LMFDB - Embedded newform 6045.2.a.bg.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-1.89108$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.89108 q^{2} -1.00000 q^{3} +1.57618 q^{4} -1.00000 q^{5} -1.89108 q^{6} -4.66347 q^{7} -0.801474 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.89108 q^{2} -1.00000 q^{3} +1.57618 q^{4} -1.00000 q^{5} -1.89108 q^{6} -4.66347 q^{7} -0.801474 q^{8} +1.00000 q^{9} -1.89108 q^{10} -1.18887 q^{11} -1.57618 q^{12} -1.00000 q^{13} -8.81900 q^{14} +1.00000 q^{15} -4.66801 q^{16} -3.45934 q^{17} +1.89108 q^{18} -6.98703 q^{19} -1.57618 q^{20} +4.66347 q^{21} -2.24826 q^{22} +3.07928 q^{23} +0.801474 q^{24} +1.00000 q^{25} -1.89108 q^{26} -1.00000 q^{27} -7.35048 q^{28} -6.06614 q^{29} +1.89108 q^{30} +1.00000 q^{31} -7.22464 q^{32} +1.18887 q^{33} -6.54189 q^{34} +4.66347 q^{35} +1.57618 q^{36} +8.79048 q^{37} -13.2130 q^{38} +1.00000 q^{39} +0.801474 q^{40} +8.93189 q^{41} +8.81900 q^{42} +4.12119 q^{43} -1.87388 q^{44} -1.00000 q^{45} +5.82317 q^{46} +1.75423 q^{47} +4.66801 q^{48} +14.7480 q^{49} +1.89108 q^{50} +3.45934 q^{51} -1.57618 q^{52} -11.1664 q^{53} -1.89108 q^{54} +1.18887 q^{55} +3.73765 q^{56} +6.98703 q^{57} -11.4715 q^{58} +5.71527 q^{59} +1.57618 q^{60} +10.7254 q^{61} +1.89108 q^{62} -4.66347 q^{63} -4.32634 q^{64} +1.00000 q^{65} +2.24826 q^{66} -7.50222 q^{67} -5.45255 q^{68} -3.07928 q^{69} +8.81900 q^{70} -6.83059 q^{71} -0.801474 q^{72} -12.3097 q^{73} +16.6235 q^{74} -1.00000 q^{75} -11.0128 q^{76} +5.54428 q^{77} +1.89108 q^{78} +9.72740 q^{79} +4.66801 q^{80} +1.00000 q^{81} +16.8909 q^{82} +15.5753 q^{83} +7.35048 q^{84} +3.45934 q^{85} +7.79350 q^{86} +6.06614 q^{87} +0.952852 q^{88} +6.71923 q^{89} -1.89108 q^{90} +4.66347 q^{91} +4.85351 q^{92} -1.00000 q^{93} +3.31738 q^{94} +6.98703 q^{95} +7.22464 q^{96} -13.9548 q^{97} +27.8896 q^{98} -1.18887 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * 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 $

$2$	1.89108	1.33720	0.668598	−	0.743624i	$-0.266895\pi$
$2$	1.89108	1.33720	0.668598	+	0.743624i	$0.266895\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.57618	0.788091
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.89108	−0.772030
$7$	−4.66347	−1.76263	−0.881313	−	0.472532i	$-0.843340\pi$
$7$	−4.66347	−1.76263	−0.881313	+	0.472532i	$0.843340\pi$
$8$	−0.801474	−0.283364
$9$	1.00000	0.333333
$10$	−1.89108	−0.598012
$11$	−1.18887	−0.358459	−0.179230	−	0.983807i	$-0.557360\pi$
$11$	−1.18887	−0.358459	−0.179230	+	0.983807i	$0.557360\pi$
$12$	−1.57618	−0.455004
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−8.81900	−2.35698
$15$	1.00000	0.258199
$16$	−4.66801	−1.16700
$17$	−3.45934	−0.839014	−0.419507	−	0.907752i	$-0.637797\pi$
$17$	−3.45934	−0.839014	−0.419507	+	0.907752i	$0.637797\pi$
$18$	1.89108	0.445732
$19$	−6.98703	−1.60293	−0.801467	−	0.598039i	$-0.795947\pi$
$19$	−6.98703	−1.60293	−0.801467	+	0.598039i	$0.795947\pi$
$20$	−1.57618	−0.352445
$21$	4.66347	1.01765
$22$	−2.24826	−0.479330
$23$	3.07928	0.642075	0.321037	−	0.947067i	$-0.395969\pi$
$23$	3.07928	0.642075	0.321037	+	0.947067i	$0.395969\pi$
$24$	0.801474	0.163600
$25$	1.00000	0.200000
$26$	−1.89108	−0.370871
$27$	−1.00000	−0.192450
$28$	−7.35048	−1.38911
$29$	−6.06614	−1.12645	−0.563227	−	0.826302i	$-0.690440\pi$
$29$	−6.06614	−1.12645	−0.563227	+	0.826302i	$0.690440\pi$
$30$	1.89108	0.345262
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−7.22464	−1.27715
$33$	1.18887	0.206956
$34$	−6.54189	−1.12193
$35$	4.66347	0.788271
$36$	1.57618	0.262697
$37$	8.79048	1.44515	0.722573	−	0.691295i	$-0.242959\pi$
$37$	8.79048	1.44515	0.722573	+	0.691295i	$0.242959\pi$
$38$	−13.2130	−2.14344
$39$	1.00000	0.160128
$40$	0.801474	0.126724
$41$	8.93189	1.39493	0.697463	−	0.716621i	$-0.254312\pi$
$41$	8.93189	1.39493	0.697463	+	0.716621i	$0.254312\pi$
$42$	8.81900	1.36080
$43$	4.12119	0.628476	0.314238	−	0.949344i	$-0.398251\pi$
$43$	4.12119	0.628476	0.314238	+	0.949344i	$0.398251\pi$
$44$	−1.87388	−0.282498
$45$	−1.00000	−0.149071
$46$	5.82317	0.858579
$47$	1.75423	0.255880	0.127940	−	0.991782i	$-0.459163\pi$
$47$	1.75423	0.255880	0.127940	+	0.991782i	$0.459163\pi$
$48$	4.66801	0.673770
$49$	14.7480	2.10685
$50$	1.89108	0.267439
$51$	3.45934	0.484405
$52$	−1.57618	−0.218577
$53$	−11.1664	−1.53382	−0.766910	−	0.641755i	$-0.778207\pi$
$53$	−11.1664	−1.53382	−0.766910	+	0.641755i	$0.778207\pi$
$54$	−1.89108	−0.257343
$55$	1.18887	0.160308
$56$	3.73765	0.499465
$57$	6.98703	0.925454
$58$	−11.4715	−1.50629
$59$	5.71527	0.744064	0.372032	−	0.928220i	$-0.378661\pi$
$59$	5.71527	0.744064	0.372032	+	0.928220i	$0.378661\pi$
$60$	1.57618	0.203484
$61$	10.7254	1.37325	0.686626	−	0.727010i	$-0.259091\pi$
$61$	10.7254	1.37325	0.686626	+	0.727010i	$0.259091\pi$
$62$	1.89108	0.240167
$63$	−4.66347	−0.587542
$64$	−4.32634	−0.540792
$65$	1.00000	0.124035
$66$	2.24826	0.276741
$67$	−7.50222	−0.916543	−0.458271	−	0.888812i	$-0.651531\pi$
$67$	−7.50222	−0.916543	−0.458271	+	0.888812i	$0.651531\pi$
$68$	−5.45255	−0.661219
$69$	−3.07928	−0.370702
$70$	8.81900	1.05407
$71$	−6.83059	−0.810642	−0.405321	−	0.914174i	$-0.632840\pi$
$71$	−6.83059	−0.810642	−0.405321	+	0.914174i	$0.632840\pi$
$72$	−0.801474	−0.0944546
$73$	−12.3097	−1.44074	−0.720369	−	0.693591i	$-0.756027\pi$
$73$	−12.3097	−1.44074	−0.720369	+	0.693591i	$0.756027\pi$
$74$	16.6235	1.93244
$75$	−1.00000	−0.115470
$76$	−11.0128	−1.26326
$77$	5.54428	0.631830
$78$	1.89108	0.214123
