LMFDB - Embedded newform 1849.2.a.p.1.16

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			1.16
Root			$-1.59121$ 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)

$f(q)$	$=$	$q+1.59121 q^{2} +2.69692 q^{3} +0.531955 q^{4} +1.39893 q^{5} +4.29137 q^{6} +3.40103 q^{7} -2.33597 q^{8} +4.27339 q^{9} +O(q^{10})$	\(q+1.59121 q^{2} +2.69692 q^{3} +0.531955 q^{4} +1.39893 q^{5} +4.29137 q^{6} +3.40103 q^{7} -2.33597 q^{8} +4.27339 q^{9} +2.22600 q^{10} +3.00388 q^{11} +1.43464 q^{12} +4.01281 q^{13} +5.41176 q^{14} +3.77281 q^{15} -4.78093 q^{16} -5.59257 q^{17} +6.79986 q^{18} -4.72532 q^{19} +0.744168 q^{20} +9.17231 q^{21} +4.77981 q^{22} -2.45986 q^{23} -6.29993 q^{24} -3.04299 q^{25} +6.38523 q^{26} +3.43422 q^{27} +1.80919 q^{28} -0.534027 q^{29} +6.00334 q^{30} -8.47342 q^{31} -2.93554 q^{32} +8.10123 q^{33} -8.89897 q^{34} +4.75781 q^{35} +2.27325 q^{36} +4.64697 q^{37} -7.51899 q^{38} +10.8222 q^{39} -3.26786 q^{40} -2.25975 q^{41} +14.5951 q^{42} +1.59793 q^{44} +5.97818 q^{45} -3.91416 q^{46} -0.893285 q^{47} -12.8938 q^{48} +4.56699 q^{49} -4.84204 q^{50} -15.0827 q^{51} +2.13463 q^{52} -9.68307 q^{53} +5.46457 q^{54} +4.20223 q^{55} -7.94470 q^{56} -12.7438 q^{57} -0.849750 q^{58} +10.3383 q^{59} +2.00696 q^{60} -2.23249 q^{61} -13.4830 q^{62} +14.5339 q^{63} +4.89081 q^{64} +5.61365 q^{65} +12.8908 q^{66} +3.27528 q^{67} -2.97499 q^{68} -6.63406 q^{69} +7.57068 q^{70} +7.07450 q^{71} -9.98250 q^{72} +12.7026 q^{73} +7.39431 q^{74} -8.20670 q^{75} -2.51366 q^{76} +10.2163 q^{77} +17.2205 q^{78} +2.34826 q^{79} -6.68820 q^{80} -3.55833 q^{81} -3.59573 q^{82} +3.50210 q^{83} +4.87925 q^{84} -7.82363 q^{85} -1.44023 q^{87} -7.01698 q^{88} -0.915547 q^{89} +9.51254 q^{90} +13.6477 q^{91} -1.30854 q^{92} -22.8521 q^{93} -1.42141 q^{94} -6.61041 q^{95} -7.91691 q^{96} +1.91917 q^{97} +7.26705 q^{98} +12.8367 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.59121	1.12516	0.562578	−	0.826744i	$-0.309809\pi$
$2$	1.59121	1.12516	0.562578	+	0.826744i	$0.309809\pi$
$3$	2.69692	1.55707	0.778534	−	0.627602i	$-0.215964\pi$
$3$	2.69692	1.55707	0.778534	+	0.627602i	$0.215964\pi$
$4$	0.531955	0.265977
$5$	1.39893	0.625621	0.312811	−	0.949815i	$-0.398729\pi$
$5$	1.39893	0.625621	0.312811	+	0.949815i	$0.398729\pi$
$6$	4.29137	1.75195
$7$	3.40103	1.28547	0.642734	−	0.766090i	$-0.277800\pi$
$7$	3.40103	1.28547	0.642734	+	0.766090i	$0.277800\pi$
$8$	−2.33597	−0.825890
$9$	4.27339	1.42446
$10$	2.22600	0.703922
$11$	3.00388	0.905705	0.452852	−	0.891586i	$-0.350406\pi$
$11$	3.00388	0.905705	0.452852	+	0.891586i	$0.350406\pi$
$12$	1.43464	0.414145
$13$	4.01281	1.11295	0.556477	−	0.830863i	$-0.312153\pi$
$13$	4.01281	1.11295	0.556477	+	0.830863i	$0.312153\pi$
$14$	5.41176	1.44635
$15$	3.77281	0.974135
$16$	−4.78093	−1.19523
$17$	−5.59257	−1.35640	−0.678199	−	0.734878i	$-0.737239\pi$
$17$	−5.59257	−1.35640	−0.678199	+	0.734878i	$0.737239\pi$
$18$	6.79986	1.60274
$19$	−4.72532	−1.08406	−0.542032	−	0.840358i	$-0.682345\pi$
$19$	−4.72532	−1.08406	−0.542032	+	0.840358i	$0.682345\pi$
$20$	0.744168	0.166401
$21$	9.17231	2.00156
$22$	4.77981	1.01906
$23$	−2.45986	−0.512917	−0.256459	−	0.966555i	$-0.582556\pi$
$23$	−2.45986	−0.512917	−0.256459	+	0.966555i	$0.582556\pi$
$24$	−6.29993	−1.28597
$25$	−3.04299	−0.608598
$26$	6.38523	1.25225
$27$	3.43422	0.660916
$28$	1.80919	0.341905
$29$	−0.534027	−0.0991664	−0.0495832	−	0.998770i	$-0.515789\pi$
$29$	−0.534027	−0.0991664	−0.0495832	+	0.998770i	$0.515789\pi$
$30$	6.00334	1.09605
$31$	−8.47342	−1.52187	−0.760936	−	0.648828i	$-0.775260\pi$
$31$	−8.47342	−1.52187	−0.760936	+	0.648828i	$0.775260\pi$
$32$	−2.93554	−0.518934
$33$	8.10123	1.41024
$34$	−8.89897	−1.52616
$35$	4.75781	0.804216
$36$	2.27325	0.378874
$37$	4.64697	0.763957	0.381978	−	0.924171i	$-0.375243\pi$
$37$	4.64697	0.763957	0.381978	+	0.924171i	$0.375243\pi$
$38$	−7.51899	−1.21974
$39$	10.8222	1.73295
$40$	−3.26786	−0.516695
$41$	−2.25975	−0.352913	−0.176456	−	0.984308i	$-0.556463\pi$
$41$	−2.25975	−0.352913	−0.176456	+	0.984308i	$0.556463\pi$
$42$	14.5951	2.25207
$43$	0	0
$43$	0	0
$44$	1.59793	0.240897
$45$	5.97818	0.891174
$46$	−3.91416	−0.577112
$47$	−0.893285	−0.130299	−0.0651495	−	0.997876i	$-0.520752\pi$
$47$	−0.893285	−0.130299	−0.0651495	+	0.997876i	$0.520752\pi$
$48$	−12.8938	−1.86106
$49$	4.56699	0.652427
$50$	−4.84204	−0.684768
$51$	−15.0827	−2.11200
$52$	2.13463	0.296020
$53$	−9.68307	−1.33007	−0.665036	−	0.746811i	$-0.731584\pi$
$53$	−9.68307	−1.33007	−0.665036	+	0.746811i	$0.731584\pi$
$54$	5.46457	0.743634
$55$	4.20223	0.566628
$56$	−7.94470	−1.06166
$57$	−12.7438	−1.68796
$58$	−0.849750	−0.111578
$59$	10.3383	1.34593	0.672966	−	0.739673i	$-0.265020\pi$
$59$	10.3383	1.34593	0.672966	+	0.739673i	$0.265020\pi$
$60$	2.00696	0.259098
$61$	−2.23249	−0.285841	−0.142921	−	0.989734i	$-0.545649\pi$
$61$	−2.23249	−0.285841	−0.142921	+	0.989734i	$0.545649\pi$
$62$	−13.4830	−1.71234
$63$	14.5339	1.83110
$64$	4.89081	0.611351
$65$	5.61365	0.696288
$66$	12.8908	1.58675
$67$	3.27528	0.400139	0.200070	−	0.979782i	$-0.435883\pi$
$67$	3.27528	0.400139	0.200070	+	0.979782i	$0.435883\pi$
$68$	−2.97499	−0.360771
$69$	−6.63406	−0.798647
$70$	7.57068	0.904869
$71$	7.07450	0.839589	0.419795	−	0.907619i	$-0.362102\pi$
$71$	7.07450	0.839589	0.419795	+	0.907619i	$0.362102\pi$
$72$	−9.98250	−1.17645
$73$	12.7026	1.48673	0.743366	−	0.668885i	$-0.233228\pi$
$73$	12.7026	1.48673	0.743366	+	0.668885i	$0.233228\pi$
$74$	7.39431	0.859571
