} +(3.36727 - 2.48913i) q^{35} +(-6.03820 - 0.734967i) q^{37} -4.68621i q^{38} +(-1.32920 - 1.79812i) q^{40} -4.10248 q^{41} +10.9236 q^{43} +1.51044 q^{44} +6.51179 q^{46} -5.15269i q^{47} +3.49314 q^{49} +(-1.46648 + 4.78011i) q^{50} -4.78011 q^{52} +2.22232i q^{53} +(-2.00766 - 2.71595i) q^{55} +1.87266i q^{56} +0.200007i q^{58} -13.2646i q^{59} -4.09616i q^{61} -5.79625i q^{62} +1.00000 q^{64} +(6.35370 + 8.59521i) q^{65} +5.76584i q^{67} +3.99190 q^{68} +(3.36727 - 2.48913i) q^{70} +9.17428 q^{71} -14.2905i q^{73} +(-6.03820 - 0.734967i) q^{74} -4.68621i q^{76} +2.82853i q^{77} +5.95465i q^{79} +(-1.32920 - 1.79812i) q^{80} -4.10248 q^{82} -9.42048i q^{83} +(-5.30602 - 7.17793i) q^{85} +10.9236 q^{86} +1.51044 q^{88} -5.47289i q^{89} -8.95152i q^{91} +6.51179 q^{92} -5.15269i q^{94} +(-8.42637 + 6.22888i) q^{95} -2.03303 q^{97} +3.49314 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{16} + 38 q^{17} + 2 q^{20} - 2 q^{22} + 20 q^{23} - 4 q^{25} + 16 q^{32} + 38 q^{34} + 10 q^{35} - 4 q^{37} + 2 q^{40} + 6 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 2 * q^5 + 16 * q^8 + 2 * q^10 - 2 * q^11 + 16 * q^16 + 38 * q^17 + 2 * q^20 - 2 * q^22 + 20 * q^23 - 4 * q^25 + 16 * q^32 + 38 * q^34 + 10 * q^35 - 4 * q^37 + 2 * q^40 + 6 * q^41 - 2 * q^43 - 2 * q^44 + 20 * q^46 - 18 * q^49 - 4 * q^50 + 20 * q^55 + 16 * q^64 - 12 * q^65 + 38 * q^68 + 10 * q^70 + 24 * q^71 - 4 * q^74 + 2 * q^80 + 6 * q^82 - 2 * q^86 - 2 * q^88 + 20 * q^92 + 16 * q^95 + 38 * q^97 - 18 * q^98

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.32920	−	1.79812i	−0.594434	−	0.804144i
$5$	−1.32920	−	1.79812i	−0.594434	−	0.804144i
$6$		0			0
$7$			1.87266i			0.707799i	0.935283	+	0.353900i	$0.115145\pi$
$7$			1.87266i			0.707799i	−0.935283	+	0.353900i	$0.884855\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.32920	−	1.79812i	−0.420328	−	0.568616i
$11$	1.51044			0.455413			0.227707	−	0.973730i	$-0.426877\pi$
$11$	1.51044			0.455413			0.227707	+	0.973730i	$0.426877\pi$
$12$		0			0
$13$	−4.78011			−1.32576			−0.662882	−	0.748724i	$-0.730667\pi$
$13$	−4.78011			−1.32576			−0.662882	+	0.748724i	$0.730667\pi$
$14$			1.87266i			0.500490i
$15$		0			0
$16$	1.00000			0.250000
$17$	3.99190			0.968179			0.484089	−	0.875018i	$-0.339151\pi$
$17$	3.99190			0.968179			0.484089	+	0.875018i	$0.339151\pi$
$18$		0			0
$19$		−	4.68621i		−	1.07509i	−0.843235	−	0.537545i	$-0.819352\pi$
$19$		−	4.68621i		−	1.07509i	0.843235	−	0.537545i	$-0.180648\pi$
$20$	−1.32920	−	1.79812i	−0.297217	−	0.402072i
$21$		0			0
$22$	1.51044			0.322026
$23$	6.51179			1.35780			0.678901	−	0.734230i	$-0.262457\pi$
$23$	6.51179			1.35780			0.678901	+	0.734230i	$0.262457\pi$
$24$		0			0
$25$	−1.46648	+	4.78011i	−0.293296	+	0.956022i
$26$	−4.78011			−0.937456
$27$		0			0
$28$			1.87266i			0.353900i
$29$			0.200007i			0.0371403i	0.999828	+	0.0185702i	$0.00591141\pi$
$29$			0.200007i			0.0371403i	−0.999828	+	0.0185702i	$0.994089\pi$
$30$		0			0
$31$		−	5.79625i		−	1.04104i	−0.853850	−	0.520519i	$-0.825739\pi$
$31$		−	5.79625i		−	1.04104i	0.853850	−	0.520519i	$-0.174261\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	3.99190			0.684606
$35$	3.36727	−	2.48913i	0.569173	−	0.420740i
$36$		0			0
$37$	−6.03820	−	0.734967i	−0.992673	−	0.120828i
$37$	−6.03820	−	0.734967i	−0.992673	−	0.120828i
$38$		−	4.68621i		−	0.760203i
$39$		0			0
$40$	−1.32920	−	1.79812i	−0.210164	−	0.284308i
$41$	−4.10248			−0.640700			−0.320350	−	0.947299i	$-0.603801\pi$
$41$	−4.10248			−0.640700			−0.320350	+	0.947299i	$0.603801\pi$
$42$		0			0
$43$	10.9236			1.66583			0.832916	−	0.553399i	$-0.186670\pi$
$43$	10.9236			1.66583			0.832916	+	0.553399i	$0.186670\pi$
$44$	1.51044			0.227707
$45$		0			0
$46$	6.51179			0.960110
$47$		−	5.15269i		−	0.751596i	−0.926702	−	0.375798i	$-0.877369\pi$
$47$		−	5.15269i		−	0.751596i	0.926702	−	0.375798i	$-0.122631\pi$
$48$		0			0
$49$	3.49314			0.499020
$50$	−1.46648	+	4.78011i	−0.207392	+	0.676009i
$51$		0			0
$52$	−4.78011			−0.662882
$53$			2.22232i			0.305259i	0.988284	+	0.152629i	$0.0487740\pi$
$53$			2.22232i			0.305259i	−0.988284	+	0.152629i	$0.951226\pi$
$54$		0			0
$55$	−2.00766	−	2.71595i	−0.270713	−	0.366218i
$56$			1.87266i			0.250245i
$57$		0			0
$58$			0.200007i			0.0262622i
$59$		−	13.2646i		−	1.72690i	−0.504435	−	0.863450i	$-0.668299\pi$
$59$		−	13.2646i		−	1.72690i	0.504435	−	0.863450i	$-0.331701\pi$
$60$		0			0
$61$		−	4.09616i		−	0.524459i	−0.965005	−	0.262230i	$-0.915542\pi$
$61$		−	4.09616i		−	0.524459i	0.965005	−	0.262230i	$-0.0844578\pi$
$62$		−	5.79625i		−	0.736124i
$63$		0			0
$64$	1.00000			0.125000
$65$	6.35370	+	8.59521i	0.788079	+	1.06611i
$66$		0			0
$67$			5.76584i			0.704410i	0.935923	+	0.352205i	$0.114568\pi$
$67$			5.76584i			0.704410i	−0.935923	+	0.352205i	$0.885432\pi$
$68$	3.99190			0.484089
$69$		0			0
$70$	3.36727	−	2.48913i	0.402466	−	0.297508i
$71$	9.17428			1.08879			0.544393	−	0.838830i	$-0.316760\pi$
$71$	9.17428			1.08879			0.544393	+	0.838830i	$0.316760\pi$
$72$		0			0
$73$		−	14.2905i		−	1.67257i	−0.548294	−	0.836286i	$-0.684722\pi$
$73$		−	14.2905i		−	1.67257i	0.548294	−	0.836286i	$-0.315278\pi$
$74$	−6.03820	−	0.734967i	−0.701926	−	0.0854382i
$75$		0			0
$76$		−	4.68621i		−	0.537545i
$77$			2.82853i			0.322341i
$78$		0			0
$79$			5.95465i			0.669950i	0.942227	+	0.334975i	$0.108728\pi$
$79$			5.95465i			0.669950i	−0.942227	+	0.334975i	$0.891272\pi$
$80$	−1.32920	−	1.79812i	−0.148609	−	0.201036i
$81$		0			0
$82$	−4.10248			−0.453044
$83$		−	9.42048i		−	1.03403i	−0.855976	−	0.517016i	$-0.827043\pi$
$83$		−	9.42048i		−	1.03403i	0.855976	−	0.517016i	$-0.172957\pi$
$84$		0			0
$85$	−5.30602	−	7.17793i	−0.575519	−	0.778556i
$86$	10.9236			1.17792
$87$		0			0
$88$	1.51044			0.161013
$89$		−	5.47289i		−	0.580126i	−0.957008	−	0.290063i	$-0.906324\pi$
$89$		−	5.47289i		−	0.580126i	0.957008	−	0.290063i	$-0.0936762\pi$
$90$		0			0
$91$		−	8.95152i		−	0.938375i
