LMFDB - Embedded newform 1110.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.a (trivial)

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:	yes
Analytic conductor:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.81507$ of defining polynomial
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -3.81507 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.81507 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +5.81507 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.81507 q^{26} +1.00000 q^{27} +3.00000 q^{28} -6.81507 q^{29} -1.00000 q^{30} +6.81507 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -7.81507 q^{38} -3.81507 q^{39} -1.00000 q^{40} -4.81507 q^{41} -3.00000 q^{42} -4.81507 q^{43} -1.00000 q^{44} +1.00000 q^{45} -5.81507 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -3.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +7.81507 q^{57} +6.81507 q^{58} +2.00000 q^{59} +1.00000 q^{60} +6.81507 q^{61} -6.81507 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.81507 q^{65} +1.00000 q^{66} +7.63015 q^{67} +1.00000 q^{68} +5.81507 q^{69} -3.00000 q^{70} -11.6301 q^{71} -1.00000 q^{72} -5.81507 q^{73} -1.00000 q^{74} +1.00000 q^{75} +7.81507 q^{76} -3.00000 q^{77} +3.81507 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.81507 q^{82} -3.81507 q^{83} +3.00000 q^{84} +1.00000 q^{85} +4.81507 q^{86} -6.81507 q^{87} +1.00000 q^{88} +11.8151 q^{89} -1.00000 q^{90} -11.4452 q^{91} +5.81507 q^{92} +6.81507 q^{93} +8.00000 q^{94} +7.81507 q^{95} -1.00000 q^{96} -4.81507 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 3 q^{13} - 6 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{20} + 6 q^{21} + 2 q^{22} + q^{23} - 2 q^{24} + 2 q^{25} - 3 q^{26} + 2 q^{27} + 6 q^{28} - 3 q^{29} - 2 q^{30} + 3 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + 6 q^{35} + 2 q^{36} + 2 q^{37} - 5 q^{38} + 3 q^{39} - 2 q^{40} + q^{41} - 6 q^{42} + q^{43} - 2 q^{44} + 2 q^{45} - q^{46} - 16 q^{47} + 2 q^{48} + 4 q^{49} - 2 q^{50} + 2 q^{51} + 3 q^{52} - 2 q^{54} - 2 q^{55} - 6 q^{56} + 5 q^{57} + 3 q^{58} + 4 q^{59} + 2 q^{60} + 3 q^{61} - 3 q^{62} + 6 q^{63} + 2 q^{64} + 3 q^{65} + 2 q^{66} - 6 q^{67} + 2 q^{68} + q^{69} - 6 q^{70} - 2 q^{71} - 2 q^{72} - q^{73} - 2 q^{74} + 2 q^{75} + 5 q^{76} - 6 q^{77} - 3 q^{78} + 24 q^{79} + 2 q^{80} + 2 q^{81} - q^{82} + 3 q^{83} + 6 q^{84} + 2 q^{85} - q^{86} - 3 q^{87} + 2 q^{88} + 13 q^{89} - 2 q^{90} + 9 q^{91} + q^{92} + 3 q^{93} + 16 q^{94} + 5 q^{95} - 2 q^{96} + q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 3 * q^13 - 6 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^20 + 6 * q^21 + 2 * q^22 + q^23 - 2 * q^24 + 2 * q^25 - 3 * q^26 + 2 * q^27 + 6 * q^28 - 3 * q^29 - 2 * q^30 + 3 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + 6 * q^35 + 2 * q^36 + 2 * q^37 - 5 * q^38 + 3 * q^39 - 2 * q^40 + q^41 - 6 * q^42 + q^43 - 2 * q^44 + 2 * q^45 - q^46 - 16 * q^47 + 2 * q^48 + 4 * q^49 - 2 * q^50 + 2 * q^51 + 3 * q^52 - 2 * q^54 - 2 * q^55 - 6 * q^56 + 5 * q^57 + 3 * q^58 + 4 * q^59 + 2 * q^60 + 3 * q^61 - 3 * q^62 + 6 * q^63 + 2 * q^64 + 3 * q^65 + 2 * q^66 - 6 * q^67 + 2 * q^68 + q^69 - 6 * q^70 - 2 * q^71 - 2 * q^72 - q^73 - 2 * q^74 + 2 * q^75 + 5 * q^76 - 6 * q^77 - 3 * q^78 + 24 * q^79 + 2 * q^80 + 2 * q^81 - q^82 + 3 * q^83 + 6 * q^84 + 2 * q^85 - q^86 - 3 * q^87 + 2 * q^88 + 13 * q^89 - 2 * q^90 + 9 * q^91 + q^92 + 3 * q^93 + 16 * q^94 + 5 * q^95 - 2 * q^96 + q^97 - 4 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	−3.81507	−1.05811	−0.529055	−	0.848587i	$-0.677454\pi$
$13$	−3.81507	−1.05811	−0.529055	+	0.848587i	$0.677454\pi$
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	7.81507	1.79290	0.896450	−	0.443144i	$-0.146137\pi$
$19$	7.81507	1.79290	0.896450	+	0.443144i	$0.146137\pi$
$20$	1.00000	0.223607
$21$	3.00000	0.654654
$22$	1.00000	0.213201
$23$	5.81507	1.21253	0.606263	−	0.795264i	$-0.292668\pi$
$23$	5.81507	1.21253	0.606263	+	0.795264i	$0.292668\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	3.81507	0.748197
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	−6.81507	−1.26553	−0.632764	−	0.774345i	$-0.718080\pi$
$29$	−6.81507	−1.26553	−0.632764	+	0.774345i	$0.718080\pi$
$30$	−1.00000	−0.182574
$31$	6.81507	1.22402	0.612012	−	0.790849i	$-0.290361\pi$
$31$	6.81507	1.22402	0.612012	+	0.790849i	$0.290361\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−7.81507	−1.26777
$39$	−3.81507	−0.610901
$40$	−1.00000	−0.158114
$41$	−4.81507	−0.751988	−0.375994	−	0.926622i	$-0.622699\pi$
$41$	−4.81507	−0.751988	−0.375994	+	0.926622i	$0.622699\pi$
$42$	−3.00000	−0.462910
$43$	−4.81507	−0.734292	−0.367146	−	0.930163i	$-0.619665\pi$
$43$	−4.81507	−0.734292	−0.367146	+	0.930163i	$0.619665\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−5.81507	−0.857386
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	−3.81507	−0.529055
$53$	10.6301	1.46016	0.730081	−	0.683360i	$-0.239482\pi$
$53$	10.6301	1.46016	0.730081	+	0.683360i	$0.239482\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	−3.00000	−0.400892
$57$	7.81507	1.03513
$58$	6.81507	0.894863
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	1.00000	0.129099
$61$	6.81507	0.872581	0.436290	−	0.899806i	$-0.356292\pi$
$61$	6.81507	0.872581	0.436290	+	0.899806i	$0.356292\pi$
$62$	−6.81507	−0.865515
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−3.81507	−0.473202
$66$	1.00000	0.123091
$67$	7.63015	0.932171	0.466085	−	0.884740i	$-0.345664\pi$
