LMFDB - Embedded newform 4009.2.a.d.1.2

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:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.73305 q^{2} +0.112405 q^{3} +5.46956 q^{4} +0.682101 q^{5} -0.307207 q^{6} -0.658651 q^{7} -9.48248 q^{8} -2.98737 q^{9} +O(q^{10})$	\(q-2.73305 q^{2} +0.112405 q^{3} +5.46956 q^{4} +0.682101 q^{5} -0.307207 q^{6} -0.658651 q^{7} -9.48248 q^{8} -2.98737 q^{9} -1.86422 q^{10} +1.29791 q^{11} +0.614803 q^{12} +1.76666 q^{13} +1.80013 q^{14} +0.0766712 q^{15} +14.9770 q^{16} -2.89581 q^{17} +8.16462 q^{18} -1.00000 q^{19} +3.73079 q^{20} -0.0740353 q^{21} -3.54724 q^{22} +1.86760 q^{23} -1.06587 q^{24} -4.53474 q^{25} -4.82837 q^{26} -0.673007 q^{27} -3.60253 q^{28} +1.28729 q^{29} -0.209546 q^{30} -3.74094 q^{31} -21.9678 q^{32} +0.145890 q^{33} +7.91439 q^{34} -0.449267 q^{35} -16.3396 q^{36} +11.2199 q^{37} +2.73305 q^{38} +0.198581 q^{39} -6.46801 q^{40} -2.93873 q^{41} +0.202342 q^{42} +11.6048 q^{43} +7.09897 q^{44} -2.03768 q^{45} -5.10424 q^{46} +0.467044 q^{47} +1.68348 q^{48} -6.56618 q^{49} +12.3937 q^{50} -0.325502 q^{51} +9.66286 q^{52} -0.146296 q^{53} +1.83936 q^{54} +0.885302 q^{55} +6.24564 q^{56} -0.112405 q^{57} -3.51824 q^{58} +11.1435 q^{59} +0.419358 q^{60} +8.10366 q^{61} +10.2242 q^{62} +1.96763 q^{63} +30.0852 q^{64} +1.20504 q^{65} -0.398726 q^{66} -0.316953 q^{67} -15.8388 q^{68} +0.209926 q^{69} +1.22787 q^{70} -4.15162 q^{71} +28.3276 q^{72} -13.1183 q^{73} -30.6645 q^{74} -0.509725 q^{75} -5.46956 q^{76} -0.854866 q^{77} -0.542731 q^{78} -1.30074 q^{79} +10.2158 q^{80} +8.88645 q^{81} +8.03169 q^{82} -12.9463 q^{83} -0.404941 q^{84} -1.97523 q^{85} -31.7165 q^{86} +0.144698 q^{87} -12.3074 q^{88} -9.50542 q^{89} +5.56909 q^{90} -1.16361 q^{91} +10.2149 q^{92} -0.420499 q^{93} -1.27646 q^{94} -0.682101 q^{95} -2.46928 q^{96} +8.60361 q^{97} +17.9457 q^{98} -3.87732 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * 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 $

$2$	−2.73305	−1.93256	−0.966279	−	0.257498i	$-0.917102\pi$
$2$	−2.73305	−1.93256	−0.966279	+	0.257498i	$0.917102\pi$
$3$	0.112405	0.0648968	0.0324484	−	0.999473i	$-0.489670\pi$
$3$	0.112405	0.0648968	0.0324484	+	0.999473i	$0.489670\pi$
$4$	5.46956	2.73478
$5$	0.682101	0.305045	0.152522	−	0.988300i	$-0.451260\pi$
$5$	0.682101	0.305045	0.152522	+	0.988300i	$0.451260\pi$
$6$	−0.307207	−0.125417
$7$	−0.658651	−0.248947	−0.124473	−	0.992223i	$-0.539724\pi$
$7$	−0.658651	−0.248947	−0.124473	+	0.992223i	$0.539724\pi$
$8$	−9.48248	−3.35256
$9$	−2.98737	−0.995788
$10$	−1.86422	−0.589517
$11$	1.29791	0.391333	0.195667	−	0.980670i	$-0.437313\pi$
$11$	1.29791	0.391333	0.195667	+	0.980670i	$0.437313\pi$
$12$	0.614803	0.177478
$13$	1.76666	0.489984	0.244992	−	0.969525i	$-0.421215\pi$
$13$	1.76666	0.489984	0.244992	+	0.969525i	$0.421215\pi$
$14$	1.80013	0.481104
$15$	0.0766712	0.0197964
$16$	14.9770	3.74424
$17$	−2.89581	−0.702337	−0.351168	−	0.936312i	$-0.614216\pi$
$17$	−2.89581	−0.702337	−0.351168	+	0.936312i	$0.614216\pi$
$18$	8.16462	1.92442
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.73079	0.834230
$21$	−0.0740353	−0.0161558
$22$	−3.54724	−0.756274
$23$	1.86760	0.389421	0.194711	−	0.980861i	$-0.437623\pi$
$23$	1.86760	0.389421	0.194711	+	0.980861i	$0.437623\pi$
$24$	−1.06587	−0.217570
$25$	−4.53474	−0.906948
$26$	−4.82837	−0.946922
$27$	−0.673007	−0.129520
$28$	−3.60253	−0.680814
$29$	1.28729	0.239044	0.119522	−	0.992832i	$-0.461864\pi$
$29$	1.28729	0.239044	0.119522	+	0.992832i	$0.461864\pi$
$30$	−0.209546	−0.0382577
$31$	−3.74094	−0.671893	−0.335947	−	0.941881i	$-0.609056\pi$
$31$	−3.74094	−0.671893	−0.335947	+	0.941881i	$0.609056\pi$
$32$	−21.9678	−3.88340
$33$	0.145890	0.0253963
$34$	7.91439	1.35731
$35$	−0.449267	−0.0759399
$36$	−16.3396	−2.72326
$37$	11.2199	1.84454	0.922268	−	0.386551i	$-0.126334\pi$
$37$	11.2199	1.84454	0.922268	+	0.386551i	$0.126334\pi$
$38$	2.73305	0.443359
$39$	0.198581	0.0317984
$40$	−6.46801	−1.02268
$41$	−2.93873	−0.458952	−0.229476	−	0.973314i	$-0.573701\pi$
$41$	−2.93873	−0.458952	−0.229476	+	0.973314i	$0.573701\pi$
$42$	0.202342	0.0312221
$43$	11.6048	1.76972	0.884859	−	0.465859i	$-0.154254\pi$
$43$	11.6048	1.76972	0.884859	+	0.465859i	$0.154254\pi$
$44$	7.09897	1.07021
$45$	−2.03768	−0.303760
$46$	−5.10424	−0.752579
$47$	0.467044	0.0681254	0.0340627	−	0.999420i	$-0.489155\pi$
$47$	0.467044	0.0681254	0.0340627	+	0.999420i	$0.489155\pi$
$48$	1.68348	0.242989
$49$	−6.56618	−0.938026
$50$	12.3937	1.75273
$51$	−0.325502	−0.0455794
$52$	9.66286	1.34000
$53$	−0.146296	−0.0200953	−0.0100476	−	0.999950i	$-0.503198\pi$
$53$	−0.146296	−0.0200953	−0.0100476	+	0.999950i	$0.503198\pi$
$54$	1.83936	0.250305
$55$	0.885302	0.119374
$56$	6.24564	0.834609
$57$	−0.112405	−0.0148883
$58$	−3.51824	−0.461967
$59$	11.1435	1.45076	0.725378	−	0.688351i	$-0.241665\pi$
$59$	11.1435	1.45076	0.725378	+	0.688351i	$0.241665\pi$
$60$	0.419358	0.0541389
$61$	8.10366	1.03757	0.518784	−	0.854906i	$-0.326385\pi$
$61$	8.10366	1.03757	0.518784	+	0.854906i	$0.326385\pi$
$62$	10.2242	1.29847
$63$	1.96763	0.247898
$64$	30.0852	3.76065
$65$	1.20504	0.149467
$66$	−0.398726	−0.0490797
$67$	−0.316953	−0.0387220	−0.0193610	−	0.999813i	$-0.506163\pi$
$67$	−0.316953	−0.0387220	−0.0193610	+	0.999813i	$0.506163\pi$
$68$	−15.8388	−1.92074
$69$	0.209926	0.0252722
$70$	1.22787	0.146758
$71$	−4.15162	−0.492706	−0.246353	−	0.969180i	$-0.579232\pi$
$71$	−4.15162	−0.492706	−0.246353	+	0.969180i	$0.579232\pi$
$72$	28.3276	3.33844
$73$	−13.1183	−1.53538	−0.767690	−	0.640822i	$-0.778594\pi$
