LMFDB - Embedded newform 1849.2.a.p.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 $ x^20 - 5x^19 - 19x^18 + 127x^17 + 95x^16 - 1293x^15 + 329x^14 + 6765x^13 - 5068x^12 - 19360x^11 + 20357x^10 + 29618x^9 - 37889x^8 - 21700x^7 + 32885x^6 + 6093x^5 - 11330x^4 - 1004x^3 + 1386x^2 + 87*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-0.528833$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.528833 q^{2} -0.931794 q^{3} -1.72034 q^{4} -0.425701 q^{5} -0.492764 q^{6} -2.15726 q^{7} -1.96744 q^{8} -2.13176 q^{9} +O(q^{10})$	\(q+0.528833 q^{2} -0.931794 q^{3} -1.72034 q^{4} -0.425701 q^{5} -0.492764 q^{6} -2.15726 q^{7} -1.96744 q^{8} -2.13176 q^{9} -0.225125 q^{10} -2.72076 q^{11} +1.60300 q^{12} +4.70548 q^{13} -1.14083 q^{14} +0.396666 q^{15} +2.40022 q^{16} -1.13312 q^{17} -1.12735 q^{18} -4.22913 q^{19} +0.732349 q^{20} +2.01012 q^{21} -1.43883 q^{22} -0.880811 q^{23} +1.83325 q^{24} -4.81878 q^{25} +2.48842 q^{26} +4.78174 q^{27} +3.71121 q^{28} -10.6343 q^{29} +0.209770 q^{30} -0.386801 q^{31} +5.20419 q^{32} +2.53519 q^{33} -0.599233 q^{34} +0.918348 q^{35} +3.66734 q^{36} +7.52028 q^{37} -2.23650 q^{38} -4.38454 q^{39} +0.837540 q^{40} +2.51176 q^{41} +1.06302 q^{42} +4.68063 q^{44} +0.907492 q^{45} -0.465802 q^{46} +9.33716 q^{47} -2.23652 q^{48} -2.34623 q^{49} -2.54833 q^{50} +1.05584 q^{51} -8.09501 q^{52} +10.3740 q^{53} +2.52875 q^{54} +1.15823 q^{55} +4.24427 q^{56} +3.94068 q^{57} -5.62377 q^{58} +13.6325 q^{59} -0.682398 q^{60} -10.2581 q^{61} -0.204553 q^{62} +4.59876 q^{63} -2.04830 q^{64} -2.00313 q^{65} +1.34069 q^{66} +10.6618 q^{67} +1.94935 q^{68} +0.820735 q^{69} +0.485653 q^{70} +13.0737 q^{71} +4.19410 q^{72} +5.62550 q^{73} +3.97697 q^{74} +4.49011 q^{75} +7.27552 q^{76} +5.86939 q^{77} -2.31869 q^{78} -1.39915 q^{79} -1.02178 q^{80} +1.93968 q^{81} +1.32830 q^{82} -1.35689 q^{83} -3.45808 q^{84} +0.482371 q^{85} +9.90897 q^{87} +5.35293 q^{88} -11.9220 q^{89} +0.479912 q^{90} -10.1509 q^{91} +1.51529 q^{92} +0.360419 q^{93} +4.93780 q^{94} +1.80034 q^{95} -4.84924 q^{96} +5.81407 q^{97} -1.24077 q^{98} +5.80001 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) $ 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9} + 21 q^{11} + 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} - 9 q^{18} + 7 q^{19} + 17 q^{20} + 8 q^{21} - 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} - 58 q^{26} + 22 q^{27} + 28 q^{28} + 12 q^{29} + 24 q^{30} + 26 q^{31} - 3 q^{32} + 17 q^{33} - 54 q^{34} + 26 q^{35} + 14 q^{36} + 19 q^{37} + 5 q^{38} + 7 q^{39} - 16 q^{40} + 27 q^{41} + 52 q^{42} + 32 q^{44} - 75 q^{45} - 59 q^{46} + 45 q^{47} + 66 q^{48} + 20 q^{49} - 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} + 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} - 30 q^{61} - 72 q^{62} + 21 q^{63} - 7 q^{64} + 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} + 35 q^{69} + 80 q^{70} + 31 q^{71} + 15 q^{72} + 26 q^{73} + 87 q^{74} - 73 q^{75} + 68 q^{76} + 21 q^{77} - 50 q^{78} + 39 q^{79} + 60 q^{80} + 16 q^{81} - 45 q^{82} + 13 q^{83} + 31 q^{84} + 22 q^{85} + 61 q^{87} - 50 q^{88} + 4 q^{89} - 13 q^{90} + 25 q^{91} + 5 q^{92} - 67 q^{93} - 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} + 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9 + 21 * q^11 + 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 - 9 * q^18 + 7 * q^19 + 17 * q^20 + 8 * q^21 - 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 - 58 * q^26 + 22 * q^27 + 28 * q^28 + 12 * q^29 + 24 * q^30 + 26 * q^31 - 3 * q^32 + 17 * q^33 - 54 * q^34 + 26 * q^35 + 14 * q^36 + 19 * q^37 + 5 * q^38 + 7 * q^39 - 16 * q^40 + 27 * q^41 + 52 * q^42 + 32 * q^44 - 75 * q^45 - 59 * q^46 + 45 * q^47 + 66 * q^48 + 20 * q^49 - 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 + 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 - 30 * q^61 - 72 * q^62 + 21 * q^63 - 7 * q^64 + 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 + 35 * q^69 + 80 * q^70 + 31 * q^71 + 15 * q^72 + 26 * q^73 + 87 * q^74 - 73 * q^75 + 68 * q^76 + 21 * q^77 - 50 * q^78 + 39 * q^79 + 60 * q^80 + 16 * q^81 - 45 * q^82 + 13 * q^83 + 31 * q^84 + 22 * q^85 + 61 * q^87 - 50 * q^88 + 4 * q^89 - 13 * q^90 + 25 * q^91 + 5 * q^92 - 67 * q^93 - 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 + 35 * q^98 + 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.528833	0.373942	0.186971	−	0.982365i	$-0.440133\pi$
$2$	0.528833	0.373942	0.186971	+	0.982365i	$0.440133\pi$
$3$	−0.931794	−0.537972	−0.268986	−	0.963144i	$-0.586688\pi$
$3$	−0.931794	−0.537972	−0.268986	+	0.963144i	$0.586688\pi$
$4$	−1.72034	−0.860168
$5$	−0.425701	−0.190379	−0.0951897	−	0.995459i	$-0.530346\pi$
$5$	−0.425701	−0.190379	−0.0951897	+	0.995459i	$0.530346\pi$
$6$	−0.492764	−0.201170
$7$	−2.15726	−0.815367	−0.407684	−	0.913123i	$-0.633663\pi$
$7$	−2.15726	−0.815367	−0.407684	+	0.913123i	$0.633663\pi$
$8$	−1.96744	−0.695594
$9$	−2.13176	−0.710586
$10$	−0.225125	−0.0711908
$11$	−2.72076	−0.820341	−0.410171	−	0.912009i	$-0.634531\pi$
$11$	−2.72076	−0.820341	−0.410171	+	0.912009i	$0.634531\pi$
$12$	1.60300	0.462746
$13$	4.70548	1.30507	0.652533	−	0.757760i	$-0.273706\pi$
$13$	4.70548	1.30507	0.652533	+	0.757760i	$0.273706\pi$
$14$	−1.14083	−0.304900
$15$	0.396666	0.102419
$16$	2.40022	0.600056
$17$	−1.13312	−0.274822	−0.137411	−	0.990514i	$-0.543878\pi$
$17$	−1.13312	−0.274822	−0.137411	+	0.990514i	$0.543878\pi$
$18$	−1.12735	−0.265718
$19$	−4.22913	−0.970229	−0.485114	−	0.874451i	$-0.661222\pi$
$19$	−4.22913	−0.970229	−0.485114	+	0.874451i	$0.661222\pi$
$20$	0.732349	0.163758
$21$	2.01012	0.438645
$22$	−1.43883	−0.306760
$23$	−0.880811	−0.183662	−0.0918309	−	0.995775i	$-0.529272\pi$
$23$	−0.880811	−0.183662	−0.0918309	+	0.995775i	$0.529272\pi$
$24$	1.83325	0.374210
$25$	−4.81878	−0.963756
$26$	2.48842	0.488019
$27$	4.78174	0.920247
$28$	3.71121	0.701353
$29$	−10.6343	−1.97474	−0.987369	−	0.158438i	$-0.949354\pi$
$29$	−10.6343	−1.97474	−0.987369	+	0.158438i	$0.949354\pi$
$30$	0.209770	0.0382986
$31$	−0.386801	−0.0694715	−0.0347358	−	0.999397i	$-0.511059\pi$
