LMFDB - Embedded newform 1689.2.a.c.1.16

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			1.16
Root			$1.87162$ 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)

$f(q)$	$=$	$q+1.87162 q^{2} -1.00000 q^{3} +1.50296 q^{4} -1.48997 q^{5} -1.87162 q^{6} +0.477684 q^{7} -0.930266 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.87162 q^{2} -1.00000 q^{3} +1.50296 q^{4} -1.48997 q^{5} -1.87162 q^{6} +0.477684 q^{7} -0.930266 q^{8} +1.00000 q^{9} -2.78865 q^{10} +4.18687 q^{11} -1.50296 q^{12} -4.36116 q^{13} +0.894042 q^{14} +1.48997 q^{15} -4.74703 q^{16} -1.38704 q^{17} +1.87162 q^{18} -1.78753 q^{19} -2.23936 q^{20} -0.477684 q^{21} +7.83623 q^{22} -0.497622 q^{23} +0.930266 q^{24} -2.78000 q^{25} -8.16243 q^{26} -1.00000 q^{27} +0.717940 q^{28} -10.2643 q^{29} +2.78865 q^{30} +2.12343 q^{31} -7.02410 q^{32} -4.18687 q^{33} -2.59601 q^{34} -0.711732 q^{35} +1.50296 q^{36} +1.76002 q^{37} -3.34558 q^{38} +4.36116 q^{39} +1.38607 q^{40} -0.227561 q^{41} -0.894042 q^{42} +10.0988 q^{43} +6.29271 q^{44} -1.48997 q^{45} -0.931359 q^{46} -8.29013 q^{47} +4.74703 q^{48} -6.77182 q^{49} -5.20310 q^{50} +1.38704 q^{51} -6.55466 q^{52} -4.55689 q^{53} -1.87162 q^{54} -6.23830 q^{55} -0.444373 q^{56} +1.78753 q^{57} -19.2109 q^{58} +1.38667 q^{59} +2.23936 q^{60} -10.5843 q^{61} +3.97425 q^{62} +0.477684 q^{63} -3.65240 q^{64} +6.49798 q^{65} -7.83623 q^{66} -4.84823 q^{67} -2.08467 q^{68} +0.497622 q^{69} -1.33209 q^{70} -5.79051 q^{71} -0.930266 q^{72} -4.56352 q^{73} +3.29408 q^{74} +2.78000 q^{75} -2.68659 q^{76} +2.00000 q^{77} +8.16243 q^{78} +2.90862 q^{79} +7.07291 q^{80} +1.00000 q^{81} -0.425907 q^{82} +5.12063 q^{83} -0.717940 q^{84} +2.06664 q^{85} +18.9011 q^{86} +10.2643 q^{87} -3.89490 q^{88} +2.49866 q^{89} -2.78865 q^{90} -2.08325 q^{91} -0.747907 q^{92} -2.12343 q^{93} -15.5160 q^{94} +2.66336 q^{95} +7.02410 q^{96} +2.38748 q^{97} -12.6743 q^{98} +4.18687 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.87162	1.32344	0.661718	−	0.749753i	$-0.269828\pi$
$2$	1.87162	1.32344	0.661718	+	0.749753i	$0.269828\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.50296	0.751481
$5$	−1.48997	−0.666333	−0.333167	−	0.942868i	$-0.608117\pi$
$5$	−1.48997	−0.666333	−0.333167	+	0.942868i	$0.608117\pi$
$6$	−1.87162	−0.764086
$7$	0.477684	0.180547	0.0902737	−	0.995917i	$-0.471226\pi$
$7$	0.477684	0.180547	0.0902737	+	0.995917i	$0.471226\pi$
$8$	−0.930266	−0.328899
$9$	1.00000	0.333333
$10$	−2.78865	−0.881849
$11$	4.18687	1.26239	0.631195	−	0.775625i	$-0.282565\pi$
$11$	4.18687	1.26239	0.631195	+	0.775625i	$0.282565\pi$
$12$	−1.50296	−0.433868
$13$	−4.36116	−1.20957	−0.604784	−	0.796390i	$-0.706741\pi$
$13$	−4.36116	−1.20957	−0.604784	+	0.796390i	$0.706741\pi$
$14$	0.894042	0.238943
$15$	1.48997	0.384708
$16$	−4.74703	−1.18676
$17$	−1.38704	−0.336407	−0.168203	−	0.985752i	$-0.553797\pi$
$17$	−1.38704	−0.336407	−0.168203	+	0.985752i	$0.553797\pi$
$18$	1.87162	0.441145
$19$	−1.78753	−0.410088	−0.205044	−	0.978753i	$-0.565734\pi$
$19$	−1.78753	−0.410088	−0.205044	+	0.978753i	$0.565734\pi$
$20$	−2.23936	−0.500737
$21$	−0.477684	−0.104239
$22$	7.83623	1.67069
$23$	−0.497622	−0.103761	−0.0518806	−	0.998653i	$-0.516522\pi$
$23$	−0.497622	−0.103761	−0.0518806	+	0.998653i	$0.516522\pi$
$24$	0.930266	0.189890
$25$	−2.78000	−0.556000
$26$	−8.16243	−1.60079
$27$	−1.00000	−0.192450
$28$	0.717940	0.135678
$29$	−10.2643	−1.90604	−0.953018	−	0.302915i	$-0.902040\pi$
$29$	−10.2643	−1.90604	−0.953018	+	0.302915i	$0.902040\pi$
$30$	2.78865	0.509136
$31$	2.12343	0.381379	0.190690	−	0.981650i	$-0.438928\pi$
$31$	2.12343	0.381379	0.190690	+	0.981650i	$0.438928\pi$
$32$	−7.02410	−1.24170
$33$	−4.18687	−0.728841
$34$	−2.59601	−0.445212
$35$	−0.711732	−0.120305
$36$	1.50296	0.250494
$37$	1.76002	0.289345	0.144672	−	0.989480i	$-0.453787\pi$
$37$	1.76002	0.289345	0.144672	+	0.989480i	$0.453787\pi$
$38$	−3.34558	−0.542725
$39$	4.36116	0.698344
$40$	1.38607	0.219156
$41$	−0.227561	−0.0355390	−0.0177695	−	0.999842i	$-0.505657\pi$
$41$	−0.227561	−0.0355390	−0.0177695	+	0.999842i	$0.505657\pi$
$42$	−0.894042	−0.137954
$43$	10.0988	1.54005	0.770025	−	0.638013i	$-0.220244\pi$
$43$	10.0988	1.54005	0.770025	+	0.638013i	$0.220244\pi$
$44$	6.29271	0.948662
$45$	−1.48997	−0.222111
$46$	−0.931359	−0.137321
$47$	−8.29013	−1.20924	−0.604620	−	0.796514i	$-0.706675\pi$
$47$	−8.29013	−1.20924	−0.604620	+	0.796514i	$0.706675\pi$
$48$	4.74703	0.685175
$49$	−6.77182	−0.967403
$50$	−5.20310	−0.735830
$51$	1.38704	0.194224
$52$	−6.55466	−0.908968
$53$	−4.55689	−0.625937	−0.312969	−	0.949763i	$-0.601323\pi$
$53$	−4.55689	−0.625937	−0.312969	+	0.949763i	$0.601323\pi$
$54$	−1.87162	−0.254695
$55$	−6.23830	−0.841172
$56$	−0.444373	−0.0593818
$57$	1.78753	0.236764
$58$	−19.2109	−2.52251
$59$	1.38667	0.180529	0.0902647	−	0.995918i	$-0.471229\pi$
$59$	1.38667	0.180529	0.0902647	+	0.995918i	$0.471229\pi$
$60$	2.23936	0.289101
$61$	−10.5843	−1.35518	−0.677590	−	0.735440i	$-0.736976\pi$
$61$	−10.5843	−1.35518	−0.677590	+	0.735440i	$0.736976\pi$
$62$	3.97425	0.504731
$63$	0.477684	0.0601825
$64$	−3.65240	−0.456549
$65$	6.49798	0.805975
$66$	−7.83623	−0.964574
$67$	−4.84823	−0.592306	−0.296153	−	0.955140i	$-0.595704\pi$
$67$	−4.84823	−0.592306	−0.296153	+	0.955140i	$0.595704\pi$
$68$	−2.08467	−0.252803
$69$	0.497622	0.0599066
$70$	−1.33209	−0.159216
$71$	−5.79051	−0.687207	−0.343604	−	0.939115i	$-0.611648\pi$
$71$	−5.79051	−0.687207	−0.343604	+	0.939115i	$0.611648\pi$
$72$	−0.930266	−0.109633
$73$	−4.56352	−0.534119	−0.267060	−	0.963680i	$-0.586052\pi$