$73$	−13.1183	−1.53538	−0.767690	+	0.640822i	$0.778594\pi$
$74$	−30.6645	−3.56467
$75$	−0.509725	−0.0588580
$76$	−5.46956	−0.627401
$77$	−0.854866	−0.0974211
$78$	−0.542731	−0.0614522
$79$	−1.30074	−0.146345	−0.0731724	−	0.997319i	$-0.523312\pi$
$79$	−1.30074	−0.146345	−0.0731724	+	0.997319i	$0.523312\pi$
$80$	10.2158	1.14216
$81$	8.88645	0.987383
$82$	8.03169	0.886951
$83$	−12.9463	−1.42104	−0.710522	−	0.703675i	$-0.751541\pi$
$83$	−12.9463	−1.42104	−0.710522	+	0.703675i	$0.751541\pi$
$84$	−0.404941	−0.0441826
$85$	−1.97523	−0.214244
$86$	−31.7165	−3.42008
$87$	0.144698	0.0155132
$88$	−12.3074	−1.31197
$89$	−9.50542	−1.00757	−0.503786	−	0.863828i	$-0.668060\pi$
$89$	−9.50542	−1.00757	−0.503786	+	0.863828i	$0.668060\pi$
$90$	5.56909	0.587034
$91$	−1.16361	−0.121980
$92$	10.2149	1.06498
$93$	−0.420499	−0.0436037
$94$	−1.27646	−0.131656
$95$	−0.682101	−0.0699821
$96$	−2.46928	−0.252020
$97$	8.60361	0.873564	0.436782	−	0.899567i	$-0.356118\pi$
$97$	8.60361	0.873564	0.436782	+	0.899567i	$0.356118\pi$
$98$	17.9457	1.81279
$99$	−3.87732	−0.389685
$100$	−24.8030	−2.48030
$101$	−12.1695	−1.21091	−0.605454	−	0.795880i	$-0.707008\pi$
$101$	−12.1695	−1.21091	−0.605454	+	0.795880i	$0.707008\pi$
$102$	0.889613	0.0880848
$103$	16.8978	1.66499	0.832497	−	0.554029i	$-0.186910\pi$
$103$	16.8978	1.66499	0.832497	+	0.554029i	$0.186910\pi$
$104$	−16.7523	−1.64270
$105$	−0.0504996	−0.00492825
$106$	0.399834	0.0388353
$107$	−14.7097	−1.42204	−0.711020	−	0.703172i	$-0.751766\pi$
$107$	−14.7097	−1.42204	−0.711020	+	0.703172i	$0.751766\pi$
$108$	−3.68105	−0.354209
$109$	−2.94674	−0.282246	−0.141123	−	0.989992i	$-0.545071\pi$
$109$	−2.94674	−0.282246	−0.141123	+	0.989992i	$0.545071\pi$
$110$	−2.41958	−0.230697
$111$	1.26116	0.119704
$112$	−9.86459	−0.932116
$113$	−10.3884	−0.977254	−0.488627	−	0.872493i	$-0.662502\pi$
$113$	−10.3884	−0.977254	−0.488627	+	0.872493i	$0.662502\pi$
$114$	0.307207	0.0287726
$115$	1.27389	0.118791
$116$	7.04093	0.653734
$117$	−5.27766	−0.487920
$118$	−30.4556	−2.80367
$119$	1.90733	0.174844
$120$	−0.727033	−0.0663687
$121$	−9.31544	−0.846858
$122$	−22.1477	−2.00516
$123$	−0.330326	−0.0297845
$124$	−20.4613	−1.83748
$125$	−6.50365	−0.581705
$126$	−5.37763	−0.479078
$127$	3.23225	0.286816	0.143408	−	0.989664i	$-0.454194\pi$
$127$	3.23225	0.286816	0.143408	+	0.989664i	$0.454194\pi$
$128$	−38.2887	−3.38428
$129$	1.30443	0.114849
$130$	−3.29344	−0.288854
$131$	−14.4687	−1.26414	−0.632070	−	0.774911i	$-0.717794\pi$
$131$	−14.4687	−1.26414	−0.632070	+	0.774911i	$0.717794\pi$
$132$	0.797956	0.0694532
$133$	0.658651	0.0571123
$134$	0.866250	0.0748326
$135$	−0.459059	−0.0395095
$136$	27.4594	2.35463
$137$	9.94480	0.849642	0.424821	−	0.905277i	$-0.360337\pi$
$137$	9.94480	0.849642	0.424821	+	0.905277i	$0.360337\pi$
$138$	−0.573739	−0.0488399
$139$	−9.85829	−0.836170	−0.418085	−	0.908408i	$-0.637298\pi$
$139$	−9.85829	−0.836170	−0.418085	+	0.908408i	$0.637298\pi$
$140$	−2.45729	−0.207679
$141$	0.0524979	0.00442112
$142$	11.3466	0.952183
$143$	2.29296	0.191747
$144$	−44.7416	−3.72847
$145$	0.878064	0.0729192
$146$	35.8529	2.96721
$147$	−0.738068	−0.0608748
$148$	61.3678	5.04440
$149$	−19.1687	−1.57036	−0.785179	−	0.619268i	$-0.787429\pi$
$149$	−19.1687	−1.57036	−0.785179	+	0.619268i	$0.787429\pi$
$150$	1.39310	0.113746
$151$	22.1441	1.80206	0.901032	−	0.433752i	$-0.142811\pi$
$151$	22.1441	1.80206	0.901032	+	0.433752i	$0.142811\pi$
$152$	9.48248	0.769130
$153$	8.65084	0.699379
$154$	2.33639	0.188272
$155$	−2.55170	−0.204958
$156$	1.08615	0.0869615
$157$	14.6869	1.17214	0.586070	−	0.810260i	$-0.300674\pi$
$157$	14.6869	1.17214	0.586070	+	0.810260i	$0.300674\pi$
$158$	3.55499	0.282820
$159$	−0.0164443	−0.00130412
$160$	−14.9843	−1.18461
$161$	−1.23010	−0.0969451
$162$	−24.2871	−1.90817
$163$	−19.4842	−1.52612	−0.763061	−	0.646327i	$-0.776304\pi$
$163$	−19.4842	−1.52612	−0.763061	+	0.646327i	$0.776304\pi$
$164$	−16.0735	−1.25513
$165$	0.0995120	0.00774700
$166$	35.3830	2.74625
$167$	−19.9224	−1.54164	−0.770820	−	0.637052i	$-0.780153\pi$
$167$	−19.9224	−1.54164	−0.770820	+	0.637052i	$0.780153\pi$
$168$	0.702038	0.0541634
$169$	−9.87891	−0.759916
$170$	5.39841	0.414039
$171$	2.98737	0.228450
$172$	63.4732	4.83979
$173$	19.2914	1.46670	0.733348	−	0.679854i	$-0.237957\pi$
$173$	19.2914	1.46670	0.733348	+	0.679854i	$0.237957\pi$
$174$	−0.395466	−0.0299802
$175$	2.98681	0.225782
$176$	19.4387	1.46524
$177$	1.25258	0.0941494
$178$	25.9788	1.94719
$179$	−16.2031	−1.21107	−0.605537	−	0.795817i	$-0.707041\pi$
$179$	−16.2031	−1.21107	−0.605537	+	0.795817i	$0.707041\pi$
$180$	−11.1452	−0.830717
$181$	−4.77684	−0.355060	−0.177530	−	0.984115i	$-0.556811\pi$
$181$	−4.77684	−0.355060	−0.177530	+	0.984115i	$0.556811\pi$
$182$	3.18021	0.235733
$183$	0.910888	0.0673348
$184$	−17.7095	−1.30556
$185$	7.65309	0.562666
$186$	1.14924	0.0842667
$187$	−3.75849	−0.274848
$188$	2.55453	0.186308
$189$	0.443277	0.0322436
$190$	1.86422	0.135244
$191$	11.3834	0.823676	0.411838	−	0.911257i	$-0.364887\pi$
$191$	11.3834	0.823676	0.411838	+	0.911257i	$0.364887\pi$
$192$	3.38171	0.244054
$193$	2.44932	0.176306	0.0881530	−	0.996107i	$-0.471904\pi$
$193$	2.44932	0.176306	0.0881530	+	0.996107i	$0.471904\pi$
$194$	−23.5141	−1.68821
$195$	0.135452	0.00969993
$196$	−35.9141	−2.56529
$197$	11.5275	0.821297	0.410649	−	0.911794i	$-0.365302\pi$
$197$	11.5275	0.821297	0.410649	+	0.911794i	$0.365302\pi$
$198$	10.5969	0.753089
$199$	16.6299	1.17887	0.589433	−	0.807817i	$-0.299351\pi$
$199$	16.6299	1.17887	0.589433	+	0.807817i	$0.299351\pi$
$200$	43.0005	3.04060
