LMFDB - Embedded newform 4009.2.a.c.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.48400 q^{2} +0.221625 q^{3} +4.17025 q^{4} -2.50863 q^{5} -0.550515 q^{6} +2.33062 q^{7} -5.39089 q^{8} -2.95088 q^{9} +O(q^{10})$	\(q-2.48400 q^{2} +0.221625 q^{3} +4.17025 q^{4} -2.50863 q^{5} -0.550515 q^{6} +2.33062 q^{7} -5.39089 q^{8} -2.95088 q^{9} +6.23143 q^{10} -4.94355 q^{11} +0.924230 q^{12} -3.44352 q^{13} -5.78925 q^{14} -0.555974 q^{15} +5.05047 q^{16} +4.24355 q^{17} +7.32999 q^{18} +1.00000 q^{19} -10.4616 q^{20} +0.516522 q^{21} +12.2798 q^{22} -3.69856 q^{23} -1.19475 q^{24} +1.29322 q^{25} +8.55371 q^{26} -1.31886 q^{27} +9.71925 q^{28} +3.97886 q^{29} +1.38104 q^{30} +8.72405 q^{31} -1.76358 q^{32} -1.09561 q^{33} -10.5410 q^{34} -5.84666 q^{35} -12.3059 q^{36} +9.05184 q^{37} -2.48400 q^{38} -0.763170 q^{39} +13.5238 q^{40} +10.8766 q^{41} -1.28304 q^{42} +1.63320 q^{43} -20.6158 q^{44} +7.40267 q^{45} +9.18722 q^{46} -6.41415 q^{47} +1.11931 q^{48} -1.56822 q^{49} -3.21237 q^{50} +0.940474 q^{51} -14.3603 q^{52} +10.2742 q^{53} +3.27605 q^{54} +12.4015 q^{55} -12.5641 q^{56} +0.221625 q^{57} -9.88347 q^{58} -2.69761 q^{59} -2.31855 q^{60} +14.5808 q^{61} -21.6705 q^{62} -6.87738 q^{63} -5.72022 q^{64} +8.63853 q^{65} +2.72150 q^{66} -0.0512046 q^{67} +17.6966 q^{68} -0.819693 q^{69} +14.5231 q^{70} -12.5130 q^{71} +15.9079 q^{72} +0.953806 q^{73} -22.4848 q^{74} +0.286610 q^{75} +4.17025 q^{76} -11.5215 q^{77} +1.89571 q^{78} +0.539584 q^{79} -12.6698 q^{80} +8.56036 q^{81} -27.0175 q^{82} +4.69541 q^{83} +2.15403 q^{84} -10.6455 q^{85} -4.05687 q^{86} +0.881813 q^{87} +26.6502 q^{88} -11.0504 q^{89} -18.3882 q^{90} -8.02554 q^{91} -15.4239 q^{92} +1.93347 q^{93} +15.9327 q^{94} -2.50863 q^{95} -0.390853 q^{96} -17.2966 q^{97} +3.89546 q^{98} +14.5878 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.48400	−1.75645	−0.878226	−	0.478246i	$-0.841273\pi$
$2$	−2.48400	−1.75645	−0.878226	+	0.478246i	$0.841273\pi$
$3$	0.221625	0.127955	0.0639775	−	0.997951i	$-0.479621\pi$
$3$	0.221625	0.127955	0.0639775	+	0.997951i	$0.479621\pi$
$4$	4.17025	2.08512
$5$	−2.50863	−1.12189	−0.560947	−	0.827852i	$-0.689563\pi$
$5$	−2.50863	−1.12189	−0.560947	+	0.827852i	$0.689563\pi$
$6$	−0.550515	−0.224747
$7$	2.33062	0.880891	0.440445	−	0.897779i	$-0.354821\pi$
$7$	2.33062	0.880891	0.440445	+	0.897779i	$0.354821\pi$
$8$	−5.39089	−1.90597
$9$	−2.95088	−0.983628
$10$	6.23143	1.97055
$11$	−4.94355	−1.49054	−0.745268	−	0.666765i	$-0.767679\pi$
$11$	−4.94355	−1.49054	−0.745268	+	0.666765i	$0.767679\pi$
$12$	0.924230	0.266802
$13$	−3.44352	−0.955062	−0.477531	−	0.878615i	$-0.658468\pi$
$13$	−3.44352	−0.955062	−0.477531	+	0.878615i	$0.658468\pi$
$14$	−5.78925	−1.54724
$15$	−0.555974	−0.143552
$16$	5.05047	1.26262
$17$	4.24355	1.02921	0.514605	−	0.857427i	$-0.327938\pi$
$17$	4.24355	1.02921	0.514605	+	0.857427i	$0.327938\pi$
$18$	7.32999	1.72769
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−10.4616	−2.33929
$21$	0.516522	0.112714
$22$	12.2798	2.61806
$23$	−3.69856	−0.771204	−0.385602	−	0.922665i	$-0.626006\pi$
$23$	−3.69856	−0.771204	−0.385602	+	0.922665i	$0.626006\pi$
$24$	−1.19475	−0.243878
$25$	1.29322	0.258645
$26$	8.55371	1.67752
$27$	−1.31886	−0.253815
$28$	9.71925	1.83677
$29$	3.97886	0.738855	0.369427	−	0.929260i	$-0.379554\pi$
$29$	3.97886	0.738855	0.369427	+	0.929260i	$0.379554\pi$
$30$	1.38104	0.252142
$31$	8.72405	1.56689	0.783443	−	0.621464i	$-0.213462\pi$
$31$	8.72405	1.56689	0.783443	+	0.621464i	$0.213462\pi$
$32$	−1.76358	−0.311760
$33$	−1.09561	−0.190722
$34$	−10.5410	−1.80776
$35$	−5.84666	−0.988265
$36$	−12.3059	−2.05099
$37$	9.05184	1.48811	0.744057	−	0.668117i	$-0.232899\pi$
$37$	9.05184	1.48811	0.744057	+	0.668117i	$0.232899\pi$
$38$	−2.48400	−0.402958
$39$	−0.763170	−0.122205
$40$	13.5238	2.13829
$41$	10.8766	1.69864	0.849322	−	0.527875i	$-0.177011\pi$
$41$	10.8766	1.69864	0.849322	+	0.527875i	$0.177011\pi$
$42$	−1.28304	−0.197977
$43$	1.63320	0.249061	0.124531	−	0.992216i	$-0.460258\pi$
$43$	1.63320	0.249061	0.124531	+	0.992216i	$0.460258\pi$
$44$	−20.6158	−3.10795
$45$	7.40267	1.10353
$46$	9.18722	1.35458
$47$	−6.41415	−0.935600	−0.467800	−	0.883834i	$-0.654953\pi$
$47$	−6.41415	−0.935600	−0.467800	+	0.883834i	$0.654953\pi$
$48$	1.11931	0.161558
$49$	−1.56822	−0.224032
$50$	−3.21237	−0.454297
$51$	0.940474	0.131693
$52$	−14.3603	−1.99142
$53$	10.2742	1.41127	0.705633	−	0.708577i	$-0.250663\pi$
$53$	10.2742	1.41127	0.705633	+	0.708577i	$0.250663\pi$
$54$	3.27605	0.445814
$55$	12.4015	1.67222
$56$	−12.5641	−1.67895
$57$	0.221625	0.0293549
$58$	−9.88347	−1.29776
$59$	−2.69761	−0.351199	−0.175599	−	0.984462i	$-0.556186\pi$
$59$	−2.69761	−0.351199	−0.175599	+	0.984462i	$0.556186\pi$
$60$	−2.31855	−0.299324
$61$	14.5808	1.86689	0.933443	−	0.358726i	$-0.116789\pi$
$61$	14.5808	1.86689	0.933443	+	0.358726i	$0.116789\pi$
$62$	−21.6705	−2.75216
$63$	−6.87738	−0.866468
$64$	−5.72022	−0.715027
$65$	8.63853	1.07148
$66$	2.72150	0.334994
$67$	−0.0512046	−0.00625564	−0.00312782	−	0.999995i	$-0.500996\pi$
$67$	−0.0512046	−0.00625564	−0.00312782	+	0.999995i	$0.500996\pi$
$68$	17.6966	2.14603
$69$	−0.819693	−0.0986794
$70$	14.5231	1.73584
$71$	−12.5130	−1.48503	−0.742513	−	0.669832i	$-0.766366\pi$
$71$	−12.5130	−1.48503	−0.742513	+	0.669832i	$0.766366\pi$
$72$	15.9079	1.87476
$73$	0.953806	0.111635	0.0558173	−	0.998441i	$-0.482224\pi$
$73$	0.953806	0.111635	0.0558173	+	0.998441i	$0.482224\pi$
$74$	−22.4848	−2.61380
