LMFDB - Embedded newform 109.2.b.a.108.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [109,2,Mod(108,109)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(109, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("109.108");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	109.b (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:	$0.870369382032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			108.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	109.108
Dual form			109.2.b.a.108.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} -3.46410i q^{6} +2.00000 q^{7} -1.73205i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} -3.46410i q^{6} +2.00000 q^{7} -1.73205i q^{8} +1.00000 q^{9} +5.19615i q^{10} +5.19615i q^{11} -2.00000 q^{12} -3.46410i q^{14} -6.00000 q^{15} -5.00000 q^{16} -3.46410i q^{17} -1.73205i q^{18} +5.19615i q^{19} +3.00000 q^{20} +4.00000 q^{21} +9.00000 q^{22} +1.73205i q^{23} -3.46410i q^{24} +4.00000 q^{25} -4.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +10.3923i q^{30} +10.0000 q^{31} +5.19615i q^{32} +10.3923i q^{33} -6.00000 q^{34} -6.00000 q^{35} -1.00000 q^{36} -10.3923i q^{37} +9.00000 q^{38} +5.19615i q^{40} +3.46410i q^{41} -6.92820i q^{42} -2.00000 q^{43} -5.19615i q^{44} -3.00000 q^{45} +3.00000 q^{46} -12.1244i q^{47} -10.0000 q^{48} -3.00000 q^{49} -6.92820i q^{50} -6.92820i q^{51} +6.92820i q^{54} -15.5885i q^{55} -3.46410i q^{56} +10.3923i q^{57} +5.19615i q^{58} -3.46410i q^{59} +6.00000 q^{60} -7.00000 q^{61} -17.3205i q^{62} +2.00000 q^{63} -1.00000 q^{64} +18.0000 q^{66} +10.3923i q^{67} +3.46410i q^{68} +3.46410i q^{69} +10.3923i q^{70} -6.00000 q^{71} -1.73205i q^{72} -5.00000 q^{73} -18.0000 q^{74} +8.00000 q^{75} -5.19615i q^{76} +10.3923i q^{77} -10.3923i q^{79} +15.0000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +10.3923i q^{85} +3.46410i q^{86} -6.00000 q^{87} +9.00000 q^{88} +15.0000 q^{89} +5.19615i q^{90} -1.73205i q^{92} +20.0000 q^{93} -21.0000 q^{94} -15.5885i q^{95} +10.3923i q^{96} +7.00000 q^{97} +5.19615i q^{98} +5.19615i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{4} - 6 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 2 * q^4 - 6 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 4 q^{3} - 2 q^{4} - 6 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{12} - 12 q^{15} - 10 q^{16} + 6 q^{20} + 8 q^{21} + 18 q^{22} + 8 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{29} + 20 q^{31} - 12 q^{34} - 12 q^{35} - 2 q^{36} + 18 q^{38} - 4 q^{43} - 6 q^{45} + 6 q^{46} - 20 q^{48} - 6 q^{49} + 12 q^{60} - 14 q^{61} + 4 q^{63} - 2 q^{64} + 36 q^{66} - 12 q^{71} - 10 q^{73} - 36 q^{74} + 16 q^{75} + 30 q^{80} - 22 q^{81} + 12 q^{82} + 24 q^{83} - 8 q^{84} - 12 q^{87} + 18 q^{88} + 30 q^{89} + 40 q^{93} - 42 q^{94} + 14 q^{97}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^4 - 6 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^12 - 12 * q^15 - 10 * q^16 + 6 * q^20 + 8 * q^21 + 18 * q^22 + 8 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^29 + 20 * q^31 - 12 * q^34 - 12 * q^35 - 2 * q^36 + 18 * q^38 - 4 * q^43 - 6 * q^45 + 6 * q^46 - 20 * q^48 - 6 * q^49 + 12 * q^60 - 14 * q^61 + 4 * q^63 - 2 * q^64 + 36 * q^66 - 12 * q^71 - 10 * q^73 - 36 * q^74 + 16 * q^75 + 30 * q^80 - 22 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 12 * q^87 + 18 * q^88 + 30 * q^89 + 40 * q^93 - 42 * q^94 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/109\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	2.00000			1.15470			0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000			1.15470			0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000			−0.500000
$5$	−3.00000			−1.34164			−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000			−1.34164			−0.670820	+	0.741620i	$0.734058\pi$
$6$		−	3.46410i		−	1.41421i
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		−	1.73205i		−	0.612372i
$9$	1.00000			0.333333
$10$			5.19615i			1.64317i
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$	−2.00000			−0.577350
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		−	3.46410i		−	0.925820i
$15$	−6.00000			−1.54919
$16$	−5.00000			−1.25000
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		−	1.73205i		−	0.408248i
$19$			5.19615i			1.19208i	0.802955	+	0.596040i	$0.203260\pi$
$19$			5.19615i			1.19208i	−0.802955	+	0.596040i	$0.796740\pi$
$20$	3.00000			0.670820
$21$	4.00000			0.872872
$22$	9.00000			1.91881
$23$			1.73205i			0.361158i	0.983561	+	0.180579i	$0.0577971\pi$
$23$			1.73205i			0.361158i	−0.983561	+	0.180579i	$0.942203\pi$
$24$		−	3.46410i		−	0.707107i
$25$	4.00000			0.800000
$26$		0			0
$27$	−4.00000			−0.769800
$28$	−2.00000			−0.377964
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$			10.3923i			1.89737i
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$			5.19615i			0.918559i
$33$			10.3923i			1.80907i
$34$	−6.00000			−1.02899
$35$	−6.00000			−1.01419
$36$	−1.00000			−0.166667
$37$		−	10.3923i		−	1.70848i	−0.519875	−	0.854242i	$-0.674022\pi$
$37$		−	10.3923i		−	1.70848i	0.519875	−	0.854242i	$-0.325978\pi$
$38$	9.00000			1.45999
$39$		0			0
$40$			5.19615i			0.821584i
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		−	6.92820i		−	1.06904i
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		−	5.19615i		−	0.783349i
$45$	−3.00000			−0.447214
$46$	3.00000			0.442326
$47$		−	12.1244i		−	1.76852i	−0.466996	−	0.884260i	$-0.654664\pi$
$47$		−	12.1244i		−	1.76852i	0.466996	−	0.884260i	$-0.345336\pi$
$48$	−10.0000			−1.44338
$49$	−3.00000			−0.428571
$50$		−	6.92820i		−	0.979796i
$51$		−	6.92820i		−	0.970143i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$			6.92820i			0.942809i
$55$		−	15.5885i		−	2.10195i
$56$		−	3.46410i		−	0.462910i
$57$			10.3923i			1.37649i
$58$			5.19615i			0.682288i
