LMFDB - Embedded newform 1689.2.a.c.1.3

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			1.3
Root			$-2.14263$ 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)

$f(q)$	$=$	$q-2.14263 q^{2} -1.00000 q^{3} +2.59086 q^{4} -0.408504 q^{5} +2.14263 q^{6} +1.65028 q^{7} -1.26600 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.14263 q^{2} -1.00000 q^{3} +2.59086 q^{4} -0.408504 q^{5} +2.14263 q^{6} +1.65028 q^{7} -1.26600 q^{8} +1.00000 q^{9} +0.875272 q^{10} +5.86332 q^{11} -2.59086 q^{12} -2.44607 q^{13} -3.53594 q^{14} +0.408504 q^{15} -2.46916 q^{16} -4.30924 q^{17} -2.14263 q^{18} -6.97130 q^{19} -1.05838 q^{20} -1.65028 q^{21} -12.5629 q^{22} +1.58811 q^{23} +1.26600 q^{24} -4.83312 q^{25} +5.24103 q^{26} -1.00000 q^{27} +4.27564 q^{28} +1.81730 q^{29} -0.875272 q^{30} -2.40212 q^{31} +7.82249 q^{32} -5.86332 q^{33} +9.23310 q^{34} -0.674145 q^{35} +2.59086 q^{36} +5.50553 q^{37} +14.9369 q^{38} +2.44607 q^{39} +0.517164 q^{40} -6.22817 q^{41} +3.53594 q^{42} +3.16440 q^{43} +15.1910 q^{44} -0.408504 q^{45} -3.40274 q^{46} +10.6576 q^{47} +2.46916 q^{48} -4.27658 q^{49} +10.3556 q^{50} +4.30924 q^{51} -6.33743 q^{52} +4.05277 q^{53} +2.14263 q^{54} -2.39519 q^{55} -2.08925 q^{56} +6.97130 q^{57} -3.89380 q^{58} -11.4612 q^{59} +1.05838 q^{60} -11.7144 q^{61} +5.14685 q^{62} +1.65028 q^{63} -11.8224 q^{64} +0.999229 q^{65} +12.5629 q^{66} -5.02241 q^{67} -11.1646 q^{68} -1.58811 q^{69} +1.44444 q^{70} -1.84996 q^{71} -1.26600 q^{72} +4.51249 q^{73} -11.7963 q^{74} +4.83312 q^{75} -18.0617 q^{76} +9.67610 q^{77} -5.24103 q^{78} +4.30307 q^{79} +1.00866 q^{80} +1.00000 q^{81} +13.3447 q^{82} +7.27200 q^{83} -4.27564 q^{84} +1.76034 q^{85} -6.78013 q^{86} -1.81730 q^{87} -7.42293 q^{88} -4.41928 q^{89} +0.875272 q^{90} -4.03670 q^{91} +4.11458 q^{92} +2.40212 q^{93} -22.8353 q^{94} +2.84780 q^{95} -7.82249 q^{96} -13.0992 q^{97} +9.16313 q^{98} +5.86332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.14263	−1.51507	−0.757534	−	0.652796i	$-0.773596\pi$
$2$	−2.14263	−1.51507	−0.757534	+	0.652796i	$0.773596\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.59086	1.29543
$5$	−0.408504	−0.182688	−0.0913442	−	0.995819i	$-0.529116\pi$
$5$	−0.408504	−0.182688	−0.0913442	+	0.995819i	$0.529116\pi$
$6$	2.14263	0.874725
$7$	1.65028	0.623747	0.311873	−	0.950124i	$-0.399044\pi$
$7$	1.65028	0.623747	0.311873	+	0.950124i	$0.399044\pi$
$8$	−1.26600	−0.447597
$9$	1.00000	0.333333
$10$	0.875272	0.276785
$11$	5.86332	1.76786	0.883928	−	0.467623i	$-0.154889\pi$
$11$	5.86332	1.76786	0.883928	+	0.467623i	$0.154889\pi$
$12$	−2.59086	−0.747917
$13$	−2.44607	−0.678418	−0.339209	−	0.940711i	$-0.610159\pi$
$13$	−2.44607	−0.678418	−0.339209	+	0.940711i	$0.610159\pi$
$14$	−3.53594	−0.945019
$15$	0.408504	0.105475
$16$	−2.46916	−0.617290
$17$	−4.30924	−1.04514	−0.522572	−	0.852595i	$-0.675027\pi$
$17$	−4.30924	−1.04514	−0.522572	+	0.852595i	$0.675027\pi$
$18$	−2.14263	−0.505023
$19$	−6.97130	−1.59933	−0.799663	−	0.600450i	$-0.794988\pi$
$19$	−6.97130	−1.59933	−0.799663	+	0.600450i	$0.794988\pi$
$20$	−1.05838	−0.236660
$21$	−1.65028	−0.360120
$22$	−12.5629	−2.67842
$23$	1.58811	0.331145	0.165572	−	0.986198i	$-0.447053\pi$
$23$	1.58811	0.331145	0.165572	+	0.986198i	$0.447053\pi$
$24$	1.26600	0.258420
$25$	−4.83312	−0.966625
$26$	5.24103	1.02785
$27$	−1.00000	−0.192450
$28$	4.27564	0.808020
$29$	1.81730	0.337464	0.168732	−	0.985662i	$-0.446033\pi$
$29$	1.81730	0.337464	0.168732	+	0.985662i	$0.446033\pi$
$30$	−0.875272	−0.159802
$31$	−2.40212	−0.431433	−0.215717	−	0.976456i	$-0.569209\pi$
$31$	−2.40212	−0.431433	−0.215717	+	0.976456i	$0.569209\pi$
$32$	7.82249	1.38283
$33$	−5.86332	−1.02067
$34$	9.23310	1.58346
$35$	−0.674145	−0.113951
$36$	2.59086	0.431810
$37$	5.50553	0.905103	0.452551	−	0.891738i	$-0.350514\pi$
$37$	5.50553	0.905103	0.452551	+	0.891738i	$0.350514\pi$
$38$	14.9369	2.42309
$39$	2.44607	0.391685
$40$	0.517164	0.0817708
$41$	−6.22817	−0.972677	−0.486338	−	0.873771i	$-0.661668\pi$
$41$	−6.22817	−0.972677	−0.486338	+	0.873771i	$0.661668\pi$
$42$	3.53594	0.545607
$43$	3.16440	0.482566	0.241283	−	0.970455i	$-0.422432\pi$
$43$	3.16440	0.482566	0.241283	+	0.970455i	$0.422432\pi$
$44$	15.1910	2.29013
$45$	−0.408504	−0.0608961
$46$	−3.40274	−0.501706
$47$	10.6576	1.55457	0.777285	−	0.629149i	$-0.216596\pi$
$47$	10.6576	1.55457	0.777285	+	0.629149i	$0.216596\pi$
$48$	2.46916	0.356393
$49$	−4.27658	−0.610940
$50$	10.3556	1.46450
$51$	4.30924	0.603414
$52$	−6.33743	−0.878844
$53$	4.05277	0.556690	0.278345	−	0.960481i	$-0.410214\pi$
$53$	4.05277	0.556690	0.278345	+	0.960481i	$0.410214\pi$
$54$	2.14263	0.291575
$55$	−2.39519	−0.322967
$56$	−2.08925	−0.279187
$57$	6.97130	0.923371
$58$	−3.89380	−0.511281
$59$	−11.4612	−1.49213	−0.746063	−	0.665875i	$-0.768058\pi$
$59$	−11.4612	−1.49213	−0.746063	+	0.665875i	$0.768058\pi$
$60$	1.05838	0.136636
$61$	−11.7144	−1.49987	−0.749935	−	0.661511i	$-0.769915\pi$
$61$	−11.7144	−1.49987	−0.749935	+	0.661511i	$0.769915\pi$
$62$	5.14685	0.653650
$63$	1.65028	0.207916
$64$	−11.8224	−1.47780
$65$	0.999229	0.123939
$66$	12.5629	1.54639
$67$	−5.02241	−0.613586	−0.306793	−	0.951776i	$-0.599256\pi$
$67$	−5.02241	−0.613586	−0.306793	+	0.951776i	$0.599256\pi$
$68$	−11.1646	−1.35391
$69$	−1.58811	−0.191186
$70$	1.44444	0.172644
$71$	−1.84996	−0.219550	−0.109775	−	0.993956i	$-0.535013\pi$
$71$	−1.84996	−0.219550	−0.109775	+	0.993956i	$0.535013\pi$
$72$	−1.26600	−0.149199
$73$	4.51249	0.528147	0.264073	−	0.964503i	$-0.414934\pi$
