LMFDB - Embedded newform 4009.2.a.c.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.51184 q^{2} -2.51408 q^{3} +4.30932 q^{4} +3.86977 q^{5} +6.31496 q^{6} +2.24136 q^{7} -5.80064 q^{8} +3.32060 q^{9} +O(q^{10})$	\(q-2.51184 q^{2} -2.51408 q^{3} +4.30932 q^{4} +3.86977 q^{5} +6.31496 q^{6} +2.24136 q^{7} -5.80064 q^{8} +3.32060 q^{9} -9.72023 q^{10} -3.66849 q^{11} -10.8340 q^{12} -4.71240 q^{13} -5.62993 q^{14} -9.72891 q^{15} +5.95162 q^{16} +5.95455 q^{17} -8.34081 q^{18} +1.00000 q^{19} +16.6761 q^{20} -5.63496 q^{21} +9.21464 q^{22} -5.33707 q^{23} +14.5833 q^{24} +9.97511 q^{25} +11.8368 q^{26} -0.806021 q^{27} +9.65875 q^{28} -4.89281 q^{29} +24.4374 q^{30} +1.00382 q^{31} -3.34822 q^{32} +9.22288 q^{33} -14.9569 q^{34} +8.67355 q^{35} +14.3096 q^{36} +5.73851 q^{37} -2.51184 q^{38} +11.8474 q^{39} -22.4472 q^{40} -8.57893 q^{41} +14.1541 q^{42} -1.69839 q^{43} -15.8087 q^{44} +12.8500 q^{45} +13.4058 q^{46} +0.701681 q^{47} -14.9629 q^{48} -1.97630 q^{49} -25.0559 q^{50} -14.9702 q^{51} -20.3073 q^{52} +0.437181 q^{53} +2.02459 q^{54} -14.1962 q^{55} -13.0013 q^{56} -2.51408 q^{57} +12.2899 q^{58} +13.5962 q^{59} -41.9250 q^{60} -0.0150536 q^{61} -2.52144 q^{62} +7.44267 q^{63} -3.49307 q^{64} -18.2359 q^{65} -23.1664 q^{66} +1.84947 q^{67} +25.6601 q^{68} +13.4178 q^{69} -21.7865 q^{70} -13.3069 q^{71} -19.2616 q^{72} -8.08633 q^{73} -14.4142 q^{74} -25.0782 q^{75} +4.30932 q^{76} -8.22241 q^{77} -29.7586 q^{78} -4.80577 q^{79} +23.0314 q^{80} -7.93541 q^{81} +21.5489 q^{82} -15.3828 q^{83} -24.2829 q^{84} +23.0427 q^{85} +4.26607 q^{86} +12.3009 q^{87} +21.2796 q^{88} +8.34046 q^{89} -32.2770 q^{90} -10.5622 q^{91} -22.9992 q^{92} -2.52369 q^{93} -1.76251 q^{94} +3.86977 q^{95} +8.41769 q^{96} +0.165235 q^{97} +4.96414 q^{98} -12.1816 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.51184	−1.77614	−0.888068	−	0.459711i	$-0.847953\pi$
$2$	−2.51184	−1.77614	−0.888068	+	0.459711i	$0.847953\pi$
$3$	−2.51408	−1.45151	−0.725753	−	0.687956i	$-0.758508\pi$
$3$	−2.51408	−1.45151	−0.725753	+	0.687956i	$0.758508\pi$
$4$	4.30932	2.15466
$5$	3.86977	1.73061	0.865307	−	0.501243i	$-0.167124\pi$
$5$	3.86977	1.73061	0.865307	+	0.501243i	$0.167124\pi$
$6$	6.31496	2.57807
$7$	2.24136	0.847155	0.423578	−	0.905860i	$-0.360774\pi$
$7$	2.24136	0.847155	0.423578	+	0.905860i	$0.360774\pi$
$8$	−5.80064	−2.05084
$9$	3.32060	1.10687
$10$	−9.72023	−3.07381
$11$	−3.66849	−1.10609	−0.553045	−	0.833151i	$-0.686534\pi$
$11$	−3.66849	−1.10609	−0.553045	+	0.833151i	$0.686534\pi$
$12$	−10.8340	−3.12750
$13$	−4.71240	−1.30699	−0.653493	−	0.756933i	$-0.726697\pi$
$13$	−4.71240	−1.30699	−0.653493	+	0.756933i	$0.726697\pi$
$14$	−5.62993	−1.50466
$15$	−9.72891	−2.51199
$16$	5.95162	1.48791
$17$	5.95455	1.44419	0.722095	−	0.691794i	$-0.243179\pi$
$17$	5.95455	1.44419	0.722095	+	0.691794i	$0.243179\pi$
$18$	−8.34081	−1.96595
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	16.6761	3.72889
$21$	−5.63496	−1.22965
$22$	9.21464	1.96457
$23$	−5.33707	−1.11286	−0.556428	−	0.830896i	$-0.687828\pi$
$23$	−5.33707	−1.11286	−0.556428	+	0.830896i	$0.687828\pi$
$24$	14.5833	2.97680
$25$	9.97511	1.99502
$26$	11.8368	2.32138
$27$	−0.806021	−0.155119
$28$	9.65875	1.82533
$29$	−4.89281	−0.908572	−0.454286	−	0.890856i	$-0.650106\pi$
$29$	−4.89281	−0.908572	−0.454286	+	0.890856i	$0.650106\pi$
$30$	24.4374	4.46165
$31$	1.00382	0.180292	0.0901460	−	0.995929i	$-0.471267\pi$
$31$	1.00382	0.180292	0.0901460	+	0.995929i	$0.471267\pi$
$32$	−3.34822	−0.591887
$33$	9.22288	1.60550
$34$	−14.9569	−2.56508
$35$	8.67355	1.46610
$36$	14.3096	2.38493
$37$	5.73851	0.943405	0.471703	−	0.881758i	$-0.343640\pi$
$37$	5.73851	0.943405	0.471703	+	0.881758i	$0.343640\pi$
$38$	−2.51184	−0.407474
$39$	11.8474	1.89710
$40$	−22.4472	−3.54921
$41$	−8.57893	−1.33980	−0.669902	−	0.742449i	$-0.733664\pi$
$41$	−8.57893	−1.33980	−0.669902	+	0.742449i	$0.733664\pi$
$42$	14.1541	2.18403
$43$	−1.69839	−0.259002	−0.129501	−	0.991579i	$-0.541337\pi$
$43$	−1.69839	−0.259002	−0.129501	+	0.991579i	$0.541337\pi$
$44$	−15.8087	−2.38325
$45$	12.8500	1.91556
$46$	13.4058	1.97658
$47$	0.701681	0.102351	0.0511753	−	0.998690i	$-0.483703\pi$
$47$	0.701681	0.102351	0.0511753	+	0.998690i	$0.483703\pi$
$48$	−14.9629	−2.15970
$49$	−1.97630	−0.282328
$50$	−25.0559	−3.54343
$51$	−14.9702	−2.09625
$52$	−20.3073	−2.81611
$53$	0.437181	0.0600514	0.0300257	−	0.999549i	$-0.490441\pi$
$53$	0.437181	0.0600514	0.0300257	+	0.999549i	$0.490441\pi$
$54$	2.02459	0.275512
$55$	−14.1962	−1.91422
$56$	−13.0013	−1.73738
$57$	−2.51408	−0.332998
$58$	12.2899	1.61375
$59$	13.5962	1.77008	0.885040	−	0.465515i	$-0.154131\pi$
$59$	13.5962	1.77008	0.885040	+	0.465515i	$0.154131\pi$
$60$	−41.9250	−5.41250
$61$	−0.0150536	−0.00192742	−0.000963711	−	1.00000i	$-0.500307\pi$
$61$	−0.0150536	−0.00192742	−0.000963711		1.00000i	$0.500307\pi$
$62$	−2.52144	−0.320223
$63$	7.44267	0.937688
$64$	−3.49307	−0.436634
$65$	−18.2359	−2.26189
$66$	−23.1664	−2.85158
$67$	1.84947	0.225948	0.112974	−	0.993598i	$-0.463962\pi$
$67$	1.84947	0.225948	0.112974	+	0.993598i	$0.463962\pi$
$68$	25.6601	3.11174
$69$	13.4178	1.61532
$70$	−21.7865	−2.60399
$71$	−13.3069	−1.57924	−0.789621	−	0.613595i	$-0.789723\pi$
$71$	−13.3069	−1.57924	−0.789621	+	0.613595i	$0.789723\pi$
$72$	−19.2616	−2.27001
$73$	−8.08633	−0.946434	−0.473217	−	0.880946i	$-0.656907\pi$
$73$	−8.08633	−0.946434	−0.473217	+	0.880946i	$0.656907\pi$
$74$	−14.4142	−1.67562