$31$	−0.386801	−0.0694715	−0.0347358	+	0.999397i	$0.511059\pi$
$32$	5.20419	0.919980
$33$	2.53519	0.441320
$34$	−0.599233	−0.102768
$35$	0.918348	0.155229
$36$	3.66734	0.611223
$37$	7.52028	1.23633	0.618163	−	0.786050i	$-0.287877\pi$
$37$	7.52028	1.23633	0.618163	+	0.786050i	$0.287877\pi$
$38$	−2.23650	−0.362809
$39$	−4.38454	−0.702089
$40$	0.837540	0.132427
$41$	2.51176	0.392270	0.196135	−	0.980577i	$-0.437161\pi$
$41$	2.51176	0.392270	0.196135	+	0.980577i	$0.437161\pi$
$42$	1.06302	0.164027
$43$	0	0
$43$	0	0
$44$	4.68063	0.705631
$45$	0.907492	0.135281
$46$	−0.465802	−0.0686788
$47$	9.33716	1.36196	0.680982	−	0.732300i	$-0.261553\pi$
$47$	9.33716	1.36196	0.680982	+	0.732300i	$0.261553\pi$
$48$	−2.23652	−0.322813
$49$	−2.34623	−0.335176
$50$	−2.54833	−0.360388
$51$	1.05584	0.147847
$52$	−8.09501	−1.12258
$53$	10.3740	1.42498	0.712488	−	0.701684i	$-0.247568\pi$
$53$	10.3740	1.42498	0.712488	+	0.701684i	$0.247568\pi$
$54$	2.52875	0.344119
$55$	1.15823	0.156176
$56$	4.24427	0.567165
$57$	3.94068	0.521956
$58$	−5.62377	−0.738437
$59$	13.6325	1.77480	0.887402	−	0.460997i	$-0.152508\pi$
$59$	13.6325	1.77480	0.887402	+	0.460997i	$0.152508\pi$
$60$	−0.682398	−0.0880972
$61$	−10.2581	−1.31341	−0.656705	−	0.754147i	$-0.728050\pi$
$61$	−10.2581	−1.31341	−0.656705	+	0.754147i	$0.728050\pi$
$62$	−0.204553	−0.0259783
$63$	4.59876	0.579389
$64$	−2.04830	−0.256037
$65$	−2.00313	−0.248458
$66$	1.34069	0.165028
$67$	10.6618	1.30254	0.651270	−	0.758846i	$-0.274236\pi$
$67$	10.6618	1.30254	0.651270	+	0.758846i	$0.274236\pi$
$68$	1.94935	0.236393
$69$	0.820735	0.0988048
$70$	0.485653	0.0580466
$71$	13.0737	1.55156	0.775780	−	0.631003i	$-0.217357\pi$
$71$	13.0737	1.55156	0.775780	+	0.631003i	$0.217357\pi$
$72$	4.19410	0.494280
$73$	5.62550	0.658415	0.329208	−	0.944258i	$-0.393218\pi$
$73$	5.62550	0.658415	0.329208	+	0.944258i	$0.393218\pi$
$74$	3.97697	0.462314
$75$	4.49011	0.518473
$76$	7.27552	0.834559
$77$	5.86939	0.668879
$78$	−2.31869	−0.262540
$79$	−1.39915	−0.157416	−0.0787082	−	0.996898i	$-0.525080\pi$
$79$	−1.39915	−0.157416	−0.0787082	+	0.996898i	$0.525080\pi$
$80$	−1.02178	−0.114238
$81$	1.93968	0.215520
$82$	1.32830	0.146686
$83$	−1.35689	−0.148938	−0.0744691	−	0.997223i	$-0.523726\pi$
$83$	−1.35689	−0.148938	−0.0744691	+	0.997223i	$0.523726\pi$
$84$	−3.45808	−0.377308
$85$	0.482371	0.0523205
$86$	0	0
$87$	9.90897	1.06235
$88$	5.35293	0.570625
$89$	−11.9220	−1.26373	−0.631867	−	0.775077i	$-0.717711\pi$
$89$	−11.9220	−1.26373	−0.631867	+	0.775077i	$0.717711\pi$
$90$	0.479912	0.0505872
$91$	−10.1509	−1.06411
$92$	1.51529	0.157980
$93$	0.360419	0.0373737
$94$	4.93780	0.509295
$95$	1.80034	0.184711
$96$	−4.84924	−0.494923
$97$	5.81407	0.590329	0.295165	−	0.955446i	$-0.404626\pi$
$97$	5.81407	0.590329	0.295165	+	0.955446i	$0.404626\pi$
$98$	−1.24077	−0.125336
$99$	5.80001	0.582923
$100$	8.28991	0.828991
$101$	−4.96054	−0.493592	−0.246796	−	0.969067i	$-0.579378\pi$
$101$	−4.96054	−0.493592	−0.246796	+	0.969067i	$0.579378\pi$
$102$	0.558362	0.0552860
$103$	−13.7333	−1.35319	−0.676593	−	0.736357i	$-0.736544\pi$
$103$	−13.7333	−1.35319	−0.676593	+	0.736357i	$0.736544\pi$
$104$	−9.25774	−0.907796
$105$	−0.855711	−0.0835089
$106$	5.48611	0.532858
$107$	16.7459	1.61889	0.809445	−	0.587196i	$-0.199768\pi$
$107$	16.7459	1.61889	0.809445	+	0.587196i	$0.199768\pi$
$108$	−8.22620	−0.791567
$109$	−6.38948	−0.612001	−0.306001	−	0.952031i	$-0.598991\pi$
$109$	−6.38948	−0.612001	−0.306001	+	0.952031i	$0.598991\pi$
$110$	0.612512	0.0584007
$111$	−7.00735	−0.665108
$112$	−5.17791	−0.489266
$113$	−4.75212	−0.447042	−0.223521	−	0.974699i	$-0.571755\pi$
$113$	−4.75212	−0.447042	−0.223521	+	0.974699i	$0.571755\pi$
$114$	2.08396	0.195181
$115$	0.374962	0.0349654
$116$	18.2945	1.69861
$117$	−10.0310	−0.927362
$118$	7.20933	0.663673
$119$	2.44444	0.224081
$120$	−0.780415	−0.0712418
$121$	−3.59744	−0.327040
$122$	−5.42481	−0.491139
$123$	−2.34044	−0.211030
$124$	0.665428	0.0597572
$125$	4.17986	0.373858
$126$	2.43198	0.216658
$127$	15.2863	1.35644	0.678220	−	0.734859i	$-0.262751\pi$
$127$	15.2863	1.35644	0.678220	+	0.734859i	$0.262751\pi$
$128$	−11.4916	−1.01572
$129$	0	0
$130$	−1.05932	−0.0929086
$131$	−15.9163	−1.39061	−0.695307	−	0.718713i	$-0.744732\pi$
$131$	−15.9163	−1.39061	−0.695307	+	0.718713i	$0.744732\pi$
$132$	−4.36138	−0.379609
$133$	9.12333	0.791093
$134$	5.63829	0.487074
$135$	−2.03559	−0.175196
$136$	2.22935	0.191165
$137$	−3.55172	−0.303444	−0.151722	−	0.988423i	$-0.548482\pi$
$137$	−3.55172	−0.303444	−0.151722	+	0.988423i	$0.548482\pi$
$138$	0.434032	0.0369472
$139$	11.3328	0.961236	0.480618	−	0.876930i	$-0.340412\pi$
$139$	11.3328	0.961236	0.480618	+	0.876930i	$0.340412\pi$
$140$	−1.57987	−0.133523
$141$	−8.70031	−0.732698
$142$	6.91380	0.580193
$143$	−12.8025	−1.07060
$144$	−5.11670	−0.426392
$145$	4.52703	0.375949
$146$	2.97495	0.246209
$147$	2.18621	0.180315
$148$	−12.9374	−1.06345
$149$	−9.80007	−0.802853	−0.401427	−	0.915891i	$-0.631486\pi$
$149$	−9.80007	−0.802853	−0.401427	+	0.915891i	$0.631486\pi$
$150$	2.37452	0.193879
$151$	−1.70695	−0.138910	−0.0694549	−	0.997585i	$-0.522126\pi$
$151$	−1.70695	−0.138910	−0.0694549	+	0.997585i	$0.522126\pi$
$152$	8.32055	0.674885
$153$	2.41554	0.195285
$154$	3.10393	0.250122
$155$	0.164662	0.0132259
$156$	7.54288	0.603914
$157$	−10.3160	−0.823309	−0.411654	−	0.911340i	$-0.635049\pi$
$157$	−10.3160	−0.823309	−0.411654	+	0.911340i	$0.635049\pi$
$158$	−0.739916	−0.0588646
$159$	−9.66642	−0.766597
$160$	−2.21543	−0.175145
$161$	1.90014	0.149752
$162$	1.02577	0.0805917
$163$	10.9407	0.856944	0.428472	−	0.903555i	$-0.359052\pi$
$163$	10.9407	0.856944	0.428472	+	0.903555i	$0.359052\pi$
$164$	−4.32106	−0.337418
$165$	−1.07923	−0.0840183
$166$	−0.717570	−0.0556942
$167$	−1.98101	−0.153295	−0.0766476	−	0.997058i	$-0.524422\pi$
$167$	−1.98101	−0.153295	−0.0766476	+	0.997058i	$0.524422\pi$
$168$	−3.95479	−0.305119
$169$	9.14157	0.703198
$170$	0.255094	0.0195648
$171$	9.01549	0.689431
$172$	0	0
$173$	4.61430	0.350819	0.175409	−	0.984496i	$-0.443875\pi$
$173$	4.61430	0.350819	0.175409	+	0.984496i	$0.443875\pi$