$73$	4.51249	0.528147	0.264073	+	0.964503i	$0.414934\pi$
$74$	−11.7963	−1.37129
$75$	4.83312	0.558081
$76$	−18.0617	−2.07181
$77$	9.67610	1.10269
$78$	−5.24103	−0.593429
$79$	4.30307	0.484133	0.242066	−	0.970260i	$-0.422175\pi$
$79$	4.30307	0.484133	0.242066	+	0.970260i	$0.422175\pi$
$80$	1.00866	0.112772
$81$	1.00000	0.111111
$82$	13.3447	1.47367
$83$	7.27200	0.798205	0.399103	−	0.916906i	$-0.369322\pi$
$83$	7.27200	0.798205	0.399103	+	0.916906i	$0.369322\pi$
$84$	−4.27564	−0.466511
$85$	1.76034	0.190936
$86$	−6.78013	−0.731120
$87$	−1.81730	−0.194835
$88$	−7.42293	−0.791287
$89$	−4.41928	−0.468443	−0.234221	−	0.972183i	$-0.575254\pi$
$89$	−4.41928	−0.468443	−0.234221	+	0.972183i	$0.575254\pi$
$90$	0.875272	0.0922617
$91$	−4.03670	−0.423161
$92$	4.11458	0.428975
$93$	2.40212	0.249088
$94$	−22.8353	−2.35528
$95$	2.84780	0.292178
$96$	−7.82249	−0.798380
$97$	−13.0992	−1.33002	−0.665012	−	0.746833i	$-0.731574\pi$
$97$	−13.0992	−1.33002	−0.665012	+	0.746833i	$0.731574\pi$
$98$	9.16313	0.925616
$99$	5.86332	0.589285
$100$	−12.5220	−1.25220
$101$	12.8481	1.27843	0.639217	−	0.769026i	$-0.279259\pi$
$101$	12.8481	1.27843	0.639217	+	0.769026i	$0.279259\pi$
$102$	−9.23310	−0.914214
$103$	14.7260	1.45099	0.725497	−	0.688226i	$-0.241610\pi$
$103$	14.7260	1.45099	0.725497	+	0.688226i	$0.241610\pi$
$104$	3.09672	0.303658
$105$	0.674145	0.0657898
$106$	−8.68358	−0.843423
$107$	−12.3237	−1.19137	−0.595687	−	0.803217i	$-0.703120\pi$
$107$	−12.3237	−1.19137	−0.595687	+	0.803217i	$0.703120\pi$
$108$	−2.59086	−0.249306
$109$	−20.7631	−1.98874	−0.994372	−	0.105944i	$-0.966214\pi$
$109$	−20.7631	−1.98874	−0.994372	+	0.105944i	$0.966214\pi$
$110$	5.13199	0.489317
$111$	−5.50553	−0.522561
$112$	−4.07480	−0.385033
$113$	−11.7116	−1.10174	−0.550869	−	0.834591i	$-0.685704\pi$
$113$	−11.7116	−1.10174	−0.550869	+	0.834591i	$0.685704\pi$
$114$	−14.9369	−1.39897
$115$	−0.648750	−0.0604963
$116$	4.70837	0.437161
$117$	−2.44607	−0.226139
$118$	24.5572	2.26067
$119$	−7.11145	−0.651905
$120$	−0.517164	−0.0472104
$121$	23.3785	2.12532
$122$	25.0995	2.27241
$123$	6.22817	0.561575
$124$	−6.22355	−0.558891
$125$	4.01687	0.359279
$126$	−3.53594	−0.315006
$127$	4.81980	0.427688	0.213844	−	0.976868i	$-0.431402\pi$
$127$	4.81980	0.427688	0.213844	+	0.976868i	$0.431402\pi$
$128$	9.68599	0.856128
$129$	−3.16440	−0.278610
$130$	−2.14098	−0.187776
$131$	−18.7589	−1.63897	−0.819485	−	0.573100i	$-0.805741\pi$
$131$	−18.7589	−1.63897	−0.819485	+	0.573100i	$0.805741\pi$
$132$	−15.1910	−1.32221
$133$	−11.5046	−0.997574
$134$	10.7612	0.929624
$135$	0.408504	0.0351584
$136$	5.45548	0.467804
$137$	−17.2639	−1.47495	−0.737476	−	0.675373i	$-0.763983\pi$
$137$	−17.2639	−1.47495	−0.737476	+	0.675373i	$0.763983\pi$
$138$	3.40274	0.289660
$139$	8.47371	0.718731	0.359365	−	0.933197i	$-0.382993\pi$
$139$	8.47371	0.718731	0.359365	+	0.933197i	$0.382993\pi$
$140$	−1.74662	−0.147616
$141$	−10.6576	−0.897531
$142$	3.96379	0.332634
$143$	−14.3421	−1.19935
$144$	−2.46916	−0.205763
$145$	−0.742373	−0.0616508
$146$	−9.66859	−0.800178
$147$	4.27658	0.352726
$148$	14.2641	1.17250
$149$	17.8182	1.45972	0.729862	−	0.683595i	$-0.239584\pi$
$149$	17.8182	1.45972	0.729862	+	0.683595i	$0.239584\pi$
$150$	−10.3556	−0.845531
$151$	11.8538	0.964651	0.482325	−	0.875992i	$-0.339792\pi$
$151$	11.8538	0.964651	0.482325	+	0.875992i	$0.339792\pi$
$152$	8.82563	0.715853
$153$	−4.30924	−0.348381
$154$	−20.7323	−1.67066
$155$	0.981274	0.0788178
$156$	6.33743	0.507401
$157$	−5.10296	−0.407261	−0.203630	−	0.979048i	$-0.565274\pi$
$157$	−5.10296	−0.407261	−0.203630	+	0.979048i	$0.565274\pi$
$158$	−9.21988	−0.733494
$159$	−4.05277	−0.321405
$160$	−3.19552	−0.252628
$161$	2.62083	0.206550
$162$	−2.14263	−0.168341
$163$	10.8684	0.851283	0.425641	−	0.904892i	$-0.360049\pi$
$163$	10.8684	0.851283	0.425641	+	0.904892i	$0.360049\pi$
$164$	−16.1363	−1.26003
$165$	2.39519	0.186465
$166$	−15.5812	−1.20934
$167$	−9.99387	−0.773349	−0.386675	−	0.922216i	$-0.626376\pi$
$167$	−9.99387	−0.773349	−0.386675	+	0.922216i	$0.626376\pi$
$168$	2.08925	0.161189
$169$	−7.01673	−0.539749
$170$	−3.77176	−0.289280
$171$	−6.97130	−0.533108
$172$	8.19851	0.625131
$173$	10.4135	0.791727	0.395864	−	0.918309i	$-0.370445\pi$
$173$	10.4135	0.791727	0.395864	+	0.918309i	$0.370445\pi$
$174$	3.89380	0.295188
$175$	−7.97600	−0.602929
$176$	−14.4775	−1.09128
$177$	11.4612	0.861479
$178$	9.46888	0.709722
$179$	−16.7206	−1.24976	−0.624878	−	0.780722i	$-0.714851\pi$
$179$	−16.7206	−1.24976	−0.624878	+	0.780722i	$0.714851\pi$
$180$	−1.05838	−0.0788867
$181$	−13.2369	−0.983893	−0.491947	−	0.870625i	$-0.663715\pi$
$181$	−13.2369	−0.983893	−0.491947	+	0.870625i	$0.663715\pi$
$182$	8.64915	0.641118
$183$	11.7144	0.865951
$184$	−2.01055	−0.148219
$185$	−2.24903	−0.165352
$186$	−5.14685	−0.377385
$187$	−25.2664	−1.84766
$188$	27.6123	2.01384
$189$	−1.65028	−0.120040
$190$	−6.10178	−0.442670
$191$	−5.71465	−0.413497	−0.206749	−	0.978394i	$-0.566288\pi$
$191$	−5.71465	−0.413497	−0.206749	+	0.978394i	$0.566288\pi$
$192$	11.8224	0.853206
$193$	−2.12692	−0.153099	−0.0765495	−	0.997066i	$-0.524390\pi$
$193$	−2.12692	−0.153099	−0.0765495	+	0.997066i	$0.524390\pi$
$194$	28.0668	2.01508
$195$	−0.999229	−0.0715563
$196$	−11.0800	−0.791430
$197$	−6.08434	−0.433491	−0.216745	−	0.976228i	$-0.569544\pi$
$197$	−6.08434	−0.433491	−0.216745	+	0.976228i	$0.569544\pi$
$198$	−12.5629	−0.892807
$199$	−24.4545	−1.73354	−0.866768	−	0.498711i	$-0.833807\pi$
$199$	−24.4545	−1.73354	−0.866768	+	0.498711i	$0.833807\pi$
$200$	6.11872	0.432659
$201$	5.02241	0.354254
