LMFDB - Embedded newform 4009.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35519 q^{2} +0.815593 q^{3} +3.54692 q^{4} +1.05865 q^{5} -1.92088 q^{6} -4.06376 q^{7} -3.64328 q^{8} -2.33481 q^{9} +O(q^{10})$	\(q-2.35519 q^{2} +0.815593 q^{3} +3.54692 q^{4} +1.05865 q^{5} -1.92088 q^{6} -4.06376 q^{7} -3.64328 q^{8} -2.33481 q^{9} -2.49333 q^{10} +3.34628 q^{11} +2.89284 q^{12} +2.90167 q^{13} +9.57091 q^{14} +0.863432 q^{15} +1.48678 q^{16} +7.81825 q^{17} +5.49891 q^{18} +1.00000 q^{19} +3.75496 q^{20} -3.31437 q^{21} -7.88113 q^{22} -5.53700 q^{23} -2.97144 q^{24} -3.87925 q^{25} -6.83397 q^{26} -4.35103 q^{27} -14.4138 q^{28} +4.90550 q^{29} -2.03354 q^{30} -10.2234 q^{31} +3.78490 q^{32} +2.72921 q^{33} -18.4135 q^{34} -4.30212 q^{35} -8.28137 q^{36} -4.34401 q^{37} -2.35519 q^{38} +2.36658 q^{39} -3.85698 q^{40} -2.46182 q^{41} +7.80597 q^{42} -5.12426 q^{43} +11.8690 q^{44} -2.47176 q^{45} +13.0407 q^{46} -8.47533 q^{47} +1.21261 q^{48} +9.51411 q^{49} +9.13637 q^{50} +6.37651 q^{51} +10.2920 q^{52} +8.14686 q^{53} +10.2475 q^{54} +3.54256 q^{55} +14.8054 q^{56} +0.815593 q^{57} -11.5534 q^{58} +4.16450 q^{59} +3.06252 q^{60} +11.1105 q^{61} +24.0781 q^{62} +9.48809 q^{63} -11.8877 q^{64} +3.07186 q^{65} -6.42780 q^{66} +4.88499 q^{67} +27.7307 q^{68} -4.51594 q^{69} +10.1323 q^{70} -15.7871 q^{71} +8.50636 q^{72} +5.09224 q^{73} +10.2310 q^{74} -3.16389 q^{75} +3.54692 q^{76} -13.5985 q^{77} -5.57374 q^{78} -6.78921 q^{79} +1.57399 q^{80} +3.45575 q^{81} +5.79806 q^{82} -7.77213 q^{83} -11.7558 q^{84} +8.27683 q^{85} +12.0686 q^{86} +4.00089 q^{87} -12.1915 q^{88} +1.21312 q^{89} +5.82145 q^{90} -11.7917 q^{91} -19.6393 q^{92} -8.33814 q^{93} +19.9610 q^{94} +1.05865 q^{95} +3.08694 q^{96} +4.15615 q^{97} -22.4075 q^{98} -7.81293 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.35519	−1.66537	−0.832685	−	0.553747i	$-0.813198\pi$
$2$	−2.35519	−1.66537	−0.832685	+	0.553747i	$0.813198\pi$
$3$	0.815593	0.470883	0.235441	−	0.971889i	$-0.424346\pi$
$3$	0.815593	0.470883	0.235441	+	0.971889i	$0.424346\pi$
$4$	3.54692	1.77346
$5$	1.05865	0.473445	0.236722	−	0.971577i	$-0.423927\pi$
$5$	1.05865	0.473445	0.236722	+	0.971577i	$0.423927\pi$
$6$	−1.92088	−0.784194
$7$	−4.06376	−1.53596	−0.767978	−	0.640477i	$-0.778737\pi$
$7$	−4.06376	−1.53596	−0.767978	+	0.640477i	$0.778737\pi$
$8$	−3.64328	−1.28809
$9$	−2.33481	−0.778269
$10$	−2.49333	−0.788461
$11$	3.34628	1.00894	0.504471	−	0.863428i	$-0.331687\pi$
$11$	3.34628	1.00894	0.504471	+	0.863428i	$0.331687\pi$
$12$	2.89284	0.835091
$13$	2.90167	0.804777	0.402389	−	0.915469i	$-0.368180\pi$
$13$	2.90167	0.804777	0.402389	+	0.915469i	$0.368180\pi$
$14$	9.57091	2.55793
$15$	0.863432	0.222937
$16$	1.48678	0.371696
$17$	7.81825	1.89620	0.948102	−	0.317966i	$-0.103000\pi$
$17$	7.81825	1.89620	0.948102	+	0.317966i	$0.103000\pi$
$18$	5.49891	1.29611
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.75496	0.839635
$21$	−3.31437	−0.723255
$22$	−7.88113	−1.68026
$23$	−5.53700	−1.15455	−0.577273	−	0.816551i	$-0.695883\pi$
$23$	−5.53700	−1.15455	−0.577273	+	0.816551i	$0.695883\pi$
$24$	−2.97144	−0.606542
$25$	−3.87925	−0.775850
$26$	−6.83397	−1.34025
$27$	−4.35103	−0.837357
$28$	−14.4138	−2.72395
$29$	4.90550	0.910928	0.455464	−	0.890254i	$-0.349473\pi$
$29$	4.90550	0.910928	0.455464	+	0.890254i	$0.349473\pi$
$30$	−2.03354	−0.371273
$31$	−10.2234	−1.83618	−0.918089	−	0.396374i	$-0.870268\pi$
$31$	−10.2234	−1.83618	−0.918089	+	0.396374i	$0.870268\pi$
$32$	3.78490	0.669083
$33$	2.72921	0.475094
$34$	−18.4135	−3.15788
$35$	−4.30212	−0.727190
$36$	−8.28137	−1.38023
$37$	−4.34401	−0.714150	−0.357075	−	0.934076i	$-0.616226\pi$
$37$	−4.34401	−0.714150	−0.357075	+	0.934076i	$0.616226\pi$
$38$	−2.35519	−0.382062
$39$	2.36658	0.378956
$40$	−3.85698	−0.609842
$41$	−2.46182	−0.384472	−0.192236	−	0.981349i	$-0.561574\pi$
$41$	−2.46182	−0.384472	−0.192236	+	0.981349i	$0.561574\pi$
$42$	7.80597	1.20449
$43$	−5.12426	−0.781443	−0.390721	−	0.920509i	$-0.627774\pi$
$43$	−5.12426	−0.781443	−0.390721	+	0.920509i	$0.627774\pi$
$44$	11.8690	1.78932
$45$	−2.47176	−0.368468
$46$	13.0407	1.92275
$47$	−8.47533	−1.23625	−0.618127	−	0.786078i	$-0.712108\pi$
$47$	−8.47533	−1.23625	−0.618127	+	0.786078i	$0.712108\pi$
$48$	1.21261	0.175025
$49$	9.51411	1.35916
$50$	9.13637	1.29208
$51$	6.37651	0.892890
$52$	10.2920	1.42724
$53$	8.14686	1.11906	0.559529	−	0.828811i	$-0.310982\pi$
$53$	8.14686	1.11906	0.559529	+	0.828811i	$0.310982\pi$
$54$	10.2475	1.39451
$55$	3.54256	0.477679
$56$	14.8054	1.97846
$57$	0.815593	0.108028
$58$	−11.5534	−1.51703
$59$	4.16450	0.542172	0.271086	−	0.962555i	$-0.412617\pi$
$59$	4.16450	0.542172	0.271086	+	0.962555i	$0.412617\pi$
$60$	3.06252	0.395370
$61$	11.1105	1.42256	0.711279	−	0.702910i	$-0.248116\pi$
$61$	11.1105	1.42256	0.711279	+	0.702910i	$0.248116\pi$
$62$	24.0781	3.05792
$63$	9.48809	1.19539
$64$	−11.8877	−1.48597
$65$	3.07186	0.381018
$66$	−6.42780	−0.791207
$67$	4.88499	0.596797	0.298398	−	0.954441i	$-0.403548\pi$
$67$	4.88499	0.596797	0.298398	+	0.954441i	$0.403548\pi$
$68$	27.7307	3.36284
$69$	−4.51594	−0.543656
$70$	10.1323	1.21104
$71$	−15.7871	−1.87359	−0.936793	−	0.349885i	$-0.886221\pi$
$71$	−15.7871	−1.87359	−0.936793	+	0.349885i	$0.886221\pi$
$72$	8.50636	1.00248
$73$	5.09224	0.596002	0.298001	−	0.954566i	$-0.403680\pi$
$73$	5.09224	0.596002	0.298001	+	0.954566i	$0.403680\pi$
$74$	10.2310	1.18932
$75$	−3.16389	−0.365334
$76$	3.54692	0.406859