$75$	−8.20670	−0.947628
$76$	−2.51366	−0.288336
$77$	10.2163	1.16425
$78$	17.2205	1.94983
$79$	2.34826	0.264200	0.132100	−	0.991236i	$-0.457828\pi$
$79$	2.34826	0.264200	0.132100	+	0.991236i	$0.457828\pi$
$80$	−6.68820	−0.747764
$81$	−3.55833	−0.395370
$82$	−3.59573	−0.397082
$83$	3.50210	0.384406	0.192203	−	0.981355i	$-0.438437\pi$
$83$	3.50210	0.384406	0.192203	+	0.981355i	$0.438437\pi$
$84$	4.87925	0.532370
$85$	−7.82363	−0.848592
$86$	0	0
$87$	−1.44023	−0.154409
$88$	−7.01698	−0.748013
$89$	−0.915547	−0.0970477	−0.0485239	−	0.998822i	$-0.515452\pi$
$89$	−0.915547	−0.0970477	−0.0485239	+	0.998822i	$0.515452\pi$
$90$	9.51254	1.00271
$91$	13.6477	1.43067
$92$	−1.30854	−0.136424
$93$	−22.8521	−2.36966
$94$	−1.42141	−0.146607
$95$	−6.61041	−0.678213
$96$	−7.91691	−0.808016
$97$	1.91917	0.194862	0.0974309	−	0.995242i	$-0.468938\pi$
$97$	1.91917	0.194862	0.0974309	+	0.995242i	$0.468938\pi$
$98$	7.26705	0.734083
$99$	12.8367	1.29014
$100$	−1.61873	−0.161873
$101$	7.67146	0.763339	0.381670	−	0.924299i	$-0.375349\pi$
$101$	7.67146	0.763339	0.381670	+	0.924299i	$0.375349\pi$
$102$	−23.9998	−2.37634
$103$	−0.699112	−0.0688855	−0.0344428	−	0.999407i	$-0.510966\pi$
$103$	−0.699112	−0.0688855	−0.0344428	+	0.999407i	$0.510966\pi$
$104$	−9.37381	−0.919178
$105$	12.8314	1.25222
$106$	−15.4078	−1.49654
$107$	12.7477	1.23236	0.616181	−	0.787604i	$-0.288679\pi$
$107$	12.7477	1.23236	0.616181	+	0.787604i	$0.288679\pi$
$108$	1.82685	0.175789
$109$	−7.83181	−0.750151	−0.375076	−	0.926994i	$-0.622383\pi$
$109$	−7.83181	−0.750151	−0.375076	+	0.926994i	$0.622383\pi$
$110$	6.68663	0.637545
$111$	12.5325	1.18953
$112$	−16.2601	−1.53643
$113$	7.54717	0.709978	0.354989	−	0.934871i	$-0.384485\pi$
$113$	7.54717	0.709978	0.354989	+	0.934871i	$0.384485\pi$
$114$	−20.2781	−1.89922
$115$	−3.44118	−0.320892
$116$	−0.284078	−0.0263760
$117$	17.1483	1.58536
$118$	16.4504	1.51439
$119$	−19.0205	−1.74361
$120$	−8.81317	−0.804529
$121$	−1.97669	−0.179699
$122$	−3.55237	−0.321616
$123$	−6.09435	−0.549509
$124$	−4.50747	−0.404783
$125$	−11.2516	−1.00637
$126$	23.1265	2.06027
$127$	−3.05508	−0.271094	−0.135547	−	0.990771i	$-0.543279\pi$
$127$	−3.05508	−0.271094	−0.135547	+	0.990771i	$0.543279\pi$
$128$	13.6534	1.20680
$129$	0	0
$130$	8.93251	0.783433
$131$	10.9528	0.956948	0.478474	−	0.878102i	$-0.341190\pi$
$131$	10.9528	0.956948	0.478474	+	0.878102i	$0.341190\pi$
$132$	4.30949	0.375093
$133$	−16.0710	−1.39353
$134$	5.21166	0.450219
$135$	4.80424	0.413483
$136$	13.0641	1.12024
$137$	18.3639	1.56894	0.784468	−	0.620170i	$-0.212936\pi$
$137$	18.3639	1.56894	0.784468	+	0.620170i	$0.212936\pi$
$138$	−10.5562	−0.898603
$139$	−20.6344	−1.75019	−0.875093	−	0.483955i	$-0.839200\pi$
$139$	−20.6344	−1.75019	−0.875093	+	0.483955i	$0.839200\pi$
$140$	2.53094	0.213903
$141$	−2.40912	−0.202884
$142$	11.2570	0.944669
$143$	12.0540	1.00801
$144$	−20.4308	−1.70256
$145$	−0.747068	−0.0620406
$146$	20.2126	1.67281
$147$	12.3168	1.01587
$148$	2.47198	0.203195
$149$	−11.5188	−0.943658	−0.471829	−	0.881690i	$-0.656406\pi$
$149$	−11.5188	−0.943658	−0.471829	+	0.881690i	$0.656406\pi$
$150$	−13.0586	−1.06623
$151$	−19.1074	−1.55494	−0.777470	−	0.628920i	$-0.783497\pi$
$151$	−19.1074	−1.55494	−0.777470	+	0.628920i	$0.783497\pi$
$152$	11.0382	0.895318
$153$	−23.8992	−1.93214
$154$	16.2563	1.30997
$155$	−11.8537	−0.952115
$156$	5.75694	0.460924
$157$	2.95930	0.236178	0.118089	−	0.993003i	$-0.462323\pi$
$157$	2.95930	0.236178	0.118089	+	0.993003i	$0.462323\pi$
$158$	3.73658	0.297267
$159$	−26.1145	−2.07101
$160$	−4.10661	−0.324656
$161$	−8.36607	−0.659338
$162$	−5.66206	−0.444854
$163$	−12.5553	−0.983410	−0.491705	−	0.870762i	$-0.663626\pi$
$163$	−12.5553	−0.983410	−0.491705	+	0.870762i	$0.663626\pi$
$164$	−1.20208	−0.0938668
$165$	11.3331	0.882279
$166$	5.57259	0.432517
$167$	5.30051	0.410166	0.205083	−	0.978745i	$-0.434254\pi$
$167$	5.30051	0.410166	0.205083	+	0.978745i	$0.434254\pi$
$168$	−21.4262	−1.65307
$169$	3.10266	0.238666
$170$	−12.4490	−0.954798
$171$	−20.1931	−1.54421
$172$	0	0
$173$	−4.59027	−0.348992	−0.174496	−	0.984658i	$-0.555830\pi$
$173$	−4.59027	−0.348992	−0.174496	+	0.984658i	$0.555830\pi$
$174$	−2.29171	−0.173734
$175$	−10.3493	−0.782333
$176$	−14.3614	−1.08253
$177$	27.8816	2.09571
$178$	−1.45683	−0.109194
$179$	−1.69814	−0.126925	−0.0634624	−	0.997984i	$-0.520214\pi$
$179$	−1.69814	−0.126925	−0.0634624	+	0.997984i	$0.520214\pi$
$180$	3.18012	0.237032
$181$	21.6860	1.61191	0.805953	−	0.591979i	$-0.201653\pi$
$181$	21.6860	1.61191	0.805953	+	0.591979i	$0.201653\pi$
$182$	21.7164	1.60972
$183$	−6.02086	−0.445075
$184$	5.74617	0.423613
$185$	6.50079	0.477948
$186$	−36.3626	−2.66624
$187$	−16.7994	−1.22850
$188$	−0.475187	−0.0346566
$189$	11.6799	0.849586
$190$	−10.5186	−0.763096
$191$	−14.5080	−1.04976	−0.524881	−	0.851176i	$-0.675890\pi$
$191$	−14.5080	−1.04976	−0.524881	+	0.851176i	$0.675890\pi$
$192$	13.1901	0.951916
$193$	−12.7206	−0.915647	−0.457824	−	0.889043i	$-0.651371\pi$
$193$	−12.7206	−0.915647	−0.457824	+	0.889043i	$0.651371\pi$
$194$	3.05380	0.219250
$195$	15.1396	1.08417
$196$	2.42943	0.173531
$197$	15.2633	1.08746	0.543732	−	0.839259i	$-0.317011\pi$
$197$	15.2633	1.08746	0.543732	+	0.839259i	$0.317011\pi$
$198$	20.4260	1.45161
$199$	−19.2754	−1.36640	−0.683199	−	0.730232i	$-0.739412\pi$
$199$	−19.2754	−1.36640	−0.683199	+	0.730232i	$0.739412\pi$
$200$	7.10834	0.502635
$201$	8.83317	0.623044
$202$	12.2069	0.858876
$203$	−1.81624	−0.127475