$92$	6.51179			0.678901
$93$		0			0
$94$		−	5.15269i		−	0.531459i
$95$	−8.42637	+	6.22888i	−0.864527	+	0.639070i
$96$		0			0
$97$	−2.03303			−0.206423			−0.103211	−	0.994659i	$-0.532912\pi$
$97$	−2.03303			−0.206423			−0.103211	+	0.994659i	$0.532912\pi$
$98$	3.49314			0.352860
$99$		0			0
$100$	−1.46648	+	4.78011i	−0.146648	+	0.478011i
$101$	9.70730			0.965913			0.482956	−	0.875644i	$-0.339563\pi$
$101$	9.70730			0.965913			0.482956	+	0.875644i	$0.339563\pi$
$102$		0			0
$103$	2.66390			0.262482			0.131241	−	0.991351i	$-0.458104\pi$
$103$	2.66390			0.262482			0.131241	+	0.991351i	$0.458104\pi$
$104$	−4.78011			−0.468728
$105$		0			0
$106$			2.22232i			0.215850i
$107$		−	0.823199i		−	0.0795816i	−0.999208	−	0.0397908i	$-0.987331\pi$
$107$		−	0.823199i		−	0.0795816i	0.999208	−	0.0397908i	$-0.0126692\pi$
$108$		0			0
$109$			0.478427i			0.0458250i	0.999737	+	0.0229125i	$0.00729391\pi$
$109$			0.478427i			0.0458250i	−0.999737	+	0.0229125i	$0.992706\pi$
$110$	−2.00766	−	2.71595i	−0.191423	−	0.258955i
$111$		0			0
$112$			1.87266i			0.176950i
$113$	19.8469			1.86704			0.933522	−	0.358521i	$-0.116719\pi$
$113$	19.8469			1.86704			0.933522	+	0.358521i	$0.116719\pi$
$114$		0			0
$115$	−8.65543	−	11.7090i	−0.807123	−	1.09187i
$116$			0.200007i			0.0185702i
$117$		0			0
$118$		−	13.2646i		−	1.22110i
$119$			7.47548i			0.685277i
$120$		0			0
$121$	−8.71858			−0.792599
$122$		−	4.09616i		−	0.370849i
$123$		0			0
$124$		−	5.79625i		−	0.520519i
$125$	10.5445	−	3.71679i	0.943124	−	0.332440i
$126$		0			0
$127$			6.85655i			0.608420i	0.952605	+	0.304210i	$0.0983925\pi$
$127$			6.85655i			0.608420i	−0.952605	+	0.304210i	$0.901608\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	6.35370	+	8.59521i	0.557256	+	0.753850i
$131$		−	3.26950i		−	0.285658i	−0.989747	−	0.142829i	$-0.954380\pi$
$131$		−	3.26950i		−	0.285658i	0.989747	−	0.142829i	$-0.0456198\pi$
$132$		0			0
$133$	8.77568			0.760948
$134$			5.76584i			0.498093i
$135$		0			0
$136$	3.99190			0.342303
$137$		−	2.11793i		−	0.180947i	−0.995899	−	0.0904733i	$-0.971162\pi$
$137$		−	2.11793i		−	0.180947i	0.995899	−	0.0904733i	$-0.0288380\pi$
$138$		0			0
$139$	−15.7132			−1.33277			−0.666387	−	0.745606i	$-0.732160\pi$
$139$	−15.7132			−1.33277			−0.666387	+	0.745606i	$0.732160\pi$
$140$	3.36727	−	2.48913i	0.284586	−	0.210370i
$141$		0			0
$142$	9.17428			0.769889
$143$	−7.22005			−0.603771
$144$		0			0
$145$	0.359637	−	0.265848i	0.0298662	−	0.0220775i
$146$		−	14.2905i		−	1.18269i
$147$		0			0
$148$	−6.03820	−	0.734967i	−0.496337	−	0.0604139i
$149$	4.47152			0.366321			0.183161	−	0.983083i	$-0.441367\pi$
$149$	4.47152			0.366321			0.183161	+	0.983083i	$0.441367\pi$
$150$		0			0
$151$	3.25118			0.264577			0.132289	−	0.991211i	$-0.457767\pi$
$151$	3.25118			0.264577			0.132289	+	0.991211i	$0.457767\pi$
$152$		−	4.68621i		−	0.380101i
$153$		0			0
$154$			2.82853i			0.227930i
$155$	−10.4224	+	7.70435i	−0.837144	+	0.618828i
$156$		0			0
$157$		−	0.0713960i		−	0.00569802i	−0.999996	−	0.00284901i	$-0.999093\pi$
$157$		−	0.0713960i		−	0.00569802i	0.999996	−	0.00284901i	$-0.000906870\pi$
$158$			5.95465i			0.473726i
$159$		0			0
$160$	−1.32920	−	1.79812i	−0.105082	−	0.142154i
$161$			12.1944i			0.961051i
$162$		0			0
$163$	−1.06145			−0.0831389			−0.0415695	−	0.999136i	$-0.513236\pi$
$163$	−1.06145			−0.0831389			−0.0415695	+	0.999136i	$0.513236\pi$
$164$	−4.10248			−0.320350
$165$		0			0
$166$		−	9.42048i		−	0.731171i
$167$	−20.4097			−1.57935			−0.789673	−	0.613528i	$-0.789750\pi$
$167$	−20.4097			−1.57935			−0.789673	+	0.613528i	$0.789750\pi$
$168$		0			0
$169$	9.84944			0.757649
$170$	−5.30602	−	7.17793i	−0.406953	−	0.550522i
$171$		0			0
$172$	10.9236			0.832916
$173$			16.9195i			1.28636i	0.765713	+	0.643182i	$0.222386\pi$
$173$			16.9195i			1.28636i	−0.765713	+	0.643182i	$0.777614\pi$
$174$		0			0
$175$	−8.95152	−	2.74622i	−0.676672	−	0.207595i
$176$	1.51044			0.113853
$177$		0			0
$178$		−	5.47289i		−	0.410211i
$179$		−	26.6575i		−	1.99247i	−0.0866723	−	0.996237i	$-0.527623\pi$
$179$		−	26.6575i		−	1.99247i	0.0866723	−	0.996237i	$-0.472377\pi$
$180$		0			0
$181$	−13.4428			−0.999196			−0.499598	−	0.866257i	$-0.666519\pi$
$181$	−13.4428			−0.999196			−0.499598	+	0.866257i	$0.666519\pi$
$182$		−	8.95152i		−	0.663531i
$183$		0			0
$184$	6.51179			0.480055
$185$	6.70438	+	11.8343i	0.492916	+	0.870077i
$186$		0			0
$187$	6.02951			0.440922
$188$		−	5.15269i		−	0.375798i
$189$		0			0
$190$	−8.42637	+	6.22888i	−0.611313	+	0.451891i
$191$			6.76465i			0.489473i	0.969590	+	0.244736i	$0.0787014\pi$
$191$			6.76465i			0.489473i	−0.969590	+	0.244736i	$0.921299\pi$
$192$		0			0
$193$	−15.4592			−1.11278			−0.556389	−	0.830922i	$-0.687814\pi$
$193$	−15.4592			−1.11278			−0.556389	+	0.830922i	$0.687814\pi$
$194$	−2.03303			−0.145963
$195$		0			0
$196$	3.49314			0.249510
$197$		−	12.5068i		−	0.891072i	−0.895264	−	0.445536i	$-0.853013\pi$
$197$		−	12.5068i		−	0.891072i	0.895264	−	0.445536i	$-0.146987\pi$
$198$		0			0
$199$			13.4172i			0.951119i	0.879684	+	0.475559i	$0.157754\pi$
$199$			13.4172i			0.951119i	−0.879684	+	0.475559i	$0.842246\pi$
$200$	−1.46648	+	4.78011i	−0.103696	+	0.338005i
$201$		0			0
$202$	9.70730			0.683003
$203$	−0.374545			−0.0262879
$204$		0			0
$205$	5.45300	+	7.37676i	0.380854	+	0.515216i
$206$	2.66390			0.185603
$207$		0			0
$208$	−4.78011			−0.331441
$209$		−	7.07821i		−	0.489610i
$210$		0			0
$211$	6.46040			0.444752			0.222376	−	0.974961i	$-0.428619\pi$
$211$	6.46040			0.444752			0.222376	+	0.974961i	$0.428619\pi$
$212$			2.22232i			0.152629i
$213$		0			0
$214$		−	0.823199i		−	0.0562727i
$215$	−14.5196	−	19.6419i	−0.990228	−	1.33957i
$216$		0			0
$217$	10.8544			0.736845
$218$			0.478427i			0.0324031i
$219$		0			0
$220$	−2.00766	−	2.71595i	−0.135357	−	0.183109i
$221$	−19.0817			−1.28358
$222$		0			0
$223$		−	8.15634i		−	0.546189i	−0.961987	−	0.273095i	$-0.911953\pi$