$59$		−	3.46410i		−	0.450988i	−0.974245	−	0.225494i	$-0.927600\pi$
$59$		−	3.46410i		−	0.450988i	0.974245	−	0.225494i	$-0.0723995\pi$
$60$	6.00000			0.774597
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		−	17.3205i		−	2.19971i
$63$	2.00000			0.251976
$64$	−1.00000			−0.125000
$65$		0			0
$66$	18.0000			2.21565
$67$			10.3923i			1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$			10.3923i			1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$			3.46410i			0.420084i
$69$			3.46410i			0.417029i
$70$			10.3923i			1.24212i
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		−	1.73205i		−	0.204124i
$73$	−5.00000			−0.585206			−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000			−0.585206			−0.292603	+	0.956234i	$0.594521\pi$
$74$	−18.0000			−2.09246
$75$	8.00000			0.923760
$76$		−	5.19615i		−	0.596040i
$77$			10.3923i			1.18431i
$78$		0			0
$79$		−	10.3923i		−	1.16923i	−0.811312	−	0.584613i	$-0.801246\pi$
$79$		−	10.3923i		−	1.16923i	0.811312	−	0.584613i	$-0.198754\pi$
$80$	15.0000			1.67705
$81$	−11.0000			−1.22222
$82$	6.00000			0.662589
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000			−0.436436
$85$			10.3923i			1.12720i
$86$			3.46410i			0.373544i
$87$	−6.00000			−0.643268
$88$	9.00000			0.959403
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$			5.19615i			0.547723i
$91$		0			0
$92$		−	1.73205i		−	0.180579i
$93$	20.0000			2.07390
$94$	−21.0000			−2.16598
$95$		−	15.5885i		−	1.59934i
$96$			10.3923i			1.06066i
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$			5.19615i			0.524891i
$99$			5.19615i			0.522233i
$100$	−4.00000			−0.400000
$101$			10.3923i			1.03407i	0.855963	+	0.517036i	$0.172965\pi$
$101$			10.3923i			1.03407i	−0.855963	+	0.517036i	$0.827035\pi$
$102$	−12.0000			−1.18818
$103$			15.5885i			1.53598i	0.640464	+	0.767988i	$0.278742\pi$
$103$			15.5885i			1.53598i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$	−12.0000			−1.17108
$106$		0			0
$107$		−	1.73205i		−	0.167444i	−0.996489	−	0.0837218i	$-0.973319\pi$
$107$		−	1.73205i		−	0.167444i	0.996489	−	0.0837218i	$-0.0266807\pi$
$108$	4.00000			0.384900
$109$	−1.00000	−	10.3923i	−0.0957826	−	0.995402i
$109$	−1.00000	−	10.3923i	−0.0957826	−	0.995402i
$110$	−27.0000			−2.57435
$111$		−	20.7846i		−	1.97279i
$112$	−10.0000			−0.944911
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$	18.0000			1.68585
$115$		−	5.19615i		−	0.484544i
$116$	3.00000			0.278543
$117$		0			0
$118$	−6.00000			−0.552345
$119$		−	6.92820i		−	0.635107i
$120$			10.3923i			0.948683i
$121$	−16.0000			−1.45455
$122$			12.1244i			1.09769i
$123$			6.92820i			0.624695i
$124$	−10.0000			−0.898027
$125$	3.00000			0.268328
$126$		−	3.46410i		−	0.308607i
$127$		−	5.19615i		−	0.461084i	−0.973062	−	0.230542i	$-0.925950\pi$
$127$		−	5.19615i		−	0.461084i	0.973062	−	0.230542i	$-0.0740499\pi$
$128$			12.1244i			1.07165i
$129$	−4.00000			−0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		−	10.3923i		−	0.904534i
$133$			10.3923i			0.901127i
$134$	18.0000			1.55496
$135$	12.0000			1.03280
$136$	−6.00000			−0.514496
$137$	−9.00000			−0.768922			−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000			−0.768922			−0.384461	+	0.923141i	$0.625613\pi$
$138$	6.00000			0.510754
$139$		−	10.3923i		−	0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$139$		−	10.3923i		−	0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$	6.00000			0.507093
$141$		−	24.2487i		−	2.04211i
$142$			10.3923i			0.872103i
$143$		0			0
$144$	−5.00000			−0.416667
$145$	9.00000			0.747409
$146$			8.66025i			0.716728i
$147$	−6.00000			−0.494872
$148$			10.3923i			0.854242i
$149$			17.3205i			1.41895i	0.704730	+	0.709476i	$0.251068\pi$
$149$			17.3205i			1.41895i	−0.704730	+	0.709476i	$0.748932\pi$
$150$		−	13.8564i		−	1.13137i
$151$		−	5.19615i		−	0.422857i	−0.977393	−	0.211428i	$-0.932188\pi$
$151$		−	5.19615i		−	0.422857i	0.977393	−	0.211428i	$-0.0678115\pi$
$152$	9.00000			0.729996
$153$		−	3.46410i		−	0.280056i
$154$	18.0000			1.45048
$155$	−30.0000			−2.40966
$156$		0			0
$157$	13.0000			1.03751			0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000			1.03751			0.518756	+	0.854922i	$0.326395\pi$
$158$	−18.0000			−1.43200
$159$		0			0
$160$		−	15.5885i		−	1.23238i
$161$			3.46410i			0.273009i
$162$			19.0526i			1.49691i
$163$			5.19615i			0.406994i	0.979076	+	0.203497i	$0.0652307\pi$
$163$			5.19615i			0.406994i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		−	3.46410i		−	0.270501i
$165$		−	31.1769i		−	2.42712i
$166$		−	20.7846i		−	1.61320i
$167$			10.3923i			0.804181i	0.915600	+	0.402090i	$0.131716\pi$
$167$			10.3923i			0.804181i	−0.915600	+	0.402090i	$0.868284\pi$
$168$		−	6.92820i		−	0.534522i
$169$	13.0000			1.00000
$170$	18.0000			1.38054
$171$			5.19615i			0.397360i
$172$	2.00000			0.152499
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$			10.3923i			0.787839i
$175$	8.00000			0.604743
$176$		−	25.9808i		−	1.95837i
$177$		−	6.92820i		−	0.520756i
$178$		−	25.9808i		−	1.94734i
$179$			12.1244i			0.906217i	0.891455	+	0.453108i	$0.149685\pi$
$179$			12.1244i			0.906217i	−0.891455	+	0.453108i	$0.850315\pi$
$180$	3.00000			0.223607
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−14.0000			−1.03491
$184$	3.00000			0.221163
$185$			31.1769i			2.29217i
$186$		−	34.6410i		−	2.54000i
$187$	18.0000			1.31629
$188$			12.1244i			0.884260i
$189$	−8.00000			−0.581914
$190$	−27.0000			−1.95879
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−2.00000			−0.144338
$193$	−1.00000			−0.0719816			−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000			−0.0719816			−0.0359908	+	0.999352i	$0.511459\pi$
$194$		−	12.1244i		−	0.870478i
$195$		0			0
$196$	3.00000			0.214286
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$	9.00000			0.639602
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		−	6.92820i		−	0.489898i
$201$			20.7846i			1.46603i