$77$	−13.5985	−1.54969
$78$	−5.57374	−0.631102
$79$	−6.78921	−0.763846	−0.381923	−	0.924194i	$-0.624738\pi$
$79$	−6.78921	−0.763846	−0.381923	+	0.924194i	$0.624738\pi$
$80$	1.57399	0.175978
$81$	3.45575	0.383973
$82$	5.79806	0.640288
$83$	−7.77213	−0.853102	−0.426551	−	0.904463i	$-0.640272\pi$
$83$	−7.77213	−0.853102	−0.426551	+	0.904463i	$0.640272\pi$
$84$	−11.7558	−1.28266
$85$	8.27683	0.897748
$86$	12.0686	1.30139
$87$	4.00089	0.428940
$88$	−12.1915	−1.29961
$89$	1.21312	0.128591	0.0642954	−	0.997931i	$-0.479520\pi$
$89$	1.21312	0.128591	0.0642954	+	0.997931i	$0.479520\pi$
$90$	5.82145	0.613635
$91$	−11.7917	−1.23610
$92$	−19.6393	−2.04754
$93$	−8.33814	−0.864625
$94$	19.9610	2.05882
$95$	1.05865	0.108616
$96$	3.08694	0.315060
$97$	4.15615	0.421993	0.210996	−	0.977487i	$-0.432329\pi$
$97$	4.15615	0.421993	0.210996	+	0.977487i	$0.432329\pi$
$98$	−22.4075	−2.26350
$99$	−7.81293	−0.785229
$100$	−13.7594	−1.37594
$101$	−2.33289	−0.232131	−0.116065	−	0.993242i	$-0.537028\pi$
$101$	−2.33289	−0.232131	−0.116065	+	0.993242i	$0.537028\pi$
$102$	−15.0179	−1.48699
$103$	8.58403	0.845810	0.422905	−	0.906174i	$-0.361010\pi$
$103$	8.58403	0.845810	0.422905	+	0.906174i	$0.361010\pi$
$104$	−10.5716	−1.03663
$105$	−3.50878	−0.342421
$106$	−19.1874	−1.86364
$107$	17.4673	1.68863	0.844316	−	0.535846i	$-0.180007\pi$
$107$	17.4673	1.68863	0.844316	+	0.535846i	$0.180007\pi$
$108$	−15.4327	−1.48502
$109$	−2.51329	−0.240729	−0.120365	−	0.992730i	$-0.538406\pi$
$109$	−2.51329	−0.240729	−0.120365	+	0.992730i	$0.538406\pi$
$110$	−8.34340	−0.795512
$111$	−3.54294	−0.336281
$112$	−6.04193	−0.570909
$113$	10.4302	0.981189	0.490594	−	0.871388i	$-0.336780\pi$
$113$	10.4302	0.981189	0.490594	+	0.871388i	$0.336780\pi$
$114$	−1.92088	−0.179907
$115$	−5.86178	−0.546614
$116$	17.3994	1.61549
$117$	−6.77483	−0.626334
$118$	−9.80819	−0.902917
$119$	−31.7715	−2.91249
$120$	−3.14572	−0.287164
$121$	0.197616	0.0179651
$122$	−26.1674	−2.36909
$123$	−2.00784	−0.181041
$124$	−36.2616	−3.25639
$125$	−9.40006	−0.840767
$126$	−22.3462	−1.99076
$127$	−15.3610	−1.36307	−0.681536	−	0.731785i	$-0.738688\pi$
$127$	−15.3610	−1.36307	−0.681536	+	0.731785i	$0.738688\pi$
$128$	20.4281	1.80560
$129$	−4.17931	−0.367968
$130$	−7.23482	−0.634536
$131$	−7.82477	−0.683653	−0.341827	−	0.939763i	$-0.611046\pi$
$131$	−7.82477	−0.683653	−0.341827	+	0.939763i	$0.611046\pi$
$132$	9.68027	0.842559
$133$	−4.06376	−0.352372
$134$	−11.5051	−0.993888
$135$	−4.60624	−0.396442
$136$	−28.4841	−2.44249
$137$	14.0762	1.20261	0.601306	−	0.799019i	$-0.294647\pi$
$137$	14.0762	1.20261	0.601306	+	0.799019i	$0.294647\pi$
$138$	10.6359	0.905388
$139$	14.1465	1.19989	0.599947	−	0.800040i	$-0.295188\pi$
$139$	14.1465	1.19989	0.599947	+	0.800040i	$0.295188\pi$
$140$	−15.2592	−1.28964
$141$	−6.91242	−0.582131
$142$	37.1816	3.12021
$143$	9.70980	0.811974
$144$	−3.47136	−0.289280
$145$	5.19323	0.431274
$146$	−11.9932	−0.992564
$147$	7.75964	0.640005
$148$	−15.4078	−1.26652
$149$	−20.1160	−1.64797	−0.823984	−	0.566613i	$-0.808254\pi$
$149$	−20.1160	−1.64797	−0.823984	+	0.566613i	$0.808254\pi$
$150$	7.45156	0.608417
$151$	−12.8877	−1.04879	−0.524395	−	0.851475i	$-0.675708\pi$
$151$	−12.8877	−1.04879	−0.524395	+	0.851475i	$0.675708\pi$
$152$	−3.64328	−0.295509
$153$	−18.2541	−1.47576
$154$	32.0270	2.58081
$155$	−10.8231	−0.869329
$156$	8.39406	0.672063
$157$	2.28696	0.182519	0.0912594	−	0.995827i	$-0.470911\pi$
$157$	2.28696	0.182519	0.0912594	+	0.995827i	$0.470911\pi$
$158$	15.9899	1.27209
$159$	6.64452	0.526945
$160$	4.00691	0.316774
$161$	22.5010	1.77333
$162$	−8.13895	−0.639456
$163$	6.88038	0.538913	0.269457	−	0.963013i	$-0.413156\pi$
$163$	6.88038	0.538913	0.269457	+	0.963013i	$0.413156\pi$
$164$	−8.73188	−0.681845
$165$	2.88929	0.224931
$166$	18.3048	1.42073
$167$	−6.36028	−0.492173	−0.246087	−	0.969248i	$-0.579145\pi$
$167$	−6.36028	−0.492173	−0.246087	+	0.969248i	$0.579145\pi$
$168$	12.0752	0.931621
$169$	−4.58033	−0.352333
$170$	−19.4935	−1.49508
$171$	−2.33481	−0.178547
$172$	−18.1753	−1.38586
$173$	−21.2737	−1.61741	−0.808706	−	0.588213i	$-0.799832\pi$
$173$	−21.2737	−1.61741	−0.808706	+	0.588213i	$0.799832\pi$
$174$	−9.42285	−0.714344
$175$	15.7643	1.19167
$176$	4.97520	0.375020
$177$	3.39654	0.255300
$178$	−2.85714	−0.214151
$179$	−9.63670	−0.720281	−0.360140	−	0.932898i	$-0.617271\pi$
$179$	−9.63670	−0.720281	−0.360140	+	0.932898i	$0.617271\pi$
$180$	−8.76711	−0.653462
$181$	23.4314	1.74164	0.870819	−	0.491603i	$-0.163589\pi$
$181$	23.4314	1.74164	0.870819	+	0.491603i	$0.163589\pi$
$182$	27.7716	2.05857
$183$	9.06167	0.669858
$184$	20.1729	1.48716
$185$	−4.59880	−0.338111
$186$	19.6379	1.43992
$187$	26.1621	1.91316
$188$	−30.0613	−2.19244
$189$	17.6815	1.28614
$190$	−2.49333	−0.180885
$191$	3.01536	0.218184	0.109092	−	0.994032i	$-0.465206\pi$
$191$	3.01536	0.218184	0.109092	+	0.994032i	$0.465206\pi$
$192$	−9.69555	−0.699716
$193$	−17.5746	−1.26505	−0.632523	−	0.774541i	$-0.717981\pi$
$193$	−17.5746	−1.26505	−0.632523	+	0.774541i	$0.717981\pi$
$194$	−9.78851	−0.702774
$195$	2.50539	0.179415
$196$	33.7458	2.41041
$197$	8.88922	0.633331	0.316665	−	0.948537i	$-0.397437\pi$
$197$	8.88922	0.633331	0.316665	+	0.948537i	$0.397437\pi$
$198$	18.4009	1.30770
$199$	−25.9490	−1.83948	−0.919738	−	0.392534i	$-0.871599\pi$
$199$	−25.9490	−1.83948	−0.919738	+	0.392534i	$0.871599\pi$
$200$	14.1332	0.999368
$201$	3.98417	0.281021
$202$	5.49439	0.386584
$203$	−19.9347	−1.39914
$204$	22.6170	1.58350
$205$	−2.60622	−0.182026