$67$	7.63015	0.932171	0.466085	+	0.884740i	$0.345664\pi$
$68$	1.00000	0.121268
$69$	5.81507	0.700053
$70$	−3.00000	−0.358569
$71$	−11.6301	−1.38024	−0.690122	−	0.723693i	$-0.742443\pi$
$71$	−11.6301	−1.38024	−0.690122	+	0.723693i	$0.742443\pi$
$72$	−1.00000	−0.117851
$73$	−5.81507	−0.680603	−0.340301	−	0.940316i	$-0.610529\pi$
$73$	−5.81507	−0.680603	−0.340301	+	0.940316i	$0.610529\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	7.81507	0.896450
$77$	−3.00000	−0.341882
$78$	3.81507	0.431972
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	4.81507	0.531736
$83$	−3.81507	−0.418759	−0.209379	−	0.977834i	$-0.567144\pi$
$83$	−3.81507	−0.418759	−0.209379	+	0.977834i	$0.567144\pi$
$84$	3.00000	0.327327
$85$	1.00000	0.108465
$86$	4.81507	0.519223
$87$	−6.81507	−0.730653
$88$	1.00000	0.106600
$89$	11.8151	1.25240	0.626198	−	0.779664i	$-0.284610\pi$
$89$	11.8151	1.25240	0.626198	+	0.779664i	$0.284610\pi$
$90$	−1.00000	−0.105409
$91$	−11.4452	−1.19978
$92$	5.81507	0.606263
$93$	6.81507	0.706690
$94$	8.00000	0.825137
$95$	7.81507	0.801810
$96$	−1.00000	−0.102062
$97$	−4.81507	−0.488897	−0.244448	−	0.969662i	$-0.578607\pi$
$97$	−4.81507	−0.488897	−0.244448	+	0.969662i	$0.578607\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	11.6301	1.15724	0.578621	−	0.815596i	$-0.303591\pi$
$101$	11.6301	1.15724	0.578621	+	0.815596i	$0.303591\pi$
$102$	−1.00000	−0.0990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	3.81507	0.374099
$105$	3.00000	0.292770
$106$	−10.6301	−1.03249
$107$	−8.18493	−0.791267	−0.395633	−	0.918409i	$-0.629475\pi$
$107$	−8.18493	−0.791267	−0.395633	+	0.918409i	$0.629475\pi$
$108$	1.00000	0.0962250
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	1.00000	0.0953463
$111$	1.00000	0.0949158
$112$	3.00000	0.283473
$113$	−6.81507	−0.641108	−0.320554	−	0.947230i	$-0.603869\pi$
$113$	−6.81507	−0.641108	−0.320554	+	0.947230i	$0.603869\pi$
$114$	−7.81507	−0.731949
$115$	5.81507	0.542258
$116$	−6.81507	−0.632764
$117$	−3.81507	−0.352704
$118$	−2.00000	−0.184115
$119$	3.00000	0.275010
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	−6.81507	−0.617008
$123$	−4.81507	−0.434161
$124$	6.81507	0.612012
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	−21.4452	−1.90296	−0.951478	−	0.307718i	$-0.900435\pi$
$127$	−21.4452	−1.90296	−0.951478	+	0.307718i	$0.900435\pi$
$128$	−1.00000	−0.0883883
$129$	−4.81507	−0.423944
$130$	3.81507	0.334604
$131$	−9.63015	−0.841390	−0.420695	−	0.907202i	$-0.638214\pi$
$131$	−9.63015	−0.841390	−0.420695	+	0.907202i	$0.638214\pi$
$132$	−1.00000	−0.0870388
$133$	23.4452	2.03296
$134$	−7.63015	−0.659144
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−5.81507	−0.495012
$139$	22.8151	1.93515	0.967575	−	0.252585i	$-0.0812809\pi$
$139$	22.8151	1.93515	0.967575	+	0.252585i	$0.0812809\pi$
$140$	3.00000	0.253546
$141$	−8.00000	−0.673722
$142$	11.6301	0.975980
$143$	3.81507	0.319032
$144$	1.00000	0.0833333
$145$	−6.81507	−0.565961
$146$	5.81507	0.481259
$147$	2.00000	0.164957
$148$	1.00000	0.0821995
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−1.00000	−0.0816497
$151$	3.81507	0.310466	0.155233	−	0.987878i	$-0.450387\pi$
$151$	3.81507	0.310466	0.155233	+	0.987878i	$0.450387\pi$
$152$	−7.81507	−0.633886
$153$	1.00000	0.0808452
$154$	3.00000	0.241747
$155$	6.81507	0.547400
$156$	−3.81507	−0.305450
$157$	−12.8151	−1.02275	−0.511377	−	0.859356i	$-0.670864\pi$
$157$	−12.8151	−1.02275	−0.511377	+	0.859356i	$0.670864\pi$
$158$	−12.0000	−0.954669
$159$	10.6301	0.843025
$160$	−1.00000	−0.0790569
$161$	17.4452	1.37488
$162$	−1.00000	−0.0785674
$163$	16.6301	1.30257	0.651287	−	0.758832i	$-0.274230\pi$
$163$	16.6301	1.30257	0.651287	+	0.758832i	$0.274230\pi$
$164$	−4.81507	−0.375994
$165$	−1.00000	−0.0778499
$166$	3.81507	0.296107
$167$	−21.8151	−1.68810	−0.844051	−	0.536264i	$-0.819835\pi$
$167$	−21.8151	−1.68810	−0.844051	+	0.536264i	$0.819835\pi$
$168$	−3.00000	−0.231455
$169$	1.55478	0.119599
$170$	−1.00000	−0.0766965
$171$	7.81507	0.597634
$172$	−4.81507	−0.367146
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	6.81507	0.516649
$175$	3.00000	0.226779
$176$	−1.00000	−0.0753778
$177$	2.00000	0.150329
$178$	−11.8151	−0.885577
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	1.00000	0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	11.4452	0.848376
$183$	6.81507	0.503785
$184$	−5.81507	−0.428693
$185$	1.00000	0.0735215
$186$	−6.81507	−0.499705
$187$	−1.00000	−0.0731272
$188$	−8.00000	−0.583460
$189$	3.00000	0.218218
$190$	−7.81507	−0.566965
$191$	4.63015	0.335026	0.167513	−	0.985870i	$-0.446426\pi$
$191$	4.63015	0.335026	0.167513	+	0.985870i	$0.446426\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	4.81507	0.345702
$195$	−3.81507	−0.273203
$196$	2.00000	0.142857
$197$	−4.18493	−0.298164	−0.149082	−	0.988825i	$-0.547632\pi$
$197$	−4.18493	−0.298164	−0.149082	+	0.988825i	$0.547632\pi$
$198$	1.00000	0.0710669
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	7.63015	0.538189
$202$	−11.6301	−0.818294
$203$	−20.4452	−1.43497
$204$	1.00000	0.0700140
$205$	−4.81507	−0.336299
$206$	0	0
$207$	5.81507	0.404176
$208$	−3.81507	−0.264528
$209$	−7.81507	−0.540580
$210$	−3.00000	−0.207020
$211$	8.44522	0.581393	0.290696	−	0.956815i	$-0.406113\pi$
$211$	8.44522	0.581393	0.290696	+	0.956815i	$0.406113\pi$
$212$	10.6301	0.730081
$213$	−11.6301	−0.796884
$214$	8.18493	0.559510
$215$	−4.81507	−0.328385
$216$	−1.00000	−0.0680414
$217$	20.4452	1.38791
$218$	−13.0000	−0.880471