$174$	5.24019	0.397258
$175$	10.3954	0.785815
$176$	−6.53044	−0.492251
$177$	−12.7027	−0.954794
$178$	−6.30478	−0.472563
$179$	0.990918	0.0740647	0.0370324	−	0.999314i	$-0.488210\pi$
$179$	0.990918	0.0740647	0.0370324	+	0.999314i	$0.488210\pi$
$180$	−1.56119	−0.116364
$181$	8.61309	0.640206	0.320103	−	0.947383i	$-0.396282\pi$
$181$	8.61309	0.640206	0.320103	+	0.947383i	$0.396282\pi$
$182$	−5.36816	−0.397914
$183$	9.55840	0.706578
$184$	1.73294	0.127754
$185$	−3.20139	−0.235371
$186$	0.190602	0.0139756
$187$	3.08296	0.225448
$188$	−16.0630	−1.17152
$189$	−10.3155	−0.750339
$190$	0.952082	0.0690713
$191$	14.9525	1.08193	0.540963	−	0.841046i	$-0.318060\pi$
$191$	14.9525	1.08193	0.540963	+	0.841046i	$0.318060\pi$
$192$	1.90859	0.137741
$193$	−3.60127	−0.259225	−0.129612	−	0.991565i	$-0.541373\pi$
$193$	−3.60127	−0.259225	−0.129612	+	0.991565i	$0.541373\pi$
$194$	3.07467	0.220749
$195$	1.86650	0.133663
$196$	4.03631	0.288308
$197$	13.5798	0.967520	0.483760	−	0.875201i	$-0.339271\pi$
$197$	13.5798	0.967520	0.483760	+	0.875201i	$0.339271\pi$
$198$	3.06724	0.217979
$199$	1.44808	0.102652	0.0513259	−	0.998682i	$-0.483655\pi$
$199$	1.44808	0.102652	0.0513259	+	0.998682i	$0.483655\pi$
$200$	9.48065	0.670383
$201$	−9.93456	−0.700730
$202$	−2.62330	−0.184575
$203$	22.9409	1.61014
$204$	−1.81639	−0.127173
$205$	−1.06926	−0.0746802
$206$	−7.26265	−0.506013
$207$	1.87768	0.130508
$208$	11.2942	0.783113
$209$	11.5065	0.795919
$210$	−0.452529	−0.0312274
$211$	−3.05662	−0.210426	−0.105213	−	0.994450i	$-0.533553\pi$
$211$	−3.05662	−0.210426	−0.105213	+	0.994450i	$0.533553\pi$
$212$	−17.8467	−1.22572
$213$	−12.1820	−0.834695
$214$	8.85580	0.605370
$215$	0	0
$216$	−9.40778	−0.640118
$217$	0.834430	0.0566448
$218$	−3.37897	−0.228853
$219$	−5.24181	−0.354209
$220$	−1.99255	−0.134338
$221$	−5.33189	−0.358662
$222$	−3.70572	−0.248712
$223$	−14.0644	−0.941819	−0.470909	−	0.882182i	$-0.656074\pi$
$223$	−14.0644	−0.941819	−0.470909	+	0.882182i	$0.656074\pi$
$224$	−11.2268	−0.750122
$225$	10.2725	0.684832
$226$	−2.51308	−0.167168
$227$	16.5418	1.09792	0.548958	−	0.835850i	$-0.315025\pi$
$227$	16.5418	1.09792	0.548958	+	0.835850i	$0.315025\pi$
$228$	−6.77929	−0.448969
$229$	−6.98662	−0.461689	−0.230844	−	0.972991i	$-0.574149\pi$
$229$	−6.98662	−0.461689	−0.230844	+	0.972991i	$0.574149\pi$
$230$	0.198292	0.0130750
$231$	−5.46907	−0.359838
$232$	20.9223	1.37362
$233$	9.04219	0.592373	0.296187	−	0.955130i	$-0.404285\pi$
$233$	9.04219	0.592373	0.296187	+	0.955130i	$0.404285\pi$
$234$	−5.30471	−0.346779
$235$	−3.97484	−0.259290
$236$	−23.4525	−1.52663
$237$	1.30372	0.0846856
$238$	1.29270	0.0837933
$239$	−9.92538	−0.642020	−0.321010	−	0.947076i	$-0.604022\pi$
$239$	−9.92538	−0.642020	−0.321010	+	0.947076i	$0.604022\pi$
$240$	0.952087	0.0614569
$241$	25.5745	1.64740	0.823699	−	0.567027i	$-0.191906\pi$
$241$	25.5745	1.64740	0.823699	+	0.567027i	$0.191906\pi$
$242$	−1.90245	−0.122294
$243$	−16.1526	−1.03619
$244$	17.6473	1.12975
$245$	0.998794	0.0638106
$246$	−1.23770	−0.0789130
$247$	−19.9001	−1.26621
$248$	0.761007	0.0483240
$249$	1.26434	0.0801246
$250$	2.21045	0.139801
$251$	13.6458	0.861317	0.430658	−	0.902515i	$-0.358281\pi$
$251$	13.6458	0.861317	0.430658	+	0.902515i	$0.358281\pi$
$252$	−7.91141	−0.498372
$253$	2.39648	0.150665
$254$	8.08391	0.507230
$255$	−0.449471	−0.0281470
$256$	−1.98054	−0.123784
$257$	−13.2175	−0.824484	−0.412242	−	0.911074i	$-0.635254\pi$
$257$	−13.2175	−0.824484	−0.412242	+	0.911074i	$0.635254\pi$
$258$	0	0
$259$	−16.2232	−1.00806
$260$	3.44605	0.213715
$261$	22.6697	1.40322
$262$	−8.41707	−0.520008
$263$	14.0424	0.865889	0.432944	−	0.901421i	$-0.357475\pi$
$263$	14.0424	0.865889	0.432944	+	0.901421i	$0.357475\pi$
$264$	−4.98783	−0.306980
$265$	−4.41622	−0.271286
$266$	4.82472	0.295823
$267$	11.1089	0.679853
$268$	−18.3418	−1.12040
$269$	−24.8363	−1.51429	−0.757147	−	0.653245i	$-0.773407\pi$
$269$	−24.8363	−1.51429	−0.757147	+	0.653245i	$0.773407\pi$
$270$	−1.07649	−0.0655131
$271$	−12.6799	−0.770249	−0.385125	−	0.922865i	$-0.625842\pi$
$271$	−12.6799	−0.770249	−0.385125	+	0.922865i	$0.625842\pi$
$272$	−2.71975	−0.164909
$273$	9.45860	0.572460
$274$	−1.87827	−0.113470
$275$	13.1108	0.790608
$276$	−1.41194	−0.0849887
$277$	7.53499	0.452734	0.226367	−	0.974042i	$-0.427315\pi$
$277$	7.53499	0.452734	0.226367	+	0.974042i	$0.427315\pi$
$278$	5.99316	0.359446
$279$	0.824567	0.0493655
$280$	−1.80679	−0.107976
$281$	7.96411	0.475099	0.237549	−	0.971375i	$-0.423656\pi$
$281$	7.96411	0.475099	0.237549	+	0.971375i	$0.423656\pi$
$282$	−4.60101	−0.273986
$283$	−17.9030	−1.06422	−0.532112	−	0.846674i	$-0.678601\pi$
$283$	−17.9030	−1.06422	−0.532112	+	0.846674i	$0.678601\pi$
$284$	−22.4911	−1.33460
$285$	−1.67755	−0.0993695
$286$	−6.77039	−0.400342
$287$	−5.41851	−0.319844
$288$	−11.0941	−0.653725
$289$	−15.7160	−0.924473
$290$	2.39404	0.140583
$291$	−5.41751	−0.317580
$292$	−9.67775	−0.566348
$293$	4.77756	0.279108	0.139554	−	0.990214i	$-0.455433\pi$
$293$	4.77756	0.279108	0.139554	+	0.990214i	$0.455433\pi$
$294$	1.15614	0.0674274
$295$	−5.80338	−0.337886
$296$	−14.7957	−0.859981
$297$	−13.0100	−0.754917
$298$	−5.18261	−0.300220
$299$	−4.14464	−0.239691
$300$	−7.72450	−0.445974
$301$	0	0
$302$	−0.902694	−0.0519442
$303$	4.62221	0.265539
$304$	−10.1509	−0.582192
$305$	4.36687	0.250046
$306$	1.27742	0.0730252
$307$	27.3627	1.56167	0.780836	−	0.624736i	$-0.214793\pi$
$307$	27.3627	1.56167	0.780836	+	0.624736i	$0.214793\pi$
$308$	−10.0973	−0.575348
$309$	12.7966	0.727976
$310$	0.0870786	0.00494573
$311$	−25.4339	−1.44223	−0.721113	−	0.692818i	$-0.756369\pi$
$311$	−25.4339	−1.44223	−0.721113	+	0.692818i	$0.756369\pi$
$312$	8.62631	0.488369
$313$	−29.1090	−1.64534	−0.822669	−	0.568520i	$-0.807516\pi$
$313$	−29.1090	−1.64534	−0.822669	+	0.568520i	$0.807516\pi$
$314$	−5.45546	−0.307869
$315$	−1.95770	−0.110304
$316$	2.40700	0.135405
$317$	−8.37450	−0.470359	−0.235179	−	0.971952i	$-0.575568\pi$
$317$	−8.37450	−0.470359	−0.235179	+	0.971952i	$0.575568\pi$
$318$	−5.11192	−0.286662
$319$	28.9334	1.61996
$320$	0.871962	0.0487442
