LMFDB - Embedded newform 1849.2.a.p.1.12

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			1.12
Root			$-0.277120$ 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)

$f(q)$	$=$	$q+0.277120 q^{2} +3.15485 q^{3} -1.92320 q^{4} -0.330136 q^{5} +0.874272 q^{6} +1.68241 q^{7} -1.08720 q^{8} +6.95305 q^{9} +O(q^{10})$	\(q+0.277120 q^{2} +3.15485 q^{3} -1.92320 q^{4} -0.330136 q^{5} +0.874272 q^{6} +1.68241 q^{7} -1.08720 q^{8} +6.95305 q^{9} -0.0914873 q^{10} +4.70976 q^{11} -6.06741 q^{12} -2.45403 q^{13} +0.466229 q^{14} -1.04153 q^{15} +3.54512 q^{16} -2.98833 q^{17} +1.92683 q^{18} +3.21031 q^{19} +0.634918 q^{20} +5.30773 q^{21} +1.30517 q^{22} -4.77558 q^{23} -3.42995 q^{24} -4.89101 q^{25} -0.680063 q^{26} +12.4713 q^{27} -3.23561 q^{28} +6.73807 q^{29} -0.288628 q^{30} +7.27801 q^{31} +3.15683 q^{32} +14.8585 q^{33} -0.828128 q^{34} -0.555422 q^{35} -13.3721 q^{36} +4.80685 q^{37} +0.889642 q^{38} -7.74210 q^{39} +0.358924 q^{40} -0.0870863 q^{41} +1.47088 q^{42} -9.05782 q^{44} -2.29545 q^{45} -1.32341 q^{46} +6.96882 q^{47} +11.1843 q^{48} -4.16951 q^{49} -1.35540 q^{50} -9.42772 q^{51} +4.71961 q^{52} +1.51425 q^{53} +3.45604 q^{54} -1.55486 q^{55} -1.82911 q^{56} +10.1280 q^{57} +1.86726 q^{58} -1.29165 q^{59} +2.00307 q^{60} -6.16047 q^{61} +2.01689 q^{62} +11.6979 q^{63} -6.21543 q^{64} +0.810164 q^{65} +4.11761 q^{66} -2.64081 q^{67} +5.74717 q^{68} -15.0662 q^{69} -0.153919 q^{70} -0.474031 q^{71} -7.55935 q^{72} -6.86360 q^{73} +1.33207 q^{74} -15.4304 q^{75} -6.17408 q^{76} +7.92372 q^{77} -2.14549 q^{78} +6.15013 q^{79} -1.17037 q^{80} +18.4857 q^{81} -0.0241334 q^{82} -4.71008 q^{83} -10.2079 q^{84} +0.986555 q^{85} +21.2576 q^{87} -5.12045 q^{88} +6.17384 q^{89} -0.636116 q^{90} -4.12868 q^{91} +9.18441 q^{92} +22.9610 q^{93} +1.93120 q^{94} -1.05984 q^{95} +9.95930 q^{96} +1.51108 q^{97} -1.15546 q^{98} +32.7472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.277120	0.195954	0.0979769	−	0.995189i	$-0.468763\pi$
$2$	0.277120	0.195954	0.0979769	+	0.995189i	$0.468763\pi$
$3$	3.15485	1.82145	0.910725	−	0.413012i	$-0.135523\pi$
$3$	3.15485	1.82145	0.910725	+	0.413012i	$0.135523\pi$
$4$	−1.92320	−0.961602
$5$	−0.330136	−0.147641	−0.0738206	−	0.997272i	$-0.523519\pi$
$5$	−0.330136	−0.147641	−0.0738206	+	0.997272i	$0.523519\pi$
$6$	0.874272	0.356920
$7$	1.68241	0.635890	0.317945	−	0.948109i	$-0.397007\pi$
$7$	1.68241	0.635890	0.317945	+	0.948109i	$0.397007\pi$
$8$	−1.08720	−0.384383
$9$	6.95305	2.31768
$10$	−0.0914873	−0.0289308
$11$	4.70976	1.42004	0.710022	−	0.704179i	$-0.248685\pi$
$11$	4.70976	1.42004	0.710022	+	0.704179i	$0.248685\pi$
$12$	−6.06741	−1.75151
$13$	−2.45403	−0.680626	−0.340313	−	0.940312i	$-0.610533\pi$
$13$	−2.45403	−0.680626	−0.340313	+	0.940312i	$0.610533\pi$
$14$	0.466229	0.124605
$15$	−1.04153	−0.268921
$16$	3.54512	0.886281
$17$	−2.98833	−0.724777	−0.362388	−	0.932027i	$-0.618039\pi$
$17$	−2.98833	−0.724777	−0.362388	+	0.932027i	$0.618039\pi$
$18$	1.92683	0.454159
$19$	3.21031	0.736495	0.368248	−	0.929728i	$-0.379958\pi$
$19$	3.21031	0.736495	0.368248	+	0.929728i	$0.379958\pi$
$20$	0.634918	0.141972
$21$	5.30773	1.15824
$22$	1.30517	0.278263
$23$	−4.77558	−0.995776	−0.497888	−	0.867241i	$-0.665891\pi$
$23$	−4.77558	−0.995776	−0.497888	+	0.867241i	$0.665891\pi$
$24$	−3.42995	−0.700135
$25$	−4.89101	−0.978202
$26$	−0.680063	−0.133371
$27$	12.4713	2.40009
$28$	−3.23561	−0.611473
$29$	6.73807	1.25123	0.625614	−	0.780133i	$-0.284848\pi$
$29$	6.73807	1.25123	0.625614	+	0.780133i	$0.284848\pi$
$30$	−0.288628	−0.0526961
$31$	7.27801	1.30717	0.653585	−	0.756853i	$-0.273264\pi$
$31$	7.27801	1.30717	0.653585	+	0.756853i	$0.273264\pi$
$32$	3.15683	0.558053
$33$	14.8585	2.58654
$34$	−0.828128	−0.142023
$35$	−0.555422	−0.0938835
$36$	−13.3721	−2.22869
$37$	4.80685	0.790240	0.395120	−	0.918629i	$-0.370703\pi$
$37$	4.80685	0.790240	0.395120	+	0.918629i	$0.370703\pi$
$38$	0.889642	0.144319
$39$	−7.74210	−1.23973
$40$	0.358924	0.0567508
$41$	−0.0870863	−0.0136006	−0.00680029	−	0.999977i	$-0.502165\pi$
$41$	−0.0870863	−0.0136006	−0.00680029	+	0.999977i	$0.502165\pi$
$42$	1.47088	0.226962
$43$	0	0
$43$	0	0
$44$	−9.05782	−1.36552
$45$	−2.29545	−0.342185
$46$	−1.32341	−0.195126
$47$	6.96882	1.01651	0.508254	−	0.861207i	$-0.330291\pi$
$47$	6.96882	1.01651	0.508254	+	0.861207i	$0.330291\pi$
$48$	11.1843	1.61432
$49$	−4.16951	−0.595644
$50$	−1.35540	−0.191682
$51$	−9.42772	−1.32015
$52$	4.71961	0.654492
$53$	1.51425	0.207998	0.103999	−	0.994577i	$-0.466836\pi$
$53$	1.51425	0.207998	0.103999	+	0.994577i	$0.466836\pi$
$54$	3.45604	0.470307
$55$	−1.55486	−0.209657
$56$	−1.82911	−0.244425
$57$	10.1280	1.34149
$58$	1.86726	0.245183
$59$	−1.29165	−0.168159	−0.0840796	−	0.996459i	$-0.526795\pi$
$59$	−1.29165	−0.168159	−0.0840796	+	0.996459i	$0.526795\pi$
$60$	2.00307	0.258595
$61$	−6.16047	−0.788768	−0.394384	−	0.918946i	$-0.629042\pi$
$61$	−6.16047	−0.788768	−0.394384	+	0.918946i	$0.629042\pi$
$62$	2.01689	0.256145
$63$	11.6979	1.47379
$64$	−6.21543	−0.776928
$65$	0.810164	0.100488
$66$	4.11761	0.506842
$67$	−2.64081	−0.322626	−0.161313	−	0.986903i	$-0.551573\pi$
$67$	−2.64081	−0.322626	−0.161313	+	0.986903i	$0.551573\pi$
$68$	5.74717	0.696947
$69$	−15.0662	−1.81376
$70$	−0.153919	−0.0183968
$71$	−0.474031	−0.0562572	−0.0281286	−	0.999604i	$-0.508955\pi$
$71$	−0.474031	−0.0562572	−0.0281286	+	0.999604i	$0.508955\pi$
$72$	−7.55935	−0.890878
$73$	−6.86360	−0.803324	−0.401662	−	0.915788i	$-0.631567\pi$
$73$	−6.86360	−0.803324	−0.401662	+	0.915788i	$0.631567\pi$