$75$	−25.0782	−2.89579
$76$	4.30932	0.494313
$77$	−8.22241	−0.937030
$78$	−29.7586	−3.36950
$79$	−4.80577	−0.540691	−0.270346	−	0.962763i	$-0.587138\pi$
$79$	−4.80577	−0.540691	−0.270346	+	0.962763i	$0.587138\pi$
$80$	23.0314	2.57499
$81$	−7.93541	−0.881712
$82$	21.5489	2.37968
$83$	−15.3828	−1.68848	−0.844241	−	0.535964i	$-0.819948\pi$
$83$	−15.3828	−1.68848	−0.844241	+	0.535964i	$0.819948\pi$
$84$	−24.2829	−2.64948
$85$	23.0427	2.49933
$86$	4.26607	0.460022
$87$	12.3009	1.31880
$88$	21.2796	2.26841
$89$	8.34046	0.884087	0.442044	−	0.896994i	$-0.354254\pi$
$89$	8.34046	0.884087	0.442044	+	0.896994i	$0.354254\pi$
$90$	−32.2770	−3.40230
$91$	−10.5622	−1.10722
$92$	−22.9992	−2.39783
$93$	−2.52369	−0.261695
$94$	−1.76251	−0.181789
$95$	3.86977	0.397030
$96$	8.41769	0.859127
$97$	0.165235	0.0167770	0.00838852	−	0.999965i	$-0.497330\pi$
$97$	0.165235	0.0167770	0.00838852	+	0.999965i	$0.497330\pi$
$98$	4.96414	0.501454
$99$	−12.1816	−1.22430
$100$	42.9860	4.29860
$101$	9.50154	0.945438	0.472719	−	0.881213i	$-0.343273\pi$
$101$	9.50154	0.945438	0.472719	+	0.881213i	$0.343273\pi$
$102$	37.6027	3.72323
$103$	3.86988	0.381311	0.190656	−	0.981657i	$-0.438939\pi$
$103$	3.86988	0.381311	0.190656	+	0.981657i	$0.438939\pi$
$104$	27.3350	2.68041
$105$	−21.8060	−2.12805
$106$	−1.09813	−0.106659
$107$	−5.54097	−0.535665	−0.267833	−	0.963465i	$-0.586307\pi$
$107$	−5.54097	−0.535665	−0.267833	+	0.963465i	$0.586307\pi$
$108$	−3.47341	−0.334229
$109$	1.31470	0.125926	0.0629628	−	0.998016i	$-0.479945\pi$
$109$	1.31470	0.125926	0.0629628	+	0.998016i	$0.479945\pi$
$110$	35.6585	3.39991
$111$	−14.4271	−1.36936
$112$	13.3397	1.26049
$113$	5.08727	0.478570	0.239285	−	0.970949i	$-0.423087\pi$
$113$	5.08727	0.478570	0.239285	+	0.970949i	$0.423087\pi$
$114$	6.31496	0.591450
$115$	−20.6532	−1.92592
$116$	−21.0847	−1.95766
$117$	−15.6480	−1.44666
$118$	−34.1515	−3.14390
$119$	13.3463	1.22345
$120$	56.4340	5.15169
$121$	2.45780	0.223437
$122$	0.0378123	0.00342336
$123$	21.5681	1.94473
$124$	4.32580	0.388468
$125$	19.2525	1.72200
$126$	−18.6948	−1.66546
$127$	−8.44650	−0.749506	−0.374753	−	0.927125i	$-0.622272\pi$
$127$	−8.44650	−0.749506	−0.374753	+	0.927125i	$0.622272\pi$
$128$	15.4705	1.36741
$129$	4.26988	0.375942
$130$	45.8056	4.01742
$131$	−19.3216	−1.68814	−0.844068	−	0.536235i	$-0.819846\pi$
$131$	−19.3216	−1.68814	−0.844068	+	0.536235i	$0.819846\pi$
$132$	39.7444	3.45930
$133$	2.24136	0.194351
$134$	−4.64556	−0.401315
$135$	−3.11912	−0.268451
$136$	−34.5402	−2.96180
$137$	−11.2507	−0.961210	−0.480605	−	0.876937i	$-0.659583\pi$
$137$	−11.2507	−0.961210	−0.480605	+	0.876937i	$0.659583\pi$
$138$	−33.7034	−2.86902
$139$	−17.6209	−1.49459	−0.747294	−	0.664494i	$-0.768647\pi$
$139$	−17.6209	−1.49459	−0.747294	+	0.664494i	$0.768647\pi$
$140$	37.3771	3.15895
$141$	−1.76408	−0.148563
$142$	33.4248	2.80495
$143$	17.2874	1.44564
$144$	19.7630	1.64691
$145$	−18.9340	−1.57239
$146$	20.3115	1.68100
$147$	4.96857	0.409801
$148$	24.7291	2.03272
$149$	−2.69684	−0.220934	−0.110467	−	0.993880i	$-0.535235\pi$
$149$	−2.69684	−0.220934	−0.110467	+	0.993880i	$0.535235\pi$
$150$	62.9924	5.14331
$151$	6.78790	0.552391	0.276196	−	0.961101i	$-0.410926\pi$
$151$	6.78790	0.552391	0.276196	+	0.961101i	$0.410926\pi$
$152$	−5.80064	−0.470494
$153$	19.7727	1.59853
$154$	20.6533	1.66429
$155$	3.88456	0.312016
$156$	51.0541	4.08760
$157$	21.5192	1.71742	0.858710	−	0.512461i	$-0.171266\pi$
$157$	21.5192	1.71742	0.858710	+	0.512461i	$0.171266\pi$
$158$	12.0713	0.960342
$159$	−1.09911	−0.0871649
$160$	−12.9568	−1.02433
$161$	−11.9623	−0.942761
$162$	19.9324	1.56604
$163$	10.4381	0.817571	0.408786	−	0.912630i	$-0.365952\pi$
$163$	10.4381	0.817571	0.408786	+	0.912630i	$0.365952\pi$
$164$	−36.9694	−2.88683
$165$	35.6904	2.77849
$166$	38.6391	2.99897
$167$	−7.03352	−0.544270	−0.272135	−	0.962259i	$-0.587730\pi$
$167$	−7.03352	−0.544270	−0.272135	+	0.962259i	$0.587730\pi$
$168$	32.6864	2.52181
$169$	9.20673	0.708210
$170$	−57.8796	−4.43916
$171$	3.32060	0.253933
$172$	−7.31890	−0.558061
$173$	16.2628	1.23644	0.618218	−	0.786007i	$-0.287855\pi$
$173$	16.2628	1.23644	0.618218	+	0.786007i	$0.287855\pi$
$174$	−30.8979	−2.34236
$175$	22.3578	1.69009
$176$	−21.8335	−1.64576
$177$	−34.1821	−2.56928
$178$	−20.9499	−1.57026
$179$	−16.3585	−1.22269	−0.611344	−	0.791365i	$-0.709371\pi$
$179$	−16.3585	−1.22269	−0.611344	+	0.791365i	$0.709371\pi$
$180$	55.3747	4.12738
$181$	−20.9874	−1.55998	−0.779990	−	0.625792i	$-0.784776\pi$
$181$	−20.9874	−1.55998	−0.779990	+	0.625792i	$0.784776\pi$
$182$	26.5305	1.96657
$183$	0.0378461	0.00279766
$184$	30.9584	2.28229
$185$	22.2067	1.63267
$186$	6.33910	0.464806
$187$	−21.8442	−1.59741
$188$	3.02377	0.220531
$189$	−1.80659	−0.131410
$190$	−9.72023	−0.705179
$191$	−9.80803	−0.709684	−0.354842	−	0.934926i	$-0.615465\pi$
$191$	−9.80803	−0.709684	−0.354842	+	0.934926i	$0.615465\pi$
$192$	8.78186	0.633776
$193$	15.2445	1.09732	0.548660	−	0.836046i	$-0.315138\pi$
$193$	15.2445	1.09732	0.548660	+	0.836046i	$0.315138\pi$
$194$	−0.415043	−0.0297983
$195$	45.8465	3.28314
$196$	−8.51651	−0.608322
$197$	20.1164	1.43323	0.716616	−	0.697468i	$-0.245690\pi$
$197$	20.1164	1.43323	0.716616	+	0.697468i	$0.245690\pi$
$198$	30.5982	2.17452
$199$	17.9659	1.27357	0.636785	−	0.771042i	$-0.280264\pi$
$199$	17.9659	1.27357	0.636785	+	0.771042i	$0.280264\pi$
$200$	−57.8621	−4.09147
$201$	−4.64971	−0.327965
$202$	−23.8663	−1.67923
$203$	−10.9666	−0.769701
$204$	−64.5115	−4.51671