$79$	9.72740	1.09442	0.547209	−	0.836996i	$-0.315690\pi$
$79$	9.72740	1.09442	0.547209	+	0.836996i	$0.315690\pi$
$80$	4.66801	0.521900
$81$	1.00000	0.111111
$82$	16.8909	1.86529
$83$	15.5753	1.70961	0.854807	−	0.518946i	$-0.173676\pi$
$83$	15.5753	1.70961	0.854807	+	0.518946i	$0.173676\pi$
$84$	7.35048	0.802003
$85$	3.45934	0.375218
$86$	7.79350	0.840395
$87$	6.06614	0.650358
$88$	0.952852	0.101574
$89$	6.71923	0.712237	0.356118	−	0.934441i	$-0.384100\pi$
$89$	6.71923	0.712237	0.356118	+	0.934441i	$0.384100\pi$
$90$	−1.89108	−0.199337
$91$	4.66347	0.488865
$92$	4.85351	0.506013
$93$	−1.00000	−0.103695
$94$	3.31738	0.342162
$95$	6.98703	0.716854
$96$	7.22464	0.737362
$97$	−13.9548	−1.41690	−0.708449	−	0.705762i	$-0.750605\pi$
$97$	−13.9548	−1.41690	−0.708449	+	0.705762i	$0.750605\pi$
$98$	27.8896	2.81727
$99$	−1.18887	−0.119486
$100$	1.57618	0.157618
$101$	−11.3311	−1.12749	−0.563743	−	0.825950i	$-0.690639\pi$
$101$	−11.3311	−1.12749	−0.563743	+	0.825950i	$0.690639\pi$
$102$	6.54189	0.647744
$103$	6.47881	0.638376	0.319188	−	0.947691i	$-0.396590\pi$
$103$	6.47881	0.638376	0.319188	+	0.947691i	$0.396590\pi$
$104$	0.801474	0.0785910
$105$	−4.66347	−0.455108
$106$	−21.1165	−2.05102
$107$	17.4549	1.68743	0.843715	−	0.536791i	$-0.180363\pi$
$107$	17.4549	1.68743	0.843715	+	0.536791i	$0.180363\pi$
$108$	−1.57618	−0.151668
$109$	−18.2331	−1.74641	−0.873206	−	0.487351i	$-0.837963\pi$
$109$	−18.2331	−1.74641	−0.873206	+	0.487351i	$0.837963\pi$
$110$	2.24826	0.214363
$111$	−8.79048	−0.834355
$112$	21.7692	2.05699
$113$	−11.7136	−1.10192	−0.550960	−	0.834532i	$-0.685738\pi$
$113$	−11.7136	−1.10192	−0.550960	+	0.834532i	$0.685738\pi$
$114$	13.2130	1.23751
$115$	−3.07928	−0.287144
$116$	−9.56133	−0.887748
$117$	−1.00000	−0.0924500
$118$	10.8080	0.994959
$119$	16.1325	1.47887
$120$	−0.801474	−0.0731642
$121$	−9.58658	−0.871507
$122$	20.2827	1.83631
$123$	−8.93189	−0.805361
$124$	1.57618	0.141545
$125$	−1.00000	−0.0894427
$126$	−8.81900	−0.785659
$127$	4.71176	0.418101	0.209051	−	0.977905i	$-0.432963\pi$
$127$	4.71176	0.418101	0.209051	+	0.977905i	$0.432963\pi$
$128$	6.26783	0.554003
$129$	−4.12119	−0.362851
$130$	1.89108	0.165859
$131$	−0.477381	−0.0417090	−0.0208545	−	0.999783i	$-0.506639\pi$
$131$	−0.477381	−0.0417090	−0.0208545	+	0.999783i	$0.506639\pi$
$132$	1.87388	0.163101
$133$	32.5838	2.82537
$134$	−14.1873	−1.22560
$135$	1.00000	0.0860663
$136$	2.77257	0.237746
$137$	−8.24034	−0.704020	−0.352010	−	0.935996i	$-0.614502\pi$
$137$	−8.24034	−0.704020	−0.352010	+	0.935996i	$0.614502\pi$
$138$	−5.82317	−0.495701
$139$	−19.2587	−1.63351	−0.816753	−	0.576988i	$-0.804228\pi$
$139$	−19.2587	−1.63351	−0.816753	+	0.576988i	$0.804228\pi$
$140$	7.35048	0.621229
$141$	−1.75423	−0.147733
$142$	−12.9172	−1.08399
$143$	1.18887	0.0994187
$144$	−4.66801	−0.389001
$145$	6.06614	0.503765
$146$	−23.2786	−1.92655
$147$	−14.7480	−1.21639
$148$	13.8554	1.13891
$149$	−4.50660	−0.369195	−0.184597	−	0.982814i	$-0.559098\pi$
$149$	−4.50660	−0.369195	−0.184597	+	0.982814i	$0.559098\pi$
$150$	−1.89108	−0.154406
$151$	10.8913	0.886321	0.443160	−	0.896442i	$-0.353857\pi$
$151$	10.8913	0.886321	0.443160	+	0.896442i	$0.353857\pi$
$152$	5.59992	0.454214
$153$	−3.45934	−0.279671
$154$	10.4847	0.844880
$155$	−1.00000	−0.0803219
$156$	1.57618	0.126196
$157$	−7.28462	−0.581376	−0.290688	−	0.956818i	$-0.593884\pi$
$157$	−7.28462	−0.581376	−0.290688	+	0.956818i	$0.593884\pi$
$158$	18.3953	1.46345
$159$	11.1664	0.885552
$160$	7.22464	0.571158
$161$	−14.3601	−1.13174
$162$	1.89108	0.148577
$163$	−9.84982	−0.771497	−0.385749	−	0.922604i	$-0.626057\pi$
$163$	−9.84982	−0.771497	−0.385749	+	0.922604i	$0.626057\pi$
$164$	14.0783	1.09933
$165$	−1.18887	−0.0925538
$166$	29.4542	2.28609
$167$	−5.19323	−0.401864	−0.200932	−	0.979605i	$-0.564397\pi$
$167$	−5.19323	−0.401864	−0.200932	+	0.979605i	$0.564397\pi$
$168$	−3.73765	−0.288366
$169$	1.00000	0.0769231
$170$	6.54189	0.501740
$171$	−6.98703	−0.534311
$172$	6.49575	0.495296
$173$	20.6576	1.57057	0.785283	−	0.619137i	$-0.212517\pi$
$173$	20.6576	1.57057	0.785283	+	0.619137i	$0.212517\pi$
$174$	11.4715	0.869656
$175$	−4.66347	−0.352525
$176$	5.54968	0.418323
$177$	−5.71527	−0.429586
$178$	12.7066	0.952399
$179$	10.4282	0.779438	0.389719	−	0.920934i	$-0.372572\pi$
$179$	10.4282	0.779438	0.389719	+	0.920934i	$0.372572\pi$
$180$	−1.57618	−0.117482
$181$	5.76884	0.428795	0.214397	−	0.976747i	$-0.431221\pi$
$181$	5.76884	0.428795	0.214397	+	0.976747i	$0.431221\pi$
$182$	8.81900	0.653707
$183$	−10.7254	−0.792848
$184$	−2.46796	−0.181941
$185$	−8.79048	−0.646289
$186$	−1.89108	−0.138661
$187$	4.11273	0.300752
$188$	2.76498	0.201657
$189$	4.66347	0.339218
$190$	13.2130	0.958573
$191$	12.0951	0.875173	0.437587	−	0.899176i	$-0.355833\pi$
$191$	12.0951	0.875173	0.437587	+	0.899176i	$0.355833\pi$
$192$	4.32634	0.312226
$193$	8.41269	0.605559	0.302779	−	0.953061i	$-0.402085\pi$
$193$	8.41269	0.605559	0.302779	+	0.953061i	$0.402085\pi$
$194$	−26.3897	−1.89467
$195$	−1.00000	−0.0716115
$196$	23.2455	1.66039
$197$	−7.90206	−0.562999	−0.281499	−	0.959561i	$-0.590832\pi$
$197$	−7.90206	−0.562999	−0.281499	+	0.959561i	$0.590832\pi$
$198$	−2.24826	−0.159777
$199$	18.7700	1.33057	0.665286	−	0.746589i	$-0.268310\pi$
$199$	18.7700	1.33057	0.665286	+	0.746589i	$0.268310\pi$
$200$	−0.801474	−0.0566728
$201$	7.50222	0.529166
$202$	−21.4280	−1.50767
$203$	28.2893	1.98552
$204$	5.45255	0.381755
$205$	−8.93189	−0.623830
$206$	12.2519	0.853634
$207$	3.07928	0.214025
$208$	4.66801	0.323669