$74$	1.33207	0.154851
$75$	−15.4304	−1.78175
$76$	−6.17408	−0.708215
$77$	7.92372	0.902992
$78$	−2.14549	−0.242929
$79$	6.15013	0.691943	0.345972	−	0.938245i	$-0.387549\pi$
$79$	6.15013	0.691943	0.345972	+	0.938245i	$0.387549\pi$
$80$	−1.17037	−0.130852
$81$	18.4857	2.05397
$82$	−0.0241334	−0.00266509
$83$	−4.71008	−0.516999	−0.258499	−	0.966011i	$-0.583228\pi$
$83$	−4.71008	−0.516999	−0.258499	+	0.966011i	$0.583228\pi$
$84$	−10.2079	−1.11377
$85$	0.986555	0.107007
$86$	0	0
$87$	21.2576	2.27905
$88$	−5.12045	−0.545841
$89$	6.17384	0.654426	0.327213	−	0.944951i	$-0.393891\pi$
$89$	6.17384	0.654426	0.327213	+	0.944951i	$0.393891\pi$
$90$	−0.636116	−0.0670525
$91$	−4.12868	−0.432803
$92$	9.18441	0.957541
$93$	22.9610	2.38095
$94$	1.93120	0.199188
$95$	−1.05984	−0.108737
$96$	9.95930	1.01647
$97$	1.51108	0.153427	0.0767136	−	0.997053i	$-0.475557\pi$
$97$	1.51108	0.153427	0.0767136	+	0.997053i	$0.475557\pi$
$98$	−1.15546	−0.116719
$99$	32.7472	3.29121
$100$	9.40641	0.940641
$101$	1.89488	0.188547	0.0942737	−	0.995546i	$-0.469947\pi$
$101$	1.89488	0.188547	0.0942737	+	0.995546i	$0.469947\pi$
$102$	−2.61261	−0.258687
$103$	−2.41685	−0.238139	−0.119070	−	0.992886i	$-0.537991\pi$
$103$	−2.41685	−0.238139	−0.119070	+	0.992886i	$0.537991\pi$
$104$	2.66802	0.261621
$105$	−1.75227	−0.171004
$106$	0.419628	0.0407579
$107$	−18.5194	−1.79033	−0.895167	−	0.445731i	$-0.852944\pi$
$107$	−18.5194	−1.79033	−0.895167	+	0.445731i	$0.852944\pi$
$108$	−23.9848	−2.30794
$109$	16.3101	1.56222	0.781112	−	0.624392i	$-0.214653\pi$
$109$	16.3101	1.56222	0.781112	+	0.624392i	$0.214653\pi$
$110$	−0.430883	−0.0410831
$111$	15.1649	1.43938
$112$	5.96434	0.563577
$113$	−11.9502	−1.12418	−0.562092	−	0.827075i	$-0.690003\pi$
$113$	−11.9502	−1.12418	−0.562092	+	0.827075i	$0.690003\pi$
$114$	2.80668	0.262870
$115$	1.57659	0.147018
$116$	−12.9587	−1.20318
$117$	−17.0630	−1.57748
$118$	−0.357944	−0.0329514
$119$	−5.02759	−0.460878
$120$	1.13235	0.103369
$121$	11.1818	1.01653
$122$	−1.70719	−0.154562
$123$	−0.274744	−0.0247728
$124$	−13.9971	−1.25698
$125$	3.26538	0.292064
$126$	3.24171	0.288795
$127$	−0.334512	−0.0296832	−0.0148416	−	0.999890i	$-0.504724\pi$
$127$	−0.334512	−0.0296832	−0.0148416	+	0.999890i	$0.504724\pi$
$128$	−8.03607	−0.710295
$129$	0	0
$130$	0.224513	0.0196911
$131$	−12.5235	−1.09419	−0.547093	−	0.837072i	$-0.684266\pi$
$131$	−12.5235	−1.09419	−0.547093	+	0.837072i	$0.684266\pi$
$132$	−28.5760	−2.48722
$133$	5.40104	0.468330
$134$	−0.731822	−0.0632198
$135$	−4.11721	−0.354353
$136$	3.24891	0.278592
$137$	17.0229	1.45436	0.727181	−	0.686445i	$-0.240830\pi$
$137$	17.0229	1.45436	0.727181	+	0.686445i	$0.240830\pi$
$138$	−4.17515	−0.355413
$139$	16.9845	1.44061	0.720305	−	0.693658i	$-0.244002\pi$
$139$	16.9845	1.44061	0.720305	+	0.693658i	$0.244002\pi$
$140$	1.06819	0.0902786
$141$	21.9856	1.85152
$142$	−0.131364	−0.0110238
$143$	−11.5579	−0.966520
$144$	24.6494	2.05412
$145$	−2.22448	−0.184733
$146$	−1.90204	−0.157414
$147$	−13.1542	−1.08494
$148$	−9.24455	−0.759897
$149$	20.1214	1.64841	0.824204	−	0.566293i	$-0.191623\pi$
$149$	20.1214	1.64841	0.824204	+	0.566293i	$0.191623\pi$
$150$	−4.27607	−0.349140
$151$	11.7694	0.957782	0.478891	−	0.877874i	$-0.341039\pi$
$151$	11.7694	0.957782	0.478891	+	0.877874i	$0.341039\pi$
$152$	−3.49025	−0.283096
$153$	−20.7780	−1.67980
$154$	2.19583	0.176945
$155$	−2.40273	−0.192992
$156$	14.8896	1.19212
$157$	−13.5051	−1.07783	−0.538913	−	0.842361i	$-0.681165\pi$
$157$	−13.5051	−1.07783	−0.538913	+	0.842361i	$0.681165\pi$
$158$	1.70433	0.135589
$159$	4.77721	0.378857
$160$	−1.04218	−0.0823916
$161$	−8.03446	−0.633204
$162$	5.12277	0.402483
$163$	−7.31784	−0.573178	−0.286589	−	0.958054i	$-0.592521\pi$
$163$	−7.31784	−0.573178	−0.286589	+	0.958054i	$0.592521\pi$
$164$	0.167485	0.0130784
$165$	−4.90534	−0.381880
$166$	−1.30526	−0.101308
$167$	18.2930	1.41556	0.707778	−	0.706435i	$-0.249698\pi$
$167$	18.2930	1.41556	0.707778	+	0.706435i	$0.249698\pi$
$168$	−5.77057	−0.445209
$169$	−6.97772	−0.536748
$170$	0.273394	0.0209684
$171$	22.3214	1.70696
$172$	0	0
$173$	−9.47235	−0.720169	−0.360085	−	0.932920i	$-0.617252\pi$
$173$	−9.47235	−0.720169	−0.360085	+	0.932920i	$0.617252\pi$
$174$	5.89091	0.446589
$175$	−8.22867	−0.622029
$176$	16.6967	1.25856
$177$	−4.07497	−0.306294
$178$	1.71090	0.128237
$179$	−17.2757	−1.29124	−0.645622	−	0.763657i	$-0.723402\pi$
$179$	−17.2757	−1.29124	−0.645622	+	0.763657i	$0.723402\pi$
$180$	4.41462	0.329046
$181$	−8.88341	−0.660299	−0.330149	−	0.943929i	$-0.607099\pi$
$181$	−8.88341	−0.660299	−0.330149	+	0.943929i	$0.607099\pi$
$182$	−1.14414	−0.0848094
$183$	−19.4353	−1.43670
$184$	5.19201	0.382760
$185$	−1.58691	−0.116672
$186$	6.36296	0.466555
$187$	−14.0743	−1.02922
$188$	−13.4025	−0.977476
$189$	20.9817	1.52620
$190$	−0.293702	−0.0213074
$191$	−11.8494	−0.857393	−0.428697	−	0.903448i	$-0.641027\pi$
$191$	−11.8494	−0.857393	−0.428697	+	0.903448i	$0.641027\pi$
$192$	−19.6087	−1.41514
$193$	−0.606593	−0.0436635	−0.0218318	−	0.999762i	$-0.506950\pi$
$193$	−0.606593	−0.0436635	−0.0218318	+	0.999762i	$0.506950\pi$
$194$	0.418752	0.0300646
$195$	2.55594	0.183035
$196$	8.01882	0.572773
$197$	−22.3390	−1.59159	−0.795795	−	0.605566i	$-0.792947\pi$
$197$	−22.3390	−1.59159	−0.795795	+	0.605566i	$0.792947\pi$
$198$	9.07490	0.644925
$199$	−12.2596	−0.869059	−0.434530	−	0.900658i	$-0.643085\pi$
$199$	−12.2596	−0.869059	−0.434530	+	0.900658i	$0.643085\pi$
$200$	5.31751	0.376004
$201$	−8.33135	−0.587648