$206$	−20.2170	−1.40859
$207$	12.9278	0.898547
$208$	4.31415	0.299133
$209$	3.34628	0.231467
$210$	8.26383	0.570258
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	28.8962	1.98460
$213$	−12.8759	−0.882239
$214$	−41.1389	−2.81220
$215$	−5.42482	−0.369970
$216$	15.8520	1.07859
$217$	41.5454	2.82029
$218$	5.91927	0.400903
$219$	4.15320	0.280647
$220$	12.5652	0.847143
$221$	22.6860	1.52602
$222$	8.34430	0.560033
$223$	−26.7138	−1.78889	−0.894444	−	0.447180i	$-0.852428\pi$
$223$	−26.7138	−1.78889	−0.894444	+	0.447180i	$0.852428\pi$
$224$	−15.3809	−1.02768
$225$	9.05730	0.603820
$226$	−24.5650	−1.63404
$227$	2.97136	0.197216	0.0986080	−	0.995126i	$-0.468561\pi$
$227$	2.97136	0.197216	0.0986080	+	0.995126i	$0.468561\pi$
$228$	2.89284	0.191583
$229$	−6.05167	−0.399906	−0.199953	−	0.979806i	$-0.564079\pi$
$229$	−6.05167	−0.399906	−0.199953	+	0.979806i	$0.564079\pi$
$230$	13.8056	0.910314
$231$	−11.0908	−0.729723
$232$	−17.8721	−1.17336
$233$	−23.3870	−1.53213	−0.766066	−	0.642762i	$-0.777788\pi$
$233$	−23.3870	−1.53213	−0.766066	+	0.642762i	$0.777788\pi$
$234$	15.9560	1.04308
$235$	−8.97245	−0.585298
$236$	14.7711	0.961520
$237$	−5.53724	−0.359682
$238$	74.8278	4.85037
$239$	0.891708	0.0576798	0.0288399	−	0.999584i	$-0.490819\pi$
$239$	0.891708	0.0576798	0.0288399	+	0.999584i	$0.490819\pi$
$240$	1.28374	0.0828649
$241$	−0.680321	−0.0438234	−0.0219117	−	0.999760i	$-0.506975\pi$
$241$	−0.680321	−0.0438234	−0.0219117	+	0.999760i	$0.506975\pi$
$242$	−0.465423	−0.0299185
$243$	15.8716	1.01816
$244$	39.4081	2.52285
$245$	10.0722	0.643487
$246$	4.72885	0.301501
$247$	2.90167	0.184629
$248$	37.2467	2.36517
$249$	−6.33890	−0.401711
$250$	22.1389	1.40019
$251$	−22.9201	−1.44671	−0.723353	−	0.690479i	$-0.757400\pi$
$251$	−22.9201	−1.44671	−0.723353	+	0.690479i	$0.757400\pi$
$252$	33.6535	2.11997
$253$	−18.5284	−1.16487
$254$	36.1781	2.27002
$255$	6.75052	0.422734
$256$	−24.3365	−1.52103
$257$	−27.3317	−1.70490	−0.852452	−	0.522806i	$-0.824885\pi$
$257$	−27.3317	−1.70490	−0.852452	+	0.522806i	$0.824885\pi$
$258$	9.84307	0.612803
$259$	17.6530	1.09690
$260$	10.8956	0.675719
$261$	−11.4534	−0.708947
$262$	18.4288	1.13854
$263$	13.2661	0.818021	0.409010	−	0.912530i	$-0.365874\pi$
$263$	13.2661	0.818021	0.409010	+	0.912530i	$0.365874\pi$
$264$	−9.94327	−0.611966
$265$	8.62471	0.529812
$266$	9.57091	0.586830
$267$	0.989415	0.0605512
$268$	17.3267	1.05839
$269$	4.79322	0.292248	0.146124	−	0.989266i	$-0.453320\pi$
$269$	4.79322	0.292248	0.146124	+	0.989266i	$0.453320\pi$
$270$	10.8486	0.660223
$271$	−0.696460	−0.0423069	−0.0211535	−	0.999776i	$-0.506734\pi$
$271$	−0.696460	−0.0423069	−0.0211535	+	0.999776i	$0.506734\pi$
$272$	11.6241	0.704812
$273$	−9.61720	−0.582059
$274$	−33.1521	−2.00279
$275$	−12.9811	−0.782788
$276$	−16.0177	−0.964151
$277$	−3.51672	−0.211299	−0.105650	−	0.994403i	$-0.533692\pi$
$277$	−3.51672	−0.211299	−0.105650	+	0.994403i	$0.533692\pi$
$278$	−33.3178	−1.99827
$279$	23.8697	1.42904
$280$	15.6738	0.936690
$281$	−33.4025	−1.99263	−0.996314	−	0.0857861i	$-0.972660\pi$
$281$	−33.4025	−1.99263	−0.996314	+	0.0857861i	$0.972660\pi$
$282$	16.2801	0.969463
$283$	−7.93631	−0.471765	−0.235882	−	0.971782i	$-0.575798\pi$
$283$	−7.93631	−0.471765	−0.235882	+	0.971782i	$0.575798\pi$
$284$	−55.9956	−3.32273
$285$	0.863432	0.0511453
$286$	−22.8684	−1.35224
$287$	10.0042	0.590532
$288$	−8.83702	−0.520727
$289$	44.1251	2.59559
$290$	−12.2310	−0.718231
$291$	3.38972	0.198709
$292$	18.0618	1.05698
$293$	−1.05604	−0.0616945	−0.0308473	−	0.999524i	$-0.509821\pi$
$293$	−1.05604	−0.0616945	−0.0308473	+	0.999524i	$0.509821\pi$
$294$	−18.2754	−1.06584
$295$	4.40877	0.256689
$296$	15.8264	0.919893
$297$	−14.5598	−0.844845
$298$	47.3770	2.74448
$299$	−16.0665	−0.929152
$300$	−11.2221	−0.647905
$301$	20.8237	1.20026
$302$	30.3531	1.74662
$303$	−1.90269	−0.109306
$304$	1.48678	0.0852730
$305$	11.7622	0.673503
$306$	42.9919	2.45768
$307$	11.9040	0.679395	0.339697	−	0.940535i	$-0.389675\pi$
$307$	11.9040	0.679395	0.339697	+	0.940535i	$0.389675\pi$
$308$	−48.2327	−2.74831
$309$	7.00107	0.398277
$310$	25.4904	1.44775
$311$	−19.6767	−1.11576	−0.557881	−	0.829921i	$-0.688385\pi$
$311$	−19.6767	−1.11576	−0.557881	+	0.829921i	$0.688385\pi$
$312$	−8.62211	−0.488131
$313$	2.40808	0.136113	0.0680564	−	0.997681i	$-0.478320\pi$
$313$	2.40808	0.136113	0.0680564	+	0.997681i	$0.478320\pi$
$314$	−5.38621	−0.303962
$315$	10.0446	0.565950
$316$	−24.0808	−1.35465
$317$	−5.62913	−0.316163	−0.158082	−	0.987426i	$-0.550531\pi$
$317$	−5.62913	−0.316163	−0.158082	+	0.987426i	$0.550531\pi$
$318$	−15.6491	−0.877558
$319$	16.4152	0.919074
$320$	−12.5850	−0.703523
$321$	14.2462	0.795148
$322$	−52.9942	−2.95325
$323$	7.81825	0.435019
$324$	12.2573	0.680959
$325$	−11.2563	−0.624387
$326$	−16.2046	−0.897490
$327$	−2.04982	−0.113355
$328$	8.96911	0.495236
$329$	34.4417	1.89883
$330$	−6.80482	−0.374593
$331$	11.2081	0.616052	0.308026	−	0.951378i	$-0.400332\pi$
$331$	11.2081	0.616052	0.308026	+	0.951378i	$0.400332\pi$
$332$	−27.5671	−1.51294
$333$	10.1424	0.555801
$334$	14.9797	0.819651
$335$	5.17152	0.282550
$336$	−4.92776	−0.268831
$337$	−23.1766	−1.26251	−0.631256	−	0.775575i	$-0.717460\pi$
$337$	−23.1766	−1.26251	−0.631256	+	0.775575i	$0.717460\pi$
$338$	10.7875	0.586765
$339$	8.50678	0.462025
$340$	29.3572	1.59212
$341$	−34.2104	−1.85260
$342$	5.49891	0.297347
$343$	−10.2167	−0.551652
$344$	18.6691	1.00657
$345$	−4.78082	−0.257391
$346$	50.1037	2.69359