$73$	−4.56352	−0.534119	−0.267060	+	0.963680i	$0.586052\pi$
$74$	3.29408	0.382929
$75$	2.78000	0.321007
$76$	−2.68659	−0.308173
$77$	2.00000	0.227921
$78$	8.16243	0.924214
$79$	2.90862	0.327245	0.163623	−	0.986523i	$-0.447682\pi$
$79$	2.90862	0.327245	0.163623	+	0.986523i	$0.447682\pi$
$80$	7.07291	0.790776
$81$	1.00000	0.111111
$82$	−0.425907	−0.0470336
$83$	5.12063	0.562063	0.281031	−	0.959699i	$-0.409324\pi$
$83$	5.12063	0.562063	0.281031	+	0.959699i	$0.409324\pi$
$84$	−0.717940	−0.0783337
$85$	2.06664	0.224159
$86$	18.9011	2.03816
$87$	10.2643	1.10045
$88$	−3.89490	−0.415198
$89$	2.49866	0.264857	0.132428	−	0.991193i	$-0.457723\pi$
$89$	2.49866	0.264857	0.132428	+	0.991193i	$0.457723\pi$
$90$	−2.78865	−0.293950
$91$	−2.08325	−0.218384
$92$	−0.747907	−0.0779746
$93$	−2.12343	−0.220189
$94$	−15.5160	−1.60035
$95$	2.66336	0.273255
$96$	7.02410	0.716895
$97$	2.38748	0.242412	0.121206	−	0.992627i	$-0.461324\pi$
$97$	2.38748	0.242412	0.121206	+	0.992627i	$0.461324\pi$
$98$	−12.6743	−1.28029
$99$	4.18687	0.420796
$100$	−4.17823	−0.417823
$101$	−9.72751	−0.967923	−0.483962	−	0.875089i	$-0.660803\pi$
$101$	−9.72751	−0.967923	−0.483962	+	0.875089i	$0.660803\pi$
$102$	2.59601	0.257043
$103$	−18.9232	−1.86456	−0.932278	−	0.361742i	$-0.882182\pi$
$103$	−18.9232	−1.86456	−0.932278	+	0.361742i	$0.882182\pi$
$104$	4.05704	0.397825
$105$	0.711732	0.0694580
$106$	−8.52877	−0.828387
$107$	7.62305	0.736948	0.368474	−	0.929638i	$-0.379880\pi$
$107$	7.62305	0.736948	0.368474	+	0.929638i	$0.379880\pi$
$108$	−1.50296	−0.144623
$109$	0.730882	0.0700058	0.0350029	−	0.999387i	$-0.488856\pi$
$109$	0.730882	0.0700058	0.0350029	+	0.999387i	$0.488856\pi$
$110$	−11.6757	−1.11324
$111$	−1.76002	−0.167053
$112$	−2.26758	−0.214266
$113$	9.50701	0.894344	0.447172	−	0.894448i	$-0.352431\pi$
$113$	9.50701	0.894344	0.447172	+	0.894448i	$0.352431\pi$
$114$	3.34558	0.313342
$115$	0.741440	0.0691396
$116$	−15.4269	−1.43235
$117$	−4.36116	−0.403189
$118$	2.59532	0.238919
$119$	−0.662566	−0.0607373
$120$	−1.38607	−0.126530
$121$	6.52989	0.593626
$122$	−19.8098	−1.79349
$123$	0.227561	0.0205185
$124$	3.19143	0.286599
$125$	11.5919	1.03681
$126$	0.894042	0.0796476
$127$	7.13410	0.633049	0.316524	−	0.948584i	$-0.397484\pi$
$127$	7.13410	0.633049	0.316524	+	0.948584i	$0.397484\pi$
$128$	7.21231	0.637484
$129$	−10.0988	−0.889148
$130$	12.1618	1.06666
$131$	19.8428	1.73367	0.866837	−	0.498591i	$-0.166149\pi$
$131$	19.8428	1.73367	0.866837	+	0.498591i	$0.166149\pi$
$132$	−6.29271	−0.547710
$133$	−0.853875	−0.0740403
$134$	−9.07405	−0.783879
$135$	1.48997	0.128236
$136$	1.29032	0.110644
$137$	0.444496	0.0379759	0.0189879	−	0.999820i	$-0.493956\pi$
$137$	0.444496	0.0379759	0.0189879	+	0.999820i	$0.493956\pi$
$138$	0.931359	0.0792825
$139$	−6.18281	−0.524419	−0.262210	−	0.965011i	$-0.584451\pi$
$139$	−6.18281	−0.524419	−0.262210	+	0.965011i	$0.584451\pi$
$140$	−1.06971	−0.0904067
$141$	8.29013	0.698155
$142$	−10.8376	−0.909475
$143$	−18.2596	−1.52695
$144$	−4.74703	−0.395586
$145$	15.2935	1.27005
$146$	−8.54117	−0.706872
$147$	6.77182	0.558530
$148$	2.64524	0.217437
$149$	8.33987	0.683229	0.341614	−	0.939840i	$-0.389026\pi$
$149$	8.33987	0.683229	0.341614	+	0.939840i	$0.389026\pi$
$150$	5.20310	0.424832
$151$	−5.46461	−0.444703	−0.222352	−	0.974967i	$-0.571373\pi$
$151$	−5.46461	−0.444703	−0.222352	+	0.974967i	$0.571373\pi$
$152$	1.66288	0.134877
$153$	−1.38704	−0.112136
$154$	3.74324	0.301639
$155$	−3.16384	−0.254126
$156$	6.55466	0.524793
$157$	23.9822	1.91399	0.956993	−	0.290112i	$-0.0936925\pi$
$157$	23.9822	1.91399	0.956993	+	0.290112i	$0.0936925\pi$
$158$	5.44383	0.433088
$159$	4.55689	0.361385
$160$	10.4657	0.827385
$161$	−0.237706	−0.0187338
$162$	1.87162	0.147048
$163$	11.4277	0.895083	0.447542	−	0.894263i	$-0.352300\pi$
$163$	11.4277	0.895083	0.447542	+	0.894263i	$0.352300\pi$
$164$	−0.342015	−0.0267069
$165$	6.23830	0.485651
$166$	9.58388	0.743854
$167$	9.80479	0.758718	0.379359	−	0.925250i	$-0.376145\pi$
$167$	9.80479	0.758718	0.379359	+	0.925250i	$0.376145\pi$
$168$	0.444373	0.0342841
$169$	6.01971	0.463055
$170$	3.86797	0.296660
$171$	−1.78753	−0.136696
$172$	15.1781	1.15732
$173$	0.0467554	0.00355475	0.00177737	−	0.999998i	$-0.499434\pi$
$173$	0.0467554	0.00355475	0.00177737	+	0.999998i	$0.499434\pi$
$174$	19.2109	1.45637
$175$	−1.32796	−0.100384
$176$	−19.8752	−1.49815
$177$	−1.38667	−0.104229
$178$	4.67653	0.350521
$179$	15.7734	1.17896	0.589479	−	0.807784i	$-0.299333\pi$
$179$	15.7734	1.17896	0.589479	+	0.807784i	$0.299333\pi$
$180$	−2.23936	−0.166912
$181$	−17.0865	−1.27003	−0.635016	−	0.772499i	$-0.719006\pi$
$181$	−17.0865	−1.27003	−0.635016	+	0.772499i	$0.719006\pi$
$182$	−3.89906	−0.289018
$183$	10.5843	0.782414
$184$	0.462920	0.0341270
$185$	−2.62236	−0.192800
$186$	−3.97425	−0.291406
$187$	−5.80736	−0.424676
$188$	−12.4598	−0.908721
$189$	−0.477684	−0.0347464
$190$	4.98480	0.361636
$191$	−3.24230	−0.234604	−0.117302	−	0.993096i	$-0.537425\pi$
$191$	−3.24230	−0.234604	−0.117302	+	0.993096i	$0.537425\pi$
$192$	3.65240	0.263589
$193$	−6.22638	−0.448185	−0.224092	−	0.974568i	$-0.571942\pi$
$193$	−6.22638	−0.448185	−0.224092	+	0.974568i	$0.571942\pi$
$194$	4.46846	0.320817
$195$	−6.49798	−0.465330
$196$	−10.1778	−0.726985
$197$	20.4663	1.45816	0.729082	−	0.684426i	$-0.239947\pi$
$197$	20.4663	1.45816	0.729082	+	0.684426i	$0.239947\pi$
$198$	7.83623	0.556897
$199$	14.2050	1.00697	0.503484	−	0.864004i	$-0.332051\pi$
$199$	14.2050	1.00697	0.503484	+	0.864004i	$0.332051\pi$
$200$	2.58614	0.182868
$201$	4.84823	0.341968
$202$	−18.2062	−1.28098
$203$	−4.90309	−0.344130