$219$	−5.81507	−0.392946
$220$	−1.00000	−0.0674200
$221$	−3.81507	−0.256630
$222$	−1.00000	−0.0671156
$223$	−16.8151	−1.12602	−0.563010	−	0.826450i	$-0.690357\pi$
$223$	−16.8151	−1.12602	−0.563010	+	0.826450i	$0.690357\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	6.81507	0.453332
$227$	−24.8151	−1.64703	−0.823517	−	0.567291i	$-0.807991\pi$
$227$	−24.8151	−1.64703	−0.823517	+	0.567291i	$0.807991\pi$
$228$	7.81507	0.517566
$229$	0.369854	0.0244407	0.0122203	−	0.999925i	$-0.496110\pi$
$229$	0.369854	0.0244407	0.0122203	+	0.999925i	$0.496110\pi$
$230$	−5.81507	−0.383435
$231$	−3.00000	−0.197386
$232$	6.81507	0.447431
$233$	−7.63015	−0.499867	−0.249934	−	0.968263i	$-0.580409\pi$
$233$	−7.63015	−0.499867	−0.249934	+	0.968263i	$0.580409\pi$
$234$	3.81507	0.249399
$235$	−8.00000	−0.521862
$236$	2.00000	0.130189
$237$	12.0000	0.779484
$238$	−3.00000	−0.194461
$239$	6.44522	0.416907	0.208453	−	0.978032i	$-0.433157\pi$
$239$	6.44522	0.416907	0.208453	+	0.978032i	$0.433157\pi$
$240$	1.00000	0.0645497
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	10.0000	0.642824
$243$	1.00000	0.0641500
$244$	6.81507	0.436290
$245$	2.00000	0.127775
$246$	4.81507	0.306998
$247$	−29.8151	−1.89709
$248$	−6.81507	−0.432758
$249$	−3.81507	−0.241770
$250$	−1.00000	−0.0632456
$251$	−11.6301	−0.734088	−0.367044	−	0.930204i	$-0.619630\pi$
$251$	−11.6301	−0.734088	−0.367044	+	0.930204i	$0.619630\pi$
$252$	3.00000	0.188982
$253$	−5.81507	−0.365591
$254$	21.4452	1.34559
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−3.81507	−0.237978	−0.118989	−	0.992896i	$-0.537965\pi$
$257$	−3.81507	−0.237978	−0.118989	+	0.992896i	$0.537965\pi$
$258$	4.81507	0.299773
$259$	3.00000	0.186411
$260$	−3.81507	−0.236601
$261$	−6.81507	−0.421842
$262$	9.63015	0.594952
$263$	−30.4452	−1.87733	−0.938666	−	0.344827i	$-0.887938\pi$
$263$	−30.4452	−1.87733	−0.938666	+	0.344827i	$0.887938\pi$
$264$	1.00000	0.0615457
$265$	10.6301	0.653005
$266$	−23.4452	−1.43752
$267$	11.8151	0.723071
$268$	7.63015	0.466085
$269$	9.81507	0.598436	0.299218	−	0.954185i	$-0.403274\pi$
$269$	9.81507	0.598436	0.299218	+	0.954185i	$0.403274\pi$
$270$	−1.00000	−0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	1.00000	0.0606339
$273$	−11.4452	−0.692696
$274$	−2.00000	−0.120824
$275$	−1.00000	−0.0603023
$276$	5.81507	0.350026
$277$	21.4452	1.28852	0.644259	−	0.764807i	$-0.277166\pi$
$277$	21.4452	1.28852	0.644259	+	0.764807i	$0.277166\pi$
$278$	−22.8151	−1.36836
$279$	6.81507	0.408008
$280$	−3.00000	−0.179284
$281$	−19.4452	−1.16000	−0.580002	−	0.814615i	$-0.696948\pi$
$281$	−19.4452	−1.16000	−0.580002	+	0.814615i	$0.696948\pi$
$282$	8.00000	0.476393
$283$	−21.8151	−1.29677	−0.648386	−	0.761312i	$-0.724556\pi$
$283$	−21.8151	−1.29677	−0.648386	+	0.761312i	$0.724556\pi$
$284$	−11.6301	−0.690122
$285$	7.81507	0.462925
$286$	−3.81507	−0.225590
$287$	−14.4452	−0.852674
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	6.81507	0.400195
$291$	−4.81507	−0.282265
$292$	−5.81507	−0.340301
$293$	28.2603	1.65098	0.825492	−	0.564414i	$-0.190898\pi$
$293$	28.2603	1.65098	0.825492	+	0.564414i	$0.190898\pi$
$294$	−2.00000	−0.116642
$295$	2.00000	0.116445
$296$	−1.00000	−0.0581238
$297$	−1.00000	−0.0580259
$298$	−2.00000	−0.115857
$299$	−22.1849	−1.28299
$300$	1.00000	0.0577350
$301$	−14.4452	−0.832609
$302$	−3.81507	−0.219533
$303$	11.6301	0.668134
$304$	7.81507	0.448225
$305$	6.81507	0.390230
$306$	−1.00000	−0.0571662
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	−6.81507	−0.387070
$311$	10.4452	0.592294	0.296147	−	0.955142i	$-0.404298\pi$
$311$	10.4452	0.592294	0.296147	+	0.955142i	$0.404298\pi$
$312$	3.81507	0.215986
$313$	32.8904	1.85908	0.929539	−	0.368725i	$-0.120205\pi$
$313$	32.8904	1.85908	0.929539	+	0.368725i	$0.120205\pi$
$314$	12.8151	0.723196
$315$	3.00000	0.169031
$316$	12.0000	0.675053
$317$	−2.81507	−0.158110	−0.0790551	−	0.996870i	$-0.525190\pi$
$317$	−2.81507	−0.158110	−0.0790551	+	0.996870i	$0.525190\pi$
$318$	−10.6301	−0.596109
$319$	6.81507	0.381571
$320$	1.00000	0.0559017
$321$	−8.18493	−0.456838
$322$	−17.4452	−0.972184
$323$	7.81507	0.434842
$324$	1.00000	0.0555556
$325$	−3.81507	−0.211622
$326$	−16.6301	−0.921059
$327$	13.0000	0.718902
$328$	4.81507	0.265868
$329$	−24.0000	−1.32316
$330$	1.00000	0.0550482
$331$	−23.2603	−1.27850	−0.639251	−	0.768998i	$-0.720755\pi$
$331$	−23.2603	−1.27850	−0.639251	+	0.768998i	$0.720755\pi$
$332$	−3.81507	−0.209379
$333$	1.00000	0.0547997
$334$	21.8151	1.19367
$335$	7.63015	0.416879
$336$	3.00000	0.163663
$337$	−4.18493	−0.227968	−0.113984	−	0.993483i	$-0.536361\pi$
$337$	−4.18493	−0.227968	−0.113984	+	0.993483i	$0.536361\pi$
$338$	−1.55478	−0.0845690
$339$	−6.81507	−0.370144
$340$	1.00000	0.0542326
$341$	−6.81507	−0.369057
$342$	−7.81507	−0.422591
$343$	−15.0000	−0.809924
$344$	4.81507	0.259611
$345$	5.81507	0.313073
$346$	1.00000	0.0537603
$347$	11.2603	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$347$	11.2603	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$348$	−6.81507	−0.365326
$349$	7.26029	0.388635	0.194317	−	0.980939i	$-0.437751\pi$
$349$	7.26029	0.388635	0.194317	+	0.980939i	$0.437751\pi$
$350$	−3.00000	−0.160357
$351$	−3.81507	−0.203634
$352$	1.00000	0.0533002
$353$	−13.1849	−0.701763	−0.350881	−	0.936420i	$-0.614118\pi$
$353$	−13.1849	−0.701763	−0.350881	+	0.936420i	$0.614118\pi$
$354$	−2.00000	−0.106299
$355$	−11.6301	−0.617264
$356$	11.8151	0.626198
$357$	3.00000	0.158777
$358$	−16.0000	−0.845626