$347$	−7.00451	−0.376022	−0.188011	−	0.982167i	$-0.560204\pi$
$347$	−7.00451	−0.376022	−0.188011	+	0.982167i	$0.560204\pi$
$348$	14.1908	0.760708
$349$	−19.8934	−1.06487	−0.532436	−	0.846471i	$-0.678723\pi$
$349$	−19.8934	−1.06487	−0.532436	+	0.846471i	$0.678723\pi$
$350$	−37.1280	−1.98457
$351$	−12.6252	−0.673886
$352$	12.6654	0.675066
$353$	−15.7660	−0.839142	−0.419571	−	0.907723i	$-0.637819\pi$
$353$	−15.7660	−0.839142	−0.419571	+	0.907723i	$0.637819\pi$
$354$	−7.99949	−0.425168
$355$	−16.7131	−0.887039
$356$	4.30285	0.228051
$357$	−25.9126	−1.37144
$358$	22.6963	1.19953
$359$	0.780277	0.0411815	0.0205907	−	0.999788i	$-0.493445\pi$
$359$	0.780277	0.0411815	0.0205907	+	0.999788i	$0.493445\pi$
$360$	9.00530	0.474621
$361$	1.00000	0.0526316
$362$	−55.1853	−2.90047
$363$	0.161174	0.00845945
$364$	−41.8240	−2.19218
$365$	5.39093	0.282174
$366$	−21.3420	−1.11556
$367$	12.4372	0.649218	0.324609	−	0.945848i	$-0.394767\pi$
$367$	12.4372	0.649218	0.324609	+	0.945848i	$0.394767\pi$
$368$	−8.23233	−0.429140
$369$	5.74788	0.299223
$370$	10.8311	0.563080
$371$	−33.1069	−1.71882
$372$	−29.5747	−1.53338
$373$	−36.3071	−1.87991	−0.939955	−	0.341298i	$-0.889134\pi$
$373$	−36.3071	−1.87991	−0.939955	+	0.341298i	$0.889134\pi$
$374$	−61.6167	−3.18612
$375$	−7.66662	−0.395903
$376$	30.8780	1.59241
$377$	14.2341	0.733094
$378$	−41.6434	−2.14190
$379$	−33.8596	−1.73925	−0.869626	−	0.493711i	$-0.835640\pi$
$379$	−33.8596	−1.73925	−0.869626	+	0.493711i	$0.835640\pi$
$380$	3.75496	0.192625
$381$	−12.5284	−0.641847
$382$	−7.10175	−0.363357
$383$	25.9094	1.32391	0.661954	−	0.749544i	$-0.269727\pi$
$383$	25.9094	1.32391	0.661954	+	0.749544i	$0.269727\pi$
$384$	16.6610	0.850227
$385$	−14.3961	−0.733693
$386$	41.3915	2.10677
$387$	11.9642	0.608173
$388$	14.7415	0.748387
$389$	−23.3379	−1.18328	−0.591640	−	0.806202i	$-0.701519\pi$
$389$	−23.3379	−1.18328	−0.591640	+	0.806202i	$0.701519\pi$
$390$	−5.90067	−0.298792
$391$	−43.2897	−2.18925
$392$	−34.6626	−1.75073
$393$	−6.38183	−0.321921
$394$	−20.9358	−1.05473
$395$	−7.18744	−0.361639
$396$	−27.7118	−1.39257
$397$	31.6373	1.58783	0.793915	−	0.608028i	$-0.208039\pi$
$397$	31.6373	1.58783	0.793915	+	0.608028i	$0.208039\pi$
$398$	61.1148	3.06341
$399$	−3.31437	−0.165926
$400$	−5.76761	−0.288380
$401$	−5.11584	−0.255473	−0.127736	−	0.991808i	$-0.540771\pi$
$401$	−5.11584	−0.255473	−0.127736	+	0.991808i	$0.540771\pi$
$402$	−9.38347	−0.468005
$403$	−29.6649	−1.47771
$404$	−8.27455	−0.411674
$405$	3.65845	0.181790
$406$	46.9501	2.33009
$407$	−14.5363	−0.720537
$408$	−23.2314	−1.15013
$409$	−4.36186	−0.215680	−0.107840	−	0.994168i	$-0.534393\pi$
$409$	−4.36186	−0.215680	−0.107840	+	0.994168i	$0.534393\pi$
$410$	6.13814	0.303141
$411$	11.4805	0.566289
$412$	30.4468	1.50001
$413$	−16.9235	−0.832752
$414$	−30.4475	−1.49641
$415$	−8.22800	−0.403897
$416$	10.9825	0.538463
$417$	11.5378	0.565010
$418$	−7.88113	−0.385479
$419$	−11.7294	−0.573019	−0.286509	−	0.958077i	$-0.592495\pi$
$419$	−11.7294	−0.573019	−0.286509	+	0.958077i	$0.592495\pi$
$420$	−12.4453	−0.607270
$421$	14.8381	0.723164	0.361582	−	0.932340i	$-0.382237\pi$
$421$	14.8381	0.723164	0.361582	+	0.932340i	$0.382237\pi$
$422$	−2.35519	−0.114649
$423$	19.7883	0.962138
$424$	−29.6813	−1.44145
$425$	−30.3289	−1.47117
$426$	30.3251	1.46925
$427$	−45.1505	−2.18499
$428$	61.9552	2.99472
$429$	7.91924	0.382345
$430$	12.7765	0.616137
$431$	−26.5767	−1.28015	−0.640077	−	0.768311i	$-0.721098\pi$
$431$	−26.5767	−1.28015	−0.640077	+	0.768311i	$0.721098\pi$
$432$	−6.46905	−0.311242
$433$	−32.3118	−1.55281	−0.776403	−	0.630237i	$-0.782958\pi$
$433$	−32.3118	−1.55281	−0.776403	+	0.630237i	$0.782958\pi$
$434$	−97.8473	−4.69682
$435$	4.23556	0.203080
$436$	−8.91442	−0.426923
$437$	−5.53700	−0.264871
$438$	−9.78157	−0.467381
$439$	−0.252507	−0.0120515	−0.00602574	−	0.999982i	$-0.501918\pi$
$439$	−0.252507	−0.0120515	−0.00602574	+	0.999982i	$0.501918\pi$
$440$	−12.9065	−0.615295
$441$	−22.2136	−1.05779
$442$	−53.4297	−2.54139
$443$	−29.0251	−1.37902	−0.689511	−	0.724275i	$-0.742175\pi$
$443$	−29.0251	−1.37902	−0.689511	+	0.724275i	$0.742175\pi$
$444$	−12.5665	−0.596381
$445$	1.28428	0.0608807
$446$	62.9160	2.97916
$447$	−16.4065	−0.776000
$448$	48.3089	2.28238
$449$	8.10323	0.382415	0.191208	−	0.981550i	$-0.438760\pi$
$449$	8.10323	0.382415	0.191208	+	0.981550i	$0.438760\pi$
$450$	−21.3317	−1.00558
$451$	−8.23795	−0.387910
$452$	36.9950	1.74010
$453$	−10.5111	−0.493857
$454$	−6.99811	−0.328438
$455$	−12.4833	−0.585226
$456$	−2.97144	−0.139150
$457$	17.1469	0.802099	0.401050	−	0.916056i	$-0.368645\pi$
$457$	17.1469	0.802099	0.401050	+	0.916056i	$0.368645\pi$
$458$	14.2528	0.665991
$459$	−34.0175	−1.58780
$460$	−20.7912	−0.969396
$461$	17.9364	0.835381	0.417691	−	0.908589i	$-0.362840\pi$
$461$	17.9364	0.835381	0.417691	+	0.908589i	$0.362840\pi$
$462$	26.1210	1.21526
$463$	−3.68266	−0.171148	−0.0855739	−	0.996332i	$-0.527272\pi$
$463$	−3.68266	−0.171148	−0.0855739	+	0.996332i	$0.527272\pi$
$464$	7.29342	0.338588
$465$	−8.82721	−0.409352
$466$	55.0808	2.55157
$467$	31.3913	1.45262	0.726308	−	0.687369i	$-0.241234\pi$
$467$	31.3913	1.45262	0.726308	+	0.687369i	$0.241234\pi$
$468$	−24.0298	−1.11078
$469$	−19.8514	−0.916653
$470$	21.1318	0.974738
$471$	1.86522	0.0859450
$472$	−15.1725	−0.698369
$473$	−17.1472	−0.788431
$474$	13.0412	0.599004
$475$	−3.87925	−0.177992
$476$	−112.691	−5.16517
$477$	−19.0214	−0.870928
$478$	−2.10014	−0.0960582
$479$	14.9656	0.683796	0.341898	−	0.939737i	$-0.388930\pi$