$201$	−0.0356270	−0.00251293
$202$	33.2598	2.34015
$203$	−0.847877	−0.0595093
$204$	−1.78035	−0.124650
$205$	−2.00451	−0.140001
$206$	−46.1827	−3.21770
$207$	−5.57920	−0.387781
$208$	26.4592	1.83462
$209$	−1.29791	−0.0897780
$210$	0.138018	0.00952414
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−0.800173	−0.0549561
$213$	−0.466660	−0.0319750
$214$	40.2023	2.74817
$215$	7.91566	0.539843
$216$	6.38177	0.434225
$217$	2.46398	0.167266
$218$	8.05358	0.545457
$219$	−1.47455	−0.0996412
$220$	4.84221	0.326462
$221$	−5.11591	−0.344134
$222$	−3.44682	−0.231336
$223$	−24.1317	−1.61598	−0.807989	−	0.589197i	$-0.799444\pi$
$223$	−24.1317	−1.61598	−0.807989	+	0.589197i	$0.799444\pi$
$224$	14.4691	0.966759
$225$	13.5469	0.903128
$226$	28.3919	1.88860
$227$	7.39843	0.491051	0.245525	−	0.969390i	$-0.421039\pi$
$227$	7.39843	0.491051	0.245525	+	0.969390i	$0.421039\pi$
$228$	−0.614803	−0.0407163
$229$	−22.3876	−1.47941	−0.739706	−	0.672930i	$-0.765036\pi$
$229$	−22.3876	−1.47941	−0.739706	+	0.672930i	$0.765036\pi$
$230$	−3.48161	−0.229570
$231$	−0.0960908	−0.00632231
$232$	−12.2067	−0.801411
$233$	−24.2463	−1.58843	−0.794214	−	0.607639i	$-0.792117\pi$
$233$	−24.2463	−1.58843	−0.794214	+	0.607639i	$0.792117\pi$
$234$	14.4241	0.942934
$235$	0.318571	0.0207813
$236$	60.9498	3.96750
$237$	−0.146209	−0.00949731
$238$	−5.21282	−0.337897
$239$	19.8342	1.28297	0.641485	−	0.767135i	$-0.278318\pi$
$239$	19.8342	1.28297	0.641485	+	0.767135i	$0.278318\pi$
$240$	1.14830	0.0741226
$241$	9.14928	0.589357	0.294679	−	0.955596i	$-0.404787\pi$
$241$	9.14928	0.589357	0.294679	+	0.955596i	$0.404787\pi$
$242$	25.4596	1.63660
$243$	3.01790	0.193598
$244$	44.3234	2.83752
$245$	−4.47880	−0.286140
$246$	0.902798	0.0575603
$247$	−1.76666	−0.112410
$248$	35.4734	2.25256
$249$	−1.45523	−0.0922212
$250$	17.7748	1.12418
$251$	−12.3886	−0.781963	−0.390981	−	0.920399i	$-0.627864\pi$
$251$	−12.3886	−0.781963	−0.390981	+	0.920399i	$0.627864\pi$
$252$	10.7621	0.677947
$253$	2.42397	0.152393
$254$	−8.83389	−0.554288
$255$	−0.222025	−0.0139038
$256$	44.4746	2.77966
$257$	26.3988	1.64671	0.823356	−	0.567525i	$-0.192099\pi$
$257$	26.3988	1.64671	0.823356	+	0.567525i	$0.192099\pi$
$258$	−3.56508	−0.221952
$259$	−7.38998	−0.459191
$260$	6.59105	0.408759
$261$	−3.84561	−0.238038
$262$	39.5438	2.44302
$263$	3.89554	0.240209	0.120105	−	0.992761i	$-0.461677\pi$
$263$	3.89554	0.240209	0.120105	+	0.992761i	$0.461677\pi$
$264$	−1.38340	−0.0851425
$265$	−0.0997885	−0.00612996
$266$	−1.80013	−0.110373
$267$	−1.06845	−0.0653882
$268$	−1.73360	−0.105896
$269$	−26.0128	−1.58603	−0.793016	−	0.609201i	$-0.791490\pi$
$269$	−26.0128	−1.58603	−0.793016	+	0.609201i	$0.791490\pi$
$270$	1.25463	0.0763543
$271$	−19.7563	−1.20011	−0.600054	−	0.799959i	$-0.704854\pi$
$271$	−19.7563	−1.20011	−0.600054	+	0.799959i	$0.704854\pi$
$272$	−43.3704	−2.62972
$273$	−0.130795	−0.00791610
$274$	−27.1796	−1.64198
$275$	−5.88566	−0.354919
$276$	1.14821	0.0691138
$277$	11.7845	0.708061	0.354031	−	0.935234i	$-0.384811\pi$
$277$	11.7845	0.708061	0.354031	+	0.935234i	$0.384811\pi$
$278$	26.9432	1.61595
$279$	11.1756	0.669064
$280$	4.26016	0.254593
$281$	−7.31863	−0.436593	−0.218296	−	0.975883i	$-0.570050\pi$
$281$	−7.31863	−0.436593	−0.218296	+	0.975883i	$0.570050\pi$
$282$	−0.143479	−0.00854407
$283$	−25.4971	−1.51565	−0.757823	−	0.652460i	$-0.773737\pi$
$283$	−25.4971	−1.51565	−0.757823	+	0.652460i	$0.773737\pi$
$284$	−22.7075	−1.34744
$285$	−0.0766712	−0.00454161
$286$	−6.26677	−0.370562
$287$	1.93560	0.114255
$288$	65.6259	3.86704
$289$	−8.61429	−0.506723
$290$	−2.39979	−0.140921
$291$	0.967085	0.0566915
$292$	−71.7512	−4.19892
$293$	2.48402	0.145118	0.0725589	−	0.997364i	$-0.476883\pi$
$293$	2.48402	0.145118	0.0725589	+	0.997364i	$0.476883\pi$
$294$	2.01718	0.117644
$295$	7.60097	0.442546
$296$	−106.392	−6.18392
$297$	−0.873499	−0.0506856
$298$	52.3889	3.03481
$299$	3.29941	0.190810
$300$	−2.78797	−0.160964
$301$	−7.64353	−0.440565
$302$	−60.5210	−3.48259
$303$	−1.36790	−0.0785840
$304$	−14.9770	−0.858987
$305$	5.52751	0.316505
$306$	−23.6432	−1.35159
$307$	22.0269	1.25714	0.628570	−	0.777753i	$-0.283641\pi$
$307$	22.0269	1.25714	0.628570	+	0.777753i	$0.283641\pi$
$308$	−4.67574	−0.266425
$309$	1.89939	0.108053
$310$	6.97393	0.396092
$311$	10.5671	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$311$	10.5671	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$312$	−1.88304	−0.106606
$313$	20.1016	1.13621	0.568105	−	0.822956i	$-0.307677\pi$
$313$	20.1016	1.13621	0.568105	+	0.822956i	$0.307677\pi$
$314$	−40.1400	−2.26523
$315$	1.34212	0.0756201
$316$	−7.11448	−0.400221
$317$	−0.0780818	−0.00438551	−0.00219276	−	0.999998i	$-0.500698\pi$
$317$	−0.0780818	−0.00438551	−0.00219276	+	0.999998i	$0.500698\pi$
$318$	0.0449431	0.00252028
$319$	1.67078	0.0935460
$320$	20.5211	1.14717
$321$	−1.65344	−0.0922858
$322$	3.36191	0.187352
$323$	2.89581	0.161127
$324$	48.6049	2.70027
$325$	−8.01135	−0.444390
$326$	53.2513	2.94932
$327$	−0.331227	−0.0183169
$328$	27.8664	1.53866
$329$	−0.307619	−0.0169596
$330$	−0.271971	−0.0149715
$331$	−27.7833	−1.52711	−0.763553	−	0.645745i	$-0.776547\pi$
$331$	−27.7833	−1.52711	−0.763553	+	0.645745i	$0.776547\pi$
$332$	−70.8107	−3.88624
$333$	−33.5179	−1.83677
$334$	54.4489	2.97931
$335$	−0.216194	−0.0118120
$336$	−1.10882	−0.0604913
$337$	−6.98711	−0.380612	−0.190306	−	0.981725i	$-0.560948\pi$
$337$	−6.98711	−0.380612	−0.190306	+	0.981725i	$0.560948\pi$
$338$	26.9995	1.46858
$339$	−1.16770	−0.0634206
$340$	−10.8037	−0.585911
$341$	−4.85539	−0.262934
$342$	−8.16462	−0.441492