$359$	6.36985	0.336188	0.168094	−	0.985771i	$-0.446239\pi$
$359$	6.36985	0.336188	0.168094	+	0.985771i	$0.446239\pi$
$360$	−1.00000	−0.0527046
$361$	42.0754	2.21449
$362$	10.0000	0.525588
$363$	−10.0000	−0.524864
$364$	−11.4452	−0.599892
$365$	−5.81507	−0.304375
$366$	−6.81507	−0.356230
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	5.81507	0.303132
$369$	−4.81507	−0.250663
$370$	−1.00000	−0.0519875
$371$	31.8904	1.65567
$372$	6.81507	0.353345
$373$	−17.2603	−0.893704	−0.446852	−	0.894608i	$-0.647455\pi$
$373$	−17.2603	−0.893704	−0.446852	+	0.894608i	$0.647455\pi$
$374$	1.00000	0.0517088
$375$	1.00000	0.0516398
$376$	8.00000	0.412568
$377$	26.0000	1.33907
$378$	−3.00000	−0.154303
$379$	−19.6301	−1.00833	−0.504166	−	0.863607i	$-0.668200\pi$
$379$	−19.6301	−1.00833	−0.504166	+	0.863607i	$0.668200\pi$
$380$	7.81507	0.400905
$381$	−21.4452	−1.09867
$382$	−4.63015	−0.236899
$383$	33.4452	1.70897	0.854485	−	0.519475i	$-0.173873\pi$
$383$	33.4452	1.70897	0.854485	+	0.519475i	$0.173873\pi$
$384$	−1.00000	−0.0510310
$385$	−3.00000	−0.152894
$386$	2.00000	0.101797
$387$	−4.81507	−0.244764
$388$	−4.81507	−0.244448
$389$	−35.7055	−1.81034	−0.905171	−	0.425048i	$-0.860257\pi$
$389$	−35.7055	−1.81034	−0.905171	+	0.425048i	$0.860257\pi$
$390$	3.81507	0.193184
$391$	5.81507	0.294081
$392$	−2.00000	−0.101015
$393$	−9.63015	−0.485777
$394$	4.18493	0.210834
$395$	12.0000	0.603786
$396$	−1.00000	−0.0502519
$397$	−5.63015	−0.282569	−0.141284	−	0.989969i	$-0.545123\pi$
$397$	−5.63015	−0.282569	−0.141284	+	0.989969i	$0.545123\pi$
$398$	4.00000	0.200502
$399$	23.4452	1.17373
$400$	1.00000	0.0500000
$401$	0.554781	0.0277045	0.0138522	−	0.999904i	$-0.495591\pi$
$401$	0.554781	0.0277045	0.0138522	+	0.999904i	$0.495591\pi$
$402$	−7.63015	−0.380557
$403$	−26.0000	−1.29515
$404$	11.6301	0.578621
$405$	1.00000	0.0496904
$406$	20.4452	1.01468
$407$	−1.00000	−0.0495682
$408$	−1.00000	−0.0495074
$409$	6.36985	0.314969	0.157485	−	0.987521i	$-0.449662\pi$
$409$	6.36985	0.314969	0.157485	+	0.987521i	$0.449662\pi$
$410$	4.81507	0.237800
$411$	2.00000	0.0986527
$412$	0	0
$413$	6.00000	0.295241
$414$	−5.81507	−0.285795
$415$	−3.81507	−0.187275
$416$	3.81507	0.187049
$417$	22.8151	1.11726
$418$	7.81507	0.382248
$419$	−25.4452	−1.24308	−0.621540	−	0.783382i	$-0.713493\pi$
$419$	−25.4452	−1.24308	−0.621540	+	0.783382i	$0.713493\pi$
$420$	3.00000	0.146385
$421$	−29.2603	−1.42606	−0.713030	−	0.701134i	$-0.752678\pi$
$421$	−29.2603	−1.42606	−0.713030	+	0.701134i	$0.752678\pi$
$422$	−8.44522	−0.411107
$423$	−8.00000	−0.388973
$424$	−10.6301	−0.516246
$425$	1.00000	0.0485071
$426$	11.6301	0.563482
$427$	20.4452	0.989413
$428$	−8.18493	−0.395633
$429$	3.81507	0.184193
$430$	4.81507	0.232203
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	1.00000	0.0481125
$433$	1.81507	0.0872268	0.0436134	−	0.999048i	$-0.486113\pi$
$433$	1.81507	0.0872268	0.0436134	+	0.999048i	$0.486113\pi$
$434$	−20.4452	−0.981402
$435$	−6.81507	−0.326758
$436$	13.0000	0.622587
$437$	45.4452	2.17394
$438$	5.81507	0.277855
$439$	−18.4452	−0.880342	−0.440171	−	0.897914i	$-0.645082\pi$
$439$	−18.4452	−0.880342	−0.440171	+	0.897914i	$0.645082\pi$
$440$	1.00000	0.0476731
$441$	2.00000	0.0952381
$442$	3.81507	0.181465
$443$	13.6301	0.647588	0.323794	−	0.946128i	$-0.395042\pi$
$443$	13.6301	0.647588	0.323794	+	0.946128i	$0.395042\pi$
$444$	1.00000	0.0474579
$445$	11.8151	0.560088
$446$	16.8151	0.796217
$447$	2.00000	0.0945968
$448$	3.00000	0.141737
$449$	30.8904	1.45781	0.728905	−	0.684615i	$-0.240030\pi$
$449$	30.8904	1.45781	0.728905	+	0.684615i	$0.240030\pi$
$450$	−1.00000	−0.0471405
$451$	4.81507	0.226733
$452$	−6.81507	−0.320554
$453$	3.81507	0.179248
$454$	24.8151	1.16463
$455$	−11.4452	−0.536560
$456$	−7.81507	−0.365974
$457$	−28.0754	−1.31331	−0.656655	−	0.754191i	$-0.728029\pi$
$457$	−28.0754	−1.31331	−0.656655	+	0.754191i	$0.728029\pi$
$458$	−0.369854	−0.0172822
$459$	1.00000	0.0466760
$460$	5.81507	0.271129
$461$	34.8151	1.62150	0.810750	−	0.585393i	$-0.199060\pi$
$461$	34.8151	1.62150	0.810750	+	0.585393i	$0.199060\pi$
$462$	3.00000	0.139573
$463$	−17.6301	−0.819342	−0.409671	−	0.912233i	$-0.634357\pi$
$463$	−17.6301	−0.819342	−0.409671	+	0.912233i	$0.634357\pi$
$464$	−6.81507	−0.316382
$465$	6.81507	0.316041
$466$	7.63015	0.353460
$467$	28.8151	1.33340	0.666701	−	0.745325i	$-0.267706\pi$
$467$	28.8151	1.33340	0.666701	+	0.745325i	$0.267706\pi$
$468$	−3.81507	−0.176352
$469$	22.8904	1.05698
$470$	8.00000	0.369012
$471$	−12.8151	−0.590487
$472$	−2.00000	−0.0920575
$473$	4.81507	0.221397
$474$	−12.0000	−0.551178
$475$	7.81507	0.358580
$476$	3.00000	0.137505
$477$	10.6301	0.486721
$478$	−6.44522	−0.294797
$479$	9.81507	0.448462	0.224231	−	0.974536i	$-0.428013\pi$
$479$	9.81507	0.448462	0.224231	+	0.974536i	$0.428013\pi$
$480$	−1.00000	−0.0456435
$481$	−3.81507	−0.173952
$482$	26.0000	1.18427
$483$	17.4452	0.793785
$484$	−10.0000	−0.454545
$485$	−4.81507	−0.218641
$486$	−1.00000	−0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−6.81507	−0.308504
$489$	16.6301	0.752041
$490$	−2.00000	−0.0903508
$491$	2.18493	0.0986044	0.0493022	−	0.998784i	$-0.484300\pi$
$491$	2.18493	0.0986044	0.0493022	+	0.998784i	$0.484300\pi$
$492$	−4.81507	−0.217080
$493$	−6.81507	−0.306935
$494$	29.8151	1.34144
$495$	−1.00000	−0.0449467
$496$	6.81507	0.306006
$497$	−34.8904	−1.56505
$498$	3.81507	0.170958
$499$	−17.4452	−0.780955	−0.390478	−	0.920612i	$-0.627690\pi$
$499$	−17.4452	−0.780955	−0.390478	+	0.920612i	$0.627690\pi$