$209$	8.30670	0.574586
$210$	−8.81900	−0.608568
$211$	22.3870	1.54118	0.770591	−	0.637330i	$-0.219961\pi$
$211$	22.3870	1.54118	0.770591	+	0.637330i	$0.219961\pi$
$212$	−17.6002	−1.20879
$213$	6.83059	0.468025
$214$	33.0086	2.25642
$215$	−4.12119	−0.281063
$216$	0.801474	0.0545334
$217$	−4.66347	−0.316577
$218$	−34.4802	−2.33529
$219$	12.3097	0.831810
$220$	1.87388	0.126337
$221$	3.45934	0.232701
$222$	−16.6235	−1.11570
$223$	−6.66898	−0.446588	−0.223294	−	0.974751i	$-0.571681\pi$
$223$	−6.66898	−0.446588	−0.223294	+	0.974751i	$0.571681\pi$
$224$	33.6919	2.25113
$225$	1.00000	0.0666667
$226$	−22.1513	−1.47348
$227$	13.4391	0.891985	0.445992	−	0.895037i	$-0.352851\pi$
$227$	13.4391	0.891985	0.445992	+	0.895037i	$0.352851\pi$
$228$	11.0128	0.729342
$229$	10.1594	0.671351	0.335675	−	0.941978i	$-0.391036\pi$
$229$	10.1594	0.671351	0.335675	+	0.941978i	$0.391036\pi$
$230$	−5.82317	−0.383968
$231$	−5.54428	−0.364787
$232$	4.86185	0.319196
$233$	−12.1373	−0.795141	−0.397570	−	0.917572i	$-0.630147\pi$
$233$	−12.1373	−0.795141	−0.397570	+	0.917572i	$0.630147\pi$
$234$	−1.89108	−0.123624
$235$	−1.75423	−0.114433
$236$	9.00830	0.586390
$237$	−9.72740	−0.631862
$238$	30.5079	1.97754
$239$	−8.25563	−0.534012	−0.267006	−	0.963695i	$-0.586034\pi$
$239$	−8.25563	−0.534012	−0.267006	+	0.963695i	$0.586034\pi$
$240$	−4.66801	−0.301319
$241$	−3.33009	−0.214510	−0.107255	−	0.994232i	$-0.534206\pi$
$241$	−3.33009	−0.214510	−0.107255	+	0.994232i	$0.534206\pi$
$242$	−18.1290	−1.16537
$243$	−1.00000	−0.0641500
$244$	16.9053	1.08225
$245$	−14.7480	−0.942213
$246$	−16.8909	−1.07693
$247$	6.98703	0.444574
$248$	−0.801474	−0.0508937
$249$	−15.5753	−0.987046
$250$	−1.89108	−0.119602
$251$	14.5127	0.916034	0.458017	−	0.888943i	$-0.348560\pi$
$251$	14.5127	0.916034	0.458017	+	0.888943i	$0.348560\pi$
$252$	−7.35048	−0.463037
$253$	−3.66088	−0.230158
$254$	8.91031	0.559083
$255$	−3.45934	−0.216632
$256$	20.5056	1.28160
$257$	−5.65093	−0.352495	−0.176248	−	0.984346i	$-0.556396\pi$
$257$	−5.65093	−0.352495	−0.176248	+	0.984346i	$0.556396\pi$
$258$	−7.79350	−0.485202
$259$	−40.9941	−2.54725
$260$	1.57618	0.0977506
$261$	−6.06614	−0.375484
$262$	−0.902766	−0.0557730
$263$	−2.97379	−0.183372	−0.0916858	−	0.995788i	$-0.529226\pi$
$263$	−2.97379	−0.183372	−0.0916858	+	0.995788i	$0.529226\pi$
$264$	−0.952852	−0.0586440
$265$	11.1664	0.685945
$266$	61.6186	3.77808
$267$	−6.71923	−0.411210
$268$	−11.8249	−0.722319
$269$	17.7190	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$269$	17.7190	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$270$	1.89108	0.115087
$271$	25.0020	1.51876	0.759382	−	0.650645i	$-0.225502\pi$
$271$	25.0020	1.51876	0.759382	+	0.650645i	$0.225502\pi$
$272$	16.1483	0.979132
$273$	−4.66347	−0.282246
$274$	−15.5831	−0.941412
$275$	−1.18887	−0.0716918
$276$	−4.85351	−0.292147
$277$	−20.1077	−1.20815	−0.604077	−	0.796926i	$-0.706458\pi$
$277$	−20.1077	−1.20815	−0.604077	+	0.796926i	$0.706458\pi$
$278$	−36.4198	−2.18432
$279$	1.00000	0.0598684
$280$	−3.73765	−0.223367
$281$	−22.7432	−1.35675	−0.678373	−	0.734718i	$-0.737315\pi$
$281$	−22.7432	−1.35675	−0.678373	+	0.734718i	$0.737315\pi$
$282$	−3.31738	−0.197547
$283$	−19.6471	−1.16790	−0.583950	−	0.811790i	$-0.698493\pi$
$283$	−19.6471	−1.16790	−0.583950	+	0.811790i	$0.698493\pi$
$284$	−10.7663	−0.638860
$285$	−6.98703	−0.413876
$286$	2.24826	0.132942
$287$	−41.6536	−2.45873
$288$	−7.22464	−0.425716
$289$	−5.03294	−0.296055
$290$	11.4715	0.673632
$291$	13.9548	0.818047
$292$	−19.4023	−1.13543
$293$	9.59812	0.560728	0.280364	−	0.959894i	$-0.409545\pi$
$293$	9.59812	0.560728	0.280364	+	0.959894i	$0.409545\pi$
$294$	−27.8896	−1.62655
$295$	−5.71527	−0.332756
$296$	−7.04534	−0.409502
$297$	1.18887	0.0689855
$298$	−8.52233	−0.493685
$299$	−3.07928	−0.178079
$300$	−1.57618	−0.0910009
$301$	−19.2191	−1.10777
$302$	20.5963	1.18518
$303$	11.3311	0.650954
$304$	32.6155	1.87063
$305$	−10.7254	−0.614137
$306$	−6.54189	−0.373975
$307$	−11.6527	−0.665052	−0.332526	−	0.943094i	$-0.607901\pi$
$307$	−11.6527	−0.665052	−0.332526	+	0.943094i	$0.607901\pi$
$308$	8.73880	0.497939
$309$	−6.47881	−0.368567
$310$	−1.89108	−0.107406
$311$	31.8466	1.80586	0.902928	−	0.429792i	$-0.141413\pi$
$311$	31.8466	1.80586	0.902928	+	0.429792i	$0.141413\pi$
$312$	−0.801474	−0.0453745
$313$	10.4465	0.590469	0.295234	−	0.955425i	$-0.404602\pi$
$313$	10.4465	0.590469	0.295234	+	0.955425i	$0.404602\pi$
$314$	−13.7758	−0.777414
$315$	4.66347	0.262757
$316$	15.3321	0.862501
$317$	16.6200	0.933473	0.466736	−	0.884397i	$-0.345430\pi$
$317$	16.6200	0.933473	0.466736	+	0.884397i	$0.345430\pi$
$318$	21.1165	1.18416
$319$	7.21188	0.403788
$320$	4.32634	0.241850
$321$	−17.4549	−0.974239
$322$	−27.1562	−1.51335
$323$	24.1705	1.34488
$324$	1.57618	0.0875657
$325$	−1.00000	−0.0554700
$326$	−18.6268	−1.03164
$327$	18.2331	1.00829
$328$	−7.15868	−0.395272
$329$	−8.18079	−0.451022
$330$	−2.24826	−0.123762
$331$	28.9163	1.58938	0.794691	−	0.607014i	$-0.207633\pi$
$331$	28.9163	1.58938	0.794691	+	0.607014i	$0.207633\pi$
$332$	24.5495	1.34733
$333$	8.79048	0.481715
$334$	−9.82080	−0.537370
$335$	7.50222	0.409890
$336$	−21.7692	−1.18760
$337$	−15.0921	−0.822120	−0.411060	−	0.911608i	$-0.634841\pi$
$337$	−15.0921	−0.822120	−0.411060	+	0.911608i	$0.634841\pi$
$338$	1.89108	0.102861
$339$	11.7136	0.636193
$340$	5.45255	0.295706
$341$	−1.18887	−0.0643812
$342$	−13.2130	−0.714478
$343$	−36.1324	−1.95097
$344$	−3.30303	−0.178087
$345$	3.07928	0.165783