$202$	18.0000			1.26648
$203$	−6.00000			−0.421117
$204$			6.92820i			0.485071i
$205$		−	10.3923i		−	0.725830i
$206$	27.0000			1.88118
$207$			1.73205i			0.120386i
$208$		0			0
$209$	−27.0000			−1.86763
$210$			20.7846i			1.43427i
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$		0			0
$213$	−12.0000			−0.822226
$214$	−3.00000			−0.205076
$215$	6.00000			0.409197
$216$			6.92820i			0.471405i
$217$	20.0000			1.35769
$218$	−18.0000	+	1.73205i	−1.21911	+	0.117309i
$219$	−10.0000			−0.675737
$220$			15.5885i			1.05097i
$221$		0			0
$222$	−36.0000			−2.41616
$223$	−2.00000			−0.133930			−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000			−0.133930			−0.0669650	+	0.997755i	$0.521332\pi$
$224$			10.3923i			0.694365i
$225$	4.00000			0.266667
$226$		−	10.3923i		−	0.691286i
$227$	18.0000			1.19470			0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000			1.19470			0.597351	+	0.801980i	$0.296220\pi$
$228$		−	10.3923i		−	0.688247i
$229$			10.3923i			0.686743i	0.939200	+	0.343371i	$0.111569\pi$
$229$			10.3923i			0.686743i	−0.939200	+	0.343371i	$0.888431\pi$
$230$	−9.00000			−0.593442
$231$			20.7846i			1.36753i
$232$			5.19615i			0.341144i
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$			36.3731i			2.37272i
$236$			3.46410i			0.225494i
$237$		−	20.7846i		−	1.35011i
$238$	−12.0000			−0.777844
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$	30.0000			1.93649
$241$			10.3923i			0.669427i	0.942320	+	0.334714i	$0.108640\pi$
$241$			10.3923i			0.669427i	−0.942320	+	0.334714i	$0.891360\pi$
$242$			27.7128i			1.78145i
$243$	−10.0000			−0.641500
$244$	7.00000			0.448129
$245$	9.00000			0.574989
$246$	12.0000			0.765092
$247$		0			0
$248$		−	17.3205i		−	1.09985i
$249$	24.0000			1.52094
$250$		−	5.19615i		−	0.328634i
$251$		−	22.5167i		−	1.42124i	−0.703577	−	0.710620i	$-0.748415\pi$
$251$		−	22.5167i		−	1.42124i	0.703577	−	0.710620i	$-0.251585\pi$
$252$	−2.00000			−0.125988
$253$	−9.00000			−0.565825
$254$	−9.00000			−0.564710
$255$			20.7846i			1.30158i
$256$	19.0000			1.18750
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$			6.92820i			0.431331i
$259$		−	20.7846i		−	1.29149i
$260$		0			0
$261$	−3.00000			−0.185695
$262$		−	20.7846i		−	1.28408i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$	18.0000			1.10782
$265$		0			0
$266$	18.0000			1.10365
$267$	30.0000			1.83597
$268$		−	10.3923i		−	0.634811i
$269$		−	10.3923i		−	0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$		−	10.3923i		−	0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$		−	20.7846i		−	1.26491i
$271$		−	25.9808i		−	1.57822i	−0.614253	−	0.789109i	$-0.710542\pi$
$271$		−	25.9808i		−	1.57822i	0.614253	−	0.789109i	$-0.289458\pi$
$272$			17.3205i			1.05021i
$273$		0			0
$274$			15.5885i			0.941733i
$275$			20.7846i			1.25336i
$276$		−	3.46410i		−	0.208514i
$277$		−	20.7846i		−	1.24883i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$		−	20.7846i		−	1.24883i	0.781094	−	0.624413i	$-0.214662\pi$
$278$	−18.0000			−1.07957
$279$	10.0000			0.598684
$280$			10.3923i			0.621059i
$281$	−9.00000			−0.536895			−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000			−0.536895			−0.268447	+	0.963294i	$0.586511\pi$
$282$	−42.0000			−2.50106
$283$			25.9808i			1.54440i	0.635382	+	0.772198i	$0.280843\pi$
$283$			25.9808i			1.54440i	−0.635382	+	0.772198i	$0.719157\pi$
$284$	6.00000			0.356034
$285$		−	31.1769i		−	1.84676i
$286$		0			0
$287$			6.92820i			0.408959i
$288$			5.19615i			0.306186i
$289$	5.00000			0.294118
$290$		−	15.5885i		−	0.915386i
$291$	14.0000			0.820695
$292$	5.00000			0.292603
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$			10.3923i			0.606092i
$295$			10.3923i			0.605063i
$296$	−18.0000			−1.04623
$297$		−	20.7846i		−	1.20605i
$298$	30.0000			1.73785
$299$		0			0
$300$	−8.00000			−0.461880
$301$	−4.00000			−0.230556
$302$	−9.00000			−0.517892
$303$			20.7846i			1.19404i
$304$		−	25.9808i		−	1.49010i
$305$	21.0000			1.20246
$306$	−6.00000			−0.342997
$307$	−22.0000			−1.25561			−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000			−1.25561			−0.627803	+	0.778372i	$0.716046\pi$
$308$		−	10.3923i		−	0.592157i
$309$			31.1769i			1.77359i
$310$			51.9615i			2.95122i
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$		−	20.7846i		−	1.17482i	−0.809291	−	0.587408i	$-0.800148\pi$
$313$		−	20.7846i		−	1.17482i	0.809291	−	0.587408i	$-0.199852\pi$
$314$		−	22.5167i		−	1.27069i
$315$	−6.00000			−0.338062
$316$			10.3923i			0.584613i
$317$			24.2487i			1.36194i	0.732310	+	0.680972i	$0.238442\pi$
$317$			24.2487i			1.36194i	−0.732310	+	0.680972i	$0.761558\pi$
$318$		0			0
$319$		−	15.5885i		−	0.872786i
$320$	3.00000			0.167705
$321$		−	3.46410i		−	0.193347i
$322$	6.00000			0.334367
$323$	18.0000			1.00155
$324$	11.0000			0.611111
$325$		0			0
$326$	9.00000			0.498464
$327$	−2.00000	−	20.7846i	−0.110600	−	1.14939i
$328$	6.00000			0.331295
$329$		−	24.2487i		−	1.33687i
$330$	−54.0000			−2.97260
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$	−12.0000			−0.658586
$333$		−	10.3923i		−	0.569495i
$334$	18.0000			0.984916
$335$		−	31.1769i		−	1.70338i
$336$	−20.0000			−1.09109
$337$		−	10.3923i		−	0.566105i	−0.959104	−	0.283052i	$-0.908653\pi$
$337$		−	10.3923i		−	0.566105i	0.959104	−	0.283052i	$-0.0913471\pi$
$338$		−	22.5167i		−	1.22474i
$339$	12.0000			0.651751
$340$		−	10.3923i		−	0.563602i
$341$			51.9615i			2.81387i
$342$	9.00000			0.486664
$343$	−20.0000			−1.07990
$344$			3.46410i			0.186772i
$345$		−	10.3923i		−	0.559503i
$346$		−	10.3923i		−	0.558694i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	6.00000			0.321634
$349$	−7.00000			−0.374701			−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000			−0.374701			−0.187351	+	0.982293i	$0.559990\pi$
$350$		−	13.8564i		−	0.740656i
$351$		0			0
$352$	−27.0000			−1.43910