$75$	0.286610	0.0330949
$76$	4.17025	0.478360
$77$	−11.5215	−1.31300
$78$	1.89571	0.214647
$79$	0.539584	0.0607079	0.0303540	−	0.999539i	$-0.490337\pi$
$79$	0.539584	0.0607079	0.0303540	+	0.999539i	$0.490337\pi$
$80$	−12.6698	−1.41652
$81$	8.56036	0.951151
$82$	−27.0175	−2.98359
$83$	4.69541	0.515388	0.257694	−	0.966227i	$-0.417037\pi$
$83$	4.69541	0.515388	0.257694	+	0.966227i	$0.417037\pi$
$84$	2.15403	0.235024
$85$	−10.6455	−1.15466
$86$	−4.05687	−0.437464
$87$	0.881813	0.0945402
$88$	26.6502	2.84092
$89$	−11.0504	−1.17134	−0.585671	−	0.810549i	$-0.699169\pi$
$89$	−11.0504	−1.17134	−0.585671	+	0.810549i	$0.699169\pi$
$90$	−18.3882	−1.93829
$91$	−8.02554	−0.841305
$92$	−15.4239	−1.60805
$93$	1.93347	0.200491
$94$	15.9327	1.64334
$95$	−2.50863	−0.257380
$96$	−0.390853	−0.0398912
$97$	−17.2966	−1.75621	−0.878103	−	0.478472i	$-0.841191\pi$
$97$	−17.2966	−1.75621	−0.878103	+	0.478472i	$0.841191\pi$
$98$	3.89546	0.393501
$99$	14.5878	1.46613
$100$	5.39306	0.539306
$101$	−14.3360	−1.42649	−0.713243	−	0.700916i	$-0.752775\pi$
$101$	−14.3360	−1.42649	−0.713243	+	0.700916i	$0.752775\pi$
$102$	−2.33614	−0.231312
$103$	−4.26223	−0.419970	−0.209985	−	0.977705i	$-0.567342\pi$
$103$	−4.26223	−0.419970	−0.209985	+	0.977705i	$0.567342\pi$
$104$	18.5637	1.82032
$105$	−1.29576	−0.126454
$106$	−25.5210	−2.47882
$107$	−13.4756	−1.30274	−0.651370	−	0.758760i	$-0.725805\pi$
$107$	−13.4756	−1.30274	−0.651370	+	0.758760i	$0.725805\pi$
$108$	−5.49998	−0.529236
$109$	15.7658	1.51009	0.755044	−	0.655674i	$-0.227616\pi$
$109$	15.7658	1.51009	0.755044	+	0.655674i	$0.227616\pi$
$110$	−30.8054	−2.93718
$111$	2.00611	0.190412
$112$	11.7707	1.11223
$113$	−12.6181	−1.18701	−0.593505	−	0.804830i	$-0.702256\pi$
$113$	−12.6181	−1.18701	−0.593505	+	0.804830i	$0.702256\pi$
$114$	−0.550515	−0.0515605
$115$	9.27832	0.865208
$116$	16.5928	1.54060
$117$	10.1614	0.939425
$118$	6.70086	0.616864
$119$	9.89008	0.906622
$120$	2.99720	0.273605
$121$	13.4387	1.22170
$122$	−36.2188	−3.27910
$123$	2.41053	0.217350
$124$	36.3815	3.26715
$125$	9.29893	0.831722
$126$	17.0834	1.52191
$127$	−9.26718	−0.822330	−0.411165	−	0.911561i	$-0.634878\pi$
$127$	−9.26718	−0.822330	−0.411165	+	0.911561i	$0.634878\pi$
$128$	17.7362	1.56767
$129$	0.361958	0.0318686
$130$	−21.4581	−1.88200
$131$	5.98666	0.523057	0.261528	−	0.965196i	$-0.415773\pi$
$131$	5.98666	0.523057	0.261528	+	0.965196i	$0.415773\pi$
$132$	−4.56898	−0.397678
$133$	2.33062	0.202090
$134$	0.127192	0.0109877
$135$	3.30854	0.284754
$136$	−22.8765	−1.96164
$137$	−0.504017	−0.0430610	−0.0215305	−	0.999768i	$-0.506854\pi$
$137$	−0.504017	−0.0430610	−0.0215305	+	0.999768i	$0.506854\pi$
$138$	2.03612	0.173326
$139$	−2.31136	−0.196047	−0.0980236	−	0.995184i	$-0.531252\pi$
$139$	−2.31136	−0.196047	−0.0980236	+	0.995184i	$0.531252\pi$
$140$	−24.3820	−2.06066
$141$	−1.42153	−0.119715
$142$	31.0824	2.60838
$143$	17.0232	1.42355
$144$	−14.9033	−1.24195
$145$	−9.98148	−0.828916
$146$	−2.36925	−0.196081
$147$	−0.347557	−0.0286660
$148$	37.7484	3.10290
$149$	15.9651	1.30792	0.653958	−	0.756531i	$-0.273107\pi$
$149$	15.9651	1.30792	0.653958	+	0.756531i	$0.273107\pi$
$150$	−0.711939	−0.0581296
$151$	−15.5455	−1.26508	−0.632539	−	0.774528i	$-0.717987\pi$
$151$	−15.5455	−1.26508	−0.632539	+	0.774528i	$0.717987\pi$
$152$	−5.39089	−0.437259
$153$	−12.5222	−1.01236
$154$	28.6195	2.30622
$155$	−21.8854	−1.75788
$156$	−3.18261	−0.254813
$157$	5.86492	0.468072	0.234036	−	0.972228i	$-0.424807\pi$
$157$	5.86492	0.468072	0.234036	+	0.972228i	$0.424807\pi$
$158$	−1.34033	−0.106631
$159$	2.27701	0.180579
$160$	4.42417	0.349761
$161$	−8.61993	−0.679346
$162$	−21.2639	−1.67065
$163$	−13.7970	−1.08067	−0.540333	−	0.841451i	$-0.681702\pi$
$163$	−13.7970	−1.08067	−0.540333	+	0.841451i	$0.681702\pi$
$164$	45.3582	3.54188
$165$	2.74849	0.213969
$166$	−11.6634	−0.905254
$167$	−1.56502	−0.121105	−0.0605524	−	0.998165i	$-0.519286\pi$
$167$	−1.56502	−0.121105	−0.0605524	+	0.998165i	$0.519286\pi$
$168$	−2.78452	−0.214830
$169$	−1.14215	−0.0878574
$170$	26.4434	2.02811
$171$	−2.95088	−0.225660
$172$	6.81086	0.519323
$173$	−8.92206	−0.678332	−0.339166	−	0.940727i	$-0.610145\pi$
$173$	−8.92206	−0.678332	−0.339166	+	0.940727i	$0.610145\pi$
$174$	−2.19042	−0.166055
$175$	3.01401	0.227838
$176$	−24.9673	−1.88198
$177$	−0.597857	−0.0449377
$178$	27.4492	2.05740
$179$	−22.4025	−1.67444	−0.837220	−	0.546865i	$-0.815821\pi$
$179$	−22.4025	−1.67444	−0.837220	+	0.546865i	$0.815821\pi$
$180$	30.8710	2.30099
$181$	3.50650	0.260636	0.130318	−	0.991472i	$-0.458400\pi$
$181$	3.50650	0.260636	0.130318	+	0.991472i	$0.458400\pi$
$182$	19.9354	1.47771
$183$	3.23147	0.238877
$184$	19.9386	1.46989
$185$	−22.7077	−1.66950
$186$	−4.80273	−0.352153
$187$	−20.9782	−1.53408
$188$	−26.7486	−1.95084
$189$	−3.07376	−0.223583
$190$	6.23143	0.452076
$191$	9.43273	0.682528	0.341264	−	0.939967i	$-0.389145\pi$
$191$	9.43273	0.682528	0.341264	+	0.939967i	$0.389145\pi$
$192$	−1.26774	−0.0914913
$193$	9.19990	0.662223	0.331112	−	0.943592i	$-0.392576\pi$
$193$	9.19990	0.662223	0.331112	+	0.943592i	$0.392576\pi$
$194$	42.9648	3.08469
$195$	1.91451	0.137101
$196$	−6.53988	−0.467134
$197$	−23.4229	−1.66882	−0.834408	−	0.551147i	$-0.814190\pi$
$197$	−23.4229	−1.66882	−0.834408	+	0.551147i	$0.814190\pi$
$198$	−36.2362	−2.57519
$199$	−6.96553	−0.493773	−0.246887	−	0.969044i	$-0.579408\pi$
$199$	−6.96553	−0.493773	−0.246887	+	0.969044i	$0.579408\pi$
$200$	−6.97163	−0.492969
$201$	−0.0113482	−0.000800440 0
$202$	35.6106	2.50556