$204$	−8.02333	−0.561745
$205$	−3.16123	−0.220790
$206$	−1.11243	−0.0775070
$207$	−10.5119	−0.730631
$208$	−19.1850	−1.33024
$209$	−14.1943	−0.981842
$210$	20.4175	1.40894
$211$	21.3719	1.47130	0.735649	−	0.677362i	$-0.236877\pi$
$211$	21.3719	1.47130	0.735649	+	0.677362i	$0.236877\pi$
$212$	−5.15095	−0.353769
$213$	19.0794	1.30730
$214$	20.2842	1.38660
$215$	0	0
$216$	−8.02224	−0.545844
$217$	−28.8183	−1.95632
$218$	−12.4621	−0.844038
$219$	34.2580	2.31494
$220$	2.23539	0.150710
$221$	−22.4419	−1.50961
$222$	19.9419	1.33841
$223$	13.3355	0.893012	0.446506	−	0.894781i	$-0.352668\pi$
$223$	13.3355	0.893012	0.446506	+	0.894781i	$0.352668\pi$
$224$	−9.98384	−0.667073
$225$	−13.0039	−0.866924
$226$	12.0091	0.798836
$227$	−5.05601	−0.335579	−0.167790	−	0.985823i	$-0.553663\pi$
$227$	−5.05601	−0.335579	−0.167790	+	0.985823i	$0.553663\pi$
$228$	−6.77914	−0.448959
$229$	22.0663	1.45818	0.729092	−	0.684415i	$-0.239942\pi$
$229$	22.0663	1.45818	0.729092	+	0.684415i	$0.239942\pi$
$230$	−5.47565	−0.361054
$231$	27.5525	1.81282
$232$	1.24747	0.0819006
$233$	−9.68779	−0.634668	−0.317334	−	0.948314i	$-0.602788\pi$
$233$	−9.68779	−0.634668	−0.317334	+	0.948314i	$0.602788\pi$
$234$	27.2866	1.78378
$235$	−1.24965	−0.0815179
$236$	5.49951	0.357988
$237$	6.33308	0.411378
$238$	−30.2656	−1.96183
$239$	11.0134	0.712395	0.356197	−	0.934411i	$-0.384073\pi$
$239$	11.0134	0.712395	0.356197	+	0.934411i	$0.384073\pi$
$240$	−18.0376	−1.16432
$241$	13.5694	0.874081	0.437040	−	0.899442i	$-0.356027\pi$
$241$	13.5694	0.874081	0.437040	+	0.899442i	$0.356027\pi$
$242$	−3.14533	−0.202190
$243$	−19.8992	−1.27653
$244$	−1.18758	−0.0760273
$245$	6.38891	0.408173
$246$	−9.69741	−0.618284
$247$	−18.9618	−1.20651
$248$	19.7937	1.25690
$249$	9.44490	0.598546
$250$	−17.9037	−1.13233
$251$	19.8413	1.25237	0.626186	−	0.779673i	$-0.284615\pi$
$251$	19.8413	1.25237	0.626186	+	0.779673i	$0.284615\pi$
$252$	7.73138	0.487031
$253$	−7.38914	−0.464551
$254$	−4.86127	−0.305023
$255$	−21.0997	−1.32132
$256$	11.9438	0.746488
$257$	−15.2857	−0.953497	−0.476749	−	0.879040i	$-0.658185\pi$
$257$	−15.2857	−0.953497	−0.476749	+	0.879040i	$0.658185\pi$
$258$	0	0
$259$	15.8045	0.982042
$260$	2.98621	0.185197
$261$	−2.28210	−0.141259
$262$	17.4282	1.07672
$263$	−19.2163	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$263$	−19.2163	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$264$	−18.9243	−1.16471
$265$	−13.5460	−0.832122
$266$	−25.5723	−1.56794
$267$	−2.46916	−0.151110
$268$	1.74230	0.106428
$269$	−20.6980	−1.26198	−0.630988	−	0.775792i	$-0.717350\pi$
$269$	−20.6980	−1.26198	−0.630988	+	0.775792i	$0.717350\pi$
$270$	7.64456	0.465233
$271$	−19.8545	−1.20607	−0.603037	−	0.797713i	$-0.706043\pi$
$271$	−19.8545	−1.20607	−0.603037	+	0.797713i	$0.706043\pi$
$272$	26.7377	1.62121
$273$	36.8067	2.22765
$274$	29.2209	1.76530
$275$	−9.14078	−0.551210
$276$	−3.52902	−0.212422
$277$	3.79499	0.228019	0.114009	−	0.993480i	$-0.463631\pi$
$277$	3.79499	0.228019	0.114009	+	0.993480i	$0.463631\pi$
$278$	−32.8337	−1.96923
$279$	−36.2102	−2.16785
$280$	−11.1141	−0.664194
$281$	6.63552	0.395842	0.197921	−	0.980218i	$-0.436581\pi$
$281$	6.63552	0.395842	0.197921	+	0.980218i	$0.436581\pi$
$282$	−3.83342	−0.228277
$283$	−27.9623	−1.66218	−0.831092	−	0.556134i	$-0.812284\pi$
$283$	−27.9623	−1.66218	−0.831092	+	0.556134i	$0.812284\pi$
$284$	3.76331	0.223312
$285$	−17.8277	−1.05602
$286$	19.1805	1.13417
$287$	−7.68546	−0.453658
$288$	−12.5447	−0.739202
$289$	14.2769	0.839816
$290$	−1.18874	−0.0698054
$291$	5.17584	0.303413
$292$	6.75723	0.395437
$293$	6.14946	0.359255	0.179628	−	0.983735i	$-0.442511\pi$
$293$	6.14946	0.359255	0.179628	+	0.983735i	$0.442511\pi$
$294$	19.5987	1.14302
$295$	14.4626	0.842044
$296$	−10.8552	−0.630945
$297$	10.3160	0.598595
$298$	−18.3289	−1.06176
$299$	−9.87097	−0.570853
$300$	−4.36559	−0.252048
$301$	0	0
$302$	−30.4040	−1.74955
$303$	20.6893	1.18857
$304$	22.5915	1.29571
$305$	−3.12311	−0.178829
$306$	−38.0287	−2.17396
$307$	16.6202	0.948566	0.474283	−	0.880372i	$-0.342707\pi$
$307$	16.6202	0.948566	0.474283	+	0.880372i	$0.342707\pi$
$308$	5.43460	0.309665
$309$	−1.88545	−0.107259
$310$	−18.8618	−1.07128
$311$	29.7732	1.68828	0.844141	−	0.536121i	$-0.180111\pi$
$311$	29.7732	1.68828	0.844141	+	0.536121i	$0.180111\pi$
$312$	−25.2804	−1.43122
$313$	6.59385	0.372707	0.186353	−	0.982483i	$-0.440333\pi$
$313$	6.59385	0.372707	0.186353	+	0.982483i	$0.440333\pi$
$314$	4.70888	0.265737
$315$	20.3319	1.14558
$316$	1.24917	0.0702713
$317$	5.30076	0.297720	0.148860	−	0.988858i	$-0.452440\pi$
$317$	5.30076	0.297720	0.148860	+	0.988858i	$0.452440\pi$
$318$	−41.5537	−2.33021
$319$	−1.60415	−0.0898154
$320$	6.84191	0.382474
$321$	34.3794	1.91887
$322$	−13.3122	−0.741859
$323$	26.4267	1.47042
$324$	−1.89287	−0.105160
$325$	−12.2109	−0.677341
$326$	−19.9782	−1.10649
$327$	−21.1218	−1.16804
$328$	5.27870	0.291467
$329$	−3.03809	−0.167495
$330$	18.0333	0.992702
$331$	29.5150	1.62229	0.811146	−	0.584843i	$-0.198844\pi$
$331$	29.5150	1.62229	0.811146	+	0.584843i	$0.198844\pi$
$332$	1.86296	0.102243
$333$	19.8583	1.08823
$334$	8.43424	0.461501
$335$	4.58189	0.250336
$336$	−43.8522	−2.39233
$337$	−30.3713	−1.65443	−0.827215	−	0.561885i	$-0.810076\pi$
$337$	−30.3713	−1.65443	−0.827215	+	0.561885i	$0.810076\pi$
$338$	4.93699	0.268537
$339$	20.3541	1.10548
$340$	−4.16181	−0.225706
$341$	−25.4532	−1.37837
$342$	−32.1315	−1.73748
$343$	−8.27473	−0.446793
$344$	0	0
