LMFDB - Embedded newform 1339.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
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} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 $ x^19 - 9x^18 + 13x^17 + 106x^16 - 351x^15 - 279x^14 + 2337x^13 - 1079x^12 - 6826x^11 + 7199x^10 + 9364x^9 - 14841x^8 - 5183x^7 + 14037x^6 - 108x^5 - 6099x^4 + 855x^3 + 1062x^2 - 135x - 63
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.979850$ of defining polynomial
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.979850 q^{2} +0.723336 q^{3} -1.03989 q^{4} -1.69086 q^{5} -0.708761 q^{6} +1.13749 q^{7} +2.97864 q^{8} -2.47678 q^{9} +O(q^{10})$	\(q-0.979850 q^{2} +0.723336 q^{3} -1.03989 q^{4} -1.69086 q^{5} -0.708761 q^{6} +1.13749 q^{7} +2.97864 q^{8} -2.47678 q^{9} +1.65679 q^{10} +4.15988 q^{11} -0.752193 q^{12} +1.00000 q^{13} -1.11457 q^{14} -1.22306 q^{15} -0.838830 q^{16} -6.66688 q^{17} +2.42688 q^{18} -5.77679 q^{19} +1.75831 q^{20} +0.822786 q^{21} -4.07606 q^{22} +6.49646 q^{23} +2.15456 q^{24} -2.14100 q^{25} -0.979850 q^{26} -3.96156 q^{27} -1.18287 q^{28} +1.65075 q^{29} +1.19841 q^{30} +6.30875 q^{31} -5.13535 q^{32} +3.00899 q^{33} +6.53254 q^{34} -1.92333 q^{35} +2.57560 q^{36} +8.32880 q^{37} +5.66039 q^{38} +0.723336 q^{39} -5.03645 q^{40} +2.39580 q^{41} -0.806207 q^{42} -6.46226 q^{43} -4.32583 q^{44} +4.18789 q^{45} -6.36555 q^{46} +2.63737 q^{47} -0.606756 q^{48} -5.70612 q^{49} +2.09786 q^{50} -4.82240 q^{51} -1.03989 q^{52} -7.00281 q^{53} +3.88173 q^{54} -7.03376 q^{55} +3.38817 q^{56} -4.17856 q^{57} -1.61748 q^{58} -14.8854 q^{59} +1.27185 q^{60} +4.97790 q^{61} -6.18162 q^{62} -2.81731 q^{63} +6.70953 q^{64} -1.69086 q^{65} -2.94836 q^{66} -12.1321 q^{67} +6.93286 q^{68} +4.69912 q^{69} +1.88457 q^{70} -16.1909 q^{71} -7.37745 q^{72} +3.50476 q^{73} -8.16097 q^{74} -1.54866 q^{75} +6.00726 q^{76} +4.73181 q^{77} -0.708761 q^{78} -16.2696 q^{79} +1.41834 q^{80} +4.56482 q^{81} -2.34752 q^{82} -10.0354 q^{83} -0.855611 q^{84} +11.2727 q^{85} +6.33204 q^{86} +1.19405 q^{87} +12.3908 q^{88} -6.25788 q^{89} -4.10350 q^{90} +1.13749 q^{91} -6.75563 q^{92} +4.56334 q^{93} -2.58422 q^{94} +9.76773 q^{95} -3.71459 q^{96} -6.88905 q^{97} +5.59114 q^{98} -10.3031 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) $ 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * 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 $

$2$	−0.979850	−0.692858	−0.346429	−	0.938076i	$-0.612606\pi$
$2$	−0.979850	−0.692858	−0.346429	+	0.938076i	$0.612606\pi$
$3$	0.723336	0.417618	0.208809	−	0.977956i	$-0.433041\pi$
$3$	0.723336	0.417618	0.208809	+	0.977956i	$0.433041\pi$
$4$	−1.03989	−0.519947
$5$	−1.69086	−0.756174	−0.378087	−	0.925770i	$-0.623418\pi$
$5$	−1.69086	−0.756174	−0.378087	+	0.925770i	$0.623418\pi$
$6$	−0.708761	−0.289350
$7$	1.13749	0.429930	0.214965	−	0.976622i	$-0.431036\pi$
$7$	1.13749	0.429930	0.214965	+	0.976622i	$0.431036\pi$
$8$	2.97864	1.05311
$9$	−2.47678	−0.825595
$10$	1.65679	0.523922
$11$	4.15988	1.25425	0.627125	−	0.778918i	$-0.284231\pi$
$11$	4.15988	1.25425	0.627125	+	0.778918i	$0.284231\pi$
$12$	−0.752193	−0.217140
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.11457	−0.297881
$15$	−1.22306	−0.315792
$16$	−0.838830	−0.209708
$17$	−6.66688	−1.61696	−0.808478	−	0.588526i	$-0.799708\pi$
$17$	−6.66688	−1.61696	−0.808478	+	0.588526i	$0.799708\pi$
$18$	2.42688	0.572020
$19$	−5.77679	−1.32529	−0.662644	−	0.748935i	$-0.730566\pi$
$19$	−5.77679	−1.32529	−0.662644	+	0.748935i	$0.730566\pi$
$20$	1.75831	0.393171
$21$	0.822786	0.179547
$22$	−4.07606	−0.869018
$23$	6.49646	1.35461	0.677303	−	0.735704i	$-0.263149\pi$
$23$	6.49646	1.35461	0.677303	+	0.735704i	$0.263149\pi$
$24$	2.15456	0.439797
$25$	−2.14100	−0.428200
$26$	−0.979850	−0.192164
$27$	−3.96156	−0.762402
$28$	−1.18287	−0.223541
$29$	1.65075	0.306536	0.153268	−	0.988185i	$-0.451020\pi$
$29$	1.65075	0.306536	0.153268	+	0.988185i	$0.451020\pi$
$30$	1.19841	0.218799
$31$	6.30875	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$31$	6.30875	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$32$	−5.13535	−0.907811
$33$	3.00899	0.523798
$34$	6.53254	1.12032
$35$	−1.92333	−0.325102
$36$	2.57560	0.429266
$37$	8.32880	1.36925	0.684623	−	0.728897i	$-0.259967\pi$
$37$	8.32880	1.36925	0.684623	+	0.728897i	$0.259967\pi$
$38$	5.66039	0.918237
$39$	0.723336	0.115826
$40$	−5.03645	−0.796333
$41$	2.39580	0.374161	0.187080	−	0.982345i	$-0.440098\pi$
$41$	2.39580	0.374161	0.187080	+	0.982345i	$0.440098\pi$
$42$	−0.806207	−0.124400
$43$	−6.46226	−0.985485	−0.492743	−	0.870175i	$-0.664006\pi$
$43$	−6.46226	−0.985485	−0.492743	+	0.870175i	$0.664006\pi$
$44$	−4.32583	−0.652144
$45$	4.18789	0.624294
$46$	−6.36555	−0.938550
$47$	2.63737	0.384699	0.192350	−	0.981326i	$-0.438389\pi$
$47$	2.63737	0.384699	0.192350	+	0.981326i	$0.438389\pi$
$48$	−0.606756	−0.0875777
$49$	−5.70612	−0.815160
$50$	2.09786	0.296682
$51$	−4.82240	−0.675271
$52$	−1.03989	−0.144207
$53$	−7.00281	−0.961910	−0.480955	−	0.876745i	$-0.659710\pi$
$53$	−7.00281	−0.961910	−0.480955	+	0.876745i	$0.659710\pi$
$54$	3.88173	0.528237
$55$	−7.03376	−0.948432
$56$	3.38817	0.452763
$57$	−4.17856	−0.553464
$58$	−1.61748	−0.212386
$59$	−14.8854	−1.93792	−0.968960	−	0.247217i	$-0.920484\pi$
$59$	−14.8854	−1.93792	−0.968960	+	0.247217i	$0.920484\pi$
$60$	1.27185	0.164195
$61$	4.97790	0.637355	0.318677	−	0.947863i	$-0.396761\pi$
$61$	4.97790	0.637355	0.318677	+	0.947863i	$0.396761\pi$
$62$	−6.18162	−0.785067
$63$	−2.81731	−0.354948
$64$	6.70953	0.838692
$65$	−1.69086	−0.209725