$205$	−33.1985	−2.31868
$206$	−9.72052	−0.677261
$207$	−17.7223	−1.23178
$208$	−28.0464	−1.94467
$209$	−3.66849	−0.253755
$210$	54.7731	3.77971
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	1.88395	0.129390
$213$	33.4547	2.29228
$214$	13.9180	0.951415
$215$	−6.57236	−0.448231
$216$	4.67544	0.318124
$217$	2.24993	0.152735
$218$	−3.30232	−0.223661
$219$	20.3297	1.37375
$220$	−61.1760	−4.12449
$221$	−28.0602	−1.88753
$222$	36.2385	2.43217
$223$	7.87139	0.527107	0.263554	−	0.964645i	$-0.415105\pi$
$223$	7.87139	0.527107	0.263554	+	0.964645i	$0.415105\pi$
$224$	−7.50457	−0.501420
$225$	33.1234	2.20823
$226$	−12.7784	−0.850006
$227$	−23.4109	−1.55384	−0.776918	−	0.629602i	$-0.783218\pi$
$227$	−23.4109	−1.55384	−0.776918	+	0.629602i	$0.783218\pi$
$228$	−10.8340	−0.717498
$229$	3.65273	0.241379	0.120690	−	0.992690i	$-0.461489\pi$
$229$	3.65273	0.241379	0.120690	+	0.992690i	$0.461489\pi$
$230$	51.8775	3.42070
$231$	20.6718	1.36010
$232$	28.3814	1.86333
$233$	24.4712	1.60316	0.801581	−	0.597887i	$-0.203993\pi$
$233$	24.4712	1.60316	0.801581	+	0.597887i	$0.203993\pi$
$234$	39.3053	2.56947
$235$	2.71534	0.177129
$236$	58.5906	3.81392
$237$	12.0821	0.784816
$238$	−33.5237	−2.17302
$239$	−19.1928	−1.24148	−0.620740	−	0.784016i	$-0.713168\pi$
$239$	−19.1928	−1.24148	−0.620740	+	0.784016i	$0.713168\pi$
$240$	−57.9028	−3.73761
$241$	−15.0736	−0.970977	−0.485489	−	0.874243i	$-0.661358\pi$
$241$	−15.0736	−0.970977	−0.485489	+	0.874243i	$0.661358\pi$
$242$	−6.17360	−0.396854
$243$	22.3683	1.43493
$244$	−0.0648710	−0.00415294
$245$	−7.64782	−0.488601
$246$	−54.1756	−3.45411
$247$	−4.71240	−0.299843
$248$	−5.82282	−0.369749
$249$	38.6736	2.45084
$250$	−48.3592	−3.05851
$251$	−26.0152	−1.64207	−0.821034	−	0.570879i	$-0.806602\pi$
$251$	−26.0152	−1.64207	−0.821034	+	0.570879i	$0.806602\pi$
$252$	32.0729	2.02040
$253$	19.5790	1.23092
$254$	21.2162	1.33122
$255$	−57.9313	−3.62780
$256$	−31.8731	−1.99207
$257$	−29.6834	−1.85160	−0.925799	−	0.378017i	$-0.876606\pi$
$257$	−29.6834	−1.85160	−0.925799	+	0.378017i	$0.876606\pi$
$258$	−10.7252	−0.667725
$259$	12.8621	0.799211
$260$	−78.5844	−4.87360
$261$	−16.2471	−1.00567
$262$	48.5327	2.99836
$263$	−23.3222	−1.43811	−0.719053	−	0.694955i	$-0.755424\pi$
$263$	−23.3222	−1.43811	−0.719053	+	0.694955i	$0.755424\pi$
$264$	−53.4986	−3.29261
$265$	1.69179	0.103926
$266$	−5.62993	−0.345193
$267$	−20.9686	−1.28326
$268$	7.96996	0.486842
$269$	−19.1515	−1.16769	−0.583843	−	0.811866i	$-0.698452\pi$
$269$	−19.1515	−1.16769	−0.583843	+	0.811866i	$0.698452\pi$
$270$	7.83471	0.476805
$271$	−5.71778	−0.347331	−0.173665	−	0.984805i	$-0.555561\pi$
$271$	−5.71778	−0.347331	−0.173665	+	0.984805i	$0.555561\pi$
$272$	35.4392	2.14882
$273$	26.5542	1.60713
$274$	28.2599	1.70724
$275$	−36.5936	−2.20668
$276$	57.8217	3.48046
$277$	16.8896	1.01480	0.507399	−	0.861711i	$-0.330607\pi$
$277$	16.8896	1.01480	0.507399	+	0.861711i	$0.330607\pi$
$278$	44.2609	2.65459
$279$	3.33330	0.199559
$280$	−50.3122	−3.00673
$281$	−15.4220	−0.920002	−0.460001	−	0.887918i	$-0.652151\pi$
$281$	−15.4220	−0.920002	−0.460001	+	0.887918i	$0.652151\pi$
$282$	4.43109	0.263867
$283$	−10.9939	−0.653517	−0.326758	−	0.945108i	$-0.605956\pi$
$283$	−10.9939	−0.653517	−0.326758	+	0.945108i	$0.605956\pi$
$284$	−57.3438	−3.40273
$285$	−9.72891	−0.576291
$286$	−43.4231	−2.56766
$287$	−19.2285	−1.13502
$288$	−11.1181	−0.655141
$289$	18.4566	1.08568
$290$	47.5592	2.79277
$291$	−0.415413	−0.0243520
$292$	−34.8466	−2.03924
$293$	−1.44814	−0.0846012	−0.0423006	−	0.999105i	$-0.513469\pi$
$293$	−1.44814	−0.0846012	−0.0423006	+	0.999105i	$0.513469\pi$
$294$	−12.4802	−0.727863
$295$	52.6143	3.06332
$296$	−33.2871	−1.93477
$297$	2.95688	0.171576
$298$	6.77403	0.392409
$299$	25.1504	1.45449
$300$	−108.070	−6.23944
$301$	−3.80670	−0.219414
$302$	−17.0501	−0.981123
$303$	−23.8876	−1.37231
$304$	5.95162	0.341349
$305$	−0.0582541	−0.00333562
$306$	−49.6658	−2.83920
$307$	3.45110	0.196965	0.0984823	−	0.995139i	$-0.468601\pi$
$307$	3.45110	0.196965	0.0984823	+	0.995139i	$0.468601\pi$
$308$	−35.4330	−2.01898
$309$	−9.72920	−0.553475
$310$	−9.75739	−0.554182
$311$	−3.20625	−0.181810	−0.0909049	−	0.995860i	$-0.528976\pi$
$311$	−3.20625	−0.181810	−0.0909049	+	0.995860i	$0.528976\pi$
$312$	−68.7223	−3.89064
$313$	−4.24418	−0.239895	−0.119948	−	0.992780i	$-0.538273\pi$
$313$	−4.24418	−0.239895	−0.119948	+	0.992780i	$0.538273\pi$
$314$	−54.0528	−3.05037
$315$	28.8014	1.62278
$316$	−20.7096	−1.16501
$317$	26.1975	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$317$	26.1975	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$318$	2.76078	0.154817
$319$	17.9492	1.00496
$320$	−13.5174	−0.755644
$321$	13.9304	0.777521
$322$	30.0473	1.67447
$323$	5.95455	0.331320
$324$	−34.1962	−1.89979
$325$	−47.0067	−2.60746
$326$	−26.2187	−1.45212
$327$	−3.30527	−0.182782
$328$	49.7633	2.74772
$329$	1.57272	0.0867069
$330$	−89.6485	−4.93499
$331$	17.7740	0.976945	0.488472	−	0.872579i	$-0.337554\pi$
$331$	17.7740	0.976945	0.488472	+	0.872579i	$0.337554\pi$
$332$	−66.2894	−3.63811
$333$	19.0553	1.04422
$334$	17.6670	0.966698
$335$	7.15701	0.391029
$336$	−33.5372	−1.82960
$337$	18.5094	1.00827	0.504135	−	0.863625i	$-0.331811\pi$
$337$	18.5094	1.00827	0.504135	+	0.863625i	$0.331811\pi$
$338$	−23.1258	−1.25788
$339$	−12.7898	−0.694647
$340$	99.2986	5.38522
$341$	−3.68251	−0.199419
$342$	−8.34081	−0.451019
$343$	−20.1191	−1.08633
$344$	9.85174	0.531170
$345$	51.9239	2.79549