$345$	−9.28060	−0.499651
$346$	−7.30410	−0.392671
$347$	9.38404	0.503762	0.251881	−	0.967758i	$-0.418951\pi$
$347$	9.38404	0.503762	0.251881	+	0.967758i	$0.418951\pi$
$348$	−0.766137	−0.0410692
$349$	10.8935	0.583114	0.291557	−	0.956554i	$-0.405827\pi$
$349$	10.8935	0.583114	0.291557	+	0.956554i	$0.405827\pi$
$350$	−16.4679	−0.880247
$351$	13.7809	0.735569
$352$	−8.81800	−0.470001
$353$	28.7437	1.52987	0.764935	−	0.644107i	$-0.222771\pi$
$353$	28.7437	1.52987	0.764935	+	0.644107i	$0.222771\pi$
$354$	44.3655	2.35800
$355$	9.89675	0.525265
$356$	−0.487029	−0.0258125
$357$	−51.2968	−2.71491
$358$	−2.70210	−0.142810
$359$	−9.21850	−0.486534	−0.243267	−	0.969959i	$-0.578219\pi$
$359$	−9.21850	−0.486534	−0.243267	+	0.969959i	$0.578219\pi$
$360$	−13.9648	−0.736012
$361$	3.32869	0.175194
$362$	34.5070	1.81365
$363$	−5.33098	−0.279804
$364$	7.25995	0.380525
$365$	17.7701	0.930132
$366$	−9.58046	−0.500779
$367$	−18.3320	−0.956921	−0.478460	−	0.878109i	$-0.658805\pi$
$367$	−18.3320	−0.956921	−0.478460	+	0.878109i	$0.658805\pi$
$368$	11.7604	0.613056
$369$	−9.65676	−0.502711
$370$	10.3441	0.537766
$371$	−32.9324	−1.70976
$372$	−12.1563	−0.630275
$373$	27.5423	1.42609	0.713043	−	0.701121i	$-0.247317\pi$
$373$	27.5423	1.42609	0.713043	+	0.701121i	$0.247317\pi$
$374$	−26.7314	−1.38225
$375$	−30.3447	−1.56699
$376$	2.08669	0.107613
$377$	−2.14295	−0.110368
$378$	18.5852	0.955917
$379$	9.09979	0.467425	0.233713	−	0.972306i	$-0.424913\pi$
$379$	9.09979	0.467425	0.233713	+	0.972306i	$0.424913\pi$
$380$	−3.51644	−0.180389
$381$	−8.23930	−0.422112
$382$	−23.0853	−1.18115
$383$	−18.1462	−0.927225	−0.463613	−	0.886038i	$-0.653447\pi$
$383$	−18.1462	−0.927225	−0.463613	+	0.886038i	$0.653447\pi$
$384$	36.8221	1.87907
$385$	14.2919	0.728382
$386$	−20.2411	−1.03025
$387$	0	0
$388$	1.02091	0.0518288
$389$	12.4180	0.629618	0.314809	−	0.949155i	$-0.398060\pi$
$389$	12.4180	0.629618	0.314809	+	0.949155i	$0.398060\pi$
$390$	24.0903	1.21986
$391$	13.7570	0.695720
$392$	−10.6684	−0.538834
$393$	29.5387	1.49003
$394$	24.2871	1.22357
$395$	3.28506	0.165289
$396$	6.82857	0.343148
$397$	32.5998	1.63614	0.818068	−	0.575121i	$-0.195045\pi$
$397$	32.5998	1.63614	0.818068	+	0.575121i	$0.195045\pi$
$398$	−30.6713	−1.53741
$399$	−43.3421	−2.16982
$400$	14.5483	0.727417
$401$	−28.4192	−1.41919	−0.709594	−	0.704611i	$-0.751122\pi$
$401$	−28.4192	−1.41919	−0.709594	+	0.704611i	$0.751122\pi$
$402$	14.0554	0.701022
$403$	−34.0022	−1.69377
$404$	4.08087	0.203031
$405$	−4.97787	−0.247352
$406$	−2.89002	−0.143430
$407$	13.9589	0.691919
$408$	35.2328	1.74428
$409$	−23.6424	−1.16904	−0.584521	−	0.811379i	$-0.698718\pi$
$409$	−23.6424	−1.16904	−0.584521	+	0.811379i	$0.698718\pi$
$410$	−5.03019	−0.248423
$411$	49.5260	2.44294
$412$	−0.371896	−0.0183220
$413$	35.1609	1.73015
$414$	−16.7267	−0.822074
$415$	4.89920	0.240493
$416$	−11.7798	−0.577550
$417$	−55.6493	−2.72516
$418$	−22.5862	−1.10473
$419$	3.07048	0.150003	0.0750014	−	0.997183i	$-0.476104\pi$
$419$	3.07048	0.150003	0.0750014	+	0.997183i	$0.476104\pi$
$420$	6.82574	0.333062
$421$	7.88852	0.384463	0.192231	−	0.981350i	$-0.438428\pi$
$421$	7.88852	0.384463	0.192231	+	0.981350i	$0.438428\pi$
$422$	34.0071	1.65544
$423$	−3.81735	−0.185606
$424$	22.6194	1.09849
$425$	17.0181	0.825501
$426$	30.3593	1.47091
$427$	−7.59277	−0.367440
$428$	6.78117	0.327780
$429$	32.5087	1.56954
$430$	0	0
$431$	25.7332	1.23952	0.619762	−	0.784790i	$-0.287229\pi$
$431$	25.7332	1.23952	0.619762	+	0.784790i	$0.287229\pi$
$432$	−16.4188	−0.789949
$433$	−9.31965	−0.447874	−0.223937	−	0.974604i	$-0.571891\pi$
$433$	−9.31965	−0.447874	−0.223937	+	0.974604i	$0.571891\pi$
$434$	−45.8561	−2.20116
$435$	−2.01478	−0.0966014
$436$	−4.16617	−0.199523
$437$	11.6237	0.556035
$438$	54.5118	2.60467
$439$	−11.5668	−0.552051	−0.276025	−	0.961150i	$-0.589017\pi$
$439$	−11.5668	−0.552051	−0.276025	+	0.961150i	$0.589017\pi$
$440$	−9.81628	−0.467973
$441$	19.5165	0.929358
$442$	−35.7099	−1.69855
$443$	38.9254	1.84940	0.924701	−	0.380694i	$-0.124315\pi$
$443$	38.9254	1.84940	0.924701	+	0.380694i	$0.124315\pi$
$444$	6.66673	0.316389
$445$	−1.28079	−0.0607151
$446$	21.2196	1.00478
$447$	−31.0653	−1.46934
$448$	16.6338	0.785872
$449$	−16.2249	−0.765701	−0.382850	−	0.923810i	$-0.625057\pi$
$449$	−16.2249	−0.765701	−0.382850	+	0.923810i	$0.625057\pi$
$450$	−20.6919	−0.975426
$451$	−6.78801	−0.319635
$452$	4.01475	0.188838
$453$	−51.5312	−2.42115
$454$	−8.04519	−0.377579
$455$	19.0922	0.895055
$456$	29.7692	1.39407
$457$	32.0054	1.49715	0.748574	−	0.663051i	$-0.230739\pi$
$457$	32.0054	1.49715	0.748574	+	0.663051i	$0.230739\pi$
$458$	35.1122	1.64069
$459$	−19.2061	−0.896465
$460$	−1.83055	−0.0853499
$461$	31.3240	1.45890	0.729451	−	0.684033i	$-0.239775\pi$
$461$	31.3240	1.45890	0.729451	+	0.684033i	$0.239775\pi$
$462$	43.8419	2.03971
$463$	19.9993	0.929448	0.464724	−	0.885456i	$-0.346154\pi$
$463$	19.9993	0.929448	0.464724	+	0.885456i	$0.346154\pi$
$464$	2.55315	0.118527
$465$	−31.9686	−1.48251
$466$	−15.4153	−0.714101
$467$	3.31678	0.153482	0.0767412	−	0.997051i	$-0.475548\pi$
$467$	3.31678	0.153482	0.0767412	+	0.997051i	$0.475548\pi$
$468$	9.12211	0.421670
$469$	11.1393	0.514366
$470$	−1.98845	−0.0917203
$471$	7.98101	0.367746
$472$	−24.1500	−1.11159
$473$	0	0
$474$	10.0773	0.462865
$475$	14.3791	0.659759
$476$	−10.1180	−0.463760
$477$	−41.3795	−1.89464
$478$	17.5246	0.801555
$479$	21.1522	0.966468	0.483234	−	0.875491i	$-0.339462\pi$
