LMFDB - Embedded newform 670.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [670,2,Mod(1,670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(670, 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("670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 670 = 2 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	670.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:	$5.34997693543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.490689$ of defining polynomial
Character	$\chi$	$=$	670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.658339 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.658339 q^{6} -3.43618 q^{7} -1.00000 q^{8} -2.56659 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.658339 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.658339 q^{6} -3.43618 q^{7} -1.00000 q^{8} -2.56659 q^{9} +1.00000 q^{10} -0.0945233 q^{11} -0.658339 q^{12} -0.0731262 q^{13} +3.43618 q^{14} +0.658339 q^{15} +1.00000 q^{16} +7.49346 q^{17} +2.56659 q^{18} +6.53707 q^{19} -1.00000 q^{20} +2.26217 q^{21} +0.0945233 q^{22} -6.15180 q^{23} +0.658339 q^{24} +1.00000 q^{25} +0.0731262 q^{26} +3.66470 q^{27} -3.43618 q^{28} +8.90825 q^{29} -0.658339 q^{30} -0.945496 q^{31} -1.00000 q^{32} +0.0622284 q^{33} -7.49346 q^{34} +3.43618 q^{35} -2.56659 q^{36} -5.98415 q^{37} -6.53707 q^{38} +0.0481418 q^{39} +1.00000 q^{40} +4.95639 q^{41} -2.26217 q^{42} -2.36028 q^{43} -0.0945233 q^{44} +2.56659 q^{45} +6.15180 q^{46} +10.1159 q^{47} -0.658339 q^{48} +4.80737 q^{49} -1.00000 q^{50} -4.93324 q^{51} -0.0731262 q^{52} -6.46394 q^{53} -3.66470 q^{54} +0.0945233 q^{55} +3.43618 q^{56} -4.30361 q^{57} -8.90825 q^{58} +13.1582 q^{59} +0.658339 q^{60} +10.1732 q^{61} +0.945496 q^{62} +8.81928 q^{63} +1.00000 q^{64} +0.0731262 q^{65} -0.0622284 q^{66} +1.00000 q^{67} +7.49346 q^{68} +4.04997 q^{69} -3.43618 q^{70} -5.53985 q^{71} +2.56659 q^{72} +8.81014 q^{73} +5.98415 q^{74} -0.658339 q^{75} +6.53707 q^{76} +0.324800 q^{77} -0.0481418 q^{78} -10.5371 q^{79} -1.00000 q^{80} +5.28716 q^{81} -4.95639 q^{82} -4.97779 q^{83} +2.26217 q^{84} -7.49346 q^{85} +2.36028 q^{86} -5.86465 q^{87} +0.0945233 q^{88} +1.90548 q^{89} -2.56659 q^{90} +0.251275 q^{91} -6.15180 q^{92} +0.622456 q^{93} -10.1159 q^{94} -6.53707 q^{95} +0.658339 q^{96} -4.44255 q^{97} -4.80737 q^{98} +0.242603 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9} + 4 q^{10} + 9 q^{11} - 2 q^{12} - 12 q^{13} + 5 q^{14} + 2 q^{15} + 4 q^{16} - 8 q^{18} + 4 q^{19} - 4 q^{20} + 2 q^{21} - 9 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{25} + 12 q^{26} + 10 q^{27} - 5 q^{28} + 18 q^{29} - 2 q^{30} + 2 q^{31} - 4 q^{32} + 14 q^{33} + 5 q^{35} + 8 q^{36} + 9 q^{37} - 4 q^{38} + 10 q^{39} + 4 q^{40} + 12 q^{41} - 2 q^{42} - 16 q^{43} + 9 q^{44} - 8 q^{45} - 6 q^{46} + 10 q^{47} - 2 q^{48} + 15 q^{49} - 4 q^{50} - 4 q^{51} - 12 q^{52} + 8 q^{53} - 10 q^{54} - 9 q^{55} + 5 q^{56} + 44 q^{57} - 18 q^{58} + 18 q^{59} + 2 q^{60} - 11 q^{61} - 2 q^{62} - 27 q^{63} + 4 q^{64} + 12 q^{65} - 14 q^{66} + 4 q^{67} + 20 q^{69} - 5 q^{70} + 27 q^{71} - 8 q^{72} + 4 q^{73} - 9 q^{74} - 2 q^{75} + 4 q^{76} + 25 q^{77} - 10 q^{78} - 20 q^{79} - 4 q^{80} + 16 q^{81} - 12 q^{82} + 9 q^{83} + 2 q^{84} + 16 q^{86} + 2 q^{87} - 9 q^{88} + 17 q^{89} + 8 q^{90} - 4 q^{91} + 6 q^{92} + 2 q^{93} - 10 q^{94} - 4 q^{95} + 2 q^{96} - 5 q^{97} - 15 q^{98} + 9 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9 + 4 * q^10 + 9 * q^11 - 2 * q^12 - 12 * q^13 + 5 * q^14 + 2 * q^15 + 4 * q^16 - 8 * q^18 + 4 * q^19 - 4 * q^20 + 2 * q^21 - 9 * q^22 + 6 * q^23 + 2 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 5 * q^28 + 18 * q^29 - 2 * q^30 + 2 * q^31 - 4 * q^32 + 14 * q^33 + 5 * q^35 + 8 * q^36 + 9 * q^37 - 4 * q^38 + 10 * q^39 + 4 * q^40 + 12 * q^41 - 2 * q^42 - 16 * q^43 + 9 * q^44 - 8 * q^45 - 6 * q^46 + 10 * q^47 - 2 * q^48 + 15 * q^49 - 4 * q^50 - 4 * q^51 - 12 * q^52 + 8 * q^53 - 10 * q^54 - 9 * q^55 + 5 * q^56 + 44 * q^57 - 18 * q^58 + 18 * q^59 + 2 * q^60 - 11 * q^61 - 2 * q^62 - 27 * q^63 + 4 * q^64 + 12 * q^65 - 14 * q^66 + 4 * q^67 + 20 * q^69 - 5 * q^70 + 27 * q^71 - 8 * q^72 + 4 * q^73 - 9 * q^74 - 2 * q^75 + 4 * q^76 + 25 * q^77 - 10 * q^78 - 20 * q^79 - 4 * q^80 + 16 * q^81 - 12 * q^82 + 9 * q^83 + 2 * q^84 + 16 * q^86 + 2 * q^87 - 9 * q^88 + 17 * q^89 + 8 * q^90 - 4 * q^91 + 6 * q^92 + 2 * q^93 - 10 * q^94 - 4 * q^95 + 2 * q^96 - 5 * q^97 - 15 * q^98 + 9 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.658339	−0.380092	−0.190046	−	0.981775i	$-0.560864\pi$
$3$	−0.658339	−0.380092	−0.190046	+	0.981775i	$0.560864\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.658339	0.268766
$7$	−3.43618	−1.29876	−0.649378	−	0.760466i	$-0.724971\pi$
$7$	−3.43618	−1.29876	−0.649378	+	0.760466i	$0.724971\pi$
$8$	−1.00000	−0.353553
$9$	−2.56659	−0.855530
$10$	1.00000	0.316228
$11$	−0.0945233	−0.0284999	−0.0142499	−	0.999898i	$-0.504536\pi$
$11$	−0.0945233	−0.0284999	−0.0142499	+	0.999898i	$0.504536\pi$
$12$	−0.658339	−0.190046
$13$	−0.0731262	−0.0202816	−0.0101408	−	0.999949i	$-0.503228\pi$
$13$	−0.0731262	−0.0202816	−0.0101408	+	0.999949i	$0.503228\pi$
$14$	3.43618	0.918359
$15$	0.658339	0.169982
$16$	1.00000	0.250000
$17$	7.49346	1.81743	0.908716	−	0.417415i	$-0.137064\pi$
$17$	7.49346	1.81743	0.908716	+	0.417415i	$0.137064\pi$
$18$	2.56659	0.604951
$19$	6.53707	1.49971	0.749853	−	0.661604i	$-0.230124\pi$
$19$	6.53707	1.49971	0.749853	+	0.661604i	$0.230124\pi$
$20$	−1.00000	−0.223607
$21$	2.26217	0.493647
$22$	0.0945233	0.0201524
$23$	−6.15180	−1.28274	−0.641370	−	0.767232i	$-0.721634\pi$
$23$	−6.15180	−1.28274	−0.641370	+	0.767232i	$0.721634\pi$
$24$	0.658339	0.134383
$25$	1.00000	0.200000
$26$	0.0731262	0.0143412
$27$	3.66470	0.705272
$28$	−3.43618	−0.649378
$29$	8.90825	1.65422	0.827110	−	0.562039i	$-0.189983\pi$
$29$	8.90825	1.65422	0.827110	+	0.562039i	$0.189983\pi$
$30$	−0.658339	−0.120196
$31$	−0.945496	−0.169816	−0.0849080	−	0.996389i	$-0.527060\pi$
$31$	−0.945496	−0.169816	−0.0849080	+	0.996389i	$0.527060\pi$
$32$	−1.00000	−0.176777
$33$	0.0622284	0.0108326
$34$	−7.49346	−1.28512
$35$	3.43618	0.580821
$36$	−2.56659	−0.427765