$202$	0.525110	0.0369466
$203$	11.3362	0.795644
$204$	18.1314	1.26945
$205$	0.0287503	0.00200801
$206$	−0.669758	−0.0466642
$207$	−33.2048	−2.30789
$208$	−8.69985	−0.603226
$209$	15.1198	1.04586
$210$	−0.485590	−0.0335089
$211$	−22.6896	−1.56202	−0.781008	−	0.624521i	$-0.785294\pi$
$211$	−22.6896	−1.56202	−0.781008	+	0.624521i	$0.785294\pi$
$212$	−2.91220	−0.200011
$213$	−1.49550	−0.102470
$214$	−5.13209	−0.350823
$215$	0	0
$216$	−13.5587	−0.922556
$217$	12.2446	0.831216
$218$	4.51986	0.306123
$219$	−21.6536	−1.46321
$220$	2.99031	0.201607
$221$	7.33346	0.493302
$222$	4.20249	0.282053
$223$	−20.2297	−1.35468	−0.677339	−	0.735671i	$-0.736867\pi$
$223$	−20.2297	−1.35468	−0.677339	+	0.735671i	$0.736867\pi$
$224$	5.31106	0.354860
$225$	−34.0074	−2.26716
$226$	−3.31165	−0.220288
$227$	−23.3251	−1.54814	−0.774071	−	0.633099i	$-0.781783\pi$
$227$	−23.3251	−1.54814	−0.774071	+	0.633099i	$0.781783\pi$
$228$	−19.4783	−1.28998
$229$	−7.10045	−0.469211	−0.234605	−	0.972091i	$-0.575380\pi$
$229$	−7.10045	−0.469211	−0.234605	+	0.972091i	$0.575380\pi$
$230$	0.436905	0.0288086
$231$	24.9981	1.64476
$232$	−7.32563	−0.480951
$233$	−8.21943	−0.538473	−0.269236	−	0.963074i	$-0.586771\pi$
$233$	−8.21943	−0.538473	−0.269236	+	0.963074i	$0.586771\pi$
$234$	−4.72851	−0.309112
$235$	−2.30066	−0.150078
$236$	2.48412	0.161702
$237$	19.4027	1.26034
$238$	−1.39325	−0.0903108
$239$	12.2552	0.792723	0.396361	−	0.918095i	$-0.370273\pi$
$239$	12.2552	0.792723	0.396361	+	0.918095i	$0.370273\pi$
$240$	−3.69234	−0.238340
$241$	−21.7581	−1.40156	−0.700782	−	0.713376i	$-0.747165\pi$
$241$	−21.7581	−1.40156	−0.700782	+	0.713376i	$0.747165\pi$
$242$	3.09870	0.199192
$243$	20.9059	1.34111
$244$	11.8479	0.758481
$245$	1.37650	0.0879416
$246$	−0.0761371	−0.00485432
$247$	−7.87820	−0.501278
$248$	−7.91266	−0.502454
$249$	−14.8596	−0.941687
$250$	0.904902	0.0572310
$251$	23.7231	1.49739	0.748694	−	0.662916i	$-0.230681\pi$
$251$	23.7231	1.49739	0.748694	+	0.662916i	$0.230681\pi$
$252$	−22.4974	−1.41720
$253$	−22.4918	−1.41405
$254$	−0.0927002	−0.00581653
$255$	3.11243	0.194908
$256$	10.2039	0.637743
$257$	−22.1466	−1.38147	−0.690735	−	0.723108i	$-0.742713\pi$
$257$	−22.1466	−1.38147	−0.690735	+	0.723108i	$0.742713\pi$
$258$	0	0
$259$	8.08707	0.502506
$260$	−1.55811	−0.0966299
$261$	46.8501	2.89995
$262$	−3.47053	−0.214410
$263$	−16.7892	−1.03527	−0.517633	−	0.855603i	$-0.673187\pi$
$263$	−16.7892	−1.03527	−0.517633	+	0.855603i	$0.673187\pi$
$264$	−16.1542	−0.994223
$265$	−0.499906	−0.0307090
$266$	1.49674	0.0917709
$267$	19.4775	1.19200
$268$	5.07882	0.310238
$269$	22.5689	1.37605	0.688024	−	0.725688i	$-0.258478\pi$
$269$	22.5689	1.37605	0.688024	+	0.725688i	$0.258478\pi$
$270$	−1.14096	−0.0694367
$271$	−18.3993	−1.11768	−0.558840	−	0.829276i	$-0.688753\pi$
$271$	−18.3993	−1.11768	−0.558840	+	0.829276i	$0.688753\pi$
$272$	−10.5940	−0.642356
$273$	−13.0253	−0.788330
$274$	4.71739	0.284988
$275$	−23.0355	−1.38909
$276$	28.9754	1.74411
$277$	22.8424	1.37247	0.686234	−	0.727381i	$-0.259263\pi$
$277$	22.8424	1.37247	0.686234	+	0.727381i	$0.259263\pi$
$278$	4.70676	0.282293
$279$	50.6044	3.02961
$280$	0.603855	0.0360873
$281$	−8.28280	−0.494111	−0.247055	−	0.969001i	$-0.579463\pi$
$281$	−8.28280	−0.494111	−0.247055	+	0.969001i	$0.579463\pi$
$282$	6.09265	0.362812
$283$	−4.03859	−0.240069	−0.120035	−	0.992770i	$-0.538301\pi$
$283$	−4.03859	−0.240069	−0.120035	+	0.992770i	$0.538301\pi$
$284$	0.911659	0.0540970
$285$	−3.34362	−0.198059
$286$	−3.20293	−0.189393
$287$	−0.146514	−0.00864848
$288$	21.9496	1.29339
$289$	−8.06988	−0.474699
$290$	−0.616448	−0.0361991
$291$	4.76723	0.279460
$292$	13.2001	0.772478
$293$	27.8582	1.62749	0.813746	−	0.581221i	$-0.197425\pi$
$293$	27.8582	1.62749	0.813746	+	0.581221i	$0.197425\pi$
$294$	−3.64528	−0.212597
$295$	0.426421	0.0248272
$296$	−5.22600	−0.303755
$297$	58.7366	3.40824
$298$	5.57604	0.323012
$299$	11.7194	0.677752
$300$	29.6758	1.71333
$301$	0	0
$302$	3.26155	0.187681
$303$	5.97805	0.343430
$304$	11.3809	0.652741
$305$	2.03379	0.116455
$306$	−5.75801	−0.329164
$307$	25.2714	1.44231	0.721157	−	0.692772i	$-0.243611\pi$
$307$	25.2714	1.44231	0.721157	+	0.692772i	$0.243611\pi$
$308$	−15.2389	−0.868319
$309$	−7.62478	−0.433759
$310$	−0.665846	−0.0378175
$311$	−15.0489	−0.853346	−0.426673	−	0.904406i	$-0.640315\pi$
$311$	−15.0489	−0.853346	−0.426673	+	0.904406i	$0.640315\pi$
$312$	8.41720	0.476530
$313$	−8.60352	−0.486300	−0.243150	−	0.969989i	$-0.578181\pi$
$313$	−8.60352	−0.486300	−0.243150	+	0.969989i	$0.578181\pi$
$314$	−3.74255	−0.211204
$315$	−3.86188	−0.217592
$316$	−11.8279	−0.665374
$317$	1.86247	0.104607	0.0523034	−	0.998631i	$-0.483344\pi$
$317$	1.86247	0.104607	0.0523034	+	0.998631i	$0.483344\pi$
$318$	1.32386	0.0742385
$319$	31.7347	1.77680
$320$	2.05193	0.114707
$321$	−58.4257	−3.26100
$322$	−2.22651	−0.124079
$323$	−9.59346	−0.533795
$324$	−35.5518	−1.97510
$325$	12.0027	0.665790
$326$	−2.02792	−0.112316
$327$	51.4558	2.84551
$328$	0.0946802	0.00522784
$329$	11.7244	0.646387
$330$	−1.35937	−0.0748308
$331$	3.22312	0.177159	0.0885793	−	0.996069i	$-0.471767\pi$
$331$	3.22312	0.177159	0.0885793	+	0.996069i	$0.471767\pi$
$332$	9.05845	0.497147
$333$	33.4222	1.83153
$334$	5.06937	0.277384
$335$	0.871826	0.0476329
$336$	18.8166	1.02653
$337$	−4.68934	−0.255445	−0.127722	−	0.991810i	$-0.540767\pi$
$337$	−4.68934	−0.255445	−0.127722	+	0.991810i	$0.540767\pi$
$338$	−1.93367	−0.105178
$339$	−37.7011	−2.04764
$340$	−1.89735	−0.102898
$341$	34.2777	1.85624