$343$	8.93538	0.482465
$344$	−110.042	−5.93309
$345$	0.143191	0.00770915
$346$	−52.7243	−2.83447
$347$	−32.0269	−1.71930	−0.859648	−	0.510887i	$-0.829317\pi$
$347$	−32.0269	−1.71930	−0.859648	+	0.510887i	$0.829317\pi$
$348$	0.791432	0.0424252
$349$	−1.45574	−0.0779241	−0.0389620	−	0.999241i	$-0.512405\pi$
$349$	−1.45574	−0.0779241	−0.0389620	+	0.999241i	$0.512405\pi$
$350$	−8.16310	−0.436336
$351$	−1.18898	−0.0634628
$352$	−28.5121	−1.51970
$353$	15.6362	0.832232	0.416116	−	0.909312i	$-0.363391\pi$
$353$	15.6362	0.832232	0.416116	+	0.909312i	$0.363391\pi$
$354$	−3.42335	−0.181949
$355$	−2.83182	−0.150298
$356$	−51.9904	−2.75549
$357$	0.214392	0.0113468
$358$	44.2838	2.34047
$359$	19.6711	1.03820	0.519101	−	0.854713i	$-0.326267\pi$
$359$	19.6711	1.03820	0.519101	+	0.854713i	$0.326267\pi$
$360$	19.3223	1.01837
$361$	1.00000	0.0526316
$362$	13.0553	0.686173
$363$	−1.04710	−0.0549584
$364$	−6.36445	−0.333588
$365$	−8.94800	−0.468360
$366$	−2.48950	−0.130128
$367$	−22.5886	−1.17912	−0.589558	−	0.807726i	$-0.700698\pi$
$367$	−22.5886	−1.17912	−0.589558	+	0.807726i	$0.700698\pi$
$368$	27.9709	1.45809
$369$	8.77905	0.457019
$370$	−20.9163	−1.08738
$371$	0.0963578	0.00500265
$372$	−2.29994	−0.119247
$373$	12.4396	0.644096	0.322048	−	0.946723i	$-0.395629\pi$
$373$	12.4396	0.644096	0.322048	+	0.946723i	$0.395629\pi$
$374$	10.2721	0.531159
$375$	−0.731040	−0.0377507
$376$	−4.42874	−0.228395
$377$	2.27421	0.117128
$378$	−1.21150	−0.0623127
$379$	18.5691	0.953829	0.476915	−	0.878950i	$-0.341755\pi$
$379$	18.5691	0.953829	0.476915	+	0.878950i	$0.341755\pi$
$380$	−3.73079	−0.191386
$381$	0.363319	0.0186134
$382$	−31.1115	−1.59180
$383$	2.09458	0.107028	0.0535140	−	0.998567i	$-0.482958\pi$
$383$	2.09458	0.107028	0.0535140	+	0.998567i	$0.482958\pi$
$384$	−4.30382	−0.219629
$385$	−0.583105	−0.0297178
$386$	−6.69412	−0.340722
$387$	−34.6678	−1.76227
$388$	47.0580	2.38901
$389$	−31.9835	−1.62163	−0.810814	−	0.585303i	$-0.800975\pi$
$389$	−31.9835	−1.62163	−0.810814	+	0.585303i	$0.800975\pi$
$390$	−0.370197	−0.0187457
$391$	−5.40821	−0.273505
$392$	62.2636	3.14479
$393$	−1.62635	−0.0820386
$394$	−31.5051	−1.58720
$395$	−0.887237	−0.0446417
$396$	−21.2072	−1.06570
$397$	−27.7131	−1.39088	−0.695441	−	0.718583i	$-0.744791\pi$
$397$	−27.7131	−1.39088	−0.695441	+	0.718583i	$0.744791\pi$
$398$	−45.4505	−2.27823
$399$	0.0740353	0.00370640
$400$	−67.9166	−3.39583
$401$	−21.6682	−1.08206	−0.541028	−	0.841004i	$-0.681965\pi$
$401$	−21.6682	−1.08206	−0.541028	+	0.841004i	$0.681965\pi$
$402$	0.0973703	0.00485639
$403$	−6.60898	−0.329217
$404$	−66.5617	−3.31157
$405$	6.06145	0.301196
$406$	2.31729	0.115005
$407$	14.5623	0.721828
$408$	3.08656	0.152808
$409$	−18.0172	−0.890895	−0.445448	−	0.895308i	$-0.646955\pi$
$409$	−18.0172	−0.890895	−0.445448	+	0.895308i	$0.646955\pi$
$410$	5.47842	0.270560
$411$	1.11784	0.0551390
$412$	92.4238	4.55339
$413$	−7.33966	−0.361161
$414$	15.2482	0.749409
$415$	−8.83071	−0.433482
$416$	−38.8097	−1.90280
$417$	−1.10812	−0.0542647
$418$	3.54724	0.173501
$419$	13.5395	0.661449	0.330724	−	0.943727i	$-0.392707\pi$
$419$	13.5395	0.661449	0.330724	+	0.943727i	$0.392707\pi$
$420$	−0.276210	−0.0134777
$421$	−16.3670	−0.797678	−0.398839	−	0.917021i	$-0.630587\pi$
$421$	−16.3670	−0.797678	−0.398839	+	0.917021i	$0.630587\pi$
$422$	2.73305	0.133043
$423$	−1.39523	−0.0678385
$424$	1.38725	0.0673706
$425$	13.1317	0.636983
$426$	1.27541	0.0617936
$427$	−5.33748	−0.258299
$428$	−80.4555	−3.88896
$429$	0.257739	0.0124438
$430$	−21.6339	−1.04328
$431$	−9.91086	−0.477389	−0.238695	−	0.971095i	$-0.576719\pi$
$431$	−9.91086	−0.477389	−0.238695	+	0.971095i	$0.576719\pi$
$432$	−10.0796	−0.484955
$433$	4.70066	0.225899	0.112950	−	0.993601i	$-0.463970\pi$
$433$	4.70066	0.225899	0.112950	+	0.993601i	$0.463970\pi$
$434$	−6.73417	−0.323250
$435$	0.0986983	0.00473222
$436$	−16.1174	−0.771881
$437$	−1.86760	−0.0893393
$438$	4.03003	0.192562
$439$	−13.5599	−0.647180	−0.323590	−	0.946197i	$-0.604890\pi$
$439$	−13.5599	−0.647180	−0.323590	+	0.946197i	$0.604890\pi$
$440$	−8.39486	−0.400209
$441$	19.6156	0.934075
$442$	13.9820	0.665058
$443$	−13.1375	−0.624183	−0.312091	−	0.950052i	$-0.601030\pi$
$443$	−13.1375	−0.624183	−0.312091	+	0.950052i	$0.601030\pi$
$444$	6.89801	0.327365
$445$	−6.48365	−0.307355
$446$	65.9531	3.12297
$447$	−2.15464	−0.101911
$448$	−19.8156	−0.936201
$449$	−0.300888	−0.0141998	−0.00709988	−	0.999975i	$-0.502260\pi$
$449$	−0.300888	−0.0141998	−0.00709988	+	0.999975i	$0.502260\pi$
$450$	−37.0244	−1.74535
$451$	−3.81419	−0.179603
$452$	−56.8197	−2.67257
$453$	2.48910	0.116948
$454$	−20.2203	−0.948984
$455$	−0.793702	−0.0372093
$456$	1.06587	0.0499141
$457$	7.92800	0.370856	0.185428	−	0.982658i	$-0.440633\pi$
$457$	7.92800	0.370856	0.185428	+	0.982658i	$0.440633\pi$
$458$	61.1863	2.85905
$459$	1.94890	0.0909668
$460$	6.96762	0.324867
$461$	−5.91553	−0.275514	−0.137757	−	0.990466i	$-0.543989\pi$
$461$	−5.91553	−0.275514	−0.137757	+	0.990466i	$0.543989\pi$
$462$	0.262621	0.0122182
$463$	−19.7290	−0.916886	−0.458443	−	0.888724i	$-0.651593\pi$
$463$	−19.7290	−0.916886	−0.458443	+	0.888724i	$0.651593\pi$
$464$	19.2797	0.895039
$465$	−0.286823	−0.0133011
$466$	66.2663	3.06973
$467$	5.60582	0.259407	0.129703	−	0.991553i	$-0.458598\pi$
$467$	5.60582	0.259407	0.129703	+	0.991553i	$0.458598\pi$
$468$	−28.8665	−1.33435
$469$	0.208762	0.00963972
$470$	−0.870672	−0.0401611
$471$	1.65087	0.0760681
$472$	−105.668	−4.86375
$473$	15.0620	0.692549
$474$	0.399597	0.0183541
$475$	4.53474	0.208068
$476$	10.4322	0.478161
$477$	0.437039	0.0200106