$353$	3.00000			0.159674			0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000			0.159674			0.0798369	+	0.996808i	$0.474560\pi$
$354$	−12.0000			−0.637793
$355$	18.0000			0.955341
$356$	−15.0000			−0.794998
$357$		−	13.8564i		−	0.733359i
$358$	21.0000			1.10988
$359$			1.73205i			0.0914141i	0.998955	+	0.0457071i	$0.0145541\pi$
$359$			1.73205i			0.0914141i	−0.998955	+	0.0457071i	$0.985446\pi$
$360$			5.19615i			0.273861i
$361$	−8.00000			−0.421053
$362$		0			0
$363$	−32.0000			−1.67956
$364$		0			0
$365$	15.0000			0.785136
$366$			24.2487i			1.26750i
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$		−	8.66025i		−	0.451447i
$369$			3.46410i			0.180334i
$370$	54.0000			2.80733
$371$		0			0
$372$	−20.0000			−1.03695
$373$	1.00000			0.0517780			0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000			0.0517780			0.0258890	+	0.999665i	$0.491758\pi$
$374$		−	31.1769i		−	1.61212i
$375$	6.00000			0.309839
$376$	−21.0000			−1.08299
$377$		0			0
$378$			13.8564i			0.712697i
$379$			25.9808i			1.33454i	0.744815	+	0.667271i	$0.232538\pi$
$379$			25.9808i			1.33454i	−0.744815	+	0.667271i	$0.767462\pi$
$380$			15.5885i			0.799671i
$381$		−	10.3923i		−	0.532414i
$382$		0			0
$383$			8.66025i			0.442518i	0.975215	+	0.221259i	$0.0710167\pi$
$383$			8.66025i			0.442518i	−0.975215	+	0.221259i	$0.928983\pi$
$384$			24.2487i			1.23744i
$385$		−	31.1769i		−	1.58892i
$386$			1.73205i			0.0881591i
$387$	−2.00000			−0.101666
$388$	−7.00000			−0.355371
$389$			13.8564i			0.702548i	0.936273	+	0.351274i	$0.114251\pi$
$389$			13.8564i			0.702548i	−0.936273	+	0.351274i	$0.885749\pi$
$390$		0			0
$391$	6.00000			0.303433
$392$			5.19615i			0.262445i
$393$	24.0000			1.21064
$394$			25.9808i			1.30889i
$395$			31.1769i			1.56868i
$396$		−	5.19615i		−	0.261116i
$397$		−	10.3923i		−	0.521575i	−0.965396	−	0.260787i	$-0.916018\pi$
$397$		−	10.3923i		−	0.521575i	0.965396	−	0.260787i	$-0.0839822\pi$
$398$	18.0000			0.902258
$399$			20.7846i			1.04053i
$400$	−20.0000			−1.00000
$401$	27.0000			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$402$	36.0000			1.79552
$403$		0			0
$404$		−	10.3923i		−	0.517036i
$405$	33.0000			1.63978
$406$			10.3923i			0.515761i
$407$	54.0000			2.67668
$408$	−12.0000			−0.594089
$409$	−5.00000			−0.247234			−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000			−0.247234			−0.123617	+	0.992330i	$0.539449\pi$
$410$	−18.0000			−0.888957
$411$	−18.0000			−0.887875
$412$		−	15.5885i		−	0.767988i
$413$		−	6.92820i		−	0.340915i
$414$	3.00000			0.147442
$415$	−36.0000			−1.76717
$416$		0			0
$417$		−	20.7846i		−	1.01783i
$418$			46.7654i			2.28737i
$419$		−	17.3205i		−	0.846162i	−0.906092	−	0.423081i	$-0.860949\pi$
$419$		−	17.3205i		−	0.846162i	0.906092	−	0.423081i	$-0.139051\pi$
$420$	12.0000			0.585540
$421$	−7.00000			−0.341159			−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000			−0.341159			−0.170580	+	0.985344i	$0.554564\pi$
$422$			48.4974i			2.36082i
$423$		−	12.1244i		−	0.589506i
$424$		0			0
$425$		−	13.8564i		−	0.672134i
$426$			20.7846i			1.00702i
$427$	−14.0000			−0.677507
$428$			1.73205i			0.0837218i
$429$		0			0
$430$		−	10.3923i		−	0.501161i
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$	20.0000			0.962250
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		−	34.6410i		−	1.66282i
$435$	18.0000			0.863034
$436$	1.00000	+	10.3923i	0.0478913	+	0.497701i
$437$	−9.00000			−0.430528
$438$			17.3205i			0.827606i
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$	−27.0000			−1.28717
$441$	−3.00000			−0.142857
$442$		0			0
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$			20.7846i			0.986394i
$445$	−45.0000			−2.13320
$446$			3.46410i			0.164030i
$447$			34.6410i			1.63846i
$448$	−2.00000			−0.0944911
$449$		−	13.8564i		−	0.653924i	−0.945037	−	0.326962i	$-0.893975\pi$
$449$		−	13.8564i		−	0.653924i	0.945037	−	0.326962i	$-0.106025\pi$
$450$		−	6.92820i		−	0.326599i
$451$	−18.0000			−0.847587
$452$	−6.00000			−0.282216
$453$		−	10.3923i		−	0.488273i
$454$		−	31.1769i		−	1.46321i
$455$		0			0
$456$	18.0000			0.842927
$457$	−25.0000			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$458$	18.0000			0.841085
$459$			13.8564i			0.646762i
$460$			5.19615i			0.242272i
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$	36.0000			1.67487
$463$	22.0000			1.02243			0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000			1.02243			0.511213	+	0.859454i	$0.329196\pi$
$464$	15.0000			0.696358
$465$	−60.0000			−2.78243
$466$			36.3731i			1.68495i
$467$	−24.0000			−1.11059			−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		0			0
$469$			20.7846i			0.959744i
$470$	63.0000			2.90597
$471$	26.0000			1.19802
$472$	−6.00000			−0.276172
$473$		−	10.3923i		−	0.477839i
$474$	−36.0000			−1.65353
$475$			20.7846i			0.953663i
$476$			6.92820i			0.317554i
$477$		0			0
$478$		−	41.5692i		−	1.90133i
$479$	−18.0000			−0.822441			−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000			−0.822441			−0.411220	+	0.911536i	$0.634897\pi$
$480$		−	31.1769i		−	1.42302i
$481$		0			0
$482$	18.0000			0.819878
$483$			6.92820i			0.315244i
$484$	16.0000			0.727273
$485$	−21.0000			−0.953561
$486$			17.3205i			0.785674i
$487$			15.5885i			0.706380i	0.935552	+	0.353190i	$0.114903\pi$
$487$			15.5885i			0.706380i	−0.935552	+	0.353190i	$0.885097\pi$
$488$			12.1244i			0.548844i
$489$			10.3923i			0.469956i
$490$		−	15.5885i		−	0.704215i
$491$			10.3923i			0.468998i	0.972116	+	0.234499i	$0.0753450\pi$
$491$			10.3923i			0.468998i	−0.972116	+	0.234499i	$0.924655\pi$
$492$		−	6.92820i		−	0.312348i
$493$			10.3923i			0.468046i
$494$		0			0
$495$		−	15.5885i		−	0.700649i
$496$	−50.0000			−2.24507
$497$	−12.0000			−0.538274
$498$		−	41.5692i		−	1.86276i
$499$	−8.00000			−0.358129			−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000			−0.358129			−0.179065	+	0.983837i	$0.557307\pi$