$66$	−2.94836	−0.362918
$67$	−12.1321	−1.48218	−0.741088	−	0.671407i	$-0.765690\pi$
$67$	−12.1321	−1.48218	−0.741088	+	0.671407i	$0.765690\pi$
$68$	6.93286	0.840732
$69$	4.69912	0.565708
$70$	1.88457	0.225250
$71$	−16.1909	−1.92151	−0.960755	−	0.277399i	$-0.910527\pi$
$71$	−16.1909	−1.92151	−0.960755	+	0.277399i	$0.910527\pi$
$72$	−7.37745	−0.869441
$73$	3.50476	0.410201	0.205101	−	0.978741i	$-0.434248\pi$
$73$	3.50476	0.410201	0.205101	+	0.978741i	$0.434248\pi$
$74$	−8.16097	−0.948693
$75$	−1.54866	−0.178824
$76$	6.00726	0.689080
$77$	4.73181	0.539240
$78$	−0.708761	−0.0802513
$79$	−16.2696	−1.83047	−0.915235	−	0.402919i	$-0.867996\pi$
$79$	−16.2696	−1.83047	−0.915235	+	0.402919i	$0.867996\pi$
$80$	1.41834	0.158575
$81$	4.56482	0.507202
$82$	−2.34752	−0.259240
$83$	−10.0354	−1.10153	−0.550765	−	0.834661i	$-0.685664\pi$
$83$	−10.0354	−1.10153	−0.550765	+	0.834661i	$0.685664\pi$
$84$	−0.855611	−0.0933548
$85$	11.2727	1.22270
$86$	6.33204	0.682802
$87$	1.19405	0.128015
$88$	12.3908	1.32086
$89$	−6.25788	−0.663334	−0.331667	−	0.943397i	$-0.607611\pi$
$89$	−6.25788	−0.663334	−0.331667	+	0.943397i	$0.607611\pi$
$90$	−4.10350	−0.432547
$91$	1.13749	0.119241
$92$	−6.75563	−0.704323
$93$	4.56334	0.473197
$94$	−2.58422	−0.266542
$95$	9.76773	1.00215
$96$	−3.71459	−0.379118
$97$	−6.88905	−0.699477	−0.349738	−	0.936847i	$-0.613730\pi$
$97$	−6.88905	−0.699477	−0.349738	+	0.936847i	$0.613730\pi$
$98$	5.59114	0.564791
$99$	−10.3031	−1.03550
$100$	2.22642	0.222642
$101$	−14.6278	−1.45552	−0.727762	−	0.685830i	$-0.759439\pi$
$101$	−14.6278	−1.45552	−0.727762	+	0.685830i	$0.759439\pi$
$102$	4.72522	0.467867
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	2.97864	0.292080
$105$	−1.39121	−0.135769
$106$	6.86171	0.666468
$107$	−10.3453	−1.00012	−0.500058	−	0.865992i	$-0.666688\pi$
$107$	−10.3453	−1.00012	−0.500058	+	0.865992i	$0.666688\pi$
$108$	4.11960	0.396409
$109$	3.28887	0.315017	0.157508	−	0.987518i	$-0.449654\pi$
$109$	3.28887	0.315017	0.157508	+	0.987518i	$0.449654\pi$
$110$	6.89203	0.657129
$111$	6.02452	0.571822
$112$	−0.954159	−0.0901596
$113$	−9.25674	−0.870801	−0.435401	−	0.900237i	$-0.643393\pi$
$113$	−9.25674	−0.870801	−0.435401	+	0.900237i	$0.643393\pi$
$114$	4.09436	0.383472
$115$	−10.9846	−1.02432
$116$	−1.71660	−0.159383
$117$	−2.47678	−0.228979
$118$	14.5855	1.34270
$119$	−7.58350	−0.695178
$120$	−3.64305	−0.332563
$121$	6.30459	0.573144
$122$	−4.87759	−0.441597
$123$	1.73297	0.156256
$124$	−6.56043	−0.589144
$125$	12.0744	1.07997
$126$	2.76054	0.245929
$127$	12.3052	1.09191	0.545953	−	0.837816i	$-0.316168\pi$
$127$	12.3052	1.09191	0.545953	+	0.837816i	$0.316168\pi$
$128$	3.69637	0.326716
$129$	−4.67438	−0.411557
$130$	1.65679	0.145310
$131$	22.6151	1.97589	0.987945	−	0.154806i	$-0.0494753\pi$
$131$	22.6151	1.97589	0.987945	+	0.154806i	$0.0494753\pi$
$132$	−3.12903	−0.272347
$133$	−6.57103	−0.569781
$134$	11.8877	1.02694
$135$	6.69843	0.576509
$136$	−19.8582	−1.70283
$137$	−1.16616	−0.0996318	−0.0498159	−	0.998758i	$-0.515863\pi$
$137$	−1.16616	−0.0996318	−0.0498159	+	0.998758i	$0.515863\pi$
$138$	−4.60444	−0.391956
$139$	17.4830	1.48289	0.741446	−	0.671013i	$-0.234140\pi$
$139$	17.4830	1.48289	0.741446	+	0.671013i	$0.234140\pi$
$140$	2.00006	0.169036
$141$	1.90770	0.160658
$142$	15.8647	1.33133
$143$	4.15988	0.347866
$144$	2.07760	0.173133
$145$	−2.79118	−0.231795
$146$	−3.43414	−0.284211
$147$	−4.12744	−0.340426
$148$	−8.66107	−0.711936
$149$	9.59309	0.785897	0.392948	−	0.919561i	$-0.371455\pi$
$149$	9.59309	0.785897	0.392948	+	0.919561i	$0.371455\pi$
$150$	1.51746	0.123900
$151$	−4.72061	−0.384158	−0.192079	−	0.981379i	$-0.561523\pi$
$151$	−4.72061	−0.384158	−0.192079	+	0.981379i	$0.561523\pi$
$152$	−17.2070	−1.39567
$153$	16.5124	1.33495
$154$	−4.63646	−0.373617
$155$	−10.6672	−0.856809
$156$	−0.752193	−0.0602237
$157$	−7.28178	−0.581149	−0.290575	−	0.956852i	$-0.593846\pi$
$157$	−7.28178	−0.581149	−0.290575	+	0.956852i	$0.593846\pi$
$158$	15.9417	1.26826
$159$	−5.06539	−0.401711
$160$	8.68315	0.686463
$161$	7.38964	0.582386
$162$	−4.47284	−0.351419
$163$	2.42283	0.189771	0.0948854	−	0.995488i	$-0.469752\pi$
$163$	2.42283	0.189771	0.0948854	+	0.995488i	$0.469752\pi$
$164$	−2.49138	−0.194544
$165$	−5.08777	−0.396083
$166$	9.83319	0.763204
$167$	−15.7097	−1.21565	−0.607827	−	0.794070i	$-0.707959\pi$
$167$	−15.7097	−1.21565	−0.607827	+	0.794070i	$0.707959\pi$
$168$	2.45078	0.189082
$169$	1.00000	0.0769231
$170$	−11.0456	−0.847159
$171$	14.3079	1.09415
$172$	6.72007	0.512400
$173$	−17.5041	−1.33081	−0.665405	−	0.746483i	$-0.731741\pi$
$173$	−17.5041	−1.33081	−0.665405	+	0.746483i	$0.731741\pi$
$174$	−1.16999	−0.0886963
$175$	−2.43536	−0.184096
$176$	−3.48943	−0.263026
$177$	−10.7672	−0.809311
$178$	6.13178	0.459596
$179$	−13.8156	−1.03263	−0.516313	−	0.856400i	$-0.672696\pi$
$179$	−13.8156	−1.03263	−0.516313	+	0.856400i	$0.672696\pi$
$180$	−4.35496	−0.324600
$181$	9.64854	0.717171	0.358585	−	0.933497i	$-0.383259\pi$
$181$	9.64854	0.717171	0.358585	+	0.933497i	$0.383259\pi$
$182$	−1.11457	−0.0826172
$183$	3.60069	0.266171
$184$	19.3506	1.42655
$185$	−14.0828	−1.03539
$186$	−4.47139	−0.327858
$187$	−27.7334	−2.02807
$188$	−2.74258	−0.200023
$189$	−4.50622	−0.327779
$190$	−9.57091	−0.694347
$191$	12.3820	0.895932	0.447966	−	0.894051i	$-0.352149\pi$
$191$	12.3820	0.895932	0.447966	+	0.894051i	$0.352149\pi$
$192$	4.85325	0.350253
$193$	−8.74845	−0.629727	−0.314864	−	0.949137i	$-0.601959\pi$
$193$	−8.74845	−0.629727	−0.314864	+	0.949137i	$0.601959\pi$
$194$	6.75023	0.484638
$195$	−1.22306	−0.0875850
$196$	5.93376	0.423840
