LMFDB - Embedded newform 4009.2.a.c.1.37

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			1.37
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)

$f(q)$	$=$	$q-0.167979 q^{2} -1.05293 q^{3} -1.97178 q^{4} -3.01771 q^{5} +0.176869 q^{6} -4.42982 q^{7} +0.667175 q^{8} -1.89135 q^{9} +O(q^{10})$	\(q-0.167979 q^{2} -1.05293 q^{3} -1.97178 q^{4} -3.01771 q^{5} +0.176869 q^{6} -4.42982 q^{7} +0.667175 q^{8} -1.89135 q^{9} +0.506912 q^{10} -1.01851 q^{11} +2.07614 q^{12} -5.01175 q^{13} +0.744115 q^{14} +3.17743 q^{15} +3.83150 q^{16} +6.40726 q^{17} +0.317706 q^{18} +1.00000 q^{19} +5.95028 q^{20} +4.66427 q^{21} +0.171089 q^{22} -0.181281 q^{23} -0.702486 q^{24} +4.10660 q^{25} +0.841867 q^{26} +5.15023 q^{27} +8.73464 q^{28} +9.82241 q^{29} -0.533741 q^{30} +3.12072 q^{31} -1.97796 q^{32} +1.07242 q^{33} -1.07628 q^{34} +13.3679 q^{35} +3.72933 q^{36} -9.27053 q^{37} -0.167979 q^{38} +5.27700 q^{39} -2.01334 q^{40} -12.4270 q^{41} -0.783498 q^{42} +2.76896 q^{43} +2.00829 q^{44} +5.70755 q^{45} +0.0304514 q^{46} +7.83803 q^{47} -4.03428 q^{48} +12.6233 q^{49} -0.689822 q^{50} -6.74637 q^{51} +9.88208 q^{52} +0.131565 q^{53} -0.865128 q^{54} +3.07358 q^{55} -2.95546 q^{56} -1.05293 q^{57} -1.64996 q^{58} -5.16143 q^{59} -6.26520 q^{60} +9.92246 q^{61} -0.524214 q^{62} +8.37832 q^{63} -7.33073 q^{64} +15.1240 q^{65} -0.180144 q^{66} -14.9820 q^{67} -12.6337 q^{68} +0.190876 q^{69} -2.24553 q^{70} -3.98850 q^{71} -1.26186 q^{72} +7.36540 q^{73} +1.55725 q^{74} -4.32395 q^{75} -1.97178 q^{76} +4.51183 q^{77} -0.886424 q^{78} -8.38814 q^{79} -11.5624 q^{80} +0.251233 q^{81} +2.08747 q^{82} +9.82119 q^{83} -9.19693 q^{84} -19.3353 q^{85} -0.465127 q^{86} -10.3423 q^{87} -0.679527 q^{88} +3.45717 q^{89} -0.958746 q^{90} +22.2011 q^{91} +0.357447 q^{92} -3.28588 q^{93} -1.31662 q^{94} -3.01771 q^{95} +2.08265 q^{96} -11.8827 q^{97} -2.12044 q^{98} +1.92636 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.167979	−0.118779	−0.0593894	−	0.998235i	$-0.518915\pi$
$2$	−0.167979	−0.118779	−0.0593894	+	0.998235i	$0.518915\pi$
$3$	−1.05293	−0.607907	−0.303954	−	0.952687i	$-0.598307\pi$
$3$	−1.05293	−0.607907	−0.303954	+	0.952687i	$0.598307\pi$
$4$	−1.97178	−0.985892
$5$	−3.01771	−1.34956	−0.674782	−	0.738018i	$-0.735762\pi$
$5$	−3.01771	−1.34956	−0.674782	+	0.738018i	$0.735762\pi$
$6$	0.176869	0.0722065
$7$	−4.42982	−1.67431	−0.837157	−	0.546963i	$-0.815784\pi$
$7$	−4.42982	−1.67431	−0.837157	+	0.546963i	$0.815784\pi$
$8$	0.667175	0.235882
$9$	−1.89135	−0.630449
$10$	0.506912	0.160300
$11$	−1.01851	−0.307093	−0.153547	−	0.988141i	$-0.549070\pi$
$11$	−1.01851	−0.307093	−0.153547	+	0.988141i	$0.549070\pi$
$12$	2.07614	0.599331
$13$	−5.01175	−1.39001	−0.695005	−	0.719005i	$-0.744598\pi$
$13$	−5.01175	−1.39001	−0.695005	+	0.719005i	$0.744598\pi$
$14$	0.744115	0.198873
$15$	3.17743	0.820409
$16$	3.83150	0.957874
$17$	6.40726	1.55399	0.776994	−	0.629508i	$-0.216743\pi$
$17$	6.40726	1.55399	0.776994	+	0.629508i	$0.216743\pi$
$18$	0.317706	0.0748840
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.95028	1.33052
$21$	4.66427	1.01783
$22$	0.171089	0.0364762
$23$	−0.181281	−0.0377997	−0.0188999	−	0.999821i	$-0.506016\pi$
$23$	−0.181281	−0.0377997	−0.0188999	+	0.999821i	$0.506016\pi$
$24$	−0.702486	−0.143394
$25$	4.10660	0.821321
$26$	0.841867	0.165104
$27$	5.15023	0.991162
$28$	8.73464	1.65069
$29$	9.82241	1.82398	0.911988	−	0.410216i	$-0.134547\pi$
$29$	9.82241	1.82398	0.911988	+	0.410216i	$0.134547\pi$
$30$	−0.533741	−0.0974473
$31$	3.12072	0.560497	0.280249	−	0.959927i	$-0.409583\pi$
$31$	3.12072	0.560497	0.280249	+	0.959927i	$0.409583\pi$
$32$	−1.97796	−0.349657
$33$	1.07242	0.186684
$34$	−1.07628	−0.184581
$35$	13.3679	2.25959
$36$	3.72933	0.621554
$37$	−9.27053	−1.52406	−0.762032	−	0.647539i	$-0.775798\pi$
$37$	−9.27053	−1.52406	−0.762032	+	0.647539i	$0.775798\pi$
$38$	−0.167979	−0.0272497
$39$	5.27700	0.844997
$40$	−2.01334	−0.318338
$41$	−12.4270	−1.94077	−0.970385	−	0.241565i	$-0.922339\pi$
$41$	−12.4270	−1.94077	−0.970385	+	0.241565i	$0.922339\pi$
$42$	−0.783498	−0.120896
$43$	2.76896	0.422263	0.211131	−	0.977458i	$-0.432285\pi$
$43$	2.76896	0.422263	0.211131	+	0.977458i	$0.432285\pi$
$44$	2.00829	0.302761
$45$	5.70755	0.850831
$46$	0.0304514	0.00448981
$47$	7.83803	1.14329	0.571647	−	0.820499i	$-0.306305\pi$
$47$	7.83803	1.14329	0.571647	+	0.820499i	$0.306305\pi$
$48$	−4.03428	−0.582298
$49$	12.6233	1.80333
$50$	−0.689822	−0.0975555
$51$	−6.74637	−0.944681
$52$	9.88208	1.37040
$53$	0.131565	0.0180718	0.00903592	−	0.999959i	$-0.497124\pi$
$53$	0.131565	0.0180718	0.00903592	+	0.999959i	$0.497124\pi$
$54$	−0.865128	−0.117729
$55$	3.07358	0.414442
$56$	−2.95546	−0.394940
$57$	−1.05293	−0.139463
$58$	−1.64996	−0.216650
$59$	−5.16143	−0.671961	−0.335981	−	0.941869i	$-0.609068\pi$
$59$	−5.16143	−0.671961	−0.335981	+	0.941869i	$0.609068\pi$
$60$	−6.26520	−0.808834
$61$	9.92246	1.27044	0.635220	−	0.772331i	$-0.280909\pi$
$61$	9.92246	1.27044	0.635220	+	0.772331i	$0.280909\pi$
$62$	−0.524214	−0.0665753
$63$	8.37832	1.05557
$64$	−7.33073	−0.916342
$65$	15.1240	1.87591
$66$	−0.180144	−0.0221741
$67$	−14.9820	−1.83035	−0.915173	−	0.403062i	$-0.867946\pi$
$67$	−14.9820	−1.83035	−0.915173	+	0.403062i	$0.867946\pi$
$68$	−12.6337	−1.53206
$69$	0.190876	0.0229787
$70$	−2.24553	−0.268392
$71$	−3.98850	−0.473347	−0.236674	−	0.971589i	$-0.576057\pi$
$71$	−3.98850	−0.473347	−0.236674	+	0.971589i	$0.576057\pi$
$72$	−1.26186	−0.148712
$73$	7.36540	0.862055	0.431027	−	0.902339i	$-0.358151\pi$
