LMFDB - Embedded newform 1689.2.a.c.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1689,2,Mod(1,1689)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1689, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1689.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1689 = 3 \cdot 563 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1689.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:	$13.4867329014$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - x^{18} - 24 x^{17} + 22 x^{16} + 237 x^{15} - 196 x^{14} - 1247 x^{13} + 905 x^{12} + 3782 x^{11} + \cdots - 5 $ x^19 - x^18 - 24x^17 + 22x^16 + 237x^15 - 196x^14 - 1247x^13 + 905x^12 + 3782x^11 - 2298x^10 - 6680x^9 + 3118x^8 + 6620x^7 - 1953x^6 - 3386x^5 + 294x^4 + 773x^3 + 75x^2 - 39*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.29235$ of defining polynomial
Character	$\chi$	$=$	1689.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29235 q^{2} -1.00000 q^{3} -0.329835 q^{4} +2.95294 q^{5} -1.29235 q^{6} -2.75729 q^{7} -3.01096 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.29235 q^{2} -1.00000 q^{3} -0.329835 q^{4} +2.95294 q^{5} -1.29235 q^{6} -2.75729 q^{7} -3.01096 q^{8} +1.00000 q^{9} +3.81623 q^{10} +1.11973 q^{11} +0.329835 q^{12} +0.0324787 q^{13} -3.56338 q^{14} -2.95294 q^{15} -3.23154 q^{16} -2.27376 q^{17} +1.29235 q^{18} -5.28970 q^{19} -0.973984 q^{20} +2.75729 q^{21} +1.44709 q^{22} -5.54756 q^{23} +3.01096 q^{24} +3.71986 q^{25} +0.0419738 q^{26} -1.00000 q^{27} +0.909452 q^{28} +4.80451 q^{29} -3.81623 q^{30} -5.71745 q^{31} +1.84564 q^{32} -1.11973 q^{33} -2.93848 q^{34} -8.14212 q^{35} -0.329835 q^{36} +6.80190 q^{37} -6.83614 q^{38} -0.0324787 q^{39} -8.89119 q^{40} -3.71413 q^{41} +3.56338 q^{42} -0.412178 q^{43} -0.369327 q^{44} +2.95294 q^{45} -7.16939 q^{46} -8.17581 q^{47} +3.23154 q^{48} +0.602659 q^{49} +4.80736 q^{50} +2.27376 q^{51} -0.0107126 q^{52} -8.11023 q^{53} -1.29235 q^{54} +3.30651 q^{55} +8.30209 q^{56} +5.28970 q^{57} +6.20910 q^{58} -2.49226 q^{59} +0.973984 q^{60} -7.61268 q^{61} -7.38894 q^{62} -2.75729 q^{63} +8.84829 q^{64} +0.0959077 q^{65} -1.44709 q^{66} +2.58148 q^{67} +0.749964 q^{68} +5.54756 q^{69} -10.5225 q^{70} -1.78817 q^{71} -3.01096 q^{72} +8.54410 q^{73} +8.79043 q^{74} -3.71986 q^{75} +1.74473 q^{76} -3.08743 q^{77} -0.0419738 q^{78} -7.38055 q^{79} -9.54254 q^{80} +1.00000 q^{81} -4.79995 q^{82} -15.5328 q^{83} -0.909452 q^{84} -6.71427 q^{85} -0.532678 q^{86} -4.80451 q^{87} -3.37147 q^{88} -2.16822 q^{89} +3.81623 q^{90} -0.0895532 q^{91} +1.82978 q^{92} +5.71745 q^{93} -10.5660 q^{94} -15.6202 q^{95} -1.84564 q^{96} +10.7955 q^{97} +0.778846 q^{98} +1.11973 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10}) $ 19 * q + q^2 - 19 * q^3 + 11 * q^4 - q^5 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9	\( 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 13 q^{10} - 8 q^{11} - 11 q^{12} - 11 q^{13} - 12 q^{14} + q^{15} - 5 q^{16} + 3 q^{17} + q^{18} - 20 q^{19} - 7 q^{20} + 11 q^{21} - 10 q^{22} - 9 q^{23} - 3 q^{24} - 4 q^{25} - 19 q^{27} - 27 q^{28} + 3 q^{29} + 13 q^{30} - 52 q^{31} + 2 q^{32} + 8 q^{33} - 26 q^{34} - 5 q^{35} + 11 q^{36} - 17 q^{37} - q^{38} + 11 q^{39} - 31 q^{40} - 22 q^{41} + 12 q^{42} - 15 q^{43} - 17 q^{44} - q^{45} - 38 q^{46} - 6 q^{47} + 5 q^{48} - 26 q^{49} + 11 q^{50} - 3 q^{51} - 15 q^{52} + 33 q^{53} - q^{54} - 51 q^{55} - 22 q^{56} + 20 q^{57} - 32 q^{58} - 42 q^{59} + 7 q^{60} - 26 q^{61} - 13 q^{62} - 11 q^{63} - 37 q^{64} - 6 q^{65} + 10 q^{66} - 12 q^{67} - 12 q^{68} + 9 q^{69} + 3 q^{70} - 26 q^{71} + 3 q^{72} - 19 q^{73} - 17 q^{74} + 4 q^{75} - 52 q^{76} + 44 q^{77} - 56 q^{79} - 30 q^{80} + 19 q^{81} - 22 q^{82} - 9 q^{83} + 27 q^{84} - 29 q^{85} - 10 q^{86} - 3 q^{87} - 32 q^{88} - q^{89} - 13 q^{90} - 60 q^{91} + 10 q^{92} + 52 q^{93} - 29 q^{94} - 27 q^{95} - 2 q^{96} - 44 q^{97} - 9 q^{98} - 8 q^{99}+O(q^{100}) \) 19 * q + q^2 - 19 * q^3 + 11 * q^4 - q^5 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 - 13 * q^10 - 8 * q^11 - 11 * q^12 - 11 * q^13 - 12 * q^14 + q^15 - 5 * q^16 + 3 * q^17 + q^18 - 20 * q^19 - 7 * q^20 + 11 * q^21 - 10 * q^22 - 9 * q^23 - 3 * q^24 - 4 * q^25 - 19 * q^27 - 27 * q^28 + 3 * q^29 + 13 * q^30 - 52 * q^31 + 2 * q^32 + 8 * q^33 - 26 * q^34 - 5 * q^35 + 11 * q^36 - 17 * q^37 - q^38 + 11 * q^39 - 31 * q^40 - 22 * q^41 + 12 * q^42 - 15 * q^43 - 17 * q^44 - q^45 - 38 * q^46 - 6 * q^47 + 5 * q^48 - 26 * q^49 + 11 * q^50 - 3 * q^51 - 15 * q^52 + 33 * q^53 - q^54 - 51 * q^55 - 22 * q^56 + 20 * q^57 - 32 * q^58 - 42 * q^59 + 7 * q^60 - 26 * q^61 - 13 * q^62 - 11 * q^63 - 37 * q^64 - 6 * q^65 + 10 * q^66 - 12 * q^67 - 12 * q^68 + 9 * q^69 + 3 * q^70 - 26 * q^71 + 3 * q^72 - 19 * q^73 - 17 * q^74 + 4 * q^75 - 52 * q^76 + 44 * q^77 - 56 * q^79 - 30 * q^80 + 19 * q^81 - 22 * q^82 - 9 * q^83 + 27 * q^84 - 29 * q^85 - 10 * q^86 - 3 * q^87 - 32 * q^88 - q^89 - 13 * q^90 - 60 * q^91 + 10 * q^92 + 52 * q^93 - 29 * q^94 - 27 * q^95 - 2 * q^96 - 44 * q^97 - 9 * q^98 - 8 * 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 $

$2$	1.29235	0.913828	0.456914	−	0.889511i	$-0.348955\pi$
$2$	1.29235	0.913828	0.456914	+	0.889511i	$0.348955\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.329835	−0.164918
$5$	2.95294	1.32060	0.660298	−	0.751004i	$-0.270430\pi$
$5$	2.95294	1.32060	0.660298	+	0.751004i	$0.270430\pi$
$6$	−1.29235	−0.527599
$7$	−2.75729	−1.04216	−0.521079	−	0.853508i	$-0.674470\pi$
$7$	−2.75729	−1.04216	−0.521079	+	0.853508i	$0.674470\pi$
$8$	−3.01096	−1.06453
$9$	1.00000	0.333333
$10$	3.81623	1.20680
$11$	1.11973	0.337612	0.168806	−	0.985649i	$-0.446009\pi$
$11$	1.11973	0.337612	0.168806	+	0.985649i	$0.446009\pi$
$12$	0.329835	0.0952152
$13$	0.0324787	0.00900797	0.00450398	−	0.999990i	$-0.498566\pi$
$13$	0.0324787	0.00900797	0.00450398	+	0.999990i	$0.498566\pi$
$14$	−3.56338	−0.952354
$15$	−2.95294	−0.762446
$16$	−3.23154	−0.807885
$17$	−2.27376	−0.551467	−0.275733	−	0.961234i	$-0.588921\pi$
$17$	−2.27376	−0.551467	−0.275733	+	0.961234i	$0.588921\pi$
$18$	1.29235	0.304609
$19$	−5.28970	−1.21354	−0.606771	−	0.794877i	$-0.707535\pi$
$19$	−5.28970	−1.21354	−0.606771	+	0.794877i	$0.707535\pi$
$20$	−0.973984	−0.217789
$21$	2.75729	0.601690
$22$	1.44709	0.308520