$197$	6.31275	0.449765	0.224882	−	0.974386i	$-0.427800\pi$
$197$	6.31275	0.449765	0.224882	+	0.974386i	$0.427800\pi$
$198$	10.0955	0.717457
$199$	20.9032	1.48179	0.740896	−	0.671620i	$-0.234401\pi$
$199$	20.9032	1.48179	0.740896	+	0.671620i	$0.234401\pi$
$200$	−6.37727	−0.450941
$201$	−8.77562	−0.618984
$202$	14.3331	1.00847
$203$	1.87771	0.131789
$204$	5.01478	0.351105
$205$	−4.05095	−0.282931
$206$	−0.979850	−0.0682694
$207$	−16.0903	−1.11836
$208$	−0.838830	−0.0581624
$209$	−24.0308	−1.66224
$210$	1.36318	0.0940684
$211$	11.5301	0.793766	0.396883	−	0.917869i	$-0.370092\pi$
$211$	11.5301	0.793766	0.396883	+	0.917869i	$0.370092\pi$
$212$	7.28219	0.500143
$213$	−11.7115	−0.802457
$214$	10.1368	0.692939
$215$	10.9268	0.745199
$216$	−11.8000	−0.802892
$217$	7.17612	0.487147
$218$	−3.22260	−0.218262
$219$	2.53512	0.171307
$220$	7.31437	0.493135
$221$	−6.66688	−0.448463
$222$	−5.90312	−0.396192
$223$	−4.87890	−0.326716	−0.163358	−	0.986567i	$-0.552232\pi$
$223$	−4.87890	−0.326716	−0.163358	+	0.986567i	$0.552232\pi$
$224$	−5.84140	−0.390295
$225$	5.30280	0.353520
$226$	9.07022	0.603342
$227$	−11.4159	−0.757702	−0.378851	−	0.925458i	$-0.623681\pi$
$227$	−11.4159	−0.757702	−0.378851	+	0.925458i	$0.623681\pi$
$228$	4.34527	0.287772
$229$	15.8785	1.04928	0.524642	−	0.851323i	$-0.324199\pi$
$229$	15.8785	1.04928	0.524642	+	0.851323i	$0.324199\pi$
$230$	10.7632	0.709707
$231$	3.42269	0.225197
$232$	4.91698	0.322816
$233$	9.57654	0.627380	0.313690	−	0.949526i	$-0.398435\pi$
$233$	9.57654	0.627380	0.313690	+	0.949526i	$0.398435\pi$
$234$	2.42688	0.158650
$235$	−4.45941	−0.290900
$236$	15.4793	1.00762
$237$	−11.7684	−0.764438
$238$	7.43069	0.481660
$239$	13.7394	0.888726	0.444363	−	0.895847i	$-0.353430\pi$
$239$	13.7394	0.888726	0.444363	+	0.895847i	$0.353430\pi$
$240$	1.02594	0.0662240
$241$	4.83615	0.311524	0.155762	−	0.987795i	$-0.450217\pi$
$241$	4.83615	0.311524	0.155762	+	0.987795i	$0.450217\pi$
$242$	−6.17755	−0.397108
$243$	15.1866	0.974219
$244$	−5.17649	−0.331391
$245$	9.64824	0.616403
$246$	−1.69805	−0.108264
$247$	−5.77679	−0.367569
$248$	18.7915	1.19326
$249$	−7.25897	−0.460019
$250$	−11.8311	−0.748265
$251$	4.72795	0.298426	0.149213	−	0.988805i	$-0.452326\pi$
$251$	4.72795	0.298426	0.149213	+	0.988805i	$0.452326\pi$
$252$	2.92971	0.184554
$253$	27.0245	1.69901
$254$	−12.0572	−0.756536
$255$	8.15399	0.510622
$256$	−17.0410	−1.06506
$257$	11.0158	0.687145	0.343572	−	0.939126i	$-0.388363\pi$
$257$	11.0158	0.687145	0.343572	+	0.939126i	$0.388363\pi$
$258$	4.58019	0.285150
$259$	9.47391	0.588680
$260$	1.75831	0.109046
$261$	−4.08855	−0.253075
$262$	−22.1594	−1.36901
$263$	−8.16564	−0.503515	−0.251757	−	0.967790i	$-0.581008\pi$
$263$	−8.16564	−0.503515	−0.251757	+	0.967790i	$0.581008\pi$
$264$	8.96270	0.551616
$265$	11.8408	0.727372
$266$	6.43863	0.394777
$267$	−4.52655	−0.277020
$268$	12.6161	0.770654
$269$	−32.0789	−1.95589	−0.977943	−	0.208872i	$-0.933021\pi$
$269$	−32.0789	−1.95589	−0.977943	+	0.208872i	$0.933021\pi$
$270$	−6.56345	−0.399439
$271$	−27.6887	−1.68197	−0.840983	−	0.541062i	$-0.818023\pi$
$271$	−27.6887	−1.68197	−0.840983	+	0.541062i	$0.818023\pi$
$272$	5.59238	0.339088
$273$	0.822786	0.0497973
$274$	1.14266	0.0690307
$275$	−8.90631	−0.537070
$276$	−4.88659	−0.294138
$277$	23.2139	1.39479	0.697394	−	0.716688i	$-0.254343\pi$
$277$	23.2139	1.39479	0.697394	+	0.716688i	$0.254343\pi$
$278$	−17.1307	−1.02743
$279$	−15.6254	−0.935469
$280$	−5.72891	−0.342368
$281$	−4.18822	−0.249848	−0.124924	−	0.992166i	$-0.539869\pi$
$281$	−4.18822	−0.249848	−0.124924	+	0.992166i	$0.539869\pi$
$282$	−1.86926	−0.111313
$283$	−28.7852	−1.71110	−0.855552	−	0.517717i	$-0.826782\pi$
$283$	−28.7852	−1.71110	−0.855552	+	0.517717i	$0.826782\pi$
$284$	16.8369	0.999084
$285$	7.06536	0.418516
$286$	−4.07606	−0.241022
$287$	2.72519	0.160863
$288$	12.7192	0.749484
$289$	27.4473	1.61455
$290$	2.73494	0.160601
$291$	−4.98310	−0.292114
$292$	−3.64458	−0.213283
$293$	−24.1941	−1.41343	−0.706716	−	0.707497i	$-0.749824\pi$
$293$	−24.1941	−1.41343	−0.706716	+	0.707497i	$0.749824\pi$
$294$	4.04427	0.235867
$295$	25.1692	1.46541
$296$	24.8085	1.44196
$297$	−16.4796	−0.956243
$298$	−9.39979	−0.544515
$299$	6.49646	0.375700
$300$	1.61045	0.0929792
$301$	−7.35074	−0.423690
$302$	4.62549	0.266167
$303$	−10.5808	−0.607853
$304$	4.84575	0.277923
$305$	−8.41692	−0.481951
$306$	−16.1797	−0.924932
$307$	10.5341	0.601212	0.300606	−	0.953748i	$-0.402811\pi$
$307$	10.5341	0.601212	0.300606	+	0.953748i	$0.402811\pi$
$308$	−4.92059	−0.280376
$309$	0.723336	0.0411492
$310$	10.4522	0.593648
$311$	−33.8427	−1.91905	−0.959523	−	0.281630i	$-0.909125\pi$
$311$	−33.8427	−1.91905	−0.959523	+	0.281630i	$0.909125\pi$
$312$	2.15456	0.121978
$313$	5.29137	0.299086	0.149543	−	0.988755i	$-0.452220\pi$
$313$	5.29137	0.299086	0.149543	+	0.988755i	$0.452220\pi$
$314$	7.13505	0.402654
$315$	4.76367	0.268403
$316$	16.9186	0.951748
$317$	25.6938	1.44311	0.721553	−	0.692359i	$-0.243429\pi$
$317$	25.6938	1.44311	0.721553	+	0.692359i	$0.243429\pi$
$318$	4.96332	0.278329
$319$	6.86691	0.384473
$320$	−11.3449	−0.634197
$321$	−7.48312	−0.417667
$322$	−7.24074	−0.403511
$323$	38.5132	2.14293
$324$	−4.74693	−0.263718
$325$	−2.14100	−0.118761
$326$	−2.37401	−0.131484
$327$	2.37896	0.131557
$328$	7.13622	0.394032
$329$	2.99997	0.165394
$330$	4.98525	0.274429
$331$	−13.8331	−0.760338	−0.380169	−	0.924917i	$-0.624134\pi$
$331$	−13.8331	−0.760338	−0.380169	+	0.924917i	$0.624134\pi$
$332$	10.4358	0.572737
$333$	−20.6286	−1.13044
$334$	15.3932	0.842276
$335$	20.5137	1.12078