$223$		−	8.15634i		−	0.546189i	0.961987	−	0.273095i	$-0.0880472\pi$
$224$			1.87266i			0.125122i
$225$		0			0
$226$	19.8469			1.32020
$227$	13.0456			0.865866			0.432933	−	0.901426i	$-0.357479\pi$
$227$	13.0456			0.865866			0.432933	+	0.901426i	$0.357479\pi$
$228$		0			0
$229$	13.3621			0.882995			0.441498	−	0.897262i	$-0.354447\pi$
$229$	13.3621			0.882995			0.441498	+	0.897262i	$0.354447\pi$
$230$	−8.65543	−	11.7090i	−0.570722	−	0.772067i
$231$		0			0
$232$			0.200007i			0.0131311i
$233$		−	27.2277i		−	1.78375i	−0.452283	−	0.891875i	$-0.649390\pi$
$233$		−	27.2277i		−	1.78375i	0.452283	−	0.891875i	$-0.350610\pi$
$234$		0			0
$235$	−9.26515	+	6.84893i	−0.604392	+	0.446775i
$236$		−	13.2646i		−	0.863450i
$237$		0			0
$238$			7.47548i			0.484564i
$239$		−	10.5417i		−	0.681888i	−0.940084	−	0.340944i	$-0.889253\pi$
$239$		−	10.5417i		−	0.681888i	0.940084	−	0.340944i	$-0.110747\pi$
$240$		0			0
$241$			13.3732i			0.861446i	0.902484	+	0.430723i	$0.141741\pi$
$241$			13.3732i			0.861446i	−0.902484	+	0.430723i	$0.858259\pi$
$242$	−8.71858			−0.560452
$243$		0			0
$244$		−	4.09616i		−	0.262230i
$245$	−4.64307	−	6.28109i	−0.296635	−	0.401284i
$246$		0			0
$247$			22.4006i			1.42531i
$248$		−	5.79625i		−	0.368062i
$249$		0			0
$250$	10.5445	−	3.71679i	0.666890	−	0.235070i
$251$		−	6.68149i		−	0.421732i	−0.977515	−	0.210866i	$-0.932372\pi$
$251$		−	6.68149i		−	0.421732i	0.977515	−	0.210866i	$-0.0676283\pi$
$252$		0			0
$253$	9.83563			0.618361
$254$			6.85655i			0.430218i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−4.63516			−0.289133			−0.144567	−	0.989495i	$-0.546179\pi$
$257$	−4.63516			−0.289133			−0.144567	+	0.989495i	$0.546179\pi$
$258$		0			0
$259$	1.37634	−	11.3075i	0.0855219	−	0.702614i
$260$	6.35370	+	8.59521i	0.394040	+	0.533053i
$261$		0			0
$262$		−	3.26950i		−	0.201990i
$263$			29.5118i			1.81977i	0.414857	+	0.909887i	$0.363832\pi$
$263$			29.5118i			1.81977i	−0.414857	+	0.909887i	$0.636168\pi$
$264$		0			0
$265$	3.99599	−	2.95389i	0.245472	−	0.181456i
$266$	8.77568			0.538071
$267$		0			0
$268$			5.76584i			0.352205i
$269$	21.3385			1.30103			0.650516	−	0.759493i	$-0.274553\pi$
$269$	21.3385			1.30103			0.650516	+	0.759493i	$0.274553\pi$
$270$		0			0
$271$	16.5121			1.00304			0.501521	−	0.865146i	$-0.332774\pi$
$271$	16.5121			1.00304			0.501521	+	0.865146i	$0.332774\pi$
$272$	3.99190			0.242045
$273$		0			0
$274$		−	2.11793i		−	0.127949i
$275$	−2.21502	+	7.22005i	−0.133571	+	0.435385i
$276$		0			0
$277$	11.0446			0.663605			0.331802	−	0.943349i	$-0.392343\pi$
$277$	11.0446			0.663605			0.331802	+	0.943349i	$0.392343\pi$
$278$	−15.7132			−0.942414
$279$		0			0
$280$	3.36727	−	2.48913i	0.201233	−	0.148754i
$281$		−	17.9190i		−	1.06896i	−0.845182	−	0.534478i	$-0.820508\pi$
$281$		−	17.9190i		−	1.06896i	0.845182	−	0.534478i	$-0.179492\pi$
$282$		0			0
$283$	14.0543			0.835440			0.417720	−	0.908576i	$-0.362829\pi$
$283$	14.0543			0.835440			0.417720	+	0.908576i	$0.362829\pi$
$284$	9.17428			0.544393
$285$		0			0
$286$	−7.22005			−0.426930
$287$		−	7.68256i		−	0.453487i
$288$		0			0
$289$	−1.06470			−0.0626295
$290$	0.359637	−	0.265848i	0.0211186	−	0.0156111i
$291$		0			0
$292$		−	14.2905i		−	0.836286i
$293$			32.2561i			1.88442i	0.335018	+	0.942212i	$0.391258\pi$
$293$			32.2561i			1.88442i	−0.335018	+	0.942212i	$0.608742\pi$
$294$		0			0
$295$	−23.8513	+	17.6312i	−1.38868	+	1.02653i
$296$	−6.03820	−	0.734967i	−0.350963	−	0.0427191i
$297$		0			0
$298$	4.47152			0.259028
$299$	−31.1270			−1.80012
$300$		0			0
$301$			20.4562i			1.17908i
$302$	3.25118			0.187084
$303$		0			0
$304$		−	4.68621i		−	0.268772i
$305$	−7.36539	+	5.44459i	−0.421741	+	0.311757i
$306$		0			0
$307$			27.5771i			1.57391i	0.617012	+	0.786953i	$0.288343\pi$
$307$			27.5771i			1.57391i	−0.617012	+	0.786953i	$0.711657\pi$
$308$			2.82853i			0.161171i
$309$		0			0
$310$	−10.4224	+	7.70435i	−0.591950	+	0.437578i
$311$			13.8537i			0.785570i	0.919630	+	0.392785i	$0.128488\pi$
$311$			13.8537i			0.785570i	−0.919630	+	0.392785i	$0.871512\pi$
$312$		0			0
$313$	24.3768			1.37786			0.688930	−	0.724828i	$-0.258081\pi$
$313$	24.3768			1.37786			0.688930	+	0.724828i	$0.258081\pi$
$314$		−	0.0713960i		−	0.00402911i
$315$		0			0
$316$			5.95465i			0.334975i
$317$			8.91522i			0.500729i	0.968152	+	0.250364i	$0.0805504\pi$
$317$			8.91522i			0.500729i	−0.968152	+	0.250364i	$0.919450\pi$
$318$		0			0
$319$			0.302097i			0.0169142i
$320$	−1.32920	−	1.79812i	−0.0743043	−	0.100518i
$321$		0			0
$322$			12.1944i			0.679566i
$323$		−	18.7069i		−	1.04088i
$324$		0			0
$325$	7.00993	−	22.8494i	0.388841	−	1.26746i
$326$	−1.06145			−0.0587881
$327$		0			0
$328$	−4.10248			−0.226522
$329$	9.64923			0.531979
$330$		0			0
$331$		−	25.6563i		−	1.41020i	−0.709109	−	0.705099i	$-0.750903\pi$
$331$		−	25.6563i		−	1.41020i	0.709109	−	0.705099i	$-0.249097\pi$
$332$		−	9.42048i		−	0.517016i
$333$		0			0
$334$	−20.4097			−1.11677
$335$	10.3677	−	7.66393i	0.566447	−	0.418725i
$336$		0			0
$337$			18.5473i			1.01033i	0.863021	+	0.505167i	$0.168569\pi$
$337$			18.5473i			1.01033i	−0.863021	+	0.505167i	$0.831431\pi$
$338$	9.84944			0.535739
$339$		0			0
$340$	−5.30602	−	7.17793i	−0.287759	−	0.389278i
$341$		−	8.75486i		−	0.474102i
$342$		0			0
$343$			19.6501i			1.06101i
$344$	10.9236			0.588961
$345$		0			0
$346$			16.9195i			0.909597i
$347$	−9.07304			−0.487066			−0.243533	−	0.969893i	$-0.578306\pi$
$347$	−9.07304			−0.487066			−0.243533	+	0.969893i	$0.578306\pi$
$348$		0			0
$349$	−31.1022			−1.66486			−0.832431	−	0.554129i	$-0.813052\pi$
$349$	−31.1022			−1.66486			−0.832431	+	0.554129i	$0.813052\pi$
$350$	−8.95152	−	2.74622i	−0.478479	−	0.146792i
$351$		0			0
$352$	1.51044			0.0805065
$353$	−16.3289			−0.869098			−0.434549	−	0.900648i	$-0.643092\pi$
$353$	−16.3289			−0.869098			−0.434549	+	0.900648i	$0.643092\pi$
$354$		0			0
$355$	−12.1944	−	16.4965i	−0.647212	−	0.875542i