$204$	2.08467	0.145956
$205$	0.339058	0.0236808
$206$	−35.4170	−2.46762
$207$	−0.497622	−0.0345871
$208$	20.7026	1.43546
$209$	−7.48417	−0.517691
$210$	1.33209	0.0919231
$211$	−2.46236	−0.169516	−0.0847581	−	0.996402i	$-0.527012\pi$
$211$	−2.46236	−0.169516	−0.0847581	+	0.996402i	$0.527012\pi$
$212$	−6.84884	−0.470380
$213$	5.79051	0.396759
$214$	14.2675	0.975303
$215$	−15.0469	−1.02619
$216$	0.930266	0.0632966
$217$	1.01433	0.0688570
$218$	1.36793	0.0926481
$219$	4.56352	0.308374
$220$	−9.37593	−0.632125
$221$	6.04910	0.406907
$222$	−3.29408	−0.221084
$223$	−14.6649	−0.982034	−0.491017	−	0.871150i	$-0.663375\pi$
$223$	−14.6649	−0.982034	−0.491017	+	0.871150i	$0.663375\pi$
$224$	−3.35530	−0.224185
$225$	−2.78000	−0.185333
$226$	17.7935	1.18361
$227$	−19.9511	−1.32420	−0.662102	−	0.749414i	$-0.730335\pi$
$227$	−19.9511	−1.32420	−0.662102	+	0.749414i	$0.730335\pi$
$228$	2.68659	0.177924
$229$	−8.34211	−0.551262	−0.275631	−	0.961263i	$-0.588887\pi$
$229$	−8.34211	−0.551262	−0.275631	+	0.961263i	$0.588887\pi$
$230$	1.38769	0.0915018
$231$	−2.00000	−0.131590
$232$	9.54854	0.626892
$233$	19.7559	1.29425	0.647127	−	0.762382i	$-0.275970\pi$
$233$	19.7559	1.29425	0.647127	+	0.762382i	$0.275970\pi$
$234$	−8.16243	−0.533595
$235$	12.3520	0.805757
$236$	2.08412	0.135664
$237$	−2.90862	−0.188935
$238$	−1.24007	−0.0803819
$239$	11.0570	0.715215	0.357607	−	0.933872i	$-0.383593\pi$
$239$	11.0570	0.715215	0.357607	+	0.933872i	$0.383593\pi$
$240$	−7.07291	−0.456555
$241$	−13.5809	−0.874824	−0.437412	−	0.899261i	$-0.644105\pi$
$241$	−13.5809	−0.874824	−0.437412	+	0.899261i	$0.644105\pi$
$242$	12.2215	0.785626
$243$	−1.00000	−0.0641500
$244$	−15.9078	−1.01839
$245$	10.0898	0.644613
$246$	0.425907	0.0271549
$247$	7.79571	0.496029
$248$	−1.97535	−0.125435
$249$	−5.12063	−0.324507
$250$	21.6957	1.37216
$251$	−4.26125	−0.268968	−0.134484	−	0.990916i	$-0.542938\pi$
$251$	−4.26125	−0.268968	−0.134484	+	0.990916i	$0.542938\pi$
$252$	0.717940	0.0452260
$253$	−2.08348	−0.130987
$254$	13.3523	0.837799
$255$	−2.06664	−0.129418
$256$	20.8035	1.30022
$257$	5.89520	0.367732	0.183866	−	0.982951i	$-0.441139\pi$
$257$	5.89520	0.367732	0.183866	+	0.982951i	$0.441139\pi$
$258$	−18.9011	−1.17673
$259$	0.840730	0.0522404
$260$	9.76622	0.605675
$261$	−10.2643	−0.635345
$262$	37.1382	2.29441
$263$	−16.5087	−1.01797	−0.508984	−	0.860776i	$-0.669979\pi$
$263$	−16.5087	−1.01797	−0.508984	+	0.860776i	$0.669979\pi$
$264$	3.89490	0.239715
$265$	6.78962	0.417083
$266$	−1.59813	−0.0979876
$267$	−2.49866	−0.152915
$268$	−7.28671	−0.445107
$269$	−7.66106	−0.467103	−0.233551	−	0.972344i	$-0.575035\pi$
$269$	−7.66106	−0.467103	−0.233551	+	0.972344i	$0.575035\pi$
$270$	2.78865	0.169712
$271$	−8.44947	−0.513269	−0.256634	−	0.966509i	$-0.582614\pi$
$271$	−8.44947	−0.513269	−0.256634	+	0.966509i	$0.582614\pi$
$272$	6.58432	0.399233
$273$	2.08325	0.126084
$274$	0.831928	0.0502586
$275$	−11.6395	−0.701888
$276$	0.747907	0.0450187
$277$	16.3916	0.984875	0.492438	−	0.870348i	$-0.336106\pi$
$277$	16.3916	0.984875	0.492438	+	0.870348i	$0.336106\pi$
$278$	−11.5719	−0.694035
$279$	2.12343	0.127126
$280$	0.662100	0.0395681
$281$	3.04348	0.181559	0.0907794	−	0.995871i	$-0.471064\pi$
$281$	3.04348	0.181559	0.0907794	+	0.995871i	$0.471064\pi$
$282$	15.5160	0.923963
$283$	−20.4480	−1.21551	−0.607754	−	0.794126i	$-0.707929\pi$
$283$	−20.4480	−1.21551	−0.607754	+	0.794126i	$0.707929\pi$
$284$	−8.70292	−0.516423
$285$	−2.66336	−0.157764
$286$	−34.1751	−2.02081
$287$	−0.108702	−0.00641648
$288$	−7.02410	−0.413899
$289$	−15.0761	−0.886831
$290$	28.6236	1.68084
$291$	−2.38748	−0.139957
$292$	−6.85879	−0.401380
$293$	−12.9485	−0.756461	−0.378231	−	0.925711i	$-0.623467\pi$
$293$	−12.9485	−0.756461	−0.378231	+	0.925711i	$0.623467\pi$
$294$	12.6743	0.739179
$295$	−2.06609	−0.120293
$296$	−1.63728	−0.0951651
$297$	−4.18687	−0.242947
$298$	15.6091	0.904209
$299$	2.17021	0.125506
$300$	4.17823	0.241231
$301$	4.82402	0.278052
$302$	−10.2277	−0.588536
$303$	9.72751	0.558831
$304$	8.48547	0.486675
$305$	15.7702	0.903002
$306$	−2.59601	−0.148404
$307$	12.5861	0.718327	0.359164	−	0.933275i	$-0.383062\pi$
$307$	12.5861	0.718327	0.359164	+	0.933275i	$0.383062\pi$
$308$	3.00592	0.171278
$309$	18.9232	1.07650
$310$	−5.92150	−0.336319
$311$	−9.36853	−0.531241	−0.265620	−	0.964078i	$-0.585577\pi$
$311$	−9.36853	−0.531241	−0.265620	+	0.964078i	$0.585577\pi$
$312$	−4.05704	−0.229685
$313$	−12.2244	−0.690963	−0.345482	−	0.938426i	$-0.612284\pi$
$313$	−12.2244	−0.690963	−0.345482	+	0.938426i	$0.612284\pi$
$314$	44.8855	2.53304
$315$	−0.711732	−0.0401016
$316$	4.37154	0.245919
$317$	24.5863	1.38091	0.690453	−	0.723377i	$-0.257411\pi$
$317$	24.5863	1.38091	0.690453	+	0.723377i	$0.257411\pi$
$318$	8.52877	0.478270
$319$	−42.9754	−2.40616
$320$	5.44195	0.304214
$321$	−7.62305	−0.425477
$322$	−0.444895	−0.0247930
$323$	2.47938	0.137956
$324$	1.50296	0.0834979
$325$	12.1240	0.672520
$326$	21.3882	1.18458
$327$	−0.730882	−0.0404179
$328$	0.211692	0.0116887
$329$	−3.96006	−0.218325
$330$	11.6757	0.642727
$331$	−22.5684	−1.24047	−0.620235	−	0.784416i	$-0.712963\pi$
$331$	−22.5684	−1.24047	−0.620235	+	0.784416i	$0.712963\pi$
$332$	7.69612	0.422379
$333$	1.76002	0.0964482
$334$	18.3508	1.00411
$335$	7.22371	0.394673
$336$	2.26758	0.123706
$337$	−25.5652	−1.39263	−0.696314	−	0.717737i	$-0.745178\pi$
$337$	−25.5652	−1.39263	−0.696314	+	0.717737i	$0.745178\pi$
$338$	11.2666	0.612823
$339$	−9.50701	−0.516350
$340$	3.10609	0.168451
$341$	8.89052	0.481449
$342$	−3.34558	−0.180908
$343$	−6.57857	−0.355209
$344$	−9.39456	−0.506521