$336$	−0.690178	−0.0376523
$337$	30.4384	1.65809	0.829044	−	0.559184i	$-0.188885\pi$
$337$	30.4384	1.65809	0.829044	+	0.559184i	$0.188885\pi$
$338$	−0.979850	−0.0532968
$339$	−6.69574	−0.363662
$340$	−11.7225	−0.635740
$341$	26.2436	1.42117
$342$	−14.0196	−0.758091
$343$	−14.4531	−0.780392
$344$	−19.2487	−1.03782
$345$	−7.94555	−0.427774
$346$	17.1514	0.922063
$347$	17.6187	0.945820	0.472910	−	0.881111i	$-0.343204\pi$
$347$	17.6187	0.945820	0.472910	+	0.881111i	$0.343204\pi$
$348$	−1.24168	−0.0665611
$349$	−10.9907	−0.588318	−0.294159	−	0.955757i	$-0.595039\pi$
$349$	−10.9907	−0.588318	−0.294159	+	0.955757i	$0.595039\pi$
$350$	2.38629	0.127553
$351$	−3.96156	−0.211452
$352$	−21.3624	−1.13862
$353$	−22.0539	−1.17381	−0.586907	−	0.809655i	$-0.699654\pi$
$353$	−22.0539	−1.17381	−0.586907	+	0.809655i	$0.699654\pi$
$354$	10.5502	0.560738
$355$	27.3765	1.45300
$356$	6.50753	0.344899
$357$	−5.48542	−0.290319
$358$	13.5372	0.715464
$359$	4.31686	0.227835	0.113918	−	0.993490i	$-0.463660\pi$
$359$	4.31686	0.227835	0.113918	+	0.993490i	$0.463660\pi$
$360$	12.4742	0.657449
$361$	14.3713	0.756387
$362$	−9.45412	−0.496898
$363$	4.56034	0.239356
$364$	−1.18287	−0.0619991
$365$	−5.92605	−0.310183
$366$	−3.52814	−0.184419
$367$	28.7427	1.50036	0.750178	−	0.661236i	$-0.229968\pi$
$367$	28.7427	1.50036	0.750178	+	0.661236i	$0.229968\pi$
$368$	−5.44943	−0.284071
$369$	−5.93388	−0.308905
$370$	13.7990	0.717378
$371$	−7.96562	−0.413554
$372$	−4.74540	−0.246037
$373$	−9.99261	−0.517398	−0.258699	−	0.965958i	$-0.583294\pi$
$373$	−9.99261	−0.517398	−0.258699	+	0.965958i	$0.583294\pi$
$374$	27.1746	1.40516
$375$	8.73386	0.451015
$376$	7.85576	0.405130
$377$	1.65075	0.0850178
$378$	4.41542	0.227105
$379$	13.4947	0.693177	0.346589	−	0.938017i	$-0.387340\pi$
$379$	13.4947	0.693177	0.346589	+	0.938017i	$0.387340\pi$
$380$	−10.1574	−0.521064
$381$	8.90076	0.456000
$382$	−12.1325	−0.620754
$383$	16.2006	0.827814	0.413907	−	0.910319i	$-0.364164\pi$
$383$	16.2006	0.827814	0.413907	+	0.910319i	$0.364164\pi$
$384$	2.67372	0.136443
$385$	−8.00082	−0.407759
$386$	8.57217	0.436312
$387$	16.0056	0.813611
$388$	7.16388	0.363691
$389$	20.3539	1.03198	0.515992	−	0.856593i	$-0.327423\pi$
$389$	20.3539	1.03198	0.515992	+	0.856593i	$0.327423\pi$
$390$	1.19841	0.0606840
$391$	−43.3111	−2.19034
$392$	−16.9965	−0.858452
$393$	16.3583	0.825168
$394$	−6.18554	−0.311623
$395$	27.5095	1.38416
$396$	10.7142	0.538407
$397$	−0.922450	−0.0462965	−0.0231482	−	0.999732i	$-0.507369\pi$
$397$	−0.922450	−0.0462965	−0.0231482	+	0.999732i	$0.507369\pi$
$398$	−20.4820	−1.02667
$399$	−4.75307	−0.237951
$400$	1.79594	0.0897968
$401$	−29.2854	−1.46244	−0.731221	−	0.682140i	$-0.761049\pi$
$401$	−29.2854	−1.46244	−0.731221	+	0.682140i	$0.761049\pi$
$402$	8.59878	0.428868
$403$	6.30875	0.314261
$404$	15.2114	0.756795
$405$	−7.71846	−0.383533
$406$	−1.83987	−0.0913112
$407$	34.6468	1.71738
$408$	−14.3642	−0.711133
$409$	−12.7361	−0.629761	−0.314881	−	0.949131i	$-0.601964\pi$
$409$	−12.7361	−0.629761	−0.314881	+	0.949131i	$0.601964\pi$
$410$	3.96932	0.196031
$411$	−0.843526	−0.0416081
$412$	−1.03989	−0.0512319
$413$	−16.9320	−0.833170
$414$	15.7661	0.774862
$415$	16.9684	0.832948
$416$	−5.13535	−0.251781
$417$	12.6461	0.619283
$418$	23.5465	1.15170
$419$	23.4145	1.14387	0.571937	−	0.820297i	$-0.306192\pi$
$419$	23.4145	1.14387	0.571937	+	0.820297i	$0.306192\pi$
$420$	1.44672	0.0705925
$421$	−9.41987	−0.459096	−0.229548	−	0.973297i	$-0.573725\pi$
$421$	−9.41987	−0.459096	−0.229548	+	0.973297i	$0.573725\pi$
$422$	−11.2978	−0.549967
$423$	−6.53219	−0.317606
$424$	−20.8589	−1.01300
$425$	14.2738	0.692381
$426$	11.4755	0.555989
$427$	5.66230	0.274018
$428$	10.7580	0.520008
$429$	3.00899	0.145275
$430$	−10.7066	−0.516317
$431$	20.1628	0.971208	0.485604	−	0.874179i	$-0.338600\pi$
$431$	20.1628	0.971208	0.485604	+	0.874179i	$0.338600\pi$
$432$	3.32307	0.159881
$433$	6.33205	0.304299	0.152149	−	0.988357i	$-0.451380\pi$
$433$	6.33205	0.304299	0.152149	+	0.988357i	$0.451380\pi$
$434$	−7.03152	−0.337524
$435$	−2.01896	−0.0968017
$436$	−3.42008	−0.163792
$437$	−37.5287	−1.79524
$438$	−2.48404	−0.118692
$439$	12.3170	0.587857	0.293928	−	0.955827i	$-0.405037\pi$
$439$	12.3170	0.587857	0.293928	+	0.955827i	$0.405037\pi$
$440$	−20.9510	−0.998802
$441$	14.1328	0.672992
$442$	6.53254	0.310721
$443$	10.7673	0.511568	0.255784	−	0.966734i	$-0.417666\pi$
$443$	10.7673	0.511568	0.255784	+	0.966734i	$0.417666\pi$
$444$	−6.26487	−0.297317
$445$	10.5812	0.501596
$446$	4.78059	0.226368
$447$	6.93903	0.328205
$448$	7.63201	0.360579
$449$	−31.8202	−1.50169	−0.750845	−	0.660479i	$-0.770353\pi$
$449$	−31.8202	−1.50169	−0.750845	+	0.660479i	$0.770353\pi$
$450$	−5.19595	−0.244939
$451$	9.96623	0.469291
$452$	9.62604	0.452771
$453$	−3.41459	−0.160431
$454$	11.1859	0.524980
$455$	−1.92333	−0.0901671
$456$	−12.4464	−0.582858
$457$	40.7254	1.90506	0.952528	−	0.304451i	$-0.0984730\pi$
$457$	40.7254	1.90506	0.952528	+	0.304451i	$0.0984730\pi$
$458$	−15.5586	−0.727005
$459$	26.4112	1.23277
$460$	11.4228	0.532591
$461$	18.0008	0.838380	0.419190	−	0.907899i	$-0.362314\pi$
$461$	18.0008	0.838380	0.419190	+	0.907899i	$0.362314\pi$
$462$	−3.35372	−0.156029
$463$	−9.72636	−0.452022	−0.226011	−	0.974125i	$-0.572569\pi$
$463$	−9.72636	−0.452022	−0.226011	+	0.974125i	$0.572569\pi$
$464$	−1.38470	−0.0642829
$465$	−7.71596	−0.357819
$466$	−9.38357	−0.434685
$467$	−5.50640	−0.254806	−0.127403	−	0.991851i	$-0.540664\pi$
$467$	−5.50640	−0.254806	−0.127403	+	0.991851i	$0.540664\pi$
$468$	2.57560	0.119057
$469$	−13.8002	−0.637232
$470$	4.36955	0.201552