$346$	39.0651	2.10015
$347$	13.8081	0.741258	0.370629	−	0.928781i	$-0.379142\pi$
$347$	13.8081	0.741258	0.370629	+	0.928781i	$0.379142\pi$
$348$	9.56133	0.512541
$349$	19.1632	1.02578	0.512892	−	0.858453i	$-0.328574\pi$
$349$	19.1632	1.02578	0.512892	+	0.858453i	$0.328574\pi$
$350$	−8.81900	−0.471395
$351$	1.00000	0.0533761
$352$	8.58919	0.457805
$353$	10.6446	0.566555	0.283277	−	0.959038i	$-0.408578\pi$
$353$	10.6446	0.566555	0.283277	+	0.959038i	$0.408578\pi$
$354$	−10.8080	−0.574440
$355$	6.83059	0.362530
$356$	10.5907	0.561307
$357$	−16.1325	−0.853825
$358$	19.7205	1.04226
$359$	−17.4894	−0.923054	−0.461527	−	0.887126i	$-0.652698\pi$
$359$	−17.4894	−0.923054	−0.461527	+	0.887126i	$0.652698\pi$
$360$	0.801474	0.0422414
$361$	29.8185	1.56940
$362$	10.9093	0.573382
$363$	9.58658	0.503165
$364$	7.35048	0.385270
$365$	12.3097	0.644318
$366$	−20.2827	−1.06019
$367$	−4.54521	−0.237258	−0.118629	−	0.992939i	$-0.537850\pi$
$367$	−4.54521	−0.237258	−0.118629	+	0.992939i	$0.537850\pi$
$368$	−14.3741	−0.749303
$369$	8.93189	0.464975
$370$	−16.6235	−0.864214
$371$	52.0741	2.70355
$372$	−1.57618	−0.0817212
$373$	13.4549	0.696669	0.348334	−	0.937370i	$-0.386747\pi$
$373$	13.4549	0.696669	0.348334	+	0.937370i	$0.386747\pi$
$374$	7.77749	0.402164
$375$	1.00000	0.0516398
$376$	−1.40597	−0.0725073
$377$	6.06614	0.312422
$378$	8.81900	0.453600
$379$	36.8708	1.89392	0.946962	−	0.321344i	$-0.104135\pi$
$379$	36.8708	1.89392	0.946962	+	0.321344i	$0.104135\pi$
$380$	11.0128	0.564946
$381$	−4.71176	−0.241391
$382$	22.8729	1.17028
$383$	16.5830	0.847354	0.423677	−	0.905813i	$-0.360739\pi$
$383$	16.5830	0.847354	0.423677	+	0.905813i	$0.360739\pi$
$384$	−6.26783	−0.319854
$385$	−5.54428	−0.282563
$386$	15.9091	0.809751
$387$	4.12119	0.209492
$388$	−21.9954	−1.11665
$389$	−14.7724	−0.748989	−0.374494	−	0.927229i	$-0.622184\pi$
$389$	−14.7724	−0.748989	−0.374494	+	0.927229i	$0.622184\pi$
$390$	−1.89108	−0.0957585
$391$	−10.6523	−0.538710
$392$	−11.8201	−0.597006
$393$	0.477381	0.0240807
$394$	−14.9434	−0.752839
$395$	−9.72740	−0.489439
$396$	−1.87388	−0.0941661
$397$	−24.0479	−1.20693	−0.603464	−	0.797390i	$-0.706213\pi$
$397$	−24.0479	−1.20693	−0.603464	+	0.797390i	$0.706213\pi$
$398$	35.4956	1.77923
$399$	−32.5838	−1.63123
$400$	−4.66801	−0.233401
$401$	−16.4133	−0.819639	−0.409820	−	0.912167i	$-0.634408\pi$
$401$	−16.4133	−0.819639	−0.409820	+	0.912167i	$0.634408\pi$
$402$	14.1873	0.707598
$403$	−1.00000	−0.0498135
$404$	−17.8599	−0.888561
$405$	−1.00000	−0.0496904
$406$	53.4972	2.65502
$407$	−10.4508	−0.518026
$408$	−2.77257	−0.137263
$409$	7.86536	0.388917	0.194459	−	0.980911i	$-0.437705\pi$
$409$	7.86536	0.388917	0.194459	+	0.980911i	$0.437705\pi$
$410$	−16.8909	−0.834183
$411$	8.24034	0.406466
$412$	10.2118	0.503098
$413$	−26.6530	−1.31151
$414$	5.82317	0.286193
$415$	−15.5753	−0.764562
$416$	7.22464	0.354217
$417$	19.2587	0.943105
$418$	15.7086	0.768334
$419$	−14.6308	−0.714764	−0.357382	−	0.933958i	$-0.616330\pi$
$419$	−14.6308	−0.714764	−0.357382	+	0.933958i	$0.616330\pi$
$420$	−7.35048	−0.358667
$421$	14.6040	0.711754	0.355877	−	0.934533i	$-0.384182\pi$
$421$	14.6040	0.711754	0.355877	+	0.934533i	$0.384182\pi$
$422$	42.3355	2.06086
$423$	1.75423	0.0852935
$424$	8.94956	0.434629
$425$	−3.45934	−0.167803
$426$	12.9172	0.625840
$427$	−50.0178	−2.42053
$428$	27.5121	1.32985
$429$	−1.18887	−0.0573994
$430$	−7.79350	−0.375836
$431$	−36.0277	−1.73539	−0.867696	−	0.497095i	$-0.834400\pi$
$431$	−36.0277	−1.73539	−0.867696	+	0.497095i	$0.834400\pi$
$432$	4.66801	0.224590
$433$	14.2782	0.686168	0.343084	−	0.939305i	$-0.388528\pi$
$433$	14.2782	0.686168	0.343084	+	0.939305i	$0.388528\pi$
$434$	−8.81900	−0.423325
$435$	−6.06614	−0.290849
$436$	−28.7387	−1.37633
$437$	−21.5150	−1.02920
$438$	23.2786	1.11229
$439$	−36.7407	−1.75354	−0.876770	−	0.480910i	$-0.840306\pi$
$439$	−36.7407	−1.75354	−0.876770	+	0.480910i	$0.840306\pi$
$440$	−0.952852	−0.0454254
$441$	14.7480	0.702284
$442$	6.54189	0.311166
$443$	−8.29270	−0.393998	−0.196999	−	0.980404i	$-0.563120\pi$
$443$	−8.29270	−0.393998	−0.196999	+	0.980404i	$0.563120\pi$
$444$	−13.8554	−0.657548
$445$	−6.71923	−0.318522
$446$	−12.6116	−0.597175
$447$	4.50660	0.213155
$448$	20.1757	0.953215
$449$	−13.6226	−0.642888	−0.321444	−	0.946929i	$-0.604168\pi$
$449$	−13.6226	−0.642888	−0.321444	+	0.946929i	$0.604168\pi$
$450$	1.89108	0.0891463
$451$	−10.6189	−0.500024
$452$	−18.4627	−0.868413
$453$	−10.8913	−0.511717
$454$	25.4144	1.19276
$455$	−4.66347	−0.218627
$456$	−5.59992	−0.262240
$457$	−16.1134	−0.753754	−0.376877	−	0.926263i	$-0.623002\pi$
$457$	−16.1134	−0.753754	−0.376877	+	0.926263i	$0.623002\pi$
$458$	19.2122	0.897727
$459$	3.45934	0.161468
$460$	−4.85351	−0.226296
$461$	−12.9412	−0.602730	−0.301365	−	0.953509i	$-0.597442\pi$
$461$	−12.9412	−0.602730	−0.301365	+	0.953509i	$0.597442\pi$
$462$	−10.4847	−0.487791
$463$	−21.9668	−1.02088	−0.510441	−	0.859913i	$-0.670518\pi$
$463$	−21.9668	−1.02088	−0.510441	+	0.859913i	$0.670518\pi$
$464$	28.3168	1.31458
$465$	1.00000	0.0463739
$466$	−22.9526	−1.06326
$467$	−12.9513	−0.599316	−0.299658	−	0.954047i	$-0.596873\pi$
$467$	−12.9513	−0.599316	−0.299658	+	0.954047i	$0.596873\pi$
$468$	−1.57618	−0.0728590
$469$	34.9864	1.61552
$470$	−3.31738	−0.153020
$471$	7.28462	0.335658
$472$	−4.58064	−0.210841
$473$	−4.89958	−0.225283
$474$	−18.3953	−0.844923
$475$	−6.98703	−0.320587
$476$	25.4278	1.16548
$477$	−11.1664	−0.511273
$478$	−15.6121	−0.714079