$346$	−40.8495	−2.19608
$347$	4.93695	0.265030	0.132515	−	0.991181i	$-0.457695\pi$
$347$	4.93695	0.265030	0.132515	+	0.991181i	$0.457695\pi$
$348$	53.0086	2.84156
$349$	21.4593	1.14869	0.574346	−	0.818613i	$-0.305257\pi$
$349$	21.4593	1.14869	0.574346	+	0.818613i	$0.305257\pi$
$350$	−56.1592	−3.00184
$351$	3.79830	0.202738
$352$	12.2829	0.654681
$353$	−19.8723	−1.05770	−0.528848	−	0.848717i	$-0.677376\pi$
$353$	−19.8723	−1.05770	−0.528848	+	0.848717i	$0.677376\pi$
$354$	85.8597	4.56339
$355$	−51.4947	−2.73306
$356$	35.9418	1.90491
$357$	−33.5537	−1.77585
$358$	41.0898	2.17166
$359$	−23.3055	−1.23002	−0.615009	−	0.788520i	$-0.710848\pi$
$359$	−23.3055	−1.23002	−0.615009	+	0.788520i	$0.710848\pi$
$360$	−74.5381	−3.92850
$361$	1.00000	0.0526316
$362$	52.7169	2.77074
$363$	−6.17912	−0.324320
$364$	−45.5159	−2.38568
$365$	−31.2922	−1.63791
$366$	−0.0950631	−0.00496903
$367$	−5.44203	−0.284071	−0.142036	−	0.989862i	$-0.545365\pi$
$367$	−5.44203	−0.284071	−0.142036	+	0.989862i	$0.545365\pi$
$368$	−31.7642	−1.65582
$369$	−28.4872	−1.48299
$370$	−55.7796	−2.89985
$371$	0.979880	0.0508728
$372$	−10.8754	−0.563864
$373$	10.1229	0.524144	0.262072	−	0.965048i	$-0.415594\pi$
$373$	10.1229	0.524144	0.262072	+	0.965048i	$0.415594\pi$
$374$	54.8690	2.83721
$375$	−48.4024	−2.49949
$376$	−4.07020	−0.209905
$377$	23.0569	1.18749
$378$	4.53785	0.233402
$379$	25.3608	1.30270	0.651348	−	0.758779i	$-0.274204\pi$
$379$	25.3608	1.30270	0.651348	+	0.758779i	$0.274204\pi$
$380$	16.6761	0.855465
$381$	21.2352	1.08791
$382$	24.6362	1.26050
$383$	−6.28404	−0.321100	−0.160550	−	0.987028i	$-0.551327\pi$
$383$	−6.28404	−0.321100	−0.160550	+	0.987028i	$0.551327\pi$
$384$	−38.8940	−1.98480
$385$	−31.8188	−1.62164
$386$	−38.2916	−1.94899
$387$	−5.63967	−0.286680
$388$	0.712050	0.0361489
$389$	4.92909	0.249915	0.124958	−	0.992162i	$-0.460121\pi$
$389$	4.92909	0.249915	0.124958	+	0.992162i	$0.460121\pi$
$390$	−115.159	−5.83130
$391$	−31.7798	−1.60717
$392$	11.4638	0.579009
$393$	48.5761	2.45034
$394$	−50.5290	−2.54562
$395$	−18.5972	−0.935728
$396$	−52.4944	−2.63794
$397$	30.2060	1.51599	0.757997	−	0.652258i	$-0.226178\pi$
$397$	30.2060	1.51599	0.757997	+	0.652258i	$0.226178\pi$
$398$	−45.1274	−2.26203
$399$	−5.63496	−0.282101
$400$	59.3681	2.96841
$401$	−9.21321	−0.460086	−0.230043	−	0.973181i	$-0.573887\pi$
$401$	−9.21321	−0.460086	−0.230043	+	0.973181i	$0.573887\pi$
$402$	11.6793	0.582511
$403$	−4.73042	−0.235639
$404$	40.9452	2.03710
$405$	−30.7082	−1.52590
$406$	27.5462	1.36709
$407$	−21.0517	−1.04349
$408$	86.8369	4.29907
$409$	3.70221	0.183062	0.0915312	−	0.995802i	$-0.470824\pi$
$409$	3.70221	0.183062	0.0915312	+	0.995802i	$0.470824\pi$
$410$	83.3892	4.11830
$411$	28.2851	1.39520
$412$	16.6766	0.821596
$413$	30.4741	1.49953
$414$	44.5155	2.18782
$415$	−59.5279	−2.92211
$416$	15.7782	0.773587
$417$	44.3004	2.16940
$418$	9.21464	0.450703
$419$	−31.9262	−1.55970	−0.779849	−	0.625967i	$-0.784704\pi$
$419$	−31.9262	−1.55970	−0.779849	+	0.625967i	$0.784704\pi$
$420$	−93.9692	−4.58523
$421$	19.1099	0.931360	0.465680	−	0.884953i	$-0.345810\pi$
$421$	19.1099	0.931360	0.465680	+	0.884953i	$0.345810\pi$
$422$	−2.51184	−0.122274
$423$	2.33000	0.113289
$424$	−2.53593	−0.123156
$425$	59.3973	2.88119
$426$	−84.0327	−4.07140
$427$	−0.0337407	−0.00163282
$428$	−23.8778	−1.15418
$429$	−43.4619	−2.09836
$430$	16.5087	0.796120
$431$	33.5101	1.61412	0.807062	−	0.590467i	$-0.201056\pi$
$431$	33.5101	1.61412	0.807062	+	0.590467i	$0.201056\pi$
$432$	−4.79714	−0.230802
$433$	−18.0900	−0.869351	−0.434675	−	0.900587i	$-0.643137\pi$
$433$	−18.0900	−0.869351	−0.434675	+	0.900587i	$0.643137\pi$
$434$	−5.65146	−0.271279
$435$	47.6017	2.28233
$436$	5.66548	0.271327
$437$	−5.33707	−0.255307
$438$	−51.0649	−2.43997
$439$	7.49629	0.357778	0.178889	−	0.983869i	$-0.442750\pi$
$439$	7.49629	0.357778	0.178889	+	0.983869i	$0.442750\pi$
$440$	82.3471	3.92574
$441$	−6.56250	−0.312500
$442$	70.4827	3.35252
$443$	−10.0173	−0.475935	−0.237967	−	0.971273i	$-0.576481\pi$
$443$	−10.0173	−0.475935	−0.237967	+	0.971273i	$0.576481\pi$
$444$	−62.1710	−2.95050
$445$	32.2757	1.53001
$446$	−19.7716	−0.936215
$447$	6.78008	0.320687
$448$	−7.82923	−0.369896
$449$	−2.44501	−0.115387	−0.0576937	−	0.998334i	$-0.518375\pi$
$449$	−2.44501	−0.115387	−0.0576937	+	0.998334i	$0.518375\pi$
$450$	−83.2005	−3.92211
$451$	31.4717	1.48195
$452$	21.9227	1.03116
$453$	−17.0653	−0.801799
$454$	58.8044	2.75983
$455$	−40.8733	−1.91617
$456$	14.5833	0.682925
$457$	−26.7420	−1.25094	−0.625470	−	0.780248i	$-0.715093\pi$
$457$	−26.7420	−1.25094	−0.625470	+	0.780248i	$0.715093\pi$
$458$	−9.17506	−0.428723
$459$	−4.79949	−0.224021
$460$	−89.0014	−4.14971
$461$	3.88863	0.181112	0.0905558	−	0.995891i	$-0.471136\pi$
$461$	3.88863	0.181112	0.0905558	+	0.995891i	$0.471136\pi$
$462$	−51.9242	−2.41573
$463$	2.74342	0.127497	0.0637487	−	0.997966i	$-0.479694\pi$
$463$	2.74342	0.127497	0.0637487	+	0.997966i	$0.479694\pi$
$464$	−29.1202	−1.35187
$465$	−9.76611	−0.452892
$466$	−61.4676	−2.84743
$467$	41.1650	1.90489	0.952444	−	0.304715i	$-0.0985612\pi$
$467$	41.1650	1.90489	0.952444	+	0.304715i	$0.0985612\pi$
$468$	−67.4324	−3.11706
$469$	4.14533	0.191413
$470$	−6.82050	−0.314606
$471$	−54.1011	−2.49284
$472$	−78.8670	−3.63015
$473$	6.23051	0.286479
$474$	−30.3482	−1.39394
$475$	9.97511	0.457690
$476$	57.5135	2.63613
$477$	1.45170	0.0664689
$478$	48.2092	2.20504
$479$	−19.9855	−0.913159	−0.456580	−	0.889683i	$-0.650926\pi$