$37$	−5.98415	−0.983789	−0.491894	−	0.870655i	$-0.663695\pi$
$37$	−5.98415	−0.983789	−0.491894	+	0.870655i	$0.663695\pi$
$38$	−6.53707	−1.06045
$39$	0.0481418	0.00770885
$40$	1.00000	0.158114
$41$	4.95639	0.774059	0.387029	−	0.922067i	$-0.373501\pi$
$41$	4.95639	0.774059	0.387029	+	0.922067i	$0.373501\pi$
$42$	−2.26217	−0.349061
$43$	−2.36028	−0.359940	−0.179970	−	0.983672i	$-0.557600\pi$
$43$	−2.36028	−0.359940	−0.179970	+	0.983672i	$0.557600\pi$
$44$	−0.0945233	−0.0142499
$45$	2.56659	0.382605
$46$	6.15180	0.907034
$47$	10.1159	1.47556	0.737779	−	0.675042i	$-0.235874\pi$
$47$	10.1159	1.47556	0.737779	+	0.675042i	$0.235874\pi$
$48$	−0.658339	−0.0950230
$49$	4.80737	0.686767
$50$	−1.00000	−0.141421
$51$	−4.93324	−0.690791
$52$	−0.0731262	−0.0101408
$53$	−6.46394	−0.887891	−0.443946	−	0.896054i	$-0.646422\pi$
$53$	−6.46394	−0.887891	−0.443946	+	0.896054i	$0.646422\pi$
$54$	−3.66470	−0.498703
$55$	0.0945233	0.0127455
$56$	3.43618	0.459180
$57$	−4.30361	−0.570026
$58$	−8.90825	−1.16971
$59$	13.1582	1.71305	0.856524	−	0.516108i	$-0.172620\pi$
$59$	13.1582	1.71305	0.856524	+	0.516108i	$0.172620\pi$
$60$	0.658339	0.0849911
$61$	10.1732	1.30254	0.651272	−	0.758844i	$-0.274236\pi$
$61$	10.1732	1.30254	0.651272	+	0.758844i	$0.274236\pi$
$62$	0.945496	0.120078
$63$	8.81928	1.11112
$64$	1.00000	0.125000
$65$	0.0731262	0.00907019
$66$	−0.0622284	−0.00765978
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	7.49346	0.908716
$69$	4.04997	0.487559
$70$	−3.43618	−0.410703
$71$	−5.53985	−0.657459	−0.328729	−	0.944424i	$-0.606620\pi$
$71$	−5.53985	−0.657459	−0.328729	+	0.944424i	$0.606620\pi$
$72$	2.56659	0.302476
$73$	8.81014	1.03115	0.515575	−	0.856845i	$-0.327579\pi$
$73$	8.81014	1.03115	0.515575	+	0.856845i	$0.327579\pi$
$74$	5.98415	0.695644
$75$	−0.658339	−0.0760184
$76$	6.53707	0.749853
$77$	0.324800	0.0370144
$78$	−0.0481418	−0.00545098
$79$	−10.5371	−1.18551	−0.592757	−	0.805382i	$-0.701960\pi$
$79$	−10.5371	−1.18551	−0.592757	+	0.805382i	$0.701960\pi$
$80$	−1.00000	−0.111803
$81$	5.28716	0.587462
$82$	−4.95639	−0.547342
$83$	−4.97779	−0.546384	−0.273192	−	0.961960i	$-0.588079\pi$
$83$	−4.97779	−0.546384	−0.273192	+	0.961960i	$0.588079\pi$
$84$	2.26217	0.246823
$85$	−7.49346	−0.812780
$86$	2.36028	0.254516
$87$	−5.86465	−0.628756
$88$	0.0945233	0.0100762
$89$	1.90548	0.201980	0.100990	−	0.994887i	$-0.467799\pi$
$89$	1.90548	0.201980	0.100990	+	0.994887i	$0.467799\pi$
$90$	−2.56659	−0.270542
$91$	0.251275	0.0263408
$92$	−6.15180	−0.641370
$93$	0.622456	0.0645457
$94$	−10.1159	−1.04338
$95$	−6.53707	−0.670689
$96$	0.658339	0.0671914
$97$	−4.44255	−0.451072	−0.225536	−	0.974235i	$-0.572413\pi$
$97$	−4.44255	−0.451072	−0.225536	+	0.974235i	$0.572413\pi$
$98$	−4.80737	−0.485617
$99$	0.242603	0.0243825
$100$	1.00000	0.100000
$101$	5.62523	0.559731	0.279866	−	0.960039i	$-0.409710\pi$
$101$	5.62523	0.559731	0.279866	+	0.960039i	$0.409710\pi$
$102$	4.93324	0.488463
$103$	−12.9455	−1.27556	−0.637779	−	0.770220i	$-0.720147\pi$
$103$	−12.9455	−1.27556	−0.637779	+	0.770220i	$0.720147\pi$
$104$	0.0731262	0.00717061
$105$	−2.26217	−0.220765
$106$	6.46394	0.627834
$107$	15.9947	1.54626	0.773131	−	0.634247i	$-0.218690\pi$
$107$	15.9947	1.54626	0.773131	+	0.634247i	$0.218690\pi$
$108$	3.66470	0.352636
$109$	11.2527	1.07781	0.538906	−	0.842366i	$-0.318838\pi$
$109$	11.2527	1.07781	0.538906	+	0.842366i	$0.318838\pi$
$110$	−0.0945233	−0.00901245
$111$	3.93960	0.373930
$112$	−3.43618	−0.324689
$113$	18.1609	1.70844	0.854219	−	0.519914i	$-0.174036\pi$
$113$	18.1609	1.70844	0.854219	+	0.519914i	$0.174036\pi$
$114$	4.30361	0.403070
$115$	6.15180	0.573659
$116$	8.90825	0.827110
$117$	0.187685	0.0173515
$118$	−13.1582	−1.21131
$119$	−25.7489	−2.36040
$120$	−0.658339	−0.0600978
$121$	−10.9911	−0.999188
$122$	−10.1732	−0.921038
$123$	−3.26298	−0.294213
$124$	−0.945496	−0.0849080
$125$	−1.00000	−0.0894427
$126$	−8.81928	−0.785684
$127$	7.16906	0.636151	0.318076	−	0.948065i	$-0.396963\pi$
$127$	7.16906	0.636151	0.318076	+	0.948065i	$0.396963\pi$
$128$	−1.00000	−0.0883883
$129$	1.55387	0.136810
$130$	−0.0731262	−0.00641359
$131$	−13.0242	−1.13793	−0.568964	−	0.822363i	$-0.692655\pi$
$131$	−13.0242	−1.13793	−0.568964	+	0.822363i	$0.692655\pi$
$132$	0.0622284	0.00541628
$133$	−22.4626	−1.94775
$134$	−1.00000	−0.0863868
$135$	−3.66470	−0.315407
$136$	−7.49346	−0.642559
$137$	11.6984	0.999458	0.499729	−	0.866182i	$-0.333433\pi$
$137$	11.6984	0.999458	0.499729	+	0.866182i	$0.333433\pi$
$138$	−4.04997	−0.344756
$139$	−14.6998	−1.24682	−0.623409	−	0.781896i	$-0.714253\pi$
$139$	−14.6998	−1.24682	−0.623409	+	0.781896i	$0.714253\pi$
$140$	3.43618	0.290411
$141$	−6.65970	−0.560848
$142$	5.53985	0.464894
$143$	0.00691213	0.000578021 0
$144$	−2.56659	−0.213883
$145$	−8.90825	−0.739790
$146$	−8.81014	−0.729132
$147$	−3.16487	−0.261034
$148$	−5.98415	−0.491894
$149$	8.98693	0.736238	0.368119	−	0.929779i	$-0.380002\pi$
$149$	8.98693	0.736238	0.368119	+	0.929779i	$0.380002\pi$
$150$	0.658339	0.0537531
$151$	−6.76376	−0.550427	−0.275213	−	0.961383i	$-0.588749\pi$
$151$	−6.76376	−0.550427	−0.275213	+	0.961383i	$0.588749\pi$
$152$	−6.53707	−0.530226
$153$	−19.2327	−1.55487
$154$	−0.324800	−0.0261731
$155$	0.945496	0.0759441
$156$	0.0481418	0.00385443
$157$	−9.15953	−0.731010	−0.365505	−	0.930809i	$-0.619104\pi$
$157$	−9.15953	−0.731010	−0.365505	+	0.930809i	$0.619104\pi$
$158$	10.5371	0.838284
$159$	4.25546	0.337480
$160$	1.00000	0.0790569
$161$	21.1387	1.66597
$162$	−5.28716	−0.415398
$163$	−6.05728	−0.474443	−0.237221	−	0.971456i	$-0.576237\pi$
$163$	−6.05728	−0.474443	−0.237221	+	0.971456i	$0.576237\pi$
$164$	4.95639	0.387029
$165$	−0.0622284	−0.00484447
$166$	4.97779	0.386351
$167$	18.1946	1.40794	0.703970	−	0.710230i	$-0.251409\pi$
$167$	18.1946	1.40794	0.703970	+	0.710230i	$0.251409\pi$
$168$	−2.26217	−0.174530
$169$	−12.9947	−0.999589
$170$	7.49346	0.574722
$171$	−16.7780	−1.28304
$172$	−2.36028	−0.179970
$173$	−11.8252	−0.899051	−0.449526	−	0.893267i	$-0.648407\pi$
$173$	−11.8252	−0.899051	−0.449526	+	0.893267i	$0.648407\pi$
$174$	5.86465	0.444598
$175$	−3.43618	−0.259751
$176$	−0.0945233	−0.00712497
$177$	−8.66253	−0.651115
$178$	−1.90548	−0.142822