$479$	−5.56048	−0.254065	−0.127033	−	0.991899i	$-0.540545\pi$
$479$	−5.56048	−0.254065	−0.127033	+	0.991899i	$0.540545\pi$
$480$	−7.22464	−0.329758
$481$	−8.79048	−0.400811
$482$	−6.29747	−0.286842
$483$	14.3601	0.653409
$484$	−15.1102	−0.686827
$485$	13.9548	0.633656
$486$	−1.89108	−0.0857811
$487$	−21.6231	−0.979838	−0.489919	−	0.871768i	$-0.662974\pi$
$487$	−21.6231	−0.979838	−0.489919	+	0.871768i	$0.662974\pi$
$488$	−8.59617	−0.389130
$489$	9.84982	0.445424
$490$	−27.8896	−1.25992
$491$	24.1889	1.09163	0.545815	−	0.837906i	$-0.316220\pi$
$491$	24.1889	1.09163	0.545815	+	0.837906i	$0.316220\pi$
$492$	−14.0783	−0.634698
$493$	20.9849	0.945110
$494$	13.2130	0.594482
$495$	1.18887	0.0534359
$496$	−4.66801	−0.209600
$497$	31.8543	1.42886
$498$	−29.4542	−1.31987
$499$	−3.38300	−0.151444	−0.0757219	−	0.997129i	$-0.524126\pi$
$499$	−3.38300	−0.151444	−0.0757219	+	0.997129i	$0.524126\pi$
$500$	−1.57618	−0.0704890
$501$	5.19323	0.232016
$502$	27.4447	1.22492
$503$	−13.5390	−0.603676	−0.301838	−	0.953359i	$-0.597600\pi$
$503$	−13.5390	−0.603676	−0.301838	+	0.953359i	$0.597600\pi$
$504$	3.73765	0.166488
$505$	11.3311	0.504227
$506$	−6.92301	−0.307766
$507$	−1.00000	−0.0444116
$508$	7.42659	0.329502
$509$	25.7180	1.13993	0.569966	−	0.821668i	$-0.306956\pi$
$509$	25.7180	1.13993	0.569966	+	0.821668i	$0.306956\pi$
$510$	−6.54189	−0.289680
$511$	57.4058	2.53948
$512$	26.2421	1.15975
$513$	6.98703	0.308485
$514$	−10.6864	−0.471355
$515$	−6.47881	−0.285491
$516$	−6.49575	−0.285959
$517$	−2.08556	−0.0917227
$518$	−77.5232	−3.40617
$519$	−20.6576	−0.906767
$520$	−0.801474	−0.0351470
$521$	13.2451	0.580279	0.290139	−	0.956984i	$-0.406298\pi$
$521$	13.2451	0.580279	0.290139	+	0.956984i	$0.406298\pi$
$522$	−11.4715	−0.502096
$523$	18.7371	0.819314	0.409657	−	0.912240i	$-0.365648\pi$
$523$	18.7371	0.819314	0.409657	+	0.912240i	$0.365648\pi$
$524$	−0.752439	−0.0328705
$525$	4.66347	0.203531
$526$	−5.62367	−0.245204
$527$	−3.45934	−0.150691
$528$	−5.54968	−0.241519
$529$	−13.5180	−0.587740
$530$	21.1165	0.917243
$531$	5.71527	0.248021
$532$	51.3580	2.22665
$533$	−8.93189	−0.386883
$534$	−12.7066	−0.549868
$535$	−17.4549	−0.754642
$536$	6.01284	0.259715
$537$	−10.4282	−0.450009
$538$	33.5080	1.44463
$539$	−17.5335	−0.755221
$540$	1.57618	0.0678281
$541$	−18.9190	−0.813392	−0.406696	−	0.913564i	$-0.633319\pi$
$541$	−18.9190	−0.813392	−0.406696	+	0.913564i	$0.633319\pi$
$542$	47.2808	2.03088
$543$	−5.76884	−0.247565
$544$	24.9925	1.07154
$545$	18.2331	0.781020
$546$	−8.81900	−0.377418
$547$	−9.53044	−0.407492	−0.203746	−	0.979024i	$-0.565312\pi$
$547$	−9.53044	−0.407492	−0.203746	+	0.979024i	$0.565312\pi$
$548$	−12.9883	−0.554832
$549$	10.7254	0.457751
$550$	−2.24826	−0.0958660
$551$	42.3843	1.80563
$552$	2.46796	0.105044
$553$	−45.3634	−1.92905
$554$	−38.0252	−1.61554
$555$	8.79048	0.373135
$556$	−30.3553	−1.28735
$557$	−42.7272	−1.81041	−0.905205	−	0.424976i	$-0.860283\pi$
$557$	−42.7272	−1.81041	−0.905205	+	0.424976i	$0.860283\pi$
$558$	1.89108	0.0800558
$559$	−4.12119	−0.174308
$560$	−21.7692	−0.919915
$561$	−4.11273	−0.173639
$562$	−43.0092	−1.81423
$563$	−4.16912	−0.175708	−0.0878538	−	0.996133i	$-0.528001\pi$
$563$	−4.16912	−0.175708	−0.0878538	+	0.996133i	$0.528001\pi$
$564$	−2.76498	−0.116427
$565$	11.7136	0.492793
$566$	−37.1542	−1.56171
$567$	−4.66347	−0.195847
$568$	5.47454	0.229707
$569$	−6.39053	−0.267905	−0.133953	−	0.990988i	$-0.542767\pi$
$569$	−6.39053	−0.267905	−0.133953	+	0.990988i	$0.542767\pi$
$570$	−13.2130	−0.553433
$571$	−31.2947	−1.30964	−0.654821	−	0.755784i	$-0.727256\pi$
$571$	−31.2947	−1.30964	−0.654821	+	0.755784i	$0.727256\pi$
$572$	1.87388	0.0783510
$573$	−12.0951	−0.505281
$574$	−78.7703	−3.28781
$575$	3.07928	0.128415
$576$	−4.32634	−0.180264
$577$	39.2727	1.63495	0.817473	−	0.575968i	$-0.195375\pi$
$577$	39.2727	1.63495	0.817473	+	0.575968i	$0.195375\pi$
$578$	−9.51769	−0.395884
$579$	−8.41269	−0.349620
$580$	9.56133	0.397013
$581$	−72.6351	−3.01341
$582$	26.3897	1.09389
$583$	13.2754	0.549812
$584$	9.86588	0.408253
$585$	1.00000	0.0413449
$586$	18.1508	0.749803
$587$	−17.6829	−0.729851	−0.364925	−	0.931037i	$-0.618906\pi$
$587$	−17.6829	−0.729851	−0.364925	+	0.931037i	$0.618906\pi$
$588$	−23.2455	−0.958627
$589$	−6.98703	−0.287895
$590$	−10.8080	−0.444959
$591$	7.90206	0.325047
$592$	−41.0341	−1.68649
$593$	−21.4693	−0.881638	−0.440819	−	0.897596i	$-0.645312\pi$
$593$	−21.4693	−0.881638	−0.440819	+	0.897596i	$0.645312\pi$
$594$	2.24826	0.0922471
$595$	−16.1325	−0.661370
$596$	−7.10321	−0.290959
$597$	−18.7700	−0.768206
$598$	−5.82317	−0.238127
$599$	−8.75941	−0.357900	−0.178950	−	0.983858i	$-0.557270\pi$
$599$	−8.75941	−0.357900	−0.178950	+	0.983858i	$0.557270\pi$
$600$	0.801474	0.0327200
$601$	28.7111	1.17115	0.585574	−	0.810619i	$-0.300869\pi$
$601$	28.7111	1.17115	0.585574	+	0.810619i	$0.300869\pi$
$602$	−36.3448	−1.48130
$603$	−7.50222	−0.305514
$604$	17.1667	0.698501
$605$	9.58658	0.389750
$606$	21.4280	0.870453
$607$	−22.2060	−0.901315	−0.450657	−	0.892697i	$-0.648810\pi$
$607$	−22.2060	−0.901315	−0.450657	+	0.892697i	$0.648810\pi$
$608$	50.4787	2.04718
$609$	−28.2893	−1.14634
$610$	−20.2827	−0.821221
$611$	−1.75423	−0.0709685
$612$	−5.45255	−0.220406
$613$	−13.2122	−0.533634	−0.266817	−	0.963747i	$-0.585972\pi$
$613$	−13.2122	−0.533634	−0.266817	+	0.963747i	$0.585972\pi$
$614$	−22.0361	−0.889304
$615$	8.93189	0.360168
$616$	−4.44360	−0.179038
$617$	16.1419	0.649847	0.324923	−	0.945740i	$-0.394661\pi$
$617$	16.1419	0.649847	0.324923	+	0.945740i	$0.394661\pi$