$478$	−54.2080	−2.47942
$479$	−30.6619	−1.40098	−0.700488	−	0.713664i	$-0.747034\pi$
$479$	−30.6619	−1.40098	−0.700488	+	0.713664i	$0.747034\pi$
$480$	−1.68430	−0.0768774
$481$	19.8217	0.903792
$482$	−25.0054	−1.13897
$483$	−0.138268	−0.00629142
$484$	−50.9514	−2.31597
$485$	5.86853	0.266476
$486$	−8.24806	−0.374140
$487$	28.0713	1.27203	0.636017	−	0.771675i	$-0.280581\pi$
$487$	28.0713	1.27203	0.636017	+	0.771675i	$0.280581\pi$
$488$	−76.8428	−3.47851
$489$	−2.19011	−0.0990403
$490$	12.2408	0.552982
$491$	−38.1934	−1.72364	−0.861822	−	0.507212i	$-0.830676\pi$
$491$	−38.1934	−1.72364	−0.861822	+	0.507212i	$0.830676\pi$
$492$	−1.80674	−0.0814541
$493$	−3.72776	−0.167890
$494$	4.82837	0.217239
$495$	−2.64472	−0.118871
$496$	−56.0280	−2.51573
$497$	2.73447	0.122658
$498$	3.97720	0.178223
$499$	35.5100	1.58964	0.794822	−	0.606842i	$-0.207564\pi$
$499$	35.5100	1.58964	0.794822	+	0.606842i	$0.207564\pi$
$500$	−35.5721	−1.59083
$501$	−2.23937	−0.100048
$502$	33.8587	1.51119
$503$	25.7596	1.14856	0.574282	−	0.818657i	$-0.305281\pi$
$503$	25.7596	1.14856	0.574282	+	0.818657i	$0.305281\pi$
$504$	−18.6580	−0.831094
$505$	−8.30081	−0.369381
$506$	−6.62482	−0.294509
$507$	−1.11043	−0.0493161
$508$	17.6790	0.784377
$509$	35.3535	1.56702	0.783509	−	0.621380i	$-0.213428\pi$
$509$	35.3535	1.56702	0.783509	+	0.621380i	$0.213428\pi$
$510$	0.606806	0.0268698
$511$	8.64037	0.382228
$512$	−44.9737	−1.98758
$513$	0.673007	0.0297140
$514$	−72.1492	−3.18237
$515$	11.5260	0.507898
$516$	7.13468	0.314087
$517$	0.606179	0.0266597
$518$	20.1972	0.887413
$519$	2.16844	0.0951838
$520$	−11.4268	−0.501097
$521$	1.79447	0.0786170	0.0393085	−	0.999227i	$-0.487484\pi$
$521$	1.79447	0.0786170	0.0393085	+	0.999227i	$0.487484\pi$
$522$	10.5103	0.460021
$523$	12.5504	0.548792	0.274396	−	0.961617i	$-0.411522\pi$
$523$	12.5504	0.548792	0.274396	+	0.961617i	$0.411522\pi$
$524$	−79.1376	−3.45714
$525$	0.335731	0.0146525
$526$	−10.6467	−0.464218
$527$	10.8331	0.471895
$528$	2.18499	0.0950897
$529$	−19.5121	−0.848351
$530$	0.272727	0.0118465
$531$	−33.2896	−1.44465
$532$	3.60253	0.156190
$533$	−5.19174	−0.224879
$534$	2.92013	0.126366
$535$	−10.0335	−0.433786
$536$	3.00550	0.129818
$537$	−1.82130	−0.0785947
$538$	71.0944	3.06510
$539$	−8.52228	−0.367080
$540$	−2.51085	−0.108050
$541$	11.0638	0.475671	0.237836	−	0.971305i	$-0.423562\pi$
$541$	11.0638	0.475671	0.237836	+	0.971305i	$0.423562\pi$
$542$	53.9949	2.31928
$543$	−0.536938	−0.0230422
$544$	63.6146	2.72745
$545$	−2.00997	−0.0860978
$546$	0.357470	0.0152983
$547$	8.70286	0.372107	0.186054	−	0.982540i	$-0.440430\pi$
$547$	8.70286	0.372107	0.186054	+	0.982540i	$0.440430\pi$
$548$	54.3937	2.32358
$549$	−24.2086	−1.03320
$550$	16.0858	0.685901
$551$	−1.28729	−0.0548405
$552$	−1.99062	−0.0847265
$553$	0.856734	0.0364321
$554$	−32.2076	−1.36837
$555$	0.860241	0.0365152
$556$	−53.9205	−2.28674
$557$	6.25272	0.264936	0.132468	−	0.991187i	$-0.457710\pi$
$557$	6.25272	0.264936	0.132468	+	0.991187i	$0.457710\pi$
$558$	−30.5434	−1.29300
$559$	20.5018	0.867133
$560$	−6.72865	−0.284337
$561$	−0.422471	−0.0178367
$562$	20.0022	0.843741
$563$	−35.9179	−1.51376	−0.756880	−	0.653554i	$-0.773277\pi$
$563$	−35.9179	−1.51376	−0.756880	+	0.653554i	$0.773277\pi$
$564$	0.287140	0.0120908
$565$	−7.08591	−0.298106
$566$	69.6848	2.92907
$567$	−5.85307	−0.245806
$568$	39.3676	1.65183
$569$	−22.9346	−0.961470	−0.480735	−	0.876866i	$-0.659630\pi$
$569$	−22.9346	−0.961470	−0.480735	+	0.876866i	$0.659630\pi$
$570$	0.209546	0.00877693
$571$	−30.6427	−1.28236	−0.641178	−	0.767392i	$-0.721554\pi$
$571$	−30.6427	−1.28236	−0.641178	+	0.767392i	$0.721554\pi$
$572$	12.5415	0.524385
$573$	1.27955	0.0534539
$574$	−5.29008	−0.220804
$575$	−8.46907	−0.353185
$576$	−89.8755	−3.74481
$577$	24.9590	1.03905	0.519527	−	0.854454i	$-0.326108\pi$
$577$	24.9590	1.03905	0.519527	+	0.854454i	$0.326108\pi$
$578$	23.5433	0.979271
$579$	0.275315	0.0114417
$580$	4.80262	0.199418
$581$	8.52711	0.353764
$582$	−2.64309	−0.109560
$583$	−0.189878	−0.00786394
$584$	124.394	5.14745
$585$	−3.59990	−0.148838
$586$	−6.78894	−0.280448
$587$	−27.2499	−1.12473	−0.562363	−	0.826891i	$-0.690108\pi$
$587$	−27.2499	−1.12473	−0.562363	+	0.826891i	$0.690108\pi$
$588$	−4.03691	−0.166479
$589$	3.74094	0.154143
$590$	−20.7738	−0.855245
$591$	1.29574	0.0532995
$592$	168.040	6.90638
$593$	10.9459	0.449492	0.224746	−	0.974417i	$-0.427845\pi$
$593$	10.9459	0.449492	0.224746	+	0.974417i	$0.427845\pi$
$594$	2.38732	0.0979528
$595$	1.30099	0.0533354
$596$	−104.844	−4.29458
$597$	1.86928	0.0765046
$598$	−9.01746	−0.368751
$599$	0.698822	0.0285531	0.0142766	−	0.999898i	$-0.495455\pi$
$599$	0.698822	0.0285531	0.0142766	+	0.999898i	$0.495455\pi$
$600$	4.83346	0.197325
$601$	−22.4276	−0.914843	−0.457421	−	0.889250i	$-0.651227\pi$
$601$	−22.4276	−0.914843	−0.457421	+	0.889250i	$0.651227\pi$
$602$	20.8901	0.851418
$603$	0.946856	0.0385589
$604$	121.119	4.92825
$605$	−6.35407	−0.258330
$606$	3.73855	0.151868
$607$	6.11086	0.248032	0.124016	−	0.992280i	$-0.460423\pi$
$607$	6.11086	0.248032	0.124016	+	0.992280i	$0.460423\pi$
$608$	21.9678	0.890912
$609$	−0.0953052	−0.00386196
$610$	−15.1070	−0.611663
$611$	0.825109	0.0333803
$612$	47.3163	1.91265
$613$	−7.09282	−0.286476	−0.143238	−	0.989688i	$-0.545752\pi$
$613$	−7.09282	−0.286476	−0.143238	+	0.989688i	$0.545752\pi$
$614$	−60.2005	−2.42950
$615$	−0.225316	−0.00908561
$616$	8.10625	0.326610
$617$	−42.5337	−1.71234	−0.856171	−	0.516693i	$-0.827163\pi$
$617$	−42.5337	−1.71234	−0.856171	+	0.516693i	$0.827163\pi$
$618$	−5.19114	−0.208818