$23$	−5.54756	−1.15675	−0.578373	−	0.815772i	$-0.696312\pi$
$23$	−5.54756	−1.15675	−0.578373	+	0.815772i	$0.696312\pi$
$24$	3.01096	0.614609
$25$	3.71986	0.743973
$26$	0.0419738	0.00823174
$27$	−1.00000	−0.192450
$28$	0.909452	0.171870
$29$	4.80451	0.892174	0.446087	−	0.894989i	$-0.352817\pi$
$29$	4.80451	0.892174	0.446087	+	0.894989i	$0.352817\pi$
$30$	−3.81623	−0.696745
$31$	−5.71745	−1.02688	−0.513442	−	0.858124i	$-0.671630\pi$
$31$	−5.71745	−1.02688	−0.513442	+	0.858124i	$0.671630\pi$
$32$	1.84564	0.326267
$33$	−1.11973	−0.194921
$34$	−2.93848	−0.503946
$35$	−8.14212	−1.37627
$36$	−0.329835	−0.0549725
$37$	6.80190	1.11823	0.559113	−	0.829092i	$-0.311142\pi$
$37$	6.80190	1.11823	0.559113	+	0.829092i	$0.311142\pi$
$38$	−6.83614	−1.10897
$39$	−0.0324787	−0.00520075
$40$	−8.89119	−1.40582
$41$	−3.71413	−0.580050	−0.290025	−	0.957019i	$-0.593664\pi$
$41$	−3.71413	−0.580050	−0.290025	+	0.957019i	$0.593664\pi$
$42$	3.56338	0.549842
$43$	−0.412178	−0.0628566	−0.0314283	−	0.999506i	$-0.510006\pi$
$43$	−0.412178	−0.0628566	−0.0314283	+	0.999506i	$0.510006\pi$
$44$	−0.369327	−0.0556782
$45$	2.95294	0.440199
$46$	−7.16939	−1.05707
$47$	−8.17581	−1.19256	−0.596282	−	0.802775i	$-0.703356\pi$
$47$	−8.17581	−1.19256	−0.596282	+	0.802775i	$0.703356\pi$
$48$	3.23154	0.466432
$49$	0.602659	0.0860942
$50$	4.80736	0.679863
$51$	2.27376	0.318389
$52$	−0.0107126	−0.00148557
$53$	−8.11023	−1.11403	−0.557013	−	0.830504i	$-0.688053\pi$
$53$	−8.11023	−1.11403	−0.557013	+	0.830504i	$0.688053\pi$
$54$	−1.29235	−0.175866
$55$	3.30651	0.445849
$56$	8.30209	1.10941
$57$	5.28970	0.700638
$58$	6.20910	0.815294
$59$	−2.49226	−0.324464	−0.162232	−	0.986753i	$-0.551869\pi$
$59$	−2.49226	−0.324464	−0.162232	+	0.986753i	$0.551869\pi$
$60$	0.973984	0.125741
$61$	−7.61268	−0.974703	−0.487352	−	0.873206i	$-0.662037\pi$
$61$	−7.61268	−0.974703	−0.487352	+	0.873206i	$0.662037\pi$
$62$	−7.38894	−0.938396
$63$	−2.75729	−0.347386
$64$	8.84829	1.10604
$65$	0.0959077	0.0118959
$66$	−1.44709	−0.178124
$67$	2.58148	0.315378	0.157689	−	0.987489i	$-0.449596\pi$
$67$	2.58148	0.315378	0.157689	+	0.987489i	$0.449596\pi$
$68$	0.749964	0.0909465
$69$	5.54756	0.667848
$70$	−10.5225	−1.25767
$71$	−1.78817	−0.212217	−0.106108	−	0.994355i	$-0.533839\pi$
$71$	−1.78817	−0.212217	−0.106108	+	0.994355i	$0.533839\pi$
$72$	−3.01096	−0.354845
$73$	8.54410	1.00001	0.500006	−	0.866022i	$-0.333331\pi$
$73$	8.54410	1.00001	0.500006	+	0.866022i	$0.333331\pi$
$74$	8.79043	1.02187
$75$	−3.71986	−0.429533
$76$	1.74473	0.200134
$77$	−3.08743	−0.351845
$78$	−0.0419738	−0.00475260
$79$	−7.38055	−0.830377	−0.415188	−	0.909735i	$-0.636284\pi$
$79$	−7.38055	−0.830377	−0.415188	+	0.909735i	$0.636284\pi$
$80$	−9.54254	−1.06689
$81$	1.00000	0.111111
$82$	−4.79995	−0.530066
$83$	−15.5328	−1.70495	−0.852475	−	0.522768i	$-0.824899\pi$
$83$	−15.5328	−1.70495	−0.852475	+	0.522768i	$0.824899\pi$
$84$	−0.909452	−0.0992293
$85$	−6.71427	−0.728264
$86$	−0.532678	−0.0574401
$87$	−4.80451	−0.515097
$88$	−3.37147	−0.359400
$89$	−2.16822	−0.229831	−0.114915	−	0.993375i	$-0.536660\pi$
$89$	−2.16822	−0.229831	−0.114915	+	0.993375i	$0.536660\pi$
$90$	3.81623	0.402266
$91$	−0.0895532	−0.00938773
$92$	1.82978	0.190768
$93$	5.71745	0.592872
$94$	−10.5660	−1.08980
$95$	−15.6202	−1.60260
$96$	−1.84564	−0.188370
$97$	10.7955	1.09612	0.548059	−	0.836440i	$-0.315367\pi$
$97$	10.7955	1.09612	0.548059	+	0.836440i	$0.315367\pi$
$98$	0.778846	0.0786753
$99$	1.11973	0.112537
$100$	−1.22694	−0.122694
$101$	−10.0430	−0.999313	−0.499656	−	0.866224i	$-0.666540\pi$
$101$	−10.0430	−0.999313	−0.499656	+	0.866224i	$0.666540\pi$
$102$	2.93848	0.290953
$103$	9.92327	0.977769	0.488884	−	0.872349i	$-0.337404\pi$
$103$	9.92327	0.977769	0.488884	+	0.872349i	$0.337404\pi$
$104$	−0.0977920	−0.00958930
$105$	8.14212	0.794590
$106$	−10.4812	−1.01803
$107$	6.06586	0.586409	0.293204	−	0.956050i	$-0.405278\pi$
$107$	6.06586	0.586409	0.293204	+	0.956050i	$0.405278\pi$
$108$	0.329835	0.0317384
$109$	−8.31356	−0.796294	−0.398147	−	0.917322i	$-0.630347\pi$
$109$	−8.31356	−0.796294	−0.398147	+	0.917322i	$0.630347\pi$
$110$	4.27316	0.407430
$111$	−6.80190	−0.645608
$112$	8.91030	0.841944
$113$	5.70780	0.536945	0.268472	−	0.963287i	$-0.413481\pi$
$113$	5.70780	0.536945	0.268472	+	0.963287i	$0.413481\pi$
$114$	6.83614	0.640263
$115$	−16.3816	−1.52759
$116$	−1.58470	−0.147135
$117$	0.0324787	0.00300266
$118$	−3.22087	−0.296505
$119$	6.26941	0.574716
$120$	8.89119	0.811651
$121$	−9.74620	−0.886018
$122$	−9.83823	−0.890712
$123$	3.71413	0.334892
$124$	1.88582	0.169351
$125$	−3.78017	−0.338109
$126$	−3.56338	−0.317451
$127$	3.35855	0.298023	0.149011	−	0.988835i	$-0.452391\pi$
$127$	3.35855	0.298023	0.149011	+	0.988835i	$0.452391\pi$
$128$	7.74379	0.684461
$129$	0.412178	0.0362903
$130$	0.123946	0.0108708
$131$	−3.60181	−0.314692	−0.157346	−	0.987544i	$-0.550294\pi$
$131$	−3.60181	−0.314692	−0.157346	+	0.987544i	$0.550294\pi$
$132$	0.369327	0.0321458
$133$	14.5853	1.26470
$134$	3.33618	0.288202
$135$	−2.95294	−0.254149
$136$	6.84618	0.587055
$137$	2.08782	0.178374	0.0891872	−	0.996015i	$-0.471573\pi$
$137$	2.08782	0.178374	0.0891872	+	0.996015i	$0.471573\pi$
$138$	7.16939	0.610299
$139$	6.76479	0.573782	0.286891	−	0.957963i	$-0.407378\pi$
$139$	6.76479	0.573782	0.286891	+	0.957963i	$0.407378\pi$
$140$	2.68556	0.226971
$141$	8.17581	0.688527
$142$	−2.31094	−0.193930
$143$	0.0363675	0.00304120
$144$	−3.23154	−0.269295
$145$	14.1874	1.17820
$146$	11.0420	0.913839
$147$	−0.602659	−0.0497065
$148$	−2.24351	−0.184415
$149$	12.9511	1.06100	0.530498	−	0.847686i	$-0.322005\pi$
$149$	12.9511	1.06100	0.530498	+	0.847686i	$0.322005\pi$
$150$	−4.80736	−0.392519
$151$	22.9359	1.86650	0.933249	−	0.359231i	$-0.116961\pi$
$151$	22.9359	1.86650	0.933249	+	0.359231i	$0.116961\pi$
$152$	15.9271	1.29186
$153$	−2.27376	−0.183822
$154$	−3.99004	−0.321526
$155$	−16.8833	−1.35610
$156$	0.0107126	0.000857696 0
$157$	−13.4796	−1.07579	−0.537894	−	0.843012i	$-0.680780\pi$
$157$	−13.4796	−1.07579	−0.537894	+	0.843012i	$0.680780\pi$
$158$	−9.53824	−0.758822
$159$	8.11023	0.643183
$160$	5.45008	0.430867
$161$	15.2963	1.20551
$162$	1.29235	0.101536
$163$	2.58773	0.202686	0.101343	−	0.994852i	$-0.467686\pi$
$163$	2.58773	0.202686	0.101343	+	0.994852i	$0.467686\pi$
$164$	1.22505	0.0956605