$342$	6.18572	0.334485
$343$	−18.7917	−1.01465
$344$	0	0
$345$	4.97389	0.267785
$346$	−2.62498	−0.141120
$347$	−8.17679	−0.438953	−0.219477	−	0.975618i	$-0.570435\pi$
$347$	−8.17679	−0.438953	−0.219477	+	0.975618i	$0.570435\pi$
$348$	−40.8827	−2.19154
$349$	21.9077	1.17269	0.586345	−	0.810061i	$-0.300566\pi$
$349$	21.9077	1.17269	0.586345	+	0.810061i	$0.300566\pi$
$350$	−2.28033	−0.121889
$351$	−30.6049	−1.63357
$352$	14.8679	0.792461
$353$	11.5744	0.616044	0.308022	−	0.951379i	$-0.400333\pi$
$353$	11.5744	0.616044	0.308022	+	0.951379i	$0.400333\pi$
$354$	−1.12926	−0.0600194
$355$	0.156495	0.00830588
$356$	−11.8736	−0.629297
$357$	−15.8613	−0.839467
$358$	−4.78744	−0.253024
$359$	−13.7479	−0.725587	−0.362793	−	0.931870i	$-0.618177\pi$
$359$	−13.7479	−0.725587	−0.362793	+	0.931870i	$0.618177\pi$
$360$	2.49561	0.131530
$361$	−8.69393	−0.457575
$362$	−2.46177	−0.129388
$363$	35.2768	1.85155
$364$	7.94030	0.416185
$365$	2.26592	0.118604
$366$	−5.38593	−0.281527
$367$	−14.0031	−0.730954	−0.365477	−	0.930820i	$-0.619094\pi$
$367$	−14.0031	−0.730954	−0.365477	+	0.930820i	$0.619094\pi$
$368$	−16.9300	−0.882538
$369$	−0.605515	−0.0315218
$370$	−0.439765	−0.0228623
$371$	2.54758	0.132264
$372$	−44.1587	−2.28952
$373$	9.14022	0.473262	0.236631	−	0.971600i	$-0.423957\pi$
$373$	9.14022	0.473262	0.236631	+	0.971600i	$0.423957\pi$
$374$	−3.90028	−0.201679
$375$	10.3018	0.531980
$376$	−7.57650	−0.390728
$377$	−16.5355	−0.851619
$378$	5.81446	0.299064
$379$	−16.2291	−0.833634	−0.416817	−	0.908990i	$-0.636854\pi$
$379$	−16.2291	−0.833634	−0.416817	+	0.908990i	$0.636854\pi$
$380$	2.03828	0.104562
$381$	−1.05534	−0.0540665
$382$	−3.28371	−0.168009
$383$	4.62623	0.236389	0.118195	−	0.992990i	$-0.462289\pi$
$383$	4.62623	0.236389	0.118195	+	0.992990i	$0.462289\pi$
$384$	−25.3526	−1.29377
$385$	−2.61590	−0.133319
$386$	−0.168099	−0.00855603
$387$	0	0
$388$	−2.90612	−0.147536
$389$	8.25092	0.418338	0.209169	−	0.977879i	$-0.432924\pi$
$389$	8.25092	0.418338	0.209169	+	0.977879i	$0.432924\pi$
$390$	0.708304	0.0358664
$391$	14.2710	0.721716
$392$	4.53309	0.228956
$393$	−39.5098	−1.99301
$394$	−6.19060	−0.311878
$395$	−2.03038	−0.102159
$396$	−62.9795	−3.16484
$397$	−10.1545	−0.509638	−0.254819	−	0.966989i	$-0.582016\pi$
$397$	−10.1545	−0.509638	−0.254819	+	0.966989i	$0.582016\pi$
$398$	−3.39738	−0.170295
$399$	17.0395	0.853039
$400$	−17.3392	−0.866962
$401$	4.22369	0.210921	0.105460	−	0.994423i	$-0.466368\pi$
$401$	4.22369	0.210921	0.105460	+	0.994423i	$0.466368\pi$
$402$	−2.30879	−0.115152
$403$	−17.8605	−0.889694
$404$	−3.64424	−0.181308
$405$	−6.10280	−0.303251
$406$	3.14149	0.155909
$407$	22.6391	1.12218
$408$	10.2498	0.507442
$409$	13.6200	0.673467	0.336733	−	0.941600i	$-0.390678\pi$
$409$	13.6200	0.673467	0.336733	+	0.941600i	$0.390678\pi$
$410$	0.00796729	0.000393476 0
$411$	53.7046	2.64905
$412$	4.64809	0.228995
$413$	−2.17309	−0.106931
$414$	−9.20173	−0.452240
$415$	1.55497	0.0763303
$416$	−7.74696	−0.379826
$417$	53.5836	2.62400
$418$	4.18999	0.204939
$419$	35.2336	1.72128	0.860638	−	0.509218i	$-0.170065\pi$
$419$	35.2336	1.72128	0.860638	+	0.509218i	$0.170065\pi$
$420$	3.36998	0.164438
$421$	−11.2735	−0.549438	−0.274719	−	0.961524i	$-0.588585\pi$
$421$	−11.2735	−0.549438	−0.274719	+	0.961524i	$0.588585\pi$
$422$	−6.28775	−0.306083
$423$	48.4546	2.35594
$424$	−1.64629	−0.0799508
$425$	14.6160	0.708978
$426$	−0.414432	−0.0200793
$427$	−10.3644	−0.501570
$428$	35.6165	1.72159
$429$	−36.4634	−1.76047
$430$	0	0
$431$	−34.5576	−1.66458	−0.832291	−	0.554339i	$-0.812971\pi$
$431$	−34.5576	−1.66458	−0.832291	+	0.554339i	$0.812971\pi$
$432$	44.2121	2.12716
$433$	−19.9503	−0.958750	−0.479375	−	0.877610i	$-0.659137\pi$
$433$	−19.9503	−0.958750	−0.479375	+	0.877610i	$0.659137\pi$
$434$	3.39322	0.162880
$435$	−7.01789	−0.336482
$436$	−31.3676	−1.50224
$437$	−15.3311	−0.733384
$438$	−6.00065	−0.286722
$439$	−4.30315	−0.205378	−0.102689	−	0.994714i	$-0.532745\pi$
$439$	−4.30315	−0.205378	−0.102689	+	0.994714i	$0.532745\pi$
$440$	1.69044	0.0805887
$441$	−28.9908	−1.38051
$442$	2.03225	0.0966644
$443$	14.1621	0.672864	0.336432	−	0.941708i	$-0.390780\pi$
$443$	14.1621	0.672864	0.336432	+	0.941708i	$0.390780\pi$
$444$	−29.1651	−1.38411
$445$	−2.03821	−0.0966202
$446$	−5.60605	−0.265454
$447$	63.4798	3.00249
$448$	−10.4569	−0.494041
$449$	−37.4853	−1.76904	−0.884521	−	0.466499i	$-0.845515\pi$
$449$	−37.4853	−1.76904	−0.884521	+	0.466499i	$0.845515\pi$
$450$	−9.42415	−0.444259
$451$	−0.410155	−0.0193134
$452$	22.9827	1.08102
$453$	37.1307	1.74455
$454$	−6.46386	−0.303364
$455$	1.36303	0.0638996
$456$	−11.0112	−0.515646
$457$	−24.2546	−1.13458	−0.567291	−	0.823517i	$-0.692009\pi$
$457$	−24.2546	−1.13458	−0.567291	+	0.823517i	$0.692009\pi$
$458$	−1.96768	−0.0919436
$459$	−37.2682	−1.73953
$460$	−3.03210	−0.141372
$461$	12.7583	0.594214	0.297107	−	0.954844i	$-0.403978\pi$
$461$	12.7583	0.594214	0.297107	+	0.954844i	$0.403978\pi$
$462$	6.92749	0.322296
$463$	−21.3861	−0.993898	−0.496949	−	0.867780i	$-0.665546\pi$
$463$	−21.3861	−0.993898	−0.496949	+	0.867780i	$0.665546\pi$
$464$	23.8873	1.10894
$465$	−7.58025	−0.351526
$466$	−2.27777	−0.105516
$467$	7.31804	0.338638	0.169319	−	0.985561i	$-0.445843\pi$
$467$	7.31804	0.338638	0.169319	+	0.985561i	$0.445843\pi$
$468$	32.8157	1.51690
$469$	−4.44291	−0.205155
$470$	−0.637559	−0.0294084
$471$	−42.6066	−1.96321
$472$	1.40429	0.0646376
$473$	0	0
$474$	5.37688	0.246968
$475$	−15.7016	−0.720441
$476$	9.66908	0.443181
$477$	10.5286	0.482072
$478$	3.39617	0.155337