$202$	−27.5287	−1.93691
$203$	2.99905	0.210492
$204$	11.1646	0.781681
$205$	2.54423	0.177697
$206$	−31.5523	−2.19835
$207$	1.58811	0.110382
$208$	6.03975	0.418781
$209$	−40.8749	−2.82738
$210$	−1.44444	−0.0996760
$211$	−15.8773	−1.09304	−0.546519	−	0.837446i	$-0.684047\pi$
$211$	−15.8773	−1.09304	−0.546519	+	0.837446i	$0.684047\pi$
$212$	10.5002	0.721154
$213$	1.84996	0.126757
$214$	26.4050	1.80501
$215$	−1.29267	−0.0881592
$216$	1.26600	0.0861401
$217$	−3.96416	−0.269105
$218$	44.4876	3.01308
$219$	−4.51249	−0.304926
$220$	−6.20559	−0.418381
$221$	10.5407	0.709045
$222$	11.7963	0.791716
$223$	−18.6301	−1.24756	−0.623780	−	0.781600i	$-0.714404\pi$
$223$	−18.6301	−1.24756	−0.623780	+	0.781600i	$0.714404\pi$
$224$	12.9093	0.862538
$225$	−4.83312	−0.322208
$226$	25.0937	1.66921
$227$	1.67712	0.111314	0.0556572	−	0.998450i	$-0.482275\pi$
$227$	1.67712	0.111314	0.0556572	+	0.998450i	$0.482275\pi$
$228$	18.0617	1.19616
$229$	−8.94886	−0.591357	−0.295679	−	0.955287i	$-0.595546\pi$
$229$	−8.94886	−0.591357	−0.295679	+	0.955287i	$0.595546\pi$
$230$	1.39003	0.0916559
$231$	−9.67610	−0.636641
$232$	−2.30069	−0.151048
$233$	20.0055	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$233$	20.0055	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$234$	5.24103	0.342617
$235$	−4.35367	−0.284002
$236$	−29.6945	−1.93295
$237$	−4.30307	−0.279514
$238$	15.2372	0.987681
$239$	19.1062	1.23588	0.617939	−	0.786226i	$-0.287968\pi$
$239$	19.1062	1.23588	0.617939	+	0.786226i	$0.287968\pi$
$240$	−1.00866	−0.0651088
$241$	−9.63786	−0.620829	−0.310414	−	0.950601i	$-0.600468\pi$
$241$	−9.63786	−0.620829	−0.310414	+	0.950601i	$0.600468\pi$
$242$	−50.0914	−3.22000
$243$	−1.00000	−0.0641500
$244$	−30.3503	−1.94298
$245$	1.74700	0.111612
$246$	−13.3447	−0.850824
$247$	17.0523	1.08501
$248$	3.04107	0.193108
$249$	−7.27200	−0.460844
$250$	−8.60666	−0.544333
$251$	−13.9822	−0.882550	−0.441275	−	0.897372i	$-0.645474\pi$
$251$	−13.9822	−0.882550	−0.441275	+	0.897372i	$0.645474\pi$
$252$	4.27564	0.269340
$253$	9.31161	0.585416
$254$	−10.3270	−0.647977
$255$	−1.76034	−0.110237
$256$	2.89127	0.180704
$257$	14.9010	0.929497	0.464748	−	0.885443i	$-0.346145\pi$
$257$	14.9010	0.929497	0.464748	+	0.885443i	$0.346145\pi$
$258$	6.78013	0.422112
$259$	9.08565	0.564555
$260$	2.58886	0.160555
$261$	1.81730	0.112488
$262$	40.1933	2.48315
$263$	−9.38950	−0.578981	−0.289491	−	0.957181i	$-0.593486\pi$
$263$	−9.38950	−0.578981	−0.289491	+	0.957181i	$0.593486\pi$
$264$	7.42293	0.456850
$265$	−1.65557	−0.101701
$266$	24.6501	1.51139
$267$	4.41928	0.270455
$268$	−13.0124	−0.794858
$269$	25.4306	1.55053	0.775265	−	0.631636i	$-0.217616\pi$
$269$	25.4306	1.55053	0.775265	+	0.631636i	$0.217616\pi$
$270$	−0.875272	−0.0532673
$271$	6.54878	0.397810	0.198905	−	0.980019i	$-0.436261\pi$
$271$	6.54878	0.397810	0.198905	+	0.980019i	$0.436261\pi$
$272$	10.6402	0.645157
$273$	4.03670	0.244312
$274$	36.9901	2.23465
$275$	−28.3381	−1.70885
$276$	−4.11458	−0.247669
$277$	−26.1342	−1.57025	−0.785125	−	0.619337i	$-0.787401\pi$
$277$	−26.1342	−1.57025	−0.785125	+	0.619337i	$0.787401\pi$
$278$	−18.1560	−1.08893
$279$	−2.40212	−0.143811
$280$	0.853465	0.0510043
$281$	9.61512	0.573590	0.286795	−	0.957992i	$-0.407410\pi$
$281$	9.61512	0.573590	0.286795	+	0.957992i	$0.407410\pi$
$282$	22.8353	1.35982
$283$	14.3560	0.853375	0.426687	−	0.904399i	$-0.359681\pi$
$283$	14.3560	0.853375	0.426687	+	0.904399i	$0.359681\pi$
$284$	−4.79300	−0.284412
$285$	−2.84780	−0.168689
$286$	30.7298	1.81709
$287$	−10.2782	−0.606704
$288$	7.82249	0.460945
$289$	1.56955	0.0923263
$290$	1.59063	0.0934051
$291$	13.0992	0.767889
$292$	11.6912	0.684177
$293$	−15.8410	−0.925444	−0.462722	−	0.886504i	$-0.653127\pi$
$293$	−15.8410	−0.925444	−0.462722	+	0.886504i	$0.653127\pi$
$294$	−9.16313	−0.534404
$295$	4.68196	0.272594
$296$	−6.96997	−0.405121
$297$	−5.86332	−0.340224
$298$	−38.1778	−2.21158
$299$	−3.88464	−0.224655
$300$	12.5220	0.722955
$301$	5.22214	0.300999
$302$	−25.3984	−1.46151
$303$	−12.8481	−0.738104
$304$	17.2133	0.987248
$305$	4.78536	0.274009
$306$	9.23310	0.527821
$307$	−16.2796	−0.929128	−0.464564	−	0.885540i	$-0.653789\pi$
$307$	−16.2796	−0.929128	−0.464564	+	0.885540i	$0.653789\pi$
$308$	25.0694	1.42846
$309$	−14.7260	−0.837731
$310$	−2.10251	−0.119414
$311$	−2.23944	−0.126987	−0.0634934	−	0.997982i	$-0.520224\pi$
$311$	−2.23944	−0.126987	−0.0634934	+	0.997982i	$0.520224\pi$
$312$	−3.09672	−0.175317
$313$	−4.64993	−0.262830	−0.131415	−	0.991327i	$-0.541952\pi$
$313$	−4.64993	−0.262830	−0.131415	+	0.991327i	$0.541952\pi$
$314$	10.9338	0.617028
$315$	−0.674145	−0.0379838
$316$	11.1487	0.627161
$317$	0.416099	0.0233704	0.0116852	−	0.999932i	$-0.496280\pi$
$317$	0.416099	0.0233704	0.0116852	+	0.999932i	$0.496280\pi$
$318$	8.68358	0.486951
$319$	10.6554	0.596588
$320$	4.82948	0.269976
$321$	12.3237	0.687840
$322$	−5.61547	−0.312938
$323$	30.0410	1.67153
$324$	2.59086	0.143937
$325$	11.8222	0.655776
$326$	−23.2871	−1.28975
$327$	20.7631	1.14820
$328$	7.88484	0.435367
$329$	17.5880	0.969658
$330$	−5.13199	−0.282507
$331$	3.36185	0.184784	0.0923919	−	0.995723i	$-0.470549\pi$
$331$	3.36185	0.184784	0.0923919	+	0.995723i	$0.470549\pi$
$332$	18.8407	1.03402
$333$	5.50553	0.301701
$334$	21.4132	1.17168
$335$	2.05167	0.112095
$336$	4.07480	0.222299
$337$	−12.2001	−0.664581	−0.332290	−	0.943177i	$-0.607821\pi$
$337$	−12.2001	−0.664581	−0.332290	+	0.943177i	$0.607821\pi$
$338$	15.0343	0.817756
$339$	11.7116	0.636089
$340$	4.56080	0.247344
$341$	−14.0844	−0.762712
$342$	14.9369	0.807695