$618$	−12.2519	−0.492846
$619$	35.0971	1.41067	0.705336	−	0.708874i	$-0.250796\pi$
$619$	35.0971	1.41067	0.705336	+	0.708874i	$0.250796\pi$
$620$	−1.57618	−0.0633010
$621$	−3.07928	−0.123567
$622$	60.2245	2.41478
$623$	−31.3349	−1.25541
$624$	−4.66801	−0.186870
$625$	1.00000	0.0400000
$626$	19.7551	0.789572
$627$	−8.30670	−0.331738
$628$	−11.4819	−0.458177
$629$	−30.4093	−1.21250
$630$	8.81900	0.351357
$631$	41.8618	1.66649	0.833246	−	0.552902i	$-0.186480\pi$
$631$	41.8618	1.66649	0.833246	+	0.552902i	$0.186480\pi$
$632$	−7.79626	−0.310118
$633$	−22.3870	−0.889802
$634$	31.4298	1.24823
$635$	−4.71176	−0.186980
$636$	17.6002	0.697895
$637$	−14.7480	−0.584336
$638$	13.6382	0.539943
$639$	−6.83059	−0.270214
$640$	−6.26783	−0.247758
$641$	9.72633	0.384167	0.192083	−	0.981379i	$-0.438476\pi$
$641$	9.72633	0.384167	0.192083	+	0.981379i	$0.438476\pi$
$642$	−33.0086	−1.30275
$643$	13.3923	0.528141	0.264071	−	0.964503i	$-0.414935\pi$
$643$	13.3923	0.528141	0.264071	+	0.964503i	$0.414935\pi$
$644$	−22.6342	−0.891912
$645$	4.12119	0.162272
$646$	45.7084	1.79837
$647$	−38.8397	−1.52695	−0.763473	−	0.645840i	$-0.776507\pi$
$647$	−38.8397	−1.52695	−0.763473	+	0.645840i	$0.776507\pi$
$648$	−0.801474	−0.0314849
$649$	−6.79473	−0.266717
$650$	−1.89108	−0.0741742
$651$	4.66347	0.182776
$652$	−15.5251	−0.608010
$653$	4.46218	0.174618	0.0873092	−	0.996181i	$-0.472173\pi$
$653$	4.46218	0.174618	0.0873092	+	0.996181i	$0.472173\pi$
$654$	34.4802	1.34828
$655$	0.477381	0.0186528
$656$	−41.6942	−1.62788
$657$	−12.3097	−0.480246
$658$	−15.4705	−0.603104
$659$	31.4152	1.22376	0.611880	−	0.790950i	$-0.290413\pi$
$659$	31.4152	1.22376	0.611880	+	0.790950i	$0.290413\pi$
$660$	−1.87388	−0.0729408
$661$	18.1073	0.704292	0.352146	−	0.935945i	$-0.385452\pi$
$661$	18.1073	0.704292	0.352146	+	0.935945i	$0.385452\pi$
$662$	54.6830	2.12531
$663$	−3.45934	−0.134350
$664$	−12.4832	−0.484443
$665$	−32.5838	−1.26355
$666$	16.6235	0.644147
$667$	−18.6793	−0.723267
$668$	−8.18547	−0.316705
$669$	6.66898	0.257838
$670$	14.1873	0.548103
$671$	−12.7512	−0.492255
$672$	−33.6919	−1.29969
$673$	42.0678	1.62159	0.810797	−	0.585328i	$-0.199034\pi$
$673$	42.0678	1.62159	0.810797	+	0.585328i	$0.199034\pi$
$674$	−28.5404	−1.09934
$675$	−1.00000	−0.0384900
$676$	1.57618	0.0606224
$677$	33.5705	1.29022	0.645109	−	0.764090i	$-0.276812\pi$
$677$	33.5705	1.29022	0.645109	+	0.764090i	$0.276812\pi$
$678$	22.1513	0.850715
$679$	65.0780	2.49746
$680$	−2.77257	−0.106323
$681$	−13.4391	−0.514988
$682$	−2.24826	−0.0860902
$683$	3.08038	0.117868	0.0589338	−	0.998262i	$-0.481230\pi$
$683$	3.08038	0.117868	0.0589338	+	0.998262i	$0.481230\pi$
$684$	−11.0128	−0.421086
$685$	8.24034	0.314847
$686$	−68.3293	−2.60882
$687$	−10.1594	−0.387604
$688$	−19.2378	−0.733434
$689$	11.1664	0.425405
$690$	5.82317	0.221684
$691$	−17.3272	−0.659156	−0.329578	−	0.944128i	$-0.606907\pi$
$691$	−17.3272	−0.659156	−0.329578	+	0.944128i	$0.606907\pi$
$692$	32.5601	1.23775
$693$	5.54428	0.210610
$694$	26.1122	0.991206
$695$	19.2587	0.730526
$696$	−4.86185	−0.184288
$697$	−30.8985	−1.17036
$698$	36.2392	1.37167
$699$	12.1373	0.459075
$700$	−7.35048	−0.277822
$701$	−41.4143	−1.56420	−0.782098	−	0.623156i	$-0.785850\pi$
$701$	−41.4143	−1.56420	−0.782098	+	0.623156i	$0.785850\pi$
$702$	1.89108	0.0713742
$703$	−61.4193	−2.31647
$704$	5.14347	0.193852
$705$	1.75423	0.0660680
$706$	20.1298	0.757594
$707$	52.8422	1.98734
$708$	−9.00830	−0.338553
$709$	−22.0414	−0.827782	−0.413891	−	0.910326i	$-0.635831\pi$
$709$	−22.0414	−0.827782	−0.413891	+	0.910326i	$0.635831\pi$
$710$	12.9172	0.484774
$711$	9.72740	0.364806
$712$	−5.38529	−0.201822
$713$	3.07928	0.115320
$714$	−30.5079	−1.14173
$715$	−1.18887	−0.0444614
$716$	16.4367	0.614268
$717$	8.25563	0.308312
$718$	−33.0738	−1.23430
$719$	−38.5634	−1.43817	−0.719085	−	0.694922i	$-0.755439\pi$
$719$	−38.5634	−1.43817	−0.719085	+	0.694922i	$0.755439\pi$
$720$	4.66801	0.173967
$721$	−30.2138	−1.12522
$722$	56.3892	2.09859
$723$	3.33009	0.123847
$724$	9.09274	0.337929
$725$	−6.06614	−0.225291
$726$	18.1290	0.672830
$727$	−39.1495	−1.45198	−0.725988	−	0.687707i	$-0.758617\pi$
$727$	−39.1495	−1.45198	−0.725988	+	0.687707i	$0.758617\pi$
$728$	−3.73765	−0.138527
$729$	1.00000	0.0370370
$730$	23.2786	0.861578
$731$	−14.2566	−0.527300
$732$	−16.9053	−0.624836
$733$	45.5462	1.68229	0.841143	−	0.540812i	$-0.181883\pi$
$733$	45.5462	1.68229	0.841143	+	0.540812i	$0.181883\pi$
$734$	−8.59536	−0.317260
$735$	14.7480	0.543987
$736$	−22.2467	−0.820024
$737$	8.91920	0.328543
$738$	16.8909	0.621763
$739$	14.1845	0.521786	0.260893	−	0.965368i	$-0.415983\pi$
$739$	14.1845	0.521786	0.260893	+	0.965368i	$0.415983\pi$
$740$	−13.8554	−0.509334
$741$	−6.98703	−0.256675
$742$	98.4763	3.61518
$743$	9.73056	0.356980	0.178490	−	0.983942i	$-0.442879\pi$
$743$	9.73056	0.356980	0.178490	+	0.983942i	$0.442879\pi$
$744$	0.801474	0.0293835
$745$	4.50660	0.165109
$746$	25.4443	0.931582
$747$	15.5753	0.569871
$748$	6.48240	0.237020
$749$	−81.4005	−2.97431
$750$	1.89108	0.0690525
$751$	37.8828	1.38236	0.691181	−	0.722681i	$-0.257091\pi$
$751$	37.8828	1.38236	0.691181	+	0.722681i	$0.257091\pi$
$752$	−8.18876	−0.298613
$753$	−14.5127	−0.528873
$754$	11.4715	0.417769
$755$	−10.8913	−0.396375
$756$	7.35048	0.267334
$757$	32.3448	1.17559	0.587796	−	0.809009i	$-0.299996\pi$
$757$	32.3448	1.17559	0.587796	+	0.809009i	$0.299996\pi$
$758$	69.7256	2.53255
$759$	3.66088	0.132882
$760$	−5.59992	−0.203130