$479$	23.9089	1.09243	0.546214	−	0.837646i	$-0.316069\pi$
$479$	23.9089	1.09243	0.546214	+	0.837646i	$0.316069\pi$
$480$	−3.28792	−0.150072
$481$	−11.7962	−0.537859
$482$	−6.02962	−0.274642
$483$	−25.3475	−1.15335
$484$	−21.5049	−0.977495
$485$	−0.498862	−0.0226522
$486$	5.79344	0.262796
$487$	21.0344	0.953161	0.476581	−	0.879131i	$-0.341876\pi$
$487$	21.0344	0.953161	0.476581	+	0.879131i	$0.341876\pi$
$488$	6.69767	0.303189
$489$	−23.0867	−1.04402
$490$	0.381457	0.0172325
$491$	−31.2479	−1.41020	−0.705099	−	0.709109i	$-0.749097\pi$
$491$	−31.2479	−1.41020	−0.705099	+	0.709109i	$0.749097\pi$
$492$	0.528388	0.0238216
$493$	−20.1356	−0.906862
$494$	−2.18321	−0.0982273
$495$	−10.8110	−0.485919
$496$	25.8015	1.15852
$497$	−0.797514	−0.0357734
$498$	−4.11789	−0.184527
$499$	24.2579	1.08593	0.542967	−	0.839754i	$-0.317301\pi$
$499$	24.2579	1.08593	0.542967	+	0.839754i	$0.317301\pi$
$500$	−6.27999	−0.280849
$501$	57.7117	2.57837
$502$	6.57415	0.293419
$503$	−43.6157	−1.94473	−0.972364	−	0.233470i	$-0.924992\pi$
$503$	−43.6157	−1.94473	−0.972364	+	0.233470i	$0.924992\pi$
$504$	−12.7179	−0.566500
$505$	−0.625567	−0.0278374
$506$	−6.23293	−0.277088
$507$	−22.0136	−0.977660
$508$	0.643336	0.0285434
$509$	6.58138	0.291714	0.145857	−	0.989306i	$-0.453406\pi$
$509$	6.58138	0.291714	0.145857	+	0.989306i	$0.453406\pi$
$510$	0.862517	0.0381929
$511$	−11.5474	−0.510825
$512$	18.8999	0.835263
$513$	40.0366	1.76766
$514$	−6.13728	−0.270704
$515$	0.797888	0.0351591
$516$	0	0
$517$	32.8215	1.44349
$518$	2.24109	0.0984679
$519$	−29.8838	−1.31175
$520$	−0.880810	−0.0386261
$521$	12.9451	0.567137	0.283569	−	0.958952i	$-0.408482\pi$
$521$	12.9451	0.567137	0.283569	+	0.958952i	$0.408482\pi$
$522$	12.9831	0.568256
$523$	20.0741	0.877779	0.438890	−	0.898541i	$-0.355372\pi$
$523$	20.0741	0.877779	0.438890	+	0.898541i	$0.355372\pi$
$524$	24.0853	1.05217
$525$	−25.9602	−1.13299
$526$	−4.65263	−0.202864
$527$	−21.7491	−0.947406
$528$	52.6754	2.29240
$529$	−0.193876	−0.00842940
$530$	−0.138534	−0.00601754
$531$	−8.98094	−0.389740
$532$	−10.3873	−0.450347
$533$	0.213713	0.00925692
$534$	5.39762	0.233578
$535$	6.11390	0.264327
$536$	2.87109	0.124012
$537$	−54.5021	−2.35194
$538$	6.25429	0.269642
$539$	−19.6374	−0.845841
$540$	7.91823	0.340746
$541$	14.0745	0.605109	0.302554	−	0.953132i	$-0.402161\pi$
$541$	14.0745	0.605109	0.302554	+	0.953132i	$0.402161\pi$
$542$	−5.09883	−0.219013
$543$	−28.0258	−1.20270
$544$	−9.43364	−0.404464
$545$	−5.38454	−0.230648
$546$	−3.60959	−0.154476
$547$	−32.4873	−1.38906	−0.694528	−	0.719465i	$-0.744387\pi$
$547$	−32.4873	−1.38906	−0.694528	+	0.719465i	$0.744387\pi$
$548$	−32.7385	−1.39852
$549$	−42.8341	−1.82811
$550$	−6.38360	−0.272197
$551$	21.6313	0.921524
$552$	16.3800	0.697178
$553$	10.3470	0.440000
$554$	6.33010	0.268940
$555$	−5.00646	−0.212512
$556$	−32.6647	−1.38529
$557$	4.65754	0.197346	0.0986731	−	0.995120i	$-0.468540\pi$
$557$	4.65754	0.197346	0.0986731	+	0.995120i	$0.468540\pi$
$558$	14.0235	0.593662
$559$	0	0
$560$	−1.96904	−0.0832072
$561$	−44.4023	−1.87467
$562$	−2.29533	−0.0968228
$563$	−11.1458	−0.469738	−0.234869	−	0.972027i	$-0.575466\pi$
$563$	−11.1458	−0.469738	−0.234869	+	0.972027i	$0.575466\pi$
$564$	−42.2827	−1.78042
$565$	3.94520	0.165976
$566$	−1.11918	−0.0470425
$567$	31.1005	1.30610
$568$	0.515367	0.0216243
$569$	38.9746	1.63390	0.816950	−	0.576708i	$-0.195663\pi$
$569$	38.9746	1.63390	0.816950	+	0.576708i	$0.195663\pi$
$570$	−0.926586	−0.0388104
$571$	17.3762	0.727171	0.363585	−	0.931561i	$-0.381553\pi$
$571$	17.3762	0.727171	0.363585	+	0.931561i	$0.381553\pi$
$572$	22.2282	0.929408
$573$	−37.3831	−1.56170
$574$	−0.0406021	−0.00169470
$575$	23.3574	0.974071
$576$	−43.2162	−1.80067
$577$	−5.68461	−0.236654	−0.118327	−	0.992975i	$-0.537753\pi$
$577$	−5.68461	−0.236654	−0.118327	+	0.992975i	$0.537753\pi$
$578$	−2.23633	−0.0930189
$579$	−1.91371	−0.0795310
$580$	4.27813	0.177640
$581$	−7.92427	−0.328754
$582$	1.32110	0.0547612
$583$	7.13172	0.295366
$584$	7.46211	0.308784
$585$	5.63311	0.232900
$586$	7.72007	0.318913
$587$	14.1858	0.585510	0.292755	−	0.956188i	$-0.405428\pi$
$587$	14.1858	0.585510	0.292755	+	0.956188i	$0.405428\pi$
$588$	25.2981	1.04328
$589$	23.3647	0.962724
$590$	0.118170	0.00486498
$591$	−70.4762	−2.89900
$592$	17.0409	0.700375
$593$	2.99513	0.122995	0.0614977	−	0.998107i	$-0.480412\pi$
$593$	2.99513	0.122995	0.0614977	+	0.998107i	$0.480412\pi$
$594$	16.2771	0.667857
$595$	1.65979	0.0680446
$596$	−38.6975	−1.58511
$597$	−38.6771	−1.58295
$598$	3.24769	0.132808
$599$	25.6516	1.04809	0.524047	−	0.851689i	$-0.324421\pi$
$599$	25.6516	1.04809	0.524047	+	0.851689i	$0.324421\pi$
$600$	16.7759	0.684874
$601$	32.6995	1.33384	0.666921	−	0.745129i	$-0.267612\pi$
$601$	32.6995	1.33384	0.666921	+	0.745129i	$0.267612\pi$
$602$	0	0
$603$	−18.3617	−0.747745
$604$	−22.6350	−0.921006
$605$	−3.69151	−0.150081
$606$	1.65664	0.0672964
$607$	−46.4051	−1.88353	−0.941763	−	0.336278i	$-0.890832\pi$
$607$	−46.4051	−1.88353	−0.941763	+	0.336278i	$0.890832\pi$
$608$	10.1344	0.411003
$609$	35.7639	1.44923
$610$	0.563605	0.0228197
$611$	−17.1017	−0.691862
$612$	39.9604	1.61530
$613$	22.8751	0.923916	0.461958	−	0.886902i	$-0.347147\pi$
$613$	22.8751	0.923916	0.461958	+	0.886902i	$0.347147\pi$
$614$	7.00321	0.282627
$615$	0.0907027	0.00365749
$616$	−8.61467	−0.347095
$617$	36.5536	1.47159	0.735795	−	0.677204i	$-0.236808\pi$
$617$	36.5536	1.47159	0.735795	+	0.677204i	$0.236808\pi$
$618$	−2.11298	−0.0849966