$343$	−18.6095	−1.00482
$344$	−4.00611	−0.215995
$345$	0.648750	0.0349275
$346$	−22.3124	−1.19952
$347$	−6.81244	−0.365711	−0.182856	−	0.983140i	$-0.558534\pi$
$347$	−6.81244	−0.365711	−0.182856	+	0.983140i	$0.558534\pi$
$348$	−4.70837	−0.252395
$349$	16.8228	0.900502	0.450251	−	0.892902i	$-0.351334\pi$
$349$	16.8228	0.900502	0.450251	+	0.892902i	$0.351334\pi$
$350$	17.0896	0.913478
$351$	2.44607	0.130562
$352$	45.8657	2.44465
$353$	−24.9005	−1.32532	−0.662659	−	0.748921i	$-0.730572\pi$
$353$	−24.9005	−1.32532	−0.662659	+	0.748921i	$0.730572\pi$
$354$	−24.5572	−1.30520
$355$	0.755717	0.0401093
$356$	−11.4497	−0.606835
$357$	7.11145	0.376378
$358$	35.8260	1.89347
$359$	−17.0800	−0.901449	−0.450724	−	0.892663i	$-0.648834\pi$
$359$	−17.0800	−0.901449	−0.450724	+	0.892663i	$0.648834\pi$
$360$	0.517164	0.0272569
$361$	29.5990	1.55784
$362$	28.3618	1.49067
$363$	−23.3785	−1.22705
$364$	−10.4585	−0.548176
$365$	−1.84337	−0.0964862
$366$	−25.0995	−1.31197
$367$	21.2709	1.11033	0.555166	−	0.831739i	$-0.312655\pi$
$367$	21.2709	1.11033	0.555166	+	0.831739i	$0.312655\pi$
$368$	−3.92131	−0.204412
$369$	−6.22817	−0.324226
$370$	4.81883	0.250519
$371$	6.68819	0.347234
$372$	6.22355	0.322676
$373$	25.6173	1.32642	0.663208	−	0.748435i	$-0.269194\pi$
$373$	25.6173	1.32642	0.663208	+	0.748435i	$0.269194\pi$
$374$	54.1366	2.79934
$375$	−4.01687	−0.207430
$376$	−13.4925	−0.695821
$377$	−4.44525	−0.228942
$378$	3.53594	0.181869
$379$	−18.6928	−0.960183	−0.480092	−	0.877218i	$-0.659397\pi$
$379$	−18.6928	−0.960183	−0.480092	+	0.877218i	$0.659397\pi$
$380$	7.37825	0.378496
$381$	−4.81980	−0.246926
$382$	12.2444	0.626477
$383$	−34.2959	−1.75244	−0.876220	−	0.481911i	$-0.839943\pi$
$383$	−34.2959	−1.75244	−0.876220	+	0.481911i	$0.839943\pi$
$384$	−9.68599	−0.494286
$385$	−3.95272	−0.201449
$386$	4.55720	0.231955
$387$	3.16440	0.160855
$388$	−33.9382	−1.72295
$389$	−6.07599	−0.308065	−0.154033	−	0.988066i	$-0.549226\pi$
$389$	−6.07599	−0.308065	−0.154033	+	0.988066i	$0.549226\pi$
$390$	2.14098	0.108413
$391$	−6.84356	−0.346094
$392$	5.41413	0.273455
$393$	18.7589	0.946260
$394$	13.0365	0.656768
$395$	−1.75782	−0.0884455
$396$	15.1910	0.763378
$397$	−2.85051	−0.143063	−0.0715314	−	0.997438i	$-0.522789\pi$
$397$	−2.85051	−0.143063	−0.0715314	+	0.997438i	$0.522789\pi$
$398$	52.3970	2.62643
$399$	11.5046	0.575949
$400$	11.9338	0.596688
$401$	23.8253	1.18978	0.594889	−	0.803808i	$-0.297196\pi$
$401$	23.8253	1.18978	0.594889	+	0.803808i	$0.297196\pi$
$402$	−10.7612	−0.536719
$403$	5.87575	0.292692
$404$	33.2876	1.65612
$405$	−0.408504	−0.0202987
$406$	−6.42585	−0.318910
$407$	32.2806	1.60009
$408$	−5.45548	−0.270087
$409$	−15.2168	−0.752422	−0.376211	−	0.926534i	$-0.622773\pi$
$409$	−15.2168	−0.752422	−0.376211	+	0.926534i	$0.622773\pi$
$410$	−5.45134	−0.269223
$411$	17.2639	0.851564
$412$	38.1530	1.87966
$413$	−18.9142	−0.930708
$414$	−3.40274	−0.167235
$415$	−2.97064	−0.145823
$416$	−19.1344	−0.938140
$417$	−8.47371	−0.414959
$418$	87.5798	4.28367
$419$	33.1174	1.61789	0.808945	−	0.587885i	$-0.200039\pi$
$419$	33.1174	1.61789	0.808945	+	0.587885i	$0.200039\pi$
$420$	1.74662	0.0852261
$421$	21.1437	1.03048	0.515239	−	0.857046i	$-0.327703\pi$
$421$	21.1437	1.03048	0.515239	+	0.857046i	$0.327703\pi$
$422$	34.0192	1.65603
$423$	10.6576	0.518190
$424$	−5.13079	−0.249173
$425$	20.8271	1.01026
$426$	−3.96379	−0.192046
$427$	−19.3320	−0.935539
$428$	−31.9289	−1.54334
$429$	14.3421	0.692443
$430$	2.76971	0.133567
$431$	−9.82396	−0.473204	−0.236602	−	0.971607i	$-0.576034\pi$
$431$	−9.82396	−0.473204	−0.236602	+	0.971607i	$0.576034\pi$
$432$	2.46916	0.118798
$433$	16.5277	0.794271	0.397135	−	0.917760i	$-0.370004\pi$
$433$	16.5277	0.794271	0.397135	+	0.917760i	$0.370004\pi$
$434$	8.49373	0.407712
$435$	0.742373	0.0355941
$436$	−53.7943	−2.57628
$437$	−11.0712	−0.529608
$438$	9.66859	0.461983
$439$	−34.0443	−1.62485	−0.812424	−	0.583067i	$-0.801852\pi$
$439$	−34.0443	−1.62485	−0.812424	+	0.583067i	$0.801852\pi$
$440$	3.03230	0.144559
$441$	−4.27658	−0.203647
$442$	−22.5848	−1.07425
$443$	−30.7426	−1.46062	−0.730311	−	0.683114i	$-0.760625\pi$
$443$	−30.7426	−1.46062	−0.730311	+	0.683114i	$0.760625\pi$
$444$	−14.2641	−0.676942
$445$	1.80529	0.0855790
$446$	39.9173	1.89014
$447$	−17.8182	−0.842772
$448$	−19.5102	−0.921771
$449$	−6.15866	−0.290645	−0.145322	−	0.989384i	$-0.546422\pi$
$449$	−6.15866	−0.290645	−0.145322	+	0.989384i	$0.546422\pi$
$450$	10.3556	0.488167
$451$	−36.5177	−1.71955
$452$	−30.3432	−1.42723
$453$	−11.8538	−0.556941
$454$	−3.59345	−0.168649
$455$	1.64901	0.0773066
$456$	−8.82563	−0.413298
$457$	10.9616	0.512761	0.256381	−	0.966576i	$-0.417470\pi$
$457$	10.9616	0.512761	0.256381	+	0.966576i	$0.417470\pi$
$458$	19.1741	0.895946
$459$	4.30924	0.201138
$460$	−1.68082	−0.0783687
$461$	−20.0595	−0.934262	−0.467131	−	0.884188i	$-0.654712\pi$
$461$	−20.0595	−0.934262	−0.467131	+	0.884188i	$0.654712\pi$
$462$	20.7323	0.964554
$463$	−34.0566	−1.58274	−0.791372	−	0.611335i	$-0.790633\pi$
$463$	−34.0566	−1.58274	−0.791372	+	0.611335i	$0.790633\pi$
$464$	−4.48721	−0.208313
$465$	−0.981274	−0.0455055
$466$	−42.8644	−1.98565
$467$	19.8640	0.919197	0.459598	−	0.888127i	$-0.347993\pi$
$467$	19.8640	0.919197	0.459598	+	0.888127i	$0.347993\pi$
$468$	−6.33743	−0.292948
$469$	−8.28838	−0.382722
$470$	9.32829	0.430282
$471$	5.10296	0.235132
$472$	14.5099	0.667871
$473$	18.5539	0.853107
$474$	9.21988	0.423483
$475$	33.6931	1.54595
$476$	−18.4248	−0.844498
$477$	4.05277	0.185563
$478$	−40.9375	−1.87244