$479$	−19.9855	−0.913159	−0.456580	+	0.889683i	$0.650926\pi$
$480$	32.5745	1.48682
$481$	−27.0422	−1.23302
$482$	37.8625	1.72459
$483$	30.0742	1.36842
$484$	10.5915	0.481431
$485$	0.639420	0.0290346
$486$	−56.1856	−2.54863
$487$	−22.1530	−1.00385	−0.501923	−	0.864912i	$-0.667374\pi$
$487$	−22.1530	−1.00385	−0.501923	+	0.864912i	$0.667374\pi$
$488$	0.0873208	0.00395283
$489$	−26.2421	−1.18671
$490$	19.2101	0.867822
$491$	28.4675	1.28472	0.642360	−	0.766403i	$-0.277956\pi$
$491$	28.4675	1.28472	0.642360	+	0.766403i	$0.277956\pi$
$492$	92.9441	4.19024
$493$	−29.1345	−1.31215
$494$	11.8368	0.532562
$495$	−47.1399	−2.11878
$496$	5.97438	0.268257
$497$	−29.8256	−1.33786
$498$	−97.1417	−4.35303
$499$	−5.14118	−0.230151	−0.115075	−	0.993357i	$-0.536711\pi$
$499$	−5.14118	−0.230151	−0.115075	+	0.993357i	$0.536711\pi$
$500$	82.9654	3.71033
$501$	17.6828	0.790011
$502$	65.3461	2.91654
$503$	4.36152	0.194471	0.0972353	−	0.995261i	$-0.469000\pi$
$503$	4.36152	0.194471	0.0972353	+	0.995261i	$0.469000\pi$
$504$	−43.1723	−1.92305
$505$	36.7688	1.63619
$506$	−49.1792	−2.18628
$507$	−23.1465	−1.02797
$508$	−36.3987	−1.61493
$509$	1.86957	0.0828670	0.0414335	−	0.999141i	$-0.486808\pi$
$509$	1.86957	0.0828670	0.0414335	+	0.999141i	$0.486808\pi$
$510$	145.514	6.44346
$511$	−18.1244	−0.801776
$512$	49.1192	2.17078
$513$	−0.806021	−0.0355867
$514$	74.5598	3.28869
$515$	14.9756	0.659902
$516$	18.4003	0.810028
$517$	−2.57411	−0.113209
$518$	−32.3074	−1.41951
$519$	−40.8859	−1.79469
$520$	105.780	4.63876
$521$	−27.0090	−1.18329	−0.591644	−	0.806200i	$-0.701521\pi$
$521$	−27.0090	−1.18329	−0.591644	+	0.806200i	$0.701521\pi$
$522$	40.8100	1.78620
$523$	23.3456	1.02083	0.510416	−	0.859928i	$-0.329492\pi$
$523$	23.3456	1.02083	0.510416	+	0.859928i	$0.329492\pi$
$524$	−83.2631	−3.63736
$525$	−56.2094	−2.45318
$526$	58.5815	2.55427
$527$	5.97731	0.260376
$528$	54.8911	2.38883
$529$	5.48429	0.238447
$530$	−4.24950	−0.184586
$531$	45.1477	1.95924
$532$	9.65875	0.418760
$533$	40.4274	1.75110
$534$	52.6697	2.27924
$535$	−21.4423	−0.927030
$536$	−10.7281	−0.463384
$537$	41.1265	1.77474
$538$	48.1054	2.07397
$539$	7.25003	0.312281
$540$	−13.4413	−0.578421
$541$	−36.8758	−1.58541	−0.792706	−	0.609604i	$-0.791329\pi$
$541$	−36.8758	−1.58541	−0.792706	+	0.609604i	$0.791329\pi$
$542$	14.3621	0.616907
$543$	52.7640	2.26432
$544$	−19.9371	−0.854797
$545$	5.08759	0.217929
$546$	−66.6999	−2.85449
$547$	−46.6362	−1.99402	−0.997009	−	0.0772843i	$-0.975375\pi$
$547$	−46.6362	−1.99402	−0.997009	+	0.0772843i	$0.975375\pi$
$548$	−48.4828	−2.07108
$549$	−0.0499872	−0.00213340
$550$	91.9171	3.91936
$551$	−4.89281	−0.208441
$552$	−77.8320	−3.31275
$553$	−10.7715	−0.458049
$554$	−42.4239	−1.80242
$555$	−55.8295	−2.36983
$556$	−75.9343	−3.22033
$557$	20.8434	0.883165	0.441582	−	0.897221i	$-0.354417\pi$
$557$	20.8434	0.883165	0.441582	+	0.897221i	$0.354417\pi$
$558$	−8.37270	−0.354445
$559$	8.00348	0.338511
$560$	51.6217	2.18142
$561$	54.9181	2.31864
$562$	38.7377	1.63405
$563$	−25.4415	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$563$	−25.4415	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$564$	−7.60200	−0.320102
$565$	19.6866	0.828219
$566$	27.6148	1.16074
$567$	−17.7861	−0.746947
$568$	77.1887	3.23877
$569$	−28.6096	−1.19938	−0.599689	−	0.800233i	$-0.704709\pi$
$569$	−28.6096	−1.19938	−0.599689	+	0.800233i	$0.704709\pi$
$570$	24.4374	1.02357
$571$	37.2341	1.55820	0.779099	−	0.626901i	$-0.215677\pi$
$571$	37.2341	1.55820	0.779099	+	0.626901i	$0.215677\pi$
$572$	74.4970	3.11487
$573$	24.6582	1.03011
$574$	48.2988	2.01595
$575$	−53.2379	−2.22017
$576$	−11.5991	−0.483296
$577$	13.9850	0.582204	0.291102	−	0.956692i	$-0.405978\pi$
$577$	13.9850	0.582204	0.291102	+	0.956692i	$0.405978\pi$
$578$	−46.3601	−1.92832
$579$	−38.3258	−1.59277
$580$	−81.5929	−3.38796
$581$	−34.4784	−1.43041
$582$	1.04345	0.0432524
$583$	−1.60379	−0.0664223
$584$	46.9059	1.94098
$585$	−60.5542	−2.50361
$586$	3.63749	0.150263
$587$	23.5859	0.973494	0.486747	−	0.873543i	$-0.338183\pi$
$587$	23.5859	0.973494	0.486747	+	0.873543i	$0.338183\pi$
$588$	21.4112	0.882983
$589$	1.00382	0.0413618
$590$	−132.159	−5.44088
$591$	−50.5742	−2.08034
$592$	34.1535	1.40370
$593$	1.45578	0.0597816	0.0298908	−	0.999553i	$-0.490484\pi$
$593$	1.45578	0.0597816	0.0298908	+	0.999553i	$0.490484\pi$
$594$	−7.42720	−0.304742
$595$	51.6471	2.11732
$596$	−11.6216	−0.476038
$597$	−45.1677	−1.84859
$598$	−63.1737	−2.58337
$599$	28.6379	1.17011	0.585057	−	0.810992i	$-0.301072\pi$
$599$	28.6379	1.17011	0.585057	+	0.810992i	$0.301072\pi$
$600$	145.470	5.93879
$601$	5.63985	0.230054	0.115027	−	0.993362i	$-0.463304\pi$
$601$	5.63985	0.230054	0.115027	+	0.993362i	$0.463304\pi$
$602$	9.56180	0.389710
$603$	6.14135	0.250095
$604$	29.2513	1.19022
$605$	9.51114	0.386683
$606$	60.0018	2.43741
$607$	−2.85271	−0.115788	−0.0578940	−	0.998323i	$-0.518439\pi$
$607$	−2.85271	−0.115788	−0.0578940	+	0.998323i	$0.518439\pi$
$608$	−3.34822	−0.135788
$609$	27.5708	1.11723
$610$	0.146325	0.00592452
$611$	−3.30660	−0.133771
$612$	85.2069	3.44429
$613$	−38.7463	−1.56495	−0.782474	−	0.622684i	$-0.786042\pi$
$613$	−38.7463	−1.56495	−0.782474	+	0.622684i	$0.786042\pi$
$614$	−8.66860	−0.349836
$615$	83.4637	3.36558
$616$	47.6953	1.92170
$617$	−3.84843	−0.154932	−0.0774660	−	0.996995i	$-0.524683\pi$
$617$	−3.84843	−0.154932	−0.0774660	+	0.996995i	$0.524683\pi$
$618$	24.4382	0.983047