$165$	−3.30651	−0.257411
$166$	−20.0738	−1.55803
$167$	9.59034	0.742123	0.371061	−	0.928608i	$-0.378994\pi$
$167$	9.59034	0.742123	0.371061	+	0.928608i	$0.378994\pi$
$168$	−8.30209	−0.640520
$169$	−12.9989	−0.999919
$170$	−8.67717	−0.665509
$171$	−5.28970	−0.404514
$172$	0.135951	0.0103662
$173$	21.1498	1.60799	0.803996	−	0.594635i	$-0.202704\pi$
$173$	21.1498	1.60799	0.803996	+	0.594635i	$0.202704\pi$
$174$	−6.20910	−0.470710
$175$	−10.2567	−0.775337
$176$	−3.61846	−0.272752
$177$	2.49226	0.187330
$178$	−2.80209	−0.210026
$179$	−0.425786	−0.0318247	−0.0159124	−	0.999873i	$-0.505065\pi$
$179$	−0.425786	−0.0318247	−0.0159124	+	0.999873i	$0.505065\pi$
$180$	−0.973984	−0.0725965
$181$	−6.11341	−0.454406	−0.227203	−	0.973847i	$-0.572958\pi$
$181$	−6.11341	−0.454406	−0.227203	+	0.973847i	$0.572958\pi$
$182$	−0.115734	−0.00857878
$183$	7.61268	0.562745
$184$	16.7035	1.23140
$185$	20.0856	1.47672
$186$	7.38894	0.541783
$187$	−2.54600	−0.186182
$188$	2.69667	0.196675
$189$	2.75729	0.200563
$190$	−20.1867	−1.46450
$191$	7.37329	0.533513	0.266756	−	0.963764i	$-0.414048\pi$
$191$	7.37329	0.533513	0.266756	+	0.963764i	$0.414048\pi$
$192$	−8.84829	−0.638571
$193$	5.13641	0.369727	0.184863	−	0.982764i	$-0.440816\pi$
$193$	5.13641	0.369727	0.184863	+	0.982764i	$0.440816\pi$
$194$	13.9516	1.00166
$195$	−0.0959077	−0.00686809
$196$	−0.198778	−0.0141984
$197$	−26.2508	−1.87029	−0.935146	−	0.354263i	$-0.884732\pi$
$197$	−26.2508	−1.87029	−0.935146	+	0.354263i	$0.884732\pi$
$198$	1.44709	0.102840
$199$	4.18107	0.296388	0.148194	−	0.988958i	$-0.452654\pi$
$199$	4.18107	0.296388	0.148194	+	0.988958i	$0.452654\pi$
$200$	−11.2004	−0.791985
$201$	−2.58148	−0.182084
$202$	−12.9790	−0.913201
$203$	−13.2474	−0.929787
$204$	−0.749964	−0.0525080
$205$	−10.9676	−0.766012
$206$	12.8243	0.893513
$207$	−5.54756	−0.385582
$208$	−0.104956	−0.00727740
$209$	−5.92306	−0.409706
$210$	10.5225	0.726119
$211$	−2.38990	−0.164528	−0.0822638	−	0.996611i	$-0.526215\pi$
$211$	−2.38990	−0.164528	−0.0822638	+	0.996611i	$0.526215\pi$
$212$	2.67504	0.183722
$213$	1.78817	0.122523
$214$	7.83921	0.535877
$215$	−1.21714	−0.0830081
$216$	3.01096	0.204870
$217$	15.7647	1.07018
$218$	−10.7440	−0.727676
$219$	−8.54410	−0.577357
$220$	−1.09060	−0.0735284
$221$	−0.0738486	−0.00496759
$222$	−8.79043	−0.589975
$223$	−9.45284	−0.633009	−0.316504	−	0.948591i	$-0.602509\pi$
$223$	−9.45284	−0.633009	−0.316504	+	0.948591i	$0.602509\pi$
$224$	−5.08898	−0.340022
$225$	3.71986	0.247991
$226$	7.37647	0.490675
$227$	29.0745	1.92974	0.964871	−	0.262724i	$-0.0846210\pi$
$227$	29.0745	1.92974	0.964871	+	0.262724i	$0.0846210\pi$
$228$	−1.74473	−0.115548
$229$	−6.87083	−0.454037	−0.227019	−	0.973890i	$-0.572898\pi$
$229$	−6.87083	−0.454037	−0.227019	+	0.973890i	$0.572898\pi$
$230$	−21.1708	−1.39596
$231$	3.08743	0.203138
$232$	−14.4662	−0.949751
$233$	−9.04986	−0.592876	−0.296438	−	0.955052i	$-0.595799\pi$
$233$	−9.04986	−0.592876	−0.296438	+	0.955052i	$0.595799\pi$
$234$	0.0419738	0.00274391
$235$	−24.1427	−1.57490
$236$	0.822035	0.0535099
$237$	7.38055	0.479418
$238$	8.10226	0.525191
$239$	14.2853	0.924038	0.462019	−	0.886870i	$-0.347125\pi$
$239$	14.2853	0.924038	0.462019	+	0.886870i	$0.347125\pi$
$240$	9.54254	0.615969
$241$	−22.8028	−1.46886	−0.734428	−	0.678687i	$-0.762549\pi$
$241$	−22.8028	−1.46886	−0.734428	+	0.678687i	$0.762549\pi$
$242$	−12.5955	−0.809668
$243$	−1.00000	−0.0641500
$244$	2.51093	0.160746
$245$	1.77962	0.113696
$246$	4.79995	0.306034
$247$	−0.171803	−0.0109315
$248$	17.2150	1.09315
$249$	15.5328	0.984353
$250$	−4.88530	−0.308973
$251$	−0.885596	−0.0558983	−0.0279492	−	0.999609i	$-0.508898\pi$
$251$	−0.885596	−0.0558983	−0.0279492	+	0.999609i	$0.508898\pi$
$252$	0.909452	0.0572901
$253$	−6.21179	−0.390532
$254$	4.34041	0.272342
$255$	6.71427	0.420464
$256$	−7.68891	−0.480557
$257$	10.4406	0.651264	0.325632	−	0.945497i	$-0.394423\pi$
$257$	10.4406	0.651264	0.325632	+	0.945497i	$0.394423\pi$
$258$	0.532678	0.0331631
$259$	−18.7548	−1.16537
$260$	−0.0316337	−0.00196184
$261$	4.80451	0.297391
$262$	−4.65479	−0.287574
$263$	−4.23209	−0.260962	−0.130481	−	0.991451i	$-0.541652\pi$
$263$	−4.23209	−0.260962	−0.130481	+	0.991451i	$0.541652\pi$
$264$	3.37147	0.207500
$265$	−23.9490	−1.47118
$266$	18.8492	1.15572
$267$	2.16822	0.132693
$268$	−0.851464	−0.0520115
$269$	14.8618	0.906140	0.453070	−	0.891475i	$-0.350329\pi$
$269$	14.8618	0.906140	0.453070	+	0.891475i	$0.350329\pi$
$270$	−3.81623	−0.232248
$271$	7.15960	0.434915	0.217457	−	0.976070i	$-0.430224\pi$
$271$	7.15960	0.434915	0.217457	+	0.976070i	$0.430224\pi$
$272$	7.34773	0.445521
$273$	0.0895532	0.00542001
$274$	2.69819	0.163004
$275$	4.16525	0.251174
$276$	−1.82978	−0.110140
$277$	14.6978	0.883104	0.441552	−	0.897236i	$-0.354428\pi$
$277$	14.6978	0.883104	0.441552	+	0.897236i	$0.354428\pi$
$278$	8.74246	0.524338
$279$	−5.71745	−0.342295
$280$	24.5156	1.46509
$281$	0.259588	0.0154857	0.00774286	−	0.999970i	$-0.497535\pi$
$281$	0.259588	0.0154857	0.00774286	+	0.999970i	$0.497535\pi$
$282$	10.5660	0.629196
$283$	27.4229	1.63012	0.815061	−	0.579375i	$-0.196703\pi$
$283$	27.4229	1.63012	0.815061	+	0.579375i	$0.196703\pi$
$284$	0.589802	0.0349983
$285$	15.6202	0.925260
$286$	0.0469994	0.00277914
$287$	10.2409	0.604504
$288$	1.84564	0.108756
$289$	−11.8300	−0.695885
$290$	18.3351	1.07667
$291$	−10.7955	−0.632844
$292$	−2.81814	−0.164919
$293$	5.95377	0.347823	0.173912	−	0.984761i	$-0.444359\pi$
$293$	5.95377	0.347823	0.173912	+	0.984761i	$0.444359\pi$
$294$	−0.778846	−0.0454232
$295$	−7.35949	−0.428486
$296$	−20.4802	−1.19039
$297$	−1.11973	−0.0649735
$298$	16.7373	0.969569
$299$	−0.180178	−0.0104199
$300$	1.22694	0.0708375
$301$	1.13650	0.0655065
$302$	29.6412	1.70566
$303$	10.0430	0.576954
$304$	17.0939	0.980401
$305$	−22.4798	−1.28719
$306$	−2.93848	−0.167982
$307$	−18.3993	−1.05010	−0.525051	−	0.851071i	$-0.675954\pi$
$307$	−18.3993	−1.05010	−0.525051	+	0.851071i	$0.675954\pi$
$308$	1.01834	0.0580255
$309$	−9.92327	−0.564515
$310$	−21.8191	−1.23924
$311$	4.00072	0.226860	0.113430	−	0.993546i	$-0.463816\pi$
$311$	4.00072	0.226860	0.113430	+	0.993546i	$0.463816\pi$
$312$	0.0977920	0.00553638
$313$	13.0739	0.738980	0.369490	−	0.929235i	$-0.379532\pi$
$313$	13.0739	0.738980	0.369490	+	0.929235i	$0.379532\pi$