$73$	7.36540	0.862055	0.431027	+	0.902339i	$0.358151\pi$
$74$	1.55725	0.181027
$75$	−4.32395	−0.499287
$76$	−1.97178	−0.226179
$77$	4.51183	0.514170
$78$	−0.886424	−0.100368
$79$	−8.38814	−0.943739	−0.471870	−	0.881668i	$-0.656421\pi$
$79$	−8.38814	−0.943739	−0.471870	+	0.881668i	$0.656421\pi$
$80$	−11.5624	−1.29271
$81$	0.251233	0.0279147
$82$	2.08747	0.230522
$83$	9.82119	1.07802	0.539008	−	0.842301i	$-0.318799\pi$
$83$	9.82119	1.07802	0.539008	+	0.842301i	$0.318799\pi$
$84$	−9.19693	−1.00347
$85$	−19.3353	−2.09721
$86$	−0.465127	−0.0501559
$87$	−10.3423	−1.10881
$88$	−0.679527	−0.0724378
$89$	3.45717	0.366459	0.183229	−	0.983070i	$-0.441345\pi$
$89$	3.45717	0.366459	0.183229	+	0.983070i	$0.441345\pi$
$90$	−0.958746	−0.101061
$91$	22.2011	2.32731
$92$	0.357447	0.0372664
$93$	−3.28588	−0.340730
$94$	−1.31662	−0.135799
$95$	−3.01771	−0.309611
$96$	2.08265	0.212559
$97$	−11.8827	−1.20650	−0.603252	−	0.797551i	$-0.706129\pi$
$97$	−11.8827	−1.20650	−0.603252	+	0.797551i	$0.706129\pi$
$98$	−2.12044	−0.214197
$99$	1.92636	0.193607
$100$	−8.09733	−0.809733
$101$	−12.8344	−1.27707	−0.638537	−	0.769591i	$-0.720460\pi$
$101$	−12.8344	−1.27707	−0.638537	+	0.769591i	$0.720460\pi$
$102$	1.13325	0.112208
$103$	−12.9331	−1.27434	−0.637169	−	0.770724i	$-0.719895\pi$
$103$	−12.9331	−1.27434	−0.637169	+	0.770724i	$0.719895\pi$
$104$	−3.34371	−0.327878
$105$	−14.0754	−1.37362
$106$	−0.0221001	−0.00214655
$107$	11.8906	1.14950	0.574752	−	0.818328i	$-0.305099\pi$
$107$	11.8906	1.14950	0.574752	+	0.818328i	$0.305099\pi$
$108$	−10.1551	−0.977178
$109$	−8.42650	−0.807112	−0.403556	−	0.914955i	$-0.632226\pi$
$109$	−8.42650	−0.807112	−0.403556	+	0.914955i	$0.632226\pi$
$110$	−0.516296	−0.0492269
$111$	9.76118	0.926490
$112$	−16.9728	−1.60378
$113$	15.2425	1.43389	0.716945	−	0.697130i	$-0.245540\pi$
$113$	15.2425	1.43389	0.716945	+	0.697130i	$0.245540\pi$
$114$	0.176869	0.0165653
$115$	0.547055	0.0510131
$116$	−19.3677	−1.79824
$117$	9.47896	0.876330
$118$	0.867011	0.0798148
$119$	−28.3830	−2.60186
$120$	2.11990	0.193520
$121$	−9.96263	−0.905694
$122$	−1.66676	−0.150902
$123$	13.0847	1.17981
$124$	−6.15338	−0.552590
$125$	2.69602	0.241139
$126$	−1.40738	−0.125379
$127$	13.9355	1.23657	0.618287	−	0.785953i	$-0.287827\pi$
$127$	13.9355	1.23657	0.618287	+	0.785953i	$0.287827\pi$
$128$	5.18733	0.458499
$129$	−2.91551	−0.256696
$130$	−2.54052	−0.222818
$131$	16.2405	1.41894	0.709471	−	0.704734i	$-0.248934\pi$
$131$	16.2405	1.41894	0.709471	+	0.704734i	$0.248934\pi$
$132$	−2.11458	−0.184050
$133$	−4.42982	−0.384114
$134$	2.51666	0.217406
$135$	−15.5419	−1.33764
$136$	4.27476	0.366558
$137$	16.6411	1.42175	0.710873	−	0.703321i	$-0.248300\pi$
$137$	16.6411	1.42175	0.710873	+	0.703321i	$0.248300\pi$
$138$	−0.0320630	−0.00272939
$139$	7.55073	0.640444	0.320222	−	0.947342i	$-0.396242\pi$
$139$	7.55073	0.640444	0.320222	+	0.947342i	$0.396242\pi$
$140$	−26.3586	−2.22771
$141$	−8.25287	−0.695017
$142$	0.669982	0.0562237
$143$	5.10453	0.426863
$144$	−7.24669	−0.603890
$145$	−29.6412	−2.46157
$146$	−1.23723	−0.102394
$147$	−13.2914	−1.09625
$148$	18.2795	1.50256
$149$	19.8950	1.62987	0.814933	−	0.579556i	$-0.196774\pi$
$149$	19.8950	1.62987	0.814933	+	0.579556i	$0.196774\pi$
$150$	0.726331	0.0593047
$151$	−17.9871	−1.46377	−0.731883	−	0.681430i	$-0.761358\pi$
$151$	−17.9871	−1.46377	−0.731883	+	0.681430i	$0.761358\pi$
$152$	0.667175	0.0541150
$153$	−12.1183	−0.979710
$154$	−0.757891	−0.0610726
$155$	−9.41744	−0.756427
$156$	−10.4051	−0.833075
$157$	−5.75020	−0.458916	−0.229458	−	0.973319i	$-0.573695\pi$
$157$	−5.75020	−0.458916	−0.229458	+	0.973319i	$0.573695\pi$
$158$	1.40903	0.112096
$159$	−0.138528	−0.0109860
$160$	5.96892	0.471884
$161$	0.803042	0.0632886
$162$	−0.0422017	−0.00331568
$163$	16.4793	1.29076	0.645379	−	0.763862i	$-0.276699\pi$
$163$	16.4793	1.29076	0.645379	+	0.763862i	$0.276699\pi$
$164$	24.5033	1.91339
$165$	−3.23626	−0.251942
$166$	−1.64975	−0.128046
$167$	8.17513	0.632611	0.316305	−	0.948657i	$-0.397558\pi$
$167$	8.17513	0.632611	0.316305	+	0.948657i	$0.397558\pi$
$168$	3.11188	0.240087
$169$	12.1176	0.932126
$170$	3.24792	0.249104
$171$	−1.89135	−0.144635
$172$	−5.45979	−0.416305
$173$	−17.8784	−1.35927	−0.679636	−	0.733549i	$-0.737862\pi$
$173$	−17.8784	−1.35927	−0.679636	+	0.733549i	$0.737862\pi$
$174$	1.73728	0.131703
$175$	−18.1915	−1.37515
$176$	−3.90243	−0.294157
$177$	5.43460	0.408490
$178$	−0.580730	−0.0435276
$179$	8.22543	0.614798	0.307399	−	0.951581i	$-0.400541\pi$
$179$	8.22543	0.614798	0.307399	+	0.951581i	$0.400541\pi$
$180$	−11.2540	−0.838827
$181$	4.18910	0.311374	0.155687	−	0.987806i	$-0.450241\pi$
$181$	4.18910	0.311374	0.155687	+	0.987806i	$0.450241\pi$
$182$	−3.72932	−0.276435
$183$	−10.4476	−0.772310
$184$	−0.120946	−0.00891628
$185$	27.9758	2.05682
$186$	0.551959	0.0404716
$187$	−6.52588	−0.477219
$188$	−15.4549	−1.12716
$189$	−22.8146	−1.65952
$190$	0.506912	0.0367753
$191$	1.69592	0.122713	0.0613563	−	0.998116i	$-0.480457\pi$
$191$	1.69592	0.122713	0.0613563	+	0.998116i	$0.480457\pi$
$192$	7.71872	0.557051
$193$	21.0864	1.51783	0.758917	−	0.651188i	$-0.225729\pi$
$193$	21.0864	1.51783	0.758917	+	0.651188i	$0.225729\pi$
$194$	1.99604	0.143307
$195$	−15.9245	−1.14038
$196$	−24.8904	−1.77788
$197$	7.78008	0.554308	0.277154	−	0.960826i	$-0.410609\pi$
$197$	7.78008	0.554308	0.277154	+	0.960826i	$0.410609\pi$
$198$	−0.323588	−0.0229964
$199$	−11.3260	−0.802881	−0.401441	−	0.915885i	$-0.631490\pi$
$199$	−11.3260	−0.802881	−0.401441	+	0.915885i	$0.631490\pi$
$200$	2.73982	0.193735
$201$	15.7750	1.11268