$619$	3.89880	0.156706	0.0783531	−	0.996926i	$-0.475034\pi$
$619$	3.89880	0.156706	0.0783531	+	0.996926i	$0.475034\pi$
$620$	4.62095	0.185582
$621$	−59.5574	−2.38996
$622$	−4.17036	−0.167216
$623$	10.3869	0.416143
$624$	−27.4467	−1.09875
$625$	23.3770	0.935081
$626$	−2.38421	−0.0952922
$627$	47.7005	1.90497
$628$	25.9731	1.03644
$629$	−14.3644	−0.572748
$630$	−1.07021	−0.0426380
$631$	−13.8145	−0.549945	−0.274973	−	0.961452i	$-0.588669\pi$
$631$	−13.8145	−0.549945	−0.274973	+	0.961452i	$0.588669\pi$
$632$	−6.68642	−0.265971
$633$	−71.5822	−2.84514
$634$	0.516128	0.0204981
$635$	0.110435	0.00438246
$636$	−9.18755	−0.364310
$637$	10.2321	0.405411
$638$	8.79433	0.348171
$639$	−3.29596	−0.130386
$640$	2.65299	0.104869
$641$	10.5318	0.415982	0.207991	−	0.978131i	$-0.433308\pi$
$641$	10.5318	0.415982	0.207991	+	0.978131i	$0.433308\pi$
$642$	−16.1910	−0.639006
$643$	−26.8974	−1.06073	−0.530366	−	0.847769i	$-0.677945\pi$
$643$	−26.8974	−1.06073	−0.530366	+	0.847769i	$0.677945\pi$
$644$	15.4519	0.608890
$645$	0	0
$646$	−2.65854	−0.104599
$647$	43.9827	1.72914	0.864570	−	0.502513i	$-0.167591\pi$
$647$	43.9827	1.72914	0.864570	+	0.502513i	$0.167591\pi$
$648$	−20.0977	−0.789512
$649$	−6.08338	−0.238793
$650$	3.32619	0.130464
$651$	38.6297	1.51402
$652$	14.0737	0.551169
$653$	39.7369	1.55502	0.777512	−	0.628868i	$-0.216481\pi$
$653$	39.7369	1.55502	0.777512	+	0.628868i	$0.216481\pi$
$654$	14.2595	0.557589
$655$	4.13447	0.161547
$656$	−0.308732	−0.0120539
$657$	−47.7229	−1.86185
$658$	3.24907	0.126662
$659$	14.4512	0.562938	0.281469	−	0.959570i	$-0.409178\pi$
$659$	14.4512	0.562938	0.281469	+	0.959570i	$0.409178\pi$
$660$	9.43397	0.367217
$661$	41.2901	1.60600	0.803000	−	0.595979i	$-0.203236\pi$
$661$	41.2901	1.60600	0.803000	+	0.595979i	$0.203236\pi$
$662$	0.893191	0.0347149
$663$	23.1359	0.898526
$664$	5.12080	0.198726
$665$	−1.78308	−0.0691448
$666$	9.26198	0.358894
$667$	−32.1782	−1.24594
$668$	−35.1812	−1.36120
$669$	−63.8214	−2.46748
$670$	0.241601	0.00933385
$671$	−29.0143	−1.12009
$672$	16.7556	0.646361
$673$	1.37373	0.0529533	0.0264767	−	0.999649i	$-0.491571\pi$
$673$	1.37373	0.0529533	0.0264767	+	0.999649i	$0.491571\pi$
$674$	−1.29951	−0.0500553
$675$	−60.9970	−2.34778
$676$	13.4196	0.516138
$677$	36.0293	1.38472	0.692359	−	0.721554i	$-0.256572\pi$
$677$	36.0293	1.38472	0.692359	+	0.721554i	$0.256572\pi$
$678$	−10.4478	−0.401244
$679$	2.54225	0.0975628
$680$	−1.07258	−0.0411317
$681$	−73.5871	−2.81986
$682$	9.49904	0.363737
$683$	17.4967	0.669493	0.334746	−	0.942308i	$-0.391349\pi$
$683$	17.4967	0.669493	0.334746	+	0.942308i	$0.391349\pi$
$684$	−42.9287	−1.64142
$685$	−5.61986	−0.214724
$686$	−5.20755	−0.198825
$687$	−22.4008	−0.854644
$688$	0	0
$689$	−3.71601	−0.141569
$690$	1.37837	0.0524735
$691$	20.1302	0.765790	0.382895	−	0.923792i	$-0.374927\pi$
$691$	20.1302	0.765790	0.382895	+	0.923792i	$0.374927\pi$
$692$	18.2173	0.692517
$693$	55.0940	2.09285
$694$	−2.26596	−0.0860145
$695$	−5.60720	−0.212693
$696$	−23.1112	−0.876029
$697$	0.260243	0.00985739
$698$	6.07106	0.229793
$699$	−25.9310	−0.980802
$700$	15.8254	0.598144
$701$	6.32349	0.238835	0.119417	−	0.992844i	$-0.461897\pi$
$701$	6.32349	0.238835	0.119417	+	0.992844i	$0.461897\pi$
$702$	−8.48123	−0.320104
$703$	15.4315	0.582008
$704$	−29.2731	−1.10327
$705$	−7.25822	−0.273360
$706$	3.20751	0.120716
$707$	3.18796	0.119895
$708$	7.83700	0.294533
$709$	−42.3779	−1.59154	−0.795768	−	0.605601i	$-0.792933\pi$
$709$	−42.3779	−1.59154	−0.795768	+	0.605601i	$0.792933\pi$
$710$	0.0433679	0.00162757
$711$	42.7621	1.60370
$712$	−6.71220	−0.251550
$713$	−34.7567	−1.30165
$714$	−4.39548	−0.164497
$715$	3.81567	0.142698
$716$	33.2247	1.24166
$717$	38.6633	1.44391
$718$	−3.80983	−0.142181
$719$	34.3385	1.28061	0.640306	−	0.768120i	$-0.278808\pi$
$719$	34.3385	1.28061	0.640306	+	0.768120i	$0.278808\pi$
$720$	−8.13765	−0.303272
$721$	−4.06612	−0.151430
$722$	−2.40926	−0.0896635
$723$	−68.6435	−2.55288
$724$	17.0846	0.634945
$725$	−32.9560	−1.22395
$726$	9.77593	0.362819
$727$	−20.7180	−0.768389	−0.384195	−	0.923252i	$-0.625521\pi$
$727$	−20.7180	−0.768389	−0.384195	+	0.923252i	$0.625521\pi$
$728$	4.48870	0.166362
$729$	10.4976	0.388799
$730$	0.627933	0.0232408
$731$	0	0
$732$	37.3781	1.38154
$733$	20.2283	0.747148	0.373574	−	0.927600i	$-0.378132\pi$
$733$	20.2283	0.747148	0.373574	+	0.927600i	$0.378132\pi$
$734$	−3.88054	−0.143233
$735$	4.34266	0.160181
$736$	−15.0757	−0.555696
$737$	−12.4376	−0.458144
$738$	−0.167801	−0.00617682
$739$	−5.38694	−0.198162	−0.0990810	−	0.995079i	$-0.531590\pi$
$739$	−5.38694	−0.198162	−0.0990810	+	0.995079i	$0.531590\pi$
$740$	3.05195	0.112192
$741$	−24.8545	−0.913053
$742$	0.705985	0.0259175
$743$	−19.8517	−0.728290	−0.364145	−	0.931342i	$-0.618639\pi$
$743$	−19.8517	−0.728290	−0.364145	+	0.931342i	$0.618639\pi$
$744$	−24.9632	−0.915196
$745$	−6.64279	−0.243373
$746$	2.53294	0.0927375
$747$	−32.7494	−1.19824
$748$	27.0678	0.989696
$749$	−31.1571	−1.13845
$750$	2.85483	0.104244
$751$	11.1535	0.406998	0.203499	−	0.979075i	$-0.434769\pi$
$751$	11.1535	0.406998	0.203499	+	0.979075i	$0.434769\pi$
$752$	24.7053	0.900911
$753$	74.8426	2.72742
$754$	−4.58231	−0.166878
$755$	−3.88551	−0.141408
$756$	−40.3521	−1.46759
$757$	−13.8616	−0.503808	−0.251904	−	0.967752i	$-0.581057\pi$
$757$	−13.8616	−0.503808	−0.251904	+	0.967752i	$0.581057\pi$
$758$	−4.49742	−0.163354
$759$	−70.9581	−2.57562
$760$	1.15225	0.0417967
$761$	32.5418	1.17964	0.589819	−	0.807536i	$-0.299199\pi$
$761$	32.5418	1.17964	0.589819	+	0.807536i	$0.299199\pi$