$619$	15.3763	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$619$	15.3763	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$620$	16.7398	0.672288
$621$	4.30179	0.172625
$622$	8.05358	0.322919
$623$	18.6940	0.748959
$624$	70.5110	2.82270
$625$	24.6273	0.985093
$626$	10.6607	0.426087
$627$	9.22288	0.368326
$628$	92.7333	3.70046
$629$	34.1702	1.36246
$630$	−72.3445	−2.88227
$631$	−21.5442	−0.857662	−0.428831	−	0.903385i	$-0.641074\pi$
$631$	−21.5442	−0.857662	−0.428831	+	0.903385i	$0.641074\pi$
$632$	27.8766	1.10887
$633$	−2.51408	−0.0999257
$634$	−65.8038	−2.61340
$635$	−32.6860	−1.29710
$636$	−4.73641	−0.187811
$637$	9.31311	0.368999
$638$	−45.0855	−1.78495
$639$	−44.1870	−1.74801
$640$	59.8671	2.36645
$641$	−5.09331	−0.201174	−0.100587	−	0.994928i	$-0.532072\pi$
$641$	−5.09331	−0.201174	−0.100587	+	0.994928i	$0.532072\pi$
$642$	−34.9910	−1.38098
$643$	43.2382	1.70515	0.852573	−	0.522607i	$-0.175041\pi$
$643$	43.2382	1.70515	0.852573	+	0.522607i	$0.175041\pi$
$644$	−51.5494	−2.03133
$645$	16.5235	0.650610
$646$	−14.9569	−0.588469
$647$	−25.1713	−0.989587	−0.494793	−	0.869011i	$-0.664756\pi$
$647$	−25.1713	−0.989587	−0.494793	+	0.869011i	$0.664756\pi$
$648$	46.0305	1.80825
$649$	−49.8777	−1.95787
$650$	118.073	4.63121
$651$	−5.65651	−0.221696
$652$	44.9810	1.76159
$653$	−45.8672	−1.79492	−0.897461	−	0.441094i	$-0.854591\pi$
$653$	−45.8672	−1.79492	−0.897461	+	0.441094i	$0.854591\pi$
$654$	8.30229	0.324645
$655$	−74.7702	−2.92151
$656$	−51.0586	−1.99350
$657$	−26.8515	−1.04758
$658$	−3.95042	−0.154003
$659$	−5.20393	−0.202716	−0.101358	−	0.994850i	$-0.532319\pi$
$659$	−5.20393	−0.202716	−0.101358	+	0.994850i	$0.532319\pi$
$660$	153.801	5.98671
$661$	−32.1290	−1.24967	−0.624837	−	0.780755i	$-0.714835\pi$
$661$	−32.1290	−1.24967	−0.624837	+	0.780755i	$0.714835\pi$
$662$	−44.6453	−1.73519
$663$	70.5457	2.73977
$664$	89.2301	3.46280
$665$	8.67355	0.336346
$666$	−47.8638	−1.85469
$667$	26.1132	1.01111
$668$	−30.3097	−1.17272
$669$	−19.7893	−0.765099
$670$	−17.9772	−0.694522
$671$	0.0552241	0.00213190
$672$	18.8671	0.727814
$673$	−19.9867	−0.770432	−0.385216	−	0.922826i	$-0.625873\pi$
$673$	−19.9867	−0.770432	−0.385216	+	0.922826i	$0.625873\pi$
$674$	−46.4925	−1.79083
$675$	−8.04016	−0.309466
$676$	39.6748	1.52595
$677$	−27.9200	−1.07305	−0.536526	−	0.843884i	$-0.680264\pi$
$677$	−27.9200	−1.07305	−0.536526	+	0.843884i	$0.680264\pi$
$678$	32.1259	1.23379
$679$	0.370351	0.0142128
$680$	−133.663	−5.12573
$681$	58.8569	2.25540
$682$	9.24987	0.354196
$683$	−36.0837	−1.38071	−0.690353	−	0.723473i	$-0.742545\pi$
$683$	−36.0837	−1.38071	−0.690353	+	0.723473i	$0.742545\pi$
$684$	14.3096	0.547139
$685$	−43.5375	−1.66348
$686$	50.5360	1.92947
$687$	−9.18326	−0.350363
$688$	−10.1082	−0.385370
$689$	−2.06017	−0.0784863
$690$	−130.424	−4.96517
$691$	−47.9756	−1.82508	−0.912539	−	0.408990i	$-0.865881\pi$
$691$	−47.9756	−1.82508	−0.912539	+	0.408990i	$0.865881\pi$
$692$	70.0816	2.66410
$693$	−27.3034	−1.03717
$694$	−12.4008	−0.470729
$695$	−68.1889	−2.58655
$696$	−71.3532	−2.70464
$697$	−51.0837	−1.93493
$698$	−53.9023	−2.04023
$699$	−61.5226	−2.32700
$700$	96.3472	3.64158
$701$	19.7008	0.744090	0.372045	−	0.928215i	$-0.378657\pi$
$701$	19.7008	0.744090	0.372045	+	0.928215i	$0.378657\pi$
$702$	−9.54070	−0.360091
$703$	5.73851	0.216432
$704$	12.8143	0.482956
$705$	−6.82659	−0.257104
$706$	49.9160	1.87861
$707$	21.2964	0.800933
$708$	−147.302	−5.53593
$709$	−43.7682	−1.64375	−0.821874	−	0.569669i	$-0.807071\pi$
$709$	−43.7682	−1.64375	−0.821874	+	0.569669i	$0.807071\pi$
$710$	129.346	4.85428
$711$	−15.9581	−0.598474
$712$	−48.3801	−1.81312
$713$	−5.35747	−0.200639
$714$	84.2813	3.15415
$715$	66.8982	2.50185
$716$	−70.4939	−2.63448
$717$	48.2523	1.80202
$718$	58.5397	2.18468
$719$	−9.88300	−0.368574	−0.184287	−	0.982872i	$-0.558998\pi$
$719$	−9.88300	−0.368574	−0.184287	+	0.982872i	$0.558998\pi$
$720$	76.4782	2.85017
$721$	8.67381	0.323030
$722$	−2.51184	−0.0934809
$723$	37.8963	1.40938
$724$	−90.4414	−3.36123
$725$	−48.8063	−1.81262
$726$	15.5209	0.576036
$727$	45.9201	1.70308	0.851540	−	0.524289i	$-0.175669\pi$
$727$	45.9201	1.70308	0.851540	+	0.524289i	$0.175669\pi$
$728$	61.2675	2.27073
$729$	−32.4295	−1.20109
$730$	78.6010	2.90915
$731$	−10.1131	−0.374047
$732$	0.163091	0.00602802
$733$	16.6835	0.616218	0.308109	−	0.951351i	$-0.400304\pi$
$733$	16.6835	0.616218	0.308109	+	0.951351i	$0.400304\pi$
$734$	13.6695	0.504550
$735$	19.2272	0.709207
$736$	17.8697	0.658685
$737$	−6.78475	−0.249919
$738$	71.5553	2.63399
$739$	18.8710	0.694182	0.347091	−	0.937831i	$-0.387169\pi$
$739$	18.8710	0.694182	0.347091	+	0.937831i	$0.387169\pi$
$740$	95.6959	3.51785
$741$	11.8474	0.435224
$742$	−2.46130	−0.0903571
$743$	14.3438	0.526221	0.263111	−	0.964766i	$-0.415252\pi$
$743$	14.3438	0.526221	0.263111	+	0.964766i	$0.415252\pi$
$744$	14.6390	0.536693
$745$	−10.4362	−0.382351
$746$	−25.4271	−0.930952
$747$	−51.0802	−1.86892
$748$	−94.1337	−3.44187
$749$	−12.4193	−0.453792
$750$	121.579	4.43944
$751$	−29.9408	−1.09255	−0.546277	−	0.837605i	$-0.683955\pi$
$751$	−29.9408	−1.09255	−0.546277	+	0.837605i	$0.683955\pi$
$752$	4.17614	0.152288
$753$	65.4044	2.38347
$754$	−57.9151	−2.10914
$755$	26.2676	0.955976
$756$	−7.78516	−0.283144
$757$	−1.44824	−0.0526371	−0.0263185	−	0.999654i	$-0.508378\pi$
$757$	−1.44824	−0.0526371	−0.0263185	+	0.999654i	$0.508378\pi$
$758$	−63.7022	−2.31377
$759$	−49.2231	−1.78669
$760$	−22.4472	−0.814244
$761$	−48.5606	−1.76032	−0.880159	−	0.474678i	$-0.842564\pi$