$202$	2.15591	0.151690
$203$	−43.5115	−3.05391
$204$	13.3024	0.931353
$205$	37.5011	2.61919
$206$	2.17249	0.151364
$207$	0.342866	0.0238308
$208$	−19.2025	−1.33145
$209$	−1.01851	−0.0704520
$210$	2.36437	0.163157
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−0.259418	−0.0178169
$213$	4.19959	0.287751
$214$	−1.99736	−0.136537
$215$	−8.35594	−0.569870
$216$	3.43610	0.233797
$217$	−13.8242	−0.938448
$218$	1.41547	0.0958679
$219$	−7.75522	−0.524049
$220$	−6.06044	−0.408595
$221$	−32.1116	−2.16006
$222$	−1.63967	−0.110047
$223$	2.89828	0.194083	0.0970415	−	0.995280i	$-0.469062\pi$
$223$	2.89828	0.194083	0.0970415	+	0.995280i	$0.469062\pi$
$224$	8.76200	0.585436
$225$	−7.76701	−0.517801
$226$	−2.56041	−0.170316
$227$	10.4055	0.690634	0.345317	−	0.938486i	$-0.387771\pi$
$227$	10.4055	0.690634	0.345317	+	0.938486i	$0.387771\pi$
$228$	2.07614	0.137496
$229$	−4.95495	−0.327432	−0.163716	−	0.986507i	$-0.552348\pi$
$229$	−4.95495	−0.327432	−0.163716	+	0.986507i	$0.552348\pi$
$230$	−0.0918936	−0.00605928
$231$	−4.75062	−0.312568
$232$	6.55327	0.430243
$233$	24.5350	1.60734	0.803671	−	0.595074i	$-0.202877\pi$
$233$	24.5350	1.60734	0.803671	+	0.595074i	$0.202877\pi$
$234$	−1.59226	−0.104089
$235$	−23.6530	−1.54295
$236$	10.1772	0.662481
$237$	8.83209	0.573706
$238$	4.76774	0.309047
$239$	−7.43971	−0.481235	−0.240618	−	0.970620i	$-0.577350\pi$
$239$	−7.43971	−0.481235	−0.240618	+	0.970620i	$0.577350\pi$
$240$	12.1743	0.785848
$241$	1.04242	0.0671481	0.0335740	−	0.999436i	$-0.489311\pi$
$241$	1.04242	0.0671481	0.0335740	+	0.999436i	$0.489311\pi$
$242$	1.67351	0.107577
$243$	−15.7152	−1.00813
$244$	−19.5649	−1.25252
$245$	−38.0935	−2.43370
$246$	−2.19795	−0.140136
$247$	−5.01175	−0.318890
$248$	2.08206	0.132211
$249$	−10.3410	−0.655334
$250$	−0.452874	−0.0286423
$251$	3.98837	0.251744	0.125872	−	0.992046i	$-0.459827\pi$
$251$	3.98837	0.251744	0.125872	+	0.992046i	$0.459827\pi$
$252$	−16.5202	−1.04068
$253$	0.184637	0.0116080
$254$	−2.34086	−0.146879
$255$	20.3586	1.27491
$256$	13.7901	0.861882
$257$	−9.74064	−0.607605	−0.303802	−	0.952735i	$-0.598256\pi$
$257$	−9.74064	−0.607605	−0.303802	+	0.952735i	$0.598256\pi$
$258$	0.489744	0.0304901
$259$	41.0667	2.55176
$260$	−29.8213	−1.84944
$261$	−18.5776	−1.14992
$262$	−2.72807	−0.168540
$263$	13.2172	0.815010	0.407505	−	0.913203i	$-0.366399\pi$
$263$	13.2172	0.815010	0.407505	+	0.913203i	$0.366399\pi$
$264$	0.715491	0.0440354
$265$	−0.397026	−0.0243891
$266$	0.744115	0.0456246
$267$	−3.64014	−0.222773
$268$	29.5413	1.80452
$269$	−17.7769	−1.08388	−0.541939	−	0.840418i	$-0.682310\pi$
$269$	−17.7769	−1.08388	−0.541939	+	0.840418i	$0.682310\pi$
$270$	2.61071	0.158883
$271$	−6.55526	−0.398204	−0.199102	−	0.979979i	$-0.563802\pi$
$271$	−6.55526	−0.398204	−0.199102	+	0.979979i	$0.563802\pi$
$272$	24.5494	1.48852
$273$	−23.3762	−1.41479
$274$	−2.79535	−0.168873
$275$	−4.18263	−0.252222
$276$	−0.376365	−0.0226545
$277$	14.2136	0.854010	0.427005	−	0.904249i	$-0.359569\pi$
$277$	14.2136	0.854010	0.427005	+	0.904249i	$0.359569\pi$
$278$	−1.26836	−0.0760713
$279$	−5.90236	−0.353365
$280$	8.91875	0.532997
$281$	8.17255	0.487533	0.243767	−	0.969834i	$-0.421617\pi$
$281$	8.17255	0.487533	0.243767	+	0.969834i	$0.421617\pi$
$282$	1.38631	0.0825534
$283$	9.84337	0.585127	0.292564	−	0.956246i	$-0.405492\pi$
$283$	9.84337	0.585127	0.292564	+	0.956246i	$0.405492\pi$
$284$	7.86445	0.466669
$285$	3.17743	0.188215
$286$	−0.857453	−0.0507023
$287$	55.0493	3.24946
$288$	3.74101	0.220441
$289$	24.0530	1.41488
$290$	4.97910	0.292383
$291$	12.5116	0.733442
$292$	−14.5230	−0.849893
$293$	−8.57238	−0.500804	−0.250402	−	0.968142i	$-0.580563\pi$
$293$	−8.57238	−0.500804	−0.250402	+	0.968142i	$0.580563\pi$
$294$	2.23267	0.130212
$295$	15.5757	0.906854
$296$	−6.18506	−0.359500
$297$	−5.24557	−0.304379
$298$	−3.34194	−0.193594
$299$	0.908536	0.0525420
$300$	8.52589	0.492242
$301$	−12.2660	−0.707000
$302$	3.02144	0.173865
$303$	13.5137	0.776343
$304$	3.83150	0.219751
$305$	−29.9432	−1.71454
$306$	2.03562	0.116369
$307$	−5.00580	−0.285696	−0.142848	−	0.989745i	$-0.545626\pi$
$307$	−5.00580	−0.285696	−0.142848	+	0.989745i	$0.545626\pi$
$308$	−8.89635	−0.506916
$309$	13.6176	0.774679
$310$	1.58193	0.0898475
$311$	−11.1597	−0.632810	−0.316405	−	0.948624i	$-0.602476\pi$
$311$	−11.1597	−0.632810	−0.316405	+	0.948624i	$0.602476\pi$
$312$	3.52068	0.199319
$313$	−27.5048	−1.55466	−0.777332	−	0.629091i	$-0.783427\pi$
$313$	−27.5048	−1.55466	−0.777332	+	0.629091i	$0.783427\pi$
$314$	0.965912	0.0545095
$315$	−25.2834	−1.42456
$316$	16.5396	0.930425
$317$	−18.4002	−1.03346	−0.516730	−	0.856148i	$-0.672851\pi$
$317$	−18.4002	−1.03346	−0.516730	+	0.856148i	$0.672851\pi$
$318$	0.0232698	0.00130491
$319$	−10.0043	−0.560131
$320$	22.1221	1.23666
$321$	−12.5199	−0.698792
$322$	−0.134894	−0.00751735
$323$	6.40726	0.356509
$324$	−0.495376	−0.0275209
$325$	−20.5813	−1.14164
$326$	−2.76817	−0.153315
$327$	8.87248	0.490649
$328$	−8.29098	−0.457793
$329$	−34.7211	−1.91423
$330$	0.543622	0.0299254
$331$	−17.9463	−0.986416	−0.493208	−	0.869911i	$-0.664176\pi$
$331$	−17.9463	−0.986416	−0.493208	+	0.869911i	$0.664176\pi$
$332$	−19.3653	−1.06281
$333$	17.5338	0.960845
$334$	−1.37325	−0.0751408
$335$	45.2115	2.47017
$336$	17.8711	0.974950
$337$	−14.3991	−0.784367	−0.392183	−	0.919887i	$-0.628280\pi$
$337$	−14.3991	−0.784367	−0.392183	+	0.919887i	$0.628280\pi$
$338$	−2.03550	−0.110717
$339$	−16.0492	−0.871671
$340$	38.1250	2.06762
$341$	−3.17849	−0.172125
$342$	0.317706	0.0171796
$343$	−24.9101	−1.34502
$344$	1.84738	0.0996042