$762$	−0.292455	−0.0105945
$763$	27.4402	0.993402
$764$	22.7888	0.824471
$765$	6.85956	0.248008
$766$	1.28202	0.0463214
$767$	3.16976	0.114454
$768$	32.1917	1.16162
$769$	42.7581	1.54190	0.770949	−	0.636897i	$-0.219783\pi$
$769$	42.7581	1.54190	0.770949	+	0.636897i	$0.219783\pi$
$770$	−0.724920	−0.0261243
$771$	−69.8692	−2.51628
$772$	1.16660	0.0419869
$773$	25.1380	0.904152	0.452076	−	0.891979i	$-0.350684\pi$
$773$	25.1380	0.904152	0.452076	+	0.891979i	$0.350684\pi$
$774$	0	0
$775$	−35.5968	−1.27868
$776$	−1.64285	−0.0589748
$777$	25.5134	0.915290
$778$	2.28650	0.0819749
$779$	−0.279574	−0.0100168
$780$	−4.91560	−0.176007
$781$	−2.23257	−0.0798877
$782$	3.95479	0.141423
$783$	84.0322	3.00307
$784$	−14.7814	−0.527908
$785$	4.45853	0.159132
$786$	−10.9490	−0.390537
$787$	1.54450	0.0550553	0.0275277	−	0.999621i	$-0.491237\pi$
$787$	1.54450	0.0550553	0.0275277	+	0.999621i	$0.491237\pi$
$788$	42.9625	1.53048
$789$	−52.9673	−1.88569
$790$	−0.562659	−0.0200185
$791$	−20.1051	−0.714857
$792$	−35.6027	−1.26509
$793$	15.1180	0.536856
$794$	−2.81401	−0.0998654
$795$	−1.57713	−0.0559349
$796$	23.5777	0.835689
$797$	−20.1477	−0.713667	−0.356833	−	0.934168i	$-0.616144\pi$
$797$	−20.1477	−0.713667	−0.356833	+	0.934168i	$0.616144\pi$
$798$	4.72198	0.167156
$799$	−20.8252	−0.736741
$800$	−15.4401	−0.545889
$801$	42.9270	1.51675
$802$	1.17047	0.0413307
$803$	−32.3259	−1.14076
$804$	16.0229	0.565083
$805$	2.65246	0.0934870
$806$	−4.94951	−0.174339
$807$	71.2013	2.50640
$808$	−2.06011	−0.0724745
$809$	−26.5658	−0.934004	−0.467002	−	0.884256i	$-0.654666\pi$
$809$	−26.5658	−0.934004	−0.467002	+	0.884256i	$0.654666\pi$
$810$	−1.69121	−0.0594231
$811$	40.3943	1.41843	0.709217	−	0.704990i	$-0.249049\pi$
$811$	40.3943	1.41843	0.709217	+	0.704990i	$0.249049\pi$
$812$	−21.8018	−0.765093
$813$	−58.0470	−2.03580
$814$	6.27375	0.219895
$815$	2.41588	0.0846247
$816$	−33.4224	−1.17002
$817$	0	0
$818$	3.77438	0.131968
$819$	−28.7069	−1.00310
$820$	−0.0552927	−0.00193090
$821$	20.2695	0.707409	0.353705	−	0.935357i	$-0.384922\pi$
$821$	20.2695	0.707409	0.353705	+	0.935357i	$0.384922\pi$
$822$	14.8826	0.519091
$823$	−27.9276	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$823$	−27.9276	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$824$	2.62760	0.0915366
$825$	−72.6733	−2.53016
$826$	−0.602207	−0.0209535
$827$	13.5783	0.472164	0.236082	−	0.971733i	$-0.424137\pi$
$827$	13.5783	0.472164	0.236082	+	0.971733i	$0.424137\pi$
$828$	63.8596	2.21928
$829$	−16.8903	−0.586625	−0.293313	−	0.956017i	$-0.594758\pi$
$829$	−16.8903	−0.586625	−0.293313	+	0.956017i	$0.594758\pi$
$830$	0.430913	0.0149572
$831$	72.0643	2.49988
$832$	15.2529	0.528798
$833$	12.4599	0.431709
$834$	14.8491	0.514182
$835$	−6.03918	−0.208994
$836$	−29.0784	−1.00570
$837$	90.7660	3.13733
$838$	9.76396	0.337290
$839$	−32.4216	−1.11932	−0.559659	−	0.828723i	$-0.689068\pi$
$839$	−32.4216	−1.11932	−0.559659	+	0.828723i	$0.689068\pi$
$840$	1.90507	0.0657312
$841$	16.4016	0.565573
$842$	−3.12413	−0.107664
$843$	−26.1310	−0.899998
$844$	43.6367	1.50204
$845$	2.30359	0.0792461
$846$	13.4277	0.461655
$847$	18.8123	0.646399
$848$	5.36819	0.184344
$849$	−12.7411	−0.437275
$850$	4.05038	0.138927
$851$	−22.9555	−0.786903
$852$	2.87614	0.0985351
$853$	−40.5438	−1.38819	−0.694097	−	0.719881i	$-0.744196\pi$
$853$	−40.5438	−1.38819	−0.694097	+	0.719881i	$0.744196\pi$
$854$	−2.87219	−0.0982844
$855$	−7.36910	−0.252018
$856$	20.1342	0.688174
$857$	28.2862	0.966237	0.483118	−	0.875555i	$-0.339504\pi$
$857$	28.2862	0.966237	0.483118	+	0.875555i	$0.339504\pi$
$858$	−10.1047	−0.344970
$859$	−7.11076	−0.242616	−0.121308	−	0.992615i	$-0.538709\pi$
$859$	−7.11076	−0.242616	−0.121308	+	0.992615i	$0.538709\pi$
$860$	0	0
$861$	−0.462230	−0.0157528
$862$	−9.57662	−0.326181
$863$	−10.2232	−0.348002	−0.174001	−	0.984745i	$-0.555670\pi$
$863$	−10.2232	−0.348002	−0.174001	+	0.984745i	$0.555670\pi$
$864$	39.3696	1.33938
$865$	3.12716	0.106327
$866$	−5.52864	−0.187871
$867$	−25.4592	−0.864640
$868$	−23.5488	−0.799299
$869$	28.9656	0.982590
$870$	−1.94480	−0.0659349
$871$	6.48063	0.219588
$872$	−17.7323	−0.600492
$873$	10.5066	0.355596
$874$	−4.24855	−0.143709
$875$	5.49369	0.185721
$876$	41.6443	1.40703
$877$	47.7597	1.61273	0.806365	−	0.591418i	$-0.201432\pi$
$877$	47.7597	1.61273	0.806365	+	0.591418i	$0.201432\pi$
$878$	−1.19249	−0.0402446
$879$	87.8882	2.96440
$880$	−5.51217	−0.185815
$881$	43.5236	1.46635	0.733173	−	0.680042i	$-0.238038\pi$
$881$	43.5236	1.46635	0.733173	+	0.680042i	$0.238038\pi$
$882$	−8.03394	−0.270517
$883$	−23.1874	−0.780319	−0.390159	−	0.920747i	$-0.627580\pi$
$883$	−23.1874	−0.780319	−0.390159	+	0.920747i	$0.627580\pi$
$884$	−14.1038	−0.474360
$885$	1.34529	0.0452215
$886$	3.92462	0.131850
$887$	18.2513	0.612818	0.306409	−	0.951900i	$-0.400872\pi$
$887$	18.2513	0.612818	0.306409	+	0.951900i	$0.400872\pi$
$888$	−16.4872	−0.553275
$889$	−0.562786	−0.0188752
$890$	−0.564828	−0.0189331
$891$	87.0633	2.91673
$892$	38.9058	1.30266
$893$	22.3721	0.748653
$894$	17.5916	0.588350
$895$	5.70332	0.190641
$896$	−13.5199	−0.451669
$897$	36.9730	1.23449
$898$	−10.3880	−0.346651
$899$	49.0398	1.63557
$900$	65.4032	2.18011
$901$	−4.52507	−0.150752
$902$	−0.113662	−0.00378454
$903$	0	0
$904$	12.9923	0.432117
$905$	2.93273	0.0974873
$906$	10.2897	0.341852
$907$	−55.1763	−1.83210	−0.916049	−	0.401066i	$-0.868640\pi$
$907$	−55.1763	−1.83210	−0.916049	+	0.401066i	$0.868640\pi$
$908$	44.8590	1.48870
$909$	13.1752	0.436993
$910$	0.377722	0.0125214