$479$	−42.5361	−1.94352	−0.971761	−	0.235966i	$-0.924175\pi$
$479$	−42.5361	−1.94352	−0.971761	+	0.235966i	$0.924175\pi$
$480$	3.19552	0.145855
$481$	−13.4669	−0.614038
$482$	20.6504	0.940598
$483$	−2.62083	−0.119252
$484$	60.5704	2.75320
$485$	5.35108	0.242980
$486$	2.14263	0.0971916
$487$	−2.04297	−0.0925760	−0.0462880	−	0.998928i	$-0.514739\pi$
$487$	−2.04297	−0.0925760	−0.0462880	+	0.998928i	$0.514739\pi$
$488$	14.8303	0.671338
$489$	−10.8684	−0.491488
$490$	−3.74317	−0.169099
$491$	28.7192	1.29608	0.648039	−	0.761607i	$-0.275589\pi$
$491$	28.7192	1.29608	0.648039	+	0.761607i	$0.275589\pi$
$492$	16.1363	0.727482
$493$	−7.83118	−0.352699
$494$	−36.5367	−1.64387
$495$	−2.39519	−0.107656
$496$	5.93122	0.266319
$497$	−3.05296	−0.136944
$498$	15.5812	0.698210
$499$	−18.2337	−0.816251	−0.408126	−	0.912926i	$-0.633817\pi$
$499$	−18.2337	−0.816251	−0.408126	+	0.912926i	$0.633817\pi$
$500$	10.4071	0.465422
$501$	9.99387	0.446493
$502$	29.9587	1.33712
$503$	−16.4478	−0.733373	−0.366687	−	0.930344i	$-0.619508\pi$
$503$	−16.4478	−0.733373	−0.366687	+	0.930344i	$0.619508\pi$
$504$	−2.08925	−0.0930624
$505$	−5.24850	−0.233555
$506$	−19.9513	−0.886945
$507$	7.01673	0.311624
$508$	12.4874	0.554040
$509$	24.1917	1.07228	0.536139	−	0.844130i	$-0.319882\pi$
$509$	24.1917	1.07228	0.536139	+	0.844130i	$0.319882\pi$
$510$	3.77176	0.167016
$511$	7.44686	0.329430
$512$	−25.5669	−1.12991
$513$	6.97130	0.307790
$514$	−31.9273	−1.40825
$515$	−6.01561	−0.265080
$516$	−8.19851	−0.360919
$517$	62.4888	2.74826
$518$	−19.4672	−0.855339
$519$	−10.4135	−0.457104
$520$	−1.26502	−0.0554748
$521$	44.0421	1.92952	0.964760	−	0.263131i	$-0.0847553\pi$
$521$	44.0421	1.92952	0.964760	+	0.263131i	$0.0847553\pi$
$522$	−3.89380	−0.170427
$523$	23.3221	1.01980	0.509902	−	0.860233i	$-0.329682\pi$
$523$	23.3221	1.01980	0.509902	+	0.860233i	$0.329682\pi$
$524$	−48.6016	−2.12317
$525$	7.97600	0.348101
$526$	20.1182	0.877196
$527$	10.3513	0.450910
$528$	14.4775	0.630051
$529$	−20.4779	−0.890343
$530$	3.54727	0.154084
$531$	−11.4612	−0.497375
$532$	−29.8068	−1.29229
$533$	15.2346	0.659882
$534$	−9.46888	−0.409758
$535$	5.03426	0.217650
$536$	6.35836	0.274639
$537$	16.7206	0.721547
$538$	−54.4883	−2.34916
$539$	−25.0749	−1.08005
$540$	1.05838	0.0455453
$541$	−8.73066	−0.375360	−0.187680	−	0.982230i	$-0.560097\pi$
$541$	−8.73066	−0.375360	−0.187680	+	0.982230i	$0.560097\pi$
$542$	−14.0316	−0.602709
$543$	13.2369	0.568051
$544$	−33.7090	−1.44526
$545$	8.48180	0.363320
$546$	−8.64915	−0.370150
$547$	−6.19336	−0.264809	−0.132404	−	0.991196i	$-0.542270\pi$
$547$	−6.19336	−0.264809	−0.132404	+	0.991196i	$0.542270\pi$
$548$	−44.7283	−1.91070
$549$	−11.7144	−0.499957
$550$	60.7181	2.58903
$551$	−12.6689	−0.539715
$552$	2.01055	0.0855745
$553$	7.10126	0.301976
$554$	55.9959	2.37904
$555$	2.24903	0.0954659
$556$	21.9542	0.931065
$557$	15.7975	0.669363	0.334682	−	0.942331i	$-0.391371\pi$
$557$	15.7975	0.669363	0.334682	+	0.942331i	$0.391371\pi$
$558$	5.14685	0.217883
$559$	−7.74034	−0.327382
$560$	1.66457	0.0703410
$561$	25.2664	1.06675
$562$	−20.6016	−0.869028
$563$	−1.00000	−0.0421450
$563$	−1.00000	−0.0421450
$564$	−27.6123	−1.16269
$565$	4.78425	0.201275
$566$	−30.7595	−1.29292
$567$	1.65028	0.0693052
$568$	2.34205	0.0982701
$569$	7.66425	0.321302	0.160651	−	0.987011i	$-0.448641\pi$
$569$	7.66425	0.321302	0.160651	+	0.987011i	$0.448641\pi$
$570$	6.10178	0.255575
$571$	41.8589	1.75174	0.875871	−	0.482545i	$-0.160288\pi$
$571$	41.8589	1.75174	0.875871	+	0.482545i	$0.160288\pi$
$572$	−37.1584	−1.55367
$573$	5.71465	0.238733
$574$	22.0224	0.919197
$575$	−7.67555	−0.320093
$576$	−11.8224	−0.492599
$577$	7.21960	0.300556	0.150278	−	0.988644i	$-0.451983\pi$
$577$	7.21960	0.300556	0.150278	+	0.988644i	$0.451983\pi$
$578$	−3.36296	−0.139881
$579$	2.12692	0.0883918
$580$	−1.92339	−0.0798643
$581$	12.0008	0.497878
$582$	−28.0668	−1.16340
$583$	23.7626	0.984148
$584$	−5.71279	−0.236397
$585$	0.999229	0.0413130
$586$	33.9415	1.40211
$587$	−11.4219	−0.471433	−0.235717	−	0.971822i	$-0.575744\pi$
$587$	−11.4219	−0.471433	−0.235717	+	0.971822i	$0.575744\pi$
$588$	11.0800	0.456933
$589$	16.7459	0.690002
$590$	−10.0317	−0.412998
$591$	6.08434	0.250276
$592$	−13.5940	−0.558711
$593$	16.9488	0.696002	0.348001	−	0.937494i	$-0.386861\pi$
$593$	16.9488	0.696002	0.348001	+	0.937494i	$0.386861\pi$
$594$	12.5629	0.515463
$595$	2.90505	0.119095
$596$	46.1645	1.89097
$597$	24.4545	1.00086
$598$	8.32334	0.340367
$599$	−19.3311	−0.789847	−0.394923	−	0.918714i	$-0.629229\pi$
$599$	−19.3311	−0.789847	−0.394923	+	0.918714i	$0.629229\pi$
$600$	−6.11872	−0.249796
$601$	−3.50510	−0.142976	−0.0714880	−	0.997441i	$-0.522775\pi$
$601$	−3.50510	−0.142976	−0.0714880	+	0.997441i	$0.522775\pi$
$602$	−11.1891	−0.456034
$603$	−5.02241	−0.204529
$604$	30.7116	1.24964
$605$	−9.55019	−0.388270
$606$	27.5287	1.11828
$607$	−13.7858	−0.559547	−0.279774	−	0.960066i	$-0.590259\pi$
$607$	−13.7858	−0.559547	−0.279774	+	0.960066i	$0.590259\pi$
$608$	−54.5329	−2.21160
$609$	−2.99905	−0.121528
$610$	−10.2533	−0.415142
$611$	−26.0692	−1.05465
$612$	−11.1646	−0.451304
$613$	−30.0255	−1.21272	−0.606360	−	0.795190i	$-0.707371\pi$
$613$	−30.0255	−1.21272	−0.606360	+	0.795190i	$0.707371\pi$
$614$	34.8812	1.40769
$615$	−2.54423	−0.102593
$616$	−12.2499	−0.493563
$617$	−1.41223	−0.0568544	−0.0284272	−	0.999596i	$-0.509050\pi$
$617$	−1.41223	−0.0568544	−0.0284272	+	0.999596i	$0.509050\pi$
$618$	31.5523	1.26922
$619$	−44.2211	−1.77739	−0.888697	−	0.458494i	$-0.848389\pi$
$619$	−44.2211	−1.77739	−0.888697	+	0.458494i	$0.848389\pi$