$345$	−0.741440	−0.0399178
$346$	0.0875084	0.00470448
$347$	23.9200	1.28409	0.642047	−	0.766665i	$-0.278085\pi$
$347$	23.9200	1.28409	0.642047	+	0.766665i	$0.278085\pi$
$348$	15.4269	0.826967
$349$	−30.2963	−1.62172	−0.810861	−	0.585239i	$-0.801001\pi$
$349$	−30.2963	−1.62172	−0.810861	+	0.585239i	$0.801001\pi$
$350$	−2.48544	−0.132852
$351$	4.36116	0.232781
$352$	−29.4090	−1.56751
$353$	11.1640	0.594202	0.297101	−	0.954846i	$-0.403980\pi$
$353$	11.1640	0.594202	0.297101	+	0.954846i	$0.403980\pi$
$354$	−2.59532	−0.137940
$355$	8.62767	0.457909
$356$	3.75539	0.199035
$357$	0.662566	0.0350667
$358$	29.5218	1.56027
$359$	−9.28228	−0.489900	−0.244950	−	0.969536i	$-0.578772\pi$
$359$	−9.28228	−0.489900	−0.244950	+	0.969536i	$0.578772\pi$
$360$	1.38607	0.0730520
$361$	−15.8047	−0.831828
$362$	−31.9795	−1.68080
$363$	−6.52989	−0.342730
$364$	−3.13105	−0.164112
$365$	6.79949	0.355901
$366$	19.8098	1.03547
$367$	24.0071	1.25316	0.626580	−	0.779357i	$-0.284454\pi$
$367$	24.0071	1.25316	0.626580	+	0.779357i	$0.284454\pi$
$368$	2.36222	0.123139
$369$	−0.227561	−0.0118463
$370$	−4.90807	−0.255158
$371$	−2.17675	−0.113011
$372$	−3.19143	−0.165468
$373$	−32.9735	−1.70730	−0.853651	−	0.520846i	$-0.825617\pi$
$373$	−32.9735	−1.70730	−0.853651	+	0.520846i	$0.825617\pi$
$374$	−10.8692	−0.562031
$375$	−11.5919	−0.598605
$376$	7.71203	0.397718
$377$	44.7643	2.30548
$378$	−0.894042	−0.0459846
$379$	13.9615	0.717156	0.358578	−	0.933500i	$-0.383262\pi$
$379$	13.9615	0.717156	0.358578	+	0.933500i	$0.383262\pi$
$380$	4.00293	0.205346
$381$	−7.13410	−0.365491
$382$	−6.06835	−0.310484
$383$	−24.7218	−1.26323	−0.631613	−	0.775284i	$-0.717607\pi$
$383$	−24.7218	−1.26323	−0.631613	+	0.775284i	$0.717607\pi$
$384$	−7.21231	−0.368052
$385$	−2.97993	−0.151871
$386$	−11.6534	−0.593144
$387$	10.0988	0.513350
$388$	3.58830	0.182168
$389$	−17.9451	−0.909850	−0.454925	−	0.890530i	$-0.650334\pi$
$389$	−17.9451	−0.909850	−0.454925	+	0.890530i	$0.650334\pi$
$390$	−12.1618	−0.615834
$391$	0.690221	0.0349060
$392$	6.29959	0.318177
$393$	−19.8428	−1.00094
$394$	38.3052	1.92979
$395$	−4.33374	−0.218054
$396$	6.29271	0.316221
$397$	20.4183	1.02477	0.512383	−	0.858757i	$-0.328763\pi$
$397$	20.4183	1.02477	0.512383	+	0.858757i	$0.328763\pi$
$398$	26.5864	1.33266
$399$	0.853875	0.0427472
$400$	13.1967	0.659837
$401$	6.42498	0.320848	0.160424	−	0.987048i	$-0.448714\pi$
$401$	6.42498	0.320848	0.160424	+	0.987048i	$0.448714\pi$
$402$	9.07405	0.452573
$403$	−9.26061	−0.461304
$404$	−14.6201	−0.727376
$405$	−1.48997	−0.0740370
$406$	−9.17673	−0.455433
$407$	7.36896	0.365266
$408$	−1.29032	−0.0638802
$409$	−21.0960	−1.04313	−0.521565	−	0.853211i	$-0.674652\pi$
$409$	−21.0960	−1.04313	−0.521565	+	0.853211i	$0.674652\pi$
$410$	0.634588	0.0313400
$411$	−0.444496	−0.0219254
$412$	−28.4408	−1.40118
$413$	0.662390	0.0325941
$414$	−0.931359	−0.0457738
$415$	−7.62957	−0.374521
$416$	30.6332	1.50192
$417$	6.18281	0.302773
$418$	−14.0075	−0.685130
$419$	−5.05671	−0.247036	−0.123518	−	0.992342i	$-0.539418\pi$
$419$	−5.05671	−0.247036	−0.123518	+	0.992342i	$0.539418\pi$
$420$	1.06971	0.0521964
$421$	5.72245	0.278895	0.139447	−	0.990229i	$-0.455467\pi$
$421$	5.72245	0.278895	0.139447	+	0.990229i	$0.455467\pi$
$422$	−4.60861	−0.224344
$423$	−8.29013	−0.403080
$424$	4.23912	0.205870
$425$	3.85597	0.187042
$426$	10.8376	0.525085
$427$	−5.05594	−0.244674
$428$	11.4572	0.553802
$429$	18.2596	0.881582
$430$	−28.1620	−1.35809
$431$	−22.0533	−1.06227	−0.531136	−	0.847287i	$-0.678235\pi$
$431$	−22.0533	−1.06227	−0.531136	+	0.847287i	$0.678235\pi$
$432$	4.74703	0.228392
$433$	−2.74488	−0.131911	−0.0659553	−	0.997823i	$-0.521009\pi$
$433$	−2.74488	−0.131911	−0.0659553	+	0.997823i	$0.521009\pi$
$434$	1.89843	0.0911278
$435$	−15.2935	−0.733266
$436$	1.09849	0.0526080
$437$	0.889515	0.0425513
$438$	8.54117	0.408113
$439$	24.0508	1.14788	0.573941	−	0.818896i	$-0.305414\pi$
$439$	24.0508	1.14788	0.573941	+	0.818896i	$0.305414\pi$
$440$	5.80328	0.276660
$441$	−6.77182	−0.322468
$442$	11.3216	0.538515
$443$	18.5138	0.879615	0.439808	−	0.898092i	$-0.355047\pi$
$443$	18.5138	0.879615	0.439808	+	0.898092i	$0.355047\pi$
$444$	−2.64524	−0.125537
$445$	−3.72291	−0.176483
$446$	−27.4471	−1.29966
$447$	−8.33987	−0.394462
$448$	−1.74469	−0.0824288
$449$	20.9980	0.990958	0.495479	−	0.868620i	$-0.334993\pi$
$449$	20.9980	0.990958	0.495479	+	0.868620i	$0.334993\pi$
$450$	−5.20310	−0.245277
$451$	−0.952768	−0.0448641
$452$	14.2887	0.672083
$453$	5.46461	0.256750
$454$	−37.3410	−1.75250
$455$	3.10398	0.145517
$456$	−1.66288	−0.0778715
$457$	3.74136	0.175013	0.0875066	−	0.996164i	$-0.472110\pi$
$457$	3.74136	0.175013	0.0875066	+	0.996164i	$0.472110\pi$
$458$	−15.6133	−0.729560
$459$	1.38704	0.0647415
$460$	1.11436	0.0519571
$461$	−30.2396	−1.40840	−0.704199	−	0.710003i	$-0.748694\pi$
$461$	−30.2396	−1.40840	−0.704199	+	0.710003i	$0.748694\pi$
$462$	−3.74324	−0.174151
$463$	8.90010	0.413623	0.206811	−	0.978381i	$-0.433691\pi$
$463$	8.90010	0.413623	0.206811	+	0.978381i	$0.433691\pi$
$464$	48.7250	2.26200
$465$	3.16384	0.146719
$466$	36.9756	1.71286
$467$	23.2566	1.07619	0.538094	−	0.842885i	$-0.319145\pi$
$467$	23.2566	1.07619	0.538094	+	0.842885i	$0.319145\pi$
$468$	−6.55466	−0.302989
$469$	−2.31592	−0.106939
$470$	23.1183	1.06637
$471$	−23.9822	−1.10504
$472$	−1.28997	−0.0593759
$473$	42.2823	1.94414
$474$	−5.44383	−0.250043
$475$	4.96934	0.228009
$476$	−0.995812	−0.0456430
$477$	−4.55689	−0.208646
$478$	20.6944	0.946541
$479$	−24.3028	−1.11042	−0.555212	−	0.831709i	$-0.687363\pi$
$479$	−24.3028	−1.11042	−0.555212	+	0.831709i	$0.687363\pi$