$345$	−0.576008	−0.0310112
$346$	3.00320	0.161453
$347$	−34.5534	−1.85492	−0.927461	−	0.373920i	$-0.878013\pi$
$347$	−34.5534	−1.85492	−0.927461	+	0.373920i	$0.878013\pi$
$348$	20.3927	1.09316
$349$	24.2300	1.29700	0.648502	−	0.761213i	$-0.275396\pi$
$349$	24.2300	1.29700	0.648502	+	0.761213i	$0.275396\pi$
$350$	3.05578	0.163339
$351$	−25.8116	−1.37772
$352$	2.01458	0.107377
$353$	21.1378	1.12505	0.562526	−	0.826780i	$-0.309830\pi$
$353$	21.1378	1.12505	0.562526	+	0.826780i	$0.309830\pi$
$354$	−0.912898	−0.0485200
$355$	12.0361	0.638812
$356$	−6.81678	−0.361289
$357$	29.8852	1.58169
$358$	−1.38170	−0.0730250
$359$	16.3519	0.863018	0.431509	−	0.902109i	$-0.357981\pi$
$359$	16.3519	0.863018	0.431509	+	0.902109i	$0.357981\pi$
$360$	3.80793	0.200696
$361$	1.00000	0.0526316
$362$	−0.703680	−0.0369846
$363$	10.4899	0.550578
$364$	−43.7758	−2.29448
$365$	−22.2267	−1.16340
$366$	1.75498	0.0917341
$367$	−14.0660	−0.734238	−0.367119	−	0.930174i	$-0.619656\pi$
$367$	−14.0660	−0.734238	−0.367119	+	0.930174i	$0.619656\pi$
$368$	−0.694578	−0.0362074
$369$	23.5037	1.22356
$370$	−4.69934	−0.244307
$371$	−0.582809	−0.0302579
$372$	6.47905	0.335923
$373$	30.0716	1.55705	0.778524	−	0.627614i	$-0.215969\pi$
$373$	30.0716	1.55705	0.778524	+	0.627614i	$0.215969\pi$
$374$	1.09621	0.0566836
$375$	−2.83871	−0.146590
$376$	5.22934	0.269683
$377$	−49.2275	−2.53534
$378$	3.83236	0.197115
$379$	−0.913028	−0.0468991	−0.0234495	−	0.999725i	$-0.507465\pi$
$379$	−0.913028	−0.0468991	−0.0234495	+	0.999725i	$0.507465\pi$
$380$	5.95028	0.305243
$381$	−14.6730	−0.751722
$382$	−0.284879	−0.0145757
$383$	−26.1895	−1.33822	−0.669110	−	0.743163i	$-0.733324\pi$
$383$	−26.1895	−1.33822	−0.669110	+	0.743163i	$0.733324\pi$
$384$	−5.46187	−0.278725
$385$	−13.6154	−0.693906
$386$	−3.54207	−0.180287
$387$	−5.23707	−0.266215
$388$	23.4301	1.18948
$389$	−8.40845	−0.426325	−0.213163	−	0.977017i	$-0.568376\pi$
$389$	−8.40845	−0.426325	−0.213163	+	0.977017i	$0.568376\pi$
$390$	2.67497	0.135453
$391$	−1.16152	−0.0587403
$392$	8.42194	0.425372
$393$	−17.1001	−0.862586
$394$	−1.30689	−0.0658401
$395$	25.3130	1.27364
$396$	−3.79837	−0.190875
$397$	26.7715	1.34363	0.671813	−	0.740721i	$-0.265516\pi$
$397$	26.7715	1.34363	0.671813	+	0.740721i	$0.265516\pi$
$398$	1.90253	0.0953653
$399$	4.66427	0.233506
$400$	15.7344	0.786721
$401$	−24.1567	−1.20633	−0.603163	−	0.797618i	$-0.706093\pi$
$401$	−24.1567	−1.20633	−0.603163	+	0.797618i	$0.706093\pi$
$402$	−2.64986	−0.132163
$403$	−15.6403	−0.779097
$404$	25.3067	1.25906
$405$	−0.758149	−0.0376727
$406$	7.30901	0.362740
$407$	9.44215	0.468030
$408$	−4.50101	−0.222833
$409$	3.26647	0.161517	0.0807583	−	0.996734i	$-0.474266\pi$
$409$	3.26647	0.161517	0.0807583	+	0.996734i	$0.474266\pi$
$410$	−6.29939	−0.311105
$411$	−17.5219	−0.864289
$412$	25.5013	1.25636
$413$	22.8642	1.12507
$414$	−0.0575941	−0.00283060
$415$	−29.6376	−1.45485
$416$	9.91304	0.486027
$417$	−7.95036	−0.389331
$418$	0.171089	0.00836822
$419$	−26.9394	−1.31608	−0.658038	−	0.752984i	$-0.728614\pi$
$419$	−26.9394	−1.31608	−0.658038	+	0.752984i	$0.728614\pi$
$420$	27.7537	1.35424
$421$	20.2670	0.987754	0.493877	−	0.869532i	$-0.335579\pi$
$421$	20.2670	0.987754	0.493877	+	0.869532i	$0.335579\pi$
$422$	−0.167979	−0.00817708
$423$	−14.8244	−0.720789
$424$	0.0877769	0.00426282
$425$	26.3121	1.27632
$426$	−0.705442	−0.0341788
$427$	−43.9547	−2.12712
$428$	−23.4456	−1.13329
$429$	−5.37470	−0.259493
$430$	1.40362	0.0676885
$431$	−34.5121	−1.66239	−0.831194	−	0.555982i	$-0.812342\pi$
$431$	−34.5121	−1.66239	−0.831194	+	0.555982i	$0.812342\pi$
$432$	19.7331	0.949408
$433$	12.9190	0.620846	0.310423	−	0.950598i	$-0.399529\pi$
$433$	12.9190	0.620846	0.310423	+	0.950598i	$0.399529\pi$
$434$	2.32217	0.111468
$435$	31.2100	1.49641
$436$	16.6152	0.795725
$437$	−0.181281	−0.00867185
$438$	1.30271	0.0622460
$439$	−4.55086	−0.217201	−0.108600	−	0.994085i	$-0.534637\pi$
$439$	−4.55086	−0.217201	−0.108600	+	0.994085i	$0.534637\pi$
$440$	2.05062	0.0977594
$441$	−23.8750	−1.13690
$442$	5.39406	0.256569
$443$	−28.0155	−1.33106	−0.665529	−	0.746372i	$-0.731794\pi$
$443$	−28.0155	−1.33106	−0.665529	+	0.746372i	$0.731794\pi$
$444$	−19.2469	−0.913419
$445$	−10.4327	−0.494559
$446$	−0.486849	−0.0230530
$447$	−20.9480	−0.990807
$448$	32.4738	1.53424
$449$	−2.16539	−0.102191	−0.0510954	−	0.998694i	$-0.516271\pi$
$449$	−2.16539	−0.102191	−0.0510954	+	0.998694i	$0.516271\pi$
$450$	1.30469	0.0615038
$451$	12.6571	0.595997
$452$	−30.0548	−1.41366
$453$	18.9391	0.889834
$454$	−1.74789	−0.0820328
$455$	−66.9967	−3.14085
$456$	−0.702486	−0.0328969
$457$	−33.1662	−1.55145	−0.775725	−	0.631072i	$-0.782615\pi$
$457$	−33.1662	−1.55145	−0.775725	+	0.631072i	$0.782615\pi$
$458$	0.832326	0.0388920
$459$	32.9988	1.54025
$460$	−1.07867	−0.0502934
$461$	−0.316197	−0.0147268	−0.00736338	−	0.999973i	$-0.502344\pi$
$461$	−0.316197	−0.0147268	−0.00736338	+	0.999973i	$0.502344\pi$
$462$	0.798003	0.0371265
$463$	8.43542	0.392027	0.196014	−	0.980601i	$-0.437200\pi$
$463$	8.43542	0.392027	0.196014	+	0.980601i	$0.437200\pi$
$464$	37.6345	1.74714
$465$	9.91586	0.459837
$466$	−4.12136	−0.190918
$467$	6.56939	0.303995	0.151998	−	0.988381i	$-0.451429\pi$
$467$	6.56939	0.303995	0.151998	+	0.988381i	$0.451429\pi$
$468$	−18.6904	−0.863966
$469$	66.3676	3.06457
$470$	3.97319	0.183270
$471$	6.05454	0.278978
$472$	−3.44358	−0.158504
$473$	−2.82022	−0.129674
$474$	−1.48360	−0.0681442
$475$	4.10660	0.188424
$476$	55.9651	2.56516
$477$	−0.248835	−0.0113934
$478$	1.24971	0.0571606