$471$	−5.26718	−0.242699
$472$	−44.3384	−2.04084
$473$	−26.8822	−1.23605
$474$	11.5312	0.529647
$475$	12.3681	0.567488
$476$	7.88604	0.361456
$477$	17.3445	0.794148
$478$	−13.4625	−0.615762
$479$	−42.5894	−1.94596	−0.972981	−	0.230886i	$-0.925837\pi$
$479$	−42.5894	−1.94596	−0.972981	+	0.230886i	$0.925837\pi$
$480$	6.28083	0.286680
$481$	8.32880	0.379760
$482$	−4.73870	−0.215842
$483$	5.34520	0.243215
$484$	−6.55611	−0.298005
$485$	11.6484	0.528926
$486$	−14.8806	−0.674996
$487$	9.37700	0.424912	0.212456	−	0.977171i	$-0.431854\pi$
$487$	9.37700	0.424912	0.212456	+	0.977171i	$0.431854\pi$
$488$	14.8274	0.671203
$489$	1.75252	0.0792517
$490$	−9.45382	−0.427080
$491$	−33.9083	−1.53026	−0.765130	−	0.643876i	$-0.777325\pi$
$491$	−33.9083	−1.53026	−0.765130	+	0.643876i	$0.777325\pi$
$492$	−1.80210	−0.0812451
$493$	−11.0053	−0.495656
$494$	5.66039	0.254673
$495$	17.4211	0.783021
$496$	−5.29197	−0.237616
$497$	−18.4170	−0.826115
$498$	7.11270	0.318728
$499$	12.9156	0.578181	0.289090	−	0.957302i	$-0.406647\pi$
$499$	12.9156	0.578181	0.289090	+	0.957302i	$0.406647\pi$
$500$	−12.5561	−0.561527
$501$	−11.3634	−0.507679
$502$	−4.63268	−0.206767
$503$	12.9754	0.578544	0.289272	−	0.957247i	$-0.406587\pi$
$503$	12.9754	0.578544	0.289272	+	0.957247i	$0.406587\pi$
$504$	−8.39176	−0.373799
$505$	24.7336	1.10063
$506$	−26.4799	−1.17718
$507$	0.723336	0.0321245
$508$	−12.7961	−0.567734
$509$	22.5916	1.00135	0.500677	−	0.865634i	$-0.333084\pi$
$509$	22.5916	1.00135	0.500677	+	0.865634i	$0.333084\pi$
$510$	−7.98968	−0.353789
$511$	3.98662	0.176358
$512$	9.30483	0.411219
$513$	22.8851	1.01040
$514$	−10.7938	−0.476094
$515$	−1.69086	−0.0745081
$516$	4.86087	0.213988
$517$	10.9711	0.482509
$518$	−9.28301	−0.407872
$519$	−12.6613	−0.555771
$520$	−5.03645	−0.220863
$521$	−17.6180	−0.771857	−0.385928	−	0.922529i	$-0.626119\pi$
$521$	−17.6180	−0.771857	−0.385928	+	0.922529i	$0.626119\pi$
$522$	4.00616	0.175345
$523$	18.1906	0.795420	0.397710	−	0.917511i	$-0.369805\pi$
$523$	18.1906	0.795420	0.397710	+	0.917511i	$0.369805\pi$
$524$	−23.5173	−1.02736
$525$	−1.76159	−0.0768819
$526$	8.00110	0.348864
$527$	−42.0597	−1.83215
$528$	−2.52403	−0.109844
$529$	19.2040	0.834956
$530$	−11.6022	−0.503966
$531$	36.8681	1.59994
$532$	6.83318	0.296256
$533$	2.39580	0.103774
$534$	4.43534	0.191936
$535$	17.4924	0.756262
$536$	−36.1373	−1.56089
$537$	−9.99332	−0.431244
$538$	31.4325	1.35515
$539$	−23.7368	−1.02241
$540$	−6.96566	−0.299754
$541$	−21.3841	−0.919375	−0.459688	−	0.888081i	$-0.652039\pi$
$541$	−21.3841	−0.919375	−0.459688	+	0.888081i	$0.652039\pi$
$542$	27.1307	1.16536
$543$	6.97914	0.299504
$544$	34.2368	1.46789
$545$	−5.56101	−0.238208
$546$	−0.806207	−0.0345025
$547$	−33.6998	−1.44090	−0.720449	−	0.693508i	$-0.756064\pi$
$547$	−33.6998	−1.44090	−0.720449	+	0.693508i	$0.756064\pi$
$548$	1.21268	0.0518033
$549$	−12.3292	−0.526197
$550$	8.72684	0.372114
$551$	−9.53603	−0.406248
$552$	13.9970	0.595752
$553$	−18.5065	−0.786975
$554$	−22.7461	−0.966391
$555$	−10.1866	−0.432397
$556$	−18.1805	−0.771025
$557$	−21.2048	−0.898477	−0.449238	−	0.893412i	$-0.648305\pi$
$557$	−21.2048	−0.898477	−0.449238	+	0.893412i	$0.648305\pi$
$558$	15.3106	0.648147
$559$	−6.46226	−0.273324
$560$	1.61335	0.0681764
$561$	−20.0606	−0.846959
$562$	4.10382	0.173109
$563$	−9.22849	−0.388934	−0.194467	−	0.980909i	$-0.562298\pi$
$563$	−9.22849	−0.388934	−0.194467	+	0.980909i	$0.562298\pi$
$564$	−1.98381	−0.0835334
$565$	15.6518	0.658477
$566$	28.2052	1.18555
$567$	5.19243	0.218061
$568$	−48.2269	−2.02356
$569$	−31.6427	−1.32653	−0.663265	−	0.748385i	$-0.730830\pi$
$569$	−31.6427	−1.32653	−0.663265	+	0.748385i	$0.730830\pi$
$570$	−6.92299	−0.289972
$571$	30.5461	1.27832	0.639158	−	0.769076i	$-0.279283\pi$
$571$	30.5461	1.27832	0.639158	+	0.769076i	$0.279283\pi$
$572$	−4.32583	−0.180872
$573$	8.95637	0.374158
$574$	−2.67028	−0.111455
$575$	−13.9089	−0.580042
$576$	−16.6181	−0.692420
$577$	12.3957	0.516041	0.258020	−	0.966139i	$-0.416930\pi$
$577$	12.3957	0.516041	0.258020	+	0.966139i	$0.416930\pi$
$578$	−26.8943	−1.11865
$579$	−6.32807	−0.262986
$580$	2.90253	0.120521
$581$	−11.4152	−0.473581
$582$	4.88269	0.202394
$583$	−29.1309	−1.20648
$584$	10.4394	0.431986
$585$	4.18789	0.173148
$586$	23.7066	0.979309
$587$	−27.3126	−1.12731	−0.563656	−	0.826010i	$-0.690606\pi$
$587$	−27.3126	−1.12731	−0.563656	+	0.826010i	$0.690606\pi$
$588$	4.29211	0.177003
$589$	−36.4443	−1.50166
$590$	−24.6620	−1.01532
$591$	4.56624	0.187830
$592$	−6.98645	−0.287141
$593$	18.0116	0.739649	0.369824	−	0.929102i	$-0.379418\pi$
$593$	18.0116	0.739649	0.369824	+	0.929102i	$0.379418\pi$
$594$	16.1475	0.662541
$595$	12.8226	0.525676
$596$	−9.97580	−0.408625
$597$	15.1201	0.618823
$598$	−6.36555	−0.260307
$599$	−38.6709	−1.58005	−0.790026	−	0.613073i	$-0.789933\pi$
$599$	−38.6709	−1.58005	−0.790026	+	0.613073i	$0.789933\pi$
$600$	−4.61291	−0.188321
$601$	−14.6592	−0.597963	−0.298981	−	0.954259i	$-0.596647\pi$
$601$	−14.6592	−0.597963	−0.298981	+	0.954259i	$0.596647\pi$
$602$	7.20262	0.293557
$603$	30.0487	1.22368
$604$	4.90894	0.199742
$605$	−10.6602	−0.433397
$606$	10.3676	0.421156
$607$	−30.1587	−1.22411	−0.612053	−	0.790817i	$-0.709656\pi$
$607$	−30.1587	−1.22411	−0.612053	+	0.790817i	$0.709656\pi$
$608$	29.6659	1.20311
$609$	1.35821	0.0550375
$610$	8.24731	0.333924
$611$	2.63737	0.106696
$612$	−17.1712	−0.694104
$613$	−16.8913	−0.682233	−0.341117	−	0.940021i	$-0.610805\pi$
$613$	−16.8913	−0.682233	−0.341117	+	0.940021i	$0.610805\pi$
$614$	−10.3218	−0.416555
$615$	−2.93020	−0.118157
$616$	14.0944	0.567878