$179$	−12.0664	−0.901886	−0.450943	−	0.892553i	$-0.648912\pi$
$179$	−12.0664	−0.901886	−0.450943	+	0.892553i	$0.648912\pi$
$180$	2.56659	0.191302
$181$	18.1932	1.35229	0.676146	−	0.736767i	$-0.263649\pi$
$181$	18.1932	1.35229	0.676146	+	0.736767i	$0.263649\pi$
$182$	−0.251275	−0.0186257
$183$	−6.69741	−0.495087
$184$	6.15180	0.453517
$185$	5.98415	0.439964
$186$	−0.622456	−0.0456407
$187$	−0.708307	−0.0517966
$188$	10.1159	0.737779
$189$	−12.5926	−0.915976
$190$	6.53707	0.474249
$191$	−1.01726	−0.0736064	−0.0368032	−	0.999323i	$-0.511717\pi$
$191$	−1.01726	−0.0736064	−0.0368032	+	0.999323i	$0.511717\pi$
$192$	−0.658339	−0.0475115
$193$	9.58068	0.689632	0.344816	−	0.938670i	$-0.387941\pi$
$193$	9.58068	0.689632	0.344816	+	0.938670i	$0.387941\pi$
$194$	4.44255	0.318956
$195$	−0.0481418	−0.00344750
$196$	4.80737	0.343383
$197$	15.2632	1.08746	0.543729	−	0.839261i	$-0.317012\pi$
$197$	15.2632	1.08746	0.543729	+	0.839261i	$0.317012\pi$
$198$	−0.242603	−0.0172410
$199$	17.9233	1.27055	0.635274	−	0.772287i	$-0.280887\pi$
$199$	17.9233	1.27055	0.635274	+	0.772287i	$0.280887\pi$
$200$	−1.00000	−0.0707107
$201$	−0.658339	−0.0464356
$202$	−5.62523	−0.395790
$203$	−30.6104	−2.14843
$204$	−4.93324	−0.345396
$205$	−4.95639	−0.346170
$206$	12.9455	0.901955
$207$	15.7892	1.09742
$208$	−0.0731262	−0.00507039
$209$	−0.617906	−0.0427414
$210$	2.26217	0.156105
$211$	5.18268	0.356791	0.178395	−	0.983959i	$-0.442909\pi$
$211$	5.18268	0.356791	0.178395	+	0.983959i	$0.442909\pi$
$212$	−6.46394	−0.443946
$213$	3.64709	0.249895
$214$	−15.9947	−1.09337
$215$	2.36028	0.160970
$216$	−3.66470	−0.249351
$217$	3.24890	0.220550
$218$	−11.2527	−0.762128
$219$	−5.80006	−0.391931
$220$	0.0945233	0.00637276
$221$	−0.547968	−0.0368603
$222$	−3.93960	−0.264409
$223$	−17.5557	−1.17562	−0.587808	−	0.809000i	$-0.700009\pi$
$223$	−17.5557	−1.17562	−0.587808	+	0.809000i	$0.700009\pi$
$224$	3.43618	0.229590
$225$	−2.56659	−0.171106
$226$	−18.1609	−1.20805
$227$	15.0832	1.00111	0.500554	−	0.865705i	$-0.333130\pi$
$227$	15.0832	1.00111	0.500554	+	0.865705i	$0.333130\pi$
$228$	−4.30361	−0.285013
$229$	−10.5803	−0.699163	−0.349582	−	0.936906i	$-0.613676\pi$
$229$	−10.5803	−0.699163	−0.349582	+	0.936906i	$0.613676\pi$
$230$	−6.15180	−0.405638
$231$	−0.213828	−0.0140689
$232$	−8.90825	−0.584855
$233$	10.6906	0.700367	0.350183	−	0.936681i	$-0.386119\pi$
$233$	10.6906	0.700367	0.350183	+	0.936681i	$0.386119\pi$
$234$	−0.187685	−0.0122693
$235$	−10.1159	−0.659890
$236$	13.1582	0.856524
$237$	6.93696	0.450604
$238$	25.7489	1.66906
$239$	19.0587	1.23280	0.616402	−	0.787431i	$-0.288590\pi$
$239$	19.0587	1.23280	0.616402	+	0.787431i	$0.288590\pi$
$240$	0.658339	0.0424956
$241$	15.8874	1.02340	0.511699	−	0.859165i	$-0.329016\pi$
$241$	15.8874	1.02340	0.511699	+	0.859165i	$0.329016\pi$
$242$	10.9911	0.706532
$243$	−14.4748	−0.928562
$244$	10.1732	0.651272
$245$	−4.80737	−0.307131
$246$	3.26298	0.208040
$247$	−0.478031	−0.0304164
$248$	0.945496	0.0600390
$249$	3.27707	0.207676
$250$	1.00000	0.0632456
$251$	−6.82599	−0.430853	−0.215426	−	0.976520i	$-0.569114\pi$
$251$	−6.82599	−0.430853	−0.215426	+	0.976520i	$0.569114\pi$
$252$	8.81928	0.555562
$253$	0.581489	0.0365579
$254$	−7.16906	−0.449827
$255$	4.93324	0.308931
$256$	1.00000	0.0625000
$257$	−9.50619	−0.592980	−0.296490	−	0.955036i	$-0.595816\pi$
$257$	−9.50619	−0.592980	−0.296490	+	0.955036i	$0.595816\pi$
$258$	−1.55387	−0.0967394
$259$	20.5627	1.27770
$260$	0.0731262	0.00453509
$261$	−22.8638	−1.41524
$262$	13.0242	0.804636
$263$	−11.7862	−0.726766	−0.363383	−	0.931640i	$-0.618378\pi$
$263$	−11.7862	−0.726766	−0.363383	+	0.931640i	$0.618378\pi$
$264$	−0.0622284	−0.00382989
$265$	6.46394	0.397077
$266$	22.4626	1.37727
$267$	−1.25445	−0.0767710
$268$	1.00000	0.0610847
$269$	−15.1704	−0.924957	−0.462479	−	0.886630i	$-0.653040\pi$
$269$	−15.1704	−0.924957	−0.462479	+	0.886630i	$0.653040\pi$
$270$	3.66470	0.223027
$271$	15.0614	0.914916	0.457458	−	0.889231i	$-0.348760\pi$
$271$	15.0614	0.914916	0.457458	+	0.889231i	$0.348760\pi$
$272$	7.49346	0.454358
$273$	−0.165424	−0.0100119
$274$	−11.6984	−0.706724
$275$	−0.0945233	−0.00569997
$276$	4.04997	0.243779
$277$	4.19127	0.251829	0.125915	−	0.992041i	$-0.459813\pi$
$277$	4.19127	0.251829	0.125915	+	0.992041i	$0.459813\pi$
$278$	14.6998	0.881634
$279$	2.42670	0.145283
$280$	−3.43618	−0.205351
$281$	−29.8720	−1.78202	−0.891008	−	0.453988i	$-0.850001\pi$
$281$	−29.8720	−1.78202	−0.891008	+	0.453988i	$0.850001\pi$
$282$	6.65970	0.396579
$283$	−0.766948	−0.0455904	−0.0227952	−	0.999740i	$-0.507257\pi$
$283$	−0.766948	−0.0455904	−0.0227952	+	0.999740i	$0.507257\pi$
$284$	−5.53985	−0.328729
$285$	4.30361	0.254924
$286$	−0.00691213	−0.000408723 0
$287$	−17.0311	−1.00531
$288$	2.56659	0.151238
$289$	39.1520	2.30306
$290$	8.90825	0.523111
$291$	2.92470	0.171449
$292$	8.81014	0.515575
$293$	−11.0137	−0.643426	−0.321713	−	0.946837i	$-0.604259\pi$
$293$	−11.0137	−0.643426	−0.321713	+	0.946837i	$0.604259\pi$
$294$	3.16487	0.184579
$295$	−13.1582	−0.766098
$296$	5.98415	0.347822
$297$	−0.346400	−0.0201002
$298$	−8.98693	−0.520599
$299$	0.449858	0.0260159
$300$	−0.658339	−0.0380092
$301$	8.11037	0.467474
$302$	6.76376	0.389211
$303$	−3.70331	−0.212749
$304$	6.53707	0.374927
$305$	−10.1732	−0.582516
$306$	19.2327	1.09946
$307$	−10.4407	−0.595883	−0.297942	−	0.954584i	$-0.596300\pi$
$307$	−10.4407	−0.595883	−0.297942	+	0.954584i	$0.596300\pi$
$308$	0.324800	0.0185072
$309$	8.52252	0.484829
$310$	−0.945496	−0.0537006
$311$	12.5785	0.713261	0.356631	−	0.934245i	$-0.383925\pi$
$311$	12.5785	0.713261	0.356631	+	0.934245i	$0.383925\pi$
$312$	−0.0481418	−0.00272549
$313$	32.6480	1.84538	0.922688	−	0.385547i	$-0.125987\pi$
$313$	32.6480	1.84538	0.922688	+	0.385547i	$0.125987\pi$
$314$	9.15953	0.516902
$315$	−8.81928	−0.496910
$316$	−10.5371	−0.592757
$317$	25.8038	1.44928	0.724642	−	0.689125i	$-0.242005\pi$
$317$	25.8038	1.44928	0.724642	+	0.689125i	$0.242005\pi$
$318$	−4.25546	−0.238635
$319$	−0.842038	−0.0471451
$320$	−1.00000	−0.0559017
$321$	−10.5299	−0.587721
$322$	−21.1387	−1.17802
$323$	48.9853	2.72562
$324$	5.28716	0.293731
$325$	−0.0731262	−0.00405631
$326$	6.05728	0.335482
$327$	−7.40808	−0.409668
$328$	−4.95639	−0.273671