$314$	−17.4203	−0.983086
$315$	−8.14212	−0.458757
$316$	2.43436	0.136944
$317$	20.5945	1.15670	0.578352	−	0.815787i	$-0.303696\pi$
$317$	20.5945	1.15670	0.578352	+	0.815787i	$0.303696\pi$
$318$	10.4812	0.587759
$319$	5.37977	0.301209
$320$	26.1285	1.46063
$321$	−6.06586	−0.338563
$322$	19.7681	1.10163
$323$	12.0275	0.669227
$324$	−0.329835	−0.0183242
$325$	0.120816	0.00670168
$326$	3.34424	0.185221
$327$	8.31356	0.459741
$328$	11.1831	0.617484
$329$	22.5431	1.24284
$330$	−4.27316	−0.235230
$331$	−26.0395	−1.43126	−0.715630	−	0.698479i	$-0.753860\pi$
$331$	−26.0395	−1.43126	−0.715630	+	0.698479i	$0.753860\pi$
$332$	5.12327	0.281176
$333$	6.80190	0.372742
$334$	12.3941	0.678173
$335$	7.62297	0.416487
$336$	−8.91030	−0.486096
$337$	20.8810	1.13746	0.568731	−	0.822524i	$-0.307435\pi$
$337$	20.8810	1.13746	0.568731	+	0.822524i	$0.307435\pi$
$338$	−16.7992	−0.913754
$339$	−5.70780	−0.310005
$340$	2.21460	0.120104
$341$	−6.40202	−0.346689
$342$	−6.83614	−0.369656
$343$	17.6393	0.952435
$344$	1.24105	0.0669130
$345$	16.3816	0.881957
$346$	27.3329	1.46943
$347$	11.2252	0.602599	0.301299	−	0.953530i	$-0.402580\pi$
$347$	11.2252	0.602599	0.301299	+	0.953530i	$0.402580\pi$
$348$	1.58470	0.0849486
$349$	−26.0616	−1.39505	−0.697523	−	0.716563i	$-0.745714\pi$
$349$	−26.0616	−1.39505	−0.697523	+	0.716563i	$0.745714\pi$
$350$	−13.2553	−0.708525
$351$	−0.0324787	−0.00173358
$352$	2.06663	0.110152
$353$	22.9698	1.22256	0.611279	−	0.791415i	$-0.290655\pi$
$353$	22.9698	1.22256	0.611279	+	0.791415i	$0.290655\pi$
$354$	3.22087	0.171187
$355$	−5.28036	−0.280253
$356$	0.715155	0.0379031
$357$	−6.26941	−0.331812
$358$	−0.550264	−0.0290823
$359$	−2.51223	−0.132590	−0.0662952	−	0.997800i	$-0.521118\pi$
$359$	−2.51223	−0.132590	−0.0662952	+	0.997800i	$0.521118\pi$
$360$	−8.89119	−0.468607
$361$	8.98096	0.472682
$362$	−7.90066	−0.415249
$363$	9.74620	0.511543
$364$	0.0295378	0.00154820
$365$	25.2302	1.32061
$366$	9.83823	0.514253
$367$	−34.8819	−1.82082	−0.910409	−	0.413708i	$-0.864233\pi$
$367$	−34.8819	−1.82082	−0.910409	+	0.413708i	$0.864233\pi$
$368$	17.9272	0.934518
$369$	−3.71413	−0.193350
$370$	25.9576	1.34947
$371$	22.3623	1.16099
$372$	−1.88582	−0.0977750
$373$	−2.72482	−0.141086	−0.0705428	−	0.997509i	$-0.522473\pi$
$373$	−2.72482	−0.141086	−0.0705428	+	0.997509i	$0.522473\pi$
$374$	−3.29032	−0.170138
$375$	3.78017	0.195207
$376$	24.6170	1.26953
$377$	0.156044	0.00803668
$378$	3.56338	0.183281
$379$	27.9650	1.43646	0.718232	−	0.695804i	$-0.244952\pi$
$379$	27.9650	1.43646	0.718232	+	0.695804i	$0.244952\pi$
$380$	5.15209	0.264296
$381$	−3.35855	−0.172064
$382$	9.52886	0.487539
$383$	10.4124	0.532049	0.266025	−	0.963966i	$-0.414290\pi$
$383$	10.4124	0.532049	0.266025	+	0.963966i	$0.414290\pi$
$384$	−7.74379	−0.395174
$385$	−9.11700	−0.464646
$386$	6.63803	0.337867
$387$	−0.412178	−0.0209522
$388$	−3.56074	−0.180769
$389$	30.8732	1.56533	0.782666	−	0.622442i	$-0.213860\pi$
$389$	30.8732	1.56533	0.782666	+	0.622442i	$0.213860\pi$
$390$	−0.123946	−0.00627626
$391$	12.6138	0.637907
$392$	−1.81458	−0.0916502
$393$	3.60181	0.181687
$394$	−33.9252	−1.70913
$395$	−21.7943	−1.09659
$396$	−0.369327	−0.0185594
$397$	−4.13707	−0.207634	−0.103817	−	0.994596i	$-0.533106\pi$
$397$	−4.13707	−0.207634	−0.103817	+	0.994596i	$0.533106\pi$
$398$	5.40340	0.270848
$399$	−14.5853	−0.730176
$400$	−12.0209	−0.601044
$401$	−11.7886	−0.588695	−0.294347	−	0.955698i	$-0.595102\pi$
$401$	−11.7886	−0.588695	−0.294347	+	0.955698i	$0.595102\pi$
$402$	−3.33618	−0.166393
$403$	−0.185695	−0.00925014
$404$	3.31252	0.164804
$405$	2.95294	0.146733
$406$	−17.1203	−0.849666
$407$	7.61631	0.377527
$408$	−6.84618	−0.338937
$409$	24.5311	1.21298	0.606492	−	0.795090i	$-0.292576\pi$
$409$	24.5311	1.21298	0.606492	+	0.795090i	$0.292576\pi$
$410$	−14.1740	−0.700003
$411$	−2.08782	−0.102984
$412$	−3.27304	−0.161251
$413$	6.87188	0.338143
$414$	−7.16939	−0.352356
$415$	−45.8675	−2.25155
$416$	0.0599441	0.00293900
$417$	−6.76479	−0.331273
$418$	−7.65465	−0.374401
$419$	−39.8249	−1.94557	−0.972787	−	0.231701i	$-0.925571\pi$
$419$	−39.8249	−1.94557	−0.972787	+	0.231701i	$0.925571\pi$
$420$	−2.68556	−0.131042
$421$	−9.72422	−0.473929	−0.236965	−	0.971518i	$-0.576153\pi$
$421$	−9.72422	−0.473929	−0.236965	+	0.971518i	$0.576153\pi$
$422$	−3.08859	−0.150350
$423$	−8.17581	−0.397521
$424$	24.4196	1.18592
$425$	−8.45806	−0.410276
$426$	2.31094	0.111965
$427$	20.9904	1.01580
$428$	−2.00073	−0.0967091
$429$	−0.0363675	−0.00175584
$430$	−1.57297	−0.0758552
$431$	−19.1413	−0.922005	−0.461002	−	0.887399i	$-0.652510\pi$
$431$	−19.1413	−0.922005	−0.461002	+	0.887399i	$0.652510\pi$
$432$	3.23154	0.155477
$433$	8.86690	0.426116	0.213058	−	0.977040i	$-0.431658\pi$
$433$	8.86690	0.426116	0.213058	+	0.977040i	$0.431658\pi$
$434$	20.3735	0.977958
$435$	−14.1874	−0.680235
$436$	2.74210	0.131323
$437$	29.3450	1.40376
$438$	−11.0420	−0.527605
$439$	20.3568	0.971576	0.485788	−	0.874077i	$-0.338533\pi$
$439$	20.3568	0.971576	0.485788	+	0.874077i	$0.338533\pi$
$440$	−9.95576	−0.474622
$441$	0.602659	0.0286981
$442$	−0.0954381	−0.00453953
$443$	−27.0747	−1.28636	−0.643178	−	0.765717i	$-0.722384\pi$
$443$	−27.0747	−1.28636	−0.643178	+	0.765717i	$0.722384\pi$
$444$	2.24351	0.106472
$445$	−6.40262	−0.303513
$446$	−12.2164	−0.578462
$447$	−12.9511	−0.612566
$448$	−24.3973	−1.15267
$449$	−9.61610	−0.453812	−0.226906	−	0.973917i	$-0.572861\pi$
$449$	−9.61610	−0.453812	−0.226906	+	0.973917i	$0.572861\pi$
$450$	4.80736	0.226621
$451$	−4.15884	−0.195832
$452$	−1.88263	−0.0885516
$453$	−22.9359	−1.07762
$454$	37.5744	1.76345
$455$	−0.264445	−0.0123974
$456$	−15.9271	−0.745854
$457$	22.7159	1.06260	0.531302	−	0.847182i	$-0.321703\pi$
$457$	22.7159	1.06260	0.531302	+	0.847182i	$0.321703\pi$
$458$	−8.87951	−0.414912
$459$	2.27376	0.106130
$460$	5.40324	0.251927
$461$	0.156909	0.00730798	0.00365399	−	0.999993i	$-0.498837\pi$
$461$	0.156909	0.00730798	0.00365399	+	0.999993i	$0.498837\pi$
$462$	3.99004	0.185633
$463$	−37.7771	−1.75565	−0.877825	−	0.478981i	$-0.841006\pi$
$463$	−37.7771	−1.75565	−0.877825	+	0.478981i	$0.841006\pi$
$464$	−15.5259	−0.720774
$465$	16.8833	0.782944
$466$	−11.6956	−0.541787
$467$	−2.53969	−0.117523	−0.0587614	−	0.998272i	$-0.518715\pi$
$467$	−2.53969	−0.117523	−0.0587614	+	0.998272i	$0.518715\pi$