$479$	25.8895	1.18292	0.591461	−	0.806333i	$-0.298551\pi$
$479$	25.8895	1.18292	0.591461	+	0.806333i	$0.298551\pi$
$480$	−6.28483	−0.286862
$481$	46.4616	2.11846
$482$	−0.175104	−0.00797577
$483$	−0.845544	−0.0384736
$484$	19.6441	0.892916
$485$	35.8585	1.62825
$486$	2.63982	0.119745
$487$	−15.9873	−0.724455	−0.362228	−	0.932090i	$-0.617984\pi$
$487$	−15.9873	−0.724455	−0.362228	+	0.932090i	$0.617984\pi$
$488$	6.62002	0.299674
$489$	−17.3515	−0.784661
$490$	6.39889	0.289072
$491$	−19.8825	−0.897287	−0.448643	−	0.893711i	$-0.648093\pi$
$491$	−19.8825	−0.897287	−0.448643	+	0.893711i	$0.648093\pi$
$492$	−25.8002	−1.16316
$493$	62.9348	2.83444
$494$	0.841867	0.0378774
$495$	−5.81321	−0.261284
$496$	11.9570	0.536886
$497$	17.6683	0.792532
$498$	1.73707	0.0778398
$499$	−10.4861	−0.469424	−0.234712	−	0.972065i	$-0.575415\pi$
$499$	−10.4861	−0.469424	−0.234712	+	0.972065i	$0.575415\pi$
$500$	−5.31596	−0.237737
$501$	−8.60781	−0.384568
$502$	−0.669962	−0.0299019
$503$	35.5745	1.58619	0.793095	−	0.609099i	$-0.208469\pi$
$503$	35.5745	1.58619	0.793095	+	0.609099i	$0.208469\pi$
$504$	5.58981	0.248990
$505$	38.7307	1.72349
$506$	−0.0310151	−0.00137879
$507$	−12.7590	−0.566646
$508$	−27.4777	−1.21913
$509$	29.8193	1.32172	0.660859	−	0.750510i	$-0.270192\pi$
$509$	29.8193	1.32172	0.660859	+	0.750510i	$0.270192\pi$
$510$	−3.41982	−0.151432
$511$	−32.6274	−1.44335
$512$	−12.6911	−0.560873
$513$	5.15023	0.227388
$514$	1.63622	0.0721706
$515$	39.0285	1.71980
$516$	5.74876	0.253075
$517$	−7.98314	−0.351098
$518$	−6.89834	−0.303096
$519$	18.8247	0.826311
$520$	10.0904	0.442492
$521$	6.30191	0.276092	0.138046	−	0.990426i	$-0.455918\pi$
$521$	6.30191	0.276092	0.138046	+	0.990426i	$0.455918\pi$
$522$	3.12064	0.136587
$523$	5.39521	0.235916	0.117958	−	0.993019i	$-0.462365\pi$
$523$	5.39521	0.235916	0.117958	+	0.993019i	$0.462365\pi$
$524$	−32.0228	−1.39892
$525$	19.1543	0.835962
$526$	−2.22022	−0.0968060
$527$	19.9952	0.871007
$528$	4.10897	0.178820
$529$	−22.9671	−0.998571
$530$	0.0666919	0.00289691
$531$	9.76205	0.423637
$532$	8.73464	0.378695
$533$	62.2810	2.69769
$534$	0.611466	0.0264607
$535$	−35.8823	−1.55133
$536$	−9.99563	−0.431746
$537$	−8.66077	−0.373740
$538$	2.98615	0.128742
$539$	−12.8570	−0.553789
$540$	30.6453	1.31876
$541$	−2.73667	−0.117658	−0.0588292	−	0.998268i	$-0.518737\pi$
$541$	−2.73667	−0.117658	−0.0588292	+	0.998268i	$0.518737\pi$
$542$	1.10114	0.0472982
$543$	−4.41082	−0.189286
$544$	−12.6733	−0.543363
$545$	25.4288	1.08925
$546$	3.92670	0.168047
$547$	7.11290	0.304126	0.152063	−	0.988371i	$-0.451408\pi$
$547$	7.11290	0.304126	0.152063	+	0.988371i	$0.451408\pi$
$548$	−32.8127	−1.40169
$549$	−18.7668	−0.800948
$550$	0.702593	0.0299587
$551$	9.82241	0.418449
$552$	0.127347	0.00542027
$553$	37.1579	1.58012
$554$	−2.38757	−0.101438
$555$	−29.4564	−1.25036
$556$	−14.8884	−0.631409
$557$	−3.99784	−0.169394	−0.0846969	−	0.996407i	$-0.526992\pi$
$557$	−3.99784	−0.169394	−0.0846969	+	0.996407i	$0.526992\pi$
$558$	0.991471	0.0419723
$559$	−13.8773	−0.586949
$560$	51.2191	2.16440
$561$	6.87127	0.290105
$562$	−1.37281	−0.0579087
$563$	7.66498	0.323040	0.161520	−	0.986869i	$-0.448360\pi$
$563$	7.66498	0.323040	0.161520	+	0.986869i	$0.448360\pi$
$564$	16.2729	0.685212
$565$	−45.9974	−1.93512
$566$	−1.65348	−0.0695008
$567$	−1.11291	−0.0467380
$568$	−2.66102	−0.111654
$569$	10.4772	0.439227	0.219614	−	0.975587i	$-0.429520\pi$
$569$	10.4772	0.439227	0.219614	+	0.975587i	$0.429520\pi$
$570$	−0.533741	−0.0223559
$571$	0.146671	0.00613798	0.00306899	−	0.999995i	$-0.499023\pi$
$571$	0.146671	0.00613798	0.00306899	+	0.999995i	$0.499023\pi$
$572$	−10.0650	−0.420840
$573$	−1.78568	−0.0745979
$574$	−9.24711	−0.385967
$575$	−0.744450	−0.0310457
$576$	13.8650	0.577707
$577$	−19.3727	−0.806494	−0.403247	−	0.915091i	$-0.632119\pi$
$577$	−19.3727	−0.806494	−0.403247	+	0.915091i	$0.632119\pi$
$578$	−4.04039	−0.168058
$579$	−22.2024	−0.922702
$580$	58.4461	2.42684
$581$	−43.5061	−1.80494
$582$	−2.10168	−0.0871174
$583$	−0.134001	−0.00554974
$584$	4.91401	0.203343
$585$	−28.6048	−1.18266
$586$	1.43998	0.0594849
$587$	−6.87498	−0.283761	−0.141880	−	0.989884i	$-0.545315\pi$
$587$	−6.87498	−0.283761	−0.141880	+	0.989884i	$0.545315\pi$
$588$	26.2077	1.08079
$589$	3.12072	0.128587
$590$	−2.61639	−0.107715
$591$	−8.19185	−0.336968
$592$	−35.5200	−1.45986
$593$	5.05982	0.207782	0.103891	−	0.994589i	$-0.466871\pi$
$593$	5.05982	0.207782	0.103891	+	0.994589i	$0.466871\pi$
$594$	0.881145	0.0361538
$595$	85.6518	3.51138
$596$	−39.2287	−1.60687
$597$	11.9255	0.488077
$598$	−0.152615	−0.00624088
$599$	−39.1426	−1.59932	−0.799662	−	0.600451i	$-0.794988\pi$
$599$	−39.1426	−1.59932	−0.799662	+	0.600451i	$0.794988\pi$
$600$	−2.88483	−0.117773
$601$	13.0118	0.530761	0.265381	−	0.964144i	$-0.414502\pi$
$601$	13.0118	0.530761	0.265381	+	0.964144i	$0.414502\pi$
$602$	2.06043	0.0839767
$603$	28.3362	1.15394
$604$	35.4666	1.44312
$605$	30.0644	1.22229
$606$	−2.27002	−0.0922131
$607$	−11.3929	−0.462424	−0.231212	−	0.972903i	$-0.574269\pi$
$607$	−11.3929	−0.462424	−0.231212	+	0.972903i	$0.574269\pi$
$608$	−1.97796	−0.0802169
$609$	45.8144	1.85649
$610$	5.02981	0.203651
$611$	−39.2823	−1.58919
$612$	23.8948	0.965888
$613$	−13.1728	−0.532044	−0.266022	−	0.963967i	$-0.585709\pi$
$613$	−13.1728	−0.532044	−0.266022	+	0.963967i	$0.585709\pi$
$614$	0.840869	0.0339347
$615$	−39.4859	−1.59222
$616$	3.01018	0.121284
$617$	−23.3233	−0.938960	−0.469480	−	0.882943i	$-0.655559\pi$
$617$	−23.3233	−0.938960	−0.469480	+	0.882943i	$0.655559\pi$
$618$	−2.28747	−0.0920155