$321$	−15.6038	−0.870917
$322$	1.00486	0.0559984
$323$	4.79212	0.266641
$324$	−3.33689	−0.185383
$325$	−22.6747	−1.25777
$326$	5.78582	0.320447
$327$	5.95368	0.329239
$328$	−4.94172	−0.272861
$329$	−20.1427	−1.11050
$330$	−0.570735	−0.0314179
$331$	−20.3881	−1.12063	−0.560317	−	0.828279i	$-0.689321\pi$
$331$	−20.3881	−1.12063	−0.560317	+	0.828279i	$0.689321\pi$
$332$	2.33431	0.128112
$333$	−16.0314	−0.878516
$334$	−1.04762	−0.0573235
$335$	−4.53872	−0.247977
$336$	4.82474	0.263211
$337$	−23.2139	−1.26454	−0.632271	−	0.774748i	$-0.717877\pi$
$337$	−23.2139	−1.26454	−0.632271	+	0.774748i	$0.717877\pi$
$338$	4.83437	0.262955
$339$	4.42800	0.240496
$340$	−0.829840	−0.0450044
$341$	1.05239	0.0569904
$342$	4.76769	0.257807
$343$	20.1622	1.08866
$344$	0	0
$345$	−0.349388	−0.0188104
$346$	2.44020	0.131186
$347$	3.98347	0.213844	0.106922	−	0.994267i	$-0.465900\pi$
$347$	3.98347	0.213844	0.106922	+	0.994267i	$0.465900\pi$
$348$	−17.0467	−0.913802
$349$	32.3496	1.73164	0.865818	−	0.500358i	$-0.166798\pi$
$349$	32.3496	1.73164	0.865818	+	0.500358i	$0.166798\pi$
$350$	5.49741	0.293849
$351$	22.5004	1.20098
$352$	−14.1594	−0.754698
$353$	23.3943	1.24515	0.622577	−	0.782558i	$-0.286086\pi$
$353$	23.3943	1.24515	0.622577	+	0.782558i	$0.286086\pi$
$354$	−6.71761	−0.357037
$355$	−5.56548	−0.295385
$356$	20.5099	1.08702
$357$	−2.27771	−0.120549
$358$	0.524031	0.0276959
$359$	21.4751	1.13341	0.566707	−	0.823919i	$-0.308217\pi$
$359$	21.4751	1.13341	0.566707	+	0.823919i	$0.308217\pi$
$360$	−1.78543	−0.0941006
$361$	−1.11447	−0.0586562
$362$	4.55489	0.239400
$363$	3.35208	0.175938
$364$	17.4630	0.915312
$365$	−2.39478	−0.125349
$366$	5.05480	0.264219
$367$	−22.8439	−1.19244	−0.596220	−	0.802821i	$-0.703331\pi$
$367$	−22.8439	−1.19244	−0.596220	+	0.802821i	$0.703331\pi$
$368$	−2.11414	−0.110207
$369$	−5.35446	−0.278742
$370$	−1.69300	−0.0880150
$371$	−22.3794	−1.16188
$372$	−0.620042	−0.0321477
$373$	13.6363	0.706063	0.353031	−	0.935611i	$-0.385151\pi$
$373$	13.6363	0.706063	0.353031	+	0.935611i	$0.385151\pi$
$374$	1.63037	0.0843045
$375$	−3.89477	−0.201125
$376$	−18.3703	−0.947374
$377$	−50.0395	−2.57716
$378$	−5.45516	−0.280583
$379$	10.0729	0.517413	0.258706	−	0.965956i	$-0.416704\pi$
$379$	10.0729	0.517413	0.258706	+	0.965956i	$0.416704\pi$
$380$	−3.09720	−0.158883
$381$	−14.2437	−0.729727
$382$	7.90739	0.404577
$383$	37.6506	1.92386	0.961929	−	0.273300i	$-0.0881153\pi$
$383$	37.6506	1.92386	0.961929	+	0.273300i	$0.0881153\pi$
$384$	10.7078	0.546430
$385$	−2.49861	−0.127341
$386$	−1.90447	−0.0969349
$387$	0	0
$388$	−10.0021	−0.507782
$389$	−24.0820	−1.22101	−0.610503	−	0.792014i	$-0.709033\pi$
$389$	−24.0820	−1.22101	−0.610503	+	0.792014i	$0.709033\pi$
$390$	0.987070	0.0499822
$391$	0.998066	0.0504744
$392$	4.61606	0.233146
$393$	14.8307	0.748111
$394$	7.18145	0.361796
$395$	0.595619	0.0299688
$396$	−9.97797	−0.501412
$397$	8.62297	0.432775	0.216387	−	0.976308i	$-0.430573\pi$
$397$	8.62297	0.432775	0.216387	+	0.976308i	$0.430573\pi$
$398$	0.765793	0.0383858
$399$	−8.50107	−0.425586
$400$	−11.5661	−0.578307
$401$	16.8631	0.842102	0.421051	−	0.907037i	$-0.361661\pi$
$401$	16.8631	0.842102	0.421051	+	0.907037i	$0.361661\pi$
$402$	−5.25373	−0.262032
$403$	−1.82009	−0.0906650
$404$	8.53380	0.424572
$405$	−0.825722	−0.0410305
$406$	12.1319	0.602097
$407$	−20.4609	−1.01421
$408$	−2.07729	−0.102841
$409$	36.7451	1.81693	0.908463	−	0.417965i	$-0.137256\pi$
$409$	36.7451	1.81693	0.908463	+	0.417965i	$0.137256\pi$
$410$	−0.565459	−0.0279260
$411$	3.30947	0.163244
$412$	23.6260	1.16397
$413$	−29.4089	−1.44712
$414$	0.992978	0.0488022
$415$	0.577630	0.0283548
$416$	24.4882	1.20063
$417$	−10.5598	−0.517118
$418$	6.08500	0.297627
$419$	9.07765	0.443472	0.221736	−	0.975107i	$-0.428828\pi$
$419$	9.07765	0.443472	0.221736	+	0.975107i	$0.428828\pi$
$420$	1.47211	0.0718316
$421$	−1.03935	−0.0506547	−0.0253274	−	0.999679i	$-0.508063\pi$
$421$	−1.03935	−0.0506547	−0.0253274	+	0.999679i	$0.508063\pi$
$422$	−1.61644	−0.0786872
$423$	−19.9046	−0.967793
$424$	−20.4102	−0.991205
$425$	5.46026	0.264862
$426$	−6.44224	−0.312127
$427$	22.1293	1.07091
$428$	−28.8086	−1.39252
$429$	11.9293	0.575952
$430$	0	0
$431$	1.78620	0.0860384	0.0430192	−	0.999074i	$-0.486302\pi$
$431$	1.78620	0.0860384	0.0430192	+	0.999074i	$0.486302\pi$
$432$	11.4773	0.552200
$433$	25.7117	1.23563	0.617814	−	0.786324i	$-0.288019\pi$
$433$	25.7117	1.23563	0.617814	+	0.786324i	$0.288019\pi$
$434$	0.441275	0.0211819
$435$	−4.21826	−0.202250
$436$	10.9921	0.526424
$437$	3.72506	0.178194
$438$	−2.77204	−0.132453
$439$	−3.76841	−0.179856	−0.0899282	−	0.995948i	$-0.528664\pi$
$439$	−3.76841	−0.179856	−0.0899282	+	0.995948i	$0.528664\pi$
$440$	−2.27875	−0.108635
$441$	5.00160	0.238172
$442$	−2.81968	−0.134118
$443$	−20.2175	−0.960561	−0.480281	−	0.877115i	$-0.659465\pi$
$443$	−20.2175	−0.960561	−0.480281	+	0.877115i	$0.659465\pi$
$444$	12.0550	0.572105
$445$	5.07523	0.240589
$446$	−7.43770	−0.352185
$447$	9.13165	0.431912
$448$	4.41871	0.208764
$449$	18.5647	0.876123	0.438061	−	0.898945i	$-0.355665\pi$
$449$	18.5647	0.876123	0.438061	+	0.898945i	$0.355665\pi$
$450$	5.43243	0.256087
$451$	−6.83389	−0.321796
$452$	8.17524	0.384531
$453$	1.59053	0.0747296
$454$	8.74785	0.410557
$455$	4.32127	0.202584
$456$	−7.75304	−0.363069
$457$	10.8601	0.508014	0.254007	−	0.967202i	$-0.418251\pi$
$457$	10.8601	0.508014	0.254007	+	0.967202i	$0.418251\pi$
$458$	−3.69476	−0.172645
$459$	−5.41830	−0.252905
$460$	−0.645061	−0.0300761
$461$	7.15608	0.333292	0.166646	−	0.986017i	$-0.446706\pi$
$461$	7.15608	0.333292	0.166646	+	0.986017i	$0.446706\pi$
$462$	−2.89223	−0.134558
$463$	−5.42636	−0.252184	−0.126092	−	0.992019i	$-0.540244\pi$
$463$	−5.42636	−0.252184	−0.126092	+	0.992019i	$0.540244\pi$
$464$	−25.5247	−1.18495
$465$	−0.153431	−0.00711518
$466$	4.78181	0.221513
$467$	−3.83542	−0.177482	−0.0887411	−	0.996055i	$-0.528284\pi$
$467$	−3.83542	−0.177482	−0.0887411	+	0.996055i	$0.528284\pi$
$468$	17.2566	0.797687
$469$	−23.0002	−1.06205
$470$	−2.10203	−0.0969593
$471$	9.61242	0.442917
$472$	−26.8211	−1.23454