$479$	14.9656	0.683796	0.341898	+	0.939737i	$0.388930\pi$
$480$	3.26801	0.149163
$481$	−12.6049	−0.574732
$482$	1.60229	0.0729821
$483$	18.3517	0.835031
$484$	0.700928	0.0318603
$485$	4.39993	0.199790
$486$	−37.3806	−1.69562
$487$	19.5342	0.885178	0.442589	−	0.896725i	$-0.354060\pi$
$487$	19.5342	0.885178	0.442589	+	0.896725i	$0.354060\pi$
$488$	−40.4788	−1.83239
$489$	5.61159	0.253765
$490$	−23.7218	−1.07164
$491$	22.7050	1.02466	0.512332	−	0.858787i	$-0.328782\pi$
$491$	22.7050	1.02466	0.512332	+	0.858787i	$0.328782\pi$
$492$	−7.12166	−0.321069
$493$	38.3524	1.72730
$494$	−6.83397	−0.307475
$495$	−8.27120	−0.371763
$496$	−15.2000	−0.682500
$497$	64.1550	2.87774
$498$	14.9293	0.668998
$499$	44.5243	1.99318	0.996590	−	0.0825176i	$-0.0262961\pi$
$499$	44.5243	1.99318	0.996590	+	0.0825176i	$0.0262961\pi$
$500$	−33.3412	−1.49107
$501$	−5.18740	−0.231756
$502$	53.9812	2.40930
$503$	29.7657	1.32719	0.663594	−	0.748093i	$-0.269030\pi$
$503$	29.7657	1.32719	0.663594	+	0.748093i	$0.269030\pi$
$504$	−34.5678	−1.53977
$505$	−2.46972	−0.109901
$506$	43.6379	1.93994
$507$	−3.73569	−0.165908
$508$	−54.4843	−2.41735
$509$	−35.7821	−1.58602	−0.793008	−	0.609212i	$-0.791486\pi$
$509$	−35.7821	−1.58602	−0.793008	+	0.609212i	$0.791486\pi$
$510$	−15.8988	−0.704009
$511$	−20.6936	−0.915433
$512$	16.4609	0.727475
$513$	−4.35103	−0.192103
$514$	64.3713	2.83930
$515$	9.08753	0.400444
$516$	−14.8237	−0.652576
$517$	−28.3609	−1.24731
$518$	−41.5761	−1.82675
$519$	−17.3507	−0.761612
$520$	−11.1917	−0.490787
$521$	15.2653	0.668785	0.334393	−	0.942434i	$-0.391469\pi$
$521$	15.2653	0.668785	0.334393	+	0.942434i	$0.391469\pi$
$522$	26.9749	1.18066
$523$	−0.665276	−0.0290905	−0.0145453	−	0.999894i	$-0.504630\pi$
$523$	−0.665276	−0.0290905	−0.0145453	+	0.999894i	$0.504630\pi$
$524$	−27.7538	−1.21243
$525$	12.8573	0.561137
$526$	−31.2441	−1.36231
$527$	−79.9292	−3.48177
$528$	4.05774	0.176591
$529$	7.65842	0.332975
$530$	−20.3128	−0.882333
$531$	−9.72332	−0.421956
$532$	−14.4138	−0.624918
$533$	−7.14338	−0.309414
$534$	−2.33026	−0.100840
$535$	18.4919	0.799474
$536$	−17.7974	−0.768731
$537$	−7.85963	−0.339168
$538$	−11.2889	−0.486701
$539$	31.8369	1.37131
$540$	−16.3380	−0.703074
$541$	−4.60724	−0.198081	−0.0990403	−	0.995083i	$-0.531577\pi$
$541$	−4.60724	−0.198081	−0.0990403	+	0.995083i	$0.531577\pi$
$542$	1.64029	0.0704567
$543$	19.1104	0.820108
$544$	29.5913	1.26872
$545$	−2.66070	−0.113972
$546$	22.6503	0.969344
$547$	−12.0929	−0.517054	−0.258527	−	0.966004i	$-0.583237\pi$
$547$	−12.0929	−0.517054	−0.258527	+	0.966004i	$0.583237\pi$
$548$	49.9271	2.13278
$549$	−25.9410	−1.10713
$550$	30.5729	1.30363
$551$	4.90550	0.208981
$552$	16.4529	0.700280
$553$	27.5897	1.17323
$554$	8.28254	0.351891
$555$	−3.75075	−0.159211
$556$	50.1766	2.12796
$557$	24.5146	1.03872	0.519359	−	0.854556i	$-0.326171\pi$
$557$	24.5146	1.03872	0.519359	+	0.854556i	$0.326171\pi$
$558$	−56.2176	−2.37988
$559$	−14.8689	−0.628887
$560$	−6.39632	−0.270294
$561$	21.3376	0.900875
$562$	78.6693	3.31846
$563$	−15.3476	−0.646823	−0.323412	−	0.946258i	$-0.604830\pi$
$563$	−15.3476	−0.646823	−0.323412	+	0.946258i	$0.604830\pi$
$564$	−24.5178	−1.03238
$565$	11.0420	0.464539
$566$	18.6915	0.785663
$567$	−14.0433	−0.589765
$568$	57.5169	2.41336
$569$	−9.28783	−0.389366	−0.194683	−	0.980866i	$-0.562368\pi$
$569$	−9.28783	−0.389366	−0.194683	+	0.980866i	$0.562368\pi$
$570$	−2.03354	−0.0851758
$571$	15.9374	0.666960	0.333480	−	0.942757i	$-0.391777\pi$
$571$	15.9374	0.666960	0.333480	+	0.942757i	$0.391777\pi$
$572$	34.4399	1.44000
$573$	2.45931	0.102739
$574$	−23.5619	−0.983454
$575$	21.4794	0.895754
$576$	27.7556	1.15648
$577$	−39.7523	−1.65491	−0.827455	−	0.561532i	$-0.810212\pi$
$577$	−39.7523	−1.65491	−0.827455	+	0.561532i	$0.810212\pi$
$578$	−103.923	−4.32262
$579$	−14.3337	−0.595689
$580$	18.4199	0.764846
$581$	31.5840	1.31033
$582$	−7.98344	−0.330924
$583$	27.2617	1.12906
$584$	−18.5525	−0.767707
$585$	−7.17221	−0.296534
$586$	2.48717	0.102744
$587$	17.4164	0.718852	0.359426	−	0.933174i	$-0.382973\pi$
$587$	17.4164	0.718852	0.359426	+	0.933174i	$0.382973\pi$
$588$	27.5228	1.13502
$589$	−10.2234	−0.421248
$590$	−10.3835	−0.427482
$591$	7.24999	0.298225
$592$	−6.45860	−0.265447
$593$	17.9171	0.735768	0.367884	−	0.929872i	$-0.380082\pi$
$593$	17.9171	0.735768	0.367884	+	0.929872i	$0.380082\pi$
$594$	34.2911	1.40698
$595$	−33.6350	−1.37890
$596$	−71.3498	−2.92260
$597$	−21.1638	−0.866177
$598$	37.8397	1.54738
$599$	4.65533	0.190212	0.0951059	−	0.995467i	$-0.469681\pi$
$599$	4.65533	0.190212	0.0951059	+	0.995467i	$0.469681\pi$
$600$	11.5269	0.470585
$601$	−34.9283	−1.42476	−0.712378	−	0.701796i	$-0.752382\pi$
$601$	−34.9283	−1.42476	−0.712378	+	0.701796i	$0.752382\pi$
$602$	−49.0439	−1.99888
$603$	−11.4055	−0.464469
$604$	−45.7117	−1.85998
$605$	0.209207	0.00850548
$606$	4.48118	0.182036
$607$	45.9974	1.86698	0.933489	−	0.358607i	$-0.116748\pi$
$607$	45.9974	1.86698	0.933489	+	0.358607i	$0.116748\pi$
$608$	3.78490	0.153498
$609$	−16.2586	−0.658833
$610$	−27.7023	−1.12163
$611$	−24.5926	−0.994909
$612$	−64.7458	−2.61720
$613$	0.916594	0.0370209	0.0185104	−	0.999829i	$-0.494108\pi$
$613$	0.916594	0.0370209	0.0185104	+	0.999829i	$0.494108\pi$
$614$	−28.0361	−1.13144
$615$	−2.12561	−0.0857130
$616$	49.5431	1.99615
$617$	−40.2900	−1.62201	−0.811007	−	0.585036i	$-0.801080\pi$
$617$	−40.2900	−1.62201	−0.811007	+	0.585036i	$0.801080\pi$
$618$	−16.4889	−0.663279