$468$	−0.0107126	−0.000495191 0
$469$	−7.11791	−0.328674
$470$	−31.2008	−1.43918
$471$	13.4796	0.621107
$472$	7.50409	0.345404
$473$	−0.461529	−0.0212211
$474$	9.53824	0.438106
$475$	−19.6770	−0.902841
$476$	−2.06787	−0.0947807
$477$	−8.11023	−0.371342
$478$	18.4616	0.844412
$479$	−31.1355	−1.42262	−0.711309	−	0.702880i	$-0.751897\pi$
$479$	−31.1355	−1.42262	−0.711309	+	0.702880i	$0.751897\pi$
$480$	−5.45008	−0.248761
$481$	0.220917	0.0100729
$482$	−29.4691	−1.34228
$483$	−15.2963	−0.696004
$484$	3.21464	0.146120
$485$	31.8785	1.44753
$486$	−1.29235	−0.0586221
$487$	−1.34130	−0.0607801	−0.0303901	−	0.999538i	$-0.509675\pi$
$487$	−1.34130	−0.0607801	−0.0303901	+	0.999538i	$0.509675\pi$
$488$	22.9215	1.03761
$489$	−2.58773	−0.117021
$490$	2.29989	0.103898
$491$	−40.1663	−1.81268	−0.906340	−	0.422548i	$-0.861136\pi$
$491$	−40.1663	−1.81268	−0.906340	+	0.422548i	$0.861136\pi$
$492$	−1.22505	−0.0552296
$493$	−10.9243	−0.492004
$494$	−0.222029	−0.00998955
$495$	3.30651	0.148616
$496$	18.4762	0.829604
$497$	4.93051	0.221164
$498$	20.0738	0.899530
$499$	−2.28026	−0.102079	−0.0510393	−	0.998697i	$-0.516253\pi$
$499$	−2.28026	−0.102079	−0.0510393	+	0.998697i	$0.516253\pi$
$500$	1.24683	0.0557600
$501$	−9.59034	−0.428465
$502$	−1.14450	−0.0510815
$503$	−21.8043	−0.972206	−0.486103	−	0.873902i	$-0.661582\pi$
$503$	−21.8043	−0.972206	−0.486103	+	0.873902i	$0.661582\pi$
$504$	8.30209	0.369805
$505$	−29.6563	−1.31969
$506$	−8.02780	−0.356879
$507$	12.9989	0.577303
$508$	−1.10777	−0.0491492
$509$	−28.4411	−1.26063	−0.630316	−	0.776339i	$-0.717075\pi$
$509$	−28.4411	−1.26063	−0.630316	+	0.776339i	$0.717075\pi$
$510$	8.67717	0.384232
$511$	−23.5586	−1.04217
$512$	−25.4243	−1.12361
$513$	5.28970	0.233546
$514$	13.4928	0.595144
$515$	29.3028	1.29124
$516$	−0.135951	−0.00598490
$517$	−9.15473	−0.402624
$518$	−24.2378	−1.06495
$519$	−21.1498	−0.928374
$520$	−0.288774	−0.0126636
$521$	−16.3230	−0.715123	−0.357562	−	0.933890i	$-0.616392\pi$
$521$	−16.3230	−0.715123	−0.357562	+	0.933890i	$0.616392\pi$
$522$	6.20910	0.271765
$523$	−30.3431	−1.32681	−0.663406	−	0.748259i	$-0.730890\pi$
$523$	−30.3431	−1.32681	−0.663406	+	0.748259i	$0.730890\pi$
$524$	1.18800	0.0518982
$525$	10.2567	0.447641
$526$	−5.46934	−0.238474
$527$	13.0001	0.566293
$528$	3.61846	0.157473
$529$	7.77546	0.338063
$530$	−30.9505	−1.34440
$531$	−2.49226	−0.108155
$532$	−4.81073	−0.208572
$533$	−0.120630	−0.00522507
$534$	2.80209	0.121258
$535$	17.9121	0.774409
$536$	−7.77274	−0.335731
$537$	0.425786	0.0183740
$538$	19.2066	0.828056
$539$	0.674817	0.0290664
$540$	0.973984	0.0419136
$541$	−5.88817	−0.253152	−0.126576	−	0.991957i	$-0.540399\pi$
$541$	−5.88817	−0.253152	−0.126576	+	0.991957i	$0.540399\pi$
$542$	9.25270	0.397438
$543$	6.11341	0.262352
$544$	−4.19654	−0.179925
$545$	−24.5494	−1.05158
$546$	0.115734	0.00495296
$547$	−15.1088	−0.646006	−0.323003	−	0.946398i	$-0.604692\pi$
$547$	−15.1088	−0.646006	−0.323003	+	0.946398i	$0.604692\pi$
$548$	−0.688636	−0.0294171
$549$	−7.61268	−0.324901
$550$	5.38296	0.229530
$551$	−25.4144	−1.08269
$552$	−16.7035	−0.710948
$553$	20.3503	0.865384
$554$	18.9947	0.807005
$555$	−20.0856	−0.852587
$556$	−2.23126	−0.0946267
$557$	15.0752	0.638755	0.319378	−	0.947628i	$-0.396526\pi$
$557$	15.0752	0.638755	0.319378	+	0.947628i	$0.396526\pi$
$558$	−7.38894	−0.312799
$559$	−0.0133870	−0.000566210 0
$560$	26.3116	1.11187
$561$	2.54600	0.107492
$562$	0.335478	0.0141513
$563$	−1.00000	−0.0421450
$563$	−1.00000	−0.0421450
$564$	−2.69667	−0.113550
$565$	16.8548	0.709087
$566$	35.4399	1.48965
$567$	−2.75729	−0.115795
$568$	5.38411	0.225912
$569$	26.5031	1.11107	0.555533	−	0.831494i	$-0.312514\pi$
$569$	26.5031	1.11107	0.555533	+	0.831494i	$0.312514\pi$
$570$	20.1867	0.845529
$571$	−29.7480	−1.24492	−0.622458	−	0.782653i	$-0.713866\pi$
$571$	−29.7480	−1.24492	−0.622458	+	0.782653i	$0.713866\pi$
$572$	−0.0119953	−0.000501547 0
$573$	−7.37329	−0.308024
$574$	13.2349	0.552413
$575$	−20.6362	−0.860588
$576$	8.84829	0.368679
$577$	−5.84557	−0.243354	−0.121677	−	0.992570i	$-0.538827\pi$
$577$	−5.84557	−0.243354	−0.121677	+	0.992570i	$0.538827\pi$
$578$	−15.2885	−0.635919
$579$	−5.13641	−0.213462
$580$	−4.67951	−0.194306
$581$	42.8285	1.77683
$582$	−13.9516	−0.578311
$583$	−9.08129	−0.376109
$584$	−25.7259	−1.06455
$585$	0.0959077	0.00396529
$586$	7.69435	0.317851
$587$	15.0611	0.621638	0.310819	−	0.950469i	$-0.399397\pi$
$587$	15.0611	0.621638	0.310819	+	0.950469i	$0.399397\pi$
$588$	0.198778	0.00819747
$589$	30.2436	1.24617
$590$	−9.51103	−0.391563
$591$	26.2508	1.07981
$592$	−21.9806	−0.903397
$593$	0.443990	0.0182325	0.00911625	−	0.999958i	$-0.497098\pi$
$593$	0.443990	0.0182325	0.00911625	+	0.999958i	$0.497098\pi$
$594$	−1.44709	−0.0593746
$595$	18.5132	0.758967
$596$	−4.27173	−0.174977
$597$	−4.18107	−0.171120
$598$	−0.232852	−0.00952204
$599$	15.8036	0.645717	0.322858	−	0.946447i	$-0.395356\pi$
$599$	15.8036	0.645717	0.322858	+	0.946447i	$0.395356\pi$
$600$	11.2004	0.457253
$601$	−22.0522	−0.899527	−0.449764	−	0.893148i	$-0.648492\pi$
$601$	−22.0522	−0.899527	−0.449764	+	0.893148i	$0.648492\pi$
$602$	1.46875	0.0598617
$603$	2.58148	0.105126
$604$	−7.56507	−0.307818
$605$	−28.7800	−1.17007
$606$	12.9790	0.527237
$607$	21.0160	0.853014	0.426507	−	0.904484i	$-0.359744\pi$
$607$	21.0160	0.853014	0.426507	+	0.904484i	$0.359744\pi$
$608$	−9.76291	−0.395938
$609$	13.2474	0.536813
$610$	−29.0517	−1.17627
$611$	−0.265540	−0.0107426
$612$	0.749964	0.0303155
$613$	−12.3580	−0.499135	−0.249568	−	0.968357i	$-0.580288\pi$
$613$	−12.3580	−0.499135	−0.249568	+	0.968357i	$0.580288\pi$
$614$	−23.7783	−0.959613
$615$	10.9676	0.442257
$616$	9.29613	0.374552
$617$	27.6008	1.11117	0.555583	−	0.831461i	$-0.312495\pi$
$617$	27.6008	1.11117	0.555583	+	0.831461i	$0.312495\pi$
$618$	−12.8243	−0.515870
$619$	19.3496	0.777725	0.388862	−	0.921296i	$-0.372868\pi$
$619$	19.3496	0.777725	0.388862	+	0.921296i	$0.372868\pi$
$620$	5.56871	0.223645
$621$	5.54756	0.222616
$622$	5.17033	0.207311
$623$	5.97841	0.239520
$624$	0.104956	0.00420161
$625$	−29.7619	−1.19048
$626$	16.8960	0.675301
$627$	5.92306	0.236544
$628$	4.44604	0.177416
$629$	−15.4659	−0.616664
$630$	−10.5225	−0.419225
$631$	−29.1480	−1.16037	−0.580183	−	0.814486i	$-0.697019\pi$