$480$	−10.4657	−0.477691
$481$	−7.67571	−0.349982
$482$	−25.4183	−1.15777
$483$	0.237706	0.0108160
$484$	9.81418	0.446099
$485$	−3.55727	−0.161527
$486$	−1.87162	−0.0848984
$487$	−2.13779	−0.0968725	−0.0484363	−	0.998826i	$-0.515424\pi$
$487$	−2.13779	−0.0968725	−0.0484363	+	0.998826i	$0.515424\pi$
$488$	9.84621	0.445717
$489$	−11.4277	−0.516777
$490$	18.8842	0.853103
$491$	−22.4686	−1.01399	−0.506996	−	0.861948i	$-0.669244\pi$
$491$	−22.4686	−1.01399	−0.506996	+	0.861948i	$0.669244\pi$
$492$	0.342015	0.0154192
$493$	14.2370	0.641203
$494$	14.5906	0.656463
$495$	−6.23830	−0.280391
$496$	−10.0800	−0.452604
$497$	−2.76603	−0.124074
$498$	−9.58388	−0.429464
$499$	−24.7079	−1.10608	−0.553039	−	0.833156i	$-0.686532\pi$
$499$	−24.7079	−1.10608	−0.553039	+	0.833156i	$0.686532\pi$
$500$	17.4222	0.779147
$501$	−9.80479	−0.438046
$502$	−7.97545	−0.355962
$503$	−3.75183	−0.167286	−0.0836430	−	0.996496i	$-0.526656\pi$
$503$	−3.75183	−0.167286	−0.0836430	+	0.996496i	$0.526656\pi$
$504$	−0.444373	−0.0197939
$505$	14.4937	0.644959
$506$	−3.89948	−0.173353
$507$	−6.01971	−0.267345
$508$	10.7223	0.475724
$509$	5.26823	0.233510	0.116755	−	0.993161i	$-0.462751\pi$
$509$	5.26823	0.233510	0.116755	+	0.993161i	$0.462751\pi$
$510$	−3.86797	−0.171277
$511$	−2.17992	−0.0964338
$512$	24.5116	1.08327
$513$	1.78753	0.0789215
$514$	11.0336	0.486670
$515$	28.1949	1.24242
$516$	−15.1781	−0.668178
$517$	−34.7097	−1.52653
$518$	1.57353	0.0691368
$519$	−0.0467554	−0.00205233
$520$	−6.04485	−0.265084
$521$	−6.44779	−0.282483	−0.141242	−	0.989975i	$-0.545109\pi$
$521$	−6.44779	−0.282483	−0.141242	+	0.989975i	$0.545109\pi$
$522$	−19.2109	−0.840838
$523$	−30.4320	−1.33070	−0.665350	−	0.746531i	$-0.731718\pi$
$523$	−30.4320	−1.33070	−0.665350	+	0.746531i	$0.731718\pi$
$524$	29.8230	1.30282
$525$	1.32796	0.0579569
$526$	−30.8979	−1.34721
$527$	−2.94528	−0.128298
$528$	19.8752	0.864957
$529$	−22.7524	−0.989234
$530$	12.7076	0.551982
$531$	1.38667	0.0601764
$532$	−1.28334	−0.0556399
$533$	0.992429	0.0429869
$534$	−4.67653	−0.202373
$535$	−11.3581	−0.491053
$536$	4.51015	0.194809
$537$	−15.7734	−0.680672
$538$	−14.3386	−0.618181
$539$	−28.3527	−1.22124
$540$	2.23936	0.0963669
$541$	42.8702	1.84313	0.921567	−	0.388220i	$-0.126910\pi$
$541$	42.8702	1.84313	0.921567	+	0.388220i	$0.126910\pi$
$542$	−15.8142	−0.679278
$543$	17.0865	0.733253
$544$	9.74271	0.417715
$545$	−1.08899	−0.0466472
$546$	3.89906	0.166864
$547$	−7.04655	−0.301289	−0.150644	−	0.988588i	$-0.548135\pi$
$547$	−7.04655	−0.301289	−0.150644	+	0.988588i	$0.548135\pi$
$548$	0.668061	0.0285382
$549$	−10.5843	−0.451727
$550$	−21.7847	−0.928904
$551$	18.3478	0.781642
$552$	−0.462920	−0.0197032
$553$	1.38940	0.0590833
$554$	30.6788	1.30342
$555$	2.62236	0.111313
$556$	−9.29253	−0.394091
$557$	21.5559	0.913354	0.456677	−	0.889633i	$-0.349040\pi$
$557$	21.5559	0.913354	0.456677	+	0.889633i	$0.349040\pi$
$558$	3.97425	0.168244
$559$	−44.0424	−1.86280
$560$	3.37861	0.142773
$561$	5.80736	0.245187
$562$	5.69624	0.240281
$563$	−1.00000	−0.0421450
$563$	−1.00000	−0.0421450
$564$	12.4598	0.524651
$565$	−14.1651	−0.595931
$566$	−38.2709	−1.60865
$567$	0.477684	0.0200608
$568$	5.38672	0.226022
$569$	−29.8952	−1.25327	−0.626635	−	0.779313i	$-0.715568\pi$
$569$	−29.8952	−1.25327	−0.626635	+	0.779313i	$0.715568\pi$
$570$	−4.98480	−0.208790
$571$	−1.40700	−0.0588810	−0.0294405	−	0.999567i	$-0.509373\pi$
$571$	−1.40700	−0.0588810	−0.0294405	+	0.999567i	$0.509373\pi$
$572$	−27.4435	−1.14747
$573$	3.24230	0.135449
$574$	−0.203449	−0.00849179
$575$	1.38339	0.0576913
$576$	−3.65240	−0.152183
$577$	10.8786	0.452881	0.226440	−	0.974025i	$-0.427291\pi$
$577$	10.8786	0.452881	0.226440	+	0.974025i	$0.427291\pi$
$578$	−28.2168	−1.17366
$579$	6.22638	0.258760
$580$	22.9855	0.954422
$581$	2.44604	0.101479
$582$	−4.46846	−0.185224
$583$	−19.0791	−0.790176
$584$	4.24528	0.175671
$585$	6.49798	0.268658
$586$	−24.2347	−1.00113
$587$	−32.6273	−1.34667	−0.673336	−	0.739336i	$-0.735139\pi$
$587$	−32.6273	−1.34667	−0.673336	+	0.739336i	$0.735139\pi$
$588$	10.1778	0.419725
$589$	−3.79570	−0.156399
$590$	−3.86694	−0.159200
$591$	−20.4663	−0.841872
$592$	−8.35484	−0.343382
$593$	−11.2413	−0.461624	−0.230812	−	0.972998i	$-0.574138\pi$
$593$	−11.2413	−0.461624	−0.230812	+	0.972998i	$0.574138\pi$
$594$	−7.83623	−0.321525
$595$	0.987201	0.0404713
$596$	12.5345	0.513433
$597$	−14.2050	−0.581374
$598$	4.06180	0.166100
$599$	−31.7255	−1.29627	−0.648135	−	0.761526i	$-0.724451\pi$
$599$	−31.7255	−1.29627	−0.648135	+	0.761526i	$0.724451\pi$
$600$	−2.58614	−0.105579
$601$	38.1499	1.55617	0.778084	−	0.628161i	$-0.216192\pi$
$601$	38.1499	1.55617	0.778084	+	0.628161i	$0.216192\pi$
$602$	9.02874	0.367984
$603$	−4.84823	−0.197435
$604$	−8.21310	−0.334186
$605$	−9.72932	−0.395553
$606$	18.2062	0.739576
$607$	−35.4348	−1.43825	−0.719127	−	0.694879i	$-0.755458\pi$
$607$	−35.4348	−1.43825	−0.719127	+	0.694879i	$0.755458\pi$
$608$	12.5558	0.509205
$609$	4.90309	0.198683
$610$	29.5159	1.19506
$611$	36.1546	1.46266
$612$	−2.08467	−0.0842677
$613$	10.6720	0.431038	0.215519	−	0.976500i	$-0.430856\pi$
$613$	10.6720	0.431038	0.215519	+	0.976500i	$0.430856\pi$
$614$	23.5564	0.950660
$615$	−0.339058	−0.0136721
$616$	−1.86053	−0.0749629
$617$	13.2135	0.531955	0.265977	−	0.963979i	$-0.414305\pi$
$617$	13.2135	0.531955	0.265977	+	0.963979i	$0.414305\pi$
$618$	35.4170	1.42468
$619$	−23.9075	−0.960924	−0.480462	−	0.877015i	$-0.659531\pi$
$619$	−23.9075	−0.960924	−0.480462	+	0.877015i	$0.659531\pi$
$620$	−4.75513	−0.190971
$621$	0.497622	0.0199689