$203$	9.27319	0.650850
$204$	3.92201	0.274596
$205$	−27.2854	−1.90570
$206$	10.5874	0.737658
$207$	10.9140	0.758577
$208$	−17.3914	−1.20588
$209$	−4.94355	−0.341953
$210$	3.21867	0.222110
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	42.8459	2.94267
$213$	−2.77320	−0.190017
$214$	33.4735	2.28820
$215$	−4.09710	−0.279420
$216$	7.10985	0.483764
$217$	20.3324	1.38026
$218$	−39.1622	−2.65240
$219$	0.211387	0.0142842
$220$	51.7175	3.48679
$221$	−14.6127	−0.982960
$222$	−4.98318	−0.334449
$223$	22.5475	1.50989	0.754946	−	0.655787i	$-0.227663\pi$
$223$	22.5475	1.50989	0.754946	+	0.655787i	$0.227663\pi$
$224$	−4.11023	−0.274626
$225$	−3.81615	−0.254410
$226$	31.3433	2.08493
$227$	−19.9149	−1.32180	−0.660900	−	0.750474i	$-0.729825\pi$
$227$	−19.9149	−1.32180	−0.660900	+	0.750474i	$0.729825\pi$
$228$	0.924230	0.0612086
$229$	−24.8757	−1.64383	−0.821917	−	0.569607i	$-0.807095\pi$
$229$	−24.8757	−1.64383	−0.821917	+	0.569607i	$0.807095\pi$
$230$	−23.0473	−1.51970
$231$	−2.55345	−0.168005
$232$	−21.4496	−1.40823
$233$	13.5434	0.887259	0.443630	−	0.896210i	$-0.353691\pi$
$233$	13.5434	0.887259	0.443630	+	0.896210i	$0.353691\pi$
$234$	−25.2410	−1.65005
$235$	16.0907	1.04964
$236$	−11.2497	−0.732293
$237$	0.119585	0.00776789
$238$	−24.5669	−1.59244
$239$	23.1409	1.49686	0.748432	−	0.663212i	$-0.230807\pi$
$239$	23.1409	1.49686	0.748432	+	0.663212i	$0.230807\pi$
$240$	−2.80793	−0.181251
$241$	9.30424	0.599339	0.299669	−	0.954043i	$-0.403124\pi$
$241$	9.30424	0.599339	0.299669	+	0.954043i	$0.403124\pi$
$242$	−33.3817	−2.14586
$243$	5.85377	0.375520
$244$	60.8057	3.89269
$245$	3.93409	0.251340
$246$	−5.98775	−0.381765
$247$	−3.44352	−0.219106
$248$	−47.0304	−2.98644
$249$	1.04062	0.0659465
$250$	−23.0985	−1.46088
$251$	−15.5023	−0.978498	−0.489249	−	0.872144i	$-0.662729\pi$
$251$	−15.5023	−0.978498	−0.489249	+	0.872144i	$0.662729\pi$
$252$	−28.6804	−1.80669
$253$	18.2840	1.14951
$254$	23.0197	1.44438
$255$	−2.35930	−0.147745
$256$	−32.6162	−2.03851
$257$	6.49144	0.404925	0.202462	−	0.979290i	$-0.435106\pi$
$257$	6.49144	0.404925	0.202462	+	0.979290i	$0.435106\pi$
$258$	−0.899103	−0.0559757
$259$	21.0964	1.31086
$260$	36.0248	2.23416
$261$	−11.7411	−0.726758
$262$	−14.8709	−0.918724
$263$	4.57133	0.281880	0.140940	−	0.990018i	$-0.454987\pi$
$263$	4.57133	0.281880	0.140940	+	0.990018i	$0.454987\pi$
$264$	5.90633	0.363510
$265$	−25.7741	−1.58329
$266$	−5.78925	−0.354962
$267$	−2.44904	−0.149879
$268$	−0.213536	−0.0130438
$269$	−12.3664	−0.753994	−0.376997	−	0.926214i	$-0.623043\pi$
$269$	−12.3664	−0.753994	−0.376997	+	0.926214i	$0.623043\pi$
$270$	−8.21840	−0.500156
$271$	−12.7536	−0.774723	−0.387362	−	0.921928i	$-0.626613\pi$
$271$	−12.7536	−0.774723	−0.387362	+	0.921928i	$0.626613\pi$
$272$	21.4319	1.29950
$273$	−1.77866	−0.107649
$274$	1.25198	0.0756347
$275$	−6.39312	−0.385519
$276$	−3.41832	−0.205759
$277$	32.3341	1.94277	0.971383	−	0.237520i	$-0.0763345\pi$
$277$	32.3341	1.94277	0.971383	+	0.237520i	$0.0763345\pi$
$278$	5.74142	0.344348
$279$	−25.7437	−1.54123
$280$	31.5187	1.88360
$281$	−15.8695	−0.946693	−0.473346	−	0.880876i	$-0.656954\pi$
$281$	−15.8695	−0.946693	−0.473346	+	0.880876i	$0.656954\pi$
$282$	3.53109	0.210273
$283$	30.0476	1.78614	0.893072	−	0.449913i	$-0.148545\pi$
$283$	30.0476	1.78614	0.893072	+	0.449913i	$0.148545\pi$
$284$	−52.1825	−3.09646
$285$	−0.555974	−0.0329331
$286$	−42.2857	−2.50041
$287$	25.3493	1.49632
$288$	5.20412	0.306655
$289$	1.00768	0.0592751
$290$	24.7940	1.45595
$291$	−3.83336	−0.224715
$292$	3.97761	0.232772
$293$	19.7465	1.15360	0.576802	−	0.816884i	$-0.304300\pi$
$293$	19.7465	1.15360	0.576802	+	0.816884i	$0.304300\pi$
$294$	0.863331	0.0503505
$295$	6.76731	0.394008
$296$	−48.7975	−2.83630
$297$	6.51986	0.378321
$298$	−39.6574	−2.29729
$299$	12.7361	0.736547
$300$	1.19524	0.0690070
$301$	3.80637	0.219395
$302$	38.6151	2.22205
$303$	−3.17722	−0.182526
$304$	5.05047	0.289664
$305$	−36.5779	−2.09445
$306$	31.1051	1.77816
$307$	−3.84824	−0.219631	−0.109815	−	0.993952i	$-0.535026\pi$
$307$	−3.84824	−0.219631	−0.109815	+	0.993952i	$0.535026\pi$
$308$	−48.0476	−2.73777
$309$	−0.944616	−0.0537373
$310$	54.3633	3.08763
$311$	1.07263	0.0608234	0.0304117	−	0.999537i	$-0.490318\pi$
$311$	1.07263	0.0608234	0.0304117	+	0.999537i	$0.490318\pi$
$312$	4.11417	0.232919
$313$	−16.2107	−0.916285	−0.458143	−	0.888879i	$-0.651485\pi$
$313$	−16.2107	−0.916285	−0.458143	+	0.888879i	$0.651485\pi$
$314$	−14.5685	−0.822146
$315$	17.2528	0.972085
$316$	2.25020	0.126584
$317$	−19.3172	−1.08496	−0.542480	−	0.840069i	$-0.682515\pi$
$317$	−19.3172	−1.08496	−0.542480	+	0.840069i	$0.682515\pi$
$318$	−5.65609	−0.317178
$319$	−19.6697	−1.10129
$320$	14.3499	0.802184
$321$	−2.98654	−0.166692
$322$	21.4119	1.19324
$323$	4.24355	0.236117
$324$	35.6988	1.98327
$325$	−4.45325	−0.247022
$326$	34.2718	1.89814
$327$	3.49409	0.193223
$328$	−58.6347	−3.23756
$329$	−14.9489	−0.824161
$330$	−6.82724	−0.375827
$331$	−6.39605	−0.351559	−0.175779	−	0.984430i	$-0.556245\pi$
$331$	−6.39605	−0.351559	−0.175779	+	0.984430i	$0.556245\pi$
$332$	19.5810	1.07465
$333$	−26.7109	−1.46375
$334$	3.88751	0.212715
$335$	0.128453	0.00701816
$336$	2.60868	0.142315
$337$	17.5536	0.956204	0.478102	−	0.878304i	$-0.341325\pi$
$337$	17.5536	0.956204	0.478102	+	0.878304i	$0.341325\pi$
$338$	2.83709	0.154317
$339$	−2.79648	−0.151884
$340$	−44.3943	−2.40762
$341$	−43.1278	−2.33550
$342$	7.32999	0.396360
$343$	−19.9692	−1.07824
$344$	−8.80442	−0.474702