$356$		−	5.47289i		−	0.290063i
$357$		0			0
$358$		−	26.6575i		−	1.40889i
$359$	−17.1135			−0.903216			−0.451608	−	0.892217i	$-0.649149\pi$
$359$	−17.1135			−0.903216			−0.451608	+	0.892217i	$0.649149\pi$
$360$		0			0
$361$	−2.96053			−0.155817
$362$	−13.4428			−0.706538
$363$		0			0
$364$		−	8.95152i		−	0.469187i
$365$	−25.6960	+	18.9948i	−1.34499	+	0.994234i
$366$		0			0
$367$			23.2045i			1.21126i	0.795745	+	0.605632i	$0.207080\pi$
$367$			23.2045i			1.21126i	−0.795745	+	0.605632i	$0.792920\pi$
$368$	6.51179			0.339450
$369$		0			0
$370$	6.70438	+	11.8343i	0.348544	+	0.615237i
$371$	−4.16164			−0.216062
$372$		0			0
$373$		−	9.69704i		−	0.502094i	−0.967975	−	0.251047i	$-0.919225\pi$
$373$		−	9.69704i		−	0.502094i	0.967975	−	0.251047i	$-0.0807748\pi$
$374$	6.02951			0.311779
$375$		0			0
$376$		−	5.15269i		−	0.265729i
$377$		−	0.956055i		−	0.0492393i
$378$		0			0
$379$	−9.40066			−0.482880			−0.241440	−	0.970416i	$-0.577620\pi$
$379$	−9.40066			−0.482880			−0.241440	+	0.970416i	$0.577620\pi$
$380$	−8.42637	+	6.22888i	−0.432263	+	0.319535i
$381$		0			0
$382$			6.76465i			0.346109i
$383$	32.3874			1.65492			0.827459	−	0.561526i	$-0.189785\pi$
$383$	32.3874			1.65492			0.827459	+	0.561526i	$0.189785\pi$
$384$		0			0
$385$	5.08605	−	3.75967i	0.259209	−	0.191611i
$386$	−15.4592			−0.786853
$387$		0			0
$388$	−2.03303			−0.103211
$389$			20.8774i			1.05853i	0.848457	+	0.529264i	$0.177532\pi$
$389$			20.8774i			1.05853i	−0.848457	+	0.529264i	$0.822468\pi$
$390$		0			0
$391$	25.9944			1.31459
$392$	3.49314			0.176430
$393$		0			0
$394$		−	12.5068i		−	0.630083i
$395$	10.7072	−	7.91489i	0.538737	−	0.398241i
$396$		0			0
$397$		−	12.2447i		−	0.614546i	−0.951621	−	0.307273i	$-0.900584\pi$
$397$		−	12.2447i		−	0.614546i	0.951621	−	0.307273i	$-0.0994164\pi$
$398$			13.4172i			0.672543i
$399$		0			0
$400$	−1.46648	+	4.78011i	−0.0733240	+	0.239005i
$401$			13.7736i			0.687820i	0.939003	+	0.343910i	$0.111752\pi$
$401$			13.7736i			0.687820i	−0.939003	+	0.343910i	$0.888248\pi$
$402$		0			0
$403$			27.7067i			1.38017i
$404$	9.70730			0.482956
$405$		0			0
$406$	−0.374545			−0.0185884
$407$	−9.12031	−	1.11012i	−0.452077	−	0.0550266i
$408$		0			0
$409$			34.5667i			1.70921i	0.519276	+	0.854606i	$0.326202\pi$
$409$			34.5667i			1.70921i	−0.519276	+	0.854606i	$0.673798\pi$
$410$	5.45300	+	7.37676i	0.269305	+	0.364312i
$411$		0			0
$412$	2.66390			0.131241
$413$	24.8400			1.22230
$414$		0			0
$415$	−16.9392	+	12.5217i	−0.831511	+	0.614664i
$416$	−4.78011			−0.234364
$417$		0			0
$418$		−	7.07821i		−	0.346207i
$419$	−22.9413			−1.12076			−0.560378	−	0.828237i	$-0.689344\pi$
$419$	−22.9413			−1.12076			−0.560378	+	0.828237i	$0.689344\pi$
$420$		0			0
$421$		−	12.1679i		−	0.593026i	−0.955029	−	0.296513i	$-0.904176\pi$
$421$		−	12.1679i		−	0.593026i	0.955029	−	0.296513i	$-0.0958239\pi$
$422$	6.46040			0.314487
$423$		0			0
$424$			2.22232i			0.107925i
$425$	−5.85405	+	19.0817i	−0.283963	+	0.925600i
$426$		0			0
$427$	7.67072			0.371212
$428$		−	0.823199i		−	0.0397908i
$429$		0			0
$430$	−14.5196	−	19.6419i	−0.700197	−	0.947219i
$431$			11.2076i			0.539853i	0.962881	+	0.269927i	$0.0869995\pi$
$431$			11.2076i			0.539853i	−0.962881	+	0.269927i	$0.913001\pi$
$432$		0			0
$433$			31.6559i			1.52128i	0.649172	+	0.760642i	$0.275116\pi$
$433$			31.6559i			1.52128i	−0.649172	+	0.760642i	$0.724884\pi$
$434$	10.8544			0.521028
$435$		0			0
$436$			0.478427i			0.0229125i
$437$		−	30.5156i		−	1.45976i
$438$		0			0
$439$			33.9912i			1.62231i	0.584831	+	0.811155i	$0.301161\pi$
$439$			33.9912i			1.62231i	−0.584831	+	0.811155i	$0.698839\pi$
$440$	−2.00766	−	2.71595i	−0.0957116	−	0.129478i
$441$		0			0
$442$	−19.0817			−0.907626
$443$			34.9477i			1.66042i	0.557454	+	0.830208i	$0.311778\pi$
$443$			34.9477i			1.66042i	−0.557454	+	0.830208i	$0.688222\pi$
$444$		0			0
$445$	−9.84093	+	7.27454i	−0.466505	+	0.344846i
$446$		−	8.15634i		−	0.386214i
$447$		0			0
$448$			1.87266i			0.0884749i
$449$			17.2274i			0.813012i	0.913648	+	0.406506i	$0.133253\pi$
$449$			17.2274i			0.813012i	−0.913648	+	0.406506i	$0.866747\pi$
$450$		0			0
$451$	−6.19654			−0.291784
$452$	19.8469			0.933522
$453$		0			0
$454$	13.0456			0.612260
$455$	−16.0959	+	11.8983i	−0.754589	+	0.557802i
$456$		0			0
$457$	−24.5044			−1.14627			−0.573133	−	0.819463i	$-0.694272\pi$
$457$	−24.5044			−1.14627			−0.573133	+	0.819463i	$0.694272\pi$
$458$	13.3621			0.624372
$459$		0			0
$460$	−8.65543	−	11.7090i	−0.403562	−	0.545934i
$461$		−	9.05274i		−	0.421628i	−0.977526	−	0.210814i	$-0.932389\pi$
$461$		−	9.05274i		−	0.421628i	0.977526	−	0.210814i	$-0.0676115\pi$
$462$		0			0
$463$	2.10640			0.0978927			0.0489463	−	0.998801i	$-0.484414\pi$
$463$	2.10640			0.0978927			0.0489463	+	0.998801i	$0.484414\pi$
$464$			0.200007i			0.00928509i
$465$		0			0
$466$		−	27.2277i		−	1.26130i
$467$	24.0814			1.11435			0.557177	−	0.830394i	$-0.311884\pi$
$467$	24.0814			1.11435			0.557177	+	0.830394i	$0.311884\pi$
$468$		0			0
$469$	−10.7975			−0.498581
$470$	−9.26515	+	6.84893i	−0.427370	+	0.315917i
$471$		0			0
$472$		−	13.2646i		−	0.610551i
$473$	16.4994			0.758642
$474$		0			0
$475$	22.4006	+	6.87223i	1.02781	+	0.315319i
$476$			7.47548i			0.342638i
$477$		0			0
$478$		−	10.5417i		−	0.482168i
$479$		−	5.46224i		−	0.249576i	−0.992183	−	0.124788i	$-0.960175\pi$
$479$		−	5.46224i		−	0.249576i	0.992183	−	0.124788i	$-0.0398251\pi$
$480$		0			0
$481$	28.8632	+	3.51322i	1.31605	+	0.160189i
$482$			13.3732i			0.609134i
$483$		0			0
$484$	−8.71858			−0.396299
$485$	2.70229	+	3.65563i	0.122705	+	0.165994i
$486$		0			0
$487$	−11.8494			−0.536949			−0.268475	−	0.963287i	$-0.586520\pi$
$487$	−11.8494			−0.536949			−0.268475	+	0.963287i	$0.586520\pi$
$488$		−	4.09616i		−	0.185424i
$489$		0			0
$490$	−4.64307	−	6.28109i	−0.209752	−	0.283751i
$491$	7.19955			0.324911			0.162456	−	0.986716i	$-0.448059\pi$
$491$	7.19955			0.324911			0.162456	+	0.986716i	$0.448059\pi$