$479$	21.1522	0.966468	0.483234	+	0.875491i	$0.339462\pi$
$480$	−11.0752	−0.505512
$481$	18.6474	0.850249
$482$	21.5918	0.983478
$483$	−22.5626	−1.02663
$484$	−1.05151	−0.0477959
$485$	2.68478	0.121910
$486$	−31.6639	−1.43630
$487$	19.1829	0.869260	0.434630	−	0.900609i	$-0.356879\pi$
$487$	19.1829	0.869260	0.434630	+	0.900609i	$0.356879\pi$
$488$	5.21504	0.236074
$489$	−33.8608	−1.53124
$490$	10.1661	0.459258
$491$	−11.3550	−0.512442	−0.256221	−	0.966618i	$-0.582478\pi$
$491$	−11.3550	−0.512442	−0.256221	+	0.966618i	$0.582478\pi$
$492$	−3.24192	−0.146157
$493$	2.98659	0.134509
$494$	−30.1723	−1.35752
$495$	17.9577	0.807140
$496$	40.5109	1.81899
$497$	24.0606	1.07926
$498$	15.0288	0.673458
$499$	−32.2406	−1.44329	−0.721643	−	0.692265i	$-0.756613\pi$
$499$	−32.2406	−1.44329	−0.721643	+	0.692265i	$0.756613\pi$
$500$	−5.98534	−0.267672
$501$	14.2951	0.638656
$502$	31.5717	1.40912
$503$	−29.2740	−1.30526	−0.652632	−	0.757675i	$-0.726335\pi$
$503$	−29.2740	−1.30526	−0.652632	+	0.757675i	$0.726335\pi$
$504$	−33.9508	−1.51229
$505$	10.7319	0.477561
$506$	−11.7577	−0.522693
$507$	8.36764	0.371620
$508$	−1.62516	−0.0721049
$509$	−19.8904	−0.881628	−0.440814	−	0.897599i	$-0.645310\pi$
$509$	−19.8904	−0.881628	−0.440814	+	0.897599i	$0.645310\pi$
$510$	−33.5741	−1.48669
$511$	43.2021	1.91115
$512$	−8.30165	−0.366884
$513$	−16.2278	−0.716475
$514$	−24.3228	−1.07283
$515$	−0.978009	−0.0430962
$516$	0	0
$517$	−2.68332	−0.118012
$518$	25.1483	1.10495
$519$	−12.3796	−0.543405
$520$	−13.1133	−0.575057
$521$	−38.3269	−1.67913	−0.839566	−	0.543257i	$-0.817191\pi$
$521$	−38.3269	−1.67913	−0.839566	+	0.543257i	$0.817191\pi$
$522$	−3.63131	−0.158938
$523$	3.95763	0.173055	0.0865275	−	0.996249i	$-0.472423\pi$
$523$	3.95763	0.173055	0.0865275	+	0.996249i	$0.472423\pi$
$524$	5.82637	0.254526
$525$	−27.9112	−1.21815
$526$	−30.5772	−1.33323
$527$	47.3882	2.06426
$528$	−38.7315	−1.68557
$529$	−16.9491	−0.736916
$530$	−21.5545	−0.936267
$531$	44.1796	1.91723
$532$	−8.54902	−0.370647
$533$	−9.06793	−0.392776
$534$	−3.92895	−0.170022
$535$	17.8331	0.770992
$536$	−7.65096	−0.330471
$537$	−4.57974	−0.197631
$538$	−32.9348	−1.41992
$539$	13.7187	0.590907
$540$	2.55564	0.109977
$541$	27.3590	1.17626	0.588128	−	0.808768i	$-0.299865\pi$
$541$	27.3590	1.17626	0.588128	+	0.808768i	$0.299865\pi$
$542$	−31.5927	−1.35702
$543$	58.4854	2.50985
$544$	16.4172	0.703881
$545$	−10.9562	−0.469311
$546$	58.5673	2.50645
$547$	−18.6376	−0.796885	−0.398443	−	0.917193i	$-0.630449\pi$
$547$	−18.6376	−0.796885	−0.398443	+	0.917193i	$0.630449\pi$
$548$	9.76877	0.417301
$549$	−9.54030	−0.407170
$550$	−14.5449	−0.620197
$551$	2.52345	0.107503
$552$	15.4970	0.659595
$553$	7.98651	0.339621
$554$	6.03863	0.256557
$555$	17.5321	0.744197
$556$	−10.9766	−0.465510
$557$	33.3451	1.41288	0.706439	−	0.707774i	$-0.250300\pi$
$557$	33.3451	1.41288	0.706439	+	0.707774i	$0.250300\pi$
$558$	−57.6181	−2.43917
$559$	0	0
$560$	−22.7468	−0.961226
$561$	−45.3067	−1.91285
$562$	10.5585	0.445384
$563$	−34.3483	−1.44761	−0.723805	−	0.690005i	$-0.757608\pi$
$563$	−34.3483	−1.44761	−0.723805	+	0.690005i	$0.757608\pi$
$564$	−1.28154	−0.0539627
$565$	10.5580	0.444177
$566$	−44.4939	−1.87022
$567$	−12.1020	−0.508236
$568$	−16.5258	−0.693409
$569$	2.84152	0.119123	0.0595613	−	0.998225i	$-0.481030\pi$
$569$	2.84152	0.119123	0.0595613	+	0.998225i	$0.481030\pi$
$570$	−28.3677	−1.18819
$571$	−23.0580	−0.964948	−0.482474	−	0.875910i	$-0.660262\pi$
$571$	−23.0580	−0.964948	−0.482474	+	0.875910i	$0.660262\pi$
$572$	6.41219	0.268107
$573$	−39.1269	−1.63455
$574$	−12.2292	−0.510436
$575$	7.48534	0.312160
$576$	20.9003	0.870846
$577$	−20.8477	−0.867900	−0.433950	−	0.900937i	$-0.642881\pi$
$577$	−20.8477	−0.867900	−0.433950	+	0.900937i	$0.642881\pi$
$578$	22.7175	0.944924
$579$	−34.3064	−1.42573
$580$	−0.397406	−0.0165014
$581$	11.9108	0.494141
$582$	8.23586	0.341387
$583$	−29.0868	−1.20465
$584$	−29.6730	−1.22788
$585$	23.9893	0.991835
$586$	9.78509	0.404218
$587$	−26.6929	−1.10173	−0.550867	−	0.834593i	$-0.685703\pi$
$587$	−26.6929	−1.10173	−0.550867	+	0.834593i	$0.685703\pi$
$588$	6.55199	0.270199
$589$	40.0397	1.64981
$590$	23.0130	0.947432
$591$	41.1639	1.69326
$592$	−22.2168	−0.913107
$593$	42.3372	1.73858	0.869290	−	0.494302i	$-0.164576\pi$
$593$	42.3372	1.73858	0.869290	+	0.494302i	$0.164576\pi$
$594$	16.4149	0.673513
$595$	−26.6084	−1.09084
$596$	−6.12749	−0.250992
$597$	−51.9843	−2.12758
$598$	−15.7068	−0.642299
$599$	6.44069	0.263159	0.131580	−	0.991306i	$-0.457995\pi$
$599$	6.44069	0.263159	0.131580	+	0.991306i	$0.457995\pi$
$600$	19.1706	0.782637
$601$	−20.6424	−0.842023	−0.421012	−	0.907055i	$-0.638325\pi$
$601$	−20.6424	−0.842023	−0.421012	+	0.907055i	$0.638325\pi$
$602$	0	0
$603$	13.9965	0.569983
$604$	−10.1643	−0.413579
$605$	−2.76526	−0.112424
$606$	32.9211	1.33733
$607$	−20.7341	−0.841570	−0.420785	−	0.907160i	$-0.638245\pi$
$607$	−20.7341	−0.841570	−0.420785	+	0.907160i	$0.638245\pi$
$608$	13.8714	0.562558
$609$	−4.89826	−0.198488
$610$	−4.96952	−0.201210
$611$	−3.58459	−0.145017
$612$	−12.7133	−0.513905
$613$	31.9337	1.28979	0.644895	−	0.764271i	$-0.276901\pi$
$613$	31.9337	1.28979	0.644895	+	0.764271i	$0.276901\pi$
$614$	26.4463	1.06729
$615$	−8.52559	−0.343785
$616$	−23.8650	−0.961546
$617$	−43.3901	−1.74682	−0.873410	−	0.486986i	$-0.838096\pi$
$617$	−43.3901	−1.74682	−0.873410	+	0.486986i	$0.838096\pi$
$618$	−3.00015	−0.120684
$619$	13.5577	0.544930	0.272465	−	0.962166i	$-0.412161\pi$