$619$	−40.7316	−1.63714	−0.818571	−	0.574405i	$-0.805234\pi$
$619$	−40.7316	−1.63714	−0.818571	+	0.574405i	$0.805234\pi$
$620$	−13.9567	−0.560514
$621$	−1.25691	−0.0504379
$622$	−28.8804	−1.15800
$623$	6.26075	0.250832
$624$	2.97413	0.119061
$625$	18.2375	0.729502
$626$	−54.9387	−2.19579
$627$	−0.145890	−0.00582630
$628$	80.3308	3.20555
$629$	−32.4906	−1.29549
$630$	−3.66809	−0.146140
$631$	−30.8735	−1.22905	−0.614527	−	0.788895i	$-0.710653\pi$
$631$	−30.8735	−1.22905	−0.614527	+	0.788895i	$0.710653\pi$
$632$	12.3342	0.490630
$633$	−0.112405	−0.00446768
$634$	0.213401	0.00847526
$635$	2.20472	0.0874916
$636$	−0.0899431	−0.00356648
$637$	−11.6002	−0.459617
$638$	−4.56634	−0.180783
$639$	12.4024	0.490631
$640$	−26.1168	−1.03236
$641$	12.7752	0.504588	0.252294	−	0.967651i	$-0.418815\pi$
$641$	12.7752	0.504588	0.252294	+	0.967651i	$0.418815\pi$
$642$	4.51892	0.178348
$643$	−24.8761	−0.981016	−0.490508	−	0.871437i	$-0.663189\pi$
$643$	−24.8761	−0.981016	−0.490508	+	0.871437i	$0.663189\pi$
$644$	−6.72808	−0.265123
$645$	0.889756	0.0350341
$646$	−7.91439	−0.311387
$647$	4.08655	0.160659	0.0803293	−	0.996768i	$-0.474403\pi$
$647$	4.08655	0.160659	0.0803293	+	0.996768i	$0.474403\pi$
$648$	−84.2655	−3.31026
$649$	14.4632	0.567729
$650$	21.8954	0.858808
$651$	0.276962	0.0108550
$652$	−106.570	−4.17360
$653$	32.8671	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$653$	32.8671	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$654$	0.905259	0.0353984
$655$	−9.86914	−0.385619
$656$	−44.0132	−1.71843
$657$	39.1891	1.52891
$658$	0.840739	0.0327754
$659$	23.9707	0.933766	0.466883	−	0.884319i	$-0.345377\pi$
$659$	23.9707	0.933766	0.466883	+	0.884319i	$0.345377\pi$
$660$	0.544287	0.0211863
$661$	41.9200	1.63050	0.815250	−	0.579109i	$-0.196599\pi$
$661$	41.9200	1.63050	0.815250	+	0.579109i	$0.196599\pi$
$662$	75.9330	2.95122
$663$	−0.575052	−0.0223332
$664$	122.763	4.76414
$665$	0.449267	0.0174218
$666$	91.6060	3.54966
$667$	2.40415	0.0930889
$668$	−108.967	−4.21605
$669$	−2.71251	−0.104872
$670$	0.590870	0.0228273
$671$	10.5178	0.406034
$672$	1.62639	0.0627395
$673$	−42.6343	−1.64343	−0.821717	−	0.569896i	$-0.806983\pi$
$673$	−42.6343	−1.64343	−0.821717	+	0.569896i	$0.806983\pi$
$674$	19.0961	0.735555
$675$	3.05191	0.117468
$676$	−54.0333	−2.07820
$677$	−30.9384	−1.18906	−0.594529	−	0.804074i	$-0.702662\pi$
$677$	−30.9384	−1.18906	−0.594529	+	0.804074i	$0.702662\pi$
$678$	3.19138	0.122564
$679$	−5.66678	−0.217471
$680$	18.7301	0.718267
$681$	0.831617	0.0318676
$682$	13.2700	0.508135
$683$	−44.4406	−1.70047	−0.850237	−	0.526400i	$-0.823541\pi$
$683$	−44.4406	−1.70047	−0.850237	+	0.526400i	$0.823541\pi$
$684$	16.3396	0.624759
$685$	6.78336	0.259179
$686$	−24.4208	−0.932392
$687$	−2.51646	−0.0960091
$688$	173.805	6.62625
$689$	−0.258455	−0.00984635
$690$	−0.391348	−0.0148984
$691$	24.9148	0.947804	0.473902	−	0.880578i	$-0.342845\pi$
$691$	24.9148	0.947804	0.473902	+	0.880578i	$0.342845\pi$
$692$	105.515	4.01109
$693$	2.55380	0.0970108
$694$	87.5312	3.32264
$695$	−6.72435	−0.255069
$696$	−1.37209	−0.0520090
$697$	8.50999	0.322339
$698$	3.97862	0.150593
$699$	−2.72539	−0.103084
$700$	16.3365	0.617463
$701$	1.55805	0.0588467	0.0294234	−	0.999567i	$-0.490633\pi$
$701$	1.55805	0.0588467	0.0294234	+	0.999567i	$0.490633\pi$
$702$	3.24953	0.122646
$703$	−11.2199	−0.423166
$704$	39.0477	1.47167
$705$	0.0358089	0.00134864
$706$	−42.7346	−1.60834
$707$	8.01544	0.301451
$708$	6.85104	0.257478
$709$	−36.4795	−1.37002	−0.685008	−	0.728535i	$-0.740201\pi$
$709$	−36.4795	−1.37002	−0.685008	+	0.728535i	$0.740201\pi$
$710$	7.73951	0.290459
$711$	3.88579	0.145728
$712$	90.1349	3.37795
$713$	−6.98658	−0.261649
$714$	−0.585944	−0.0219284
$715$	1.56403	0.0584914
$716$	−88.6236	−3.31202
$717$	2.22946	0.0832607
$718$	−53.7621	−2.00638
$719$	−7.35082	−0.274139	−0.137070	−	0.990561i	$-0.543768\pi$
$719$	−7.35082	−0.274139	−0.137070	+	0.990561i	$0.543768\pi$
$720$	−30.5183	−1.13735
$721$	−11.1298	−0.414495
$722$	−2.73305	−0.101714
$723$	1.02842	0.0382474
$724$	−26.1272	−0.971010
$725$	−5.83754	−0.216801
$726$	2.86177	0.106210
$727$	35.6279	1.32136	0.660682	−	0.750666i	$-0.270267\pi$
$727$	35.6279	1.32136	0.660682	+	0.750666i	$0.270267\pi$
$728$	11.0339	0.408945
$729$	−26.3201	−0.974819
$730$	24.4553	0.905132
$731$	−33.6053	−1.24294
$732$	4.98215	0.184146
$733$	−23.6344	−0.872956	−0.436478	−	0.899715i	$-0.643774\pi$
$733$	−23.6344	−0.872956	−0.436478	+	0.899715i	$0.643774\pi$
$734$	61.7357	2.27871
$735$	−0.503437	−0.0185696
$736$	−41.0270	−1.51228
$737$	−0.411376	−0.0151532
$738$	−23.9936	−0.883216
$739$	−35.5529	−1.30783	−0.653917	−	0.756566i	$-0.726876\pi$
$739$	−35.5529	−1.30783	−0.653917	+	0.756566i	$0.726876\pi$
$740$	41.8590	1.53877
$741$	−0.198581	−0.00729504
$742$	−0.263351	−0.00966791
$743$	−43.9878	−1.61376	−0.806878	−	0.590718i	$-0.798844\pi$
$743$	−43.9878	−1.61376	−0.806878	+	0.590718i	$0.798844\pi$
$744$	3.98737	0.146184
$745$	−13.0750	−0.479030
$746$	−33.9979	−1.24475
$747$	38.6754	1.41506
$748$	−20.5573	−0.751648
$749$	9.68855	0.354012
$750$	1.99797	0.0729555
$751$	30.4391	1.11074	0.555369	−	0.831604i	$-0.312577\pi$
$751$	30.4391	1.11074	0.555369	+	0.831604i	$0.312577\pi$
$752$	6.99490	0.255078
$753$	−1.39254	−0.0507468
$754$	−6.21553	−0.226356
$755$	15.1045	0.549710
$756$	2.42453	0.0881792
$757$	4.89089	0.177763	0.0888813	−	0.996042i	$-0.471671\pi$
$757$	4.89089	0.177763	0.0888813	+	0.996042i	$0.471671\pi$
$758$	−50.7502	−1.84333
$759$	0.272465	0.00988984
$760$	6.46801	0.234619
$761$	8.15961	0.295786	0.147893	−	0.989003i	$-0.452751\pi$