$631$	−29.1480	−1.16037	−0.580183	+	0.814486i	$0.697019\pi$
$632$	22.2225	0.883965
$633$	2.38990	0.0949901
$634$	26.6153	1.05703
$635$	9.91759	0.393568
$636$	−2.67504	−0.106072
$637$	0.0195736	0.000775534 0
$638$	6.95253	0.275253
$639$	−1.78817	−0.0707389
$640$	22.8670	0.903896
$641$	−7.26302	−0.286872	−0.143436	−	0.989660i	$-0.545815\pi$
$641$	−7.26302	−0.286872	−0.143436	+	0.989660i	$0.545815\pi$
$642$	−7.83921	−0.309389
$643$	−26.5076	−1.04536	−0.522679	−	0.852529i	$-0.675068\pi$
$643$	−26.5076	−1.04536	−0.522679	+	0.852529i	$0.675068\pi$
$644$	−5.04524	−0.198810
$645$	1.21714	0.0479247
$646$	15.5437	0.611559
$647$	18.4533	0.725475	0.362738	−	0.931891i	$-0.381842\pi$
$647$	18.4533	0.725475	0.362738	+	0.931891i	$0.381842\pi$
$648$	−3.01096	−0.118282
$649$	−2.79066	−0.109543
$650$	0.156137	0.00612419
$651$	−15.7647	−0.617867
$652$	−0.853523	−0.0334265
$653$	−6.56598	−0.256947	−0.128473	−	0.991713i	$-0.541008\pi$
$653$	−6.56598	−0.256947	−0.128473	+	0.991713i	$0.541008\pi$
$654$	10.7440	0.420124
$655$	−10.6359	−0.415580
$656$	12.0024	0.468614
$657$	8.54410	0.333337
$658$	29.1335	1.13574
$659$	38.9576	1.51757	0.758787	−	0.651339i	$-0.225792\pi$
$659$	38.9576	1.51757	0.758787	+	0.651339i	$0.225792\pi$
$660$	1.09060	0.0424516
$661$	−36.4623	−1.41822	−0.709110	−	0.705098i	$-0.750903\pi$
$661$	−36.4623	−1.41822	−0.709110	+	0.705098i	$0.750903\pi$
$662$	−33.6521	−1.30793
$663$	0.0738486	0.00286804
$664$	46.7687	1.81498
$665$	43.0694	1.67016
$666$	8.79043	0.340622
$667$	−26.6533	−1.03202
$668$	−3.16323	−0.122389
$669$	9.45284	0.365468
$670$	9.85154	0.380598
$671$	−8.52417	−0.329072
$672$	5.08898	0.196312
$673$	−23.8638	−0.919882	−0.459941	−	0.887950i	$-0.652129\pi$
$673$	−23.8638	−0.919882	−0.459941	+	0.887950i	$0.652129\pi$
$674$	26.9856	1.03944
$675$	−3.71986	−0.143178
$676$	4.28751	0.164904
$677$	2.06741	0.0794572	0.0397286	−	0.999211i	$-0.487351\pi$
$677$	2.06741	0.0794572	0.0397286	+	0.999211i	$0.487351\pi$
$678$	−7.37647	−0.283292
$679$	−29.7664	−1.14233
$680$	20.2164	0.775263
$681$	−29.0745	−1.11414
$682$	−8.27364	−0.316814
$683$	−2.61604	−0.100100	−0.0500501	−	0.998747i	$-0.515938\pi$
$683$	−2.61604	−0.100100	−0.0500501	+	0.998747i	$0.515938\pi$
$684$	1.74473	0.0667114
$685$	6.16521	0.235560
$686$	22.7962	0.870362
$687$	6.87083	0.262139
$688$	1.33197	0.0507808
$689$	−0.263410	−0.0100351
$690$	21.1708	0.805958
$691$	−15.4329	−0.587097	−0.293548	−	0.955944i	$-0.594836\pi$
$691$	−15.4329	−0.587097	−0.293548	+	0.955944i	$0.594836\pi$
$692$	−6.97595	−0.265186
$693$	−3.08743	−0.117282
$694$	14.5068	0.550672
$695$	19.9760	0.757734
$696$	14.4662	0.548339
$697$	8.44503	0.319878
$698$	−33.6807	−1.27483
$699$	9.04986	0.342297
$700$	3.38304	0.127867
$701$	−16.9534	−0.640319	−0.320160	−	0.947364i	$-0.603737\pi$
$701$	−16.9534	−0.640319	−0.320160	+	0.947364i	$0.603737\pi$
$702$	−0.0419738	−0.00158420
$703$	−35.9800	−1.35701
$704$	9.90773	0.373411
$705$	24.1427	0.909266
$706$	29.6850	1.11721
$707$	27.6914	1.04144
$708$	−0.822035	−0.0308940
$709$	−27.1727	−1.02049	−0.510245	−	0.860029i	$-0.670445\pi$
$709$	−27.1727	−1.02049	−0.510245	+	0.860029i	$0.670445\pi$
$710$	−6.82407	−0.256103
$711$	−7.38055	−0.276792
$712$	6.52842	0.244663
$713$	31.7179	1.18785
$714$	−8.10226	−0.303219
$715$	0.107391	0.00401620
$716$	0.140439	0.00524846
$717$	−14.2853	−0.533493
$718$	−3.24668	−0.121165
$719$	19.6527	0.732923	0.366461	−	0.930433i	$-0.380569\pi$
$719$	19.6527	0.732923	0.366461	+	0.930433i	$0.380569\pi$
$720$	−9.54254	−0.355630
$721$	−27.3614	−1.01899
$722$	11.6065	0.431950
$723$	22.8028	0.848044
$724$	2.01642	0.0749396
$725$	17.8721	0.663753
$726$	12.5955	0.467462
$727$	0.946373	0.0350990	0.0175495	−	0.999846i	$-0.494414\pi$
$727$	0.946373	0.0350990	0.0175495	+	0.999846i	$0.494414\pi$
$728$	0.269641	0.00999357
$729$	1.00000	0.0370370
$730$	32.6062	1.20681
$731$	0.937192	0.0346633
$732$	−2.51093	−0.0928066
$733$	−5.94689	−0.219653	−0.109827	−	0.993951i	$-0.535030\pi$
$733$	−5.94689	−0.219653	−0.109827	+	0.993951i	$0.535030\pi$
$734$	−45.0795	−1.66392
$735$	−1.77962	−0.0656422
$736$	−10.2388	−0.377408
$737$	2.89057	0.106476
$738$	−4.79995	−0.176689
$739$	37.8182	1.39116	0.695582	−	0.718447i	$-0.255147\pi$
$739$	37.8182	1.39116	0.695582	+	0.718447i	$0.255147\pi$
$740$	−6.62494	−0.243538
$741$	0.171803	0.00631133
$742$	28.8999	1.06095
$743$	−31.3872	−1.15148	−0.575742	−	0.817631i	$-0.695287\pi$
$743$	−31.3872	−1.15148	−0.575742	+	0.817631i	$0.695287\pi$
$744$	−17.2150	−0.631133
$745$	38.2439	1.40115
$746$	−3.52141	−0.128928
$747$	−15.5328	−0.568317
$748$	0.839760	0.0307047
$749$	−16.7253	−0.611131
$750$	4.88530	0.178386
$751$	−17.8416	−0.651050	−0.325525	−	0.945533i	$-0.605541\pi$
$751$	−17.8416	−0.651050	−0.325525	+	0.945533i	$0.605541\pi$
$752$	26.4204	0.963454
$753$	0.885596	0.0322729
$754$	0.201663	0.00734415
$755$	67.7284	2.46489
$756$	−0.909452	−0.0330764
$757$	−40.7728	−1.48191	−0.740956	−	0.671553i	$-0.765627\pi$
$757$	−40.7728	−1.48191	−0.740956	+	0.671553i	$0.765627\pi$
$758$	36.1405	1.31268
$759$	6.21179	0.225474
$760$	47.0317	1.70602
$761$	−51.5253	−1.86779	−0.933896	−	0.357546i	$-0.883614\pi$
$761$	−51.5253	−1.86779	−0.933896	+	0.357546i	$0.883614\pi$
$762$	−4.34041	−0.157237
$763$	22.9229	0.829865
$764$	−2.43197	−0.0879856
$765$	−6.71427	−0.242755
$766$	13.4565	0.486202
$767$	−0.0809453	−0.00292277
$768$	7.68891	0.277450
$769$	−18.6105	−0.671113	−0.335557	−	0.942020i	$-0.608924\pi$
$769$	−18.6105	−0.671113	−0.335557	+	0.942020i	$0.608924\pi$
$770$	−11.7823	−0.424606
$771$	−10.4406	−0.376008
$772$	−1.69417	−0.0609744
$773$	−30.8108	−1.10819	−0.554094	−	0.832454i	$-0.686935\pi$
$773$	−30.8108	−1.10819	−0.554094	+	0.832454i	$0.686935\pi$
$774$	−0.532678	−0.0191467
$775$	−21.2681	−0.763974
$776$	−32.5048	−1.16686
$777$	18.7548	0.672826
$778$	39.8989	1.43044
$779$	19.6467	0.703915
$780$	0.0316337	0.00113267
$781$	−2.00227	−0.0716470
$782$	16.3014	0.582938
$783$	−4.80451	−0.171699
$784$	−1.94752	−0.0695542
$785$	−39.8044	−1.42068
$786$	4.65479	0.166031
$787$	−2.39446	−0.0853534	−0.0426767	−	0.999089i	$-0.513589\pi$
$787$	−2.39446	−0.0853534	−0.0426767	+	0.999089i	$0.513589\pi$
$788$	8.65844	0.308444
$789$	4.23209	0.150666
$790$	−28.1659	−1.00210
$791$	−15.7381	−0.559582
$792$	−3.37147	−0.119800
$793$	−0.247250	−0.00878010