$761$	29.7909	1.07992	0.539960	−	0.841691i	$-0.318439\pi$
$761$	29.7909	1.07992	0.539960	+	0.841691i	$0.318439\pi$
$762$	−8.91031	−0.322787
$763$	85.0295	3.07827
$764$	19.0641	0.689716
$765$	3.45934	0.125073
$766$	31.3598	1.13308
$767$	−5.71527	−0.206366
$768$	−20.5056	−0.739934
$769$	−8.13159	−0.293233	−0.146616	−	0.989193i	$-0.546838\pi$
$769$	−8.13159	−0.293233	−0.146616	+	0.989193i	$0.546838\pi$
$770$	−10.4847	−0.377842
$771$	5.65093	0.203513
$772$	13.2599	0.477235
$773$	1.63962	0.0589730	0.0294865	−	0.999565i	$-0.490613\pi$
$773$	1.63962	0.0589730	0.0294865	+	0.999565i	$0.490613\pi$
$774$	7.79350	0.280132
$775$	1.00000	0.0359211
$776$	11.1844	0.401498
$777$	40.9941	1.47066
$778$	−27.9357	−1.00154
$779$	−62.4073	−2.23597
$780$	−1.57618	−0.0564364
$781$	8.12072	0.290582
$782$	−20.1443	−0.720360
$783$	6.06614	0.216786
$784$	−68.8437	−2.45870
$785$	7.28462	0.259999
$786$	0.902766	0.0322006
$787$	−49.1268	−1.75118	−0.875590	−	0.483055i	$-0.839527\pi$
$787$	−49.1268	−1.75118	−0.875590	+	0.483055i	$0.839527\pi$
$788$	−12.4551	−0.443694
$789$	2.97379	0.105870
$790$	−18.3953	−0.654475
$791$	54.6259	1.94227
$792$	0.952852	0.0338581
$793$	−10.7254	−0.380872
$794$	−45.4764	−1.61390
$795$	−11.1664	−0.396031
$796$	29.5850	1.04861
$797$	−46.2961	−1.63989	−0.819947	−	0.572440i	$-0.805997\pi$
$797$	−46.2961	−1.63989	−0.819947	+	0.572440i	$0.805997\pi$
$798$	−61.6186	−2.18127
$799$	−6.06848	−0.214687
$800$	−7.22464	−0.255430
$801$	6.71923	0.237412
$802$	−31.0388	−1.09602
$803$	14.6347	0.516446
$804$	11.8249	0.417031
$805$	14.3601	0.506129
$806$	−1.89108	−0.0666104
$807$	−17.7190	−0.623738
$808$	9.08157	0.319489
$809$	17.4273	0.612711	0.306355	−	0.951917i	$-0.400890\pi$
$809$	17.4273	0.612711	0.306355	+	0.951917i	$0.400890\pi$
$810$	−1.89108	−0.0664458
$811$	36.8768	1.29492	0.647459	−	0.762100i	$-0.275832\pi$
$811$	36.8768	1.29492	0.647459	+	0.762100i	$0.275832\pi$
$812$	44.5890	1.56477
$813$	−25.0020	−0.876859
$814$	−19.7632	−0.692702
$815$	9.84982	0.345024
$816$	−16.1483	−0.565302
$817$	−28.7949	−1.00741
$818$	14.8740	0.520058
$819$	4.66347	0.162955
$820$	−14.0783	−0.491635
$821$	8.65857	0.302186	0.151093	−	0.988520i	$-0.451721\pi$
$821$	8.65857	0.302186	0.151093	+	0.988520i	$0.451721\pi$
$822$	15.5831	0.543524
$823$	−20.8141	−0.725534	−0.362767	−	0.931880i	$-0.618168\pi$
$823$	−20.8141	−0.725534	−0.362767	+	0.931880i	$0.618168\pi$
$824$	−5.19260	−0.180893
$825$	1.18887	0.0413913
$826$	−50.4029	−1.75374
$827$	43.5836	1.51555	0.757775	−	0.652516i	$-0.226286\pi$
$827$	43.5836	1.51555	0.757775	+	0.652516i	$0.226286\pi$
$828$	4.85351	0.168671
$829$	43.0701	1.49589	0.747943	−	0.663763i	$-0.231042\pi$
$829$	43.0701	1.49589	0.747943	+	0.663763i	$0.231042\pi$
$830$	−29.4542	−1.02237
$831$	20.1077	0.697528
$832$	4.32634	0.149989
$833$	−51.0183	−1.76768
$834$	36.4198	1.26112
$835$	5.19323	0.179719
$836$	13.0929	0.452826
$837$	−1.00000	−0.0345651
$838$	−27.6681	−0.955778
$839$	24.4791	0.845112	0.422556	−	0.906337i	$-0.361133\pi$
$839$	24.4791	0.845112	0.422556	+	0.906337i	$0.361133\pi$
$840$	3.73765	0.128961
$841$	7.79802	0.268897
$842$	27.6173	0.951754
$843$	22.7432	0.783317
$844$	35.2859	1.21459
$845$	−1.00000	−0.0344010
$846$	3.31738	0.114054
$847$	44.7067	1.53614
$848$	52.1248	1.78997
$849$	19.6471	0.674287
$850$	−6.54189	−0.224385
$851$	27.0684	0.927891
$852$	10.7663	0.368846
$853$	−1.52149	−0.0520948	−0.0260474	−	0.999661i	$-0.508292\pi$
$853$	−1.52149	−0.0520948	−0.0260474	+	0.999661i	$0.508292\pi$
$854$	−94.5877	−3.23672
$855$	6.98703	0.238951
$856$	−13.9897	−0.478157
$857$	46.3300	1.58260	0.791301	−	0.611427i	$-0.209404\pi$
$857$	46.3300	1.58260	0.791301	+	0.611427i	$0.209404\pi$
$858$	−2.24826	−0.0767542
$859$	13.9576	0.476227	0.238113	−	0.971237i	$-0.423471\pi$
$859$	13.9576	0.476227	0.238113	+	0.971237i	$0.423471\pi$
$860$	−6.49575	−0.221503
$861$	41.6536	1.41955
$862$	−68.1312	−2.32056
$863$	−40.8557	−1.39074	−0.695372	−	0.718650i	$-0.744760\pi$
$863$	−40.8557	−1.39074	−0.695372	+	0.718650i	$0.744760\pi$
$864$	7.22464	0.245787
$865$	−20.6576	−0.702379
$866$	27.0013	0.917541
$867$	5.03294	0.170928
$868$	−7.35048	−0.249491
$869$	−11.5647	−0.392304
$870$	−11.4715	−0.388922
$871$	7.50222	0.254203
$872$	14.6133	0.494870
$873$	−13.9548	−0.472300
$874$	−40.6866	−1.37625
$875$	4.66347	0.157654
$876$	19.4023	0.655542
$877$	27.2516	0.920223	0.460111	−	0.887861i	$-0.347810\pi$
$877$	27.2516	0.920223	0.460111	+	0.887861i	$0.347810\pi$
$878$	−69.4797	−2.34482
$879$	−9.59812	−0.323737
$880$	−5.54968	−0.187080
$881$	12.5621	0.423229	0.211614	−	0.977353i	$-0.432128\pi$
$881$	12.5621	0.423229	0.211614	+	0.977353i	$0.432128\pi$
$882$	27.8896	0.939091
$883$	58.1492	1.95688	0.978439	−	0.206537i	$-0.0662193\pi$
$883$	58.1492	1.95688	0.978439	+	0.206537i	$0.0662193\pi$
$884$	5.45255	0.183389
$885$	5.71527	0.192117
$886$	−15.6821	−0.526852
$887$	57.4798	1.92998	0.964991	−	0.262282i	$-0.0844750\pi$
$887$	57.4798	1.92998	0.964991	+	0.262282i	$0.0844750\pi$
$888$	7.04534	0.236426
$889$	−21.9732	−0.736956
$890$	−12.7066	−0.425926
$891$	−1.18887	−0.0398288
$892$	−10.5115	−0.351952
$893$	−12.2568	−0.410159
$894$	8.52233	0.285029
$895$	−10.4282	−0.348575
$896$	−29.2298	−0.976501
$897$	3.07928	0.102814
$898$	−25.7613	−0.859667
$899$	−6.06614	−0.202317
$900$	1.57618	0.0525394
$901$	38.6283	1.28690
$902$	−20.0812	−0.668630
$903$	19.2191	0.639570
$904$	9.38812	0.312244
$905$	−5.76884	−0.191763
$906$	−20.5963	−0.684266
$907$	25.3944	0.843206	0.421603	−	0.906781i	$-0.361468\pi$