$473$	0	0
$474$	0.689450	0.0316675
$475$	20.3792	0.935064
$476$	−4.20525	−0.192747
$477$	−22.1148	−1.01257
$478$	−5.24887	−0.240078
$479$	−30.8277	−1.40856	−0.704278	−	0.709925i	$-0.748729\pi$
$479$	−30.8277	−1.40856	−0.704278	+	0.709925i	$0.748729\pi$
$480$	2.06433	0.0942231
$481$	35.3865	1.61349
$482$	13.5246	0.616031
$483$	−1.77054	−0.0805622
$484$	6.18881	0.281310
$485$	−2.47505	−0.112386
$486$	−8.54204	−0.387475
$487$	9.55874	0.433148	0.216574	−	0.976266i	$-0.430512\pi$
$487$	9.55874	0.433148	0.216574	+	0.976266i	$0.430512\pi$
$488$	20.1821	0.913601
$489$	−10.1945	−0.461012
$490$	0.528195	0.0238614
$491$	16.0645	0.724980	0.362490	−	0.931988i	$-0.381927\pi$
$491$	16.0645	0.724980	0.362490	+	0.931988i	$0.381927\pi$
$492$	4.02634	0.181521
$493$	12.0499	0.542702
$494$	−10.5238	−0.473490
$495$	−2.46907	−0.110977
$496$	−0.928409	−0.0416868
$497$	−28.2033	−1.26509
$498$	0.668628	0.0299619
$499$	−3.54262	−0.158589	−0.0792947	−	0.996851i	$-0.525267\pi$
$499$	−3.54262	−0.158589	−0.0792947	+	0.996851i	$0.525267\pi$
$500$	−7.19077	−0.321581
$501$	1.84589	0.0824685
$502$	7.21637	0.322082
$503$	−8.74118	−0.389750	−0.194875	−	0.980828i	$-0.562430\pi$
$503$	−8.74118	−0.389750	−0.194875	+	0.980828i	$0.562430\pi$
$504$	−9.04777	−0.403020
$505$	2.11171	0.0939698
$506$	1.26734	0.0563400
$507$	−8.51807	−0.378301
$508$	−26.2976	−1.16677
$509$	−1.61908	−0.0717644	−0.0358822	−	0.999356i	$-0.511424\pi$
$509$	−1.61908	−0.0717644	−0.0358822	+	0.999356i	$0.511424\pi$
$510$	−0.237695	−0.0105253
$511$	−12.1357	−0.536850
$512$	21.9358	0.969435
$513$	−20.2226	−0.892850
$514$	−6.98985	−0.308309
$515$	5.84630	0.257619
$516$	0	0
$517$	−25.4042	−1.11728
$518$	−8.57936	−0.376955
$519$	−4.29958	−0.188730
$520$	3.94103	0.172826
$521$	27.9422	1.22417	0.612085	−	0.790792i	$-0.290331\pi$
$521$	27.9422	1.22417	0.612085	+	0.790792i	$0.290331\pi$
$522$	11.9885	0.524723
$523$	38.9815	1.70454	0.852271	−	0.523101i	$-0.175225\pi$
$523$	38.9815	1.70454	0.852271	+	0.523101i	$0.175225\pi$
$524$	27.3814	1.19616
$525$	−9.68633	−0.422746
$526$	7.42607	0.323792
$527$	0.438293	0.0190923
$528$	6.08503	0.264817
$529$	−22.2242	−0.966268
$530$	−2.33544	−0.101445
$531$	−29.0613	−1.26115
$532$	−15.6952	−0.680473
$533$	11.8190	0.511939
$534$	5.87475	0.254225
$535$	−7.12876	−0.308203
$536$	−20.9763	−0.906039
$537$	−0.923332	−0.0398447
$538$	−13.1342	−0.566258
$539$	6.38354	0.274959
$540$	3.50190	0.150698
$541$	10.1926	0.438212	0.219106	−	0.975701i	$-0.429686\pi$
$541$	10.1926	0.438212	0.219106	+	0.975701i	$0.429686\pi$
$542$	−6.70556	−0.288028
$543$	−8.02563	−0.344413
$544$	−5.89699	−0.252831
$545$	2.72001	0.116512
$546$	5.00202	0.214067
$547$	−16.3912	−0.700838	−0.350419	−	0.936593i	$-0.613961\pi$
$547$	−16.3912	−0.700838	−0.350419	+	0.936593i	$0.613961\pi$
$548$	6.11014	0.261012
$549$	21.8677	0.933292
$550$	6.93341	0.295641
$551$	44.9738	1.91595
$552$	−1.61474	−0.0687281
$553$	3.01833	0.128352
$554$	3.98475	0.169296
$555$	2.98304	0.126623
$556$	−19.4962	−0.826824
$557$	16.1766	0.685423	0.342711	−	0.939441i	$-0.388655\pi$
$557$	16.1766	0.685423	0.342711	+	0.939441i	$0.388655\pi$
$558$	0.436058	0.0184598
$559$	0	0
$560$	2.20424	0.0931461
$561$	−2.87268	−0.121285
$562$	4.21169	0.177659
$563$	35.9870	1.51667	0.758336	−	0.651863i	$-0.226012\pi$
$563$	35.9870	1.51667	0.758336	+	0.651863i	$0.226012\pi$
$564$	14.9674	0.630243
$565$	2.02298	0.0851075
$566$	−9.46771	−0.397957
$567$	−4.18438	−0.175728
$568$	−25.7216	−1.07926
$569$	−6.11277	−0.256260	−0.128130	−	0.991757i	$-0.540898\pi$
$569$	−6.11277	−0.256260	−0.128130	+	0.991757i	$0.540898\pi$
$570$	−0.887145	−0.0371584
$571$	26.2404	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$571$	26.2404	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$572$	22.0246	0.920895
$573$	−13.9327	−0.582046
$574$	−2.86549	−0.119603
$575$	4.24443	0.177005
$576$	4.36648	0.181937
$577$	−10.0639	−0.418968	−0.209484	−	0.977812i	$-0.567178\pi$
$577$	−10.0639	−0.418968	−0.209484	+	0.977812i	$0.567178\pi$
$578$	−8.31116	−0.345699
$579$	3.35564	0.139456
$580$	−7.78801	−0.323379
$581$	2.92717	0.121439
$582$	−2.86496	−0.118757
$583$	−28.2252	−1.16897
$584$	−11.0678	−0.457990
$585$	4.27019	0.176551
$586$	2.52653	0.104370
$587$	14.8643	0.613517	0.306758	−	0.951787i	$-0.400756\pi$
$587$	14.8643	0.613517	0.306758	+	0.951787i	$0.400756\pi$
$588$	−3.76101	−0.155101
$589$	1.63583	0.0674033
$590$	−3.06902	−0.126350
$591$	−12.6536	−0.520499
$592$	18.0503	0.741865
$593$	38.2736	1.57171	0.785854	−	0.618412i	$-0.212224\pi$
$593$	38.2736	1.57171	0.785854	+	0.618412i	$0.212224\pi$
$594$	−6.88012	−0.282295
$595$	−1.04060	−0.0426604
$596$	16.8594	0.690589
$597$	−1.34931	−0.0552237
$598$	−2.19182	−0.0896304
$599$	24.1292	0.985891	0.492945	−	0.870060i	$-0.335920\pi$
$599$	24.1292	0.985891	0.492945	+	0.870060i	$0.335920\pi$
$600$	−8.83401	−0.360647
$601$	16.0402	0.654293	0.327147	−	0.944974i	$-0.393913\pi$
$601$	16.0402	0.654293	0.327147	+	0.944974i	$0.393913\pi$
$602$	0	0
$603$	−22.7283	−0.925567
$604$	2.93653	0.119486
$605$	1.53144	0.0622617
$606$	2.44438	0.0992960
$607$	−31.9069	−1.29506	−0.647530	−	0.762040i	$-0.724198\pi$
$607$	−31.9069	−1.29506	−0.647530	+	0.762040i	$0.724198\pi$
$608$	−22.0092	−0.892591
$609$	−21.3762	−0.866208
$610$	2.30935	0.0935027
$611$	43.9358	1.77745
$612$	−4.15554	−0.167978
$613$	−10.4129	−0.420574	−0.210287	−	0.977640i	$-0.567440\pi$
$613$	−10.4129	−0.420574	−0.210287	+	0.977640i	$0.567440\pi$
$614$	14.4703	0.583974
$615$	0.996328	0.0401758
$616$	−11.5477	−0.465269
$617$	38.2973	1.54179	0.770895	−	0.636962i	$-0.219809\pi$
$617$	38.2973	1.54179	0.770895	+	0.636962i	$0.219809\pi$
$618$	6.76730	0.272221
$619$	−12.3128	−0.494894	−0.247447	−	0.968901i	$-0.579592\pi$
$619$	−12.3128	−0.494894	−0.247447	+	0.968901i	$0.579592\pi$
$620$	−0.283273	−0.0113765
$621$	−4.21181	−0.169014
$622$	−13.4503	−0.539308
$623$	25.7189	1.03041
$624$	−10.5239	−0.421293
$625$	22.3145	0.892581
$626$	−15.3938	−0.615261
$627$	−10.7217	−0.428182
$628$	17.7470	0.708184
$629$	−8.52139	−0.339770
$630$	−1.03530	−0.0412471
$631$	−6.94499	−0.276476	−0.138238	−	0.990399i	$-0.544144\pi$