$794$	−5.34654	−0.189741
$795$	23.9490	0.849385
$796$	−1.37906	−0.0488796
$797$	−25.0555	−0.887510	−0.443755	−	0.896148i	$-0.646354\pi$
$797$	−25.0555	−0.887510	−0.443755	+	0.896148i	$0.646354\pi$
$798$	−18.8492	−0.667256
$799$	18.5898	0.657659
$800$	6.86554	0.242734
$801$	−2.16822	−0.0766102
$802$	−15.2350	−0.537966
$803$	9.56711	0.337616
$804$	0.851464	0.0300288
$805$	45.1689	1.59200
$806$	−0.239983	−0.00845305
$807$	−14.8618	−0.523160
$808$	30.2390	1.06380
$809$	33.8100	1.18870	0.594348	−	0.804208i	$-0.297410\pi$
$809$	33.8100	1.18870	0.594348	+	0.804208i	$0.297410\pi$
$810$	3.81623	0.134089
$811$	−5.97357	−0.209760	−0.104880	−	0.994485i	$-0.533446\pi$
$811$	−5.97357	−0.209760	−0.104880	+	0.994485i	$0.533446\pi$
$812$	4.36947	0.153338
$813$	−7.15960	−0.251098
$814$	9.84293	0.344995
$815$	7.64141	0.267667
$816$	−7.34773	−0.257222
$817$	2.18030	0.0762790
$818$	31.7027	1.10846
$819$	−0.0895532	−0.00312924
$820$	3.61751	0.126329
$821$	36.6076	1.27761	0.638807	−	0.769367i	$-0.279428\pi$
$821$	36.6076	1.27761	0.638807	+	0.769367i	$0.279428\pi$
$822$	−2.69819	−0.0941102
$823$	12.3274	0.429708	0.214854	−	0.976646i	$-0.431072\pi$
$823$	12.3274	0.429708	0.214854	+	0.976646i	$0.431072\pi$
$824$	−29.8786	−1.04087
$825$	−4.16525	−0.145016
$826$	8.88087	0.309005
$827$	−30.4622	−1.05927	−0.529637	−	0.848225i	$-0.677672\pi$
$827$	−30.4622	−1.05927	−0.529637	+	0.848225i	$0.677672\pi$
$828$	1.82978	0.0635893
$829$	18.4439	0.640582	0.320291	−	0.947319i	$-0.396219\pi$
$829$	18.4439	0.640582	0.320291	+	0.947319i	$0.396219\pi$
$830$	−59.2768	−2.05753
$831$	−14.6978	−0.509860
$832$	0.287381	0.00996314
$833$	−1.37030	−0.0474781
$834$	−8.74246	−0.302727
$835$	28.3197	0.980044
$836$	1.95363	0.0675678
$837$	5.71745	0.197624
$838$	−51.4677	−1.77792
$839$	−28.3751	−0.979617	−0.489808	−	0.871830i	$-0.662933\pi$
$839$	−28.3751	−0.979617	−0.489808	+	0.871830i	$0.662933\pi$
$840$	−24.5156	−0.845868
$841$	−5.91672	−0.204025
$842$	−12.5671	−0.433090
$843$	−0.259588	−0.00894069
$844$	0.788274	0.0271335
$845$	−38.3851	−1.32049
$846$	−10.5660	−0.363266
$847$	26.8731	0.923371
$848$	26.2085	0.900004
$849$	−27.4229	−0.941151
$850$	−10.9308	−0.374922
$851$	−37.7340	−1.29350
$852$	−0.589802	−0.0202063
$853$	21.3718	0.731758	0.365879	−	0.930662i	$-0.380768\pi$
$853$	21.3718	0.731758	0.365879	+	0.930662i	$0.380768\pi$
$854$	27.1269	0.928263
$855$	−15.6202	−0.534199
$856$	−18.2641	−0.624253
$857$	−10.6800	−0.364822	−0.182411	−	0.983222i	$-0.558390\pi$
$857$	−10.6800	−0.364822	−0.182411	+	0.983222i	$0.558390\pi$
$858$	−0.0469994	−0.00160453
$859$	−49.3000	−1.68209	−0.841047	−	0.540962i	$-0.818060\pi$
$859$	−49.3000	−1.68209	−0.841047	+	0.540962i	$0.818060\pi$
$860$	0.401455	0.0136895
$861$	−10.2409	−0.349011
$862$	−24.7372	−0.842554
$863$	−28.0667	−0.955403	−0.477701	−	0.878522i	$-0.658530\pi$
$863$	−28.0667	−0.955403	−0.477701	+	0.878522i	$0.658530\pi$
$864$	−1.84564	−0.0627901
$865$	62.4542	2.12351
$866$	11.4591	0.389397
$867$	11.8300	0.401769
$868$	−5.19975	−0.176491
$869$	−8.26425	−0.280345
$870$	−18.3351	−0.621618
$871$	0.0838432	0.00284092
$872$	25.0318	0.847683
$873$	10.7955	0.365373
$874$	37.9239	1.28280
$875$	10.4230	0.352363
$876$	2.81814	0.0952163
$877$	−42.6659	−1.44072	−0.720362	−	0.693599i	$-0.756024\pi$
$877$	−42.6659	−1.44072	−0.720362	+	0.693599i	$0.756024\pi$
$878$	26.3081	0.887854
$879$	−5.95377	−0.200816
$880$	−10.6851	−0.360195
$881$	−16.8724	−0.568447	−0.284224	−	0.958758i	$-0.591736\pi$
$881$	−16.8724	−0.568447	−0.284224	+	0.958758i	$0.591736\pi$
$882$	0.778846	0.0262251
$883$	25.6615	0.863578	0.431789	−	0.901975i	$-0.357882\pi$
$883$	25.6615	0.863578	0.431789	+	0.901975i	$0.357882\pi$
$884$	0.0243579	0.000819244 0
$885$	7.35949	0.247387
$886$	−34.9899	−1.17551
$887$	15.2260	0.511240	0.255620	−	0.966777i	$-0.417720\pi$
$887$	15.2260	0.511240	0.255620	+	0.966777i	$0.417720\pi$
$888$	20.4802	0.687272
$889$	−9.26049	−0.310587
$890$	−8.27442	−0.277359
$891$	1.11973	0.0375125
$892$	3.11788	0.104394
$893$	43.2476	1.44723
$894$	−16.7373	−0.559781
$895$	−1.25732	−0.0420276
$896$	−21.3519	−0.713317
$897$	0.180178	0.00601595
$898$	−12.4274	−0.414706
$899$	−27.4695	−0.916160
$900$	−1.22694	−0.0408981
$901$	18.4407	0.614348
$902$	−5.37467	−0.178957
$903$	−1.13650	−0.0378202
$904$	−17.1860	−0.571596
$905$	−18.0525	−0.600087
$906$	−29.6412	−0.984762
$907$	2.13992	0.0710547	0.0355274	−	0.999369i	$-0.488689\pi$
$907$	2.13992	0.0710547	0.0355274	+	0.999369i	$0.488689\pi$
$908$	−9.58979	−0.318248
$909$	−10.0430	−0.333104
$910$	−0.341756	−0.0113291
$911$	35.5814	1.17887	0.589433	−	0.807818i	$-0.299351\pi$
$911$	35.5814	1.17887	0.589433	+	0.807818i	$0.299351\pi$
$912$	−17.0939	−0.566035
$913$	−17.3926	−0.575612
$914$	29.3568	0.971038
$915$	22.4798	0.743159
$916$	2.26624	0.0748787
$917$	9.93124	0.327958
$918$	2.93848	0.0969844
$919$	−15.1763	−0.500620	−0.250310	−	0.968166i	$-0.580533\pi$
$919$	−15.1763	−0.500620	−0.250310	+	0.968166i	$0.580533\pi$
$920$	49.3244	1.62618
$921$	18.3993	0.606277
$922$	0.202781	0.00667824
$923$	−0.0580774	−0.00191164
$924$	−1.01834	−0.0335010
$925$	25.3021	0.831929
$926$	−48.8212	−1.60436
$927$	9.92327	0.325923
$928$	8.86741	0.291087
$929$	−6.88177	−0.225783	−0.112892	−	0.993607i	$-0.536011\pi$
$929$	−6.88177	−0.225783	−0.112892	+	0.993607i	$0.536011\pi$
$930$	21.8191	0.715477
$931$	−3.18789	−0.104479
$932$	2.98496	0.0977757
$933$	−4.00072	−0.130978
$934$	−3.28216	−0.107396
$935$	−7.51819	−0.245871
$936$	−0.0977920	−0.00319643
$937$	−3.76184	−0.122894	−0.0614469	−	0.998110i	$-0.519571\pi$
$937$	−3.76184	−0.122894	−0.0614469	+	0.998110i	$0.519571\pi$
$938$	−9.19881	−0.300352
$939$	−13.0739	−0.426650
$940$	7.96311	0.259728
$941$	−9.87753	−0.321998	−0.160999	−	0.986955i	$-0.551472\pi$
$941$	−9.87753	−0.321998	−0.160999	+	0.986955i	$0.551472\pi$
$942$	17.4203	0.567585
$943$	20.6044	0.670971
$944$	8.05383	0.262130
$945$	8.14212	0.264863
$946$	−0.596457	−0.0193925
$947$	30.8496	1.00248	0.501239	−	0.865309i	$-0.332878\pi$
$947$	30.8496	1.00248	0.501239	+	0.865309i	$0.332878\pi$
$948$	−2.43436	−0.0790645
$949$	0.277501	0.00900807
$950$	−25.4295	−0.825042
$951$	−20.5945	−0.667824
$952$	−18.8769	−0.611805
$953$	−39.9535	−1.29422	−0.647111	−	0.762396i	$-0.724023\pi$
$953$	−39.9535	−1.29422	−0.647111	+	0.762396i	$0.724023\pi$