$329$	−34.7602	−1.91639
$330$	0.0622284	0.00342556
$331$	−19.8720	−1.09226	−0.546131	−	0.837700i	$-0.683900\pi$
$331$	−19.8720	−1.09226	−0.546131	+	0.837700i	$0.683900\pi$
$332$	−4.97779	−0.273192
$333$	15.3589	0.841661
$334$	−18.1946	−0.995564
$335$	−1.00000	−0.0546358
$336$	2.26217	0.123412
$337$	21.9202	1.19407	0.597034	−	0.802216i	$-0.296346\pi$
$337$	21.9202	1.19407	0.597034	+	0.802216i	$0.296346\pi$
$338$	12.9947	0.706816
$339$	−11.9560	−0.649363
$340$	−7.49346	−0.406390
$341$	0.0893714	0.00483973
$342$	16.7780	0.907249
$343$	7.53430	0.406814
$344$	2.36028	0.127258
$345$	−4.04997	−0.218043
$346$	11.8252	0.635725
$347$	−21.7665	−1.16849	−0.584244	−	0.811578i	$-0.698609\pi$
$347$	−21.7665	−1.16849	−0.584244	+	0.811578i	$0.698609\pi$
$348$	−5.86465	−0.314378
$349$	11.3350	0.606746	0.303373	−	0.952872i	$-0.401887\pi$
$349$	11.3350	0.606746	0.303373	+	0.952872i	$0.401887\pi$
$350$	3.43618	0.183672
$351$	−0.267986	−0.0143040
$352$	0.0945233	0.00503811
$353$	32.7109	1.74102	0.870512	−	0.492147i	$-0.163788\pi$
$353$	32.7109	1.74102	0.870512	+	0.492147i	$0.163788\pi$
$354$	8.66253	0.460408
$355$	5.53985	0.294024
$356$	1.90548	0.100990
$357$	16.9515	0.897169
$358$	12.0664	0.637730
$359$	8.46570	0.446803	0.223401	−	0.974727i	$-0.428284\pi$
$359$	8.46570	0.446803	0.223401	+	0.974727i	$0.428284\pi$
$360$	−2.56659	−0.135271
$361$	23.7333	1.24912
$362$	−18.1932	−0.956215
$363$	7.23584	0.379783
$364$	0.251275	0.0131704
$365$	−8.81014	−0.461144
$366$	6.69741	0.350079
$367$	−26.0720	−1.36095	−0.680473	−	0.732773i	$-0.738226\pi$
$367$	−26.0720	−1.36095	−0.680473	+	0.732773i	$0.738226\pi$
$368$	−6.15180	−0.320685
$369$	−12.7210	−0.662230
$370$	−5.98415	−0.311101
$371$	22.2113	1.15315
$372$	0.622456	0.0322729
$373$	−29.4078	−1.52268	−0.761340	−	0.648353i	$-0.775458\pi$
$373$	−29.4078	−1.52268	−0.761340	+	0.648353i	$0.775458\pi$
$374$	0.708307	0.0366257
$375$	0.658339	0.0339965
$376$	−10.1159	−0.521689
$377$	−0.651426	−0.0335502
$378$	12.5926	0.647693
$379$	9.30023	0.477721	0.238860	−	0.971054i	$-0.423226\pi$
$379$	9.30023	0.477721	0.238860	+	0.971054i	$0.423226\pi$
$380$	−6.53707	−0.335345
$381$	−4.71967	−0.241796
$382$	1.01726	0.0520476
$383$	25.6789	1.31213	0.656066	−	0.754704i	$-0.272219\pi$
$383$	25.6789	1.31213	0.656066	+	0.754704i	$0.272219\pi$
$384$	0.658339	0.0335957
$385$	−0.324800	−0.0165533
$386$	−9.58068	−0.487644
$387$	6.05788	0.307939
$388$	−4.44255	−0.225536
$389$	33.4333	1.69513	0.847567	−	0.530689i	$-0.178067\pi$
$389$	33.4333	1.69513	0.847567	+	0.530689i	$0.178067\pi$
$390$	0.0481418	0.00243775
$391$	−46.0983	−2.33129
$392$	−4.80737	−0.242809
$393$	8.57431	0.432517
$394$	−15.2632	−0.768948
$395$	10.5371	0.530178
$396$	0.242603	0.0121912
$397$	18.7311	0.940085	0.470043	−	0.882644i	$-0.344239\pi$
$397$	18.7311	0.940085	0.470043	+	0.882644i	$0.344239\pi$
$398$	−17.9233	−0.898413
$399$	14.7880	0.740325
$400$	1.00000	0.0500000
$401$	15.8787	0.792946	0.396473	−	0.918046i	$-0.370234\pi$
$401$	15.8787	0.792946	0.396473	+	0.918046i	$0.370234\pi$
$402$	0.658339	0.0328349
$403$	0.0691405	0.00344413
$404$	5.62523	0.279866
$405$	−5.28716	−0.262721
$406$	30.6104	1.51917
$407$	0.565642	0.0280378
$408$	4.93324	0.244232
$409$	−6.03771	−0.298546	−0.149273	−	0.988796i	$-0.547693\pi$
$409$	−6.03771	−0.298546	−0.149273	+	0.988796i	$0.547693\pi$
$410$	4.95639	0.244779
$411$	−7.70148	−0.379886
$412$	−12.9455	−0.637779
$413$	−45.2139	−2.22483
$414$	−15.7892	−0.775995
$415$	4.97779	0.244350
$416$	0.0731262	0.00358531
$417$	9.67743	0.473906
$418$	0.617906	0.0302228
$419$	−17.2632	−0.843362	−0.421681	−	0.906744i	$-0.638560\pi$
$419$	−17.2632	−0.843362	−0.421681	+	0.906744i	$0.638560\pi$
$420$	−2.26217	−0.110383
$421$	−2.18213	−0.106351	−0.0531754	−	0.998585i	$-0.516934\pi$
$421$	−2.18213	−0.106351	−0.0531754	+	0.998585i	$0.516934\pi$
$422$	−5.18268	−0.252289
$423$	−25.9634	−1.26238
$424$	6.46394	0.313917
$425$	7.49346	0.363486
$426$	−3.64709	−0.176702
$427$	−34.9570	−1.69169
$428$	15.9947	0.773131
$429$	−0.00455052	−0.000219701 0
$430$	−2.36028	−0.113823
$431$	33.1011	1.59442	0.797212	−	0.603700i	$-0.206307\pi$
$431$	33.1011	1.59442	0.797212	+	0.603700i	$0.206307\pi$
$432$	3.66470	0.176318
$433$	0.324401	0.0155897	0.00779486	−	0.999970i	$-0.497519\pi$
$433$	0.324401	0.0155897	0.00779486	+	0.999970i	$0.497519\pi$
$434$	−3.24890	−0.155952
$435$	5.86465	0.281188
$436$	11.2527	0.538906
$437$	−40.2148	−1.92373
$438$	5.80006	0.277137
$439$	−15.7398	−0.751219	−0.375610	−	0.926778i	$-0.622567\pi$
$439$	−15.7398	−0.751219	−0.375610	+	0.926778i	$0.622567\pi$
$440$	−0.0945233	−0.00450622
$441$	−12.3385	−0.587549
$442$	0.547968	0.0260642
$443$	26.8407	1.27524	0.637620	−	0.770351i	$-0.279919\pi$
$443$	26.8407	1.27524	0.637620	+	0.770351i	$0.279919\pi$
$444$	3.93960	0.186965
$445$	−1.90548	−0.0903283
$446$	17.5557	0.831286
$447$	−5.91644	−0.279838
$448$	−3.43618	−0.162344
$449$	−38.7464	−1.82855	−0.914277	−	0.405090i	$-0.867240\pi$
$449$	−38.7464	−1.82855	−0.914277	+	0.405090i	$0.867240\pi$
$450$	2.56659	0.120990
$451$	−0.468495	−0.0220606
$452$	18.1609	0.854219
$453$	4.45284	0.209213
$454$	−15.0832	−0.707890
$455$	−0.251275	−0.0117800
$456$	4.30361	0.201535
$457$	19.4740	0.910957	0.455478	−	0.890247i	$-0.349468\pi$
$457$	19.4740	0.910957	0.455478	+	0.890247i	$0.349468\pi$
$458$	10.5803	0.494383
$459$	27.4613	1.28178
$460$	6.15180	0.286829
$461$	−28.6017	−1.33211	−0.666056	−	0.745902i	$-0.732019\pi$
$461$	−28.6017	−1.33211	−0.666056	+	0.745902i	$0.732019\pi$
$462$	0.213828	0.00994819
$463$	10.6088	0.493034	0.246517	−	0.969138i	$-0.420714\pi$
$463$	10.6088	0.493034	0.246517	+	0.969138i	$0.420714\pi$
$464$	8.90825	0.413555
$465$	−0.622456	−0.0288657
$466$	−10.6906	−0.495234
$467$	5.84461	0.270456	0.135228	−	0.990814i	$-0.456823\pi$
$467$	5.84461	0.270456	0.135228	+	0.990814i	$0.456823\pi$
$468$	0.187685	0.00867574
$469$	−3.43618	−0.158668
$470$	10.1159	0.466613
$471$	6.03007	0.277851
$472$	−13.1582	−0.605654
$473$	0.223102	0.0102582
$474$	−6.93696	−0.318625
$475$	6.53707	0.299941
$476$	−25.7489	−1.18020
$477$	16.5903	0.759618
$478$	−19.0587	−0.871724
$479$	−0.228516	−0.0104412	−0.00522058	−	0.999986i	$-0.501662\pi$