$492$		0			0
$493$			0.798408i			0.0359585i
$494$			22.4006i			1.00785i
$495$		0			0
$496$		−	5.79625i		−	0.260259i
$497$			17.1803i			0.770643i
$498$		0			0
$499$		−	3.32481i		−	0.148839i	−0.997227	−	0.0744194i	$-0.976290\pi$
$499$		−	3.32481i		−	0.148839i	0.997227	−	0.0744194i	$-0.0237104\pi$
$500$	10.5445	−	3.71679i	0.471562	−	0.166220i
$501$		0			0
$502$		−	6.68149i		−	0.298209i
$503$	−9.09539			−0.405544			−0.202772	−	0.979226i	$-0.564995\pi$
$503$	−9.09539			−0.405544			−0.202772	+	0.979226i	$0.564995\pi$
$504$		0			0
$505$	−12.9029	−	17.4549i	−0.574172	−	0.776733i
$506$	9.83563			0.437247
$507$		0			0
$508$			6.85655i			0.304210i
$509$	12.7781			0.566380			0.283190	−	0.959064i	$-0.408607\pi$
$509$	12.7781			0.566380			0.283190	+	0.959064i	$0.408607\pi$
$510$		0			0
$511$	26.7612			1.18385
$512$	1.00000			0.0441942
$513$		0			0
$514$	−4.63516			−0.204448
$515$	−3.54084	−	4.79001i	−0.156028	−	0.211073i
$516$		0			0
$517$		−	7.78280i		−	0.342287i
$518$	1.37634	−	11.3075i	0.0604731	−	0.496823i
$519$		0			0
$520$	6.35370	+	8.59521i	0.278628	+	0.376925i
$521$	37.3185			1.63495			0.817476	−	0.575963i	$-0.195373\pi$
$521$	37.3185			1.63495			0.817476	+	0.575963i	$0.195373\pi$
$522$		0			0
$523$	−1.83536			−0.0802549			−0.0401274	−	0.999195i	$-0.512776\pi$
$523$	−1.83536			−0.0802549			−0.0401274	+	0.999195i	$0.512776\pi$
$524$		−	3.26950i		−	0.142829i
$525$		0			0
$526$			29.5118i			1.28677i
$527$		−	23.1381i		−	1.00791i
$528$		0			0
$529$	19.4034			0.843624
$530$	3.99599	−	2.95389i	0.173575	−	0.128309i
$531$		0			0
$532$	8.77568			0.380474
$533$	19.6103			0.849417
$534$		0			0
$535$	−1.48021	+	1.09419i	−0.0639951	+	0.0473060i
$536$			5.76584i			0.249047i
$537$		0			0
$538$	21.3385			0.919968
$539$	5.27616			0.227260
$540$		0			0
$541$		−	37.3336i		−	1.60510i	−0.596587	−	0.802548i	$-0.703477\pi$
$541$		−	37.3336i		−	1.60510i	0.596587	−	0.802548i	$-0.296523\pi$
$542$	16.5121			0.709257
$543$		0			0
$544$	3.99190			0.171151
$545$	0.860269	−	0.635922i	0.0368499	−	0.0272399i
$546$		0			0
$547$	−17.5386			−0.749897			−0.374949	−	0.927046i	$-0.622340\pi$
$547$	−17.5386			−0.749897			−0.374949	+	0.927046i	$0.622340\pi$
$548$		−	2.11793i		−	0.0904733i
$549$		0			0
$550$	−2.21502	+	7.22005i	−0.0944489	+	0.307864i
$551$	0.937273			0.0399292
$552$		0			0
$553$	−11.1510			−0.474191
$554$	11.0446			0.469239
$555$		0			0
$556$	−15.7132			−0.666387
$557$	−40.7617			−1.72713			−0.863565	−	0.504237i	$-0.831774\pi$
$557$	−40.7617			−1.72713			−0.863565	+	0.504237i	$0.831774\pi$
$558$		0			0
$559$	−52.2160			−2.20850
$560$	3.36727	−	2.48913i	0.142293	−	0.105185i
$561$		0			0
$562$		−	17.9190i		−	0.755867i
$563$	4.47929			0.188779			0.0943897	−	0.995535i	$-0.469910\pi$
$563$	4.47929			0.188779			0.0943897	+	0.995535i	$0.469910\pi$
$564$		0			0
$565$	−26.3805	−	35.6872i	−1.10983	−	1.50137i
$566$	14.0543			0.590746
$567$		0			0
$568$	9.17428			0.384944
$569$			0.243177i			0.0101945i	0.999987	+	0.00509725i	$0.00162251\pi$
$569$			0.243177i			0.0101945i	−0.999987	+	0.00509725i	$0.998377\pi$
$570$		0			0
$571$	34.2552			1.43353			0.716767	−	0.697313i	$-0.245621\pi$
$571$	34.2552			1.43353			0.716767	+	0.697313i	$0.245621\pi$
$572$	−7.22005			−0.301885
$573$		0			0
$574$		−	7.68256i		−	0.320664i
$575$	−9.54940	+	31.1270i	−0.398238	+	1.29809i
$576$		0			0
$577$	−10.5028			−0.437239			−0.218620	−	0.975810i	$-0.570155\pi$
$577$	−10.5028			−0.437239			−0.218620	+	0.975810i	$0.570155\pi$
$578$	−1.06470			−0.0442857
$579$		0			0
$580$	0.359637	−	0.265848i	0.0149331	−	0.0110387i
$581$	17.6414			0.731887
$582$		0			0
$583$			3.35666i			0.139019i
$584$		−	14.2905i		−	0.591343i
$585$		0			0
$586$			32.2561i			1.33249i
$587$	−27.0890			−1.11808			−0.559042	−	0.829139i	$-0.688831\pi$
$587$	−27.0890			−1.11808			−0.559042	+	0.829139i	$0.688831\pi$
$588$		0			0
$589$	−27.1624			−1.11921
$590$	−23.8513	+	17.6312i	−0.981942	+	0.725865i
$591$		0			0
$592$	−6.03820	−	0.734967i	−0.248168	−	0.0302070i
$593$		−	24.2823i		−	0.997156i	−0.866845	−	0.498578i	$-0.833856\pi$
$593$		−	24.2823i		−	0.997156i	0.866845	−	0.498578i	$-0.166144\pi$
$594$		0			0
$595$	13.4418	−	9.93638i	0.551061	−	0.407352i
$596$	4.47152			0.183161
$597$		0			0
$598$	−31.1270			−1.27288
$599$	16.0532			0.655916			0.327958	−	0.944692i	$-0.393640\pi$
$599$	16.0532			0.655916			0.327958	+	0.944692i	$0.393640\pi$
$600$		0			0
$601$	42.5213			1.73448			0.867240	−	0.497891i	$-0.165892\pi$
$601$	42.5213			1.73448			0.867240	+	0.497891i	$0.165892\pi$
$602$			20.4562i			0.833732i
$603$		0			0
$604$	3.25118			0.132289
$605$	11.5887	+	15.6771i	0.471148	+	0.637364i
$606$		0			0
$607$	−28.4432			−1.15447			−0.577237	−	0.816577i	$-0.695869\pi$
$607$	−28.4432			−1.15447			−0.577237	+	0.816577i	$0.695869\pi$
$608$		−	4.68621i		−	0.190051i
$609$		0			0
$610$	−7.36539	+	5.44459i	−0.298216	+	0.220445i
$611$			24.6304i			0.996439i
$612$		0			0
$613$		−	20.9056i		−	0.844370i	−0.906510	−	0.422185i	$-0.861263\pi$
$613$		−	20.9056i		−	0.844370i	0.906510	−	0.422185i	$-0.138737\pi$
$614$			27.5771i			1.11292i
$615$		0			0
$616$			2.82853i			0.113965i
$617$			4.08940i			0.164633i	0.996606	+	0.0823165i	$0.0262318\pi$
$617$			4.08940i			0.164633i	−0.996606	+	0.0823165i	$0.973768\pi$
$618$		0			0
$619$	11.4039			0.458361			0.229181	−	0.973384i	$-0.426395\pi$
$619$	11.4039			0.458361			0.229181	+	0.973384i	$0.426395\pi$
$620$	−10.4224	+	7.70435i	−0.418572	+	0.309414i
$621$		0			0
$622$			13.8537i			0.555482i
$623$	10.2489			0.410613
$624$		0			0
$625$	−20.6989	−	14.0199i	−0.827955	−	0.560795i
$626$	24.3768			0.974294
$627$		0			0
$628$		−	0.0713960i		−	0.00284901i
$629$	−24.1039	−	2.93392i	−0.961086	−	0.116983i
$630$		0			0
$631$			28.1305i			1.11986i	0.828541	+	0.559929i	$0.189171\pi$
$631$			28.1305i			1.11986i	−0.828541	+	0.559929i	$0.810829\pi$
$632$			5.95465i			0.236863i
$633$		0			0
$634$			8.91522i			0.354069i