$500$	−3.00000			−0.134164
$501$			20.7846i			0.928588i
$502$	−39.0000			−1.74066
$503$		−	31.1769i		−	1.39011i	−0.718957	−	0.695055i	$-0.755380\pi$
$503$		−	31.1769i		−	1.39011i	0.718957	−	0.695055i	$-0.244620\pi$
$504$		−	3.46410i		−	0.154303i
$505$		−	31.1769i		−	1.38735i
$506$			15.5885i			0.692991i
$507$	26.0000			1.15470
$508$			5.19615i			0.230542i
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$	36.0000			1.59411
$511$	−10.0000			−0.442374
$512$		−	8.66025i		−	0.382733i
$513$		−	20.7846i		−	0.917663i
$514$		0			0
$515$		−	46.7654i		−	2.06073i
$516$	4.00000			0.176090
$517$	63.0000			2.77074
$518$	−36.0000			−1.58175
$519$	12.0000			0.526742
$520$		0			0
$521$			3.46410i			0.151765i	0.997117	+	0.0758825i	$0.0241774\pi$
$521$			3.46410i			0.151765i	−0.997117	+	0.0758825i	$0.975823\pi$
$522$			5.19615i			0.227429i
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000			−0.524222
$525$	16.0000			0.698297
$526$			41.5692i			1.81250i
$527$		−	34.6410i		−	1.50899i
$528$		−	51.9615i		−	2.26134i
$529$	20.0000			0.869565
$530$		0			0
$531$		−	3.46410i		−	0.150329i
$532$		−	10.3923i		−	0.450564i
$533$		0			0
$534$		−	51.9615i		−	2.24860i
$535$			5.19615i			0.224649i
$536$	18.0000			0.777482
$537$			24.2487i			1.04641i
$538$	−18.0000			−0.776035
$539$		−	15.5885i		−	0.671442i
$540$	−12.0000			−0.516398
$541$	−11.0000			−0.472927			−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000			−0.472927			−0.236463	+	0.971640i	$0.575988\pi$
$542$	−45.0000			−1.93292
$543$		0			0
$544$	18.0000			0.771744
$545$	3.00000	+	31.1769i	0.128506	+	1.33547i
$546$		0			0
$547$		−	15.5885i		−	0.666514i	−0.942836	−	0.333257i	$-0.891852\pi$
$547$		−	15.5885i		−	0.666514i	0.942836	−	0.333257i	$-0.108148\pi$
$548$	9.00000			0.384461
$549$	−7.00000			−0.298753
$550$	36.0000			1.53505
$551$		−	15.5885i		−	0.664091i
$552$	6.00000			0.255377
$553$		−	20.7846i		−	0.883852i
$554$	−36.0000			−1.52949
$555$			62.3538i			2.64677i
$556$			10.3923i			0.440732i
$557$	6.00000			0.254228			0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000			0.254228			0.127114	+	0.991888i	$0.459429\pi$
$558$		−	17.3205i		−	0.733236i
$559$		0			0
$560$	30.0000			1.26773
$561$	36.0000			1.51992
$562$			15.5885i			0.657559i
$563$		−	15.5885i		−	0.656975i	−0.944508	−	0.328488i	$-0.893461\pi$
$563$		−	15.5885i		−	0.656975i	0.944508	−	0.328488i	$-0.106539\pi$
$564$			24.2487i			1.02105i
$565$	−18.0000			−0.757266
$566$	45.0000			1.89149
$567$	−22.0000			−0.923913
$568$			10.3923i			0.436051i
$569$		−	27.7128i		−	1.16178i	−0.813982	−	0.580891i	$-0.802704\pi$
$569$		−	27.7128i		−	1.16178i	0.813982	−	0.580891i	$-0.197296\pi$
$570$	−54.0000			−2.26181
$571$	10.0000			0.418487			0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000			0.418487			0.209243	+	0.977864i	$0.432900\pi$
$572$		0			0
$573$		0			0
$574$	12.0000			0.500870
$575$			6.92820i			0.288926i
$576$	−1.00000			−0.0416667
$577$			10.3923i			0.432637i	0.976323	+	0.216319i	$0.0694050\pi$
$577$			10.3923i			0.432637i	−0.976323	+	0.216319i	$0.930595\pi$
$578$		−	8.66025i		−	0.360219i
$579$	−2.00000			−0.0831172
$580$	−9.00000			−0.373705
$581$	24.0000			0.995688
$582$		−	24.2487i		−	1.00514i
$583$		0			0
$584$			8.66025i			0.358364i
$585$		0			0
$586$		−	15.5885i		−	0.643953i
$587$		−	15.5885i		−	0.643404i	−0.946841	−	0.321702i	$-0.895745\pi$
$587$		−	15.5885i		−	0.643404i	0.946841	−	0.321702i	$-0.104255\pi$
$588$	6.00000			0.247436
$589$			51.9615i			2.14104i
$590$	18.0000			0.741048
$591$	−30.0000			−1.23404
$592$			51.9615i			2.13561i
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$	−36.0000			−1.47710
$595$			20.7846i			0.852086i
$596$		−	17.3205i		−	0.709476i
$597$			20.7846i			0.850657i
$598$		0			0
$599$			24.2487i			0.990775i	0.868672	+	0.495388i	$0.164974\pi$
$599$			24.2487i			0.990775i	−0.868672	+	0.495388i	$0.835026\pi$
$600$		−	13.8564i		−	0.565685i
$601$			31.1769i			1.27173i	0.771799	+	0.635866i	$0.219357\pi$
$601$			31.1769i			1.27173i	−0.771799	+	0.635866i	$0.780643\pi$
$602$			6.92820i			0.282372i
$603$			10.3923i			0.423207i
$604$			5.19615i			0.211428i
$605$	48.0000			1.95148
$606$	36.0000			1.46240
$607$		−	25.9808i		−	1.05453i	−0.849702	−	0.527263i	$-0.823218\pi$
$607$		−	25.9808i		−	1.05453i	0.849702	−	0.527263i	$-0.176782\pi$
$608$	−27.0000			−1.09499
$609$	−12.0000			−0.486265
$610$		−	36.3731i		−	1.47270i
$611$		0			0
$612$			3.46410i			0.140028i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			38.1051i			1.53780i
$615$		−	20.7846i		−	0.838116i
$616$	18.0000			0.725241
$617$		−	41.5692i		−	1.67351i	−0.547575	−	0.836757i	$-0.684449\pi$
$617$		−	41.5692i		−	1.67351i	0.547575	−	0.836757i	$-0.315551\pi$
$618$	54.0000			2.17220
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$	30.0000			1.20483
$621$		−	6.92820i		−	0.278019i
$622$			31.1769i			1.25008i
$623$	30.0000			1.20192
$624$		0			0
$625$	−29.0000			−1.16000
$626$	−36.0000			−1.43885
$627$	−54.0000			−2.15655
$628$	−13.0000			−0.518756
$629$	−36.0000			−1.43541
$630$			10.3923i			0.414039i
$631$		−	5.19615i		−	0.206856i	−0.994637	−	0.103428i	$-0.967019\pi$
$631$		−	5.19615i		−	0.206856i	0.994637	−	0.103428i	$-0.0329811\pi$
$632$	−18.0000			−0.716002
$633$	−56.0000			−2.22580
$634$	42.0000			1.66803
$635$			15.5885i			0.618609i
$636$		0			0
$637$		0			0
$638$	−27.0000			−1.06894
$639$	−6.00000			−0.237356
$640$		−	36.3731i		−	1.43777i
$641$			31.1769i			1.23141i	0.787975	+	0.615707i	$0.211130\pi$
$641$			31.1769i			1.23141i	−0.787975	+	0.615707i	$0.788870\pi$
$642$	−6.00000			−0.236801
$643$		−	15.5885i		−	0.614749i	−0.951589	−	0.307374i	$-0.900550\pi$
$643$		−	15.5885i		−	0.614749i	0.951589	−	0.307374i	$-0.0994504\pi$
$644$		−	3.46410i		−	0.136505i
$645$	12.0000			0.472500
$646$		−	31.1769i		−	1.22664i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$			19.0526i			0.748455i