$479$	−0.228516	−0.0104412	−0.00522058	+	0.999986i	$0.501662\pi$
$480$	−0.658339	−0.0300489
$481$	0.437598	0.0199528
$482$	−15.8874	−0.723651
$483$	−13.9164	−0.633220
$484$	−10.9911	−0.499594
$485$	4.44255	0.201726
$486$	14.4748	0.656592
$487$	−6.52657	−0.295747	−0.147874	−	0.989006i	$-0.547243\pi$
$487$	−6.52657	−0.295747	−0.147874	+	0.989006i	$0.547243\pi$
$488$	−10.1732	−0.460519
$489$	3.98774	0.180332
$490$	4.80737	0.217175
$491$	−29.3227	−1.32331	−0.661657	−	0.749806i	$-0.730147\pi$
$491$	−29.3227	−1.32331	−0.661657	+	0.749806i	$0.730147\pi$
$492$	−3.26298	−0.147107
$493$	66.7537	3.00643
$494$	0.478031	0.0215076
$495$	−0.242603	−0.0109042
$496$	−0.945496	−0.0424540
$497$	19.0359	0.853878
$498$	−3.27707	−0.146849
$499$	−9.70371	−0.434398	−0.217199	−	0.976127i	$-0.569692\pi$
$499$	−9.70371	−0.434398	−0.217199	+	0.976127i	$0.569692\pi$
$500$	−1.00000	−0.0447214
$501$	−11.9782	−0.535147
$502$	6.82599	0.304659
$503$	5.63978	0.251466	0.125733	−	0.992064i	$-0.459872\pi$
$503$	5.63978	0.251466	0.125733	+	0.992064i	$0.459872\pi$
$504$	−8.81928	−0.392842
$505$	−5.62523	−0.250320
$506$	−0.581489	−0.0258503
$507$	8.55488	0.379936
$508$	7.16906	0.318076
$509$	−21.4754	−0.951880	−0.475940	−	0.879478i	$-0.657892\pi$
$509$	−21.4754	−0.951880	−0.475940	+	0.879478i	$0.657892\pi$
$510$	−4.93324	−0.218447
$511$	−30.2733	−1.33921
$512$	−1.00000	−0.0441942
$513$	23.9564	1.05770
$514$	9.50619	0.419300
$515$	12.9455	0.570447
$516$	1.55387	0.0684051
$517$	−0.956191	−0.0420532
$518$	−20.5627	−0.903471
$519$	7.78497	0.341722
$520$	−0.0731262	−0.00320680
$521$	−14.4086	−0.631253	−0.315627	−	0.948883i	$-0.602215\pi$
$521$	−14.4086	−0.631253	−0.315627	+	0.948883i	$0.602215\pi$
$522$	22.8638	1.00072
$523$	25.8111	1.12864	0.564320	−	0.825556i	$-0.309138\pi$
$523$	25.8111	1.12864	0.564320	+	0.825556i	$0.309138\pi$
$524$	−13.0242	−0.568964
$525$	2.26217	0.0987293
$526$	11.7862	0.513902
$527$	−7.08504	−0.308629
$528$	0.0622284	0.00270814
$529$	14.8447	0.645421
$530$	−6.46394	−0.280776
$531$	−33.7716	−1.46556
$532$	−22.4626	−0.973876
$533$	−0.362442	−0.0156991
$534$	1.25445	0.0542853
$535$	−15.9947	−0.691509
$536$	−1.00000	−0.0431934
$537$	7.94379	0.342800
$538$	15.1704	0.654043
$539$	−0.454408	−0.0195728
$540$	−3.66470	−0.157704
$541$	12.7671	0.548900	0.274450	−	0.961601i	$-0.411504\pi$
$541$	12.7671	0.548900	0.274450	+	0.961601i	$0.411504\pi$
$542$	−15.0614	−0.646943
$543$	−11.9773	−0.513995
$544$	−7.49346	−0.321280
$545$	−11.2527	−0.482012
$546$	0.165424	0.00707950
$547$	25.6814	1.09806	0.549028	−	0.835804i	$-0.314998\pi$
$547$	25.6814	1.09806	0.549028	+	0.835804i	$0.314998\pi$
$548$	11.6984	0.499729
$549$	−26.1104	−1.11437
$550$	0.0945233	0.00403049
$551$	58.2339	2.48085
$552$	−4.04997	−0.172378
$553$	36.2073	1.53969
$554$	−4.19127	−0.178070
$555$	−3.93960	−0.167227
$556$	−14.6998	−0.623409
$557$	−18.5961	−0.787942	−0.393971	−	0.919123i	$-0.628899\pi$
$557$	−18.5961	−0.787942	−0.393971	+	0.919123i	$0.628899\pi$
$558$	−2.42670	−0.102730
$559$	0.172599	0.00730014
$560$	3.43618	0.145205
$561$	0.466306	0.0196875
$562$	29.8720	1.26008
$563$	4.92905	0.207735	0.103867	−	0.994591i	$-0.466878\pi$
$563$	4.92905	0.207735	0.103867	+	0.994591i	$0.466878\pi$
$564$	−6.65970	−0.280424
$565$	−18.1609	−0.764037
$566$	0.766948	0.0322372
$567$	−18.1676	−0.762970
$568$	5.53985	0.232447
$569$	5.96472	0.250054	0.125027	−	0.992153i	$-0.460098\pi$
$569$	5.96472	0.250054	0.125027	+	0.992153i	$0.460098\pi$
$570$	−4.30361	−0.180258
$571$	−29.7080	−1.24324	−0.621620	−	0.783319i	$-0.713525\pi$
$571$	−29.7080	−1.24324	−0.621620	+	0.783319i	$0.713525\pi$
$572$	0.00691213	0.000289011 0
$573$	0.669702	0.0279772
$574$	17.0311	0.710864
$575$	−6.15180	−0.256548
$576$	−2.56659	−0.106941
$577$	−26.7317	−1.11285	−0.556427	−	0.830897i	$-0.687828\pi$
$577$	−26.7317	−1.11285	−0.556427	+	0.830897i	$0.687828\pi$
$578$	−39.1520	−1.62851
$579$	−6.30733	−0.262124
$580$	−8.90825	−0.369895
$581$	17.1046	0.709619
$582$	−2.92470	−0.121233
$583$	0.610994	0.0253048
$584$	−8.81014	−0.364566
$585$	−0.187685	−0.00775982
$586$	11.0137	0.454971
$587$	−42.4764	−1.75319	−0.876594	−	0.481230i	$-0.840190\pi$
$587$	−42.4764	−1.75319	−0.876594	+	0.481230i	$0.840190\pi$
$588$	−3.16487	−0.130517
$589$	−6.18077	−0.254674
$590$	13.1582	0.541713
$591$	−10.0483	−0.413334
$592$	−5.98415	−0.245947
$593$	−17.3450	−0.712273	−0.356137	−	0.934434i	$-0.615906\pi$
$593$	−17.3450	−0.712273	−0.356137	+	0.934434i	$0.615906\pi$
$594$	0.346400	0.0142130
$595$	25.7489	1.05560
$596$	8.98693	0.368119
$597$	−11.7996	−0.482925
$598$	−0.449858	−0.0183961
$599$	−47.8508	−1.95513	−0.977565	−	0.210636i	$-0.932447\pi$
$599$	−47.8508	−1.95513	−0.977565	+	0.210636i	$0.932447\pi$
$600$	0.658339	0.0268766
$601$	−3.85571	−0.157278	−0.0786389	−	0.996903i	$-0.525057\pi$
$601$	−3.85571	−0.157278	−0.0786389	+	0.996903i	$0.525057\pi$
$602$	−8.11037	−0.330554
$603$	−2.56659	−0.104520
$604$	−6.76376	−0.275213
$605$	10.9911	0.446850
$606$	3.70331	0.150437
$607$	13.6631	0.554567	0.277284	−	0.960788i	$-0.410566\pi$
$607$	13.6631	0.554567	0.277284	+	0.960788i	$0.410566\pi$
$608$	−6.53707	−0.265113
$609$	20.1520	0.816600
$610$	10.1732	0.411901
$611$	−0.739739	−0.0299266
$612$	−19.2327	−0.777434
$613$	18.9150	0.763968	0.381984	−	0.924169i	$-0.375241\pi$
$613$	18.9150	0.763968	0.381984	+	0.924169i	$0.375241\pi$
$614$	10.4407	0.421353
$615$	3.26298	0.131576
$616$	−0.324800	−0.0130866
$617$	1.55250	0.0625014	0.0312507	−	0.999512i	$-0.490051\pi$
$617$	1.55250	0.0625014	0.0312507	+	0.999512i	$0.490051\pi$
$618$	−8.52252	−0.342826
$619$	37.6913	1.51494	0.757470	−	0.652870i	$-0.226435\pi$
$619$	37.6913	1.51494	0.757470	+	0.652870i	$0.226435\pi$
$620$	0.945496	0.0379720
$621$	−22.5445	−0.904680
$622$	−12.5785	−0.504352
$623$	−6.54757	−0.262323
$624$	0.0481418	0.00192721
$625$	1.00000	0.0400000
$626$	−32.6480	−1.30488
$627$	0.406791	0.0162457
$628$	−9.15953	−0.365505
$629$	−44.8420	−1.78797
$630$	8.81928	0.351368
$631$	8.26772	0.329133	0.164566	−	0.986366i	$-0.447378\pi$
$631$	8.26772	0.329133	0.164566	+	0.986366i	$0.447378\pi$
$632$	10.5371	0.419142
$633$	−3.41196	−0.135613
$634$	−25.8038	−1.02480
$635$	−7.16906	−0.284496
$636$	4.25546	0.168740
$637$	−0.351544	−0.0139287