$500$	1.00000	0.0447214
$501$	−21.8151	−0.974626
$502$	11.6301	0.519079
$503$	0.739708	0.0329820	0.0164910	−	0.999864i	$-0.494751\pi$
$503$	0.739708	0.0329820	0.0164910	+	0.999864i	$0.494751\pi$
$504$	−3.00000	−0.133631
$505$	11.6301	0.517535
$506$	5.81507	0.258512
$507$	1.55478	0.0690503
$508$	−21.4452	−0.951478
$509$	17.4452	0.773246	0.386623	−	0.922238i	$-0.373642\pi$
$509$	17.4452	0.773246	0.386623	+	0.922238i	$0.373642\pi$
$510$	−1.00000	−0.0442807
$511$	−17.4452	−0.771731
$512$	−1.00000	−0.0441942
$513$	7.81507	0.345044
$514$	3.81507	0.168276
$515$	0	0
$516$	−4.81507	−0.211972
$517$	8.00000	0.351840
$518$	−3.00000	−0.131812
$519$	−1.00000	−0.0438951
$520$	3.81507	0.167302
$521$	0.815073	0.0357090	0.0178545	−	0.999841i	$-0.494316\pi$
$521$	0.815073	0.0357090	0.0178545	+	0.999841i	$0.494316\pi$
$522$	6.81507	0.298288
$523$	−0.739708	−0.0323452	−0.0161726	−	0.999869i	$-0.505148\pi$
$523$	−0.739708	−0.0323452	−0.0161726	+	0.999869i	$0.505148\pi$
$524$	−9.63015	−0.420695
$525$	3.00000	0.130931
$526$	30.4452	1.32747
$527$	6.81507	0.296869
$528$	−1.00000	−0.0435194
$529$	10.8151	0.470221
$530$	−10.6301	−0.461744
$531$	2.00000	0.0867926
$532$	23.4452	1.01648
$533$	18.3699	0.795687
$534$	−11.8151	−0.511288
$535$	−8.18493	−0.353865
$536$	−7.63015	−0.329572
$537$	16.0000	0.690451
$538$	−9.81507	−0.423158
$539$	−2.00000	−0.0861461
$540$	1.00000	0.0430331
$541$	27.8151	1.19586	0.597932	−	0.801547i	$-0.295989\pi$
$541$	27.8151	1.19586	0.597932	+	0.801547i	$0.295989\pi$
$542$	20.0000	0.859074
$543$	−10.0000	−0.429141
$544$	−1.00000	−0.0428746
$545$	13.0000	0.556859
$546$	11.4452	0.489810
$547$	23.8904	1.02148	0.510741	−	0.859735i	$-0.329371\pi$
$547$	23.8904	1.02148	0.510741	+	0.859735i	$0.329371\pi$
$548$	2.00000	0.0854358
$549$	6.81507	0.290860
$550$	1.00000	0.0426401
$551$	−53.2603	−2.26896
$552$	−5.81507	−0.247506
$553$	36.0000	1.53088
$554$	−21.4452	−0.911120
$555$	1.00000	0.0424476
$556$	22.8151	0.967575
$557$	−21.6301	−0.916499	−0.458249	−	0.888824i	$-0.651523\pi$
$557$	−21.6301	−0.916499	−0.458249	+	0.888824i	$0.651523\pi$
$558$	−6.81507	−0.288505
$559$	18.3699	0.776962
$560$	3.00000	0.126773
$561$	−1.00000	−0.0422200
$562$	19.4452	0.820247
$563$	−8.44522	−0.355924	−0.177962	−	0.984037i	$-0.556950\pi$
$563$	−8.44522	−0.355924	−0.177962	+	0.984037i	$0.556950\pi$
$564$	−8.00000	−0.336861
$565$	−6.81507	−0.286712
$566$	21.8151	0.916956
$567$	3.00000	0.125988
$568$	11.6301	0.487990
$569$	−35.8151	−1.50145	−0.750723	−	0.660617i	$-0.770295\pi$
$569$	−35.8151	−1.50145	−0.750723	+	0.660617i	$0.770295\pi$
$570$	−7.81507	−0.327337
$571$	−14.8151	−0.619992	−0.309996	−	0.950738i	$-0.600328\pi$
$571$	−14.8151	−0.619992	−0.309996	+	0.950738i	$0.600328\pi$
$572$	3.81507	0.159516
$573$	4.63015	0.193427
$574$	14.4452	0.602932
$575$	5.81507	0.242505
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	16.0000	0.665512
$579$	−2.00000	−0.0831172
$580$	−6.81507	−0.282980
$581$	−11.4452	−0.474828
$582$	4.81507	0.199591
$583$	−10.6301	−0.440256
$584$	5.81507	0.240629
$585$	−3.81507	−0.157734
$586$	−28.2603	−1.16742
$587$	20.8151	0.859130	0.429565	−	0.903036i	$-0.358667\pi$
$587$	20.8151	0.859130	0.429565	+	0.903036i	$0.358667\pi$
$588$	2.00000	0.0824786
$589$	53.2603	2.19455
$590$	−2.00000	−0.0823387
$591$	−4.18493	−0.172145
$592$	1.00000	0.0410997
$593$	−13.6301	−0.559723	−0.279862	−	0.960040i	$-0.590289\pi$
$593$	−13.6301	−0.559723	−0.279862	+	0.960040i	$0.590289\pi$
$594$	1.00000	0.0410305
$595$	3.00000	0.122988
$596$	2.00000	0.0819232
$597$	−4.00000	−0.163709
$598$	22.1849	0.907209
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−1.00000	−0.0408248
$601$	−19.3699	−0.790113	−0.395056	−	0.918657i	$-0.629275\pi$
$601$	−19.3699	−0.790113	−0.395056	+	0.918657i	$0.629275\pi$
$602$	14.4452	0.588743
$603$	7.63015	0.310724
$604$	3.81507	0.155233
$605$	−10.0000	−0.406558
$606$	−11.6301	−0.472442
$607$	−37.2603	−1.51235	−0.756174	−	0.654370i	$-0.772934\pi$
$607$	−37.2603	−1.51235	−0.756174	+	0.654370i	$0.772934\pi$
$608$	−7.81507	−0.316943
$609$	−20.4452	−0.828482
$610$	−6.81507	−0.275934
$611$	30.5206	1.23473
$612$	1.00000	0.0404226
$613$	−6.07536	−0.245382	−0.122691	−	0.992445i	$-0.539152\pi$
$613$	−6.07536	−0.245382	−0.122691	+	0.992445i	$0.539152\pi$
$614$	18.0000	0.726421
$615$	−4.81507	−0.194162
$616$	3.00000	0.120873
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−13.5548	−0.544813	−0.272406	−	0.962182i	$-0.587819\pi$
$619$	−13.5548	−0.544813	−0.272406	+	0.962182i	$0.587819\pi$
$620$	6.81507	0.273700
$621$	5.81507	0.233351
$622$	−10.4452	−0.418815
$623$	35.4452	1.42008
$624$	−3.81507	−0.152725
$625$	1.00000	0.0400000
$626$	−32.8904	−1.31457
$627$	−7.81507	−0.312104
$628$	−12.8151	−0.511377
$629$	1.00000	0.0398726
$630$	−3.00000	−0.119523
$631$	33.7055	1.34180	0.670898	−	0.741550i	$-0.265909\pi$
$631$	33.7055	1.34180	0.670898	+	0.741550i	$0.265909\pi$
$632$	−12.0000	−0.477334
$633$	8.44522	0.335667
$634$	2.81507	0.111801
$635$	−21.4452	−0.851028
$636$	10.6301	0.421513
$637$	−7.63015	−0.302317
$638$	−6.81507	−0.269811
$639$	−11.6301	−0.460081
$640$	−1.00000	−0.0395285
$641$	−13.5548	−0.535382	−0.267691	−	0.963505i	$-0.586261\pi$
$641$	−13.5548	−0.535382	−0.267691	+	0.963505i	$0.586261\pi$
$642$	8.18493	0.323033
$643$	17.0000	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$643$	17.0000	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$644$	17.4452	0.687438
$645$	−4.81507	−0.189593
$646$	−7.81507	−0.307480