$649$	18.0000			0.706562
$650$		0			0
$651$	40.0000			1.56772
$652$		−	5.19615i		−	0.203497i
$653$	−15.0000			−0.586995			−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000			−0.586995			−0.293498	+	0.955960i	$0.594819\pi$
$654$	−36.0000	+	3.46410i	−1.40771	+	0.135457i
$655$	−36.0000			−1.40664
$656$		−	17.3205i		−	0.676252i
$657$	−5.00000			−0.195069
$658$	−42.0000			−1.63733
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$			31.1769i			1.21356i
$661$	−26.0000			−1.01128			−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000			−1.01128			−0.505641	+	0.862744i	$0.668744\pi$
$662$		−	13.8564i		−	0.538545i
$663$		0			0
$664$		−	20.7846i		−	0.806599i
$665$		−	31.1769i		−	1.20899i
$666$	−18.0000			−0.697486
$667$		−	5.19615i		−	0.201196i
$668$		−	10.3923i		−	0.402090i
$669$	−4.00000			−0.154649
$670$	−54.0000			−2.08620
$671$		−	36.3731i		−	1.40417i
$672$			20.7846i			0.801784i
$673$			31.1769i			1.20178i	0.799331	+	0.600891i	$0.205187\pi$
$673$			31.1769i			1.20178i	−0.799331	+	0.600891i	$0.794813\pi$
$674$	−18.0000			−0.693334
$675$	−16.0000			−0.615840
$676$	−13.0000			−0.500000
$677$			38.1051i			1.46450i	0.681037	+	0.732249i	$0.261529\pi$
$677$			38.1051i			1.46450i	−0.681037	+	0.732249i	$0.738471\pi$
$678$		−	20.7846i		−	0.798228i
$679$	14.0000			0.537271
$680$	18.0000			0.690268
$681$	36.0000			1.37952
$682$	90.0000			3.44628
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		−	5.19615i		−	0.198680i
$685$	27.0000			1.03162
$686$			34.6410i			1.32260i
$687$			20.7846i			0.792982i
$688$	10.0000			0.381246
$689$		0			0
$690$	−18.0000			−0.685248
$691$			5.19615i			0.197671i	0.995104	+	0.0988355i	$0.0315118\pi$
$691$			5.19615i			0.197671i	−0.995104	+	0.0988355i	$0.968488\pi$
$692$	−6.00000			−0.228086
$693$			10.3923i			0.394771i
$694$		0			0
$695$			31.1769i			1.18261i
$696$			10.3923i			0.393919i
$697$	12.0000			0.454532
$698$			12.1244i			0.458914i
$699$	−42.0000			−1.58859
$700$	−8.00000			−0.302372
$701$		−	13.8564i		−	0.523349i	−0.965156	−	0.261675i	$-0.915725\pi$
$701$		−	13.8564i		−	0.523349i	0.965156	−	0.261675i	$-0.0842747\pi$
$702$		0			0
$703$	54.0000			2.03665
$704$		−	5.19615i		−	0.195837i
$705$			72.7461i			2.73978i
$706$		−	5.19615i		−	0.195560i
$707$			20.7846i			0.781686i
$708$			6.92820i			0.260378i
$709$		−	20.7846i		−	0.780582i	−0.920691	−	0.390291i	$-0.872374\pi$
$709$		−	20.7846i		−	0.780582i	0.920691	−	0.390291i	$-0.127626\pi$
$710$		−	31.1769i		−	1.17005i
$711$		−	10.3923i		−	0.389742i
$712$		−	25.9808i		−	0.973670i
$713$			17.3205i			0.648658i
$714$	−24.0000			−0.898177
$715$		0			0
$716$		−	12.1244i		−	0.453108i
$717$	48.0000			1.79259
$718$	3.00000			0.111959
$719$		−	12.1244i		−	0.452162i	−0.974108	−	0.226081i	$-0.927409\pi$
$719$		−	12.1244i		−	0.452162i	0.974108	−	0.226081i	$-0.0725914\pi$
$720$	15.0000			0.559017
$721$			31.1769i			1.16109i
$722$			13.8564i			0.515682i
$723$			20.7846i			0.772988i
$724$		0			0
$725$	−12.0000			−0.445669
$726$			55.4256i			2.05704i
$727$	44.0000			1.63187			0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000			1.63187			0.815935	+	0.578144i	$0.196223\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		−	25.9808i		−	0.961591i
$731$			6.92820i			0.256249i
$732$	14.0000			0.517455
$733$			31.1769i			1.15155i	0.817610	+	0.575773i	$0.195299\pi$
$733$			31.1769i			1.15155i	−0.817610	+	0.575773i	$0.804701\pi$
$734$	−18.0000			−0.664392
$735$	18.0000			0.663940
$736$	−9.00000			−0.331744
$737$	−54.0000			−1.98912
$738$	6.00000			0.220863
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$		−	31.1769i		−	1.14609i
$741$		0			0
$742$		0			0
$743$	−24.0000			−0.880475			−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000			−0.880475			−0.440237	+	0.897881i	$0.645106\pi$
$744$		−	34.6410i		−	1.27000i
$745$		−	51.9615i		−	1.90372i
$746$		−	1.73205i		−	0.0634149i
$747$	12.0000			0.439057
$748$	−18.0000			−0.658145
$749$		−	3.46410i		−	0.126576i
$750$		−	10.3923i		−	0.379473i
$751$	−10.0000			−0.364905			−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000			−0.364905			−0.182453	+	0.983215i	$0.558404\pi$
$752$			60.6218i			2.21065i
$753$		−	45.0333i		−	1.64111i
$754$		0			0
$755$			15.5885i			0.567322i
$756$	8.00000			0.290957
$757$			31.1769i			1.13314i	0.824012	+	0.566572i	$0.191731\pi$
$757$			31.1769i			1.13314i	−0.824012	+	0.566572i	$0.808269\pi$
$758$	45.0000			1.63447
$759$	−18.0000			−0.653359
$760$	−27.0000			−0.979393
$761$			31.1769i			1.13016i	0.825035	+	0.565081i	$0.191155\pi$
$761$			31.1769i			1.13016i	−0.825035	+	0.565081i	$0.808845\pi$
$762$	−18.0000			−0.652071
$763$	−2.00000	−	20.7846i	−0.0724049	−	0.752453i
$764$		0			0
$765$			10.3923i			0.375735i
$766$	15.0000			0.541972
$767$		0			0
$768$	38.0000			1.37121
$769$			10.3923i			0.374756i	0.982288	+	0.187378i	$0.0599989\pi$
$769$			10.3923i			0.374756i	−0.982288	+	0.187378i	$0.940001\pi$
$770$	−54.0000			−1.94602
$771$		0			0
$772$	1.00000			0.0359908
$773$		−	34.6410i		−	1.24595i	−0.782241	−	0.622975i	$-0.785924\pi$
$773$		−	34.6410i		−	1.24595i	0.782241	−	0.622975i	$-0.214076\pi$
$774$			3.46410i			0.124515i
$775$	40.0000			1.43684
$776$		−	12.1244i		−	0.435239i
$777$		−	41.5692i		−	1.49129i
$778$	24.0000			0.860442
$779$	−18.0000			−0.644917
$780$		0			0
$781$		−	31.1769i		−	1.11560i
$782$		−	10.3923i		−	0.371628i
$783$	12.0000			0.428845
$784$	15.0000			0.535714
$785$	−39.0000			−1.39197
$786$		−	41.5692i		−	1.48272i
$787$			25.9808i			0.926114i	0.886328	+	0.463057i	$0.153248\pi$
$787$			25.9808i			0.926114i	−0.886328	+	0.463057i	$0.846752\pi$
$788$	15.0000			0.534353
$789$	−48.0000			−1.70885
$790$	54.0000			1.92123
$791$	12.0000			0.426671
$792$	9.00000			0.319801
$793$		0			0
$794$	−18.0000			−0.638796
$795$		0			0
$796$		−	10.3923i		−	0.368345i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	36.0000			1.27439