$619$	13.5577	0.544930	0.272465	+	0.962166i	$0.412161\pi$
$620$	−6.30565	−0.253241
$621$	−8.44771	−0.338995
$622$	47.3754	1.89958
$623$	−3.11380	−0.124752
$624$	−51.7404	−2.07127
$625$	−0.525268	−0.0210107
$626$	10.4922	0.419353
$627$	−38.2810	−1.52879
$628$	1.57422	0.0628180
$629$	−25.9885	−1.03623
$630$	32.3524	1.28895
$631$	−28.9685	−1.15322	−0.576608	−	0.817021i	$-0.695624\pi$
$631$	−28.9685	−1.15322	−0.576608	+	0.817021i	$0.695624\pi$
$632$	−5.48548	−0.218201
$633$	57.6382	2.29091
$634$	8.43463	0.334982
$635$	−4.27384	−0.169602
$636$	−13.8917	−0.550842
$637$	18.3265	0.726122
$638$	−2.55255	−0.101056
$639$	30.2321	1.19596
$640$	19.1002	0.755000
$641$	14.1942	0.560636	0.280318	−	0.959907i	$-0.409560\pi$
$641$	14.1942	0.560636	0.280318	+	0.959907i	$0.409560\pi$
$642$	54.7049	2.15903
$643$	−0.787730	−0.0310650	−0.0155325	−	0.999879i	$-0.504944\pi$
$643$	−0.787730	−0.0310650	−0.0155325	+	0.999879i	$0.504944\pi$
$644$	−4.45037	−0.175369
$645$	0	0
$646$	42.0505	1.65445
$647$	−1.14447	−0.0449937	−0.0224969	−	0.999747i	$-0.507162\pi$
$647$	−1.14447	−0.0449937	−0.0224969	+	0.999747i	$0.507162\pi$
$648$	8.31217	0.326533
$649$	31.0551	1.21902
$650$	−19.4302	−0.762115
$651$	−77.7208	−3.04612
$652$	−6.67887	−0.261565
$653$	−41.8841	−1.63905	−0.819525	−	0.573044i	$-0.805763\pi$
$653$	−41.8841	−1.63905	−0.819525	+	0.573044i	$0.805763\pi$
$654$	−33.6092	−1.31422
$655$	15.3222	0.598687
$656$	10.8037	0.421813
$657$	54.2833	2.11779
$658$	−4.83424	−0.188458
$659$	26.2488	1.02251	0.511255	−	0.859429i	$-0.329181\pi$
$659$	26.2488	1.02251	0.511255	+	0.859429i	$0.329181\pi$
$660$	6.02868	0.234666
$661$	12.1269	0.471682	0.235841	−	0.971792i	$-0.424216\pi$
$661$	12.1269	0.471682	0.235841	+	0.971792i	$0.424216\pi$
$662$	46.9647	1.82533
$663$	−60.5242	−2.35056
$664$	−8.18081	−0.317477
$665$	−22.4822	−0.871822
$666$	31.5987	1.22443
$667$	1.31363	0.0508641
$668$	2.81963	0.109095
$669$	35.9648	1.39048
$670$	7.29076	0.281667
$671$	−6.70615	−0.258888
$672$	−26.9256	−1.03868
$673$	−29.0911	−1.12138	−0.560689	−	0.828027i	$-0.689464\pi$
$673$	−29.0911	−1.12138	−0.560689	+	0.828027i	$0.689464\pi$
$674$	−48.3272	−1.86149
$675$	−10.4503	−0.402232
$676$	1.65048	0.0634798
$677$	−2.47305	−0.0950469	−0.0475235	−	0.998870i	$-0.515133\pi$
$677$	−2.47305	−0.0950469	−0.0475235	+	0.998870i	$0.515133\pi$
$678$	32.3877	1.24384
$679$	6.52714	0.250489
$680$	18.2758	0.700844
$681$	−13.6357	−0.522520
$682$	−40.5014	−1.55088
$683$	−5.38825	−0.206176	−0.103088	−	0.994672i	$-0.532872\pi$
$683$	−5.38825	−0.206176	−0.103088	+	0.994672i	$0.532872\pi$
$684$	−10.7418	−0.410724
$685$	25.6899	0.981559
$686$	−13.1668	−0.502712
$687$	59.5112	2.27049
$688$	0	0
$689$	−38.8563	−1.48031
$690$	−14.7674	−0.562185
$691$	−4.72188	−0.179629	−0.0898143	−	0.995959i	$-0.528627\pi$
$691$	−4.72188	−0.179629	−0.0898143	+	0.995959i	$0.528627\pi$
$692$	−2.44182	−0.0928240
$693$	43.6581	1.65844
$694$	14.9320	0.566811
$695$	−28.8661	−1.09495
$696$	3.36433	0.127525
$697$	12.6378	0.478690
$698$	17.3338	0.656094
$699$	−26.1272	−0.988222
$700$	−5.50535	−0.208083
$701$	−26.8815	−1.01530	−0.507650	−	0.861564i	$-0.669486\pi$
$701$	−26.8815	−1.01530	−0.507650	+	0.861564i	$0.669486\pi$
$702$	21.9283	0.827630
$703$	−21.9584	−0.828178
$704$	14.6914	0.553704
$705$	−3.37019	−0.126929
$706$	45.7373	1.72134
$707$	26.0909	0.981248
$708$	14.8317	0.557411
$709$	−7.05891	−0.265103	−0.132551	−	0.991176i	$-0.542317\pi$
$709$	−7.05891	−0.265103	−0.132551	+	0.991176i	$0.542317\pi$
$710$	15.7478	0.591005
$711$	10.0350	0.376343
$712$	2.13869	0.0801508
$713$	20.8435	0.780594
$714$	−81.6240	−3.05470
$715$	16.8627	0.630631
$716$	−0.903332	−0.0337591
$717$	29.7021	1.10925
$718$	−14.6686	−0.547426
$719$	−21.3692	−0.796936	−0.398468	−	0.917182i	$-0.630458\pi$
$719$	−21.3692	−0.796936	−0.398468	+	0.917182i	$0.630458\pi$
$720$	−28.5813	−1.06516
$721$	−2.37770	−0.0885501
$722$	5.29665	0.197121
$723$	36.5956	1.36100
$724$	11.5360	0.428730
$725$	1.62504	0.0603524
$726$	−8.48272	−0.314823
$727$	−5.21292	−0.193336	−0.0966682	−	0.995317i	$-0.530819\pi$
$727$	−5.21292	−0.193336	−0.0966682	+	0.995317i	$0.530819\pi$
$728$	−31.8806	−1.18157
$729$	−42.9916	−1.59228
$730$	28.2761	1.04654
$731$	0	0
$732$	−3.20282	−0.118380
$733$	−25.3899	−0.937796	−0.468898	−	0.883252i	$-0.655349\pi$
$733$	−25.3899	−0.937796	−0.468898	+	0.883252i	$0.655349\pi$
$734$	−29.1700	−1.07669
$735$	17.2304	0.635553
$736$	7.22102	0.266170
$737$	9.83855	0.362408
$738$	−15.3660	−0.565628
$739$	7.02353	0.258365	0.129182	−	0.991621i	$-0.458765\pi$
$739$	7.02353	0.258365	0.129182	+	0.991621i	$0.458765\pi$
$740$	3.45813	0.127123
$741$	−51.1386	−1.87862
$742$	−52.4024	−1.92375
$743$	34.9579	1.28248	0.641240	−	0.767341i	$-0.278420\pi$
$743$	34.9579	1.28248	0.641240	+	0.767341i	$0.278420\pi$
$744$	53.3820	1.95708
$745$	−16.1140	−0.590373
$746$	43.8256	1.60457
$747$	14.9658	0.547571
$748$	−8.93653	−0.326752
$749$	43.3551	1.58416
$750$	−48.2848	−1.76311
$751$	−10.8777	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$751$	−10.8777	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$752$	4.27074	0.155738
$753$	53.5105	1.95003
$754$	−3.40989	−0.124181
$755$	−26.7300	−0.972804
$756$	6.21316	0.225971
$757$	−3.07537	−0.111776	−0.0558881	−	0.998437i	$-0.517799\pi$
$757$	−3.07537	−0.111776	−0.0558881	+	0.998437i	$0.517799\pi$
$758$	14.4797	0.525926
$759$	−19.9279	−0.723338
$760$	15.4417	0.560130
$761$	−20.2784	−0.735090	−0.367545	−	0.930006i	$-0.619802\pi$