$647$	25.0754	0.985814	0.492907	−	0.870082i	$-0.335934\pi$
$647$	25.0754	0.985814	0.492907	+	0.870082i	$0.335934\pi$
$648$	−1.00000	−0.0392837
$649$	−2.00000	−0.0785069
$650$	3.81507	0.149639
$651$	20.4452	0.801311
$652$	16.6301	0.651287
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	−13.0000	−0.508340
$655$	−9.63015	−0.376281
$656$	−4.81507	−0.187997
$657$	−5.81507	−0.226868
$658$	24.0000	0.935617
$659$	−31.2603	−1.21773	−0.608864	−	0.793275i	$-0.708375\pi$
$659$	−31.2603	−1.21773	−0.608864	+	0.793275i	$0.708375\pi$
$660$	−1.00000	−0.0389249
$661$	3.73971	0.145458	0.0727289	−	0.997352i	$-0.476829\pi$
$661$	3.73971	0.145458	0.0727289	+	0.997352i	$0.476829\pi$
$662$	23.2603	0.904037
$663$	−3.81507	−0.148165
$664$	3.81507	0.148054
$665$	23.4452	0.909167
$666$	−1.00000	−0.0387492
$667$	−39.6301	−1.53449
$668$	−21.8151	−0.844051
$669$	−16.8151	−0.650108
$670$	−7.63015	−0.294778
$671$	−6.81507	−0.263093
$672$	−3.00000	−0.115728
$673$	6.18493	0.238411	0.119206	−	0.992870i	$-0.461965\pi$
$673$	6.18493	0.238411	0.119206	+	0.992870i	$0.461965\pi$
$674$	4.18493	0.161197
$675$	1.00000	0.0384900
$676$	1.55478	0.0597993
$677$	47.0754	1.80925	0.904627	−	0.426205i	$-0.140150\pi$
$677$	47.0754	1.80925	0.904627	+	0.426205i	$0.140150\pi$
$678$	6.81507	0.261731
$679$	−14.4452	−0.554357
$680$	−1.00000	−0.0383482
$681$	−24.8151	−0.950916
$682$	6.81507	0.260963
$683$	−47.3357	−1.81125	−0.905624	−	0.424081i	$-0.860597\pi$
$683$	−47.3357	−1.81125	−0.905624	+	0.424081i	$0.860597\pi$
$684$	7.81507	0.298817
$685$	2.00000	0.0764161
$686$	15.0000	0.572703
$687$	0.369854	0.0141108
$688$	−4.81507	−0.183573
$689$	−40.5548	−1.54501
$690$	−5.81507	−0.221376
$691$	−34.0754	−1.29629	−0.648144	−	0.761518i	$-0.724455\pi$
$691$	−34.0754	−1.29629	−0.648144	+	0.761518i	$0.724455\pi$
$692$	−1.00000	−0.0380143
$693$	−3.00000	−0.113961
$694$	−11.2603	−0.427435
$695$	22.8151	0.865425
$696$	6.81507	0.258325
$697$	−4.81507	−0.182384
$698$	−7.26029	−0.274806
$699$	−7.63015	−0.288599
$700$	3.00000	0.113389
$701$	33.6301	1.27019	0.635097	−	0.772433i	$-0.280960\pi$
$701$	33.6301	1.27019	0.635097	+	0.772433i	$0.280960\pi$
$702$	3.81507	0.143991
$703$	7.81507	0.294751
$704$	−1.00000	−0.0376889
$705$	−8.00000	−0.301297
$706$	13.1849	0.496221
$707$	34.8904	1.31219
$708$	2.00000	0.0751646
$709$	20.6301	0.774781	0.387391	−	0.921916i	$-0.373376\pi$
$709$	20.6301	0.774781	0.387391	+	0.921916i	$0.373376\pi$
$710$	11.6301	0.436472
$711$	12.0000	0.450035
$712$	−11.8151	−0.442789
$713$	39.6301	1.48416
$714$	−3.00000	−0.112272
$715$	3.81507	0.142676
$716$	16.0000	0.597948
$717$	6.44522	0.240701
$718$	−6.36985	−0.237721
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−42.0754	−1.56588
$723$	−26.0000	−0.966950
$724$	−10.0000	−0.371647
$725$	−6.81507	−0.253105
$726$	10.0000	0.371135
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	11.4452	0.424188
$729$	1.00000	0.0370370
$730$	5.81507	0.215226
$731$	−4.81507	−0.178092
$732$	6.81507	0.251892
$733$	40.4452	1.49388	0.746939	−	0.664892i	$-0.231523\pi$
$733$	40.4452	1.49388	0.746939	+	0.664892i	$0.231523\pi$
$734$	21.0000	0.775124
$735$	2.00000	0.0737711
$736$	−5.81507	−0.214346
$737$	−7.63015	−0.281060
$738$	4.81507	0.177245
$739$	34.0754	1.25348	0.626741	−	0.779227i	$-0.284388\pi$
$739$	34.0754	1.25348	0.626741	+	0.779227i	$0.284388\pi$
$740$	1.00000	0.0367607
$741$	−29.8151	−1.09528
$742$	−31.8904	−1.17073
$743$	4.44522	0.163079	0.0815396	−	0.996670i	$-0.474016\pi$
$743$	4.44522	0.163079	0.0815396	+	0.996670i	$0.474016\pi$
$744$	−6.81507	−0.249853
$745$	2.00000	0.0732743
$746$	17.2603	0.631944
$747$	−3.81507	−0.139586
$748$	−1.00000	−0.0365636
$749$	−24.5548	−0.897212
$750$	−1.00000	−0.0365148
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	−8.00000	−0.291730
$753$	−11.6301	−0.423826
$754$	−26.0000	−0.946864
$755$	3.81507	0.138845
$756$	3.00000	0.109109
$757$	23.8151	0.865574	0.432787	−	0.901496i	$-0.357530\pi$
$757$	23.8151	0.865574	0.432787	+	0.901496i	$0.357530\pi$
$758$	19.6301	0.712999
$759$	−5.81507	−0.211074
$760$	−7.81507	−0.283482
$761$	−9.18493	−0.332953	−0.166477	−	0.986045i	$-0.553239\pi$
$761$	−9.18493	−0.332953	−0.166477	+	0.986045i	$0.553239\pi$
$762$	21.4452	0.776878
$763$	39.0000	1.41189
$764$	4.63015	0.167513
$765$	1.00000	0.0361551
$766$	−33.4452	−1.20842
$767$	−7.63015	−0.275509
$768$	1.00000	0.0360844
$769$	49.2603	1.77637	0.888186	−	0.459485i	$-0.151966\pi$
$769$	49.2603	1.77637	0.888186	+	0.459485i	$0.151966\pi$
$770$	3.00000	0.108112
$771$	−3.81507	−0.137396
$772$	−2.00000	−0.0719816
$773$	−7.36985	−0.265075	−0.132538	−	0.991178i	$-0.542313\pi$
$773$	−7.36985	−0.265075	−0.132538	+	0.991178i	$0.542313\pi$
$774$	4.81507	0.173074
$775$	6.81507	0.244805
$776$	4.81507	0.172851
$777$	3.00000	0.107624
$778$	35.7055	1.28010
$779$	−37.6301	−1.34824
$780$	−3.81507	−0.136602
$781$	11.6301	0.416159
$782$	−5.81507	−0.207947
$783$	−6.81507	−0.243551
$784$	2.00000	0.0714286
$785$	−12.8151	−0.457390
$786$	9.63015	0.343496
$787$	−39.6301	−1.41266	−0.706331	−	0.707882i	$-0.749651\pi$
$787$	−39.6301	−1.41266	−0.706331	+	0.707882i	$0.749651\pi$
$788$	−4.18493	−0.149082
$789$	−30.4452	−1.08388
$790$	−12.0000	−0.426941
$791$	−20.4452	−0.726948
$792$	1.00000	0.0355335
$793$	−26.0000	−0.923287
$794$	5.63015	0.199806
$795$	10.6301	0.377012
$796$	−4.00000	−0.141776
$797$	−34.8904	−1.23588	−0.617941	−	0.786224i	$-0.712033\pi$
$797$	−34.8904	−1.23588	−0.617941	+	0.786224i	$0.712033\pi$
$798$	−23.4452	−0.829952