$619$	35.4905	1.42648	0.713241	−	0.700919i	$-0.247226\pi$
$619$	35.4905	1.42648	0.713241	+	0.700919i	$0.247226\pi$
$620$	18.5691	0.745755
$621$	−0.933639	−0.0374656
$622$	1.87460	0.0751645
$623$	−15.3146	−0.613567
$624$	20.2188	0.809400
$625$	−28.6688	−1.14675
$626$	4.62022	0.184661
$627$	1.07242	0.0428283
$628$	11.3382	0.452441
$629$	−59.3987	−2.36838
$630$	4.24707	0.169207
$631$	−23.0187	−0.916359	−0.458179	−	0.888860i	$-0.651498\pi$
$631$	−23.0187	−0.916359	−0.458179	+	0.888860i	$0.651498\pi$
$632$	−5.59636	−0.222611
$633$	−1.05293	−0.0418501
$634$	3.09085	0.122753
$635$	−42.0533	−1.66883
$636$	0.273148	0.0108310
$637$	−63.2647	−2.50664
$638$	1.68050	0.0665317
$639$	7.54363	0.298421
$640$	−15.6539	−0.618774
$641$	−0.215143	−0.00849766	−0.00424883	−	0.999991i	$-0.501352\pi$
$641$	−0.215143	−0.00849766	−0.00424883	+	0.999991i	$0.501352\pi$
$642$	2.10307	0.0830017
$643$	30.4303	1.20006	0.600028	−	0.799979i	$-0.295156\pi$
$643$	30.4303	1.20006	0.600028	+	0.799979i	$0.295156\pi$
$644$	−1.58343	−0.0623957
$645$	8.79818	0.346428
$646$	−1.07628	−0.0423458
$647$	17.0109	0.668769	0.334384	−	0.942437i	$-0.391472\pi$
$647$	17.0109	0.668769	0.334384	+	0.942437i	$0.391472\pi$
$648$	0.167616	0.00658459
$649$	5.25699	0.206355
$650$	3.45721	0.135603
$651$	14.5559	0.570489
$652$	−32.4936	−1.27255
$653$	23.5709	0.922401	0.461200	−	0.887296i	$-0.347419\pi$
$653$	23.5709	0.922401	0.461200	+	0.887296i	$0.347419\pi$
$654$	−1.49039	−0.0582788
$655$	−49.0093	−1.91495
$656$	−47.6139	−1.85901
$657$	−13.9305	−0.543482
$658$	5.83240	0.227371
$659$	16.3244	0.635910	0.317955	−	0.948106i	$-0.397004\pi$
$659$	16.3244	0.635910	0.317955	+	0.948106i	$0.397004\pi$
$660$	6.38119	0.248388
$661$	−9.04158	−0.351677	−0.175838	−	0.984419i	$-0.556264\pi$
$661$	−9.04158	−0.351677	−0.175838	+	0.984419i	$0.556264\pi$
$662$	3.01459	0.117165
$663$	33.8111	1.31311
$664$	6.55245	0.254285
$665$	13.3679	0.518386
$666$	−2.94530	−0.114128
$667$	−1.78062	−0.0689458
$668$	−16.1196	−0.623685
$669$	−3.05167	−0.117984
$670$	−7.59457	−0.293404
$671$	−10.1062	−0.390144
$672$	−9.22574	−0.355891
$673$	11.5145	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$673$	11.5145	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$674$	2.41874	0.0931662
$675$	21.1499	0.814061
$676$	−23.8933	−0.918975
$677$	15.0573	0.578698	0.289349	−	0.957224i	$-0.406561\pi$
$677$	15.0573	0.578698	0.289349	+	0.957224i	$0.406561\pi$
$678$	2.69592	0.103536
$679$	52.6381	2.02006
$680$	−12.9000	−0.494693
$681$	−10.9562	−0.419841
$682$	0.533919	0.0204448
$683$	2.35071	0.0899474	0.0449737	−	0.998988i	$-0.485680\pi$
$683$	2.35071	0.0899474	0.0449737	+	0.998988i	$0.485680\pi$
$684$	3.72933	0.142594
$685$	−50.2181	−1.91874
$686$	4.18437	0.159760
$687$	5.21720	0.199048
$688$	10.6093	0.404474
$689$	−0.659371	−0.0251200
$690$	0.0967571	0.00368348
$691$	−27.6164	−1.05058	−0.525288	−	0.850924i	$-0.676043\pi$
$691$	−27.6164	−1.05058	−0.525288	+	0.850924i	$0.676043\pi$
$692$	35.2524	1.34009
$693$	−8.53343	−0.324158
$694$	5.80423	0.220326
$695$	−22.7859	−0.864320
$696$	−6.90011	−0.261548
$697$	−79.6229	−3.01593
$698$	−4.07013	−0.154057
$699$	−25.8336	−0.977115
$700$	35.8697	1.35575
$701$	−2.96557	−0.112008	−0.0560041	−	0.998431i	$-0.517836\pi$
$701$	−2.96557	−0.112008	−0.0560041	+	0.998431i	$0.517836\pi$
$702$	4.33581	0.163645
$703$	−9.27053	−0.349644
$704$	7.46645	0.281402
$705$	24.9048	0.937969
$706$	−3.55070	−0.133632
$707$	56.8542	2.13822
$708$	−10.7159	−0.402727
$709$	−10.2928	−0.386554	−0.193277	−	0.981144i	$-0.561912\pi$
$709$	−10.2928	−0.386554	−0.193277	+	0.981144i	$0.561912\pi$
$710$	−2.02182	−0.0758774
$711$	15.8649	0.594980
$712$	2.30654	0.0864411
$713$	−0.565727	−0.0211867
$714$	−5.02008	−0.187872
$715$	−15.4040	−0.576078
$716$	−16.2188	−0.606124
$717$	7.83347	0.292546
$718$	−2.74676	−0.102508
$719$	−32.3729	−1.20731	−0.603653	−	0.797247i	$-0.706289\pi$
$719$	−32.3729	−1.20731	−0.603653	+	0.797247i	$0.706289\pi$
$720$	21.8684	0.814988
$721$	57.2913	2.13364
$722$	−0.167979	−0.00625152
$723$	−1.09759	−0.0408198
$724$	−8.26000	−0.306981
$725$	40.3368	1.49807
$726$	−1.76208	−0.0653970
$727$	18.9869	0.704184	0.352092	−	0.935965i	$-0.385470\pi$
$727$	18.9869	0.704184	0.352092	+	0.935965i	$0.385470\pi$
$728$	14.8120	0.548971
$729$	15.7933	0.584935
$730$	3.73361	0.138187
$731$	17.7415	0.656191
$732$	20.6004	0.761414
$733$	34.2516	1.26511	0.632556	−	0.774514i	$-0.282006\pi$
$733$	34.2516	1.26511	0.632556	+	0.774514i	$0.282006\pi$
$734$	2.36278	0.0872119
$735$	40.1096	1.47946
$736$	0.358567	0.0132169
$737$	15.2594	0.562087
$738$	−3.94813	−0.145333
$739$	15.2461	0.560838	0.280419	−	0.959878i	$-0.409527\pi$
$739$	15.2461	0.560838	0.280419	+	0.959878i	$0.409527\pi$
$740$	−55.1622	−2.02780
$741$	5.27700	0.193856
$742$	0.0978995	0.00359400
$743$	−35.7770	−1.31253	−0.656265	−	0.754531i	$-0.727864\pi$
$743$	−35.7770	−1.31253	−0.656265	+	0.754531i	$0.727864\pi$
$744$	−2.19226	−0.0803722
$745$	−60.0376	−2.19961
$746$	−5.05139	−0.184945
$747$	−18.5753	−0.679634
$748$	12.8676	0.470487
$749$	−52.6730	−1.92463
$750$	0.476843	0.0174118
$751$	16.7121	0.609832	0.304916	−	0.952379i	$-0.401372\pi$
$751$	16.7121	0.609832	0.304916	+	0.952379i	$0.401372\pi$
$752$	30.0314	1.09513
$753$	−4.19946	−0.153037
$754$	8.26917	0.301145
$755$	54.2798	1.97545
$756$	44.9854	1.63610
$757$	−29.5783	−1.07504	−0.537521	−	0.843250i	$-0.680639\pi$
$757$	−29.5783	−1.07504	−0.537521	+	0.843250i	$0.680639\pi$
$758$	0.153369	0.00557062
$759$	−0.194409	−0.00705661
$760$	−2.01334	−0.0730317
$761$	−11.1490	−0.404152	−0.202076	−	0.979370i	$-0.564769\pi$