$635$	12.3289	−	9.11369i	0.489258	−	0.361666i
$636$		0			0
$637$	−16.6976			−0.661583
$638$			0.302097i			0.0119602i
$639$		0			0
$640$	−1.32920	−	1.79812i	−0.0525411	−	0.0710770i
$641$	−26.3364			−1.04023			−0.520113	−	0.854098i	$-0.674110\pi$
$641$	−26.3364			−1.04023			−0.520113	+	0.854098i	$0.674110\pi$
$642$		0			0
$643$	−32.4390			−1.27927			−0.639636	−	0.768678i	$-0.720915\pi$
$643$	−32.4390			−1.27927			−0.639636	+	0.768678i	$0.720915\pi$
$644$			12.1944i			0.480525i
$645$		0			0
$646$		−	18.7069i		−	0.736013i
$647$	−40.5401			−1.59380			−0.796898	−	0.604114i	$-0.793527\pi$
$647$	−40.5401			−1.59380			−0.796898	+	0.604114i	$0.793527\pi$
$648$		0			0
$649$		−	20.0353i		−	0.786453i
$650$	7.00993	−	22.8494i	0.274952	−	0.896229i
$651$		0			0
$652$	−1.06145			−0.0415695
$653$	1.15329			0.0451319			0.0225659	−	0.999745i	$-0.492816\pi$
$653$	1.15329			0.0451319			0.0225659	+	0.999745i	$0.492816\pi$
$654$		0			0
$655$	−5.87896	+	4.34581i	−0.229710	+	0.169805i
$656$	−4.10248			−0.160175
$657$		0			0
$658$	9.64923			0.376166
$659$	−46.3614			−1.80598			−0.902992	−	0.429657i	$-0.858634\pi$
$659$	−46.3614			−1.80598			−0.902992	+	0.429657i	$0.858634\pi$
$660$		0			0
$661$			40.8589i			1.58923i	0.607117	+	0.794613i	$0.292326\pi$
$661$			40.8589i			1.58923i	−0.607117	+	0.794613i	$0.707674\pi$
$662$		−	25.6563i		−	0.997160i
$663$		0			0
$664$		−	9.42048i		−	0.365585i
$665$	−11.6646	−	15.7797i	−0.452333	−	0.611912i
$666$		0			0
$667$			1.30240i			0.0504292i
$668$	−20.4097			−0.789673
$669$		0			0
$670$	10.3677	−	7.66393i	0.400539	−	0.296084i
$671$		−	6.18698i		−	0.238846i
$672$		0			0
$673$		−	31.1524i		−	1.20084i	−0.799686	−	0.600419i	$-0.795001\pi$
$673$		−	31.1524i		−	1.20084i	0.799686	−	0.600419i	$-0.204999\pi$
$674$			18.5473i			0.714415i
$675$		0			0
$676$	9.84944			0.378825
$677$		−	19.8870i		−	0.764321i	−0.924096	−	0.382160i	$-0.875180\pi$
$677$		−	19.8870i		−	0.764321i	0.924096	−	0.382160i	$-0.124820\pi$
$678$		0			0
$679$		−	3.80718i		−	0.146106i
$680$	−5.30602	−	7.17793i	−0.203477	−	0.275261i
$681$		0			0
$682$		−	8.75486i		−	0.335241i
$683$	−11.3646			−0.434855			−0.217428	−	0.976076i	$-0.569767\pi$
$683$	−11.3646			−0.434855			−0.217428	+	0.976076i	$0.569767\pi$
$684$		0			0
$685$	−3.80829	+	2.81514i	−0.145507	+	0.107561i
$686$			19.6501i			0.750244i
$687$		0			0
$688$	10.9236			0.416458
$689$		−	10.6229i		−	0.404701i
$690$		0			0
$691$	−27.8449			−1.05927			−0.529635	−	0.848226i	$-0.677671\pi$
$691$	−27.8449			−1.05927			−0.529635	+	0.848226i	$0.677671\pi$
$692$			16.9195i			0.643182i
$693$		0			0
$694$	−9.07304			−0.344408
$695$	20.8859	+	28.2542i	0.792247	+	1.07174i
$696$		0			0
$697$	−16.3767			−0.620313
$698$	−31.1022			−1.17724
$699$		0			0
$700$	−8.95152	−	2.74622i	−0.338336	−	0.103797i
$701$		−	30.8041i		−	1.16346i	−0.813384	−	0.581728i	$-0.802377\pi$
$701$		−	30.8041i		−	1.16346i	0.813384	−	0.581728i	$-0.197623\pi$
$702$		0			0
$703$	−3.44421	+	28.2962i	−0.129901	+	1.06721i
$704$	1.51044			0.0569267
$705$		0			0
$706$	−16.3289			−0.614545
$707$			18.1785i			0.683672i
$708$		0			0
$709$		−	48.0105i		−	1.80307i	−0.432704	−	0.901536i	$-0.642441\pi$
$709$		−	48.0105i		−	1.80307i	0.432704	−	0.901536i	$-0.357559\pi$
$710$	−12.1944	−	16.4965i	−0.457648	−	0.619102i
$711$		0			0
$712$		−	5.47289i		−	0.205105i
$713$		−	37.7439i		−	1.41352i
$714$		0			0
$715$	9.59685	+	12.9825i	0.358902	+	0.485519i
$716$		−	26.6575i		−	0.996237i
$717$		0			0
$718$	−17.1135			−0.638670
$719$	−10.5990			−0.395275			−0.197638	−	0.980275i	$-0.563327\pi$
$719$	−10.5990			−0.395275			−0.197638	+	0.980275i	$0.563327\pi$
$720$		0			0
$721$			4.98858i			0.185784i
$722$	−2.96053			−0.110179
$723$		0			0
$724$	−13.4428			−0.499598
$725$	−0.956055	−	0.293306i	−0.0355070	−	0.0108931i
$726$		0			0
$727$	38.9977			1.44634			0.723172	−	0.690668i	$-0.242683\pi$
$727$	38.9977			1.44634			0.723172	+	0.690668i	$0.242683\pi$
$728$		−	8.95152i		−	0.331766i
$729$		0			0
$730$	−25.6960	+	18.9948i	−0.951051	+	0.703029i
$731$	43.6059			1.61282
$732$		0			0
$733$		−	16.0616i		−	0.593250i	−0.954994	−	0.296625i	$-0.904139\pi$
$733$		−	16.0616i		−	0.593250i	0.954994	−	0.296625i	$-0.0958611\pi$
$734$			23.2045i			0.856493i
$735$		0			0
$736$	6.51179			0.240028
$737$			8.70894i			0.320798i
$738$		0			0
$739$	28.6759			1.05486			0.527430	−	0.849599i	$-0.323156\pi$
$739$	28.6759			1.05486			0.527430	+	0.849599i	$0.323156\pi$
$740$	6.70438	+	11.8343i	0.246458	+	0.435038i
$741$		0			0
$742$	−4.16164			−0.152779
$743$		−	23.4444i		−	0.860090i	−0.902807	−	0.430045i	$-0.858498\pi$
$743$		−	23.4444i		−	0.860090i	0.902807	−	0.430045i	$-0.141502\pi$
$744$		0			0
$745$	−5.94353	−	8.04034i	−0.217754	−	0.294575i
$746$		−	9.69704i		−	0.355034i
$747$		0			0
$748$	6.02951			0.220461
$749$	1.54157			0.0563278
$750$		0			0
$751$	−20.3229			−0.741592			−0.370796	−	0.928714i	$-0.620915\pi$
$751$	−20.3229			−0.741592			−0.370796	+	0.928714i	$0.620915\pi$
$752$		−	5.15269i		−	0.187899i
$753$		0			0
$754$		−	0.956055i		−	0.0348175i
$755$	−4.32145	−	5.84601i	−0.157274	−	0.212758i
$756$		0			0
$757$	−37.8496			−1.37567			−0.687834	−	0.725868i	$-0.741438\pi$
$757$	−37.8496			−1.37567			−0.687834	+	0.725868i	$0.741438\pi$
$758$	−9.40066			−0.341448
$759$		0			0
$760$	−8.42637	+	6.22888i	−0.305656	+	0.225945i
$761$	13.4618			0.487990			0.243995	−	0.969777i	$-0.421542\pi$
$761$	13.4618			0.487990			0.243995	+	0.969777i	$0.421542\pi$
$762$		0			0
$763$	−0.895931			−0.0324349
$764$			6.76465i			0.244736i
$765$		0			0
$766$	32.3874			1.17020
$767$			63.4061i			2.28946i
$768$		0			0
$769$			19.7800i			0.713285i	0.934241	+	0.356642i	$0.116079\pi$
$769$			19.7800i			0.713285i	−0.934241	+	0.356642i	$0.883921\pi$
$770$	5.08605	−	3.75967i	0.183288	−	0.135489i
$771$		0			0
$772$	−15.4592			−0.556389
$773$			25.4792i			0.916423i	0.888843	+	0.458212i	$0.151510\pi$
$773$			25.4792i			0.916423i	−0.888843	+	0.458212i	$0.848490\pi$