$799$	−42.0000			−1.48585
$800$			20.7846i			0.734847i
$801$	15.0000			0.529999
$802$		−	46.7654i		−	1.65134i
$803$		−	25.9808i		−	0.916841i
$804$		−	20.7846i		−	0.733017i
$805$		−	10.3923i		−	0.366281i
$806$		0			0
$807$		−	20.7846i		−	0.731653i
$808$	18.0000			0.633238
$809$	51.0000			1.79306			0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000			1.79306			0.896532	+	0.442978i	$0.146078\pi$
$810$		−	57.1577i		−	2.00832i
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$	6.00000			0.210559
$813$		−	51.9615i		−	1.82237i
$814$		−	93.5307i		−	3.27825i
$815$		−	15.5885i		−	0.546040i
$816$			34.6410i			1.21268i
$817$		−	10.3923i		−	0.363581i
$818$			8.66025i			0.302799i
$819$		0			0
$820$			10.3923i			0.362915i
$821$			3.46410i			0.120898i	0.998171	+	0.0604490i	$0.0192532\pi$
$821$			3.46410i			0.120898i	−0.998171	+	0.0604490i	$0.980747\pi$
$822$			31.1769i			1.08742i
$823$	8.00000			0.278862			0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000			0.278862			0.139431	+	0.990232i	$0.455473\pi$
$824$	27.0000			0.940590
$825$			41.5692i			1.44725i
$826$	−12.0000			−0.417533
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$		−	1.73205i		−	0.0601929i
$829$	−31.0000			−1.07667			−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000			−1.07667			−0.538337	+	0.842729i	$0.680947\pi$
$830$			62.3538i			2.16433i
$831$		−	41.5692i		−	1.44202i
$832$		0			0
$833$			10.3923i			0.360072i
$834$	−36.0000			−1.24658
$835$		−	31.1769i		−	1.07892i
$836$	27.0000			0.933815
$837$	−40.0000			−1.38260
$838$	−30.0000			−1.03633
$839$		−	5.19615i		−	0.179391i	−0.995969	−	0.0896956i	$-0.971411\pi$
$839$		−	5.19615i		−	0.179391i	0.995969	−	0.0896956i	$-0.0285894\pi$
$840$			20.7846i			0.717137i
$841$	−20.0000			−0.689655
$842$			12.1244i			0.417833i
$843$	−18.0000			−0.619953
$844$	28.0000			0.963800
$845$	−39.0000			−1.34164
$846$	−21.0000			−0.721995
$847$	−32.0000			−1.09953
$848$		0			0
$849$			51.9615i			1.78331i
$850$	−24.0000			−0.823193
$851$	18.0000			0.617032
$852$	12.0000			0.411113
$853$		−	20.7846i		−	0.711651i	−0.934552	−	0.355826i	$-0.884200\pi$
$853$		−	20.7846i		−	0.711651i	0.934552	−	0.355826i	$-0.115800\pi$
$854$			24.2487i			0.829774i
$855$		−	15.5885i		−	0.533114i
$856$	−3.00000			−0.102538
$857$	−33.0000			−1.12726			−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000			−1.12726			−0.563629	+	0.826028i	$0.690595\pi$
$858$		0			0
$859$			46.7654i			1.59561i	0.602913	+	0.797807i	$0.294007\pi$
$859$			46.7654i			1.59561i	−0.602913	+	0.797807i	$0.705993\pi$
$860$	−6.00000			−0.204598
$861$			13.8564i			0.472225i
$862$			10.3923i			0.353963i
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		−	20.7846i		−	0.707107i
$865$	−18.0000			−0.612018
$866$		−	24.2487i		−	0.824005i
$867$	10.0000			0.339618
$868$	−20.0000			−0.678844
$869$	54.0000			1.83182
$870$		−	31.1769i		−	1.05700i
$871$		0			0
$872$	−18.0000	+	1.73205i	−0.609557	+	0.0586546i
$873$	7.00000			0.236914
$874$			15.5885i			0.527287i
$875$	6.00000			0.202837
$876$	10.0000			0.337869
$877$	−14.0000			−0.472746			−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000			−0.472746			−0.236373	+	0.971662i	$0.575959\pi$
$878$			13.8564i			0.467631i
$879$	18.0000			0.607125
$880$			77.9423i			2.62743i
$881$	3.00000			0.101073			0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000			0.101073			0.0505363	+	0.998722i	$0.483907\pi$
$882$			5.19615i			0.174964i
$883$		−	36.3731i		−	1.22405i	−0.790838	−	0.612026i	$-0.790355\pi$
$883$		−	36.3731i		−	1.22405i	0.790838	−	0.612026i	$-0.209645\pi$
$884$		0			0
$885$			20.7846i			0.698667i
$886$		−	10.3923i		−	0.349136i
$887$	−30.0000			−1.00730			−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000			−1.00730			−0.503651	+	0.863907i	$0.668010\pi$
$888$	−36.0000			−1.20808
$889$		−	10.3923i		−	0.348547i
$890$			77.9423i			2.61263i
$891$		−	57.1577i		−	1.91485i
$892$	2.00000			0.0669650
$893$	63.0000			2.10821
$894$	60.0000			2.00670
$895$		−	36.3731i		−	1.21582i
$896$			24.2487i			0.810093i
$897$		0			0
$898$	−24.0000			−0.800890
$899$	−30.0000			−1.00056
$900$	−4.00000			−0.133333
$901$		0			0
$902$			31.1769i			1.03808i
$903$	−8.00000			−0.266223
$904$		−	10.3923i		−	0.345643i
$905$		0			0
$906$	−18.0000			−0.598010
$907$	2.00000			0.0664089			0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000			0.0664089			0.0332045	+	0.999449i	$0.489429\pi$
$908$	−18.0000			−0.597351
$909$			10.3923i			0.344691i
$910$		0			0
$911$		−	45.0333i		−	1.49202i	−0.665934	−	0.746010i	$-0.731967\pi$
$911$		−	45.0333i		−	1.49202i	0.665934	−	0.746010i	$-0.268033\pi$
$912$		−	51.9615i		−	1.72062i
$913$			62.3538i			2.06361i
$914$			43.3013i			1.43228i
$915$	42.0000			1.38848
$916$		−	10.3923i		−	0.343371i
$917$	24.0000			0.792550
$918$	24.0000			0.792118
$919$			36.3731i			1.19984i	0.800061	+	0.599918i	$0.204800\pi$
$919$			36.3731i			1.19984i	−0.800061	+	0.599918i	$0.795200\pi$
$920$	−9.00000			−0.296721
$921$	−44.0000			−1.44985
$922$			51.9615i			1.71126i
$923$		0			0
$924$		−	20.7846i		−	0.683763i
$925$		−	41.5692i		−	1.36679i
$926$		−	38.1051i		−	1.25221i
$927$			15.5885i			0.511992i
$928$		−	15.5885i		−	0.511716i
$929$		−	48.4974i		−	1.59115i	−0.605856	−	0.795574i	$-0.707169\pi$
$929$		−	48.4974i		−	1.59115i	0.605856	−	0.795574i	$-0.292831\pi$
$930$			103.923i			3.40777i
$931$		−	15.5885i		−	0.510891i
$932$	21.0000			0.687878
$933$	−36.0000			−1.17859
$934$			41.5692i			1.36019i
$935$	−54.0000			−1.76599
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	36.0000			1.17544
$939$		−	41.5692i		−	1.35656i
$940$		−	36.3731i		−	1.18636i
$941$			38.1051i			1.24219i	0.783735	+	0.621096i	$0.213312\pi$
$941$			38.1051i			1.24219i	−0.783735	+	0.621096i	$0.786688\pi$
$942$		−	45.0333i		−	1.46726i
$943$	−6.00000			−0.195387
$944$			17.3205i			0.563735i