$761$	−20.2784	−0.735090	−0.367545	+	0.930006i	$0.619802\pi$
$762$	−13.1105	−0.474942
$763$	−26.6362	−0.964295
$764$	−7.71759	−0.279213
$765$	−33.4334	−1.20879
$766$	−28.8744	−1.04327
$767$	41.4857	1.49796
$768$	32.2115	1.16233
$769$	−19.6319	−0.707946	−0.353973	−	0.935256i	$-0.615169\pi$
$769$	−19.6319	−0.707946	−0.353973	+	0.935256i	$0.615169\pi$
$770$	22.7414	0.819544
$771$	−41.2244	−1.48466
$772$	−6.76677	−0.243541
$773$	−15.6254	−0.562006	−0.281003	−	0.959707i	$-0.590667\pi$
$773$	−15.6254	−0.562006	−0.281003	+	0.959707i	$0.590667\pi$
$774$	0	0
$775$	25.7845	0.926208
$776$	−4.48312	−0.160935
$777$	42.6234	1.52911
$778$	19.7597	0.708419
$779$	10.6780	0.382580
$780$	8.05357	0.288364
$781$	21.2510	0.760420
$782$	21.8902	0.782794
$783$	−1.83397	−0.0655406
$784$	−21.8345	−0.779803
$785$	4.13987	0.147758
$786$	47.0024	1.67652
$787$	23.8861	0.851448	0.425724	−	0.904853i	$-0.360019\pi$
$787$	23.8861	0.851448	0.425724	+	0.904853i	$0.360019\pi$
$788$	8.11938	0.289241
$789$	−51.8249	−1.84501
$790$	5.22723	0.185976
$791$	25.6681	0.912654
$792$	−29.9863	−1.06552
$793$	−8.95858	−0.318128
$794$	51.8732	1.84091
$795$	−36.5324	−1.29567
$796$	−10.2536	−0.363431
$797$	−6.40006	−0.226702	−0.113351	−	0.993555i	$-0.536158\pi$
$797$	−6.40006	−0.226702	−0.113351	+	0.993555i	$0.536158\pi$
$798$	−68.9665	−2.44139
$799$	4.99576	0.176737
$800$	8.93280	0.315822
$801$	−3.91248	−0.138241
$802$	−45.2210	−1.59681
$803$	38.1573	1.34654
$804$	4.69885	0.165716
$805$	−11.7036	−0.412496
$806$	−54.1048	−1.90576
$807$	−55.8208	−1.96498
$808$	−17.9203	−0.630435
$809$	−37.6885	−1.32506	−0.662528	−	0.749037i	$-0.730517\pi$
$809$	−37.6885	−1.32506	−0.662528	+	0.749037i	$0.730517\pi$
$810$	−7.92084	−0.278310
$811$	−3.96460	−0.139216	−0.0696079	−	0.997574i	$-0.522175\pi$
$811$	−3.96460	−0.139216	−0.0696079	+	0.997574i	$0.522175\pi$
$812$	−0.966158	−0.0339055
$813$	−53.5460	−1.87794
$814$	22.2116	0.778518
$815$	−17.5641	−0.615242
$816$	72.1095	2.52434
$817$	0	0
$818$	−37.6201	−1.31536
$819$	58.3218	2.03793
$820$	−1.68163	−0.0587251
$821$	−5.50054	−0.191970	−0.0959851	−	0.995383i	$-0.530600\pi$
$821$	−5.50054	−0.191970	−0.0959851	+	0.995383i	$0.530600\pi$
$822$	78.8064	2.74869
$823$	−17.2937	−0.602820	−0.301410	−	0.953495i	$-0.597457\pi$
$823$	−17.2937	−0.602820	−0.301410	+	0.953495i	$0.597457\pi$
$824$	1.63310	0.0568919
$825$	−24.6520	−0.858272
$826$	55.9484	1.94669
$827$	23.1815	0.806101	0.403051	−	0.915178i	$-0.367950\pi$
$827$	23.1815	0.806101	0.403051	+	0.915178i	$0.367950\pi$
$828$	−5.59188	−0.194331
$829$	−37.8108	−1.31322	−0.656611	−	0.754229i	$-0.728011\pi$
$829$	−37.8108	−1.31322	−0.656611	+	0.754229i	$0.728011\pi$
$830$	7.79567	0.270592
$831$	10.2348	0.355041
$832$	19.6259	0.680406
$833$	−25.5412	−0.884951
$834$	−88.5498	−3.06623
$835$	7.41505	0.256609
$836$	−7.55073	−0.261148
$837$	−29.0996	−1.00583
$838$	4.88579	0.168777
$839$	39.5304	1.36474	0.682370	−	0.731007i	$-0.260949\pi$
$839$	39.5304	1.36474	0.682370	+	0.731007i	$0.260949\pi$
$840$	−29.9739	−1.03420
$841$	−28.7148	−0.990166
$842$	12.5523	0.432581
$843$	17.8955	0.616353
$844$	11.3689	0.391332
$845$	4.34041	0.149315
$846$	−6.07421	−0.208836
$847$	−6.72278	−0.230997
$848$	46.2941	1.58975
$849$	−75.4121	−2.58814
$850$	27.0795	0.928818
$851$	−11.4309	−0.391847
$852$	10.1494	0.347712
$853$	53.1969	1.82143	0.910713	−	0.413040i	$-0.135533\pi$
$853$	53.1969	1.82143	0.910713	+	0.413040i	$0.135533\pi$
$854$	−12.0817	−0.413427
$855$	−28.2488	−0.966089
$856$	−29.7782	−1.01780
$857$	32.7837	1.11987	0.559935	−	0.828536i	$-0.310826\pi$
$857$	32.7837	1.11987	0.559935	+	0.828536i	$0.310826\pi$
$858$	51.7283	1.76597
$859$	18.7249	0.638885	0.319442	−	0.947606i	$-0.396504\pi$
$859$	18.7249	0.638885	0.319442	+	0.947606i	$0.396504\pi$
$860$	0	0
$861$	−20.7271	−0.706377
$862$	40.9469	1.39466
$863$	−18.8899	−0.643019	−0.321510	−	0.946906i	$-0.604190\pi$
$863$	−18.8899	−0.643019	−0.321510	+	0.946906i	$0.604190\pi$
$864$	−10.0813	−0.342972
$865$	−6.42148	−0.218337
$866$	−14.8295	−0.503928
$867$	38.5036	1.30765
$868$	−15.3300	−0.520336
$869$	7.05391	0.239287
$870$	−3.20595	−0.108692
$871$	13.1431	0.445336
$872$	18.2949	0.619543
$873$	8.20134	0.277573
$874$	18.4957	0.625626
$875$	−38.2670	−1.29366
$876$	18.2237	0.615722
$877$	22.6957	0.766378	0.383189	−	0.923670i	$-0.374826\pi$
$877$	22.6957	0.766378	0.383189	+	0.923670i	$0.374826\pi$
$878$	−18.4052	−0.621144
$879$	16.5846	0.559385
$880$	−20.0906	−0.677253
$881$	37.8496	1.27519	0.637593	−	0.770373i	$-0.279930\pi$
$881$	37.8496	1.27519	0.637593	+	0.770373i	$0.279930\pi$
$882$	31.0549	1.04567
$883$	6.97263	0.234648	0.117324	−	0.993094i	$-0.462568\pi$
$883$	6.97263	0.234648	0.117324	+	0.993094i	$0.462568\pi$
$884$	−11.9381	−0.401522
$885$	39.0045	1.31112
$886$	61.9386	2.08087
$887$	12.7605	0.428456	0.214228	−	0.976784i	$-0.431277\pi$
$887$	12.7605	0.428456	0.214228	+	0.976784i	$0.431277\pi$
$888$	−29.2756	−0.982424
$889$	−10.3904	−0.348483
$890$	−2.03800	−0.0683140
$891$	−10.6888	−0.358089
$892$	7.09389	0.237521
$893$	4.22106	0.141252
$894$	−49.4315	−1.65324
$895$	−2.37558	−0.0794068
$896$	46.4355	1.55130
$897$	−26.6212	−0.888857
$898$	−25.8173	−0.861533
$899$	4.52504	0.150918
$900$	−6.91747	−0.230582
$901$	54.1533	1.80411
$902$	−10.8012	−0.359639
$903$	0	0
$904$	−17.6300	−0.586364
$905$	30.3372	1.00844
$906$	−81.9971	−2.72417
$907$	32.6147	1.08295	0.541476	−	0.840716i	$-0.317866\pi$
$907$	32.6147	1.08295	0.541476	+	0.840716i	$0.317866\pi$
$908$	−2.68957	−0.0892565
$909$	32.7831	1.08735