$617$	−24.1757	−0.973278	−0.486639	−	0.873603i	$-0.661777\pi$
$617$	−24.1757	−0.973278	−0.486639	+	0.873603i	$0.661777\pi$
$618$	−0.708761	−0.0285105
$619$	−3.58409	−0.144057	−0.0720284	−	0.997403i	$-0.522947\pi$
$619$	−3.58409	−0.144057	−0.0720284	+	0.997403i	$0.522947\pi$
$620$	11.0928	0.445496
$621$	−25.7361	−1.03275
$622$	33.1608	1.32963
$623$	−7.11826	−0.285187
$624$	−0.606756	−0.0242897
$625$	−9.71111	−0.388444
$626$	−5.18475	−0.207224
$627$	−17.3823	−0.694183
$628$	7.57229	0.302167
$629$	−55.5271	−2.21401
$630$	−4.66769	−0.185965
$631$	23.4763	0.934579	0.467289	−	0.884104i	$-0.345231\pi$
$631$	23.4763	0.934579	0.467289	+	0.884104i	$0.345231\pi$
$632$	−48.4612	−1.92768
$633$	8.34015	0.331491
$634$	−25.1760	−0.999868
$635$	−20.8063	−0.825671
$636$	5.26747	0.208869
$637$	−5.70612	−0.226085
$638$	−6.72854	−0.266385
$639$	40.1014	1.58639
$640$	−6.25003	−0.247054
$641$	23.8332	0.941353	0.470676	−	0.882306i	$-0.344010\pi$
$641$	23.8332	0.941353	0.470676	+	0.882306i	$0.344010\pi$
$642$	7.33233	0.289384
$643$	−42.9870	−1.69524	−0.847622	−	0.530601i	$-0.821966\pi$
$643$	−42.9870	−1.69524	−0.847622	+	0.530601i	$0.821966\pi$
$644$	−7.68445	−0.302810
$645$	7.90372	0.311209
$646$	−37.7372	−1.48475
$647$	20.5228	0.806834	0.403417	−	0.915016i	$-0.367822\pi$
$647$	20.5228	0.806834	0.403417	+	0.915016i	$0.367822\pi$
$648$	13.5969	0.534139
$649$	−61.9217	−2.43064
$650$	2.09786	0.0822848
$651$	5.19075	0.203442
$652$	−2.51949	−0.0986708
$653$	4.87166	0.190643	0.0953213	−	0.995447i	$-0.469612\pi$
$653$	4.87166	0.190643	0.0953213	+	0.995447i	$0.469612\pi$
$654$	−2.33102	−0.0911502
$655$	−38.2389	−1.49412
$656$	−2.00967	−0.0784643
$657$	−8.68053	−0.338660
$658$	−2.93952	−0.114595
$659$	44.7381	1.74275	0.871374	−	0.490620i	$-0.163230\pi$
$659$	44.7381	1.74275	0.871374	+	0.490620i	$0.163230\pi$
$660$	5.29075	0.205942
$661$	6.13482	0.238617	0.119308	−	0.992857i	$-0.461932\pi$
$661$	6.13482	0.238617	0.119308	+	0.992857i	$0.461932\pi$
$662$	13.5544	0.526806
$663$	−4.82240	−0.187286
$664$	−29.8919	−1.16003
$665$	11.1107	0.430854
$666$	20.2130	0.783237
$667$	10.7240	0.415236
$668$	16.3364	0.632076
$669$	−3.52909	−0.136442
$670$	−20.1004	−0.776545
$671$	20.7075	0.799402
$672$	−4.22530	−0.162994
$673$	51.4589	1.98360	0.991798	−	0.127815i	$-0.0407965\pi$
$673$	51.4589	1.98360	0.991798	+	0.127815i	$0.0407965\pi$
$674$	−29.8251	−1.14882
$675$	8.48170	0.326461
$676$	−1.03989	−0.0399959
$677$	10.5970	0.407275	0.203638	−	0.979046i	$-0.434724\pi$
$677$	10.5970	0.407275	0.203638	+	0.979046i	$0.434724\pi$
$678$	6.56082	0.251967
$679$	−7.83621	−0.300726
$680$	33.5775	1.28764
$681$	−8.25756	−0.316430
$682$	−25.7148	−0.984671
$683$	13.7509	0.526165	0.263082	−	0.964773i	$-0.415261\pi$
$683$	13.7509	0.526165	0.263082	+	0.964773i	$0.415261\pi$
$684$	−14.8787	−0.568901
$685$	1.97181	0.0753390
$686$	14.1618	0.540701
$687$	11.4855	0.438200
$688$	5.42074	0.206664
$689$	−7.00281	−0.266786
$690$	7.78544	0.296387
$691$	−34.9594	−1.32992	−0.664959	−	0.746880i	$-0.731551\pi$
$691$	−34.9594	−1.32992	−0.664959	+	0.746880i	$0.731551\pi$
$692$	18.2024	0.691951
$693$	−11.7197	−0.445194
$694$	−17.2637	−0.655320
$695$	−29.5613	−1.12132
$696$	3.55663	0.134814
$697$	−15.9725	−0.605002
$698$	10.7692	0.407621
$699$	6.92705	0.262005
$700$	2.53252	0.0957203
$701$	−9.61504	−0.363155	−0.181578	−	0.983377i	$-0.558120\pi$
$701$	−9.61504	−0.363155	−0.181578	+	0.983377i	$0.558120\pi$
$702$	3.88173	0.146506
$703$	−48.1137	−1.81464
$704$	27.9108	1.05193
$705$	−3.22565	−0.121485
$706$	21.6096	0.813286
$707$	−16.6390	−0.625773
$708$	11.1967	0.420799
$709$	27.4114	1.02946	0.514728	−	0.857354i	$-0.327893\pi$
$709$	27.4114	1.02946	0.514728	+	0.857354i	$0.327893\pi$
$710$	−26.8249	−1.00672
$711$	40.2963	1.51123
$712$	−18.6400	−0.698562
$713$	40.9845	1.53488
$714$	5.37489	0.201150
$715$	−7.03376	−0.263048
$716$	14.3668	0.536911
$717$	9.93819	0.371148
$718$	−4.22988	−0.157858
$719$	−51.8965	−1.93541	−0.967706	−	0.252080i	$-0.918885\pi$
$719$	−51.8965	−1.93541	−0.967706	+	0.252080i	$0.918885\pi$
$720$	−3.51293	−0.130919
$721$	1.13749	0.0423623
$722$	−14.0818	−0.524069
$723$	3.49816	0.130098
$724$	−10.0335	−0.372891
$725$	−3.53425	−0.131259
$726$	−4.46844	−0.165839
$727$	47.4126	1.75844	0.879219	−	0.476418i	$-0.158065\pi$
$727$	47.4126	1.75844	0.879219	+	0.476418i	$0.158065\pi$
$728$	3.38817	0.125574
$729$	−2.70946	−0.100350
$730$	5.80664	0.214913
$731$	43.0831	1.59349
$732$	−3.74434	−0.138395
$733$	8.52715	0.314957	0.157479	−	0.987522i	$-0.449663\pi$
$733$	8.52715	0.314957	0.157479	+	0.987522i	$0.449663\pi$
$734$	−28.1635	−1.03953
$735$	6.97892	0.257421
$736$	−33.3616	−1.22973
$737$	−50.4682	−1.85902
$738$	5.81431	0.214028
$739$	41.6029	1.53039	0.765194	−	0.643800i	$-0.222643\pi$
$739$	41.6029	1.53039	0.765194	+	0.643800i	$0.222643\pi$
$740$	14.6446	0.538348
$741$	−4.17856	−0.153503
$742$	7.80511	0.286535
$743$	21.3593	0.783598	0.391799	−	0.920051i	$-0.371853\pi$
$743$	21.3593	0.783598	0.391799	+	0.920051i	$0.371853\pi$
$744$	13.5926	0.498327
$745$	−16.2205	−0.594275
$746$	9.79126	0.358483
$747$	24.8555	0.909417
$748$	28.8398	1.05449
$749$	−11.7676	−0.429980
$750$	−8.55787	−0.312489
$751$	−25.2396	−0.921005	−0.460502	−	0.887659i	$-0.652331\pi$
$751$	−25.2396	−0.921005	−0.460502	+	0.887659i	$0.652331\pi$
$752$	−2.21230	−0.0806744
$753$	3.41990	0.124628
$754$	−1.61748	−0.0589053
$755$	7.98188	0.290490
$756$	4.68600	0.170428
$757$	23.2735	0.845888	0.422944	−	0.906156i	$-0.360997\pi$
$757$	23.2735	0.845888	0.422944	+	0.906156i	$0.360997\pi$
$758$	−13.2228	−0.480274
$759$	19.5478	0.709540