$638$	0.842038	0.0333366
$639$	14.2185	0.562476
$640$	1.00000	0.0395285
$641$	33.3528	1.31735	0.658677	−	0.752425i	$-0.271116\pi$
$641$	33.3528	1.31735	0.658677	+	0.752425i	$0.271116\pi$
$642$	10.5299	0.415582
$643$	11.2023	0.441776	0.220888	−	0.975299i	$-0.429104\pi$
$643$	11.2023	0.441776	0.220888	+	0.975299i	$0.429104\pi$
$644$	21.1387	0.832983
$645$	−1.55387	−0.0611834
$646$	−48.9853	−1.92730
$647$	40.6195	1.59692	0.798459	−	0.602049i	$-0.205649\pi$
$647$	40.6195	1.59692	0.798459	+	0.602049i	$0.205649\pi$
$648$	−5.28716	−0.207699
$649$	−1.24375	−0.0488216
$650$	0.0731262	0.00286824
$651$	−2.13887	−0.0838291
$652$	−6.05728	−0.237221
$653$	−18.0287	−0.705518	−0.352759	−	0.935714i	$-0.614756\pi$
$653$	−18.0287	−0.705518	−0.352759	+	0.935714i	$0.614756\pi$
$654$	7.40808	0.289679
$655$	13.0242	0.508897
$656$	4.95639	0.193515
$657$	−22.6120	−0.882179
$658$	34.7602	1.35509
$659$	−22.7567	−0.886473	−0.443237	−	0.896405i	$-0.646170\pi$
$659$	−22.7567	−0.886473	−0.443237	+	0.896405i	$0.646170\pi$
$660$	−0.0622284	−0.00242224
$661$	−9.27943	−0.360928	−0.180464	−	0.983582i	$-0.557760\pi$
$661$	−9.27943	−0.360928	−0.180464	+	0.983582i	$0.557760\pi$
$662$	19.8720	0.772346
$663$	0.360749	0.0140103
$664$	4.97779	0.193176
$665$	22.4626	0.871062
$666$	−15.3589	−0.595144
$667$	−54.8018	−2.12193
$668$	18.1946	0.703970
$669$	11.5576	0.446842
$670$	1.00000	0.0386334
$671$	−0.961605	−0.0371223
$672$	−2.26217	−0.0872652
$673$	−37.5777	−1.44851	−0.724257	−	0.689530i	$-0.757817\pi$
$673$	−37.5777	−1.44851	−0.724257	+	0.689530i	$0.757817\pi$
$674$	−21.9202	−0.844333
$675$	3.66470	0.141054
$676$	−12.9947	−0.499794
$677$	−24.8835	−0.956349	−0.478175	−	0.878265i	$-0.658701\pi$
$677$	−24.8835	−0.956349	−0.478175	+	0.878265i	$0.658701\pi$
$678$	11.9560	0.459169
$679$	15.2654	0.585833
$680$	7.49346	0.287361
$681$	−9.92986	−0.380513
$682$	−0.0893714	−0.00342221
$683$	28.4499	1.08860	0.544302	−	0.838889i	$-0.316795\pi$
$683$	28.4499	1.08860	0.544302	+	0.838889i	$0.316795\pi$
$684$	−16.7780	−0.641522
$685$	−11.6984	−0.446971
$686$	−7.53430	−0.287661
$687$	6.96540	0.265746
$688$	−2.36028	−0.0899850
$689$	0.472684	0.0180078
$690$	4.04997	0.154180
$691$	−2.83831	−0.107975	−0.0539873	−	0.998542i	$-0.517193\pi$
$691$	−2.83831	−0.107975	−0.0539873	+	0.998542i	$0.517193\pi$
$692$	−11.8252	−0.449526
$693$	−0.833628	−0.0316669
$694$	21.7665	0.826246
$695$	14.6998	0.557594
$696$	5.86465	0.222299
$697$	37.1406	1.40680
$698$	−11.3350	−0.429034
$699$	−7.03806	−0.266204
$700$	−3.43618	−0.129876
$701$	−43.8171	−1.65495	−0.827475	−	0.561503i	$-0.810223\pi$
$701$	−43.8171	−1.65495	−0.827475	+	0.561503i	$0.810223\pi$
$702$	0.267986	0.0101145
$703$	−39.1188	−1.47539
$704$	−0.0945233	−0.00356248
$705$	6.65970	0.250819
$706$	−32.7109	−1.23109
$707$	−19.3293	−0.726954
$708$	−8.66253	−0.325558
$709$	−14.8638	−0.558223	−0.279111	−	0.960259i	$-0.590040\pi$
$709$	−14.8638	−0.558223	−0.279111	+	0.960259i	$0.590040\pi$
$710$	−5.53985	−0.207907
$711$	27.0443	1.01424
$712$	−1.90548	−0.0714108
$713$	5.81650	0.217830
$714$	−16.9515	−0.634394
$715$	−0.00691213	−0.000258499 0
$716$	−12.0664	−0.450943
$717$	−12.5471	−0.468579
$718$	−8.46570	−0.315937
$719$	38.0265	1.41815	0.709075	−	0.705133i	$-0.249113\pi$
$719$	38.0265	1.41815	0.709075	+	0.705133i	$0.249113\pi$
$720$	2.56659	0.0956512
$721$	44.4831	1.65664
$722$	−23.7333	−0.883262
$723$	−10.4593	−0.388985
$724$	18.1932	0.676146
$725$	8.90825	0.330844
$726$	−7.23584	−0.268547
$727$	46.5718	1.72725	0.863626	−	0.504133i	$-0.168188\pi$
$727$	46.5718	1.72725	0.863626	+	0.504133i	$0.168188\pi$
$728$	−0.251275	−0.00931287
$729$	−6.33213	−0.234523
$730$	8.81014	0.326078
$731$	−17.6867	−0.654166
$732$	−6.69741	−0.247543
$733$	−13.5971	−0.502221	−0.251111	−	0.967958i	$-0.580796\pi$
$733$	−13.5971	−0.502221	−0.251111	+	0.967958i	$0.580796\pi$
$734$	26.0720	0.962334
$735$	3.16487	0.116738
$736$	6.15180	0.226758
$737$	−0.0945233	−0.00348181
$738$	12.7210	0.468268
$739$	52.7370	1.93996	0.969981	−	0.243180i	$-0.0781905\pi$
$739$	52.7370	1.93996	0.969981	+	0.243180i	$0.0781905\pi$
$740$	5.98415	0.219982
$741$	0.314706	0.0115610
$742$	−22.2113	−0.815403
$743$	20.3204	0.745484	0.372742	−	0.927935i	$-0.378418\pi$
$743$	20.3204	0.745484	0.372742	+	0.927935i	$0.378418\pi$
$744$	−0.622456	−0.0228204
$745$	−8.98693	−0.329256
$746$	29.4078	1.07670
$747$	12.7759	0.467448
$748$	−0.708307	−0.0258983
$749$	−54.9606	−2.00822
$750$	−0.658339	−0.0240391
$751$	−9.22039	−0.336457	−0.168228	−	0.985748i	$-0.553805\pi$
$751$	−9.22039	−0.336457	−0.168228	+	0.985748i	$0.553805\pi$
$752$	10.1159	0.368890
$753$	4.49381	0.163764
$754$	0.651426	0.0237236
$755$	6.76376	0.246158
$756$	−12.5926	−0.457988
$757$	−9.74338	−0.354129	−0.177065	−	0.984199i	$-0.556660\pi$
$757$	−9.74338	−0.354129	−0.177065	+	0.984199i	$0.556660\pi$
$758$	−9.30023	−0.337800
$759$	−0.382817	−0.0138954
$760$	6.53707	0.237124
$761$	32.7707	1.18794	0.593968	−	0.804489i	$-0.297561\pi$
$761$	32.7707	1.18794	0.593968	+	0.804489i	$0.297561\pi$
$762$	4.71967	0.170976
$763$	−38.6663	−1.39981
$764$	−1.01726	−0.0368032
$765$	19.2327	0.695358
$766$	−25.6789	−0.927817
$767$	−0.962206	−0.0347433
$768$	−0.658339	−0.0237557
$769$	−13.6663	−0.492818	−0.246409	−	0.969166i	$-0.579251\pi$
$769$	−13.6663	−0.492818	−0.246409	+	0.969166i	$0.579251\pi$
$770$	0.324800	0.0117050
$771$	6.25829	0.225387
$772$	9.58068	0.344816
$773$	−21.3064	−0.766337	−0.383169	−	0.923678i	$-0.625167\pi$
$773$	−21.3064	−0.766337	−0.383169	+	0.923678i	$0.625167\pi$
$774$	−6.05788	−0.217746
$775$	−0.945496	−0.0339632
$776$	4.44255	0.159478
$777$	−13.5372	−0.485644
$778$	−33.4333	−1.19864
$779$	32.4003	1.16086
$780$	−0.0481418	−0.00172375
$781$	0.523645	0.0187375
$782$	46.0983	1.64847
$783$	32.6461	1.16668
$784$	4.80737	0.171692
$785$	9.15953	0.326918
$786$	−8.57431	−0.305836
$787$	−18.0151	−0.642168	−0.321084	−	0.947051i	$-0.604047\pi$
$787$	−18.0151	−0.642168	−0.321084	+	0.947051i	$0.604047\pi$
$788$	15.2632	0.543729
$789$	7.75929	0.276238
$790$	−10.5371	−0.374892
$791$	−62.4043	−2.21884
$792$	−0.242603	−0.00862051
$793$	−0.743927	−0.0264176
$794$	−18.7311	−0.664741
$795$	−4.25546	−0.150926
$796$	17.9233	0.635274
$797$	−2.28776	−0.0810366	−0.0405183	−	0.999179i	$-0.512901\pi$