$761$	−11.1490	−0.404152	−0.202076	+	0.979370i	$0.564769\pi$
$762$	2.46476	0.0892887
$763$	37.3279	1.35136
$764$	−3.34399	−0.120981
$765$	36.5697	1.32218
$766$	4.39928	0.158952
$767$	25.8678	0.934032
$768$	−14.5200	−0.523944
$769$	−49.3210	−1.77856	−0.889281	−	0.457362i	$-0.848795\pi$
$769$	−49.3210	−1.77856	−0.889281	+	0.457362i	$0.848795\pi$
$770$	2.28710	0.0824213
$771$	10.2562	0.369367
$772$	−41.5778	−1.49642
$773$	−48.0354	−1.72771	−0.863856	−	0.503738i	$-0.831958\pi$
$773$	−48.0354	−1.72771	−0.863856	+	0.503738i	$0.831958\pi$
$774$	0.879716	0.0316207
$775$	12.8155	0.460348
$776$	−7.92783	−0.284592
$777$	−43.2402	−1.55123
$778$	1.41244	0.0506385
$779$	−12.4270	−0.445243
$780$	31.3996	1.12429
$781$	4.06234	0.145362
$782$	0.195110	0.00697711
$783$	50.5877	1.80786
$784$	48.3660	1.72736
$785$	17.3525	0.619336
$786$	2.87245	0.102457
$787$	30.7118	1.09476	0.547379	−	0.836885i	$-0.315626\pi$
$787$	30.7118	1.09476	0.547379	+	0.836885i	$0.315626\pi$
$788$	−15.3406	−0.546488
$789$	−13.9168	−0.495451
$790$	−4.25205	−0.151281
$791$	−67.5213	−2.40078
$792$	1.28522	0.0456683
$793$	−49.7289	−1.76592
$794$	−4.49705	−0.159594
$795$	0.418039	0.0148263
$796$	22.3325	0.791554
$797$	19.2699	0.682576	0.341288	−	0.939959i	$-0.389137\pi$
$797$	19.2699	0.682576	0.341288	+	0.939959i	$0.389137\pi$
$798$	−0.783498	−0.0277355
$799$	50.2203	1.77667
$800$	−8.12269	−0.287181
$801$	−6.53870	−0.231034
$802$	4.05781	0.143286
$803$	−7.50176	−0.264731
$804$	−31.1048	−1.09698
$805$	−2.42335	−0.0854120
$806$	2.62723	0.0925402
$807$	18.7178	0.658897
$808$	−8.56282	−0.301239
$809$	11.5383	0.405665	0.202833	−	0.979213i	$-0.434985\pi$
$809$	11.5383	0.405665	0.202833	+	0.979213i	$0.434985\pi$
$810$	0.127353	0.00447472
$811$	13.6147	0.478075	0.239038	−	0.971010i	$-0.423168\pi$
$811$	13.6147	0.478075	0.239038	+	0.971010i	$0.423168\pi$
$812$	85.7952	3.01082
$813$	6.90220	0.242071
$814$	−1.58608	−0.0555921
$815$	−49.7298	−1.74196
$816$	−25.8487	−0.904885
$817$	2.76896	0.0968737
$818$	−0.548697	−0.0191848
$819$	−41.9900	−1.46725
$820$	−73.9440	−2.58224
$821$	−5.60034	−0.195453	−0.0977266	−	0.995213i	$-0.531157\pi$
$821$	−5.60034	−0.195453	−0.0977266	+	0.995213i	$0.531157\pi$
$822$	2.94330	0.102659
$823$	3.00934	0.104899	0.0524495	−	0.998624i	$-0.483297\pi$
$823$	3.00934	0.104899	0.0524495	+	0.998624i	$0.483297\pi$
$824$	−8.62865	−0.300593
$825$	4.40400	0.153328
$826$	−3.84070	−0.133635
$827$	3.57139	0.124189	0.0620946	−	0.998070i	$-0.480222\pi$
$827$	3.57139	0.124189	0.0620946	+	0.998070i	$0.480222\pi$
$828$	−0.676056	−0.0234946
$829$	24.3381	0.845298	0.422649	−	0.906293i	$-0.361100\pi$
$829$	24.3381	0.845298	0.422649	+	0.906293i	$0.361100\pi$
$830$	4.97848	0.172806
$831$	−14.9658	−0.519159
$832$	36.7398	1.27372
$833$	80.8806	2.80235
$834$	1.33549	0.0462443
$835$	−24.6702	−0.853748
$836$	2.00829	0.0694581
$837$	16.0724	0.555543
$838$	4.52525	0.156322
$839$	−32.6591	−1.12752	−0.563758	−	0.825940i	$-0.690645\pi$
$839$	−32.6591	−1.12752	−0.563758	+	0.825940i	$0.690645\pi$
$840$	−9.39078	−0.324013
$841$	67.4798	2.32689
$842$	−3.40443	−0.117324
$843$	−8.60509	−0.296375
$844$	−1.97178	−0.0678716
$845$	−36.5676	−1.25796
$846$	2.49019	0.0856145
$847$	44.1326	1.51642
$848$	0.504091	0.0173105
$849$	−10.3643	−0.355703
$850$	−4.41987	−0.151600
$851$	1.68057	0.0576092
$852$	−8.28068	−0.283691
$853$	38.3163	1.31193	0.655963	−	0.754793i	$-0.272263\pi$
$853$	38.3163	1.31193	0.655963	+	0.754793i	$0.272263\pi$
$854$	7.38345	0.252657
$855$	5.70755	0.195194
$856$	7.93309	0.271147
$857$	−15.0085	−0.512679	−0.256339	−	0.966587i	$-0.582517\pi$
$857$	−15.0085	−0.512679	−0.256339	+	0.966587i	$0.582517\pi$
$858$	0.902835	0.0308223
$859$	40.4096	1.37876	0.689379	−	0.724401i	$-0.257884\pi$
$859$	40.4096	1.37876	0.689379	+	0.724401i	$0.257884\pi$
$860$	16.4761	0.561830
$861$	−57.9628	−1.97537
$862$	5.79730	0.197457
$863$	−20.8831	−0.710869	−0.355434	−	0.934701i	$-0.615667\pi$
$863$	−20.8831	−0.710869	−0.355434	+	0.934701i	$0.615667\pi$
$864$	−10.1869	−0.346567
$865$	53.9520	1.83442
$866$	−2.17011	−0.0737434
$867$	−25.3260	−0.860116
$868$	27.2583	0.925208
$869$	8.54343	0.289816
$870$	−5.24262	−0.177742
$871$	75.0862	2.54420
$872$	−5.62195	−0.190383
$873$	22.4743	0.760639
$874$	0.0304514	0.00103003
$875$	−11.9429	−0.403743
$876$	15.2916	0.516656
$877$	−33.9682	−1.14702	−0.573512	−	0.819197i	$-0.694419\pi$
$877$	−33.9682	−1.14702	−0.573512	+	0.819197i	$0.694419\pi$
$878$	0.764448	0.0257989
$879$	9.02608	0.304442
$880$	11.7764	0.396983
$881$	53.9913	1.81901	0.909506	−	0.415691i	$-0.136460\pi$
$881$	53.9913	1.81901	0.909506	+	0.415691i	$0.136460\pi$
$882$	4.01049	0.135040
$883$	−34.3357	−1.15549	−0.577744	−	0.816218i	$-0.696067\pi$
$883$	−34.3357	−1.15549	−0.577744	+	0.816218i	$0.696067\pi$
$884$	63.3171	2.12958
$885$	−16.4001	−0.551283
$886$	4.70602	0.158102
$887$	−22.0253	−0.739536	−0.369768	−	0.929124i	$-0.620563\pi$
$887$	−22.0253	−0.739536	−0.369768	+	0.929124i	$0.620563\pi$
$888$	6.51241	0.218542
$889$	−61.7316	−2.07041
$890$	1.75248	0.0587432
$891$	−0.255884	−0.00857243
$892$	−5.71477	−0.191345
$893$	7.83803	0.262290
$894$	3.51882	0.117687
$895$	−24.8220	−0.829708
$896$	−22.9789	−0.767672
$897$	−0.956621	−0.0319406
$898$	0.363739	0.0121381
$899$	30.6530	1.02233
$900$	15.3149	0.510495
$901$	0.842971	0.0280834
$902$	−2.12612	−0.0707919
$903$	12.9152	0.429790
$904$	10.1694	0.338229
$905$	−12.6415	−0.420218
$906$	−3.18136	−0.105694
$907$	1.47295	0.0489086	0.0244543	−	0.999701i	$-0.492215\pi$
$907$	1.47295	0.0489086	0.0244543	+	0.999701i	$0.492215\pi$