$631$	−6.94499	−0.276476	−0.138238	+	0.990399i	$0.544144\pi$
$632$	2.75274	0.109498
$633$	2.84814	0.113203
$634$	−4.42872	−0.175887
$635$	−6.50740	−0.258238
$636$	16.6295	0.659402
$637$	−11.0402	−0.437427
$638$	15.3009	0.605770
$639$	−27.8699	−1.10252
$640$	4.89198	0.193373
$641$	−28.2101	−1.11423	−0.557116	−	0.830435i	$-0.688092\pi$
$641$	−28.2101	−1.11423	−0.557116	+	0.830435i	$0.688092\pi$
$642$	−8.25179	−0.325672
$643$	2.29706	0.0905870	0.0452935	−	0.998974i	$-0.485578\pi$
$643$	2.29706	0.0905870	0.0452935	+	0.998974i	$0.485578\pi$
$644$	−3.26887	−0.128812
$645$	0	0
$646$	2.53423	0.0997080
$647$	−6.73543	−0.264797	−0.132399	−	0.991197i	$-0.542268\pi$
$647$	−6.73543	−0.264797	−0.132399	+	0.991197i	$0.542268\pi$
$648$	−3.81619	−0.149914
$649$	−37.0909	−1.45594
$650$	−11.9911	−0.470331
$651$	−0.777517	−0.0304733
$652$	−18.8217	−0.737116
$653$	−10.3579	−0.405337	−0.202668	−	0.979247i	$-0.564961\pi$
$653$	−10.3579	−0.405337	−0.202668	+	0.979247i	$0.564961\pi$
$654$	3.14851	0.123116
$655$	6.77559	0.264744
$656$	6.02878	0.235384
$657$	−11.9922	−0.467861
$658$	−10.6521	−0.415263
$659$	−13.7179	−0.534372	−0.267186	−	0.963645i	$-0.586094\pi$
$659$	−13.7179	−0.534372	−0.267186	+	0.963645i	$0.586094\pi$
$660$	1.85664	0.0722698
$661$	4.41072	0.171557	0.0857785	−	0.996314i	$-0.472662\pi$
$661$	4.41072	0.171557	0.0857785	+	0.996314i	$0.472662\pi$
$662$	−10.7819	−0.419051
$663$	4.96822	0.192950
$664$	2.66960	0.103601
$665$	−3.88381	−0.150608
$666$	−8.47795	−0.328514
$667$	9.36680	0.362684
$668$	3.40800	0.131860
$669$	13.1051	0.506672
$670$	−2.40023	−0.0927288
$671$	27.9098	1.07744
$672$	10.4611	0.403544
$673$	6.06191	0.233669	0.116835	−	0.993151i	$-0.462725\pi$
$673$	6.06191	0.233669	0.116835	+	0.993151i	$0.462725\pi$
$674$	−12.2763	−0.472865
$675$	−23.0422	−0.886893
$676$	−15.7266	−0.604868
$677$	19.8647	0.763464	0.381732	−	0.924273i	$-0.375328\pi$
$677$	19.8647	0.763464	0.381732	+	0.924273i	$0.375328\pi$
$678$	2.34167	0.0899314
$679$	−12.5425	−0.481335
$680$	−0.949035	−0.0363938
$681$	−15.4135	−0.590648
$682$	0.556541	0.0213111
$683$	21.2903	0.814649	0.407325	−	0.913283i	$-0.366462\pi$
$683$	21.2903	0.814649	0.407325	+	0.913283i	$0.366462\pi$
$684$	−15.5097	−0.593027
$685$	1.51197	0.0577694
$686$	10.6625	0.407095
$687$	6.51009	0.248375
$688$	0	0
$689$	48.8146	1.85969
$690$	−0.184768	−0.00703399
$691$	−28.6949	−1.09161	−0.545803	−	0.837913i	$-0.683775\pi$
$691$	−28.6949	−1.09161	−0.545803	+	0.837913i	$0.683775\pi$
$692$	−7.93814	−0.301763
$693$	−12.5121	−0.475297
$694$	2.10659	0.0799652
$695$	−4.82439	−0.182999
$696$	−19.4953	−0.738967
$697$	−2.84613	−0.107805
$698$	17.1076	0.647531
$699$	−8.42546	−0.318680
$700$	−17.8835	−0.675933
$701$	44.2512	1.67134	0.835672	−	0.549228i	$-0.185078\pi$
$701$	44.2512	1.67134	0.835672	+	0.549228i	$0.185078\pi$
$702$	11.8990	0.449098
$703$	−31.8042	−1.19952
$704$	5.57293	0.210038
$705$	3.70373	0.139491
$706$	12.3717	0.465615
$707$	10.7012	0.402459
$708$	21.8529	0.821283
$709$	−5.53107	−0.207724	−0.103862	−	0.994592i	$-0.533120\pi$
$709$	−5.53107	−0.207724	−0.103862	+	0.994592i	$0.533120\pi$
$710$	−2.94321	−0.110457
$711$	2.98265	0.111858
$712$	23.4559	0.879046
$713$	0.340699	0.0127593
$714$	−1.20453	−0.0450784
$715$	5.45004	0.203820
$716$	−1.70471	−0.0637081
$717$	9.24842	0.345388
$718$	11.3568	0.423831
$719$	40.5818	1.51344	0.756722	−	0.653737i	$-0.226800\pi$
$719$	40.5818	1.51344	0.756722	+	0.653737i	$0.226800\pi$
$720$	2.17818	0.0811762
$721$	29.6264	1.10334
$722$	−0.589367	−0.0219340
$723$	−23.8302	−0.886254
$724$	−14.8174	−0.550685
$725$	51.2443	1.90316
$726$	1.77269	0.0657907
$727$	30.7096	1.13896	0.569478	−	0.822007i	$-0.307146\pi$
$727$	30.7096	1.13896	0.569478	+	0.822007i	$0.307146\pi$
$728$	19.9714	0.740188
$729$	9.23188	0.341922
$730$	−1.26644	−0.0468731
$731$	0	0
$732$	−16.4437	−0.607775
$733$	−5.45093	−0.201335	−0.100667	−	0.994920i	$-0.532098\pi$
$733$	−5.45093	−0.201335	−0.100667	+	0.994920i	$0.532098\pi$
$734$	−12.0806	−0.445903
$735$	−0.930670	−0.0343283
$736$	−4.58391	−0.168965
$737$	−29.0081	−1.06853
$738$	−2.83162	−0.104233
$739$	−44.1283	−1.62329	−0.811643	−	0.584153i	$-0.801427\pi$
$739$	−44.1283	−1.62329	−0.811643	+	0.584153i	$0.801427\pi$
$740$	5.50746	0.202458
$741$	18.5428	0.681187
$742$	−11.8350	−0.434475
$743$	−4.77882	−0.175318	−0.0876589	−	0.996151i	$-0.527939\pi$
$743$	−4.77882	−0.175318	−0.0876589	+	0.996151i	$0.527939\pi$
$744$	−0.709102	−0.0259969
$745$	4.17190	0.152847
$746$	7.21135	0.264026
$747$	2.89257	0.105834
$748$	−5.30372	−0.193923
$749$	−36.1253	−1.31999
$750$	−2.05969	−0.0752091
$751$	−12.3001	−0.448838	−0.224419	−	0.974493i	$-0.572048\pi$
$751$	−12.3001	−0.448838	−0.224419	+	0.974493i	$0.572048\pi$
$752$	22.4113	0.817255
$753$	−12.7151	−0.463364
$754$	−26.4625	−0.963709
$755$	0.726652	0.0264456
$756$	17.7461	0.645418
$757$	21.1439	0.768489	0.384244	−	0.923231i	$-0.374462\pi$
$757$	21.1439	0.768489	0.384244	+	0.923231i	$0.374462\pi$
$758$	5.32691	0.193482
$759$	−2.23303	−0.0810537
$760$	−3.54207	−0.128484
$761$	36.1072	1.30888	0.654442	−	0.756112i	$-0.272904\pi$
$761$	36.1072	1.30888	0.654442	+	0.756112i	$0.272904\pi$
$762$	−7.53254	−0.272875
$763$	13.7838	0.499006
$764$	−25.7233	−0.930638
$765$	−1.02830	−0.0371782
$766$	19.9109	0.719410
$767$	64.1476	2.31624
$768$	1.84546	0.0665923
$769$	−14.2293	−0.513120	−0.256560	−	0.966528i	$-0.582589\pi$
$769$	−14.2293	−0.513120	−0.256560	+	0.966528i	$0.582589\pi$
$770$	−1.32135	−0.0476180
$771$	12.3160	0.443549
$772$	6.19538	0.222977
$773$	3.67071	0.132026	0.0660131	−	0.997819i	$-0.478972\pi$
$773$	3.67071	0.132026	0.0660131	+	0.997819i	$0.478972\pi$
$774$	0	0
$775$	1.86391	0.0669536
$776$	−11.4388	−0.410629
$777$	15.1167	0.542308
$778$	−12.7354	−0.456585
$779$	−10.6225	−0.380592
$780$	−3.21101	−0.114973
$781$	−35.5704	−1.27281
$782$	0.527811	0.0188745
$783$	−50.8504	−1.81725
$784$	−5.63148	−0.201124
$785$	4.39155	0.156741
$786$	7.84298	0.279750
$787$	46.5193	1.65824	0.829118	−	0.559074i	$-0.188843\pi$
$787$	46.5193	1.65824	0.829118	+	0.559074i	$0.188843\pi$
$788$	−23.3618	−0.832230