$761$	−48.5606	−1.76032	−0.880159	+	0.474678i	$0.842564\pi$
$762$	−53.3393	−1.93228
$763$	2.94672	0.106679
$764$	−42.2660	−1.52913
$765$	76.5157	2.76643
$766$	15.7845	0.570317
$767$	−64.0710	−2.31347
$768$	80.1316	2.89150
$769$	7.63086	0.275176	0.137588	−	0.990490i	$-0.456065\pi$
$769$	7.63086	0.275176	0.137588	+	0.990490i	$0.456065\pi$
$770$	79.9237	2.88025
$771$	74.6264	2.68760
$772$	65.6933	2.36435
$773$	15.6049	0.561270	0.280635	−	0.959815i	$-0.409455\pi$
$773$	15.6049	0.561270	0.280635	+	0.959815i	$0.409455\pi$
$774$	14.1659	0.509184
$775$	10.0132	0.359687
$776$	−0.958468	−0.0344070
$777$	−32.3363	−1.16006
$778$	−12.3811	−0.443883
$779$	−8.57893	−0.307372
$780$	197.568	7.07406
$781$	48.8163	1.74678
$782$	79.8257	2.85456
$783$	3.94371	0.140937
$784$	−11.7622	−0.420078
$785$	83.2744	2.97219
$786$	−122.015	−4.35214
$787$	−38.7918	−1.38278	−0.691389	−	0.722483i	$-0.743001\pi$
$787$	−38.7918	−1.38278	−0.691389	+	0.722483i	$0.743001\pi$
$788$	86.6879	3.08813
$789$	58.6338	2.08742
$790$	46.7132	1.66198
$791$	11.4024	0.405423
$792$	70.6611	2.51083
$793$	0.0709388	0.00251911
$794$	−75.8725	−2.69261
$795$	−4.25329	−0.150849
$796$	77.4209	2.74411
$797$	28.7351	1.01785	0.508925	−	0.860811i	$-0.330043\pi$
$797$	28.7351	1.01785	0.508925	+	0.860811i	$0.330043\pi$
$798$	14.1541	0.501050
$799$	4.17819	0.147814
$800$	−33.3989	−1.18083
$801$	27.6954	0.978568
$802$	23.1421	0.817175
$803$	29.6646	1.04684
$804$	−20.0371	−0.706654
$805$	−46.2913	−1.63156
$806$	11.8820	0.418527
$807$	48.1484	1.69490
$808$	−55.1150	−1.93894
$809$	13.4142	0.471619	0.235809	−	0.971799i	$-0.424226\pi$
$809$	13.4142	0.471619	0.235809	+	0.971799i	$0.424226\pi$
$810$	77.1340	2.71021
$811$	51.1579	1.79640	0.898199	−	0.439589i	$-0.144876\pi$
$811$	51.1579	1.79640	0.898199	+	0.439589i	$0.144876\pi$
$812$	−47.2584	−1.65845
$813$	14.3750	0.504152
$814$	52.8783	1.85338
$815$	40.3929	1.41490
$816$	−89.0971	−3.11902
$817$	−1.69839	−0.0594190
$818$	−9.29934	−0.325144
$819$	−35.0729	−1.22554
$820$	−143.063	−4.99598
$821$	17.7760	0.620386	0.310193	−	0.950674i	$-0.399606\pi$
$821$	17.7760	0.620386	0.310193	+	0.950674i	$0.399606\pi$
$822$	−71.0476	−2.47807
$823$	−24.5914	−0.857201	−0.428600	−	0.903494i	$-0.640993\pi$
$823$	−24.5914	−0.857201	−0.428600	+	0.903494i	$0.640993\pi$
$824$	−22.4478	−0.782007
$825$	91.9992	3.20300
$826$	−76.5460	−2.66337
$827$	−39.2562	−1.36507	−0.682536	−	0.730852i	$-0.739123\pi$
$827$	−39.2562	−1.36507	−0.682536	+	0.730852i	$0.739123\pi$
$828$	−76.3711	−2.65408
$829$	−32.1395	−1.11625	−0.558126	−	0.829756i	$-0.688479\pi$
$829$	−32.1395	−1.11625	−0.558126	+	0.829756i	$0.688479\pi$
$830$	149.524	5.19006
$831$	−42.4619	−1.47299
$832$	16.4607	0.570674
$833$	−11.7680	−0.407736
$834$	−111.275	−3.85316
$835$	−27.2181	−0.941921
$836$	−15.8087	−0.546755
$837$	−0.809103	−0.0279667
$838$	80.1935	2.77024
$839$	−34.0142	−1.17430	−0.587150	−	0.809478i	$-0.699750\pi$
$839$	−34.0142	−1.17430	−0.587150	+	0.809478i	$0.699750\pi$
$840$	126.489	4.36428
$841$	−5.06043	−0.174497
$842$	−48.0010	−1.65422
$843$	38.7723	1.33539
$844$	4.30932	0.148333
$845$	35.6279	1.22564
$846$	−5.85259	−0.201216
$847$	5.50883	0.189286
$848$	2.60193	0.0893508
$849$	27.6394	0.948583
$850$	−149.196	−5.11739
$851$	−30.6268	−1.04987
$852$	144.167	4.93908
$853$	−51.8556	−1.77550	−0.887750	−	0.460325i	$-0.847733\pi$
$853$	−51.8556	−1.77550	−0.887750	+	0.460325i	$0.847733\pi$
$854$	0.0847510	0.00290012
$855$	12.8500	0.439460
$856$	32.1412	1.09856
$857$	19.0989	0.652405	0.326202	−	0.945300i	$-0.394231\pi$
$857$	19.0989	0.652405	0.326202	+	0.945300i	$0.394231\pi$
$858$	109.169	3.72697
$859$	−16.9110	−0.576996	−0.288498	−	0.957480i	$-0.593156\pi$
$859$	−16.9110	−0.576996	−0.288498	+	0.957480i	$0.593156\pi$
$860$	−28.3224	−0.965787
$861$	48.3420	1.64749
$862$	−84.1719	−2.86690
$863$	−29.6949	−1.01082	−0.505412	−	0.862878i	$-0.668660\pi$
$863$	−29.6949	−1.01082	−0.505412	+	0.862878i	$0.668660\pi$
$864$	2.69874	0.0918129
$865$	62.9332	2.13979
$866$	45.4392	1.54409
$867$	−46.4015	−1.57588
$868$	9.69568	0.329093
$869$	17.6299	0.598054
$870$	−119.568	−4.05373
$871$	−8.71544	−0.295311
$872$	−7.62612	−0.258253
$873$	0.548679	0.0185700
$874$	13.4058	0.453459
$875$	43.1519	1.45880
$876$	87.6072	2.95997
$877$	−39.8740	−1.34645	−0.673224	−	0.739439i	$-0.735091\pi$
$877$	−39.8740	−1.34645	−0.673224	+	0.739439i	$0.735091\pi$
$878$	−18.8295	−0.635463
$879$	3.64074	0.122799
$880$	−84.4905	−2.84817
$881$	−10.3416	−0.348416	−0.174208	−	0.984709i	$-0.555736\pi$
$881$	−10.3416	−0.348416	−0.174208	+	0.984709i	$0.555736\pi$
$882$	16.4839	0.555043
$883$	−23.4206	−0.788167	−0.394083	−	0.919075i	$-0.628938\pi$
$883$	−23.4206	−0.788167	−0.394083	+	0.919075i	$0.628938\pi$
$884$	−120.921	−4.06700
$885$	−132.277	−4.44643
$886$	25.1617	0.845326
$887$	−35.6887	−1.19831	−0.599155	−	0.800633i	$-0.704497\pi$
$887$	−35.6887	−1.19831	−0.599155	+	0.800633i	$0.704497\pi$
$888$	83.6864	2.80833
$889$	−18.9317	−0.634947
$890$	−81.0712	−2.71751
$891$	29.1109	0.975253
$892$	33.9204	1.13574
$893$	0.701681	0.0234809
$894$	−17.0305	−0.569584
$895$	−63.3035	−2.11600
$896$	34.6749	1.15841
$897$	−63.2302	−2.11119
$898$	6.14148	0.204944
$899$	−4.91151	−0.163808
$900$	142.739	4.75798
$901$	2.60321	0.0867256
$902$	−79.0518	−2.63214
$903$	9.57035	0.318481
$904$	−29.5094	−0.981469
$905$	−81.2163	−2.69972
$906$	42.8653	1.42410
$907$	45.9570	1.52598	0.762988	−	0.646413i	$-0.223731\pi$
$907$	45.9570	1.52598	0.762988	+	0.646413i	$0.223731\pi$