$620$	2.54234	0.102103
$621$	−1.58811	−0.0637288
$622$	4.79828	0.192394
$623$	−7.29304	−0.292189
$624$	−6.03975	−0.241783
$625$	22.5247	0.900989
$626$	9.96309	0.398205
$627$	40.8749	1.63239
$628$	−13.2211	−0.527578
$629$	−23.7246	−0.945963
$630$	1.44444	0.0575480
$631$	10.8920	0.433602	0.216801	−	0.976216i	$-0.430438\pi$
$631$	10.8920	0.433602	0.216801	+	0.976216i	$0.430438\pi$
$632$	−5.44767	−0.216697
$633$	15.8773	0.631066
$634$	−0.891545	−0.0354078
$635$	−1.96891	−0.0781337
$636$	−10.5002	−0.416358
$637$	10.4608	0.414473
$638$	−22.8306	−0.903871
$639$	−1.84996	−0.0731835
$640$	−3.95676	−0.156405
$641$	−5.78938	−0.228667	−0.114333	−	0.993442i	$-0.536473\pi$
$641$	−5.78938	−0.228667	−0.114333	+	0.993442i	$0.536473\pi$
$642$	−26.4050	−1.04212
$643$	35.8341	1.41316	0.706580	−	0.707634i	$-0.250237\pi$
$643$	35.8341	1.41316	0.706580	+	0.707634i	$0.250237\pi$
$644$	6.79020	0.267572
$645$	1.29267	0.0508987
$646$	−64.3667	−2.53247
$647$	−9.54741	−0.375347	−0.187674	−	0.982231i	$-0.560095\pi$
$647$	−9.54741	−0.375347	−0.187674	+	0.982231i	$0.560095\pi$
$648$	−1.26600	−0.0497330
$649$	−67.2008	−2.63786
$650$	−25.3305	−0.993545
$651$	3.96416	0.155368
$652$	28.1586	1.10278
$653$	33.0408	1.29299	0.646494	−	0.762919i	$-0.276235\pi$
$653$	33.0408	1.29299	0.646494	+	0.762919i	$0.276235\pi$
$654$	−44.4876	−1.73960
$655$	7.66307	0.299421
$656$	15.3784	0.600424
$657$	4.51249	0.176049
$658$	−37.6846	−1.46910
$659$	37.0903	1.44483	0.722417	−	0.691458i	$-0.243031\pi$
$659$	37.0903	1.44483	0.722417	+	0.691458i	$0.243031\pi$
$660$	6.20559	0.241552
$661$	6.59940	0.256687	0.128344	−	0.991730i	$-0.459034\pi$
$661$	6.59940	0.256687	0.128344	+	0.991730i	$0.459034\pi$
$662$	−7.20319	−0.279960
$663$	−10.5407	−0.409367
$664$	−9.20632	−0.357274
$665$	4.69966	0.182245
$666$	−11.7963	−0.457097
$667$	2.88608	0.111749
$668$	−25.8927	−1.00182
$669$	18.6301	0.720280
$670$	−4.39598	−0.169831
$671$	−68.6850	−2.65156
$672$	−12.9093	−0.497987
$673$	38.7318	1.49300	0.746501	−	0.665384i	$-0.231732\pi$
$673$	38.7318	1.49300	0.746501	+	0.665384i	$0.231732\pi$
$674$	26.1403	1.00689
$675$	4.83312	0.186027
$676$	−18.1794	−0.699207
$677$	−1.87432	−0.0720359	−0.0360180	−	0.999351i	$-0.511467\pi$
$677$	−1.87432	−0.0720359	−0.0360180	+	0.999351i	$0.511467\pi$
$678$	−25.0937	−0.963718
$679$	−21.6174	−0.829598
$680$	−2.22858	−0.0854623
$681$	−1.67712	−0.0642674
$682$	30.1776	1.15556
$683$	−48.0603	−1.83898	−0.919488	−	0.393117i	$-0.871397\pi$
$683$	−48.0603	−1.83898	−0.919488	+	0.393117i	$0.871397\pi$
$684$	−18.0617	−0.690605
$685$	7.05236	0.269457
$686$	39.8733	1.52237
$687$	8.94886	0.341420
$688$	−7.81341	−0.297883
$689$	−9.91336	−0.377669
$690$	−1.39003	−0.0529176
$691$	2.88040	0.109576	0.0547879	−	0.998498i	$-0.482552\pi$
$691$	2.88040	0.109576	0.0547879	+	0.998498i	$0.482552\pi$
$692$	26.9801	1.02563
$693$	9.67610	0.367565
$694$	14.5965	0.554077
$695$	−3.46154	−0.131304
$696$	2.30069	0.0872076
$697$	26.8387	1.01659
$698$	−36.0450	−1.36432
$699$	−20.0055	−0.756678
$700$	−20.6647	−0.781053
$701$	8.95163	0.338098	0.169049	−	0.985608i	$-0.445930\pi$
$701$	8.95163	0.338098	0.169049	+	0.985608i	$0.445930\pi$
$702$	−5.24103	−0.197810
$703$	−38.3806	−1.44755
$704$	−69.3183	−2.61253
$705$	4.35367	0.163969
$706$	53.3525	2.00795
$707$	21.2029	0.797419
$708$	29.6945	1.11599
$709$	22.3493	0.839347	0.419673	−	0.907675i	$-0.362145\pi$
$709$	22.3493	0.839347	0.419673	+	0.907675i	$0.362145\pi$
$710$	−1.61922	−0.0607683
$711$	4.30307	0.161378
$712$	5.59479	0.209674
$713$	−3.81483	−0.142867
$714$	−15.2372	−0.570238
$715$	5.85880	0.219107
$716$	−43.3207	−1.61897
$717$	−19.1062	−0.713534
$718$	36.5961	1.36576
$719$	−4.81757	−0.179665	−0.0898326	−	0.995957i	$-0.528633\pi$
$719$	−4.81757	−0.179665	−0.0898326	+	0.995957i	$0.528633\pi$
$720$	1.00866	0.0375906
$721$	24.3020	0.905052
$722$	−63.4196	−2.36023
$723$	9.63786	0.358436
$724$	−34.2951	−1.27457
$725$	−8.78324	−0.326201
$726$	50.0914	1.85907
$727$	−6.29071	−0.233309	−0.116655	−	0.993173i	$-0.537217\pi$
$727$	−6.29071	−0.233309	−0.116655	+	0.993173i	$0.537217\pi$
$728$	5.11045	0.189406
$729$	1.00000	0.0370370
$730$	3.94965	0.146183
$731$	−13.6361	−0.504351
$732$	30.3503	1.12178
$733$	−0.145324	−0.00536766	−0.00268383	−	0.999996i	$-0.500854\pi$
$733$	−0.145324	−0.00536766	−0.00268383	+	0.999996i	$0.500854\pi$
$734$	−45.5757	−1.68223
$735$	−1.74700	−0.0644390
$736$	12.4230	0.457918
$737$	−29.4480	−1.08473
$738$	13.3447	0.491224
$739$	−35.8851	−1.32006	−0.660028	−	0.751241i	$-0.729456\pi$
$739$	−35.8851	−1.32006	−0.660028	+	0.751241i	$0.729456\pi$
$740$	−5.82692	−0.214202
$741$	−17.0523	−0.626432
$742$	−14.3303	−0.526083
$743$	13.5490	0.497064	0.248532	−	0.968624i	$-0.420052\pi$
$743$	13.5490	0.497064	0.248532	+	0.968624i	$0.420052\pi$
$744$	−3.04107	−0.111491
$745$	−7.27880	−0.266675
$746$	−54.8885	−2.00961
$747$	7.27200	0.266068
$748$	−65.4618	−2.39352
$749$	−20.3375	−0.743115
$750$	8.60666	0.314271
$751$	−19.4754	−0.710668	−0.355334	−	0.934739i	$-0.615633\pi$
$751$	−19.4754	−0.710668	−0.355334	+	0.934739i	$0.615633\pi$
$752$	−26.3153	−0.959621
$753$	13.9822	0.509541
$754$	9.52451	0.346862
$755$	−4.84233	−0.176230
$756$	−4.27564	−0.155504
$757$	−10.2572	−0.372804	−0.186402	−	0.982474i	$-0.559683\pi$
$757$	−10.2572	−0.372804	−0.186402	+	0.982474i	$0.559683\pi$
$758$	40.0517	1.45474
$759$	−9.31161	−0.337990
$760$	−3.60530	−0.130778
$761$	−34.0868	−1.23565	−0.617823	−	0.786317i	$-0.711985\pi$
$761$	−34.0868	−1.23565	−0.617823	+	0.786317i	$0.711985\pi$
$762$	10.3270	0.374109