$345$	2.05631	0.110708
$346$	22.1624	1.19146
$347$	34.5541	1.85496	0.927481	−	0.373870i	$-0.121969\pi$
$347$	34.5541	1.85496	0.927481	+	0.373870i	$0.121969\pi$
$348$	3.67738	0.197128
$349$	−18.4862	−0.989545	−0.494772	−	0.869023i	$-0.664749\pi$
$349$	−18.4862	−0.989545	−0.494772	+	0.869023i	$0.664749\pi$
$350$	−7.48679	−0.400186
$351$	4.54153	0.242409
$352$	8.71834	0.464689
$353$	16.8822	0.898551	0.449275	−	0.893393i	$-0.351682\pi$
$353$	16.8822	0.898551	0.449275	+	0.893393i	$0.351682\pi$
$354$	1.48508	0.0789309
$355$	31.3906	1.66604
$356$	−46.0830	−2.44239
$357$	2.19189	0.116007
$358$	55.6478	2.94108
$359$	−5.96952	−0.315059	−0.157530	−	0.987514i	$-0.550353\pi$
$359$	−5.96952	−0.315059	−0.157530	+	0.987514i	$0.550353\pi$
$360$	−39.9070	−2.10328
$361$	1.00000	0.0526316
$362$	−8.71013	−0.457795
$363$	2.97835	0.156323
$364$	−33.4685	−1.75422
$365$	−2.39275	−0.125242
$366$	−8.02698	−0.419577
$367$	20.0412	1.04614	0.523071	−	0.852289i	$-0.324786\pi$
$367$	20.0412	1.04614	0.523071	+	0.852289i	$0.324786\pi$
$368$	−18.6795	−0.973735
$369$	−32.0957	−1.67083
$370$	56.4059	2.93240
$371$	23.9452	1.24317
$372$	8.06303	0.418049
$373$	22.1700	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$373$	22.1700	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$374$	52.1098	2.69453
$375$	2.06087	0.106423
$376$	34.5780	1.78322
$377$	−13.7013	−0.705652
$378$	7.63522	0.392714
$379$	−23.3293	−1.19835	−0.599173	−	0.800620i	$-0.704504\pi$
$379$	−23.3293	−1.19835	−0.599173	+	0.800620i	$0.704504\pi$
$380$	−10.4616	−0.536669
$381$	−2.05384	−0.105221
$382$	−23.4309	−1.19883
$383$	15.9142	0.813180	0.406590	−	0.913611i	$-0.366718\pi$
$383$	15.9142	0.813180	0.406590	+	0.913611i	$0.366718\pi$
$384$	3.93077	0.200591
$385$	28.9032	1.47305
$386$	−22.8525	−1.16316
$387$	−4.81939	−0.244983
$388$	−72.1312	−3.66191
$389$	−23.6738	−1.20031	−0.600155	−	0.799884i	$-0.704894\pi$
$389$	−23.6738	−1.20031	−0.600155	+	0.799884i	$0.704894\pi$
$390$	−4.75564	−0.240811
$391$	−15.6950	−0.793731
$392$	8.45412	0.426998
$393$	1.32679	0.0669278
$394$	58.1826	2.93120
$395$	−1.35362	−0.0681078
$396$	60.8349	3.05707
$397$	−36.3576	−1.82473	−0.912367	−	0.409373i	$-0.865748\pi$
$397$	−36.3576	−1.82473	−0.912367	+	0.409373i	$0.865748\pi$
$398$	17.3024	0.867289
$399$	0.516522	0.0258585
$400$	6.53139	0.326569
$401$	22.3963	1.11842	0.559208	−	0.829027i	$-0.311105\pi$
$401$	22.3963	1.11842	0.559208	+	0.829027i	$0.311105\pi$
$402$	0.0281889	0.00140594
$403$	−30.0415	−1.49647
$404$	−59.7847	−2.97440
$405$	−21.4748	−1.06709
$406$	−23.0346	−1.14319
$407$	−44.7482	−2.21809
$408$	−5.07000	−0.251002
$409$	−27.1435	−1.34216	−0.671079	−	0.741386i	$-0.734169\pi$
$409$	−27.1435	−1.34216	−0.671079	+	0.741386i	$0.734169\pi$
$410$	67.7770	3.34727
$411$	−0.111703	−0.00550988
$412$	−17.7746	−0.875690
$413$	−6.28710	−0.309368
$414$	−27.1104	−1.33240
$415$	−11.7790	−0.578210
$416$	6.07293	0.297750
$417$	−0.512255	−0.0250852
$418$	12.2798	0.600623
$419$	15.4957	0.757015	0.378508	−	0.925598i	$-0.376437\pi$
$419$	15.4957	0.757015	0.378508	+	0.925598i	$0.376437\pi$
$420$	−5.40365	−0.263671
$421$	−14.6595	−0.714462	−0.357231	−	0.934016i	$-0.616279\pi$
$421$	−14.6595	−0.714462	−0.357231	+	0.934016i	$0.616279\pi$
$422$	−2.48400	−0.120919
$423$	18.9274	0.920282
$424$	−55.3870	−2.68983
$425$	5.48785	0.266200
$426$	6.88863	0.333755
$427$	33.9824	1.64452
$428$	−56.1968	−2.71637
$429$	3.77277	0.182151
$430$	10.1772	0.490788
$431$	−32.7896	−1.57942	−0.789711	−	0.613479i	$-0.789769\pi$
$431$	−32.7896	−1.57942	−0.789711	+	0.613479i	$0.789769\pi$
$432$	−6.66088	−0.320472
$433$	−25.6682	−1.23354	−0.616768	−	0.787145i	$-0.711558\pi$
$433$	−25.6682	−1.23354	−0.616768	+	0.787145i	$0.711558\pi$
$434$	−50.5057	−2.42435
$435$	−2.21214	−0.106064
$436$	65.7472	3.14872
$437$	−3.69856	−0.176926
$438$	−0.525085	−0.0250895
$439$	−31.6150	−1.50890	−0.754451	−	0.656356i	$-0.772097\pi$
$439$	−31.6150	−1.50890	−0.754451	+	0.656356i	$0.772097\pi$
$440$	−66.8554	−3.18720
$441$	4.62764	0.220364
$442$	36.2980	1.72652
$443$	26.4232	1.25541	0.627703	−	0.778453i	$-0.283995\pi$
$443$	26.4232	1.25541	0.627703	+	0.778453i	$0.283995\pi$
$444$	8.36598	0.397032
$445$	27.7214	1.31412
$446$	−56.0079	−2.65205
$447$	3.53827	0.167354
$448$	−13.3316	−0.629860
$449$	−4.55193	−0.214819	−0.107409	−	0.994215i	$-0.534256\pi$
$449$	−4.55193	−0.214819	−0.107409	+	0.994215i	$0.534256\pi$
$450$	9.47931	0.446859
$451$	−53.7692	−2.53189
$452$	−52.6206	−2.47506
$453$	−3.44528	−0.161873
$454$	49.4686	2.32168
$455$	20.1331	0.943854
$456$	−1.19475	−0.0559495
$457$	−23.0502	−1.07824	−0.539121	−	0.842228i	$-0.681244\pi$
$457$	−23.0502	−1.07824	−0.539121	+	0.842228i	$0.681244\pi$
$458$	61.7913	2.88732
$459$	−5.59665	−0.261229
$460$	38.6929	1.80407
$461$	−5.06797	−0.236039	−0.118020	−	0.993011i	$-0.537655\pi$
$461$	−5.06797	−0.236039	−0.118020	+	0.993011i	$0.537655\pi$
$462$	6.34278	0.295093
$463$	−25.6071	−1.19006	−0.595032	−	0.803702i	$-0.702861\pi$
$463$	−25.6071	−1.19006	−0.595032	+	0.803702i	$0.702861\pi$
$464$	20.0951	0.932892
$465$	−4.85035	−0.224930
$466$	−33.6418	−1.55843
$467$	−33.1025	−1.53180	−0.765901	−	0.642958i	$-0.777707\pi$
$467$	−33.1025	−1.53180	−0.765901	+	0.642958i	$0.777707\pi$
$468$	42.3757	1.95882
$469$	−0.119338	−0.00551053
$470$	−39.9694	−1.84365
$471$	1.29981	0.0598922
$472$	14.5425	0.669374
$473$	−8.07382	−0.371235
$474$	−0.297049	−0.0136439
$475$	1.29322	0.0593372
$476$	41.2441	1.89042
$477$	−30.3179	−1.38816
$478$	−57.4821	−2.62917