$797$	−2.28776	−0.0810366	−0.0405183	+	0.999179i	$0.512901\pi$
$798$	−14.7880	−0.523489
$799$	75.8033	2.68173
$800$	−1.00000	−0.0353553
$801$	−4.89058	−0.172800
$802$	−15.8787	−0.560698
$803$	−0.832764	−0.0293876
$804$	−0.658339	−0.0232178
$805$	−21.1387	−0.745042
$806$	−0.0691405	−0.00243537
$807$	9.98728	0.351569
$808$	−5.62523	−0.197895
$809$	47.1693	1.65838	0.829192	−	0.558965i	$-0.188801\pi$
$809$	47.1693	1.65838	0.829192	+	0.558965i	$0.188801\pi$
$810$	5.28716	0.185772
$811$	−28.4463	−0.998884	−0.499442	−	0.866347i	$-0.666462\pi$
$811$	−28.4463	−0.998884	−0.499442	+	0.866347i	$0.666462\pi$
$812$	−30.6104	−1.07421
$813$	−9.91551	−0.347752
$814$	−0.565642	−0.0198257
$815$	6.05728	0.212177
$816$	−4.93324	−0.172698
$817$	−15.4293	−0.539804
$818$	6.03771	0.211104
$819$	−0.644920	−0.0225353
$820$	−4.95639	−0.173085
$821$	−47.0248	−1.64118	−0.820589	−	0.571519i	$-0.806355\pi$
$821$	−47.0248	−1.64118	−0.820589	+	0.571519i	$0.806355\pi$
$822$	7.70148	0.268620
$823$	−33.4453	−1.16583	−0.582915	−	0.812533i	$-0.698088\pi$
$823$	−33.4453	−1.16583	−0.582915	+	0.812533i	$0.698088\pi$
$824$	12.9455	0.450978
$825$	0.0622284	0.00216651
$826$	45.2139	1.57319
$827$	11.4239	0.397249	0.198624	−	0.980076i	$-0.436353\pi$
$827$	11.4239	0.397249	0.198624	+	0.980076i	$0.436353\pi$
$828$	15.7892	0.548711
$829$	30.0084	1.04223	0.521117	−	0.853485i	$-0.325516\pi$
$829$	30.0084	1.04223	0.521117	+	0.853485i	$0.325516\pi$
$830$	−4.97779	−0.172782
$831$	−2.75928	−0.0957182
$832$	−0.0731262	−0.00253519
$833$	36.0238	1.24815
$834$	−9.67743	−0.335102
$835$	−18.1946	−0.629650
$836$	−0.617906	−0.0213707
$837$	−3.46496	−0.119767
$838$	17.2632	0.596347
$839$	31.0225	1.07102	0.535509	−	0.844530i	$-0.320120\pi$
$839$	31.0225	1.07102	0.535509	+	0.844530i	$0.320120\pi$
$840$	2.26217	0.0780524
$841$	50.3570	1.73645
$842$	2.18213	0.0752013
$843$	19.6659	0.677330
$844$	5.18268	0.178395
$845$	12.9947	0.447030
$846$	25.9634	0.892641
$847$	37.7673	1.29770
$848$	−6.46394	−0.221973
$849$	0.504912	0.0173285
$850$	−7.49346	−0.257024
$851$	36.8133	1.26194
$852$	3.64709	0.124947
$853$	21.7293	0.743997	0.371998	−	0.928233i	$-0.378673\pi$
$853$	21.7293	0.743997	0.371998	+	0.928233i	$0.378673\pi$
$854$	34.9570	1.19620
$855$	16.7780	0.573795
$856$	−15.9947	−0.546686
$857$	31.9466	1.09128	0.545638	−	0.838021i	$-0.316287\pi$
$857$	31.9466	1.09128	0.545638	+	0.838021i	$0.316287\pi$
$858$	0.00455052	0.000155352 0
$859$	40.3377	1.37630	0.688152	−	0.725567i	$-0.258422\pi$
$859$	40.3377	1.37630	0.688152	+	0.725567i	$0.258422\pi$
$860$	2.36028	0.0804850
$861$	11.2122	0.382111
$862$	−33.1011	−1.12743
$863$	42.7654	1.45575	0.727875	−	0.685709i	$-0.240508\pi$
$863$	42.7654	1.45575	0.727875	+	0.685709i	$0.240508\pi$
$864$	−3.66470	−0.124676
$865$	11.8252	0.402068
$866$	−0.324401	−0.0110236
$867$	−25.7753	−0.875374
$868$	3.24890	0.110275
$869$	0.995999	0.0337870
$870$	−5.86465	−0.198830
$871$	−0.0731262	−0.00247779
$872$	−11.2527	−0.381064
$873$	11.4022	0.385906
$874$	40.2148	1.36028
$875$	3.43618	0.116164
$876$	−5.80006	−0.195966
$877$	−42.1110	−1.42199	−0.710994	−	0.703198i	$-0.751755\pi$
$877$	−42.1110	−1.42199	−0.710994	+	0.703198i	$0.751755\pi$
$878$	15.7398	0.531192
$879$	7.25073	0.244561
$880$	0.0945233	0.00318638
$881$	−17.4186	−0.586847	−0.293423	−	0.955983i	$-0.594795\pi$
$881$	−17.4186	−0.586847	−0.293423	+	0.955983i	$0.594795\pi$
$882$	12.3385	0.415460
$883$	3.43951	0.115749	0.0578744	−	0.998324i	$-0.481568\pi$
$883$	3.43951	0.115749	0.0578744	+	0.998324i	$0.481568\pi$
$884$	−0.547968	−0.0184302
$885$	8.66253	0.291188
$886$	−26.8407	−0.901730
$887$	9.51343	0.319430	0.159715	−	0.987163i	$-0.448943\pi$
$887$	9.51343	0.319430	0.159715	+	0.987163i	$0.448943\pi$
$888$	−3.93960	−0.132204
$889$	−24.6342	−0.826205
$890$	1.90548	0.0638717
$891$	−0.499760	−0.0167426
$892$	−17.5557	−0.587808
$893$	66.1285	2.21291
$894$	5.91644	0.197875
$895$	12.0664	0.403336
$896$	3.43618	0.114795
$897$	−0.296159	−0.00988845
$898$	38.7464	1.29298
$899$	−8.42271	−0.280913
$900$	−2.56659	−0.0855530
$901$	−48.4373	−1.61368
$902$	0.468495	0.0155992
$903$	−5.33937	−0.177683
$904$	−18.1609	−0.604024
$905$	−18.1932	−0.604764
$906$	−4.45284	−0.147936
$907$	−4.81326	−0.159822	−0.0799109	−	0.996802i	$-0.525464\pi$
$907$	−4.81326	−0.159822	−0.0799109	+	0.996802i	$0.525464\pi$
$908$	15.0832	0.500554
$909$	−14.4377	−0.478867
$910$	0.251275	0.00832969
$911$	−12.3088	−0.407807	−0.203904	−	0.978991i	$-0.565363\pi$
$911$	−12.3088	−0.407807	−0.203904	+	0.978991i	$0.565363\pi$
$912$	−4.30361	−0.142507
$913$	0.470517	0.0155719
$914$	−19.4740	−0.644144
$915$	6.69741	0.221410
$916$	−10.5803	−0.349582
$917$	44.7535	1.47789
$918$	−27.4613	−0.906358
$919$	−33.3461	−1.09998	−0.549992	−	0.835170i	$-0.685369\pi$
$919$	−33.3461	−1.09998	−0.549992	+	0.835170i	$0.685369\pi$
$920$	−6.15180	−0.202819
$921$	6.87353	0.226490
$922$	28.6017	0.941946
$923$	0.405108	0.0133343
$924$	−0.213828	−0.00703443
$925$	−5.98415	−0.196758
$926$	−10.6088	−0.348628
$927$	33.2258	1.09128
$928$	−8.90825	−0.292428
$929$	−17.0832	−0.560482	−0.280241	−	0.959930i	$-0.590414\pi$
$929$	−17.0832	−0.560482	−0.280241	+	0.959930i	$0.590414\pi$
$930$	0.622456	0.0204111
$931$	31.4261	1.02995
$932$	10.6906	0.350183
$933$	−8.28091	−0.271105
$934$	−5.84461	−0.191241
$935$	0.708307	0.0231641
$936$	−0.187685	−0.00613467
$937$	39.1152	1.27784	0.638919	−	0.769274i	$-0.279382\pi$
$937$	39.1152	1.27784	0.638919	+	0.769274i	$0.279382\pi$
$938$	3.43618	0.112195
$939$	−21.4935	−0.701413
$940$	−10.1159	−0.329945
$941$	22.0111	0.717541	0.358771	−	0.933426i	$-0.383196\pi$
$941$	22.0111	0.717541	0.358771	+	0.933426i	$0.383196\pi$
$942$	−6.03007	−0.196470
$943$	−30.4908	−0.992916
$944$	13.1582	0.428262
$945$	12.5926	0.409637
$946$	−0.223102	−0.00725367
$947$	4.33523	0.140876	0.0704381	−	0.997516i	$-0.477560\pi$
$947$	4.33523	0.140876	0.0704381	+	0.997516i	$0.477560\pi$
$948$	6.93696	0.225302
$949$	−0.644252	−0.0209133
$950$	−6.53707	−0.212091
$951$	−16.9876	−0.550861
$952$	25.7489	0.834528
$953$	−58.6148	−1.89872	−0.949360	−	0.314191i	$-0.898267\pi$
$953$	−58.6148	−1.89872	−0.949360	+	0.314191i	$0.898267\pi$
$954$	−16.5903	−0.537131