$954$	−10.4812	−0.339343
$955$	21.7729	0.704555
$956$	−4.71179	−0.152390
$957$	−5.37977	−0.173903
$958$	−40.2379	−1.30003
$959$	−5.75673	−0.185894
$960$	−26.1285	−0.843293
$961$	1.68925	0.0544920
$962$	0.285502	0.00920494
$963$	6.06586	0.195470
$964$	7.52116	0.242240
$965$	15.1675	0.488259
$966$	−19.7681	−0.636028
$967$	3.87908	0.124743	0.0623713	−	0.998053i	$-0.480134\pi$
$967$	3.87908	0.124743	0.0623713	+	0.998053i	$0.480134\pi$
$968$	29.3454	0.943197
$969$	−12.0275	−0.386379
$970$	41.1981	1.32279
$971$	13.8879	0.445683	0.222841	−	0.974855i	$-0.428467\pi$
$971$	13.8879	0.445683	0.222841	+	0.974855i	$0.428467\pi$
$972$	0.329835	0.0105795
$973$	−18.6525	−0.597972
$974$	−1.73343	−0.0555426
$975$	−0.120816	−0.00386922
$976$	24.6007	0.787448
$977$	−18.2430	−0.583645	−0.291823	−	0.956472i	$-0.594262\pi$
$977$	−18.2430	−0.583645	−0.291823	+	0.956472i	$0.594262\pi$
$978$	−3.34424	−0.106937
$979$	−2.42783	−0.0775936
$980$	−0.586980	−0.0187504
$981$	−8.31356	−0.265431
$982$	−51.9089	−1.65648
$983$	−59.4428	−1.89593	−0.947966	−	0.318372i	$-0.896864\pi$
$983$	−59.4428	−1.89593	−0.947966	+	0.318372i	$0.896864\pi$
$984$	−11.1831	−0.356504
$985$	−77.5171	−2.46990
$986$	−14.1180	−0.449608
$987$	−22.5431	−0.717555
$988$	0.0566665	0.00180280
$989$	2.28658	0.0727091
$990$	4.27316	0.135810
$991$	1.55982	0.0495492	0.0247746	−	0.999693i	$-0.492113\pi$
$991$	1.55982	0.0495492	0.0247746	+	0.999693i	$0.492113\pi$
$992$	−10.5524	−0.335038
$993$	26.0395	0.826339
$994$	6.37194	0.202106
$995$	12.3464	0.391409
$996$	−5.12327	−0.162337
$997$	24.0172	0.760631	0.380316	−	0.924857i	$-0.375815\pi$
$997$	24.0172	0.760631	0.380316	+	0.924857i	$0.375815\pi$
$998$	−2.94689	−0.0932823
$999$	−6.80190	−0.215203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1689.2.a.c.1.14	✓	19
3.2	odd	2		5067.2.a.h.1.6		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1689.2.a.c.1.14	✓	19	1.1	even	1	trivial
5067.2.a.h.1.6		19	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1689 = 3 \cdot 563 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1689.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29235 q^{2} -1.00000 q^{3} -0.329835 q^{4} +2.95294 q^{5} -1.29235 q^{6} -2.75729 q^{7} -3.01096 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.29235 q^{2} -1.00000 q^{3} -0.329835 q^{4} +2.95294 q^{5} -1.29235 q^{6} -2.75729 q^{7} -3.01096 q^{8} +1.00000 q^{9} +3.81623 q^{10} +1.11973 q^{11} +0.329835 q^{12} +0.0324787 q^{13} -3.56338 q^{14} -2.95294 q^{15} -3.23154 q^{16} -2.27376 q^{17} +1.29235 q^{18} -5.28970 q^{19} -0.973984 q^{20} +2.75729 q^{21} +1.44709 q^{22} -5.54756 q^{23} +3.01096 q^{24} +3.71986 q^{25} +0.0419738 q^{26} -1.00000 q^{27} +0.909452 q^{28} +4.80451 q^{29} -3.81623 q^{30} -5.71745 q^{31} +1.84564 q^{32} -1.11973 q^{33} -2.93848 q^{34} -8.14212 q^{35} -0.329835 q^{36} +6.80190 q^{37} -6.83614 q^{38} -0.0324787 q^{39} -8.89119 q^{40} -3.71413 q^{41} +3.56338 q^{42} -0.412178 q^{43} -0.369327 q^{44} +2.95294 q^{45} -7.16939 q^{46} -8.17581 q^{47} +3.23154 q^{48} +0.602659 q^{49} +4.80736 q^{50} +2.27376 q^{51} -0.0107126 q^{52} -8.11023 q^{53} -1.29235 q^{54} +3.30651 q^{55} +8.30209 q^{56} +5.28970 q^{57} +6.20910 q^{58} -2.49226 q^{59} +0.973984 q^{60} -7.61268 q^{61} -7.38894 q^{62} -2.75729 q^{63} +8.84829 q^{64} +0.0959077 q^{65} -1.44709 q^{66} +2.58148 q^{67} +0.749964 q^{68} +5.54756 q^{69} -10.5225 q^{70} -1.78817 q^{71} -3.01096 q^{72} +8.54410 q^{73} +8.79043 q^{74} -3.71986 q^{75} +1.74473 q^{76} -3.08743 q^{77} -0.0419738 q^{78} -7.38055 q^{79} -9.54254 q^{80} +1.00000 q^{81} -4.79995 q^{82} -15.5328 q^{83} -0.909452 q^{84} -6.71427 q^{85} -0.532678 q^{86} -4.80451 q^{87} -3.37147 q^{88} -2.16822 q^{89} +3.81623 q^{90} -0.0895532 q^{91} +1.82978 q^{92} +5.71745 q^{93} -10.5660 q^{94} -15.6202 q^{95} -1.84564 q^{96} +10.7955 q^{97} +0.778846 q^{98} +1.11973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10}) \) 19 * q + q^2 - 19 * q^3 + 11 * q^4 - q^5 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9	\( 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 13 q^{10} - 8 q^{11} - 11 q^{12} - 11 q^{13} - 12 q^{14} + q^{15} - 5 q^{16} + 3 q^{17} + q^{18} - 20 q^{19} - 7 q^{20} + 11 q^{21} - 10 q^{22} - 9 q^{23} - 3 q^{24} - 4 q^{25} - 19 q^{27} - 27 q^{28} + 3 q^{29} + 13 q^{30} - 52 q^{31} + 2 q^{32} + 8 q^{33} - 26 q^{34} - 5 q^{35} + 11 q^{36} - 17 q^{37} - q^{38} + 11 q^{39} - 31 q^{40} - 22 q^{41} + 12 q^{42} - 15 q^{43} - 17 q^{44} - q^{45} - 38 q^{46} - 6 q^{47} + 5 q^{48} - 26 q^{49} + 11 q^{50} - 3 q^{51} - 15 q^{52} + 33 q^{53} - q^{54} - 51 q^{55} - 22 q^{56} + 20 q^{57} - 32 q^{58} - 42 q^{59} + 7 q^{60} - 26 q^{61} - 13 q^{62} - 11 q^{63} - 37 q^{64} - 6 q^{65} + 10 q^{66} - 12 q^{67} - 12 q^{68} + 9 q^{69} + 3 q^{70} - 26 q^{71} + 3 q^{72} - 19 q^{73} - 17 q^{74} + 4 q^{75} - 52 q^{76} + 44 q^{77} - 56 q^{79} - 30 q^{80} + 19 q^{81} - 22 q^{82} - 9 q^{83} + 27 q^{84} - 29 q^{85} - 10 q^{86} - 3 q^{87} - 32 q^{88} - q^{89} - 13 q^{90} - 60 q^{91} + 10 q^{92} + 52 q^{93} - 29 q^{94} - 27 q^{95} - 2 q^{96} - 44 q^{97} - 9 q^{98} - 8 q^{99}+O(q^{100}) \) 19 * q + q^2 - 19 * q^3 + 11 * q^4 - q^5 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 - 13 * q^10 - 8 * q^11 - 11 * q^12 - 11 * q^13 - 12 * q^14 + q^15 - 5 * q^16 + 3 * q^17 + q^18 - 20 * q^19 - 7 * q^20 + 11 * q^21 - 10 * q^22 - 9 * q^23 - 3 * q^24 - 4 * q^25 - 19 * q^27 - 27 * q^28 + 3 * q^29 + 13 * q^30 - 52 * q^31 + 2 * q^32 + 8 * q^33 - 26 * q^34 - 5 * q^35 + 11 * q^36 - 17 * q^37 - q^38 + 11 * q^39 - 31 * q^40 - 22 * q^41 + 12 * q^42 - 15 * q^43 - 17 * q^44 - q^45 - 38 * q^46 - 6 * q^47 + 5 * q^48 - 26 * q^49 + 11 * q^50 - 3 * q^51 - 15 * q^52 + 33 * q^53 - q^54 - 51 * q^55 - 22 * q^56 + 20 * q^57 - 32 * q^58 - 42 * q^59 + 7 * q^60 - 26 * q^61 - 13 * q^62 - 11 * q^63 - 37 * q^64 - 6 * q^65 + 10 * q^66 - 12 * q^67 - 12 * q^68 + 9 * q^69 + 3 * q^70 - 26 * q^71 + 3 * q^72 - 19 * q^73 - 17 * q^74 + 4 * q^75 - 52 * q^76 + 44 * q^77 - 56 * q^79 - 30 * q^80 + 19 * q^81 - 22 * q^82 - 9 * q^83 + 27 * q^84 - 29 * q^85 - 10 * q^86 - 3 * q^87 - 32 * q^88 - q^89 - 13 * q^90 - 60 * q^91 + 10 * q^92 + 52 * q^93 - 29 * q^94 - 27 * q^95 - 2 * q^96 - 44 * q^97 - 9 * q^98 - 8 * q^99

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