$619$	−13.0772	−0.525617	−0.262808	−	0.964848i	$-0.584649\pi$
$619$	−13.0772	−0.525617	−0.262808	+	0.964848i	$0.584649\pi$
$620$	−38.3885	−1.54172
$621$	24.0917	0.966766
$622$	46.3423	1.85816
$623$	−4.92984	−0.197510
$624$	3.51859	0.140856
$625$	9.44483	0.377793
$626$	−5.67148	−0.226678
$627$	2.72921	0.108994
$628$	8.11164	0.323690
$629$	−33.9625	−1.35418
$630$	−23.6570	−0.942516
$631$	−1.30780	−0.0520626	−0.0260313	−	0.999661i	$-0.508287\pi$
$631$	−1.30780	−0.0520626	−0.0260313	+	0.999661i	$0.508287\pi$
$632$	24.7350	0.983906
$633$	0.815593	0.0324169
$634$	13.2577	0.526529
$635$	−16.2620	−0.645339
$636$	23.5676	0.934515
$637$	27.6068	1.09382
$638$	−38.6609	−1.53060
$639$	36.8599	1.45815
$640$	21.6263	0.854853
$641$	−8.07067	−0.318772	−0.159386	−	0.987216i	$-0.550951\pi$
$641$	−8.07067	−0.318772	−0.159386	+	0.987216i	$0.550951\pi$
$642$	−33.5526	−1.32422
$643$	14.3952	0.567691	0.283845	−	0.958870i	$-0.408390\pi$
$643$	14.3952	0.567691	0.283845	+	0.958870i	$0.408390\pi$
$644$	79.8093	3.14493
$645$	−4.42445	−0.174213
$646$	−18.4135	−0.724468
$647$	13.6290	0.535810	0.267905	−	0.963445i	$-0.413669\pi$
$647$	13.6290	0.535810	0.267905	+	0.963445i	$0.413669\pi$
$648$	−12.5903	−0.494593
$649$	13.9356	0.547021
$650$	26.5107	1.03983
$651$	33.8842	1.32802
$652$	24.4042	0.955740
$653$	29.9937	1.17374	0.586872	−	0.809680i	$-0.300359\pi$
$653$	29.9937	1.17374	0.586872	+	0.809680i	$0.300359\pi$
$654$	4.82771	0.188779
$655$	−8.28373	−0.323672
$656$	−3.66020	−0.142907
$657$	−11.8894	−0.463850
$658$	−81.1166	−3.16226
$659$	−33.1760	−1.29235	−0.646176	−	0.763188i	$-0.723633\pi$
$659$	−33.1760	−1.29235	−0.646176	+	0.763188i	$0.723633\pi$
$660$	10.2481	0.398905
$661$	−27.9290	−1.08631	−0.543155	−	0.839632i	$-0.682771\pi$
$661$	−27.9290	−1.08631	−0.543155	+	0.839632i	$0.682771\pi$
$662$	−26.3972	−1.02595
$663$	18.5025	0.718578
$664$	28.3161	1.09888
$665$	−4.30212	−0.166829
$666$	−23.8873	−0.925615
$667$	−27.1617	−1.05171
$668$	−22.5594	−0.872849
$669$	−21.7876	−0.842357
$670$	−12.1799	−0.470551
$671$	37.1790	1.43528
$672$	−12.5446	−0.483917
$673$	44.6394	1.72072	0.860361	−	0.509685i	$-0.170238\pi$
$673$	44.6394	1.72072	0.860361	+	0.509685i	$0.170238\pi$
$674$	54.5854	2.10255
$675$	16.8787	0.649663
$676$	−16.2461	−0.624848
$677$	−46.8823	−1.80183	−0.900917	−	0.433991i	$-0.857105\pi$
$677$	−46.8823	−1.80183	−0.900917	+	0.433991i	$0.857105\pi$
$678$	−20.0351	−0.769443
$679$	−16.8896	−0.648162
$680$	−30.1548	−1.15638
$681$	2.42342	0.0928656
$682$	80.5720	3.08526
$683$	−11.1401	−0.426264	−0.213132	−	0.977023i	$-0.568366\pi$
$683$	−11.1401	−0.426264	−0.213132	+	0.977023i	$0.568366\pi$
$684$	−8.28137	−0.316646
$685$	14.9018	0.569370
$686$	24.0623	0.918705
$687$	−4.93570	−0.188309
$688$	−7.61867	−0.290459
$689$	23.6395	0.900592
$690$	11.2597	0.428651
$691$	−33.4231	−1.27148	−0.635738	−	0.771905i	$-0.719304\pi$
$691$	−33.4231	−1.27148	−0.635738	+	0.771905i	$0.719304\pi$
$692$	−75.4562	−2.86841
$693$	31.7498	1.20608
$694$	16.4970	0.626216
$695$	14.9763	0.568084
$696$	−14.5764	−0.552516
$697$	−19.2471	−0.729037
$698$	46.8528	1.77340
$699$	−19.0743	−0.721455
$700$	55.9147	2.11338
$701$	1.17001	0.0441907	0.0220953	−	0.999756i	$-0.492966\pi$
$701$	1.17001	0.0441907	0.0220953	+	0.999756i	$0.492966\pi$
$702$	29.7348	1.12227
$703$	−4.34401	−0.163837
$704$	−39.7797	−1.49926
$705$	−7.31787	−0.275607
$706$	37.1320	1.39748
$707$	9.48028	0.356543
$708$	12.0472	0.452763
$709$	−29.1309	−1.09403	−0.547016	−	0.837122i	$-0.684236\pi$
$709$	−29.1309	−1.09403	−0.547016	+	0.837122i	$0.684236\pi$
$710$	39.3625	1.47725
$711$	15.8515	0.594478
$712$	−4.41975	−0.165637
$713$	56.6070	2.11995
$714$	61.0290	2.28395
$715$	10.2793	0.384425
$716$	−34.1806	−1.27739
$717$	0.727271	0.0271604
$718$	−1.83770	−0.0685824
$719$	2.98339	0.111262	0.0556308	−	0.998451i	$-0.482283\pi$
$719$	2.98339	0.111262	0.0556308	+	0.998451i	$0.482283\pi$
$720$	−3.67497	−0.136958
$721$	−34.8834	−1.29913
$722$	−2.35519	−0.0876511
$723$	−0.554865	−0.0206357
$724$	83.1091	3.08872
$725$	−19.0296	−0.706743
$726$	−0.379596	−0.0140881
$727$	1.04362	0.0387056	0.0193528	−	0.999813i	$-0.493839\pi$
$727$	1.04362	0.0387056	0.0193528	+	0.999813i	$0.493839\pi$
$728$	42.9604	1.59222
$729$	2.57750	0.0954628
$730$	−12.6967	−0.469924
$731$	−40.0628	−1.48177
$732$	32.1410	1.18797
$733$	−3.94867	−0.145847	−0.0729237	−	0.997338i	$-0.523233\pi$
$733$	−3.94867	−0.145847	−0.0729237	+	0.997338i	$0.523233\pi$
$734$	−29.2920	−1.08119
$735$	8.21478	0.303007
$736$	−20.9570	−0.772486
$737$	16.3466	0.602134
$738$	−13.5373	−0.498317
$739$	8.68242	0.319388	0.159694	−	0.987167i	$-0.448949\pi$
$739$	8.68242	0.319388	0.159694	+	0.987167i	$0.448949\pi$
$740$	−16.3116	−0.599626
$741$	2.36658	0.0869384
$742$	77.9729	2.86248
$743$	2.95345	0.108351	0.0541757	−	0.998531i	$-0.482747\pi$
$743$	2.95345	0.108351	0.0541757	+	0.998531i	$0.482747\pi$
$744$	30.3782	1.11372
$745$	−21.2959	−0.780222
$746$	85.5101	3.13075
$747$	18.1464	0.663943
$748$	92.7947	3.39291
$749$	−70.9830	−2.59366
$750$	18.0564	0.659325
$751$	−38.1300	−1.39138	−0.695691	−	0.718341i	$-0.744902\pi$
$751$	−38.1300	−1.39138	−0.695691	+	0.718341i	$0.744902\pi$
$752$	−12.6010	−0.459511
$753$	−18.6935	−0.681229
$754$	−33.5240	−1.22087
$755$	−13.6437	−0.496544
$756$	62.7149	2.28092
$757$	5.07316	0.184387	0.0921937	−	0.995741i	$-0.470612\pi$
$757$	5.07316	0.184387	0.0921937	+	0.995741i	$0.470612\pi$
$758$	79.7458	2.89650
$759$	−15.1116	−0.548517
$760$	−3.85698	−0.139907