$761$	8.15961	0.295786	0.147893	+	0.989003i	$0.452751\pi$
$762$	−0.992969	−0.0359715
$763$	1.94087	0.0702643
$764$	62.2624	2.25257
$765$	5.90075	0.213342
$766$	−5.72459	−0.206838
$767$	19.6867	0.710847
$768$	4.99914	0.180391
$769$	−7.86639	−0.283669	−0.141835	−	0.989890i	$-0.545300\pi$
$769$	−7.86639	−0.283669	−0.141835	+	0.989890i	$0.545300\pi$
$770$	1.59366	0.0574314
$771$	2.96734	0.106866
$772$	13.3967	0.482158
$773$	−13.1331	−0.472366	−0.236183	−	0.971709i	$-0.575896\pi$
$773$	−13.1331	−0.472366	−0.236183	+	0.971709i	$0.575896\pi$
$774$	94.7489	3.40568
$775$	16.9642	0.609372
$776$	−81.5835	−2.92868
$777$	−0.830667	−0.0298000
$778$	87.4125	3.13389
$779$	2.93873	0.105291
$780$	0.740863	0.0265272
$781$	−5.38840	−0.192812
$782$	14.7809	0.528564
$783$	−0.866357	−0.0309611
$784$	−98.3414	−3.51219
$785$	10.0179	0.357555
$786$	4.44490	0.158544
$787$	29.5835	1.05454	0.527269	−	0.849699i	$-0.323216\pi$
$787$	29.5835	1.05454	0.527269	+	0.849699i	$0.323216\pi$
$788$	63.0501	2.24607
$789$	0.437876	0.0155888
$790$	2.42486	0.0862727
$791$	6.84230	0.243284
$792$	36.7666	1.30644
$793$	14.3164	0.508391
$794$	75.7414	2.68796
$795$	−0.0112167	−0.000397814 0
$796$	90.9585	3.22394
$797$	1.84660	0.0654101	0.0327050	−	0.999465i	$-0.489588\pi$
$797$	1.84660	0.0654101	0.0327050	+	0.999465i	$0.489588\pi$
$798$	−0.202342	−0.00716284
$799$	−1.35247	−0.0478470
$800$	99.6183	3.52204
$801$	28.3961	1.00333
$802$	59.2202	2.09114
$803$	−17.0263	−0.600845
$804$	−0.194864	−0.00687232
$805$	−0.839049	−0.0295726
$806$	18.0627	0.636230
$807$	−2.92396	−0.102928
$808$	115.397	4.05964
$809$	−20.6331	−0.725421	−0.362710	−	0.931902i	$-0.618149\pi$
$809$	−20.6331	−0.725421	−0.362710	+	0.931902i	$0.618149\pi$
$810$	−16.5663	−0.582079
$811$	−1.68539	−0.0591821	−0.0295911	−	0.999562i	$-0.509421\pi$
$811$	−1.68539	−0.0591821	−0.0295911	+	0.999562i	$0.509421\pi$
$812$	−4.63751	−0.162745
$813$	−2.22069	−0.0778831
$814$	−39.7996	−1.39497
$815$	−13.2902	−0.465535
$816$	−4.87503	−0.170660
$817$	−11.6048	−0.406001
$818$	49.2420	1.72171
$819$	3.47614	0.121466
$820$	−10.9638	−0.382872
$821$	54.1803	1.89091	0.945453	−	0.325759i	$-0.105620\pi$
$821$	54.1803	1.89091	0.945453	+	0.325759i	$0.105620\pi$
$822$	−3.05511	−0.106559
$823$	−30.5843	−1.06610	−0.533051	−	0.846083i	$-0.678955\pi$
$823$	−30.5843	−1.06610	−0.533051	+	0.846083i	$0.678955\pi$
$824$	−160.233	−5.58200
$825$	−0.661575	−0.0230331
$826$	20.0596	0.697964
$827$	−46.8038	−1.62753	−0.813763	−	0.581197i	$-0.802585\pi$
$827$	−46.8038	−1.62753	−0.813763	+	0.581197i	$0.802585\pi$
$828$	−30.5158	−1.06050
$829$	51.9904	1.80570	0.902850	−	0.429955i	$-0.141470\pi$
$829$	51.9904	1.80570	0.902850	+	0.429955i	$0.141470\pi$
$830$	24.1348	0.837730
$831$	1.32463	0.0459509
$832$	53.1504	1.84266
$833$	19.0144	0.658810
$834$	3.02854	0.104870
$835$	−13.5891	−0.470270
$836$	−7.09897	−0.245523
$837$	2.51768	0.0870238
$838$	−37.0042	−1.27829
$839$	−18.0116	−0.621829	−0.310915	−	0.950438i	$-0.600635\pi$
$839$	−18.0116	−0.621829	−0.310915	+	0.950438i	$0.600635\pi$
$840$	0.478861	0.0165223
$841$	−27.3429	−0.942858
$842$	44.7318	1.54156
$843$	−0.822647	−0.0283335
$844$	−5.46956	−0.188270
$845$	−6.73841	−0.231808
$846$	3.81324	0.131102
$847$	6.13563	0.210823
$848$	−2.19107	−0.0752415
$849$	−2.86599	−0.0983605
$850$	−35.8897	−1.23101
$851$	20.9542	0.718301
$852$	−2.55243	−0.0874447
$853$	−16.8976	−0.578564	−0.289282	−	0.957244i	$-0.593417\pi$
$853$	−16.8976	−0.578564	−0.289282	+	0.957244i	$0.593417\pi$
$854$	14.5876	0.499178
$855$	2.03768	0.0696874
$856$	139.484	4.76748
$857$	−4.09986	−0.140049	−0.0700243	−	0.997545i	$-0.522308\pi$
$857$	−4.09986	−0.140049	−0.0700243	+	0.997545i	$0.522308\pi$
$858$	−0.704413	−0.0240483
$859$	18.1760	0.620156	0.310078	−	0.950711i	$-0.399645\pi$
$859$	18.1760	0.620156	0.310078	+	0.950711i	$0.399645\pi$
$860$	43.2952	1.47635
$861$	0.217570	0.00741475
$862$	27.0869	0.922582
$863$	−11.3160	−0.385202	−0.192601	−	0.981277i	$-0.561692\pi$
$863$	−11.3160	−0.385202	−0.192601	+	0.981277i	$0.561692\pi$
$864$	14.7845	0.502978
$865$	13.1587	0.447408
$866$	−12.8471	−0.436564
$867$	−0.968285	−0.0328847
$868$	13.4769	0.457435
$869$	−1.68824	−0.0572696
$870$	−0.269747	−0.00914530
$871$	−0.559949	−0.0189732
$872$	27.9424	0.946248
$873$	−25.7021	−0.869885
$874$	5.10424	0.172653
$875$	4.28364	0.144813
$876$	−8.06516	−0.272497
$877$	2.83579	0.0957579	0.0478790	−	0.998853i	$-0.484754\pi$
$877$	2.83579	0.0957579	0.0478790	+	0.998853i	$0.484754\pi$
$878$	37.0599	1.25071
$879$	0.279215	0.00941767
$880$	13.2591	0.446965
$881$	9.84093	0.331549	0.165775	−	0.986164i	$-0.446988\pi$
$881$	9.84093	0.331549	0.165775	+	0.986164i	$0.446988\pi$
$882$	−53.6103	−1.80515
$883$	−2.63309	−0.0886105	−0.0443052	−	0.999018i	$-0.514107\pi$
$883$	−2.63309	−0.0886105	−0.0443052	+	0.999018i	$0.514107\pi$
$884$	−27.9818	−0.941130
$885$	0.854383	0.0287198
$886$	35.9055	1.20627
$887$	47.5318	1.59596	0.797981	−	0.602682i	$-0.205901\pi$
$887$	47.5318	1.59596	0.797981	+	0.602682i	$0.205901\pi$
$888$	−11.9590	−0.401316
$889$	−2.12892	−0.0714018
$890$	17.7201	0.593981
$891$	11.5338	0.386396
$892$	−131.990	−4.41934
$893$	−0.467044	−0.0156290
$894$	5.88875	0.196949
$895$	−11.0521	−0.369432
$896$	25.2189	0.842504
$897$	0.370869	0.0123830
$898$	0.822340	0.0274419
$899$	−4.81569	−0.160612
$900$	74.0957	2.46986
$901$	0.423645	0.0141136
$902$	10.4244	0.347093
$903$	−0.859167	−0.0285913
$904$	98.5073	3.27630
$905$	−3.25829	−0.108309
$906$	−6.80284	−0.226009
$907$	19.0185	0.631499	0.315749	−	0.948843i	$-0.397744\pi$
$907$	19.0185	0.631499	0.315749	+	0.948843i	$0.397744\pi$