$908$	−20.5173	−0.680890
$909$	24.2744	0.805130
$910$	11.2540	0.373067
$911$	17.2951	0.573012	0.286506	−	0.958078i	$-0.407506\pi$
$911$	17.2951	0.573012	0.286506	+	0.958078i	$0.407506\pi$
$912$	−4.03428	−0.133588
$913$	−10.0030	−0.331051
$914$	5.57122	0.184279
$915$	31.5279	1.04228
$916$	9.77009	0.322813
$917$	−71.9426	−2.37576
$918$	−5.54310	−0.182950
$919$	−13.6752	−0.451102	−0.225551	−	0.974231i	$-0.572418\pi$
$919$	−13.6752	−0.451102	−0.225551	+	0.974231i	$0.572418\pi$
$920$	0.364981	0.0120331
$921$	5.27074	0.173677
$922$	0.0531143	0.00174923
$923$	19.9893	0.657957
$924$	9.36719	0.308158
$925$	−38.0704	−1.25175
$926$	−1.41697	−0.0465645
$927$	24.4610	0.803405
$928$	−19.4283	−0.637767
$929$	47.5213	1.55912	0.779562	−	0.626325i	$-0.215442\pi$
$929$	47.5213	1.55912	0.779562	+	0.626325i	$0.215442\pi$
$930$	−1.66565	−0.0546189
$931$	12.6233	0.413711
$932$	−48.3777	−1.58467
$933$	11.7504	0.384690
$934$	−1.10352	−0.0361082
$935$	19.6932	0.644038
$936$	6.32412	0.206710
$937$	49.7683	1.62586	0.812929	−	0.582363i	$-0.197872\pi$
$937$	49.7683	1.62586	0.812929	+	0.582363i	$0.197872\pi$
$938$	−11.1484	−0.364007
$939$	28.9605	0.945091
$940$	46.6385	1.52118
$941$	−46.9795	−1.53149	−0.765744	−	0.643146i	$-0.777629\pi$
$941$	−46.9795	−1.53149	−0.765744	+	0.643146i	$0.777629\pi$
$942$	−1.01703	−0.0331367
$943$	2.25278	0.0733606
$944$	−19.7760	−0.643654
$945$	68.8478	2.23962
$946$	0.473738	0.0154025
$947$	−16.9254	−0.550002	−0.275001	−	0.961444i	$-0.588678\pi$
$947$	−16.9254	−0.550002	−0.275001	+	0.961444i	$0.588678\pi$
$948$	−17.4150	−0.565612
$949$	−36.9135	−1.19826
$950$	−0.689822	−0.0223808
$951$	19.3741	0.628248
$952$	−18.9364	−0.613733
$953$	32.5553	1.05457	0.527286	−	0.849688i	$-0.323210\pi$
$953$	32.5553	1.05457	0.527286	+	0.849688i	$0.323210\pi$
$954$	0.0417990	0.00135329
$955$	−5.11781	−0.165608
$956$	14.6695	0.474446
$957$	10.5337	0.340508
$958$	−4.34889	−0.140506
$959$	−73.7171	−2.38045
$960$	−23.2929	−0.751775
$961$	−21.2611	−0.685843
$962$	−7.80455	−0.251629
$963$	−22.4892	−0.724704
$964$	−2.05542	−0.0662007
$965$	−63.6328	−2.04841
$966$	0.142033	0.00456985
$967$	10.8169	0.347848	0.173924	−	0.984759i	$-0.444355\pi$
$967$	10.8169	0.347848	0.173924	+	0.984759i	$0.444355\pi$
$968$	−6.64682	−0.213637
$969$	−6.74637	−0.216725
$970$	−6.02347	−0.193402
$971$	−30.0910	−0.965665	−0.482832	−	0.875713i	$-0.660392\pi$
$971$	−30.0910	−0.965665	−0.482832	+	0.875713i	$0.660392\pi$
$972$	30.9870	0.993908
$973$	−33.4483	−1.07230
$974$	2.68553	0.0860500
$975$	21.6705	0.694013
$976$	38.0179	1.21692
$977$	44.5385	1.42491	0.712457	−	0.701716i	$-0.247582\pi$
$977$	44.5385	1.42491	0.712457	+	0.701716i	$0.247582\pi$
$978$	2.91468	0.0932012
$979$	−3.52117	−0.112537
$980$	75.1120	2.39937
$981$	15.9374	0.508843
$982$	3.33984	0.106579
$983$	−21.1972	−0.676087	−0.338043	−	0.941131i	$-0.609765\pi$
$983$	−21.1972	−0.676087	−0.338043	+	0.941131i	$0.609765\pi$
$984$	8.72979	0.278295
$985$	−23.4781	−0.748074
$986$	−10.5717	−0.336671
$987$	36.5587	1.16368
$988$	9.88208	0.314391
$989$	−0.501960	−0.0159614
$990$	0.976496	0.0310351
$991$	−33.2198	−1.05526	−0.527631	−	0.849473i	$-0.676920\pi$
$991$	−33.2198	−1.05526	−0.527631	+	0.849473i	$0.676920\pi$
$992$	−6.17265	−0.195982
$993$	18.8961	0.599649
$994$	−2.96790	−0.0941361
$995$	34.1787	1.08354
$996$	20.3902	0.646088
$997$	27.0647	0.857148	0.428574	−	0.903507i	$-0.359016\pi$
$997$	27.0647	0.857148	0.428574	+	0.903507i	$0.359016\pi$
$998$	1.76145	0.0557577
$999$	−47.7453	−1.51059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.37	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.37	✓	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)

\(f(q)\)	\(=\)	\(q-0.167979 q^{2} -1.05293 q^{3} -1.97178 q^{4} -3.01771 q^{5} +0.176869 q^{6} -4.42982 q^{7} +0.667175 q^{8} -1.89135 q^{9} +O(q^{10})\)	\(q-0.167979 q^{2} -1.05293 q^{3} -1.97178 q^{4} -3.01771 q^{5} +0.176869 q^{6} -4.42982 q^{7} +0.667175 q^{8} -1.89135 q^{9} +0.506912 q^{10} -1.01851 q^{11} +2.07614 q^{12} -5.01175 q^{13} +0.744115 q^{14} +3.17743 q^{15} +3.83150 q^{16} +6.40726 q^{17} +0.317706 q^{18} +1.00000 q^{19} +5.95028 q^{20} +4.66427 q^{21} +0.171089 q^{22} -0.181281 q^{23} -0.702486 q^{24} +4.10660 q^{25} +0.841867 q^{26} +5.15023 q^{27} +8.73464 q^{28} +9.82241 q^{29} -0.533741 q^{30} +3.12072 q^{31} -1.97796 q^{32} +1.07242 q^{33} -1.07628 q^{34} +13.3679 q^{35} +3.72933 q^{36} -9.27053 q^{37} -0.167979 q^{38} +5.27700 q^{39} -2.01334 q^{40} -12.4270 q^{41} -0.783498 q^{42} +2.76896 q^{43} +2.00829 q^{44} +5.70755 q^{45} +0.0304514 q^{46} +7.83803 q^{47} -4.03428 q^{48} +12.6233 q^{49} -0.689822 q^{50} -6.74637 q^{51} +9.88208 q^{52} +0.131565 q^{53} -0.865128 q^{54} +3.07358 q^{55} -2.95546 q^{56} -1.05293 q^{57} -1.64996 q^{58} -5.16143 q^{59} -6.26520 q^{60} +9.92246 q^{61} -0.524214 q^{62} +8.37832 q^{63} -7.33073 q^{64} +15.1240 q^{65} -0.180144 q^{66} -14.9820 q^{67} -12.6337 q^{68} +0.190876 q^{69} -2.24553 q^{70} -3.98850 q^{71} -1.26186 q^{72} +7.36540 q^{73} +1.55725 q^{74} -4.32395 q^{75} -1.97178 q^{76} +4.51183 q^{77} -0.886424 q^{78} -8.38814 q^{79} -11.5624 q^{80} +0.251233 q^{81} +2.08747 q^{82} +9.82119 q^{83} -9.19693 q^{84} -19.3353 q^{85} -0.465127 q^{86} -10.3423 q^{87} -0.679527 q^{88} +3.45717 q^{89} -0.958746 q^{90} +22.2011 q^{91} +0.357447 q^{92} -3.28588 q^{93} -1.31662 q^{94} -3.01771 q^{95} +2.08265 q^{96} -11.8827 q^{97} -2.12044 q^{98} +1.92636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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