$760$	29.0946	1.05537
$761$	19.7858	0.717234	0.358617	−	0.933485i	$-0.383248\pi$
$761$	19.7858	0.717234	0.358617	+	0.933485i	$0.383248\pi$
$762$	−8.72141	−0.315943
$763$	3.74105	0.135435
$764$	−12.8760	−0.465838
$765$	−27.9202	−1.00946
$766$	−15.8742	−0.573558
$767$	−14.8854	−0.537482
$768$	−12.3263	−0.444788
$769$	0.235304	0.00848527	0.00424264	−	0.999991i	$-0.498650\pi$
$769$	0.235304	0.00848527	0.00424264	+	0.999991i	$0.498650\pi$
$770$	7.83960	0.282520
$771$	7.96810	0.286964
$772$	9.09747	0.327425
$773$	13.0881	0.470745	0.235372	−	0.971905i	$-0.424369\pi$
$773$	13.0881	0.470745	0.235372	+	0.971905i	$0.424369\pi$
$774$	−15.6831	−0.563718
$775$	−13.5070	−0.485187
$776$	−20.5200	−0.736625
$777$	6.85282	0.245844
$778$	−19.9438	−0.715019
$779$	−13.8400	−0.495871
$780$	1.27185	0.0455396
$781$	−67.3523	−2.41005
$782$	42.4384	1.51759
$783$	−6.53953	−0.233704
$784$	4.78647	0.170945
$785$	12.3125	0.439450
$786$	−16.0287	−0.571724
$787$	2.17533	0.0775421	0.0387710	−	0.999248i	$-0.487656\pi$
$787$	2.17533	0.0775421	0.0387710	+	0.999248i	$0.487656\pi$
$788$	−6.56459	−0.233854
$789$	−5.90650	−0.210277
$790$	−26.9552	−0.959024
$791$	−10.5294	−0.374384
$792$	−30.6893	−1.09050
$793$	4.97790	0.176770
$794$	0.903863	0.0320769
$795$	8.56485	0.303764
$796$	−21.7372	−0.770454
$797$	−0.313833	−0.0111165	−0.00555826	−	0.999985i	$-0.501769\pi$
$797$	−0.313833	−0.0111165	−0.00555826	+	0.999985i	$0.501769\pi$
$798$	4.65729	0.164866
$799$	−17.5830	−0.622042
$800$	10.9948	0.388725
$801$	15.4994	0.547645
$802$	28.6953	1.01327
$803$	14.5794	0.514495
$804$	9.12571	0.321839
$805$	−12.4948	−0.440385
$806$	−6.18162	−0.217738
$807$	−23.2038	−0.816814
$808$	−43.5710	−1.53282
$809$	11.8944	0.418185	0.209092	−	0.977896i	$-0.432949\pi$
$809$	11.8944	0.418185	0.209092	+	0.977896i	$0.432949\pi$
$810$	7.56293	0.265734
$811$	−30.1587	−1.05902	−0.529508	−	0.848305i	$-0.677623\pi$
$811$	−30.1587	−1.05902	−0.529508	+	0.848305i	$0.677623\pi$
$812$	−1.95262	−0.0685234
$813$	−20.0282	−0.702420
$814$	−33.9486	−1.18990
$815$	−4.09666	−0.143500
$816$	4.04517	0.141609
$817$	37.3311	1.30605
$818$	12.4795	0.436335
$819$	−2.81731	−0.0984449
$820$	4.21256	0.147109
$821$	−18.3250	−0.639548	−0.319774	−	0.947494i	$-0.603607\pi$
$821$	−18.3250	−0.639548	−0.319774	+	0.947494i	$0.603607\pi$
$822$	0.826529	0.0288285
$823$	3.94568	0.137538	0.0687689	−	0.997633i	$-0.478093\pi$
$823$	3.94568	0.137538	0.0687689	+	0.997633i	$0.478093\pi$
$824$	2.97864	0.103766
$825$	−6.44225	−0.224290
$826$	16.5908	0.577269
$827$	28.9701	1.00739	0.503695	−	0.863882i	$-0.331974\pi$
$827$	28.9701	1.00739	0.503695	+	0.863882i	$0.331974\pi$
$828$	16.7322	0.581486
$829$	−31.6825	−1.10038	−0.550190	−	0.835040i	$-0.685445\pi$
$829$	−31.6825	−1.10038	−0.550190	+	0.835040i	$0.685445\pi$
$830$	−16.6265	−0.577115
$831$	16.7915	0.582489
$832$	6.70953	0.232611
$833$	38.0420	1.31808
$834$	−12.3913	−0.429075
$835$	26.5629	0.919246
$836$	24.9895	0.864278
$837$	−24.9925	−0.863866
$838$	−22.9427	−0.792543
$839$	41.4769	1.43194	0.715970	−	0.698131i	$-0.245985\pi$
$839$	41.4769	1.43194	0.715970	+	0.698131i	$0.245985\pi$
$840$	−4.14393	−0.142979
$841$	−26.2750	−0.906036
$842$	9.23006	0.318089
$843$	−3.02949	−0.104341
$844$	−11.9901	−0.412716
$845$	−1.69086	−0.0581673
$846$	6.40056	0.220056
$847$	7.17139	0.246412
$848$	5.87417	0.201720
$849$	−20.8214	−0.714589
$850$	−13.9862	−0.479722
$851$	54.1077	1.85479
$852$	12.1787	0.417236
$853$	22.6674	0.776117	0.388059	−	0.921635i	$-0.373146\pi$
$853$	22.6674	0.776117	0.388059	+	0.921635i	$0.373146\pi$
$854$	−5.54820	−0.189856
$855$	−24.1926	−0.827369
$856$	−30.8149	−1.05323
$857$	−19.6765	−0.672136	−0.336068	−	0.941838i	$-0.609097\pi$
$857$	−19.6765	−0.672136	−0.336068	+	0.941838i	$0.609097\pi$
$858$	−2.94836	−0.100655
$859$	11.1167	0.379296	0.189648	−	0.981852i	$-0.439265\pi$
$859$	11.1167	0.379296	0.189648	+	0.981852i	$0.439265\pi$
$860$	−11.3627	−0.387464
$861$	1.97123	0.0671793
$862$	−19.7565	−0.672910
$863$	22.0171	0.749471	0.374735	−	0.927132i	$-0.377734\pi$
$863$	22.0171	0.749471	0.374735	+	0.927132i	$0.377734\pi$
$864$	20.3440	0.692116
$865$	29.5969	1.00632
$866$	−6.20446	−0.210836
$867$	19.8536	0.674265
$868$	−7.46241	−0.253291
$869$	−67.6795	−2.29587
$870$	1.97828	0.0670699
$871$	−12.1321	−0.411082
$872$	9.79637	0.331747
$873$	17.0627	0.577485
$874$	36.7725	1.24385
$875$	13.7345	0.464311
$876$	−2.63626	−0.0890708
$877$	−14.1944	−0.479312	−0.239656	−	0.970858i	$-0.577035\pi$
$877$	−14.1944	−0.479312	−0.239656	+	0.970858i	$0.577035\pi$
$878$	−12.0688	−0.407302
$879$	−17.5004	−0.590275
$880$	5.90013	0.198893
$881$	0.0435137	0.00146601	0.000733007	−	1.00000i	$-0.499767\pi$
$881$	0.0435137	0.00146601	0.000733007		1.00000i	$0.499767\pi$
$882$	−13.8481	−0.466288
$883$	30.6163	1.03032	0.515160	−	0.857094i	$-0.327733\pi$
$883$	30.6163	1.03032	0.515160	+	0.857094i	$0.327733\pi$
$884$	6.93286	0.233177
$885$	18.2058	0.611980
$886$	−10.5503	−0.354444
$887$	43.2601	1.45253	0.726266	−	0.687413i	$-0.241254\pi$
$887$	43.2601	1.45253	0.726266	+	0.687413i	$0.241254\pi$
$888$	17.9449	0.602191
$889$	13.9970	0.469443
$890$	−10.3680	−0.347535
$891$	18.9891	0.636158
$892$	5.07354	0.169875
$893$	−15.2355	−0.509837
$894$	−6.79921	−0.227399
$895$	23.3602	0.780846
$896$	4.20458	0.140465
$897$	4.69912	0.156899
$898$	31.1790	1.04046
$899$	10.4141	0.347331
$900$	−5.51435	−0.183812
$901$	46.6869	1.55537
$902$	−9.76541	−0.325152
$903$	−5.31706	−0.176941
$904$	−27.5725	−0.917048
$905$	−16.3143	−0.542306
$906$	3.34578	0.111156
$907$	−3.51297	−0.116646	−0.0583232	−	0.998298i	$-0.518575\pi$