$761$	32.2741	1.16993	0.584967	−	0.811057i	$-0.301107\pi$
$761$	32.2741	1.16993	0.584967	+	0.811057i	$0.301107\pi$
$762$	29.5066	1.06891
$763$	10.2134	0.369749
$764$	10.6952	0.386940
$765$	−19.3248	−0.698690
$766$	−61.0216	−2.20480
$767$	12.0840	0.436328
$768$	−19.8487	−0.716227
$769$	50.4795	1.82034	0.910169	−	0.414237i	$-0.135952\pi$
$769$	50.4795	1.82034	0.910169	+	0.414237i	$0.135952\pi$
$770$	33.9055	1.22187
$771$	−22.2915	−0.802810
$772$	−62.3356	−2.24351
$773$	9.63174	0.346430	0.173215	−	0.984884i	$-0.444584\pi$
$773$	9.63174	0.346430	0.173215	+	0.984884i	$0.444584\pi$
$774$	−28.1779	−1.01283
$775$	39.6591	1.42460
$776$	−15.1420	−0.543567
$777$	14.3977	0.516513
$778$	54.9653	1.97060
$779$	−2.46182	−0.0882039
$780$	8.88641	0.318185
$781$	−52.8282	−1.89034
$782$	101.955	3.64592
$783$	−21.3440	−0.762771
$784$	14.1454	0.505194
$785$	2.42110	0.0864126
$786$	15.0304	0.536117
$787$	−30.8820	−1.10082	−0.550412	−	0.834893i	$-0.685529\pi$
$787$	−30.8820	−1.10082	−0.550412	+	0.834893i	$0.685529\pi$
$788$	31.5293	1.12319
$789$	10.8197	0.385192
$790$	16.9278	0.602263
$791$	−42.3857	−1.50706
$792$	28.4647	1.01145
$793$	32.2391	1.14484
$794$	−74.5118	−2.64433
$795$	7.03426	0.249479
$796$	−92.0389	−3.26223
$797$	4.67012	0.165424	0.0827121	−	0.996573i	$-0.473642\pi$
$797$	4.67012	0.165424	0.0827121	+	0.996573i	$0.473642\pi$
$798$	7.80597	0.276328
$799$	−66.2622	−2.34419
$800$	−14.6826	−0.519108
$801$	−2.83241	−0.100078
$802$	12.0488	0.425457
$803$	17.0401	0.601332
$804$	14.1315	0.498380
$805$	23.8208	0.839574
$806$	69.8665	2.46094
$807$	3.90932	0.137614
$808$	8.49936	0.299006
$809$	40.6331	1.42858	0.714292	−	0.699847i	$-0.246749\pi$
$809$	40.6331	1.42858	0.714292	+	0.699847i	$0.246749\pi$
$810$	−8.61634	−0.302747
$811$	−3.17941	−0.111644	−0.0558221	−	0.998441i	$-0.517778\pi$
$811$	−3.17941	−0.111644	−0.0558221	+	0.998441i	$0.517778\pi$
$812$	−70.7068	−2.48132
$813$	−0.568028	−0.0199216
$814$	34.2357	1.19996
$815$	7.28395	0.255146
$816$	9.48050	0.331884
$817$	−5.12426	−0.179275
$818$	10.2730	0.359187
$819$	27.5313	0.962020
$820$	−9.24404	−0.322816
$821$	−51.9477	−1.81299	−0.906493	−	0.422220i	$-0.861251\pi$
$821$	−51.9477	−1.81299	−0.906493	+	0.422220i	$0.861251\pi$
$822$	−27.0386	−0.943081
$823$	26.6951	0.930534	0.465267	−	0.885170i	$-0.345958\pi$
$823$	26.6951	0.930534	0.465267	+	0.885170i	$0.345958\pi$
$824$	−31.2740	−1.08948
$825$	−10.5873	−0.368601
$826$	39.8581	1.38684
$827$	22.5541	0.784284	0.392142	−	0.919905i	$-0.371734\pi$
$827$	22.5541	0.784284	0.392142	+	0.919905i	$0.371734\pi$
$828$	45.8540	1.59354
$829$	−34.3884	−1.19436	−0.597180	−	0.802107i	$-0.703712\pi$
$829$	−34.3884	−1.19436	−0.597180	+	0.802107i	$0.703712\pi$
$830$	19.3785	0.672638
$831$	−2.86821	−0.0994972
$832$	−34.4942	−1.19587
$833$	74.3837	2.57724
$834$	−27.1738	−0.940950
$835$	−6.73334	−0.233017
$836$	11.8690	0.410498
$837$	44.4824	1.53754
$838$	27.6250	0.954289
$839$	−23.8833	−0.824543	−0.412272	−	0.911061i	$-0.635265\pi$
$839$	−23.8833	−0.824543	−0.412272	+	0.911061i	$0.635265\pi$
$840$	12.7835	0.441071
$841$	−4.93612	−0.170211
$842$	−34.9465	−1.20434
$843$	−27.2429	−0.938294
$844$	3.54692	0.122090
$845$	−4.84899	−0.166810
$846$	−46.6051	−1.60232
$847$	−0.803063	−0.0275936
$848$	12.1126	0.415949
$849$	−6.47280	−0.222146
$850$	71.4304	2.45004
$851$	24.0528	0.824519
$852$	−45.6696	−1.56461
$853$	13.5779	0.464900	0.232450	−	0.972608i	$-0.425326\pi$
$853$	13.5779	0.464900	0.232450	+	0.972608i	$0.425326\pi$
$854$	106.338	3.63881
$855$	−2.47176	−0.0845323
$856$	−63.6384	−2.17512
$857$	42.6481	1.45683	0.728415	−	0.685136i	$-0.240257\pi$
$857$	42.6481	1.45683	0.728415	+	0.685136i	$0.240257\pi$
$858$	−18.6513	−0.636746
$859$	−28.4263	−0.969894	−0.484947	−	0.874544i	$-0.661161\pi$
$859$	−28.4263	−0.969894	−0.484947	+	0.874544i	$0.661161\pi$
$860$	−19.2414	−0.656126
$861$	8.15939	0.278071
$862$	62.5932	2.13193
$863$	−30.8627	−1.05058	−0.525288	−	0.850924i	$-0.676043\pi$
$863$	−30.8627	−1.05058	−0.525288	+	0.850924i	$0.676043\pi$
$864$	−16.4682	−0.560261
$865$	−22.5216	−0.765756
$866$	76.1004	2.58600
$867$	35.9881	1.22222
$868$	147.358	5.00166
$869$	−22.7186	−0.770677
$870$	−9.97554	−0.338203
$871$	14.1746	0.480289
$872$	9.15661	0.310082
$873$	−9.70381	−0.328424
$874$	13.0407	0.441108
$875$	38.1996	1.29138
$876$	14.7311	0.497716
$877$	−6.78441	−0.229093	−0.114547	−	0.993418i	$-0.536542\pi$
$877$	−6.78441	−0.229093	−0.114547	+	0.993418i	$0.536542\pi$
$878$	0.594701	0.0200702
$879$	−0.861299	−0.0290509
$880$	5.26702	0.177551
$881$	18.7865	0.632932	0.316466	−	0.948604i	$-0.397504\pi$
$881$	18.7865	0.632932	0.316466	+	0.948604i	$0.397504\pi$
$882$	52.3173	1.76161
$883$	10.9100	0.367152	0.183576	−	0.983006i	$-0.441233\pi$
$883$	10.9100	0.367152	0.183576	+	0.983006i	$0.441233\pi$
$884$	80.4652	2.70634
$885$	3.59576	0.120870
$886$	68.3595	2.29658
$887$	17.9859	0.603906	0.301953	−	0.953323i	$-0.402361\pi$
$887$	17.9859	0.603906	0.301953	+	0.953323i	$0.402361\pi$
$888$	12.9079	0.433162
$889$	62.4235	2.09362
$890$	−3.02472	−0.101389
$891$	11.5639	0.387406
$892$	−94.7516	−3.17252
$893$	−8.47533	−0.283616
$894$	38.6404	1.29233
$895$	−10.2019	−0.341013
$896$	−83.0146	−2.77332
$897$	−13.1038	−0.437522
$898$	−19.0846	−0.636863
$899$	−50.1509	−1.67263
$900$	32.1255	1.07085
$901$	63.6942	2.12196
$902$	19.4019	0.646014
$903$	16.9837	0.565182
$904$	−38.0001	−1.26386
$905$	24.8057	0.824570
$906$	24.7557	0.822455
$907$	38.6298	1.28268	0.641341	−	0.767256i	$-0.278378\pi$