$774$		0			0
$775$	27.7067	+	8.50008i	0.995254	+	0.305332i
$776$	−2.03303			−0.0729815
$777$		0			0
$778$			20.8774i			0.748492i
$779$			19.2251i			0.688810i
$780$		0			0
$781$	13.8572			0.495848
$782$	25.9944			0.929559
$783$		0			0
$784$	3.49314			0.124755
$785$	−0.128379	+	0.0948993i	−0.00458203	+	0.00338710i
$786$		0			0
$787$		−	50.1801i		−	1.78873i	−0.447339	−	0.894364i	$-0.647628\pi$
$787$		−	50.1801i		−	1.78873i	0.447339	−	0.894364i	$-0.352372\pi$
$788$		−	12.5068i		−	0.445536i
$789$		0			0
$790$	10.7072	−	7.91489i	0.380944	−	0.281599i
$791$			37.1666i			1.32149i
$792$		0			0
$793$			19.5801i			0.695309i
$794$		−	12.2447i		−	0.434549i
$795$		0			0
$796$			13.4172i			0.475559i
$797$	−26.4099			−0.935485			−0.467743	−	0.883865i	$-0.654933\pi$
$797$	−26.4099			−0.935485			−0.467743	+	0.883865i	$0.654933\pi$
$798$		0			0
$799$		−	20.5690i		−	0.727680i
$800$	−1.46648	+	4.78011i	−0.0518479	+	0.169002i
$801$		0			0
$802$			13.7736i			0.486362i
$803$		−	21.5848i		−	0.761711i
$804$		0			0
$805$	21.9270	−	16.2087i	0.772823	−	0.571282i
$806$			27.7067i			0.975927i
$807$		0			0
$808$	9.70730			0.341502
$809$			8.11189i			0.285199i	0.989780	+	0.142599i	$0.0455461\pi$
$809$			8.11189i			0.285199i	−0.989780	+	0.142599i	$0.954454\pi$
$810$		0			0
$811$	27.0320			0.949223			0.474611	−	0.880195i	$-0.342589\pi$
$811$	27.0320			0.949223			0.474611	+	0.880195i	$0.342589\pi$
$812$	−0.374545			−0.0131440
$813$		0			0
$814$	−9.12031	−	1.11012i	−0.319667	−	0.0389097i
$815$	1.41087	+	1.90861i	0.0494206	+	0.0668557i
$816$		0			0
$817$		−	51.1902i		−	1.79092i
$818$			34.5667i			1.20860i
$819$		0			0
$820$	5.45300	+	7.37676i	0.190427	+	0.257608i
$821$	19.7666			0.689859			0.344930	−	0.938629i	$-0.387903\pi$
$821$	19.7666			0.689859			0.344930	+	0.938629i	$0.387903\pi$
$822$		0			0
$823$			41.4991i			1.44657i	0.690550	+	0.723285i	$0.257368\pi$
$823$			41.4991i			1.44657i	−0.690550	+	0.723285i	$0.742632\pi$
$824$	2.66390			0.0928013
$825$		0			0
$826$	24.8400			0.864295
$827$	−39.6665			−1.37934			−0.689671	−	0.724123i	$-0.742245\pi$
$827$	−39.6665			−1.37934			−0.689671	+	0.724123i	$0.742245\pi$
$828$		0			0
$829$			27.0227i			0.938538i	0.883055	+	0.469269i	$0.155483\pi$
$829$			27.0227i			0.938538i	−0.883055	+	0.469269i	$0.844517\pi$
$830$	−16.9392	+	12.5217i	−0.587967	+	0.434633i
$831$		0			0
$832$	−4.78011			−0.165720
$833$	13.9443			0.483141
$834$		0			0
$835$	27.1284	+	36.6990i	0.938818	+	1.27002i
$836$		−	7.07821i		−	0.244805i
$837$		0			0
$838$	−22.9413			−0.792495
$839$	−6.29148			−0.217206			−0.108603	−	0.994085i	$-0.534638\pi$
$839$	−6.29148			−0.217206			−0.108603	+	0.994085i	$0.534638\pi$
$840$		0			0
$841$	28.9600			0.998621
$842$		−	12.1679i		−	0.419333i
$843$		0			0
$844$	6.46040			0.222376
$845$	−13.0918	−	17.7105i	−0.450373	−	0.609259i
$846$		0			0
$847$		−	16.3270i		−	0.561001i
$848$			2.22232i			0.0763146i
$849$		0			0
$850$	−5.85405	+	19.0817i	−0.200792	+	0.654498i
$851$	−39.3194	−	4.78595i	−1.34785	−	0.164060i
$852$		0			0
$853$	−1.33464			−0.0456971			−0.0228486	−	0.999739i	$-0.507274\pi$
$853$	−1.33464			−0.0456971			−0.0228486	+	0.999739i	$0.507274\pi$
$854$	7.67072			0.262487
$855$		0			0
$856$		−	0.823199i		−	0.0281364i
$857$	−22.7053			−0.775597			−0.387799	−	0.921744i	$-0.626764\pi$
$857$	−22.7053			−0.775597			−0.387799	+	0.921744i	$0.626764\pi$
$858$		0			0
$859$			8.92509i			0.304520i	0.988340	+	0.152260i	$0.0486551\pi$
$859$			8.92509i			0.304520i	−0.988340	+	0.152260i	$0.951345\pi$
$860$	−14.5196	−	19.6419i	−0.495114	−	0.669785i
$861$		0			0
$862$			11.2076i			0.381734i
$863$		−	8.98998i		−	0.306022i	−0.988224	−	0.153011i	$-0.951103\pi$
$863$		−	8.98998i		−	0.306022i	0.988224	−	0.153011i	$-0.0488971\pi$
$864$		0			0
$865$	30.4233	−	22.4893i	1.03442	−	0.764659i
$866$			31.6559i			1.07571i
$867$		0			0
$868$	10.8544			0.368423
$869$			8.99411i			0.305104i
$870$		0			0
$871$		−	27.5614i		−	0.933881i
$872$			0.478427i			0.0162016i
$873$		0			0
$874$		−	30.5156i		−	1.03220i
$875$	6.96029	+	19.7462i	0.235301	+	0.667543i
$876$		0			0
$877$			9.40084i			0.317444i	0.987323	+	0.158722i	$0.0507373\pi$
$877$			9.40084i			0.317444i	−0.987323	+	0.158722i	$0.949263\pi$
$878$			33.9912i			1.14715i
$879$		0			0
$880$	−2.00766	−	2.71595i	−0.0676783	−	0.0915545i
$881$	19.9044			0.670596			0.335298	−	0.942112i	$-0.391163\pi$
$881$	19.9044			0.670596			0.335298	+	0.942112i	$0.391163\pi$
$882$		0			0
$883$	48.8646			1.64442			0.822212	−	0.569181i	$-0.192740\pi$
$883$	48.8646			1.64442			0.822212	+	0.569181i	$0.192740\pi$
$884$	−19.0817			−0.641788
$885$		0			0
$886$			34.9477i			1.17409i
$887$			32.4651i			1.09007i	0.838413	+	0.545036i	$0.183484\pi$
$887$			32.4651i			1.09007i	−0.838413	+	0.545036i	$0.816516\pi$
$888$		0			0
$889$	−12.8400			−0.430639
$890$	−9.84093	+	7.27454i	−0.329869	+	0.243843i
$891$		0			0
$892$		−	8.15634i		−	0.273095i
$893$	−24.1465			−0.808033
$894$		0			0
$895$	−47.9334	+	35.4330i	−1.60224	+	1.18439i
$896$			1.87266i			0.0625612i
$897$		0			0
$898$			17.2274i			0.574886i
$899$	1.15929			0.0386645
$900$		0			0
$901$			8.87127i			0.295545i
$902$	−6.19654			−0.206322
$903$		0			0
$904$	19.8469			0.660099
$905$	17.8681	+	24.1718i	0.593956	+	0.803497i
$906$		0			0
$907$	−6.47916			−0.215137			−0.107568	−	0.994198i	$-0.534306\pi$
$907$	−6.47916			−0.215137			−0.107568	+	0.994198i	$0.534306\pi$
$908$	13.0456			0.432933
$909$		0			0
$910$	−16.0959	+	11.8983i	−0.533575	+	0.394426i
$911$			17.1972i			0.569767i	0.958562	+	0.284884i	$0.0919550\pi$
$911$			17.1972i			0.569767i	−0.958562	+	0.284884i	$0.908045\pi$
$912$		0			0
$913$		−	14.2290i		−	0.470912i
$914$	−24.5044			−0.810532
$915$		0			0
$916$	13.3621			0.441498
$917$	6.12267			0.202188
$918$		0			0
$919$			39.5341i			1.30411i	0.758172	+	0.652054i	$0.226093\pi$
$919$			39.5341i			1.30411i	−0.758172	+	0.652054i	$0.773907\pi$