$789$	−13.0846	−0.465824
$790$	0.314983	0.0112066
$791$	10.2516	0.364503
$792$	−11.4112	−0.405478
$793$	−48.2691	−1.71409
$794$	4.56011	0.161832
$795$	4.11500	0.145944
$796$	−2.49118	−0.0882977
$797$	−16.5824	−0.587378	−0.293689	−	0.955901i	$-0.594883\pi$
$797$	−16.5824	−0.587378	−0.293689	+	0.955901i	$0.594883\pi$
$798$	−4.49565	−0.159144
$799$	−10.5801	−0.374298
$800$	−25.0779	−0.886636
$801$	25.4149	0.897992
$802$	8.91776	0.314897
$803$	−15.3057	−0.540125
$804$	17.0908	0.602745
$805$	−0.808891	−0.0285096
$806$	−0.962522	−0.0339034
$807$	23.1423	0.814647
$808$	9.75956	0.343340
$809$	−40.4462	−1.42201	−0.711006	−	0.703186i	$-0.751760\pi$
$809$	−40.4462	−1.42201	−0.711006	+	0.703186i	$0.751760\pi$
$810$	−0.436669	−0.0153430
$811$	−14.3292	−0.503167	−0.251584	−	0.967836i	$-0.580951\pi$
$811$	−14.3292	−0.503167	−0.251584	+	0.967836i	$0.580951\pi$
$812$	−39.4661	−1.38499
$813$	11.8151	0.414372
$814$	−10.8204	−0.379255
$815$	−4.65748	−0.163144
$816$	2.53424	0.0887163
$817$	0	0
$818$	19.4320	0.679424
$819$	21.6394	0.756141
$820$	1.83948	0.0642375
$821$	39.1812	1.36743	0.683716	−	0.729748i	$-0.260363\pi$
$821$	39.1812	1.36743	0.683716	+	0.729748i	$0.260363\pi$
$822$	1.75016	0.0610438
$823$	17.5364	0.611280	0.305640	−	0.952147i	$-0.401130\pi$
$823$	17.5364	0.611280	0.305640	+	0.952147i	$0.401130\pi$
$824$	27.0195	0.941269
$825$	−12.2165	−0.425325
$826$	−15.5524	−0.541137
$827$	−23.9801	−0.833871	−0.416936	−	0.908936i	$-0.636896\pi$
$827$	−23.9801	−0.833871	−0.416936	+	0.908936i	$0.636896\pi$
$828$	−3.23023	−0.112258
$829$	−22.1957	−0.770889	−0.385445	−	0.922731i	$-0.625952\pi$
$829$	−22.1957	−0.770889	−0.385445	+	0.922731i	$0.625952\pi$
$830$	0.305470	0.0106030
$831$	−7.02106	−0.243558
$832$	−9.63823	−0.334145
$833$	2.65857	0.0921139
$834$	−5.58440	−0.193372
$835$	0.843318	0.0291842
$836$	−19.7950	−0.684623
$837$	−1.84958	−0.0639310
$838$	4.80056	0.165833
$839$	−31.1307	−1.07475	−0.537376	−	0.843343i	$-0.680584\pi$
$839$	−31.1307	−1.07475	−0.537376	+	0.843343i	$0.680584\pi$
$840$	1.68356	0.0580883
$841$	84.0881	2.89959
$842$	−0.549642	−0.0189419
$843$	−7.42091	−0.255590
$844$	5.25841	0.181002
$845$	−3.89158	−0.133874
$846$	−10.5262	−0.361898
$847$	7.76062	0.266658
$848$	24.8999	0.855065
$849$	16.6819	0.572522
$850$	2.88757	0.0990428
$851$	−6.62394	−0.227066
$852$	20.9571	0.717978
$853$	−36.9048	−1.26360	−0.631799	−	0.775132i	$-0.717683\pi$
$853$	−36.9048	−1.26360	−0.631799	+	0.775132i	$0.717683\pi$
$854$	11.7027	0.400459
$855$	−3.83790	−0.131253
$856$	−32.9466	−1.12609
$857$	−20.3295	−0.694442	−0.347221	−	0.937783i	$-0.612875\pi$
$857$	−20.3295	−0.694442	−0.347221	+	0.937783i	$0.612875\pi$
$858$	6.30861	0.215373
$859$	−52.0566	−1.77615	−0.888074	−	0.459700i	$-0.847957\pi$
$859$	−52.0566	−1.77615	−0.888074	+	0.459700i	$0.847957\pi$
$860$	0	0
$861$	5.04894	0.172067
$862$	0.944604	0.0321733
$863$	−27.9473	−0.951337	−0.475669	−	0.879625i	$-0.657794\pi$
$863$	−27.9473	−0.951337	−0.475669	+	0.879625i	$0.657794\pi$
$864$	24.8851	0.846609
$865$	−1.96431	−0.0667886
$866$	13.5972	0.462053
$867$	14.6441	0.497340
$868$	−1.43550	−0.0487240
$869$	3.80675	0.129135
$870$	−2.23076	−0.0756297
$871$	50.1687	1.69990
$872$	12.5709	0.425705
$873$	−12.3942	−0.419480
$874$	1.96994	0.0666341
$875$	−9.01705	−0.304832
$876$	9.01767	0.304679
$877$	−43.6670	−1.47453	−0.737266	−	0.675603i	$-0.763883\pi$
$877$	−43.6670	−1.47453	−0.737266	+	0.675603i	$0.763883\pi$
$878$	−1.99286	−0.0672558
$879$	−4.45170	−0.150152
$880$	2.78002	0.0937143
$881$	−8.47337	−0.285475	−0.142738	−	0.989761i	$-0.545590\pi$
$881$	−8.47337	−0.285475	−0.142738	+	0.989761i	$0.545590\pi$
$882$	2.64501	0.0890623
$883$	37.9899	1.27846	0.639231	−	0.769015i	$-0.279253\pi$
$883$	37.9899	1.27846	0.639231	+	0.769015i	$0.279253\pi$
$884$	9.17263	0.308509
$885$	5.40756	0.181773
$886$	−10.6917	−0.359194
$887$	28.0721	0.942569	0.471285	−	0.881981i	$-0.343790\pi$
$887$	28.0721	0.942569	0.471285	+	0.881981i	$0.343790\pi$
$888$	13.7865	0.462645
$889$	−32.9765	−1.10600
$890$	2.68395	0.0899662
$891$	−5.27740	−0.176800
$892$	24.1954	0.810122
$893$	−39.4880	−1.32142
$894$	4.82912	0.161510
$895$	−0.421835	−0.0141004
$896$	24.7903	0.828187
$897$	3.86195	0.128947
$898$	9.81764	0.327619
$899$	4.11335	0.137188
$900$	−17.6721	−0.589070
$901$	−11.7550	−0.391615
$902$	−3.61399	−0.120333
$903$	0	0
$904$	9.34950	0.310960
$905$	−3.66660	−0.121882
$906$	0.841125	0.0279445
$907$	11.2324	0.372966	0.186483	−	0.982458i	$-0.440291\pi$
$907$	11.2324	0.372966	0.186483	+	0.982458i	$0.440291\pi$
$908$	−28.4574	−0.944392
$909$	10.5747	0.350740
$910$	2.28523	0.0757547
$911$	34.3109	1.13677	0.568385	−	0.822763i	$-0.307569\pi$
$911$	34.3109	1.13677	0.568385	+	0.822763i	$0.307569\pi$
$912$	9.45851	0.313203
$913$	3.69178	0.122180
$914$	5.74318	0.189967
$915$	−4.06902	−0.134518
$916$	12.0193	0.397130
$917$	34.3356	1.13386
$918$	−2.86538	−0.0945716
$919$	42.5166	1.40249	0.701246	−	0.712919i	$-0.252627\pi$
$919$	42.5166	1.40249	0.701246	+	0.712919i	$0.252627\pi$
$920$	−0.737715	−0.0243217
$921$	−25.4964	−0.840135
$922$	3.78437	0.124632
$923$	61.5180	2.02489
$924$	9.40863	0.309521
$925$	−36.2385	−1.19152
$926$	−2.86964	−0.0943023
$927$	29.2762	0.961556
$928$	−55.3429	−1.81672
$929$	21.0903	0.691951	0.345976	−	0.938243i	$-0.387548\pi$
$929$	21.0903	0.691951	0.345976	+	0.938243i	$0.387548\pi$
$930$	−0.0811393	−0.00266066
$931$	9.92252	0.325197
$932$	−15.5556	−0.509541
$933$	23.6992	0.775876
$934$	−2.02830	−0.0663680
$935$	−1.31242	−0.0429207
$936$	19.7353	0.645068
$937$	−4.95024	−0.161717	−0.0808587	−	0.996726i	$-0.525766\pi$
$937$	−4.95024	−0.161717	−0.0808587	+	0.996726i	$0.525766\pi$
$938$	−12.1633	−0.397144
$939$	27.1236	0.885146
$940$	6.83805	0.223033
$941$	−42.0028	−1.36925	−0.684627	−	0.728894i	$-0.740035\pi$
$941$	−42.0028	−1.36925	−0.684627	+	0.728894i	$0.740035\pi$
$942$	5.08337	0.165625
$943$	−2.21238	−0.0720451
$944$	32.7211	1.06498
$945$	4.39130	0.142849
$946$	0	0
$947$	38.2562	1.24316	0.621580	−	0.783351i	$-0.286491\pi$
$947$	38.2562	1.24316	0.621580	+	0.783351i	$0.286491\pi$
$948$	−2.24283	−0.0728438