$955$	1.01726	0.0329178
$956$	19.0587	0.616402
$957$	0.554346	0.0179195
$958$	0.228516	0.00738302
$959$	−40.1977	−1.29805
$960$	0.658339	0.0212478
$961$	−30.1060	−0.971163
$962$	−0.437598	−0.0141087
$963$	−41.0517	−1.32287
$964$	15.8874	0.511699
$965$	−9.58068	−0.308413
$966$	13.9164	0.447754
$967$	−12.5602	−0.403910	−0.201955	−	0.979395i	$-0.564729\pi$
$967$	−12.5602	−0.403910	−0.201955	+	0.979395i	$0.564729\pi$
$968$	10.9911	0.353266
$969$	−32.2489	−1.03598
$970$	−4.44255	−0.142642
$971$	−7.75029	−0.248719	−0.124359	−	0.992237i	$-0.539688\pi$
$971$	−7.75029	−0.248719	−0.124359	+	0.992237i	$0.539688\pi$
$972$	−14.4748	−0.464281
$973$	50.5111	1.61931
$974$	6.52657	0.209125
$975$	0.0481418	0.00154177
$976$	10.1732	0.325636
$977$	−40.4570	−1.29433	−0.647167	−	0.762349i	$-0.724046\pi$
$977$	−40.4570	−1.29433	−0.647167	+	0.762349i	$0.724046\pi$
$978$	−3.98774	−0.127514
$979$	−0.180112	−0.00575641
$980$	−4.80737	−0.153566
$981$	−28.8810	−0.922101
$982$	29.3227	0.935725
$983$	−26.5019	−0.845278	−0.422639	−	0.906298i	$-0.638896\pi$
$983$	−26.5019	−0.845278	−0.422639	+	0.906298i	$0.638896\pi$
$984$	3.26298	0.104020
$985$	−15.2632	−0.486326
$986$	−66.7537	−2.12587
$987$	22.8840	0.728405
$988$	−0.478031	−0.0152082
$989$	14.5200	0.461709
$990$	0.242603	0.00771042
$991$	9.50220	0.301847	0.150924	−	0.988545i	$-0.451775\pi$
$991$	9.50220	0.301847	0.150924	+	0.988545i	$0.451775\pi$
$992$	0.945496	0.0300195
$993$	13.0825	0.415160
$994$	−19.0359	−0.603783
$995$	−17.9233	−0.568206
$996$	3.27707	0.103838
$997$	−25.2436	−0.799473	−0.399737	−	0.916630i	$-0.630898\pi$
$997$	−25.2436	−0.799473	−0.399737	+	0.916630i	$0.630898\pi$
$998$	9.70371	0.307165
$999$	−21.9301	−0.693839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	670.2.a.j.1.3	✓	4
3.2	odd	2		6030.2.a.bt.1.2		4
4.3	odd	2		5360.2.a.be.1.2		4
5.2	odd	4		3350.2.c.m.2949.2		8
5.3	odd	4		3350.2.c.m.2949.7		8
5.4	even	2		3350.2.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
670.2.a.j.1.3	✓	4	1.1	even	1	trivial
3350.2.a.n.1.2		4	5.4	even	2
3350.2.c.m.2949.2		8	5.2	odd	4
3350.2.c.m.2949.7		8	5.3	odd	4
5360.2.a.be.1.2		4	4.3	odd	2
6030.2.a.bt.1.2		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 670 = 2 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.658339 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.658339 q^{6} -3.43618 q^{7} -1.00000 q^{8} -2.56659 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.658339 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.658339 q^{6} -3.43618 q^{7} -1.00000 q^{8} -2.56659 q^{9} +1.00000 q^{10} -0.0945233 q^{11} -0.658339 q^{12} -0.0731262 q^{13} +3.43618 q^{14} +0.658339 q^{15} +1.00000 q^{16} +7.49346 q^{17} +2.56659 q^{18} +6.53707 q^{19} -1.00000 q^{20} +2.26217 q^{21} +0.0945233 q^{22} -6.15180 q^{23} +0.658339 q^{24} +1.00000 q^{25} +0.0731262 q^{26} +3.66470 q^{27} -3.43618 q^{28} +8.90825 q^{29} -0.658339 q^{30} -0.945496 q^{31} -1.00000 q^{32} +0.0622284 q^{33} -7.49346 q^{34} +3.43618 q^{35} -2.56659 q^{36} -5.98415 q^{37} -6.53707 q^{38} +0.0481418 q^{39} +1.00000 q^{40} +4.95639 q^{41} -2.26217 q^{42} -2.36028 q^{43} -0.0945233 q^{44} +2.56659 q^{45} +6.15180 q^{46} +10.1159 q^{47} -0.658339 q^{48} +4.80737 q^{49} -1.00000 q^{50} -4.93324 q^{51} -0.0731262 q^{52} -6.46394 q^{53} -3.66470 q^{54} +0.0945233 q^{55} +3.43618 q^{56} -4.30361 q^{57} -8.90825 q^{58} +13.1582 q^{59} +0.658339 q^{60} +10.1732 q^{61} +0.945496 q^{62} +8.81928 q^{63} +1.00000 q^{64} +0.0731262 q^{65} -0.0622284 q^{66} +1.00000 q^{67} +7.49346 q^{68} +4.04997 q^{69} -3.43618 q^{70} -5.53985 q^{71} +2.56659 q^{72} +8.81014 q^{73} +5.98415 q^{74} -0.658339 q^{75} +6.53707 q^{76} +0.324800 q^{77} -0.0481418 q^{78} -10.5371 q^{79} -1.00000 q^{80} +5.28716 q^{81} -4.95639 q^{82} -4.97779 q^{83} +2.26217 q^{84} -7.49346 q^{85} +2.36028 q^{86} -5.86465 q^{87} +0.0945233 q^{88} +1.90548 q^{89} -2.56659 q^{90} +0.251275 q^{91} -6.15180 q^{92} +0.622456 q^{93} -10.1159 q^{94} -6.53707 q^{95} +0.658339 q^{96} -4.44255 q^{97} -4.80737 q^{98} +0.242603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9} + 4 q^{10} + 9 q^{11} - 2 q^{12} - 12 q^{13} + 5 q^{14} + 2 q^{15} + 4 q^{16} - 8 q^{18} + 4 q^{19} - 4 q^{20} + 2 q^{21} - 9 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{25} + 12 q^{26} + 10 q^{27} - 5 q^{28} + 18 q^{29} - 2 q^{30} + 2 q^{31} - 4 q^{32} + 14 q^{33} + 5 q^{35} + 8 q^{36} + 9 q^{37} - 4 q^{38} + 10 q^{39} + 4 q^{40} + 12 q^{41} - 2 q^{42} - 16 q^{43} + 9 q^{44} - 8 q^{45} - 6 q^{46} + 10 q^{47} - 2 q^{48} + 15 q^{49} - 4 q^{50} - 4 q^{51} - 12 q^{52} + 8 q^{53} - 10 q^{54} - 9 q^{55} + 5 q^{56} + 44 q^{57} - 18 q^{58} + 18 q^{59} + 2 q^{60} - 11 q^{61} - 2 q^{62} - 27 q^{63} + 4 q^{64} + 12 q^{65} - 14 q^{66} + 4 q^{67} + 20 q^{69} - 5 q^{70} + 27 q^{71} - 8 q^{72} + 4 q^{73} - 9 q^{74} - 2 q^{75} + 4 q^{76} + 25 q^{77} - 10 q^{78} - 20 q^{79} - 4 q^{80} + 16 q^{81} - 12 q^{82} + 9 q^{83} + 2 q^{84} + 16 q^{86} + 2 q^{87} - 9 q^{88} + 17 q^{89} + 8 q^{90} - 4 q^{91} + 6 q^{92} + 2 q^{93} - 10 q^{94} - 4 q^{95} + 2 q^{96} - 5 q^{97} - 15 q^{98} + 9 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9 + 4 * q^10 + 9 * q^11 - 2 * q^12 - 12 * q^13 + 5 * q^14 + 2 * q^15 + 4 * q^16 - 8 * q^18 + 4 * q^19 - 4 * q^20 + 2 * q^21 - 9 * q^22 + 6 * q^23 + 2 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 5 * q^28 + 18 * q^29 - 2 * q^30 + 2 * q^31 - 4 * q^32 + 14 * q^33 + 5 * q^35 + 8 * q^36 + 9 * q^37 - 4 * q^38 + 10 * q^39 + 4 * q^40 + 12 * q^41 - 2 * q^42 - 16 * q^43 + 9 * q^44 - 8 * q^45 - 6 * q^46 + 10 * q^47 - 2 * q^48 + 15 * q^49 - 4 * q^50 - 4 * q^51 - 12 * q^52 + 8 * q^53 - 10 * q^54 - 9 * q^55 + 5 * q^56 + 44 * q^57 - 18 * q^58 + 18 * q^59 + 2 * q^60 - 11 * q^61 - 2 * q^62 - 27 * q^63 + 4 * q^64 + 12 * q^65 - 14 * q^66 + 4 * q^67 + 20 * q^69 - 5 * q^70 + 27 * q^71 - 8 * q^72 + 4 * q^73 - 9 * q^74 - 2 * q^75 + 4 * q^76 + 25 * q^77 - 10 * q^78 - 20 * q^79 - 4 * q^80 + 16 * q^81 - 12 * q^82 + 9 * q^83 + 2 * q^84 + 16 * q^86 + 2 * q^87 - 9 * q^88 + 17 * q^89 + 8 * q^90 - 4 * q^91 + 6 * q^92 + 2 * q^93 - 10 * q^94 - 4 * q^95 + 2 * q^96 - 5 * q^97 - 15 * q^98 + 9 * q^99

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