$622$	−17.5343	−0.703063
$623$	1.19357	0.0478192
$624$	−20.7026	−0.828765
$625$	−3.37160	−0.134864
$626$	−22.8794	−0.914445
$627$	7.48417	0.298889
$628$	36.0443	1.43832
$629$	−2.44121	−0.0973375
$630$	−1.33209	−0.0530718
$631$	−40.6527	−1.61836	−0.809179	−	0.587563i	$-0.800088\pi$
$631$	−40.6527	−1.61836	−0.809179	+	0.587563i	$0.800088\pi$
$632$	−2.70579	−0.107631
$633$	2.46236	0.0978702
$634$	46.0163	1.82754
$635$	−10.6296	−0.421821
$636$	6.84884	0.271574
$637$	29.5330	1.17014
$638$	−80.4335	−3.18439
$639$	−5.79051	−0.229069
$640$	−10.7461	−0.424777
$641$	−1.34569	−0.0531516	−0.0265758	−	0.999647i	$-0.508460\pi$
$641$	−1.34569	−0.0531516	−0.0265758	+	0.999647i	$0.508460\pi$
$642$	−14.2675	−0.563091
$643$	22.4353	0.884763	0.442382	−	0.896827i	$-0.354134\pi$
$643$	22.4353	0.884763	0.442382	+	0.896827i	$0.354134\pi$
$644$	−0.357263	−0.0140781
$645$	15.0469	0.592469
$646$	4.64045	0.182576
$647$	46.7839	1.83926	0.919632	−	0.392781i	$-0.128487\pi$
$647$	46.7839	1.83926	0.919632	+	0.392781i	$0.128487\pi$
$648$	−0.930266	−0.0365443
$649$	5.80582	0.227898
$650$	22.6916	0.890037
$651$	−1.01433	−0.0397546
$652$	17.1753	0.672638
$653$	20.8887	0.817438	0.408719	−	0.912660i	$-0.365976\pi$
$653$	20.8887	0.817438	0.408719	+	0.912660i	$0.365976\pi$
$654$	−1.36793	−0.0534904
$655$	−29.5651	−1.15520
$656$	1.08024	0.0421762
$657$	−4.56352	−0.178040
$658$	−7.41173	−0.288939
$659$	48.5318	1.89053	0.945265	−	0.326303i	$-0.105803\pi$
$659$	48.5318	1.89053	0.945265	+	0.326303i	$0.105803\pi$
$660$	9.37593	0.364957
$661$	11.3155	0.440121	0.220061	−	0.975486i	$-0.429375\pi$
$661$	11.3155	0.440121	0.220061	+	0.975486i	$0.429375\pi$
$662$	−42.2394	−1.64168
$663$	−6.04910	−0.234928
$664$	−4.76355	−0.184862
$665$	1.27224	0.0493355
$666$	3.29408	0.127643
$667$	5.10774	0.197773
$668$	14.7362	0.570162
$669$	14.6649	0.566978
$670$	13.5200	0.522325
$671$	−44.3151	−1.71077
$672$	3.35530	0.129433
$673$	−38.7526	−1.49380	−0.746902	−	0.664935i	$-0.768459\pi$
$673$	−38.7526	−1.49380	−0.746902	+	0.664935i	$0.768459\pi$
$674$	−47.8484	−1.84305
$675$	2.78000	0.107002
$676$	9.04740	0.347977
$677$	25.2092	0.968868	0.484434	−	0.874828i	$-0.339026\pi$
$677$	25.2092	0.968868	0.484434	+	0.874828i	$0.339026\pi$
$678$	−17.7935	−0.683356
$679$	1.14046	0.0437669
$680$	−1.92253	−0.0737256
$681$	19.9511	0.764530
$682$	16.6397	0.637166
$683$	−15.4382	−0.590727	−0.295363	−	0.955385i	$-0.595441\pi$
$683$	−15.4382	−0.590727	−0.295363	+	0.955385i	$0.595441\pi$
$684$	−2.68659	−0.102724
$685$	−0.662284	−0.0253046
$686$	−12.3126	−0.470097
$687$	8.34211	0.318271
$688$	−47.9392	−1.82767
$689$	19.8733	0.757114
$690$	−1.38769	−0.0528286
$691$	26.4705	1.00699	0.503493	−	0.863999i	$-0.332048\pi$
$691$	26.4705	1.00699	0.503493	+	0.863999i	$0.332048\pi$
$692$	0.0702716	0.00267133
$693$	2.00000	0.0759737
$694$	44.7692	1.69942
$695$	9.21218	0.349438
$696$	−9.54854	−0.361937
$697$	0.315636	0.0119556
$698$	−56.7031	−2.14624
$699$	−19.7559	−0.747238
$700$	−1.99587	−0.0754369
$701$	39.5850	1.49511	0.747553	−	0.664203i	$-0.231229\pi$
$701$	39.5850	1.49511	0.747553	+	0.664203i	$0.231229\pi$
$702$	8.16243	0.308071
$703$	−3.14608	−0.118657
$704$	−15.2921	−0.576343
$705$	−12.3520	−0.465204
$706$	20.8948	0.786388
$707$	−4.64667	−0.174756
$708$	−2.08412	−0.0783259
$709$	−9.66953	−0.363147	−0.181573	−	0.983377i	$-0.558119\pi$
$709$	−9.66953	−0.363147	−0.181573	+	0.983377i	$0.558119\pi$
$710$	16.1477	0.606013
$711$	2.90862	0.109082
$712$	−2.32441	−0.0871111
$713$	−1.05666	−0.0395724
$714$	1.24007	0.0464085
$715$	27.2062	1.01745
$716$	23.7068	0.885965
$717$	−11.0570	−0.412929
$718$	−17.3729	−0.648351
$719$	1.14528	0.0427119	0.0213560	−	0.999772i	$-0.493202\pi$
$719$	1.14528	0.0427119	0.0213560	+	0.999772i	$0.493202\pi$
$720$	7.07291	0.263592
$721$	−9.03929	−0.336641
$722$	−29.5805	−1.10087
$723$	13.5809	0.505080
$724$	−25.6804	−0.954405
$725$	28.5348	1.05976
$726$	−12.2215	−0.453581
$727$	13.9573	0.517646	0.258823	−	0.965925i	$-0.416665\pi$
$727$	13.9573	0.517646	0.258823	+	0.965925i	$0.416665\pi$
$728$	1.93798	0.0718263
$729$	1.00000	0.0370370
$730$	12.7261	0.471012
$731$	−14.0074	−0.518083
$732$	15.9078	0.587969
$733$	−39.0654	−1.44291	−0.721457	−	0.692459i	$-0.756527\pi$
$733$	−39.0654	−1.44291	−0.721457	+	0.692459i	$0.756527\pi$
$734$	44.9322	1.65848
$735$	−10.0898	−0.372167
$736$	3.49535	0.128840
$737$	−20.2989	−0.747721
$738$	−0.425907	−0.0156779
$739$	11.0647	0.407022	0.203511	−	0.979073i	$-0.434765\pi$
$739$	11.0647	0.407022	0.203511	+	0.979073i	$0.434765\pi$
$740$	−3.94131	−0.144886
$741$	−7.79571	−0.286383
$742$	−4.07405	−0.149563
$743$	−9.43405	−0.346102	−0.173051	−	0.984913i	$-0.555363\pi$
$743$	−9.43405	−0.346102	−0.173051	+	0.984913i	$0.555363\pi$
$744$	1.97535	0.0724200
$745$	−12.4261	−0.455258
$746$	−61.7138	−2.25950
$747$	5.12063	0.187354
$748$	−8.72824	−0.319136
$749$	3.64140	0.133054
$750$	−21.6957	−0.792215
$751$	−42.1364	−1.53758	−0.768789	−	0.639502i	$-0.779141\pi$
$751$	−42.1364	−1.53758	−0.768789	+	0.639502i	$0.779141\pi$
$752$	39.3535	1.43507
$753$	4.26125	0.155289
$754$	83.7818	3.05115
$755$	8.14208	0.296321
$756$	−0.717940	−0.0261112
$757$	−15.3145	−0.556615	−0.278307	−	0.960492i	$-0.589773\pi$
$757$	−15.3145	−0.556615	−0.278307	+	0.960492i	$0.589773\pi$
$758$	26.1307	0.949109
$759$	2.08348	0.0756255
$760$	−2.47764	−0.0898733
$761$	42.2044	1.52991	0.764954	−	0.644085i	$-0.222762\pi$
$761$	42.2044	1.52991	0.764954	+	0.644085i	$0.222762\pi$
$762$	−13.3523	−0.483703
$763$	0.349130	0.0126394
$764$	−4.87305	−0.176301
$765$	2.06664	0.0747196
$766$	−46.2699	−1.67180
$767$	−6.04750	−0.218362