$763$	−34.2649	−1.24047
$764$	−14.8059	−0.535657
$765$	1.76034	0.0636452
$766$	73.4835	2.65507
$767$	28.0350	1.01229
$768$	−2.89127	−0.104330
$769$	16.4814	0.594336	0.297168	−	0.954825i	$-0.403958\pi$
$769$	16.4814	0.594336	0.297168	+	0.954825i	$0.403958\pi$
$770$	8.46922	0.305210
$771$	−14.9010	−0.536645
$772$	−5.51056	−0.198329
$773$	13.5952	0.488985	0.244493	−	0.969651i	$-0.421379\pi$
$773$	13.5952	0.488985	0.244493	+	0.969651i	$0.421379\pi$
$774$	−6.78013	−0.243707
$775$	11.6097	0.417034
$776$	16.5836	0.595315
$777$	−9.08565	−0.325946
$778$	13.0186	0.466740
$779$	43.4184	1.55563
$780$	−2.58886	−0.0926962
$781$	−10.8469	−0.388133
$782$	14.6632	0.524356
$783$	−1.81730	−0.0649450
$784$	10.5596	0.377127
$785$	2.08458	0.0744018
$786$	−40.1933	−1.43365
$787$	53.1072	1.89307	0.946533	−	0.322607i	$-0.104559\pi$
$787$	53.1072	1.89307	0.946533	+	0.322607i	$0.104559\pi$
$788$	−15.7637	−0.561557
$789$	9.38950	0.334275
$790$	3.76635	0.134001
$791$	−19.3275	−0.687206
$792$	−7.42293	−0.263762
$793$	28.6542	1.01754
$794$	6.10758	0.216750
$795$	1.65557	0.0587170
$796$	−63.3583	−2.24568
$797$	−11.1817	−0.396075	−0.198037	−	0.980194i	$-0.563457\pi$
$797$	−11.1817	−0.396075	−0.198037	+	0.980194i	$0.563457\pi$
$798$	−24.6501	−0.872602
$799$	−45.9261	−1.62475
$800$	−37.8071	−1.33668
$801$	−4.41928	−0.156148
$802$	−51.0488	−1.80259
$803$	26.4581	0.933687
$804$	13.0124	0.458911
$805$	−1.07062	−0.0377343
$806$	−12.5896	−0.443448
$807$	−25.4306	−0.895199
$808$	−16.2656	−0.572223
$809$	20.4289	0.718242	0.359121	−	0.933291i	$-0.383076\pi$
$809$	20.4289	0.718242	0.359121	+	0.933291i	$0.383076\pi$
$810$	0.875272	0.0307539
$811$	−43.5565	−1.52948	−0.764738	−	0.644341i	$-0.777132\pi$
$811$	−43.5565	−1.52948	−0.764738	+	0.644341i	$0.777132\pi$
$812$	7.77012	0.272678
$813$	−6.54878	−0.229676
$814$	−69.1654	−2.42425
$815$	−4.43980	−0.155519
$816$	−10.6402	−0.372482
$817$	−22.0599	−0.771780
$818$	32.6039	1.13997
$819$	−4.03670	−0.141054
$820$	6.59174	0.230194
$821$	13.8764	0.484289	0.242145	−	0.970240i	$-0.422149\pi$
$821$	13.8764	0.484289	0.242145	+	0.970240i	$0.422149\pi$
$822$	−36.9901	−1.29018
$823$	19.8683	0.692564	0.346282	−	0.938130i	$-0.387444\pi$
$823$	19.8683	0.692564	0.346282	+	0.938130i	$0.387444\pi$
$824$	−18.6430	−0.649461
$825$	28.3381	0.986607
$826$	40.5262	1.41009
$827$	12.8331	0.446249	0.223125	−	0.974790i	$-0.428374\pi$
$827$	12.8331	0.446249	0.223125	+	0.974790i	$0.428374\pi$
$828$	4.11458	0.142992
$829$	−31.2274	−1.08457	−0.542287	−	0.840194i	$-0.682441\pi$
$829$	−31.2274	−1.08457	−0.542287	+	0.840194i	$0.682441\pi$
$830$	6.36497	0.220931
$831$	26.1342	0.906584
$832$	28.9184	1.00256
$833$	18.4288	0.638520
$834$	18.1560	0.628691
$835$	4.08253	0.141282
$836$	−105.901	−3.66267
$837$	2.40212	0.0830293
$838$	−70.9582	−2.45121
$839$	8.85999	0.305881	0.152940	−	0.988235i	$-0.451126\pi$
$839$	8.85999	0.305881	0.152940	+	0.988235i	$0.451126\pi$
$840$	−0.853465	−0.0294473
$841$	−25.6974	−0.886118
$842$	−45.3030	−1.56125
$843$	−9.61512	−0.331162
$844$	−41.1359	−1.41596
$845$	2.86636	0.0986058
$846$	−22.8353	−0.785093
$847$	38.5810	1.32566
$848$	−10.0069	−0.343640
$849$	−14.3560	−0.492696
$850$	−44.6247	−1.53062
$851$	8.74340	0.299720
$852$	4.79300	0.164205
$853$	53.8133	1.84253	0.921266	−	0.388933i	$-0.127156\pi$
$853$	53.8133	1.84253	0.921266	+	0.388933i	$0.127156\pi$
$854$	41.4212	1.41741
$855$	2.84780	0.0973927
$856$	15.6017	0.533255
$857$	−44.4691	−1.51904	−0.759518	−	0.650486i	$-0.774565\pi$
$857$	−44.4691	−1.51904	−0.759518	+	0.650486i	$0.774565\pi$
$858$	−30.7298	−1.04910
$859$	53.0285	1.80931	0.904654	−	0.426147i	$-0.140129\pi$
$859$	53.0285	1.80931	0.904654	+	0.426147i	$0.140129\pi$
$860$	−3.34912	−0.114204
$861$	10.2782	0.350281
$862$	21.0491	0.716936
$863$	−47.5071	−1.61716	−0.808581	−	0.588385i	$-0.799764\pi$
$863$	−47.5071	−1.61716	−0.808581	+	0.588385i	$0.799764\pi$
$864$	−7.82249	−0.266127
$865$	−4.25397	−0.144639
$866$	−35.4128	−1.20337
$867$	−1.56955	−0.0533046
$868$	−10.2706	−0.348607
$869$	25.2302	0.855877
$870$	−1.59063	−0.0539274
$871$	12.2852	0.416268
$872$	26.2860	0.890156
$873$	−13.0992	−0.443341
$874$	23.7215	0.802392
$875$	6.62895	0.224099
$876$	−11.6912	−0.395010
$877$	22.7825	0.769310	0.384655	−	0.923060i	$-0.374320\pi$
$877$	22.7825	0.769310	0.384655	+	0.923060i	$0.374320\pi$
$878$	72.9444	2.46175
$879$	15.8410	0.534305
$880$	5.91410	0.199364
$881$	−51.7917	−1.74491	−0.872453	−	0.488698i	$-0.837472\pi$
$881$	−51.7917	−1.74491	−0.872453	+	0.488698i	$0.837472\pi$
$882$	9.16313	0.308539
$883$	−18.2752	−0.615010	−0.307505	−	0.951546i	$-0.599494\pi$
$883$	−18.2752	−0.615010	−0.307505	+	0.951546i	$0.599494\pi$
$884$	27.3095	0.918518
$885$	−4.68196	−0.157382
$886$	65.8699	2.21294
$887$	−40.8555	−1.37180	−0.685898	−	0.727698i	$-0.740590\pi$
$887$	−40.8555	−1.37180	−0.685898	+	0.727698i	$0.740590\pi$
$888$	6.96997	0.233897
$889$	7.95401	0.266769
$890$	−3.86807	−0.129658
$891$	5.86332	0.196428
$892$	−48.2679	−1.61613
$893$	−74.2972	−2.48626
$894$	38.1778	1.27686
$895$	6.83042	0.228316
$896$	15.9846	0.534007
$897$	3.88464	0.129704
$898$	13.1957	0.440347
$899$	−4.36537	−0.145593
$900$	−12.5220	−0.417398
$901$	−17.4643	−0.581822
$902$	78.2439	2.60524
$903$	−5.22214	−0.173782
$904$	14.8269	0.493135
$905$	5.40733	0.179746
$906$	25.3984	0.843804
$907$	−17.1003	−0.567806	−0.283903	−	0.958853i	$-0.591629\pi$
$907$	−17.1003	−0.567806	−0.283903	+	0.958853i	$0.591629\pi$
$908$	4.34519	0.144200
$909$	12.8481	0.426145
$910$	−3.53321	−0.117125
$911$	52.8801	1.75199	0.875997	−	0.482316i	$-0.160204\pi$