$907$	25.3944	0.843206	0.421603	+	0.906781i	$0.361468\pi$
$908$	21.1825	0.702965
$909$	−11.3311	−0.375829
$910$	−8.81900	−0.292347
$911$	−1.73853	−0.0576002	−0.0288001	−	0.999585i	$-0.509169\pi$
$911$	−1.73853	−0.0576002	−0.0288001	+	0.999585i	$0.509169\pi$
$912$	−32.6155	−1.08001
$913$	−18.5171	−0.612827
$914$	−30.4717	−1.00792
$915$	10.7254	0.354572
$916$	16.0130	0.529085
$917$	2.22625	0.0735174
$918$	6.54189	0.215915
$919$	20.6101	0.679864	0.339932	−	0.940450i	$-0.389596\pi$
$919$	20.6101	0.679864	0.339932	+	0.940450i	$0.389596\pi$
$920$	2.46796	0.0813664
$921$	11.6527	0.383968
$922$	−24.4728	−0.805968
$923$	6.83059	0.224832
$924$	−8.73880	−0.287485
$925$	8.79048	0.289029
$926$	−41.5409	−1.36512
$927$	6.47881	0.212792
$928$	43.8257	1.43865
$929$	32.7996	1.07612	0.538060	−	0.842907i	$-0.319157\pi$
$929$	32.7996	1.07612	0.538060	+	0.842907i	$0.319157\pi$
$930$	1.89108	0.0620109
$931$	−103.044	−3.37715
$932$	−19.1306	−0.626643
$933$	−31.8466	−1.04261
$934$	−24.4920	−0.801403
$935$	−4.11273	−0.134501
$936$	0.801474	0.0261970
$937$	25.1903	0.822933	0.411466	−	0.911425i	$-0.365017\pi$
$937$	25.1903	0.822933	0.411466	+	0.911425i	$0.365017\pi$
$938$	66.1621	2.16027
$939$	−10.4465	−0.340907
$940$	−2.76498	−0.0901838
$941$	−51.4596	−1.67754	−0.838768	−	0.544489i	$-0.816724\pi$
$941$	−51.4596	−1.67754	−0.838768	+	0.544489i	$0.816724\pi$
$942$	13.7758	0.448840
$943$	27.5038	0.895647
$944$	−26.6789	−0.868326
$945$	−4.66347	−0.151703
$946$	−9.26550	−0.301247
$947$	43.8190	1.42393	0.711963	−	0.702217i	$-0.247806\pi$
$947$	43.8190	1.42393	0.711963	+	0.702217i	$0.247806\pi$
$948$	−15.3321	−0.497965
$949$	12.3097	0.399589
$950$	−13.2130	−0.428687
$951$	−16.6200	−0.538941
$952$	−12.9298	−0.419058
$953$	−44.0492	−1.42689	−0.713446	−	0.700710i	$-0.752867\pi$
$953$	−44.0492	−1.42689	−0.713446	+	0.700710i	$0.752867\pi$
$954$	−21.1165	−0.683672
$955$	−12.0951	−0.391389
$956$	−13.0124	−0.420850
$957$	−7.21188	−0.233127
$958$	−10.5153	−0.339734
$959$	38.4286	1.24092
$960$	−4.32634	−0.139632
$961$	1.00000	0.0322581
$962$	−16.6235	−0.535963
$963$	17.4549	0.562477
$964$	−5.24883	−0.169053
$965$	−8.41269	−0.270814
$966$	27.1562	0.873735
$967$	−34.8630	−1.12112	−0.560560	−	0.828114i	$-0.689414\pi$
$967$	−34.8630	−1.12112	−0.560560	+	0.828114i	$0.689414\pi$
$968$	7.68339	0.246954
$969$	−24.1705	−0.776469
$970$	26.3897	0.847322
$971$	36.1509	1.16014	0.580068	−	0.814568i	$-0.303026\pi$
$971$	36.1509	1.16014	0.580068	+	0.814568i	$0.303026\pi$
$972$	−1.57618	−0.0505561
$973$	89.8126	2.87926
$974$	−40.8911	−1.31023
$975$	1.00000	0.0320256
$976$	−50.0665	−1.60259
$977$	−21.4448	−0.686080	−0.343040	−	0.939321i	$-0.611457\pi$
$977$	−21.4448	−0.686080	−0.343040	+	0.939321i	$0.611457\pi$
$978$	18.6268	0.595619
$979$	−7.98832	−0.255308
$980$	−23.2455	−0.742549
$981$	−18.2331	−0.582138
$982$	45.7432	1.45972
$983$	−31.2372	−0.996313	−0.498157	−	0.867087i	$-0.665990\pi$
$983$	−31.2372	−0.996313	−0.498157	+	0.867087i	$0.665990\pi$
$984$	7.15868	0.228210
$985$	7.90206	0.251781
$986$	39.6840	1.26380
$987$	8.18079	0.260397
$988$	11.0128	0.350365
$989$	12.6903	0.403529
$990$	2.24826	0.0714543
$991$	46.9728	1.49214	0.746070	−	0.665868i	$-0.231938\pi$
$991$	46.9728	1.49214	0.746070	+	0.665868i	$0.231938\pi$
$992$	−7.22464	−0.229383
$993$	−28.9163	−0.917630
$994$	60.2390	1.91066
$995$	−18.7700	−0.595050
$996$	−24.5495	−0.777882
$997$	−3.10708	−0.0984021	−0.0492011	−	0.998789i	$-0.515668\pi$
$997$	−3.10708	−0.0984021	−0.0492011	+	0.998789i	$0.515668\pi$
$998$	−6.39752	−0.202510
$999$	−8.79048	−0.278118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.13	✓	16	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.89108 q^{2} -1.00000 q^{3} +1.57618 q^{4} -1.00000 q^{5} -1.89108 q^{6} -4.66347 q^{7} -0.801474 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.89108 q^{2} -1.00000 q^{3} +1.57618 q^{4} -1.00000 q^{5} -1.89108 q^{6} -4.66347 q^{7} -0.801474 q^{8} +1.00000 q^{9} -1.89108 q^{10} -1.18887 q^{11} -1.57618 q^{12} -1.00000 q^{13} -8.81900 q^{14} +1.00000 q^{15} -4.66801 q^{16} -3.45934 q^{17} +1.89108 q^{18} -6.98703 q^{19} -1.57618 q^{20} +4.66347 q^{21} -2.24826 q^{22} +3.07928 q^{23} +0.801474 q^{24} +1.00000 q^{25} -1.89108 q^{26} -1.00000 q^{27} -7.35048 q^{28} -6.06614 q^{29} +1.89108 q^{30} +1.00000 q^{31} -7.22464 q^{32} +1.18887 q^{33} -6.54189 q^{34} +4.66347 q^{35} +1.57618 q^{36} +8.79048 q^{37} -13.2130 q^{38} +1.00000 q^{39} +0.801474 q^{40} +8.93189 q^{41} +8.81900 q^{42} +4.12119 q^{43} -1.87388 q^{44} -1.00000 q^{45} +5.82317 q^{46} +1.75423 q^{47} +4.66801 q^{48} +14.7480 q^{49} +1.89108 q^{50} +3.45934 q^{51} -1.57618 q^{52} -11.1664 q^{53} -1.89108 q^{54} +1.18887 q^{55} +3.73765 q^{56} +6.98703 q^{57} -11.4715 q^{58} +5.71527 q^{59} +1.57618 q^{60} +10.7254 q^{61} +1.89108 q^{62} -4.66347 q^{63} -4.32634 q^{64} +1.00000 q^{65} +2.24826 q^{66} -7.50222 q^{67} -5.45255 q^{68} -3.07928 q^{69} +8.81900 q^{70} -6.83059 q^{71} -0.801474 q^{72} -12.3097 q^{73} +16.6235 q^{74} -1.00000 q^{75} -11.0128 q^{76} +5.54428 q^{77} +1.89108 q^{78} +9.72740 q^{79} +4.66801 q^{80} +1.00000 q^{81} +16.8909 q^{82} +15.5753 q^{83} +7.35048 q^{84} +3.45934 q^{85} +7.79350 q^{86} +6.06614 q^{87} +0.952852 q^{88} +6.71923 q^{89} -1.89108 q^{90} +4.66347 q^{91} +4.85351 q^{92} -1.00000 q^{93} +3.31738 q^{94} +6.98703 q^{95} +7.22464 q^{96} -13.9548 q^{97} +27.8896 q^{98} -1.18887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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