$911$	−3.32587	−0.110191	−0.0550955	−	0.998481i	$-0.517546\pi$
$911$	−3.32587	−0.110191	−0.0550955	+	0.998481i	$0.517546\pi$
$912$	35.9051	1.18894
$913$	−22.1833	−0.734161
$914$	−6.72145	−0.222326
$915$	6.41630	0.212116
$916$	13.6556	0.451194
$917$	−21.0697	−0.695782
$918$	−10.3278	−0.340868
$919$	32.8210	1.08267	0.541333	−	0.840808i	$-0.317920\pi$
$919$	32.8210	1.08267	0.541333	+	0.840808i	$0.317920\pi$
$920$	−1.71407	−0.0565111
$921$	79.7273	2.62710
$922$	3.53559	0.116438
$923$	1.16329	0.0382901
$924$	−48.0765	−1.58160
$925$	−23.5103	−0.773015
$926$	−5.92653	−0.194758
$927$	−16.8045	−0.551931
$928$	21.2709	0.698252
$929$	3.23330	0.106081	0.0530406	−	0.998592i	$-0.483109\pi$
$929$	3.23330	0.106081	0.0530406	+	0.998592i	$0.483109\pi$
$930$	−2.10064	−0.0688828
$931$	−13.3854	−0.438689
$932$	15.8076	0.517797
$933$	−47.4770	−1.55433
$934$	2.02798	0.0663574
$935$	4.64643	0.151955
$936$	18.5509	0.606355
$937$	4.99621	0.163219	0.0816095	−	0.996664i	$-0.473994\pi$
$937$	4.99621	0.163219	0.0816095	+	0.996664i	$0.473994\pi$
$938$	−1.23122	−0.0402008
$939$	−27.1428	−0.885771
$940$	4.42463	0.144316
$941$	31.0314	1.01159	0.505797	−	0.862652i	$-0.331198\pi$
$941$	31.0314	1.01159	0.505797	+	0.862652i	$0.331198\pi$
$942$	−11.8072	−0.384698
$943$	0.415887	0.0135431
$944$	−4.57908	−0.149036
$945$	−6.92682	−0.225329
$946$	0	0
$947$	24.8621	0.807911	0.403956	−	0.914779i	$-0.367635\pi$
$947$	24.8621	0.807911	0.403956	+	0.914779i	$0.367635\pi$
$948$	−37.3153	−1.21195
$949$	16.8435	0.546763
$950$	−4.35125	−0.141173
$951$	5.87580	0.190536
$952$	5.46599	0.177154
$953$	24.7810	0.802734	0.401367	−	0.915917i	$-0.368535\pi$
$953$	24.7810	0.802734	0.401367	+	0.915917i	$0.368535\pi$
$954$	2.91769	0.0944639
$955$	3.91191	0.126587
$956$	−23.5693	−0.762284
$957$	100.118	3.23635
$958$	6.62565	0.214065
$959$	28.6394	0.924815
$960$	6.47353	0.208932
$961$	21.9695	0.708693
$962$	−3.26896	−0.105395
$963$	−128.766	−4.14943
$964$	41.8453	1.34775
$965$	0.200258	0.00644654
$966$	−7.02430	−0.226003
$967$	44.8178	1.44124	0.720621	−	0.693329i	$-0.243857\pi$
$967$	44.8178	1.44124	0.720621	+	0.693329i	$0.243857\pi$
$968$	−12.1568	−0.390736
$969$	−30.2659	−0.972280
$970$	−0.138245	−0.00443878
$971$	2.65395	0.0851694	0.0425847	−	0.999093i	$-0.486441\pi$
$971$	2.65395	0.0851694	0.0425847	+	0.999093i	$0.486441\pi$
$972$	−40.2062	−1.28962
$973$	28.5749	0.916069
$974$	5.82907	0.186775
$975$	37.8667	1.21270
$976$	−21.8396	−0.699070
$977$	40.4223	1.29322	0.646612	−	0.762819i	$-0.276185\pi$
$977$	40.4223	1.29322	0.646612	+	0.762819i	$0.276185\pi$
$978$	−6.39779	−0.204579
$979$	29.0773	0.929314
$980$	−2.64730	−0.0845648
$981$	113.405	3.62074
$982$	−8.65942	−0.276333
$983$	−10.9910	−0.350559	−0.175279	−	0.984519i	$-0.556083\pi$
$983$	−10.9910	−0.350559	−0.175279	+	0.984519i	$0.556083\pi$
$984$	0.298701	0.00952225
$985$	7.37491	0.234984
$986$	−5.57998	−0.177703
$987$	36.9886	1.17736
$988$	15.1514	0.482030
$989$	0	0
$990$	−2.99595	−0.0952176
$991$	−8.74235	−0.277710	−0.138855	−	0.990313i	$-0.544342\pi$
$991$	−8.74235	−0.277710	−0.138855	+	0.990313i	$0.544342\pi$
$992$	22.9754	0.729470
$993$	10.1684	0.322686
$994$	−0.221007	−0.00700992
$995$	4.04733	0.128309
$996$	28.5780	0.905529
$997$	−14.7508	−0.467164	−0.233582	−	0.972337i	$-0.575045\pi$
$997$	−14.7508	−0.467164	−0.233582	+	0.972337i	$0.575045\pi$
$998$	6.72236	0.212793
$999$	59.9474	1.89665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.12	✓	20
43.42	odd	2		1849.2.a.r.1.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.12	✓	20	1.1	even	1	trivial
1849.2.a.r.1.9	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)

\(f(q)\)	\(=\)	\(q+0.277120 q^{2} +3.15485 q^{3} -1.92320 q^{4} -0.330136 q^{5} +0.874272 q^{6} +1.68241 q^{7} -1.08720 q^{8} +6.95305 q^{9} +O(q^{10})\)	\(q+0.277120 q^{2} +3.15485 q^{3} -1.92320 q^{4} -0.330136 q^{5} +0.874272 q^{6} +1.68241 q^{7} -1.08720 q^{8} +6.95305 q^{9} -0.0914873 q^{10} +4.70976 q^{11} -6.06741 q^{12} -2.45403 q^{13} +0.466229 q^{14} -1.04153 q^{15} +3.54512 q^{16} -2.98833 q^{17} +1.92683 q^{18} +3.21031 q^{19} +0.634918 q^{20} +5.30773 q^{21} +1.30517 q^{22} -4.77558 q^{23} -3.42995 q^{24} -4.89101 q^{25} -0.680063 q^{26} +12.4713 q^{27} -3.23561 q^{28} +6.73807 q^{29} -0.288628 q^{30} +7.27801 q^{31} +3.15683 q^{32} +14.8585 q^{33} -0.828128 q^{34} -0.555422 q^{35} -13.3721 q^{36} +4.80685 q^{37} +0.889642 q^{38} -7.74210 q^{39} +0.358924 q^{40} -0.0870863 q^{41} +1.47088 q^{42} -9.05782 q^{44} -2.29545 q^{45} -1.32341 q^{46} +6.96882 q^{47} +11.1843 q^{48} -4.16951 q^{49} -1.35540 q^{50} -9.42772 q^{51} +4.71961 q^{52} +1.51425 q^{53} +3.45604 q^{54} -1.55486 q^{55} -1.82911 q^{56} +10.1280 q^{57} +1.86726 q^{58} -1.29165 q^{59} +2.00307 q^{60} -6.16047 q^{61} +2.01689 q^{62} +11.6979 q^{63} -6.21543 q^{64} +0.810164 q^{65} +4.11761 q^{66} -2.64081 q^{67} +5.74717 q^{68} -15.0662 q^{69} -0.153919 q^{70} -0.474031 q^{71} -7.55935 q^{72} -6.86360 q^{73} +1.33207 q^{74} -15.4304 q^{75} -6.17408 q^{76} +7.92372 q^{77} -2.14549 q^{78} +6.15013 q^{79} -1.17037 q^{80} +18.4857 q^{81} -0.0241334 q^{82} -4.71008 q^{83} -10.2079 q^{84} +0.986555 q^{85} +21.2576 q^{87} -5.12045 q^{88} +6.17384 q^{89} -0.636116 q^{90} -4.12868 q^{91} +9.18441 q^{92} +22.9610 q^{93} +1.93120 q^{94} -1.05984 q^{95} +9.95930 q^{96} +1.51108 q^{97} -1.15546 q^{98} +32.7472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 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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