$479$	−1.66981	−0.0762954	−0.0381477	−	0.999272i	$-0.512146\pi$
$479$	−1.66981	−0.0762954	−0.0381477	+	0.999272i	$0.512146\pi$
$480$	0.980505	0.0447537
$481$	−31.1702	−1.42124
$482$	−23.1117	−1.05271
$483$	−1.91039	−0.0869257
$484$	56.0427	2.54740
$485$	43.3908	1.97028
$486$	−14.5408	−0.659582
$487$	−14.3500	−0.650263	−0.325131	−	0.945669i	$-0.605409\pi$
$487$	−14.3500	−0.650263	−0.325131	+	0.945669i	$0.605409\pi$
$488$	−78.6038	−3.55822
$489$	−3.05776	−0.138277
$490$	−9.77228	−0.441466
$491$	25.3987	1.14623	0.573113	−	0.819477i	$-0.305736\pi$
$491$	25.3987	1.14623	0.573113	+	0.819477i	$0.305736\pi$
$492$	10.0525	0.453202
$493$	16.8845	0.760438
$494$	8.55371	0.384849
$495$	−36.5955	−1.64484
$496$	44.0606	1.97838
$497$	−29.1631	−1.30815
$498$	−2.58489	−0.115832
$499$	0.531444	0.0237907	0.0118954	−	0.999929i	$-0.496214\pi$
$499$	0.531444	0.0237907	0.0118954	+	0.999929i	$0.496214\pi$
$500$	38.7788	1.73424
$501$	−0.346847	−0.0154960
$502$	38.5078	1.71869
$503$	2.66162	0.118676	0.0593378	−	0.998238i	$-0.481101\pi$
$503$	2.66162	0.118676	0.0593378	+	0.998238i	$0.481101\pi$
$504$	37.0752	1.65146
$505$	35.9638	1.60037
$506$	−45.4175	−2.01905
$507$	−0.253128	−0.0112418
$508$	−38.6465	−1.71466
$509$	−2.09432	−0.0928289	−0.0464144	−	0.998922i	$-0.514779\pi$
$509$	−2.09432	−0.0928289	−0.0464144	+	0.998922i	$0.514779\pi$
$510$	5.86050	0.259507
$511$	2.22296	0.0983378
$512$	45.5462	2.01288
$513$	−1.31886	−0.0582292
$514$	−16.1247	−0.711231
$515$	10.6924	0.471162
$516$	1.50945	0.0664500
$517$	31.7087	1.39455
$518$	−52.4034	−2.30247
$519$	−1.97735	−0.0867960
$520$	−46.5694	−2.04220
$521$	−8.73101	−0.382512	−0.191256	−	0.981540i	$-0.561256\pi$
$521$	−8.73101	−0.382512	−0.191256	+	0.981540i	$0.561256\pi$
$522$	29.1650	1.27652
$523$	13.9520	0.610080	0.305040	−	0.952339i	$-0.401330\pi$
$523$	13.9520	0.610080	0.305040	+	0.952339i	$0.401330\pi$
$524$	24.9659	1.09064
$525$	0.667979	0.0291530
$526$	−11.3552	−0.495109
$527$	37.0209	1.61266
$528$	−5.53336	−0.240809
$529$	−9.32064	−0.405245
$530$	64.0228	2.78097
$531$	7.96033	0.345449
$532$	9.71925	0.421383
$533$	−37.4539	−1.62231
$534$	6.08342	0.263255
$535$	33.8054	1.46154
$536$	0.276038	0.0119230
$537$	−4.96495	−0.214253
$538$	30.7182	1.32435
$539$	7.75259	0.333928
$540$	13.7974	0.593747
$541$	33.1935	1.42710	0.713550	−	0.700604i	$-0.247086\pi$
$541$	33.1935	1.42710	0.713550	+	0.700604i	$0.247086\pi$
$542$	31.6798	1.36076
$543$	0.777126	0.0333497
$544$	−7.48383	−0.320867
$545$	−39.5505	−1.69416
$546$	4.41818	0.189081
$547$	−12.7655	−0.545813	−0.272907	−	0.962041i	$-0.587985\pi$
$547$	−12.7655	−0.545813	−0.272907	+	0.962041i	$0.587985\pi$
$548$	−2.10187	−0.0897876
$549$	−43.0264	−1.83632
$550$	15.8805	0.677146
$551$	3.97886	0.169505
$552$	4.41887	0.188080
$553$	1.25756	0.0534771
$554$	−80.3178	−3.41237
$555$	−5.03259	−0.213622
$556$	−9.63896	−0.408783
$557$	26.3592	1.11688	0.558438	−	0.829546i	$-0.311401\pi$
$557$	26.3592	1.11688	0.558438	+	0.829546i	$0.311401\pi$
$558$	63.9472	2.70710
$559$	−5.62397	−0.237869
$560$	−29.5284	−1.24780
$561$	−4.64928	−0.196293
$562$	39.4197	1.66282
$563$	−4.98489	−0.210088	−0.105044	−	0.994468i	$-0.533498\pi$
$563$	−4.98489	−0.210088	−0.105044	+	0.994468i	$0.533498\pi$
$564$	−5.92815	−0.249620
$565$	31.6541	1.33170
$566$	−74.6382	−3.13728
$567$	19.9509	0.837860
$568$	67.4565	2.83041
$569$	34.9624	1.46570	0.732851	−	0.680389i	$-0.238189\pi$
$569$	34.9624	1.46570	0.732851	+	0.680389i	$0.238189\pi$
$570$	1.38104	0.0578454
$571$	−2.19496	−0.0918562	−0.0459281	−	0.998945i	$-0.514625\pi$
$571$	−2.19496	−0.0918562	−0.0459281	+	0.998945i	$0.514625\pi$
$572$	70.9911	2.96829
$573$	2.09053	0.0873329
$574$	−62.9675	−2.62821
$575$	−4.78307	−0.199468
$576$	16.8797	0.703320
$577$	−16.4455	−0.684633	−0.342317	−	0.939585i	$-0.611212\pi$
$577$	−16.4455	−0.684633	−0.342317	+	0.939585i	$0.611212\pi$
$578$	−2.50307	−0.104114
$579$	2.03892	0.0847348
$580$	−41.6252	−1.72839
$581$	10.9432	0.454000
$582$	9.52206	0.394702
$583$	−50.7909	−2.10354
$584$	−5.14186	−0.212772
$585$	−25.4913	−1.05393
$586$	−49.0504	−2.02625
$587$	−9.47010	−0.390873	−0.195436	−	0.980716i	$-0.562612\pi$
$587$	−9.47010	−0.390873	−0.195436	+	0.980716i	$0.562612\pi$
$588$	−1.44940	−0.0597722
$589$	8.72405	0.359468
$590$	−16.8100	−0.692056
$591$	−5.19110	−0.213533
$592$	45.7161	1.87892
$593$	0.549011	0.0225452	0.0112726	−	0.999936i	$-0.496412\pi$
$593$	0.549011	0.0225452	0.0112726	+	0.999936i	$0.496412\pi$
$594$	−16.1953	−0.664502
$595$	−24.8105	−1.01713
$596$	66.5786	2.72717
$597$	−1.54373	−0.0631808
$598$	−31.6364	−1.29371
$599$	−39.6248	−1.61903	−0.809513	−	0.587103i	$-0.800269\pi$
$599$	−39.6248	−1.61903	−0.809513	+	0.587103i	$0.800269\pi$
$600$	−1.54509	−0.0630778
$601$	−30.7723	−1.25523	−0.627613	−	0.778525i	$-0.715968\pi$
$601$	−30.7723	−1.25523	−0.627613	+	0.778525i	$0.715968\pi$
$602$	−9.45502	−0.385358
$603$	0.151099	0.00615322
$604$	−64.8288	−2.63785
$605$	−33.7127	−1.37062
$606$	7.89220	0.320599
$607$	18.6210	0.755801	0.377901	−	0.925846i	$-0.376646\pi$
$607$	18.6210	0.755801	0.377901	+	0.925846i	$0.376646\pi$
$608$	−1.76358	−0.0715226
$609$	2.05517	0.0832796
$610$	90.8595	3.67880
$611$	22.0873	0.893556
$612$	−52.2207	−2.11090
$613$	−14.2846	−0.576951	−0.288475	−	0.957487i	$-0.593148\pi$
$613$	−14.2846	−0.576951	−0.288475	+	0.957487i	$0.593148\pi$
$614$	9.55902	0.385771
$615$	−6.04713	−0.243844
$616$	62.1113	2.50254
$617$	−34.9741	−1.40800	−0.704002	−	0.710198i	$-0.748606\pi$
$617$	−34.9741	−1.40800	−0.704002	+	0.710198i	$0.748606\pi$