$949$	26.4707	0.859276
$950$	10.7772	0.349659
$951$	7.80331	0.253040
$952$	−4.80928	−0.155870
$953$	−20.6900	−0.670216	−0.335108	−	0.942180i	$-0.608773\pi$
$953$	−20.6900	−0.670216	−0.335108	+	0.942180i	$0.608773\pi$
$954$	−11.6951	−0.378642
$955$	−6.36530	−0.205976
$956$	17.0750	0.552245
$957$	−26.9600	−0.871492
$958$	−16.3027	−0.526718
$959$	7.66198	0.247418
$960$	−0.812490	−0.0262230
$961$	−30.8504	−0.995174
$962$	18.7136	0.603350
$963$	−35.6983	−1.15036
$964$	−43.9967	−1.41704
$965$	1.53306	0.0493510
$966$	−0.936319	−0.0301256
$967$	28.2565	0.908668	0.454334	−	0.890831i	$-0.349877\pi$
$967$	28.2565	0.908668	0.454334	+	0.890831i	$0.349877\pi$
$968$	7.07775	0.227487
$969$	−4.46527	−0.143445
$970$	−1.30889	−0.0420260
$971$	12.9184	0.414573	0.207286	−	0.978280i	$-0.433537\pi$
$971$	12.9184	0.414573	0.207286	+	0.978280i	$0.433537\pi$
$972$	27.7879	0.891298
$973$	−24.4478	−0.783760
$974$	5.05498	0.161972
$975$	21.1281	0.676642
$976$	−24.6216	−0.788120
$977$	−16.7934	−0.537268	−0.268634	−	0.963242i	$-0.586572\pi$
$977$	−16.7934	−0.537268	−0.268634	+	0.963242i	$0.586572\pi$
$978$	−5.39120	−0.172391
$979$	32.4371	1.03669
$980$	−1.71826	−0.0548878
$981$	13.6208	0.434880
$982$	8.49544	0.271100
$983$	26.2001	0.835653	0.417826	−	0.908527i	$-0.362792\pi$
$983$	26.2001	0.835653	0.417826	+	0.908527i	$0.362792\pi$
$984$	4.60467	0.146791
$985$	−5.78093	−0.184196
$986$	6.37241	0.202939
$987$	18.7688	0.597418
$988$	34.2348	1.08916
$989$	0	0
$990$	−1.30573	−0.0414988
$991$	11.1921	0.355528	0.177764	−	0.984073i	$-0.443114\pi$
$991$	11.1921	0.355528	0.177764	+	0.984073i	$0.443114\pi$
$992$	−2.01299	−0.0639124
$993$	18.9975	0.602869
$994$	−14.9149	−0.473070
$995$	−0.616450	−0.0195428
$996$	−2.17510	−0.0689206
$997$	−3.53130	−0.111837	−0.0559186	−	0.998435i	$-0.517809\pi$
$997$	−3.53130	−0.111837	−0.0559186	+	0.998435i	$0.517809\pi$
$998$	−1.87345	−0.0593032
$999$	35.9600	1.13773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.14	✓	20
43.42	odd	2		1849.2.a.r.1.7	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.14	✓	20	1.1	even	1	trivial
1849.2.a.r.1.7	yes	20	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+0.528833 q^{2} -0.931794 q^{3} -1.72034 q^{4} -0.425701 q^{5} -0.492764 q^{6} -2.15726 q^{7} -1.96744 q^{8} -2.13176 q^{9} +O(q^{10})\)	\(q+0.528833 q^{2} -0.931794 q^{3} -1.72034 q^{4} -0.425701 q^{5} -0.492764 q^{6} -2.15726 q^{7} -1.96744 q^{8} -2.13176 q^{9} -0.225125 q^{10} -2.72076 q^{11} +1.60300 q^{12} +4.70548 q^{13} -1.14083 q^{14} +0.396666 q^{15} +2.40022 q^{16} -1.13312 q^{17} -1.12735 q^{18} -4.22913 q^{19} +0.732349 q^{20} +2.01012 q^{21} -1.43883 q^{22} -0.880811 q^{23} +1.83325 q^{24} -4.81878 q^{25} +2.48842 q^{26} +4.78174 q^{27} +3.71121 q^{28} -10.6343 q^{29} +0.209770 q^{30} -0.386801 q^{31} +5.20419 q^{32} +2.53519 q^{33} -0.599233 q^{34} +0.918348 q^{35} +3.66734 q^{36} +7.52028 q^{37} -2.23650 q^{38} -4.38454 q^{39} +0.837540 q^{40} +2.51176 q^{41} +1.06302 q^{42} +4.68063 q^{44} +0.907492 q^{45} -0.465802 q^{46} +9.33716 q^{47} -2.23652 q^{48} -2.34623 q^{49} -2.54833 q^{50} +1.05584 q^{51} -8.09501 q^{52} +10.3740 q^{53} +2.52875 q^{54} +1.15823 q^{55} +4.24427 q^{56} +3.94068 q^{57} -5.62377 q^{58} +13.6325 q^{59} -0.682398 q^{60} -10.2581 q^{61} -0.204553 q^{62} +4.59876 q^{63} -2.04830 q^{64} -2.00313 q^{65} +1.34069 q^{66} +10.6618 q^{67} +1.94935 q^{68} +0.820735 q^{69} +0.485653 q^{70} +13.0737 q^{71} +4.19410 q^{72} +5.62550 q^{73} +3.97697 q^{74} +4.49011 q^{75} +7.27552 q^{76} +5.86939 q^{77} -2.31869 q^{78} -1.39915 q^{79} -1.02178 q^{80} +1.93968 q^{81} +1.32830 q^{82} -1.35689 q^{83} -3.45808 q^{84} +0.482371 q^{85} +9.90897 q^{87} +5.35293 q^{88} -11.9220 q^{89} +0.479912 q^{90} -10.1509 q^{91} +1.51529 q^{92} +0.360419 q^{93} +4.93780 q^{94} +1.80034 q^{95} -4.84924 q^{96} +5.81407 q^{97} -1.24077 q^{98} +5.80001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9} + 21 q^{11} + 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} - 9 q^{18} + 7 q^{19} + 17 q^{20} + 8 q^{21} - 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} - 58 q^{26} + 22 q^{27} + 28 q^{28} + 12 q^{29} + 24 q^{30} + 26 q^{31} - 3 q^{32} + 17 q^{33} - 54 q^{34} + 26 q^{35} + 14 q^{36} + 19 q^{37} + 5 q^{38} + 7 q^{39} - 16 q^{40} + 27 q^{41} + 52 q^{42} + 32 q^{44} - 75 q^{45} - 59 q^{46} + 45 q^{47} + 66 q^{48} + 20 q^{49} - 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} + 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} - 30 q^{61} - 72 q^{62} + 21 q^{63} - 7 q^{64} + 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} + 35 q^{69} + 80 q^{70} + 31 q^{71} + 15 q^{72} + 26 q^{73} + 87 q^{74} - 73 q^{75} + 68 q^{76} + 21 q^{77} - 50 q^{78} + 39 q^{79} + 60 q^{80} + 16 q^{81} - 45 q^{82} + 13 q^{83} + 31 q^{84} + 22 q^{85} + 61 q^{87} - 50 q^{88} + 4 q^{89} - 13 q^{90} + 25 q^{91} + 5 q^{92} - 67 q^{93} - 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} + 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9 + 21 * q^11 + 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 - 9 * q^18 + 7 * q^19 + 17 * q^20 + 8 * q^21 - 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 - 58 * q^26 + 22 * q^27 + 28 * q^28 + 12 * q^29 + 24 * q^30 + 26 * q^31 - 3 * q^32 + 17 * q^33 - 54 * q^34 + 26 * q^35 + 14 * q^36 + 19 * q^37 + 5 * q^38 + 7 * q^39 - 16 * q^40 + 27 * q^41 + 52 * q^42 + 32 * q^44 - 75 * q^45 - 59 * q^46 + 45 * q^47 + 66 * q^48 + 20 * q^49 - 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 + 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 - 30 * q^61 - 72 * q^62 + 21 * q^63 - 7 * q^64 + 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 + 35 * q^69 + 80 * q^70 + 31 * q^71 + 15 * q^72 + 26 * q^73 + 87 * q^74 - 73 * q^75 + 68 * q^76 + 21 * q^77 - 50 * q^78 + 39 * q^79 + 60 * q^80 + 16 * q^81 - 45 * q^82 + 13 * q^83 + 31 * q^84 + 22 * q^85 + 61 * q^87 - 50 * q^88 + 4 * q^89 - 13 * q^90 + 25 * q^91 + 5 * q^92 - 67 * q^93 - 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 + 35 * q^98 + 13 * q^99

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