$920$	−8.65543	−	11.7090i	−0.285361	−	0.386034i
$921$		0			0
$922$		−	9.05274i		−	0.298136i
$923$	−43.8541			−1.44347
$924$		0			0
$925$	12.3681	−	27.7854i	0.406661	−	0.913579i
$926$	2.10640			0.0692206
$927$		0			0
$928$			0.200007i			0.00656555i
$929$	−53.4172			−1.75256			−0.876281	−	0.481800i	$-0.839983\pi$
$929$	−53.4172			−1.75256			−0.876281	+	0.481800i	$0.839983\pi$
$930$		0			0
$931$		−	16.3696i		−	0.536491i
$932$		−	27.2277i		−	0.891875i
$933$		0			0
$934$	24.0814			0.787967
$935$	−8.01440	−	10.8418i	−0.262099	−	0.354565i
$936$		0			0
$937$			15.1507i			0.494953i	0.968894	+	0.247477i	$0.0796013\pi$
$937$			15.1507i			0.494953i	−0.968894	+	0.247477i	$0.920399\pi$
$938$	−10.7975			−0.352550
$939$		0			0
$940$	−9.26515	+	6.84893i	−0.302196	+	0.223387i
$941$	−24.9479			−0.813279			−0.406639	−	0.913589i	$-0.633299\pi$
$941$	−24.9479			−0.813279			−0.406639	+	0.913589i	$0.633299\pi$
$942$		0			0
$943$	−26.7145			−0.869944
$944$		−	13.2646i		−	0.431725i
$945$		0			0
$946$	16.4994			0.536441
$947$	3.44085			0.111813			0.0559063	−	0.998436i	$-0.482195\pi$
$947$	3.44085			0.111813			0.0559063	+	0.998436i	$0.482195\pi$
$948$		0			0
$949$			68.3099i			2.21743i
$950$	22.4006	+	6.87223i	0.726771	+	0.222964i
$951$		0			0
$952$			7.47548i			0.242282i
$953$		−	16.3551i		−	0.529794i	−0.964277	−	0.264897i	$-0.914662\pi$
$953$		−	16.3551i		−	0.529794i	0.964277	−	0.264897i	$-0.0853380\pi$
$954$		0			0
$955$	12.1637	−	8.99153i	0.393607	−	0.290959i
$956$		−	10.5417i		−	0.340944i
$957$		0			0
$958$		−	5.46224i		−	0.176477i
$959$	3.96616			0.128074
$960$		0			0
$961$	−2.59651			−0.0837582
$962$	28.8632	+	3.51322i	0.930588	+	0.113271i
$963$		0			0
$964$			13.3732i			0.430723i
$965$	20.5483	+	27.7975i	0.661473	+	0.894834i
$966$		0			0
$967$	40.2993			1.29594			0.647968	−	0.761667i	$-0.275619\pi$
$967$	40.2993			1.29594			0.647968	+	0.761667i	$0.275619\pi$
$968$	−8.71858			−0.280226
$969$		0			0
$970$	2.70229	+	3.65563i	0.0867654	+	0.117375i
$971$	45.6611			1.46533			0.732667	−	0.680587i	$-0.238275\pi$
$971$	45.6611			1.46533			0.732667	+	0.680587i	$0.238275\pi$
$972$		0			0
$973$		−	29.4255i		−	0.943337i
$974$	−11.8494			−0.379681
$975$		0			0
$976$		−	4.09616i		−	0.131115i
$977$	4.88784			0.156376			0.0781879	−	0.996939i	$-0.475087\pi$
$977$	4.88784			0.156376			0.0781879	+	0.996939i	$0.475087\pi$
$978$		0			0
$979$		−	8.26645i		−	0.264197i
$980$	−4.64307	−	6.28109i	−0.148317	−	0.200642i
$981$		0			0
$982$	7.19955			0.229747
$983$		−	0.259395i		−	0.00827341i	−0.999991	−	0.00413670i	$-0.998683\pi$
$983$		−	0.259395i		−	0.00827341i	0.999991	−	0.00413670i	$-0.00131676\pi$
$984$		0			0
$985$	−22.4887	+	16.6240i	−0.716550	+	0.529684i
$986$			0.798408i			0.0254265i
$987$		0			0
$988$			22.4006i			0.712657i
$989$	71.1321			2.26187
$990$		0			0
$991$		−	29.6464i		−	0.941750i	−0.882200	−	0.470875i	$-0.843938\pi$
$991$		−	29.6464i		−	0.941750i	0.882200	−	0.470875i	$-0.156062\pi$
$992$		−	5.79625i		−	0.184031i
$993$		0			0
$994$			17.1803i			0.544927i
$995$	24.1257	−	17.8341i	0.764837	−	0.565378i
$996$		0			0
$997$	33.3064			1.05482			0.527411	−	0.849610i	$-0.323163\pi$
$997$	33.3064			1.05482			0.527411	+	0.849610i	$0.323163\pi$
$998$		−	3.32481i		−	0.105245i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.e.f.739.3		16
3.2	odd	2		1110.2.e.c.739.15	yes	16
5.4	even	2		3330.2.e.e.739.13		16
15.14	odd	2		1110.2.e.d.739.2	yes	16
37.36	even	2		3330.2.e.e.739.14		16
111.110	odd	2		1110.2.e.d.739.10	yes	16
185.184	even	2	inner	3330.2.e.f.739.4		16
555.554	odd	2		1110.2.e.c.739.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.e.c.739.7	✓	16	555.554	odd	2
1110.2.e.c.739.15	yes	16	3.2	odd	2
1110.2.e.d.739.2	yes	16	15.14	odd	2
1110.2.e.d.739.10	yes	16	111.110	odd	2
3330.2.e.e.739.13		16	5.4	even	2
3330.2.e.e.739.14		16	37.36	even	2
3330.2.e.f.739.3		16	1.1	even	1	trivial
3330.2.e.f.739.4		16	185.184	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(-1.32920 - 1.79812i) q^{5} +1.87266i q^{7} +1.00000 q^{8} +(-1.32920 - 1.79812i) q^{10} +1.51044 q^{11} -4.78011 q^{13} +1.87266i q^{14} +1.00000 q^{16} +3.99190 q^{17} -4.68621i q^{19} +(-1.32920 - 1.79812i) q^{20} +1.51044 q^{22} +6.51179 q^{23} +(-1.46648 + 4.78011i) q^{25} -4.78011 q^{26} +1.87266i q^{28} +0.200007i q^{29} -5.79625i q^{31} +1.00000 q^{32} +3.99190 q^{34} +(3.36727 - 2.48913i) q^{35} +(-6.03820 - 0.734967i) q^{37} -4.68621i q^{38} +(-1.32920 - 1.79812i) q^{40} -4.10248 q^{41} +10.9236 q^{43} +1.51044 q^{44} +6.51179 q^{46} -5.15269i q^{47} +3.49314 q^{49} +(-1.46648 + 4.78011i) q^{50} -4.78011 q^{52} +2.22232i q^{53} +(-2.00766 - 2.71595i) q^{55} +1.87266i q^{56} +0.200007i q^{58} -13.2646i q^{59} -4.09616i q^{61} -5.79625i q^{62} +1.00000 q^{64} +(6.35370 + 8.59521i) q^{65} +5.76584i q^{67} +3.99190 q^{68} +(3.36727 - 2.48913i) q^{70} +9.17428 q^{71} -14.2905i q^{73} +(-6.03820 - 0.734967i) q^{74} -4.68621i q^{76} +2.82853i q^{77} +5.95465i q^{79} +(-1.32920 - 1.79812i) q^{80} -4.10248 q^{82} -9.42048i q^{83} +(-5.30602 - 7.17793i) q^{85} +10.9236 q^{86} +1.51044 q^{88} -5.47289i q^{89} -8.95152i q^{91} +6.51179 q^{92} -5.15269i q^{94} +(-8.42637 + 6.22888i) q^{95} -2.03303 q^{97} +3.49314 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{16} + 38 q^{17} + 2 q^{20} - 2 q^{22} + 20 q^{23} - 4 q^{25} + 16 q^{32} + 38 q^{34} + 10 q^{35} - 4 q^{37} + 2 q^{40} + 6 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 2 * q^5 + 16 * q^8 + 2 * q^10 - 2 * q^11 + 16 * q^16 + 38 * q^17 + 2 * q^20 - 2 * q^22 + 20 * q^23 - 4 * q^25 + 16 * q^32 + 38 * q^34 + 10 * q^35 - 4 * q^37 + 2 * q^40 + 6 * q^41 - 2 * q^43 - 2 * q^44 + 20 * q^46 - 18 * q^49 - 4 * q^50 + 20 * q^55 + 16 * q^64 - 12 * q^65 + 38 * q^68 + 10 * q^70 + 24 * q^71 - 4 * q^74 + 2 * q^80 + 6 * q^82 - 2 * q^86 - 2 * q^88 + 20 * q^92 + 16 * q^95 + 38 * q^97 - 18 * q^98

Introduction

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