$618$	2.34643	0.0943871
$619$	−17.2219	−0.692206	−0.346103	−	0.938197i	$-0.612495\pi$
$619$	−17.2219	−0.692206	−0.346103	+	0.938197i	$0.612495\pi$
$620$	−91.2676	−3.66540
$621$	4.87789	0.195743
$622$	−2.66442	−0.106833
$623$	−25.7543	−1.03182
$624$	−3.85437	−0.154298
$625$	−29.7937	−1.19175
$626$	40.2675	1.60941
$627$	−1.09561	−0.0437546
$628$	24.4582	0.975988
$629$	38.4119	1.53158
$630$	−42.8559	−1.70742
$631$	1.30626	0.0520014	0.0260007	−	0.999662i	$-0.491723\pi$
$631$	1.30626	0.0520014	0.0260007	+	0.999662i	$0.491723\pi$
$632$	−2.90884	−0.115707
$633$	0.221625	0.00880879
$634$	47.9838	1.90568
$635$	23.2479	0.922566
$636$	9.49570	0.376529
$637$	5.40021	0.213964
$638$	48.8594	1.93436
$639$	36.9245	1.46071
$640$	−44.4935	−1.75876
$641$	38.0780	1.50399	0.751995	−	0.659169i	$-0.229092\pi$
$641$	38.0780	1.50399	0.751995	+	0.659169i	$0.229092\pi$
$642$	7.41855	0.292787
$643$	−37.1455	−1.46487	−0.732437	−	0.680835i	$-0.761617\pi$
$643$	−37.1455	−1.46487	−0.732437	+	0.680835i	$0.761617\pi$
$644$	−35.9473	−1.41652
$645$	−0.908018	−0.0357532
$646$	−10.5410	−0.414729
$647$	12.4874	0.490931	0.245465	−	0.969405i	$-0.421059\pi$
$647$	12.4874	0.490931	0.245465	+	0.969405i	$0.421059\pi$
$648$	−46.1480	−1.81286
$649$	13.3358	0.523475
$650$	11.0619	0.433882
$651$	4.50617	0.176611
$652$	−57.5370	−2.25332
$653$	31.8988	1.24830	0.624149	−	0.781305i	$-0.285446\pi$
$653$	31.8988	1.24830	0.624149	+	0.781305i	$0.285446\pi$
$654$	−8.67930	−0.339388
$655$	−15.0183	−0.586814
$656$	54.9321	2.14474
$657$	−2.81457	−0.109807
$658$	37.1331	1.44760
$659$	−28.1454	−1.09639	−0.548195	−	0.836351i	$-0.684685\pi$
$659$	−28.1454	−1.09639	−0.548195	+	0.836351i	$0.684685\pi$
$660$	11.4619	0.446153
$661$	−11.4205	−0.444206	−0.222103	−	0.975023i	$-0.571292\pi$
$661$	−11.4205	−0.444206	−0.222103	+	0.975023i	$0.571292\pi$
$662$	15.8878	0.617496
$663$	−3.23855	−0.125775
$664$	−25.3124	−0.982313
$665$	−5.84666	−0.226724
$666$	66.3499	2.57101
$667$	−14.7160	−0.569808
$668$	−6.52652	−0.252519
$669$	4.99708	0.193198
$670$	−0.319078	−0.0123271
$671$	−72.0811	−2.78266
$672$	−0.910928	−0.0351398
$673$	34.8198	1.34221	0.671103	−	0.741364i	$-0.265821\pi$
$673$	34.8198	1.34221	0.671103	+	0.741364i	$0.265821\pi$
$674$	−43.6031	−1.67953
$675$	−1.70558	−0.0656480
$676$	−4.76303	−0.183194
$677$	24.5439	0.943299	0.471650	−	0.881786i	$-0.343659\pi$
$677$	24.5439	0.943299	0.471650	+	0.881786i	$0.343659\pi$
$678$	6.94645	0.266777
$679$	−40.3118	−1.54703
$680$	57.3887	2.20075
$681$	−4.41364	−0.169131
$682$	107.129	4.10220
$683$	25.1092	0.960778	0.480389	−	0.877055i	$-0.340495\pi$
$683$	25.1092	0.960778	0.480389	+	0.877055i	$0.340495\pi$
$684$	−12.3059	−0.470528
$685$	1.26439	0.0483099
$686$	49.6036	1.89387
$687$	−5.51308	−0.210337
$688$	8.24844	0.314469
$689$	−35.3794	−1.34785
$690$	−5.10786	−0.194453
$691$	−14.5623	−0.553975	−0.276987	−	0.960874i	$-0.589336\pi$
$691$	−14.5623	−0.553975	−0.276987	+	0.960874i	$0.589336\pi$
$692$	−37.2072	−1.41441
$693$	33.9987	1.29150
$694$	−85.8324	−3.25815
$695$	5.79835	0.219944
$696$	−4.75376	−0.180191
$697$	46.1555	1.74826
$698$	45.9197	1.73809
$699$	3.00156	0.113529
$700$	12.5692	0.475070
$701$	−24.1765	−0.913136	−0.456568	−	0.889689i	$-0.650921\pi$
$701$	−24.1765	−0.913136	−0.456568	+	0.889689i	$0.650921\pi$
$702$	−11.2812	−0.425780
$703$	9.05184	0.341397
$704$	28.2782	1.06577
$705$	3.56610	0.134307
$706$	−41.9354	−1.57826
$707$	−33.4118	−1.25658
$708$	−2.49321	−0.0937006
$709$	25.6750	0.964244	0.482122	−	0.876104i	$-0.339866\pi$
$709$	25.6750	0.964244	0.482122	+	0.876104i	$0.339866\pi$
$710$	−77.9742	−2.92632
$711$	−1.59225	−0.0597140
$712$	59.5716	2.23254
$713$	−32.2665	−1.20839
$714$	−5.44464	−0.203761
$715$	−42.7050	−1.59708
$716$	−93.4240	−3.49142
$717$	5.12860	0.191531
$718$	14.8283	0.553386
$719$	23.8298	0.888703	0.444352	−	0.895852i	$-0.353434\pi$
$719$	23.8298	0.888703	0.444352	+	0.895852i	$0.353434\pi$
$720$	37.3870	1.39333
$721$	−9.93364	−0.369948
$722$	−2.48400	−0.0924448
$723$	2.06205	0.0766884
$724$	14.6230	0.543458
$725$	5.14555	0.191101
$726$	−7.39821	−0.274573
$727$	−50.8608	−1.88632	−0.943161	−	0.332336i	$-0.892163\pi$
$727$	−50.8608	−1.88632	−0.943161	+	0.332336i	$0.892163\pi$
$728$	43.2648	1.60350
$729$	−24.3837	−0.903101
$730$	5.94358	0.219982
$731$	6.93057	0.256336
$732$	13.4761	0.498089
$733$	−12.9203	−0.477222	−0.238611	−	0.971115i	$-0.576692\pi$
$733$	−12.9203	−0.477222	−0.238611	+	0.971115i	$0.576692\pi$
$734$	−49.7823	−1.83750
$735$	0.871892	0.0321602
$736$	6.52271	0.240430
$737$	0.253133	0.00932426
$738$	79.7256	2.93474
$739$	23.2148	0.853972	0.426986	−	0.904258i	$-0.359575\pi$
$739$	23.2148	0.853972	0.426986	+	0.904258i	$0.359575\pi$
$740$	−94.6968	−3.48112
$741$	−0.763170	−0.0280357
$742$	−59.4798	−2.18357
$743$	−15.6594	−0.574489	−0.287245	−	0.957857i	$-0.592739\pi$
$743$	−15.6594	−0.574489	−0.287245	+	0.957857i	$0.592739\pi$
$744$	−10.4231	−0.382130
$745$	−40.0506	−1.46734
$746$	−55.0702	−2.01626
$747$	−13.8556	−0.506950
$748$	−87.4842	−3.19874
$749$	−31.4066	−1.14757
$750$	−5.11920	−0.186927
$751$	−38.3303	−1.39869	−0.699346	−	0.714783i	$-0.746525\pi$
$751$	−38.3303	−1.39869	−0.699346	+	0.714783i	$0.746525\pi$
$752$	−32.3945	−1.18131
$753$	−3.43570	−0.125204
$754$	34.0340	1.23944
$755$	38.9980	1.41928
$756$	−12.8184	−0.466199
$757$	36.7843	1.33695	0.668473	−	0.743736i	$-0.266948\pi$
$757$	36.7843	1.33695	0.668473	+	0.743736i	$0.266948\pi$
$758$	57.9499	2.10484
$759$	4.05219	0.147085
$760$	13.5238	0.490558