$911$	52.8801	1.75199	0.875997	+	0.482316i	$0.160204\pi$
$912$	−17.2133	−0.569988
$913$	42.6380	1.41111
$914$	−23.4866	−0.776868
$915$	−4.78536	−0.158199
$916$	−23.1852	−0.766062
$917$	−30.9574	−1.02230
$918$	−9.23310	−0.304738
$919$	−54.5335	−1.79889	−0.899447	−	0.437031i	$-0.856030\pi$
$919$	−54.5335	−1.79889	−0.899447	+	0.437031i	$0.856030\pi$
$920$	0.821315	0.0270780
$921$	16.2796	0.536432
$922$	42.9800	1.41547
$923$	4.52515	0.148947
$924$	−25.0694	−0.824724
$925$	−26.6089	−0.874895
$926$	72.9706	2.39796
$927$	14.7260	0.483664
$928$	14.2158	0.466657
$929$	13.6453	0.447689	0.223845	−	0.974625i	$-0.428139\pi$
$929$	13.6453	0.447689	0.223845	+	0.974625i	$0.428139\pi$
$930$	2.10251	0.0689439
$931$	29.8133	0.977092
$932$	51.8315	1.69780
$933$	2.23944	0.0733159
$934$	−42.5612	−1.39265
$935$	10.3214	0.337547
$936$	3.09672	0.101219
$937$	−42.4174	−1.38571	−0.692857	−	0.721075i	$-0.743648\pi$
$937$	−42.4174	−1.38571	−0.692857	+	0.721075i	$0.743648\pi$
$938$	17.7589	0.579850
$939$	4.64993	0.151745
$940$	−11.2797	−0.367905
$941$	23.2831	0.759008	0.379504	−	0.925190i	$-0.376095\pi$
$941$	23.2831	0.759008	0.379504	+	0.925190i	$0.376095\pi$
$942$	−10.9338	−0.356241
$943$	−9.89104	−0.322097
$944$	28.2996	0.921075
$945$	0.674145	0.0219299
$946$	−39.7540	−1.29252
$947$	42.2028	1.37141	0.685703	−	0.727881i	$-0.259495\pi$
$947$	42.2028	1.37141	0.685703	+	0.727881i	$0.259495\pi$
$948$	−11.1487	−0.362091
$949$	−11.0379	−0.358304
$950$	−72.1919	−2.34222
$951$	−0.416099	−0.0134929
$952$	9.00306	0.291791
$953$	−6.76522	−0.219147	−0.109573	−	0.993979i	$-0.534948\pi$
$953$	−6.76522	−0.219147	−0.109573	+	0.993979i	$0.534948\pi$
$954$	−8.68358	−0.281141
$955$	2.33445	0.0755412
$956$	49.5015	1.60099
$957$	−10.6554	−0.344440
$958$	91.1390	2.94457
$959$	−28.4902	−0.919997
$960$	−4.82948	−0.155871
$961$	−25.2298	−0.813866
$962$	28.8546	0.930310
$963$	−12.3237	−0.397124
$964$	−24.9703	−0.804241
$965$	0.868855	0.0279694
$966$	5.61547	0.180675
$967$	26.0783	0.838621	0.419310	−	0.907843i	$-0.362272\pi$
$967$	26.0783	0.838621	0.419310	+	0.907843i	$0.362272\pi$
$968$	−29.5970	−0.951285
$969$	−30.0410	−0.965056
$970$	−11.4654	−0.368131
$971$	32.8185	1.05320	0.526598	−	0.850114i	$-0.323467\pi$
$971$	32.8185	1.05320	0.526598	+	0.850114i	$0.323467\pi$
$972$	−2.59086	−0.0831019
$973$	13.9840	0.448306
$974$	4.37733	0.140259
$975$	−11.8222	−0.378612
$976$	28.9247	0.925856
$977$	33.0972	1.05887	0.529437	−	0.848349i	$-0.322403\pi$
$977$	33.0972	1.05887	0.529437	+	0.848349i	$0.322403\pi$
$978$	23.2871	0.744638
$979$	−25.9116	−0.828139
$980$	4.52623	0.144585
$981$	−20.7631	−0.662915
$982$	−61.5345	−1.96365
$983$	48.5038	1.54703	0.773516	−	0.633777i	$-0.218496\pi$
$983$	48.5038	1.54703	0.773516	+	0.633777i	$0.218496\pi$
$984$	−7.88484	−0.251359
$985$	2.48547	0.0791938
$986$	16.7793	0.534362
$987$	−17.5880	−0.559832
$988$	44.1801	1.40556
$989$	5.02542	0.159799
$990$	5.13199	0.163106
$991$	−21.0825	−0.669706	−0.334853	−	0.942270i	$-0.608687\pi$
$991$	−21.0825	−0.669706	−0.334853	+	0.942270i	$0.608687\pi$
$992$	−18.7905	−0.596600
$993$	−3.36185	−0.106685
$994$	6.54135	0.207479
$995$	9.98977	0.316697
$996$	−18.8407	−0.596991
$997$	34.0039	1.07692	0.538458	−	0.842653i	$-0.319007\pi$
$997$	34.0039	1.07692	0.538458	+	0.842653i	$0.319007\pi$
$998$	39.0680	1.23668
$999$	−5.50553	−0.174187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1689.2.a.c.1.3	✓	19
3.2	odd	2		5067.2.a.h.1.17		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1689.2.a.c.1.3	✓	19	1.1	even	1	trivial
5067.2.a.h.1.17		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)

\(f(q)\)	\(=\)	\(q-2.14263 q^{2} -1.00000 q^{3} +2.59086 q^{4} -0.408504 q^{5} +2.14263 q^{6} +1.65028 q^{7} -1.26600 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.14263 q^{2} -1.00000 q^{3} +2.59086 q^{4} -0.408504 q^{5} +2.14263 q^{6} +1.65028 q^{7} -1.26600 q^{8} +1.00000 q^{9} +0.875272 q^{10} +5.86332 q^{11} -2.59086 q^{12} -2.44607 q^{13} -3.53594 q^{14} +0.408504 q^{15} -2.46916 q^{16} -4.30924 q^{17} -2.14263 q^{18} -6.97130 q^{19} -1.05838 q^{20} -1.65028 q^{21} -12.5629 q^{22} +1.58811 q^{23} +1.26600 q^{24} -4.83312 q^{25} +5.24103 q^{26} -1.00000 q^{27} +4.27564 q^{28} +1.81730 q^{29} -0.875272 q^{30} -2.40212 q^{31} +7.82249 q^{32} -5.86332 q^{33} +9.23310 q^{34} -0.674145 q^{35} +2.59086 q^{36} +5.50553 q^{37} +14.9369 q^{38} +2.44607 q^{39} +0.517164 q^{40} -6.22817 q^{41} +3.53594 q^{42} +3.16440 q^{43} +15.1910 q^{44} -0.408504 q^{45} -3.40274 q^{46} +10.6576 q^{47} +2.46916 q^{48} -4.27658 q^{49} +10.3556 q^{50} +4.30924 q^{51} -6.33743 q^{52} +4.05277 q^{53} +2.14263 q^{54} -2.39519 q^{55} -2.08925 q^{56} +6.97130 q^{57} -3.89380 q^{58} -11.4612 q^{59} +1.05838 q^{60} -11.7144 q^{61} +5.14685 q^{62} +1.65028 q^{63} -11.8224 q^{64} +0.999229 q^{65} +12.5629 q^{66} -5.02241 q^{67} -11.1646 q^{68} -1.58811 q^{69} +1.44444 q^{70} -1.84996 q^{71} -1.26600 q^{72} +4.51249 q^{73} -11.7963 q^{74} +4.83312 q^{75} -18.0617 q^{76} +9.67610 q^{77} -5.24103 q^{78} +4.30307 q^{79} +1.00866 q^{80} +1.00000 q^{81} +13.3447 q^{82} +7.27200 q^{83} -4.27564 q^{84} +1.76034 q^{85} -6.78013 q^{86} -1.81730 q^{87} -7.42293 q^{88} -4.41928 q^{89} +0.875272 q^{90} -4.03670 q^{91} +4.11458 q^{92} +2.40212 q^{93} -22.8353 q^{94} +2.84780 q^{95} -7.82249 q^{96} -13.0992 q^{97} +9.16313 q^{98} +5.86332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10}) \) 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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