$768$	−20.8035	−0.750681
$769$	−11.7838	−0.424936	−0.212468	−	0.977168i	$-0.568150\pi$
$769$	−11.7838	−0.424936	−0.212468	+	0.977168i	$0.568150\pi$
$770$	−5.57730	−0.200992
$771$	−5.89520	−0.212310
$772$	−9.35802	−0.336802
$773$	25.2821	0.909334	0.454667	−	0.890662i	$-0.349758\pi$
$773$	25.2821	0.909334	0.454667	+	0.890662i	$0.349758\pi$
$774$	18.9011	0.679386
$775$	−5.90313	−0.212047
$776$	−2.22099	−0.0797290
$777$	−0.840730	−0.0301610
$778$	−33.5863	−1.20413
$779$	0.406772	0.0145741
$780$	−9.76622	−0.349687
$781$	−24.2441	−0.867523
$782$	1.29183	0.0461958
$783$	10.2643	0.366817
$784$	32.1460	1.14807
$785$	−35.7326	−1.27535
$786$	−37.1382	−1.32468
$787$	−8.34104	−0.297326	−0.148663	−	0.988888i	$-0.547497\pi$
$787$	−8.34104	−0.297326	−0.148663	+	0.988888i	$0.547497\pi$
$788$	30.7601	1.09578
$789$	16.5087	0.587724
$790$	−8.11112	−0.288581
$791$	4.54134	0.161472
$792$	−3.89490	−0.138399
$793$	46.1598	1.63918
$794$	38.2154	1.35621
$795$	−6.78962	−0.240803
$796$	21.3496	0.756718
$797$	−9.84333	−0.348669	−0.174334	−	0.984687i	$-0.555777\pi$
$797$	−9.84333	−0.348669	−0.174334	+	0.984687i	$0.555777\pi$
$798$	1.59813	0.0565731
$799$	11.4987	0.406796
$800$	19.5270	0.690384
$801$	2.49866	0.0882857
$802$	12.0251	0.424622
$803$	−19.1069	−0.674266
$804$	7.28671	0.256983
$805$	0.354173	0.0124830
$806$	−17.3324	−0.610506
$807$	7.66106	0.269682
$808$	9.04917	0.318349
$809$	−15.5294	−0.545986	−0.272993	−	0.962016i	$-0.588014\pi$
$809$	−15.5294	−0.545986	−0.272993	+	0.962016i	$0.588014\pi$
$810$	−2.78865	−0.0979832
$811$	4.60648	0.161755	0.0808777	−	0.996724i	$-0.474228\pi$
$811$	4.60648	0.161755	0.0808777	+	0.996724i	$0.474228\pi$
$812$	−7.36916	−0.258607
$813$	8.44947	0.296336
$814$	13.7919	0.483405
$815$	−17.0268	−0.596424
$816$	−6.58432	−0.230497
$817$	−18.0519	−0.631556
$818$	−39.4837	−1.38052
$819$	−2.08325	−0.0727948
$820$	0.509591	0.0177957
$821$	−51.4234	−1.79469	−0.897344	−	0.441331i	$-0.854506\pi$
$821$	−51.4234	−1.79469	−0.897344	+	0.441331i	$0.854506\pi$
$822$	−0.831928	−0.0290168
$823$	−4.14000	−0.144311	−0.0721556	−	0.997393i	$-0.522988\pi$
$823$	−4.14000	−0.144311	−0.0721556	+	0.997393i	$0.522988\pi$
$824$	17.6036	0.613250
$825$	11.6395	0.405235
$826$	1.23974	0.0431362
$827$	16.6983	0.580656	0.290328	−	0.956927i	$-0.406236\pi$
$827$	16.6983	0.580656	0.290328	+	0.956927i	$0.406236\pi$
$828$	−0.747907	−0.0259915
$829$	34.3721	1.19379	0.596896	−	0.802319i	$-0.296401\pi$
$829$	34.3721	1.19379	0.596896	+	0.802319i	$0.296401\pi$
$830$	−14.2797	−0.495654
$831$	−16.3916	−0.568618
$832$	15.9287	0.552228
$833$	9.39278	0.325441
$834$	11.5719	0.400701
$835$	−14.6088	−0.505559
$836$	−11.2484	−0.389035
$837$	−2.12343	−0.0733964
$838$	−9.46423	−0.326936
$839$	20.0609	0.692579	0.346289	−	0.938128i	$-0.387442\pi$
$839$	20.0609	0.692579	0.346289	+	0.938128i	$0.387442\pi$
$840$	−0.662100	−0.0228446
$841$	76.3561	2.63297
$842$	10.7102	0.369099
$843$	−3.04348	−0.104823
$844$	−3.70084	−0.127388
$845$	−8.96917	−0.308549
$846$	−15.5160	−0.533451
$847$	3.11922	0.107178
$848$	21.6317	0.742836
$849$	20.4480	0.701773
$850$	7.21691	0.247538
$851$	−0.875822	−0.0300228
$852$	8.70292	0.298157
$853$	−13.4410	−0.460211	−0.230106	−	0.973166i	$-0.573907\pi$
$853$	−13.4410	−0.460211	−0.230106	+	0.973166i	$0.573907\pi$
$854$	−9.46281	−0.323811
$855$	2.66336	0.0910851
$856$	−7.09146	−0.242381
$857$	−25.3769	−0.866857	−0.433429	−	0.901188i	$-0.642696\pi$
$857$	−25.3769	−0.866857	−0.433429	+	0.901188i	$0.642696\pi$
$858$	34.1751	1.16672
$859$	18.0919	0.617289	0.308644	−	0.951178i	$-0.400125\pi$
$859$	18.0919	0.617289	0.308644	+	0.951178i	$0.400125\pi$
$860$	−22.6148	−0.771160
$861$	0.108702	0.00370456
$862$	−41.2755	−1.40585
$863$	−17.8620	−0.608028	−0.304014	−	0.952668i	$-0.598327\pi$
$863$	−17.8620	−0.608028	−0.304014	+	0.952668i	$0.598327\pi$
$864$	7.02410	0.238965
$865$	−0.0696640	−0.00236865
$866$	−5.13738	−0.174575
$867$	15.0761	0.512012
$868$	1.52450	0.0517447
$869$	12.1780	0.413111
$870$	−28.6236	−0.970431
$871$	21.1439	0.716435
$872$	−0.679915	−0.0230248
$873$	2.38748	0.0808041
$874$	1.66483	0.0563138
$875$	5.53728	0.187194
$876$	6.85879	0.231737
$877$	10.6370	0.359186	0.179593	−	0.983741i	$-0.442522\pi$
$877$	10.6370	0.359186	0.179593	+	0.983741i	$0.442522\pi$
$878$	45.0140	1.51915
$879$	12.9485	0.436743
$880$	29.6134	0.998267
$881$	3.91394	0.131864	0.0659320	−	0.997824i	$-0.478998\pi$
$881$	3.91394	0.131864	0.0659320	+	0.997824i	$0.478998\pi$
$882$	−12.6743	−0.426765
$883$	−32.0632	−1.07901	−0.539506	−	0.841982i	$-0.681389\pi$
$883$	−32.0632	−1.07901	−0.539506	+	0.841982i	$0.681389\pi$
$884$	9.09157	0.305783
$885$	2.06609	0.0694510
$886$	34.6507	1.16411
$887$	−11.0534	−0.371136	−0.185568	−	0.982631i	$-0.559412\pi$
$887$	−11.0534	−0.371136	−0.185568	+	0.982631i	$0.559412\pi$
$888$	1.63728	0.0549436
$889$	3.40784	0.114295
$890$	−6.96788	−0.233564
$891$	4.18687	0.140265
$892$	−22.0408	−0.737980
$893$	14.8189	0.495895
$894$	−15.6091	−0.522045
$895$	−23.5018	−0.785579
$896$	3.44520	0.115096
$897$	−2.17021	−0.0724611
$898$	39.3003	1.31147
$899$	−21.7955	−0.726922
$900$	−4.17823	−0.139274
$901$	6.32059	0.210569
$902$	−1.78322	−0.0593747
$903$	−4.82402	−0.160533
$904$	−8.84405	−0.294149
$905$	25.4584	0.846264
$906$	10.2277	0.339791
$907$	−24.2700	−0.805873	−0.402937	−	0.915228i	$-0.632010\pi$
$907$	−24.2700	−0.805873	−0.402937	+	0.915228i	$0.632010\pi$
$908$	−29.9858	−0.995114
$909$	−9.72751	−0.322641
$910$	5.80947	0.192582
$911$	6.00150	0.198839	0.0994193	−	0.995046i	$-0.468301\pi$
$911$	6.00150	0.198839	0.0994193	+	0.995046i	$0.468301\pi$