$761$	14.5656	0.528004	0.264002	−	0.964522i	$-0.414957\pi$
$761$	14.5656	0.528004	0.264002	+	0.964522i	$0.414957\pi$
$762$	5.10173	0.184816
$763$	36.7440	1.33022
$764$	39.3368	1.42316
$765$	31.4136	1.13576
$766$	−39.5309	−1.42831
$767$	9.28929	0.335417
$768$	−7.22855	−0.260838
$769$	17.6657	0.637041	0.318520	−	0.947916i	$-0.396814\pi$
$769$	17.6657	0.637041	0.318520	+	0.947916i	$0.396814\pi$
$770$	−71.7956	−2.58733
$771$	1.43866	0.0518122
$772$	38.3659	1.38082
$773$	−3.79132	−0.136364	−0.0681822	−	0.997673i	$-0.521720\pi$
$773$	−3.79132	−0.136364	−0.0681822	+	0.997673i	$0.521720\pi$
$774$	11.9714	0.430301
$775$	11.2822	0.405267
$776$	93.2442	3.34727
$777$	4.67548	0.167732
$778$	58.8057	2.10829
$779$	10.8766	0.389696
$780$	7.98398	0.285872
$781$	61.8589	2.21349
$782$	38.9864	1.39415
$783$	−5.24756	−0.187533
$784$	−7.92027	−0.282867
$785$	−14.7129	−0.525127
$786$	−3.29575	−0.117555
$787$	−7.62789	−0.271905	−0.135952	−	0.990715i	$-0.543409\pi$
$787$	−7.62789	−0.271905	−0.135952	+	0.990715i	$0.543409\pi$
$788$	−97.6795	−3.47969
$789$	1.01312	0.0360680
$790$	3.36238	0.119628
$791$	−29.4079	−1.04563
$792$	−78.6415	−2.79440
$793$	−50.2095	−1.78299
$794$	90.3122	3.20506
$795$	−5.71218	−0.202590
$796$	−29.0480	−1.02958
$797$	28.6141	1.01356	0.506781	−	0.862075i	$-0.330835\pi$
$797$	28.6141	1.01356	0.506781	+	0.862075i	$0.330835\pi$
$798$	−1.28304	−0.0454191
$799$	−27.2187	−0.962930
$800$	−2.28070	−0.0806350
$801$	32.6085	1.15216
$802$	−55.6323	−1.96444
$803$	−4.71519	−0.166395
$804$	−0.0473248	−0.00166902
$805$	21.6242	0.762154
$806$	74.6230	2.62848
$807$	−2.74070	−0.0964774
$808$	77.2839	2.71884
$809$	−38.7781	−1.36337	−0.681683	−	0.731647i	$-0.738752\pi$
$809$	−38.7781	−1.36337	−0.681683	+	0.731647i	$0.738752\pi$
$810$	53.3433	1.87429
$811$	55.0715	1.93382	0.966910	−	0.255116i	$-0.0821137\pi$
$811$	55.0715	1.93382	0.966910	+	0.255116i	$0.0821137\pi$
$812$	38.6715	1.35710
$813$	−2.82650	−0.0991298
$814$	111.155	3.89596
$815$	34.6116	1.21239
$816$	4.74984	0.166278
$817$	1.63320	0.0571385
$818$	67.4243	2.35744
$819$	23.6824	0.827530
$820$	−113.787	−3.97362
$821$	−2.36477	−0.0825310	−0.0412655	−	0.999148i	$-0.513139\pi$
$821$	−2.36477	−0.0825310	−0.0412655	+	0.999148i	$0.513139\pi$
$822$	0.277469	0.00967784
$823$	33.0978	1.15372	0.576859	−	0.816844i	$-0.304278\pi$
$823$	33.0978	1.15372	0.576859	+	0.816844i	$0.304278\pi$
$824$	22.9772	0.800450
$825$	−1.41687	−0.0493292
$826$	15.6171	0.543390
$827$	−11.1874	−0.389025	−0.194513	−	0.980900i	$-0.562313\pi$
$827$	−11.1874	−0.389025	−0.194513	+	0.980900i	$0.562313\pi$
$828$	45.5142	1.58173
$829$	−37.1760	−1.29118	−0.645589	−	0.763685i	$-0.723388\pi$
$829$	−37.1760	−1.29118	−0.645589	+	0.763685i	$0.723388\pi$
$830$	29.2591	1.01560
$831$	7.16603	0.248587
$832$	19.6977	0.682895
$833$	−6.65483	−0.230576
$834$	1.27244	0.0440610
$835$	3.92605	0.135867
$836$	−20.6158	−0.713014
$837$	−11.5058	−0.397700
$838$	−38.4913	−1.32966
$839$	24.0132	0.829029	0.414515	−	0.910043i	$-0.363951\pi$
$839$	24.0132	0.829029	0.414515	+	0.910043i	$0.363951\pi$
$840$	6.98532	0.241016
$841$	−13.1687	−0.454093
$842$	36.4143	1.25492
$843$	−3.51706	−0.121134
$844$	4.17025	0.143546
$845$	2.86522	0.0985666
$846$	−47.0156	−1.61643
$847$	31.3205	1.07618
$848$	51.8894	1.78189
$849$	6.65929	0.228546
$850$	−13.6318	−0.467567
$851$	−33.4788	−1.14764
$852$	−11.5649	−0.396208
$853$	33.9166	1.16128	0.580641	−	0.814160i	$-0.302802\pi$
$853$	33.9166	1.16128	0.580641	+	0.814160i	$0.302802\pi$
$854$	−84.4121	−2.88852
$855$	7.40267	0.253166
$856$	72.6458	2.48298
$857$	−3.91309	−0.133669	−0.0668343	−	0.997764i	$-0.521290\pi$
$857$	−3.91309	−0.133669	−0.0668343	+	0.997764i	$0.521290\pi$
$858$	−9.37155	−0.319940
$859$	51.2580	1.74890	0.874451	−	0.485114i	$-0.161222\pi$
$859$	51.2580	1.74890	0.874451	+	0.485114i	$0.161222\pi$
$860$	−17.0859	−0.582625
$861$	5.61802	0.191462
$862$	81.4494	2.77418
$863$	25.5023	0.868109	0.434055	−	0.900887i	$-0.357082\pi$
$863$	25.5023	0.868109	0.434055	+	0.900887i	$0.357082\pi$
$864$	2.32592	0.0791294
$865$	22.3821	0.761016
$866$	63.7598	2.16665
$867$	0.223326	0.00758455
$868$	84.7913	2.87800
$869$	−2.66746	−0.0904874
$870$	5.49496	0.186296
$871$	0.176324	0.00597452
$872$	−84.9916	−2.87818
$873$	51.0403	1.72745
$874$	9.18722	0.310762
$875$	21.6722	0.732656
$876$	0.881536	0.0297843
$877$	−17.4490	−0.589211	−0.294606	−	0.955619i	$-0.595188\pi$
$877$	−17.4490	−0.589211	−0.294606	+	0.955619i	$0.595188\pi$
$878$	78.5317	2.65032
$879$	4.37632	0.147610
$880$	62.6336	2.11138
$881$	29.5358	0.995087	0.497544	−	0.867439i	$-0.334235\pi$
$881$	29.5358	0.995087	0.497544	+	0.867439i	$0.334235\pi$
$882$	−11.4951	−0.387059
$883$	−11.6888	−0.393359	−0.196679	−	0.980468i	$-0.563016\pi$
$883$	−11.6888	−0.393359	−0.196679	+	0.980468i	$0.563016\pi$
$884$	−60.9388	−2.04959
$885$	1.49980	0.0504153
$886$	−65.6353	−2.20506
$887$	32.2008	1.08120	0.540599	−	0.841281i	$-0.318198\pi$
$887$	32.2008	1.08120	0.540599	+	0.841281i	$0.318198\pi$
$888$	−10.8147	−0.362918
$889$	−21.5983	−0.724382
$890$	−68.8599	−2.30819
$891$	−42.3186	−1.41772
$892$	94.0286	3.14831
$893$	−6.41415	−0.214641
$894$	−8.78906	−0.293950
$895$	56.1996	1.87854
$896$	41.3362	1.38095
$897$	2.82263	0.0942449
$898$	11.3070	0.377319
$899$	34.7117	1.15770
$900$	−15.9143	−0.530477
$901$	43.5989	1.45249
$902$	133.563	4.44715
$903$	0.843585	0.0280728
$904$	68.0228	2.26240
$905$	−8.79651	−0.292406
$906$	8.55806	0.284323
$907$	7.45024	0.247381	0.123691	−	0.992321i	$-0.460527\pi$