$907$	38.6298	1.28268	0.641341	+	0.767256i	$0.278378\pi$
$908$	10.5392	0.349754
$909$	5.44684	0.180660
$910$	29.4005	0.974618
$911$	−18.4358	−0.610805	−0.305402	−	0.952223i	$-0.598791\pi$
$911$	−18.4358	−0.610805	−0.305402	+	0.952223i	$0.598791\pi$
$912$	1.21261	0.0401536
$913$	−26.0078	−0.860731
$914$	−40.3842	−1.33579
$915$	9.59319	0.317141
$916$	−21.4648	−0.709216
$917$	31.7980	1.05006
$918$	80.1176	2.64427
$919$	14.0343	0.462948	0.231474	−	0.972841i	$-0.425645\pi$
$919$	14.0343	0.462948	0.231474	+	0.972841i	$0.425645\pi$
$920$	21.3561	0.704090
$921$	9.70878	0.319915
$922$	−42.2436	−1.39122
$923$	−45.8089	−1.50782
$924$	−39.3382	−1.29413
$925$	16.8515	0.554074
$926$	8.67336	0.285024
$927$	−20.0421	−0.658268
$928$	18.5668	0.609486
$929$	−5.79232	−0.190040	−0.0950199	−	0.995475i	$-0.530291\pi$
$929$	−5.79232	−0.190040	−0.0950199	+	0.995475i	$0.530291\pi$
$930$	20.7898	0.681723
$931$	9.51411	0.311812
$932$	−82.9517	−2.71717
$933$	−16.0482	−0.525393
$934$	−73.9325	−2.41914
$935$	27.6966	0.905776
$936$	24.6826	0.806777
$937$	−39.6431	−1.29508	−0.647542	−	0.762030i	$-0.724203\pi$
$937$	−39.6431	−1.29508	−0.647542	+	0.762030i	$0.724203\pi$
$938$	46.7539	1.52657
$939$	1.96401	0.0640932
$940$	−31.8245	−1.03800
$941$	29.3255	0.955985	0.477992	−	0.878364i	$-0.341365\pi$
$941$	29.3255	0.955985	0.477992	+	0.878364i	$0.341365\pi$
$942$	−4.39296	−0.143130
$943$	13.6311	0.443890
$944$	6.19172	0.201523
$945$	18.7186	0.608917
$946$	40.3850	1.31303
$947$	41.5570	1.35042	0.675211	−	0.737625i	$-0.264053\pi$
$947$	41.5570	1.35042	0.675211	+	0.737625i	$0.264053\pi$
$948$	−19.6401	−0.637881
$949$	14.7760	0.479649
$950$	9.13637	0.296423
$951$	−4.59108	−0.148876
$952$	115.752	3.75156
$953$	−57.7073	−1.86932	−0.934661	−	0.355540i	$-0.884297\pi$
$953$	−57.7073	−1.86932	−0.934661	+	0.355540i	$0.884297\pi$
$954$	44.7989	1.45042
$955$	3.19223	0.103298
$956$	3.16281	0.102293
$957$	13.3881	0.432776
$958$	−35.2468	−1.13877
$959$	−57.2023	−1.84716
$960$	−10.2642	−0.331277
$961$	73.5180	2.37155
$962$	29.6868	0.957142
$963$	−40.7829	−1.31421
$964$	−2.41304	−0.0777189
$965$	−18.6054	−0.598930
$966$	−43.2217	−1.39064
$967$	−16.3938	−0.527188	−0.263594	−	0.964634i	$-0.584908\pi$
$967$	−16.3938	−0.527188	−0.263594	+	0.964634i	$0.584908\pi$
$968$	−0.719971	−0.0231407
$969$	6.37651	0.204843
$970$	−10.3627	−0.332725
$971$	5.62352	0.180467	0.0902337	−	0.995921i	$-0.471239\pi$
$971$	5.62352	0.180467	0.0902337	+	0.995921i	$0.471239\pi$
$972$	56.2952	1.80567
$973$	−57.4881	−1.84298
$974$	−46.0067	−1.47415
$975$	−9.18055	−0.294013
$976$	16.5190	0.528760
$977$	48.0619	1.53764	0.768818	−	0.639467i	$-0.220845\pi$
$977$	48.0619	1.53764	0.768818	+	0.639467i	$0.220845\pi$
$978$	−13.2164	−0.422613
$979$	4.05946	0.129741
$980$	35.7251	1.14120
$981$	5.86804	0.187352
$982$	−53.4747	−1.70645
$983$	−21.2619	−0.678151	−0.339075	−	0.940759i	$-0.610114\pi$
$983$	−21.2619	−0.678151	−0.339075	+	0.940759i	$0.610114\pi$
$984$	7.31514	0.233198
$985$	9.41062	0.299847
$986$	−90.3271	−2.87660
$987$	28.0904	0.894127
$988$	10.2920	0.327431
$989$	28.3731	0.902211
$990$	19.4802	0.619123
$991$	16.5905	0.527015	0.263507	−	0.964657i	$-0.415121\pi$
$991$	16.5905	0.527015	0.263507	+	0.964657i	$0.415121\pi$
$992$	−38.6946	−1.22856
$993$	9.14123	0.290088
$994$	−151.097	−4.79251
$995$	−27.4710	−0.870890
$996$	−22.4835	−0.712418
$997$	16.5687	0.524737	0.262369	−	0.964968i	$-0.415496\pi$
$997$	16.5687	0.524737	0.262369	+	0.964968i	$0.415496\pi$
$998$	−104.863	−3.31938
$999$	18.9009	0.597999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.9	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.9	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35519 q^{2} +0.815593 q^{3} +3.54692 q^{4} +1.05865 q^{5} -1.92088 q^{6} -4.06376 q^{7} -3.64328 q^{8} -2.33481 q^{9} +O(q^{10})\)	\(q-2.35519 q^{2} +0.815593 q^{3} +3.54692 q^{4} +1.05865 q^{5} -1.92088 q^{6} -4.06376 q^{7} -3.64328 q^{8} -2.33481 q^{9} -2.49333 q^{10} +3.34628 q^{11} +2.89284 q^{12} +2.90167 q^{13} +9.57091 q^{14} +0.863432 q^{15} +1.48678 q^{16} +7.81825 q^{17} +5.49891 q^{18} +1.00000 q^{19} +3.75496 q^{20} -3.31437 q^{21} -7.88113 q^{22} -5.53700 q^{23} -2.97144 q^{24} -3.87925 q^{25} -6.83397 q^{26} -4.35103 q^{27} -14.4138 q^{28} +4.90550 q^{29} -2.03354 q^{30} -10.2234 q^{31} +3.78490 q^{32} +2.72921 q^{33} -18.4135 q^{34} -4.30212 q^{35} -8.28137 q^{36} -4.34401 q^{37} -2.35519 q^{38} +2.36658 q^{39} -3.85698 q^{40} -2.46182 q^{41} +7.80597 q^{42} -5.12426 q^{43} +11.8690 q^{44} -2.47176 q^{45} +13.0407 q^{46} -8.47533 q^{47} +1.21261 q^{48} +9.51411 q^{49} +9.13637 q^{50} +6.37651 q^{51} +10.2920 q^{52} +8.14686 q^{53} +10.2475 q^{54} +3.54256 q^{55} +14.8054 q^{56} +0.815593 q^{57} -11.5534 q^{58} +4.16450 q^{59} +3.06252 q^{60} +11.1105 q^{61} +24.0781 q^{62} +9.48809 q^{63} -11.8877 q^{64} +3.07186 q^{65} -6.42780 q^{66} +4.88499 q^{67} +27.7307 q^{68} -4.51594 q^{69} +10.1323 q^{70} -15.7871 q^{71} +8.50636 q^{72} +5.09224 q^{73} +10.2310 q^{74} -3.16389 q^{75} +3.54692 q^{76} -13.5985 q^{77} -5.57374 q^{78} -6.78921 q^{79} +1.57399 q^{80} +3.45575 q^{81} +5.79806 q^{82} -7.77213 q^{83} -11.7558 q^{84} +8.27683 q^{85} +12.0686 q^{86} +4.00089 q^{87} -12.1915 q^{88} +1.21312 q^{89} +5.82145 q^{90} -11.7917 q^{91} -19.6393 q^{92} -8.33814 q^{93} +19.9610 q^{94} +1.05865 q^{95} +3.08694 q^{96} +4.15615 q^{97} -22.4075 q^{98} -7.81293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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