$908$	40.4661	1.34292
$909$	36.3547	1.20581
$910$	2.16923	0.0719091
$911$	24.7835	0.821113	0.410556	−	0.911835i	$-0.365334\pi$
$911$	24.7835	0.821113	0.410556	+	0.911835i	$0.365334\pi$
$912$	−1.68348	−0.0557455
$913$	−16.8031	−0.556102
$914$	−21.6676	−0.716701
$915$	0.621318	0.0205401
$916$	−122.450	−4.04587
$917$	9.52985	0.314703
$918$	−5.32644	−0.175799
$919$	38.7632	1.27868	0.639340	−	0.768924i	$-0.279208\pi$
$919$	38.7632	1.27868	0.639340	+	0.768924i	$0.279208\pi$
$920$	−12.0796	−0.398254
$921$	2.47592	0.0815843
$922$	16.1674	0.532446
$923$	−7.33450	−0.241418
$924$	−0.525575	−0.0172901
$925$	−50.8792	−1.67290
$926$	53.9204	1.77193
$927$	−50.4800	−1.65798
$928$	−28.2790	−0.928304
$929$	−25.0709	−0.822550	−0.411275	−	0.911511i	$-0.634916\pi$
$929$	−25.0709	−0.822550	−0.411275	+	0.911511i	$0.634916\pi$
$930$	0.783901	0.0257051
$931$	6.56618	0.215198
$932$	−132.617	−4.34400
$933$	1.18779	0.0388865
$934$	−15.3210	−0.501318
$935$	−2.56367	−0.0838409
$936$	50.0453	1.63578
$937$	−9.47570	−0.309558	−0.154779	−	0.987949i	$-0.549466\pi$
$937$	−9.47570	−0.309558	−0.154779	+	0.987949i	$0.549466\pi$
$938$	−0.570556	−0.0186293
$939$	2.25951	0.0737364
$940$	1.74245	0.0568323
$941$	16.5874	0.540735	0.270367	−	0.962757i	$-0.412855\pi$
$941$	16.5874	0.540735	0.270367	+	0.962757i	$0.412855\pi$
$942$	−4.51191	−0.147006
$943$	−5.48836	−0.178726
$944$	166.895	5.43198
$945$	0.302359	0.00983575
$946$	−41.1651	−1.33839
$947$	−23.9199	−0.777291	−0.388645	−	0.921387i	$-0.627057\pi$
$947$	−23.9199	−0.777291	−0.388645	+	0.921387i	$0.627057\pi$
$948$	−0.799700	−0.0259730
$949$	−23.1756	−0.752311
$950$	−12.3937	−0.402104
$951$	−0.00877675	−0.000284606 0
$952$	−18.0862	−0.586177
$953$	−7.42110	−0.240393	−0.120197	−	0.992750i	$-0.538352\pi$
$953$	−7.42110	−0.240393	−0.120197	+	0.992750i	$0.538352\pi$
$954$	−1.19445	−0.0386717
$955$	7.76465	0.251258
$956$	108.485	3.50864
$957$	0.187804	0.00607083
$958$	83.8004	2.70747
$959$	−6.55015	−0.211515
$960$	2.30667	0.0744474
$961$	−17.0053	−0.548559
$962$	−54.1737	−1.74663
$963$	43.9432	1.41605
$964$	50.0426	1.61176
$965$	1.67069	0.0537813
$966$	0.377894	0.0121585
$967$	3.24046	0.104206	0.0521031	−	0.998642i	$-0.483408\pi$
$967$	3.24046	0.104206	0.0521031	+	0.998642i	$0.483408\pi$
$968$	88.3335	2.83914
$969$	0.325502	0.0104566
$970$	−16.0390	−0.514981
$971$	17.8123	0.571624	0.285812	−	0.958286i	$-0.407737\pi$
$971$	17.8123	0.571624	0.285812	+	0.958286i	$0.407737\pi$
$972$	16.5066	0.529448
$973$	6.49317	0.208162
$974$	−76.7204	−2.45828
$975$	−0.900511	−0.0288394
$976$	121.368	3.88490
$977$	52.2255	1.67084	0.835422	−	0.549610i	$-0.185224\pi$
$977$	52.2255	1.67084	0.835422	+	0.549610i	$0.185224\pi$
$978$	5.98569	0.191401
$979$	−12.3371	−0.394296
$980$	−24.4970	−0.782529
$981$	8.80298	0.281058
$982$	104.384	3.33104
$983$	−1.53581	−0.0489848	−0.0244924	−	0.999700i	$-0.507797\pi$
$983$	−1.53581	−0.0489848	−0.0244924	+	0.999700i	$0.507797\pi$
$984$	3.13231	0.0998544
$985$	7.86289	0.250533
$986$	10.1881	0.324456
$987$	−0.0345778	−0.00110062
$988$	−9.66286	−0.307416
$989$	21.6731	0.689166
$990$	7.22816	0.229726
$991$	−31.3693	−0.996480	−0.498240	−	0.867039i	$-0.666020\pi$
$991$	−31.3693	−0.996480	−0.498240	+	0.867039i	$0.666020\pi$
$992$	82.1803	2.60923
$993$	−3.12296	−0.0991043
$994$	−7.47343	−0.237043
$995$	11.3433	0.359607
$996$	−7.95944	−0.252205
$997$	34.9640	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$997$	34.9640	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$998$	−97.0505	−3.07208
$999$	−7.55105	−0.238905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.2	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.2	✓	75	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)

\(f(q)\)	\(=\)	\(q-2.73305 q^{2} +0.112405 q^{3} +5.46956 q^{4} +0.682101 q^{5} -0.307207 q^{6} -0.658651 q^{7} -9.48248 q^{8} -2.98737 q^{9} +O(q^{10})\)	\(q-2.73305 q^{2} +0.112405 q^{3} +5.46956 q^{4} +0.682101 q^{5} -0.307207 q^{6} -0.658651 q^{7} -9.48248 q^{8} -2.98737 q^{9} -1.86422 q^{10} +1.29791 q^{11} +0.614803 q^{12} +1.76666 q^{13} +1.80013 q^{14} +0.0766712 q^{15} +14.9770 q^{16} -2.89581 q^{17} +8.16462 q^{18} -1.00000 q^{19} +3.73079 q^{20} -0.0740353 q^{21} -3.54724 q^{22} +1.86760 q^{23} -1.06587 q^{24} -4.53474 q^{25} -4.82837 q^{26} -0.673007 q^{27} -3.60253 q^{28} +1.28729 q^{29} -0.209546 q^{30} -3.74094 q^{31} -21.9678 q^{32} +0.145890 q^{33} +7.91439 q^{34} -0.449267 q^{35} -16.3396 q^{36} +11.2199 q^{37} +2.73305 q^{38} +0.198581 q^{39} -6.46801 q^{40} -2.93873 q^{41} +0.202342 q^{42} +11.6048 q^{43} +7.09897 q^{44} -2.03768 q^{45} -5.10424 q^{46} +0.467044 q^{47} +1.68348 q^{48} -6.56618 q^{49} +12.3937 q^{50} -0.325502 q^{51} +9.66286 q^{52} -0.146296 q^{53} +1.83936 q^{54} +0.885302 q^{55} +6.24564 q^{56} -0.112405 q^{57} -3.51824 q^{58} +11.1435 q^{59} +0.419358 q^{60} +8.10366 q^{61} +10.2242 q^{62} +1.96763 q^{63} +30.0852 q^{64} +1.20504 q^{65} -0.398726 q^{66} -0.316953 q^{67} -15.8388 q^{68} +0.209926 q^{69} +1.22787 q^{70} -4.15162 q^{71} +28.3276 q^{72} -13.1183 q^{73} -30.6645 q^{74} -0.509725 q^{75} -5.46956 q^{76} -0.854866 q^{77} -0.542731 q^{78} -1.30074 q^{79} +10.2158 q^{80} +8.88645 q^{81} +8.03169 q^{82} -12.9463 q^{83} -0.404941 q^{84} -1.97523 q^{85} -31.7165 q^{86} +0.144698 q^{87} -12.3074 q^{88} -9.50542 q^{89} +5.56909 q^{90} -1.16361 q^{91} +10.2149 q^{92} -0.420499 q^{93} -1.27646 q^{94} -0.682101 q^{95} -2.46928 q^{96} +8.60361 q^{97} +17.9457 q^{98} -3.87732 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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