$912$	−8.48547	−0.280982
$913$	21.4394	0.709542
$914$	7.00240	0.231619
$915$	−15.7702	−0.521348
$916$	−12.5379	−0.414263
$917$	9.47858	0.313010
$918$	2.59601	0.0856812
$919$	51.6963	1.70530	0.852652	−	0.522480i	$-0.174993\pi$
$919$	51.6963	1.70530	0.852652	+	0.522480i	$0.174993\pi$
$920$	−0.689736	−0.0227399
$921$	−12.5861	−0.414726
$922$	−56.5970	−1.86392
$923$	25.2534	0.831224
$924$	−3.00592	−0.0988876
$925$	−4.89284	−0.160876
$926$	16.6576	0.547403
$927$	−18.9232	−0.621519
$928$	72.0976	2.36672
$929$	−25.8215	−0.847176	−0.423588	−	0.905855i	$-0.639230\pi$
$929$	−25.8215	−0.847176	−0.423588	+	0.905855i	$0.639230\pi$
$930$	5.92150	0.194174
$931$	12.1048	0.396720
$932$	29.6924	0.972607
$933$	9.36853	0.306712
$934$	43.5275	1.42426
$935$	8.65277	0.282976
$936$	4.05704	0.132608
$937$	32.9418	1.07616	0.538081	−	0.842893i	$-0.319150\pi$
$937$	32.9418	1.07616	0.538081	+	0.842893i	$0.319150\pi$
$938$	−4.33453	−0.141527
$939$	12.2244	0.398928
$940$	18.5646	0.605511
$941$	47.8081	1.55850	0.779249	−	0.626714i	$-0.215601\pi$
$941$	47.8081	1.55850	0.779249	+	0.626714i	$0.215601\pi$
$942$	−44.8855	−1.46245
$943$	0.113239	0.00368757
$944$	−6.58257	−0.214244
$945$	0.711732	0.0231527
$946$	79.1364	2.57295
$947$	23.8490	0.774987	0.387494	−	0.921872i	$-0.373341\pi$
$947$	23.8490	0.774987	0.387494	+	0.921872i	$0.373341\pi$
$948$	−4.37154	−0.141981
$949$	19.9022	0.646053
$950$	9.30071	0.301755
$951$	−24.5863	−0.797266
$952$	0.616363	0.0199764
$953$	−3.99520	−0.129417	−0.0647087	−	0.997904i	$-0.520612\pi$
$953$	−3.99520	−0.129417	−0.0647087	+	0.997904i	$0.520612\pi$
$954$	−8.52877	−0.276129
$955$	4.83091	0.156325
$956$	16.6182	0.537470
$957$	42.9754	1.38920
$958$	−45.4857	−1.46958
$959$	0.212329	0.00685645
$960$	−5.44195	−0.175638
$961$	−26.4910	−0.854550
$962$	−14.3660	−0.463179
$963$	7.62305	0.245649
$964$	−20.4116	−0.657414
$965$	9.27710	0.298640
$966$	0.444895	0.0143143
$967$	26.9746	0.867444	0.433722	−	0.901047i	$-0.357200\pi$
$967$	26.9746	0.867444	0.433722	+	0.901047i	$0.357200\pi$
$968$	−6.07453	−0.195243
$969$	−2.47938	−0.0796491
$970$	−6.65786	−0.213771
$971$	−8.39224	−0.269320	−0.134660	−	0.990892i	$-0.542994\pi$
$971$	−8.39224	−0.269320	−0.134660	+	0.990892i	$0.542994\pi$
$972$	−1.50296	−0.0482075
$973$	−2.95343	−0.0946825
$974$	−4.00113	−0.128205
$975$	−12.1240	−0.388280
$976$	50.2440	1.60827
$977$	56.5256	1.80842	0.904208	−	0.427093i	$-0.140462\pi$
$977$	56.5256	1.80842	0.904208	+	0.427093i	$0.140462\pi$
$978$	−21.3882	−0.683920
$979$	10.4615	0.334353
$980$	15.1646	0.484414
$981$	0.730882	0.0233353
$982$	−42.0526	−1.34195
$983$	−60.6393	−1.93409	−0.967046	−	0.254601i	$-0.918056\pi$
$983$	−60.6393	−1.93409	−0.967046	+	0.254601i	$0.918056\pi$
$984$	−0.211692	−0.00674850
$985$	−30.4941	−0.971624
$986$	26.6463	0.848590
$987$	3.96006	0.126050
$988$	11.7167	0.372757
$989$	−5.02537	−0.159798
$990$	−11.6757	−0.371079
$991$	22.8832	0.726909	0.363455	−	0.931612i	$-0.381597\pi$
$991$	22.8832	0.726909	0.363455	+	0.931612i	$0.381597\pi$
$992$	−14.9152	−0.473558
$993$	22.5684	0.716185
$994$	−5.17696	−0.164203
$995$	−21.1650	−0.670977
$996$	−7.69612	−0.243861
$997$	−43.2517	−1.36979	−0.684897	−	0.728640i	$-0.740153\pi$
$997$	−43.2517	−1.36979	−0.684897	+	0.728640i	$0.740153\pi$
$998$	−46.2438	−1.46382
$999$	−1.76002	−0.0556844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1689.2.a.c.1.16	✓	19
3.2	odd	2		5067.2.a.h.1.4		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1689.2.a.c.1.16	✓	19	1.1	even	1	trivial
5067.2.a.h.1.4		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)

\(f(q)\)	\(=\)	\(q+1.87162 q^{2} -1.00000 q^{3} +1.50296 q^{4} -1.48997 q^{5} -1.87162 q^{6} +0.477684 q^{7} -0.930266 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.87162 q^{2} -1.00000 q^{3} +1.50296 q^{4} -1.48997 q^{5} -1.87162 q^{6} +0.477684 q^{7} -0.930266 q^{8} +1.00000 q^{9} -2.78865 q^{10} +4.18687 q^{11} -1.50296 q^{12} -4.36116 q^{13} +0.894042 q^{14} +1.48997 q^{15} -4.74703 q^{16} -1.38704 q^{17} +1.87162 q^{18} -1.78753 q^{19} -2.23936 q^{20} -0.477684 q^{21} +7.83623 q^{22} -0.497622 q^{23} +0.930266 q^{24} -2.78000 q^{25} -8.16243 q^{26} -1.00000 q^{27} +0.717940 q^{28} -10.2643 q^{29} +2.78865 q^{30} +2.12343 q^{31} -7.02410 q^{32} -4.18687 q^{33} -2.59601 q^{34} -0.711732 q^{35} +1.50296 q^{36} +1.76002 q^{37} -3.34558 q^{38} +4.36116 q^{39} +1.38607 q^{40} -0.227561 q^{41} -0.894042 q^{42} +10.0988 q^{43} +6.29271 q^{44} -1.48997 q^{45} -0.931359 q^{46} -8.29013 q^{47} +4.74703 q^{48} -6.77182 q^{49} -5.20310 q^{50} +1.38704 q^{51} -6.55466 q^{52} -4.55689 q^{53} -1.87162 q^{54} -6.23830 q^{55} -0.444373 q^{56} +1.78753 q^{57} -19.2109 q^{58} +1.38667 q^{59} +2.23936 q^{60} -10.5843 q^{61} +3.97425 q^{62} +0.477684 q^{63} -3.65240 q^{64} +6.49798 q^{65} -7.83623 q^{66} -4.84823 q^{67} -2.08467 q^{68} +0.497622 q^{69} -1.33209 q^{70} -5.79051 q^{71} -0.930266 q^{72} -4.56352 q^{73} +3.29408 q^{74} +2.78000 q^{75} -2.68659 q^{76} +2.00000 q^{77} +8.16243 q^{78} +2.90862 q^{79} +7.07291 q^{80} +1.00000 q^{81} -0.425907 q^{82} +5.12063 q^{83} -0.717940 q^{84} +2.06664 q^{85} +18.9011 q^{86} +10.2643 q^{87} -3.89490 q^{88} +2.49866 q^{89} -2.78865 q^{90} -2.08325 q^{91} -0.747907 q^{92} -2.12343 q^{93} -15.5160 q^{94} +2.66336 q^{95} +7.02410 q^{96} +2.38748 q^{97} -12.6743 q^{98} +4.18687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + q^{2} - 19 q^{3} + 11 q^{4} - q^{5} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10}) \) 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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