$799$	−8.00000	−0.283020
$800$	−1.00000	−0.0353553
$801$	11.8151	0.417465
$802$	−0.554781	−0.0195900
$803$	5.81507	0.205209
$804$	7.63015	0.269094
$805$	17.4452	0.614863
$806$	26.0000	0.915811
$807$	9.81507	0.345507
$808$	−11.6301	−0.409147
$809$	−13.0754	−0.459705	−0.229853	−	0.973225i	$-0.573824\pi$
$809$	−13.0754	−0.459705	−0.229853	+	0.973225i	$0.573824\pi$
$810$	−1.00000	−0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−20.4452	−0.717487
$813$	−20.0000	−0.701431
$814$	1.00000	0.0350500
$815$	16.6301	0.582529
$816$	1.00000	0.0350070
$817$	−37.6301	−1.31651
$818$	−6.36985	−0.222717
$819$	−11.4452	−0.399928
$820$	−4.81507	−0.168150
$821$	11.0754	0.386533	0.193266	−	0.981146i	$-0.438092\pi$
$821$	11.0754	0.386533	0.193266	+	0.981146i	$0.438092\pi$
$822$	−2.00000	−0.0697580
$823$	41.0754	1.43180	0.715899	−	0.698204i	$-0.246017\pi$
$823$	41.0754	1.43180	0.715899	+	0.698204i	$0.246017\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	−6.00000	−0.208767
$827$	−28.0754	−0.976276	−0.488138	−	0.872766i	$-0.662324\pi$
$827$	−28.0754	−0.976276	−0.488138	+	0.872766i	$0.662324\pi$
$828$	5.81507	0.202088
$829$	−53.5206	−1.85885	−0.929423	−	0.369015i	$-0.879695\pi$
$829$	−53.5206	−1.85885	−0.929423	+	0.369015i	$0.879695\pi$
$830$	3.81507	0.132423
$831$	21.4452	0.743926
$832$	−3.81507	−0.132264
$833$	2.00000	0.0692959
$834$	−22.8151	−0.790021
$835$	−21.8151	−0.754942
$836$	−7.81507	−0.270290
$837$	6.81507	0.235563
$838$	25.4452	0.878990
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	−3.00000	−0.103510
$841$	17.4452	0.601559
$842$	29.2603	1.00838
$843$	−19.4452	−0.669729
$844$	8.44522	0.290696
$845$	1.55478	0.0534861
$846$	8.00000	0.275046
$847$	−30.0000	−1.03081
$848$	10.6301	0.365041
$849$	−21.8151	−0.748691
$850$	−1.00000	−0.0342997
$851$	5.81507	0.199338
$852$	−11.6301	−0.398442
$853$	−11.0754	−0.379213	−0.189607	−	0.981860i	$-0.560721\pi$
$853$	−11.0754	−0.379213	−0.189607	+	0.981860i	$0.560721\pi$
$854$	−20.4452	−0.699621
$855$	7.81507	0.267270
$856$	8.18493	0.279755
$857$	30.2603	1.03367	0.516836	−	0.856084i	$-0.327110\pi$
$857$	30.2603	1.03367	0.516836	+	0.856084i	$0.327110\pi$
$858$	−3.81507	−0.130244
$859$	31.4452	1.07290	0.536449	−	0.843933i	$-0.319766\pi$
$859$	31.4452	1.07290	0.536449	+	0.843933i	$0.319766\pi$
$860$	−4.81507	−0.164193
$861$	−14.4452	−0.492292
$862$	−21.0000	−0.715263
$863$	−1.55478	−0.0529254	−0.0264627	−	0.999650i	$-0.508424\pi$
$863$	−1.55478	−0.0529254	−0.0264627	+	0.999650i	$0.508424\pi$
$864$	−1.00000	−0.0340207
$865$	−1.00000	−0.0340010
$866$	−1.81507	−0.0616787
$867$	−16.0000	−0.543388
$868$	20.4452	0.693956
$869$	−12.0000	−0.407072
$870$	6.81507	0.231053
$871$	−29.1096	−0.986340
$872$	−13.0000	−0.440236
$873$	−4.81507	−0.162966
$874$	−45.4452	−1.53721
$875$	3.00000	0.101419
$876$	−5.81507	−0.196473
$877$	49.7055	1.67844	0.839218	−	0.543795i	$-0.183013\pi$
$877$	49.7055	1.67844	0.839218	+	0.543795i	$0.183013\pi$
$878$	18.4452	0.622496
$879$	28.2603	0.953196
$880$	−1.00000	−0.0337100
$881$	43.1849	1.45494	0.727469	−	0.686141i	$-0.240697\pi$
$881$	43.1849	1.45494	0.727469	+	0.686141i	$0.240697\pi$
$882$	−2.00000	−0.0673435
$883$	27.0000	0.908622	0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000	0.908622	0.454311	+	0.890843i	$0.349885\pi$
$884$	−3.81507	−0.128315
$885$	2.00000	0.0672293
$886$	−13.6301	−0.457914
$887$	−14.0754	−0.472604	−0.236302	−	0.971680i	$-0.575936\pi$
$887$	−14.0754	−0.472604	−0.236302	+	0.971680i	$0.575936\pi$
$888$	−1.00000	−0.0335578
$889$	−64.3357	−2.15775
$890$	−11.8151	−0.396042
$891$	−1.00000	−0.0335013
$892$	−16.8151	−0.563010
$893$	−62.5206	−2.09217
$894$	−2.00000	−0.0668900
$895$	16.0000	0.534821
$896$	−3.00000	−0.100223
$897$	−22.1849	−0.740733
$898$	−30.8904	−1.03083
$899$	−46.4452	−1.54903
$900$	1.00000	0.0333333
$901$	10.6301	0.354142
$902$	−4.81507	−0.160324
$903$	−14.4452	−0.480707
$904$	6.81507	0.226666
$905$	−10.0000	−0.332411
$906$	−3.81507	−0.126747
$907$	−45.8151	−1.52126	−0.760632	−	0.649183i	$-0.775111\pi$
$907$	−45.8151	−1.52126	−0.760632	+	0.649183i	$0.775111\pi$
$908$	−24.8151	−0.823517
$909$	11.6301	0.385748
$910$	11.4452	0.379405
$911$	−54.5206	−1.80635	−0.903174	−	0.429275i	$-0.858769\pi$
$911$	−54.5206	−1.80635	−0.903174	+	0.429275i	$0.858769\pi$
$912$	7.81507	0.258783
$913$	3.81507	0.126260
$914$	28.0754	0.928651
$915$	6.81507	0.225299
$916$	0.369854	0.0122203
$917$	−28.8904	−0.954046
$918$	−1.00000	−0.0330049
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−5.81507	−0.191717
$921$	−18.0000	−0.593120
$922$	−34.8151	−1.14657
$923$	44.3699	1.46045
$924$	−3.00000	−0.0986928
$925$	1.00000	0.0328798
$926$	17.6301	0.579363
$927$	0	0
$928$	6.81507	0.223716
$929$	24.4452	0.802022	0.401011	−	0.916073i	$-0.368659\pi$
$929$	24.4452	0.802022	0.401011	+	0.916073i	$0.368659\pi$
$930$	−6.81507	−0.223475
$931$	15.6301	0.512257
$932$	−7.63015	−0.249934
$933$	10.4452	0.341961
$934$	−28.8151	−0.942858
$935$	−1.00000	−0.0327035
$936$	3.81507	0.124700
$937$	−11.6301	−0.379940	−0.189970	−	0.981790i	$-0.560839\pi$
$937$	−11.6301	−0.379940	−0.189970	+	0.981790i	$0.560839\pi$
$938$	−22.8904	−0.747399
$939$	32.8904	1.07334
$940$	−8.00000	−0.260931
$941$	−1.26029	−0.0410843	−0.0205422	−	0.999789i	$-0.506539\pi$
$941$	−1.26029	−0.0410843	−0.0205422	+	0.999789i	$0.506539\pi$
$942$	12.8151	0.417538
$943$	−28.0000	−0.911805
$944$	2.00000	0.0650945
$945$	3.00000	0.0975900
$946$	−4.81507	−0.156552
$947$	−25.5548	−0.830419	−0.415209	−	0.909726i	$-0.636292\pi$