$945$	24.0000			0.780720
$946$	−18.0000			−0.585230
$947$	36.0000			1.16984			0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000			1.16984			0.584921	+	0.811090i	$0.301125\pi$
$948$			20.7846i			0.675053i
$949$		0			0
$950$	36.0000			1.16799
$951$			48.4974i			1.57264i
$952$	−12.0000			−0.388922
$953$	3.00000			0.0971795			0.0485898	−	0.998819i	$-0.484527\pi$
$953$	3.00000			0.0971795			0.0485898	+	0.998819i	$0.484527\pi$
$954$		0			0
$955$		0			0
$956$	−24.0000			−0.776215
$957$		−	31.1769i		−	1.00781i
$958$			31.1769i			1.00728i
$959$	−18.0000			−0.581250
$960$	6.00000			0.193649
$961$	69.0000			2.22581
$962$		0			0
$963$		−	1.73205i		−	0.0558146i
$964$		−	10.3923i		−	0.334714i
$965$	3.00000			0.0965734
$966$	12.0000			0.386094
$967$		−	5.19615i		−	0.167097i	−0.996504	−	0.0835485i	$-0.973375\pi$
$967$		−	5.19615i		−	0.167097i	0.996504	−	0.0835485i	$-0.0266254\pi$
$968$			27.7128i			0.890724i
$969$	36.0000			1.15649
$970$			36.3731i			1.16787i
$971$		−	45.0333i		−	1.44519i	−0.691273	−	0.722594i	$-0.742950\pi$
$971$		−	45.0333i		−	1.44519i	0.691273	−	0.722594i	$-0.257050\pi$
$972$	10.0000			0.320750
$973$		−	20.7846i		−	0.666324i
$974$	27.0000			0.865136
$975$		0			0
$976$	35.0000			1.12032
$977$	3.00000			0.0959785			0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000			0.0959785			0.0479893	+	0.998848i	$0.484719\pi$
$978$	18.0000			0.575577
$979$			77.9423i			2.49105i
$980$	−9.00000			−0.287494
$981$	−1.00000	−	10.3923i	−0.0319275	−	0.331801i
$982$	18.0000			0.574403
$983$			22.5167i			0.718170i	0.933305	+	0.359085i	$0.116911\pi$
$983$			22.5167i			0.718170i	−0.933305	+	0.359085i	$0.883089\pi$
$984$	12.0000			0.382546
$985$	45.0000			1.43382
$986$	18.0000			0.573237
$987$		−	48.4974i		−	1.54369i
$988$		0			0
$989$		−	3.46410i		−	0.110152i
$990$	−27.0000			−0.858116
$991$			15.5885i			0.495184i	0.968864	+	0.247592i	$0.0796392\pi$
$991$			15.5885i			0.495184i	−0.968864	+	0.247592i	$0.920361\pi$
$992$			51.9615i			1.64978i
$993$	16.0000			0.507745
$994$			20.7846i			0.659248i
$995$		−	31.1769i		−	0.988375i
$996$	−24.0000			−0.760469
$997$	−38.0000			−1.20347			−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000			−1.20347			−0.601736	+	0.798695i	$0.705524\pi$
$998$			13.8564i			0.438617i
$999$			41.5692i			1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.b.a.108.1	✓	2
3.2	odd	2		981.2.c.a.217.2		2
4.3	odd	2		1744.2.h.a.1089.1		2
109.108	even	2	inner	109.2.b.a.108.2	yes	2
327.326	odd	2		981.2.c.a.217.1		2
436.435	odd	2		1744.2.h.a.1089.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.b.a.108.1	✓	2	1.1	even	1	trivial
109.2.b.a.108.2	yes	2	109.108	even	2	inner
981.2.c.a.217.1		2	327.326	odd	2
981.2.c.a.217.2		2	3.2	odd	2
1744.2.h.a.1089.1		2	4.3	odd	2
1744.2.h.a.1089.2		2	436.435	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:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	109.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} -3.46410i q^{6} +2.00000 q^{7} -1.73205i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} -3.46410i q^{6} +2.00000 q^{7} -1.73205i q^{8} +1.00000 q^{9} +5.19615i q^{10} +5.19615i q^{11} -2.00000 q^{12} -3.46410i q^{14} -6.00000 q^{15} -5.00000 q^{16} -3.46410i q^{17} -1.73205i q^{18} +5.19615i q^{19} +3.00000 q^{20} +4.00000 q^{21} +9.00000 q^{22} +1.73205i q^{23} -3.46410i q^{24} +4.00000 q^{25} -4.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +10.3923i q^{30} +10.0000 q^{31} +5.19615i q^{32} +10.3923i q^{33} -6.00000 q^{34} -6.00000 q^{35} -1.00000 q^{36} -10.3923i q^{37} +9.00000 q^{38} +5.19615i q^{40} +3.46410i q^{41} -6.92820i q^{42} -2.00000 q^{43} -5.19615i q^{44} -3.00000 q^{45} +3.00000 q^{46} -12.1244i q^{47} -10.0000 q^{48} -3.00000 q^{49} -6.92820i q^{50} -6.92820i q^{51} +6.92820i q^{54} -15.5885i q^{55} -3.46410i q^{56} +10.3923i q^{57} +5.19615i q^{58} -3.46410i q^{59} +6.00000 q^{60} -7.00000 q^{61} -17.3205i q^{62} +2.00000 q^{63} -1.00000 q^{64} +18.0000 q^{66} +10.3923i q^{67} +3.46410i q^{68} +3.46410i q^{69} +10.3923i q^{70} -6.00000 q^{71} -1.73205i q^{72} -5.00000 q^{73} -18.0000 q^{74} +8.00000 q^{75} -5.19615i q^{76} +10.3923i q^{77} -10.3923i q^{79} +15.0000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +10.3923i q^{85} +3.46410i q^{86} -6.00000 q^{87} +9.00000 q^{88} +15.0000 q^{89} +5.19615i q^{90} -1.73205i q^{92} +20.0000 q^{93} -21.0000 q^{94} -15.5885i q^{95} +10.3923i q^{96} +7.00000 q^{97} +5.19615i q^{98} +5.19615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{4} - 6 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 2 * q^4 - 6 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 4 q^{3} - 2 q^{4} - 6 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{12} - 12 q^{15} - 10 q^{16} + 6 q^{20} + 8 q^{21} + 18 q^{22} + 8 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{29} + 20 q^{31} - 12 q^{34} - 12 q^{35} - 2 q^{36} + 18 q^{38} - 4 q^{43} - 6 q^{45} + 6 q^{46} - 20 q^{48} - 6 q^{49} + 12 q^{60} - 14 q^{61} + 4 q^{63} - 2 q^{64} + 36 q^{66} - 12 q^{71} - 10 q^{73} - 36 q^{74} + 16 q^{75} + 30 q^{80} - 22 q^{81} + 12 q^{82} + 24 q^{83} - 8 q^{84} - 12 q^{87} + 18 q^{88} + 30 q^{89} + 40 q^{93} - 42 q^{94} + 14 q^{97}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^4 - 6 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^12 - 12 * q^15 - 10 * q^16 + 6 * q^20 + 8 * q^21 + 18 * q^22 + 8 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^29 + 20 * q^31 - 12 * q^34 - 12 * q^35 - 2 * q^36 + 18 * q^38 - 4 * q^43 - 6 * q^45 + 6 * q^46 - 20 * q^48 - 6 * q^49 + 12 * q^60 - 14 * q^61 + 4 * q^63 - 2 * q^64 + 36 * q^66 - 12 * q^71 - 10 * q^73 - 36 * q^74 + 16 * q^75 + 30 * q^80 - 22 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 12 * q^87 + 18 * q^88 + 30 * q^89 + 40 * q^93 - 42 * q^94 + 14 * q^97

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