$910$	30.3797	1.00708
$911$	16.9707	0.562264	0.281132	−	0.959669i	$-0.409290\pi$
$911$	16.9707	0.562264	0.281132	+	0.959669i	$0.409290\pi$
$912$	60.9274	2.01751
$913$	10.5199	0.348158
$914$	50.9273	1.68453
$915$	−8.42277	−0.278448
$916$	11.7383	0.387844
$917$	37.2507	1.23013
$918$	−30.5610	−1.00866
$919$	−20.0436	−0.661178	−0.330589	−	0.943775i	$-0.607247\pi$
$919$	−20.0436	−0.661178	−0.330589	+	0.943775i	$0.607247\pi$
$920$	8.03850	0.265022
$921$	44.8234	1.47698
$922$	49.8431	1.64149
$923$	28.3887	0.934424
$924$	14.6567	0.482170
$925$	−14.1407	−0.464943
$926$	31.8232	1.04577
$927$	−2.98757	−0.0981248
$928$	1.56766	0.0514608
$929$	39.5173	1.29652	0.648260	−	0.761419i	$-0.275497\pi$
$929$	39.5173	1.29652	0.648260	+	0.761419i	$0.275497\pi$
$930$	−50.8688	−1.66805
$931$	−21.5805	−0.707273
$932$	−5.15346	−0.168807
$933$	80.2959	2.62877
$934$	5.27770	0.172692
$935$	−23.5013	−0.768573
$936$	−40.0579	−1.30933
$937$	−32.5538	−1.06349	−0.531744	−	0.846905i	$-0.678463\pi$
$937$	−32.5538	−1.06349	−0.531744	+	0.846905i	$0.678463\pi$
$938$	17.7250	0.578742
$939$	17.7831	0.580330
$940$	−0.664754	−0.0216819
$941$	33.7288	1.09953	0.549764	−	0.835320i	$-0.314718\pi$
$941$	33.7288	1.09953	0.549764	+	0.835320i	$0.314718\pi$
$942$	12.6995	0.413771
$943$	5.55866	0.181015
$944$	−49.4268	−1.60870
$945$	16.3394	0.531519
$946$	0	0
$947$	41.2833	1.34153	0.670763	−	0.741672i	$-0.265967\pi$
$947$	41.2833	1.34153	0.670763	+	0.741672i	$0.265967\pi$
$948$	3.36891	0.109417
$949$	50.9733	1.65466
$950$	22.8802	0.742332
$951$	14.2957	0.463571
$952$	44.4313	1.44003
$953$	37.8902	1.22738	0.613692	−	0.789546i	$-0.289684\pi$
$953$	37.8902	1.22738	0.613692	+	0.789546i	$0.289684\pi$
$954$	−65.8435	−2.13176
$955$	−20.2957	−0.656753
$956$	5.85860	0.189481
$957$	−4.32628	−0.139849
$958$	33.6576	1.08743
$959$	62.4562	2.01682
$960$	18.4521	0.595539
$961$	40.7988	1.31609
$962$	29.6720	0.956663
$963$	54.4756	1.75545
$964$	7.21829	0.232486
$965$	−17.7952	−0.572849
$966$	−35.9019	−1.15512
$967$	39.6951	1.27651	0.638254	−	0.769826i	$-0.279657\pi$
$967$	39.6951	1.27651	0.638254	+	0.769826i	$0.279657\pi$
$968$	4.61749	0.148412
$969$	71.2708	2.28955
$970$	4.27206	0.137168
$971$	3.14256	0.100849	0.0504247	−	0.998728i	$-0.483942\pi$
$971$	3.14256	0.100849	0.0504247	+	0.998728i	$0.483942\pi$
$972$	−10.5855	−0.339529
$973$	−70.1781	−2.24981
$974$	30.5241	0.978054
$975$	−32.9320	−1.05467
$976$	10.6734	0.341647
$977$	18.1280	0.579966	0.289983	−	0.957032i	$-0.406350\pi$
$977$	18.1280	0.579966	0.289983	+	0.957032i	$0.406350\pi$
$978$	−53.8796	−1.72288
$979$	−2.75019	−0.0878966
$980$	3.39861	0.108565
$981$	−33.4683	−1.06856
$982$	−18.0681	−0.576578
$983$	−0.540449	−0.0172376	−0.00861882	−	0.999963i	$-0.502743\pi$
$983$	−0.540449	−0.0172376	−0.00861882	+	0.999963i	$0.502743\pi$
$984$	14.2362	0.453835
$985$	21.3523	0.680341
$986$	4.75229	0.151344
$987$	−8.19348	−0.260801
$988$	−10.0868	−0.320905
$989$	0	0
$990$	28.5746	0.908159
$991$	−2.80194	−0.0890065	−0.0445032	−	0.999009i	$-0.514171\pi$
$991$	−2.80194	−0.0890065	−0.0445032	+	0.999009i	$0.514171\pi$
$992$	24.8740	0.789751
$993$	79.5997	2.52602
$994$	38.2855	1.21434
$995$	−26.9650	−0.854848
$996$	5.02426	0.159200
$997$	6.73119	0.213179	0.106590	−	0.994303i	$-0.466007\pi$
$997$	6.73119	0.213179	0.106590	+	0.994303i	$0.466007\pi$
$998$	−51.3016	−1.62392
$999$	15.9587	0.504911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.16	✓	20
43.42	odd	2		1849.2.a.r.1.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.16	✓	20	1.1	even	1	trivial
1849.2.a.r.1.5	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)

\(f(q)\)	\(=\)	\(q+1.59121 q^{2} +2.69692 q^{3} +0.531955 q^{4} +1.39893 q^{5} +4.29137 q^{6} +3.40103 q^{7} -2.33597 q^{8} +4.27339 q^{9} +O(q^{10})\)	\(q+1.59121 q^{2} +2.69692 q^{3} +0.531955 q^{4} +1.39893 q^{5} +4.29137 q^{6} +3.40103 q^{7} -2.33597 q^{8} +4.27339 q^{9} +2.22600 q^{10} +3.00388 q^{11} +1.43464 q^{12} +4.01281 q^{13} +5.41176 q^{14} +3.77281 q^{15} -4.78093 q^{16} -5.59257 q^{17} +6.79986 q^{18} -4.72532 q^{19} +0.744168 q^{20} +9.17231 q^{21} +4.77981 q^{22} -2.45986 q^{23} -6.29993 q^{24} -3.04299 q^{25} +6.38523 q^{26} +3.43422 q^{27} +1.80919 q^{28} -0.534027 q^{29} +6.00334 q^{30} -8.47342 q^{31} -2.93554 q^{32} +8.10123 q^{33} -8.89897 q^{34} +4.75781 q^{35} +2.27325 q^{36} +4.64697 q^{37} -7.51899 q^{38} +10.8222 q^{39} -3.26786 q^{40} -2.25975 q^{41} +14.5951 q^{42} +1.59793 q^{44} +5.97818 q^{45} -3.91416 q^{46} -0.893285 q^{47} -12.8938 q^{48} +4.56699 q^{49} -4.84204 q^{50} -15.0827 q^{51} +2.13463 q^{52} -9.68307 q^{53} +5.46457 q^{54} +4.20223 q^{55} -7.94470 q^{56} -12.7438 q^{57} -0.849750 q^{58} +10.3383 q^{59} +2.00696 q^{60} -2.23249 q^{61} -13.4830 q^{62} +14.5339 q^{63} +4.89081 q^{64} +5.61365 q^{65} +12.8908 q^{66} +3.27528 q^{67} -2.97499 q^{68} -6.63406 q^{69} +7.57068 q^{70} +7.07450 q^{71} -9.98250 q^{72} +12.7026 q^{73} +7.39431 q^{74} -8.20670 q^{75} -2.51366 q^{76} +10.2163 q^{77} +17.2205 q^{78} +2.34826 q^{79} -6.68820 q^{80} -3.55833 q^{81} -3.59573 q^{82} +3.50210 q^{83} +4.87925 q^{84} -7.82363 q^{85} -1.44023 q^{87} -7.01698 q^{88} -0.915547 q^{89} +9.51254 q^{90} +13.6477 q^{91} -1.30854 q^{92} -22.8521 q^{93} -1.42141 q^{94} -6.61041 q^{95} -7.91691 q^{96} +1.91917 q^{97} +7.26705 q^{98} +12.8367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 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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