$947$	−25.5548	−0.830419	−0.415209	+	0.909726i	$0.636292\pi$
$948$	12.0000	0.389742
$949$	22.1849	0.720153
$950$	−7.81507	−0.253554
$951$	−2.81507	−0.0912850
$952$	−3.00000	−0.0972306
$953$	14.8904	0.482349	0.241174	−	0.970482i	$-0.422467\pi$
$953$	14.8904	0.482349	0.241174	+	0.970482i	$0.422467\pi$
$954$	−10.6301	−0.344164
$955$	4.63015	0.149828
$956$	6.44522	0.208453
$957$	6.81507	0.220300
$958$	−9.81507	−0.317111
$959$	6.00000	0.193750
$960$	1.00000	0.0322749
$961$	15.4452	0.498233
$962$	3.81507	0.123003
$963$	−8.18493	−0.263756
$964$	−26.0000	−0.837404
$965$	−2.00000	−0.0643823
$966$	−17.4452	−0.561291
$967$	7.63015	0.245369	0.122684	−	0.992446i	$-0.460850\pi$
$967$	7.63015	0.245369	0.122684	+	0.992446i	$0.460850\pi$
$968$	10.0000	0.321412
$969$	7.81507	0.251056
$970$	4.81507	0.154603
$971$	−48.0754	−1.54281	−0.771406	−	0.636343i	$-0.780446\pi$
$971$	−48.0754	−1.54281	−0.771406	+	0.636343i	$0.780446\pi$
$972$	1.00000	0.0320750
$973$	68.4452	2.19425
$974$	−2.00000	−0.0640841
$975$	−3.81507	−0.122180
$976$	6.81507	0.218145
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	−16.6301	−0.531773
$979$	−11.8151	−0.377611
$980$	2.00000	0.0638877
$981$	13.0000	0.415058
$982$	−2.18493	−0.0697238
$983$	57.3357	1.82872	0.914362	−	0.404898i	$-0.132693\pi$
$983$	57.3357	1.82872	0.914362	+	0.404898i	$0.132693\pi$
$984$	4.81507	0.153499
$985$	−4.18493	−0.133343
$986$	6.81507	0.217036
$987$	−24.0000	−0.763928
$988$	−29.8151	−0.948544
$989$	−28.0000	−0.890348
$990$	1.00000	0.0317821
$991$	−38.4452	−1.22125	−0.610626	−	0.791919i	$-0.709082\pi$
$991$	−38.4452	−1.22125	−0.610626	+	0.791919i	$0.709082\pi$
$992$	−6.81507	−0.216379
$993$	−23.2603	−0.738143
$994$	34.8904	1.10666
$995$	−4.00000	−0.126809
$996$	−3.81507	−0.120885
$997$	55.0754	1.74425	0.872127	−	0.489279i	$-0.162740\pi$
$997$	55.0754	1.74425	0.872127	+	0.489279i	$0.162740\pi$
$998$	17.4452	0.552219
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.q.1.1	✓	2
3.2	odd	2		3330.2.a.bf.1.1		2
4.3	odd	2		8880.2.a.bh.1.1		2
5.4	even	2		5550.2.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.q.1.1	✓	2	1.1	even	1	trivial
3330.2.a.bf.1.1		2	3.2	odd	2
5550.2.a.bx.1.2		2	5.4	even	2
8880.2.a.bh.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -3.81507 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.81507 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +5.81507 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.81507 q^{26} +1.00000 q^{27} +3.00000 q^{28} -6.81507 q^{29} -1.00000 q^{30} +6.81507 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -7.81507 q^{38} -3.81507 q^{39} -1.00000 q^{40} -4.81507 q^{41} -3.00000 q^{42} -4.81507 q^{43} -1.00000 q^{44} +1.00000 q^{45} -5.81507 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -3.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +7.81507 q^{57} +6.81507 q^{58} +2.00000 q^{59} +1.00000 q^{60} +6.81507 q^{61} -6.81507 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.81507 q^{65} +1.00000 q^{66} +7.63015 q^{67} +1.00000 q^{68} +5.81507 q^{69} -3.00000 q^{70} -11.6301 q^{71} -1.00000 q^{72} -5.81507 q^{73} -1.00000 q^{74} +1.00000 q^{75} +7.81507 q^{76} -3.00000 q^{77} +3.81507 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.81507 q^{82} -3.81507 q^{83} +3.00000 q^{84} +1.00000 q^{85} +4.81507 q^{86} -6.81507 q^{87} +1.00000 q^{88} +11.8151 q^{89} -1.00000 q^{90} -11.4452 q^{91} +5.81507 q^{92} +6.81507 q^{93} +8.00000 q^{94} +7.81507 q^{95} -1.00000 q^{96} -4.81507 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 3 q^{13} - 6 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{20} + 6 q^{21} + 2 q^{22} + q^{23} - 2 q^{24} + 2 q^{25} - 3 q^{26} + 2 q^{27} + 6 q^{28} - 3 q^{29} - 2 q^{30} + 3 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + 6 q^{35} + 2 q^{36} + 2 q^{37} - 5 q^{38} + 3 q^{39} - 2 q^{40} + q^{41} - 6 q^{42} + q^{43} - 2 q^{44} + 2 q^{45} - q^{46} - 16 q^{47} + 2 q^{48} + 4 q^{49} - 2 q^{50} + 2 q^{51} + 3 q^{52} - 2 q^{54} - 2 q^{55} - 6 q^{56} + 5 q^{57} + 3 q^{58} + 4 q^{59} + 2 q^{60} + 3 q^{61} - 3 q^{62} + 6 q^{63} + 2 q^{64} + 3 q^{65} + 2 q^{66} - 6 q^{67} + 2 q^{68} + q^{69} - 6 q^{70} - 2 q^{71} - 2 q^{72} - q^{73} - 2 q^{74} + 2 q^{75} + 5 q^{76} - 6 q^{77} - 3 q^{78} + 24 q^{79} + 2 q^{80} + 2 q^{81} - q^{82} + 3 q^{83} + 6 q^{84} + 2 q^{85} - q^{86} - 3 q^{87} + 2 q^{88} + 13 q^{89} - 2 q^{90} + 9 q^{91} + q^{92} + 3 q^{93} + 16 q^{94} + 5 q^{95} - 2 q^{96} + q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 3 * q^13 - 6 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^20 + 6 * q^21 + 2 * q^22 + q^23 - 2 * q^24 + 2 * q^25 - 3 * q^26 + 2 * q^27 + 6 * q^28 - 3 * q^29 - 2 * q^30 + 3 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + 6 * q^35 + 2 * q^36 + 2 * q^37 - 5 * q^38 + 3 * q^39 - 2 * q^40 + q^41 - 6 * q^42 + q^43 - 2 * q^44 + 2 * q^45 - q^46 - 16 * q^47 + 2 * q^48 + 4 * q^49 - 2 * q^50 + 2 * q^51 + 3 * q^52 - 2 * q^54 - 2 * q^55 - 6 * q^56 + 5 * q^57 + 3 * q^58 + 4 * q^59 + 2 * q^60 + 3 * q^61 - 3 * q^62 + 6 * q^63 + 2 * q^64 + 3 * q^65 + 2 * q^66 - 6 * q^67 + 2 * q^68 + q^69 - 6 * q^70 - 2 * q^71 - 2 * q^72 - q^73 - 2 * q^74 + 2 * q^75 + 5 * q^76 - 6 * q^77 - 3 * q^78 + 24 * q^79 + 2 * q^80 + 2 * q^81 - q^82 + 3 * q^83 + 6 * q^84 + 2 * q^85 - q^86 - 3 * q^87 + 2 * q^88 + 13 * q^89 - 2 * q^90 + 9 * q^91 + q^92 + 3 * q^93 + 16 * q^94 + 5 * q^95 - 2 * q^96 + q^97 - 4 * q^98 - 2 * q^99

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