$907$	−3.51297	−0.116646	−0.0583232	+	0.998298i	$0.518575\pi$
$908$	11.8714	0.393965
$909$	36.2300	1.20167
$910$	1.88457	0.0624730
$911$	−19.0364	−0.630706	−0.315353	−	0.948975i	$-0.602123\pi$
$911$	−19.0364	−0.630706	−0.315353	+	0.948975i	$0.602123\pi$
$912$	3.50511	0.116066
$913$	−41.7461	−1.38159
$914$	−39.9048	−1.31993
$915$	−6.08826	−0.201272
$916$	−16.5120	−0.545572
$917$	25.7244	0.849494
$918$	−25.8790	−0.854136
$919$	−12.9946	−0.428652	−0.214326	−	0.976762i	$-0.568756\pi$
$919$	−12.9946	−0.428652	−0.214326	+	0.976762i	$0.568756\pi$
$920$	−32.7191	−1.07872
$921$	7.61969	0.251077
$922$	−17.6381	−0.580879
$923$	−16.1909	−0.532931
$924$	−3.55924	−0.117090
$925$	−17.8320	−0.586311
$926$	9.53037	0.313187
$927$	−2.47678	−0.0813483
$928$	−8.47717	−0.278277
$929$	21.7623	0.713999	0.356999	−	0.934105i	$-0.383800\pi$
$929$	21.7623	0.713999	0.356999	+	0.934105i	$0.383800\pi$
$930$	7.56049	0.247918
$931$	32.9631	1.08032
$932$	−9.95859	−0.326204
$933$	−24.4797	−0.801429
$934$	5.39545	0.176544
$935$	46.8933	1.53357
$936$	−7.37745	−0.241139
$937$	−45.4571	−1.48502	−0.742509	−	0.669836i	$-0.766364\pi$
$937$	−45.4571	−1.48502	−0.742509	+	0.669836i	$0.766364\pi$
$938$	13.5221	0.441512
$939$	3.82744	0.124904
$940$	4.63732	0.151253
$941$	−56.9156	−1.85540	−0.927698	−	0.373332i	$-0.878215\pi$
$941$	−56.9156	−1.85540	−0.927698	+	0.373332i	$0.878215\pi$
$942$	5.16104	0.168156
$943$	15.5642	0.506840
$944$	12.4864	0.406397
$945$	7.61938	0.247858
$946$	26.3405	0.856404
$947$	−34.5053	−1.12127	−0.560635	−	0.828063i	$-0.689443\pi$
$947$	−34.5053	−1.12127	−0.560635	+	0.828063i	$0.689443\pi$
$948$	12.2379	0.397468
$949$	3.50476	0.113769
$950$	−12.1189	−0.393189
$951$	18.5852	0.602668
$952$	−22.5885	−0.732098
$953$	33.3108	1.07904	0.539521	−	0.841972i	$-0.318605\pi$
$953$	33.3108	1.07904	0.539521	+	0.841972i	$0.318605\pi$
$954$	−16.9950	−0.550232
$955$	−20.9362	−0.677481
$956$	−14.2875	−0.462091
$957$	4.96708	0.160563
$958$	41.7313	1.34828
$959$	−1.32649	−0.0428347
$960$	−8.20615	−0.264852
$961$	8.80029	0.283880
$962$	−8.16097	−0.263120
$963$	25.6230	0.825691
$964$	−5.02909	−0.161976
$965$	14.7924	0.476184
$966$	−5.23749	−0.168513
$967$	−4.53237	−0.145751	−0.0728755	−	0.997341i	$-0.523218\pi$
$967$	−4.53237	−0.145751	−0.0728755	+	0.997341i	$0.523218\pi$
$968$	18.7791	0.603583
$969$	27.8580	0.894928
$970$	−11.4137	−0.366471
$971$	−13.2598	−0.425528	−0.212764	−	0.977104i	$-0.568247\pi$
$971$	−13.2598	−0.425528	−0.212764	+	0.977104i	$0.568247\pi$
$972$	−15.7924	−0.506542
$973$	19.8867	0.637540
$974$	−9.18805	−0.294404
$975$	−1.54866	−0.0495969
$976$	−4.17561	−0.133658
$977$	−17.0221	−0.544584	−0.272292	−	0.962215i	$-0.587782\pi$
$977$	−17.0221	−0.544584	−0.272292	+	0.962215i	$0.587782\pi$
$978$	−1.71721	−0.0549102
$979$	−26.0320	−0.831987
$980$	−10.0331	−0.320497
$981$	−8.14583	−0.260076
$982$	33.2250	1.06025
$983$	35.5227	1.13300	0.566500	−	0.824062i	$-0.308297\pi$
$983$	35.5227	1.13300	0.566500	+	0.824062i	$0.308297\pi$
$984$	5.16189	0.164555
$985$	−10.6740	−0.340101
$986$	10.7836	0.343419
$987$	2.16999	0.0690715
$988$	6.00726	0.191116
$989$	−41.9818	−1.33494
$990$	−17.0701	−0.542522
$991$	28.8405	0.916147	0.458074	−	0.888914i	$-0.348540\pi$
$991$	28.8405	0.916147	0.458074	+	0.888914i	$0.348540\pi$
$992$	−32.3976	−1.02863
$993$	−10.0060	−0.317531
$994$	18.0459	0.572380
$995$	−35.3444	−1.12049
$996$	7.54857	0.239185
$997$	17.7782	0.563040	0.281520	−	0.959555i	$-0.409161\pi$
$997$	17.7782	0.563040	0.281520	+	0.959555i	$0.409161\pi$
$998$	−12.6553	−0.400597
$999$	−32.9950	−1.04392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.d.1.9	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.d.1.9	✓	19	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-0.979850 q^{2} +0.723336 q^{3} -1.03989 q^{4} -1.69086 q^{5} -0.708761 q^{6} +1.13749 q^{7} +2.97864 q^{8} -2.47678 q^{9} +O(q^{10})\)	\(q-0.979850 q^{2} +0.723336 q^{3} -1.03989 q^{4} -1.69086 q^{5} -0.708761 q^{6} +1.13749 q^{7} +2.97864 q^{8} -2.47678 q^{9} +1.65679 q^{10} +4.15988 q^{11} -0.752193 q^{12} +1.00000 q^{13} -1.11457 q^{14} -1.22306 q^{15} -0.838830 q^{16} -6.66688 q^{17} +2.42688 q^{18} -5.77679 q^{19} +1.75831 q^{20} +0.822786 q^{21} -4.07606 q^{22} +6.49646 q^{23} +2.15456 q^{24} -2.14100 q^{25} -0.979850 q^{26} -3.96156 q^{27} -1.18287 q^{28} +1.65075 q^{29} +1.19841 q^{30} +6.30875 q^{31} -5.13535 q^{32} +3.00899 q^{33} +6.53254 q^{34} -1.92333 q^{35} +2.57560 q^{36} +8.32880 q^{37} +5.66039 q^{38} +0.723336 q^{39} -5.03645 q^{40} +2.39580 q^{41} -0.806207 q^{42} -6.46226 q^{43} -4.32583 q^{44} +4.18789 q^{45} -6.36555 q^{46} +2.63737 q^{47} -0.606756 q^{48} -5.70612 q^{49} +2.09786 q^{50} -4.82240 q^{51} -1.03989 q^{52} -7.00281 q^{53} +3.88173 q^{54} -7.03376 q^{55} +3.38817 q^{56} -4.17856 q^{57} -1.61748 q^{58} -14.8854 q^{59} +1.27185 q^{60} +4.97790 q^{61} -6.18162 q^{62} -2.81731 q^{63} +6.70953 q^{64} -1.69086 q^{65} -2.94836 q^{66} -12.1321 q^{67} +6.93286 q^{68} +4.69912 q^{69} +1.88457 q^{70} -16.1909 q^{71} -7.37745 q^{72} +3.50476 q^{73} -8.16097 q^{74} -1.54866 q^{75} +6.00726 q^{76} +4.73181 q^{77} -0.708761 q^{78} -16.2696 q^{79} +1.41834 q^{80} +4.56482 q^{81} -2.34752 q^{82} -10.0354 q^{83} -0.855611 q^{84} +11.2727 q^{85} +6.33204 q^{86} +1.19405 q^{87} +12.3908 q^{88} -6.25788 q^{89} -4.10350 q^{90} +1.13749 q^{91} -6.75563 q^{92} +4.56334 q^{93} -2.58422 q^{94} +9.76773 q^{95} -3.71459 q^{96} -6.88905 q^{97} +5.59114 q^{98} -10.3031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * q^99

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