$907$	7.45024	0.247381	0.123691	+	0.992321i	$0.460527\pi$
$908$	−83.0501	−2.75612
$909$	42.3039	1.40313
$910$	−50.0106	−1.65783
$911$	39.6783	1.31460	0.657300	−	0.753629i	$-0.271699\pi$
$911$	39.6783	1.31460	0.657300	+	0.753629i	$0.271699\pi$
$912$	1.11931	0.0370640
$913$	−23.2120	−0.768204
$914$	57.2567	1.89388
$915$	−8.10657	−0.267995
$916$	−103.738	−3.42760
$917$	13.9526	0.460756
$918$	13.9021	0.458837
$919$	28.7383	0.947990	0.473995	−	0.880528i	$-0.342811\pi$
$919$	28.7383	0.947990	0.473995	+	0.880528i	$0.342811\pi$
$920$	−50.0184	−1.64906
$921$	−0.852865	−0.0281028
$922$	12.5888	0.414591
$923$	43.0890	1.41829
$924$	−10.6485	−0.350311
$925$	11.7061	0.384893
$926$	63.6081	2.09029
$927$	12.5774	0.413094
$928$	−7.01703	−0.230345
$929$	1.27637	0.0418764	0.0209382	−	0.999781i	$-0.493335\pi$
$929$	1.27637	0.0418764	0.0209382	+	0.999781i	$0.493335\pi$
$930$	12.0483	0.395078
$931$	−1.56822	−0.0513964
$932$	56.4794	1.85005
$933$	0.237722	0.00778266
$934$	82.2266	2.69054
$935$	52.6265	1.72107
$936$	−54.7792	−1.79051
$937$	−20.8315	−0.680534	−0.340267	−	0.940329i	$-0.610517\pi$
$937$	−20.8315	−0.680534	−0.340267	+	0.940329i	$0.610517\pi$
$938$	0.296436	0.00967898
$939$	−3.59270	−0.117243
$940$	67.1023	2.18864
$941$	−21.9126	−0.714331	−0.357165	−	0.934041i	$-0.616257\pi$
$941$	−21.9126	−0.714331	−0.357165	+	0.934041i	$0.616257\pi$
$942$	−3.22873	−0.105198
$943$	−40.2279	−1.31000
$944$	−13.6242	−0.443430
$945$	7.71093	0.250837
$946$	20.0554	0.652056
$947$	−3.85162	−0.125161	−0.0625804	−	0.998040i	$-0.519933\pi$
$947$	−3.85162	−0.125161	−0.0625804	+	0.998040i	$0.519933\pi$
$948$	0.498700	0.0161970
$949$	−3.28445	−0.106618
$950$	−3.21237	−0.104223
$951$	−4.28116	−0.138826
$952$	−53.3164	−1.72799
$953$	−54.6977	−1.77183	−0.885916	−	0.463845i	$-0.846470\pi$
$953$	−54.6977	−1.77183	−0.885916	+	0.463845i	$0.846470\pi$
$954$	75.3096	2.43824
$955$	−23.6632	−0.765724
$956$	96.5035	3.12115
$957$	−4.35929	−0.140916
$958$	4.14780	0.134009
$959$	−1.17467	−0.0379321
$960$	3.18029	0.102644
$961$	45.1091	1.45513
$962$	77.4268	2.49634
$963$	39.7651	1.28141
$964$	38.8010	1.24970
$965$	−23.0791	−0.742944
$966$	4.74541	0.152681
$967$	−31.3609	−1.00850	−0.504250	−	0.863558i	$-0.668231\pi$
$967$	−31.3609	−1.00850	−0.504250	+	0.863558i	$0.668231\pi$
$968$	−72.4466	−2.32852
$969$	0.940474	0.0302124
$970$	−107.783	−3.46070
$971$	−29.9121	−0.959923	−0.479962	−	0.877289i	$-0.659349\pi$
$971$	−29.9121	−0.959923	−0.479962	+	0.877289i	$0.659349\pi$
$972$	24.4117	0.783005
$973$	−5.38690	−0.172696
$974$	35.6455	1.14216
$975$	−0.986949	−0.0316077
$976$	73.6401	2.35716
$977$	−19.0711	−0.610139	−0.305070	−	0.952330i	$-0.598680\pi$
$977$	−19.0711	−0.610139	−0.305070	+	0.952330i	$0.598680\pi$
$978$	7.59547	0.242876
$979$	54.6283	1.74593
$980$	16.4061	0.524075
$981$	−46.5230	−1.48536
$982$	−63.0902	−2.01329
$983$	22.1866	0.707644	0.353822	−	0.935313i	$-0.384882\pi$
$983$	22.1866	0.707644	0.353822	+	0.935313i	$0.384882\pi$
$984$	−12.9949	−0.414262
$985$	58.7595	1.87223
$986$	−41.9410	−1.33567
$987$	−3.31305	−0.105456
$988$	−14.3603	−0.456863
$989$	−6.04050	−0.192077
$990$	90.9031	2.88909
$991$	−20.0570	−0.637132	−0.318566	−	0.947901i	$-0.603201\pi$
$991$	−20.0570	−0.637132	−0.318566	+	0.947901i	$0.603201\pi$
$992$	−15.3856	−0.488492
$993$	−1.41752	−0.0449837
$994$	72.4412	2.29769
$995$	17.4739	0.553961
$996$	4.33963	0.137507
$997$	−16.4914	−0.522287	−0.261143	−	0.965300i	$-0.584100\pi$
$997$	−16.4914	−0.522287	−0.261143	+	0.965300i	$0.584100\pi$
$998$	−1.32011	−0.0417872
$999$	−11.9381	−0.377706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.8	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.48400 q^{2} +0.221625 q^{3} +4.17025 q^{4} -2.50863 q^{5} -0.550515 q^{6} +2.33062 q^{7} -5.39089 q^{8} -2.95088 q^{9} +O(q^{10})\)	\(q-2.48400 q^{2} +0.221625 q^{3} +4.17025 q^{4} -2.50863 q^{5} -0.550515 q^{6} +2.33062 q^{7} -5.39089 q^{8} -2.95088 q^{9} +6.23143 q^{10} -4.94355 q^{11} +0.924230 q^{12} -3.44352 q^{13} -5.78925 q^{14} -0.555974 q^{15} +5.05047 q^{16} +4.24355 q^{17} +7.32999 q^{18} +1.00000 q^{19} -10.4616 q^{20} +0.516522 q^{21} +12.2798 q^{22} -3.69856 q^{23} -1.19475 q^{24} +1.29322 q^{25} +8.55371 q^{26} -1.31886 q^{27} +9.71925 q^{28} +3.97886 q^{29} +1.38104 q^{30} +8.72405 q^{31} -1.76358 q^{32} -1.09561 q^{33} -10.5410 q^{34} -5.84666 q^{35} -12.3059 q^{36} +9.05184 q^{37} -2.48400 q^{38} -0.763170 q^{39} +13.5238 q^{40} +10.8766 q^{41} -1.28304 q^{42} +1.63320 q^{43} -20.6158 q^{44} +7.40267 q^{45} +9.18722 q^{46} -6.41415 q^{47} +1.11931 q^{48} -1.56822 q^{49} -3.21237 q^{50} +0.940474 q^{51} -14.3603 q^{52} +10.2742 q^{53} +3.27605 q^{54} +12.4015 q^{55} -12.5641 q^{56} +0.221625 q^{57} -9.88347 q^{58} -2.69761 q^{59} -2.31855 q^{60} +14.5808 q^{61} -21.6705 q^{62} -6.87738 q^{63} -5.72022 q^{64} +8.63853 q^{65} +2.72150 q^{66} -0.0512046 q^{67} +17.6966 q^{68} -0.819693 q^{69} +14.5231 q^{70} -12.5130 q^{71} +15.9079 q^{72} +0.953806 q^{73} -22.4848 q^{74} +0.286610 q^{75} +4.17025 q^{76} -11.5215 q^{77} +1.89571 q^{78} +0.539584 q^{79} -12.6698 q^{80} +8.56036 q^{81} -27.0175 q^{82} +4.69541 q^{83} +2.15403 q^{84} -10.6455 q^{85} -4.05687 q^{86} +0.881813 q^{87} +26.6502 q^{88} -11.0504 q^{89} -18.3882 q^{90} -8.02554 q^{91} -15.4239 q^{92} +1.93347 q^{93} +15.9327 q^{94} -2.50863 q^{95} -0.390853 q^{96} -17.2966 q^{97} +3.89546 q^{98} +14.5878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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