$908$	−100.885	−3.34799
$909$	31.5508	1.04648
$910$	102.667	3.40338
$911$	37.8284	1.25331	0.626655	−	0.779297i	$-0.284424\pi$
$911$	37.8284	1.25331	0.626655	+	0.779297i	$0.284424\pi$
$912$	−14.9629	−0.495470
$913$	56.4316	1.86761
$914$	67.1717	2.22184
$915$	0.146456	0.00484167
$916$	15.7408	0.520091
$917$	−43.3067	−1.43011
$918$	12.0555	0.397892
$919$	51.1854	1.68845	0.844225	−	0.535988i	$-0.180061\pi$
$919$	51.1854	1.68845	0.844225	+	0.535988i	$0.180061\pi$
$920$	119.802	3.94975
$921$	−8.67634	−0.285895
$922$	−9.76761	−0.321679
$923$	62.7076	2.06404
$924$	89.0815	2.93057
$925$	57.2423	1.88212
$926$	−6.89101	−0.226453
$927$	12.8503	0.422061
$928$	16.3822	0.537772
$929$	0.523188	0.0171652	0.00858262	−	0.999963i	$-0.497268\pi$
$929$	0.523188	0.0171652	0.00858262	+	0.999963i	$0.497268\pi$
$930$	24.5309	0.804399
$931$	−1.97630	−0.0647706
$932$	105.454	3.45427
$933$	8.06077	0.263898
$934$	−103.400	−3.38334
$935$	−84.5320	−2.76449
$936$	90.7686	2.96686
$937$	−12.6585	−0.413536	−0.206768	−	0.978390i	$-0.566294\pi$
$937$	−12.6585	−0.413536	−0.206768	+	0.978390i	$0.566294\pi$
$938$	−10.4124	−0.339976
$939$	10.6702	0.348209
$940$	11.7013	0.381654
$941$	−29.5087	−0.961957	−0.480978	−	0.876732i	$-0.659718\pi$
$941$	−29.5087	−0.961957	−0.480978	+	0.876732i	$0.659718\pi$
$942$	135.893	4.42763
$943$	45.7863	1.49101
$944$	80.9197	2.63371
$945$	−6.99107	−0.227419
$946$	−15.6500	−0.508826
$947$	−5.55968	−0.180665	−0.0903327	−	0.995912i	$-0.528793\pi$
$947$	−5.55968	−0.180665	−0.0903327	+	0.995912i	$0.528793\pi$
$948$	52.0656	1.69101
$949$	38.1060	1.23697
$950$	−25.0559	−0.812919
$951$	−65.8626	−2.13574
$952$	−77.4171	−2.50910
$953$	−38.6940	−1.25342	−0.626711	−	0.779252i	$-0.715599\pi$
$953$	−38.6940	−1.25342	−0.626711	+	0.779252i	$0.715599\pi$
$954$	−3.64644	−0.118058
$955$	−37.9548	−1.22819
$956$	−82.7081	−2.67497
$957$	−45.1258	−1.45871
$958$	50.2002	1.62190
$959$	−25.2168	−0.814294
$960$	33.9838	1.09682
$961$	−29.9923	−0.967495
$962$	67.9255	2.19001
$963$	−18.3993	−0.592911
$964$	−64.9571	−2.09213
$965$	58.9925	1.89904
$966$	−75.5414	−2.43051
$967$	45.5036	1.46330	0.731649	−	0.681682i	$-0.238751\pi$
$967$	45.5036	1.46330	0.731649	+	0.681682i	$0.238751\pi$
$968$	−14.2569	−0.458233
$969$	−14.9702	−0.480913
$970$	−1.60612	−0.0515694
$971$	39.9132	1.28087	0.640437	−	0.768010i	$-0.278753\pi$
$971$	39.9132	1.28087	0.640437	+	0.768010i	$0.278753\pi$
$972$	96.3923	3.09178
$973$	−39.4949	−1.26615
$974$	55.6446	1.78297
$975$	118.179	3.78475
$976$	−0.0895936	−0.00286782
$977$	−21.2315	−0.679255	−0.339628	−	0.940560i	$-0.610301\pi$
$977$	−21.2315	−0.679255	−0.339628	+	0.940560i	$0.610301\pi$
$978$	65.9159	2.10776
$979$	−30.5969	−0.977881
$980$	−32.9569	−1.05277
$981$	4.36560	0.139383
$982$	−71.5056	−2.28184
$983$	7.98369	0.254640	0.127320	−	0.991862i	$-0.459362\pi$
$983$	7.98369	0.254640	0.127320	+	0.991862i	$0.459362\pi$
$984$	−125.109	−3.98833
$985$	77.8457	2.48037
$986$	73.1810	2.33056
$987$	−3.95395	−0.125855
$988$	−20.3073	−0.646060
$989$	9.06440	0.288231
$990$	118.408	3.76325
$991$	17.0428	0.541381	0.270691	−	0.962666i	$-0.412748\pi$
$991$	17.0428	0.541381	0.270691	+	0.962666i	$0.412748\pi$
$992$	−3.36102	−0.106712
$993$	−44.6851	−1.41804
$994$	74.9171	2.37623
$995$	69.5239	2.20406
$996$	166.657	5.28073
$997$	17.1096	0.541866	0.270933	−	0.962598i	$-0.412668\pi$
$997$	17.1096	0.541866	0.270933	+	0.962598i	$0.412668\pi$
$998$	12.9138	0.408779
$999$	−4.62536	−0.146340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.7	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.7	✓	71	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51184 q^{2} -2.51408 q^{3} +4.30932 q^{4} +3.86977 q^{5} +6.31496 q^{6} +2.24136 q^{7} -5.80064 q^{8} +3.32060 q^{9} +O(q^{10})\)	\(q-2.51184 q^{2} -2.51408 q^{3} +4.30932 q^{4} +3.86977 q^{5} +6.31496 q^{6} +2.24136 q^{7} -5.80064 q^{8} +3.32060 q^{9} -9.72023 q^{10} -3.66849 q^{11} -10.8340 q^{12} -4.71240 q^{13} -5.62993 q^{14} -9.72891 q^{15} +5.95162 q^{16} +5.95455 q^{17} -8.34081 q^{18} +1.00000 q^{19} +16.6761 q^{20} -5.63496 q^{21} +9.21464 q^{22} -5.33707 q^{23} +14.5833 q^{24} +9.97511 q^{25} +11.8368 q^{26} -0.806021 q^{27} +9.65875 q^{28} -4.89281 q^{29} +24.4374 q^{30} +1.00382 q^{31} -3.34822 q^{32} +9.22288 q^{33} -14.9569 q^{34} +8.67355 q^{35} +14.3096 q^{36} +5.73851 q^{37} -2.51184 q^{38} +11.8474 q^{39} -22.4472 q^{40} -8.57893 q^{41} +14.1541 q^{42} -1.69839 q^{43} -15.8087 q^{44} +12.8500 q^{45} +13.4058 q^{46} +0.701681 q^{47} -14.9629 q^{48} -1.97630 q^{49} -25.0559 q^{50} -14.9702 q^{51} -20.3073 q^{52} +0.437181 q^{53} +2.02459 q^{54} -14.1962 q^{55} -13.0013 q^{56} -2.51408 q^{57} +12.2899 q^{58} +13.5962 q^{59} -41.9250 q^{60} -0.0150536 q^{61} -2.52144 q^{62} +7.44267 q^{63} -3.49307 q^{64} -18.2359 q^{65} -23.1664 q^{66} +1.84947 q^{67} +25.6601 q^{68} +13.4178 q^{69} -21.7865 q^{70} -13.3069 q^{71} -19.2616 q^{72} -8.08633 q^{73} -14.4142 q^{74} -25.0782 q^{75} +4.30932 q^{76} -8.22241 q^{77} -29.7586 q^{78} -4.80577 q^{79} +23.0314 q^{80} -7.93541 q^{81} +21.5489 q^{82} -15.3828 q^{83} -24.2829 q^{84} +23.0427 q^{85} +4.26607 q^{86} +12.3009 q^{87} +21.2796 q^{88} +8.34046 q^{89} -32.2770 q^{90} -10.5622 q^{91} -22.9992 q^{92} -2.52369 q^{93} -1.76251 q^{94} +3.86977 q^{95} +8.41769 q^{96} +0.165235 q^{97} +4.96414 q^{98} -12.1816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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